id	sid	tid	token	lemma	pos
ejpam-6559	1	1	european	european	PROPN
ejpam-6559	1	2	journal	journal	PROPN
ejpam-6559	1	3	of	of	ADP
ejpam-6559	1	4	pure	pure	ADJ
ejpam-6559	1	5	and	and	CCONJ
ejpam-6559	1	6	applied	applied	ADJ
ejpam-6559	1	7	mathematics	mathematic	NOUN
ejpam-6559	1	8	2025	2025	NUM
ejpam-6559	1	9	,	,	PUNCT
ejpam-6559	1	10	vol	vol	NOUN
ejpam-6559	1	11	.	.	PROPN
ejpam-6559	1	12	18	18	NUM
ejpam-6559	1	13	,	,	PUNCT
ejpam-6559	1	14	issue	issue	NOUN
ejpam-6559	1	15	4	4	NUM
ejpam-6559	1	16	,	,	PUNCT
ejpam-6559	1	17	article	article	NOUN
ejpam-6559	1	18	number	number	NOUN
ejpam-6559	1	19	6559	6559	NUM
ejpam-6559	1	20	issn	issn	VERB
ejpam-6559	1	21	1307	1307	NUM
ejpam-6559	1	22	-	-	SYM
ejpam-6559	1	23	5543	5543	NUM
ejpam-6559	1	24	–	–	PUNCT
ejpam-6559	1	25	ejpam.com	ejpam.com	X
ejpam-6559	1	26	published	publish	VERB
ejpam-6559	1	27	by	by	ADP
ejpam-6559	1	28	new	new	PROPN
ejpam-6559	1	29	york	york	PROPN
ejpam-6559	1	30	business	business	PROPN
ejpam-6559	1	31	global	global	ADJ
ejpam-6559	1	32	second	second	ADJ
ejpam-6559	1	33	-	-	PUNCT
ejpam-6559	1	34	order	order	NOUN
ejpam-6559	1	35	abstract	abstract	ADJ
ejpam-6559	1	36	cauchy	cauchy	ADJ
ejpam-6559	1	37	problem	problem	NOUN
ejpam-6559	1	38	in	in	ADP
ejpam-6559	1	39	two	two	NUM
ejpam-6559	1	40	variables	variable	NOUN
ejpam-6559	1	41	waseem	waseem	PROPN
ejpam-6559	1	42	ghazi	ghazi	PROPN
ejpam-6559	1	43	alshanti1,∗	alshanti1,∗	PROPN
ejpam-6559	1	44	,	,	PUNCT
ejpam-6559	1	45	ma′mon	ma′mon	PROPN
ejpam-6559	1	46	abu	abu	PROPN
ejpam-6559	1	47	hammad1	hammad1	PROPN
ejpam-6559	1	48	,	,	PUNCT
ejpam-6559	1	49	roshdi	roshdi	ADJ
ejpam-6559	1	50	khalil2	khalil2	PROPN
ejpam-6559	1	51	1	1	NUM
ejpam-6559	1	52	department	department	NOUN
ejpam-6559	1	53	of	of	ADP
ejpam-6559	1	54	mathematics	mathematic	NOUN
ejpam-6559	1	55	,	,	PUNCT
ejpam-6559	1	56	al	al	PROPN
ejpam-6559	1	57	zaytoonah	zaytoonah	PROPN
ejpam-6559	1	58	university	university	PROPN
ejpam-6559	1	59	of	of	ADP
ejpam-6559	1	60	jordan	jordan	PROPN
ejpam-6559	1	61	,	,	PUNCT
ejpam-6559	1	62	amman	amman	PROPN
ejpam-6559	1	63	,	,	PUNCT
ejpam-6559	1	64	jordan	jordan	PROPN
ejpam-6559	1	65	2	2	NUM
ejpam-6559	1	66	department	department	NOUN
ejpam-6559	1	67	of	of	ADP
ejpam-6559	1	68	mathematics	mathematic	NOUN
ejpam-6559	1	69	,	,	PUNCT
ejpam-6559	1	70	the	the	DET
ejpam-6559	1	71	university	university	PROPN
ejpam-6559	1	72	of	of	ADP
ejpam-6559	1	73	jordan	jordan	PROPN
ejpam-6559	1	74	,	,	PUNCT
ejpam-6559	1	75	amman	amman	PROPN
ejpam-6559	1	76	,	,	PUNCT
ejpam-6559	1	77	jordan	jordan	PROPN
ejpam-6559	1	78	abstract	abstract	PROPN
ejpam-6559	1	79	.	.	PUNCT
ejpam-6559	2	1	in	in	ADP
ejpam-6559	2	2	this	this	DET
ejpam-6559	2	3	paper	paper	NOUN
ejpam-6559	2	4	,	,	PUNCT
ejpam-6559	2	5	we	we	PRON
ejpam-6559	2	6	consider	consider	VERB
ejpam-6559	2	7	the	the	DET
ejpam-6559	2	8	following	follow	VERB
ejpam-6559	2	9	second	second	ADJ
ejpam-6559	2	10	-	-	PUNCT
ejpam-6559	2	11	order	order	NOUN
ejpam-6559	2	12	abstract	abstract	ADJ
ejpam-6559	2	13	cauchy	cauchy	ADJ
ejpam-6559	2	14	problem	problem	NOUN
ejpam-6559	2	15	in	in	ADP
ejpam-6559	2	16	two	two	NUM
ejpam-6559	2	17	variables	variable	NOUN
ejpam-6559	2	18	d2	d2	PROPN
ejpam-6559	2	19	ssu(s	ssu(s	PROPN
ejpam-6559	2	20	,	,	PUNCT
ejpam-6559	2	21	t	t	PROPN
ejpam-6559	2	22	)	)	PUNCT
ejpam-6559	2	23	+	+	NOUN
ejpam-6559	2	24	d2	d2	PROPN
ejpam-6559	2	25	ttu(s	ttu(s	PROPN
ejpam-6559	2	26	,	,	PUNCT
ejpam-6559	2	27	t	t	PROPN
ejpam-6559	2	28	)	)	PUNCT
ejpam-6559	2	29	+	+	CCONJ
ejpam-6559	2	30	2d2	2d2	NUM
ejpam-6559	2	31	stu(s	stu(s	NOUN
ejpam-6559	2	32	,	,	PUNCT
ejpam-6559	2	33	t	t	PROPN
ejpam-6559	2	34	)	)	PUNCT
ejpam-6559	3	1	+	+	ADP
ejpam-6559	3	2	a	a	DET
ejpam-6559	3	3	[	[	X
ejpam-6559	3	4	dsu(s	dsu(s	PROPN
ejpam-6559	3	5	,	,	PUNCT
ejpam-6559	3	6	t	t	PROPN
ejpam-6559	3	7	)	)	PUNCT
ejpam-6559	3	8	+	+	NOUN
ejpam-6559	3	9	dtu(s	dtu(s	PROPN
ejpam-6559	3	10	,	,	PUNCT
ejpam-6559	3	11	t	t	PROPN
ejpam-6559	3	12	)	)	PUNCT
ejpam-6559	3	13	]	]	PUNCT
ejpam-6559	4	1	=	=	PUNCT
ejpam-6559	4	2	bu(s	bu(s	PROPN
ejpam-6559	4	3	,	,	PUNCT
ejpam-6559	4	4	t	t	PROPN
ejpam-6559	4	5	)	)	PUNCT
ejpam-6559	4	6	,	,	PUNCT
ejpam-6559	4	7	where	where	SCONJ
ejpam-6559	4	8	a	a	DET
ejpam-6559	4	9	,	,	PUNCT
ejpam-6559	4	10	b	b	NOUN
ejpam-6559	4	11	are	be	AUX
ejpam-6559	4	12	closed	close	VERB
ejpam-6559	4	13	linear	linear	ADJ
ejpam-6559	4	14	operators	operator	NOUN
ejpam-6559	4	15	on	on	ADP
ejpam-6559	4	16	a	a	DET
ejpam-6559	4	17	banach	banach	NOUN
ejpam-6559	4	18	space	space	NOUN
ejpam-6559	4	19	x	x	NOUN
ejpam-6559	4	20	such	such	ADJ
ejpam-6559	4	21	that	that	SCONJ
ejpam-6559	4	22	a	a	DET
ejpam-6559	4	23	:	:	PUNCT
ejpam-6559	4	24	dom(a	dom(a	PROPN
ejpam-6559	4	25	)	)	PUNCT
ejpam-6559	4	26	⊆	⊆	NUM
ejpam-6559	4	27	x	x	SYM
ejpam-6559	4	28	→	→	SYM
ejpam-6559	4	29	x	x	PROPN
ejpam-6559	4	30	,	,	PUNCT
ejpam-6559	4	31	b	b	NOUN
ejpam-6559	4	32	:	:	PUNCT
ejpam-6559	4	33	dom(b	dom(b	PROPN
ejpam-6559	4	34	)	)	PUNCT
ejpam-6559	4	35	⊆	⊆	NUM
ejpam-6559	4	36	x	x	SYM
ejpam-6559	4	37	→	→	SYM
ejpam-6559	4	38	x	x	SYM
ejpam-6559	4	39	,	,	PUNCT
ejpam-6559	4	40	u(s	u(s	PROPN
ejpam-6559	4	41	,	,	PUNCT
ejpam-6559	4	42	t	t	PROPN
ejpam-6559	4	43	)	)	PUNCT
ejpam-6559	4	44	:	:	PUNCT
ejpam-6559	5	1	[	[	X
ejpam-6559	5	2	0	0	NUM
ejpam-6559	5	3	,	,	PUNCT
ejpam-6559	5	4	1	1	NUM
ejpam-6559	5	5	]	]	SYM
ejpam-6559	5	6	×	×	NOUN
ejpam-6559	6	1	[	[	X
ejpam-6559	6	2	0	0	NUM
ejpam-6559	6	3	,	,	PUNCT
ejpam-6559	6	4	1	1	NUM
ejpam-6559	6	5	]	]	PUNCT
ejpam-6559	6	6	→	→	PUNCT
ejpam-6559	6	7	x	x	X
ejpam-6559	6	8	is	be	AUX
ejpam-6559	6	9	an	an	DET
ejpam-6559	6	10	unknown	unknown	ADJ
ejpam-6559	6	11	twice	twice	ADV
ejpam-6559	6	12	-	-	PUNCT
ejpam-6559	6	13	continuously	continuously	ADV
ejpam-6559	6	14	partially	partially	ADV
ejpam-6559	6	15	differentiable	differentiable	ADJ
ejpam-6559	6	16	function	function	NOUN
ejpam-6559	6	17	on	on	ADP
ejpam-6559	6	18	[	[	X
ejpam-6559	6	19	0	0	NUM
ejpam-6559	6	20	,	,	PUNCT
ejpam-6559	6	21	1]×	1]×	NUM
ejpam-6559	6	22	[	[	X
ejpam-6559	6	23	0	0	NUM
ejpam-6559	6	24	,	,	PUNCT
ejpam-6559	6	25	1	1	NUM
ejpam-6559	6	26	]	]	SYM
ejpam-6559	6	27	⊆	⊆	NUM
ejpam-6559	6	28	r2	r2	NOUN
ejpam-6559	6	29	where	where	SCONJ
ejpam-6559	6	30	rang(u	rang(u	ADJ
ejpam-6559	6	31	)	)	PUNCT
ejpam-6559	6	32	⊆	⊆	NUM
ejpam-6559	6	33	dom(a)∩dom(b	dom(a)∩dom(b	NOUN
ejpam-6559	6	34	)	)	PUNCT
ejpam-6559	6	35	.	.	PUNCT
ejpam-6559	7	1	utilizing	utilize	VERB
ejpam-6559	7	2	properties	property	NOUN
ejpam-6559	7	3	of	of	ADP
ejpam-6559	7	4	atomic	atomic	ADJ
ejpam-6559	7	5	operators	operator	NOUN
ejpam-6559	7	6	,	,	PUNCT
ejpam-6559	7	7	an	an	DET
ejpam-6559	7	8	atomic	atomic	ADJ
ejpam-6559	7	9	solution	solution	NOUN
ejpam-6559	7	10	is	be	AUX
ejpam-6559	7	11	obtained	obtain	VERB
ejpam-6559	7	12	.	.	PUNCT
ejpam-6559	8	1	2020	2020	NUM
ejpam-6559	8	2	mathematics	mathematic	NOUN
ejpam-6559	8	3	subject	subject	NOUN
ejpam-6559	8	4	classifications	classification	NOUN
ejpam-6559	8	5	:	:	PUNCT
ejpam-6559	8	6	34k30	34k30	NUM
ejpam-6559	8	7	,	,	PUNCT
ejpam-6559	8	8	46b10	46b10	NUM
ejpam-6559	8	9	key	key	ADJ
ejpam-6559	8	10	words	word	NOUN
ejpam-6559	8	11	and	and	CCONJ
ejpam-6559	8	12	phrases	phrase	NOUN
ejpam-6559	8	13	:	:	PUNCT
ejpam-6559	8	14	abstract	abstract	ADJ
ejpam-6559	8	15	cauchy	cauchy	ADJ
ejpam-6559	8	16	problem	problem	NOUN
ejpam-6559	8	17	in	in	ADP
ejpam-6559	8	18	two	two	NUM
ejpam-6559	8	19	variables	variable	NOUN
ejpam-6559	8	20	,	,	PUNCT
ejpam-6559	8	21	atoms	atom	NOUN
ejpam-6559	8	22	operators	operator	NOUN
ejpam-6559	8	23	,	,	PUNCT
ejpam-6559	8	24	dual	dual	ADJ
ejpam-6559	8	25	space	space	NOUN
ejpam-6559	8	26	1	1	NUM
ejpam-6559	8	27	.	.	PUNCT
ejpam-6559	9	1	introduction	introduction	NOUN
ejpam-6559	9	2	ordinary	ordinary	ADJ
ejpam-6559	9	3	and	and	CCONJ
ejpam-6559	9	4	partial	partial	ADJ
ejpam-6559	9	5	differential	differential	ADJ
ejpam-6559	9	6	equations	equation	NOUN
ejpam-6559	9	7	in	in	ADP
ejpam-6559	9	8	which	which	PRON
ejpam-6559	9	9	the	the	DET
ejpam-6559	9	10	unknown	unknown	ADJ
ejpam-6559	9	11	function	function	NOUN
ejpam-6559	9	12	and	and	CCONJ
ejpam-6559	9	13	its	its	PRON
ejpam-6559	9	14	derivatives	derivative	NOUN
ejpam-6559	9	15	take	take	VERB
ejpam-6559	9	16	values	value	NOUN
ejpam-6559	9	17	in	in	ADP
ejpam-6559	9	18	some	some	DET
ejpam-6559	9	19	abstract	abstract	ADJ
ejpam-6559	9	20	space	space	NOUN
ejpam-6559	9	21	such	such	ADJ
ejpam-6559	9	22	as	as	ADP
ejpam-6559	9	23	hilbert	hilbert	NOUN
ejpam-6559	9	24	space	space	NOUN
ejpam-6559	9	25	or	or	CCONJ
ejpam-6559	9	26	banach	banach	NOUN
ejpam-6559	9	27	space	space	NOUN
ejpam-6559	9	28	are	be	AUX
ejpam-6559	9	29	called	call	VERB
ejpam-6559	9	30	abstract	abstract	ADJ
ejpam-6559	9	31	differential	differential	ADJ
ejpam-6559	9	32	equations	equation	NOUN
ejpam-6559	9	33	[	[	X
ejpam-6559	9	34	1	1	NUM
ejpam-6559	9	35	,	,	PUNCT
ejpam-6559	9	36	2	2	NUM
ejpam-6559	9	37	]	]	PUNCT
ejpam-6559	9	38	.	.	PUNCT
ejpam-6559	10	1	among	among	ADP
ejpam-6559	10	2	these	these	DET
ejpam-6559	10	3	classical	classical	ADJ
ejpam-6559	10	4	vector	vector	NOUN
ejpam-6559	10	5	valued	value	VERB
ejpam-6559	10	6	abstract	abstract	ADJ
ejpam-6559	10	7	differential	differential	NOUN
ejpam-6559	10	8	equations	equation	NOUN
ejpam-6559	10	9	,	,	PUNCT
ejpam-6559	10	10	the	the	DET
ejpam-6559	10	11	abstract	abstract	ADJ
ejpam-6559	10	12	cauchy	cauchy	ADJ
ejpam-6559	10	13	problem	problem	NOUN
ejpam-6559	10	14	(	(	PUNCT
ejpam-6559	10	15	acp	acp	PROPN
ejpam-6559	10	16	)	)	PUNCT
ejpam-6559	10	17	is	be	AUX
ejpam-6559	10	18	considered	consider	VERB
ejpam-6559	10	19	as	as	ADP
ejpam-6559	10	20	the	the	DET
ejpam-6559	10	21	most	most	ADV
ejpam-6559	10	22	famous	famous	ADJ
ejpam-6559	10	23	one	one	NUM
ejpam-6559	10	24	.	.	PUNCT
ejpam-6559	11	1	the	the	DET
ejpam-6559	11	2	classical	classical	ADJ
ejpam-6559	11	3	form	form	NOUN
ejpam-6559	11	4	of	of	ADP
ejpam-6559	11	5	the	the	DET
ejpam-6559	11	6	acp	acp	PROPN
ejpam-6559	11	7	can	can	AUX
ejpam-6559	11	8	be	be	AUX
ejpam-6559	11	9	presented	present	VERB
ejpam-6559	11	10	as	as	SCONJ
ejpam-6559	11	11	follows	follow	VERB
ejpam-6559	11	12	:	:	PUNCT
ejpam-6559	11	13	du	du	PROPN
ejpam-6559	11	14	dt	dt	NOUN
ejpam-6559	11	15	=	=	SYM
ejpam-6559	11	16	lu(t	lu(t	PROPN
ejpam-6559	11	17	)	)	PUNCT
ejpam-6559	11	18	,	,	PUNCT
ejpam-6559	11	19	t	t	PROPN
ejpam-6559	11	20	≥	≥	NUM
ejpam-6559	11	21	0	0	NUM
ejpam-6559	11	22	,	,	PUNCT
ejpam-6559	11	23	u(0	u(0	NOUN
ejpam-6559	11	24	)	)	PUNCT
ejpam-6559	11	25	=	=	SYM
ejpam-6559	11	26	t	t	PROPN
ejpam-6559	11	27	◦	◦	NOUN
ejpam-6559	11	28	,	,	PUNCT
ejpam-6559	11	29	(	(	PUNCT
ejpam-6559	11	30	1	1	X
ejpam-6559	11	31	)	)	PUNCT
ejpam-6559	11	32	∗corresponding	∗corresponde	VERB
ejpam-6559	11	33	author	author	NOUN
ejpam-6559	11	34	.	.	PUNCT
ejpam-6559	12	1	doi	doi	NOUN
ejpam-6559	12	2	:	:	PUNCT
ejpam-6559	12	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6559	https://doi.org/10.29020/nybg.ejpam.v18i4.6559	PRON
ejpam-6559	12	4	email	email	NOUN
ejpam-6559	12	5	addresses	address	VERB
ejpam-6559	12	6	:	:	PUNCT
ejpam-6559	12	7	w.alshanti@zuj.edu.jo	w.alshanti@zuj.edu.jo	ADP
ejpam-6559	12	8	(	(	PUNCT
ejpam-6559	12	9	w.	w.	PROPN
ejpam-6559	12	10	g.	g.	PROPN
ejpam-6559	12	11	alshanti	alshanti	PROPN
ejpam-6559	12	12	)	)	PUNCT
ejpam-6559	12	13	,	,	PUNCT
ejpam-6559	12	14	m.abuhammad@zuj.edu.jo	m.abuhammad@zuj.edu.jo	NOUN
ejpam-6559	12	15	(	(	PUNCT
ejpam-6559	12	16	m.	m.	NOUN
ejpam-6559	12	17	abu	abu	PROPN
ejpam-6559	12	18	hammad	hammad	PROPN
ejpam-6559	12	19	)	)	PUNCT
ejpam-6559	12	20	,	,	PUNCT
ejpam-6559	12	21	roshdi@ju.edu.jo	roshdi@ju.edu.jo	PROPN
ejpam-6559	12	22	(	(	PUNCT
ejpam-6559	12	23	r.	r.	PROPN
ejpam-6559	12	24	khalil	khalil	PROPN
ejpam-6559	12	25	)	)	PUNCT
ejpam-6559	12	26	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6559	13	1	1	1	NUM
ejpam-6559	13	2	copyright	copyright	NOUN
ejpam-6559	13	3	:	:	PUNCT
ejpam-6559	13	4	©	©	PROPN
ejpam-6559	13	5	2025	2025	NUM
ejpam-6559	13	6	the	the	DET
ejpam-6559	13	7	author(s	author(s	NOUN
ejpam-6559	13	8	)	)	PUNCT
ejpam-6559	13	9	.	.	PUNCT
ejpam-6559	14	1	(	(	PUNCT
ejpam-6559	14	2	cc	cc	NOUN
ejpam-6559	14	3	by	by	ADP
ejpam-6559	14	4	-	-	PUNCT
ejpam-6559	14	5	nc	nc	PROPN
ejpam-6559	14	6	4.0	4.0	NUM
ejpam-6559	14	7	)	)	PUNCT
ejpam-6559	14	8	w.	w.	PROPN
ejpam-6559	14	9	g.	g.	PROPN
ejpam-6559	14	10	alshanti	alshanti	PROPN
ejpam-6559	14	11	,	,	PUNCT
ejpam-6559	14	12	m.	m.	PROPN
ejpam-6559	14	13	abu	abu	PROPN
ejpam-6559	14	14	hammad	hammad	PROPN
ejpam-6559	14	15	,	,	PUNCT
ejpam-6559	14	16	r.	r.	PROPN
ejpam-6559	14	17	khalil	khalil	PROPN
ejpam-6559	14	18	/	/	SYM
ejpam-6559	14	19	eur	eur	PROPN
ejpam-6559	14	20	.	.	PUNCT
ejpam-6559	15	1	j.	j.	PROPN
ejpam-6559	15	2	pure	pure	PROPN
ejpam-6559	15	3	appl	appl	PROPN
ejpam-6559	15	4	.	.	PROPN
ejpam-6559	15	5	math	math	PROPN
ejpam-6559	15	6	,	,	PUNCT
ejpam-6559	15	7	18	18	NUM
ejpam-6559	15	8	(	(	PUNCT
ejpam-6559	15	9	4	4	NUM
ejpam-6559	15	10	)	)	PUNCT
ejpam-6559	15	11	(	(	PUNCT
ejpam-6559	15	12	2025	2025	NUM
ejpam-6559	15	13	)	)	PUNCT
ejpam-6559	15	14	,	,	PUNCT
ejpam-6559	15	15	6559	6559	NUM
ejpam-6559	15	16	2	2	NUM
ejpam-6559	15	17	of	of	ADP
ejpam-6559	15	18	8	8	NUM
ejpam-6559	15	19	where	where	SCONJ
ejpam-6559	15	20	l	l	NOUN
ejpam-6559	15	21	:	:	PUNCT
ejpam-6559	15	22	dom(l	dom(l	NOUN
ejpam-6559	15	23	)	)	PUNCT
ejpam-6559	16	1	⊆	⊆	NUM
ejpam-6559	16	2	x	x	SYM
ejpam-6559	16	3	→	→	PUNCT
ejpam-6559	16	4	x	x	X
ejpam-6559	16	5	is	be	AUX
ejpam-6559	16	6	a	a	DET
ejpam-6559	16	7	linear	linear	ADJ
ejpam-6559	16	8	operator	operator	NOUN
ejpam-6559	16	9	of	of	ADP
ejpam-6559	16	10	an	an	DET
ejpam-6559	16	11	appropriate	appropriate	ADJ
ejpam-6559	16	12	type	type	NOUN
ejpam-6559	16	13	such	such	ADJ
ejpam-6559	16	14	that	that	DET
ejpam-6559	16	15	rang(u	rang(u	NOUN
ejpam-6559	16	16	)	)	PUNCT
ejpam-6559	16	17	⊆	⊆	NUM
ejpam-6559	16	18	dom(l	dom(l	NOUN
ejpam-6559	16	19	)	)	PUNCT
ejpam-6559	16	20	,	,	PUNCT
ejpam-6559	16	21	t	t	X
ejpam-6559	16	22	◦	◦	NOUN
ejpam-6559	16	23	∈	∈	PROPN
ejpam-6559	16	24	x	x	PUNCT
ejpam-6559	16	25	is	be	AUX
ejpam-6559	16	26	given	give	VERB
ejpam-6559	16	27	and	and	CCONJ
ejpam-6559	16	28	u	u	NOUN
ejpam-6559	16	29	:	:	PUNCT
ejpam-6559	16	30	[	[	X
ejpam-6559	16	31	0,∞	0,∞	NOUN
ejpam-6559	16	32	)	)	PUNCT
ejpam-6559	16	33	→	→	PUNCT
ejpam-6559	16	34	x	x	X
ejpam-6559	16	35	is	be	AUX
ejpam-6559	16	36	the	the	DET
ejpam-6559	16	37	unknown	unknown	ADJ
ejpam-6559	16	38	function	function	NOUN
ejpam-6559	16	39	.	.	PUNCT
ejpam-6559	17	1	for	for	ADP
ejpam-6559	17	2	both	both	CCONJ
ejpam-6559	17	3	linear	linear	ADJ
ejpam-6559	17	4	and	and	CCONJ
ejpam-6559	17	5	nonlinear	nonlinear	ADJ
ejpam-6559	17	6	acps	acp	NOUN
ejpam-6559	17	7	,	,	PUNCT
ejpam-6559	17	8	there	there	PRON
ejpam-6559	17	9	are	be	VERB
ejpam-6559	17	10	many	many	ADJ
ejpam-6559	17	11	applications	application	NOUN
ejpam-6559	17	12	in	in	ADP
ejpam-6559	17	13	engineering	engineering	NOUN
ejpam-6559	17	14	and	and	CCONJ
ejpam-6559	17	15	applied	apply	VERB
ejpam-6559	17	16	sciences	science	NOUN
ejpam-6559	18	1	[	[	X
ejpam-6559	18	2	3	3	NUM
ejpam-6559	18	3	,	,	PUNCT
ejpam-6559	18	4	4	4	NUM
ejpam-6559	18	5	]	]	PUNCT
ejpam-6559	18	6	.	.	PUNCT
ejpam-6559	19	1	in	in	ADP
ejpam-6559	19	2	spite	spite	NOUN
ejpam-6559	19	3	of	of	ADP
ejpam-6559	19	4	the	the	DET
ejpam-6559	19	5	fact	fact	NOUN
ejpam-6559	19	6	that	that	SCONJ
ejpam-6559	19	7	,	,	PUNCT
ejpam-6559	19	8	the	the	DET
ejpam-6559	19	9	acps	acp	NOUN
ejpam-6559	19	10	have	have	AUX
ejpam-6559	19	11	a	a	DET
ejpam-6559	19	12	standing	standing	NOUN
ejpam-6559	19	13	investigate	investigate	VERB
ejpam-6559	19	14	history	history	NOUN
ejpam-6559	19	15	that	that	PRON
ejpam-6559	19	16	has	have	VERB
ejpam-6559	19	17	profound	profound	ADJ
ejpam-6559	19	18	roots	root	NOUN
ejpam-6559	19	19	in	in	ADP
ejpam-6559	19	20	time	time	NOUN
ejpam-6559	19	21	,	,	PUNCT
