id	sid	tid	token	lemma	pos
ejpam-6699	1	1	european	european	PROPN
ejpam-6699	1	2	journal	journal	PROPN
ejpam-6699	1	3	of	of	ADP
ejpam-6699	1	4	pure	pure	ADJ
ejpam-6699	1	5	and	and	CCONJ
ejpam-6699	1	6	applied	applied	ADJ
ejpam-6699	1	7	mathematics	mathematic	NOUN
ejpam-6699	1	8	2025	2025	NUM
ejpam-6699	1	9	,	,	PUNCT
ejpam-6699	1	10	vol	vol	NOUN
ejpam-6699	1	11	.	.	PROPN
ejpam-6699	1	12	18	18	NUM
ejpam-6699	1	13	,	,	PUNCT
ejpam-6699	1	14	issue	issue	NOUN
ejpam-6699	1	15	4	4	NUM
ejpam-6699	1	16	,	,	PUNCT
ejpam-6699	1	17	article	article	NOUN
ejpam-6699	1	18	number	number	NOUN
ejpam-6699	1	19	6699	6699	NUM
ejpam-6699	1	20	issn	issn	PROPN
ejpam-6699	1	21	1307	1307	NUM
ejpam-6699	1	22	-	-	SYM
ejpam-6699	1	23	5543	5543	NUM
ejpam-6699	1	24	–	–	PUNCT
ejpam-6699	1	25	ejpam.com	ejpam.com	X
ejpam-6699	1	26	published	publish	VERB
ejpam-6699	1	27	by	by	ADP
ejpam-6699	1	28	new	new	PROPN
ejpam-6699	1	29	york	york	PROPN
ejpam-6699	1	30	business	business	PROPN
ejpam-6699	1	31	global	global	PROPN
ejpam-6699	1	32	hyers	hyers	PROPN
ejpam-6699	1	33	-	-	PUNCT
ejpam-6699	1	34	ulam	ulam	PROPN
ejpam-6699	1	35	stability	stability	NOUN
ejpam-6699	1	36	of	of	ADP
ejpam-6699	1	37	generalized	generalized	ADJ
ejpam-6699	1	38	quartic	quartic	ADJ
ejpam-6699	1	39	mapping	mapping	NOUN
ejpam-6699	1	40	in	in	ADP
ejpam-6699	1	41	non	non	ADJ
ejpam-6699	1	42	-	-	ADJ
ejpam-6699	1	43	archimedean	archimedean	ADJ
ejpam-6699	1	44	(	(	PUNCT
ejpam-6699	1	45	n	n	CCONJ
ejpam-6699	1	46	,	,	PUNCT
ejpam-6699	1	47	β)-normed	β)-normed	PUNCT
ejpam-6699	1	48	spaces	space	VERB
ejpam-6699	1	49	senthil	senthil	PROPN
ejpam-6699	1	50	gowri1	gowri1	PROPN
ejpam-6699	1	51	,	,	PUNCT
ejpam-6699	1	52	siriluk	siriluk	PROPN
ejpam-6699	1	53	donganont2,∗	donganont2,∗	PROPN
ejpam-6699	1	54	,	,	PUNCT
ejpam-6699	1	55	s.	s.	PROPN
ejpam-6699	1	56	karthick3	karthick3	PROPN
ejpam-6699	1	57	,	,	PUNCT
ejpam-6699	1	58	radhakrishnan	radhakrishnan	PROPN
ejpam-6699	1	59	balaanandhan4	balaanandhan4	PROPN
ejpam-6699	1	60	,	,	PUNCT
ejpam-6699	1	61	kandhasamy	kandhasamy	ADJ
ejpam-6699	1	62	tamilvanan5	tamilvanan5	PROPN
ejpam-6699	1	63	1	1	NUM
ejpam-6699	1	64	department	department	NOUN
ejpam-6699	1	65	of	of	ADP
ejpam-6699	1	66	mathematics	mathematic	NOUN
ejpam-6699	1	67	,	,	PUNCT
ejpam-6699	1	68	saveetha	saveetha	PROPN
ejpam-6699	1	69	school	school	PROPN
ejpam-6699	1	70	of	of	ADP
ejpam-6699	1	71	engineering	engineering	PROPN
ejpam-6699	1	72	,	,	PUNCT
ejpam-6699	1	73	saveetha	saveetha	PROPN
ejpam-6699	1	74	institute	institute	PROPN
ejpam-6699	1	75	of	of	ADP
ejpam-6699	1	76	medical	medical	ADJ
ejpam-6699	1	77	and	and	CCONJ
ejpam-6699	1	78	technical	technical	ADJ
ejpam-6699	1	79	sciences	science	NOUN
ejpam-6699	1	80	,	,	PUNCT
ejpam-6699	1	81	saveetha	saveetha	PROPN
ejpam-6699	1	82	university	university	PROPN
ejpam-6699	1	83	,	,	PUNCT
ejpam-6699	1	84	tandalam	tandalam	PROPN
ejpam-6699	1	85	,	,	PUNCT
ejpam-6699	1	86	chennai	chennai	PROPN
ejpam-6699	1	87	602105	602105	NUM
ejpam-6699	1	88	,	,	PUNCT
ejpam-6699	1	89	tamil	tamil	PROPN
ejpam-6699	1	90	nadu	nadu	PROPN
ejpam-6699	1	91	,	,	PUNCT
ejpam-6699	1	92	india	india	PROPN
ejpam-6699	1	93	2	2	NUM
ejpam-6699	1	94	school	school	NOUN
ejpam-6699	1	95	of	of	ADP
ejpam-6699	1	96	science	science	NOUN
ejpam-6699	1	97	,	,	PUNCT
ejpam-6699	1	98	university	university	NOUN
ejpam-6699	1	99	of	of	ADP
ejpam-6699	1	100	phayao	phayao	NOUN
ejpam-6699	1	101	,	,	PUNCT
ejpam-6699	1	102	phayao	phayao	NOUN
ejpam-6699	1	103	56000	56000	NUM
ejpam-6699	1	104	,	,	PUNCT
ejpam-6699	1	105	thailand	thailand	PROPN
ejpam-6699	1	106	3	3	NUM
ejpam-6699	1	107	department	department	NOUN
ejpam-6699	1	108	of	of	ADP
ejpam-6699	1	109	mathematics	mathematics	PROPN
ejpam-6699	1	110	,	,	PUNCT
ejpam-6699	1	111	college	college	NOUN
ejpam-6699	1	112	of	of	ADP
ejpam-6699	1	113	engineering	engineering	NOUN
ejpam-6699	1	114	and	and	CCONJ
ejpam-6699	1	115	technology	technology	NOUN
ejpam-6699	1	116	,	,	PUNCT
ejpam-6699	1	117	srm	srm	PROPN
ejpam-6699	1	118	institute	institute	PROPN
ejpam-6699	1	119	of	of	ADP
ejpam-6699	1	120	science	science	NOUN
ejpam-6699	1	121	and	and	CCONJ
ejpam-6699	1	122	technology	technology	NOUN
ejpam-6699	1	123	,	,	PUNCT
ejpam-6699	1	124	kattankulathur	kattankulathur	PROPN
ejpam-6699	1	125	603203	603203	NUM
ejpam-6699	1	126	,	,	PUNCT
ejpam-6699	1	127	tamil	tamil	PROPN
ejpam-6699	1	128	nadu	nadu	PROPN
ejpam-6699	1	129	,	,	PUNCT
ejpam-6699	1	130	india	india	PROPN
ejpam-6699	1	131	4	4	NUM
ejpam-6699	1	132	department	department	NOUN
ejpam-6699	1	133	of	of	ADP
ejpam-6699	1	134	mathematics	mathematic	NOUN
ejpam-6699	1	135	,	,	PUNCT
ejpam-6699	1	136	sri	sri	PROPN
ejpam-6699	1	137	sankara	sankara	PROPN
ejpam-6699	1	138	arts	arts	PROPN
ejpam-6699	1	139	and	and	CCONJ
ejpam-6699	1	140	science	science	PROPN
ejpam-6699	1	141	college	college	PROPN
ejpam-6699	1	142	(	(	PUNCT
ejpam-6699	1	143	autonomous	autonomous	ADJ
ejpam-6699	1	144	)	)	PUNCT
ejpam-6699	1	145	,	,	PUNCT
ejpam-6699	1	146	enathur	enathur	NOUN
ejpam-6699	1	147	631561	631561	NUM
ejpam-6699	1	148	,	,	PUNCT
ejpam-6699	1	149	kanchipuram	kanchipuram	PROPN
ejpam-6699	1	150	,	,	PUNCT
ejpam-6699	1	151	tamil	tamil	PROPN
ejpam-6699	1	152	nadu	nadu	PROPN
ejpam-6699	1	153	,	,	PUNCT
ejpam-6699	1	154	india	india	PROPN
ejpam-6699	1	155	5	5	NUM
ejpam-6699	1	156	department	department	NOUN
ejpam-6699	1	157	of	of	ADP
ejpam-6699	1	158	mathematics	mathematic	NOUN
ejpam-6699	1	159	,	,	PUNCT
ejpam-6699	1	160	faculty	faculty	NOUN
ejpam-6699	1	161	of	of	ADP
ejpam-6699	1	162	science	science	NOUN
ejpam-6699	1	163	and	and	CCONJ
ejpam-6699	1	164	humanities	humanity	NOUN
ejpam-6699	1	165	,	,	PUNCT
ejpam-6699	1	166	r.m.k	r.m.k	NOUN
ejpam-6699	1	167	.	.	PUNCT
ejpam-6699	2	1	engineering	engineering	PROPN
ejpam-6699	2	2	college	college	PROPN
ejpam-6699	2	3	,	,	PUNCT
ejpam-6699	2	4	kavaraipettai	kavaraipettai	NOUN
ejpam-6699	2	5	,	,	PUNCT
ejpam-6699	2	6	tiruvallur	tiruvallur	NOUN
ejpam-6699	2	7	601206	601206	NUM
ejpam-6699	2	8	,	,	PUNCT
ejpam-6699	2	9	tamil	tamil	PROPN
ejpam-6699	2	10	nadu	nadu	PROPN
ejpam-6699	2	11	,	,	PUNCT
ejpam-6699	2	12	india	india	PROPN
ejpam-6699	2	13	abstract	abstract	NOUN
ejpam-6699	2	14	.	.	PUNCT
ejpam-6699	3	1	in	in	ADP
ejpam-6699	3	2	this	this	DET
ejpam-6699	3	3	article	article	NOUN
ejpam-6699	3	4	,	,	PUNCT
ejpam-6699	3	5	we	we	PRON
ejpam-6699	3	6	introduce	introduce	VERB
ejpam-6699	3	7	a	a	DET
ejpam-6699	3	8	novel	novel	ADJ
ejpam-6699	3	9	structure	structure	NOUN
ejpam-6699	3	10	termed	term	VERB
ejpam-6699	3	11	as	as	ADP
ejpam-6699	3	12	the	the	DET
ejpam-6699	3	13	non	non	ADJ
ejpam-6699	3	14	-	-	ADJ
ejpam-6699	3	15	archimedean	archimedean	ADJ
ejpam-6699	3	16	(	(	PUNCT
ejpam-6699	3	17	n	n	CCONJ
ejpam-6699	3	18	,	,	PUNCT
ejpam-6699	3	19	β)normed	β)normed	ADJ
ejpam-6699	3	20	space	space	NOUN
ejpam-6699	3	21	,	,	PUNCT
ejpam-6699	3	22	formulated	formulate	VERB
ejpam-6699	3	23	over	over	ADP
ejpam-6699	3	24	a	a	DET
ejpam-6699	3	25	non	non	ADJ
ejpam-6699	3	26	-	-	ADJ
ejpam-6699	3	27	archimedean	archimedean	ADJ
ejpam-6699	3	28	field	field	NOUN
ejpam-6699	3	29	.	.	PUNCT
ejpam-6699	4	1	this	this	DET
ejpam-6699	4	2	generalization	generalization	NOUN
ejpam-6699	4	3	extends	extend	VERB
ejpam-6699	4	4	the	the	DET
ejpam-6699	4	5	concept	concept	NOUN
ejpam-6699	4	6	of	of	ADP
ejpam-6699	4	7	classical	classical	ADJ
ejpam-6699	4	8	normed	norme	VERB
ejpam-6699	4	9	spaces	space	NOUN
ejpam-6699	4	10	by	by	ADP
ejpam-6699	4	11	integrating	integrate	VERB
ejpam-6699	4	12	a	a	DET
ejpam-6699	4	13	parameterized	parameterized	ADJ
ejpam-6699	4	14	framework	framework	NOUN
ejpam-6699	4	15	involving	involve	VERB
ejpam-6699	4	16	n	n	CCONJ
ejpam-6699	4	17	-	-	PUNCT
ejpam-6699	4	18	tuples	tuple	NOUN
ejpam-6699	4	19	and	and	CCONJ
ejpam-6699	4	20	an	an	DET
ejpam-6699	4	21	exponent	exponent	NOUN
ejpam-6699	4	22	β	β	X
ejpam-6699	4	23	.	.	PUNCT
ejpam-6699	5	1	we	we	PRON
ejpam-6699	5	2	delve	delve	VERB
ejpam-6699	5	3	into	into	ADP
ejpam-6699	5	4	the	the	DET
ejpam-6699	5	5	fundamental	fundamental	ADJ
ejpam-6699	5	6	characteristics	characteristic	NOUN
ejpam-6699	5	7	of	of	ADP
ejpam-6699	5	8	these	these	DET
ejpam-6699	5	9	spaces	space	NOUN
ejpam-6699	5	10	,	,	PUNCT
ejpam-6699	5	11	demonstrating	demonstrate	VERB
ejpam-6699	5	12	how	how	SCONJ
ejpam-6699	5	13	they	they	PRON
ejpam-6699	5	14	connect	connect	VERB
ejpam-6699	5	15	to	to	ADP
ejpam-6699	5	16	standard	standard	ADJ
ejpam-6699	5	17	non	non	ADJ
ejpam-6699	5	18	-	-	ADJ
ejpam-6699	5	19	archimedean	archimedean	ADJ
ejpam-6699	5	20	n	n	CCONJ
ejpam-6699	5	21	-	-	PUNCT
ejpam-6699	5	22	normed	normed	ADJ
ejpam-6699	5	23	and	and	CCONJ
ejpam-6699	5	24	n	n	CCONJ
ejpam-6699	5	25	-	-	PUNCT
ejpam-6699	5	26	quasi	quasi	ADJ
ejpam-6699	5	27	-	-	ADJ
ejpam-6699	5	28	normed	normed	ADJ
ejpam-6699	5	29	structures	structure	NOUN
ejpam-6699	5	30	.	.	PUNCT
ejpam-6699	6	1	moreover	moreover	ADV
ejpam-6699	6	2	,	,	PUNCT
ejpam-6699	6	3	we	we	PRON
ejpam-6699	6	4	provide	provide	VERB
ejpam-6699	6	5	examples	example	NOUN
ejpam-6699	6	6	that	that	PRON
ejpam-6699	6	7	support	support	VERB
ejpam-6699	6	8	the	the	DET
ejpam-6699	6	9	theory	theory	NOUN
ejpam-6699	6	10	and	and	CCONJ
ejpam-6699	6	11	help	help	VERB
ejpam-6699	6	12	show	show	VERB
ejpam-6699	6	13	some	some	DET
ejpam-6699	6	14	fixed	fix	VERB
ejpam-6699	6	15	point	point	NOUN
ejpam-6699	6	16	results	result	NOUN
ejpam-6699	6	17	,	,	PUNCT
ejpam-6699	6	18	making	make	VERB
ejpam-6699	6	19	these	these	DET
ejpam-6699	6	20	spaces	space	NOUN
ejpam-6699	6	21	easier	easy	ADJ
ejpam-6699	6	22	to	to	PART
ejpam-6699	6	23	use	use	VERB
ejpam-6699	6	24	in	in	ADP
ejpam-6699	6	25	real	real	ADJ
ejpam-6699	6	26	problems	problem	NOUN
ejpam-6699	6	27	.	.	PUNCT
ejpam-6699	7	1	2020	2020	NUM
ejpam-6699	7	2	mathematics	mathematic	NOUN
ejpam-6699	7	3	subject	subject	NOUN
ejpam-6699	7	4	classifications	classification	NOUN
ejpam-6699	7	5	:	:	PUNCT
ejpam-6699	7	6	39b52	39b52	NUM
ejpam-6699	7	7	,	,	PUNCT
ejpam-6699	7	8	39b72	39b72	NUM
ejpam-6699	7	9	,	,	PUNCT
ejpam-6699	7	10	46s40	46s40	PRON
ejpam-6699	7	11	key	key	ADJ
ejpam-6699	7	12	words	word	NOUN
ejpam-6699	7	13	and	and	CCONJ
ejpam-6699	7	14	phrases	phrase	NOUN
ejpam-6699	7	15	:	:	PUNCT
ejpam-6699	7	16	quartic	quartic	ADJ
ejpam-6699	7	17	functional	functional	ADJ
ejpam-6699	7	18	equation	equation	NOUN
ejpam-6699	7	19	,	,	PUNCT
ejpam-6699	7	20	hyers	hyers	PROPN
ejpam-6699	7	21	-	-	PUNCT
ejpam-6699	7	22	ulam	ulam	PROPN
ejpam-6699	7	23	stability	stability	PROPN
ejpam-6699	7	24	,	,	PUNCT
ejpam-6699	7	25	non	non	ADJ
ejpam-6699	7	26	-	-	ADJ
ejpam-6699	7	27	archimedean	archimedean	ADJ
ejpam-6699	7	28	(	(	PUNCT
ejpam-6699	7	29	n	n	CCONJ
ejpam-6699	7	30	,	,	PUNCT
ejpam-6699	7	31	β)-normed	β)-normed	PUNCT
ejpam-6699	7	32	spaces	space	NOUN
ejpam-6699	7	33	,	,	PUNCT
ejpam-6699	7	34	generalized	generalized	ADJ
ejpam-6699	7	35	control	control	NOUN
ejpam-6699	7	36	function	function	NOUN
ejpam-6699	7	37	1	1	NUM
ejpam-6699	7	38	.	.	PUNCT
ejpam-6699	7	39	introduction	introduction	NOUN
ejpam-6699	7	40	due	due	ADP
ejpam-6699	7	41	to	to	ADP
ejpam-6699	7	42	its	its	PRON
ejpam-6699	7	43	extensive	extensive	ADJ
ejpam-6699	7	44	applicability	applicability	NOUN
ejpam-6699	7	45	in	in	ADP
ejpam-6699	7	46	many	many	ADJ
ejpam-6699	7	47	other	other	ADJ
ejpam-6699	7	48	domains	domain	NOUN
ejpam-6699	7	49	and	and	CCONJ
ejpam-6699	7	50	its	its	PRON
ejpam-6699	7	51	deep	deep	ADJ
ejpam-6699	7	52	consequences	consequence	NOUN
ejpam-6699	7	53	in	in	ADP
ejpam-6699	7	54	mathematical	mathematical	ADJ
ejpam-6699	7	55	analysis	analysis	NOUN
ejpam-6699	8	1	,	,	PUNCT
ejpam-6699	8	2	the	the	DET
ejpam-6699	8	3	study	study	NOUN
ejpam-6699	8	4	of	of	ADP
ejpam-6699	8	5	functional	functional	ADJ
ejpam-6699	8	6	equations	equation	NOUN
ejpam-6699	8	7	and	and	CCONJ
ejpam-6699	8	8	their	their	PRON
ejpam-6699	8	9	stability	stability	NOUN
ejpam-6699	8	10	qualities	quality	NOUN
ejpam-6699	8	11	has	have	AUX
ejpam-6699	8	12	attracted	attract	VERB
ejpam-6699	8	13	a	a	DET
ejpam-6699	8	14	lot	lot	NOUN
ejpam-6699	8	15	of	of	ADP
ejpam-6699	8	16	attention	attention	NOUN
ejpam-6699	8	17	.	.	PUNCT
ejpam-6699	9	1	ulam	ulam	X
ejpam-6699	10	1	[	[	X
ejpam-6699	10	2	1	1	NUM
ejpam-6699	10	3	]	]	PUNCT
ejpam-6699	10	4	raised	raise	VERB
ejpam-6699	10	5	a	a	DET
ejpam-6699	10	6	basic	basic	ADJ
ejpam-6699	10	7	query	query	NOUN
ejpam-6699	10	8	about	about	ADP
ejpam-6699	10	9	the	the	DET
ejpam-6699	10	10	stability	stability	NOUN
ejpam-6699	10	11	of	of	ADP
ejpam-6699	10	12	group	group	NOUN
ejpam-6699	10	13	homomorphisms	homomorphism	NOUN
ejpam-6699	10	14	in	in	ADP
ejpam-6699	10	15	1940	1940	NUM
ejpam-6699	10	16	,	,	PUNCT
ejpam-6699	10	17	which	which	PRON
ejpam-6699	10	18	is	be	AUX
ejpam-6699	10	19	when	when	SCONJ
ejpam-6699	10	20	this	this	DET
ejpam-6699	10	21	topic	topic	NOUN
ejpam-6699	10	22	first	first	ADV
ejpam-6699	10	23	emerged	emerge	VERB
ejpam-6699	10	24	.	.	PUNCT
ejpam-6699	11	1	the	the	DET
ejpam-6699	11	2	theory	theory	NOUN
ejpam-6699	11	3	of	of	ADP
ejpam-6699	11	4	functional	functional	ADJ
ejpam-6699	11	5	∗corresponding	∗corresponding	NOUN
ejpam-6699	11	6	author	author	NOUN
ejpam-6699	11	7	.	.	PUNCT
ejpam-6699	12	1	doi	doi	NOUN
ejpam-6699	12	2	:	:	PUNCT
ejpam-6699	12	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6699	https://doi.org/10.29020/nybg.ejpam.v18i4.6699	NUM
ejpam-6699	12	4	email	email	NOUN
ejpam-6699	12	5	addresses	address	NOUN
ejpam-6699	12	6	:	:	PUNCT
ejpam-6699	12	7	gowrisenthil.sse@saveetha.com	gowrisenthil.sse@saveetha.com	X
ejpam-6699	12	8	(	(	PUNCT
ejpam-6699	12	9	s.	s.	PROPN
ejpam-6699	12	10	gowri	gowri	PROPN
ejpam-6699	12	11	)	)	PUNCT
ejpam-6699	12	12	,	,	PUNCT
ejpam-6699	12	13	siriluk.pa@up.ac.th	siriluk.pa@up.ac.th	PROPN
ejpam-6699	12	14	(	(	PUNCT
ejpam-6699	12	15	s.	s.	PROPN
ejpam-6699	12	16	donganont	donganont	PROPN
ejpam-6699	12	17	)	)	PUNCT
ejpam-6699	12	18	,	,	PUNCT
ejpam-6699	12	19	karthickmaths007@gmail.com	karthickmaths007@gmail.com	X
ejpam-6699	12	20	(	(	PUNCT
ejpam-6699	12	21	s.	s.	PROPN
ejpam-6699	12	22	karthick	karthick	PROPN
ejpam-6699	12	23	)	)	PUNCT
ejpam-6699	12	24	,	,	PUNCT
ejpam-6699	12	25	balaanandhanmaths@gmail.com	balaanandhanmaths@gmail.com	X
ejpam-6699	12	26	(	(	PUNCT
ejpam-6699	12	27	r.	r.	PROPN
ejpam-6699	12	28	balaanandhan	balaanandhan	PROPN
ejpam-6699	12	29	)	)	PUNCT
ejpam-6699	12	30	,	,	PUNCT
ejpam-6699	12	31	tamiltamilk7@gmail.com	tamiltamilk7@gmail.com	X
ejpam-6699	12	32	(	(	PUNCT
ejpam-6699	12	33	k.	k.	PROPN
ejpam-6699	12	34	tamilvanan	tamilvanan	PROPN
ejpam-6699	12	35	)	)	PUNCT
ejpam-6699	12	36	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6699	13	1	1	1	NUM
ejpam-6699	13	2	copyright	copyright	NOUN
ejpam-6699	13	3	:	:	PUNCT
ejpam-6699	13	4	©	©	PROPN
ejpam-6699	13	5	2025	2025	NUM
ejpam-6699	13	6	the	the	DET
ejpam-6699	13	7	author(s	author(s	NOUN
ejpam-6699	13	8	)	)	PUNCT
ejpam-6699	13	9	.	.	PUNCT
ejpam-6699	14	1	(	(	PUNCT
ejpam-6699	14	2	cc	cc	NOUN
ejpam-6699	14	3	by	by	ADP
ejpam-6699	14	4	-	-	PUNCT
ejpam-6699	14	5	nc	nc	PROPN
ejpam-6699	14	6	4.0	4.0	NUM
ejpam-6699	14	7	)	)	PUNCT
ejpam-6699	14	8	s.	s.	PROPN
ejpam-6699	14	9	gowri	gowri	PROPN
ejpam-6699	14	10	et	et	PROPN
ejpam-6699	14	11	al	al	PROPN
ejpam-6699	14	12	.	.	PUNCT
ejpam-6699	14	13	/	/	SYM
ejpam-6699	14	14	eur	eur	PROPN
ejpam-6699	14	15	.	.	PUNCT
ejpam-6699	15	1	j.	j.	PROPN
ejpam-6699	15	2	pure	pure	PROPN
ejpam-6699	15	3	appl	appl	PROPN
ejpam-6699	15	4	.	.	PROPN
ejpam-6699	15	5	math	math	PROPN
ejpam-6699	15	6	,	,	PUNCT
ejpam-6699	15	7	18	18	NUM
ejpam-6699	15	8	(	(	PUNCT
ejpam-6699	15	9	4	4	NUM
ejpam-6699	15	10	)	)	PUNCT
ejpam-6699	15	11	(	(	PUNCT
ejpam-6699	15	12	2025	2025	NUM
ejpam-6699	15	13	)	)	PUNCT
ejpam-6699	15	14	,	,	PUNCT
ejpam-6699	15	15	6699	6699	NUM
ejpam-6699	15	16	2	2	NUM
ejpam-6699	15	17	of	of	ADP
ejpam-6699	15	18	14	14	NUM
ejpam-6699	15	19	equation	equation	NOUN
ejpam-6699	15	20	stability	stability	NOUN
ejpam-6699	15	21	was	be	AUX
ejpam-6699	15	22	developed	develop	VERB
ejpam-6699	15	23	as	as	ADP
ejpam-6699	15	24	a	a	DET
ejpam-6699	15	25	result	result	NOUN
ejpam-6699	15	26	of	of	ADP
ejpam-6699	15	27	this	this	DET
ejpam-6699	15	28	investigation	investigation	NOUN
ejpam-6699	15	29	.	.	PUNCT
ejpam-6699	16	1	hyers	hyer	NOUN
ejpam-6699	17	1	[	[	X
ejpam-6699	17	2	2	2	X
ejpam-6699	17	3	]	]	PUNCT
ejpam-6699	17	4	in	in	ADP
ejpam-6699	17	5	1941	1941	NUM
ejpam-6699	17	6	provided	provide	VERB
ejpam-6699	17	7	a	a	DET
ejpam-6699	17	8	crucial	crucial	ADJ
ejpam-6699	17	9	answer	answer	NOUN
ejpam-6699	17	10	by	by	ADP
ejpam-6699	17	11	establishing	establish	VERB
ejpam-6699	17	12	the	the	DET
ejpam-6699	17	13	first	first	ADJ
ejpam-6699	17	14	stability	stability	NOUN
ejpam-6699	17	15	result	result	VERB
ejpam-6699	17	16	for	for	ADP
ejpam-6699	17	17	linear	linear	ADJ
ejpam-6699	17	18	functional	functional	ADJ
ejpam-6699	17	19	equations	equation	NOUN
ejpam-6699	17	20	,	,	PUNCT
ejpam-6699	17	21	which	which	PRON
ejpam-6699	17	22	is	be	AUX
ejpam-6699	17	23	now	now	ADV
ejpam-6699	17	24	known	know	VERB
ejpam-6699	17	25	as	as	ADP
ejpam-6699	17	26	the	the	DET
ejpam-6699	17	27	hyers	hyers	PROPN
ejpam-6699	17	28	-	-	PUNCT
ejpam-6699	17	29	ulam	ulam	PROPN
ejpam-6699	17	30	stability	stability	NOUN
ejpam-6699	17	31	.	.	PUNCT
ejpam-6699	18	1	later	later	ADV
ejpam-6699	18	2	,	,	PUNCT
ejpam-6699	18	3	rassias	rassias	PROPN
ejpam-6699	18	4	developed	develop	VERB
ejpam-6699	18	5	hyers	hyer	NOUN
ejpam-6699	18	6	-	-	PUNCT
ejpam-6699	18	7	ulamrassias	ulamrassias	ADJ
ejpam-6699	18	8	stability	stability	NOUN
ejpam-6699	18	9	theory	theory	NOUN
ejpam-6699	18	10	by	by	ADP
ejpam-6699	18	11	adding	add	VERB
ejpam-6699	18	12	a	a	DET
ejpam-6699	18	13	perturbation	perturbation	NOUN
ejpam-6699	18	14	term	term	NOUN
ejpam-6699	18	15	that	that	PRON
ejpam-6699	18	16	was	be	AUX
ejpam-6699	18	17	dependent	dependent	ADJ
ejpam-6699	18	18	on	on	ADP
ejpam-6699	18	19	the	the	DET
ejpam-6699	18	20	norm	norm	NOUN
ejpam-6699	18	21	,	,	PUNCT
ejpam-6699	18	22	which	which	PRON
ejpam-6699	18	23	was	be	AUX
ejpam-6699	18	24	a	a	DET
ejpam-6699	18	25	substantial	substantial	ADJ
ejpam-6699	18	26	generalization	generalization	NOUN
ejpam-6699	18	27	.	.	PUNCT
ejpam-6699	19	1	notable	notable	ADJ
ejpam-6699	19	2	among	among	ADP
ejpam-6699	19	3	the	the	DET
ejpam-6699	19	4	many	many	ADJ
ejpam-6699	19	5	functional	functional	ADJ
ejpam-6699	19	6	equation	equation	NOUN
ejpam-6699	19	7	classes	class	NOUN
ejpam-6699	19	8	are	be	AUX
ejpam-6699	19	9	quartic	quartic	ADJ
ejpam-6699	19	10	functional	functional	ADJ
ejpam-6699	19	11	equations	equation	NOUN
ejpam-6699	19	12	,	,	PUNCT
ejpam-6699	19	13	which	which	PRON
ejpam-6699	19	14	generalize	generalize	VERB
ejpam-6699	19	15	the	the	DET
ejpam-6699	19	16	behavior	behavior	NOUN
ejpam-6699	19	17	of	of	ADP
ejpam-6699	19	18	fourth	fourth	ADJ
ejpam-6699	19	19	-	-	PUNCT
ejpam-6699	19	20	degree	degree	NOUN
ejpam-6699	19	21	polynomials	polynomial	NOUN
ejpam-6699	19	22	.	.	PUNCT
ejpam-6699	20	1	approximation	approximation	NOUN
ejpam-6699	20	2	theory	theory	NOUN
ejpam-6699	20	3	,	,	PUNCT
ejpam-6699	20	4	information	information	NOUN
ejpam-6699	20	5	theory	theory	NOUN
ejpam-6699	20	6	,	,	PUNCT
ejpam-6699	20	7	theoretical	theoretical	ADJ
ejpam-6699	20	8	physics	physics	NOUN
ejpam-6699	20	9	,	,	PUNCT
ejpam-6699	20	10	and	and	CCONJ
ejpam-6699	20	11	differential	differential	ADJ
ejpam-6699	20	12	equations	equation	NOUN
ejpam-6699	20	13	all	all	PRON
ejpam-6699	20	14	naturally	naturally	ADV
ejpam-6699	20	15	produce	produce	VERB
ejpam-6699	20	16	these	these	DET
ejpam-6699	20	17	types	type	NOUN
ejpam-6699	20	18	of	of	ADP
ejpam-6699	20	19	equations	equation	NOUN
ejpam-6699	20	20	.	.	PUNCT
ejpam-6699	21	1	the	the	DET
ejpam-6699	21	2	quartic	quartic	ADJ
ejpam-6699	21	3	functional	functional	ADJ
ejpam-6699	21	4	equation	equation	NOUN
ejpam-6699	21	5	τ(t1	τ(t1	ADP
ejpam-6699	21	6	+	+	CCONJ
ejpam-6699	21	7	2t2	2t2	NUM
ejpam-6699	21	8	)	)	PUNCT
ejpam-6699	22	1	+	+	CCONJ
ejpam-6699	22	2	τ(t1	τ(t1	ADP
ejpam-6699	22	3	−	−	PROPN
ejpam-6699	22	4	2t2	2t2	NUM
ejpam-6699	22	5	)	)	PUNCT
ejpam-6699	22	6	=	=	SYM
ejpam-6699	23	1	4[τ(t1	4[τ(t1	NUM
ejpam-6699	24	1	+	+	NUM
ejpam-6699	24	2	t2	t2	NOUN
ejpam-6699	24	3	)	)	PUNCT
ejpam-6699	25	1	+	+	CCONJ
ejpam-6699	25	2	τ(t1	τ(t1	ADP
ejpam-6699	25	3	−	−	PROPN
ejpam-6699	25	4	t2	t2	PROPN
ejpam-6699	25	5	)	)	PUNCT
ejpam-6699	25	6	]	]	PUNCT
ejpam-6699	26	1	+	+	CCONJ
ejpam-6699	26	2	6τ(t1)−	6τ(t1)−	NUM
ejpam-6699	26	3	24τ(t2	24τ(t2	NUM
ejpam-6699	26	4	)	)	PUNCT
ejpam-6699	26	5	serves	serve	VERB
ejpam-6699	26	6	as	as	ADP
ejpam-6699	26	7	a	a	DET
ejpam-6699	26	8	classical	classical	ADJ
ejpam-6699	26	9	example	example	NOUN
ejpam-6699	26	10	whose	whose	DET
ejpam-6699	26	11	general	general	ADJ
ejpam-6699	26	12	solution	solution	NOUN
ejpam-6699	26	13	often	often	ADV
ejpam-6699	26	14	involves	involve	VERB
ejpam-6699	26	15	quartic	quartic	ADJ
ejpam-6699	26	16	mappings	mapping	NOUN
ejpam-6699	26	17	such	such	ADJ
ejpam-6699	26	18	as	as	ADP
ejpam-6699	26	19	τ(t	τ(t	NOUN
ejpam-6699	26	20	)	)	PUNCT
ejpam-6699	26	21	=	=	SYM
ejpam-6699	26	22	at4	at4	PROPN
ejpam-6699	26	23	.	.	PUNCT
ejpam-6699	27	1	a	a	DET
ejpam-6699	27	2	lot	lot	NOUN
ejpam-6699	27	3	of	of	ADP
ejpam-6699	27	4	research	research	NOUN
ejpam-6699	27	5	has	have	AUX
ejpam-6699	27	6	been	be	AUX
ejpam-6699	27	7	done	do	VERB
