id	sid	tid	token	lemma	pos
ejpam-6461	1	1	european	european	PROPN
ejpam-6461	1	2	journal	journal	PROPN
ejpam-6461	1	3	of	of	ADP
ejpam-6461	1	4	pure	pure	ADJ
ejpam-6461	1	5	and	and	CCONJ
ejpam-6461	1	6	applied	applied	ADJ
ejpam-6461	1	7	mathematics	mathematic	NOUN
ejpam-6461	1	8	2025	2025	NUM
ejpam-6461	1	9	,	,	PUNCT
ejpam-6461	1	10	vol	vol	NOUN
ejpam-6461	1	11	.	.	PROPN
ejpam-6461	1	12	18	18	NUM
ejpam-6461	1	13	,	,	PUNCT
ejpam-6461	1	14	issue	issue	NOUN
ejpam-6461	1	15	3	3	NUM
ejpam-6461	1	16	,	,	PUNCT
ejpam-6461	1	17	article	article	NOUN
ejpam-6461	1	18	number	number	NOUN
ejpam-6461	1	19	6461	6461	NUM
ejpam-6461	1	20	issn	issn	PROPN
ejpam-6461	1	21	1307	1307	NUM
ejpam-6461	1	22	-	-	SYM
ejpam-6461	1	23	5543	5543	NUM
ejpam-6461	1	24	–	–	PUNCT
ejpam-6461	1	25	ejpam.com	ejpam.com	X
ejpam-6461	1	26	published	publish	VERB
ejpam-6461	1	27	by	by	ADP
ejpam-6461	1	28	new	new	PROPN
ejpam-6461	1	29	york	york	PROPN
ejpam-6461	1	30	business	business	PROPN
ejpam-6461	1	31	global	global	ADJ
ejpam-6461	1	32	some	some	DET
ejpam-6461	1	33	refinements	refinement	NOUN
ejpam-6461	1	34	for	for	ADP
ejpam-6461	1	35	numerical	numerical	ADJ
ejpam-6461	1	36	radius	radius	PROPN
ejpam-6461	1	37	inequalities	inequality	NOUN
ejpam-6461	1	38	of	of	ADP
ejpam-6461	1	39	operators	operators	PROPN
ejpam-6461	1	40	manal	manal	PROPN
ejpam-6461	1	41	al	al	PROPN
ejpam-6461	1	42	-	-	PUNCT
ejpam-6461	1	43	labadi	labadi	PROPN
ejpam-6461	1	44	,	,	PUNCT
ejpam-6461	1	45	wasim	wasim	PROPN
ejpam-6461	1	46	audeh	audeh	PROPN
ejpam-6461	1	47	,	,	PUNCT
ejpam-6461	1	48	raja’a	raja’a	PROPN
ejpam-6461	1	49	al	al	PROPN
ejpam-6461	1	50	-	-	PUNCT
ejpam-6461	1	51	naimi	naimi	PROPN
ejpam-6461	1	52	,	,	PUNCT
ejpam-6461	1	53	jamal	jamal	PROPN
ejpam-6461	1	54	oudetallah1	oudetallah1	PROPN
ejpam-6461	1	55	,	,	PUNCT
ejpam-6461	1	56	eman	eman	PROPN
ejpam-6461	1	57	almuhur2	almuhur2	PROPN
ejpam-6461	1	58	,	,	PUNCT
ejpam-6461	1	59	nazneen	nazneen	PROPN
ejpam-6461	1	60	khan3	khan3	PROPN
ejpam-6461	1	61	1	1	NUM
ejpam-6461	1	62	department	department	NOUN
ejpam-6461	1	63	of	of	ADP
ejpam-6461	1	64	mathematics	mathematics	PROPN
ejpam-6461	1	65	,	,	PUNCT
ejpam-6461	1	66	university	university	PROPN
ejpam-6461	1	67	of	of	ADP
ejpam-6461	1	68	petra	petra	PROPN
ejpam-6461	1	69	,	,	PUNCT
ejpam-6461	1	70	amman	amman	PROPN
ejpam-6461	1	71	,	,	PUNCT
ejpam-6461	1	72	jordan	jordan	PROPN
ejpam-6461	1	73	2	2	NUM
ejpam-6461	1	74	department	department	NOUN
ejpam-6461	1	75	of	of	ADP
ejpam-6461	1	76	mathematics	mathematic	NOUN
ejpam-6461	1	77	,	,	PUNCT
ejpam-6461	1	78	applied	apply	VERB
ejpam-6461	1	79	science	science	NOUN
ejpam-6461	1	80	private	private	ADJ
ejpam-6461	1	81	university	university	NOUN
ejpam-6461	1	82	,	,	PUNCT
ejpam-6461	1	83	amman	amman	PROPN
ejpam-6461	1	84	,	,	PUNCT
ejpam-6461	1	85	jordan	jordan	PROPN
ejpam-6461	1	86	3	3	NUM
ejpam-6461	1	87	department	department	PROPN
ejpam-6461	1	88	of	of	ADP
ejpam-6461	1	89	mathematics	mathematics	PROPN
ejpam-6461	1	90	,	,	PUNCT
ejpam-6461	1	91	taibah	taibah	PROPN
ejpam-6461	1	92	university	university	PROPN
ejpam-6461	1	93	,	,	PUNCT
ejpam-6461	1	94	madina	madina	PROPN
ejpam-6461	1	95	munawwara	munawwara	PROPN
ejpam-6461	1	96	,	,	PUNCT
ejpam-6461	1	97	chandigarh	chandigarh	ADJ
ejpam-6461	1	98	,	,	PUNCT
ejpam-6461	1	99	saudi	saudi	PROPN
ejpam-6461	1	100	arabia	arabia	PROPN
ejpam-6461	1	101	abstract	abstract	NOUN
ejpam-6461	1	102	.	.	PUNCT
ejpam-6461	2	1	several	several	ADJ
ejpam-6461	2	2	recent	recent	ADJ
ejpam-6461	2	3	papers	paper	NOUN
ejpam-6461	2	4	gave	give	VERB
ejpam-6461	2	5	numerical	numerical	ADJ
ejpam-6461	2	6	radius	radius	NOUN
ejpam-6461	2	7	inequalities	inequality	NOUN
ejpam-6461	2	8	for	for	ADP
ejpam-6461	2	9	sums	sum	NOUN
ejpam-6461	2	10	and	and	CCONJ
ejpam-6461	2	11	products	product	NOUN
ejpam-6461	2	12	of	of	ADP
ejpam-6461	2	13	operators	operator	NOUN
ejpam-6461	2	14	which	which	PRON
ejpam-6461	2	15	are	be	AUX
ejpam-6461	2	16	defined	define	VERB
ejpam-6461	2	17	on	on	ADP
ejpam-6461	2	18	a	a	DET
ejpam-6461	2	19	complex	complex	ADJ
ejpam-6461	2	20	separable	separable	ADJ
ejpam-6461	2	21	hilbert	hilbert	PROPN
ejpam-6461	2	22	space	space	PROPN
ejpam-6461	2	23	h.	h.	PROPN
ejpam-6461	2	24	in	in	ADP
ejpam-6461	2	25	this	this	DET
ejpam-6461	2	26	paper	paper	NOUN
ejpam-6461	2	27	,	,	PUNCT
ejpam-6461	2	28	we	we	PRON
ejpam-6461	2	29	prove	prove	VERB
ejpam-6461	2	30	a	a	DET
ejpam-6461	2	31	numerical	numerical	ADJ
ejpam-6461	2	32	radius	radius	NOUN
ejpam-6461	2	33	inequality	inequality	NOUN
ejpam-6461	2	34	which	which	PRON
ejpam-6461	2	35	generalizes	generalize	VERB
ejpam-6461	2	36	and	and	CCONJ
ejpam-6461	2	37	refines	refine	VERB
ejpam-6461	2	38	a	a	DET
ejpam-6461	2	39	recent	recent	ADJ
ejpam-6461	2	40	inequality	inequality	NOUN
ejpam-6461	2	41	proved	prove	VERB
ejpam-6461	2	42	by	by	ADP
ejpam-6461	2	43	kittaneh	kittaneh	PROPN
ejpam-6461	2	44	.	.	PUNCT
ejpam-6461	3	1	2020	2020	NUM
ejpam-6461	3	2	mathematics	mathematic	NOUN
ejpam-6461	3	3	subject	subject	NOUN
ejpam-6461	3	4	classifications	classification	NOUN
ejpam-6461	3	5	:	:	PUNCT
ejpam-6461	3	6	47a30	47a30	NUM
ejpam-6461	3	7	,	,	PUNCT
ejpam-6461	3	8	15a18	15a18	NUM
ejpam-6461	3	9	,	,	PUNCT
ejpam-6461	3	10	47a12	47a12	NUM
ejpam-6461	3	11	,	,	PUNCT
ejpam-6461	3	12	15a60	15a60	NUM
ejpam-6461	3	13	,	,	PUNCT
ejpam-6461	3	14	47b15	47b15	DET
ejpam-6461	3	15	key	key	ADJ
ejpam-6461	3	16	words	word	NOUN
ejpam-6461	3	17	and	and	CCONJ
ejpam-6461	3	18	phrases	phrase	NOUN
ejpam-6461	3	19	:	:	PUNCT
ejpam-6461	3	20	inequality	inequality	NOUN
ejpam-6461	3	21	,	,	PUNCT
ejpam-6461	3	22	numerical	numerical	ADJ
ejpam-6461	3	23	radius	radius	NOUN
ejpam-6461	3	24	,	,	PUNCT
ejpam-6461	3	25	operator	operator	NOUN
ejpam-6461	3	26	,	,	PUNCT
ejpam-6461	3	27	norm	norm	NOUN
ejpam-6461	3	28	1	1	NUM
ejpam-6461	3	29	.	.	PUNCT
ejpam-6461	4	1	introduction	introduction	NOUN
ejpam-6461	4	2	let	let	VERB
ejpam-6461	4	3	h	h	PRON
ejpam-6461	4	4	be	be	AUX
ejpam-6461	4	5	a	a	DET
ejpam-6461	4	6	hilbert	hilbert	NOUN
ejpam-6461	4	7	space	space	NOUN
ejpam-6461	4	8	over	over	ADP
ejpam-6461	4	9	the	the	DET
ejpam-6461	4	10	field	field	NOUN
ejpam-6461	4	11	of	of	ADP
ejpam-6461	4	12	complex	complex	ADJ
ejpam-6461	4	13	numbers	number	NOUN
ejpam-6461	4	14	with	with	ADP
ejpam-6461	4	15	inner	inner	ADJ
ejpam-6461	4	16	product	product	NOUN
ejpam-6461	4	17	⟨.	⟨.	NOUN
ejpam-6461	4	18	,	,	PUNCT
ejpam-6461	4	19	.⟩.	.⟩.	PUNCT
ejpam-6461	5	1	the	the	DET
ejpam-6461	5	2	set	set	NOUN
ejpam-6461	5	3	of	of	ADP
ejpam-6461	5	4	all	all	DET
ejpam-6461	5	5	operators	operator	NOUN
ejpam-6461	5	6	in	in	ADP
ejpam-6461	5	7	h	h	NOUN
ejpam-6461	5	8	is	be	AUX
ejpam-6461	5	9	denoted	denote	VERB
ejpam-6461	5	10	by	by	ADP
ejpam-6461	5	11	b(h	b(h	NOUN
ejpam-6461	5	12	)	)	PUNCT
ejpam-6461	5	13	.	.	PUNCT
ejpam-6461	6	1	upper	upper	ADJ
ejpam-6461	6	2	case	case	NOUN
ejpam-6461	6	3	letters	letter	NOUN
ejpam-6461	6	4	will	will	AUX
ejpam-6461	6	5	denote	denote	VERB
ejpam-6461	6	6	elements	element	NOUN
ejpam-6461	6	7	of	of	ADP
ejpam-6461	6	8	b(h	b(h	NOUN
ejpam-6461	6	9	)	)	PUNCT
ejpam-6461	6	10	.	.	PUNCT
ejpam-6461	7	1	if	if	SCONJ
ejpam-6461	7	2	a	a	DET
ejpam-6461	7	3	∈	∈	PROPN
ejpam-6461	7	4	b(h	b(h	NOUN
ejpam-6461	7	5	)	)	PUNCT
ejpam-6461	7	6	such	such	ADJ
ejpam-6461	7	7	that	that	SCONJ
ejpam-6461	7	8	the	the	DET
ejpam-6461	7	9	operator	operator	NOUN
ejpam-6461	7	10	a	a	PRON
ejpam-6461	7	11	is	be	AUX
ejpam-6461	7	12	equal	equal	ADJ
ejpam-6461	7	13	to	to	ADP
ejpam-6461	7	14	its	its	PRON
ejpam-6461	7	15	conjugate	conjugate	NOUN
ejpam-6461	7	16	transpose	transpose	NOUN
ejpam-6461	7	17	,	,	PUNCT
ejpam-6461	7	18	then	then	ADV
ejpam-6461	7	19	we	we	PRON
ejpam-6461	7	20	say	say	VERB
ejpam-6461	7	21	that	that	SCONJ
ejpam-6461	7	22	the	the	DET
ejpam-6461	7	23	operator	operator	NOUN
ejpam-6461	7	24	a	a	PRON
ejpam-6461	7	25	is	be	AUX
ejpam-6461	7	26	self	self	NOUN
ejpam-6461	7	27	-	-	PUNCT
ejpam-6461	7	28	adjoint	adjoint	NOUN
ejpam-6461	7	29	and	and	CCONJ
ejpam-6461	7	30	in	in	ADP
ejpam-6461	7	31	this	this	DET
ejpam-6461	7	32	case	case	NOUN
ejpam-6461	7	33	all	all	DET
ejpam-6461	7	34	its	its	PRON
ejpam-6461	7	35	eigenvalues	eigenvalue	NOUN
ejpam-6461	7	36	are	be	AUX
ejpam-6461	7	37	real	real	ADJ
ejpam-6461	7	38	numbers	number	NOUN
ejpam-6461	7	39	.	.	PUNCT
ejpam-6461	8	1	the	the	DET
ejpam-6461	8	2	set	set	NOUN
ejpam-6461	8	3	of	of	ADP
ejpam-6461	8	4	all	all	DET
ejpam-6461	8	5	eigenvalues	eigenvalue	NOUN
ejpam-6461	8	6	of	of	ADP
ejpam-6461	8	7	a	a	DET
ejpam-6461	8	8	∈	∈	PROPN
ejpam-6461	8	9	b(h	b(h	PROPN
ejpam-6461	8	10	)	)	PUNCT
ejpam-6461	8	11	is	be	AUX
ejpam-6461	8	12	denoted	denote	VERB
ejpam-6461	8	13	by	by	ADP
ejpam-6461	8	14	σ(a	σ(a	PROPN
ejpam-6461	8	15	)	)	PUNCT
ejpam-6461	8	16	.	.	PUNCT
ejpam-6461	9	1	the	the	DET
ejpam-6461	9	2	conjugate	conjugate	ADJ
ejpam-6461	9	3	transpose	transpose	NOUN
ejpam-6461	9	4	(	(	PUNCT
ejpam-6461	9	5	adjoint	adjoint	NOUN
ejpam-6461	9	6	)	)	PUNCT
ejpam-6461	9	7	of	of	ADP
ejpam-6461	9	8	a	a	PRON
ejpam-6461	9	9	is	be	AUX
ejpam-6461	9	10	denoted	denote	VERB
ejpam-6461	9	11	by	by	ADP
ejpam-6461	9	12	a∗.	a∗.	NOUN
ejpam-6461	9	13	a	a	DET
ejpam-6461	9	14	self	self	NOUN
ejpam-6461	9	15	-	-	PUNCT
ejpam-6461	9	16	adjoint	adjoint	NOUN
ejpam-6461	9	17	operator	operator	NOUN
ejpam-6461	9	18	a	a	DET
ejpam-6461	9	19	∈	∈	PROPN
ejpam-6461	9	20	b(h	b(h	PROPN
ejpam-6461	9	21	)	)	PUNCT
ejpam-6461	9	22	is	be	AUX
ejpam-6461	9	23	called	call	VERB
ejpam-6461	9	24	positive	positive	ADJ
ejpam-6461	9	25	semi	semi	ADJ
ejpam-6461	9	26	-	-	ADJ
ejpam-6461	9	27	definite	definite	ADJ
ejpam-6461	9	28	if	if	SCONJ
ejpam-6461	9	29	⟨ax	⟨ax	VERB
ejpam-6461	9	30	,	,	PUNCT
ejpam-6461	9	31	x⟩	x⟩	PUNCT
ejpam-6461	9	32	≥	≥	X
ejpam-6461	9	33	0	0	NUM
ejpam-6461	9	34	for	for	ADP
ejpam-6461	9	35	all	all	DET
ejpam-6461	9	36	x	x	SYM
ejpam-6461	9	37	∈	∈	PROPN
ejpam-6461	9	38	h.	h.	NOUN
ejpam-6461	9	39	for	for	ADP
ejpam-6461	9	40	any	any	DET
ejpam-6461	9	41	operator	operator	NOUN
ejpam-6461	9	42	a	a	DET
ejpam-6461	9	43	∈	∈	PROPN
ejpam-6461	9	44	b(h	b(h	PROPN
ejpam-6461	9	45	)	)	PUNCT
ejpam-6461	9	46	,	,	PUNCT
ejpam-6461	9	47	the	the	DET
ejpam-6461	9	48	operator	operator	NOUN
ejpam-6461	9	49	a∗a	a∗a	PUNCT
ejpam-6461	9	50	is	be	AUX
ejpam-6461	9	51	positive	positive	ADJ
ejpam-6461	9	52	semi	semi	ADJ
ejpam-6461	9	53	-	-	ADJ
ejpam-6461	9	54	definite	definite	ADJ
ejpam-6461	9	55	.	.	PUNCT
ejpam-6461	10	1	the	the	DET
ejpam-6461	10	2	square	square	ADJ
ejpam-6461	10	3	root	root	NOUN
ejpam-6461	10	4	of	of	ADP
ejpam-6461	10	5	a∗a	a∗a	PROPN
ejpam-6461	10	6	,	,	PUNCT
ejpam-6461	10	7	denoted	denote	VERB
ejpam-6461	10	8	by	by	ADP
ejpam-6461	10	9	|a|	|a|	PROPN
ejpam-6461	10	10	,	,	PUNCT
ejpam-6461	10	11	is	be	AUX
ejpam-6461	10	12	defined	define	VERB
ejpam-6461	10	13	as	as	ADP
ejpam-6461	10	14	|a|	|a|	PROPN
ejpam-6461	10	15	=	=	SYM
ejpam-6461	10	16	(	(	PUNCT
ejpam-6461	10	17	a∗a	a∗a	X
ejpam-6461	10	18	)	)	PUNCT
ejpam-6461	10	19	1	1	NUM
ejpam-6461	10	20	2	2	NUM
ejpam-6461	10	21	.	.	PUNCT
ejpam-6461	11	1	the	the	DET
ejpam-6461	11	2	singular	singular	ADJ
ejpam-6461	11	3	values	value	NOUN
ejpam-6461	11	4	of	of	ADP
ejpam-6461	11	5	a	a	DET
ejpam-6461	11	6	∈	∈	PROPN
ejpam-6461	11	7	b(h	b(h	PROPN
ejpam-6461	11	8	)	)	PUNCT
ejpam-6461	11	9	are	be	AUX
ejpam-6461	11	10	ordered	order	VERB
ejpam-6461	11	11	descendingly	descendingly	ADV
ejpam-6461	11	12	as	as	SCONJ
ejpam-6461	11	13	follows	follow	VERB
ejpam-6461	11	14	,	,	PUNCT
ejpam-6461	11	15	s1(a	s1(a	ADP
ejpam-6461	11	16	)	)	PUNCT
ejpam-6461	11	17	≥	≥	NOUN
ejpam-6461	11	18	s2(a	s2(a	NOUN
ejpam-6461	11	19	)	)	PUNCT
ejpam-6461	11	20	≥	≥	NOUN
ejpam-6461	11	21	·	·	PUNCT
ejpam-6461	11	22	·	·	PUNCT
ejpam-6461	11	23	·	·	PUNCT
ejpam-6461	11	24	and	and	CCONJ
ejpam-6461	11	25	they	they	PRON
ejpam-6461	11	26	are	be	AUX
ejpam-6461	11	27	the	the	DET
ejpam-6461	11	28	eigenvalues	eigenvalue	NOUN
ejpam-6461	11	29	of	of	ADP
ejpam-6461	11	30	|a|	|a|	NOUN
ejpam-6461	11	31	.	.	PROPN
ejpam-6461	12	1	in	in	ADP
ejpam-6461	12	2	fact	fact	NOUN
ejpam-6461	12	3	sj(a	sj(a	NUM
ejpam-6461	12	4	)	)	PUNCT
ejpam-6461	12	5	=	=	PUNCT
ejpam-6461	12	6	λj(|a|	λj(|a|	ADJ
ejpam-6461	12	7	)	)	PUNCT
ejpam-6461	12	8	=	=	SYM
ejpam-6461	12	9	sj(|a|	sj(|a|	NOUN
ejpam-6461	12	10	)	)	PUNCT
ejpam-6461	12	11	for	for	ADP
ejpam-6461	12	12	j	j	PROPN
ejpam-6461	12	13	=	=	SYM
ejpam-6461	12	14	1	1	NUM
ejpam-6461	12	15	,	,	PUNCT
ejpam-6461	12	16	2	2	NUM
ejpam-6461	12	17	,	,	PUNCT
ejpam-6461	12	18	....	....	PUNCT
ejpam-6461	12	19	for	for	ADP
ejpam-6461	12	20	recent	recent	ADJ
ejpam-6461	12	21	studies	study	NOUN
ejpam-6461	12	22	about	about	ADP
ejpam-6461	12	23	singular	singular	ADJ
ejpam-6461	12	24	values	value	NOUN
ejpam-6461	12	25	,	,	PUNCT
ejpam-6461	12	26	we	we	PRON
ejpam-6461	12	27	advise	advise	VERB
ejpam-6461	12	28	the	the	DET
ejpam-6461	12	29	readers	reader	NOUN
ejpam-6461	12	30	to	to	PART
ejpam-6461	12	31	read	read	VERB
ejpam-6461	12	32	[	[	X
ejpam-6461	12	33	[	[	X
ejpam-6461	12	34	1]-[4	1]-[4	NOUN
ejpam-6461	12	35	]	]	X
ejpam-6461	12	36	]	]	PUNCT
ejpam-6461	12	37	,	,	PUNCT
ejpam-6461	13	1	[	[	X
ejpam-6461	13	2	[	[	X
ejpam-6461	13	3	8]-[10	8]-[10	NUM
ejpam-6461	13	4	]	]	X
ejpam-6461	13	5	]	]	PUNCT
ejpam-6461	13	6	,	,	PUNCT
ejpam-6461	14	1	[	[	X
ejpam-6461	14	2	[	[	X
ejpam-6461	14	3	13	13	NUM
ejpam-6461	14	4	]	]	X
ejpam-6461	14	5	]	]	PUNCT
ejpam-6461	14	6	and	and	CCONJ
ejpam-6461	14	7	[	[	X
ejpam-6461	14	8	[	[	X
ejpam-6461	14	9	14]-[18	14]-[18	X
ejpam-6461	14	10	]	]	X
ejpam-6461	14	11	]	]	PUNCT
ejpam-6461	14	12	.	.	PUNCT
ejpam-6461	15	1	we	we	PRON
ejpam-6461	15	2	denote	denote	VERB
ejpam-6461	15	3	the	the	DET
ejpam-6461	15	4	identity	identity	NOUN
ejpam-6461	15	5	operator	operator	NOUN
ejpam-6461	15	6	on	on	ADP
ejpam-6461	15	7	h	h	NOUN
ejpam-6461	15	8	by	by	ADP
ejpam-6461	15	9	i	i	PROPN
ejpam-6461	15	10	∈	∈	PROPN
ejpam-6461	15	11	b(h	b(h	PROPN
ejpam-6461	15	12	)	)	PUNCT
ejpam-6461	15	13	and	and	CCONJ
ejpam-6461	15	14	we	we	PRON
ejpam-6461	15	15	denote	denote	VERB
ejpam-6461	15	16	the	the	DET
ejpam-6461	15	17	zero	zero	NUM
ejpam-6461	15	18	operator	operator	NOUN
ejpam-6461	15	19	on	on	ADP
ejpam-6461	15	20	h	h	NOUN
ejpam-6461	15	21	by	by	ADP
ejpam-6461	15	22	o	o	PROPN
ejpam-6461	15	23	∈	∈	PROPN
ejpam-6461	15	24	b(h	b(h	PROPN
ejpam-6461	15	25	)	)	PUNCT
ejpam-6461	15	26	.	.	PUNCT
ejpam-6461	16	1	each	each	DET
ejpam-6461	16	2	operator	operator	NOUN
ejpam-6461	16	3	a	a	PRON
ejpam-6461	16	4	can	can	AUX
ejpam-6461	16	5	be	be	AUX
ejpam-6461	16	6	written	write	VERB
ejpam-6461	16	7	as	as	ADP
ejpam-6461	16	8	a	a	DET
ejpam-6461	16	9	sum	sum	NOUN
ejpam-6461	16	10	of	of	ADP
ejpam-6461	16	11	its	its	PRON
ejpam-6461	16	12	real	real	ADJ
ejpam-6461	16	13	and	and	CCONJ
ejpam-6461	16	14	imaginary	imaginary	ADJ
ejpam-6461	16	15	parts	part	NOUN
ejpam-6461	16	16	,	,	PUNCT
ejpam-6461	16	17	as	as	SCONJ
ejpam-6461	16	18	follows	follow	VERB
ejpam-6461	16	19	,	,	PUNCT
ejpam-6461	16	20	a	a	DET
ejpam-6461	16	21	=	=	SYM
ejpam-6461	16	22	b	b	PROPN
ejpam-6461	16	23	+	+	CCONJ
ejpam-6461	16	24	ic	ic	PROPN
ejpam-6461	16	25	,	,	PUNCT
ejpam-6461	16	26	where	where	SCONJ
ejpam-6461	16	27	b	b	NOUN
ejpam-6461	16	28	=	=	SYM
ejpam-6461	16	29	re(a	re(a	X
ejpam-6461	16	30	)	)	PUNCT
ejpam-6461	16	31	=	=	PUNCT
ejpam-6461	16	32	a+a∗	a+a∗	NUM
ejpam-6461	16	33	2	2	NUM
ejpam-6461	16	34	doi	doi	NOUN
ejpam-6461	16	35	:	:	PUNCT
ejpam-6461	16	36	https://doi.org/10.29020/nybg.ejpam.v18i3.6461	https://doi.org/10.29020/nybg.ejpam.v18i3.6461	DET
ejpam-6461	16	37	email	email	NOUN
ejpam-6461	16	38	addresses	address	VERB
ejpam-6461	16	39	:	:	PUNCT
ejpam-6461	16	40	manal.allabadi@uop.edu.jo	manal.allabadi@uop.edu.jo	ADJ
ejpam-6461	16	41	(	(	PUNCT
ejpam-6461	16	42	m.	m.	NOUN
ejpam-6461	16	43	al	al	PROPN
ejpam-6461	16	44	-	-	PUNCT
ejpam-6461	16	45	labadi	labadi	NOUN
ejpam-6461	16	46	)	)	PUNCT
ejpam-6461	16	47	,	,	PUNCT
ejpam-6461	16	48	waudeh@uop.edu.jo	waudeh@uop.edu.jo	NOUN
ejpam-6461	16	49	(	(	PUNCT
ejpam-6461	16	50	w.	w.	PROPN
ejpam-6461	16	51	audeh	audeh	PROPN
ejpam-6461	16	52	)	)	PUNCT
ejpam-6461	16	53	,	,	PUNCT
ejpam-6461	16	54	rajaa.alnaimi@uop.edu.jo	rajaa.alnaimi@uop.edu.jo	PROPN
ejpam-6461	16	55	(	(	PUNCT
ejpam-6461	16	56	r.	r.	PROPN
ejpam-6461	16	57	al	al	PROPN
ejpam-6461	16	58	-	-	PUNCT
ejpam-6461	16	59	naimi	naimi	NOUN
ejpam-6461	16	60	)	)	PUNCT
ejpam-6461	16	61	,	,	PUNCT
ejpam-6461	16	62	jamal.oudetallah@uop.edu.jo	jamal.oudetallah@uop.edu.jo	PROPN
ejpam-6461	16	63	(	(	PUNCT
ejpam-6461	16	64	j.	j.	PROPN
ejpam-6461	16	65	oudetallah	oudetallah	PROPN
ejpam-6461	16	66	)	)	PUNCT
ejpam-6461	16	67	,	,	PUNCT
ejpam-6461	16	68	e	e	PROPN
ejpam-6461	16	69	almuhur@asu.edu.jo	almuhur@asu.edu.jo	NOUN
ejpam-6461	16	70	(	(	PUNCT
ejpam-6461	16	71	e.	e.	PROPN
ejpam-6461	16	72	almuhur	almuhur	PROPN
ejpam-6461	16	73	)	)	PUNCT
ejpam-6461	16	74	,	,	PUNCT
ejpam-6461	16	75	nkkhan@taibahu.edu.sa	nkkhan@taibahu.edu.sa	PROPN
ejpam-6461	16	76	(	(	PUNCT
ejpam-6461	16	77	n.	n.	PROPN
ejpam-6461	16	78	khan	khan	PROPN
ejpam-6461	16	79	)	)	PUNCT
ejpam-6461	16	80	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6461	16	81	1	1	NUM
ejpam-6461	16	82	copyright	copyright	NOUN
ejpam-6461	16	83	:	:	PUNCT
ejpam-6461	16	84	©	©	PROPN
ejpam-6461	16	85	2025	2025	NUM
ejpam-6461	16	86	the	the	DET
ejpam-6461	16	87	author(s	author(s	NOUN
ejpam-6461	16	88	)	)	PUNCT
ejpam-6461	16	89	.	.	PUNCT
ejpam-6461	17	1	(	(	PUNCT
ejpam-6461	17	2	cc	cc	NOUN
ejpam-6461	17	3	by	by	ADP
ejpam-6461	17	4	-	-	PUNCT
ejpam-6461	17	5	nc	nc	PROPN
ejpam-6461	17	6	4.0	4.0	NUM
ejpam-6461	17	7	)	)	PUNCT
ejpam-6461	17	8	m.	m.	NOUN
ejpam-6461	17	9	al	al	PROPN
ejpam-6461	17	10	-	-	PUNCT
ejpam-6461	17	11	labadi	labadi	PROPN
ejpam-6461	17	12	et	et	PROPN
ejpam-6461	17	13	al	al	PROPN
ejpam-6461	17	14	.	.	PUNCT
ejpam-6461	17	15	/	/	SYM
ejpam-6461	17	16	eur	eur	PROPN
ejpam-6461	17	17	.	.	PUNCT
ejpam-6461	18	1	j.	j.	PROPN
ejpam-6461	18	2	pure	pure	PROPN
ejpam-6461	18	3	appl	appl	PROPN
ejpam-6461	18	4	.	.	PROPN
ejpam-6461	18	5	math	math	PROPN
ejpam-6461	18	6	,	,	PUNCT
ejpam-6461	18	7	18	18	NUM
ejpam-6461	18	8	(	(	PUNCT
ejpam-6461	18	9	3	3	NUM
ejpam-6461	18	10	)	)	PUNCT
ejpam-6461	18	11	(	(	PUNCT
ejpam-6461	18	12	2025	2025	NUM
ejpam-6461	18	13	)	)	PUNCT
ejpam-6461	18	14	,	,	PUNCT
ejpam-6461	18	15	6461	6461	NUM
ejpam-6461	18	16	2	2	NUM
ejpam-6461	18	17	of	of	ADP
ejpam-6461	18	18	13	13	NUM
ejpam-6461	18	19	and	and	CCONJ
ejpam-6461	18	20	c	c	NOUN
ejpam-6461	18	21	=	=	SYM
ejpam-6461	18	22	im(a	im(a	X
ejpam-6461	18	23	)	)	PUNCT
ejpam-6461	19	1	=	=	SYM
ejpam-6461	19	2	a−a∗	a−a∗	PROPN
ejpam-6461	19	3	2i	2i	NOUN
ejpam-6461	19	4	.	.	PUNCT
ejpam-6461	20	1	note	note	VERB
ejpam-6461	20	2	that	that	SCONJ
ejpam-6461	20	3	re(a	re(a	NOUN
ejpam-6461	20	4	)	)	PUNCT
ejpam-6461	20	5	and	and	CCONJ
ejpam-6461	20	6	im(a	im(a	NOUN
ejpam-6461	20	7	)	)	PUNCT
ejpam-6461	20	8	are	be	AUX
ejpam-6461	20	9	self	self	NOUN
ejpam-6461	20	10	-	-	PUNCT
ejpam-6461	20	11	adjoint	adjoint	NOUN
ejpam-6461	20	12	operators	operator	NOUN
ejpam-6461	20	13	.	.	PUNCT
ejpam-6461	21	1	for	for	ADP
ejpam-6461	21	2	a	a	DET
ejpam-6461	21	3	∈	∈	PROPN
ejpam-6461	21	4	b(h	b(h	PROPN
ejpam-6461	21	5	)	)	PUNCT
ejpam-6461	21	6	,	,	PUNCT
ejpam-6461	21	7	the	the	DET
ejpam-6461	21	8	spectral	spectral	ADJ
ejpam-6461	21	9	radius	radius	NOUN
ejpam-6461	21	10	,	,	PUNCT
ejpam-6461	21	11	numerical	numerical	ADJ
ejpam-6461	21	12	radius	radius	NOUN
ejpam-6461	21	13	and	and	CCONJ
ejpam-6461	21	14	the	the	DET
ejpam-6461	21	15	usual	usual	ADJ
ejpam-6461	21	16	operator	operator	NOUN
ejpam-6461	21	17	norm	norm	NOUN
ejpam-6461	21	18	of	of	ADP
ejpam-6461	21	19	a	a	PRON
ejpam-6461	21	20	are	be	AUX
ejpam-6461	21	21	given	give	VERB
ejpam-6461	21	22	,	,	PUNCT
ejpam-6461	21	23	respectively	respectively	ADV
ejpam-6461	21	24	,	,	PUNCT
ejpam-6461	21	25	by	by	ADP
ejpam-6461	21	26	r(a	r(a	VERB
ejpam-6461	21	27	)	)	PUNCT
ejpam-6461	21	28	=	=	SYM
ejpam-6461	21	29	max	max	PROPN
ejpam-6461	21	30	λ∈σ(a	λ∈σ(a	PROPN
ejpam-6461	21	31	)	)	PUNCT
ejpam-6461	21	32	{	{	PUNCT
ejpam-6461	21	33	|λ|	|λ|	NOUN
ejpam-6461	21	34	}	}	PUNCT
ejpam-6461	21	35	,	,	PUNCT
ejpam-6461	21	36	w(a	w(a	NOUN
ejpam-6461	21	37	)	)	PUNCT
ejpam-6461	22	1	=	=	SYM
ejpam-6461	22	2	sup	sup	NOUN
ejpam-6461	22	3	||x||=1	||x||=1	NOUN
ejpam-6461	22	4	|⟨ax	|⟨ax	VERB
ejpam-6461	22	5	,	,	PUNCT
ejpam-6461	22	6	x⟩|	x⟩|	PROPN
ejpam-6461	22	7	and	and	CCONJ
ejpam-6461	22	8	∥a∥	∥a∥	NOUN
ejpam-6461	22	9	=	=	SYM
ejpam-6461	22	10	sup	sup	NOUN
ejpam-6461	22	11	||x||=||y||=1	||x||=||y||=1	NOUN
ejpam-6461	22	12	|⟨ax	|⟨ax	VERB
ejpam-6461	22	13	,	,	PUNCT
ejpam-6461	22	14	y⟩|	y⟩|	NUM
ejpam-6461	22	15	.	.	PUNCT
ejpam-6461	23	1	the	the	DET
ejpam-6461	23	2	computation	computation	NOUN
ejpam-6461	23	3	process	process	NOUN
ejpam-6461	23	4	to	to	PART
ejpam-6461	23	5	reach	reach	VERB
ejpam-6461	23	6	the	the	DET
ejpam-6461	23	7	exact	exact	ADJ
ejpam-6461	23	8	value	value	NOUN
ejpam-6461	23	9	of	of	ADP
ejpam-6461	23	10	w(a	w(a	PROPN
ejpam-6461	23	11	)	)	PUNCT
ejpam-6461	23	12	is	be	AUX
ejpam-6461	23	13	not	not	PART
ejpam-6461	23	14	always	always	ADV
ejpam-6461	23	15	of	of	ADP
ejpam-6461	23	16	that	that	DET
ejpam-6461	23	17	simplicity	simplicity	NOUN
ejpam-6461	23	18	,	,	PUNCT
ejpam-6461	23	19	this	this	PRON
ejpam-6461	23	20	gave	give	VERB
ejpam-6461	23	21	the	the	DET
ejpam-6461	23	22	authors	author	NOUN
ejpam-6461	23	23	the	the	DET
ejpam-6461	23	24	area	area	NOUN
ejpam-6461	23	25	to	to	PART
ejpam-6461	23	26	obtain	obtain	VERB
ejpam-6461	23	27	lower	low	ADJ
ejpam-6461	23	28	and	and	CCONJ
ejpam-6461	23	29	upper	upper	ADJ
ejpam-6461	23	30	bounds	bound	NOUN
ejpam-6461	23	31	of	of	ADP
ejpam-6461	23	32	w(a	w(a	PROPN
ejpam-6461	23	33	)	)	PUNCT
ejpam-6461	23	34	,	,	PUNCT
ejpam-6461	23	35	these	these	DET
ejpam-6461	23	36	bounds	bound	NOUN
ejpam-6461	23	37	are	be	AUX
ejpam-6461	23	38	usually	usually	ADV
ejpam-6461	23	39	in	in	ADP
ejpam-6461	23	40	terms	term	NOUN
ejpam-6461	23	41	of	of	ADP
ejpam-6461	23	42	||a||	||a||	NOUN
ejpam-6461	23	43	.	.	PUNCT
ejpam-6461	24	1	for	for	ADP
ejpam-6461	24	2	recent	recent	ADJ
ejpam-6461	24	3	studies	study	NOUN
ejpam-6461	24	4	of	of	ADP
ejpam-6461	24	5	numerical	numerical	ADJ
ejpam-6461	24	6	radius	radius	PROPN
ejpam-6461	24	7	inequalities	inequality	NOUN
ejpam-6461	24	8	of	of	ADP
ejpam-6461	24	9	operaors	operaor	NOUN
ejpam-6461	24	10	,	,	PUNCT
