id	sid	tid	token	lemma	pos
ejpam-6098	1	1	european	european	PROPN
ejpam-6098	1	2	journal	journal	PROPN
ejpam-6098	1	3	of	of	ADP
ejpam-6098	1	4	pure	pure	ADJ
ejpam-6098	1	5	and	and	CCONJ
ejpam-6098	1	6	applied	applied	ADJ
ejpam-6098	1	7	mathematics	mathematic	NOUN
ejpam-6098	1	8	2025	2025	NUM
ejpam-6098	1	9	,	,	PUNCT
ejpam-6098	1	10	vol	vol	NOUN
ejpam-6098	1	11	.	.	PROPN
ejpam-6098	1	12	18	18	NUM
ejpam-6098	1	13	,	,	PUNCT
ejpam-6098	1	14	issue	issue	NOUN
ejpam-6098	1	15	2	2	NUM
ejpam-6098	1	16	,	,	PUNCT
ejpam-6098	1	17	article	article	NOUN
ejpam-6098	1	18	number	number	NOUN
ejpam-6098	1	19	6098	6098	NUM
ejpam-6098	1	20	issn	issn	VERB
ejpam-6098	1	21	1307	1307	NUM
ejpam-6098	1	22	-	-	SYM
ejpam-6098	1	23	5543	5543	NUM
ejpam-6098	1	24	–	–	PUNCT
ejpam-6098	1	25	ejpam.com	ejpam.com	X
ejpam-6098	1	26	published	publish	VERB
ejpam-6098	1	27	by	by	ADP
ejpam-6098	1	28	new	new	PROPN
ejpam-6098	1	29	york	york	PROPN
ejpam-6098	1	30	business	business	PROPN
ejpam-6098	1	31	global	global	ADJ
ejpam-6098	1	32	comparative	comparative	ADJ
ejpam-6098	1	33	results	result	NOUN
ejpam-6098	1	34	on	on	ADP
ejpam-6098	1	35	stability	stability	NOUN
ejpam-6098	1	36	analysis	analysis	NOUN
ejpam-6098	1	37	for	for	ADP
ejpam-6098	1	38	three	three	NUM
ejpam-6098	1	39	dimensional	dimensional	ADJ
ejpam-6098	1	40	functional	functional	ADJ
ejpam-6098	1	41	equations	equation	NOUN
ejpam-6098	1	42	jagjeet	jagjeet	PROPN
ejpam-6098	1	43	jakhar1	jakhar1	PROPN
ejpam-6098	1	44	,	,	PUNCT
ejpam-6098	1	45	shalu	shalu	PROPN
ejpam-6098	1	46	sharma1	sharma1	PROPN
ejpam-6098	1	47	,	,	PUNCT
ejpam-6098	1	48	dumitru	dumitru	PROPN
ejpam-6098	1	49	baleanu2	baleanu2	NOUN
ejpam-6098	1	50	,	,	PUNCT
ejpam-6098	1	51	majeed	majeed	PROPN
ejpam-6098	1	52	a.	a.	NOUN
ejpam-6098	1	53	yousif3	yousif3	PROPN
ejpam-6098	1	54	,	,	PUNCT
ejpam-6098	1	55	jyotsana	jyotsana	PROPN
ejpam-6098	1	56	jakhar4	jakhar4	PROPN
ejpam-6098	1	57	,	,	PUNCT
ejpam-6098	1	58	nejmeddine	nejmeddine	ADJ
ejpam-6098	1	59	chorfi5	chorfi5	NOUN
ejpam-6098	1	60	,	,	PUNCT
ejpam-6098	1	61	pshtiwan	pshtiwan	PROPN
ejpam-6098	1	62	othman	othman	PROPN
ejpam-6098	1	63	mohammed6,7,8,∗	mohammed6,7,8,∗	PROPN
ejpam-6098	1	64	1	1	NUM
ejpam-6098	1	65	department	department	NOUN
ejpam-6098	1	66	of	of	ADP
ejpam-6098	1	67	mathematics	mathematic	NOUN
ejpam-6098	1	68	,	,	PUNCT
ejpam-6098	1	69	central	central	ADJ
ejpam-6098	1	70	university	university	PROPN
ejpam-6098	1	71	of	of	ADP
ejpam-6098	1	72	haryana	haryana	PROPN
ejpam-6098	1	73	,	,	PUNCT
ejpam-6098	1	74	jant	jant	ADJ
ejpam-6098	1	75	-	-	PUNCT
ejpam-6098	1	76	pali	pali	ADJ
ejpam-6098	1	77	,	,	PUNCT
ejpam-6098	1	78	mahendergarh	mahendergarh	NOUN
ejpam-6098	1	79	,	,	PUNCT
ejpam-6098	1	80	123031	123031	NUM
ejpam-6098	1	81	,	,	PUNCT
ejpam-6098	1	82	india	india	PROPN
ejpam-6098	1	83	2	2	NUM
ejpam-6098	1	84	department	department	NOUN
ejpam-6098	1	85	of	of	ADP
ejpam-6098	1	86	computer	computer	NOUN
ejpam-6098	1	87	science	science	NOUN
ejpam-6098	1	88	and	and	CCONJ
ejpam-6098	1	89	mathematics	mathematic	NOUN
ejpam-6098	1	90	,	,	PUNCT
ejpam-6098	1	91	lebanese	lebanese	ADJ
ejpam-6098	1	92	american	american	PROPN
ejpam-6098	1	93	university	university	PROPN
ejpam-6098	1	94	,	,	PUNCT
ejpam-6098	1	95	beirut	beirut	PROPN
ejpam-6098	1	96	11022801	11022801	NUM
ejpam-6098	1	97	,	,	PUNCT
ejpam-6098	1	98	lebanon	lebanon	PROPN
ejpam-6098	1	99	3	3	NUM
ejpam-6098	1	100	department	department	PROPN
ejpam-6098	1	101	of	of	ADP
ejpam-6098	1	102	mathematics	mathematics	PROPN
ejpam-6098	1	103	,	,	PUNCT
ejpam-6098	1	104	college	college	NOUN
ejpam-6098	1	105	of	of	ADP
ejpam-6098	1	106	education	education	NOUN
ejpam-6098	1	107	,	,	PUNCT
ejpam-6098	1	108	university	university	NOUN
ejpam-6098	1	109	of	of	ADP
ejpam-6098	1	110	zakho	zakho	PROPN
ejpam-6098	1	111	,	,	PUNCT
ejpam-6098	1	112	zakho	zakho	PROPN
ejpam-6098	1	113	42002	42002	NUM
ejpam-6098	1	114	,	,	PUNCT
ejpam-6098	1	115	iraq	iraq	PROPN
ejpam-6098	1	116	4	4	NUM
ejpam-6098	1	117	department	department	NOUN
ejpam-6098	1	118	of	of	ADP
ejpam-6098	1	119	mathematics	mathematic	NOUN
ejpam-6098	1	120	,	,	PUNCT
ejpam-6098	1	121	pandit	pandit	PROPN
ejpam-6098	1	122	neki	neki	PROPN
ejpam-6098	1	123	ram	ram	PROPN
ejpam-6098	1	124	sharma	sharma	PROPN
ejpam-6098	1	125	government	government	PROPN
ejpam-6098	1	126	college	college	PROPN
ejpam-6098	1	127	,	,	PUNCT
ejpam-6098	1	128	rohtak	rohtak	PROPN
ejpam-6098	1	129	,	,	PUNCT
ejpam-6098	1	130	haryana	haryana	PROPN
ejpam-6098	1	131	,	,	PUNCT
ejpam-6098	1	132	123031	123031	NUM
ejpam-6098	1	133	,	,	PUNCT
ejpam-6098	1	134	india	india	PROPN
ejpam-6098	1	135	5	5	NUM
ejpam-6098	1	136	department	department	NOUN
ejpam-6098	1	137	of	of	ADP
ejpam-6098	1	138	mathematics	mathematic	NOUN
ejpam-6098	1	139	,	,	PUNCT
ejpam-6098	1	140	college	college	NOUN
ejpam-6098	1	141	of	of	ADP
ejpam-6098	1	142	science	science	NOUN
ejpam-6098	1	143	,	,	PUNCT
ejpam-6098	1	144	king	king	PROPN
ejpam-6098	1	145	saud	saud	PROPN
ejpam-6098	1	146	university	university	PROPN
ejpam-6098	1	147	,	,	PUNCT
ejpam-6098	1	148	p.o	p.o	PROPN
ejpam-6098	1	149	.	.	PROPN
ejpam-6098	1	150	box	box	PROPN
ejpam-6098	1	151	2455	2455	NUM
ejpam-6098	1	152	,	,	PUNCT
ejpam-6098	1	153	riyadh	riyadh	PROPN
ejpam-6098	1	154	11451	11451	NUM
ejpam-6098	1	155	,	,	PUNCT
ejpam-6098	1	156	saudi	saudi	PROPN
ejpam-6098	1	157	arabia	arabia	PROPN
ejpam-6098	1	158	6	6	NUM
ejpam-6098	1	159	research	research	NOUN
ejpam-6098	1	160	and	and	CCONJ
ejpam-6098	1	161	development	development	NOUN
ejpam-6098	1	162	center	center	NOUN
ejpam-6098	1	163	,	,	PUNCT
ejpam-6098	1	164	university	university	NOUN
ejpam-6098	1	165	of	of	ADP
ejpam-6098	1	166	sulaimani	sulaimani	PROPN
ejpam-6098	1	167	,	,	PUNCT
ejpam-6098	1	168	sulaymaniyah	sulaymaniyah	NOUN
ejpam-6098	1	169	46001	46001	NUM
ejpam-6098	1	170	,	,	PUNCT
ejpam-6098	1	171	iraq	iraq	PROPN
ejpam-6098	1	172	7	7	NUM
ejpam-6098	1	173	research	research	NOUN
ejpam-6098	1	174	center	center	NOUN
ejpam-6098	1	175	,	,	PUNCT
ejpam-6098	1	176	university	university	NOUN
ejpam-6098	1	177	of	of	ADP
ejpam-6098	1	178	halabja	halabja	PROPN
ejpam-6098	1	179	,	,	PUNCT
ejpam-6098	1	180	halabja	halabja	PROPN
ejpam-6098	1	181	46018	46018	NUM
ejpam-6098	1	182	,	,	PUNCT
ejpam-6098	1	183	iraq	iraq	PROPN
ejpam-6098	1	184	8	8	NUM
ejpam-6098	1	185	associate	associate	NOUN
ejpam-6098	1	186	member	member	NOUN
ejpam-6098	1	187	of	of	ADP
ejpam-6098	1	188	section	section	NOUN
ejpam-6098	1	189	of	of	ADP
ejpam-6098	1	190	mathematics	mathematic	NOUN
ejpam-6098	1	191	,	,	PUNCT
ejpam-6098	1	192	international	international	ADJ
ejpam-6098	1	193	telematic	telematic	ADJ
ejpam-6098	1	194	university	university	NOUN
ejpam-6098	1	195	uninettuno	uninettuno	NOUN
ejpam-6098	1	196	,	,	PUNCT
ejpam-6098	1	197	corso	corso	PROPN
ejpam-6098	1	198	vittorio	vittorio	PROPN
ejpam-6098	1	199	emanuele	emanuele	PROPN
ejpam-6098	1	200	ii	ii	PROPN
ejpam-6098	1	201	,	,	PUNCT
ejpam-6098	1	202	39	39	NUM
ejpam-6098	1	203	,	,	PUNCT
ejpam-6098	1	204	00186	00186	NUM
ejpam-6098	1	205	roma	roma	PROPN
ejpam-6098	1	206	,	,	PUNCT
ejpam-6098	1	207	italy	italy	PROPN
ejpam-6098	1	208	abstract	abstract	NOUN
ejpam-6098	1	209	.	.	PUNCT
ejpam-6098	2	1	this	this	DET
ejpam-6098	2	2	study	study	NOUN
ejpam-6098	2	3	investigates	investigate	VERB
ejpam-6098	2	4	the	the	DET
ejpam-6098	2	5	stability	stability	NOUN
ejpam-6098	2	6	of	of	ADP
ejpam-6098	2	7	a	a	DET
ejpam-6098	2	8	three	three	NUM
ejpam-6098	2	9	-	-	PUNCT
ejpam-6098	2	10	dimensional	dimensional	ADJ
ejpam-6098	2	11	cubic	cubic	ADJ
ejpam-6098	2	12	functional	functional	ADJ
ejpam-6098	2	13	equation	equation	NOUN
ejpam-6098	2	14	within	within	ADP
ejpam-6098	2	15	several	several	ADJ
ejpam-6098	2	16	mathematical	mathematical	ADJ
ejpam-6098	2	17	frameworks	framework	NOUN
ejpam-6098	2	18	,	,	PUNCT
ejpam-6098	2	19	including	include	VERB
ejpam-6098	2	20	(	(	PUNCT
ejpam-6098	2	21	n	n	CCONJ
ejpam-6098	2	22	,	,	PUNCT
ejpam-6098	2	23	β)-normed	β)-normed	PUNCT
ejpam-6098	2	24	spaces	space	NOUN
ejpam-6098	2	25	,	,	PUNCT
ejpam-6098	2	26	non	non	ADJ
ejpam-6098	2	27	-	-	ADJ
ejpam-6098	2	28	archimedean	archimedean	ADJ
ejpam-6098	2	29	(	(	PUNCT
ejpam-6098	2	30	n	n	CCONJ
ejpam-6098	2	31	,	,	PUNCT
ejpam-6098	2	32	β)normed	β)normed	ADJ
ejpam-6098	2	33	spaces	space	NOUN
ejpam-6098	2	34	,	,	PUNCT
ejpam-6098	2	35	and	and	CCONJ
ejpam-6098	2	36	random	random	ADJ
ejpam-6098	2	37	normed	normed	ADJ
ejpam-6098	2	38	spaces	space	NOUN
ejpam-6098	2	39	.	.	PUNCT
ejpam-6098	3	1	the	the	DET
ejpam-6098	3	2	theoretical	theoretical	ADJ
ejpam-6098	3	3	stability	stability	NOUN
ejpam-6098	3	4	results	result	NOUN
ejpam-6098	3	5	are	be	AUX
ejpam-6098	3	6	validated	validate	VERB
ejpam-6098	3	7	through	through	ADP
ejpam-6098	3	8	experimental	experimental	ADJ
ejpam-6098	3	9	approaches	approach	NOUN
ejpam-6098	3	10	,	,	PUNCT
ejpam-6098	3	11	offering	offer	VERB
ejpam-6098	3	12	practical	practical	ADJ
ejpam-6098	3	13	insight	insight	NOUN
ejpam-6098	3	14	into	into	ADP
ejpam-6098	3	15	the	the	DET
ejpam-6098	3	16	behavior	behavior	NOUN
ejpam-6098	3	17	of	of	ADP
ejpam-6098	3	18	these	these	DET
ejpam-6098	3	19	functional	functional	ADJ
ejpam-6098	3	20	equations	equation	NOUN
ejpam-6098	3	21	.	.	PUNCT
ejpam-6098	4	1	a	a	DET
ejpam-6098	4	2	comparative	comparative	ADJ
ejpam-6098	4	3	analysis	analysis	NOUN
ejpam-6098	4	4	is	be	AUX
ejpam-6098	4	5	provided	provide	VERB
ejpam-6098	4	6	,	,	PUNCT
ejpam-6098	4	7	highlighting	highlight	VERB
ejpam-6098	4	8	differences	difference	NOUN
ejpam-6098	4	9	in	in	ADP
ejpam-6098	4	10	stability	stability	NOUN
ejpam-6098	4	11	dynamics	dynamic	NOUN
ejpam-6098	4	12	across	across	ADP
ejpam-6098	4	13	the	the	DET
ejpam-6098	4	14	various	various	ADJ
ejpam-6098	4	15	spaces	space	NOUN
ejpam-6098	4	16	.	.	PUNCT
ejpam-6098	5	1	notably	notably	ADV
ejpam-6098	5	2	,	,	PUNCT
ejpam-6098	5	3	the	the	DET
ejpam-6098	5	4	introduction	introduction	NOUN
ejpam-6098	5	5	of	of	ADP
ejpam-6098	5	6	(	(	PUNCT
ejpam-6098	5	7	n	n	CCONJ
ejpam-6098	5	8	,	,	PUNCT
ejpam-6098	5	9	β)-normed	β)-normed	PUNCT
ejpam-6098	5	10	spaces	space	NOUN
ejpam-6098	5	11	and	and	CCONJ
ejpam-6098	5	12	their	their	PRON
ejpam-6098	5	13	non	non	ADJ
ejpam-6098	5	14	-	-	ADJ
ejpam-6098	5	15	archimedean	archimedean	ADJ
ejpam-6098	5	16	counterparts	counterpart	NOUN
ejpam-6098	5	17	presents	present	VERB
ejpam-6098	5	18	a	a	DET
ejpam-6098	5	19	novel	novel	ADJ
ejpam-6098	5	20	framework	framework	NOUN
ejpam-6098	5	21	for	for	ADP
ejpam-6098	5	22	analyzing	analyze	VERB
ejpam-6098	5	23	stability	stability	NOUN
ejpam-6098	5	24	,	,	PUNCT
ejpam-6098	5	25	while	while	SCONJ
ejpam-6098	5	26	the	the	DET
ejpam-6098	5	27	inclusion	inclusion	NOUN
ejpam-6098	5	28	of	of	ADP
ejpam-6098	5	29	random	random	ADJ
ejpam-6098	5	30	normed	normed	ADJ
ejpam-6098	5	31	spaces	space	NOUN
ejpam-6098	5	32	adds	add	VERB
ejpam-6098	5	33	a	a	DET
ejpam-6098	5	34	stochastic	stochastic	ADJ
ejpam-6098	5	35	dimension	dimension	NOUN
ejpam-6098	5	36	to	to	ADP
ejpam-6098	5	37	the	the	DET
ejpam-6098	5	38	analysis	analysis	NOUN
ejpam-6098	5	39	.	.	PUNCT
ejpam-6098	6	1	the	the	DET
ejpam-6098	6	2	experimental	experimental	ADJ
ejpam-6098	6	3	validation	validation	NOUN
ejpam-6098	6	4	further	far	ADV
ejpam-6098	6	5	strengthens	strengthen	VERB
ejpam-6098	6	6	the	the	DET
ejpam-6098	6	7	practical	practical	ADJ
ejpam-6098	6	8	application	application	NOUN
ejpam-6098	6	9	of	of	ADP
ejpam-6098	6	10	the	the	DET
ejpam-6098	6	11	stability	stability	NOUN
ejpam-6098	6	12	results	result	VERB
ejpam-6098	6	13	,	,	PUNCT
ejpam-6098	6	14	distinguishing	distinguish	VERB
ejpam-6098	6	15	this	this	DET
ejpam-6098	6	16	study	study	NOUN
ejpam-6098	6	17	from	from	ADP
ejpam-6098	6	18	traditional	traditional	ADJ
ejpam-6098	6	19	approaches	approach	NOUN
ejpam-6098	6	20	.	.	PUNCT
ejpam-6098	7	1	2020	2020	NUM
ejpam-6098	7	2	mathematics	mathematic	NOUN
ejpam-6098	7	3	subject	subject	NOUN
ejpam-6098	7	4	classifications	classification	NOUN
ejpam-6098	7	5	:	:	PUNCT
ejpam-6098	7	6	39b82	39b82	NUM
ejpam-6098	7	7	,	,	PUNCT
ejpam-6098	7	8	46b40	46b40	ADV
ejpam-6098	7	9	key	key	ADJ
ejpam-6098	7	10	words	word	NOUN
ejpam-6098	7	11	and	and	CCONJ
ejpam-6098	7	12	phrases	phrase	NOUN
ejpam-6098	7	13	:	:	PUNCT
ejpam-6098	7	14	functional	functional	ADJ
ejpam-6098	7	15	equation	equation	NOUN
ejpam-6098	7	16	,	,	PUNCT
ejpam-6098	7	17	stability	stability	NOUN
ejpam-6098	7	18	,	,	PUNCT
ejpam-6098	7	19	(	(	PUNCT
ejpam-6098	7	20	n	n	CCONJ
ejpam-6098	7	21	,	,	PUNCT
ejpam-6098	7	22	β)-normed	β)-normed	PUNCT
ejpam-6098	7	23	space	space	NOUN
ejpam-6098	7	24	,	,	PUNCT
ejpam-6098	7	25	random	random	ADJ
ejpam-6098	7	26	normed	normed	ADJ
ejpam-6098	7	27	space	space	NOUN
ejpam-6098	7	28	∗corresponding	∗corresponde	VERB
ejpam-6098	7	29	author	author	NOUN
ejpam-6098	7	30	.	.	PUNCT
ejpam-6098	8	1	doi	doi	NOUN
ejpam-6098	8	2	:	:	PUNCT
ejpam-6098	8	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6098	https://doi.org/10.29020/nybg.ejpam.v18i2.6098	PROPN
ejpam-6098	8	4	email	email	NOUN
ejpam-6098	8	5	addresses	address	NOUN
ejpam-6098	8	6	:	:	PUNCT
ejpam-6098	8	7	jagjeet@cuh.ac.in	jagjeet@cuh.ac.in	PROPN
ejpam-6098	8	8	(	(	PUNCT
ejpam-6098	8	9	j.	j.	PROPN
ejpam-6098	8	10	jakhar	jakhar	PROPN
ejpam-6098	8	11	)	)	PUNCT
ejpam-6098	8	12	,	,	PUNCT
ejpam-6098	8	13	dumitru.baleanu@lau.edu.lb	dumitru.baleanu@lau.edu.lb	PROPN
ejpam-6098	8	14	(	(	PUNCT
ejpam-6098	8	15	d.	d.	PROPN
ejpam-6098	8	16	baleanu	baleanu	PROPN
ejpam-6098	8	17	)	)	PUNCT
ejpam-6098	8	18	,	,	PUNCT
ejpam-6098	8	19	shalu211940@cuh.ac.in	shalu211940@cuh.ac.in	NOUN
ejpam-6098	8	20	(	(	PUNCT
ejpam-6098	8	21	s.	s.	PROPN
ejpam-6098	8	22	sharma	sharma	PROPN
ejpam-6098	8	23	)	)	PUNCT
ejpam-6098	8	24	,	,	PUNCT
ejpam-6098	9	1	majeed.yousif@uoz.edu.krd	majeed.yousif@uoz.edu.krd	PROPN
ejpam-6098	9	2	(	(	PUNCT
ejpam-6098	9	3	m.a	m.a	PROPN
ejpam-6098	9	4	.	.	PROPN
ejpam-6098	9	5	yousif	yousif	PROPN
ejpam-6098	9	6	)	)	PUNCT
ejpam-6098	9	7	,	,	PUNCT
ejpam-6098	9	8	dahiya.jyotsana.j@gmail.com	dahiya.jyotsana.j@gmail.com	X
ejpam-6098	9	9	(	(	PUNCT
ejpam-6098	9	10	j.	j.	PROPN
ejpam-6098	9	11	jakhar	jakhar	PROPN
ejpam-6098	9	12	)	)	PUNCT
ejpam-6098	9	13	,	,	PUNCT
ejpam-6098	9	14	nchorfi@ksu.edu.sa	nchorfi@ksu.edu.sa	PROPN
ejpam-6098	9	15	(	(	PUNCT
ejpam-6098	9	16	n.	n.	PROPN
ejpam-6098	9	17	chorfi	chorfi	PROPN
ejpam-6098	9	18	)	)	PUNCT
ejpam-6098	9	19	,	,	PUNCT
ejpam-6098	9	20	pshtiwansangawi@gmail.com	pshtiwansangawi@gmail.com	X
ejpam-6098	9	21	(	(	PUNCT
ejpam-6098	9	22	p.o	p.o	PROPN
ejpam-6098	9	23	.	.	PROPN
ejpam-6098	9	24	mohammed	mohammed	PROPN
ejpam-6098	9	25	)	)	PUNCT
ejpam-6098	9	26	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6098	9	27	1	1	NUM
ejpam-6098	9	28	copyright	copyright	NOUN
ejpam-6098	9	29	:	:	PUNCT
ejpam-6098	9	30	©	©	PROPN
ejpam-6098	9	31	2025	2025	NUM
ejpam-6098	9	32	the	the	DET
ejpam-6098	9	33	author(s	author(s	NOUN
ejpam-6098	9	34	)	)	PUNCT
ejpam-6098	9	35	.	.	PUNCT
ejpam-6098	10	1	(	(	PUNCT
ejpam-6098	10	2	cc	cc	NOUN
ejpam-6098	10	3	by	by	ADP
ejpam-6098	10	4	-	-	PUNCT
ejpam-6098	10	5	nc	nc	PROPN
ejpam-6098	10	6	4.0	4.0	NUM
ejpam-6098	10	7	)	)	PUNCT
ejpam-6098	10	8	j.	j.	PROPN
ejpam-6098	10	9	jakhar	jakhar	PROPN
ejpam-6098	10	10	et	et	PROPN
ejpam-6098	10	11	al	al	PROPN
ejpam-6098	10	12	.	.	PUNCT
ejpam-6098	10	13	/	/	SYM
ejpam-6098	10	14	eur	eur	PROPN
ejpam-6098	10	15	.	.	PUNCT
ejpam-6098	11	1	j.	j.	PROPN
ejpam-6098	11	2	pure	pure	PROPN
ejpam-6098	11	3	appl	appl	PROPN
ejpam-6098	11	4	.	.	PROPN
ejpam-6098	11	5	math	math	PROPN
ejpam-6098	11	6	,	,	PUNCT
ejpam-6098	11	7	18	18	NUM
ejpam-6098	11	8	(	(	PUNCT
ejpam-6098	11	9	2	2	NUM
ejpam-6098	11	10	)	)	PUNCT
ejpam-6098	11	11	(	(	PUNCT
ejpam-6098	11	12	2025	2025	NUM
ejpam-6098	11	13	)	)	PUNCT
ejpam-6098	11	14	,	,	PUNCT
ejpam-6098	11	15	6098	6098	NUM
ejpam-6098	11	16	2	2	NUM
ejpam-6098	11	17	of	of	ADP
ejpam-6098	11	18	23	23	NUM
ejpam-6098	11	19	1	1	NUM
ejpam-6098	11	20	.	.	PUNCT
ejpam-6098	11	21	introduction	introduction	NOUN
ejpam-6098	11	22	in	in	ADP
ejpam-6098	11	23	nearly	nearly	ADV
ejpam-6098	11	24	every	every	PRON
ejpam-6098	11	25	branch	branch	NOUN
ejpam-6098	11	26	of	of	ADP
ejpam-6098	11	27	mathematical	mathematical	ADJ
ejpam-6098	11	28	analysis	analysis	NOUN
ejpam-6098	11	29	,	,	PUNCT
ejpam-6098	11	30	a	a	DET
ejpam-6098	11	31	fundamental	fundamental	ADJ
ejpam-6098	11	32	question	question	NOUN
ejpam-6098	11	33	arises	arise	VERB
ejpam-6098	11	34	:	:	PUNCT
ejpam-6098	11	35	under	under	ADP
ejpam-6098	11	36	what	what	DET
ejpam-6098	11	37	conditions	condition	NOUN
ejpam-6098	11	38	does	do	AUX
ejpam-6098	11	39	an	an	DET
ejpam-6098	11	40	object	object	NOUN
ejpam-6098	11	41	that	that	PRON
ejpam-6098	11	42	approximately	approximately	ADV
ejpam-6098	11	43	satisfies	satisfy	VERB
ejpam-6098	11	44	a	a	DET
ejpam-6098	11	45	particular	particular	ADJ
ejpam-6098	11	46	property	property	NOUN
ejpam-6098	11	47	also	also	ADV
ejpam-6098	11	48	lie	lie	VERB
ejpam-6098	11	49	close	close	ADV
ejpam-6098	11	50	to	to	ADP
ejpam-6098	11	51	an	an	DET
ejpam-6098	11	52	object	object	NOUN
ejpam-6098	11	53	that	that	PRON
ejpam-6098	11	54	exactly	exactly	ADV
ejpam-6098	11	55	satisfies	satisfy	VERB
ejpam-6098	11	56	that	that	DET
ejpam-6098	11	57	same	same	ADJ
ejpam-6098	11	58	property	property	NOUN
ejpam-6098	11	59	?	?	PUNCT
ejpam-6098	12	1	when	when	SCONJ
ejpam-6098	12	2	this	this	DET
ejpam-6098	12	3	question	question	NOUN
ejpam-6098	12	4	is	be	AUX
ejpam-6098	12	5	explicitly	explicitly	ADV
ejpam-6098	12	6	applied	apply	VERB
ejpam-6098	12	7	to	to	ADP
ejpam-6098	12	8	functional	functional	ADJ
ejpam-6098	12	9	equations	equation	NOUN
ejpam-6098	12	10	,	,	PUNCT
ejpam-6098	12	11	it	it	PRON
ejpam-6098	12	12	leads	lead	VERB
ejpam-6098	12	13	to	to	ADP
ejpam-6098	12	14	a	a	DET
ejpam-6098	12	15	crucial	crucial	ADJ
ejpam-6098	12	16	inquiry	inquiry	NOUN
ejpam-6098	12	17	:	:	PUNCT
ejpam-6098	12	18	if	if	SCONJ
ejpam-6098	12	19	a	a	DET
ejpam-6098	12	20	solution	solution	NOUN
ejpam-6098	12	21	to	to	ADP
ejpam-6098	12	22	a	a	DET
ejpam-6098	12	23	slightly	slightly	ADV
ejpam-6098	12	24	perturbed	perturb	VERB
ejpam-6098	12	25	version	version	NOUN
ejpam-6098	12	26	of	of	ADP
ejpam-6098	12	27	a	a	DET
ejpam-6098	12	28	functional	functional	ADJ
ejpam-6098	12	29	equation	equation	NOUN
ejpam-6098	12	30	is	be	AUX
ejpam-6098	12	31	found	find	VERB
ejpam-6098	12	32	,	,	PUNCT
ejpam-6098	12	33	under	under	ADP
ejpam-6098	12	34	what	what	DET
ejpam-6098	12	35	circumstances	circumstance	NOUN
ejpam-6098	12	36	can	can	AUX
ejpam-6098	12	37	we	we	PRON
ejpam-6098	12	38	guarantee	guarantee	VERB
ejpam-6098	12	39	that	that	SCONJ
ejpam-6098	12	40	this	this	DET
ejpam-6098	12	41	solution	solution	NOUN
ejpam-6098	12	42	remains	remain	VERB
ejpam-6098	12	43	close	close	ADJ
ejpam-6098	12	44	to	to	ADP
ejpam-6098	12	45	the	the	DET
ejpam-6098	12	46	exact	exact	ADJ
ejpam-6098	12	47	solution	solution	NOUN
ejpam-6098	12	48	of	of	ADP
ejpam-6098	12	49	the	the	DET
ejpam-6098	12	50	original	original	ADJ
ejpam-6098	12	51	equation	equation	NOUN
ejpam-6098	12	52	?	?	PUNCT
ejpam-6098	13	1	this	this	DET
ejpam-6098	13	2	question	question	NOUN
ejpam-6098	13	3	of	of	ADP
ejpam-6098	13	4	stability	stability	NOUN
ejpam-6098	13	5	was	be	AUX
ejpam-6098	13	6	first	first	ADV
ejpam-6098	13	7	raised	raise	VERB
ejpam-6098	13	8	by	by	ADP
ejpam-6098	13	9	ulam	ulam	PROPN
ejpam-6098	13	10	[	[	X
ejpam-6098	13	11	1	1	NUM
ejpam-6098	13	12	]	]	PUNCT
ejpam-6098	13	13	in	in	ADP
ejpam-6098	13	14	the	the	DET
ejpam-6098	13	15	context	context	NOUN
ejpam-6098	13	16	of	of	ADP
ejpam-6098	13	17	group	group	NOUN
ejpam-6098	13	18	homomorphisms	homomorphism	NOUN
ejpam-6098	13	19	in	in	ADP
ejpam-6098	13	20	1940	1940	NUM
ejpam-6098	13	21	.	.	PUNCT
ejpam-6098	14	1	it	it	PRON
ejpam-6098	14	2	has	have	AUX
ejpam-6098	14	3	since	since	ADV
ejpam-6098	14	4	been	be	AUX
ejpam-6098	14	5	central	central	ADJ
ejpam-6098	14	6	to	to	ADP
ejpam-6098	14	7	many	many	ADJ
ejpam-6098	14	8	developments	development	NOUN
ejpam-6098	14	9	in	in	ADP
ejpam-6098	14	10	the	the	DET
ejpam-6098	14	11	theory	theory	NOUN
ejpam-6098	14	12	of	of	ADP
ejpam-6098	14	13	functional	functional	ADJ
ejpam-6098	14	14	equations	equation	NOUN
ejpam-6098	14	15	,	,	PUNCT
ejpam-6098	14	16	particularly	particularly	ADV
ejpam-6098	14	17	in	in	ADP
ejpam-6098	14	18	the	the	DET
ejpam-6098	14	19	study	study	NOUN
ejpam-6098	14	20	of	of	ADP
ejpam-6098	14	21	their	their	PRON
ejpam-6098	14	22	stability	stability	NOUN
ejpam-6098	14	23	.	.	PUNCT
ejpam-6098	15	1	in	in	ADP
ejpam-6098	15	2	1941	1941	NUM
ejpam-6098	15	3	,	,	PUNCT
ejpam-6098	15	4	hyers	hyer	NOUN
ejpam-6098	15	5	provided	provide	VERB
ejpam-6098	15	6	an	an	DET
ejpam-6098	15	7	initial	initial	ADJ
ejpam-6098	15	8	answer	answer	NOUN
ejpam-6098	15	9	to	to	ADP
ejpam-6098	15	10	ulam	ulam	PROPN
ejpam-6098	15	11	’s	’s	PART
ejpam-6098	15	12	question	question	NOUN
ejpam-6098	15	13	regarding	regard	VERB
ejpam-6098	15	14	the	the	DET
ejpam-6098	15	15	stability	stability	NOUN
ejpam-6098	15	16	of	of	ADP
ejpam-6098	15	17	functional	functional	ADJ
ejpam-6098	15	18	equations	equation	NOUN
ejpam-6098	15	19	in	in	ADP
ejpam-6098	15	20	banach	banach	NOUN
ejpam-6098	15	21	spaces	space	NOUN
ejpam-6098	15	22	[	[	X
ejpam-6098	15	23	2	2	NUM
ejpam-6098	15	24	]	]	PUNCT
ejpam-6098	15	25	.	.	PUNCT
ejpam-6098	16	1	rassias	rassias	PROPN
ejpam-6098	16	2	extended	extend	VERB
ejpam-6098	16	3	this	this	DET
ejpam-6098	16	4	foundational	foundational	ADJ
ejpam-6098	16	5	work	work	NOUN
ejpam-6098	16	6	in	in	ADP
ejpam-6098	16	7	1978	1978	NUM
ejpam-6098	16	8	,	,	PUNCT
ejpam-6098	16	9	introducing	introduce	VERB
ejpam-6098	16	10	a	a	DET
ejpam-6098	16	11	generalized	generalized	ADJ
ejpam-6098	16	12	version	version	NOUN
ejpam-6098	16	13	of	of	ADP
ejpam-6098	16	14	hyers	hyer	NOUN
ejpam-6098	16	15	’	'	PUNCT
ejpam-6098	16	16	theorem	theorem	VERB
ejpam-6098	16	17	by	by	ADP
ejpam-6098	16	18	considering	consider	VERB
ejpam-6098	16	19	additive	additive	ADJ
ejpam-6098	16	20	mappings	mapping	NOUN
ejpam-6098	16	21	with	with	ADP
ejpam-6098	16	22	unbounded	unbounded	ADJ
ejpam-6098	16	23	control	control	NOUN
ejpam-6098	16	24	functions	function	NOUN
ejpam-6098	16	25	[	[	X
ejpam-6098	16	26	3	3	NUM
ejpam-6098	16	27	]	]	PUNCT
ejpam-6098	16	28	.	.	PUNCT
ejpam-6098	17	1	rassias	rassias	PROPN
ejpam-6098	17	2	’	'	PUNCT
ejpam-6098	17	3	extension	extension	NOUN
ejpam-6098	17	4	significantly	significantly	ADV
ejpam-6098	17	5	contributed	contribute	VERB
ejpam-6098	17	6	to	to	ADP
ejpam-6098	17	7	the	the	DET
ejpam-6098	17	8	development	development	NOUN
ejpam-6098	17	9	of	of	ADP
ejpam-6098	17	10	what	what	PRON
ejpam-6098	17	11	is	be	AUX
ejpam-6098	17	12	now	now	ADV
ejpam-6098	17	13	called	call	VERB
ejpam-6098	17	14	hyers	hyer	NOUN
ejpam-6098	17	15	-	-	PUNCT
ejpam-6098	17	16	ulam	ulam	ADJ
ejpam-6098	17	17	-	-	PUNCT
ejpam-6098	17	18	rassias	rassias	PROPN
ejpam-6098	17	19	stability	stability	NOUN
ejpam-6098	17	20	for	for	ADP
ejpam-6098	17	21	functional	functional	ADJ
ejpam-6098	17	22	equations	equation	NOUN
ejpam-6098	17	23	.	.	PUNCT
ejpam-6098	18	1	over	over	ADP
ejpam-6098	18	2	time	time	NOUN
ejpam-6098	18	3	,	,	PUNCT
ejpam-6098	18	4	several	several	ADJ
ejpam-6098	18	5	mathematicians	mathematician	NOUN
ejpam-6098	18	6	have	have	AUX
ejpam-6098	18	7	explored	explore	VERB
ejpam-6098	18	8	related	related	ADJ
ejpam-6098	18	9	problems	problem	NOUN
ejpam-6098	18	10	in	in	ADP
ejpam-6098	18	11	various	various	ADJ
ejpam-6098	18	12	spaces	space	NOUN
ejpam-6098	18	13	,	,	PUNCT
ejpam-6098	18	14	further	far	ADV
ejpam-6098	18	15	enriching	enrich	VERB
ejpam-6098	18	16	the	the	DET
ejpam-6098	18	17	study	study	NOUN
ejpam-6098	18	18	of	of	ADP
ejpam-6098	18	19	functional	functional	ADJ
ejpam-6098	18	20	equations	equation	NOUN
ejpam-6098	18	21	and	and	CCONJ
ejpam-6098	18	22	their	their	PRON
ejpam-6098	18	23	stability	stability	NOUN
ejpam-6098	18	24	.	.	PUNCT
ejpam-6098	19	1	notable	notable	ADJ
ejpam-6098	19	2	contributions	contribution	NOUN
ejpam-6098	19	3	include	include	VERB
ejpam-6098	19	4	the	the	DET
ejpam-6098	19	5	investigation	investigation	NOUN
ejpam-6098	19	6	of	of	ADP
ejpam-6098	19	7	intuitionistic	intuitionistic	ADJ
ejpam-6098	19	8	fuzzy	fuzzy	ADJ
ejpam-6098	19	9	stability	stability	NOUN
ejpam-6098	19	10	[	[	X
ejpam-6098	19	11	4	4	NUM
ejpam-6098	19	12	]	]	PUNCT
ejpam-6098	19	13	,	,	PUNCT
ejpam-6098	19	14	functional	functional	ADJ
ejpam-6098	19	15	equations	equation	NOUN
ejpam-6098	19	16	in	in	ADP
ejpam-6098	19	17	menger-ϕ	menger-ϕ	NOUN
ejpam-6098	19	18	normed	norme	VERB
ejpam-6098	19	19	spaces	space	NOUN
ejpam-6098	19	20	[	[	X
ejpam-6098	19	21	5	5	NUM
ejpam-6098	19	22	]	]	PUNCT
ejpam-6098	19	23	,	,	PUNCT
ejpam-6098	19	24	and	and	CCONJ
ejpam-6098	19	25	functional	functional	ADJ
ejpam-6098	19	26	equations	equation	NOUN
ejpam-6098	19	27	in	in	ADP
ejpam-6098	19	28	modular	modular	ADJ
ejpam-6098	19	29	spaces	space	NOUN
ejpam-6098	19	30	[	[	X
ejpam-6098	19	31	6	6	NUM
ejpam-6098	19	32	]	]	PUNCT
ejpam-6098	19	33	.	.	PUNCT
ejpam-6098	20	1	additionally	additionally	ADV
ejpam-6098	20	2	,	,	PUNCT
ejpam-6098	20	3	research	research	NOUN
ejpam-6098	20	4	has	have	AUX
ejpam-6098	20	5	addressed	address	VERB
ejpam-6098	20	6	quartic	quartic	ADJ
ejpam-6098	20	7	functional	functional	ADJ
ejpam-6098	20	8	equations	equation	NOUN
ejpam-6098	20	9	[	[	X
ejpam-6098	20	10	7	7	NUM
ejpam-6098	20	11	]	]	PUNCT
ejpam-6098	20	12	,	,	PUNCT
ejpam-6098	20	13	fractional	fractional	ADJ
ejpam-6098	20	14	differential	differential	ADJ
ejpam-6098	20	15	equations	equation	NOUN
ejpam-6098	20	16	[	[	X
ejpam-6098	20	17	8	8	NUM
ejpam-6098	20	18	,	,	PUNCT
ejpam-6098	20	19	9	9	NUM
ejpam-6098	20	20	]	]	PUNCT
ejpam-6098	20	21	,	,	PUNCT
ejpam-6098	20	22	and	and	CCONJ
ejpam-6098	20	23	fuzzy	fuzzy	ADJ
ejpam-6098	20	24	approximately	approximately	ADV
ejpam-6098	20	25	cubic	cubic	ADJ
ejpam-6098	20	26	mappings	mapping	NOUN
ejpam-6098	20	27	[	[	X
ejpam-6098	20	28	10	10	NUM
ejpam-6098	20	29	]	]	PUNCT
ejpam-6098	20	30	.	.	PUNCT
ejpam-6098	21	1	further	further	ADJ
ejpam-6098	21	2	studies	study	NOUN
ejpam-6098	21	3	have	have	AUX
ejpam-6098	21	4	examined	examine	VERB
ejpam-6098	21	5	additive	additive	ADJ
ejpam-6098	21	6	mappings	mapping	NOUN
ejpam-6098	21	7	in	in	ADP
ejpam-6098	21	8	2	2	NUM
ejpam-6098	21	9	-	-	PUNCT
ejpam-6098	21	10	banach	banach	NOUN
ejpam-6098	21	11	spaces	space	NOUN
ejpam-6098	21	12	and	and	CCONJ
ejpam-6098	21	13	other	other	ADJ
ejpam-6098	21	14	related	related	ADJ
ejpam-6098	21	15	topics	topic	NOUN
ejpam-6098	21	16	[	[	X
ejpam-6098	21	17	11	11	NUM
ejpam-6098	21	18	]	]	PUNCT
ejpam-6098	21	19	,	,	PUNCT
ejpam-6098	21	20	broadening	broaden	VERB
ejpam-6098	21	21	the	the	DET
ejpam-6098	21	22	understanding	understanding	NOUN
ejpam-6098	21	23	of	of	ADP
ejpam-6098	21	24	functional	functional	ADJ
ejpam-6098	21	25	equations	equation	NOUN
ejpam-6098	21	26	in	in	ADP
ejpam-6098	21	27	diverse	diverse	ADJ
ejpam-6098	21	28	mathematical	mathematical	ADJ
ejpam-6098	21	29	frameworks	framework	NOUN
ejpam-6098	21	30	.	.	PUNCT
ejpam-6098	22	1	on	on	ADP
ejpam-6098	22	2	the	the	DET
ejpam-6098	22	3	other	other	ADJ
ejpam-6098	22	4	hand	hand	NOUN
ejpam-6098	22	5	,	,	PUNCT
ejpam-6098	22	6	some	some	DET
ejpam-6098	22	7	mathematicians	mathematician	NOUN
ejpam-6098	22	8	have	have	AUX
ejpam-6098	22	9	extended	extend	VERB
ejpam-6098	22	10	the	the	DET
ejpam-6098	22	11	concept	concept	NOUN
ejpam-6098	22	12	of	of	ADP
ejpam-6098	22	13	normed	normed	ADJ
ejpam-6098	22	14	linear	linear	PROPN
ejpam-6098	22	15	spaces	space	NOUN
ejpam-6098	22	16	.	.	PUNCT
ejpam-6098	23	1	gähler	gähler	NOUN
ejpam-6098	24	1	[	[	X
ejpam-6098	24	2	12	12	NUM
ejpam-6098	24	3	,	,	PUNCT
ejpam-6098	24	4	13	13	NUM
ejpam-6098	24	5	]	]	PUNCT
ejpam-6098	24	6	initiated	initiate	VERB
ejpam-6098	24	7	the	the	DET
ejpam-6098	24	8	exploration	exploration	NOUN
ejpam-6098	24	9	of	of	ADP
ejpam-6098	24	10	multi	multi	NOUN
ejpam-6098	24	11	-	-	NOUN
ejpam-6098	24	12	norms	norm	NOUN
ejpam-6098	24	13	within	within	ADP
ejpam-6098	24	14	linear	linear	ADJ
ejpam-6098	24	15	spaces	space	NOUN
ejpam-6098	24	16	.	.	PUNCT
ejpam-6098	25	1	significant	significant	ADJ
ejpam-6098	25	2	contributions	contribution	NOUN
ejpam-6098	25	3	to	to	ADP
ejpam-6098	25	4	the	the	DET
ejpam-6098	25	5	development	development	NOUN
ejpam-6098	25	6	of	of	ADP
ejpam-6098	25	7	n	n	ADV
ejpam-6098	25	8	-	-	PUNCT
ejpam-6098	25	9	normed	norme	VERB
ejpam-6098	25	10	spaces	space	NOUN
ejpam-6098	25	11	have	have	AUX
ejpam-6098	25	12	been	be	AUX
ejpam-6098	25	13	made	make	VERB
ejpam-6098	25	14	by	by	ADP
ejpam-6098	25	15	gunawan	gunawan	NOUN
ejpam-6098	25	16	and	and	CCONJ
ejpam-6098	25	17	mashadi	mashadi	VERB
ejpam-6098	25	18	[	[	X
ejpam-6098	25	19	14	14	NUM
ejpam-6098	25	20	]	]	PUNCT
ejpam-6098	25	21	,	,	PUNCT
ejpam-6098	25	22	malčeski	malčeski	PROPN
ejpam-6098	25	23	[	[	X
ejpam-6098	25	24	15	15	NUM
ejpam-6098	25	25	]	]	PUNCT
ejpam-6098	25	26	,	,	PUNCT
ejpam-6098	25	27	kim	kim	PROPN
ejpam-6098	25	28	and	and	CCONJ
ejpam-6098	25	29	cho	cho	PROPN
ejpam-6098	26	1	[	[	X
ejpam-6098	26	2	16	16	NUM
ejpam-6098	26	3	]	]	PUNCT
ejpam-6098	26	4	,	,	PUNCT
ejpam-6098	26	5	and	and	CCONJ
ejpam-6098	26	6	misiak	misiak	NOUN
ejpam-6098	27	1	[	[	X
ejpam-6098	27	2	17	17	NUM
ejpam-6098	27	3	]	]	PUNCT
ejpam-6098	27	4	.	.	PUNCT
ejpam-6098	28	1	park	park	NOUN
ejpam-6098	29	1	[	[	X
ejpam-6098	29	2	11	11	NUM
ejpam-6098	29	3	]	]	PUNCT
ejpam-6098	29	4	studied	study	VERB
ejpam-6098	29	5	approximate	approximate	ADJ
ejpam-6098	29	6	additive	additive	ADJ
ejpam-6098	29	7	mappings	mapping	NOUN
ejpam-6098	29	8	,	,	PUNCT
ejpam-6098	29	9	jensen	jensen	PROPN
ejpam-6098	29	10	mappings	mapping	NOUN
ejpam-6098	29	11	,	,	PUNCT
ejpam-6098	29	12	and	and	CCONJ
ejpam-6098	29	13	related	related	ADJ
ejpam-6098	29	14	topics	topic	NOUN
ejpam-6098	29	15	in	in	ADP
ejpam-6098	29	16	2	2	NUM
ejpam-6098	29	17	-	-	PUNCT
ejpam-6098	29	18	banach	banach	NOUN
ejpam-6098	29	19	spaces	space	NOUN
ejpam-6098	29	20	.	.	PUNCT
ejpam-6098	30	1	in	in	ADP
ejpam-6098	30	2	2012	2012	NUM
ejpam-6098	30	3	,	,	PUNCT
ejpam-6098	30	4	xu	xu	PROPN
ejpam-6098	30	5	and	and	CCONJ
ejpam-6098	30	6	rassias	rassia	VERB
ejpam-6098	30	7	[	[	X
ejpam-6098	30	8	18	18	NUM
ejpam-6098	30	9	]	]	PUNCT
ejpam-6098	30	10	investigated	investigate	VERB
ejpam-6098	30	11	the	the	DET
ejpam-6098	30	12	stability	stability	NOUN
ejpam-6098	30	13	of	of	ADP
ejpam-6098	30	14	cubic	cubic	ADJ
ejpam-6098	30	15	and	and	CCONJ
ejpam-6098	30	16	general	general	ADJ
ejpam-6098	30	17	mixed	mixed	ADJ
ejpam-6098	30	18	additive	additive	ADJ
ejpam-6098	30	19	functional	functional	ADJ
ejpam-6098	30	20	equations	equation	NOUN
ejpam-6098	30	21	in	in	ADP
ejpam-6098	30	22	n	n	NOUN
ejpam-6098	30	23	-	-	PUNCT
ejpam-6098	30	24	banach	banach	NOUN
ejpam-6098	30	25	spaces	space	NOUN
ejpam-6098	30	26	.	.	PUNCT
ejpam-6098	31	1	in	in	ADP
ejpam-6098	31	2	2015	2015	NUM
ejpam-6098	31	3	,	,	PUNCT
ejpam-6098	31	4	yang	yang	PROPN
ejpam-6098	31	5	et	et	PROPN
ejpam-6098	31	6	al	al	PROPN
ejpam-6098	31	7	.	.	PROPN
ejpam-6098	31	8	introduced	introduce	VERB
ejpam-6098	31	9	the	the	DET
ejpam-6098	31	10	concepts	concept	NOUN
ejpam-6098	31	11	of	of	ADP
ejpam-6098	31	12	non	non	ADJ
ejpam-6098	31	13	-	-	ADJ
ejpam-6098	31	14	archimedean	archimedean	ADJ
ejpam-6098	31	15	(	(	PUNCT
ejpam-6098	31	16	n	n	CCONJ
ejpam-6098	31	17	,	,	PUNCT
ejpam-6098	31	18	β)-normed	β)-normed	PUNCT
ejpam-6098	31	19	space	space	NOUN
ejpam-6098	31	20	(	(	PUNCT
ejpam-6098	31	21	na-(n	na-(n	NUM
ejpam-6098	31	22	,	,	PUNCT
ejpam-6098	31	23	β)-ns	β)-ns	NOUN
ejpam-6098	31	24	)	)	PUNCT
ejpam-6098	31	25	and	and	CCONJ
ejpam-6098	31	26	(	(	PUNCT
ejpam-6098	31	27	n	n	CCONJ
ejpam-6098	31	28	,	,	PUNCT
ejpam-6098	31	29	β)-normed	β)-normed	PUNCT
ejpam-6098	31	30	space	space	NOUN
ejpam-6098	31	31	[	[	X
ejpam-6098	31	32	19	19	NUM
ejpam-6098	31	33	]	]	PUNCT
ejpam-6098	31	34	.	.	PUNCT
ejpam-6098	32	1	in	in	ADP
ejpam-6098	32	2	2022	2022	NUM
ejpam-6098	32	3	,	,	PUNCT
ejpam-6098	32	4	jyotsana	jyotsana	PROPN
ejpam-6098	32	5	et	et	PROPN
ejpam-6098	32	6	al	al	PROPN
ejpam-6098	32	7	.	.	PROPN
ejpam-6098	32	8	examined	examine	VERB
ejpam-6098	32	9	the	the	DET
ejpam-6098	32	10	stability	stability	NOUN
ejpam-6098	32	11	of	of	ADP
ejpam-6098	32	12	additive	additive	ADJ
ejpam-6098	32	13	functional	functional	ADJ
ejpam-6098	32	14	equations	equation	NOUN
ejpam-6098	32	15	,	,	PUNCT
ejpam-6098	32	16	quartic	quartic	ADJ
ejpam-6098	32	17	functional	functional	ADJ
ejpam-6098	32	18	equations	equation	NOUN
ejpam-6098	32	19	,	,	PUNCT
ejpam-6098	32	20	and	and	CCONJ
ejpam-6098	32	21	a	a	X
ejpam-6098	32	22	-	-	PUNCT
ejpam-6098	32	23	cubic	cubic	ADJ
ejpam-6098	32	24	and	and	CCONJ
ejpam-6098	32	25	b	b	NOUN
ejpam-6098	32	26	-	-	PUNCT
ejpam-6098	32	27	cubic	cubic	ADJ
ejpam-6098	32	28	functional	functional	ADJ
ejpam-6098	32	29	equations	equation	NOUN
ejpam-6098	32	30	in	in	ADP
ejpam-6098	32	31	na-(n	na-(n	PROPN
ejpam-6098	32	32	,	,	PUNCT
ejpam-6098	32	33	β)-normed	β)-normed	PUNCT
ejpam-6098	32	34	spaces	space	NOUN
ejpam-6098	32	35	[	[	X
ejpam-6098	32	36	20	20	NUM
ejpam-6098	32	37	]	]	PUNCT
ejpam-6098	32	38	.	.	PUNCT
ejpam-6098	33	1	the	the	DET
ejpam-6098	33	2	theory	theory	NOUN
ejpam-6098	33	3	of	of	ADP
ejpam-6098	33	4	random	random	ADJ
ejpam-6098	33	5	normed	norme	VERB
ejpam-6098	33	6	spaces	space	NOUN
ejpam-6098	33	7	(	(	PUNCT
ejpam-6098	33	8	rns	rn	NOUN
ejpam-6098	33	9	)	)	PUNCT
ejpam-6098	33	10	is	be	AUX
ejpam-6098	33	11	significant	significant	ADJ
ejpam-6098	33	12	because	because	SCONJ
ejpam-6098	33	13	it	it	PRON
ejpam-6098	33	14	generalizes	generalize	VERB
ejpam-6098	33	15	the	the	DET
ejpam-6098	33	16	deterministic	deterministic	ADJ
ejpam-6098	33	17	results	result	NOUN
ejpam-6098	33	18	observed	observe	VERB
ejpam-6098	33	19	in	in	ADP
ejpam-6098	33	20	linear	linear	PROPN
ejpam-6098	33	21	normed	norme	VERB
ejpam-6098	33	22	spaces	space	NOUN
ejpam-6098	33	23	and	and	CCONJ
ejpam-6098	33	24	has	have	VERB
ejpam-6098	33	25	applications	application	NOUN
ejpam-6098	33	26	in	in	ADP
ejpam-6098	33	27	the	the	DET
ejpam-6098	33	28	study	study	NOUN
ejpam-6098	33	29	of	of	ADP
ejpam-6098	33	30	random	random	ADJ
ejpam-6098	33	31	operators	operator	NOUN
ejpam-6098	33	32	and	and	CCONJ
ejpam-6098	33	33	functional	functional	ADJ
ejpam-6098	33	34	equations	equation	NOUN
ejpam-6098	33	35	.	.	PUNCT
ejpam-6098	34	1	recent	recent	ADJ
ejpam-6098	34	2	studies	study	NOUN
ejpam-6098	34	3	have	have	AUX
ejpam-6098	34	4	extensively	extensively	ADV
ejpam-6098	34	5	examined	examine	VERB
ejpam-6098	34	6	the	the	DET
ejpam-6098	34	7	stability	stability	NOUN
ejpam-6098	34	8	and	and	CCONJ
ejpam-6098	34	9	related	related	ADJ
ejpam-6098	34	10	properties	property	NOUN
ejpam-6098	34	11	of	of	ADP
ejpam-6098	34	12	functional	functional	ADJ
ejpam-6098	34	13	equations	equation	NOUN
ejpam-6098	34	14	in	in	ADP
ejpam-6098	34	15	various	various	ADJ
ejpam-6098	34	16	mathematical	mathematical	ADJ
ejpam-6098	34	17	spaces	space	NOUN
ejpam-6098	34	18	.	.	PUNCT
ejpam-6098	35	1	researchers	researcher	NOUN
ejpam-6098	35	2	have	have	AUX
ejpam-6098	35	3	explored	explore	VERB
ejpam-6098	35	4	fuzzy	fuzzy	ADJ
ejpam-6098	35	5	approximately	approximately	ADV
ejpam-6098	35	6	cubic	cubic	ADJ
ejpam-6098	35	7	mappings	mapping	NOUN
ejpam-6098	36	1	[	[	X
ejpam-6098	36	2	10	10	NUM
ejpam-6098	36	3	]	]	PUNCT
ejpam-6098	36	4	,	,	PUNCT
ejpam-6098	36	5	the	the	DET
ejpam-6098	36	6	σ	σ	NOUN
ejpam-6098	36	7	-	-	ADJ
ejpam-6098	36	8	quadratic	quadratic	ADJ
ejpam-6098	36	9	functional	functional	ADJ
ejpam-6098	36	10	equation	equation	NOUN
ejpam-6098	36	11	[	[	X
ejpam-6098	36	12	21	21	NUM
ejpam-6098	36	13	]	]	PUNCT
ejpam-6098	36	14	,	,	PUNCT
ejpam-6098	36	15	and	and	CCONJ
ejpam-6098	36	16	quadratic	quadratic	ADJ
ejpam-6098	36	17	equations	equation	NOUN
ejpam-6098	37	1	[	[	X
ejpam-6098	37	2	22	22	NUM
ejpam-6098	37	3	]	]	PUNCT
ejpam-6098	37	4	.	.	PUNCT
ejpam-6098	38	1	the	the	DET
ejpam-6098	38	2	cauchy	cauchy	ADJ
ejpam-6098	38	3	functional	functional	ADJ
ejpam-6098	38	4	equation	equation	NOUN
ejpam-6098	38	5	has	have	AUX
ejpam-6098	38	6	also	also	ADV
ejpam-6098	38	7	been	be	AUX
ejpam-6098	38	8	investigated	investigate	VERB
ejpam-6098	38	9	in	in	ADP
ejpam-6098	38	10	random	random	ADJ
ejpam-6098	38	11	normed	normed	ADJ
ejpam-6098	38	12	spaces	space	NOUN
ejpam-6098	38	13	[	[	X
ejpam-6098	38	14	23	23	NUM
ejpam-6098	38	15	]	]	PUNCT
ejpam-6098	38	16	,	,	PUNCT
ejpam-6098	38	17	alongside	alongside	ADP
ejpam-6098	38	18	analyses	analysis	NOUN
ejpam-6098	38	19	of	of	ADP
ejpam-6098	38	20	cubic	cubic	ADJ
ejpam-6098	38	21	and	and	CCONJ
ejpam-6098	38	22	quadratic	quadratic	ADJ
ejpam-6098	38	23	mappings	mapping	NOUN
ejpam-6098	38	24	[	[	X
ejpam-6098	38	25	24	24	NUM
ejpam-6098	38	26	]	]	PUNCT
ejpam-6098	38	27	,	,	PUNCT
ejpam-6098	38	28	as	as	ADV
ejpam-6098	38	29	well	well	ADV
ejpam-6098	38	30	as	as	ADP
ejpam-6098	38	31	cubic	cubic	ADJ
ejpam-6098	38	32	and	and	CCONJ
ejpam-6098	38	33	quartic	quartic	ADJ
ejpam-6098	38	34	mappings	mapping	NOUN
ejpam-6098	38	35	[	[	X
ejpam-6098	38	36	25	25	NUM
ejpam-6098	38	37	]	]	PUNCT
ejpam-6098	38	38	.	.	PUNCT
ejpam-6098	39	1	additionally	additionally	ADV
ejpam-6098	39	2	,	,	PUNCT
ejpam-6098	39	3	felj	felj	NOUN
ejpam-6098	39	4	.	.	PUNCT
ejpam-6098	40	1	jakhar	jakhar	PROPN
ejpam-6098	40	2	et	et	PROPN
ejpam-6098	40	3	al	al	PROPN
ejpam-6098	40	4	.	.	PUNCT
ejpam-6098	40	5	/	/	SYM
ejpam-6098	40	6	eur	eur	PROPN
ejpam-6098	40	7	.	.	PUNCT
ejpam-6098	41	1	j.	j.	PROPN
ejpam-6098	41	2	pure	pure	PROPN
ejpam-6098	41	3	appl	appl	PROPN
ejpam-6098	41	4	.	.	PROPN
ejpam-6098	41	5	math	math	PROPN
ejpam-6098	41	6	,	,	PUNCT
ejpam-6098	41	7	18	18	NUM
ejpam-6098	41	8	(	(	PUNCT
ejpam-6098	41	9	2	2	NUM
ejpam-6098	41	10	)	)	PUNCT
ejpam-6098	41	11	(	(	PUNCT
ejpam-6098	41	12	2025	2025	NUM
ejpam-6098	41	13	)	)	PUNCT
ejpam-6098	41	14	,	,	PUNCT
ejpam-6098	41	15	6098	6098	NUM
ejpam-6098	41	16	3	3	NUM
ejpam-6098	41	17	of	of	ADP
ejpam-6098	41	18	23	23	NUM
ejpam-6098	41	19	bin	bin	NOUN
ejpam-6098	41	20	’s	’s	PART
ejpam-6098	41	21	type	type	NOUN
ejpam-6098	41	22	non	non	ADJ
ejpam-6098	41	23	-	-	ADJ
ejpam-6098	41	24	archimedean	archimedean	ADJ
ejpam-6098	41	25	fuzzy	fuzzy	ADJ
ejpam-6098	41	26	normed	norme	VERB
ejpam-6098	41	27	spaces	space	NOUN
ejpam-6098	41	28	[	[	X
ejpam-6098	41	29	26	26	NUM
ejpam-6098	41	30	]	]	PUNCT
ejpam-6098	41	31	have	have	AUX
ejpam-6098	41	32	been	be	AUX
ejpam-6098	41	33	studied	study	VERB
ejpam-6098	41	34	,	,	PUNCT
ejpam-6098	41	35	further	far	ADV
ejpam-6098	41	36	enriching	enrich	VERB
ejpam-6098	41	37	the	the	DET
ejpam-6098	41	38	field	field	NOUN
ejpam-6098	41	39	and	and	CCONJ
ejpam-6098	41	40	expanding	expand	VERB
ejpam-6098	41	41	its	its	PRON
ejpam-6098	41	42	theoretical	theoretical	ADJ
ejpam-6098	41	43	foundations	foundation	NOUN
ejpam-6098	41	44	.	.	PUNCT
ejpam-6098	42	1	in	in	ADP
ejpam-6098	42	2	2020	2020	NUM
ejpam-6098	42	3	,	,	PUNCT
ejpam-6098	42	4	govindan	govindan	PROPN
ejpam-6098	42	5	et	et	PROPN
ejpam-6098	42	6	al	al	PROPN
ejpam-6098	42	7	.	.	PUNCT
ejpam-6098	43	1	[	[	X
ejpam-6098	43	2	27	27	NUM
ejpam-6098	43	3	]	]	PUNCT
ejpam-6098	43	4	investigated	investigate	VERB
ejpam-6098	43	5	the	the	DET
ejpam-6098	43	6	stability	stability	NOUN
ejpam-6098	43	7	and	and	CCONJ
ejpam-6098	43	8	solutions	solution	NOUN
ejpam-6098	43	9	of	of	ADP
ejpam-6098	43	10	the	the	DET
ejpam-6098	43	11	cubic	cubic	ADJ
ejpam-6098	43	12	functional	functional	ADJ
ejpam-6098	43	13	equation	equation	NOUN
ejpam-6098	43	14	.	.	PUNCT
ejpam-6098	44	1	g(2u+	g(2u+	PROPN
ejpam-6098	44	2	w	w	PROPN
ejpam-6098	44	3	+	+	NOUN
ejpam-6098	44	4	v	v	NOUN
ejpam-6098	44	5	)	)	PUNCT
ejpam-6098	44	6	=	=	SYM
ejpam-6098	45	1	3g(w	3g(w	PROPN
ejpam-6098	45	2	+	+	NUM
ejpam-6098	45	3	u+	u+	NOUN
ejpam-6098	45	4	v	v	NOUN
ejpam-6098	45	5	)	)	PUNCT
ejpam-6098	46	1	+	+	CCONJ
ejpam-6098	46	2	g(w	g(w	ADJ
ejpam-6098	46	3	−	−	NOUN
ejpam-6098	46	4	u+	u+	NOUN
ejpam-6098	46	5	v	v	NOUN
ejpam-6098	46	6	)	)	PUNCT
ejpam-6098	47	1	+	+	CCONJ
ejpam-6098	47	2	2g(w	2g(w	NUM
ejpam-6098	47	3	+	+	CCONJ
ejpam-6098	47	4	u	u	NOUN
ejpam-6098	47	5	)	)	PUNCT
ejpam-6098	48	1	+	+	CCONJ
ejpam-6098	49	1	2g(u+	2g(u+	PROPN
ejpam-6098	49	2	v)−	v)−	PROPN
ejpam-6098	49	3	6g(u−	6g(u−	NUM
ejpam-6098	49	4	w	w	PROPN
ejpam-6098	49	5	)	)	PUNCT
ejpam-6098	49	6	−	−	PROPN
ejpam-6098	49	7	6g(u−	6g(u−	NUM
ejpam-6098	49	8	v)−	v)−	PROPN
ejpam-6098	49	9	3g(v	3g(v	NUM
ejpam-6098	49	10	+	+	CCONJ
ejpam-6098	49	11	w	w	X
ejpam-6098	49	12	)	)	PUNCT
ejpam-6098	49	13	+	+	CCONJ
ejpam-6098	49	14	2g(2u−	2g(2u−	NUM
ejpam-6098	49	15	v	v	NOUN
ejpam-6098	49	16	)	)	PUNCT
ejpam-6098	49	17	+	+	NUM
ejpam-6098	49	18	2g(2u−	2g(2u−	NUM
ejpam-6098	49	19	w)−	w)−	PROPN
ejpam-6098	49	20	6g(v)−	6g(v)−	NUM
ejpam-6098	49	21	6g(w	6g(w	NUM
ejpam-6098	49	22	)	)	PUNCT
ejpam-6098	50	1	−	−	NOUN
ejpam-6098	50	2	18g(u	18g(u	NUM
ejpam-6098	50	3	)	)	PUNCT
ejpam-6098	50	4	.	.	PUNCT
ejpam-6098	51	1	(	(	PUNCT