ejpam-6699	27	8	on	on	ADP
ejpam-6699	27	9	how	how	SCONJ
ejpam-6699	27	10	stable	stable	ADJ
ejpam-6699	27	11	these	these	DET
ejpam-6699	27	12	kinds	kind	NOUN
ejpam-6699	27	13	of	of	ADP
ejpam-6699	27	14	equations	equation	NOUN
ejpam-6699	27	15	are	be	AUX
ejpam-6699	27	16	in	in	ADP
ejpam-6699	27	17	both	both	CCONJ
ejpam-6699	27	18	standard	standard	ADJ
ejpam-6699	27	19	and	and	CCONJ
ejpam-6699	27	20	extended	extend	VERB
ejpam-6699	27	21	normed	normed	ADJ
ejpam-6699	27	22	settings	setting	NOUN
ejpam-6699	27	23	[	[	X
ejpam-6699	27	24	3–9	3–9	NUM
ejpam-6699	27	25	]	]	PUNCT
ejpam-6699	27	26	.	.	PUNCT
ejpam-6699	28	1	a	a	DET
ejpam-6699	28	2	notable	notable	ADJ
ejpam-6699	28	3	development	development	NOUN
ejpam-6699	28	4	in	in	ADP
ejpam-6699	28	5	this	this	DET
ejpam-6699	28	6	field	field	NOUN
ejpam-6699	28	7	pertains	pertain	VERB
ejpam-6699	28	8	to	to	ADP
ejpam-6699	28	9	the	the	DET
ejpam-6699	28	10	application	application	NOUN
ejpam-6699	28	11	of	of	ADP
ejpam-6699	28	12	non	non	ADJ
ejpam-6699	28	13	-	-	ADJ
ejpam-6699	28	14	archimedean	archimedean	ADJ
ejpam-6699	28	15	normed	norme	VERB
ejpam-6699	28	16	spaces	space	NOUN
ejpam-6699	28	17	.	.	PUNCT
ejpam-6699	29	1	these	these	DET
ejpam-6699	29	2	spaces	space	NOUN
ejpam-6699	29	3	satisfy	satisfy	VERB
ejpam-6699	29	4	the	the	DET
ejpam-6699	29	5	strong	strong	ADJ
ejpam-6699	29	6	triangle	triangle	NOUN
ejpam-6699	29	7	inequality	inequality	NOUN
ejpam-6699	29	8	:	:	PUNCT
ejpam-6699	29	9	∥t1	∥t1	X
ejpam-6699	29	10	+	+	CCONJ
ejpam-6699	29	11	t2∥	t2∥	VERB
ejpam-6699	29	12	≤	≤	ADJ
ejpam-6699	29	13	max{∥t1∥	max{∥t1∥	NOUN
ejpam-6699	29	14	,	,	PUNCT
ejpam-6699	29	15	∥t2∥	∥t2∥	PROPN
ejpam-6699	29	16	}	}	PUNCT
ejpam-6699	29	17	,	,	PUNCT
ejpam-6699	29	18	which	which	PRON
ejpam-6699	29	19	endows	endow	VERB
ejpam-6699	29	20	them	they	PRON
ejpam-6699	29	21	with	with	ADP
ejpam-6699	29	22	distinct	distinct	ADJ
ejpam-6699	29	23	topological	topological	ADJ
ejpam-6699	29	24	and	and	CCONJ
ejpam-6699	29	25	algebraic	algebraic	ADJ
ejpam-6699	29	26	properties	property	NOUN
ejpam-6699	29	27	(	(	PUNCT
ejpam-6699	29	28	see	see	VERB
ejpam-6699	29	29	,	,	PUNCT
ejpam-6699	29	30	[	[	X
ejpam-6699	29	31	10	10	NUM
ejpam-6699	29	32	,	,	PUNCT
ejpam-6699	29	33	11	11	NUM
ejpam-6699	29	34	]	]	NUM
ejpam-6699	29	35	)	)	PUNCT
ejpam-6699	29	36	.	.	PUNCT
ejpam-6699	30	1	such	such	ADJ
ejpam-6699	30	2	spaces	space	NOUN
ejpam-6699	30	3	frequently	frequently	ADV
ejpam-6699	30	4	occur	occur	VERB
ejpam-6699	30	5	in	in	ADP
ejpam-6699	30	6	p	p	NOUN
ejpam-6699	30	7	-	-	PUNCT
ejpam-6699	30	8	adic	adic	ADJ
ejpam-6699	30	9	analysis	analysis	NOUN
ejpam-6699	30	10	,	,	PUNCT
ejpam-6699	30	11	number	number	NOUN
ejpam-6699	30	12	theory	theory	NOUN
ejpam-6699	30	13	,	,	PUNCT
ejpam-6699	30	14	and	and	CCONJ
ejpam-6699	30	15	information	information	NOUN
ejpam-6699	30	16	theory	theory	NOUN
ejpam-6699	30	17	.	.	PUNCT
ejpam-6699	31	1	building	build	VERB
ejpam-6699	31	2	on	on	ADP
ejpam-6699	31	3	this	this	DET
ejpam-6699	31	4	framework	framework	NOUN
ejpam-6699	31	5	,	,	PUNCT
ejpam-6699	31	6	the	the	DET
ejpam-6699	31	7	notion	notion	NOUN
ejpam-6699	31	8	of	of	ADP
ejpam-6699	31	9	(	(	PUNCT
ejpam-6699	31	10	n	n	CCONJ
ejpam-6699	31	11	,	,	PUNCT
ejpam-6699	31	12	β)-normed	β)-normed	PUNCT
ejpam-6699	31	13	spaces	space	NOUN
ejpam-6699	31	14	was	be	AUX
ejpam-6699	31	15	introduced	introduce	VERB
ejpam-6699	31	16	to	to	PART
ejpam-6699	31	17	capture	capture	VERB
ejpam-6699	31	18	a	a	DET
ejpam-6699	31	19	broader	broad	ADJ
ejpam-6699	31	20	class	class	NOUN
ejpam-6699	31	21	of	of	ADP
ejpam-6699	31	22	normed	normed	ADJ
ejpam-6699	31	23	structures	structure	NOUN
ejpam-6699	31	24	(	(	PUNCT
ejpam-6699	31	25	ref	ref	NOUN
ejpam-6699	31	26	.	.	PUNCT
ejpam-6699	32	1	[	[	X
ejpam-6699	32	2	12–14]).these	12–14]).these	ADJ
ejpam-6699	32	3	spaces	space	NOUN
ejpam-6699	32	4	provide	provide	VERB
ejpam-6699	32	5	a	a	DET
ejpam-6699	32	6	more	more	ADV
ejpam-6699	32	7	comprehensive	comprehensive	ADJ
ejpam-6699	32	8	analytical	analytical	ADJ
ejpam-6699	32	9	investigation	investigation	NOUN
ejpam-6699	32	10	by	by	ADP
ejpam-6699	32	11	generalizing	generalize	VERB
ejpam-6699	32	12	n	n	CCONJ
ejpam-6699	32	13	-	-	PUNCT
ejpam-6699	32	14	normed	norme	VERB
ejpam-6699	32	15	spaces	space	NOUN
ejpam-6699	32	16	(	(	PUNCT
ejpam-6699	32	17	β	β	X
ejpam-6699	32	18	=	=	SYM
ejpam-6699	32	19	1	1	NUM
ejpam-6699	32	20	)	)	PUNCT
ejpam-6699	32	21	and	and	CCONJ
ejpam-6699	32	22	β	β	X
ejpam-6699	32	23	-	-	ADJ
ejpam-6699	32	24	normed	norme	VERB
ejpam-6699	32	25	spaces	space	NOUN
ejpam-6699	32	26	(	(	PUNCT
ejpam-6699	32	27	n	n	NOUN
ejpam-6699	32	28	=	=	SYM
ejpam-6699	32	29	1	1	NUM
ejpam-6699	32	30	)	)	PUNCT
ejpam-6699	32	31	(	(	PUNCT
ejpam-6699	32	32	ref	ref	NOUN
ejpam-6699	32	33	.	.	PUNCT
ejpam-6699	33	1	[	[	X
ejpam-6699	33	2	15–17	15–17	NUM
ejpam-6699	33	3	]	]	PUNCT
ejpam-6699	33	4	)	)	PUNCT
ejpam-6699	33	5	.	.	PUNCT
ejpam-6699	34	1	recent	recent	ADJ
ejpam-6699	34	2	research	research	NOUN
ejpam-6699	34	3	has	have	AUX
ejpam-6699	34	4	expanded	expand	VERB
ejpam-6699	34	5	classical	classical	ADJ
ejpam-6699	34	6	stability	stability	NOUN
ejpam-6699	34	7	results	result	NOUN
ejpam-6699	34	8	to	to	ADP
ejpam-6699	34	9	non	non	ADJ
ejpam-6699	34	10	-	-	ADJ
ejpam-6699	34	11	archimedean	archimedean	ADJ
ejpam-6699	34	12	(	(	PUNCT
ejpam-6699	34	13	n	n	CCONJ
ejpam-6699	34	14	,	,	PUNCT
ejpam-6699	34	15	β)normed	β)normed	ADJ
ejpam-6699	34	16	spaces	space	NOUN
ejpam-6699	34	17	,	,	PUNCT
ejpam-6699	34	18	analyzing	analyze	VERB
ejpam-6699	34	19	quadratic	quadratic	ADJ
ejpam-6699	34	20	,	,	PUNCT
ejpam-6699	34	21	cubic	cubic	ADJ
ejpam-6699	34	22	,	,	PUNCT
ejpam-6699	34	23	and	and	CCONJ
ejpam-6699	34	24	quartic	quartic	ADJ
ejpam-6699	34	25	equations	equation	NOUN
ejpam-6699	34	26	through	through	ADP
ejpam-6699	34	27	novel	novel	ADJ
ejpam-6699	34	28	fixed	fix	VERB
ejpam-6699	34	29	point	point	NOUN
ejpam-6699	34	30	methods	method	NOUN
ejpam-6699	34	31	and	and	CCONJ
ejpam-6699	34	32	contractive	contractive	ADJ
ejpam-6699	34	33	conditions	condition	NOUN
ejpam-6699	34	34	.	.	PUNCT
ejpam-6699	35	1	these	these	DET
ejpam-6699	35	2	advancements	advancement	NOUN
ejpam-6699	35	3	offer	offer	VERB
ejpam-6699	35	4	enhanced	enhance	VERB
ejpam-6699	35	5	understanding	understanding	NOUN
ejpam-6699	35	6	of	of	ADP
ejpam-6699	35	7	the	the	DET
ejpam-6699	35	8	structure	structure	NOUN
ejpam-6699	35	9	of	of	ADP
ejpam-6699	35	10	functional	functional	ADJ
ejpam-6699	35	11	equations	equation	NOUN
ejpam-6699	35	12	in	in	ADP
ejpam-6699	35	13	the	the	DET
ejpam-6699	35	14	context	context	NOUN
ejpam-6699	35	15	of	of	ADP
ejpam-6699	35	16	perturbations	perturbation	NOUN
ejpam-6699	35	17	within	within	ADP
ejpam-6699	35	18	ultrametric	ultrametric	ADJ
ejpam-6699	35	19	environments	environment	NOUN
ejpam-6699	35	20	(	(	PUNCT
ejpam-6699	35	21	[	[	X
ejpam-6699	35	22	18–27	18–27	NUM
ejpam-6699	35	23	]	]	NUM
ejpam-6699	35	24	)	)	PUNCT
ejpam-6699	35	25	.	.	PUNCT
ejpam-6699	36	1	this	this	DET
ejpam-6699	36	2	paper	paper	NOUN
ejpam-6699	36	3	aims	aim	VERB
ejpam-6699	36	4	to	to	PART
ejpam-6699	36	5	investigate	investigate	VERB
ejpam-6699	36	6	the	the	DET
ejpam-6699	36	7	hyers	hyer	NOUN
ejpam-6699	36	8	-	-	PUNCT
ejpam-6699	36	9	ulam	ulam	ADJ
ejpam-6699	36	10	stability	stability	NOUN
ejpam-6699	36	11	of	of	ADP
ejpam-6699	36	12	a	a	DET
ejpam-6699	36	13	generalized	generalize	VERB
ejpam-6699	36	14	quartic	quartic	ADJ
ejpam-6699	36	15	functional	functional	ADJ
ejpam-6699	36	16	equation	equation	NOUN
ejpam-6699	36	17	within	within	ADP
ejpam-6699	36	18	non	non	ADJ
ejpam-6699	36	19	-	-	ADJ
ejpam-6699	36	20	archimedean	archimedean	ADJ
ejpam-6699	36	21	(	(	PUNCT
ejpam-6699	36	22	n	n	CCONJ
ejpam-6699	36	23	,	,	PUNCT
ejpam-6699	36	24	β)-normed	β)-normed	PUNCT
ejpam-6699	36	25	spaces	space	NOUN
ejpam-6699	36	26	.	.	PUNCT
ejpam-6699	37	1	by	by	ADP
ejpam-6699	37	2	employing	employ	VERB
ejpam-6699	37	3	direct	direct	ADJ
ejpam-6699	37	4	analytical	analytical	ADJ
ejpam-6699	37	5	methods	method	NOUN
ejpam-6699	37	6	alongside	alongside	ADP
ejpam-6699	37	7	fixed	fix	VERB
ejpam-6699	37	8	point	point	NOUN
ejpam-6699	37	9	techniques	technique	NOUN
ejpam-6699	37	10	,	,	PUNCT
ejpam-6699	37	11	we	we	PRON
ejpam-6699	37	12	derive	derive	VERB
ejpam-6699	37	13	new	new	ADJ
ejpam-6699	37	14	stability	stability	NOUN
ejpam-6699	37	15	results	result	NOUN
ejpam-6699	37	16	and	and	CCONJ
ejpam-6699	37	17	outline	outline	VERB
ejpam-6699	37	18	conditions	condition	NOUN
ejpam-6699	37	19	for	for	ADP
ejpam-6699	37	20	the	the	DET
ejpam-6699	37	21	existence	existence	NOUN
ejpam-6699	37	22	and	and	CCONJ
ejpam-6699	37	23	uniqueness	uniqueness	NOUN
ejpam-6699	37	24	of	of	ADP
ejpam-6699	37	25	quartic	quartic	ADJ
ejpam-6699	37	26	solutions	solution	NOUN
ejpam-6699	37	27	.	.	PUNCT
ejpam-6699	38	1	the	the	DET
ejpam-6699	38	2	findings	finding	NOUN
ejpam-6699	38	3	present	present	VERB
ejpam-6699	38	4	a	a	DET
ejpam-6699	38	5	notable	notable	ADJ
ejpam-6699	38	6	extension	extension	NOUN
ejpam-6699	38	7	of	of	ADP
ejpam-6699	38	8	current	current	ADJ
ejpam-6699	38	9	research	research	NOUN
ejpam-6699	38	10	and	and	CCONJ
ejpam-6699	38	11	pave	pave	VERB
ejpam-6699	38	12	the	the	DET
ejpam-6699	38	13	way	way	NOUN
ejpam-6699	38	14	for	for	ADP
ejpam-6699	38	15	future	future	ADJ
ejpam-6699	38	16	investigations	investigation	NOUN
ejpam-6699	38	17	in	in	ADP
ejpam-6699	38	18	abstract	abstract	ADJ
ejpam-6699	38	19	analysis	analysis	NOUN
ejpam-6699	38	20	and	and	CCONJ
ejpam-6699	38	21	p	p	NOUN
ejpam-6699	38	22	-	-	PUNCT
ejpam-6699	38	23	adic	adic	ADJ
ejpam-6699	38	24	functional	functional	ADJ
ejpam-6699	38	25	theory	theory	NOUN
ejpam-6699	38	26	.	.	PUNCT
ejpam-6699	39	1	the	the	DET
ejpam-6699	39	2	purpose	purpose	NOUN
ejpam-6699	39	3	of	of	ADP
ejpam-6699	39	4	this	this	DET
ejpam-6699	39	5	study	study	NOUN
ejpam-6699	39	6	is	be	AUX
ejpam-6699	39	7	to	to	PART
ejpam-6699	39	8	explore	explore	VERB
ejpam-6699	39	9	the	the	DET
ejpam-6699	39	10	hyers	hyer	NOUN
ejpam-6699	39	11	-	-	PUNCT
ejpam-6699	39	12	ulam	ulam	ADJ
ejpam-6699	39	13	stability	stability	NOUN
ejpam-6699	39	14	of	of	ADP
ejpam-6699	39	15	the	the	DET
ejpam-6699	39	16	generalized	generalize	VERB
ejpam-6699	39	17	quartic	quartic	ADJ
ejpam-6699	39	18	functional	functional	ADJ
ejpam-6699	39	19	equation	equation	NOUN
ejpam-6699	39	20	ϕ	ϕ	NOUN
ejpam-6699	39	21	(	(	PUNCT
ejpam-6699	39	22	r∑	r∑	NOUN
ejpam-6699	39	23	i=1	i=1	X
ejpam-6699	39	24	ti	ti	NOUN
ejpam-6699	39	25	)	)	PUNCT
ejpam-6699	39	26	=	=	PUNCT
ejpam-6699	40	1	∑	∑	PUNCT
ejpam-6699	40	2	1≤i	1≤i	X
ejpam-6699	40	3	<	<	X
ejpam-6699	40	4	j	j	X
ejpam-6699	40	5	<	<	X
ejpam-6699	40	6	k	k	X
ejpam-6699	40	7	<	<	X
ejpam-6699	40	8	l≤r	l≤r	NOUN
ejpam-6699	40	9	ϕ(ti	ϕ(ti	NOUN
ejpam-6699	40	10	+	+	CCONJ
ejpam-6699	40	11	tj	tj	PROPN
ejpam-6699	40	12	+	+	PROPN
ejpam-6699	40	13	tk	tk	PROPN
ejpam-6699	40	14	+	+	CCONJ
ejpam-6699	40	15	tl	tl	PROPN
ejpam-6699	40	16	)	)	PUNCT
ejpam-6699	41	1	+	+	CCONJ
ejpam-6699	41	2	(	(	PUNCT
ejpam-6699	41	3	−r	−r	ADJ
ejpam-6699	41	4	+	+	X
ejpam-6699	41	5	4	4	NUM
ejpam-6699	41	6	)	)	PUNCT
ejpam-6699	41	7	∑	∑	PUNCT
ejpam-6699	41	8	1≤i	1≤i	PROPN
ejpam-6699	41	9	<	<	X
ejpam-6699	41	10	j	j	X
ejpam-6699	41	11	<	<	X
ejpam-6699	41	12	k≤r	k≤r	PROPN
ejpam-6699	41	13	ϕ(ti	ϕ(ti	PROPN
ejpam-6699	41	14	+	+	CCONJ
ejpam-6699	41	15	tj	tj	PROPN
ejpam-6699	41	16	+	+	X
ejpam-6699	41	17	tk	tk	PROPN
ejpam-6699	41	18	)	)	PUNCT
ejpam-6699	41	19	+	+	CCONJ
ejpam-6699	41	20	(	(	PUNCT
ejpam-6699	41	21	r2−7r+12	r2−7r+12	X
ejpam-6699	41	22	2	2	NUM
ejpam-6699	41	23	)	)	PUNCT
ejpam-6699	41	24	∑r	∑r	NOUN
ejpam-6699	41	25	1	1	NUM
ejpam-6699	41	26	=	=	NOUN
ejpam-6699	41	27	i;i	i;i	PROPN
ejpam-6699	41	28	̸=j	̸=j	PROPN
ejpam-6699	41	29	ϕ(ti	ϕ(ti	PROPN
ejpam-6699	41	30	+	+	PUNCT
ejpam-6699	41	31	tj)−	tj)−	NOUN
ejpam-6699	41	32	∑r	∑r	PROPN
ejpam-6699	41	33	i=1	i=1	PROPN
ejpam-6699	41	34	ϕ(2ti	ϕ(2ti	PRON
ejpam-6699	41	35	)	)	PUNCT
ejpam-6699	41	36	s.	s.	PROPN
ejpam-6699	41	37	gowri	gowri	PROPN
ejpam-6699	41	38	et	et	PROPN
ejpam-6699	41	39	al	al	PROPN
ejpam-6699	41	40	.	.	PUNCT
ejpam-6699	41	41	/	/	SYM
ejpam-6699	41	42	eur	eur	PROPN
ejpam-6699	41	43	.	.	PUNCT
ejpam-6699	42	1	j.	j.	PROPN
ejpam-6699	42	2	pure	pure	PROPN
ejpam-6699	42	3	appl	appl	PROPN
ejpam-6699	42	4	.	.	PROPN
ejpam-6699	42	5	math	math	PROPN
ejpam-6699	42	6	,	,	PUNCT
ejpam-6699	42	7	18	18	NUM
ejpam-6699	42	8	(	(	PUNCT
ejpam-6699	42	9	4	4	NUM
ejpam-6699	42	10	)	)	PUNCT
ejpam-6699	42	11	(	(	PUNCT
ejpam-6699	42	12	2025	2025	NUM
ejpam-6699	42	13	)	)	PUNCT
ejpam-6699	42	14	,	,	PUNCT
ejpam-6699	42	15	6699	6699	NUM
ejpam-6699	42	16	3	3	NUM
ejpam-6699	42	17	of	of	ADP
ejpam-6699	42	18	14	14	NUM
ejpam-6699	42	19	+	+	CCONJ
ejpam-6699	42	20	(	(	PUNCT
ejpam-6699	42	21	−r3	−r3	PROPN
ejpam-6699	42	22	+	+	PROPN
ejpam-6699	42	23	9r2−26r+120	9r2−26r+120	NUM
ejpam-6699	42	24	6	6	NUM
ejpam-6699	42	25	)	)	PUNCT
ejpam-6699	43	1	∑r	∑r	PROPN
ejpam-6699	43	2	i=1	i=1	PROPN
ejpam-6699	43	3	(	(	PUNCT
ejpam-6699	43	4	ϕ(ti)+ϕ(−ti	ϕ(ti)+ϕ(−ti	PROPN
ejpam-6699	43	5	)	)	PUNCT
ejpam-6699	43	6	2	2	NUM
ejpam-6699	43	7	)	)	PUNCT
ejpam-6699	43	8	,	,	PUNCT
ejpam-6699	43	9	(	(	PUNCT
ejpam-6699	43	10	1	1	X
ejpam-6699	43	11	)	)	PUNCT
ejpam-6699	43	12	where	where	SCONJ
ejpam-6699	43	13	r	r	NOUN
ejpam-6699	43	14	≥	≥	NOUN
ejpam-6699	43	15	4	4	NUM
ejpam-6699	43	16	,	,	PUNCT
ejpam-6699	43	17	in	in	ADP
ejpam-6699	43	18	non	non	ADJ
ejpam-6699	43	19	-	-	ADJ
ejpam-6699	43	20	archimedean	archimedean	ADJ
ejpam-6699	43	21	(	(	PUNCT
ejpam-6699	43	22	n	n	CCONJ
ejpam-6699	43	23	,	,	PUNCT
ejpam-6699	43	24	β)-normed	β)-normed	PUNCT
ejpam-6699	43	25	spaces	space	NOUN
ejpam-6699	43	26	.	.	PUNCT
ejpam-6699	44	1	2	2	X
ejpam-6699	44	2	.	.	NUM
ejpam-6699	44	3	preliminaries	preliminary	NOUN
ejpam-6699	44	4	the	the	DET
ejpam-6699	44	5	following	follow	VERB
ejpam-6699	44	6	are	be	AUX
ejpam-6699	44	7	some	some	DET
ejpam-6699	44	8	ideas	idea	NOUN
ejpam-6699	44	9	and	and	CCONJ
ejpam-6699	44	10	findings	finding	NOUN
ejpam-6699	44	11	that	that	PRON
ejpam-6699	44	12	will	will	AUX
ejpam-6699	44	13	be	be	AUX
ejpam-6699	44	14	utilized	utilize	VERB
ejpam-6699	44	15	in	in	ADP
ejpam-6699	44	16	the	the	DET
ejpam-6699	44	17	upcoming	upcoming	ADJ
ejpam-6699	44	18	sections	section	NOUN
ejpam-6699	44	19	.	.	PUNCT
ejpam-6699	45	1	here	here	ADV
ejpam-6699	45	2	,	,	PUNCT
ejpam-6699	45	3	we	we	PRON
ejpam-6699	45	4	denote	denote	VERB
ejpam-6699	45	5	n	n	CCONJ
ejpam-6699	45	6	as	as	ADP
ejpam-6699	45	7	the	the	DET
ejpam-6699	45	8	set	set	NOUN
ejpam-6699	45	9	of	of	ADP
ejpam-6699	45	10	non	non	ADJ
ejpam-6699	45	11	-	-	ADJ
ejpam-6699	45	12	negative	negative	ADJ
ejpam-6699	45	13	integers	integer	NOUN
ejpam-6699	45	14	,	,	PUNCT
ejpam-6699	45	15	with	with	ADP
ejpam-6699	45	16	n	n	CCONJ
ejpam-6699	45	17	,	,	PUNCT
ejpam-6699	45	18	t	t	PROPN
ejpam-6699	45	19	,	,	PUNCT
ejpam-6699	45	20	p	p	X
ejpam-6699	45	21	,	,	PUNCT
ejpam-6699	45	22	i	i	PROPN
ejpam-6699	45	23	∈	∈	PROPN
ejpam-6699	45	24	n	n	CCONJ
ejpam-6699	45	25	,	,	PUNCT
ejpam-6699	45	26	and	and	CCONJ
ejpam-6699	45	27	fix	fix	VERB
ejpam-6699	45	28	2	2	NUM
ejpam-6699	45	29	≤	≤	NOUN
ejpam-6699	45	30	n.	n.	NOUN
ejpam-6699	45	31	definition	definition	NOUN
ejpam-6699	45	32	1	1	NUM
ejpam-6699	45	33	.	.	PUNCT
ejpam-6699	46	1	[	[	X
ejpam-6699	46	2	28	28	NUM
ejpam-6699	46	3	]	]	X
ejpam-6699	46	4	let	let	VERB
ejpam-6699	46	5	e	e	PRON
ejpam-6699	46	6	be	be	AUX
ejpam-6699	46	7	a	a	DET
ejpam-6699	46	8	linear	linear	ADJ
ejpam-6699	46	9	space	space	NOUN
ejpam-6699	46	10	with	with	ADP
ejpam-6699	46	11	dim	dim	ADJ
ejpam-6699	46	12	e	e	X
ejpam-6699	46	13	≥	≥	NOUN
ejpam-6699	46	14	n	n	CCONJ
ejpam-6699	46	15	,	,	PUNCT
ejpam-6699	46	16	and	and	CCONJ
ejpam-6699	46	17	let	let	VERB
ejpam-6699	46	18	0	0	NUM
ejpam-6699	46	19	<	<	X
ejpam-6699	46	20	β	β	X
ejpam-6699	46	21	≤	≤	NUM
ejpam-6699	46	22	1	1	NUM
ejpam-6699	46	23	.	.	PUNCT
ejpam-6699	47	1	a	a	DET
ejpam-6699	47	2	mapping	mapping	NOUN
ejpam-6699	47	3	∥	∥	NOUN
ejpam-6699	47	4	·	·	PUNCT
ejpam-6699	47	5	,	,	PUNCT
ejpam-6699	47	6	·	·	PUNCT
ejpam-6699	47	7	·	·	PUNCT
ejpam-6699	47	8	·	·	PUNCT
ejpam-6699	47	9	,	,	PUNCT
ejpam-6699	47	10	·	·	PUNCT
ejpam-6699	47	11	∥β	∥β	NOUN
ejpam-6699	47	12	:	:	PUNCT
ejpam-6699	47	13	en	en	X
ejpam-6699	47	14	→	→	SYM
ejpam-6699	47	15	r	r	NOUN
ejpam-6699	47	16	is	be	AUX
ejpam-6699	47	17	called	call	VERB
ejpam-6699	47	18	an	an	DET
ejpam-6699	47	19	(	(	PUNCT
ejpam-6699	47	20	n	n	CCONJ
ejpam-6699	47	21	,	,	PUNCT
ejpam-6699	47	22	β)-norm	β)-norm	PUNCT
ejpam-6699	47	23	on	on	ADP
ejpam-6699	47	24	e	e	NOUN
ejpam-6699	47	25	if	if	SCONJ
ejpam-6699	47	26	it	it	PRON
ejpam-6699	47	27	satisfies	satisfy	VERB
ejpam-6699	47	28	the	the	DET
ejpam-6699	47	29	following	follow	VERB
ejpam-6699	47	30	conditions	condition	NOUN
ejpam-6699	47	31	for	for	ADP
ejpam-6699	47	32	every	every	DET
ejpam-6699	47	33	ν1	ν1	NOUN
ejpam-6699	47	34	,	,	PUNCT
ejpam-6699	47	35	·	·	PUNCT
ejpam-6699	47	36	·	·	PUNCT
ejpam-6699	47	37	·	·	PUNCT
ejpam-6699	47	38	,	,	PUNCT
ejpam-6699	47	39	νn	νn	PROPN
ejpam-6699	47	40	,	,	PUNCT
ejpam-6699	47	41	u	u	NOUN
ejpam-6699	47	42	,	,	PUNCT
ejpam-6699	47	43	t	t	PROPN
ejpam-6699	47	44	∈	∈	PROPN
ejpam-6699	47	45	e	e	PROPN
ejpam-6699	47	46	and	and	CCONJ
ejpam-6699	47	47	every	every	DET
ejpam-6699	47	48	λ	λ	X
ejpam-6699	47	49	∈	∈	PROPN
ejpam-6699	47	50	r	r	NOUN
ejpam-6699	47	51	:	:	PUNCT
ejpam-6699	47	52	(	(	PUNCT
ejpam-6699	47	53	i	i	NOUN
ejpam-6699	47	54	)	)	PUNCT
ejpam-6699	47	55	∥ν1	∥ν1	PROPN
ejpam-6699	47	56	,	,	PUNCT
ejpam-6699	47	57	·	·	PUNCT
ejpam-6699	47	58	·	·	PUNCT
ejpam-6699	47	59	·	·	PUNCT
ejpam-6699	47	60	,	,	PUNCT
ejpam-6699	47	61	νn∥β	νn∥β	VERB
ejpam-6699	47	62	=	=	SYM
ejpam-6699	47	63	0	0	PUNCT
ejpam-6699	48	1	if	if	SCONJ
ejpam-6699	48	2	and	and	CCONJ
ejpam-6699	48	3	only	only	ADV
ejpam-6699	48	4	if	if	SCONJ
ejpam-6699	48	5	ν1	ν1	NOUN
ejpam-6699	48	6	,	,	PUNCT
ejpam-6699	48	7	·	·	PUNCT
ejpam-6699	48	8	·	·	PUNCT
ejpam-6699	48	9	·	·	PUNCT
ejpam-6699	48	10	,	,	PUNCT
ejpam-6699	48	11	νn	νn	AUX
ejpam-6699	48	12	are	be	AUX
ejpam-6699	48	13	linearly	linearly	ADV
ejpam-6699	48	14	dependent	dependent	ADJ
ejpam-6699	48	15	;	;	PUNCT
ejpam-6699	48	16	(	(	PUNCT
ejpam-6699	48	17	ii	ii	NOUN
ejpam-6699	48	18	)	)	PUNCT
ejpam-6699	48	19	∥ν1	∥ν1	NOUN
ejpam-6699	48	20	,	,	PUNCT
ejpam-6699	48	21	·	·	PUNCT
ejpam-6699	48	22	·	·	PUNCT
ejpam-6699	48	23	·	·	PUNCT
ejpam-6699	48	24	,	,	PUNCT
ejpam-6699	48	25	νn∥β	νn∥β	NOUN
ejpam-6699	48	26	is	be	AUX
ejpam-6699	48	27	invariant	invariant	ADJ
ejpam-6699	48	28	under	under	ADP
ejpam-6699	48	29	any	any	DET
ejpam-6699	48	30	permutations	permutation	NOUN
ejpam-6699	48	31	of	of	ADP
ejpam-6699	48	32	its	its	PRON
ejpam-6699	48	33	arguments	argument	NOUN
ejpam-6699	48	34	;	;	PUNCT
ejpam-6699	48	35	(	(	PUNCT
ejpam-6699	48	36	iii	iii	X
ejpam-6699	48	37	)	)	PUNCT
ejpam-6699	48	38	∥λν1	∥λν1	PROPN
ejpam-6699	48	39	,	,	PUNCT
ejpam-6699	48	40	·	·	PUNCT
ejpam-6699	48	41	·	·	PUNCT
ejpam-6699	48	42	·	·	PUNCT
ejpam-6699	48	43	,	,	PUNCT
ejpam-6699	48	44	νn∥β	νn∥β	X
ejpam-6699	48	45	=	=	SYM
ejpam-6699	48	46	|λ|β	|λ|β	NOUN
ejpam-6699	48	47	∥ν1	∥ν1	NOUN
ejpam-6699	48	48	,	,	PUNCT
ejpam-6699	48	49	·	·	PUNCT
ejpam-6699	48	50	·	·	PUNCT
ejpam-6699	48	51	·	·	PUNCT
ejpam-6699	48	52	,	,	PUNCT
ejpam-6699	48	53	νn∥β	νn∥β	NOUN
ejpam-6699	48	54	;	;	PUNCT
ejpam-6699	48	55	(	(	PUNCT
ejpam-6699	48	56	iv	iv	X
ejpam-6699	48	57	)	)	PUNCT
ejpam-6699	48	58	∥ν1	∥ν1	NOUN
ejpam-6699	48	59	,	,	PUNCT
ejpam-6699	48	60	·	·	PUNCT
ejpam-6699	48	61	·	·	PUNCT
ejpam-6699	48	62	·	·	PUNCT
ejpam-6699	48	63	,	,	PUNCT
ejpam-6699	48	64	νn−1	νn−1	PROPN
ejpam-6699	48	65	,	,	PUNCT
ejpam-6699	48	66	u+	u+	NOUN
ejpam-6699	48	67	t∥β	t∥β	NOUN
ejpam-6699	48	68	≤	≤	NUM
ejpam-6699	48	69	∥ν1	∥ν1	NOUN
ejpam-6699	48	70	,	,	PUNCT
ejpam-6699	48	71	·	·	PUNCT
ejpam-6699	48	72	·	·	PUNCT
ejpam-6699	48	73	·	·	PUNCT
ejpam-6699	48	74	,	,	PUNCT
ejpam-6699	48	75	νn−1	νn−1	PROPN
ejpam-6699	48	76	,	,	PUNCT
ejpam-6699	48	77	u∥β	u∥β	ADV
ejpam-6699	48	78	+	+	CCONJ
ejpam-6699	48	79	∥ν1	∥ν1	ADJ
ejpam-6699	48	80	,	,	PUNCT
ejpam-6699	48	81	·	·	PUNCT
ejpam-6699	48	82	·	·	PUNCT
ejpam-6699	48	83	·	·	PUNCT
ejpam-6699	48	84	,	,	PUNCT
ejpam-6699	48	85	νn−1	νn−1	PROPN
ejpam-6699	48	86	,	,	PUNCT
ejpam-6699	48	87	t∥β	t∥β	NOUN
ejpam-6699	48	88	.	.	PUNCT
ejpam-6699	49	1	the	the	DET
ejpam-6699	49	2	pair	pair	NOUN
ejpam-6699	49	3	(	(	PUNCT
ejpam-6699	49	4	e	e	NOUN
ejpam-6699	49	5	,	,	PUNCT
ejpam-6699	49	6	∥	∥	X
ejpam-6699	49	7	·	·	PUNCT
ejpam-6699	49	8	,	,	PUNCT
ejpam-6699	49	9	·	·	PUNCT
ejpam-6699	49	10	·	·	PUNCT
ejpam-6699	49	11	·	·	PUNCT
ejpam-6699	49	12	,	,	PUNCT
ejpam-6699	49	13	·	·	PUNCT
ejpam-6699	49	14	∥β	∥β	PROPN
ejpam-6699	49	15	)	)	PUNCT
ejpam-6699	49	16	is	be	AUX
ejpam-6699	49	17	then	then	ADV
ejpam-6699	49	18	called	call	VERB
ejpam-6699	49	19	a	a	DET
ejpam-6699	49	20	linear	linear	NOUN
ejpam-6699	49	21	(	(	PUNCT
ejpam-6699	49	22	n	n	CCONJ
ejpam-6699	49	23	,	,	PUNCT
ejpam-6699	49	24	β)-normed	β)-normed	PUNCT
ejpam-6699	49	25	spaces	space	NOUN
ejpam-6699	49	26	or	or	CCONJ
ejpam-6699	49	27	simply	simply	ADV
ejpam-6699	49	28	(	(	PUNCT
ejpam-6699	49	29	n	n	CCONJ
ejpam-6699	49	30	,	,	PUNCT
ejpam-6699	49	31	β)normed	β)normed	ADJ
ejpam-6699	49	32	space	space	NOUN
ejpam-6699	49	33	.	.	PUNCT
ejpam-6699	50	1	the	the	DET
ejpam-6699	50	2	linear	linear	ADJ