ejpam-6461	24	11	we	we	PRON
ejpam-6461	24	12	refer	refer	VERB
ejpam-6461	24	13	the	the	DET
ejpam-6461	24	14	reader	reader	NOUN
ejpam-6461	24	15	to	to	ADP
ejpam-6461	24	16	[	[	X
ejpam-6461	24	17	5]-[6	5]-[6	NUM
ejpam-6461	24	18	]	]	PUNCT
ejpam-6461	24	19	,	,	PUNCT
ejpam-6461	25	1	[	[	X
ejpam-6461	25	2	11]-[12	11]-[12	X
ejpam-6461	25	3	]	]	X
ejpam-6461	25	4	and	and	CCONJ
ejpam-6461	25	5	[	[	X
ejpam-6461	25	6	7	7	NUM
ejpam-6461	25	7	]	]	PUNCT
ejpam-6461	25	8	.	.	PUNCT
ejpam-6461	26	1	the	the	DET
ejpam-6461	26	2	most	most	ADV
ejpam-6461	26	3	known	know	VERB
ejpam-6461	26	4	famous	famous	ADJ
ejpam-6461	26	5	lower	low	ADJ
ejpam-6461	26	6	and	and	CCONJ
ejpam-6461	26	7	upper	upper	ADJ
ejpam-6461	26	8	bounds	bound	NOUN
ejpam-6461	26	9	for	for	ADP
ejpam-6461	26	10	w(a	w(a	NOUN
ejpam-6461	26	11	)	)	PUNCT
ejpam-6461	26	12	is	be	AUX
ejpam-6461	26	13	the	the	DET
ejpam-6461	26	14	following	following	ADJ
ejpam-6461	26	15	equivalence	equivalence	NOUN
ejpam-6461	26	16	between	between	ADP
ejpam-6461	26	17	w(a	w(a	PROPN
ejpam-6461	26	18	)	)	PUNCT
ejpam-6461	26	19	and	and	CCONJ
ejpam-6461	26	20	||a||	||a||	PROPN
ejpam-6461	26	21	,	,	PUNCT
ejpam-6461	26	22	see	see	VERB
ejpam-6461	26	23	[	[	X
ejpam-6461	26	24	22	22	NUM
ejpam-6461	26	25	]	]	PUNCT
ejpam-6461	26	26	,	,	PUNCT
ejpam-6461	26	27	∥a∥	∥a∥	VERB
ejpam-6461	26	28	2	2	NUM
ejpam-6461	26	29	≤	≤	NOUN
ejpam-6461	26	30	w(a	w(a	PUNCT
ejpam-6461	26	31	)	)	PUNCT
ejpam-6461	26	32	≤	≤	NUM
ejpam-6461	26	33	∥a∥	∥a∥	NOUN
ejpam-6461	26	34	.	.	PUNCT
ejpam-6461	27	1	(	(	PUNCT
ejpam-6461	27	2	1	1	X
ejpam-6461	27	3	)	)	PUNCT
ejpam-6461	27	4	several	several	ADJ
ejpam-6461	27	5	improvements	improvement	NOUN
ejpam-6461	27	6	and	and	CCONJ
ejpam-6461	27	7	generalizations	generalization	NOUN
ejpam-6461	27	8	of	of	ADP
ejpam-6461	27	9	inequality	inequality	NOUN
ejpam-6461	27	10	(	(	PUNCT
ejpam-6461	27	11	1	1	NUM
ejpam-6461	27	12	)	)	PUNCT
ejpam-6461	27	13	has	have	AUX
ejpam-6461	27	14	been	be	AUX
ejpam-6461	27	15	given	give	VERB
ejpam-6461	27	16	.	.	PUNCT
ejpam-6461	28	1	for	for	ADP
ejpam-6461	28	2	example	example	NOUN
ejpam-6461	28	3	,	,	PUNCT
ejpam-6461	28	4	in	in	ADP
ejpam-6461	28	5	[	[	PUNCT
ejpam-6461	28	6	26	26	NUM
ejpam-6461	28	7	]	]	PUNCT
ejpam-6461	28	8	,	,	PUNCT
ejpam-6461	28	9	it	it	PRON
ejpam-6461	28	10	is	be	AUX
ejpam-6461	28	11	shown	show	VERB
ejpam-6461	28	12	that	that	SCONJ
ejpam-6461	28	13	w(a	w(a	NOUN
ejpam-6461	28	14	)	)	PUNCT
ejpam-6461	28	15	≤	≤	NUM
ejpam-6461	28	16	1	1	NUM
ejpam-6461	28	17	2	2	NUM
ejpam-6461	28	18	||	||	NOUN
ejpam-6461	28	19	|a|+	|a|+	NOUN
ejpam-6461	28	20	|a∗|	|a∗|	PUNCT
ejpam-6461	28	21	||	||	PUNCT
ejpam-6461	29	1	(	(	PUNCT
ejpam-6461	29	2	2	2	NUM
ejpam-6461	29	3	)	)	PUNCT
ejpam-6461	29	4	and	and	CCONJ
ejpam-6461	29	5	w(a	w(a	NUM
ejpam-6461	29	6	)	)	PUNCT
ejpam-6461	29	7	≤	≤	NUM
ejpam-6461	29	8	1	1	NUM
ejpam-6461	29	9	2	2	NUM
ejpam-6461	29	10	(	(	PUNCT
ejpam-6461	29	11	||	||	NOUN
ejpam-6461	29	12	a	a	DET
ejpam-6461	29	13	||+	||+	NOUN
ejpam-6461	29	14	||	||	PROPN
ejpam-6461	29	15	a2	a2	PROPN
ejpam-6461	30	1	||	||	NOUN
ejpam-6461	31	1	1	1	NUM
ejpam-6461	31	2	2	2	NUM
ejpam-6461	31	3	)	)	PUNCT
ejpam-6461	31	4	,	,	PUNCT
ejpam-6461	31	5	(	(	PUNCT
ejpam-6461	31	6	3	3	X
ejpam-6461	31	7	)	)	PUNCT
ejpam-6461	31	8	as	as	ADP
ejpam-6461	31	9	a	a	DET
ejpam-6461	31	10	refinement	refinement	NOUN
ejpam-6461	31	11	of	of	ADP
ejpam-6461	31	12	the	the	DET
ejpam-6461	31	13	second	second	ADJ
ejpam-6461	31	14	inequality	inequality	NOUN
ejpam-6461	31	15	in	in	ADP
ejpam-6461	31	16	(	(	PUNCT
ejpam-6461	31	17	1	1	NUM
ejpam-6461	31	18	)	)	PUNCT
ejpam-6461	31	19	.	.	PUNCT
ejpam-6461	32	1	in	in	ADP
ejpam-6461	32	2	[	[	X
ejpam-6461	32	3	25	25	NUM
ejpam-6461	32	4	]	]	PUNCT
ejpam-6461	32	5	,	,	PUNCT
ejpam-6461	32	6	another	another	DET
ejpam-6461	32	7	refinement	refinement	NOUN
ejpam-6461	32	8	of	of	ADP
ejpam-6461	32	9	the	the	DET
ejpam-6461	32	10	second	second	ADJ
ejpam-6461	32	11	inequality	inequality	NOUN
ejpam-6461	32	12	in	in	ADP
ejpam-6461	32	13	(	(	PUNCT
ejpam-6461	32	14	1	1	X
ejpam-6461	32	15	)	)	PUNCT
ejpam-6461	32	16	has	have	AUX
ejpam-6461	32	17	been	be	AUX
ejpam-6461	32	18	given	give	VERB
ejpam-6461	32	19	as	as	SCONJ
ejpam-6461	32	20	follows	follow	VERB
ejpam-6461	32	21	:	:	PUNCT
ejpam-6461	32	22	w2(a	w2(a	NOUN
ejpam-6461	32	23	)	)	PUNCT
ejpam-6461	32	24	≤	≤	NOUN
ejpam-6461	32	25	1	1	NUM
ejpam-6461	32	26	2	2	NUM
ejpam-6461	32	27	||	||	NOUN
ejpam-6461	33	1	a∗a+aa∗	a∗a+aa∗	NOUN
ejpam-6461	33	2	||	||	NOUN
ejpam-6461	33	3	.	.	PUNCT
ejpam-6461	34	1	(	(	PUNCT
ejpam-6461	34	2	4	4	X
ejpam-6461	34	3	)	)	PUNCT
ejpam-6461	34	4	this	this	PRON
ejpam-6461	34	5	bound	bind	VERB
ejpam-6461	34	6	for	for	ADP
ejpam-6461	34	7	the	the	DET
ejpam-6461	34	8	numerical	numerical	ADJ
ejpam-6461	34	9	radius	radius	NOUN
ejpam-6461	34	10	is	be	AUX
ejpam-6461	34	11	given	give	VERB
ejpam-6461	34	12	as	as	ADP
ejpam-6461	34	13	one	one	NUM
ejpam-6461	34	14	of	of	ADP
ejpam-6461	34	15	the	the	DET
ejpam-6461	34	16	sharpest	sharp	ADJ
ejpam-6461	34	17	simple	simple	ADJ
ejpam-6461	34	18	bounds	bound	NOUN
ejpam-6461	34	19	in	in	ADP
ejpam-6461	34	20	the	the	DET
ejpam-6461	34	21	literature	literature	NOUN
ejpam-6461	34	22	.	.	PUNCT
ejpam-6461	35	1	the	the	DET
ejpam-6461	35	2	author	author	NOUN
ejpam-6461	35	3	in	in	ADP
ejpam-6461	35	4	[	[	X
ejpam-6461	35	5	20	20	NUM
ejpam-6461	35	6	]	]	PUNCT
ejpam-6461	35	7	refines	refine	VERB
ejpam-6461	35	8	inequality	inequality	NOUN
ejpam-6461	35	9	(	(	PUNCT
ejpam-6461	35	10	3	3	NUM
ejpam-6461	35	11	)	)	PUNCT
ejpam-6461	35	12	as	as	SCONJ
ejpam-6461	35	13	follows	follow	VERB
ejpam-6461	35	14	,	,	PUNCT
ejpam-6461	35	15	w(a	w(a	NOUN
ejpam-6461	35	16	)	)	PUNCT
ejpam-6461	35	17	≤	≤	NUM
ejpam-6461	35	18	1	1	NUM
ejpam-6461	35	19	2	2	NUM
ejpam-6461	35	20	(	(	PUNCT
ejpam-6461	35	21	||	||	NOUN
ejpam-6461	35	22	a	a	DET
ejpam-6461	35	23	||+	||+	NOUN
ejpam-6461	35	24	√	√	PUNCT
ejpam-6461	35	25	r(|a||a∗|	r(|a||a∗|	PROPN
ejpam-6461	35	26	)	)	PUNCT
ejpam-6461	35	27	)	)	PUNCT
ejpam-6461	35	28	.	.	PUNCT
ejpam-6461	36	1	(	(	PUNCT
ejpam-6461	36	2	5	5	X
ejpam-6461	36	3	)	)	PUNCT
ejpam-6461	36	4	recently	recently	ADV
ejpam-6461	36	5	,	,	PUNCT
ejpam-6461	36	6	in	in	ADP
ejpam-6461	36	7	[	[	X
ejpam-6461	36	8	23	23	NUM
ejpam-6461	36	9	]	]	PUNCT
ejpam-6461	36	10	,	,	PUNCT
ejpam-6461	36	11	it	it	PRON
ejpam-6461	36	12	is	be	AUX
ejpam-6461	36	13	shown	show	VERB
ejpam-6461	36	14	that	that	SCONJ
ejpam-6461	36	15	:	:	PUNCT
ejpam-6461	36	16	w(ab∗	w(ab∗	X
ejpam-6461	36	17	)	)	PUNCT
ejpam-6461	36	18	≤	≤	NUM
ejpam-6461	37	1	1	1	NUM
ejpam-6461	37	2	4	4	NUM
ejpam-6461	37	3	||	||	NOUN
ejpam-6461	37	4	|a|+	|a|+	NOUN
ejpam-6461	37	5	|b|	|b|	X
ejpam-6461	37	6	||	||	NOUN
ejpam-6461	38	1	||	||	NUM
ejpam-6461	38	2	|a∗|+	|a∗|+	NOUN
ejpam-6461	38	3	|b∗|	|b∗|	PUNCT
ejpam-6461	39	1	||	||	NOUN
ejpam-6461	39	2	.	.	PUNCT
ejpam-6461	40	1	(	(	PUNCT
ejpam-6461	40	2	6	6	NUM
ejpam-6461	40	3	)	)	PUNCT
ejpam-6461	40	4	in	in	ADP
ejpam-6461	40	5	this	this	DET
ejpam-6461	40	6	paper	paper	NOUN
ejpam-6461	40	7	,	,	PUNCT
ejpam-6461	40	8	we	we	PRON
ejpam-6461	40	9	give	give	VERB
ejpam-6461	40	10	a	a	DET
ejpam-6461	40	11	remarkable	remarkable	ADJ
ejpam-6461	40	12	refinement	refinement	NOUN
ejpam-6461	40	13	and	and	CCONJ
ejpam-6461	40	14	a	a	DET
ejpam-6461	40	15	generalization	generalization	NOUN
ejpam-6461	40	16	of	of	ADP
ejpam-6461	40	17	inequality	inequality	NOUN
ejpam-6461	40	18	(	(	PUNCT
ejpam-6461	40	19	4	4	NUM
ejpam-6461	40	20	)	)	PUNCT
ejpam-6461	40	21	.	.	PUNCT
ejpam-6461	41	1	an	an	DET
ejpam-6461	41	2	attractive	attractive	ADJ
ejpam-6461	41	3	generalization	generalization	NOUN
ejpam-6461	41	4	of	of	ADP
ejpam-6461	41	5	inequality	inequality	NOUN
ejpam-6461	41	6	(	(	PUNCT
ejpam-6461	41	7	5	5	NUM
ejpam-6461	41	8	)	)	PUNCT
ejpam-6461	41	9	is	be	AUX
ejpam-6461	41	10	also	also	ADV
ejpam-6461	41	11	given	give	VERB
ejpam-6461	41	12	.	.	PUNCT
ejpam-6461	42	1	moreover	moreover	ADV
ejpam-6461	42	2	,	,	PUNCT
ejpam-6461	42	3	we	we	PRON
ejpam-6461	42	4	give	give	VERB
ejpam-6461	42	5	an	an	DET
ejpam-6461	42	6	inequality	inequality	NOUN
ejpam-6461	42	7	that	that	PRON
ejpam-6461	42	8	is	be	AUX
ejpam-6461	42	9	equivalent	equivalent	ADJ
ejpam-6461	42	10	,	,	PUNCT
ejpam-6461	42	11	if	if	SCONJ
ejpam-6461	42	12	a	a	PRON
ejpam-6461	42	13	and	and	CCONJ
ejpam-6461	42	14	b	b	NOUN
ejpam-6461	42	15	are	be	AUX
ejpam-6461	42	16	self	self	NOUN
ejpam-6461	42	17	-	-	PUNCT
ejpam-6461	42	18	adjoint	adjoint	NOUN
ejpam-6461	42	19	,	,	PUNCT
ejpam-6461	42	20	to	to	ADP
ejpam-6461	42	21	inequality	inequality	NOUN
ejpam-6461	42	22	(	(	PUNCT
ejpam-6461	42	23	6	6	NUM
ejpam-6461	42	24	)	)	PUNCT
ejpam-6461	42	25	.	.	PUNCT
ejpam-6461	43	1	a	a	DET
ejpam-6461	43	2	new	new	ADJ
ejpam-6461	43	3	proof	proof	NOUN
ejpam-6461	43	4	of	of	ADP
ejpam-6461	43	5	inequality	inequality	NOUN
ejpam-6461	43	6	(	(	PUNCT
ejpam-6461	43	7	2	2	NUM
ejpam-6461	43	8	)	)	PUNCT
ejpam-6461	43	9	is	be	AUX
ejpam-6461	43	10	obtained	obtain	VERB
ejpam-6461	43	11	.	.	PUNCT
ejpam-6461	44	1	several	several	ADJ
ejpam-6461	44	2	numerical	numerical	PROPN
ejpam-6461	44	3	radius	radius	PROPN
ejpam-6461	44	4	inequalities	inequality	NOUN
ejpam-6461	44	5	are	be	AUX
ejpam-6461	44	6	included	include	VERB
ejpam-6461	44	7	.	.	PUNCT
ejpam-6461	45	1	m.	m.	NOUN
ejpam-6461	45	2	al	al	PROPN
ejpam-6461	45	3	-	-	PUNCT
ejpam-6461	45	4	labadi	labadi	PROPN
ejpam-6461	45	5	et	et	PROPN
ejpam-6461	45	6	al	al	PROPN
ejpam-6461	45	7	.	.	PUNCT
ejpam-6461	45	8	/	/	SYM
ejpam-6461	45	9	eur	eur	PROPN
ejpam-6461	45	10	.	.	PUNCT
ejpam-6461	46	1	j.	j.	PROPN
ejpam-6461	46	2	pure	pure	PROPN
ejpam-6461	46	3	appl	appl	PROPN
ejpam-6461	46	4	.	.	PROPN
ejpam-6461	46	5	math	math	PROPN
ejpam-6461	46	6	,	,	PUNCT
ejpam-6461	46	7	18	18	NUM
ejpam-6461	46	8	(	(	PUNCT
ejpam-6461	46	9	3	3	NUM
ejpam-6461	46	10	)	)	PUNCT
ejpam-6461	46	11	(	(	PUNCT
ejpam-6461	46	12	2025	2025	NUM
ejpam-6461	46	13	)	)	PUNCT
ejpam-6461	46	14	,	,	PUNCT
ejpam-6461	46	15	6461	6461	NUM
ejpam-6461	46	16	3	3	NUM
ejpam-6461	46	17	of	of	ADP
ejpam-6461	46	18	13	13	NUM
ejpam-6461	46	19	2	2	NUM
ejpam-6461	46	20	.	.	PUNCT
ejpam-6461	46	21	numerical	numerical	PROPN
ejpam-6461	46	22	radius	radius	PROPN
ejpam-6461	46	23	inequalities	inequality	NOUN
ejpam-6461	46	24	via	via	ADP
ejpam-6461	46	25	inner	inner	ADJ
ejpam-6461	46	26	product	product	NOUN
ejpam-6461	46	27	the	the	DET
ejpam-6461	46	28	main	main	ADJ
ejpam-6461	46	29	result	result	NOUN
ejpam-6461	46	30	in	in	ADP
ejpam-6461	46	31	this	this	DET
ejpam-6461	46	32	section	section	NOUN
ejpam-6461	46	33	is	be	AUX
ejpam-6461	46	34	a	a	DET
ejpam-6461	46	35	numerical	numerical	ADJ
ejpam-6461	46	36	radius	radius	NOUN
ejpam-6461	46	37	inequality	inequality	NOUN
ejpam-6461	46	38	for	for	ADP
ejpam-6461	46	39	finite	finite	ADJ
ejpam-6461	46	40	sums	sum	NOUN
ejpam-6461	46	41	of	of	ADP
ejpam-6461	46	42	operators	operator	NOUN
ejpam-6461	46	43	.	.	PUNCT
ejpam-6461	47	1	to	to	PART
ejpam-6461	47	2	prove	prove	VERB
ejpam-6461	47	3	this	this	DET
ejpam-6461	47	4	result	result	NOUN
ejpam-6461	47	5	and	and	CCONJ
ejpam-6461	47	6	to	to	PART
ejpam-6461	47	7	make	make	VERB
ejpam-6461	47	8	some	some	DET
ejpam-6461	47	9	comparisons	comparison	NOUN
ejpam-6461	47	10	between	between	ADP
ejpam-6461	47	11	some	some	PRON
ejpam-6461	47	12	of	of	ADP
ejpam-6461	47	13	its	its	PRON
ejpam-6461	47	14	special	special	ADJ
ejpam-6461	47	15	cases	case	NOUN
ejpam-6461	47	16	and	and	CCONJ
ejpam-6461	47	17	recent	recent	ADJ
ejpam-6461	47	18	inequalities	inequality	NOUN
ejpam-6461	47	19	proved	prove	VERB
ejpam-6461	47	20	by	by	ADP
ejpam-6461	47	21	different	different	ADJ
ejpam-6461	47	22	authors	author	NOUN
ejpam-6461	47	23	,	,	PUNCT
ejpam-6461	47	24	we	we	PRON
ejpam-6461	47	25	need	need	VERB
ejpam-6461	47	26	the	the	DET
ejpam-6461	47	27	following	follow	VERB
ejpam-6461	47	28	lemmas	lemmas	PROPN
ejpam-6461	47	29	,	,	PUNCT
ejpam-6461	47	30	which	which	PRON
ejpam-6461	47	31	are	be	AUX
ejpam-6461	47	32	proved	prove	VERB
ejpam-6461	47	33	in	in	ADP
ejpam-6461	47	34	[	[	X
ejpam-6461	47	35	28	28	NUM
ejpam-6461	47	36	]	]	PUNCT
ejpam-6461	47	37	,	,	PUNCT
ejpam-6461	47	38	and	and	CCONJ
ejpam-6461	47	39	they	they	PRON
ejpam-6461	47	40	are	be	AUX
ejpam-6461	47	41	essential	essential	ADJ
ejpam-6461	47	42	in	in	ADP
ejpam-6461	47	43	our	our	PRON
ejpam-6461	47	44	analysis	analysis	NOUN
ejpam-6461	47	45	.	.	PUNCT
ejpam-6461	48	1	lemma	lemma	PROPN
ejpam-6461	48	2	1	1	X
ejpam-6461	48	3	.	.	PUNCT
ejpam-6461	49	1	let	let	VERB
ejpam-6461	49	2	a	a	DET
ejpam-6461	49	3	∈	∈	PROPN
ejpam-6461	49	4	b(h	b(h	PROPN
ejpam-6461	49	5	)	)	PUNCT
ejpam-6461	49	6	be	be	AUX
ejpam-6461	49	7	positive	positive	ADJ
ejpam-6461	49	8	semidefinite	semidefinite	NOUN
ejpam-6461	49	9	and	and	CCONJ
ejpam-6461	49	10	x	x	PUNCT
ejpam-6461	49	11	∈	∈	NOUN
ejpam-6461	49	12	h	h	NOUN
ejpam-6461	49	13	such	such	ADJ
ejpam-6461	49	14	that	that	SCONJ
ejpam-6461	49	15	∥x∥	∥x∥	NOUN
ejpam-6461	49	16	≤	≤	ADV
ejpam-6461	49	17	1	1	NUM
ejpam-6461	49	18	.	.	PUNCT
ejpam-6461	50	1	then	then	ADV
ejpam-6461	50	2	(	(	PUNCT
ejpam-6461	50	3	i	i	NOUN
ejpam-6461	50	4	)	)	PUNCT
ejpam-6461	50	5	⟨ax	⟨ax	VERB
ejpam-6461	50	6	,	,	PUNCT
ejpam-6461	50	7	x⟩r	x⟩r	NUM
ejpam-6461	50	8	≤	≤	NUM
ejpam-6461	50	9	⟨arx	⟨arx	PROPN
ejpam-6461	50	10	,	,	PUNCT
ejpam-6461	50	11	x⟩	x⟩	PUNCT
ejpam-6461	50	12	for	for	ADP
ejpam-6461	50	13	r	r	NOUN
ejpam-6461	50	14	≥	≥	NUM
ejpam-6461	50	15	1	1	NUM
ejpam-6461	50	16	.	.	PUNCT
ejpam-6461	50	17	(	(	PUNCT
ejpam-6461	50	18	ii	ii	NOUN
ejpam-6461	50	19	)	)	PUNCT
ejpam-6461	50	20	⟨arx	⟨arx	PROPN
ejpam-6461	50	21	,	,	PUNCT
ejpam-6461	50	22	x⟩	x⟩	PUNCT
ejpam-6461	50	23	≤	≤	PROPN
ejpam-6461	50	24	⟨ax	⟨ax	VERB
ejpam-6461	50	25	,	,	PUNCT
ejpam-6461	50	26	x⟩r	x⟩r	NUM
ejpam-6461	50	27	for	for	ADP
ejpam-6461	50	28	0	0	NUM
ejpam-6461	50	29	<	<	X
ejpam-6461	50	30	r	r	NOUN
ejpam-6461	50	31	≤	≤	NUM
ejpam-6461	50	32	1	1	NUM
ejpam-6461	50	33	.	.	PUNCT
ejpam-6461	51	1	lemma	lemma	PROPN
ejpam-6461	51	2	2	2	X
ejpam-6461	51	3	.	.	PUNCT
ejpam-6461	51	4	let	let	VERB
ejpam-6461	51	5	a	a	DET
ejpam-6461	51	6	∈	∈	PROPN
ejpam-6461	51	7	b(h	b(h	PROPN
ejpam-6461	51	8	)	)	PUNCT
ejpam-6461	51	9	be	be	VERB
ejpam-6461	51	10	self	self	NOUN
ejpam-6461	51	11	-	-	PUNCT
ejpam-6461	51	12	adjoint	adjoint	NOUN
ejpam-6461	51	13	operator	operator	NOUN
ejpam-6461	51	14	and	and	CCONJ
ejpam-6461	51	15	x	x	SYM
ejpam-6461	51	16	∈	∈	PROPN
ejpam-6461	51	17	h.	h.	NOUN
ejpam-6461	51	18	then	then	ADV
ejpam-6461	51	19	|⟨ax	|⟨ax	VERB
ejpam-6461	51	20	,	,	PUNCT
ejpam-6461	51	21	x⟩|	x⟩|	VERB
ejpam-6461	51	22	≤	≤	NOUN
ejpam-6461	51	23	⟨|a|x	⟨|a|x	NOUN
ejpam-6461	51	24	,	,	PUNCT
ejpam-6461	51	25	x⟩	x⟩	X
ejpam-6461	51	26	.	.	PUNCT
ejpam-6461	52	1	(	(	PUNCT
ejpam-6461	52	2	7	7	X
ejpam-6461	52	3	)	)	PUNCT
ejpam-6461	52	4	lemma	lemma	PROPN
ejpam-6461	52	5	3	3	X
ejpam-6461	52	6	.	.	PUNCT
ejpam-6461	53	1	let	let	VERB
ejpam-6461	53	2	a	a	DET
ejpam-6461	53	3	∈	∈	PROPN
ejpam-6461	53	4	b(h	b(h	PROPN
ejpam-6461	53	5	)	)	PUNCT
ejpam-6461	53	6	and	and	CCONJ
ejpam-6461	53	7	x	x	X
ejpam-6461	53	8	,	,	PUNCT
ejpam-6461	53	9	y	y	PROPN
ejpam-6461	53	10	∈	∈	PROPN
ejpam-6461	53	11	h	h	NOUN
ejpam-6461	53	12	be	be	AUX
ejpam-6461	53	13	any	any	DET
ejpam-6461	53	14	vectors	vector	NOUN
ejpam-6461	53	15	.	.	PUNCT
ejpam-6461	54	1	if	if	SCONJ
ejpam-6461	54	2	f	f	PROPN
ejpam-6461	54	3	and	and	CCONJ
ejpam-6461	54	4	g	g	PROPN
ejpam-6461	54	5	are	be	AUX
ejpam-6461	54	6	nonnegative	nonnegative	ADJ
ejpam-6461	54	7	continuous	continuous	ADJ
ejpam-6461	54	8	functions	function	NOUN
ejpam-6461	54	9	on	on	ADP
ejpam-6461	54	10	[	[	X
ejpam-6461	54	11	0,∞	0,∞	NOUN
ejpam-6461	54	12	)	)	PUNCT
ejpam-6461	54	13	satisfying	satisfy	VERB
ejpam-6461	54	14	the	the	DET
ejpam-6461	54	15	relation	relation	NOUN
ejpam-6461	54	16	f(a)g(a	f(a)g(a	NOUN
ejpam-6461	54	17	)	)	PUNCT
ejpam-6461	54	18	=	=	SYM
ejpam-6461	55	1	a	a	PRON
ejpam-6461	55	2	(	(	PUNCT
ejpam-6461	55	3	a	a	DET
ejpam-6461	55	4	∈	∈	PROPN
ejpam-6461	56	1	[	[	X
ejpam-6461	56	2	0,∞	0,∞	NOUN
ejpam-6461	56	3	)	)	PUNCT
ejpam-6461	56	4	)	)	PUNCT
ejpam-6461	56	5	,	,	PUNCT
ejpam-6461	56	6	then	then	ADV
ejpam-6461	56	7	|⟨ax	|⟨ax	VERB
ejpam-6461	56	8	,	,	PUNCT
ejpam-6461	56	9	y⟩|2	y⟩|2	NUM
ejpam-6461	56	10	≤	≤	NUM
ejpam-6461	56	11	⟨|a|x	⟨|a|x	NOUN
ejpam-6461	56	12	,	,	PUNCT
ejpam-6461	56	13	x⟩	x⟩	PUNCT
ejpam-6461	56	14	⟨|a∗|	⟨|a∗|	PUNCT
ejpam-6461	57	1	y	y	PROPN
ejpam-6461	57	2	,	,	PUNCT
ejpam-6461	57	3	y⟩	y⟩	NOUN
ejpam-6461	57	4	(	(	PUNCT
ejpam-6461	57	5	8)	8)	NUM
ejpam-6461	57	6	and	and	CCONJ
ejpam-6461	57	7	more	more	ADV
ejpam-6461	57	8	general	general	ADJ
ejpam-6461	57	9	|⟨ax	|⟨ax	VERB
ejpam-6461	57	10	,	,	PUNCT
ejpam-6461	57	11	y⟩|2	y⟩|2	NUM
ejpam-6461	57	12	≤	≤	NUM
ejpam-6461	57	13	〈	〈	PRON
ejpam-6461	57	14	f2	f2	PROPN
ejpam-6461	57	15	(	(	PUNCT
ejpam-6461	57	16	|a|)x	|a|)x	PROPN
ejpam-6461	57	17	,	,	PUNCT
ejpam-6461	57	18	x	x	SYM
ejpam-6461	57	19	〉	〉	NOUN
ejpam-6461	57	20	〈	〈	PROPN
ejpam-6461	57	21	g2	g2	PROPN
ejpam-6461	57	22	(	(	PUNCT
ejpam-6461	57	23	|a∗|	|a∗|	NUM
ejpam-6461	57	24	)	)	PUNCT
ejpam-6461	57	25	y	y	PROPN
ejpam-6461	57	26	,	,	PUNCT
ejpam-6461	57	27	y	y	PROPN
ejpam-6461	57	28	〉	〉	NOUN
ejpam-6461	57	29	.	.	PUNCT
ejpam-6461	58	1	(	(	PUNCT
ejpam-6461	58	2	9	9	X
ejpam-6461	58	3	)	)	PUNCT
ejpam-6461	58	4	the	the	DET
ejpam-6461	58	5	following	follow	VERB
ejpam-6461	58	6	lemma	lemma	PROPN
ejpam-6461	58	7	follows	follow	VERB
ejpam-6461	58	8	by	by	ADP
ejpam-6461	58	9	weyl	weyl	PROPN
ejpam-6461	58	10	’s	’s	PART
ejpam-6461	58	11	monotonocity	monotonocity	NOUN
ejpam-6461	58	12	principle	principle	NOUN
ejpam-6461	58	13	(	(	PUNCT
ejpam-6461	58	14	see	see	VERB
ejpam-6461	58	15	,	,	PUNCT
ejpam-6461	58	16	e.g.	e.g.	ADV
ejpam-6461	58	17	,	,	PUNCT
ejpam-6461	58	18	[	[	X
ejpam-6461	58	19	5	5	NUM
ejpam-6461	58	20	,	,	PUNCT
ejpam-6461	58	21	p.63	p.63	X
ejpam-6461	58	22	]	]	X
ejpam-6461	58	23	or	or	CCONJ
ejpam-6461	58	24	[	[	X
ejpam-6461	58	25	5	5	NUM
ejpam-6461	58	26	,	,	PUNCT
ejpam-6461	58	27	p.	p.	NOUN
ejpam-6461	58	28	20	20	NUM
ejpam-6461	58	29	]	]	PUNCT
ejpam-6461	58	30	)	)	PUNCT
ejpam-6461	58	31	.	.	PUNCT
ejpam-6461	59	1	lemma	lemma	PROPN
ejpam-6461	59	2	4	4	X
ejpam-6461	59	3	.	.	PUNCT
ejpam-6461	60	1	if	if	SCONJ
ejpam-6461	60	2	a	a	PRON
ejpam-6461	60	3	,	,	PUNCT
ejpam-6461	60	4	b	b	PROPN
ejpam-6461	60	5	∈	∈	PROPN
ejpam-6461	60	6	b(h	b(h	PROPN
ejpam-6461	60	7	)	)	PUNCT
ejpam-6461	60	8	are	be	AUX
ejpam-6461	60	9	positive	positive	ADJ
ejpam-6461	60	10	semidefinite	semidefinite	NOUN
ejpam-6461	60	11	such	such	ADJ
ejpam-6461	60	12	that	that	SCONJ
ejpam-6461	60	13	a	a	DET
ejpam-6461	60	14	≤	≤	PROPN
ejpam-6461	60	15	b.	b.	PROPN
ejpam-6461	60	16	then	then	ADV
ejpam-6461	60	17	||	||	VERB
ejpam-6461	60	18	a	a	DET
ejpam-6461	60	19	||	||	NOUN
ejpam-6461	60	20	≤	≤	NUM
ejpam-6461	60	21	||	||	PUNCT
ejpam-6461	61	1	b	b	X
ejpam-6461	61	2	||	||	NOUN
ejpam-6461	61	3	.	.	PUNCT
ejpam-6461	62	1	(	(	PUNCT
ejpam-6461	62	2	10	10	NUM
ejpam-6461	62	3	)	)	PUNCT
ejpam-6461	62	4	lemma	lemma	PROPN
ejpam-6461	62	5	5	5	X
ejpam-6461	62	6	.	.	PUNCT
ejpam-6461	63	1	let	let	VERB
ejpam-6461	63	2	a	a	DET
ejpam-6461	63	3	,	,	PUNCT
ejpam-6461	63	4	b	b	PROPN
ejpam-6461	63	5	≥	≥	NOUN
ejpam-6461	64	1	o.	o.	INTJ
ejpam-6461	64	2	then	then	ADV
ejpam-6461	64	3	||	||	PROPN
ejpam-6461	64	4	a+b	a+b	NUM
ejpam-6461	64	5	||2	||2	NOUN
ejpam-6461	64	6	≤	≤	NOUN
ejpam-6461	64	7	2||	2||	NUM
ejpam-6461	64	8	a2	a2	NOUN
ejpam-6461	64	9	+	+	NOUN
ejpam-6461	64	10	b2	b2	NOUN
ejpam-6461	64	11	||	||	NOUN
ejpam-6461	64	12	.	.	PUNCT
ejpam-6461	65	1	(	(	PUNCT
ejpam-6461	65	2	11	11	NUM
ejpam-6461	65	3	)	)	PUNCT
ejpam-6461	65	4	equality	equality	NOUN
ejpam-6461	65	5	holds	hold	VERB
ejpam-6461	65	6	iff	iff	PROPN
ejpam-6461	65	7	a	a	DET
ejpam-6461	65	8	=	=	X
ejpam-6461	65	9	b.	b.	NOUN
ejpam-6461	65	10	proof	proof	NOUN
ejpam-6461	65	11	.	.	PUNCT
ejpam-6461	66	1	it	it	PRON
ejpam-6461	66	2	is	be	AUX
ejpam-6461	66	3	well	well	ADV
ejpam-6461	66	4	known	know	VERB
ejpam-6461	67	1	that	that	SCONJ
ejpam-6461	67	2	||	||	NOUN
ejpam-6461	68	1	(	(	PUNCT
ejpam-6461	68	2	a	a	DET
ejpam-6461	68	3	+	+	X
ejpam-6461	68	4	b)2	b)2	ADJ
ejpam-6461	68	5	||	||	PUNCT
ejpam-6461	69	1	=	=	PUNCT
ejpam-6461	69	2	||	||	NOUN
ejpam-6461	69	3	a	a	DET
ejpam-6461	69	4	+	+	NOUN
ejpam-6461	69	5	b	b	NOUN
ejpam-6461	69	6	||2	||2	NOUN
ejpam-6461	69	7	(	(	PUNCT
ejpam-6461	69	8	since	since	SCONJ
ejpam-6461	69	9	a	a	DET
ejpam-6461	69	10	,	,	PUNCT
ejpam-6461	69	11	b	b	PROPN
ejpam-6461	69	12	≥	≥	NOUN
ejpam-6461	69	13	0	0	NUM
ejpam-6461	69	14	)	)	PUNCT
ejpam-6461	69	15	.	.	PUNCT
ejpam-6461	70	1	to	to	PART
ejpam-6461	70	2	reach	reach	VERB
ejpam-6461	70	3	inequality	inequality	NOUN
ejpam-6461	70	4	(	(	PUNCT
ejpam-6461	70	5	11	11	NUM
ejpam-6461	70	6	)	)	PUNCT
ejpam-6461	70	7	,	,	PUNCT
ejpam-6461	70	8	it	it	PRON
ejpam-6461	70	9	is	be	AUX
ejpam-6461	70	10	enough	enough	ADJ
ejpam-6461	70	11	to	to	PART
ejpam-6461	70	12	prove	prove	VERB
ejpam-6461	70	13	that	that	SCONJ
ejpam-6461	70	14	(	(	PUNCT
ejpam-6461	70	15	a	a	DET
ejpam-6461	70	16	+	+	NUM
ejpam-6461	70	17	b)2	b)2	ADJ
ejpam-6461	70	18	≤	≤	PROPN
ejpam-6461	70	19	2(a2	2(a2	NUM
ejpam-6461	70	20	+	+	NUM
ejpam-6461	70	21	b2	b2	NOUN
ejpam-6461	70	22	)	)	PUNCT
ejpam-6461	70	23	and	and	CCONJ
ejpam-6461	70	24	then	then	ADV
ejpam-6461	70	25	applying	apply	VERB
ejpam-6461	70	26	inequality	inequality	NOUN
ejpam-6461	70	27	(	(	PUNCT
ejpam-6461	70	28	10	10	NUM
ejpam-6461	70	29	)	)	PUNCT
ejpam-6461	70	30	.	.	PUNCT
ejpam-6461	71	1	now	now	ADV
ejpam-6461	71	2	,	,	PUNCT
ejpam-6461	71	3	2a2	2a2	NUM
ejpam-6461	71	4	+	+	CCONJ
ejpam-6461	71	5	2b2	2b2	NUM
ejpam-6461	71	6	−	−	NOUN
ejpam-6461	71	7	(	(	PUNCT
ejpam-6461	71	8	a+b)2	a+b)2	NOUN
ejpam-6461	71	9	=	=	PUNCT
ejpam-6461	71	10	2a2	2a2	NUM
ejpam-6461	71	11	+	+	CCONJ
ejpam-6461	71	12	2b2	2b2	NUM
ejpam-6461	71	13	−	−	NOUN
ejpam-6461	71	14	(	(	PUNCT
ejpam-6461	71	15	a2	a2	PROPN
ejpam-6461	71	16	+	+	NOUN
ejpam-6461	71	17	b2	b2	NOUN
ejpam-6461	71	18	+	+	PROPN
ejpam-6461	71	19	ab	ab	PROPN
ejpam-6461	71	20	+	+	PROPN
ejpam-6461	71	21	ba	ba	NOUN
ejpam-6461	71	22	)	)	PUNCT
ejpam-6461	71	23	=	=	SYM
ejpam-6461	71	24	a2	a2	PROPN
ejpam-6461	71	25	+	+	NOUN
ejpam-6461	71	26	b2	b2	NOUN
ejpam-6461	71	27	−ab	−ab	NOUN
ejpam-6461	71	28	−ba	−ba	NOUN
ejpam-6461	71	29	=	=	PUNCT
ejpam-6461	71	30	(	(	PUNCT
ejpam-6461	71	31	a−b)2	a−b)2	NUM
