id	sid	tid	token	lemma	pos
ejpam-854	1	1	7_854_xu.dvi	7_854_xu.dvi	NUM
ejpam-854	1	2	european	european	ADJ
ejpam-854	1	3	journal	journal	NOUN
ejpam-854	1	4	of	of	ADP
ejpam-854	1	5	pure	pure	ADJ
ejpam-854	1	6	and	and	CCONJ
ejpam-854	1	7	applied	apply	VERB
ejpam-854	1	8	mathematics	mathematic	NOUN
ejpam-854	1	9	vol	vol	NOUN
ejpam-854	1	10	.	.	PUNCT
ejpam-854	2	1	3	3	NUM
ejpam-854	2	2	,	,	PUNCT
ejpam-854	2	3	no	no	INTJ
ejpam-854	2	4	.	.	NOUN
ejpam-854	2	5	6	6	NUM
ejpam-854	2	6	,	,	PUNCT
ejpam-854	2	7	2010	2010	NUM
ejpam-854	2	8	,	,	PUNCT
ejpam-854	2	9	1032	1032	NUM
ejpam-854	2	10	-	-	SYM
ejpam-854	2	11	1047	1047	NUM
ejpam-854	2	12	issn	issn	PROPN
ejpam-854	2	13	1307	1307	NUM
ejpam-854	2	14	-	-	SYM
ejpam-854	2	15	5543	5543	NUM
ejpam-854	2	16	–	–	PUNCT
ejpam-854	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-854	2	18	special	special	ADJ
ejpam-854	2	19	issue	issue	NOUN
ejpam-854	2	20	on	on	ADP
ejpam-854	2	21	complex	complex	ADJ
ejpam-854	2	22	analysis	analysis	NOUN
ejpam-854	2	23	:	:	PUNCT
ejpam-854	2	24	theory	theory	NOUN
ejpam-854	2	25	and	and	CCONJ
ejpam-854	2	26	applications	application	NOUN
ejpam-854	2	27	dedicated	dedicate	VERB
ejpam-854	2	28	to	to	ADP
ejpam-854	2	29	professor	professor	PROPN
ejpam-854	2	30	hari	hari	PROPN
ejpam-854	2	31	m.	m.	PROPN
ejpam-854	2	32	srivastava	srivastava	PROPN
ejpam-854	2	33	,	,	PUNCT
ejpam-854	2	34	on	on	ADP
ejpam-854	2	35	the	the	DET
ejpam-854	2	36	occasion	occasion	NOUN
ejpam-854	2	37	of	of	ADP
ejpam-854	2	38	his	his	PRON
ejpam-854	2	39	70th	70th	ADJ
ejpam-854	2	40	birthday	birthday	NOUN
ejpam-854	2	41	generalized	generalize	VERB
ejpam-854	2	42	ulam	ulam	X
ejpam-854	2	43	-	-	PUNCT
ejpam-854	2	44	hyers	hyer	NOUN
ejpam-854	2	45	stability	stability	NOUN
ejpam-854	2	46	of	of	ADP
ejpam-854	2	47	a	a	DET
ejpam-854	2	48	general	general	ADJ
ejpam-854	2	49	mixed	mixed	ADJ
ejpam-854	2	50	aqcq	aqcq	NOUN
ejpam-854	2	51	-	-	PUNCT
ejpam-854	2	52	functional	functional	ADJ
ejpam-854	2	53	equation	equation	NOUN
ejpam-854	2	54	in	in	ADP
ejpam-854	2	55	multi	multi	ADJ
ejpam-854	2	56	-	-	ADJ
ejpam-854	2	57	banach	banach	ADJ
ejpam-854	2	58	spaces	space	VERB
ejpam-854	2	59	:	:	PUNCT
ejpam-854	2	60	a	a	DET
ejpam-854	2	61	fixed	fix	VERB
ejpam-854	2	62	point	point	NOUN
ejpam-854	2	63	approach	approach	NOUN
ejpam-854	3	1	tian	tian	PROPN
ejpam-854	3	2	zhou	zhou	PROPN
ejpam-854	3	3	xu1,∗	xu1,∗	PROPN
ejpam-854	3	4	,	,	PUNCT
ejpam-854	3	5	john	john	PROPN
ejpam-854	3	6	michael	michael	PROPN
ejpam-854	3	7	rassias	rassias	PROPN
ejpam-854	3	8	2	2	NUM
ejpam-854	3	9	,	,	PUNCT
ejpam-854	3	10	wan	wan	PROPN
ejpam-854	3	11	xin	xin	PROPN
ejpam-854	3	12	xu	xu	PROPN
ejpam-854	4	1	3	3	NUM
ejpam-854	4	2	1	1	NUM
ejpam-854	4	3	department	department	NOUN
ejpam-854	4	4	of	of	ADP
ejpam-854	4	5	mathematics	mathematic	NOUN
ejpam-854	4	6	,	,	PUNCT
ejpam-854	4	7	school	school	NOUN
ejpam-854	4	8	of	of	ADP
ejpam-854	4	9	science	science	NOUN
ejpam-854	4	10	,	,	PUNCT
ejpam-854	4	11	beijing	beijing	PROPN
ejpam-854	4	12	institute	institute	PROPN
ejpam-854	4	13	of	of	ADP
ejpam-854	4	14	technology	technology	PROPN
ejpam-854	4	15	,	,	PUNCT
ejpam-854	4	16	beijing	beijing	PROPN
ejpam-854	4	17	100081	100081	NUM
ejpam-854	4	18	,	,	PUNCT
ejpam-854	4	19	peoples	people	NOUN
ejpam-854	4	20	republic	republic	NOUN
ejpam-854	4	21	of	of	ADP
ejpam-854	4	22	china	china	PROPN
ejpam-854	4	23	2	2	NUM
ejpam-854	4	24	pedagogical	pedagogical	PROPN
ejpam-854	4	25	department	department	NOUN
ejpam-854	4	26	e.e	e.e	PROPN
ejpam-854	4	27	.	.	PROPN
ejpam-854	4	28	,	,	PUNCT
ejpam-854	4	29	section	section	NOUN
ejpam-854	4	30	of	of	ADP
ejpam-854	4	31	mathematics	mathematic	NOUN
ejpam-854	4	32	and	and	CCONJ
ejpam-854	4	33	informatics	informatic	NOUN
ejpam-854	4	34	,	,	PUNCT
ejpam-854	4	35	national	national	ADJ
ejpam-854	4	36	and	and	CCONJ
ejpam-854	4	37	capodistrian	capodistrian	ADJ
ejpam-854	4	38	university	university	PROPN
ejpam-854	4	39	of	of	ADP
ejpam-854	4	40	athens	athens	PROPN
ejpam-854	4	41	,	,	PUNCT
ejpam-854	4	42	4	4	NUM
ejpam-854	4	43	,	,	PUNCT
ejpam-854	4	44	agamemnonos	agamemnono	NOUN
ejpam-854	4	45	str	str	PRON
ejpam-854	4	46	.	.	PUNCT
ejpam-854	4	47	,	,	PUNCT
ejpam-854	4	48	aghia	aghia	VERB
ejpam-854	4	49	paraskevi	paraskevi	ADJ
ejpam-854	4	50	,	,	PUNCT
ejpam-854	4	51	athens	athens	PROPN
ejpam-854	4	52	15342	15342	NUM
ejpam-854	4	53	,	,	PUNCT
ejpam-854	4	54	greece	greece	PROPN
ejpam-854	4	55	3	3	NUM
ejpam-854	4	56	school	school	NOUN
ejpam-854	4	57	of	of	ADP
ejpam-854	4	58	communication	communication	NOUN
ejpam-854	4	59	and	and	CCONJ
ejpam-854	4	60	information	information	NOUN
ejpam-854	4	61	engineering	engineering	NOUN
ejpam-854	4	62	,	,	PUNCT
ejpam-854	4	63	university	university	NOUN
ejpam-854	4	64	of	of	ADP
ejpam-854	4	65	electronic	electronic	ADJ
ejpam-854	4	66	science	science	NOUN
ejpam-854	4	67	and	and	CCONJ
ejpam-854	4	68	technology	technology	NOUN
ejpam-854	4	69	of	of	ADP
ejpam-854	4	70	china	china	PROPN
ejpam-854	4	71	,	,	PUNCT
ejpam-854	4	72	chengdu	chengdu	PROPN
ejpam-854	4	73	611731	611731	NUM
ejpam-854	4	74	,	,	PUNCT
ejpam-854	4	75	peoples	people	NOUN
ejpam-854	4	76	republic	republic	NOUN
ejpam-854	4	77	of	of	ADP
ejpam-854	4	78	china	china	PROPN
ejpam-854	4	79	abstract	abstract	PROPN
ejpam-854	4	80	.	.	PUNCT
ejpam-854	5	1	using	use	VERB
ejpam-854	5	2	the	the	DET
ejpam-854	5	3	fixed	fix	VERB
ejpam-854	5	4	point	point	NOUN
ejpam-854	5	5	method	method	NOUN
ejpam-854	5	6	,	,	PUNCT
ejpam-854	5	7	we	we	PRON
ejpam-854	5	8	investigate	investigate	VERB
ejpam-854	5	9	the	the	DET
ejpam-854	5	10	generalized	generalize	VERB
ejpam-854	5	11	hyers	hyer	NOUN
ejpam-854	5	12	-	-	PUNCT
ejpam-854	5	13	ulam	ulam	ADJ
ejpam-854	5	14	stability	stability	NOUN
ejpam-854	5	15	of	of	ADP
ejpam-854	5	16	the	the	DET
ejpam-854	5	17	general	general	ADJ
ejpam-854	5	18	mixed	mix	VERB
ejpam-854	5	19	additive	additive	ADJ
ejpam-854	5	20	-	-	PUNCT
ejpam-854	5	21	quadratic	quadratic	ADJ
ejpam-854	5	22	-	-	PUNCT
ejpam-854	5	23	cubic	cubic	ADJ
ejpam-854	5	24	-	-	PUNCT
ejpam-854	5	25	quartic	quartic	ADJ
ejpam-854	5	26	functional	functional	ADJ
ejpam-854	5	27	equation	equation	NOUN
ejpam-854	5	28	f	f	X
ejpam-854	5	29	(	(	PUNCT
ejpam-854	5	30	x	x	PROPN
ejpam-854	5	31	+	+	NUM
ejpam-854	5	32	ny	ny	NOUN
ejpam-854	5	33	)	)	PUNCT
ejpam-854	6	1	+	+	NUM
ejpam-854	6	2	f	f	X
ejpam-854	6	3	(	(	PUNCT
ejpam-854	6	4	x	x	X
ejpam-854	6	5	−	−	PROPN
ejpam-854	6	6	ny	ny	NOUN
ejpam-854	6	7	)	)	PUNCT
ejpam-854	6	8	=	=	SYM
ejpam-854	6	9	n2	n2	PROPN
ejpam-854	6	10	f	f	PROPN
ejpam-854	6	11	(	(	PUNCT
ejpam-854	6	12	x	x	PROPN
ejpam-854	6	13	+	+	NUM
ejpam-854	6	14	y	y	NOUN
ejpam-854	6	15	)	)	PUNCT
ejpam-854	7	1	+	+	CCONJ
ejpam-854	7	2	n2	n2	PROPN
ejpam-854	7	3	f	f	PROPN
ejpam-854	7	4	(	(	PUNCT
ejpam-854	7	5	x	x	PROPN
ejpam-854	7	6	−	−	PROPN
ejpam-854	7	7	y	y	PROPN
ejpam-854	7	8	)	)	PUNCT
ejpam-854	7	9	+	+	CCONJ
ejpam-854	7	10	2(1−	2(1−	NUM
ejpam-854	7	11	n2	n2	ADJ
ejpam-854	7	12	)	)	PUNCT
ejpam-854	7	13	f	f	PROPN
ejpam-854	7	14	(	(	PUNCT
ejpam-854	7	15	x	x	X
ejpam-854	7	16	)	)	PUNCT
ejpam-854	8	1	+	+	CCONJ
ejpam-854	8	2	n4	n4	PROPN
ejpam-854	8	3	−	−	PROPN
ejpam-854	8	4	n2	n2	NOUN
ejpam-854	8	5	12	12	NUM
ejpam-854	8	6	[	[	PUNCT
ejpam-854	8	7	f	f	X
ejpam-854	8	8	(	(	PUNCT
ejpam-854	8	9	2y	2y	NUM
ejpam-854	8	10	)	)	PUNCT
ejpam-854	9	1	+	+	NUM
ejpam-854	9	2	f	f	X
ejpam-854	9	3	(	(	PUNCT
ejpam-854	9	4	−2y)−	−2y)−	NUM
ejpam-854	9	5	4	4	NUM
ejpam-854	9	6	f	f	NOUN
ejpam-854	9	7	(	(	PUNCT
ejpam-854	9	8	y)−	y)−	PROPN
ejpam-854	9	9	4	4	NUM
ejpam-854	9	10	f	f	NOUN
ejpam-854	9	11	(	(	PUNCT
ejpam-854	9	12	−y	−y	NOUN
ejpam-854	9	13	)	)	PUNCT
ejpam-854	9	14	]	]	PUNCT
ejpam-854	9	15	for	for	ADP
ejpam-854	9	16	fixed	fix	VERB
ejpam-854	9	17	integers	integer	NOUN
ejpam-854	9	18	n	n	X
ejpam-854	9	19	with	with	ADP
ejpam-854	9	20	n	n	PROPN
ejpam-854	9	21	6=	6=	NUM
ejpam-854	9	22	0,±1	0,±1	NOUN
ejpam-854	9	23	in	in	ADP
ejpam-854	9	24	multi	multi	ADJ
ejpam-854	9	25	-	-	ADJ
ejpam-854	9	26	banach	banach	ADJ
ejpam-854	9	27	spaces	space	NOUN
ejpam-854	9	28	.	.	PUNCT
ejpam-854	10	1	2000	2000	NUM
ejpam-854	10	2	mathematics	mathematic	NOUN
ejpam-854	10	3	subject	subject	NOUN
ejpam-854	10	4	classifications	classification	NOUN
ejpam-854	10	5	:	:	PUNCT
ejpam-854	10	6	39b82	39b82	NUM
ejpam-854	10	7	,	,	PUNCT
ejpam-854	10	8	39b52	39b52	NUM
ejpam-854	10	9	,	,	PUNCT
ejpam-854	10	10	46b99	46b99	NOUN
ejpam-854	10	11	key	key	ADJ
ejpam-854	10	12	words	word	NOUN
ejpam-854	10	13	and	and	CCONJ
ejpam-854	10	14	phrases	phrase	NOUN
ejpam-854	10	15	:	:	PUNCT
ejpam-854	10	16	fixed	fix	VERB
ejpam-854	10	17	point	point	NOUN
ejpam-854	10	18	alternative	alternative	NOUN
ejpam-854	10	19	,	,	PUNCT
ejpam-854	10	20	stability	stability	NOUN
ejpam-854	10	21	,	,	PUNCT
ejpam-854	10	22	additive	additive	ADJ
ejpam-854	10	23	function	function	NOUN
ejpam-854	10	24	,	,	PUNCT
ejpam-854	10	25	quadratic	quadratic	ADJ
ejpam-854	10	26	function	function	NOUN
ejpam-854	10	27	,	,	PUNCT
ejpam-854	10	28	cubic	cubic	ADJ
ejpam-854	10	29	function	function	NOUN
ejpam-854	10	30	,	,	PUNCT
ejpam-854	10	31	quartic	quartic	ADJ
ejpam-854	10	32	function	function	NOUN
ejpam-854	10	33	,	,	PUNCT
ejpam-854	10	34	multi	multi	ADJ
ejpam-854	10	35	-	-	ADJ
ejpam-854	10	36	banach	banach	ADJ
ejpam-854	10	37	space	space	NOUN
ejpam-854	10	38	∗corresponding	∗corresponde	VERB
ejpam-854	10	39	author	author	NOUN
ejpam-854	10	40	.	.	PUNCT
ejpam-854	11	1	email	email	NOUN
ejpam-854	11	2	addresses	address	NOUN
ejpam-854	11	3	:	:	PUNCT
ejpam-854	11	4	xutianzhou�bit.edu	xutianzhou�bit.edu	PROPN
ejpam-854	11	5	.	.	PROPN
ejpam-854	12	1	n	n	CCONJ
ejpam-854	12	2	,	,	PUNCT
ejpam-854	12	3	tzxubit	tzxubit	NOUN
ejpam-854	12	4	�	�	NOUN
ejpam-854	12	5	gmail	gmail	NOUN
ejpam-854	12	6	.	.	PUNCT
ejpam-854	13	1	om	om	PROPN
ejpam-854	13	2	(	(	PUNCT
ejpam-854	13	3	t.	t.	PROPN
ejpam-854	13	4	xu	xu	PROPN
ejpam-854	13	5	)	)	PUNCT
ejpam-854	13	6	,	,	PUNCT
ejpam-854	13	7	jrassias	jrassias	PROPN
ejpam-854	13	8	�	�	PROPN
ejpam-854	13	9	primedu.uoa.gr	primedu.uoa.gr	NUM
ejpam-854	13	10	,	,	PUNCT
ejpam-854	13	11	ioannis.rassias	ioannis.rassias	PROPN
ejpam-854	13	12	�	�	NOUN
ejpam-854	13	13	primedu.uoa.gr	primedu.uoa.gr	NUM
ejpam-854	13	14	,	,	PUNCT
ejpam-854	13	15	jrass�otenet.gr	jrass�otenet.gr	PROPN
ejpam-854	13	16	(	(	PUNCT
ejpam-854	13	17	j.	j.	PROPN
ejpam-854	13	18	rassias	rassias	PROPN
ejpam-854	13	19	)	)	PUNCT
ejpam-854	13	20	,	,	PUNCT
ejpam-854	13	21	wxbit0930	wxbit0930	PROPN
ejpam-854	13	22	�	�	NOUN
ejpam-854	13	23	gmail	gmail	NOUN
ejpam-854	13	24	.	.	PUNCT
ejpam-854	14	1	om	om	PROPN
ejpam-854	14	2	,	,	PUNCT
ejpam-854	14	3	wxbit	wxbit	PROPN
ejpam-854	14	4	�	�	PROPN
ejpam-854	14	5	sina	sina	PROPN
ejpam-854	14	6	.	.	PUNCT
ejpam-854	15	1	om	om	PROPN
ejpam-854	15	2	(	(	PUNCT
ejpam-854	15	3	w.	w.	PROPN
ejpam-854	15	4	xu	xu	PROPN
ejpam-854	15	5	)	)	PUNCT
ejpam-854	15	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-854	16	1	1032	1032	NUM
ejpam-854	16	2	c	c	X
ejpam-854	16	3	©	©	PROPN
ejpam-854	16	4	2010	2010	NUM
ejpam-854	16	5	ejpam	ejpam	NOUN
ejpam-854	16	6	all	all	DET
ejpam-854	16	7	rights	right	NOUN
ejpam-854	16	8	reserved	reserve	VERB
ejpam-854	16	9	.	.	PUNCT
ejpam-854	17	1	t.	t.	PROPN
ejpam-854	17	2	xu	xu	PROPN
ejpam-854	17	3	,	,	PUNCT
ejpam-854	17	4	j.	j.	PROPN
ejpam-854	17	5	rassias	rassias	PROPN
ejpam-854	17	6	,	,	PUNCT
ejpam-854	17	7	w.	w.	PROPN
ejpam-854	17	8	xu	xu	PROPN
ejpam-854	17	9	/	/	SYM
ejpam-854	17	10	eur	eur	PROPN
ejpam-854	17	11	.	.	PUNCT
ejpam-854	18	1	j.	j.	PROPN
ejpam-854	18	2	pure	pure	PROPN
ejpam-854	18	3	appl	appl	PROPN
ejpam-854	18	4	.	.	PROPN
ejpam-854	18	5	math	math	PROPN
ejpam-854	18	6	,	,	PUNCT
ejpam-854	18	7	3	3	NUM
ejpam-854	18	8	(	(	PUNCT
ejpam-854	18	9	2010	2010	NUM
ejpam-854	18	10	)	)	PUNCT
ejpam-854	18	11	,	,	PUNCT
ejpam-854	18	12	1032	1032	NUM
ejpam-854	18	13	-	-	SYM
ejpam-854	18	14	1047	1047	NUM
ejpam-854	18	15	1033	1033	NUM
ejpam-854	18	16	1	1	NUM
ejpam-854	18	17	.	.	PUNCT
ejpam-854	18	18	introduction	introduction	NOUN
ejpam-854	18	19	the	the	DET
ejpam-854	18	20	concept	concept	NOUN
ejpam-854	18	21	of	of	ADP
ejpam-854	18	22	stability	stability	NOUN
ejpam-854	18	23	for	for	ADP
ejpam-854	18	24	a	a	DET
ejpam-854	18	25	functional	functional	ADJ
ejpam-854	18	26	equation	equation	NOUN
ejpam-854	18	27	arises	arise	VERB
ejpam-854	18	28	when	when	SCONJ
ejpam-854	18	29	one	one	PRON
ejpam-854	18	30	replaces	replace	VERB
ejpam-854	18	31	a	a	DET
ejpam-854	18	32	functional	functional	ADJ
ejpam-854	18	33	equation	equation	NOUN
ejpam-854	18	34	by	by	ADP
ejpam-854	18	35	an	an	DET
ejpam-854	18	36	inequality	inequality	NOUN
ejpam-854	18	37	which	which	PRON
ejpam-854	18	38	acts	act	VERB
ejpam-854	18	39	as	as	ADP
ejpam-854	18	40	a	a	DET
ejpam-854	18	41	perturbation	perturbation	NOUN
ejpam-854	18	42	of	of	ADP
ejpam-854	18	43	the	the	DET
ejpam-854	18	44	equation	equation	NOUN
ejpam-854	18	45	.	.	PUNCT
ejpam-854	19	1	the	the	DET
ejpam-854	19	2	first	first	ADJ
ejpam-854	19	3	stability	stability	NOUN
ejpam-854	19	4	problem	problem	NOUN
ejpam-854	19	5	concerning	concern	VERB
ejpam-854	19	6	group	group	NOUN
ejpam-854	19	7	homomorphisms	homomorphism	NOUN
ejpam-854	19	8	was	be	AUX
ejpam-854	19	9	raised	raise	VERB
ejpam-854	19	10	by	by	ADP
ejpam-854	19	11	ulam	ulam	PROPN
ejpam-854	19	12	[	[	X
ejpam-854	19	13	35	35	NUM
ejpam-854	19	14	]	]	PUNCT
ejpam-854	19	15	in	in	ADP
ejpam-854	19	16	1940	1940	NUM
ejpam-854	19	17	and	and	CCONJ
ejpam-854	19	18	affirmatively	affirmatively	ADV
ejpam-854	19	19	solved	solve	VERB
ejpam-854	19	20	by	by	ADP
ejpam-854	19	21	hyers	hyer	NOUN
ejpam-854	19	22	[	[	X
ejpam-854	19	23	17	17	NUM
ejpam-854	19	24	]	]	PUNCT
ejpam-854	19	25	.	.	PUNCT
ejpam-854	20	1	the	the	DET
ejpam-854	20	2	result	result	NOUN
ejpam-854	20	3	of	of	ADP
ejpam-854	20	4	hyers	hyer	NOUN
ejpam-854	20	5	was	be	AUX
ejpam-854	20	6	generalized	generalize	VERB
ejpam-854	20	7	by	by	ADP
ejpam-854	20	8	rassias	rassias	PROPN
ejpam-854	21	1	[	[	X
ejpam-854	21	2	28	28	NUM
ejpam-854	21	3	]	]	PUNCT
ejpam-854	21	4	for	for	ADP
ejpam-854	21	5	approximate	approximate	ADJ
ejpam-854	21	6	linear	linear	NOUN
ejpam-854	21	7	mappings	mapping	NOUN
ejpam-854	21	8	by	by	ADP
ejpam-854	21	9	allowing	allow	VERB
ejpam-854	21	10	the	the	DET
ejpam-854	21	11	cauchy	cauchy	ADJ
ejpam-854	21	12	difference	difference	NOUN
ejpam-854	21	13	operator	operator	NOUN
ejpam-854	21	14	c	c	PROPN
ejpam-854	21	15	d	d	X
ejpam-854	21	16	f	f	X
ejpam-854	21	17	(	(	PUNCT
ejpam-854	21	18	x	x	PROPN
ejpam-854	21	19	,	,	PUNCT
ejpam-854	21	20	y	y	PROPN
ejpam-854	21	21	)	)	PUNCT
ejpam-854	22	1	=	=	SYM
ejpam-854	22	2	f	f	PROPN
ejpam-854	22	3	(	(	PUNCT
ejpam-854	22	4	x+	x+	X
ejpam-854	22	5	y)−	y)−	PROPN
ejpam-854	22	6	[	[	PUNCT
ejpam-854	22	7	f	f	X
ejpam-854	22	8	(	(	PUNCT
ejpam-854	22	9	x)+	x)+	PROPN
ejpam-854	22	10	f	f	PROPN
ejpam-854	22	11	(	(	PUNCT
ejpam-854	22	12	y	y	PROPN
ejpam-854	22	13	)	)	PUNCT
ejpam-854	22	14	]	]	PUNCT
ejpam-854	22	15	to	to	PART
ejpam-854	22	16	be	be	AUX
ejpam-854	22	17	controlled	control	VERB
ejpam-854	22	18	by	by	ADP
ejpam-854	22	19	ε(‖x‖p+‖y‖p	ε(‖x‖p+‖y‖p	NOUN
ejpam-854	22	20	)	)	PUNCT
ejpam-854	22	21	.	.	PUNCT
ejpam-854	23	1	in	in	ADP
ejpam-854	23	2	1994	1994	NUM
ejpam-854	23	3	,	,	PUNCT
ejpam-854	23	4	a	a	DET
ejpam-854	23	5	generalization	generalization	NOUN
ejpam-854	23	6	of	of	ADP
ejpam-854	23	7	rassias	rassias	PROPN
ejpam-854	23	8	’	'	PUNCT
ejpam-854	23	9	theorem	theorem	NOUN
ejpam-854	23	10	was	be	AUX
ejpam-854	23	11	obtained	obtain	VERB
ejpam-854	23	12	by	by	ADP
ejpam-854	23	13	găvruţa	găvruţa	NOUN
ejpam-854	24	1	[	[	X
ejpam-854	24	2	11	11	NUM
ejpam-854	24	3	]	]	PUNCT
ejpam-854	24	4	,	,	PUNCT
ejpam-854	24	5	who	who	PRON
ejpam-854	24	6	replaced	replace	VERB
ejpam-854	24	7	ε(‖x‖p+	ε(‖x‖p+	ADJ
ejpam-854	24	8	‖y‖p	‖y‖p	NOUN
ejpam-854	24	9	)	)	PUNCT
ejpam-854	24	10	by	by	ADP
ejpam-854	24	11	a	a	DET
ejpam-854	24	12	general	general	ADJ
ejpam-854	24	13	control	control	NOUN
ejpam-854	24	14	function	function	NOUN
ejpam-854	24	15	ϕ(x	ϕ(x	PROPN
ejpam-854	24	16	,	,	PUNCT
ejpam-854	24	17	y	y	PROPN
ejpam-854	24	18	)	)	PUNCT
ejpam-854	24	19	.	.	PUNCT
ejpam-854	25	1	in	in	ADP
ejpam-854	25	2	addition	addition	NOUN
ejpam-854	25	3	,	,	PUNCT
ejpam-854	25	4	j.	j.	PROPN
ejpam-854	25	5	m.	m.	PROPN
ejpam-854	25	6	rassias	rassias	PROPN
ejpam-854	25	7	et	et	PROPN
ejpam-854	25	8	al.([29]-[32	al.([29]-[32	PROPN
ejpam-854	25	9	]	]	X
ejpam-854	25	10	,	,	PUNCT
ejpam-854	25	11	[	[	X
ejpam-854	25	12	37]-[39	37]-[39	NUM
ejpam-854	25	13	]	]	PUNCT
ejpam-854	25	14	)	)	PUNCT
ejpam-854	25	15	generalized	generalize	VERB
ejpam-854	25	16	the	the	DET
ejpam-854	25	17	hyers	hyer	NOUN
ejpam-854	25	18	stability	stability	NOUN
ejpam-854	25	19	result	result	VERB
ejpam-854	25	20	by	by	ADP
ejpam-854	25	21	introducing	introduce	VERB
ejpam-854	25	22	two	two	NUM
ejpam-854	25	23	weaker	weak	ADJ
ejpam-854	25	24	conditions	condition	NOUN
ejpam-854	25	25	controlled	control	VERB
ejpam-854	25	26	by	by	ADP
ejpam-854	25	27	a	a	DET
ejpam-854	25	28	product	product	NOUN
ejpam-854	25	29	of	of	ADP
ejpam-854	25	30	different	different	ADJ
ejpam-854	25	31	powers	power	NOUN
ejpam-854	25	32	of	of	ADP
ejpam-854	25	33	norms	norm	NOUN
ejpam-854	25	34	and	and	CCONJ
ejpam-854	25	35	a	a	DET
ejpam-854	25	36	mixed	mixed	ADJ
ejpam-854	25	37	product	product	NOUN
ejpam-854	25	38	-	-	PUNCT
ejpam-854	25	39	sum	sum	NOUN
ejpam-854	25	40	of	of	ADP
ejpam-854	25	41	powers	power	NOUN
ejpam-854	25	42	of	of	ADP
ejpam-854	25	43	norms	norm	NOUN
ejpam-854	25	44	,	,	PUNCT
ejpam-854	25	45	respectively	respectively	ADV
ejpam-854	25	46	.	.	PUNCT
ejpam-854	26	1	recently	recently	ADV
ejpam-854	26	2	,	,	PUNCT
ejpam-854	26	3	several	several	ADJ
ejpam-854	26	4	further	further	ADJ
ejpam-854	26	5	interesting	interesting	ADJ
ejpam-854	26	6	discussions	discussion	NOUN
ejpam-854	26	7	,	,	PUNCT
ejpam-854	26	8	modifications	modification	NOUN
ejpam-854	26	9	,	,	PUNCT
ejpam-854	26	10	extensions	extension	NOUN
ejpam-854	26	11	,	,	PUNCT
ejpam-854	26	12	and	and	CCONJ
ejpam-854	26	13	generalizations	generalization	NOUN
ejpam-854	26	14	of	of	ADP
ejpam-854	26	15	the	the	DET
ejpam-854	26	16	original	original	ADJ
ejpam-854	26	17	problem	problem	NOUN
ejpam-854	26	18	of	of	ADP
ejpam-854	26	19	ulam	ulam	PROPN
ejpam-854	26	20	have	have	AUX
ejpam-854	26	21	been	be	AUX
ejpam-854	26	22	proposed	propose	VERB
ejpam-854	26	23	(	(	PUNCT
ejpam-854	26	24	see	see	VERB
ejpam-854	26	25	,	,	PUNCT
ejpam-854	26	26	e.g.	e.g.	ADV
ejpam-854	26	27	,	,	PUNCT
ejpam-854	26	28	[	[	X
ejpam-854	26	29	2]-[3	2]-[3	NUM
ejpam-854	26	30	]	]	PUNCT
ejpam-854	26	31	,	,	PUNCT
ejpam-854	26	32	[	[	X
ejpam-854	26	33	6	6	NUM
ejpam-854	26	34	]	]	PUNCT
ejpam-854	26	35	,	,	PUNCT
ejpam-854	27	1	[	[	X
ejpam-854	27	2	8]-[16	8]-[16	NUM
ejpam-854	27	3	]	]	PUNCT
ejpam-854	27	4	,	,	PUNCT
ejpam-854	27	5	[	[	X
ejpam-854	27	6	18	18	NUM
ejpam-854	27	7	]	]	PUNCT
ejpam-854	27	8	,	,	PUNCT
ejpam-854	28	1	[	[	X
ejpam-854	28	2	20]-[27	20]-[27	NOUN
ejpam-854	28	3	]	]	PUNCT
ejpam-854	28	4	,	,	PUNCT
ejpam-854	29	1	[	[	X
ejpam-854	29	2	33	33	NUM
ejpam-854	29	3	]	]	PUNCT
ejpam-854	29	4	,	,	PUNCT
ejpam-854	30	1	[	[	X
ejpam-854	30	2	36]-[42	36]-[42	X
ejpam-854	30	3	]	]	PUNCT
ejpam-854	30	4	and	and	CCONJ
ejpam-854	30	5	the	the	DET
ejpam-854	30	6	references	reference	NOUN
ejpam-854	30	7	therein	therein	ADV
ejpam-854	30	8	)	)	PUNCT
ejpam-854	30	9	.	.	PUNCT
ejpam-854	31	1	the	the	DET
ejpam-854	31	2	historical	historical	ADJ
ejpam-854	31	3	background	background	NOUN
ejpam-854	31	4	and	and	CCONJ
ejpam-854	31	5	many	many	ADJ
ejpam-854	31	6	important	important	ADJ
ejpam-854	31	7	results	result	NOUN
ejpam-854	31	8	for	for	ADP
ejpam-854	31	9	the	the	DET
ejpam-854	31	10	ulam	ulam	NOUN
ejpam-854	31	11	-	-	PUNCT
ejpam-854	31	12	hyers	hyer	NOUN
ejpam-854	31	13	stability	stability	NOUN
ejpam-854	31	14	of	of	ADP
ejpam-854	31	15	various	various	ADJ
ejpam-854	31	16	functional	functional	ADJ
ejpam-854	31	17	equations	equation	NOUN
ejpam-854	31	18	are	be	AUX
ejpam-854	31	19	surveyed	survey	VERB
ejpam-854	31	20	in	in	ADP
ejpam-854	31	21	[	[	X
ejpam-854	31	22	4	4	NUM
ejpam-854	31	23	]	]	PUNCT
ejpam-854	31	24	(	(	PUNCT
ejpam-854	31	25	see	see	VERB
ejpam-854	31	26	also	also	ADV
ejpam-854	31	27	[	[	X
ejpam-854	31	28	19	19	NUM
ejpam-854	31	29	]	]	PUNCT
ejpam-854	31	30	.	.	PUNCT
ejpam-854	32	1	there	there	PRON
ejpam-854	32	2	are	be	VERB
ejpam-854	32	3	applications	application	NOUN
ejpam-854	32	4	in	in	ADP
ejpam-854	32	5	actuarial	actuarial	ADJ
ejpam-854	32	6	and	and	CCONJ
ejpam-854	32	7	financial	financial	ADJ
ejpam-854	32	8	mathematics	mathematic	NOUN
ejpam-854	32	9	,	,	PUNCT
ejpam-854	32	10	sociology	sociology	NOUN
ejpam-854	32	11	and	and	CCONJ
ejpam-854	32	12	psychology	psychology	NOUN
ejpam-854	32	13	,	,	PUNCT
ejpam-854	32	14	as	as	ADV
ejpam-854	32	15	well	well	ADV
ejpam-854	32	16	as	as	ADP
ejpam-854	32	17	in	in	ADP
ejpam-854	32	18	algebra	algebra	NOUN
ejpam-854	32	19	and	and	CCONJ
ejpam-854	32	20	geometry	geometry	NOUN
ejpam-854	32	21	[	[	X
ejpam-854	32	22	1	1	NUM
ejpam-854	32	23	,	,	PUNCT
ejpam-854	32	24	4	4	NUM
ejpam-854	32	25	,	,	PUNCT
ejpam-854	32	26	19	19	NUM
ejpam-854	32	27	]	]	PUNCT
ejpam-854	32	28	.	.	PUNCT
ejpam-854	33	1	in	in	ADP
ejpam-854	33	2	addition	addition	NOUN
ejpam-854	33	3	,	,	PUNCT
ejpam-854	33	4	the	the	DET
ejpam-854	33	5	motivation	motivation	NOUN
ejpam-854	33	6	for	for	ADP
ejpam-854	33	7	studying	study	VERB
ejpam-854	33	8	these	these	DET
ejpam-854	33	9	functional	functional	ADJ
ejpam-854	33	10	equations	equation	NOUN
ejpam-854	33	11	came	come	VERB
ejpam-854	33	12	from	from	ADP
ejpam-854	33	13	the	the	DET
ejpam-854	33	14	fact	fact	NOUN
ejpam-854	33	15	that	that	SCONJ
ejpam-854	33	16	recently	recently	ADV
ejpam-854	33	17	polynomial	polynomial	ADJ
ejpam-854	33	18	equations	equation	NOUN
ejpam-854	33	19	have	have	AUX
ejpam-854	33	20	found	find	VERB
ejpam-854	33	21	applications	application	NOUN
ejpam-854	33	22	in	in	ADP
ejpam-854	33	23	approximate	approximate	ADJ
ejpam-854	33	24	checking	checking	NOUN
ejpam-854	33	25	,	,	PUNCT
ejpam-854	33	26	selftesting	selftesting	NOUN
ejpam-854	33	27	,	,	PUNCT
ejpam-854	33	28	and	and	CCONJ
ejpam-854	33	29	self	self	NOUN
ejpam-854	33	30	-	-	PUNCT
ejpam-854	33	31	correcting	correcting	NOUN
ejpam-854	33	32	of	of	ADP
ejpam-854	33	33	computer	computer	NOUN
ejpam-854	33	34	programs	program	NOUN
ejpam-854	33	35	that	that	PRON
ejpam-854	33	36	compute	compute	NOUN
ejpam-854	33	37	polynomials	polynomial	NOUN
ejpam-854	33	38	.	.	PUNCT
ejpam-854	34	1	the	the	DET
ejpam-854	34	2	interested	interested	ADJ
ejpam-854	34	3	reader	reader	NOUN
ejpam-854	34	4	should	should	AUX
ejpam-854	34	5	refer	refer	VERB
ejpam-854	34	6	to	to	ADP
ejpam-854	34	7	[	[	X
ejpam-854	34	8	34	34	NUM
ejpam-854	34	9	]	]	PUNCT
ejpam-854	34	10	and	and	CCONJ
ejpam-854	34	11	[	[	X
ejpam-854	34	12	40	40	NUM
ejpam-854	34	13	]	]	PUNCT
ejpam-854	34	14	and	and	CCONJ
ejpam-854	34	15	references	reference	NOUN
ejpam-854	34	16	therein	therein	ADV
ejpam-854	34	17	.	.	PUNCT
ejpam-854	35	1	the	the	DET
ejpam-854	35	2	functional	functional	ADJ
ejpam-854	35	3	equation	equation	NOUN
ejpam-854	35	4	f	f	X
ejpam-854	35	5	(	(	PUNCT
ejpam-854	35	6	x	x	PROPN
ejpam-854	35	7	+	+	NUM
ejpam-854	35	8	y	y	NOUN
ejpam-854	35	9	)	)	PUNCT
ejpam-854	36	1	+	+	NOUN
ejpam-854	36	2	f	f	X
ejpam-854	36	3	(	(	PUNCT
ejpam-854	36	4	x	x	INTJ
ejpam-854	36	5	−	−	PROPN
ejpam-854	36	6	y	y	PROPN
ejpam-854	36	7	)	)	PUNCT
ejpam-854	36	8	=	=	SYM
ejpam-854	36	9	2	2	NUM
ejpam-854	36	10	f	f	NOUN
ejpam-854	36	11	(	(	PUNCT
ejpam-854	36	12	x)+	x)+	PROPN
ejpam-854	36	13	2	2	NUM
ejpam-854	36	14	f	f	NOUN
ejpam-854	36	15	(	(	PUNCT
ejpam-854	36	16	y	y	NOUN
ejpam-854	36	17	)	)	PUNCT
ejpam-854	36	18	(	(	PUNCT
ejpam-854	36	19	1	1	X
ejpam-854	36	20	)	)	PUNCT
ejpam-854	36	21	is	be	AUX
ejpam-854	36	22	said	say	VERB
ejpam-854	36	23	to	to	PART
ejpam-854	36	24	be	be	AUX
ejpam-854	36	25	a	a	DET
ejpam-854	36	26	quadratic	quadratic	ADJ
ejpam-854	36	27	functional	functional	ADJ
ejpam-854	36	28	equation	equation	NOUN
ejpam-854	36	29	because	because	SCONJ
ejpam-854	36	30	the	the	DET
ejpam-854	36	31	quadratic	quadratic	ADJ
ejpam-854	36	32	function	function	NOUN
ejpam-854	36	33	f	f	PROPN
ejpam-854	36	34	(	(	PUNCT
ejpam-854	36	35	x	x	X
ejpam-854	36	36	)	)	PUNCT
ejpam-854	36	37	=	=	SYM
ejpam-854	37	1	x2	x2	PROPN
ejpam-854	37	2	is	be	AUX
ejpam-854	37	3	a	a	DET
ejpam-854	37	4	solution	solution	NOUN
ejpam-854	37	5	of	of	ADP
ejpam-854	37	6	the	the	DET
ejpam-854	37	7	functional	functional	ADJ
ejpam-854	37	8	equation	equation	NOUN
ejpam-854	37	9	(	(	PUNCT
ejpam-854	37	10	1	1	NUM
ejpam-854	37	11	)	)	PUNCT
ejpam-854	37	12	.	.	PUNCT
ejpam-854	38	1	every	every	DET
ejpam-854	38	2	solution	solution	NOUN
ejpam-854	38	3	of	of	ADP
ejpam-854	38	4	the	the	DET
ejpam-854	38	5	quadratic	quadratic	ADJ
ejpam-854	38	6	functional	functional	ADJ
ejpam-854	38	7	equation	equation	NOUN
ejpam-854	38	8	is	be	AUX
ejpam-854	38	9	said	say	VERB
ejpam-854	38	10	to	to	PART
ejpam-854	38	11	be	be	AUX
ejpam-854	38	12	a	a	DET
ejpam-854	38	13	quadratic	quadratic	ADJ
ejpam-854	38	14	mapping	mapping	NOUN
ejpam-854	38	15	.	.	PUNCT
ejpam-854	39	1	a	a	DET
ejpam-854	39	2	quadratic	quadratic	ADJ
ejpam-854	39	3	functional	functional	ADJ
ejpam-854	39	4	equation	equation	NOUN
ejpam-854	39	5	was	be	AUX
ejpam-854	39	6	used	use	VERB
ejpam-854	39	7	to	to	PART
ejpam-854	39	8	characterize	characterize	VERB
ejpam-854	39	9	inner	inner	ADJ
ejpam-854	39	10	product	product	NOUN
ejpam-854	39	11	spaces	space	NOUN
ejpam-854	39	12	.	.	PUNCT
ejpam-854	40	1	in	in	ADP
ejpam-854	40	2	2001	2001	NUM
ejpam-854	40	3	,	,	PUNCT
ejpam-854	40	4	j.	j.	PROPN
ejpam-854	40	5	m.	m.	PROPN
ejpam-854	40	6	rassias	rassias	PROPN
ejpam-854	41	1	[	[	X
ejpam-854	41	2	29	29	NUM
ejpam-854	41	3	]	]	PUNCT
ejpam-854	41	4	introduced	introduce	VERB
ejpam-854	41	5	the	the	DET
ejpam-854	41	6	cubic	cubic	ADJ
ejpam-854	41	7	functional	functional	ADJ
ejpam-854	41	8	equation	equation	NOUN
ejpam-854	41	9	f	f	X
ejpam-854	41	10	(	(	PUNCT
ejpam-854	41	11	x	x	X
ejpam-854	41	12	+	+	PUNCT
ejpam-854	41	13	2y)−	2y)−	NUM
ejpam-854	41	14	3	3	NUM
ejpam-854	41	15	f	f	NOUN
ejpam-854	41	16	(	(	PUNCT
ejpam-854	41	17	x	x	PROPN
ejpam-854	41	18	+	+	NUM
ejpam-854	41	19	y	y	NOUN
ejpam-854	41	20	)	)	PUNCT
ejpam-854	42	1	+	+	CCONJ
ejpam-854	42	2	3	3	NUM
ejpam-854	42	3	f	f	X
ejpam-854	42	4	(	(	PUNCT
ejpam-854	42	5	x)−	x)−	PROPN
ejpam-854	42	6	f	f	PROPN
ejpam-854	42	7	(	(	PUNCT
ejpam-854	42	8	x	x	PROPN
ejpam-854	42	9	−	−	PROPN
ejpam-854	42	10	y	y	PROPN
ejpam-854	42	11	)	)	PUNCT
ejpam-854	42	12	=	=	PUNCT
ejpam-854	42	13	6	6	NUM
ejpam-854	42	14	f	f	NOUN
ejpam-854	42	15	(	(	PUNCT
ejpam-854	42	16	y	y	NOUN
ejpam-854	42	17	)	)	PUNCT
ejpam-854	42	18	(	(	PUNCT
ejpam-854	42	19	2	2	X
ejpam-854	42	20	)	)	PUNCT
ejpam-854	42	21	and	and	CCONJ
ejpam-854	42	22	established	establish	VERB
ejpam-854	42	23	the	the	DET
ejpam-854	42	24	solution	solution	NOUN
ejpam-854	42	25	of	of	ADP
ejpam-854	42	26	the	the	DET
ejpam-854	42	27	ulam	ulam	PROPN
ejpam-854	42	28	stability	stability	PROPN
ejpam-854	42	29	problem	problem	NOUN
ejpam-854	42	30	for	for	ADP
ejpam-854	42	31	these	these	DET
ejpam-854	42	32	cubic	cubic	ADJ
ejpam-854	42	33	mappings	mapping	NOUN
ejpam-854	42	34	.	.	PUNCT
ejpam-854	43	1	it	it	PRON
ejpam-854	43	2	is	be	AUX
ejpam-854	43	3	easy	easy	ADJ
ejpam-854	43	4	to	to	PART
ejpam-854	43	5	show	show	VERB
ejpam-854	43	6	that	that	SCONJ
ejpam-854	43	7	the	the	DET
ejpam-854	43	8	function	function	NOUN
ejpam-854	43	9	f	f	X
ejpam-854	43	10	(	(	PUNCT
ejpam-854	43	11	x	x	X
ejpam-854	43	12	)	)	PUNCT
ejpam-854	43	13	=	=	SYM
ejpam-854	43	14	x3	x3	ADJ
ejpam-854	43	15	satisfies	satisfy	VERB
ejpam-854	43	16	the	the	DET
ejpam-854	43	17	functional	functional	ADJ
ejpam-854	43	18	equation	equation	NOUN
ejpam-854	43	19	(	(	PUNCT
ejpam-854	43	20	2	2	NUM
ejpam-854	43	21	)	)	PUNCT
ejpam-854	43	22	,	,	PUNCT
ejpam-854	43	23	which	which	PRON
ejpam-854	43	24	is	be	AUX
ejpam-854	43	25	called	call	VERB
ejpam-854	43	26	a	a	DET
ejpam-854	43	27	cubic	cubic	ADJ
ejpam-854	43	28	functional	functional	ADJ
ejpam-854	43	29	equation	equation	NOUN
ejpam-854	43	30	and	and	CCONJ
ejpam-854	43	31	every	every	DET
ejpam-854	43	32	solution	solution	NOUN
ejpam-854	43	33	of	of	ADP
ejpam-854	43	34	the	the	DET
ejpam-854	43	35	cubic	cubic	ADJ
ejpam-854	43	36	functional	functional	ADJ
ejpam-854	43	37	equation	equation	NOUN
ejpam-854	43	38	is	be	AUX
ejpam-854	43	39	said	say	VERB
ejpam-854	43	40	to	to	PART
ejpam-854	43	41	be	be	AUX
ejpam-854	43	42	a	a	DET
ejpam-854	43	43	cubic	cubic	ADJ
ejpam-854	43	44	mapping	mapping	NOUN
ejpam-854	43	45	.	.	PUNCT
ejpam-854	44	1	the	the	DET
ejpam-854	44	2	quartic	quartic	ADJ
ejpam-854	44	3	functional	functional	ADJ
ejpam-854	44	4	equation	equation	NOUN
ejpam-854	44	5	f	f	X
ejpam-854	44	6	(	(	PUNCT
ejpam-854	44	7	x	x	X
ejpam-854	44	8	+	+	PUNCT
ejpam-854	44	9	2y	2y	NUM
ejpam-854	44	10	)	)	PUNCT
ejpam-854	45	1	+	+	NUM
ejpam-854	45	2	f	f	X
ejpam-854	45	3	(	(	PUNCT
ejpam-854	45	4	x	x	X
ejpam-854	45	5	−	−	NOUN
ejpam-854	45	6	2y	2y	NUM
ejpam-854	45	7	)	)	PUNCT
ejpam-854	45	8	=	=	SYM
ejpam-854	45	9	4	4	NUM
ejpam-854	45	10	f	f	X
ejpam-854	45	11	(	(	PUNCT
ejpam-854	45	12	x	x	PROPN
ejpam-854	45	13	+	+	NUM
ejpam-854	45	14	y	y	NOUN
ejpam-854	45	15	)	)	PUNCT
ejpam-854	45	16	+	+	CCONJ
ejpam-854	45	17	4	4	NUM
ejpam-854	45	18	f	f	X
ejpam-854	45	19	(	(	PUNCT
ejpam-854	45	20	x	x	PROPN
ejpam-854	45	21	−	−	PROPN
ejpam-854	45	22	y	y	PROPN
ejpam-854	45	23	)	)	PUNCT
ejpam-854	45	24	+	+	CCONJ
ejpam-854	45	25	6	6	NUM
ejpam-854	45	26	f	f	NOUN
ejpam-854	45	27	(	(	PUNCT
ejpam-854	45	28	x)+	x)+	PROPN
ejpam-854	45	29	24	24	NUM
ejpam-854	45	30	f	f	PROPN
ejpam-854	45	31	(	(	PUNCT
ejpam-854	45	32	y	y	NOUN
ejpam-854	45	33	)	)	PUNCT
ejpam-854	45	34	(	(	PUNCT
ejpam-854	45	35	3	3	X
ejpam-854	45	36	)	)	PUNCT
ejpam-854	45	37	was	be	AUX
ejpam-854	45	38	introduced	introduce	VERB
ejpam-854	45	39	by	by	ADP
ejpam-854	45	40	j.	j.	PROPN
ejpam-854	45	41	m.	m.	PROPN
ejpam-854	45	42	rassias	rassias	PROPN
ejpam-854	46	1	[	[	X
ejpam-854	46	2	31	31	NUM
ejpam-854	46	3	]	]	PUNCT
ejpam-854	46	4	.	.	PUNCT
ejpam-854	47	1	it	it	PRON
ejpam-854	47	2	is	be	AUX
ejpam-854	47	3	easy	easy	ADJ
ejpam-854	47	4	to	to	PART
ejpam-854	47	5	show	show	VERB
ejpam-854	47	6	that	that	SCONJ
ejpam-854	47	7	the	the	DET
ejpam-854	47	8	function	function	NOUN
ejpam-854	47	9	f	f	X
ejpam-854	47	10	(	(	PUNCT
ejpam-854	47	11	x	x	X
ejpam-854	47	12	)	)	PUNCT
ejpam-854	47	13	=	=	PUNCT
ejpam-854	47	14	x4	x4	PROPN
ejpam-854	47	15	is	be	AUX
ejpam-854	47	16	the	the	DET
ejpam-854	47	17	solution	solution	NOUN
ejpam-854	47	18	of	of	ADP
ejpam-854	47	19	(	(	PUNCT
ejpam-854	47	20	3	3	NUM
ejpam-854	47	21	)	)	PUNCT
ejpam-854	47	22	.	.	PUNCT
ejpam-854	48	1	every	every	DET
ejpam-854	48	2	solution	solution	NOUN
ejpam-854	48	3	of	of	ADP
ejpam-854	48	4	the	the	DET
ejpam-854	48	5	quartic	quartic	ADJ
ejpam-854	48	6	functional	functional	ADJ
ejpam-854	48	7	equation	equation	NOUN
ejpam-854	48	8	is	be	AUX
ejpam-854	48	9	said	say	VERB
ejpam-854	48	10	to	to	PART
ejpam-854	48	11	be	be	AUX
ejpam-854	48	12	a	a	DET
ejpam-854	48	13	quartic	quartic	ADJ
ejpam-854	48	14	mapping	mapping	NOUN
ejpam-854	48	15	.	.	PUNCT
ejpam-854	49	1	c.	c.	PROPN
ejpam-854	49	2	park	park	PROPN
ejpam-854	50	1	[	[	X
ejpam-854	50	2	25	25	NUM
ejpam-854	50	3	]	]	PUNCT
ejpam-854	50	4	proved	prove	VERB
ejpam-854	50	5	the	the	DET
ejpam-854	50	6	generalized	generalize	VERB
ejpam-854	50	7	hyers	hyer	NOUN
ejpam-854	50	8	-	-	PUNCT
ejpam-854	50	9	ulam	ulam	ADJ
ejpam-854	50	10	stability	stability	NOUN
ejpam-854	50	11	of	of	ADP
ejpam-854	50	12	the	the	DET
ejpam-854	50	13	following	follow	VERB
ejpam-854	50	14	additivequadratic	additivequadratic	ADJ
ejpam-854	50	15	-	-	PUNCT
ejpam-854	50	16	cubic	cubic	ADJ
ejpam-854	50	17	-	-	PUNCT
ejpam-854	50	18	quartic	quartic	ADJ
ejpam-854	50	19	functional	functional	ADJ
ejpam-854	50	20	equation	equation	NOUN
ejpam-854	50	21	(	(	PUNCT
ejpam-854	50	22	briefly	briefly	ADV
ejpam-854	50	23	,	,	PUNCT
ejpam-854	50	24	aqcq	aqcq	NOUN
ejpam-854	50	25	-	-	PUNCT
ejpam-854	50	26	functional	functional	ADJ
ejpam-854	50	27	equation	equation	NOUN
ejpam-854	50	28	)	)	PUNCT
ejpam-854	51	1	f	f	PROPN
ejpam-854	52	1	(	(	PUNCT
ejpam-854	52	2	x+2y)+	x+2y)+	PROPN
ejpam-854	52	3	f	f	PROPN
ejpam-854	52	4	(	(	PUNCT
ejpam-854	52	5	x−2y	x−2y	PROPN
ejpam-854	52	6	)	)	PUNCT
ejpam-854	52	7	=	=	SYM
ejpam-854	52	8	4	4	NUM
ejpam-854	52	9	f	f	X
ejpam-854	52	10	(	(	PUNCT
ejpam-854	52	11	x+	x+	PROPN
ejpam-854	52	12	y)+4	y)+4	PROPN
ejpam-854	52	13	f	f	PROPN
ejpam-854	52	14	(	(	PUNCT
ejpam-854	52	15	x−	x−	PROPN
ejpam-854	52	16	y)−6	y)−6	PROPN
ejpam-854	52	17	f	f	PROPN
ejpam-854	52	18	(	(	PUNCT
ejpam-854	52	19	x)+	x)+	PROPN
ejpam-854	52	20	f	f	PROPN
ejpam-854	52	21	(	(	PUNCT
ejpam-854	52	22	2y)+	2y)+	NUM
ejpam-854	52	23	f	f	X
ejpam-854	52	24	(	(	PUNCT
ejpam-854	52	25	−2y)−4	−2y)−4	PROPN
ejpam-854	52	26	f	f	PROPN
ejpam-854	52	27	(	(	PUNCT
ejpam-854	52	28	y)−4(−y	y)−4(−y	PROPN
ejpam-854	52	29	)	)	PUNCT
ejpam-854	52	30	(	(	PUNCT
ejpam-854	52	31	4	4	X
ejpam-854	52	32	)	)	PUNCT
ejpam-854	52	33	t.	t.	NOUN
ejpam-854	52	34	xu	xu	PROPN
ejpam-854	52	35	,	,	PUNCT
ejpam-854	52	36	j.	j.	PROPN
ejpam-854	52	37	rassias	rassias	PROPN
ejpam-854	52	38	,	,	PUNCT
ejpam-854	52	39	w.	w.	PROPN
ejpam-854	52	40	xu	xu	PROPN
ejpam-854	52	41	/	/	SYM
ejpam-854	52	42	eur	eur	PROPN
ejpam-854	52	43	.	.	PUNCT
ejpam-854	53	1	j.	j.	PROPN
ejpam-854	53	2	pure	pure	PROPN
ejpam-854	53	3	appl	appl	PROPN
ejpam-854	53	4	.	.	PROPN
ejpam-854	53	5	math	math	PROPN
ejpam-854	53	6	,	,	PUNCT
ejpam-854	53	7	3	3	NUM
ejpam-854	53	8	(	(	PUNCT
ejpam-854	53	9	2010	2010	NUM
ejpam-854	53	10	)	)	PUNCT
ejpam-854	53	11	,	,	PUNCT
ejpam-854	53	12	1032	1032	NUM
ejpam-854	53	13	-	-	SYM
ejpam-854	53	14	1047	1047	NUM
ejpam-854	53	15	1034	1034	NUM
ejpam-854	53	16	in	in	ADP
ejpam-854	53	17	non	non	ADJ
ejpam-854	53	18	-	-	ADJ
ejpam-854	53	19	archimedean	archimedean	ADJ
ejpam-854	53	20	normed	norme	VERB
ejpam-854	53	21	spaces	space	NOUN
ejpam-854	53	22	.	.	PUNCT
ejpam-854	54	1	in	in	ADP
ejpam-854	54	2	[	[	X
ejpam-854	54	3	9	9	NUM
ejpam-854	54	4	,	,	PUNCT
ejpam-854	54	5	33	33	NUM
ejpam-854	54	6	]	]	PUNCT
ejpam-854	54	7	,	,	PUNCT
ejpam-854	54	8	the	the	DET
ejpam-854	54	9	authors	author	NOUN
ejpam-854	54	10	introduced	introduce	VERB
ejpam-854	54	11	a	a	DET
ejpam-854	54	12	general	general	ADJ
ejpam-854	54	13	mixed	mixed	ADJ
ejpam-854	54	14	type	type	NOUN
ejpam-854	54	15	functional	functional	ADJ
ejpam-854	54	16	equation	equation	NOUN
ejpam-854	54	17	f	f	X
ejpam-854	54	18	(	(	PUNCT
ejpam-854	54	19	x	x	PROPN
ejpam-854	54	20	+	+	NUM
ejpam-854	54	21	ny	ny	NOUN
ejpam-854	54	22	)	)	PUNCT
ejpam-854	55	1	+	+	NUM
ejpam-854	55	2	f	f	X
ejpam-854	55	3	(	(	PUNCT
ejpam-854	55	4	x	x	X
ejpam-854	55	5	−	−	PROPN
ejpam-854	55	6	ny	ny	NOUN
ejpam-854	55	7	)	)	PUNCT
ejpam-854	55	8	=	=	SYM
ejpam-854	55	9	n2	n2	PROPN
ejpam-854	55	10	f	f	PROPN
ejpam-854	55	11	(	(	PUNCT
ejpam-854	55	12	x	x	PROPN
ejpam-854	55	13	+	+	NUM
ejpam-854	55	14	y	y	NOUN
ejpam-854	55	15	)	)	PUNCT
ejpam-854	56	1	+	+	CCONJ
ejpam-854	56	2	n2	n2	PROPN
ejpam-854	56	3	f	f	PROPN
ejpam-854	56	4	(	(	PUNCT
ejpam-854	56	5	x	x	PROPN
ejpam-854	56	6	−	−	PROPN
ejpam-854	56	7	y	y	PROPN
ejpam-854	56	8	)	)	PUNCT
ejpam-854	56	9	+	+	CCONJ
ejpam-854	56	10	2(1−	2(1−	NUM
ejpam-854	56	11	n2	n2	ADJ
ejpam-854	56	12	)	)	PUNCT
ejpam-854	56	13	f	f	PROPN
ejpam-854	56	14	(	(	PUNCT
ejpam-854	56	15	x	x	X
ejpam-854	56	16	)	)	PUNCT
ejpam-854	57	1	+	+	CCONJ
ejpam-854	57	2	n4	n4	PROPN
ejpam-854	57	3	−	−	PROPN