ejpam-6559	19	22	a	a	DET
ejpam-6559	19	23	common	common	ADJ
ejpam-6559	19	24	hypothesis	hypothesis	NOUN
ejpam-6559	19	25	that	that	PRON
ejpam-6559	19	26	controls	control	VERB
ejpam-6559	19	27	the	the	DET
ejpam-6559	19	28	solutions	solution	NOUN
ejpam-6559	19	29	of	of	ADP
ejpam-6559	19	30	such	such	ADJ
ejpam-6559	19	31	problems	problem	NOUN
ejpam-6559	19	32	has	have	AUX
ejpam-6559	19	33	not	not	PART
ejpam-6559	19	34	,	,	PUNCT
ejpam-6559	19	35	however	however	ADV
ejpam-6559	19	36	,	,	PUNCT
ejpam-6559	19	37	been	be	AUX
ejpam-6559	19	38	confirmed	confirm	VERB
ejpam-6559	19	39	yet	yet	ADV
ejpam-6559	19	40	.	.	PUNCT
ejpam-6559	20	1	early	early	ADJ
ejpam-6559	20	2	studies	study	NOUN
ejpam-6559	20	3	,	,	PUNCT
ejpam-6559	20	4	principally	principally	ADV
ejpam-6559	20	5	,	,	PUNCT
ejpam-6559	20	6	were	be	AUX
ejpam-6559	20	7	conducted	conduct	VERB
ejpam-6559	20	8	by	by	ADP
ejpam-6559	20	9	means	mean	NOUN
ejpam-6559	20	10	of	of	ADP
ejpam-6559	20	11	both	both	CCONJ
ejpam-6559	20	12	laplace	laplace	NOUN
ejpam-6559	20	13	transform	transform	VERB
ejpam-6559	20	14	techniques	technique	NOUN
ejpam-6559	20	15	[	[	X
ejpam-6559	20	16	5	5	NUM
ejpam-6559	20	17	]	]	PUNCT
ejpam-6559	20	18	and	and	CCONJ
ejpam-6559	20	19	the	the	DET
ejpam-6559	20	20	concept	concept	NOUN
ejpam-6559	20	21	of	of	ADP
ejpam-6559	20	22	semigroup	semigroup	NOUN
ejpam-6559	20	23	of	of	ADP
ejpam-6559	20	24	linear	linear	PROPN
ejpam-6559	20	25	operators	operator	NOUN
ejpam-6559	21	1	[	[	X
ejpam-6559	21	2	6	6	NUM
ejpam-6559	21	3	]	]	PUNCT
ejpam-6559	21	4	,	,	PUNCT
ejpam-6559	21	5	[	[	X
ejpam-6559	21	6	7	7	NUM
ejpam-6559	21	7	]	]	PUNCT
ejpam-6559	21	8	.	.	PUNCT
ejpam-6559	22	1	lately	lately	ADV
ejpam-6559	22	2	,	,	PUNCT
ejpam-6559	22	3	in	in	ADP
ejpam-6559	22	4	2010	2010	NUM
ejpam-6559	22	5	,	,	PUNCT
ejpam-6559	22	6	modern	modern	ADJ
ejpam-6559	22	7	strategy	strategy	NOUN
ejpam-6559	22	8	has	have	AUX
ejpam-6559	22	9	been	be	AUX
ejpam-6559	22	10	suggested	suggest	VERB
ejpam-6559	22	11	[	[	X
ejpam-6559	22	12	8	8	NUM
ejpam-6559	22	13	]	]	PUNCT
ejpam-6559	22	14	,	,	PUNCT
ejpam-6559	22	15	[	[	X
ejpam-6559	22	16	9	9	NUM
ejpam-6559	22	17	]	]	PUNCT
ejpam-6559	22	18	,	,	PUNCT
ejpam-6559	22	19	[	[	X
ejpam-6559	22	20	10	10	NUM
ejpam-6559	22	21	]	]	PUNCT
ejpam-6559	22	22	,	,	PUNCT
ejpam-6559	22	23	[	[	X
ejpam-6559	22	24	11	11	NUM
ejpam-6559	22	25	]	]	PUNCT
ejpam-6559	22	26	to	to	PART
ejpam-6559	22	27	manipulate	manipulate	VERB
ejpam-6559	22	28	specific	specific	ADJ
ejpam-6559	22	29	types	type	NOUN
ejpam-6559	22	30	of	of	ADP
ejpam-6559	22	31	ordinary	ordinary	ADJ
ejpam-6559	22	32	and	and	CCONJ
ejpam-6559	22	33	fractional	fractional	ADJ
ejpam-6559	22	34	acps	acp	NOUN
ejpam-6559	22	35	.	.	PUNCT
ejpam-6559	23	1	the	the	DET
ejpam-6559	23	2	new	new	ADJ
ejpam-6559	23	3	approach	approach	NOUN
ejpam-6559	23	4	is	be	AUX
ejpam-6559	23	5	carried	carry	VERB
ejpam-6559	23	6	out	out	ADP
ejpam-6559	23	7	by	by	ADP
ejpam-6559	23	8	utilizing	utilize	VERB
ejpam-6559	23	9	the	the	DET
ejpam-6559	23	10	theory	theory	NOUN
ejpam-6559	23	11	of	of	ADP
ejpam-6559	23	12	atoms	atom	NOUN
ejpam-6559	23	13	operators	operator	NOUN
ejpam-6559	23	14	in	in	ADP
ejpam-6559	23	15	banach	banach	NOUN
ejpam-6559	23	16	spaces	space	NOUN
ejpam-6559	23	17	and	and	CCONJ
ejpam-6559	23	18	the	the	DET
ejpam-6559	23	19	obtained	obtain	VERB
ejpam-6559	23	20	solutions	solution	NOUN
ejpam-6559	23	21	in	in	ADP
ejpam-6559	23	22	such	such	ADJ
ejpam-6559	23	23	cases	case	NOUN
ejpam-6559	23	24	are	be	AUX
ejpam-6559	23	25	called	call	VERB
ejpam-6559	23	26	atomic	atomic	ADJ
ejpam-6559	23	27	solutions	solution	NOUN
ejpam-6559	23	28	.	.	PUNCT
ejpam-6559	24	1	let	let	VERB
ejpam-6559	24	2	x	x	PRON
ejpam-6559	24	3	is	be	AUX
ejpam-6559	24	4	a	a	DET
ejpam-6559	24	5	banach	banach	NOUN
ejpam-6559	24	6	space	space	NOUN
ejpam-6559	24	7	and	and	CCONJ
ejpam-6559	24	8	a	a	DET
ejpam-6559	24	9	,	,	PUNCT
ejpam-6559	24	10	b	b	NOUN
ejpam-6559	24	11	are	be	AUX
ejpam-6559	24	12	closed	close	VERB
ejpam-6559	24	13	linear	linear	ADJ
ejpam-6559	24	14	operators	operator	NOUN
ejpam-6559	24	15	on	on	ADP
ejpam-6559	24	16	x	x	SYM
ejpam-6559	24	17	such	such	ADJ
ejpam-6559	24	18	that	that	SCONJ
ejpam-6559	24	19	a	a	DET
ejpam-6559	24	20	:	:	PUNCT
ejpam-6559	24	21	dom(a	dom(a	PROPN
ejpam-6559	24	22	)	)	PUNCT
ejpam-6559	24	23	⊆	⊆	NUM
ejpam-6559	24	24	x	x	SYM
ejpam-6559	24	25	→	→	SYM
ejpam-6559	24	26	x	x	PROPN
ejpam-6559	24	27	,	,	PUNCT
ejpam-6559	24	28	b	b	NOUN
ejpam-6559	24	29	:	:	PUNCT
ejpam-6559	24	30	dom(b	dom(b	PROPN
ejpam-6559	24	31	)	)	PUNCT
ejpam-6559	24	32	⊆	⊆	NUM
ejpam-6559	24	33	x	x	SYM
ejpam-6559	24	34	→	→	SYM
ejpam-6559	24	35	x.	x.	NOUN
ejpam-6559	24	36	moreover	moreover	ADV
ejpam-6559	24	37	,	,	PUNCT
ejpam-6559	24	38	assume	assume	VERB
ejpam-6559	24	39	u(s	u(s	ADJ
ejpam-6559	24	40	,	,	PUNCT
ejpam-6559	24	41	t	t	PROPN
ejpam-6559	24	42	)	)	PUNCT
ejpam-6559	24	43	:	:	PUNCT
ejpam-6559	25	1	[	[	X
ejpam-6559	25	2	0	0	NUM
ejpam-6559	25	3	,	,	PUNCT
ejpam-6559	25	4	1]×	1]×	NUM
ejpam-6559	26	1	[	[	X
ejpam-6559	26	2	0	0	NUM
ejpam-6559	26	3	,	,	PUNCT
ejpam-6559	26	4	1	1	NUM
ejpam-6559	26	5	]	]	PUNCT
ejpam-6559	26	6	→	→	PUNCT
ejpam-6559	26	7	x	x	X
ejpam-6559	26	8	is	be	AUX
ejpam-6559	26	9	an	an	DET
ejpam-6559	26	10	unknown	unknown	ADJ
ejpam-6559	26	11	twice	twice	ADV
ejpam-6559	26	12	-	-	PUNCT
ejpam-6559	26	13	continuously	continuously	ADV
ejpam-6559	26	14	partially	partially	ADV
ejpam-6559	26	15	differentiable	differentiable	ADJ
ejpam-6559	26	16	function	function	NOUN
ejpam-6559	26	17	on	on	ADP
ejpam-6559	26	18	[	[	X
ejpam-6559	26	19	0	0	NUM
ejpam-6559	26	20	,	,	PUNCT
ejpam-6559	26	21	1	1	NUM
ejpam-6559	26	22	]	]	SYM
ejpam-6559	26	23	×	×	NOUN
ejpam-6559	27	1	[	[	X
ejpam-6559	27	2	0	0	NUM
ejpam-6559	27	3	,	,	PUNCT
ejpam-6559	27	4	1	1	NUM
ejpam-6559	27	5	]	]	SYM
ejpam-6559	27	6	⊆	⊆	NUM
ejpam-6559	27	7	r2	r2	NOUN
ejpam-6559	27	8	where	where	SCONJ
ejpam-6559	27	9	rang(u	rang(u	ADJ
ejpam-6559	27	10	)	)	PUNCT
ejpam-6559	27	11	⊆	⊆	NUM
ejpam-6559	27	12	dom(a	dom(a	PROPN
ejpam-6559	27	13	)	)	PUNCT
ejpam-6559	27	14	∩dom(b	∩dom(b	NOUN
ejpam-6559	27	15	)	)	PUNCT
ejpam-6559	27	16	.	.	PUNCT
ejpam-6559	28	1	our	our	PRON
ejpam-6559	28	2	main	main	ADJ
ejpam-6559	28	3	result	result	NOUN
ejpam-6559	28	4	discusses	discuss	VERB
ejpam-6559	28	5	the	the	DET
ejpam-6559	28	6	atomic	atomic	ADJ
ejpam-6559	28	7	solution	solution	NOUN
ejpam-6559	28	8	of	of	ADP
ejpam-6559	28	9	the	the	DET
ejpam-6559	28	10	following	follow	VERB
ejpam-6559	28	11	second	second	ADJ
ejpam-6559	28	12	-	-	PUNCT
ejpam-6559	28	13	order	order	NOUN
ejpam-6559	28	14	acp	acp	PROPN
ejpam-6559	28	15	in	in	ADP
ejpam-6559	28	16	two	two	NUM
ejpam-6559	28	17	variables	variable	NOUN
ejpam-6559	28	18	,	,	PUNCT
ejpam-6559	28	19	d2	d2	PROPN
ejpam-6559	28	20	ssu(s	ssu(s	PROPN
ejpam-6559	28	21	,	,	PUNCT
ejpam-6559	28	22	t	t	PROPN
ejpam-6559	28	23	)	)	PUNCT
ejpam-6559	28	24	+	+	NOUN
ejpam-6559	28	25	d2	d2	PROPN
ejpam-6559	28	26	ttu(s	ttu(s	PROPN
ejpam-6559	28	27	,	,	PUNCT
ejpam-6559	28	28	t	t	PROPN
ejpam-6559	28	29	)	)	PUNCT
ejpam-6559	28	30	+	+	CCONJ
ejpam-6559	28	31	2d2	2d2	NUM
ejpam-6559	28	32	stu(s	stu(s	NOUN
ejpam-6559	28	33	,	,	PUNCT
ejpam-6559	28	34	t	t	PROPN
ejpam-6559	28	35	)	)	PUNCT
ejpam-6559	29	1	+	+	ADP
ejpam-6559	29	2	a	a	DET
ejpam-6559	29	3	[	[	X
ejpam-6559	29	4	dsu(s	dsu(s	PROPN
ejpam-6559	29	5	,	,	PUNCT
ejpam-6559	29	6	t	t	PROPN
ejpam-6559	29	7	)	)	PUNCT
ejpam-6559	29	8	+	+	NOUN
ejpam-6559	29	9	dtu(s	dtu(s	PROPN
ejpam-6559	29	10	,	,	PUNCT
ejpam-6559	29	11	t	t	PROPN
ejpam-6559	29	12	)	)	PUNCT
ejpam-6559	29	13	]	]	PUNCT
ejpam-6559	30	1	=	=	PUNCT
ejpam-6559	30	2	bu(s	bu(s	PROPN
ejpam-6559	30	3	,	,	PUNCT
ejpam-6559	30	4	t	t	PROPN
ejpam-6559	30	5	)	)	PUNCT
ejpam-6559	30	6	.	.	PUNCT
ejpam-6559	31	1	before	before	SCONJ
ejpam-6559	31	2	we	we	PRON
ejpam-6559	31	3	present	present	VERB
ejpam-6559	31	4	the	the	DET
ejpam-6559	31	5	main	main	ADJ
ejpam-6559	31	6	result	result	NOUN
ejpam-6559	31	7	,	,	PUNCT
ejpam-6559	31	8	some	some	DET
ejpam-6559	31	9	definitions	definition	NOUN
ejpam-6559	31	10	and	and	CCONJ
ejpam-6559	31	11	results	result	NOUN
ejpam-6559	31	12	from	from	ADP
ejpam-6559	31	13	atoms	atom	NOUN
ejpam-6559	31	14	operators	operator	NOUN
ejpam-6559	31	15	concept	concept	NOUN
ejpam-6559	31	16	are	be	AUX
ejpam-6559	31	17	presented	present	VERB
ejpam-6559	31	18	.	.	PUNCT
ejpam-6559	32	1	2	2	X
ejpam-6559	32	2	.	.	X
ejpam-6559	32	3	preliminaries	preliminary	NOUN
ejpam-6559	32	4	definition	definition	NOUN
ejpam-6559	32	5	1	1	NUM
ejpam-6559	32	6	.	.	PUNCT
ejpam-6559	33	1	[	[	X
ejpam-6559	33	2	12	12	NUM
ejpam-6559	33	3	]	]	X
ejpam-6559	33	4	let	let	VERB
ejpam-6559	33	5	x	x	PRON
ejpam-6559	33	6	and	and	CCONJ
ejpam-6559	33	7	y	y	PROPN
ejpam-6559	33	8	be	be	AUX
ejpam-6559	33	9	any	any	DET
ejpam-6559	33	10	two	two	NUM
ejpam-6559	33	11	banach	banach	NOUN
ejpam-6559	33	12	spaces	space	NOUN
ejpam-6559	33	13	and	and	CCONJ
ejpam-6559	33	14	x∗	x∗	PROPN
ejpam-6559	33	15	is	be	AUX
ejpam-6559	33	16	the	the	DET
ejpam-6559	33	17	dual	dual	ADJ
ejpam-6559	33	18	space	space	NOUN
ejpam-6559	33	19	of	of	ADP
ejpam-6559	33	20	x.	x.	NOUN
ejpam-6559	33	21	for	for	ADP
ejpam-6559	33	22	ζ	ζ	PROPN
ejpam-6559	33	23	∈	∈	PROPN
ejpam-6559	33	24	x	x	X
ejpam-6559	33	25	and	and	CCONJ
ejpam-6559	33	26	η	η	PROPN
ejpam-6559	33	27	∈	∈	PROPN
ejpam-6559	33	28	y	y	PROPN
ejpam-6559	33	29	,	,	PUNCT
ejpam-6559	33	30	the	the	DET
ejpam-6559	33	31	operator	operator	NOUN
ejpam-6559	33	32	ζ	ζ	PROPN
ejpam-6559	33	33	⊗	⊗	PROPN
ejpam-6559	33	34	η	η	PROPN
ejpam-6559	33	35	:	:	PUNCT
ejpam-6559	33	36	x∗	x∗	PROPN
ejpam-6559	33	37	→	→	SYM
ejpam-6559	33	38	y	y	PROPN
ejpam-6559	33	39	,	,	PUNCT
ejpam-6559	33	40	defined	define	VERB
ejpam-6559	33	41	by	by	ADP
ejpam-6559	33	42	ζ	ζ	PROPN
ejpam-6559	33	43	⊗	⊗	PROPN
ejpam-6559	33	44	η(z∗	η(z∗	X
ejpam-6559	33	45	)	)	PUNCT
ejpam-6559	33	46	=	=	SYM
ejpam-6559	33	47	z∗(ζ)η	z∗(ζ)η	NOUN
ejpam-6559	33	48	=	=	SYM
ejpam-6559	33	49	⟨ζ	⟨ζ	X
ejpam-6559	33	50	,	,	PUNCT
ejpam-6559	33	51	z∗⟩	z∗⟩	PROPN
ejpam-6559	33	52	η	η	PROPN
ejpam-6559	33	53	,	,	PUNCT
ejpam-6559	33	54	is	be	AUX
ejpam-6559	33	55	a	a	DET
ejpam-6559	33	56	bounded	bounded	ADJ
ejpam-6559	33	57	one	one	NUM
ejpam-6559	33	58	rank	rank	NOUN
ejpam-6559	33	59	linear	linear	NOUN
ejpam-6559	33	60	operator	operator	NOUN
ejpam-6559	33	61	.	.	PUNCT
ejpam-6559	34	1	such	such	ADJ
ejpam-6559	34	2	operators	operator	NOUN
ejpam-6559	34	3	are	be	AUX
ejpam-6559	34	4	called	call	VERB
ejpam-6559	34	5	atoms	atom	NOUN
ejpam-6559	34	6	.	.	PUNCT
ejpam-6559	35	1	now	now	ADV
ejpam-6559	35	2	,	,	PUNCT
ejpam-6559	35	3	if	if	SCONJ
ejpam-6559	35	4	we	we	PRON
ejpam-6559	35	5	assume	assume	VERB
ejpam-6559	35	6	two	two	NUM
ejpam-6559	35	7	equal	equal	ADJ
ejpam-6559	35	8	atoms	atom	NOUN
ejpam-6559	35	9	,	,	PUNCT
ejpam-6559	35	10	namely	namely	ADV
ejpam-6559	35	11	,	,	PUNCT
ejpam-6559	35	12	ζ1	ζ1	PROPN
ejpam-6559	35	13	⊗	⊗	NOUN
ejpam-6559	35	14	η1	η1	NOUN
ejpam-6559	35	15	=	=	SYM
ejpam-6559	35	16	ζ2	ζ2	NOUN
ejpam-6559	35	17	⊗	⊗	PROPN
ejpam-6559	35	18	η2	η2	PROPN
ejpam-6559	35	19	,	,	PUNCT
ejpam-6559	35	20	then	then	ADV
ejpam-6559	35	21	for	for	ADP
ejpam-6559	35	22	any	any	DET
ejpam-6559	35	23	z∗	z∗	NOUN
ejpam-6559	35	24	∈	∈	PROPN
ejpam-6559	35	25	x∗	x∗	NOUN
ejpam-6559	35	26	we	we	PRON
ejpam-6559	35	27	have	have	VERB
ejpam-6559	35	28	⟨ζ1	⟨ζ1	PROPN
ejpam-6559	35	29	,	,	PUNCT
ejpam-6559	35	30	z∗⟩	z∗⟩	NOUN
ejpam-6559	35	31	η1	η1	NOUN
ejpam-6559	35	32	=	=	SYM
ejpam-6559	35	33	⟨ζ2	⟨ζ2	PROPN
ejpam-6559	35	34	,	,	PUNCT
ejpam-6559	35	35	z∗⟩	z∗⟩	NOUN
ejpam-6559	35	36	η2	η2	NOUN
ejpam-6559	35	37	.	.	PUNCT
ejpam-6559	36	1	thus	thus	ADV
ejpam-6559	36	2	,	,	PUNCT
ejpam-6559	36	3	η1	η1	NOUN
ejpam-6559	36	4	=	=	SYM
ejpam-6559	36	5	η2	η2	PROPN
ejpam-6559	36	6	.	.	PUNCT
ejpam-6559	37	1	similarly	similarly	ADV
ejpam-6559	37	2	,	,	PUNCT
ejpam-6559	37	3	one	one	PRON
ejpam-6559	37	4	can	can	AUX
ejpam-6559	37	5	prove	prove	VERB
ejpam-6559	37	6	ζ1	ζ1	NOUN
ejpam-6559	37	7	=	=	NOUN
ejpam-6559	37	8	ζ2	ζ2	NOUN
ejpam-6559	37	9	.	.	PUNCT
ejpam-6559	38	1	this	this	PRON
ejpam-6559	38	2	leads	lead	VERB
ejpam-6559	38	3	us	we	PRON
ejpam-6559	38	4	to	to	ADP
ejpam-6559	38	5	the	the	DET
ejpam-6559	38	6	following	follow	VERB
ejpam-6559	38	7	interesting	interesting	ADJ
ejpam-6559	38	8	lemma	lemma	PROPN
ejpam-6559	38	9	.	.	PUNCT
ejpam-6559	39	1	lemma	lemma	PROPN
ejpam-6559	39	2	1	1	NUM
ejpam-6559	39	3	.	.	PUNCT
ejpam-6559	40	1	[	[	X
ejpam-6559	40	2	13	13	NUM
ejpam-6559	40	3	]	]	PUNCT
ejpam-6559	40	4	let	let	VERB
ejpam-6559	40	5	ζ1⊗η1	ζ1⊗η1	NOUN
ejpam-6559	40	6	and	and	CCONJ
ejpam-6559	40	7	ζ2⊗η2	ζ2⊗η2	NOUN
ejpam-6559	40	8	be	be	AUX
ejpam-6559	40	9	two	two	NUM
ejpam-6559	40	10	nonzero	nonzero	ADJ
ejpam-6559	40	11	atoms	atom	NOUN
ejpam-6559	40	12	in	in	ADP
ejpam-6559	40	13	x⊗y	x⊗y	PROPN
ejpam-6559	40	14	such	such	ADJ
ejpam-6559	40	15	that	that	SCONJ
ejpam-6559	40	16	ζ1⊗η1	ζ1⊗η1	ADJ
ejpam-6559	40	17	+	+	CCONJ
ejpam-6559	40	18	ζ2	ζ2	NOUN
ejpam-6559	40	19	⊗	⊗	NOUN
ejpam-6559	40	20	η2	η2	VERB
ejpam-6559	40	21	=	=	SYM
ejpam-6559	40	22	ζ3	ζ3	PROPN
ejpam-6559	40	23	⊗	⊗	PROPN
ejpam-6559	40	24	η3	η3	PROPN
ejpam-6559	40	25	.	.	PUNCT
ejpam-6559	41	1	then	then	ADV
ejpam-6559	41	2	either	either	CCONJ
ejpam-6559	41	3	ζ1	ζ1	NOUN
ejpam-6559	41	4	=	=	PUNCT
ejpam-6559	41	5	ζ2	ζ2	NOUN
ejpam-6559	41	6	=	=	SYM
ejpam-6559	41	7	ζ3	ζ3	NOUN
ejpam-6559	41	8	or	or	CCONJ
ejpam-6559	41	9	η1	η1	NOUN
ejpam-6559	41	10	=	=	SYM
ejpam-6559	41	11	η2	η2	PROPN
ejpam-6559	41	12	=	=	SYM
ejpam-6559	41	13	η3	η3	NOUN
ejpam-6559	41	14	.	.	PUNCT
ejpam-6559	42	1	finally	finally	ADV
ejpam-6559	42	2	,	,	PUNCT
ejpam-6559	42	3	one	one	NUM
ejpam-6559	42	4	of	of	ADP
ejpam-6559	42	5	the	the	DET
ejpam-6559	42	6	most	most	ADV
ejpam-6559	42	7	interesting	interesting	ADJ
ejpam-6559	42	8	theorem	theorem	NOUN
ejpam-6559	42	9	that	that	PRON
ejpam-6559	42	10	lies	lie	VERB
ejpam-6559	42	11	in	in	ADP
ejpam-6559	42	12	the	the	DET
ejpam-6559	42	13	heart	heart	NOUN
ejpam-6559	42	14	of	of	ADP
ejpam-6559	42	15	functional	functional	ADJ
ejpam-6559	42	16	analysis	analysis	NOUN
ejpam-6559	42	17	as	as	ADV
ejpam-6559	42	18	well	well	ADV
ejpam-6559	42	19	as	as	ADP
ejpam-6559	42	20	approximation	approximation	NOUN
ejpam-6559	42	21	theory	theory	NOUN
ejpam-6559	42	22	and	and	CCONJ
ejpam-6559	42	23	guarantees	guarantee	NOUN
ejpam-6559	42	24	that	that	SCONJ
ejpam-6559	42	25	any	any	DET
ejpam-6559	42	26	continuous	continuous	ADJ
ejpam-6559	42	27	function	function	NOUN
ejpam-6559	42	28	of	of	ADP
ejpam-6559	42	29	several	several	ADJ
ejpam-6559	42	30	variables	variable	NOUN
ejpam-6559	42	31	can	can	AUX
ejpam-6559	42	32	be	be	AUX
ejpam-6559	42	33	written	write	VERB
ejpam-6559	42	34	as	as	ADP
ejpam-6559	42	35	a	a	DET
ejpam-6559	42	36	sum	sum	NOUN
ejpam-6559	42	37	of	of	ADP
ejpam-6559	42	38	products	product	NOUN
ejpam-6559	42	39	of	of	ADP
ejpam-6559	42	40	continuous	continuous	ADJ
ejpam-6559	42	41	separated	separate	VERB
ejpam-6559	42	42	functions	function	NOUN
ejpam-6559	42	43	.	.	PUNCT
ejpam-6559	42	44	theorem	theorem	NOUN
ejpam-6559	42	45	1	1	NUM
ejpam-6559	42	46	.	.	PUNCT
ejpam-6559	43	1	[	[	X
ejpam-6559	43	2	12	12	NUM
ejpam-6559	43	3	]	]	PUNCT
ejpam-6559	43	4	let	let	VERB
ejpam-6559	43	5	i	i	PRON
ejpam-6559	43	6	,	,	PUNCT
ejpam-6559	43	7	j	j	PROPN
ejpam-6559	43	8	be	be	VERB
ejpam-6559	43	9	two	two	NUM
ejpam-6559	43	10	compact	compact	ADJ
ejpam-6559	43	11	intervals	interval	NOUN
ejpam-6559	43	12	,	,	PUNCT
ejpam-6559	43	13	and	and	CCONJ
ejpam-6559	43	14	c	c	X
ejpam-6559	43	15	(	(	PUNCT
ejpam-6559	43	16	i	i	NOUN
ejpam-6559	43	17	)	)	PUNCT
ejpam-6559	43	18	,	,	PUNCT
ejpam-6559	43	19	c	c	PROPN
ejpam-6559	43	20	(	(	PUNCT
ejpam-6559	43	21	j	j	NOUN
ejpam-6559	43	22	)	)	PUNCT
ejpam-6559	43	23	,	,	PUNCT
ejpam-6559	43	24	and	and	CCONJ
ejpam-6559	43	25	c	c	X
ejpam-6559	44	1	(	(	PUNCT