ejpam-6461	71	32	≥	≥	NOUN
ejpam-6461	71	33	0	0	NUM
ejpam-6461	71	34	(	(	PUNCT
ejpam-6461	71	35	since	since	SCONJ
ejpam-6461	71	36	a	a	DET
ejpam-6461	71	37	-	-	PUNCT
ejpam-6461	71	38	b	b	NOUN
ejpam-6461	71	39	is	be	AUX
ejpam-6461	71	40	hermitian	hermitian	ADJ
ejpam-6461	71	41	)	)	PUNCT
ejpam-6461	71	42	.	.	PUNCT
ejpam-6461	72	1	this	this	PRON
ejpam-6461	72	2	implies	imply	VERB
ejpam-6461	72	3	that	that	SCONJ
ejpam-6461	72	4	(	(	PUNCT
ejpam-6461	72	5	a+b)2	a+b)2	NOUN
ejpam-6461	72	6	≤	≤	ADJ
ejpam-6461	72	7	2(a2	2(a2	NUM
ejpam-6461	72	8	+	+	NUM
ejpam-6461	72	9	b2	b2	NOUN
ejpam-6461	72	10	)	)	PUNCT
ejpam-6461	72	11	,	,	PUNCT
ejpam-6461	72	12	and	and	CCONJ
ejpam-6461	72	13	so	so	ADV
ejpam-6461	72	14	we	we	PRON
ejpam-6461	72	15	reach	reach	VERB
ejpam-6461	72	16	our	our	PRON
ejpam-6461	72	17	claim	claim	NOUN
ejpam-6461	72	18	.	.	PUNCT
ejpam-6461	73	1	m.	m.	NOUN
ejpam-6461	73	2	al	al	PROPN
ejpam-6461	73	3	-	-	PUNCT
ejpam-6461	73	4	labadi	labadi	PROPN
ejpam-6461	73	5	et	et	PROPN
ejpam-6461	73	6	al	al	PROPN
ejpam-6461	73	7	.	.	PUNCT
ejpam-6461	73	8	/	/	SYM
ejpam-6461	73	9	eur	eur	PROPN
ejpam-6461	73	10	.	.	PUNCT
ejpam-6461	74	1	j.	j.	PROPN
ejpam-6461	74	2	pure	pure	PROPN
ejpam-6461	74	3	appl	appl	PROPN
ejpam-6461	74	4	.	.	PROPN
ejpam-6461	74	5	math	math	PROPN
ejpam-6461	74	6	,	,	PUNCT
ejpam-6461	74	7	18	18	NUM
ejpam-6461	74	8	(	(	PUNCT
ejpam-6461	74	9	3	3	NUM
ejpam-6461	74	10	)	)	PUNCT
ejpam-6461	74	11	(	(	PUNCT
ejpam-6461	74	12	2025	2025	NUM
ejpam-6461	74	13	)	)	PUNCT
ejpam-6461	74	14	,	,	PUNCT
ejpam-6461	74	15	6461	6461	NUM
ejpam-6461	74	16	4	4	NUM
ejpam-6461	74	17	of	of	ADP
ejpam-6461	74	18	13	13	NUM
ejpam-6461	74	19	theorem	theorem	NOUN
ejpam-6461	74	20	1	1	NUM
ejpam-6461	74	21	.	.	PUNCT
ejpam-6461	75	1	let	let	VERB
ejpam-6461	75	2	a1	a1	NOUN
ejpam-6461	75	3	,	,	PUNCT
ejpam-6461	75	4	a2	a2	PROPN
ejpam-6461	75	5	,	,	PUNCT
ejpam-6461	75	6	...	...	PUNCT
ejpam-6461	75	7	,	,	PUNCT
ejpam-6461	75	8	an	an	PRON
ejpam-6461	75	9	,	,	PUNCT
ejpam-6461	75	10	b1	b1	NOUN
ejpam-6461	75	11	,	,	PUNCT
ejpam-6461	75	12	b2	b2	NOUN
ejpam-6461	75	13	,	,	PUNCT
ejpam-6461	75	14	...	...	PUNCT
ejpam-6461	75	15	,	,	PUNCT
ejpam-6461	75	16	bn	bn	PART
ejpam-6461	75	17	be	be	AUX
ejpam-6461	75	18	operators	operator	NOUN
ejpam-6461	75	19	in	in	ADP
ejpam-6461	75	20	b(h	b(h	PROPN
ejpam-6461	75	21	)	)	PUNCT
ejpam-6461	75	22	.	.	PUNCT
ejpam-6461	76	1	if	if	SCONJ
ejpam-6461	76	2	f	f	PROPN
ejpam-6461	76	3	and	and	CCONJ
ejpam-6461	76	4	g	g	PROPN
ejpam-6461	76	5	are	be	AUX
ejpam-6461	76	6	nonnegative	nonnegative	ADJ
ejpam-6461	76	7	continuous	continuous	ADJ
ejpam-6461	76	8	functions	function	NOUN
ejpam-6461	76	9	on	on	ADP
ejpam-6461	76	10	[	[	X
ejpam-6461	76	11	0,∞	0,∞	NOUN
ejpam-6461	76	12	)	)	PUNCT
ejpam-6461	76	13	satisfying	satisfy	VERB
ejpam-6461	76	14	the	the	DET
ejpam-6461	76	15	relation	relation	NOUN
ejpam-6461	76	16	f(a)g(a	f(a)g(a	NOUN
ejpam-6461	76	17	)	)	PUNCT
ejpam-6461	76	18	=	=	SYM
ejpam-6461	77	1	a	a	PRON
ejpam-6461	77	2	(	(	PUNCT
ejpam-6461	77	3	a	a	DET
ejpam-6461	77	4	∈	∈	PROPN
ejpam-6461	78	1	[	[	X
ejpam-6461	78	2	0,∞	0,∞	NOUN
ejpam-6461	78	3	)	)	PUNCT
ejpam-6461	78	4	)	)	PUNCT
ejpam-6461	78	5	,	,	PUNCT
ejpam-6461	78	6	then	then	ADV
ejpam-6461	78	7	w	w	PROPN
ejpam-6461	78	8	(	(	PUNCT
ejpam-6461	78	9	n∑	n∑	NOUN
ejpam-6461	78	10	i=1	i=1	PROPN
ejpam-6461	78	11	(	(	PUNCT
ejpam-6461	78	12	ai	ai	VERB
ejpam-6461	78	13	+	+	PROPN
ejpam-6461	78	14	bi	bi	NOUN
ejpam-6461	78	15	)	)	PUNCT
ejpam-6461	78	16	)	)	PUNCT
ejpam-6461	79	1	≤	≤	NUM
ejpam-6461	79	2	1	1	NUM
ejpam-6461	79	3	2	2	NUM
ejpam-6461	79	4	n∑	n∑	NOUN
ejpam-6461	79	5	i=1	i=1	PROPN
ejpam-6461	79	6	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6461	79	7	f2(|ai|	f2(|ai|	PROPN
ejpam-6461	79	8	)	)	PUNCT
ejpam-6461	80	1	+	+	CCONJ
ejpam-6461	80	2	g2(|a∗	g2(|a∗	VERB
ejpam-6461	80	3	i	i	PRON
ejpam-6461	80	4	|	|	NOUN
ejpam-6461	80	5	)	)	PUNCT
ejpam-6461	81	1	+	+	NUM
ejpam-6461	81	2	f2(|bi|	f2(|bi|	NOUN
ejpam-6461	81	3	)	)	PUNCT
ejpam-6461	82	1	+	+	CCONJ
ejpam-6461	82	2	g2(|b∗	g2(|b∗	VERB
ejpam-6461	82	3	i	i	PRON
ejpam-6461	82	4	|	|	NOUN
ejpam-6461	82	5	)	)	PUNCT
ejpam-6461	82	6	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6461	82	7	.	.	PUNCT
ejpam-6461	83	1	(	(	PUNCT
ejpam-6461	83	2	12	12	NUM
ejpam-6461	83	3	)	)	PUNCT
ejpam-6461	83	4	proof.∣∣∣∣∣	proof.∣∣∣∣∣	PROPN
ejpam-6461	83	5	〈	〈	PROPN
ejpam-6461	83	6	n∑	n∑	NOUN
ejpam-6461	83	7	i=1	i=1	PROPN
ejpam-6461	83	8	(	(	PUNCT
ejpam-6461	83	9	ai	ai	VERB
ejpam-6461	83	10	+	+	NOUN
ejpam-6461	83	11	bi)x	bi)x	ADJ
ejpam-6461	83	12	,	,	PUNCT
ejpam-6461	83	13	x	x	SYM
ejpam-6461	83	14	〉	〉	NOUN
ejpam-6461	83	15	∣∣∣∣∣	∣∣∣∣∣	PROPN
ejpam-6461	83	16	=	=	SYM
ejpam-6461	83	17	|⟨((a1	|⟨((a1	PROPN
ejpam-6461	83	18	+	+	PROPN
ejpam-6461	83	19	b1	b1	NOUN
ejpam-6461	83	20	)	)	PUNCT
ejpam-6461	83	21	+	+	CCONJ
ejpam-6461	83	22	(	(	PUNCT
ejpam-6461	83	23	a2	a2	PROPN
ejpam-6461	83	24	+	+	NOUN
ejpam-6461	83	25	b2	b2	NOUN
ejpam-6461	83	26	)	)	PUNCT
ejpam-6461	83	27	+	+	CCONJ
ejpam-6461	83	28	·	·	PUNCT
ejpam-6461	83	29	·	·	PUNCT
ejpam-6461	83	30	·	·	PUNCT
ejpam-6461	84	1	+	+	CCONJ
ejpam-6461	84	2	(	(	PUNCT
ejpam-6461	84	3	an	an	DET
ejpam-6461	84	4	+	+	PROPN
ejpam-6461	84	5	bn))x	bn))x	PROPN
ejpam-6461	84	6	,	,	PUNCT
ejpam-6461	84	7	x⟩|	x⟩|	X
ejpam-6461	84	8	=	=	SYM
ejpam-6461	84	9	|⟨(a1	|⟨(a1	PROPN
ejpam-6461	85	1	+	+	ADJ
ejpam-6461	85	2	b1)x	b1)x	NOUN
ejpam-6461	85	3	,	,	PUNCT
ejpam-6461	85	4	x⟩+	x⟩+	X
ejpam-6461	85	5	⟨(a2	⟨(a2	PROPN
ejpam-6461	86	1	+	+	NUM
ejpam-6461	86	2	b2)x	b2)x	NOUN
ejpam-6461	86	3	,	,	PUNCT
ejpam-6461	86	4	x⟩+	x⟩+	PROPN
ejpam-6461	86	5	·	·	PUNCT
ejpam-6461	86	6	·	·	PUNCT
ejpam-6461	86	7	·	·	PUNCT
ejpam-6461	87	1	+	+	CCONJ
ejpam-6461	88	1	⟨(an	⟨(an	PROPN
ejpam-6461	88	2	+	+	NOUN
ejpam-6461	88	3	bn)x	bn)x	PROPN
ejpam-6461	88	4	,	,	PUNCT
ejpam-6461	88	5	x⟩|	x⟩|	ADP
ejpam-6461	88	6	≤	≤	X
ejpam-6461	88	7	(	(	PUNCT
ejpam-6461	88	8	|⟨(a1	|⟨(a1	NOUN
ejpam-6461	88	9	+	+	NOUN
ejpam-6461	88	10	b1)x	b1)x	NOUN
ejpam-6461	88	11	,	,	PUNCT
ejpam-6461	88	12	x⟩|+	x⟩|+	VERB
ejpam-6461	88	13	|⟨(a2	|⟨(a2	PROPN
ejpam-6461	89	1	+	+	NUM
ejpam-6461	89	2	b2)x	b2)x	NOUN
ejpam-6461	89	3	,	,	PUNCT
ejpam-6461	89	4	x⟩|+	x⟩|+	NOUN
ejpam-6461	89	5	·	·	PUNCT
ejpam-6461	89	6	·	·	PUNCT
ejpam-6461	89	7	·	·	PUNCT
ejpam-6461	90	1	+	+	NUM
ejpam-6461	90	2	|⟨(an	|⟨(an	PROPN
ejpam-6461	90	3	+	+	SYM
ejpam-6461	90	4	bn)x	bn)x	PROPN
ejpam-6461	90	5	,	,	PUNCT
ejpam-6461	90	6	x⟩|	x⟩|	PROPN
ejpam-6461	90	7	)	)	PUNCT
ejpam-6461	90	8	(	(	PUNCT
ejpam-6461	90	9	by	by	ADP
ejpam-6461	90	10	using	use	VERB
ejpam-6461	90	11	triangle	triangle	NOUN
ejpam-6461	90	12	inequality	inequality	NOUN
ejpam-6461	90	13	)	)	PUNCT
ejpam-6461	91	1	=	=	PUNCT
ejpam-6461	92	1	n∑	n∑	NOUN
ejpam-6461	92	2	i=1	i=1	X
ejpam-6461	92	3	|⟨(ai	|⟨(ai	PROPN
ejpam-6461	93	1	+	+	NOUN
ejpam-6461	93	2	bi)x	bi)x	ADJ
ejpam-6461	93	3	,	,	PUNCT
ejpam-6461	93	4	x⟩|	x⟩|	PUNCT
ejpam-6461	94	1	=	=	PUNCT
ejpam-6461	94	2	n∑	n∑	PROPN
ejpam-6461	94	3	i=1	i=1	PROPN
ejpam-6461	94	4	|⟨aix	|⟨aix	PROPN
ejpam-6461	94	5	,	,	PUNCT
ejpam-6461	94	6	x⟩+	x⟩+	PROPN
ejpam-6461	94	7	⟨bix	⟨bix	PROPN
ejpam-6461	94	8	,	,	PUNCT
ejpam-6461	94	9	x⟩|	x⟩|	VERB
ejpam-6461	94	10	≤	≤	NUM
ejpam-6461	95	1	n∑	n∑	X
ejpam-6461	95	2	i=1	i=1	PROPN
ejpam-6461	95	3	(	(	PUNCT
ejpam-6461	95	4	|⟨aix	|⟨aix	PROPN
ejpam-6461	95	5	,	,	PUNCT
ejpam-6461	95	6	x⟩|+	x⟩|+	PROPN
ejpam-6461	95	7	|⟨bix	|⟨bix	PROPN
ejpam-6461	95	8	,	,	PUNCT
ejpam-6461	95	9	x⟩|	x⟩|	PROPN
ejpam-6461	95	10	)	)	PUNCT
ejpam-6461	95	11	(	(	PUNCT
ejpam-6461	95	12	by	by	ADP
ejpam-6461	95	13	using	use	VERB
ejpam-6461	95	14	triangle	triangle	NOUN
ejpam-6461	95	15	inequality	inequality	NOUN
ejpam-6461	95	16	)	)	PUNCT
ejpam-6461	95	17	≤	≤	NUM
ejpam-6461	96	1	n∑	n∑	X
ejpam-6461	96	2	i=1	i=1	PROPN
ejpam-6461	96	3	(	(	PUNCT
ejpam-6461	96	4	〈	〈	PROPN
ejpam-6461	96	5	f2(|ai|)x	f2(|ai|)x	PROPN
ejpam-6461	96	6	,	,	PUNCT
ejpam-6461	96	7	x	x	SYM
ejpam-6461	96	8	〉	〉	NOUN
ejpam-6461	96	9	1	1	NUM
ejpam-6461	96	10	2	2	NUM
ejpam-6461	96	11	〈	〈	NOUN
ejpam-6461	96	12	g2(|a∗	g2(|a∗	NOUN
ejpam-6461	96	13	i	i	PRON
ejpam-6461	96	14	|)x	|)x	X
ejpam-6461	96	15	,	,	PUNCT
ejpam-6461	96	16	x	x	SYM
ejpam-6461	96	17	〉	〉	NOUN
ejpam-6461	96	18	1	1	NUM
ejpam-6461	96	19	2	2	NUM
ejpam-6461	96	20	+	+	NUM
ejpam-6461	96	21	〈	〈	PROPN
ejpam-6461	96	22	f2(|bi|)x	f2(|bi|)x	PROPN
ejpam-6461	96	23	,	,	PUNCT
ejpam-6461	96	24	x	x	SYM
ejpam-6461	96	25	〉	〉	NOUN
ejpam-6461	96	26	1	1	NUM
ejpam-6461	96	27	2	2	NUM
ejpam-6461	96	28	〈	〈	PROPN
ejpam-6461	96	29	g2(|b∗	g2(|b∗	NOUN
ejpam-6461	96	30	i	i	PRON
ejpam-6461	96	31	|)x	|)x	X
ejpam-6461	96	32	,	,	PUNCT
ejpam-6461	96	33	x	x	SYM
ejpam-6461	96	34	〉	〉	NOUN
ejpam-6461	96	35	1	1	NUM
ejpam-6461	96	36	2	2	NUM
ejpam-6461	96	37	)	)	PUNCT
ejpam-6461	96	38	(	(	PUNCT
ejpam-6461	96	39	by	by	ADP
ejpam-6461	96	40	using	use	VERB
ejpam-6461	96	41	inequality	inequality	NOUN
ejpam-6461	96	42	(	(	PUNCT
ejpam-6461	96	43	9	9	NUM
ejpam-6461	96	44	)	)	PUNCT
ejpam-6461	96	45	)	)	PUNCT
ejpam-6461	96	46	≤	≤	NUM
ejpam-6461	96	47	1	1	NUM
ejpam-6461	96	48	2	2	NUM
ejpam-6461	96	49	n∑	n∑	NOUN
ejpam-6461	96	50	i=1	i=1	PROPN
ejpam-6461	96	51	(	(	PUNCT
ejpam-6461	96	52	〈	〈	PROPN
ejpam-6461	96	53	f2(|ai|)x	f2(|ai|)x	PROPN
ejpam-6461	96	54	,	,	PUNCT
ejpam-6461	96	55	x	x	SYM
ejpam-6461	96	56	〉	〉	NOUN
ejpam-6461	96	57	+	+	NUM
ejpam-6461	96	58	〈	〈	PROPN
ejpam-6461	96	59	g2(|a∗	g2(|a∗	NOUN
ejpam-6461	96	60	i	i	PRON
ejpam-6461	96	61	|)x	|)x	X
ejpam-6461	96	62	,	,	PUNCT
ejpam-6461	96	63	x	x	SYM
ejpam-6461	96	64	〉	〉	NOUN
ejpam-6461	96	65	+	+	NUM
ejpam-6461	96	66	〈	〈	PROPN
ejpam-6461	96	67	f2(|bi|)x	f2(|bi|)x	PROPN
ejpam-6461	96	68	,	,	PUNCT
ejpam-6461	96	69	x	x	SYM
ejpam-6461	96	70	〉	〉	NOUN
ejpam-6461	96	71	+	+	CCONJ
ejpam-6461	96	72	〈	〈	PROPN
ejpam-6461	96	73	g2(|b∗	g2(|b∗	NOUN
ejpam-6461	96	74	i	i	PRON
ejpam-6461	96	75	|)x	|)x	X
ejpam-6461	96	76	,	,	PUNCT
ejpam-6461	96	77	x	x	SYM
ejpam-6461	96	78	〉	〉	NOUN
ejpam-6461	96	79	)	)	PUNCT
ejpam-6461	96	80	(	(	PUNCT
ejpam-6461	96	81	by	by	ADP
ejpam-6461	96	82	using	use	VERB
ejpam-6461	96	83	arithmetic	arithmetic	ADJ
ejpam-6461	96	84	geometric	geometric	ADJ
ejpam-6461	96	85	mean	mean	NOUN
ejpam-6461	96	86	inequality	inequality	NOUN
ejpam-6461	96	87	)	)	PUNCT
ejpam-6461	96	88	=	=	SYM
ejpam-6461	96	89	1	1	NUM
ejpam-6461	96	90	2	2	NUM
ejpam-6461	96	91	n∑	n∑	NOUN
ejpam-6461	96	92	i=1	i=1	PROPN
ejpam-6461	96	93	〈	〈	PROPN
ejpam-6461	96	94	(	(	PUNCT
ejpam-6461	96	95	f2(|ai|	f2(|ai|	PROPN
ejpam-6461	96	96	)	)	PUNCT
ejpam-6461	96	97	+	+	CCONJ
ejpam-6461	96	98	g2(|a∗	g2(|a∗	VERB
ejpam-6461	96	99	i	i	PRON
ejpam-6461	96	100	|	|	NOUN
ejpam-6461	96	101	)	)	PUNCT
ejpam-6461	96	102	+	+	NUM
ejpam-6461	96	103	f2(|bi|	f2(|bi|	NOUN
ejpam-6461	96	104	)	)	PUNCT
ejpam-6461	97	1	+	+	CCONJ
ejpam-6461	97	2	g2(|b∗	g2(|b∗	VERB
ejpam-6461	97	3	i	i	PRON
ejpam-6461	97	4	|	|	NOUN
ejpam-6461	97	5	)	)	PUNCT
ejpam-6461	97	6	)	)	PUNCT
ejpam-6461	98	1	x	x	X
ejpam-6461	98	2	,	,	PUNCT
ejpam-6461	98	3	x	x	SYM
ejpam-6461	98	4	〉	〉	NOUN
ejpam-6461	98	5	.	.	PUNCT
ejpam-6461	99	1	taking	take	VERB
ejpam-6461	99	2	the	the	DET
ejpam-6461	99	3	supremum	supremum	ADJ
ejpam-6461	99	4	over	over	ADP
ejpam-6461	99	5	all	all	DET
ejpam-6461	99	6	unit	unit	NOUN
ejpam-6461	99	7	vectors	vector	NOUN
ejpam-6461	99	8	x	x	SYM
ejpam-6461	99	9	∈	∈	PROPN
ejpam-6461	99	10	h	h	NOUN
ejpam-6461	99	11	,	,	PUNCT
ejpam-6461	99	12	we	we	PRON
ejpam-6461	99	13	obtain	obtain	VERB
ejpam-6461	99	14	the	the	DET
ejpam-6461	99	15	inequality	inequality	NOUN
ejpam-6461	99	16	(	(	PUNCT
ejpam-6461	99	17	12	12	NUM
ejpam-6461	99	18	)	)	PUNCT
ejpam-6461	99	19	.	.	PUNCT
ejpam-6461	100	1	corollary	corollary	ADJ
ejpam-6461	100	2	1	1	NUM
ejpam-6461	100	3	.	.	PUNCT
ejpam-6461	101	1	let	let	VERB
ejpam-6461	101	2	a1	a1	NOUN
ejpam-6461	101	3	,	,	PUNCT
ejpam-6461	101	4	a2	a2	PROPN
ejpam-6461	101	5	,	,	PUNCT
ejpam-6461	101	6	...	...	PUNCT
ejpam-6461	101	7	,	,	PUNCT
ejpam-6461	101	8	an	an	PRON
ejpam-6461	101	9	,	,	PUNCT
ejpam-6461	101	10	b1	b1	NOUN
ejpam-6461	101	11	,	,	PUNCT
ejpam-6461	101	12	b2	b2	NOUN
ejpam-6461	101	13	,	,	PUNCT
ejpam-6461	101	14	...	...	PUNCT
ejpam-6461	101	15	,	,	PUNCT
ejpam-6461	101	16	bn	bn	PART
ejpam-6461	101	17	be	be	AUX
ejpam-6461	101	18	operators	operator	NOUN
ejpam-6461	101	19	in	in	ADP
ejpam-6461	101	20	b(h	b(h	PROPN
ejpam-6461	101	21	)	)	PUNCT
ejpam-6461	101	22	.	.	PUNCT
ejpam-6461	102	1	then	then	ADV
ejpam-6461	102	2	w	w	PROPN
ejpam-6461	102	3	(	(	PUNCT
ejpam-6461	102	4	n∑	n∑	NOUN
ejpam-6461	102	5	i=1	i=1	PROPN
ejpam-6461	102	6	(	(	PUNCT
ejpam-6461	102	7	ai	ai	VERB
ejpam-6461	102	8	+	+	PROPN
ejpam-6461	102	9	bi	bi	NOUN
ejpam-6461	102	10	)	)	PUNCT
ejpam-6461	102	11	)	)	PUNCT
ejpam-6461	103	1	≤	≤	NUM
ejpam-6461	103	2	1	1	NUM
ejpam-6461	103	3	2	2	NUM
ejpam-6461	103	4	n∑	n∑	NOUN
ejpam-6461	103	5	i=1	i=1	PROPN
ejpam-6461	103	6	||	||	NOUN
ejpam-6461	103	7	|ai|+	|ai|+	NUM
ejpam-6461	104	1	|a∗	|a∗	PROPN
ejpam-6461	104	2	i	i	PRON
ejpam-6461	104	3	|+	|+	VERB
ejpam-6461	104	4	|bi|+	|bi|+	PROPN
ejpam-6461	104	5	|b∗	|b∗	PROPN
ejpam-6461	104	6	i	i	PRON
ejpam-6461	105	1	|	|	ADV
ejpam-6461	105	2	||	||	INTJ
ejpam-6461	105	3	.	.	PUNCT
ejpam-6461	106	1	(	(	PUNCT
ejpam-6461	106	2	13	13	NUM
ejpam-6461	106	3	)	)	PUNCT
ejpam-6461	106	4	m.	m.	NOUN
ejpam-6461	106	5	al	al	PROPN
ejpam-6461	106	6	-	-	PUNCT
ejpam-6461	106	7	labadi	labadi	PROPN
ejpam-6461	106	8	et	et	PROPN
ejpam-6461	106	9	al	al	PROPN
ejpam-6461	106	10	.	.	PUNCT
ejpam-6461	106	11	/	/	SYM
ejpam-6461	106	12	eur	eur	PROPN
ejpam-6461	106	13	.	.	PUNCT
ejpam-6461	107	1	j.	j.	PROPN
ejpam-6461	107	2	pure	pure	PROPN
ejpam-6461	107	3	appl	appl	PROPN
ejpam-6461	107	4	.	.	PROPN
ejpam-6461	107	5	math	math	PROPN
ejpam-6461	107	6	,	,	PUNCT
ejpam-6461	107	7	18	18	NUM
ejpam-6461	107	8	(	(	PUNCT
ejpam-6461	107	9	3	3	NUM
ejpam-6461	107	10	)	)	PUNCT
ejpam-6461	107	11	(	(	PUNCT
ejpam-6461	107	12	2025	2025	NUM
ejpam-6461	107	13	)	)	PUNCT
ejpam-6461	107	14	,	,	PUNCT
ejpam-6461	107	15	6461	6461	NUM
ejpam-6461	107	16	5	5	NUM
ejpam-6461	107	17	of	of	ADP
ejpam-6461	107	18	13	13	NUM
ejpam-6461	107	19	proof	proof	NOUN
ejpam-6461	107	20	.	.	PUNCT
ejpam-6461	108	1	letting	let	VERB
ejpam-6461	108	2	f(t	f(t	NOUN
ejpam-6461	108	3	)	)	PUNCT
ejpam-6461	108	4	=	=	SYM
ejpam-6461	108	5	g(t	g(t	PROPN
ejpam-6461	108	6	)	)	PUNCT
ejpam-6461	109	1	=	=	SYM
ejpam-6461	109	2	√	√	PROPN
ejpam-6461	109	3	t	t	PROPN
ejpam-6461	109	4	in	in	ADP
ejpam-6461	109	5	inequality	inequality	NOUN
ejpam-6461	109	6	(	(	PUNCT
ejpam-6461	109	7	12	12	NUM
ejpam-6461	109	8	)	)	PUNCT
ejpam-6461	109	9	,	,	PUNCT
ejpam-6461	109	10	we	we	PRON
ejpam-6461	109	11	obtain	obtain	VERB
ejpam-6461	109	12	the	the	DET
ejpam-6461	109	13	inequality	inequality	NOUN
ejpam-6461	109	14	(	(	PUNCT
ejpam-6461	109	15	13	13	NUM
ejpam-6461	109	16	)	)	PUNCT
ejpam-6461	109	17	.	.	PUNCT
ejpam-6461	110	1	corollary	corollary	ADJ
ejpam-6461	110	2	2	2	NUM
ejpam-6461	110	3	.	.	PUNCT
ejpam-6461	111	1	let	let	VERB
ejpam-6461	111	2	a	a	DET
ejpam-6461	111	3	,	,	PUNCT
ejpam-6461	111	4	b	b	NOUN
ejpam-6461	111	5	be	be	AUX
ejpam-6461	111	6	operators	operator	NOUN
ejpam-6461	111	7	in	in	ADP
ejpam-6461	111	8	b(h	b(h	PROPN
ejpam-6461	111	9	)	)	PUNCT
ejpam-6461	111	10	.	.	PUNCT
ejpam-6461	112	1	then	then	ADV
ejpam-6461	112	2	w(a+b	w(a+b	NUM
ejpam-6461	112	3	)	)	PUNCT
ejpam-6461	112	4	≤	≤	NUM
ejpam-6461	112	5	1	1	NUM
ejpam-6461	112	6	2	2	NUM
ejpam-6461	112	7	||	||	NOUN
ejpam-6461	112	8	|a|+	|a|+	NOUN
ejpam-6461	112	9	|a∗|+	|a∗|+	ADJ
ejpam-6461	112	10	|b|+	|b|+	NOUN
ejpam-6461	112	11	|b∗|	|b∗|	PROPN
ejpam-6461	112	12	||	||	NOUN
ejpam-6461	112	13	.	.	PUNCT
ejpam-6461	113	1	(	(	PUNCT
ejpam-6461	113	2	14	14	NUM
ejpam-6461	113	3	)	)	PUNCT
ejpam-6461	113	4	proof	proof	NOUN
ejpam-6461	113	5	.	.	PUNCT
ejpam-6461	114	1	letting	let	VERB
ejpam-6461	114	2	ai	ai	VERB
ejpam-6461	114	3	=	=	ADJ
ejpam-6461	114	4	bi	bi	NOUN
ejpam-6461	114	5	=	=	PROPN
ejpam-6461	114	6	o	o	PROPN
ejpam-6461	114	7	for	for	ADP
ejpam-6461	114	8	i	i	PRON
ejpam-6461	114	9	=	=	SYM
ejpam-6461	114	10	2	2	NUM
ejpam-6461	114	11	,	,	PUNCT
ejpam-6461	114	12	3	3	NUM
ejpam-6461	114	13	,	,	PUNCT
ejpam-6461	114	14	4	4	NUM
ejpam-6461	114	15	,	,	PUNCT
ejpam-6461	114	16	...	...	PUNCT
ejpam-6461	114	17	,	,	PUNCT
ejpam-6461	114	18	n	n	CCONJ
ejpam-6461	114	19	in	in	ADP
ejpam-6461	114	20	inequality	inequality	NOUN
ejpam-6461	114	21	(	(	PUNCT
ejpam-6461	114	22	13	13	NUM
ejpam-6461	114	23	)	)	PUNCT
ejpam-6461	114	24	,	,	PUNCT
ejpam-6461	114	25	we	we	PRON
ejpam-6461	114	26	obtain	obtain	VERB
ejpam-6461	114	27	the	the	DET
ejpam-6461	114	28	inequality	inequality	NOUN
ejpam-6461	114	29	(	(	PUNCT
ejpam-6461	114	30	14	14	NUM
ejpam-6461	114	31	)	)	PUNCT
ejpam-6461	114	32	.	.	PUNCT
ejpam-6461	115	1	corollary	corollary	ADJ
ejpam-6461	115	2	3	3	X
ejpam-6461	115	3	.	.	PUNCT
ejpam-6461	116	1	let	let	VERB
ejpam-6461	116	2	a	a	DET
ejpam-6461	116	3	,	,	PUNCT
ejpam-6461	116	4	b	b	NOUN
ejpam-6461	116	5	be	be	AUX
ejpam-6461	116	6	operators	operator	NOUN
ejpam-6461	116	7	in	in	ADP
ejpam-6461	116	8	b(h	b(h	PROPN
ejpam-6461	116	9	)	)	PUNCT
ejpam-6461	116	10	.	.	PUNCT
ejpam-6461	117	1	then	then	ADV
ejpam-6461	117	2	w2(a+b	w2(a+b	NOUN
ejpam-6461	117	3	)	)	PUNCT
ejpam-6461	117	4	≤	≤	NUM
ejpam-6461	117	5	1	1	NUM
ejpam-6461	117	6	4	4	NUM
ejpam-6461	117	7	||	||	NOUN
ejpam-6461	117	8	|a|+	|a|+	NOUN
ejpam-6461	117	9	|a∗|+	|a∗|+	ADJ
ejpam-6461	117	10	|b|+	|b|+	PROPN
ejpam-6461	117	11	|b∗|	|b∗|	PROPN
ejpam-6461	117	12	||2	||2	NOUN
ejpam-6461	117	13	.	.	PUNCT
ejpam-6461	118	1	(	(	PUNCT
ejpam-6461	118	2	15	15	X
ejpam-6461	118	3	)	)	PUNCT
ejpam-6461	118	4	proof	proof	NOUN
ejpam-6461	118	5	.	.	PUNCT
ejpam-6461	119	1	by	by	ADP
ejpam-6461	119	2	squaring	square	VERB
ejpam-6461	119	3	both	both	DET
ejpam-6461	119	4	sides	side	NOUN
ejpam-6461	119	5	of	of	ADP
ejpam-6461	119	6	inequality	inequality	NOUN
ejpam-6461	119	7	(	(	PUNCT
ejpam-6461	119	8	14	14	NUM
ejpam-6461	119	9	)	)	PUNCT
ejpam-6461	119	10	,	,	PUNCT
ejpam-6461	119	11	we	we	PRON
ejpam-6461	119	12	obtain	obtain	VERB
ejpam-6461	119	13	the	the	DET
ejpam-6461	119	14	inequality	inequality	NOUN
ejpam-6461	119	15	(	(	PUNCT
ejpam-6461	119	16	15	15	NUM
ejpam-6461	119	17	)	)	PUNCT
ejpam-6461	119	18	.	.	PUNCT
ejpam-6461	120	1	remark	remark	PROPN
ejpam-6461	120	2	1	1	NUM
ejpam-6461	120	3	.	.	PUNCT
ejpam-6461	121	1	letting	let	VERB
ejpam-6461	121	2	b	b	NOUN
ejpam-6461	121	3	=	=	SYM
ejpam-6461	121	4	o	o	PROPN
ejpam-6461	121	5	in	in	ADP
ejpam-6461	121	6	inequality	inequality	NOUN
ejpam-6461	121	7	(	(	PUNCT
ejpam-6461	121	8	14	14	NUM
ejpam-6461	121	9	)	)	PUNCT
ejpam-6461	121	10	,	,	PUNCT
ejpam-6461	121	11	we	we	PRON
ejpam-6461	121	12	derive	derive	VERB
ejpam-6461	121	13	inequality	inequality	NOUN
ejpam-6461	121	14	(	(	PUNCT
ejpam-6461	121	15	2	2	NUM
ejpam-6461	121	16	)	)	PUNCT
ejpam-6461	121	17	.	.	PUNCT
ejpam-6461	122	1	this	this	PRON
ejpam-6461	122	2	is	be	AUX
ejpam-6461	122	3	considered	consider	VERB
ejpam-6461	122	4	as	as	ADP
ejpam-6461	122	5	a	a	DET
ejpam-6461	122	6	new	new	ADJ
ejpam-6461	122	7	proof	proof	NOUN
ejpam-6461	122	8	of	of	ADP
ejpam-6461	122	9	inequality	inequality	NOUN
ejpam-6461	122	10	(	(	PUNCT
ejpam-6461	122	11	2	2	NUM
ejpam-6461	122	12	)	)	PUNCT
ejpam-6461	122	13	.	.	PUNCT
ejpam-6461	123	1	corollary	corollary	ADJ
ejpam-6461	123	2	4	4	NUM
ejpam-6461	123	3	.	.	PUNCT
ejpam-6461	123	4	let	let	VERB
ejpam-6461	123	5	a	a	DET
ejpam-6461	123	6	∈	∈	PROPN
ejpam-6461	123	7	b(h	b(h	PROPN
ejpam-6461	123	8	)	)	PUNCT
ejpam-6461	123	9	.	.	PUNCT
ejpam-6461	124	1	then	then	ADV
ejpam-6461	124	2	w2(a	w2(a	NOUN
ejpam-6461	124	3	)	)	PUNCT
ejpam-6461	124	4	≤	≤	NOUN
ejpam-6461	124	5	1	1	NUM
ejpam-6461	124	6	4	4	NUM
ejpam-6461	124	7	||	||	NOUN
ejpam-6461	124	8	|a|+	|a|+	NOUN
ejpam-6461	124	9	|a∗|	|a∗|	PRON
ejpam-6461	124	10	||2	||2	NOUN
ejpam-6461	124	11	.	.	PUNCT
ejpam-6461	125	1	(	(	PUNCT
ejpam-6461	125	2	16	16	X
ejpam-6461	125	3	)	)	PUNCT
ejpam-6461	125	4	proof	proof	NOUN
ejpam-6461	125	5	.	.	PUNCT
ejpam-6461	126	1	letting	let	VERB
ejpam-6461	126	2	b	b	X
ejpam-6461	126	3	=	=	SYM
ejpam-6461	126	4	o	o	PROPN
ejpam-6461	126	5	in	in	ADP
ejpam-6461	126	6	inequality	inequality	NOUN
ejpam-6461	126	7	(	(	PUNCT
ejpam-6461	126	8	15	15	NUM
ejpam-6461	126	9	)	)	PUNCT
ejpam-6461	126	10	,	,	PUNCT
ejpam-6461	126	11	we	we	PRON
ejpam-6461	126	12	obtain	obtain	VERB
ejpam-6461	126	13	the	the	DET
ejpam-6461	126	14	inequality	inequality	NOUN
ejpam-6461	126	15	(	(	PUNCT
ejpam-6461	126	16	16	16	NUM
ejpam-6461	126	17	)	)	PUNCT
ejpam-6461	126	18	.	.	PUNCT
ejpam-6461	127	1	remark	remark	PROPN
ejpam-6461	127	2	2	2	NUM
ejpam-6461	127	3	.	.	X
ejpam-6461	128	1	inequality	inequality	NOUN
ejpam-6461	128	2	(	(	PUNCT
ejpam-6461	128	3	16	16	NUM
ejpam-6461	128	4	)	)	PUNCT
ejpam-6461	128	5	refines	refine	VERB
ejpam-6461	128	6	inequality	inequality	NOUN
ejpam-6461	128	7	(	(	PUNCT
ejpam-6461	128	8	4	4	NUM
ejpam-6461	128	9	)	)	PUNCT
ejpam-6461	128	10	.	.	PUNCT
ejpam-6461	129	1	to	to	PART
ejpam-6461	129	2	show	show	VERB
ejpam-6461	129	3	this	this	PRON
ejpam-6461	129	4	,	,	PUNCT
ejpam-6461	129	5	note	note	VERB
ejpam-6461	129	6	that	that	SCONJ
ejpam-6461	129	7	while	while	SCONJ
ejpam-6461	129	8	the	the	DET
ejpam-6461	129	9	left	left	ADJ
ejpam-6461	129	10	sides	side	NOUN
ejpam-6461	129	11	of	of	ADP
ejpam-6461	129	12	inequalities	inequality	NOUN
ejpam-6461	129	13	(	(	PUNCT
ejpam-6461	129	14	4	4	NUM
ejpam-6461	129	15	)	)	PUNCT
ejpam-6461	129	16	and	and	CCONJ
ejpam-6461	129	17	(	(	PUNCT
ejpam-6461	129	18	16	16	NUM
ejpam-6461	129	19	)	)	PUNCT
ejpam-6461	129	20	are	be	AUX
ejpam-6461	129	21	the	the	DET
ejpam-6461	129	22	same	same	ADJ
ejpam-6461	129	23	.	.	PUNCT
ejpam-6461	130	1	the	the	DET
ejpam-6461	130	2	right	right	ADJ
ejpam-6461	130	3	side	side	NOUN
ejpam-6461	130	4	of	of	ADP
ejpam-6461	130	5	inequality	inequality	NOUN
ejpam-6461	130	6	(	(	PUNCT
ejpam-6461	130	7	16	16	NUM
ejpam-6461	130	8	)	)	PUNCT
ejpam-6461	130	9	is	be	AUX
ejpam-6461	130	10	1	1	NUM
ejpam-6461	130	11	4	4	NUM
ejpam-6461	130	12	||	||	NOUN
ejpam-6461	130	13	|a|+	|a|+	NOUN
ejpam-6461	130	14	|a∗|	|a∗|	PUNCT
ejpam-6461	130	15	||2	||2	ADV
ejpam-6461	130	16	≤	≤	ADV
ejpam-6461	130	17	1	1	NUM
ejpam-6461	130	18	2	2	NUM
ejpam-6461	130	19	||	||	NOUN
ejpam-6461	130	20	|a|2	|a|2	PROPN
ejpam-6461	130	21	+	+	PROPN
ejpam-6461	131	1	|a∗|2	|a∗|2	NUM
ejpam-6461	131	2	||	||	NOUN