ejpam-6699	50	3	(	(	PUNCT
ejpam-6699	50	4	n	n	CCONJ
ejpam-6699	50	5	,	,	PUNCT
ejpam-6699	50	6	β)-normed	β)-normed	PUNCT
ejpam-6699	50	7	space	space	NOUN
ejpam-6699	50	8	is	be	AUX
ejpam-6699	50	9	defined	define	VERB
ejpam-6699	50	10	as	as	ADP
ejpam-6699	50	11	an	an	DET
ejpam-6699	50	12	integration	integration	NOUN
ejpam-6699	50	13	of	of	ADP
ejpam-6699	50	14	a	a	DET
ejpam-6699	50	15	linear	linear	ADJ
ejpam-6699	50	16	n	n	CCONJ
ejpam-6699	50	17	-	-	PUNCT
ejpam-6699	50	18	normed	norme	VERB
ejpam-6699	50	19	space	space	NOUN
ejpam-6699	50	20	,	,	PUNCT
ejpam-6699	50	21	applicable	applicable	ADJ
ejpam-6699	50	22	when	when	SCONJ
ejpam-6699	50	23	β	β	X
ejpam-6699	50	24	=	=	SYM
ejpam-6699	50	25	1	1	NUM
ejpam-6699	50	26	,	,	PUNCT
ejpam-6699	50	27	and	and	CCONJ
ejpam-6699	50	28	a	a	DET
ejpam-6699	50	29	β	β	X
ejpam-6699	50	30	-	-	PUNCT
ejpam-6699	50	31	normed	normed	ADJ
ejpam-6699	50	32	space	space	NOUN
ejpam-6699	50	33	,	,	PUNCT
ejpam-6699	50	34	relevant	relevant	ADJ
ejpam-6699	50	35	when	when	SCONJ
ejpam-6699	50	36	n	n	X
ejpam-6699	50	37	=	=	SYM
ejpam-6699	50	38	1	1	X
ejpam-6699	50	39	.	.	PUNCT
ejpam-6699	51	1	definition	definition	NOUN
ejpam-6699	51	2	2	2	NUM
ejpam-6699	51	3	.	.	PUNCT
ejpam-6699	52	1	[	[	X
ejpam-6699	52	2	28	28	NUM
ejpam-6699	52	3	]	]	X
ejpam-6699	52	4	let	let	VERB
ejpam-6699	52	5	e	e	PRON
ejpam-6699	52	6	be	be	AUX
ejpam-6699	52	7	a	a	DET
ejpam-6699	52	8	real	real	ADJ
ejpam-6699	52	9	vector	vector	NOUN
ejpam-6699	52	10	space	space	NOUN
ejpam-6699	52	11	over	over	ADP
ejpam-6699	52	12	a	a	DET
ejpam-6699	52	13	scalar	scalar	ADJ
ejpam-6699	52	14	field	field	NOUN
ejpam-6699	52	15	k	k	PROPN
ejpam-6699	52	16	equipped	equip	VERB
ejpam-6699	52	17	with	with	ADP
ejpam-6699	52	18	a	a	DET
ejpam-6699	52	19	nonarchimedean	nonarchimedean	ADJ
ejpam-6699	52	20	non	non	ADJ
ejpam-6699	52	21	-	-	ADJ
ejpam-6699	52	22	trivial	trivial	ADJ
ejpam-6699	52	23	valuation	valuation	NOUN
ejpam-6699	52	24	|	|	ADV
ejpam-6699	52	25	·	·	PUNCT
ejpam-6699	53	1	|	|	ADV
ejpam-6699	53	2	,	,	PUNCT
ejpam-6699	53	3	and	and	CCONJ
ejpam-6699	53	4	assume	assume	VERB
ejpam-6699	53	5	that	that	SCONJ
ejpam-6699	53	6	dim	dim	ADJ
ejpam-6699	53	7	e	e	X
ejpam-6699	53	8	≥	≥	X
ejpam-6699	53	9	n	n	CCONJ
ejpam-6699	53	10	,	,	PUNCT
ejpam-6699	54	1	where	where	SCONJ
ejpam-6699	54	2	n	n	X
ejpam-6699	54	3	∈	∈	PROPN
ejpam-6699	54	4	z+	z+	PUNCT
ejpam-6699	54	5	.	.	PUNCT
ejpam-6699	54	6	let	let	VERB
ejpam-6699	54	7	0	0	NUM
ejpam-6699	54	8	<	<	X
ejpam-6699	54	9	β	β	X
ejpam-6699	54	10	≤	≤	NUM
ejpam-6699	54	11	1	1	NUM
ejpam-6699	54	12	be	be	AUX
ejpam-6699	54	13	a	a	DET
ejpam-6699	54	14	fixed	fix	VERB
ejpam-6699	54	15	constant	constant	ADJ
ejpam-6699	54	16	.	.	PUNCT
ejpam-6699	55	1	a	a	DET
ejpam-6699	55	2	function	function	NOUN
ejpam-6699	55	3	∥	∥	NOUN
ejpam-6699	55	4	·	·	PUNCT
ejpam-6699	55	5	,	,	PUNCT
ejpam-6699	55	6	·	·	PUNCT
ejpam-6699	55	7	·	·	PUNCT
ejpam-6699	55	8	·	·	PUNCT
ejpam-6699	55	9	,	,	PUNCT
ejpam-6699	55	10	·	·	PUNCT
ejpam-6699	55	11	∥β	∥β	NOUN
ejpam-6699	55	12	:	:	PUNCT
ejpam-6699	55	13	en	en	X
ejpam-6699	55	14	→	→	SYM
ejpam-6699	55	15	r	r	NOUN
ejpam-6699	55	16	is	be	AUX
ejpam-6699	55	17	called	call	VERB
ejpam-6699	55	18	an	an	DET
ejpam-6699	55	19	(	(	PUNCT
ejpam-6699	55	20	n	n	CCONJ
ejpam-6699	55	21	,	,	PUNCT
ejpam-6699	55	22	β)-norm	β)-norm	PUNCT
ejpam-6699	55	23	on	on	ADP
ejpam-6699	55	24	e	e	NOUN
ejpam-6699	55	25	if	if	SCONJ
ejpam-6699	55	26	it	it	PRON
ejpam-6699	55	27	satisfies	satisfy	VERB
ejpam-6699	55	28	the	the	DET
ejpam-6699	55	29	following	follow	VERB
ejpam-6699	55	30	conditions	condition	NOUN
ejpam-6699	55	31	for	for	ADP
ejpam-6699	55	32	all	all	DET
ejpam-6699	55	33	λ	λ	PROPN
ejpam-6699	55	34	∈	∈	PROPN
ejpam-6699	55	35	k	k	NOUN
ejpam-6699	55	36	and	and	CCONJ
ejpam-6699	55	37	all	all	DET
ejpam-6699	55	38	ν0	ν0	PROPN
ejpam-6699	55	39	,	,	PUNCT
ejpam-6699	55	40	ν1	ν1	NOUN
ejpam-6699	55	41	,	,	PUNCT
ejpam-6699	55	42	·	·	PUNCT
ejpam-6699	55	43	·	·	PUNCT
ejpam-6699	55	44	·	·	PUNCT
ejpam-6699	55	45	,	,	PUNCT
ejpam-6699	55	46	νn	νn	X
ejpam-6699	55	47	∈	∈	PROPN
ejpam-6699	55	48	e.	e.	PROPN
ejpam-6699	55	49	(	(	PUNCT
ejpam-6699	55	50	a	a	NOUN
ejpam-6699	55	51	)	)	PUNCT
ejpam-6699	55	52	∥ν1	∥ν1	NOUN
ejpam-6699	55	53	,	,	PUNCT
ejpam-6699	55	54	·	·	PUNCT
ejpam-6699	55	55	·	·	PUNCT
ejpam-6699	55	56	·	·	PUNCT
ejpam-6699	55	57	,	,	PUNCT
ejpam-6699	55	58	νn∥β	νn∥β	VERB
ejpam-6699	55	59	=	=	SYM
ejpam-6699	55	60	0	0	PUNCT
ejpam-6699	56	1	if	if	SCONJ
ejpam-6699	56	2	and	and	CCONJ
ejpam-6699	56	3	only	only	ADV
ejpam-6699	56	4	if	if	SCONJ
ejpam-6699	56	5	ν1	ν1	NOUN
ejpam-6699	56	6	,	,	PUNCT
ejpam-6699	56	7	·	·	PUNCT
ejpam-6699	56	8	·	·	PUNCT
ejpam-6699	56	9	·	·	PUNCT
ejpam-6699	56	10	,	,	PUNCT
ejpam-6699	56	11	νn	νn	AUX
ejpam-6699	56	12	are	be	AUX
ejpam-6699	56	13	linearly	linearly	ADV
ejpam-6699	56	14	dependent	dependent	ADJ
ejpam-6699	56	15	;	;	PUNCT
ejpam-6699	56	16	(	(	PUNCT
ejpam-6699	56	17	b	b	X
ejpam-6699	56	18	)	)	PUNCT
ejpam-6699	56	19	∥ν1	∥ν1	NOUN
ejpam-6699	56	20	,	,	PUNCT
ejpam-6699	56	21	·	·	PUNCT
ejpam-6699	56	22	·	·	PUNCT
ejpam-6699	56	23	·	·	PUNCT
ejpam-6699	56	24	,	,	PUNCT
ejpam-6699	56	25	νn∥β	νn∥β	NOUN
ejpam-6699	56	26	is	be	AUX
ejpam-6699	56	27	invariant	invariant	ADJ
ejpam-6699	56	28	under	under	ADP
ejpam-6699	56	29	permutations	permutation	NOUN
ejpam-6699	56	30	of	of	ADP
ejpam-6699	56	31	its	its	PRON
ejpam-6699	56	32	arguments	argument	NOUN
ejpam-6699	56	33	;	;	PUNCT
ejpam-6699	56	34	(	(	PUNCT
ejpam-6699	56	35	c	c	X
ejpam-6699	56	36	)	)	PUNCT
ejpam-6699	56	37	∥λν1	∥λν1	PROPN
ejpam-6699	56	38	,	,	PUNCT
ejpam-6699	56	39	·	·	PUNCT
ejpam-6699	56	40	·	·	PUNCT
ejpam-6699	56	41	·	·	PUNCT
ejpam-6699	56	42	,	,	PUNCT
ejpam-6699	56	43	νn∥β	νn∥β	X
ejpam-6699	56	44	=	=	SYM
ejpam-6699	56	45	|λ|β	|λ|β	NOUN
ejpam-6699	56	46	∥ν1	∥ν1	NOUN
ejpam-6699	56	47	,	,	PUNCT
ejpam-6699	56	48	·	·	PUNCT
ejpam-6699	56	49	·	·	PUNCT
ejpam-6699	56	50	·	·	PUNCT
ejpam-6699	56	51	,	,	PUNCT
ejpam-6699	56	52	νn∥β	νn∥β	NOUN
ejpam-6699	56	53	;	;	PUNCT
ejpam-6699	56	54	(	(	PUNCT
ejpam-6699	56	55	d	d	X
ejpam-6699	56	56	)	)	PUNCT
ejpam-6699	56	57	∥ν0	∥ν0	NOUN
ejpam-6699	56	58	+	+	NUM
ejpam-6699	56	59	ν1	ν1	NOUN
ejpam-6699	56	60	,	,	PUNCT
ejpam-6699	56	61	·	·	PUNCT
ejpam-6699	56	62	·	·	PUNCT
ejpam-6699	56	63	·	·	PUNCT
ejpam-6699	56	64	,	,	PUNCT
ejpam-6699	56	65	νn∥β	νn∥β	VERB
ejpam-6699	56	66	≤	≤	X
ejpam-6699	56	67	max	max	PROPN
ejpam-6699	56	68	{	{	PUNCT
ejpam-6699	56	69	∥ν0	∥ν0	NOUN
ejpam-6699	56	70	,	,	PUNCT
ejpam-6699	56	71	ν2	ν2	NOUN
ejpam-6699	56	72	,	,	PUNCT
ejpam-6699	56	73	·	·	PUNCT
ejpam-6699	56	74	·	·	PUNCT
ejpam-6699	56	75	·	·	PUNCT
ejpam-6699	56	76	,	,	PUNCT
ejpam-6699	56	77	νn∥β	νn∥β	PROPN
ejpam-6699	56	78	,	,	PUNCT
ejpam-6699	56	79	∥ν1	∥ν1	NOUN
ejpam-6699	56	80	,	,	PUNCT
ejpam-6699	56	81	ν2	ν2	NOUN
ejpam-6699	56	82	·	·	PUNCT
ejpam-6699	56	83	·	·	PUNCT
ejpam-6699	56	84	·	·	PUNCT
ejpam-6699	56	85	,	,	PUNCT
ejpam-6699	56	86	νn∥β	νn∥β	PROPN
ejpam-6699	56	87	}	}	PUNCT
ejpam-6699	56	88	.	.	PUNCT
ejpam-6699	57	1	if	if	SCONJ
ejpam-6699	57	2	these	these	DET
ejpam-6699	57	3	conditions	condition	NOUN
ejpam-6699	57	4	are	be	AUX
ejpam-6699	57	5	satisfies	satisfie	NOUN
ejpam-6699	57	6	,	,	PUNCT
ejpam-6699	57	7	then	then	ADV
ejpam-6699	57	8	the	the	DET
ejpam-6699	57	9	pair	pair	NOUN
ejpam-6699	57	10	(	(	PUNCT
ejpam-6699	57	11	e	e	NOUN
ejpam-6699	57	12	,	,	PUNCT
ejpam-6699	57	13	∥	∥	X
ejpam-6699	57	14	·	·	PUNCT
ejpam-6699	57	15	,	,	PUNCT
ejpam-6699	57	16	·	·	PUNCT
ejpam-6699	57	17	·	·	PUNCT
ejpam-6699	57	18	·	·	PUNCT
ejpam-6699	57	19	,	,	PUNCT
ejpam-6699	57	20	·	·	PUNCT
ejpam-6699	57	21	∥β	∥β	PROPN
ejpam-6699	57	22	)	)	PUNCT
ejpam-6699	57	23	is	be	AUX
ejpam-6699	57	24	called	call	VERB
ejpam-6699	57	25	a	a	DET
ejpam-6699	57	26	non	non	ADJ
ejpam-6699	57	27	-	-	ADJ
ejpam-6699	57	28	archimedean	archimedean	ADJ
ejpam-6699	57	29	(	(	PUNCT
ejpam-6699	57	30	n	n	CCONJ
ejpam-6699	57	31	,	,	PUNCT
ejpam-6699	57	32	β)-normed	β)-normed	PUNCT
ejpam-6699	57	33	space	space	NOUN
ejpam-6699	57	34	.	.	PUNCT
ejpam-6699	57	35	example	example	NOUN
ejpam-6699	58	1	1	1	NUM
ejpam-6699	58	2	.	.	PUNCT
ejpam-6699	59	1	[	[	X
ejpam-6699	59	2	25	25	NUM
ejpam-6699	59	3	]	]	PUNCT
ejpam-6699	59	4	let	let	VERB
ejpam-6699	59	5	p	p	PRON
ejpam-6699	59	6	denote	denote	VERB
ejpam-6699	59	7	a	a	DET
ejpam-6699	59	8	prime	prime	ADJ
ejpam-6699	59	9	integer	integer	NOUN
ejpam-6699	59	10	.	.	PUNCT
ejpam-6699	60	1	if	if	SCONJ
ejpam-6699	60	2	x	x	PRON
ejpam-6699	60	3	=	=	PUNCT
ejpam-6699	60	4	a	a	DET
ejpam-6699	60	5	bp	bp	PROPN
ejpam-6699	60	6	r	r	NOUN
ejpam-6699	60	7	is	be	AUX
ejpam-6699	60	8	a	a	DET
ejpam-6699	60	9	nonzero	nonzero	ADJ
ejpam-6699	60	10	rational	rational	ADJ
ejpam-6699	60	11	number	number	NOUN
ejpam-6699	60	12	,	,	PUNCT
ejpam-6699	60	13	where	where	SCONJ
ejpam-6699	60	14	a	a	PRON
ejpam-6699	60	15	and	and	CCONJ
ejpam-6699	60	16	b	b	NOUN
ejpam-6699	60	17	are	be	AUX
ejpam-6699	60	18	coprime	coprime	ADJ
ejpam-6699	60	19	to	to	ADP
ejpam-6699	60	20	the	the	DET
ejpam-6699	60	21	prime	prime	ADJ
ejpam-6699	60	22	number	number	NOUN
ejpam-6699	60	23	p	p	NOUN
ejpam-6699	60	24	,	,	PUNCT
ejpam-6699	60	25	then	then	ADV
ejpam-6699	60	26	the	the	DET
ejpam-6699	60	27	p	p	ADJ
ejpam-6699	60	28	-	-	PUNCT
ejpam-6699	60	29	adic	adic	ADJ
ejpam-6699	60	30	absolute	absolute	ADJ
ejpam-6699	60	31	value	value	NOUN
ejpam-6699	60	32	is	be	AUX
ejpam-6699	60	33	defined	define	VERB
ejpam-6699	60	34	as	as	ADP
ejpam-6699	60	35	∥x∥p	∥x∥p	ADJ
ejpam-6699	60	36	:	:	PUNCT
ejpam-6699	61	1	=	=	SYM
ejpam-6699	61	2	pr	pr	NOUN
ejpam-6699	61	3	,	,	PUNCT
ejpam-6699	61	4	and	and	CCONJ
ejpam-6699	61	5	r	r	NOUN
ejpam-6699	61	6	∈	∈	PROPN
ejpam-6699	61	7	z.	z.	PROPN
ejpam-6699	61	8	on	on	ADP
ejpam-6699	61	9	q	q	PROPN
ejpam-6699	61	10	,	,	PUNCT
ejpam-6699	61	11	the	the	DET
ejpam-6699	61	12	norm	norm	NOUN
ejpam-6699	61	13	∥	∥	X
ejpam-6699	61	14	·	·	PUNCT
ejpam-6699	62	1	∥p	∥p	NOUN
ejpam-6699	62	2	is	be	AUX
ejpam-6699	62	3	classified	classify	VERB
ejpam-6699	62	4	as	as	ADP
ejpam-6699	62	5	a	a	DET
ejpam-6699	62	6	non	non	ADJ
ejpam-6699	62	7	-	-	ADJ
ejpam-6699	62	8	archimedean	archimedean	ADJ
ejpam-6699	62	9	norm	norm	NOUN
ejpam-6699	62	10	.	.	PUNCT
ejpam-6699	63	1	the	the	DET
ejpam-6699	63	2	field	field	NOUN
ejpam-6699	63	3	qp	qp	ADV
ejpam-6699	63	4	represents	represent	VERB
ejpam-6699	63	5	the	the	DET
ejpam-6699	63	6	completion	completion	NOUN
ejpam-6699	63	7	of	of	ADP
ejpam-6699	63	8	the	the	DET
ejpam-6699	63	9	rational	rational	ADJ
ejpam-6699	63	10	numbers	number	NOUN
ejpam-6699	63	11	q	q	NOUN
ejpam-6699	63	12	under	under	ADP
ejpam-6699	63	13	the	the	DET
ejpam-6699	63	14	p	p	NOUN
ejpam-6699	63	15	-	-	PUNCT
ejpam-6699	63	16	adic	adic	ADJ
ejpam-6699	63	17	norm	norm	NOUN
ejpam-6699	63	18	∥	∥	X
ejpam-6699	64	1	·	·	PUNCT
ejpam-6699	64	2	∥p	∥p	ADJ
ejpam-6699	64	3	.	.	PUNCT
ejpam-6699	65	1	the	the	DET
ejpam-6699	65	2	p	p	NOUN
ejpam-6699	65	3	-	-	PUNCT
ejpam-6699	65	4	adic	adic	ADJ
ejpam-6699	65	5	number	number	NOUN
ejpam-6699	65	6	field	field	NOUN
ejpam-6699	65	7	is	be	AUX
ejpam-6699	65	8	also	also	ADV
ejpam-6699	65	9	referred	refer	VERB
ejpam-6699	65	10	to	to	ADP
ejpam-6699	65	11	as	as	ADV
ejpam-6699	65	12	such	such	ADJ
ejpam-6699	65	13	.	.	PUNCT
ejpam-6699	66	1	if	if	SCONJ
ejpam-6699	66	2	p	p	PROPN
ejpam-6699	66	3	>	>	X
ejpam-6699	66	4	3	3	NUM
ejpam-6699	66	5	,	,	PUNCT
ejpam-6699	66	6	then	then	ADV
ejpam-6699	66	7	∥2n∥	∥2n∥	PUNCT
ejpam-6699	66	8	=	=	PUNCT
ejpam-6699	66	9	1	1	NUM
ejpam-6699	66	10	for	for	ADP
ejpam-6699	66	11	all	all	DET
ejpam-6699	66	12	integer	integer	PROPN
ejpam-6699	66	13	n.	n.	PROPN
ejpam-6699	66	14	s.	s.	PROPN
ejpam-6699	66	15	gowri	gowri	PROPN
ejpam-6699	66	16	et	et	PROPN
ejpam-6699	66	17	al	al	PROPN
ejpam-6699	66	18	.	.	PUNCT
ejpam-6699	66	19	/	/	SYM
ejpam-6699	66	20	eur	eur	PROPN
ejpam-6699	66	21	.	.	PUNCT
ejpam-6699	67	1	j.	j.	PROPN
ejpam-6699	67	2	pure	pure	PROPN
ejpam-6699	67	3	appl	appl	PROPN
ejpam-6699	67	4	.	.	PROPN
ejpam-6699	67	5	math	math	PROPN
ejpam-6699	67	6	,	,	PUNCT
ejpam-6699	67	7	18	18	NUM
ejpam-6699	67	8	(	(	PUNCT
ejpam-6699	67	9	4	4	NUM
ejpam-6699	67	10	)	)	PUNCT
ejpam-6699	67	11	(	(	PUNCT
ejpam-6699	67	12	2025	2025	NUM
ejpam-6699	67	13	)	)	PUNCT
ejpam-6699	67	14	,	,	PUNCT
ejpam-6699	67	15	6699	6699	NUM
ejpam-6699	67	16	4	4	NUM
ejpam-6699	67	17	of	of	ADP
ejpam-6699	67	18	14	14	NUM
ejpam-6699	67	19	remark	remark	NOUN
ejpam-6699	67	20	1	1	NUM
ejpam-6699	67	21	.	.	PUNCT
ejpam-6699	68	1	[	[	X
ejpam-6699	68	2	28	28	NUM
ejpam-6699	68	3	]	]	X
ejpam-6699	68	4	a	a	DET
ejpam-6699	68	5	non	non	ADJ
ejpam-6699	68	6	-	-	ADJ
ejpam-6699	68	7	archimedean	archimedean	ADJ
ejpam-6699	68	8	(	(	PUNCT
ejpam-6699	68	9	n	n	X
ejpam-6699	68	10	,	,	PUNCT
ejpam-6699	68	11	β	β	NOUN
ejpam-6699	68	12	)	)	PUNCT
ejpam-6699	68	13	containing	contain	VERB
ejpam-6699	68	14	a	a	DET
ejpam-6699	68	15	sequence	sequence	NOUN
ejpam-6699	68	16	{	{	PUNCT
ejpam-6699	68	17	tm	tm	NOUN
ejpam-6699	68	18	}	}	PUNCT
ejpam-6699	68	19	if	if	SCONJ
ejpam-6699	68	20	and	and	CCONJ
ejpam-6699	68	21	only	only	ADV
ejpam-6699	68	22	if	if	SCONJ
ejpam-6699	68	23	the	the	DET
ejpam-6699	68	24	negative	negative	ADJ
ejpam-6699	68	25	absolute	absolute	ADJ
ejpam-6699	68	26	value	value	NOUN
ejpam-6699	68	27	of	of	ADP
ejpam-6699	68	28	tm+1	tm+1	PROPN
ejpam-6699	68	29	converges	converge	NOUN
ejpam-6699	68	30	to	to	ADP
ejpam-6699	68	31	zero	zero	NUM
ejpam-6699	68	32	,	,	PUNCT
ejpam-6699	68	33	then	then	ADV
ejpam-6699	68	34	normed	normed	PROPN
ejpam-6699	68	35	space	space	NOUN
ejpam-6699	68	36	e	e	PROPN
ejpam-6699	68	37	is	be	AUX
ejpam-6699	68	38	a	a	DET
ejpam-6699	68	39	cauchy	cauchy	ADJ
ejpam-6699	68	40	sequence	sequence	NOUN
ejpam-6699	68	41	.	.	PUNCT
ejpam-6699	69	1	lemma	lemma	PROPN
ejpam-6699	69	2	1	1	NUM
ejpam-6699	69	3	.	.	PUNCT
ejpam-6699	70	1	[	[	X
ejpam-6699	70	2	28	28	NUM
ejpam-6699	70	3	]	]	PUNCT
ejpam-6699	70	4	consider	consider	VERB
ejpam-6699	70	5	{	{	PUNCT
ejpam-6699	70	6	tp	tp	PART
ejpam-6699	70	7	}	}	PUNCT
ejpam-6699	70	8	is	be	AUX
ejpam-6699	70	9	a	a	DET
ejpam-6699	70	10	convergent	convergent	NOUN
ejpam-6699	70	11	sequence	sequence	NOUN
ejpam-6699	70	12	in	in	ADP
ejpam-6699	70	13	a	a	DET
ejpam-6699	70	14	linear	linear	NOUN
ejpam-6699	70	15	(	(	PUNCT
ejpam-6699	70	16	n	n	CCONJ
ejpam-6699	70	17	,	,	PUNCT
ejpam-6699	70	18	β)-normed	β)-normed	PUNCT
ejpam-6699	70	19	space	space	NOUN
ejpam-6699	70	20	e	e	NOUN
ejpam-6699	70	21	,	,	PUNCT
ejpam-6699	70	22	lim	lim	PROPN
ejpam-6699	70	23	p→∞	p→∞	NOUN
ejpam-6699	70	24	∥tp	∥tp	PROPN
ejpam-6699	70	25	,	,	PUNCT
ejpam-6699	70	26	κ1	κ1	NOUN
ejpam-6699	70	27	,	,	PUNCT
ejpam-6699	70	28	κ2	κ2	NOUN
ejpam-6699	70	29	,	,	PUNCT
ejpam-6699	70	30	·	·	PUNCT
ejpam-6699	70	31	·	·	PUNCT
ejpam-6699	70	32	·	·	PUNCT
ejpam-6699	70	33	,	,	PUNCT
ejpam-6699	70	34	κn−1∥β	κn−1∥β	NOUN
ejpam-6699	70	35	=	=	SYM
ejpam-6699	70	36	∥∥∥∥	∥∥∥∥	PROPN
ejpam-6699	70	37	lim	lim	PROPN
ejpam-6699	70	38	p→∞	p→∞	ADJ
ejpam-6699	70	39	tp	tp	PROPN
ejpam-6699	70	40	,	,	PUNCT
ejpam-6699	70	41	κ1	κ1	NOUN
ejpam-6699	70	42	,	,	PUNCT
ejpam-6699	70	43	κ2	κ2	NOUN
ejpam-6699	70	44	,	,	PUNCT
ejpam-6699	70	45	·	·	PUNCT
ejpam-6699	70	46	·	·	PUNCT
ejpam-6699	70	47	·	·	PUNCT
ejpam-6699	70	48	,	,	PUNCT
ejpam-6699	70	49	κn−1	κn−1	PROPN
ejpam-6699	70	50	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	70	51	β	β	NOUN
ejpam-6699	70	52	for	for	ADP
ejpam-6699	70	53	all	all	DET
ejpam-6699	70	54	κ1	κ1	NOUN
ejpam-6699	70	55	,	,	PUNCT
ejpam-6699	70	56	κ2	κ2	NOUN
ejpam-6699	70	57	,	,	PUNCT
ejpam-6699	70	58	·	·	PUNCT
ejpam-6699	70	59	·	·	PUNCT
ejpam-6699	70	60	·	·	PUNCT
ejpam-6699	70	61	,	,	PUNCT
ejpam-6699	70	62	κn−1	κn−1	PROPN
ejpam-6699	70	63	∈	∈	PROPN
ejpam-6699	70	64	e.	e.	PROPN
ejpam-6699	70	65	lemma	lemma	PROPN
ejpam-6699	71	1	2	2	X
ejpam-6699	71	2	.	.	PUNCT
ejpam-6699	72	1	[	[	X
ejpam-6699	72	2	28	28	NUM
ejpam-6699	72	3	]	]	X
ejpam-6699	72	4	let	let	VERB
ejpam-6699	72	5	(	(	PUNCT
ejpam-6699	72	6	e	e	NOUN
ejpam-6699	72	7	,	,	PUNCT
ejpam-6699	72	8	∥	∥	X
ejpam-6699	72	9	·	·	PUNCT
ejpam-6699	72	10	,	,	PUNCT
ejpam-6699	72	11	·	·	PUNCT
ejpam-6699	72	12	·	·	PUNCT
ejpam-6699	72	13	·	·	PUNCT
ejpam-6699	72	14	,	,	PUNCT
ejpam-6699	72	15	·	·	PUNCT
ejpam-6699	72	16	∥β	∥β	PROPN
ejpam-6699	72	17	)	)	PUNCT
ejpam-6699	72	18	be	be	VERB
ejpam-6699	72	19	a	a	DET
ejpam-6699	72	20	linear	linear	ADJ
ejpam-6699	72	21	(	(	PUNCT
ejpam-6699	72	22	n	n	CCONJ
ejpam-6699	72	23	,	,	PUNCT
ejpam-6699	72	24	β)-normed	β)-normed	PUNCT
ejpam-6699	72	25	space	space	NOUN
ejpam-6699	72	26	,	,	PUNCT
ejpam-6699	72	27	0	0	PUNCT
ejpam-6699	72	28	<	<	X
ejpam-6699	72	29	β	β	X
ejpam-6699	72	30	≤	≤	NUM
ejpam-6699	72	31	1	1	NUM
ejpam-6699	72	32	and	and	CCONJ
ejpam-6699	72	33	n	n	PRON
ejpam-6699	72	34	≥	≥	NOUN
ejpam-6699	72	35	2	2	NUM
ejpam-6699	72	36	.	.	PUNCT
ejpam-6699	73	1	if	if	SCONJ
ejpam-6699	73	2	t1	t1	NOUN
ejpam-6699	73	3	∈	∈	PROPN
ejpam-6699	73	4	e	e	NOUN
ejpam-6699	73	5	and	and	CCONJ
ejpam-6699	73	6	∥t1	∥t1	NOUN
ejpam-6699	73	7	,	,	PUNCT
ejpam-6699	73	8	κ1	κ1	NOUN
ejpam-6699	73	9	,	,	PUNCT
ejpam-6699	73	10	·	·	PUNCT
ejpam-6699	73	11	·	·	PUNCT
ejpam-6699	73	12	·	·	PUNCT
ejpam-6699	73	13	,	,	PUNCT
ejpam-6699	73	14	κn−1∥β	κn−1∥β	NOUN
ejpam-6699	73	15	=	=	SYM
ejpam-6699	73	16	0	0	NUM
ejpam-6699	73	17	for	for	ADP
ejpam-6699	73	18	all	all	DET
ejpam-6699	73	19	κ1	κ1	NOUN
ejpam-6699	73	20	,	,	PUNCT
ejpam-6699	73	21	·	·	PUNCT
ejpam-6699	73	22	·	·	PUNCT
ejpam-6699	73	23	·	·	PUNCT
ejpam-6699	73	24	,	,	PUNCT
ejpam-6699	73	25	κn−1	κn−1	PROPN
ejpam-6699	73	26	∈	∈	PROPN
ejpam-6699	73	27	e	e	NOUN
ejpam-6699	73	28	,	,	PUNCT
ejpam-6699	73	29	then	then	ADV
ejpam-6699	73	30	t1	t1	NOUN
ejpam-6699	73	31	=	=	SYM
ejpam-6699	73	32	0	0	X
ejpam-6699	73	33	.	.	PUNCT
ejpam-6699	73	34	theorem	theorem	NOUN
ejpam-6699	73	35	1	1	NUM
ejpam-6699	73	36	.	.	PUNCT
ejpam-6699	74	1	[	[	X
ejpam-6699	74	2	29	29	NUM
ejpam-6699	74	3	]	]	X
ejpam-6699	74	4	if	if	SCONJ
ejpam-6699	74	5	a	a	DET
ejpam-6699	74	6	mapping	mapping	NOUN
ejpam-6699	74	7	ϕ	ϕ	NOUN
ejpam-6699	74	8	:	:	PUNCT
ejpam-6699	74	9	e	e	X
ejpam-6699	74	10	→	→	SYM
ejpam-6699	74	11	f	f	PROPN
ejpam-6699	74	12	satisfies	satisfy	VERB
ejpam-6699	74	13	the	the	DET
ejpam-6699	74	14	functional	functional	ADJ
ejpam-6699	74	15	equation	equation	NOUN
ejpam-6699	74	16	(	(	PUNCT
ejpam-6699	74	17	1	1	NUM
ejpam-6699	74	18	)	)	PUNCT
ejpam-6699	74	19	for	for	ADP
ejpam-6699	74	20	all	all	DET
ejpam-6699	74	21	t1	t1	NOUN
ejpam-6699	74	22	,	,	PUNCT
ejpam-6699	74	23	t2	t2	NOUN
ejpam-6699	74	24	,	,	PUNCT
ejpam-6699	74	25	·	·	PUNCT
ejpam-6699	74	26	·	·	PUNCT
ejpam-6699	74	27	·	·	PUNCT
ejpam-6699	74	28	,	,	PUNCT
ejpam-6699	74	29	tr	tr	NOUN
ejpam-6699	74	30	∈	∈	NOUN
ejpam-6699	74	31	e	e	NOUN
ejpam-6699	74	32	,	,	PUNCT
ejpam-6699	74	33	then	then	ADV
ejpam-6699	74	34	the	the	DET
ejpam-6699	74	35	function	function	NOUN
ejpam-6699	74	36	ϕ	ϕ	NOUN
ejpam-6699	74	37	:	:	PUNCT
ejpam-6699	74	38	e	e	X
ejpam-6699	74	39	→	→	SYM
ejpam-6699	74	40	f	f	PROPN
ejpam-6699	74	41	is	be	AUX
ejpam-6699	74	42	quartic	quartic	ADJ
ejpam-6699	74	43	.	.	PUNCT
ejpam-6699	75	1	2.1	2.1	NUM
ejpam-6699	75	2	.	.	PUNCT
ejpam-6699	76	1	structural	structural	ADJ
ejpam-6699	76	2	examples	example	NOUN
ejpam-6699	76	3	and	and	CCONJ
ejpam-6699	76	4	fundamental	fundamental	ADJ
ejpam-6699	76	5	properties	property	NOUN
ejpam-6699	76	6	of	of	ADP
ejpam-6699	76	7	non	non	ADJ
ejpam-6699	76	8	-	-	ADJ
ejpam-6699	76	9	archimedean	archimedean	ADJ
ejpam-6699	76	10	(	(	PUNCT
ejpam-6699	76	11	n	n	CCONJ
ejpam-6699	76	12	,	,	PUNCT
ejpam-6699	76	13	β)-normed	β)-normed	PUNCT
ejpam-6699	76	14	spaces	space	NOUN
ejpam-6699	76	15	in	in	ADP
ejpam-6699	76	16	this	this	DET
ejpam-6699	76	17	subsection	subsection	NOUN
ejpam-6699	76	18	,	,	PUNCT
ejpam-6699	76	19	we	we	PRON
ejpam-6699	76	20	provide	provide	VERB
ejpam-6699	76	21	illustrative	illustrative	ADJ
ejpam-6699	76	22	examples	example	NOUN
ejpam-6699	76	23	to	to	PART
ejpam-6699	76	24	demonstrate	demonstrate	VERB
ejpam-6699	76	25	the	the	DET
ejpam-6699	76	26	structure	structure	NOUN
ejpam-6699	76	27	and	and	CCONJ
ejpam-6699	76	28	behavior	behavior	NOUN
ejpam-6699	76	29	of	of	ADP
ejpam-6699	76	30	non	non	ADJ
ejpam-6699	76	31	-	-	ADJ
ejpam-6699	76	32	archimedean	archimedean	ADJ
ejpam-6699	76	33	(	(	PUNCT
ejpam-6699	76	34	n	n	CCONJ
ejpam-6699	76	35	,	,	PUNCT
ejpam-6699	76	36	β)-normed	β)-normed	PUNCT
ejpam-6699	76	37	spaces	space	NOUN
ejpam-6699	76	38	.	.	PUNCT
ejpam-6699	77	1	these	these	DET
ejpam-6699	77	2	examples	example	NOUN
ejpam-6699	77	3	underline	underline	VERB
ejpam-6699	77	4	how	how	SCONJ
ejpam-6699	77	5	such	such	ADJ
ejpam-6699	77	6	spaces	space	NOUN
ejpam-6699	77	7	extend	extend	VERB
ejpam-6699	77	8	the	the	DET
ejpam-6699	77	9	classical	classical	ADJ
ejpam-6699	77	10	notions	notion	NOUN
ejpam-6699	77	11	of	of	ADP
ejpam-6699	77	12	normed	normed	ADJ
ejpam-6699	77	13	vector	vector	NOUN
ejpam-6699	77	14	spaces	space	NOUN
ejpam-6699	77	15	under	under	ADP
ejpam-6699	77	16	ultrametric	ultrametric	ADJ
ejpam-6699	77	17	constraints	constraint	NOUN
ejpam-6699	77	18	.	.	PUNCT
ejpam-6699	78	1	example	example	NOUN
ejpam-6699	79	1	2	2	NUM
ejpam-6699	79	2	.	.	PUNCT
ejpam-6699	80	1	[	[	X
ejpam-6699	80	2	28	28	NUM
ejpam-6699	80	3	,	,	PUNCT
ejpam-6699	80	4	30	30	NUM
ejpam-6699	80	5	,	,	PUNCT
ejpam-6699	80	6	31	31	NUM
ejpam-6699	80	7	]	]	PUNCT
ejpam-6699	80	8	let	let	VERB