ejpam-6559	44	2	i	i	PRON
ejpam-6559	44	3	×	×	PROPN
ejpam-6559	44	4	j	j	NOUN
ejpam-6559	44	5	)	)	PUNCT
ejpam-6559	44	6	be	be	AUX
ejpam-6559	44	7	,	,	PUNCT
ejpam-6559	44	8	respectively	respectively	ADV
ejpam-6559	44	9	,	,	PUNCT
ejpam-6559	44	10	the	the	DET
ejpam-6559	44	11	spaces	space	NOUN
ejpam-6559	44	12	of	of	ADP
ejpam-6559	44	13	continuous	continuous	ADJ
ejpam-6559	44	14	functions	function	NOUN
ejpam-6559	44	15	on	on	ADP
ejpam-6559	44	16	i	i	PROPN
ejpam-6559	44	17	,	,	PUNCT
ejpam-6559	44	18	j	j	PROPN
ejpam-6559	44	19	,	,	PUNCT
ejpam-6559	44	20	and	and	CCONJ
ejpam-6559	44	21	i	i	PRON
ejpam-6559	45	1	×	×	PROPN
ejpam-6559	45	2	j	j	PROPN
ejpam-6559	45	3	.	.	PUNCT
ejpam-6559	46	1	then	then	ADV
ejpam-6559	46	2	every	every	DET
ejpam-6559	46	3	f	f	PROPN
ejpam-6559	46	4	∈	∈	PROPN
ejpam-6559	46	5	w.	w.	PROPN
ejpam-6559	46	6	g.	g.	PROPN
ejpam-6559	46	7	alshanti	alshanti	PROPN
ejpam-6559	46	8	,	,	PUNCT
ejpam-6559	46	9	m.	m.	PROPN
ejpam-6559	46	10	abu	abu	PROPN
ejpam-6559	46	11	hammad	hammad	PROPN
ejpam-6559	46	12	,	,	PUNCT
ejpam-6559	46	13	r.	r.	PROPN
ejpam-6559	46	14	khalil	khalil	PROPN
ejpam-6559	46	15	/	/	SYM
ejpam-6559	46	16	eur	eur	PROPN
ejpam-6559	46	17	.	.	PUNCT
ejpam-6559	47	1	j.	j.	PROPN
ejpam-6559	47	2	pure	pure	PROPN
ejpam-6559	47	3	appl	appl	PROPN
ejpam-6559	47	4	.	.	PROPN
ejpam-6559	47	5	math	math	PROPN
ejpam-6559	47	6	,	,	PUNCT
ejpam-6559	47	7	18	18	NUM
ejpam-6559	47	8	(	(	PUNCT
ejpam-6559	47	9	4	4	NUM
ejpam-6559	47	10	)	)	PUNCT
ejpam-6559	47	11	(	(	PUNCT
ejpam-6559	47	12	2025	2025	NUM
ejpam-6559	47	13	)	)	PUNCT
ejpam-6559	47	14	,	,	PUNCT
ejpam-6559	47	15	6559	6559	NUM
ejpam-6559	47	16	3	3	NUM
ejpam-6559	47	17	of	of	ADP
ejpam-6559	47	18	8	8	NUM
ejpam-6559	47	19	c	c	NOUN
ejpam-6559	47	20	(	(	PUNCT
ejpam-6559	47	21	i	i	PRON
ejpam-6559	47	22	×	×	PROPN
ejpam-6559	47	23	j	j	NOUN
ejpam-6559	47	24	)	)	PUNCT
ejpam-6559	47	25	can	can	AUX
ejpam-6559	47	26	be	be	AUX
ejpam-6559	47	27	written	write	VERB
ejpam-6559	47	28	in	in	ADP
ejpam-6559	47	29	the	the	DET
ejpam-6559	47	30	form	form	NOUN
ejpam-6559	47	31	f(x	f(x	PROPN
ejpam-6559	47	32	,	,	PUNCT
ejpam-6559	47	33	y	y	NOUN
ejpam-6559	47	34	)	)	PUNCT
ejpam-6559	47	35	=	=	PUNCT
ejpam-6559	48	1	∞∑	∞∑	NUM
ejpam-6559	48	2	i=1	i=1	X
ejpam-6559	48	3	ui	ui	PROPN
ejpam-6559	49	1	(	(	PUNCT
ejpam-6559	49	2	x	x	X
ejpam-6559	49	3	)	)	PUNCT
ejpam-6559	49	4	vi	vi	PROPN
ejpam-6559	49	5	(	(	PUNCT
ejpam-6559	49	6	y	y	NOUN
ejpam-6559	49	7	)	)	PUNCT
ejpam-6559	49	8	,	,	PUNCT
ejpam-6559	49	9	where	where	SCONJ
ejpam-6559	49	10	ui	ui	PROPN
ejpam-6559	49	11	(	(	PUNCT
ejpam-6559	49	12	x	x	X
ejpam-6559	49	13	)	)	PUNCT
ejpam-6559	49	14	∈	∈	PROPN
ejpam-6559	49	15	c	c	NOUN
ejpam-6559	49	16	(	(	PUNCT
ejpam-6559	49	17	i	i	NOUN
ejpam-6559	49	18	)	)	PUNCT
ejpam-6559	49	19	and	and	CCONJ
ejpam-6559	49	20	vi	vi	PROPN
ejpam-6559	49	21	(	(	PUNCT
ejpam-6559	49	22	y	y	NOUN
ejpam-6559	49	23	)	)	PUNCT
ejpam-6559	49	24	∈	∈	PROPN
ejpam-6559	50	1	c	c	PROPN
ejpam-6559	50	2	(	(	PUNCT
ejpam-6559	50	3	j	j	PROPN
ejpam-6559	50	4	)	)	PUNCT
ejpam-6559	50	5	.	.	PUNCT
ejpam-6559	51	1	hahn	hahn	VERB
ejpam-6559	51	2	-	-	PUNCT
ejpam-6559	51	3	banach	banach	NOUN
ejpam-6559	51	4	theorem	theorem	NOUN
ejpam-6559	51	5	is	be	AUX
ejpam-6559	51	6	known	know	VERB
ejpam-6559	51	7	as	as	ADP
ejpam-6559	51	8	one	one	NUM
ejpam-6559	51	9	of	of	ADP
ejpam-6559	51	10	the	the	DET
ejpam-6559	51	11	most	most	ADV
ejpam-6559	51	12	important	important	ADJ
ejpam-6559	51	13	results	result	NOUN
ejpam-6559	51	14	in	in	ADP
ejpam-6559	51	15	functional	functional	ADJ
ejpam-6559	51	16	analysis	analysis	NOUN
ejpam-6559	51	17	.	.	PUNCT
ejpam-6559	52	1	this	this	DET
ejpam-6559	52	2	theorem	theorem	NOUN
ejpam-6559	52	3	enables	enable	VERB
ejpam-6559	52	4	us	we	PRON
ejpam-6559	52	5	to	to	PART
ejpam-6559	52	6	extend	extend	VERB
ejpam-6559	52	7	bounded	bound	VERB
ejpam-6559	52	8	linear	linear	PROPN
ejpam-6559	52	9	functionals	functional	NOUN
ejpam-6559	52	10	that	that	PRON
ejpam-6559	52	11	is	be	AUX
ejpam-6559	52	12	defined	define	VERB
ejpam-6559	52	13	on	on	ADP
ejpam-6559	52	14	a	a	DET
ejpam-6559	52	15	subspace	subspace	NOUN
ejpam-6559	52	16	of	of	ADP
ejpam-6559	52	17	a	a	DET
ejpam-6559	52	18	vector	vector	NOUN
ejpam-6559	52	19	space	space	NOUN
ejpam-6559	52	20	to	to	ADP
ejpam-6559	52	21	the	the	DET
ejpam-6559	52	22	whole	whole	ADJ
ejpam-6559	52	23	space	space	NOUN
ejpam-6559	52	24	.	.	PUNCT
ejpam-6559	53	1	consequently	consequently	ADV
ejpam-6559	53	2	,	,	PUNCT
ejpam-6559	53	3	the	the	DET
ejpam-6559	53	4	dual	dual	ADJ
ejpam-6559	53	5	space	space	NOUN
ejpam-6559	53	6	x∗	x∗	PROPN
ejpam-6559	53	7	is	be	AUX
ejpam-6559	53	8	a	a	DET
ejpam-6559	53	9	non	non	ADJ
ejpam-6559	53	10	-	-	ADJ
ejpam-6559	53	11	trivial	trivial	ADJ
ejpam-6559	53	12	space	space	NOUN
ejpam-6559	53	13	associated	associate	VERB
ejpam-6559	53	14	with	with	ADP
ejpam-6559	53	15	x.	x.	NOUN
ejpam-6559	53	16	theorem	theorem	NOUN
ejpam-6559	53	17	2	2	NUM
ejpam-6559	53	18	.	.	PUNCT
ejpam-6559	54	1	[	[	X
ejpam-6559	54	2	14	14	NUM
ejpam-6559	54	3	]	]	X
ejpam-6559	54	4	(	(	PUNCT
ejpam-6559	54	5	hahn	hahn	NOUN
ejpam-6559	54	6	-	-	PUNCT
ejpam-6559	54	7	banach	banach	NOUN
ejpam-6559	54	8	theorem	theorem	VERB
ejpam-6559	54	9	)	)	PUNCT
ejpam-6559	54	10	let	let	VERB
ejpam-6559	54	11	m	m	PRON
ejpam-6559	54	12	be	be	AUX
ejpam-6559	54	13	a	a	DET
ejpam-6559	54	14	linear	linear	ADJ
ejpam-6559	54	15	subspace	subspace	NOUN
ejpam-6559	54	16	of	of	ADP
ejpam-6559	54	17	a	a	DET
ejpam-6559	54	18	normed	normed	ADJ
ejpam-6559	54	19	linear	linear	ADJ
ejpam-6559	54	20	space	space	NOUN
ejpam-6559	54	21	n	n	NOUN
ejpam-6559	54	22	and	and	CCONJ
ejpam-6559	54	23	let	let	VERB
ejpam-6559	54	24	f	f	PROPN
ejpam-6559	54	25	∈	∈	PROPN
ejpam-6559	54	26	m∗.	m∗.	PROPN
ejpam-6559	54	27	then	then	ADV
ejpam-6559	54	28	f	f	PROPN
ejpam-6559	54	29	can	can	AUX
ejpam-6559	54	30	be	be	AUX
ejpam-6559	54	31	extended	extend	VERB
ejpam-6559	54	32	to	to	ADP
ejpam-6559	54	33	a	a	DET
ejpam-6559	54	34	bounded	bounded	ADJ
ejpam-6559	54	35	linear	linear	ADJ
ejpam-6559	54	36	functional	functional	PROPN
ejpam-6559	54	37	f	f	PROPN
ejpam-6559	54	38	∈	∈	PROPN
ejpam-6559	54	39	n∗	n∗	PROPN
ejpam-6559	54	40	(	(	PUNCT
ejpam-6559	54	41	defined	define	VERB
ejpam-6559	54	42	on	on	ADP
ejpam-6559	54	43	the	the	DET
ejpam-6559	54	44	whole	whole	ADJ
ejpam-6559	54	45	linear	linear	ADJ
ejpam-6559	54	46	space	space	NOUN
ejpam-6559	54	47	n	n	CCONJ
ejpam-6559	54	48	)	)	PUNCT
ejpam-6559	54	49	such	such	ADJ
ejpam-6559	54	50	that	that	SCONJ
ejpam-6559	54	51	f	f	PROPN
ejpam-6559	54	52	|m	|m	VERB
ejpam-6559	54	53	=	=	SYM
ejpam-6559	54	54	f	f	PROPN
ejpam-6559	54	55	and	and	CCONJ
ejpam-6559	54	56	∥f∥m∗	∥f∥m∗	PROPN
ejpam-6559	54	57	=	=	SYM
ejpam-6559	54	58	∥f∥n∗.	∥f∥n∗.	PROPN
ejpam-6559	54	59	a	a	DET
ejpam-6559	54	60	consequence	consequence	NOUN
ejpam-6559	54	61	of	of	ADP
ejpam-6559	54	62	hahn	hahn	NOUN
ejpam-6559	54	63	-	-	PUNCT
ejpam-6559	54	64	banach	banach	NOUN
ejpam-6559	54	65	theorem	theorem	VERB
ejpam-6559	54	66	,	,	PUNCT
ejpam-6559	54	67	is	be	AUX
ejpam-6559	54	68	the	the	DET
ejpam-6559	54	69	following	follow	VERB
ejpam-6559	54	70	corollary	corollary	NOUN
ejpam-6559	54	71	.	.	PUNCT
ejpam-6559	55	1	corollary	corollary	ADJ
ejpam-6559	55	2	1	1	NUM
ejpam-6559	55	3	.	.	PUNCT
ejpam-6559	56	1	[	[	X
ejpam-6559	56	2	14	14	NUM
ejpam-6559	56	3	]	]	PUNCT
ejpam-6559	56	4	let	let	VERB
ejpam-6559	56	5	n	n	PRON
ejpam-6559	56	6	be	be	AUX
ejpam-6559	56	7	a	a	DET
ejpam-6559	56	8	normed	normed	ADJ
ejpam-6559	56	9	linear	linear	ADJ
ejpam-6559	56	10	space	space	NOUN
ejpam-6559	56	11	.	.	PUNCT
ejpam-6559	57	1	then	then	ADV
ejpam-6559	57	2	for	for	ADP
ejpam-6559	57	3	all	all	DET
ejpam-6559	57	4	x	x	SYM
ejpam-6559	57	5	∈	∈	NOUN
ejpam-6559	57	6	n	n	CCONJ
ejpam-6559	57	7	there	there	ADV
ejpam-6559	57	8	exists	exist	VERB
ejpam-6559	57	9	x∗	x∗	PROPN
ejpam-6559	57	10	∈	∈	PROPN
ejpam-6559	57	11	n∗	n∗	VERB
ejpam-6559	57	12	such	such	ADJ
ejpam-6559	57	13	that	that	SCONJ
ejpam-6559	57	14	∥x∗∥n∗	∥x∗∥n∗	PROPN
ejpam-6559	57	15	=	=	SYM
ejpam-6559	57	16	1	1	NUM
ejpam-6559	57	17	and	and	CCONJ
ejpam-6559	57	18	x∗	x∗	PROPN
ejpam-6559	57	19	(	(	PUNCT
ejpam-6559	57	20	x	x	X
ejpam-6559	57	21	)	)	PUNCT
ejpam-6559	57	22	=	=	PUNCT
ejpam-6559	58	1	∥x∥n	∥x∥n	NOUN
ejpam-6559	58	2	.	.	PUNCT
ejpam-6559	59	1	3	3	X
ejpam-6559	59	2	.	.	X
ejpam-6559	59	3	main	main	ADJ
ejpam-6559	59	4	results	result	NOUN
ejpam-6559	59	5	theorem	theorem	VERB
ejpam-6559	59	6	3	3	X
ejpam-6559	59	7	.	.	PUNCT
ejpam-6559	59	8	consider	consider	VERB
ejpam-6559	59	9	the	the	DET
ejpam-6559	59	10	following	follow	VERB
ejpam-6559	59	11	acp	acp	PROPN
ejpam-6559	59	12	in	in	ADP
ejpam-6559	59	13	two	two	NUM
ejpam-6559	59	14	variables	variable	NOUN
ejpam-6559	59	15	d2	d2	PROPN
ejpam-6559	59	16	ssu(s	ssu(s	PROPN
ejpam-6559	59	17	,	,	PUNCT
ejpam-6559	59	18	t	t	PROPN
ejpam-6559	59	19	)	)	PUNCT
ejpam-6559	60	1	+	+	NOUN
ejpam-6559	60	2	d2	d2	PROPN
ejpam-6559	60	3	ttu(s	ttu(s	PROPN
ejpam-6559	60	4	,	,	PUNCT
ejpam-6559	60	5	t	t	PROPN
ejpam-6559	60	6	)	)	PUNCT
ejpam-6559	60	7	+	+	CCONJ
ejpam-6559	60	8	2d2	2d2	NUM
ejpam-6559	60	9	stu(s	stu(s	NOUN
ejpam-6559	60	10	,	,	PUNCT
ejpam-6559	60	11	t	t	PROPN
ejpam-6559	60	12	)	)	PUNCT
ejpam-6559	61	1	+	+	ADP
ejpam-6559	61	2	a	a	DET
ejpam-6559	61	3	[	[	X
ejpam-6559	61	4	dsu(s	dsu(s	PROPN
ejpam-6559	61	5	,	,	PUNCT
ejpam-6559	61	6	t	t	PROPN
ejpam-6559	61	7	)	)	PUNCT
ejpam-6559	61	8	+	+	NOUN
ejpam-6559	61	9	dtu(s	dtu(s	PROPN
ejpam-6559	61	10	,	,	PUNCT
ejpam-6559	61	11	t	t	PROPN
ejpam-6559	61	12	)	)	PUNCT
ejpam-6559	61	13	]	]	PUNCT
ejpam-6559	62	1	=	=	PUNCT
ejpam-6559	62	2	bu(s	bu(s	PROPN
ejpam-6559	62	3	,	,	PUNCT
ejpam-6559	62	4	t	t	PROPN
ejpam-6559	62	5	)	)	PUNCT
ejpam-6559	62	6	,	,	PUNCT
ejpam-6559	62	7	(	(	PUNCT
ejpam-6559	62	8	2	2	X
ejpam-6559	62	9	)	)	PUNCT
ejpam-6559	62	10	where	where	SCONJ
ejpam-6559	62	11	a	a	DET
ejpam-6559	62	12	,	,	PUNCT
ejpam-6559	62	13	b	b	NOUN
ejpam-6559	62	14	are	be	AUX
ejpam-6559	62	15	closed	close	VERB
ejpam-6559	62	16	linear	linear	ADJ
ejpam-6559	62	17	operators	operator	NOUN
ejpam-6559	62	18	on	on	ADP
ejpam-6559	62	19	a	a	DET
ejpam-6559	62	20	banach	banach	NOUN
ejpam-6559	62	21	space	space	NOUN
ejpam-6559	62	22	x	x	NOUN
ejpam-6559	62	23	such	such	ADJ
ejpam-6559	62	24	that	that	SCONJ
ejpam-6559	62	25	a	a	DET
ejpam-6559	62	26	:	:	PUNCT
ejpam-6559	62	27	dom(a	dom(a	PROPN
ejpam-6559	62	28	)	)	PUNCT
ejpam-6559	62	29	⊆	⊆	NUM
ejpam-6559	62	30	x	x	SYM
ejpam-6559	62	31	→	→	SYM
ejpam-6559	62	32	x	x	PROPN
ejpam-6559	62	33	,	,	PUNCT
ejpam-6559	62	34	b	b	NOUN
ejpam-6559	62	35	:	:	PUNCT
ejpam-6559	62	36	dom(b	dom(b	PROPN
ejpam-6559	62	37	)	)	PUNCT
ejpam-6559	62	38	⊆	⊆	NUM
ejpam-6559	62	39	x	x	SYM
ejpam-6559	62	40	→	→	SYM
ejpam-6559	62	41	x	x	SYM
ejpam-6559	62	42	,	,	PUNCT
ejpam-6559	62	43	u(s	u(s	PROPN
ejpam-6559	62	44	,	,	PUNCT
ejpam-6559	62	45	t	t	PROPN
ejpam-6559	62	46	)	)	PUNCT
ejpam-6559	62	47	:	:	PUNCT
ejpam-6559	63	1	[	[	X
ejpam-6559	63	2	0	0	NUM
ejpam-6559	63	3	,	,	PUNCT
ejpam-6559	63	4	1]×[0	1]×[0	NUM
ejpam-6559	63	5	,	,	PUNCT
ejpam-6559	63	6	1	1	NUM
ejpam-6559	63	7	]	]	PUNCT
ejpam-6559	63	8	→	→	PUNCT
ejpam-6559	63	9	x	x	X
ejpam-6559	63	10	is	be	AUX
ejpam-6559	63	11	an	an	DET
ejpam-6559	63	12	unknown	unknown	ADJ
ejpam-6559	63	13	twice	twice	ADV
ejpam-6559	63	14	-	-	PUNCT
ejpam-6559	63	15	continuously	continuously	ADV
ejpam-6559	63	16	partially	partially	ADV
ejpam-6559	63	17	differentiable	differentiable	ADJ
ejpam-6559	63	18	function	function	NOUN
ejpam-6559	63	19	on	on	ADP
ejpam-6559	63	20	[	[	X
ejpam-6559	63	21	0	0	NUM
ejpam-6559	63	22	,	,	PUNCT
ejpam-6559	63	23	1]×	1]×	NUM
ejpam-6559	63	24	[	[	X
ejpam-6559	63	25	0	0	NUM
ejpam-6559	63	26	,	,	PUNCT
ejpam-6559	63	27	1	1	NUM
ejpam-6559	63	28	]	]	SYM
ejpam-6559	63	29	⊆	⊆	NUM
ejpam-6559	63	30	r2	r2	NOUN
ejpam-6559	63	31	where	where	SCONJ
ejpam-6559	63	32	rang(u	rang(u	ADJ
ejpam-6559	63	33	)	)	PUNCT
ejpam-6559	63	34	⊆	⊆	NUM
ejpam-6559	63	35	dom(a	dom(a	PROPN
ejpam-6559	63	36	)	)	PUNCT
ejpam-6559	63	37	∩dom(b	∩dom(b	NOUN
ejpam-6559	63	38	)	)	PUNCT
ejpam-6559	63	39	.	.	PUNCT
ejpam-6559	64	1	then	then	ADV
ejpam-6559	64	2	,	,	PUNCT
ejpam-6559	64	3	for	for	ADP
ejpam-6559	64	4	all	all	DET
ejpam-6559	64	5	ω	ω	NUM
ejpam-6559	64	6	̸=	̸=	PROPN
ejpam-6559	64	7	0	0	NUM
ejpam-6559	64	8	∈	∈	PROPN
ejpam-6559	64	9	x	x	X
ejpam-6559	64	10	,	,	PUNCT
ejpam-6559	64	11	(	(	PUNCT
ejpam-6559	64	12	2	2	X
ejpam-6559	64	13	)	)	PUNCT
ejpam-6559	64	14	has	have	VERB
ejpam-6559	64	15	non	non	ADJ
ejpam-6559	64	16	-	-	ADJ
ejpam-6559	64	17	trivial	trivial	ADJ
ejpam-6559	64	18	atomic	atomic	ADJ
ejpam-6559	64	19	solution	solution	NOUN
ejpam-6559	64	20	of	of	ADP
ejpam-6559	64	21	the	the	DET
ejpam-6559	64	22	form	form	NOUN
ejpam-6559	64	23	u(s	u(s	ADJ
ejpam-6559	64	24	,	,	PUNCT
ejpam-6559	64	25	t	t	PROPN
ejpam-6559	64	26	)	)	PUNCT
ejpam-6559	64	27	=	=	PUNCT
ejpam-6559	64	28	es−3	es−3	PROPN
ejpam-6559	64	29	t	t	NOUN
ejpam-6559	64	30	2	2	NUM
ejpam-6559	64	31	[	[	X
ejpam-6559	64	32	(	(	PUNCT
ejpam-6559	64	33	1−	1−	NUM
ejpam-6559	64	34	√	√	NUM
ejpam-6559	64	35	5	5	NUM
ejpam-6559	64	36	)	)	PUNCT
ejpam-6559	64	37	e	e	NOUN
ejpam-6559	64	38	3−	3−	NUM
ejpam-6559	64	39	√	√	NUM
ejpam-6559	64	40	5	5	NUM
ejpam-6559	64	41	2	2	NUM
ejpam-6559	64	42	t	t	NOUN
ejpam-6559	65	1	−	−	PROPN
ejpam-6559	65	2	(	(	PUNCT
ejpam-6559	65	3	1	1	NUM
ejpam-6559	65	4	+	+	CCONJ
ejpam-6559	65	5	√	√	NUM
ejpam-6559	65	6	5	5	NUM
ejpam-6559	65	7	)	)	PUNCT
ejpam-6559	65	8	e	e	NOUN
ejpam-6559	65	9	3	3	NUM
ejpam-6559	65	10	+	+	NUM
ejpam-6559	65	11	√	√	NUM
ejpam-6559	65	12	5	5	NUM
ejpam-6559	65	13	2	2	NUM
ejpam-6559	65	14	t	t	NOUN
ejpam-6559	65	15	]	]	PUNCT
ejpam-6559	65	16	⊗	⊗	PROPN
ejpam-6559	65	17	ω	ω	PROPN
ejpam-6559	65	18	.	.	PUNCT
ejpam-6559	66	1	proof	proof	NOUN
ejpam-6559	66	2	.	.	PUNCT
ejpam-6559	67	1	let	let	VERB
ejpam-6559	67	2	p	p	PROPN
ejpam-6559	67	3	∈	∈	PROPN
ejpam-6559	67	4	c2	c2	PROPN
ejpam-6559	68	1	[	[	X
ejpam-6559	68	2	0	0	NUM
ejpam-6559	68	3	,	,	PUNCT
ejpam-6559	68	4	1	1	NUM
ejpam-6559	68	5	]	]	PUNCT
ejpam-6559	68	6	and	and	CCONJ
ejpam-6559	68	7	q	q	PROPN
ejpam-6559	68	8	∈	∈	PROPN
ejpam-6559	68	9	c2	c2	PROPN
ejpam-6559	68	10	[	[	X
ejpam-6559	68	11	0	0	NUM
ejpam-6559	68	12	,	,	PUNCT
ejpam-6559	68	13	1	1	NUM
ejpam-6559	68	14	]	]	PUNCT
ejpam-6559	68	15	such	such	ADJ
ejpam-6559	68	16	that	that	SCONJ
ejpam-6559	68	17	p	p	NOUN
ejpam-6559	68	18	=	=	X
ejpam-6559	68	19	p	p	X
ejpam-6559	68	20	(	(	PUNCT
ejpam-6559	68	21	s	s	NOUN
ejpam-6559	68	22	)	)	PUNCT
ejpam-6559	68	23	:	:	PUNCT
ejpam-6559	69	1	[	[	X
ejpam-6559	69	2	0	0	NUM
ejpam-6559	69	3	,	,	PUNCT
ejpam-6559	69	4	1	1	NUM
ejpam-6559	69	5	]	]	PUNCT
ejpam-6559	69	6	→	→	SYM
ejpam-6559	69	7	x	x	SYM
ejpam-6559	69	8	,	,	PUNCT
ejpam-6559	69	9	q	q	X
ejpam-6559	69	10	=	=	PUNCT
ejpam-6559	69	11	q(t	q(t	PROPN
ejpam-6559	69	12	)	)	PUNCT
ejpam-6559	69	13	:	:	PUNCT
ejpam-6559	70	1	[	[	X
ejpam-6559	70	2	0	0	NUM
ejpam-6559	70	3	,	,	PUNCT
ejpam-6559	70	4	1	1	NUM
ejpam-6559	70	5	]	]	PUNCT
ejpam-6559	70	6	→	→	PUNCT
ejpam-6559	70	7	x.	x.	NOUN
ejpam-6559	70	8	suppose	suppose	VERB
ejpam-6559	70	9	u(s	u(s	ADJ
ejpam-6559	70	10	,	,	PUNCT
ejpam-6559	70	11	t	t	PROPN
ejpam-6559	70	12	)	)	PUNCT
ejpam-6559	70	13	,	,	PUNCT
ejpam-6559	70	14	the	the	DET
ejpam-6559	70	15	solution	solution	NOUN
ejpam-6559	70	16	to	to	ADP
ejpam-6559	70	17	(	(	PUNCT
ejpam-6559	70	18	2	2	NUM
ejpam-6559	70	19	)	)	PUNCT
ejpam-6559	70	20	,	,	PUNCT
ejpam-6559	70	21	be	be	AUX
ejpam-6559	70	22	in	in	ADP
ejpam-6559	70	23	the	the	DET
ejpam-6559	70	24	following	follow	VERB
ejpam-6559	70	25	atom	atom	NOUN
ejpam-6559	70	26	form	form	NOUN
ejpam-6559	70	27	u	u	NOUN
ejpam-6559	70	28	=	=	PROPN
ejpam-6559	70	29	p	p	PROPN
ejpam-6559	70	30	⊗	⊗	PROPN
ejpam-6559	70	31	q	q	PROPN
ejpam-6559	70	32	⊗	⊗	PROPN
ejpam-6559	70	33	ω	ω	PROPN
ejpam-6559	70	34	,	,	PUNCT