ejpam-6098	51	2	1	1	X
ejpam-6098	51	3	)	)	PUNCT
ejpam-6098	51	4	1.1	1.1	NUM
ejpam-6098	51	5	.	.	PUNCT
ejpam-6098	52	1	preliminaries	preliminary	NOUN
ejpam-6098	52	2	and	and	CCONJ
ejpam-6098	52	3	definitions	definition	NOUN
ejpam-6098	52	4	in	in	ADP
ejpam-6098	52	5	this	this	DET
ejpam-6098	52	6	subsection	subsection	NOUN
ejpam-6098	52	7	,	,	PUNCT
ejpam-6098	52	8	we	we	PRON
ejpam-6098	52	9	generalize	generalize	VERB
ejpam-6098	52	10	the	the	DET
ejpam-6098	52	11	basic	basic	ADJ
ejpam-6098	52	12	definitions	definition	NOUN
ejpam-6098	52	13	,	,	PUNCT
ejpam-6098	52	14	terminology	terminology	NOUN
ejpam-6098	52	15	,	,	PUNCT
ejpam-6098	52	16	notations	notation	NOUN
ejpam-6098	52	17	,	,	PUNCT
ejpam-6098	52	18	and	and	CCONJ
ejpam-6098	52	19	typical	typical	ADJ
ejpam-6098	52	20	characteristics	characteristic	NOUN
ejpam-6098	52	21	of	of	ADP
ejpam-6098	52	22	(	(	PUNCT
ejpam-6098	52	23	n	n	CCONJ
ejpam-6098	52	24	,	,	PUNCT
ejpam-6098	52	25	β)-ns	β)-ns	NOUN
ejpam-6098	52	26	,	,	PUNCT
ejpam-6098	52	27	na-(n	na-(n	NUM
ejpam-6098	52	28	,	,	PUNCT
ejpam-6098	52	29	β)-ns	β)-ns	NOUN
ejpam-6098	52	30	,	,	PUNCT
ejpam-6098	52	31	and	and	CCONJ
ejpam-6098	52	32	rns	rns	PROPN
ejpam-6098	52	33	.	.	PUNCT
ejpam-6098	53	1	definition	definition	NOUN
ejpam-6098	53	2	1	1	NUM
ejpam-6098	53	3	.	.	PUNCT
ejpam-6098	54	1	[	[	X
ejpam-6098	54	2	20	20	NUM
ejpam-6098	54	3	]	]	PUNCT
ejpam-6098	54	4	”	"	PUNCT
ejpam-6098	54	5	let	let	VERB
ejpam-6098	54	6	u(r	u(r	NOUN
ejpam-6098	54	7	)	)	PUNCT
ejpam-6098	54	8	be	be	AUX
ejpam-6098	54	9	a	a	DET
ejpam-6098	54	10	vector	vector	NOUN
ejpam-6098	54	11	space	space	NOUN
ejpam-6098	54	12	with	with	ADP
ejpam-6098	54	13	dimu	dimu	NOUN
ejpam-6098	54	14	≥	≥	X
ejpam-6098	54	15	n	n	CCONJ
ejpam-6098	54	16	,	,	PUNCT
ejpam-6098	54	17	and	and	CCONJ
ejpam-6098	54	18	let	let	VERB
ejpam-6098	54	19	||	||	NUM
ejpam-6098	54	20	·	·	PUNCT
ejpam-6098	54	21	,	,	PUNCT
ejpam-6098	54	22	.	.	PUNCT
ejpam-6098	54	23	.	.	PUNCT
ejpam-6098	55	1	.	.	PUNCT
ejpam-6098	56	1	,	,	PUNCT
ejpam-6098	56	2	·	·	PUNCT
ejpam-6098	56	3	||β	||β	NOUN
ejpam-6098	56	4	:	:	PUNCT
ejpam-6098	56	5	un	un	PROPN
ejpam-6098	56	6	→	→	SYM
ejpam-6098	56	7	r	r	NOUN
ejpam-6098	56	8	be	be	AUX
ejpam-6098	56	9	a	a	DET
ejpam-6098	56	10	mapping	mapping	NOUN
ejpam-6098	56	11	that	that	PRON
ejpam-6098	56	12	satisfies	satisfy	VERB
ejpam-6098	56	13	the	the	DET
ejpam-6098	56	14	following	follow	VERB
ejpam-6098	56	15	properties	property	NOUN
ejpam-6098	56	16	:	:	PUNCT
ejpam-6098	56	17	(	(	PUNCT
ejpam-6098	56	18	i	i	NOUN
ejpam-6098	56	19	)	)	PUNCT
ejpam-6098	56	20	||u1	||u1	PROPN
ejpam-6098	56	21	,	,	PUNCT
ejpam-6098	56	22	.	.	PUNCT
ejpam-6098	56	23	.	.	PUNCT
ejpam-6098	57	1	.	.	PUNCT
ejpam-6098	58	1	,	,	PUNCT
ejpam-6098	58	2	un||β	un||β	PROPN
ejpam-6098	59	1	=	=	SYM
ejpam-6098	59	2	0	0	PUNCT
ejpam-6098	60	1	if	if	SCONJ
ejpam-6098	60	2	and	and	CCONJ
ejpam-6098	60	3	only	only	ADV
ejpam-6098	60	4	if	if	SCONJ
ejpam-6098	60	5	u1	u1	NOUN
ejpam-6098	60	6	,	,	PUNCT
ejpam-6098	60	7	.	.	PUNCT
ejpam-6098	60	8	.	.	PUNCT
ejpam-6098	60	9	.	.	PUNCT
ejpam-6098	61	1	,	,	PUNCT
ejpam-6098	61	2	un	un	PROPN
ejpam-6098	61	3	are	be	AUX
ejpam-6098	61	4	linearly	linearly	ADV
ejpam-6098	61	5	dependent	dependent	ADJ
ejpam-6098	61	6	,	,	PUNCT
ejpam-6098	61	7	(	(	PUNCT
ejpam-6098	61	8	ii	ii	NOUN
ejpam-6098	61	9	)	)	PUNCT
ejpam-6098	61	10	||u1	||u1	PROPN
ejpam-6098	61	11	,	,	PUNCT
ejpam-6098	61	12	.	.	PUNCT
ejpam-6098	61	13	.	.	PUNCT
ejpam-6098	62	1	.	.	PUNCT
ejpam-6098	63	1	,	,	PUNCT
ejpam-6098	63	2	un||β	un||β	PROPN
ejpam-6098	63	3	remains	remain	VERB
ejpam-6098	63	4	unchanged	unchanged	ADJ
ejpam-6098	63	5	under	under	ADP
ejpam-6098	63	6	any	any	DET
ejpam-6098	63	7	permutation	permutation	NOUN
ejpam-6098	63	8	of	of	ADP
ejpam-6098	63	9	the	the	DET
ejpam-6098	63	10	elements	element	NOUN
ejpam-6098	63	11	u1	u1	NOUN
ejpam-6098	63	12	,	,	PUNCT
ejpam-6098	63	13	.	.	PUNCT
ejpam-6098	63	14	.	.	PUNCT
ejpam-6098	64	1	.	.	PUNCT
ejpam-6098	65	1	,	,	PUNCT
ejpam-6098	65	2	un	un	PROPN
ejpam-6098	65	3	,	,	PUNCT
ejpam-6098	65	4	(	(	PUNCT
ejpam-6098	65	5	iii	iii	NOUN
ejpam-6098	65	6	)	)	PUNCT
ejpam-6098	65	7	||cu1	||cu1	PROPN
ejpam-6098	65	8	,	,	PUNCT
ejpam-6098	65	9	.	.	PUNCT
ejpam-6098	65	10	.	.	PUNCT
ejpam-6098	66	1	.	.	PUNCT
ejpam-6098	67	1	,	,	PUNCT
ejpam-6098	67	2	un||β	un||β	PROPN
ejpam-6098	67	3	=	=	PUNCT
ejpam-6098	67	4	|c|β||u1	|c|β||u1	PROPN
ejpam-6098	67	5	,	,	PUNCT
ejpam-6098	67	6	.	.	PUNCT
ejpam-6098	67	7	.	.	PUNCT
ejpam-6098	68	1	.	.	PUNCT
ejpam-6098	69	1	,	,	PUNCT
ejpam-6098	69	2	un||β	un||β	PROPN
ejpam-6098	69	3	for	for	ADP
ejpam-6098	69	4	any	any	DET
ejpam-6098	69	5	scalar	scalar	NOUN
ejpam-6098	69	6	c	c	NOUN
ejpam-6098	69	7	∈	∈	PROPN
ejpam-6098	69	8	r	r	NOUN
ejpam-6098	69	9	,	,	PUNCT
ejpam-6098	69	10	(	(	PUNCT
ejpam-6098	69	11	iv	iv	X
ejpam-6098	69	12	)	)	PUNCT
ejpam-6098	69	13	||u1	||u1	NOUN
ejpam-6098	69	14	,	,	PUNCT
ejpam-6098	69	15	.	.	PUNCT
ejpam-6098	69	16	.	.	PUNCT
ejpam-6098	70	1	.	.	PUNCT
ejpam-6098	71	1	,	,	PUNCT
ejpam-6098	71	2	un−1	un−1	PROPN
ejpam-6098	71	3	,	,	PUNCT
ejpam-6098	71	4	un	un	PROPN
ejpam-6098	71	5	+	+	PROPN
ejpam-6098	71	6	un+1||β	un+1||β	PROPN
ejpam-6098	71	7	≤	≤	NOUN
ejpam-6098	71	8	||u1	||u1	NOUN
ejpam-6098	71	9	,	,	PUNCT
ejpam-6098	71	10	.	.	PUNCT
ejpam-6098	71	11	.	.	PUNCT
ejpam-6098	72	1	.	.	PUNCT
ejpam-6098	73	1	,	,	PUNCT
ejpam-6098	73	2	un−1	un−1	PROPN
ejpam-6098	73	3	,	,	PUNCT
ejpam-6098	73	4	un||β	un||β	PROPN
ejpam-6098	73	5	+	+	CCONJ
ejpam-6098	73	6	||u1	||u1	NOUN
ejpam-6098	73	7	,	,	PUNCT
ejpam-6098	73	8	.	.	PUNCT
ejpam-6098	73	9	.	.	PUNCT
ejpam-6098	74	1	.	.	PUNCT
ejpam-6098	75	1	,	,	PUNCT
ejpam-6098	75	2	un−1	un−1	PROPN
ejpam-6098	75	3	,	,	PUNCT
ejpam-6098	75	4	un+1||β	un+1||β	PROPN
ejpam-6098	75	5	for	for	ADP
ejpam-6098	75	6	all	all	DET
ejpam-6098	75	7	u1	u1	NOUN
ejpam-6098	75	8	,	,	PUNCT
ejpam-6098	75	9	.	.	PUNCT
ejpam-6098	75	10	.	.	PUNCT
ejpam-6098	75	11	.	.	PUNCT
ejpam-6098	76	1	,	,	PUNCT
ejpam-6098	76	2	un+1	un+1	PROPN
ejpam-6098	76	3	∈	∈	PROPN
ejpam-6098	76	4	u	u	NOUN
ejpam-6098	76	5	,	,	PUNCT
ejpam-6098	76	6	and	and	CCONJ
ejpam-6098	76	7	for	for	ADP
ejpam-6098	76	8	0	0	NUM
ejpam-6098	76	9	<	<	X
ejpam-6098	76	10	β	β	X
ejpam-6098	76	11	≤	≤	NUM
ejpam-6098	76	12	1	1	NUM
ejpam-6098	76	13	.	.	PUNCT
ejpam-6098	77	1	the	the	DET
ejpam-6098	77	2	mapping	mapping	NOUN
ejpam-6098	77	3	||	||	PROPN
ejpam-6098	77	4	·	·	PUNCT
ejpam-6098	77	5	,	,	PUNCT
ejpam-6098	77	6	.	.	PUNCT
ejpam-6098	77	7	.	.	PUNCT
ejpam-6098	77	8	.	.	PUNCT
ejpam-6098	78	1	,	,	PUNCT
ejpam-6098	78	2	·	·	PUNCT
ejpam-6098	78	3	||β	||β	NOUN
ejpam-6098	78	4	is	be	AUX
ejpam-6098	78	5	referred	refer	VERB
ejpam-6098	78	6	to	to	ADP
ejpam-6098	78	7	as	as	ADP
ejpam-6098	78	8	the	the	DET
ejpam-6098	78	9	(	(	PUNCT
ejpam-6098	78	10	n	n	CCONJ
ejpam-6098	78	11	,	,	PUNCT
ejpam-6098	78	12	β)-norm	β)-norm	PUNCT
ejpam-6098	78	13	,	,	PUNCT
ejpam-6098	78	14	and	and	CCONJ
ejpam-6098	78	15	the	the	DET
ejpam-6098	78	16	pair	pair	NOUN
ejpam-6098	78	17	(	(	PUNCT
ejpam-6098	78	18	u	u	NOUN
ejpam-6098	78	19	,	,	PUNCT
ejpam-6098	78	20	||	||	PROPN
ejpam-6098	78	21	·	·	PUNCT
ejpam-6098	78	22	,	,	PUNCT
ejpam-6098	78	23	.	.	PUNCT
ejpam-6098	78	24	.	.	PUNCT
ejpam-6098	79	1	.	.	PUNCT
ejpam-6098	80	1	,	,	PUNCT
ejpam-6098	80	2	·	·	PUNCT
ejpam-6098	80	3	||β	||β	NOUN
ejpam-6098	80	4	)	)	PUNCT
ejpam-6098	80	5	is	be	AUX
ejpam-6098	80	6	called	call	VERB
ejpam-6098	80	7	an	an	PRON
ejpam-6098	80	8	(	(	PUNCT
ejpam-6098	80	9	n	n	NUM
ejpam-6098	80	10	,	,	PUNCT
ejpam-6098	80	11	β)-normed	β)-normed	PUNCT
ejpam-6098	80	12	space	space	NOUN
ejpam-6098	80	13	(	(	PUNCT
ejpam-6098	80	14	ns	ns	NUM
ejpam-6098	80	15	)	)	PUNCT
ejpam-6098	80	16	.	.	PUNCT
ejpam-6098	80	17	”	"	PUNCT
ejpam-6098	81	1	lemma	lemma	PROPN
ejpam-6098	81	2	1	1	NUM
ejpam-6098	81	3	.	.	PUNCT
ejpam-6098	82	1	[	[	X
ejpam-6098	82	2	20	20	NUM
ejpam-6098	82	3	]	]	PUNCT
ejpam-6098	82	4	“	"	PUNCT
ejpam-6098	82	5	suppose	suppose	VERB
ejpam-6098	82	6	(	(	PUNCT
ejpam-6098	82	7	u	u	NOUN
ejpam-6098	82	8	,	,	PUNCT
ejpam-6098	82	9	||	||	PROPN
ejpam-6098	82	10	.	.	PROPN
ejpam-6098	82	11	,	,	PUNCT
ejpam-6098	82	12	...	...	PUNCT
ejpam-6098	82	13	,	,	PUNCT
ejpam-6098	82	14	.||β	.||β	PRON
ejpam-6098	82	15	)	)	PUNCT
ejpam-6098	82	16	is	be	AUX
ejpam-6098	82	17	an	an	DET
ejpam-6098	82	18	(	(	PUNCT
ejpam-6098	82	19	n	n	CCONJ
ejpam-6098	82	20	,	,	PUNCT
ejpam-6098	82	21	β)-ns	β)-ns	NOUN
ejpam-6098	82	22	,	,	PUNCT
ejpam-6098	82	23	n	n	PRON
ejpam-6098	82	24	≥	≥	NOUN
ejpam-6098	82	25	2	2	NUM
ejpam-6098	82	26	,	,	PUNCT
ejpam-6098	82	27	0	0	PUNCT
ejpam-6098	82	28	<	<	X
ejpam-6098	82	29	β	β	X
ejpam-6098	82	30	≤	≤	NUM
ejpam-6098	82	31	1	1	NUM
ejpam-6098	82	32	.	.	PUNCT
ejpam-6098	83	1	if	if	SCONJ
ejpam-6098	83	2	v	v	NUM
ejpam-6098	83	3	∈	∈	PROPN
ejpam-6098	83	4	u	u	NOUN
ejpam-6098	83	5	and	and	CCONJ
ejpam-6098	83	6	||v	||v	NOUN
ejpam-6098	83	7	,	,	PUNCT
ejpam-6098	83	8	u1	u1	NOUN
ejpam-6098	83	9	,	,	PUNCT
ejpam-6098	83	10	...	...	PUNCT
ejpam-6098	83	11	,	,	PUNCT
ejpam-6098	83	12	un−1||β	un−1||β	PROPN
ejpam-6098	83	13	=	=	SYM
ejpam-6098	83	14	0	0	NUM
ejpam-6098	83	15	for	for	ADP
ejpam-6098	83	16	all	all	DET
ejpam-6098	83	17	linearly	linearly	ADV
ejpam-6098	83	18	independent	independent	ADJ
ejpam-6098	83	19	vectors	vector	NOUN
ejpam-6098	83	20	u1	u1	NOUN
ejpam-6098	83	21	,	,	PUNCT
ejpam-6098	83	22	...	...	PUNCT
ejpam-6098	83	23	,	,	PUNCT
ejpam-6098	83	24	un−1	un−1	PROPN
ejpam-6098	83	25	∈	∈	PROPN
ejpam-6098	83	26	u	u	NOUN
ejpam-6098	83	27	,	,	PUNCT
ejpam-6098	83	28	then	then	ADV
ejpam-6098	83	29	v	v	ADP
ejpam-6098	83	30	=	=	SYM
ejpam-6098	83	31	0	0	PROPN
ejpam-6098	83	32	.	.	PUNCT
ejpam-6098	83	33	”	"	PUNCT
ejpam-6098	84	1	definition	definition	NOUN
ejpam-6098	84	2	2	2	NUM
ejpam-6098	84	3	.	.	PUNCT
ejpam-6098	85	1	[	[	X
ejpam-6098	85	2	20	20	NUM
ejpam-6098	85	3	]	]	PUNCT
ejpam-6098	85	4	“	"	PUNCT
ejpam-6098	85	5	a	a	DET
ejpam-6098	85	6	sequence	sequence	NOUN
ejpam-6098	85	7	{	{	PUNCT
ejpam-6098	85	8	vm	vm	NOUN
ejpam-6098	85	9	}	}	PUNCT
ejpam-6098	85	10	in	in	ADP
ejpam-6098	85	11	a	a	DET
ejpam-6098	85	12	(	(	PUNCT
ejpam-6098	85	13	n	n	CCONJ
ejpam-6098	85	14	,	,	PUNCT
ejpam-6098	85	15	β)-ns	β)-ns	SCONJ
ejpam-6098	85	16	u	u	NOUN
ejpam-6098	85	17	is	be	AUX
ejpam-6098	85	18	called	call	VERB
ejpam-6098	85	19	convergent	convergent	ADJ
ejpam-6098	85	20	sequence	sequence	NOUN
ejpam-6098	85	21	if	if	SCONJ
ejpam-6098	85	22	limm	limm	NOUN
ejpam-6098	85	23	→∞	→∞	X
ejpam-6098	85	24	||vm	||vm	PROPN
ejpam-6098	85	25	−	−	PROPN
ejpam-6098	85	26	v	v	PROPN
ejpam-6098	85	27	,	,	PUNCT
ejpam-6098	85	28	u1	u1	NOUN
ejpam-6098	85	29	,	,	PUNCT
ejpam-6098	85	30	...	...	PUNCT
ejpam-6098	85	31	,	,	PUNCT
ejpam-6098	85	32	un−1||β	un−1||β	PROPN
ejpam-6098	85	33	=	=	SYM
ejpam-6098	85	34	0	0	PUNCT
ejpam-6098	86	1	and	and	CCONJ
ejpam-6098	86	2	it	it	PRON
ejpam-6098	86	3	is	be	AUX
ejpam-6098	86	4	called	call	VERB
ejpam-6098	86	5	the	the	DET
ejpam-6098	86	6	cauchy	cauchy	ADJ
ejpam-6098	86	7	sequence	sequence	NOUN
ejpam-6098	86	8	if	if	SCONJ
ejpam-6098	86	9	limm	limm	NOUN
ejpam-6098	86	10	,	,	PUNCT
ejpam-6098	86	11	k	k	PROPN
ejpam-6098	86	12	→∞	→∞	PROPN
ejpam-6098	86	13	||vm	||vm	PROPN
ejpam-6098	86	14	−	−	PROPN
ejpam-6098	86	15	vk	vk	PROPN
ejpam-6098	86	16	,	,	PUNCT
ejpam-6098	86	17	u1	u1	NOUN
ejpam-6098	86	18	,	,	PUNCT
ejpam-6098	86	19	...	...	PUNCT
ejpam-6098	86	20	,	,	PUNCT
ejpam-6098	86	21	un−1||β	un−1||β	PROPN
ejpam-6098	86	22	=	=	SYM
ejpam-6098	86	23	0	0	NUM
ejpam-6098	86	24	,	,	PUNCT
ejpam-6098	86	25	for	for	ADP
ejpam-6098	86	26	all	all	DET
ejpam-6098	86	27	u1	u1	NOUN
ejpam-6098	86	28	,	,	PUNCT
ejpam-6098	86	29	...	...	PUNCT
ejpam-6098	86	30	,	,	PUNCT
ejpam-6098	86	31	un−1	un−1	PROPN
ejpam-6098	86	32	∈	∈	PROPN
ejpam-6098	86	33	u	u	NOUN
ejpam-6098	86	34	.	.	PUNCT
ejpam-6098	87	1	if	if	SCONJ
ejpam-6098	87	2	every	every	DET
ejpam-6098	87	3	cauchy	cauchy	ADJ
ejpam-6098	87	4	sequence	sequence	NOUN
ejpam-6098	87	5	converges	converge	VERB
ejpam-6098	87	6	in	in	ADP
ejpam-6098	87	7	linear	linear	PROPN
ejpam-6098	87	8	(	(	PUNCT
ejpam-6098	87	9	n	n	CCONJ
ejpam-6098	87	10	,	,	PUNCT
ejpam-6098	87	11	β)-ns	β)-ns	NOUN
ejpam-6098	87	12	,	,	PUNCT
ejpam-6098	87	13	it	it	PRON
ejpam-6098	87	14	is	be	AUX
ejpam-6098	87	15	called	call	VERB
ejpam-6098	87	16	complete	complete	ADJ
ejpam-6098	87	17	(	(	PUNCT
ejpam-6098	87	18	n	n	CCONJ
ejpam-6098	87	19	,	,	PUNCT
ejpam-6098	87	20	β)-ns	β)-ns	NOUN
ejpam-6098	87	21	.	.	PUNCT
ejpam-6098	87	22	”	"	PUNCT
ejpam-6098	88	1	definition	definition	NOUN
ejpam-6098	88	2	3	3	NUM
ejpam-6098	88	3	.	.	PUNCT
ejpam-6098	89	1	[	[	X
ejpam-6098	89	2	20	20	NUM
ejpam-6098	89	3	]	]	PUNCT
ejpam-6098	89	4	”	"	PUNCT
ejpam-6098	89	5	let	let	VERB
ejpam-6098	89	6	v	v	NOUN
ejpam-6098	89	7	(	(	PUNCT
ejpam-6098	89	8	r	r	NOUN
ejpam-6098	89	9	)	)	PUNCT
ejpam-6098	89	10	be	be	AUX
ejpam-6098	89	11	a	a	DET
ejpam-6098	89	12	vector	vector	NOUN
ejpam-6098	89	13	space	space	NOUN
ejpam-6098	89	14	with	with	ADP
ejpam-6098	89	15	dim(v	dim(v	PROPN
ejpam-6098	89	16	)	)	PUNCT
ejpam-6098	89	17	≥	≥	PROPN
ejpam-6098	89	18	n	n	CCONJ
ejpam-6098	89	19	,	,	PUNCT
ejpam-6098	89	20	equipped	equip	VERB
ejpam-6098	89	21	with	with	ADP
ejpam-6098	89	22	a	a	DET
ejpam-6098	89	23	nonarchimedean	nonarchimedean	ADJ
ejpam-6098	89	24	(	(	PUNCT
ejpam-6098	89	25	na	na	NOUN
ejpam-6098	89	26	)	)	PUNCT
ejpam-6098	89	27	nonzero	nonzero	NOUN
ejpam-6098	89	28	valuation	valuation	NOUN
ejpam-6098	89	29	|	|	ADV
ejpam-6098	89	30	·	·	PUNCT
ejpam-6098	90	1	|	|	ADV
ejpam-6098	90	2	,	,	PUNCT
ejpam-6098	90	3	and	and	CCONJ
ejpam-6098	90	4	let	let	VERB
ejpam-6098	90	5	β	β	NOUN
ejpam-6098	90	6	be	be	AUX
ejpam-6098	90	7	a	a	DET
ejpam-6098	90	8	constant	constant	ADJ
ejpam-6098	90	9	such	such	ADJ
ejpam-6098	90	10	that	that	SCONJ
ejpam-6098	90	11	0	0	NUM
ejpam-6098	90	12	<	<	X
ejpam-6098	90	13	β	β	X
ejpam-6098	90	14	≤	≤	NUM
ejpam-6098	90	15	1	1	NUM
ejpam-6098	90	16	.	.	PUNCT
ejpam-6098	91	1	a	a	DET
ejpam-6098	91	2	mapping	mapping	NOUN
ejpam-6098	91	3	||	||	NOUN
ejpam-6098	91	4	·	·	PUNCT
ejpam-6098	91	5	,	,	PUNCT
ejpam-6098	91	6	.	.	PUNCT
ejpam-6098	91	7	.	.	PUNCT
ejpam-6098	91	8	.	.	PUNCT
ejpam-6098	92	1	,	,	PUNCT
ejpam-6098	92	2	·	·	PUNCT
ejpam-6098	92	3	||β	||β	NOUN
ejpam-6098	92	4	:	:	PUNCT
ejpam-6098	92	5	v	v	NUM
ejpam-6098	92	6	n	n	NOUN
ejpam-6098	92	7	→	→	SYM
ejpam-6098	92	8	r	r	NOUN
ejpam-6098	92	9	is	be	AUX
ejpam-6098	92	10	referred	refer	VERB
ejpam-6098	92	11	to	to	ADP
ejpam-6098	92	12	as	as	ADP
ejpam-6098	92	13	a	a	DET
ejpam-6098	92	14	na-(n	na-(n	NUM
ejpam-6098	92	15	,	,	PUNCT
ejpam-6098	92	16	β)-norm	β)-norm	PUNCT
ejpam-6098	92	17	on	on	ADP
ejpam-6098	92	18	v	v	PRON
ejpam-6098	92	19	if	if	SCONJ
ejpam-6098	92	20	it	it	PRON
ejpam-6098	92	21	satisfies	satisfy	VERB
ejpam-6098	92	22	the	the	DET
ejpam-6098	92	23	following	follow	VERB
ejpam-6098	92	24	conditions	condition	NOUN
ejpam-6098	92	25	:	:	PUNCT
ejpam-6098	92	26	(	(	PUNCT
ejpam-6098	92	27	i	i	NOUN
ejpam-6098	92	28	)	)	PUNCT
ejpam-6098	92	29	||u1	||u1	PROPN
ejpam-6098	92	30	,	,	PUNCT
ejpam-6098	92	31	.	.	PUNCT
ejpam-6098	92	32	.	.	PUNCT
ejpam-6098	93	1	.	.	PUNCT
ejpam-6098	94	1	,	,	PUNCT
ejpam-6098	94	2	un||β	un||β	PROPN
ejpam-6098	95	1	=	=	SYM
ejpam-6098	95	2	0	0	PUNCT
ejpam-6098	96	1	if	if	SCONJ
ejpam-6098	96	2	and	and	CCONJ
ejpam-6098	96	3	only	only	ADV
ejpam-6098	96	4	if	if	SCONJ
ejpam-6098	96	5	u1	u1	NOUN
ejpam-6098	96	6	,	,	PUNCT
ejpam-6098	96	7	.	.	PUNCT
ejpam-6098	96	8	.	.	PUNCT
ejpam-6098	96	9	.	.	PUNCT
ejpam-6098	97	1	,	,	PUNCT
ejpam-6098	97	2	un	un	PROPN
ejpam-6098	97	3	are	be	AUX
ejpam-6098	97	4	linearly	linearly	ADV
ejpam-6098	97	5	dependent	dependent	ADJ
ejpam-6098	97	6	,	,	PUNCT
ejpam-6098	97	7	j.	j.	PROPN
ejpam-6098	97	8	jakhar	jakhar	PROPN
ejpam-6098	97	9	et	et	PROPN
ejpam-6098	97	10	al	al	PROPN
ejpam-6098	97	11	.	.	PUNCT
ejpam-6098	97	12	/	/	SYM
ejpam-6098	97	13	eur	eur	PROPN
ejpam-6098	97	14	.	.	PUNCT
ejpam-6098	98	1	j.	j.	PROPN
ejpam-6098	98	2	pure	pure	PROPN
ejpam-6098	98	3	appl	appl	PROPN
ejpam-6098	98	4	.	.	PROPN
ejpam-6098	98	5	math	math	PROPN
ejpam-6098	98	6	,	,	PUNCT
ejpam-6098	98	7	18	18	NUM
ejpam-6098	98	8	(	(	PUNCT
ejpam-6098	98	9	2	2	NUM
ejpam-6098	98	10	)	)	PUNCT
ejpam-6098	98	11	(	(	PUNCT
ejpam-6098	98	12	2025	2025	NUM
ejpam-6098	98	13	)	)	PUNCT
ejpam-6098	98	14	,	,	PUNCT
ejpam-6098	98	15	6098	6098	NUM
ejpam-6098	98	16	4	4	NUM
ejpam-6098	98	17	of	of	ADP
ejpam-6098	98	18	23	23	NUM
ejpam-6098	98	19	(	(	PUNCT
ejpam-6098	98	20	ii	ii	NOUN
ejpam-6098	98	21	)	)	PUNCT
ejpam-6098	98	22	||u1	||u1	PROPN
ejpam-6098	98	23	,	,	PUNCT
ejpam-6098	98	24	.	.	PUNCT
ejpam-6098	98	25	.	.	PUNCT
ejpam-6098	99	1	.	.	PUNCT
ejpam-6098	100	1	,	,	PUNCT
ejpam-6098	100	2	un||β	un||β	PROPN
ejpam-6098	100	3	is	be	AUX
ejpam-6098	100	4	invariant	invariant	ADJ
ejpam-6098	100	5	under	under	ADP
ejpam-6098	100	6	any	any	DET
ejpam-6098	100	7	permutation	permutation	NOUN
ejpam-6098	100	8	of	of	ADP
ejpam-6098	100	9	the	the	DET
ejpam-6098	100	10	elements	element	NOUN
ejpam-6098	100	11	u1	u1	NOUN
ejpam-6098	100	12	,	,	PUNCT
ejpam-6098	100	13	.	.	PUNCT
ejpam-6098	100	14	.	.	PUNCT
ejpam-6098	101	1	.	.	PUNCT
ejpam-6098	102	1	,	,	PUNCT
ejpam-6098	102	2	un	un	PROPN
ejpam-6098	102	3	,	,	PUNCT
ejpam-6098	102	4	(	(	PUNCT
ejpam-6098	102	5	iii	iii	NOUN
ejpam-6098	102	6	)	)	PUNCT
ejpam-6098	102	7	||cu1	||cu1	PROPN
ejpam-6098	102	8	,	,	PUNCT
ejpam-6098	102	9	.	.	PUNCT
ejpam-6098	102	10	.	.	PUNCT
ejpam-6098	103	1	.	.	PUNCT
ejpam-6098	104	1	,	,	PUNCT
ejpam-6098	104	2	un||β	un||β	PROPN
ejpam-6098	104	3	=	=	PUNCT
ejpam-6098	104	4	|c|β||u1	|c|β||u1	PROPN
ejpam-6098	104	5	,	,	PUNCT
ejpam-6098	104	6	.	.	PUNCT
ejpam-6098	104	7	.	.	PUNCT
ejpam-6098	105	1	.	.	PUNCT
ejpam-6098	106	1	,	,	PUNCT
ejpam-6098	106	2	un||β	un||β	PROPN
ejpam-6098	106	3	for	for	ADP
ejpam-6098	106	4	all	all	DET
ejpam-6098	106	5	c	c	NOUN
ejpam-6098	106	6	∈	∈	PROPN
ejpam-6098	106	7	r	r	NOUN
ejpam-6098	106	8	,	,	PUNCT
ejpam-6098	106	9	(	(	PUNCT
ejpam-6098	106	10	iv	iv	X
ejpam-6098	106	11	)	)	PUNCT
ejpam-6098	106	12	||v1	||v1	PROPN
ejpam-6098	106	13	+	+	CCONJ
ejpam-6098	106	14	v2	v2	NOUN
ejpam-6098	106	15	,	,	PUNCT
ejpam-6098	106	16	u1	u1	NOUN
ejpam-6098	106	17	,	,	PUNCT
ejpam-6098	106	18	.	.	PUNCT
ejpam-6098	106	19	.	.	PUNCT
ejpam-6098	107	1	.	.	PUNCT
ejpam-6098	108	1	,	,	PUNCT
ejpam-6098	108	2	un−1||β	un−1||β	PROPN
ejpam-6098	108	3	≤	≤	NUM
ejpam-6098	108	4	max{||v1	max{||v1	PROPN
ejpam-6098	108	5	,	,	PUNCT
ejpam-6098	108	6	u1	u1	NOUN
ejpam-6098	108	7	,	,	PUNCT
ejpam-6098	108	8	.	.	PUNCT
ejpam-6098	108	9	.	.	PUNCT
ejpam-6098	108	10	.	.	PUNCT
ejpam-6098	109	1	,	,	PUNCT
ejpam-6098	109	2	un−1||β	un−1||β	PROPN
ejpam-6098	109	3	,	,	PUNCT
ejpam-6098	109	4	||v2	||v2	PROPN
ejpam-6098	109	5	,	,	PUNCT
ejpam-6098	109	6	u1	u1	NOUN
ejpam-6098	109	7	,	,	PUNCT
ejpam-6098	109	8	.	.	PUNCT
ejpam-6098	109	9	.	.	PUNCT
ejpam-6098	109	10	.	.	PUNCT
ejpam-6098	110	1	,	,	PUNCT
ejpam-6098	110	2	un−1||β	un−1||β	NOUN
ejpam-6098	110	3	}	}	PUNCT
ejpam-6098	110	4	for	for	ADP
ejpam-6098	110	5	all	all	DET
ejpam-6098	110	6	v1	v1	NOUN
ejpam-6098	110	7	,	,	PUNCT
ejpam-6098	110	8	v2	v2	NOUN
ejpam-6098	110	9	,	,	PUNCT
ejpam-6098	110	10	u1	u1	NOUN
ejpam-6098	110	11	,	,	PUNCT
ejpam-6098	110	12	.	.	PUNCT
ejpam-6098	110	13	.	.	PUNCT
ejpam-6098	110	14	.	.	PUNCT
ejpam-6098	111	1	,	,	PUNCT
ejpam-6098	111	2	un	un	PROPN
ejpam-6098	111	3	∈	∈	PROPN
ejpam-6098	111	4	v	v	PROPN
ejpam-6098	111	5	.	.	PUNCT
ejpam-6098	112	1	the	the	DET
ejpam-6098	112	2	pair	pair	NOUN
ejpam-6098	112	3	(	(	PUNCT
ejpam-6098	112	4	v	v	NOUN
ejpam-6098	112	5	,	,	PUNCT
ejpam-6098	112	6	||	||	PROPN
ejpam-6098	112	7	·	·	PUNCT
ejpam-6098	112	8	,	,	PUNCT
ejpam-6098	112	9	.	.	PUNCT
ejpam-6098	112	10	.	.	PUNCT
ejpam-6098	112	11	.	.	PUNCT
ejpam-6098	113	1	,	,	PUNCT
ejpam-6098	113	2	·	·	PUNCT
ejpam-6098	113	3	||β	||β	NOUN
ejpam-6098	113	4	)	)	PUNCT
ejpam-6098	113	5	is	be	AUX
ejpam-6098	113	6	called	call	VERB
ejpam-6098	113	7	a	a	DET
ejpam-6098	113	8	na-(n	na-(n	PROPN
ejpam-6098	113	9	,	,	PUNCT
ejpam-6098	113	10	β)-normed	β)-normed	PUNCT
ejpam-6098	113	11	space	space	NOUN
ejpam-6098	113	12	(	(	PUNCT
ejpam-6098	113	13	ns	ns	NUM
ejpam-6098	113	14	)	)	PUNCT
ejpam-6098	113	15	.	.	PUNCT
ejpam-6098	113	16	”	"	PUNCT
ejpam-6098	114	1	definition	definition	NOUN
ejpam-6098	114	2	4	4	NUM
ejpam-6098	114	3	.	.	PUNCT
ejpam-6098	115	1	[	[	X
ejpam-6098	115	2	28	28	NUM
ejpam-6098	115	3	]	]	PUNCT
ejpam-6098	115	4	”	"	PUNCT
ejpam-6098	115	5	a	a	DET
ejpam-6098	115	6	function	function	NOUN
ejpam-6098	115	7	t	t	NOUN
ejpam-6098	115	8	:	:	PUNCT
ejpam-6098	116	1	[	[	X
ejpam-6098	116	2	0	0	NUM
ejpam-6098	116	3	,	,	PUNCT
ejpam-6098	116	4	1	1	NUM
ejpam-6098	116	5	]	]	SYM
ejpam-6098	116	6	×	×	NOUN
ejpam-6098	116	7	[	[	X
ejpam-6098	116	8	0	0	NUM
ejpam-6098	116	9	,	,	PUNCT
ejpam-6098	116	10	1	1	NUM
ejpam-6098	116	11	]	]	PUNCT
ejpam-6098	116	12	→	→	PUNCT
ejpam-6098	116	13	[	[	X
ejpam-6098	116	14	0	0	NUM
ejpam-6098	116	15	,	,	PUNCT
ejpam-6098	116	16	1	1	NUM
ejpam-6098	116	17	]	]	PUNCT
ejpam-6098	116	18	is	be	AUX
ejpam-6098	116	19	called	call	VERB
ejpam-6098	116	20	a	a	DET
ejpam-6098	116	21	continuous	continuous	ADJ
ejpam-6098	116	22	t	t	NOUN
ejpam-6098	116	23	-	-	PUNCT
ejpam-6098	116	24	norm	norm	NOUN
ejpam-6098	116	25	if	if	SCONJ
ejpam-6098	116	26	it	it	PRON
ejpam-6098	116	27	satisfies	satisfy	VERB
ejpam-6098	116	28	the	the	DET
ejpam-6098	116	29	following	follow	VERB
ejpam-6098	116	30	properties	property	NOUN
ejpam-6098	116	31	:	:	PUNCT
ejpam-6098	116	32	(	(	PUNCT
ejpam-6098	116	33	i	i	NOUN
ejpam-6098	116	34	)	)	PUNCT
ejpam-6098	116	35	t	t	PROPN
ejpam-6098	116	36	is	be	AUX
ejpam-6098	116	37	commutative	commutative	ADJ
ejpam-6098	116	38	,	,	PUNCT
ejpam-6098	116	39	associative	associative	ADJ
ejpam-6098	116	40	,	,	PUNCT
ejpam-6098	116	41	and	and	CCONJ
ejpam-6098	116	42	continuous	continuous	ADJ
ejpam-6098	116	43	.	.	PUNCT
ejpam-6098	117	1	(	(	PUNCT
ejpam-6098	117	2	ii	ii	PROPN
ejpam-6098	117	3	)	)	PUNCT
ejpam-6098	117	4	t	t	PROPN
ejpam-6098	117	5	(	(	PUNCT
ejpam-6098	117	6	α1	α1	PROPN
ejpam-6098	117	7	,	,	PUNCT
ejpam-6098	117	8	1	1	NUM
ejpam-6098	117	9	)	)	PUNCT
ejpam-6098	117	10	=	=	SYM
ejpam-6098	117	11	α1	α1	NOUN
ejpam-6098	117	12	∀	∀	X
ejpam-6098	117	13	α1	α1	PROPN
ejpam-6098	117	14	∈	∈	PROPN
ejpam-6098	117	15	[	[	X
ejpam-6098	117	16	0	0	NUM
ejpam-6098	117	17	,	,	PUNCT
ejpam-6098	117	18	1	1	NUM
ejpam-6098	117	19	]	]	PUNCT
ejpam-6098	117	20	.	.	PUNCT
ejpam-6098	118	1	(	(	PUNCT
ejpam-6098	118	2	iii	iii	X
ejpam-6098	118	3	)	)	PUNCT
ejpam-6098	118	4	t	t	PROPN
ejpam-6098	118	5	(	(	PUNCT
ejpam-6098	118	6	α1	α1	PROPN
ejpam-6098	118	7	,	,	PUNCT
ejpam-6098	118	8	α2	α2	ADJ
ejpam-6098	118	9	)	)	PUNCT
ejpam-6098	118	10	≤	≤	NOUN
ejpam-6098	118	11	t	t	PROPN
ejpam-6098	118	12	(	(	PUNCT
ejpam-6098	118	13	α3	α3	PROPN
ejpam-6098	118	14	,	,	PUNCT
ejpam-6098	118	15	α4	α4	NOUN
ejpam-6098	118	16	)	)	PUNCT
ejpam-6098	118	17	whenever	whenever	SCONJ
ejpam-6098	118	18	α1	α1	PROPN
ejpam-6098	118	19	≤	≤	ADV
ejpam-6098	118	20	α3	α3	NOUN
ejpam-6098	118	21	and	and	CCONJ
ejpam-6098	118	22	α2	α2	ADJ
ejpam-6098	118	23	≤	≤	NUM
ejpam-6098	118	24	α4	α4	NOUN
ejpam-6098	118	25	,	,	PUNCT
ejpam-6098	118	26	for	for	ADP
ejpam-6098	118	27	all	all	DET
ejpam-6098	118	28	α1	α1	PROPN
ejpam-6098	118	29	,	,	PUNCT
ejpam-6098	118	30	α2	α2	ADJ
ejpam-6098	118	31	,	,	PUNCT
ejpam-6098	118	32	α3	α3	PROPN
ejpam-6098	118	33	,	,	PUNCT
ejpam-6098	118	34	α4	α4	NOUN
ejpam-6098	118	35	∈	∈	PROPN
ejpam-6098	119	1	[	[	X
ejpam-6098	119	2	0	0	NUM
ejpam-6098	119	3	,	,	PUNCT
ejpam-6098	119	4	1	1	NUM
ejpam-6098	119	5	]	]	PUNCT
ejpam-6098	119	6	.	.	PUNCT
ejpam-6098	120	1	remark	remark	PROPN
ejpam-6098	120	2	1	1	NUM
ejpam-6098	120	3	.	.	PUNCT
ejpam-6098	121	1	let	let	VERB
ejpam-6098	121	2	t	t	PROPN
ejpam-6098	121	3	be	be	AUX
ejpam-6098	121	4	a	a	DET
ejpam-6098	121	5	t	t	NOUN
ejpam-6098	121	6	-	-	PUNCT
ejpam-6098	121	7	norm	norm	NOUN
ejpam-6098	121	8	and	and	CCONJ
ejpam-6098	121	9	{	{	PUNCT
ejpam-6098	121	10	um	um	INTJ
ejpam-6098	121	11	}	}	PUNCT
ejpam-6098	121	12	be	be	AUX
ejpam-6098	121	13	a	a	DET
ejpam-6098	121	14	sequence	sequence	NOUN
ejpam-6098	121	15	then	then	ADV
ejpam-6098	121	16	tm	tm	PROPN
ejpam-6098	121	17	i=1ui	i=1ui	PROPN
ejpam-6098	121	18	is	be	AUX
ejpam-6098	121	19	defined	define	VERB
ejpam-6098	121	20	as	as	ADP
ejpam-6098	121	21	tm	tm	PROPN
ejpam-6098	121	22	i=1ui	i=1ui	X
ejpam-6098	122	1	=	=	NOUN
ejpam-6098	122	2	{	{	PUNCT
ejpam-6098	122	3	u1	u1	NOUN
ejpam-6098	122	4	,	,	PUNCT
ejpam-6098	122	5	if	if	SCONJ
ejpam-6098	122	6	m	m	ADV
ejpam-6098	122	7	=	=	NOUN
ejpam-6098	122	8	1	1	X
ejpam-6098	122	9	.	.	X
ejpam-6098	122	10	t	t	PROPN
ejpam-6098	122	11	(	(	PUNCT
ejpam-6098	122	12	tm−1	tm−1	PROPN
ejpam-6098	122	13	i=1	i=1	PROPN
ejpam-6098	122	14	ui	ui	PROPN
ejpam-6098	122	15	,	,	PUNCT
ejpam-6098	122	16	um	um	INTJ
ejpam-6098	122	17	)	)	PUNCT
ejpam-6098	122	18	,	,	PUNCT
ejpam-6098	122	19	if	if	SCONJ
ejpam-6098	122	20	m	m	PROPN
ejpam-6098	122	21	≥	≥	NOUN
ejpam-6098	122	22	2	2	NUM
ejpam-6098	122	23	.	.	PUNCT
ejpam-6098	122	24	here	here	ADV
ejpam-6098	122	25	,	,	PUNCT
ejpam-6098	122	26	∆+	∆+	PUNCT
ejpam-6098	122	27	denotes	denote	VERB
ejpam-6098	122	28	the	the	DET
ejpam-6098	122	29	set	set	NOUN
ejpam-6098	122	30	of	of	ADP
ejpam-6098	122	31	distribution	distribution	NOUN
ejpam-6098	122	32	functions	function	NOUN
ejpam-6098	122	33	,	,	PUNCT
ejpam-6098	122	34	which	which	PRON
ejpam-6098	122	35	are	be	AUX
ejpam-6098	122	36	functions	function	NOUN
ejpam-6098	122	37	f	f	NOUN
ejpam-6098	122	38	:	:	PUNCT
ejpam-6098	123	1	r	r	X
ejpam-6098	123	2	→	→	SYM
ejpam-6098	123	3	[	[	X
ejpam-6098	123	4	0	0	NUM
ejpam-6098	123	5	,	,	PUNCT
ejpam-6098	123	6	1	1	NUM
ejpam-6098	123	7	]	]	PUNCT
ejpam-6098	123	8	with	with	ADP
ejpam-6098	123	9	the	the	DET
ejpam-6098	123	10	following	follow	VERB
ejpam-6098	123	11	properties	property	NOUN
ejpam-6098	123	12	:	:	PUNCT
ejpam-6098	123	13	they	they	PRON
ejpam-6098	123	14	are	be	AUX
ejpam-6098	123	15	left	leave	VERB
ejpam-6098	123	16	-	-	PUNCT
ejpam-6098	123	17	continuous	continuous	ADJ
ejpam-6098	123	18	,	,	PUNCT
ejpam-6098	123	19	non	non	ADJ
ejpam-6098	123	20	-	-	ADJ
ejpam-6098	123	21	decreasing	decrease	VERB
ejpam-6098	123	22	over	over	ADP
ejpam-6098	123	23	the	the	DET
ejpam-6098	123	24	real	real	ADJ
ejpam-6098	123	25	numbers	number	NOUN
ejpam-6098	123	26	,	,	PUNCT
ejpam-6098	123	27	and	and	CCONJ
ejpam-6098	123	28	satisfy	satisfy	VERB
ejpam-6098	123	29	f	f	PROPN
ejpam-6098	123	30	(	(	PUNCT
ejpam-6098	123	31	0	0	NUM
ejpam-6098	123	32	)	)	PUNCT
ejpam-6098	123	33	=	=	NOUN
ejpam-6098	124	1	0	0	X
ejpam-6098	124	2	.	.	PUNCT
ejpam-6098	125	1	a	a	DET
ejpam-6098	125	2	subset	subset	NOUN
ejpam-6098	125	3	of	of	ADP
ejpam-6098	125	4	∆+	∆+	NOUN
ejpam-6098	125	5	,	,	PUNCT
ejpam-6098	125	6	denoted	denote	VERB
ejpam-6098	125	7	d+	d+	NOUN
ejpam-6098	125	8	,	,	PUNCT
ejpam-6098	125	9	consists	consist	VERB
ejpam-6098	125	10	of	of	ADP
ejpam-6098	125	11	all	all	DET
ejpam-6098	125	12	functions	function	NOUN
ejpam-6098	125	13	f	f	PRON
ejpam-6098	126	1	such	such	ADJ
ejpam-6098	126	2	that	that	PRON
ejpam-6098	126	3	l−f	l−f	NOUN
ejpam-6098	126	4	(	(	PUNCT
ejpam-6098	126	5	+	+	NOUN
ejpam-6098	126	6	∞	∞	NOUN
ejpam-6098	126	7	)	)	PUNCT
ejpam-6098	126	8	=	=	SYM
ejpam-6098	126	9	1	1	NUM
ejpam-6098	126	10	,	,	PUNCT
ejpam-6098	126	11	where	where	SCONJ
ejpam-6098	126	12	l−f	l−f	NOUN
ejpam-6098	126	13	(	(	PUNCT
ejpam-6098	126	14	u	u	NOUN
ejpam-6098	126	15	)	)	PUNCT
ejpam-6098	126	16	=	=	SYM
ejpam-6098	126	17	limt→u−	limt→u−	PROPN
ejpam-6098	126	18	f	f	PROPN
ejpam-6098	126	19	(	(	PUNCT
ejpam-6098	126	20	t	t	PROPN
ejpam-6098	126	21	)	)	PUNCT
ejpam-6098	126	22	.	.	PUNCT
ejpam-6098	127	1	the	the	DET
ejpam-6098	127	2	function	function	NOUN
ejpam-6098	127	3	ϵ0	ϵ0	NOUN
ejpam-6098	127	4	is	be	AUX
ejpam-6098	127	5	a	a	DET
ejpam-6098	127	6	particular	particular	ADJ
ejpam-6098	127	7	distribution	distribution	NOUN
ejpam-6098	127	8	function	function	NOUN
ejpam-6098	127	9	defined	define	VERB
ejpam-6098	127	10	as	as	SCONJ
ejpam-6098	127	11	follows	follow	VERB
ejpam-6098	127	12	:	:	PUNCT
ejpam-6098	127	13	ϵ0(t	ϵ0(t	X
ejpam-6098	127	14	)	)	PUNCT
ejpam-6098	127	15	=	=	NOUN
ejpam-6098	127	16	{	{	PUNCT
ejpam-6098	127	17	1	1	NUM
ejpam-6098	127	18	,	,	PUNCT
ejpam-6098	127	19	if	if	SCONJ
ejpam-6098	127	20	t	t	PROPN
ejpam-6098	127	21	>	>	X
ejpam-6098	127	22	0	0	NUM
ejpam-6098	127	23	0	0	PUNCT
ejpam-6098	128	1	if	if	SCONJ
ejpam-6098	128	2	t	t	PRON
ejpam-6098	128	3	≤	≤	NOUN
ejpam-6098	128	4	0	0	NUM
ejpam-6098	128	5	.	.	PUNCT
ejpam-6098	129	1	definition	definition	NOUN
ejpam-6098	129	2	5	5	NUM
ejpam-6098	129	3	.	.	PUNCT
ejpam-6098	130	1	[	[	X
ejpam-6098	130	2	29	29	NUM
ejpam-6098	130	3	]	]	PUNCT
ejpam-6098	130	4	“	"	PUNCT
ejpam-6098	130	5	a	a	DET
ejpam-6098	130	6	triplet	triplet	NOUN
ejpam-6098	130	7	(	(	PUNCT
ejpam-6098	130	8	z	z	NOUN
ejpam-6098	130	9	,	,	PUNCT
ejpam-6098	130	10	µ	µ	NUM
ejpam-6098	130	11	,	,	PUNCT
ejpam-6098	130	12	t	t	PROPN
ejpam-6098	130	13	)	)	PUNCT
ejpam-6098	130	14	is	be	AUX
ejpam-6098	130	15	called	call	VERB
ejpam-6098	130	16	rns	rn	NOUN
ejpam-6098	130	17	if	if	SCONJ
ejpam-6098	130	18	it	it	PRON
ejpam-6098	130	19	satisfies	satisfy	VERB
ejpam-6098	130	20	the	the	DET
ejpam-6098	130	21	below	below	ADJ
ejpam-6098	130	22	conditions	condition	NOUN
ejpam-6098	130	23	:	:	PUNCT
ejpam-6098	130	24	(	(	PUNCT
ejpam-6098	130	25	i	i	NOUN
ejpam-6098	130	26	)	)	PUNCT
ejpam-6098	130	27	µu(t	µu(t	PUNCT
ejpam-6098	130	28	)	)	PUNCT
ejpam-6098	130	29	=	=	SYM
ejpam-6098	131	1	ϵ0(t	ϵ0(t	NOUN
ejpam-6098	131	2	)	)	PUNCT
ejpam-6098	131	3	∀	∀	PUNCT
ejpam-6098	132	1	t	t	NOUN
ejpam-6098	132	2	>	>	X
ejpam-6098	132	3	0	0	PUNCT
ejpam-6098	133	1	⇐	⇐	ADJ
ejpam-6098	133	2	⇒	⇒	PROPN
ejpam-6098	133	3	u	u	NOUN
ejpam-6098	133	4	=	=	PROPN
ejpam-6098	133	5	0	0	NUM
ejpam-6098	133	6	;	;	PUNCT
ejpam-6098	133	7	(	(	PUNCT
ejpam-6098	133	8	ii	ii	NOUN
ejpam-6098	133	9	)	)	PUNCT
ejpam-6098	133	10	µαu(t	µαu(t	PROPN
ejpam-6098	133	11	)	)	PUNCT
ejpam-6098	133	12	=	=	SYM
ejpam-6098	133	13	µu	µu	PROPN
ejpam-6098	133	14	(	(	PUNCT
ejpam-6098	133	15	t	t	PROPN
ejpam-6098	133	16	|α|	|α|	PROPN
ejpam-6098	133	17	)	)	PUNCT
ejpam-6098	133	18	∀	∀	PUNCT
ejpam-6098	133	19	u	u	NOUN
ejpam-6098	133	20	∈	∈	PROPN
ejpam-6098	133	21	z	z	PROPN
ejpam-6098	133	22	,	,	PUNCT
ejpam-6098	133	23	α	α	PROPN
ejpam-6098	133	24	̸=	̸=	PROPN
ejpam-6098	133	25	0	0	NUM
ejpam-6098	133	26	;	;	PUNCT
ejpam-6098	133	27	(	(	PUNCT
ejpam-6098	133	28	iii	iii	X
ejpam-6098	133	29	)	)	PUNCT
ejpam-6098	133	30	µu1+u2(t+	µu1+u2(t+	PROPN
ejpam-6098	133	31	s	s	PART
ejpam-6098	133	32	)	)	PUNCT
ejpam-6098	133	33	≥	≥	PROPN
ejpam-6098	133	34	t	t	PROPN
ejpam-6098	133	35	(	(	PUNCT
ejpam-6098	133	36	µu1(t	µu1(t	ADJ
ejpam-6098	133	37	)	)	PUNCT
ejpam-6098	133	38	,	,	PUNCT
ejpam-6098	133	39	µu2(s	µu2(s	PROPN
ejpam-6098	133	40	)	)	PUNCT
ejpam-6098	133	41	)	)	PUNCT
ejpam-6098	133	42	∀	∀	NOUN
ejpam-6098	133	43	u1	u1	NOUN
ejpam-6098	133	44	,	,	PUNCT
ejpam-6098	133	45	u2	u2	PROPN
ejpam-6098	133	46	∈	∈	PROPN
ejpam-6098	133	47	z	z	NOUN
ejpam-6098	133	48	and	and	CCONJ
ejpam-6098	133	49	all	all	DET
ejpam-6098	133	50	t	t	PROPN
ejpam-6098	133	51	,	,	PUNCT
ejpam-6098	133	52	s	s	VERB
ejpam-6098	133	53	≥	≥	NOUN
ejpam-6098	133	54	0	0	NUM
ejpam-6098	133	55	.	.	PUNCT
ejpam-6098	133	56	”	"	PUNCT
ejpam-6098	134	1	definition	definition	NOUN
ejpam-6098	134	2	6	6	NUM
ejpam-6098	134	3	.	.	PUNCT
ejpam-6098	135	1	[	[	X
ejpam-6098	135	2	30	30	NUM
ejpam-6098	135	3	]	]	PUNCT
ejpam-6098	135	4	consider	consider	VERB
ejpam-6098	135	5	a	a	DET
ejpam-6098	135	6	rns	rn	NOUN
ejpam-6098	135	7	(	(	PUNCT
ejpam-6098	135	8	z	z	PROPN
ejpam-6098	135	9	,	,	PUNCT
ejpam-6098	135	10	µ	µ	NOUN
ejpam-6098	135	11	,	,	PUNCT
ejpam-6098	135	12	t	t	NOUN
ejpam-6098	135	13	)	)	PUNCT
ejpam-6098	135	14	.	.	PUNCT
ejpam-6098	136	1	we	we	PRON
ejpam-6098	136	2	define	define	VERB
ejpam-6098	136	3	the	the	DET
ejpam-6098	136	4	following	following	NOUN
ejpam-6098	136	5	:	:	PUNCT
ejpam-6098	136	6	(	(	PUNCT
ejpam-6098	136	7	i	i	NOUN
ejpam-6098	136	8	)	)	PUNCT
ejpam-6098	136	9	a	a	DET
ejpam-6098	136	10	sequence	sequence	NOUN
ejpam-6098	136	11	{	{	PUNCT
ejpam-6098	136	12	uα	uα	NOUN
ejpam-6098	136	13	}	}	PUNCT
ejpam-6098	136	14	in	in	ADP
ejpam-6098	136	15	z	z	NOUN
ejpam-6098	136	16	converges	converge	NOUN
ejpam-6098	136	17	to	to	ADP
ejpam-6098	136	18	an	an	DET
ejpam-6098	136	19	element	element	NOUN
ejpam-6098	136	20	u	u	NOUN
ejpam-6098	136	21	∈	∈	PROPN
ejpam-6098	136	22	z	z	NOUN
ejpam-6098	136	23	if	if	SCONJ
ejpam-6098	136	24	,	,	PUNCT
ejpam-6098	136	25	for	for	ADP
ejpam-6098	136	26	any	any	DET
ejpam-6098	136	27	s	s	X
ejpam-6098	136	28	>	>	X
ejpam-6098	136	29	0	0	PUNCT
ejpam-6098	136	30	and	and	CCONJ
ejpam-6098	136	31	λ	λ	X
ejpam-6098	136	32	>	>	X
ejpam-6098	136	33	0	0	NUM
ejpam-6098	136	34	,	,	PUNCT
ejpam-6098	136	35	there	there	PRON
ejpam-6098	136	36	exists	exist	VERB
ejpam-6098	136	37	a	a	DET
ejpam-6098	136	38	positive	positive	ADJ
ejpam-6098	136	39	integer	integer	NOUN
ejpam-6098	136	40	n0	n0	NOUN
ejpam-6098	136	41	such	such	ADJ
ejpam-6098	136	42	that	that	SCONJ
ejpam-6098	136	43	µun−u(s	µun−u(s	PROPN
ejpam-6098	136	44	)	)	PUNCT
ejpam-6098	136	45	>	>	X
ejpam-6098	136	46	1−	1−	NUM
ejpam-6098	136	47	λ	λ	NOUN
ejpam-6098	136	48	for	for	ADP
ejpam-6098	136	49	all	all	DET
ejpam-6098	136	50	n	n	PRON
ejpam-6098	136	51	≥	≥	NOUN
ejpam-6098	136	52	n0	n0	NUM
ejpam-6098	136	53	.	.	PUNCT
ejpam-6098	137	1	(	(	PUNCT
ejpam-6098	137	2	ii	ii	NOUN
ejpam-6098	137	3	)	)	PUNCT
ejpam-6098	137	4	a	a	DET
ejpam-6098	137	5	sequence	sequence	NOUN
ejpam-6098	137	6	{	{	PUNCT
ejpam-6098	137	7	uα	uα	NOUN
ejpam-6098	137	8	}	}	PUNCT
ejpam-6098	137	9	in	in	ADP
ejpam-6098	137	10	z	z	PROPN
ejpam-6098	137	11	is	be	AUX
ejpam-6098	137	12	termed	term	VERB
ejpam-6098	137	13	cauchy	cauchy	ADJ
ejpam-6098	137	14	if	if	SCONJ
ejpam-6098	137	15	,	,	PUNCT
ejpam-6098	137	16	for	for	ADP
ejpam-6098	137	17	any	any	DET
ejpam-6098	137	18	s	s	X
ejpam-6098	137	19	>	>	X
ejpam-6098	137	20	0	0	PUNCT
ejpam-6098	137	21	and	and	CCONJ
ejpam-6098	137	22	λ	λ	X
ejpam-6098	137	23	>	>	X
ejpam-6098	137	24	0	0	NUM
ejpam-6098	137	25	,	,	PUNCT
ejpam-6098	137	26	there	there	PRON
ejpam-6098	137	27	exists	exist	VERB
ejpam-6098	137	28	a	a	DET
ejpam-6098	137	29	positive	positive	ADJ
ejpam-6098	137	30	integer	integer	NOUN
ejpam-6098	137	31	n0	n0	NOUN
ejpam-6098	137	32	such	such	ADJ
ejpam-6098	137	33	that	that	SCONJ
ejpam-6098	137	34	µun−um(s	µun−um(s	PROPN
ejpam-6098	137	35	)	)	PUNCT
ejpam-6098	137	36	>	>	X
ejpam-6098	138	1	1−	1−	NUM
ejpam-6098	138	2	λ	λ	NOUN
ejpam-6098	138	3	for	for	ADP
ejpam-6098	138	4	all	all	DET
ejpam-6098	138	5	n	n	DET
ejpam-6098	138	6	≥	≥	NOUN
ejpam-6098	138	7	m	m	PROPN
ejpam-6098	138	8	≥	≥	PROPN
ejpam-6098	138	9	n0	n0	NUM
ejpam-6098	138	10	.	.	PUNCT
ejpam-6098	139	1	j.	j.	PROPN
ejpam-6098	139	2	jakhar	jakhar	PROPN
ejpam-6098	139	3	et	et	PROPN
ejpam-6098	139	4	al	al	PROPN
ejpam-6098	139	5	.	.	PUNCT
ejpam-6098	139	6	/	/	SYM
ejpam-6098	139	7	eur	eur	PROPN
ejpam-6098	139	8	.	.	PUNCT
ejpam-6098	140	1	j.	j.	PROPN
ejpam-6098	140	2	pure	pure	PROPN
ejpam-6098	140	3	appl	appl	PROPN
ejpam-6098	140	4	.	.	PROPN
ejpam-6098	140	5	math	math	PROPN
ejpam-6098	140	6	,	,	PUNCT
ejpam-6098	140	7	18	18	NUM
ejpam-6098	140	8	(	(	PUNCT
ejpam-6098	140	9	2	2	NUM
ejpam-6098	140	10	)	)	PUNCT
ejpam-6098	140	11	(	(	PUNCT
ejpam-6098	140	12	2025	2025	NUM
ejpam-6098	140	13	)	)	PUNCT
ejpam-6098	140	14	,	,	PUNCT
ejpam-6098	140	15	6098	6098	NUM
ejpam-6098	140	16	5	5	NUM
ejpam-6098	140	17	of	of	ADP
ejpam-6098	140	18	23	23	NUM
ejpam-6098	140	19	a	a	DET
ejpam-6098	140	20	rns	rn	NOUN
ejpam-6098	140	21	(	(	PUNCT
ejpam-6098	140	22	z	z	PROPN
ejpam-6098	140	23	,	,	PUNCT
ejpam-6098	140	24	µ	µ	NUM
ejpam-6098	140	25	,	,	PUNCT
ejpam-6098	140	26	t	t	PROPN
ejpam-6098	140	27	)	)	PUNCT
ejpam-6098	140	28	is	be	AUX
ejpam-6098	140	29	called	call	VERB
ejpam-6098	140	30	complete	complete	ADJ
ejpam-6098	140	31	if	if	SCONJ
ejpam-6098	140	32	every	every	DET
ejpam-6098	140	33	cauchy	cauchy	ADJ
ejpam-6098	140	34	sequence	sequence	NOUN
ejpam-6098	140	35	in	in	ADP
ejpam-6098	140	36	z	z	NOUN
ejpam-6098	140	37	converges	converge	NOUN
ejpam-6098	140	38	.	.	PUNCT
ejpam-6098	141	1	the	the	DET
ejpam-6098	141	2	contributions	contribution	NOUN
ejpam-6098	141	3	of	of	ADP
ejpam-6098	141	4	this	this	DET
ejpam-6098	141	5	paper	paper	NOUN
ejpam-6098	141	6	lie	lie	NOUN
ejpam-6098	141	7	in	in	ADP
ejpam-6098	141	8	its	its	PRON
ejpam-6098	141	9	comprehensive	comprehensive	ADJ
ejpam-6098	141	10	analysis	analysis	NOUN
ejpam-6098	141	11	of	of	ADP
ejpam-6098	141	12	the	the	DET
ejpam-6098	141	13	stability	stability	NOUN
ejpam-6098	141	14	of	of	ADP
ejpam-6098	141	15	three	three	NUM
ejpam-6098	141	16	-	-	PUNCT
ejpam-6098	141	17	dimensional	dimensional	ADJ
ejpam-6098	141	18	cubic	cubic	ADJ
ejpam-6098	141	19	functional	functional	ADJ
ejpam-6098	141	20	equations	equation	NOUN
ejpam-6098	141	21	across	across	ADP
ejpam-6098	141	22	distinct	distinct	ADJ
ejpam-6098	141	23	mathematical	mathematical	ADJ
ejpam-6098	141	24	frameworks	framework	NOUN
ejpam-6098	141	25	.	.	PUNCT
ejpam-6098	142	1	the	the	DET
ejpam-6098	142	2	study	study	NOUN
ejpam-6098	142	3	introduces	introduce	NOUN
ejpam-6098	142	4	(	(	PUNCT
ejpam-6098	142	5	n	n	CCONJ
ejpam-6098	142	6	,	,	PUNCT
ejpam-6098	142	7	β)-normed	β)-normed	PUNCT
ejpam-6098	142	8	spaces	space	NOUN
ejpam-6098	142	9	and	and	CCONJ
ejpam-6098	142	10	non	non	ADJ
ejpam-6098	142	11	-	-	ADJ
ejpam-6098	142	12	archimedean	archimedean	ADJ
ejpam-6098	142	13	(	(	PUNCT
ejpam-6098	142	14	n	n	CCONJ
ejpam-6098	142	15	,	,	PUNCT
ejpam-6098	142	16	β)-normed	β)-normed	PUNCT
ejpam-6098	142	17	spaces	space	NOUN
ejpam-6098	142	18	as	as	ADP
ejpam-6098	142	19	novel	novel	ADJ
ejpam-6098	142	20	settings	setting	NOUN
ejpam-6098	142	21	for	for	ADP
ejpam-6098	142	22	conducting	conduct	VERB
ejpam-6098	142	23	stability	stability	NOUN
ejpam-6098	142	24	investigations	investigation	NOUN
ejpam-6098	142	25	,	,	PUNCT
ejpam-6098	142	26	thereby	thereby	ADV
ejpam-6098	142	27	extending	extend	VERB
ejpam-6098	142	28	classical	classical	ADJ
ejpam-6098	142	29	approaches	approach	NOUN
ejpam-6098	142	30	into	into	ADP
ejpam-6098	142	31	new	new	ADJ
ejpam-6098	142	32	theoretical	theoretical	ADJ
ejpam-6098	142	33	domains	domain	NOUN
ejpam-6098	142	34	.	.	PUNCT
ejpam-6098	143	1	furthermore	furthermore	ADV
ejpam-6098	143	2	,	,	PUNCT
ejpam-6098	143	3	the	the	DET
ejpam-6098	143	4	inclusion	inclusion	NOUN
ejpam-6098	143	5	of	of	ADP
ejpam-6098	143	6	random	random	ADJ
ejpam-6098	143	7	normed	normed	ADJ
ejpam-6098	143	8	spaces	space	NOUN
ejpam-6098	143	9	adds	add	VERB
ejpam-6098	143	10	a	a	DET
ejpam-6098	143	11	stochastic	stochastic	ADJ
ejpam-6098	143	12	perspective	perspective	NOUN
ejpam-6098	143	13	,	,	PUNCT
ejpam-6098	143	14	enhancing	enhance	VERB
ejpam-6098	143	15	the	the	DET
ejpam-6098	143	16	scope	scope	NOUN
ejpam-6098	143	17	and	and	CCONJ
ejpam-6098	143	18	relevance	relevance	NOUN
ejpam-6098	143	19	of	of	ADP
ejpam-6098	143	20	the	the	DET
ejpam-6098	143	21	analysis	analysis	NOUN
ejpam-6098	143	22	in	in	ADP
ejpam-6098	143	23	probabilistic	probabilistic	ADJ
ejpam-6098	143	24	contexts	contexts	NOUN
ejpam-6098	143	25	.	.	PUNCT
ejpam-6098	144	1	a	a	DET
ejpam-6098	144	2	key	key	ADJ
ejpam-6098	144	3	feature	feature	NOUN
ejpam-6098	144	4	of	of	ADP
ejpam-6098	144	5	this	this	DET
ejpam-6098	144	6	work	work	NOUN
ejpam-6098	144	7	is	be	AUX
ejpam-6098	144	8	the	the	DET
ejpam-6098	144	9	comparative	comparative	ADJ
ejpam-6098	144	10	evaluation	evaluation	NOUN
ejpam-6098	144	11	of	of	ADP
ejpam-6098	144	12	stability	stability	NOUN
ejpam-6098	144	13	behavior	behavior	NOUN
ejpam-6098	144	14	across	across	ADP
ejpam-6098	144	15	these	these	DET
ejpam-6098	144	16	frameworks	framework	NOUN
ejpam-6098	144	17	,	,	PUNCT
ejpam-6098	144	18	which	which	PRON
ejpam-6098	144	19	uncovers	uncover	NOUN
ejpam-6098	144	20	both	both	DET
ejpam-6098	144	21	similarities	similarity	NOUN
ejpam-6098	144	22	and	and	CCONJ
ejpam-6098	144	23	critical	critical	ADJ
ejpam-6098	144	24	differences	difference	NOUN
ejpam-6098	144	25	.	.	PUNCT
ejpam-6098	145	1	in	in	ADP
ejpam-6098	145	2	addition	addition	NOUN
ejpam-6098	145	3	,	,	PUNCT
ejpam-6098	145	4	the	the	DET
ejpam-6098	145	5	theoretical	theoretical	ADJ
ejpam-6098	145	6	results	result	NOUN
ejpam-6098	145	7	are	be	AUX
ejpam-6098	145	8	supported	support	VERB
ejpam-6098	145	9	by	by	ADP
ejpam-6098	145	10	experimental	experimental	ADJ
ejpam-6098	145	11	validation	validation	NOUN
ejpam-6098	145	12	,	,	PUNCT
ejpam-6098	145	13	demonstrating	demonstrate	VERB
ejpam-6098	145	14	the	the	DET
ejpam-6098	145	15	practical	practical	ADJ
ejpam-6098	145	16	applicability	applicability	NOUN
ejpam-6098	145	17	of	of	ADP
ejpam-6098	145	18	the	the	DET
ejpam-6098	145	19	proposed	propose	VERB
ejpam-6098	145	20	stability	stability	NOUN
ejpam-6098	145	21	conditions	condition	NOUN