ejpam-854	57	4	n2	n2	NOUN
ejpam-854	57	5	12	12	NUM
ejpam-854	57	6	[	[	PUNCT
ejpam-854	57	7	f	f	X
ejpam-854	57	8	(	(	PUNCT
ejpam-854	57	9	2y	2y	NUM
ejpam-854	57	10	)	)	PUNCT
ejpam-854	58	1	+	+	NUM
ejpam-854	58	2	f	f	X
ejpam-854	58	3	(	(	PUNCT
ejpam-854	58	4	−2y)−	−2y)−	NUM
ejpam-854	58	5	4	4	NUM
ejpam-854	58	6	f	f	NOUN
ejpam-854	58	7	(	(	PUNCT
ejpam-854	58	8	y)−	y)−	PROPN
ejpam-854	58	9	4	4	NUM
ejpam-854	58	10	f	f	NOUN
ejpam-854	58	11	(	(	PUNCT
ejpam-854	58	12	−y	−y	NOUN
ejpam-854	58	13	)	)	PUNCT
ejpam-854	58	14	]	]	PUNCT
ejpam-854	58	15	(	(	PUNCT
ejpam-854	58	16	5	5	X
ejpam-854	58	17	)	)	PUNCT
ejpam-854	58	18	which	which	PRON
ejpam-854	58	19	is	be	AUX
ejpam-854	58	20	a	a	DET
ejpam-854	58	21	generalized	generalized	ADJ
ejpam-854	58	22	form	form	NOUN
ejpam-854	58	23	of	of	ADP
ejpam-854	58	24	the	the	DET
ejpam-854	58	25	additive	additive	ADJ
ejpam-854	58	26	-	-	PUNCT
ejpam-854	58	27	quadratic	quadratic	ADJ
ejpam-854	58	28	-	-	PUNCT
ejpam-854	58	29	cubic	cubic	NOUN
ejpam-854	58	30	-	-	PUNCT
ejpam-854	58	31	quartic	quartic	ADJ
ejpam-854	58	32	(	(	PUNCT
ejpam-854	58	33	4	4	NUM
ejpam-854	58	34	)	)	PUNCT
ejpam-854	58	35	and	and	CCONJ
ejpam-854	58	36	obtained	obtain	VERB
ejpam-854	58	37	its	its	PRON
ejpam-854	58	38	general	general	ADJ
ejpam-854	58	39	solution	solution	NOUN
ejpam-854	58	40	and	and	CCONJ
ejpam-854	58	41	generalized	generalized	ADJ
ejpam-854	58	42	hyers	hyer	NOUN
ejpam-854	58	43	-	-	PUNCT
ejpam-854	58	44	ulam	ulam	PROPN
ejpam-854	58	45	stability	stability	NOUN
ejpam-854	58	46	for	for	ADP
ejpam-854	58	47	fixed	fix	VERB
ejpam-854	58	48	integers	integer	NOUN
ejpam-854	58	49	n	n	X
ejpam-854	58	50	with	with	ADP
ejpam-854	58	51	n	n	PROPN
ejpam-854	58	52	6=	6=	NUM
ejpam-854	58	53	0,±1	0,±1	NOUN
ejpam-854	58	54	in	in	ADP
ejpam-854	58	55	banach	banach	NOUN
ejpam-854	58	56	spaces	space	NOUN
ejpam-854	58	57	.	.	PUNCT
ejpam-854	59	1	the	the	DET
ejpam-854	59	2	notion	notion	NOUN
ejpam-854	59	3	of	of	ADP
ejpam-854	59	4	multi	multi	ADJ
ejpam-854	59	5	-	-	ADJ
ejpam-854	59	6	normed	normed	ADJ
ejpam-854	59	7	space	space	NOUN
ejpam-854	59	8	was	be	AUX
ejpam-854	59	9	introduced	introduce	VERB
ejpam-854	59	10	by	by	ADP
ejpam-854	59	11	h.	h.	PROPN
ejpam-854	59	12	g.	g.	PROPN
ejpam-854	59	13	dales	dales	PROPN
ejpam-854	59	14	and	and	CCONJ
ejpam-854	59	15	m.	m.	PROPN
ejpam-854	59	16	e.	e.	PROPN
ejpam-854	59	17	polyakov	polyakov	PROPN
ejpam-854	59	18	[	[	X
ejpam-854	59	19	5	5	NUM
ejpam-854	59	20	]	]	PUNCT
ejpam-854	59	21	.	.	PUNCT
ejpam-854	60	1	this	this	DET
ejpam-854	60	2	concept	concept	NOUN
ejpam-854	60	3	is	be	AUX
ejpam-854	60	4	somewhat	somewhat	ADV
ejpam-854	60	5	similar	similar	ADJ
ejpam-854	60	6	to	to	PART
ejpam-854	60	7	operator	operator	NOUN
ejpam-854	60	8	sequence	sequence	NOUN
ejpam-854	60	9	space	space	NOUN
ejpam-854	60	10	and	and	CCONJ
ejpam-854	60	11	has	have	VERB
ejpam-854	60	12	some	some	DET
ejpam-854	60	13	connections	connection	NOUN
ejpam-854	60	14	with	with	ADP
ejpam-854	60	15	operator	operator	NOUN
ejpam-854	60	16	spaces	space	NOUN
ejpam-854	60	17	and	and	CCONJ
ejpam-854	60	18	banach	banach	NOUN
ejpam-854	60	19	lattices	lattice	NOUN
ejpam-854	60	20	.	.	PUNCT
ejpam-854	61	1	motivations	motivation	NOUN
ejpam-854	61	2	for	for	ADP
ejpam-854	61	3	the	the	DET
ejpam-854	61	4	study	study	NOUN
ejpam-854	61	5	of	of	ADP
ejpam-854	61	6	multi	multi	ADJ
ejpam-854	61	7	-	-	ADJ
ejpam-854	61	8	normed	normed	ADJ
ejpam-854	61	9	spaces	space	NOUN
ejpam-854	61	10	and	and	CCONJ
ejpam-854	61	11	many	many	ADJ
ejpam-854	61	12	examples	example	NOUN
ejpam-854	61	13	were	be	AUX
ejpam-854	61	14	given	give	VERB
ejpam-854	61	15	in	in	ADP
ejpam-854	61	16	[	[	X
ejpam-854	61	17	5	5	NUM
ejpam-854	61	18	]	]	PUNCT
ejpam-854	61	19	.	.	PUNCT
ejpam-854	62	1	also	also	ADV
ejpam-854	62	2	,	,	PUNCT
ejpam-854	62	3	the	the	DET
ejpam-854	62	4	stability	stability	NOUN
ejpam-854	62	5	problems	problem	VERB
ejpam-854	62	6	in	in	ADP
ejpam-854	62	7	multi	multi	ADJ
ejpam-854	62	8	-	-	ADJ
ejpam-854	62	9	banach	banach	ADJ
ejpam-854	62	10	spaces	space	NOUN
ejpam-854	62	11	are	be	AUX
ejpam-854	62	12	studied	study	VERB
ejpam-854	62	13	by	by	ADP
ejpam-854	62	14	dales	dale	NOUN
ejpam-854	62	15	and	and	CCONJ
ejpam-854	62	16	moslehian	moslehian	NOUN
ejpam-854	63	1	[	[	X
ejpam-854	63	2	6	6	NUM
ejpam-854	63	3	]	]	PUNCT
ejpam-854	63	4	,	,	PUNCT
ejpam-854	63	5	moslehian	moslehian	PROPN
ejpam-854	63	6	et	et	PROPN
ejpam-854	63	7	al	al	PROPN
ejpam-854	63	8	.	.	PUNCT
ejpam-854	64	1	(	(	PUNCT
ejpam-854	64	2	[	[	X
ejpam-854	64	3	21]-[23	21]-[23	NOUN
ejpam-854	64	4	]	]	PUNCT
ejpam-854	64	5	)	)	PUNCT
ejpam-854	64	6	and	and	CCONJ
ejpam-854	64	7	wang	wang	PROPN
ejpam-854	64	8	et	et	PROPN
ejpam-854	64	9	al	al	PROPN
ejpam-854	64	10	.	.	PUNCT
ejpam-854	65	1	[	[	X
ejpam-854	65	2	36	36	NUM
ejpam-854	65	3	]	]	PUNCT
ejpam-854	65	4	.	.	PUNCT
ejpam-854	66	1	in	in	ADP
ejpam-854	66	2	1996	1996	NUM
ejpam-854	66	3	,	,	PUNCT
ejpam-854	66	4	isac	isac	NOUN
ejpam-854	66	5	and	and	CCONJ
ejpam-854	66	6	rassias	rassia	VERB
ejpam-854	67	1	[	[	X
ejpam-854	67	2	18	18	NUM
ejpam-854	67	3	]	]	PUNCT
ejpam-854	67	4	were	be	AUX
ejpam-854	67	5	the	the	DET
ejpam-854	67	6	first	first	ADJ
ejpam-854	67	7	to	to	PART
ejpam-854	67	8	provide	provide	VERB
ejpam-854	67	9	applications	application	NOUN
ejpam-854	67	10	of	of	ADP
ejpam-854	67	11	stability	stability	NOUN
ejpam-854	67	12	theory	theory	NOUN
ejpam-854	67	13	of	of	ADP
ejpam-854	67	14	functional	functional	ADJ
ejpam-854	67	15	equations	equation	NOUN
ejpam-854	67	16	for	for	ADP
ejpam-854	67	17	the	the	DET
ejpam-854	67	18	proof	proof	NOUN
ejpam-854	67	19	of	of	ADP
ejpam-854	67	20	new	new	ADJ
ejpam-854	67	21	fixed	fix	VERB
ejpam-854	67	22	point	point	NOUN
ejpam-854	67	23	theorems	theorem	NOUN
ejpam-854	67	24	with	with	ADP
ejpam-854	67	25	applications	application	NOUN
ejpam-854	67	26	.	.	PUNCT
ejpam-854	68	1	the	the	DET
ejpam-854	68	2	stability	stability	NOUN
ejpam-854	68	3	problems	problem	NOUN
ejpam-854	68	4	of	of	ADP
ejpam-854	68	5	several	several	ADJ
ejpam-854	68	6	various	various	ADJ
ejpam-854	68	7	functional	functional	ADJ
ejpam-854	68	8	equations	equation	NOUN
ejpam-854	68	9	have	have	AUX
ejpam-854	68	10	been	be	AUX
ejpam-854	68	11	extensively	extensively	ADV
ejpam-854	68	12	investigated	investigate	VERB
ejpam-854	68	13	by	by	ADP
ejpam-854	68	14	a	a	DET
ejpam-854	68	15	number	number	NOUN
ejpam-854	68	16	of	of	ADP
ejpam-854	68	17	authors	author	NOUN
ejpam-854	68	18	using	use	VERB
ejpam-854	68	19	the	the	DET
ejpam-854	68	20	fixed	fix	VERB
ejpam-854	68	21	point	point	NOUN
ejpam-854	68	22	method	method	NOUN
ejpam-854	68	23	(	(	PUNCT
ejpam-854	68	24	see	see	VERB
ejpam-854	68	25	[	[	X
ejpam-854	68	26	2]-[3	2]-[3	NUM
ejpam-854	68	27	]	]	PUNCT
ejpam-854	68	28	,	,	PUNCT
ejpam-854	69	1	[	[	X
ejpam-854	69	2	6]-[7	6]-[7	NOUN
ejpam-854	69	3	]	]	X
ejpam-854	69	4	,	,	PUNCT
ejpam-854	69	5	[	[	X
ejpam-854	69	6	20	20	NUM
ejpam-854	69	7	]	]	PUNCT
ejpam-854	69	8	,	,	PUNCT
ejpam-854	70	1	[	[	X
ejpam-854	70	2	25]-[27	25]-[27	NUM
ejpam-854	70	3	]	]	PUNCT
ejpam-854	70	4	,	,	PUNCT
ejpam-854	71	1	[	[	X
ejpam-854	71	2	36	36	NUM
ejpam-854	71	3	]	]	PUNCT
ejpam-854	71	4	,	,	PUNCT
ejpam-854	72	1	[	[	X
ejpam-854	72	2	38	38	NUM
ejpam-854	72	3	]	]	PUNCT
ejpam-854	72	4	.	.	PUNCT
ejpam-854	73	1	in	in	ADP
ejpam-854	73	2	this	this	DET
ejpam-854	73	3	paper	paper	NOUN
ejpam-854	73	4	,	,	PUNCT
ejpam-854	73	5	we	we	PRON
ejpam-854	73	6	prove	prove	VERB
ejpam-854	73	7	the	the	DET
ejpam-854	73	8	generalized	generalize	VERB
ejpam-854	73	9	hyers	hyer	NOUN
ejpam-854	73	10	-	-	PUNCT
ejpam-854	73	11	ulam	ulam	ADJ
ejpam-854	73	12	stability	stability	NOUN
ejpam-854	73	13	of	of	ADP
ejpam-854	73	14	the	the	DET
ejpam-854	73	15	general	general	ADJ
ejpam-854	73	16	mixed	mixed	ADJ
ejpam-854	73	17	aqcqfunctional	aqcqfunctional	ADJ
ejpam-854	73	18	equation	equation	NOUN
ejpam-854	73	19	(	(	PUNCT
ejpam-854	73	20	5	5	NUM
ejpam-854	73	21	)	)	PUNCT
ejpam-854	73	22	in	in	ADP
ejpam-854	73	23	multi	multi	ADJ
ejpam-854	73	24	-	-	ADJ
ejpam-854	73	25	banach	banach	ADJ
ejpam-854	73	26	spaces	space	NOUN
ejpam-854	73	27	using	use	VERB
ejpam-854	73	28	the	the	DET
ejpam-854	73	29	fixed	fix	VERB
ejpam-854	73	30	point	point	NOUN
ejpam-854	73	31	method	method	NOUN
ejpam-854	73	32	.	.	PUNCT
ejpam-854	74	1	2	2	X
ejpam-854	74	2	.	.	X
ejpam-854	74	3	preliminaries	preliminary	NOUN
ejpam-854	74	4	we	we	PRON
ejpam-854	74	5	recall	recall	VERB
ejpam-854	74	6	some	some	DET
ejpam-854	74	7	preliminaries	preliminary	NOUN
ejpam-854	74	8	concerning	concern	VERB
ejpam-854	74	9	multi	multi	ADJ
ejpam-854	74	10	-	-	ADJ
ejpam-854	74	11	banach	banach	ADJ
ejpam-854	74	12	space	space	NOUN
ejpam-854	74	13	(	(	PUNCT
ejpam-854	74	14	see	see	VERB
ejpam-854	74	15	[	[	X
ejpam-854	74	16	5]-[6	5]-[6	X
ejpam-854	74	17	]	]	PUNCT
ejpam-854	74	18	,	,	PUNCT
ejpam-854	75	1	[	[	X
ejpam-854	75	2	21]-[23	21]-[23	NOUN
ejpam-854	75	3	]	]	PUNCT
ejpam-854	75	4	)	)	PUNCT
ejpam-854	75	5	.	.	PUNCT
ejpam-854	76	1	let	let	VERB
ejpam-854	76	2	(	(	PUNCT
ejpam-854	76	3	e,‖	e,‖	X
ejpam-854	76	4	·	·	PUNCT
ejpam-854	76	5	‖	‖	X
ejpam-854	76	6	)	)	PUNCT
ejpam-854	76	7	be	be	AUX
ejpam-854	76	8	a	a	DET
ejpam-854	76	9	complex	complex	ADJ
ejpam-854	76	10	linear	linear	ADJ
ejpam-854	76	11	space	space	NOUN
ejpam-854	76	12	,	,	PUNCT
ejpam-854	76	13	and	and	CCONJ
ejpam-854	76	14	let	let	VERB
ejpam-854	76	15	k	k	PROPN
ejpam-854	76	16	∈	∈	PROPN
ejpam-854	76	17	n.	n.	NOUN
ejpam-854	76	18	we	we	PRON
ejpam-854	76	19	denote	denote	VERB
ejpam-854	76	20	by	by	ADP
ejpam-854	76	21	ek	ek	NOUN
ejpam-854	76	22	the	the	DET
ejpam-854	76	23	linear	linear	ADJ
ejpam-854	76	24	space	space	NOUN
ejpam-854	76	25	e	e	PROPN
ejpam-854	76	26	⊕	⊕	PROPN
ejpam-854	76	27	·	·	PUNCT
ejpam-854	76	28	·	·	PUNCT
ejpam-854	76	29	·	·	PUNCT
ejpam-854	77	1	⊕	⊕	NOUN
ejpam-854	78	1	e	e	X
ejpam-854	78	2	consisting	consist	VERB
ejpam-854	78	3	of	of	ADP
ejpam-854	78	4	k	k	NOUN
ejpam-854	78	5	-	-	PUNCT
ejpam-854	78	6	tuples	tuples	PROPN
ejpam-854	78	7	(	(	PUNCT
ejpam-854	78	8	x1	x1	PROPN
ejpam-854	78	9	,	,	PUNCT
ejpam-854	78	10	.	.	PUNCT
ejpam-854	78	11	.	.	PUNCT
ejpam-854	78	12	.	.	PUNCT
ejpam-854	79	1	,	,	PUNCT
ejpam-854	79	2	xk	xk	PROPN
ejpam-854	79	3	)	)	PUNCT
ejpam-854	79	4	,	,	PUNCT
ejpam-854	79	5	where	where	SCONJ
ejpam-854	79	6	x1	x1	ADJ
ejpam-854	79	7	,	,	PUNCT
ejpam-854	79	8	.	.	PUNCT
ejpam-854	79	9	.	.	PUNCT
ejpam-854	79	10	.	.	PUNCT
ejpam-854	80	1	,	,	PUNCT
ejpam-854	80	2	xk	xk	PROPN
ejpam-854	80	3	∈	∈	PROPN
ejpam-854	80	4	e.	e.	PROPN
ejpam-854	81	1	the	the	DET
ejpam-854	81	2	linear	linear	PROPN
ejpam-854	81	3	operations	operation	NOUN
ejpam-854	81	4	on	on	ADP
ejpam-854	81	5	ek	ek	PROPN
ejpam-854	81	6	are	be	AUX
ejpam-854	81	7	defined	define	VERB
ejpam-854	81	8	coordinate	coordinate	NOUN
ejpam-854	81	9	-	-	PUNCT
ejpam-854	81	10	wise	wise	ADJ
ejpam-854	81	11	.	.	PUNCT
ejpam-854	82	1	when	when	SCONJ
ejpam-854	82	2	we	we	PRON
ejpam-854	82	3	write	write	VERB
ejpam-854	82	4	(	(	PUNCT
ejpam-854	82	5	0	0	NUM
ejpam-854	82	6	,	,	PUNCT
ejpam-854	82	7	.	.	PUNCT
ejpam-854	82	8	.	.	PUNCT
ejpam-854	83	1	.	.	PUNCT
ejpam-854	84	1	,	,	PUNCT
ejpam-854	84	2	0	0	NUM
ejpam-854	84	3	,	,	PUNCT
ejpam-854	84	4	x	x	PROPN
ejpam-854	85	1	i	i	PROPN
ejpam-854	85	2	,	,	PUNCT
ejpam-854	85	3	0	0	NUM
ejpam-854	85	4	,	,	PUNCT
ejpam-854	85	5	.	.	PUNCT
ejpam-854	85	6	.	.	PUNCT
ejpam-854	85	7	.	.	PUNCT
ejpam-854	86	1	,	,	PUNCT
ejpam-854	86	2	0	0	X
ejpam-854	86	3	)	)	PUNCT
ejpam-854	86	4	for	for	ADP
ejpam-854	86	5	an	an	DET
ejpam-854	86	6	element	element	NOUN
ejpam-854	86	7	in	in	ADP
ejpam-854	86	8	ek	ek	PROPN
ejpam-854	86	9	,	,	PUNCT
ejpam-854	86	10	we	we	PRON
ejpam-854	86	11	understand	understand	VERB
ejpam-854	86	12	that	that	SCONJ
ejpam-854	86	13	x	x	PRON
ejpam-854	86	14	i	i	PRON
ejpam-854	86	15	appears	appear	VERB
ejpam-854	86	16	in	in	ADP
ejpam-854	86	17	the	the	DET
ejpam-854	86	18	ith	ith	PROPN
ejpam-854	86	19	coordinate	coordinate	NOUN
ejpam-854	86	20	.	.	PUNCT
ejpam-854	87	1	the	the	DET
ejpam-854	87	2	zero	zero	NUM
ejpam-854	87	3	elements	element	NOUN
ejpam-854	87	4	of	of	ADP
ejpam-854	87	5	either	either	CCONJ
ejpam-854	87	6	e	e	NOUN
ejpam-854	87	7	or	or	CCONJ
ejpam-854	87	8	ek	ek	PROPN
ejpam-854	87	9	are	be	AUX
ejpam-854	87	10	both	both	PRON
ejpam-854	87	11	denoted	denote	VERB
ejpam-854	87	12	by	by	ADP
ejpam-854	87	13	0	0	NUM
ejpam-854	87	14	when	when	SCONJ
ejpam-854	87	15	there	there	PRON
ejpam-854	87	16	is	be	VERB
ejpam-854	87	17	no	no	DET
ejpam-854	87	18	confusion	confusion	NOUN
ejpam-854	87	19	.	.	PUNCT
ejpam-854	88	1	we	we	PRON
ejpam-854	88	2	denote	denote	VERB
ejpam-854	88	3	by	by	ADP
ejpam-854	88	4	nk	nk	PROPN
ejpam-854	88	5	the	the	DET
ejpam-854	88	6	set	set	NOUN
ejpam-854	88	7	{	{	PUNCT
ejpam-854	88	8	1,2	1,2	NUM
ejpam-854	88	9	,	,	PUNCT
ejpam-854	88	10	.	.	PUNCT
ejpam-854	88	11	.	.	PUNCT
ejpam-854	88	12	.	.	PUNCT
ejpam-854	89	1	,	,	PUNCT
ejpam-854	89	2	k	k	X
ejpam-854	89	3	}	}	PUNCT
ejpam-854	89	4	and	and	CCONJ
ejpam-854	89	5	by	by	ADP
ejpam-854	89	6	bk	bk	ADP
ejpam-854	89	7	the	the	DET
ejpam-854	89	8	group	group	NOUN
ejpam-854	89	9	of	of	ADP
ejpam-854	89	10	permutations	permutation	NOUN
ejpam-854	89	11	on	on	ADP
ejpam-854	89	12	nk	nk	PROPN
ejpam-854	89	13	.	.	PUNCT
ejpam-854	89	14	definition	definition	NOUN
ejpam-854	89	15	1	1	NUM
ejpam-854	89	16	.	.	PUNCT
ejpam-854	90	1	a	a	DET
ejpam-854	90	2	multi	multi	NOUN
ejpam-854	90	3	-	-	ADJ
ejpam-854	90	4	norm	norm	NOUN
ejpam-854	90	5	on	on	ADP
ejpam-854	90	6	{	{	PUNCT
ejpam-854	90	7	ek	ek	X
ejpam-854	90	8	,	,	PUNCT
ejpam-854	90	9	k	k	PROPN
ejpam-854	90	10	∈	∈	PROPN
ejpam-854	90	11	n	n	CCONJ
ejpam-854	90	12	}	}	PUNCT
ejpam-854	90	13	is	be	AUX
ejpam-854	90	14	a	a	DET
ejpam-854	90	15	sequence	sequence	NOUN
ejpam-854	90	16	(	(	PUNCT
ejpam-854	90	17	‖	‖	PROPN
ejpam-854	90	18	·	·	SYM
ejpam-854	90	19	‖k	‖k	PROPN
ejpam-854	90	20	)	)	PUNCT
ejpam-854	90	21	=	=	SYM
ejpam-854	90	22	(	(	PUNCT
ejpam-854	90	23	‖	‖	PROPN
ejpam-854	90	24	·	·	PUNCT
ejpam-854	90	25	‖k	‖k	NOUN
ejpam-854	90	26	:	:	PUNCT
ejpam-854	90	27	k	k	PROPN
ejpam-854	90	28	∈	∈	PROPN
ejpam-854	90	29	n	n	CCONJ
ejpam-854	90	30	)	)	PUNCT
ejpam-854	90	31	such	such	ADJ
ejpam-854	90	32	that	that	PRON
ejpam-854	90	33	‖	‖	PROPN
ejpam-854	90	34	·	·	PUNCT
ejpam-854	90	35	‖k	‖k	NOUN
ejpam-854	90	36	is	be	AUX
ejpam-854	90	37	a	a	DET
ejpam-854	90	38	norm	norm	NOUN
ejpam-854	90	39	on	on	ADP
ejpam-854	90	40	ek	ek	PROPN
ejpam-854	90	41	for	for	ADP
ejpam-854	90	42	each	each	DET
ejpam-854	90	43	k	k	PROPN
ejpam-854	90	44	∈	∈	PROPN
ejpam-854	90	45	n	n	CCONJ
ejpam-854	90	46	,	,	PUNCT
ejpam-854	90	47	such	such	ADJ
ejpam-854	90	48	that	that	DET
ejpam-854	90	49	‖x‖1	‖x‖1	NOUN
ejpam-854	90	50	=	=	PUNCT
ejpam-854	90	51	‖x‖	‖x‖	PROPN
ejpam-854	90	52	for	for	ADP
ejpam-854	90	53	each	each	DET
ejpam-854	90	54	x	x	SYM
ejpam-854	90	55	∈	∈	PROPN
ejpam-854	90	56	e	e	NOUN
ejpam-854	90	57	,	,	PUNCT
ejpam-854	90	58	and	and	CCONJ
ejpam-854	90	59	the	the	DET
ejpam-854	90	60	following	following	ADJ
ejpam-854	90	61	axioms	axiom	NOUN
ejpam-854	90	62	are	be	AUX
ejpam-854	90	63	satisfied	satisfied	ADJ
ejpam-854	90	64	for	for	ADP
ejpam-854	90	65	each	each	DET
ejpam-854	90	66	k	k	PROPN
ejpam-854	90	67	∈	∈	PROPN
ejpam-854	90	68	n	n	X
ejpam-854	90	69	with	with	ADP
ejpam-854	90	70	k	k	PROPN
ejpam-854	90	71	≥	≥	NUM
ejpam-854	90	72	2	2	NUM
ejpam-854	90	73	:	:	PUNCT
ejpam-854	90	74	(	(	PUNCT
ejpam-854	90	75	a1	a1	NOUN
ejpam-854	90	76	)	)	PUNCT
ejpam-854	90	77	‖(xσ(1	‖(xσ(1	NUM
ejpam-854	90	78	)	)	PUNCT
ejpam-854	90	79	,	,	PUNCT
ejpam-854	90	80	.	.	PUNCT
ejpam-854	90	81	.	.	PUNCT
ejpam-854	90	82	.	.	PUNCT
ejpam-854	91	1	,	,	PUNCT
ejpam-854	91	2	xσ(k))‖k	xσ(k))‖k	PUNCT
ejpam-854	92	1	=	=	SYM
ejpam-854	92	2	‖(x1	‖(x1	PROPN
ejpam-854	92	3	,	,	PUNCT
ejpam-854	92	4	.	.	PUNCT
ejpam-854	92	5	.	.	PUNCT
ejpam-854	92	6	.	.	PUNCT
ejpam-854	93	1	,	,	PUNCT
ejpam-854	93	2	xk)‖k	xk)‖k	PROPN
ejpam-854	93	3	(	(	PUNCT
ejpam-854	93	4	σ	σ	PROPN
ejpam-854	93	5	∈	∈	PROPN
ejpam-854	93	6	bk	bk	PROPN
ejpam-854	93	7	,	,	PUNCT
ejpam-854	93	8	x1	x1	PROPN
ejpam-854	93	9	,	,	PUNCT
ejpam-854	93	10	.	.	PUNCT
ejpam-854	93	11	.	.	PUNCT
ejpam-854	93	12	.	.	PUNCT
ejpam-854	94	1	,	,	PUNCT
ejpam-854	94	2	xk	xk	PROPN
ejpam-854	94	3	∈	∈	PROPN
ejpam-854	94	4	e	e	PROPN
ejpam-854	94	5	)	)	PUNCT
ejpam-854	94	6	;	;	PUNCT
ejpam-854	94	7	(	(	PUNCT
ejpam-854	94	8	a2	a2	NOUN
ejpam-854	94	9	)	)	PUNCT
ejpam-854	94	10	‖(α1	‖(α1	NUM
ejpam-854	94	11	x1	x1	PROPN
ejpam-854	94	12	,	,	PUNCT
ejpam-854	94	13	.	.	PUNCT
ejpam-854	94	14	.	.	PUNCT
ejpam-854	95	1	.	.	PUNCT
ejpam-854	96	1	,	,	PUNCT
ejpam-854	96	2	αk	αk	X
ejpam-854	96	3	xk)‖k	xk)‖k	PROPN
ejpam-854	96	4	≤	≤	NUM
ejpam-854	96	5	(	(	PUNCT
ejpam-854	96	6	maxi∈nk	maxi∈nk	PROPN
ejpam-854	96	7	|αi|)‖(x1	|αi|)‖(x1	PROPN
ejpam-854	96	8	,	,	PUNCT
ejpam-854	96	9	.	.	PUNCT
ejpam-854	96	10	.	.	PUNCT
ejpam-854	97	1	.	.	PUNCT
ejpam-854	98	1	,	,	PUNCT
ejpam-854	98	2	xk)‖k(x	xk)‖k(x	PROPN
ejpam-854	99	1	i	i	PRON
ejpam-854	99	2	∈	∈	PROPN
ejpam-854	99	3	e	e	NOUN
ejpam-854	99	4	,	,	PUNCT
ejpam-854	99	5	αi	αi	VERB
ejpam-854	99	6	∈	∈	PROPN
ejpam-854	100	1	c	c	X
ejpam-854	100	2	,	,	PUNCT
ejpam-854	100	3	i	i	PRON
ejpam-854	100	4	=	=	NOUN
ejpam-854	100	5	1	1	NUM
ejpam-854	100	6	,	,	PUNCT
ejpam-854	100	7	.	.	PUNCT
ejpam-854	100	8	.	.	PUNCT
ejpam-854	100	9	.	.	PUNCT
ejpam-854	101	1	,	,	PUNCT
ejpam-854	101	2	k	k	X
ejpam-854	101	3	)	)	PUNCT
ejpam-854	101	4	;	;	PUNCT
ejpam-854	101	5	(	(	PUNCT
ejpam-854	101	6	a3	a3	NOUN
ejpam-854	101	7	)	)	PUNCT
ejpam-854	101	8	‖(x1	‖(x1	PROPN
ejpam-854	101	9	,	,	PUNCT
ejpam-854	101	10	.	.	PUNCT
ejpam-854	101	11	.	.	PUNCT
ejpam-854	102	1	.	.	PUNCT
ejpam-854	103	1	,	,	PUNCT
ejpam-854	103	2	xk−1	xk−1	PROPN
ejpam-854	103	3	,	,	PUNCT
ejpam-854	103	4	0)‖k	0)‖k	X
ejpam-854	103	5	=	=	SYM
ejpam-854	103	6	‖(x1	‖(x1	PROPN
ejpam-854	103	7	,	,	PUNCT
ejpam-854	103	8	.	.	PUNCT
ejpam-854	103	9	.	.	PUNCT
ejpam-854	103	10	.	.	PUNCT
ejpam-854	104	1	,	,	PUNCT
ejpam-854	104	2	xk−1)‖k−1	xk−1)‖k−1	PROPN
ejpam-854	104	3	(	(	PUNCT
ejpam-854	104	4	x1	x1	PROPN
ejpam-854	104	5	,	,	PUNCT
ejpam-854	104	6	.	.	PUNCT
ejpam-854	104	7	.	.	PUNCT
ejpam-854	105	1	.	.	PUNCT
ejpam-854	106	1	,	,	PUNCT
ejpam-854	106	2	xk−1	xk−1	PROPN
ejpam-854	106	3	∈	∈	PROPN
ejpam-854	106	4	e	e	PROPN
ejpam-854	106	5	)	)	PUNCT
ejpam-854	106	6	;	;	PUNCT
ejpam-854	106	7	(	(	PUNCT
ejpam-854	106	8	a4	a4	NOUN
ejpam-854	106	9	)	)	PUNCT
ejpam-854	106	10	‖(x1	‖(x1	PROPN
ejpam-854	106	11	,	,	PUNCT
ejpam-854	106	12	.	.	PUNCT
ejpam-854	106	13	.	.	PUNCT
ejpam-854	107	1	.	.	PUNCT
ejpam-854	108	1	,	,	PUNCT
ejpam-854	108	2	xk−1	xk−1	PROPN
ejpam-854	108	3	,	,	PUNCT
ejpam-854	108	4	xk−1)‖k	xk−1)‖k	PUNCT
ejpam-854	109	1	=	=	SYM
ejpam-854	109	2	‖(x1	‖(x1	PROPN
ejpam-854	109	3	,	,	PUNCT
ejpam-854	109	4	.	.	PUNCT
ejpam-854	109	5	.	.	PUNCT
ejpam-854	109	6	.	.	PUNCT
ejpam-854	110	1	,	,	PUNCT
ejpam-854	110	2	xk−1)‖k−1	xk−1)‖k−1	PROPN
ejpam-854	110	3	(	(	PUNCT
ejpam-854	110	4	x1	x1	PROPN
ejpam-854	110	5	,	,	PUNCT
ejpam-854	110	6	.	.	PUNCT
ejpam-854	110	7	.	.	PUNCT
ejpam-854	111	1	.	.	PUNCT
ejpam-854	112	1	,	,	PUNCT
ejpam-854	112	2	xk−1	xk−1	PROPN
ejpam-854	112	3	∈	∈	PROPN
ejpam-854	112	4	e	e	NOUN
ejpam-854	112	5	)	)	PUNCT
ejpam-854	112	6	.	.	PUNCT
ejpam-854	113	1	t.	t.	PROPN
ejpam-854	113	2	xu	xu	PROPN
ejpam-854	113	3	,	,	PUNCT
ejpam-854	113	4	j.	j.	PROPN
ejpam-854	113	5	rassias	rassias	PROPN
ejpam-854	113	6	,	,	PUNCT
ejpam-854	113	7	w.	w.	PROPN
ejpam-854	113	8	xu	xu	PROPN
ejpam-854	113	9	/	/	SYM
ejpam-854	113	10	eur	eur	PROPN
ejpam-854	113	11	.	.	PUNCT
ejpam-854	114	1	j.	j.	PROPN
ejpam-854	114	2	pure	pure	PROPN
ejpam-854	114	3	appl	appl	PROPN
ejpam-854	114	4	.	.	PROPN
ejpam-854	114	5	math	math	PROPN
ejpam-854	114	6	,	,	PUNCT
ejpam-854	114	7	3	3	NUM
ejpam-854	114	8	(	(	PUNCT
ejpam-854	114	9	2010	2010	NUM
ejpam-854	114	10	)	)	PUNCT
ejpam-854	114	11	,	,	PUNCT
ejpam-854	114	12	1032	1032	NUM
ejpam-854	114	13	-	-	SYM
ejpam-854	114	14	1047	1047	NUM
ejpam-854	114	15	1035	1035	NUM
ejpam-854	114	16	in	in	ADP
ejpam-854	114	17	this	this	DET
ejpam-854	114	18	case	case	NOUN
ejpam-854	114	19	,	,	PUNCT
ejpam-854	114	20	we	we	PRON
ejpam-854	114	21	say	say	VERB
ejpam-854	114	22	that	that	SCONJ
ejpam-854	114	23	(	(	PUNCT
ejpam-854	114	24	(	(	PUNCT
ejpam-854	114	25	ek,‖	ek,‖	NOUN
ejpam-854	114	26	·	·	PUNCT
ejpam-854	114	27	‖k	‖k	PROPN
ejpam-854	114	28	)	)	PUNCT
ejpam-854	114	29	:	:	PUNCT
ejpam-854	115	1	k	k	PROPN
ejpam-854	115	2	∈	∈	PROPN
ejpam-854	115	3	n	n	CCONJ
ejpam-854	115	4	)	)	PUNCT
ejpam-854	115	5	is	be	AUX
ejpam-854	115	6	a	a	DET
ejpam-854	115	7	multi	multi	ADJ
ejpam-854	115	8	-	-	ADJ
ejpam-854	115	9	normed	normed	ADJ
ejpam-854	115	10	space	space	NOUN
ejpam-854	115	11	.	.	PUNCT
ejpam-854	116	1	suppose	suppose	VERB
ejpam-854	116	2	that	that	SCONJ
ejpam-854	116	3	(	(	PUNCT
ejpam-854	116	4	(	(	PUNCT
ejpam-854	116	5	ek,‖	ek,‖	NOUN
ejpam-854	116	6	·	·	PUNCT
ejpam-854	116	7	‖k	‖k	PROPN
ejpam-854	116	8	)	)	PUNCT
ejpam-854	116	9	:	:	PUNCT
ejpam-854	116	10	k	k	PROPN
ejpam-854	116	11	∈	∈	PROPN
ejpam-854	116	12	n	n	CCONJ
ejpam-854	116	13	)	)	PUNCT
ejpam-854	116	14	is	be	AUX
ejpam-854	116	15	a	a	DET
ejpam-854	116	16	multi	multi	ADJ
ejpam-854	116	17	-	-	ADJ
ejpam-854	116	18	normed	normed	ADJ
ejpam-854	116	19	space	space	NOUN
ejpam-854	116	20	and	and	CCONJ
ejpam-854	116	21	take	take	VERB
ejpam-854	116	22	k	k	PROPN
ejpam-854	116	23	∈	∈	PROPN
ejpam-854	116	24	n.	n.	NOUN
ejpam-854	116	25	it	it	PRON
ejpam-854	116	26	is	be	AUX
ejpam-854	116	27	easy	easy	ADJ
ejpam-854	116	28	to	to	PART
ejpam-854	116	29	show	show	VERB
ejpam-854	116	30	that	that	SCONJ
ejpam-854	116	31	(	(	PUNCT
ejpam-854	116	32	a	a	X
ejpam-854	116	33	)	)	PUNCT
ejpam-854	116	34	‖(x	‖(x	ADJ
ejpam-854	116	35	,	,	PUNCT
ejpam-854	116	36	.	.	PUNCT
ejpam-854	116	37	.	.	PUNCT
ejpam-854	117	1	.	.	PUNCT
ejpam-854	118	1	,	,	PUNCT
ejpam-854	118	2	x)‖k	x)‖k	NOUN
ejpam-854	118	3	=	=	PUNCT
ejpam-854	118	4	‖x‖	‖x‖	PROPN
ejpam-854	118	5	(	(	PUNCT
ejpam-854	118	6	x	x	SYM
ejpam-854	118	7	∈	∈	PROPN
ejpam-854	118	8	e	e	NOUN
ejpam-854	118	9	)	)	PUNCT
ejpam-854	118	10	;	;	PUNCT
ejpam-854	118	11	(	(	PUNCT
ejpam-854	118	12	b	b	X
ejpam-854	118	13	)	)	PUNCT
ejpam-854	118	14	max	max	PROPN
ejpam-854	118	15	i∈nk	i∈nk	PROPN
ejpam-854	118	16	‖x	‖x	PROPN
ejpam-854	118	17	i‖	i‖	PROPN
ejpam-854	118	18	≤	≤	PROPN
ejpam-854	118	19	‖(x1	‖(x1	PROPN
ejpam-854	118	20	,	,	PUNCT
ejpam-854	118	21	.	.	PUNCT
ejpam-854	118	22	.	.	PUNCT
ejpam-854	118	23	.	.	PUNCT
ejpam-854	119	1	,	,	PUNCT
ejpam-854	119	2	xk)‖k	xk)‖k	PROPN
ejpam-854	119	3	≤	≤	NUM
ejpam-854	119	4	k∑	k∑	VERB
ejpam-854	119	5	i=1	i=1	PROPN
ejpam-854	120	1	‖x	‖x	PROPN
ejpam-854	121	1	i‖	i‖	PROPN
ejpam-854	121	2	≤	≤	PUNCT
ejpam-854	121	3	k	k	PROPN
ejpam-854	121	4	max	max	PROPN
ejpam-854	121	5	i∈nk	i∈nk	PROPN
ejpam-854	121	6	‖x	‖x	PROPN
ejpam-854	121	7	i‖	i‖	PROPN
ejpam-854	121	8	(	(	PUNCT
ejpam-854	121	9	x1	x1	PROPN
ejpam-854	121	10	,	,	PUNCT
ejpam-854	121	11	.	.	PUNCT
ejpam-854	121	12	.	.	PUNCT
ejpam-854	121	13	.	.	PUNCT
ejpam-854	122	1	,	,	PUNCT
ejpam-854	122	2	xk	xk	PROPN
ejpam-854	122	3	∈	∈	PROPN
ejpam-854	122	4	e	e	PROPN
ejpam-854	122	5	)	)	PUNCT
ejpam-854	122	6	.	.	PUNCT
ejpam-854	123	1	it	it	PRON
ejpam-854	123	2	follows	follow	VERB
ejpam-854	123	3	from	from	ADP
ejpam-854	123	4	(	(	PUNCT
ejpam-854	123	5	b	b	NOUN
ejpam-854	123	6	)	)	PUNCT
ejpam-854	123	7	that	that	SCONJ
ejpam-854	123	8	if	if	SCONJ
ejpam-854	123	9	(	(	PUNCT
ejpam-854	123	10	e,‖	e,‖	X
ejpam-854	123	11	·	·	PUNCT
ejpam-854	123	12	‖	‖	NUM
ejpam-854	123	13	)	)	PUNCT
ejpam-854	123	14	is	be	AUX
ejpam-854	123	15	a	a	DET
ejpam-854	123	16	banach	banach	NOUN
ejpam-854	123	17	space	space	NOUN
ejpam-854	123	18	,	,	PUNCT
ejpam-854	123	19	then	then	ADV
ejpam-854	123	20	(	(	PUNCT
ejpam-854	123	21	ek,‖	ek,‖	PROPN
ejpam-854	123	22	·	·	PUNCT
ejpam-854	123	23	‖k	‖k	PROPN
ejpam-854	123	24	)	)	PUNCT
ejpam-854	123	25	is	be	AUX
ejpam-854	123	26	a	a	DET
ejpam-854	123	27	banach	banach	NOUN
ejpam-854	123	28	space	space	NOUN
ejpam-854	123	29	for	for	ADP
ejpam-854	123	30	each	each	DET
ejpam-854	123	31	k	k	PROPN
ejpam-854	123	32	∈	∈	PROPN
ejpam-854	123	33	n	n	CCONJ
ejpam-854	123	34	;	;	PUNCT
ejpam-854	123	35	in	in	ADP
ejpam-854	123	36	this	this	DET
ejpam-854	123	37	case	case	NOUN
ejpam-854	123	38	(	(	PUNCT
ejpam-854	123	39	(	(	PUNCT
ejpam-854	123	40	ek,‖	ek,‖	PROPN
ejpam-854	123	41	·	·	PUNCT
ejpam-854	123	42	‖k	‖k	PROPN
ejpam-854	123	43	)	)	PUNCT
ejpam-854	123	44	:	:	PUNCT
ejpam-854	124	1	k	k	PROPN
ejpam-854	124	2	∈	∈	PROPN
ejpam-854	124	3	n	n	CCONJ
ejpam-854	124	4	)	)	PUNCT
ejpam-854	124	5	is	be	AUX
ejpam-854	124	6	said	say	VERB
ejpam-854	124	7	to	to	PART
ejpam-854	124	8	be	be	AUX
ejpam-854	124	9	a	a	DET
ejpam-854	124	10	multi	multi	ADJ
ejpam-854	124	11	-	-	ADJ
ejpam-854	124	12	banach	banach	ADJ
ejpam-854	124	13	space	space	NOUN
ejpam-854	124	14	.	.	PUNCT
ejpam-854	125	1	now	now	ADV
ejpam-854	125	2	we	we	PRON
ejpam-854	125	3	state	state	VERB
ejpam-854	125	4	two	two	NUM
ejpam-854	125	5	important	important	ADJ
ejpam-854	125	6	examples	example	NOUN
ejpam-854	125	7	of	of	ADP
ejpam-854	125	8	multi	multi	NOUN
ejpam-854	125	9	-	-	NOUN
ejpam-854	125	10	norms	norm	NOUN
ejpam-854	125	11	for	for	ADP
ejpam-854	125	12	arbitrary	arbitrary	ADJ
ejpam-854	125	13	normed	normed	ADJ
ejpam-854	125	14	space	space	NOUN
ejpam-854	125	15	e	e	NOUN
ejpam-854	125	16	(	(	PUNCT
ejpam-854	125	17	see	see	VERB
ejpam-854	125	18	[	[	X
ejpam-854	125	19	5]-[6	5]-[6	X
ejpam-854	125	20	]	]	PUNCT
ejpam-854	125	21	,	,	PUNCT
ejpam-854	125	22	[	[	X
ejpam-854	125	23	21]-[23	21]-[23	NOUN
ejpam-854	125	24	]	]	PUNCT
ejpam-854	125	25	)	)	PUNCT
ejpam-854	125	26	.	.	PUNCT
ejpam-854	126	1	example	example	NOUN
ejpam-854	127	1	1	1	X
ejpam-854	127	2	.	.	PUNCT
ejpam-854	128	1	let	let	VERB
ejpam-854	128	2	e	e	PRON
ejpam-854	128	3	be	be	AUX
ejpam-854	128	4	an	an	DET
ejpam-854	128	5	arbitrary	arbitrary	ADJ
ejpam-854	128	6	normed	normed	ADJ
ejpam-854	128	7	space	space	NOUN
ejpam-854	128	8	.	.	PUNCT
ejpam-854	129	1	the	the	DET
ejpam-854	129	2	sequence	sequence	NOUN
ejpam-854	129	3	(	(	PUNCT
ejpam-854	129	4	‖	‖	PROPN
ejpam-854	129	5	·	·	PUNCT
ejpam-854	129	6	‖k	‖k	NOUN
ejpam-854	129	7	:	:	PUNCT
ejpam-854	129	8	k	k	PROPN
ejpam-854	129	9	∈	∈	PROPN
ejpam-854	129	10	n	n	CCONJ
ejpam-854	129	11	)	)	PUNCT
ejpam-854	129	12	on	on	ADP
ejpam-854	129	13	{	{	PUNCT
ejpam-854	129	14	ek	ek	X
ejpam-854	129	15	:	:	PUNCT
ejpam-854	129	16	k	k	PROPN
ejpam-854	129	17	∈	∈	PROPN
ejpam-854	129	18	n	n	CCONJ
ejpam-854	129	19	}	}	PUNCT
ejpam-854	129	20	defined	define	VERB
ejpam-854	129	21	by	by	ADP
ejpam-854	129	22	‖(x1	‖(x1	PROPN
ejpam-854	129	23	,	,	PUNCT
ejpam-854	129	24	.	.	PUNCT
ejpam-854	129	25	.	.	PUNCT
ejpam-854	129	26	.	.	PUNCT
ejpam-854	130	1	,	,	PUNCT
ejpam-854	130	2	xk)‖k	xk)‖k	NOUN
ejpam-854	130	3	:	:	PUNCT
ejpam-854	131	1	=	=	NUM
ejpam-854	131	2	max	max	PROPN
ejpam-854	131	3	i∈nk	i∈nk	PROPN
ejpam-854	131	4	‖x	‖x	PROPN
ejpam-854	131	5	i‖	i‖	PROPN
ejpam-854	131	6	(	(	PUNCT
ejpam-854	131	7	x1	x1	PROPN
ejpam-854	131	8	,	,	PUNCT
ejpam-854	131	9	.	.	PUNCT
ejpam-854	131	10	.	.	PUNCT
ejpam-854	131	11	.	.	PUNCT
ejpam-854	132	1	,	,	PUNCT
ejpam-854	132	2	xk	xk	PROPN
ejpam-854	132	3	∈	∈	PROPN
ejpam-854	132	4	e	e	X
ejpam-854	132	5	)	)	PUNCT
ejpam-854	132	6	is	be	AUX
ejpam-854	132	7	a	a	DET
ejpam-854	132	8	multi	multi	ADJ
ejpam-854	132	9	-	-	ADJ
ejpam-854	132	10	norm	norm	NOUN
ejpam-854	132	11	called	call	VERB
ejpam-854	132	12	the	the	DET
ejpam-854	132	13	minimum	minimum	ADJ
ejpam-854	132	14	multi	multi	NOUN
ejpam-854	132	15	-	-	NOUN
ejpam-854	132	16	norm	norm	NOUN
ejpam-854	132	17	.	.	PUNCT
ejpam-854	133	1	the	the	DET
ejpam-854	133	2	terminology	terminology	NOUN
ejpam-854	133	3	minimum	minimum	NOUN
ejpam-854	133	4	is	be	AUX
ejpam-854	133	5	justified	justify	VERB
ejpam-854	133	6	by	by	ADP
ejpam-854	133	7	property	property	NOUN
ejpam-854	133	8	(	(	PUNCT
ejpam-854	133	9	b	b	NOUN
ejpam-854	133	10	)	)	PUNCT
ejpam-854	133	11	.	.	PUNCT
ejpam-854	134	1	example	example	NOUN
ejpam-854	135	1	2	2	NUM
ejpam-854	135	2	.	.	PUNCT
ejpam-854	135	3	let	let	VERB
ejpam-854	135	4	e	e	PRON
ejpam-854	135	5	be	be	AUX
ejpam-854	135	6	an	an	DET
ejpam-854	135	7	arbitrary	arbitrary	ADJ
ejpam-854	135	8	normed	normed	ADJ
ejpam-854	135	9	space	space	NOUN
ejpam-854	135	10	and	and	CCONJ
ejpam-854	135	11	let	let	VERB
ejpam-854	135	12	{	{	PUNCT
ejpam-854	135	13	(	(	PUNCT
ejpam-854	135	14	‖	‖	PROPN
ejpam-854	135	15	·	·	PUNCT
ejpam-854	136	1	‖α	‖α	PROPN
ejpam-854	136	2	k	k	NOUN
ejpam-854	136	3	:	:	PUNCT
ejpam-854	136	4	k	k	PROPN
ejpam-854	136	5	∈	∈	PROPN
ejpam-854	136	6	n	n	CCONJ
ejpam-854	136	7	)	)	PUNCT
ejpam-854	136	8	:	:	PUNCT
ejpam-854	137	1	α	α	PROPN
ejpam-854	137	2	∈	∈	PROPN
ejpam-854	137	3	a	a	DET
ejpam-854	137	4	}	}	PUNCT
ejpam-854	137	5	be	be	AUX
ejpam-854	137	6	the	the	DET
ejpam-854	137	7	(	(	PUNCT
ejpam-854	137	8	non	non	ADJ
ejpam-854	137	9	-	-	ADJ
ejpam-854	137	10	empty	empty	ADJ
ejpam-854	137	11	)	)	PUNCT
ejpam-854	137	12	family	family	NOUN
ejpam-854	137	13	of	of	ADP
ejpam-854	137	14	all	all	DET
ejpam-854	137	15	multi	multi	NOUN
ejpam-854	137	16	-	-	NOUN
ejpam-854	137	17	norms	norm	NOUN
ejpam-854	137	18	on	on	ADP
ejpam-854	137	19	{	{	PUNCT
ejpam-854	137	20	ek	ek	NOUN
ejpam-854	137	21	:	:	PUNCT
ejpam-854	137	22	k	k	PROPN
ejpam-854	137	23	∈	∈	PROPN
ejpam-854	137	24	n	n	CCONJ
ejpam-854	137	25	}	}	PUNCT
ejpam-854	137	26	.	.	PUNCT
ejpam-854	138	1	for	for	ADP
ejpam-854	138	2	k	k	PROPN
ejpam-854	138	3	∈	∈	PROPN
ejpam-854	138	4	n	n	CCONJ
ejpam-854	138	5	,	,	PUNCT
ejpam-854	138	6	consider	consider	VERB
ejpam-854	138	7	‖|(x1	‖|(x1	PRON
ejpam-854	138	8	,	,	PUNCT
ejpam-854	138	9	.	.	PUNCT
ejpam-854	138	10	.	.	PUNCT
ejpam-854	139	1	.	.	PUNCT
ejpam-854	140	1	,	,	PUNCT
ejpam-854	140	2	xk)‖|k	xk)‖|k	PROPN
ejpam-854	140	3	:	:	PUNCT
ejpam-854	141	1	=	=	PUNCT
ejpam-854	141	2	sup	sup	NOUN
ejpam-854	141	3	α∈a	α∈a	NOUN
ejpam-854	141	4	‖(x1	‖(x1	PROPN
ejpam-854	141	5	,	,	PUNCT
ejpam-854	141	6	.	.	PUNCT
ejpam-854	141	7	.	.	PUNCT
ejpam-854	141	8	.	.	PUNCT
ejpam-854	142	1	,	,	PUNCT
ejpam-854	143	1	xk)‖	xk)‖	PROPN
ejpam-854	143	2	α	α	PROPN
ejpam-854	143	3	k	k	PROPN
ejpam-854	143	4	(	(	PUNCT
ejpam-854	143	5	x1	x1	PROPN
ejpam-854	143	6	,	,	PUNCT
ejpam-854	143	7	.	.	PUNCT
ejpam-854	143	8	.	.	PUNCT
ejpam-854	143	9	.	.	PUNCT
ejpam-854	144	1	,	,	PUNCT
ejpam-854	144	2	xk	xk	PROPN
ejpam-854	144	3	∈	∈	PROPN
ejpam-854	144	4	e	e	PROPN
ejpam-854	144	5	)	)	PUNCT
ejpam-854	144	6	.	.	PUNCT
ejpam-854	145	1	then	then	ADV
ejpam-854	145	2	(	(	PUNCT
ejpam-854	145	3	‖|	‖|	PROPN
ejpam-854	145	4	·	·	PUNCT
ejpam-854	145	5	‖|k	‖|k	NOUN
ejpam-854	145	6	:	:	PUNCT
ejpam-854	145	7	k	k	PROPN
ejpam-854	145	8	∈	∈	PROPN
ejpam-854	145	9	n	n	CCONJ
ejpam-854	145	10	)	)	PUNCT
ejpam-854	145	11	is	be	AUX
ejpam-854	145	12	a	a	DET
ejpam-854	145	13	multi	multi	ADJ
ejpam-854	145	14	-	-	ADJ
ejpam-854	145	15	norm	norm	NOUN
ejpam-854	145	16	on	on	ADP
ejpam-854	145	17	{	{	PUNCT
ejpam-854	145	18	ek	ek	NOUN
ejpam-854	145	19	:	:	PUNCT
ejpam-854	145	20	k	k	PROPN
ejpam-854	145	21	∈	∈	PROPN
ejpam-854	145	22	n	n	CCONJ
ejpam-854	145	23	}	}	PUNCT
ejpam-854	145	24	,	,	PUNCT
ejpam-854	145	25	called	call	VERB
ejpam-854	145	26	the	the	DET
ejpam-854	145	27	maximum	maximum	ADJ
ejpam-854	145	28	multi	multi	NOUN
ejpam-854	145	29	-	-	NOUN
ejpam-854	145	30	norm	norm	ADJ
ejpam-854	145	31	.	.	PUNCT
ejpam-854	146	1	definition	definition	NOUN
ejpam-854	146	2	2	2	NUM
ejpam-854	146	3	.	.	PUNCT
ejpam-854	147	1	let	let	VERB
ejpam-854	147	2	(	(	PUNCT
ejpam-854	147	3	(	(	PUNCT
ejpam-854	147	4	ek,‖	ek,‖	PROPN
ejpam-854	147	5	·	·	PUNCT
ejpam-854	147	6	‖k	‖k	PROPN
ejpam-854	147	7	)	)	PUNCT
ejpam-854	147	8	:	:	PUNCT
ejpam-854	148	1	k	k	PROPN
ejpam-854	148	2	∈	∈	PROPN
ejpam-854	148	3	n	n	CCONJ
ejpam-854	148	4	)	)	PUNCT
ejpam-854	148	5	be	be	AUX
ejpam-854	148	6	a	a	DET
ejpam-854	148	7	multi	multi	ADJ
ejpam-854	148	8	-	-	ADJ
ejpam-854	148	9	normed	normed	ADJ
ejpam-854	148	10	space	space	NOUN
ejpam-854	148	11	.	.	PUNCT
ejpam-854	149	1	a	a	DET
ejpam-854	149	2	sequence	sequence	NOUN
ejpam-854	149	3	{	{	PUNCT
ejpam-854	149	4	xn	xn	NOUN
ejpam-854	149	5	}	}	PUNCT
ejpam-854	149	6	in	in	ADP
ejpam-854	149	7	e	e	NOUN
ejpam-854	149	8	is	be	AUX
ejpam-854	149	9	a	a	DET
ejpam-854	149	10	multi	multi	ADJ
ejpam-854	149	11	-	-	ADJ
ejpam-854	149	12	null	null	ADJ
ejpam-854	149	13	sequence	sequence	NOUN
ejpam-854	149	14	if	if	SCONJ
ejpam-854	149	15	,	,	PUNCT
ejpam-854	149	16	for	for	ADP
ejpam-854	149	17	each	each	DET
ejpam-854	149	18	ǫ	ǫ	PRON
ejpam-854	149	19	>	>	X
ejpam-854	149	20	0	0	NUM
ejpam-854	149	21	,	,	PUNCT
ejpam-854	149	22	there	there	PRON
ejpam-854	149	23	exists	exist	VERB
ejpam-854	149	24	n0	n0	PROPN
ejpam-854	149	25	∈	∈	PROPN
ejpam-854	149	26	n	n	PRON
ejpam-854	149	27	such	such	ADJ
ejpam-854	149	28	that	that	DET
ejpam-854	149	29	sup	sup	NOUN
ejpam-854	149	30	k∈n	k∈n	PROPN
ejpam-854	149	31	‖(xn	‖(xn	NOUN
ejpam-854	149	32	,	,	PUNCT
ejpam-854	149	33	.	.	PUNCT
ejpam-854	149	34	.	.	PUNCT
ejpam-854	149	35	.	.	PUNCT
ejpam-854	150	1	,	,	PUNCT
ejpam-854	150	2	xn+k−1)‖k	xn+k−1)‖k	NOUN
ejpam-854	150	3	<	<	X
ejpam-854	150	4	ǫ(n≥	ǫ(n≥	NUM
ejpam-854	150	5	n0	n0	NUM
ejpam-854	150	6	)	)	PUNCT
ejpam-854	150	7	.	.	PUNCT
ejpam-854	151	1	let	let	VERB
ejpam-854	151	2	x	x	SYM
ejpam-854	151	3	∈	∈	PROPN
ejpam-854	151	4	e.	e.	PROPN
ejpam-854	151	5	we	we	PRON
ejpam-854	151	6	say	say	VERB
ejpam-854	151	7	that	that	SCONJ
ejpam-854	151	8	the	the	DET
ejpam-854	151	9	sequence	sequence	NOUN
ejpam-854	151	10	{	{	PUNCT
ejpam-854	151	11	xn	xn	NOUN
ejpam-854	151	12	}	}	PUNCT
ejpam-854	151	13	is	be	AUX
ejpam-854	151	14	multi	multi	ADJ
ejpam-854	151	15	-	-	ADJ
ejpam-854	151	16	convergent	convergent	ADJ
ejpam-854	151	17	to	to	ADP
ejpam-854	151	18	x	x	VERB
ejpam-854	151	19	in	in	ADP
ejpam-854	151	20	e	e	NOUN
ejpam-854	151	21	if	if	SCONJ
ejpam-854	151	22	{	{	PUNCT
ejpam-854	151	23	xn−	xn−	PUNCT
ejpam-854	151	24	x	x	PRON
ejpam-854	151	25	}	}	PUNCT
ejpam-854	151	26	is	be	AUX
ejpam-854	151	27	a	a	DET
ejpam-854	151	28	multi	multi	ADJ
ejpam-854	151	29	-	-	ADJ
ejpam-854	151	30	null	null	ADJ
ejpam-854	151	31	sequence	sequence	NOUN
ejpam-854	151	32	.	.	PUNCT
ejpam-854	152	1	in	in	ADP
ejpam-854	152	2	this	this	DET
ejpam-854	152	3	case	case	NOUN
ejpam-854	152	4	,	,	PUNCT
ejpam-854	152	5	x	x	PRON
ejpam-854	152	6	is	be	AUX
ejpam-854	152	7	called	call	VERB
ejpam-854	152	8	the	the	DET
ejpam-854	152	9	limit	limit	NOUN
ejpam-854	152	10	of	of	ADP
ejpam-854	152	11	the	the	DET
ejpam-854	152	12	sequence	sequence	NOUN
ejpam-854	152	13	{	{	PUNCT
ejpam-854	152	14	xn	xn	PUNCT
ejpam-854	152	15	}	}	PUNCT
ejpam-854	152	16	and	and	CCONJ
ejpam-854	152	17	we	we	PRON
ejpam-854	152	18	denote	denote	VERB
ejpam-854	152	19	it	it	PRON
ejpam-854	152	20	by	by	ADP
ejpam-854	152	21	lim	lim	PROPN
ejpam-854	152	22	n→∞	n→∞	X
ejpam-854	152	23	xn	xn	PROPN
ejpam-854	153	1	=	=	PUNCT
ejpam-854	153	2	x.	x.	NOUN
ejpam-854	153	3	for	for	ADP
ejpam-854	153	4	explicitly	explicitly	ADV
ejpam-854	153	5	later	later	ADV
ejpam-854	153	6	use	use	NOUN
ejpam-854	153	7	,	,	PUNCT
ejpam-854	153	8	we	we	PRON
ejpam-854	153	9	recall	recall	VERB
ejpam-854	153	10	a	a	DET
ejpam-854	153	11	fundamental	fundamental	ADJ
ejpam-854	153	12	result	result	NOUN
ejpam-854	153	13	in	in	ADP
ejpam-854	153	14	fixed	fix	VERB
ejpam-854	153	15	point	point	NOUN
ejpam-854	153	16	theory	theory	NOUN
ejpam-854	153	17	.	.	PUNCT
ejpam-854	154	1	let	let	VERB
ejpam-854	154	2	x	x	PRON
ejpam-854	154	3	be	be	AUX
ejpam-854	154	4	a	a	DET
ejpam-854	154	5	set	set	NOUN
ejpam-854	154	6	.	.	PUNCT
ejpam-854	155	1	a	a	DET
ejpam-854	155	2	function	function	NOUN