ejpam-6461	131	3	(	(	PUNCT
ejpam-6461	131	4	by	by	ADP
ejpam-6461	131	5	using	use	VERB
ejpam-6461	131	6	inequality	inequality	NOUN
ejpam-6461	131	7	(	(	PUNCT
ejpam-6461	131	8	11	11	NUM
ejpam-6461	131	9	)	)	PUNCT
ejpam-6461	131	10	)	)	PUNCT
ejpam-6461	132	1	=	=	SYM
ejpam-6461	132	2	1	1	NUM
ejpam-6461	132	3	2	2	NUM
ejpam-6461	132	4	||	||	NOUN
ejpam-6461	133	1	a∗a+aa∗	a∗a+aa∗	NOUN
ejpam-6461	133	2	||	||	PROPN
ejpam-6461	133	3	,	,	PUNCT
ejpam-6461	133	4	which	which	PRON
ejpam-6461	133	5	is	be	AUX
ejpam-6461	133	6	the	the	DET
ejpam-6461	133	7	right	right	ADJ
ejpam-6461	133	8	side	side	NOUN
ejpam-6461	133	9	of	of	ADP
ejpam-6461	133	10	inequality	inequality	NOUN
ejpam-6461	133	11	(	(	PUNCT
ejpam-6461	133	12	4	4	NUM
ejpam-6461	133	13	)	)	PUNCT
ejpam-6461	133	14	.	.	PUNCT
ejpam-6461	134	1	the	the	DET
ejpam-6461	134	2	following	follow	VERB
ejpam-6461	134	3	lemma	lemma	PROPN
ejpam-6461	134	4	[	[	X
ejpam-6461	134	5	27	27	NUM
ejpam-6461	134	6	]	]	X
ejpam-6461	134	7	is	be	AUX
ejpam-6461	134	8	essential	essential	ADJ
ejpam-6461	134	9	to	to	PART
ejpam-6461	134	10	prove	prove	VERB
ejpam-6461	134	11	the	the	DET
ejpam-6461	134	12	following	follow	VERB
ejpam-6461	134	13	theorem	theorem	NOUN
ejpam-6461	134	14	which	which	PRON
ejpam-6461	134	15	is	be	AUX
ejpam-6461	134	16	a	a	DET
ejpam-6461	134	17	generalization	generalization	NOUN
ejpam-6461	134	18	of	of	ADP
ejpam-6461	134	19	inequality	inequality	NOUN
ejpam-6461	134	20	(	(	PUNCT
ejpam-6461	134	21	5	5	NUM
ejpam-6461	134	22	)	)	PUNCT
ejpam-6461	134	23	.	.	PUNCT
ejpam-6461	135	1	lemma	lemma	PROPN
ejpam-6461	135	2	6	6	NUM
ejpam-6461	135	3	.	.	PUNCT
ejpam-6461	136	1	let	let	VERB
ejpam-6461	136	2	a	a	DET
ejpam-6461	136	3	,	,	PUNCT
ejpam-6461	136	4	b	b	NOUN
ejpam-6461	136	5	be	be	AUX
ejpam-6461	136	6	positive	positive	ADJ
ejpam-6461	136	7	semidefinite	semidefinite	NOUN
ejpam-6461	136	8	operators	operator	NOUN
ejpam-6461	136	9	in	in	ADP
ejpam-6461	136	10	b(h	b(h	PROPN
ejpam-6461	136	11	)	)	PUNCT
ejpam-6461	136	12	.	.	PUNCT
ejpam-6461	137	1	then	then	ADV
ejpam-6461	137	2	||	||	PROPN
ejpam-6461	138	1	a+b	a+b	NUM
ejpam-6461	138	2	||	||	X
ejpam-6461	139	1	≤	≤	NUM
ejpam-6461	139	2	max{||	max{||	PROPN
ejpam-6461	139	3	a	a	DET
ejpam-6461	139	4	||	||	NOUN
ejpam-6461	139	5	,	,	PUNCT
ejpam-6461	139	6	||	||	PROPN
ejpam-6461	140	1	b	b	PROPN
ejpam-6461	140	2	||}+	||}+	PROPN
ejpam-6461	140	3	||	||	NOUN
ejpam-6461	141	1	a	a	DET
ejpam-6461	141	2	1	1	NUM
ejpam-6461	141	3	2b	2b	NUM
ejpam-6461	141	4	1	1	NUM
ejpam-6461	141	5	2	2	NUM
ejpam-6461	141	6	||	||	NOUN
ejpam-6461	141	7	(	(	PUNCT
ejpam-6461	141	8	17	17	NUM
ejpam-6461	141	9	)	)	PUNCT
ejpam-6461	141	10	theorem	theorem	NOUN
ejpam-6461	141	11	2	2	NUM
ejpam-6461	141	12	.	.	PUNCT
ejpam-6461	142	1	let	let	VERB
ejpam-6461	142	2	a	a	DET
ejpam-6461	142	3	,	,	PUNCT
ejpam-6461	142	4	b	b	NOUN
ejpam-6461	142	5	be	be	AUX
ejpam-6461	142	6	operators	operator	NOUN
ejpam-6461	142	7	in	in	ADP
ejpam-6461	142	8	b(h	b(h	PROPN
ejpam-6461	142	9	)	)	PUNCT
ejpam-6461	142	10	.	.	PUNCT
ejpam-6461	143	1	then	then	ADV
ejpam-6461	143	2	w(a+b	w(a+b	NUM
ejpam-6461	143	3	)	)	PUNCT
ejpam-6461	143	4	≤	≤	NUM
ejpam-6461	143	5	1	1	NUM
ejpam-6461	143	6	2	2	NUM
ejpam-6461	143	7	(	(	PUNCT
ejpam-6461	143	8	max{||	max{||	PROPN
ejpam-6461	143	9	|a|+	|a|+	NOUN
ejpam-6461	143	10	|b|	|b|	X
ejpam-6461	143	11	||	||	NOUN
ejpam-6461	143	12	,	,	PUNCT
ejpam-6461	143	13	||	||	NUM
ejpam-6461	143	14	|a∗|+	|a∗|+	NOUN
ejpam-6461	143	15	|b∗|	|b∗|	PUNCT
ejpam-6461	144	1	||}+	||}+	NOUN
ejpam-6461	144	2	√	√	NOUN
ejpam-6461	144	3	r	r	NOUN
ejpam-6461	144	4	(	(	PUNCT
ejpam-6461	144	5	(	(	PUNCT
ejpam-6461	144	6	|a|+	|a|+	NOUN
ejpam-6461	144	7	|b|)(|a∗|+	|b|)(|a∗|+	PROPN
ejpam-6461	144	8	|b∗|	|b∗|	NUM
ejpam-6461	144	9	)	)	PUNCT
ejpam-6461	144	10	)	)	PUNCT
ejpam-6461	144	11	)	)	PUNCT
ejpam-6461	144	12	.	.	PUNCT
ejpam-6461	145	1	(	(	PUNCT
ejpam-6461	145	2	18	18	NUM
ejpam-6461	145	3	)	)	PUNCT
ejpam-6461	145	4	m.	m.	NOUN
ejpam-6461	145	5	al	al	PROPN
ejpam-6461	145	6	-	-	PUNCT
ejpam-6461	145	7	labadi	labadi	PROPN
ejpam-6461	145	8	et	et	PROPN
ejpam-6461	145	9	al	al	PROPN
ejpam-6461	145	10	.	.	PUNCT
ejpam-6461	145	11	/	/	SYM
ejpam-6461	145	12	eur	eur	PROPN
ejpam-6461	145	13	.	.	PUNCT
ejpam-6461	146	1	j.	j.	PROPN
ejpam-6461	146	2	pure	pure	PROPN
ejpam-6461	146	3	appl	appl	PROPN
ejpam-6461	146	4	.	.	PROPN
ejpam-6461	146	5	math	math	PROPN
ejpam-6461	146	6	,	,	PUNCT
ejpam-6461	146	7	18	18	NUM
ejpam-6461	146	8	(	(	PUNCT
ejpam-6461	146	9	3	3	NUM
ejpam-6461	146	10	)	)	PUNCT
ejpam-6461	146	11	(	(	PUNCT
ejpam-6461	146	12	2025	2025	NUM
ejpam-6461	146	13	)	)	PUNCT
ejpam-6461	146	14	,	,	PUNCT
ejpam-6461	146	15	6461	6461	NUM
ejpam-6461	146	16	6	6	NUM
ejpam-6461	146	17	of	of	ADP
ejpam-6461	146	18	13	13	NUM
ejpam-6461	146	19	proof	proof	NOUN
ejpam-6461	146	20	.	.	PUNCT
ejpam-6461	147	1	w(a+b	w(a+b	NOUN
ejpam-6461	147	2	)	)	PUNCT
ejpam-6461	147	3	≤	≤	NUM
ejpam-6461	147	4	1	1	NUM
ejpam-6461	147	5	2	2	NUM
ejpam-6461	147	6	||	||	NOUN
ejpam-6461	147	7	|a|+	|a|+	NOUN
ejpam-6461	147	8	|a∗|+	|a∗|+	ADJ
ejpam-6461	147	9	|b|+	|b|+	NOUN
ejpam-6461	147	10	|b∗|	|b∗|	PROPN
ejpam-6461	147	11	||	||	PROPN
ejpam-6461	147	12	(	(	PUNCT
ejpam-6461	147	13	by	by	ADP
ejpam-6461	147	14	inequality	inequality	NOUN
ejpam-6461	147	15	(	(	PUNCT
ejpam-6461	147	16	14	14	NUM
ejpam-6461	147	17	)	)	PUNCT
ejpam-6461	147	18	)	)	PUNCT
ejpam-6461	148	1	=	=	SYM
ejpam-6461	148	2	1	1	NUM
ejpam-6461	148	3	2	2	NUM
ejpam-6461	148	4	||	||	NOUN
ejpam-6461	148	5	(	(	PUNCT
ejpam-6461	148	6	|a|+	|a|+	NOUN
ejpam-6461	148	7	|b|	|b|	VERB
ejpam-6461	148	8	)	)	PUNCT
ejpam-6461	148	9	+	+	CCONJ
ejpam-6461	148	10	(	(	PUNCT
ejpam-6461	148	11	|a∗|+	|a∗|+	NOUN
ejpam-6461	148	12	|b∗|	|b∗|	NUM
ejpam-6461	148	13	)	)	PUNCT
ejpam-6461	148	14	||	||	NOUN
ejpam-6461	149	1	≤	≤	NUM
ejpam-6461	149	2	1	1	NUM
ejpam-6461	149	3	2	2	NUM
ejpam-6461	149	4	(	(	PUNCT
ejpam-6461	149	5	max{||	max{||	PROPN
ejpam-6461	149	6	|a|+	|a|+	NOUN
ejpam-6461	149	7	|b|	|b|	X
ejpam-6461	149	8	||	||	NOUN
ejpam-6461	149	9	,	,	PUNCT
ejpam-6461	149	10	||	||	NUM
ejpam-6461	149	11	|a∗|+	|a∗|+	NOUN
ejpam-6461	149	12	|b∗|	|b∗|	PROPN
ejpam-6461	149	13	||}+	||}+	PROPN
ejpam-6461	149	14	||	||	NOUN
ejpam-6461	149	15	(	(	PUNCT
ejpam-6461	149	16	|a|+	|a|+	NOUN
ejpam-6461	149	17	|b|	|b|	VERB
ejpam-6461	149	18	)	)	PUNCT
ejpam-6461	149	19	1	1	NUM
ejpam-6461	149	20	2	2	NUM
ejpam-6461	149	21	(	(	PUNCT
ejpam-6461	149	22	|a∗|+	|a∗|+	NOUN
ejpam-6461	149	23	|b∗|	|b∗|	NUM
ejpam-6461	149	24	)	)	PUNCT
ejpam-6461	149	25	1	1	NUM
ejpam-6461	149	26	2	2	NUM
ejpam-6461	149	27	||	||	NOUN
ejpam-6461	149	28	)	)	PUNCT
ejpam-6461	149	29	(	(	PUNCT
ejpam-6461	149	30	by	by	ADP
ejpam-6461	149	31	inequality	inequality	NOUN
ejpam-6461	149	32	(	(	PUNCT
ejpam-6461	149	33	17	17	NUM
ejpam-6461	149	34	)	)	PUNCT
ejpam-6461	149	35	=	=	SYM
ejpam-6461	149	36	1	1	NUM
ejpam-6461	149	37	2	2	NUM
ejpam-6461	149	38	(	(	PUNCT
ejpam-6461	149	39	max{||	max{||	PROPN
ejpam-6461	149	40	|a|+	|a|+	NOUN
ejpam-6461	149	41	|b|	|b|	X
ejpam-6461	149	42	||	||	NOUN
ejpam-6461	149	43	,	,	PUNCT
ejpam-6461	149	44	||	||	NUM
ejpam-6461	149	45	|a∗|+	|a∗|+	NOUN
ejpam-6461	149	46	|b∗|	|b∗|	PUNCT
ejpam-6461	150	1	||}+	||}+	NOUN
ejpam-6461	150	2	√	√	NOUN
ejpam-6461	150	3	r	r	NOUN
ejpam-6461	150	4	(	(	PUNCT
ejpam-6461	150	5	(	(	PUNCT
ejpam-6461	150	6	|a|+	|a|+	NOUN
ejpam-6461	150	7	|b|)(|a∗|+	|b|)(|a∗|+	PROPN
ejpam-6461	150	8	|b∗|	|b∗|	NUM
ejpam-6461	150	9	)	)	PUNCT
ejpam-6461	150	10	)	)	PUNCT
ejpam-6461	150	11	)	)	PUNCT
ejpam-6461	151	1	(	(	PUNCT
ejpam-6461	151	2	since	since	SCONJ
ejpam-6461	151	3	||	||	NUM
ejpam-6461	151	4	a	a	DET
ejpam-6461	151	5	1	1	NUM
ejpam-6461	151	6	2b	2b	NUM
ejpam-6461	151	7	1	1	NUM
ejpam-6461	151	8	2	2	NUM
ejpam-6461	151	9	||2	||2	NOUN
ejpam-6461	151	10	=	=	SYM
ejpam-6461	151	11	r(ab	r(ab	NOUN
ejpam-6461	151	12	)	)	PUNCT
ejpam-6461	151	13	)	)	PUNCT
ejpam-6461	151	14	.	.	PUNCT
ejpam-6461	151	15	remark	remark	PROPN
ejpam-6461	151	16	3	3	NUM
ejpam-6461	151	17	.	.	PUNCT
ejpam-6461	151	18	letting	let	VERB
ejpam-6461	151	19	b	b	NOUN
ejpam-6461	151	20	=	=	SYM
ejpam-6461	151	21	o	o	PROPN
ejpam-6461	151	22	in	in	ADP
ejpam-6461	151	23	inequality	inequality	NOUN
ejpam-6461	151	24	(	(	PUNCT
ejpam-6461	151	25	18	18	NUM
ejpam-6461	151	26	)	)	PUNCT
ejpam-6461	151	27	,	,	PUNCT
ejpam-6461	151	28	we	we	PRON
ejpam-6461	151	29	obtain	obtain	VERB
ejpam-6461	151	30	the	the	DET
ejpam-6461	151	31	inequality	inequality	NOUN
ejpam-6461	151	32	(	(	PUNCT
ejpam-6461	151	33	5	5	NUM
ejpam-6461	151	34	)	)	PUNCT
ejpam-6461	151	35	.	.	PUNCT
ejpam-6461	152	1	3	3	X
ejpam-6461	152	2	.	.	X
ejpam-6461	152	3	numerical	numerical	PROPN
ejpam-6461	152	4	radius	radius	PROPN
ejpam-6461	152	5	inequalities	inequality	NOUN
ejpam-6461	152	6	via	via	ADP
ejpam-6461	152	7	block	block	NOUN
ejpam-6461	152	8	matrices	matrix	NOUN
ejpam-6461	152	9	in	in	ADP
ejpam-6461	152	10	this	this	DET
ejpam-6461	152	11	section	section	NOUN
ejpam-6461	152	12	,	,	PUNCT
ejpam-6461	152	13	new	new	PROPN
ejpam-6461	152	14	numerical	numerical	PROPN
ejpam-6461	152	15	radius	radius	PROPN
ejpam-6461	152	16	inequalities	inequality	NOUN
ejpam-6461	152	17	are	be	AUX
ejpam-6461	152	18	proved	prove	VERB
ejpam-6461	152	19	by	by	ADP
ejpam-6461	152	20	using	use	VERB
ejpam-6461	152	21	block	block	NOUN
ejpam-6461	152	22	matrices	matrix	NOUN
ejpam-6461	152	23	.	.	PUNCT
ejpam-6461	153	1	we	we	PRON
ejpam-6461	153	2	begin	begin	VERB
ejpam-6461	153	3	by	by	ADP
ejpam-6461	153	4	proving	prove	VERB
ejpam-6461	153	5	the	the	DET
ejpam-6461	153	6	following	follow	VERB
ejpam-6461	153	7	inequality	inequality	NOUN
ejpam-6461	153	8	which	which	PRON
ejpam-6461	153	9	is	be	AUX
ejpam-6461	153	10	a	a	DET
ejpam-6461	153	11	direct	direct	ADJ
ejpam-6461	153	12	application	application	NOUN
ejpam-6461	153	13	of	of	ADP
ejpam-6461	153	14	inequality	inequality	NOUN
ejpam-6461	153	15	(	(	PUNCT
ejpam-6461	153	16	15	15	NUM
ejpam-6461	153	17	)	)	PUNCT
ejpam-6461	153	18	.	.	PUNCT
ejpam-6461	154	1	theorem	theorem	NOUN
ejpam-6461	154	2	3	3	X
ejpam-6461	154	3	.	.	PUNCT
ejpam-6461	155	1	let	let	VERB
ejpam-6461	155	2	x	x	PRON
ejpam-6461	155	3	,	,	PUNCT
ejpam-6461	155	4	y	y	PROPN
ejpam-6461	155	5	,	,	PUNCT
ejpam-6461	155	6	w	w	PROPN
ejpam-6461	155	7	,	,	PUNCT
ejpam-6461	155	8	z	z	NOUN
ejpam-6461	155	9	be	be	AUX
ejpam-6461	155	10	operators	operator	NOUN
ejpam-6461	155	11	in	in	ADP
ejpam-6461	155	12	b(h	b(h	PROPN
ejpam-6461	155	13	)	)	PUNCT
ejpam-6461	155	14	.	.	PUNCT
ejpam-6461	156	1	then	then	ADV
ejpam-6461	156	2	w2	w2	NOUN
ejpam-6461	156	3	[	[	PUNCT
ejpam-6461	156	4	x	x	PROPN
ejpam-6461	156	5	z	z	PROPN
ejpam-6461	156	6	w	w	PROPN
ejpam-6461	156	7	y	y	PROPN
ejpam-6461	156	8	]	]	PUNCT
ejpam-6461	156	9	≤	≤	NUM
ejpam-6461	156	10	1	1	NUM
ejpam-6461	156	11	4	4	NUM
ejpam-6461	156	12	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADJ
ejpam-6461	156	13	[	[	PUNCT
ejpam-6461	156	14	|x|+	|x|+	NOUN
ejpam-6461	156	15	|x∗|+	|x∗|+	ADJ
ejpam-6461	156	16	|w	|w	NOUN
ejpam-6461	156	17	|+	|+	X
ejpam-6461	156	18	|z∗|	|z∗|	PUNCT
ejpam-6461	157	1	o	o	NOUN
ejpam-6461	157	2	o	o	X
ejpam-6461	157	3	|y	|y	NOUN
ejpam-6461	157	4	|+	|+	X
ejpam-6461	157	5	|y	|y	VERB
ejpam-6461	157	6	∗|+	∗|+	PROPN
ejpam-6461	157	7	|w	|w	ADJ
ejpam-6461	157	8	∗|+	∗|+	PROPN
ejpam-6461	157	9	|z|	|z|	NOUN
ejpam-6461	157	10	]	]	X
ejpam-6461	157	11	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	157	12	.	.	PUNCT
ejpam-6461	158	1	(	(	PUNCT
ejpam-6461	158	2	19	19	NUM
ejpam-6461	158	3	)	)	PUNCT
ejpam-6461	158	4	proof	proof	NOUN
ejpam-6461	158	5	.	.	PUNCT
ejpam-6461	159	1	letting	let	VERB
ejpam-6461	159	2	a	a	PRON
ejpam-6461	159	3	=	=	PUNCT
ejpam-6461	159	4	[	[	PUNCT
ejpam-6461	159	5	x	x	PUNCT
ejpam-6461	159	6	o	o	X
ejpam-6461	159	7	o	o	X
ejpam-6461	159	8	y	y	X
ejpam-6461	159	9	]	]	PUNCT
ejpam-6461	159	10	and	and	CCONJ
ejpam-6461	159	11	b	b	X
ejpam-6461	159	12	=	=	PUNCT
ejpam-6461	160	1	[	[	PUNCT
ejpam-6461	160	2	o	o	X
ejpam-6461	160	3	z	z	NOUN
ejpam-6461	160	4	w	w	NOUN
ejpam-6461	160	5	o	o	X
ejpam-6461	160	6	]	]	PUNCT
ejpam-6461	160	7	in	in	ADP
ejpam-6461	160	8	inequality	inequality	NOUN
ejpam-6461	160	9	(	(	PUNCT
ejpam-6461	160	10	15	15	NUM
ejpam-6461	160	11	)	)	PUNCT
ejpam-6461	160	12	,	,	PUNCT
ejpam-6461	160	13	we	we	PRON
ejpam-6461	160	14	obtain	obtain	VERB
ejpam-6461	160	15	the	the	DET
ejpam-6461	160	16	inequality	inequality	NOUN
ejpam-6461	160	17	(	(	PUNCT
ejpam-6461	160	18	19	19	NUM
ejpam-6461	160	19	)	)	PUNCT
ejpam-6461	160	20	.	.	PUNCT
ejpam-6461	161	1	corollary	corollary	ADJ
ejpam-6461	161	2	5	5	NUM
ejpam-6461	161	3	.	.	PUNCT
ejpam-6461	162	1	let	let	VERB
ejpam-6461	162	2	x	x	PRON
ejpam-6461	162	3	,	,	PUNCT
ejpam-6461	162	4	y	y	PROPN
ejpam-6461	162	5	be	be	VERB
ejpam-6461	162	6	operators	operator	NOUN
ejpam-6461	162	7	in	in	ADP
ejpam-6461	162	8	b(h	b(h	PROPN
ejpam-6461	162	9	)	)	PUNCT
ejpam-6461	162	10	.	.	PUNCT
ejpam-6461	163	1	then	then	ADV
ejpam-6461	163	2	w2	w2	NOUN
ejpam-6461	163	3	[	[	PUNCT
ejpam-6461	163	4	x	x	PUNCT
ejpam-6461	163	5	o	o	X
ejpam-6461	163	6	o	o	X
ejpam-6461	163	7	y	y	X
ejpam-6461	163	8	]	]	PUNCT
ejpam-6461	163	9	≤	≤	NUM
ejpam-6461	163	10	1	1	NUM
ejpam-6461	163	11	4	4	NUM
ejpam-6461	163	12	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADJ
ejpam-6461	163	13	[	[	PUNCT
ejpam-6461	163	14	|x|+	|x|+	NOUN
ejpam-6461	163	15	|x∗|	|x∗|	NUM
ejpam-6461	163	16	o	o	NOUN
ejpam-6461	163	17	o	o	X
ejpam-6461	163	18	|y	|y	NOUN
ejpam-6461	163	19	|+	|+	X
ejpam-6461	163	20	|y	|y	VERB
ejpam-6461	163	21	∗|	∗|	NOUN
ejpam-6461	163	22	]	]	PUNCT
ejpam-6461	163	23	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	163	24	.	.	PUNCT
ejpam-6461	164	1	(	(	PUNCT
ejpam-6461	164	2	20	20	X
ejpam-6461	164	3	)	)	PUNCT
ejpam-6461	164	4	proof	proof	NOUN
ejpam-6461	164	5	.	.	PUNCT
ejpam-6461	165	1	letting	let	VERB
ejpam-6461	165	2	z	z	NOUN
ejpam-6461	165	3	=	=	PUNCT
ejpam-6461	165	4	w	w	PROPN
ejpam-6461	165	5	=	=	SYM
ejpam-6461	165	6	o	o	PROPN
ejpam-6461	165	7	in	in	ADP
ejpam-6461	165	8	inequality	inequality	NOUN
ejpam-6461	165	9	(	(	PUNCT
ejpam-6461	165	10	19	19	NUM
ejpam-6461	165	11	)	)	PUNCT
ejpam-6461	165	12	,	,	PUNCT
ejpam-6461	165	13	we	we	PRON
ejpam-6461	165	14	obtain	obtain	VERB
ejpam-6461	165	15	the	the	DET
ejpam-6461	165	16	inequality	inequality	NOUN
ejpam-6461	165	17	(	(	PUNCT
ejpam-6461	165	18	20	20	NUM
ejpam-6461	165	19	)	)	PUNCT
ejpam-6461	165	20	.	.	PUNCT
ejpam-6461	166	1	remark	remark	PROPN
ejpam-6461	166	2	4	4	NUM
ejpam-6461	166	3	.	.	PUNCT
ejpam-6461	167	1	letting	let	VERB
ejpam-6461	167	2	y	y	PROPN
ejpam-6461	167	3	=	=	PUNCT
ejpam-6461	167	4	o	o	PROPN
ejpam-6461	167	5	in	in	ADP
ejpam-6461	167	6	inequality	inequality	NOUN
ejpam-6461	167	7	(	(	PUNCT
ejpam-6461	167	8	20	20	NUM
ejpam-6461	167	9	)	)	PUNCT
ejpam-6461	167	10	,	,	PUNCT
ejpam-6461	167	11	we	we	PRON
ejpam-6461	167	12	obtain	obtain	VERB
ejpam-6461	167	13	the	the	DET
ejpam-6461	167	14	inequality	inequality	NOUN
ejpam-6461	167	15	(	(	PUNCT
ejpam-6461	167	16	16	16	NUM
ejpam-6461	167	17	)	)	PUNCT
ejpam-6461	167	18	.	.	PUNCT
ejpam-6461	168	1	in	in	ADP
ejpam-6461	168	2	that	that	DET
ejpam-6461	168	3	sense	sense	NOUN
ejpam-6461	168	4	,	,	PUNCT
ejpam-6461	168	5	inequality	inequality	NOUN
ejpam-6461	168	6	(	(	PUNCT
ejpam-6461	168	7	20	20	NUM
ejpam-6461	168	8	)	)	PUNCT
ejpam-6461	168	9	is	be	AUX
ejpam-6461	168	10	a	a	DET
ejpam-6461	168	11	generalization	generalization	NOUN
ejpam-6461	168	12	of	of	ADP
ejpam-6461	168	13	inequality	inequality	NOUN
ejpam-6461	168	14	(	(	PUNCT
ejpam-6461	168	15	16	16	NUM
ejpam-6461	168	16	)	)	PUNCT
ejpam-6461	168	17	.	.	PUNCT
ejpam-6461	169	1	corollary	corollary	ADJ
ejpam-6461	169	2	6	6	NUM
ejpam-6461	169	3	.	.	PUNCT
ejpam-6461	170	1	let	let	VERB
ejpam-6461	170	2	w	w	X
ejpam-6461	170	3	,	,	PUNCT
ejpam-6461	170	4	z	z	NOUN
ejpam-6461	170	5	be	be	AUX
ejpam-6461	170	6	operators	operator	NOUN
ejpam-6461	170	7	in	in	ADP
ejpam-6461	170	8	b(h	b(h	PROPN
ejpam-6461	170	9	)	)	PUNCT
ejpam-6461	170	10	.	.	PUNCT
ejpam-6461	171	1	then	then	ADV
ejpam-6461	171	2	w2	w2	NOUN
ejpam-6461	171	3	[	[	PUNCT
ejpam-6461	171	4	o	o	X
ejpam-6461	171	5	z	z	PROPN
ejpam-6461	171	6	w	w	PROPN
ejpam-6461	171	7	o	o	X
ejpam-6461	171	8	]	]	PUNCT
ejpam-6461	171	9	≤	≤	NUM
ejpam-6461	171	10	1	1	NUM
ejpam-6461	171	11	4	4	NUM
ejpam-6461	171	12	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADJ
ejpam-6461	171	13	[	[	PUNCT
ejpam-6461	171	14	|w	|w	ADJ
ejpam-6461	171	15	|+	|+	X
ejpam-6461	171	16	|z∗|	|z∗|	PUNCT
ejpam-6461	172	1	o	o	NOUN
ejpam-6461	172	2	o	o	X
ejpam-6461	172	3	|w	|w	ADJ
ejpam-6461	172	4	∗|+	∗|+	PROPN
ejpam-6461	172	5	|z|	|z|	NOUN
ejpam-6461	172	6	]	]	X
ejpam-6461	172	7	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	172	8	.	.	PUNCT
ejpam-6461	173	1	(	(	PUNCT
ejpam-6461	173	2	21	21	NUM
ejpam-6461	173	3	)	)	PUNCT
ejpam-6461	173	4	m.	m.	NOUN
ejpam-6461	173	5	al	al	PROPN
ejpam-6461	173	6	-	-	PUNCT
ejpam-6461	173	7	labadi	labadi	PROPN
ejpam-6461	173	8	et	et	PROPN
ejpam-6461	173	9	al	al	PROPN
ejpam-6461	173	10	.	.	PUNCT
ejpam-6461	173	11	/	/	SYM
ejpam-6461	173	12	eur	eur	PROPN
ejpam-6461	173	13	.	.	PUNCT
ejpam-6461	174	1	j.	j.	PROPN
ejpam-6461	174	2	pure	pure	PROPN
ejpam-6461	174	3	appl	appl	PROPN
ejpam-6461	174	4	.	.	PROPN
ejpam-6461	174	5	math	math	PROPN
ejpam-6461	174	6	,	,	PUNCT
ejpam-6461	174	7	18	18	NUM
ejpam-6461	174	8	(	(	PUNCT
ejpam-6461	174	9	3	3	NUM
ejpam-6461	174	10	)	)	PUNCT
ejpam-6461	174	11	(	(	PUNCT
ejpam-6461	174	12	2025	2025	NUM
ejpam-6461	174	13	)	)	PUNCT
ejpam-6461	174	14	,	,	PUNCT
ejpam-6461	174	15	6461	6461	NUM
ejpam-6461	174	16	7	7	NUM
ejpam-6461	174	17	of	of	ADP
ejpam-6461	174	18	13	13	NUM
ejpam-6461	174	19	proof	proof	NOUN
ejpam-6461	174	20	.	.	PUNCT
ejpam-6461	175	1	letting	let	VERB
ejpam-6461	175	2	x	x	PUNCT
ejpam-6461	175	3	=	=	PUNCT
ejpam-6461	175	4	y	y	PROPN
ejpam-6461	175	5	=	=	SYM
ejpam-6461	175	6	o	o	PROPN
ejpam-6461	175	7	in	in	ADP
ejpam-6461	175	8	inequality	inequality	NOUN
ejpam-6461	175	9	(	(	PUNCT
ejpam-6461	175	10	19	19	NUM
ejpam-6461	175	11	)	)	PUNCT
ejpam-6461	175	12	,	,	PUNCT
ejpam-6461	175	13	we	we	PRON
ejpam-6461	175	14	obtain	obtain	VERB
ejpam-6461	175	15	the	the	DET
ejpam-6461	175	16	inequality	inequality	NOUN
ejpam-6461	175	17	(	(	PUNCT
ejpam-6461	175	18	21	21	NUM
ejpam-6461	175	19	)	)	PUNCT
ejpam-6461	175	20	.	.	PUNCT
ejpam-6461	176	1	remark	remark	PROPN
ejpam-6461	176	2	5	5	NUM
ejpam-6461	176	3	.	.	PUNCT
ejpam-6461	177	1	replacing	replace	VERB
ejpam-6461	177	2	w	w	ADV
ejpam-6461	177	3	by	by	ADP
ejpam-6461	177	4	w	w	NOUN
ejpam-6461	177	5	∗	∗	NOUN
ejpam-6461	177	6	in	in	ADP
ejpam-6461	177	7	inequality	inequality	NOUN
ejpam-6461	177	8	(	(	PUNCT
ejpam-6461	177	9	21	21	NUM
ejpam-6461	177	10	)	)	PUNCT
ejpam-6461	177	11	,	,	PUNCT
ejpam-6461	177	12	we	we	PRON
ejpam-6461	177	13	obtain	obtain	VERB
ejpam-6461	177	14	w2	w2	NOUN
ejpam-6461	177	15	[	[	PUNCT
ejpam-6461	177	16	o	o	X
ejpam-6461	177	17	z	z	X
ejpam-6461	177	18	w	w	PROPN
ejpam-6461	177	19	∗	∗	X
ejpam-6461	177	20	o	o	X
ejpam-6461	177	21	]	]	PUNCT
ejpam-6461	177	22	≤	≤	NUM
ejpam-6461	177	23	1	1	NUM
ejpam-6461	177	24	4	4	NUM
ejpam-6461	177	25	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADJ
ejpam-6461	177	26	[	[	PUNCT
ejpam-6461	177	27	|w	|w	ADJ
ejpam-6461	177	28	∗|+	∗|+	PROPN
ejpam-6461	177	29	|z∗|	|z∗|	PROPN
ejpam-6461	178	1	o	o	NOUN
ejpam-6461	178	2	o	o	X
ejpam-6461	178	3	|w	|w	NOUN
ejpam-6461	178	4	|+	|+	NOUN
ejpam-6461	178	5	|z|	|z|	NOUN
ejpam-6461	178	6	]	]	PUNCT
ejpam-6461	178	7	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	178	8	=	=	SYM
ejpam-6461	178	9	1	1	NUM
ejpam-6461	178	10	4	4	NUM
ejpam-6461	178	11	max	max	PROPN
ejpam-6461	178	12	{	{	PUNCT
ejpam-6461	178	13	||	||	NOUN
ejpam-6461	178	14	|w	|w	NOUN
ejpam-6461	178	15	∗|+|z∗|	∗|+|z∗|	NOUN
ejpam-6461	178	16	||	||	PROPN
ejpam-6461	178	17	,	,	PUNCT
ejpam-6461	178	18	||	||	PROPN
ejpam-6461	179	1	|w	|w	PROPN
ejpam-6461	179	2	|+|z|	|+|z|	NUM
ejpam-6461	179	3	||}2	||}2	PROPN
ejpam-6461	179	4	.	.	PUNCT
ejpam-6461	180	1	(	(	PUNCT
ejpam-6461	180	2	22	22	NUM
ejpam-6461	180	3	)	)	PUNCT
ejpam-6461	180	4	but	but	CCONJ
ejpam-6461	180	5	max{w(zw	max{w(zw	NOUN
ejpam-6461	180	6	∗	∗	NOUN
ejpam-6461	180	7	)	)	PUNCT
ejpam-6461	180	8	,	,	PUNCT
ejpam-6461	180	9	w(w	w(w	PROPN
ejpam-6461	180	10	∗z	∗z	X
ejpam-6461	180	11	)	)	PUNCT
ejpam-6461	180	12	}	}	PUNCT
ejpam-6461	181	1	=	=	SYM
ejpam-6461	181	2	w	w	PROPN
ejpam-6461	181	3	[	[	PUNCT
ejpam-6461	181	4	zw	zw	X
ejpam-6461	181	5	∗	∗	NOUN
ejpam-6461	181	6	o	o	X
ejpam-6461	182	1	o	o	X
ejpam-6461	182	2	w	w	NOUN
ejpam-6461	182	3	∗z	∗z	PROPN
ejpam-6461	182	4	]	]	PUNCT
ejpam-6461	183	1	=	=	PUNCT
ejpam-6461	183	2	w	w	PROPN
ejpam-6461	183	3	[	[	PUNCT
ejpam-6461	183	4	o	o	X
ejpam-6461	183	5	z	z	X
ejpam-6461	183	6	w	w	NOUN
ejpam-6461	183	7	∗	∗	X
ejpam-6461	183	8	o	o	X
ejpam-6461	183	9	]	]	X
ejpam-6461	183	10	2	2	NUM
ejpam-6461	183	11	≤	≤	NOUN
ejpam-6461	183	12	w2	w2	NOUN
ejpam-6461	183	13	[	[	PUNCT
ejpam-6461	183	14	o	o	X
ejpam-6461	183	15	z	z	X
ejpam-6461	183	16	w	w	PROPN
ejpam-6461	183	17	∗	∗	X
ejpam-6461	183	18	o	o	X
ejpam-6461	183	19	]	]	PUNCT
ejpam-6461	183	20	.	.	PUNCT
ejpam-6461	184	1	thus	thus	ADV
ejpam-6461	184	2	max{w(zw	max{w(zw	VERB
ejpam-6461	184	3	∗	∗	NOUN
ejpam-6461	184	4	)	)	PUNCT
ejpam-6461	184	5	,	,	PUNCT
ejpam-6461	184	6	w(w	w(w	PROPN
ejpam-6461	184	7	∗z	∗z	X
ejpam-6461	184	8	)	)	PUNCT
ejpam-6461	184	9	}	}	PUNCT
ejpam-6461	184	10	≤	≤	NUM
ejpam-6461	184	11	w2	w2	NOUN
ejpam-6461	184	12	[	[	PUNCT
ejpam-6461	184	13	o	o	X
ejpam-6461	184	14	z	z	X
ejpam-6461	184	15	w	w	PROPN
ejpam-6461	184	16	∗	∗	X
ejpam-6461	184	17	o	o	X
ejpam-6461	184	18	]	]	PUNCT
ejpam-6461	184	19	.	.	PUNCT
ejpam-6461	185	1	(	(	PUNCT
ejpam-6461	185	2	23	23	X
ejpam-6461	185	3	)	)	PUNCT
ejpam-6461	185	4	combining	combine	VERB
ejpam-6461	185	5	inequalities	inequality	NOUN
ejpam-6461	185	6	(	(	PUNCT
ejpam-6461	185	7	22	22	NUM
ejpam-6461	185	8	)	)	PUNCT
ejpam-6461	185	9	and	and	CCONJ
ejpam-6461	185	10	(	(	PUNCT
ejpam-6461	185	11	23	23	NUM
ejpam-6461	185	12	)	)	PUNCT
ejpam-6461	185	13	,	,	PUNCT
ejpam-6461	185	14	we	we	PRON
ejpam-6461	185	15	obtain	obtain	VERB
ejpam-6461	185	16	:	:	PUNCT
ejpam-6461	185	17	max{w(zw	max{w(zw	NOUN
ejpam-6461	185	18	∗	∗	NOUN
ejpam-6461	185	19	)	)	PUNCT
ejpam-6461	185	20	,	,	PUNCT
ejpam-6461	185	21	w(w	w(w	PROPN
ejpam-6461	185	22	∗z	∗z	X