ejpam-6699	80	9	x	x	SYM
ejpam-6699	80	10	=	=	SYM
ejpam-6699	80	11	kn	kn	PROPN
ejpam-6699	80	12	be	be	AUX
ejpam-6699	80	13	the	the	DET
ejpam-6699	80	14	n	n	ADV
ejpam-6699	80	15	-	-	PUNCT
ejpam-6699	80	16	dimensional	dimensional	ADJ
ejpam-6699	80	17	vector	vector	NOUN
ejpam-6699	80	18	space	space	NOUN
ejpam-6699	80	19	over	over	ADP
ejpam-6699	80	20	a	a	DET
ejpam-6699	80	21	nonarchimedean	nonarchimedean	ADJ
ejpam-6699	80	22	field	field	NOUN
ejpam-6699	80	23	k.	k.	NOUN
ejpam-6699	80	24	define	define	VERB
ejpam-6699	80	25	the	the	DET
ejpam-6699	80	26	mapping	mapping	NOUN
ejpam-6699	80	27	∥	∥	NOUN
ejpam-6699	80	28	·	·	PUNCT
ejpam-6699	80	29	,	,	PUNCT
ejpam-6699	80	30	.	.	PUNCT
ejpam-6699	80	31	.	.	PUNCT
ejpam-6699	81	1	.	.	PUNCT
ejpam-6699	82	1	,	,	PUNCT
ejpam-6699	82	2	·	·	PUNCT
ejpam-6699	82	3	∥β	∥β	NOUN
ejpam-6699	82	4	:	:	PUNCT
ejpam-6699	82	5	xn	xn	PROPN
ejpam-6699	83	1	→	→	NOUN
ejpam-6699	83	2	r+	r+	NOUN
ejpam-6699	83	3	as	as	ADP
ejpam-6699	83	4	∥x1	∥x1	NOUN
ejpam-6699	83	5	,	,	PUNCT
ejpam-6699	83	6	x2	x2	PROPN
ejpam-6699	83	7	,	,	PUNCT
ejpam-6699	83	8	.	.	PUNCT
ejpam-6699	83	9	.	.	PUNCT
ejpam-6699	83	10	.	.	PUNCT
ejpam-6699	84	1	,	,	PUNCT
ejpam-6699	84	2	xn∥β	xn∥β	VERB
ejpam-6699	85	1	:	:	PUNCT
ejpam-6699	85	2	=	=	SYM
ejpam-6699	85	3	|det(x1	|det(x1	PROPN
ejpam-6699	85	4	,	,	PUNCT
ejpam-6699	85	5	x2	x2	PROPN
ejpam-6699	85	6	,	,	PUNCT
ejpam-6699	85	7	.	.	PUNCT
ejpam-6699	85	8	.	.	PUNCT
ejpam-6699	85	9	.	.	PUNCT
ejpam-6699	86	1	,	,	PUNCT
ejpam-6699	86	2	xn)|β	xn)|β	PROPN
ejpam-6699	86	3	,	,	PUNCT
ejpam-6699	86	4	where	where	SCONJ
ejpam-6699	86	5	x1	x1	ADJ
ejpam-6699	86	6	,	,	PUNCT
ejpam-6699	86	7	x2	x2	PROPN
ejpam-6699	86	8	,	,	PUNCT
ejpam-6699	86	9	.	.	PUNCT
ejpam-6699	86	10	.	.	PUNCT
ejpam-6699	86	11	.	.	PUNCT
ejpam-6699	87	1	,	,	PUNCT
ejpam-6699	87	2	xn	xn	PROPN
ejpam-6699	87	3	are	be	AUX
ejpam-6699	87	4	vectors	vector	NOUN
ejpam-6699	87	5	in	in	ADP
ejpam-6699	87	6	x	x	PUNCT
ejpam-6699	87	7	and	and	CCONJ
ejpam-6699	87	8	the	the	DET
ejpam-6699	87	9	determinant	determinant	ADJ
ejpam-6699	87	10	is	be	AUX
ejpam-6699	87	11	computed	compute	VERB
ejpam-6699	87	12	by	by	ADP
ejpam-6699	87	13	treating	treat	VERB
ejpam-6699	87	14	them	they	PRON
ejpam-6699	87	15	as	as	ADP
ejpam-6699	87	16	rows	row	NOUN
ejpam-6699	87	17	of	of	ADP
ejpam-6699	87	18	an	an	DET
ejpam-6699	87	19	n×	n×	NOUN
ejpam-6699	87	20	n	n	NOUN
ejpam-6699	87	21	matrix	matrix	NOUN
ejpam-6699	87	22	.	.	PUNCT
ejpam-6699	88	1	it	it	PRON
ejpam-6699	88	2	can	can	AUX
ejpam-6699	88	3	be	be	AUX
ejpam-6699	88	4	verified	verify	VERB
ejpam-6699	88	5	that	that	SCONJ
ejpam-6699	88	6	this	this	DET
ejpam-6699	88	7	function	function	NOUN
ejpam-6699	88	8	satisfies	satisfy	VERB
ejpam-6699	88	9	all	all	DET
ejpam-6699	88	10	the	the	DET
ejpam-6699	88	11	conditions	condition	NOUN
ejpam-6699	88	12	of	of	ADP
ejpam-6699	88	13	a	a	DET
ejpam-6699	88	14	non	non	ADJ
ejpam-6699	88	15	-	-	ADJ
ejpam-6699	88	16	archimedean	archimedean	ADJ
ejpam-6699	88	17	(	(	PUNCT
ejpam-6699	88	18	n	n	CCONJ
ejpam-6699	88	19	,	,	PUNCT
ejpam-6699	89	1	β)-norm	β)-norm	PUNCT
ejpam-6699	89	2	.	.	PUNCT
ejpam-6699	89	3	example	example	NOUN
ejpam-6699	90	1	3	3	NUM
ejpam-6699	90	2	.	.	PUNCT
ejpam-6699	91	1	[	[	X
ejpam-6699	91	2	28	28	NUM
ejpam-6699	91	3	,	,	PUNCT
ejpam-6699	91	4	30	30	NUM
ejpam-6699	91	5	,	,	PUNCT
ejpam-6699	91	6	31	31	NUM
ejpam-6699	91	7	]	]	PUNCT
ejpam-6699	91	8	consider	consider	VERB
ejpam-6699	91	9	the	the	DET
ejpam-6699	91	10	vector	vector	NOUN
ejpam-6699	91	11	space	space	NOUN
ejpam-6699	91	12	x	x	PUNCT
ejpam-6699	91	13	=	=	SYM
ejpam-6699	91	14	c0(k	c0(k	NOUN
ejpam-6699	91	15	)	)	PUNCT
ejpam-6699	91	16	,	,	PUNCT
ejpam-6699	91	17	the	the	DET
ejpam-6699	91	18	space	space	NOUN
ejpam-6699	91	19	of	of	ADP
ejpam-6699	91	20	sequences	sequence	NOUN
ejpam-6699	91	21	converging	converge	VERB
ejpam-6699	91	22	to	to	ADP
ejpam-6699	91	23	zero	zero	NUM
ejpam-6699	91	24	over	over	ADP
ejpam-6699	91	25	a	a	DET
ejpam-6699	91	26	non	non	ADJ
ejpam-6699	91	27	-	-	ADJ
ejpam-6699	91	28	archimedean	archimedean	ADJ
ejpam-6699	91	29	field	field	NOUN
ejpam-6699	91	30	k.	k.	PROPN
ejpam-6699	91	31	define	define	VERB
ejpam-6699	91	32	the	the	DET
ejpam-6699	91	33	(	(	PUNCT
ejpam-6699	91	34	n	n	CCONJ
ejpam-6699	91	35	,	,	PUNCT
ejpam-6699	91	36	β)-norm	β)-norm	PUNCT
ejpam-6699	91	37	by	by	ADP
ejpam-6699	91	38	∥x1	∥x1	NOUN
ejpam-6699	91	39	,	,	PUNCT
ejpam-6699	91	40	.	.	PUNCT
ejpam-6699	91	41	.	.	PUNCT
ejpam-6699	92	1	.	.	PUNCT
ejpam-6699	93	1	,	,	PUNCT
ejpam-6699	93	2	xn∥β	xn∥β	PUNCT
ejpam-6699	94	1	:	:	PUNCT
ejpam-6699	94	2	=	=	SYM
ejpam-6699	94	3	sup	sup	NOUN
ejpam-6699	94	4	m∈n	m∈n	NOUN
ejpam-6699	94	5	∣∣∣det(x(m	∣∣∣det(x(m	PROPN
ejpam-6699	94	6	)	)	PUNCT
ejpam-6699	94	7	1	1	NUM
ejpam-6699	94	8	,	,	PUNCT
ejpam-6699	94	9	x	x	X
ejpam-6699	94	10	(	(	PUNCT
ejpam-6699	94	11	m	m	NOUN
ejpam-6699	94	12	)	)	PUNCT
ejpam-6699	94	13	2	2	NUM
ejpam-6699	94	14	,	,	PUNCT
ejpam-6699	94	15	.	.	PUNCT
ejpam-6699	94	16	.	.	PUNCT
ejpam-6699	95	1	.	.	PUNCT
ejpam-6699	96	1	,	,	PUNCT
ejpam-6699	96	2	x	x	X
ejpam-6699	96	3	(	(	PUNCT
ejpam-6699	96	4	m	m	NOUN
ejpam-6699	96	5	)	)	PUNCT
ejpam-6699	96	6	n	n	CCONJ
ejpam-6699	96	7	)	)	PUNCT
ejpam-6699	96	8	∣∣∣β	∣∣∣β	NOUN
ejpam-6699	96	9	,	,	PUNCT
ejpam-6699	96	10	where	where	SCONJ
ejpam-6699	96	11	x	x	X
ejpam-6699	96	12	(	(	PUNCT
ejpam-6699	96	13	m	m	NOUN
ejpam-6699	96	14	)	)	PUNCT
ejpam-6699	97	1	i	i	PRON
ejpam-6699	97	2	denotes	denote	VERB
ejpam-6699	97	3	the	the	DET
ejpam-6699	97	4	m	m	PROPN
ejpam-6699	97	5	-	-	PUNCT
ejpam-6699	97	6	th	th	VERB
ejpam-6699	97	7	component	component	NOUN
ejpam-6699	97	8	of	of	ADP
ejpam-6699	97	9	the	the	DET
ejpam-6699	97	10	sequence	sequence	NOUN
ejpam-6699	97	11	xi	xi	INTJ
ejpam-6699	97	12	.	.	PUNCT
ejpam-6699	98	1	this	this	DET
ejpam-6699	98	2	function	function	NOUN
ejpam-6699	98	3	defines	define	VERB
ejpam-6699	98	4	a	a	DET
ejpam-6699	98	5	valid	valid	ADJ
ejpam-6699	98	6	(	(	PUNCT
ejpam-6699	98	7	n	n	CCONJ
ejpam-6699	98	8	,	,	PUNCT
ejpam-6699	98	9	β)-norm	β)-norm	PUNCT
ejpam-6699	98	10	due	due	ADP
ejpam-6699	98	11	to	to	ADP
ejpam-6699	98	12	the	the	DET
ejpam-6699	98	13	ultrametric	ultrametric	ADJ
ejpam-6699	98	14	inequality	inequality	NOUN
ejpam-6699	98	15	and	and	CCONJ
ejpam-6699	98	16	properties	property	NOUN
ejpam-6699	98	17	of	of	ADP
ejpam-6699	98	18	determinants	determinant	NOUN
ejpam-6699	98	19	over	over	ADP
ejpam-6699	98	20	k.	k.	PROPN
ejpam-6699	99	1	we	we	PRON
ejpam-6699	99	2	now	now	ADV
ejpam-6699	99	3	list	list	VERB
ejpam-6699	99	4	some	some	DET
ejpam-6699	99	5	fundamental	fundamental	ADJ
ejpam-6699	99	6	properties	property	NOUN
ejpam-6699	99	7	that	that	PRON
ejpam-6699	99	8	hold	hold	VERB
ejpam-6699	99	9	in	in	ADP
ejpam-6699	99	10	any	any	DET
ejpam-6699	99	11	non	non	ADJ
ejpam-6699	99	12	-	-	ADJ
ejpam-6699	99	13	archimedean	archimedean	ADJ
ejpam-6699	99	14	(	(	PUNCT
ejpam-6699	99	15	n	n	CCONJ
ejpam-6699	99	16	,	,	PUNCT
ejpam-6699	99	17	β)normed	β)normed	ADJ
ejpam-6699	99	18	space	space	NOUN
ejpam-6699	99	19	(	(	PUNCT
ejpam-6699	99	20	x	x	X
ejpam-6699	99	21	,	,	PUNCT
ejpam-6699	99	22	∥	∥	PROPN
ejpam-6699	99	23	·	·	PUNCT
ejpam-6699	99	24	,	,	PUNCT
ejpam-6699	99	25	.	.	PUNCT
ejpam-6699	99	26	.	.	PUNCT
ejpam-6699	100	1	.	.	PUNCT
ejpam-6699	101	1	,	,	PUNCT
ejpam-6699	101	2	·	·	PUNCT
ejpam-6699	101	3	∥β	∥β	PROPN
ejpam-6699	101	4	)	)	PUNCT
ejpam-6699	101	5	.	.	PUNCT
ejpam-6699	102	1	proposition	proposition	NOUN
ejpam-6699	102	2	1	1	NUM
ejpam-6699	102	3	.	.	PUNCT
ejpam-6699	103	1	[	[	X
ejpam-6699	103	2	28	28	NUM
ejpam-6699	103	3	]	]	X
ejpam-6699	103	4	let	let	VERB
ejpam-6699	103	5	x1	x1	PROPN
ejpam-6699	103	6	,	,	PUNCT
ejpam-6699	103	7	.	.	PUNCT
ejpam-6699	103	8	.	.	PUNCT
ejpam-6699	104	1	.	.	PUNCT
ejpam-6699	105	1	,	,	PUNCT
ejpam-6699	105	2	xn	xn	PROPN
ejpam-6699	105	3	,	,	PUNCT
ejpam-6699	105	4	y	y	PROPN
ejpam-6699	105	5	∈	∈	PROPN
ejpam-6699	105	6	x.	x.	NOUN
ejpam-6699	105	7	suppose	suppose	VERB
ejpam-6699	105	8	that	that	SCONJ
ejpam-6699	105	9	x	x	PRON
ejpam-6699	105	10	is	be	AUX
ejpam-6699	105	11	non	non	ADJ
ejpam-6699	105	12	-	-	ADJ
ejpam-6699	105	13	archimedean	archimedean	ADJ
ejpam-6699	105	14	(	(	PUNCT
ejpam-6699	105	15	n	n	CCONJ
ejpam-6699	105	16	,	,	PUNCT
ejpam-6699	105	17	β)normed	β)normed	ADJ
ejpam-6699	105	18	space	space	NOUN
ejpam-6699	105	19	.	.	PUNCT
ejpam-6699	106	1	then	then	ADV
ejpam-6699	106	2	:	:	PUNCT
ejpam-6699	106	3	s.	s.	PROPN
ejpam-6699	106	4	gowri	gowri	PROPN
ejpam-6699	106	5	et	et	PROPN
ejpam-6699	106	6	al	al	PROPN
ejpam-6699	106	7	.	.	PUNCT
ejpam-6699	106	8	/	/	SYM
ejpam-6699	106	9	eur	eur	PROPN
ejpam-6699	106	10	.	.	PUNCT
ejpam-6699	107	1	j.	j.	PROPN
ejpam-6699	107	2	pure	pure	PROPN
ejpam-6699	107	3	appl	appl	PROPN
ejpam-6699	107	4	.	.	PROPN
ejpam-6699	107	5	math	math	PROPN
ejpam-6699	107	6	,	,	PUNCT
ejpam-6699	107	7	18	18	NUM
ejpam-6699	107	8	(	(	PUNCT
ejpam-6699	107	9	4	4	NUM
ejpam-6699	107	10	)	)	PUNCT
ejpam-6699	107	11	(	(	PUNCT
ejpam-6699	107	12	2025	2025	NUM
ejpam-6699	107	13	)	)	PUNCT
ejpam-6699	107	14	,	,	PUNCT
ejpam-6699	107	15	6699	6699	NUM
ejpam-6699	107	16	5	5	NUM
ejpam-6699	107	17	of	of	ADP
ejpam-6699	107	18	14	14	NUM
ejpam-6699	107	19	(	(	PUNCT
ejpam-6699	107	20	i	i	NOUN
ejpam-6699	107	21	)	)	PUNCT
ejpam-6699	107	22	if	if	SCONJ
ejpam-6699	107	23	x1	x1	PROPN
ejpam-6699	107	24	is	be	AUX
ejpam-6699	107	25	linearly	linearly	ADV
ejpam-6699	107	26	dependent	dependent	ADJ
ejpam-6699	107	27	on	on	ADP
ejpam-6699	107	28	{	{	PUNCT
ejpam-6699	107	29	x2	x2	PROPN
ejpam-6699	107	30	,	,	PUNCT
ejpam-6699	107	31	.	.	PUNCT
ejpam-6699	107	32	.	.	PUNCT
ejpam-6699	107	33	.	.	PUNCT
ejpam-6699	108	1	,	,	PUNCT
ejpam-6699	108	2	xn	xn	PROPN
ejpam-6699	108	3	}	}	PUNCT
ejpam-6699	108	4	,	,	PUNCT
ejpam-6699	108	5	then	then	ADV
ejpam-6699	108	6	∥x1	∥x1	NOUN
ejpam-6699	108	7	,	,	PUNCT
ejpam-6699	108	8	.	.	PUNCT
ejpam-6699	108	9	.	.	PUNCT
ejpam-6699	108	10	.	.	PUNCT
ejpam-6699	109	1	,	,	PUNCT
ejpam-6699	109	2	xn∥β	xn∥β	PUNCT
ejpam-6699	110	1	=	=	NOUN
ejpam-6699	110	2	0	0	PROPN
ejpam-6699	110	3	.	.	PUNCT
ejpam-6699	110	4	(	(	PUNCT
ejpam-6699	110	5	ii	ii	NOUN
ejpam-6699	110	6	)	)	PUNCT
ejpam-6699	110	7	if	if	SCONJ
ejpam-6699	110	8	all	all	DET
ejpam-6699	110	9	vectors	vector	NOUN
ejpam-6699	110	10	x1	x1	PRON
ejpam-6699	110	11	,	,	PUNCT
ejpam-6699	110	12	x2	x2	PROPN
ejpam-6699	110	13	,	,	PUNCT
ejpam-6699	110	14	.	.	PUNCT
ejpam-6699	110	15	.	.	PUNCT
ejpam-6699	110	16	.	.	PUNCT
ejpam-6699	111	1	,	,	PUNCT
ejpam-6699	111	2	xn	xn	PROPN
ejpam-6699	111	3	are	be	AUX
ejpam-6699	111	4	linearly	linearly	ADV
ejpam-6699	111	5	independent	independent	ADJ
ejpam-6699	111	6	,	,	PUNCT
ejpam-6699	111	7	then	then	ADV
ejpam-6699	111	8	∥x1	∥x1	NOUN
ejpam-6699	111	9	,	,	PUNCT
ejpam-6699	111	10	x2	x2	PROPN
ejpam-6699	111	11	,	,	PUNCT
ejpam-6699	111	12	.	.	PUNCT
ejpam-6699	111	13	.	.	PUNCT
ejpam-6699	111	14	.	.	PUNCT
ejpam-6699	112	1	,	,	PUNCT
ejpam-6699	112	2	xn∥β	xn∥β	X
ejpam-6699	112	3	>	>	X
ejpam-6699	112	4	0	0	X
ejpam-6699	112	5	.	.	PUNCT
ejpam-6699	112	6	(	(	PUNCT
ejpam-6699	112	7	iii	iii	X
ejpam-6699	112	8	)	)	PUNCT
ejpam-6699	112	9	the	the	PRON
ejpam-6699	112	10	(	(	PUNCT
ejpam-6699	112	11	n	n	CCONJ
ejpam-6699	112	12	,	,	PUNCT
ejpam-6699	112	13	β)-norm	β)-norm	PUNCT
ejpam-6699	112	14	is	be	AUX
ejpam-6699	112	15	symmetric	symmetric	ADJ
ejpam-6699	112	16	in	in	ADP
ejpam-6699	112	17	all	all	DET
ejpam-6699	112	18	arguments	argument	NOUN
ejpam-6699	112	19	.	.	PUNCT
ejpam-6699	113	1	(	(	PUNCT
ejpam-6699	113	2	iv	iv	X
ejpam-6699	113	3	)	)	PUNCT
ejpam-6699	113	4	for	for	ADP
ejpam-6699	113	5	any	any	DET
ejpam-6699	113	6	scalar	scalar	ADJ
ejpam-6699	113	7	ζ	ζ	NOUN
ejpam-6699	113	8	∈	∈	PROPN
ejpam-6699	113	9	k	k	NOUN
ejpam-6699	113	10	,	,	PUNCT
ejpam-6699	113	11	∥ζx1	∥ζx1	PROPN
ejpam-6699	113	12	,	,	PUNCT
ejpam-6699	113	13	x2	x2	PROPN
ejpam-6699	113	14	,	,	PUNCT
ejpam-6699	113	15	x3	x3	ADJ
ejpam-6699	113	16	,	,	PUNCT
ejpam-6699	113	17	.	.	PUNCT
ejpam-6699	113	18	.	.	PUNCT
ejpam-6699	114	1	.	.	PUNCT
ejpam-6699	115	1	,	,	PUNCT
ejpam-6699	115	2	xn−1	xn−1	PROPN
ejpam-6699	115	3	,	,	PUNCT
ejpam-6699	115	4	xn∥β	xn∥β	X
ejpam-6699	116	1	=	=	SYM
ejpam-6699	116	2	|ζ|β	|ζ|β	ADV
ejpam-6699	116	3	·	·	PUNCT
ejpam-6699	116	4	∥x1	∥x1	NOUN
ejpam-6699	116	5	,	,	PUNCT
ejpam-6699	116	6	x2	x2	PROPN
ejpam-6699	116	7	,	,	PUNCT
ejpam-6699	116	8	x3	x3	ADJ
ejpam-6699	116	9	,	,	PUNCT
ejpam-6699	116	10	.	.	PUNCT
ejpam-6699	116	11	.	.	PUNCT
ejpam-6699	117	1	.	.	PUNCT
ejpam-6699	118	1	,	,	PUNCT
ejpam-6699	118	2	xn−1	xn−1	PROPN
ejpam-6699	118	3	,	,	PUNCT
ejpam-6699	118	4	xn∥β	xn∥β	PROPN
ejpam-6699	118	5	.	.	PUNCT
ejpam-6699	119	1	(	(	PUNCT
ejpam-6699	119	2	v	v	NOUN
ejpam-6699	119	3	)	)	PUNCT
ejpam-6699	119	4	the	the	DET
ejpam-6699	119	5	strong	strong	ADJ
ejpam-6699	119	6	triangle	triangle	NOUN
ejpam-6699	119	7	inequality	inequality	NOUN
ejpam-6699	119	8	holds	hold	VERB
ejpam-6699	119	9	:	:	PUNCT
ejpam-6699	119	10	∥x1	∥x1	NOUN
ejpam-6699	119	11	+	+	CCONJ
ejpam-6699	119	12	y	y	PROPN
ejpam-6699	119	13	,	,	PUNCT
ejpam-6699	119	14	x2	x2	PROPN
ejpam-6699	119	15	,	,	PUNCT
ejpam-6699	119	16	x3	x3	ADJ
ejpam-6699	119	17	,	,	PUNCT
ejpam-6699	119	18	.	.	PUNCT
ejpam-6699	119	19	.	.	PUNCT
ejpam-6699	120	1	.	.	PUNCT
ejpam-6699	121	1	,	,	PUNCT
ejpam-6699	121	2	xn−1	xn−1	PROPN
ejpam-6699	121	3	,	,	PUNCT
ejpam-6699	121	4	xn∥β	xn∥β	VERB
ejpam-6699	121	5	≤	≤	PROPN
ejpam-6699	121	6	max	max	PROPN
ejpam-6699	121	7	{	{	PUNCT
ejpam-6699	121	8	∥x1	∥x1	NOUN
ejpam-6699	121	9	,	,	PUNCT
ejpam-6699	121	10	x2	x2	PROPN
ejpam-6699	121	11	,	,	PUNCT
ejpam-6699	121	12	x3	x3	ADJ
ejpam-6699	121	13	,	,	PUNCT
ejpam-6699	121	14	.	.	PUNCT
ejpam-6699	121	15	.	.	PUNCT
ejpam-6699	122	1	.	.	PUNCT
ejpam-6699	123	1	,	,	PUNCT
ejpam-6699	123	2	xn−1	xn−1	PROPN
ejpam-6699	123	3	,	,	PUNCT
ejpam-6699	123	4	xn∥β	xn∥β	X
ejpam-6699	123	5	,	,	PUNCT
ejpam-6699	123	6	∥y	∥y	ADJ
ejpam-6699	123	7	,	,	PUNCT
ejpam-6699	123	8	x2	x2	PROPN
ejpam-6699	123	9	,	,	PUNCT
ejpam-6699	123	10	x3	x3	ADJ
ejpam-6699	123	11	,	,	PUNCT
ejpam-6699	123	12	.	.	PUNCT
ejpam-6699	123	13	.	.	PUNCT
ejpam-6699	124	1	.	.	PUNCT
ejpam-6699	125	1	,	,	PUNCT
ejpam-6699	125	2	xn−1	xn−1	PROPN
ejpam-6699	125	3	,	,	PUNCT
ejpam-6699	125	4	xn∥β	xn∥β	VERB
ejpam-6699	125	5	}	}	PUNCT
ejpam-6699	125	6	.	.	PUNCT
ejpam-6699	126	1	3	3	X
ejpam-6699	126	2	.	.	X
ejpam-6699	126	3	stability	stability	NOUN
ejpam-6699	126	4	of	of	ADP
ejpam-6699	126	5	the	the	DET
ejpam-6699	126	6	generalized	generalize	VERB
ejpam-6699	126	7	quartic	quartic	ADJ
ejpam-6699	126	8	functional	functional	ADJ
ejpam-6699	126	9	equation	equation	NOUN
ejpam-6699	126	10	consider	consider	VERB
ejpam-6699	126	11	e	e	NOUN
ejpam-6699	126	12	as	as	ADP
ejpam-6699	126	13	a	a	DET
ejpam-6699	126	14	vector	vector	NOUN
ejpam-6699	126	15	space	space	NOUN
ejpam-6699	126	16	and	and	CCONJ
ejpam-6699	126	17	(	(	PUNCT
ejpam-6699	126	18	f	f	X
ejpam-6699	126	19	,	,	PUNCT
ejpam-6699	126	20	∥	∥	PROPN
ejpam-6699	126	21	·	·	PUNCT
ejpam-6699	126	22	,	,	PUNCT
ejpam-6699	126	23	.	.	PUNCT
ejpam-6699	126	24	.	.	PUNCT
ejpam-6699	127	1	.	.	PUNCT
ejpam-6699	128	1	,	,	PUNCT
ejpam-6699	128	2	·	·	PUNCT
ejpam-6699	128	3	∥β	∥β	PROPN
ejpam-6699	128	4	)	)	PUNCT
ejpam-6699	128	5	as	as	ADP
ejpam-6699	128	6	an	an	DET
ejpam-6699	128	7	element	element	NOUN
ejpam-6699	128	8	of	of	ADP
ejpam-6699	128	9	it	it	PRON
ejpam-6699	128	10	.	.	PUNCT
ejpam-6699	129	1	rest	rest	NOUN
ejpam-6699	129	2	assured	assure	VERB
ejpam-6699	129	3	that	that	SCONJ
ejpam-6699	129	4	the	the	DET
ejpam-6699	129	5	space	space	NOUN
ejpam-6699	129	6	(	(	PUNCT
ejpam-6699	129	7	n	n	X
ejpam-6699	129	8	,	,	PUNCT
ejpam-6699	129	9	β	β	NOUN
ejpam-6699	129	10	)	)	PUNCT
ejpam-6699	129	11	is	be	AUX
ejpam-6699	129	12	non	non	ADJ
ejpam-6699	129	13	-	-	ADJ
ejpam-6699	129	14	archimedean	archimedean	ADJ
ejpam-6699	129	15	,	,	PUNCT
ejpam-6699	129	16	with	with	ADP
ejpam-6699	129	17	n	n	PRON
ejpam-6699	129	18	≥	≥	NUM
ejpam-6699	129	19	2	2	NUM
ejpam-6699	129	20	and	and	CCONJ
ejpam-6699	129	21	0	0	NUM
ejpam-6699	129	22	<	<	X
ejpam-6699	129	23	β	β	X
ejpam-6699	129	24	,	,	PUNCT
ejpam-6699	129	25	β1	β1	VERB
ejpam-6699	129	26	≤	≤	NUM
ejpam-6699	129	27	1	1	NUM
ejpam-6699	129	28	.	.	PUNCT
ejpam-6699	130	1	we	we	PRON
ejpam-6699	130	2	consider	consider	VERB
ejpam-6699	130	3	the	the	DET
ejpam-6699	130	4	generalized	generalized	ADJ
ejpam-6699	130	5	quartic	quartic	ADJ
ejpam-6699	130	6	functional	functional	ADJ
ejpam-6699	130	7	equation	equation	NOUN
ejpam-6699	130	8	(	(	PUNCT
ejpam-6699	130	9	1	1	X
ejpam-6699	130	10	)	)	PUNCT
ejpam-6699	130	11	defined	define	VERB
ejpam-6699	130	12	via	via	ADP
ejpam-6699	130	13	the	the	DET
ejpam-6699	130	14	following	follow	VERB
ejpam-6699	130	15	difference	difference	NOUN
ejpam-6699	130	16	operator	operator	NOUN
ejpam-6699	130	17	:	:	PUNCT
ejpam-6699	130	18	∆ϕ(t1	∆ϕ(t1	NOUN
ejpam-6699	130	19	,	,	PUNCT
ejpam-6699	130	20	.	.	PUNCT
ejpam-6699	130	21	.	.	PUNCT
ejpam-6699	131	1	.	.	PUNCT
ejpam-6699	132	1	,	,	PUNCT
ejpam-6699	132	2	tr	tr	VERB
ejpam-6699	132	3	)	)	PUNCT
ejpam-6699	132	4	=	=	PUNCT
ejpam-6699	133	1	−ϕ	−ϕ	ADV
ejpam-6699	133	2	(	(	PUNCT
ejpam-6699	133	3	r∑	r∑	NOUN
ejpam-6699	133	4	i=1	i=1	X
ejpam-6699	133	5	ti	ti	NOUN
ejpam-6699	133	6	)	)	PUNCT
ejpam-6699	134	1	+	+	CCONJ
ejpam-6699	134	2	∑	∑	PUNCT
ejpam-6699	134	3	1≤i	1≤i	PROPN
ejpam-6699	134	4	<	<	X
ejpam-6699	134	5	j	j	X
ejpam-6699	134	6	<	<	X
ejpam-6699	134	7	k	k	X
ejpam-6699	134	8	<	<	X
ejpam-6699	134	9	l≤r	l≤r	NOUN
ejpam-6699	134	10	ϕ(ti	ϕ(ti	NOUN
ejpam-6699	134	11	+	+	CCONJ
ejpam-6699	134	12	tj	tj	PROPN
ejpam-6699	134	13	+	+	PROPN
ejpam-6699	134	14	tk	tk	PROPN
ejpam-6699	134	15	+	+	CCONJ
ejpam-6699	134	16	tl	tl	PROPN
ejpam-6699	134	17	)	)	PUNCT
ejpam-6699	135	1	+	+	PROPN
ejpam-6699	135	2	(	(	PUNCT
ejpam-6699	135	3	−r	−r	ADJ
ejpam-6699	135	4	+	+	X
ejpam-6699	135	5	4	4	NUM
ejpam-6699	135	6	)	)	PUNCT
ejpam-6699	135	7	∑	∑	PUNCT
ejpam-6699	135	8	1≤i	1≤i	PROPN
ejpam-6699	135	9	<	<	X
ejpam-6699	135	10	j	j	X
ejpam-6699	135	11	<	<	X
ejpam-6699	135	12	k≤r	k≤r	PROPN
ejpam-6699	135	13	ϕ(ti	ϕ(ti	PROPN
ejpam-6699	135	14	+	+	CCONJ
ejpam-6699	135	15	tj	tj	PROPN
ejpam-6699	135	16	+	+	X
ejpam-6699	135	17	tk	tk	PROPN
ejpam-6699	135	18	)	)	PUNCT
ejpam-6699	135	19	+	+	CCONJ
ejpam-6699	135	20	(	(	PUNCT
ejpam-6699	135	21	r2	r2	PROPN
ejpam-6699	135	22	−	−	PROPN
ejpam-6699	135	23	7r	7r	NUM
ejpam-6699	135	24	+	+	CCONJ
ejpam-6699	135	25	12	12	NUM
ejpam-6699	135	26	2	2	NUM
ejpam-6699	135	27	)	)	PUNCT
ejpam-6699	136	1	r∑	r∑	ADV
ejpam-6699	136	2	i=1	i=1	ADP
ejpam-6699	136	3	,	,	PUNCT
ejpam-6699	136	4	i	i	PRON
ejpam-6699	136	5	̸=j	̸=j	VERB
ejpam-6699	136	6	ϕ(ti	ϕ(ti	PROPN
ejpam-6699	136	7	+	+	CCONJ
ejpam-6699	136	8	tj)−	tj)−	NOUN
ejpam-6699	136	9	r∑	r∑	NOUN
ejpam-6699	136	10	i=1	i=1	PROPN
ejpam-6699	136	11	ϕ(2ti	ϕ(2ti	NUM
ejpam-6699	136	12	)	)	PUNCT
ejpam-6699	137	1	+	+	CCONJ
ejpam-6699	137	2	(	(	PUNCT
ejpam-6699	137	3	−r3	−r3	NOUN
ejpam-6699	137	4	+	+	CCONJ
ejpam-6699	137	5	9r2	9r2	NUM
ejpam-6699	137	6	−	−	NOUN
ejpam-6699	137	7	26r	26r	NOUN
ejpam-6699	137	8	+	+	CCONJ
ejpam-6699	137	9	120	120	NUM
ejpam-6699	137	10	6	6	NUM
ejpam-6699	137	11	)	)	PUNCT
ejpam-6699	137	12	r∑	r∑	NOUN
ejpam-6699	138	1	i=1	i=1	PROPN
ejpam-6699	138	2	(	(	PUNCT
ejpam-6699	138	3	ϕ(ti	ϕ(ti	PROPN
ejpam-6699	138	4	)	)	PUNCT
ejpam-6699	139	1	+	+	NUM
ejpam-6699	139	2	ϕ(−ti	ϕ(−ti	NOUN
ejpam-6699	139	3	)	)	PUNCT
ejpam-6699	139	4	2	2	NUM
ejpam-6699	139	5	)	)	PUNCT
ejpam-6699	139	6	for	for	ADP
ejpam-6699	139	7	any	any	DET
ejpam-6699	139	8	t1	t1	NOUN
ejpam-6699	139	9	,	,	PUNCT
ejpam-6699	139	10	.	.	PUNCT
ejpam-6699	139	11	.	.	PUNCT
ejpam-6699	139	12	.	.	PUNCT
ejpam-6699	140	1	,	,	PUNCT
ejpam-6699	140	2	tr	tr	NOUN
ejpam-6699	140	3	∈	∈	NOUN
ejpam-6699	140	4	e	e	NOUN
ejpam-6699	140	5	,	,	PUNCT
ejpam-6699	140	6	with	with	ADP
ejpam-6699	140	7	r	r	NOUN
ejpam-6699	140	8	≥	≥	NUM
ejpam-6699	140	9	4	4	NUM
ejpam-6699	140	10	.	.	PUNCT
ejpam-6699	140	11	theorem	theorem	NOUN
ejpam-6699	140	12	2	2	NUM
ejpam-6699	140	13	.	.	PUNCT
ejpam-6699	140	14	let	let	VERB
ejpam-6699	140	15	µ	µ	PRON
ejpam-6699	140	16	∈	∈	NOUN
ejpam-6699	140	17	[	[	X
ejpam-6699	140	18	0,∞	0,∞	NOUN
ejpam-6699	140	19	)	)	PUNCT
ejpam-6699	140	20	and	and	CCONJ
ejpam-6699	140	21	s	s	PROPN
ejpam-6699	140	22	∈	∈	PROPN
ejpam-6699	140	23	(	(	PUNCT
ejpam-6699	140	24	0,∞	0,∞	NOUN
ejpam-6699	140	25	)	)	PUNCT
ejpam-6699	140	26	with	with	ADP
ejpam-6699	140	27	sβ1	sβ1	PROPN
ejpam-6699	140	28	>	>	X
ejpam-6699	140	29	β	β	X
ejpam-6699	140	30	,	,	PUNCT
ejpam-6699	140	31	and	and	CCONJ
ejpam-6699	140	32	let	let	VERB
ejpam-6699	140	33	ϖ	ϖ	PRON
ejpam-6699	140	34	:	:	PUNCT
ejpam-6699	140	35	fn−1	fn−1	ADJ
ejpam-6699	140	36	→	→	SYM
ejpam-6699	140	37	[	[	X
ejpam-6699	140	38	0,∞	0,∞	X
ejpam-6699	140	39	)	)	PUNCT
ejpam-6699	140	40	be	be	VERB
ejpam-6699	140	41	a	a	DET
ejpam-6699	140	42	control	control	NOUN
ejpam-6699	140	43	function	function	NOUN
ejpam-6699	140	44	.	.	PUNCT
ejpam-6699	141	1	assume	assume	VERB
ejpam-6699	141	2	that	that	SCONJ
ejpam-6699	141	3	ϕ	ϕ	X
ejpam-6699	141	4	:	:	PUNCT
ejpam-6699	141	5	e	e	X
ejpam-6699	141	6	→	→	SYM
ejpam-6699	141	7	f	f	PROPN
ejpam-6699	141	8	is	be	AUX
ejpam-6699	141	9	a	a	DET
ejpam-6699	141	10	function	function	NOUN
ejpam-6699	141	11	such	such	ADJ
ejpam-6699	141	12	that	that	SCONJ
ejpam-6699	141	13	∥∆ϕ	∥∆ϕ	PROPN
ejpam-6699	141	14	(	(	PUNCT
ejpam-6699	141	15	t1	t1	NOUN