ejpam-6559	70	35	where	where	SCONJ
ejpam-6559	70	36	ω	ω	NOUN
ejpam-6559	70	37	̸=	̸=	PROPN
ejpam-6559	70	38	0	0	NUM
ejpam-6559	70	39	∈	∈	PROPN
ejpam-6559	70	40	x.	x.	NOUN
ejpam-6559	70	41	on	on	ADP
ejpam-6559	70	42	substituting	substitute	VERB
ejpam-6559	70	43	this	this	DET
ejpam-6559	70	44	atom	atom	NOUN
ejpam-6559	70	45	operator	operator	NOUN
ejpam-6559	70	46	form	form	NOUN
ejpam-6559	70	47	into	into	ADP
ejpam-6559	70	48	(	(	PUNCT
ejpam-6559	70	49	2	2	X
ejpam-6559	70	50	)	)	PUNCT
ejpam-6559	70	51	we	we	PRON
ejpam-6559	70	52	obtain	obtain	VERB
ejpam-6559	70	53	the	the	DET
ejpam-6559	70	54	following	follow	VERB
ejpam-6559	70	55	tensor	tensor	NOUN
ejpam-6559	70	56	product	product	NOUN
ejpam-6559	70	57	equation	equation	NOUN
ejpam-6559	70	58	[	[	X
ejpam-6559	70	59	p	p	X
ejpam-6559	70	60	′′	′′	PROPN
ejpam-6559	70	61	(	(	PUNCT
ejpam-6559	70	62	s)⊗q	s)⊗q	PROPN
ejpam-6559	70	63	(	(	PUNCT
ejpam-6559	70	64	t	t	PROPN
ejpam-6559	70	65	)	)	PUNCT
ejpam-6559	70	66	+	+	NOUN
ejpam-6559	71	1	p	p	X
ejpam-6559	71	2	(	(	PUNCT
ejpam-6559	71	3	s)⊗q′′	s)⊗q′′	PROPN
ejpam-6559	71	4	(	(	PUNCT
ejpam-6559	71	5	t	t	PROPN
ejpam-6559	71	6	)	)	PUNCT
ejpam-6559	71	7	+	+	CCONJ
ejpam-6559	71	8	2p	2p	NUM
ejpam-6559	71	9	′	′	NUM
ejpam-6559	71	10	(	(	PUNCT
ejpam-6559	71	11	s)⊗q′	s)⊗q′	X
ejpam-6559	71	12	(	(	PUNCT
ejpam-6559	71	13	t)]⊗	t)]⊗	PROPN
ejpam-6559	71	14	ω	ω	NUM
ejpam-6559	72	1	+	+	X
ejpam-6559	73	1	[	[	X
ejpam-6559	73	2	p	p	X
ejpam-6559	73	3	′	′	X
ejpam-6559	73	4	(	(	PUNCT
ejpam-6559	73	5	s)⊗q	s)⊗q	PROPN
ejpam-6559	73	6	(	(	PUNCT
ejpam-6559	73	7	t	t	PROPN
ejpam-6559	73	8	)	)	PUNCT
ejpam-6559	73	9	+	+	NOUN
ejpam-6559	74	1	p	p	X
ejpam-6559	74	2	(	(	PUNCT
ejpam-6559	74	3	s)⊗q′	s)⊗q′	PROPN
ejpam-6559	74	4	(	(	PUNCT
ejpam-6559	74	5	t)]⊗aω	t)]⊗aω	PROPN
ejpam-6559	74	6	=	=	SYM
ejpam-6559	74	7	p	p	X
ejpam-6559	74	8	(	(	PUNCT
ejpam-6559	74	9	s)⊗q	s)⊗q	PROPN
ejpam-6559	74	10	(	(	PUNCT
ejpam-6559	74	11	t)⊗bω	t)⊗bω	NOUN
ejpam-6559	74	12	.	.	PUNCT
ejpam-6559	75	1	(	(	PUNCT
ejpam-6559	75	2	3	3	X
ejpam-6559	75	3	)	)	PUNCT
ejpam-6559	75	4	w.	w.	PROPN
ejpam-6559	75	5	g.	g.	PROPN
ejpam-6559	75	6	alshanti	alshanti	PROPN
ejpam-6559	75	7	,	,	PUNCT
ejpam-6559	75	8	m.	m.	PROPN
ejpam-6559	75	9	abu	abu	PROPN
ejpam-6559	75	10	hammad	hammad	PROPN
ejpam-6559	75	11	,	,	PUNCT
ejpam-6559	75	12	r.	r.	PROPN
ejpam-6559	75	13	khalil	khalil	PROPN
ejpam-6559	75	14	/	/	SYM
ejpam-6559	75	15	eur	eur	PROPN
ejpam-6559	75	16	.	.	PUNCT
ejpam-6559	76	1	j.	j.	PROPN
ejpam-6559	76	2	pure	pure	PROPN
ejpam-6559	76	3	appl	appl	PROPN
ejpam-6559	76	4	.	.	PROPN
ejpam-6559	76	5	math	math	PROPN
ejpam-6559	76	6	,	,	PUNCT
ejpam-6559	76	7	18	18	NUM
ejpam-6559	76	8	(	(	PUNCT
ejpam-6559	76	9	4	4	NUM
ejpam-6559	76	10	)	)	PUNCT
ejpam-6559	76	11	(	(	PUNCT
ejpam-6559	76	12	2025	2025	NUM
ejpam-6559	76	13	)	)	PUNCT
ejpam-6559	76	14	,	,	PUNCT
ejpam-6559	76	15	6559	6559	NUM
ejpam-6559	76	16	4	4	NUM
ejpam-6559	76	17	of	of	ADP
ejpam-6559	76	18	8	8	NUM
ejpam-6559	76	19	now	now	ADV
ejpam-6559	76	20	,	,	PUNCT
ejpam-6559	76	21	let	let	VERB
ejpam-6559	76	22	us	we	PRON
ejpam-6559	76	23	assume	assume	VERB
ejpam-6559	76	24	that	that	SCONJ
ejpam-6559	76	25	{	{	PUNCT
ejpam-6559	76	26	ω	ω	NOUN
ejpam-6559	76	27	,	,	PUNCT
ejpam-6559	76	28	aω	aω	PROPN
ejpam-6559	76	29	,	,	PUNCT
ejpam-6559	76	30	bω	bω	VERB
ejpam-6559	76	31	}	}	PUNCT
ejpam-6559	76	32	are	be	AUX
ejpam-6559	76	33	independent	independent	ADJ
ejpam-6559	76	34	in	in	ADP
ejpam-6559	76	35	x.	x.	NOUN
ejpam-6559	76	36	hence	hence	ADV
ejpam-6559	76	37	,	,	PUNCT
ejpam-6559	76	38	by	by	ADP
ejpam-6559	76	39	corollary	corollary	ADJ
ejpam-6559	76	40	1	1	NUM
ejpam-6559	76	41	,	,	PUNCT
ejpam-6559	76	42	one	one	PRON
ejpam-6559	76	43	can	can	AUX
ejpam-6559	76	44	assume	assume	VERB
ejpam-6559	76	45	that	that	SCONJ
ejpam-6559	76	46	there	there	PRON
ejpam-6559	76	47	exists	exist	VERB
ejpam-6559	76	48	x∗	x∗	PROPN
ejpam-6559	76	49	∈	∈	PROPN
ejpam-6559	76	50	x	x	PUNCT
ejpam-6559	76	51	such	such	ADJ
ejpam-6559	76	52	that	that	SCONJ
ejpam-6559	76	53	⟨x∗	⟨x∗	PROPN
ejpam-6559	76	54	,	,	PUNCT
ejpam-6559	76	55	ω⟩	ω⟩	PUNCT
ejpam-6559	76	56	=	=	SYM
ejpam-6559	76	57	⟨x∗	⟨x∗	PROPN
ejpam-6559	76	58	,	,	PUNCT
ejpam-6559	76	59	aω⟩	aω⟩	PUNCT
ejpam-6559	76	60	=	=	SYM
ejpam-6559	76	61	⟨x∗	⟨x∗	PROPN
ejpam-6559	76	62	,	,	PUNCT
ejpam-6559	76	63	bω⟩	bω⟩	ADJ
ejpam-6559	76	64	=	=	SYM
ejpam-6559	76	65	1	1	NUM
ejpam-6559	76	66	,	,	PUNCT
ejpam-6559	76	67	where	where	SCONJ
ejpam-6559	76	68	⟨x∗	⟨x∗	PROPN
ejpam-6559	76	69	,	,	PUNCT
ejpam-6559	76	70	·	·	PUNCT
ejpam-6559	76	71	⟩	⟩	NOUN
ejpam-6559	76	72	:	:	PUNCT
ejpam-6559	76	73	=	=	SYM
ejpam-6559	76	74	x∗	x∗	X
ejpam-6559	76	75	(	(	PUNCT
ejpam-6559	76	76	·	·	PUNCT
ejpam-6559	76	77	)	)	PUNCT
ejpam-6559	76	78	.	.	PUNCT
ejpam-6559	77	1	therefore	therefore	ADV
ejpam-6559	77	2	,	,	PUNCT
ejpam-6559	77	3	on	on	ADP
ejpam-6559	77	4	applying	apply	VERB
ejpam-6559	77	5	such	such	ADJ
ejpam-6559	77	6	x∗	x∗	PROPN
ejpam-6559	77	7	to	to	ADP
ejpam-6559	77	8	both	both	DET
ejpam-6559	77	9	sides	side	NOUN
ejpam-6559	77	10	of	of	ADP
ejpam-6559	77	11	(	(	PUNCT
ejpam-6559	77	12	3	3	NUM
ejpam-6559	77	13	)	)	PUNCT
ejpam-6559	77	14	,	,	PUNCT
ejpam-6559	77	15	we	we	PRON
ejpam-6559	77	16	get	get	VERB
ejpam-6559	77	17	[	[	X
ejpam-6559	77	18	p	p	X
ejpam-6559	77	19	′′	′′	PROPN
ejpam-6559	77	20	(	(	PUNCT
ejpam-6559	77	21	s)⊗q	s)⊗q	PROPN
ejpam-6559	77	22	(	(	PUNCT
ejpam-6559	77	23	t	t	PROPN
ejpam-6559	77	24	)	)	PUNCT
ejpam-6559	77	25	+	+	NOUN
ejpam-6559	78	1	p	p	X
ejpam-6559	78	2	(	(	PUNCT
ejpam-6559	78	3	s)⊗q′′	s)⊗q′′	PROPN
ejpam-6559	78	4	(	(	PUNCT
ejpam-6559	78	5	t	t	PROPN
ejpam-6559	78	6	)	)	PUNCT
ejpam-6559	78	7	+	+	CCONJ
ejpam-6559	78	8	2p	2p	NUM
ejpam-6559	78	9	′	′	NUM
ejpam-6559	78	10	(	(	PUNCT
ejpam-6559	78	11	s)⊗q′	s)⊗q′	PROPN
ejpam-6559	78	12	(	(	PUNCT
ejpam-6559	78	13	t	t	NOUN
ejpam-6559	78	14	)	)	PUNCT
ejpam-6559	78	15	]	]	PUNCT
ejpam-6559	79	1	+	+	CCONJ
ejpam-6559	80	1	[	[	X
ejpam-6559	80	2	p	p	X
ejpam-6559	80	3	′	′	X
ejpam-6559	80	4	(	(	PUNCT
ejpam-6559	80	5	s)⊗q	s)⊗q	PROPN
ejpam-6559	80	6	(	(	PUNCT
ejpam-6559	80	7	t	t	PROPN
ejpam-6559	80	8	)	)	PUNCT
ejpam-6559	80	9	+	+	NOUN
ejpam-6559	80	10	p	p	X
ejpam-6559	80	11	(	(	PUNCT
ejpam-6559	80	12	s)⊗q′	s)⊗q′	PROPN
ejpam-6559	80	13	(	(	PUNCT
ejpam-6559	80	14	t	t	NOUN
ejpam-6559	80	15	)	)	PUNCT
ejpam-6559	80	16	]	]	PUNCT
ejpam-6559	81	1	=	=	PUNCT
ejpam-6559	81	2	p	p	X
ejpam-6559	81	3	(	(	PUNCT
ejpam-6559	81	4	s)⊗q	s)⊗q	PROPN
ejpam-6559	81	5	(	(	PUNCT
ejpam-6559	81	6	t	t	PROPN
ejpam-6559	81	7	)	)	PUNCT
ejpam-6559	81	8	,	,	PUNCT
ejpam-6559	81	9	(	(	PUNCT
ejpam-6559	81	10	4	4	X
ejpam-6559	81	11	)	)	PUNCT
ejpam-6559	81	12	which	which	PRON
ejpam-6559	81	13	can	can	AUX
ejpam-6559	81	14	be	be	AUX
ejpam-6559	81	15	simplified	simplify	VERB
ejpam-6559	81	16	to	to	ADP
ejpam-6559	81	17	p	p	PROPN
ejpam-6559	81	18	′′	′′	PROPN
ejpam-6559	81	19	(	(	PUNCT
ejpam-6559	81	20	s)⊗q	s)⊗q	PROPN
ejpam-6559	81	21	(	(	PUNCT
ejpam-6559	81	22	t	t	PROPN
ejpam-6559	81	23	)	)	PUNCT
ejpam-6559	81	24	+	+	NOUN
ejpam-6559	81	25	p	p	NOUN
ejpam-6559	82	1	′	′	NUM
ejpam-6559	82	2	(	(	PUNCT
ejpam-6559	82	3	s)⊗	s)⊗	PROPN
ejpam-6559	82	4	[	[	PUNCT
ejpam-6559	82	5	2q′	2q′	NUM
ejpam-6559	82	6	(	(	PUNCT
ejpam-6559	82	7	t	t	PROPN
ejpam-6559	82	8	)	)	PUNCT
ejpam-6559	83	1	+	+	NOUN
ejpam-6559	83	2	q	q	X
ejpam-6559	83	3	(	(	PUNCT
ejpam-6559	83	4	t	t	PROPN
ejpam-6559	83	5	)	)	PUNCT
ejpam-6559	83	6	]	]	PUNCT
ejpam-6559	84	1	=	=	PUNCT
ejpam-6559	84	2	p	p	X
ejpam-6559	84	3	(	(	PUNCT
ejpam-6559	84	4	s)⊗	s)⊗	PROPN
ejpam-6559	84	5	[	[	PUNCT
ejpam-6559	84	6	q	q	X
ejpam-6559	84	7	(	(	PUNCT
ejpam-6559	84	8	t)−q′	t)−q′	NUM
ejpam-6559	84	9	(	(	PUNCT
ejpam-6559	84	10	t)−q′′	t)−q′′	NUM
ejpam-6559	84	11	(	(	PUNCT
ejpam-6559	84	12	t	t	PROPN
ejpam-6559	84	13	)	)	PUNCT
ejpam-6559	84	14	]	]	PUNCT
ejpam-6559	84	15	.	.	PUNCT
ejpam-6559	85	1	(	(	PUNCT
ejpam-6559	85	2	5	5	NUM
ejpam-6559	85	3	)	)	PUNCT
ejpam-6559	85	4	but	but	CCONJ
ejpam-6559	85	5	,	,	PUNCT
ejpam-6559	85	6	equation	equation	NOUN
ejpam-6559	85	7	(	(	PUNCT
ejpam-6559	85	8	5	5	NUM
ejpam-6559	85	9	)	)	PUNCT
ejpam-6559	85	10	reveals	reveal	VERB
ejpam-6559	85	11	that	that	SCONJ
ejpam-6559	85	12	the	the	DET
ejpam-6559	85	13	sum	sum	NOUN
ejpam-6559	85	14	of	of	ADP
ejpam-6559	85	15	two	two	NUM
ejpam-6559	85	16	atoms	atom	NOUN
ejpam-6559	85	17	is	be	AUX
ejpam-6559	85	18	an	an	DET
ejpam-6559	85	19	atom	atom	NOUN
ejpam-6559	85	20	.	.	PUNCT
ejpam-6559	86	1	hence	hence	ADV
ejpam-6559	86	2	,	,	PUNCT
ejpam-6559	86	3	by	by	ADP
ejpam-6559	86	4	lemma	lemma	PROPN
ejpam-6559	86	5	1	1	NUM
ejpam-6559	86	6	,	,	PUNCT
ejpam-6559	86	7	we	we	PRON
ejpam-6559	86	8	get	get	VERB
ejpam-6559	86	9	the	the	DET
ejpam-6559	86	10	following	follow	VERB
ejpam-6559	86	11	two	two	NUM
ejpam-6559	86	12	cases	case	NOUN
ejpam-6559	86	13	either	either	CCONJ
ejpam-6559	86	14	(	(	PUNCT
ejpam-6559	86	15	i	i	NOUN
ejpam-6559	86	16	)	)	PUNCT
ejpam-6559	86	17	p	p	PROPN
ejpam-6559	86	18	′′	′′	PROPN
ejpam-6559	86	19	(	(	PUNCT
ejpam-6559	86	20	s	s	X
ejpam-6559	86	21	)	)	PUNCT
ejpam-6559	86	22	=	=	SYM
ejpam-6559	87	1	p	p	NOUN
ejpam-6559	87	2	′	′	NUM
ejpam-6559	87	3	(	(	PUNCT
ejpam-6559	87	4	s	s	X
ejpam-6559	87	5	)	)	PUNCT
ejpam-6559	88	1	=	=	SYM
ejpam-6559	88	2	p	p	X
ejpam-6559	88	3	(	(	PUNCT
ejpam-6559	88	4	s	s	NOUN
ejpam-6559	88	5	)	)	PUNCT
ejpam-6559	88	6	,	,	PUNCT
ejpam-6559	88	7	or	or	CCONJ
ejpam-6559	88	8	(	(	PUNCT
ejpam-6559	88	9	ii	ii	NOUN
ejpam-6559	88	10	)	)	PUNCT
ejpam-6559	88	11	q	q	NOUN
ejpam-6559	88	12	(	(	PUNCT
ejpam-6559	88	13	t	t	NOUN
ejpam-6559	88	14	)	)	PUNCT
ejpam-6559	88	15	=	=	SYM
ejpam-6559	88	16	2q′	2q′	NUM
ejpam-6559	88	17	(	(	PUNCT
ejpam-6559	88	18	t	t	PROPN
ejpam-6559	88	19	)	)	PUNCT
ejpam-6559	89	1	+	+	NOUN
ejpam-6559	89	2	q	q	X
ejpam-6559	89	3	(	(	PUNCT
ejpam-6559	89	4	t	t	NOUN
ejpam-6559	89	5	)	)	PUNCT
ejpam-6559	89	6	=	=	PUNCT
ejpam-6559	90	1	q	q	X
ejpam-6559	90	2	(	(	PUNCT
ejpam-6559	90	3	t)−q′	t)−q′	NUM
ejpam-6559	90	4	(	(	PUNCT
ejpam-6559	90	5	t)−q′′	t)−q′′	X
ejpam-6559	90	6	(	(	PUNCT
ejpam-6559	90	7	t	t	PROPN
ejpam-6559	90	8	)	)	PUNCT
ejpam-6559	90	9	.	.	PUNCT
ejpam-6559	91	1	(	(	PUNCT
ejpam-6559	91	2	6	6	NUM
ejpam-6559	91	3	)	)	PUNCT
ejpam-6559	91	4	before	before	SCONJ
ejpam-6559	91	5	we	we	PRON
ejpam-6559	91	6	start	start	VERB
ejpam-6559	91	7	handling	handle	VERB
ejpam-6559	91	8	the	the	DET
ejpam-6559	91	9	two	two	NUM
ejpam-6559	91	10	cases	case	NOUN
ejpam-6559	91	11	,	,	PUNCT
ejpam-6559	91	12	let	let	VERB
ejpam-6559	91	13	us	we	PRON
ejpam-6559	91	14	consider	consider	VERB
ejpam-6559	91	15	,	,	PUNCT
ejpam-6559	91	16	without	without	ADP
ejpam-6559	91	17	loss	loss	NOUN
ejpam-6559	91	18	of	of	ADP
ejpam-6559	91	19	generality	generality	NOUN
ejpam-6559	91	20	,	,	PUNCT
ejpam-6559	91	21	the	the	DET
ejpam-6559	91	22	following	follow	VERB
ejpam-6559	91	23	initial	initial	ADJ
ejpam-6559	91	24	conditions	condition	NOUN
ejpam-6559	91	25	,	,	PUNCT
ejpam-6559	91	26	p	p	X
ejpam-6559	91	27	(	(	PUNCT
ejpam-6559	91	28	0	0	NUM
ejpam-6559	91	29	)	)	PUNCT
ejpam-6559	91	30	=	=	PUNCT
ejpam-6559	91	31	p	p	X
ejpam-6559	91	32	′(0	′(0	PROPN
ejpam-6559	91	33	)	)	PUNCT
ejpam-6559	91	34	=	=	SYM
ejpam-6559	92	1	1	1	NUM
ejpam-6559	92	2	,	,	PUNCT
ejpam-6559	92	3	(	(	PUNCT
ejpam-6559	92	4	7	7	X
ejpam-6559	92	5	)	)	PUNCT
ejpam-6559	92	6	q(0	q(0	NOUN
ejpam-6559	92	7	)	)	PUNCT
ejpam-6559	93	1	=	=	SYM
ejpam-6559	93	2	q′(0	q′(0	PROPN
ejpam-6559	93	3	)	)	PUNCT
ejpam-6559	93	4	=	=	SYM
ejpam-6559	93	5	1	1	X
ejpam-6559	93	6	.	.	X
ejpam-6559	93	7	case	case	NOUN
ejpam-6559	93	8	(	(	PUNCT
ejpam-6559	93	9	i	i	NOUN
ejpam-6559	93	10	):	):	PUNCT
ejpam-6559	93	11	for	for	ADP
ejpam-6559	93	12	this	this	DET
ejpam-6559	93	13	case	case	NOUN
ejpam-6559	93	14	,	,	PUNCT
ejpam-6559	93	15	there	there	PRON
ejpam-6559	93	16	are	be	VERB
ejpam-6559	93	17	three	three	NUM
ejpam-6559	93	18	sub	sub	NOUN
ejpam-6559	93	19	-	-	NOUN
ejpam-6559	93	20	cases	case	NOUN
ejpam-6559	93	21	which	which	PRON
ejpam-6559	93	22	are	be	AUX
ejpam-6559	93	23	p	p	ADJ
ejpam-6559	93	24	′′	′′	PROPN
ejpam-6559	93	25	=	=	PROPN
ejpam-6559	93	26	p	p	PROPN
ejpam-6559	93	27	′	′	NOUN
ejpam-6559	93	28	,	,	PUNCT
ejpam-6559	93	29	p	p	X
ejpam-6559	93	30	′′	′′	PROPN
ejpam-6559	93	31	=	=	PROPN
ejpam-6559	93	32	p	p	X
ejpam-6559	93	33	,	,	PUNCT
ejpam-6559	93	34	and	and	CCONJ
ejpam-6559	93	35	p	p	NOUN
ejpam-6559	94	1	′	′	NOUN
ejpam-6559	95	1	=	=	PUNCT
ejpam-6559	95	2	p	p	NOUN
ejpam-6559	95	3	together	together	ADV
ejpam-6559	95	4	with	with	ADP
ejpam-6559	95	5	appropriate	appropriate	ADJ
ejpam-6559	95	6	conditions	condition	NOUN
ejpam-6559	95	7	from	from	ADP
ejpam-6559	95	8	(	(	PUNCT
ejpam-6559	95	9	7	7	X
ejpam-6559	95	10	)	)	PUNCT
ejpam-6559	95	11	are	be	AUX
ejpam-6559	95	12	,	,	PUNCT
ejpam-6559	95	13	simply	simply	ADV
ejpam-6559	95	14	,	,	PUNCT
ejpam-6559	95	15	the	the	DET
ejpam-6559	95	16	following	follow	VERB
ejpam-6559	95	17	classical	classical	ADJ
ejpam-6559	95	18	initial	initial	ADJ
ejpam-6559	95	19	value	value	NOUN
ejpam-6559	95	20	problems	problem	NOUN
ejpam-6559	95	21	p	p	PROPN
ejpam-6559	95	22	′′	′′	PROPN
ejpam-6559	95	23	(	(	PUNCT
ejpam-6559	95	24	s)−	s)−	PROPN
ejpam-6559	95	25	p	p	NOUN
ejpam-6559	95	26	′	′	NOUN
ejpam-6559	95	27	(	(	PUNCT
ejpam-6559	95	28	s	s	X
ejpam-6559	95	29	)	)	PUNCT
ejpam-6559	95	30	=	=	SYM
ejpam-6559	95	31	0	0	NUM
ejpam-6559	95	32	,	,	PUNCT
ejpam-6559	95	33	p	p	NOUN
ejpam-6559	95	34	′(0	′(0	PROPN
ejpam-6559	95	35	)	)	PUNCT
ejpam-6559	96	1	=	=	SYM
ejpam-6559	96	2	p	p	X
ejpam-6559	96	3	(	(	PUNCT
ejpam-6559	96	4	0	0	NUM
ejpam-6559	96	5	)	)	PUNCT
ejpam-6559	96	6	=	=	SYM
ejpam-6559	96	7	1	1	NUM
ejpam-6559	96	8	,	,	PUNCT
ejpam-6559	96	9	p	p	PROPN
ejpam-6559	96	10	′′	′′	PROPN
ejpam-6559	96	11	(	(	PUNCT
ejpam-6559	96	12	s)−	s)−	PROPN
ejpam-6559	96	13	p	p	PROPN
ejpam-6559	96	14	(	(	PUNCT
ejpam-6559	96	15	s	s	NOUN
ejpam-6559	96	16	)	)	PUNCT
ejpam-6559	96	17	=	=	SYM
ejpam-6559	96	18	0	0	NUM
ejpam-6559	96	19	,	,	PUNCT
ejpam-6559	96	20	p	p	NOUN
ejpam-6559	96	21	′(0	′(0	PROPN
ejpam-6559	96	22	)	)	PUNCT
ejpam-6559	97	1	=	=	SYM
ejpam-6559	97	2	p	p	X
ejpam-6559	97	3	(	(	PUNCT
ejpam-6559	97	4	0	0	NUM
ejpam-6559	97	5	)	)	PUNCT
ejpam-6559	97	6	=	=	SYM
ejpam-6559	97	7	1	1	NUM
ejpam-6559	97	8	,	,	PUNCT
ejpam-6559	97	9	p	p	NOUN
ejpam-6559	97	10	′	′	NOUN
ejpam-6559	97	11	(	(	PUNCT
ejpam-6559	97	12	s)−	s)−	PROPN
ejpam-6559	97	13	p	p	X
ejpam-6559	97	14	(	(	PUNCT
ejpam-6559	97	15	s	s	NOUN
ejpam-6559	97	16	)	)	PUNCT
ejpam-6559	97	17	=	=	SYM
ejpam-6559	97	18	0	0	NUM
ejpam-6559	97	19	,	,	PUNCT
ejpam-6559	97	20	p	p	X
ejpam-6559	97	21	(	(	PUNCT
ejpam-6559	97	22	0	0	NUM
ejpam-6559	97	23	)	)	PUNCT
ejpam-6559	97	24	=	=	SYM
ejpam-6559	97	25	1	1	X
ejpam-6559	97	26	.	.	PUNCT
ejpam-6559	97	27	(	(	PUNCT
ejpam-6559	97	28	8)	8)	NUM
ejpam-6559	97	29	all	all	PRON
ejpam-6559	97	30	intial	intial	ADJ
ejpam-6559	97	31	value	value	NOUN
ejpam-6559	97	32	problems	problem	NOUN
ejpam-6559	97	33	listed	list	VERB
ejpam-6559	97	34	in	in	ADP
ejpam-6559	97	35	(	(	PUNCT
ejpam-6559	97	36	8)	8)	NUM
ejpam-6559	97	37	have	have	VERB
ejpam-6559	97	38	the	the	DET
ejpam-6559	97	39	same	same	ADJ
ejpam-6559	97	40	solution	solution	NOUN
ejpam-6559	97	41	which	which	PRON
ejpam-6559	97	42	is	be	AUX
ejpam-6559	97	43	p	p	X
ejpam-6559	97	44	(	(	PUNCT
ejpam-6559	97	45	s	s	NOUN
ejpam-6559	97	46	)	)	PUNCT
ejpam-6559	97	47	=	=	SYM
ejpam-6559	97	48	es	es	PROPN
ejpam-6559	97	49	.	.	PUNCT
ejpam-6559	98	1	(	(	PUNCT
ejpam-6559	98	2	9	9	X
ejpam-6559	98	3	)	)	PUNCT
ejpam-6559	98	4	this	this	PRON
ejpam-6559	98	5	implies	imply	VERB
ejpam-6559	98	6	that	that	SCONJ
ejpam-6559	98	7	an	an	DET
ejpam-6559	98	8	atomic	atomic	ADJ
ejpam-6559	98	9	solution	solution	NOUN
ejpam-6559	98	10	can	can	AUX
ejpam-6559	98	11	be	be	AUX
ejpam-6559	98	12	obtained	obtain	VERB
ejpam-6559	98	13	for	for	ADP
ejpam-6559	98	14	this	this	DET
ejpam-6559	98	15	particular	particular	ADJ