ejpam-6461	185	23	)	)	PUNCT
ejpam-6461	185	24	}	}	PUNCT
ejpam-6461	185	25	≤	≤	NUM
ejpam-6461	185	26	1	1	NUM
ejpam-6461	185	27	4	4	NUM
ejpam-6461	185	28	max{||	max{||	PROPN
ejpam-6461	185	29	|w	|w	ADJ
ejpam-6461	185	30	∗|+	∗|+	PROPN
ejpam-6461	185	31	|z∗|	|z∗|	PROPN
ejpam-6461	186	1	||	||	PROPN
ejpam-6461	186	2	,	,	PUNCT
ejpam-6461	186	3	||	||	PROPN
ejpam-6461	187	1	|w	|w	ADJ
ejpam-6461	187	2	|+	|+	NOUN
ejpam-6461	187	3	|z|	|z|	VERB
ejpam-6461	187	4	||}2	||}2	PROPN
ejpam-6461	187	5	.	.	PUNCT
ejpam-6461	188	1	(	(	PUNCT
ejpam-6461	188	2	24	24	NUM
ejpam-6461	188	3	)	)	PUNCT
ejpam-6461	188	4	note	note	VERB
ejpam-6461	188	5	that	that	SCONJ
ejpam-6461	188	6	inequality	inequality	NOUN
ejpam-6461	188	7	(	(	PUNCT
ejpam-6461	188	8	24	24	NUM
ejpam-6461	188	9	)	)	PUNCT
ejpam-6461	188	10	is	be	AUX
ejpam-6461	188	11	equivalent	equivalent	ADJ
ejpam-6461	188	12	to	to	ADP
ejpam-6461	188	13	inequality	inequality	NOUN
ejpam-6461	188	14	(	(	PUNCT
ejpam-6461	188	15	6	6	NUM
ejpam-6461	188	16	)	)	PUNCT
ejpam-6461	188	17	if	if	SCONJ
ejpam-6461	188	18	w	w	PROPN
ejpam-6461	188	19	and	and	CCONJ
ejpam-6461	188	20	z	z	NOUN
ejpam-6461	188	21	are	be	AUX
ejpam-6461	188	22	self	self	NOUN
ejpam-6461	188	23	-	-	PUNCT
ejpam-6461	188	24	adjoint	adjoint	NOUN
ejpam-6461	188	25	operators	operator	NOUN
ejpam-6461	188	26	.	.	PUNCT
ejpam-6461	189	1	in	in	ADP
ejpam-6461	189	2	that	that	DET
ejpam-6461	189	3	sense	sense	NOUN
ejpam-6461	189	4	,	,	PUNCT
ejpam-6461	189	5	inequality	inequality	NOUN
ejpam-6461	189	6	(	(	PUNCT
ejpam-6461	189	7	19	19	NUM
ejpam-6461	189	8	)	)	PUNCT
ejpam-6461	189	9	is	be	AUX
ejpam-6461	189	10	a	a	DET
ejpam-6461	189	11	generalization	generalization	NOUN
ejpam-6461	189	12	of	of	ADP
ejpam-6461	189	13	inequality	inequality	NOUN
ejpam-6461	189	14	(	(	PUNCT
ejpam-6461	189	15	6	6	NUM
ejpam-6461	189	16	)	)	PUNCT
ejpam-6461	189	17	when	when	SCONJ
ejpam-6461	189	18	w	w	NOUN
ejpam-6461	189	19	and	and	CCONJ
ejpam-6461	189	20	z	z	NOUN
ejpam-6461	189	21	are	be	AUX
ejpam-6461	189	22	self	self	NOUN
ejpam-6461	189	23	-	-	PUNCT
ejpam-6461	189	24	adjoint	adjoint	NOUN
ejpam-6461	189	25	operators	operator	NOUN
ejpam-6461	189	26	.	.	PUNCT
ejpam-6461	190	1	lemma	lemma	PROPN
ejpam-6461	190	2	7	7	NUM
ejpam-6461	190	3	.	.	PUNCT
ejpam-6461	191	1	[	[	X
ejpam-6461	191	2	7	7	X
ejpam-6461	191	3	]	]	PUNCT
ejpam-6461	191	4	let	let	VERB
ejpam-6461	191	5	a	a	DET
ejpam-6461	191	6	∈	∈	PROPN
ejpam-6461	191	7	b(h	b(h	PROPN
ejpam-6461	191	8	)	)	PUNCT
ejpam-6461	191	9	and	and	CCONJ
ejpam-6461	191	10	r	r	NOUN
ejpam-6461	191	11	≥	≥	NUM
ejpam-6461	191	12	2	2	NUM
ejpam-6461	191	13	.	.	PUNCT
ejpam-6461	192	1	then	then	ADV
ejpam-6461	192	2	wr(a	wr(a	X
ejpam-6461	192	3	)	)	PUNCT
ejpam-6461	192	4	≤	≤	NOUN
ejpam-6461	192	5	2r−3||	2r−3||	NUM
ejpam-6461	192	6	|a|r	|a|r	ADP
ejpam-6461	192	7	+	+	NOUN
ejpam-6461	192	8	|a∗|r	|a∗|r	NUM
ejpam-6461	192	9	||	||	NOUN
ejpam-6461	192	10	.	.	PUNCT
ejpam-6461	193	1	(	(	PUNCT
ejpam-6461	193	2	25	25	NUM
ejpam-6461	193	3	)	)	PUNCT
ejpam-6461	193	4	the	the	DET
ejpam-6461	193	5	following	follow	VERB
ejpam-6461	193	6	theorem	theorem	NOUN
ejpam-6461	193	7	is	be	AUX
ejpam-6461	193	8	another	another	DET
ejpam-6461	193	9	generalization	generalization	NOUN
ejpam-6461	193	10	of	of	ADP
ejpam-6461	193	11	inequality	inequality	NOUN
ejpam-6461	193	12	(	(	PUNCT
ejpam-6461	193	13	4	4	NUM
ejpam-6461	193	14	)	)	PUNCT
ejpam-6461	193	15	.	.	PUNCT
ejpam-6461	194	1	theorem	theorem	ADJ
ejpam-6461	194	2	4	4	NUM
ejpam-6461	194	3	.	.	PUNCT
ejpam-6461	195	1	let	let	VERB
ejpam-6461	195	2	x	x	PRON
ejpam-6461	195	3	,	,	PUNCT
ejpam-6461	195	4	y	y	PROPN
ejpam-6461	195	5	be	be	VERB
ejpam-6461	195	6	operators	operator	NOUN
ejpam-6461	195	7	in	in	ADP
ejpam-6461	195	8	b(h	b(h	PROPN
ejpam-6461	195	9	)	)	PUNCT
ejpam-6461	195	10	,	,	PUNCT
ejpam-6461	196	1	r	r	NOUN
ejpam-6461	196	2	≥	≥	NOUN
ejpam-6461	196	3	2	2	NUM
ejpam-6461	196	4	.	.	PUNCT
ejpam-6461	196	5	then	then	ADV
ejpam-6461	196	6	wr	wr	X
ejpam-6461	196	7	[	[	PUNCT
ejpam-6461	196	8	o	o	NOUN
ejpam-6461	196	9	x	x	X
ejpam-6461	196	10	y	y	NOUN
ejpam-6461	196	11	∗	∗	X
ejpam-6461	196	12	o	o	X
ejpam-6461	196	13	]	]	PUNCT
ejpam-6461	196	14	≤	≤	NUM
ejpam-6461	196	15	2r−3max{||	2r−3max{||	NUM
ejpam-6461	196	16	|x∗|r	|x∗|r	NUM
ejpam-6461	196	17	+	+	PUNCT
ejpam-6461	196	18	|y	|y	ADJ
ejpam-6461	196	19	∗|r	∗|r	NOUN
ejpam-6461	196	20	||	||	NOUN
ejpam-6461	196	21	,	,	PUNCT
ejpam-6461	196	22	||	||	PROPN
ejpam-6461	196	23	|x|r	|x|r	PROPN
ejpam-6461	196	24	+	+	CCONJ
ejpam-6461	196	25	|y	|y	NOUN
ejpam-6461	196	26	|r	|r	X
ejpam-6461	196	27	||	||	NUM
ejpam-6461	196	28	}	}	PUNCT
ejpam-6461	196	29	.	.	PUNCT
ejpam-6461	197	1	(	(	PUNCT
ejpam-6461	197	2	26	26	NUM
ejpam-6461	197	3	)	)	PUNCT
ejpam-6461	197	4	in	in	ADP
ejpam-6461	197	5	particular	particular	ADJ
ejpam-6461	197	6	,	,	PUNCT
ejpam-6461	197	7	w2	w2	NOUN
ejpam-6461	197	8	[	[	PUNCT
ejpam-6461	197	9	o	o	X
ejpam-6461	197	10	x	x	X
ejpam-6461	197	11	y	y	NOUN
ejpam-6461	197	12	∗	∗	X
ejpam-6461	197	13	o	o	X
ejpam-6461	197	14	]	]	X
ejpam-6461	197	15	≤	≤	NUM
ejpam-6461	197	16	1	1	NUM
ejpam-6461	197	17	2	2	NUM
ejpam-6461	197	18	max{||	max{||	NOUN
ejpam-6461	197	19	|x|2	|x|2	NOUN
ejpam-6461	197	20	+	+	CCONJ
ejpam-6461	197	21	|y	|y	NOUN
ejpam-6461	197	22	|2	|2	NUM
ejpam-6461	197	23	||	||	NOUN
ejpam-6461	197	24	,	,	PUNCT
ejpam-6461	197	25	||	||	NOUN
ejpam-6461	198	1	|x∗|2	|x∗|2	NUM
ejpam-6461	198	2	+	+	CCONJ
ejpam-6461	198	3	|y	|y	X
ejpam-6461	198	4	∗|2	∗|2	PUNCT
ejpam-6461	198	5	||	||	NUM
ejpam-6461	198	6	}	}	PUNCT
ejpam-6461	198	7	(	(	PUNCT
ejpam-6461	198	8	27	27	NUM
ejpam-6461	198	9	)	)	PUNCT
ejpam-6461	198	10	proof	proof	NOUN
ejpam-6461	198	11	.	.	PUNCT
ejpam-6461	199	1	letting	let	VERB
ejpam-6461	199	2	a	a	PRON
ejpam-6461	199	3	=	=	PUNCT
ejpam-6461	199	4	[	[	PUNCT
ejpam-6461	199	5	o	o	X
ejpam-6461	199	6	x	x	X
ejpam-6461	199	7	y	y	NOUN
ejpam-6461	199	8	∗	∗	X
ejpam-6461	199	9	o	o	X
ejpam-6461	199	10	]	]	PUNCT
ejpam-6461	199	11	in	in	ADP
ejpam-6461	199	12	inequality	inequality	NOUN
ejpam-6461	199	13	(	(	PUNCT
ejpam-6461	199	14	25	25	NUM
ejpam-6461	199	15	)	)	PUNCT
ejpam-6461	199	16	,	,	PUNCT
ejpam-6461	199	17	implies	imply	VERB
ejpam-6461	199	18	that	that	PRON
ejpam-6461	199	19	wr	wr	NOUN
ejpam-6461	199	20	[	[	PUNCT
ejpam-6461	199	21	o	o	X
ejpam-6461	199	22	x	x	X
ejpam-6461	199	23	y	y	NOUN
ejpam-6461	199	24	∗	∗	X
ejpam-6461	199	25	o	o	X
ejpam-6461	199	26	]	]	PUNCT
ejpam-6461	199	27	≤	≤	NUM
ejpam-6461	199	28	2r−3	2r−3	NOUN
ejpam-6461	199	29	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	NOUN
ejpam-6461	199	30	[	[	PUNCT
ejpam-6461	199	31	|y	|y	NOUN
ejpam-6461	199	32	∗|r	∗|r	NOUN
ejpam-6461	199	33	o	o	NOUN
ejpam-6461	199	34	o	o	X
ejpam-6461	199	35	|x|r	|x|r	PROPN
ejpam-6461	199	36	]	]	PUNCT
ejpam-6461	200	1	+	+	CCONJ
ejpam-6461	200	2	[	[	PUNCT
ejpam-6461	200	3	|x∗|r	|x∗|r	NUM
ejpam-6461	200	4	o	o	NOUN
ejpam-6461	200	5	o	o	X
ejpam-6461	200	6	|y	|y	X
ejpam-6461	200	7	|r	|r	X
ejpam-6461	200	8	]	]	PUNCT
ejpam-6461	200	9	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	PROPN
ejpam-6461	200	10	=	=	PUNCT
ejpam-6461	200	11	2r−3	2r−3	NOUN
ejpam-6461	200	12	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	NOUN
ejpam-6461	200	13	[	[	PUNCT
ejpam-6461	200	14	|x∗|r	|x∗|r	NOUN
ejpam-6461	200	15	+	+	CCONJ
ejpam-6461	200	16	|y	|y	NOUN
ejpam-6461	200	17	∗|r	∗|r	NOUN
ejpam-6461	200	18	o	o	NOUN
ejpam-6461	200	19	o	o	X
ejpam-6461	200	20	|x|r	|x|r	PROPN
ejpam-6461	200	21	+	+	CCONJ
ejpam-6461	200	22	|y	|y	X
ejpam-6461	200	23	|r	|r	X
ejpam-6461	200	24	]	]	PUNCT
ejpam-6461	200	25	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	PROPN
ejpam-6461	200	26	=	=	SYM
ejpam-6461	200	27	2r−3max{||	2r−3max{||	NUM
ejpam-6461	200	28	|x∗|r	|x∗|r	NUM
ejpam-6461	200	29	+	+	PUNCT
ejpam-6461	200	30	|y	|y	ADJ
ejpam-6461	200	31	∗|r	∗|r	NOUN
ejpam-6461	200	32	||	||	NOUN
ejpam-6461	200	33	,	,	PUNCT
ejpam-6461	200	34	||	||	PROPN
ejpam-6461	201	1	|x|r	|x|r	PROPN
ejpam-6461	201	2	+	+	CCONJ
ejpam-6461	201	3	|y	|y	NOUN
ejpam-6461	201	4	|r	|r	X
ejpam-6461	201	5	||	||	NUM
ejpam-6461	201	6	}	}	PUNCT
ejpam-6461	201	7	.	.	PUNCT
ejpam-6461	202	1	m.	m.	PROPN
ejpam-6461	202	2	al	al	PROPN
ejpam-6461	202	3	-	-	PUNCT
ejpam-6461	202	4	labadi	labadi	PROPN
ejpam-6461	202	5	et	et	PROPN
ejpam-6461	202	6	al	al	PROPN
ejpam-6461	202	7	.	.	PUNCT
ejpam-6461	202	8	/	/	SYM
ejpam-6461	202	9	eur	eur	PROPN
ejpam-6461	202	10	.	.	PUNCT
ejpam-6461	203	1	j.	j.	PROPN
ejpam-6461	203	2	pure	pure	PROPN
ejpam-6461	203	3	appl	appl	PROPN
ejpam-6461	203	4	.	.	PROPN
ejpam-6461	203	5	math	math	PROPN
ejpam-6461	203	6	,	,	PUNCT
ejpam-6461	203	7	18	18	NUM
ejpam-6461	203	8	(	(	PUNCT
ejpam-6461	203	9	3	3	NUM
ejpam-6461	203	10	)	)	PUNCT
ejpam-6461	203	11	(	(	PUNCT
ejpam-6461	203	12	2025	2025	NUM
ejpam-6461	203	13	)	)	PUNCT
ejpam-6461	203	14	,	,	PUNCT
ejpam-6461	203	15	6461	6461	NUM
ejpam-6461	203	16	8	8	NUM
ejpam-6461	203	17	of	of	ADP
ejpam-6461	203	18	13	13	NUM
ejpam-6461	203	19	remark	remark	NOUN
ejpam-6461	203	20	6	6	NUM
ejpam-6461	203	21	.	.	PUNCT
ejpam-6461	204	1	inequality	inequality	NOUN
ejpam-6461	204	2	(	(	PUNCT
ejpam-6461	204	3	24	24	NUM
ejpam-6461	204	4	)	)	PUNCT
ejpam-6461	204	5	is	be	AUX
ejpam-6461	204	6	sharper	sharp	ADJ
ejpam-6461	204	7	than	than	ADP
ejpam-6461	204	8	inequality	inequality	NOUN
ejpam-6461	204	9	(	(	PUNCT
ejpam-6461	204	10	29	29	NUM
ejpam-6461	204	11	)	)	PUNCT
ejpam-6461	204	12	.	.	PUNCT
ejpam-6461	205	1	to	to	PART
ejpam-6461	205	2	show	show	VERB
ejpam-6461	205	3	this	this	PRON
ejpam-6461	205	4	,	,	PUNCT
ejpam-6461	205	5	note	note	VERB
ejpam-6461	205	6	that	that	SCONJ
ejpam-6461	205	7	max{w(xy	max{w(xy	PROPN
ejpam-6461	205	8	∗	∗	NOUN
ejpam-6461	205	9	)	)	PUNCT
ejpam-6461	205	10	,	,	PUNCT
ejpam-6461	205	11	w(y	w(y	AUX
ejpam-6461	205	12	∗x	∗x	NOUN
ejpam-6461	205	13	)	)	PUNCT
ejpam-6461	205	14	}	}	PUNCT
ejpam-6461	206	1	=	=	SYM
ejpam-6461	206	2	w	w	PROPN
ejpam-6461	206	3	[	[	PUNCT
ejpam-6461	206	4	xy	xy	PROPN
ejpam-6461	206	5	∗	∗	NOUN
ejpam-6461	206	6	o	o	X
ejpam-6461	206	7	o	o	X
ejpam-6461	206	8	y	y	NOUN
ejpam-6461	206	9	∗x	∗x	PRON
ejpam-6461	206	10	]	]	PUNCT
ejpam-6461	207	1	=	=	PUNCT
ejpam-6461	207	2	w	w	PROPN
ejpam-6461	207	3	[	[	PUNCT
ejpam-6461	207	4	o	o	X
ejpam-6461	207	5	x	x	X
ejpam-6461	207	6	y	y	NOUN
ejpam-6461	207	7	∗	∗	X
ejpam-6461	207	8	o	o	X
ejpam-6461	207	9	]	]	X
ejpam-6461	207	10	2	2	NUM
ejpam-6461	207	11	≤	≤	NOUN
ejpam-6461	207	12	w2	w2	NOUN
ejpam-6461	207	13	[	[	PUNCT
ejpam-6461	207	14	o	o	X
ejpam-6461	207	15	x	x	X
ejpam-6461	207	16	y	y	NOUN
ejpam-6461	207	17	∗	∗	X
ejpam-6461	207	18	o	o	X
ejpam-6461	207	19	]	]	PUNCT
ejpam-6461	207	20	.	.	PUNCT
ejpam-6461	208	1	(	(	PUNCT
ejpam-6461	208	2	28	28	X
ejpam-6461	208	3	)	)	PUNCT
ejpam-6461	208	4	combining	combine	VERB
ejpam-6461	208	5	inequalities	inequality	NOUN
ejpam-6461	208	6	(	(	PUNCT
ejpam-6461	208	7	27	27	NUM
ejpam-6461	208	8	)	)	PUNCT
ejpam-6461	208	9	and	and	CCONJ
ejpam-6461	208	10	(	(	PUNCT
ejpam-6461	208	11	28	28	NUM
ejpam-6461	208	12	)	)	PUNCT
ejpam-6461	208	13	,	,	PUNCT
ejpam-6461	208	14	it	it	PRON
ejpam-6461	208	15	follows	follow	VERB
ejpam-6461	208	16	that	that	SCONJ
ejpam-6461	209	1	max	max	PROPN
ejpam-6461	209	2	{	{	PUNCT
ejpam-6461	209	3	w(xy	w(xy	PROPN
ejpam-6461	209	4	∗	∗	NOUN
ejpam-6461	209	5	)	)	PUNCT
ejpam-6461	209	6	,	,	PUNCT
ejpam-6461	209	7	w(y	w(y	AUX
ejpam-6461	209	8	∗x	∗x	NOUN
ejpam-6461	209	9	)	)	PUNCT
ejpam-6461	209	10	}	}	PUNCT
ejpam-6461	209	11	≤	≤	NUM
ejpam-6461	209	12	max{1	max{1	NOUN
ejpam-6461	209	13	2	2	NUM
ejpam-6461	209	14	||	||	NOUN
ejpam-6461	209	15	|x|2	|x|2	NOUN
ejpam-6461	209	16	+	+	CCONJ
ejpam-6461	209	17	|y	|y	NOUN
ejpam-6461	209	18	|2	|2	NUM
ejpam-6461	209	19	||	||	NOUN
ejpam-6461	209	20	,	,	PUNCT
ejpam-6461	209	21	1	1	NUM
ejpam-6461	209	22	2	2	NUM
ejpam-6461	209	23	||	||	NOUN
ejpam-6461	209	24	|x∗|2	|x∗|2	NUM
ejpam-6461	209	25	+	+	CCONJ
ejpam-6461	209	26	|y	|y	X
ejpam-6461	209	27	∗|2	∗|2	PUNCT
ejpam-6461	209	28	||	||	NUM
ejpam-6461	209	29	}	}	PUNCT
ejpam-6461	209	30	.	.	PUNCT
ejpam-6461	210	1	(	(	PUNCT
ejpam-6461	210	2	29	29	NUM
ejpam-6461	210	3	)	)	PUNCT
ejpam-6461	210	4	by	by	ADP
ejpam-6461	210	5	comparing	compare	VERB
ejpam-6461	210	6	inequality	inequality	NOUN
ejpam-6461	210	7	(	(	PUNCT
ejpam-6461	210	8	24	24	NUM
ejpam-6461	210	9	)	)	PUNCT
ejpam-6461	210	10	and	and	CCONJ
ejpam-6461	210	11	inequality	inequality	NOUN
ejpam-6461	210	12	(	(	PUNCT
ejpam-6461	210	13	29	29	NUM
ejpam-6461	210	14	)	)	PUNCT
ejpam-6461	210	15	,	,	PUNCT
ejpam-6461	210	16	we	we	PRON
ejpam-6461	210	17	note	note	VERB
ejpam-6461	210	18	that	that	SCONJ
ejpam-6461	210	19	while	while	SCONJ
ejpam-6461	210	20	the	the	DET
ejpam-6461	210	21	left	left	ADJ
ejpam-6461	210	22	sides	side	NOUN
ejpam-6461	210	23	of	of	ADP
ejpam-6461	210	24	both	both	DET
ejpam-6461	210	25	inequalities	inequality	NOUN
ejpam-6461	210	26	are	be	AUX
ejpam-6461	210	27	the	the	DET
ejpam-6461	210	28	same	same	ADJ
ejpam-6461	210	29	,	,	PUNCT
ejpam-6461	210	30	the	the	DET
ejpam-6461	210	31	right	right	ADJ
ejpam-6461	210	32	side	side	NOUN
ejpam-6461	210	33	in	in	ADP
ejpam-6461	210	34	inequality	inequality	NOUN
ejpam-6461	210	35	(	(	PUNCT
ejpam-6461	210	36	24	24	NUM
ejpam-6461	210	37	)	)	PUNCT
ejpam-6461	210	38	is	be	AUX
ejpam-6461	210	39	1	1	NUM
ejpam-6461	210	40	4	4	NUM
ejpam-6461	210	41	max{||	max{||	PROPN
ejpam-6461	210	42	|w	|w	ADJ
ejpam-6461	210	43	|+	|+	NOUN
ejpam-6461	210	44	|z|	|z|	NOUN
ejpam-6461	210	45	||	||	PROPN
ejpam-6461	210	46	,	,	PUNCT
ejpam-6461	210	47	||	||	NOUN
ejpam-6461	211	1	|w	|w	ADJ
ejpam-6461	211	2	∗|+	∗|+	PROPN
ejpam-6461	211	3	|z∗|	|z∗|	PROPN
ejpam-6461	211	4	||}2	||}2	PRON
ejpam-6461	212	1	≤	≤	ADV
ejpam-6461	212	2	1	1	NUM
ejpam-6461	212	3	2	2	NUM
ejpam-6461	212	4	max{||	max{||	PROPN
ejpam-6461	212	5	|w	|w	ADJ
ejpam-6461	212	6	|2	|2	NUM
ejpam-6461	212	7	+	+	NUM
ejpam-6461	212	8	|z|2	|z|2	NOUN
ejpam-6461	212	9	||	||	NOUN
ejpam-6461	212	10	,	,	PUNCT
ejpam-6461	212	11	||	||	NOUN
ejpam-6461	212	12	|w	|w	ADJ
ejpam-6461	212	13	∗|2	∗|2	PUNCT
ejpam-6461	213	1	+	+	CCONJ
ejpam-6461	213	2	|z∗|2	|z∗|2	NUM
ejpam-6461	213	3	||	||	NUM
ejpam-6461	213	4	}	}	PUNCT
ejpam-6461	213	5	(	(	PUNCT
ejpam-6461	213	6	by	by	ADP
ejpam-6461	213	7	inequality	inequality	NOUN
ejpam-6461	213	8	(	(	PUNCT
ejpam-6461	213	9	11	11	NUM
ejpam-6461	213	10	)	)	PUNCT
ejpam-6461	213	11	)	)	PUNCT
ejpam-6461	213	12	,	,	PUNCT
ejpam-6461	213	13	which	which	PRON
ejpam-6461	213	14	is	be	AUX
ejpam-6461	213	15	the	the	DET
ejpam-6461	213	16	right	right	ADJ
ejpam-6461	213	17	side	side	NOUN
ejpam-6461	213	18	of	of	ADP
ejpam-6461	213	19	inequality	inequality	NOUN
ejpam-6461	213	20	(	(	PUNCT
ejpam-6461	213	21	29	29	NUM
ejpam-6461	213	22	)	)	PUNCT
ejpam-6461	213	23	.	.	PUNCT
ejpam-6461	214	1	this	this	PRON
ejpam-6461	214	2	implies	imply	VERB
ejpam-6461	214	3	that	that	SCONJ
ejpam-6461	214	4	inequality	inequality	NOUN
ejpam-6461	214	5	(	(	PUNCT
ejpam-6461	214	6	24	24	NUM
ejpam-6461	214	7	)	)	PUNCT
ejpam-6461	214	8	is	be	AUX
ejpam-6461	214	9	sharper	sharp	ADJ
ejpam-6461	214	10	than	than	ADP
ejpam-6461	214	11	inequality	inequality	NOUN
ejpam-6461	214	12	(	(	PUNCT
ejpam-6461	214	13	29	29	NUM
ejpam-6461	214	14	)	)	PUNCT
ejpam-6461	214	15	.	.	PUNCT
ejpam-6461	215	1	corollary	corollary	ADJ
ejpam-6461	215	2	7	7	NUM
ejpam-6461	215	3	.	.	PUNCT
ejpam-6461	216	1	let	let	VERB
ejpam-6461	216	2	x	x	PRON
ejpam-6461	216	3	be	be	AUX
ejpam-6461	216	4	operators	operator	NOUN
ejpam-6461	216	5	in	in	ADP
ejpam-6461	216	6	b(h	b(h	PROPN
ejpam-6461	216	7	)	)	PUNCT
ejpam-6461	216	8	,	,	PUNCT
ejpam-6461	216	9	r	r	NOUN
ejpam-6461	216	10	≥	≥	NOUN
ejpam-6461	216	11	2	2	NUM
ejpam-6461	216	12	.	.	PUNCT
ejpam-6461	217	1	then	then	ADV
ejpam-6461	217	2	wr	wr	X
ejpam-6461	217	3	[	[	PUNCT
ejpam-6461	217	4	o	o	NOUN
ejpam-6461	217	5	x	x	X
ejpam-6461	217	6	o	o	X
ejpam-6461	217	7	o	o	X
ejpam-6461	217	8	]	]	PUNCT
ejpam-6461	217	9	≤	≤	NUM
ejpam-6461	217	10	2r−3max{||	2r−3max{||	NUM
ejpam-6461	217	11	|x∗|r	|x∗|r	NUM
ejpam-6461	217	12	||	||	NOUN
ejpam-6461	217	13	,	,	PUNCT
ejpam-6461	217	14	||	||	PROPN
ejpam-6461	217	15	|x|r	|x|r	PROPN
ejpam-6461	217	16	||	||	PROPN
ejpam-6461	217	17	}	}	PUNCT
ejpam-6461	217	18	.	.	PUNCT
ejpam-6461	218	1	(	(	PUNCT
ejpam-6461	218	2	30	30	X
ejpam-6461	218	3	)	)	PUNCT
ejpam-6461	218	4	proof	proof	NOUN
ejpam-6461	218	5	.	.	PUNCT
ejpam-6461	219	1	letting	let	VERB
ejpam-6461	219	2	y	y	PRON
ejpam-6461	219	3	∗	∗	NOUN
ejpam-6461	219	4	=	=	SYM
ejpam-6461	219	5	o	o	NOUN
ejpam-6461	219	6	in	in	ADP
ejpam-6461	219	7	inequality	inequality	NOUN
ejpam-6461	219	8	(	(	PUNCT
ejpam-6461	219	9	26	26	NUM
ejpam-6461	219	10	)	)	PUNCT
ejpam-6461	219	11	,	,	PUNCT
ejpam-6461	219	12	we	we	PRON
ejpam-6461	219	13	obtain	obtain	VERB
ejpam-6461	219	14	the	the	DET
ejpam-6461	219	15	inequality	inequality	NOUN
ejpam-6461	219	16	(	(	PUNCT
ejpam-6461	219	17	30	30	NUM
ejpam-6461	219	18	)	)	PUNCT
ejpam-6461	219	19	.	.	PUNCT
ejpam-6461	220	1	remark	remark	PROPN
ejpam-6461	220	2	7	7	NUM
ejpam-6461	220	3	.	.	PUNCT
ejpam-6461	221	1	letting	let	VERB
ejpam-6461	221	2	r	r	NOUN
ejpam-6461	221	3	=	=	SYM
ejpam-6461	221	4	2	2	NUM
ejpam-6461	221	5	in	in	ADP
ejpam-6461	221	6	inequality	inequality	NOUN
ejpam-6461	221	7	(	(	PUNCT
ejpam-6461	221	8	30	30	NUM
ejpam-6461	221	9	)	)	PUNCT
ejpam-6461	221	10	,	,	PUNCT
ejpam-6461	221	11	we	we	PRON
ejpam-6461	221	12	obtain	obtain	VERB
ejpam-6461	221	13	w2	w2	NOUN
ejpam-6461	221	14	(	(	PUNCT
ejpam-6461	221	15	[	[	PUNCT
ejpam-6461	221	16	o	o	X
ejpam-6461	221	17	x	x	X
ejpam-6461	221	18	o	o	NOUN
ejpam-6461	221	19	o	o	X
ejpam-6461	221	20	]	]	X
ejpam-6461	221	21	)	)	PUNCT
ejpam-6461	221	22	≤	≤	NUM
ejpam-6461	221	23	1	1	NUM
ejpam-6461	221	24	2	2	NUM
ejpam-6461	221	25	max{||	max{||	NOUN
ejpam-6461	221	26	xx∗	xx∗	NOUN
ejpam-6461	221	27	||	||	PROPN
ejpam-6461	221	28	,	,	PUNCT
ejpam-6461	221	29	||	||	NOUN
ejpam-6461	222	1	x∗x	x∗x	NUM
ejpam-6461	222	2	||	||	NUM
ejpam-6461	222	3	}	}	PUNCT
ejpam-6461	222	4	=	=	SYM
ejpam-6461	223	1	1	1	NUM
ejpam-6461	223	2	2	2	NUM
ejpam-6461	223	3	||	||	NOUN
ejpam-6461	223	4	x∗x	x∗x	NUM
ejpam-6461	223	5	||	||	PUNCT
ejpam-6461	224	1	=	=	SYM
ejpam-6461	224	2	1	1	NUM
ejpam-6461	224	3	2	2	NUM
ejpam-6461	224	4	||	||	NOUN
ejpam-6461	224	5	x	x	PUNCT
ejpam-6461	224	6	||2	||2	NOUN
ejpam-6461	224	7	.	.	PUNCT
ejpam-6461	225	1	(	(	PUNCT
ejpam-6461	225	2	31	31	NUM
ejpam-6461	225	3	)	)	PUNCT
ejpam-6461	225	4	theorem	theorem	NOUN
ejpam-6461	225	5	5	5	NUM
ejpam-6461	225	6	.	.	PUNCT
ejpam-6461	226	1	let	let	VERB
ejpam-6461	226	2	x	x	PRON
ejpam-6461	226	3	and	and	CCONJ
ejpam-6461	226	4	y	y	PROPN
ejpam-6461	226	5	be	be	AUX
ejpam-6461	226	6	operators	operator	NOUN
ejpam-6461	226	7	in	in	ADP
ejpam-6461	226	8	b(h	b(h	PROPN
ejpam-6461	226	9	)	)	PUNCT
ejpam-6461	226	10	,	,	PUNCT
ejpam-6461	226	11	r	r	NOUN
ejpam-6461	226	12	≥	≥	NOUN
ejpam-6461	226	13	2	2	NUM
ejpam-6461	226	14	.	.	PUNCT
ejpam-6461	227	1	then	then	ADV
ejpam-6461	227	2	wr	wr	X
ejpam-6461	227	3	[	[	PUNCT
ejpam-6461	227	4	x	x	PUNCT
ejpam-6461	227	5	o	o	X
ejpam-6461	227	6	o	o	X
ejpam-6461	227	7	y	y	NOUN
ejpam-6461	227	8	∗	∗	NOUN
ejpam-6461	227	9	]	]	PUNCT
ejpam-6461	227	10	≤	≤	NUM
ejpam-6461	227	11	1	1	NUM
ejpam-6461	227	12	2	2	NUM
ejpam-6461	227	13	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	PROPN
ejpam-6461	227	14	[	[	PUNCT
ejpam-6461	227	15	|x|r	|x|r	PROPN
ejpam-6461	227	16	+	+	SYM
ejpam-6461	227	17	|x∗|r	|x∗|r	NUM
ejpam-6461	227	18	o	o	NOUN
ejpam-6461	227	19	o	o	X
ejpam-6461	227	20	|y	|y	NOUN
ejpam-6461	227	21	|r	|r	X
ejpam-6461	227	22	+	+	CCONJ
ejpam-6461	227	23	|y	|y	ADJ
ejpam-6461	227	24	∗|r	∗|r	NOUN
ejpam-6461	227	25	]	]	PUNCT
ejpam-6461	227	26	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	PROPN
ejpam-6461	227	27	.	.	PUNCT
ejpam-6461	228	1	(	(	PUNCT
ejpam-6461	228	2	32	32	NUM
ejpam-6461	228	3	)	)	PUNCT
ejpam-6461	228	4	in	in	ADP
ejpam-6461	228	5	particular	particular	ADJ
ejpam-6461	228	6	,	,	PUNCT
ejpam-6461	228	7	if	if	SCONJ
ejpam-6461	228	8	r	r	NOUN
ejpam-6461	228	9	=	=	SYM
ejpam-6461	228	10	2	2	NUM
ejpam-6461	228	11	,	,	PUNCT
ejpam-6461	228	12	we	we	PRON
ejpam-6461	228	13	obtain	obtain	VERB
ejpam-6461	228	14	w2	w2	NOUN
ejpam-6461	228	15	[	[	PUNCT
ejpam-6461	228	16	x	x	PUNCT
ejpam-6461	228	17	o	o	X
ejpam-6461	228	18	o	o	X
ejpam-6461	228	19	y	y	NOUN
ejpam-6461	228	20	∗	∗	NOUN
ejpam-6461	228	21	]	]	PUNCT
ejpam-6461	228	22	≤	≤	NUM
ejpam-6461	228	23	1	1	NUM
ejpam-6461	228	24	2	2	NUM
ejpam-6461	228	25	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	NOUN
ejpam-6461	228	26	[	[	PUNCT
ejpam-6461	228	27	|x|2	|x|2	NOUN
ejpam-6461	228	28	+	+	CCONJ
ejpam-6461	228	29	|x∗|2	|x∗|2	NUM
ejpam-6461	228	30	o	o	NOUN
ejpam-6461	228	31	o	o	NOUN
ejpam-6461	228	32	|y	|y	NOUN
ejpam-6461	228	33	|2	|2	X
ejpam-6461	228	34	+	+	X
ejpam-6461	228	35	|y	|y	ADJ
ejpam-6461	228	36	∗|2	∗|2	X
ejpam-6461	228	37	]	]	PUNCT
ejpam-6461	228	38	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	PROPN
ejpam-6461	228	39	.	.	PUNCT
ejpam-6461	229	1	(	(	PUNCT
ejpam-6461	229	2	33	33	NUM
ejpam-6461	229	3	)	)	PUNCT
ejpam-6461	229	4	proof	proof	NOUN
ejpam-6461	229	5	.	.	PUNCT
ejpam-6461	230	1	let	let	VERB
ejpam-6461	230	2	a	a	PRON
ejpam-6461	230	3	=	=	PUNCT
ejpam-6461	230	4	[	[	PUNCT
ejpam-6461	230	5	x	x	PUNCT
ejpam-6461	230	6	o	o	X
ejpam-6461	230	7	o	o	X
ejpam-6461	230	8	y	y	NOUN
ejpam-6461	230	9	∗	∗	NOUN
ejpam-6461	230	10	]	]	PUNCT
ejpam-6461	230	11	in	in	ADP
ejpam-6461	230	12	inequality	inequality	NOUN
ejpam-6461	230	13	(	(	PUNCT
ejpam-6461	230	14	25	25	NUM
ejpam-6461	230	15	)	)	PUNCT
ejpam-6461	230	16	,	,	PUNCT
ejpam-6461	230	17	we	we	PRON
ejpam-6461	230	18	obtain	obtain	VERB
ejpam-6461	230	19	the	the	DET
ejpam-6461	230	20	inequality	inequality	NOUN
ejpam-6461	230	21	(	(	PUNCT
ejpam-6461	230	22	32	32	NUM
ejpam-6461	230	23	)	)	PUNCT
ejpam-6461	230	24	.	.	PUNCT
ejpam-6461	231	1	remark	remark	PROPN
ejpam-6461	231	2	8	8	NUM
ejpam-6461	231	3	.	.	PUNCT
ejpam-6461	232	1	letting	let	VERB
ejpam-6461	232	2	y	y	PROPN
ejpam-6461	232	3	=	=	PUNCT
ejpam-6461	232	4	o	o	PROPN
ejpam-6461	232	5	in	in	ADP
ejpam-6461	232	6	inequality	inequality	NOUN
ejpam-6461	232	7	(	(	PUNCT
ejpam-6461	232	8	33	33	NUM
ejpam-6461	232	9	)	)	PUNCT
ejpam-6461	232	10	,	,	PUNCT
ejpam-6461	232	11	we	we	PRON
ejpam-6461	232	12	obtain	obtain	VERB
ejpam-6461	232	13	the	the	DET
ejpam-6461	232	14	inequality	inequality	NOUN