ejpam-6699	141	16	,	,	PUNCT
ejpam-6699	141	17	t2	t2	NOUN
ejpam-6699	141	18	,	,	PUNCT
ejpam-6699	141	19	.	.	PUNCT
ejpam-6699	141	20	.	.	PUNCT
ejpam-6699	142	1	.	.	PUNCT
ejpam-6699	143	1	,	,	PUNCT
ejpam-6699	143	2	tr	tr	VERB
ejpam-6699	143	3	)	)	PUNCT
ejpam-6699	143	4	,	,	PUNCT
ejpam-6699	143	5	ν1	ν1	NOUN
ejpam-6699	143	6	,	,	PUNCT
ejpam-6699	143	7	.	.	PUNCT
ejpam-6699	143	8	.	.	PUNCT
ejpam-6699	144	1	.	.	PUNCT
ejpam-6699	145	1	,	,	PUNCT
ejpam-6699	145	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	145	3	≤	≤	NOUN
ejpam-6699	145	4	µ	µ	PRON
ejpam-6699	145	5	r∑	r∑	NOUN
ejpam-6699	145	6	j=1	j=1	PROPN
ejpam-6699	145	7	∥tj∥sβ1	∥tj∥sβ1	PROPN
ejpam-6699	145	8	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	145	9	,	,	PUNCT
ejpam-6699	145	10	.	.	PUNCT
ejpam-6699	145	11	.	.	PUNCT
ejpam-6699	146	1	.	.	PUNCT
ejpam-6699	147	1	,	,	PUNCT
ejpam-6699	147	2	νn−1	νn−1	PROPN
ejpam-6699	147	3	)	)	PUNCT
ejpam-6699	147	4	(	(	PUNCT
ejpam-6699	147	5	2	2	X
ejpam-6699	147	6	)	)	PUNCT
ejpam-6699	147	7	for	for	ADP
ejpam-6699	147	8	all	all	DET
ejpam-6699	147	9	t1	t1	NOUN
ejpam-6699	147	10	,	,	PUNCT
ejpam-6699	147	11	.	.	PUNCT
ejpam-6699	147	12	.	.	PUNCT
ejpam-6699	148	1	.	.	PUNCT
ejpam-6699	149	1	,	,	PUNCT
ejpam-6699	149	2	tr	tr	NOUN
ejpam-6699	149	3	∈	∈	PROPN
ejpam-6699	149	4	e	e	NOUN
ejpam-6699	149	5	and	and	CCONJ
ejpam-6699	149	6	ν1	ν1	NOUN
ejpam-6699	149	7	,	,	PUNCT
ejpam-6699	149	8	.	.	PUNCT
ejpam-6699	149	9	.	.	PUNCT
ejpam-6699	149	10	.	.	PUNCT
ejpam-6699	150	1	,	,	PUNCT
ejpam-6699	150	2	νn−1	νn−1	PROPN
ejpam-6699	150	3	∈	∈	PROPN
ejpam-6699	150	4	f	f	X
ejpam-6699	150	5	.	.	PUNCT
ejpam-6699	151	1	then	then	ADV
ejpam-6699	151	2	there	there	PRON
ejpam-6699	151	3	exists	exist	VERB
ejpam-6699	151	4	a	a	DET
ejpam-6699	151	5	unique	unique	ADJ
ejpam-6699	151	6	quartic	quartic	ADJ
ejpam-6699	151	7	mapping	mapping	NOUN
ejpam-6699	151	8	q4	q4	NOUN
ejpam-6699	151	9	:	:	PUNCT
ejpam-6699	151	10	e	e	X
ejpam-6699	151	11	→	→	SYM
ejpam-6699	151	12	f	f	X
ejpam-6699	151	13	satisfying	satisfy	VERB
ejpam-6699	151	14	∥ϕ(t)−q4(t	∥ϕ(t)−q4(t	NOUN
ejpam-6699	151	15	)	)	PUNCT
ejpam-6699	151	16	,	,	PUNCT
ejpam-6699	151	17	ν1	ν1	NOUN
ejpam-6699	151	18	,	,	PUNCT
ejpam-6699	151	19	.	.	PUNCT
ejpam-6699	151	20	.	.	PUNCT
ejpam-6699	151	21	.	.	PUNCT
ejpam-6699	152	1	,	,	PUNCT
ejpam-6699	152	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	152	3	≤	≤	NUM
ejpam-6699	152	4	µ|2−4β|	µ|2−4β|	PROPN
ejpam-6699	152	5	∥t∥sβ1	∥t∥sβ1	PROPN
ejpam-6699	152	6	ϖ(ν1	ϖ(ν1	PROPN
ejpam-6699	152	7	,	,	PUNCT
ejpam-6699	152	8	.	.	PUNCT
ejpam-6699	152	9	.	.	PUNCT
ejpam-6699	153	1	.	.	PUNCT
ejpam-6699	154	1	,	,	PUNCT
ejpam-6699	154	2	νn−1	νn−1	PROPN
ejpam-6699	154	3	)	)	PUNCT
ejpam-6699	154	4	(	(	PUNCT
ejpam-6699	154	5	3	3	X
ejpam-6699	154	6	)	)	PUNCT
ejpam-6699	154	7	for	for	ADP
ejpam-6699	154	8	all	all	DET
ejpam-6699	154	9	t	t	NOUN
ejpam-6699	154	10	∈	∈	NOUN
ejpam-6699	154	11	e	e	NOUN
ejpam-6699	154	12	and	and	CCONJ
ejpam-6699	154	13	all	all	DET
ejpam-6699	154	14	ν1	ν1	NOUN
ejpam-6699	154	15	,	,	PUNCT
ejpam-6699	154	16	.	.	PUNCT
ejpam-6699	154	17	.	.	PUNCT
ejpam-6699	155	1	.	.	PUNCT
ejpam-6699	156	1	,	,	PUNCT
ejpam-6699	156	2	νn−1	νn−1	PROPN
ejpam-6699	156	3	∈	∈	PROPN
ejpam-6699	156	4	f	f	PROPN
ejpam-6699	156	5	.	.	PUNCT
ejpam-6699	157	1	s.	s.	PROPN
ejpam-6699	157	2	gowri	gowri	PROPN
ejpam-6699	157	3	et	et	PROPN
ejpam-6699	157	4	al	al	PROPN
ejpam-6699	157	5	.	.	PUNCT
ejpam-6699	157	6	/	/	SYM
ejpam-6699	157	7	eur	eur	PROPN
ejpam-6699	157	8	.	.	PUNCT
ejpam-6699	158	1	j.	j.	PROPN
ejpam-6699	158	2	pure	pure	PROPN
ejpam-6699	158	3	appl	appl	PROPN
ejpam-6699	158	4	.	.	PROPN
ejpam-6699	158	5	math	math	PROPN
ejpam-6699	158	6	,	,	PUNCT
ejpam-6699	158	7	18	18	NUM
ejpam-6699	158	8	(	(	PUNCT
ejpam-6699	158	9	4	4	NUM
ejpam-6699	158	10	)	)	PUNCT
ejpam-6699	158	11	(	(	PUNCT
ejpam-6699	158	12	2025	2025	NUM
ejpam-6699	158	13	)	)	PUNCT
ejpam-6699	158	14	,	,	PUNCT
ejpam-6699	158	15	6699	6699	NUM
ejpam-6699	158	16	6	6	NUM
ejpam-6699	158	17	of	of	ADP
ejpam-6699	158	18	14	14	NUM
ejpam-6699	158	19	proof	proof	NOUN
ejpam-6699	158	20	.	.	PUNCT
ejpam-6699	159	1	replacing	replace	VERB
ejpam-6699	159	2	(	(	PUNCT
ejpam-6699	159	3	t1	t1	NOUN
ejpam-6699	159	4	,	,	PUNCT
ejpam-6699	159	5	t2	t2	NOUN
ejpam-6699	159	6	,	,	PUNCT
ejpam-6699	159	7	.	.	PUNCT
ejpam-6699	159	8	.	.	PUNCT
ejpam-6699	159	9	.	.	PUNCT
ejpam-6699	160	1	,	,	PUNCT
ejpam-6699	160	2	tr	tr	VERB
ejpam-6699	160	3	)	)	PUNCT
ejpam-6699	160	4	by	by	ADP
ejpam-6699	160	5	(	(	PUNCT
ejpam-6699	160	6	t	t	PROPN
ejpam-6699	160	7	,	,	PUNCT
ejpam-6699	160	8	0	0	NUM
ejpam-6699	160	9	,	,	PUNCT
ejpam-6699	160	10	.	.	PUNCT
ejpam-6699	160	11	.	.	PUNCT
ejpam-6699	161	1	.	.	PUNCT
ejpam-6699	162	1	,	,	PUNCT
ejpam-6699	162	2	0	0	X
ejpam-6699	162	3	)	)	PUNCT
ejpam-6699	162	4	in	in	ADP
ejpam-6699	162	5	(	(	PUNCT
ejpam-6699	162	6	2	2	NUM
ejpam-6699	162	7	)	)	PUNCT
ejpam-6699	162	8	,	,	PUNCT
ejpam-6699	162	9	we	we	PRON
ejpam-6699	162	10	obtain∥∥ϕ(2t)−	obtain∥∥ϕ(2t)−	PROPN
ejpam-6699	162	11	24ϕ(t	24ϕ(t	NUM
ejpam-6699	162	12	)	)	PUNCT
ejpam-6699	162	13	,	,	PUNCT
ejpam-6699	162	14	ν1	ν1	NOUN
ejpam-6699	162	15	,	,	PUNCT
ejpam-6699	162	16	.	.	PUNCT
ejpam-6699	162	17	.	.	PUNCT
ejpam-6699	162	18	.	.	PUNCT
ejpam-6699	163	1	,	,	PUNCT
ejpam-6699	163	2	νn−1	νn−1	VERB
ejpam-6699	163	3	∥∥	∥∥	X
ejpam-6699	163	4	β	β	X
ejpam-6699	163	5	≤	≤	X
ejpam-6699	163	6	µ	µ	NUM
ejpam-6699	163	7	∥t∥sβ1	∥t∥sβ1	PRON
ejpam-6699	163	8	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	163	9	,	,	PUNCT
ejpam-6699	163	10	.	.	PUNCT
ejpam-6699	163	11	.	.	PUNCT
ejpam-6699	164	1	.	.	PUNCT
ejpam-6699	165	1	,	,	PUNCT
ejpam-6699	165	2	νn−1	νn−1	PROPN
ejpam-6699	165	3	)	)	PUNCT
ejpam-6699	165	4	.	.	PUNCT
ejpam-6699	166	1	(	(	PUNCT
ejpam-6699	166	2	4	4	X
ejpam-6699	166	3	)	)	PUNCT
ejpam-6699	166	4	dividing	divide	VERB
ejpam-6699	166	5	both	both	DET
ejpam-6699	166	6	sides	side	NOUN
ejpam-6699	166	7	of	of	ADP
ejpam-6699	166	8	(	(	PUNCT
ejpam-6699	166	9	4	4	NUM
ejpam-6699	166	10	)	)	PUNCT
ejpam-6699	166	11	by	by	ADP
ejpam-6699	166	12	|24β|	|24β|	PROPN
ejpam-6699	166	13	gives∥∥∥∥ϕ(2t)24	gives∥∥∥∥ϕ(2t)24	NOUN
ejpam-6699	166	14	−	−	PROPN
ejpam-6699	166	15	ϕ(t	ϕ(t	NUM
ejpam-6699	166	16	)	)	PUNCT
ejpam-6699	166	17	,	,	PUNCT
ejpam-6699	166	18	ν1	ν1	NOUN
ejpam-6699	166	19	,	,	PUNCT
ejpam-6699	166	20	.	.	PUNCT
ejpam-6699	166	21	.	.	PUNCT
ejpam-6699	167	1	.	.	PUNCT
ejpam-6699	168	1	,	,	PUNCT
ejpam-6699	168	2	νn−1	νn−1	PROPN
ejpam-6699	168	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	168	4	β	β	NOUN
ejpam-6699	168	5	≤	≤	ADJ
ejpam-6699	168	6	|2−4β|µ	|2−4β|µ	NOUN
ejpam-6699	168	7	∥t∥sβ1	∥t∥sβ1	PROPN
ejpam-6699	168	8	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	168	9	,	,	PUNCT
ejpam-6699	168	10	.	.	PUNCT
ejpam-6699	168	11	.	.	PUNCT
ejpam-6699	169	1	.	.	PUNCT
ejpam-6699	170	1	,	,	PUNCT
ejpam-6699	170	2	νn−1	νn−1	PROPN
ejpam-6699	170	3	)	)	PUNCT
ejpam-6699	170	4	.	.	PUNCT
ejpam-6699	171	1	(	(	PUNCT
ejpam-6699	171	2	5	5	X
ejpam-6699	171	3	)	)	PUNCT
ejpam-6699	171	4	replacing	replace	VERB
ejpam-6699	171	5	t	t	NOUN
ejpam-6699	171	6	by	by	ADP
ejpam-6699	171	7	2pt	2pt	NOUN
ejpam-6699	171	8	in	in	ADP
ejpam-6699	171	9	(	(	PUNCT
ejpam-6699	171	10	5	5	NUM
ejpam-6699	171	11	)	)	PUNCT
ejpam-6699	171	12	yields∥∥∥∥ϕ(2p+1	yields∥∥∥∥ϕ(2p+1	PROPN
ejpam-6699	171	13	t	t	NOUN
ejpam-6699	171	14	)	)	PUNCT
ejpam-6699	171	15	24(p+1	24(p+1	NUM
ejpam-6699	171	16	)	)	PUNCT
ejpam-6699	172	1	−	−	ADP
ejpam-6699	172	2	ϕ(2pt	ϕ(2pt	NOUN
ejpam-6699	172	3	)	)	PUNCT
ejpam-6699	172	4	24p	24p	NOUN
ejpam-6699	172	5	,	,	PUNCT
ejpam-6699	172	6	ν1	ν1	NOUN
ejpam-6699	172	7	,	,	PUNCT
ejpam-6699	172	8	.	.	PUNCT
ejpam-6699	172	9	.	.	PUNCT
ejpam-6699	173	1	.	.	PUNCT
ejpam-6699	174	1	,	,	PUNCT
ejpam-6699	174	2	νn−1	νn−1	PROPN
ejpam-6699	174	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	174	4	β	β	X
ejpam-6699	174	5	≤	≤	NUM
ejpam-6699	174	6	|2−4(p+1)β|µ∥2pt∥sβ1	|2−4(p+1)β|µ∥2pt∥sβ1	ADP
ejpam-6699	174	7	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	174	8	,	,	PUNCT
ejpam-6699	174	9	.	.	PUNCT
ejpam-6699	174	10	.	.	PUNCT
ejpam-6699	175	1	.	.	PUNCT
ejpam-6699	176	1	,	,	PUNCT
ejpam-6699	176	2	νn−1	νn−1	PROPN
ejpam-6699	176	3	)	)	PUNCT
ejpam-6699	176	4	≤	≤	NOUN
ejpam-6699	176	5	|2−4β|	|2−4β|	PUNCT
ejpam-6699	176	6	∣∣∣2sβ1−4β	∣∣∣2sβ1−4β	NOUN
ejpam-6699	176	7	∣∣∣p	∣∣∣p	NOUN
ejpam-6699	176	8	µ∥t∥sβ1	µ∥t∥sβ1	PRON
ejpam-6699	176	9	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	176	10	,	,	PUNCT
ejpam-6699	176	11	.	.	PUNCT
ejpam-6699	176	12	.	.	PUNCT
ejpam-6699	177	1	.	.	PUNCT
ejpam-6699	178	1	,	,	PUNCT
ejpam-6699	178	2	νn−1	νn−1	PROPN
ejpam-6699	178	3	)	)	PUNCT
ejpam-6699	178	4	.	.	PUNCT
ejpam-6699	179	1	(	(	PUNCT
ejpam-6699	179	2	6	6	NUM
ejpam-6699	179	3	)	)	PUNCT
ejpam-6699	179	4	since	since	SCONJ
ejpam-6699	179	5	sβ1	sβ1	PROPN
ejpam-6699	179	6	>	>	X
ejpam-6699	179	7	β	β	X
ejpam-6699	179	8	and	and	CCONJ
ejpam-6699	179	9	|2|	|2|	PROPN
ejpam-6699	179	10	=	=	NOUN
ejpam-6699	179	11	̸	̸	NUM
ejpam-6699	179	12	1	1	NUM
ejpam-6699	179	13	,	,	PUNCT
ejpam-6699	179	14	the	the	DET
ejpam-6699	179	15	r.h.s	r.h.s	NOUN
ejpam-6699	179	16	.	.	PUNCT
ejpam-6699	180	1	of	of	ADP
ejpam-6699	180	2	(	(	PUNCT
ejpam-6699	180	3	6	6	NUM
ejpam-6699	180	4	)	)	PUNCT
ejpam-6699	180	5	tends	tend	VERB
ejpam-6699	180	6	to	to	ADP
ejpam-6699	180	7	zero	zero	NUM
ejpam-6699	180	8	as	as	ADP
ejpam-6699	180	9	p→	p→	VERB
ejpam-6699	180	10	∞.	∞.	PROPN
ejpam-6699	180	11	hence	hence	ADV
ejpam-6699	180	12	,	,	PUNCT
ejpam-6699	180	13	the	the	DET
ejpam-6699	180	14	sequence	sequence	NOUN
ejpam-6699	180	15	{	{	PUNCT
ejpam-6699	180	16	ϕ(2pt	ϕ(2pt	PROPN
ejpam-6699	180	17	)	)	PUNCT
ejpam-6699	180	18	24p	24p	NOUN
ejpam-6699	180	19	}	}	PUNCT
ejpam-6699	180	20	is	be	AUX
ejpam-6699	180	21	cauchy	cauchy	ADJ
ejpam-6699	180	22	in	in	ADP
ejpam-6699	180	23	f	f	PROPN
ejpam-6699	180	24	,	,	PUNCT
ejpam-6699	180	25	which	which	PRON
ejpam-6699	180	26	is	be	AUX
ejpam-6699	180	27	complete	complete	ADJ
ejpam-6699	180	28	.	.	PUNCT
ejpam-6699	181	1	therefore	therefore	ADV
ejpam-6699	181	2	,	,	PUNCT
ejpam-6699	181	3	we	we	PRON
ejpam-6699	181	4	define	define	VERB
ejpam-6699	181	5	q4(t	q4(t	PROPN
ejpam-6699	181	6	)	)	PUNCT
ejpam-6699	181	7	:	:	PUNCT
ejpam-6699	182	1	=	=	PUNCT
ejpam-6699	182	2	lim	lim	PROPN
ejpam-6699	182	3	p→∞	p→∞	CCONJ
ejpam-6699	182	4	ϕ(2pt	ϕ(2pt	PROPN
ejpam-6699	182	5	)	)	PUNCT
ejpam-6699	182	6	24p	24p	NOUN
ejpam-6699	182	7	for	for	ADP
ejpam-6699	182	8	all	all	DET
ejpam-6699	182	9	t	t	PROPN
ejpam-6699	182	10	∈	∈	PROPN
ejpam-6699	182	11	e.	e.	PROPN
ejpam-6699	182	12	(	(	PUNCT
ejpam-6699	182	13	7	7	NUM
ejpam-6699	182	14	)	)	PUNCT
ejpam-6699	182	15	to	to	PART
ejpam-6699	182	16	prove	prove	VERB
ejpam-6699	182	17	that	that	SCONJ
ejpam-6699	182	18	q4	q4	PROPN
ejpam-6699	182	19	is	be	AUX
ejpam-6699	182	20	quartic	quartic	ADJ
ejpam-6699	182	21	,	,	PUNCT
ejpam-6699	182	22	apply	apply	VERB
ejpam-6699	182	23	(	(	PUNCT
ejpam-6699	182	24	2	2	NUM
ejpam-6699	182	25	)	)	PUNCT
ejpam-6699	182	26	and	and	CCONJ
ejpam-6699	182	27	lemma	lemma	PROPN
ejpam-6699	182	28	1	1	NUM
ejpam-6699	182	29	:	:	PUNCT
ejpam-6699	182	30	∥∆q4(t1	∥∆q4(t1	NOUN
ejpam-6699	182	31	,	,	PUNCT
ejpam-6699	182	32	.	.	PUNCT
ejpam-6699	182	33	.	.	PUNCT
ejpam-6699	182	34	.	.	PUNCT
ejpam-6699	183	1	,	,	PUNCT
ejpam-6699	183	2	tr	tr	VERB
ejpam-6699	183	3	)	)	PUNCT
ejpam-6699	183	4	,	,	PUNCT
ejpam-6699	183	5	ν1	ν1	NOUN
ejpam-6699	183	6	,	,	PUNCT
ejpam-6699	183	7	.	.	PUNCT
ejpam-6699	183	8	.	.	PUNCT
ejpam-6699	184	1	.	.	PUNCT
ejpam-6699	185	1	,	,	PUNCT
ejpam-6699	185	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	185	3	=	=	PUNCT
ejpam-6699	185	4	lim	lim	PROPN
ejpam-6699	185	5	p→∞	p→∞	CCONJ
ejpam-6699	185	6	∣∣∣2−4pβ	∣∣∣2−4pβ	PROPN
ejpam-6699	185	7	∣∣∣	∣∣∣	ADJ
ejpam-6699	185	8	∥∆ϕ(2pt1	∥∆ϕ(2pt1	PROPN
ejpam-6699	185	9	,	,	PUNCT
ejpam-6699	185	10	.	.	PUNCT
ejpam-6699	185	11	.	.	PUNCT
ejpam-6699	186	1	.	.	PUNCT
ejpam-6699	187	1	,	,	PUNCT
ejpam-6699	187	2	2ptr	2ptr	NUM
ejpam-6699	187	3	)	)	PUNCT
ejpam-6699	187	4	,	,	PUNCT
ejpam-6699	187	5	ν1	ν1	NOUN
ejpam-6699	187	6	,	,	PUNCT
ejpam-6699	187	7	.	.	PUNCT
ejpam-6699	187	8	.	.	PUNCT
ejpam-6699	187	9	.	.	PUNCT
ejpam-6699	188	1	,	,	PUNCT
ejpam-6699	188	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	188	3	≤	≤	NUM
ejpam-6699	188	4	lim	lim	PROPN
ejpam-6699	188	5	p→∞	p→∞	NOUN
ejpam-6699	188	6	µ	µ	PRON
ejpam-6699	188	7	∣∣∣2sβ1−4β	∣∣∣2sβ1−4β	NOUN
ejpam-6699	188	8	∣∣∣p	∣∣∣p	NOUN
ejpam-6699	188	9	r∑	r∑	NOUN
ejpam-6699	188	10	j=1	j=1	PROPN
ejpam-6699	188	11	∥tj∥sβ1	∥tj∥sβ1	PROPN
ejpam-6699	188	12	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	188	13	,	,	PUNCT
ejpam-6699	188	14	.	.	PUNCT
ejpam-6699	188	15	.	.	PUNCT
ejpam-6699	189	1	.	.	PUNCT
ejpam-6699	190	1	,	,	PUNCT
ejpam-6699	190	2	νn−1	νn−1	PROPN
ejpam-6699	190	3	)	)	PUNCT
ejpam-6699	190	4	=	=	SYM
ejpam-6699	191	1	0	0	X
ejpam-6699	191	2	.	.	PUNCT
ejpam-6699	192	1	thus	thus	ADV
ejpam-6699	192	2	,	,	PUNCT
ejpam-6699	192	3	by	by	ADP
ejpam-6699	192	4	lemma	lemma	PROPN
ejpam-6699	192	5	2	2	NUM
ejpam-6699	192	6	,	,	PUNCT
ejpam-6699	192	7	q4	q4	PROPN
ejpam-6699	192	8	satisfies	satisfy	VERB
ejpam-6699	192	9	∆q4	∆q4	NOUN
ejpam-6699	192	10	=	=	SYM
ejpam-6699	192	11	0	0	NUM
ejpam-6699	192	12	,	,	PUNCT
ejpam-6699	192	13	and	and	CCONJ
ejpam-6699	192	14	so	so	ADV
ejpam-6699	192	15	q4	q4	PROPN
ejpam-6699	192	16	is	be	AUX
ejpam-6699	192	17	quartic	quartic	ADJ
ejpam-6699	192	18	.	.	PUNCT
ejpam-6699	193	1	to	to	PART
ejpam-6699	193	2	estimate	estimate	VERB
ejpam-6699	193	3	the	the	DET
ejpam-6699	193	4	difference	difference	NOUN
ejpam-6699	193	5	between	between	ADP
ejpam-6699	193	6	ϕ	ϕ	PROPN
ejpam-6699	193	7	and	and	CCONJ
ejpam-6699	193	8	q4	q4	PROPN
ejpam-6699	193	9	,	,	PUNCT
ejpam-6699	193	10	we	we	PRON
ejpam-6699	193	11	observe	observe	VERB
ejpam-6699	193	12	from	from	ADP
ejpam-6699	193	13	(	(	PUNCT
ejpam-6699	193	14	5	5	NUM
ejpam-6699	193	15	)	)	PUNCT
ejpam-6699	193	16	and	and	CCONJ
ejpam-6699	193	17	similar	similar	ADJ
ejpam-6699	193	18	recursive	recursive	ADJ
ejpam-6699	193	19	steps	step	NOUN
ejpam-6699	193	20	that	that	PRON
ejpam-6699	193	21	∥∥∥∥ϕ(t)−	∥∥∥∥ϕ(t)−	PROPN
ejpam-6699	193	22	ϕ(2pt	ϕ(2pt	PROPN
ejpam-6699	193	23	)	)	PUNCT
ejpam-6699	193	24	24p	24p	NOUN
ejpam-6699	193	25	,	,	PUNCT
ejpam-6699	193	26	ν1	ν1	NOUN
ejpam-6699	193	27	,	,	PUNCT
ejpam-6699	193	28	.	.	PUNCT
ejpam-6699	193	29	.	.	PUNCT
ejpam-6699	194	1	.	.	PUNCT
ejpam-6699	195	1	,	,	PUNCT
ejpam-6699	195	2	νn−1	νn−1	PROPN
ejpam-6699	195	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	195	4	β	β	X
ejpam-6699	195	5	≤	≤	NUM
ejpam-6699	195	6	|2−4β|µ∥t∥sβ1	|2−4β|µ∥t∥sβ1	X
ejpam-6699	195	7	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	195	8	,	,	PUNCT
ejpam-6699	195	9	.	.	PUNCT
ejpam-6699	195	10	.	.	PUNCT
ejpam-6699	196	1	.	.	PUNCT
ejpam-6699	197	1	,	,	PUNCT
ejpam-6699	197	2	νn−1	νn−1	PROPN
ejpam-6699	197	3	)	)	PUNCT
ejpam-6699	197	4	.	.	PUNCT
ejpam-6699	198	1	(	(	PUNCT
ejpam-6699	198	2	8)	8)	NUM
ejpam-6699	198	3	letting	let	VERB
ejpam-6699	198	4	p→	p→	VERB
ejpam-6699	198	5	∞	∞	PROPN
ejpam-6699	198	6	in	in	ADP
ejpam-6699	198	7	(	(	PUNCT
ejpam-6699	198	8	8)	8)	NUM
ejpam-6699	198	9	and	and	CCONJ
ejpam-6699	198	10	applying	apply	VERB
ejpam-6699	198	11	the	the	DET
ejpam-6699	198	12	definition	definition	NOUN
ejpam-6699	198	13	of	of	ADP
ejpam-6699	198	14	q4	q4	PROPN
ejpam-6699	198	15	,	,	PUNCT
ejpam-6699	198	16	we	we	PRON
ejpam-6699	198	17	obtain	obtain	VERB
ejpam-6699	198	18	the	the	DET
ejpam-6699	198	19	inequality	inequality	NOUN
ejpam-6699	198	20	(	(	PUNCT
ejpam-6699	198	21	3	3	NUM
ejpam-6699	198	22	)	)	PUNCT
ejpam-6699	198	23	.	.	PUNCT
ejpam-6699	199	1	finally	finally	ADV
ejpam-6699	199	2	,	,	PUNCT
ejpam-6699	199	3	we	we	PRON
ejpam-6699	199	4	show	show	VERB
ejpam-6699	199	5	uniqueness	uniqueness	NOUN
ejpam-6699	199	6	.	.	PUNCT
ejpam-6699	200	1	consider	consider	VERB
ejpam-6699	200	2	another	another	DET
ejpam-6699	200	3	quartic	quartic	ADJ
ejpam-6699	200	4	mapping	mapping	NOUN
ejpam-6699	200	5	q′	q′	NOUN
ejpam-6699	200	6	4	4	NUM
ejpam-6699	200	7	satisfying	satisfying	ADJ
ejpam-6699	200	8	(	(	PUNCT
ejpam-6699	200	9	3	3	NUM
ejpam-6699	200	10	)	)	PUNCT
ejpam-6699	200	11	.	.	PUNCT
ejpam-6699	201	1	then∥∥q4(t)−q′	then∥∥q4(t)−q′	PROPN
ejpam-6699	201	2	4(t	4(t	NUM
ejpam-6699	201	3	)	)	PUNCT
ejpam-6699	201	4	,	,	PUNCT
ejpam-6699	201	5	ν1	ν1	NOUN
ejpam-6699	201	6	,	,	PUNCT
ejpam-6699	201	7	.	.	PUNCT
ejpam-6699	201	8	.	.	PUNCT
ejpam-6699	202	1	.	.	PUNCT
ejpam-6699	203	1	,	,	PUNCT
ejpam-6699	203	2	νn−1	νn−1	VERB
ejpam-6699	203	3	∥∥	∥∥	X
ejpam-6699	203	4	β	β	X
ejpam-6699	203	5	=	=	SYM
ejpam-6699	203	6	∣∣∣2−4pβ	∣∣∣2−4pβ	ADJ
ejpam-6699	203	7	∣∣∣	∣∣∣	NOUN
ejpam-6699	203	8	∥∥q4(2	∥∥q4(2	NOUN
ejpam-6699	203	9	pt)−q′	pt)−q′	PROPN
ejpam-6699	203	10	4(2	4(2	NUM
ejpam-6699	203	11	pt	pt	NOUN
ejpam-6699	203	12	)	)	PUNCT
ejpam-6699	203	13	,	,	PUNCT
ejpam-6699	203	14	ν1	ν1	NOUN
ejpam-6699	203	15	,	,	PUNCT
ejpam-6699	203	16	.	.	PUNCT
ejpam-6699	203	17	.	.	PUNCT
ejpam-6699	204	1	.	.	PUNCT
ejpam-6699	205	1	,	,	PUNCT
ejpam-6699	205	2	νn−1	νn−1	VERB
ejpam-6699	205	3	∥∥	∥∥	X
ejpam-6699	205	4	β	β	X
ejpam-6699	205	5	≤	≤	X
ejpam-6699	205	6	∣∣∣2−4pβ	∣∣∣2−4pβ	ADJ
ejpam-6699	205	7	∣∣∣	∣∣∣	NOUN
ejpam-6699	205	8	·	·	PUNCT
ejpam-6699	205	9	max	max	PROPN
ejpam-6699	205	10	{	{	PUNCT
ejpam-6699	205	11	∥q4(2	∥q4(2	PROPN
ejpam-6699	205	12	pt)−	pt)−	NOUN
ejpam-6699	205	13	ϕ(2pt	ϕ(2pt	NOUN
ejpam-6699	205	14	)	)	PUNCT
ejpam-6699	205	15	,	,	PUNCT
ejpam-6699	205	16	ν1	ν1	NOUN
ejpam-6699	205	17	,	,	PUNCT
ejpam-6699	205	18	.	.	PUNCT
ejpam-6699	205	19	.	.	PUNCT
ejpam-6699	206	1	.	.	PUNCT
ejpam-6699	207	1	,	,	PUNCT
ejpam-6699	207	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	207	3	,	,	PUNCT
ejpam-6699	207	4	∥∥ϕ(2pt)−q′	∥∥ϕ(2pt)−q′	NOUN
ejpam-6699	207	5	4(2	4(2	NUM
ejpam-6699	207	6	pt	pt	NOUN
ejpam-6699	207	7	)	)	PUNCT
ejpam-6699	207	8	,	,	PUNCT
ejpam-6699	207	9	ν1	ν1	NOUN
ejpam-6699	207	10	,	,	PUNCT
ejpam-6699	207	11	.	.	PUNCT
ejpam-6699	207	12	.	.	PUNCT
ejpam-6699	208	1	.	.	PUNCT
ejpam-6699	209	1	,	,	PUNCT
ejpam-6699	209	2	νn−1	νn−1	VERB
ejpam-6699	209	3	∥∥	∥∥	X
ejpam-6699	209	4	β	β	X
ejpam-6699	209	5	}	}	PUNCT
ejpam-6699	209	6	≤	≤	X
ejpam-6699	209	7	µ	µ	NUM
ejpam-6699	209	8	∣∣∣2−4β	∣∣∣2−4β	NOUN
ejpam-6699	209	9	∣∣∣	∣∣∣	ADJ
ejpam-6699	209	10	∣∣∣2sβ1−4β	∣∣∣2sβ1−4β	NOUN
ejpam-6699	209	11	∣∣∣p	∣∣∣p	NOUN
ejpam-6699	209	12	∥t∥sβ1	∥t∥sβ1	PROPN
ejpam-6699	209	13	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	209	14	,	,	PUNCT
ejpam-6699	209	15	.	.	PUNCT
ejpam-6699	209	16	.	.	PUNCT
ejpam-6699	210	1	.	.	PUNCT
ejpam-6699	211	1	,	,	PUNCT
ejpam-6699	211	2	νn−1	νn−1	PROPN
ejpam-6699	211	3	)	)	PUNCT
ejpam-6699	211	4	.	.	PUNCT
ejpam-6699	212	1	taking	take	VERB
ejpam-6699	212	2	the	the	DET
ejpam-6699	212	3	limit	limit	NOUN
ejpam-6699	212	4	as	as	ADP
ejpam-6699	212	5	p	p	NOUN
ejpam-6699	212	6	→	→	SYM
ejpam-6699	212	7	∞	∞	PROPN
ejpam-6699	212	8	,	,	PUNCT
ejpam-6699	212	9	we	we	PRON
ejpam-6699	212	10	conclude	conclude	VERB
ejpam-6699	212	11	q4(t	q4(t	PROPN
ejpam-6699	212	12	)	)	PUNCT
ejpam-6699	212	13	=	=	SYM
ejpam-6699	212	14	q′	q′	NOUN
ejpam-6699	212	15	4(t	4(t	NUM
ejpam-6699	212	16	)	)	PUNCT
ejpam-6699	212	17	for	for	ADP
ejpam-6699	212	18	all	all	DET
ejpam-6699	212	19	t	t	PROPN
ejpam-6699	212	20	∈	∈	PROPN
ejpam-6699	212	21	e.	e.	PROPN
ejpam-6699	212	22	hence	hence	PROPN
ejpam-6699	212	23	,	,	PUNCT
ejpam-6699	212	24	q4	q4	PROPN
ejpam-6699	212	25	is	be	AUX
ejpam-6699	212	26	the	the	DET
ejpam-6699	212	27	only	only	ADJ
ejpam-6699	212	28	one	one	NUM
ejpam-6699	212	29	quartic	quartic	ADJ
ejpam-6699	212	30	function	function	NOUN
ejpam-6699	212	31	satisfying	satisfy	VERB
ejpam-6699	212	32	(	(	PUNCT
ejpam-6699	212	33	3	3	NUM
ejpam-6699	212	34	)	)	PUNCT
ejpam-6699	212	35	.	.	PUNCT
ejpam-6699	213	1	s.	s.	PROPN
ejpam-6699	213	2	gowri	gowri	PROPN
ejpam-6699	213	3	et	et	PROPN
ejpam-6699	213	4	al	al	PROPN
ejpam-6699	213	5	.	.	PUNCT
ejpam-6699	213	6	/	/	SYM
ejpam-6699	213	7	eur	eur	PROPN
ejpam-6699	213	8	.	.	PUNCT
ejpam-6699	214	1	j.	j.	PROPN
ejpam-6699	214	2	pure	pure	PROPN
ejpam-6699	214	3	appl	appl	PROPN
ejpam-6699	214	4	.	.	PROPN
ejpam-6699	214	5	math	math	PROPN
ejpam-6699	214	6	,	,	PUNCT
ejpam-6699	214	7	18	18	NUM
ejpam-6699	214	8	(	(	PUNCT
ejpam-6699	214	9	4	4	NUM
ejpam-6699	214	10	)	)	PUNCT
ejpam-6699	214	11	(	(	PUNCT
ejpam-6699	214	12	2025	2025	NUM
ejpam-6699	214	13	)	)	PUNCT
ejpam-6699	214	14	,	,	PUNCT
ejpam-6699	214	15	6699	6699	NUM
ejpam-6699	214	16	7	7	NUM
ejpam-6699	214	17	of	of	ADP
ejpam-6699	214	18	14	14	NUM
ejpam-6699	214	19	theorem	theorem	NOUN
ejpam-6699	214	20	3	3	X
ejpam-6699	214	21	.	.	PUNCT
ejpam-6699	215	1	let	let	VERB
ejpam-6699	215	2	µ	µ	PRON
ejpam-6699	215	3	∈	∈	NOUN
ejpam-6699	215	4	[	[	X
ejpam-6699	215	5	0,∞	0,∞	NOUN
ejpam-6699	215	6	)	)	PUNCT
ejpam-6699	215	7	and	and	CCONJ
ejpam-6699	215	8	s	s	PROPN
ejpam-6699	215	9	∈	∈	PROPN
ejpam-6699	215	10	(	(	PUNCT
ejpam-6699	215	11	0,∞	0,∞	NOUN
ejpam-6699	215	12	)	)	PUNCT
ejpam-6699	215	13	with	with	ADP
ejpam-6699	215	14	sβ1	sβ1	PROPN
ejpam-6699	215	15	<	<	X
ejpam-6699	215	16	β	β	X
ejpam-6699	215	17	.	.	PUNCT
ejpam-6699	216	1	let	let	VERB
ejpam-6699	216	2	ϖ	ϖ	PRON
ejpam-6699	216	3	:	:	PUNCT
ejpam-6699	216	4	fn−1	fn−1	ADJ
ejpam-6699	216	5	→	→	SYM
ejpam-6699	216	6	[	[	X
ejpam-6699	216	7	0,∞	0,∞	X