ejpam-6559	98	16	case	case	NOUN
ejpam-6559	98	17	.	.	PUNCT
ejpam-6559	99	1	this	this	DET
ejpam-6559	99	2	atomic	atomic	ADJ
ejpam-6559	99	3	solution	solution	NOUN
ejpam-6559	99	4	can	can	AUX
ejpam-6559	99	5	be	be	AUX
ejpam-6559	99	6	found	find	VERB
ejpam-6559	99	7	by	by	ADP
ejpam-6559	99	8	substituting	substitute	VERB
ejpam-6559	99	9	the	the	DET
ejpam-6559	99	10	solution	solution	NOUN
ejpam-6559	99	11	p	p	X
ejpam-6559	99	12	(	(	PUNCT
ejpam-6559	99	13	s	s	NOUN
ejpam-6559	99	14	)	)	PUNCT
ejpam-6559	99	15	=	=	SYM
ejpam-6559	99	16	es	es	NOUN
ejpam-6559	99	17	into	into	ADP
ejpam-6559	99	18	(	(	PUNCT
ejpam-6559	99	19	5	5	NUM
ejpam-6559	99	20	)	)	PUNCT
ejpam-6559	99	21	we	we	PRON
ejpam-6559	99	22	have	have	VERB
ejpam-6559	99	23	q′′	q′′	ADV
ejpam-6559	99	24	(	(	PUNCT
ejpam-6559	99	25	t	t	NOUN
ejpam-6559	99	26	)	)	PUNCT
ejpam-6559	100	1	+	+	NUM
ejpam-6559	100	2	3q′	3q′	NUM
ejpam-6559	100	3	(	(	PUNCT
ejpam-6559	100	4	t	t	PROPN
ejpam-6559	100	5	)	)	PUNCT
ejpam-6559	100	6	+	+	NOUN
ejpam-6559	100	7	q(t	q(t	X
ejpam-6559	100	8	)	)	PUNCT
ejpam-6559	100	9	=	=	SYM
ejpam-6559	101	1	0	0	X
ejpam-6559	101	2	.	.	PUNCT
ejpam-6559	102	1	(	(	PUNCT
ejpam-6559	102	2	10	10	NUM
ejpam-6559	102	3	)	)	PUNCT
ejpam-6559	102	4	equation	equation	NOUN
ejpam-6559	102	5	(	(	PUNCT
ejpam-6559	102	6	10	10	NUM
ejpam-6559	102	7	)	)	PUNCT
ejpam-6559	102	8	is	be	AUX
ejpam-6559	102	9	a	a	DET
ejpam-6559	102	10	second	second	ADJ
ejpam-6559	102	11	order	order	NOUN
ejpam-6559	102	12	ordinary	ordinary	ADJ
ejpam-6559	102	13	differential	differential	ADJ
ejpam-6559	102	14	equation	equation	NOUN
ejpam-6559	102	15	with	with	ADP
ejpam-6559	102	16	constant	constant	ADJ
ejpam-6559	102	17	coefficients	coefficient	NOUN
ejpam-6559	102	18	.	.	PUNCT
ejpam-6559	103	1	together	together	ADV
ejpam-6559	103	2	with	with	ADP
ejpam-6559	103	3	appropriate	appropriate	ADJ
ejpam-6559	103	4	initial	initial	ADJ
ejpam-6559	103	5	conditions	condition	NOUN
ejpam-6559	103	6	form	form	NOUN
ejpam-6559	103	7	(	(	PUNCT
ejpam-6559	103	8	7	7	NUM
ejpam-6559	103	9	)	)	PUNCT
ejpam-6559	103	10	namely	namely	ADV
ejpam-6559	103	11	,	,	PUNCT
ejpam-6559	103	12	q(0	q(0	PROPN
ejpam-6559	103	13	)	)	PUNCT
ejpam-6559	103	14	=	=	SYM
ejpam-6559	103	15	q′(0	q′(0	PROPN
ejpam-6559	103	16	)	)	PUNCT
ejpam-6559	103	17	=	=	SYM
ejpam-6559	103	18	1	1	NUM
ejpam-6559	103	19	,	,	PUNCT
ejpam-6559	103	20	can	can	AUX
ejpam-6559	103	21	be	be	AUX
ejpam-6559	103	22	solved	solve	VERB
ejpam-6559	103	23	to	to	PART
ejpam-6559	103	24	give	give	VERB
ejpam-6559	103	25	q(t	q(t	PROPN
ejpam-6559	103	26	)	)	PUNCT
ejpam-6559	103	27	=	=	PUNCT
ejpam-6559	104	1	e−3	e−3	PROPN
ejpam-6559	104	2	t	t	NOUN
ejpam-6559	104	3	2	2	NUM
ejpam-6559	105	1	[	[	X
ejpam-6559	105	2	(	(	PUNCT
ejpam-6559	105	3	1−	1−	NUM
ejpam-6559	105	4	√	√	NUM
ejpam-6559	105	5	5	5	NUM
ejpam-6559	105	6	)	)	PUNCT
ejpam-6559	105	7	e	e	NOUN
ejpam-6559	105	8	3−	3−	NUM
ejpam-6559	105	9	√	√	NUM
ejpam-6559	105	10	5	5	NUM
ejpam-6559	105	11	2	2	NUM
ejpam-6559	105	12	t	t	NOUN
ejpam-6559	105	13	−	−	PROPN
ejpam-6559	105	14	(	(	PUNCT
ejpam-6559	105	15	1	1	NUM
ejpam-6559	105	16	+	+	CCONJ
ejpam-6559	105	17	√	√	NUM
ejpam-6559	105	18	5	5	NUM
ejpam-6559	105	19	)	)	PUNCT
ejpam-6559	105	20	e	e	NOUN
ejpam-6559	105	21	3	3	NUM
ejpam-6559	105	22	+	+	NUM
ejpam-6559	105	23	√	√	NUM
ejpam-6559	105	24	5	5	NUM
ejpam-6559	105	25	2	2	NUM
ejpam-6559	105	26	t	t	NOUN
ejpam-6559	105	27	]	]	PUNCT
ejpam-6559	105	28	.	.	PUNCT
ejpam-6559	106	1	(	(	PUNCT
ejpam-6559	106	2	11	11	X
ejpam-6559	106	3	)	)	PUNCT
ejpam-6559	106	4	w.	w.	PROPN
ejpam-6559	106	5	g.	g.	PROPN
ejpam-6559	106	6	alshanti	alshanti	PROPN
ejpam-6559	106	7	,	,	PUNCT
ejpam-6559	106	8	m.	m.	PROPN
ejpam-6559	106	9	abu	abu	PROPN
ejpam-6559	106	10	hammad	hammad	PROPN
ejpam-6559	106	11	,	,	PUNCT
ejpam-6559	106	12	r.	r.	PROPN
ejpam-6559	106	13	khalil	khalil	PROPN
ejpam-6559	106	14	/	/	SYM
ejpam-6559	106	15	eur	eur	PROPN
ejpam-6559	106	16	.	.	PUNCT
ejpam-6559	107	1	j.	j.	PROPN
ejpam-6559	107	2	pure	pure	PROPN
ejpam-6559	107	3	appl	appl	PROPN
ejpam-6559	107	4	.	.	PROPN
ejpam-6559	107	5	math	math	PROPN
ejpam-6559	107	6	,	,	PUNCT
ejpam-6559	107	7	18	18	NUM
ejpam-6559	107	8	(	(	PUNCT
ejpam-6559	107	9	4	4	NUM
ejpam-6559	107	10	)	)	PUNCT
ejpam-6559	107	11	(	(	PUNCT
ejpam-6559	107	12	2025	2025	NUM
ejpam-6559	107	13	)	)	PUNCT
ejpam-6559	107	14	,	,	PUNCT
ejpam-6559	107	15	6559	6559	NUM
ejpam-6559	107	16	5	5	NUM
ejpam-6559	107	17	of	of	ADP
ejpam-6559	107	18	8	8	NUM
ejpam-6559	107	19	therefore	therefore	ADV
ejpam-6559	107	20	,	,	PUNCT
ejpam-6559	107	21	for	for	ADP
ejpam-6559	107	22	all	all	DET
ejpam-6559	107	23	ω	ω	NUM
ejpam-6559	107	24	̸=	̸=	PROPN
ejpam-6559	107	25	0	0	NUM
ejpam-6559	107	26	∈	∈	PROPN
ejpam-6559	107	27	x	x	NOUN
ejpam-6559	107	28	,	,	PUNCT
ejpam-6559	107	29	an	an	DET
ejpam-6559	107	30	atomic	atomic	ADJ
ejpam-6559	107	31	solution	solution	NOUN
ejpam-6559	107	32	,	,	PUNCT
ejpam-6559	107	33	of	of	ADP
ejpam-6559	107	34	the	the	DET
ejpam-6559	107	35	form	form	NOUN
ejpam-6559	107	36	u	u	NOUN
ejpam-6559	107	37	=	=	PROPN
ejpam-6559	107	38	p	p	PROPN
ejpam-6559	107	39	⊗q	⊗q	PROPN
ejpam-6559	107	40	⊗	⊗	PROPN
ejpam-6559	107	41	ω	ω	PROPN
ejpam-6559	107	42	to	to	ADP
ejpam-6559	107	43	(	(	PUNCT
ejpam-6559	107	44	2	2	NUM
ejpam-6559	107	45	)	)	PUNCT
ejpam-6559	107	46	,	,	PUNCT
ejpam-6559	107	47	can	can	AUX
ejpam-6559	107	48	be	be	AUX
ejpam-6559	107	49	obtained	obtain	VERB
ejpam-6559	107	50	,	,	PUNCT
ejpam-6559	107	51	by	by	ADP
ejpam-6559	107	52	considering	consider	VERB
ejpam-6559	107	53	(	(	PUNCT
ejpam-6559	107	54	9	9	NUM
ejpam-6559	107	55	)	)	PUNCT
ejpam-6559	107	56	and	and	CCONJ
ejpam-6559	107	57	(	(	PUNCT
ejpam-6559	107	58	11	11	NUM
ejpam-6559	107	59	)	)	PUNCT
ejpam-6559	107	60	,	,	PUNCT
ejpam-6559	107	61	as	as	ADP
ejpam-6559	107	62	u(s	u(s	PROPN
ejpam-6559	107	63	,	,	PUNCT
ejpam-6559	107	64	t	t	NOUN
ejpam-6559	107	65	)	)	PUNCT
ejpam-6559	107	66	=	=	PUNCT
ejpam-6559	107	67	es−3	es−3	PROPN
ejpam-6559	107	68	t	t	NOUN
ejpam-6559	107	69	2	2	NUM
ejpam-6559	108	1	[	[	X
ejpam-6559	108	2	(	(	PUNCT
ejpam-6559	108	3	1−	1−	NUM
ejpam-6559	108	4	√	√	NUM
ejpam-6559	108	5	5	5	NUM
ejpam-6559	108	6	)	)	PUNCT
ejpam-6559	108	7	e	e	NOUN
ejpam-6559	108	8	3−	3−	NUM
ejpam-6559	108	9	√	√	NUM
ejpam-6559	108	10	5	5	NUM
ejpam-6559	108	11	2	2	NUM
ejpam-6559	108	12	t	t	NOUN
ejpam-6559	108	13	−	−	PROPN
ejpam-6559	108	14	(	(	PUNCT
ejpam-6559	108	15	1	1	NUM
ejpam-6559	108	16	+	+	CCONJ
ejpam-6559	108	17	√	√	NUM
ejpam-6559	108	18	5	5	NUM
ejpam-6559	108	19	)	)	PUNCT
ejpam-6559	108	20	e	e	NOUN
ejpam-6559	108	21	3	3	NUM
ejpam-6559	108	22	+	+	NUM
ejpam-6559	108	23	√	√	NUM
ejpam-6559	108	24	5	5	NUM
ejpam-6559	108	25	2	2	NUM
ejpam-6559	108	26	t	t	NOUN
ejpam-6559	108	27	]	]	PUNCT
ejpam-6559	108	28	⊗	⊗	PROPN
ejpam-6559	108	29	ω	ω	PROPN
ejpam-6559	108	30	.	.	PUNCT
ejpam-6559	109	1	(	(	PUNCT
ejpam-6559	109	2	12	12	NUM
ejpam-6559	109	3	)	)	PUNCT
ejpam-6559	109	4	case	case	NOUN
ejpam-6559	109	5	(	(	PUNCT
ejpam-6559	109	6	ii	ii	NOUN
ejpam-6559	109	7	):	):	PUNCT
ejpam-6559	109	8	this	this	DET
ejpam-6559	109	9	case	case	NOUN
ejpam-6559	109	10	reveals	reveal	VERB
ejpam-6559	109	11	three	three	NUM
ejpam-6559	109	12	sub	sub	NOUN
ejpam-6559	109	13	-	-	NOUN
ejpam-6559	109	14	cases	case	NOUN
ejpam-6559	109	15	,	,	PUNCT
ejpam-6559	109	16	namely	namely	ADV
ejpam-6559	109	17	,	,	PUNCT
ejpam-6559	109	18	q	q	PROPN
ejpam-6559	109	19	(	(	PUNCT
ejpam-6559	109	20	t	t	NOUN
ejpam-6559	109	21	)	)	PUNCT
ejpam-6559	109	22	=	=	SYM
ejpam-6559	110	1	2q′	2q′	NUM
ejpam-6559	110	2	(	(	PUNCT
ejpam-6559	110	3	t	t	PROPN
ejpam-6559	110	4	)	)	PUNCT
ejpam-6559	110	5	+	+	NOUN
ejpam-6559	110	6	q	q	X
ejpam-6559	110	7	(	(	PUNCT
ejpam-6559	110	8	t	t	PROPN
ejpam-6559	110	9	)	)	PUNCT
ejpam-6559	110	10	,	,	PUNCT
ejpam-6559	110	11	q	q	PROPN
ejpam-6559	110	12	(	(	PUNCT
ejpam-6559	110	13	t	t	NOUN
ejpam-6559	110	14	)	)	PUNCT
ejpam-6559	110	15	=	=	SYM
ejpam-6559	111	1	q	q	X
ejpam-6559	111	2	(	(	PUNCT
ejpam-6559	111	3	t	t	NOUN
ejpam-6559	111	4	)	)	PUNCT
ejpam-6559	111	5	−	−	PROPN
ejpam-6559	111	6	q′	q′	NOUN
ejpam-6559	111	7	(	(	PUNCT
ejpam-6559	111	8	t	t	PROPN
ejpam-6559	111	9	)	)	PUNCT
ejpam-6559	111	10	−	−	PROPN
ejpam-6559	112	1	q′′	q′′	PROPN
ejpam-6559	112	2	(	(	PUNCT
ejpam-6559	112	3	t	t	NOUN
ejpam-6559	112	4	)	)	PUNCT
ejpam-6559	112	5	,	,	PUNCT
ejpam-6559	112	6	and	and	CCONJ
ejpam-6559	112	7	2q′	2q′	NUM
ejpam-6559	112	8	(	(	PUNCT
ejpam-6559	112	9	t	t	PROPN
ejpam-6559	112	10	)	)	PUNCT
ejpam-6559	112	11	+	+	NOUN
ejpam-6559	112	12	q	q	X
ejpam-6559	112	13	(	(	PUNCT
ejpam-6559	112	14	t	t	NOUN
ejpam-6559	112	15	)	)	PUNCT
ejpam-6559	112	16	=	=	SYM
ejpam-6559	112	17	q	q	X
ejpam-6559	112	18	(	(	PUNCT
ejpam-6559	112	19	t	t	NOUN
ejpam-6559	112	20	)	)	PUNCT
ejpam-6559	113	1	−	−	PROPN
ejpam-6559	113	2	q′	q′	NOUN
ejpam-6559	113	3	(	(	PUNCT
ejpam-6559	113	4	t	t	PROPN
ejpam-6559	113	5	)	)	PUNCT
ejpam-6559	113	6	−	−	PROPN
ejpam-6559	114	1	q′′	q′′	PROPN
ejpam-6559	114	2	(	(	PUNCT
ejpam-6559	114	3	t	t	PROPN
ejpam-6559	114	4	)	)	PUNCT
ejpam-6559	114	5	.	.	PUNCT
ejpam-6559	115	1	correspondingly	correspondingly	ADV
ejpam-6559	115	2	,	,	PUNCT
ejpam-6559	115	3	we	we	PRON
ejpam-6559	115	4	have	have	VERB
ejpam-6559	115	5	q′	q′	NOUN
ejpam-6559	115	6	(	(	PUNCT
ejpam-6559	115	7	t	t	PROPN
ejpam-6559	115	8	)	)	PUNCT
ejpam-6559	115	9	=	=	SYM
ejpam-6559	115	10	0	0	NUM
ejpam-6559	115	11	,	,	PUNCT
ejpam-6559	115	12	q′′	q′′	ADP
ejpam-6559	115	13	(	(	PUNCT
ejpam-6559	115	14	t	t	NOUN
ejpam-6559	115	15	)	)	PUNCT
ejpam-6559	116	1	+	+	NUM
ejpam-6559	116	2	q′	q′	NOUN
ejpam-6559	116	3	(	(	PUNCT
ejpam-6559	116	4	t	t	NOUN
ejpam-6559	116	5	)	)	PUNCT
ejpam-6559	116	6	=	=	SYM
ejpam-6559	116	7	0	0	NUM
ejpam-6559	116	8	,	,	PUNCT
ejpam-6559	116	9	and	and	CCONJ
ejpam-6559	116	10	q′′	q′′	ADV
ejpam-6559	116	11	(	(	PUNCT
ejpam-6559	116	12	t	t	NOUN
ejpam-6559	116	13	)	)	PUNCT
ejpam-6559	116	14	+	+	NUM
ejpam-6559	116	15	3q′	3q′	NUM
ejpam-6559	116	16	(	(	PUNCT
ejpam-6559	116	17	t	t	NOUN
ejpam-6559	116	18	)	)	PUNCT
ejpam-6559	116	19	=	=	NOUN
ejpam-6559	117	1	0	0	X
ejpam-6559	117	2	.	.	PUNCT
ejpam-6559	118	1	hence	hence	ADV
ejpam-6559	118	2	,	,	PUNCT
ejpam-6559	118	3	together	together	ADV
ejpam-6559	118	4	with	with	ADP
ejpam-6559	118	5	appropriate	appropriate	ADJ
ejpam-6559	118	6	conditions	condition	NOUN
ejpam-6559	118	7	from	from	ADP
ejpam-6559	118	8	(	(	PUNCT
ejpam-6559	118	9	7	7	NUM
ejpam-6559	118	10	)	)	PUNCT
ejpam-6559	118	11	,	,	PUNCT
ejpam-6559	118	12	we	we	PRON
ejpam-6559	118	13	get	get	VERB
ejpam-6559	118	14	the	the	DET
ejpam-6559	118	15	following	follow	VERB
ejpam-6559	118	16	classical	classical	ADJ
ejpam-6559	118	17	initial	initial	ADJ
ejpam-6559	118	18	value	value	NOUN
ejpam-6559	118	19	problems	problem	NOUN
ejpam-6559	118	20	q′	q′	VERB
ejpam-6559	118	21	(	(	PUNCT
ejpam-6559	118	22	t	t	PROPN
ejpam-6559	118	23	)	)	PUNCT
ejpam-6559	118	24	=	=	SYM
ejpam-6559	119	1	0	0	NUM
ejpam-6559	119	2	,	,	PUNCT
ejpam-6559	119	3	q(0	q(0	PROPN
ejpam-6559	119	4	)	)	PUNCT
ejpam-6559	119	5	=	=	SYM
ejpam-6559	120	1	1	1	NUM
ejpam-6559	120	2	,	,	PUNCT
ejpam-6559	120	3	q′′	q′′	PROPN
ejpam-6559	120	4	(	(	PUNCT
ejpam-6559	120	5	t	t	NOUN
ejpam-6559	120	6	)	)	PUNCT
ejpam-6559	120	7	+	+	NOUN
ejpam-6559	120	8	q′	q′	NOUN
ejpam-6559	120	9	(	(	PUNCT
ejpam-6559	120	10	t	t	PROPN
ejpam-6559	120	11	)	)	PUNCT
ejpam-6559	120	12	=	=	SYM
ejpam-6559	120	13	0	0	NUM
ejpam-6559	120	14	,	,	PUNCT
ejpam-6559	120	15	q′(0	q′(0	PROPN
ejpam-6559	120	16	)	)	PUNCT
ejpam-6559	121	1	=	=	SYM
ejpam-6559	121	2	q(0	q(0	NOUN
ejpam-6559	121	3	)	)	PUNCT
ejpam-6559	121	4	=	=	SYM
ejpam-6559	121	5	1	1	NUM
ejpam-6559	121	6	,	,	PUNCT
ejpam-6559	121	7	q′′	q′′	PROPN
ejpam-6559	121	8	(	(	PUNCT
ejpam-6559	121	9	t	t	NOUN
ejpam-6559	121	10	)	)	PUNCT
ejpam-6559	122	1	+	+	NUM
ejpam-6559	122	2	3q′	3q′	NUM
ejpam-6559	122	3	(	(	PUNCT
ejpam-6559	122	4	t	t	NOUN
ejpam-6559	122	5	)	)	PUNCT
ejpam-6559	122	6	=	=	SYM
ejpam-6559	122	7	0	0	NUM
ejpam-6559	122	8	,	,	PUNCT
ejpam-6559	122	9	q′(0	q′(0	PROPN
ejpam-6559	122	10	)	)	PUNCT
ejpam-6559	123	1	=	=	SYM
ejpam-6559	123	2	q(0	q(0	NOUN
ejpam-6559	123	3	)	)	PUNCT
ejpam-6559	123	4	=	=	SYM
ejpam-6559	124	1	1	1	X
ejpam-6559	124	2	.	.	PUNCT
ejpam-6559	124	3	(	(	PUNCT
ejpam-6559	124	4	13	13	NUM
ejpam-6559	124	5	)	)	PUNCT
ejpam-6559	124	6	the	the	DET
ejpam-6559	124	7	corresponding	corresponding	ADJ
ejpam-6559	124	8	solution	solution	NOUN
ejpam-6559	124	9	to	to	ADP
ejpam-6559	124	10	each	each	DET
ejpam-6559	124	11	initial	initial	ADJ
ejpam-6559	124	12	value	value	NOUN
ejpam-6559	124	13	problem	problem	NOUN
ejpam-6559	124	14	,	,	PUNCT
ejpam-6559	124	15	respectively	respectively	ADV
ejpam-6559	124	16	,	,	PUNCT
ejpam-6559	124	17	is	be	AUX
ejpam-6559	124	18	q(t	q(t	NOUN
ejpam-6559	124	19	)	)	PUNCT
ejpam-6559	124	20	=	=	SYM
ejpam-6559	124	21	1	1	NUM
ejpam-6559	124	22	,	,	PUNCT
ejpam-6559	124	23	q(t	q(t	ADJ
ejpam-6559	124	24	)	)	PUNCT
ejpam-6559	124	25	=	=	SYM
ejpam-6559	124	26	2	2	NUM
ejpam-6559	124	27	−	−	NOUN
ejpam-6559	124	28	e−t	e−t	NOUN
ejpam-6559	124	29	,	,	PUNCT
ejpam-6559	124	30	and	and	CCONJ
ejpam-6559	124	31	q(t	q(t	X
ejpam-6559	124	32	)	)	PUNCT
ejpam-6559	124	33	=	=	SYM
ejpam-6559	124	34	4	4	NUM
ejpam-6559	124	35	3	3	NUM
ejpam-6559	124	36	−	−	NOUN
ejpam-6559	125	1	e−3	e−3	PROPN
ejpam-6559	125	2	t	t	PROPN
ejpam-6559	125	3	3	3	NUM
ejpam-6559	125	4	.	.	PUNCT
ejpam-6559	126	1	different	different	ADJ
ejpam-6559	126	2	form	form	NOUN
ejpam-6559	126	3	of	of	ADP
ejpam-6559	126	4	q(t	q(t	NOUN
ejpam-6559	126	5	)	)	PUNCT
ejpam-6559	126	6	for	for	ADP
ejpam-6559	126	7	each	each	PRON
ejpam-6559	126	8	particular	particular	ADJ
ejpam-6559	126	9	sub	sub	NOUN
ejpam-6559	126	10	-	-	NOUN
ejpam-6559	126	11	case	case	NOUN
ejpam-6559	126	12	means	mean	VERB
ejpam-6559	126	13	that	that	SCONJ
ejpam-6559	126	14	case(ii	case(ii	NOUN
ejpam-6559	126	15	)	)	PUNCT
ejpam-6559	126	16	does	do	AUX
ejpam-6559	126	17	not	not	PART
ejpam-6559	126	18	admit	admit	VERB
ejpam-6559	126	19	an	an	DET
ejpam-6559	126	20	atomic	atomic	ADJ
ejpam-6559	126	21	solution	solution	NOUN
ejpam-6559	126	22	.	.	PUNCT
ejpam-6559	127	1	so	so	ADV
ejpam-6559	127	2	,	,	PUNCT
ejpam-6559	127	3	the	the	DET
ejpam-6559	127	4	only	only	ADJ
ejpam-6559	127	5	atomic	atomic	ADJ
ejpam-6559	127	6	solution	solution	NOUN
ejpam-6559	127	7	of	of	ADP
ejpam-6559	127	8	the	the	DET
ejpam-6559	127	9	form	form	NOUN
ejpam-6559	127	10	u	u	NOUN
ejpam-6559	127	11	=	=	PROPN
ejpam-6559	127	12	pq⊗	pq⊗	PROPN
ejpam-6559	127	13	ω	ω	PROPN
ejpam-6559	127	14	,	,	PUNCT
ejpam-6559	127	15	where	where	SCONJ
ejpam-6559	127	16	ω	ω	NOUN
ejpam-6559	127	17	̸=	̸=	NOUN
ejpam-6559	127	18	0	0	NUM
ejpam-6559	127	19	∈	∈	PROPN
ejpam-6559	127	20	x	x	PUNCT
ejpam-6559	127	21	to	to	ADP
ejpam-6559	127	22	(	(	PUNCT
ejpam-6559	127	23	2	2	NUM
ejpam-6559	127	24	)	)	PUNCT
ejpam-6559	127	25	is	be	AUX
ejpam-6559	127	26	the	the	DET
ejpam-6559	127	27	one	one	NUM
ejpam-6559	127	28	obtained	obtain	VERB
ejpam-6559	127	29	by	by	ADP
ejpam-6559	127	30	case(i	case(i	PROPN
ejpam-6559	127	31	)	)	PUNCT
ejpam-6559	127	32	which	which	PRON
ejpam-6559	127	33	is	be	AUX
ejpam-6559	127	34	described	describe	VERB
ejpam-6559	127	35	in	in	ADP
ejpam-6559	127	36	(	(	PUNCT
ejpam-6559	127	37	12	12	NUM
ejpam-6559	127	38	)	)	PUNCT
ejpam-6559	127	39	.	.	PUNCT
ejpam-6559	128	1	therefore	therefore	ADV
ejpam-6559	128	2	,	,	PUNCT
ejpam-6559	128	3	the	the	DET
ejpam-6559	128	4	proof	proof	NOUN
ejpam-6559	128	5	is	be	AUX
ejpam-6559	128	6	complete	complete	ADJ
ejpam-6559	128	7	.	.	PUNCT
ejpam-6559	129	1	4	4	X
ejpam-6559	129	2	.	.	X
ejpam-6559	129	3	application	application	NOUN
ejpam-6559	129	4	in	in	ADP
ejpam-6559	129	5	this	this	DET
ejpam-6559	129	6	section	section	NOUN
ejpam-6559	129	7	,	,	PUNCT
ejpam-6559	129	8	we	we	PRON
ejpam-6559	129	9	present	present	VERB
ejpam-6559	129	10	an	an	DET
ejpam-6559	129	11	example	example	NOUN