ejpam-6461	232	15	(	(	PUNCT
ejpam-6461	232	16	4	4	NUM
ejpam-6461	232	17	)	)	PUNCT
ejpam-6461	232	18	.	.	PUNCT
ejpam-6461	233	1	in	in	ADP
ejpam-6461	233	2	that	that	DET
ejpam-6461	233	3	sense	sense	NOUN
ejpam-6461	233	4	,	,	PUNCT
ejpam-6461	233	5	inequality	inequality	NOUN
ejpam-6461	233	6	(	(	PUNCT
ejpam-6461	233	7	33	33	NUM
ejpam-6461	233	8	)	)	PUNCT
ejpam-6461	233	9	is	be	AUX
ejpam-6461	233	10	a	a	DET
ejpam-6461	233	11	generalization	generalization	NOUN
ejpam-6461	233	12	of	of	ADP
ejpam-6461	233	13	inequality	inequality	NOUN
ejpam-6461	233	14	(	(	PUNCT
ejpam-6461	233	15	4	4	NUM
ejpam-6461	233	16	)	)	PUNCT
ejpam-6461	233	17	.	.	PUNCT
ejpam-6461	234	1	m.	m.	PROPN
ejpam-6461	234	2	al	al	PROPN
ejpam-6461	234	3	-	-	PUNCT
ejpam-6461	234	4	labadi	labadi	PROPN
ejpam-6461	234	5	et	et	PROPN
ejpam-6461	234	6	al	al	PROPN
ejpam-6461	234	7	.	.	PUNCT
ejpam-6461	234	8	/	/	SYM
ejpam-6461	234	9	eur	eur	PROPN
ejpam-6461	234	10	.	.	PUNCT
ejpam-6461	235	1	j.	j.	PROPN
ejpam-6461	235	2	pure	pure	PROPN
ejpam-6461	235	3	appl	appl	PROPN
ejpam-6461	235	4	.	.	PROPN
ejpam-6461	235	5	math	math	PROPN
ejpam-6461	235	6	,	,	PUNCT
ejpam-6461	235	7	18	18	NUM
ejpam-6461	235	8	(	(	PUNCT
ejpam-6461	235	9	3	3	NUM
ejpam-6461	235	10	)	)	PUNCT
ejpam-6461	235	11	(	(	PUNCT
ejpam-6461	235	12	2025	2025	NUM
ejpam-6461	235	13	)	)	PUNCT
ejpam-6461	235	14	,	,	PUNCT
ejpam-6461	235	15	6461	6461	NUM
ejpam-6461	235	16	9	9	NUM
ejpam-6461	235	17	of	of	ADP
ejpam-6461	235	18	13	13	NUM
ejpam-6461	235	19	recall	recall	NOUN
ejpam-6461	235	20	that	that	SCONJ
ejpam-6461	235	21	the	the	DET
ejpam-6461	235	22	cartesian	cartesian	ADJ
ejpam-6461	235	23	decomposition	decomposition	NOUN
ejpam-6461	235	24	of	of	ADP
ejpam-6461	235	25	the	the	DET
ejpam-6461	235	26	operator	operator	NOUN
ejpam-6461	235	27	x	x	PUNCT
ejpam-6461	235	28	is	be	AUX
ejpam-6461	235	29	given	give	VERB
ejpam-6461	235	30	by	by	ADP
ejpam-6461	235	31	x	x	X
ejpam-6461	235	32	=	=	PUNCT
ejpam-6461	235	33	a	a	PROPN
ejpam-6461	236	1	+	+	NOUN
ejpam-6461	236	2	ib	ib	NOUN
ejpam-6461	236	3	where	where	SCONJ
ejpam-6461	236	4	a	a	DET
ejpam-6461	236	5	=	=	NOUN
ejpam-6461	236	6	re(x	re(x	X
ejpam-6461	236	7	)	)	PUNCT
ejpam-6461	236	8	and	and	CCONJ
ejpam-6461	236	9	b	b	X
ejpam-6461	236	10	=	=	PUNCT
ejpam-6461	236	11	im(x	im(x	PROPN
ejpam-6461	236	12	)	)	PUNCT
ejpam-6461	236	13	.	.	PUNCT
ejpam-6461	237	1	the	the	DET
ejpam-6461	237	2	author	author	NOUN
ejpam-6461	237	3	in	in	ADP
ejpam-6461	237	4	[	[	X
ejpam-6461	237	5	24	24	NUM
ejpam-6461	237	6	]	]	PUNCT
ejpam-6461	237	7	proved	prove	VERB
ejpam-6461	237	8	that	that	SCONJ
ejpam-6461	237	9	if	if	SCONJ
ejpam-6461	237	10	x	x	X
ejpam-6461	237	11	=	=	PRON
ejpam-6461	237	12	a+	a+	PUNCT
ejpam-6461	237	13	ib	ib	NOUN
ejpam-6461	237	14	and	and	CCONJ
ejpam-6461	237	15	0	0	NUM
ejpam-6461	237	16	<	<	X
ejpam-6461	237	17	r	r	NOUN
ejpam-6461	237	18	≤	≤	NUM
ejpam-6461	237	19	2	2	NUM
ejpam-6461	237	20	,	,	PUNCT
ejpam-6461	237	21	then	then	ADV
ejpam-6461	237	22	1	1	NUM
ejpam-6461	237	23	2	2	NUM
ejpam-6461	237	24	||	||	NOUN
ejpam-6461	237	25	|a|r	|a|r	ADP
ejpam-6461	237	26	+	+	PROPN
ejpam-6461	237	27	|b|r	|b|r	PROPN
ejpam-6461	237	28	||	||	NOUN
ejpam-6461	237	29	≤	≤	NUM
ejpam-6461	237	30	wr(x	wr(x	X
ejpam-6461	237	31	)	)	PUNCT
ejpam-6461	237	32	≤	≤	NOUN
ejpam-6461	237	33	||	||	NOUN
ejpam-6461	238	1	|a|r	|a|r	ADP
ejpam-6461	238	2	+	+	PROPN
ejpam-6461	238	3	|b|r	|b|r	PROPN
ejpam-6461	238	4	||	||	PROPN
ejpam-6461	238	5	.	.	PUNCT
ejpam-6461	239	1	(	(	PUNCT
ejpam-6461	239	2	34	34	NUM
ejpam-6461	239	3	)	)	PUNCT
ejpam-6461	239	4	this	this	PRON
ejpam-6461	239	5	implies	imply	VERB
ejpam-6461	239	6	,	,	PUNCT
ejpam-6461	239	7	if	if	SCONJ
ejpam-6461	239	8	r	r	NOUN
ejpam-6461	239	9	=	=	SYM
ejpam-6461	239	10	2	2	NUM
ejpam-6461	239	11	,	,	PUNCT
ejpam-6461	239	12	1	1	NUM
ejpam-6461	239	13	2	2	NUM
ejpam-6461	239	14	||	||	NOUN
ejpam-6461	239	15	|a|2	|a|2	PROPN
ejpam-6461	239	16	+	+	PROPN
ejpam-6461	240	1	|b|2	|b|2	PROPN
ejpam-6461	240	2	||	||	NOUN
ejpam-6461	240	3	≤	≤	NUM
ejpam-6461	240	4	w2(x	w2(x	NOUN
ejpam-6461	240	5	)	)	PUNCT
ejpam-6461	240	6	≤	≤	NOUN
ejpam-6461	240	7	||	||	PUNCT
ejpam-6461	241	1	|a|2	|a|2	PROPN
ejpam-6461	241	2	+	+	PROPN
ejpam-6461	242	1	|b|2	|b|2	PROPN
ejpam-6461	242	2	||	||	PUNCT
ejpam-6461	243	1	=	=	SYM
ejpam-6461	243	2	||	||	NUM
ejpam-6461	243	3	a2	a2	PROPN
ejpam-6461	243	4	+	+	NOUN
ejpam-6461	243	5	b2	b2	PROPN
ejpam-6461	243	6	||	||	NOUN
ejpam-6461	243	7	.	.	PUNCT
ejpam-6461	244	1	(	(	PUNCT
ejpam-6461	244	2	35	35	NUM
ejpam-6461	244	3	)	)	PUNCT
ejpam-6461	244	4	in	in	ADP
ejpam-6461	244	5	the	the	DET
ejpam-6461	244	6	following	following	NOUN
ejpam-6461	244	7	,	,	PUNCT
ejpam-6461	244	8	we	we	PRON
ejpam-6461	244	9	provide	provide	VERB
ejpam-6461	244	10	a	a	DET
ejpam-6461	244	11	new	new	ADJ
ejpam-6461	244	12	proof	proof	NOUN
ejpam-6461	244	13	of	of	ADP
ejpam-6461	244	14	the	the	DET
ejpam-6461	244	15	second	second	ADJ
ejpam-6461	244	16	inequality	inequality	NOUN
ejpam-6461	244	17	in	in	ADP
ejpam-6461	244	18	(	(	PUNCT
ejpam-6461	244	19	35	35	NUM
ejpam-6461	244	20	)	)	PUNCT
ejpam-6461	244	21	.	.	PUNCT
ejpam-6461	245	1	theorem	theorem	VERB
ejpam-6461	245	2	6	6	NUM
ejpam-6461	245	3	.	.	PUNCT
ejpam-6461	246	1	let	let	VERB
ejpam-6461	246	2	x	x	SYM
ejpam-6461	246	3	∈	∈	PROPN
ejpam-6461	246	4	b(h	b(h	PROPN
ejpam-6461	246	5	)	)	PUNCT
ejpam-6461	246	6	with	with	ADP
ejpam-6461	246	7	cartesian	cartesian	ADJ
ejpam-6461	246	8	decomposition	decomposition	NOUN
ejpam-6461	247	1	x	x	PUNCT
ejpam-6461	247	2	=	=	PRON
ejpam-6461	247	3	a+	a+	PUNCT
ejpam-6461	247	4	ib	ib	NOUN
ejpam-6461	247	5	.	.	PUNCT
ejpam-6461	248	1	then	then	ADV
ejpam-6461	248	2	w2(x	w2(x	X
ejpam-6461	248	3	)	)	PUNCT
ejpam-6461	248	4	≤	≤	NOUN
ejpam-6461	248	5	||	||	PUNCT
ejpam-6461	249	1	a2	a2	PROPN
ejpam-6461	249	2	+	+	NOUN
ejpam-6461	249	3	b2	b2	PROPN
ejpam-6461	249	4	||	||	NOUN
ejpam-6461	249	5	.	.	PUNCT
ejpam-6461	250	1	(	(	PUNCT
ejpam-6461	250	2	36	36	NUM
ejpam-6461	250	3	)	)	PUNCT
ejpam-6461	250	4	proof	proof	NOUN
ejpam-6461	250	5	.	.	PUNCT
ejpam-6461	251	1	|⟨xx	|⟨xx	ADJ
ejpam-6461	251	2	,	,	PUNCT
ejpam-6461	251	3	x⟩|2	x⟩|2	PUNCT
ejpam-6461	251	4	=	=	PUNCT
ejpam-6461	251	5	|⟨(a+	|⟨(a+	PROPN
ejpam-6461	251	6	ib)x	ib)x	PROPN
ejpam-6461	251	7	,	,	PUNCT
ejpam-6461	251	8	x⟩|2	x⟩|2	PUNCT
ejpam-6461	251	9	=	=	PUNCT
ejpam-6461	251	10	|⟨ax	|⟨ax	VERB
ejpam-6461	251	11	,	,	PUNCT
ejpam-6461	251	12	x⟩+	x⟩+	PROPN
ejpam-6461	251	13	i	i	PRON
ejpam-6461	251	14	⟨bx	⟨bx	PROPN
ejpam-6461	251	15	,	,	PUNCT
ejpam-6461	251	16	x⟩|2	x⟩|2	PUNCT
ejpam-6461	252	1	=	=	PUNCT
ejpam-6461	252	2	|⟨ax	|⟨ax	VERB
ejpam-6461	252	3	,	,	PUNCT
ejpam-6461	252	4	x⟩|2	x⟩|2	PUNCT
ejpam-6461	252	5	+	+	CCONJ
ejpam-6461	252	6	|⟨bx	|⟨bx	NOUN
ejpam-6461	252	7	,	,	PUNCT
ejpam-6461	252	8	x⟩|2	x⟩|2	PUNCT
ejpam-6461	252	9	=	=	PUNCT
ejpam-6461	252	10	⟨ax	⟨ax	ADJ
ejpam-6461	252	11	,	,	PUNCT
ejpam-6461	252	12	x⟩2	x⟩2	VERB
ejpam-6461	253	1	+	+	CCONJ
ejpam-6461	253	2	⟨bx	⟨bx	NUM
ejpam-6461	253	3	,	,	PUNCT
ejpam-6461	254	1	x⟩2	x⟩2	VERB
ejpam-6461	255	1	(	(	PUNCT
ejpam-6461	255	2	since	since	SCONJ
ejpam-6461	255	3	<	<	X
ejpam-6461	255	4	ax	ax	NOUN
ejpam-6461	255	5	,	,	PUNCT
ejpam-6461	255	6	x	x	X
ejpam-6461	255	7	>	>	PUNCT
ejpam-6461	255	8	∈	∈	PROPN
ejpam-6461	255	9	r	r	NOUN
ejpam-6461	255	10	)	)	PUNCT
ejpam-6461	255	11	≤	≤	NOUN
ejpam-6461	255	12	〈	〈	VERB
ejpam-6461	255	13	a2x	a2x	PROPN
ejpam-6461	255	14	,	,	PUNCT
ejpam-6461	255	15	x	x	SYM
ejpam-6461	255	16	〉	〉	NOUN
ejpam-6461	255	17	+	+	NUM
ejpam-6461	255	18	〈	〈	PROPN
ejpam-6461	255	19	b2x	b2x	NOUN
ejpam-6461	255	20	,	,	PUNCT
ejpam-6461	255	21	x	x	SYM
ejpam-6461	255	22	〉	〉	NOUN
ejpam-6461	255	23	(	(	PUNCT
ejpam-6461	255	24	by	by	ADP
ejpam-6461	255	25	lemma	lemma	PROPN
ejpam-6461	255	26	(	(	PUNCT
ejpam-6461	255	27	1	1	NUM
ejpam-6461	255	28	)	)	PUNCT
ejpam-6461	255	29	(	(	PUNCT
ejpam-6461	255	30	i	i	NOUN
ejpam-6461	255	31	)	)	PUNCT
ejpam-6461	255	32	)	)	PUNCT
ejpam-6461	256	1	=	=	PUNCT
ejpam-6461	256	2	〈	〈	PROPN
ejpam-6461	256	3	(	(	PUNCT
ejpam-6461	256	4	a2	a2	PROPN
ejpam-6461	256	5	+	+	NOUN
ejpam-6461	256	6	b2	b2	NOUN
ejpam-6461	256	7	)	)	PUNCT
ejpam-6461	256	8	x	x	X
ejpam-6461	256	9	,	,	PUNCT
ejpam-6461	256	10	x	x	SYM
ejpam-6461	256	11	〉	〉	NOUN
ejpam-6461	256	12	.	.	PUNCT
ejpam-6461	257	1	taking	take	VERB
ejpam-6461	257	2	the	the	DET
ejpam-6461	257	3	supremum	supremum	ADJ
ejpam-6461	257	4	over	over	ADP
ejpam-6461	257	5	all	all	DET
ejpam-6461	257	6	unit	unit	NOUN
ejpam-6461	257	7	vectors	vector	NOUN
ejpam-6461	257	8	x	x	SYM
ejpam-6461	257	9	∈	∈	PROPN
ejpam-6461	257	10	h	h	NOUN
ejpam-6461	257	11	,	,	PUNCT
ejpam-6461	257	12	we	we	PRON
ejpam-6461	257	13	obtain	obtain	VERB
ejpam-6461	257	14	the	the	DET
ejpam-6461	257	15	inequality	inequality	NOUN
ejpam-6461	257	16	(	(	PUNCT
ejpam-6461	257	17	36	36	NUM
ejpam-6461	257	18	)	)	PUNCT
ejpam-6461	257	19	.	.	PUNCT
ejpam-6461	258	1	4	4	X
ejpam-6461	258	2	.	.	X
ejpam-6461	258	3	numerical	numerical	PROPN
ejpam-6461	258	4	radius	radius	PROPN
ejpam-6461	258	5	inequalities	inequality	NOUN
ejpam-6461	258	6	via	via	ADP
ejpam-6461	258	7	singular	singular	ADJ
ejpam-6461	258	8	values	value	NOUN
ejpam-6461	258	9	and	and	CCONJ
ejpam-6461	258	10	aluthge	aluthge	ADJ
ejpam-6461	258	11	transform	transform	NOUN
ejpam-6461	258	12	in	in	ADP
ejpam-6461	258	13	this	this	DET
ejpam-6461	258	14	section	section	NOUN
ejpam-6461	258	15	,	,	PUNCT
ejpam-6461	258	16	we	we	PRON
ejpam-6461	258	17	prove	prove	VERB
ejpam-6461	258	18	numerical	numerical	ADJ
ejpam-6461	258	19	radius	radius	PROPN
ejpam-6461	258	20	inequalities	inequality	NOUN
ejpam-6461	258	21	using	use	VERB
ejpam-6461	258	22	recent	recent	ADJ
ejpam-6461	258	23	singular	singular	ADJ
ejpam-6461	258	24	values	value	NOUN
ejpam-6461	258	25	inequalities	inequality	NOUN
ejpam-6461	258	26	and	and	CCONJ
ejpam-6461	258	27	aluthge	aluthge	ADJ
ejpam-6461	258	28	transform	transform	NOUN
ejpam-6461	258	29	.	.	PUNCT
ejpam-6461	259	1	the	the	DET
ejpam-6461	259	2	author	author	NOUN
ejpam-6461	259	3	in	in	ADP
ejpam-6461	259	4	[	[	X
ejpam-6461	259	5	15	15	NUM
ejpam-6461	259	6	]	]	PUNCT
ejpam-6461	259	7	proves	prove	VERB
ejpam-6461	259	8	that	that	SCONJ
ejpam-6461	259	9	if	if	SCONJ
ejpam-6461	259	10	a	a	PRON
ejpam-6461	259	11	,	,	PUNCT
ejpam-6461	259	12	b	b	PROPN
ejpam-6461	259	13	∈	∈	PROPN
ejpam-6461	259	14	b(h	b(h	PROPN
ejpam-6461	259	15	)	)	PUNCT
ejpam-6461	259	16	,	,	PUNCT
ejpam-6461	259	17	then	then	ADV
ejpam-6461	259	18	2sj(ab	2sj(ab	NUM
ejpam-6461	259	19	∗	∗	NOUN
ejpam-6461	259	20	+	+	NOUN
ejpam-6461	259	21	ba∗	ba∗	NOUN
ejpam-6461	259	22	)	)	PUNCT
ejpam-6461	259	23	≤	≤	NOUN
ejpam-6461	259	24	s2j	s2j	NOUN
ejpam-6461	259	25	[	[	PUNCT
ejpam-6461	259	26	a	a	DET
ejpam-6461	259	27	b	b	PROPN
ejpam-6461	259	28	b	b	PROPN
ejpam-6461	259	29	a	a	NOUN
ejpam-6461	259	30	]	]	PUNCT
ejpam-6461	259	31	.	.	PUNCT
ejpam-6461	260	1	(	(	PUNCT
ejpam-6461	260	2	37	37	NUM
ejpam-6461	260	3	)	)	PUNCT
ejpam-6461	260	4	this	this	DET
ejpam-6461	260	5	inequality	inequality	NOUN
ejpam-6461	260	6	implies	imply	VERB
ejpam-6461	260	7	,	,	PUNCT
ejpam-6461	260	8	since	since	SCONJ
ejpam-6461	260	9	unitarily	unitarily	ADV
ejpam-6461	260	10	invariant	invariant	ADJ
ejpam-6461	260	11	norms	norm	NOUN
ejpam-6461	260	12	and	and	CCONJ
ejpam-6461	260	13	in	in	ADP
ejpam-6461	260	14	particular	particular	ADJ
ejpam-6461	260	15	the	the	DET
ejpam-6461	260	16	spectral	spectral	ADJ
ejpam-6461	260	17	norm	norm	NOUN
ejpam-6461	260	18	,	,	PUNCT
ejpam-6461	260	19	are	be	AUX
ejpam-6461	260	20	increasing	increase	VERB
ejpam-6461	260	21	functions	function	NOUN
ejpam-6461	260	22	of	of	ADP
ejpam-6461	260	23	singular	singular	ADJ
ejpam-6461	260	24	values	value	NOUN
ejpam-6461	260	25	,	,	PUNCT
ejpam-6461	260	26	that	that	SCONJ
ejpam-6461	260	27	||ab∗	||ab∗	X
ejpam-6461	260	28	+	+	ADJ
ejpam-6461	260	29	ba∗||	ba∗||	NOUN
ejpam-6461	260	30	≤	≤	ADV
ejpam-6461	260	31	1	1	NUM
ejpam-6461	260	32	2	2	NUM
ejpam-6461	260	33	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	PROPN
ejpam-6461	260	34	[	[	PUNCT
ejpam-6461	260	35	a	a	DET
ejpam-6461	260	36	b	b	PROPN
ejpam-6461	260	37	b	b	PROPN
ejpam-6461	260	38	a	a	DET
ejpam-6461	260	39	]	]	X
ejpam-6461	260	40	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	260	41	.	.	PUNCT
ejpam-6461	261	1	(	(	PUNCT
ejpam-6461	261	2	38	38	NUM
ejpam-6461	261	3	)	)	PUNCT
ejpam-6461	261	4	m.	m.	NOUN
ejpam-6461	261	5	al	al	PROPN
ejpam-6461	261	6	-	-	PUNCT
ejpam-6461	261	7	labadi	labadi	PROPN
ejpam-6461	261	8	et	et	PROPN
ejpam-6461	261	9	al	al	PROPN
ejpam-6461	261	10	.	.	PUNCT
ejpam-6461	261	11	/	/	SYM
ejpam-6461	261	12	eur	eur	PROPN
ejpam-6461	261	13	.	.	PUNCT
ejpam-6461	262	1	j.	j.	PROPN
ejpam-6461	262	2	pure	pure	PROPN
ejpam-6461	262	3	appl	appl	PROPN
ejpam-6461	262	4	.	.	PROPN
ejpam-6461	262	5	math	math	PROPN
ejpam-6461	262	6	,	,	PUNCT
ejpam-6461	262	7	18	18	NUM
ejpam-6461	262	8	(	(	PUNCT
ejpam-6461	262	9	3	3	NUM
ejpam-6461	262	10	)	)	PUNCT
ejpam-6461	262	11	(	(	PUNCT
ejpam-6461	262	12	2025	2025	NUM
ejpam-6461	262	13	)	)	PUNCT
ejpam-6461	262	14	,	,	PUNCT
ejpam-6461	262	15	6461	6461	NUM
ejpam-6461	262	16	10	10	NUM
ejpam-6461	262	17	of	of	ADP
ejpam-6461	262	18	13	13	NUM
ejpam-6461	262	19	theorem	theorem	NOUN
ejpam-6461	262	20	7	7	NUM
ejpam-6461	262	21	.	.	PUNCT
ejpam-6461	263	1	let	let	VERB
ejpam-6461	263	2	a	a	PRON
ejpam-6461	263	3	and	and	CCONJ
ejpam-6461	263	4	b	b	NOUN
ejpam-6461	263	5	be	be	AUX
ejpam-6461	263	6	operators	operator	NOUN
ejpam-6461	263	7	in	in	ADP
ejpam-6461	263	8	b(h	b(h	PROPN
ejpam-6461	263	9	)	)	PUNCT
ejpam-6461	263	10	.	.	PUNCT
ejpam-6461	264	1	then	then	ADV
ejpam-6461	264	2	w(ab∗	w(ab∗	X
ejpam-6461	264	3	)	)	PUNCT
ejpam-6461	264	4	≤	≤	NUM
ejpam-6461	264	5	1	1	NUM
ejpam-6461	264	6	4	4	NUM
ejpam-6461	264	7	||	||	NOUN
ejpam-6461	264	8	|a|+	|a|+	NOUN
ejpam-6461	264	9	|b|	|b|	PROPN
ejpam-6461	264	10	||2	||2	NOUN
ejpam-6461	264	11	(	(	PUNCT
ejpam-6461	264	12	39	39	NUM
ejpam-6461	264	13	)	)	PUNCT
ejpam-6461	264	14	proof	proof	NOUN
ejpam-6461	264	15	.	.	PUNCT
ejpam-6461	265	1	if	if	SCONJ
ejpam-6461	265	2	a	a	PRON
ejpam-6461	265	3	and	and	CCONJ
ejpam-6461	265	4	b	b	NOUN
ejpam-6461	265	5	are	be	AUX
ejpam-6461	265	6	operators	operator	NOUN
ejpam-6461	265	7	in	in	ADP
ejpam-6461	265	8	b(h	b(h	PROPN
ejpam-6461	265	9	)	)	PUNCT
ejpam-6461	265	10	,	,	PUNCT
ejpam-6461	265	11	then	then	ADV
ejpam-6461	265	12	||re(eiθab∗)||	||re(eiθab∗)||	NOUN
ejpam-6461	265	13	=	=	NOUN
ejpam-6461	265	14	1	1	NUM
ejpam-6461	265	15	2	2	NUM
ejpam-6461	265	16	||eiθab∗	||eiθab∗	NOUN
ejpam-6461	265	17	+	+	X
ejpam-6461	265	18	e−iθba∗||	e−iθba∗||	NOUN
ejpam-6461	265	19	=	=	SYM
ejpam-6461	265	20	1	1	NUM
ejpam-6461	265	21	2	2	NUM
ejpam-6461	265	22	||aeiθb∗	||aeiθb∗	NOUN
ejpam-6461	265	23	+	+	CCONJ
ejpam-6461	265	24	e−iθba∗||	e−iθba∗||	ADJ
ejpam-6461	265	25	≤	≤	NUM
ejpam-6461	265	26	1	1	NUM
ejpam-6461	265	27	4	4	NUM
ejpam-6461	265	28	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADP
ejpam-6461	265	29	[	[	PUNCT
ejpam-6461	265	30	a	a	DET
ejpam-6461	265	31	e−iθb	e−iθb	NOUN
ejpam-6461	265	32	e−iθb	e−iθb	NOUN
ejpam-6461	265	33	a	a	DET
ejpam-6461	265	34	]	]	X
ejpam-6461	265	35	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	265	36	(	(	PUNCT
ejpam-6461	265	37	by	by	ADP
ejpam-6461	265	38	inequality	inequality	NOUN
ejpam-6461	265	39	(	(	PUNCT
ejpam-6461	265	40	38	38	NUM
ejpam-6461	265	41	)	)	PUNCT
ejpam-6461	265	42	)	)	PUNCT
ejpam-6461	265	43	=	=	SYM
ejpam-6461	266	1	1	1	NUM
ejpam-6461	266	2	4	4	NUM
ejpam-6461	266	3	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADP
ejpam-6461	266	4	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6461	266	5	[	[	PUNCT
ejpam-6461	266	6	a	a	DET
ejpam-6461	266	7	e−iθb	e−iθb	NOUN
ejpam-6461	266	8	e−iθb	e−iθb	NOUN
ejpam-6461	266	9	a	a	DET
ejpam-6461	266	10	]	]	X
ejpam-6461	266	11	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6461	266	12	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	266	13	(	(	PUNCT
ejpam-6461	266	14	since	since	SCONJ
ejpam-6461	266	15	||a||	||a||	PROPN
ejpam-6461	266	16	=	=	SYM
ejpam-6461	266	17	||	||	NUM
ejpam-6461	266	18	|a|	|a|	NUM
ejpam-6461	266	19	||	||	NOUN
ejpam-6461	266	20	)	)	PUNCT
ejpam-6461	266	21	=	=	SYM
ejpam-6461	266	22	1	1	NUM
ejpam-6461	266	23	4	4	NUM
ejpam-6461	266	24	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADP
ejpam-6461	266	25	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6461	266	26	[	[	PUNCT
ejpam-6461	266	27	a	a	DET
ejpam-6461	266	28	o	o	NOUN
ejpam-6461	266	29	o	o	X
ejpam-6461	266	30	a	a	X
ejpam-6461	266	31	]	]	PUNCT
ejpam-6461	267	1	+	+	PUNCT
ejpam-6461	267	2	[	[	PUNCT
ejpam-6461	267	3	o	o	X
ejpam-6461	267	4	e−iθb	e−iθb	NOUN
ejpam-6461	267	5	e−iθb	e−iθb	NOUN
ejpam-6461	267	6	o	o	X
ejpam-6461	267	7	]	]	X
ejpam-6461	267	8	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6461	267	9	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	ADV
ejpam-6461	267	10	≤	≤	NUM
ejpam-6461	267	11	1	1	NUM
ejpam-6461	267	12	4	4	NUM
ejpam-6461	267	13	∣∣∣∣∣∣	∣∣∣∣∣∣	ADJ
ejpam-6461	267	14	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ejpam-6461	267	15	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6461	267	16	[	[	PUNCT
ejpam-6461	267	17	a	a	DET
ejpam-6461	267	18	o	o	NOUN
ejpam-6461	267	19	o	o	NOUN
ejpam-6461	267	20	a	a	X
ejpam-6461	267	21	]	]	X
ejpam-6461	267	22	∣∣∣∣+	∣∣∣∣+	PROPN
ejpam-6461	267	23	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ejpam-6461	267	24			PROPN
ejpam-6461	267	25	o	o	INTJ
ejpam-6461	267	26	e−iθb	e−iθb	NOUN
ejpam-6461	267	27	e−iθb	e−iθb	NOUN
ejpam-6461	267	28	o	o	PROPN
ejpam-6461	267	29	∣∣∣∣∣∣	∣∣∣∣∣∣	PROPN
ejpam-6461	267	30	∣∣∣∣∣∣	∣∣∣∣∣∣	PROPN
ejpam-6461	267	31	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ejpam-6461	267	32	2	2	NUM
ejpam-6461	267	33	≤	≤	NUM
ejpam-6461	267	34	1	1	NUM
ejpam-6461	267	35	4	4	NUM
ejpam-6461	267	36	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADP
ejpam-6461	267	37	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6461	267	38	[	[	PUNCT
ejpam-6461	267	39	a	a	DET
ejpam-6461	267	40	o	o	NOUN
ejpam-6461	267	41	o	o	X
ejpam-6461	267	42	a	a	X
ejpam-6461	267	43	]	]	X
ejpam-6461	267	44	∣∣∣∣+	∣∣∣∣+	PROPN
ejpam-6461	267	45	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6461	267	46	[	[	PUNCT
ejpam-6461	267	47	|e−iθb|	|e−iθb|	NOUN
ejpam-6461	267	48	o	o	NOUN
ejpam-6461	267	49	o	o	X
ejpam-6461	267	50	|e−iθb|	|e−iθb|	NOUN
ejpam-6461	267	51	]	]	PUNCT
ejpam-6461	267	52	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6461	267	53	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	267	54	(	(	PUNCT
ejpam-6461	267	55	by	by	ADP
ejpam-6461	267	56	triangle	triangle	NOUN
ejpam-6461	267	57	inequality	inequality	NOUN
ejpam-6461	267	58	)	)	PUNCT
ejpam-6461	267	59	=	=	SYM
ejpam-6461	267	60	1	1	NUM
ejpam-6461	267	61	4	4	NUM
ejpam-6461	267	62	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	NOUN
ejpam-6461	267	63	[	[	PUNCT
ejpam-6461	267	64	|a|	|a|	NOUN
ejpam-6461	267	65	o	o	NOUN
ejpam-6461	267	66	o	o	X
ejpam-6461	267	67	|a|	|a|	PROPN
ejpam-6461	267	68	]	]	PUNCT
ejpam-6461	267	69	+	+	CCONJ
ejpam-6461	267	70	[	[	PUNCT
ejpam-6461	267	71	|b|	|b|	X
ejpam-6461	267	72	o	o	NOUN
ejpam-6461	267	73	o	o	X
ejpam-6461	267	74	|b|	|b|	X
ejpam-6461	267	75	]	]	PUNCT
ejpam-6461	267	76	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	267	77	=	=	SYM
ejpam-6461	267	78	1	1	NUM
ejpam-6461	267	79	4	4	NUM
ejpam-6461	267	80	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADJ
ejpam-6461	267	81	[	[	PUNCT
ejpam-6461	267	82	|a|+	|a|+	NOUN
ejpam-6461	267	83	|b|	|b|	PUNCT
ejpam-6461	267	84	o	o	NOUN
ejpam-6461	267	85	o	o	X
ejpam-6461	267	86	|a|+	|a|+	VERB
ejpam-6461	267	87	|b|	|b|	X
ejpam-6461	267	88	]	]	PUNCT
ejpam-6461	267	89	∣∣∣∣∣∣∣∣2	∣∣∣∣∣∣∣∣2	NOUN
ejpam-6461	267	90	=	=	SYM
ejpam-6461	267	91	1	1	NUM
ejpam-6461	267	92	4	4	NUM
ejpam-6461	267	93	||	||	NOUN
ejpam-6461	267	94	|a|+	|a|+	NOUN
ejpam-6461	267	95	|b|	|b|	PROPN
ejpam-6461	267	96	||2	||2	NOUN
ejpam-6461	267	97	.	.	PUNCT
ejpam-6461	268	1	taking	take	VERB
ejpam-6461	268	2	the	the	DET
ejpam-6461	268	3	supremum	supremum	ADJ
ejpam-6461	268	4	over	over	ADP
ejpam-6461	268	5	all	all	PRON
ejpam-6461	268	6	θ	θ	NOUN
ejpam-6461	268	7	∈	∈	PROPN
ejpam-6461	268	8	r	r	NOUN
ejpam-6461	268	9	,	,	PUNCT
ejpam-6461	268	10	we	we	PRON
ejpam-6461	268	11	obtain	obtain	VERB
ejpam-6461	268	12	the	the	DET
ejpam-6461	268	13	inequality	inequality	NOUN
ejpam-6461	268	14	(	(	PUNCT
ejpam-6461	268	15	39	39	NUM
ejpam-6461	268	16	)	)	PUNCT
ejpam-6461	268	17	.	.	PUNCT
ejpam-6461	269	1	it	it	PRON
ejpam-6461	269	2	can	can	AUX
ejpam-6461	269	3	be	be	AUX
ejpam-6461	269	4	shown	show	VERB
ejpam-6461	269	5	easily	easily	ADV
ejpam-6461	269	6	that	that	SCONJ
ejpam-6461	269	7	inequalities	inequality	NOUN
ejpam-6461	269	8	(	(	PUNCT
ejpam-6461	269	9	6	6	NUM
ejpam-6461	269	10	)	)	PUNCT
ejpam-6461	269	11	and	and	CCONJ
ejpam-6461	269	12	(	(	PUNCT
ejpam-6461	269	13	39	39	NUM
ejpam-6461	269	14	)	)	PUNCT
ejpam-6461	269	15	,	,	PUNCT
ejpam-6461	269	16	if	if	SCONJ
ejpam-6461	269	17	a	a	PRON
ejpam-6461	269	18	and	and	CCONJ
ejpam-6461	269	19	b	b	NOUN
ejpam-6461	269	20	are	be	AUX
ejpam-6461	269	21	self	self	NOUN
ejpam-6461	269	22	-	-	PUNCT
ejpam-6461	269	23	adjoint	adjoint	NOUN
ejpam-6461	269	24	operators	operator	NOUN
ejpam-6461	269	25	,	,	PUNCT
ejpam-6461	269	26	are	be	AUX
ejpam-6461	269	27	the	the	DET
ejpam-6461	269	28	same	same	ADJ
ejpam-6461	269	29	.	.	PUNCT
ejpam-6461	270	1	it	it	PRON
ejpam-6461	270	2	is	be	AUX
ejpam-6461	270	3	obvious	obvious	ADJ
ejpam-6461	270	4	that	that	SCONJ
ejpam-6461	270	5	in	in	ADP
ejpam-6461	270	6	the	the	DET
ejpam-6461	270	7	general	general	ADJ
ejpam-6461	270	8	case	case	NOUN
ejpam-6461	270	9	if	if	SCONJ
ejpam-6461	270	10	a	a	PRON
ejpam-6461	270	11	and	and	CCONJ
ejpam-6461	270	12	b	b	NOUN
ejpam-6461	270	13	are	be	AUX
ejpam-6461	270	14	not	not	PART
ejpam-6461	270	15	selfadjoint	selfadjoint	VERB
ejpam-6461	270	16	operators	operator	NOUN
ejpam-6461	270	17	,	,	PUNCT
ejpam-6461	270	18	when	when	SCONJ
ejpam-6461	270	19	inequality	inequality	NOUN
ejpam-6461	270	20	(	(	PUNCT
ejpam-6461	270	21	6	6	NUM
ejpam-6461	270	22	)	)	PUNCT
ejpam-6461	270	23	is	be	AUX
ejpam-6461	270	24	sharper	sharp	ADJ
ejpam-6461	270	25	than	than	ADP
ejpam-6461	270	26	inequality	inequality	NOUN
ejpam-6461	270	27	(	(	PUNCT
ejpam-6461	270	28	39	39	NUM