ejpam-6699	216	8	)	)	PUNCT
ejpam-6699	216	9	be	be	VERB
ejpam-6699	216	10	a	a	DET
ejpam-6699	216	11	control	control	NOUN
ejpam-6699	216	12	function	function	NOUN
ejpam-6699	216	13	.	.	PUNCT
ejpam-6699	217	1	assume	assume	VERB
ejpam-6699	217	2	that	that	SCONJ
ejpam-6699	217	3	the	the	DET
ejpam-6699	217	4	mapping	mapping	NOUN
ejpam-6699	217	5	ϕ	ϕ	X
ejpam-6699	217	6	:	:	PUNCT
ejpam-6699	217	7	e	e	X
ejpam-6699	217	8	→	→	SYM
ejpam-6699	217	9	f	f	PROPN
ejpam-6699	217	10	satisfies	satisfy	VERB
ejpam-6699	217	11	∥∆ϕ	∥∆ϕ	PROPN
ejpam-6699	217	12	(	(	PUNCT
ejpam-6699	217	13	t1	t1	NOUN
ejpam-6699	217	14	,	,	PUNCT
ejpam-6699	217	15	.	.	PUNCT
ejpam-6699	217	16	.	.	PUNCT
ejpam-6699	217	17	.	.	PUNCT
ejpam-6699	218	1	,	,	PUNCT
ejpam-6699	218	2	tr	tr	VERB
ejpam-6699	218	3	)	)	PUNCT
ejpam-6699	218	4	,	,	PUNCT
ejpam-6699	218	5	ν1	ν1	NOUN
ejpam-6699	218	6	,	,	PUNCT
ejpam-6699	218	7	.	.	PUNCT
ejpam-6699	218	8	.	.	PUNCT
ejpam-6699	219	1	.	.	PUNCT
ejpam-6699	220	1	,	,	PUNCT
ejpam-6699	220	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	220	3	≤	≤	NOUN
ejpam-6699	220	4	µ	µ	PRON
ejpam-6699	220	5	r∑	r∑	NOUN
ejpam-6699	220	6	j=1	j=1	PROPN
ejpam-6699	220	7	∥tj∥sβ1	∥tj∥sβ1	PROPN
ejpam-6699	220	8	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	220	9	,	,	PUNCT
ejpam-6699	220	10	.	.	PUNCT
ejpam-6699	220	11	.	.	PUNCT
ejpam-6699	221	1	.	.	PUNCT
ejpam-6699	222	1	,	,	PUNCT
ejpam-6699	222	2	νn−1	νn−1	PROPN
ejpam-6699	222	3	)	)	PUNCT
ejpam-6699	222	4	(	(	PUNCT
ejpam-6699	222	5	9	9	NUM
ejpam-6699	222	6	)	)	PUNCT
ejpam-6699	222	7	for	for	ADP
ejpam-6699	222	8	all	all	DET
ejpam-6699	222	9	t1	t1	NOUN
ejpam-6699	222	10	,	,	PUNCT
ejpam-6699	222	11	.	.	PUNCT
ejpam-6699	222	12	.	.	PUNCT
ejpam-6699	223	1	.	.	PUNCT
ejpam-6699	224	1	,	,	PUNCT
ejpam-6699	224	2	tr	tr	NOUN
ejpam-6699	224	3	∈	∈	NOUN
ejpam-6699	224	4	e	e	NOUN
ejpam-6699	224	5	and	and	CCONJ
ejpam-6699	224	6	all	all	DET
ejpam-6699	224	7	ν1	ν1	NOUN
ejpam-6699	224	8	,	,	PUNCT
ejpam-6699	224	9	.	.	PUNCT
ejpam-6699	224	10	.	.	PUNCT
ejpam-6699	225	1	.	.	PUNCT
ejpam-6699	226	1	,	,	PUNCT
ejpam-6699	226	2	νn−1	νn−1	PROPN
ejpam-6699	226	3	∈	∈	PROPN
ejpam-6699	226	4	f	f	X
ejpam-6699	226	5	.	.	PUNCT
ejpam-6699	227	1	then	then	ADV
ejpam-6699	227	2	there	there	PRON
ejpam-6699	227	3	exists	exist	VERB
ejpam-6699	227	4	a	a	DET
ejpam-6699	227	5	unique	unique	ADJ
ejpam-6699	227	6	quartic	quartic	ADJ
ejpam-6699	227	7	function	function	NOUN
ejpam-6699	227	8	q4	q4	NOUN
ejpam-6699	227	9	:	:	PUNCT
ejpam-6699	227	10	e	e	X
ejpam-6699	227	11	→	→	SYM
ejpam-6699	227	12	f	f	X
ejpam-6699	227	13	satisfying	satisfy	VERB
ejpam-6699	227	14	∥ϕ(t)−q4(t	∥ϕ(t)−q4(t	NOUN
ejpam-6699	227	15	)	)	PUNCT
ejpam-6699	227	16	,	,	PUNCT
ejpam-6699	227	17	ν1	ν1	NOUN
ejpam-6699	227	18	,	,	PUNCT
ejpam-6699	227	19	.	.	PUNCT
ejpam-6699	227	20	.	.	PUNCT
ejpam-6699	227	21	.	.	PUNCT
ejpam-6699	228	1	,	,	PUNCT
ejpam-6699	228	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	228	3	≤	≤	NOUN
ejpam-6699	228	4	µ|2−sβ1	µ|2−sβ1	PUNCT
ejpam-6699	228	5	|	|	ADV
ejpam-6699	228	6	∥t∥sβ1	∥t∥sβ1	PROPN
ejpam-6699	228	7	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	228	8	,	,	PUNCT
ejpam-6699	228	9	.	.	PUNCT
ejpam-6699	228	10	.	.	PUNCT
ejpam-6699	229	1	.	.	PUNCT
ejpam-6699	230	1	,	,	PUNCT
ejpam-6699	230	2	νn−1	νn−1	PROPN
ejpam-6699	230	3	)	)	PUNCT
ejpam-6699	230	4	(	(	PUNCT
ejpam-6699	230	5	10	10	NUM
ejpam-6699	230	6	)	)	PUNCT
ejpam-6699	230	7	for	for	ADP
ejpam-6699	230	8	all	all	DET
ejpam-6699	230	9	t	t	NOUN
ejpam-6699	230	10	∈	∈	NOUN
ejpam-6699	230	11	e	e	NOUN
ejpam-6699	230	12	and	and	CCONJ
ejpam-6699	230	13	all	all	DET
ejpam-6699	230	14	ν1	ν1	NOUN
ejpam-6699	230	15	,	,	PUNCT
ejpam-6699	230	16	.	.	PUNCT
ejpam-6699	230	17	.	.	PUNCT
ejpam-6699	231	1	.	.	PUNCT
ejpam-6699	232	1	,	,	PUNCT
ejpam-6699	232	2	νn−1	νn−1	PROPN
ejpam-6699	232	3	∈	∈	PROPN
ejpam-6699	232	4	f	f	X
ejpam-6699	232	5	.	.	PUNCT
ejpam-6699	233	1	proof	proof	NOUN
ejpam-6699	233	2	.	.	PUNCT
ejpam-6699	234	1	replacing	replace	VERB
ejpam-6699	234	2	(	(	PUNCT
ejpam-6699	234	3	t1	t1	NOUN
ejpam-6699	234	4	,	,	PUNCT
ejpam-6699	234	5	.	.	PUNCT
ejpam-6699	234	6	.	.	PUNCT
ejpam-6699	234	7	.	.	PUNCT
ejpam-6699	235	1	,	,	PUNCT
ejpam-6699	235	2	tr	tr	VERB
ejpam-6699	235	3	)	)	PUNCT
ejpam-6699	235	4	by	by	ADP
ejpam-6699	235	5	(	(	PUNCT
ejpam-6699	235	6	t	t	PROPN
ejpam-6699	235	7	,	,	PUNCT
ejpam-6699	235	8	0	0	NUM
ejpam-6699	235	9	,	,	PUNCT
ejpam-6699	235	10	.	.	PUNCT
ejpam-6699	235	11	.	.	PUNCT
ejpam-6699	236	1	.	.	PUNCT
ejpam-6699	237	1	,	,	PUNCT
ejpam-6699	237	2	0	0	X
ejpam-6699	237	3	)	)	PUNCT
ejpam-6699	237	4	in	in	ADP
ejpam-6699	237	5	(	(	PUNCT
ejpam-6699	237	6	9	9	NUM
ejpam-6699	237	7	)	)	PUNCT
ejpam-6699	237	8	,	,	PUNCT
ejpam-6699	237	9	we	we	PRON
ejpam-6699	237	10	have∥∥ϕ(2t)−	have∥∥ϕ(2t)−	PROPN
ejpam-6699	237	11	24ϕ(t	24ϕ(t	NUM
ejpam-6699	237	12	)	)	PUNCT
ejpam-6699	237	13	,	,	PUNCT
ejpam-6699	237	14	ν1	ν1	NOUN
ejpam-6699	237	15	,	,	PUNCT
ejpam-6699	237	16	.	.	PUNCT
ejpam-6699	237	17	.	.	PUNCT
ejpam-6699	238	1	.	.	PUNCT
ejpam-6699	239	1	,	,	PUNCT
ejpam-6699	239	2	νn−1	νn−1	VERB
ejpam-6699	239	3	∥∥	∥∥	X
ejpam-6699	239	4	β	β	X
ejpam-6699	239	5	≤	≤	X
ejpam-6699	239	6	µ	µ	NUM
ejpam-6699	239	7	∥t∥sβ1	∥t∥sβ1	PRON
ejpam-6699	239	8	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	239	9	,	,	PUNCT
ejpam-6699	239	10	.	.	PUNCT
ejpam-6699	239	11	.	.	PUNCT
ejpam-6699	240	1	.	.	PUNCT
ejpam-6699	241	1	,	,	PUNCT
ejpam-6699	241	2	νn−1	νn−1	PROPN
ejpam-6699	241	3	)	)	PUNCT
ejpam-6699	241	4	.	.	PUNCT
ejpam-6699	242	1	(	(	PUNCT
ejpam-6699	242	2	11	11	X
ejpam-6699	242	3	)	)	PUNCT
ejpam-6699	242	4	replacing	replace	VERB
ejpam-6699	242	5	t	t	NOUN
ejpam-6699	242	6	by	by	ADP
ejpam-6699	242	7	t/2	t/2	NUM
ejpam-6699	242	8	in	in	ADP
ejpam-6699	242	9	(	(	PUNCT
ejpam-6699	242	10	11	11	NUM
ejpam-6699	242	11	)	)	PUNCT
ejpam-6699	242	12	,	,	PUNCT
ejpam-6699	242	13	we	we	PRON
ejpam-6699	242	14	obtain∥∥∥ϕ(t)−	obtain∥∥∥ϕ(t)−	VERB
ejpam-6699	242	15	24ϕ	24ϕ	NOUN
ejpam-6699	242	16	(	(	PUNCT
ejpam-6699	242	17	t	t	PROPN
ejpam-6699	242	18	2	2	NUM
ejpam-6699	242	19	)	)	PUNCT
ejpam-6699	242	20	,	,	PUNCT
ejpam-6699	242	21	ν1	ν1	NOUN
ejpam-6699	242	22	,	,	PUNCT
ejpam-6699	242	23	.	.	PUNCT
ejpam-6699	242	24	.	.	PUNCT
ejpam-6699	243	1	.	.	PUNCT
ejpam-6699	244	1	,	,	PUNCT
ejpam-6699	244	2	νn−1	νn−1	PROPN
ejpam-6699	244	3	∥∥∥	∥∥∥	PROPN
ejpam-6699	244	4	β	β	NOUN
ejpam-6699	244	5	≤	≤	NOUN
ejpam-6699	244	6	µ|2−sβ1	µ|2−sβ1	PUNCT
ejpam-6699	244	7	|	|	ADV
ejpam-6699	244	8	∥t∥sβ1	∥t∥sβ1	PROPN
ejpam-6699	244	9	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	244	10	,	,	PUNCT
ejpam-6699	244	11	.	.	PUNCT
ejpam-6699	244	12	.	.	PUNCT
ejpam-6699	245	1	.	.	PUNCT
ejpam-6699	246	1	,	,	PUNCT
ejpam-6699	246	2	νn−1	νn−1	PROPN
ejpam-6699	246	3	)	)	PUNCT
ejpam-6699	246	4	.	.	PUNCT
ejpam-6699	247	1	(	(	PUNCT
ejpam-6699	247	2	12	12	X
ejpam-6699	247	3	)	)	PUNCT
ejpam-6699	247	4	switching	switch	VERB
ejpam-6699	247	5	t	t	PROPN
ejpam-6699	247	6	by	by	ADP
ejpam-6699	247	7	t/2p	t/2p	PROPN
ejpam-6699	247	8	in	in	ADP
ejpam-6699	247	9	(	(	PUNCT
ejpam-6699	247	10	12	12	NUM
ejpam-6699	247	11	)	)	PUNCT
ejpam-6699	247	12	,	,	PUNCT
ejpam-6699	247	13	we	we	PRON
ejpam-6699	247	14	have∥∥∥24pϕ	have∥∥∥24pϕ	NOUN
ejpam-6699	247	15	(	(	PUNCT
ejpam-6699	247	16	t	t	NOUN
ejpam-6699	247	17	2p	2p	NUM
ejpam-6699	247	18	)	)	PUNCT
ejpam-6699	248	1	−	−	PROPN
ejpam-6699	248	2	24(p+1)ϕ	24(p+1)ϕ	NUM
ejpam-6699	248	3	(	(	PUNCT
ejpam-6699	248	4	t	t	PROPN
ejpam-6699	248	5	2p+1	2p+1	PROPN
ejpam-6699	248	6	)	)	PUNCT
ejpam-6699	248	7	,	,	PUNCT
ejpam-6699	248	8	ν1	ν1	NOUN
ejpam-6699	248	9	,	,	PUNCT
ejpam-6699	248	10	.	.	PUNCT
ejpam-6699	248	11	.	.	PUNCT
ejpam-6699	248	12	.	.	PUNCT
ejpam-6699	249	1	,	,	PUNCT
ejpam-6699	249	2	νn−1	νn−1	PROPN
ejpam-6699	249	3	∥∥∥	∥∥∥	PROPN
ejpam-6699	249	4	β	β	NOUN
ejpam-6699	249	5	≤	≤	NOUN
ejpam-6699	249	6	µ|2−sβ1	µ|2−sβ1	PUNCT
ejpam-6699	249	7	|	|	ADV
ejpam-6699	249	8	∣∣∣24β−sβ1	∣∣∣24β−sβ1	ADP
ejpam-6699	249	9	∣∣∣p	∣∣∣p	NOUN
ejpam-6699	249	10	∥t∥sβ1	∥t∥sβ1	PROPN
ejpam-6699	249	11	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	249	12	,	,	PUNCT
ejpam-6699	249	13	.	.	PUNCT
ejpam-6699	249	14	.	.	PUNCT
ejpam-6699	250	1	.	.	PUNCT
ejpam-6699	251	1	,	,	PUNCT
ejpam-6699	251	2	νn−1	νn−1	PROPN
ejpam-6699	251	3	)	)	PUNCT
ejpam-6699	251	4	.	.	PUNCT
ejpam-6699	252	1	(	(	PUNCT
ejpam-6699	252	2	13	13	NUM
ejpam-6699	252	3	)	)	PUNCT
ejpam-6699	252	4	since	since	SCONJ
ejpam-6699	252	5	sβ1	sβ1	PROPN
ejpam-6699	252	6	<	<	X
ejpam-6699	252	7	β	β	X
ejpam-6699	252	8	and	and	CCONJ
ejpam-6699	252	9	|2|	|2|	PROPN
ejpam-6699	252	10	=	=	NOUN
ejpam-6699	252	11	̸	̸	NUM
ejpam-6699	252	12	1	1	NUM
ejpam-6699	252	13	,	,	PUNCT
ejpam-6699	252	14	the	the	DET
ejpam-6699	252	15	r.h.s	r.h.s	NOUN
ejpam-6699	252	16	.	.	PUNCT
ejpam-6699	253	1	of	of	ADP
ejpam-6699	253	2	(	(	PUNCT
ejpam-6699	253	3	13	13	NUM
ejpam-6699	253	4	)	)	PUNCT
ejpam-6699	253	5	tends	tend	VERB
ejpam-6699	253	6	to	to	ADP
ejpam-6699	253	7	zero	zero	NUM
ejpam-6699	253	8	as	as	ADP
ejpam-6699	253	9	p→	p→	PROPN
ejpam-6699	253	10	∞.	∞.	PROPN
ejpam-6699	253	11	thus	thus	ADV
ejpam-6699	253	12	,	,	PUNCT
ejpam-6699	253	13	the	the	DET
ejpam-6699	253	14	sequence	sequence	NOUN
ejpam-6699	253	15	{	{	PUNCT
ejpam-6699	253	16	24pϕ(t/2p	24pϕ(t/2p	NUM
ejpam-6699	253	17	)	)	PUNCT
ejpam-6699	253	18	}	}	PUNCT
ejpam-6699	253	19	is	be	AUX
ejpam-6699	253	20	cauchy	cauchy	ADJ
ejpam-6699	253	21	in	in	ADP
ejpam-6699	253	22	f	f	PROPN
ejpam-6699	253	23	,	,	PUNCT
ejpam-6699	253	24	which	which	PRON
ejpam-6699	253	25	is	be	AUX
ejpam-6699	253	26	complete	complete	ADJ
ejpam-6699	253	27	.	.	PUNCT
ejpam-6699	254	1	hence	hence	ADV
ejpam-6699	254	2	,	,	PUNCT
ejpam-6699	254	3	define	define	VERB
ejpam-6699	254	4	q4(t	q4(t	PROPN
ejpam-6699	254	5	)	)	PUNCT
ejpam-6699	254	6	:	:	PUNCT
ejpam-6699	255	1	=	=	PUNCT
ejpam-6699	255	2	lim	lim	PROPN
ejpam-6699	255	3	p→∞	p→∞	ADJ
ejpam-6699	255	4	24pϕ	24pϕ	NOUN
ejpam-6699	255	5	(	(	PUNCT
ejpam-6699	255	6	t	t	NOUN
ejpam-6699	255	7	2p	2p	NUM
ejpam-6699	255	8	)	)	PUNCT
ejpam-6699	255	9	(	(	PUNCT
ejpam-6699	255	10	14	14	NUM
ejpam-6699	255	11	)	)	PUNCT
ejpam-6699	255	12	for	for	ADP
ejpam-6699	255	13	all	all	PRON
ejpam-6699	255	14	t	t	PROPN
ejpam-6699	255	15	∈	∈	PROPN
ejpam-6699	255	16	e.	e.	PROPN
ejpam-6699	256	1	we	we	PRON
ejpam-6699	256	2	now	now	ADV
ejpam-6699	256	3	show	show	VERB
ejpam-6699	256	4	that	that	SCONJ
ejpam-6699	256	5	q4	q4	PROPN
ejpam-6699	256	6	is	be	AUX
ejpam-6699	256	7	quartic	quartic	ADJ
ejpam-6699	256	8	.	.	PUNCT
ejpam-6699	257	1	from	from	ADP
ejpam-6699	257	2	(	(	PUNCT
ejpam-6699	257	3	9	9	NUM
ejpam-6699	257	4	)	)	PUNCT
ejpam-6699	257	5	and	and	CCONJ
ejpam-6699	257	6	lemma	lemma	PROPN
ejpam-6699	257	7	1	1	NUM
ejpam-6699	257	8	,	,	PUNCT
ejpam-6699	257	9	we	we	PRON
ejpam-6699	257	10	get	get	VERB
ejpam-6699	257	11	∥∆q4(t1	∥∆q4(t1	NOUN
ejpam-6699	257	12	,	,	PUNCT
ejpam-6699	257	13	.	.	PUNCT
ejpam-6699	257	14	.	.	PUNCT
ejpam-6699	257	15	.	.	PUNCT
ejpam-6699	258	1	,	,	PUNCT
ejpam-6699	258	2	tr	tr	VERB
ejpam-6699	258	3	)	)	PUNCT
ejpam-6699	258	4	,	,	PUNCT
ejpam-6699	258	5	ν1	ν1	NOUN
ejpam-6699	258	6	,	,	PUNCT
ejpam-6699	258	7	.	.	PUNCT
ejpam-6699	258	8	.	.	PUNCT
ejpam-6699	259	1	.	.	PUNCT
ejpam-6699	260	1	,	,	PUNCT
ejpam-6699	260	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	260	3	=	=	SYM
ejpam-6699	260	4	lim	lim	PROPN
ejpam-6699	260	5	p→∞	p→∞	PROPN
ejpam-6699	260	6	∣∣∣24pβ∣∣∣	∣∣∣24pβ∣∣∣	PROPN
ejpam-6699	260	7	∥∥∥∆ϕ	∥∥∥∆ϕ	PROPN
ejpam-6699	260	8	(	(	PUNCT
ejpam-6699	260	9	t1	t1	NOUN
ejpam-6699	260	10	2p	2p	NUM
ejpam-6699	260	11	,	,	PUNCT
ejpam-6699	260	12	.	.	PUNCT
ejpam-6699	260	13	.	.	PUNCT
ejpam-6699	261	1	.	.	PUNCT
ejpam-6699	262	1	,	,	PUNCT
ejpam-6699	262	2	tr	tr	VERB
ejpam-6699	262	3	2p	2p	NUM
ejpam-6699	262	4	)	)	PUNCT
ejpam-6699	262	5	,	,	PUNCT
ejpam-6699	262	6	ν1	ν1	NOUN
ejpam-6699	262	7	,	,	PUNCT
ejpam-6699	262	8	.	.	PUNCT
ejpam-6699	262	9	.	.	PUNCT
ejpam-6699	263	1	.	.	PUNCT
ejpam-6699	264	1	,	,	PUNCT
ejpam-6699	264	2	νn−1	νn−1	PROPN
ejpam-6699	264	3	∥∥∥	∥∥∥	PROPN
ejpam-6699	264	4	β	β	NOUN
ejpam-6699	264	5	≤	≤	PROPN
ejpam-6699	264	6	lim	lim	PROPN
ejpam-6699	264	7	p→∞	p→∞	NOUN
ejpam-6699	264	8	µ	µ	X
ejpam-6699	264	9	∣∣∣24β−sβ1	∣∣∣24β−sβ1	X
ejpam-6699	264	10	∣∣∣p	∣∣∣p	NOUN
ejpam-6699	264	11	r∑	r∑	NOUN
ejpam-6699	264	12	j=1	j=1	PROPN
ejpam-6699	264	13	∥tj∥sβ1	∥tj∥sβ1	PROPN
ejpam-6699	264	14	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	264	15	,	,	PUNCT
ejpam-6699	264	16	.	.	PUNCT
ejpam-6699	264	17	.	.	PUNCT
ejpam-6699	265	1	.	.	PUNCT
ejpam-6699	266	1	,	,	PUNCT
ejpam-6699	266	2	νn−1	νn−1	PROPN
ejpam-6699	266	3	)	)	PUNCT
ejpam-6699	266	4	=	=	SYM
ejpam-6699	267	1	0	0	X
ejpam-6699	267	2	.	.	PUNCT
ejpam-6699	268	1	hence	hence	ADV
ejpam-6699	268	2	,	,	PUNCT
ejpam-6699	268	3	by	by	ADP
ejpam-6699	268	4	lemma	lemma	PROPN
ejpam-6699	268	5	2	2	NUM
ejpam-6699	268	6	,	,	PUNCT
ejpam-6699	268	7	the	the	DET
ejpam-6699	268	8	function	function	NOUN
ejpam-6699	268	9	q4	q4	PROPN
ejpam-6699	268	10	is	be	AUX
ejpam-6699	268	11	quartic	quartic	ADJ
ejpam-6699	268	12	.	.	PUNCT
ejpam-6699	269	1	to	to	PART
ejpam-6699	269	2	prove	prove	VERB
ejpam-6699	269	3	inequality	inequality	NOUN
ejpam-6699	269	4	(	(	PUNCT
ejpam-6699	269	5	10	10	NUM
ejpam-6699	269	6	)	)	PUNCT
ejpam-6699	269	7	,	,	PUNCT
ejpam-6699	269	8	note	note	VERB
ejpam-6699	269	9	from	from	ADP
ejpam-6699	269	10	(	(	PUNCT
ejpam-6699	269	11	12	12	NUM
ejpam-6699	269	12	)	)	PUNCT
ejpam-6699	269	13	and	and	CCONJ
ejpam-6699	269	14	similar	similar	ADJ
ejpam-6699	269	15	reasoning	reasoning	NOUN
ejpam-6699	269	16	(	(	PUNCT
ejpam-6699	269	17	induction	induction	NOUN
ejpam-6699	269	18	or	or	CCONJ
ejpam-6699	269	19	recursion	recursion	NOUN
ejpam-6699	269	20	)	)	PUNCT
ejpam-6699	270	1	that	that	SCONJ
ejpam-6699	270	2	∥∥∥ϕ(t)−	∥∥∥ϕ(t)−	PROPN
ejpam-6699	270	3	24pϕ	24pϕ	PROPN
ejpam-6699	270	4	(	(	PUNCT
ejpam-6699	270	5	t	t	PROPN
ejpam-6699	270	6	2p	2p	NUM
ejpam-6699	270	7	)	)	PUNCT
ejpam-6699	270	8	,	,	PUNCT
ejpam-6699	270	9	ν1	ν1	NOUN
ejpam-6699	270	10	,	,	PUNCT
ejpam-6699	270	11	.	.	PUNCT
ejpam-6699	270	12	.	.	PUNCT
ejpam-6699	271	1	.	.	PUNCT
ejpam-6699	272	1	,	,	PUNCT
ejpam-6699	272	2	νn−1	νn−1	PROPN
ejpam-6699	272	3	∥∥∥	∥∥∥	PROPN
ejpam-6699	272	4	β	β	NOUN
ejpam-6699	272	5	≤	≤	NOUN
ejpam-6699	272	6	µ|2−sβ1	µ|2−sβ1	PUNCT
ejpam-6699	272	7	|	|	ADV
ejpam-6699	272	8	∥t∥sβ1	∥t∥sβ1	PROPN
ejpam-6699	272	9	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	272	10	,	,	PUNCT
ejpam-6699	272	11	.	.	PUNCT
ejpam-6699	272	12	.	.	PUNCT
ejpam-6699	273	1	.	.	PUNCT
ejpam-6699	274	1	,	,	PUNCT
ejpam-6699	274	2	νn−1	νn−1	PROPN
ejpam-6699	274	3	)	)	PUNCT
ejpam-6699	274	4	.	.	PUNCT
ejpam-6699	275	1	(	(	PUNCT
ejpam-6699	275	2	15	15	X
ejpam-6699	275	3	)	)	PUNCT
ejpam-6699	275	4	s.	s.	PROPN
ejpam-6699	275	5	gowri	gowri	PROPN
ejpam-6699	275	6	et	et	PROPN
ejpam-6699	275	7	al	al	PROPN
ejpam-6699	275	8	.	.	PUNCT
ejpam-6699	275	9	/	/	SYM
ejpam-6699	275	10	eur	eur	PROPN
ejpam-6699	275	11	.	.	PUNCT
ejpam-6699	276	1	j.	j.	PROPN
ejpam-6699	276	2	pure	pure	PROPN
ejpam-6699	276	3	appl	appl	PROPN
ejpam-6699	276	4	.	.	PROPN
ejpam-6699	276	5	math	math	PROPN
ejpam-6699	276	6	,	,	PUNCT
ejpam-6699	276	7	18	18	NUM
ejpam-6699	276	8	(	(	PUNCT
ejpam-6699	276	9	4	4	NUM
ejpam-6699	276	10	)	)	PUNCT
ejpam-6699	276	11	(	(	PUNCT
ejpam-6699	276	12	2025	2025	NUM
ejpam-6699	276	13	)	)	PUNCT
ejpam-6699	276	14	,	,	PUNCT
ejpam-6699	276	15	6699	6699	NUM
ejpam-6699	276	16	8	8	NUM
ejpam-6699	276	17	of	of	ADP
ejpam-6699	276	18	14	14	NUM
ejpam-6699	276	19	taking	take	VERB
ejpam-6699	276	20	the	the	DET
ejpam-6699	276	21	limit	limit	NOUN
ejpam-6699	276	22	as	as	ADP
ejpam-6699	276	23	p→	p→	VERB
ejpam-6699	276	24	∞	∞	PROPN
ejpam-6699	276	25	in	in	ADP
ejpam-6699	276	26	(	(	PUNCT
ejpam-6699	276	27	15	15	NUM
ejpam-6699	276	28	)	)	PUNCT
ejpam-6699	276	29	and	and	CCONJ
ejpam-6699	276	30	using	use	VERB
ejpam-6699	276	31	the	the	DET
ejpam-6699	276	32	definition	definition	NOUN
ejpam-6699	276	33	of	of	ADP
ejpam-6699	276	34	q4	q4	PROPN
ejpam-6699	276	35	in	in	ADP
ejpam-6699	276	36	(	(	PUNCT
ejpam-6699	276	37	14	14	NUM
ejpam-6699	276	38	)	)	PUNCT
ejpam-6699	276	39	,	,	PUNCT
ejpam-6699	276	40	we	we	PRON
ejpam-6699	276	41	obtain	obtain	VERB
ejpam-6699	276	42	(	(	PUNCT
ejpam-6699	276	43	10	10	NUM
ejpam-6699	276	44	)	)	PUNCT
ejpam-6699	276	45	.	.	PUNCT
ejpam-6699	277	1	to	to	PART
ejpam-6699	277	2	prove	prove	VERB
ejpam-6699	277	3	uniqueness	uniqueness	NOUN
ejpam-6699	277	4	,	,	PUNCT
ejpam-6699	277	5	assume	assume	VERB
ejpam-6699	277	6	that	that	SCONJ
ejpam-6699	277	7	another	another	DET
ejpam-6699	277	8	quartic	quartic	ADJ
ejpam-6699	277	9	function	function	NOUN
ejpam-6699	277	10	q′	q′	NOUN
ejpam-6699	277	11	4	4	NUM
ejpam-6699	277	12	:	:	PUNCT
ejpam-6699	277	13	e	e	X
ejpam-6699	277	14	→	→	SYM
ejpam-6699	277	15	f	f	X
ejpam-6699	277	16	satisfying	satisfying	NOUN
ejpam-6699	277	17	(	(	PUNCT
ejpam-6699	277	18	10	10	NUM
ejpam-6699	277	19	)	)	PUNCT
ejpam-6699	277	20	.	.	PUNCT
ejpam-6699	278	1	then∥∥q4(t)−q′	then∥∥q4(t)−q′	PROPN
ejpam-6699	278	2	4(t	4(t	NUM
ejpam-6699	278	3	)	)	PUNCT
ejpam-6699	278	4	,	,	PUNCT
ejpam-6699	278	5	ν1	ν1	NOUN
ejpam-6699	278	6	,	,	PUNCT
ejpam-6699	278	7	.	.	PUNCT
ejpam-6699	278	8	.	.	PUNCT
ejpam-6699	279	1	.	.	PUNCT
ejpam-6699	280	1	,	,	PUNCT
ejpam-6699	280	2	νn−1	νn−1	VERB
ejpam-6699	280	3	∥∥	∥∥	X
ejpam-6699	280	4	β	β	X
ejpam-6699	280	5	=	=	PUNCT
ejpam-6699	280	6	∣∣∣24pβ∣∣∣	∣∣∣24pβ∣∣∣	PROPN
ejpam-6699	280	7	∥∥∥q4	∥∥∥q4	PROPN
ejpam-6699	280	8	(	(	PUNCT
ejpam-6699	280	9	t	t	NOUN
ejpam-6699	280	10	2p	2p	NUM
ejpam-6699	280	11	)	)	PUNCT
ejpam-6699	281	1	−q′	−q′	NOUN
ejpam-6699	281	2	4	4	NUM
ejpam-6699	281	3	(	(	PUNCT
ejpam-6699	281	4	t	t	NOUN
ejpam-6699	281	5	2p	2p	NUM
ejpam-6699	281	6	)	)	PUNCT
ejpam-6699	281	7	,	,	PUNCT
ejpam-6699	281	8	ν1	ν1	NOUN
ejpam-6699	281	9	,	,	PUNCT
ejpam-6699	281	10	.	.	PUNCT
ejpam-6699	281	11	.	.	PUNCT
ejpam-6699	282	1	.	.	PUNCT
ejpam-6699	283	1	,	,	PUNCT
ejpam-6699	283	2	νn−1	νn−1	PROPN
ejpam-6699	283	3	∥∥∥	∥∥∥	PROPN
ejpam-6699	283	4	β	β	NOUN
ejpam-6699	283	5	≤	≤	X
ejpam-6699	283	6	∣∣∣24pβ∣∣∣max	∣∣∣24pβ∣∣∣max	NOUN
ejpam-6699	283	7	{	{	PUNCT
ejpam-6699	283	8	∥∥∥q4	∥∥∥q4	PROPN
ejpam-6699	283	9	(	(	PUNCT
ejpam-6699	283	10	t	t	NOUN
ejpam-6699	283	11	2p	2p	NUM
ejpam-6699	283	12	)	)	PUNCT
ejpam-6699	284	1	−	−	PROPN
ejpam-6699	284	2	ϕ	ϕ	NOUN
ejpam-6699	284	3	(	(	PUNCT
ejpam-6699	284	4	t	t	PROPN
ejpam-6699	284	5	2p	2p	NUM
ejpam-6699	284	6	)	)	PUNCT
ejpam-6699	284	7	∥∥∥	∥∥∥	PROPN
ejpam-6699	284	8	β	β	X
ejpam-6699	284	9	,	,	PUNCT
ejpam-6699	284	10	∥∥∥ϕ	∥∥∥ϕ	PROPN
ejpam-6699	284	11	(	(	PUNCT
ejpam-6699	284	12	t	t	NOUN
ejpam-6699	284	13	2p	2p	NUM
ejpam-6699	284	14	)	)	PUNCT
ejpam-6699	284	15	−q′	−q′	NOUN
ejpam-6699	284	16	4	4	NUM
ejpam-6699	284	17	(	(	PUNCT
ejpam-6699	284	18	t	t	NOUN
ejpam-6699	284	19	2p	2p	NUM
ejpam-6699	284	20	)	)	PUNCT
ejpam-6699	284	21	∥∥∥	∥∥∥	NOUN
ejpam-6699	284	22	β	β	X
ejpam-6699	284	23	}	}	PUNCT
ejpam-6699	284	24	≤	≤	NOUN
ejpam-6699	284	25	µ|2−sβ1	µ|2−sβ1	PUNCT
ejpam-6699	284	26	|	|	ADV
ejpam-6699	284	27	∣∣∣24β−sβ1	∣∣∣24β−sβ1	ADP
ejpam-6699	284	28	∣∣∣p	∣∣∣p	NOUN
ejpam-6699	284	29	∥t∥sβ1	∥t∥sβ1	PROPN
ejpam-6699	284	30	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	284	31	,	,	PUNCT
ejpam-6699	284	32	.	.	PUNCT
ejpam-6699	284	33	.	.	PUNCT
ejpam-6699	284	34	.	.	PUNCT
ejpam-6699	285	1	,	,	PUNCT
ejpam-6699	285	2	νn−1	νn−1	PROPN
ejpam-6699	285	3	)	)	PUNCT
ejpam-6699	285	4	.	.	PUNCT
ejpam-6699	286	1	as	as	SCONJ
ejpam-6699	286	2	p→	p→	NOUN
ejpam-6699	286	3	∞	∞	PROPN
ejpam-6699	286	4	,	,	PUNCT
ejpam-6699	286	5	the	the	DET
ejpam-6699	286	6	r.h.s	r.h.s	NOUN
ejpam-6699	286	7	.	.	PUNCT
ejpam-6699	286	8	tends	tend	VERB
ejpam-6699	286	9	to	to	ADP
ejpam-6699	286	10	zero	zero	NUM
ejpam-6699	286	11	.	.	PUNCT
ejpam-6699	287	1	thus,∥∥q4(t)−q′	thus,∥∥q4(t)−q′	X
ejpam-6699	287	2	4(t	4(t	NUM
ejpam-6699	287	3	)	)	PUNCT
ejpam-6699	288	1	,	,	PUNCT
ejpam-6699	288	2	ν1	ν1	NOUN
ejpam-6699	288	3	,	,	PUNCT
ejpam-6699	288	4	.	.	PUNCT
ejpam-6699	288	5	.	.	PUNCT
ejpam-6699	289	1	.	.	PUNCT
ejpam-6699	290	1	,	,	PUNCT
ejpam-6699	290	2	νn−1	νn−1	VERB
ejpam-6699	290	3	∥∥	∥∥	X
ejpam-6699	290	4	β	β	X
ejpam-6699	290	5	=	=	SYM
ejpam-6699	290	6	0	0	PROPN
ejpam-6699	290	7	,	,	PUNCT
ejpam-6699	290	8	which	which	PRON
ejpam-6699	290	9	implies	imply	VERB
ejpam-6699	290	10	,	,	PUNCT
ejpam-6699	290	11	by	by	ADP
ejpam-6699	290	12	lemma	lemma	PROPN
ejpam-6699	290	13	2	2	NUM
ejpam-6699	290	14	,	,	PUNCT
ejpam-6699	290	15	that	that	DET
ejpam-6699	290	16	q4	q4	PROPN
ejpam-6699	290	17	=	=	PUNCT
ejpam-6699	290	18	q′	q′	NOUN
ejpam-6699	290	19	4	4	NUM
ejpam-6699	290	20	.	.	PUNCT
ejpam-6699	291	1	hence	hence	ADV
ejpam-6699	291	2	,	,	PUNCT
ejpam-6699	291	3	q4	q4	PROPN
ejpam-6699	291	4	is	be	AUX
ejpam-6699	291	5	unique	unique	ADJ
ejpam-6699	291	6	.	.	PUNCT
ejpam-6699	292	1	theorem	theorem	ADJ
ejpam-6699	292	2	4	4	NUM
ejpam-6699	292	3	.	.	PUNCT
ejpam-6699	293	1	let	let	VERB
ejpam-6699	293	2	a	a	DET
ejpam-6699	293	3	function	function	NOUN
ejpam-6699	293	4	ψ	ψ	NOUN
ejpam-6699	293	5	:	:	PUNCT
ejpam-6699	293	6	er	er	INTJ
ejpam-6699	293	7	→	→	X
ejpam-6699	293	8	[	[	X
ejpam-6699	293	9	0,∞	0,∞	NOUN
ejpam-6699	293	10	)	)	PUNCT
ejpam-6699	293	11	such	such	ADJ
ejpam-6699	293	12	that	that	AUX
ejpam-6699	293	13	lim	lim	PROPN
ejpam-6699	293	14	p→∞	p→∞	NOUN