ejpam-854	155	3	d	d	NOUN
ejpam-854	155	4	:	:	PUNCT
ejpam-854	155	5	x	x	SYM
ejpam-854	155	6	×	×	NOUN
ejpam-854	155	7	x	x	INTJ
ejpam-854	155	8	→	→	X
ejpam-854	155	9	[	[	X
ejpam-854	155	10	0,∞	0,∞	X
ejpam-854	155	11	]	]	PUNCT
ejpam-854	155	12	is	be	AUX
ejpam-854	155	13	called	call	VERB
ejpam-854	155	14	a	a	DET
ejpam-854	155	15	generalized	generalize	VERB
ejpam-854	155	16	metric	metric	NOUN
ejpam-854	155	17	on	on	ADP
ejpam-854	155	18	x	x	SYM
ejpam-854	155	19	if	if	SCONJ
ejpam-854	155	20	d	d	NOUN
ejpam-854	155	21	satisfies	satisfy	VERB
ejpam-854	155	22	:	:	PUNCT
ejpam-854	155	23	(	(	PUNCT
ejpam-854	155	24	1	1	NUM
ejpam-854	155	25	)	)	PUNCT
ejpam-854	155	26	d(x	d(x	NOUN
ejpam-854	155	27	,	,	PUNCT
ejpam-854	155	28	y	y	X
ejpam-854	155	29	)	)	PUNCT
ejpam-854	155	30	=	=	SYM
ejpam-854	155	31	0	0	PUNCT
ejpam-854	156	1	if	if	SCONJ
ejpam-854	156	2	and	and	CCONJ
ejpam-854	156	3	only	only	ADV
ejpam-854	156	4	if	if	SCONJ
ejpam-854	156	5	x	x	X
ejpam-854	156	6	=	=	SYM
ejpam-854	156	7	y	y	PROPN
ejpam-854	156	8	;	;	PUNCT
ejpam-854	156	9	(	(	PUNCT
ejpam-854	156	10	2	2	NUM
ejpam-854	156	11	)	)	PUNCT
ejpam-854	156	12	d(x	d(x	NOUN
ejpam-854	156	13	,	,	PUNCT
ejpam-854	156	14	y	y	X
ejpam-854	156	15	)	)	PUNCT
ejpam-854	156	16	=	=	SYM
ejpam-854	156	17	d(y	d(y	NOUN
ejpam-854	156	18	,	,	PUNCT
ejpam-854	156	19	x	x	NOUN
ejpam-854	156	20	)	)	PUNCT
ejpam-854	156	21	for	for	ADP
ejpam-854	156	22	all	all	DET
ejpam-854	156	23	x	x	SYM
ejpam-854	156	24	,	,	PUNCT
ejpam-854	156	25	y	y	PROPN
ejpam-854	156	26	∈	∈	PROPN
ejpam-854	156	27	x	x	X
ejpam-854	156	28	;	;	PUNCT
ejpam-854	156	29	(	(	PUNCT
ejpam-854	156	30	3	3	X
ejpam-854	156	31	)	)	PUNCT
ejpam-854	156	32	d(x	d(x	NOUN
ejpam-854	156	33	,	,	PUNCT
ejpam-854	156	34	y)≤	y)≤	ADJ
ejpam-854	156	35	d(x	d(x	NOUN
ejpam-854	156	36	,	,	PUNCT
ejpam-854	156	37	z	z	X
ejpam-854	156	38	)	)	PUNCT
ejpam-854	157	1	+	+	CCONJ
ejpam-854	157	2	d(y	d(y	NOUN
ejpam-854	157	3	,	,	PUNCT
ejpam-854	157	4	z	z	NOUN
ejpam-854	157	5	)	)	PUNCT
ejpam-854	157	6	for	for	ADP
ejpam-854	157	7	all	all	DET
ejpam-854	157	8	x	x	SYM
ejpam-854	157	9	,	,	PUNCT
ejpam-854	157	10	y	y	PROPN
ejpam-854	157	11	,	,	PUNCT
ejpam-854	157	12	z	z	NOUN
ejpam-854	157	13	∈	∈	PROPN
ejpam-854	157	14	x	x	X
ejpam-854	157	15	.	.	PUNCT
ejpam-854	158	1	theorem	theorem	NOUN
ejpam-854	158	2	1	1	NUM
ejpam-854	158	3	(	(	PUNCT
ejpam-854	158	4	the	the	DET
ejpam-854	158	5	fixed	fix	VERB
ejpam-854	158	6	point	point	NOUN
ejpam-854	158	7	alternative	alternative	NOUN
ejpam-854	158	8	theorem	theorem	ADJ
ejpam-854	158	9	,	,	PUNCT
ejpam-854	158	10	see	see	VERB
ejpam-854	158	11	[	[	X
ejpam-854	158	12	2	2	NUM
ejpam-854	158	13	,	,	PUNCT
ejpam-854	158	14	7	7	NUM
ejpam-854	158	15	,	,	PUNCT
ejpam-854	158	16	20	20	NUM
ejpam-854	158	17	,	,	PUNCT
ejpam-854	158	18	25	25	NUM
ejpam-854	158	19	,	,	PUNCT
ejpam-854	158	20	38	38	NUM
ejpam-854	158	21	]	]	PUNCT
ejpam-854	158	22	)	)	PUNCT
ejpam-854	158	23	.	.	PUNCT
ejpam-854	159	1	let	let	VERB
ejpam-854	159	2	(	(	PUNCT
ejpam-854	159	3	ω	ω	NOUN
ejpam-854	159	4	,	,	PUNCT
ejpam-854	159	5	d	d	NOUN
ejpam-854	159	6	)	)	PUNCT
ejpam-854	159	7	be	be	AUX
ejpam-854	159	8	a	a	DET
ejpam-854	159	9	complete	complete	ADJ
ejpam-854	159	10	generalized	generalize	VERB
ejpam-854	159	11	metric	metric	ADJ
ejpam-854	159	12	space	space	NOUN
ejpam-854	159	13	and	and	CCONJ
ejpam-854	159	14	j	j	NOUN
ejpam-854	159	15	:	:	PUNCT
ejpam-854	159	16	ω→	ω→	PUNCT
ejpam-854	159	17	ω	ω	PROPN
ejpam-854	159	18	be	be	AUX
ejpam-854	159	19	a	a	DET
ejpam-854	159	20	strictly	strictly	ADV
ejpam-854	159	21	contractive	contractive	ADJ
ejpam-854	159	22	mapping	mapping	NOUN
ejpam-854	159	23	with	with	ADP
ejpam-854	159	24	lipschitz	lipschitz	NOUN
ejpam-854	160	1	constant	constant	ADJ
ejpam-854	160	2	0≤	0≤	ADJ
ejpam-854	160	3	l	l	NOUN
ejpam-854	160	4	<	<	X
ejpam-854	160	5	1	1	NUM
ejpam-854	160	6	,	,	PUNCT
ejpam-854	160	7	that	that	PRON
ejpam-854	160	8	is	be	AUX
ejpam-854	160	9	d(j	d(j	PROPN
ejpam-854	160	10	x	x	SYM
ejpam-854	160	11	,	,	PUNCT
ejpam-854	160	12	j	j	PROPN
ejpam-854	160	13	y)≤	y)≤	PROPN
ejpam-854	160	14	ld(x	ld(x	PUNCT
ejpam-854	160	15	,	,	PUNCT
ejpam-854	160	16	y	y	X
ejpam-854	160	17	)	)	PUNCT
ejpam-854	160	18	for	for	ADP
ejpam-854	160	19	all	all	PRON
ejpam-854	160	20	x	x	SYM
ejpam-854	160	21	∈	∈	NOUN
ejpam-854	160	22	x	x	X
ejpam-854	160	23	.	.	PUNCT
ejpam-854	161	1	t.	t.	PROPN
ejpam-854	161	2	xu	xu	PROPN
ejpam-854	161	3	,	,	PUNCT
ejpam-854	161	4	j.	j.	PROPN
ejpam-854	161	5	rassias	rassias	PROPN
ejpam-854	161	6	,	,	PUNCT
ejpam-854	161	7	w.	w.	PROPN
ejpam-854	161	8	xu	xu	PROPN
ejpam-854	161	9	/	/	SYM
ejpam-854	161	10	eur	eur	PROPN
ejpam-854	161	11	.	.	PUNCT
ejpam-854	162	1	j.	j.	PROPN
ejpam-854	162	2	pure	pure	PROPN
ejpam-854	162	3	appl	appl	PROPN
ejpam-854	162	4	.	.	PROPN
ejpam-854	162	5	math	math	PROPN
ejpam-854	162	6	,	,	PUNCT
ejpam-854	162	7	3	3	NUM
ejpam-854	162	8	(	(	PUNCT
ejpam-854	162	9	2010	2010	NUM
ejpam-854	162	10	)	)	PUNCT
ejpam-854	162	11	,	,	PUNCT
ejpam-854	162	12	1032	1032	NUM
ejpam-854	162	13	-	-	SYM
ejpam-854	162	14	1047	1047	NUM
ejpam-854	162	15	1036	1036	NUM
ejpam-854	162	16	then	then	ADV
ejpam-854	162	17	,	,	PUNCT
ejpam-854	162	18	for	for	ADP
ejpam-854	162	19	each	each	DET
ejpam-854	162	20	given	give	VERB
ejpam-854	162	21	x	x	PROPN
ejpam-854	162	22	∈	∈	PROPN
ejpam-854	162	23	ω	ω	PROPN
ejpam-854	162	24	,	,	PUNCT
ejpam-854	162	25	either	either	CCONJ
ejpam-854	162	26	d(j	d(j	PROPN
ejpam-854	162	27	mx	mx	PROPN
ejpam-854	162	28	,	,	PUNCT
ejpam-854	162	29	j	j	PROPN
ejpam-854	162	30	m+1x	m+1x	PROPN
ejpam-854	162	31	)	)	PUNCT
ejpam-854	163	1	=	=	NOUN
ejpam-854	163	2	∞	∞	NOUN
ejpam-854	163	3	for	for	ADP
ejpam-854	163	4	all	all	DET
ejpam-854	163	5	m	m	PROPN
ejpam-854	163	6	≥	≥	NOUN
ejpam-854	163	7	0	0	NUM
ejpam-854	163	8	,	,	PUNCT
ejpam-854	163	9	or	or	CCONJ
ejpam-854	163	10	d(j	d(j	PROPN
ejpam-854	163	11	mx	mx	PROPN
ejpam-854	163	12	,	,	PUNCT
ejpam-854	163	13	j	j	PROPN
ejpam-854	164	1	m+1	m+1	X
ejpam-854	164	2	x)<∞	x)<∞	PROPN
ejpam-854	164	3	for	for	ADP
ejpam-854	164	4	all	all	DET
ejpam-854	164	5	m≥	m≥	PROPN
ejpam-854	164	6	m0	m0	NOUN
ejpam-854	164	7	,	,	PUNCT
ejpam-854	164	8	for	for	ADP
ejpam-854	164	9	some	some	DET
ejpam-854	164	10	nonnegative	nonnegative	ADJ
ejpam-854	164	11	integer	integer	NOUN
ejpam-854	164	12	m0	m0	NOUN
ejpam-854	164	13	.	.	PUNCT
ejpam-854	165	1	actually	actually	ADV
ejpam-854	165	2	,	,	PUNCT
ejpam-854	165	3	if	if	SCONJ
ejpam-854	165	4	the	the	DET
ejpam-854	165	5	second	second	ADJ
ejpam-854	165	6	alternative	alternative	NOUN
ejpam-854	165	7	holds	hold	VERB
ejpam-854	165	8	,	,	PUNCT
ejpam-854	165	9	then	then	ADV
ejpam-854	165	10	the	the	DET
ejpam-854	165	11	sequence	sequence	NOUN
ejpam-854	165	12	{	{	PUNCT
ejpam-854	165	13	j	j	PROPN
ejpam-854	165	14	m	m	PROPN
ejpam-854	165	15	x	x	NOUN
ejpam-854	165	16	}	}	PUNCT
ejpam-854	165	17	converges	converge	VERB
ejpam-854	165	18	to	to	ADP
ejpam-854	165	19	a	a	DET
ejpam-854	165	20	fixed	fix	VERB
ejpam-854	165	21	point	point	NOUN
ejpam-854	165	22	y∗	y∗	ADV
ejpam-854	165	23	of	of	ADP
ejpam-854	165	24	j	j	PROPN
ejpam-854	165	25	and	and	CCONJ
ejpam-854	165	26	(	(	PUNCT
ejpam-854	165	27	i	i	NOUN
ejpam-854	165	28	)	)	PUNCT
ejpam-854	165	29	y∗	y∗	ADV
ejpam-854	165	30	is	be	AUX
ejpam-854	165	31	the	the	DET
ejpam-854	165	32	unique	unique	ADJ
ejpam-854	165	33	fixed	fix	VERB
ejpam-854	165	34	point	point	NOUN
ejpam-854	165	35	of	of	ADP
ejpam-854	165	36	j	j	PROPN
ejpam-854	165	37	in	in	ADP
ejpam-854	165	38	the	the	DET
ejpam-854	165	39	set	set	NOUN
ejpam-854	165	40	∆=	∆=	NOUN
ejpam-854	165	41	{	{	PUNCT
ejpam-854	165	42	y	y	PROPN
ejpam-854	165	43	∈	∈	PROPN
ejpam-854	165	44	ω	ω	NOUN
ejpam-854	165	45	:	:	PUNCT
ejpam-854	165	46	d(j	d(j	PROPN
ejpam-854	165	47	m0	m0	NOUN
ejpam-854	165	48	x	x	PUNCT
ejpam-854	165	49	,	,	PUNCT
ejpam-854	165	50	y)<∞	y)<∞	PROPN
ejpam-854	165	51	}	}	PUNCT
ejpam-854	165	52	;	;	PUNCT
ejpam-854	165	53	(	(	PUNCT
ejpam-854	165	54	ii	ii	NOUN
ejpam-854	165	55	)	)	PUNCT
ejpam-854	165	56	d(y	d(y	PROPN
ejpam-854	165	57	,	,	PUNCT
ejpam-854	165	58	y∗)≤	y∗)≤	PROPN
ejpam-854	165	59	1	1	NUM
ejpam-854	165	60	1−l	1−l	NUM
ejpam-854	165	61	d(y	d(y	NOUN
ejpam-854	165	62	,	,	PUNCT
ejpam-854	165	63	j	j	PROPN
ejpam-854	165	64	y	y	PROPN
ejpam-854	165	65	)	)	PUNCT
ejpam-854	165	66	for	for	ADP
ejpam-854	165	67	all	all	PRON
ejpam-854	165	68	y	y	PROPN
ejpam-854	165	69	∈∆.	∈∆.	PROPN
ejpam-854	165	70	3	3	NUM
ejpam-854	165	71	.	.	PUNCT
ejpam-854	165	72	generalized	generalize	VERB
ejpam-854	165	73	hyers	hyers	PROPN
ejpam-854	165	74	-	-	PUNCT
ejpam-854	165	75	ulam	ulam	PROPN
ejpam-854	165	76	stability	stability	NOUN
ejpam-854	165	77	of	of	ADP
ejpam-854	165	78	the	the	DET
ejpam-854	165	79	functional	functional	ADJ
ejpam-854	165	80	equation	equation	NOUN
ejpam-854	165	81	in	in	ADP
ejpam-854	165	82	this	this	DET
ejpam-854	165	83	section	section	NOUN
ejpam-854	165	84	,	,	PUNCT
ejpam-854	165	85	we	we	PRON
ejpam-854	165	86	investigate	investigate	VERB
ejpam-854	165	87	the	the	DET
ejpam-854	165	88	stability	stability	NOUN
ejpam-854	165	89	of	of	ADP
ejpam-854	165	90	the	the	DET
ejpam-854	165	91	mixed	mixed	ADJ
ejpam-854	165	92	type	type	NOUN
ejpam-854	165	93	functional	functional	ADJ
ejpam-854	165	94	equation	equation	NOUN
ejpam-854	165	95	(	(	PUNCT
ejpam-854	165	96	5	5	NUM
ejpam-854	165	97	)	)	PUNCT
ejpam-854	165	98	in	in	ADP
ejpam-854	165	99	multi	multi	ADJ
ejpam-854	165	100	-	-	ADJ
ejpam-854	165	101	banach	banach	ADJ
ejpam-854	165	102	spaces	space	NOUN
ejpam-854	165	103	.	.	PUNCT
ejpam-854	166	1	for	for	ADP
ejpam-854	166	2	convenience	convenience	NOUN
ejpam-854	166	3	,	,	PUNCT
ejpam-854	166	4	we	we	PRON
ejpam-854	166	5	use	use	VERB
ejpam-854	166	6	the	the	DET
ejpam-854	166	7	following	follow	VERB
ejpam-854	166	8	abbreviation	abbreviation	NOUN
ejpam-854	166	9	for	for	ADP
ejpam-854	166	10	a	a	DET
ejpam-854	166	11	given	give	VERB
ejpam-854	166	12	mapping	mapping	NOUN
ejpam-854	166	13	f	f	NOUN
ejpam-854	166	14	:	:	PUNCT
ejpam-854	167	1	e→	e→	PROPN
ejpam-854	167	2	f	f	PROPN
ejpam-854	167	3	:	:	PUNCT
ejpam-854	167	4	d	d	X
ejpam-854	167	5	f	f	X
ejpam-854	167	6	(	(	PUNCT
ejpam-854	167	7	x	x	PROPN
ejpam-854	167	8	,	,	PUNCT
ejpam-854	167	9	y	y	PROPN
ejpam-854	167	10	)	)	PUNCT
ejpam-854	167	11	:	:	PUNCT
ejpam-854	168	1	=	=	SYM
ejpam-854	168	2	f	f	X
ejpam-854	168	3	(	(	PUNCT
ejpam-854	168	4	x	x	PROPN
ejpam-854	168	5	+	+	NUM
ejpam-854	168	6	ny	ny	NOUN
ejpam-854	168	7	)	)	PUNCT
ejpam-854	169	1	+	+	NUM
ejpam-854	169	2	f	f	X
ejpam-854	169	3	(	(	PUNCT
ejpam-854	169	4	x	x	NOUN
ejpam-854	169	5	−	−	PROPN
ejpam-854	169	6	ny)−	ny)−	PROPN
ejpam-854	169	7	n2	n2	PROPN
ejpam-854	169	8	f	f	X
ejpam-854	169	9	(	(	PUNCT
ejpam-854	169	10	x	x	PROPN
ejpam-854	169	11	+	+	PUNCT
ejpam-854	169	12	y)−	y)−	PROPN
ejpam-854	169	13	n2	n2	NOUN
ejpam-854	169	14	f	f	PROPN
ejpam-854	169	15	(	(	PUNCT
ejpam-854	169	16	x	x	SYM
ejpam-854	169	17	−	−	PUNCT
ejpam-854	169	18	y)−	y)−	PROPN
ejpam-854	169	19	2(1−	2(1−	NUM
ejpam-854	169	20	n2	n2	NOUN
ejpam-854	169	21	)	)	PUNCT
ejpam-854	169	22	f	f	PROPN
ejpam-854	169	23	(	(	PUNCT
ejpam-854	169	24	x	x	NOUN
ejpam-854	169	25	)	)	PUNCT
ejpam-854	169	26	−	−	PROPN
ejpam-854	169	27	n4	n4	PROPN
ejpam-854	169	28	−	−	PROPN
ejpam-854	169	29	n2	n2	NOUN
ejpam-854	169	30	12	12	NUM
ejpam-854	169	31	[	[	PUNCT
ejpam-854	169	32	f	f	X
ejpam-854	169	33	(	(	PUNCT
ejpam-854	169	34	2y	2y	NUM
ejpam-854	169	35	)	)	PUNCT
ejpam-854	170	1	+	+	NUM
ejpam-854	170	2	f	f	X
ejpam-854	170	3	(	(	PUNCT
ejpam-854	170	4	−2y)−	−2y)−	NUM
ejpam-854	170	5	4	4	NUM
ejpam-854	170	6	f	f	NOUN
ejpam-854	170	7	(	(	PUNCT
ejpam-854	170	8	y)−	y)−	PROPN
ejpam-854	170	9	4	4	NUM
ejpam-854	170	10	f	f	NOUN
ejpam-854	170	11	(	(	PUNCT
ejpam-854	170	12	−y	−y	NOUN
ejpam-854	170	13	)	)	PUNCT
ejpam-854	170	14	]	]	PUNCT
ejpam-854	170	15	for	for	ADP
ejpam-854	170	16	all	all	PRON
ejpam-854	170	17	x	x	SYM
ejpam-854	170	18	,	,	PUNCT
ejpam-854	170	19	y	y	PROPN
ejpam-854	170	20	∈	∈	PROPN
ejpam-854	170	21	x	x	X
ejpam-854	170	22	.	.	PUNCT
ejpam-854	171	1	theorem	theorem	NOUN
ejpam-854	171	2	2	2	NUM
ejpam-854	171	3	.	.	PUNCT
ejpam-854	172	1	let	let	VERB
ejpam-854	172	2	e	e	PRON
ejpam-854	172	3	be	be	AUX
ejpam-854	172	4	a	a	DET
ejpam-854	172	5	linear	linear	ADJ
ejpam-854	172	6	space	space	NOUN
ejpam-854	172	7	and	and	CCONJ
ejpam-854	172	8	let	let	VERB
ejpam-854	172	9	(	(	PUNCT
ejpam-854	172	10	(	(	PUNCT
ejpam-854	172	11	f	f	X
ejpam-854	172	12	k,‖	k,‖	PROPN
ejpam-854	172	13	·	·	PUNCT
ejpam-854	172	14	‖k	‖k	PROPN
ejpam-854	172	15	)	)	PUNCT
ejpam-854	172	16	:	:	PUNCT
ejpam-854	173	1	k	k	PROPN
ejpam-854	173	2	∈	∈	PROPN
ejpam-854	173	3	n	n	CCONJ
ejpam-854	173	4	)	)	PUNCT
ejpam-854	173	5	be	be	AUX
ejpam-854	173	6	a	a	DET
ejpam-854	173	7	multi	multi	ADJ
ejpam-854	173	8	-	-	ADJ
ejpam-854	173	9	banach	banach	ADJ
ejpam-854	173	10	space	space	NOUN
ejpam-854	173	11	.	.	PUNCT
ejpam-854	174	1	suppose	suppose	VERB
ejpam-854	174	2	that	that	SCONJ
ejpam-854	174	3	ǫ	ǫ	PRON
ejpam-854	174	4	≥	≥	NOUN
ejpam-854	174	5	0	0	NUM
ejpam-854	174	6	and	and	CCONJ
ejpam-854	174	7	f	f	NOUN
ejpam-854	174	8	:	:	PUNCT
ejpam-854	174	9	e→	e→	PROPN
ejpam-854	174	10	f	f	PROPN
ejpam-854	174	11	is	be	AUX
ejpam-854	174	12	an	an	DET
ejpam-854	174	13	odd	odd	ADJ
ejpam-854	174	14	mapping	mapping	NOUN
ejpam-854	174	15	satisfying	satisfying	ADJ
ejpam-854	174	16	sup	sup	NOUN
ejpam-854	174	17	k∈n	k∈n	PROPN
ejpam-854	174	18	‖(d	‖(d	PROPN
ejpam-854	175	1	f	f	PROPN
ejpam-854	175	2	(	(	PUNCT
ejpam-854	175	3	x1	x1	PROPN
ejpam-854	175	4	,	,	PUNCT
ejpam-854	175	5	y1	y1	PROPN
ejpam-854	175	6	)	)	PUNCT
ejpam-854	175	7	,	,	PUNCT
ejpam-854	175	8	.	.	PUNCT
ejpam-854	175	9	.	.	PUNCT
ejpam-854	175	10	.	.	PUNCT
ejpam-854	176	1	,	,	PUNCT
ejpam-854	177	1	d	d	X
ejpam-854	177	2	f	f	X
ejpam-854	177	3	(	(	PUNCT
ejpam-854	177	4	xk	xk	PROPN
ejpam-854	177	5	,	,	PUNCT
ejpam-854	177	6	yk))‖k	yk))‖k	INTJ
ejpam-854	177	7	≤	≤	ADJ
ejpam-854	177	8	ǫ	ǫ	PRON
ejpam-854	177	9	(	(	PUNCT
ejpam-854	177	10	6	6	NUM
ejpam-854	177	11	)	)	PUNCT
ejpam-854	177	12	for	for	ADP
ejpam-854	177	13	all	all	DET
ejpam-854	177	14	x1	x1	PROPN
ejpam-854	177	15	,	,	PUNCT
ejpam-854	177	16	.	.	PUNCT
ejpam-854	177	17	.	.	PUNCT
ejpam-854	177	18	.	.	PUNCT
ejpam-854	178	1	,	,	PUNCT
ejpam-854	178	2	xk	xk	PROPN
ejpam-854	178	3	,	,	PUNCT
ejpam-854	178	4	y1	y1	PROPN
ejpam-854	178	5	,	,	PUNCT
ejpam-854	178	6	.	.	PUNCT
ejpam-854	178	7	.	.	PUNCT
ejpam-854	178	8	.	.	PUNCT
ejpam-854	179	1	,	,	PUNCT
ejpam-854	179	2	yk	yk	PROPN
ejpam-854	179	3	∈	∈	PROPN
ejpam-854	179	4	e.	e.	PROPN
ejpam-854	179	5	then	then	ADV
ejpam-854	179	6	there	there	PRON
ejpam-854	179	7	exists	exist	VERB
ejpam-854	179	8	a	a	DET
ejpam-854	179	9	unique	unique	ADJ
ejpam-854	179	10	additive	additive	NOUN
ejpam-854	179	11	mapping	mapping	NOUN
ejpam-854	179	12	a	a	PRON
ejpam-854	179	13	:	:	PUNCT
ejpam-854	179	14	e	e	X
ejpam-854	179	15	→	→	SYM
ejpam-854	179	16	f	f	PROPN
ejpam-854	179	17	such	such	ADJ
ejpam-854	179	18	that	that	DET
ejpam-854	179	19	sup	sup	NOUN
ejpam-854	179	20	k∈n	k∈n	PROPN
ejpam-854	179	21	‖	‖	PROPN
ejpam-854	179	22	(	(	PUNCT
ejpam-854	179	23	f	f	X
ejpam-854	179	24	(	(	PUNCT
ejpam-854	179	25	2x1)−	2x1)−	NUM
ejpam-854	179	26	8	8	NUM
ejpam-854	179	27	f	f	NOUN
ejpam-854	179	28	(	(	PUNCT
ejpam-854	179	29	x1)−	x1)−	PROPN
ejpam-854	179	30	a(x1	a(x1	PROPN
ejpam-854	179	31	)	)	PUNCT
ejpam-854	179	32	,	,	PUNCT
ejpam-854	179	33	.	.	PUNCT
ejpam-854	179	34	.	.	PUNCT
ejpam-854	180	1	.	.	PUNCT
ejpam-854	181	1	,	,	PUNCT
ejpam-854	181	2	f	f	PROPN
ejpam-854	181	3	(	(	PUNCT
ejpam-854	181	4	2xk)−	2xk)−	PROPN
ejpam-854	181	5	8	8	NUM
ejpam-854	181	6	f	f	NOUN
ejpam-854	181	7	(	(	PUNCT
ejpam-854	181	8	xk)−	xk)−	PROPN
ejpam-854	181	9	a(xk))‖k	a(xk))‖k	PROPN
ejpam-854	181	10	≤	≤	NOUN
ejpam-854	181	11	9n2	9n2	NUM
ejpam-854	181	12	+	+	SYM
ejpam-854	181	13	4	4	NUM
ejpam-854	181	14	n4	n4	PROPN
ejpam-854	181	15	−	−	PROPN
ejpam-854	181	16	n2	n2	ADJ
ejpam-854	181	17	ǫ	ǫ	X
ejpam-854	181	18	(	(	PUNCT
ejpam-854	181	19	7	7	NUM
ejpam-854	181	20	)	)	PUNCT
ejpam-854	181	21	for	for	ADP
ejpam-854	181	22	all	all	DET
ejpam-854	181	23	x1	x1	PROPN
ejpam-854	181	24	,	,	PUNCT
ejpam-854	181	25	.	.	PUNCT
ejpam-854	181	26	.	.	PUNCT
ejpam-854	181	27	.	.	PUNCT
ejpam-854	182	1	,	,	PUNCT
ejpam-854	182	2	xk	xk	PROPN
ejpam-854	182	3	∈	∈	PROPN
ejpam-854	182	4	e.	e.	PROPN
ejpam-854	182	5	proof	proof	PROPN
ejpam-854	182	6	.	.	PUNCT
ejpam-854	183	1	let	let	VERB
ejpam-854	183	2	x1	x1	NUM
ejpam-854	183	3	,	,	PUNCT
ejpam-854	183	4	.	.	PUNCT
ejpam-854	183	5	.	.	PUNCT
ejpam-854	184	1	.	.	PUNCT
ejpam-854	185	1	,	,	PUNCT
ejpam-854	185	2	xk	xk	PROPN
ejpam-854	185	3	,	,	PUNCT
ejpam-854	185	4	y1	y1	PROPN
ejpam-854	185	5	,	,	PUNCT
ejpam-854	185	6	.	.	PUNCT
ejpam-854	185	7	.	.	PUNCT
ejpam-854	185	8	.	.	PUNCT
ejpam-854	186	1	,	,	PUNCT
ejpam-854	186	2	yk	yk	PROPN
ejpam-854	186	3	∈	∈	PROPN
ejpam-854	186	4	e.	e.	PROPN
ejpam-854	186	5	using	use	VERB
ejpam-854	186	6	the	the	DET
ejpam-854	186	7	oddness	oddness	NOUN
ejpam-854	186	8	of	of	ADP
ejpam-854	186	9	f	f	PROPN
ejpam-854	186	10	and	and	CCONJ
ejpam-854	186	11	(	(	PUNCT
ejpam-854	186	12	6	6	NUM
ejpam-854	186	13	)	)	PUNCT
ejpam-854	186	14	,	,	PUNCT
ejpam-854	186	15	we	we	PRON
ejpam-854	186	16	have	have	VERB
ejpam-854	186	17	sup	sup	NOUN
ejpam-854	186	18	k∈n	k∈n	PROPN
ejpam-854	186	19	‖	‖	PROPN
ejpam-854	186	20	(	(	PUNCT
ejpam-854	186	21	f	f	PROPN
ejpam-854	186	22	(	(	PUNCT
ejpam-854	186	23	x1	x1	PROPN
ejpam-854	186	24	+	+	CCONJ
ejpam-854	186	25	ny1	ny1	NOUN
ejpam-854	186	26	)	)	PUNCT
ejpam-854	187	1	+	+	NUM
ejpam-854	187	2	f	f	X
ejpam-854	187	3	(	(	PUNCT
ejpam-854	187	4	x1−	x1−	PROPN
ejpam-854	187	5	ny1)−	ny1)−	PROPN
ejpam-854	187	6	n2	n2	PROPN
ejpam-854	187	7	f	f	PROPN
ejpam-854	187	8	(	(	PUNCT
ejpam-854	187	9	x1	x1	PROPN
ejpam-854	187	10	+	+	PROPN
ejpam-854	187	11	y1)−	y1)−	PROPN
ejpam-854	187	12	n2	n2	PROPN
ejpam-854	187	13	f	f	PROPN
ejpam-854	187	14	(	(	PUNCT
ejpam-854	187	15	x1−	x1−	PROPN
ejpam-854	187	16	y1	y1	PROPN
ejpam-854	187	17	)	)	PUNCT
ejpam-854	187	18	−2(1−	−2(1−	PROPN
ejpam-854	187	19	n2	n2	NOUN
ejpam-854	187	20	)	)	PUNCT
ejpam-854	187	21	f	f	PROPN
ejpam-854	187	22	(	(	PUNCT
ejpam-854	187	23	x1	x1	PROPN
ejpam-854	187	24	)	)	PUNCT
ejpam-854	187	25	,	,	PUNCT
ejpam-854	187	26	.	.	PUNCT
ejpam-854	187	27	.	.	PUNCT
ejpam-854	187	28	.	.	PUNCT
ejpam-854	188	1	,	,	PUNCT
ejpam-854	188	2	f	f	PROPN
ejpam-854	188	3	(	(	PUNCT
ejpam-854	188	4	xk	xk	PROPN
ejpam-854	188	5	+	+	CCONJ
ejpam-854	188	6	nyk	nyk	PROPN
ejpam-854	188	7	)	)	PUNCT
ejpam-854	189	1	+	+	CCONJ
ejpam-854	189	2	f	f	X
ejpam-854	189	3	(	(	PUNCT
ejpam-854	189	4	xk	xk	INTJ
ejpam-854	189	5	−	−	PROPN
ejpam-854	189	6	nyk)−	nyk)−	PROPN
ejpam-854	189	7	n2	n2	PROPN
ejpam-854	189	8	f	f	PROPN
ejpam-854	189	9	(	(	PUNCT
ejpam-854	189	10	xk	xk	PROPN
ejpam-854	189	11	+	+	CCONJ
ejpam-854	189	12	yk	yk	PROPN
ejpam-854	189	13	)	)	PUNCT
ejpam-854	190	1	−n2	−n2	PROPN
ejpam-854	190	2	f	f	PROPN
ejpam-854	190	3	(	(	PUNCT
ejpam-854	190	4	xk	xk	INTJ
ejpam-854	190	5	−	−	PROPN
ejpam-854	190	6	yk)−	yk)−	PROPN
ejpam-854	190	7	2(1−	2(1−	NUM
ejpam-854	190	8	n2	n2	ADJ
ejpam-854	190	9	)	)	PUNCT
ejpam-854	190	10	f	f	NOUN
ejpam-854	190	11	(	(	PUNCT
ejpam-854	190	12	xk))‖k	xk))‖k	PROPN
ejpam-854	190	13	≤	≤	PROPN
ejpam-854	190	14	ǫ	ǫ	X
ejpam-854	190	15	.	.	PUNCT
ejpam-854	191	1	(	(	PUNCT
ejpam-854	191	2	8)	8)	NUM
ejpam-854	191	3	replacing	replace	VERB
ejpam-854	191	4	yi	yi	NOUN
ejpam-854	191	5	by	by	ADP
ejpam-854	191	6	x	x	PROPN
ejpam-854	191	7	i(i	i(i	PROPN
ejpam-854	191	8	∈	∈	PROPN
ejpam-854	191	9	nk	nk	PROPN
ejpam-854	191	10	)	)	PUNCT
ejpam-854	191	11	in	in	ADP
ejpam-854	191	12	(	(	PUNCT
ejpam-854	191	13	8)	8)	NUM
ejpam-854	191	14	,	,	PUNCT
ejpam-854	191	15	we	we	PRON
ejpam-854	191	16	get	get	VERB
ejpam-854	191	17	sup	sup	NOUN
ejpam-854	191	18	k∈n	k∈n	PROPN
ejpam-854	191	19	‖	‖	PROPN
ejpam-854	191	20	(	(	PUNCT
ejpam-854	191	21	f	f	X
ejpam-854	191	22	(	(	PUNCT
ejpam-854	191	23	(	(	PUNCT
ejpam-854	191	24	1	1	NUM
ejpam-854	191	25	+	+	NUM
ejpam-854	191	26	n)x1	n)x1	NOUN
ejpam-854	191	27	)	)	PUNCT
ejpam-854	192	1	+	+	NUM
ejpam-854	192	2	f	f	X
ejpam-854	192	3	(	(	PUNCT
ejpam-854	192	4	(	(	PUNCT
ejpam-854	192	5	1−	1−	NUM
ejpam-854	192	6	n)x1)−	n)x1)−	NOUN
ejpam-854	192	7	n2	n2	PROPN
ejpam-854	192	8	f	f	PROPN
ejpam-854	192	9	(	(	PUNCT
ejpam-854	192	10	2x1)−	2x1)−	PROPN
ejpam-854	192	11	2(1−	2(1−	NUM
ejpam-854	192	12	n2	n2	ADJ
ejpam-854	192	13	)	)	PUNCT
ejpam-854	192	14	f	f	PROPN
ejpam-854	192	15	(	(	PUNCT
ejpam-854	192	16	x1	x1	PROPN
ejpam-854	192	17	)	)	PUNCT
ejpam-854	192	18	,	,	PUNCT
ejpam-854	192	19	.	.	PUNCT
ejpam-854	192	20	.	.	PUNCT
ejpam-854	193	1	.	.	PUNCT
ejpam-854	194	1	,	,	PUNCT
ejpam-854	194	2	f	f	X
ejpam-854	194	3	(	(	PUNCT
ejpam-854	194	4	(	(	PUNCT
ejpam-854	194	5	1	1	NUM
ejpam-854	194	6	+	+	CCONJ
ejpam-854	194	7	n)xk	n)xk	ADJ
ejpam-854	194	8	)	)	PUNCT
ejpam-854	195	1	+	+	NUM
ejpam-854	195	2	f	f	X
ejpam-854	195	3	(	(	PUNCT
ejpam-854	195	4	(	(	PUNCT
ejpam-854	195	5	1−	1−	NUM
ejpam-854	195	6	n)xk)−	n)xk)−	PROPN
ejpam-854	195	7	n2	n2	ADJ
ejpam-854	195	8	f	f	PROPN
ejpam-854	195	9	(	(	PUNCT
ejpam-854	195	10	2xk)−	2xk)−	PROPN
ejpam-854	195	11	2(1−	2(1−	NUM
ejpam-854	195	12	n2	n2	NOUN
ejpam-854	195	13	)	)	PUNCT
ejpam-854	195	14	f	f	NOUN
ejpam-854	195	15	(	(	PUNCT
ejpam-854	195	16	xk))‖k	xk))‖k	PROPN
ejpam-854	195	17	≤	≤	PROPN
ejpam-854	195	18	ǫ	ǫ	X
ejpam-854	195	19	.	.	PUNCT
ejpam-854	196	1	(	(	PUNCT
ejpam-854	196	2	9	9	X
ejpam-854	196	3	)	)	PUNCT
ejpam-854	196	4	t.	t.	NOUN
ejpam-854	196	5	xu	xu	PROPN
ejpam-854	196	6	,	,	PUNCT
ejpam-854	196	7	j.	j.	PROPN
ejpam-854	196	8	rassias	rassias	PROPN
ejpam-854	196	9	,	,	PUNCT
ejpam-854	196	10	w.	w.	PROPN
ejpam-854	196	11	xu	xu	PROPN
ejpam-854	196	12	/	/	SYM
ejpam-854	196	13	eur	eur	PROPN
ejpam-854	196	14	.	.	PUNCT
ejpam-854	197	1	j.	j.	PROPN
ejpam-854	197	2	pure	pure	PROPN
ejpam-854	197	3	appl	appl	PROPN
ejpam-854	197	4	.	.	PROPN
ejpam-854	197	5	math	math	PROPN
ejpam-854	197	6	,	,	PUNCT
ejpam-854	197	7	3	3	NUM
ejpam-854	197	8	(	(	PUNCT
ejpam-854	197	9	2010	2010	NUM
ejpam-854	197	10	)	)	PUNCT
ejpam-854	197	11	,	,	PUNCT
ejpam-854	197	12	1032	1032	NUM
ejpam-854	197	13	-	-	SYM
ejpam-854	197	14	1047	1047	NUM
ejpam-854	197	15	1037	1037	NUM
ejpam-854	197	16	replacing	replace	VERB
ejpam-854	197	17	x	x	PUNCT
ejpam-854	197	18	i	i	PRON
ejpam-854	197	19	by	by	ADP
ejpam-854	197	20	2x	2x	NUM
ejpam-854	197	21	i(i	i(i	PROPN
ejpam-854	197	22	∈	∈	PROPN
ejpam-854	197	23	nk	nk	PROPN
ejpam-854	197	24	)	)	PUNCT
ejpam-854	197	25	in	in	ADP
ejpam-854	197	26	(	(	PUNCT
ejpam-854	197	27	9	9	NUM
ejpam-854	197	28	)	)	PUNCT
ejpam-854	197	29	,	,	PUNCT
ejpam-854	197	30	we	we	PRON
ejpam-854	197	31	get	get	VERB
ejpam-854	197	32	sup	sup	NOUN
ejpam-854	197	33	k∈n	k∈n	PROPN
ejpam-854	197	34	‖	‖	PROPN
ejpam-854	197	35	(	(	PUNCT
ejpam-854	197	36	f	f	X
ejpam-854	197	37	(	(	PUNCT
ejpam-854	197	38	2(1	2(1	NUM
ejpam-854	197	39	+	+	NUM
ejpam-854	197	40	n)x1	n)x1	NOUN
ejpam-854	197	41	)	)	PUNCT
ejpam-854	198	1	+	+	NUM
ejpam-854	198	2	f	f	X
ejpam-854	198	3	(	(	PUNCT
ejpam-854	198	4	2(1−	2(1−	NUM
ejpam-854	198	5	n)x1)−	n)x1)−	ADJ
ejpam-854	198	6	n2	n2	PROPN
ejpam-854	198	7	f	f	PROPN
ejpam-854	198	8	(	(	PUNCT
ejpam-854	198	9	4x1)−	4x1)−	NOUN
ejpam-854	198	10	2(1−	2(1−	NUM
ejpam-854	198	11	n2	n2	ADJ
ejpam-854	198	12	)	)	PUNCT
ejpam-854	198	13	f	f	PROPN
ejpam-854	198	14	(	(	PUNCT
ejpam-854	198	15	2x1	2x1	NUM
ejpam-854	198	16	)	)	PUNCT
ejpam-854	198	17	,	,	PUNCT
ejpam-854	198	18	.	.	PUNCT
ejpam-854	198	19	.	.	PUNCT
ejpam-854	198	20	.	.	PUNCT
ejpam-854	198	21	,	,	PUNCT
ejpam-854	198	22	f	f	X
ejpam-854	198	23	(	(	PUNCT
ejpam-854	198	24	2(1	2(1	NUM
ejpam-854	198	25	+	+	CCONJ
ejpam-854	198	26	n)xk	n)xk	ADJ
ejpam-854	198	27	)	)	PUNCT
ejpam-854	199	1	+	+	NUM
ejpam-854	199	2	f	f	X
ejpam-854	199	3	(	(	PUNCT
ejpam-854	199	4	2(1−	2(1−	NUM
ejpam-854	199	5	n)xk)−	n)xk)−	PROPN
ejpam-854	199	6	n2	n2	ADJ
ejpam-854	199	7	f	f	PROPN
ejpam-854	199	8	(	(	PUNCT
ejpam-854	199	9	4xk)−	4xk)−	PROPN
ejpam-854	199	10	2(1−	2(1−	NUM
ejpam-854	199	11	n2	n2	ADJ
ejpam-854	199	12	)	)	PUNCT
ejpam-854	199	13	f	f	NOUN
ejpam-854	199	14	(	(	PUNCT
ejpam-854	199	15	2xk))‖k	2xk))‖k	PROPN
ejpam-854	199	16	≤	≤	NOUN
ejpam-854	199	17	ǫ	ǫ	PRON
ejpam-854	199	18	.	.	PUNCT
ejpam-854	200	1	(	(	PUNCT
ejpam-854	200	2	10	10	NUM
ejpam-854	200	3	)	)	PUNCT
ejpam-854	200	4	replacing	replace	VERB
ejpam-854	200	5	x	x	PUNCT
ejpam-854	200	6	i	i	PRON
ejpam-854	200	7	and	and	CCONJ
ejpam-854	200	8	yi	yi	VERB
ejpam-854	200	9	by	by	ADP
ejpam-854	200	10	2x	2x	NUM
ejpam-854	200	11	i	i	PRON
ejpam-854	200	12	and	and	CCONJ
ejpam-854	200	13	x	x	SYM
ejpam-854	200	14	i(i	i(i	PROPN
ejpam-854	200	15	∈	∈	PROPN
ejpam-854	200	16	nk	nk	PROPN
ejpam-854	200	17	)	)	PUNCT
ejpam-854	200	18	in	in	ADP
ejpam-854	200	19	(	(	PUNCT
ejpam-854	200	20	8)	8)	NUM
ejpam-854	200	21	,	,	PUNCT
ejpam-854	200	22	respectively	respectively	ADV
ejpam-854	200	23	,	,	PUNCT
ejpam-854	200	24	we	we	PRON
ejpam-854	200	25	get	get	VERB
ejpam-854	200	26	sup	sup	NOUN
ejpam-854	200	27	k∈n	k∈n	PROPN
ejpam-854	200	28	‖	‖	PROPN
ejpam-854	201	1	(	(	PUNCT
ejpam-854	201	2	f	f	X
ejpam-854	201	3	(	(	PUNCT
ejpam-854	201	4	(	(	PUNCT
ejpam-854	201	5	2	2	NUM
ejpam-854	201	6	+	+	NUM
ejpam-854	201	7	n)x1	n)x1	NOUN
ejpam-854	201	8	)	)	PUNCT
ejpam-854	202	1	+	+	NUM
ejpam-854	202	2	f	f	X
ejpam-854	202	3	(	(	PUNCT
ejpam-854	202	4	(	(	PUNCT
ejpam-854	202	5	2−	2−	NUM
ejpam-854	202	6	n)x1)−	n)x1)−	ADJ
ejpam-854	202	7	n2	n2	PROPN
ejpam-854	202	8	f	f	PROPN
ejpam-854	202	9	(	(	PUNCT
ejpam-854	202	10	3x1)−	3x1)−	PROPN
ejpam-854	202	11	n2	n2	PROPN
ejpam-854	202	12	f	f	PROPN
ejpam-854	202	13	(	(	PUNCT
ejpam-854	202	14	x1)−	x1)−	PROPN
ejpam-854	202	15	2(1−	2(1−	PROPN
ejpam-854	202	16	k2	k2	PROPN
ejpam-854	202	17	)	)	PUNCT
ejpam-854	202	18	f	f	PROPN
ejpam-854	202	19	(	(	PUNCT
ejpam-854	202	20	2x1	2x1	NUM
ejpam-854	202	21	)	)	PUNCT
ejpam-854	202	22	,	,	PUNCT
ejpam-854	202	23	.	.	PUNCT
ejpam-854	202	24	.	.	PUNCT
ejpam-854	203	1	.	.	PUNCT
ejpam-854	204	1	,	,	PUNCT
ejpam-854	204	2	f	f	X
ejpam-854	204	3	(	(	PUNCT
ejpam-854	204	4	(	(	PUNCT
ejpam-854	204	5	2	2	NUM
ejpam-854	204	6	+	+	ADJ
ejpam-854	204	7	n)xk	n)xk	ADJ
ejpam-854	204	8	)	)	PUNCT
ejpam-854	205	1	+	+	NUM
ejpam-854	205	2	f	f	X
ejpam-854	205	3	(	(	PUNCT
ejpam-854	205	4	(	(	PUNCT
ejpam-854	205	5	2−	2−	NUM
ejpam-854	205	6	n)xk)−	n)xk)−	PROPN
ejpam-854	205	7	n2	n2	ADJ
ejpam-854	205	8	f	f	PROPN
ejpam-854	205	9	(	(	PUNCT
ejpam-854	205	10	3xk)−	3xk)−	PROPN
ejpam-854	205	11	n2	n2	ADJ
ejpam-854	205	12	f	f	PROPN
ejpam-854	205	13	(	(	PUNCT
ejpam-854	205	14	xk)−	xk)−	PROPN
ejpam-854	205	15	2(1−	2(1−	PROPN
ejpam-854	205	16	k2	k2	ADJ
ejpam-854	205	17	)	)	PUNCT
ejpam-854	205	18	f	f	PROPN
ejpam-854	205	19	(	(	PUNCT
ejpam-854	205	20	2xk))‖k	2xk))‖k	PROPN
ejpam-854	205	21	≤	≤	NOUN
ejpam-854	205	22	ǫ	ǫ	PRON
ejpam-854	205	23	.	.	PUNCT
ejpam-854	206	1	(	(	PUNCT
ejpam-854	206	2	11	11	NUM
ejpam-854	206	3	)	)	PUNCT
ejpam-854	206	4	replacing	replace	VERB
ejpam-854	206	5	yi	yi	NOUN
ejpam-854	206	6	by	by	ADP
ejpam-854	206	7	2x	2x	NUM
ejpam-854	206	8	i(i	i(i	PROPN
ejpam-854	206	9	∈	∈	PROPN
ejpam-854	206	10	nk	nk	PROPN
ejpam-854	206	11	)	)	PUNCT
ejpam-854	206	12	in	in	ADP
ejpam-854	206	13	(	(	PUNCT
ejpam-854	206	14	8)	8)	NUM
ejpam-854	206	15	,	,	PUNCT
ejpam-854	206	16	we	we	PRON
ejpam-854	206	17	get	get	VERB
ejpam-854	206	18	sup	sup	NOUN
ejpam-854	206	19	k∈n	k∈n	PROPN
ejpam-854	206	20	‖	‖	PROPN
ejpam-854	207	1	(	(	PUNCT
ejpam-854	207	2	f	f	X
ejpam-854	207	3	(	(	PUNCT
ejpam-854	207	4	(	(	PUNCT
ejpam-854	207	5	1	1	NUM
ejpam-854	207	6	+	+	NUM
ejpam-854	207	7	2n)x1	2n)x1	NUM
ejpam-854	207	8	)	)	PUNCT
ejpam-854	208	1	+	+	CCONJ
ejpam-854	208	2	f	f	X
ejpam-854	208	3	(	(	PUNCT
ejpam-854	208	4	(	(	PUNCT
ejpam-854	208	5	1−	1−	NUM
ejpam-854	208	6	2n)x1)−	2n)x1)−	NUM
ejpam-854	208	7	n2	n2	PROPN
ejpam-854	208	8	f	f	PROPN
ejpam-854	208	9	(	(	PUNCT
ejpam-854	208	10	3x1	3x1	NUM
ejpam-854	208	11	)	)	PUNCT
ejpam-854	208	12	+	+	CCONJ
ejpam-854	208	13	n2	n2	PROPN
ejpam-854	208	14	f	f	PROPN
ejpam-854	208	15	(	(	PUNCT
ejpam-854	208	16	x1)−	x1)−	PROPN
ejpam-854	208	17	2(1−	2(1−	PROPN
ejpam-854	208	18	n2	n2	PROPN
ejpam-854	208	19	)	)	PUNCT
ejpam-854	208	20	f	f	PROPN
ejpam-854	208	21	(	(	PUNCT
ejpam-854	208	22	x1	x1	PROPN
ejpam-854	208	23	)	)	PUNCT
ejpam-854	208	24	,	,	PUNCT
ejpam-854	208	25	.	.	PUNCT
ejpam-854	208	26	.	.	PUNCT
ejpam-854	208	27	.	.	PUNCT
ejpam-854	209	1	,	,	PUNCT
ejpam-854	209	2	f	f	X
ejpam-854	209	3	(	(	PUNCT
ejpam-854	209	4	(	(	PUNCT
ejpam-854	209	5	1	1	NUM
ejpam-854	209	6	+	+	NUM
ejpam-854	209	7	2n)xk	2n)xk	NUM
ejpam-854	209	8	)	)	PUNCT
ejpam-854	210	1	+	+	NUM
ejpam-854	210	2	f	f	X
ejpam-854	210	3	(	(	PUNCT
ejpam-854	210	4	(	(	PUNCT
ejpam-854	210	5	1−	1−	NUM
ejpam-854	210	6	2n)xk)−	2n)xk)−	NUM
ejpam-854	210	7	n2	n2	ADJ
ejpam-854	210	8	f	f	X
ejpam-854	210	9	(	(	PUNCT
ejpam-854	210	10	3xk	3xk	ADJ
ejpam-854	210	11	)	)	PUNCT
ejpam-854	210	12	+	+	CCONJ
ejpam-854	210	13	n2	n2	PROPN
ejpam-854	210	14	f	f	PROPN
ejpam-854	210	15	(	(	PUNCT
ejpam-854	210	16	xk)−	xk)−	PROPN
ejpam-854	210	17	2(1−	2(1−	NUM
ejpam-854	210	18	n2	n2	ADJ
ejpam-854	210	19	)	)	PUNCT
ejpam-854	210	20	f	f	PROPN
ejpam-854	210	21	(	(	PUNCT
ejpam-854	210	22	xk))‖k	xk))‖k	PROPN
ejpam-854	210	23	≤	≤	PROPN
ejpam-854	210	24	ǫ	ǫ	X
ejpam-854	210	25	.	.	PUNCT
ejpam-854	211	1	(	(	PUNCT
ejpam-854	211	2	12	12	NUM
ejpam-854	211	3	)	)	PUNCT
ejpam-854	211	4	replacing	replace	VERB
ejpam-854	211	5	yi	yi	NOUN
ejpam-854	211	6	by	by	ADP
ejpam-854	211	7	3x	3x	PROPN
ejpam-854	211	8	i(i	i(i	PROPN
ejpam-854	211	9	∈	∈	PROPN
ejpam-854	211	10	nk	nk	PROPN
ejpam-854	211	11	)	)	PUNCT
ejpam-854	211	12	in	in	ADP
ejpam-854	211	13	(	(	PUNCT
ejpam-854	211	14	8)	8)	NUM
ejpam-854	211	15	,	,	PUNCT
ejpam-854	211	16	we	we	PRON
ejpam-854	211	17	get	get	VERB
ejpam-854	211	18	sup	sup	NOUN
ejpam-854	211	19	k∈n	k∈n	PROPN
ejpam-854	211	20	‖	‖	PROPN
ejpam-854	212	1	(	(	PUNCT
ejpam-854	212	2	f	f	X
ejpam-854	212	3	(	(	PUNCT
ejpam-854	212	4	(	(	PUNCT
ejpam-854	212	5	1	1	NUM
ejpam-854	212	6	+	+	NUM
ejpam-854	212	7	3n)x1	3n)x1	NUM
ejpam-854	212	8	)	)	PUNCT
ejpam-854	213	1	+	+	NUM
ejpam-854	213	2	f	f	X
ejpam-854	213	3	(	(	PUNCT
ejpam-854	213	4	(	(	PUNCT
ejpam-854	213	5	1−	1−	NUM
ejpam-854	213	6	3n)x1)−	3n)x1)−	NUM
ejpam-854	213	7	n2	n2	PROPN
ejpam-854	213	8	f	f	PROPN
ejpam-854	213	9	(	(	PUNCT
ejpam-854	213	10	4x1	4x1	NUM
ejpam-854	213	11	)	)	PUNCT
ejpam-854	213	12	+	+	CCONJ
ejpam-854	213	13	n2	n2	PROPN
ejpam-854	213	14	f	f	PROPN
ejpam-854	213	15	(	(	PUNCT
ejpam-854	213	16	2x1)−	2x1)−	PROPN
ejpam-854	213	17	2(1−	2(1−	NUM
ejpam-854	213	18	n2	n2	ADJ
ejpam-854	213	19	)	)	PUNCT
ejpam-854	213	20	f	f	PROPN
ejpam-854	213	21	(	(	PUNCT
ejpam-854	213	22	x1	x1	PROPN
ejpam-854	213	23	)	)	PUNCT
ejpam-854	213	24	,	,	PUNCT
ejpam-854	213	25	.	.	PUNCT
ejpam-854	213	26	.	.	PUNCT
ejpam-854	213	27	.	.	PUNCT
ejpam-854	214	1	,	,	PUNCT
ejpam-854	214	2	f	f	X
ejpam-854	214	3	(	(	PUNCT
ejpam-854	214	4	(	(	PUNCT
ejpam-854	214	5	1	1	NUM
ejpam-854	214	6	+	+	NUM
ejpam-854	214	7	3n)xk	3n)xk	NUM
ejpam-854	214	8	)	)	PUNCT
ejpam-854	215	1	+	+	NUM
ejpam-854	215	2	f	f	X
ejpam-854	215	3	(	(	PUNCT
ejpam-854	215	4	(	(	PUNCT
ejpam-854	215	5	1−	1−	NUM
ejpam-854	215	6	3n)xk)−	3n)xk)−	NUM
ejpam-854	215	7	n2	n2	ADJ
ejpam-854	215	8	f	f	PROPN
ejpam-854	215	9	(	(	PUNCT
ejpam-854	215	10	4xk	4xk	ADJ
ejpam-854	215	11	)	)	PUNCT
ejpam-854	215	12	+	+	CCONJ
ejpam-854	215	13	n2	n2	PROPN
ejpam-854	215	14	f	f	PROPN
ejpam-854	215	15	(	(	PUNCT
ejpam-854	215	16	2xk)−	2xk)−	PROPN
ejpam-854	215	17	2(1−	2(1−	NUM
ejpam-854	215	18	n2	n2	NOUN
ejpam-854	215	19	)	)	PUNCT
ejpam-854	215	20	f	f	NOUN
ejpam-854	215	21	(	(	PUNCT
ejpam-854	215	22	xk))‖k	xk))‖k	PROPN
ejpam-854	215	23	≤	≤	PROPN
ejpam-854	215	24	ǫ	ǫ	X
ejpam-854	215	25	.	.	PUNCT
ejpam-854	216	1	(	(	PUNCT
ejpam-854	216	2	13	13	NUM
ejpam-854	216	3	)	)	PUNCT
ejpam-854	216	4	replacing	replace	VERB
ejpam-854	216	5	x	x	PUNCT
ejpam-854	216	6	i	i	PRON
ejpam-854	216	7	and	and	CCONJ
ejpam-854	216	8	yi	yi	VERB
ejpam-854	216	9	by	by	ADP
ejpam-854	216	10	(	(	PUNCT
ejpam-854	216	11	1	1	NUM
ejpam-854	216	12	+	+	NUM
ejpam-854	216	13	n)x	n)x	ADP
ejpam-854	217	1	i	i	PRON
ejpam-854	217	2	and	and	CCONJ
ejpam-854	217	3	x	x	SYM
ejpam-854	217	4	i(i	i(i	PROPN
ejpam-854	217	5	∈	∈	PROPN
ejpam-854	217	6	nk	nk	PROPN
ejpam-854	217	7	)	)	PUNCT
ejpam-854	217	8	in	in	ADP
ejpam-854	217	9	(	(	PUNCT
ejpam-854	217	10	8)	8)	NUM
ejpam-854	217	11	,	,	PUNCT
ejpam-854	217	12	respectively	respectively	ADV
ejpam-854	217	13	,	,	PUNCT
ejpam-854	217	14	we	we	PRON
ejpam-854	217	15	have	have	VERB
ejpam-854	217	16	sup	sup	NOUN
ejpam-854	217	17	k∈n	k∈n	PROPN
ejpam-854	217	18	‖	‖	PROPN
ejpam-854	217	19	(	(	PUNCT
ejpam-854	217	20	f	f	X
ejpam-854	217	21	(	(	PUNCT
ejpam-854	217	22	(	(	PUNCT
ejpam-854	217	23	1	1	NUM
ejpam-854	217	24	+	+	NUM
ejpam-854	217	25	2n)x1	2n)x1	NUM
ejpam-854	217	26	)	)	PUNCT
ejpam-854	218	1	+	+	NUM
ejpam-854	218	2	f	f	X
ejpam-854	218	3	(	(	PUNCT
ejpam-854	218	4	x1)−	x1)−	PROPN
ejpam-854	218	5	n2	n2	PROPN
ejpam-854	218	6	f	f	PROPN
ejpam-854	218	7	(	(	PUNCT
ejpam-854	218	8	(	(	PUNCT
ejpam-854	218	9	2	2	NUM
ejpam-854	218	10	+	+	NUM
ejpam-854	218	11	n)x1)−	n)x1)−	ADJ
ejpam-854	218	12	n2	n2	PROPN
ejpam-854	218	13	f	f	PROPN
ejpam-854	218	14	(	(	PUNCT
ejpam-854	218	15	nx1	nx1	ADJ
ejpam-854	218	16	)	)	PUNCT
ejpam-854	218	17	−2(1−	−2(1−	PROPN
ejpam-854	218	18	n2	n2	NOUN
ejpam-854	218	19	)	)	PUNCT
ejpam-854	218	20	f	f	NOUN
ejpam-854	218	21	(	(	PUNCT
ejpam-854	218	22	(	(	PUNCT
ejpam-854	218	23	1	1	NUM
ejpam-854	218	24	+	+	NUM
ejpam-854	218	25	n)x1	n)x1	NOUN
ejpam-854	218	26	)	)	PUNCT
ejpam-854	218	27	,	,	PUNCT
ejpam-854	218	28	.	.	PUNCT
ejpam-854	218	29	.	.	PUNCT
ejpam-854	218	30	.	.	PUNCT
ejpam-854	219	1	,	,	PUNCT
ejpam-854	219	2	f	f	X
ejpam-854	219	3	(	(	PUNCT
ejpam-854	219	4	(	(	PUNCT
ejpam-854	219	5	1	1	NUM
ejpam-854	219	6	+	+	NUM
ejpam-854	219	7	2n)xk	2n)xk	NUM
ejpam-854	219	8	)	)	PUNCT
ejpam-854	220	1	+	+	NUM
ejpam-854	220	2	f	f	X
ejpam-854	220	3	(	(	PUNCT
ejpam-854	220	4	xk	xk	PROPN
ejpam-854	220	5	)	)	PUNCT
ejpam-854	220	6	−n2	−n2	PROPN
ejpam-854	220	7	f	f	NOUN
ejpam-854	220	8	(	(	PUNCT
ejpam-854	220	9	(	(	PUNCT
ejpam-854	220	10	2	2	NUM
ejpam-854	220	11	+	+	NUM
ejpam-854	220	12	n)xk)−	n)xk)−	PROPN
ejpam-854	220	13	n2	n2	ADJ
ejpam-854	220	14	f	f	X
ejpam-854	220	15	(	(	PUNCT
ejpam-854	220	16	nxk)−	nxk)−	PROPN
ejpam-854	220	17	2(1−	2(1−	PROPN
ejpam-854	220	18	n2	n2	ADJ
ejpam-854	220	19	)	)	PUNCT
ejpam-854	220	20	f	f	NOUN
ejpam-854	220	21	(	(	PUNCT
ejpam-854	220	22	(	(	PUNCT
ejpam-854	220	23	1	1	NUM
ejpam-854	220	24	+	+	X
ejpam-854	220	25	n)xk))‖k	n)xk))‖k	PROPN
ejpam-854	220	26	≤	≤	NUM
ejpam-854	220	27	ǫ	ǫ	X
ejpam-854	220	28	.	.	PUNCT
ejpam-854	221	1	(	(	PUNCT
ejpam-854	221	2	14	14	NUM
ejpam-854	221	3	)	)	PUNCT
ejpam-854	221	4	again	again	ADV
ejpam-854	221	5	replacing	replace	VERB
ejpam-854	221	6	x	x	PUNCT
ejpam-854	221	7	i	i	PRON
ejpam-854	221	8	and	and	CCONJ
ejpam-854	221	9	yi	yi	PROPN
ejpam-854	221	10	by	by	ADP
ejpam-854	221	11	(	(	PUNCT
ejpam-854	221	12	1−	1−	NUM
ejpam-854	221	13	n)x	n)x	ADV
ejpam-854	222	1	i	i	PRON
ejpam-854	222	2	and	and	CCONJ
ejpam-854	222	3	x	x	SYM
ejpam-854	222	4	i(i	i(i	PROPN
ejpam-854	222	5	∈	∈	PROPN
ejpam-854	222	6	nk	nk	PROPN
ejpam-854	222	7	)	)	PUNCT
ejpam-854	222	8	in	in	ADP
ejpam-854	222	9	(	(	PUNCT
ejpam-854	222	10	8)	8)	NUM
ejpam-854	222	11	,	,	PUNCT
ejpam-854	222	12	respectively	respectively	ADV
ejpam-854	222	13	,	,	PUNCT
ejpam-854	222	14	we	we	PRON
ejpam-854	222	15	have	have	VERB
ejpam-854	222	16	sup	sup	NOUN
ejpam-854	222	17	k∈n	k∈n	PROPN
ejpam-854	222	18	‖	‖	PROPN
ejpam-854	222	19	(	(	PUNCT