ejpam-6461	270	29	)	)	PUNCT
ejpam-6461	270	30	then	then	ADV
ejpam-6461	270	31	replacing	replace	VERB
ejpam-6461	270	32	a	a	PRON
ejpam-6461	270	33	by	by	ADP
ejpam-6461	270	34	a∗	a∗	NOUN
ejpam-6461	270	35	and	and	CCONJ
ejpam-6461	270	36	b	b	NOUN
ejpam-6461	270	37	by	by	ADP
ejpam-6461	270	38	b∗	b∗	ADJ
ejpam-6461	270	39	in	in	ADP
ejpam-6461	270	40	the	the	DET
ejpam-6461	270	41	same	same	ADJ
ejpam-6461	270	42	example	example	NOUN
ejpam-6461	270	43	will	will	AUX
ejpam-6461	270	44	make	make	VERB
ejpam-6461	270	45	inequality	inequality	NOUN
ejpam-6461	270	46	(	(	PUNCT
ejpam-6461	270	47	39	39	NUM
ejpam-6461	270	48	)	)	PUNCT
ejpam-6461	270	49	is	be	AUX
ejpam-6461	270	50	sharper	sharp	ADJ
ejpam-6461	270	51	than	than	ADP
ejpam-6461	270	52	inequality	inequality	NOUN
ejpam-6461	270	53	(	(	PUNCT
ejpam-6461	270	54	6	6	NUM
ejpam-6461	270	55	)	)	PUNCT
ejpam-6461	270	56	.	.	PUNCT
ejpam-6461	271	1	the	the	DET
ejpam-6461	271	2	following	follow	VERB
ejpam-6461	271	3	example	example	NOUN
ejpam-6461	271	4	shows	show	VERB
ejpam-6461	271	5	that	that	SCONJ
ejpam-6461	271	6	inequality	inequality	NOUN
ejpam-6461	271	7	(	(	PUNCT
ejpam-6461	271	8	39	39	NUM
ejpam-6461	271	9	)	)	PUNCT
ejpam-6461	271	10	is	be	AUX
ejpam-6461	271	11	sharper	sharp	ADJ
ejpam-6461	271	12	than	than	ADP
ejpam-6461	271	13	inequality	inequality	NOUN
ejpam-6461	271	14	(	(	PUNCT
ejpam-6461	271	15	6	6	NUM
ejpam-6461	271	16	)	)	PUNCT
ejpam-6461	271	17	.	.	PUNCT
ejpam-6461	272	1	example	example	NOUN
ejpam-6461	273	1	1	1	NUM
ejpam-6461	273	2	.	.	PUNCT
ejpam-6461	273	3	let	let	VERB
ejpam-6461	273	4	a	a	PRON
ejpam-6461	273	5	=	=	X
ejpam-6461	273	6	[	[	PUNCT
ejpam-6461	273	7	0	0	NUM
ejpam-6461	273	8	4	4	NUM
ejpam-6461	273	9	2	2	NUM
ejpam-6461	273	10	0	0	NUM
ejpam-6461	273	11	]	]	PUNCT
ejpam-6461	273	12	,	,	PUNCT
ejpam-6461	273	13	b	b	X
ejpam-6461	273	14	=	=	PUNCT
ejpam-6461	274	1	[	[	PUNCT
ejpam-6461	274	2	9	9	NUM
ejpam-6461	274	3	0	0	NUM
ejpam-6461	274	4	0	0	NUM
ejpam-6461	274	5	1	1	NUM
ejpam-6461	274	6	]	]	PUNCT
ejpam-6461	274	7	.	.	PUNCT
ejpam-6461	275	1	then	then	ADV
ejpam-6461	275	2	||	||	PUNCT
ejpam-6461	276	1	|a|+	|a|+	NOUN
ejpam-6461	276	2	|b|	|b|	X
ejpam-6461	276	3	||	||	PUNCT
ejpam-6461	277	1	=	=	SYM
ejpam-6461	277	2	11	11	NUM
ejpam-6461	277	3	and	and	CCONJ
ejpam-6461	277	4	||	||	ADJ
ejpam-6461	277	5	|a∗|+	|a∗|+	NOUN
ejpam-6461	277	6	|b∗|	|b∗|	PUNCT
ejpam-6461	277	7	||	||	NOUN
ejpam-6461	278	1	=	=	SYM
ejpam-6461	278	2	13	13	NUM
ejpam-6461	278	3	.	.	PUNCT
ejpam-6461	279	1	m.	m.	PROPN
ejpam-6461	279	2	al	al	PROPN
ejpam-6461	279	3	-	-	PUNCT
ejpam-6461	279	4	labadi	labadi	PROPN
ejpam-6461	279	5	et	et	PROPN
ejpam-6461	279	6	al	al	PROPN
ejpam-6461	279	7	.	.	PUNCT
ejpam-6461	279	8	/	/	SYM
ejpam-6461	279	9	eur	eur	PROPN
ejpam-6461	279	10	.	.	PUNCT
ejpam-6461	280	1	j.	j.	PROPN
ejpam-6461	280	2	pure	pure	PROPN
ejpam-6461	280	3	appl	appl	PROPN
ejpam-6461	280	4	.	.	PROPN
ejpam-6461	280	5	math	math	PROPN
ejpam-6461	280	6	,	,	PUNCT
ejpam-6461	280	7	18	18	NUM
ejpam-6461	280	8	(	(	PUNCT
ejpam-6461	280	9	3	3	NUM
ejpam-6461	280	10	)	)	PUNCT
ejpam-6461	280	11	(	(	PUNCT
ejpam-6461	280	12	2025	2025	NUM
ejpam-6461	280	13	)	)	PUNCT
ejpam-6461	280	14	,	,	PUNCT
ejpam-6461	280	15	6461	6461	NUM
ejpam-6461	280	16	11	11	NUM
ejpam-6461	280	17	of	of	ADP
ejpam-6461	280	18	13	13	NUM
ejpam-6461	280	19	this	this	PRON
ejpam-6461	280	20	implies	imply	VERB
ejpam-6461	280	21	that	that	SCONJ
ejpam-6461	280	22	inequality	inequality	NOUN
ejpam-6461	280	23	(	(	PUNCT
ejpam-6461	280	24	39	39	NUM
ejpam-6461	280	25	)	)	PUNCT
ejpam-6461	280	26	is	be	AUX
ejpam-6461	280	27	sharper	sharp	ADJ
ejpam-6461	280	28	than	than	ADP
ejpam-6461	280	29	inequality	inequality	NOUN
ejpam-6461	280	30	(	(	PUNCT
ejpam-6461	280	31	6	6	NUM
ejpam-6461	280	32	)	)	PUNCT
ejpam-6461	280	33	,	,	PUNCT
ejpam-6461	280	34	since	since	SCONJ
ejpam-6461	280	35	||	||	NUM
ejpam-6461	280	36	|a|+	|a|+	NOUN
ejpam-6461	281	1	|b|	|b|	X
ejpam-6461	281	2	||	||	PUNCT
ejpam-6461	281	3	<	<	X
ejpam-6461	281	4	||	||	PROPN
ejpam-6461	281	5	|a∗|+	|a∗|+	NOUN
ejpam-6461	281	6	|b∗|	|b∗|	PUNCT
ejpam-6461	282	1	||	||	NOUN
ejpam-6461	282	2	.	.	PUNCT
ejpam-6461	283	1	this	this	PRON
ejpam-6461	283	2	implies	imply	VERB
ejpam-6461	283	3	that	that	SCONJ
ejpam-6461	283	4	||	||	PUNCT
ejpam-6461	284	1	|a|+	|a|+	NOUN
ejpam-6461	284	2	|b|	|b|	VERB
ejpam-6461	284	3	||2	||2	NOUN
ejpam-6461	284	4	<	<	X
ejpam-6461	284	5	||	||	PUNCT
ejpam-6461	285	1	|a|+	|a|+	NOUN
ejpam-6461	285	2	|b|	|b|	X
ejpam-6461	285	3	||	||	NOUN
ejpam-6461	286	1	||	||	NUM
ejpam-6461	286	2	|a∗|+	|a∗|+	NOUN
ejpam-6461	286	3	|b∗|	|b∗|	PUNCT
ejpam-6461	287	1	||	||	NOUN
ejpam-6461	287	2	.	.	PUNCT
ejpam-6461	288	1	remark	remark	PROPN
ejpam-6461	288	2	9	9	NUM
ejpam-6461	288	3	.	.	PUNCT
ejpam-6461	289	1	if	if	SCONJ
ejpam-6461	289	2	we	we	PRON
ejpam-6461	289	3	replace	replace	VERB
ejpam-6461	289	4	a	a	PRON
ejpam-6461	289	5	by	by	ADP
ejpam-6461	289	6	a∗	a∗	NOUN
ejpam-6461	289	7	and	and	CCONJ
ejpam-6461	289	8	b	b	NOUN
ejpam-6461	289	9	by	by	ADP
ejpam-6461	289	10	b∗	b∗	ADJ
ejpam-6461	289	11	,	,	PUNCT
ejpam-6461	289	12	in	in	ADP
ejpam-6461	289	13	example	example	NOUN
ejpam-6461	289	14	1	1	NUM
ejpam-6461	289	15	,	,	PUNCT
ejpam-6461	289	16	we	we	PRON
ejpam-6461	289	17	obtain	obtain	VERB
ejpam-6461	289	18	||	||	ADJ
ejpam-6461	289	19	|a∗|+	|a∗|+	NOUN
ejpam-6461	289	20	|b∗|	|b∗|	PUNCT
ejpam-6461	290	1	||	||	NOUN
ejpam-6461	290	2	<	<	X
ejpam-6461	290	3	||	||	PUNCT
ejpam-6461	291	1	|a|+	|a|+	NOUN
ejpam-6461	291	2	|b|	|b|	PROPN
ejpam-6461	291	3	||	||	NOUN
ejpam-6461	291	4	.	.	PUNCT
ejpam-6461	292	1	this	this	PRON
ejpam-6461	292	2	implies	imply	VERB
ejpam-6461	292	3	that	that	SCONJ
ejpam-6461	292	4	inequality	inequality	NOUN
ejpam-6461	292	5	(	(	PUNCT
ejpam-6461	292	6	6	6	NUM
ejpam-6461	292	7	)	)	PUNCT
ejpam-6461	292	8	is	be	AUX
ejpam-6461	292	9	sharper	sharp	ADJ
ejpam-6461	292	10	than	than	ADP
ejpam-6461	292	11	inequality	inequality	NOUN
ejpam-6461	292	12	(	(	PUNCT
ejpam-6461	292	13	39	39	NUM
ejpam-6461	292	14	)	)	PUNCT
ejpam-6461	292	15	.	.	PUNCT
ejpam-6461	293	1	the	the	DET
ejpam-6461	293	2	aluthge	aluthge	ADJ
ejpam-6461	293	3	transform	transform	NOUN
ejpam-6461	293	4	is	be	AUX
ejpam-6461	293	5	used	use	VERB
ejpam-6461	293	6	to	to	PART
ejpam-6461	293	7	study	study	VERB
ejpam-6461	293	8	numerical	numerical	PROPN
ejpam-6461	293	9	radius	radius	PROPN
ejpam-6461	293	10	bounds	bound	NOUN
ejpam-6461	293	11	.	.	PUNCT
ejpam-6461	294	1	recall	recall	VERB
ejpam-6461	294	2	that	that	SCONJ
ejpam-6461	294	3	if	if	SCONJ
ejpam-6461	294	4	a	a	DET
ejpam-6461	294	5	∈	∈	PROPN
ejpam-6461	294	6	b(h	b(h	PROPN
ejpam-6461	294	7	)	)	PUNCT
ejpam-6461	294	8	,	,	PUNCT
ejpam-6461	294	9	then	then	ADV
ejpam-6461	294	10	the	the	DET
ejpam-6461	294	11	polar	polar	ADJ
ejpam-6461	294	12	decomposition	decomposition	NOUN
ejpam-6461	294	13	of	of	ADP
ejpam-6461	294	14	a	a	PRON
ejpam-6461	294	15	is	be	AUX
ejpam-6461	294	16	given	give	VERB
ejpam-6461	294	17	by	by	ADP
ejpam-6461	294	18	a	a	DET
ejpam-6461	294	19	=	=	X
ejpam-6461	294	20	u	u	NOUN
ejpam-6461	294	21	|a|	|a|	NOUN
ejpam-6461	294	22	,	,	PUNCT
ejpam-6461	294	23	where	where	SCONJ
ejpam-6461	294	24	u	u	NOUN
ejpam-6461	294	25	is	be	AUX
ejpam-6461	294	26	a	a	DET
ejpam-6461	294	27	partial	partial	ADJ
ejpam-6461	294	28	isometry	isometry	NOUN
ejpam-6461	294	29	.	.	PUNCT
ejpam-6461	295	1	for	for	ADP
ejpam-6461	295	2	0	0	NUM
ejpam-6461	295	3	≤	≤	NOUN
ejpam-6461	295	4	t	t	NOUN
ejpam-6461	295	5	≤	≤	NUM
ejpam-6461	295	6	1	1	NUM
ejpam-6461	295	7	,	,	PUNCT
ejpam-6461	295	8	the	the	DET
ejpam-6461	295	9	weighted	weight	VERB
ejpam-6461	295	10	aluthge	aluthge	ADJ
ejpam-6461	295	11	transform	transform	NOUN
ejpam-6461	295	12	is	be	AUX
ejpam-6461	295	13	defined	define	VERB
ejpam-6461	295	14	by	by	ADP
ejpam-6461	295	15	ãt	ãt	NOUN
ejpam-6461	295	16	=	=	SYM
ejpam-6461	295	17	|a|1−tu	|a|1−tu	PROPN
ejpam-6461	295	18	|a|t	|a|t	NOUN
ejpam-6461	295	19	.	.	PUNCT
ejpam-6461	296	1	if	if	SCONJ
ejpam-6461	296	2	t	t	NOUN
ejpam-6461	296	3	=	=	SYM
ejpam-6461	296	4	2	2	NUM
ejpam-6461	296	5	,	,	PUNCT
ejpam-6461	296	6	we	we	PRON
ejpam-6461	296	7	write	write	VERB
ejpam-6461	296	8	ã	ã	PROPN
ejpam-6461	296	9	instead	instead	ADV
ejpam-6461	296	10	of	of	ADP
ejpam-6461	296	11	ã	ã	PROPN
ejpam-6461	296	12	1	1	NUM
ejpam-6461	296	13	2	2	NUM
ejpam-6461	296	14	.	.	PUNCT
ejpam-6461	297	1	the	the	DET
ejpam-6461	297	2	following	follow	VERB
ejpam-6461	297	3	corollary	corollary	NOUN
ejpam-6461	297	4	is	be	AUX
ejpam-6461	297	5	an	an	DET
ejpam-6461	297	6	application	application	NOUN
ejpam-6461	297	7	of	of	ADP
ejpam-6461	297	8	theorem	theorem	NOUN
ejpam-6461	297	9	7	7	NUM
ejpam-6461	297	10	,	,	PUNCT
ejpam-6461	297	11	which	which	PRON
ejpam-6461	297	12	inturns	inturn	VERB
ejpam-6461	297	13	a	a	DET
ejpam-6461	297	14	generalization	generalization	NOUN
ejpam-6461	297	15	of	of	ADP
ejpam-6461	297	16	the	the	DET
ejpam-6461	297	17	second	second	ADJ
ejpam-6461	297	18	inequality	inequality	NOUN
ejpam-6461	297	19	of	of	ADP
ejpam-6461	297	20	(	(	PUNCT
ejpam-6461	297	21	1	1	NUM
ejpam-6461	297	22	)	)	PUNCT
ejpam-6461	297	23	.	.	PUNCT
ejpam-6461	298	1	corollary	corollary	ADJ
ejpam-6461	298	2	8	8	NUM
ejpam-6461	298	3	.	.	PUNCT
ejpam-6461	299	1	let	let	VERB
ejpam-6461	299	2	t	t	PROPN
ejpam-6461	299	3	∈	∈	PROPN
ejpam-6461	299	4	b(h	b(h	PROPN
ejpam-6461	299	5	)	)	PUNCT
ejpam-6461	299	6	.	.	PUNCT
ejpam-6461	300	1	then	then	ADV
ejpam-6461	300	2	w(t	w(t	PROPN
ejpam-6461	300	3	)	)	PUNCT
ejpam-6461	300	4	≤	≤	NOUN
ejpam-6461	300	5	1	1	NUM
ejpam-6461	300	6	4	4	NUM
ejpam-6461	300	7	||	||	NOUN
ejpam-6461	300	8	|t	|t	VERB
ejpam-6461	300	9	|t	|t	PROPN
ejpam-6461	301	1	+	+	CCONJ
ejpam-6461	301	2	|t	|t	PROPN
ejpam-6461	301	3	|1−t	|1−t	PROPN
ejpam-6461	301	4	||2	||2	NOUN
ejpam-6461	301	5	,	,	PUNCT
ejpam-6461	301	6	0	0	NUM
ejpam-6461	301	7	≤	≤	NUM
ejpam-6461	301	8	t	t	X
ejpam-6461	301	9	≤	≤	NUM
ejpam-6461	301	10	1	1	NUM
ejpam-6461	301	11	.	.	PUNCT
ejpam-6461	302	1	(	(	PUNCT
ejpam-6461	302	2	40	40	NUM
ejpam-6461	302	3	)	)	PUNCT
ejpam-6461	302	4	in	in	ADP
ejpam-6461	302	5	particular	particular	ADJ
ejpam-6461	302	6	,	,	PUNCT
ejpam-6461	302	7	if	if	SCONJ
ejpam-6461	302	8	t	t	NOUN
ejpam-6461	302	9	=	=	SYM
ejpam-6461	302	10	1	1	NUM
ejpam-6461	302	11	2	2	NUM
ejpam-6461	302	12	,	,	PUNCT
ejpam-6461	302	13	we	we	PRON
ejpam-6461	302	14	obtain	obtain	VERB
ejpam-6461	302	15	the	the	DET
ejpam-6461	302	16	second	second	ADJ
ejpam-6461	302	17	inequality	inequality	NOUN
ejpam-6461	302	18	of	of	ADP
ejpam-6461	302	19	(	(	PUNCT
ejpam-6461	302	20	1	1	NUM
ejpam-6461	302	21	)	)	PUNCT
ejpam-6461	302	22	.	.	PUNCT
ejpam-6461	303	1	proof	proof	NOUN
ejpam-6461	303	2	.	.	PUNCT
ejpam-6461	304	1	letting	let	VERB
ejpam-6461	304	2	a	a	DET
ejpam-6461	304	3	=	=	PUNCT
ejpam-6461	304	4	|t	|t	NOUN
ejpam-6461	304	5	|t	|t	ADJ
ejpam-6461	304	6	and	and	CCONJ
ejpam-6461	304	7	b∗	b∗	ADJ
ejpam-6461	304	8	=	=	PRON
ejpam-6461	304	9	|t	|t	VERB
ejpam-6461	304	10	|1−tu∗	|1−tu∗	ADP
ejpam-6461	304	11	in	in	ADP
ejpam-6461	304	12	inequality	inequality	NOUN
ejpam-6461	304	13	(	(	PUNCT
ejpam-6461	304	14	39	39	NUM
ejpam-6461	304	15	)	)	PUNCT
ejpam-6461	304	16	,	,	PUNCT
ejpam-6461	304	17	we	we	PRON
ejpam-6461	304	18	obtain	obtain	VERB
ejpam-6461	304	19	the	the	DET
ejpam-6461	304	20	inequality	inequality	NOUN
ejpam-6461	304	21	(	(	PUNCT
ejpam-6461	304	22	40	40	NUM
ejpam-6461	304	23	)	)	PUNCT
ejpam-6461	304	24	.	.	PUNCT
ejpam-6461	305	1	5	5	X
ejpam-6461	305	2	.	.	X
ejpam-6461	305	3	conclusions	conclusion	NOUN
ejpam-6461	305	4	several	several	ADJ
ejpam-6461	305	5	numerical	numerical	ADJ
ejpam-6461	305	6	radius	radius	PROPN
ejpam-6461	305	7	inequalities	inequality	NOUN
ejpam-6461	305	8	of	of	ADP
ejpam-6461	305	9	operators	operator	NOUN
ejpam-6461	305	10	are	be	AUX
ejpam-6461	305	11	proved	prove	VERB
ejpam-6461	305	12	.	.	PUNCT
ejpam-6461	306	1	we	we	PRON
ejpam-6461	306	2	compare	compare	VERB
ejpam-6461	306	3	these	these	DET
ejpam-6461	306	4	new	new	ADJ
ejpam-6461	306	5	inequalities	inequality	NOUN
ejpam-6461	306	6	with	with	ADP
ejpam-6461	306	7	recent	recent	ADJ
ejpam-6461	306	8	inequalities	inequality	NOUN
ejpam-6461	306	9	proved	prove	VERB
ejpam-6461	306	10	by	by	ADP
ejpam-6461	306	11	kittaneh	kittaneh	PROPN
ejpam-6461	306	12	.	.	PUNCT
ejpam-6461	307	1	our	our	PRON
ejpam-6461	307	2	new	new	ADJ
ejpam-6461	307	3	inequalities	inequality	NOUN
ejpam-6461	307	4	refine	refine	VERB
ejpam-6461	307	5	and	and	CCONJ
ejpam-6461	307	6	generalize	generalize	VERB
ejpam-6461	307	7	kittaneh	kittaneh	PROPN
ejpam-6461	307	8	inequalities	inequality	NOUN
ejpam-6461	307	9	.	.	PUNCT
ejpam-6461	308	1	we	we	PRON
ejpam-6461	308	2	use	use	VERB
ejpam-6461	308	3	several	several	ADJ
ejpam-6461	308	4	techniques	technique	NOUN
ejpam-6461	308	5	to	to	PART
ejpam-6461	308	6	reach	reach	VERB
ejpam-6461	308	7	our	our	PRON
ejpam-6461	308	8	new	new	ADJ
ejpam-6461	308	9	bounds	bound	NOUN
ejpam-6461	308	10	for	for	ADP
ejpam-6461	308	11	numerical	numerical	ADJ
ejpam-6461	308	12	radius	radius	PROPN
ejpam-6461	308	13	inequalities	inequality	NOUN
ejpam-6461	308	14	of	of	ADP
ejpam-6461	308	15	operators	operator	NOUN
ejpam-6461	308	16	.	.	PUNCT
ejpam-6461	309	1	these	these	DET
ejpam-6461	309	2	techniques	technique	NOUN
ejpam-6461	309	3	included	include	VERB
ejpam-6461	309	4	inner	inner	ADJ
ejpam-6461	309	5	products	product	NOUN
ejpam-6461	309	6	,	,	PUNCT
ejpam-6461	309	7	block	block	NOUN
ejpam-6461	309	8	matrices	matrix	NOUN
ejpam-6461	309	9	,	,	PUNCT
ejpam-6461	309	10	singular	singular	ADJ
ejpam-6461	309	11	values	value	NOUN
ejpam-6461	309	12	and	and	CCONJ
ejpam-6461	309	13	aluthge	aluthge	ADJ
ejpam-6461	309	14	transform	transform	NOUN
ejpam-6461	309	15	.	.	PUNCT
ejpam-6461	310	1	acknowledgements	acknowledgement	NOUN
ejpam-6461	310	2	the	the	DET
ejpam-6461	310	3	authors	author	NOUN
ejpam-6461	310	4	are	be	AUX
ejpam-6461	310	5	grateful	grateful	ADJ
ejpam-6461	310	6	to	to	ADP
ejpam-6461	310	7	the	the	DET
ejpam-6461	310	8	editor	editor	NOUN
ejpam-6461	310	9	and	and	CCONJ
ejpam-6461	310	10	referees	referee	NOUN
ejpam-6461	310	11	for	for	ADP
ejpam-6461	310	12	their	their	PRON
ejpam-6461	310	13	comments	comment	NOUN
ejpam-6461	310	14	and	and	CCONJ
ejpam-6461	310	15	suggestions	suggestion	NOUN
ejpam-6461	310	16	.	.	PUNCT
ejpam-6461	311	1	the	the	DET
ejpam-6461	311	2	authors	author	NOUN
ejpam-6461	311	3	are	be	AUX
ejpam-6461	311	4	indebted	indebte	VERB
ejpam-6461	311	5	to	to	ADP
ejpam-6461	311	6	university	university	PROPN
ejpam-6461	311	7	of	of	ADP
ejpam-6461	311	8	petra	petra	PROPN
ejpam-6461	311	9	for	for	ADP
ejpam-6461	311	10	its	its	PRON
ejpam-6461	311	11	support	support	NOUN
ejpam-6461	311	12	.	.	PUNCT
ejpam-6461	312	1	m.	m.	NOUN
ejpam-6461	312	2	al	al	PROPN
ejpam-6461	312	3	-	-	PUNCT
ejpam-6461	312	4	labadi	labadi	PROPN
ejpam-6461	312	5	et	et	PROPN
ejpam-6461	312	6	al	al	PROPN
ejpam-6461	312	7	.	.	PUNCT
ejpam-6461	312	8	/	/	SYM
ejpam-6461	312	9	eur	eur	PROPN
ejpam-6461	312	10	.	.	PUNCT
ejpam-6461	313	1	j.	j.	PROPN
ejpam-6461	313	2	pure	pure	PROPN
ejpam-6461	313	3	appl	appl	PROPN
ejpam-6461	313	4	.	.	PROPN
ejpam-6461	313	5	math	math	PROPN
ejpam-6461	313	6	,	,	PUNCT
ejpam-6461	313	7	18	18	NUM
ejpam-6461	313	8	(	(	PUNCT
ejpam-6461	313	9	3	3	NUM
ejpam-6461	313	10	)	)	PUNCT
ejpam-6461	313	11	(	(	PUNCT
ejpam-6461	313	12	2025	2025	NUM
ejpam-6461	313	13	)	)	PUNCT
ejpam-6461	313	14	,	,	PUNCT
ejpam-6461	313	15	6461	6461	NUM
ejpam-6461	313	16	12	12	NUM
ejpam-6461	313	17	of	of	ADP
ejpam-6461	313	18	13	13	NUM
ejpam-6461	313	19	references	reference	NOUN
ejpam-6461	313	20	[	[	X
ejpam-6461	313	21	1	1	NUM
ejpam-6461	313	22	]	]	PUNCT
ejpam-6461	313	23	m.	m.	NOUN
ejpam-6461	313	24	al	al	PROPN
ejpam-6461	313	25	-	-	PUNCT
ejpam-6461	313	26	labadi	labadi	PROPN
ejpam-6461	313	27	,	,	PUNCT
ejpam-6461	313	28	r.	r.	PROPN
ejpam-6461	313	29	al	al	PROPN
ejpam-6461	313	30	-	-	PROPN
ejpam-6461	313	31	naimi	naimi	PROPN
ejpam-6461	313	32	and	and	CCONJ
ejpam-6461	313	33	w.	w.	PROPN
ejpam-6461	313	34	audeh	audeh	PROPN
ejpam-6461	313	35	,	,	PUNCT
ejpam-6461	313	36	singular	singular	ADJ
ejpam-6461	313	37	value	value	NOUN
ejpam-6461	313	38	inequalities	inequality	NOUN
ejpam-6461	313	39	for	for	ADP
ejpam-6461	313	40	generalized	generalized	ADJ
ejpam-6461	313	41	anticommutators	anticommutator	NOUN
ejpam-6461	313	42	,	,	PUNCT
ejpam-6461	313	43	j	j	PROPN
ejpam-6461	313	44	inequal	inequal	PROPN
ejpam-6461	313	45	appl	appl	PROPN
ejpam-6461	313	46	,	,	PUNCT
ejpam-6461	313	47	15(2025	15(2025	NUM
ejpam-6461	313	48	)	)	PUNCT
ejpam-6461	313	49	.	.	PUNCT
ejpam-6461	314	1	[	[	X
ejpam-6461	314	2	2	2	X
ejpam-6461	314	3	]	]	PUNCT
ejpam-6461	314	4	w.	w.	PROPN
ejpam-6461	314	5	audeh	audeh	PROPN
ejpam-6461	314	6	,	,	PUNCT
ejpam-6461	314	7	h.r	h.r	PROPN
ejpam-6461	314	8	.	.	PROPN
ejpam-6461	314	9	moradi	moradi	PROPN
ejpam-6461	314	10	and	and	CCONJ
ejpam-6461	314	11	m.	m.	NOUN
ejpam-6461	314	12	sababheh	sababheh	PROPN
ejpam-6461	314	13	,	,	PUNCT
ejpam-6461	314	14	commutator	commutator	NOUN
ejpam-6461	314	15	bounds	bound	NOUN
ejpam-6461	314	16	via	via	ADP
ejpam-6461	314	17	singular	singular	ADJ
ejpam-6461	314	18	values	value	NOUN
ejpam-6461	314	19	with	with	ADP
ejpam-6461	314	20	applications	application	NOUN
ejpam-6461	314	21	to	to	ADP
ejpam-6461	314	22	the	the	DET
ejpam-6461	314	23	numerical	numerical	ADJ
ejpam-6461	314	24	radius	radius	NOUN
ejpam-6461	314	25	,	,	PUNCT
ejpam-6461	314	26	mediterr	mediterr	PROPN
ejpam-6461	314	27	.	.	PUNCT
ejpam-6461	315	1	j.	j.	PROPN
ejpam-6461	315	2	math	math	PROPN
ejpam-6461	315	3	.	.	PUNCT
ejpam-6461	316	1	22,8(2025	22,8(2025	NUM
ejpam-6461	316	2	)	)	PUNCT
ejpam-6461	316	3	.	.	PUNCT
ejpam-6461	317	1	[	[	X
ejpam-6461	317	2	3	3	X
ejpam-6461	317	3	]	]	PUNCT
ejpam-6461	317	4	w.	w.	NOUN
ejpam-6461	317	5	audeh	audeh	PROPN
ejpam-6461	317	6	,	,	PUNCT
ejpam-6461	317	7	a.	a.	PROPN
ejpam-6461	317	8	al	al	PROPN
ejpam-6461	317	9	-	-	PUNCT
ejpam-6461	317	10	boustanji	boustanji	X
ejpam-6461	317	11	,	,	PUNCT
ejpam-6461	317	12	m.	m.	NOUN
ejpam-6461	317	13	al	al	PROPN
ejpam-6461	317	14	-	-	PUNCT
ejpam-6461	317	15	labadi	labadi	PROPN
ejpam-6461	317	16	and	and	CCONJ
ejpam-6461	317	17	r.	r.	PROPN
ejpam-6461	317	18	al	al	PROPN
ejpam-6461	317	19	-	-	PUNCT
ejpam-6461	317	20	naimi	naimi	PROPN
ejpam-6461	317	21	,	,	PUNCT
ejpam-6461	317	22	singular	singular	ADJ
ejpam-6461	317	23	value	value	NOUN
ejpam-6461	317	24	inequalities	inequality	NOUN
ejpam-6461	317	25	of	of	ADP
ejpam-6461	317	26	matrices	matrix	NOUN
ejpam-6461	317	27	via	via	ADP
ejpam-6461	317	28	increasing	increase	VERB
ejpam-6461	317	29	functions	function	NOUN
ejpam-6461	317	30	,	,	PUNCT
ejpam-6461	317	31	j	j	PROPN
ejpam-6461	317	32	inequal	inequal	PROPN
ejpam-6461	317	33	appl	appl	PROPN
ejpam-6461	317	34	,	,	PUNCT
ejpam-6461	317	35	114(2024	114(2024	NUM
ejpam-6461	317	36	)	)	PUNCT
ejpam-6461	317	37	.	.	PUNCT
ejpam-6461	318	1	[	[	X
ejpam-6461	318	2	4	4	X
ejpam-6461	318	3	]	]	PUNCT
ejpam-6461	318	4	w.	w.	PROPN
ejpam-6461	318	5	audeh	audeh	PROPN
ejpam-6461	318	6	,	,	PUNCT
ejpam-6461	318	7	h.r	h.r	PROPN
ejpam-6461	318	8	.	.	PROPN
ejpam-6461	318	9	moradi	moradi	PROPN
ejpam-6461	318	10	and	and	CCONJ
ejpam-6461	318	11	m.	m.	NOUN
ejpam-6461	318	12	sababheh	sababheh	NOUN
ejpam-6461	318	13	,	,	PUNCT
ejpam-6461	318	14	generalizations	generalization	NOUN
ejpam-6461	318	15	of	of	ADP
ejpam-6461	318	16	recent	recent	ADJ
ejpam-6461	318	17	singular	singular	ADJ
ejpam-6461	318	18	value	value	NOUN
ejpam-6461	318	19	inequalities	inequality	NOUN
ejpam-6461	318	20	for	for	ADP
ejpam-6461	318	21	sums	sum	NOUN
ejpam-6461	318	22	of	of	ADP
ejpam-6461	318	23	products	product	NOUN
ejpam-6461	318	24	of	of	ADP
ejpam-6461	318	25	matrices	matrix	NOUN
ejpam-6461	318	26	,	,	PUNCT
ejpam-6461	318	27	filomat	filomat	NOUN
ejpam-6461	318	28	,	,	PUNCT
ejpam-6461	318	29	38,28(2024	38,28(2024	NOUN
ejpam-6461	318	30	)	)	PUNCT
ejpam-6461	318	31	,	,	PUNCT
ejpam-6461	318	32	9905−	9905−	NUM
ejpam-6461	318	33	9919	9919	NUM
ejpam-6461	318	34	.	.	PUNCT
ejpam-6461	319	1	[	[	X
ejpam-6461	319	2	5	5	X
ejpam-6461	319	3	]	]	PUNCT
ejpam-6461	319	4	w.	w.	PROPN
ejpam-6461	319	5	audeh	audeh	PROPN
ejpam-6461	319	6	,	,	PUNCT
ejpam-6461	319	7	h.r	h.r	PROPN
ejpam-6461	319	8	.	.	PROPN
ejpam-6461	319	9	moradi	moradi	PROPN
ejpam-6461	319	10	and	and	CCONJ
ejpam-6461	319	11	m.	m.	NOUN
ejpam-6461	319	12	sababheh	sababheh	NOUN
ejpam-6461	319	13	,	,	PUNCT
ejpam-6461	319	14	matrix	matrix	NOUN
ejpam-6461	319	15	holder	holder	NOUN
ejpam-6461	319	16	inequalities	inequality	NOUN
ejpam-6461	319	17	and	and	CCONJ
ejpam-6461	319	18	numerical	numerical	PROPN
ejpam-6461	319	19	radius	radius	PROPN
ejpam-6461	319	20	applications	application	NOUN
ejpam-6461	319	21	,	,	PUNCT
ejpam-6461	319	22	linear	linear	PROPN
ejpam-6461	319	23	algebra	algebra	PROPN
ejpam-6461	319	24	appl	appl	NOUN
ejpam-6461	319	25	.	.	PROPN
ejpam-6461	319	26	,	,	PUNCT
ejpam-6461	319	27	696(2024	696(2024	NUM
ejpam-6461	319	28	)	)	PUNCT
ejpam-6461	319	29	,	,	PUNCT
ejpam-6461	319	30	68−	68−	NOUN
ejpam-6461	319	31	84	84	NUM
ejpam-6461	319	32	.	.	PUNCT
ejpam-6461	320	1	[	[	X
ejpam-6461	320	2	6	6	NUM
ejpam-6461	320	3	]	]	PUNCT
ejpam-6461	320	4	w.	w.	PROPN
ejpam-6461	320	5	audeh	audeh	PROPN
ejpam-6461	320	6	,	,	PUNCT
ejpam-6461	320	7	m.	m.	PROPN
ejpam-6461	320	8	al	al	PROPN
ejpam-6461	320	9	-	-	PUNCT
ejpam-6461	320	10	labadi	labadi	PROPN
ejpam-6461	320	11	and	and	CCONJ
ejpam-6461	320	12	r.	r.	PROPN
ejpam-6461	320	13	al	al	PROPN
ejpam-6461	320	14	-	-	PUNCT
ejpam-6461	320	15	naimi	naimi	PROPN
ejpam-6461	320	16	,	,	PUNCT
ejpam-6461	320	17	numerical	numerical	ADJ
ejpam-6461	320	18	radius	radius	PROPN
ejpam-6461	320	19	inequalities	inequality	NOUN
ejpam-6461	320	20	via	via	ADP
ejpam-6461	320	21	block	block	NOUN
ejpam-6461	320	22	matrices	matrix	NOUN
ejpam-6461	320	23	,	,	PUNCT
ejpam-6461	320	24	acta	acta	PROPN
ejpam-6461	320	25	sci	sci	PROPN
ejpam-6461	320	26	.	.	PUNCT
ejpam-6461	320	27	math	math	PROPN
ejpam-6461	320	28	.	.	PUNCT
ejpam-6461	321	1	(	(	PUNCT
ejpam-6461	321	2	2024	2024	NUM
ejpam-6461	321	3	)	)	PUNCT
ejpam-6461	321	4	.	.	PUNCT
ejpam-6461	322	1	[	[	X
ejpam-6461	322	2	7	7	X
ejpam-6461	322	3	]	]	PUNCT
ejpam-6461	322	4	w.	w.	NOUN
ejpam-6461	322	5	audeh	audeh	PROPN
ejpam-6461	322	6	and	and	CCONJ
ejpam-6461	322	7	m.	m.	PROPN
ejpam-6461	322	8	al	al	PROPN