ejpam-6699	293	15	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6699	293	16	1	1	NUM
ejpam-6699	293	17	24pβ	24pβ	NOUN
ejpam-6699	293	18	∣∣∣∣ψ	∣∣∣∣ψ	NOUN
ejpam-6699	293	19	(	(	PUNCT
ejpam-6699	293	20	2pt1	2pt1	NUM
ejpam-6699	293	21	,	,	PUNCT
ejpam-6699	293	22	.	.	PUNCT
ejpam-6699	293	23	.	.	PUNCT
ejpam-6699	294	1	.	.	PUNCT
ejpam-6699	295	1	,	,	PUNCT
ejpam-6699	295	2	2	2	NUM
ejpam-6699	295	3	ptr	ptr	NOUN
ejpam-6699	295	4	)	)	PUNCT
ejpam-6699	296	1	=	=	SYM
ejpam-6699	296	2	0	0	PUNCT
ejpam-6699	297	1	(	(	PUNCT
ejpam-6699	297	2	16	16	NUM
ejpam-6699	297	3	)	)	PUNCT
ejpam-6699	297	4	for	for	ADP
ejpam-6699	297	5	all	all	DET
ejpam-6699	297	6	t1	t1	NOUN
ejpam-6699	297	7	,	,	PUNCT
ejpam-6699	297	8	.	.	PUNCT
ejpam-6699	297	9	.	.	PUNCT
ejpam-6699	297	10	.	.	PUNCT
ejpam-6699	298	1	,	,	PUNCT
ejpam-6699	298	2	tr	tr	NOUN
ejpam-6699	298	3	∈	∈	NOUN
ejpam-6699	298	4	e	e	NOUN
ejpam-6699	298	5	,	,	PUNCT
ejpam-6699	298	6	and	and	CCONJ
ejpam-6699	298	7	let	let	VERB
ejpam-6699	298	8	ϖ	ϖ	PRON
ejpam-6699	298	9	:	:	PUNCT
ejpam-6699	298	10	fn−1	fn−1	ADJ
ejpam-6699	298	11	→	→	SYM
ejpam-6699	298	12	[	[	X
ejpam-6699	298	13	0,∞	0,∞	X
ejpam-6699	298	14	)	)	PUNCT
ejpam-6699	298	15	be	be	VERB
ejpam-6699	298	16	a	a	DET
ejpam-6699	298	17	control	control	NOUN
ejpam-6699	298	18	function	function	NOUN
ejpam-6699	298	19	.	.	PUNCT
ejpam-6699	299	1	assume	assume	VERB
ejpam-6699	299	2	that	that	SCONJ
ejpam-6699	299	3	the	the	DET
ejpam-6699	299	4	mapping	mapping	NOUN
ejpam-6699	299	5	ϕ	ϕ	X
ejpam-6699	299	6	:	:	PUNCT
ejpam-6699	299	7	e	e	X
ejpam-6699	299	8	→	→	SYM
ejpam-6699	299	9	f	f	PROPN
ejpam-6699	299	10	satisfies	satisfie	NOUN
ejpam-6699	299	11	∥∆ϕ(t1	∥∆ϕ(t1	PROPN
ejpam-6699	299	12	,	,	PUNCT
ejpam-6699	299	13	.	.	PUNCT
ejpam-6699	299	14	.	.	PUNCT
ejpam-6699	299	15	.	.	PUNCT
ejpam-6699	300	1	,	,	PUNCT
ejpam-6699	300	2	tr	tr	VERB
ejpam-6699	300	3	)	)	PUNCT
ejpam-6699	300	4	,	,	PUNCT
ejpam-6699	300	5	ν1	ν1	NOUN
ejpam-6699	300	6	,	,	PUNCT
ejpam-6699	300	7	.	.	PUNCT
ejpam-6699	300	8	.	.	PUNCT
ejpam-6699	301	1	.	.	PUNCT
ejpam-6699	302	1	,	,	PUNCT
ejpam-6699	302	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	302	3	≤	≤	NUM
ejpam-6699	302	4	ψ(t1	ψ(t1	VERB
ejpam-6699	302	5	,	,	PUNCT
ejpam-6699	302	6	.	.	PUNCT
ejpam-6699	302	7	.	.	PUNCT
ejpam-6699	303	1	.	.	PUNCT
ejpam-6699	304	1	,	,	PUNCT
ejpam-6699	305	1	tr)ϖ(ν1	tr)ϖ(ν1	PROPN
ejpam-6699	305	2	,	,	PUNCT
ejpam-6699	305	3	.	.	PUNCT
ejpam-6699	305	4	.	.	PUNCT
ejpam-6699	306	1	.	.	PUNCT
ejpam-6699	307	1	,	,	PUNCT
ejpam-6699	307	2	νn−1	νn−1	PROPN
ejpam-6699	307	3	)	)	PUNCT
ejpam-6699	307	4	(	(	PUNCT
ejpam-6699	307	5	17	17	NUM
ejpam-6699	307	6	)	)	PUNCT
ejpam-6699	307	7	for	for	ADP
ejpam-6699	307	8	all	all	DET
ejpam-6699	307	9	t1	t1	NOUN
ejpam-6699	307	10	,	,	PUNCT
ejpam-6699	307	11	.	.	PUNCT
ejpam-6699	307	12	.	.	PUNCT
ejpam-6699	308	1	.	.	PUNCT
ejpam-6699	309	1	,	,	PUNCT
ejpam-6699	309	2	tr	tr	NOUN
ejpam-6699	309	3	∈	∈	PROPN
ejpam-6699	309	4	e	e	NOUN
ejpam-6699	309	5	and	and	CCONJ
ejpam-6699	309	6	ν1	ν1	NOUN
ejpam-6699	309	7	,	,	PUNCT
ejpam-6699	309	8	.	.	PUNCT
ejpam-6699	309	9	.	.	PUNCT
ejpam-6699	309	10	.	.	PUNCT
ejpam-6699	310	1	,	,	PUNCT
ejpam-6699	310	2	νn−1	νn−1	PROPN
ejpam-6699	310	3	∈	∈	PROPN
ejpam-6699	310	4	f	f	X
ejpam-6699	310	5	.	.	PUNCT
ejpam-6699	311	1	then	then	ADV
ejpam-6699	311	2	there	there	PRON
ejpam-6699	311	3	exists	exist	VERB
ejpam-6699	311	4	a	a	DET
ejpam-6699	311	5	unique	unique	ADJ
ejpam-6699	311	6	quartic	quartic	ADJ
ejpam-6699	311	7	mapping	mapping	NOUN
ejpam-6699	311	8	q4	q4	NOUN
ejpam-6699	311	9	:	:	PUNCT
ejpam-6699	311	10	e	e	X
ejpam-6699	311	11	→	→	SYM
ejpam-6699	311	12	f	f	X
ejpam-6699	311	13	satisfying	satisfy	VERB
ejpam-6699	311	14	∥ϕ(t)−q4(t	∥ϕ(t)−q4(t	NOUN
ejpam-6699	311	15	)	)	PUNCT
ejpam-6699	311	16	,	,	PUNCT
ejpam-6699	311	17	ν1	ν1	NOUN
ejpam-6699	311	18	,	,	PUNCT
ejpam-6699	311	19	.	.	PUNCT
ejpam-6699	311	20	.	.	PUNCT
ejpam-6699	311	21	.	.	PUNCT
ejpam-6699	312	1	,	,	PUNCT
ejpam-6699	312	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	312	3	≤	≤	NUM
ejpam-6699	312	4	ψ̃(t)ϖ(ν1	ψ̃(t)ϖ(ν1	NOUN
ejpam-6699	312	5	,	,	PUNCT
ejpam-6699	312	6	.	.	PUNCT
ejpam-6699	312	7	.	.	PUNCT
ejpam-6699	313	1	.	.	PUNCT
ejpam-6699	314	1	,	,	PUNCT
ejpam-6699	314	2	νn−1	νn−1	PROPN
ejpam-6699	314	3	)	)	PUNCT
ejpam-6699	314	4	(	(	PUNCT
ejpam-6699	314	5	18	18	NUM
ejpam-6699	314	6	)	)	PUNCT
ejpam-6699	314	7	where	where	SCONJ
ejpam-6699	314	8	ψ̃(t	ψ̃(t	PROPN
ejpam-6699	314	9	)	)	PUNCT
ejpam-6699	314	10	:	:	PUNCT
ejpam-6699	315	1	=	=	SYM
ejpam-6699	315	2	lim	lim	PROPN
ejpam-6699	315	3	p→∞	p→∞	ADJ
ejpam-6699	315	4	max	max	PROPN
ejpam-6699	315	5	{	{	PUNCT
ejpam-6699	315	6	∣∣∣2−4iβ	∣∣∣2−4iβ	PROPN
ejpam-6699	315	7	∣∣∣ψ	∣∣∣ψ	PROPN
ejpam-6699	315	8	(	(	PUNCT
ejpam-6699	315	9	2i−1	2i−1	NUM
ejpam-6699	315	10	t	t	PROPN
ejpam-6699	315	11	,	,	PUNCT
ejpam-6699	315	12	0	0	NUM
ejpam-6699	315	13	,	,	PUNCT
ejpam-6699	315	14	0	0	NUM
ejpam-6699	315	15	,	,	PUNCT
ejpam-6699	315	16	.	.	PUNCT
ejpam-6699	315	17	.	.	PUNCT
ejpam-6699	315	18	.	.	PUNCT
ejpam-6699	316	1	,	,	PUNCT
ejpam-6699	316	2	0	0	NUM
ejpam-6699	316	3	)	)	PUNCT
ejpam-6699	316	4	:	:	PUNCT
ejpam-6699	317	1	1	1	NUM
ejpam-6699	317	2	≤	≤	NUM
ejpam-6699	317	3	i	i	PRON
ejpam-6699	317	4	≤	≤	ADV
ejpam-6699	317	5	p	p	X
ejpam-6699	317	6	}	}	PUNCT
ejpam-6699	317	7	(	(	PUNCT
ejpam-6699	317	8	19	19	NUM
ejpam-6699	317	9	)	)	PUNCT
ejpam-6699	317	10	for	for	ADP
ejpam-6699	317	11	all	all	DET
ejpam-6699	317	12	t	t	PROPN
ejpam-6699	317	13	∈	∈	PROPN
ejpam-6699	317	14	e.	e.	PROPN
ejpam-6699	318	1	moreover	moreover	ADV
ejpam-6699	318	2	,	,	PUNCT
ejpam-6699	318	3	if	if	SCONJ
ejpam-6699	318	4	lim	lim	PROPN
ejpam-6699	318	5	t→∞	t→∞	NUM
ejpam-6699	318	6	lim	lim	PROPN
ejpam-6699	318	7	p→∞	p→∞	ADJ
ejpam-6699	318	8	max	max	PROPN
ejpam-6699	318	9	{	{	PUNCT
ejpam-6699	318	10	∣∣∣2−4iβ	∣∣∣2−4iβ	PROPN
ejpam-6699	318	11	∣∣∣ψ	∣∣∣ψ	PROPN
ejpam-6699	318	12	(	(	PUNCT
ejpam-6699	318	13	2i−1	2i−1	NUM
ejpam-6699	318	14	t	t	PROPN
ejpam-6699	318	15	,	,	PUNCT
ejpam-6699	318	16	0	0	NUM
ejpam-6699	318	17	,	,	PUNCT
ejpam-6699	318	18	0	0	NUM
ejpam-6699	318	19	,	,	PUNCT
ejpam-6699	318	20	.	.	PUNCT
ejpam-6699	318	21	.	.	PUNCT
ejpam-6699	318	22	.	.	PUNCT
ejpam-6699	319	1	,	,	PUNCT
ejpam-6699	319	2	0	0	NUM
ejpam-6699	319	3	)	)	PUNCT
ejpam-6699	319	4	:	:	PUNCT
ejpam-6699	320	1	1	1	NUM
ejpam-6699	320	2	+	+	NUM
ejpam-6699	320	3	t	t	NOUN
ejpam-6699	320	4	≤	≤	NUM
ejpam-6699	320	5	i	i	PRON
ejpam-6699	320	6	≤	≤	PROPN
ejpam-6699	320	7	p+	p+	VERB
ejpam-6699	320	8	t	t	NOUN
ejpam-6699	320	9	}	}	PUNCT
ejpam-6699	320	10	=	=	SYM
ejpam-6699	320	11	0	0	NUM
ejpam-6699	320	12	(	(	PUNCT
ejpam-6699	320	13	20	20	NUM
ejpam-6699	320	14	)	)	PUNCT
ejpam-6699	320	15	for	for	ADP
ejpam-6699	320	16	every	every	DET
ejpam-6699	320	17	t	t	NOUN
ejpam-6699	320	18	∈	∈	PROPN
ejpam-6699	320	19	e	e	NOUN
ejpam-6699	320	20	,	,	PUNCT
ejpam-6699	320	21	then	then	ADV
ejpam-6699	320	22	the	the	DET
ejpam-6699	320	23	function	function	NOUN
ejpam-6699	320	24	q4	q4	PROPN
ejpam-6699	320	25	is	be	AUX
ejpam-6699	320	26	unique	unique	ADJ
ejpam-6699	320	27	.	.	PUNCT
ejpam-6699	321	1	proof	proof	NOUN
ejpam-6699	321	2	.	.	PUNCT
ejpam-6699	322	1	replacing	replace	VERB
ejpam-6699	322	2	(	(	PUNCT
ejpam-6699	322	3	t1	t1	NOUN
ejpam-6699	322	4	,	,	PUNCT
ejpam-6699	322	5	.	.	PUNCT
ejpam-6699	322	6	.	.	PUNCT
ejpam-6699	322	7	.	.	PUNCT
ejpam-6699	323	1	,	,	PUNCT
ejpam-6699	323	2	tr	tr	VERB
ejpam-6699	323	3	)	)	PUNCT
ejpam-6699	323	4	by	by	ADP
ejpam-6699	323	5	(	(	PUNCT
ejpam-6699	323	6	t	t	PROPN
ejpam-6699	323	7	,	,	PUNCT
ejpam-6699	323	8	0	0	NUM
ejpam-6699	323	9	,	,	PUNCT
ejpam-6699	323	10	.	.	PUNCT
ejpam-6699	323	11	.	.	PUNCT
ejpam-6699	324	1	.	.	PUNCT
ejpam-6699	325	1	,	,	PUNCT
ejpam-6699	325	2	0	0	X
ejpam-6699	325	3	)	)	PUNCT
ejpam-6699	325	4	in	in	ADP
ejpam-6699	325	5	(	(	PUNCT
ejpam-6699	325	6	17	17	NUM
ejpam-6699	325	7	)	)	PUNCT
ejpam-6699	325	8	and	and	CCONJ
ejpam-6699	325	9	dividing	divide	VERB
ejpam-6699	325	10	both	both	DET
ejpam-6699	325	11	sides	side	NOUN
ejpam-6699	325	12	by	by	ADP
ejpam-6699	325	13	|24β|	|24β|	PROPN
ejpam-6699	325	14	gives	give	VERB
ejpam-6699	325	15	∥∥∥∥ϕ(2t)24	∥∥∥∥ϕ(2t)24	NUM
ejpam-6699	325	16	−	−	NOUN
ejpam-6699	325	17	ϕ(t	ϕ(t	NUM
ejpam-6699	325	18	)	)	PUNCT
ejpam-6699	325	19	,	,	PUNCT
ejpam-6699	325	20	ν1	ν1	NOUN
ejpam-6699	325	21	,	,	PUNCT
ejpam-6699	325	22	.	.	PUNCT
ejpam-6699	325	23	.	.	PUNCT
ejpam-6699	326	1	.	.	PUNCT
ejpam-6699	327	1	,	,	PUNCT
ejpam-6699	327	2	νn−1	νn−1	PROPN
ejpam-6699	327	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	327	4	β	β	NOUN
ejpam-6699	327	5	≤	≤	NUM
ejpam-6699	327	6	|2−4β|ψ(t	|2−4β|ψ(t	NOUN
ejpam-6699	327	7	,	,	PUNCT
ejpam-6699	327	8	0	0	NUM
ejpam-6699	327	9	,	,	PUNCT
ejpam-6699	327	10	.	.	PUNCT
ejpam-6699	327	11	.	.	PUNCT
ejpam-6699	328	1	.	.	PUNCT
ejpam-6699	329	1	,	,	PUNCT
ejpam-6699	329	2	0)ϖ(ν1	0)ϖ(ν1	PROPN
ejpam-6699	329	3	,	,	PUNCT
ejpam-6699	329	4	.	.	PUNCT
ejpam-6699	329	5	.	.	PUNCT
ejpam-6699	330	1	.	.	PUNCT
ejpam-6699	331	1	,	,	PUNCT
ejpam-6699	331	2	νn−1	νn−1	PROPN
ejpam-6699	331	3	)	)	PUNCT
ejpam-6699	331	4	.	.	PUNCT
ejpam-6699	332	1	(	(	PUNCT
ejpam-6699	332	2	21	21	NUM
ejpam-6699	332	3	)	)	PUNCT
ejpam-6699	332	4	s.	s.	PROPN
ejpam-6699	332	5	gowri	gowri	PROPN
ejpam-6699	332	6	et	et	PROPN
ejpam-6699	332	7	al	al	PROPN
ejpam-6699	332	8	.	.	PUNCT
ejpam-6699	332	9	/	/	SYM
ejpam-6699	332	10	eur	eur	PROPN
ejpam-6699	332	11	.	.	PUNCT
ejpam-6699	333	1	j.	j.	PROPN
ejpam-6699	333	2	pure	pure	PROPN
ejpam-6699	333	3	appl	appl	PROPN
ejpam-6699	333	4	.	.	PROPN
ejpam-6699	333	5	math	math	PROPN
ejpam-6699	333	6	,	,	PUNCT
ejpam-6699	333	7	18	18	NUM
ejpam-6699	333	8	(	(	PUNCT
ejpam-6699	333	9	4	4	NUM
ejpam-6699	333	10	)	)	PUNCT
ejpam-6699	333	11	(	(	PUNCT
ejpam-6699	333	12	2025	2025	NUM
ejpam-6699	333	13	)	)	PUNCT
ejpam-6699	333	14	,	,	PUNCT
ejpam-6699	333	15	6699	6699	NUM
ejpam-6699	333	16	9	9	NUM
ejpam-6699	333	17	of	of	ADP
ejpam-6699	333	18	14	14	NUM
ejpam-6699	333	19	replacing	replace	VERB
ejpam-6699	333	20	t	t	NOUN
ejpam-6699	333	21	with	with	ADP
ejpam-6699	333	22	2it	2it	NOUN
ejpam-6699	333	23	and	and	CCONJ
ejpam-6699	333	24	dividing	dividing	NOUN
ejpam-6699	333	25	by	by	ADP
ejpam-6699	333	26	|24iβ|	|24iβ|	PROPN
ejpam-6699	333	27	,	,	PUNCT
ejpam-6699	333	28	we	we	PRON
ejpam-6699	333	29	get∥∥∥∥ϕ(2i+1	get∥∥∥∥ϕ(2i+1	PROPN
ejpam-6699	333	30	t	t	PROPN
ejpam-6699	333	31	)	)	PUNCT
ejpam-6699	333	32	24(i+1	24(i+1	NUM
ejpam-6699	333	33	)	)	PUNCT
ejpam-6699	333	34	−	−	ADP
ejpam-6699	333	35	ϕ(2it	ϕ(2it	NOUN
ejpam-6699	333	36	)	)	PUNCT
ejpam-6699	333	37	24i	24i	NOUN
ejpam-6699	333	38	,	,	PUNCT
ejpam-6699	333	39	ν1	ν1	NOUN
ejpam-6699	333	40	,	,	PUNCT
ejpam-6699	333	41	.	.	PUNCT
ejpam-6699	333	42	.	.	PUNCT
ejpam-6699	334	1	.	.	PUNCT
ejpam-6699	335	1	,	,	PUNCT
ejpam-6699	335	2	νn−1	νn−1	PROPN
ejpam-6699	335	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	335	4	β	β	X
ejpam-6699	335	5	≤	≤	NUM
ejpam-6699	335	6	|2−4(i+1)β|ψ	|2−4(i+1)β|ψ	ADV
ejpam-6699	335	7	(	(	PUNCT
ejpam-6699	335	8	2it	2it	NOUN
ejpam-6699	335	9	,	,	PUNCT
ejpam-6699	335	10	0	0	NUM
ejpam-6699	335	11	,	,	PUNCT
ejpam-6699	335	12	.	.	PUNCT
ejpam-6699	335	13	.	.	PUNCT
ejpam-6699	336	1	.	.	PUNCT
ejpam-6699	337	1	,	,	PUNCT
ejpam-6699	337	2	0	0	X
ejpam-6699	337	3	)	)	PUNCT
ejpam-6699	337	4	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	337	5	,	,	PUNCT
ejpam-6699	337	6	.	.	PUNCT
ejpam-6699	337	7	.	.	PUNCT
ejpam-6699	337	8	.	.	PUNCT
ejpam-6699	338	1	,	,	PUNCT
ejpam-6699	338	2	νn−1	νn−1	PROPN
ejpam-6699	338	3	)	)	PUNCT
ejpam-6699	338	4	.	.	PUNCT
ejpam-6699	339	1	(	(	PUNCT
ejpam-6699	339	2	22	22	NUM
ejpam-6699	339	3	)	)	PUNCT
ejpam-6699	339	4	by	by	ADP
ejpam-6699	339	5	(	(	PUNCT
ejpam-6699	339	6	16	16	NUM
ejpam-6699	339	7	)	)	PUNCT
ejpam-6699	339	8	,	,	PUNCT
ejpam-6699	339	9	the	the	DET
ejpam-6699	339	10	right	right	ADJ
ejpam-6699	339	11	-	-	PUNCT
ejpam-6699	339	12	hand	hand	NOUN
ejpam-6699	339	13	side	side	NOUN
ejpam-6699	339	14	tends	tend	VERB
ejpam-6699	339	15	to	to	ADP
ejpam-6699	339	16	zero	zero	NUM
ejpam-6699	339	17	as	as	ADP
ejpam-6699	339	18	i→	i→	PROPN
ejpam-6699	339	19	∞	∞	PROPN
ejpam-6699	339	20	,	,	PUNCT
ejpam-6699	339	21	so	so	CCONJ
ejpam-6699	339	22	the	the	DET
ejpam-6699	339	23	sequence	sequence	NOUN
ejpam-6699	339	24	{	{	PUNCT
ejpam-6699	339	25	ϕ(2mt	ϕ(2mt	NOUN
ejpam-6699	339	26	)	)	PUNCT
ejpam-6699	339	27	24	24	NUM
ejpam-6699	339	28	m	m	NOUN
ejpam-6699	339	29	}	}	PUNCT
ejpam-6699	339	30	is	be	AUX
ejpam-6699	339	31	cauchy	cauchy	PROPN
ejpam-6699	339	32	.	.	PUNCT
ejpam-6699	340	1	as	as	SCONJ
ejpam-6699	340	2	f	f	PROPN
ejpam-6699	340	3	is	be	AUX
ejpam-6699	340	4	complete	complete	ADJ
ejpam-6699	340	5	,	,	PUNCT
ejpam-6699	340	6	define	define	VERB
ejpam-6699	340	7	q4(t	q4(t	PROPN
ejpam-6699	340	8	)	)	PUNCT
ejpam-6699	340	9	:	:	PUNCT
ejpam-6699	341	1	=	=	PUNCT
ejpam-6699	341	2	lim	lim	PROPN
ejpam-6699	341	3	m→∞	m→∞	NOUN
ejpam-6699	341	4	ϕ(2mt	ϕ(2mt	NOUN
ejpam-6699	341	5	)	)	PUNCT
ejpam-6699	341	6	24	24	NUM
ejpam-6699	341	7	m	m	NOUN
ejpam-6699	341	8	.	.	PUNCT
ejpam-6699	342	1	we	we	PRON
ejpam-6699	342	2	next	next	ADV
ejpam-6699	342	3	show	show	VERB
ejpam-6699	342	4	q4	q4	PROPN
ejpam-6699	342	5	is	be	AUX
ejpam-6699	342	6	quartic	quartic	ADJ
ejpam-6699	342	7	.	.	PUNCT
ejpam-6699	343	1	from	from	ADP
ejpam-6699	343	2	(	(	PUNCT
ejpam-6699	343	3	17	17	NUM
ejpam-6699	343	4	)	)	PUNCT
ejpam-6699	343	5	,	,	PUNCT
ejpam-6699	343	6	lemma	lemma	PROPN
ejpam-6699	343	7	1	1	NUM
ejpam-6699	343	8	,	,	PUNCT
ejpam-6699	343	9	and	and	CCONJ
ejpam-6699	343	10	the	the	DET
ejpam-6699	343	11	definition	definition	NOUN
ejpam-6699	343	12	of	of	ADP
ejpam-6699	343	13	q4	q4	PROPN
ejpam-6699	343	14	,	,	PUNCT
ejpam-6699	343	15	we	we	PRON
ejpam-6699	343	16	have	have	VERB
ejpam-6699	343	17	:	:	PUNCT
ejpam-6699	343	18	∥∆q4(t1	∥∆q4(t1	NOUN
ejpam-6699	343	19	,	,	PUNCT
ejpam-6699	343	20	.	.	PUNCT
ejpam-6699	343	21	.	.	PUNCT
ejpam-6699	343	22	.	.	PUNCT
ejpam-6699	344	1	,	,	PUNCT
ejpam-6699	344	2	tr	tr	VERB
ejpam-6699	344	3	)	)	PUNCT
ejpam-6699	344	4	,	,	PUNCT
ejpam-6699	344	5	ν1	ν1	NOUN
ejpam-6699	344	6	,	,	PUNCT
ejpam-6699	344	7	.	.	PUNCT
ejpam-6699	344	8	.	.	PUNCT
ejpam-6699	345	1	.	.	PUNCT
ejpam-6699	346	1	,	,	PUNCT
ejpam-6699	346	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	346	3	=	=	PUNCT
ejpam-6699	346	4	lim	lim	PROPN
ejpam-6699	346	5	p→∞	p→∞	ADV
ejpam-6699	346	6	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	346	7	1	1	NUM
ejpam-6699	346	8	24p	24p	NOUN
ejpam-6699	346	9	∆ϕ	∆ϕ	PROPN
ejpam-6699	346	10	(	(	PUNCT
ejpam-6699	346	11	2pt1	2pt1	NUM
ejpam-6699	346	12	,	,	PUNCT
ejpam-6699	346	13	.	.	PUNCT
ejpam-6699	346	14	.	.	PUNCT
ejpam-6699	347	1	.	.	PUNCT
ejpam-6699	348	1	,	,	PUNCT
ejpam-6699	348	2	2	2	NUM
ejpam-6699	348	3	ptr	ptr	NOUN
ejpam-6699	348	4	)	)	PUNCT
ejpam-6699	348	5	,	,	PUNCT
ejpam-6699	348	6	ν1	ν1	NOUN
ejpam-6699	348	7	,	,	PUNCT
ejpam-6699	348	8	.	.	PUNCT
ejpam-6699	348	9	.	.	PUNCT
ejpam-6699	348	10	.	.	PUNCT
ejpam-6699	349	1	,	,	PUNCT
ejpam-6699	349	2	νn−1	νn−1	PROPN
ejpam-6699	349	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	349	4	β	β	X
ejpam-6699	349	5	≤	≤	ADJ
ejpam-6699	349	6	lim	lim	PROPN
ejpam-6699	349	7	p→∞	p→∞	ADJ
ejpam-6699	349	8	|2−4pβ|ψ	|2−4pβ|ψ	PROPN
ejpam-6699	349	9	(	(	PUNCT
ejpam-6699	349	10	2pt1	2pt1	NUM
ejpam-6699	349	11	,	,	PUNCT
ejpam-6699	349	12	.	.	PUNCT
ejpam-6699	349	13	.	.	PUNCT
ejpam-6699	350	1	.	.	PUNCT
ejpam-6699	351	1	,	,	PUNCT
ejpam-6699	351	2	2	2	NUM
ejpam-6699	351	3	ptr	ptr	NOUN
ejpam-6699	351	4	)	)	PUNCT
ejpam-6699	351	5	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	351	6	,	,	PUNCT
ejpam-6699	351	7	.	.	PUNCT
ejpam-6699	351	8	.	.	PUNCT
ejpam-6699	351	9	.	.	PUNCT
ejpam-6699	352	1	,	,	PUNCT
ejpam-6699	352	2	νn−1	νn−1	PROPN
ejpam-6699	352	3	)	)	PUNCT
ejpam-6699	352	4	=	=	SYM
ejpam-6699	353	1	0	0	X
ejpam-6699	353	2	.	.	PUNCT
ejpam-6699	354	1	hence	hence	ADV
ejpam-6699	354	2	,	,	PUNCT
ejpam-6699	354	3	by	by	ADP
ejpam-6699	354	4	lemma	lemma	PROPN
ejpam-6699	354	5	2	2	NUM
ejpam-6699	354	6	,	,	PUNCT
ejpam-6699	354	7	q4	q4	PROPN
ejpam-6699	354	8	is	be	AUX
ejpam-6699	354	9	quartic	quartic	ADJ
ejpam-6699	354	10	.	.	PUNCT
ejpam-6699	355	1	from	from	ADP
ejpam-6699	355	2	(	(	PUNCT
ejpam-6699	355	3	21	21	NUM
ejpam-6699	355	4	)	)	PUNCT
ejpam-6699	355	5	,	,	PUNCT
ejpam-6699	355	6	we	we	PRON
ejpam-6699	355	7	obtain:∥∥∥∥ϕ(t)−	obtain:∥∥∥∥ϕ(t)−	PROPN
ejpam-6699	355	8	ϕ(24	ϕ(24	VERB
ejpam-6699	355	9	t	t	PROPN
ejpam-6699	355	10	)	)	PUNCT
ejpam-6699	355	11	28	28	NUM
ejpam-6699	355	12	,	,	PUNCT
ejpam-6699	355	13	ν1	ν1	NOUN
ejpam-6699	355	14	,	,	PUNCT
ejpam-6699	355	15	.	.	PUNCT
ejpam-6699	355	16	.	.	PUNCT
ejpam-6699	355	17	.	.	PUNCT
ejpam-6699	356	1	,	,	PUNCT
ejpam-6699	356	2	νn−1	νn−1	PROPN
ejpam-6699	356	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	356	4	β	β	NOUN
ejpam-6699	356	5	≤	≤	ADJ
ejpam-6699	356	6	max	max	PROPN
ejpam-6699	356	7	{	{	PUNCT
ejpam-6699	356	8	|2−4β|ψ(t	|2−4β|ψ(t	PROPN
ejpam-6699	356	9	,	,	PUNCT
ejpam-6699	356	10	0	0	NUM
ejpam-6699	356	11	,	,	PUNCT
ejpam-6699	356	12	.	.	PUNCT
ejpam-6699	356	13	.	.	PUNCT
ejpam-6699	356	14	.	.	PUNCT
ejpam-6699	357	1	,	,	PUNCT
ejpam-6699	357	2	0	0	NUM
ejpam-6699	357	3	)	)	PUNCT
ejpam-6699	357	4	,	,	PUNCT
ejpam-6699	357	5	|2−8β|ψ(2	|2−8β|ψ(2	X
ejpam-6699	357	6	t	t	PROPN
ejpam-6699	357	7	,	,	PUNCT
ejpam-6699	357	8	0	0	NUM
ejpam-6699	357	9	,	,	PUNCT
ejpam-6699	357	10	.	.	PUNCT
ejpam-6699	357	11	.	.	PUNCT
ejpam-6699	358	1	.	.	PUNCT
ejpam-6699	359	1	,	,	PUNCT
ejpam-6699	359	2	0	0	NUM
ejpam-6699	359	3	)	)	PUNCT
ejpam-6699	359	4	}	}	PUNCT
ejpam-6699	359	5	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	359	6	,	,	PUNCT
ejpam-6699	359	7	.	.	PUNCT
ejpam-6699	359	8	.	.	PUNCT
ejpam-6699	360	1	.	.	PUNCT
ejpam-6699	361	1	,	,	PUNCT
ejpam-6699	361	2	νn−1	νn−1	PROPN
ejpam-6699	361	3	)	)	PUNCT
ejpam-6699	361	4	.	.	PUNCT
ejpam-6699	362	1	inductively	inductively	ADV
ejpam-6699	362	2	,	,	PUNCT
ejpam-6699	362	3	for	for	ADP
ejpam-6699	362	4	all	all	PRON
ejpam-6699	362	5	p	p	PROPN
ejpam-6699	362	6	∈	∈	PROPN
ejpam-6699	362	7	n	n	CCONJ
ejpam-6699	362	8	,	,	PUNCT
ejpam-6699	362	9	we	we	PRON
ejpam-6699	362	10	get:∥∥∥∥ϕ(t)−	get:∥∥∥∥ϕ(t)−	PROPN
ejpam-6699	362	11	ϕ(2pt	ϕ(2pt	PROPN
ejpam-6699	362	12	)	)	PUNCT
ejpam-6699	362	13	24p	24p	NOUN
ejpam-6699	362	14	,	,	PUNCT
ejpam-6699	362	15	ν1	ν1	NOUN
ejpam-6699	362	16	,	,	PUNCT
ejpam-6699	362	17	.	.	PUNCT
ejpam-6699	362	18	.	.	PUNCT
ejpam-6699	362	19	.	.	PUNCT
ejpam-6699	363	1	,	,	PUNCT
ejpam-6699	363	2	νn−1	νn−1	PROPN
ejpam-6699	363	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-6699	363	4	β	β	NOUN
ejpam-6699	363	5	≤	≤	ADJ
ejpam-6699	363	6	max	max	PROPN
ejpam-6699	363	7	{	{	PUNCT
ejpam-6699	363	8	∣∣∣2−4tβ	∣∣∣2−4tβ	PROPN
ejpam-6699	363	9	∣∣∣	∣∣∣	NOUN
ejpam-6699	363	10	ψ	ψ	X
ejpam-6699	363	11	(	(	PUNCT
ejpam-6699	363	12	2t−1	2t−1	NUM
ejpam-6699	363	13	t	t	PROPN
ejpam-6699	363	14	,	,	PUNCT
ejpam-6699	363	15	0	0	NUM
ejpam-6699	363	16	,	,	PUNCT
ejpam-6699	363	17	.	.	PUNCT
ejpam-6699	363	18	.	.	PUNCT
ejpam-6699	364	1	.	.	PUNCT
ejpam-6699	365	1	,	,	PUNCT
ejpam-6699	365	2	0	0	NUM
ejpam-6699	365	3	)	)	PUNCT
ejpam-6699	365	4	:	:	PUNCT
ejpam-6699	365	5	1	1	NUM
ejpam-6699	365	6	≤	≤	NUM
ejpam-6699	365	7	t	t	X
ejpam-6699	365	8	≤	≤	NOUN
ejpam-6699	365	9	p	p	X
ejpam-6699	365	10	}	}	PUNCT
ejpam-6699	365	11	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	365	12	,	,	PUNCT
ejpam-6699	365	13	.	.	PUNCT
ejpam-6699	365	14	.	.	PUNCT
ejpam-6699	366	1	.	.	PUNCT
ejpam-6699	367	1	,	,	PUNCT
ejpam-6699	367	2	νn−1	νn−1	PROPN
ejpam-6699	367	3	)	)	PUNCT
ejpam-6699	367	4	.	.	PUNCT
ejpam-6699	368	1	(	(	PUNCT
ejpam-6699	368	2	23	23	X
ejpam-6699	368	3	)	)	PUNCT
ejpam-6699	368	4	letting	let	VERB
ejpam-6699	368	5	p→	p→	VERB
ejpam-6699	368	6	∞	∞	PROPN
ejpam-6699	368	7	in	in	ADP
ejpam-6699	368	8	(	(	PUNCT
ejpam-6699	368	9	23	23	NUM
ejpam-6699	368	10	)	)	PUNCT
ejpam-6699	368	11	,	,	PUNCT
ejpam-6699	368	12	we	we	PRON
ejpam-6699	368	13	obtain	obtain	VERB
ejpam-6699	368	14	(	(	PUNCT
ejpam-6699	368	15	18	18	NUM
ejpam-6699	368	16	)	)	PUNCT
ejpam-6699	368	17	by	by	ADP
ejpam-6699	368	18	the	the	DET
ejpam-6699	368	19	definition	definition	NOUN
ejpam-6699	368	20	of	of	ADP
ejpam-6699	368	21	ψ̃(t	ψ̃(t	PROPN
ejpam-6699	368	22	)	)	PUNCT
ejpam-6699	368	23	in	in	ADP
ejpam-6699	368	24	(	(	PUNCT
ejpam-6699	368	25	19	19	NUM
ejpam-6699	368	26	)	)	PUNCT
ejpam-6699	368	27	.	.	PUNCT
ejpam-6699	369	1	to	to	PART
ejpam-6699	369	2	prove	prove	VERB
ejpam-6699	369	3	uniqueness	uniqueness	NOUN
ejpam-6699	369	4	,	,	PUNCT
ejpam-6699	369	5	suppose	suppose	VERB
ejpam-6699	369	6	another	another	DET
ejpam-6699	369	7	quartic	quartic	ADJ
ejpam-6699	369	8	mapping	mapping	NOUN
ejpam-6699	369	9	q′	q′	NOUN
ejpam-6699	369	10	4	4	NUM
ejpam-6699	369	11	also	also	ADV
ejpam-6699	369	12	satisfies	satisfy	VERB
ejpam-6699	369	13	(	(	PUNCT
ejpam-6699	369	14	18	18	NUM
ejpam-6699	369	15	)	)	PUNCT
ejpam-6699	369	16	.	.	PUNCT
ejpam-6699	370	1	then∥∥q4(t)−q′	then∥∥q4(t)−q′	PROPN
ejpam-6699	370	2	4(t	4(t	NUM
ejpam-6699	370	3	)	)	PUNCT
ejpam-6699	370	4	,	,	PUNCT
ejpam-6699	370	5	ν1	ν1	NOUN
ejpam-6699	370	6	,	,	PUNCT
ejpam-6699	370	7	.	.	PUNCT
ejpam-6699	370	8	.	.	PUNCT
ejpam-6699	371	1	.	.	PUNCT
ejpam-6699	372	1	,	,	PUNCT
ejpam-6699	372	2	νn−1	νn−1	VERB
ejpam-6699	372	3	∥∥	∥∥	X
ejpam-6699	372	4	β	β	X
ejpam-6699	372	5	=	=	PUNCT