ejpam-854	222	20	f	f	PROPN
ejpam-854	222	21	(	(	PUNCT
ejpam-854	222	22	x1	x1	PROPN
ejpam-854	222	23	)	)	PUNCT
ejpam-854	223	1	+	+	NUM
ejpam-854	223	2	f	f	X
ejpam-854	223	3	(	(	PUNCT
ejpam-854	223	4	(	(	PUNCT
ejpam-854	223	5	1−	1−	NUM
ejpam-854	223	6	2n)x1)−	2n)x1)−	NUM
ejpam-854	223	7	n2	n2	ADJ
ejpam-854	223	8	f	f	PROPN
ejpam-854	223	9	(	(	PUNCT
ejpam-854	223	10	(	(	PUNCT
ejpam-854	223	11	2−	2−	NUM
ejpam-854	223	12	n)x1	n)x1	NOUN
ejpam-854	223	13	)	)	PUNCT
ejpam-854	223	14	+	+	NUM
ejpam-854	223	15	n2	n2	PROPN
ejpam-854	223	16	f	f	PROPN
ejpam-854	223	17	(	(	PUNCT
ejpam-854	223	18	nx1	nx1	ADJ
ejpam-854	223	19	)	)	PUNCT
ejpam-854	223	20	−2(1−	−2(1−	PROPN
ejpam-854	223	21	n2	n2	NOUN
ejpam-854	223	22	)	)	PUNCT
ejpam-854	223	23	f	f	PROPN
ejpam-854	223	24	(	(	PUNCT
ejpam-854	223	25	(	(	PUNCT
ejpam-854	223	26	1−	1−	NUM
ejpam-854	223	27	n)x1	n)x1	NOUN
ejpam-854	223	28	)	)	PUNCT
ejpam-854	223	29	,	,	PUNCT
ejpam-854	223	30	.	.	PUNCT
ejpam-854	223	31	.	.	PUNCT
ejpam-854	223	32	.	.	PUNCT
ejpam-854	224	1	,	,	PUNCT
ejpam-854	224	2	f	f	PROPN
ejpam-854	224	3	(	(	PUNCT
ejpam-854	224	4	xk	xk	PROPN
ejpam-854	224	5	)	)	PUNCT
ejpam-854	225	1	+	+	NUM
ejpam-854	225	2	f	f	X
ejpam-854	225	3	(	(	PUNCT
ejpam-854	225	4	(	(	PUNCT
ejpam-854	225	5	1−	1−	NUM
ejpam-854	225	6	2n)xk	2n)xk	NUM
ejpam-854	225	7	)	)	PUNCT
ejpam-854	226	1	−n2	−n2	PROPN
ejpam-854	226	2	f	f	NOUN
ejpam-854	226	3	(	(	PUNCT
ejpam-854	226	4	(	(	PUNCT
ejpam-854	226	5	2−	2−	NUM
ejpam-854	226	6	n)xk	n)xk	PROPN
ejpam-854	226	7	)	)	PUNCT
ejpam-854	226	8	+	+	SYM
ejpam-854	226	9	n2	n2	PROPN
ejpam-854	226	10	f	f	X
ejpam-854	226	11	(	(	PUNCT
ejpam-854	226	12	nxk)−	nxk)−	PROPN
ejpam-854	226	13	2(1−	2(1−	PROPN
ejpam-854	226	14	n2	n2	ADJ
ejpam-854	226	15	)	)	PUNCT
ejpam-854	226	16	f	f	NOUN
ejpam-854	226	17	(	(	PUNCT
ejpam-854	226	18	(	(	PUNCT
ejpam-854	226	19	1−	1−	NUM
ejpam-854	226	20	n)xk))‖k	n)xk))‖k	PROPN
ejpam-854	226	21	≤	≤	PUNCT
ejpam-854	226	22	ǫ	ǫ	X
ejpam-854	226	23	.	.	PUNCT
ejpam-854	226	24	(	(	PUNCT
ejpam-854	226	25	15	15	NUM
ejpam-854	226	26	)	)	PUNCT
ejpam-854	226	27	by	by	ADP
ejpam-854	226	28	(	(	PUNCT
ejpam-854	226	29	14	14	NUM
ejpam-854	226	30	)	)	PUNCT
ejpam-854	226	31	and	and	CCONJ
ejpam-854	226	32	(	(	PUNCT
ejpam-854	226	33	15	15	NUM
ejpam-854	226	34	)	)	PUNCT
ejpam-854	226	35	,	,	PUNCT
ejpam-854	226	36	we	we	PRON
ejpam-854	226	37	have	have	VERB
ejpam-854	226	38	sup	sup	NOUN
ejpam-854	226	39	k∈n	k∈n	PROPN
ejpam-854	226	40	‖	‖	PROPN
ejpam-854	226	41	(	(	PUNCT
ejpam-854	226	42	f	f	X
ejpam-854	226	43	(	(	PUNCT
ejpam-854	226	44	(	(	PUNCT
ejpam-854	226	45	1	1	NUM
ejpam-854	226	46	+	+	NUM
ejpam-854	226	47	2n)x1	2n)x1	NUM
ejpam-854	226	48	)	)	PUNCT
ejpam-854	227	1	+	+	CCONJ
ejpam-854	227	2	f	f	X
ejpam-854	227	3	(	(	PUNCT
ejpam-854	227	4	(	(	PUNCT
ejpam-854	227	5	1−	1−	NUM
ejpam-854	227	6	2n)x1	2n)x1	NUM
ejpam-854	227	7	)	)	PUNCT
ejpam-854	227	8	+	+	CCONJ
ejpam-854	227	9	2	2	NUM
ejpam-854	227	10	f	f	NOUN
ejpam-854	227	11	(	(	PUNCT
ejpam-854	227	12	x1)−	x1)−	PROPN
ejpam-854	227	13	n2	n2	PROPN
ejpam-854	227	14	f	f	PROPN
ejpam-854	227	15	(	(	PUNCT
ejpam-854	227	16	(	(	PUNCT
ejpam-854	227	17	2	2	NUM
ejpam-854	227	18	+	+	NUM
ejpam-854	227	19	n)x1	n)x1	NOUN
ejpam-854	227	20	)	)	PUNCT
ejpam-854	227	21	−2(1−	−2(1−	PROPN
ejpam-854	227	22	n2	n2	NOUN
ejpam-854	227	23	)	)	PUNCT
ejpam-854	227	24	f	f	NOUN
ejpam-854	227	25	(	(	PUNCT
ejpam-854	227	26	(	(	PUNCT
ejpam-854	227	27	1	1	NUM
ejpam-854	227	28	+	+	NUM
ejpam-854	227	29	n)x1)−	n)x1)−	ADJ
ejpam-854	227	30	n2	n2	PROPN
ejpam-854	227	31	f	f	PROPN
ejpam-854	227	32	(	(	PUNCT
ejpam-854	227	33	(	(	PUNCT
ejpam-854	227	34	2−	2−	NUM
ejpam-854	227	35	n)x1)−	n)x1)−	NOUN
ejpam-854	227	36	2(1−	2(1−	NUM
ejpam-854	227	37	n2	n2	ADJ
ejpam-854	227	38	)	)	PUNCT
ejpam-854	227	39	f	f	NOUN
ejpam-854	227	40	(	(	PUNCT
ejpam-854	227	41	(	(	PUNCT
ejpam-854	227	42	1−	1−	NUM
ejpam-854	227	43	n)x1	n)x1	NOUN
ejpam-854	227	44	)	)	PUNCT
ejpam-854	227	45	,	,	PUNCT
ejpam-854	227	46	.	.	PUNCT
ejpam-854	227	47	.	.	PUNCT
ejpam-854	227	48	.	.	PUNCT
ejpam-854	228	1	,	,	PUNCT
ejpam-854	228	2	f	f	X
ejpam-854	228	3	(	(	PUNCT
ejpam-854	228	4	(	(	PUNCT
ejpam-854	228	5	1	1	NUM
ejpam-854	228	6	+	+	NUM
ejpam-854	228	7	2n)xk	2n)xk	NUM
ejpam-854	228	8	)	)	PUNCT
ejpam-854	229	1	+	+	NUM
ejpam-854	229	2	f	f	X
ejpam-854	229	3	(	(	PUNCT
ejpam-854	229	4	(	(	PUNCT
ejpam-854	229	5	1−	1−	NUM
ejpam-854	229	6	2n)xk)+	2n)xk)+	NUM
ejpam-854	229	7	2	2	NUM
ejpam-854	229	8	f	f	X
ejpam-854	229	9	(	(	PUNCT
ejpam-854	229	10	xk)−	xk)−	PROPN
ejpam-854	229	11	n2	n2	PROPN
ejpam-854	229	12	f	f	PROPN
ejpam-854	229	13	(	(	PUNCT
ejpam-854	229	14	(	(	PUNCT
ejpam-854	229	15	2	2	NUM
ejpam-854	229	16	+	+	ADJ
ejpam-854	229	17	n)xk	n)xk	ADJ
ejpam-854	229	18	)	)	PUNCT
ejpam-854	229	19	−n2	−n2	PROPN
ejpam-854	229	20	f	f	NOUN
ejpam-854	229	21	(	(	PUNCT
ejpam-854	229	22	(	(	PUNCT
ejpam-854	229	23	2−	2−	NUM
ejpam-854	229	24	n)xk)−	n)xk)−	NOUN
ejpam-854	229	25	2(1−	2(1−	NUM
ejpam-854	229	26	n2	n2	NOUN
ejpam-854	229	27	)	)	PUNCT
ejpam-854	229	28	f	f	NOUN
ejpam-854	229	29	(	(	PUNCT
ejpam-854	229	30	(	(	PUNCT
ejpam-854	229	31	1	1	NUM
ejpam-854	229	32	+	+	NUM
ejpam-854	229	33	n)xk)−	n)xk)−	PROPN
ejpam-854	229	34	2(1−	2(1−	NUM
ejpam-854	229	35	n2	n2	ADJ
ejpam-854	229	36	)	)	PUNCT
ejpam-854	229	37	f	f	NOUN
ejpam-854	229	38	(	(	PUNCT
ejpam-854	229	39	(	(	PUNCT
ejpam-854	229	40	1−	1−	NUM
ejpam-854	229	41	n)xk))‖k	n)xk))‖k	PROPN
ejpam-854	229	42	≤	≤	NUM
ejpam-854	229	43	2ǫ	2ǫ	NOUN
ejpam-854	229	44	.	.	PUNCT
ejpam-854	230	1	(	(	PUNCT
ejpam-854	230	2	16	16	NUM
ejpam-854	230	3	)	)	PUNCT
ejpam-854	230	4	replacing	replace	VERB
ejpam-854	230	5	x	x	PUNCT
ejpam-854	230	6	i	i	PRON
ejpam-854	230	7	and	and	CCONJ
ejpam-854	230	8	yi	yi	VERB
ejpam-854	230	9	by	by	ADP
ejpam-854	230	10	(	(	PUNCT
ejpam-854	230	11	1	1	NUM
ejpam-854	230	12	+	+	NUM
ejpam-854	230	13	2n)x	2n)x	NUM
ejpam-854	230	14	i	i	PRON
ejpam-854	230	15	and	and	CCONJ
ejpam-854	230	16	x	x	SYM
ejpam-854	230	17	i(i	i(i	PROPN
ejpam-854	230	18	∈	∈	PROPN
ejpam-854	230	19	nk	nk	PROPN
ejpam-854	230	20	)	)	PUNCT
ejpam-854	230	21	in	in	ADP
ejpam-854	230	22	(	(	PUNCT
ejpam-854	230	23	8)	8)	NUM
ejpam-854	230	24	,	,	PUNCT
ejpam-854	230	25	respectively	respectively	ADV
ejpam-854	230	26	,	,	PUNCT
ejpam-854	230	27	we	we	PRON
ejpam-854	230	28	have	have	VERB
ejpam-854	230	29	sup	sup	NOUN
ejpam-854	230	30	k∈n	k∈n	PROPN
ejpam-854	230	31	‖	‖	PROPN
ejpam-854	230	32	(	(	PUNCT
ejpam-854	230	33	f	f	X
ejpam-854	230	34	(	(	PUNCT
ejpam-854	230	35	(	(	PUNCT
ejpam-854	230	36	1	1	NUM
ejpam-854	230	37	+	+	NUM
ejpam-854	230	38	3n)x1	3n)x1	NUM
ejpam-854	230	39	)	)	PUNCT
ejpam-854	231	1	+	+	NUM
ejpam-854	231	2	f	f	X
ejpam-854	231	3	(	(	PUNCT
ejpam-854	231	4	(	(	PUNCT
ejpam-854	231	5	1	1	NUM
ejpam-854	231	6	+	+	NUM
ejpam-854	231	7	n)x1)−	n)x1)−	ADJ
ejpam-854	231	8	n2	n2	PROPN
ejpam-854	231	9	f	f	PROPN
ejpam-854	231	10	(	(	PUNCT
ejpam-854	231	11	2(1	2(1	NUM
ejpam-854	231	12	+	+	CCONJ
ejpam-854	231	13	n)x1)−	n)x1)−	ADJ
ejpam-854	231	14	n2	n2	PROPN
ejpam-854	231	15	f	f	PROPN
ejpam-854	231	16	(	(	PUNCT
ejpam-854	231	17	2nx1	2nx1	NUM
ejpam-854	231	18	)	)	PUNCT
ejpam-854	231	19	−2(1−	−2(1−	PROPN
ejpam-854	231	20	n2	n2	NOUN
ejpam-854	231	21	)	)	PUNCT
ejpam-854	231	22	f	f	NOUN
ejpam-854	231	23	(	(	PUNCT
ejpam-854	231	24	(	(	PUNCT
ejpam-854	231	25	1	1	NUM
ejpam-854	231	26	+	+	NUM
ejpam-854	231	27	2n)x1	2n)x1	NUM
ejpam-854	231	28	)	)	PUNCT
ejpam-854	231	29	,	,	PUNCT
ejpam-854	231	30	.	.	PUNCT
ejpam-854	231	31	.	.	PUNCT
ejpam-854	231	32	.	.	PUNCT
ejpam-854	232	1	,	,	PUNCT
ejpam-854	232	2	f	f	X
ejpam-854	232	3	(	(	PUNCT
ejpam-854	232	4	(	(	PUNCT
ejpam-854	232	5	1	1	NUM
ejpam-854	232	6	+	+	NUM
ejpam-854	232	7	3n)xk	3n)xk	NUM
ejpam-854	232	8	)	)	PUNCT
ejpam-854	233	1	+	+	NUM
ejpam-854	233	2	f	f	X
ejpam-854	233	3	(	(	PUNCT
ejpam-854	233	4	(	(	PUNCT
ejpam-854	233	5	1	1	NUM
ejpam-854	233	6	+	+	NUM
ejpam-854	233	7	n)xk	n)xk	ADJ
ejpam-854	233	8	)	)	PUNCT
ejpam-854	233	9	−n2	−n2	PROPN
ejpam-854	233	10	f	f	X
ejpam-854	233	11	(	(	PUNCT
ejpam-854	233	12	2(1	2(1	NUM
ejpam-854	233	13	+	+	CCONJ
ejpam-854	233	14	n)xk)−	n)xk)−	PROPN
ejpam-854	233	15	n2	n2	ADJ
ejpam-854	233	16	f	f	PROPN
ejpam-854	233	17	(	(	PUNCT
ejpam-854	233	18	2nxk)−	2nxk)−	PROPN
ejpam-854	233	19	2(1−	2(1−	NUM
ejpam-854	233	20	n2	n2	NOUN
ejpam-854	233	21	)	)	PUNCT
ejpam-854	233	22	f	f	NOUN
ejpam-854	233	23	(	(	PUNCT
ejpam-854	233	24	(	(	PUNCT
ejpam-854	233	25	1	1	NUM
ejpam-854	233	26	+	+	NUM
ejpam-854	233	27	2n)xk))‖k	2n)xk))‖k	NUM
ejpam-854	233	28	≤	≤	ADJ
ejpam-854	233	29	ǫ	ǫ	NOUN
ejpam-854	233	30	.	.	PUNCT
ejpam-854	234	1	(	(	PUNCT
ejpam-854	234	2	17	17	NUM
ejpam-854	234	3	)	)	PUNCT
ejpam-854	234	4	t.	t.	PROPN
ejpam-854	234	5	xu	xu	PROPN
ejpam-854	234	6	,	,	PUNCT
ejpam-854	234	7	j.	j.	PROPN
ejpam-854	234	8	rassias	rassias	PROPN
ejpam-854	234	9	,	,	PUNCT
ejpam-854	234	10	w.	w.	PROPN
ejpam-854	234	11	xu	xu	PROPN
ejpam-854	234	12	/	/	SYM
ejpam-854	234	13	eur	eur	PROPN
ejpam-854	234	14	.	.	PUNCT
ejpam-854	235	1	j.	j.	PROPN
ejpam-854	235	2	pure	pure	PROPN
ejpam-854	235	3	appl	appl	PROPN
ejpam-854	235	4	.	.	PROPN
ejpam-854	235	5	math	math	PROPN
ejpam-854	235	6	,	,	PUNCT
ejpam-854	235	7	3	3	NUM
ejpam-854	235	8	(	(	PUNCT
ejpam-854	235	9	2010	2010	NUM
ejpam-854	235	10	)	)	PUNCT
ejpam-854	235	11	,	,	PUNCT
ejpam-854	235	12	1032	1032	NUM
ejpam-854	235	13	-	-	SYM
ejpam-854	235	14	1047	1047	NUM
ejpam-854	235	15	1038	1038	NUM
ejpam-854	235	16	again	again	ADV
ejpam-854	235	17	replacing	replace	VERB
ejpam-854	235	18	x	x	PUNCT
ejpam-854	235	19	i	i	PRON
ejpam-854	235	20	and	and	CCONJ
ejpam-854	235	21	yi	yi	PROPN
ejpam-854	235	22	by	by	ADP
ejpam-854	235	23	(	(	PUNCT
ejpam-854	235	24	1−	1−	NUM
ejpam-854	235	25	2n)x	2n)x	NUM
ejpam-854	235	26	i	i	PRON
ejpam-854	235	27	and	and	CCONJ
ejpam-854	235	28	x	x	SYM
ejpam-854	235	29	i(i	i(i	PROPN
ejpam-854	235	30	∈	∈	PROPN
ejpam-854	235	31	nk	nk	PROPN
ejpam-854	235	32	)	)	PUNCT
ejpam-854	235	33	in	in	ADP
ejpam-854	235	34	(	(	PUNCT
ejpam-854	235	35	8)	8)	NUM
ejpam-854	235	36	,	,	PUNCT
ejpam-854	235	37	respectively	respectively	ADV
ejpam-854	235	38	,	,	PUNCT
ejpam-854	235	39	we	we	PRON
ejpam-854	235	40	obtain	obtain	VERB
ejpam-854	235	41	sup	sup	NOUN
ejpam-854	235	42	k∈n	k∈n	PROPN
ejpam-854	235	43	‖	‖	PROPN
ejpam-854	236	1	(	(	PUNCT
ejpam-854	236	2	f	f	X
ejpam-854	236	3	(	(	PUNCT
ejpam-854	236	4	(	(	PUNCT
ejpam-854	236	5	1−	1−	NUM
ejpam-854	236	6	3n)x1	3n)x1	NUM
ejpam-854	236	7	)	)	PUNCT
ejpam-854	237	1	+	+	NUM
ejpam-854	237	2	f	f	X
ejpam-854	237	3	(	(	PUNCT
ejpam-854	237	4	(	(	PUNCT
ejpam-854	237	5	1−	1−	NUM
ejpam-854	237	6	n)x1)−	n)x1)−	NOUN
ejpam-854	237	7	n2	n2	PROPN
ejpam-854	237	8	f	f	PROPN
ejpam-854	237	9	(	(	PUNCT
ejpam-854	237	10	2(1−	2(1−	PROPN
ejpam-854	237	11	n)x1	n)x1	NOUN
ejpam-854	237	12	)	)	PUNCT
ejpam-854	237	13	+	+	NUM
ejpam-854	237	14	n2	n2	PROPN
ejpam-854	237	15	f	f	PROPN
ejpam-854	237	16	(	(	PUNCT
ejpam-854	237	17	2nx1	2nx1	NUM
ejpam-854	237	18	)	)	PUNCT
ejpam-854	237	19	−2(1−	−2(1−	PROPN
ejpam-854	237	20	n2	n2	NOUN
ejpam-854	237	21	)	)	PUNCT
ejpam-854	237	22	f	f	NOUN
ejpam-854	237	23	(	(	PUNCT
ejpam-854	237	24	(	(	PUNCT
ejpam-854	237	25	1−	1−	NUM
ejpam-854	237	26	2n)x1	2n)x1	NUM
ejpam-854	237	27	)	)	PUNCT
ejpam-854	237	28	,	,	PUNCT
ejpam-854	237	29	.	.	PUNCT
ejpam-854	237	30	.	.	PUNCT
ejpam-854	237	31	.	.	PUNCT
ejpam-854	238	1	,	,	PUNCT
ejpam-854	238	2	f	f	X
ejpam-854	238	3	(	(	PUNCT
ejpam-854	238	4	(	(	PUNCT
ejpam-854	238	5	1−	1−	NUM
ejpam-854	238	6	3n)xk	3n)xk	NUM
ejpam-854	238	7	)	)	PUNCT
ejpam-854	239	1	+	+	NUM
ejpam-854	239	2	f	f	X
ejpam-854	239	3	(	(	PUNCT
ejpam-854	239	4	(	(	PUNCT
ejpam-854	239	5	1−	1−	NUM
ejpam-854	239	6	n)xk	n)xk	PROPN
ejpam-854	239	7	)	)	PUNCT
ejpam-854	239	8	−n2	−n2	PROPN
ejpam-854	239	9	f	f	X
ejpam-854	239	10	(	(	PUNCT
ejpam-854	239	11	2(1−	2(1−	NUM
ejpam-854	239	12	n)xk	n)xk	PROPN
ejpam-854	239	13	)	)	PUNCT
ejpam-854	239	14	+	+	SYM
ejpam-854	239	15	n2	n2	PROPN
ejpam-854	239	16	f	f	PROPN
ejpam-854	239	17	(	(	PUNCT
ejpam-854	239	18	2nxk)−	2nxk)−	PROPN
ejpam-854	239	19	2(1−	2(1−	NUM
ejpam-854	239	20	n2	n2	NOUN
ejpam-854	239	21	)	)	PUNCT
ejpam-854	239	22	f	f	NOUN
ejpam-854	239	23	(	(	PUNCT
ejpam-854	239	24	(	(	PUNCT
ejpam-854	239	25	1−	1−	NUM
ejpam-854	239	26	2n)xk))‖k	2n)xk))‖k	NUM
ejpam-854	239	27	≤	≤	ADJ
ejpam-854	239	28	ǫ	ǫ	NOUN
ejpam-854	239	29	.	.	PUNCT
ejpam-854	239	30	(	(	PUNCT
ejpam-854	239	31	18	18	NUM
ejpam-854	239	32	)	)	PUNCT
ejpam-854	239	33	by	by	ADP
ejpam-854	239	34	(	(	PUNCT
ejpam-854	239	35	17	17	NUM
ejpam-854	239	36	)	)	PUNCT
ejpam-854	239	37	and	and	CCONJ
ejpam-854	239	38	(	(	PUNCT
ejpam-854	239	39	18	18	NUM
ejpam-854	239	40	)	)	PUNCT
ejpam-854	239	41	,	,	PUNCT
ejpam-854	239	42	we	we	PRON
ejpam-854	239	43	have	have	VERB
ejpam-854	239	44	sup	sup	NOUN
ejpam-854	239	45	k∈n	k∈n	PROPN
ejpam-854	239	46	‖	‖	PROPN
ejpam-854	239	47	(	(	PUNCT
ejpam-854	239	48	f	f	X
ejpam-854	239	49	(	(	PUNCT
ejpam-854	239	50	(	(	PUNCT
ejpam-854	239	51	1	1	NUM
ejpam-854	239	52	+	+	NUM
ejpam-854	239	53	3n)x1	3n)x1	NUM
ejpam-854	239	54	)	)	PUNCT
ejpam-854	240	1	+	+	NUM
ejpam-854	240	2	f	f	X
ejpam-854	240	3	(	(	PUNCT
ejpam-854	240	4	(	(	PUNCT
ejpam-854	240	5	1−	1−	NUM
ejpam-854	240	6	3n)x1	3n)x1	NUM
ejpam-854	240	7	)	)	PUNCT
ejpam-854	241	1	+	+	NUM
ejpam-854	241	2	f	f	X
ejpam-854	241	3	(	(	PUNCT
ejpam-854	241	4	(	(	PUNCT
ejpam-854	241	5	1	1	NUM
ejpam-854	241	6	+	+	NUM
ejpam-854	241	7	n)x1	n)x1	NOUN
ejpam-854	241	8	)	)	PUNCT
ejpam-854	242	1	+	+	NUM
ejpam-854	242	2	f	f	X
ejpam-854	242	3	(	(	PUNCT
ejpam-854	242	4	(	(	PUNCT
ejpam-854	242	5	1−	1−	NUM
ejpam-854	242	6	n)x1	n)x1	NOUN
ejpam-854	242	7	)	)	PUNCT
ejpam-854	242	8	−n2	−n2	PROPN
ejpam-854	242	9	f	f	NOUN
ejpam-854	242	10	(	(	PUNCT
ejpam-854	242	11	2(1	2(1	NUM
ejpam-854	242	12	+	+	CCONJ
ejpam-854	242	13	n)x1)−	n)x1)−	ADJ
ejpam-854	242	14	n2	n2	PROPN
ejpam-854	242	15	f	f	PROPN
ejpam-854	242	16	(	(	PUNCT
ejpam-854	242	17	2(1−	2(1−	NUM
ejpam-854	242	18	n)x1)−	n)x1)−	NOUN
ejpam-854	242	19	2(1−	2(1−	NUM
ejpam-854	242	20	n2	n2	ADJ
ejpam-854	242	21	)	)	PUNCT
ejpam-854	242	22	f	f	NOUN
ejpam-854	242	23	(	(	PUNCT
ejpam-854	242	24	(	(	PUNCT
ejpam-854	242	25	1	1	NUM
ejpam-854	242	26	+	+	NUM
ejpam-854	242	27	2n)x1	2n)x1	NUM
ejpam-854	242	28	)	)	PUNCT
ejpam-854	242	29	−2(1−	−2(1−	PROPN
ejpam-854	242	30	n2	n2	NOUN
ejpam-854	242	31	)	)	PUNCT
ejpam-854	242	32	f	f	NOUN
ejpam-854	242	33	(	(	PUNCT
ejpam-854	242	34	(	(	PUNCT
ejpam-854	242	35	1−	1−	NUM
ejpam-854	242	36	2n)x1	2n)x1	NUM
ejpam-854	242	37	)	)	PUNCT
ejpam-854	242	38	,	,	PUNCT
ejpam-854	242	39	.	.	PUNCT
ejpam-854	242	40	.	.	PUNCT
ejpam-854	242	41	.	.	PUNCT
ejpam-854	243	1	,	,	PUNCT
ejpam-854	243	2	f	f	X
ejpam-854	243	3	(	(	PUNCT
ejpam-854	243	4	(	(	PUNCT
ejpam-854	243	5	1	1	NUM
ejpam-854	243	6	+	+	NUM
ejpam-854	243	7	3n)xk	3n)xk	NUM
ejpam-854	243	8	)	)	PUNCT
ejpam-854	244	1	+	+	NUM
ejpam-854	244	2	f	f	X
ejpam-854	244	3	(	(	PUNCT
ejpam-854	244	4	(	(	PUNCT
ejpam-854	244	5	1−	1−	NUM
ejpam-854	244	6	3n)xk	3n)xk	NUM
ejpam-854	244	7	)	)	PUNCT
ejpam-854	245	1	+	+	NUM
ejpam-854	245	2	f	f	X
ejpam-854	245	3	(	(	PUNCT
ejpam-854	245	4	(	(	PUNCT
ejpam-854	245	5	1	1	NUM
ejpam-854	245	6	+	+	CCONJ
ejpam-854	245	7	n)xk	n)xk	ADJ
ejpam-854	245	8	)	)	PUNCT
ejpam-854	246	1	+	+	NUM
ejpam-854	246	2	f	f	X
ejpam-854	246	3	(	(	PUNCT
ejpam-854	246	4	(	(	PUNCT
ejpam-854	246	5	1−	1−	NUM
ejpam-854	246	6	n)xk)−	n)xk)−	PROPN
ejpam-854	246	7	n2	n2	ADJ
ejpam-854	246	8	f	f	PROPN
ejpam-854	246	9	(	(	PUNCT
ejpam-854	246	10	2(1	2(1	NUM
ejpam-854	246	11	+	+	CCONJ
ejpam-854	246	12	n)xk)−	n)xk)−	PROPN
ejpam-854	246	13	n2	n2	ADJ
ejpam-854	246	14	f	f	PROPN
ejpam-854	246	15	(	(	PUNCT
ejpam-854	246	16	2(1−	2(1−	NUM
ejpam-854	246	17	n)xk	n)xk	PROPN
ejpam-854	246	18	)	)	PUNCT
ejpam-854	246	19	−2(1−	−2(1−	PROPN
ejpam-854	246	20	n2	n2	NOUN
ejpam-854	246	21	)	)	PUNCT
ejpam-854	246	22	f	f	NOUN
ejpam-854	246	23	(	(	PUNCT
ejpam-854	246	24	(	(	PUNCT
ejpam-854	246	25	1	1	NUM
ejpam-854	246	26	+	+	NUM
ejpam-854	246	27	2n)xk)−	2n)xk)−	NUM
ejpam-854	246	28	2(1−	2(1−	NUM
ejpam-854	246	29	n2	n2	ADJ
ejpam-854	246	30	)	)	PUNCT
ejpam-854	246	31	f	f	NOUN
ejpam-854	246	32	(	(	PUNCT
ejpam-854	246	33	(	(	PUNCT
ejpam-854	246	34	1−	1−	NUM
ejpam-854	246	35	2n)xk))‖k	2n)xk))‖k	NUM
ejpam-854	246	36	≤	≤	ADJ
ejpam-854	246	37	2ǫ	2ǫ	NOUN
ejpam-854	246	38	.	.	PUNCT
ejpam-854	247	1	(	(	PUNCT
ejpam-854	247	2	19	19	NUM
ejpam-854	247	3	)	)	PUNCT
ejpam-854	247	4	by	by	ADP
ejpam-854	247	5	(	(	PUNCT
ejpam-854	247	6	9	9	NUM
ejpam-854	247	7	)	)	PUNCT
ejpam-854	247	8	,	,	PUNCT
ejpam-854	247	9	(	(	PUNCT
ejpam-854	247	10	11	11	NUM
ejpam-854	247	11	)	)	PUNCT
ejpam-854	247	12	,	,	PUNCT
ejpam-854	247	13	(	(	PUNCT
ejpam-854	247	14	12	12	NUM
ejpam-854	247	15	)	)	PUNCT
ejpam-854	247	16	and	and	CCONJ
ejpam-854	247	17	(	(	PUNCT
ejpam-854	247	18	16	16	NUM
ejpam-854	247	19	)	)	PUNCT
ejpam-854	247	20	,	,	PUNCT
ejpam-854	247	21	we	we	PRON
ejpam-854	247	22	get	get	VERB
ejpam-854	247	23	sup	sup	NOUN
ejpam-854	247	24	k∈n	k∈n	PROPN
ejpam-854	247	25	‖	‖	PROPN
ejpam-854	247	26	(	(	PUNCT
ejpam-854	247	27	f	f	PROPN
ejpam-854	247	28	(	(	PUNCT
ejpam-854	247	29	3x1)−	3x1)−	NOUN
ejpam-854	247	30	4	4	NUM
ejpam-854	247	31	f	f	NOUN
ejpam-854	247	32	(	(	PUNCT
ejpam-854	247	33	2x1	2x1	NUM
ejpam-854	247	34	)	)	PUNCT
ejpam-854	248	1	+	+	CCONJ
ejpam-854	248	2	5	5	NUM
ejpam-854	248	3	f	f	NOUN
ejpam-854	248	4	(	(	PUNCT
ejpam-854	248	5	x1	x1	PROPN
ejpam-854	248	6	)	)	PUNCT
ejpam-854	248	7	,	,	PUNCT
ejpam-854	248	8	.	.	PUNCT
ejpam-854	248	9	.	.	PUNCT
ejpam-854	248	10	.	.	PUNCT
ejpam-854	249	1	,	,	PUNCT
ejpam-854	249	2	f	f	PROPN
ejpam-854	249	3	(	(	PUNCT
ejpam-854	249	4	3xk)−	3xk)−	PROPN
ejpam-854	249	5	4	4	NUM
ejpam-854	249	6	f	f	NOUN
ejpam-854	249	7	(	(	PUNCT
ejpam-854	249	8	2xk	2xk	NOUN
ejpam-854	249	9	)	)	PUNCT
ejpam-854	250	1	+	+	CCONJ
ejpam-854	250	2	5	5	NUM
ejpam-854	250	3	f	f	NOUN
ejpam-854	250	4	(	(	PUNCT
ejpam-854	250	5	xk))‖k	xk))‖k	PROPN
ejpam-854	250	6	≤	≤	PROPN
ejpam-854	250	7	3n2	3n2	NUM
ejpam-854	250	8	+	+	CCONJ
ejpam-854	250	9	1	1	NUM
ejpam-854	250	10	n4	n4	PROPN
ejpam-854	250	11	−	−	PROPN
ejpam-854	250	12	n2	n2	PROPN
ejpam-854	250	13	ǫ	ǫ	X
ejpam-854	250	14	.	.	PUNCT
ejpam-854	250	15	(	(	PUNCT
ejpam-854	250	16	20	20	NUM
ejpam-854	250	17	)	)	PUNCT
ejpam-854	250	18	by	by	ADP
ejpam-854	250	19	(	(	PUNCT
ejpam-854	250	20	9	9	NUM
ejpam-854	250	21	)	)	PUNCT
ejpam-854	250	22	,	,	PUNCT
ejpam-854	250	23	(	(	PUNCT
ejpam-854	250	24	10	10	NUM
ejpam-854	250	25	)	)	PUNCT
ejpam-854	250	26	,	,	PUNCT
ejpam-854	250	27	(	(	PUNCT
ejpam-854	250	28	12	12	NUM
ejpam-854	250	29	)	)	PUNCT
ejpam-854	250	30	,	,	PUNCT
ejpam-854	250	31	(	(	PUNCT
ejpam-854	250	32	13	13	NUM
ejpam-854	250	33	)	)	PUNCT
ejpam-854	250	34	and	and	CCONJ
ejpam-854	250	35	(	(	PUNCT
ejpam-854	250	36	19	19	NUM
ejpam-854	250	37	)	)	PUNCT
ejpam-854	250	38	,	,	PUNCT
ejpam-854	250	39	we	we	PRON
ejpam-854	250	40	get	get	VERB
ejpam-854	250	41	sup	sup	NOUN
ejpam-854	250	42	k∈n	k∈n	PROPN
ejpam-854	250	43	‖	‖	PROPN
ejpam-854	250	44	(	(	PUNCT
ejpam-854	250	45	f	f	X
ejpam-854	250	46	(	(	PUNCT
ejpam-854	250	47	4x1)−	4x1)−	NOUN
ejpam-854	250	48	2	2	NUM
ejpam-854	250	49	f	f	NOUN
ejpam-854	250	50	(	(	PUNCT
ejpam-854	250	51	3x1)−	3x1)−	NOUN
ejpam-854	250	52	2	2	NUM
ejpam-854	250	53	f	f	NOUN
ejpam-854	250	54	(	(	PUNCT
ejpam-854	250	55	2x1	2x1	NUM
ejpam-854	250	56	)	)	PUNCT
ejpam-854	251	1	+	+	CCONJ
ejpam-854	251	2	6	6	NUM
ejpam-854	251	3	f	f	NOUN
ejpam-854	251	4	(	(	PUNCT
ejpam-854	251	5	x1	x1	PROPN
ejpam-854	251	6	)	)	PUNCT
ejpam-854	251	7	,	,	PUNCT
ejpam-854	251	8	.	.	PUNCT
ejpam-854	251	9	.	.	PUNCT
ejpam-854	251	10	.	.	PUNCT
ejpam-854	252	1	,	,	PUNCT
ejpam-854	252	2	f	f	PROPN
ejpam-854	252	3	(	(	PUNCT
ejpam-854	252	4	4xk)−	4xk)−	PROPN
ejpam-854	252	5	2	2	NUM
ejpam-854	252	6	f	f	NOUN
ejpam-854	252	7	(	(	PUNCT
ejpam-854	252	8	3xk	3xk	ADJ
ejpam-854	252	9	)	)	PUNCT
ejpam-854	252	10	−2	−2	PROPN
ejpam-854	252	11	f	f	PROPN
ejpam-854	252	12	(	(	PUNCT
ejpam-854	252	13	2xk	2xk	NOUN
ejpam-854	252	14	)	)	PUNCT
ejpam-854	253	1	+	+	CCONJ
ejpam-854	253	2	6	6	NUM
ejpam-854	253	3	f	f	NOUN
ejpam-854	253	4	(	(	PUNCT
ejpam-854	253	5	xk))‖k	xk))‖k	PROPN
ejpam-854	253	6	≤	≤	PROPN
ejpam-854	253	7	3n2	3n2	NUM
ejpam-854	253	8	+	+	SYM
ejpam-854	253	9	2	2	NUM
ejpam-854	253	10	n4−n2	n4−n2	NUM
ejpam-854	253	11	ǫ	ǫ	NOUN
ejpam-854	253	12	.	.	PUNCT
ejpam-854	254	1	(	(	PUNCT
ejpam-854	254	2	21	21	NUM
ejpam-854	254	3	)	)	PUNCT
ejpam-854	254	4	by	by	ADP
ejpam-854	254	5	(	(	PUNCT
ejpam-854	254	6	20	20	NUM
ejpam-854	254	7	)	)	PUNCT
ejpam-854	254	8	and	and	CCONJ
ejpam-854	254	9	(	(	PUNCT
ejpam-854	254	10	21	21	NUM
ejpam-854	254	11	)	)	PUNCT
ejpam-854	254	12	,	,	PUNCT
ejpam-854	254	13	we	we	PRON
ejpam-854	254	14	get	get	VERB
ejpam-854	254	15	sup	sup	NOUN
ejpam-854	254	16	k∈n	k∈n	PROPN
ejpam-854	254	17	‖	‖	PROPN
ejpam-854	254	18	(	(	PUNCT
ejpam-854	254	19	f	f	X
ejpam-854	254	20	(	(	PUNCT
ejpam-854	254	21	4x1)−	4x1)−	NOUN
ejpam-854	254	22	10	10	NUM
ejpam-854	254	23	f	f	NOUN
ejpam-854	254	24	(	(	PUNCT
ejpam-854	254	25	2x1)+	2x1)+	NUM
ejpam-854	254	26	16	16	NUM
ejpam-854	254	27	f	f	X
ejpam-854	254	28	(	(	PUNCT
ejpam-854	254	29	x1	x1	PROPN
ejpam-854	254	30	)	)	PUNCT
ejpam-854	254	31	,	,	PUNCT
ejpam-854	254	32	.	.	PUNCT
ejpam-854	254	33	.	.	PUNCT
ejpam-854	254	34	.	.	PUNCT
ejpam-854	255	1	,	,	PUNCT
ejpam-854	255	2	f	f	PROPN
ejpam-854	255	3	(	(	PUNCT
ejpam-854	255	4	4xk)−	4xk)−	PROPN
ejpam-854	255	5	10	10	NUM
ejpam-854	255	6	f	f	NOUN
ejpam-854	255	7	(	(	PUNCT
ejpam-854	255	8	2xk)+	2xk)+	NUM
ejpam-854	255	9	16	16	NUM
ejpam-854	255	10	f	f	NOUN
ejpam-854	255	11	(	(	PUNCT
ejpam-854	255	12	xk))‖k	xk))‖k	PROPN
ejpam-854	255	13	≤	≤	PROPN
ejpam-854	255	14	9n2	9n2	NUM
ejpam-854	255	15	+	+	SYM
ejpam-854	255	16	4	4	NUM
ejpam-854	255	17	n4	n4	PROPN
ejpam-854	255	18	−	−	PROPN
ejpam-854	255	19	n2	n2	PROPN
ejpam-854	255	20	ǫ	ǫ	X
ejpam-854	255	21	.	.	PUNCT
ejpam-854	256	1	(	(	PUNCT
ejpam-854	256	2	22	22	NUM
ejpam-854	256	3	)	)	PUNCT
ejpam-854	256	4	consider	consider	VERB
ejpam-854	256	5	the	the	DET
ejpam-854	256	6	set	set	NOUN
ejpam-854	256	7	ω	ω	NOUN
ejpam-854	256	8	:	:	PUNCT
ejpam-854	256	9	=	=	SYM
ejpam-854	256	10	{	{	PUNCT
ejpam-854	256	11	g	g	NOUN
ejpam-854	256	12	|	|	ADV
ejpam-854	256	13	g	g	PROPN
ejpam-854	256	14	:	:	PUNCT
ejpam-854	256	15	e→	e→	PROPN
ejpam-854	256	16	f	f	PROPN
ejpam-854	256	17	,	,	PUNCT
ejpam-854	256	18	g(0	g(0	PROPN
ejpam-854	256	19	)	)	PUNCT
ejpam-854	256	20	=	=	SYM
ejpam-854	256	21	0	0	NUM
ejpam-854	256	22	}	}	PUNCT
ejpam-854	256	23	and	and	CCONJ
ejpam-854	256	24	introduce	introduce	VERB
ejpam-854	256	25	the	the	DET
ejpam-854	256	26	generalized	generalize	VERB
ejpam-854	256	27	metric	metric	NOUN
ejpam-854	256	28	on	on	ADP
ejpam-854	256	29	ω	ω	PROPN
ejpam-854	256	30	,	,	PUNCT
ejpam-854	256	31	d(g	d(g	PROPN
ejpam-854	256	32	,	,	PUNCT
ejpam-854	256	33	h	h	NOUN
ejpam-854	256	34	)	)	PUNCT
ejpam-854	257	1	=	=	SYM
ejpam-854	257	2	inf{α	inf{α	NOUN
ejpam-854	257	3	>	>	X
ejpam-854	257	4	0|	0|	NUM
ejpam-854	257	5	sup	sup	NOUN
ejpam-854	257	6	k∈n	k∈n	PROPN
ejpam-854	257	7	‖(g(x1)−	‖(g(x1)−	PROPN
ejpam-854	257	8	h(x1	h(x1	PROPN
ejpam-854	257	9	)	)	PUNCT
ejpam-854	257	10	,	,	PUNCT
ejpam-854	257	11	.	.	PUNCT
ejpam-854	257	12	.	.	PUNCT
ejpam-854	258	1	.	.	PUNCT
ejpam-854	259	1	,	,	PUNCT
ejpam-854	259	2	g(xk)−	g(xk)−	NOUN
ejpam-854	259	3	h(xk))‖k	h(xk))‖k	NOUN
ejpam-854	259	4	≤	≤	NUM
ejpam-854	259	5	α,∀x1	α,∀x1	NOUN
ejpam-854	259	6	,	,	PUNCT
ejpam-854	259	7	.	.	PUNCT
ejpam-854	259	8	.	.	PUNCT
ejpam-854	259	9	.	.	PUNCT
ejpam-854	260	1	,	,	PUNCT
ejpam-854	260	2	xk	xk	PROPN
ejpam-854	260	3	∈	∈	PROPN
ejpam-854	260	4	e	e	PROPN
ejpam-854	260	5	,	,	PUNCT
ejpam-854	260	6	k	k	PROPN
ejpam-854	260	7	∈	∈	PROPN
ejpam-854	260	8	n	n	CCONJ
ejpam-854	260	9	}	}	PUNCT
ejpam-854	260	10	.	.	PUNCT
ejpam-854	261	1	it	it	PRON
ejpam-854	261	2	is	be	AUX
ejpam-854	261	3	easy	easy	ADJ
ejpam-854	261	4	to	to	PART
ejpam-854	261	5	show	show	VERB
ejpam-854	261	6	that	that	SCONJ
ejpam-854	261	7	(	(	PUNCT
ejpam-854	261	8	ω	ω	NOUN
ejpam-854	261	9	,	,	PUNCT
ejpam-854	261	10	d	d	NOUN
ejpam-854	261	11	)	)	PUNCT
ejpam-854	261	12	is	be	AUX
ejpam-854	261	13	a	a	DET
ejpam-854	261	14	generalized	generalize	VERB
ejpam-854	261	15	complete	complete	ADJ
ejpam-854	261	16	metric	metric	ADJ
ejpam-854	261	17	space	space	NOUN
ejpam-854	261	18	[	[	X
ejpam-854	261	19	see	see	VERB
ejpam-854	261	20	20	20	NUM
ejpam-854	261	21	,	,	PUNCT
ejpam-854	261	22	lemma	lemma	PROPN
ejpam-854	261	23	2.1	2.1	NUM
ejpam-854	261	24	]	]	PUNCT
ejpam-854	261	25	.	.	PUNCT
ejpam-854	262	1	define	define	VERB
ejpam-854	262	2	j	j	PROPN
ejpam-854	262	3	:	:	PUNCT
ejpam-854	262	4	ω	ω	PROPN
ejpam-854	262	5	→	→	SYM
ejpam-854	262	6	ω	ω	PROPN
ejpam-854	262	7	by	by	ADP
ejpam-854	262	8	j	j	PROPN
ejpam-854	262	9	g(x	g(x	NOUN
ejpam-854	262	10	)	)	PUNCT
ejpam-854	263	1	=	=	SYM
ejpam-854	263	2	g(2x)/2	g(2x)/2	NOUN
ejpam-854	263	3	for	for	ADP
ejpam-854	263	4	all	all	PRON
ejpam-854	263	5	x	x	SYM
ejpam-854	263	6	∈	∈	PROPN
ejpam-854	263	7	e.	e.	PROPN
ejpam-854	263	8	let	let	VERB
ejpam-854	263	9	g	g	NOUN
ejpam-854	263	10	,	,	PUNCT
ejpam-854	263	11	h	h	PROPN
ejpam-854	263	12	∈	∈	PROPN
ejpam-854	263	13	ω	ω	PROPN
ejpam-854	263	14	be	be	AUX
ejpam-854	263	15	given	give	VERB
ejpam-854	263	16	such	such	ADJ
ejpam-854	263	17	that	that	DET
ejpam-854	263	18	d(g	d(g	PROPN
ejpam-854	263	19	,	,	PUNCT
ejpam-854	263	20	h	h	NOUN
ejpam-854	263	21	)	)	PUNCT
ejpam-854	263	22	<	<	X
ejpam-854	263	23	β	β	X
ejpam-854	263	24	,	,	PUNCT
ejpam-854	263	25	by	by	ADP
ejpam-854	263	26	the	the	DET
ejpam-854	263	27	definition	definition	NOUN
ejpam-854	263	28	,	,	PUNCT
ejpam-854	263	29	sup	sup	NOUN
ejpam-854	263	30	k∈n	k∈n	PROPN
ejpam-854	263	31	‖(g(x1)−	‖(g(x1)−	PROPN
ejpam-854	263	32	h(x1	h(x1	PROPN
ejpam-854	263	33	)	)	PUNCT
ejpam-854	263	34	,	,	PUNCT
ejpam-854	263	35	.	.	PUNCT
ejpam-854	263	36	.	.	PUNCT
ejpam-854	263	37	.	.	PUNCT
ejpam-854	264	1	,	,	PUNCT
ejpam-854	264	2	g(xk)−	g(xk)−	NOUN
ejpam-854	264	3	h(xk))‖k	h(xk))‖k	NOUN
ejpam-854	264	4	≤	≤	NUM
ejpam-854	264	5	β	β	X
ejpam-854	264	6	for	for	ADP
ejpam-854	264	7	all	all	DET
ejpam-854	264	8	x1	x1	PROPN
ejpam-854	264	9	,	,	PUNCT
ejpam-854	264	10	.	.	PUNCT
ejpam-854	264	11	.	.	PUNCT
ejpam-854	264	12	.	.	PUNCT
ejpam-854	265	1	,	,	PUNCT
ejpam-854	265	2	xk	xk	PROPN
ejpam-854	265	3	∈	∈	PROPN
ejpam-854	265	4	e	e	PROPN
ejpam-854	265	5	,	,	PUNCT
ejpam-854	265	6	k	k	PROPN
ejpam-854	265	7	∈	∈	PROPN
ejpam-854	265	8	n.	n.	NOUN
ejpam-854	265	9	hence	hence	ADV
ejpam-854	265	10	sup	sup	PROPN
ejpam-854	265	11	k∈n	k∈n	PROPN
ejpam-854	265	12	‖(j	‖(j	PROPN
ejpam-854	265	13	g(x1)−	g(x1)−	PROPN
ejpam-854	265	14	jh(x1	jh(x1	PROPN
ejpam-854	265	15	)	)	PUNCT
ejpam-854	265	16	,	,	PUNCT
ejpam-854	265	17	.	.	PUNCT
ejpam-854	265	18	.	.	PUNCT
ejpam-854	265	19	.	.	PUNCT
ejpam-854	266	1	,	,	PUNCT
ejpam-854	266	2	j	j	PROPN
ejpam-854	266	3	g(xk)−	g(xk)−	NOUN
ejpam-854	266	4	jh(xk))‖k	jh(xk))‖k	PROPN
ejpam-854	266	5	≤	≤	NOUN
ejpam-854	266	6	1	1	NUM
ejpam-854	266	7	2	2	NUM
ejpam-854	266	8	sup	sup	NOUN
ejpam-854	266	9	k∈n	k∈n	NOUN
ejpam-854	266	10	‖(g(2x1)−	‖(g(2x1)−	PROPN
ejpam-854	267	1	h(2x1	h(2x1	NUM
ejpam-854	267	2	)	)	PUNCT
ejpam-854	267	3	,	,	PUNCT
ejpam-854	267	4	.	.	PUNCT
ejpam-854	267	5	.	.	PUNCT
ejpam-854	268	1	.	.	PUNCT
ejpam-854	269	1	,	,	PUNCT
ejpam-854	269	2	g(2xk)−	g(2xk)−	PROPN
ejpam-854	269	3	h(2xk))‖k	h(2xk))‖k	VERB
ejpam-854	269	4	≤	≤	NUM
ejpam-854	269	5	β	β	X
ejpam-854	269	6	2	2	NUM
ejpam-854	269	7	t.	t.	NOUN
ejpam-854	269	8	xu	xu	PROPN
ejpam-854	269	9	,	,	PUNCT
ejpam-854	269	10	j.	j.	PROPN
ejpam-854	269	11	rassias	rassias	PROPN
ejpam-854	269	12	,	,	PUNCT
ejpam-854	269	13	w.	w.	PROPN
ejpam-854	269	14	xu	xu	PROPN
ejpam-854	269	15	/	/	SYM
ejpam-854	269	16	eur	eur	PROPN
ejpam-854	269	17	.	.	PUNCT
ejpam-854	270	1	j.	j.	PROPN
ejpam-854	270	2	pure	pure	PROPN
ejpam-854	270	3	appl	appl	PROPN
ejpam-854	270	4	.	.	PROPN
ejpam-854	270	5	math	math	PROPN
ejpam-854	270	6	,	,	PUNCT
ejpam-854	270	7	3	3	NUM
ejpam-854	270	8	(	(	PUNCT
ejpam-854	270	9	2010	2010	NUM
ejpam-854	270	10	)	)	PUNCT
ejpam-854	270	11	,	,	PUNCT
ejpam-854	270	12	1032	1032	NUM
ejpam-854	270	13	-	-	SYM
ejpam-854	270	14	1047	1047	NUM
ejpam-854	270	15	1039	1039	NUM
ejpam-854	270	16	for	for	ADP
ejpam-854	270	17	all	all	DET
ejpam-854	270	18	x1	x1	PROPN
ejpam-854	270	19	,	,	PUNCT
ejpam-854	270	20	.	.	PUNCT
ejpam-854	270	21	.	.	PUNCT
ejpam-854	270	22	.	.	PUNCT
ejpam-854	271	1	,	,	PUNCT
ejpam-854	271	2	xk	xk	PROPN
ejpam-854	271	3	∈	∈	PROPN
ejpam-854	271	4	e	e	PROPN
ejpam-854	271	5	,	,	PUNCT
ejpam-854	271	6	k	k	PROPN
ejpam-854	271	7	∈	∈	PROPN
ejpam-854	271	8	n.	n.	NOUN
ejpam-854	271	9	by	by	ADP
ejpam-854	271	10	definition	definition	NOUN
ejpam-854	271	11	,	,	PUNCT
ejpam-854	271	12	d(j	d(j	PROPN
ejpam-854	271	13	g	g	PROPN
ejpam-854	271	14	,	,	PUNCT
ejpam-854	271	15	jh	jh	PROPN
ejpam-854	271	16	)	)	PUNCT
ejpam-854	271	17	≤	≤	NOUN
ejpam-854	271	18	β/2	β/2	NOUN
ejpam-854	271	19	.	.	PUNCT
ejpam-854	272	1	therefore	therefore	ADV
ejpam-854	272	2	,	,	PUNCT
ejpam-854	272	3	d(j	d(j	PROPN
ejpam-854	272	4	g	g	PROPN
ejpam-854	272	5	,	,	PUNCT
ejpam-854	272	6	jh	jh	PROPN
ejpam-854	272	7	)	)	PUNCT
ejpam-854	272	8	≤	≤	NOUN
ejpam-854	272	9	1	1	NUM
ejpam-854	272	10	2	2	NUM
ejpam-854	272	11	d(g	d(g	PROPN
ejpam-854	272	12	,	,	PUNCT
ejpam-854	272	13	h	h	NOUN
ejpam-854	272	14	)	)	PUNCT
ejpam-854	272	15	for	for	ADP
ejpam-854	272	16	all	all	DET
ejpam-854	272	17	g	g	NOUN
ejpam-854	272	18	,	,	PUNCT
ejpam-854	272	19	h	h	NOUN
ejpam-854	272	20	∈	∈	PROPN
ejpam-854	272	21	ω	ω	PROPN
ejpam-854	272	22	.	.	PUNCT
ejpam-854	273	1	this	this	PRON
ejpam-854	273	2	means	mean	VERB
ejpam-854	273	3	that	that	SCONJ
ejpam-854	273	4	j	j	PROPN
ejpam-854	273	5	is	be	AUX
ejpam-854	273	6	a	a	DET
ejpam-854	273	7	strictly	strictly	ADV
ejpam-854	273	8	contractive	contractive	ADJ
ejpam-854	273	9	self	self	NOUN
ejpam-854	273	10	-	-	PUNCT
ejpam-854	273	11	mapping	mapping	NOUN
ejpam-854	273	12	of	of	ADP
ejpam-854	273	13	ω	ω	PROPN
ejpam-854	273	14	with	with	ADP
ejpam-854	273	15	lipschitz	lipschitz	NOUN
ejpam-854	273	16	constant	constant	ADJ
ejpam-854	273	17	1/2	1/2	NUM
ejpam-854	273	18	.	.	PUNCT
ejpam-854	274	1	now	now	ADV
ejpam-854	274	2	,	,	PUNCT
ejpam-854	274	3	let	let	VERB
ejpam-854	274	4	f̃	f̃	PROPN
ejpam-854	274	5	:	:	PUNCT
ejpam-854	274	6	e	e	X
ejpam-854	274	7	→	→	PUNCT
ejpam-854	274	8	f	f	X
ejpam-854	274	9	be	be	AUX
ejpam-854	274	10	the	the	DET
ejpam-854	274	11	mapping	mapping	NOUN
ejpam-854	274	12	defined	define	VERB
ejpam-854	274	13	by	by	ADP
ejpam-854	274	14	f̃	f̃	PROPN
ejpam-854	274	15	(	(	PUNCT
ejpam-854	274	16	x	x	NOUN
ejpam-854	274	17	)	)	PUNCT
ejpam-854	274	18	:	:	PUNCT
ejpam-854	275	1	=	=	SYM
ejpam-854	275	2	f	f	X
ejpam-854	275	3	(	(	PUNCT
ejpam-854	275	4	2x)−	2x)−	NUM
ejpam-854	275	5	8	8	NUM
ejpam-854	275	6	f	f	PROPN
ejpam-854	275	7	(	(	PUNCT
ejpam-854	275	8	x	x	NOUN
ejpam-854	275	9	)	)	PUNCT
ejpam-854	275	10	for	for	ADP
ejpam-854	275	11	each	each	DET
ejpam-854	275	12	x	x	SYM
ejpam-854	275	13	∈	∈	PROPN
ejpam-854	275	14	e.	e.	PROPN
ejpam-854	275	15	by	by	ADP
ejpam-854	275	16	(	(	PUNCT
ejpam-854	275	17	22	22	NUM
ejpam-854	275	18	)	)	PUNCT
ejpam-854	275	19	,	,	PUNCT
ejpam-854	275	20	we	we	PRON
ejpam-854	275	21	get	get	VERB
ejpam-854	275	22	sup	sup	NOUN
ejpam-854	275	23	k∈n	k∈n	PROPN
ejpam-854	275	24	‖	‖	PROPN
ejpam-854	275	25	(	(	PUNCT
ejpam-854	275	26	f̃	f̃	PROPN
ejpam-854	275	27	(	(	PUNCT
ejpam-854	275	28	2x1)−	2x1)−	NUM
ejpam-854	275	29	2	2	NUM
ejpam-854	275	30	f̃	f̃	PROPN
ejpam-854	275	31	(	(	PUNCT
ejpam-854	275	32	x1	x1	PROPN
ejpam-854	275	33	)	)	PUNCT
ejpam-854	275	34	,	,	PUNCT
ejpam-854	275	35	.	.	PUNCT
ejpam-854	275	36	.	.	PUNCT
ejpam-854	275	37	.	.	PUNCT
ejpam-854	275	38	,	,	PUNCT
ejpam-854	275	39	f̃	f̃	PROPN
ejpam-854	275	40	(	(	PUNCT
ejpam-854	275	41	2xk)−	2xk)−	PROPN
ejpam-854	275	42	2	2	NUM
ejpam-854	275	43	f̃	f̃	PROPN
ejpam-854	275	44	(	(	PUNCT
ejpam-854	275	45	xk))‖k	xk))‖k	PROPN
ejpam-854	275	46	≤	≤	PROPN
ejpam-854	275	47	9n2	9n2	NUM
ejpam-854	275	48	+	+	SYM
ejpam-854	275	49	4	4	NUM
ejpam-854	275	50	n4	n4	PROPN
ejpam-854	275	51	−	−	PROPN
ejpam-854	275	52	n2	n2	PROPN
ejpam-854	275	53	ǫ	ǫ	X
ejpam-854	275	54	.	.	PUNCT
ejpam-854	276	1	(	(	PUNCT
ejpam-854	276	2	23	23	NUM
ejpam-854	276	3	)	)	PUNCT
ejpam-854	276	4	multiplying	multiplying	NOUN
ejpam-854	276	5	(	(	PUNCT
ejpam-854	276	6	23	23	NUM
ejpam-854	276	7	)	)	PUNCT
ejpam-854	276	8	by	by	ADP
ejpam-854	276	9	1/2	1/2	NUM
ejpam-854	276	10	,	,	PUNCT
ejpam-854	276	11	we	we	PRON
ejpam-854	276	12	obtain	obtain	VERB
ejpam-854	276	13	sup	sup	NOUN
ejpam-854	276	14	k∈n	k∈n	NOUN
ejpam-854	276	15	‖(j	‖(j	PROPN
ejpam-854	277	1	f̃	f̃	PROPN
ejpam-854	278	1	(	(	PUNCT
ejpam-854	278	2	x1)−	x1)−	PROPN
ejpam-854	278	3	f̃	f̃	PROPN
ejpam-854	278	4	(	(	PUNCT
ejpam-854	278	5	x1	x1	PROPN
ejpam-854	278	6	)	)	PUNCT
ejpam-854	278	7	,	,	PUNCT
ejpam-854	278	8	.	.	PUNCT
ejpam-854	278	9	.	.	PUNCT
ejpam-854	279	1	.	.	PUNCT
ejpam-854	280	1	,	,	PUNCT
ejpam-854	280	2	j	j	PROPN
ejpam-854	280	3	f̃	f̃	PROPN
ejpam-854	280	4	(	(	PUNCT
ejpam-854	280	5	xk)−	xk)−	PROPN
ejpam-854	280	6	f̃	f̃	PROPN
ejpam-854	280	7	(	(	PUNCT
ejpam-854	280	8	xk))‖k	xk))‖k	PROPN
ejpam-854	280	9	≤	≤	PROPN
ejpam-854	280	10	9n2	9n2	NUM
ejpam-854	280	11	+	+	CCONJ
ejpam-854	280	12	4	4	NUM
ejpam-854	280	13	2(n4−	2(n4−	NUM
ejpam-854	280	14	n2	n2	ADJ
ejpam-854	280	15	)	)	PUNCT
ejpam-854	280	16	ǫ	ǫ	NOUN
ejpam-854	280	17	.	.	PUNCT
ejpam-854	281	1	(	(	PUNCT
ejpam-854	281	2	24	24	NUM
ejpam-854	281	3	)	)	PUNCT
ejpam-854	281	4	then	then	ADV
ejpam-854	281	5	d(j	d(j	PROPN
ejpam-854	281	6	f̃	f̃	PROPN
ejpam-854	281	7	,	,	PUNCT
ejpam-854	281	8	f̃	f̃	PROPN
ejpam-854	281	9	)	)	PUNCT
ejpam-854	281	10	≤	≤	NOUN
ejpam-854	281	11	(	(	PUNCT
ejpam-854	281	12	9n2	9n2	NUM
ejpam-854	281	13	+	+	CCONJ
ejpam-854	281	14	4)/(2(n4−	4)/(2(n4−	NUM
ejpam-854	281	15	n2))ǫ	n2))ǫ	NOUN
ejpam-854	281	16	and	and	CCONJ
ejpam-854	281	17	therefore	therefore	ADV
ejpam-854	281	18	,	,	PUNCT
ejpam-854	281	19	by	by	ADP
ejpam-854	281	20	theorem	theorem	NOUN
ejpam-854	281	21	1	1	NUM
ejpam-854	281	22	,	,	PUNCT
ejpam-854	281	23	j	j	PROPN
ejpam-854	281	24	has	have	VERB
ejpam-854	281	25	a	a	DET
ejpam-854	281	26	unique	unique	ADJ
ejpam-854	281	27	fixed	fix	VERB
ejpam-854	281	28	point	point	NOUN
ejpam-854	281	29	a	a	PRON
ejpam-854	281	30	:	:	PUNCT
ejpam-854	281	31	e→	e→	PROPN
ejpam-854	281	32	f	f	PROPN
ejpam-854	281	33	in	in	ADP
ejpam-854	281	34	the	the	DET
ejpam-854	281	35	set	set	NOUN
ejpam-854	281	36	∆=	∆=	NOUN
ejpam-854	281	37	{	{	PUNCT
ejpam-854	281	38	h	h	NOUN
ejpam-854	281	39	∈	∈	PROPN
ejpam-854	281	40	ω	ω	NOUN
ejpam-854	281	41	:	:	PUNCT
ejpam-854	281	42	d	d	X
ejpam-854	281	43	(	(	PUNCT
ejpam-854	281	44	f̃	f̃	PROPN
ejpam-854	281	45	,	,	PUNCT
ejpam-854	281	46	h	h	NOUN
ejpam-854	281	47	)	)	PUNCT
ejpam-854	281	48	<	<	X
ejpam-854	281	49	∞	∞	NUM
ejpam-854	281	50	}	}	PUNCT
ejpam-854	281	51	.	.	PUNCT
ejpam-854	282	1	this	this	PRON
ejpam-854	282	2	implies	imply	VERB
ejpam-854	282	3	that	that	PRON
ejpam-854	282	4	a(2x	a(2x	VERB
ejpam-854	282	5	)	)	PUNCT
ejpam-854	282	6	=	=	SYM
ejpam-854	282	7	2a(x	2a(x	PROPN
ejpam-854	282	8	)	)	PUNCT
ejpam-854	282	9	and	and	CCONJ
ejpam-854	282	10	a(x	a(x	PROPN
ejpam-854	282	11	)	)	PUNCT
ejpam-854	282	12	=	=	PROPN
ejpam-854	282	13	lim	lim	PROPN
ejpam-854	282	14	m→∞	m→∞	NUM
ejpam-854	282	15	j	j	PROPN
ejpam-854	282	16	m	m	PROPN
ejpam-854	282	17	f̃	f̃	PROPN
ejpam-854	282	18	(	(	PUNCT
ejpam-854	282	19	x	x	NOUN
ejpam-854	282	20	)	)	PUNCT
ejpam-854	282	21	=	=	SYM
ejpam-854	282	22	lim	lim	PROPN
ejpam-854	282	23	m→∞	m→∞	NOUN
ejpam-854	282	24	1	1	NUM
ejpam-854	282	25	2	2	NUM
ejpam-854	282	26	m	m	NOUN