ejpam-6559	129	12	to	to	PART
ejpam-6559	129	13	illustrate	illustrate	VERB
ejpam-6559	129	14	the	the	DET
ejpam-6559	129	15	analytical	analytical	ADJ
ejpam-6559	129	16	method	method	NOUN
ejpam-6559	129	17	described	describe	VERB
ejpam-6559	129	18	in	in	ADP
ejpam-6559	129	19	the	the	DET
ejpam-6559	129	20	previous	previous	ADJ
ejpam-6559	129	21	section	section	NOUN
ejpam-6559	129	22	.	.	PUNCT
ejpam-6559	130	1	let	let	VERB
ejpam-6559	130	2	x	x	PRON
ejpam-6559	130	3	is	be	AUX
ejpam-6559	130	4	a	a	DET
ejpam-6559	130	5	banach	banach	NOUN
ejpam-6559	130	6	space	space	NOUN
ejpam-6559	130	7	and	and	CCONJ
ejpam-6559	130	8	a	a	DET
ejpam-6559	130	9	,	,	PUNCT
ejpam-6559	130	10	b	b	NOUN
ejpam-6559	130	11	are	be	AUX
ejpam-6559	130	12	closed	close	VERB
ejpam-6559	130	13	linear	linear	ADJ
ejpam-6559	130	14	operators	operator	NOUN
ejpam-6559	130	15	on	on	ADP
ejpam-6559	130	16	x	x	SYM
ejpam-6559	130	17	such	such	ADJ
ejpam-6559	130	18	that	that	SCONJ
ejpam-6559	130	19	a	a	DET
ejpam-6559	130	20	:	:	PUNCT
ejpam-6559	130	21	dom(a	dom(a	PROPN
ejpam-6559	130	22	)	)	PUNCT
ejpam-6559	130	23	⊆	⊆	NUM
ejpam-6559	130	24	x	x	SYM
ejpam-6559	130	25	→	→	SYM
ejpam-6559	130	26	x	x	PROPN
ejpam-6559	130	27	,	,	PUNCT
ejpam-6559	130	28	b	b	NOUN
ejpam-6559	130	29	:	:	PUNCT
ejpam-6559	130	30	dom(b	dom(b	PROPN
ejpam-6559	130	31	)	)	PUNCT
ejpam-6559	130	32	⊆	⊆	NUM
ejpam-6559	130	33	x	x	SYM
ejpam-6559	130	34	→	→	SYM
ejpam-6559	130	35	x.	x.	NOUN
ejpam-6559	130	36	moreover	moreover	ADV
ejpam-6559	130	37	,	,	PUNCT
ejpam-6559	130	38	assume	assume	VERB
ejpam-6559	130	39	u(t	u(t	NOUN
ejpam-6559	130	40	)	)	PUNCT
ejpam-6559	130	41	:	:	PUNCT
ejpam-6559	131	1	[	[	X
ejpam-6559	131	2	0	0	NUM
ejpam-6559	131	3	,	,	PUNCT
ejpam-6559	131	4	1	1	NUM
ejpam-6559	131	5	]	]	PUNCT
ejpam-6559	131	6	→	→	PUNCT
ejpam-6559	131	7	x	x	X
ejpam-6559	131	8	is	be	AUX
ejpam-6559	131	9	an	an	DET
ejpam-6559	131	10	unknown	unknown	ADJ
ejpam-6559	131	11	twice	twice	ADV
ejpam-6559	131	12	-	-	PUNCT
ejpam-6559	131	13	continuously	continuously	ADV
ejpam-6559	131	14	differentiable	differentiable	ADJ
ejpam-6559	131	15	function	function	NOUN
ejpam-6559	131	16	on	on	ADP
ejpam-6559	131	17	[	[	X
ejpam-6559	131	18	0	0	NUM
ejpam-6559	131	19	,	,	PUNCT
ejpam-6559	131	20	1	1	NUM
ejpam-6559	131	21	]	]	PUNCT
ejpam-6559	131	22	⊂	⊂	PROPN
ejpam-6559	131	23	r	r	X
ejpam-6559	131	24	where	where	SCONJ
ejpam-6559	131	25	rang(u	rang(u	ADJ
ejpam-6559	131	26	)	)	PUNCT
ejpam-6559	131	27	⊆	⊆	NUM
ejpam-6559	131	28	dom(a	dom(a	PROPN
ejpam-6559	131	29	)	)	PUNCT
ejpam-6559	131	30	∩dom(b	∩dom(b	NOUN
ejpam-6559	131	31	)	)	PUNCT
ejpam-6559	131	32	.	.	PUNCT
ejpam-6559	132	1	now	now	ADV
ejpam-6559	132	2	,	,	PUNCT
ejpam-6559	132	3	consider	consider	VERB
ejpam-6559	132	4	the	the	DET
ejpam-6559	132	5	following	follow	VERB
ejpam-6559	132	6	first	first	ADJ
ejpam-6559	132	7	-	-	PUNCT
ejpam-6559	132	8	order	order	NOUN
ejpam-6559	132	9	acp	acp	PROPN
ejpam-6559	132	10	in	in	ADP
ejpam-6559	132	11	one	one	NUM
ejpam-6559	132	12	variable	variable	NOUN
ejpam-6559	132	13	,	,	PUNCT
ejpam-6559	132	14	u′′(t	u′′(t	NOUN
ejpam-6559	132	15	)	)	PUNCT
ejpam-6559	133	1	+	+	X
ejpam-6559	133	2	au′(t	au′(t	NOUN
ejpam-6559	133	3	)	)	PUNCT
ejpam-6559	133	4	+	+	NOUN
ejpam-6559	133	5	bu(t	bu(t	NOUN
ejpam-6559	133	6	)	)	PUNCT
ejpam-6559	133	7	=	=	SYM
ejpam-6559	133	8	0	0	NUM
ejpam-6559	133	9	,	,	PUNCT
ejpam-6559	133	10	(	(	PUNCT
ejpam-6559	133	11	14	14	NUM
ejpam-6559	133	12	)	)	PUNCT
ejpam-6559	133	13	let	let	VERB
ejpam-6559	133	14	v	v	X
ejpam-6559	133	15	∈	∈	PROPN
ejpam-6559	133	16	c2	c2	PROPN
ejpam-6559	134	1	[	[	X
ejpam-6559	134	2	0	0	NUM
ejpam-6559	134	3	,	,	PUNCT
ejpam-6559	134	4	1	1	NUM
ejpam-6559	134	5	]	]	PUNCT
ejpam-6559	134	6	such	such	ADJ
ejpam-6559	134	7	that	that	DET
ejpam-6559	134	8	v	v	NOUN
ejpam-6559	134	9	:	:	PUNCT
ejpam-6559	134	10	=	=	SYM
ejpam-6559	134	11	v(t	v(t	NUM
ejpam-6559	134	12	)	)	PUNCT
ejpam-6559	134	13	:	:	PUNCT
ejpam-6559	135	1	[	[	X
ejpam-6559	135	2	0	0	NUM
ejpam-6559	135	3	,	,	PUNCT
ejpam-6559	135	4	1	1	NUM
ejpam-6559	135	5	]	]	PUNCT
ejpam-6559	135	6	→	→	PUNCT
ejpam-6559	135	7	x.	x.	NOUN
ejpam-6559	135	8	suppose	suppose	VERB
ejpam-6559	135	9	u(t	u(t	NOUN
ejpam-6559	135	10	)	)	PUNCT
ejpam-6559	135	11	,	,	PUNCT
ejpam-6559	135	12	the	the	DET
ejpam-6559	135	13	solution	solution	NOUN
ejpam-6559	135	14	to	to	ADP
ejpam-6559	135	15	(	(	PUNCT
ejpam-6559	135	16	14	14	NUM
ejpam-6559	135	17	)	)	PUNCT
ejpam-6559	135	18	,	,	PUNCT
ejpam-6559	135	19	be	be	AUX
ejpam-6559	135	20	in	in	ADP
ejpam-6559	135	21	the	the	DET
ejpam-6559	135	22	following	follow	VERB
ejpam-6559	135	23	atom	atom	NOUN
ejpam-6559	135	24	form	form	NOUN
ejpam-6559	135	25	u	u	NOUN
ejpam-6559	135	26	=	=	NOUN
ejpam-6559	135	27	v⊗ω	v⊗ω	NOUN
ejpam-6559	135	28	,	,	PUNCT
ejpam-6559	135	29	where	where	SCONJ
ejpam-6559	135	30	ω	ω	NOUN
ejpam-6559	135	31	̸=	̸=	PROPN
ejpam-6559	135	32	0	0	NUM
ejpam-6559	135	33	∈	∈	PROPN
ejpam-6559	135	34	x.	x.	NOUN
ejpam-6559	135	35	on	on	ADP
ejpam-6559	135	36	substituting	substitute	VERB
ejpam-6559	135	37	this	this	DET
ejpam-6559	135	38	atom	atom	NOUN
ejpam-6559	135	39	operator	operator	NOUN
ejpam-6559	135	40	form	form	NOUN
ejpam-6559	135	41	into	into	ADP
ejpam-6559	135	42	(	(	PUNCT
ejpam-6559	135	43	14	14	NUM
ejpam-6559	135	44	)	)	PUNCT
ejpam-6559	135	45	we	we	PRON
ejpam-6559	135	46	obtain	obtain	VERB
ejpam-6559	135	47	the	the	DET
ejpam-6559	135	48	following	follow	VERB
ejpam-6559	135	49	tensor	tensor	NOUN
ejpam-6559	135	50	product	product	NOUN
ejpam-6559	135	51	equation	equation	NOUN
ejpam-6559	135	52	v′′(t)⊗	v′′(t)⊗	X
ejpam-6559	135	53	ω	ω	PROPN
ejpam-6559	136	1	+	+	CCONJ
ejpam-6559	136	2	v′(t)⊗aω	v′(t)⊗aω	NOUN
ejpam-6559	136	3	+	+	CCONJ
ejpam-6559	136	4	v(t)⊗bω	v(t)⊗bω	NOUN
ejpam-6559	136	5	=	=	NOUN
ejpam-6559	136	6	0	0	PROPN
ejpam-6559	136	7	.	.	PUNCT
ejpam-6559	137	1	(	(	PUNCT
ejpam-6559	137	2	15	15	NUM
ejpam-6559	137	3	)	)	PUNCT
ejpam-6559	137	4	assume	assume	VERB
ejpam-6559	137	5	that	that	SCONJ
ejpam-6559	137	6	{	{	PUNCT
ejpam-6559	137	7	ω	ω	NOUN
ejpam-6559	137	8	,	,	PUNCT
ejpam-6559	137	9	aω	aω	PROPN
ejpam-6559	137	10	,	,	PUNCT
ejpam-6559	137	11	bω	bω	VERB
ejpam-6559	137	12	}	}	PUNCT
ejpam-6559	137	13	are	be	AUX
ejpam-6559	137	14	independent	independent	ADJ
ejpam-6559	137	15	in	in	ADP
ejpam-6559	137	16	x.	x.	NOUN
ejpam-6559	137	17	hence	hence	ADV
ejpam-6559	137	18	,	,	PUNCT
ejpam-6559	137	19	by	by	ADP
ejpam-6559	137	20	corollary	corollary	ADJ
ejpam-6559	137	21	1	1	NUM
ejpam-6559	137	22	,	,	PUNCT
ejpam-6559	137	23	one	one	PRON
ejpam-6559	137	24	can	can	AUX
ejpam-6559	137	25	assume	assume	VERB
ejpam-6559	137	26	that	that	SCONJ
ejpam-6559	137	27	there	there	PRON
ejpam-6559	137	28	exists	exist	VERB
ejpam-6559	137	29	x∗	x∗	PROPN
ejpam-6559	137	30	∈	∈	PROPN
ejpam-6559	137	31	x	x	PUNCT
ejpam-6559	137	32	such	such	ADJ
ejpam-6559	137	33	that	that	SCONJ
ejpam-6559	137	34	⟨x∗	⟨x∗	PROPN
ejpam-6559	137	35	,	,	PUNCT
ejpam-6559	137	36	ω⟩	ω⟩	PUNCT
ejpam-6559	137	37	=	=	SYM
ejpam-6559	137	38	⟨x∗	⟨x∗	PROPN
ejpam-6559	137	39	,	,	PUNCT
ejpam-6559	137	40	aω⟩	aω⟩	PUNCT
ejpam-6559	137	41	=	=	SYM
ejpam-6559	137	42	⟨x∗	⟨x∗	PROPN
ejpam-6559	137	43	,	,	PUNCT
ejpam-6559	137	44	bω⟩	bω⟩	ADJ
ejpam-6559	137	45	=	=	SYM
ejpam-6559	137	46	1	1	NUM
ejpam-6559	137	47	,	,	PUNCT
ejpam-6559	137	48	where	where	SCONJ
ejpam-6559	137	49	⟨x∗	⟨x∗	PROPN
ejpam-6559	137	50	,	,	PUNCT
ejpam-6559	137	51	·	·	PUNCT
ejpam-6559	137	52	⟩	⟩	NOUN
ejpam-6559	137	53	:	:	PUNCT
ejpam-6559	137	54	=	=	SYM
ejpam-6559	137	55	x∗	x∗	X
ejpam-6559	137	56	(	(	PUNCT
ejpam-6559	137	57	·	·	PUNCT
ejpam-6559	137	58	)	)	PUNCT
ejpam-6559	137	59	.	.	PUNCT
ejpam-6559	138	1	therefore	therefore	ADV
ejpam-6559	138	2	,	,	PUNCT
ejpam-6559	138	3	on	on	ADP
ejpam-6559	138	4	applying	apply	VERB
ejpam-6559	138	5	such	such	ADJ
ejpam-6559	138	6	x∗	x∗	PROPN
ejpam-6559	138	7	to	to	ADP
ejpam-6559	138	8	both	both	DET
ejpam-6559	138	9	sides	side	NOUN
ejpam-6559	138	10	of	of	ADP
ejpam-6559	138	11	(	(	PUNCT
ejpam-6559	138	12	15	15	NUM
ejpam-6559	138	13	)	)	PUNCT
ejpam-6559	138	14	,	,	PUNCT
ejpam-6559	138	15	we	we	PRON
ejpam-6559	138	16	get	get	VERB
ejpam-6559	138	17	v′′(t	v′′(t	VERB
ejpam-6559	138	18	)	)	PUNCT
ejpam-6559	139	1	+	+	CCONJ
ejpam-6559	139	2	v′(t	v′(t	ADJ
ejpam-6559	139	3	)	)	PUNCT
ejpam-6559	139	4	+	+	CCONJ
ejpam-6559	139	5	v(t	v(t	VERB
ejpam-6559	139	6	)	)	PUNCT
ejpam-6559	139	7	=	=	SYM
ejpam-6559	140	1	0	0	X
ejpam-6559	140	2	.	.	PUNCT
ejpam-6559	141	1	(	(	PUNCT
ejpam-6559	141	2	16	16	NUM
ejpam-6559	141	3	)	)	PUNCT
ejpam-6559	141	4	by	by	ADP
ejpam-6559	141	5	assuming	assume	VERB
ejpam-6559	141	6	v(0	v(0	NOUN
ejpam-6559	141	7	)	)	PUNCT
ejpam-6559	141	8	=	=	SYM
ejpam-6559	142	1	1	1	NUM
ejpam-6559	142	2	,	,	PUNCT
ejpam-6559	142	3	we	we	PRON
ejpam-6559	142	4	have	have	VERB
ejpam-6559	142	5	v(t	v(t	NOUN
ejpam-6559	142	6	)	)	PUNCT
ejpam-6559	142	7	=	=	PUNCT
ejpam-6559	142	8	2e	2e	PROPN
ejpam-6559	142	9	−t	−t	NOUN
ejpam-6559	142	10	2	2	NUM
ejpam-6559	142	11	sin	sin	NOUN
ejpam-6559	142	12	(	(	PUNCT
ejpam-6559	142	13	√	√	NUM
ejpam-6559	142	14	3	3	NUM
ejpam-6559	142	15	2	2	NUM
ejpam-6559	142	16	t+	t+	NOUN
ejpam-6559	142	17	π	π	PROPN
ejpam-6559	142	18	6	6	NUM
ejpam-6559	142	19	)	)	PUNCT
ejpam-6559	142	20	and	and	CCONJ
ejpam-6559	142	21	therefore	therefore	ADV
ejpam-6559	142	22	,	,	PUNCT
ejpam-6559	142	23	(	(	PUNCT
ejpam-6559	142	24	14	14	NUM
ejpam-6559	142	25	)	)	PUNCT
ejpam-6559	142	26	has	have	VERB
ejpam-6559	142	27	the	the	DET
ejpam-6559	142	28	following	follow	VERB
ejpam-6559	142	29	atomic	atomic	ADJ
ejpam-6559	142	30	solution	solution	NOUN
ejpam-6559	142	31	u	u	NOUN
ejpam-6559	142	32	=	=	PROPN
ejpam-6559	142	33	v	v	PROPN
ejpam-6559	142	34	⊗	⊗	PROPN
ejpam-6559	142	35	ω	ω	PROPN
ejpam-6559	142	36	w.	w.	PROPN
ejpam-6559	142	37	g.	g.	PROPN
ejpam-6559	142	38	alshanti	alshanti	PROPN
ejpam-6559	142	39	,	,	PUNCT
ejpam-6559	142	40	m.	m.	PROPN
ejpam-6559	142	41	abu	abu	PROPN
ejpam-6559	142	42	hammad	hammad	PROPN
ejpam-6559	142	43	,	,	PUNCT
ejpam-6559	142	44	r.	r.	PROPN
ejpam-6559	142	45	khalil	khalil	PROPN
ejpam-6559	142	46	/	/	SYM
ejpam-6559	142	47	eur	eur	PROPN
ejpam-6559	142	48	.	.	PUNCT
ejpam-6559	143	1	j.	j.	PROPN
ejpam-6559	143	2	pure	pure	PROPN
ejpam-6559	143	3	appl	appl	PROPN
ejpam-6559	143	4	.	.	PROPN
ejpam-6559	143	5	math	math	PROPN
ejpam-6559	143	6	,	,	PUNCT
ejpam-6559	143	7	18	18	NUM
ejpam-6559	143	8	(	(	PUNCT
ejpam-6559	143	9	4	4	NUM
ejpam-6559	143	10	)	)	PUNCT
ejpam-6559	143	11	(	(	PUNCT
ejpam-6559	143	12	2025	2025	NUM
ejpam-6559	143	13	)	)	PUNCT
ejpam-6559	143	14	,	,	PUNCT
ejpam-6559	143	15	6559	6559	NUM
ejpam-6559	143	16	6	6	NUM
ejpam-6559	143	17	of	of	ADP
ejpam-6559	143	18	8	8	NUM
ejpam-6559	143	19	=	=	SYM
ejpam-6559	143	20	2e	2e	NOUN
ejpam-6559	143	21	−t	−t	NOUN
ejpam-6559	143	22	2	2	NUM
ejpam-6559	143	23	sin	sin	NOUN
ejpam-6559	143	24	(	(	PUNCT
ejpam-6559	143	25	√	√	NUM
ejpam-6559	143	26	3	3	NUM
ejpam-6559	143	27	2	2	NUM
ejpam-6559	143	28	t+	t+	NOUN
ejpam-6559	143	29	π	π	PROPN
ejpam-6559	143	30	6	6	NUM
ejpam-6559	143	31	)	)	PUNCT
ejpam-6559	143	32	⊗	⊗	PROPN
ejpam-6559	143	33	ω	ω	PROPN
ejpam-6559	143	34	,	,	PUNCT
ejpam-6559	143	35	where	where	SCONJ
ejpam-6559	143	36	ω	ω	NOUN
ejpam-6559	143	37	̸=	̸=	PROPN
ejpam-6559	143	38	0	0	NUM
ejpam-6559	143	39	∈	∈	PROPN
ejpam-6559	143	40	x.	x.	NOUN
ejpam-6559	143	41	apart	apart	ADV
ejpam-6559	143	42	from	from	ADP
ejpam-6559	143	43	the	the	DET
ejpam-6559	143	44	example	example	NOUN
ejpam-6559	143	45	discussed	discuss	VERB
ejpam-6559	143	46	above	above	ADV
ejpam-6559	143	47	,	,	PUNCT
ejpam-6559	143	48	the	the	DET
ejpam-6559	143	49	ideas	idea	NOUN
ejpam-6559	143	50	developed	develop	VERB
ejpam-6559	143	51	here	here	ADV
ejpam-6559	143	52	may	may	AUX
ejpam-6559	143	53	be	be	AUX
ejpam-6559	143	54	applied	apply	VERB
ejpam-6559	143	55	in	in	ADP
ejpam-6559	143	56	several	several	ADJ
ejpam-6559	143	57	directions	direction	NOUN
ejpam-6559	143	58	.	.	PUNCT
ejpam-6559	144	1	in	in	ADP
ejpam-6559	144	2	control	control	NOUN
ejpam-6559	144	3	theory	theory	NOUN
ejpam-6559	144	4	,	,	PUNCT
ejpam-6559	144	5	many	many	ADJ
ejpam-6559	144	6	systems	system	NOUN
ejpam-6559	144	7	described	describe	VERB
ejpam-6559	144	8	by	by	ADP
ejpam-6559	144	9	partial	partial	ADJ
ejpam-6559	144	10	differential	differential	ADJ
ejpam-6559	144	11	equations	equation	NOUN
ejpam-6559	144	12	(	(	PUNCT
ejpam-6559	144	13	pdes	pde	NOUN
ejpam-6559	144	14	)	)	PUNCT
ejpam-6559	144	15	can	can	AUX
ejpam-6559	144	16	be	be	AUX
ejpam-6559	144	17	formulated	formulate	VERB
ejpam-6559	144	18	as	as	ADP
ejpam-6559	144	19	acps	acp	NOUN
ejpam-6559	144	20	in	in	ADP
ejpam-6559	144	21	banach	banach	NOUN
ejpam-6559	144	22	or	or	CCONJ
ejpam-6559	144	23	hilbert	hilbert	NOUN
ejpam-6559	144	24	spaces	space	NOUN
ejpam-6559	144	25	.	.	PUNCT
ejpam-6559	145	1	this	this	PRON
ejpam-6559	145	2	is	be	AUX
ejpam-6559	145	3	often	often	ADV
ejpam-6559	145	4	the	the	DET
ejpam-6559	145	5	case	case	NOUN
ejpam-6559	145	6	for	for	ADP
ejpam-6559	145	7	distributed	distribute	VERB
ejpam-6559	145	8	parameter	parameter	NOUN
ejpam-6559	145	9	systems	system	NOUN
ejpam-6559	145	10	such	such	ADJ
ejpam-6559	145	11	as	as	ADP
ejpam-6559	145	12	vibrating	vibrating	NOUN
ejpam-6559	145	13	beams	beam	NOUN
ejpam-6559	145	14	,	,	PUNCT
ejpam-6559	145	15	heat	heat	NOUN
ejpam-6559	145	16	exchangers	exchanger	NOUN
ejpam-6559	145	17	,	,	PUNCT
ejpam-6559	145	18	or	or	CCONJ
ejpam-6559	145	19	models	model	NOUN
ejpam-6559	145	20	involving	involve	VERB
ejpam-6559	145	21	fluid	fluid	ADJ
ejpam-6559	145	22	structure	structure	NOUN
ejpam-6559	145	23	interaction	interaction	NOUN
ejpam-6559	145	24	[	[	X
ejpam-6559	145	25	15	15	NUM
ejpam-6559	145	26	]	]	PUNCT
ejpam-6559	145	27	,	,	PUNCT
ejpam-6559	145	28	[	[	X
ejpam-6559	145	29	16	16	NUM
ejpam-6559	145	30	]	]	PUNCT
ejpam-6559	145	31	.	.	PUNCT
ejpam-6559	146	1	by	by	ADP
ejpam-6559	146	2	expressing	express	VERB
ejpam-6559	146	3	solutions	solution	NOUN
ejpam-6559	146	4	in	in	ADP
ejpam-6559	146	5	the	the	DET
ejpam-6559	146	6	atomic	atomic	ADJ
ejpam-6559	146	7	form	form	NOUN
ejpam-6559	146	8	used	use	VERB
ejpam-6559	146	9	here	here	ADV
ejpam-6559	146	10	,	,	PUNCT
ejpam-6559	146	11	the	the	DET
ejpam-6559	146	12	governing	govern	VERB
ejpam-6559	146	13	equations	equation	NOUN
ejpam-6559	146	14	can	can	AUX
ejpam-6559	146	15	sometimes	sometimes	ADV
ejpam-6559	146	16	be	be	AUX
ejpam-6559	146	17	reduced	reduce	VERB
ejpam-6559	146	18	to	to	ADP
ejpam-6559	146	19	simpler	simple	ADJ
ejpam-6559	146	20	,	,	PUNCT
ejpam-6559	146	21	separable	separable	ADJ
ejpam-6559	146	22	problems	problem	NOUN
ejpam-6559	146	23	.	.	PUNCT
ejpam-6559	147	1	such	such	ADJ
ejpam-6559	147	2	reductions	reduction	NOUN
ejpam-6559	147	3	may	may	AUX
ejpam-6559	147	4	help	help	VERB
ejpam-6559	147	5	in	in	ADP
ejpam-6559	147	6	studying	study	VERB
ejpam-6559	147	7	controllability	controllability	NOUN
ejpam-6559	147	8	,	,	PUNCT
ejpam-6559	147	9	stability	stability	NOUN
ejpam-6559	147	10	,	,	PUNCT
ejpam-6559	147	11	and	and	CCONJ
ejpam-6559	147	12	in	in	ADP
ejpam-6559	147	13	designing	design	VERB
ejpam-6559	147	14	feedback	feedback	NOUN
ejpam-6559	147	15	controls	control	NOUN
ejpam-6559	147	16	for	for	ADP
ejpam-6559	147	17	complex	complex	ADJ
ejpam-6559	147	18	systems	system	NOUN
ejpam-6559	147	19	.	.	PUNCT
ejpam-6559	148	1	problems	problem	NOUN
ejpam-6559	148	2	of	of	ADP
ejpam-6559	148	3	a	a	DET
ejpam-6559	148	4	similar	similar	ADJ
ejpam-6559	148	5	nature	nature	NOUN
ejpam-6559	148	6	appear	appear	VERB
ejpam-6559	148	7	in	in	ADP
ejpam-6559	148	8	mathematical	mathematical	ADJ
ejpam-6559	148	9	physics	physics	NOUN
ejpam-6559	148	10	,	,	PUNCT
ejpam-6559	148	11	where	where	SCONJ
ejpam-6559	148	12	the	the	DET
ejpam-6559	148	13	evolution	evolution	NOUN
ejpam-6559	148	14	of	of	ADP
ejpam-6559	148	15	a	a	DET