ejpam-6098	145	22	.	.	PUNCT
ejpam-6098	146	1	collectively	collectively	ADV
ejpam-6098	146	2	,	,	PUNCT
ejpam-6098	146	3	these	these	DET
ejpam-6098	146	4	contributions	contribution	NOUN
ejpam-6098	146	5	offer	offer	VERB
ejpam-6098	146	6	a	a	DET
ejpam-6098	146	7	broader	broad	ADJ
ejpam-6098	146	8	and	and	CCONJ
ejpam-6098	146	9	deeper	deep	ADJ
ejpam-6098	146	10	understanding	understanding	NOUN
ejpam-6098	146	11	of	of	ADP
ejpam-6098	146	12	functional	functional	ADJ
ejpam-6098	146	13	equation	equation	NOUN
ejpam-6098	146	14	stability	stability	NOUN
ejpam-6098	146	15	,	,	PUNCT
ejpam-6098	146	16	bridging	bridge	VERB
ejpam-6098	146	17	deterministic	deterministic	ADJ
ejpam-6098	146	18	and	and	CCONJ
ejpam-6098	146	19	stochastic	stochastic	ADJ
ejpam-6098	146	20	approaches	approach	NOUN
ejpam-6098	146	21	in	in	ADP
ejpam-6098	146	22	a	a	DET
ejpam-6098	146	23	unified	unified	ADJ
ejpam-6098	146	24	framework	framework	NOUN
ejpam-6098	146	25	.	.	PUNCT
ejpam-6098	147	1	the	the	DET
ejpam-6098	147	2	motivation	motivation	NOUN
ejpam-6098	147	3	for	for	ADP
ejpam-6098	147	4	this	this	DET
ejpam-6098	147	5	study	study	NOUN
ejpam-6098	147	6	stems	stem	VERB
ejpam-6098	147	7	from	from	ADP
ejpam-6098	147	8	the	the	DET
ejpam-6098	147	9	need	need	NOUN
ejpam-6098	147	10	to	to	PART
ejpam-6098	147	11	explore	explore	VERB
ejpam-6098	147	12	how	how	SCONJ
ejpam-6098	147	13	the	the	DET
ejpam-6098	147	14	nature	nature	NOUN
ejpam-6098	147	15	of	of	ADP
ejpam-6098	147	16	the	the	DET
ejpam-6098	147	17	underlying	underlie	VERB
ejpam-6098	147	18	space	space	NOUN
ejpam-6098	147	19	influences	influence	VERB
ejpam-6098	147	20	the	the	DET
ejpam-6098	147	21	stability	stability	NOUN
ejpam-6098	147	22	of	of	ADP
ejpam-6098	147	23	functional	functional	ADJ
ejpam-6098	147	24	equations	equation	NOUN
ejpam-6098	147	25	,	,	PUNCT
ejpam-6098	147	26	particularly	particularly	ADV
ejpam-6098	147	27	in	in	ADP
ejpam-6098	147	28	higher	higher	ADV
ejpam-6098	147	29	-	-	PUNCT
ejpam-6098	147	30	dimensional	dimensional	ADJ
ejpam-6098	147	31	and	and	CCONJ
ejpam-6098	147	32	nontraditional	nontraditional	ADJ
ejpam-6098	147	33	settings	setting	NOUN
ejpam-6098	147	34	.	.	PUNCT
ejpam-6098	148	1	the	the	DET
ejpam-6098	148	2	novelty	novelty	NOUN
ejpam-6098	148	3	lies	lie	VERB
ejpam-6098	148	4	not	not	PART
ejpam-6098	148	5	only	only	ADV
ejpam-6098	148	6	in	in	ADP
ejpam-6098	148	7	the	the	DET
ejpam-6098	148	8	introduction	introduction	NOUN
ejpam-6098	148	9	of	of	ADP
ejpam-6098	148	10	the	the	DET
ejpam-6098	148	11	(	(	PUNCT
ejpam-6098	148	12	n	n	CCONJ
ejpam-6098	148	13	,	,	PUNCT
ejpam-6098	148	14	β)-normed	β)-normed	PUNCT
ejpam-6098	148	15	and	and	CCONJ
ejpam-6098	148	16	non	non	ADJ
ejpam-6098	148	17	-	-	ADJ
ejpam-6098	148	18	archimedean	archimedean	ADJ
ejpam-6098	148	19	variants	variant	NOUN
ejpam-6098	148	20	but	but	CCONJ
ejpam-6098	148	21	also	also	ADV
ejpam-6098	148	22	in	in	ADP
ejpam-6098	148	23	the	the	DET
ejpam-6098	148	24	integration	integration	NOUN
ejpam-6098	148	25	of	of	ADP
ejpam-6098	148	26	a	a	DET
ejpam-6098	148	27	stochastic	stochastic	ADJ
ejpam-6098	148	28	viewpoint	viewpoint	NOUN
ejpam-6098	148	29	through	through	ADP
ejpam-6098	148	30	random	random	ADJ
ejpam-6098	148	31	normed	normed	ADJ
ejpam-6098	148	32	spaces	space	NOUN
ejpam-6098	148	33	,	,	PUNCT
ejpam-6098	148	34	which	which	PRON
ejpam-6098	148	35	has	have	AUX
ejpam-6098	148	36	not	not	PART
ejpam-6098	148	37	been	be	AUX
ejpam-6098	148	38	extensively	extensively	ADV
ejpam-6098	148	39	considered	consider	VERB
ejpam-6098	148	40	in	in	ADP
ejpam-6098	148	41	previous	previous	ADJ
ejpam-6098	148	42	stability	stability	NOUN
ejpam-6098	148	43	analyses	analysis	NOUN
ejpam-6098	148	44	.	.	PUNCT
ejpam-6098	149	1	this	this	DET
ejpam-6098	149	2	unified	unified	ADJ
ejpam-6098	149	3	and	and	CCONJ
ejpam-6098	149	4	comparative	comparative	ADJ
ejpam-6098	149	5	approach	approach	NOUN
ejpam-6098	149	6	opens	open	VERB
ejpam-6098	149	7	new	new	ADJ
ejpam-6098	149	8	avenues	avenue	NOUN
ejpam-6098	149	9	for	for	ADP
ejpam-6098	149	10	theoretical	theoretical	ADJ
ejpam-6098	149	11	advancements	advancement	NOUN
ejpam-6098	149	12	and	and	CCONJ
ejpam-6098	149	13	practical	practical	ADJ
ejpam-6098	149	14	applications	application	NOUN
ejpam-6098	149	15	in	in	ADP
ejpam-6098	149	16	the	the	DET
ejpam-6098	149	17	study	study	NOUN
ejpam-6098	149	18	of	of	ADP
ejpam-6098	149	19	functional	functional	ADJ
ejpam-6098	149	20	equations	equation	NOUN
ejpam-6098	149	21	.	.	PUNCT
ejpam-6098	150	1	this	this	DET
ejpam-6098	150	2	article	article	NOUN
ejpam-6098	150	3	is	be	AUX
ejpam-6098	150	4	organized	organize	VERB
ejpam-6098	150	5	into	into	ADP
ejpam-6098	150	6	seven	seven	NUM
ejpam-6098	150	7	sections	section	NOUN
ejpam-6098	150	8	.	.	PUNCT
ejpam-6098	151	1	section	section	NOUN
ejpam-6098	151	2	1	1	NUM
ejpam-6098	151	3	presents	present	VERB
ejpam-6098	151	4	the	the	DET
ejpam-6098	151	5	introduction	introduction	NOUN
ejpam-6098	151	6	and	and	CCONJ
ejpam-6098	151	7	background	background	NOUN
ejpam-6098	151	8	of	of	ADP
ejpam-6098	151	9	the	the	DET
ejpam-6098	151	10	study	study	NOUN
ejpam-6098	151	11	.	.	PUNCT
ejpam-6098	152	1	in	in	ADP
ejpam-6098	152	2	sections	section	NOUN
ejpam-6098	152	3	2	2	NUM
ejpam-6098	152	4	and	and	CCONJ
ejpam-6098	152	5	3	3	NUM
ejpam-6098	152	6	,	,	PUNCT
ejpam-6098	152	7	we	we	PRON
ejpam-6098	152	8	investigate	investigate	VERB
ejpam-6098	152	9	the	the	DET
ejpam-6098	152	10	stability	stability	NOUN
ejpam-6098	152	11	of	of	ADP
ejpam-6098	152	12	the	the	DET
ejpam-6098	152	13	cubic	cubic	ADJ
ejpam-6098	152	14	functional	functional	ADJ
ejpam-6098	152	15	equation	equation	NOUN
ejpam-6098	152	16	(	(	PUNCT
ejpam-6098	152	17	1	1	NUM
ejpam-6098	152	18	)	)	PUNCT
ejpam-6098	152	19	within	within	ADP
ejpam-6098	152	20	the	the	DET
ejpam-6098	152	21	frameworks	framework	NOUN
ejpam-6098	152	22	of	of	ADP
ejpam-6098	152	23	(	(	PUNCT
ejpam-6098	152	24	n	n	CCONJ
ejpam-6098	152	25	,	,	PUNCT
ejpam-6098	152	26	β)-normed	β)-normed	PUNCT
ejpam-6098	152	27	spaces	space	NOUN
ejpam-6098	152	28	and	and	CCONJ
ejpam-6098	152	29	non	non	ADJ
ejpam-6098	152	30	-	-	ADJ
ejpam-6098	152	31	archimedean	archimedean	ADJ
ejpam-6098	152	32	(	(	PUNCT
ejpam-6098	152	33	n	n	CCONJ
ejpam-6098	152	34	,	,	PUNCT
ejpam-6098	152	35	β)-normed	β)-normed	PUNCT
ejpam-6098	152	36	spaces	space	NOUN
ejpam-6098	152	37	,	,	PUNCT
ejpam-6098	152	38	respectively	respectively	ADV
ejpam-6098	152	39	.	.	PUNCT
ejpam-6098	153	1	section	section	NOUN
ejpam-6098	153	2	4	4	NUM
ejpam-6098	153	3	is	be	AUX
ejpam-6098	153	4	devoted	devote	VERB
ejpam-6098	153	5	to	to	ADP
ejpam-6098	153	6	the	the	DET
ejpam-6098	153	7	analysis	analysis	NOUN
ejpam-6098	153	8	of	of	ADP
ejpam-6098	153	9	stability	stability	NOUN
ejpam-6098	153	10	in	in	ADP
ejpam-6098	153	11	random	random	ADJ
ejpam-6098	153	12	normed	normed	ADJ
ejpam-6098	153	13	spaces	space	NOUN
ejpam-6098	153	14	.	.	PUNCT
ejpam-6098	154	1	sections	section	NOUN
ejpam-6098	154	2	5	5	NUM
ejpam-6098	154	3	and	and	CCONJ
ejpam-6098	154	4	6	6	NUM
ejpam-6098	154	5	provide	provide	VERB
ejpam-6098	154	6	the	the	DET
ejpam-6098	154	7	experimental	experimental	ADJ
ejpam-6098	154	8	results	result	NOUN
ejpam-6098	154	9	and	and	CCONJ
ejpam-6098	154	10	a	a	DET
ejpam-6098	154	11	comparative	comparative	ADJ
ejpam-6098	154	12	evaluation	evaluation	NOUN
ejpam-6098	154	13	of	of	ADP
ejpam-6098	154	14	the	the	DET
ejpam-6098	154	15	findings	finding	NOUN
ejpam-6098	154	16	across	across	ADP
ejpam-6098	154	17	the	the	DET
ejpam-6098	154	18	different	different	ADJ
ejpam-6098	154	19	spaces	space	NOUN
ejpam-6098	154	20	.	.	PUNCT
ejpam-6098	155	1	finally	finally	ADV
ejpam-6098	155	2	,	,	PUNCT
ejpam-6098	155	3	the	the	DET
ejpam-6098	155	4	conclusions	conclusion	NOUN
ejpam-6098	155	5	of	of	ADP
ejpam-6098	155	6	the	the	DET
ejpam-6098	155	7	study	study	NOUN
ejpam-6098	155	8	are	be	AUX
ejpam-6098	155	9	presented	present	VERB
ejpam-6098	155	10	in	in	ADP
ejpam-6098	155	11	section	section	NOUN
ejpam-6098	155	12	7	7	NUM
ejpam-6098	155	13	.	.	PUNCT
ejpam-6098	156	1	throughout	throughout	ADP
ejpam-6098	156	2	this	this	DET
ejpam-6098	156	3	article	article	NOUN
ejpam-6098	156	4	,	,	PUNCT
ejpam-6098	156	5	u	u	PROPN
ejpam-6098	156	6	is	be	AUX
ejpam-6098	156	7	a	a	DET
ejpam-6098	156	8	linear	linear	ADJ
ejpam-6098	156	9	sapce	sapce	NOUN
ejpam-6098	156	10	,	,	PUNCT
ejpam-6098	156	11	v	v	PRON
ejpam-6098	156	12	is	be	AUX
ejpam-6098	156	13	a	a	DET
ejpam-6098	156	14	complete	complete	ADJ
ejpam-6098	156	15	(	(	PUNCT
ejpam-6098	156	16	n	n	CCONJ
ejpam-6098	156	17	,	,	PUNCT
ejpam-6098	156	18	β)-ns	β)-ns	NOUN
ejpam-6098	156	19	,	,	PUNCT
ejpam-6098	156	20	w	w	PROPN
ejpam-6098	156	21	is	be	AUX
ejpam-6098	156	22	a	a	DET
ejpam-6098	156	23	na(n	na(n	NOUN
ejpam-6098	156	24	,	,	PUNCT
ejpam-6098	156	25	β)-ns	β)-ns	NOUN
ejpam-6098	156	26	,	,	PUNCT
ejpam-6098	156	27	(	(	PUNCT
ejpam-6098	156	28	x,µ,min	x,µ,min	NOUN
ejpam-6098	156	29	)	)	PUNCT
ejpam-6098	156	30	is	be	AUX
ejpam-6098	156	31	a	a	DET
ejpam-6098	156	32	complete	complete	ADJ
ejpam-6098	156	33	rns	rn	NOUN
ejpam-6098	156	34	and	and	CCONJ
ejpam-6098	156	35	(	(	PUNCT
ejpam-6098	156	36	z	z	NOUN
ejpam-6098	156	37	,	,	PUNCT
ejpam-6098	156	38	µ′,min	µ′,min	PROPN
ejpam-6098	156	39	)	)	PUNCT
ejpam-6098	156	40	is	be	AUX
ejpam-6098	156	41	a	a	DET
ejpam-6098	156	42	rns	rn	NOUN
ejpam-6098	156	43	.	.	PUNCT
ejpam-6098	157	1	pmi	pmi	PROPN
ejpam-6098	157	2	stands	stand	VERB
ejpam-6098	157	3	for	for	ADP
ejpam-6098	157	4	the	the	DET
ejpam-6098	157	5	principle	principle	NOUN
ejpam-6098	157	6	of	of	ADP
ejpam-6098	157	7	mathematical	mathematical	ADJ
ejpam-6098	157	8	induction	induction	NOUN
ejpam-6098	157	9	.	.	PUNCT
ejpam-6098	158	1	2	2	X
ejpam-6098	158	2	.	.	X
ejpam-6098	158	3	stability	stability	NOUN
ejpam-6098	158	4	in	in	ADP
ejpam-6098	158	5	(	(	PUNCT
ejpam-6098	158	6	n	n	CCONJ
ejpam-6098	158	7	,	,	PUNCT
ejpam-6098	158	8	β)-normed	β)-normed	PUNCT
ejpam-6098	158	9	space	space	NOUN
ejpam-6098	158	10	the	the	PRON
ejpam-6098	158	11	(	(	PUNCT
ejpam-6098	158	12	n	n	X
ejpam-6098	158	13	,	,	PUNCT
ejpam-6098	158	14	β)-normed	β)-normed	PUNCT
ejpam-6098	158	15	space	space	NOUN
ejpam-6098	158	16	(	(	PUNCT
ejpam-6098	158	17	(	(	PUNCT
ejpam-6098	158	18	n	n	CCONJ
ejpam-6098	158	19	,	,	PUNCT
ejpam-6098	158	20	β)-ns	β)-ns	NOUN
ejpam-6098	158	21	)	)	PUNCT
ejpam-6098	158	22	is	be	AUX
ejpam-6098	158	23	a	a	DET
ejpam-6098	158	24	generalization	generalization	NOUN
ejpam-6098	158	25	of	of	ADP
ejpam-6098	158	26	normed	normed	ADJ
ejpam-6098	158	27	linear	linear	PROPN
ejpam-6098	158	28	spaces	space	NOUN
ejpam-6098	158	29	,	,	PUNCT
ejpam-6098	158	30	where	where	SCONJ
ejpam-6098	158	31	instead	instead	ADV
ejpam-6098	158	32	of	of	ADP
ejpam-6098	158	33	a	a	DET
ejpam-6098	158	34	single	single	ADJ
ejpam-6098	158	35	norm	norm	NOUN
ejpam-6098	158	36	,	,	PUNCT
ejpam-6098	158	37	a	a	DET
ejpam-6098	158	38	family	family	NOUN
ejpam-6098	158	39	of	of	ADP
ejpam-6098	158	40	norms	norm	NOUN
ejpam-6098	158	41	is	be	AUX
ejpam-6098	158	42	considered	consider	VERB
ejpam-6098	158	43	.	.	PUNCT
ejpam-6098	159	1	the	the	DET
ejpam-6098	159	2	parameters	parameter	NOUN
ejpam-6098	159	3	n	n	ADV
ejpam-6098	159	4	and	and	CCONJ
ejpam-6098	159	5	β	β	NOUN
ejpam-6098	159	6	provide	provide	VERB
ejpam-6098	159	7	flexibility	flexibility	NOUN
ejpam-6098	159	8	in	in	ADP
ejpam-6098	159	9	defining	define	VERB
ejpam-6098	159	10	the	the	DET
ejpam-6098	159	11	size	size	NOUN
ejpam-6098	159	12	metric	metric	NOUN
ejpam-6098	159	13	and	and	CCONJ
ejpam-6098	159	14	influence	influence	VERB
ejpam-6098	159	15	the	the	DET
ejpam-6098	159	16	behavior	behavior	NOUN
ejpam-6098	159	17	of	of	ADP
ejpam-6098	159	18	the	the	DET
ejpam-6098	159	19	norms	norm	NOUN
ejpam-6098	159	20	.	.	PUNCT
ejpam-6098	160	1	(	(	PUNCT
ejpam-6098	160	2	n	n	CCONJ
ejpam-6098	160	3	,	,	PUNCT
ejpam-6098	160	4	β)-ns	β)-ns	ADP
ejpam-6098	160	5	is	be	AUX
ejpam-6098	160	6	used	use	VERB
ejpam-6098	160	7	to	to	PART
ejpam-6098	160	8	explore	explore	VERB
ejpam-6098	160	9	the	the	DET
ejpam-6098	160	10	stability	stability	NOUN
ejpam-6098	160	11	and	and	CCONJ
ejpam-6098	160	12	convergence	convergence	NOUN
ejpam-6098	160	13	of	of	ADP
ejpam-6098	160	14	solutions	solution	NOUN
ejpam-6098	160	15	in	in	ADP
ejpam-6098	160	16	functional	functional	ADJ
ejpam-6098	160	17	equations	equation	NOUN
ejpam-6098	160	18	.	.	PUNCT
ejpam-6098	161	1	these	these	DET
ejpam-6098	161	2	spaces	space	NOUN
ejpam-6098	161	3	help	help	AUX
ejpam-6098	161	4	analyze	analyze	VERB
ejpam-6098	161	5	the	the	DET
ejpam-6098	161	6	characteristics	characteristic	NOUN
ejpam-6098	161	7	of	of	ADP
ejpam-6098	161	8	the	the	DET
ejpam-6098	161	9	functions	function	NOUN
ejpam-6098	161	10	involved	involve	VERB
ejpam-6098	161	11	,	,	PUNCT
ejpam-6098	161	12	allowing	allow	VERB
ejpam-6098	161	13	us	we	PRON
ejpam-6098	161	14	to	to	PART
ejpam-6098	161	15	determine	determine	VERB
ejpam-6098	161	16	how	how	SCONJ
ejpam-6098	161	17	small	small	ADJ
ejpam-6098	161	18	perturbations	perturbation	NOUN
ejpam-6098	161	19	affect	affect	VERB
ejpam-6098	161	20	the	the	DET
ejpam-6098	161	21	solutions	solution	NOUN
ejpam-6098	161	22	.	.	PUNCT
ejpam-6098	162	1	using	use	VERB
ejpam-6098	162	2	the	the	DET
ejpam-6098	162	3	approach	approach	NOUN
ejpam-6098	162	4	from	from	ADP
ejpam-6098	162	5	[	[	X
ejpam-6098	162	6	19	19	NUM
ejpam-6098	162	7	]	]	PUNCT
ejpam-6098	162	8	,	,	PUNCT
ejpam-6098	162	9	we	we	PRON
ejpam-6098	162	10	address	address	VERB
ejpam-6098	162	11	the	the	DET
ejpam-6098	162	12	stability	stability	NOUN
ejpam-6098	162	13	problem	problem	NOUN
ejpam-6098	162	14	for	for	ADP
ejpam-6098	162	15	the	the	DET
ejpam-6098	162	16	given	give	VERB
ejpam-6098	162	17	functional	functional	ADJ
ejpam-6098	162	18	equation	equation	NOUN
ejpam-6098	162	19	(	(	PUNCT
ejpam-6098	162	20	1	1	NUM
ejpam-6098	162	21	)	)	PUNCT
ejpam-6098	162	22	in	in	ADP
ejpam-6098	162	23	(	(	PUNCT
ejpam-6098	162	24	n	n	CCONJ
ejpam-6098	162	25	,	,	PUNCT
ejpam-6098	162	26	β)-ns	β)-ns	NOUN
ejpam-6098	162	27	.	.	PUNCT
ejpam-6098	163	1	j.	j.	PROPN
ejpam-6098	163	2	jakhar	jakhar	PROPN
ejpam-6098	163	3	et	et	PROPN
ejpam-6098	163	4	al	al	PROPN
ejpam-6098	163	5	.	.	PUNCT
ejpam-6098	163	6	/	/	SYM
ejpam-6098	163	7	eur	eur	PROPN
ejpam-6098	163	8	.	.	PUNCT
ejpam-6098	164	1	j.	j.	PROPN
ejpam-6098	164	2	pure	pure	PROPN
ejpam-6098	164	3	appl	appl	PROPN
ejpam-6098	164	4	.	.	PROPN
ejpam-6098	164	5	math	math	PROPN
ejpam-6098	164	6	,	,	PUNCT
ejpam-6098	164	7	18	18	NUM
ejpam-6098	164	8	(	(	PUNCT
ejpam-6098	164	9	2	2	NUM
ejpam-6098	164	10	)	)	PUNCT
ejpam-6098	164	11	(	(	PUNCT
ejpam-6098	164	12	2025	2025	NUM
ejpam-6098	164	13	)	)	PUNCT
ejpam-6098	164	14	,	,	PUNCT
ejpam-6098	164	15	6098	6098	NUM
ejpam-6098	164	16	6	6	NUM
ejpam-6098	164	17	of	of	ADP
ejpam-6098	164	18	23	23	NUM
ejpam-6098	164	19	theorem	theorem	NOUN
ejpam-6098	164	20	1	1	NUM
ejpam-6098	164	21	.	.	PUNCT
ejpam-6098	165	1	let	let	VERB
ejpam-6098	165	2	ω	ω	NOUN
ejpam-6098	165	3	:	:	PUNCT
ejpam-6098	165	4	u	u	PRON
ejpam-6098	165	5	×	×	NOUN
ejpam-6098	165	6	u	u	NOUN
ejpam-6098	165	7	×	×	PROPN
ejpam-6098	165	8	u	u	X
ejpam-6098	165	9	→	→	SYM
ejpam-6098	165	10	[	[	X
ejpam-6098	165	11	0,∞	0,∞	X
ejpam-6098	165	12	)	)	PUNCT
ejpam-6098	165	13	be	be	VERB
ejpam-6098	165	14	a	a	DET
ejpam-6098	165	15	mapping	mapping	NOUN
ejpam-6098	165	16	holds	hold	VERB
ejpam-6098	165	17	φ(u	φ(u	NOUN
ejpam-6098	165	18	)	)	PUNCT
ejpam-6098	165	19	=	=	PUNCT
ejpam-6098	166	1	∞∑	∞∑	NUM
ejpam-6098	166	2	j=1	j=1	NOUN
ejpam-6098	166	3	1	1	NUM
ejpam-6098	166	4	23jβ	23jβ	NOUN
ejpam-6098	166	5	ω(2j−1u	ω(2j−1u	NUM
ejpam-6098	166	6	,	,	PUNCT
ejpam-6098	166	7	0	0	NUM
ejpam-6098	166	8	,	,	PUNCT
ejpam-6098	166	9	0	0	NUM
ejpam-6098	166	10	)	)	PUNCT
ejpam-6098	166	11	<	<	X
ejpam-6098	166	12	∞	∞	NUM
ejpam-6098	166	13	(	(	PUNCT
ejpam-6098	166	14	2	2	NUM
ejpam-6098	166	15	)	)	PUNCT
ejpam-6098	166	16	and	and	CCONJ
ejpam-6098	166	17	limm→∞	limm→∞	PROPN
ejpam-6098	166	18	1	1	NUM
ejpam-6098	166	19	23mβ	23mβ	ADJ
ejpam-6098	166	20	ω(2mu	ω(2mu	NOUN
ejpam-6098	166	21	,	,	PUNCT
ejpam-6098	166	22	0	0	NUM
ejpam-6098	166	23	,	,	PUNCT
ejpam-6098	166	24	0	0	NUM
ejpam-6098	166	25	)	)	PUNCT
ejpam-6098	166	26	=	=	SYM
ejpam-6098	166	27	0	0	X
ejpam-6098	166	28	.	.	PUNCT
ejpam-6098	167	1	(	(	PUNCT
ejpam-6098	167	2	3	3	X
ejpam-6098	167	3	)	)	PUNCT
ejpam-6098	167	4	if	if	SCONJ
ejpam-6098	167	5	ψ	ψ	X
ejpam-6098	167	6	:	:	PUNCT
ejpam-6098	167	7	v	v	NUM
ejpam-6098	167	8	n−1	n−1	PROPN
ejpam-6098	167	9	→	→	SYM
ejpam-6098	167	10	[	[	X
ejpam-6098	167	11	0,∞	0,∞	NUM
ejpam-6098	167	12	)	)	PUNCT
ejpam-6098	167	13	is	be	AUX
ejpam-6098	167	14	a	a	DET
ejpam-6098	167	15	function	function	NOUN
ejpam-6098	167	16	and	and	CCONJ
ejpam-6098	167	17	f	f	NOUN
ejpam-6098	167	18	:	:	PUNCT
ejpam-6098	167	19	u	u	X
ejpam-6098	167	20	→	→	SYM
ejpam-6098	167	21	v	v	PROPN
ejpam-6098	167	22	is	be	AUX
ejpam-6098	167	23	an	an	DET
ejpam-6098	167	24	odd	odd	ADJ
ejpam-6098	167	25	mapping	mapping	NOUN
ejpam-6098	167	26	holding	holding	NOUN
ejpam-6098	167	27	∥f(2u+	∥f(2u+	PROPN
ejpam-6098	168	1	w	w	PROPN
ejpam-6098	169	1	+	+	CCONJ
ejpam-6098	169	2	v)−	v)−	PROPN
ejpam-6098	169	3	3f(v	3f(v	NUM
ejpam-6098	170	1	+	+	CCONJ
ejpam-6098	170	2	w	w	PROPN
ejpam-6098	171	1	+	+	CCONJ
ejpam-6098	171	2	u)−	u)−	PROPN
ejpam-6098	171	3	f(w	f(w	PROPN
ejpam-6098	171	4	−	−	NOUN
ejpam-6098	171	5	u+	u+	NOUN
ejpam-6098	171	6	v)−	v)−	PROPN
ejpam-6098	171	7	2f(u+	2f(u+	PROPN
ejpam-6098	171	8	v	v	NOUN
ejpam-6098	171	9	)	)	PUNCT
ejpam-6098	171	10	−2f(w	−2f(w	NOUN
ejpam-6098	171	11	+	+	CCONJ
ejpam-6098	171	12	u	u	NOUN
ejpam-6098	171	13	)	)	PUNCT
ejpam-6098	171	14	+	+	CCONJ
ejpam-6098	171	15	6f(u−	6f(u−	NUM
ejpam-6098	171	16	v	v	NOUN
ejpam-6098	171	17	)	)	PUNCT
ejpam-6098	171	18	+	+	CCONJ
ejpam-6098	171	19	6f(u−	6f(u−	NUM
ejpam-6098	171	20	w	w	NOUN
ejpam-6098	171	21	)	)	PUNCT
ejpam-6098	171	22	+	+	CCONJ
ejpam-6098	171	23	3f(w	3f(w	NOUN
ejpam-6098	171	24	+	+	CCONJ
ejpam-6098	171	25	v)−	v)−	PROPN
ejpam-6098	171	26	2f(2u−	2f(2u−	NUM
ejpam-6098	171	27	v	v	NOUN
ejpam-6098	171	28	)	)	PUNCT
ejpam-6098	171	29	−2f(2u−	−2f(2u−	PROPN
ejpam-6098	171	30	w	w	PROPN
ejpam-6098	171	31	)	)	PUNCT
ejpam-6098	171	32	+	+	CCONJ
ejpam-6098	171	33	6f(v	6f(v	NUM
ejpam-6098	171	34	)	)	PUNCT
ejpam-6098	171	35	+	+	CCONJ
ejpam-6098	171	36	6f(w	6f(w	NUM
ejpam-6098	171	37	)	)	PUNCT
ejpam-6098	172	1	+	+	CCONJ
ejpam-6098	172	2	18f(u	18f(u	NUM
ejpam-6098	172	3	)	)	PUNCT
ejpam-6098	172	4	,	,	PUNCT
ejpam-6098	172	5	δ1	δ1	NOUN
ejpam-6098	172	6	,	,	PUNCT
ejpam-6098	172	7	...	...	PUNCT
ejpam-6098	172	8	,	,	PUNCT
ejpam-6098	172	9	δn−1∥β	δn−1∥β	NOUN
ejpam-6098	172	10	≤	≤	NUM
ejpam-6098	172	11	ω(u	ω(u	PROPN
ejpam-6098	172	12	,	,	PUNCT
ejpam-6098	172	13	v	v	NOUN
ejpam-6098	172	14	,	,	PUNCT
ejpam-6098	172	15	w)ψ(δ1	w)ψ(δ1	NOUN
ejpam-6098	172	16	,	,	PUNCT
ejpam-6098	172	17	...	...	PUNCT
ejpam-6098	172	18	,	,	PUNCT
ejpam-6098	172	19	δn−1	δn−1	PROPN
ejpam-6098	172	20	)	)	PUNCT
ejpam-6098	172	21	(	(	PUNCT
ejpam-6098	172	22	4	4	X
ejpam-6098	172	23	)	)	PUNCT
ejpam-6098	172	24	for	for	ADP
ejpam-6098	172	25	all	all	DET
ejpam-6098	172	26	u	u	NOUN
ejpam-6098	172	27	,	,	PUNCT
ejpam-6098	172	28	v	v	NOUN
ejpam-6098	172	29	,	,	PUNCT
ejpam-6098	172	30	w	w	PROPN
ejpam-6098	172	31	∈	∈	PROPN
ejpam-6098	172	32	u	u	NOUN
ejpam-6098	172	33	and	and	CCONJ
ejpam-6098	172	34	δ1	δ1	NOUN
ejpam-6098	172	35	,	,	PUNCT
ejpam-6098	172	36	...	...	PUNCT
ejpam-6098	172	37	,	,	PUNCT
ejpam-6098	172	38	δn−1	δn−1	PROPN
ejpam-6098	172	39	∈	∈	PROPN
ejpam-6098	172	40	v	v	NOUN
ejpam-6098	172	41	,	,	PUNCT
ejpam-6098	172	42	then	then	ADV
ejpam-6098	172	43	there	there	PRON
ejpam-6098	172	44	is	be	VERB
ejpam-6098	172	45	a	a	DET
ejpam-6098	172	46	unique	unique	ADJ
ejpam-6098	172	47	cubic	cubic	ADJ
ejpam-6098	172	48	mapping	mapping	NOUN
ejpam-6098	172	49	q	q	NOUN
ejpam-6098	172	50	:	:	PUNCT
ejpam-6098	172	51	u	u	X
ejpam-6098	172	52	→	→	SYM
ejpam-6098	172	53	v	v	NOUN
ejpam-6098	172	54	satisfying	satisfy	VERB
ejpam-6098	172	55	||f(u)−q(u	||f(u)−q(u	NOUN
ejpam-6098	172	56	)	)	PUNCT
ejpam-6098	172	57	,	,	PUNCT
ejpam-6098	172	58	δ1	δ1	NOUN
ejpam-6098	172	59	,	,	PUNCT
ejpam-6098	172	60	...	...	PUNCT
ejpam-6098	172	61	,	,	PUNCT
ejpam-6098	172	62	δn−1||β	δn−1||β	PROPN
ejpam-6098	172	63	≤	≤	NUM
ejpam-6098	172	64	φ(u)ψ(δ1	φ(u)ψ(δ1	PROPN
ejpam-6098	172	65	,	,	PUNCT
ejpam-6098	172	66	...	...	PUNCT
ejpam-6098	172	67	,	,	PUNCT
ejpam-6098	172	68	δn−1	δn−1	PROPN
ejpam-6098	172	69	)	)	PUNCT
ejpam-6098	172	70	.	.	PUNCT
ejpam-6098	173	1	(	(	PUNCT
ejpam-6098	173	2	5	5	X
ejpam-6098	173	3	)	)	PUNCT
ejpam-6098	173	4	proof	proof	NOUN
ejpam-6098	173	5	.	.	PUNCT
ejpam-6098	174	1	putting	put	VERB
ejpam-6098	174	2	(	(	PUNCT
ejpam-6098	174	3	u	u	NOUN
ejpam-6098	174	4	,	,	PUNCT
ejpam-6098	174	5	v	v	NOUN
ejpam-6098	174	6	,	,	PUNCT
ejpam-6098	174	7	w	w	NOUN
ejpam-6098	174	8	)	)	PUNCT
ejpam-6098	174	9	=	=	SYM
ejpam-6098	174	10	(	(	PUNCT
ejpam-6098	174	11	u	u	NOUN
ejpam-6098	174	12	,	,	PUNCT
ejpam-6098	174	13	0	0	NUM
ejpam-6098	174	14	,	,	PUNCT
ejpam-6098	174	15	0	0	NUM
ejpam-6098	174	16	)	)	PUNCT
ejpam-6098	174	17	in	in	ADP
ejpam-6098	174	18	(	(	PUNCT
ejpam-6098	174	19	4	4	NUM
ejpam-6098	174	20	)	)	PUNCT
ejpam-6098	174	21	and	and	CCONJ
ejpam-6098	174	22	multiplying	multiply	VERB
ejpam-6098	174	23	both	both	DET
ejpam-6098	174	24	side	side	NOUN
ejpam-6098	174	25	by	by	ADP
ejpam-6098	174	26	1	1	NUM
ejpam-6098	174	27	3−β	3−β	NUM
ejpam-6098	174	28	,	,	PUNCT
ejpam-6098	174	29	we	we	PRON
ejpam-6098	174	30	have	have	VERB
ejpam-6098	174	31	∥f(2u)−	∥f(2u)−	PROPN
ejpam-6098	174	32	23f(u	23f(u	NUM
ejpam-6098	174	33	)	)	PUNCT
ejpam-6098	174	34	,	,	PUNCT
ejpam-6098	174	35	δ1	δ1	NOUN
ejpam-6098	174	36	,	,	PUNCT
ejpam-6098	174	37	...	...	PUNCT
ejpam-6098	174	38	,	,	PUNCT
ejpam-6098	174	39	δn−1∥β	δn−1∥β	NOUN
ejpam-6098	174	40	≤	≤	NOUN
ejpam-6098	174	41	3−βℓ(u	3−βℓ(u	NUM
ejpam-6098	174	42	,	,	PUNCT
ejpam-6098	174	43	0	0	NUM
ejpam-6098	174	44	,	,	PUNCT
ejpam-6098	174	45	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	174	46	,	,	PUNCT
ejpam-6098	174	47	...	...	PUNCT
ejpam-6098	174	48	,	,	PUNCT
ejpam-6098	174	49	δn−1	δn−1	PROPN
ejpam-6098	174	50	)	)	PUNCT
ejpam-6098	174	51	.	.	PUNCT
ejpam-6098	175	1	let	let	VERB
ejpam-6098	175	2	ω(u	ω(u	PROPN
ejpam-6098	175	3	,	,	PUNCT
ejpam-6098	175	4	0	0	NUM
ejpam-6098	175	5	,	,	PUNCT
ejpam-6098	175	6	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	175	7	,	,	PUNCT
ejpam-6098	175	8	...	...	PUNCT
ejpam-6098	175	9	,	,	PUNCT
ejpam-6098	175	10	δn−1	δn−1	NUM
ejpam-6098	175	11	)	)	PUNCT
ejpam-6098	175	12	=	=	SYM
ejpam-6098	175	13	m(u	m(u	PROPN
ejpam-6098	175	14	,	,	PUNCT
ejpam-6098	175	15	δ1	δ1	NOUN
ejpam-6098	175	16	,	,	PUNCT
ejpam-6098	175	17	...	...	PUNCT
ejpam-6098	175	18	,	,	PUNCT
ejpam-6098	175	19	δn−1	δn−1	PROPN
ejpam-6098	175	20	)	)	PUNCT
ejpam-6098	175	21	.	.	PUNCT
ejpam-6098	176	1	this	this	PRON
ejpam-6098	176	2	implies	imply	VERB
ejpam-6098	176	3	that	that	SCONJ
ejpam-6098	176	4	∥f(2u)−	∥f(2u)−	PROPN
ejpam-6098	176	5	23f(u	23f(u	NUM
ejpam-6098	176	6	)	)	PUNCT
ejpam-6098	176	7	,	,	PUNCT
ejpam-6098	176	8	δ1	δ1	NOUN
ejpam-6098	176	9	,	,	PUNCT
ejpam-6098	176	10	...	...	PUNCT
ejpam-6098	176	11	,	,	PUNCT
ejpam-6098	176	12	δn−1∥β	δn−1∥β	VERB
ejpam-6098	176	13	≤	≤	NOUN
ejpam-6098	176	14	3−βm(u	3−βm(u	NUM
ejpam-6098	176	15	,	,	PUNCT
ejpam-6098	176	16	δ1	δ1	NOUN
ejpam-6098	176	17	,	,	PUNCT
ejpam-6098	176	18	...	...	PUNCT
ejpam-6098	176	19	,	,	PUNCT
ejpam-6098	176	20	δn−1	δn−1	PROPN
ejpam-6098	176	21	)	)	PUNCT
ejpam-6098	176	22	.	.	PUNCT
ejpam-6098	177	1	(	(	PUNCT
ejpam-6098	177	2	6	6	X
ejpam-6098	177	3	)	)	PUNCT
ejpam-6098	177	4	replace	replace	VERB
ejpam-6098	177	5	u	u	NOUN
ejpam-6098	177	6	by	by	ADP
ejpam-6098	177	7	2u	2u	NOUN
ejpam-6098	177	8	in	in	ADP
ejpam-6098	177	9	(	(	PUNCT
ejpam-6098	177	10	6	6	NUM
ejpam-6098	177	11	)	)	PUNCT
ejpam-6098	177	12	||f(22u)−	||f(22u)−	PROPN
ejpam-6098	177	13	23f(2u	23f(2u	PROPN
ejpam-6098	177	14	)	)	PUNCT
ejpam-6098	177	15	,	,	PUNCT
ejpam-6098	177	16	δ1	δ1	NOUN
ejpam-6098	177	17	,	,	PUNCT
ejpam-6098	177	18	...	...	PUNCT
ejpam-6098	177	19	,	,	PUNCT
ejpam-6098	177	20	δn−1||β	δn−1||β	NOUN
ejpam-6098	177	21	≤	≤	NOUN
ejpam-6098	177	22	3−βm(2u	3−βm(2u	NUM
ejpam-6098	177	23	,	,	PUNCT
ejpam-6098	177	24	δ1	δ1	NOUN
ejpam-6098	177	25	,	,	PUNCT
ejpam-6098	177	26	...	...	PUNCT
ejpam-6098	177	27	,	,	PUNCT
ejpam-6098	177	28	δn−1	δn−1	PROPN
ejpam-6098	177	29	)	)	PUNCT
ejpam-6098	177	30	.	.	PUNCT
ejpam-6098	178	1	(	(	PUNCT
ejpam-6098	178	2	7	7	X
ejpam-6098	178	3	)	)	PUNCT
ejpam-6098	178	4	now	now	ADV
ejpam-6098	178	5	,	,	PUNCT
ejpam-6098	178	6	using	use	VERB
ejpam-6098	178	7	(	(	PUNCT
ejpam-6098	178	8	6	6	NUM
ejpam-6098	178	9	)	)	PUNCT
ejpam-6098	178	10	and	and	CCONJ
ejpam-6098	178	11	(	(	PUNCT
ejpam-6098	178	12	7	7	X
ejpam-6098	178	13	)	)	PUNCT
ejpam-6098	178	14	∥f(22u)−	∥f(22u)−	PROPN
ejpam-6098	178	15	(	(	PUNCT
ejpam-6098	178	16	23)2f(u	23)2f(u	NUM
ejpam-6098	178	17	)	)	PUNCT
ejpam-6098	178	18	,	,	PUNCT
ejpam-6098	178	19	δ1	δ1	NOUN
ejpam-6098	178	20	,	,	PUNCT
ejpam-6098	178	21	...	...	PUNCT
ejpam-6098	178	22	,	,	PUNCT
ejpam-6098	178	23	δn−1∥β	δn−1∥β	NOUN
ejpam-6098	178	24	=	=	SYM
ejpam-6098	178	25	||f(22u)−	||f(22u)−	PROPN
ejpam-6098	178	26	23f(2u	23f(2u	PROPN
ejpam-6098	178	27	)	)	PUNCT
ejpam-6098	179	1	+	+	NUM
ejpam-6098	179	2	23f(2u)−	23f(2u)−	NUM
ejpam-6098	179	3	(	(	PUNCT
ejpam-6098	179	4	23)2f(u	23)2f(u	NUM
ejpam-6098	179	5	)	)	PUNCT
ejpam-6098	179	6	,	,	PUNCT
ejpam-6098	179	7	δ1	δ1	NOUN
ejpam-6098	179	8	,	,	PUNCT
ejpam-6098	179	9	...	...	PUNCT
ejpam-6098	179	10	,	,	PUNCT
ejpam-6098	179	11	δn−1||β	δn−1||β	PROPN
ejpam-6098	179	12	≤	≤	NUM
ejpam-6098	179	13	||f(22u)−	||f(22u)−	PROPN
ejpam-6098	179	14	23f(2u	23f(2u	PROPN
ejpam-6098	179	15	)	)	PUNCT
ejpam-6098	179	16	,	,	PUNCT
ejpam-6098	179	17	δ1	δ1	NOUN
ejpam-6098	179	18	,	,	PUNCT
ejpam-6098	179	19	...	...	PUNCT
ejpam-6098	179	20	,	,	PUNCT
ejpam-6098	179	21	δn−1||β	δn−1||β	X
ejpam-6098	179	22	+	+	PUNCT
ejpam-6098	179	23	23β||f(2u)−	23β||f(2u)−	NUM
ejpam-6098	179	24	(	(	PUNCT
ejpam-6098	179	25	23)f(u	23)f(u	NUM
ejpam-6098	179	26	)	)	PUNCT
ejpam-6098	179	27	,	,	PUNCT
ejpam-6098	179	28	δ1	δ1	NOUN
ejpam-6098	179	29	,	,	PUNCT
ejpam-6098	179	30	...	...	PUNCT
ejpam-6098	179	31	,	,	PUNCT
ejpam-6098	179	32	δn−1||β	δn−1||β	NOUN
ejpam-6098	179	33	≤	≤	NOUN
ejpam-6098	179	34	3−βm(2u	3−βm(2u	NUM
ejpam-6098	179	35	,	,	PUNCT
ejpam-6098	179	36	δ1	δ1	NOUN
ejpam-6098	179	37	,	,	PUNCT
ejpam-6098	179	38	...	...	PUNCT
ejpam-6098	179	39	,	,	PUNCT
ejpam-6098	179	40	δn−1	δn−1	PROPN
ejpam-6098	179	41	)	)	PUNCT
ejpam-6098	179	42	+	+	CCONJ
ejpam-6098	179	43	3−β23βm(u	3−β23βm(u	NUM
ejpam-6098	179	44	,	,	PUNCT
ejpam-6098	179	45	δ1	δ1	NOUN
ejpam-6098	179	46	,	,	PUNCT
ejpam-6098	179	47	...	...	PUNCT
ejpam-6098	179	48	,	,	PUNCT
ejpam-6098	179	49	δn−1	δn−1	PROPN
ejpam-6098	179	50	)	)	PUNCT
ejpam-6098	179	51	.	.	PUNCT
ejpam-6098	180	1	j.	j.	PROPN
ejpam-6098	180	2	jakhar	jakhar	PROPN
ejpam-6098	180	3	et	et	PROPN
ejpam-6098	180	4	al	al	PROPN
ejpam-6098	180	5	.	.	PUNCT
ejpam-6098	180	6	/	/	SYM
ejpam-6098	180	7	eur	eur	PROPN
ejpam-6098	180	8	.	.	PUNCT
ejpam-6098	181	1	j.	j.	PROPN
ejpam-6098	181	2	pure	pure	PROPN
ejpam-6098	181	3	appl	appl	PROPN
ejpam-6098	181	4	.	.	PROPN
ejpam-6098	181	5	math	math	PROPN
ejpam-6098	181	6	,	,	PUNCT
ejpam-6098	181	7	18	18	NUM
ejpam-6098	181	8	(	(	PUNCT
ejpam-6098	181	9	2	2	NUM
ejpam-6098	181	10	)	)	PUNCT
ejpam-6098	181	11	(	(	PUNCT
ejpam-6098	181	12	2025	2025	NUM
ejpam-6098	181	13	)	)	PUNCT
ejpam-6098	181	14	,	,	PUNCT
ejpam-6098	181	15	6098	6098	NUM
ejpam-6098	181	16	7	7	NUM
ejpam-6098	181	17	of	of	ADP
ejpam-6098	181	18	23	23	NUM
ejpam-6098	181	19	by	by	ADP
ejpam-6098	181	20	using	use	VERB
ejpam-6098	181	21	pmi	pmi	PROPN
ejpam-6098	181	22	on	on	ADP
ejpam-6098	181	23	m	m	PROPN
ejpam-6098	181	24	,	,	PUNCT
ejpam-6098	181	25	we	we	PRON
ejpam-6098	181	26	assume	assume	VERB
ejpam-6098	181	27	that	that	SCONJ
ejpam-6098	181	28	||f(2mu)−	||f(2mu)−	PROPN
ejpam-6098	181	29	(	(	PUNCT
ejpam-6098	181	30	23)mf(u	23)mf(u	NUM
ejpam-6098	181	31	)	)	PUNCT
ejpam-6098	181	32	,	,	PUNCT
ejpam-6098	181	33	δ1	δ1	NOUN
ejpam-6098	181	34	,	,	PUNCT
ejpam-6098	181	35	...	...	PUNCT
ejpam-6098	181	36	,	,	PUNCT
ejpam-6098	181	37	δn−1||β	δn−1||β	PROPN
ejpam-6098	181	38	≤	≤	NUM
ejpam-6098	181	39	3−β	3−β	NUM
ejpam-6098	181	40	m∑	m∑	VERB
ejpam-6098	181	41	j=1	j=1	PROPN
ejpam-6098	181	42	23(j−1)βm(2m−ju	23(j−1)βm(2m−ju	PROPN
ejpam-6098	181	43	,	,	PUNCT
ejpam-6098	181	44	δ1	δ1	NOUN
ejpam-6098	181	45	,	,	PUNCT
ejpam-6098	181	46	...	...	PUNCT
ejpam-6098	181	47	,	,	PUNCT
ejpam-6098	181	48	δn−1	δn−1	PROPN
ejpam-6098	181	49	)	)	PUNCT
ejpam-6098	181	50	(	(	PUNCT
ejpam-6098	181	51	8)	8)	NUM
ejpam-6098	181	52	for	for	ADP
ejpam-6098	181	53	all	all	DET
ejpam-6098	181	54	u	u	PROPN
ejpam-6098	181	55	∈	∈	PROPN
ejpam-6098	181	56	u	u	NOUN
ejpam-6098	181	57	,	,	PUNCT
ejpam-6098	181	58	δ1	δ1	NOUN
ejpam-6098	181	59	,	,	PUNCT
ejpam-6098	181	60	...	...	PUNCT
ejpam-6098	181	61	,	,	PUNCT
ejpam-6098	181	62	δn−1	δn−1	PROPN
ejpam-6098	181	63	∈	∈	PROPN
ejpam-6098	181	64	v	v	NOUN
ejpam-6098	181	65	and	and	CCONJ
ejpam-6098	181	66	m	m	PROPN
ejpam-6098	181	67	∈	∈	PROPN
ejpam-6098	181	68	n	n	NOUN
ejpam-6098	181	69	.	.	PUNCT
ejpam-6098	182	1	from	from	ADP
ejpam-6098	182	2	(	(	PUNCT
ejpam-6098	182	3	6	6	NUM
ejpam-6098	182	4	)	)	PUNCT
ejpam-6098	182	5	,	,	PUNCT
ejpam-6098	182	6	we	we	PRON
ejpam-6098	182	7	can	can	AUX
ejpam-6098	182	8	say	say	VERB
ejpam-6098	182	9	that	that	DET
ejpam-6098	182	10	inequality	inequality	NOUN
ejpam-6098	182	11	(	(	PUNCT
ejpam-6098	182	12	8)	8)	NUM
ejpam-6098	182	13	holds	hold	VERB
ejpam-6098	182	14	for	for	ADP
ejpam-6098	182	15	m	m	NOUN
ejpam-6098	182	16	=	=	SYM
ejpam-6098	182	17	1	1	X
ejpam-6098	182	18	.	.	PUNCT
ejpam-6098	182	19	suppose	suppose	VERB
ejpam-6098	182	20	(	(	PUNCT
ejpam-6098	182	21	8)	8)	NUM
ejpam-6098	182	22	is	be	AUX
ejpam-6098	182	23	true	true	ADJ
ejpam-6098	182	24	for	for	ADP
ejpam-6098	182	25	certain	certain	ADJ
ejpam-6098	182	26	m	m	NOUN
ejpam-6098	182	27	>	>	X
ejpam-6098	182	28	1	1	NUM
ejpam-6098	182	29	.	.	PUNCT
ejpam-6098	183	1	changing	change	VERB
ejpam-6098	183	2	u	u	NOUN
ejpam-6098	183	3	by	by	ADP
ejpam-6098	183	4	2u	2u	NOUN
ejpam-6098	183	5	in	in	ADP
ejpam-6098	183	6	(	(	PUNCT
ejpam-6098	183	7	8)	8)	NUM
ejpam-6098	183	8	,	,	PUNCT
ejpam-6098	183	9	we	we	PRON
ejpam-6098	183	10	get	get	VERB
ejpam-6098	183	11	||f(2m+1u)−	||f(2m+1u)−	NOUN
ejpam-6098	183	12	(	(	PUNCT
ejpam-6098	183	13	23)mf(2u	23)mf(2u	NUM
ejpam-6098	183	14	)	)	PUNCT
ejpam-6098	183	15	,	,	PUNCT
ejpam-6098	183	16	δ1	δ1	NOUN
ejpam-6098	183	17	,	,	PUNCT
ejpam-6098	183	18	...	...	PUNCT
ejpam-6098	183	19	,	,	PUNCT
ejpam-6098	183	20	δn−1||β	δn−1||β	PROPN
ejpam-6098	183	21	≤	≤	NUM
ejpam-6098	183	22	3−β	3−β	NUM
ejpam-6098	183	23	m∑	m∑	VERB
ejpam-6098	183	24	j=1	j=1	PROPN
ejpam-6098	183	25	23(j−1)βm(2m+1−ju	23(j−1)βm(2m+1−ju	ADJ
ejpam-6098	183	26	,	,	PUNCT
ejpam-6098	183	27	δ1	δ1	NOUN
ejpam-6098	183	28	,	,	PUNCT
ejpam-6098	183	29	...	...	PUNCT
ejpam-6098	183	30	,	,	PUNCT
ejpam-6098	183	31	δn−1	δn−1	PROPN
ejpam-6098	183	32	)	)	PUNCT
ejpam-6098	183	33	.	.	PUNCT
ejpam-6098	184	1	hence	hence	ADV
ejpam-6098	184	2	,	,	PUNCT
ejpam-6098	184	3	it	it	PRON
ejpam-6098	184	4	follows	follow	VERB
ejpam-6098	184	5	from	from	ADP
ejpam-6098	184	6	(	(	PUNCT
ejpam-6098	184	7	8)	8)	NUM
ejpam-6098	184	8	||f(2m+1u)−	||f(2m+1u)−	NOUN
ejpam-6098	184	9	(	(	PUNCT
ejpam-6098	184	10	23)m+1f(u	23)m+1f(u	NUM
ejpam-6098	184	11	)	)	PUNCT
ejpam-6098	184	12	,	,	PUNCT
ejpam-6098	184	13	δ1	δ1	NOUN
ejpam-6098	184	14	,	,	PUNCT
ejpam-6098	184	15	...	...	PUNCT
ejpam-6098	184	16	,	,	PUNCT
ejpam-6098	184	17	δn−1||β	δn−1||β	NOUN
ejpam-6098	184	18	≤	≤	NUM
ejpam-6098	184	19	||f(2m+1u)−	||f(2m+1u)−	NOUN
ejpam-6098	184	20	(	(	PUNCT
ejpam-6098	184	21	23)mf(2u	23)mf(2u	NUM
ejpam-6098	184	22	)	)	PUNCT
ejpam-6098	184	23	,	,	PUNCT
ejpam-6098	184	24	δ1	δ1	NOUN
ejpam-6098	184	25	,	,	PUNCT
ejpam-6098	184	26	...	...	PUNCT
ejpam-6098	184	27	,	,	PUNCT
ejpam-6098	184	28	δn−1||β	δn−1||β	PROPN
ejpam-6098	184	29	+	+	CCONJ
ejpam-6098	184	30	23mβ||f(2u)−	23mβ||f(2u)−	NUM
ejpam-6098	184	31	23f(u	23f(u	NUM
ejpam-6098	184	32	)	)	PUNCT
ejpam-6098	184	33	,	,	PUNCT
ejpam-6098	184	34	δ1	δ1	NOUN
ejpam-6098	184	35	,	,	PUNCT
ejpam-6098	184	36	...	...	PUNCT
ejpam-6098	184	37	,	,	PUNCT
ejpam-6098	184	38	δn−1||β	δn−1||β	PROPN
ejpam-6098	184	39	≤	≤	NUM
ejpam-6098	184	40	3−β	3−β	NUM
ejpam-6098	184	41	m∑	m∑	VERB
ejpam-6098	184	42	j=1	j=1	PROPN
ejpam-6098	184	43	23(j−1)βm(2m+1−ju	23(j−1)βm(2m+1−ju	ADJ
ejpam-6098	184	44	,	,	PUNCT
ejpam-6098	184	45	δ1	δ1	NOUN
ejpam-6098	184	46	,	,	PUNCT
ejpam-6098	184	47	...	...	PUNCT
ejpam-6098	184	48	,	,	PUNCT
ejpam-6098	184	49	δn−1	δn−1	PROPN
ejpam-6098	184	50	)	)	PUNCT
ejpam-6098	185	1	+	+	CCONJ
ejpam-6098	185	2	3−β23mβm(u	3−β23mβm(u	NUM
ejpam-6098	185	3	,	,	PUNCT
ejpam-6098	185	4	δ1	δ1	NOUN
ejpam-6098	185	5	,	,	PUNCT
ejpam-6098	185	6	...	...	PUNCT
ejpam-6098	185	7	,	,	PUNCT
ejpam-6098	185	8	δn−1	δn−1	PROPN
ejpam-6098	185	9	)	)	PUNCT
ejpam-6098	185	10	=	=	SYM
ejpam-6098	186	1	3−β	3−β	NUM
ejpam-6098	186	2	m+1∑	m+1∑	NOUN
ejpam-6098	186	3	j=1	j=1	NOUN
ejpam-6098	187	1	23(j−1)βm(2m+1−ju	23(j−1)βm(2m+1−ju	NUM
ejpam-6098	187	2	,	,	PUNCT
ejpam-6098	187	3	δ1	δ1	NOUN
ejpam-6098	187	4	,	,	PUNCT
ejpam-6098	187	5	...	...	PUNCT
ejpam-6098	187	6	,	,	PUNCT
ejpam-6098	187	7	δn−1	δn−1	PROPN
ejpam-6098	187	8	)	)	PUNCT
ejpam-6098	187	9	.	.	PUNCT
ejpam-6098	188	1	by	by	ADP
ejpam-6098	188	2	using	use	VERB
ejpam-6098	188	3	(	(	PUNCT
ejpam-6098	188	4	8)	8)	NUM
ejpam-6098	188	5	,	,	PUNCT
ejpam-6098	188	6	we	we	PRON
ejpam-6098	188	7	get	get	VERB
ejpam-6098	188	8	||2−3mf(2mu)−	||2−3mf(2mu)−	NOUN
ejpam-6098	188	9	f(u	f(u	PROPN
ejpam-6098	188	10	)	)	PUNCT
ejpam-6098	188	11	,	,	PUNCT
ejpam-6098	188	12	δ1	δ1	NOUN
ejpam-6098	188	13	,	,	PUNCT
ejpam-6098	188	14	...	...	PUNCT
ejpam-6098	188	15	,	,	PUNCT
ejpam-6098	188	16	δn−1||β	δn−1||β	PROPN
ejpam-6098	188	17	≤	≤	NUM
ejpam-6098	188	18	3−β	3−β	NUM
ejpam-6098	188	19	m∑	m∑	VERB
ejpam-6098	188	20	j=1	j=1	PROPN
ejpam-6098	188	21	23(j−m−1)βm(2m−ju	23(j−m−1)βm(2m−ju	PROPN
ejpam-6098	188	22	,	,	PUNCT
ejpam-6098	188	23	δ1	δ1	NOUN
ejpam-6098	188	24	,	,	PUNCT
ejpam-6098	188	25	...	...	PUNCT
ejpam-6098	188	26	,	,	PUNCT
ejpam-6098	188	27	δn−1	δn−1	PROPN
ejpam-6098	188	28	)	)	PUNCT
ejpam-6098	188	29	.	.	PUNCT
ejpam-6098	189	1	(	(	PUNCT
ejpam-6098	189	2	9	9	X
ejpam-6098	189	3	)	)	PUNCT
ejpam-6098	189	4	if	if	SCONJ
ejpam-6098	189	5	k	k	PROPN
ejpam-6098	189	6	,	,	PUNCT
ejpam-6098	189	7	m	m	VERB
ejpam-6098	189	8	∈	∈	NOUN
ejpam-6098	189	9	n	n	NOUN
ejpam-6098	189	10	with	with	ADP
ejpam-6098	189	11	k	k	PROPN
ejpam-6098	189	12	>	>	X
ejpam-6098	189	13	m	m	VERB
ejpam-6098	189	14	then	then	ADV
ejpam-6098	189	15	authors	author	NOUN
ejpam-6098	189	16	observe	observe	VERB
ejpam-6098	189	17	from	from	ADP
ejpam-6098	189	18	(	(	PUNCT
ejpam-6098	189	19	6	6	NUM
ejpam-6098	189	20	)	)	PUNCT
ejpam-6098	189	21	that	that	PRON
ejpam-6098	189	22	||2−3kf(2ku)−	||2−3kf(2ku)−	NOUN
ejpam-6098	189	23	2−3mf(2mu	2−3mf(2mu	NUM
ejpam-6098	189	24	)	)	PUNCT
ejpam-6098	189	25	,	,	PUNCT
ejpam-6098	189	26	δ1	δ1	NOUN
ejpam-6098	189	27	,	,	PUNCT
ejpam-6098	189	28	...	...	PUNCT
ejpam-6098	189	29	,	,	PUNCT
ejpam-6098	189	30	δn−1||β	δn−1||β	PROPN
ejpam-6098	189	31	≤	≤	NOUN
ejpam-6098	189	32	3−β	3−β	NUM
ejpam-6098	190	1	k−1∑	k−1∑	PROPN
ejpam-6098	190	2	j	j	PROPN
ejpam-6098	190	3	=	=	PROPN
ejpam-6098	190	4	m	m	PROPN
ejpam-6098	190	5	||2−3jf(2ju)−	||2−3jf(2ju)−	NOUN
ejpam-6098	190	6	2−3(j+1)f(2j+1u	2−3(j+1)f(2j+1u	NUM
ejpam-6098	190	7	)	)	PUNCT
ejpam-6098	190	8	,	,	PUNCT
ejpam-6098	190	9	δ1	δ1	NOUN
ejpam-6098	190	10	,	,	PUNCT
ejpam-6098	190	11	...	...	PUNCT
ejpam-6098	190	12	,	,	PUNCT
ejpam-6098	190	13	δn−1||β	δn−1||β	X
ejpam-6098	190	14	=	=	SYM
ejpam-6098	190	15	3−β	3−β	PROPN
ejpam-6098	190	16	k−1∑	k−1∑	PROPN
ejpam-6098	190	17	j	j	PROPN
ejpam-6098	190	18	=	=	PROPN
ejpam-6098	190	19	m	m	PROPN
ejpam-6098	190	20	2−3(j+1)β||23f(2ju)−	2−3(j+1)β||23f(2ju)−	NUM
ejpam-6098	190	21	f(2j+1u	f(2j+1u	NOUN
ejpam-6098	190	22	)	)	PUNCT
ejpam-6098	190	23	,	,	PUNCT
ejpam-6098	190	24	δ1	δ1	NOUN
ejpam-6098	190	25	,	,	PUNCT
ejpam-6098	190	26	...	...	PUNCT
ejpam-6098	190	27	,	,	PUNCT
ejpam-6098	190	28	δn−1||β	δn−1||β	PROPN
ejpam-6098	190	29	≤	≤	NOUN
ejpam-6098	190	30	3−β	3−β	NUM
ejpam-6098	191	1	k−1∑	k−1∑	PROPN
ejpam-6098	191	2	j	j	PROPN
ejpam-6098	191	3	=	=	PROPN
ejpam-6098	191	4	m	m	NOUN
ejpam-6098	191	5	2−3(j+1)βm(2ju	2−3(j+1)βm(2ju	NUM
ejpam-6098	191	6	,	,	PUNCT
ejpam-6098	191	7	δ1	δ1	NOUN
ejpam-6098	191	8	,	,	PUNCT
ejpam-6098	191	9	...	...	PUNCT
ejpam-6098	191	10	,	,	PUNCT
ejpam-6098	191	11	δn−1	δn−1	PROPN
ejpam-6098	191	12	)	)	PUNCT
ejpam-6098	191	13	=	=	SYM
ejpam-6098	192	1	3−β	3−β	NUM
ejpam-6098	193	1	k−1∑	k−1∑	PROPN
ejpam-6098	193	2	j	j	PROPN
ejpam-6098	193	3	=	=	PROPN
ejpam-6098	193	4	m	m	PROPN
ejpam-6098	193	5	2−3(j+1)βω(u	2−3(j+1)βω(u	NUM
ejpam-6098	193	6	,	,	PUNCT
ejpam-6098	193	7	0	0	NUM
ejpam-6098	193	8	,	,	PUNCT
ejpam-6098	193	9	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	193	10	,	,	PUNCT
ejpam-6098	193	11	...	...	PUNCT
ejpam-6098	193	12	,	,	PUNCT
ejpam-6098	193	13	δn−1	δn−1	PROPN
ejpam-6098	193	14	)	)	PUNCT
ejpam-6098	193	15	.	.	PUNCT
ejpam-6098	194	1	j.	j.	PROPN
ejpam-6098	194	2	jakhar	jakhar	PROPN
ejpam-6098	194	3	et	et	PROPN
ejpam-6098	194	4	al	al	PROPN
ejpam-6098	194	5	.	.	PUNCT
ejpam-6098	194	6	/	/	SYM
ejpam-6098	194	7	eur	eur	PROPN
ejpam-6098	194	8	.	.	PUNCT
ejpam-6098	195	1	j.	j.	PROPN
ejpam-6098	195	2	pure	pure	PROPN
ejpam-6098	195	3	appl	appl	PROPN
ejpam-6098	195	4	.	.	PROPN
ejpam-6098	195	5	math	math	PROPN
ejpam-6098	195	6	,	,	PUNCT
ejpam-6098	195	7	18	18	NUM
ejpam-6098	195	8	(	(	PUNCT
ejpam-6098	195	9	2	2	NUM
ejpam-6098	195	10	)	)	PUNCT
ejpam-6098	195	11	(	(	PUNCT
ejpam-6098	195	12	2025	2025	NUM
ejpam-6098	195	13	)	)	PUNCT
ejpam-6098	195	14	,	,	PUNCT
ejpam-6098	195	15	6098	6098	NUM
ejpam-6098	195	16	8	8	NUM
ejpam-6098	195	17	of	of	ADP
ejpam-6098	195	18	23	23	NUM
ejpam-6098	195	19	by	by	ADP
ejpam-6098	195	20	using	use	VERB
ejpam-6098	195	21	lemma(1	lemma(1	PROPN
ejpam-6098	195	22	)	)	PUNCT
ejpam-6098	196	1	and	and	CCONJ
ejpam-6098	196	2	applying	apply	VERB
ejpam-6098	196	3	limit	limit	NOUN
ejpam-6098	196	4	as	as	ADP
ejpam-6098	196	5	m	m	PROPN
ejpam-6098	196	6	,	,	PUNCT
ejpam-6098	196	7	k	k	PROPN
ejpam-6098	196	8	→	→	SYM
ejpam-6098	196	9	∞	∞	PROPN
ejpam-6098	196	10	,	,	PUNCT
ejpam-6098	196	11	we	we	PRON
ejpam-6098	196	12	get	get	VERB
ejpam-6098	196	13	limm	limm	NOUN
ejpam-6098	196	14	,	,	PUNCT
ejpam-6098	196	15	k→∞||2−3kf(2ku)−	k→∞||2−3kf(2ku)−	VERB
ejpam-6098	196	16	2−3mf(2mu	2−3mf(2mu	NUM
ejpam-6098	196	17	)	)	PUNCT
ejpam-6098	196	18	,	,	PUNCT
ejpam-6098	196	19	δ1	δ1	NOUN
ejpam-6098	196	20	,	,	PUNCT
ejpam-6098	196	21	...	...	PUNCT
ejpam-6098	196	22	,	,	PUNCT
ejpam-6098	196	23	δn−1||β	δn−1||β	X
ejpam-6098	196	24	=	=	SYM
ejpam-6098	196	25	0	0	X
ejpam-6098	196	26	.	.	PUNCT
ejpam-6098	197	1	according	accord	VERB
ejpam-6098	197	2	to	to	ADP
ejpam-6098	197	3	definition(2	definition(2	NUM
ejpam-6098	197	4	)	)	PUNCT
ejpam-6098	197	5	,	,	PUNCT
ejpam-6098	197	6	we	we	PRON
ejpam-6098	197	7	say	say	VERB
ejpam-6098	197	8	that	that	SCONJ
ejpam-6098	197	9	{	{	PUNCT
ejpam-6098	197	10	2−3mf(2mu	2−3mf(2mu	NUM
ejpam-6098	197	11	)	)	PUNCT
ejpam-6098	197	12	}	}	PUNCT
ejpam-6098	197	13	is	be	AUX
ejpam-6098	197	14	a	a	DET
ejpam-6098	197	15	cauchy	cauchy	ADJ
ejpam-6098	197	16	sequence	sequence	NOUN
ejpam-6098	197	17	in	in	ADP
ejpam-6098	197	18	v	v	NOUN
ejpam-6098	197	19	(	(	PUNCT
ejpam-6098	197	20	complete	complete	ADJ
ejpam-6098	197	21	(	(	PUNCT
ejpam-6098	197	22	n	n	CCONJ
ejpam-6098	197	23	,	,	PUNCT
ejpam-6098	197	24	β)-ns	β)-ns	NOUN
ejpam-6098	197	25	)	)	PUNCT
ejpam-6098	197	26	,	,	PUNCT
ejpam-6098	197	27	therefore	therefore	ADV
ejpam-6098	197	28	this	this	DET
ejpam-6098	197	29	sequence	sequence	NOUN
ejpam-6098	197	30	converges	converge	VERB
ejpam-6098	197	31	to	to	ADP
ejpam-6098	197	32	the	the	DET
ejpam-6098	197	33	point	point	NOUN
ejpam-6098	197	34	q(u	q(u	NOUN
ejpam-6098	197	35	)	)	PUNCT
ejpam-6098	197	36	∈	∈	NOUN
ejpam-6098	197	37	v	v	NOUN
ejpam-6098	197	38	.	.	PUNCT
ejpam-6098	198	1	now	now	ADV
ejpam-6098	198	2	,	,	PUNCT
ejpam-6098	198	3	we	we	PRON
ejpam-6098	198	4	consider	consider	VERB
ejpam-6098	198	5	a	a	DET
ejpam-6098	198	6	mapping	mapping	NOUN
ejpam-6098	198	7	q	q	NOUN
ejpam-6098	198	8	:	:	PUNCT
ejpam-6098	198	9	u	u	X
ejpam-6098	198	10	→	→	SYM
ejpam-6098	198	11	v	v	ADP
ejpam-6098	198	12	such	such	ADJ
ejpam-6098	198	13	that	that	DET
ejpam-6098	198	14	q(u	q(u	NOUN
ejpam-6098	198	15	)	)	PUNCT
ejpam-6098	198	16	=	=	SYM
ejpam-6098	198	17	limm→∞2−3mf(2mu	limm→∞2−3mf(2mu	PROPN
ejpam-6098	198	18	)	)	PUNCT
ejpam-6098	198	19	.	.	PUNCT
ejpam-6098	199	1	replacing	replace	VERB
ejpam-6098	199	2	(	(	PUNCT