ejpam-854	282	27	f̃	f̃	PROPN
ejpam-854	282	28	(	(	PUNCT
ejpam-854	282	29	2mx	2mx	PROPN
ejpam-854	282	30	)	)	PUNCT
ejpam-854	282	31	(	(	PUNCT
ejpam-854	282	32	25	25	NUM
ejpam-854	282	33	)	)	PUNCT
ejpam-854	282	34	for	for	ADP
ejpam-854	282	35	all	all	DET
ejpam-854	282	36	x	x	SYM
ejpam-854	282	37	∈	∈	PROPN
ejpam-854	282	38	e.	e.	PROPN
ejpam-854	282	39	since	since	SCONJ
ejpam-854	282	40	f̃	f̃	PROPN
ejpam-854	282	41	:	:	PUNCT
ejpam-854	282	42	e→	e→	PROPN
ejpam-854	282	43	f	f	PROPN
ejpam-854	282	44	is	be	AUX
ejpam-854	282	45	odd	odd	ADJ
ejpam-854	282	46	,	,	PUNCT
ejpam-854	282	47	a	a	PRON
ejpam-854	282	48	:	:	PUNCT
ejpam-854	282	49	e→	e→	NOUN
ejpam-854	282	50	f	f	PROPN
ejpam-854	282	51	is	be	AUX
ejpam-854	282	52	an	an	DET
ejpam-854	282	53	odd	odd	ADJ
ejpam-854	282	54	mapping	mapping	NOUN
ejpam-854	282	55	.	.	PUNCT
ejpam-854	283	1	moreover	moreover	ADV
ejpam-854	283	2	,	,	PUNCT
ejpam-854	283	3	d	d	X
ejpam-854	283	4	(	(	PUNCT
ejpam-854	283	5	f̃	f̃	PROPN
ejpam-854	283	6	,	,	PUNCT
ejpam-854	283	7	a)≤	a)≤	PROPN
ejpam-854	283	8	1	1	NUM
ejpam-854	283	9	1−	1−	NUM
ejpam-854	283	10	l	l	NOUN
ejpam-854	284	1	d	d	PROPN
ejpam-854	284	2	(	(	PUNCT
ejpam-854	284	3	f̃	f̃	PROPN
ejpam-854	284	4	,	,	PUNCT
ejpam-854	284	5	j	j	PROPN
ejpam-854	284	6	f̃	f̃	PROPN
ejpam-854	284	7	)	)	PUNCT
ejpam-854	284	8	≤	≤	PROPN
ejpam-854	284	9	9n2	9n2	NUM
ejpam-854	284	10	+	+	SYM
ejpam-854	284	11	4	4	NUM
ejpam-854	284	12	n4−	n4−	NOUN
ejpam-854	284	13	n2	n2	NOUN
ejpam-854	284	14	ǫ	ǫ	NOUN
ejpam-854	284	15	.	.	PUNCT
ejpam-854	285	1	this	this	PRON
ejpam-854	285	2	implies	imply	VERB
ejpam-854	285	3	that	that	SCONJ
ejpam-854	285	4	the	the	DET
ejpam-854	285	5	inequality	inequality	NOUN
ejpam-854	285	6	(	(	PUNCT
ejpam-854	285	7	7	7	X
ejpam-854	285	8	)	)	PUNCT
ejpam-854	285	9	holds	hold	NOUN
ejpam-854	285	10	.	.	PUNCT
ejpam-854	286	1	also	also	ADV
ejpam-854	286	2	we	we	PRON
ejpam-854	286	3	have	have	VERB
ejpam-854	286	4	‖da(x	‖da(x	NOUN
ejpam-854	286	5	,	,	PUNCT
ejpam-854	286	6	y)‖=	y)‖=	PROPN
ejpam-854	286	7	lim	lim	PROPN
ejpam-854	286	8	m→∞	m→∞	NUM
ejpam-854	287	1	1	1	NUM
ejpam-854	287	2	2	2	NUM
ejpam-854	287	3	m	m	NUM
ejpam-854	287	4	‖d	‖d	ADJ
ejpam-854	287	5	f	f	X
ejpam-854	287	6	(	(	PUNCT
ejpam-854	287	7	2m+1	2m+1	PROPN
ejpam-854	287	8	x	x	X
ejpam-854	287	9	,	,	PUNCT
ejpam-854	287	10	2m+1	2m+1	PROPN
ejpam-854	287	11	y)−	y)−	PROPN
ejpam-854	287	12	8d	8d	NOUN
ejpam-854	287	13	f	f	X
ejpam-854	288	1	(	(	PUNCT
ejpam-854	288	2	2mx	2mx	ADJ
ejpam-854	288	3	,	,	PUNCT
ejpam-854	288	4	2	2	NUM
ejpam-854	288	5	m	m	NOUN
ejpam-854	288	6	y)‖	y)‖	ADJ
ejpam-854	288	7	≤	≤	ADJ
ejpam-854	288	8	lim	lim	PROPN
ejpam-854	288	9	m→∞	m→∞	NOUN
ejpam-854	288	10	9ǫ	9ǫ	NOUN
ejpam-854	288	11	2	2	NUM
ejpam-854	288	12	m	m	NOUN
ejpam-854	288	13	=	=	SYM
ejpam-854	288	14	0	0	NUM
ejpam-854	288	15	,	,	PUNCT
ejpam-854	288	16	and	and	CCONJ
ejpam-854	288	17	a	a	DET
ejpam-854	288	18	satisfies	satisfie	NOUN
ejpam-854	288	19	(	(	PUNCT
ejpam-854	288	20	5	5	NUM
ejpam-854	288	21	)	)	PUNCT
ejpam-854	288	22	.	.	PUNCT
ejpam-854	289	1	by	by	ADP
ejpam-854	289	2	theorem	theorem	NOUN
ejpam-854	289	3	2.2	2.2	NUM
ejpam-854	289	4	of	of	ADP
ejpam-854	289	5	[	[	X
ejpam-854	289	6	33	33	NUM
ejpam-854	289	7	]	]	PUNCT
ejpam-854	289	8	,	,	PUNCT
ejpam-854	289	9	the	the	DET
ejpam-854	289	10	function	function	NOUN
ejpam-854	289	11	x	x	INTJ
ejpam-854	289	12	→	→	SYM
ejpam-854	289	13	a(2x)−8a(x	a(2x)−8a(x	ADV
ejpam-854	289	14	)	)	PUNCT
ejpam-854	289	15	is	be	AUX
ejpam-854	289	16	additive	additive	ADJ
ejpam-854	289	17	.	.	PUNCT
ejpam-854	290	1	hence	hence	ADV
ejpam-854	290	2	a(2x	a(2x	VERB
ejpam-854	290	3	)	)	PUNCT
ejpam-854	290	4	=	=	SYM
ejpam-854	290	5	2a(x	2a(x	X
ejpam-854	290	6	)	)	PUNCT
ejpam-854	290	7	implies	imply	VERB
ejpam-854	290	8	that	that	SCONJ
ejpam-854	290	9	a	a	PRON
ejpam-854	290	10	is	be	AUX
ejpam-854	290	11	an	an	DET
ejpam-854	290	12	additive	additive	ADJ
ejpam-854	290	13	mapping	mapping	NOUN
ejpam-854	290	14	.	.	PUNCT
ejpam-854	291	1	if	if	SCONJ
ejpam-854	291	2	t	t	PROPN
ejpam-854	291	3	is	be	AUX
ejpam-854	291	4	another	another	DET
ejpam-854	291	5	additive	additive	ADJ
ejpam-854	291	6	mapping	mapping	NOUN
ejpam-854	291	7	satisfying	satisfy	VERB
ejpam-854	291	8	(	(	PUNCT
ejpam-854	291	9	7	7	NUM
ejpam-854	291	10	)	)	PUNCT
ejpam-854	291	11	.	.	PUNCT
ejpam-854	292	1	then	then	ADV
ejpam-854	292	2	t	t	PROPN
ejpam-854	292	3	is	be	AUX
ejpam-854	292	4	a	a	DET
ejpam-854	292	5	fixed	fix	VERB
ejpam-854	292	6	point	point	NOUN
ejpam-854	292	7	of	of	ADP
ejpam-854	292	8	j	j	PROPN
ejpam-854	292	9	in∆.	in∆.	NOUN
ejpam-854	292	10	however	however	ADV
ejpam-854	292	11	,	,	PUNCT
ejpam-854	292	12	by	by	ADP
ejpam-854	292	13	theorem	theorem	NOUN
ejpam-854	292	14	1	1	NUM
ejpam-854	292	15	,	,	PUNCT
ejpam-854	292	16	j	j	PROPN
ejpam-854	292	17	has	have	VERB
ejpam-854	292	18	only	only	ADV
ejpam-854	292	19	one	one	NUM
ejpam-854	292	20	fixed	fix	VERB
ejpam-854	292	21	point	point	NOUN
ejpam-854	292	22	in	in	ADP
ejpam-854	292	23	∆	∆	PROPN
ejpam-854	292	24	,	,	PUNCT
ejpam-854	292	25	hence	hence	ADV
ejpam-854	292	26	a=	a=	VERB
ejpam-854	292	27	t	t	PROPN
ejpam-854	292	28	.	.	PUNCT
ejpam-854	293	1	this	this	PRON
ejpam-854	293	2	completes	complete	VERB
ejpam-854	293	3	the	the	DET
ejpam-854	293	4	proof	proof	NOUN
ejpam-854	293	5	.	.	PUNCT
ejpam-854	294	1	theorem	theorem	NOUN
ejpam-854	294	2	3	3	X
ejpam-854	294	3	.	.	PUNCT
ejpam-854	295	1	let	let	VERB
ejpam-854	295	2	e	e	PRON
ejpam-854	295	3	be	be	AUX
ejpam-854	295	4	a	a	DET
ejpam-854	295	5	linear	linear	ADJ
ejpam-854	295	6	space	space	NOUN
ejpam-854	295	7	and	and	CCONJ
ejpam-854	295	8	let	let	VERB
ejpam-854	295	9	(	(	PUNCT
ejpam-854	295	10	(	(	PUNCT
ejpam-854	295	11	f	f	X
ejpam-854	295	12	k,‖	k,‖	PROPN
ejpam-854	295	13	·	·	PUNCT
ejpam-854	295	14	‖k	‖k	PROPN
ejpam-854	295	15	)	)	PUNCT
ejpam-854	295	16	:	:	PUNCT
ejpam-854	296	1	k	k	PROPN
ejpam-854	296	2	∈	∈	PROPN
ejpam-854	296	3	n	n	CCONJ
ejpam-854	296	4	)	)	PUNCT
ejpam-854	296	5	be	be	AUX
ejpam-854	296	6	a	a	DET
ejpam-854	296	7	multi	multi	ADJ
ejpam-854	296	8	-	-	ADJ
ejpam-854	296	9	banach	banach	ADJ
ejpam-854	296	10	space	space	NOUN
ejpam-854	296	11	.	.	PUNCT
ejpam-854	297	1	suppose	suppose	VERB
ejpam-854	297	2	that	that	SCONJ
ejpam-854	297	3	ǫ	ǫ	PRON
ejpam-854	297	4	≥	≥	NOUN
ejpam-854	297	5	0	0	NUM
ejpam-854	297	6	and	and	CCONJ
ejpam-854	297	7	f	f	NOUN
ejpam-854	297	8	:	:	PUNCT
ejpam-854	297	9	e→	e→	PROPN
ejpam-854	297	10	f	f	PROPN
ejpam-854	297	11	is	be	AUX
ejpam-854	297	12	an	an	DET
ejpam-854	297	13	odd	odd	ADJ
ejpam-854	297	14	mapping	mapping	NOUN
ejpam-854	297	15	satisfying	satisfying	ADJ
ejpam-854	297	16	sup	sup	NOUN
ejpam-854	297	17	k∈n	k∈n	PROPN
ejpam-854	297	18	‖(d	‖(d	PROPN
ejpam-854	298	1	f	f	PROPN
ejpam-854	298	2	(	(	PUNCT
ejpam-854	298	3	x1	x1	PROPN
ejpam-854	298	4	,	,	PUNCT
ejpam-854	298	5	y1	y1	PROPN
ejpam-854	298	6	)	)	PUNCT
ejpam-854	298	7	,	,	PUNCT
ejpam-854	298	8	.	.	PUNCT
ejpam-854	298	9	.	.	PUNCT
ejpam-854	298	10	.	.	PUNCT
ejpam-854	299	1	,	,	PUNCT
ejpam-854	300	1	d	d	X
ejpam-854	300	2	f	f	X
ejpam-854	300	3	(	(	PUNCT
ejpam-854	300	4	xk	xk	PROPN
ejpam-854	300	5	,	,	PUNCT
ejpam-854	300	6	yk))‖k	yk))‖k	INTJ
ejpam-854	300	7	≤	≤	ADJ
ejpam-854	300	8	ǫ	ǫ	NOUN
ejpam-854	300	9	for	for	ADP
ejpam-854	300	10	all	all	DET
ejpam-854	300	11	x1	x1	PROPN
ejpam-854	300	12	,	,	PUNCT
ejpam-854	300	13	.	.	PUNCT
ejpam-854	300	14	.	.	PUNCT
ejpam-854	300	15	.	.	PUNCT
ejpam-854	301	1	,	,	PUNCT
ejpam-854	301	2	xk	xk	PROPN
ejpam-854	301	3	,	,	PUNCT
ejpam-854	301	4	y1	y1	PROPN
ejpam-854	301	5	,	,	PUNCT
ejpam-854	301	6	.	.	PUNCT
ejpam-854	301	7	.	.	PUNCT
ejpam-854	301	8	.	.	PUNCT
ejpam-854	302	1	,	,	PUNCT
ejpam-854	302	2	yk	yk	PROPN
ejpam-854	302	3	∈	∈	PROPN
ejpam-854	302	4	e.	e.	PROPN
ejpam-854	302	5	then	then	ADV
ejpam-854	302	6	there	there	PRON
ejpam-854	302	7	exists	exist	VERB
ejpam-854	302	8	a	a	DET
ejpam-854	302	9	unique	unique	ADJ
ejpam-854	302	10	cubic	cubic	ADJ
ejpam-854	302	11	mapping	mapping	NOUN
ejpam-854	302	12	c	c	NOUN
ejpam-854	302	13	:	:	PUNCT
ejpam-854	303	1	e→	e→	PROPN
ejpam-854	303	2	f	f	PROPN
ejpam-854	303	3	such	such	ADJ
ejpam-854	303	4	that	that	DET
ejpam-854	303	5	sup	sup	NOUN
ejpam-854	303	6	k∈n	k∈n	PROPN
ejpam-854	303	7	‖	‖	PROPN
ejpam-854	303	8	(	(	PUNCT
ejpam-854	303	9	f	f	X
ejpam-854	303	10	(	(	PUNCT
ejpam-854	303	11	2x1)−	2x1)−	NUM
ejpam-854	303	12	2	2	NUM
ejpam-854	303	13	f	f	NOUN
ejpam-854	303	14	(	(	PUNCT
ejpam-854	303	15	x1)−	x1)−	PROPN
ejpam-854	303	16	c(x1	c(x1	PROPN
ejpam-854	303	17	)	)	PUNCT
ejpam-854	303	18	,	,	PUNCT
ejpam-854	303	19	.	.	PUNCT
ejpam-854	303	20	.	.	PUNCT
ejpam-854	303	21	.	.	PUNCT
ejpam-854	304	1	,	,	PUNCT
ejpam-854	304	2	f	f	PROPN
ejpam-854	304	3	(	(	PUNCT
ejpam-854	304	4	2xk)−	2xk)−	PROPN
ejpam-854	304	5	2	2	NUM
ejpam-854	304	6	f	f	X
ejpam-854	304	7	(	(	PUNCT
ejpam-854	304	8	xk)−	xk)−	X
ejpam-854	304	9	c(xk))‖k	c(xk))‖k	PROPN
ejpam-854	304	10	≤	≤	NOUN
ejpam-854	304	11	9n2	9n2	NUM
ejpam-854	304	12	+	+	CCONJ
ejpam-854	304	13	4	4	NUM
ejpam-854	304	14	7(n4−	7(n4−	NUM
ejpam-854	304	15	n2	n2	ADJ
ejpam-854	304	16	)	)	PUNCT
ejpam-854	304	17	ǫ	ǫ	NOUN
ejpam-854	304	18	for	for	ADP
ejpam-854	304	19	all	all	DET
ejpam-854	304	20	x1	x1	PROPN
ejpam-854	304	21	,	,	PUNCT
ejpam-854	304	22	.	.	PUNCT
ejpam-854	304	23	.	.	PUNCT
ejpam-854	305	1	.	.	PUNCT
ejpam-854	306	1	,	,	PUNCT
ejpam-854	306	2	xk	xk	PROPN
ejpam-854	306	3	∈	∈	PROPN
ejpam-854	306	4	e.	e.	PROPN
ejpam-854	306	5	t.	t.	PROPN
ejpam-854	306	6	xu	xu	PROPN
ejpam-854	306	7	,	,	PUNCT
ejpam-854	306	8	j.	j.	PROPN
ejpam-854	306	9	rassias	rassias	PROPN
ejpam-854	306	10	,	,	PUNCT
ejpam-854	306	11	w.	w.	PROPN
ejpam-854	306	12	xu	xu	PROPN
ejpam-854	306	13	/	/	SYM
ejpam-854	306	14	eur	eur	PROPN
ejpam-854	306	15	.	.	PUNCT
ejpam-854	307	1	j.	j.	PROPN
ejpam-854	307	2	pure	pure	PROPN
ejpam-854	307	3	appl	appl	PROPN
ejpam-854	307	4	.	.	PROPN
ejpam-854	307	5	math	math	PROPN
ejpam-854	307	6	,	,	PUNCT
ejpam-854	307	7	3	3	NUM
ejpam-854	307	8	(	(	PUNCT
ejpam-854	307	9	2010	2010	NUM
ejpam-854	307	10	)	)	PUNCT
ejpam-854	307	11	,	,	PUNCT
ejpam-854	307	12	1032	1032	NUM
ejpam-854	307	13	-	-	SYM
ejpam-854	307	14	1047	1047	NUM
ejpam-854	307	15	1040	1040	NUM
ejpam-854	307	16	proof	proof	NOUN
ejpam-854	307	17	.	.	PUNCT
ejpam-854	308	1	the	the	DET
ejpam-854	308	2	proof	proof	NOUN
ejpam-854	308	3	is	be	AUX
ejpam-854	308	4	similar	similar	ADJ
ejpam-854	308	5	to	to	ADP
ejpam-854	308	6	that	that	PRON
ejpam-854	308	7	of	of	ADP
ejpam-854	308	8	theorem	theorem	ADJ
ejpam-854	308	9	2	2	NUM
ejpam-854	308	10	.	.	PUNCT
ejpam-854	308	11	theorem	theorem	NOUN
ejpam-854	308	12	4	4	NUM
ejpam-854	308	13	.	.	PUNCT
ejpam-854	309	1	let	let	VERB
ejpam-854	309	2	e	e	PRON
ejpam-854	309	3	be	be	AUX
ejpam-854	309	4	a	a	DET
ejpam-854	309	5	linear	linear	ADJ
ejpam-854	309	6	space	space	NOUN
ejpam-854	309	7	and	and	CCONJ
ejpam-854	309	8	let	let	VERB
ejpam-854	309	9	(	(	PUNCT
ejpam-854	309	10	(	(	PUNCT
ejpam-854	309	11	f	f	X
ejpam-854	309	12	k,‖	k,‖	PROPN
ejpam-854	309	13	·	·	PUNCT
ejpam-854	309	14	‖k	‖k	PROPN
ejpam-854	309	15	)	)	PUNCT
ejpam-854	309	16	:	:	PUNCT
ejpam-854	310	1	k	k	PROPN
ejpam-854	310	2	∈	∈	PROPN
ejpam-854	310	3	n	n	CCONJ
ejpam-854	310	4	)	)	PUNCT
ejpam-854	310	5	be	be	AUX
ejpam-854	310	6	a	a	DET
ejpam-854	310	7	multi	multi	ADJ
ejpam-854	310	8	-	-	ADJ
ejpam-854	310	9	banach	banach	ADJ
ejpam-854	310	10	space	space	NOUN
ejpam-854	310	11	.	.	PUNCT
ejpam-854	311	1	suppose	suppose	VERB
ejpam-854	311	2	that	that	SCONJ
ejpam-854	311	3	ǫ	ǫ	PRON
ejpam-854	311	4	≥	≥	NOUN
ejpam-854	311	5	0	0	NUM
ejpam-854	311	6	and	and	CCONJ
ejpam-854	311	7	f	f	NOUN
ejpam-854	311	8	:	:	PUNCT
ejpam-854	311	9	e→	e→	PROPN
ejpam-854	311	10	f	f	PROPN
ejpam-854	311	11	is	be	AUX
ejpam-854	311	12	an	an	DET
ejpam-854	311	13	even	even	ADV
ejpam-854	311	14	mapping	mapping	NOUN
ejpam-854	311	15	with	with	ADP
ejpam-854	311	16	f	f	PROPN
ejpam-854	311	17	(	(	PUNCT
ejpam-854	311	18	0	0	NUM
ejpam-854	311	19	)	)	PUNCT
ejpam-854	311	20	=	=	SYM
ejpam-854	311	21	0	0	NUM
ejpam-854	311	22	,	,	PUNCT
ejpam-854	311	23	satisfying	satisfy	VERB
ejpam-854	311	24	condition	condition	NOUN
ejpam-854	311	25	sup	sup	NOUN
ejpam-854	311	26	k∈n	k∈n	PROPN
ejpam-854	311	27	‖(d	‖(d	PROPN
ejpam-854	312	1	f	f	PROPN
ejpam-854	312	2	(	(	PUNCT
ejpam-854	312	3	x1	x1	PROPN
ejpam-854	312	4	,	,	PUNCT
ejpam-854	312	5	y1	y1	PROPN
ejpam-854	312	6	)	)	PUNCT
ejpam-854	312	7	,	,	PUNCT
ejpam-854	312	8	.	.	PUNCT
ejpam-854	312	9	.	.	PUNCT
ejpam-854	312	10	.	.	PUNCT
ejpam-854	313	1	,	,	PUNCT
ejpam-854	314	1	d	d	X
ejpam-854	314	2	f	f	X
ejpam-854	314	3	(	(	PUNCT
ejpam-854	314	4	xk	xk	PROPN
ejpam-854	314	5	,	,	PUNCT
ejpam-854	314	6	yk))‖k	yk))‖k	INTJ
ejpam-854	314	7	≤	≤	ADJ
ejpam-854	314	8	ǫ	ǫ	PRON
ejpam-854	314	9	(	(	PUNCT
ejpam-854	314	10	26	26	NUM
ejpam-854	314	11	)	)	PUNCT
ejpam-854	314	12	for	for	ADP
ejpam-854	314	13	all	all	DET
ejpam-854	314	14	x1	x1	PROPN
ejpam-854	314	15	,	,	PUNCT
ejpam-854	314	16	.	.	PUNCT
ejpam-854	314	17	.	.	PUNCT
ejpam-854	314	18	.	.	PUNCT
ejpam-854	315	1	,	,	PUNCT
ejpam-854	315	2	xk	xk	PROPN
ejpam-854	315	3	,	,	PUNCT
ejpam-854	315	4	y1	y1	PROPN
ejpam-854	315	5	,	,	PUNCT
ejpam-854	315	6	.	.	PUNCT
ejpam-854	315	7	.	.	PUNCT
ejpam-854	315	8	.	.	PUNCT
ejpam-854	316	1	,	,	PUNCT
ejpam-854	316	2	yk	yk	PROPN
ejpam-854	316	3	∈	∈	PROPN
ejpam-854	316	4	e.	e.	PROPN
ejpam-854	316	5	then	then	ADV
ejpam-854	316	6	there	there	PRON
ejpam-854	316	7	exists	exist	VERB
ejpam-854	316	8	a	a	DET
ejpam-854	316	9	unique	unique	ADJ
ejpam-854	316	10	quadratic	quadratic	ADJ
ejpam-854	316	11	mapping	mapping	NOUN
ejpam-854	316	12	b	b	NOUN
ejpam-854	316	13	:	:	PUNCT
ejpam-854	316	14	e	e	X
ejpam-854	316	15	→	→	SYM
ejpam-854	316	16	f	f	PROPN
ejpam-854	316	17	such	such	ADJ
ejpam-854	316	18	that	that	DET
ejpam-854	316	19	sup	sup	NOUN
ejpam-854	316	20	k∈n	k∈n	PROPN
ejpam-854	316	21	‖	‖	PROPN
ejpam-854	317	1	(	(	PUNCT
ejpam-854	317	2	f	f	X
ejpam-854	317	3	(	(	PUNCT
ejpam-854	317	4	2x1)−	2x1)−	NUM
ejpam-854	317	5	16	16	NUM
ejpam-854	317	6	f	f	NOUN
ejpam-854	317	7	(	(	PUNCT
ejpam-854	317	8	x1)−	x1)−	PROPN
ejpam-854	317	9	b(x1	b(x1	NOUN
ejpam-854	317	10	)	)	PUNCT
ejpam-854	317	11	,	,	PUNCT
ejpam-854	317	12	.	.	PUNCT
ejpam-854	317	13	.	.	PUNCT
ejpam-854	317	14	.	.	PUNCT
ejpam-854	318	1	,	,	PUNCT
ejpam-854	318	2	f	f	PROPN
ejpam-854	318	3	(	(	PUNCT
ejpam-854	318	4	2xk)−	2xk)−	PROPN
ejpam-854	318	5	16	16	NUM
ejpam-854	318	6	f	f	X
ejpam-854	318	7	(	(	PUNCT
ejpam-854	318	8	xk)−	xk)−	PROPN
ejpam-854	318	9	b(xk))‖k	b(xk))‖k	NOUN
ejpam-854	318	10	≤	≤	NOUN
ejpam-854	318	11	8n2	8n2	NUM
ejpam-854	318	12	+	+	SYM
ejpam-854	318	13	2	2	NUM
ejpam-854	318	14	n4	n4	PROPN
ejpam-854	318	15	−	−	PROPN
ejpam-854	318	16	n2	n2	ADJ
ejpam-854	318	17	ǫ	ǫ	X
ejpam-854	318	18	(	(	PUNCT
ejpam-854	318	19	27	27	NUM
ejpam-854	318	20	)	)	PUNCT
ejpam-854	318	21	for	for	ADP
ejpam-854	318	22	all	all	DET
ejpam-854	318	23	x1	x1	PROPN
ejpam-854	318	24	,	,	PUNCT
ejpam-854	318	25	.	.	PUNCT
ejpam-854	318	26	.	.	PUNCT
ejpam-854	319	1	.	.	PUNCT
ejpam-854	320	1	,	,	PUNCT
ejpam-854	320	2	xk	xk	PROPN
ejpam-854	320	3	∈	∈	PROPN
ejpam-854	320	4	e.	e.	PROPN
ejpam-854	320	5	proof	proof	PROPN
ejpam-854	320	6	.	.	PUNCT
ejpam-854	321	1	let	let	VERB
ejpam-854	321	2	x1	x1	NUM
ejpam-854	321	3	,	,	PUNCT
ejpam-854	321	4	.	.	PUNCT
ejpam-854	321	5	.	.	PUNCT
ejpam-854	322	1	.	.	PUNCT
ejpam-854	323	1	,	,	PUNCT
ejpam-854	323	2	xk	xk	PROPN
ejpam-854	323	3	,	,	PUNCT
ejpam-854	323	4	y1	y1	PROPN
ejpam-854	323	5	,	,	PUNCT
ejpam-854	323	6	.	.	PUNCT
ejpam-854	323	7	.	.	PUNCT
ejpam-854	323	8	.	.	PUNCT
ejpam-854	324	1	,	,	PUNCT
ejpam-854	324	2	yk	yk	PROPN
ejpam-854	324	3	∈	∈	PROPN
ejpam-854	324	4	e.	e.	PROPN
ejpam-854	324	5	using	use	VERB
ejpam-854	324	6	the	the	DET
ejpam-854	324	7	evenness	evenness	NOUN
ejpam-854	324	8	of	of	ADP
ejpam-854	324	9	f	f	PROPN
ejpam-854	324	10	and	and	CCONJ
ejpam-854	324	11	from	from	ADP
ejpam-854	324	12	(	(	PUNCT
ejpam-854	324	13	26	26	NUM
ejpam-854	324	14	)	)	PUNCT
ejpam-854	324	15	,	,	PUNCT
ejpam-854	324	16	we	we	PRON
ejpam-854	324	17	have	have	VERB
ejpam-854	324	18	sup	sup	NOUN
ejpam-854	324	19	k∈n	k∈n	PROPN
ejpam-854	324	20	‖	‖	PROPN
ejpam-854	324	21	(	(	PUNCT
ejpam-854	324	22	f	f	PROPN
ejpam-854	324	23	(	(	PUNCT
ejpam-854	324	24	x1	x1	PROPN
ejpam-854	324	25	+	+	X
ejpam-854	324	26	ny1	ny1	NOUN
ejpam-854	324	27	)	)	PUNCT
ejpam-854	325	1	+	+	NUM
ejpam-854	325	2	f	f	X
ejpam-854	325	3	(	(	PUNCT
ejpam-854	325	4	x1−	x1−	PROPN
ejpam-854	325	5	ny1)−	ny1)−	PROPN
ejpam-854	325	6	n2	n2	PROPN
ejpam-854	325	7	f	f	PROPN
ejpam-854	325	8	(	(	PUNCT
ejpam-854	325	9	x1	x1	PROPN
ejpam-854	325	10	+	+	PROPN
ejpam-854	325	11	y1)−	y1)−	PROPN
ejpam-854	325	12	n2	n2	PROPN
ejpam-854	325	13	f	f	PROPN
ejpam-854	325	14	(	(	PUNCT
ejpam-854	325	15	x1−	x1−	PROPN
ejpam-854	325	16	y1)−	y1)−	PROPN
ejpam-854	325	17	2(1−	2(1−	NUM
ejpam-854	325	18	n2	n2	ADJ
ejpam-854	325	19	)	)	PUNCT
ejpam-854	325	20	f	f	PROPN
ejpam-854	325	21	(	(	PUNCT
ejpam-854	325	22	x1	x1	PROPN
ejpam-854	325	23	)	)	PUNCT
ejpam-854	325	24	−	−	PROPN
ejpam-854	326	1	n4−n2	n4−n2	NUM
ejpam-854	326	2	12	12	NUM
ejpam-854	326	3	[	[	SYM
ejpam-854	326	4	2	2	NUM
ejpam-854	326	5	f	f	NOUN
ejpam-854	326	6	(	(	PUNCT
ejpam-854	326	7	2y1)−	2y1)−	NUM
ejpam-854	326	8	8	8	NUM
ejpam-854	326	9	f	f	NOUN
ejpam-854	326	10	(	(	PUNCT
ejpam-854	326	11	y1	y1	PROPN
ejpam-854	326	12	)	)	PUNCT
ejpam-854	326	13	]	]	PUNCT
ejpam-854	326	14	,	,	PUNCT
ejpam-854	326	15	.	.	PUNCT
ejpam-854	326	16	.	.	PUNCT
ejpam-854	326	17	.	.	PUNCT
ejpam-854	327	1	,	,	PUNCT
ejpam-854	327	2	f	f	PROPN
ejpam-854	327	3	(	(	PUNCT
ejpam-854	327	4	xk+	xk+	PROPN
ejpam-854	327	5	nyk	nyk	PROPN
ejpam-854	327	6	)	)	PUNCT
ejpam-854	328	1	+	+	CCONJ
ejpam-854	328	2	f	f	X
ejpam-854	328	3	(	(	PUNCT
ejpam-854	328	4	xk	xk	INTJ
ejpam-854	328	5	−	−	PROPN
ejpam-854	328	6	nyk)−	nyk)−	PROPN
ejpam-854	328	7	n2	n2	PROPN
ejpam-854	328	8	f	f	PROPN
ejpam-854	328	9	(	(	PUNCT
ejpam-854	328	10	xk	xk	PROPN
ejpam-854	328	11	+	+	CCONJ
ejpam-854	328	12	yk	yk	PROPN
ejpam-854	328	13	)	)	PUNCT
ejpam-854	329	1	−n2	−n2	PROPN
ejpam-854	329	2	f	f	X
ejpam-854	329	3	(	(	PUNCT
ejpam-854	329	4	xk−	xk−	NUM
ejpam-854	329	5	yk)−	yk)−	PROPN
ejpam-854	329	6	2(1−	2(1−	NUM
ejpam-854	329	7	n2	n2	ADJ
ejpam-854	329	8	)	)	PUNCT
ejpam-854	329	9	f	f	NOUN
ejpam-854	329	10	(	(	PUNCT
ejpam-854	329	11	xk)−	xk)−	X
ejpam-854	329	12	n4−n2	n4−n2	PROPN
ejpam-854	329	13	12	12	NUM
ejpam-854	329	14	[	[	SYM
ejpam-854	329	15	2	2	NUM
ejpam-854	329	16	f	f	NOUN
ejpam-854	329	17	(	(	PUNCT
ejpam-854	329	18	2yk)−	2yk)−	NUM
ejpam-854	329	19	8	8	NUM
ejpam-854	329	20	f	f	NOUN
ejpam-854	329	21	(	(	PUNCT
ejpam-854	329	22	yk)])‖k	yk)])‖k	PROPN
ejpam-854	329	23	≤	≤	PROPN
ejpam-854	329	24	ǫ	ǫ	PRON
ejpam-854	329	25	.	.	PUNCT
ejpam-854	330	1	(	(	PUNCT
ejpam-854	330	2	28	28	NUM
ejpam-854	330	3	)	)	PUNCT
ejpam-854	330	4	interchanging	interchange	VERB
ejpam-854	330	5	x	x	PUNCT
ejpam-854	331	1	i	i	PRON
ejpam-854	331	2	and	and	CCONJ
ejpam-854	331	3	yi(i	yi(i	NUM
ejpam-854	331	4	∈	∈	PROPN
ejpam-854	331	5	nk	nk	PROPN
ejpam-854	331	6	)	)	PUNCT
ejpam-854	331	7	in	in	ADP
ejpam-854	331	8	(	(	PUNCT
ejpam-854	331	9	28	28	NUM
ejpam-854	331	10	)	)	PUNCT
ejpam-854	331	11	,	,	PUNCT
ejpam-854	331	12	we	we	PRON
ejpam-854	331	13	get	get	VERB
ejpam-854	331	14	sup	sup	NOUN
ejpam-854	331	15	k∈n	k∈n	PROPN
ejpam-854	331	16	‖	‖	PROPN
ejpam-854	331	17	(	(	PUNCT
ejpam-854	331	18	f	f	PROPN
ejpam-854	331	19	(	(	PUNCT
ejpam-854	331	20	nx1	nx1	PROPN
ejpam-854	331	21	+	+	X
ejpam-854	331	22	y1	y1	NOUN
ejpam-854	331	23	)	)	PUNCT
ejpam-854	332	1	+	+	NUM
ejpam-854	332	2	f	f	X
ejpam-854	332	3	(	(	PUNCT
ejpam-854	332	4	nx1−	nx1−	PROPN
ejpam-854	332	5	y1)−	y1)−	PROPN
ejpam-854	332	6	n2	n2	PROPN
ejpam-854	332	7	f	f	PROPN
ejpam-854	332	8	(	(	PUNCT
ejpam-854	332	9	x1	x1	PROPN
ejpam-854	332	10	+	+	PROPN
ejpam-854	332	11	y1)−	y1)−	PROPN
ejpam-854	332	12	n2	n2	PROPN
ejpam-854	332	13	f	f	PROPN
ejpam-854	332	14	(	(	PUNCT
ejpam-854	332	15	x1−	x1−	PROPN
ejpam-854	332	16	y1)−	y1)−	PROPN
ejpam-854	332	17	2(1−	2(1−	NUM
ejpam-854	332	18	n2	n2	ADJ
ejpam-854	332	19	)	)	PUNCT
ejpam-854	332	20	f	f	PROPN
ejpam-854	332	21	(	(	PUNCT
ejpam-854	332	22	y1	y1	PROPN
ejpam-854	332	23	)	)	PUNCT
ejpam-854	332	24	−	−	PROPN
ejpam-854	333	1	n4−n2	n4−n2	NUM
ejpam-854	333	2	12	12	NUM
ejpam-854	333	3	[	[	SYM
ejpam-854	333	4	2	2	NUM
ejpam-854	333	5	f	f	NOUN
ejpam-854	333	6	(	(	PUNCT
ejpam-854	333	7	2x1)−	2x1)−	NUM
ejpam-854	333	8	8	8	NUM
ejpam-854	333	9	f	f	NOUN
ejpam-854	333	10	(	(	PUNCT
ejpam-854	333	11	x1	x1	PROPN
ejpam-854	333	12	)	)	PUNCT
ejpam-854	333	13	]	]	PUNCT
ejpam-854	333	14	,	,	PUNCT
ejpam-854	333	15	.	.	PUNCT
ejpam-854	333	16	.	.	PUNCT
ejpam-854	333	17	.	.	PUNCT
ejpam-854	334	1	,	,	PUNCT
ejpam-854	334	2	f	f	PROPN
ejpam-854	334	3	(	(	PUNCT
ejpam-854	334	4	nxk	nxk	PROPN
ejpam-854	334	5	+	+	CCONJ
ejpam-854	334	6	yk	yk	PROPN
ejpam-854	334	7	)	)	PUNCT
ejpam-854	335	1	+	+	NUM
ejpam-854	335	2	f	f	X
ejpam-854	335	3	(	(	PUNCT
ejpam-854	335	4	nxk−	nxk−	PROPN
ejpam-854	335	5	yk)−	yk)−	NOUN
ejpam-854	335	6	n2	n2	PROPN
ejpam-854	335	7	f	f	PROPN
ejpam-854	335	8	(	(	PUNCT
ejpam-854	335	9	xk+	xk+	PROPN
ejpam-854	335	10	yk	yk	PROPN
ejpam-854	335	11	)	)	PUNCT
ejpam-854	336	1	−n2	−n2	PROPN
ejpam-854	336	2	f	f	X
ejpam-854	336	3	(	(	PUNCT
ejpam-854	336	4	xk−	xk−	NUM
ejpam-854	336	5	yk)−	yk)−	PROPN
ejpam-854	336	6	2(1−	2(1−	NUM
ejpam-854	336	7	n2	n2	ADJ
ejpam-854	336	8	)	)	PUNCT
ejpam-854	336	9	f	f	NOUN
ejpam-854	336	10	(	(	PUNCT
ejpam-854	336	11	yk)−	yk)−	PROPN
ejpam-854	336	12	n4−n2	n4−n2	NUM
ejpam-854	336	13	12	12	NUM
ejpam-854	336	14	[	[	SYM
ejpam-854	336	15	2	2	NUM
ejpam-854	336	16	f	f	NOUN
ejpam-854	336	17	(	(	PUNCT
ejpam-854	336	18	2xk)−	2xk)−	NOUN
ejpam-854	336	19	8	8	NUM
ejpam-854	336	20	f	f	X
ejpam-854	336	21	(	(	PUNCT
ejpam-854	336	22	xk)])‖k	xk)])‖k	PROPN
ejpam-854	336	23	≤	≤	NUM
ejpam-854	336	24	ǫ	ǫ	X
ejpam-854	336	25	.	.	PUNCT
ejpam-854	337	1	(	(	PUNCT
ejpam-854	337	2	29	29	NUM
ejpam-854	337	3	)	)	PUNCT
ejpam-854	337	4	letting	let	VERB
ejpam-854	337	5	yi	yi	NOUN
ejpam-854	337	6	=	=	PUNCT
ejpam-854	337	7	0(i	0(i	NUM
ejpam-854	337	8	∈	∈	PROPN
ejpam-854	337	9	nk	nk	PROPN
ejpam-854	337	10	)	)	PUNCT
ejpam-854	337	11	in	in	ADP
ejpam-854	337	12	(	(	PUNCT
ejpam-854	337	13	29	29	NUM
ejpam-854	337	14	)	)	PUNCT
ejpam-854	337	15	,	,	PUNCT
ejpam-854	337	16	we	we	PRON
ejpam-854	337	17	get	get	VERB
ejpam-854	337	18	sup	sup	NOUN
ejpam-854	337	19	k∈n	k∈n	NOUN
ejpam-854	338	1	‖(2	‖(2	NOUN
ejpam-854	338	2	f	f	X
ejpam-854	339	1	(	(	PUNCT
ejpam-854	339	2	nx1)−	nx1)−	NOUN
ejpam-854	339	3	2n2	2n2	NUM
ejpam-854	339	4	f	f	NOUN
ejpam-854	340	1	(	(	PUNCT
ejpam-854	340	2	x1)−	x1)−	PROPN
ejpam-854	340	3	n4−n2	n4−n2	NUM
ejpam-854	340	4	12	12	NUM
ejpam-854	340	5	[	[	SYM
ejpam-854	340	6	2	2	NUM
ejpam-854	340	7	f	f	NOUN
ejpam-854	340	8	(	(	PUNCT
ejpam-854	340	9	2x1)−	2x1)−	NUM
ejpam-854	340	10	8	8	NUM
ejpam-854	340	11	f	f	NOUN
ejpam-854	340	12	(	(	PUNCT
ejpam-854	340	13	x1	x1	PROPN
ejpam-854	340	14	)	)	PUNCT
ejpam-854	340	15	]	]	PUNCT
ejpam-854	340	16	,	,	PUNCT
ejpam-854	340	17	.	.	PUNCT
ejpam-854	340	18	.	.	PUNCT
ejpam-854	340	19	.	.	PUNCT
ejpam-854	341	1	,	,	PUNCT
ejpam-854	341	2	2	2	NUM
ejpam-854	341	3	f	f	X
ejpam-854	341	4	(	(	PUNCT
ejpam-854	341	5	nxk)−	nxk)−	PROPN
ejpam-854	341	6	2n2	2n2	NUM
ejpam-854	341	7	f	f	X
ejpam-854	341	8	(	(	PUNCT
ejpam-854	341	9	xk)−	xk)−	PROPN
ejpam-854	341	10	n4−n2	n4−n2	PROPN
ejpam-854	341	11	12	12	NUM
ejpam-854	341	12	[	[	SYM
ejpam-854	341	13	2	2	NUM
ejpam-854	341	14	f	f	NOUN
ejpam-854	341	15	(	(	PUNCT
ejpam-854	341	16	2xk)−	2xk)−	NOUN
ejpam-854	341	17	8	8	NUM
ejpam-854	341	18	f	f	X
ejpam-854	341	19	(	(	PUNCT
ejpam-854	341	20	xk)])‖k	xk)])‖k	PROPN
ejpam-854	341	21	≤	≤	NUM
ejpam-854	341	22	ǫ	ǫ	X
ejpam-854	341	23	.	.	PUNCT
ejpam-854	342	1	(	(	PUNCT
ejpam-854	342	2	30	30	X
ejpam-854	342	3	)	)	PUNCT
ejpam-854	342	4	putting	put	VERB
ejpam-854	342	5	yi	yi	NOUN
ejpam-854	342	6	=	=	PUNCT
ejpam-854	342	7	x	x	SYM
ejpam-854	342	8	i(i	i(i	PROPN
ejpam-854	342	9	∈	∈	PROPN
ejpam-854	342	10	nk	nk	PROPN
ejpam-854	342	11	)	)	PUNCT
ejpam-854	342	12	in	in	ADP
ejpam-854	342	13	(	(	PUNCT
ejpam-854	342	14	29	29	NUM
ejpam-854	342	15	)	)	PUNCT
ejpam-854	342	16	,	,	PUNCT
ejpam-854	342	17	we	we	PRON
ejpam-854	342	18	have	have	VERB
ejpam-854	342	19	sup	sup	NOUN
ejpam-854	342	20	k∈n	k∈n	PROPN
ejpam-854	342	21	‖	‖	PROPN
ejpam-854	342	22	(	(	PUNCT
ejpam-854	342	23	f	f	X
ejpam-854	342	24	(	(	PUNCT
ejpam-854	342	25	(	(	PUNCT
ejpam-854	342	26	n+	n+	X
ejpam-854	342	27	1)x1	1)x1	NUM
ejpam-854	342	28	)	)	PUNCT
ejpam-854	343	1	+	+	NUM
ejpam-854	343	2	f	f	X
ejpam-854	343	3	(	(	PUNCT
ejpam-854	343	4	(	(	PUNCT
ejpam-854	343	5	n−	n−	NOUN
ejpam-854	343	6	1)x1)−	1)x1)−	PROPN
ejpam-854	343	7	n2	n2	PROPN
ejpam-854	343	8	f	f	PROPN
ejpam-854	343	9	(	(	PUNCT
ejpam-854	343	10	2x1)−	2x1)−	PROPN
ejpam-854	343	11	2(1−	2(1−	NUM
ejpam-854	343	12	n2	n2	ADJ
ejpam-854	343	13	)	)	PUNCT
ejpam-854	343	14	f	f	PROPN
ejpam-854	343	15	(	(	PUNCT
ejpam-854	343	16	x1	x1	PROPN
ejpam-854	343	17	)	)	PUNCT
ejpam-854	343	18	−	−	PROPN
ejpam-854	344	1	n4−n2	n4−n2	NUM
ejpam-854	344	2	12	12	NUM
ejpam-854	344	3	[	[	SYM
ejpam-854	344	4	2	2	NUM
ejpam-854	344	5	f	f	NOUN
ejpam-854	344	6	(	(	PUNCT
ejpam-854	344	7	2x1)−	2x1)−	NUM
ejpam-854	344	8	8	8	NUM
ejpam-854	344	9	f	f	NOUN
ejpam-854	344	10	(	(	PUNCT
ejpam-854	344	11	x1	x1	PROPN
ejpam-854	344	12	)	)	PUNCT
ejpam-854	344	13	]	]	PUNCT
ejpam-854	344	14	,	,	PUNCT
ejpam-854	344	15	.	.	PUNCT
ejpam-854	344	16	.	.	PUNCT
ejpam-854	344	17	.	.	PUNCT
ejpam-854	345	1	,	,	PUNCT
ejpam-854	345	2	f	f	X
ejpam-854	345	3	(	(	PUNCT
ejpam-854	345	4	(	(	PUNCT
ejpam-854	345	5	n+	n+	X
ejpam-854	345	6	1)xk	1)xk	NUM
ejpam-854	345	7	)	)	PUNCT
ejpam-854	346	1	+	+	NUM
ejpam-854	346	2	f	f	X
ejpam-854	346	3	(	(	PUNCT
ejpam-854	346	4	(	(	PUNCT
ejpam-854	346	5	n−	n−	NOUN
ejpam-854	346	6	1)xk	1)xk	PROPN
ejpam-854	346	7	)	)	PUNCT
ejpam-854	347	1	−n2	−n2	PROPN
ejpam-854	347	2	f	f	X
ejpam-854	347	3	(	(	PUNCT
ejpam-854	347	4	2xk)−	2xk)−	NOUN
ejpam-854	347	5	2(1−	2(1−	NUM
ejpam-854	347	6	n2	n2	NOUN
ejpam-854	347	7	)	)	PUNCT
ejpam-854	347	8	f	f	NOUN
ejpam-854	347	9	(	(	PUNCT
ejpam-854	347	10	xk)−	xk)−	X
ejpam-854	348	1	n4−n2	n4−n2	PROPN
ejpam-854	348	2	12	12	NUM
ejpam-854	348	3	[	[	SYM
ejpam-854	348	4	2	2	NUM
ejpam-854	348	5	f	f	NOUN
ejpam-854	348	6	(	(	PUNCT
ejpam-854	348	7	2xk)−	2xk)−	NOUN
ejpam-854	348	8	8	8	NUM
ejpam-854	348	9	f	f	X
ejpam-854	348	10	(	(	PUNCT
ejpam-854	348	11	xk)])‖k	xk)])‖k	PROPN
ejpam-854	348	12	≤	≤	NUM
ejpam-854	348	13	ǫ	ǫ	X
ejpam-854	348	14	.	.	PUNCT
ejpam-854	349	1	(	(	PUNCT
ejpam-854	349	2	31	31	NUM
ejpam-854	349	3	)	)	PUNCT
ejpam-854	349	4	replacing	replace	VERB
ejpam-854	349	5	x	x	PUNCT
ejpam-854	349	6	i	i	PRON
ejpam-854	349	7	by	by	ADP
ejpam-854	349	8	2x	2x	NUM
ejpam-854	349	9	i(i	i(i	PROPN
ejpam-854	349	10	∈	∈	PROPN
ejpam-854	349	11	nk	nk	PROPN
ejpam-854	349	12	)	)	PUNCT
ejpam-854	349	13	in	in	ADP
ejpam-854	349	14	(	(	PUNCT
ejpam-854	349	15	30	30	NUM
ejpam-854	349	16	)	)	PUNCT
ejpam-854	349	17	,	,	PUNCT
ejpam-854	349	18	we	we	PRON
ejpam-854	349	19	get	get	VERB
ejpam-854	349	20	sup	sup	NOUN
ejpam-854	349	21	k∈n	k∈n	NOUN
ejpam-854	350	1	‖(2	‖(2	NOUN
ejpam-854	350	2	f	f	X
ejpam-854	350	3	(	(	PUNCT
ejpam-854	350	4	2nx1)−	2nx1)−	NUM
ejpam-854	350	5	2n2	2n2	NUM
ejpam-854	350	6	f	f	NOUN
ejpam-854	350	7	(	(	PUNCT
ejpam-854	350	8	2x1)−	2x1)−	NUM
ejpam-854	350	9	n4−n2	n4−n2	NUM
ejpam-854	350	10	12	12	NUM
ejpam-854	350	11	[	[	SYM
ejpam-854	350	12	2	2	NUM
ejpam-854	350	13	f	f	NOUN
ejpam-854	350	14	(	(	PUNCT
ejpam-854	350	15	4x1)−	4x1)−	NOUN
ejpam-854	350	16	8	8	NUM
ejpam-854	350	17	f	f	NOUN
ejpam-854	350	18	(	(	PUNCT
ejpam-854	350	19	2x1	2x1	NUM
ejpam-854	350	20	)	)	PUNCT
ejpam-854	350	21	]	]	PUNCT
ejpam-854	350	22	,	,	PUNCT
ejpam-854	350	23	.	.	PUNCT
ejpam-854	350	24	.	.	PUNCT
ejpam-854	351	1	.	.	PUNCT
ejpam-854	352	1	,	,	PUNCT
ejpam-854	352	2	2	2	NUM
ejpam-854	352	3	f	f	X
ejpam-854	352	4	(	(	PUNCT
ejpam-854	352	5	2nxk)−	2nxk)−	NOUN
ejpam-854	352	6	2n2	2n2	NUM
ejpam-854	352	7	f	f	NOUN
ejpam-854	352	8	(	(	PUNCT
ejpam-854	352	9	2xk)−	2xk)−	NOUN
ejpam-854	352	10	n4−n2	n4−n2	NUM
ejpam-854	352	11	12	12	NUM
ejpam-854	352	12	[	[	SYM
ejpam-854	352	13	2	2	NUM
ejpam-854	352	14	f	f	NOUN
ejpam-854	352	15	(	(	PUNCT
ejpam-854	352	16	4xk)−	4xk)−	PROPN
ejpam-854	352	17	8	8	NUM
ejpam-854	352	18	f	f	NOUN
ejpam-854	352	19	(	(	PUNCT
ejpam-854	352	20	2xk)])‖k	2xk)])‖k	NUM
ejpam-854	352	21	≤	≤	ADJ
ejpam-854	352	22	ǫ	ǫ	NOUN
ejpam-854	352	23	.	.	PUNCT
ejpam-854	353	1	(	(	PUNCT
ejpam-854	353	2	32	32	NUM
ejpam-854	353	3	)	)	PUNCT
ejpam-854	353	4	t.	t.	PROPN
ejpam-854	353	5	xu	xu	PROPN
ejpam-854	353	6	,	,	PUNCT
ejpam-854	353	7	j.	j.	PROPN
ejpam-854	353	8	rassias	rassias	PROPN
ejpam-854	353	9	,	,	PUNCT
ejpam-854	353	10	w.	w.	PROPN
ejpam-854	353	11	xu	xu	PROPN
ejpam-854	353	12	/	/	SYM
ejpam-854	353	13	eur	eur	PROPN
ejpam-854	353	14	.	.	PUNCT
ejpam-854	354	1	j.	j.	PROPN
ejpam-854	354	2	pure	pure	PROPN
ejpam-854	354	3	appl	appl	PROPN
ejpam-854	354	4	.	.	PROPN
ejpam-854	354	5	math	math	PROPN
ejpam-854	354	6	,	,	PUNCT
ejpam-854	354	7	3	3	NUM
ejpam-854	354	8	(	(	PUNCT
ejpam-854	354	9	2010	2010	NUM
ejpam-854	354	10	)	)	PUNCT
ejpam-854	354	11	,	,	PUNCT
ejpam-854	354	12	1032	1032	NUM
ejpam-854	354	13	-	-	SYM
ejpam-854	354	14	1047	1047	NUM
ejpam-854	354	15	1041	1041	NUM
ejpam-854	354	16	letting	let	VERB
ejpam-854	354	17	yi	yi	NOUN
ejpam-854	354	18	=	=	SYM
ejpam-854	354	19	nx	nx	PROPN
ejpam-854	354	20	i(i	i(i	PROPN
ejpam-854	354	21	∈	∈	PROPN
ejpam-854	354	22	nk	nk	PROPN
ejpam-854	354	23	)	)	PUNCT
ejpam-854	354	24	in	in	ADP
ejpam-854	354	25	(	(	PUNCT
ejpam-854	354	26	29	29	NUM
ejpam-854	354	27	)	)	PUNCT
ejpam-854	354	28	,	,	PUNCT
ejpam-854	354	29	we	we	PRON
ejpam-854	354	30	get	get	VERB
ejpam-854	354	31	sup	sup	NOUN
ejpam-854	354	32	k∈n	k∈n	PROPN
ejpam-854	354	33	‖	‖	PROPN
ejpam-854	355	1	(	(	PUNCT
ejpam-854	355	2	f	f	PROPN
ejpam-854	355	3	(	(	PUNCT
ejpam-854	355	4	2nx1)−	2nx1)−	NUM
ejpam-854	355	5	n2	n2	PROPN
ejpam-854	355	6	f	f	PROPN
ejpam-854	355	7	(	(	PUNCT
ejpam-854	355	8	(	(	PUNCT
ejpam-854	355	9	1	1	NUM
ejpam-854	355	10	+	+	NUM
ejpam-854	355	11	n)x1)−	n)x1)−	ADJ
ejpam-854	355	12	n2	n2	PROPN
ejpam-854	355	13	f	f	PROPN
ejpam-854	355	14	(	(	PUNCT
ejpam-854	355	15	(	(	PUNCT
ejpam-854	355	16	1−	1−	NUM
ejpam-854	355	17	n)x1)−	n)x1)−	NOUN
ejpam-854	355	18	2(1−	2(1−	NUM
ejpam-854	355	19	n2	n2	ADJ
ejpam-854	355	20	)	)	PUNCT
ejpam-854	355	21	f	f	PROPN
ejpam-854	355	22	(	(	PUNCT
ejpam-854	355	23	nx1	nx1	PROPN
ejpam-854	355	24	)	)	PUNCT
ejpam-854	355	25	−	−	PROPN
ejpam-854	356	1	n4−n2	n4−n2	NUM
ejpam-854	356	2	12	12	NUM
ejpam-854	356	3	[	[	SYM
ejpam-854	356	4	2	2	NUM
ejpam-854	356	5	f	f	NOUN
ejpam-854	356	6	(	(	PUNCT
ejpam-854	356	7	2x1)−	2x1)−	NUM
ejpam-854	356	8	8	8	NUM
ejpam-854	356	9	f	f	NOUN
ejpam-854	356	10	(	(	PUNCT
ejpam-854	356	11	x1	x1	PROPN
ejpam-854	356	12	)	)	PUNCT
ejpam-854	356	13	]	]	PUNCT
ejpam-854	356	14	,	,	PUNCT
ejpam-854	356	15	.	.	PUNCT
ejpam-854	356	16	.	.	PUNCT
ejpam-854	356	17	.	.	PUNCT
ejpam-854	357	1	,	,	PUNCT
ejpam-854	357	2	f	f	PROPN
ejpam-854	357	3	(	(	PUNCT
ejpam-854	357	4	2nxk)−	2nxk)−	PROPN
ejpam-854	357	5	n2	n2	PROPN
ejpam-854	357	6	f	f	PROPN
ejpam-854	357	7	(	(	PUNCT
ejpam-854	357	8	(	(	PUNCT
ejpam-854	357	9	1	1	NUM
ejpam-854	357	10	+	+	NUM
ejpam-854	357	11	n)xk	n)xk	ADJ
ejpam-854	357	12	)	)	PUNCT
ejpam-854	357	13	−n2	−n2	PROPN
ejpam-854	357	14	f	f	NOUN
ejpam-854	357	15	(	(	PUNCT
ejpam-854	357	16	(	(	PUNCT
ejpam-854	357	17	1−	1−	NUM
ejpam-854	357	18	n)xk)−	n)xk)−	PROPN
ejpam-854	357	19	2(1−	2(1−	PROPN
ejpam-854	357	20	n2	n2	NOUN
ejpam-854	357	21	)	)	PUNCT
ejpam-854	357	22	f	f	NOUN
ejpam-854	357	23	(	(	PUNCT
ejpam-854	357	24	nxk)−	nxk)−	PROPN
ejpam-854	357	25	n4−n2	n4−n2	NUM
ejpam-854	357	26	12	12	NUM
ejpam-854	357	27	[	[	SYM
ejpam-854	357	28	2	2	NUM
ejpam-854	357	29	f	f	NOUN
ejpam-854	357	30	(	(	PUNCT
ejpam-854	357	31	2xk)−	2xk)−	NOUN
ejpam-854	357	32	8	8	NUM
ejpam-854	357	33	f	f	X
ejpam-854	357	34	(	(	PUNCT
ejpam-854	357	35	xk)])‖k	xk)])‖k	PROPN
ejpam-854	357	36	≤	≤	NUM
ejpam-854	357	37	ǫ	ǫ	X
ejpam-854	357	38	.	.	PUNCT
ejpam-854	358	1	(	(	PUNCT
ejpam-854	358	2	33	33	NUM
ejpam-854	358	3	)	)	PUNCT
ejpam-854	358	4	by	by	ADP
ejpam-854	358	5	(	(	PUNCT
ejpam-854	358	6	30)-(33	30)-(33	PROPN
ejpam-854	358	7	)	)	PUNCT
ejpam-854	358	8	,	,	PUNCT
ejpam-854	358	9	we	we	PRON
ejpam-854	358	10	obtain	obtain	VERB
ejpam-854	358	11	sup	sup	NOUN
ejpam-854	358	12	k∈n	k∈n	PROPN
ejpam-854	358	13	‖	‖	PROPN
ejpam-854	358	14	(	(	PUNCT
ejpam-854	358	15	f	f	PROPN
ejpam-854	358	16	(	(	PUNCT
ejpam-854	358	17	4x1)−20	4x1)−20	PROPN
ejpam-854	358	18	f	f	X
ejpam-854	358	19	(	(	PUNCT
ejpam-854	358	20	2x1)+64	2x1)+64	NUM
ejpam-854	358	21	f	f	NOUN
ejpam-854	358	22	(	(	PUNCT
ejpam-854	358	23	x1	x1	PROPN
ejpam-854	358	24	)	)	PUNCT
ejpam-854	358	25	,	,	PUNCT
ejpam-854	358	26	.	.	PUNCT
ejpam-854	358	27	.	.	PUNCT
ejpam-854	358	28	.	.	PUNCT
ejpam-854	359	1	,	,	PUNCT
ejpam-854	359	2	f	f	PROPN
ejpam-854	359	3	(	(	PUNCT
ejpam-854	359	4	4x1)−20	4x1)−20	PROPN
ejpam-854	359	5	f	f	X
ejpam-854	359	6	(	(	PUNCT
ejpam-854	359	7	2x1)+64	2x1)+64	NUM
ejpam-854	359	8	f	f	X
ejpam-854	359	9	(	(	PUNCT
ejpam-854	359	10	x1))‖k	x1))‖k	PROPN
ejpam-854	359	11	≤	≤	PROPN
ejpam-854	359	12	24n2	24n2	NUM
ejpam-854	359	13	+	+	CCONJ
ejpam-854	359	14	6	6	NUM
ejpam-854	359	15	n4−	n4−	NOUN
ejpam-854	359	16	n2	n2	NOUN
ejpam-854	359	17	ǫ	ǫ	NOUN
ejpam-854	359	18	.	.	PUNCT
ejpam-854	360	1	(	(	PUNCT
ejpam-854	360	2	34	34	NUM
ejpam-854	360	3	)	)	PUNCT
ejpam-854	360	4	consider	consider	VERB
ejpam-854	360	5	the	the	DET
ejpam-854	360	6	set	set	NOUN
ejpam-854	360	7	ω	ω	NOUN
ejpam-854	360	8	:	:	PUNCT
ejpam-854	360	9	=	=	SYM
ejpam-854	360	10	{	{	PUNCT
ejpam-854	360	11	g	g	NOUN
ejpam-854	360	12	|	|	ADV
ejpam-854	360	13	g	g	PROPN
ejpam-854	360	14	:	:	PUNCT
ejpam-854	360	15	e→	e→	PROPN
ejpam-854	360	16	f	f	PROPN
ejpam-854	360	17	,	,	PUNCT
ejpam-854	360	18	g(0	g(0	PROPN
ejpam-854	360	19	)	)	PUNCT
ejpam-854	360	20	=	=	SYM
ejpam-854	360	21	0	0	NUM
ejpam-854	360	22	}	}	PUNCT
ejpam-854	360	23	and	and	CCONJ
ejpam-854	360	24	introduce	introduce	VERB
ejpam-854	360	25	the	the	DET
ejpam-854	360	26	generalized	generalize	VERB
ejpam-854	360	27	metric	metric	NOUN
ejpam-854	360	28	on	on	ADP
ejpam-854	360	29	ω	ω	PROPN
ejpam-854	360	30	,	,	PUNCT
ejpam-854	360	31	d(g	d(g	PROPN
ejpam-854	360	32	,	,	PUNCT
ejpam-854	360	33	h	h	NOUN
ejpam-854	360	34	)	)	PUNCT
ejpam-854	361	1	=	=	SYM
ejpam-854	361	2	inf{α	inf{α	NOUN
ejpam-854	361	3	>	>	X
ejpam-854	361	4	0|	0|	NUM
ejpam-854	361	5	sup	sup	NOUN
ejpam-854	361	6	k∈n	k∈n	PROPN
ejpam-854	361	7	‖(g(x1)−	‖(g(x1)−	PROPN
ejpam-854	361	8	h(x1	h(x1	PROPN
ejpam-854	361	9	)	)	PUNCT
ejpam-854	361	10	,	,	PUNCT
ejpam-854	361	11	.	.	PUNCT
ejpam-854	361	12	.	.	PUNCT
ejpam-854	362	1	.	.	PUNCT
ejpam-854	363	1	,	,	PUNCT
ejpam-854	363	2	g(xk)−	g(xk)−	NOUN
ejpam-854	363	3	h(xk))‖k	h(xk))‖k	NOUN
ejpam-854	363	4	≤	≤	NUM
ejpam-854	363	5	α,∀x1	α,∀x1	NOUN
ejpam-854	363	6	,	,	PUNCT
ejpam-854	363	7	.	.	PUNCT
ejpam-854	363	8	.	.	PUNCT
ejpam-854	363	9	.	.	PUNCT
ejpam-854	364	1	,	,	PUNCT
ejpam-854	364	2	xk	xk	PROPN
ejpam-854	364	3	∈	∈	PROPN
ejpam-854	364	4	e	e	PROPN
ejpam-854	364	5	,	,	PUNCT
ejpam-854	364	6	k	k	PROPN
ejpam-854	364	7	∈	∈	PROPN
ejpam-854	364	8	n	n	CCONJ
ejpam-854	364	9	}	}	PUNCT
ejpam-854	364	10	.	.	PUNCT
ejpam-854	365	1	it	it	PRON
ejpam-854	365	2	is	be	AUX
ejpam-854	365	3	easy	easy	ADJ
ejpam-854	365	4	to	to	PART
ejpam-854	365	5	show	show	VERB
ejpam-854	365	6	that	that	SCONJ
ejpam-854	365	7	(	(	PUNCT
ejpam-854	365	8	ω	ω	NOUN
ejpam-854	365	9	,	,	PUNCT