ejpam-6461	322	9	-	-	PUNCT
ejpam-6461	322	10	labadi	labadi	PROPN
ejpam-6461	322	11	,	,	PUNCT
ejpam-6461	322	12	numerical	numerical	ADJ
ejpam-6461	322	13	radius	radius	PROPN
ejpam-6461	322	14	inequalities	inequality	NOUN
ejpam-6461	322	15	for	for	ADP
ejpam-6461	322	16	finite	finite	ADJ
ejpam-6461	322	17	sums	sum	NOUN
ejpam-6461	322	18	of	of	ADP
ejpam-6461	322	19	operators	operator	NOUN
ejpam-6461	322	20	,	,	PUNCT
ejpam-6461	322	21	complex	complex	ADJ
ejpam-6461	322	22	anal	anal	NOUN
ejpam-6461	322	23	.	.	PUNCT
ejpam-6461	322	24	oper	oper	PROPN
ejpam-6461	322	25	.	.	PROPN
ejpam-6461	322	26	theory	theory	NOUN
ejpam-6461	322	27	,	,	PUNCT
ejpam-6461	322	28	17	17	NUM
ejpam-6461	322	29	(	(	PUNCT
ejpam-6461	322	30	2023	2023	NUM
ejpam-6461	322	31	)	)	PUNCT
ejpam-6461	322	32	.	.	PUNCT
ejpam-6461	323	1	[	[	X
ejpam-6461	323	2	8	8	X
ejpam-6461	323	3	]	]	X
ejpam-6461	323	4	w.	w.	PROPN
ejpam-6461	323	5	audeh	audeh	PROPN
ejpam-6461	323	6	,	,	PUNCT
ejpam-6461	323	7	singular	singular	ADJ
ejpam-6461	323	8	value	value	NOUN
ejpam-6461	323	9	inequalities	inequality	NOUN
ejpam-6461	323	10	for	for	ADP
ejpam-6461	323	11	operators	operator	NOUN
ejpam-6461	323	12	and	and	CCONJ
ejpam-6461	323	13	matrices	matrix	NOUN
ejpam-6461	323	14	,	,	PUNCT
ejpam-6461	323	15	ann	ann	PROPN
ejpam-6461	323	16	.	.	PUNCT
ejpam-6461	323	17	funct	funct	PROPN
ejpam-6461	323	18	.	.	PUNCT
ejpam-6461	324	1	anal	anal	PROPN
ejpam-6461	324	2	.	.	PROPN
ejpam-6461	324	3	,	,	PUNCT
ejpam-6461	324	4	13,24(2022	13,24(2022	PROPN
ejpam-6461	324	5	)	)	PUNCT
ejpam-6461	324	6	.	.	PUNCT
ejpam-6461	325	1	[	[	X
ejpam-6461	325	2	9	9	NUM
ejpam-6461	325	3	]	]	PUNCT
ejpam-6461	325	4	w.	w.	NOUN
ejpam-6461	325	5	audeh	audeh	PROPN
ejpam-6461	325	6	,	,	PUNCT
ejpam-6461	325	7	singular	singular	ADJ
ejpam-6461	325	8	value	value	NOUN
ejpam-6461	325	9	inequalities	inequality	NOUN
ejpam-6461	325	10	for	for	ADP
ejpam-6461	325	11	accretive	accretive	ADJ
ejpam-6461	325	12	-	-	PUNCT
ejpam-6461	325	13	dissipative	dissipative	ADJ
ejpam-6461	325	14	normal	normal	ADJ
ejpam-6461	325	15	operators	operator	NOUN
ejpam-6461	325	16	,	,	PUNCT
ejpam-6461	325	17	j.	j.	PROPN
ejpam-6461	325	18	math	math	PROPN
ejpam-6461	325	19	.	.	PUNCT
ejpam-6461	326	1	inequal	inequal	PROPN
ejpam-6461	326	2	.	.	PUNCT
ejpam-6461	326	3	,	,	PUNCT
ejpam-6461	326	4	16(2022	16(2022	NOUN
ejpam-6461	326	5	)	)	PUNCT
ejpam-6461	326	6	,	,	PUNCT
ejpam-6461	326	7	729−	729−	PRON
ejpam-6461	327	1	737	737	NUM
ejpam-6461	327	2	.	.	PUNCT
ejpam-6461	328	1	[	[	X
ejpam-6461	328	2	10	10	NUM
ejpam-6461	328	3	]	]	PUNCT
ejpam-6461	328	4	w.audeh	w.audeh	PROPN
ejpam-6461	328	5	,	,	PUNCT
ejpam-6461	328	6	singular	singular	ADJ
ejpam-6461	328	7	value	value	NOUN
ejpam-6461	328	8	inequalities	inequality	NOUN
ejpam-6461	328	9	with	with	ADP
ejpam-6461	328	10	applications	application	NOUN
ejpam-6461	328	11	,	,	PUNCT
ejpam-6461	328	12	j.	j.	PROPN
ejpam-6461	328	13	math	math	PROPN
ejpam-6461	328	14	.	.	PUNCT
ejpam-6461	329	1	computer	computer	NOUN
ejpam-6461	329	2	sci	sci	PROPN
ejpam-6461	329	3	.	.	PROPN
ejpam-6461	329	4	,	,	PUNCT
ejpam-6461	329	5	24(2022	24(2022	NOUN
ejpam-6461	329	6	)	)	PUNCT
ejpam-6461	329	7	,	,	PUNCT
ejpam-6461	329	8	323−	323−	NOUN
ejpam-6461	329	9	329	329	NUM
ejpam-6461	329	10	.	.	PUNCT
ejpam-6461	330	1	[	[	X
ejpam-6461	330	2	11	11	NUM
ejpam-6461	330	3	]	]	PUNCT
ejpam-6461	330	4	w.	w.	NOUN
ejpam-6461	330	5	audeh	audeh	PROPN
ejpam-6461	330	6	and	and	CCONJ
ejpam-6461	330	7	m.	m.	PROPN
ejpam-6461	330	8	al	al	PROPN
ejpam-6461	330	9	-	-	PUNCT
ejpam-6461	330	10	labadi	labadi	PROPN
ejpam-6461	330	11	,	,	PUNCT
ejpam-6461	330	12	some	some	PRON
ejpam-6461	330	13	results	result	NOUN
ejpam-6461	330	14	about	about	ADP
ejpam-6461	330	15	numerical	numerical	ADJ
ejpam-6461	330	16	radius	radius	PROPN
ejpam-6461	330	17	inequalities	inequality	NOUN
ejpam-6461	330	18	,	,	PUNCT
ejpam-6461	330	19	int	int	NOUN
ejpam-6461	330	20	.	.	PUNCT
ejpam-6461	331	1	j.	j.	PROPN
ejpam-6461	331	2	math	math	PROPN
ejpam-6461	331	3	.	.	PUNCT
ejpam-6461	332	1	compute	compute	PROPN
ejpam-6461	332	2	.	.	PUNCT
ejpam-6461	333	1	sci	sci	PROPN
ejpam-6461	333	2	.	.	PROPN
ejpam-6461	333	3	,	,	PUNCT
ejpam-6461	333	4	17(2022	17(2022	NOUN
ejpam-6461	333	5	)	)	PUNCT
ejpam-6461	333	6	,	,	PUNCT
ejpam-6461	333	7	33−	33−	NOUN
ejpam-6461	333	8	39	39	NUM
ejpam-6461	333	9	.	.	PUNCT
ejpam-6461	334	1	[	[	X
ejpam-6461	334	2	12	12	NUM
ejpam-6461	334	3	]	]	PUNCT
ejpam-6461	334	4	a.	a.	PROPN
ejpam-6461	334	5	al	al	PROPN
ejpam-6461	334	6	-	-	PUNCT
ejpam-6461	334	7	boustanji	boustanji	PROPN
ejpam-6461	334	8	and	and	CCONJ
ejpam-6461	334	9	w.	w.	NOUN
ejpam-6461	334	10	audeh	audeh	PROPN
ejpam-6461	334	11	,	,	PUNCT
ejpam-6461	334	12	applications	application	NOUN
ejpam-6461	334	13	of	of	ADP
ejpam-6461	334	14	numerical	numerical	ADJ
ejpam-6461	334	15	radius	radius	PROPN
ejpam-6461	334	16	inequalities	inequality	NOUN
ejpam-6461	334	17	,	,	PUNCT
ejpam-6461	334	18	int	int	NOUN
ejpam-6461	334	19	.	.	PUNCT
ejpam-6461	335	1	j.	j.	PROPN
ejpam-6461	335	2	math	math	PROPN
ejpam-6461	335	3	.	.	PUNCT
ejpam-6461	336	1	compute	compute	PROPN
ejpam-6461	336	2	.	.	PUNCT
ejpam-6461	337	1	sci	sci	PROPN
ejpam-6461	337	2	.	.	PROPN
ejpam-6461	337	3	,	,	PUNCT
ejpam-6461	337	4	17,3(2022	17,3(2022	NUM
ejpam-6461	337	5	)	)	PUNCT
ejpam-6461	337	6	,	,	PUNCT
ejpam-6461	337	7	1305−	1305−	NUM
ejpam-6461	337	8	1312	1312	NUM
ejpam-6461	337	9	.	.	PUNCT
ejpam-6461	338	1	[	[	X
ejpam-6461	338	2	13	13	NUM
ejpam-6461	338	3	]	]	PUNCT
ejpam-6461	338	4	w.	w.	PROPN
ejpam-6461	338	5	audeh	audeh	PROPN
ejpam-6461	338	6	,	,	PUNCT
ejpam-6461	338	7	singular	singular	ADJ
ejpam-6461	338	8	value	value	NOUN
ejpam-6461	338	9	and	and	CCONJ
ejpam-6461	338	10	norm	norm	NOUN
ejpam-6461	338	11	inequalities	inequality	NOUN
ejpam-6461	338	12	of	of	ADP
ejpam-6461	338	13	davidson	davidson	NOUN
ejpam-6461	338	14	-	-	PUNCT
ejpam-6461	338	15	power	power	NOUN
ejpam-6461	338	16	type	type	NOUN
ejpam-6461	338	17	,	,	PUNCT
ejpam-6461	338	18	j.	j.	PROPN
ejpam-6461	338	19	math	math	PROPN
ejpam-6461	338	20	.	.	PUNCT
ejpam-6461	339	1	inequal	inequal	PROPN
ejpam-6461	339	2	.	.	PUNCT
ejpam-6461	339	3	,	,	PUNCT
ejpam-6461	339	4	15(2021	15(2021	NUM
ejpam-6461	339	5	)	)	PUNCT
ejpam-6461	339	6	,	,	PUNCT
ejpam-6461	339	7	1311−	1311−	NUM
ejpam-6461	339	8	1320	1320	NUM
ejpam-6461	339	9	.	.	PUNCT
ejpam-6461	340	1	[	[	X
ejpam-6461	340	2	14	14	NUM
ejpam-6461	340	3	]	]	X
ejpam-6461	340	4	w.	w.	PROPN
ejpam-6461	340	5	audeh	audeh	PROPN
ejpam-6461	340	6	,	,	PUNCT
ejpam-6461	340	7	some	some	DET
ejpam-6461	340	8	generalizations	generalization	NOUN
ejpam-6461	340	9	for	for	ADP
ejpam-6461	340	10	singular	singular	ADJ
ejpam-6461	340	11	value	value	NOUN
ejpam-6461	340	12	inequalities	inequality	NOUN
ejpam-6461	340	13	of	of	ADP
ejpam-6461	340	14	compact	compact	ADJ
ejpam-6461	340	15	operators	operator	NOUN
ejpam-6461	340	16	,	,	PUNCT
ejpam-6461	340	17	adv	adv	PROPN
ejpam-6461	340	18	.	.	PUNCT
ejpam-6461	340	19	oper	oper	PROPN
ejpam-6461	340	20	.	.	PROPN
ejpam-6461	340	21	theory	theory	NOUN
ejpam-6461	340	22	,	,	PUNCT
ejpam-6461	340	23	6	6	NUM
ejpam-6461	340	24	(	(	PUNCT
ejpam-6461	340	25	2021	2021	NUM
ejpam-6461	340	26	)	)	PUNCT
ejpam-6461	340	27	.	.	PUNCT
ejpam-6461	341	1	[	[	X
ejpam-6461	341	2	15	15	NUM
ejpam-6461	341	3	]	]	X
ejpam-6461	341	4	w.	w.	NOUN
ejpam-6461	341	5	audeh	audeh	PROPN
ejpam-6461	341	6	,	,	PUNCT
ejpam-6461	341	7	generalizations	generalization	NOUN
ejpam-6461	341	8	for	for	ADP
ejpam-6461	341	9	singular	singular	ADJ
ejpam-6461	341	10	value	value	NOUN
ejpam-6461	341	11	and	and	CCONJ
ejpam-6461	341	12	arithmetic	arithmetic	ADJ
ejpam-6461	341	13	-	-	PUNCT
ejpam-6461	341	14	geometric	geometric	ADJ
ejpam-6461	341	15	mean	mean	NOUN
ejpam-6461	341	16	inequalities	inequality	NOUN
ejpam-6461	341	17	of	of	ADP
ejpam-6461	341	18	operators	operator	NOUN
ejpam-6461	341	19	,	,	PUNCT
ejpam-6461	341	20	j.math	j.math	NOUN
ejpam-6461	341	21	.	.	PUNCT
ejpam-6461	342	1	anal	anal	PROPN
ejpam-6461	342	2	.	.	PUNCT
ejpam-6461	342	3	appl	appl	PROPN
ejpam-6461	342	4	.	.	PROPN
ejpam-6461	342	5	,	,	PUNCT
ejpam-6461	342	6	489	489	NUM
ejpam-6461	342	7	(	(	PUNCT
ejpam-6461	342	8	2020	2020	NUM
ejpam-6461	342	9	)	)	PUNCT
ejpam-6461	342	10	,	,	PUNCT
ejpam-6461	342	11	1−	1−	NUM
ejpam-6461	342	12	8	8	NUM
ejpam-6461	342	13	.	.	PUNCT
ejpam-6461	343	1	[	[	X
ejpam-6461	343	2	16	16	NUM
ejpam-6461	343	3	]	]	X
ejpam-6461	343	4	w.	w.	PROPN
ejpam-6461	343	5	audeh	audeh	PROPN
ejpam-6461	343	6	,	,	PUNCT
ejpam-6461	343	7	generalizations	generalization	NOUN
ejpam-6461	343	8	for	for	ADP
ejpam-6461	343	9	singular	singular	ADJ
ejpam-6461	343	10	value	value	NOUN
ejpam-6461	343	11	inequalities	inequality	NOUN
ejpam-6461	343	12	of	of	ADP
ejpam-6461	343	13	operators	operator	NOUN
ejpam-6461	343	14	,	,	PUNCT
ejpam-6461	343	15	adv	adv	PROPN
ejpam-6461	343	16	.	.	PUNCT
ejpam-6461	344	1	oper	oper	PROPN
ejpam-6461	344	2	.	.	PROPN
ejpam-6461	344	3	theory	theory	NOUN
ejpam-6461	344	4	,	,	PUNCT
ejpam-6461	344	5	5	5	NUM
ejpam-6461	344	6	(	(	PUNCT
ejpam-6461	344	7	2020	2020	NUM
ejpam-6461	344	8	)	)	PUNCT
ejpam-6461	344	9	,	,	PUNCT
ejpam-6461	344	10	371−	371−	NUM
ejpam-6461	344	11	381	381	NUM
ejpam-6461	344	12	.	.	PUNCT
ejpam-6461	345	1	[	[	X
ejpam-6461	345	2	17	17	NUM
ejpam-6461	345	3	]	]	PUNCT
ejpam-6461	345	4	w.	w.	PROPN
ejpam-6461	345	5	audeh	audeh	PROPN
ejpam-6461	345	6	,	,	PUNCT
ejpam-6461	345	7	singular	singular	ADJ
ejpam-6461	345	8	value	value	NOUN
ejpam-6461	345	9	inequalities	inequality	NOUN
ejpam-6461	345	10	and	and	CCONJ
ejpam-6461	345	11	applications	application	NOUN
ejpam-6461	345	12	,	,	PUNCT
ejpam-6461	345	13	positivity	positivity	NOUN
ejpam-6461	345	14	,	,	PUNCT
ejpam-6461	345	15	25	25	NUM
ejpam-6461	345	16	(	(	PUNCT
ejpam-6461	345	17	2020	2020	NUM
ejpam-6461	345	18	)	)	PUNCT
ejpam-6461	345	19	,	,	PUNCT
ejpam-6461	345	20	843852	843852	NUM
ejpam-6461	345	21	.	.	PUNCT
ejpam-6461	346	1	[	[	X
ejpam-6461	346	2	18	18	NUM
ejpam-6461	346	3	]	]	PUNCT
ejpam-6461	346	4	w.	w.	PROPN
ejpam-6461	346	5	audeh	audeh	PROPN
ejpam-6461	346	6	and	and	CCONJ
ejpam-6461	346	7	f.	f.	PROPN
ejpam-6461	346	8	kittaneh	kittaneh	PROPN
ejpam-6461	346	9	,	,	PUNCT
ejpam-6461	346	10	singular	singular	ADJ
ejpam-6461	346	11	value	value	NOUN
ejpam-6461	346	12	inequalities	inequality	NOUN
ejpam-6461	346	13	for	for	ADP
ejpam-6461	346	14	compact	compact	ADJ
ejpam-6461	346	15	operators	operator	NOUN
ejpam-6461	346	16	,	,	PUNCT
ejpam-6461	346	17	linear	linear	PROPN
ejpam-6461	346	18	algebra	algebra	NOUN
ejpam-6461	346	19	appl	appl	NOUN
ejpam-6461	346	20	,	,	PUNCT
ejpam-6461	346	21	437	437	NUM
ejpam-6461	346	22	(	(	PUNCT
ejpam-6461	346	23	2012	2012	NUM
ejpam-6461	346	24	)	)	PUNCT
ejpam-6461	346	25	,	,	PUNCT
ejpam-6461	346	26	2516	2516	NUM
ejpam-6461	346	27	-	-	SYM
ejpam-6461	346	28	2522	2522	NUM
ejpam-6461	346	29	.	.	PUNCT
ejpam-6461	347	1	[	[	X
ejpam-6461	347	2	19	19	NUM
ejpam-6461	347	3	]	]	X
ejpam-6461	347	4	r.	r.	PROPN
ejpam-6461	347	5	bhatia	bhatia	PROPN
ejpam-6461	347	6	,	,	PUNCT
ejpam-6461	347	7	matrix	matrix	VERB
ejpam-6461	347	8	analysis	analysis	NOUN
ejpam-6461	347	9	,	,	PUNCT
ejpam-6461	347	10	gtm169	gtm169	NOUN
ejpam-6461	347	11	,	,	PUNCT
ejpam-6461	347	12	springer	springer	NOUN
ejpam-6461	347	13	-	-	PUNCT
ejpam-6461	347	14	verlag	verlag	PROPN
ejpam-6461	347	15	,	,	PUNCT
ejpam-6461	347	16	new	new	PROPN
ejpam-6461	347	17	york	york	PROPN
ejpam-6461	347	18	,	,	PUNCT
ejpam-6461	347	19	(	(	PUNCT
ejpam-6461	347	20	1997	1997	NUM
ejpam-6461	347	21	)	)	PUNCT
ejpam-6461	347	22	.	.	PUNCT
ejpam-6461	348	1	[	[	X
ejpam-6461	348	2	20	20	NUM
ejpam-6461	348	3	]	]	PUNCT
ejpam-6461	348	4	p.	p.	NOUN
ejpam-6461	348	5	bhunia	bhunia	NOUN
ejpam-6461	348	6	and	and	CCONJ
ejpam-6461	348	7	k.	k.	PROPN
ejpam-6461	348	8	paul	paul	PROPN
ejpam-6461	348	9	,	,	PUNCT
ejpam-6461	348	10	furtherance	furtherance	NOUN
ejpam-6461	348	11	of	of	ADP
ejpam-6461	348	12	numerical	numerical	ADJ
ejpam-6461	348	13	radius	radius	PROPN
ejpam-6461	348	14	inequalities	inequality	NOUN
ejpam-6461	348	15	of	of	ADP
ejpam-6461	348	16	hilbert	hilbert	PROPN
ejpam-6461	348	17	space	space	NOUN
ejpam-6461	348	18	operators	operator	NOUN
ejpam-6461	348	19	.	.	PUNCT
ejpam-6461	349	1	arch	arch	PROPN
ejpam-6461	349	2	.	.	PUNCT
ejpam-6461	350	1	math	math	NOUN
ejpam-6461	350	2	.	.	PUNCT
ejpam-6461	351	1	(	(	PUNCT
ejpam-6461	351	2	basel	basel	PROPN
ejpam-6461	351	3	)	)	PUNCT
ejpam-6461	351	4	.	.	PUNCT
ejpam-6461	352	1	,	,	PUNCT
ejpam-6461	352	2	117	117	NUM
ejpam-6461	352	3	(	(	PUNCT
ejpam-6461	352	4	5	5	NUM
ejpam-6461	352	5	)	)	PUNCT
ejpam-6461	352	6	(	(	PUNCT
ejpam-6461	352	7	2021	2021	NUM
ejpam-6461	352	8	)	)	PUNCT
ejpam-6461	352	9	,	,	PUNCT
ejpam-6461	352	10	537−	537−	NUM
ejpam-6461	352	11	546	546	NUM
ejpam-6461	352	12	.	.	PUNCT
ejpam-6461	353	1	[	[	X
ejpam-6461	353	2	21	21	NUM
ejpam-6461	353	3	]	]	X
ejpam-6461	353	4	l.	l.	PROPN
ejpam-6461	353	5	c.	c.	PROPN
ejpam-6461	353	6	gohberg	gohberg	PROPN
ejpam-6461	353	7	and	and	CCONJ
ejpam-6461	353	8	m.g	m.g	PROPN
ejpam-6461	353	9	.	.	PROPN
ejpam-6461	353	10	krein	krein	PROPN
ejpam-6461	353	11	,	,	PUNCT
ejpam-6461	353	12	introduction	introduction	NOUN
ejpam-6461	353	13	to	to	ADP
ejpam-6461	353	14	the	the	DET
ejpam-6461	353	15	theory	theory	NOUN
ejpam-6461	353	16	of	of	ADP
ejpam-6461	353	17	linear	linear	PROPN
ejpam-6461	353	18	nonselfadjoint	nonselfadjoint	NOUN
ejpam-6461	353	19	operators	operator	NOUN
ejpam-6461	353	20	.	.	PUNCT
ejpam-6461	354	1	amer	amer	PROPN
ejpam-6461	354	2	.	.	PUNCT
ejpam-6461	354	3	math	math	PROPN
ejpam-6461	354	4	.	.	PUNCT
ejpam-6461	355	1	soc	soc	PROPN
ejpam-6461	355	2	,	,	PUNCT
ejpam-6461	355	3	providence	providence	NOUN
ejpam-6461	355	4	,	,	PUNCT
ejpam-6461	355	5	ri	ri	PROPN
ejpam-6461	355	6	(	(	PUNCT
ejpam-6461	355	7	1969	1969	NUM
ejpam-6461	355	8	)	)	PUNCT
ejpam-6461	355	9	.	.	PUNCT
ejpam-6461	356	1	m.	m.	PROPN
ejpam-6461	356	2	al	al	PROPN
ejpam-6461	356	3	-	-	PUNCT
ejpam-6461	356	4	labadi	labadi	PROPN
ejpam-6461	356	5	et	et	PROPN
ejpam-6461	356	6	al	al	PROPN
ejpam-6461	356	7	.	.	PUNCT
ejpam-6461	356	8	/	/	SYM
ejpam-6461	356	9	eur	eur	PROPN
ejpam-6461	356	10	.	.	PUNCT
ejpam-6461	357	1	j.	j.	PROPN
ejpam-6461	357	2	pure	pure	PROPN
ejpam-6461	357	3	appl	appl	PROPN
ejpam-6461	357	4	.	.	PROPN
ejpam-6461	357	5	math	math	PROPN
ejpam-6461	357	6	,	,	PUNCT
ejpam-6461	357	7	18	18	NUM
ejpam-6461	357	8	(	(	PUNCT
ejpam-6461	357	9	3	3	NUM
ejpam-6461	357	10	)	)	PUNCT
ejpam-6461	357	11	(	(	PUNCT
ejpam-6461	357	12	2025	2025	NUM
ejpam-6461	357	13	)	)	PUNCT
ejpam-6461	357	14	,	,	PUNCT
ejpam-6461	357	15	6461	6461	NUM
ejpam-6461	357	16	13	13	NUM
ejpam-6461	357	17	of	of	ADP
ejpam-6461	357	18	13	13	NUM
ejpam-6461	357	19	[	[	SYM
ejpam-6461	357	20	22	22	NUM
ejpam-6461	357	21	]	]	X
ejpam-6461	357	22	k.e	k.e	PROPN
ejpam-6461	357	23	.	.	PROPN
ejpam-6461	357	24	gustafson	gustafson	PROPN
ejpam-6461	357	25	and	and	CCONJ
ejpam-6461	357	26	d.k.m	d.k.m	PROPN
ejpam-6461	357	27	.	.	PROPN
ejpam-6461	357	28	rao	rao	PROPN
ejpam-6461	357	29	,	,	PUNCT
ejpam-6461	357	30	numerical	numerical	ADJ
ejpam-6461	357	31	range	range	PROPN
ejpam-6461	357	32	,	,	PUNCT
ejpam-6461	357	33	the	the	DET
ejpam-6461	357	34	field	field	NOUN
ejpam-6461	357	35	of	of	ADP
ejpam-6461	357	36	values	value	NOUN
ejpam-6461	357	37	of	of	ADP
ejpam-6461	357	38	linear	linear	PROPN
ejpam-6461	357	39	operators	operator	NOUN
ejpam-6461	357	40	and	and	CCONJ
ejpam-6461	357	41	matrices.springer	matrices.springer	NOUN
ejpam-6461	357	42	,	,	PUNCT
ejpam-6461	357	43	new	new	PROPN
ejpam-6461	357	44	york	york	PROPN
ejpam-6461	357	45	(	(	PUNCT
ejpam-6461	357	46	1997	1997	NUM
ejpam-6461	357	47	)	)	PUNCT
ejpam-6461	357	48	.	.	PUNCT
ejpam-6461	358	1	[	[	X
ejpam-6461	358	2	23	23	NUM
ejpam-6461	358	3	]	]	X
ejpam-6461	358	4	f.	f.	PROPN
ejpam-6461	358	5	kittaneh	kittaneh	PROPN
ejpam-6461	358	6	,	,	PUNCT
ejpam-6461	358	7	h.	h.	PROPN
ejpam-6461	358	8	moradi	moradi	PROPN
ejpam-6461	358	9	and	and	CCONJ
ejpam-6461	358	10	m.	m.	NOUN
ejpam-6461	358	11	sababheh	sababheh	NOUN
ejpam-6461	358	12	,	,	PUNCT
ejpam-6461	358	13	sharper	sharp	ADJ
ejpam-6461	358	14	bounds	bound	NOUN
ejpam-6461	358	15	for	for	ADP
ejpam-6461	358	16	the	the	DET
ejpam-6461	358	17	numerical	numerical	ADJ
ejpam-6461	358	18	radius	radius	NOUN
ejpam-6461	358	19	.	.	PUNCT
ejpam-6461	359	1	linear	linear	PROPN
ejpam-6461	359	2	multilinear	multilinear	PROPN
ejpam-6461	359	3	algebra	algebra	PROPN
ejpam-6461	359	4	,	,	PUNCT
ejpam-6461	359	5	(	(	PUNCT
ejpam-6461	359	6	2023	2023	NUM
ejpam-6461	359	7	)	)	PUNCT
ejpam-6461	359	8	.	.	PUNCT
ejpam-6461	360	1	[	[	X
ejpam-6461	360	2	24	24	NUM
ejpam-6461	360	3	]	]	X
ejpam-6461	360	4	f.	f.	PROPN
ejpam-6461	360	5	kittaneh	kittaneh	PROPN
ejpam-6461	360	6	,	,	PUNCT
ejpam-6461	360	7	numerical	numerical	PROPN
ejpam-6461	360	8	radius	radius	PROPN
ejpam-6461	360	9	inequalities	inequality	NOUN
ejpam-6461	360	10	,	,	PUNCT
ejpam-6461	360	11	associated	associate	VERB
ejpam-6461	360	12	with	with	ADP
ejpam-6461	360	13	the	the	DET
ejpam-6461	360	14	cartesian	cartesian	ADJ
ejpam-6461	360	15	decomposition	decomposition	NOUN
ejpam-6461	360	16	.	.	PUNCT
ejpam-6461	361	1	mia	mia	PROPN
ejpam-6461	361	2	.	.	PROPN
ejpam-6461	361	3	,	,	PUNCT
ejpam-6461	361	4	18	18	NUM
ejpam-6461	361	5	(	(	PUNCT
ejpam-6461	361	6	2015	2015	NUM
ejpam-6461	361	7	)	)	PUNCT
ejpam-6461	361	8	,	,	PUNCT
ejpam-6461	361	9	915−	915−	PROPN
ejpam-6461	361	10	922	922	NUM
ejpam-6461	361	11	.	.	PUNCT
ejpam-6461	362	1	[	[	X
ejpam-6461	362	2	25	25	NUM
ejpam-6461	362	3	]	]	X
ejpam-6461	362	4	f.	f.	PROPN
ejpam-6461	362	5	kittaneh	kittaneh	PROPN
ejpam-6461	362	6	,	,	PUNCT
ejpam-6461	362	7	numerical	numerical	PROPN
ejpam-6461	362	8	radius	radius	PROPN
ejpam-6461	362	9	inequalities	inequality	NOUN
ejpam-6461	362	10	for	for	ADP
ejpam-6461	362	11	hilbert	hilbert	NOUN
ejpam-6461	362	12	space	space	NOUN
ejpam-6461	362	13	operators	operator	NOUN
ejpam-6461	362	14	.	.	PUNCT
ejpam-6461	363	1	stud	stud	PROPN
ejpam-6461	363	2	.	.	PUNCT
ejpam-6461	364	1	math	math	NOUN
ejpam-6461	364	2	.	.	PUNCT
ejpam-6461	365	1	,	,	PUNCT
ejpam-6461	365	2	168	168	NUM
ejpam-6461	365	3	(	(	PUNCT
ejpam-6461	365	4	2003	2003	NUM
ejpam-6461	365	5	)	)	PUNCT
ejpam-6461	365	6	,	,	PUNCT
ejpam-6461	366	1	73−	73−	NUM
ejpam-6461	366	2	80	80	NUM
ejpam-6461	366	3	.	.	PUNCT
ejpam-6461	367	1	[	[	X
ejpam-6461	367	2	26	26	NUM
ejpam-6461	367	3	]	]	X
ejpam-6461	367	4	f.	f.	PROPN
ejpam-6461	367	5	kittaneh	kittaneh	PROPN
ejpam-6461	367	6	,	,	PUNCT
ejpam-6461	367	7	numerical	numerical	PROPN
ejpam-6461	367	8	radius	radius	PROPN
ejpam-6461	367	9	inequality	inequality	NOUN
ejpam-6461	367	10	and	and	CCONJ
ejpam-6461	367	11	an	an	DET
ejpam-6461	367	12	estimate	estimate	NOUN
ejpam-6461	367	13	for	for	ADP
ejpam-6461	367	14	the	the	DET
ejpam-6461	367	15	numerical	numerical	ADJ
ejpam-6461	367	16	radius	radius	NOUN
ejpam-6461	367	17	of	of	ADP
ejpam-6461	367	18	the	the	DET
ejpam-6461	367	19	frobenius	frobenius	ADJ
ejpam-6461	367	20	companion	companion	NOUN
ejpam-6461	367	21	matrix	matrix	NOUN
ejpam-6461	367	22	.	.	PUNCT
ejpam-6461	368	1	stud	stud	PROPN
ejpam-6461	368	2	math	math	PROPN
ejpam-6461	368	3	.	.	PUNCT
ejpam-6461	369	1	,	,	PUNCT
ejpam-6461	369	2	158	158	NUM
ejpam-6461	369	3	(	(	PUNCT
ejpam-6461	369	4	2003	2003	NUM
ejpam-6461	369	5	)	)	PUNCT
ejpam-6461	369	6	,	,	PUNCT
ejpam-6461	369	7	11−	11−	NUM
ejpam-6461	369	8	17	17	NUM
ejpam-6461	369	9	.	.	PUNCT
ejpam-6461	370	1	[	[	X
ejpam-6461	370	2	27	27	NUM
ejpam-6461	370	3	]	]	X
ejpam-6461	370	4	f.	f.	PROPN
ejpam-6461	370	5	kittaneh	kittaneh	PROPN
ejpam-6461	370	6	,	,	PUNCT
ejpam-6461	370	7	norm	norm	NOUN
ejpam-6461	370	8	inequalities	inequality	NOUN
ejpam-6461	370	9	for	for	ADP
ejpam-6461	370	10	certain	certain	ADJ
ejpam-6461	370	11	operators	operator	NOUN
ejpam-6461	370	12	sums	sum	NOUN
ejpam-6461	370	13	.	.	PUNCT
ejpam-6461	371	1	j.	j.	PROPN
ejpam-6461	371	2	funct	funct	PROPN
ejpam-6461	371	3	.	.	PUNCT
ejpam-6461	372	1	anal	anal	PROPN
ejpam-6461	372	2	.	.	PROPN
ejpam-6461	372	3	,	,	PUNCT
ejpam-6461	372	4	143	143	NUM
ejpam-6461	372	5	(	(	PUNCT
ejpam-6461	372	6	1997	1997	NUM
ejpam-6461	372	7	)	)	PUNCT
ejpam-6461	372	8	,	,	PUNCT
ejpam-6461	372	9	337−	337−	NUM
ejpam-6461	372	10	348	348	NUM
ejpam-6461	372	11	.	.	PUNCT
ejpam-6461	373	1	[	[	X
ejpam-6461	373	2	28	28	NUM
ejpam-6461	373	3	]	]	X
ejpam-6461	373	4	f.	f.	PROPN
ejpam-6461	373	5	kittaneh	kittaneh	PROPN
ejpam-6461	373	6	,	,	PUNCT
ejpam-6461	373	7	notes	note	NOUN
ejpam-6461	373	8	on	on	ADP
ejpam-6461	373	9	some	some	DET
ejpam-6461	373	10	inequalities	inequality	NOUN
ejpam-6461	373	11	for	for	ADP
ejpam-6461	373	12	hilbert	hilbert	NOUN
ejpam-6461	373	13	space	space	NOUN
ejpam-6461	373	14	operators	operator	NOUN
ejpam-6461	373	15	,	,	PUNCT
ejpam-6461	373	16	publications	publication	NOUN
ejpam-6461	373	17	of	of	ADP
ejpam-6461	373	18	the	the	DET
ejpam-6461	373	19	research	research	PROPN
ejpam-6461	373	20	institute	institute	NOUN
ejpam-6461	373	21	for	for	ADP
ejpam-6461	373	22	mathematical	mathematical	ADJ
ejpam-6461	373	23	sciences	science	NOUN
ejpam-6461	373	24	,	,	PUNCT
ejpam-6461	373	25	24(1988	24(1988	NUM
ejpam-6461	373	26	)	)	PUNCT
ejpam-6461	373	27	,	,	PUNCT
ejpam-6461	373	28	283−	283−	PROPN
ejpam-6461	373	29	293	293	NUM
ejpam-6461	373	30	.	.	PUNCT
ejpam-6461	374	1	[	[	X
ejpam-6461	374	2	29	29	NUM
ejpam-6461	374	3	]	]	X
ejpam-6461	374	4	h.r	h.r	PROPN
ejpam-6461	374	5	.	.	PROPN
ejpam-6461	374	6	moradi	moradi	PROPN
ejpam-6461	374	7	,	,	PUNCT
ejpam-6461	374	8	w.	w.	NOUN
ejpam-6461	374	9	audeh	audeh	PROPN
ejpam-6461	374	10	and	and	CCONJ
ejpam-6461	374	11	m.	m.	NOUN
ejpam-6461	374	12	sababheh	sababheh	PROPN
ejpam-6461	374	13	,	,	PUNCT
ejpam-6461	374	14	singular	singular	ADJ
ejpam-6461	374	15	value	value	NOUN
ejpam-6461	374	16	inequalities	inequality	NOUN
ejpam-6461	374	17	via	via	ADP
ejpam-6461	374	18	matrix	matrix	NOUN
ejpam-6461	374	19	monotone	monotone	NOUN
ejpam-6461	374	20	functions	function	NOUN
ejpam-6461	374	21	,	,	PUNCT
ejpam-6461	374	22	anal	anal	NOUN
ejpam-6461	374	23	.	.	PUNCT
ejpam-6461	374	24	math	math	NOUN
ejpam-6461	374	25	.	.	PUNCT
ejpam-6461	375	1	phys	phy	NOUN
ejpam-6461	375	2	.	.	PUNCT
ejpam-6461	376	1	13	13	NUM
ejpam-6461	376	2	,	,	PUNCT
ejpam-6461	376	3	71(2023	71(2023	NUM
ejpam-6461	376	4	)	)	PUNCT
ejpam-6461	376	5	.	.	PUNCT