ejpam-6098	199	3	u	u	NOUN
ejpam-6098	199	4	,	,	PUNCT
ejpam-6098	199	5	v	v	NOUN
ejpam-6098	199	6	,	,	PUNCT
ejpam-6098	199	7	w	w	NOUN
ejpam-6098	199	8	)	)	PUNCT
ejpam-6098	199	9	by	by	ADP
ejpam-6098	199	10	(	(	PUNCT
ejpam-6098	199	11	2mu	2mu	ADJ
ejpam-6098	199	12	,	,	PUNCT
ejpam-6098	199	13	2mv	2mv	ADJ
ejpam-6098	199	14	,	,	PUNCT
ejpam-6098	199	15	2mw	2mw	ADJ
ejpam-6098	199	16	)	)	PUNCT
ejpam-6098	199	17	in	in	ADP
ejpam-6098	199	18	(	(	PUNCT
ejpam-6098	199	19	4	4	NUM
ejpam-6098	199	20	)	)	PUNCT
ejpam-6098	199	21	and	and	CCONJ
ejpam-6098	199	22	multiplying	multiply	VERB
ejpam-6098	199	23	both	both	DET
ejpam-6098	199	24	side	side	NOUN
ejpam-6098	199	25	by	by	ADP
ejpam-6098	199	26	1	1	NUM
ejpam-6098	199	27	23mβ	23mβ	NOUN
ejpam-6098	199	28	,	,	PUNCT
ejpam-6098	199	29	we	we	PRON
ejpam-6098	199	30	have	have	VERB
ejpam-6098	199	31	2−3mβ||f(2m(2u+	2−3mβ||f(2m(2u+	NUM
ejpam-6098	199	32	w	w	NOUN
ejpam-6098	199	33	+	+	NUM
ejpam-6098	199	34	v))−	v))−	NOUN
ejpam-6098	199	35	3f(2m(v	3f(2m(v	NOUN
ejpam-6098	200	1	+	+	CCONJ
ejpam-6098	200	2	w	w	PROPN
ejpam-6098	201	1	+	+	NUM
ejpam-6098	201	2	u))−	u))−	ADJ
ejpam-6098	201	3	f(2m(w	f(2m(w	PROPN
ejpam-6098	201	4	−	−	PROPN
ejpam-6098	201	5	u+	u+	NOUN
ejpam-6098	201	6	v	v	NOUN
ejpam-6098	201	7	)	)	PUNCT
ejpam-6098	201	8	)	)	PUNCT
ejpam-6098	201	9	−2f(2m(u+	−2f(2m(u+	NOUN
ejpam-6098	202	1	v))−	v))−	NOUN
ejpam-6098	202	2	2f(2m(w	2f(2m(w	NOUN
ejpam-6098	202	3	+	+	X
ejpam-6098	202	4	u	u	NOUN
ejpam-6098	202	5	)	)	PUNCT
ejpam-6098	202	6	)	)	PUNCT
ejpam-6098	203	1	+	+	CCONJ
ejpam-6098	203	2	6f(2m(u−	6f(2m(u−	NUM
ejpam-6098	203	3	v	v	NOUN
ejpam-6098	203	4	)	)	PUNCT
ejpam-6098	203	5	)	)	PUNCT
ejpam-6098	204	1	+	+	CCONJ
ejpam-6098	204	2	6f(2m(u−	6f(2m(u−	NUM
ejpam-6098	204	3	w	w	NOUN
ejpam-6098	204	4	)	)	PUNCT
ejpam-6098	204	5	)	)	PUNCT
ejpam-6098	204	6	+3f(2m(w	+3f(2m(w	NOUN
ejpam-6098	205	1	+	+	CCONJ
ejpam-6098	205	2	v))−	v))−	NOUN
ejpam-6098	205	3	2f(2m(2u−	2f(2m(2u−	NUM
ejpam-6098	205	4	v))−	v))−	NOUN
ejpam-6098	205	5	2f(2m(2u−	2f(2m(2u−	PROPN
ejpam-6098	205	6	w	w	NOUN
ejpam-6098	205	7	)	)	PUNCT
ejpam-6098	205	8	)	)	PUNCT
ejpam-6098	206	1	+	+	CCONJ
ejpam-6098	206	2	18f(2mu	18f(2mu	NUM
ejpam-6098	206	3	)	)	PUNCT
ejpam-6098	206	4	+	+	CCONJ
ejpam-6098	206	5	6f(2mv	6f(2mv	X
ejpam-6098	206	6	)	)	PUNCT
ejpam-6098	206	7	+6f(2mw	+6f(2mw	NUM
ejpam-6098	206	8	)	)	PUNCT
ejpam-6098	206	9	,	,	PUNCT
ejpam-6098	206	10	δ1	δ1	NOUN
ejpam-6098	206	11	,	,	PUNCT
ejpam-6098	206	12	...	...	PUNCT
ejpam-6098	206	13	,	,	PUNCT
ejpam-6098	206	14	δn−1||β	δn−1||β	NOUN
ejpam-6098	206	15	≤	≤	NUM
ejpam-6098	207	1	3−β2−3mβω(2mu	3−β2−3mβω(2mu	NUM
ejpam-6098	207	2	,	,	PUNCT
ejpam-6098	207	3	2mv	2mv	ADJ
ejpam-6098	207	4	,	,	PUNCT
ejpam-6098	207	5	2mw)ψ(δ1	2mw)ψ(δ1	NUM
ejpam-6098	207	6	,	,	PUNCT
ejpam-6098	207	7	...	...	PUNCT
ejpam-6098	207	8	,	,	PUNCT
ejpam-6098	207	9	δn−1	δn−1	PROPN
ejpam-6098	207	10	)	)	PUNCT
ejpam-6098	207	11	.	.	PUNCT
ejpam-6098	208	1	thus	thus	ADV
ejpam-6098	208	2	by	by	ADP
ejpam-6098	208	3	using	use	VERB
ejpam-6098	208	4	equation	equation	NOUN
ejpam-6098	208	5	(	(	PUNCT
ejpam-6098	208	6	3	3	NUM
ejpam-6098	208	7	)	)	PUNCT
ejpam-6098	208	8	and	and	CCONJ
ejpam-6098	208	9	lemma	lemma	PROPN
ejpam-6098	208	10	(	(	PUNCT
ejpam-6098	208	11	1	1	NUM
ejpam-6098	208	12	)	)	PUNCT
ejpam-6098	208	13	,	,	PUNCT
ejpam-6098	208	14	we	we	PRON
ejpam-6098	208	15	get	get	VERB
ejpam-6098	208	16	||q(2u+	||q(2u+	ADJ
ejpam-6098	208	17	w	w	NOUN
ejpam-6098	208	18	+	+	NOUN
ejpam-6098	208	19	v)−q(w	v)−q(w	NOUN
ejpam-6098	208	20	−	−	NOUN
ejpam-6098	208	21	u+	u+	NOUN
ejpam-6098	208	22	v)−	v)−	PROPN
ejpam-6098	208	23	3q(w	3q(w	NUM
ejpam-6098	208	24	+	+	CCONJ
ejpam-6098	208	25	u+	u+	NOUN
ejpam-6098	208	26	v)−	v)−	PROPN
ejpam-6098	208	27	2q(u+	2q(u+	PROPN
ejpam-6098	208	28	v	v	NOUN
ejpam-6098	208	29	)	)	PUNCT
ejpam-6098	209	1	+6q(u−	+6q(u−	PRON
ejpam-6098	209	2	v)−	v)−	PROPN
ejpam-6098	209	3	2q(w	2q(w	NOUN
ejpam-6098	209	4	+	+	CCONJ
ejpam-6098	209	5	u	u	NOUN
ejpam-6098	209	6	)	)	PUNCT
ejpam-6098	209	7	+	+	CCONJ
ejpam-6098	209	8	6q(u−	6q(u−	NUM
ejpam-6098	209	9	w	w	NOUN
ejpam-6098	209	10	)	)	PUNCT
ejpam-6098	209	11	+	+	NUM
ejpam-6098	209	12	3q(w	3q(w	NUM
ejpam-6098	209	13	+	+	CCONJ
ejpam-6098	209	14	v)−	v)−	PROPN
ejpam-6098	209	15	2q(2u−	2q(2u−	NUM
ejpam-6098	209	16	v	v	NOUN
ejpam-6098	209	17	)	)	PUNCT
ejpam-6098	209	18	−2q(2u−	−2q(2u−	PROPN
ejpam-6098	209	19	w	w	PROPN
ejpam-6098	209	20	)	)	PUNCT
ejpam-6098	209	21	+	+	NOUN
ejpam-6098	209	22	18q(u	18q(u	X
ejpam-6098	209	23	)	)	PUNCT
ejpam-6098	209	24	+	+	CCONJ
ejpam-6098	209	25	6q(v	6q(v	NOUN
ejpam-6098	209	26	)	)	PUNCT
ejpam-6098	210	1	+	+	CCONJ
ejpam-6098	210	2	6q(w	6q(w	NUM
ejpam-6098	210	3	)	)	PUNCT
ejpam-6098	210	4	,	,	PUNCT
ejpam-6098	210	5	v1	v1	NOUN
ejpam-6098	210	6	,	,	PUNCT
ejpam-6098	210	7	...	...	PUNCT
ejpam-6098	210	8	,	,	PUNCT
ejpam-6098	210	9	vn−1||β	vn−1||β	PROPN
ejpam-6098	210	10	=	=	PUNCT
ejpam-6098	210	11	limm→∞||2−3mf(2u+	limm→∞||2−3mf(2u+	PROPN
ejpam-6098	211	1	w	w	NOUN
ejpam-6098	211	2	+	+	NUM
ejpam-6098	211	3	v))−	v))−	NOUN
ejpam-6098	211	4	3.2−3mf(2m(w	3.2−3mf(2m(w	NUM
ejpam-6098	211	5	+	+	CCONJ
ejpam-6098	211	6	u+	u+	NUM
ejpam-6098	211	7	v	v	NOUN
ejpam-6098	211	8	)	)	PUNCT
ejpam-6098	211	9	)	)	PUNCT
ejpam-6098	212	1	−2−3mf(2m(w	−2−3mf(2m(w	PROPN
ejpam-6098	212	2	−	−	NOUN
ejpam-6098	212	3	u+	u+	NOUN
ejpam-6098	212	4	v))−	v))−	NOUN
ejpam-6098	212	5	2.2−3mf(2m(u+	2.2−3mf(2m(u+	PROPN
ejpam-6098	212	6	v	v	NOUN
ejpam-6098	212	7	)	)	PUNCT
ejpam-6098	212	8	)	)	PUNCT
ejpam-6098	213	1	−2.2−3mf(2m(u+	−2.2−3mf(2m(u+	PROPN
ejpam-6098	213	2	w	w	PROPN
ejpam-6098	213	3	)	)	PUNCT
ejpam-6098	213	4	)	)	PUNCT
ejpam-6098	214	1	+	+	CCONJ
ejpam-6098	214	2	6.2−3mf(2m(u−	6.2−3mf(2m(u−	NUM
ejpam-6098	214	3	v	v	NOUN
ejpam-6098	214	4	)	)	PUNCT
ejpam-6098	214	5	)	)	PUNCT
ejpam-6098	215	1	+	+	CCONJ
ejpam-6098	215	2	6.2−3mf(2m(u−	6.2−3mf(2m(u−	NUM
ejpam-6098	215	3	w	w	NOUN
ejpam-6098	215	4	)	)	PUNCT
ejpam-6098	215	5	)	)	PUNCT
ejpam-6098	215	6	−2.2−3mf(2m(2u−	−2.2−3mf(2m(2u−	PROPN
ejpam-6098	215	7	v	v	NOUN
ejpam-6098	215	8	)	)	PUNCT
ejpam-6098	215	9	)	)	PUNCT
ejpam-6098	216	1	+	+	CCONJ
ejpam-6098	216	2	3.2−3mf(2m(w	3.2−3mf(2m(w	NUM
ejpam-6098	216	3	+	+	NUM
ejpam-6098	216	4	v))−	v))−	NOUN
ejpam-6098	216	5	2f(2−3m(2u−	2f(2−3m(2u−	NUM
ejpam-6098	216	6	w	w	NOUN
ejpam-6098	216	7	)	)	PUNCT
ejpam-6098	216	8	)	)	PUNCT
ejpam-6098	217	1	+18.2−3mf(2mu	+18.2−3mf(2mu	X
ejpam-6098	217	2	)	)	PUNCT
ejpam-6098	218	1	+	+	CCONJ
ejpam-6098	218	2	6.2−3mf(2mv	6.2−3mf(2mv	X
ejpam-6098	218	3	)	)	PUNCT
ejpam-6098	219	1	+	+	CCONJ
ejpam-6098	219	2	6.2−3mf(2mw	6.2−3mf(2mw	NOUN
ejpam-6098	219	3	)	)	PUNCT
ejpam-6098	219	4	,	,	PUNCT
ejpam-6098	219	5	δ1	δ1	NOUN
ejpam-6098	219	6	,	,	PUNCT
ejpam-6098	219	7	...	...	PUNCT
ejpam-6098	219	8	,	,	PUNCT
ejpam-6098	219	9	δn−1||β	δn−1||β	PROPN
ejpam-6098	219	10	≤	≤	NUM
ejpam-6098	219	11	limm→∞3−β.2−3mβω(2mu	limm→∞3−β.2−3mβω(2mu	PROPN
ejpam-6098	219	12	,	,	PUNCT
ejpam-6098	219	13	2mv	2mv	ADJ
ejpam-6098	219	14	,	,	PUNCT
ejpam-6098	219	15	2mw)ψ(δ1	2mw)ψ(δ1	NUM
ejpam-6098	219	16	,	,	PUNCT
ejpam-6098	219	17	...	...	PUNCT
ejpam-6098	219	18	,	,	PUNCT
ejpam-6098	219	19	δn−1	δn−1	PROPN
ejpam-6098	219	20	)	)	PUNCT
ejpam-6098	219	21	=	=	SYM
ejpam-6098	220	1	0	0	X
ejpam-6098	220	2	.	.	PUNCT
ejpam-6098	221	1	hence	hence	ADV
ejpam-6098	221	2	,	,	PUNCT
ejpam-6098	221	3	||q(2u+	||q(2u+	ADJ
ejpam-6098	221	4	w	w	PROPN
ejpam-6098	221	5	+	+	PROPN
ejpam-6098	221	6	v)−	v)−	PROPN
ejpam-6098	221	7	3q(v	3q(v	PROPN
ejpam-6098	221	8	+	+	CCONJ
ejpam-6098	221	9	w	w	PROPN
ejpam-6098	221	10	+	+	CCONJ
ejpam-6098	221	11	u)−q(w	u)−q(w	NOUN
ejpam-6098	221	12	−	−	PROPN
ejpam-6098	221	13	u+	u+	NOUN
ejpam-6098	221	14	v)−	v)−	PROPN
ejpam-6098	221	15	2q(v	2q(v	PROPN
ejpam-6098	221	16	+	+	CCONJ
ejpam-6098	221	17	w	w	NOUN
ejpam-6098	221	18	)	)	PUNCT
ejpam-6098	221	19	+6q(u−	+6q(u−	PRON
ejpam-6098	221	20	v)−	v)−	PROPN
ejpam-6098	221	21	2q(w	2q(w	NOUN
ejpam-6098	221	22	+	+	CCONJ
ejpam-6098	221	23	u	u	NOUN
ejpam-6098	221	24	)	)	PUNCT
ejpam-6098	221	25	+	+	CCONJ
ejpam-6098	221	26	6q(u−	6q(u−	NUM
ejpam-6098	221	27	w	w	NOUN
ejpam-6098	221	28	)	)	PUNCT
ejpam-6098	221	29	+	+	NUM
ejpam-6098	221	30	3q(w	3q(w	NUM
ejpam-6098	221	31	+	+	CCONJ
ejpam-6098	221	32	v)−	v)−	PROPN
ejpam-6098	221	33	2q(2u−	2q(2u−	NUM
ejpam-6098	221	34	v	v	NOUN
ejpam-6098	221	35	)	)	PUNCT
ejpam-6098	221	36	−2q(2u−	−2q(2u−	PROPN
ejpam-6098	221	37	w	w	PROPN
ejpam-6098	221	38	)	)	PUNCT
ejpam-6098	221	39	+	+	CCONJ
ejpam-6098	221	40	6q(v	6q(v	NOUN
ejpam-6098	221	41	)	)	PUNCT
ejpam-6098	222	1	+	+	CCONJ
ejpam-6098	223	1	18q(u	18q(u	X
ejpam-6098	223	2	)	)	PUNCT
ejpam-6098	223	3	+	+	CCONJ
ejpam-6098	223	4	6q(w	6q(w	NUM
ejpam-6098	223	5	)	)	PUNCT
ejpam-6098	223	6	,	,	PUNCT
ejpam-6098	223	7	δ1	δ1	NOUN
ejpam-6098	223	8	,	,	PUNCT
ejpam-6098	223	9	...	...	PUNCT
ejpam-6098	223	10	,	,	PUNCT
ejpam-6098	223	11	δn−1||β	δn−1||β	X
ejpam-6098	223	12	=	=	SYM
ejpam-6098	223	13	0	0	X
ejpam-6098	223	14	.	.	PUNCT
ejpam-6098	224	1	in	in	ADP
ejpam-6098	224	2	(	(	PUNCT
ejpam-6098	224	3	9	9	NUM
ejpam-6098	224	4	)	)	PUNCT
ejpam-6098	224	5	,	,	PUNCT
ejpam-6098	224	6	putting	put	VERB
ejpam-6098	224	7	the	the	DET
ejpam-6098	224	8	limit	limit	NOUN
ejpam-6098	224	9	as	as	ADP
ejpam-6098	224	10	m→	m→	NOUN
ejpam-6098	224	11	∞	∞	PROPN
ejpam-6098	224	12	,	,	PUNCT
ejpam-6098	224	13	we	we	PRON
ejpam-6098	224	14	get	get	VERB
ejpam-6098	224	15	||q(u)−	||q(u)−	PROPN
ejpam-6098	224	16	f(u	f(u	PROPN
ejpam-6098	224	17	)	)	PUNCT
ejpam-6098	224	18	,	,	PUNCT
ejpam-6098	224	19	δ1	δ1	NOUN
ejpam-6098	224	20	,	,	PUNCT
ejpam-6098	224	21	...	...	PUNCT
ejpam-6098	224	22	,	,	PUNCT
ejpam-6098	224	23	δn−1||β	δn−1||β	PROPN
ejpam-6098	224	24	≤	≤	NUM
ejpam-6098	224	25	limm→∞	limm→∞	PROPN
ejpam-6098	224	26	m∑	m∑	ADP
ejpam-6098	224	27	j=1	j=1	PROPN
ejpam-6098	224	28	2j−1−mω(2m−ju	2j−1−mω(2m−ju	NUM
ejpam-6098	224	29	,	,	PUNCT
ejpam-6098	224	30	0	0	NUM
ejpam-6098	224	31	,	,	PUNCT
ejpam-6098	224	32	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	224	33	,	,	PUNCT
ejpam-6098	224	34	...	...	PUNCT
ejpam-6098	224	35	,	,	PUNCT
ejpam-6098	224	36	δn−1	δn−1	PROPN
ejpam-6098	224	37	)	)	PUNCT
ejpam-6098	224	38	=	=	SYM
ejpam-6098	224	39	φ(u)ψ(δ1	φ(u)ψ(δ1	PROPN
ejpam-6098	224	40	,	,	PUNCT
ejpam-6098	224	41	...	...	PUNCT
ejpam-6098	224	42	,	,	PUNCT
ejpam-6098	224	43	δn−1	δn−1	PROPN
ejpam-6098	224	44	)	)	PUNCT
ejpam-6098	224	45	j.	j.	PROPN
ejpam-6098	224	46	jakhar	jakhar	PROPN
ejpam-6098	224	47	et	et	PROPN
ejpam-6098	224	48	al	al	PROPN
ejpam-6098	224	49	.	.	PUNCT
ejpam-6098	224	50	/	/	SYM
ejpam-6098	224	51	eur	eur	PROPN
ejpam-6098	224	52	.	.	PUNCT
ejpam-6098	225	1	j.	j.	PROPN
ejpam-6098	225	2	pure	pure	PROPN
ejpam-6098	225	3	appl	appl	PROPN
ejpam-6098	225	4	.	.	PROPN
ejpam-6098	225	5	math	math	PROPN
ejpam-6098	225	6	,	,	PUNCT
ejpam-6098	225	7	18	18	NUM
ejpam-6098	225	8	(	(	PUNCT
ejpam-6098	225	9	2	2	NUM
ejpam-6098	225	10	)	)	PUNCT
ejpam-6098	225	11	(	(	PUNCT
ejpam-6098	225	12	2025	2025	NUM
ejpam-6098	225	13	)	)	PUNCT
ejpam-6098	225	14	,	,	PUNCT
ejpam-6098	225	15	6098	6098	NUM
ejpam-6098	225	16	9	9	NUM
ejpam-6098	225	17	of	of	ADP
ejpam-6098	225	18	23	23	NUM
ejpam-6098	225	19	which	which	PRON
ejpam-6098	225	20	shows	show	VERB
ejpam-6098	225	21	(	(	PUNCT
ejpam-6098	225	22	5	5	NUM
ejpam-6098	225	23	)	)	PUNCT
ejpam-6098	225	24	.	.	PUNCT
ejpam-6098	226	1	to	to	PART
ejpam-6098	226	2	establish	establish	VERB
ejpam-6098	226	3	uniqueness	uniqueness	NOUN
ejpam-6098	226	4	of	of	ADP
ejpam-6098	226	5	the	the	DET
ejpam-6098	226	6	mapping	mapping	NOUN
ejpam-6098	226	7	q	q	NOUN
ejpam-6098	226	8	,	,	PUNCT
ejpam-6098	226	9	suppose	suppose	VERB
ejpam-6098	226	10	another	another	DET
ejpam-6098	226	11	cubic	cubic	ADJ
ejpam-6098	226	12	mapping	mapping	NOUN
ejpam-6098	226	13	q′	q′	NOUN
ejpam-6098	226	14	:	:	PUNCT
ejpam-6098	227	1	u	u	X
ejpam-6098	227	2	→	→	SYM
ejpam-6098	227	3	v	v	NUM
ejpam-6098	227	4	satisfies	satisfie	NOUN
ejpam-6098	227	5	(	(	PUNCT
ejpam-6098	227	6	5	5	NUM
ejpam-6098	227	7	)	)	PUNCT
ejpam-6098	227	8	.	.	PUNCT
ejpam-6098	228	1	we	we	PRON
ejpam-6098	228	2	get	get	VERB
ejpam-6098	228	3	||q(u)−q′(u	||q(u)−q′(u	NOUN
ejpam-6098	228	4	)	)	PUNCT
ejpam-6098	228	5	,	,	PUNCT
ejpam-6098	228	6	v1	v1	NOUN
ejpam-6098	228	7	,	,	PUNCT
ejpam-6098	228	8	...	...	PUNCT
ejpam-6098	228	9	,	,	PUNCT
ejpam-6098	228	10	vn−1||β	vn−1||β	NOUN
ejpam-6098	228	11	≤	≤	NOUN
ejpam-6098	228	12	3−β.2−mβ||q(2mu)−	3−β.2−mβ||q(2mu)−	NUM
ejpam-6098	228	13	f(2mu	f(2mu	NOUN
ejpam-6098	228	14	)	)	PUNCT
ejpam-6098	228	15	,	,	PUNCT
ejpam-6098	228	16	δ1	δ1	NOUN
ejpam-6098	228	17	,	,	PUNCT
ejpam-6098	228	18	...	...	PUNCT
ejpam-6098	228	19	,	,	PUNCT
ejpam-6098	228	20	δn−1||β	δn−1||β	X
ejpam-6098	228	21	+3−β.2−mβ||f(2mu)−q′(2mu	+3−β.2−mβ||f(2mu)−q′(2mu	ADJ
ejpam-6098	228	22	)	)	PUNCT
ejpam-6098	228	23	,	,	PUNCT
ejpam-6098	228	24	δ1	δ1	NOUN
ejpam-6098	228	25	,	,	PUNCT
ejpam-6098	228	26	...	...	PUNCT
ejpam-6098	228	27	,	,	PUNCT
ejpam-6098	228	28	δn−1||β	δn−1||β	VERB
ejpam-6098	228	29	≤	≤	NOUN
ejpam-6098	228	30	3−β.2−mβ+1φ(2mu)ψ(δ1	3−β.2−mβ+1φ(2mu)ψ(δ1	NUM
ejpam-6098	228	31	,	,	PUNCT
ejpam-6098	228	32	...	...	PUNCT
ejpam-6098	228	33	,	,	PUNCT
ejpam-6098	228	34	δn−1	δn−1	PROPN
ejpam-6098	228	35	)	)	PUNCT
ejpam-6098	228	36	→	→	SYM
ejpam-6098	228	37	0	0	PUNCT
ejpam-6098	229	1	as	as	ADP
ejpam-6098	229	2	m→	m→	NOUN
ejpam-6098	229	3	∞	∞	PROPN
ejpam-6098	229	4	,	,	PUNCT
ejpam-6098	229	5	now	now	ADV
ejpam-6098	229	6	,	,	PUNCT
ejpam-6098	229	7	with	with	ADP
ejpam-6098	229	8	the	the	DET
ejpam-6098	229	9	help	help	NOUN
ejpam-6098	229	10	of	of	ADP
ejpam-6098	229	11	lemma(1	lemma(1	PROPN
ejpam-6098	229	12	)	)	PUNCT
ejpam-6098	229	13	,	,	PUNCT
ejpam-6098	229	14	it	it	PRON
ejpam-6098	229	15	is	be	AUX
ejpam-6098	229	16	proved	prove	VERB
ejpam-6098	229	17	that	that	SCONJ
ejpam-6098	229	18	q(u	q(u	ADP
ejpam-6098	229	19	)	)	PUNCT
ejpam-6098	229	20	=	=	SYM
ejpam-6098	229	21	q′(u	q′(u	PROPN
ejpam-6098	229	22	)	)	PUNCT
ejpam-6098	229	23	.	.	PUNCT
ejpam-6098	230	1	corollary	corollary	ADJ
ejpam-6098	230	2	1	1	NUM
ejpam-6098	230	3	.	.	PUNCT
ejpam-6098	231	1	if	if	SCONJ
ejpam-6098	231	2	f	f	PROPN
ejpam-6098	231	3	:	:	PUNCT
ejpam-6098	231	4	u	u	X
ejpam-6098	231	5	→	→	SYM
ejpam-6098	231	6	v	v	PROPN
ejpam-6098	231	7	is	be	AUX
ejpam-6098	231	8	a	a	DET
ejpam-6098	231	9	function	function	NOUN
ejpam-6098	231	10	satisfying	satisfy	VERB
ejpam-6098	231	11	∥f(2u+	∥f(2u+	PROPN
ejpam-6098	231	12	v	v	NOUN
ejpam-6098	231	13	+	+	CCONJ
ejpam-6098	231	14	w)−	w)−	PROPN
ejpam-6098	231	15	3f(u+	3f(u+	PROPN
ejpam-6098	231	16	v	v	NOUN
ejpam-6098	231	17	+	+	CCONJ
ejpam-6098	231	18	w)−	w)−	PROPN
ejpam-6098	231	19	f(−u+	f(−u+	NOUN
ejpam-6098	231	20	v	v	ADP
ejpam-6098	231	21	+	+	CCONJ
ejpam-6098	231	22	w)−	w)−	PROPN
ejpam-6098	231	23	2f(u+	2f(u+	PROPN
ejpam-6098	231	24	v)−	v)−	PROPN
ejpam-6098	231	25	2f(u+	2f(u+	PROPN
ejpam-6098	231	26	w	w	PROPN
ejpam-6098	231	27	)	)	PUNCT
ejpam-6098	231	28	+6f(u−	+6f(u−	PROPN
ejpam-6098	231	29	v	v	NOUN
ejpam-6098	231	30	)	)	PUNCT
ejpam-6098	231	31	+	+	CCONJ
ejpam-6098	231	32	6f(u−	6f(u−	NUM
ejpam-6098	231	33	w	w	NOUN
ejpam-6098	231	34	)	)	PUNCT
ejpam-6098	231	35	+	+	NUM
ejpam-6098	231	36	3f(v	3f(v	NUM
ejpam-6098	232	1	+	+	CCONJ
ejpam-6098	232	2	w)−	w)−	PROPN
ejpam-6098	232	3	2f(2u−	2f(2u−	NUM
ejpam-6098	232	4	v)−	v)−	PROPN
ejpam-6098	232	5	2f(2u−	2f(2u−	NUM
ejpam-6098	232	6	w	w	NOUN
ejpam-6098	232	7	)	)	PUNCT
ejpam-6098	232	8	+	+	CCONJ
ejpam-6098	232	9	18f(u	18f(u	NUM
ejpam-6098	232	10	)	)	PUNCT
ejpam-6098	232	11	+6f(v	+6f(v	NOUN
ejpam-6098	232	12	)	)	PUNCT
ejpam-6098	233	1	+	+	CCONJ
ejpam-6098	233	2	6f(w	6f(w	NUM
ejpam-6098	233	3	)	)	PUNCT
ejpam-6098	233	4	,	,	PUNCT
ejpam-6098	233	5	δ1	δ1	NOUN
ejpam-6098	233	6	,	,	PUNCT
ejpam-6098	233	7	...	...	PUNCT
ejpam-6098	233	8	,	,	PUNCT
ejpam-6098	233	9	δn−1∥β	δn−1∥β	VERB
ejpam-6098	233	10	≤	≤	NUM
ejpam-6098	234	1	ϵ(||u||β	ϵ(||u||β	NOUN
ejpam-6098	235	1	+	+	CCONJ
ejpam-6098	235	2	||v||β	||v||β	NOUN
ejpam-6098	235	3	+	+	CCONJ
ejpam-6098	235	4	||w||β	||w||β	NOUN
ejpam-6098	235	5	)	)	PUNCT
ejpam-6098	235	6	,	,	PUNCT
ejpam-6098	235	7	then	then	ADV
ejpam-6098	235	8	there	there	PRON
ejpam-6098	235	9	is	be	VERB
ejpam-6098	235	10	a	a	DET
ejpam-6098	235	11	unique	unique	ADJ
ejpam-6098	235	12	cubic	cubic	ADJ
ejpam-6098	235	13	function	function	NOUN
ejpam-6098	235	14	q	q	NOUN
ejpam-6098	235	15	:	:	PUNCT
ejpam-6098	235	16	u	u	X
ejpam-6098	235	17	→	→	SYM
ejpam-6098	235	18	v	v	NUM
ejpam-6098	235	19	holding	hold	VERB
ejpam-6098	235	20	||f(u)−q(u	||f(u)−q(u	NOUN
ejpam-6098	235	21	)	)	PUNCT
ejpam-6098	235	22	,	,	PUNCT
ejpam-6098	235	23	δ1	δ1	NOUN
ejpam-6098	235	24	,	,	PUNCT
ejpam-6098	235	25	...	...	PUNCT
ejpam-6098	235	26	,	,	PUNCT
ejpam-6098	235	27	δn−1||β	δn−1||β	NOUN
ejpam-6098	235	28	≤	≤	NOUN
ejpam-6098	236	1	ϵ	ϵ	ADP
ejpam-6098	236	2	1	1	NUM
ejpam-6098	236	3	6β(22β	6β(22β	NUM
ejpam-6098	236	4	−	−	NOUN
ejpam-6098	236	5	1	1	X
ejpam-6098	236	6	)	)	PUNCT
ejpam-6098	236	7	||u||β	||u||β	NOUN
ejpam-6098	236	8	.	.	PUNCT
ejpam-6098	237	1	proof	proof	NOUN
ejpam-6098	237	2	.	.	PUNCT
ejpam-6098	238	1	put	put	VERB
ejpam-6098	238	2	ω(u	ω(u	PROPN
ejpam-6098	238	3	,	,	PUNCT
ejpam-6098	238	4	0	0	NUM
ejpam-6098	238	5	,	,	PUNCT
ejpam-6098	238	6	0	0	NUM
ejpam-6098	238	7	)	)	PUNCT
ejpam-6098	238	8	=	=	SYM
ejpam-6098	238	9	(	(	PUNCT
ejpam-6098	238	10	||u||β+||v||β+||w||β	||u||β+||v||β+||w||β	NOUN
ejpam-6098	238	11	)	)	PUNCT
ejpam-6098	238	12	and	and	CCONJ
ejpam-6098	238	13	ψ(δ1	ψ(δ1	VERB
ejpam-6098	238	14	,	,	PUNCT
ejpam-6098	238	15	...	...	PUNCT
ejpam-6098	238	16	,	,	PUNCT
ejpam-6098	238	17	δn−1	δn−1	PROPN
ejpam-6098	238	18	)	)	PUNCT
ejpam-6098	238	19	=	=	SYM
ejpam-6098	239	1	ϵ	ϵ	X
ejpam-6098	239	2	in	in	ADP
ejpam-6098	239	3	above	above	ADP
ejpam-6098	239	4	theorem	theorem	VERB
ejpam-6098	239	5	,	,	PUNCT
ejpam-6098	239	6	we	we	PRON
ejpam-6098	239	7	get	get	VERB
ejpam-6098	239	8	the	the	DET
ejpam-6098	239	9	intended	intended	ADJ
ejpam-6098	239	10	outcome	outcome	NOUN
ejpam-6098	239	11	.	.	PUNCT
ejpam-6098	240	1	example	example	NOUN
ejpam-6098	241	1	1	1	NUM
ejpam-6098	241	2	.	.	PUNCT
ejpam-6098	241	3	let	let	VERB
ejpam-6098	241	4	ϕ	ϕ	NOUN
ejpam-6098	241	5	:	:	PUNCT
ejpam-6098	241	6	v	v	ADP
ejpam-6098	241	7	n−1	n−1	PROPN
ejpam-6098	241	8	→	→	SYM
ejpam-6098	241	9	[	[	X
ejpam-6098	241	10	0,∞	0,∞	NOUN
ejpam-6098	241	11	)	)	PUNCT
ejpam-6098	241	12	be	be	VERB
ejpam-6098	241	13	a	a	DET
ejpam-6098	241	14	constant	constant	ADJ
ejpam-6098	241	15	mapping	mapping	NOUN
ejpam-6098	241	16	such	such	ADJ
ejpam-6098	241	17	that	that	SCONJ
ejpam-6098	241	18	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	241	19	,	,	PUNCT
ejpam-6098	241	20	...	...	PUNCT
ejpam-6098	241	21	,	,	PUNCT
ejpam-6098	241	22	δn−1	δn−1	PROPN
ejpam-6098	241	23	)	)	PUNCT
ejpam-6098	241	24	=	=	SYM
ejpam-6098	241	25	1	1	NUM
ejpam-6098	241	26	for	for	ADP
ejpam-6098	241	27	all	all	DET
ejpam-6098	241	28	δ1	δ1	NOUN
ejpam-6098	241	29	,	,	PUNCT
ejpam-6098	241	30	...	...	PUNCT
ejpam-6098	241	31	δn−1	δn−1	PROPN
ejpam-6098	241	32	∈	∈	PROPN
ejpam-6098	241	33	v	v	NOUN
ejpam-6098	241	34	and	and	CCONJ
ejpam-6098	241	35	f	f	NOUN
ejpam-6098	241	36	:	:	PUNCT
ejpam-6098	241	37	u	u	X
ejpam-6098	241	38	→	→	SYM
ejpam-6098	241	39	v	v	NUM
ejpam-6098	241	40	be	be	AUX
ejpam-6098	241	41	a	a	DET
ejpam-6098	241	42	mapping	mapping	NOUN
ejpam-6098	241	43	defined	define	VERB
ejpam-6098	241	44	as	as	ADP
ejpam-6098	241	45	f(u	f(u	PROPN
ejpam-6098	241	46	)	)	PUNCT
ejpam-6098	241	47	=	=	PROPN
ejpam-6098	241	48	u3	u3	PROPN
ejpam-6098	241	49	+	+	CCONJ
ejpam-6098	241	50	||u||βu0	||u||βu0	PROPN
ejpam-6098	241	51	,	,	PUNCT
ejpam-6098	241	52	where	where	SCONJ
ejpam-6098	241	53	u0	u0	ADJ
ejpam-6098	241	54	is	be	AUX
ejpam-6098	241	55	the	the	DET
ejpam-6098	241	56	unit	unit	NOUN
ejpam-6098	241	57	vector	vector	NOUN
ejpam-6098	241	58	in	in	ADP
ejpam-6098	241	59	u	u	PROPN
ejpam-6098	241	60	.	.	PUNCT
ejpam-6098	242	1	an	an	DET
ejpam-6098	242	2	easy	easy	ADJ
ejpam-6098	242	3	calculation	calculation	NOUN
ejpam-6098	242	4	demonstrates	demonstrate	VERB
ejpam-6098	242	5	that	that	PRON
ejpam-6098	242	6	∥f(2u+	∥f(2u+	PROPN
ejpam-6098	243	1	w	w	PROPN
ejpam-6098	244	1	+	+	CCONJ
ejpam-6098	245	1	v)−	v)−	PROPN
ejpam-6098	245	2	f(w	f(w	PROPN
ejpam-6098	245	3	−	−	PROPN
ejpam-6098	245	4	u+	u+	NOUN
ejpam-6098	245	5	v)−	v)−	PROPN
ejpam-6098	245	6	3f(w	3f(w	NOUN
ejpam-6098	245	7	+	+	CCONJ
ejpam-6098	245	8	u+	u+	PROPN
ejpam-6098	245	9	v)−	v)−	PROPN
ejpam-6098	245	10	2f(u+	2f(u+	PROPN
ejpam-6098	245	11	v)−	v)−	PROPN
ejpam-6098	245	12	2f(w	2f(w	PROPN
ejpam-6098	245	13	+	+	NUM
ejpam-6098	245	14	u	u	NOUN
ejpam-6098	245	15	)	)	PUNCT
ejpam-6098	245	16	+6f(u−	+6f(u−	PROPN
ejpam-6098	245	17	v	v	NOUN
ejpam-6098	245	18	)	)	PUNCT
ejpam-6098	245	19	+	+	CCONJ
ejpam-6098	245	20	6f(u−	6f(u−	NUM
ejpam-6098	245	21	w)−	w)−	PROPN
ejpam-6098	245	22	2f(2u−	2f(2u−	NUM
ejpam-6098	245	23	v	v	NOUN
ejpam-6098	245	24	)	)	PUNCT
ejpam-6098	246	1	+	+	CCONJ
ejpam-6098	246	2	3f(w	3f(w	NOUN
ejpam-6098	246	3	+	+	CCONJ
ejpam-6098	246	4	v)−	v)−	PROPN
ejpam-6098	246	5	2f(2u−	2f(2u−	NUM
ejpam-6098	246	6	w	w	NOUN
ejpam-6098	246	7	)	)	PUNCT
ejpam-6098	246	8	+	+	CCONJ
ejpam-6098	246	9	18f(u	18f(u	NUM
ejpam-6098	246	10	)	)	PUNCT
ejpam-6098	246	11	+6f(v	+6f(v	NOUN
ejpam-6098	246	12	)	)	PUNCT
ejpam-6098	247	1	+	+	CCONJ
ejpam-6098	247	2	6f(w	6f(w	NUM
ejpam-6098	247	3	)	)	PUNCT
ejpam-6098	247	4	,	,	PUNCT
ejpam-6098	247	5	δ1	δ1	NOUN
ejpam-6098	247	6	,	,	PUNCT
ejpam-6098	247	7	...	...	PUNCT
ejpam-6098	247	8	,	,	PUNCT
ejpam-6098	247	9	δn−1∥β	δn−1∥β	NOUN
ejpam-6098	247	10	≤	≤	NUM
ejpam-6098	247	11	(	(	PUNCT
ejpam-6098	247	12	48||u||β	48||u||β	NOUN
ejpam-6098	247	13	+	+	NOUN
ejpam-6098	247	14	24||v||β	24||v||β	NUM
ejpam-6098	247	15	+	+	CCONJ
ejpam-6098	247	16	24||w||β)u0	24||w||β)u0	ADV
ejpam-6098	247	17	,	,	PUNCT
ejpam-6098	247	18	and	and	CCONJ
ejpam-6098	247	19	ϕ(u	ϕ(u	NOUN
ejpam-6098	247	20	)	)	PUNCT
ejpam-6098	248	1	=	=	PUNCT
ejpam-6098	249	1	∞∑	∞∑	NUM
ejpam-6098	249	2	j=1	j=1	NOUN
ejpam-6098	249	3	1	1	NUM
ejpam-6098	249	4	23jβ	23jβ	NOUN
ejpam-6098	249	5	(	(	PUNCT
ejpam-6098	249	6	48||2j−1u||βu0	48||2j−1u||βu0	NUM
ejpam-6098	249	7	)	)	PUNCT
ejpam-6098	249	8	=	=	PUNCT
ejpam-6098	249	9	24β	24β	NOUN
ejpam-6098	249	10	4β	4β	NUM
ejpam-6098	249	11	−	−	PROPN
ejpam-6098	249	12	1	1	NUM
ejpam-6098	249	13	||u||βu0	||u||βu0	PROPN
ejpam-6098	249	14	,	,	PUNCT
ejpam-6098	249	15	lim	lim	PROPN
ejpam-6098	249	16	m→∞	m→∞	NOUN
ejpam-6098	249	17	1	1	NUM
ejpam-6098	249	18	23mβ	23mβ	NOUN
ejpam-6098	249	19	(	(	PUNCT
ejpam-6098	249	20	||2mu||βu0	||2mu||βu0	NOUN
ejpam-6098	249	21	)	)	PUNCT
ejpam-6098	249	22	=	=	SYM
ejpam-6098	250	1	0	0	X
ejpam-6098	250	2	.	.	PUNCT
ejpam-6098	251	1	consequently	consequently	ADV
ejpam-6098	251	2	,	,	PUNCT
ejpam-6098	251	3	all	all	DET
ejpam-6098	251	4	the	the	DET
ejpam-6098	251	5	requirements	requirement	NOUN
ejpam-6098	251	6	of	of	ADP
ejpam-6098	251	7	theorem	theorem	ADJ
ejpam-6098	251	8	2.1	2.1	NUM
ejpam-6098	251	9	are	be	AUX
ejpam-6098	251	10	satisfied	satisfied	ADJ
ejpam-6098	251	11	,	,	PUNCT
ejpam-6098	251	12	implying	imply	VERB
ejpam-6098	251	13	the	the	DET
ejpam-6098	251	14	existence	existence	NOUN
ejpam-6098	251	15	of	of	ADP
ejpam-6098	251	16	a	a	DET
ejpam-6098	251	17	unique	unique	ADJ
ejpam-6098	251	18	cubic	cubic	ADJ
ejpam-6098	251	19	mapping	mapping	NOUN
ejpam-6098	251	20	q	q	NOUN
ejpam-6098	251	21	:	:	PUNCT
ejpam-6098	251	22	u	u	X
ejpam-6098	251	23	→	→	SYM
ejpam-6098	251	24	v	v	ADP
ejpam-6098	251	25	such	such	ADJ
ejpam-6098	251	26	that	that	PRON
ejpam-6098	251	27	||f(u)−q(u	||f(u)−q(u	NOUN
ejpam-6098	251	28	)	)	PUNCT
ejpam-6098	251	29	,	,	PUNCT
ejpam-6098	251	30	δ1	δ1	NOUN
ejpam-6098	251	31	,	,	PUNCT
ejpam-6098	251	32	...	...	PUNCT
ejpam-6098	251	33	δn−1||β	δn−1||β	NOUN
ejpam-6098	251	34	≤	≤	NOUN
ejpam-6098	251	35	8β	8β	NUM
ejpam-6098	251	36	4β	4β	NOUN
ejpam-6098	251	37	−	−	PROPN
ejpam-6098	251	38	1	1	NUM
ejpam-6098	251	39	||u||βu0	||u||βu0	PROPN
ejpam-6098	251	40	.	.	PUNCT
ejpam-6098	252	1	j.	j.	PROPN
ejpam-6098	252	2	jakhar	jakhar	PROPN
ejpam-6098	252	3	et	et	PROPN
ejpam-6098	252	4	al	al	PROPN
ejpam-6098	252	5	.	.	PUNCT
ejpam-6098	252	6	/	/	SYM
ejpam-6098	252	7	eur	eur	PROPN
ejpam-6098	252	8	.	.	PUNCT
ejpam-6098	253	1	j.	j.	PROPN
ejpam-6098	253	2	pure	pure	PROPN
ejpam-6098	253	3	appl	appl	PROPN
ejpam-6098	253	4	.	.	PROPN
ejpam-6098	253	5	math	math	PROPN
ejpam-6098	253	6	,	,	PUNCT
ejpam-6098	253	7	18	18	NUM
ejpam-6098	253	8	(	(	PUNCT
ejpam-6098	253	9	2	2	NUM
ejpam-6098	253	10	)	)	PUNCT
ejpam-6098	253	11	(	(	PUNCT
ejpam-6098	253	12	2025	2025	NUM
ejpam-6098	253	13	)	)	PUNCT
ejpam-6098	253	14	,	,	PUNCT
ejpam-6098	253	15	6098	6098	NUM
ejpam-6098	253	16	10	10	NUM
ejpam-6098	253	17	of	of	ADP
ejpam-6098	253	18	23	23	NUM
ejpam-6098	253	19	3	3	NUM
ejpam-6098	253	20	.	.	PUNCT
ejpam-6098	253	21	stability	stability	NOUN
ejpam-6098	253	22	in	in	ADP
ejpam-6098	253	23	non	non	ADJ
ejpam-6098	253	24	-	-	ADJ
ejpam-6098	253	25	archimedean-(n	archimedean-(n	ADJ
ejpam-6098	253	26	,	,	PUNCT
ejpam-6098	253	27	β)-normed	β)-normed	PUNCT
ejpam-6098	253	28	spaces	space	NOUN
ejpam-6098	253	29	in	in	ADP
ejpam-6098	253	30	na-(n	na-(n	PROPN
ejpam-6098	253	31	,	,	PUNCT
ejpam-6098	253	32	β)-ns	β)-ns	NOUN
ejpam-6098	253	33	,	,	PUNCT
ejpam-6098	253	34	the	the	DET
ejpam-6098	253	35	triangular	triangular	NOUN
ejpam-6098	253	36	property	property	NOUN
ejpam-6098	253	37	is	be	AUX
ejpam-6098	253	38	modified	modify	VERB
ejpam-6098	253	39	such	such	ADJ
ejpam-6098	253	40	that	that	SCONJ
ejpam-6098	253	41	the	the	DET
ejpam-6098	253	42	distance	distance	NOUN
ejpam-6098	253	43	between	between	ADP
ejpam-6098	253	44	two	two	NUM
ejpam-6098	253	45	points	point	NOUN
ejpam-6098	253	46	can	can	AUX
ejpam-6098	253	47	be	be	AUX
ejpam-6098	253	48	dominated	dominate	VERB
ejpam-6098	253	49	by	by	ADP
ejpam-6098	253	50	the	the	DET
ejpam-6098	253	51	maximum	maximum	ADJ
ejpam-6098	253	52	magnitudes	magnitude	NOUN
ejpam-6098	253	53	of	of	ADP
ejpam-6098	253	54	the	the	DET
ejpam-6098	253	55	vectors	vector	NOUN
ejpam-6098	253	56	rather	rather	ADV
ejpam-6098	253	57	than	than	ADP
ejpam-6098	253	58	their	their	PRON
ejpam-6098	253	59	sum	sum	NOUN
ejpam-6098	253	60	of	of	ADP
ejpam-6098	253	61	two	two	NUM
ejpam-6098	253	62	distances	distance	NOUN
ejpam-6098	253	63	.	.	PUNCT
ejpam-6098	254	1	this	this	DET
ejpam-6098	254	2	space	space	NOUN
ejpam-6098	254	3	provides	provide	VERB
ejpam-6098	254	4	a	a	DET
ejpam-6098	254	5	framework	framework	NOUN
ejpam-6098	254	6	for	for	ADP
ejpam-6098	254	7	analyzing	analyze	VERB
ejpam-6098	254	8	the	the	DET
ejpam-6098	254	9	stability	stability	NOUN
ejpam-6098	254	10	of	of	ADP
ejpam-6098	254	11	solutions	solution	NOUN
ejpam-6098	254	12	to	to	PART
ejpam-6098	254	13	fes	fes	VERB
ejpam-6098	254	14	over	over	ADP
ejpam-6098	254	15	na	na	ADP
ejpam-6098	254	16	fields	field	NOUN
ejpam-6098	254	17	such	such	ADJ
ejpam-6098	254	18	as	as	ADP
ejpam-6098	254	19	the	the	DET
ejpam-6098	254	20	p	p	NOUN
ejpam-6098	254	21	-	-	PUNCT
ejpam-6098	254	22	adic	adic	ADJ
ejpam-6098	254	23	numbers	number	NOUN
ejpam-6098	254	24	.	.	PUNCT
ejpam-6098	255	1	also	also	ADV
ejpam-6098	255	2	,	,	PUNCT
ejpam-6098	255	3	the	the	DET
ejpam-6098	255	4	convergence	convergence	NOUN
ejpam-6098	255	5	behavior	behavior	NOUN
ejpam-6098	255	6	in	in	ADP
ejpam-6098	255	7	this	this	DET
ejpam-6098	255	8	space	space	NOUN
ejpam-6098	255	9	helps	help	VERB
ejpam-6098	255	10	characterize	characterize	VERB
ejpam-6098	255	11	the	the	DET
ejpam-6098	255	12	behavior	behavior	NOUN
ejpam-6098	255	13	of	of	ADP
ejpam-6098	255	14	solutions	solution	NOUN
ejpam-6098	255	15	and	and	CCONJ
ejpam-6098	255	16	their	their	PRON
ejpam-6098	255	17	limiting	limit	VERB
ejpam-6098	255	18	properties	property	NOUN
ejpam-6098	255	19	.	.	PUNCT
ejpam-6098	256	1	by	by	ADP
ejpam-6098	256	2	motivating	motivate	VERB
ejpam-6098	256	3	the	the	DET
ejpam-6098	256	4	approach	approach	NOUN
ejpam-6098	256	5	used	use	VERB
ejpam-6098	256	6	in	in	ADP
ejpam-6098	256	7	[	[	X
ejpam-6098	256	8	20	20	NUM
ejpam-6098	256	9	]	]	PUNCT
ejpam-6098	256	10	,	,	PUNCT
ejpam-6098	256	11	we	we	PRON
ejpam-6098	256	12	find	find	VERB
ejpam-6098	256	13	stability	stability	NOUN
ejpam-6098	256	14	problems	problem	NOUN
ejpam-6098	256	15	for	for	ADP
ejpam-6098	256	16	fe	fe	X
ejpam-6098	256	17	(	(	PUNCT
ejpam-6098	256	18	1	1	NUM
ejpam-6098	256	19	)	)	PUNCT
ejpam-6098	256	20	in	in	ADP
ejpam-6098	256	21	na-(n	na-(n	PROPN
ejpam-6098	256	22	,	,	PUNCT
ejpam-6098	256	23	β)ns	β)ns	PROPN
ejpam-6098	256	24	.	.	PROPN
ejpam-6098	257	1	in	in	ADP
ejpam-6098	257	2	the	the	DET
ejpam-6098	257	3	following	following	NOUN
ejpam-6098	257	4	theorem	theorem	VERB
ejpam-6098	257	5	2	2	NUM
ejpam-6098	257	6	the	the	PRON
ejpam-6098	257	7	,	,	PUNCT
ejpam-6098	257	8	θ	θ	PROPN
ejpam-6098	257	9	≥	≥	NUM
ejpam-6098	257	10	0	0	NUM
ejpam-6098	257	11	:	:	PUNCT
ejpam-6098	257	12	a	a	DET
ejpam-6098	257	13	constant	constant	ADJ
ejpam-6098	257	14	controlling	control	VERB
ejpam-6098	257	15	the	the	DET
ejpam-6098	257	16	“	"	PUNCT
ejpam-6098	257	17	degree	degree	NOUN
ejpam-6098	257	18	”	"	PUNCT
ejpam-6098	257	19	of	of	ADP
ejpam-6098	257	20	approximation	approximation	NOUN
ejpam-6098	257	21	.	.	PUNCT
ejpam-6098	258	1	p	p	X
ejpam-6098	258	2	,	,	PUNCT
ejpam-6098	258	3	r	r	NOUN
ejpam-6098	258	4	,	,	PUNCT
ejpam-6098	258	5	q	q	NOUN
ejpam-6098	258	6	>	>	X
ejpam-6098	258	7	0	0	PUNCT
ejpam-6098	258	8	:	:	PUNCT
ejpam-6098	258	9	exponents	exponent	NOUN
ejpam-6098	258	10	that	that	PRON
ejpam-6098	258	11	influence	influence	VERB
ejpam-6098	258	12	how	how	SCONJ
ejpam-6098	258	13	the	the	DET
ejpam-6098	258	14	right	right	ADJ
ejpam-6098	258	15	-	-	PUNCT
ejpam-6098	258	16	hand	hand	NOUN
ejpam-6098	258	17	side	side	NOUN
ejpam-6098	258	18	scales	scale	NOUN
ejpam-6098	258	19	with	with	ADP
ejpam-6098	258	20	respect	respect	NOUN
ejpam-6098	258	21	to	to	ADP
ejpam-6098	258	22	the	the	DET
ejpam-6098	258	23	norms	norm	NOUN
ejpam-6098	258	24	of	of	ADP
ejpam-6098	258	25	u	u	NOUN
ejpam-6098	258	26	,	,	PUNCT
ejpam-6098	258	27	v	v	PROPN
ejpam-6098	258	28	,	,	PUNCT
ejpam-6098	258	29	w.	w.	PROPN
ejpam-6098	258	30	and	and	CCONJ
ejpam-6098	258	31	ϕ	ϕ	NOUN
ejpam-6098	258	32	:	:	PUNCT
ejpam-6098	258	33	v	v	ADP
ejpam-6098	258	34	n−1	n−1	PROPN
ejpam-6098	258	35	→	→	SYM
ejpam-6098	259	1	[	[	X
ejpam-6098	259	2	0,∞	0,∞	NUM
ejpam-6098	259	3	)	)	PUNCT
ejpam-6098	259	4	is	be	AUX
ejpam-6098	259	5	a	a	DET
ejpam-6098	259	6	control	control	NOUN
ejpam-6098	259	7	function	function	NOUN
ejpam-6098	259	8	depending	depend	VERB
ejpam-6098	259	9	on	on	ADP
ejpam-6098	259	10	parameters	parameter	NOUN
ejpam-6098	259	11	δ1	δ1	NOUN
ejpam-6098	259	12	,	,	PUNCT
ejpam-6098	259	13	...	...	PUNCT
ejpam-6098	259	14	,	,	PUNCT
ejpam-6098	259	15	δn−1	δn−1	PROPN
ejpam-6098	259	16	used	use	VERB
ejpam-6098	259	17	to	to	PART
ejpam-6098	259	18	describe	describe	VERB
ejpam-6098	259	19	perturbation	perturbation	NOUN
ejpam-6098	259	20	or	or	CCONJ
ejpam-6098	259	21	control	control	NOUN
ejpam-6098	259	22	of	of	ADP
ejpam-6098	259	23	deviation	deviation	NOUN
ejpam-6098	259	24	.	.	PUNCT
ejpam-6098	260	1	theorem	theorem	NOUN
ejpam-6098	260	2	2	2	NUM
ejpam-6098	260	3	.	.	PUNCT
ejpam-6098	261	1	let	let	VERB
ejpam-6098	261	2	θ	θ	PROPN
ejpam-6098	261	3	≥	≥	X
ejpam-6098	261	4	0	0	NUM
ejpam-6098	261	5	,	,	PUNCT
ejpam-6098	261	6	p	p	X
ejpam-6098	261	7	,	,	PUNCT
ejpam-6098	261	8	r	r	NOUN
ejpam-6098	261	9	,	,	PUNCT
ejpam-6098	261	10	q	q	X
ejpam-6098	261	11	>	>	X
ejpam-6098	261	12	0	0	PUNCT
ejpam-6098	261	13	with	with	ADP
ejpam-6098	261	14	(	(	PUNCT
ejpam-6098	261	15	p	p	NOUN
ejpam-6098	261	16	+	+	NOUN
ejpam-6098	261	17	r	r	NOUN
ejpam-6098	261	18	+	+	NOUN
ejpam-6098	261	19	q)β1	q)β1	NOUN
ejpam-6098	261	20	<	<	X
ejpam-6098	261	21	3β	3β	NUM
ejpam-6098	261	22	,	,	PUNCT
ejpam-6098	261	23	0	0	PUNCT
ejpam-6098	261	24	<	<	X
ejpam-6098	261	25	β	β	X
ejpam-6098	261	26	,	,	PUNCT
ejpam-6098	261	27	β1	β1	VERB
ejpam-6098	261	28	≤	≤	NOUN
ejpam-6098	261	29	1	1	NUM
ejpam-6098	261	30	.	.	PUNCT
ejpam-6098	262	1	and	and	CCONJ
ejpam-6098	262	2	ϕ	ϕ	X
ejpam-6098	262	3	:	:	PUNCT
ejpam-6098	262	4	v	v	ADP
ejpam-6098	262	5	n−1	n−1	PROPN
ejpam-6098	262	6	→	→	SYM
ejpam-6098	263	1	[	[	X
ejpam-6098	263	2	0,∞	0,∞	NOUN
ejpam-6098	263	3	)	)	PUNCT
ejpam-6098	263	4	be	be	AUX
ejpam-6098	263	5	a	a	DET
ejpam-6098	263	6	function	function	NOUN
ejpam-6098	263	7	.	.	PUNCT
ejpam-6098	264	1	if	if	SCONJ
ejpam-6098	264	2	f	f	PROPN
ejpam-6098	264	3	:	:	PUNCT
ejpam-6098	264	4	v	v	PROPN
ejpam-6098	264	5	→w	→w	NUM
ejpam-6098	264	6	is	be	AUX
ejpam-6098	264	7	an	an	DET
ejpam-6098	264	8	odd	odd	ADJ
ejpam-6098	264	9	function	function	NOUN
ejpam-6098	264	10	that	that	PRON
ejpam-6098	264	11	fulfills	fulfill	VERB
ejpam-6098	264	12	inequality	inequality	NOUN
ejpam-6098	264	13	||f(2u+	||f(2u+	VERB
ejpam-6098	264	14	w	w	ADP
ejpam-6098	264	15	+	+	PROPN
ejpam-6098	264	16	v)−	v)−	PROPN
ejpam-6098	264	17	3f(v	3f(v	NUM
ejpam-6098	265	1	+	+	CCONJ
ejpam-6098	265	2	w	w	PROPN
ejpam-6098	266	1	+	+	CCONJ
ejpam-6098	266	2	u)−	u)−	PROPN
ejpam-6098	266	3	f(w	f(w	PROPN
ejpam-6098	266	4	−	−	NOUN
ejpam-6098	266	5	u+	u+	NOUN
ejpam-6098	266	6	v)−	v)−	PROPN
ejpam-6098	266	7	2f(u+	2f(u+	PROPN
ejpam-6098	266	8	v)−	v)−	PROPN
ejpam-6098	266	9	2f(u+	2f(u+	PROPN
ejpam-6098	266	10	w	w	PROPN
ejpam-6098	266	11	)	)	PUNCT
ejpam-6098	266	12	+6f(u−	+6f(u−	PROPN
ejpam-6098	266	13	v	v	NOUN
ejpam-6098	266	14	)	)	PUNCT
ejpam-6098	266	15	+	+	CCONJ
ejpam-6098	266	16	6f(u−	6f(u−	NUM
ejpam-6098	266	17	w)−	w)−	PROPN
ejpam-6098	266	18	2f(2u−	2f(2u−	NUM
ejpam-6098	266	19	v	v	NOUN
ejpam-6098	266	20	)	)	PUNCT
ejpam-6098	266	21	+	+	NUM
ejpam-6098	266	22	3f(v	3f(v	NUM
ejpam-6098	267	1	+	+	CCONJ
ejpam-6098	267	2	w)−	w)−	PROPN
ejpam-6098	267	3	2f(2u−	2f(2u−	NUM
ejpam-6098	267	4	w	w	NOUN
ejpam-6098	267	5	)	)	PUNCT
ejpam-6098	267	6	+	+	CCONJ
ejpam-6098	267	7	18f(u	18f(u	NUM
ejpam-6098	267	8	)	)	PUNCT
ejpam-6098	267	9	+6f(v	+6f(v	NOUN
ejpam-6098	267	10	)	)	PUNCT
ejpam-6098	268	1	+	+	CCONJ
ejpam-6098	268	2	6f(w	6f(w	NUM
ejpam-6098	268	3	)	)	PUNCT
ejpam-6098	268	4	,	,	PUNCT
ejpam-6098	268	5	δ1	δ1	NOUN
ejpam-6098	268	6	,	,	PUNCT
ejpam-6098	268	7	...	...	PUNCT
ejpam-6098	268	8	,	,	PUNCT
ejpam-6098	268	9	δn−1||β	δn−1||β	PROPN
ejpam-6098	268	10	≤	≤	PROPN
ejpam-6098	268	11	θ||u||pβ1	θ||u||pβ1	PROPN
ejpam-6098	268	12	||v||qβ1	||v||qβ1	PROPN
ejpam-6098	268	13	||w||rβ1	||w||rβ1	ADV
ejpam-6098	268	14	ϕ(δ1	ϕ(δ1	PROPN
ejpam-6098	268	15	,	,	PUNCT
ejpam-6098	268	16	...	...	PUNCT
ejpam-6098	268	17	,	,	PUNCT
ejpam-6098	268	18	δn−1	δn−1	PROPN
ejpam-6098	268	19	)	)	PUNCT
ejpam-6098	268	20	,	,	PUNCT
ejpam-6098	268	21	(	(	PUNCT
ejpam-6098	268	22	10	10	NUM
ejpam-6098	268	23	)	)	PUNCT
ejpam-6098	268	24	then	then	ADV
ejpam-6098	268	25	there	there	PRON
ejpam-6098	268	26	is	be	VERB
ejpam-6098	268	27	a	a	DET
ejpam-6098	268	28	unique	unique	ADJ
ejpam-6098	268	29	cubic	cubic	ADJ
ejpam-6098	268	30	mapping	mapping	NOUN
ejpam-6098	268	31	q	q	NOUN
ejpam-6098	269	1	:	:	PUNCT
ejpam-6098	269	2	w	w	X
ejpam-6098	269	3	→	→	SYM
ejpam-6098	269	4	v	v	NOUN
ejpam-6098	269	5	such	such	ADJ
ejpam-6098	269	6	that	that	PRON
ejpam-6098	269	7	||f(u)−q(u	||f(u)−q(u	NOUN
ejpam-6098	269	8	)	)	PUNCT
ejpam-6098	269	9	,	,	PUNCT
ejpam-6098	269	10	δ1	δ1	NOUN
ejpam-6098	269	11	,	,	PUNCT
ejpam-6098	269	12	...	...	PUNCT
ejpam-6098	269	13	,	,	PUNCT
ejpam-6098	269	14	δn−1||β	δn−1||β	PROPN
ejpam-6098	269	15	≤	≤	NUM
ejpam-6098	269	16	θ2−3β||u||pβ1	θ2−3β||u||pβ1	NOUN
ejpam-6098	269	17	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	269	18	,	,	PUNCT
ejpam-6098	269	19	...	...	PUNCT
ejpam-6098	269	20	,	,	PUNCT
ejpam-6098	269	21	δn−1	δn−1	PROPN
ejpam-6098	269	22	)	)	PUNCT
ejpam-6098	269	23	(	(	PUNCT
ejpam-6098	269	24	11	11	NUM
ejpam-6098	269	25	)	)	PUNCT
ejpam-6098	269	26	for	for	ADP
ejpam-6098	269	27	all	all	DET
ejpam-6098	269	28	u	u	NOUN
ejpam-6098	269	29	,	,	PUNCT
ejpam-6098	269	30	v	v	NOUN
ejpam-6098	269	31	,	,	PUNCT
ejpam-6098	269	32	w	w	NOUN
ejpam-6098	269	33	∈w	∈w	NOUN
ejpam-6098	269	34	and	and	CCONJ
ejpam-6098	269	35	δ1	δ1	NOUN
ejpam-6098	269	36	,	,	PUNCT
ejpam-6098	269	37	...	...	PUNCT
ejpam-6098	269	38	,	,	PUNCT
ejpam-6098	269	39	δn−1	δn−1	PROPN
ejpam-6098	269	40	∈	∈	PROPN
ejpam-6098	269	41	v	v	NOUN
ejpam-6098	269	42	.	.	PUNCT
ejpam-6098	270	1	proof	proof	NOUN
ejpam-6098	270	2	.	.	PUNCT
ejpam-6098	271	1	putting	put	VERB
ejpam-6098	271	2	(	(	PUNCT
ejpam-6098	271	3	u	u	NOUN
ejpam-6098	271	4	,	,	PUNCT
ejpam-6098	271	5	v	v	NOUN
ejpam-6098	271	6	,	,	PUNCT
ejpam-6098	271	7	w	w	NOUN
ejpam-6098	271	8	)	)	PUNCT
ejpam-6098	271	9	=	=	SYM
ejpam-6098	271	10	(	(	PUNCT
ejpam-6098	271	11	u	u	NOUN
ejpam-6098	271	12	,	,	PUNCT
ejpam-6098	271	13	0	0	NUM
ejpam-6098	271	14	,	,	PUNCT
ejpam-6098	271	15	0	0	NUM
ejpam-6098	271	16	)	)	PUNCT
ejpam-6098	271	17	in	in	ADP
ejpam-6098	271	18	(	(	PUNCT
ejpam-6098	271	19	10	10	NUM
ejpam-6098	271	20	)	)	PUNCT
ejpam-6098	271	21	and	and	CCONJ
ejpam-6098	271	22	multiplying	multiply	VERB
ejpam-6098	271	23	both	both	DET
ejpam-6098	271	24	side	side	NOUN
ejpam-6098	271	25	by	by	ADP
ejpam-6098	271	26	1	1	NUM
ejpam-6098	271	27	3β	3β	NUM
ejpam-6098	271	28	.23β	.23β	PROPN
ejpam-6098	271	29	,	,	PUNCT
ejpam-6098	271	30	we	we	PRON
ejpam-6098	271	31	get	get	VERB
ejpam-6098	271	32	||f(2u	||f(2u	PROPN
ejpam-6098	271	33	)	)	PUNCT
ejpam-6098	271	34	23	23	NUM
ejpam-6098	271	35	−	−	PROPN
ejpam-6098	271	36	f(u	f(u	PROPN
ejpam-6098	271	37	)	)	PUNCT
ejpam-6098	271	38	,	,	PUNCT
ejpam-6098	271	39	δ1	δ1	NOUN
ejpam-6098	271	40	,	,	PUNCT
ejpam-6098	271	41	...	...	PUNCT
ejpam-6098	271	42	,	,	PUNCT
ejpam-6098	271	43	δn−1||β	δn−1||β	PROPN
ejpam-6098	271	44	≤	≤	NOUN
ejpam-6098	271	45	θ3−β2−3β||u||pβ1	θ3−β2−3β||u||pβ1	NOUN
ejpam-6098	271	46	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	271	47	,	,	PUNCT
ejpam-6098	271	48	...	...	PUNCT
ejpam-6098	271	49	,	,	PUNCT
ejpam-6098	271	50	δn−1	δn−1	PROPN
ejpam-6098	271	51	)	)	PUNCT
ejpam-6098	271	52	.	.	PUNCT
ejpam-6098	272	1	(	(	PUNCT
ejpam-6098	272	2	12	12	NUM
ejpam-6098	272	3	)	)	PUNCT
ejpam-6098	272	4	by	by	ADP
ejpam-6098	272	5	putting	put	VERB
ejpam-6098	272	6	u	u	NOUN
ejpam-6098	272	7	=	=	NOUN
ejpam-6098	272	8	2mu	2mu	NOUN
ejpam-6098	272	9	in	in	ADP
ejpam-6098	272	10	(	(	PUNCT
ejpam-6098	272	11	12	12	NUM
ejpam-6098	272	12	)	)	PUNCT
ejpam-6098	272	13	and	and	CCONJ
ejpam-6098	272	14	multiplying	multiply	VERB
ejpam-6098	272	15	both	both	DET
ejpam-6098	272	16	side	side	NOUN
ejpam-6098	272	17	by	by	ADP
ejpam-6098	272	18	1	1	NUM
ejpam-6098	272	19	23mβ	23mβ	NOUN
ejpam-6098	272	20	,	,	PUNCT
ejpam-6098	272	21	we	we	PRON
ejpam-6098	272	22	get	get	VERB
ejpam-6098	272	23	||f(2	||f(2	PROPN
ejpam-6098	272	24	m+1u	m+1u	PROPN
ejpam-6098	272	25	)	)	PUNCT
ejpam-6098	272	26	23m+3	23m+3	NUM
ejpam-6098	272	27	−	−	NOUN
ejpam-6098	272	28	f(2mu	f(2mu	NOUN
ejpam-6098	272	29	)	)	PUNCT
ejpam-6098	272	30	23	23	NUM
ejpam-6098	272	31	m	m	NOUN
ejpam-6098	272	32	,	,	PUNCT
ejpam-6098	272	33	δ1	δ1	NOUN
ejpam-6098	272	34	,	,	PUNCT
ejpam-6098	272	35	...	...	PUNCT
ejpam-6098	272	36	,	,	PUNCT
ejpam-6098	272	37	δn−1||β	δn−1||β	PROPN
ejpam-6098	272	38	≤	≤	NUM
ejpam-6098	272	39	θ3−β2−3β2−3mβ||2mu||pβ1	θ3−β2−3β2−3mβ||2mu||pβ1	PROPN
ejpam-6098	272	40	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	272	41	,	,	PUNCT
ejpam-6098	272	42	...	...	PUNCT
ejpam-6098	272	43	,	,	PUNCT
ejpam-6098	272	44	δn−1	δn−1	PROPN
ejpam-6098	272	45	)	)	PUNCT
ejpam-6098	272	46	=	=	SYM
ejpam-6098	272	47	θ3−β2−3β2−3mβ2mpβ||u||pβ1	θ3−β2−3β2−3mβ2mpβ||u||pβ1	ADJ
ejpam-6098	272	48	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	272	49	,	,	PUNCT
ejpam-6098	272	50	...	...	PUNCT
ejpam-6098	272	51	,	,	PUNCT
ejpam-6098	272	52	δn−1	δn−1	PROPN
ejpam-6098	272	53	)	)	PUNCT
ejpam-6098	272	54	=	=	SYM
ejpam-6098	272	55	θ3−β2−3β2−3mβ(2(pβ1−3β	θ3−β2−3β2−3mβ(2(pβ1−3β	ADJ
ejpam-6098	272	56	)	)	PUNCT
ejpam-6098	272	57	)	)	PUNCT
ejpam-6098	272	58	m||u||pβ1	m||u||pβ1	NOUN
ejpam-6098	272	59	ϕ(δ1	ϕ(δ1	PROPN
ejpam-6098	272	60	,	,	PUNCT
ejpam-6098	272	61	...	...	PUNCT
ejpam-6098	272	62	,	,	PUNCT
ejpam-6098	272	63	δn−1	δn−1	PROPN
ejpam-6098	272	64	)	)	PUNCT
ejpam-6098	272	65	.	.	PUNCT
ejpam-6098	273	1	since	since	SCONJ
ejpam-6098	273	2	pβ1	pβ1	ADP
ejpam-6098	273	3	<	<	X
ejpam-6098	273	4	3β	3β	NUM
ejpam-6098	273	5	,	,	PUNCT
ejpam-6098	273	6	taking	take	VERB
ejpam-6098	273	7	limit	limit	NOUN
ejpam-6098	273	8	as	as	ADP
ejpam-6098	273	9	m→	m→	NOUN
ejpam-6098	273	10	∞	∞	PROPN
ejpam-6098	273	11	,	,	PUNCT
ejpam-6098	273	12	we	we	PRON
ejpam-6098	273	13	have	have	VERB
ejpam-6098	273	14	||f(2	||f(2	PROPN
ejpam-6098	273	15	m+1u	m+1u	PROPN
ejpam-6098	273	16	)	)	PUNCT
ejpam-6098	273	17	23m+3	23m+3	NUM
ejpam-6098	273	18	−	−	NOUN
ejpam-6098	273	19	f(2mu	f(2mu	NOUN
ejpam-6098	273	20	)	)	PUNCT
ejpam-6098	273	21	23	23	NUM
ejpam-6098	273	22	m	m	NOUN
ejpam-6098	273	23	,	,	PUNCT
ejpam-6098	273	24	δ1	δ1	NOUN
ejpam-6098	273	25	,	,	PUNCT