ejpam-6559	148	16	system	system	NOUN
ejpam-6559	148	17	depends	depend	VERB
ejpam-6559	148	18	on	on	ADP
ejpam-6559	148	19	both	both	DET
ejpam-6559	148	20	time	time	NOUN
ejpam-6559	148	21	and	and	CCONJ
ejpam-6559	148	22	space	space	NOUN
ejpam-6559	148	23	variables	variable	NOUN
ejpam-6559	148	24	.	.	PUNCT
ejpam-6559	149	1	examples	example	NOUN
ejpam-6559	149	2	include	include	VERB
ejpam-6559	149	3	wave	wave	NOUN
ejpam-6559	149	4	motion	motion	NOUN
ejpam-6559	149	5	in	in	ADP
ejpam-6559	149	6	elastic	elastic	ADJ
ejpam-6559	149	7	media	medium	NOUN
ejpam-6559	149	8	,	,	PUNCT
ejpam-6559	149	9	anisotropic	anisotropic	NOUN
ejpam-6559	149	10	heat	heat	NOUN
ejpam-6559	149	11	conduction	conduction	NOUN
ejpam-6559	149	12	,	,	PUNCT
ejpam-6559	149	13	or	or	CCONJ
ejpam-6559	149	14	certain	certain	ADJ
ejpam-6559	149	15	coupled	couple	VERB
ejpam-6559	149	16	schrödinger	schrödinger	NOUN
ejpam-6559	149	17	equations	equation	NOUN
ejpam-6559	149	18	[	[	X
ejpam-6559	149	19	17	17	NUM
ejpam-6559	149	20	]	]	PUNCT
ejpam-6559	149	21	,	,	PUNCT
ejpam-6559	149	22	[	[	X
ejpam-6559	149	23	18	18	NUM
ejpam-6559	149	24	]	]	PUNCT
ejpam-6559	149	25	.	.	PUNCT
ejpam-6559	150	1	the	the	DET
ejpam-6559	150	2	atomic	atomic	ADJ
ejpam-6559	150	3	representation	representation	NOUN
ejpam-6559	150	4	of	of	ADP
ejpam-6559	150	5	solutions	solution	NOUN
ejpam-6559	150	6	makes	make	VERB
ejpam-6559	150	7	it	it	PRON
ejpam-6559	150	8	possible	possible	ADJ
ejpam-6559	150	9	to	to	PART
ejpam-6559	150	10	obtain	obtain	VERB
ejpam-6559	150	11	exact	exact	ADJ
ejpam-6559	150	12	formulas	formula	NOUN
ejpam-6559	150	13	in	in	ADP
ejpam-6559	150	14	special	special	ADJ
ejpam-6559	150	15	cases	case	NOUN
ejpam-6559	150	16	,	,	PUNCT
ejpam-6559	150	17	and	and	CCONJ
ejpam-6559	150	18	in	in	ADP
ejpam-6559	150	19	more	more	ADV
ejpam-6559	150	20	complicated	complicated	ADJ
ejpam-6559	150	21	settings	setting	NOUN
ejpam-6559	150	22	it	it	PRON
ejpam-6559	150	23	can	can	AUX
ejpam-6559	150	24	lead	lead	VERB
ejpam-6559	150	25	to	to	ADP
ejpam-6559	150	26	accurate	accurate	ADJ
ejpam-6559	150	27	approximations	approximation	NOUN
ejpam-6559	150	28	while	while	SCONJ
ejpam-6559	150	29	retaining	retain	VERB
ejpam-6559	150	30	the	the	DET
ejpam-6559	150	31	essential	essential	ADJ
ejpam-6559	150	32	structure	structure	NOUN
ejpam-6559	150	33	dictated	dictate	VERB
ejpam-6559	150	34	by	by	ADP
ejpam-6559	150	35	the	the	DET
ejpam-6559	150	36	physical	physical	ADJ
ejpam-6559	150	37	model	model	NOUN
ejpam-6559	150	38	.	.	PUNCT
ejpam-6559	151	1	there	there	PRON
ejpam-6559	151	2	are	be	VERB
ejpam-6559	151	3	also	also	ADV
ejpam-6559	151	4	possible	possible	ADJ
ejpam-6559	151	5	uses	use	NOUN
ejpam-6559	151	6	in	in	ADP
ejpam-6559	151	7	abstract	abstract	ADJ
ejpam-6559	151	8	modeling	modeling	NOUN
ejpam-6559	151	9	outside	outside	ADP
ejpam-6559	151	10	traditional	traditional	ADJ
ejpam-6559	151	11	physics	physics	NOUN
ejpam-6559	151	12	or	or	CCONJ
ejpam-6559	151	13	engineering	engineering	NOUN
ejpam-6559	151	14	.	.	PUNCT
ejpam-6559	152	1	in	in	ADP
ejpam-6559	152	2	economics	economic	NOUN
ejpam-6559	152	3	,	,	PUNCT
ejpam-6559	152	4	population	population	NOUN
ejpam-6559	152	5	dynamics	dynamic	NOUN
ejpam-6559	152	6	,	,	PUNCT
ejpam-6559	152	7	or	or	CCONJ
ejpam-6559	152	8	network	network	NOUN
ejpam-6559	152	9	theory	theory	NOUN
ejpam-6559	152	10	,	,	PUNCT
ejpam-6559	152	11	models	model	NOUN
ejpam-6559	152	12	may	may	AUX
ejpam-6559	152	13	depend	depend	VERB
ejpam-6559	152	14	on	on	ADP
ejpam-6559	152	15	several	several	ADJ
ejpam-6559	152	16	independent	independent	ADJ
ejpam-6559	152	17	variables	variable	NOUN
ejpam-6559	152	18	,	,	PUNCT
ejpam-6559	152	19	and	and	CCONJ
ejpam-6559	152	20	their	their	PRON
ejpam-6559	152	21	governing	govern	VERB
ejpam-6559	152	22	equations	equation	NOUN
ejpam-6559	152	23	may	may	AUX
ejpam-6559	152	24	be	be	AUX
ejpam-6559	152	25	expressed	express	VERB
ejpam-6559	152	26	in	in	ADP
ejpam-6559	152	27	operator	operator	NOUN
ejpam-6559	152	28	form	form	NOUN
ejpam-6559	152	29	[	[	X
ejpam-6559	152	30	19	19	NUM
ejpam-6559	152	31	]	]	PUNCT
ejpam-6559	152	32	.	.	PUNCT
ejpam-6559	153	1	decomposing	decompose	VERB
ejpam-6559	153	2	these	these	DET
ejpam-6559	153	3	systems	system	NOUN
ejpam-6559	153	4	into	into	ADP
ejpam-6559	153	5	simpler	simple	ADJ
ejpam-6559	153	6	components	component	NOUN
ejpam-6559	153	7	,	,	PUNCT
ejpam-6559	153	8	as	as	ADP
ejpam-6559	153	9	in	in	ADP
ejpam-6559	153	10	the	the	DET
ejpam-6559	153	11	present	present	ADJ
ejpam-6559	153	12	approach	approach	NOUN
ejpam-6559	153	13	,	,	PUNCT
ejpam-6559	153	14	can	can	AUX
ejpam-6559	153	15	assist	assist	VERB
ejpam-6559	153	16	in	in	ADP
ejpam-6559	153	17	analysing	analyse	VERB
ejpam-6559	153	18	parameter	parameter	NOUN
ejpam-6559	153	19	effects	effect	NOUN
ejpam-6559	153	20	,	,	PUNCT
ejpam-6559	153	21	identifying	identify	VERB
ejpam-6559	153	22	model	model	NOUN
ejpam-6559	153	23	sensitivities	sensitivity	NOUN
ejpam-6559	153	24	,	,	PUNCT
ejpam-6559	153	25	and	and	CCONJ
ejpam-6559	153	26	constructing	construct	VERB
ejpam-6559	153	27	reduced	reduced	ADJ
ejpam-6559	153	28	models	model	NOUN
ejpam-6559	153	29	that	that	PRON
ejpam-6559	153	30	are	be	AUX
ejpam-6559	153	31	easier	easy	ADJ
ejpam-6559	153	32	to	to	PART
ejpam-6559	153	33	work	work	VERB
ejpam-6559	153	34	with	with	ADP
ejpam-6559	153	35	but	but	CCONJ
ejpam-6559	153	36	still	still	ADV
ejpam-6559	153	37	capture	capture	VERB
ejpam-6559	153	38	the	the	DET
ejpam-6559	153	39	main	main	ADJ
ejpam-6559	153	40	behaviour	behaviour	NOUN
ejpam-6559	153	41	of	of	ADP
ejpam-6559	153	42	the	the	DET
ejpam-6559	153	43	original	original	ADJ
ejpam-6559	153	44	system	system	NOUN
ejpam-6559	153	45	.	.	PUNCT
ejpam-6559	154	1	5	5	X
ejpam-6559	154	2	.	.	X
ejpam-6559	154	3	conclusions	conclusion	NOUN
ejpam-6559	154	4	in	in	ADP
ejpam-6559	154	5	this	this	DET
ejpam-6559	154	6	paper	paper	NOUN
ejpam-6559	154	7	,	,	PUNCT
ejpam-6559	154	8	we	we	PRON
ejpam-6559	154	9	utilize	utilize	VERB
ejpam-6559	154	10	some	some	DET
ejpam-6559	154	11	properties	property	NOUN
ejpam-6559	154	12	of	of	ADP
ejpam-6559	154	13	atoms	atom	NOUN
ejpam-6559	154	14	operators	operator	NOUN
ejpam-6559	154	15	to	to	PART
ejpam-6559	154	16	solve	solve	VERB
ejpam-6559	154	17	acp	acp	PROPN
ejpam-6559	154	18	in	in	ADP
ejpam-6559	154	19	two	two	NUM
ejpam-6559	154	20	variables	variable	NOUN
ejpam-6559	154	21	that	that	PRON
ejpam-6559	154	22	is	be	AUX
ejpam-6559	154	23	d2	d2	PROPN
ejpam-6559	154	24	ssu(s	ssu(s	PROPN
ejpam-6559	154	25	,	,	PUNCT
ejpam-6559	154	26	t	t	PROPN
ejpam-6559	154	27	)	)	PUNCT
ejpam-6559	154	28	+	+	NOUN
ejpam-6559	154	29	d2	d2	PROPN
ejpam-6559	154	30	ttu(s	ttu(s	PROPN
ejpam-6559	154	31	,	,	PUNCT
ejpam-6559	154	32	t	t	PROPN
ejpam-6559	154	33	)	)	PUNCT
ejpam-6559	154	34	+	+	CCONJ
ejpam-6559	154	35	2d2	2d2	NUM
ejpam-6559	154	36	stu(s	stu(s	NOUN
ejpam-6559	154	37	,	,	PUNCT
ejpam-6559	154	38	t	t	PROPN
ejpam-6559	154	39	)	)	PUNCT
ejpam-6559	155	1	+	+	ADP
ejpam-6559	155	2	a	a	DET
ejpam-6559	155	3	[	[	X
ejpam-6559	155	4	dsu(s	dsu(s	PROPN
ejpam-6559	155	5	,	,	PUNCT
ejpam-6559	155	6	t	t	PROPN
ejpam-6559	155	7	)	)	PUNCT
ejpam-6559	155	8	+	+	NOUN
ejpam-6559	155	9	dtu(s	dtu(s	PROPN
ejpam-6559	155	10	,	,	PUNCT
ejpam-6559	155	11	t	t	PROPN
ejpam-6559	155	12	)	)	PUNCT
ejpam-6559	155	13	]	]	PUNCT
ejpam-6559	156	1	=	=	PUNCT
ejpam-6559	156	2	bu(s	bu(s	PROPN
ejpam-6559	156	3	,	,	PUNCT
ejpam-6559	156	4	t	t	PROPN
ejpam-6559	156	5	)	)	PUNCT
ejpam-6559	156	6	.	.	PUNCT
ejpam-6559	157	1	the	the	DET
ejpam-6559	157	2	use	use	NOUN
ejpam-6559	157	3	of	of	ADP
ejpam-6559	157	4	an	an	DET
ejpam-6559	157	5	atomic	atomic	ADJ
ejpam-6559	157	6	solution	solution	NOUN
ejpam-6559	157	7	of	of	ADP
ejpam-6559	157	8	the	the	DET
ejpam-6559	157	9	form	form	NOUN
ejpam-6559	157	10	u	u	NOUN
ejpam-6559	157	11	=	=	PROPN
ejpam-6559	157	12	p	p	PROPN
ejpam-6559	157	13	⊗q	⊗q	PROPN
ejpam-6559	157	14	⊗	⊗	PROPN
ejpam-6559	157	15	ω	ω	PROPN
ejpam-6559	157	16	enables	enable	VERB
ejpam-6559	157	17	us	we	PRON
ejpam-6559	157	18	to	to	PART
ejpam-6559	157	19	reduce	reduce	VERB
ejpam-6559	157	20	the	the	DET
ejpam-6559	157	21	acp	acp	PROPN
ejpam-6559	157	22	into	into	ADP
ejpam-6559	157	23	separeted	separete	VERB
ejpam-6559	157	24	classical	classical	ADJ
ejpam-6559	157	25	ordinary	ordinary	ADJ
ejpam-6559	157	26	differential	differential	ADJ
ejpam-6559	157	27	equations	equation	NOUN
ejpam-6559	157	28	in	in	ADP
ejpam-6559	157	29	tems	tem	NOUN
ejpam-6559	157	30	of	of	ADP
ejpam-6559	157	31	the	the	DET
ejpam-6559	157	32	functions	function	NOUN
ejpam-6559	157	33	p	p	X
ejpam-6559	157	34	(	(	PUNCT
ejpam-6559	157	35	s	s	NOUN
ejpam-6559	157	36	)	)	PUNCT
ejpam-6559	157	37	and	and	CCONJ
ejpam-6559	157	38	q(t	q(t	NOUN
ejpam-6559	157	39	)	)	PUNCT
ejpam-6559	157	40	.	.	PUNCT
ejpam-6559	158	1	these	these	DET
ejpam-6559	158	2	equations	equation	NOUN
ejpam-6559	158	3	can	can	AUX
ejpam-6559	158	4	be	be	AUX
ejpam-6559	158	5	solved	solve	VERB
ejpam-6559	158	6	easily	easily	ADV
ejpam-6559	158	7	and	and	CCONJ
ejpam-6559	158	8	the	the	DET
ejpam-6559	158	9	atomic	atomic	ADJ
ejpam-6559	158	10	solution	solution	NOUN
ejpam-6559	158	11	can	can	AUX
ejpam-6559	158	12	be	be	AUX
ejpam-6559	158	13	then	then	ADV
ejpam-6559	158	14	obtained	obtain	VERB
ejpam-6559	158	15	.	.	PUNCT
ejpam-6559	159	1	an	an	DET
ejpam-6559	159	2	application	application	NOUN
ejpam-6559	159	3	to	to	ADP
ejpam-6559	159	4	the	the	DET
ejpam-6559	159	5	procedure	procedure	NOUN
ejpam-6559	159	6	is	be	AUX
ejpam-6559	159	7	also	also	ADV
ejpam-6559	159	8	provided	provide	VERB
ejpam-6559	159	9	.	.	PUNCT
ejpam-6559	160	1	acknowledgements	acknowledgement	NOUN
ejpam-6559	160	2	the	the	DET
ejpam-6559	160	3	authors	author	NOUN
ejpam-6559	160	4	would	would	AUX
ejpam-6559	160	5	like	like	VERB
ejpam-6559	160	6	to	to	PART
ejpam-6559	160	7	thank	thank	VERB
ejpam-6559	160	8	the	the	DET
ejpam-6559	160	9	editor	editor	NOUN
ejpam-6559	160	10	of	of	ADP
ejpam-6559	160	11	ejpam	ejpam	NOUN
ejpam-6559	160	12	as	as	ADV
ejpam-6559	160	13	well	well	ADV
ejpam-6559	160	14	as	as	ADP
ejpam-6559	160	15	the	the	DET
ejpam-6559	160	16	anonymous	anonymous	ADJ
ejpam-6559	160	17	reviewers	reviewer	NOUN
ejpam-6559	160	18	for	for	ADP
ejpam-6559	160	19	their	their	PRON
ejpam-6559	160	20	valuable	valuable	ADJ
ejpam-6559	160	21	comments	comment	NOUN
ejpam-6559	160	22	and	and	CCONJ
ejpam-6559	160	23	suggestions	suggestion	NOUN
ejpam-6559	160	24	.	.	PUNCT
ejpam-6559	161	1	w.	w.	PROPN
ejpam-6559	161	2	g.	g.	PROPN
ejpam-6559	161	3	alshanti	alshanti	PROPN
ejpam-6559	161	4	,	,	PUNCT
ejpam-6559	161	5	m.	m.	PROPN
ejpam-6559	161	6	abu	abu	PROPN
ejpam-6559	161	7	hammad	hammad	PROPN
ejpam-6559	161	8	,	,	PUNCT
ejpam-6559	161	9	r.	r.	PROPN
ejpam-6559	161	10	khalil	khalil	PROPN
ejpam-6559	161	11	/	/	SYM
ejpam-6559	161	12	eur	eur	PROPN
ejpam-6559	161	13	.	.	PUNCT
ejpam-6559	162	1	j.	j.	PROPN
ejpam-6559	162	2	pure	pure	PROPN
ejpam-6559	162	3	appl	appl	PROPN
ejpam-6559	162	4	.	.	PROPN
ejpam-6559	162	5	math	math	PROPN
ejpam-6559	162	6	,	,	PUNCT
ejpam-6559	162	7	18	18	NUM
ejpam-6559	162	8	(	(	PUNCT
ejpam-6559	162	9	4	4	NUM
ejpam-6559	162	10	)	)	PUNCT
ejpam-6559	162	11	(	(	PUNCT
ejpam-6559	162	12	2025	2025	NUM
ejpam-6559	162	13	)	)	PUNCT
ejpam-6559	162	14	,	,	PUNCT
ejpam-6559	162	15	6559	6559	NUM
ejpam-6559	162	16	7	7	NUM
ejpam-6559	162	17	of	of	ADP
ejpam-6559	162	18	8	8	NUM
ejpam-6559	162	19	references	reference	NOUN
ejpam-6559	162	20	[	[	X
ejpam-6559	162	21	1	1	NUM
ejpam-6559	162	22	]	]	PUNCT
ejpam-6559	162	23	e.	e.	PROPN
ejpam-6559	162	24	teixeira	teixeira	PROPN
ejpam-6559	162	25	.	.	PUNCT
ejpam-6559	163	1	strong	strong	ADJ
ejpam-6559	163	2	solutions	solution	NOUN
ejpam-6559	163	3	for	for	ADP
ejpam-6559	163	4	differential	differential	ADJ
ejpam-6559	163	5	equations	equation	NOUN
ejpam-6559	163	6	in	in	ADP
ejpam-6559	163	7	abstract	abstract	ADJ
ejpam-6559	163	8	spaces	space	NOUN
ejpam-6559	163	9	.	.	PUNCT
ejpam-6559	164	1	journal	journal	PROPN
ejpam-6559	164	2	of	of	ADP
ejpam-6559	164	3	differential	differential	ADJ
ejpam-6559	164	4	equations	equation	NOUN
ejpam-6559	164	5	,	,	PUNCT
ejpam-6559	164	6	214(1):65–91	214(1):65–91	NUM
ejpam-6559	164	7	,	,	PUNCT
ejpam-6559	164	8	2005	2005	NUM
ejpam-6559	164	9	.	.	PUNCT
ejpam-6559	165	1	[	[	X
ejpam-6559	165	2	2	2	NUM
ejpam-6559	165	3	]	]	PUNCT
ejpam-6559	165	4	f.	f.	PROPN
ejpam-6559	165	5	neubrander	neubrander	PROPN
ejpam-6559	165	6	.	.	PUNCT
ejpam-6559	166	1	well	well	ADJ
ejpam-6559	166	2	-	-	PUNCT
ejpam-6559	166	3	posedness	posedness	NOUN
ejpam-6559	166	4	of	of	ADP
ejpam-6559	166	5	higher	high	ADJ
ejpam-6559	166	6	order	order	NOUN
ejpam-6559	166	7	abstract	abstract	ADJ
ejpam-6559	166	8	cauchy	cauchy	NOUN
ejpam-6559	166	9	problems	problem	NOUN
ejpam-6559	166	10	.	.	PUNCT
ejpam-6559	167	1	transactions	transaction	NOUN
ejpam-6559	167	2	of	of	ADP
ejpam-6559	167	3	the	the	DET
ejpam-6559	167	4	american	american	PROPN
ejpam-6559	167	5	mathematical	mathematical	PROPN
ejpam-6559	167	6	society	society	NOUN
ejpam-6559	167	7	,	,	PUNCT
ejpam-6559	167	8	295(1):257–290	295(1):257–290	NUM
ejpam-6559	167	9	,	,	PUNCT
ejpam-6559	167	10	1986	1986	NUM
ejpam-6559	167	11	.	.	PUNCT
ejpam-6559	168	1	[	[	X
ejpam-6559	168	2	3	3	NUM
ejpam-6559	168	3	]	]	X
ejpam-6559	168	4	i.	i.	NOUN
ejpam-6559	168	5	batiha	batiha	PROPN
ejpam-6559	168	6	,	,	PUNCT
ejpam-6559	168	7	s.	s.	PROPN
ejpam-6559	168	8	njadat	njadat	PROPN
ejpam-6559	168	9	,	,	PUNCT
ejpam-6559	168	10	r.	r.	PROPN
ejpam-6559	168	11	batyha	batyha	PROPN
ejpam-6559	168	12	,	,	PUNCT
ejpam-6559	168	13	a.	a.	NOUN
ejpam-6559	168	14	zraiqat	zraiqat	PROPN
ejpam-6559	168	15	,	,	PUNCT
ejpam-6559	168	16	a.	a.	NOUN
ejpam-6559	168	17	dababneh	dababneh	PROPN
ejpam-6559	168	18	,	,	PUNCT
ejpam-6559	168	19	and	and	CCONJ
ejpam-6559	168	20	s.	s.	PROPN
ejpam-6559	168	21	momani	momani	PROPN
ejpam-6559	168	22	.	.	PUNCT
ejpam-6559	169	1	design	design	NOUN
ejpam-6559	169	2	fractional	fractional	ADJ
ejpam-6559	169	3	-	-	PUNCT
ejpam-6559	169	4	order	order	NOUN
ejpam-6559	169	5	pid	pid	NOUN
ejpam-6559	169	6	controllers	controller	NOUN
ejpam-6559	169	7	for	for	ADP
ejpam-6559	169	8	single	single	ADJ
ejpam-6559	169	9	-	-	PUNCT
ejpam-6559	169	10	joint	joint	ADJ
ejpam-6559	169	11	robot	robot	NOUN
ejpam-6559	169	12	arm	arm	NOUN
ejpam-6559	169	13	model	model	NOUN
ejpam-6559	169	14	.	.	PUNCT
ejpam-6559	170	1	international	international	ADJ
ejpam-6559	170	2	journal	journal	NOUN
ejpam-6559	170	3	of	of	ADP
ejpam-6559	170	4	advances	advance	NOUN
ejpam-6559	170	5	in	in	ADP
ejpam-6559	170	6	soft	soft	ADJ
ejpam-6559	170	7	computing	computing	NOUN
ejpam-6559	170	8	and	and	CCONJ
ejpam-6559	170	9	its	its	PRON
ejpam-6559	170	10	applications	application	NOUN
ejpam-6559	170	11	,	,	PUNCT
ejpam-6559	170	12	14(2):241–259	14(2):241–259	NUM
ejpam-6559	170	13	,	,	PUNCT
ejpam-6559	170	14	2022	2022	NUM
ejpam-6559	170	15	.	.	PUNCT
ejpam-6559	171	1	[	[	X
ejpam-6559	171	2	4	4	NUM
ejpam-6559	171	3	]	]	X
ejpam-6559	171	4	i.	i.	NOUN
ejpam-6559	171	5	batiha	batiha	PROPN
ejpam-6559	171	6	,	,	PUNCT
ejpam-6559	171	7	j.	j.	PROPN
ejpam-6559	171	8	oudetallah	oudetallah	PROPN
ejpam-6559	171	9	,	,	PUNCT
ejpam-6559	171	10	a.	a.	NOUN
ejpam-6559	171	11	ouannas	ouannas	PROPN
ejpam-6559	171	12	,	,	PUNCT
ejpam-6559	171	13	a.	a.	PROPN
ejpam-6559	171	14	al	al	PROPN
ejpam-6559	171	15	-	-	PUNCT
ejpam-6559	171	16	nana	nana	PROPN
ejpam-6559	171	17	,	,	PUNCT
ejpam-6559	171	18	and	and	CCONJ
ejpam-6559	171	19	i.	i.	PROPN
ejpam-6559	171	20	jebril	jebril	NOUN
ejpam-6559	171	21	.	.	PUNCT
ejpam-6559	172	1	tuning	tune	VERB
ejpam-6559	172	2	the	the	DET
ejpam-6559	172	3	fractionalorder	fractionalorder	ADJ
ejpam-6559	172	4	pid	pid	NOUN
ejpam-6559	172	5	-	-	NOUN
ejpam-6559	172	6	controller	controller	NOUN
ejpam-6559	172	7	for	for	ADP
ejpam-6559	172	8	blood	blood	NOUN