ejpam-854	365	10	d	d	NOUN
ejpam-854	365	11	)	)	PUNCT
ejpam-854	365	12	is	be	AUX
ejpam-854	365	13	a	a	DET
ejpam-854	365	14	generalized	generalize	VERB
ejpam-854	365	15	complete	complete	ADJ
ejpam-854	365	16	metric	metric	ADJ
ejpam-854	365	17	space	space	NOUN
ejpam-854	365	18	[	[	X
ejpam-854	365	19	see	see	VERB
ejpam-854	365	20	20	20	NUM
ejpam-854	365	21	,	,	PUNCT
ejpam-854	365	22	lemma	lemma	PROPN
ejpam-854	365	23	2.1	2.1	NUM
ejpam-854	365	24	]	]	PUNCT
ejpam-854	365	25	.	.	PUNCT
ejpam-854	366	1	define	define	VERB
ejpam-854	366	2	j	j	PROPN
ejpam-854	366	3	:	:	PUNCT
ejpam-854	366	4	ω	ω	PROPN
ejpam-854	366	5	→	→	SYM
ejpam-854	366	6	ω	ω	PROPN
ejpam-854	366	7	by	by	ADP
ejpam-854	366	8	j	j	PROPN
ejpam-854	366	9	g(x	g(x	NOUN
ejpam-854	366	10	)	)	PUNCT
ejpam-854	367	1	=	=	VERB
ejpam-854	367	2	g(2x)/4	g(2x)/4	NOUN
ejpam-854	367	3	for	for	ADP
ejpam-854	367	4	all	all	PRON
ejpam-854	367	5	x	x	SYM
ejpam-854	367	6	∈	∈	PROPN
ejpam-854	367	7	e.	e.	PROPN
ejpam-854	367	8	let	let	VERB
ejpam-854	367	9	g	g	NOUN
ejpam-854	367	10	,	,	PUNCT
ejpam-854	367	11	h	h	PROPN
ejpam-854	367	12	∈	∈	PROPN
ejpam-854	367	13	ω	ω	PROPN
ejpam-854	367	14	be	be	AUX
ejpam-854	367	15	given	give	VERB
ejpam-854	367	16	such	such	ADJ
ejpam-854	367	17	that	that	DET
ejpam-854	367	18	d(g	d(g	PROPN
ejpam-854	367	19	,	,	PUNCT
ejpam-854	367	20	h	h	NOUN
ejpam-854	367	21	)	)	PUNCT
ejpam-854	367	22	<	<	X
ejpam-854	367	23	β	β	X
ejpam-854	367	24	,	,	PUNCT
ejpam-854	367	25	by	by	ADP
ejpam-854	367	26	the	the	DET
ejpam-854	367	27	definition	definition	NOUN
ejpam-854	367	28	,	,	PUNCT
ejpam-854	367	29	sup	sup	NOUN
ejpam-854	367	30	k∈n	k∈n	PROPN
ejpam-854	367	31	‖(g(x1)−	‖(g(x1)−	PROPN
ejpam-854	367	32	h(x1	h(x1	PROPN
ejpam-854	367	33	)	)	PUNCT
ejpam-854	367	34	,	,	PUNCT
ejpam-854	367	35	.	.	PUNCT
ejpam-854	367	36	.	.	PUNCT
ejpam-854	367	37	.	.	PUNCT
ejpam-854	368	1	,	,	PUNCT
ejpam-854	368	2	g(xk)−	g(xk)−	NOUN
ejpam-854	368	3	h(xk))‖k	h(xk))‖k	NOUN
ejpam-854	368	4	≤	≤	NUM
ejpam-854	368	5	β	β	X
ejpam-854	368	6	for	for	ADP
ejpam-854	368	7	all	all	DET
ejpam-854	368	8	x1	x1	PROPN
ejpam-854	368	9	,	,	PUNCT
ejpam-854	368	10	.	.	PUNCT
ejpam-854	368	11	.	.	PUNCT
ejpam-854	368	12	.	.	PUNCT
ejpam-854	369	1	,	,	PUNCT
ejpam-854	369	2	xk	xk	PROPN
ejpam-854	369	3	∈	∈	PROPN
ejpam-854	369	4	e	e	PROPN
ejpam-854	369	5	,	,	PUNCT
ejpam-854	369	6	k	k	PROPN
ejpam-854	369	7	∈	∈	PROPN
ejpam-854	369	8	n.	n.	NOUN
ejpam-854	369	9	hence	hence	ADV
ejpam-854	369	10	sup	sup	PROPN
ejpam-854	369	11	k∈n	k∈n	PROPN
ejpam-854	369	12	‖(j	‖(j	PROPN
ejpam-854	369	13	g(x1)−	g(x1)−	PROPN
ejpam-854	369	14	jh(x1	jh(x1	PROPN
ejpam-854	369	15	)	)	PUNCT
ejpam-854	369	16	,	,	PUNCT
ejpam-854	369	17	.	.	PUNCT
ejpam-854	369	18	.	.	PUNCT
ejpam-854	369	19	.	.	PUNCT
ejpam-854	370	1	,	,	PUNCT
ejpam-854	370	2	j	j	PROPN
ejpam-854	370	3	g(xk)−	g(xk)−	NOUN
ejpam-854	370	4	jh(xk))‖k	jh(xk))‖k	PROPN
ejpam-854	370	5	≤	≤	NUM
ejpam-854	370	6	1	1	NUM
ejpam-854	370	7	4	4	NUM
ejpam-854	370	8	sup	sup	NOUN
ejpam-854	370	9	k∈n	k∈n	NOUN
ejpam-854	370	10	‖(g(2x1)−	‖(g(2x1)−	PROPN
ejpam-854	371	1	h(2x1	h(2x1	NUM
ejpam-854	371	2	)	)	PUNCT
ejpam-854	371	3	,	,	PUNCT
ejpam-854	371	4	.	.	PUNCT
ejpam-854	371	5	.	.	PUNCT
ejpam-854	372	1	.	.	PUNCT
ejpam-854	373	1	,	,	PUNCT
ejpam-854	373	2	g(2xk)−	g(2xk)−	PROPN
ejpam-854	373	3	h(2xk))‖k	h(2xk))‖k	VERB
ejpam-854	373	4	≤	≤	NUM
ejpam-854	373	5	β	β	X
ejpam-854	373	6	4	4	NUM
ejpam-854	373	7	for	for	ADP
ejpam-854	373	8	all	all	DET
ejpam-854	373	9	x1	x1	PROPN
ejpam-854	373	10	,	,	PUNCT
ejpam-854	373	11	.	.	PUNCT
ejpam-854	373	12	.	.	PUNCT
ejpam-854	374	1	.	.	PUNCT
ejpam-854	375	1	,	,	PUNCT
ejpam-854	375	2	xk	xk	PROPN
ejpam-854	375	3	∈	∈	PROPN
ejpam-854	375	4	e	e	PROPN
ejpam-854	375	5	,	,	PUNCT
ejpam-854	375	6	k	k	PROPN
ejpam-854	375	7	∈	∈	PROPN
ejpam-854	375	8	n.	n.	NOUN
ejpam-854	375	9	by	by	ADP
ejpam-854	375	10	definition	definition	NOUN
ejpam-854	375	11	,	,	PUNCT
ejpam-854	375	12	d(j	d(j	PROPN
ejpam-854	375	13	g	g	PROPN
ejpam-854	375	14	,	,	PUNCT
ejpam-854	375	15	jh	jh	PROPN
ejpam-854	375	16	)	)	PUNCT
ejpam-854	375	17	≤	≤	NOUN
ejpam-854	375	18	β/4	β/4	NUM
ejpam-854	375	19	.	.	PUNCT
ejpam-854	376	1	therefore	therefore	ADV
ejpam-854	376	2	,	,	PUNCT
ejpam-854	376	3	d(j	d(j	PROPN
ejpam-854	376	4	g	g	PROPN
ejpam-854	376	5	,	,	PUNCT
ejpam-854	376	6	jh	jh	PROPN
ejpam-854	376	7	)	)	PUNCT
ejpam-854	376	8	≤	≤	NOUN
ejpam-854	376	9	1	1	NUM
ejpam-854	376	10	4	4	NUM
ejpam-854	376	11	d(g	d(g	PROPN
ejpam-854	376	12	,	,	PUNCT
ejpam-854	376	13	h	h	NOUN
ejpam-854	376	14	)	)	PUNCT
ejpam-854	376	15	for	for	ADP
ejpam-854	376	16	all	all	DET
ejpam-854	376	17	g	g	NOUN
ejpam-854	376	18	,	,	PUNCT
ejpam-854	376	19	h	h	NOUN
ejpam-854	376	20	∈	∈	PROPN
ejpam-854	376	21	ω	ω	PROPN
ejpam-854	376	22	.	.	PUNCT
ejpam-854	377	1	this	this	PRON
ejpam-854	377	2	means	mean	VERB
ejpam-854	377	3	that	that	SCONJ
ejpam-854	377	4	j	j	PROPN
ejpam-854	377	5	is	be	AUX
ejpam-854	377	6	a	a	DET
ejpam-854	377	7	strictly	strictly	ADV
ejpam-854	377	8	contractive	contractive	ADJ
ejpam-854	377	9	self	self	NOUN
ejpam-854	377	10	-	-	PUNCT
ejpam-854	377	11	mapping	mapping	NOUN
ejpam-854	377	12	of	of	ADP
ejpam-854	377	13	ω	ω	PROPN
ejpam-854	377	14	with	with	ADP
ejpam-854	377	15	lipschitz	lipschitz	NOUN
ejpam-854	377	16	constant	constant	ADJ
ejpam-854	377	17	1/4	1/4	NUM
ejpam-854	377	18	.	.	PUNCT
ejpam-854	378	1	now	now	ADV
ejpam-854	378	2	,	,	PUNCT
ejpam-854	378	3	let	let	VERB
ejpam-854	378	4	f̃	f̃	PROPN
ejpam-854	378	5	:	:	PUNCT
ejpam-854	378	6	e→	e→	NOUN
ejpam-854	378	7	f	f	PROPN
ejpam-854	378	8	be	be	AUX
ejpam-854	378	9	the	the	DET
ejpam-854	378	10	mapping	mapping	NOUN
ejpam-854	378	11	defined	define	VERB
ejpam-854	378	12	by	by	ADP
ejpam-854	378	13	f̃	f̃	PROPN
ejpam-854	378	14	(	(	PUNCT
ejpam-854	378	15	x	x	NOUN
ejpam-854	378	16	)	)	PUNCT
ejpam-854	378	17	:	:	PUNCT
ejpam-854	379	1	=	=	SYM
ejpam-854	379	2	f	f	X
ejpam-854	379	3	(	(	PUNCT
ejpam-854	379	4	2x)−16	2x)−16	NUM
ejpam-854	379	5	f	f	X
ejpam-854	379	6	(	(	PUNCT
ejpam-854	379	7	x	x	X
ejpam-854	379	8	)	)	PUNCT
ejpam-854	379	9	for	for	ADP
ejpam-854	379	10	each	each	DET
ejpam-854	379	11	x	x	SYM
ejpam-854	379	12	∈	∈	PROPN
ejpam-854	379	13	e.	e.	PROPN
ejpam-854	379	14	by	by	PROPN
ejpam-854	379	15	(	(	PUNCT
ejpam-854	379	16	34	34	NUM
ejpam-854	379	17	)	)	PUNCT
ejpam-854	379	18	,	,	PUNCT
ejpam-854	379	19	we	we	PRON
ejpam-854	379	20	get	get	VERB
ejpam-854	379	21	sup	sup	NOUN
ejpam-854	379	22	k∈n	k∈n	PROPN
ejpam-854	379	23	‖	‖	PROPN
ejpam-854	379	24	(	(	PUNCT
ejpam-854	379	25	f̃	f̃	PROPN
ejpam-854	379	26	(	(	PUNCT
ejpam-854	379	27	2x1)−	2x1)−	NUM
ejpam-854	379	28	4	4	NUM
ejpam-854	379	29	f̃	f̃	PROPN
ejpam-854	379	30	(	(	PUNCT
ejpam-854	379	31	x1	x1	PROPN
ejpam-854	379	32	)	)	PUNCT
ejpam-854	379	33	,	,	PUNCT
ejpam-854	379	34	.	.	PUNCT
ejpam-854	379	35	.	.	PUNCT
ejpam-854	379	36	.	.	PUNCT
ejpam-854	379	37	,	,	PUNCT
ejpam-854	379	38	f̃	f̃	PROPN
ejpam-854	379	39	(	(	PUNCT
ejpam-854	379	40	2xk)−	2xk)−	PROPN
ejpam-854	379	41	4	4	NUM
ejpam-854	379	42	f̃	f̃	PROPN
ejpam-854	379	43	(	(	PUNCT
ejpam-854	379	44	xk))‖k	xk))‖k	PROPN
ejpam-854	379	45	≤	≤	PROPN
ejpam-854	379	46	24n2	24n2	NUM
ejpam-854	379	47	+	+	CCONJ
ejpam-854	379	48	6	6	NUM
ejpam-854	379	49	n4−	n4−	NOUN
ejpam-854	379	50	n2	n2	NOUN
ejpam-854	379	51	ǫ	ǫ	NOUN
ejpam-854	379	52	.	.	PUNCT
ejpam-854	379	53	(	(	PUNCT
ejpam-854	379	54	35	35	NUM
ejpam-854	379	55	)	)	PUNCT
ejpam-854	379	56	multiplying	multiplying	NOUN
ejpam-854	379	57	(	(	PUNCT
ejpam-854	379	58	35	35	NUM
ejpam-854	379	59	)	)	PUNCT
ejpam-854	379	60	by	by	ADP
ejpam-854	379	61	1/4	1/4	NUM
ejpam-854	379	62	,	,	PUNCT
ejpam-854	379	63	we	we	PRON
ejpam-854	379	64	obtain	obtain	VERB
ejpam-854	379	65	sup	sup	NOUN
ejpam-854	379	66	k∈n	k∈n	NOUN
ejpam-854	379	67	‖(j	‖(j	PROPN
ejpam-854	380	1	f̃	f̃	PROPN
ejpam-854	381	1	(	(	PUNCT
ejpam-854	381	2	x1)−	x1)−	PROPN
ejpam-854	381	3	f̃	f̃	PROPN
ejpam-854	381	4	(	(	PUNCT
ejpam-854	381	5	x1	x1	PROPN
ejpam-854	381	6	)	)	PUNCT
ejpam-854	381	7	,	,	PUNCT
ejpam-854	381	8	.	.	PUNCT
ejpam-854	381	9	.	.	PUNCT
ejpam-854	382	1	.	.	PUNCT
ejpam-854	383	1	,	,	PUNCT
ejpam-854	383	2	j	j	PROPN
ejpam-854	383	3	f̃	f̃	PROPN
ejpam-854	383	4	(	(	PUNCT
ejpam-854	383	5	xk)−	xk)−	PROPN
ejpam-854	383	6	f̃	f̃	PROPN
ejpam-854	383	7	(	(	PUNCT
ejpam-854	383	8	xk))‖k	xk))‖k	PROPN
ejpam-854	383	9	≤	≤	PROPN
ejpam-854	383	10	24n2	24n2	NUM
ejpam-854	383	11	+	+	CCONJ
ejpam-854	383	12	6	6	NUM
ejpam-854	383	13	4(n4−	4(n4−	NUM
ejpam-854	383	14	n2	n2	NOUN
ejpam-854	383	15	)	)	PUNCT
ejpam-854	383	16	ǫ	ǫ	NOUN
ejpam-854	383	17	.	.	PUNCT
ejpam-854	384	1	(	(	PUNCT
ejpam-854	384	2	36	36	NUM
ejpam-854	384	3	)	)	PUNCT
ejpam-854	384	4	then	then	ADV
ejpam-854	384	5	d(j	d(j	PROPN
ejpam-854	384	6	f̃	f̃	PROPN
ejpam-854	384	7	,	,	PUNCT
ejpam-854	384	8	f̃	f̃	PROPN
ejpam-854	384	9	)	)	PUNCT
ejpam-854	384	10	≤	≤	PROPN
ejpam-854	384	11	ǫ(24n2	ǫ(24n2	NUM
ejpam-854	385	1	+	+	CCONJ
ejpam-854	386	1	6)/(4(n4−	6)/(4(n4−	NUM
ejpam-854	386	2	n2	n2	NOUN
ejpam-854	386	3	)	)	PUNCT
ejpam-854	386	4	)	)	PUNCT
ejpam-854	387	1	and	and	CCONJ
ejpam-854	387	2	therefore	therefore	ADV
ejpam-854	387	3	,	,	PUNCT
ejpam-854	387	4	by	by	ADP
ejpam-854	387	5	theorem	theorem	NOUN
ejpam-854	387	6	1	1	NUM
ejpam-854	387	7	,	,	PUNCT
ejpam-854	387	8	j	j	PROPN
ejpam-854	387	9	has	have	VERB
ejpam-854	387	10	a	a	DET
ejpam-854	387	11	unique	unique	ADJ
ejpam-854	387	12	fixed	fix	VERB
ejpam-854	387	13	point	point	NOUN
ejpam-854	387	14	b	b	PROPN
ejpam-854	387	15	:	:	PUNCT
ejpam-854	387	16	e→	e→	PROPN
ejpam-854	387	17	f	f	PROPN
ejpam-854	387	18	in	in	ADP
ejpam-854	387	19	the	the	DET
ejpam-854	387	20	set	set	NOUN
ejpam-854	387	21	∆=	∆=	NOUN
ejpam-854	387	22	{	{	PUNCT
ejpam-854	387	23	h	h	NOUN
ejpam-854	387	24	∈	∈	PROPN
ejpam-854	387	25	ω	ω	NOUN
ejpam-854	387	26	:	:	PUNCT
ejpam-854	388	1	d	d	X
ejpam-854	388	2	(	(	PUNCT
ejpam-854	388	3	f̃	f̃	PROPN
ejpam-854	388	4	,	,	PUNCT
ejpam-854	388	5	h	h	NOUN
ejpam-854	388	6	)	)	PUNCT
ejpam-854	388	7	<	<	X
ejpam-854	388	8	∞	∞	NUM
ejpam-854	388	9	}	}	PUNCT
ejpam-854	388	10	.	.	PUNCT
ejpam-854	389	1	this	this	PRON
ejpam-854	389	2	implies	imply	VERB
ejpam-854	389	3	that	that	SCONJ
ejpam-854	389	4	b(2x	b(2x	VERB
ejpam-854	389	5	)	)	PUNCT
ejpam-854	389	6	=	=	SYM
ejpam-854	389	7	4b(x	4b(x	NUM
ejpam-854	389	8	)	)	PUNCT
ejpam-854	389	9	and	and	CCONJ
ejpam-854	389	10	b(x	b(x	NOUN
ejpam-854	389	11	)	)	PUNCT
ejpam-854	389	12	=	=	SYM
ejpam-854	389	13	lim	lim	PROPN
ejpam-854	389	14	m→∞	m→∞	NUM
ejpam-854	389	15	j	j	PROPN
ejpam-854	389	16	m	m	PROPN
ejpam-854	389	17	f̃	f̃	PROPN
ejpam-854	389	18	(	(	PUNCT
ejpam-854	389	19	x	x	NOUN
ejpam-854	389	20	)	)	PUNCT
ejpam-854	389	21	=	=	SYM
ejpam-854	389	22	lim	lim	PROPN
ejpam-854	389	23	m→∞	m→∞	NOUN
ejpam-854	389	24	1	1	NUM
ejpam-854	389	25	4	4	NUM
ejpam-854	389	26	m	m	NOUN
ejpam-854	389	27	f̃	f̃	PROPN
ejpam-854	389	28	(	(	PUNCT
ejpam-854	389	29	2mx	2mx	PROPN
ejpam-854	389	30	)	)	PUNCT
ejpam-854	389	31	(	(	PUNCT
ejpam-854	389	32	37	37	NUM
ejpam-854	389	33	)	)	PUNCT
ejpam-854	389	34	t.	t.	PROPN
ejpam-854	389	35	xu	xu	PROPN
ejpam-854	389	36	,	,	PUNCT
ejpam-854	389	37	j.	j.	PROPN
ejpam-854	389	38	rassias	rassias	PROPN
ejpam-854	389	39	,	,	PUNCT
ejpam-854	389	40	w.	w.	PROPN
ejpam-854	389	41	xu	xu	PROPN
ejpam-854	389	42	/	/	SYM
ejpam-854	389	43	eur	eur	PROPN
ejpam-854	389	44	.	.	PUNCT
ejpam-854	390	1	j.	j.	PROPN
ejpam-854	390	2	pure	pure	PROPN
ejpam-854	390	3	appl	appl	PROPN
ejpam-854	390	4	.	.	PROPN
ejpam-854	390	5	math	math	PROPN
ejpam-854	390	6	,	,	PUNCT
ejpam-854	390	7	3	3	NUM
ejpam-854	390	8	(	(	PUNCT
ejpam-854	390	9	2010	2010	NUM
ejpam-854	390	10	)	)	PUNCT
ejpam-854	390	11	,	,	PUNCT
ejpam-854	390	12	1032	1032	NUM
ejpam-854	390	13	-	-	SYM
ejpam-854	390	14	1047	1047	NUM
ejpam-854	390	15	1042	1042	NUM
ejpam-854	390	16	for	for	ADP
ejpam-854	390	17	all	all	DET
ejpam-854	390	18	x	x	SYM
ejpam-854	390	19	∈	∈	PROPN
ejpam-854	390	20	e.	e.	PROPN
ejpam-854	390	21	since	since	SCONJ
ejpam-854	390	22	f̃	f̃	PROPN
ejpam-854	390	23	:	:	PUNCT
ejpam-854	390	24	e→	e→	PROPN
ejpam-854	390	25	f	f	PROPN
ejpam-854	390	26	is	be	AUX
ejpam-854	390	27	even	even	ADV
ejpam-854	390	28	,	,	PUNCT
ejpam-854	390	29	b	b	X
ejpam-854	390	30	:	:	PUNCT
ejpam-854	390	31	e→	e→	PROPN
ejpam-854	390	32	f	f	PROPN
ejpam-854	390	33	is	be	AUX
ejpam-854	390	34	an	an	DET
ejpam-854	390	35	even	even	ADJ
ejpam-854	390	36	mapping	mapping	NOUN
ejpam-854	390	37	.	.	PUNCT
ejpam-854	391	1	moreover	moreover	ADV
ejpam-854	391	2	,	,	PUNCT
ejpam-854	391	3	d	d	X
ejpam-854	391	4	(	(	PUNCT
ejpam-854	391	5	f̃	f̃	PROPN
ejpam-854	391	6	,	,	PUNCT
ejpam-854	391	7	a)≤	a)≤	PROPN
ejpam-854	391	8	1	1	NUM
ejpam-854	391	9	1−	1−	NUM
ejpam-854	391	10	l	l	NOUN
ejpam-854	391	11	d	d	PROPN
ejpam-854	391	12	(	(	PUNCT
ejpam-854	391	13	f̃	f̃	PROPN
ejpam-854	391	14	,	,	PUNCT
ejpam-854	391	15	j	j	PROPN
ejpam-854	391	16	f̃	f̃	PROPN
ejpam-854	391	17	)	)	PUNCT
ejpam-854	391	18	≤	≤	NOUN
ejpam-854	391	19	8n2	8n2	NUM
ejpam-854	391	20	+	+	SYM
ejpam-854	391	21	2	2	NUM
ejpam-854	391	22	n4−	n4−	NOUN
ejpam-854	391	23	n2	n2	NOUN
ejpam-854	391	24	ǫ	ǫ	NOUN
ejpam-854	391	25	.	.	PUNCT
ejpam-854	392	1	this	this	PRON
ejpam-854	392	2	implies	imply	VERB
ejpam-854	392	3	that	that	SCONJ
ejpam-854	392	4	the	the	DET
ejpam-854	392	5	inequality	inequality	NOUN
ejpam-854	392	6	(	(	PUNCT
ejpam-854	392	7	27	27	NUM
ejpam-854	392	8	)	)	PUNCT
ejpam-854	392	9	holds	hold	VERB
ejpam-854	392	10	.	.	PUNCT
ejpam-854	393	1	also	also	ADV
ejpam-854	393	2	we	we	PRON
ejpam-854	393	3	have	have	VERB
ejpam-854	393	4	‖db(x	‖db(x	PROPN
ejpam-854	393	5	,	,	PUNCT
ejpam-854	393	6	y)‖	y)‖	PROPN
ejpam-854	393	7	=	=	PROPN
ejpam-854	393	8	lim	lim	PROPN
ejpam-854	393	9	m→∞	m→∞	NOUN
ejpam-854	393	10	1	1	NUM
ejpam-854	393	11	4	4	NUM
ejpam-854	393	12	m	m	NOUN
ejpam-854	393	13	‖d	‖d	ADJ
ejpam-854	393	14	f	f	X
ejpam-854	393	15	(	(	PUNCT
ejpam-854	393	16	2m+1	2m+1	PROPN
ejpam-854	393	17	x	x	PROPN
ejpam-854	393	18	,	,	PUNCT
ejpam-854	393	19	2m+1	2m+1	PROPN
ejpam-854	393	20	y)−	y)−	PROPN
ejpam-854	393	21	16d	16d	NUM
ejpam-854	393	22	f	f	PROPN
ejpam-854	393	23	(	(	PUNCT
ejpam-854	393	24	2mx	2mx	ADJ
ejpam-854	393	25	,	,	PUNCT
ejpam-854	393	26	2	2	NUM
ejpam-854	393	27	m	m	NOUN
ejpam-854	393	28	y)‖	y)‖	ADJ
ejpam-854	393	29	≤	≤	ADJ
ejpam-854	393	30	lim	lim	PROPN
ejpam-854	393	31	m→∞	m→∞	NUM
ejpam-854	393	32	17ǫ	17ǫ	NOUN
ejpam-854	393	33	4	4	NUM
ejpam-854	393	34	m	m	NOUN
ejpam-854	393	35	=	=	SYM
ejpam-854	393	36	0	0	NUM
ejpam-854	393	37	,	,	PUNCT
ejpam-854	393	38	and	and	CCONJ
ejpam-854	393	39	b	b	X
ejpam-854	393	40	satisfies	satisfie	NOUN
ejpam-854	393	41	(	(	PUNCT
ejpam-854	393	42	5	5	NUM
ejpam-854	393	43	)	)	PUNCT
ejpam-854	393	44	.	.	PUNCT
ejpam-854	394	1	by	by	ADP
ejpam-854	394	2	theorem	theorem	NOUN
ejpam-854	394	3	2.1	2.1	NUM
ejpam-854	394	4	of	of	ADP
ejpam-854	394	5	[	[	X
ejpam-854	394	6	33	33	NUM
ejpam-854	394	7	]	]	PUNCT
ejpam-854	394	8	,	,	PUNCT
ejpam-854	394	9	the	the	DET
ejpam-854	394	10	mapping	mapping	NOUN
ejpam-854	394	11	x	x	PUNCT
ejpam-854	394	12	→	→	SYM
ejpam-854	394	13	b(2x)−	b(2x)−	PROPN
ejpam-854	394	14	16b(x	16b(x	NUM
ejpam-854	394	15	)	)	PUNCT
ejpam-854	394	16	is	be	AUX
ejpam-854	394	17	quadratic	quadratic	ADJ
ejpam-854	394	18	.	.	PUNCT
ejpam-854	395	1	hence	hence	ADV
ejpam-854	395	2	b(2x	b(2x	VERB
ejpam-854	395	3	)	)	PUNCT
ejpam-854	395	4	=	=	SYM
ejpam-854	395	5	4b(x	4b(x	NUM
ejpam-854	395	6	)	)	PUNCT
ejpam-854	395	7	implies	imply	VERB
ejpam-854	395	8	that	that	SCONJ
ejpam-854	395	9	b	b	NOUN
ejpam-854	395	10	is	be	AUX
ejpam-854	395	11	a	a	DET
ejpam-854	395	12	quadratic	quadratic	ADJ
ejpam-854	395	13	mapping	mapping	NOUN
ejpam-854	395	14	.	.	PUNCT
ejpam-854	396	1	the	the	DET
ejpam-854	396	2	rest	rest	NOUN
ejpam-854	396	3	of	of	ADP
ejpam-854	396	4	the	the	DET
ejpam-854	396	5	proof	proof	NOUN
ejpam-854	396	6	is	be	AUX
ejpam-854	396	7	similar	similar	ADJ
ejpam-854	396	8	to	to	ADP
ejpam-854	396	9	that	that	PRON
ejpam-854	396	10	of	of	ADP
ejpam-854	396	11	theorem	theorem	ADJ
ejpam-854	396	12	2	2	NUM
ejpam-854	396	13	.	.	PUNCT
ejpam-854	396	14	theorem	theorem	NOUN
ejpam-854	396	15	5	5	NUM
ejpam-854	396	16	.	.	PUNCT
ejpam-854	397	1	let	let	VERB
ejpam-854	397	2	e	e	PRON
ejpam-854	397	3	be	be	AUX
ejpam-854	397	4	a	a	DET
ejpam-854	397	5	linear	linear	ADJ
ejpam-854	397	6	space	space	NOUN
ejpam-854	397	7	and	and	CCONJ
ejpam-854	397	8	let	let	VERB
ejpam-854	397	9	(	(	PUNCT
ejpam-854	397	10	(	(	PUNCT
ejpam-854	397	11	f	f	X
ejpam-854	397	12	k,‖	k,‖	PROPN
ejpam-854	397	13	·	·	PUNCT
ejpam-854	397	14	‖k	‖k	PROPN
ejpam-854	397	15	)	)	PUNCT
ejpam-854	397	16	:	:	PUNCT
ejpam-854	398	1	k	k	PROPN
ejpam-854	398	2	∈	∈	PROPN
ejpam-854	398	3	n	n	CCONJ
ejpam-854	398	4	)	)	PUNCT
ejpam-854	398	5	be	be	AUX
ejpam-854	398	6	a	a	DET
ejpam-854	398	7	multi	multi	ADJ
ejpam-854	398	8	-	-	ADJ
ejpam-854	398	9	banach	banach	ADJ
ejpam-854	398	10	space	space	NOUN
ejpam-854	398	11	.	.	PUNCT
ejpam-854	399	1	suppose	suppose	VERB
ejpam-854	399	2	that	that	SCONJ
ejpam-854	399	3	ǫ	ǫ	PRON
ejpam-854	399	4	≥	≥	NOUN
ejpam-854	399	5	0	0	NUM
ejpam-854	399	6	and	and	CCONJ
ejpam-854	399	7	f	f	NOUN
ejpam-854	399	8	:	:	PUNCT
ejpam-854	399	9	e→	e→	PROPN
ejpam-854	399	10	f	f	PROPN
ejpam-854	399	11	is	be	AUX
ejpam-854	399	12	an	an	DET
ejpam-854	399	13	even	even	ADV
ejpam-854	399	14	mapping	mapping	NOUN
ejpam-854	399	15	with	with	ADP
ejpam-854	399	16	f	f	PROPN
ejpam-854	399	17	(	(	PUNCT
ejpam-854	399	18	0	0	NUM
ejpam-854	399	19	)	)	PUNCT
ejpam-854	399	20	=	=	SYM
ejpam-854	399	21	0	0	NUM
ejpam-854	399	22	,	,	PUNCT
ejpam-854	399	23	satisfying	satisfy	VERB
ejpam-854	399	24	condition	condition	NOUN
ejpam-854	399	25	sup	sup	NOUN
ejpam-854	399	26	k∈n	k∈n	PROPN
ejpam-854	399	27	‖(d	‖(d	PROPN
ejpam-854	400	1	f	f	PROPN
ejpam-854	400	2	(	(	PUNCT
ejpam-854	400	3	x1	x1	PROPN
ejpam-854	400	4	,	,	PUNCT
ejpam-854	400	5	y1	y1	PROPN
ejpam-854	400	6	)	)	PUNCT
ejpam-854	400	7	,	,	PUNCT
ejpam-854	400	8	.	.	PUNCT
ejpam-854	400	9	.	.	PUNCT
ejpam-854	400	10	.	.	PUNCT
ejpam-854	401	1	,	,	PUNCT
ejpam-854	402	1	d	d	X
ejpam-854	402	2	f	f	X
ejpam-854	402	3	(	(	PUNCT
ejpam-854	402	4	xk	xk	PROPN
ejpam-854	402	5	,	,	PUNCT
ejpam-854	402	6	yk))‖k	yk))‖k	INTJ
ejpam-854	402	7	≤	≤	ADJ
ejpam-854	402	8	ǫ	ǫ	NOUN
ejpam-854	402	9	for	for	ADP
ejpam-854	402	10	all	all	DET
ejpam-854	402	11	x1	x1	PROPN
ejpam-854	402	12	,	,	PUNCT
ejpam-854	402	13	.	.	PUNCT
ejpam-854	402	14	.	.	PUNCT
ejpam-854	402	15	.	.	PUNCT
ejpam-854	403	1	,	,	PUNCT
ejpam-854	403	2	xk	xk	PROPN
ejpam-854	403	3	,	,	PUNCT
ejpam-854	403	4	y1	y1	PROPN
ejpam-854	403	5	,	,	PUNCT
ejpam-854	403	6	.	.	PUNCT
ejpam-854	403	7	.	.	PUNCT
ejpam-854	403	8	.	.	PUNCT
ejpam-854	404	1	,	,	PUNCT
ejpam-854	404	2	yk	yk	PROPN
ejpam-854	404	3	∈	∈	PROPN
ejpam-854	404	4	e.	e.	PROPN
ejpam-854	404	5	then	then	ADV
ejpam-854	404	6	there	there	PRON
ejpam-854	404	7	exists	exist	VERB
ejpam-854	404	8	a	a	DET
ejpam-854	404	9	unique	unique	ADJ
ejpam-854	404	10	quartic	quartic	ADJ
ejpam-854	404	11	mapping	mapping	NOUN
ejpam-854	404	12	q	q	NOUN
ejpam-854	404	13	:	:	PUNCT
ejpam-854	404	14	e	e	X
ejpam-854	404	15	→	→	SYM
ejpam-854	404	16	f	f	PROPN
ejpam-854	404	17	such	such	ADJ
ejpam-854	404	18	that	that	DET
ejpam-854	404	19	sup	sup	NOUN
ejpam-854	404	20	k∈n	k∈n	PROPN
ejpam-854	404	21	‖	‖	PROPN
ejpam-854	404	22	(	(	PUNCT
ejpam-854	404	23	f	f	X
ejpam-854	404	24	(	(	PUNCT
ejpam-854	404	25	2x1)−	2x1)−	NUM
ejpam-854	404	26	4	4	NUM
ejpam-854	404	27	f	f	NOUN
ejpam-854	404	28	(	(	PUNCT
ejpam-854	404	29	x1)−q(x1	x1)−q(x1	PROPN
ejpam-854	404	30	)	)	PUNCT
ejpam-854	404	31	,	,	PUNCT
ejpam-854	404	32	.	.	PUNCT
ejpam-854	404	33	.	.	PUNCT
ejpam-854	405	1	.	.	PUNCT
ejpam-854	406	1	,	,	PUNCT
ejpam-854	406	2	f	f	PROPN
ejpam-854	406	3	(	(	PUNCT
ejpam-854	406	4	2xk)−	2xk)−	NOUN
ejpam-854	406	5	4	4	NUM
ejpam-854	406	6	f	f	NOUN
ejpam-854	406	7	(	(	PUNCT
ejpam-854	406	8	xk)−q(xk))‖k	xk)−q(xk))‖k	X
ejpam-854	406	9	≤	≤	ADV
ejpam-854	406	10	8n2	8n2	NUM
ejpam-854	406	11	+	+	SYM
ejpam-854	406	12	2	2	NUM
ejpam-854	406	13	5(n4−	5(n4−	NUM
ejpam-854	406	14	n2	n2	NOUN
ejpam-854	406	15	)	)	PUNCT
ejpam-854	406	16	ǫ	ǫ	NOUN
ejpam-854	406	17	for	for	ADP
ejpam-854	406	18	all	all	DET
ejpam-854	406	19	x1	x1	PROPN
ejpam-854	406	20	,	,	PUNCT
ejpam-854	406	21	.	.	PUNCT
ejpam-854	406	22	.	.	PUNCT
ejpam-854	406	23	.	.	PUNCT
ejpam-854	407	1	,	,	PUNCT
ejpam-854	407	2	xk	xk	PROPN
ejpam-854	407	3	∈	∈	PROPN
ejpam-854	407	4	e.	e.	PROPN
ejpam-854	407	5	proof	proof	PROPN
ejpam-854	407	6	.	.	PUNCT
ejpam-854	408	1	the	the	DET
ejpam-854	408	2	proof	proof	NOUN
ejpam-854	408	3	is	be	AUX
ejpam-854	408	4	similar	similar	ADJ
ejpam-854	408	5	to	to	ADP
ejpam-854	408	6	that	that	PRON
ejpam-854	408	7	of	of	ADP
ejpam-854	408	8	theorem	theorem	ADJ
ejpam-854	408	9	4	4	NUM
ejpam-854	408	10	.	.	PUNCT
ejpam-854	408	11	theorem	theorem	NOUN
ejpam-854	408	12	6	6	NUM
ejpam-854	408	13	.	.	PUNCT
ejpam-854	409	1	let	let	VERB
ejpam-854	409	2	e	e	PRON
ejpam-854	409	3	be	be	AUX
ejpam-854	409	4	a	a	DET
ejpam-854	409	5	linear	linear	ADJ
ejpam-854	409	6	space	space	NOUN
ejpam-854	409	7	and	and	CCONJ
ejpam-854	409	8	let	let	VERB
ejpam-854	409	9	(	(	PUNCT
ejpam-854	409	10	(	(	PUNCT
ejpam-854	409	11	f	f	X
ejpam-854	409	12	k,‖	k,‖	PROPN
ejpam-854	409	13	·	·	PUNCT
ejpam-854	409	14	‖k	‖k	PROPN
ejpam-854	409	15	)	)	PUNCT
ejpam-854	409	16	:	:	PUNCT
ejpam-854	410	1	k	k	PROPN
ejpam-854	410	2	∈	∈	PROPN
ejpam-854	410	3	n	n	CCONJ
ejpam-854	410	4	)	)	PUNCT
ejpam-854	410	5	be	be	AUX
ejpam-854	410	6	a	a	DET
ejpam-854	410	7	multi	multi	ADJ
ejpam-854	410	8	-	-	ADJ
ejpam-854	410	9	banach	banach	ADJ
ejpam-854	410	10	space	space	NOUN
ejpam-854	410	11	.	.	PUNCT
ejpam-854	411	1	suppose	suppose	VERB
ejpam-854	411	2	that	that	SCONJ
ejpam-854	411	3	ǫ	ǫ	PRON
ejpam-854	411	4	≥	≥	NOUN
ejpam-854	411	5	0	0	NUM
ejpam-854	411	6	and	and	CCONJ
ejpam-854	411	7	f	f	NOUN
ejpam-854	411	8	:	:	PUNCT
ejpam-854	411	9	e→	e→	PROPN
ejpam-854	411	10	f	f	PROPN
ejpam-854	411	11	is	be	AUX
ejpam-854	411	12	an	an	DET
ejpam-854	411	13	odd	odd	ADJ
ejpam-854	411	14	mapping	mapping	NOUN
ejpam-854	411	15	satisfying	satisfying	ADJ
ejpam-854	411	16	sup	sup	NOUN
ejpam-854	411	17	k∈n	k∈n	PROPN
ejpam-854	411	18	‖(d	‖(d	PROPN
ejpam-854	412	1	f	f	PROPN
ejpam-854	412	2	(	(	PUNCT
ejpam-854	412	3	x1	x1	PROPN
ejpam-854	412	4	,	,	PUNCT
ejpam-854	412	5	y1	y1	PROPN
ejpam-854	412	6	)	)	PUNCT
ejpam-854	412	7	,	,	PUNCT
ejpam-854	412	8	.	.	PUNCT
ejpam-854	412	9	.	.	PUNCT
ejpam-854	412	10	.	.	PUNCT
ejpam-854	413	1	,	,	PUNCT
ejpam-854	414	1	d	d	X
ejpam-854	414	2	f	f	X
ejpam-854	414	3	(	(	PUNCT
ejpam-854	414	4	xk	xk	PROPN
ejpam-854	414	5	,	,	PUNCT
ejpam-854	414	6	yk))‖k	yk))‖k	INTJ
ejpam-854	414	7	≤	≤	ADJ
ejpam-854	414	8	ǫ	ǫ	PRON
ejpam-854	414	9	(	(	PUNCT
ejpam-854	414	10	38	38	NUM
ejpam-854	414	11	)	)	PUNCT
ejpam-854	414	12	for	for	ADP
ejpam-854	414	13	all	all	DET
ejpam-854	414	14	x1	x1	PROPN
ejpam-854	414	15	,	,	PUNCT
ejpam-854	414	16	.	.	PUNCT
ejpam-854	414	17	.	.	PUNCT
ejpam-854	414	18	.	.	PUNCT
ejpam-854	415	1	,	,	PUNCT
ejpam-854	415	2	xk	xk	PROPN
ejpam-854	415	3	,	,	PUNCT
ejpam-854	415	4	y1	y1	PROPN
ejpam-854	415	5	,	,	PUNCT
ejpam-854	415	6	.	.	PUNCT
ejpam-854	415	7	.	.	PUNCT
ejpam-854	415	8	.	.	PUNCT
ejpam-854	416	1	,	,	PUNCT
ejpam-854	416	2	yk	yk	PROPN
ejpam-854	416	3	∈	∈	PROPN
ejpam-854	416	4	e.	e.	PROPN
ejpam-854	416	5	then	then	ADV
ejpam-854	416	6	there	there	PRON
ejpam-854	416	7	exist	exist	VERB
ejpam-854	416	8	a	a	DET
ejpam-854	416	9	unique	unique	ADJ
ejpam-854	416	10	additive	additive	NOUN
ejpam-854	416	11	mapping	mapping	NOUN
ejpam-854	416	12	a	a	DET
ejpam-854	416	13	:	:	PUNCT
ejpam-854	416	14	e	e	X
ejpam-854	416	15	→	→	SYM
ejpam-854	416	16	f	f	PROPN
ejpam-854	416	17	and	and	CCONJ
ejpam-854	416	18	a	a	DET
ejpam-854	416	19	unique	unique	ADJ
ejpam-854	416	20	cubic	cubic	ADJ
ejpam-854	416	21	mapping	mapping	NOUN
ejpam-854	416	22	c	c	NOUN
ejpam-854	416	23	:	:	PUNCT
ejpam-854	417	1	e→	e→	PROPN
ejpam-854	417	2	f	f	PROPN
ejpam-854	417	3	such	such	ADJ
ejpam-854	417	4	that	that	DET
ejpam-854	417	5	sup	sup	NOUN
ejpam-854	417	6	k∈n	k∈n	PROPN
ejpam-854	417	7	‖	‖	PROPN
ejpam-854	417	8	(	(	PUNCT
ejpam-854	417	9	f	f	PROPN
ejpam-854	417	10	(	(	PUNCT
ejpam-854	417	11	x1)−	x1)−	PROPN
ejpam-854	417	12	a(x1)−	a(x1)−	PROPN
ejpam-854	417	13	c(x1	c(x1	NOUN
ejpam-854	417	14	)	)	PUNCT
ejpam-854	417	15	,	,	PUNCT
ejpam-854	417	16	.	.	PUNCT
ejpam-854	417	17	.	.	PUNCT
ejpam-854	417	18	.	.	PUNCT
ejpam-854	418	1	,	,	PUNCT
ejpam-854	418	2	f	f	PROPN
ejpam-854	418	3	(	(	PUNCT
ejpam-854	418	4	xk)−	xk)−	PROPN
ejpam-854	418	5	a(xk)−	a(xk)−	PROPN
ejpam-854	418	6	c(xk))‖k	c(xk))‖k	VERB
ejpam-854	418	7	≤	≤	NUM
ejpam-854	418	8	4(9n2	4(9n2	NOUN
ejpam-854	418	9	+	+	CCONJ
ejpam-854	418	10	4	4	NUM
ejpam-854	418	11	)	)	PUNCT
ejpam-854	418	12	21(n4−	21(n4−	NUM
ejpam-854	418	13	n2	n2	NOUN
ejpam-854	418	14	)	)	PUNCT
ejpam-854	418	15	ǫ	ǫ	PROPN
ejpam-854	418	16	(	(	PUNCT
ejpam-854	418	17	39	39	NUM
ejpam-854	418	18	)	)	PUNCT
ejpam-854	418	19	for	for	ADP
ejpam-854	418	20	all	all	DET
ejpam-854	418	21	x1	x1	PROPN
ejpam-854	418	22	,	,	PUNCT
ejpam-854	418	23	.	.	PUNCT
ejpam-854	418	24	.	.	PUNCT
ejpam-854	418	25	.	.	PUNCT
ejpam-854	419	1	,	,	PUNCT
ejpam-854	419	2	xk	xk	PROPN
ejpam-854	419	3	∈	∈	PROPN
ejpam-854	419	4	e.	e.	PROPN
ejpam-854	419	5	proof	proof	PROPN
ejpam-854	419	6	.	.	PUNCT
ejpam-854	420	1	by	by	ADP
ejpam-854	420	2	theorems	theorem	NOUN
ejpam-854	420	3	2	2	NUM
ejpam-854	420	4	and	and	CCONJ
ejpam-854	420	5	3	3	NUM
ejpam-854	420	6	,	,	PUNCT
ejpam-854	420	7	there	there	PRON
ejpam-854	420	8	exist	exist	VERB
ejpam-854	420	9	a	a	DET
ejpam-854	420	10	unique	unique	ADJ
ejpam-854	420	11	additive	additive	ADJ
ejpam-854	420	12	mapping	mapping	NOUN
ejpam-854	420	13	a0	a0	NOUN
ejpam-854	420	14	:	:	PUNCT
ejpam-854	420	15	e	e	X
ejpam-854	420	16	→	→	SYM
ejpam-854	420	17	f	f	PROPN
ejpam-854	420	18	and	and	CCONJ
ejpam-854	420	19	a	a	DET
ejpam-854	420	20	unique	unique	ADJ
ejpam-854	420	21	cubic	cubic	ADJ
ejpam-854	420	22	mapping	mapping	NOUN
ejpam-854	420	23	c0	c0	NOUN
ejpam-854	420	24	:	:	PUNCT
ejpam-854	420	25	e→	e→	PROPN
ejpam-854	420	26	f	f	PROPN
ejpam-854	420	27	such	such	ADJ
ejpam-854	420	28	that	that	DET
ejpam-854	420	29	sup	sup	NOUN
ejpam-854	420	30	k∈n	k∈n	PROPN
ejpam-854	420	31	‖	‖	PROPN
ejpam-854	420	32	(	(	PUNCT
ejpam-854	420	33	f	f	X
ejpam-854	420	34	(	(	PUNCT
ejpam-854	420	35	2x1)−	2x1)−	NUM
ejpam-854	420	36	8	8	NUM
ejpam-854	420	37	f	f	NOUN
ejpam-854	420	38	(	(	PUNCT
ejpam-854	420	39	x1)−	x1)−	PROPN
ejpam-854	420	40	a0(x1	a0(x1	PROPN
ejpam-854	420	41	)	)	PUNCT
ejpam-854	420	42	,	,	PUNCT
ejpam-854	420	43	.	.	PUNCT
ejpam-854	420	44	.	.	PUNCT
ejpam-854	421	1	.	.	PUNCT
ejpam-854	422	1	,	,	PUNCT
ejpam-854	422	2	f	f	PROPN
ejpam-854	422	3	(	(	PUNCT
ejpam-854	422	4	2xk)−	2xk)−	PROPN
ejpam-854	422	5	8	8	NUM
ejpam-854	422	6	f	f	X
ejpam-854	422	7	(	(	PUNCT
ejpam-854	422	8	xk)−	xk)−	PROPN
ejpam-854	422	9	a0(xk))‖k	a0(xk))‖k	PROPN
ejpam-854	422	10	≤	≤	NOUN
ejpam-854	422	11	9n2	9n2	NUM
ejpam-854	422	12	+	+	SYM
ejpam-854	422	13	4	4	NUM
ejpam-854	422	14	n4	n4	PROPN
ejpam-854	422	15	−	−	PROPN
ejpam-854	422	16	n2	n2	ADJ
ejpam-854	422	17	ǫ	ǫ	X
ejpam-854	422	18	(	(	PUNCT
ejpam-854	422	19	40	40	NUM
ejpam-854	422	20	)	)	PUNCT
ejpam-854	423	1	t.	t.	PROPN
ejpam-854	423	2	xu	xu	PROPN
ejpam-854	423	3	,	,	PUNCT
ejpam-854	423	4	j.	j.	PROPN
ejpam-854	423	5	rassias	rassias	PROPN
ejpam-854	423	6	,	,	PUNCT
ejpam-854	423	7	w.	w.	PROPN
ejpam-854	423	8	xu	xu	PROPN
ejpam-854	423	9	/	/	SYM
ejpam-854	423	10	eur	eur	PROPN
ejpam-854	423	11	.	.	PUNCT
ejpam-854	424	1	j.	j.	PROPN
ejpam-854	424	2	pure	pure	PROPN
ejpam-854	424	3	appl	appl	PROPN
ejpam-854	424	4	.	.	PROPN
ejpam-854	424	5	math	math	PROPN
ejpam-854	424	6	,	,	PUNCT
ejpam-854	424	7	3	3	NUM
ejpam-854	424	8	(	(	PUNCT
ejpam-854	424	9	2010	2010	NUM
ejpam-854	424	10	)	)	PUNCT
ejpam-854	424	11	,	,	PUNCT
ejpam-854	424	12	1032	1032	NUM
ejpam-854	424	13	-	-	SYM
ejpam-854	424	14	1047	1047	NUM
ejpam-854	424	15	1043	1043	NUM
ejpam-854	424	16	and	and	CCONJ
ejpam-854	424	17	sup	sup	NOUN
ejpam-854	424	18	k∈n	k∈n	PROPN
ejpam-854	424	19	‖	‖	PROPN
ejpam-854	424	20	(	(	PUNCT
ejpam-854	424	21	f	f	X
ejpam-854	424	22	(	(	PUNCT
ejpam-854	424	23	2x1)−	2x1)−	NUM
ejpam-854	424	24	2	2	NUM
ejpam-854	424	25	f	f	NOUN
ejpam-854	424	26	(	(	PUNCT
ejpam-854	424	27	x1)−	x1)−	PROPN
ejpam-854	424	28	c0(x1	c0(x1	PROPN
ejpam-854	424	29	)	)	PUNCT
ejpam-854	424	30	,	,	PUNCT
ejpam-854	424	31	.	.	PUNCT
ejpam-854	424	32	.	.	PUNCT
ejpam-854	425	1	.	.	PUNCT
ejpam-854	426	1	,	,	PUNCT
ejpam-854	426	2	f	f	PROPN
ejpam-854	426	3	(	(	PUNCT
ejpam-854	426	4	2xk)−	2xk)−	PROPN
ejpam-854	426	5	2	2	NUM
ejpam-854	426	6	f	f	X
ejpam-854	426	7	(	(	PUNCT
ejpam-854	426	8	xk)−	xk)−	X
ejpam-854	426	9	c0(xk))‖k	c0(xk))‖k	VERB
ejpam-854	426	10	≤	≤	NUM
ejpam-854	426	11	9n2	9n2	NUM
ejpam-854	426	12	+	+	SYM
ejpam-854	426	13	4	4	NUM
ejpam-854	426	14	7(n4−	7(n4−	NUM
ejpam-854	426	15	n2	n2	ADJ
ejpam-854	426	16	)	)	PUNCT
ejpam-854	426	17	ǫ	ǫ	PROPN
ejpam-854	426	18	(	(	PUNCT
ejpam-854	426	19	41	41	NUM
ejpam-854	426	20	)	)	PUNCT
ejpam-854	426	21	for	for	ADP
ejpam-854	426	22	all	all	DET
ejpam-854	426	23	x1	x1	PROPN
ejpam-854	426	24	,	,	PUNCT
ejpam-854	426	25	.	.	PUNCT
ejpam-854	426	26	.	.	PUNCT
ejpam-854	427	1	.	.	PUNCT
ejpam-854	428	1	,	,	PUNCT
ejpam-854	428	2	xk	xk	PROPN
ejpam-854	428	3	∈	∈	PROPN
ejpam-854	428	4	e.	e.	PROPN
ejpam-854	428	5	now	now	ADV
ejpam-854	428	6	from	from	ADP
ejpam-854	428	7	(	(	PUNCT
ejpam-854	428	8	40	40	NUM
ejpam-854	428	9	)	)	PUNCT
ejpam-854	428	10	and	and	CCONJ
ejpam-854	428	11	(	(	PUNCT
ejpam-854	428	12	41	41	NUM
ejpam-854	428	13	)	)	PUNCT
ejpam-854	428	14	,	,	PUNCT
ejpam-854	428	15	one	one	PRON
ejpam-854	428	16	can	can	AUX
ejpam-854	428	17	see	see	VERB
ejpam-854	428	18	that	that	DET
ejpam-854	428	19	sup	sup	NOUN
ejpam-854	428	20	k∈n	k∈n	PROPN
ejpam-854	428	21	‖(6	‖(6	PROPN
ejpam-854	428	22	f	f	PROPN
ejpam-854	428	23	(	(	PUNCT
ejpam-854	428	24	x1	x1	PROPN
ejpam-854	428	25	)	)	PUNCT
ejpam-854	428	26	+	+	CCONJ
ejpam-854	428	27	a0(x1)−	a0(x1)−	NOUN
ejpam-854	428	28	c0(x1	c0(x1	NOUN
ejpam-854	428	29	)	)	PUNCT
ejpam-854	428	30	,	,	PUNCT
ejpam-854	428	31	.	.	PUNCT
ejpam-854	428	32	.	.	PUNCT
ejpam-854	428	33	.	.	PUNCT
ejpam-854	429	1	,	,	PUNCT
ejpam-854	429	2	6	6	NUM
ejpam-854	429	3	f	f	NOUN
ejpam-854	429	4	(	(	PUNCT
ejpam-854	429	5	xk	xk	PROPN
ejpam-854	429	6	)	)	PUNCT
ejpam-854	429	7	+	+	CCONJ
ejpam-854	429	8	a0(xk)−	a0(xk)−	ADJ
ejpam-854	429	9	c0(xk))‖k	c0(xk))‖k	NOUN
ejpam-854	429	10	≤	≤	NOUN
ejpam-854	429	11	8(9n2	8(9n2	NOUN
ejpam-854	429	12	+	+	CCONJ
ejpam-854	429	13	4	4	NUM
ejpam-854	429	14	)	)	PUNCT
ejpam-854	429	15	7(n4−	7(n4−	NUM
ejpam-854	429	16	n2	n2	NOUN
ejpam-854	429	17	)	)	PUNCT
ejpam-854	429	18	ǫ	ǫ	NOUN
ejpam-854	429	19	for	for	ADP
ejpam-854	429	20	all	all	DET
ejpam-854	429	21	x1	x1	PROPN
ejpam-854	429	22	,	,	PUNCT
ejpam-854	429	23	.	.	PUNCT
ejpam-854	429	24	.	.	PUNCT
ejpam-854	430	1	.	.	PUNCT
ejpam-854	431	1	,	,	PUNCT
ejpam-854	431	2	xk	xk	PROPN
ejpam-854	431	3	∈	∈	PROPN
ejpam-854	431	4	e.	e.	PROPN
ejpam-854	431	5	thus	thus	ADV
ejpam-854	431	6	we	we	PRON
ejpam-854	431	7	obtain	obtain	VERB
ejpam-854	431	8	(	(	PUNCT
ejpam-854	431	9	39	39	NUM
ejpam-854	431	10	)	)	PUNCT
ejpam-854	431	11	by	by	ADP
ejpam-854	431	12	defining	define	VERB
ejpam-854	431	13	a(x	a(x	NOUN
ejpam-854	431	14	)	)	PUNCT
ejpam-854	431	15	=	=	SYM
ejpam-854	431	16	−a0(x)/6	−a0(x)/6	NOUN
ejpam-854	431	17	and	and	CCONJ
ejpam-854	431	18	c(x	c(x	NOUN
ejpam-854	431	19	)	)	PUNCT
ejpam-854	431	20	=	=	SYM
ejpam-854	431	21	c0(x)/6	c0(x)/6	PROPN
ejpam-854	431	22	.	.	PUNCT
ejpam-854	432	1	the	the	DET
ejpam-854	432	2	uniqueness	uniqueness	NOUN
ejpam-854	432	3	of	of	ADP
ejpam-854	432	4	a	a	PRON
ejpam-854	432	5	and	and	CCONJ
ejpam-854	432	6	c	c	NOUN
ejpam-854	432	7	is	be	AUX
ejpam-854	432	8	easy	easy	ADJ
ejpam-854	432	9	to	to	PART
ejpam-854	432	10	show	show	VERB
ejpam-854	432	11	.	.	PUNCT
ejpam-854	433	1	theorem	theorem	ADJ
ejpam-854	433	2	7	7	NUM
ejpam-854	433	3	.	.	PUNCT
ejpam-854	434	1	let	let	VERB
ejpam-854	434	2	e	e	PRON
ejpam-854	434	3	be	be	AUX
ejpam-854	434	4	a	a	DET
ejpam-854	434	5	linear	linear	ADJ
ejpam-854	434	6	space	space	NOUN
ejpam-854	434	7	and	and	CCONJ
ejpam-854	434	8	let	let	VERB
ejpam-854	434	9	(	(	PUNCT
ejpam-854	434	10	(	(	PUNCT
ejpam-854	434	11	f	f	X
ejpam-854	434	12	k,‖	k,‖	PROPN
ejpam-854	434	13	·	·	PUNCT
ejpam-854	434	14	‖k	‖k	PROPN
ejpam-854	434	15	)	)	PUNCT
ejpam-854	434	16	:	:	PUNCT
ejpam-854	435	1	k	k	PROPN
ejpam-854	435	2	∈	∈	PROPN
ejpam-854	435	3	n	n	CCONJ
ejpam-854	435	4	)	)	PUNCT
ejpam-854	435	5	be	be	AUX
ejpam-854	435	6	a	a	DET
ejpam-854	435	7	multi	multi	ADJ
ejpam-854	435	8	-	-	ADJ
ejpam-854	435	9	banach	banach	ADJ
ejpam-854	435	10	space	space	NOUN
ejpam-854	435	11	.	.	PUNCT
ejpam-854	436	1	suppose	suppose	VERB
ejpam-854	436	2	that	that	SCONJ
ejpam-854	436	3	ǫ	ǫ	PRON
ejpam-854	436	4	≥	≥	NOUN
ejpam-854	436	5	0	0	NUM
ejpam-854	436	6	and	and	CCONJ
ejpam-854	436	7	f	f	NOUN
ejpam-854	436	8	:	:	PUNCT
ejpam-854	436	9	e→	e→	PROPN
ejpam-854	436	10	f	f	PROPN
ejpam-854	436	11	is	be	AUX
ejpam-854	436	12	an	an	DET
ejpam-854	436	13	even	even	ADV
ejpam-854	436	14	mapping	mapping	NOUN
ejpam-854	436	15	with	with	ADP
ejpam-854	436	16	f	f	PROPN
ejpam-854	436	17	(	(	PUNCT
ejpam-854	436	18	0	0	NUM
ejpam-854	436	19	)	)	PUNCT
ejpam-854	436	20	=	=	SYM
ejpam-854	436	21	0	0	NUM
ejpam-854	436	22	,	,	PUNCT
ejpam-854	436	23	satisfying	satisfy	VERB
ejpam-854	436	24	condition	condition	NOUN
ejpam-854	436	25	sup	sup	NOUN
ejpam-854	436	26	k∈n	k∈n	PROPN
ejpam-854	436	27	‖(d	‖(d	PROPN
ejpam-854	437	1	f	f	PROPN
ejpam-854	437	2	(	(	PUNCT
ejpam-854	437	3	x1	x1	PROPN
ejpam-854	437	4	,	,	PUNCT
ejpam-854	437	5	y1	y1	PROPN
ejpam-854	437	6	)	)	PUNCT
ejpam-854	437	7	,	,	PUNCT
ejpam-854	437	8	.	.	PUNCT
ejpam-854	437	9	.	.	PUNCT
ejpam-854	437	10	.	.	PUNCT
ejpam-854	438	1	,	,	PUNCT
ejpam-854	439	1	d	d	X
ejpam-854	439	2	f	f	X
ejpam-854	439	3	(	(	PUNCT
ejpam-854	439	4	xk	xk	PROPN
ejpam-854	439	5	,	,	PUNCT
ejpam-854	439	6	yk))‖k	yk))‖k	INTJ
ejpam-854	439	7	≤	≤	ADJ
ejpam-854	439	8	ǫ	ǫ	PRON
ejpam-854	439	9	(	(	PUNCT
ejpam-854	439	10	42	42	NUM
ejpam-854	439	11	)	)	PUNCT
ejpam-854	439	12	for	for	ADP
ejpam-854	439	13	all	all	DET
ejpam-854	439	14	x1	x1	PROPN
ejpam-854	439	15	,	,	PUNCT
ejpam-854	439	16	.	.	PUNCT
ejpam-854	439	17	.	.	PUNCT
ejpam-854	439	18	.	.	PUNCT
ejpam-854	440	1	,	,	PUNCT
ejpam-854	440	2	xk	xk	PROPN
ejpam-854	440	3	,	,	PUNCT
ejpam-854	440	4	y1	y1	PROPN
ejpam-854	440	5	,	,	PUNCT
ejpam-854	440	6	.	.	PUNCT
ejpam-854	440	7	.	.	PUNCT
ejpam-854	440	8	.	.	PUNCT
ejpam-854	441	1	,	,	PUNCT
ejpam-854	441	2	yk	yk	PROPN
ejpam-854	441	3	∈	∈	PROPN
ejpam-854	441	4	e.	e.	PROPN
ejpam-854	441	5	then	then	ADV
ejpam-854	441	6	there	there	PRON
ejpam-854	441	7	exist	exist	VERB
ejpam-854	441	8	a	a	DET
ejpam-854	441	9	unique	unique	ADJ
ejpam-854	441	10	quadratic	quadratic	ADJ
ejpam-854	441	11	mapping	mapping	NOUN
ejpam-854	441	12	b	b	NOUN
ejpam-854	441	13	:	:	PUNCT
ejpam-854	441	14	e	e	X
ejpam-854	441	15	→	→	SYM
ejpam-854	441	16	f	f	PROPN
ejpam-854	441	17	and	and	CCONJ
ejpam-854	441	18	a	a	DET
ejpam-854	441	19	unique	unique	ADJ
ejpam-854	441	20	quartic	quartic	ADJ
ejpam-854	441	21	mapping	mapping	NOUN
ejpam-854	441	22	q	q	NOUN
ejpam-854	441	23	:	:	PUNCT
ejpam-854	441	24	e→	e→	PROPN
ejpam-854	441	25	f	f	PROPN
ejpam-854	441	26	such	such	ADJ
ejpam-854	441	27	that	that	DET
ejpam-854	441	28	sup	sup	NOUN
ejpam-854	441	29	k∈n	k∈n	PROPN
ejpam-854	441	30	‖	‖	PROPN
ejpam-854	441	31	(	(	PUNCT
ejpam-854	441	32	f	f	PROPN
ejpam-854	441	33	(	(	PUNCT
ejpam-854	441	34	x1)−	x1)−	PROPN
ejpam-854	441	35	b(x1)−q(x1	b(x1)−q(x1	PROPN