ejpam-6098	273	26	...	...	PUNCT
ejpam-6098	273	27	,	,	PUNCT
ejpam-6098	273	28	δn−1||β	δn−1||β	X
ejpam-6098	273	29	=	=	SYM
ejpam-6098	274	1	0	0	X
ejpam-6098	274	2	.	.	PUNCT
ejpam-6098	275	1	hence	hence	ADV
ejpam-6098	275	2	,	,	PUNCT
ejpam-6098	275	3	in	in	ADP
ejpam-6098	275	4	complete	complete	ADJ
ejpam-6098	275	5	space	space	NOUN
ejpam-6098	275	6	v	v	NOUN
ejpam-6098	275	7	,	,	PUNCT
ejpam-6098	275	8	the	the	DET
ejpam-6098	275	9	sequence	sequence	NOUN
ejpam-6098	275	10	<	<	X
ejpam-6098	275	11	2−3mf(2mu	2−3mf(2mu	NUM
ejpam-6098	275	12	)	)	PUNCT
ejpam-6098	275	13	>	>	X
ejpam-6098	275	14	is	be	AUX
ejpam-6098	275	15	a	a	DET
ejpam-6098	275	16	cauchy	cauchy	ADJ
ejpam-6098	275	17	sequence	sequence	NOUN
ejpam-6098	275	18	.	.	PUNCT
ejpam-6098	276	1	therefore	therefore	ADV
ejpam-6098	276	2	,	,	PUNCT
ejpam-6098	276	3	this	this	DET
ejpam-6098	276	4	sequence	sequence	NOUN
ejpam-6098	276	5	converges	converge	VERB
ejpam-6098	276	6	to	to	ADP
ejpam-6098	276	7	q(u	q(u	NOUN
ejpam-6098	276	8	)	)	PUNCT
ejpam-6098	276	9	∈	∈	NOUN
ejpam-6098	276	10	v	v	NOUN
ejpam-6098	276	11	.	.	PUNCT
ejpam-6098	277	1	now	now	ADV
ejpam-6098	277	2	,	,	PUNCT
ejpam-6098	277	3	we	we	PRON
ejpam-6098	277	4	consider	consider	VERB
ejpam-6098	277	5	q	q	NOUN
ejpam-6098	277	6	:	:	PUNCT
ejpam-6098	277	7	w	w	X
ejpam-6098	277	8	→	→	SYM
ejpam-6098	277	9	v	v	NOUN
ejpam-6098	277	10	a	a	DET
ejpam-6098	277	11	mapping	mapping	NOUN
ejpam-6098	277	12	holds	hold	VERB
ejpam-6098	277	13	q(u	q(u	NOUN
ejpam-6098	277	14	)	)	PUNCT
ejpam-6098	277	15	=	=	SYM
ejpam-6098	277	16	limm→∞2−3mf(2mu	limm→∞2−3mf(2mu	PROPN
ejpam-6098	277	17	)	)	PUNCT
ejpam-6098	277	18	.	.	PUNCT
ejpam-6098	278	1	(	(	PUNCT
ejpam-6098	278	2	13	13	NUM
ejpam-6098	278	3	)	)	PUNCT
ejpam-6098	278	4	j.	j.	PROPN
ejpam-6098	278	5	jakhar	jakhar	PROPN
ejpam-6098	278	6	et	et	PROPN
ejpam-6098	278	7	al	al	PROPN
ejpam-6098	278	8	.	.	PUNCT
ejpam-6098	278	9	/	/	SYM
ejpam-6098	278	10	eur	eur	PROPN
ejpam-6098	278	11	.	.	PUNCT
ejpam-6098	279	1	j.	j.	PROPN
ejpam-6098	279	2	pure	pure	PROPN
ejpam-6098	279	3	appl	appl	PROPN
ejpam-6098	279	4	.	.	PROPN
ejpam-6098	279	5	math	math	PROPN
ejpam-6098	279	6	,	,	PUNCT
ejpam-6098	279	7	18	18	NUM
ejpam-6098	279	8	(	(	PUNCT
ejpam-6098	279	9	2	2	NUM
ejpam-6098	279	10	)	)	PUNCT
ejpam-6098	279	11	(	(	PUNCT
ejpam-6098	279	12	2025	2025	NUM
ejpam-6098	279	13	)	)	PUNCT
ejpam-6098	279	14	,	,	PUNCT
ejpam-6098	279	15	6098	6098	NUM
ejpam-6098	279	16	11	11	NUM
ejpam-6098	279	17	of	of	ADP
ejpam-6098	279	18	23	23	NUM
ejpam-6098	279	19	now	now	ADV
ejpam-6098	279	20	,	,	PUNCT
ejpam-6098	279	21	we	we	PRON
ejpam-6098	279	22	have	have	VERB
ejpam-6098	279	23	to	to	PART
ejpam-6098	279	24	prove	prove	VERB
ejpam-6098	279	25	that	that	PRON
ejpam-6098	279	26	q	q	NOUN
ejpam-6098	279	27	is	be	AUX
ejpam-6098	279	28	a	a	DET
ejpam-6098	279	29	cubic	cubic	NOUN
ejpam-6098	279	30	.	.	PUNCT
ejpam-6098	280	1	by	by	ADP
ejpam-6098	280	2	using	use	VERB
ejpam-6098	280	3	(	(	PUNCT
ejpam-6098	280	4	10	10	NUM
ejpam-6098	280	5	)	)	PUNCT
ejpam-6098	280	6	,	,	PUNCT
ejpam-6098	280	7	(	(	PUNCT
ejpam-6098	280	8	13	13	NUM
ejpam-6098	280	9	)	)	PUNCT
ejpam-6098	280	10	and	and	CCONJ
ejpam-6098	280	11	lemma	lemma	PROPN
ejpam-6098	280	12	(	(	PUNCT
ejpam-6098	280	13	1	1	NUM
ejpam-6098	280	14	)	)	PUNCT
ejpam-6098	280	15	,	,	PUNCT
ejpam-6098	280	16	we	we	PRON
ejpam-6098	280	17	get	get	VERB
ejpam-6098	280	18	||q(2u+	||q(2u+	ADJ
ejpam-6098	280	19	v	v	ADP
ejpam-6098	280	20	+	+	CCONJ
ejpam-6098	280	21	w)−	w)−	PROPN
ejpam-6098	280	22	3q(v	3q(v	PROPN
ejpam-6098	281	1	+	+	CCONJ
ejpam-6098	281	2	w	w	PROPN
ejpam-6098	282	1	+	+	CCONJ
ejpam-6098	282	2	u)−q(w	u)−q(w	NOUN
ejpam-6098	282	3	−	−	PROPN
ejpam-6098	282	4	u+	u+	NOUN
ejpam-6098	282	5	v)−	v)−	PROPN
ejpam-6098	282	6	2q(w	2q(w	NOUN
ejpam-6098	282	7	+	+	CCONJ
ejpam-6098	282	8	u)−	u)−	PROPN
ejpam-6098	282	9	2q(u+	2q(u+	PROPN
ejpam-6098	282	10	v	v	NOUN
ejpam-6098	282	11	)	)	PUNCT
ejpam-6098	282	12	+6q(u−	+6q(u−	PROPN
ejpam-6098	282	13	w	w	NOUN
ejpam-6098	282	14	)	)	PUNCT
ejpam-6098	283	1	+	+	CCONJ
ejpam-6098	283	2	6q(u−	6q(u−	NUM
ejpam-6098	283	3	v	v	NOUN
ejpam-6098	283	4	)	)	PUNCT
ejpam-6098	284	1	+	+	NUM
ejpam-6098	284	2	3q(w	3q(w	NUM
ejpam-6098	284	3	+	+	CCONJ
ejpam-6098	284	4	v)−	v)−	PROPN
ejpam-6098	284	5	2q(2u−	2q(2u−	NUM
ejpam-6098	284	6	v)−	v)−	PROPN
ejpam-6098	284	7	2q(2u−	2q(2u−	NUM
ejpam-6098	284	8	w	w	PROPN
ejpam-6098	284	9	)	)	PUNCT
ejpam-6098	284	10	+	+	CCONJ
ejpam-6098	284	11	18q(u	18q(u	X
ejpam-6098	284	12	)	)	PUNCT
ejpam-6098	284	13	+6q(v	+6q(v	NUM
ejpam-6098	284	14	)	)	PUNCT
ejpam-6098	285	1	+	+	CCONJ
ejpam-6098	285	2	6q(w	6q(w	NUM
ejpam-6098	285	3	)	)	PUNCT
ejpam-6098	285	4	,	,	PUNCT
ejpam-6098	285	5	δ1	δ1	NOUN
ejpam-6098	285	6	,	,	PUNCT
ejpam-6098	285	7	...	...	PUNCT
ejpam-6098	285	8	,	,	PUNCT
ejpam-6098	285	9	δn−1||β	δn−1||β	X
ejpam-6098	285	10	=	=	PUNCT
ejpam-6098	285	11	limm→∞|2−3mβ|||f(2m(2u+	limm→∞|2−3mβ|||f(2m(2u+	PROPN
ejpam-6098	285	12	w	w	PROPN
ejpam-6098	286	1	+	+	NUM
ejpam-6098	286	2	v	v	NOUN
ejpam-6098	286	3	)	)	PUNCT
ejpam-6098	286	4	)	)	PUNCT
ejpam-6098	287	1	−3f(2m(w	−3f(2m(w	NOUN
ejpam-6098	288	1	+	+	CCONJ
ejpam-6098	288	2	u+	u+	NUM
ejpam-6098	288	3	v))−	v))−	NOUN
ejpam-6098	288	4	f(2m(w	f(2m(w	PROPN
ejpam-6098	288	5	−	−	PROPN
ejpam-6098	288	6	u+	u+	NOUN
ejpam-6098	288	7	v))−	v))−	NOUN
ejpam-6098	288	8	2f(2m(u+	2f(2m(u+	NOUN
ejpam-6098	288	9	v))−	v))−	NOUN
ejpam-6098	288	10	2f(2m(u+	2f(2m(u+	NOUN
ejpam-6098	288	11	w	w	NOUN
ejpam-6098	288	12	)	)	PUNCT
ejpam-6098	288	13	)	)	PUNCT
ejpam-6098	289	1	+6f(2m(u−	+6f(2m(u−	X
ejpam-6098	289	2	w	w	NOUN
ejpam-6098	289	3	)	)	PUNCT
ejpam-6098	289	4	)	)	PUNCT
ejpam-6098	290	1	+	+	CCONJ
ejpam-6098	290	2	6f(2m(u−	6f(2m(u−	NUM
ejpam-6098	290	3	v	v	NOUN
ejpam-6098	290	4	)	)	PUNCT
ejpam-6098	290	5	)	)	PUNCT
ejpam-6098	291	1	+	+	CCONJ
ejpam-6098	291	2	3f(2m(v	3f(2m(v	NUM
ejpam-6098	291	3	+	+	CCONJ
ejpam-6098	292	1	w))−	w))−	NUM
ejpam-6098	292	2	2f(2m(2u−	2f(2m(2u−	NUM
ejpam-6098	292	3	v	v	NOUN
ejpam-6098	292	4	)	)	PUNCT
ejpam-6098	292	5	)	)	PUNCT
ejpam-6098	293	1	−2f(2m(2u−	−2f(2m(2u−	PROPN
ejpam-6098	293	2	w	w	PROPN
ejpam-6098	293	3	)	)	PUNCT
ejpam-6098	293	4	)	)	PUNCT
ejpam-6098	294	1	+	+	CCONJ
ejpam-6098	294	2	18f(2mu	18f(2mu	NUM
ejpam-6098	294	3	)	)	PUNCT
ejpam-6098	294	4	+	+	CCONJ
ejpam-6098	294	5	6f(2mv	6f(2mv	X
ejpam-6098	294	6	)	)	PUNCT
ejpam-6098	294	7	+	+	CCONJ
ejpam-6098	294	8	6f(2mw	6f(2mw	NUM
ejpam-6098	294	9	)	)	PUNCT
ejpam-6098	294	10	,	,	PUNCT
ejpam-6098	294	11	δ1	δ1	NOUN
ejpam-6098	294	12	,	,	PUNCT
ejpam-6098	294	13	...	...	PUNCT
ejpam-6098	294	14	,	,	PUNCT
ejpam-6098	294	15	δn−1||β	δn−1||β	PROPN
ejpam-6098	294	16	≤	≤	NUM
ejpam-6098	294	17	limm→∞θ(2	limm→∞θ(2	NUM
ejpam-6098	294	18	−3mβ)||2mu||pβ1	−3mβ)||2mu||pβ1	NOUN
ejpam-6098	294	19	||2mv||qβ1	||2mv||qβ1	NOUN
ejpam-6098	294	20	||2w||rβ1	||2w||rβ1	VERB
ejpam-6098	294	21	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	294	22	,	,	PUNCT
ejpam-6098	294	23	...	...	PUNCT
ejpam-6098	294	24	,	,	PUNCT
ejpam-6098	294	25	δn−1	δn−1	PROPN
ejpam-6098	294	26	)	)	PUNCT
ejpam-6098	294	27	=	=	SYM
ejpam-6098	294	28	limm→∞θ(2	limm→∞θ(2	X
ejpam-6098	294	29	(	(	PUNCT
ejpam-6098	294	30	(	(	PUNCT
ejpam-6098	294	31	p+q+r)β1−3β))m||u||pβ1	p+q+r)β1−3β))m||u||pβ1	PROPN
ejpam-6098	294	32	||v||qβ1	||v||qβ1	PROPN
ejpam-6098	294	33	||w||rβ1	||w||rβ1	ADV
ejpam-6098	294	34	ϕ(δ1	ϕ(δ1	PROPN
ejpam-6098	294	35	,	,	PUNCT
ejpam-6098	294	36	...	...	PUNCT
ejpam-6098	294	37	,	,	PUNCT
ejpam-6098	294	38	δn−1	δn−1	PROPN
ejpam-6098	294	39	)	)	PUNCT
ejpam-6098	294	40	.	.	PUNCT
ejpam-6098	295	1	since	since	SCONJ
ejpam-6098	295	2	(	(	PUNCT
ejpam-6098	295	3	p+	p+	NOUN
ejpam-6098	295	4	r	r	NOUN
ejpam-6098	295	5	+	+	X
ejpam-6098	295	6	q)β1	q)β1	NOUN
ejpam-6098	295	7	<	<	X
ejpam-6098	295	8	3β	3β	NUM
ejpam-6098	295	9	,	,	PUNCT
ejpam-6098	295	10	we	we	PRON
ejpam-6098	295	11	get	get	VERB
ejpam-6098	295	12	||q(2u+	||q(2u+	ADJ
ejpam-6098	295	13	v	v	ADP
ejpam-6098	296	1	+	+	NOUN
ejpam-6098	296	2	w)−q(−u+	w)−q(−u+	PROPN
ejpam-6098	296	3	v	v	NOUN
ejpam-6098	296	4	+	+	CCONJ
ejpam-6098	296	5	w)−	w)−	PROPN
ejpam-6098	296	6	3q(u+	3q(u+	PROPN
ejpam-6098	296	7	v	v	NOUN
ejpam-6098	296	8	+	+	CCONJ
ejpam-6098	296	9	w)−	w)−	PROPN
ejpam-6098	296	10	2q(u+	2q(u+	PROPN
ejpam-6098	296	11	w)−	w)−	PROPN
ejpam-6098	296	12	2q(u+	2q(u+	PROPN
ejpam-6098	296	13	v	v	NOUN
ejpam-6098	296	14	)	)	PUNCT
ejpam-6098	296	15	+6q(u−	+6q(u−	PROPN
ejpam-6098	296	16	v	v	NOUN
ejpam-6098	296	17	)	)	PUNCT
ejpam-6098	296	18	+	+	CCONJ
ejpam-6098	296	19	6q(u−	6q(u−	NUM
ejpam-6098	296	20	w	w	NOUN
ejpam-6098	296	21	)	)	PUNCT
ejpam-6098	296	22	+	+	CCONJ
ejpam-6098	296	23	3q(v	3q(v	NUM
ejpam-6098	296	24	+	+	CCONJ
ejpam-6098	296	25	w)−	w)−	PROPN
ejpam-6098	296	26	2q(2u−	2q(2u−	NUM
ejpam-6098	296	27	w)−	w)−	PROPN
ejpam-6098	296	28	2q(2u−	2q(2u−	NUM
ejpam-6098	296	29	v	v	NOUN
ejpam-6098	296	30	)	)	PUNCT
ejpam-6098	297	1	+	+	CCONJ
ejpam-6098	297	2	18q(u	18q(u	X
ejpam-6098	297	3	)	)	PUNCT
ejpam-6098	297	4	+6q(v	+6q(v	NUM
ejpam-6098	297	5	)	)	PUNCT
ejpam-6098	298	1	+	+	CCONJ
ejpam-6098	298	2	6q(w	6q(w	NUM
ejpam-6098	298	3	)	)	PUNCT
ejpam-6098	298	4	,	,	PUNCT
ejpam-6098	298	5	δ1	δ1	NOUN
ejpam-6098	298	6	,	,	PUNCT
ejpam-6098	298	7	...	...	PUNCT
ejpam-6098	298	8	,	,	PUNCT
ejpam-6098	298	9	δn−1||β	δn−1||β	X
ejpam-6098	298	10	=	=	SYM
ejpam-6098	298	11	0	0	X
ejpam-6098	298	12	.	.	PUNCT
ejpam-6098	299	1	hence	hence	ADV
ejpam-6098	299	2	,	,	PUNCT
ejpam-6098	299	3	q	q	PROPN
ejpam-6098	299	4	is	be	AUX
ejpam-6098	299	5	a	a	DET
ejpam-6098	299	6	cubic	cubic	ADJ
ejpam-6098	299	7	mapping	mapping	NOUN
ejpam-6098	299	8	.	.	PUNCT
ejpam-6098	300	1	by	by	ADP
ejpam-6098	300	2	substituting	substitute	VERB
ejpam-6098	300	3	u	u	NOUN
ejpam-6098	300	4	with	with	ADP
ejpam-6098	300	5	2u	2u	PROPN
ejpam-6098	300	6	in	in	ADP
ejpam-6098	300	7	(	(	PUNCT
ejpam-6098	300	8	12	12	NUM
ejpam-6098	300	9	)	)	PUNCT
ejpam-6098	300	10	and	and	CCONJ
ejpam-6098	300	11	multiplying	multiply	VERB
ejpam-6098	300	12	both	both	DET
ejpam-6098	300	13	side	side	NOUN
ejpam-6098	300	14	by	by	ADP
ejpam-6098	300	15	1	1	NUM
ejpam-6098	300	16	23β	23β	NOUN
ejpam-6098	300	17	,	,	PUNCT
ejpam-6098	300	18	we	we	PRON
ejpam-6098	300	19	obtain	obtain	VERB
ejpam-6098	300	20	||f(2	||f(2	PROPN
ejpam-6098	300	21	2u	2u	NOUN
ejpam-6098	300	22	)	)	PUNCT
ejpam-6098	300	23	26	26	NUM
ejpam-6098	300	24	−	−	PROPN
ejpam-6098	300	25	f(2u	f(2u	NOUN
ejpam-6098	300	26	)	)	PUNCT
ejpam-6098	300	27	23	23	NUM
ejpam-6098	300	28	,	,	PUNCT
ejpam-6098	300	29	δ1	δ1	NOUN
ejpam-6098	300	30	,	,	PUNCT
ejpam-6098	300	31	...	...	PUNCT
ejpam-6098	300	32	,	,	PUNCT
ejpam-6098	300	33	δn−1||β	δn−1||β	NOUN
ejpam-6098	300	34	≤	≤	NOUN
ejpam-6098	300	35	θ3−β2−3β||2u||pβ1	θ3−β2−3β||2u||pβ1	DET
ejpam-6098	300	36	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	300	37	,	,	PUNCT
ejpam-6098	300	38	...	...	PUNCT
ejpam-6098	300	39	,	,	PUNCT
ejpam-6098	300	40	δn−1	δn−1	PROPN
ejpam-6098	300	41	)	)	PUNCT
ejpam-6098	300	42	.	.	PUNCT
ejpam-6098	301	1	(	(	PUNCT
ejpam-6098	301	2	14	14	NUM
ejpam-6098	301	3	)	)	PUNCT
ejpam-6098	301	4	thus	thus	ADV
ejpam-6098	301	5	,	,	PUNCT
ejpam-6098	301	6	by	by	ADP
ejpam-6098	301	7	(	(	PUNCT
ejpam-6098	301	8	12	12	NUM
ejpam-6098	301	9	)	)	PUNCT
ejpam-6098	301	10	and	and	CCONJ
ejpam-6098	301	11	(	(	PUNCT
ejpam-6098	301	12	14	14	NUM
ejpam-6098	301	13	)	)	PUNCT
ejpam-6098	301	14	,	,	PUNCT
ejpam-6098	301	15	we	we	PRON
ejpam-6098	301	16	get	get	VERB
ejpam-6098	301	17	||f(u)−	||f(u)−	NOUN
ejpam-6098	301	18	f(22u	f(22u	NOUN
ejpam-6098	301	19	)	)	PUNCT
ejpam-6098	301	20	26	26	NUM
ejpam-6098	301	21	,	,	PUNCT
ejpam-6098	301	22	δ1	δ1	NOUN
ejpam-6098	301	23	,	,	PUNCT
ejpam-6098	301	24	...	...	PUNCT
ejpam-6098	301	25	,	,	PUNCT
ejpam-6098	301	26	δn−1||β	δn−1||β	PROPN
ejpam-6098	301	27	≤	≤	PUNCT
ejpam-6098	301	28	max{||f(2u	max{||f(2u	PROPN
ejpam-6098	301	29	)	)	PUNCT
ejpam-6098	301	30	23	23	NUM
ejpam-6098	301	31	−	−	PROPN
ejpam-6098	301	32	f(u	f(u	PROPN
ejpam-6098	301	33	)	)	PUNCT
ejpam-6098	301	34	,	,	PUNCT
ejpam-6098	301	35	δ1	δ1	NOUN
ejpam-6098	301	36	,	,	PUNCT
ejpam-6098	301	37	...	...	PUNCT
ejpam-6098	301	38	,	,	PUNCT
ejpam-6098	301	39	δn−1||β	δn−1||β	PROPN
ejpam-6098	301	40	,	,	PUNCT
ejpam-6098	301	41	||	||	NOUN
ejpam-6098	301	42	f(22u	f(22u	NOUN
ejpam-6098	301	43	)	)	PUNCT
ejpam-6098	302	1	26	26	NUM
ejpam-6098	302	2	−	−	PROPN
ejpam-6098	302	3	f(2u	f(2u	NOUN
ejpam-6098	302	4	)	)	PUNCT
ejpam-6098	302	5	23	23	NUM
ejpam-6098	302	6	,	,	PUNCT
ejpam-6098	302	7	δ1	δ1	NOUN
ejpam-6098	302	8	,	,	PUNCT
ejpam-6098	302	9	...	...	PUNCT
ejpam-6098	302	10	,	,	PUNCT
ejpam-6098	302	11	δn−1||β	δn−1||β	PROPN
ejpam-6098	302	12	}	}	PUNCT
ejpam-6098	302	13	≤	≤	NUM
ejpam-6098	302	14	max{θ3−β2−3β||u||pβ1	max{θ3−β2−3β||u||pβ1	NOUN
ejpam-6098	302	15	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	302	16	,	,	PUNCT
ejpam-6098	302	17	...	...	PUNCT
ejpam-6098	302	18	,	,	PUNCT
ejpam-6098	302	19	δn−1	δn−1	PROPN
ejpam-6098	302	20	)	)	PUNCT
ejpam-6098	302	21	,	,	PUNCT
ejpam-6098	302	22	θ3	θ3	ADP
ejpam-6098	302	23	−β2−3β||2u||pβ1	−β2−3β||2u||pβ1	NOUN
ejpam-6098	302	24	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	302	25	,	,	PUNCT
ejpam-6098	302	26	...	...	PUNCT
ejpam-6098	302	27	,	,	PUNCT
ejpam-6098	302	28	δn−1	δn−1	PROPN
ejpam-6098	302	29	)	)	PUNCT
ejpam-6098	302	30	}	}	PUNCT
ejpam-6098	302	31	.	.	PUNCT
ejpam-6098	303	1	since	since	SCONJ
ejpam-6098	303	2	(	(	PUNCT
ejpam-6098	303	3	p+	p+	NOUN
ejpam-6098	303	4	r	r	NOUN
ejpam-6098	303	5	+	+	X
ejpam-6098	303	6	q)β1	q)β1	NOUN
ejpam-6098	303	7	<	<	X
ejpam-6098	303	8	3β	3β	NUM
ejpam-6098	303	9	,	,	PUNCT
ejpam-6098	303	10	we	we	PRON
ejpam-6098	303	11	get	get	VERB
ejpam-6098	303	12	||f(u)−	||f(u)−	NOUN
ejpam-6098	303	13	f(22u	f(22u	NOUN
ejpam-6098	303	14	)	)	PUNCT
ejpam-6098	303	15	26	26	NUM
ejpam-6098	303	16	,	,	PUNCT
ejpam-6098	303	17	δ1	δ1	NOUN
ejpam-6098	303	18	,	,	PUNCT
ejpam-6098	303	19	...	...	PUNCT
ejpam-6098	303	20	,	,	PUNCT
ejpam-6098	303	21	δn−1||β	δn−1||β	PROPN
ejpam-6098	303	22	≤	≤	NOUN
ejpam-6098	303	23	θ3−β2−3β||u||pβ1	θ3−β2−3β||u||pβ1	NOUN
ejpam-6098	303	24	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	303	25	,	,	PUNCT
ejpam-6098	303	26	...	...	PUNCT
ejpam-6098	303	27	,	,	PUNCT
ejpam-6098	303	28	δn−1	δn−1	PROPN
ejpam-6098	303	29	)	)	PUNCT
ejpam-6098	303	30	.	.	PUNCT
ejpam-6098	304	1	by	by	ADP
ejpam-6098	304	2	applying	apply	VERB
ejpam-6098	304	3	pmi	pmi	NOUN
ejpam-6098	304	4	on	on	ADP
ejpam-6098	304	5	m	m	PROPN
ejpam-6098	304	6	,	,	PUNCT
ejpam-6098	304	7	we	we	PRON
ejpam-6098	304	8	that	that	PRON
ejpam-6098	304	9	||f(u)−	||f(u)−	PROPN
ejpam-6098	304	10	f(2mu	f(2mu	NOUN
ejpam-6098	304	11	)	)	PUNCT
ejpam-6098	304	12	23	23	NUM
ejpam-6098	304	13	m	m	NOUN
ejpam-6098	304	14	,	,	PUNCT
ejpam-6098	304	15	δ1	δ1	NOUN
ejpam-6098	304	16	,	,	PUNCT
ejpam-6098	304	17	...	...	PUNCT
ejpam-6098	304	18	,	,	PUNCT
ejpam-6098	304	19	δn−1||β	δn−1||β	PROPN
ejpam-6098	304	20	≤	≤	NOUN
ejpam-6098	304	21	θ3−β2−3β||u||pβ1	θ3−β2−3β||u||pβ1	NOUN
ejpam-6098	304	22	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	304	23	,	,	PUNCT
ejpam-6098	304	24	...	...	PUNCT
ejpam-6098	304	25	,	,	PUNCT
ejpam-6098	304	26	δn−1	δn−1	PROPN
ejpam-6098	304	27	)	)	PUNCT
ejpam-6098	304	28	.	.	PUNCT
ejpam-6098	305	1	(	(	PUNCT
ejpam-6098	305	2	15	15	X
ejpam-6098	305	3	)	)	PUNCT
ejpam-6098	305	4	taking	take	VERB
ejpam-6098	305	5	u	u	NOUN
ejpam-6098	305	6	=	=	NOUN
ejpam-6098	305	7	2u	2u	PROPN
ejpam-6098	305	8	in	in	ADP
ejpam-6098	305	9	(	(	PUNCT
ejpam-6098	305	10	15	15	NUM
ejpam-6098	305	11	)	)	PUNCT
ejpam-6098	305	12	and	and	CCONJ
ejpam-6098	305	13	multiplying	multiply	VERB
ejpam-6098	305	14	both	both	DET
ejpam-6098	305	15	side	side	NOUN
ejpam-6098	305	16	by	by	ADP
ejpam-6098	305	17	1	1	NUM
ejpam-6098	305	18	23β	23β	NOUN
ejpam-6098	305	19	,	,	PUNCT
ejpam-6098	305	20	we	we	PRON
ejpam-6098	305	21	have	have	AUX
ejpam-6098	305	22	||f(2u	||f(2u	NOUN
ejpam-6098	305	23	)	)	PUNCT
ejpam-6098	305	24	23	23	NUM
ejpam-6098	305	25	−	−	PROPN
ejpam-6098	305	26	f(2m+1u	f(2m+1u	PROPN
ejpam-6098	305	27	)	)	PUNCT
ejpam-6098	305	28	23m+3	23m+3	NUM
ejpam-6098	305	29	,	,	PUNCT
ejpam-6098	305	30	δ1	δ1	NOUN
ejpam-6098	305	31	,	,	PUNCT
ejpam-6098	305	32	...	...	PUNCT
ejpam-6098	305	33	,	,	PUNCT
ejpam-6098	305	34	δn−1||β	δn−1||β	NOUN
ejpam-6098	305	35	≤	≤	NOUN
ejpam-6098	305	36	3−β2−6β||u||pβ1	3−β2−6β||u||pβ1	NUM
ejpam-6098	305	37	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	305	38	,	,	PUNCT
ejpam-6098	305	39	...	...	PUNCT
ejpam-6098	305	40	,	,	PUNCT
ejpam-6098	305	41	δn−1	δn−1	PROPN
ejpam-6098	305	42	)	)	PUNCT
ejpam-6098	305	43	.	.	PUNCT
ejpam-6098	306	1	(	(	PUNCT
ejpam-6098	306	2	16	16	NUM
ejpam-6098	306	3	)	)	PUNCT
ejpam-6098	306	4	j.	j.	PROPN
ejpam-6098	306	5	jakhar	jakhar	PROPN
ejpam-6098	306	6	et	et	PROPN
ejpam-6098	306	7	al	al	PROPN
ejpam-6098	306	8	.	.	PUNCT
ejpam-6098	306	9	/	/	SYM
ejpam-6098	306	10	eur	eur	PROPN
ejpam-6098	306	11	.	.	PUNCT
ejpam-6098	307	1	j.	j.	PROPN
ejpam-6098	307	2	pure	pure	PROPN
ejpam-6098	307	3	appl	appl	PROPN
ejpam-6098	307	4	.	.	PROPN
ejpam-6098	307	5	math	math	PROPN
ejpam-6098	307	6	,	,	PUNCT
ejpam-6098	307	7	18	18	NUM
ejpam-6098	307	8	(	(	PUNCT
ejpam-6098	307	9	2	2	NUM
ejpam-6098	307	10	)	)	PUNCT
ejpam-6098	307	11	(	(	PUNCT
ejpam-6098	307	12	2025	2025	NUM
ejpam-6098	307	13	)	)	PUNCT
ejpam-6098	307	14	,	,	PUNCT
ejpam-6098	307	15	6098	6098	NUM
ejpam-6098	307	16	12	12	NUM
ejpam-6098	307	17	of	of	ADP
ejpam-6098	307	18	23	23	NUM
ejpam-6098	307	19	by	by	ADP
ejpam-6098	307	20	using	use	VERB
ejpam-6098	307	21	(	(	PUNCT
ejpam-6098	307	22	12	12	NUM
ejpam-6098	307	23	)	)	PUNCT
ejpam-6098	307	24	and	and	CCONJ
ejpam-6098	307	25	(	(	PUNCT
ejpam-6098	307	26	16	16	NUM
ejpam-6098	307	27	)	)	PUNCT
ejpam-6098	308	1	,	,	PUNCT
ejpam-6098	308	2	we	we	PRON
ejpam-6098	308	3	have	have	VERB
ejpam-6098	308	4	||f(u)−	||f(u)−	PROPN
ejpam-6098	308	5	f(2m+1u	f(2m+1u	NOUN
ejpam-6098	308	6	)	)	PUNCT
ejpam-6098	308	7	23(m+1	23(m+1	NUM
ejpam-6098	308	8	)	)	PUNCT
ejpam-6098	308	9	,	,	PUNCT
ejpam-6098	308	10	δ1	δ1	NOUN
ejpam-6098	308	11	,	,	PUNCT
ejpam-6098	308	12	...	...	PUNCT
ejpam-6098	308	13	,	,	PUNCT
ejpam-6098	308	14	δn−1||β	δn−1||β	PROPN
ejpam-6098	308	15	≤	≤	NOUN
ejpam-6098	308	16	θ3−β2−3β||u||pβ1	θ3−β2−3β||u||pβ1	NOUN
ejpam-6098	308	17	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	308	18	,	,	PUNCT
ejpam-6098	308	19	...	...	PUNCT
ejpam-6098	308	20	,	,	PUNCT
ejpam-6098	308	21	δn−1	δn−1	PROPN
ejpam-6098	308	22	)	)	PUNCT
ejpam-6098	308	23	.	.	PUNCT
ejpam-6098	309	1	hence	hence	ADV
ejpam-6098	309	2	,	,	PUNCT
ejpam-6098	309	3	the	the	DET
ejpam-6098	309	4	result	result	NOUN
ejpam-6098	309	5	(	(	PUNCT
ejpam-6098	309	6	15	15	NUM
ejpam-6098	309	7	)	)	PUNCT
ejpam-6098	309	8	is	be	AUX
ejpam-6098	309	9	true	true	ADJ
ejpam-6098	309	10	for	for	ADP
ejpam-6098	309	11	all	all	DET
ejpam-6098	309	12	m.	m.	NOUN
ejpam-6098	309	13	now	now	ADV
ejpam-6098	309	14	,	,	PUNCT
ejpam-6098	309	15	taking	take	VERB
ejpam-6098	309	16	limit	limit	NOUN
ejpam-6098	309	17	as	as	ADP
ejpam-6098	309	18	m→	m→	NOUN
ejpam-6098	309	19	∞	∞	PROPN
ejpam-6098	309	20	in	in	ADP
ejpam-6098	309	21	(	(	PUNCT
ejpam-6098	309	22	15	15	NUM
ejpam-6098	309	23	)	)	PUNCT
ejpam-6098	309	24	,	,	PUNCT
ejpam-6098	309	25	we	we	PRON
ejpam-6098	309	26	obtained	obtain	VERB
ejpam-6098	309	27	(	(	PUNCT
ejpam-6098	309	28	11	11	NUM
ejpam-6098	309	29	)	)	PUNCT
ejpam-6098	309	30	.	.	PUNCT
ejpam-6098	310	1	now	now	ADV
ejpam-6098	310	2	,	,	PUNCT
ejpam-6098	310	3	to	to	PART
ejpam-6098	310	4	prove	prove	VERB
ejpam-6098	310	5	the	the	DET
ejpam-6098	310	6	q	q	NOUN
ejpam-6098	310	7	is	be	AUX
ejpam-6098	310	8	unique	unique	ADJ
ejpam-6098	310	9	,	,	PUNCT
ejpam-6098	310	10	assume	assume	VERB
ejpam-6098	310	11	q′	q′	NOUN
ejpam-6098	310	12	is	be	AUX
ejpam-6098	310	13	an	an	DET
ejpam-6098	310	14	another	another	DET
ejpam-6098	310	15	cubic	cubic	ADJ
ejpam-6098	310	16	mapping	mapping	NOUN
ejpam-6098	310	17	which	which	PRON
ejpam-6098	310	18	follows	follow	VERB
ejpam-6098	310	19	(	(	PUNCT
ejpam-6098	310	20	11	11	NUM
ejpam-6098	310	21	)	)	PUNCT
ejpam-6098	310	22	,	,	PUNCT
ejpam-6098	310	23	||q(u)−q′(u	||q(u)−q′(u	PROPN
ejpam-6098	310	24	)	)	PUNCT
ejpam-6098	310	25	,	,	PUNCT
ejpam-6098	310	26	δ1	δ1	NOUN
ejpam-6098	310	27	,	,	PUNCT
ejpam-6098	310	28	...	...	PUNCT
ejpam-6098	310	29	,	,	PUNCT
ejpam-6098	310	30	δn−1||β	δn−1||β	X
ejpam-6098	310	31	=	=	SYM
ejpam-6098	310	32	2−3mβ||q(2mu)−q′(2mu	2−3mβ||q(2mu)−q′(2mu	NUM
ejpam-6098	310	33	)	)	PUNCT
ejpam-6098	310	34	,	,	PUNCT
ejpam-6098	310	35	δ1	δ1	NOUN
ejpam-6098	310	36	,	,	PUNCT
ejpam-6098	310	37	...	...	PUNCT
ejpam-6098	310	38	,	,	PUNCT
ejpam-6098	310	39	δn−1||β	δn−1||β	NOUN
ejpam-6098	310	40	≤	≤	NOUN
ejpam-6098	310	41	2−3mβmax{||q(2mu)−	2−3mβmax{||q(2mu)−	NUM
ejpam-6098	310	42	f(2mu	f(2mu	NOUN
ejpam-6098	310	43	)	)	PUNCT
ejpam-6098	310	44	,	,	PUNCT
ejpam-6098	310	45	δ1	δ1	NOUN
ejpam-6098	310	46	,	,	PUNCT
ejpam-6098	310	47	...	...	PUNCT
ejpam-6098	310	48	,	,	PUNCT
ejpam-6098	310	49	δn−1||β	δn−1||β	PROPN
ejpam-6098	310	50	,	,	PUNCT
ejpam-6098	310	51	||f(2mu)−q′(2mu	||f(2mu)−q′(2mu	NUM
ejpam-6098	310	52	)	)	PUNCT
ejpam-6098	310	53	,	,	PUNCT
ejpam-6098	310	54	δ1	δ1	NOUN
ejpam-6098	310	55	,	,	PUNCT
ejpam-6098	310	56	...	...	PUNCT
ejpam-6098	310	57	,	,	PUNCT
ejpam-6098	310	58	δn−1||β	δn−1||β	X
ejpam-6098	310	59	}	}	PUNCT
ejpam-6098	310	60	≤	≤	NUM
ejpam-6098	310	61	3−β2−3mβ2−3βθ||2mu||pβ1	3−β2−3mβ2−3βθ||2mu||pβ1	NUM
ejpam-6098	310	62	ϕ(δ1	ϕ(δ1	NOUN
ejpam-6098	310	63	,	,	PUNCT
ejpam-6098	310	64	...	...	PUNCT
ejpam-6098	310	65	,	,	PUNCT
ejpam-6098	310	66	δn−1	δn−1	PROPN
ejpam-6098	310	67	)	)	PUNCT
ejpam-6098	310	68	.	.	PUNCT
ejpam-6098	311	1	applying	apply	VERB
ejpam-6098	311	2	limit	limit	NOUN
ejpam-6098	311	3	as	as	ADP
ejpam-6098	311	4	m→	m→	NOUN
ejpam-6098	311	5	∞	∞	PROPN
ejpam-6098	311	6	,	,	PUNCT
ejpam-6098	311	7	we	we	PRON
ejpam-6098	311	8	find	find	VERB
ejpam-6098	311	9	||q(u)−q′(u	||q(u)−q′(u	NOUN
ejpam-6098	311	10	)	)	PUNCT
ejpam-6098	311	11	,	,	PUNCT
ejpam-6098	311	12	δ1	δ1	NOUN
ejpam-6098	311	13	,	,	PUNCT
ejpam-6098	311	14	...	...	PUNCT
ejpam-6098	311	15	,	,	PUNCT
ejpam-6098	311	16	δn−1||β	δn−1||β	X
ejpam-6098	311	17	=	=	SYM
ejpam-6098	311	18	0	0	X
ejpam-6098	311	19	.	.	PUNCT
ejpam-6098	312	1	by	by	ADP
ejpam-6098	312	2	using	use	VERB
ejpam-6098	312	3	lemma	lemma	PROPN
ejpam-6098	312	4	(	(	PUNCT
ejpam-6098	312	5	1	1	NUM
ejpam-6098	312	6	)	)	PUNCT
ejpam-6098	312	7	,	,	PUNCT
ejpam-6098	312	8	we	we	PRON
ejpam-6098	312	9	find	find	VERB
ejpam-6098	312	10	the	the	DET
ejpam-6098	312	11	uniqueness	uniqueness	NOUN
ejpam-6098	312	12	of	of	ADP
ejpam-6098	312	13	mapping	mapping	NOUN
ejpam-6098	312	14	q.	q.	PROPN
ejpam-6098	312	15	theorem	theorem	VERB
ejpam-6098	312	16	3	3	X
ejpam-6098	312	17	.	.	PUNCT
ejpam-6098	313	1	let	let	VERB
ejpam-6098	313	2	ϕ	ϕ	NOUN
ejpam-6098	313	3	:	:	PUNCT
ejpam-6098	313	4	u	u	PRON
ejpam-6098	313	5	×	×	NOUN
ejpam-6098	313	6	u	u	NOUN
ejpam-6098	313	7	×	×	PROPN
ejpam-6098	313	8	u	u	X
ejpam-6098	313	9	→	→	SYM
ejpam-6098	313	10	[	[	X
ejpam-6098	313	11	0,∞	0,∞	X
ejpam-6098	313	12	)	)	PUNCT
ejpam-6098	313	13	be	be	VERB
ejpam-6098	313	14	a	a	DET
ejpam-6098	313	15	mapping	mapping	NOUN
ejpam-6098	313	16	holds	hold	VERB
ejpam-6098	313	17	limm→∞|	limm→∞|	DET
ejpam-6098	313	18	1	1	NUM
ejpam-6098	313	19	23mβ	23mβ	NUM
ejpam-6098	313	20	|ϕ(2mu	|ϕ(2mu	NOUN
ejpam-6098	313	21	,	,	PUNCT
ejpam-6098	313	22	2mv	2mv	ADJ
ejpam-6098	313	23	,	,	PUNCT
ejpam-6098	313	24	2mw	2mw	ADJ
ejpam-6098	313	25	)	)	PUNCT
ejpam-6098	314	1	=	=	SYM
ejpam-6098	314	2	0	0	PUNCT
ejpam-6098	314	3	(	(	PUNCT
ejpam-6098	314	4	17	17	NUM
ejpam-6098	314	5	)	)	PUNCT
ejpam-6098	314	6	and	and	CCONJ
ejpam-6098	314	7	ψ	ψ	X
ejpam-6098	314	8	:	:	PUNCT
ejpam-6098	314	9	v	v	X
ejpam-6098	314	10	n−1	n−1	PROPN
ejpam-6098	314	11	→	→	SYM
ejpam-6098	314	12	[	[	X
ejpam-6098	314	13	0,∞	0,∞	X
ejpam-6098	314	14	)	)	PUNCT
ejpam-6098	314	15	be	be	AUX
ejpam-6098	314	16	a	a	DET
ejpam-6098	314	17	mapping	mapping	NOUN
ejpam-6098	314	18	.	.	PUNCT
ejpam-6098	315	1	the	the	DET
ejpam-6098	315	2	limit	limit	NOUN
ejpam-6098	315	3	limm→∞max{2−3jβψ(2j−1u	limm→∞max{2−3jβψ(2j−1u	PROPN
ejpam-6098	315	4	,	,	PUNCT
ejpam-6098	315	5	0	0	NUM
ejpam-6098	315	6	,	,	PUNCT
ejpam-6098	315	7	0	0	NUM
ejpam-6098	315	8	)	)	PUNCT
ejpam-6098	315	9	:	:	PUNCT
ejpam-6098	315	10	m	m	VERB
ejpam-6098	315	11	≥	≥	VERB
ejpam-6098	315	12	j	j	NOUN
ejpam-6098	315	13	≥	≥	NUM
ejpam-6098	315	14	1	1	NUM
ejpam-6098	315	15	}	}	PUNCT
ejpam-6098	315	16	(	(	PUNCT
ejpam-6098	315	17	18	18	NUM
ejpam-6098	315	18	)	)	PUNCT
ejpam-6098	315	19	exists	exist	VERB
ejpam-6098	315	20	and	and	CCONJ
ejpam-6098	315	21	it	it	PRON
ejpam-6098	315	22	is	be	AUX
ejpam-6098	315	23	denoted	denote	VERB
ejpam-6098	315	24	by	by	ADP
ejpam-6098	315	25	ϕ̃(u	ϕ̃(u	PROPN
ejpam-6098	315	26	)	)	PUNCT
ejpam-6098	315	27	.	.	PUNCT
ejpam-6098	316	1	let	let	VERB
ejpam-6098	316	2	f	f	NOUN
ejpam-6098	316	3	:	:	PUNCT
ejpam-6098	316	4	u	u	PROPN
ejpam-6098	316	5	→	→	SYM
ejpam-6098	316	6	v	v	NUM
ejpam-6098	316	7	be	be	AUX
ejpam-6098	316	8	an	an	DET
ejpam-6098	316	9	odd	odd	ADJ
ejpam-6098	316	10	function	function	NOUN
ejpam-6098	316	11	fulfilling	fulfil	VERB
ejpam-6098	316	12	||f(2u+	||f(2u+	VERB
ejpam-6098	316	13	w	w	ADP
ejpam-6098	316	14	+	+	CCONJ
ejpam-6098	316	15	v)−	v)−	PROPN
ejpam-6098	316	16	f(v	f(v	NOUN
ejpam-6098	316	17	−	−	NOUN
ejpam-6098	316	18	u+	u+	NOUN
ejpam-6098	316	19	w)−	w)−	PROPN
ejpam-6098	316	20	3f(v	3f(v	NUM
ejpam-6098	317	1	+	+	CCONJ
ejpam-6098	317	2	w	w	PROPN
ejpam-6098	317	3	+	+	CCONJ
ejpam-6098	317	4	u)−	u)−	PROPN
ejpam-6098	317	5	2f(w	2f(w	NOUN
ejpam-6098	318	1	+	+	CCONJ
ejpam-6098	318	2	u)−	u)−	PROPN
ejpam-6098	318	3	2f(w	2f(w	PROPN
ejpam-6098	318	4	+	+	CCONJ
ejpam-6098	318	5	v	v	NOUN
ejpam-6098	318	6	)	)	PUNCT
ejpam-6098	318	7	+6f(u−	+6f(u−	PROPN
ejpam-6098	318	8	v	v	NOUN
ejpam-6098	318	9	)	)	PUNCT
ejpam-6098	319	1	+	+	CCONJ
ejpam-6098	319	2	3f(w	3f(w	NOUN
ejpam-6098	319	3	+	+	CCONJ
ejpam-6098	319	4	v	v	NOUN
ejpam-6098	319	5	)	)	PUNCT
ejpam-6098	320	1	+	+	CCONJ
ejpam-6098	320	2	6f(u−	6f(u−	NUM
ejpam-6098	320	3	w)−	w)−	PROPN
ejpam-6098	320	4	2f(2u−	2f(2u−	NUM
ejpam-6098	320	5	w)−	w)−	PROPN
ejpam-6098	320	6	2f(2u−	2f(2u−	NUM
ejpam-6098	320	7	v	v	NOUN
ejpam-6098	320	8	)	)	PUNCT
ejpam-6098	320	9	+	+	CCONJ
ejpam-6098	320	10	6f(v	6f(v	NUM
ejpam-6098	320	11	)	)	PUNCT
ejpam-6098	320	12	+18f(u	+18f(u	PROPN
ejpam-6098	320	13	)	)	PUNCT
ejpam-6098	321	1	+	+	CCONJ
ejpam-6098	321	2	6f(w	6f(w	NUM
ejpam-6098	321	3	)	)	PUNCT
ejpam-6098	321	4	,	,	PUNCT
ejpam-6098	321	5	δ1	δ1	NOUN
ejpam-6098	321	6	,	,	PUNCT
ejpam-6098	321	7	...	...	PUNCT
ejpam-6098	321	8	,	,	PUNCT
ejpam-6098	321	9	δn−1∥β	δn−1∥β	NOUN
ejpam-6098	321	10	≤	≤	NUM
ejpam-6098	321	11	ϕ(u	ϕ(u	PROPN
ejpam-6098	321	12	,	,	PUNCT
ejpam-6098	321	13	v	v	NOUN
ejpam-6098	321	14	,	,	PUNCT
ejpam-6098	321	15	w)ψ(δ1	w)ψ(δ1	NOUN
ejpam-6098	321	16	,	,	PUNCT
ejpam-6098	321	17	...	...	PUNCT
ejpam-6098	321	18	,	,	PUNCT
ejpam-6098	321	19	δn−1	δn−1	PROPN
ejpam-6098	321	20	)	)	PUNCT
ejpam-6098	321	21	.	.	PUNCT
ejpam-6098	322	1	(	(	PUNCT
ejpam-6098	322	2	19	19	NUM
ejpam-6098	322	3	)	)	PUNCT
ejpam-6098	322	4	then	then	ADV
ejpam-6098	322	5	,	,	PUNCT
ejpam-6098	322	6	there	there	PRON
ejpam-6098	322	7	is	be	VERB
ejpam-6098	322	8	exactly	exactly	ADV
ejpam-6098	322	9	one	one	NUM
ejpam-6098	322	10	cubic	cubic	ADJ
ejpam-6098	322	11	mapping	mapping	NOUN
ejpam-6098	322	12	q	q	NOUN
ejpam-6098	322	13	:	:	PUNCT
ejpam-6098	322	14	u	u	SYM
ejpam-6098	322	15	→	→	SYM
ejpam-6098	322	16	v	v	NUM
ejpam-6098	322	17	holds	hold	VERB
ejpam-6098	322	18	||f(u)−q(u	||f(u)−q(u	NOUN
ejpam-6098	322	19	)	)	PUNCT
ejpam-6098	322	20	,	,	PUNCT
ejpam-6098	322	21	δ1	δ1	NOUN
ejpam-6098	322	22	,	,	PUNCT
ejpam-6098	322	23	...	...	PUNCT
ejpam-6098	322	24	,	,	PUNCT
ejpam-6098	322	25	δn−1||β	δn−1||β	NOUN
ejpam-6098	322	26	≤	≤	NOUN
ejpam-6098	322	27	3−βϕ̃(u)ψ(δ1	3−βϕ̃(u)ψ(δ1	NUM
ejpam-6098	322	28	,	,	PUNCT
ejpam-6098	322	29	...	...	PUNCT
ejpam-6098	322	30	,	,	PUNCT
ejpam-6098	322	31	δn−1	δn−1	PROPN
ejpam-6098	322	32	)	)	PUNCT
ejpam-6098	322	33	(	(	PUNCT
ejpam-6098	322	34	20	20	NUM
ejpam-6098	322	35	)	)	PUNCT
ejpam-6098	322	36	and	and	CCONJ
ejpam-6098	322	37	limk→∞limm→∞max{2−3βϕ(u	limk→∞limm→∞max{2−3βϕ(u	PROPN
ejpam-6098	322	38	,	,	PUNCT
ejpam-6098	322	39	0	0	NUM
ejpam-6098	322	40	,	,	PUNCT
ejpam-6098	322	41	0	0	NUM
ejpam-6098	322	42	)	)	PUNCT
ejpam-6098	322	43	:	:	PUNCT
ejpam-6098	322	44	1	1	NUM
ejpam-6098	322	45	+	+	NUM
ejpam-6098	322	46	k;≤	k;≤	PROPN
ejpam-6098	322	47	j	j	PROPN
ejpam-6098	322	48	≤	≤	PROPN
ejpam-6098	322	49	k	k	PROPN
ejpam-6098	323	1	+	+	PROPN
ejpam-6098	323	2	m	m	NOUN
ejpam-6098	323	3	}	}	PUNCT
ejpam-6098	323	4	=	=	SYM
ejpam-6098	323	5	0	0	NUM
ejpam-6098	323	6	(	(	PUNCT
ejpam-6098	323	7	21	21	NUM
ejpam-6098	323	8	)	)	PUNCT
ejpam-6098	323	9	for	for	ADP
ejpam-6098	323	10	all	all	DET
ejpam-6098	323	11	u	u	PRON
ejpam-6098	323	12	∈	∈	PROPN
ejpam-6098	323	13	u	u	NOUN
ejpam-6098	323	14	and	and	CCONJ
ejpam-6098	323	15	δ1	δ1	NOUN
ejpam-6098	323	16	,	,	PUNCT
ejpam-6098	323	17	...	...	PUNCT
ejpam-6098	323	18	,	,	PUNCT
ejpam-6098	323	19	δn−1	δn−1	PROPN
ejpam-6098	323	20	∈	∈	PROPN
ejpam-6098	323	21	v	v	NOUN
ejpam-6098	323	22	.	.	PUNCT
ejpam-6098	324	1	j.	j.	PROPN
ejpam-6098	324	2	jakhar	jakhar	PROPN
ejpam-6098	324	3	et	et	PROPN
ejpam-6098	324	4	al	al	PROPN
ejpam-6098	324	5	.	.	PUNCT
ejpam-6098	324	6	/	/	SYM
ejpam-6098	324	7	eur	eur	PROPN
ejpam-6098	324	8	.	.	PUNCT
ejpam-6098	325	1	j.	j.	PROPN
ejpam-6098	325	2	pure	pure	PROPN
ejpam-6098	325	3	appl	appl	PROPN
ejpam-6098	325	4	.	.	PROPN
ejpam-6098	325	5	math	math	PROPN
ejpam-6098	325	6	,	,	PUNCT
ejpam-6098	325	7	18	18	NUM
ejpam-6098	325	8	(	(	PUNCT
ejpam-6098	325	9	2	2	NUM
ejpam-6098	325	10	)	)	PUNCT
ejpam-6098	325	11	(	(	PUNCT
ejpam-6098	325	12	2025	2025	NUM
ejpam-6098	325	13	)	)	PUNCT
ejpam-6098	325	14	,	,	PUNCT
ejpam-6098	325	15	6098	6098	NUM
ejpam-6098	325	16	13	13	NUM
ejpam-6098	325	17	of	of	ADP
ejpam-6098	325	18	23	23	NUM
ejpam-6098	325	19	proof	proof	NOUN
ejpam-6098	325	20	.	.	PUNCT
ejpam-6098	326	1	putting	put	VERB
ejpam-6098	326	2	(	(	PUNCT
ejpam-6098	326	3	u	u	NOUN
ejpam-6098	326	4	,	,	PUNCT
ejpam-6098	326	5	v	v	NOUN
ejpam-6098	326	6	,	,	PUNCT
ejpam-6098	326	7	w	w	NOUN
ejpam-6098	326	8	)	)	PUNCT
ejpam-6098	326	9	=	=	SYM
ejpam-6098	326	10	(	(	PUNCT
ejpam-6098	326	11	u	u	NOUN
ejpam-6098	326	12	,	,	PUNCT
ejpam-6098	326	13	0	0	NUM
ejpam-6098	326	14	,	,	PUNCT
ejpam-6098	326	15	0	0	NUM
ejpam-6098	326	16	)	)	PUNCT
ejpam-6098	326	17	in	in	ADP
ejpam-6098	326	18	(	(	PUNCT
ejpam-6098	326	19	19	19	NUM
ejpam-6098	326	20	)	)	PUNCT
ejpam-6098	326	21	and	and	CCONJ
ejpam-6098	326	22	multiplying	multiply	VERB
ejpam-6098	326	23	both	both	DET
ejpam-6098	326	24	side	side	NOUN
ejpam-6098	326	25	by	by	ADP
ejpam-6098	326	26	3−β2−3β	3−β2−3β	NUM
ejpam-6098	326	27	,	,	PUNCT
ejpam-6098	326	28	we	we	PRON
ejpam-6098	326	29	have	have	AUX
ejpam-6098	326	30	||f(2u	||f(2u	NOUN
ejpam-6098	326	31	)	)	PUNCT
ejpam-6098	326	32	23	23	NUM
ejpam-6098	326	33	−	−	PROPN
ejpam-6098	326	34	f(u	f(u	PROPN
ejpam-6098	326	35	)	)	PUNCT
ejpam-6098	326	36	,	,	PUNCT
ejpam-6098	326	37	δ1	δ1	NOUN
ejpam-6098	326	38	,	,	PUNCT
ejpam-6098	326	39	...	...	PUNCT
ejpam-6098	326	40	,	,	PUNCT
ejpam-6098	326	41	δn−1||β	δn−1||β	NOUN
ejpam-6098	326	42	≤	≤	NOUN
ejpam-6098	326	43	3−β2−3βϕ(u	3−β2−3βϕ(u	NUM
ejpam-6098	326	44	,	,	PUNCT
ejpam-6098	326	45	0	0	NUM
ejpam-6098	326	46	,	,	PUNCT
ejpam-6098	326	47	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	326	48	,	,	PUNCT
ejpam-6098	326	49	...	...	PUNCT
ejpam-6098	326	50	,	,	PUNCT
ejpam-6098	326	51	δn−1	δn−1	PROPN
ejpam-6098	326	52	)	)	PUNCT
ejpam-6098	326	53	.	.	PUNCT
ejpam-6098	327	1	(	(	PUNCT
ejpam-6098	327	2	22	22	X
ejpam-6098	327	3	)	)	PUNCT
ejpam-6098	327	4	changing	change	VERB
ejpam-6098	327	5	u	u	NOUN
ejpam-6098	327	6	by	by	ADP
ejpam-6098	327	7	2mu	2mu	NOUN
ejpam-6098	327	8	in	in	ADP
ejpam-6098	327	9	(	(	PUNCT
ejpam-6098	327	10	22	22	NUM
ejpam-6098	327	11	)	)	PUNCT
ejpam-6098	327	12	and	and	CCONJ
ejpam-6098	327	13	multiplying	multiply	VERB
ejpam-6098	327	14	both	both	DET
ejpam-6098	327	15	side	side	NOUN
ejpam-6098	327	16	by	by	ADP
ejpam-6098	327	17	2−3mβ	2−3mβ	PRON
ejpam-6098	327	18	,	,	PUNCT
ejpam-6098	327	19	we	we	PRON
ejpam-6098	327	20	get	get	VERB
ejpam-6098	327	21	||f(2	||f(2	PROPN
ejpam-6098	327	22	m+1u	m+1u	PROPN
ejpam-6098	327	23	)	)	PUNCT
ejpam-6098	327	24	23m+3	23m+3	NUM
ejpam-6098	327	25	−	−	NOUN
ejpam-6098	327	26	f(2mu	f(2mu	NOUN
ejpam-6098	327	27	)	)	PUNCT
ejpam-6098	327	28	23	23	NUM
ejpam-6098	327	29	m	m	NOUN
ejpam-6098	327	30	,	,	PUNCT
ejpam-6098	327	31	δ1	δ1	NOUN
ejpam-6098	327	32	,	,	PUNCT
ejpam-6098	327	33	...	...	PUNCT
ejpam-6098	327	34	,	,	PUNCT
ejpam-6098	327	35	δn−1||β	δn−1||β	NOUN
ejpam-6098	327	36	≤	≤	NOUN
ejpam-6098	327	37	3−β2−3β2−3mβϕ(2mu	3−β2−3β2−3mβϕ(2mu	NUM
ejpam-6098	327	38	,	,	PUNCT
ejpam-6098	327	39	0	0	NUM
ejpam-6098	327	40	,	,	PUNCT
ejpam-6098	327	41	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	327	42	,	,	PUNCT
ejpam-6098	327	43	...	...	PUNCT
ejpam-6098	327	44	,	,	PUNCT
ejpam-6098	327	45	δn−1	δn−1	PROPN
ejpam-6098	327	46	)	)	PUNCT
ejpam-6098	327	47	.	.	PUNCT
ejpam-6098	328	1	using	use	VERB
ejpam-6098	328	2	(	(	PUNCT
ejpam-6098	328	3	17	17	NUM
ejpam-6098	328	4	)	)	PUNCT
ejpam-6098	328	5	and	and	CCONJ
ejpam-6098	328	6	take	take	VERB
ejpam-6098	328	7	limit	limit	NOUN
ejpam-6098	328	8	m→	m→	NOUN
ejpam-6098	328	9	∞	∞	NOUN
ejpam-6098	328	10	,	,	PUNCT
ejpam-6098	328	11	we	we	PRON
ejpam-6098	328	12	get	get	VERB
ejpam-6098	328	13	||f(2	||f(2	PROPN
ejpam-6098	328	14	m+1u	m+1u	PROPN
ejpam-6098	328	15	)	)	PUNCT
ejpam-6098	328	16	23m+3	23m+3	NUM
ejpam-6098	328	17	−	−	NOUN
ejpam-6098	328	18	f(2mu	f(2mu	NOUN
ejpam-6098	328	19	)	)	PUNCT
ejpam-6098	328	20	23	23	NUM
ejpam-6098	328	21	m	m	NOUN
ejpam-6098	328	22	,	,	PUNCT
ejpam-6098	328	23	δ1	δ1	NOUN
ejpam-6098	328	24	,	,	PUNCT
ejpam-6098	328	25	...	...	PUNCT
ejpam-6098	328	26	,	,	PUNCT
ejpam-6098	328	27	δn−1||β	δn−1||β	X
ejpam-6098	328	28	=	=	SYM
ejpam-6098	329	1	0	0	X
ejpam-6098	329	2	.	.	X
ejpam-6098	329	3	using	use	VERB
ejpam-6098	329	4	definition	definition	NOUN
ejpam-6098	329	5	(	(	PUNCT
ejpam-6098	329	6	2	2	NUM
ejpam-6098	329	7	)	)	PUNCT
ejpam-6098	329	8	,	,	PUNCT
ejpam-6098	329	9	we	we	PRON
ejpam-6098	329	10	obtain	obtain	VERB
ejpam-6098	329	11	that	that	SCONJ
ejpam-6098	329	12	<	<	X
ejpam-6098	329	13	2−3mf(2mu	2−3mf(2mu	NUM
ejpam-6098	329	14	)	)	PUNCT
ejpam-6098	329	15	>	>	X
ejpam-6098	329	16	is	be	AUX
ejpam-6098	329	17	a	a	DET
ejpam-6098	329	18	cauchy	cauchy	ADJ
ejpam-6098	329	19	sequence	sequence	NOUN
ejpam-6098	329	20	in	in	ADP
ejpam-6098	329	21	complete	complete	ADJ
ejpam-6098	329	22	space	space	NOUN
ejpam-6098	329	23	v	v	NOUN
ejpam-6098	329	24	,	,	PUNCT
ejpam-6098	329	25	therefore	therefore	ADV
ejpam-6098	329	26	,	,	PUNCT
ejpam-6098	329	27	this	this	DET
ejpam-6098	329	28	sequence	sequence	NOUN
ejpam-6098	329	29	has	have	VERB
ejpam-6098	329	30	convergence	convergence	NOUN
ejpam-6098	329	31	point	point	NOUN
ejpam-6098	329	32	q(u	q(u	NOUN
ejpam-6098	329	33	)	)	PUNCT
ejpam-6098	329	34	∈	∈	NOUN
ejpam-6098	329	35	v	v	NOUN
ejpam-6098	329	36	.	.	PUNCT
ejpam-6098	330	1	now	now	ADV
ejpam-6098	330	2	,	,	PUNCT
ejpam-6098	330	3	we	we	PRON
ejpam-6098	330	4	consider	consider	VERB
ejpam-6098	330	5	a	a	DET
ejpam-6098	330	6	function	function	NOUN
ejpam-6098	330	7	q	q	NOUN
ejpam-6098	330	8	:	:	PUNCT
ejpam-6098	330	9	u	u	X
ejpam-6098	330	10	→	→	SYM
ejpam-6098	330	11	v	v	NUM
ejpam-6098	330	12	holds	hold	VERB
ejpam-6098	330	13	q(u	q(u	NOUN
ejpam-6098	330	14	)	)	PUNCT
ejpam-6098	330	15	=	=	SYM
ejpam-6098	330	16	limm→∞2−3mf(2mu	limm→∞2−3mf(2mu	PROPN
ejpam-6098	330	17	)	)	PUNCT
ejpam-6098	330	18	.	.	PUNCT
ejpam-6098	331	1	now	now	ADV
ejpam-6098	331	2	,	,	PUNCT
ejpam-6098	331	3	we	we	PRON
ejpam-6098	331	4	have	have	VERB
ejpam-6098	331	5	to	to	PART
ejpam-6098	331	6	prove	prove	VERB
ejpam-6098	331	7	that	that	PRON
ejpam-6098	331	8	q	q	NOUN
ejpam-6098	331	9	is	be	AUX
ejpam-6098	331	10	a	a	DET
ejpam-6098	331	11	cubic	cubic	NOUN
ejpam-6098	331	12	.	.	PUNCT
ejpam-6098	332	1	using	use	VERB
ejpam-6098	332	2	(	(	PUNCT
ejpam-6098	332	3	17	17	NUM
ejpam-6098	332	4	)	)	PUNCT
ejpam-6098	332	5	,	,	PUNCT
ejpam-6098	332	6	we	we	PRON
ejpam-6098	332	7	get	get	VERB
ejpam-6098	332	8	||q(2u+	||q(2u+	ADJ
ejpam-6098	333	1	w	w	PROPN
ejpam-6098	333	2	+	+	PROPN
ejpam-6098	333	3	v)−	v)−	PROPN
ejpam-6098	333	4	3q(v	3q(v	PROPN
ejpam-6098	333	5	+	+	CCONJ
ejpam-6098	333	6	w	w	PROPN
ejpam-6098	334	1	+	+	CCONJ
ejpam-6098	334	2	u)−q(w	u)−q(w	NOUN
ejpam-6098	334	3	−	−	PROPN
ejpam-6098	334	4	u+	u+	NOUN
ejpam-6098	334	5	v)−	v)−	PROPN
ejpam-6098	334	6	2q(u+	2q(u+	PROPN
ejpam-6098	334	7	v)−	v)−	PROPN
ejpam-6098	334	8	2q(w	2q(w	NOUN
ejpam-6098	334	9	+	+	CCONJ
ejpam-6098	334	10	u	u	NOUN
ejpam-6098	334	11	)	)	PUNCT
ejpam-6098	334	12	+6q(u−	+6q(u−	PROPN
ejpam-6098	334	13	v	v	NOUN
ejpam-6098	334	14	)	)	PUNCT
ejpam-6098	334	15	+	+	CCONJ
ejpam-6098	334	16	6q(u−	6q(u−	NUM
ejpam-6098	334	17	w	w	NOUN
ejpam-6098	334	18	)	)	PUNCT
ejpam-6098	334	19	+	+	CCONJ
ejpam-6098	334	20	3q(v	3q(v	NUM
ejpam-6098	334	21	+	+	CCONJ
ejpam-6098	334	22	w)−	w)−	PROPN
ejpam-6098	334	23	2q(2u−	2q(2u−	NUM
ejpam-6098	334	24	v)−	v)−	PROPN
ejpam-6098	334	25	2q(2u−	2q(2u−	NUM
ejpam-6098	334	26	w	w	PROPN
ejpam-6098	334	27	)	)	PUNCT
ejpam-6098	334	28	+	+	CCONJ
ejpam-6098	334	29	18q(u	18q(u	X
ejpam-6098	334	30	)	)	PUNCT
ejpam-6098	334	31	+6q(v	+6q(v	NUM
ejpam-6098	334	32	)	)	PUNCT
ejpam-6098	335	1	+	+	CCONJ
ejpam-6098	335	2	6q(w	6q(w	NUM
ejpam-6098	335	3	)	)	PUNCT
ejpam-6098	335	4	,	,	PUNCT
ejpam-6098	335	5	δ1	δ1	NOUN
ejpam-6098	335	6	,	,	PUNCT
ejpam-6098	335	7	...	...	PUNCT
ejpam-6098	335	8	,	,	PUNCT
ejpam-6098	335	9	δn−1||β	δn−1||β	PROPN
ejpam-6098	335	10	=	=	SYM
ejpam-6098	335	11	lim	lim	PROPN
ejpam-6098	335	12	m→∞	m→∞	NOUN
ejpam-6098	335	13	2−3mβ||f(2m(2u+	2−3mβ||f(2m(2u+	NUM
ejpam-6098	335	14	w	w	NOUN
ejpam-6098	335	15	+	+	NUM
ejpam-6098	335	16	v	v	NOUN
ejpam-6098	335	17	)	)	PUNCT
ejpam-6098	335	18	)	)	PUNCT
ejpam-6098	336	1	−3f(2m(v	−3f(2m(v	ADV
ejpam-6098	336	2	+	+	CCONJ
ejpam-6098	336	3	w	w	NOUN
ejpam-6098	336	4	+	+	NUM
ejpam-6098	336	5	u))−	u))−	ADJ
ejpam-6098	336	6	f(2m(w	f(2m(w	PROPN
ejpam-6098	336	7	−	−	PROPN
ejpam-6098	336	8	u+	u+	NOUN
ejpam-6098	336	9	v))−	v))−	NOUN
ejpam-6098	336	10	2f(2m(u+	2f(2m(u+	NOUN
ejpam-6098	336	11	v))−	v))−	NOUN
ejpam-6098	336	12	2f(2m(w	2f(2m(w	NOUN
ejpam-6098	336	13	+	+	CCONJ
ejpam-6098	336	14	u	u	NOUN
ejpam-6098	336	15	)	)	PUNCT
ejpam-6098	336	16	)	)	PUNCT
ejpam-6098	337	1	+6f(2m(u−	+6f(2m(u−	PRON
ejpam-6098	337	2	v	v	NOUN
ejpam-6098	337	3	)	)	PUNCT
ejpam-6098	337	4	)	)	PUNCT
ejpam-6098	338	1	+	+	CCONJ
ejpam-6098	338	2	6f(2m(u−	6f(2m(u−	NUM
ejpam-6098	338	3	w	w	NOUN
ejpam-6098	338	4	)	)	PUNCT
ejpam-6098	338	5	)	)	PUNCT
ejpam-6098	339	1	+	+	NUM
ejpam-6098	339	2	3f(2m(w	3f(2m(w	NUM
ejpam-6098	339	3	+	+	CCONJ
ejpam-6098	339	4	v))−	v))−	NOUN
ejpam-6098	339	5	2f(2m(2u−	2f(2m(2u−	PROPN
ejpam-6098	339	6	v	v	NOUN
ejpam-6098	339	7	)	)	PUNCT
ejpam-6098	339	8	)	)	PUNCT
ejpam-6098	340	1	−2f(2m(2u−	−2f(2m(2u−	PROPN
ejpam-6098	340	2	w	w	PROPN
ejpam-6098	340	3	)	)	PUNCT
ejpam-6098	340	4	)	)	PUNCT
ejpam-6098	341	1	+	+	CCONJ
ejpam-6098	341	2	18f(2mu	18f(2mu	NUM
ejpam-6098	341	3	)	)	PUNCT
ejpam-6098	341	4	+	+	CCONJ
ejpam-6098	341	5	6f(2mv	6f(2mv	X
ejpam-6098	341	6	)	)	PUNCT
ejpam-6098	341	7	+	+	CCONJ
ejpam-6098	341	8	6f(2mw	6f(2mw	NUM
ejpam-6098	341	9	)	)	PUNCT
ejpam-6098	341	10	,	,	PUNCT
ejpam-6098	341	11	δ1	δ1	NOUN
ejpam-6098	341	12	,	,	PUNCT
ejpam-6098	341	13	...	...	PUNCT
ejpam-6098	341	14	,	,	PUNCT
ejpam-6098	341	15	δn−1||β	δn−1||β	PROPN
ejpam-6098	341	16	≤	≤	NUM
ejpam-6098	341	17	lim	lim	PROPN
ejpam-6098	341	18	m→∞	m→∞	NOUN
ejpam-6098	341	19	2−3mβϕ(2mu	2−3mβϕ(2mu	NUM
ejpam-6098	341	20	,	,	PUNCT
ejpam-6098	341	21	2mv	2mv	ADJ
ejpam-6098	341	22	,	,	PUNCT
ejpam-6098	341	23	2mw)ψ(δ1	2mw)ψ(δ1	NUM
ejpam-6098	341	24	,	,	PUNCT
ejpam-6098	341	25	...	...	PUNCT
ejpam-6098	341	26	,	,	PUNCT
ejpam-6098	341	27	δn−1	δn−1	PROPN
ejpam-6098	341	28	)	)	PUNCT
ejpam-6098	341	29	=	=	SYM
ejpam-6098	341	30	0	0	NUM
ejpam-6098	341	31	,	,	PUNCT
ejpam-6098	341	32	hence	hence	ADV
ejpam-6098	341	33	||q(2u+	||q(2u+	VERB
ejpam-6098	341	34	v	v	ADP
ejpam-6098	341	35	+	+	CCONJ
ejpam-6098	341	36	w)−	w)−	PROPN
ejpam-6098	341	37	3q(v	3q(v	PROPN
ejpam-6098	342	1	+	+	CCONJ
ejpam-6098	342	2	w	w	PROPN
ejpam-6098	343	1	+	+	CCONJ
ejpam-6098	343	2	u)−q(w	u)−q(w	NOUN
ejpam-6098	343	3	−	−	PROPN
ejpam-6098	343	4	u+	u+	NOUN
ejpam-6098	343	5	v)−	v)−	PROPN
ejpam-6098	343	6	2q(u+	2q(u+	PROPN
ejpam-6098	343	7	v)−	v)−	PROPN
ejpam-6098	343	8	2q(w	2q(w	NOUN
ejpam-6098	343	9	+	+	CCONJ
ejpam-6098	343	10	u	u	NOUN
ejpam-6098	343	11	)	)	PUNCT
ejpam-6098	343	12	+6q(u−	+6q(u−	PROPN
ejpam-6098	343	13	v	v	NOUN
ejpam-6098	343	14	)	)	PUNCT
ejpam-6098	343	15	+	+	CCONJ