ejpam-6559	172	9	glucose	glucose	NOUN
ejpam-6559	172	10	level	level	NOUN
ejpam-6559	172	11	of	of	ADP
ejpam-6559	172	12	diabetic	diabetic	ADJ
ejpam-6559	172	13	patients	patient	NOUN
ejpam-6559	172	14	.	.	PUNCT
ejpam-6559	173	1	international	international	ADJ
ejpam-6559	173	2	journal	journal	NOUN
ejpam-6559	173	3	of	of	ADP
ejpam-6559	173	4	advances	advance	NOUN
ejpam-6559	173	5	in	in	ADP
ejpam-6559	173	6	soft	soft	ADJ
ejpam-6559	173	7	computing	computing	NOUN
ejpam-6559	173	8	and	and	CCONJ
ejpam-6559	173	9	its	its	PRON
ejpam-6559	173	10	applications	application	NOUN
ejpam-6559	173	11	,	,	PUNCT
ejpam-6559	173	12	13(2):1–10	13(2):1–10	NOUN
ejpam-6559	173	13	,	,	PUNCT
ejpam-6559	173	14	2021	2021	NUM
ejpam-6559	173	15	.	.	PUNCT
ejpam-6559	174	1	[	[	X
ejpam-6559	174	2	5	5	NUM
ejpam-6559	174	3	]	]	X
ejpam-6559	174	4	y.	y.	NOUN
ejpam-6559	174	5	lyubich	lyubich	PROPN
ejpam-6559	174	6	.	.	PUNCT
ejpam-6559	175	1	the	the	DET
ejpam-6559	175	2	classical	classical	ADJ
ejpam-6559	175	3	and	and	CCONJ
ejpam-6559	175	4	local	local	ADJ
ejpam-6559	175	5	laplace	laplace	NOUN
ejpam-6559	175	6	transformation	transformation	NOUN
ejpam-6559	175	7	in	in	ADP
ejpam-6559	175	8	an	an	DET
ejpam-6559	175	9	abstract	abstract	ADJ
ejpam-6559	175	10	cauchy	cauchy	ADJ
ejpam-6559	175	11	problem	problem	NOUN
ejpam-6559	175	12	.	.	PUNCT
ejpam-6559	176	1	russian	russian	ADJ
ejpam-6559	176	2	mathematical	mathematical	ADJ
ejpam-6559	176	3	surveys	survey	NOUN
ejpam-6559	176	4	,	,	PUNCT
ejpam-6559	176	5	21(3):1	21(3):1	NUM
ejpam-6559	176	6	,	,	PUNCT
ejpam-6559	176	7	1966	1966	NUM
ejpam-6559	176	8	.	.	PUNCT
ejpam-6559	177	1	[	[	X
ejpam-6559	177	2	6	6	NUM
ejpam-6559	177	3	]	]	X
ejpam-6559	177	4	r.	r.	NOUN
ejpam-6559	177	5	delaubenfels	delaubenfels	PROPN
ejpam-6559	177	6	.	.	PUNCT
ejpam-6559	177	7	integrated	integrate	VERB
ejpam-6559	177	8	semigroups	semigroup	NOUN
ejpam-6559	177	9	,	,	PUNCT
ejpam-6559	177	10	c	c	X
ejpam-6559	177	11	-	-	PUNCT
ejpam-6559	177	12	semigroups	semigroup	NOUN
ejpam-6559	177	13	and	and	CCONJ
ejpam-6559	177	14	the	the	DET
ejpam-6559	177	15	abstract	abstract	ADJ
ejpam-6559	177	16	cauchy	cauchy	PROPN
ejpam-6559	177	17	problem	problem	NOUN
ejpam-6559	177	18	.	.	PUNCT
ejpam-6559	178	1	semigroup	semigroup	PROPN
ejpam-6559	178	2	forum	forum	PROPN
ejpam-6559	178	3	,	,	PUNCT
ejpam-6559	178	4	41:83–95	41:83–95	NUM
ejpam-6559	178	5	,	,	PUNCT
ejpam-6559	178	6	1990	1990	NUM
ejpam-6559	178	7	.	.	PUNCT
ejpam-6559	179	1	[	[	X
ejpam-6559	179	2	7	7	X
ejpam-6559	179	3	]	]	X
ejpam-6559	179	4	m.	m.	NOUN
ejpam-6559	179	5	bj	bj	NOUN
ejpam-6559	179	6	.	.	PUNCT
ejpam-6559	179	7	operator	operator	NOUN
ejpam-6559	179	8	splitting	splitting	NOUN
ejpam-6559	179	9	for	for	ADP
ejpam-6559	179	10	abstract	abstract	ADJ
ejpam-6559	179	11	cauchy	cauchy	ADJ
ejpam-6559	179	12	problems	problem	NOUN
ejpam-6559	179	13	.	.	PUNCT
ejpam-6559	180	1	i	i	PRON
ejpam-6559	180	2	m	m	VERB
ejpam-6559	180	3	a	a	DET
ejpam-6559	180	4	journal	journal	NOUN
ejpam-6559	180	5	of	of	ADP
ejpam-6559	180	6	numerical	numerical	ADJ
ejpam-6559	180	7	analysis	analysis	NOUN
ejpam-6559	180	8	,	,	PUNCT
ejpam-6559	180	9	18(3):419–443	18(3):419–443	NUM
ejpam-6559	180	10	,	,	PUNCT
ejpam-6559	180	11	1998	1998	NUM
ejpam-6559	180	12	.	.	PUNCT
ejpam-6559	181	1	[	[	X
ejpam-6559	181	2	8	8	NUM
ejpam-6559	181	3	]	]	X
ejpam-6559	181	4	w.	w.	PROPN
ejpam-6559	181	5	alshanti	alshanti	PROPN
ejpam-6559	181	6	.	.	PUNCT
ejpam-6559	182	1	solutions	solution	NOUN
ejpam-6559	182	2	of	of	ADP
ejpam-6559	182	3	linear	linear	PROPN
ejpam-6559	182	4	and	and	CCONJ
ejpam-6559	182	5	non	non	ADJ
ejpam-6559	182	6	-	-	ADJ
ejpam-6559	182	7	linear	linear	ADJ
ejpam-6559	182	8	partial	partial	ADJ
ejpam-6559	182	9	differential	differential	NOUN
ejpam-6559	182	10	equations	equation	NOUN
ejpam-6559	182	11	by	by	ADP
ejpam-6559	182	12	means	mean	NOUN
ejpam-6559	182	13	of	of	ADP
ejpam-6559	182	14	tensor	tensor	NOUN
ejpam-6559	182	15	product	product	NOUN
ejpam-6559	182	16	theory	theory	NOUN
ejpam-6559	182	17	of	of	ADP
ejpam-6559	182	18	banach	banach	NOUN
ejpam-6559	182	19	space	space	NOUN
ejpam-6559	182	20	.	.	PUNCT
ejpam-6559	183	1	electronic	electronic	ADJ
ejpam-6559	183	2	journal	journal	NOUN
ejpam-6559	183	3	of	of	ADP
ejpam-6559	183	4	differential	differential	ADJ
ejpam-6559	183	5	equations	equation	NOUN
ejpam-6559	183	6	,	,	PUNCT
ejpam-6559	183	7	2024(28):1–8	2024(28):1–8	NUM
ejpam-6559	183	8	,	,	PUNCT
ejpam-6559	183	9	2024	2024	NUM
ejpam-6559	183	10	.	.	PUNCT
ejpam-6559	184	1	[	[	X
ejpam-6559	184	2	9	9	NUM
ejpam-6559	184	3	]	]	X
ejpam-6559	184	4	r.	r.	PROPN
ejpam-6559	184	5	khalil	khalil	PROPN
ejpam-6559	184	6	and	and	CCONJ
ejpam-6559	184	7	r.	r.	PROPN
ejpam-6559	184	8	abdelgani	abdelgani	PROPN
ejpam-6559	184	9	.	.	PUNCT
ejpam-6559	185	1	some	some	DET
ejpam-6559	185	2	solutions	solution	NOUN
ejpam-6559	185	3	of	of	ADP
ejpam-6559	185	4	fractional	fractional	ADJ
ejpam-6559	185	5	inverse	inverse	NOUN
ejpam-6559	185	6	problems	problem	NOUN
ejpam-6559	185	7	.	.	PUNCT
ejpam-6559	186	1	journal	journal	PROPN
ejpam-6559	186	2	of	of	ADP
ejpam-6559	186	3	semigroup	semigroup	PROPN
ejpam-6559	186	4	theory	theory	NOUN
ejpam-6559	186	5	and	and	CCONJ
ejpam-6559	186	6	applications	application	NOUN
ejpam-6559	186	7	,	,	PUNCT
ejpam-6559	186	8	2018(12):1–9	2018(12):1–9	NOUN
ejpam-6559	186	9	,	,	PUNCT
ejpam-6559	186	10	2018	2018	NUM
ejpam-6559	186	11	.	.	PUNCT
ejpam-6559	187	1	[	[	X
ejpam-6559	187	2	10	10	NUM
ejpam-6559	187	3	]	]	X
ejpam-6559	187	4	w.	w.	PROPN
ejpam-6559	187	5	alshanti	alshanti	PROPN
ejpam-6559	187	6	,	,	PUNCT
ejpam-6559	187	7	a.	a.	PROPN
ejpam-6559	187	8	alshanty	alshanty	PROPN
ejpam-6559	187	9	,	,	PUNCT
ejpam-6559	187	10	b.	b.	PROPN
ejpam-6559	187	11	aljawarneh	aljawarneh	PROPN
ejpam-6559	187	12	,	,	PUNCT
ejpam-6559	187	13	and	and	CCONJ
ejpam-6559	187	14	r.	r.	PROPN
ejpam-6559	187	15	khalil	khalil	PROPN
ejpam-6559	187	16	.	.	PUNCT
ejpam-6559	188	1	reliable	reliable	ADJ
ejpam-6559	188	2	analytical	analytical	ADJ
ejpam-6559	188	3	method	method	NOUN
ejpam-6559	188	4	for	for	ADP
ejpam-6559	188	5	solving	solve	VERB
ejpam-6559	188	6	klein	klein	PROPN
ejpam-6559	188	7	-	-	PUNCT
ejpam-6559	188	8	gordan	gordan	PROPN
ejpam-6559	188	9	equation	equation	NOUN
ejpam-6559	188	10	by	by	ADP
ejpam-6559	188	11	tensor	tensor	NOUN
ejpam-6559	188	12	product	product	NOUN
ejpam-6559	188	13	theory	theory	NOUN
ejpam-6559	188	14	of	of	ADP
ejpam-6559	188	15	banach	banach	NOUN
ejpam-6559	188	16	spaces	space	NOUN
ejpam-6559	188	17	.	.	PUNCT
ejpam-6559	189	1	in	in	ADP
ejpam-6559	189	2	a.	a.	PROPN
ejpam-6559	189	3	zriqat	zriqat	PROPN
ejpam-6559	189	4	,	,	PUNCT
ejpam-6559	189	5	editor	editor	NOUN
ejpam-6559	189	6	,	,	PUNCT
ejpam-6559	189	7	2023	2023	NUM
ejpam-6559	189	8	international	international	ADJ
ejpam-6559	189	9	conference	conference	NOUN
ejpam-6559	189	10	on	on	ADP
ejpam-6559	189	11	information	information	NOUN
ejpam-6559	189	12	technology	technology	NOUN
ejpam-6559	189	13	(	(	PUNCT
ejpam-6559	189	14	icit	icit	PROPN
ejpam-6559	189	15	)	)	PUNCT
ejpam-6559	189	16	,	,	PUNCT
ejpam-6559	189	17	pages	page	NOUN
ejpam-6559	189	18	674–676	674–676	NUM
ejpam-6559	189	19	,	,	PUNCT
ejpam-6559	189	20	jordan	jordan	PROPN
ejpam-6559	189	21	,	,	PUNCT
ejpam-6559	189	22	2023	2023	NUM
ejpam-6559	189	23	.	.	PUNCT
ejpam-6559	190	1	ieee	ieee	NOUN
ejpam-6559	190	2	.	.	PUNCT
ejpam-6559	191	1	[	[	X
ejpam-6559	191	2	11	11	NUM
ejpam-6559	191	3	]	]	PUNCT
ejpam-6559	191	4	r.	r.	PROPN
ejpam-6559	191	5	khalil	khalil	PROPN
ejpam-6559	191	6	,	,	PUNCT
ejpam-6559	191	7	w.	w.	PROPN
ejpam-6559	191	8	alshanti	alshanti	PROPN
ejpam-6559	191	9	,	,	PUNCT
ejpam-6559	191	10	and	and	CCONJ
ejpam-6559	191	11	m.	m.	PROPN
ejpam-6559	191	12	hammad	hammad	PROPN
ejpam-6559	191	13	.	.	PUNCT
ejpam-6559	192	1	abstract	abstract	ADJ
ejpam-6559	192	2	cauchy	cauchy	ADJ
ejpam-6559	192	3	problems	problem	NOUN
ejpam-6559	192	4	in	in	ADP
ejpam-6559	192	5	two	two	NUM
ejpam-6559	192	6	variables	variable	NOUN
ejpam-6559	192	7	and	and	CCONJ
ejpam-6559	192	8	tensor	tensor	NOUN
ejpam-6559	192	9	product	product	NOUN
ejpam-6559	192	10	of	of	ADP
ejpam-6559	192	11	banach	banach	NOUN
ejpam-6559	192	12	spaces	space	NOUN
ejpam-6559	192	13	.	.	PUNCT
ejpam-6559	193	1	journal	journal	NOUN
ejpam-6559	193	2	of	of	ADP
ejpam-6559	193	3	computational	computational	ADJ
ejpam-6559	193	4	analysis	analysis	NOUN
ejpam-6559	193	5	and	and	CCONJ
ejpam-6559	193	6	applications	application	NOUN
ejpam-6559	193	7	,	,	PUNCT
ejpam-6559	193	8	32(1):1–10	32(1):1–10	NUM
ejpam-6559	193	9	,	,	PUNCT
ejpam-6559	193	10	2024	2024	NUM
ejpam-6559	193	11	.	.	PUNCT
ejpam-6559	194	1	[	[	X
ejpam-6559	194	2	12	12	NUM
ejpam-6559	194	3	]	]	X
ejpam-6559	194	4	j.	j.	PROPN
ejpam-6559	194	5	diestel	diestel	PROPN
ejpam-6559	194	6	and	and	CCONJ
ejpam-6559	194	7	jr	jr	PROPN
ejpam-6559	194	8	.	.	PROPN
ejpam-6559	194	9	uhl	uhl	PROPN
ejpam-6559	194	10	,	,	PUNCT
ejpam-6559	194	11	j.	j.	PROPN
ejpam-6559	194	12	vector	vector	NOUN
ejpam-6559	194	13	measures	measure	NOUN
ejpam-6559	194	14	.	.	PUNCT
ejpam-6559	195	1	american	american	PROPN
ejpam-6559	195	2	mathematical	mathematical	PROPN
ejpam-6559	195	3	society	society	NOUN
ejpam-6559	195	4	,	,	PUNCT
ejpam-6559	195	5	providence	providence	NOUN
ejpam-6559	195	6	,	,	PUNCT
ejpam-6559	195	7	usa	usa	PROPN
ejpam-6559	195	8	,	,	PUNCT
ejpam-6559	195	9	1977	1977	NUM
ejpam-6559	195	10	.	.	PUNCT
ejpam-6559	196	1	[	[	X
ejpam-6559	196	2	13	13	NUM
ejpam-6559	196	3	]	]	PUNCT
ejpam-6559	196	4	r.	r.	PROPN
ejpam-6559	196	5	khalil	khalil	PROPN
ejpam-6559	196	6	.	.	PUNCT
ejpam-6559	197	1	isometries	isometry	NOUN
ejpam-6559	197	2	of	of	ADP
ejpam-6559	197	3	lp*lp	lp*lp	PROPN
ejpam-6559	197	4	.	.	PUNCT
ejpam-6559	198	1	tamkang	tamkang	PROPN
ejpam-6559	198	2	journal	journal	PROPN
ejpam-6559	198	3	of	of	ADP
ejpam-6559	198	4	mathematics	mathematic	NOUN
ejpam-6559	198	5	,	,	PUNCT
ejpam-6559	198	6	16:77–85	16:77–85	NUM
ejpam-6559	198	7	,	,	PUNCT
ejpam-6559	198	8	1985	1985	NUM
ejpam-6559	198	9	.	.	PUNCT
ejpam-6559	199	1	[	[	X
ejpam-6559	199	2	14	14	NUM
ejpam-6559	199	3	]	]	X
ejpam-6559	199	4	e.	e.	PROPN
ejpam-6559	199	5	kreyszig	kreyszig	PROPN
ejpam-6559	199	6	.	.	PUNCT
ejpam-6559	200	1	introductory	introductory	ADJ
ejpam-6559	200	2	functional	functional	ADJ
ejpam-6559	200	3	analysis	analysis	NOUN
ejpam-6559	200	4	with	with	ADP
ejpam-6559	200	5	applications	application	NOUN
ejpam-6559	200	6	.	.	PUNCT
ejpam-6559	201	1	john	john	PROPN
ejpam-6559	201	2	wiley	wiley	PROPN
ejpam-6559	201	3	and	and	CCONJ
ejpam-6559	201	4	sons	son	NOUN
ejpam-6559	201	5	,	,	PUNCT
ejpam-6559	201	6	new	new	PROPN
ejpam-6559	201	7	jersey	jersey	PROPN
ejpam-6559	201	8	,	,	PUNCT
ejpam-6559	201	9	usa	usa	PROPN
ejpam-6559	201	10	,	,	PUNCT
ejpam-6559	201	11	1991	1991	NUM
ejpam-6559	201	12	.	.	PUNCT
ejpam-6559	202	1	[	[	X
ejpam-6559	202	2	15	15	NUM
ejpam-6559	202	3	]	]	X
ejpam-6559	202	4	r.	r.	PROPN
ejpam-6559	202	5	curtain	curtain	PROPN
ejpam-6559	202	6	and	and	CCONJ
ejpam-6559	202	7	h.	h.	PROPN
ejpam-6559	202	8	zwart	zwart	PROPN
ejpam-6559	202	9	.	.	PUNCT
ejpam-6559	203	1	an	an	DET
ejpam-6559	203	2	introduction	introduction	NOUN
ejpam-6559	203	3	to	to	ADP
ejpam-6559	203	4	infinite	infinite	VERB
ejpam-6559	203	5	-	-	PUNCT
ejpam-6559	203	6	dimensional	dimensional	ADJ
ejpam-6559	203	7	linear	linear	NOUN
ejpam-6559	203	8	systems	system	NOUN
ejpam-6559	203	9	theory	theory	NOUN
ejpam-6559	203	10	.	.	PUNCT
ejpam-6559	204	1	springer	springer	NOUN
ejpam-6559	204	2	-	-	PUNCT
ejpam-6559	204	3	verlag	verlag	PROPN
ejpam-6559	204	4	,	,	PUNCT
ejpam-6559	204	5	berlin	berlin	PROPN
ejpam-6559	204	6	,	,	PUNCT
ejpam-6559	204	7	heidelberg	heidelberg	PROPN
ejpam-6559	204	8	,	,	PUNCT
ejpam-6559	204	9	1995	1995	NUM
ejpam-6559	204	10	.	.	PUNCT
ejpam-6559	205	1	[	[	X
ejpam-6559	205	2	16	16	NUM
ejpam-6559	205	3	]	]	PUNCT
ejpam-6559	205	4	m.	m.	NOUN
ejpam-6559	205	5	tucsnak	tucsnak	NOUN
ejpam-6559	205	6	and	and	CCONJ
ejpam-6559	205	7	g.	g.	PROPN
ejpam-6559	205	8	weiss	weiss	PROPN
ejpam-6559	205	9	.	.	PUNCT
ejpam-6559	206	1	observation	observation	NOUN
ejpam-6559	206	2	and	and	CCONJ
ejpam-6559	206	3	control	control	NOUN
ejpam-6559	206	4	for	for	ADP
ejpam-6559	206	5	operator	operator	NOUN
ejpam-6559	206	6	semigroups	semigroup	NOUN
ejpam-6559	206	7	.	.	PUNCT
ejpam-6559	207	1	birkhäuser	birkhäuser	X
ejpam-6559	207	2	basel	basel	PROPN
ejpam-6559	207	3	,	,	PUNCT
ejpam-6559	207	4	basel	basel	PROPN
ejpam-6559	207	5	,	,	PUNCT
ejpam-6559	207	6	switzerland	switzerland	PROPN
ejpam-6559	207	7	,	,	PUNCT
ejpam-6559	207	8	2009	2009	NUM
ejpam-6559	207	9	.	.	PUNCT
ejpam-6559	208	1	[	[	X
ejpam-6559	208	2	17	17	NUM
ejpam-6559	208	3	]	]	X
ejpam-6559	208	4	l.	l.	PROPN
ejpam-6559	208	5	evans	evans	PROPN
ejpam-6559	208	6	.	.	PUNCT
ejpam-6559	209	1	partial	partial	ADJ
ejpam-6559	209	2	differential	differential	ADJ
ejpam-6559	209	3	equations	equation	NOUN
ejpam-6559	209	4	.	.	PUNCT
ejpam-6559	210	1	american	american	PROPN
ejpam-6559	210	2	mathematical	mathematical	PROPN
ejpam-6559	210	3	society	society	NOUN
ejpam-6559	210	4	,	,	PUNCT
ejpam-6559	210	5	providence	providence	NOUN
ejpam-6559	210	6	,	,	PUNCT
ejpam-6559	210	7	usa	usa	PROPN
ejpam-6559	210	8	,	,	PUNCT
ejpam-6559	210	9	2010	2010	NUM
ejpam-6559	210	10	.	.	PUNCT
ejpam-6559	211	1	[	[	X
ejpam-6559	211	2	18	18	NUM
ejpam-6559	211	3	]	]	X
ejpam-6559	211	4	w.	w.	PROPN
ejpam-6559	211	5	strauss	strauss	PROPN
ejpam-6559	211	6	.	.	PUNCT
ejpam-6559	212	1	partial	partial	ADJ
ejpam-6559	212	2	differential	differential	ADJ
ejpam-6559	212	3	equations	equation	NOUN
ejpam-6559	212	4	:	:	PUNCT
ejpam-6559	212	5	an	an	DET
ejpam-6559	212	6	introduction	introduction	NOUN
ejpam-6559	212	7	.	.	PUNCT
ejpam-6559	213	1	john	john	PROPN
ejpam-6559	213	2	wiley	wiley	PROPN
ejpam-6559	213	3	and	and	CCONJ
ejpam-6559	213	4	sons	son	NOUN
ejpam-6559	213	5	,	,	PUNCT
ejpam-6559	213	6	new	new	PROPN
ejpam-6559	213	7	jersey	jersey	PROPN
ejpam-6559	213	8	,	,	PUNCT
ejpam-6559	213	9	usa	usa	PROPN
ejpam-6559	213	10	,	,	PUNCT
ejpam-6559	213	11	2008	2008	NUM
ejpam-6559	213	12	.	.	PUNCT
ejpam-6559	214	1	w.	w.	PROPN
ejpam-6559	214	2	g.	g.	PROPN
ejpam-6559	214	3	alshanti	alshanti	PROPN
ejpam-6559	214	4	,	,	PUNCT
ejpam-6559	214	5	m.	m.	PROPN
ejpam-6559	214	6	abu	abu	PROPN
ejpam-6559	214	7	hammad	hammad	PROPN
ejpam-6559	214	8	,	,	PUNCT
ejpam-6559	214	9	r.	r.	PROPN
ejpam-6559	214	10	khalil	khalil	PROPN
ejpam-6559	214	11	/	/	SYM
ejpam-6559	214	12	eur	eur	PROPN
ejpam-6559	214	13	.	.	PUNCT
ejpam-6559	215	1	j.	j.	PROPN
ejpam-6559	215	2	pure	pure	PROPN
ejpam-6559	215	3	appl	appl	PROPN
ejpam-6559	215	4	.	.	PROPN
ejpam-6559	215	5	math	math	PROPN
ejpam-6559	215	6	,	,	PUNCT
ejpam-6559	215	7	18	18	NUM
ejpam-6559	215	8	(	(	PUNCT
ejpam-6559	215	9	4	4	NUM
ejpam-6559	215	10	)	)	PUNCT
ejpam-6559	215	11	(	(	PUNCT
ejpam-6559	215	12	2025	2025	NUM
ejpam-6559	215	13	)	)	PUNCT
ejpam-6559	215	14	,	,	PUNCT
ejpam-6559	215	15	6559	6559	NUM
ejpam-6559	215	16	8	8	NUM
ejpam-6559	215	17	of	of	ADP
ejpam-6559	215	18	8	8	NUM
ejpam-6559	215	19	[	[	SYM
ejpam-6559	215	20	19	19	NUM
ejpam-6559	215	21	]	]	PUNCT
ejpam-6559	215	22	j.	j.	PROPN
ejpam-6559	215	23	engwerda	engwerda	PROPN
ejpam-6559	215	24	.	.	PUNCT
ejpam-6559	216	1	lq	lq	VERB
ejpam-6559	216	2	dynamic	dynamic	ADJ
ejpam-6559	216	3	optimization	optimization	NOUN
ejpam-6559	216	4	and	and	CCONJ
ejpam-6559	216	5	differential	differential	PROPN
ejpam-6559	216	6	games	game	NOUN
ejpam-6559	216	7	.	.	PUNCT
ejpam-6559	217	1	john	john	PROPN
ejpam-6559	217	2	wiley	wiley	PROPN
ejpam-6559	217	3	and	and	CCONJ
ejpam-6559	217	4	sons	son	NOUN
ejpam-6559	217	5	,	,	PUNCT
ejpam-6559	217	6	new	new	PROPN
ejpam-6559	217	7	jersey	jersey	PROPN
ejpam-6559	217	8	,	,	PUNCT
ejpam-6559	217	9	usa	usa	PROPN
ejpam-6559	217	10	,	,	PUNCT
ejpam-6559	217	11	2008	2008	NUM
ejpam-6559	217	12	.	.	PUNCT