ejpam-854	441	36	)	)	PUNCT
ejpam-854	441	37	,	,	PUNCT
ejpam-854	441	38	.	.	PUNCT
ejpam-854	441	39	.	.	PUNCT
ejpam-854	442	1	.	.	PUNCT
ejpam-854	443	1	,	,	PUNCT
ejpam-854	443	2	f	f	PROPN
ejpam-854	443	3	(	(	PUNCT
ejpam-854	443	4	xk)−	xk)−	PROPN
ejpam-854	443	5	b(xk)−q(xk))‖k	b(xk)−q(xk))‖k	PROPN
ejpam-854	443	6	≤	≤	NUM
ejpam-854	443	7	4n2	4n2	NUM
ejpam-854	444	1	+	+	CCONJ
ejpam-854	444	2	1	1	NUM
ejpam-854	444	3	5(n4−	5(n4−	NUM
ejpam-854	444	4	n2	n2	NOUN
ejpam-854	444	5	)	)	PUNCT
ejpam-854	444	6	ǫ	ǫ	PROPN
ejpam-854	444	7	(	(	PUNCT
ejpam-854	444	8	43	43	NUM
ejpam-854	444	9	)	)	PUNCT
ejpam-854	444	10	for	for	ADP
ejpam-854	444	11	all	all	DET
ejpam-854	444	12	x1	x1	PROPN
ejpam-854	444	13	,	,	PUNCT
ejpam-854	444	14	.	.	PUNCT
ejpam-854	444	15	.	.	PUNCT
ejpam-854	445	1	.	.	PUNCT
ejpam-854	446	1	,	,	PUNCT
ejpam-854	446	2	xk	xk	PROPN
ejpam-854	446	3	∈	∈	PROPN
ejpam-854	446	4	e.	e.	PROPN
ejpam-854	446	5	proof	proof	PROPN
ejpam-854	446	6	.	.	PUNCT
ejpam-854	447	1	by	by	ADP
ejpam-854	447	2	theorems	theorem	NOUN
ejpam-854	447	3	4	4	NUM
ejpam-854	447	4	and	and	CCONJ
ejpam-854	447	5	5	5	NUM
ejpam-854	447	6	,	,	PUNCT
ejpam-854	447	7	there	there	PRON
ejpam-854	447	8	exist	exist	VERB
ejpam-854	447	9	a	a	DET
ejpam-854	447	10	unique	unique	ADJ
ejpam-854	447	11	quadratic	quadratic	ADJ
ejpam-854	447	12	mapping	mapping	NOUN
ejpam-854	447	13	b0	b0	NOUN
ejpam-854	447	14	:	:	PUNCT
ejpam-854	447	15	e	e	X
ejpam-854	447	16	→	→	SYM
ejpam-854	447	17	f	f	PROPN
ejpam-854	447	18	and	and	CCONJ
ejpam-854	447	19	a	a	DET
ejpam-854	447	20	unique	unique	ADJ
ejpam-854	447	21	quartic	quartic	ADJ
ejpam-854	447	22	mapping	mapping	NOUN
ejpam-854	447	23	q0	q0	NOUN
ejpam-854	447	24	:	:	PUNCT
ejpam-854	447	25	e→	e→	PROPN
ejpam-854	447	26	f	f	PROPN
ejpam-854	447	27	such	such	ADJ
ejpam-854	447	28	that	that	DET
ejpam-854	447	29	sup	sup	NOUN
ejpam-854	447	30	k∈n	k∈n	PROPN
ejpam-854	447	31	‖	‖	PROPN
ejpam-854	447	32	(	(	PUNCT
ejpam-854	447	33	f	f	X
ejpam-854	447	34	(	(	PUNCT
ejpam-854	447	35	2x1)−	2x1)−	NUM
ejpam-854	447	36	16	16	NUM
ejpam-854	447	37	f	f	NOUN
ejpam-854	447	38	(	(	PUNCT
ejpam-854	447	39	x1)−	x1)−	PROPN
ejpam-854	447	40	b0(x1	b0(x1	PROPN
ejpam-854	447	41	)	)	PUNCT
ejpam-854	447	42	,	,	PUNCT
ejpam-854	447	43	.	.	PUNCT
ejpam-854	447	44	.	.	PUNCT
ejpam-854	448	1	.	.	PUNCT
ejpam-854	449	1	,	,	PUNCT
ejpam-854	449	2	f	f	PROPN
ejpam-854	449	3	(	(	PUNCT
ejpam-854	449	4	2xk)−	2xk)−	PROPN
ejpam-854	449	5	16	16	NUM
ejpam-854	449	6	f	f	X
ejpam-854	449	7	(	(	PUNCT
ejpam-854	449	8	xk)−	xk)−	PROPN
ejpam-854	449	9	b0(xk))‖k	b0(xk))‖k	PROPN
ejpam-854	449	10	≤	≤	NOUN
ejpam-854	449	11	8n2	8n2	NUM
ejpam-854	449	12	+	+	SYM
ejpam-854	449	13	2	2	NUM
ejpam-854	449	14	n4	n4	PROPN
ejpam-854	449	15	−	−	PROPN
ejpam-854	449	16	n2	n2	ADJ
ejpam-854	449	17	ǫ	ǫ	X
ejpam-854	449	18	(	(	PUNCT
ejpam-854	449	19	44	44	NUM
ejpam-854	449	20	)	)	PUNCT
ejpam-854	449	21	and	and	CCONJ
ejpam-854	449	22	sup	sup	NOUN
ejpam-854	449	23	k∈n	k∈n	PROPN
ejpam-854	449	24	‖	‖	PROPN
ejpam-854	449	25	(	(	PUNCT
ejpam-854	449	26	f	f	X
ejpam-854	449	27	(	(	PUNCT
ejpam-854	449	28	2x1)−	2x1)−	NUM
ejpam-854	449	29	4	4	NUM
ejpam-854	449	30	f	f	NOUN
ejpam-854	449	31	(	(	PUNCT
ejpam-854	449	32	x1)−q0(x1	x1)−q0(x1	PROPN
ejpam-854	449	33	)	)	PUNCT
ejpam-854	449	34	,	,	PUNCT
ejpam-854	449	35	.	.	PUNCT
ejpam-854	449	36	.	.	PUNCT
ejpam-854	450	1	.	.	PUNCT
ejpam-854	451	1	,	,	PUNCT
ejpam-854	451	2	f	f	PROPN
ejpam-854	451	3	(	(	PUNCT
ejpam-854	451	4	2xk)−	2xk)−	NOUN
ejpam-854	451	5	4	4	NUM
ejpam-854	451	6	f	f	NOUN
ejpam-854	451	7	(	(	PUNCT
ejpam-854	451	8	xk)−q0(xk))‖k	xk)−q0(xk))‖k	NOUN
ejpam-854	451	9	≤	≤	VERB
ejpam-854	451	10	8n2	8n2	NUM
ejpam-854	451	11	+	+	SYM
ejpam-854	451	12	2	2	NUM
ejpam-854	451	13	5(n4−	5(n4−	NUM
ejpam-854	451	14	n2	n2	NOUN
ejpam-854	451	15	)	)	PUNCT
ejpam-854	451	16	ǫ	ǫ	NOUN
ejpam-854	451	17	(	(	PUNCT
ejpam-854	451	18	45	45	NUM
ejpam-854	451	19	)	)	PUNCT
ejpam-854	451	20	for	for	ADP
ejpam-854	451	21	all	all	DET
ejpam-854	451	22	x1	x1	PROPN
ejpam-854	451	23	,	,	PUNCT
ejpam-854	451	24	.	.	PUNCT
ejpam-854	451	25	.	.	PUNCT
ejpam-854	451	26	.	.	PUNCT
ejpam-854	452	1	,	,	PUNCT
ejpam-854	452	2	xk	xk	PROPN
ejpam-854	452	3	∈	∈	PROPN
ejpam-854	452	4	e.	e.	PROPN
ejpam-854	452	5	now	now	ADV
ejpam-854	452	6	from	from	ADP
ejpam-854	452	7	(	(	PUNCT
ejpam-854	452	8	44	44	NUM
ejpam-854	452	9	)	)	PUNCT
ejpam-854	452	10	and	and	CCONJ
ejpam-854	452	11	(	(	PUNCT
ejpam-854	452	12	45	45	NUM
ejpam-854	452	13	)	)	PUNCT
ejpam-854	452	14	,	,	PUNCT
ejpam-854	452	15	one	one	PRON
ejpam-854	452	16	can	can	AUX
ejpam-854	452	17	see	see	VERB
ejpam-854	452	18	that	that	DET
ejpam-854	452	19	sup	sup	NOUN
ejpam-854	452	20	k∈n	k∈n	NOUN
ejpam-854	452	21	‖(12	‖(12	PROPN
ejpam-854	453	1	f	f	PROPN
ejpam-854	453	2	(	(	PUNCT
ejpam-854	453	3	x1	x1	PROPN
ejpam-854	453	4	)	)	PUNCT
ejpam-854	453	5	+	+	CCONJ
ejpam-854	453	6	b0(x1)−q0(x1	b0(x1)−q0(x1	NOUN
ejpam-854	453	7	)	)	PUNCT
ejpam-854	453	8	,	,	PUNCT
ejpam-854	453	9	.	.	PUNCT
ejpam-854	453	10	.	.	PUNCT
ejpam-854	454	1	.	.	PUNCT
ejpam-854	455	1	,	,	PUNCT
ejpam-854	455	2	12	12	NUM
ejpam-854	455	3	f	f	X
ejpam-854	455	4	(	(	PUNCT
ejpam-854	455	5	xk	xk	PROPN
ejpam-854	455	6	)	)	PUNCT
ejpam-854	455	7	+	+	CCONJ
ejpam-854	455	8	b0(xk)−q0(xk))‖k	b0(xk)−q0(xk))‖k	PROPN
ejpam-854	455	9	≤	≤	NOUN
ejpam-854	455	10	6(8n2	6(8n2	NOUN
ejpam-854	455	11	+	+	CCONJ
ejpam-854	455	12	2	2	X
ejpam-854	455	13	)	)	PUNCT
ejpam-854	455	14	5(n4−	5(n4−	NUM
ejpam-854	455	15	n2	n2	NOUN
ejpam-854	455	16	)	)	PUNCT
ejpam-854	455	17	ǫ	ǫ	NOUN
ejpam-854	455	18	for	for	ADP
ejpam-854	455	19	all	all	DET
ejpam-854	455	20	x1	x1	PROPN
ejpam-854	455	21	,	,	PUNCT
ejpam-854	455	22	.	.	PUNCT
ejpam-854	455	23	.	.	PUNCT
ejpam-854	456	1	.	.	PUNCT
ejpam-854	457	1	,	,	PUNCT
ejpam-854	457	2	xk	xk	PROPN
ejpam-854	457	3	∈	∈	PROPN
ejpam-854	457	4	e.	e.	PROPN
ejpam-854	457	5	thus	thus	ADV
ejpam-854	457	6	we	we	PRON
ejpam-854	457	7	obtain	obtain	VERB
ejpam-854	457	8	(	(	PUNCT
ejpam-854	457	9	43	43	NUM
ejpam-854	457	10	)	)	PUNCT
ejpam-854	457	11	by	by	ADP
ejpam-854	457	12	defining	define	VERB
ejpam-854	457	13	b(x	b(x	NOUN
ejpam-854	457	14	)	)	PUNCT
ejpam-854	457	15	=	=	SYM
ejpam-854	457	16	−b0(x)/12	−b0(x)/12	NOUN
ejpam-854	457	17	and	and	CCONJ
ejpam-854	457	18	q(x	q(x	PROPN
ejpam-854	457	19	)	)	PUNCT
ejpam-854	457	20	=	=	SYM
ejpam-854	458	1	q0(x)/12	q0(x)/12	PROPN
ejpam-854	458	2	.	.	PUNCT
ejpam-854	459	1	the	the	DET
ejpam-854	459	2	uniqueness	uniqueness	NOUN
ejpam-854	459	3	of	of	ADP
ejpam-854	459	4	b	b	PROPN
ejpam-854	459	5	and	and	CCONJ
ejpam-854	459	6	q	q	NOUN
ejpam-854	459	7	is	be	AUX
ejpam-854	459	8	easy	easy	ADJ
ejpam-854	459	9	to	to	PART
ejpam-854	459	10	show	show	VERB
ejpam-854	459	11	.	.	PUNCT
ejpam-854	460	1	t.	t.	PROPN
ejpam-854	460	2	xu	xu	PROPN
ejpam-854	460	3	,	,	PUNCT
ejpam-854	460	4	j.	j.	PROPN
ejpam-854	460	5	rassias	rassias	PROPN
ejpam-854	460	6	,	,	PUNCT
ejpam-854	460	7	w.	w.	PROPN
ejpam-854	460	8	xu	xu	PROPN
ejpam-854	460	9	/	/	SYM
ejpam-854	460	10	eur	eur	PROPN
ejpam-854	460	11	.	.	PUNCT
ejpam-854	461	1	j.	j.	PROPN
ejpam-854	461	2	pure	pure	PROPN
ejpam-854	461	3	appl	appl	PROPN
ejpam-854	461	4	.	.	PROPN
ejpam-854	461	5	math	math	PROPN
ejpam-854	461	6	,	,	PUNCT
ejpam-854	461	7	3	3	NUM
ejpam-854	461	8	(	(	PUNCT
ejpam-854	461	9	2010	2010	NUM
ejpam-854	461	10	)	)	PUNCT
ejpam-854	461	11	,	,	PUNCT
ejpam-854	461	12	1032	1032	NUM
ejpam-854	461	13	-	-	SYM
ejpam-854	461	14	1047	1047	NUM
ejpam-854	461	15	1044	1044	NUM
ejpam-854	461	16	theorem	theorem	VERB
ejpam-854	461	17	8	8	NUM
ejpam-854	461	18	.	.	PUNCT
ejpam-854	462	1	let	let	VERB
ejpam-854	462	2	e	e	PRON
ejpam-854	462	3	be	be	AUX
ejpam-854	462	4	a	a	DET
ejpam-854	462	5	linear	linear	ADJ
ejpam-854	462	6	space	space	NOUN
ejpam-854	462	7	and	and	CCONJ
ejpam-854	462	8	let	let	VERB
ejpam-854	462	9	(	(	PUNCT
ejpam-854	462	10	(	(	PUNCT
ejpam-854	462	11	f	f	X
ejpam-854	462	12	k,‖	k,‖	PROPN
ejpam-854	462	13	·	·	PUNCT
ejpam-854	462	14	‖k	‖k	PROPN
ejpam-854	462	15	)	)	PUNCT
ejpam-854	462	16	:	:	PUNCT
ejpam-854	463	1	k	k	PROPN
ejpam-854	463	2	∈	∈	PROPN
ejpam-854	463	3	n	n	CCONJ
ejpam-854	463	4	)	)	PUNCT
ejpam-854	463	5	be	be	AUX
ejpam-854	463	6	a	a	DET
ejpam-854	463	7	multi	multi	ADJ
ejpam-854	463	8	-	-	ADJ
ejpam-854	463	9	banach	banach	ADJ
ejpam-854	463	10	space	space	NOUN
ejpam-854	463	11	.	.	PUNCT
ejpam-854	464	1	suppose	suppose	VERB
ejpam-854	464	2	that	that	SCONJ
ejpam-854	464	3	ǫ	ǫ	PRON
ejpam-854	464	4	≥	≥	NOUN
ejpam-854	464	5	0	0	NUM
ejpam-854	464	6	and	and	CCONJ
ejpam-854	464	7	f	f	NOUN
ejpam-854	464	8	:	:	PUNCT
ejpam-854	464	9	e→	e→	PROPN
ejpam-854	464	10	f	f	PROPN
ejpam-854	464	11	is	be	AUX
ejpam-854	464	12	a	a	DET
ejpam-854	464	13	mapping	mapping	NOUN
ejpam-854	464	14	with	with	ADP
ejpam-854	464	15	f	f	PROPN
ejpam-854	464	16	(	(	PUNCT
ejpam-854	464	17	0	0	NUM
ejpam-854	464	18	)	)	PUNCT
ejpam-854	464	19	=	=	SYM
ejpam-854	464	20	0	0	NUM
ejpam-854	464	21	,	,	PUNCT
ejpam-854	464	22	satisfying	satisfy	VERB
ejpam-854	464	23	condition	condition	NOUN
ejpam-854	464	24	sup	sup	NOUN
ejpam-854	464	25	k∈n	k∈n	PROPN
ejpam-854	464	26	‖(d	‖(d	PROPN
ejpam-854	465	1	f	f	PROPN
ejpam-854	465	2	(	(	PUNCT
ejpam-854	465	3	x1	x1	PROPN
ejpam-854	465	4	,	,	PUNCT
ejpam-854	465	5	y1	y1	PROPN
ejpam-854	465	6	)	)	PUNCT
ejpam-854	465	7	,	,	PUNCT
ejpam-854	465	8	.	.	PUNCT
ejpam-854	465	9	.	.	PUNCT
ejpam-854	465	10	.	.	PUNCT
ejpam-854	466	1	,	,	PUNCT
ejpam-854	467	1	d	d	X
ejpam-854	467	2	f	f	X
ejpam-854	467	3	(	(	PUNCT
ejpam-854	467	4	xk	xk	PROPN
ejpam-854	467	5	,	,	PUNCT
ejpam-854	467	6	yk))‖k	yk))‖k	INTJ
ejpam-854	467	7	≤	≤	ADJ
ejpam-854	467	8	ǫ	ǫ	PRON
ejpam-854	467	9	(	(	PUNCT
ejpam-854	467	10	46	46	NUM
ejpam-854	467	11	)	)	PUNCT
ejpam-854	467	12	for	for	ADP
ejpam-854	467	13	all	all	DET
ejpam-854	467	14	x1	x1	PROPN
ejpam-854	467	15	,	,	PUNCT
ejpam-854	467	16	.	.	PUNCT
ejpam-854	467	17	.	.	PUNCT
ejpam-854	467	18	.	.	PUNCT
ejpam-854	468	1	,	,	PUNCT
ejpam-854	468	2	xk	xk	PROPN
ejpam-854	468	3	,	,	PUNCT
ejpam-854	468	4	y1	y1	PROPN
ejpam-854	468	5	,	,	PUNCT
ejpam-854	468	6	.	.	PUNCT
ejpam-854	468	7	.	.	PUNCT
ejpam-854	468	8	.	.	PUNCT
ejpam-854	469	1	,	,	PUNCT
ejpam-854	469	2	yk	yk	PROPN
ejpam-854	469	3	∈	∈	PROPN
ejpam-854	469	4	e.	e.	PROPN
ejpam-854	469	5	then	then	ADV
ejpam-854	469	6	there	there	PRON
ejpam-854	469	7	exist	exist	VERB
ejpam-854	469	8	a	a	DET
ejpam-854	469	9	unique	unique	ADJ
ejpam-854	469	10	additive	additive	NOUN
ejpam-854	469	11	mapping	mapping	NOUN
ejpam-854	469	12	a	a	PRON
ejpam-854	469	13	:	:	PUNCT
ejpam-854	469	14	e→	e→	NOUN
ejpam-854	469	15	f	f	PROPN
ejpam-854	469	16	,	,	PUNCT
ejpam-854	469	17	a	a	DET
ejpam-854	469	18	unique	unique	ADJ
ejpam-854	469	19	quadratic	quadratic	ADJ
ejpam-854	469	20	mapping	mapping	NOUN
ejpam-854	469	21	b	b	NOUN
ejpam-854	469	22	:	:	PUNCT
ejpam-854	469	23	e→	e→	PROPN
ejpam-854	469	24	f	f	PROPN
ejpam-854	469	25	,	,	PUNCT
ejpam-854	469	26	a	a	DET
ejpam-854	469	27	unique	unique	ADJ
ejpam-854	469	28	cubic	cubic	ADJ
ejpam-854	469	29	mapping	mapping	NOUN
ejpam-854	470	1	c	c	NOUN
ejpam-854	470	2	:	:	PUNCT
ejpam-854	470	3	e→	e→	NOUN
ejpam-854	470	4	f	f	PROPN
ejpam-854	470	5	,	,	PUNCT
ejpam-854	470	6	and	and	CCONJ
ejpam-854	470	7	a	a	DET
ejpam-854	470	8	unique	unique	ADJ
ejpam-854	470	9	quartic	quartic	ADJ
ejpam-854	470	10	mapping	mapping	NOUN
ejpam-854	470	11	q	q	NOUN
ejpam-854	470	12	:	:	PUNCT
ejpam-854	470	13	e→	e→	PROPN
ejpam-854	470	14	f	f	PROPN
ejpam-854	470	15	such	such	ADJ
ejpam-854	470	16	that	that	DET
ejpam-854	470	17	sup	sup	NOUN
ejpam-854	470	18	k∈n	k∈n	PROPN
ejpam-854	470	19	‖	‖	PROPN
ejpam-854	470	20	(	(	PUNCT
ejpam-854	470	21	f	f	PROPN
ejpam-854	470	22	(	(	PUNCT
ejpam-854	470	23	x1)−	x1)−	PROPN
ejpam-854	470	24	a(x1)−	a(x1)−	PROPN
ejpam-854	470	25	b(x1)−	b(x1)−	PROPN
ejpam-854	470	26	c(x1)−q(x1	c(x1)−q(x1	PROPN
ejpam-854	470	27	)	)	PUNCT
ejpam-854	470	28	,	,	PUNCT
ejpam-854	470	29	.	.	PUNCT
ejpam-854	470	30	.	.	PUNCT
ejpam-854	471	1	.	.	PUNCT
ejpam-854	472	1	,	,	PUNCT
ejpam-854	472	2	f	f	PROPN
ejpam-854	472	3	(	(	PUNCT
ejpam-854	472	4	xk)−	xk)−	PROPN
ejpam-854	472	5	a(xk)−	a(xk)−	PROPN
ejpam-854	472	6	b(xk)−	b(xk)−	NOUN
ejpam-854	472	7	c(xk)−q(xk))‖k	c(xk)−q(xk))‖k	VERB
ejpam-854	472	8	≤	≤	NUM
ejpam-854	472	9	164n2	164n2	NUM
ejpam-854	473	1	+	+	NOUN
ejpam-854	473	2	101	101	NUM
ejpam-854	473	3	105(n4−n2	105(n4−n2	NUM
ejpam-854	473	4	)	)	PUNCT
ejpam-854	473	5	ǫ	ǫ	NOUN
ejpam-854	473	6	(	(	PUNCT
ejpam-854	473	7	47	47	NUM
ejpam-854	473	8	)	)	PUNCT
ejpam-854	473	9	for	for	ADP
ejpam-854	473	10	all	all	DET
ejpam-854	473	11	x1	x1	PROPN
ejpam-854	473	12	,	,	PUNCT
ejpam-854	473	13	.	.	PUNCT
ejpam-854	473	14	.	.	PUNCT
ejpam-854	474	1	.	.	PUNCT
ejpam-854	475	1	,	,	PUNCT
ejpam-854	475	2	xk	xk	PROPN
ejpam-854	475	3	∈	∈	PROPN
ejpam-854	475	4	e.	e.	PROPN
ejpam-854	475	5	proof	proof	PROPN
ejpam-854	475	6	.	.	PUNCT
ejpam-854	476	1	let	let	VERB
ejpam-854	476	2	fo(x	fo(x	PUNCT
ejpam-854	476	3	)	)	PUNCT
ejpam-854	476	4	=	=	SYM
ejpam-854	476	5	1	1	NUM
ejpam-854	476	6	2	2	NUM
ejpam-854	476	7	[	[	PUNCT
ejpam-854	476	8	f	f	X
ejpam-854	476	9	(	(	PUNCT
ejpam-854	476	10	x)−	x)−	PROPN
ejpam-854	476	11	f	f	PROPN
ejpam-854	476	12	(	(	PUNCT
ejpam-854	476	13	−x	−x	NOUN
ejpam-854	476	14	)	)	PUNCT
ejpam-854	476	15	]	]	PUNCT
ejpam-854	476	16	for	for	ADP
ejpam-854	476	17	all	all	DET
ejpam-854	476	18	x	x	SYM
ejpam-854	476	19	∈	∈	PROPN
ejpam-854	476	20	e.	e.	PROPN
ejpam-854	476	21	then	then	ADV
ejpam-854	476	22	fo(0	fo(0	PROPN
ejpam-854	476	23	)	)	PUNCT
ejpam-854	476	24	=	=	SYM
ejpam-854	476	25	0	0	NUM
ejpam-854	476	26	,	,	PUNCT
ejpam-854	476	27	fo(x	fo(x	PUNCT
ejpam-854	476	28	)	)	PUNCT
ejpam-854	476	29	=	=	SYM
ejpam-854	476	30	−	−	PROPN
ejpam-854	476	31	fo(−x	fo(−x	NOUN
ejpam-854	476	32	)	)	PUNCT
ejpam-854	476	33	.	.	PUNCT
ejpam-854	477	1	hence	hence	ADV
ejpam-854	477	2	sup	sup	PROPN
ejpam-854	477	3	k∈n	k∈n	PROPN
ejpam-854	477	4	‖(d	‖(d	PROPN
ejpam-854	477	5	fo(x1	fo(x1	PROPN
ejpam-854	477	6	,	,	PUNCT
ejpam-854	477	7	y1	y1	NOUN
ejpam-854	477	8	)	)	PUNCT
ejpam-854	477	9	,	,	PUNCT
ejpam-854	477	10	.	.	PUNCT
ejpam-854	477	11	.	.	PUNCT
ejpam-854	478	1	.	.	PUNCT
ejpam-854	479	1	,	,	PUNCT
ejpam-854	479	2	d	d	PROPN
ejpam-854	479	3	fo(xk	fo(xk	PROPN
ejpam-854	479	4	,	,	PUNCT
ejpam-854	479	5	yk))‖k	yk))‖k	INTJ
ejpam-854	479	6	≤	≤	ADJ
ejpam-854	479	7	ǫ	ǫ	NOUN
ejpam-854	479	8	for	for	ADP
ejpam-854	479	9	all	all	DET
ejpam-854	479	10	x1	x1	PROPN
ejpam-854	479	11	,	,	PUNCT
ejpam-854	479	12	.	.	PUNCT
ejpam-854	479	13	.	.	PUNCT
ejpam-854	479	14	.	.	PUNCT
ejpam-854	480	1	,	,	PUNCT
ejpam-854	480	2	xk	xk	PROPN
ejpam-854	480	3	,	,	PUNCT
ejpam-854	480	4	y1	y1	PROPN
ejpam-854	480	5	,	,	PUNCT
ejpam-854	480	6	.	.	PUNCT
ejpam-854	480	7	.	.	PUNCT
ejpam-854	480	8	.	.	PUNCT
ejpam-854	481	1	,	,	PUNCT
ejpam-854	481	2	yk	yk	PROPN
ejpam-854	481	3	∈	∈	PROPN
ejpam-854	481	4	e.	e.	PROPN
ejpam-854	481	5	by	by	ADP
ejpam-854	481	6	theorem	theorem	NOUN
ejpam-854	481	7	6	6	NUM
ejpam-854	481	8	,	,	PUNCT
ejpam-854	481	9	there	there	PRON
ejpam-854	481	10	exist	exist	VERB
ejpam-854	481	11	a	a	DET
ejpam-854	481	12	unique	unique	ADJ
ejpam-854	481	13	additive	additive	NOUN
ejpam-854	481	14	mapping	mapping	NOUN
ejpam-854	482	1	a	a	DET
ejpam-854	482	2	:	:	PUNCT
ejpam-854	482	3	e→	e→	PROPN
ejpam-854	482	4	f	f	PROPN
ejpam-854	482	5	and	and	CCONJ
ejpam-854	482	6	a	a	DET
ejpam-854	482	7	unique	unique	ADJ
ejpam-854	482	8	cubic	cubic	ADJ
ejpam-854	482	9	mapping	mapping	NOUN
ejpam-854	482	10	c	c	NOUN
ejpam-854	482	11	:	:	PUNCT
ejpam-854	482	12	e→	e→	PROPN
ejpam-854	482	13	f	f	PROPN
ejpam-854	482	14	such	such	ADJ
ejpam-854	482	15	that	that	DET
ejpam-854	482	16	sup	sup	NOUN
ejpam-854	482	17	k∈n	k∈n	PROPN
ejpam-854	482	18	‖	‖	PROPN
ejpam-854	482	19	(	(	PUNCT
ejpam-854	482	20	f	f	PROPN
ejpam-854	482	21	(	(	PUNCT
ejpam-854	482	22	x1)−	x1)−	PROPN
ejpam-854	482	23	a(x1)−	a(x1)−	PROPN
ejpam-854	482	24	c(x1	c(x1	NOUN
ejpam-854	482	25	)	)	PUNCT
ejpam-854	482	26	,	,	PUNCT
ejpam-854	482	27	.	.	PUNCT
ejpam-854	482	28	.	.	PUNCT
ejpam-854	482	29	.	.	PUNCT
ejpam-854	483	1	,	,	PUNCT
ejpam-854	483	2	f	f	PROPN
ejpam-854	483	3	(	(	PUNCT
ejpam-854	483	4	xk)−	xk)−	PROPN
ejpam-854	483	5	a(xk)−	a(xk)−	PROPN
ejpam-854	483	6	c(xk))‖k	c(xk))‖k	VERB
ejpam-854	483	7	≤	≤	NUM
ejpam-854	483	8	4(9n2	4(9n2	NOUN
ejpam-854	483	9	+	+	CCONJ
ejpam-854	483	10	4	4	NUM
ejpam-854	483	11	)	)	PUNCT
ejpam-854	483	12	21(n4−	21(n4−	NUM
ejpam-854	483	13	n2	n2	NOUN
ejpam-854	483	14	)	)	PUNCT
ejpam-854	483	15	ǫ	ǫ	PROPN
ejpam-854	483	16	(	(	PUNCT
ejpam-854	483	17	48	48	NUM
ejpam-854	483	18	)	)	PUNCT
ejpam-854	483	19	for	for	ADP
ejpam-854	483	20	all	all	DET
ejpam-854	483	21	x1	x1	PROPN
ejpam-854	483	22	,	,	PUNCT
ejpam-854	483	23	.	.	PUNCT
ejpam-854	483	24	.	.	PUNCT
ejpam-854	483	25	.	.	PUNCT
ejpam-854	484	1	,	,	PUNCT
ejpam-854	484	2	xk	xk	PROPN
ejpam-854	484	3	∈	∈	PROPN
ejpam-854	484	4	e.	e.	PROPN
ejpam-854	484	5	let	let	VERB
ejpam-854	484	6	fe(x	fe(x	VERB
ejpam-854	484	7	)	)	PUNCT
ejpam-854	484	8	=	=	SYM
ejpam-854	484	9	1	1	NUM
ejpam-854	484	10	2	2	NUM
ejpam-854	484	11	[	[	PUNCT
ejpam-854	484	12	f	f	X
ejpam-854	484	13	(	(	PUNCT
ejpam-854	484	14	x)+	x)+	PROPN
ejpam-854	484	15	f	f	PROPN
ejpam-854	484	16	(	(	PUNCT
ejpam-854	484	17	−x	−x	NOUN
ejpam-854	484	18	)	)	PUNCT
ejpam-854	484	19	]	]	PUNCT
ejpam-854	484	20	for	for	ADP
ejpam-854	484	21	all	all	DET
ejpam-854	484	22	x	x	SYM
ejpam-854	484	23	∈	∈	PROPN
ejpam-854	484	24	e.	e.	PROPN
ejpam-854	484	25	then	then	ADV
ejpam-854	484	26	fe(0	fe(0	NOUN
ejpam-854	484	27	)	)	PUNCT
ejpam-854	484	28	=	=	SYM
ejpam-854	484	29	0	0	NUM
ejpam-854	484	30	,	,	PUNCT
ejpam-854	484	31	fe(x	fe(x	ADJ
ejpam-854	484	32	)	)	PUNCT
ejpam-854	484	33	=	=	SYM
ejpam-854	484	34	fe(−x	fe(−x	PROPN
ejpam-854	484	35	)	)	PUNCT
ejpam-854	484	36	and	and	CCONJ
ejpam-854	484	37	sup	sup	PROPN
ejpam-854	484	38	k∈n	k∈n	PROPN
ejpam-854	484	39	‖(d	‖(d	PROPN
ejpam-854	484	40	fe(x1	fe(x1	PROPN
ejpam-854	484	41	,	,	PUNCT
ejpam-854	484	42	y1	y1	NOUN
ejpam-854	484	43	)	)	PUNCT
ejpam-854	484	44	,	,	PUNCT
ejpam-854	484	45	.	.	PUNCT
ejpam-854	484	46	.	.	PUNCT
ejpam-854	485	1	.	.	PUNCT
ejpam-854	486	1	,	,	PUNCT
ejpam-854	486	2	d	d	X
ejpam-854	486	3	fe(xk	fe(xk	PROPN
ejpam-854	486	4	,	,	PUNCT
ejpam-854	486	5	yk))‖k	yk))‖k	INTJ
ejpam-854	486	6	≤	≤	ADJ
ejpam-854	486	7	ǫ	ǫ	NOUN
ejpam-854	486	8	for	for	ADP
ejpam-854	486	9	all	all	DET
ejpam-854	486	10	x1	x1	PROPN
ejpam-854	486	11	,	,	PUNCT
ejpam-854	486	12	.	.	PUNCT
ejpam-854	486	13	.	.	PUNCT
ejpam-854	486	14	.	.	PUNCT
ejpam-854	487	1	,	,	PUNCT
ejpam-854	487	2	xk	xk	PROPN
ejpam-854	487	3	,	,	PUNCT
ejpam-854	487	4	y1	y1	PROPN
ejpam-854	487	5	,	,	PUNCT
ejpam-854	487	6	.	.	PUNCT
ejpam-854	487	7	.	.	PUNCT
ejpam-854	487	8	.	.	PUNCT
ejpam-854	488	1	,	,	PUNCT
ejpam-854	488	2	yk	yk	PROPN
ejpam-854	488	3	∈	∈	PROPN
ejpam-854	488	4	e.	e.	PROPN
ejpam-854	488	5	by	by	ADP
ejpam-854	488	6	theorem	theorem	NOUN
ejpam-854	488	7	7	7	NUM
ejpam-854	488	8	,	,	PUNCT
ejpam-854	488	9	there	there	PRON
ejpam-854	488	10	exist	exist	VERB
ejpam-854	488	11	a	a	DET
ejpam-854	488	12	unique	unique	ADJ
ejpam-854	488	13	quadratic	quadratic	ADJ
ejpam-854	488	14	mapping	mapping	NOUN
ejpam-854	488	15	b	b	NOUN
ejpam-854	488	16	:	:	PUNCT
ejpam-854	488	17	e→	e→	PROPN
ejpam-854	488	18	f	f	PROPN
ejpam-854	488	19	and	and	CCONJ
ejpam-854	488	20	a	a	DET
ejpam-854	488	21	unique	unique	ADJ
ejpam-854	488	22	quartic	quartic	ADJ
ejpam-854	488	23	mapping	mapping	NOUN
ejpam-854	488	24	q	q	NOUN
ejpam-854	488	25	:	:	PUNCT
ejpam-854	488	26	e→	e→	PROPN
ejpam-854	488	27	f	f	PROPN
ejpam-854	488	28	such	such	ADJ
ejpam-854	488	29	that	that	DET
ejpam-854	488	30	sup	sup	NOUN
ejpam-854	488	31	k∈n	k∈n	PROPN
ejpam-854	488	32	‖	‖	PROPN
ejpam-854	488	33	(	(	PUNCT
ejpam-854	488	34	f	f	PROPN
ejpam-854	488	35	(	(	PUNCT
ejpam-854	488	36	x1)−	x1)−	PROPN
ejpam-854	488	37	b(x1)−q(x1	b(x1)−q(x1	PROPN
ejpam-854	488	38	)	)	PUNCT
ejpam-854	488	39	,	,	PUNCT
ejpam-854	488	40	.	.	PUNCT
ejpam-854	488	41	.	.	PUNCT
ejpam-854	489	1	.	.	PUNCT
ejpam-854	490	1	,	,	PUNCT
ejpam-854	490	2	f	f	PROPN
ejpam-854	490	3	(	(	PUNCT
ejpam-854	490	4	xk)−	xk)−	PROPN
ejpam-854	490	5	b(xk)−q(xk))‖k	b(xk)−q(xk))‖k	PROPN
ejpam-854	490	6	≤	≤	NUM
ejpam-854	490	7	4n2	4n2	NUM
ejpam-854	491	1	+	+	CCONJ
ejpam-854	491	2	1	1	NUM
ejpam-854	491	3	5(n4−	5(n4−	NUM
ejpam-854	491	4	n2	n2	NOUN
ejpam-854	491	5	)	)	PUNCT
ejpam-854	491	6	ǫ	ǫ	PROPN
ejpam-854	491	7	(	(	PUNCT
ejpam-854	491	8	49	49	NUM
ejpam-854	491	9	)	)	PUNCT
ejpam-854	491	10	for	for	ADP
ejpam-854	491	11	all	all	DET
ejpam-854	491	12	x1	x1	PROPN
ejpam-854	491	13	,	,	PUNCT
ejpam-854	491	14	.	.	PUNCT
ejpam-854	491	15	.	.	PUNCT
ejpam-854	492	1	.	.	PUNCT
ejpam-854	493	1	,	,	PUNCT
ejpam-854	493	2	xk	xk	PROPN
ejpam-854	493	3	∈	∈	PROPN
ejpam-854	493	4	e.	e.	PROPN
ejpam-854	493	5	by	by	PROPN
ejpam-854	493	6	(	(	PUNCT
ejpam-854	493	7	48	48	NUM
ejpam-854	493	8	)	)	PUNCT
ejpam-854	493	9	and	and	CCONJ
ejpam-854	493	10	(	(	PUNCT
ejpam-854	493	11	49	49	NUM
ejpam-854	493	12	)	)	PUNCT
ejpam-854	493	13	,	,	PUNCT
ejpam-854	493	14	we	we	PRON
ejpam-854	493	15	get	get	VERB
ejpam-854	493	16	(	(	PUNCT
ejpam-854	493	17	47	47	NUM
ejpam-854	493	18	)	)	PUNCT
ejpam-854	493	19	.	.	PUNCT
ejpam-854	494	1	this	this	PRON
ejpam-854	494	2	completes	complete	VERB
ejpam-854	494	3	the	the	DET
ejpam-854	494	4	proof	proof	NOUN
ejpam-854	494	5	.	.	PUNCT
ejpam-854	495	1	acknowledgements	acknowledgement	NOUN
ejpam-854	495	2	the	the	DET
ejpam-854	495	3	first	first	ADJ
ejpam-854	495	4	author	author	NOUN
ejpam-854	495	5	was	be	AUX
ejpam-854	495	6	supported	support	VERB
ejpam-854	495	7	by	by	ADP
ejpam-854	495	8	the	the	DET
ejpam-854	495	9	national	national	ADJ
ejpam-854	495	10	natural	natural	PROPN
ejpam-854	495	11	science	science	PROPN
ejpam-854	495	12	foundation	foundation	PROPN
ejpam-854	495	13	of	of	ADP
ejpam-854	495	14	china	china	PROPN
ejpam-854	495	15	(	(	PUNCT
ejpam-854	495	16	grant	grant	VERB
ejpam-854	495	17	no	no	NOUN
ejpam-854	495	18	.	.	PROPN
ejpam-854	495	19	10671013	10671013	NUM
ejpam-854	495	20	,	,	PUNCT
ejpam-854	495	21	60972089	60972089	NUM
ejpam-854	495	22	)	)	PUNCT
ejpam-854	495	23	.	.	PUNCT
ejpam-854	496	1	references	reference	NOUN
ejpam-854	496	2	1045	1045	NUM
ejpam-854	496	3	references	reference	NOUN
ejpam-854	496	4	[	[	X
ejpam-854	496	5	1	1	NUM
ejpam-854	496	6	]	]	PUNCT
ejpam-854	496	7	j	j	NOUN
ejpam-854	496	8	aczél	aczél	NOUN
ejpam-854	496	9	,	,	PUNCT
ejpam-854	496	10	j	j	PROPN
ejpam-854	496	11	c	c	PROPN
ejpam-854	496	12	falmagne	falmagne	NOUN
ejpam-854	496	13	,	,	PUNCT
ejpam-854	496	14	and	and	CCONJ
ejpam-854	496	15	r	r	NOUN
ejpam-854	496	16	d	d	NOUN
ejpam-854	496	17	luce	luce	NOUN
ejpam-854	496	18	.	.	PUNCT
ejpam-854	497	1	functional	functional	ADJ
ejpam-854	497	2	equations	equation	NOUN
ejpam-854	497	3	in	in	ADP
ejpam-854	497	4	the	the	DET
ejpam-854	497	5	behavioral	behavioral	ADJ
ejpam-854	497	6	sciences	science	NOUN
ejpam-854	497	7	.	.	PUNCT
ejpam-854	498	1	math	math	NOUN
ejpam-854	498	2	.	.	PUNCT
ejpam-854	499	1	japonica	japonica	PROPN
ejpam-854	499	2	,	,	PUNCT
ejpam-854	499	3	52	52	NUM
ejpam-854	499	4	:	:	PUNCT
ejpam-854	499	5	469	469	NUM
ejpam-854	499	6	-	-	SYM
ejpam-854	499	7	512	512	NUM
ejpam-854	499	8	,	,	PUNCT
ejpam-854	499	9	2000	2000	NUM
ejpam-854	499	10	.	.	PUNCT
ejpam-854	500	1	[	[	X
ejpam-854	500	2	2	2	NUM
ejpam-854	500	3	]	]	PUNCT
ejpam-854	500	4	l	l	NOUN
ejpam-854	500	5	cadariu	cadariu	NOUN
ejpam-854	500	6	and	and	CCONJ
ejpam-854	500	7	v	v	ADP
ejpam-854	500	8	radu	radu	PROPN
ejpam-854	500	9	.	.	PUNCT
ejpam-854	501	1	fixed	fix	VERB
ejpam-854	501	2	points	point	NOUN
ejpam-854	501	3	and	and	CCONJ
ejpam-854	501	4	the	the	DET
ejpam-854	501	5	stability	stability	NOUN
ejpam-854	501	6	of	of	ADP
ejpam-854	501	7	jensen	jensen	PROPN
ejpam-854	501	8	’s	’s	PART
ejpam-854	501	9	functional	functional	ADJ
ejpam-854	501	10	equation	equation	NOUN
ejpam-854	501	11	.	.	PUNCT
ejpam-854	502	1	journal	journal	PROPN
ejpam-854	502	2	of	of	ADP
ejpam-854	502	3	inequalities	inequality	NOUN
ejpam-854	502	4	in	in	ADP
ejpam-854	502	5	pure	pure	ADJ
ejpam-854	502	6	and	and	CCONJ
ejpam-854	502	7	applied	applied	ADJ
ejpam-854	502	8	mathematics	mathematic	NOUN
ejpam-854	502	9	,	,	PUNCT
ejpam-854	502	10	4(1	4(1	NOUN
ejpam-854	502	11	):	):	PUNCT
ejpam-854	502	12	pp7	pp7	PROPN
ejpam-854	502	13	,	,	PUNCT
ejpam-854	502	14	2003	2003	NUM
ejpam-854	502	15	.	.	PUNCT
ejpam-854	503	1	[	[	X
ejpam-854	503	2	3	3	NUM
ejpam-854	503	3	]	]	X
ejpam-854	503	4	l	l	NOUN
ejpam-854	503	5	cadariu	cadariu	NOUN
ejpam-854	503	6	and	and	CCONJ
ejpam-854	503	7	v	v	ADP
ejpam-854	503	8	radu	radu	PROPN
ejpam-854	503	9	.	.	PUNCT
ejpam-854	504	1	on	on	ADP
ejpam-854	504	2	the	the	DET
ejpam-854	504	3	stability	stability	NOUN
ejpam-854	504	4	of	of	ADP
ejpam-854	504	5	the	the	DET
ejpam-854	504	6	cauchy	cauchy	ADJ
ejpam-854	504	7	functional	functional	ADJ
ejpam-854	504	8	equation	equation	NOUN
ejpam-854	504	9	:	:	PUNCT
ejpam-854	504	10	a	a	DET
ejpam-854	504	11	fixed	fix	VERB
ejpam-854	504	12	point	point	NOUN
ejpam-854	504	13	approach	approach	NOUN
ejpam-854	504	14	,	,	PUNCT
ejpam-854	504	15	in	in	ADP
ejpam-854	504	16	iteration	iteration	NOUN
ejpam-854	504	17	theory	theory	NOUN
ejpam-854	504	18	(	(	PUNCT
ejpam-854	504	19	ecit’02	ecit’02	NOUN
ejpam-854	504	20	)	)	PUNCT
ejpam-854	504	21	,	,	PUNCT
ejpam-854	504	22	vol	vol	NOUN
ejpam-854	504	23	.	.	PUNCT
ejpam-854	504	24	346	346	NUM
ejpam-854	504	25	of	of	ADP
ejpam-854	504	26	die	die	VERB
ejpam-854	504	27	grazer	grazer	NOUN
ejpam-854	504	28	mathematischen	mathematischen	NOUN
ejpam-854	504	29	berichte	berichte	NOUN
ejpam-854	504	30	,	,	PUNCT
ejpam-854	504	31	pp	pp	PROPN
ejpam-854	504	32	.	.	PUNCT
ejpam-854	505	1	43	43	NUM
ejpam-854	505	2	-	-	SYM
ejpam-854	505	3	52	52	NUM
ejpam-854	505	4	,	,	PUNCT
ejpam-854	505	5	karl	karl	NOUN
ejpam-854	505	6	-	-	PUNCT
ejpam-854	505	7	franzens	franzens	PROPN
ejpam-854	505	8	-	-	PUNCT
ejpam-854	505	9	universitaet	universitaet	ADJ
ejpam-854	505	10	graz	graz	PROPN
ejpam-854	505	11	,	,	PUNCT
ejpam-854	505	12	graz	graz	PROPN
ejpam-854	505	13	,	,	PUNCT
ejpam-854	505	14	austria	austria	PROPN
ejpam-854	505	15	,	,	PUNCT
ejpam-854	505	16	2004	2004	NUM
ejpam-854	505	17	.	.	PUNCT
ejpam-854	506	1	[	[	X
ejpam-854	506	2	4	4	NUM
ejpam-854	506	3	]	]	X
ejpam-854	506	4	s	s	PART
ejpam-854	506	5	czerwik	czerwik	PROPN
ejpam-854	506	6	.	.	PUNCT
ejpam-854	507	1	stability	stability	NOUN
ejpam-854	507	2	of	of	ADP
ejpam-854	507	3	functional	functional	ADJ
ejpam-854	507	4	equations	equation	NOUN
ejpam-854	507	5	of	of	ADP
ejpam-854	507	6	ulam	ulam	PROPN
ejpam-854	507	7	-	-	PUNCT
ejpam-854	507	8	hyers	hyer	NOUN
ejpam-854	507	9	-	-	PUNCT
ejpam-854	507	10	rassias	rassias	PROPN
ejpam-854	507	11	type	type	NOUN
ejpam-854	507	12	.	.	PUNCT
ejpam-854	508	1	hadronic	hadronic	ADJ
ejpam-854	508	2	press	press	PROPN
ejpam-854	508	3	,	,	PUNCT
ejpam-854	508	4	inc	inc	PROPN
ejpam-854	508	5	.	.	PROPN
ejpam-854	508	6	,	,	PUNCT
ejpam-854	508	7	palm	palm	NOUN
ejpam-854	508	8	harbor	harbor	PROPN
ejpam-854	508	9	,	,	PUNCT
ejpam-854	508	10	usa	usa	PROPN
ejpam-854	508	11	,	,	PUNCT
ejpam-854	508	12	2003	2003	NUM
ejpam-854	508	13	.	.	PUNCT
ejpam-854	509	1	[	[	X
ejpam-854	509	2	5	5	NUM
ejpam-854	509	3	]	]	SYM
ejpam-854	509	4	h	h	NOUN
ejpam-854	509	5	g	g	PROPN
ejpam-854	509	6	dales	dale	NOUN
ejpam-854	509	7	and	and	CCONJ
ejpam-854	509	8	m	m	PROPN
ejpam-854	509	9	e	e	PROPN
ejpam-854	509	10	polyakov	polyakov	PROPN
ejpam-854	509	11	.	.	PUNCT
ejpam-854	510	1	multi	multi	ADJ
ejpam-854	510	2	-	-	ADJ
ejpam-854	510	3	normed	normed	ADJ
ejpam-854	510	4	spaces	space	NOUN
ejpam-854	510	5	and	and	CCONJ
ejpam-854	510	6	multi	multi	ADJ
ejpam-854	510	7	-	-	ADJ
ejpam-854	510	8	banach	banach	ADV
ejpam-854	510	9	algebras	algebra	NOUN
ejpam-854	510	10	.	.	PUNCT
ejpam-854	511	1	preprint	preprint	NOUN
ejpam-854	511	2	.	.	PUNCT
ejpam-854	512	1	[	[	X
ejpam-854	512	2	6	6	NUM
ejpam-854	512	3	]	]	SYM
ejpam-854	512	4	h	h	NOUN
ejpam-854	512	5	g	g	PROPN
ejpam-854	512	6	dales	dale	NOUN
ejpam-854	512	7	and	and	CCONJ
ejpam-854	512	8	m	m	PROPN
ejpam-854	512	9	s	s	PROPN
ejpam-854	512	10	moslehian	moslehian	NOUN
ejpam-854	512	11	.	.	PUNCT
ejpam-854	513	1	stability	stability	NOUN
ejpam-854	513	2	of	of	ADP
ejpam-854	513	3	mappings	mapping	NOUN
ejpam-854	513	4	on	on	ADP
ejpam-854	513	5	multi	multi	ADJ
ejpam-854	513	6	-	-	ADJ
ejpam-854	513	7	normed	normed	ADJ
ejpam-854	513	8	spaces	space	NOUN
ejpam-854	513	9	.	.	PUNCT
ejpam-854	514	1	glasgow	glasgow	PROPN
ejpam-854	514	2	mathematical	mathematical	ADJ
ejpam-854	514	3	journal	journal	NOUN
ejpam-854	514	4	,	,	PUNCT
ejpam-854	514	5	49	49	NUM
ejpam-854	514	6	:	:	PUNCT
ejpam-854	514	7	321	321	NUM
ejpam-854	514	8	-	-	SYM
ejpam-854	514	9	332	332	NUM
ejpam-854	514	10	,	,	PUNCT
ejpam-854	514	11	2007	2007	NUM
ejpam-854	514	12	.	.	PUNCT
ejpam-854	515	1	[	[	X
ejpam-854	515	2	7	7	X
ejpam-854	515	3	]	]	X
ejpam-854	515	4	j	j	PROPN
ejpam-854	515	5	b	b	PROPN
ejpam-854	515	6	diaz	diaz	PROPN
ejpam-854	515	7	and	and	CCONJ
ejpam-854	515	8	b	b	PROPN
ejpam-854	515	9	margolis	margolis	PROPN
ejpam-854	515	10	.	.	PUNCT
ejpam-854	516	1	a	a	DET
ejpam-854	516	2	fixed	fix	VERB
ejpam-854	516	3	point	point	NOUN
ejpam-854	516	4	theorem	theorem	NOUN
ejpam-854	516	5	of	of	ADP
ejpam-854	516	6	the	the	DET
ejpam-854	516	7	alternative	alternative	NOUN
ejpam-854	516	8	for	for	ADP
ejpam-854	516	9	the	the	DET
ejpam-854	516	10	contractions	contraction	NOUN
ejpam-854	516	11	on	on	ADP
ejpam-854	516	12	generalized	generalized	ADJ
ejpam-854	516	13	complete	complete	ADJ
ejpam-854	516	14	metric	metric	ADJ
ejpam-854	516	15	space	space	NOUN
ejpam-854	516	16	.	.	PUNCT
ejpam-854	517	1	bull	bull	NOUN
ejpam-854	517	2	.	.	PUNCT
ejpam-854	518	1	amer	amer	PROPN
ejpam-854	518	2	.	.	PUNCT
ejpam-854	518	3	math	math	PROPN
ejpam-854	518	4	.	.	PUNCT
ejpam-854	519	1	soc	soc	PROPN
ejpam-854	519	2	.	.	PUNCT
ejpam-854	519	3	,	,	PUNCT
ejpam-854	519	4	74	74	NUM
ejpam-854	519	5	:	:	SYM
ejpam-854	519	6	305	305	NUM
ejpam-854	519	7	-	-	SYM
ejpam-854	519	8	309	309	NUM
ejpam-854	519	9	,	,	PUNCT
ejpam-854	519	10	1968	1968	NUM
ejpam-854	519	11	.	.	PUNCT
ejpam-854	520	1	[	[	X
ejpam-854	520	2	8	8	NUM
ejpam-854	520	3	]	]	X
ejpam-854	520	4	m	m	PROPN
ejpam-854	520	5	eshaghi	eshaghi	PROPN
ejpam-854	520	6	gordji	gordji	PROPN
ejpam-854	520	7	,	,	PUNCT
ejpam-854	520	8	s	s	PART
ejpam-854	520	9	k	k	PROPN
ejpam-854	520	10	gharetapeh	gharetapeh	PROPN
ejpam-854	520	11	,	,	PUNCT
ejpam-854	520	12	j	j	PROPN
ejpam-854	520	13	m	m	PROPN
ejpam-854	520	14	rassias	rassias	PROPN
ejpam-854	520	15	,	,	PUNCT
ejpam-854	520	16	and	and	CCONJ
ejpam-854	520	17	s	s	VERB
ejpam-854	520	18	zolfaghari	zolfaghari	NOUN
ejpam-854	520	19	.	.	PUNCT
ejpam-854	520	20	solution	solution	NOUN
ejpam-854	520	21	and	and	CCONJ
ejpam-854	520	22	stability	stability	NOUN
ejpam-854	520	23	of	of	ADP
ejpam-854	520	24	a	a	DET
ejpam-854	520	25	mixed	mixed	ADJ
ejpam-854	520	26	type	type	NOUN
ejpam-854	520	27	additive	additive	NOUN
ejpam-854	520	28	,	,	PUNCT
ejpam-854	520	29	quadratic	quadratic	ADJ
ejpam-854	520	30	,	,	PUNCT
ejpam-854	520	31	and	and	CCONJ
ejpam-854	520	32	cubic	cubic	ADJ
ejpam-854	520	33	functional	functional	ADJ
ejpam-854	520	34	equation	equation	NOUN
ejpam-854	520	35	.	.	PUNCT
ejpam-854	521	1	advances	advance	NOUN
ejpam-854	521	2	in	in	ADP
ejpam-854	521	3	difference	difference	NOUN
ejpam-854	521	4	equations	equation	NOUN
ejpam-854	521	5	,	,	PUNCT
ejpam-854	521	6	2009	2009	NUM
ejpam-854	521	7	:	:	PUNCT
ejpam-854	521	8	1	1	NUM
ejpam-854	521	9	-	-	SYM
ejpam-854	521	10	17	17	NUM
ejpam-854	521	11	,	,	PUNCT
ejpam-854	521	12	2009	2009	NUM
ejpam-854	521	13	.	.	PUNCT
ejpam-854	522	1	[	[	X
ejpam-854	522	2	9	9	NUM
ejpam-854	522	3	]	]	SYM
ejpam-854	522	4	m	m	PROPN
ejpam-854	522	5	eshaghi	eshaghi	PROPN
ejpam-854	522	6	gordji	gordji	PROPN
ejpam-854	522	7	,	,	PUNCT
ejpam-854	522	8	h	h	NOUN
ejpam-854	522	9	khodaei	khodaei	NOUN
ejpam-854	522	10	,	,	PUNCT
ejpam-854	522	11	and	and	CCONJ
ejpam-854	522	12	th	th	X
ejpam-854	522	13	m	m	NOUN
ejpam-854	522	14	rassias	rassia	NOUN
ejpam-854	522	15	.	.	PUNCT
ejpam-854	523	1	on	on	ADP
ejpam-854	523	2	the	the	DET
ejpam-854	523	3	hyers	hyers	PROPN
ejpam-854	523	4	-	-	PUNCT
ejpam-854	523	5	ulam	ulam	ADJ
ejpam-854	523	6	-	-	PUNCT
ejpam-854	523	7	rassias	rassias	PROPN
ejpam-854	523	8	stability	stability	NOUN
ejpam-854	523	9	of	of	ADP
ejpam-854	523	10	a	a	DET
ejpam-854	523	11	generalized	generalize	VERB
ejpam-854	523	12	mixed	mixed	ADJ
ejpam-854	523	13	type	type	NOUN
ejpam-854	523	14	of	of	ADP
ejpam-854	523	15	quartic	quartic	ADJ
ejpam-854	523	16	,	,	PUNCT
ejpam-854	523	17	cubic	cubic	ADJ
ejpam-854	523	18	,	,	PUNCT
ejpam-854	523	19	quadratic	quadratic	ADJ
ejpam-854	523	20	and	and	CCONJ
ejpam-854	523	21	additive	additive	ADJ
ejpam-854	523	22	functional	functional	ADJ
ejpam-854	523	23	equations	equation	NOUN
ejpam-854	523	24	in	in	ADP
ejpam-854	523	25	quasi	quasi	ADJ
ejpam-854	523	26	-	-	ADJ
ejpam-854	523	27	banach	banach	ADJ
ejpam-854	523	28	spaces	space	NOUN
ejpam-854	523	29	.	.	PUNCT
ejpam-854	524	1	arxiv:0903.0834v2	arxiv:0903.0834v2	VERB
ejpam-854	525	1	[	[	X
ejpam-854	525	2	math.fa	math.fa	X
ejpam-854	525	3	]	]	PUNCT
ejpam-854	525	4	,	,	PUNCT
ejpam-854	525	5	24	24	NUM
ejpam-854	525	6	apr	apr	NOUN
ejpam-854	525	7	2009	2009	NUM
ejpam-854	525	8	.	.	PUNCT
ejpam-854	526	1	[	[	X
ejpam-854	526	2	10	10	NUM
ejpam-854	526	3	]	]	X
ejpam-854	526	4	m	m	PROPN
ejpam-854	526	5	eshaghi	eshaghi	PROPN
ejpam-854	526	6	gordji	gordji	PROPN
ejpam-854	526	7	and	and	CCONJ
ejpam-854	526	8	m	m	PROPN
ejpam-854	526	9	b	b	PROPN
ejpam-854	526	10	savadkouhi	savadkouhi	NOUN
ejpam-854	526	11	.	.	PUNCT
ejpam-854	527	1	stability	stability	NOUN
ejpam-854	527	2	of	of	ADP
ejpam-854	527	3	mixed	mixed	ADJ
ejpam-854	527	4	type	type	NOUN
ejpam-854	527	5	cubic	cubic	ADJ
ejpam-854	527	6	and	and	CCONJ
ejpam-854	527	7	quartic	quartic	ADJ
ejpam-854	527	8	functional	functional	ADJ
ejpam-854	527	9	equations	equation	NOUN
ejpam-854	527	10	in	in	ADP
ejpam-854	527	11	random	random	ADJ
ejpam-854	527	12	normed	normed	ADJ
ejpam-854	527	13	spaces	space	NOUN
ejpam-854	527	14	.	.	PUNCT
ejpam-854	528	1	journal	journal	PROPN
ejpam-854	528	2	of	of	ADP
ejpam-854	528	3	inequalities	inequality	NOUN
ejpam-854	528	4	and	and	CCONJ
ejpam-854	528	5	applications	application	NOUN
ejpam-854	528	6	,	,	PUNCT
ejpam-854	528	7	2009	2009	NUM
ejpam-854	528	8	:	:	PUNCT
ejpam-854	528	9	pp9	pp9	PROPN
ejpam-854	528	10	,	,	PUNCT
ejpam-854	528	11	2009	2009	NUM
ejpam-854	528	12	.	.	PUNCT
ejpam-854	529	1	[	[	X
ejpam-854	529	2	11	11	NUM
ejpam-854	529	3	]	]	X
ejpam-854	529	4	p	p	X
ejpam-854	529	5	găvruţa	găvruţa	PROPN
ejpam-854	529	6	.	.	PUNCT
ejpam-854	530	1	a	a	DET
ejpam-854	530	2	generalization	generalization	NOUN
ejpam-854	530	3	of	of	ADP
ejpam-854	530	4	the	the	DET
ejpam-854	530	5	hyers	hyers	PROPN
ejpam-854	530	6	-	-	PUNCT
ejpam-854	530	7	ulam	ulam	ADJ
ejpam-854	530	8	-	-	PUNCT
ejpam-854	530	9	rassias	rassias	PROPN
ejpam-854	530	10	stability	stability	NOUN
ejpam-854	530	11	of	of	ADP
ejpam-854	530	12	approximately	approximately	ADV
ejpam-854	530	13	additive	additive	ADJ
ejpam-854	530	14	mappings	mapping	NOUN
ejpam-854	530	15	.	.	PUNCT
ejpam-854	531	1	journal	journal	PROPN
ejpam-854	531	2	of	of	ADP
ejpam-854	531	3	mathematical	mathematical	ADJ
ejpam-854	531	4	analysis	analysis	NOUN
ejpam-854	531	5	and	and	CCONJ
ejpam-854	531	6	applications	application	NOUN
ejpam-854	531	7	,	,	PUNCT
ejpam-854	531	8	184	184	NUM
ejpam-854	531	9	:	:	SYM
ejpam-854	531	10	431	431	NUM
ejpam-854	531	11	-	-	SYM
ejpam-854	531	12	436	436	NUM
ejpam-854	531	13	,	,	PUNCT