ejpam-6098	343	16	6q(u−	6q(u−	NUM
ejpam-6098	343	17	w	w	NOUN
ejpam-6098	343	18	)	)	PUNCT
ejpam-6098	343	19	+	+	NUM
ejpam-6098	343	20	3q(w	3q(w	NUM
ejpam-6098	343	21	+	+	CCONJ
ejpam-6098	343	22	v)−	v)−	PROPN
ejpam-6098	343	23	2q(2u−	2q(2u−	NUM
ejpam-6098	343	24	v)−	v)−	PROPN
ejpam-6098	343	25	2q(2u−	2q(2u−	NUM
ejpam-6098	343	26	w	w	PROPN
ejpam-6098	343	27	)	)	PUNCT
ejpam-6098	343	28	+	+	CCONJ
ejpam-6098	343	29	18q(u	18q(u	X
ejpam-6098	343	30	)	)	PUNCT
ejpam-6098	343	31	+6q(v	+6q(v	NUM
ejpam-6098	343	32	)	)	PUNCT
ejpam-6098	344	1	+	+	CCONJ
ejpam-6098	344	2	6q(w	6q(w	NUM
ejpam-6098	344	3	)	)	PUNCT
ejpam-6098	344	4	,	,	PUNCT
ejpam-6098	344	5	δ1	δ1	NOUN
ejpam-6098	344	6	,	,	PUNCT
ejpam-6098	344	7	...	...	PUNCT
ejpam-6098	344	8	,	,	PUNCT
ejpam-6098	344	9	δn−1||β	δn−1||β	X
ejpam-6098	344	10	=	=	SYM
ejpam-6098	344	11	0	0	X
ejpam-6098	344	12	.	.	X
ejpam-6098	345	1	using	use	VERB
ejpam-6098	345	2	lemma	lemma	PROPN
ejpam-6098	345	3	(	(	PUNCT
ejpam-6098	345	4	1	1	NUM
ejpam-6098	345	5	)	)	PUNCT
ejpam-6098	345	6	,	,	PUNCT
ejpam-6098	345	7	we	we	PRON
ejpam-6098	345	8	obtain	obtain	VERB
ejpam-6098	345	9	q	q	PUNCT
ejpam-6098	345	10	is	be	AUX
ejpam-6098	345	11	cubic	cubic	ADJ
ejpam-6098	345	12	mapping	mapping	NOUN
ejpam-6098	345	13	.	.	PUNCT
ejpam-6098	346	1	next	next	ADV
ejpam-6098	346	2	,	,	PUNCT
ejpam-6098	346	3	substituting	substitute	VERB
ejpam-6098	346	4	u	u	NOUN
ejpam-6098	346	5	by	by	ADP
ejpam-6098	346	6	2u	2u	NOUN
ejpam-6098	346	7	in	in	ADP
ejpam-6098	346	8	(	(	PUNCT
ejpam-6098	346	9	22	22	NUM
ejpam-6098	346	10	)	)	PUNCT
ejpam-6098	346	11	and	and	CCONJ
ejpam-6098	346	12	multiplying	multiply	VERB
ejpam-6098	346	13	both	both	DET
ejpam-6098	346	14	side	side	NOUN
ejpam-6098	346	15	by	by	ADP
ejpam-6098	346	16	2−3β	2−3β	NUM
ejpam-6098	346	17	,	,	PUNCT
ejpam-6098	346	18	the	the	DET
ejpam-6098	346	19	authors	author	NOUN
ejpam-6098	346	20	get	get	VERB
ejpam-6098	346	21	||f(2	||f(2	PROPN
ejpam-6098	346	22	2u	2u	NOUN
ejpam-6098	346	23	)	)	PUNCT
ejpam-6098	346	24	26	26	NUM
ejpam-6098	346	25	−	−	PROPN
ejpam-6098	346	26	f(2u	f(2u	NOUN
ejpam-6098	346	27	)	)	PUNCT
ejpam-6098	346	28	23	23	NUM
ejpam-6098	346	29	,	,	PUNCT
ejpam-6098	346	30	δ1	δ1	NOUN
ejpam-6098	346	31	,	,	PUNCT
ejpam-6098	346	32	...	...	PUNCT
ejpam-6098	346	33	,	,	PUNCT
ejpam-6098	346	34	δn−1||β	δn−1||β	NOUN
ejpam-6098	346	35	≤	≤	NOUN
ejpam-6098	346	36	|3|−β|2|−6βϕ(2u	|3|−β|2|−6βϕ(2u	NOUN
ejpam-6098	346	37	,	,	PUNCT
ejpam-6098	346	38	0	0	NUM
ejpam-6098	346	39	,	,	PUNCT
ejpam-6098	346	40	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	346	41	,	,	PUNCT
ejpam-6098	346	42	...	...	PUNCT
ejpam-6098	346	43	,	,	PUNCT
ejpam-6098	346	44	δn−1	δn−1	PROPN
ejpam-6098	346	45	)	)	PUNCT
ejpam-6098	346	46	.	.	PUNCT
ejpam-6098	347	1	by	by	ADP
ejpam-6098	347	2	using	use	VERB
ejpam-6098	347	3	(	(	PUNCT
ejpam-6098	347	4	22	22	NUM
ejpam-6098	347	5	)	)	PUNCT
ejpam-6098	347	6	,	,	PUNCT
ejpam-6098	347	7	we	we	PRON
ejpam-6098	347	8	get	get	VERB
ejpam-6098	347	9	||f(u)−	||f(u)−	NOUN
ejpam-6098	347	10	f(22u	f(22u	NOUN
ejpam-6098	347	11	)	)	PUNCT
ejpam-6098	347	12	26	26	NUM
ejpam-6098	347	13	,	,	PUNCT
ejpam-6098	347	14	v1	v1	NOUN
ejpam-6098	347	15	,	,	PUNCT
ejpam-6098	347	16	...	...	PUNCT
ejpam-6098	347	17	,	,	PUNCT
ejpam-6098	347	18	vn−1||β	vn−1||β	PROPN
ejpam-6098	347	19	j.	j.	PROPN
ejpam-6098	347	20	jakhar	jakhar	PROPN
ejpam-6098	347	21	et	et	PROPN
ejpam-6098	347	22	al	al	PROPN
ejpam-6098	347	23	.	.	PUNCT
ejpam-6098	347	24	/	/	SYM
ejpam-6098	347	25	eur	eur	PROPN
ejpam-6098	347	26	.	.	PUNCT
ejpam-6098	348	1	j.	j.	PROPN
ejpam-6098	348	2	pure	pure	PROPN
ejpam-6098	348	3	appl	appl	PROPN
ejpam-6098	348	4	.	.	PROPN
ejpam-6098	348	5	math	math	PROPN
ejpam-6098	348	6	,	,	PUNCT
ejpam-6098	348	7	18	18	NUM
ejpam-6098	348	8	(	(	PUNCT
ejpam-6098	348	9	2	2	NUM
ejpam-6098	348	10	)	)	PUNCT
ejpam-6098	348	11	(	(	PUNCT
ejpam-6098	348	12	2025	2025	NUM
ejpam-6098	348	13	)	)	PUNCT
ejpam-6098	348	14	,	,	PUNCT
ejpam-6098	348	15	6098	6098	NUM
ejpam-6098	348	16	14	14	NUM
ejpam-6098	348	17	of	of	ADP
ejpam-6098	348	18	23	23	NUM
ejpam-6098	348	19	≤	≤	NUM
ejpam-6098	348	20	max{3−β2−3βϕ(u	max{3−β2−3βϕ(u	NOUN
ejpam-6098	348	21	,	,	PUNCT
ejpam-6098	348	22	0	0	NUM
ejpam-6098	348	23	,	,	PUNCT
ejpam-6098	348	24	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	348	25	,	,	PUNCT
ejpam-6098	348	26	...	...	PUNCT
ejpam-6098	348	27	,	,	PUNCT
ejpam-6098	348	28	δn−1	δn−1	PROPN
ejpam-6098	348	29	)	)	PUNCT
ejpam-6098	348	30	,	,	PUNCT
ejpam-6098	348	31	3	3	X
ejpam-6098	348	32	−β2−6βϕ(2u	−β2−6βϕ(2u	PROPN
ejpam-6098	348	33	,	,	PUNCT
ejpam-6098	348	34	0	0	NUM
ejpam-6098	348	35	,	,	PUNCT
ejpam-6098	348	36	0)ψ(δ1	0)ψ(δ1	NOUN
ejpam-6098	348	37	,	,	PUNCT
ejpam-6098	348	38	...	...	PUNCT
ejpam-6098	348	39	,	,	PUNCT
ejpam-6098	348	40	δn−1	δn−1	PROPN
ejpam-6098	348	41	)	)	PUNCT
ejpam-6098	348	42	}	}	PUNCT
ejpam-6098	348	43	.	.	PUNCT
ejpam-6098	349	1	apply	apply	VERB
ejpam-6098	349	2	pmi	pmi	PROPN
ejpam-6098	349	3	on	on	ADP
ejpam-6098	349	4	m	m	PROPN
ejpam-6098	349	5	,	,	PUNCT
ejpam-6098	349	6	we	we	PRON
ejpam-6098	349	7	have	have	VERB
ejpam-6098	349	8	||f(u)−	||f(u)−	PROPN
ejpam-6098	349	9	f(2mu	f(2mu	NOUN
ejpam-6098	349	10	)	)	PUNCT
ejpam-6098	349	11	23	23	NUM
ejpam-6098	349	12	m	m	NOUN
ejpam-6098	349	13	,	,	PUNCT
ejpam-6098	349	14	δ1	δ1	NOUN
ejpam-6098	349	15	,	,	PUNCT
ejpam-6098	349	16	...	...	PUNCT
ejpam-6098	349	17	,	,	PUNCT
ejpam-6098	349	18	δn−1||β	δn−1||β	PROPN
ejpam-6098	349	19	≤	≤	NUM
ejpam-6098	349	20	max	max	PROPN
ejpam-6098	349	21	{	{	PUNCT
ejpam-6098	349	22	3−β	3−β	NUM
ejpam-6098	349	23	ϕ(2	ϕ(2	PROPN
ejpam-6098	349	24	k−1u	k−1u	PROPN
ejpam-6098	349	25	,	,	PUNCT
ejpam-6098	349	26	0	0	NUM
ejpam-6098	349	27	,	,	PUNCT
ejpam-6098	349	28	0	0	NUM
ejpam-6098	349	29	)	)	PUNCT
ejpam-6098	349	30	23kβ	23kβ	NOUN
ejpam-6098	349	31	:	:	PUNCT
ejpam-6098	349	32	1	1	NUM
ejpam-6098	349	33	≤	≤	NUM
ejpam-6098	349	34	k	k	X
ejpam-6098	349	35	≤	≤	NUM
ejpam-6098	349	36	m	m	VERB
ejpam-6098	349	37	}	}	PUNCT
ejpam-6098	349	38	ψ(δ1	ψ(δ1	PROPN
ejpam-6098	349	39	,	,	PUNCT
ejpam-6098	349	40	...	...	PUNCT
ejpam-6098	349	41	,	,	PUNCT
ejpam-6098	349	42	δn−1	δn−1	PROPN
ejpam-6098	349	43	)	)	PUNCT
ejpam-6098	349	44	.	.	PUNCT
ejpam-6098	350	1	(	(	PUNCT
ejpam-6098	350	2	23	23	NUM
ejpam-6098	350	3	)	)	PUNCT
ejpam-6098	350	4	changing	change	VERB
ejpam-6098	350	5	u	u	NOUN
ejpam-6098	350	6	by	by	ADP
ejpam-6098	350	7	2u	2u	NOUN
ejpam-6098	350	8	in	in	ADP
ejpam-6098	350	9	(	(	PUNCT
ejpam-6098	350	10	23	23	NUM
ejpam-6098	350	11	)	)	PUNCT
ejpam-6098	350	12	and	and	CCONJ
ejpam-6098	350	13	multiplying	multiply	VERB
ejpam-6098	350	14	both	both	DET
ejpam-6098	350	15	side	side	NOUN
ejpam-6098	350	16	by	by	ADP
ejpam-6098	350	17	2−3β	2−3β	NUM
ejpam-6098	350	18	,	,	PUNCT
ejpam-6098	350	19	we	we	PRON
ejpam-6098	350	20	have	have	VERB
ejpam-6098	350	21	||f(u)−	||f(u)−	PROPN
ejpam-6098	350	22	f(2m+1u	f(2m+1u	NOUN
ejpam-6098	350	23	)	)	PUNCT
ejpam-6098	350	24	23(m+1	23(m+1	NUM
ejpam-6098	350	25	)	)	PUNCT
ejpam-6098	350	26	,	,	PUNCT
ejpam-6098	350	27	δ1	δ1	NOUN
ejpam-6098	350	28	,	,	PUNCT
ejpam-6098	350	29	...	...	PUNCT
ejpam-6098	350	30	,	,	PUNCT
ejpam-6098	350	31	δn−1||β	δn−1||β	PROPN
ejpam-6098	350	32	≤	≤	NUM
ejpam-6098	350	33	max	max	PROPN
ejpam-6098	350	34	{	{	PUNCT
ejpam-6098	350	35	3−β	3−β	NUM
ejpam-6098	350	36	ϕ(u	ϕ(u	PROPN
ejpam-6098	350	37	,	,	PUNCT
ejpam-6098	350	38	0	0	NUM
ejpam-6098	350	39	,	,	PUNCT
ejpam-6098	350	40	0	0	NUM
ejpam-6098	350	41	)	)	PUNCT
ejpam-6098	350	42	23β	23β	NOUN
ejpam-6098	350	43	,	,	PUNCT
ejpam-6098	350	44	3−β	3−β	NUM
ejpam-6098	350	45	ϕ(2	ϕ(2	PROPN
ejpam-6098	350	46	ku	ku	PROPN
ejpam-6098	350	47	,	,	PUNCT
ejpam-6098	350	48	0	0	NUM
ejpam-6098	350	49	,	,	PUNCT
ejpam-6098	350	50	0	0	NUM
ejpam-6098	350	51	)	)	PUNCT
ejpam-6098	350	52	23(k+1)β	23(k+1)β	NUM
ejpam-6098	350	53	:	:	PUNCT
ejpam-6098	351	1	m	m	VERB
ejpam-6098	351	2	≥	≥	VERB
ejpam-6098	351	3	k	k	X
ejpam-6098	351	4	≥	≥	NUM
ejpam-6098	351	5	1	1	NUM
ejpam-6098	351	6	}	}	PUNCT
ejpam-6098	351	7	ψ(δ1	ψ(δ1	PROPN
ejpam-6098	351	8	,	,	PUNCT
ejpam-6098	351	9	...	...	PUNCT
ejpam-6098	351	10	,	,	PUNCT
ejpam-6098	351	11	δn−1	δn−1	PROPN
ejpam-6098	351	12	)	)	PUNCT
ejpam-6098	351	13	=	=	SYM
ejpam-6098	351	14	max	max	PROPN
ejpam-6098	351	15	{	{	PUNCT
ejpam-6098	351	16	3−β	3−β	NUM
ejpam-6098	351	17	ϕ(2	ϕ(2	PROPN
ejpam-6098	351	18	ku	ku	PROPN
ejpam-6098	351	19	,	,	PUNCT
ejpam-6098	351	20	0	0	NUM
ejpam-6098	351	21	,	,	PUNCT
ejpam-6098	351	22	0	0	NUM
ejpam-6098	351	23	)	)	PUNCT
ejpam-6098	351	24	23(k+1)β	23(k+1)β	NUM
ejpam-6098	351	25	:	:	PUNCT
ejpam-6098	351	26	m	m	VERB
ejpam-6098	351	27	≥	≥	VERB
ejpam-6098	351	28	k	k	X
ejpam-6098	351	29	≥	≥	X
ejpam-6098	351	30	0	0	NUM
ejpam-6098	351	31	}	}	PUNCT
ejpam-6098	351	32	ψ(δ1	ψ(δ1	PROPN
ejpam-6098	351	33	,	,	PUNCT
ejpam-6098	351	34	...	...	PUNCT
ejpam-6098	351	35	,	,	PUNCT
ejpam-6098	351	36	δn−1	δn−1	PROPN
ejpam-6098	351	37	)	)	PUNCT
ejpam-6098	351	38	=	=	SYM
ejpam-6098	351	39	max	max	PROPN
ejpam-6098	351	40	{	{	PUNCT
ejpam-6098	351	41	3−β	3−β	NUM
ejpam-6098	351	42	ϕ(2	ϕ(2	PROPN
ejpam-6098	351	43	k−1u	k−1u	PROPN
ejpam-6098	351	44	,	,	PUNCT
ejpam-6098	351	45	0	0	NUM
ejpam-6098	351	46	,	,	PUNCT
ejpam-6098	351	47	0	0	NUM
ejpam-6098	351	48	)	)	PUNCT
ejpam-6098	351	49	23kβ	23kβ	NOUN
ejpam-6098	351	50	:	:	PUNCT
ejpam-6098	351	51	m+	m+	NUM
ejpam-6098	351	52	1	1	NUM
ejpam-6098	351	53	≥	≥	NOUN
ejpam-6098	351	54	k	k	X
ejpam-6098	351	55	≥	≥	NUM
ejpam-6098	351	56	1	1	NUM
ejpam-6098	351	57	}	}	PUNCT
ejpam-6098	351	58	ψ(δ1	ψ(δ1	PROPN
ejpam-6098	351	59	,	,	PUNCT
ejpam-6098	351	60	...	...	PUNCT
ejpam-6098	351	61	,	,	PUNCT
ejpam-6098	351	62	δn−1	δn−1	PROPN
ejpam-6098	351	63	)	)	PUNCT
ejpam-6098	351	64	.	.	PUNCT
ejpam-6098	352	1	hence	hence	ADV
ejpam-6098	352	2	,	,	PUNCT
ejpam-6098	352	3	the	the	DET
ejpam-6098	352	4	result	result	NOUN
ejpam-6098	352	5	(	(	PUNCT
ejpam-6098	352	6	23	23	NUM
ejpam-6098	352	7	)	)	PUNCT
ejpam-6098	352	8	is	be	AUX
ejpam-6098	352	9	true	true	ADJ
ejpam-6098	352	10	for	for	ADP
ejpam-6098	352	11	all	all	PRON
ejpam-6098	352	12	m.	m.	NOUN
ejpam-6098	352	13	taking	take	VERB
ejpam-6098	352	14	limit	limit	NOUN
ejpam-6098	352	15	as	as	ADP
ejpam-6098	352	16	m	m	PROPN
ejpam-6098	352	17	→	→	SYM
ejpam-6098	352	18	∞	∞	NUM
ejpam-6098	352	19	in	in	ADP
ejpam-6098	352	20	(	(	PUNCT
ejpam-6098	352	21	23	23	NUM
ejpam-6098	352	22	)	)	PUNCT
ejpam-6098	352	23	,	,	PUNCT
ejpam-6098	352	24	we	we	PRON
ejpam-6098	352	25	get	get	VERB
ejpam-6098	352	26	(	(	PUNCT
ejpam-6098	352	27	20	20	NUM
ejpam-6098	352	28	)	)	PUNCT
ejpam-6098	352	29	.	.	PUNCT
ejpam-6098	353	1	to	to	PART
ejpam-6098	353	2	prove	prove	VERB
ejpam-6098	353	3	q	q	NOUN
ejpam-6098	353	4	is	be	AUX
ejpam-6098	353	5	unique	unique	ADJ
ejpam-6098	353	6	mapping	mapping	NOUN
ejpam-6098	353	7	.	.	PUNCT
ejpam-6098	354	1	assume	assume	VERB
ejpam-6098	354	2	q′	q′	NOUN
ejpam-6098	354	3	is	be	AUX
ejpam-6098	354	4	another	another	DET
ejpam-6098	354	5	cubic	cubic	ADJ
ejpam-6098	354	6	mapping	mapping	NOUN
ejpam-6098	354	7	which	which	PRON
ejpam-6098	354	8	follows	follow	VERB
ejpam-6098	354	9	(	(	PUNCT
ejpam-6098	354	10	20	20	NUM
ejpam-6098	354	11	)	)	PUNCT
ejpam-6098	354	12	.	.	PUNCT
ejpam-6098	355	1	now	now	ADV
ejpam-6098	355	2	limk→∞3−β2−3kβϕ(2ku	limk→∞3−β2−3kβϕ(2ku	ADJ
ejpam-6098	355	3	)	)	PUNCT
ejpam-6098	355	4	=	=	PRON
ejpam-6098	355	5	limk→∞3−β	limk→∞3−β	NOUN
ejpam-6098	355	6	1	1	NUM
ejpam-6098	355	7	23kβ	23kβ	NOUN
ejpam-6098	355	8	limm→∞max{2−3jβψ(2j+k−1u	limm→∞max{2−3jβψ(2j+k−1u	PROPN
ejpam-6098	355	9	,	,	PUNCT
ejpam-6098	355	10	0	0	NUM
ejpam-6098	355	11	,	,	PUNCT
ejpam-6098	355	12	0	0	NUM
ejpam-6098	355	13	)	)	PUNCT
ejpam-6098	355	14	:	:	PUNCT
ejpam-6098	355	15	1	1	NUM
ejpam-6098	355	16	≤	≤	NUM
ejpam-6098	355	17	k	k	X
ejpam-6098	355	18	≤	≤	NUM
ejpam-6098	355	19	m	m	VERB
ejpam-6098	355	20	}	}	PUNCT
ejpam-6098	355	21	=	=	SYM
ejpam-6098	355	22	limk→∞limm→∞max{3−β2−3jβψ(2j−1u	limk→∞limm→∞max{3−β2−3jβψ(2j−1u	PROPN
ejpam-6098	355	23	,	,	PUNCT
ejpam-6098	355	24	0	0	NUM
ejpam-6098	355	25	,	,	PUNCT
ejpam-6098	355	26	0	0	NUM
ejpam-6098	355	27	)	)	PUNCT
ejpam-6098	355	28	:	:	PUNCT
ejpam-6098	356	1	1	1	NUM
ejpam-6098	356	2	+	+	CCONJ
ejpam-6098	356	3	k	k	PROPN
ejpam-6098	356	4	≤	≤	PROPN
ejpam-6098	356	5	j	j	PROPN
ejpam-6098	356	6	≤	≤	NUM
ejpam-6098	356	7	m+	m+	NUM
ejpam-6098	356	8	k	k	NOUN
ejpam-6098	356	9	}	}	PUNCT
ejpam-6098	356	10	,	,	PUNCT
ejpam-6098	356	11	by	by	ADP
ejpam-6098	356	12	using	use	VERB
ejpam-6098	356	13	(	(	PUNCT
ejpam-6098	356	14	21	21	NUM
ejpam-6098	356	15	)	)	PUNCT
ejpam-6098	356	16	,	,	PUNCT
ejpam-6098	356	17	we	we	PRON
ejpam-6098	356	18	have	have	VERB
ejpam-6098	356	19	||q(u)−q′(u	||q(u)−q′(u	NOUN
ejpam-6098	356	20	)	)	PUNCT
ejpam-6098	356	21	,	,	PUNCT
ejpam-6098	356	22	δ1	δ1	NOUN
ejpam-6098	356	23	,	,	PUNCT
ejpam-6098	356	24	...	...	PUNCT
ejpam-6098	356	25	,	,	PUNCT
ejpam-6098	356	26	δn−1||β	δn−1||β	X
ejpam-6098	356	27	=	=	SYM
ejpam-6098	356	28	limk→∞2−3kβ||q(2ku)−q′(2ku	limk→∞2−3kβ||q(2ku)−q′(2ku	PROPN
ejpam-6098	356	29	)	)	PUNCT
ejpam-6098	356	30	,	,	PUNCT
ejpam-6098	356	31	δ1	δ1	NOUN
ejpam-6098	356	32	,	,	PUNCT
ejpam-6098	356	33	...	...	PUNCT
ejpam-6098	356	34	,	,	PUNCT
ejpam-6098	356	35	δn−1||β	δn−1||β	NOUN
ejpam-6098	356	36	≤	≤	PUNCT
ejpam-6098	357	1	limk→∞2−3kβmax{||q(2ku)−	limk→∞2−3kβmax{||q(2ku)−	ADP
ejpam-6098	357	2	f(2ku	f(2ku	VERB
ejpam-6098	357	3	)	)	PUNCT
ejpam-6098	358	1	,	,	PUNCT
ejpam-6098	358	2	δ1	δ1	NOUN
ejpam-6098	358	3	,	,	PUNCT
ejpam-6098	358	4	...	...	PUNCT
ejpam-6098	358	5	,	,	PUNCT
ejpam-6098	358	6	δn−1||β	δn−1||β	NOUN
ejpam-6098	358	7	,	,	PUNCT
ejpam-6098	358	8	||f(2ku)−q′(2ku	||f(2ku)−q′(2ku	NOUN
ejpam-6098	358	9	)	)	PUNCT
ejpam-6098	358	10	,	,	PUNCT
ejpam-6098	358	11	δ1	δ1	NOUN
ejpam-6098	358	12	,	,	PUNCT
ejpam-6098	358	13	...	...	PUNCT
ejpam-6098	358	14	,	,	PUNCT
ejpam-6098	358	15	δn−1||β	δn−1||β	PROPN
ejpam-6098	358	16	}	}	PUNCT
ejpam-6098	358	17	≤	≤	NUM
ejpam-6098	358	18	limk→∞2−3kβϕ(2ku)ψ(δ1	limk→∞2−3kβϕ(2ku)ψ(δ1	PROPN
ejpam-6098	358	19	,	,	PUNCT
ejpam-6098	358	20	...	...	PUNCT
ejpam-6098	358	21	,	,	PUNCT
ejpam-6098	358	22	δn−1	δn−1	PROPN
ejpam-6098	358	23	)	)	PUNCT
ejpam-6098	358	24	=	=	SYM
ejpam-6098	358	25	0	0	NUM
ejpam-6098	358	26	by	by	ADP
ejpam-6098	358	27	using	use	VERB
ejpam-6098	358	28	lemma	lemma	PROPN
ejpam-6098	358	29	(	(	PUNCT
ejpam-6098	358	30	1	1	NUM
ejpam-6098	358	31	)	)	PUNCT
ejpam-6098	359	1	,	,	PUNCT
ejpam-6098	359	2	we	we	PRON
ejpam-6098	359	3	find	find	VERB
ejpam-6098	359	4	the	the	DET
ejpam-6098	359	5	uniqueness	uniqueness	NOUN
ejpam-6098	359	6	of	of	ADP
ejpam-6098	359	7	mapping	map	VERB
ejpam-6098	359	8	q.	q.	NOUN
ejpam-6098	359	9	these	these	PRON
ejpam-6098	359	10	are	be	AUX
ejpam-6098	359	11	generalized	generalized	ADJ
ejpam-6098	359	12	hyers	hyer	NOUN
ejpam-6098	359	13	-	-	PUNCT
ejpam-6098	359	14	ulam	ulam	ADJ
ejpam-6098	359	15	-	-	PUNCT
ejpam-6098	359	16	rassias	rassias	PROPN
ejpam-6098	359	17	stability	stability	NOUN
ejpam-6098	359	18	of	of	ADP
ejpam-6098	359	19	a	a	DET
ejpam-6098	359	20	cubic	cubic	ADJ
ejpam-6098	359	21	functional	functional	ADJ
ejpam-6098	359	22	equation	equation	NOUN
ejpam-6098	359	23	in	in	ADP
ejpam-6098	359	24	a	a	DET
ejpam-6098	359	25	non	non	ADJ
ejpam-6098	359	26	-	-	ADJ
ejpam-6098	359	27	archimedean	archimedean	ADJ
ejpam-6098	359	28	(	(	PUNCT
ejpam-6098	359	29	n	n	CCONJ
ejpam-6098	359	30	,	,	PUNCT
ejpam-6098	359	31	β)−	β)−	PUNCT
ejpam-6098	359	32	normed	normed	ADJ
ejpam-6098	359	33	space	space	NOUN
ejpam-6098	359	34	.	.	PUNCT
ejpam-6098	360	1	the	the	DET
ejpam-6098	360	2	original	original	ADJ
ejpam-6098	360	3	inequality	inequality	NOUN
ejpam-6098	360	4	shows	show	VERB
ejpam-6098	360	5	an	an	DET
ejpam-6098	360	6	approximate	approximate	ADJ
ejpam-6098	360	7	cubic	cubic	ADJ
ejpam-6098	360	8	behavior	behavior	NOUN
ejpam-6098	360	9	.	.	PUNCT
ejpam-6098	361	1	the	the	DET
ejpam-6098	361	2	conclusion	conclusion	NOUN
ejpam-6098	361	3	guarantees	guarantee	VERB
ejpam-6098	361	4	the	the	DET
ejpam-6098	361	5	existence	existence	NOUN
ejpam-6098	361	6	and	and	CCONJ
ejpam-6098	361	7	uniqueness	uniqueness	NOUN
ejpam-6098	361	8	of	of	ADP
ejpam-6098	361	9	a	a	DET
ejpam-6098	361	10	true	true	ADJ
ejpam-6098	361	11	cubic	cubic	ADJ
ejpam-6098	361	12	mapping	mapping	NOUN
ejpam-6098	361	13	nearby	nearby	ADV
ejpam-6098	361	14	.	.	PUNCT
ejpam-6098	362	1	the	the	DET
ejpam-6098	362	2	bound	bound	NOUN
ejpam-6098	362	3	tells	tell	VERB
ejpam-6098	362	4	how	how	SCONJ
ejpam-6098	362	5	close	close	ADJ
ejpam-6098	362	6	f	f	NOUN
ejpam-6098	362	7	is	be	AUX
ejpam-6098	362	8	to	to	ADP
ejpam-6098	362	9	this	this	DET
ejpam-6098	362	10	true	true	ADJ
ejpam-6098	362	11	cubic	cubic	ADJ
ejpam-6098	362	12	mapping	mapping	NOUN
ejpam-6098	362	13	.	.	PUNCT
ejpam-6098	363	1	corollary	corollary	ADJ
ejpam-6098	363	2	2	2	NUM
ejpam-6098	363	3	.	.	PUNCT
ejpam-6098	364	1	if	if	SCONJ
ejpam-6098	364	2	f	f	PROPN
ejpam-6098	364	3	:	:	PUNCT
ejpam-6098	364	4	u	u	X
ejpam-6098	364	5	→	→	SYM
ejpam-6098	364	6	v	v	PROPN
ejpam-6098	364	7	is	be	AUX
ejpam-6098	364	8	a	a	DET
ejpam-6098	364	9	function	function	NOUN
ejpam-6098	364	10	satisfying	satisfy	VERB
ejpam-6098	364	11	∥f(2u+	∥f(2u+	PROPN
ejpam-6098	364	12	w	w	PROPN
ejpam-6098	365	1	+	+	PROPN
ejpam-6098	365	2	v)−	v)−	PROPN
ejpam-6098	365	3	3f(u+	3f(u+	PROPN
ejpam-6098	365	4	v	v	NOUN
ejpam-6098	365	5	+	+	CCONJ
ejpam-6098	365	6	w)−	w)−	PROPN
ejpam-6098	365	7	f(w	f(w	PROPN
ejpam-6098	365	8	−	−	PROPN
ejpam-6098	365	9	u+	u+	NOUN
ejpam-6098	365	10	v)−	v)−	PROPN
ejpam-6098	365	11	2f(u+	2f(u+	PROPN
ejpam-6098	365	12	v	v	NOUN
ejpam-6098	365	13	)	)	PUNCT
ejpam-6098	365	14	j.	j.	PROPN
ejpam-6098	365	15	jakhar	jakhar	PROPN
ejpam-6098	365	16	et	et	PROPN
ejpam-6098	365	17	al	al	PROPN
ejpam-6098	365	18	.	.	PUNCT
ejpam-6098	365	19	/	/	SYM
ejpam-6098	365	20	eur	eur	PROPN
ejpam-6098	365	21	.	.	PUNCT
ejpam-6098	366	1	j.	j.	PROPN
ejpam-6098	366	2	pure	pure	PROPN
ejpam-6098	366	3	appl	appl	PROPN
ejpam-6098	366	4	.	.	PROPN
ejpam-6098	366	5	math	math	PROPN
ejpam-6098	366	6	,	,	PUNCT
ejpam-6098	366	7	18	18	NUM
ejpam-6098	366	8	(	(	PUNCT
ejpam-6098	366	9	2	2	NUM
ejpam-6098	366	10	)	)	PUNCT
ejpam-6098	366	11	(	(	PUNCT
ejpam-6098	366	12	2025	2025	NUM
ejpam-6098	366	13	)	)	PUNCT
ejpam-6098	366	14	,	,	PUNCT
ejpam-6098	366	15	6098	6098	NUM
ejpam-6098	366	16	15	15	NUM
ejpam-6098	366	17	of	of	ADP
ejpam-6098	366	18	23	23	NUM
ejpam-6098	366	19	−2f(w	−2f(w	NOUN
ejpam-6098	366	20	+	+	CCONJ
ejpam-6098	366	21	u	u	NOUN
ejpam-6098	366	22	)	)	PUNCT
ejpam-6098	367	1	+	+	CCONJ
ejpam-6098	367	2	6f(u−	6f(u−	NUM
ejpam-6098	367	3	v	v	NOUN
ejpam-6098	367	4	)	)	PUNCT
ejpam-6098	368	1	+	+	CCONJ
ejpam-6098	368	2	6f(u−	6f(u−	NUM
ejpam-6098	368	3	w	w	NOUN
ejpam-6098	368	4	)	)	PUNCT
ejpam-6098	369	1	+	+	CCONJ
ejpam-6098	369	2	3f(w	3f(w	NOUN
ejpam-6098	369	3	+	+	CCONJ
ejpam-6098	369	4	v)−	v)−	PROPN
ejpam-6098	369	5	2f(2u−	2f(2u−	NUM
ejpam-6098	369	6	v	v	NOUN
ejpam-6098	369	7	)	)	PUNCT
ejpam-6098	369	8	−2f(2u−	−2f(2u−	PROPN
ejpam-6098	369	9	w	w	PROPN
ejpam-6098	369	10	)	)	PUNCT
ejpam-6098	369	11	+	+	CCONJ
ejpam-6098	369	12	6f(v	6f(v	NUM
ejpam-6098	369	13	)	)	PUNCT
ejpam-6098	370	1	+	+	CCONJ
ejpam-6098	370	2	18f(u	18f(u	NUM
ejpam-6098	370	3	)	)	PUNCT
ejpam-6098	371	1	+	+	CCONJ
ejpam-6098	371	2	6f(w	6f(w	NUM
ejpam-6098	371	3	)	)	PUNCT
ejpam-6098	371	4	,	,	PUNCT
ejpam-6098	371	5	δ1	δ1	NOUN
ejpam-6098	371	6	,	,	PUNCT
ejpam-6098	371	7	...	...	PUNCT
ejpam-6098	371	8	,	,	PUNCT
ejpam-6098	371	9	δn−1∥β	δn−1∥β	NOUN
ejpam-6098	371	10	≤	≤	NUM
ejpam-6098	371	11	(	(	PUNCT
ejpam-6098	371	12	||u||+	||u||+	ADJ
ejpam-6098	371	13	||v||+	||v||+	NOUN
ejpam-6098	371	14	||w||)ϵ	||w||)ϵ	PROPN
ejpam-6098	371	15	,	,	PUNCT
ejpam-6098	371	16	then	then	ADV
ejpam-6098	371	17	there	there	PRON
ejpam-6098	371	18	is	be	VERB
ejpam-6098	371	19	a	a	DET
ejpam-6098	371	20	unique	unique	ADJ
ejpam-6098	371	21	cubic	cubic	ADJ
ejpam-6098	371	22	function	function	NOUN
ejpam-6098	371	23	q	q	NOUN
ejpam-6098	371	24	:	:	PUNCT
ejpam-6098	371	25	u	u	X
ejpam-6098	371	26	→	→	SYM
ejpam-6098	371	27	v	v	ADP
ejpam-6098	371	28	such	such	ADJ
ejpam-6098	371	29	that	that	PRON
ejpam-6098	371	30	||f(u)−q(u	||f(u)−q(u	NOUN
ejpam-6098	371	31	)	)	PUNCT
ejpam-6098	371	32	,	,	PUNCT
ejpam-6098	371	33	δ1	δ1	NOUN
ejpam-6098	371	34	,	,	PUNCT
ejpam-6098	371	35	...	...	PUNCT
ejpam-6098	371	36	,	,	PUNCT
ejpam-6098	372	1	δn−1||β	δn−1||β	NOUN
ejpam-6098	372	2	≤	≤	NOUN
ejpam-6098	372	3	24−β||u||ϵ.	24−β||u||ϵ.	NUM
ejpam-6098	372	4	proof	proof	NOUN
ejpam-6098	372	5	.	.	PUNCT
ejpam-6098	373	1	put	put	VERB
ejpam-6098	373	2	ϕ(u	ϕ(u	PROPN
ejpam-6098	373	3	,	,	PUNCT
ejpam-6098	373	4	0	0	NUM
ejpam-6098	373	5	,	,	PUNCT
ejpam-6098	373	6	0	0	NUM
ejpam-6098	373	7	)	)	PUNCT
ejpam-6098	373	8	=	=	SYM
ejpam-6098	374	1	||u||+	||u||+	NOUN
ejpam-6098	374	2	||v||+	||v||+	NOUN
ejpam-6098	374	3	||w||	||w||	NOUN
ejpam-6098	374	4	and	and	CCONJ
ejpam-6098	374	5	ψ(δ1	ψ(δ1	VERB
ejpam-6098	374	6	,	,	PUNCT
ejpam-6098	374	7	...	...	PUNCT
ejpam-6098	374	8	,	,	PUNCT
ejpam-6098	374	9	δn−1	δn−1	PROPN
ejpam-6098	374	10	)	)	PUNCT
ejpam-6098	374	11	=	=	SYM
ejpam-6098	375	1	ϵ	ϵ	X
ejpam-6098	375	2	in	in	ADP
ejpam-6098	375	3	above	above	ADP
ejpam-6098	375	4	theorem	theorem	VERB
ejpam-6098	375	5	,	,	PUNCT
ejpam-6098	375	6	we	we	PRON
ejpam-6098	375	7	obtained	obtain	VERB
ejpam-6098	375	8	desired	desire	VERB
ejpam-6098	375	9	outcome	outcome	NOUN
ejpam-6098	375	10	.	.	PUNCT
ejpam-6098	376	1	4	4	X
ejpam-6098	376	2	.	.	X
ejpam-6098	376	3	stability	stability	NOUN
ejpam-6098	376	4	in	in	ADP
ejpam-6098	376	5	random	random	ADJ
ejpam-6098	376	6	normed	normed	ADJ
ejpam-6098	376	7	space	space	NOUN
ejpam-6098	376	8	an	an	DET
ejpam-6098	376	9	rns	rn	NOUN
ejpam-6098	376	10	is	be	AUX
ejpam-6098	376	11	a	a	DET
ejpam-6098	376	12	generalization	generalization	NOUN
ejpam-6098	376	13	of	of	ADP
ejpam-6098	376	14	the	the	DET
ejpam-6098	376	15	concept	concept	NOUN
ejpam-6098	376	16	of	of	ADP
ejpam-6098	376	17	an	an	DET
ejpam-6098	376	18	ns	ns	ADJ
ejpam-6098	376	19	,	,	PUNCT
ejpam-6098	376	20	where	where	SCONJ
ejpam-6098	376	21	the	the	DET
ejpam-6098	376	22	norm	norm	NOUN
ejpam-6098	376	23	can	can	AUX
ejpam-6098	376	24	vary	vary	VERB
ejpam-6098	376	25	randomly	randomly	ADV
ejpam-6098	376	26	.	.	PUNCT
ejpam-6098	377	1	the	the	DET
ejpam-6098	377	2	study	study	NOUN
ejpam-6098	377	3	of	of	ADP
ejpam-6098	377	4	rns	rn	NOUN
ejpam-6098	377	5	in	in	ADP
ejpam-6098	377	6	functional	functional	ADJ
ejpam-6098	377	7	analysis	analysis	NOUN
ejpam-6098	377	8	allows	allow	VERB
ejpam-6098	377	9	for	for	ADP
ejpam-6098	377	10	investigating	investigate	VERB
ejpam-6098	377	11	random	random	ADJ
ejpam-6098	377	12	variations	variation	NOUN
ejpam-6098	377	13	in	in	ADP
ejpam-6098	377	14	normed	normed	ADJ
ejpam-6098	377	15	structures	structure	NOUN
ejpam-6098	377	16	.	.	PUNCT
ejpam-6098	378	1	in	in	ADP
ejpam-6098	378	2	the	the	DET
ejpam-6098	378	3	study	study	NOUN
ejpam-6098	378	4	of	of	ADP
ejpam-6098	378	5	the	the	DET
ejpam-6098	378	6	stability	stability	NOUN
ejpam-6098	378	7	of	of	ADP
ejpam-6098	378	8	fes	fes	NOUN
ejpam-6098	378	9	in	in	ADP
ejpam-6098	378	10	rns	rn	NOUN
ejpam-6098	378	11	,	,	PUNCT
ejpam-6098	378	12	one	one	NUM
ejpam-6098	378	13	examines	examine	VERB
ejpam-6098	378	14	how	how	SCONJ
ejpam-6098	378	15	small	small	ADJ
ejpam-6098	378	16	changes	change	NOUN
ejpam-6098	378	17	in	in	ADP
ejpam-6098	378	18	the	the	DET
ejpam-6098	378	19	fe	fe	NOUN
ejpam-6098	378	20	or	or	CCONJ
ejpam-6098	378	21	its	its	PRON
ejpam-6098	378	22	solutions	solution	NOUN
ejpam-6098	378	23	behave	behave	VERB
ejpam-6098	378	24	under	under	ADP
ejpam-6098	378	25	random	random	ADJ
ejpam-6098	378	26	variations	variation	NOUN
ejpam-6098	378	27	of	of	ADP
ejpam-6098	378	28	the	the	DET
ejpam-6098	378	29	norm	norm	NOUN
ejpam-6098	378	30	.	.	PUNCT
ejpam-6098	379	1	using	use	VERB
ejpam-6098	379	2	concepts	concept	NOUN
ejpam-6098	379	3	from	from	ADP
ejpam-6098	379	4	[	[	X
ejpam-6098	379	5	30	30	NUM
ejpam-6098	379	6	]	]	PUNCT
ejpam-6098	379	7	,	,	PUNCT
ejpam-6098	379	8	we	we	PRON
ejpam-6098	379	9	solve	solve	VERB
ejpam-6098	379	10	the	the	DET
ejpam-6098	379	11	stability	stability	NOUN
ejpam-6098	379	12	problem	problem	NOUN
ejpam-6098	379	13	for	for	ADP
ejpam-6098	379	14	fe	fe	X
ejpam-6098	379	15	(	(	PUNCT
ejpam-6098	379	16	1	1	NUM
ejpam-6098	379	17	)	)	PUNCT
ejpam-6098	379	18	in	in	ADP
ejpam-6098	379	19	rns	rns	PROPN
ejpam-6098	379	20	.	.	PUNCT
ejpam-6098	380	1	theorem	theorem	NOUN
ejpam-6098	380	2	4	4	NUM
ejpam-6098	380	3	.	.	PUNCT
ejpam-6098	380	4	suppose	suppose	VERB
ejpam-6098	380	5	ϕ	ϕ	X
ejpam-6098	380	6	:	:	PUNCT
ejpam-6098	380	7	u	u	PRON
ejpam-6098	380	8	×	×	NOUN
ejpam-6098	380	9	u	u	NOUN
ejpam-6098	380	10	×	×	PROPN
ejpam-6098	380	11	u	u	PROPN
ejpam-6098	380	12	→	→	SYM
ejpam-6098	380	13	z	z	PROPN
ejpam-6098	380	14	is	be	AUX
ejpam-6098	380	15	a	a	DET
ejpam-6098	380	16	function	function	NOUN
ejpam-6098	380	17	holds	hold	VERB
ejpam-6098	380	18	µ′ϕ(u,0,0)(s	µ′ϕ(u,0,0)(s	NUM
ejpam-6098	380	19	)	)	PUNCT
ejpam-6098	380	20	≥	≥	NOUN
ejpam-6098	380	21	µ′kϕ(u,0,0)(s	µ′kϕ(u,0,0)(s	NOUN
ejpam-6098	380	22	)	)	PUNCT
ejpam-6098	380	23	(	(	PUNCT
ejpam-6098	380	24	24	24	NUM
ejpam-6098	380	25	)	)	PUNCT
ejpam-6098	380	26	and	and	CCONJ
ejpam-6098	380	27	limm→∞µ	limm→∞µ	PROPN
ejpam-6098	380	28	′	′	NOUN
ejpam-6098	380	29	ϕ(2nu,2nv,2nw)(8	ϕ(2nu,2nv,2nw)(8	NOUN
ejpam-6098	380	30	ns	ns	NUM
ejpam-6098	380	31	)	)	PUNCT
ejpam-6098	380	32	=	=	SYM
ejpam-6098	380	33	1	1	NUM
ejpam-6098	380	34	∀	∀	NOUN
ejpam-6098	380	35	u	u	NOUN
ejpam-6098	380	36	,	,	PUNCT
ejpam-6098	380	37	w	w	PROPN
ejpam-6098	380	38	,	,	PUNCT
ejpam-6098	380	39	v	v	NOUN
ejpam-6098	380	40	∈	∈	PROPN
ejpam-6098	380	41	u	u	NOUN
ejpam-6098	380	42	,	,	PUNCT
ejpam-6098	380	43	s	s	PROPN
ejpam-6098	380	44	>	>	X
ejpam-6098	380	45	0	0	PUNCT
ejpam-6098	381	1	and	and	CCONJ
ejpam-6098	381	2	0	0	NUM
ejpam-6098	381	3	<	<	X
ejpam-6098	381	4	k	k	X
ejpam-6098	381	5	<	<	X
ejpam-6098	381	6	8	8	NUM
ejpam-6098	381	7	.	.	PUNCT
ejpam-6098	382	1	if	if	SCONJ
ejpam-6098	382	2	f	f	PROPN
ejpam-6098	382	3	:	:	PUNCT
ejpam-6098	382	4	u	u	X
ejpam-6098	382	5	→	→	X
ejpam-6098	382	6	x	x	X
ejpam-6098	382	7	is	be	AUX
ejpam-6098	382	8	an	an	DET
ejpam-6098	382	9	odd	odd	ADJ
ejpam-6098	382	10	function	function	NOUN
ejpam-6098	382	11	having	have	VERB
ejpam-6098	382	12	µθ(s	µθ(s	PUNCT
ejpam-6098	382	13	)	)	PUNCT
ejpam-6098	382	14	≥	≥	PROPN
ejpam-6098	382	15	µ′ϕ(u	µ′ϕ(u	NOUN
ejpam-6098	382	16	,	,	PUNCT
ejpam-6098	382	17	v	v	NOUN
ejpam-6098	382	18	,	,	PUNCT
ejpam-6098	382	19	w)(s	w)(s	NOUN
ejpam-6098	382	20	)	)	PUNCT
ejpam-6098	382	21	,	,	PUNCT
ejpam-6098	382	22	(	(	PUNCT
ejpam-6098	382	23	25	25	NUM
ejpam-6098	382	24	)	)	PUNCT
ejpam-6098	382	25	where	where	SCONJ
ejpam-6098	382	26	θ	θ	PROPN
ejpam-6098	382	27	=	=	PUNCT
ejpam-6098	382	28	f(2u+	f(2u+	VERB
ejpam-6098	382	29	w	w	ADP
ejpam-6098	382	30	+	+	PUNCT
ejpam-6098	382	31	v)−	v)−	PROPN
ejpam-6098	382	32	3f(v	3f(v	NUM
ejpam-6098	383	1	+	+	CCONJ
ejpam-6098	383	2	w	w	PROPN
ejpam-6098	383	3	+	+	CCONJ
ejpam-6098	383	4	u)−	u)−	PROPN
ejpam-6098	383	5	f(v	f(v	PROPN
ejpam-6098	383	6	+	+	CCONJ
ejpam-6098	383	7	w	w	PROPN
ejpam-6098	383	8	−	−	PROPN
ejpam-6098	383	9	u)−	u)−	PROPN
ejpam-6098	383	10	2f(u+	2f(u+	PROPN
ejpam-6098	384	1	w)−	w)−	PROPN
ejpam-6098	384	2	2f(v	2f(v	NUM
ejpam-6098	385	1	+	+	CCONJ
ejpam-6098	385	2	u	u	NOUN
ejpam-6098	385	3	)	)	PUNCT
ejpam-6098	386	1	+	+	CCONJ
ejpam-6098	386	2	6f(u−	6f(u−	NUM
ejpam-6098	386	3	w	w	NOUN
ejpam-6098	386	4	)	)	PUNCT
ejpam-6098	386	5	+	+	CCONJ
ejpam-6098	386	6	6f(u−	6f(u−	NUM
ejpam-6098	386	7	v	v	NOUN
ejpam-6098	386	8	)	)	PUNCT
ejpam-6098	387	1	+	+	CCONJ
ejpam-6098	387	2	3f(w	3f(w	NOUN
ejpam-6098	387	3	+	+	CCONJ
ejpam-6098	387	4	v)−	v)−	PROPN
ejpam-6098	387	5	2f(2u−	2f(2u−	NUM
ejpam-6098	387	6	v)−	v)−	PROPN
ejpam-6098	387	7	2f(2u−	2f(2u−	NUM
ejpam-6098	387	8	w	w	NOUN
ejpam-6098	387	9	)	)	PUNCT
ejpam-6098	387	10	+	+	CCONJ
ejpam-6098	387	11	6f(v	6f(v	NUM
ejpam-6098	387	12	)	)	PUNCT
ejpam-6098	388	1	+	+	CCONJ
ejpam-6098	388	2	18f(u	18f(u	NUM
ejpam-6098	388	3	)	)	PUNCT
ejpam-6098	389	1	+	+	CCONJ
ejpam-6098	389	2	6f(w	6f(w	NUM
ejpam-6098	389	3	)	)	PUNCT
ejpam-6098	390	1	then	then	ADV
ejpam-6098	390	2	there	there	PRON
ejpam-6098	390	3	is	be	VERB
ejpam-6098	390	4	exactly	exactly	ADV
ejpam-6098	390	5	one	one	NUM
ejpam-6098	390	6	cubic	cubic	ADJ
ejpam-6098	390	7	mapping	mapping	NOUN
ejpam-6098	390	8	q	q	NOUN
ejpam-6098	390	9	:	:	PUNCT
ejpam-6098	390	10	u	u	SYM
ejpam-6098	390	11	→	→	SYM
ejpam-6098	390	12	v	v	NUM
ejpam-6098	390	13	holds	hold	VERB
ejpam-6098	390	14	µf(u)−q(u)(s	µf(u)−q(u)(s	PROPN
ejpam-6098	390	15	)	)	PUNCT
ejpam-6098	390	16	≥	≥	NOUN
ejpam-6098	390	17	µ′ϕ(u,0,0)(3(8−	µ′ϕ(u,0,0)(3(8−	NOUN
ejpam-6098	390	18	k)s	k)s	NOUN
ejpam-6098	390	19	)	)	PUNCT
ejpam-6098	390	20	.	.	PUNCT
ejpam-6098	391	1	(	(	PUNCT
ejpam-6098	391	2	26	26	NUM
ejpam-6098	391	3	)	)	PUNCT
ejpam-6098	391	4	proof	proof	NOUN
ejpam-6098	391	5	.	.	PUNCT
ejpam-6098	392	1	putting	put	VERB
ejpam-6098	392	2	(	(	PUNCT
ejpam-6098	392	3	u	u	NOUN
ejpam-6098	392	4	,	,	PUNCT
ejpam-6098	392	5	v	v	NOUN
ejpam-6098	392	6	,	,	PUNCT
ejpam-6098	392	7	w	w	NOUN
ejpam-6098	392	8	)	)	PUNCT
ejpam-6098	392	9	by	by	ADP
ejpam-6098	392	10	(	(	PUNCT
ejpam-6098	392	11	u	u	NOUN
ejpam-6098	392	12	,	,	PUNCT
ejpam-6098	392	13	0	0	NUM
ejpam-6098	392	14	,	,	PUNCT
ejpam-6098	392	15	0	0	NUM
ejpam-6098	392	16	)	)	PUNCT
ejpam-6098	392	17	in	in	ADP
ejpam-6098	392	18	(	(	PUNCT
ejpam-6098	392	19	25	25	NUM
ejpam-6098	392	20	)	)	PUNCT
ejpam-6098	392	21	,	,	PUNCT
ejpam-6098	392	22	we	we	PRON
ejpam-6098	392	23	have	have	VERB
ejpam-6098	392	24	µ	µ	X
ejpam-6098	392	25	f(2u	f(2u	NOUN
ejpam-6098	392	26	)	)	PUNCT
ejpam-6098	392	27	23	23	NUM
ejpam-6098	392	28	−f(u	−f(u	NOUN
ejpam-6098	392	29	)	)	PUNCT
ejpam-6098	392	30	(	(	PUNCT
ejpam-6098	392	31	s	s	X
ejpam-6098	392	32	)	)	PUNCT
ejpam-6098	392	33	≥	≥	X
ejpam-6098	392	34	µ′ϕ(u,0,0)(24s	µ′ϕ(u,0,0)(24s	NOUN
ejpam-6098	392	35	)	)	PUNCT
ejpam-6098	392	36	.	.	PUNCT
ejpam-6098	393	1	(	(	PUNCT
ejpam-6098	393	2	27	27	NUM
ejpam-6098	393	3	)	)	PUNCT
ejpam-6098	393	4	replacing	replace	VERB
ejpam-6098	393	5	u	u	NOUN
ejpam-6098	393	6	by	by	ADP
ejpam-6098	393	7	2nu	2nu	NOUN
ejpam-6098	393	8	in	in	ADP
ejpam-6098	393	9	(	(	PUNCT
ejpam-6098	393	10	27	27	NUM
ejpam-6098	393	11	)	)	PUNCT
ejpam-6098	393	12	,	,	PUNCT
ejpam-6098	393	13	we	we	PRON
ejpam-6098	393	14	get	get	VERB
ejpam-6098	393	15	µ	µ	PRON
ejpam-6098	393	16	f(2n+1u	f(2n+1u	ADV
ejpam-6098	393	17	)	)	PUNCT
ejpam-6098	393	18	23(n+1	23(n+1	NUM
ejpam-6098	393	19	)	)	PUNCT
ejpam-6098	393	20	−	−	PROPN
ejpam-6098	393	21	f(2nu	f(2nu	NOUN
ejpam-6098	393	22	)	)	PUNCT
ejpam-6098	393	23	23n	23n	NOUN
ejpam-6098	393	24	(	(	PUNCT
ejpam-6098	393	25	s	s	X
ejpam-6098	393	26	)	)	PUNCT
ejpam-6098	393	27	≥	≥	PROPN
ejpam-6098	393	28	µ′ϕ(2nu,0,0)(3.2	µ′ϕ(2nu,0,0)(3.2	PROPN
ejpam-6098	393	29	3(n+1)s	3(n+1)s	NUM
ejpam-6098	393	30	)	)	PUNCT
ejpam-6098	393	31	.	.	PUNCT
ejpam-6098	394	1	(	(	PUNCT
ejpam-6098	394	2	28	28	NUM
ejpam-6098	394	3	)	)	PUNCT
ejpam-6098	394	4	j.	j.	PROPN
ejpam-6098	394	5	jakhar	jakhar	PROPN
ejpam-6098	394	6	et	et	PROPN
ejpam-6098	394	7	al	al	PROPN
ejpam-6098	394	8	.	.	PUNCT
ejpam-6098	394	9	/	/	SYM
ejpam-6098	394	10	eur	eur	PROPN
ejpam-6098	394	11	.	.	PUNCT
ejpam-6098	395	1	j.	j.	PROPN
ejpam-6098	395	2	pure	pure	PROPN
ejpam-6098	395	3	appl	appl	PROPN
ejpam-6098	395	4	.	.	PROPN
ejpam-6098	395	5	math	math	PROPN
ejpam-6098	395	6	,	,	PUNCT
ejpam-6098	395	7	18	18	NUM
ejpam-6098	395	8	(	(	PUNCT
ejpam-6098	395	9	2	2	NUM
ejpam-6098	395	10	)	)	PUNCT
ejpam-6098	395	11	(	(	PUNCT
ejpam-6098	395	12	2025	2025	NUM
ejpam-6098	395	13	)	)	PUNCT
ejpam-6098	395	14	,	,	PUNCT
ejpam-6098	395	15	6098	6098	NUM
ejpam-6098	395	16	16	16	NUM
ejpam-6098	395	17	of	of	ADP
ejpam-6098	395	18	23	23	NUM
ejpam-6098	395	19	since	since	SCONJ
ejpam-6098	395	20	f(2nu	f(2nu	NUM
ejpam-6098	395	21	)	)	PUNCT
ejpam-6098	395	22	23n	23n	PROPN
ejpam-6098	395	23	−	−	PROPN
ejpam-6098	395	24	f(u	f(u	PROPN
ejpam-6098	395	25	)	)	PUNCT
ejpam-6098	396	1	=	=	PRON
ejpam-6098	396	2	n−1∑	n−1∑	PROPN
ejpam-6098	396	3	i=0	i=0	PROPN
ejpam-6098	396	4	{	{	PUNCT
ejpam-6098	396	5	f(2i+1u	f(2i+1u	NOUN
ejpam-6098	396	6	)	)	PUNCT
ejpam-6098	396	7	23(i+1	23(i+1	NUM
ejpam-6098	396	8	)	)	PUNCT
ejpam-6098	396	9	−	−	ADP
ejpam-6098	396	10	f(2iu	f(2iu	NOUN
ejpam-6098	396	11	)	)	PUNCT
ejpam-6098	396	12	23i	23i	NOUN
ejpam-6098	396	13	}	}	PUNCT
ejpam-6098	396	14	.	.	PUNCT
ejpam-6098	397	1	(	(	PUNCT
ejpam-6098	397	2	29	29	NUM
ejpam-6098	397	3	)	)	PUNCT
ejpam-6098	397	4	by	by	ADP
ejpam-6098	397	5	using	use	VERB
ejpam-6098	397	6	(	(	PUNCT
ejpam-6098	397	7	28	28	NUM
ejpam-6098	397	8	)	)	PUNCT
ejpam-6098	397	9	and	and	CCONJ
ejpam-6098	397	10	(	(	PUNCT
ejpam-6098	397	11	29	29	NUM
ejpam-6098	397	12	)	)	PUNCT
ejpam-6098	397	13	,	,	PUNCT
ejpam-6098	397	14	we	we	PRON
ejpam-6098	397	15	get	get	VERB
ejpam-6098	397	16	µ	µ	PRON
ejpam-6098	397	17	f(2nu	f(2nu	NOUN
ejpam-6098	397	18	)	)	PUNCT
ejpam-6098	397	19	23n	23n	NOUN
ejpam-6098	397	20	−f(u	−f(u	NUM
ejpam-6098	397	21	)	)	PUNCT
ejpam-6098	397	22	(	(	PUNCT
ejpam-6098	397	23	n−1∑	n−1∑	NUM
ejpam-6098	397	24	i=0	i=0	PROPN
ejpam-6098	397	25	ski	ski	NOUN
ejpam-6098	397	26	3×	3×	NUM
ejpam-6098	397	27	23(n+1	23(n+1	NUM
ejpam-6098	397	28	)	)	PUNCT
ejpam-6098	397	29	)	)	PUNCT
ejpam-6098	397	30	≥	≥	NOUN
ejpam-6098	398	1	tn−1	tn−1	PROPN
ejpam-6098	398	2	i=0	i=0	PROPN
ejpam-6098	398	3	(	(	PUNCT
ejpam-6098	398	4	µ′ϕ(u,0,0)(s	µ′ϕ(u,0,0)(s	NUM
ejpam-6098	398	5	)	)	PUNCT
ejpam-6098	398	6	)	)	PUNCT
ejpam-6098	398	7	=	=	SYM
ejpam-6098	399	1	µ′ϕ(u,0,0)(s	µ′ϕ(u,0,0)(s	NUM
ejpam-6098	399	2	)	)	PUNCT
ejpam-6098	399	3	.	.	PUNCT
ejpam-6098	400	1	(	(	PUNCT
ejpam-6098	400	2	30	30	NUM
ejpam-6098	400	3	)	)	PUNCT
ejpam-6098	400	4	therefore	therefore	ADV
ejpam-6098	400	5	,	,	PUNCT
ejpam-6098	400	6	µ	µ	PROPN
ejpam-6098	400	7	f(2nu	f(2nu	NOUN
ejpam-6098	400	8	)	)	PUNCT
ejpam-6098	400	9	23n	23n	NOUN
ejpam-6098	400	10	−f(u	−f(u	NUM
ejpam-6098	400	11	)	)	PUNCT
ejpam-6098	400	12	(	(	PUNCT
ejpam-6098	400	13	s	s	X
ejpam-6098	400	14	)	)	PUNCT
ejpam-6098	400	15	≥	≥	NOUN
ejpam-6098	400	16	µ′ϕ(u,0,0	µ′ϕ(u,0,0	NOUN
ejpam-6098	400	17	)	)	PUNCT
ejpam-6098	400	18	(	(	PUNCT
ejpam-6098	400	19	s∑n−1	s∑n−1	X
ejpam-6098	400	20	i=0	i=0	PROPN
ejpam-6098	400	21	ki	ki	PROPN
ejpam-6098	400	22	3×23(i+1	3×23(i+1	NUM
ejpam-6098	400	23	)	)	PUNCT
ejpam-6098	400	24	)	)	PUNCT
ejpam-6098	400	25	.	.	PUNCT
ejpam-6098	401	1	(	(	PUNCT
ejpam-6098	401	2	31	31	NUM
ejpam-6098	401	3	)	)	PUNCT
ejpam-6098	401	4	switching	switch	VERB
ejpam-6098	401	5	u	u	NOUN
ejpam-6098	401	6	by	by	ADP
ejpam-6098	401	7	2mu	2mu	NOUN
ejpam-6098	401	8	in	in	ADP
ejpam-6098	401	9	(	(	PUNCT
ejpam-6098	401	10	31	31	NUM
ejpam-6098	401	11	)	)	PUNCT
ejpam-6098	401	12	,	,	PUNCT
ejpam-6098	401	13	we	we	PRON
ejpam-6098	401	14	have	have	VERB
ejpam-6098	401	15	µ	µ	X
ejpam-6098	401	16	(	(	PUNCT
ejpam-6098	401	17	f(2m+nu	f(2m+nu	PROPN
ejpam-6098	401	18	)	)	PUNCT
ejpam-6098	401	19	2(m+n	2(m+n	NUM
ejpam-6098	401	20	)	)	PUNCT
ejpam-6098	401	21	−	−	NOUN
ejpam-6098	401	22	f(2mu	f(2mu	NOUN
ejpam-6098	401	23	)	)	PUNCT
ejpam-6098	401	24	2	2	NUM
ejpam-6098	401	25	m	m	NOUN
ejpam-6098	401	26	)	)	PUNCT
ejpam-6098	402	1	(	(	PUNCT
ejpam-6098	402	2	s	s	X
ejpam-6098	402	3	)	)	PUNCT
ejpam-6098	402	4	≥	≥	NOUN
ejpam-6098	402	5	µ′ϕ(u,0,0	µ′ϕ(u,0,0	NOUN
ejpam-6098	402	6	)	)	PUNCT
ejpam-6098	402	7	(	(	PUNCT
ejpam-6098	402	8	s∑n+m	s∑n+m	PUNCT
ejpam-6098	402	9	i	i	PRON
ejpam-6098	402	10	=	=	NOUN
ejpam-6098	402	11	m	m	PROPN
ejpam-6098	402	12	ki	ki	PROPN
ejpam-6098	402	13	3×23(i+1	3×23(i+1	PROPN
ejpam-6098	402	14	)	)	PUNCT
ejpam-6098	402	15	)	)	PUNCT
ejpam-6098	402	16	.	.	PUNCT
ejpam-6098	403	1	(	(	PUNCT
ejpam-6098	403	2	32	32	NUM
ejpam-6098	403	3	)	)	PUNCT
ejpam-6098	403	4	as	as	ADP
ejpam-6098	403	5	,	,	PUNCT
ejpam-6098	403	6	limn	limn	ADJ
ejpam-6098	403	7	,	,	PUNCT
ejpam-6098	403	8	m→∞µ	m→∞µ	ADJ
ejpam-6098	403	9	′	′	NUM
ejpam-6098	403	10	ϕ(u,0,0	ϕ(u,0,0	NOUN
ejpam-6098	403	11	)	)	PUNCT
ejpam-6098	403	12	(	(	PUNCT
ejpam-6098	403	13	s∑n+m	s∑n+m	X
ejpam-6098	403	14	i	i	PRON
ejpam-6098	403	15	=	=	NOUN
ejpam-6098	403	16	m	m	PROPN
ejpam-6098	403	17	ki	ki	PROPN
ejpam-6098	403	18	3×23(i+1	3×23(i+1	PROPN
ejpam-6098	403	19	)	)	PUNCT
ejpam-6098	403	20	)	)	PUNCT
ejpam-6098	404	1	=	=	SYM
ejpam-6098	404	2	1	1	NUM
ejpam-6098	404	3	,	,	PUNCT
ejpam-6098	404	4	therefore	therefore	ADV
ejpam-6098	404	5	,	,	PUNCT
ejpam-6098	404	6	the	the	DET
ejpam-6098	404	7	sequence	sequence	NOUN
ejpam-6098	404	8	{	{	PUNCT
ejpam-6098	404	9	f(2nu	f(2nu	NOUN
ejpam-6098	404	10	)	)	PUNCT
ejpam-6098	404	11	23n	23n	NOUN
ejpam-6098	404	12	}	}	PUNCT
ejpam-6098	404	13	is	be	AUX
ejpam-6098	404	14	a	a	DET
ejpam-6098	404	15	cauchy	cauchy	NOUN
ejpam-6098	404	16	in	in	ADP
ejpam-6098	404	17	x(complete	x(complete	NOUN
ejpam-6098	404	18	-	-	PUNCT
ejpam-6098	404	19	rns	rn	NOUN
ejpam-6098	404	20	)	)	PUNCT
ejpam-6098	404	21	,	,	PUNCT
ejpam-6098	404	22	therefore	therefore	ADV
ejpam-6098	404	23	there	there	PRON
ejpam-6098	404	24	exists	exist	VERB
ejpam-6098	404	25	the	the	DET
ejpam-6098	404	26	convergence	convergence	NOUN
ejpam-6098	404	27	point	point	NOUN
ejpam-6098	404	28	q(u	q(u	NOUN
ejpam-6098	404	29	)	)	PUNCT
ejpam-6098	404	30	∈	∈	PROPN
ejpam-6098	404	31	x	x	PUNCT
ejpam-6098	404	32	such	such	ADJ
ejpam-6098	404	33	that	that	SCONJ
ejpam-6098	404	34	limm→∞	limm→∞	PROPN
ejpam-6098	404	35	f(2nu	f(2nu	NOUN
ejpam-6098	404	36	)	)	PUNCT
ejpam-6098	404	37	23n	23n	NUM
ejpam-6098	404	38	=	=	SYM
ejpam-6098	404	39	q(u	q(u	NOUN
ejpam-6098	404	40	)	)	PUNCT
ejpam-6098	404	41	.	.	PUNCT
ejpam-6098	405	1	fix	fix	VERB
ejpam-6098	405	2	u	u	NOUN
ejpam-6098	405	3	∈	∈	PROPN
ejpam-6098	405	4	u	u	NOUN
ejpam-6098	405	5	and	and	CCONJ
ejpam-6098	405	6	taking	take	VERB
ejpam-6098	405	7	m	m	PROPN
ejpam-6098	405	8	=	=	X
ejpam-6098	405	9	0	0	NUM
ejpam-6098	405	10	in	in	ADP
ejpam-6098	405	11	(	(	PUNCT
ejpam-6098	405	12	32	32	NUM
ejpam-6098	405	13	)	)	PUNCT
ejpam-6098	405	14	,	,	PUNCT
ejpam-6098	405	15	we	we	PRON
ejpam-6098	405	16	have	have	VERB
ejpam-6098	405	17	µ	µ	PRON
ejpam-6098	405	18	f(2nu	f(2nu	NOUN
ejpam-6098	405	19	)	)	PUNCT
ejpam-6098	405	20	23n	23n	NOUN
ejpam-6098	405	21	−f(u	−f(u	NUM
ejpam-6098	405	22	)	)	PUNCT
ejpam-6098	405	23	(	(	PUNCT
ejpam-6098	405	24	s	s	X
ejpam-6098	405	25	)	)	PUNCT
ejpam-6098	405	26	≥	≥	NOUN
ejpam-6098	405	27	µ′ϕ(u,0,0	µ′ϕ(u,0,0	NOUN
ejpam-6098	405	28	)	)	PUNCT
ejpam-6098	405	29	(	(	PUNCT
ejpam-6098	405	30	s∑n−1	s∑n−1	X
ejpam-6098	405	31	i=0	i=0	PROPN
ejpam-6098	405	32	ki	ki	PROPN
ejpam-6098	405	33	23(i+1	23(i+1	NUM
ejpam-6098	405	34	)	)	PUNCT
ejpam-6098	405	35	)	)	PUNCT
ejpam-6098	406	1	(	(	PUNCT
ejpam-6098	406	2	33	33	NUM
ejpam-6098	406	3	)	)	PUNCT
ejpam-6098	406	4	and	and	CCONJ
ejpam-6098	406	5	for	for	ADP
ejpam-6098	406	6	every	every	DET
ejpam-6098	406	7	ζ	ζ	NOUN
ejpam-6098	406	8	>	>	X
ejpam-6098	406	9	0	0	NUM
ejpam-6098	406	10	,	,	PUNCT
ejpam-6098	406	11	we	we	PRON
ejpam-6098	406	12	have	have	VERB
ejpam-6098	406	13	µq(u)−f(u)(s+	µq(u)−f(u)(s+	VERB
ejpam-6098	406	14	ζ	ζ	NOUN
ejpam-6098	406	15	)	)	PUNCT
ejpam-6098	406	16	≥	≥	NOUN
ejpam-6098	406	17	t	t	PROPN
ejpam-6098	406	18	(	(	PUNCT
ejpam-6098	406	19	µ	µ	PRON
ejpam-6098	406	20	q(u)−	q(u)−	NOUN
ejpam-6098	406	21	f(2nu	f(2nu	NUM
ejpam-6098	406	22	)	)	PUNCT
ejpam-6098	406	23	23n	23n	NOUN
ejpam-6098	406	24	(	(	PUNCT
ejpam-6098	406	25	ζ	ζ	NOUN
ejpam-6098	406	26	)	)	PUNCT
ejpam-6098	406	27	,	,	PUNCT
ejpam-6098	406	28	µ	µ	PROPN
ejpam-6098	406	29	f(2nu	f(2nu	NOUN
ejpam-6098	406	30	)	)	PUNCT
ejpam-6098	406	31	23n	23n	NOUN
ejpam-6098	406	32	−f(u	−f(u	NUM
ejpam-6098	406	33	)	)	PUNCT
ejpam-6098	406	34	(	(	PUNCT
ejpam-6098	406	35	s	s	NOUN
ejpam-6098	406	36	)	)	PUNCT
ejpam-6098	406	37	)	)	PUNCT
ejpam-6098	407	1	≥	≥	PROPN
ejpam-6098	407	2	t	t	PROPN
ejpam-6098	407	3	(	(	PUNCT
ejpam-6098	407	4	µ	µ	PRON
ejpam-6098	407	5	q(u)−	q(u)−	NOUN
ejpam-6098	407	6	f(2nu	f(2nu	NUM
ejpam-6098	407	7	)	)	PUNCT
ejpam-6098	407	8	23n	23n	NOUN
ejpam-6098	407	9	(	(	PUNCT
ejpam-6098	407	10	ζ	ζ	NOUN
ejpam-6098	407	11	)	)	PUNCT
ejpam-6098	407	12	,	,	PUNCT
ejpam-6098	407	13	µ′ϕ(u,0,0	µ′ϕ(u,0,0	NOUN
ejpam-6098	407	14	)	)	PUNCT
ejpam-6098	407	15	(	(	PUNCT
ejpam-6098	407	16	s∑n−1	s∑n−1	X
ejpam-6098	407	17	i=0	i=0	PROPN
ejpam-6098	407	18	ki	ki	PROPN
ejpam-6098	407	19	3×23(n+1	3×23(n+1	NUM
ejpam-6098	407	20	)	)	PUNCT
ejpam-6098	407	21	)	)	PUNCT
ejpam-6098	407	22	)	)	PUNCT
ejpam-6098	407	23	,	,	PUNCT
ejpam-6098	407	24	(	(	PUNCT
ejpam-6098	407	25	34	34	NUM
ejpam-6098	407	26	)	)	PUNCT
ejpam-6098	407	27	considering	consider	VERB
ejpam-6098	407	28	limit	limit	NOUN
ejpam-6098	407	29	as	as	ADP
ejpam-6098	407	30	n→	n→	ADV
ejpam-6098	407	31	∞	∞	PROPN
ejpam-6098	407	32	in	in	ADP
ejpam-6098	407	33	(	(	PUNCT
ejpam-6098	407	34	34	34	NUM
ejpam-6098	407	35	)	)	PUNCT
ejpam-6098	407	36	,	,	PUNCT
ejpam-6098	407	37	µq(u)−f(u)(s+	µq(u)−f(u)(s+	PROPN
ejpam-6098	407	38	ζ	ζ	X
ejpam-6098	407	39	)	)	PUNCT
ejpam-6098	407	40	≥	≥	NOUN
ejpam-6098	407	41	µ′ϕ(u,0,0)(3(8−	µ′ϕ(u,0,0)(3(8−	NOUN
ejpam-6098	407	42	k)s	k)s	NOUN
ejpam-6098	407	43	)	)	PUNCT
ejpam-6098	407	44	.	.	PUNCT
ejpam-6098	408	1	(	(	PUNCT
ejpam-6098	408	2	35	35	NUM
ejpam-6098	408	3	)	)	PUNCT
ejpam-6098	408	4	taking	take	VERB
ejpam-6098	408	5	arbitrary	arbitrary	ADJ
ejpam-6098	408	6	ζ	ζ	NOUN
ejpam-6098	408	7	→	→	SYM
ejpam-6098	408	8	0	0	NUM
ejpam-6098	408	9	in	in	ADP
ejpam-6098	408	10	(	(	PUNCT
ejpam-6098	408	11	35	35	NUM
ejpam-6098	408	12	)	)	PUNCT
ejpam-6098	408	13	,	,	PUNCT
ejpam-6098	408	14	we	we	PRON
ejpam-6098	408	15	have	have	VERB
ejpam-6098	408	16	µq(u)−f(u)(s	µq(u)−f(u)(s	NOUN
ejpam-6098	408	17	)	)	PUNCT
ejpam-6098	408	18	≥	≥	NOUN
ejpam-6098	408	19	µ′ϕ(u,0,0)(3(8−	µ′ϕ(u,0,0)(3(8−	NOUN
ejpam-6098	408	20	k)s	k)s	NOUN
ejpam-6098	408	21	)	)	PUNCT
ejpam-6098	408	22	.	.	PUNCT
ejpam-6098	409	1	(	(	PUNCT
ejpam-6098	409	2	36	36	NUM
ejpam-6098	409	3	)	)	PUNCT
ejpam-6098	409	4	j.	j.	PROPN
ejpam-6098	409	5	jakhar	jakhar	PROPN
ejpam-6098	409	6	et	et	PROPN
ejpam-6098	409	7	al	al	PROPN
ejpam-6098	409	8	.	.	PUNCT
ejpam-6098	409	9	/	/	SYM
ejpam-6098	409	10	eur	eur	PROPN
ejpam-6098	409	11	.	.	PUNCT
ejpam-6098	410	1	j.	j.	PROPN
ejpam-6098	410	2	pure	pure	PROPN
ejpam-6098	410	3	appl	appl	PROPN
ejpam-6098	410	4	.	.	PROPN
ejpam-6098	410	5	math	math	PROPN
ejpam-6098	410	6	,	,	PUNCT
ejpam-6098	410	7	18	18	NUM
ejpam-6098	410	8	(	(	PUNCT
ejpam-6098	410	9	2	2	NUM
ejpam-6098	410	10	)	)	PUNCT
ejpam-6098	410	11	(	(	PUNCT
ejpam-6098	410	12	2025	2025	NUM
ejpam-6098	410	13	)	)	PUNCT
ejpam-6098	410	14	,	,	PUNCT
ejpam-6098	410	15	6098	6098	NUM
ejpam-6098	410	16	17	17	NUM
ejpam-6098	410	17	of	of	ADP
ejpam-6098	410	18	23	23	NUM
ejpam-6098	410	19	replacing	replacing	NOUN
ejpam-6098	410	20	(	(	PUNCT
ejpam-6098	410	21	u	u	NOUN
ejpam-6098	410	22	,	,	PUNCT
ejpam-6098	410	23	v	v	NOUN
ejpam-6098	410	24	,	,	PUNCT
ejpam-6098	410	25	w	w	NOUN
ejpam-6098	410	26	)	)	PUNCT
ejpam-6098	410	27	by	by	ADP
ejpam-6098	410	28	(	(	PUNCT
ejpam-6098	410	29	2nu	2nu	ADJ
ejpam-6098	410	30	,	,	PUNCT
ejpam-6098	410	31	2nv	2nv	ADJ
ejpam-6098	410	32	,	,	PUNCT
ejpam-6098	410	33	2nw	2nw	ADJ
ejpam-6098	410	34	)	)	PUNCT
ejpam-6098	410	35	in	in	ADP
ejpam-6098	410	36	(	(	PUNCT
ejpam-6098	410	37	25	25	NUM
ejpam-6098	410	38	)	)	PUNCT
ejpam-6098	410	39	,	,	PUNCT
ejpam-6098	410	40	we	we	PRON
ejpam-6098	410	41	obtain	obtain	VERB
ejpam-6098	410	42	µθ1(s	µθ1(s	PRON
ejpam-6098	410	43	)	)	PUNCT
ejpam-6098	410	44	≥	≥	NOUN
ejpam-6098	410	45	µ′ϕ(2nu,2nv,2nw)(2	µ′ϕ(2nu,2nv,2nw)(2	NOUN
ejpam-6098	410	46	ns	n	VERB
ejpam-6098	410	47	)	)	PUNCT
ejpam-6098	410	48	,	,	PUNCT
ejpam-6098	410	49	(	(	PUNCT
ejpam-6098	410	50	37	37	NUM
ejpam-6098	410	51	)	)	PUNCT
ejpam-6098	410	52	where	where	SCONJ
ejpam-6098	410	53	θ1	θ1	NOUN
ejpam-6098	410	54	=	=	PRON
ejpam-6098	410	55	f(2.2nu+	f(2.2nu+	VERB
ejpam-6098	410	56	2nv	2nv	ADJ
ejpam-6098	410	57	+	+	CCONJ
ejpam-6098	410	58	2nw)−	2nw)−	NUM
ejpam-6098	410	59	3f(2nu+	3f(2nu+	NUM
ejpam-6098	410	60	2nv	2nv	NOUN
ejpam-6098	410	61	+	+	CCONJ
ejpam-6098	410	62	2nw)−	2nw)−	NUM
ejpam-6098	410	63	f(−2nu+	f(−2nu+	NOUN
ejpam-6098	410	64	2nv	2nv	PROPN
ejpam-6098	411	1	+	+	CCONJ
ejpam-6098	411	2	2nw	2nw	ADJ
ejpam-6098	411	3	)	)	PUNCT
ejpam-6098	411	4	−	−	PROPN
ejpam-6098	411	5	2f(2nu+	2f(2nu+	PROPN
ejpam-6098	411	6	2nv)−	2nv)−	NUM
ejpam-6098	411	7	2f(2nu+	2f(2nu+	NUM
ejpam-6098	411	8	2βw	2βw	NOUN
ejpam-6098	411	9	)	)	PUNCT
ejpam-6098	412	1	+	+	CCONJ
ejpam-6098	412	2	6f(2nu−	6f(2nu−	NUM
ejpam-6098	412	3	2nv	2nv	NOUN
ejpam-6098	412	4	)	)	PUNCT
ejpam-6098	413	1	+	+	CCONJ
ejpam-6098	413	2	6f(2nu−	6f(2nu−	NUM
ejpam-6098	413	3	2nw	2nw	NUM
ejpam-6098	413	4	)	)	PUNCT
ejpam-6098	414	1	+	+	CCONJ
ejpam-6098	414	2	3f(2nv	3f(2nv	NUM
ejpam-6098	414	3	+	+	CCONJ
ejpam-6098	414	4	2nw)−	2nw)−	NUM
ejpam-6098	414	5	2f(2.2nu−	2f(2.2nu−	NUM
ejpam-6098	414	6	2nv)−	2nv)−	NUM
ejpam-6098	414	7	2f(2.2nu−	2f(2.2nu−	NUM
ejpam-6098	414	8	2nw	2nw	NOUN
ejpam-6098	414	9	)	)	PUNCT
ejpam-6098	415	1	+	+	CCONJ
ejpam-6098	415	2	18f(2nu	18f(2nu	NUM
ejpam-6098	415	3	)	)	PUNCT
ejpam-6098	415	4	+	+	CCONJ
ejpam-6098	415	5	6f(2nv	6f(2nv	X
ejpam-6098	415	6	)	)	PUNCT
ejpam-6098	415	7	+	+	NUM
ejpam-6098	415	8	6f(2nw	6f(2nw	NUM
ejpam-6098	415	9	)	)	PUNCT
ejpam-6098	415	10	,	,	PUNCT
ejpam-6098	415	11	taking	take	VERB
ejpam-6098	415	12	the	the	DET
ejpam-6098	415	13	limit	limit	NOUN
ejpam-6098	415	14	as	as	ADP
ejpam-6098	415	15	n→	n→	PROPN
ejpam-6098	415	16	∞	∞	PROPN
ejpam-6098	415	17	in	in	ADP
ejpam-6098	415	18	(	(	PUNCT
ejpam-6098	415	19	37	37	NUM
ejpam-6098	415	20	)	)	PUNCT
ejpam-6098	415	21	and	and	CCONJ
ejpam-6098	415	22	using	use	VERB
ejpam-6098	415	23	the	the	DET
ejpam-6098	415	24	following	follow	VERB
ejpam-6098	415	25	equality	equality	NOUN
ejpam-6098	415	26	limm→∞µ	limm→∞µ	PROPN
ejpam-6098	415	27	′	′	NOUN
ejpam-6098	415	28	ϕ(2nu,2nv,2nw)(8	ϕ(2nu,2nv,2nw)(8	NOUN
ejpam-6098	415	29	ns	ns	NUM
ejpam-6098	415	30	)	)	PUNCT
ejpam-6098	415	31	=	=	SYM
ejpam-6098	415	32	1	1	NUM
ejpam-6098	415	33	,	,	PUNCT
ejpam-6098	415	34	(	(	PUNCT
ejpam-6098	415	35	38	38	NUM
ejpam-6098	415	36	)	)	PUNCT
ejpam-6098	415	37	we	we	PRON
ejpam-6098	415	38	conclude	conclude	VERB
ejpam-6098	415	39	that	that	SCONJ
ejpam-6098	415	40	q	q	NOUN
ejpam-6098	415	41	satisfies	satisfie	NOUN
ejpam-6098	415	42	(	(	PUNCT
ejpam-6098	415	43	1	1	NUM
ejpam-6098	415	44	)	)	PUNCT
ejpam-6098	415	45	.	.	PUNCT
ejpam-6098	416	1	uniqueness	uniqueness	NOUN
ejpam-6098	416	2	:	:	PUNCT
ejpam-6098	416	3	to	to	PART
ejpam-6098	416	4	show	show	VERB
ejpam-6098	416	5	the	the	DET
ejpam-6098	416	6	function	function	NOUN
ejpam-6098	416	7	q	q	NOUN
ejpam-6098	416	8	is	be	AUX
ejpam-6098	416	9	unique	unique	ADJ
ejpam-6098	416	10	,	,	PUNCT
ejpam-6098	416	11	we	we	PRON
ejpam-6098	416	12	suppose	suppose	VERB
ejpam-6098	416	13	that	that	SCONJ
ejpam-6098	416	14	there	there	PRON
ejpam-6098	416	15	is	be	VERB
ejpam-6098	416	16	any	any	DET
ejpam-6098	416	17	other	other	ADJ
ejpam-6098	416	18	cubic	cubic	ADJ
ejpam-6098	416	19	mapping	mapping	NOUN
ejpam-6098	416	20	q′	q′	NOUN
ejpam-6098	416	21	:	:	PUNCT
ejpam-6098	417	1	u	u	X
ejpam-6098	417	2	→	→	SYM
ejpam-6098	417	3	x	x	X
ejpam-6098	417	4	which	which	PRON
ejpam-6098	417	5	holds	hold	VERB
ejpam-6098	417	6	(	(	PUNCT
ejpam-6098	417	7	26	26	NUM
ejpam-6098	417	8	)	)	PUNCT
ejpam-6098	417	9	.	.	PUNCT
ejpam-6098	418	1	since	since	SCONJ
ejpam-6098	418	2	f	f	PROPN
ejpam-6098	418	3	,	,	PUNCT
ejpam-6098	418	4	q	q	PROPN
ejpam-6098	418	5	and	and	CCONJ
ejpam-6098	418	6	q′	q′	NOUN
ejpam-6098	418	7	all	all	PRON
ejpam-6098	418	8	are	be	AUX
ejpam-6098	418	9	cubic	cubic	ADJ
ejpam-6098	418	10	mapping	mapping	NOUN
ejpam-6098	418	11	then	then	ADV
ejpam-6098	418	12	for	for	ADP
ejpam-6098	418	13	every	every	DET
ejpam-6098	418	14	u	u	PROPN
ejpam-6098	418	15	∈	∈	PROPN
ejpam-6098	418	16	u	u	NOUN
ejpam-6098	418	17	,	,	PUNCT
ejpam-6098	418	18	we	we	PRON
ejpam-6098	418	19	can	can	AUX
ejpam-6098	418	20	write	write	VERB
ejpam-6098	418	21	q(2nu	q(2nu	PROPN
ejpam-6098	418	22	)	)	PUNCT
ejpam-6098	419	1	=	=	SYM
ejpam-6098	419	2	23nq(u	23nq(u	NUM
ejpam-6098	419	3	)	)	PUNCT
ejpam-6098	419	4	and	and	CCONJ
ejpam-6098	419	5	q′(2nu	q′(2nu	NUM
ejpam-6098	419	6	)	)	PUNCT
ejpam-6098	419	7	=	=	SYM
ejpam-6098	419	8	23nq(u	23nq(u	NUM
ejpam-6098	419	9	)	)	PUNCT
ejpam-6098	419	10	.	.	PUNCT
ejpam-6098	420	1	therefore	therefore	ADV
ejpam-6098	420	2	,	,	PUNCT
ejpam-6098	420	3	we	we	PRON
ejpam-6098	420	4	have	have	VERB
ejpam-6098	420	5	µq(u)−q′(u)(s	µq(u)−q′(u)(s	NOUN
ejpam-6098	420	6	)	)	PUNCT
ejpam-6098	421	1	=	=	SYM
ejpam-6098	421	2	limn→∞µq(2nu	limn→∞µq(2nu	PROPN
ejpam-6098	421	3	)	)	PUNCT
ejpam-6098	421	4	23n	23n	NUM
ejpam-6098	421	5	−q′(2nu	−q′(2nu	PROPN
ejpam-6098	421	6	)	)	PUNCT
ejpam-6098	421	7	23n	23n	NUM
ejpam-6098	421	8	(	(	PUNCT
ejpam-6098	421	9	s	s	NOUN
ejpam-6098	421	10	)	)	PUNCT
ejpam-6098	421	11	.	.	PUNCT
ejpam-6098	422	1	(	(	PUNCT
ejpam-6098	422	2	39	39	NUM
ejpam-6098	422	3	)	)	PUNCT
ejpam-6098	422	4	now	now	ADV
ejpam-6098	422	5	,	,	PUNCT
ejpam-6098	422	6	µq(2nu	µq(2nu	NOUN
ejpam-6098	422	7	)	)	PUNCT
ejpam-6098	422	8	23n	23n	NUM
ejpam-6098	422	9	−q′(2nu	−q′(2nu	PROPN
ejpam-6098	422	10	)	)	PUNCT
ejpam-6098	422	11	23n	23n	NUM
ejpam-6098	422	12	(	(	PUNCT
ejpam-6098	422	13	s	s	NOUN
ejpam-6098	422	14	)	)	PUNCT
ejpam-6098	422	15	≥	≥	NOUN
ejpam-6098	422	16	min	min	NOUN
ejpam-6098	422	17	{	{	PUNCT
ejpam-6098	422	18	µq(2nu	µq(2nu	NOUN
ejpam-6098	422	19	)	)	PUNCT
ejpam-6098	422	20	23n	23n	NUM
ejpam-6098	422	21	−	−	PROPN
ejpam-6098	422	22	f(2nu	f(2nu	NUM
ejpam-6098	422	23	)	)	PUNCT
ejpam-6098	422	24	23n	23n	NUM
ejpam-6098	422	25	(	(	PUNCT
ejpam-6098	422	26	s	s	NOUN
ejpam-6098	422	27	2	2	NUM
ejpam-6098	422	28	)	)	PUNCT
ejpam-6098	422	29	,	,	PUNCT
ejpam-6098	422	30	µq′(2nu	µq′(2nu	PROPN
ejpam-6098	422	31	)	)	PUNCT
ejpam-6098	422	32	23n	23n	NOUN
ejpam-6098	422	33	−	−	PROPN
ejpam-6098	422	34	f(2nu	f(2nu	NUM
ejpam-6098	422	35	)	)	PUNCT
ejpam-6098	422	36	23n	23n	NUM
ejpam-6098	422	37	(	(	PUNCT
ejpam-6098	422	38	s	s	NOUN
ejpam-6098	422	39	2	2	NUM
ejpam-6098	422	40	)	)	PUNCT
ejpam-6098	422	41	}	}	PUNCT
ejpam-6098	422	42	≥	≥	NOUN
ejpam-6098	422	43	µ′ϕ(2nu,0,0	µ′ϕ(2nu,0,0	NOUN
ejpam-6098	422	44	)	)	PUNCT
ejpam-6098	422	45	(	(	PUNCT
ejpam-6098	422	46	(	(	PUNCT
ejpam-6098	422	47	3×	3×	NUM
ejpam-6098	422	48	8n)(8−	8n)(8−	NOUN
ejpam-6098	422	49	k)s	k)s	X
ejpam-6098	422	50	2	2	NUM
ejpam-6098	422	51	)	)	PUNCT
ejpam-6098	422	52	≥	≥	NOUN
ejpam-6098	422	53	µ′ϕ(u,0,0	µ′ϕ(u,0,0	NOUN
ejpam-6098	422	54	)	)	PUNCT
ejpam-6098	422	55	(	(	PUNCT
ejpam-6098	422	56	(	(	PUNCT
ejpam-6098	422	57	3×	3×	NUM
ejpam-6098	422	58	8n)(8−	8n)(8−	NOUN
ejpam-6098	422	59	k)s	k)s	NOUN
ejpam-6098	422	60	2kn	2kn	ADV
ejpam-6098	422	61	)	)	PUNCT
ejpam-6098	422	62	.	.	PUNCT
ejpam-6098	423	1	since	since	SCONJ
ejpam-6098	423	2	,	,	PUNCT
ejpam-6098	423	3	limn→∞	limn→∞	PROPN
ejpam-6098	423	4	23n(8−k)s	23n(8−k)s	NUM
ejpam-6098	423	5	2kn	2kn	ADJ
ejpam-6098	423	6	=	=	SYM
ejpam-6098	423	7	∞	∞	NOUN
ejpam-6098	423	8	,	,	PUNCT
ejpam-6098	423	9	we	we	PRON
ejpam-6098	423	10	get	get	VERB
ejpam-6098	423	11	limn→∞µ	limn→∞µ	ADJ
ejpam-6098	423	12	′	′	NUM
ejpam-6098	423	13	ϕ(u,0,0	ϕ(u,0,0	NOUN
ejpam-6098	423	14	)	)	PUNCT
ejpam-6098	423	15	(	(	PUNCT
ejpam-6098	423	16	(	(	PUNCT
ejpam-6098	423	17	3×8n)(8−k)s	3×8n)(8−k)s	NUM
ejpam-6098	423	18	2kn	2kn	ADJ
ejpam-6098	423	19	)	)	PUNCT
ejpam-6098	424	1	=	=	SYM
ejpam-6098	424	2	1	1	X
ejpam-6098	424	3	.	.	PUNCT
ejpam-6098	425	1	hence	hence	ADV
ejpam-6098	425	2	,	,	PUNCT
ejpam-6098	425	3	it	it	PRON
ejpam-6098	425	4	follows	follow	VERB
ejpam-6098	425	5	that	that	DET
ejpam-6098	425	6	µq(2nu	µq(2nu	NOUN
ejpam-6098	425	7	)	)	PUNCT
ejpam-6098	425	8	23n	23n	PROPN
ejpam-6098	425	9	−q′(2nu	−q′(2nu	PROPN
ejpam-6098	425	10	)	)	PUNCT
ejpam-6098	425	11	8n	8n	NOUN
ejpam-6098	425	12	(	(	PUNCT
ejpam-6098	425	13	s	s	X
ejpam-6098	425	14	)	)	PUNCT
ejpam-6098	425	15	=	=	SYM
ejpam-6098	425	16	1	1	NUM
ejpam-6098	425	17	and	and	CCONJ
ejpam-6098	425	18	so	so	ADV
ejpam-6098	425	19	q(u	q(u	NOUN
ejpam-6098	425	20	)	)	PUNCT
ejpam-6098	425	21	=	=	SYM
ejpam-6098	425	22	q′(u	q′(u	PROPN
ejpam-6098	425	23	)	)	PUNCT
ejpam-6098	425	24	.	.	PUNCT
ejpam-6098	426	1	corollary	corollary	ADJ
ejpam-6098	426	2	3	3	X
ejpam-6098	426	3	.	.	PUNCT
ejpam-6098	426	4	suppose	suppose	VERB
ejpam-6098	426	5	p	p	X
ejpam-6098	426	6	∈	∈	PROPN
ejpam-6098	426	7	(	(	PUNCT
ejpam-6098	426	8	0	0	NUM
ejpam-6098	426	9	,	,	PUNCT
ejpam-6098	426	10	1	1	NUM
ejpam-6098	426	11	)	)	PUNCT
ejpam-6098	426	12	and	and	CCONJ
ejpam-6098	426	13	u0	u0	PROPN
ejpam-6098	426	14	∈	∈	PROPN
ejpam-6098	426	15	u	u	NOUN
ejpam-6098	426	16	.	.	PUNCT
ejpam-6098	427	1	if	if	SCONJ
ejpam-6098	427	2	f	f	X
ejpam-6098	427	3	:	:	PUNCT
ejpam-6098	427	4	u	u	X
ejpam-6098	427	5	→	→	X
ejpam-6098	427	6	x	x	X
ejpam-6098	427	7	is	be	AUX
ejpam-6098	427	8	a	a	DET
ejpam-6098	427	9	function	function	NOUN
ejpam-6098	427	10	such	such	ADJ
ejpam-6098	427	11	that	that	PRON
ejpam-6098	427	12	µθ(s	µθ(s	PUNCT
ejpam-6098	427	13	)	)	PUNCT
ejpam-6098	427	14	≥	≥	NUM
ejpam-6098	427	15	µ′||u||pu0	µ′||u||pu0	PUNCT
ejpam-6098	427	16	(	(	PUNCT
ejpam-6098	427	17	s	s	NOUN
ejpam-6098	427	18	)	)	PUNCT
ejpam-6098	427	19	,	,	PUNCT
ejpam-6098	427	20	where	where	SCONJ
ejpam-6098	427	21	θ	θ	PROPN
ejpam-6098	427	22	=	=	PUNCT
ejpam-6098	427	23	f(2u+	f(2u+	VERB
ejpam-6098	427	24	w	w	PROPN
ejpam-6098	427	25	+	+	PROPN
ejpam-6098	427	26	v)−	v)−	PROPN
ejpam-6098	427	27	3f(w	3f(w	NOUN
ejpam-6098	427	28	+	+	CCONJ
ejpam-6098	427	29	u+	u+	NOUN
ejpam-6098	427	30	v)−	v)−	PROPN
ejpam-6098	427	31	f(v	f(v	PROPN
ejpam-6098	427	32	+	+	CCONJ
ejpam-6098	428	1	w	w	PROPN
ejpam-6098	428	2	−	−	PROPN
ejpam-6098	428	3	u)−	u)−	PROPN
ejpam-6098	428	4	2f(u+	2f(u+	PROPN
ejpam-6098	428	5	v)−	v)−	PROPN
ejpam-6098	428	6	2f(w	2f(w	NOUN
ejpam-6098	428	7	+	+	NUM
ejpam-6098	428	8	u	u	NOUN
ejpam-6098	428	9	)	)	PUNCT
ejpam-6098	428	10	+	+	CCONJ
ejpam-6098	428	11	6f(u−	6f(u−	NUM
ejpam-6098	428	12	w	w	NOUN
ejpam-6098	428	13	)	)	PUNCT
ejpam-6098	428	14	+	+	CCONJ
ejpam-6098	428	15	6f(u−	6f(u−	NUM
ejpam-6098	428	16	v	v	NOUN
ejpam-6098	428	17	)	)	PUNCT
ejpam-6098	428	18	+	+	NUM
ejpam-6098	428	19	3f(v	3f(v	NUM
ejpam-6098	429	1	+	+	CCONJ
ejpam-6098	429	2	w)−	w)−	PROPN
ejpam-6098	429	3	2f(2u−	2f(2u−	NUM
ejpam-6098	429	4	w)−	w)−	PROPN
ejpam-6098	429	5	2f(2u−	2f(2u−	NUM
ejpam-6098	429	6	v	v	NOUN
ejpam-6098	429	7	)	)	PUNCT
ejpam-6098	429	8	+	+	CCONJ
ejpam-6098	429	9	18f(u	18f(u	NUM
ejpam-6098	429	10	)	)	PUNCT
ejpam-6098	429	11	+	+	CCONJ
ejpam-6098	430	1	6f(w	6f(w	NUM
ejpam-6098	430	2	)	)	PUNCT
ejpam-6098	431	1	+	+	CCONJ
ejpam-6098	431	2	6f(v	6f(v	NUM
ejpam-6098	431	3	)	)	PUNCT
ejpam-6098	431	4	then	then	ADV
ejpam-6098	431	5	there	there	PRON
ejpam-6098	431	6	is	be	VERB
ejpam-6098	431	7	a	a	DET
ejpam-6098	431	8	unique	unique	ADJ
ejpam-6098	431	9	cubic	cubic	ADJ
ejpam-6098	431	10	mapping	mapping	NOUN
ejpam-6098	431	11	q	q	NOUN
ejpam-6098	431	12	:	:	PUNCT
ejpam-6098	431	13	u	u	X
ejpam-6098	431	14	→	→	PUNCT
ejpam-6098	431	15	x	x	SYM
ejpam-6098	431	16	holds	hold	VERB
ejpam-6098	431	17	µq(u)−f(u)(s	µq(u)−f(u)(s	PROPN
ejpam-6098	431	18	)	)	PUNCT
ejpam-6098	431	19	≥	≥	NOUN
ejpam-6098	431	20	µ′u0||u||p(3(8−	µ′u0||u||p(3(8−	VERB
ejpam-6098	431	21	8p)s	8p)s	NUM
ejpam-6098	431	22	)	)	PUNCT
ejpam-6098	431	23	.	.	PUNCT
ejpam-6098	432	1	proof	proof	NOUN
ejpam-6098	432	2	.	.	PUNCT
ejpam-6098	433	1	let	let	VERB
ejpam-6098	433	2	k	k	NOUN
ejpam-6098	433	3	=	=	PUNCT
ejpam-6098	433	4	8p	8p	NUM
ejpam-6098	433	5	and	and	CCONJ
ejpam-6098	433	6	ϕ	ϕ	NOUN
ejpam-6098	433	7	:	:	PUNCT
ejpam-6098	433	8	u	u	NOUN
ejpam-6098	433	9	→	→	PUNCT
ejpam-6098	433	10	x	x	PART
ejpam-6098	433	11	be	be	AUX
ejpam-6098	433	12	defined	define	VERB
ejpam-6098	433	13	as	as	ADP
ejpam-6098	433	14	ϕ(u	ϕ(u	PROPN
ejpam-6098	433	15	,	,	PUNCT
ejpam-6098	433	16	v	v	NOUN
ejpam-6098	433	17	,	,	PUNCT
ejpam-6098	433	18	w	w	NOUN
ejpam-6098	433	19	)	)	PUNCT
ejpam-6098	433	20	=	=	SYM
ejpam-6098	433	21	||u||pu0	||u||pu0	NOUN
ejpam-6098	433	22	and	and	CCONJ
ejpam-6098	433	23	ϕ(u	ϕ(u	PROPN
ejpam-6098	433	24	,	,	PUNCT
ejpam-6098	433	25	v	v	NOUN
ejpam-6098	433	26	,	,	PUNCT
ejpam-6098	433	27	w	w	NOUN
ejpam-6098	433	28	)	)	PUNCT
ejpam-6098	433	29	=	=	SYM
ejpam-6098	433	30	(	(	PUNCT
ejpam-6098	433	31	||w||p	||w||p	NOUN
ejpam-6098	433	32	+	+	ADJ
ejpam-6098	433	33	||v||p	||v||p	NOUN
ejpam-6098	433	34	+	+	CCONJ
ejpam-6098	433	35	||u||p)u0	||u||p)u0	NOUN
ejpam-6098	433	36	in	in	ADP
ejpam-6098	433	37	theorem	theorem	NOUN
ejpam-6098	433	38	(	(	PUNCT
ejpam-6098	433	39	4	4	NUM
ejpam-6098	433	40	)	)	PUNCT
ejpam-6098	433	41	.	.	PUNCT
ejpam-6098	434	1	we	we	PRON
ejpam-6098	434	2	obtain	obtain	VERB
ejpam-6098	434	3	the	the	DET
ejpam-6098	434	4	desired	desire	VERB
ejpam-6098	434	5	results	result	NOUN
ejpam-6098	434	6	.	.	PUNCT
ejpam-6098	435	1	j.	j.	PROPN
ejpam-6098	435	2	jakhar	jakhar	PROPN
ejpam-6098	435	3	et	et	PROPN
ejpam-6098	435	4	al	al	PROPN
ejpam-6098	435	5	.	.	PUNCT
ejpam-6098	435	6	/	/	SYM
ejpam-6098	435	7	eur	eur	PROPN
ejpam-6098	435	8	.	.	PUNCT
ejpam-6098	436	1	j.	j.	PROPN
ejpam-6098	436	2	pure	pure	PROPN
ejpam-6098	436	3	appl	appl	PROPN
ejpam-6098	436	4	.	.	PROPN
ejpam-6098	436	5	math	math	PROPN
ejpam-6098	436	6	,	,	PUNCT
ejpam-6098	436	7	18	18	NUM
ejpam-6098	436	8	(	(	PUNCT
ejpam-6098	436	9	2	2	NUM
ejpam-6098	436	10	)	)	PUNCT
ejpam-6098	436	11	(	(	PUNCT
ejpam-6098	436	12	2025	2025	NUM
ejpam-6098	436	13	)	)	PUNCT
ejpam-6098	436	14	,	,	PUNCT
ejpam-6098	436	15	6098	6098	NUM
ejpam-6098	436	16	18	18	NUM
ejpam-6098	436	17	of	of	ADP
ejpam-6098	436	18	23	23	NUM
ejpam-6098	436	19	5	5	NUM
ejpam-6098	436	20	.	.	PUNCT
ejpam-6098	436	21	results	result	NOUN
ejpam-6098	436	22	of	of	ADP
ejpam-6098	436	23	experiment	experiment	NOUN
ejpam-6098	436	24	in	in	ADP
ejpam-6098	436	25	this	this	DET
ejpam-6098	436	26	segment	segment	NOUN
ejpam-6098	437	1	,	,	PUNCT
ejpam-6098	437	2	the	the	DET
ejpam-6098	437	3	authors	author	NOUN
ejpam-6098	437	4	discuss	discuss	VERB
ejpam-6098	437	5	the	the	DET
ejpam-6098	437	6	graphs	graph	NOUN
ejpam-6098	437	7	of	of	ADP
ejpam-6098	437	8	the	the	DET
ejpam-6098	437	9	approximate	approximate	ADJ
ejpam-6098	437	10	and	and	CCONJ
ejpam-6098	437	11	exact	exact	ADJ
ejpam-6098	437	12	solutions	solution	NOUN
ejpam-6098	437	13	for	for	ADP
ejpam-6098	437	14	the	the	DET
ejpam-6098	437	15	given	give	VERB
ejpam-6098	437	16	equation	equation	NOUN
ejpam-6098	437	17	.	.	PUNCT
ejpam-6098	438	1	it	it	PRON
ejpam-6098	438	2	is	be	AUX
ejpam-6098	438	3	simple	simple	ADJ
ejpam-6098	438	4	to	to	PART
ejpam-6098	438	5	demonstrate	demonstrate	VERB
ejpam-6098	438	6	that	that	DET
ejpam-6098	438	7	function	function	NOUN
ejpam-6098	438	8	f(u	f(u	PROPN
ejpam-6098	438	9	)	)	PUNCT
ejpam-6098	439	1	=	=	NOUN
ejpam-6098	439	2	u3	u3	NOUN
ejpam-6098	439	3	is	be	AUX
ejpam-6098	439	4	an	an	DET
ejpam-6098	439	5	exact	exact	ADJ
ejpam-6098	439	6	solution	solution	NOUN
ejpam-6098	439	7	to	to	ADP
ejpam-6098	439	8	a	a	DET
ejpam-6098	439	9	cubic	cubic	ADJ
ejpam-6098	439	10	equation	equation	NOUN
ejpam-6098	439	11	.	.	PUNCT
ejpam-6098	440	1	for	for	ADP
ejpam-6098	440	2	experimental	experimental	ADJ
ejpam-6098	440	3	purposes	purpose	NOUN
ejpam-6098	440	4	,	,	PUNCT
ejpam-6098	440	5	we	we	PRON
ejpam-6098	440	6	explored	explore	VERB
ejpam-6098	440	7	an	an	DET
ejpam-6098	440	8	alternative	alternative	ADJ
ejpam-6098	440	9	function	function	NOUN
ejpam-6098	440	10	q(u	q(u	NOUN
ejpam-6098	440	11	)	)	PUNCT
ejpam-6098	440	12	=	=	NOUN
ejpam-6098	440	13	u3	u3	NOUN
ejpam-6098	440	14	+	+	CCONJ
ejpam-6098	440	15	u3(u	u3(u	ADP
ejpam-6098	440	16	−	−	PROPN
ejpam-6098	440	17	⌊u⌋)3⌊u⌋	⌊u⌋)3⌊u⌋	PROPN
ejpam-6098	440	18	,	,	PUNCT
ejpam-6098	440	19	(	(	PUNCT
ejpam-6098	440	20	here	here	ADV
ejpam-6098	440	21	,	,	PUNCT
ejpam-6098	440	22	⌊.⌋	⌊.⌋	NOUN
ejpam-6098	440	23	represents	represent	VERB
ejpam-6098	440	24	a	a	DET
ejpam-6098	440	25	floor	floor	NOUN
ejpam-6098	440	26	function	function	NOUN
ejpam-6098	440	27	)	)	PUNCT
ejpam-6098	440	28	which	which	PRON
ejpam-6098	440	29	differs	differ	VERB
ejpam-6098	440	30	from	from	ADP
ejpam-6098	440	31	a	a	DET
ejpam-6098	440	32	cubic	cubic	ADJ
ejpam-6098	440	33	function	function	NOUN
ejpam-6098	440	34	.	.	PUNCT
ejpam-6098	441	1	these	these	DET
ejpam-6098	441	2	two	two	NUM
ejpam-6098	441	3	functions	function	NOUN
ejpam-6098	441	4	were	be	AUX
ejpam-6098	441	5	graphed	graph	VERB
ejpam-6098	441	6	using	use	VERB
ejpam-6098	441	7	matlab	matlab	PROPN
ejpam-6098	441	8	,	,	PUNCT
ejpam-6098	441	9	and	and	CCONJ
ejpam-6098	441	10	it	it	PRON
ejpam-6098	441	11	was	be	AUX
ejpam-6098	441	12	observed	observe	VERB
ejpam-6098	441	13	that	that	SCONJ
ejpam-6098	441	14	the	the	DET
ejpam-6098	441	15	graphs	graph	NOUN
ejpam-6098	441	16	of	of	ADP
ejpam-6098	441	17	both	both	DET
ejpam-6098	441	18	functions	function	NOUN
ejpam-6098	441	19	,	,	PUNCT
ejpam-6098	441	20	f(u	f(u	PROPN
ejpam-6098	441	21	)	)	PUNCT
ejpam-6098	441	22	and	and	CCONJ
ejpam-6098	441	23	q(u	q(u	NOUN
ejpam-6098	441	24	)	)	PUNCT
ejpam-6098	441	25	,	,	PUNCT
ejpam-6098	441	26	coincide	coincide	NOUN
ejpam-6098	441	27	at	at	ADP
ejpam-6098	441	28	multiple	multiple	ADJ
ejpam-6098	441	29	points	point	NOUN
ejpam-6098	441	30	.	.	PUNCT
ejpam-6098	442	1	this	this	PRON
ejpam-6098	442	2	suggests	suggest	VERB
ejpam-6098	442	3	that	that	SCONJ
ejpam-6098	442	4	q(u	q(u	X
ejpam-6098	442	5	)	)	PUNCT
ejpam-6098	442	6	is	be	AUX
ejpam-6098	442	7	an	an	DET
ejpam-6098	442	8	approximate	approximate	ADJ
ejpam-6098	442	9	solution	solution	NOUN
ejpam-6098	442	10	to	to	ADP
ejpam-6098	442	11	the	the	DET
ejpam-6098	442	12	given	give	VERB
ejpam-6098	442	13	cubic	cubic	ADJ
ejpam-6098	442	14	equation	equation	NOUN
ejpam-6098	442	15	.	.	PUNCT
ejpam-6098	443	1	the	the	DET
ejpam-6098	443	2	table	table	NOUN
ejpam-6098	443	3	1	1	NUM
ejpam-6098	443	4	shows	show	VERB
ejpam-6098	443	5	the	the	DET
ejpam-6098	443	6	behavior	behavior	NOUN
ejpam-6098	443	7	of	of	ADP
ejpam-6098	443	8	the	the	DET
ejpam-6098	443	9	exact	exact	ADJ
ejpam-6098	443	10	solution	solution	NOUN
ejpam-6098	443	11	,	,	PUNCT
ejpam-6098	443	12	the	the	DET
ejpam-6098	443	13	approximate	approximate	ADJ
ejpam-6098	443	14	solution	solution	NOUN
ejpam-6098	443	15	,	,	PUNCT
ejpam-6098	443	16	and	and	CCONJ
ejpam-6098	443	17	the	the	DET
ejpam-6098	443	18	difference	difference	NOUN
ejpam-6098	443	19	between	between	ADP
ejpam-6098	443	20	their	their	PRON
ejpam-6098	443	21	values	value	NOUN
ejpam-6098	443	22	between	between	ADP
ejpam-6098	443	23	−1	−1	NOUN
ejpam-6098	443	24	and	and	CCONJ
ejpam-6098	443	25	1	1	NUM
ejpam-6098	443	26	.	.	PUNCT
ejpam-6098	444	1	also	also	ADV
ejpam-6098	444	2	,	,	PUNCT
ejpam-6098	444	3	the	the	DET
ejpam-6098	444	4	graphs	graph	NOUN
ejpam-6098	444	5	of	of	ADP
ejpam-6098	444	6	the	the	DET
ejpam-6098	444	7	functions	function	NOUN
ejpam-6098	444	8	f(u	f(u	PROPN
ejpam-6098	444	9	)	)	PUNCT
ejpam-6098	444	10	and	and	CCONJ
ejpam-6098	444	11	q(u	q(u	NOUN
ejpam-6098	444	12	)	)	PUNCT
ejpam-6098	444	13	are	be	AUX
ejpam-6098	444	14	shown	show	VERB
ejpam-6098	444	15	in	in	ADP
ejpam-6098	444	16	figure	figure	NOUN
ejpam-6098	444	17	1	1	NUM
ejpam-6098	444	18	.	.	PUNCT
ejpam-6098	444	19	-1	-1	PUNCT
ejpam-6098	445	1	-0.5	-0.5	X
ejpam-6098	445	2	0	0	NUM
ejpam-6098	445	3	0.5	0.5	NUM
ejpam-6098	445	4	1	1	NUM
ejpam-6098	445	5	-1	-1	SYM
ejpam-6098	445	6	-0.8	-0.8	PROPN
ejpam-6098	445	7	-0.6	-0.6	X
ejpam-6098	445	8	-0.4	-0.4	X
ejpam-6098	445	9	-0.2	-0.2	PROPN
ejpam-6098	445	10	0	0	NUM
ejpam-6098	445	11	0.2	0.2	NUM
ejpam-6098	445	12	0.4	0.4	NUM
ejpam-6098	445	13	0.6	0.6	NUM
ejpam-6098	445	14	0.8	0.8	NUM
ejpam-6098	445	15	1	1	NUM
ejpam-6098	445	16	figure	figure	NOUN
ejpam-6098	445	17	1	1	NUM
ejpam-6098	445	18	:	:	PUNCT
ejpam-6098	445	19	graph	graph	NOUN
ejpam-6098	445	20	of	of	ADP
ejpam-6098	445	21	f(u	f(u	PROPN
ejpam-6098	445	22	)	)	PUNCT
ejpam-6098	445	23	and	and	CCONJ
ejpam-6098	445	24	q(u	q(u	PROPN
ejpam-6098	445	25	)	)	PUNCT
ejpam-6098	445	26	j.	j.	PROPN
ejpam-6098	445	27	jakhar	jakhar	PROPN
ejpam-6098	445	28	et	et	PROPN
ejpam-6098	446	1	al	al	PROPN
ejpam-6098	446	2	.	.	PUNCT
ejpam-6098	446	3	/	/	SYM
ejpam-6098	446	4	eur	eur	PROPN
ejpam-6098	446	5	.	.	PUNCT
ejpam-6098	447	1	j.	j.	PROPN
ejpam-6098	447	2	pure	pure	PROPN
ejpam-6098	447	3	appl	appl	PROPN
ejpam-6098	447	4	.	.	PROPN
ejpam-6098	447	5	math	math	PROPN
ejpam-6098	447	6	,	,	PUNCT
ejpam-6098	447	7	18	18	NUM
ejpam-6098	447	8	(	(	PUNCT
ejpam-6098	447	9	2	2	NUM
ejpam-6098	447	10	)	)	PUNCT
ejpam-6098	447	11	(	(	PUNCT
ejpam-6098	447	12	2025	2025	NUM
ejpam-6098	447	13	)	)	PUNCT
ejpam-6098	447	14	,	,	PUNCT
ejpam-6098	447	15	6098	6098	NUM
ejpam-6098	447	16	19	19	NUM
ejpam-6098	447	17	of	of	ADP
ejpam-6098	447	18	23	23	NUM
ejpam-6098	447	19	table	table	NOUN
ejpam-6098	447	20	1	1	NUM
ejpam-6098	447	21	:	:	PUNCT
ejpam-6098	447	22	error	error	NOUN
ejpam-6098	447	23	of	of	ADP
ejpam-6098	447	24	approximation	approximation	NOUN
ejpam-6098	447	25	values	value	NOUN
ejpam-6098	447	26	of	of	ADP
ejpam-6098	447	27	u	u	NOUN
ejpam-6098	447	28	exact	exact	ADJ
ejpam-6098	447	29	solution	solution	NOUN
ejpam-6098	447	30	f(u	f(u	PROPN
ejpam-6098	447	31	)	)	PUNCT
ejpam-6098	447	32	approximate	approximate	ADJ
ejpam-6098	447	33	solution	solution	NOUN
ejpam-6098	447	34	q(u	q(u	NOUN
ejpam-6098	447	35	)	)	PUNCT
ejpam-6098	447	36	absolute	absolute	ADJ
ejpam-6098	447	37	error	error	NOUN
ejpam-6098	447	38	|f(u)−q(u)|	|f(u)−q(u)|	NOUN
ejpam-6098	447	39	-1	-1	INTJ
ejpam-6098	447	40	-1	-1	INTJ
ejpam-6098	448	1	-1	-1	NOUN
ejpam-6098	448	2	0	0	NUM
ejpam-6098	448	3	-0.9	-0.9	NOUN
ejpam-6098	448	4	-0.729	-0.729	PROPN
ejpam-6098	449	1	-0.728	-0.728	NOUN
ejpam-6098	449	2	0.001	0.001	NUM
ejpam-6098	449	3	-0.8	-0.8	NUM
ejpam-6098	449	4	-0.512	-0.512	NUM
ejpam-6098	449	5	-0.508	-0.508	PUNCT
ejpam-6098	449	6	0.004	0.004	NUM
ejpam-6098	449	7	-0.7	-0.7	NOUN
ejpam-6098	449	8	-0.343	-0.343	NOUN
ejpam-6098	449	9	-0.333	-0.333	ADP
ejpam-6098	449	10	0.010	0.010	NUM
ejpam-6098	449	11	-0.6	-0.6	NOUN
ejpam-6098	449	12	-0.216	-0.216	PUNCT
ejpam-6098	449	13	-0.202	-0.202	PROPN
ejpam-6098	449	14	0.014	0.014	NUM
ejpam-6098	449	15	-0.5	-0.5	X
ejpam-6098	449	16	-0.125	-0.125	PROPN
ejpam-6098	449	17	-0.109	-0.109	PROPN
ejpam-6098	449	18	0.016	0.016	NUM
ejpam-6098	449	19	-0.4	-0.4	NUM
ejpam-6098	449	20	-0.064	-0.064	PROPN
ejpam-6098	449	21	-0.050	-0.050	PROPN
ejpam-6098	449	22	0.014	0.014	NUM
ejpam-6098	449	23	-0.3	-0.3	PROPN
ejpam-6098	449	24	-0.027	-0.027	PUNCT
ejpam-6098	449	25	-0.017	-0.017	PUNCT
ejpam-6098	449	26	0.010	0.010	NUM
ejpam-6098	449	27	-0.2	-0.2	NOUN
ejpam-6098	449	28	-0.008	-0.008	PROPN
ejpam-6098	449	29	-0.004	-0.004	PUNCT
ejpam-6098	449	30	0.004	0.004	NUM
ejpam-6098	449	31	-0.1	-0.1	PROPN
ejpam-6098	449	32	-0.001	-0.001	NOUN
ejpam-6098	449	33	-0.0003	-0.0003	NOUN
ejpam-6098	450	1	0.0007	0.0007	NUM
ejpam-6098	450	2	0.1	0.1	NUM
ejpam-6098	450	3	0.001	0.001	NUM
ejpam-6098	450	4	0.001	0.001	NUM
ejpam-6098	450	5	0.000	0.000	NUM
ejpam-6098	450	6	0.2	0.2	NUM
ejpam-6098	450	7	0.008	0.008	NUM
ejpam-6098	450	8	0.0008	0.0008	NUM
ejpam-6098	450	9	0.000	0.000	NUM
ejpam-6098	450	10	0.3	0.3	NUM
ejpam-6098	450	11	0.027	0.027	NUM
ejpam-6098	450	12	0.027	0.027	NUM
ejpam-6098	450	13	0.000	0.000	NUM
ejpam-6098	450	14	0.4	0.4	NUM
ejpam-6098	450	15	0.064	0.064	NUM
ejpam-6098	450	16	0.0064	0.0064	NUM
ejpam-6098	450	17	0.000	0.000	NUM
ejpam-6098	450	18	0.5	0.5	NUM
ejpam-6098	450	19	0.125	0.125	NUM
ejpam-6098	450	20	0.125	0.125	NUM
ejpam-6098	450	21	0.000	0.000	NUM
ejpam-6098	450	22	0.6	0.6	NUM
ejpam-6098	450	23	0.216	0.216	NUM
ejpam-6098	450	24	0.216	0.216	NUM
ejpam-6098	450	25	0.000	0.000	NUM
ejpam-6098	450	26	0.7	0.7	NUM
ejpam-6098	450	27	0.343	0.343	NUM
ejpam-6098	450	28	0.343	0.343	NUM
ejpam-6098	450	29	0.000	0.000	NUM
ejpam-6098	450	30	0.8	0.8	NUM
ejpam-6098	450	31	0.512	0.512	NUM
ejpam-6098	450	32	0.512	0.512	NUM
ejpam-6098	450	33	0.000	0.000	NUM
ejpam-6098	450	34	0.9	0.9	NUM
ejpam-6098	450	35	0.729	0.729	NUM
ejpam-6098	450	36	0.729	0.729	NUM
ejpam-6098	450	37	0.000	0.000	NUM
ejpam-6098	450	38	1	1	NUM
ejpam-6098	450	39	1	1	NUM
ejpam-6098	450	40	1	1	NUM
ejpam-6098	450	41	0	0	NUM
ejpam-6098	450	42	j.	j.	PROPN
ejpam-6098	450	43	jakhar	jakhar	PROPN
ejpam-6098	450	44	et	et	PROPN
ejpam-6098	450	45	al	al	PROPN
ejpam-6098	450	46	.	.	PUNCT
ejpam-6098	450	47	/	/	SYM
ejpam-6098	450	48	eur	eur	PROPN
ejpam-6098	450	49	.	.	PUNCT
ejpam-6098	451	1	j.	j.	PROPN
ejpam-6098	451	2	pure	pure	PROPN
ejpam-6098	451	3	appl	appl	PROPN
ejpam-6098	451	4	.	.	PROPN
ejpam-6098	451	5	math	math	PROPN
ejpam-6098	451	6	,	,	PUNCT
ejpam-6098	451	7	18	18	NUM
ejpam-6098	451	8	(	(	PUNCT
ejpam-6098	451	9	2	2	NUM
ejpam-6098	451	10	)	)	PUNCT
ejpam-6098	451	11	(	(	PUNCT
ejpam-6098	451	12	2025	2025	NUM
ejpam-6098	451	13	)	)	PUNCT
ejpam-6098	451	14	,	,	PUNCT
ejpam-6098	451	15	6098	6098	NUM
ejpam-6098	451	16	20	20	NUM
ejpam-6098	451	17	of	of	ADP
ejpam-6098	451	18	23	23	NUM
ejpam-6098	451	19	6	6	NUM
ejpam-6098	451	20	.	.	PUNCT
ejpam-6098	451	21	comparative	comparative	ADJ
ejpam-6098	451	22	evaluation	evaluation	NOUN
ejpam-6098	451	23	of	of	ADP
ejpam-6098	451	24	the	the	DET
ejpam-6098	451	25	results	result	NOUN
ejpam-6098	451	26	this	this	DET
ejpam-6098	451	27	study	study	NOUN
ejpam-6098	451	28	established	establish	VERB
ejpam-6098	451	29	the	the	DET
ejpam-6098	451	30	stability	stability	NOUN
ejpam-6098	451	31	of	of	ADP
ejpam-6098	451	32	given	give	VERB
ejpam-6098	451	33	cubic	cubic	ADJ
ejpam-6098	451	34	fe	fe	NOUN
ejpam-6098	451	35	across	across	ADP
ejpam-6098	451	36	the	the	DET
ejpam-6098	451	37	spaces	space	NOUN
ejpam-6098	451	38	,	,	PUNCT
ejpam-6098	451	39	including	include	VERB
ejpam-6098	451	40	(	(	PUNCT
ejpam-6098	451	41	n	n	CCONJ
ejpam-6098	451	42	,	,	PUNCT
ejpam-6098	451	43	β)-ns	β)-ns	NOUN
ejpam-6098	451	44	,	,	PUNCT
ejpam-6098	451	45	na-(n	na-(n	NUM
ejpam-6098	451	46	,	,	PUNCT
ejpam-6098	451	47	β)-ns	β)-ns	NOUN
ejpam-6098	451	48	,	,	PUNCT
ejpam-6098	451	49	and	and	CCONJ
ejpam-6098	451	50	rns	rn	NOUN
ejpam-6098	451	51	.	.	PUNCT
ejpam-6098	452	1	the	the	DET
ejpam-6098	452	2	main	main	ADJ
ejpam-6098	452	3	results	result	NOUN
ejpam-6098	452	4	obtained	obtain	VERB
ejpam-6098	452	5	are	be	AUX
ejpam-6098	452	6	summarized	summarize	VERB
ejpam-6098	452	7	as	as	SCONJ
ejpam-6098	452	8	follows	follow	VERB
ejpam-6098	452	9	:	:	PUNCT
ejpam-6098	453	1	corollary	corollary	ADJ
ejpam-6098	453	2	no	no	INTJ
ejpam-6098	453	3	.	.	PUNCT
ejpam-6098	454	1	space	space	NOUN
ejpam-6098	454	2	setting	set	VERB
ejpam-6098	454	3	stability	stability	NOUN
ejpam-6098	454	4	result	result	VERB
ejpam-6098	454	5	corollary	corollary	ADJ
ejpam-6098	454	6	1	1	NUM
ejpam-6098	454	7	(	(	PUNCT
ejpam-6098	454	8	n	n	CCONJ
ejpam-6098	454	9	,	,	PUNCT
ejpam-6098	454	10	β)-ns	β)-ns	ADJ
ejpam-6098	454	11	||f(u)−q(u	||f(u)−q(u	NOUN
ejpam-6098	454	12	)	)	PUNCT
ejpam-6098	454	13	,	,	PUNCT
ejpam-6098	454	14	δ1	δ1	NOUN
ejpam-6098	454	15	,	,	PUNCT
ejpam-6098	454	16	...	...	PUNCT
ejpam-6098	454	17	,	,	PUNCT
ejpam-6098	454	18	δn−1||β	δn−1||β	NOUN
ejpam-6098	454	19	≤	≤	NOUN
ejpam-6098	455	1	ϵ	ϵ	ADP
ejpam-6098	455	2	6β(22β	6β(22β	NUM
ejpam-6098	455	3	−	−	NOUN
ejpam-6098	455	4	1	1	X
ejpam-6098	455	5	)	)	PUNCT
ejpam-6098	455	6	||u||β	||u||β	NOUN
ejpam-6098	455	7	.	.	PUNCT
ejpam-6098	455	8	corollary	corollary	ADJ
ejpam-6098	455	9	2	2	NUM
ejpam-6098	455	10	na-(n	na-(n	NOUN
ejpam-6098	455	11	,	,	PUNCT
ejpam-6098	455	12	β)-ns	β)-ns	ADJ
ejpam-6098	455	13	||f(u)−q(u	||f(u)−q(u	NOUN
ejpam-6098	455	14	)	)	PUNCT
ejpam-6098	455	15	,	,	PUNCT
ejpam-6098	455	16	δ1	δ1	NOUN
ejpam-6098	455	17	,	,	PUNCT
ejpam-6098	455	18	...	...	PUNCT
ejpam-6098	455	19	,	,	PUNCT
ejpam-6098	455	20	δn−1||β	δn−1||β	VERB
ejpam-6098	455	21	≤	≤	NOUN
ejpam-6098	455	22	24−βϵ||u||β	24−βϵ||u||β	NOUN
ejpam-6098	455	23	.	.	PUNCT
ejpam-6098	456	1	corollary	corollary	ADJ
ejpam-6098	456	2	3	3	NUM
ejpam-6098	456	3	rns	rn	NOUN
ejpam-6098	456	4	µq(u)−f(u)(s	µq(u)−f(u)(s	PROPN
ejpam-6098	456	5	)	)	PUNCT
ejpam-6098	456	6	≥	≥	NOUN
ejpam-6098	456	7	µ′||u||pu0	µ′||u||pu0	PUNCT
ejpam-6098	456	8	(	(	PUNCT
ejpam-6098	456	9	3(8−	3(8−	NUM
ejpam-6098	456	10	8p)s	8p)s	NUM
ejpam-6098	456	11	)	)	PUNCT
ejpam-6098	456	12	upon	upon	SCONJ
ejpam-6098	456	13	comparing	compare	VERB
ejpam-6098	456	14	the	the	DET
ejpam-6098	456	15	results	result	NOUN
ejpam-6098	456	16	presented	present	VERB
ejpam-6098	456	17	in	in	ADP
ejpam-6098	456	18	the	the	DET
ejpam-6098	456	19	table	table	NOUN
ejpam-6098	456	20	,	,	PUNCT
ejpam-6098	456	21	it	it	PRON
ejpam-6098	456	22	is	be	AUX
ejpam-6098	456	23	evident	evident	ADJ
ejpam-6098	456	24	that	that	SCONJ
ejpam-6098	456	25	the	the	DET
ejpam-6098	456	26	approximate	approximate	ADJ
ejpam-6098	456	27	solution	solution	NOUN
ejpam-6098	456	28	closely	closely	ADV
ejpam-6098	456	29	aligns	align	VERB
ejpam-6098	456	30	with	with	ADP
ejpam-6098	456	31	the	the	DET
ejpam-6098	456	32	exact	exact	ADJ
ejpam-6098	456	33	solution	solution	NOUN
ejpam-6098	456	34	within	within	ADP
ejpam-6098	456	35	the	the	DET
ejpam-6098	456	36	framework	framework	NOUN
ejpam-6098	456	37	of	of	ADP
ejpam-6098	456	38	na-(n	na-(n	PROPN
ejpam-6098	456	39	,	,	PUNCT
ejpam-6098	456	40	β)-ns	β)-ns	ADP
ejpam-6098	456	41	since	since	SCONJ
ejpam-6098	456	42	the	the	DET
ejpam-6098	456	43	upper	upper	ADJ
ejpam-6098	456	44	bound	bind	VERB
ejpam-6098	456	45	24−β||u||ϵ	24−β||u||ϵ	NOUN
ejpam-6098	456	46	is	be	AUX
ejpam-6098	456	47	less	less	ADJ
ejpam-6098	456	48	when	when	SCONJ
ejpam-6098	456	49	compared	compare	VERB
ejpam-6098	456	50	with	with	ADP
ejpam-6098	456	51	upper	upper	ADJ
ejpam-6098	456	52	bound	bind	VERB
ejpam-6098	456	53	in	in	ADP
ejpam-6098	456	54	(	(	PUNCT
ejpam-6098	456	55	n	n	CCONJ
ejpam-6098	456	56	,	,	PUNCT
ejpam-6098	456	57	β)-ns	β)-ns	NOUN
ejpam-6098	456	58	and	and	CCONJ
ejpam-6098	456	59	rns	rn	NOUN
ejpam-6098	456	60	.	.	PUNCT
ejpam-6098	457	1	the	the	DET
ejpam-6098	457	2	stability	stability	NOUN
ejpam-6098	457	3	results	result	VERB
ejpam-6098	457	4	concerning	concern	VERB
ejpam-6098	457	5	hyers	hyer	NOUN
ejpam-6098	457	6	-	-	PUNCT
ejpam-6098	457	7	ulam	ulam	PROPN
ejpam-6098	457	8	stability	stability	NOUN
ejpam-6098	457	9	regarding	regard	VERB
ejpam-6098	457	10	the	the	DET
ejpam-6098	457	11	upper	upper	ADJ
ejpam-6098	457	12	bound	bind	VERB
ejpam-6098	457	13	are	be	AUX
ejpam-6098	457	14	obtained	obtain	VERB
ejpam-6098	457	15	in	in	ADP
ejpam-6098	457	16	corollaries	corollary	NOUN
ejpam-6098	457	17	(	(	PUNCT
ejpam-6098	457	18	1	1	NUM
ejpam-6098	457	19	)	)	PUNCT
ejpam-6098	457	20	,	,	PUNCT
ejpam-6098	457	21	(	(	PUNCT
ejpam-6098	457	22	2	2	NUM
ejpam-6098	457	23	)	)	PUNCT
ejpam-6098	457	24	,	,	PUNCT
ejpam-6098	457	25	and	and	CCONJ
ejpam-6098	457	26	(	(	PUNCT
ejpam-6098	457	27	3	3	NUM
ejpam-6098	457	28	)	)	PUNCT
ejpam-6098	457	29	.	.	PUNCT
ejpam-6098	458	1	this	this	DET
ejpam-6098	458	2	study	study	NOUN
ejpam-6098	458	3	is	be	AUX
ejpam-6098	458	4	limited	limit	VERB
ejpam-6098	458	5	to	to	ADP
ejpam-6098	458	6	the	the	DET
ejpam-6098	458	7	analysis	analysis	NOUN
ejpam-6098	458	8	of	of	ADP
ejpam-6098	458	9	a	a	DET
ejpam-6098	458	10	specific	specific	ADJ
ejpam-6098	458	11	three	three	NUM
ejpam-6098	458	12	-	-	PUNCT
ejpam-6098	458	13	dimensional	dimensional	ADJ
ejpam-6098	458	14	cubic	cubic	ADJ
ejpam-6098	458	15	functional	functional	ADJ
ejpam-6098	458	16	equation	equation	NOUN
ejpam-6098	458	17	,	,	PUNCT
ejpam-6098	458	18	and	and	CCONJ
ejpam-6098	458	19	the	the	DET
ejpam-6098	458	20	results	result	NOUN
ejpam-6098	458	21	may	may	AUX
ejpam-6098	458	22	not	not	PART
ejpam-6098	458	23	directly	directly	ADV
ejpam-6098	458	24	extend	extend	VERB
ejpam-6098	458	25	to	to	ADP
ejpam-6098	458	26	other	other	ADJ
ejpam-6098	458	27	types	type	NOUN
ejpam-6098	458	28	.	.	PUNCT
ejpam-6098	459	1	while	while	SCONJ
ejpam-6098	459	2	novel	novel	ADJ
ejpam-6098	459	3	spaces	space	NOUN
ejpam-6098	459	4	are	be	AUX
ejpam-6098	459	5	introduced	introduce	VERB
ejpam-6098	459	6	,	,	PUNCT
ejpam-6098	459	7	other	other	ADJ
ejpam-6098	459	8	generalized	generalize	VERB
ejpam-6098	459	9	normed	normed	ADJ
ejpam-6098	459	10	structures	structure	NOUN
ejpam-6098	459	11	remain	remain	VERB
ejpam-6098	459	12	unexplored	unexplored	ADJ
ejpam-6098	459	13	.	.	PUNCT
ejpam-6098	460	1	additionally	additionally	ADV
ejpam-6098	460	2	,	,	PUNCT
ejpam-6098	460	3	the	the	DET
ejpam-6098	460	4	experimental	experimental	ADJ
ejpam-6098	460	5	validation	validation	NOUN
ejpam-6098	460	6	is	be	AUX
ejpam-6098	460	7	based	base	VERB
ejpam-6098	460	8	on	on	ADP
ejpam-6098	460	9	limited	limited	ADJ
ejpam-6098	460	10	examples	example	NOUN
ejpam-6098	460	11	,	,	PUNCT
ejpam-6098	460	12	and	and	CCONJ
ejpam-6098	460	13	deeper	deep	ADJ
ejpam-6098	460	14	stochastic	stochastic	ADJ
ejpam-6098	460	15	modeling	modeling	NOUN
ejpam-6098	460	16	in	in	ADP
ejpam-6098	460	17	random	random	ADJ
ejpam-6098	460	18	normed	norme	VERB
ejpam-6098	460	19	spaces	space	NOUN
ejpam-6098	460	20	is	be	AUX
ejpam-6098	460	21	left	leave	VERB
ejpam-6098	460	22	for	for	ADP
ejpam-6098	460	23	future	future	ADJ
ejpam-6098	460	24	work	work	NOUN
ejpam-6098	460	25	.	.	PUNCT
ejpam-6098	461	1	7	7	X
ejpam-6098	461	2	.	.	X
ejpam-6098	461	3	conclusions	conclusion	NOUN
ejpam-6098	461	4	the	the	DET
ejpam-6098	461	5	study	study	NOUN
ejpam-6098	461	6	of	of	ADP
ejpam-6098	461	7	stability	stability	NOUN
ejpam-6098	461	8	for	for	ADP
ejpam-6098	461	9	cubic	cubic	ADJ
ejpam-6098	461	10	functional	functional	ADJ
ejpam-6098	461	11	equations	equation	NOUN
ejpam-6098	461	12	has	have	AUX
ejpam-6098	461	13	been	be	AUX
ejpam-6098	461	14	a	a	DET
ejpam-6098	461	15	focal	focal	ADJ
ejpam-6098	461	16	point	point	NOUN
ejpam-6098	461	17	for	for	ADP
ejpam-6098	461	18	many	many	ADJ
ejpam-6098	461	19	mathematicians	mathematician	NOUN
ejpam-6098	461	20	,	,	PUNCT
ejpam-6098	461	21	with	with	ADP
ejpam-6098	461	22	significant	significant	ADJ
ejpam-6098	461	23	progress	progress	NOUN
ejpam-6098	461	24	achieved	achieve	VERB
ejpam-6098	461	25	across	across	ADP
ejpam-6098	461	26	various	various	ADJ
ejpam-6098	461	27	mathematical	mathematical	ADJ
ejpam-6098	461	28	spaces	space	NOUN
ejpam-6098	461	29	.	.	PUNCT
ejpam-6098	462	1	in	in	ADP
ejpam-6098	462	2	this	this	DET
ejpam-6098	462	3	work	work	NOUN
ejpam-6098	462	4	,	,	PUNCT
ejpam-6098	462	5	we	we	PRON
ejpam-6098	462	6	investigated	investigate	VERB
ejpam-6098	462	7	the	the	DET
ejpam-6098	462	8	stability	stability	NOUN
ejpam-6098	462	9	of	of	ADP
ejpam-6098	462	10	three	three	NUM
ejpam-6098	462	11	-	-	PUNCT
ejpam-6098	462	12	dimensional	dimensional	ADJ
ejpam-6098	462	13	cubic	cubic	ADJ
ejpam-6098	462	14	functional	functional	ADJ
ejpam-6098	462	15	equations	equation	NOUN
ejpam-6098	462	16	within	within	ADP
ejpam-6098	462	17	the	the	DET
ejpam-6098	462	18	frameworks	framework	NOUN
ejpam-6098	462	19	of	of	ADP
ejpam-6098	462	20	(	(	PUNCT
ejpam-6098	462	21	n	n	CCONJ
ejpam-6098	462	22	,	,	PUNCT
ejpam-6098	462	23	β)-normed	β)-normed	PUNCT
ejpam-6098	462	24	spaces	space	NOUN
ejpam-6098	462	25	,	,	PUNCT
ejpam-6098	462	26	non	non	ADJ
ejpam-6098	462	27	-	-	ADJ
ejpam-6098	462	28	archimedean	archimedean	ADJ
ejpam-6098	462	29	(	(	PUNCT
ejpam-6098	462	30	n	n	CCONJ
ejpam-6098	462	31	,	,	PUNCT
ejpam-6098	462	32	β)-normed	β)-normed	PUNCT
ejpam-6098	462	33	spaces	space	NOUN
ejpam-6098	462	34	,	,	PUNCT
ejpam-6098	462	35	and	and	CCONJ
ejpam-6098	462	36	random	random	ADJ
ejpam-6098	462	37	normed	normed	ADJ
ejpam-6098	462	38	spaces	space	NOUN
ejpam-6098	462	39	.	.	PUNCT
ejpam-6098	463	1	through	through	ADP
ejpam-6098	463	2	theoretical	theoretical	ADJ
ejpam-6098	463	3	analysis	analysis	NOUN
ejpam-6098	463	4	and	and	CCONJ
ejpam-6098	463	5	experimental	experimental	ADJ
ejpam-6098	463	6	validation	validation	NOUN
ejpam-6098	463	7	,	,	PUNCT
ejpam-6098	463	8	we	we	PRON
ejpam-6098	463	9	confirmed	confirm	VERB
ejpam-6098	463	10	the	the	DET
ejpam-6098	463	11	stability	stability	NOUN
ejpam-6098	463	12	properties	property	NOUN
ejpam-6098	463	13	of	of	ADP
ejpam-6098	463	14	these	these	DET
ejpam-6098	463	15	functional	functional	ADJ
ejpam-6098	463	16	equations	equation	NOUN
ejpam-6098	463	17	in	in	ADP
ejpam-6098	463	18	each	each	PRON
ejpam-6098	463	19	of	of	ADP
ejpam-6098	463	20	these	these	DET
ejpam-6098	463	21	distinct	distinct	ADJ
ejpam-6098	463	22	spaces	space	NOUN
ejpam-6098	463	23	.	.	PUNCT
ejpam-6098	464	1	furthermore	furthermore	ADV
ejpam-6098	464	2	,	,	PUNCT
ejpam-6098	464	3	a	a	DET
ejpam-6098	464	4	comparative	comparative	ADJ
ejpam-6098	464	5	analysis	analysis	NOUN
ejpam-6098	464	6	was	be	AUX
ejpam-6098	464	7	performed	perform	VERB
ejpam-6098	464	8	,	,	PUNCT
ejpam-6098	464	9	highlighting	highlight	VERB
ejpam-6098	464	10	both	both	DET
ejpam-6098	464	11	the	the	DET
ejpam-6098	464	12	similarities	similarity	NOUN
ejpam-6098	464	13	and	and	CCONJ
ejpam-6098	464	14	differences	difference	NOUN
ejpam-6098	464	15	in	in	ADP
ejpam-6098	464	16	stability	stability	NOUN
ejpam-6098	464	17	behavior	behavior	NOUN
ejpam-6098	464	18	across	across	ADP
ejpam-6098	464	19	these	these	DET
ejpam-6098	464	20	spaces	space	NOUN
ejpam-6098	464	21	.	.	PUNCT
ejpam-6098	465	1	the	the	DET
ejpam-6098	465	2	introduction	introduction	NOUN
ejpam-6098	465	3	of	of	ADP
ejpam-6098	465	4	(	(	PUNCT
ejpam-6098	465	5	n	n	CCONJ
ejpam-6098	465	6	,	,	PUNCT
ejpam-6098	465	7	β)-normed	β)-normed	PUNCT
ejpam-6098	465	8	and	and	CCONJ
ejpam-6098	465	9	non	non	ADJ
ejpam-6098	465	10	-	-	ADJ
ejpam-6098	465	11	archimedean	archimedean	ADJ
ejpam-6098	465	12	(	(	PUNCT
ejpam-6098	465	13	n	n	CCONJ
ejpam-6098	465	14	,	,	PUNCT
ejpam-6098	465	15	β)-normed	β)-normed	PUNCT
ejpam-6098	465	16	spaces	space	NOUN
ejpam-6098	465	17	offers	offer	VERB
ejpam-6098	465	18	a	a	DET
ejpam-6098	465	19	novel	novel	ADJ
ejpam-6098	465	20	perspective	perspective	NOUN
ejpam-6098	465	21	for	for	ADP
ejpam-6098	465	22	j.	j.	PROPN