ejpam-854	531	14	1994	1994	NUM
ejpam-854	531	15	.	.	PUNCT
ejpam-854	532	1	[	[	X
ejpam-854	532	2	12	12	NUM
ejpam-854	532	3	]	]	X
ejpam-854	532	4	p	p	X
ejpam-854	532	5	găvruţa	găvruţa	PROPN
ejpam-854	532	6	.	.	PUNCT
ejpam-854	533	1	on	on	ADP
ejpam-854	533	2	the	the	DET
ejpam-854	533	3	hyers	hyers	PROPN
ejpam-854	533	4	-	-	PUNCT
ejpam-854	533	5	ulam	ulam	ADJ
ejpam-854	533	6	-	-	PUNCT
ejpam-854	533	7	rassias	rassias	PROPN
ejpam-854	533	8	stability	stability	NOUN
ejpam-854	533	9	of	of	ADP
ejpam-854	533	10	mappings	mapping	NOUN
ejpam-854	533	11	,	,	PUNCT
ejpam-854	533	12	in	in	ADP
ejpam-854	533	13	recent	recent	ADJ
ejpam-854	533	14	progress	progress	NOUN
ejpam-854	533	15	in	in	ADP
ejpam-854	533	16	inequalities	inequality	NOUN
ejpam-854	533	17	.	.	PUNCT
ejpam-854	534	1	vol	vol	NOUN
ejpam-854	534	2	.	.	PUNCT
ejpam-854	535	1	430	430	NUM
ejpam-854	535	2	of	of	ADP
ejpam-854	535	3	mathematics	mathematic	NOUN
ejpam-854	535	4	and	and	CCONJ
ejpam-854	535	5	its	its	PRON
ejpam-854	535	6	applications	application	NOUN
ejpam-854	535	7	,	,	PUNCT
ejpam-854	535	8	pp	pp	ADJ
ejpam-854	535	9	.	.	PUNCT
ejpam-854	536	1	465	465	NUM
ejpam-854	536	2	-	-	SYM
ejpam-854	536	3	469	469	NUM
ejpam-854	536	4	,	,	PUNCT
ejpam-854	536	5	kluwer	kluwer	NOUN
ejpam-854	536	6	academic	academic	ADJ
ejpam-854	536	7	publishers	publisher	NOUN
ejpam-854	536	8	,	,	PUNCT
ejpam-854	536	9	dordrecht	dordrecht	PROPN
ejpam-854	536	10	,	,	PUNCT
ejpam-854	536	11	the	the	DET
ejpam-854	536	12	netherlands	netherlands	PROPN
ejpam-854	536	13	,	,	PUNCT
ejpam-854	536	14	1998	1998	NUM
ejpam-854	536	15	.	.	PUNCT
ejpam-854	537	1	[	[	X
ejpam-854	537	2	13	13	NUM
ejpam-854	537	3	]	]	X
ejpam-854	537	4	p	p	X
ejpam-854	537	5	găvruţa	găvruţa	PROPN
ejpam-854	537	6	.	.	PUNCT
ejpam-854	538	1	an	an	DET
ejpam-854	538	2	answer	answer	NOUN
ejpam-854	538	3	to	to	ADP
ejpam-854	538	4	a	a	DET
ejpam-854	538	5	question	question	NOUN
ejpam-854	538	6	of	of	ADP
ejpam-854	538	7	john	john	PROPN
ejpam-854	538	8	m.	m.	PROPN
ejpam-854	538	9	rassias	rassias	PROPN
ejpam-854	538	10	concerning	concern	VERB
ejpam-854	538	11	the	the	DET
ejpam-854	538	12	stability	stability	NOUN
ejpam-854	538	13	of	of	ADP
ejpam-854	538	14	cauchy	cauchy	ADJ
ejpam-854	538	15	equation	equation	NOUN
ejpam-854	538	16	,	,	PUNCT
ejpam-854	538	17	in	in	ADP
ejpam-854	538	18	advances	advance	NOUN
ejpam-854	538	19	in	in	ADP
ejpam-854	538	20	equations	equation	NOUN
ejpam-854	538	21	and	and	CCONJ
ejpam-854	538	22	inequalities	inequality	NOUN
ejpam-854	538	23	,	,	PUNCT
ejpam-854	538	24	hadronic	hadronic	ADJ
ejpam-854	538	25	mathematics	mathematic	NOUN
ejpam-854	538	26	series	series	NOUN
ejpam-854	538	27	,	,	PUNCT
ejpam-854	538	28	pp	pp	PROPN
ejpam-854	538	29	.	.	PUNCT
ejpam-854	539	1	67	67	NUM
ejpam-854	539	2	-	-	SYM
ejpam-854	539	3	71	71	NUM
ejpam-854	539	4	,	,	PUNCT
ejpam-854	539	5	1999	1999	NUM
ejpam-854	539	6	.	.	PUNCT
ejpam-854	540	1	[	[	X
ejpam-854	540	2	14	14	NUM
ejpam-854	540	3	]	]	X
ejpam-854	540	4	p	p	X
ejpam-854	540	5	găvruţa	găvruţa	PROPN
ejpam-854	540	6	.	.	PUNCT
ejpam-854	541	1	on	on	ADP
ejpam-854	541	2	a	a	DET
ejpam-854	541	3	problem	problem	NOUN
ejpam-854	541	4	of	of	ADP
ejpam-854	541	5	g.	g.	PROPN
ejpam-854	541	6	isac	isac	PROPN
ejpam-854	541	7	and	and	CCONJ
ejpam-854	541	8	th	th	PROPN
ejpam-854	541	9	.	.	PUNCT
ejpam-854	541	10	m.	m.	NOUN
ejpam-854	541	11	rassias	rassias	PROPN
ejpam-854	541	12	concerning	concern	VERB
ejpam-854	541	13	the	the	DET
ejpam-854	541	14	stability	stability	NOUN
ejpam-854	541	15	of	of	ADP
ejpam-854	541	16	mappings	mapping	NOUN
ejpam-854	541	17	.	.	PUNCT
ejpam-854	542	1	journal	journal	PROPN
ejpam-854	542	2	of	of	ADP
ejpam-854	542	3	mathematical	mathematical	ADJ
ejpam-854	542	4	analysis	analysis	NOUN
ejpam-854	542	5	and	and	CCONJ
ejpam-854	542	6	applications	application	NOUN
ejpam-854	542	7	,	,	PUNCT
ejpam-854	542	8	261	261	NUM
ejpam-854	542	9	:	:	PUNCT
ejpam-854	542	10	543	543	NUM
ejpam-854	542	11	-	-	SYM
ejpam-854	542	12	553	553	NUM
ejpam-854	542	13	,	,	PUNCT
ejpam-854	542	14	2001	2001	NUM
ejpam-854	542	15	.	.	PUNCT
ejpam-854	543	1	references	reference	NOUN
ejpam-854	543	2	1046	1046	NUM
ejpam-854	544	1	[	[	X
ejpam-854	544	2	15	15	NUM
ejpam-854	544	3	]	]	X
ejpam-854	544	4	p	p	X
ejpam-854	544	5	găvruţa	găvruţa	PROPN
ejpam-854	544	6	.	.	PUNCT
ejpam-854	545	1	on	on	ADP
ejpam-854	545	2	the	the	DET
ejpam-854	545	3	hyers	hyers	PROPN
ejpam-854	545	4	-	-	PUNCT
ejpam-854	545	5	ulam	ulam	ADJ
ejpam-854	545	6	-	-	PUNCT
ejpam-854	545	7	rassias	rassias	PROPN
ejpam-854	545	8	stability	stability	NOUN
ejpam-854	545	9	of	of	ADP
ejpam-854	545	10	the	the	DET
ejpam-854	545	11	quadratic	quadratic	ADJ
ejpam-854	545	12	mappings	mapping	NOUN
ejpam-854	545	13	.	.	PUNCT
ejpam-854	546	1	nonlinear	nonlinear	ADJ
ejpam-854	546	2	functional	functional	ADJ
ejpam-854	546	3	analysis	analysis	NOUN
ejpam-854	546	4	and	and	CCONJ
ejpam-854	546	5	applications	application	NOUN
ejpam-854	546	6	,	,	PUNCT
ejpam-854	546	7	9	9	NUM
ejpam-854	546	8	:	:	SYM
ejpam-854	546	9	415	415	NUM
ejpam-854	546	10	-	-	SYM
ejpam-854	546	11	428	428	NUM
ejpam-854	546	12	,	,	PUNCT
ejpam-854	546	13	2004	2004	NUM
ejpam-854	546	14	.	.	PUNCT
ejpam-854	547	1	[	[	X
ejpam-854	547	2	16	16	NUM
ejpam-854	547	3	]	]	X
ejpam-854	547	4	l	l	NOUN
ejpam-854	547	5	găvruţa	găvruţa	NOUN
ejpam-854	547	6	and	and	CCONJ
ejpam-854	547	7	p	p	X
ejpam-854	547	8	găvruţa	găvruţa	PROPN
ejpam-854	547	9	.	.	PUNCT
ejpam-854	548	1	on	on	ADP
ejpam-854	548	2	a	a	DET
ejpam-854	548	3	problem	problem	NOUN
ejpam-854	548	4	of	of	ADP
ejpam-854	548	5	john	john	PROPN
ejpam-854	548	6	m.	m.	PROPN
ejpam-854	548	7	rassias	rassias	PROPN
ejpam-854	548	8	concerning	concern	VERB
ejpam-854	548	9	the	the	DET
ejpam-854	548	10	stability	stability	NOUN
ejpam-854	548	11	in	in	ADP
ejpam-854	548	12	ulam	ulam	PROPN
ejpam-854	548	13	sense	sense	NOUN
ejpam-854	548	14	of	of	ADP
ejpam-854	548	15	euler	euler	NOUN
ejpam-854	548	16	-	-	PUNCT
ejpam-854	548	17	lagrange	lagrange	NOUN
ejpam-854	548	18	equation	equation	NOUN
ejpam-854	548	19	,	,	PUNCT
ejpam-854	548	20	in	in	ADP
ejpam-854	548	21	functional	functional	ADJ
ejpam-854	548	22	equations	equation	NOUN
ejpam-854	548	23	,	,	PUNCT
ejpam-854	548	24	difference	difference	NOUN
ejpam-854	548	25	inequalities	inequality	NOUN
ejpam-854	548	26	and	and	CCONJ
ejpam-854	548	27	ulam	ulam	PROPN
ejpam-854	548	28	stability	stability	PROPN
ejpam-854	548	29	notions	notion	NOUN
ejpam-854	548	30	,	,	PUNCT
ejpam-854	548	31	pp	pp	ADJ
ejpam-854	548	32	.	.	PUNCT
ejpam-854	549	1	47	47	NUM
ejpam-854	549	2	-	-	SYM
ejpam-854	549	3	53	53	NUM
ejpam-854	549	4	,	,	PUNCT
ejpam-854	549	5	nova	nova	PROPN
ejpam-854	549	6	sciences	sciences	PROPN
ejpam-854	549	7	,	,	PUNCT
ejpam-854	549	8	2010	2010	NUM
ejpam-854	549	9	.	.	PUNCT
ejpam-854	550	1	[	[	X
ejpam-854	550	2	17	17	NUM
ejpam-854	550	3	]	]	X
ejpam-854	550	4	d	d	PROPN
ejpam-854	550	5	h	h	PROPN
ejpam-854	550	6	hyers	hyer	NOUN
ejpam-854	550	7	.	.	PUNCT
ejpam-854	551	1	on	on	ADP
ejpam-854	551	2	the	the	DET
ejpam-854	551	3	stability	stability	NOUN
ejpam-854	551	4	of	of	ADP
ejpam-854	551	5	the	the	DET
ejpam-854	551	6	linear	linear	ADJ
ejpam-854	551	7	functional	functional	ADJ
ejpam-854	551	8	equation	equation	NOUN
ejpam-854	551	9	.	.	PUNCT
ejpam-854	552	1	proc	proc	NOUN
ejpam-854	552	2	.	.	PUNCT
ejpam-854	553	1	nat	nat	PROPN
ejpam-854	553	2	.	.	PUNCT
ejpam-854	554	1	acad	acad	PROPN
ejpam-854	554	2	.	.	PUNCT
ejpam-854	555	1	sci	sci	PROPN
ejpam-854	555	2	.	.	PROPN
ejpam-854	555	3	usa	usa	PROPN
ejpam-854	555	4	,	,	PUNCT
ejpam-854	555	5	27	27	NUM
ejpam-854	555	6	:	:	PUNCT
ejpam-854	555	7	222	222	NUM
ejpam-854	555	8	-	-	SYM
ejpam-854	555	9	224	224	NUM
ejpam-854	555	10	,	,	PUNCT
ejpam-854	555	11	1941	1941	NUM
ejpam-854	555	12	.	.	PUNCT
ejpam-854	556	1	[	[	X
ejpam-854	556	2	18	18	NUM
ejpam-854	556	3	]	]	X
ejpam-854	556	4	g	g	PROPN
ejpam-854	556	5	isac	isac	PROPN
ejpam-854	556	6	and	and	CCONJ
ejpam-854	556	7	th	th	X
ejpam-854	556	8	m	m	NOUN
ejpam-854	556	9	rassias	rassias	PROPN
ejpam-854	556	10	.	.	PUNCT
ejpam-854	557	1	stability	stability	NOUN
ejpam-854	557	2	of	of	ADP
ejpam-854	557	3	ψ	ψ	ADJ
ejpam-854	557	4	-	-	ADJ
ejpam-854	557	5	additive	additive	ADJ
ejpam-854	557	6	mappings	mapping	NOUN
ejpam-854	557	7	:	:	PUNCT
ejpam-854	557	8	applications	application	NOUN
ejpam-854	557	9	to	to	PART
ejpam-854	557	10	nonlinear	nonlinear	ADJ
ejpam-854	557	11	analysis	analysis	NOUN
ejpam-854	557	12	.	.	PUNCT
ejpam-854	558	1	international	international	ADJ
ejpam-854	558	2	journal	journal	NOUN
ejpam-854	558	3	of	of	ADP
ejpam-854	558	4	mathematics	mathematics	PROPN
ejpam-854	558	5	and	and	CCONJ
ejpam-854	558	6	mathematical	mathematical	ADJ
ejpam-854	558	7	sciences	science	NOUN
ejpam-854	558	8	,	,	PUNCT
ejpam-854	558	9	19	19	NUM
ejpam-854	558	10	:	:	SYM
ejpam-854	558	11	219228	219228	NUM
ejpam-854	558	12	,	,	PUNCT
ejpam-854	558	13	1996	1996	NUM
ejpam-854	558	14	.	.	PUNCT
ejpam-854	559	1	[	[	X
ejpam-854	559	2	19	19	NUM
ejpam-854	559	3	]	]	SYM
ejpam-854	559	4	s	s	PART
ejpam-854	559	5	m	m	PROPN
ejpam-854	559	6	jung	jung	PROPN
ejpam-854	559	7	.	.	PUNCT
ejpam-854	560	1	hyers	hyer	NOUN
ejpam-854	560	2	-	-	PUNCT
ejpam-854	560	3	ulam	ulam	NOUN
ejpam-854	560	4	-	-	PUNCT
ejpam-854	560	5	rassias	rassias	PROPN
ejpam-854	560	6	stability	stability	NOUN
ejpam-854	560	7	of	of	ADP
ejpam-854	560	8	functional	functional	ADJ
ejpam-854	560	9	equations	equation	NOUN
ejpam-854	560	10	in	in	ADP
ejpam-854	560	11	mathematical	mathematical	ADJ
ejpam-854	560	12	analysis	analysis	NOUN
ejpam-854	560	13	.	.	PUNCT
ejpam-854	561	1	hadronic	hadronic	ADJ
ejpam-854	561	2	press	press	PROPN
ejpam-854	561	3	.	.	PUNCT
ejpam-854	562	1	inc	inc	PROPN
ejpam-854	562	2	.	.	PROPN
ejpam-854	562	3	,	,	PUNCT
ejpam-854	562	4	florida	florida	PROPN
ejpam-854	562	5	,	,	PUNCT
ejpam-854	562	6	2000	2000	NUM
ejpam-854	562	7	.	.	PUNCT
ejpam-854	563	1	[	[	X
ejpam-854	563	2	20	20	NUM
ejpam-854	563	3	]	]	X
ejpam-854	563	4	d	d	NOUN
ejpam-854	563	5	mihȩt	mihȩt	NOUN
ejpam-854	563	6	and	and	CCONJ
ejpam-854	563	7	v	v	ADP
ejpam-854	563	8	radu	radu	PROPN
ejpam-854	563	9	.	.	PUNCT
ejpam-854	564	1	on	on	ADP
ejpam-854	564	2	the	the	DET
ejpam-854	564	3	stability	stability	NOUN
ejpam-854	564	4	of	of	ADP
ejpam-854	564	5	the	the	DET
ejpam-854	564	6	additive	additive	ADJ
ejpam-854	564	7	cauchy	cauchy	ADJ
ejpam-854	564	8	functional	functional	ADJ
ejpam-854	564	9	equation	equation	NOUN
ejpam-854	564	10	in	in	ADP
ejpam-854	564	11	random	random	ADJ
ejpam-854	564	12	normed	normed	ADJ
ejpam-854	564	13	spaces	space	NOUN
ejpam-854	564	14	.	.	PUNCT
ejpam-854	565	1	journal	journal	PROPN
ejpam-854	565	2	of	of	ADP
ejpam-854	565	3	mathematical	mathematical	ADJ
ejpam-854	565	4	analysis	analysis	NOUN
ejpam-854	565	5	and	and	CCONJ
ejpam-854	565	6	applications	application	NOUN
ejpam-854	565	7	,	,	PUNCT
ejpam-854	565	8	343	343	NUM
ejpam-854	565	9	:	:	PUNCT
ejpam-854	565	10	567572	567572	NUM
ejpam-854	565	11	,	,	PUNCT
ejpam-854	565	12	2008	2008	NUM
ejpam-854	565	13	.	.	PUNCT
ejpam-854	566	1	[	[	X
ejpam-854	566	2	21	21	NUM
ejpam-854	566	3	]	]	X
ejpam-854	566	4	m	m	PROPN
ejpam-854	566	5	s	s	NOUN
ejpam-854	566	6	moslehian	moslehian	NOUN
ejpam-854	566	7	,	,	PUNCT
ejpam-854	566	8	k	k	PROPN
ejpam-854	566	9	nikodem	nikodem	PROPN
ejpam-854	566	10	,	,	PUNCT
ejpam-854	566	11	and	and	CCONJ
ejpam-854	566	12	d	d	ADP
ejpam-854	566	13	popa	popa	NOUN
ejpam-854	566	14	.	.	PUNCT
ejpam-854	567	1	asymptotic	asymptotic	ADJ
ejpam-854	567	2	aspect	aspect	NOUN
ejpam-854	567	3	of	of	ADP
ejpam-854	567	4	the	the	DET
ejpam-854	567	5	quadratic	quadratic	ADJ
ejpam-854	567	6	functional	functional	ADJ
ejpam-854	567	7	equation	equation	NOUN
ejpam-854	567	8	in	in	ADP
ejpam-854	567	9	multi	multi	ADJ
ejpam-854	567	10	-	-	ADJ
ejpam-854	567	11	normed	normed	ADJ
ejpam-854	567	12	spaces	space	NOUN
ejpam-854	567	13	.	.	PUNCT
ejpam-854	568	1	journal	journal	PROPN
ejpam-854	568	2	of	of	ADP
ejpam-854	568	3	mathematical	mathematical	ADJ
ejpam-854	568	4	analysis	analysis	NOUN
ejpam-854	568	5	and	and	CCONJ
ejpam-854	568	6	applications	application	NOUN
ejpam-854	568	7	,	,	PUNCT
ejpam-854	568	8	355	355	NUM
ejpam-854	568	9	:	:	PUNCT
ejpam-854	568	10	717	717	NUM
ejpam-854	568	11	-	-	SYM
ejpam-854	568	12	724	724	NUM
ejpam-854	568	13	,	,	PUNCT
ejpam-854	568	14	2009	2009	NUM
ejpam-854	568	15	.	.	PUNCT
ejpam-854	569	1	[	[	X
ejpam-854	569	2	22	22	NUM
ejpam-854	569	3	]	]	X
ejpam-854	569	4	m	m	PROPN
ejpam-854	569	5	s	s	NOUN
ejpam-854	569	6	moslehian	moslehian	PROPN
ejpam-854	569	7	.	.	PUNCT
ejpam-854	569	8	superstability	superstability	NOUN
ejpam-854	569	9	of	of	ADP
ejpam-854	569	10	higher	high	ADJ
ejpam-854	569	11	derivations	derivation	NOUN
ejpam-854	569	12	in	in	ADP
ejpam-854	569	13	multi	multi	ADJ
ejpam-854	569	14	-	-	ADJ
ejpam-854	569	15	banach	banach	ADV
ejpam-854	569	16	algebras	algebra	NOUN
ejpam-854	569	17	.	.	PUNCT
ejpam-854	570	1	tamsui	tamsui	PROPN
ejpam-854	570	2	oxford	oxford	PROPN
ejpam-854	570	3	journal	journal	PROPN
ejpam-854	570	4	of	of	ADP
ejpam-854	570	5	mathematical	mathematical	ADJ
ejpam-854	570	6	sciences	science	NOUN
ejpam-854	570	7	,	,	PUNCT
ejpam-854	570	8	24	24	NUM
ejpam-854	570	9	:	:	PUNCT
ejpam-854	570	10	417	417	NUM
ejpam-854	570	11	-	-	SYM
ejpam-854	570	12	427	427	NUM
ejpam-854	570	13	,	,	PUNCT
ejpam-854	570	14	2008	2008	NUM
ejpam-854	570	15	.	.	PUNCT
ejpam-854	571	1	[	[	X
ejpam-854	571	2	23	23	NUM
ejpam-854	571	3	]	]	X
ejpam-854	571	4	m	m	PROPN
ejpam-854	571	5	s	s	NOUN
ejpam-854	571	6	moslehian	moslehian	NOUN
ejpam-854	571	7	and	and	CCONJ
ejpam-854	571	8	h	h	NOUN
ejpam-854	571	9	m	m	PROPN
ejpam-854	571	10	srivastava	srivastava	PROPN
ejpam-854	571	11	.	.	PUNCT
ejpam-854	572	1	jensen	jensen	PROPN
ejpam-854	572	2	’s	’s	PART
ejpam-854	572	3	functional	functional	ADJ
ejpam-854	572	4	equation	equation	NOUN
ejpam-854	572	5	in	in	ADP
ejpam-854	572	6	multi	multi	ADJ
ejpam-854	572	7	-	-	ADJ
ejpam-854	572	8	normed	normed	ADJ
ejpam-854	572	9	spaces	space	NOUN
ejpam-854	572	10	.	.	PUNCT
ejpam-854	573	1	taiwanese	taiwanese	ADJ
ejpam-854	573	2	journal	journal	NOUN
ejpam-854	573	3	of	of	ADP
ejpam-854	573	4	mathematics	mathematic	NOUN
ejpam-854	573	5	,	,	PUNCT
ejpam-854	573	6	14	14	NUM
ejpam-854	573	7	:	:	PUNCT
ejpam-854	573	8	453	453	NUM
ejpam-854	573	9	-	-	SYM
ejpam-854	573	10	462	462	NUM
ejpam-854	573	11	,	,	PUNCT
ejpam-854	573	12	2010	2010	NUM
ejpam-854	573	13	.	.	PUNCT
ejpam-854	574	1	[	[	X
ejpam-854	574	2	24	24	NUM
ejpam-854	574	3	]	]	SYM
ejpam-854	574	4	b	b	X
ejpam-854	574	5	paneah	paneah	NOUN
ejpam-854	574	6	.	.	PUNCT
ejpam-854	575	1	some	some	DET
ejpam-854	575	2	remarks	remark	NOUN
ejpam-854	575	3	on	on	ADP
ejpam-854	575	4	stability	stability	NOUN
ejpam-854	575	5	and	and	CCONJ
ejpam-854	575	6	solvability	solvability	NOUN
ejpam-854	575	7	of	of	ADP
ejpam-854	575	8	linear	linear	ADJ
ejpam-854	575	9	functional	functional	ADJ
ejpam-854	575	10	equations	equation	NOUN
ejpam-854	575	11	.	.	PUNCT
ejpam-854	576	1	banach	banach	PROPN
ejpam-854	576	2	j.	j.	PROPN
ejpam-854	576	3	math	math	PROPN
ejpam-854	576	4	.	.	PUNCT
ejpam-854	577	1	anal	anal	PROPN
ejpam-854	577	2	.	.	PUNCT
ejpam-854	577	3	,	,	PUNCT
ejpam-854	577	4	1	1	NUM
ejpam-854	577	5	:	:	SYM
ejpam-854	577	6	56	56	NUM
ejpam-854	577	7	-	-	SYM
ejpam-854	577	8	65	65	NUM
ejpam-854	577	9	,	,	PUNCT
ejpam-854	577	10	2007	2007	NUM
ejpam-854	577	11	.	.	PUNCT
ejpam-854	578	1	[	[	X
ejpam-854	578	2	25	25	NUM
ejpam-854	578	3	]	]	X
ejpam-854	578	4	c	c	NOUN
ejpam-854	578	5	park	park	NOUN
ejpam-854	578	6	.	.	PUNCT
ejpam-854	579	1	fixed	fix	VERB
ejpam-854	579	2	points	point	NOUN
ejpam-854	579	3	and	and	CCONJ
ejpam-854	579	4	the	the	DET
ejpam-854	579	5	stability	stability	NOUN
ejpam-854	579	6	of	of	ADP
ejpam-854	579	7	an	an	DET
ejpam-854	579	8	aqcq	aqcq	NOUN
ejpam-854	579	9	-	-	PUNCT
ejpam-854	579	10	functional	functional	ADJ
ejpam-854	579	11	equation	equation	NOUN
ejpam-854	579	12	in	in	ADP
ejpam-854	579	13	nonarchimedean	nonarchimedean	ADJ
ejpam-854	579	14	normed	normed	ADJ
ejpam-854	579	15	spaces	space	NOUN
ejpam-854	579	16	.	.	PUNCT
ejpam-854	580	1	abstract	abstract	ADJ
ejpam-854	580	2	and	and	CCONJ
ejpam-854	580	3	applied	apply	VERB
ejpam-854	580	4	analysis	analysis	NOUN
ejpam-854	580	5	,	,	PUNCT
ejpam-854	580	6	2010	2010	NUM
ejpam-854	580	7	:	:	PUNCT
ejpam-854	580	8	pp	pp	PROPN
ejpam-854	580	9	15	15	NUM
ejpam-854	580	10	,	,	PUNCT
ejpam-854	580	11	2010	2010	NUM
ejpam-854	580	12	.	.	PUNCT
ejpam-854	581	1	[	[	X
ejpam-854	581	2	26	26	NUM
ejpam-854	581	3	]	]	X
ejpam-854	581	4	c	c	NOUN
ejpam-854	581	5	park	park	NOUN
ejpam-854	581	6	and	and	CCONJ
ejpam-854	581	7	j	j	PROPN
ejpam-854	581	8	m	m	PROPN
ejpam-854	581	9	rassias	rassias	PROPN
ejpam-854	581	10	.	.	PUNCT
ejpam-854	582	1	stability	stability	NOUN
ejpam-854	582	2	of	of	ADP
ejpam-854	582	3	the	the	DET
ejpam-854	582	4	jensen	jensen	ADJ
ejpam-854	582	5	-	-	PUNCT
ejpam-854	582	6	type	type	NOUN
ejpam-854	582	7	functional	functional	ADJ
ejpam-854	582	8	equation	equation	NOUN
ejpam-854	582	9	in	in	ADP
ejpam-854	582	10	c∗-algebras	c∗-algebra	NOUN
ejpam-854	582	11	:	:	PUNCT
ejpam-854	582	12	a	a	DET
ejpam-854	582	13	fixed	fix	VERB
ejpam-854	582	14	point	point	NOUN
ejpam-854	582	15	approach	approach	NOUN
ejpam-854	582	16	.	.	PUNCT
ejpam-854	583	1	abstract	abstract	ADJ
ejpam-854	583	2	and	and	CCONJ
ejpam-854	583	3	applied	apply	VERB
ejpam-854	583	4	analysis	analysis	NOUN
ejpam-854	583	5	,	,	PUNCT
ejpam-854	583	6	2009	2009	NUM
ejpam-854	583	7	:	:	PUNCT
ejpam-854	583	8	pp17	pp17	PROPN
ejpam-854	583	9	,	,	PUNCT
ejpam-854	583	10	2009	2009	NUM
ejpam-854	583	11	.	.	PUNCT
ejpam-854	584	1	[	[	X
ejpam-854	584	2	27	27	NUM
ejpam-854	584	3	]	]	SYM
ejpam-854	584	4	v	v	X
ejpam-854	584	5	radu	radu	PROPN
ejpam-854	584	6	.	.	PUNCT
ejpam-854	585	1	the	the	DET
ejpam-854	585	2	fixed	fixed	ADJ
ejpam-854	585	3	point	point	NOUN
ejpam-854	585	4	alternative	alternative	NOUN
ejpam-854	585	5	and	and	CCONJ
ejpam-854	585	6	the	the	DET
ejpam-854	585	7	stability	stability	NOUN
ejpam-854	585	8	of	of	ADP
ejpam-854	585	9	functional	functional	ADJ
ejpam-854	585	10	equations	equation	NOUN
ejpam-854	585	11	,	,	PUNCT
ejpam-854	585	12	in	in	ADP
ejpam-854	585	13	:	:	PUNCT
ejpam-854	585	14	seminaron	seminaron	ADJ
ejpam-854	585	15	fixed	fix	VERB
ejpam-854	585	16	point	point	NOUN
ejpam-854	585	17	theory	theory	NOUN
ejpam-854	585	18	,	,	PUNCT
ejpam-854	585	19	cluj	cluj	NOUN
ejpam-854	585	20	-	-	PUNCT
ejpam-854	585	21	napoca	napoca	NOUN
ejpam-854	585	22	,	,	PUNCT
ejpam-854	585	23	vol	vol	NOUN
ejpam-854	585	24	.	.	PROPN
ejpam-854	585	25	4	4	NUM
ejpam-854	585	26	,	,	PUNCT
ejpam-854	585	27	2003	2003	NUM
ejpam-854	585	28	.	.	PUNCT
ejpam-854	586	1	[	[	X
ejpam-854	586	2	28	28	NUM
ejpam-854	586	3	]	]	X
ejpam-854	586	4	th	th	X
ejpam-854	586	5	m	m	NOUN
ejpam-854	586	6	rassias	rassia	NOUN
ejpam-854	586	7	.	.	PUNCT
ejpam-854	587	1	on	on	ADP
ejpam-854	587	2	the	the	DET
ejpam-854	587	3	stability	stability	NOUN
ejpam-854	587	4	of	of	ADP
ejpam-854	587	5	the	the	DET
ejpam-854	587	6	linear	linear	ADJ
ejpam-854	587	7	mapping	mapping	NOUN
ejpam-854	587	8	in	in	ADP
ejpam-854	587	9	banach	banach	NOUN
ejpam-854	587	10	spaces	space	NOUN
ejpam-854	587	11	.	.	PUNCT
ejpam-854	588	1	proc	proc	NOUN
ejpam-854	588	2	.	.	PUNCT
ejpam-854	589	1	amer	amer	PROPN
ejpam-854	589	2	.	.	PUNCT
ejpam-854	589	3	math	math	PROPN
ejpam-854	589	4	.	.	PUNCT
ejpam-854	590	1	soc	soc	PROPN
ejpam-854	590	2	.	.	PUNCT
ejpam-854	590	3	,	,	PUNCT
ejpam-854	590	4	72	72	NUM
ejpam-854	590	5	:	:	SYM
ejpam-854	590	6	297	297	NUM
ejpam-854	590	7	-	-	SYM
ejpam-854	590	8	300	300	NUM
ejpam-854	590	9	,	,	PUNCT
ejpam-854	590	10	1978	1978	NUM
ejpam-854	590	11	.	.	PUNCT
ejpam-854	591	1	[	[	X
ejpam-854	591	2	29	29	NUM
ejpam-854	591	3	]	]	X
ejpam-854	591	4	j	j	PROPN
ejpam-854	591	5	m	m	NOUN
ejpam-854	591	6	rassias	rassias	PROPN
ejpam-854	591	7	.	.	PUNCT
ejpam-854	592	1	solution	solution	NOUN
ejpam-854	592	2	of	of	ADP
ejpam-854	592	3	the	the	DET
ejpam-854	592	4	ulam	ulam	PROPN
ejpam-854	592	5	stability	stability	PROPN
ejpam-854	592	6	problem	problem	NOUN
ejpam-854	592	7	for	for	ADP
ejpam-854	592	8	cubic	cubic	ADJ
ejpam-854	592	9	mapping	mapping	NOUN
ejpam-854	592	10	.	.	PUNCT
ejpam-854	593	1	glasnik	glasnik	PROPN
ejpam-854	593	2	mathematicki	mathematicki	PROPN
ejpam-854	593	3	,	,	PUNCT
ejpam-854	593	4	36	36	NUM
ejpam-854	593	5	:	:	SYM
ejpam-854	593	6	63	63	NUM
ejpam-854	593	7	-	-	SYM
ejpam-854	593	8	72	72	NUM
ejpam-854	593	9	,	,	PUNCT
ejpam-854	593	10	2001	2001	NUM
ejpam-854	593	11	.	.	PUNCT
ejpam-854	594	1	references	reference	NOUN
ejpam-854	594	2	1047	1047	NUM
ejpam-854	594	3	[	[	X
ejpam-854	594	4	30	30	NUM
ejpam-854	594	5	]	]	X
ejpam-854	594	6	j	j	PROPN
ejpam-854	594	7	m	m	NOUN
ejpam-854	594	8	rassias	rassias	PROPN
ejpam-854	594	9	.	.	PUNCT
ejpam-854	595	1	solution	solution	NOUN
ejpam-854	595	2	of	of	ADP
ejpam-854	595	3	a	a	DET
ejpam-854	595	4	problem	problem	NOUN
ejpam-854	595	5	of	of	ADP
ejpam-854	595	6	ulam	ulam	PROPN
ejpam-854	595	7	.	.	PUNCT
ejpam-854	596	1	j.	j.	PROPN
ejpam-854	596	2	approx	approx	PROPN
ejpam-854	596	3	.	.	PUNCT
ejpam-854	597	1	theory	theory	NOUN
ejpam-854	597	2	,	,	PUNCT
ejpam-854	597	3	57	57	NUM
ejpam-854	597	4	:	:	SYM
ejpam-854	597	5	268	268	NUM
ejpam-854	597	6	-	-	SYM
ejpam-854	597	7	273	273	NUM
ejpam-854	597	8	,	,	PUNCT
ejpam-854	597	9	1989	1989	NUM
ejpam-854	597	10	.	.	PUNCT
ejpam-854	598	1	[	[	X
ejpam-854	598	2	31	31	NUM
ejpam-854	598	3	]	]	X
ejpam-854	598	4	j	j	PROPN
ejpam-854	598	5	m	m	NOUN
ejpam-854	598	6	rassias	rassias	PROPN
ejpam-854	598	7	.	.	PUNCT
ejpam-854	599	1	solution	solution	NOUN
ejpam-854	599	2	of	of	ADP
ejpam-854	599	3	the	the	DET
ejpam-854	599	4	ulam	ulam	PROPN
ejpam-854	599	5	stability	stability	PROPN
ejpam-854	599	6	problem	problem	NOUN
ejpam-854	599	7	for	for	ADP
ejpam-854	599	8	quartic	quartic	ADJ
ejpam-854	599	9	mappings	mapping	NOUN
ejpam-854	599	10	.	.	PUNCT
ejpam-854	600	1	glasnik	glasnik	PROPN
ejpam-854	600	2	matematicki	matematicki	PROPN
ejpam-854	600	3	,	,	PUNCT
ejpam-854	600	4	34	34	NUM
ejpam-854	600	5	:	:	SYM
ejpam-854	600	6	243	243	NUM
ejpam-854	600	7	-	-	SYM
ejpam-854	600	8	252	252	NUM
ejpam-854	600	9	,	,	PUNCT
ejpam-854	600	10	1999	1999	NUM
ejpam-854	600	11	.	.	PUNCT
ejpam-854	601	1	[	[	X
ejpam-854	601	2	32	32	NUM
ejpam-854	601	3	]	]	X
ejpam-854	601	4	j	j	PROPN
ejpam-854	601	5	m	m	NOUN
ejpam-854	601	6	rassias	rassias	PROPN
ejpam-854	601	7	.	.	PUNCT
ejpam-854	602	1	on	on	ADP
ejpam-854	602	2	approximation	approximation	NOUN
ejpam-854	602	3	of	of	ADP
ejpam-854	602	4	approximately	approximately	ADV
ejpam-854	602	5	linear	linear	ADJ
ejpam-854	602	6	mappings	mapping	NOUN
ejpam-854	602	7	by	by	ADP
ejpam-854	602	8	linear	linear	ADJ
ejpam-854	602	9	mappings	mapping	NOUN
ejpam-854	602	10	.	.	PUNCT
ejpam-854	603	1	j.	j.	PROPN
ejpam-854	603	2	funct	funct	PROPN
ejpam-854	603	3	.	.	PUNCT
ejpam-854	604	1	anal	anal	PROPN
ejpam-854	604	2	.	.	PROPN
ejpam-854	604	3	,	,	PUNCT
ejpam-854	604	4	46	46	NUM
ejpam-854	604	5	:	:	SYM
ejpam-854	604	6	126	126	NUM
ejpam-854	604	7	-	-	SYM
ejpam-854	604	8	130	130	NUM
ejpam-854	604	9	,	,	PUNCT
ejpam-854	604	10	1982	1982	NUM
ejpam-854	604	11	.	.	PUNCT
ejpam-854	605	1	[	[	X
ejpam-854	605	2	33	33	NUM
ejpam-854	605	3	]	]	X
ejpam-854	605	4	k	k	PROPN
ejpam-854	605	5	ravi	ravi	PROPN
ejpam-854	605	6	,	,	PUNCT
ejpam-854	605	7	j	j	PROPN
ejpam-854	605	8	m	m	PROPN
ejpam-854	605	9	rassias	rassias	PROPN
ejpam-854	605	10	,	,	PUNCT
ejpam-854	605	11	m	m	PROPN
ejpam-854	605	12	arunkumar	arunkumar	PROPN
ejpam-854	605	13	,	,	PUNCT
ejpam-854	605	14	and	and	CCONJ
ejpam-854	605	15	r	r	PROPN
ejpam-854	605	16	kodandan	kodandan	PROPN
ejpam-854	605	17	.	.	PUNCT
ejpam-854	606	1	stability	stability	NOUN
ejpam-854	606	2	of	of	ADP
ejpam-854	606	3	a	a	DET
ejpam-854	606	4	generalized	generalize	VERB
ejpam-854	606	5	mixed	mixed	ADJ
ejpam-854	606	6	type	type	NOUN
ejpam-854	606	7	additive	additive	NOUN
ejpam-854	606	8	,	,	PUNCT
ejpam-854	606	9	quadratic	quadratic	ADJ
ejpam-854	606	10	,	,	PUNCT
ejpam-854	606	11	cubic	cubic	ADJ
ejpam-854	606	12	and	and	CCONJ
ejpam-854	606	13	quartic	quartic	ADJ
ejpam-854	606	14	functional	functional	ADJ
ejpam-854	606	15	equational	equational	ADJ
ejpam-854	606	16	equation	equation	NOUN
ejpam-854	606	17	.	.	PUNCT
ejpam-854	607	1	journal	journal	PROPN
ejpam-854	607	2	of	of	ADP
ejpam-854	607	3	inequalities	inequality	NOUN
ejpam-854	607	4	in	in	ADP
ejpam-854	607	5	pure	pure	ADJ
ejpam-854	607	6	and	and	CCONJ
ejpam-854	607	7	applied	applied	ADJ
ejpam-854	607	8	mathematics	mathematic	NOUN
ejpam-854	607	9	,	,	PUNCT
ejpam-854	607	10	10(4	10(4	NUM
ejpam-854	607	11	):	):	PUNCT
ejpam-854	607	12	114	114	NUM
ejpam-854	607	13	,	,	PUNCT
ejpam-854	607	14	2009	2009	NUM
ejpam-854	607	15	.	.	PUNCT
ejpam-854	608	1	[	[	X
ejpam-854	608	2	34	34	NUM
ejpam-854	608	3	]	]	X
ejpam-854	608	4	r	r	NOUN
ejpam-854	608	5	rubinfeld	rubinfeld	NOUN
ejpam-854	608	6	.	.	PUNCT
ejpam-854	609	1	on	on	ADP
ejpam-854	609	2	the	the	DET
ejpam-854	609	3	robustness	robustness	NOUN
ejpam-854	609	4	of	of	ADP
ejpam-854	609	5	functional	functional	ADJ
ejpam-854	609	6	equations	equation	NOUN
ejpam-854	609	7	.	.	PUNCT
ejpam-854	610	1	siam	siam	PROPN
ejpam-854	610	2	j.	j.	PROPN
ejpam-854	610	3	comput	comput	PROPN
ejpam-854	610	4	.	.	PUNCT
ejpam-854	610	5	,	,	PUNCT
ejpam-854	610	6	28	28	NUM
ejpam-854	610	7	:	:	SYM
ejpam-854	610	8	19721997	19721997	NUM
ejpam-854	610	9	,	,	PUNCT
ejpam-854	610	10	1999	1999	NUM
ejpam-854	610	11	.	.	PUNCT
ejpam-854	611	1	[	[	X
ejpam-854	611	2	35	35	NUM
ejpam-854	611	3	]	]	SYM
ejpam-854	611	4	s	s	NOUN
ejpam-854	611	5	m	m	NOUN
ejpam-854	611	6	ulam	ulam	NOUN
ejpam-854	611	7	.	.	PUNCT
ejpam-854	612	1	a	a	DET
ejpam-854	612	2	collection	collection	NOUN
ejpam-854	612	3	of	of	ADP
ejpam-854	612	4	the	the	DET
ejpam-854	612	5	mathematical	mathematical	ADJ
ejpam-854	612	6	problems	problem	NOUN
ejpam-854	612	7	,	,	PUNCT
ejpam-854	612	8	interscience	interscience	NOUN
ejpam-854	612	9	,	,	PUNCT
ejpam-854	612	10	new	new	PROPN
ejpam-854	612	11	york	york	PROPN
ejpam-854	612	12	,	,	PUNCT
ejpam-854	612	13	1960	1960	NUM
ejpam-854	612	14	.	.	PUNCT
ejpam-854	613	1	[	[	X
ejpam-854	613	2	36	36	NUM
ejpam-854	613	3	]	]	X
ejpam-854	613	4	l	l	PROPN
ejpam-854	613	5	wang	wang	PROPN
ejpam-854	613	6	,	,	PUNCT
ejpam-854	613	7	b	b	PROPN
ejpam-854	613	8	liu	liu	PROPN
ejpam-854	613	9	,	,	PUNCT
ejpam-854	613	10	and	and	CCONJ
ejpam-854	613	11	r	r	NOUN
ejpam-854	613	12	bai	bai	PROPN
ejpam-854	613	13	.	.	PUNCT
ejpam-854	614	1	stability	stability	NOUN
ejpam-854	614	2	of	of	ADP
ejpam-854	614	3	a	a	DET
ejpam-854	614	4	mixed	mixed	ADJ
ejpam-854	614	5	type	type	NOUN
ejpam-854	614	6	functional	functional	ADJ
ejpam-854	614	7	equation	equation	NOUN
ejpam-854	614	8	on	on	ADP
ejpam-854	614	9	multi	multi	ADJ
ejpam-854	614	10	-	-	ADJ
ejpam-854	614	11	banach	banach	ADJ
ejpam-854	614	12	spaces	space	VERB
ejpam-854	614	13	:	:	PUNCT
ejpam-854	614	14	a	a	DET
ejpam-854	614	15	fixed	fix	VERB
ejpam-854	614	16	point	point	NOUN
ejpam-854	614	17	approach	approach	NOUN
ejpam-854	614	18	.	.	PUNCT
ejpam-854	615	1	fixed	fix	VERB
ejpam-854	615	2	point	point	NOUN
ejpam-854	615	3	theory	theory	NOUN
ejpam-854	615	4	and	and	CCONJ
ejpam-854	615	5	applications	application	NOUN
ejpam-854	615	6	,	,	PUNCT
ejpam-854	615	7	2010	2010	NUM
ejpam-854	615	8	:	:	PUNCT
ejpam-854	616	1	pp9	pp9	PROPN
ejpam-854	616	2	,	,	PUNCT
ejpam-854	616	3	2010	2010	NUM
ejpam-854	616	4	.	.	PUNCT
ejpam-854	617	1	[	[	X
ejpam-854	617	2	37	37	NUM
ejpam-854	617	3	]	]	X
ejpam-854	617	4	t	t	PROPN
ejpam-854	617	5	z	z	PROPN
ejpam-854	617	6	xu	xu	PROPN
ejpam-854	617	7	,	,	PUNCT
ejpam-854	617	8	j	j	PROPN
ejpam-854	617	9	m	m	PROPN
ejpam-854	617	10	rassias	rassias	PROPN
ejpam-854	617	11	,	,	PUNCT
ejpam-854	617	12	and	and	CCONJ
ejpam-854	617	13	w	w	PROPN
ejpam-854	617	14	x	x	PROPN
ejpam-854	617	15	xu	xu	PROPN
ejpam-854	617	16	.	.	PUNCT
ejpam-854	618	1	stability	stability	NOUN
ejpam-854	618	2	of	of	ADP
ejpam-854	618	3	a	a	DET
ejpam-854	618	4	general	general	ADJ
ejpam-854	618	5	mixed	mix	VERB
ejpam-854	618	6	additive	additive	ADJ
ejpam-854	618	7	-	-	PUNCT
ejpam-854	618	8	cubic	cubic	ADJ
ejpam-854	618	9	functional	functional	ADJ
ejpam-854	618	10	equation	equation	NOUN
ejpam-854	618	11	in	in	ADP
ejpam-854	618	12	non	non	ADJ
ejpam-854	618	13	-	-	ADJ
ejpam-854	618	14	archimedean	archimedean	ADJ
ejpam-854	618	15	fuzzy	fuzzy	ADJ
ejpam-854	618	16	normed	normed	PROPN
ejpam-854	618	17	spaces	space	NOUN
ejpam-854	618	18	.	.	PUNCT
ejpam-854	619	1	journal	journal	PROPN
ejpam-854	619	2	of	of	ADP
ejpam-854	619	3	mathematical	mathematical	ADJ
ejpam-854	619	4	physics	physics	NOUN
ejpam-854	619	5	,	,	PUNCT
ejpam-854	619	6	51	51	NUM
ejpam-854	619	7	:	:	PUNCT
ejpam-854	620	1	pp19	pp19	PROPN
ejpam-854	620	2	,	,	PUNCT
ejpam-854	620	3	2010	2010	NUM
ejpam-854	620	4	.	.	PUNCT
ejpam-854	621	1	[	[	X
ejpam-854	621	2	38	38	NUM
ejpam-854	621	3	]	]	PUNCT
ejpam-854	621	4	t	t	PROPN
ejpam-854	621	5	z	z	PROPN
ejpam-854	621	6	xu	xu	PROPN
ejpam-854	621	7	,	,	PUNCT
ejpam-854	621	8	j	j	PROPN
ejpam-854	621	9	m	m	PROPN
ejpam-854	621	10	rassias	rassias	PROPN
ejpam-854	621	11	,	,	PUNCT
ejpam-854	621	12	and	and	CCONJ
ejpam-854	621	13	w	w	PROPN
ejpam-854	621	14	x	x	PROPN
ejpam-854	621	15	xu	xu	PROPN
ejpam-854	621	16	.	.	PUNCT
ejpam-854	622	1	a	a	DET
ejpam-854	622	2	fixed	fix	VERB
ejpam-854	622	3	point	point	NOUN
ejpam-854	622	4	approach	approach	NOUN
ejpam-854	622	5	to	to	ADP
ejpam-854	622	6	the	the	DET
ejpam-854	622	7	stability	stability	NOUN
ejpam-854	622	8	of	of	ADP
ejpam-854	622	9	a	a	DET
ejpam-854	622	10	general	general	ADJ
ejpam-854	622	11	mixed	mix	VERB
ejpam-854	622	12	additive	additive	ADJ
ejpam-854	622	13	-	-	PUNCT
ejpam-854	622	14	cubic	cubic	ADJ
ejpam-854	622	15	functional	functional	ADJ
ejpam-854	622	16	equation	equation	NOUN
ejpam-854	622	17	in	in	ADP
ejpam-854	622	18	quasi	quasi	X
ejpam-854	622	19	fuzzy	fuzzy	ADJ
ejpam-854	622	20	normed	normed	PROPN
ejpam-854	622	21	spaces	space	NOUN
ejpam-854	622	22	.	.	PUNCT
ejpam-854	623	1	to	to	PART
ejpam-854	623	2	appear	appear	VERB
ejpam-854	623	3	in	in	ADP
ejpam-854	623	4	international	international	ADJ
ejpam-854	623	5	journal	journal	NOUN
ejpam-854	623	6	of	of	ADP
ejpam-854	623	7	physical	physical	ADJ
ejpam-854	623	8	sciences	science	NOUN
ejpam-854	623	9	.	.	PUNCT
ejpam-854	624	1	[	[	X
ejpam-854	624	2	39	39	NUM
ejpam-854	624	3	]	]	PUNCT
ejpam-854	624	4	t	t	PROPN
ejpam-854	624	5	z	z	PROPN
ejpam-854	624	6	xu	xu	PROPN
ejpam-854	624	7	,	,	PUNCT
ejpam-854	624	8	j	j	PROPN
ejpam-854	624	9	m	m	PROPN
ejpam-854	624	10	rassias	rassias	PROPN
ejpam-854	624	11	,	,	PUNCT
ejpam-854	624	12	and	and	CCONJ
ejpam-854	624	13	w	w	PROPN
ejpam-854	624	14	x	x	PROPN
ejpam-854	624	15	xu	xu	PROPN
ejpam-854	624	16	.	.	PUNCT
ejpam-854	625	1	intuitionistic	intuitionistic	ADJ
ejpam-854	625	2	fuzzy	fuzzy	ADJ
ejpam-854	625	3	stability	stability	NOUN
ejpam-854	625	4	of	of	ADP
ejpam-854	625	5	a	a	DET
ejpam-854	625	6	general	general	ADJ
ejpam-854	625	7	mixed	mixed	ADJ
ejpam-854	625	8	additivecubic	additivecubic	ADJ
ejpam-854	625	9	equation	equation	NOUN
ejpam-854	625	10	.	.	PUNCT
ejpam-854	626	1	journal	journal	PROPN
ejpam-854	626	2	of	of	ADP
ejpam-854	626	3	mathematical	mathematical	ADJ
ejpam-854	626	4	physics	physics	NOUN
ejpam-854	626	5	,	,	PUNCT
ejpam-854	626	6	51	51	NUM
ejpam-854	626	7	:	:	PUNCT
ejpam-854	626	8	pp21	pp21	PROPN
ejpam-854	626	9	,	,	PUNCT
ejpam-854	626	10	2010	2010	NUM
ejpam-854	626	11	.	.	PUNCT
ejpam-854	627	1	[	[	X
ejpam-854	627	2	40	40	NUM
ejpam-854	627	3	]	]	X
ejpam-854	627	4	t	t	PROPN
ejpam-854	627	5	z	z	PROPN
ejpam-854	627	6	xu	xu	PROPN
ejpam-854	627	7	,	,	PUNCT
ejpam-854	627	8	j	j	PROPN
ejpam-854	627	9	m	m	PROPN
ejpam-854	627	10	rassias	rassias	PROPN
ejpam-854	627	11	,	,	PUNCT
ejpam-854	627	12	and	and	CCONJ
ejpam-854	627	13	w	w	PROPN
ejpam-854	627	14	x	x	PROPN
ejpam-854	627	15	xu	xu	PROPN
ejpam-854	627	16	.	.	PUNCT
ejpam-854	628	1	a	a	DET
ejpam-854	628	2	generalized	generalize	VERB
ejpam-854	628	3	mixed	mixed	ADJ
ejpam-854	628	4	quadratic	quadratic	ADJ
ejpam-854	628	5	-	-	PUNCT
ejpam-854	628	6	quartic	quartic	ADJ
ejpam-854	628	7	functional	functional	ADJ
ejpam-854	628	8	equation	equation	NOUN
ejpam-854	628	9	.	.	PUNCT
ejpam-854	629	1	to	to	PART
ejpam-854	629	2	appear	appear	VERB
ejpam-854	629	3	in	in	ADP
ejpam-854	629	4	bulletin	bulletin	NOUN
ejpam-854	629	5	of	of	ADP
ejpam-854	629	6	the	the	DET
ejpam-854	629	7	malaysian	malaysian	PROPN
ejpam-854	629	8	mathematical	mathematical	PROPN
ejpam-854	629	9	sciences	sciences	PROPN
ejpam-854	629	10	society	society	NOUN
ejpam-854	629	11	.	.	PUNCT
ejpam-854	630	1	[	[	X
ejpam-854	630	2	41	41	NUM
ejpam-854	630	3	]	]	X
ejpam-854	630	4	t	t	PROPN
ejpam-854	630	5	z	z	PROPN
ejpam-854	630	6	xu	xu	PROPN
ejpam-854	630	7	,	,	PUNCT
ejpam-854	630	8	j	j	PROPN
ejpam-854	630	9	m	m	PROPN
ejpam-854	630	10	rassias	rassias	PROPN
ejpam-854	630	11	,	,	PUNCT
ejpam-854	630	12	and	and	CCONJ
ejpam-854	630	13	w	w	PROPN
ejpam-854	630	14	x	x	PROPN
ejpam-854	630	15	xu	xu	PROPN
ejpam-854	630	16	.	.	PUNCT
ejpam-854	631	1	on	on	ADP
ejpam-854	631	2	the	the	DET
ejpam-854	631	3	stability	stability	NOUN
ejpam-854	631	4	of	of	ADP
ejpam-854	631	5	a	a	DET
ejpam-854	631	6	general	general	ADJ
ejpam-854	631	7	mixed	mix	VERB
ejpam-854	631	8	additive	additive	ADJ
ejpam-854	631	9	-	-	PUNCT
ejpam-854	631	10	cubic	cubic	ADJ
ejpam-854	631	11	functional	functional	ADJ
ejpam-854	631	12	equation	equation	NOUN
ejpam-854	631	13	in	in	ADP
ejpam-854	631	14	random	random	ADJ
ejpam-854	631	15	normed	normed	ADJ
ejpam-854	631	16	spaces	space	NOUN
ejpam-854	631	17	.	.	PUNCT
ejpam-854	632	1	journal	journal	PROPN
ejpam-854	632	2	of	of	ADP
ejpam-854	632	3	inequalities	inequality	NOUN
ejpam-854	632	4	and	and	CCONJ
ejpam-854	632	5	applications	application	NOUN
ejpam-854	632	6	,	,	PUNCT
ejpam-854	632	7	2010	2010	NUM
ejpam-854	632	8	:	:	PUNCT
ejpam-854	633	1	pp16	pp16	NOUN
ejpam-854	633	2	,	,	PUNCT
ejpam-854	633	3	2010	2010	NUM
ejpam-854	633	4	.	.	PUNCT
ejpam-854	634	1	[	[	X
ejpam-854	634	2	42	42	NUM
ejpam-854	634	3	]	]	X
ejpam-854	634	4	t	t	PROPN
ejpam-854	634	5	z	z	PROPN
ejpam-854	634	6	xu	xu	PROPN
ejpam-854	634	7	,	,	PUNCT
ejpam-854	634	8	j	j	PROPN
ejpam-854	634	9	m	m	PROPN
ejpam-854	634	10	rassias	rassias	PROPN
ejpam-854	634	11	,	,	PUNCT
ejpam-854	634	12	and	and	CCONJ
ejpam-854	634	13	w	w	PROPN
ejpam-854	634	14	x	x	PROPN
ejpam-854	634	15	xu	xu	PROPN
ejpam-854	634	16	.	.	PUNCT
ejpam-854	635	1	a	a	DET
ejpam-854	635	2	fixed	fix	VERB
ejpam-854	635	3	point	point	NOUN
ejpam-854	635	4	approach	approach	NOUN
ejpam-854	635	5	to	to	ADP
ejpam-854	635	6	the	the	DET
ejpam-854	635	7	stability	stability	NOUN
ejpam-854	635	8	of	of	ADP
ejpam-854	635	9	a	a	DET
ejpam-854	635	10	general	general	ADJ
ejpam-854	635	11	mixed	mixed	ADJ
ejpam-854	635	12	aqcq	aqcq	NOUN
ejpam-854	635	13	-	-	PUNCT
ejpam-854	635	14	functional	functional	ADJ
ejpam-854	635	15	equation	equation	NOUN
ejpam-854	635	16	in	in	ADP
ejpam-854	635	17	non	non	ADJ
ejpam-854	635	18	-	-	ADJ
ejpam-854	635	19	archimedean	archimedean	ADJ
ejpam-854	635	20	normed	norme	VERB
ejpam-854	635	21	spaces	space	NOUN
ejpam-854	635	22	.	.	PUNCT
ejpam-854	636	1	discrete	discrete	ADJ
ejpam-854	636	2	dynamics	dynamic	NOUN
ejpam-854	636	3	in	in	ADP
ejpam-854	636	4	nature	nature	NOUN
ejpam-854	636	5	and	and	CCONJ
ejpam-854	636	6	society	society	NOUN
ejpam-854	636	7	,	,	PUNCT
ejpam-854	636	8	2010	2010	NUM
ejpam-854	636	9	:	:	PUNCT
ejpam-854	637	1	pp24	pp24	PROPN
ejpam-854	637	2	,	,	PUNCT
ejpam-854	637	3	2010	2010	NUM
ejpam-854	637	4	.	.	PUNCT