ejpam-6098	465	23	jakhar	jakhar	PROPN
ejpam-6098	465	24	et	et	PROPN
ejpam-6098	465	25	al	al	PROPN
ejpam-6098	465	26	.	.	PUNCT
ejpam-6098	465	27	/	/	SYM
ejpam-6098	465	28	eur	eur	PROPN
ejpam-6098	465	29	.	.	PUNCT
ejpam-6098	466	1	j.	j.	PROPN
ejpam-6098	466	2	pure	pure	PROPN
ejpam-6098	466	3	appl	appl	PROPN
ejpam-6098	466	4	.	.	PROPN
ejpam-6098	466	5	math	math	PROPN
ejpam-6098	466	6	,	,	PUNCT
ejpam-6098	466	7	18	18	NUM
ejpam-6098	466	8	(	(	PUNCT
ejpam-6098	466	9	2	2	NUM
ejpam-6098	466	10	)	)	PUNCT
ejpam-6098	466	11	(	(	PUNCT
ejpam-6098	466	12	2025	2025	NUM
ejpam-6098	466	13	)	)	PUNCT
ejpam-6098	466	14	,	,	PUNCT
ejpam-6098	466	15	6098	6098	NUM
ejpam-6098	466	16	21	21	NUM
ejpam-6098	466	17	of	of	ADP
ejpam-6098	466	18	23	23	NUM
ejpam-6098	466	19	understanding	understand	VERB
ejpam-6098	466	20	stability	stability	NOUN
ejpam-6098	466	21	,	,	PUNCT
ejpam-6098	466	22	while	while	SCONJ
ejpam-6098	466	23	the	the	DET
ejpam-6098	466	24	incorporation	incorporation	NOUN
ejpam-6098	466	25	of	of	ADP
ejpam-6098	466	26	random	random	ADJ
ejpam-6098	466	27	normed	normed	ADJ
ejpam-6098	466	28	spaces	space	NOUN
ejpam-6098	466	29	adds	add	VERB
ejpam-6098	466	30	a	a	DET
ejpam-6098	466	31	stochastic	stochastic	ADJ
ejpam-6098	466	32	dimension	dimension	NOUN
ejpam-6098	466	33	,	,	PUNCT
ejpam-6098	466	34	providing	provide	VERB
ejpam-6098	466	35	a	a	DET
ejpam-6098	466	36	more	more	ADV
ejpam-6098	466	37	comprehensive	comprehensive	ADJ
ejpam-6098	466	38	understanding	understanding	NOUN
ejpam-6098	466	39	of	of	ADP
ejpam-6098	466	40	the	the	DET
ejpam-6098	466	41	stability	stability	NOUN
ejpam-6098	466	42	dynamics	dynamic	NOUN
ejpam-6098	466	43	.	.	PUNCT
ejpam-6098	467	1	this	this	DET
ejpam-6098	467	2	comparative	comparative	ADJ
ejpam-6098	467	3	evaluation	evaluation	NOUN
ejpam-6098	467	4	not	not	PART
ejpam-6098	467	5	only	only	ADV
ejpam-6098	467	6	deepens	deepen	VERB
ejpam-6098	467	7	the	the	DET
ejpam-6098	467	8	insights	insight	NOUN
ejpam-6098	467	9	into	into	ADP
ejpam-6098	467	10	stability	stability	NOUN
ejpam-6098	467	11	across	across	ADP
ejpam-6098	467	12	different	different	ADJ
ejpam-6098	467	13	normed	normed	ADJ
ejpam-6098	467	14	structures	structure	NOUN
ejpam-6098	467	15	but	but	CCONJ
ejpam-6098	467	16	also	also	ADV
ejpam-6098	467	17	bridges	bridge	VERB
ejpam-6098	467	18	deterministic	deterministic	ADJ
ejpam-6098	467	19	and	and	CCONJ
ejpam-6098	467	20	probabilistic	probabilistic	ADJ
ejpam-6098	467	21	approaches	approach	NOUN
ejpam-6098	467	22	in	in	ADP
ejpam-6098	467	23	functional	functional	ADJ
ejpam-6098	467	24	equation	equation	NOUN
ejpam-6098	467	25	theory	theory	NOUN
ejpam-6098	467	26	.	.	PUNCT
ejpam-6098	468	1	data	datum	NOUN
ejpam-6098	468	2	availability	availability	NOUN
ejpam-6098	468	3	statement	statement	NOUN
ejpam-6098	468	4	data	datum	NOUN
ejpam-6098	468	5	sharing	sharing	NOUN
ejpam-6098	468	6	is	be	AUX
ejpam-6098	468	7	not	not	PART
ejpam-6098	468	8	applicable	applicable	ADJ
ejpam-6098	468	9	to	to	ADP
ejpam-6098	468	10	this	this	DET
ejpam-6098	468	11	article	article	NOUN
ejpam-6098	468	12	as	as	SCONJ
ejpam-6098	468	13	no	no	DET
ejpam-6098	468	14	datasets	dataset	NOUN
ejpam-6098	468	15	were	be	AUX
ejpam-6098	468	16	generated	generate	VERB
ejpam-6098	468	17	or	or	CCONJ
ejpam-6098	468	18	analyzed	analyze	VERB
ejpam-6098	468	19	during	during	ADP
ejpam-6098	468	20	the	the	DET
ejpam-6098	468	21	current	current	ADJ
ejpam-6098	468	22	study	study	NOUN
ejpam-6098	468	23	.	.	PUNCT
ejpam-6098	469	1	conflicts	conflict	NOUN
ejpam-6098	469	2	of	of	ADP
ejpam-6098	469	3	interest	interest	NOUN
ejpam-6098	469	4	the	the	DET
ejpam-6098	469	5	authors	author	NOUN
ejpam-6098	469	6	declare	declare	VERB
ejpam-6098	469	7	no	no	DET
ejpam-6098	469	8	conflicts	conflict	NOUN
ejpam-6098	469	9	of	of	ADP
ejpam-6098	469	10	interest	interest	NOUN
ejpam-6098	469	11	.	.	PUNCT
ejpam-6098	470	1	author	author	NOUN
ejpam-6098	470	2	contributions	contribution	NOUN
ejpam-6098	470	3	all	all	DET
ejpam-6098	470	4	authors	author	NOUN
ejpam-6098	470	5	have	have	VERB
ejpam-6098	470	6	equal	equal	ADJ
ejpam-6098	470	7	contributions	contribution	NOUN
ejpam-6098	470	8	.	.	PUNCT
ejpam-6098	471	1	all	all	DET
ejpam-6098	471	2	authors	author	NOUN
ejpam-6098	471	3	read	read	VERB
ejpam-6098	471	4	and	and	CCONJ
ejpam-6098	471	5	approved	approve	VERB
ejpam-6098	471	6	the	the	DET
ejpam-6098	471	7	final	final	ADJ
ejpam-6098	471	8	manuscript	manuscript	NOUN
ejpam-6098	471	9	.	.	PUNCT
ejpam-6098	472	1	funding	fund	VERB
ejpam-6098	472	2	this	this	DET
ejpam-6098	472	3	research	research	NOUN
ejpam-6098	472	4	study	study	NOUN
ejpam-6098	472	5	received	receive	VERB
ejpam-6098	472	6	no	no	DET
ejpam-6098	472	7	external	external	ADJ
ejpam-6098	472	8	funding	funding	NOUN
ejpam-6098	472	9	.	.	PUNCT
ejpam-6098	473	1	acknowledgements	acknowledgement	NOUN
ejpam-6098	473	2	researchers	researcher	NOUN
ejpam-6098	473	3	supporting	support	VERB
ejpam-6098	473	4	project	project	NOUN
ejpam-6098	473	5	number	number	NOUN
ejpam-6098	473	6	(	(	PUNCT
ejpam-6098	473	7	rsp2025r153	rsp2025r153	PROPN
ejpam-6098	473	8	)	)	PUNCT
ejpam-6098	473	9	,	,	PUNCT
ejpam-6098	473	10	king	king	PROPN
ejpam-6098	473	11	saud	saud	PROPN
ejpam-6098	473	12	university	university	PROPN
ejpam-6098	473	13	,	,	PUNCT
ejpam-6098	473	14	riyadh	riyadh	PROPN
ejpam-6098	473	15	,	,	PUNCT
ejpam-6098	473	16	saudi	saudi	PROPN
ejpam-6098	473	17	arabia	arabia	PROPN
ejpam-6098	473	18	.	.	PUNCT
ejpam-6098	474	1	references	reference	NOUN
ejpam-6098	474	2	[	[	X
ejpam-6098	474	3	1	1	X
ejpam-6098	474	4	]	]	PUNCT
ejpam-6098	474	5	s.	s.	PROPN
ejpam-6098	474	6	m.	m.	PROPN
ejpam-6098	474	7	ulam	ulam	PROPN
ejpam-6098	474	8	.	.	PUNCT
ejpam-6098	475	1	a	a	DET
ejpam-6098	475	2	collection	collection	NOUN
ejpam-6098	475	3	of	of	ADP
ejpam-6098	475	4	the	the	DET
ejpam-6098	475	5	mathematical	mathematical	ADJ
ejpam-6098	475	6	problems	problem	NOUN
ejpam-6098	475	7	.	.	PUNCT
ejpam-6098	476	1	interscience	interscience	NOUN
ejpam-6098	476	2	,	,	PUNCT
ejpam-6098	476	3	new	new	PROPN
ejpam-6098	476	4	york	york	PROPN
ejpam-6098	476	5	,	,	PUNCT
ejpam-6098	476	6	1960	1960	NUM
ejpam-6098	476	7	.	.	PUNCT
ejpam-6098	477	1	[	[	X
ejpam-6098	477	2	2	2	X
ejpam-6098	477	3	]	]	PUNCT
ejpam-6098	477	4	d.	d.	PROPN
ejpam-6098	477	5	h.	h.	PROPN
ejpam-6098	477	6	hyers	hyers	PROPN
ejpam-6098	477	7	.	.	PUNCT
ejpam-6098	478	1	on	on	ADP
ejpam-6098	478	2	the	the	DET
ejpam-6098	478	3	stability	stability	NOUN
ejpam-6098	478	4	of	of	ADP
ejpam-6098	478	5	the	the	DET
ejpam-6098	478	6	linear	linear	ADJ
ejpam-6098	478	7	functional	functional	ADJ
ejpam-6098	478	8	equation	equation	NOUN
ejpam-6098	478	9	.	.	PUNCT
ejpam-6098	479	1	proc	proc	PROPN
ejpam-6098	479	2	.	.	PUNCT
ejpam-6098	480	1	natl	natl	PROPN
ejpam-6098	480	2	.	.	PUNCT
ejpam-6098	481	1	acad	acad	PROPN
ejpam-6098	481	2	.	.	PUNCT
ejpam-6098	482	1	sci	sci	PROPN
ejpam-6098	482	2	.	.	PROPN
ejpam-6098	482	3	usa	usa	PROPN
ejpam-6098	482	4	,	,	PUNCT
ejpam-6098	482	5	27:222–224	27:222–224	NUM
ejpam-6098	482	6	,	,	PUNCT
ejpam-6098	482	7	1941	1941	NUM
ejpam-6098	482	8	.	.	PUNCT
ejpam-6098	483	1	[	[	X
ejpam-6098	483	2	3	3	X
ejpam-6098	483	3	]	]	X
ejpam-6098	483	4	t.	t.	PROPN
ejpam-6098	483	5	m.	m.	NOUN
ejpam-6098	483	6	rassias	rassias	PROPN
ejpam-6098	483	7	.	.	PUNCT
ejpam-6098	484	1	on	on	ADP
ejpam-6098	484	2	the	the	DET
ejpam-6098	484	3	stability	stability	NOUN
ejpam-6098	484	4	of	of	ADP
ejpam-6098	484	5	the	the	DET
ejpam-6098	484	6	linear	linear	ADJ
ejpam-6098	484	7	mapping	mapping	NOUN
ejpam-6098	484	8	in	in	ADP
ejpam-6098	484	9	banach	banach	NOUN
ejpam-6098	484	10	spaces	space	NOUN
ejpam-6098	484	11	.	.	PUNCT
ejpam-6098	485	1	proc	proc	NOUN
ejpam-6098	485	2	.	.	PUNCT
ejpam-6098	486	1	am	be	AUX
ejpam-6098	486	2	.	.	PUNCT
ejpam-6098	487	1	math	math	NOUN
ejpam-6098	487	2	.	.	PUNCT
ejpam-6098	488	1	soc	soc	PROPN
ejpam-6098	488	2	.	.	PUNCT
ejpam-6098	488	3	,	,	PUNCT
ejpam-6098	488	4	72:297–300	72:297–300	PROPN
ejpam-6098	488	5	,	,	PUNCT
ejpam-6098	488	6	1978	1978	NUM
ejpam-6098	488	7	.	.	PUNCT
ejpam-6098	489	1	[	[	X
ejpam-6098	489	2	4	4	X
ejpam-6098	489	3	]	]	PUNCT
ejpam-6098	489	4	j.	j.	PROPN
ejpam-6098	489	5	jakhar	jakhar	PROPN
ejpam-6098	489	6	,	,	PUNCT
ejpam-6098	489	7	r.	r.	PROPN
ejpam-6098	489	8	chugh	chugh	NOUN
ejpam-6098	489	9	,	,	PUNCT
ejpam-6098	489	10	and	and	CCONJ
ejpam-6098	489	11	j.	j.	PROPN
ejpam-6098	489	12	jakhar	jakhar	PROPN
ejpam-6098	489	13	.	.	PUNCT
ejpam-6098	490	1	solution	solution	NOUN
ejpam-6098	490	2	and	and	CCONJ
ejpam-6098	490	3	intuitionistic	intuitionistic	ADJ
ejpam-6098	490	4	fuzzy	fuzzy	ADJ
ejpam-6098	490	5	stability	stability	NOUN
ejpam-6098	490	6	of	of	ADP
ejpam-6098	490	7	3dimensional	3dimensional	NUM
ejpam-6098	490	8	cubic	cubic	ADJ
ejpam-6098	490	9	functional	functional	ADJ
ejpam-6098	490	10	equation	equation	NOUN
ejpam-6098	490	11	:	:	PUNCT
ejpam-6098	490	12	using	use	VERB
ejpam-6098	490	13	two	two	NUM
ejpam-6098	490	14	different	different	ADJ
ejpam-6098	490	15	methods	method	NOUN
ejpam-6098	490	16	.	.	PUNCT
ejpam-6098	491	1	j.	j.	PROPN
ejpam-6098	491	2	math	math	PROPN
ejpam-6098	491	3	.	.	PUNCT
ejpam-6098	492	1	comput	comput	NOUN
ejpam-6098	492	2	.	.	PUNCT
ejpam-6098	493	1	sci	sci	PROPN
ejpam-6098	493	2	.	.	PROPN
ejpam-6098	493	3	,	,	PUNCT
ejpam-6098	493	4	25:103–114	25:103–114	PROPN
ejpam-6098	493	5	,	,	PUNCT
ejpam-6098	493	6	2022	2022	NUM
ejpam-6098	493	7	.	.	PUNCT
ejpam-6098	494	1	[	[	X
ejpam-6098	494	2	5	5	X
ejpam-6098	494	3	]	]	PUNCT
ejpam-6098	494	4	j.	j.	PROPN
ejpam-6098	494	5	jakhar	jakhar	PROPN
ejpam-6098	494	6	,	,	PUNCT
ejpam-6098	494	7	r.	r.	PROPN
ejpam-6098	494	8	chugh	chugh	NOUN
ejpam-6098	494	9	,	,	PUNCT
ejpam-6098	494	10	and	and	CCONJ
ejpam-6098	494	11	j.	j.	PROPN
ejpam-6098	494	12	jakhar	jakhar	PROPN
ejpam-6098	494	13	.	.	PUNCT
ejpam-6098	495	1	stability	stability	NOUN
ejpam-6098	495	2	of	of	ADP
ejpam-6098	495	3	various	various	ADJ
ejpam-6098	495	4	iterative	iterative	NOUN
ejpam-6098	495	5	type	type	NOUN
ejpam-6098	495	6	functional	functional	ADJ
ejpam-6098	495	7	equation	equation	NOUN
ejpam-6098	495	8	in	in	ADP
ejpam-6098	495	9	menger-ϕ	menger-ϕ	NOUN
ejpam-6098	495	10	normed	normed	ADJ
ejpam-6098	495	11	space	space	NOUN
ejpam-6098	495	12	.	.	PUNCT
ejpam-6098	496	1	bull	bull	NOUN
ejpam-6098	496	2	.	.	PUNCT
ejpam-6098	497	1	math	math	NOUN
ejpam-6098	497	2	.	.	PUNCT
ejpam-6098	498	1	anal	anal	PROPN
ejpam-6098	498	2	.	.	PUNCT
ejpam-6098	498	3	appl	appl	PROPN
ejpam-6098	498	4	.	.	PROPN
ejpam-6098	498	5	,	,	PUNCT
ejpam-6098	498	6	13:106–120	13:106–120	NUM
ejpam-6098	498	7	,	,	PUNCT
ejpam-6098	498	8	2021	2021	NUM
ejpam-6098	498	9	.	.	PUNCT
ejpam-6098	499	1	[	[	X
ejpam-6098	499	2	6	6	NUM
ejpam-6098	499	3	]	]	PUNCT
ejpam-6098	499	4	j.	j.	PROPN
ejpam-6098	499	5	jakhar	jakhar	PROPN
ejpam-6098	499	6	,	,	PUNCT
ejpam-6098	499	7	r.	r.	PROPN
ejpam-6098	499	8	chugh	chugh	NOUN
ejpam-6098	499	9	,	,	PUNCT
ejpam-6098	499	10	and	and	CCONJ
ejpam-6098	499	11	j.	j.	PROPN
ejpam-6098	499	12	jakhar	jakhar	PROPN
ejpam-6098	499	13	.	.	PUNCT
ejpam-6098	500	1	fuzzy	fuzzy	ADJ
ejpam-6098	500	2	stability	stability	NOUN
ejpam-6098	500	3	of	of	ADP
ejpam-6098	500	4	mixed	mixed	ADJ
ejpam-6098	500	5	type	type	NOUN
ejpam-6098	500	6	functional	functional	ADJ
ejpam-6098	500	7	equations	equation	NOUN
ejpam-6098	500	8	in	in	ADP
ejpam-6098	500	9	modular	modular	ADJ
ejpam-6098	500	10	spaces	space	NOUN
ejpam-6098	500	11	.	.	PUNCT
ejpam-6098	501	1	math	math	NOUN
ejpam-6098	501	2	.	.	PUNCT
ejpam-6098	501	3	found	find	VERB
ejpam-6098	501	4	.	.	PUNCT
ejpam-6098	502	1	comput	comput	NOUN
ejpam-6098	502	2	.	.	PUNCT
ejpam-6098	502	3	,	,	PUNCT
ejpam-6098	502	4	2023	2023	NUM
ejpam-6098	502	5	.	.	PUNCT
ejpam-6098	503	1	j.	j.	PROPN
ejpam-6098	503	2	jakhar	jakhar	PROPN
ejpam-6098	503	3	et	et	PROPN
ejpam-6098	503	4	al	al	PROPN
ejpam-6098	503	5	.	.	PUNCT
ejpam-6098	503	6	/	/	SYM
ejpam-6098	503	7	eur	eur	PROPN
ejpam-6098	503	8	.	.	PUNCT
ejpam-6098	504	1	j.	j.	PROPN
ejpam-6098	504	2	pure	pure	PROPN
ejpam-6098	504	3	appl	appl	PROPN
ejpam-6098	504	4	.	.	PROPN
ejpam-6098	504	5	math	math	PROPN
ejpam-6098	504	6	,	,	PUNCT
ejpam-6098	504	7	18	18	NUM
ejpam-6098	504	8	(	(	PUNCT
ejpam-6098	504	9	2	2	NUM
ejpam-6098	504	10	)	)	PUNCT
ejpam-6098	504	11	(	(	PUNCT
ejpam-6098	504	12	2025	2025	NUM
ejpam-6098	504	13	)	)	PUNCT
ejpam-6098	504	14	,	,	PUNCT
ejpam-6098	504	15	6098	6098	NUM
ejpam-6098	504	16	22	22	NUM
ejpam-6098	504	17	of	of	ADP
ejpam-6098	504	18	23	23	NUM
ejpam-6098	505	1	[	[	X
ejpam-6098	505	2	7	7	NUM
ejpam-6098	505	3	]	]	X
ejpam-6098	505	4	d.	d.	PROPN
ejpam-6098	505	5	mihet	mihet	PROPN
ejpam-6098	505	6	,	,	PUNCT
ejpam-6098	505	7	r.	r.	PROPN
ejpam-6098	505	8	saadati	saadati	PROPN
ejpam-6098	505	9	,	,	PUNCT
ejpam-6098	505	10	and	and	CCONJ
ejpam-6098	505	11	s.	s.	PROPN
ejpam-6098	505	12	m.	m.	PROPN
ejpam-6098	505	13	vaezpour	vaezpour	NOUN
ejpam-6098	505	14	.	.	PUNCT
ejpam-6098	506	1	the	the	DET
ejpam-6098	506	2	stability	stability	NOUN
ejpam-6098	506	3	of	of	ADP
ejpam-6098	506	4	the	the	DET
ejpam-6098	506	5	quartic	quartic	ADJ
ejpam-6098	506	6	functional	functional	ADJ
ejpam-6098	506	7	equation	equation	NOUN
ejpam-6098	506	8	in	in	ADP
ejpam-6098	506	9	random	random	ADJ
ejpam-6098	506	10	normed	normed	ADJ
ejpam-6098	506	11	spaces	space	NOUN
ejpam-6098	506	12	.	.	PUNCT
ejpam-6098	507	1	acta	acta	PROPN
ejpam-6098	507	2	appl	appl	PROPN
ejpam-6098	507	3	.	.	PROPN
ejpam-6098	507	4	math	math	PROPN
ejpam-6098	507	5	.	.	PUNCT
ejpam-6098	507	6	,	,	PUNCT
ejpam-6098	507	7	110:797–803	110:797–803	NUM
ejpam-6098	507	8	,	,	PUNCT
ejpam-6098	507	9	2010	2010	NUM
ejpam-6098	507	10	.	.	PUNCT
ejpam-6098	508	1	[	[	X
ejpam-6098	508	2	8	8	NUM
ejpam-6098	508	3	]	]	PUNCT
ejpam-6098	508	4	m.	m.	NOUN
ejpam-6098	508	5	vivas	vivas	PROPN
ejpam-6098	508	6	-	-	NOUN
ejpam-6098	508	7	cortez	cortez	PROPN
ejpam-6098	508	8	,	,	PUNCT
ejpam-6098	508	9	m.	m.	NOUN
ejpam-6098	508	10	a.	a.	PROPN
ejpam-6098	508	11	yousif	yousif	PROPN
ejpam-6098	508	12	,	,	PUNCT
ejpam-6098	508	13	b.	b.	PROPN
ejpam-6098	508	14	a.	a.	PROPN
ejpam-6098	508	15	mahmood	mahmood	PROPN
ejpam-6098	508	16	,	,	PUNCT
ejpam-6098	508	17	p.	p.	PROPN
ejpam-6098	508	18	o.	o.	PROPN
ejpam-6098	508	19	mohammed	mohammed	PROPN
ejpam-6098	508	20	,	,	PUNCT
ejpam-6098	508	21	n.	n.	PROPN
ejpam-6098	508	22	chorfi	chorfi	PROPN
ejpam-6098	508	23	,	,	PUNCT
ejpam-6098	508	24	and	and	CCONJ
ejpam-6098	508	25	a.	a.	NOUN
ejpam-6098	508	26	a.	a.	NOUN
ejpam-6098	508	27	lupas	lupas	PROPN
ejpam-6098	508	28	.	.	PUNCT
ejpam-6098	509	1	high	high	ADJ
ejpam-6098	509	2	-	-	PUNCT
ejpam-6098	509	3	accuracy	accuracy	NOUN
ejpam-6098	509	4	solutions	solution	NOUN
ejpam-6098	509	5	to	to	ADP
ejpam-6098	509	6	the	the	DET
ejpam-6098	509	7	time	time	NOUN
ejpam-6098	509	8	-	-	PUNCT
ejpam-6098	509	9	fractional	fractional	ADJ
ejpam-6098	509	10	kdv	kdv	NOUN
ejpam-6098	509	11	–	–	PUNCT
ejpam-6098	509	12	burgers	burger	NOUN
ejpam-6098	509	13	equation	equation	NOUN
ejpam-6098	509	14	using	use	VERB
ejpam-6098	509	15	rational	rational	ADJ
ejpam-6098	509	16	non	non	ADJ
ejpam-6098	509	17	-	-	ADJ
ejpam-6098	509	18	polynomial	polynomial	ADJ
ejpam-6098	509	19	splines	spline	NOUN
ejpam-6098	509	20	.	.	PUNCT
ejpam-6098	510	1	symmetry	symmetry	NOUN
ejpam-6098	510	2	,	,	PUNCT
ejpam-6098	510	3	17(1):16	17(1):16	NUM
ejpam-6098	510	4	,	,	PUNCT
ejpam-6098	510	5	2025	2025	NUM
ejpam-6098	510	6	.	.	PUNCT
ejpam-6098	511	1	[	[	X
ejpam-6098	511	2	9	9	NUM
ejpam-6098	511	3	]	]	PUNCT
ejpam-6098	511	4	m.	m.	NOUN
ejpam-6098	511	5	vivas	vivas	PROPN
ejpam-6098	511	6	-	-	NOUN
ejpam-6098	511	7	cortez	cortez	PROPN
ejpam-6098	511	8	,	,	PUNCT
ejpam-6098	511	9	m.	m.	NOUN
ejpam-6098	511	10	a.	a.	PROPN
ejpam-6098	511	11	yousif	yousif	PROPN
ejpam-6098	511	12	,	,	PUNCT
ejpam-6098	511	13	p.	p.	PROPN
ejpam-6098	511	14	o.	o.	PROPN
ejpam-6098	511	15	mohammed	mohammed	PROPN
ejpam-6098	511	16	,	,	PUNCT
ejpam-6098	511	17	a.	a.	NOUN
ejpam-6098	511	18	a.	a.	NOUN
ejpam-6098	511	19	lupas	lupas	PROPN
ejpam-6098	511	20	,	,	PUNCT
ejpam-6098	511	21	i.	i.	PROPN
ejpam-6098	511	22	s.	s.	PROPN
ejpam-6098	511	23	ibrahim	ibrahim	PROPN
ejpam-6098	511	24	,	,	PUNCT
ejpam-6098	511	25	and	and	CCONJ
ejpam-6098	511	26	n.	n.	PROPN
ejpam-6098	511	27	chorfi	chorfi	PROPN
ejpam-6098	511	28	.	.	PUNCT
ejpam-6098	512	1	hyperbolic	hyperbolic	ADJ
ejpam-6098	512	2	non	non	ADJ
ejpam-6098	512	3	-	-	ADJ
ejpam-6098	512	4	polynomial	polynomial	ADJ
ejpam-6098	512	5	spline	spline	NOUN
ejpam-6098	512	6	approach	approach	NOUN
ejpam-6098	512	7	for	for	ADP
ejpam-6098	512	8	time	time	NOUN
ejpam-6098	512	9	-	-	PUNCT
ejpam-6098	512	10	fractional	fractional	ADJ
ejpam-6098	512	11	coupled	couple	VERB
ejpam-6098	512	12	kdv	kdv	NOUN
ejpam-6098	512	13	equations	equation	NOUN
ejpam-6098	512	14	:	:	PUNCT
ejpam-6098	512	15	a	a	DET
ejpam-6098	512	16	computational	computational	ADJ
ejpam-6098	512	17	investigation	investigation	NOUN
ejpam-6098	512	18	.	.	PUNCT
ejpam-6098	513	1	symmetry	symmetry	NOUN
ejpam-6098	513	2	,	,	PUNCT
ejpam-6098	513	3	16(12):1610	16(12):1610	NUM
ejpam-6098	513	4	,	,	PUNCT
ejpam-6098	513	5	2024	2024	NUM
ejpam-6098	513	6	.	.	PUNCT
ejpam-6098	514	1	[	[	X
ejpam-6098	514	2	10	10	NUM
ejpam-6098	514	3	]	]	X
ejpam-6098	514	4	a.	a.	PROPN
ejpam-6098	514	5	k.	k.	PROPN
ejpam-6098	514	6	mirmostafaee	mirmostafaee	PROPN
ejpam-6098	514	7	and	and	CCONJ
ejpam-6098	514	8	m.	m.	PROPN
ejpam-6098	514	9	s.	s.	PROPN
ejpam-6098	514	10	moslehian	moslehian	PROPN
ejpam-6098	514	11	.	.	PUNCT
ejpam-6098	515	1	fuzzy	fuzzy	ADJ
ejpam-6098	515	2	approximately	approximately	ADV
ejpam-6098	515	3	cubic	cubic	ADJ
ejpam-6098	515	4	mappings	mapping	NOUN
ejpam-6098	515	5	.	.	PUNCT
ejpam-6098	516	1	inf	inf	PROPN
ejpam-6098	516	2	.	.	PUNCT
ejpam-6098	517	1	sci	sci	PROPN
ejpam-6098	517	2	.	.	PROPN
ejpam-6098	517	3	,	,	PUNCT
ejpam-6098	517	4	178:3791–3798	178:3791–3798	NUM
ejpam-6098	517	5	,	,	PUNCT
ejpam-6098	517	6	2008	2008	NUM
ejpam-6098	517	7	.	.	PUNCT
ejpam-6098	518	1	[	[	X
ejpam-6098	518	2	11	11	NUM
ejpam-6098	518	3	]	]	X
ejpam-6098	518	4	w.	w.	PROPN
ejpam-6098	518	5	g.	g.	PROPN
ejpam-6098	518	6	park	park	PROPN
ejpam-6098	518	7	.	.	PUNCT
ejpam-6098	519	1	approximate	approximate	ADJ
ejpam-6098	519	2	additive	additive	ADJ
ejpam-6098	519	3	mappings	mapping	NOUN
ejpam-6098	519	4	in	in	ADP
ejpam-6098	519	5	2	2	NUM
ejpam-6098	519	6	-	-	PUNCT
ejpam-6098	519	7	banach	banach	NOUN
ejpam-6098	519	8	spaces	space	NOUN
ejpam-6098	519	9	and	and	CCONJ
ejpam-6098	519	10	related	related	ADJ
ejpam-6098	519	11	topics	topic	NOUN
ejpam-6098	519	12	.	.	PUNCT
ejpam-6098	520	1	j.	j.	PROPN
ejpam-6098	520	2	math	math	PROPN
ejpam-6098	520	3	.	.	PUNCT
ejpam-6098	521	1	anal	anal	PROPN
ejpam-6098	521	2	.	.	PUNCT
ejpam-6098	522	1	appl	appl	PROPN
ejpam-6098	522	2	.	.	PROPN
ejpam-6098	522	3	,	,	PUNCT
ejpam-6098	522	4	376:193–202	376:193–202	NUM
ejpam-6098	522	5	,	,	PUNCT
ejpam-6098	522	6	2011	2011	NUM
ejpam-6098	522	7	.	.	PUNCT
ejpam-6098	523	1	[	[	X
ejpam-6098	523	2	12	12	NUM
ejpam-6098	523	3	]	]	X
ejpam-6098	523	4	s.	s.	PROPN
ejpam-6098	523	5	gähler	gähler	PROPN
ejpam-6098	523	6	.	.	PUNCT
ejpam-6098	524	1	lineare	lineare	ADJ
ejpam-6098	524	2	2	2	NUM
ejpam-6098	524	3	-	-	PUNCT
ejpam-6098	524	4	normierte	normierte	NOUN
ejpam-6098	524	5	räume	räume	PROPN
ejpam-6098	524	6	.	.	PUNCT
ejpam-6098	524	7	math	math	NOUN
ejpam-6098	524	8	.	.	PUNCT
ejpam-6098	525	1	nachr	nachr	PROPN
ejpam-6098	525	2	.	.	PROPN
ejpam-6098	525	3	,	,	PUNCT
ejpam-6098	525	4	28:1–43	28:1–43	NUM
ejpam-6098	525	5	,	,	PUNCT
ejpam-6098	525	6	1964	1964	NUM
ejpam-6098	525	7	.	.	PUNCT
ejpam-6098	526	1	[	[	X
ejpam-6098	526	2	13	13	NUM
ejpam-6098	526	3	]	]	PUNCT
ejpam-6098	526	4	s.	s.	PROPN
ejpam-6098	526	5	gähler	gähler	PROPN
ejpam-6098	526	6	.	.	PUNCT
ejpam-6098	527	1	untersuchungen	untersuchungen	PROPN
ejpam-6098	527	2	über	über	PROPN
ejpam-6098	527	3	verallgemeinerte	verallgemeinerte	PROPN
ejpam-6098	527	4	m	m	PROPN
ejpam-6098	527	5	-	-	PUNCT
ejpam-6098	527	6	metrische	metrische	NOUN
ejpam-6098	527	7	räume	räume	PROPN
ejpam-6098	527	8	.	.	PUNCT
ejpam-6098	527	9	i.	i.	PROPN
ejpam-6098	527	10	math	math	PROPN
ejpam-6098	527	11	.	.	PUNCT
ejpam-6098	528	1	nachr	nachr	PROPN
ejpam-6098	528	2	.	.	PUNCT
ejpam-6098	528	3	,	,	PUNCT
ejpam-6098	528	4	40:165–189	40:165–189	NUM
ejpam-6098	528	5	,	,	PUNCT
ejpam-6098	528	6	1969	1969	NUM
ejpam-6098	528	7	.	.	PUNCT
ejpam-6098	529	1	[	[	X
ejpam-6098	529	2	14	14	NUM
ejpam-6098	529	3	]	]	X
ejpam-6098	529	4	h.	h.	PROPN
ejpam-6098	529	5	gunawan	gunawan	PROPN
ejpam-6098	529	6	and	and	CCONJ
ejpam-6098	529	7	m.	m.	NOUN
ejpam-6098	529	8	mashadi	mashadi	NOUN
ejpam-6098	529	9	.	.	PUNCT
ejpam-6098	530	1	on	on	ADP
ejpam-6098	530	2	n	n	ADV
ejpam-6098	530	3	-	-	PUNCT
ejpam-6098	530	4	normed	norme	VERB
ejpam-6098	530	5	spaces	space	NOUN
ejpam-6098	530	6	.	.	PUNCT
ejpam-6098	531	1	int	int	NOUN
ejpam-6098	531	2	.	.	PUNCT
ejpam-6098	532	1	j.	j.	PROPN
ejpam-6098	532	2	math	math	PROPN
ejpam-6098	532	3	.	.	PUNCT
ejpam-6098	533	1	math	math	NOUN
ejpam-6098	533	2	.	.	PUNCT
ejpam-6098	534	1	sci	sci	PROPN
ejpam-6098	534	2	.	.	PROPN
ejpam-6098	534	3	,	,	PUNCT
ejpam-6098	534	4	27:631	27:631	NUM
ejpam-6098	534	5	–	–	PUNCT
ejpam-6098	534	6	639	639	NUM
ejpam-6098	534	7	,	,	PUNCT
ejpam-6098	534	8	2001	2001	NUM
ejpam-6098	534	9	.	.	PUNCT
ejpam-6098	535	1	[	[	X
ejpam-6098	535	2	15	15	NUM
ejpam-6098	535	3	]	]	X
ejpam-6098	535	4	r.	r.	PROPN
ejpam-6098	535	5	malčeski	malčeski	PROPN
ejpam-6098	535	6	.	.	PUNCT
ejpam-6098	536	1	strong	strong	ADJ
ejpam-6098	536	2	n	n	CCONJ
ejpam-6098	536	3	-	-	PUNCT
ejpam-6098	536	4	convex	convex	NOUN
ejpam-6098	536	5	n	n	ADV
ejpam-6098	536	6	-	-	PUNCT
ejpam-6098	536	7	normed	norme	VERB
ejpam-6098	536	8	spaces	space	NOUN
ejpam-6098	536	9	.	.	PUNCT
ejpam-6098	537	1	math	math	NOUN
ejpam-6098	537	2	.	.	PUNCT
ejpam-6098	538	1	bilt	bilt	PROPN
ejpam-6098	538	2	.	.	PROPN
ejpam-6098	538	3	,	,	PUNCT
ejpam-6098	538	4	21:81–102	21:81–102	NUM
ejpam-6098	538	5	,	,	PUNCT
ejpam-6098	538	6	1997	1997	NUM
ejpam-6098	538	7	.	.	PUNCT
ejpam-6098	539	1	[	[	X
ejpam-6098	539	2	16	16	NUM
ejpam-6098	539	3	]	]	PUNCT
ejpam-6098	539	4	s.	s.	PROPN
ejpam-6098	539	5	s.	s.	PROPN
ejpam-6098	539	6	kim	kim	PROPN
ejpam-6098	539	7	and	and	CCONJ
ejpam-6098	539	8	y.	y.	PROPN
ejpam-6098	539	9	j.	j.	PROPN
ejpam-6098	539	10	cho	cho	PROPN
ejpam-6098	539	11	.	.	PUNCT
ejpam-6098	540	1	strict	strict	ADJ
ejpam-6098	540	2	convexity	convexity	NOUN
ejpam-6098	540	3	in	in	ADP
ejpam-6098	540	4	linear	linear	PROPN
ejpam-6098	540	5	n	n	CCONJ
ejpam-6098	540	6	-	-	PUNCT
ejpam-6098	540	7	normed	norme	VERB
ejpam-6098	540	8	spaces	space	NOUN
ejpam-6098	540	9	.	.	PUNCT
ejpam-6098	541	1	demonstr	demonstr	PROPN
ejpam-6098	541	2	.	.	PUNCT
ejpam-6098	542	1	math	math	NOUN
ejpam-6098	542	2	.	.	PUNCT
ejpam-6098	542	3	,	,	PUNCT
ejpam-6098	543	1	29:739–744	29:739–744	NUM
ejpam-6098	543	2	,	,	PUNCT
ejpam-6098	543	3	1996	1996	NUM
ejpam-6098	543	4	.	.	PUNCT
ejpam-6098	544	1	[	[	X
ejpam-6098	544	2	17	17	NUM
ejpam-6098	544	3	]	]	PUNCT
ejpam-6098	544	4	a.	a.	NOUN
ejpam-6098	544	5	misiak	misiak	PROPN
ejpam-6098	544	6	.	.	PUNCT
ejpam-6098	545	1	n	n	CCONJ
ejpam-6098	545	2	-	-	PUNCT
ejpam-6098	545	3	inner	inner	ADJ
ejpam-6098	545	4	product	product	NOUN
ejpam-6098	545	5	spaces	space	VERB
ejpam-6098	545	6	.	.	PUNCT
ejpam-6098	546	1	math	math	NOUN
ejpam-6098	546	2	.	.	PUNCT
ejpam-6098	547	1	nachr	nachr	PROPN
ejpam-6098	547	2	.	.	PUNCT
ejpam-6098	547	3	,	,	PUNCT
ejpam-6098	547	4	140:299–319	140:299–319	NUM
ejpam-6098	547	5	,	,	PUNCT
ejpam-6098	547	6	1989	1989	NUM
ejpam-6098	547	7	.	.	PUNCT
ejpam-6098	548	1	[	[	X
ejpam-6098	548	2	18	18	NUM
ejpam-6098	548	3	]	]	PUNCT
ejpam-6098	548	4	t.	t.	PROPN
ejpam-6098	548	5	z.	z.	PROPN
ejpam-6098	548	6	xu	xu	PROPN
ejpam-6098	548	7	and	and	CCONJ
ejpam-6098	548	8	j.	j.	PROPN
ejpam-6098	548	9	m.	m.	PROPN
ejpam-6098	548	10	rassias	rassias	PROPN
ejpam-6098	548	11	.	.	PUNCT
ejpam-6098	549	1	on	on	ADP
ejpam-6098	549	2	the	the	DET
ejpam-6098	549	3	hyers	hyers	PROPN
ejpam-6098	549	4	-	-	PUNCT
ejpam-6098	549	5	ulam	ulam	ADJ
ejpam-6098	549	6	stability	stability	NOUN
ejpam-6098	549	7	of	of	ADP
ejpam-6098	549	8	a	a	DET
ejpam-6098	549	9	general	general	ADJ
ejpam-6098	549	10	mixed	mixed	ADJ
ejpam-6098	549	11	additive	additive	NOUN
ejpam-6098	549	12	and	and	CCONJ
ejpam-6098	549	13	cubic	cubic	ADJ
ejpam-6098	549	14	functional	functional	ADJ
ejpam-6098	549	15	equation	equation	NOUN
ejpam-6098	549	16	in	in	ADP
ejpam-6098	549	17	n	n	CCONJ
ejpam-6098	549	18	-	-	PUNCT
ejpam-6098	549	19	banach	banach	NOUN
ejpam-6098	549	20	spaces	space	NOUN
ejpam-6098	549	21	.	.	PUNCT
ejpam-6098	550	1	abstr	abstr	PROPN
ejpam-6098	550	2	.	.	PUNCT
ejpam-6098	550	3	appl	appl	PROPN
ejpam-6098	550	4	.	.	PUNCT
ejpam-6098	551	1	anal	anal	PROPN
ejpam-6098	551	2	.	.	PROPN
ejpam-6098	551	3	,	,	PUNCT
ejpam-6098	551	4	2012	2012	NUM
ejpam-6098	551	5	:	:	PUNCT
ejpam-6098	551	6	article	article	NOUN
ejpam-6098	551	7	i	i	PROPN
ejpam-6098	551	8	d	d	PROPN
ejpam-6098	551	9	926390	926390	NUM
ejpam-6098	551	10	,	,	PUNCT
ejpam-6098	551	11	2012	2012	NUM
ejpam-6098	551	12	.	.	PUNCT
ejpam-6098	552	1	[	[	X
ejpam-6098	552	2	19	19	NUM
ejpam-6098	552	3	]	]	PUNCT
ejpam-6098	552	4	x.	x.	PROPN
ejpam-6098	552	5	yang	yang	PROPN
ejpam-6098	552	6	,	,	PUNCT
ejpam-6098	552	7	l.	l.	PROPN
ejpam-6098	552	8	chang	chang	PROPN
ejpam-6098	552	9	,	,	PUNCT
ejpam-6098	552	10	g.	g.	PROPN
ejpam-6098	552	11	liu	liu	PROPN
ejpam-6098	552	12	,	,	PUNCT
ejpam-6098	552	13	and	and	CCONJ
ejpam-6098	552	14	g.	g.	PROPN
ejpam-6098	552	15	shen	shen	PROPN
ejpam-6098	552	16	.	.	PUNCT
ejpam-6098	553	1	stability	stability	NOUN
ejpam-6098	553	2	of	of	ADP
ejpam-6098	553	3	functional	functional	ADJ
ejpam-6098	553	4	equations	equation	NOUN
ejpam-6098	553	5	in	in	ADP
ejpam-6098	553	6	(	(	PUNCT
ejpam-6098	553	7	n	n	CCONJ
ejpam-6098	553	8	,	,	PUNCT
ejpam-6098	553	9	β)normed	β)normed	ADJ
ejpam-6098	553	10	spaces	space	NOUN
ejpam-6098	553	11	.	.	PUNCT
ejpam-6098	554	1	j.	j.	PROPN
ejpam-6098	554	2	inequal	inequal	PROPN
ejpam-6098	554	3	.	.	PUNCT
ejpam-6098	555	1	appl	appl	PROPN
ejpam-6098	555	2	.	.	PROPN
ejpam-6098	555	3	,	,	PUNCT
ejpam-6098	555	4	2015(1):112–130	2015(1):112–130	NUM
ejpam-6098	555	5	,	,	PUNCT
ejpam-6098	555	6	2015	2015	NUM
ejpam-6098	555	7	.	.	PUNCT
ejpam-6098	556	1	[	[	X
ejpam-6098	556	2	20	20	NUM
ejpam-6098	556	3	]	]	X
ejpam-6098	556	4	jyotsana	jyotsana	PROPN
ejpam-6098	556	5	,	,	PUNCT
ejpam-6098	556	6	r.	r.	PROPN
ejpam-6098	556	7	chugh	chugh	PROPN
ejpam-6098	556	8	,	,	PUNCT
ejpam-6098	556	9	s.	s.	PROPN
ejpam-6098	556	10	jaiswal	jaiswal	PROPN
ejpam-6098	556	11	,	,	PUNCT
ejpam-6098	556	12	and	and	CCONJ
ejpam-6098	556	13	r.	r.	PROPN
ejpam-6098	556	14	dubey	dubey	PROPN
ejpam-6098	556	15	.	.	PUNCT
ejpam-6098	557	1	stability	stability	NOUN
ejpam-6098	557	2	of	of	ADP
ejpam-6098	557	3	functional	functional	ADJ
ejpam-6098	557	4	equations	equation	NOUN
ejpam-6098	557	5	in	in	ADP
ejpam-6098	557	6	non	non	ADJ
ejpam-6098	557	7	-	-	ADJ
ejpam-6098	557	8	archimedean	archimedean	ADJ
ejpam-6098	557	9	(	(	PUNCT
ejpam-6098	557	10	n	n	CCONJ
ejpam-6098	557	11	,	,	PUNCT
ejpam-6098	557	12	β)-normed	β)-normed	PUNCT
ejpam-6098	557	13	spaces	space	NOUN
ejpam-6098	557	14	.	.	PUNCT
ejpam-6098	558	1	j.	j.	PROPN
ejpam-6098	558	2	anal	anal	PROPN
ejpam-6098	558	3	.	.	PROPN
ejpam-6098	558	4	,	,	PUNCT
ejpam-6098	558	5	30:1653–1669	30:1653–1669	NUM
ejpam-6098	558	6	,	,	PUNCT
ejpam-6098	558	7	2022	2022	NUM
ejpam-6098	558	8	.	.	PUNCT
ejpam-6098	559	1	[	[	X
ejpam-6098	559	2	21	21	NUM
ejpam-6098	559	3	]	]	X
ejpam-6098	559	4	el	el	PROPN
ejpam-6098	559	5	fassi	fassi	PROPN
ejpam-6098	559	6	and	and	CCONJ
ejpam-6098	559	7	s.	s.	PROPN
ejpam-6098	559	8	kabbaj	kabbaj	PROPN
ejpam-6098	559	9	.	.	PROPN
ejpam-6098	560	1	non	non	ADJ
ejpam-6098	560	2	-	-	ADJ
ejpam-6098	560	3	archimedean	archimedean	ADJ
ejpam-6098	560	4	random	random	ADJ
ejpam-6098	560	5	stability	stability	NOUN
ejpam-6098	560	6	of	of	ADP
ejpam-6098	560	7	σ	σ	NOUN
ejpam-6098	560	8	-	-	ADJ
ejpam-6098	560	9	quadratic	quadratic	ADJ
ejpam-6098	560	10	functional	functional	ADJ
ejpam-6098	560	11	equation	equation	NOUN
ejpam-6098	560	12	.	.	PUNCT
ejpam-6098	561	1	thai	thai	PROPN
ejpam-6098	561	2	j.	j.	PROPN
ejpam-6098	561	3	math	math	PROPN
ejpam-6098	561	4	.	.	PUNCT
ejpam-6098	561	5	,	,	PUNCT
ejpam-6098	561	6	14:151–165	14:151–165	NUM
ejpam-6098	561	7	,	,	PUNCT
ejpam-6098	561	8	2016	2016	NUM
ejpam-6098	561	9	.	.	PUNCT
ejpam-6098	562	1	[	[	X
ejpam-6098	562	2	22	22	NUM
ejpam-6098	562	3	]	]	PUNCT
ejpam-6098	562	4	m.	m.	NOUN
ejpam-6098	562	5	mirzavaziri	mirzavaziri	PROPN
ejpam-6098	562	6	and	and	CCONJ
ejpam-6098	562	7	m.	m.	PROPN
ejpam-6098	562	8	s.	s.	PROPN
ejpam-6098	562	9	moslehian	moslehian	PROPN
ejpam-6098	562	10	.	.	PUNCT
ejpam-6098	563	1	a	a	DET
ejpam-6098	563	2	fixed	fix	VERB
ejpam-6098	563	3	point	point	NOUN
ejpam-6098	563	4	approach	approach	NOUN
ejpam-6098	563	5	to	to	ADP
ejpam-6098	563	6	stability	stability	NOUN
ejpam-6098	563	7	of	of	ADP
ejpam-6098	563	8	a	a	DET
ejpam-6098	563	9	quadratic	quadratic	ADJ
ejpam-6098	563	10	equation	equation	NOUN
ejpam-6098	563	11	.	.	PUNCT
ejpam-6098	564	1	bull	bull	NOUN
ejpam-6098	564	2	.	.	PUNCT
ejpam-6098	565	1	braz	braz	PROPN
ejpam-6098	565	2	.	.	PUNCT
ejpam-6098	565	3	math	math	PROPN
ejpam-6098	565	4	.	.	PUNCT
ejpam-6098	566	1	soc	soc	PROPN
ejpam-6098	566	2	.	.	PUNCT
ejpam-6098	566	3	,	,	PUNCT
ejpam-6098	566	4	37:361–376	37:361–376	PROPN
ejpam-6098	566	5	,	,	PUNCT
ejpam-6098	566	6	2006	2006	NUM
ejpam-6098	566	7	.	.	PUNCT
ejpam-6098	567	1	[	[	X
ejpam-6098	567	2	23	23	NUM
ejpam-6098	567	3	]	]	X
ejpam-6098	567	4	d.	d.	PROPN
ejpam-6098	567	5	mihet	mihet	PROPN
ejpam-6098	567	6	and	and	CCONJ
ejpam-6098	567	7	v.	v.	PROPN
ejpam-6098	567	8	radu	radu	PROPN
ejpam-6098	567	9	.	.	PUNCT
ejpam-6098	568	1	on	on	ADP
ejpam-6098	568	2	the	the	DET
ejpam-6098	568	3	stability	stability	NOUN
ejpam-6098	568	4	of	of	ADP
ejpam-6098	568	5	the	the	DET
ejpam-6098	568	6	additive	additive	ADJ
ejpam-6098	568	7	cauchy	cauchy	ADJ
ejpam-6098	568	8	functional	functional	ADJ
ejpam-6098	568	9	equation	equation	NOUN
ejpam-6098	568	10	in	in	ADP
ejpam-6098	568	11	random	random	ADJ
ejpam-6098	568	12	normed	normed	ADJ
ejpam-6098	568	13	spaces	space	NOUN
ejpam-6098	568	14	.	.	PUNCT
ejpam-6098	569	1	j.	j.	PROPN
ejpam-6098	569	2	math	math	PROPN
ejpam-6098	569	3	.	.	PUNCT
ejpam-6098	570	1	anal	anal	PROPN
ejpam-6098	570	2	.	.	PUNCT
ejpam-6098	570	3	appl	appl	PROPN
ejpam-6098	570	4	.	.	PROPN
ejpam-6098	570	5	,	,	PUNCT
ejpam-6098	570	6	343:567–572	343:567–572	NUM
ejpam-6098	570	7	,	,	PUNCT
ejpam-6098	570	8	2008	2008	NUM
ejpam-6098	570	9	.	.	PUNCT
ejpam-6098	571	1	[	[	X
ejpam-6098	571	2	24	24	NUM
ejpam-6098	571	3	]	]	X
ejpam-6098	571	4	e.	e.	PROPN
ejpam-6098	571	5	baktash	baktash	PROPN
ejpam-6098	571	6	,	,	PUNCT
ejpam-6098	571	7	y.	y.	PROPN
ejpam-6098	571	8	j.	j.	PROPN
ejpam-6098	571	9	cho	cho	PROPN
ejpam-6098	571	10	,	,	PUNCT
ejpam-6098	571	11	m.	m.	NOUN
ejpam-6098	571	12	jalili	jalili	PROPN
ejpam-6098	571	13	,	,	PUNCT
ejpam-6098	571	14	r.	r.	PROPN
ejpam-6098	571	15	saadati	saadati	PROPN
ejpam-6098	571	16	,	,	PUNCT
ejpam-6098	571	17	and	and	CCONJ
ejpam-6098	571	18	s.	s.	PROPN
ejpam-6098	571	19	m.	m.	PROPN
ejpam-6098	571	20	vaezpour	vaezpour	NOUN
ejpam-6098	571	21	.	.	PUNCT
ejpam-6098	572	1	on	on	ADP
ejpam-6098	572	2	the	the	DET
ejpam-6098	572	3	stability	stability	NOUN
ejpam-6098	572	4	of	of	ADP
ejpam-6098	572	5	cubic	cubic	ADJ
ejpam-6098	572	6	mappings	mapping	NOUN
ejpam-6098	572	7	and	and	CCONJ
ejpam-6098	572	8	quadratic	quadratic	ADJ
ejpam-6098	572	9	mappings	mapping	NOUN
ejpam-6098	572	10	in	in	ADP
ejpam-6098	572	11	random	random	ADJ
ejpam-6098	572	12	normed	normed	ADJ
ejpam-6098	572	13	spaces	space	NOUN
ejpam-6098	572	14	.	.	PUNCT
ejpam-6098	573	1	j.	j.	PROPN
ejpam-6098	573	2	inequal	inequal	PROPN
ejpam-6098	573	3	.	.	PUNCT
ejpam-6098	574	1	appl	appl	PROPN
ejpam-6098	574	2	.	.	PROPN
ejpam-6098	574	3	,	,	PUNCT
ejpam-6098	574	4	2008:1–11	2008:1–11	NUM
ejpam-6098	574	5	,	,	PUNCT
ejpam-6098	574	6	2008	2008	NUM
ejpam-6098	574	7	.	.	PUNCT
ejpam-6098	575	1	[	[	X
ejpam-6098	575	2	25	25	NUM
ejpam-6098	575	3	]	]	X
ejpam-6098	575	4	r.	r.	PROPN
ejpam-6098	575	5	saadati	saadati	PROPN
ejpam-6098	575	6	,	,	PUNCT
ejpam-6098	575	7	s.	s.	PROPN
ejpam-6098	575	8	m.	m.	PROPN
ejpam-6098	575	9	vaezpour	vaezpour	NOUN
ejpam-6098	575	10	,	,	PUNCT
ejpam-6098	575	11	and	and	CCONJ
ejpam-6098	575	12	y.	y.	PROPN
ejpam-6098	575	13	j.	j.	PROPN
ejpam-6098	575	14	cho	cho	PROPN
ejpam-6098	575	15	.	.	PUNCT
ejpam-6098	576	1	erratum	erratum	PROPN
ejpam-6098	576	2	to	to	ADP
ejpam-6098	576	3	:	:	PUNCT
ejpam-6098	576	4	a	a	DET
ejpam-6098	576	5	note	note	NOUN
ejpam-6098	576	6	to	to	PART
ejpam-6098	576	7	paper	paper	VERB
ejpam-6098	576	8	“	"	PUNCT
ejpam-6098	576	9	on	on	ADP
ejpam-6098	576	10	the	the	DET
ejpam-6098	576	11	stability	stability	NOUN
ejpam-6098	576	12	of	of	ADP
ejpam-6098	576	13	cubic	cubic	ADJ
ejpam-6098	576	14	mappings	mapping	NOUN
ejpam-6098	576	15	and	and	CCONJ
ejpam-6098	576	16	quartic	quartic	ADJ
ejpam-6098	576	17	mappings	mapping	NOUN
ejpam-6098	576	18	in	in	ADP
ejpam-6098	576	19	random	random	ADJ
ejpam-6098	576	20	normed	norme	VERB
ejpam-6098	576	21	spaces	space	NOUN
ejpam-6098	576	22	”	"	PUNCT
ejpam-6098	576	23	.	.	PUNCT
ejpam-6098	577	1	j.	j.	PROPN
ejpam-6098	577	2	inequal	inequal	PROPN
ejpam-6098	577	3	.	.	PUNCT
ejpam-6098	578	1	appl	appl	PROPN
ejpam-6098	578	2	.	.	PROPN
ejpam-6098	578	3	,	,	PUNCT
ejpam-6098	578	4	2009:1–6	2009:1–6	NUM
ejpam-6098	578	5	,	,	PUNCT
ejpam-6098	578	6	2009	2009	NUM
ejpam-6098	578	7	.	.	PUNCT
ejpam-6098	579	1	[	[	X
ejpam-6098	579	2	26	26	NUM
ejpam-6098	579	3	]	]	X
ejpam-6098	579	4	j.	j.	PROPN
ejpam-6098	579	5	m.	m.	PROPN
ejpam-6098	579	6	rassias	rassias	PROPN
ejpam-6098	579	7	,	,	PUNCT
ejpam-6098	579	8	s.	s.	PROPN
ejpam-6098	579	9	sharma	sharma	PROPN
ejpam-6098	579	10	,	,	PUNCT
ejpam-6098	579	11	j.	j.	PROPN
ejpam-6098	579	12	jakhar	jakhar	PROPN
ejpam-6098	579	13	,	,	PUNCT
ejpam-6098	579	14	and	and	CCONJ
ejpam-6098	579	15	j.	j.	PROPN
ejpam-6098	579	16	jakhar	jakhar	PROPN
ejpam-6098	579	17	.	.	PUNCT
ejpam-6098	580	1	direct	direct	ADJ
ejpam-6098	580	2	approach	approach	NOUN
ejpam-6098	580	3	to	to	ADP
ejpam-6098	580	4	the	the	DET
ejpam-6098	580	5	stability	stability	NOUN
ejpam-6098	580	6	of	of	ADP
ejpam-6098	580	7	various	various	ADJ
ejpam-6098	580	8	functional	functional	ADJ
ejpam-6098	580	9	equations	equation	NOUN
ejpam-6098	580	10	in	in	ADP
ejpam-6098	580	11	felbin	felbin	NOUN
ejpam-6098	580	12	’s	’s	PART
ejpam-6098	580	13	type	type	NOUN
ejpam-6098	580	14	non	non	ADJ
ejpam-6098	580	15	-	-	ADJ
ejpam-6098	580	16	archimedean	archimedean	ADJ
ejpam-6098	580	17	fuzzy	fuzzy	ADJ
ejpam-6098	580	18	normed	normed	PROPN
ejpam-6098	580	19	spaces	space	NOUN
ejpam-6098	580	20	.	.	PUNCT
ejpam-6098	581	1	j.	j.	PROPN
ejpam-6098	581	2	comput	comput	PROPN
ejpam-6098	581	3	.	.	PUNCT
ejpam-6098	582	1	anal	anal	PROPN
ejpam-6098	582	2	.	.	PUNCT
ejpam-6098	582	3	appl	appl	PROPN
ejpam-6098	582	4	.	.	PROPN
ejpam-6098	582	5	,	,	PUNCT
ejpam-6098	583	1	32:320–352	32:320–352	NUM
ejpam-6098	583	2	,	,	PUNCT
ejpam-6098	583	3	2024	2024	NUM
ejpam-6098	583	4	.	.	PUNCT
ejpam-6098	584	1	j.	j.	PROPN
ejpam-6098	584	2	jakhar	jakhar	PROPN
ejpam-6098	584	3	et	et	PROPN
ejpam-6098	584	4	al	al	PROPN
ejpam-6098	584	5	.	.	PUNCT
ejpam-6098	584	6	/	/	SYM
ejpam-6098	584	7	eur	eur	PROPN
ejpam-6098	584	8	.	.	PUNCT
ejpam-6098	585	1	j.	j.	PROPN
ejpam-6098	585	2	pure	pure	PROPN
ejpam-6098	585	3	appl	appl	PROPN
ejpam-6098	585	4	.	.	PROPN
ejpam-6098	585	5	math	math	PROPN
ejpam-6098	585	6	,	,	PUNCT
ejpam-6098	585	7	18	18	NUM
ejpam-6098	585	8	(	(	PUNCT
ejpam-6098	585	9	2	2	NUM
ejpam-6098	585	10	)	)	PUNCT
ejpam-6098	585	11	(	(	PUNCT
ejpam-6098	585	12	2025	2025	NUM
ejpam-6098	585	13	)	)	PUNCT
ejpam-6098	585	14	,	,	PUNCT
ejpam-6098	585	15	6098	6098	NUM
ejpam-6098	585	16	23	23	NUM
ejpam-6098	585	17	of	of	ADP
ejpam-6098	585	18	23	23	NUM
ejpam-6098	586	1	[	[	SYM
ejpam-6098	586	2	27	27	NUM
ejpam-6098	586	3	]	]	X
ejpam-6098	586	4	v.	v.	ADP
ejpam-6098	586	5	govindan	govindan	PROPN
ejpam-6098	586	6	,	,	PUNCT
ejpam-6098	586	7	c.	c.	PROPN
ejpam-6098	586	8	park	park	PROPN
ejpam-6098	586	9	,	,	PUNCT
ejpam-6098	586	10	s.	s.	PROPN
ejpam-6098	586	11	pinelas	pinelas	PROPN
ejpam-6098	586	12	,	,	PUNCT
ejpam-6098	586	13	and	and	CCONJ
ejpam-6098	586	14	s.	s.	PROPN
ejpam-6098	586	15	baskaran	baskaran	PROPN
ejpam-6098	586	16	.	.	PUNCT
ejpam-6098	587	1	solution	solution	NOUN
ejpam-6098	587	2	of	of	ADP
ejpam-6098	587	3	a	a	DET
ejpam-6098	587	4	3	3	NUM
ejpam-6098	587	5	-	-	PUNCT
ejpam-6098	587	6	d	d	NOUN
ejpam-6098	587	7	cubic	cubic	ADJ
ejpam-6098	587	8	functional	functional	ADJ
ejpam-6098	587	9	equation	equation	NOUN
ejpam-6098	587	10	and	and	CCONJ
ejpam-6098	587	11	its	its	PRON
ejpam-6098	587	12	stability	stability	NOUN
ejpam-6098	587	13	.	.	PUNCT
ejpam-6098	588	1	aims	aim	VERB
ejpam-6098	588	2	math	math	NOUN
ejpam-6098	588	3	.	.	PUNCT
ejpam-6098	588	4	,	,	PUNCT
ejpam-6098	589	1	5:1693–1705	5:1693–1705	NUM
ejpam-6098	589	2	,	,	PUNCT
ejpam-6098	589	3	2020	2020	NUM
ejpam-6098	589	4	.	.	PUNCT
ejpam-6098	590	1	[	[	X
ejpam-6098	590	2	28	28	NUM
ejpam-6098	590	3	]	]	X
ejpam-6098	590	4	b.	b.	PROPN
ejpam-6098	590	5	schweizer	schweizer	PROPN
ejpam-6098	590	6	and	and	CCONJ
ejpam-6098	590	7	a.	a.	NOUN
ejpam-6098	590	8	sklar	sklar	PROPN
ejpam-6098	590	9	.	.	PUNCT
ejpam-6098	591	1	probabilistic	probabilistic	ADJ
ejpam-6098	591	2	metric	metric	ADJ
ejpam-6098	591	3	spaces	space	NOUN
ejpam-6098	591	4	.	.	PUNCT
ejpam-6098	592	1	elsevier	elsevier	NOUN
ejpam-6098	592	2	,	,	PUNCT
ejpam-6098	592	3	north	north	PROPN
ejpam-6098	592	4	holland	holland	PROPN
ejpam-6098	592	5	,	,	PUNCT
ejpam-6098	592	6	new	new	PROPN
ejpam-6098	592	7	york	york	PROPN
ejpam-6098	592	8	,	,	PUNCT
ejpam-6098	592	9	1983	1983	NUM
ejpam-6098	592	10	.	.	PUNCT
ejpam-6098	593	1	[	[	X
ejpam-6098	593	2	29	29	NUM
ejpam-6098	593	3	]	]	PUNCT
ejpam-6098	593	4	a.	a.	NOUN
ejpam-6098	593	5	n.	n.	PROPN
ejpam-6098	593	6	serstnev	serstnev	PROPN
ejpam-6098	593	7	.	.	PUNCT
ejpam-6098	594	1	the	the	DET
ejpam-6098	594	2	notion	notion	NOUN
ejpam-6098	594	3	of	of	ADP
ejpam-6098	594	4	random	random	ADJ
ejpam-6098	594	5	normed	normed	ADJ
ejpam-6098	594	6	spaces	space	NOUN
ejpam-6098	594	7	.	.	PUNCT
ejpam-6098	595	1	doki	doki	NOUN
ejpam-6098	595	2	.	.	PUNCT
ejpam-6098	596	1	acad	acad	PROPN
ejpam-6098	596	2	.	.	PUNCT
ejpam-6098	597	1	nauk	nauk	PROPN
ejpam-6098	597	2	ussr	ussr	PROPN
ejpam-6098	597	3	,	,	PUNCT
ejpam-6098	597	4	149:280–283	149:280–283	NUM
ejpam-6098	597	5	,	,	PUNCT
ejpam-6098	597	6	1963	1963	NUM
ejpam-6098	597	7	.	.	PUNCT
ejpam-6098	598	1	[	[	X
ejpam-6098	598	2	30	30	NUM
ejpam-6098	598	3	]	]	X
ejpam-6098	598	4	y.	y.	PROPN
ejpam-6098	598	5	j.	j.	PROPN
ejpam-6098	598	6	cho	cho	PROPN
ejpam-6098	598	7	,	,	PUNCT
ejpam-6098	598	8	t.	t.	PROPN
ejpam-6098	598	9	m.	m.	PROPN
ejpam-6098	598	10	rassias	rassias	PROPN
ejpam-6098	598	11	,	,	PUNCT
ejpam-6098	598	12	and	and	CCONJ
ejpam-6098	598	13	r.	r.	PROPN
ejpam-6098	598	14	saadati	saadati	PROPN
ejpam-6098	598	15	.	.	PUNCT
ejpam-6098	599	1	stability	stability	NOUN
ejpam-6098	599	2	of	of	ADP
ejpam-6098	599	3	functional	functional	ADJ
ejpam-6098	599	4	equations	equation	NOUN
ejpam-6098	599	5	in	in	ADP
ejpam-6098	599	6	random	random	ADJ
ejpam-6098	599	7	normed	normed	ADJ
ejpam-6098	599	8	spaces	space	NOUN
ejpam-6098	599	9	.	.	PUNCT
ejpam-6098	600	1	springer	springer	NOUN
ejpam-6098	600	2	science	science	PROPN
ejpam-6098	600	3	&	&	CCONJ
ejpam-6098	600	4	business	business	NOUN
ejpam-6098	600	5	media	medium	NOUN
ejpam-6098	600	6	,	,	PUNCT
ejpam-6098	600	7	2013	2013	NUM
ejpam-6098	600	8	.	.	PUNCT
ejpam-6098	601	1	introduction	introduction	NOUN
ejpam-6098	601	2	preliminaries	preliminary	NOUN
ejpam-6098	601	3	and	and	CCONJ
ejpam-6098	601	4	definitions	definition	VERB
ejpam-6098	601	5	stability	stability	NOUN
ejpam-6098	601	6	in	in	ADP
ejpam-6098	601	7	(	(	PUNCT
ejpam-6098	601	8	n,)-normed	n,)-normed	ADJ
ejpam-6098	601	9	space	space	NOUN
ejpam-6098	601	10	stability	stability	NOUN
ejpam-6098	601	11	in	in	ADP
ejpam-6098	601	12	non	non	ADJ
ejpam-6098	601	13	-	-	ADJ
ejpam-6098	601	14	archimedean-(n,)-normed	archimedean-(n,)-normed	ADJ
ejpam-6098	601	15	spaces	space	NOUN
ejpam-6098	601	16	stability	stability	NOUN
ejpam-6098	601	17	in	in	ADP
ejpam-6098	601	18	random	random	ADJ
ejpam-6098	601	19	normed	normed	ADJ
ejpam-6098	601	20	space	space	NOUN
ejpam-6098	601	21	results	result	NOUN
ejpam-6098	601	22	of	of	ADP
ejpam-6098	601	23	experiment	experiment	NOUN
ejpam-6098	601	24	comparative	comparative	ADJ
ejpam-6098	601	25	evaluation	evaluation	NOUN
ejpam-6098	601	26	of	of	ADP
ejpam-6098	601	27	the	the	DET
ejpam-6098	601	28	results	result	NOUN
ejpam-6098	601	29	conclusions	conclusion	NOUN