ejpam-6699	372	6	∣∣∣2−4tβ	∣∣∣2−4tβ	PROPN
ejpam-6699	372	7	∣∣∣	∣∣∣	ADJ
ejpam-6699	372	8	∥∥q4(2	∥∥q4(2	PROPN
ejpam-6699	372	9	tt)−q′	tt)−q′	NOUN
ejpam-6699	372	10	4(2	4(2	NUM
ejpam-6699	372	11	tt	tt	PROPN
ejpam-6699	372	12	)	)	PUNCT
ejpam-6699	372	13	,	,	PUNCT
ejpam-6699	372	14	ν1	ν1	NOUN
ejpam-6699	372	15	,	,	PUNCT
ejpam-6699	372	16	.	.	PUNCT
ejpam-6699	372	17	.	.	PUNCT
ejpam-6699	373	1	.	.	PUNCT
ejpam-6699	374	1	,	,	PUNCT
ejpam-6699	374	2	νn−1	νn−1	VERB
ejpam-6699	374	3	∥∥	∥∥	X
ejpam-6699	374	4	β	β	X
ejpam-6699	374	5	≤	≤	X
ejpam-6699	374	6	∣∣∣2−4tβ	∣∣∣2−4tβ	PROPN
ejpam-6699	374	7	∣∣∣max	∣∣∣max	PROPN
ejpam-6699	374	8	{	{	PUNCT
ejpam-6699	374	9	∥∥q4(2	∥∥q4(2	PROPN
ejpam-6699	374	10	tt)−	tt)−	PROPN
ejpam-6699	374	11	ϕ(2tt	ϕ(2tt	NOUN
ejpam-6699	374	12	)	)	PUNCT
ejpam-6699	374	13	∥∥	∥∥	PROPN
ejpam-6699	374	14	β	β	X
ejpam-6699	374	15	,	,	PUNCT
ejpam-6699	374	16	∥∥ϕ(2tt)−q′	∥∥ϕ(2tt)−q′	NOUN
ejpam-6699	375	1	4(2	4(2	NUM
ejpam-6699	375	2	tt	tt	NOUN
ejpam-6699	375	3	)	)	PUNCT
ejpam-6699	375	4	∥∥	∥∥	PROPN
ejpam-6699	375	5	β	β	X
ejpam-6699	375	6	}	}	PUNCT
ejpam-6699	375	7	≤	≤	NOUN
ejpam-6699	375	8	∣∣∣2−4tβ	∣∣∣2−4tβ	PROPN
ejpam-6699	375	9	∣∣∣	∣∣∣	ADJ
ejpam-6699	375	10	ψ̃(2tt)ϖ(ν1	ψ̃(2tt)ϖ(ν1	PROPN
ejpam-6699	375	11	,	,	PUNCT
ejpam-6699	375	12	.	.	PUNCT
ejpam-6699	375	13	.	.	PUNCT
ejpam-6699	375	14	.	.	PUNCT
ejpam-6699	376	1	,	,	PUNCT
ejpam-6699	376	2	νn−1	νn−1	PROPN
ejpam-6699	376	3	)	)	PUNCT
ejpam-6699	376	4	.	.	PUNCT
ejpam-6699	377	1	by	by	ADP
ejpam-6699	377	2	assumption	assumption	NOUN
ejpam-6699	377	3	(	(	PUNCT
ejpam-6699	377	4	20	20	NUM
ejpam-6699	377	5	)	)	PUNCT
ejpam-6699	377	6	,	,	PUNCT
ejpam-6699	377	7	the	the	DET
ejpam-6699	377	8	last	last	ADJ
ejpam-6699	377	9	term	term	NOUN
ejpam-6699	377	10	tends	tend	VERB
ejpam-6699	377	11	to	to	ADP
ejpam-6699	377	12	zero	zero	NUM
ejpam-6699	377	13	as	as	ADP
ejpam-6699	377	14	t→	t→	X
ejpam-6699	377	15	∞.	∞.	PROPN
ejpam-6699	377	16	therefore,∥∥q4(t)−q′	therefore,∥∥q4(t)−q′	PROPN
ejpam-6699	377	17	4(t	4(t	NUM
ejpam-6699	377	18	)	)	PUNCT
ejpam-6699	377	19	,	,	PUNCT
ejpam-6699	377	20	ν1	ν1	NOUN
ejpam-6699	377	21	,	,	PUNCT
ejpam-6699	377	22	.	.	PUNCT
ejpam-6699	377	23	.	.	PUNCT
ejpam-6699	378	1	.	.	PUNCT
ejpam-6699	379	1	,	,	PUNCT
ejpam-6699	379	2	νn−1	νn−1	VERB
ejpam-6699	379	3	∥∥	∥∥	X
ejpam-6699	379	4	β	β	X
ejpam-6699	379	5	=	=	SYM
ejpam-6699	379	6	0	0	NUM
ejpam-6699	379	7	,	,	PUNCT
ejpam-6699	379	8	and	and	CCONJ
ejpam-6699	379	9	by	by	ADP
ejpam-6699	379	10	lemma	lemma	PROPN
ejpam-6699	379	11	2	2	NUM
ejpam-6699	379	12	,	,	PUNCT
ejpam-6699	379	13	we	we	PRON
ejpam-6699	379	14	conclude	conclude	VERB
ejpam-6699	379	15	q4	q4	PROPN
ejpam-6699	379	16	=	=	PROPN
ejpam-6699	379	17	q′	q′	NOUN
ejpam-6699	379	18	4	4	NUM
ejpam-6699	379	19	.	.	PUNCT
ejpam-6699	380	1	thus	thus	ADV
ejpam-6699	380	2	,	,	PUNCT
ejpam-6699	380	3	q4	q4	PROPN
ejpam-6699	380	4	is	be	AUX
ejpam-6699	380	5	unique	unique	ADJ
ejpam-6699	380	6	.	.	PUNCT
ejpam-6699	381	1	s.	s.	PROPN
ejpam-6699	381	2	gowri	gowri	PROPN
ejpam-6699	381	3	et	et	PROPN
ejpam-6699	381	4	al	al	PROPN
ejpam-6699	381	5	.	.	PUNCT
ejpam-6699	381	6	/	/	SYM
ejpam-6699	381	7	eur	eur	PROPN
ejpam-6699	381	8	.	.	PUNCT
ejpam-6699	382	1	j.	j.	PROPN
ejpam-6699	382	2	pure	pure	PROPN
ejpam-6699	382	3	appl	appl	PROPN
ejpam-6699	382	4	.	.	PROPN
ejpam-6699	382	5	math	math	PROPN
ejpam-6699	382	6	,	,	PUNCT
ejpam-6699	382	7	18	18	NUM
ejpam-6699	382	8	(	(	PUNCT
ejpam-6699	382	9	4	4	NUM
ejpam-6699	382	10	)	)	PUNCT
ejpam-6699	382	11	(	(	PUNCT
ejpam-6699	382	12	2025	2025	NUM
ejpam-6699	382	13	)	)	PUNCT
ejpam-6699	382	14	,	,	PUNCT
ejpam-6699	382	15	6699	6699	NUM
ejpam-6699	382	16	10	10	NUM
ejpam-6699	382	17	of	of	ADP
ejpam-6699	382	18	14	14	NUM
ejpam-6699	382	19	theorem	theorem	NOUN
ejpam-6699	382	20	5	5	NUM
ejpam-6699	382	21	.	.	PUNCT
ejpam-6699	383	1	let	let	VERB
ejpam-6699	383	2	ψ	ψ	X
ejpam-6699	383	3	:	:	PUNCT
ejpam-6699	383	4	er	er	INTJ
ejpam-6699	383	5	→	→	X
ejpam-6699	383	6	[	[	X
ejpam-6699	383	7	0,∞	0,∞	NOUN
ejpam-6699	383	8	)	)	PUNCT
ejpam-6699	383	9	be	be	VERB
ejpam-6699	383	10	a	a	DET
ejpam-6699	383	11	control	control	NOUN
ejpam-6699	383	12	function	function	NOUN
ejpam-6699	383	13	satisfying	satisfy	VERB
ejpam-6699	383	14	lim	lim	PROPN
ejpam-6699	383	15	p→∞	p→∞	PROPN
ejpam-6699	383	16	∣∣∣24pβ∣∣∣ψ	∣∣∣24pβ∣∣∣ψ	NOUN
ejpam-6699	383	17	(	(	PUNCT
ejpam-6699	383	18	t1	t1	PROPN
ejpam-6699	383	19	2p	2p	NUM
ejpam-6699	383	20	,	,	PUNCT
ejpam-6699	383	21	t2	t2	NOUN
ejpam-6699	383	22	2p	2p	NUM
ejpam-6699	383	23	,	,	PUNCT
ejpam-6699	383	24	.	.	PUNCT
ejpam-6699	383	25	.	.	PUNCT
ejpam-6699	384	1	.	.	PUNCT
ejpam-6699	385	1	,	,	PUNCT
ejpam-6699	385	2	tr	tr	VERB
ejpam-6699	385	3	2p	2p	NUM
ejpam-6699	385	4	)	)	PUNCT
ejpam-6699	386	1	=	=	SYM
ejpam-6699	386	2	0	0	PUNCT
ejpam-6699	386	3	(	(	PUNCT
ejpam-6699	386	4	24	24	NUM
ejpam-6699	386	5	)	)	PUNCT
ejpam-6699	386	6	for	for	ADP
ejpam-6699	386	7	all	all	DET
ejpam-6699	386	8	t1	t1	NOUN
ejpam-6699	386	9	,	,	PUNCT
ejpam-6699	386	10	.	.	PUNCT
ejpam-6699	386	11	.	.	PUNCT
ejpam-6699	386	12	.	.	PUNCT
ejpam-6699	387	1	,	,	PUNCT
ejpam-6699	387	2	tr	tr	NOUN
ejpam-6699	387	3	∈	∈	NOUN
ejpam-6699	387	4	e	e	NOUN
ejpam-6699	387	5	,	,	PUNCT
ejpam-6699	387	6	and	and	CCONJ
ejpam-6699	387	7	let	let	VERB
ejpam-6699	387	8	ϖ	ϖ	PRON
ejpam-6699	387	9	:	:	PUNCT
ejpam-6699	387	10	fn−1	fn−1	ADJ
ejpam-6699	387	11	→	→	SYM
ejpam-6699	387	12	[	[	X
ejpam-6699	387	13	0,∞	0,∞	X
ejpam-6699	387	14	)	)	PUNCT
ejpam-6699	387	15	be	be	VERB
ejpam-6699	387	16	a	a	DET
ejpam-6699	387	17	control	control	NOUN
ejpam-6699	387	18	mapping	mapping	NOUN
ejpam-6699	387	19	.	.	PUNCT
ejpam-6699	388	1	assume	assume	VERB
ejpam-6699	388	2	that	that	SCONJ
ejpam-6699	388	3	a	a	DET
ejpam-6699	388	4	function	function	NOUN
ejpam-6699	388	5	ϕ	ϕ	NOUN
ejpam-6699	388	6	:	:	PUNCT
ejpam-6699	388	7	e	e	X
ejpam-6699	388	8	→	→	SYM
ejpam-6699	388	9	f	f	PROPN
ejpam-6699	388	10	satisfies	satisfie	NOUN
ejpam-6699	388	11	∥∆ϕ(t1	∥∆ϕ(t1	PROPN
ejpam-6699	388	12	,	,	PUNCT
ejpam-6699	388	13	.	.	PUNCT
ejpam-6699	388	14	.	.	PUNCT
ejpam-6699	388	15	.	.	PUNCT
ejpam-6699	389	1	,	,	PUNCT
ejpam-6699	389	2	tr	tr	VERB
ejpam-6699	389	3	)	)	PUNCT
ejpam-6699	389	4	,	,	PUNCT
ejpam-6699	389	5	ν1	ν1	NOUN
ejpam-6699	389	6	,	,	PUNCT
ejpam-6699	389	7	.	.	PUNCT
ejpam-6699	389	8	.	.	PUNCT
ejpam-6699	390	1	.	.	PUNCT
ejpam-6699	391	1	,	,	PUNCT
ejpam-6699	391	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	391	3	≤	≤	NUM
ejpam-6699	391	4	ψ(t1	ψ(t1	VERB
ejpam-6699	391	5	,	,	PUNCT
ejpam-6699	391	6	.	.	PUNCT
ejpam-6699	391	7	.	.	PUNCT
ejpam-6699	392	1	.	.	PUNCT
ejpam-6699	393	1	,	,	PUNCT
ejpam-6699	394	1	tr)ϖ(ν1	tr)ϖ(ν1	PROPN
ejpam-6699	394	2	,	,	PUNCT
ejpam-6699	394	3	.	.	PUNCT
ejpam-6699	394	4	.	.	PUNCT
ejpam-6699	395	1	.	.	PUNCT
ejpam-6699	396	1	,	,	PUNCT
ejpam-6699	396	2	νn−1	νn−1	PROPN
ejpam-6699	396	3	)	)	PUNCT
ejpam-6699	396	4	(	(	PUNCT
ejpam-6699	396	5	25	25	NUM
ejpam-6699	396	6	)	)	PUNCT
ejpam-6699	396	7	for	for	ADP
ejpam-6699	396	8	all	all	DET
ejpam-6699	396	9	t1	t1	NOUN
ejpam-6699	396	10	,	,	PUNCT
ejpam-6699	396	11	.	.	PUNCT
ejpam-6699	396	12	.	.	PUNCT
ejpam-6699	397	1	.	.	PUNCT
ejpam-6699	398	1	,	,	PUNCT
ejpam-6699	398	2	tr	tr	NOUN
ejpam-6699	398	3	∈	∈	PROPN
ejpam-6699	398	4	e	e	NOUN
ejpam-6699	398	5	and	and	CCONJ
ejpam-6699	398	6	ν1	ν1	NOUN
ejpam-6699	398	7	,	,	PUNCT
ejpam-6699	398	8	.	.	PUNCT
ejpam-6699	398	9	.	.	PUNCT
ejpam-6699	398	10	.	.	PUNCT
ejpam-6699	399	1	,	,	PUNCT
ejpam-6699	399	2	νn−1	νn−1	PROPN
ejpam-6699	399	3	∈	∈	PROPN
ejpam-6699	399	4	f	f	X
ejpam-6699	399	5	.	.	PUNCT
ejpam-6699	400	1	then	then	ADV
ejpam-6699	400	2	there	there	PRON
ejpam-6699	400	3	exists	exist	VERB
ejpam-6699	400	4	a	a	DET
ejpam-6699	400	5	unique	unique	ADJ
ejpam-6699	400	6	quartic	quartic	ADJ
ejpam-6699	400	7	mapping	mapping	NOUN
ejpam-6699	400	8	q4	q4	NOUN
ejpam-6699	400	9	:	:	PUNCT
ejpam-6699	400	10	e	e	X
ejpam-6699	400	11	→	→	SYM
ejpam-6699	400	12	f	f	X
ejpam-6699	400	13	satisfying	satisfy	VERB
ejpam-6699	400	14	∥ϕ(t)−q4(t	∥ϕ(t)−q4(t	NOUN
ejpam-6699	400	15	)	)	PUNCT
ejpam-6699	400	16	,	,	PUNCT
ejpam-6699	400	17	ν1	ν1	NOUN
ejpam-6699	400	18	,	,	PUNCT
ejpam-6699	400	19	.	.	PUNCT
ejpam-6699	400	20	.	.	PUNCT
ejpam-6699	400	21	.	.	PUNCT
ejpam-6699	401	1	,	,	PUNCT
ejpam-6699	401	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	401	3	≤	≤	NUM
ejpam-6699	401	4	ψ̃(t)ϖ(ν1	ψ̃(t)ϖ(ν1	NOUN
ejpam-6699	401	5	,	,	PUNCT
ejpam-6699	401	6	.	.	PUNCT
ejpam-6699	401	7	.	.	PUNCT
ejpam-6699	402	1	.	.	PUNCT
ejpam-6699	403	1	,	,	PUNCT
ejpam-6699	403	2	νn−1	νn−1	PROPN
ejpam-6699	403	3	)	)	PUNCT
ejpam-6699	403	4	,	,	PUNCT
ejpam-6699	403	5	(	(	PUNCT
ejpam-6699	403	6	26	26	NUM
ejpam-6699	403	7	)	)	PUNCT
ejpam-6699	404	1	where	where	SCONJ
ejpam-6699	404	2	ψ̃(t	ψ̃(t	PROPN
ejpam-6699	404	3	)	)	PUNCT
ejpam-6699	404	4	:	:	PUNCT
ejpam-6699	405	1	=	=	SYM
ejpam-6699	405	2	lim	lim	PROPN
ejpam-6699	405	3	p→∞	p→∞	PROPN
ejpam-6699	405	4	max	max	PROPN
ejpam-6699	405	5	{	{	PUNCT
ejpam-6699	405	6	∣∣∣24(i−1)β	∣∣∣24(i−1)β	NUM
ejpam-6699	405	7	∣∣∣ψ	∣∣∣ψ	NOUN
ejpam-6699	405	8	(	(	PUNCT
ejpam-6699	405	9	2−it	2−it	NUM
ejpam-6699	405	10	,	,	PUNCT
ejpam-6699	405	11	0	0	NUM
ejpam-6699	405	12	,	,	PUNCT
ejpam-6699	405	13	0	0	NUM
ejpam-6699	405	14	,	,	PUNCT
ejpam-6699	405	15	.	.	PUNCT
ejpam-6699	405	16	.	.	PUNCT
ejpam-6699	405	17	.	.	PUNCT
ejpam-6699	406	1	,	,	PUNCT
ejpam-6699	406	2	0	0	NUM
ejpam-6699	406	3	)	)	PUNCT
ejpam-6699	406	4	:	:	PUNCT
ejpam-6699	407	1	1	1	NUM
ejpam-6699	407	2	≤	≤	NUM
ejpam-6699	407	3	i	i	PRON
ejpam-6699	407	4	≤	≤	NOUN
ejpam-6699	407	5	p	p	X
ejpam-6699	407	6	}	}	PUNCT
ejpam-6699	407	7	.	.	PUNCT
ejpam-6699	408	1	(	(	PUNCT
ejpam-6699	408	2	27	27	NUM
ejpam-6699	408	3	)	)	PUNCT
ejpam-6699	408	4	moreover	moreover	ADV
ejpam-6699	408	5	,	,	PUNCT
ejpam-6699	408	6	if	if	SCONJ
ejpam-6699	408	7	lim	lim	PROPN
ejpam-6699	408	8	t→∞	t→∞	NUM
ejpam-6699	408	9	lim	lim	PROPN
ejpam-6699	408	10	p→∞	p→∞	PROPN
ejpam-6699	408	11	max	max	PROPN
ejpam-6699	408	12	{	{	PUNCT
ejpam-6699	408	13	∣∣∣24(i−1)β	∣∣∣24(i−1)β	NUM
ejpam-6699	408	14	∣∣∣ψ	∣∣∣ψ	NOUN
ejpam-6699	408	15	(	(	PUNCT
ejpam-6699	408	16	2−it	2−it	NUM
ejpam-6699	408	17	,	,	PUNCT
ejpam-6699	408	18	0	0	NUM
ejpam-6699	408	19	,	,	PUNCT
ejpam-6699	408	20	0	0	NUM
ejpam-6699	408	21	,	,	PUNCT
ejpam-6699	408	22	.	.	PUNCT
ejpam-6699	408	23	.	.	PUNCT
ejpam-6699	408	24	.	.	PUNCT
ejpam-6699	409	1	,	,	PUNCT
ejpam-6699	409	2	0	0	NUM
ejpam-6699	409	3	)	)	PUNCT
ejpam-6699	409	4	:	:	PUNCT
ejpam-6699	410	1	1	1	NUM
ejpam-6699	410	2	+	+	NUM
ejpam-6699	410	3	t	t	NOUN
ejpam-6699	410	4	≤	≤	NUM
ejpam-6699	410	5	i	i	PRON
ejpam-6699	410	6	≤	≤	PROPN
ejpam-6699	410	7	p+	p+	VERB
ejpam-6699	410	8	t	t	NOUN
ejpam-6699	410	9	}	}	PUNCT
ejpam-6699	410	10	=	=	SYM
ejpam-6699	410	11	0	0	NUM
ejpam-6699	410	12	(	(	PUNCT
ejpam-6699	410	13	28	28	NUM
ejpam-6699	410	14	)	)	PUNCT
ejpam-6699	410	15	for	for	ADP
ejpam-6699	410	16	all	all	DET
ejpam-6699	410	17	t	t	NOUN
ejpam-6699	410	18	∈	∈	PROPN
ejpam-6699	410	19	e	e	NOUN
ejpam-6699	410	20	,	,	PUNCT
ejpam-6699	410	21	then	then	ADV
ejpam-6699	410	22	the	the	DET
ejpam-6699	410	23	quartic	quartic	ADJ
ejpam-6699	410	24	mapping	mapping	NOUN
ejpam-6699	410	25	q4	q4	PROPN
ejpam-6699	410	26	is	be	AUX
ejpam-6699	410	27	unique	unique	ADJ
ejpam-6699	410	28	.	.	PUNCT
ejpam-6699	411	1	proof	proof	NOUN
ejpam-6699	411	2	.	.	PUNCT
ejpam-6699	412	1	setting	set	VERB
ejpam-6699	412	2	(	(	PUNCT
ejpam-6699	412	3	t1	t1	NOUN
ejpam-6699	412	4	,	,	PUNCT
ejpam-6699	412	5	t2	t2	NOUN
ejpam-6699	412	6	,	,	PUNCT
ejpam-6699	412	7	.	.	PUNCT
ejpam-6699	412	8	.	.	PUNCT
ejpam-6699	412	9	.	.	PUNCT
ejpam-6699	413	1	,	,	PUNCT
ejpam-6699	413	2	tr	tr	VERB
ejpam-6699	413	3	)	)	PUNCT
ejpam-6699	413	4	by	by	ADP
ejpam-6699	413	5	(	(	PUNCT
ejpam-6699	413	6	t	t	PROPN
ejpam-6699	413	7	,	,	PUNCT
ejpam-6699	413	8	0	0	NUM
ejpam-6699	413	9	,	,	PUNCT
ejpam-6699	413	10	.	.	PUNCT
ejpam-6699	413	11	.	.	PUNCT
ejpam-6699	414	1	.	.	PUNCT
ejpam-6699	415	1	,	,	PUNCT
ejpam-6699	415	2	0	0	X
ejpam-6699	415	3	)	)	PUNCT
ejpam-6699	415	4	in	in	ADP
ejpam-6699	415	5	(	(	PUNCT
ejpam-6699	415	6	25	25	NUM
ejpam-6699	415	7	)	)	PUNCT
ejpam-6699	415	8	,	,	PUNCT
ejpam-6699	415	9	we	we	PRON
ejpam-6699	415	10	obtain∥∥ϕ(2t)−	obtain∥∥ϕ(2t)−	PROPN
ejpam-6699	415	11	24ϕ(t	24ϕ(t	NUM
ejpam-6699	415	12	)	)	PUNCT
ejpam-6699	415	13	,	,	PUNCT
ejpam-6699	415	14	ν1	ν1	NOUN
ejpam-6699	415	15	,	,	PUNCT
ejpam-6699	415	16	ν2	ν2	NOUN
ejpam-6699	415	17	,	,	PUNCT
ejpam-6699	415	18	.	.	PUNCT
ejpam-6699	415	19	.	.	PUNCT
ejpam-6699	416	1	.	.	PUNCT
ejpam-6699	417	1	,	,	PUNCT
ejpam-6699	417	2	νn−1	νn−1	VERB
ejpam-6699	417	3	∥∥	∥∥	X
ejpam-6699	417	4	β	β	X
ejpam-6699	417	5	≤	≤	X
ejpam-6699	417	6	ψ(t	ψ(t	PROPN
ejpam-6699	417	7	,	,	PUNCT
ejpam-6699	417	8	0	0	NUM
ejpam-6699	417	9	,	,	PUNCT
ejpam-6699	417	10	.	.	PUNCT
ejpam-6699	417	11	.	.	PUNCT
ejpam-6699	418	1	.	.	PUNCT
ejpam-6699	419	1	,	,	PUNCT
ejpam-6699	419	2	0)ϖ(ν1	0)ϖ(ν1	NOUN
ejpam-6699	419	3	,	,	PUNCT
ejpam-6699	419	4	ν2	ν2	NOUN
ejpam-6699	419	5	,	,	PUNCT
ejpam-6699	419	6	.	.	PUNCT
ejpam-6699	419	7	.	.	PUNCT
ejpam-6699	420	1	.	.	PUNCT
ejpam-6699	421	1	,	,	PUNCT
ejpam-6699	421	2	νn−1	νn−1	PROPN
ejpam-6699	421	3	)	)	PUNCT
ejpam-6699	421	4	.	.	PUNCT
ejpam-6699	422	1	(	(	PUNCT
ejpam-6699	422	2	29	29	NUM
ejpam-6699	422	3	)	)	PUNCT
ejpam-6699	422	4	replacing	replace	VERB
ejpam-6699	422	5	t	t	NOUN
ejpam-6699	422	6	by	by	ADP
ejpam-6699	422	7	t	t	PROPN
ejpam-6699	422	8	2	2	NUM
ejpam-6699	422	9	and	and	CCONJ
ejpam-6699	422	10	multiplying	multiply	VERB
ejpam-6699	422	11	by	by	ADP
ejpam-6699	422	12	|24β|	|24β|	PROPN
ejpam-6699	422	13	repeatedly	repeatedly	ADV
ejpam-6699	422	14	,	,	PUNCT
ejpam-6699	422	15	we	we	PRON
ejpam-6699	422	16	define	define	VERB
ejpam-6699	422	17	a	a	DET
ejpam-6699	422	18	sequence	sequence	NOUN
ejpam-6699	422	19	:	:	PUNCT
ejpam-6699	422	20	{	{	PUNCT
ejpam-6699	422	21	24pϕ	24pϕ	NOUN
ejpam-6699	422	22	(	(	PUNCT
ejpam-6699	422	23	t	t	PROPN
ejpam-6699	422	24	2p	2p	NUM
ejpam-6699	422	25	)	)	PUNCT
ejpam-6699	422	26	}	}	PUNCT
ejpam-6699	422	27	.	.	PUNCT
ejpam-6699	423	1	(	(	PUNCT
ejpam-6699	423	2	30	30	X
ejpam-6699	423	3	)	)	PUNCT
ejpam-6699	423	4	using	use	VERB
ejpam-6699	423	5	(	(	PUNCT
ejpam-6699	423	6	24	24	NUM
ejpam-6699	423	7	)	)	PUNCT
ejpam-6699	423	8	and	and	CCONJ
ejpam-6699	423	9	similar	similar	ADJ
ejpam-6699	423	10	arguments	argument	NOUN
ejpam-6699	423	11	as	as	ADP
ejpam-6699	423	12	in	in	ADP
ejpam-6699	423	13	theorem	theorem	NOUN
ejpam-6699	423	14	4	4	NUM
ejpam-6699	423	15	,	,	PUNCT
ejpam-6699	423	16	we	we	PRON
ejpam-6699	423	17	conclude	conclude	VERB
ejpam-6699	423	18	this	this	DET
ejpam-6699	423	19	sequence	sequence	NOUN
ejpam-6699	423	20	is	be	AUX
ejpam-6699	423	21	cauchy	cauchy	ADJ
ejpam-6699	423	22	in	in	ADP
ejpam-6699	423	23	f	f	PROPN
ejpam-6699	423	24	and	and	CCONJ
ejpam-6699	423	25	hence	hence	ADV
ejpam-6699	423	26	convergent	convergent	NOUN
ejpam-6699	423	27	,	,	PUNCT
ejpam-6699	423	28	due	due	ADP
ejpam-6699	423	29	to	to	ADP
ejpam-6699	423	30	completeness	completeness	NOUN
ejpam-6699	423	31	.	.	PUNCT
ejpam-6699	424	1	define	define	VERB
ejpam-6699	424	2	q4(t	q4(t	PROPN
ejpam-6699	424	3	)	)	PUNCT
ejpam-6699	424	4	:	:	PUNCT
ejpam-6699	425	1	=	=	PUNCT
ejpam-6699	425	2	lim	lim	PROPN
ejpam-6699	425	3	p→∞	p→∞	ADJ
ejpam-6699	425	4	24pϕ	24pϕ	NOUN
ejpam-6699	425	5	(	(	PUNCT
ejpam-6699	425	6	t	t	NOUN
ejpam-6699	425	7	2p	2p	NUM
ejpam-6699	425	8	)	)	PUNCT
ejpam-6699	425	9	,	,	PUNCT
ejpam-6699	425	10	for	for	ADP
ejpam-6699	425	11	all	all	DET
ejpam-6699	425	12	t	t	PROPN
ejpam-6699	425	13	∈	∈	PROPN
ejpam-6699	425	14	e.	e.	PROPN
ejpam-6699	425	15	following	follow	VERB
ejpam-6699	425	16	the	the	DET
ejpam-6699	425	17	structure	structure	NOUN
ejpam-6699	425	18	of	of	ADP
ejpam-6699	425	19	(	(	PUNCT
ejpam-6699	425	20	30	30	NUM
ejpam-6699	425	21	)	)	PUNCT
ejpam-6699	425	22	,	,	PUNCT
ejpam-6699	425	23	we	we	PRON
ejpam-6699	425	24	can	can	AUX
ejpam-6699	425	25	show	show	VERB
ejpam-6699	425	26	∥q4(t)−	∥q4(t)−	PROPN
ejpam-6699	425	27	ϕ(t	ϕ(t	NUM
ejpam-6699	425	28	)	)	PUNCT
ejpam-6699	425	29	,	,	PUNCT
ejpam-6699	425	30	ν1	ν1	NOUN
ejpam-6699	425	31	,	,	PUNCT
ejpam-6699	425	32	ν2	ν2	NOUN
ejpam-6699	425	33	,	,	PUNCT
ejpam-6699	425	34	.	.	PUNCT
ejpam-6699	425	35	.	.	PUNCT
ejpam-6699	425	36	.	.	PUNCT
ejpam-6699	426	1	,	,	PUNCT
ejpam-6699	426	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	426	3	≤	≤	NUM
ejpam-6699	426	4	ψ̃(t)ϖ(ν1	ψ̃(t)ϖ(ν1	NOUN
ejpam-6699	426	5	,	,	PUNCT
ejpam-6699	426	6	ν2	ν2	NOUN
ejpam-6699	426	7	,	,	PUNCT
ejpam-6699	426	8	.	.	PUNCT
ejpam-6699	426	9	.	.	PUNCT
ejpam-6699	427	1	.	.	PUNCT
ejpam-6699	428	1	,	,	PUNCT
ejpam-6699	428	2	νn−1	νn−1	PROPN
ejpam-6699	428	3	)	)	PUNCT
ejpam-6699	428	4	,	,	PUNCT
ejpam-6699	428	5	establishing	establish	VERB
ejpam-6699	428	6	(	(	PUNCT
ejpam-6699	428	7	26	26	NUM
ejpam-6699	428	8	)	)	PUNCT
ejpam-6699	428	9	.	.	PUNCT
ejpam-6699	429	1	to	to	PART
ejpam-6699	429	2	prove	prove	VERB
ejpam-6699	429	3	that	that	SCONJ
ejpam-6699	429	4	q4	q4	PROPN
ejpam-6699	429	5	is	be	AUX
ejpam-6699	429	6	quartic	quartic	ADJ
ejpam-6699	429	7	,	,	PUNCT
ejpam-6699	429	8	observe	observe	VERB
ejpam-6699	429	9	that	that	SCONJ
ejpam-6699	429	10	:	:	PUNCT
ejpam-6699	429	11	∥∆q4(t1	∥∆q4(t1	NOUN
ejpam-6699	429	12	,	,	PUNCT
ejpam-6699	429	13	.	.	PUNCT
ejpam-6699	429	14	.	.	PUNCT
ejpam-6699	430	1	.	.	PUNCT
ejpam-6699	431	1	,	,	PUNCT
ejpam-6699	431	2	tr	tr	VERB
ejpam-6699	431	3	)	)	PUNCT
ejpam-6699	431	4	,	,	PUNCT
ejpam-6699	431	5	ν1	ν1	NOUN
ejpam-6699	431	6	,	,	PUNCT
ejpam-6699	431	7	.	.	PUNCT
ejpam-6699	431	8	.	.	PUNCT
ejpam-6699	432	1	.	.	PUNCT
ejpam-6699	433	1	,	,	PUNCT
ejpam-6699	433	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	433	3	=	=	SYM
ejpam-6699	433	4	lim	lim	PROPN
ejpam-6699	433	5	p→∞	p→∞	PROPN
ejpam-6699	433	6	∣∣∣24pβ∣∣∣	∣∣∣24pβ∣∣∣	PROPN
ejpam-6699	433	7	∥∥∥∆ϕ	∥∥∥∆ϕ	PROPN
ejpam-6699	433	8	(	(	PUNCT
ejpam-6699	433	9	t1	t1	NOUN
ejpam-6699	433	10	2p	2p	NUM
ejpam-6699	433	11	,	,	PUNCT
ejpam-6699	433	12	.	.	PUNCT
ejpam-6699	433	13	.	.	PUNCT
ejpam-6699	434	1	.	.	PUNCT
ejpam-6699	435	1	,	,	PUNCT
ejpam-6699	435	2	tr	tr	VERB
ejpam-6699	435	3	2p	2p	NUM
ejpam-6699	435	4	)	)	PUNCT
ejpam-6699	435	5	,	,	PUNCT
ejpam-6699	435	6	ν1	ν1	NOUN
ejpam-6699	435	7	,	,	PUNCT
ejpam-6699	435	8	.	.	PUNCT
ejpam-6699	435	9	.	.	PUNCT
ejpam-6699	436	1	.	.	PUNCT
ejpam-6699	437	1	,	,	PUNCT
ejpam-6699	437	2	νn−1	νn−1	PROPN
ejpam-6699	437	3	∥∥∥	∥∥∥	PROPN
ejpam-6699	437	4	β	β	PROPN
ejpam-6699	437	5	,	,	PUNCT
ejpam-6699	437	6	s.	s.	PROPN
ejpam-6699	437	7	gowri	gowri	PROPN
ejpam-6699	437	8	et	et	PROPN
ejpam-6699	437	9	al	al	PROPN
ejpam-6699	437	10	.	.	PUNCT
ejpam-6699	437	11	/	/	SYM
ejpam-6699	437	12	eur	eur	PROPN
ejpam-6699	437	13	.	.	PUNCT
ejpam-6699	438	1	j.	j.	PROPN
ejpam-6699	438	2	pure	pure	PROPN
ejpam-6699	438	3	appl	appl	PROPN
ejpam-6699	438	4	.	.	PROPN
ejpam-6699	438	5	math	math	PROPN
ejpam-6699	438	6	,	,	PUNCT
ejpam-6699	438	7	18	18	NUM
ejpam-6699	438	8	(	(	PUNCT
ejpam-6699	438	9	4	4	NUM
ejpam-6699	438	10	)	)	PUNCT
ejpam-6699	438	11	(	(	PUNCT
ejpam-6699	438	12	2025	2025	NUM
ejpam-6699	438	13	)	)	PUNCT
ejpam-6699	438	14	,	,	PUNCT
ejpam-6699	438	15	6699	6699	NUM
ejpam-6699	438	16	11	11	NUM
ejpam-6699	438	17	of	of	ADP
ejpam-6699	438	18	14	14	NUM
ejpam-6699	438	19	which	which	PRON
ejpam-6699	438	20	tends	tend	VERB
ejpam-6699	438	21	to	to	ADP
ejpam-6699	438	22	zero	zero	NUM
ejpam-6699	438	23	by	by	ADP
ejpam-6699	438	24	(	(	PUNCT
ejpam-6699	438	25	24	24	NUM
ejpam-6699	438	26	)	)	PUNCT
ejpam-6699	438	27	,	,	PUNCT
ejpam-6699	438	28	hence	hence	ADV
ejpam-6699	438	29	∆q4	∆q4	VERB
ejpam-6699	438	30	=	=	SYM
ejpam-6699	438	31	0	0	NUM
ejpam-6699	438	32	and	and	CCONJ
ejpam-6699	438	33	q4	q4	PROPN
ejpam-6699	438	34	is	be	AUX
ejpam-6699	438	35	quartic	quartic	ADJ
ejpam-6699	438	36	.	.	PUNCT
ejpam-6699	439	1	for	for	ADP
ejpam-6699	439	2	uniqueness	uniqueness	NOUN
ejpam-6699	439	3	,	,	PUNCT
ejpam-6699	439	4	suppose	suppose	VERB
ejpam-6699	439	5	another	another	DET
ejpam-6699	439	6	quartic	quartic	ADJ
ejpam-6699	439	7	mapping	mapping	NOUN
ejpam-6699	439	8	q′	q′	NOUN
ejpam-6699	439	9	4	4	NUM
ejpam-6699	439	10	satisfies	satisfie	NOUN
ejpam-6699	439	11	(	(	PUNCT
ejpam-6699	439	12	26	26	NUM
ejpam-6699	439	13	)	)	PUNCT
ejpam-6699	439	14	.	.	PUNCT
ejpam-6699	440	1	then∥∥q4(t)−q′	then∥∥q4(t)−q′	PROPN
ejpam-6699	440	2	4(t	4(t	NUM
ejpam-6699	440	3	)	)	PUNCT
ejpam-6699	440	4	,	,	PUNCT
ejpam-6699	440	5	ν1	ν1	NOUN
ejpam-6699	440	6	,	,	PUNCT
ejpam-6699	440	7	.	.	PUNCT
ejpam-6699	440	8	.	.	PUNCT
ejpam-6699	441	1	.	.	PUNCT
ejpam-6699	442	1	,	,	PUNCT
ejpam-6699	442	2	νn−1	νn−1	VERB
ejpam-6699	442	3	∥∥	∥∥	X
ejpam-6699	442	4	β	β	X
ejpam-6699	442	5	=	=	PUNCT
ejpam-6699	442	6	∣∣∣24tβ∣∣∣	∣∣∣24tβ∣∣∣	NOUN
ejpam-6699	442	7	∥∥∥q4	∥∥∥q4	PROPN
ejpam-6699	442	8	(	(	PUNCT
ejpam-6699	442	9	t	t	PROPN
ejpam-6699	442	10	2	2	NUM
ejpam-6699	442	11	t	t	NOUN
ejpam-6699	442	12	)	)	PUNCT
ejpam-6699	443	1	−q′	−q′	NOUN
ejpam-6699	443	2	4	4	NUM
ejpam-6699	443	3	(	(	PUNCT
ejpam-6699	443	4	t	t	PROPN
ejpam-6699	443	5	2	2	NUM
ejpam-6699	443	6	t	t	NOUN
ejpam-6699	443	7	)	)	PUNCT
ejpam-6699	443	8	,	,	PUNCT
ejpam-6699	443	9	ν1	ν1	NOUN
ejpam-6699	443	10	,	,	PUNCT
ejpam-6699	443	11	.	.	PUNCT
ejpam-6699	443	12	.	.	PUNCT
ejpam-6699	444	1	.	.	PUNCT
ejpam-6699	445	1	,	,	PUNCT
ejpam-6699	445	2	νn−1	νn−1	PROPN
ejpam-6699	445	3	∥∥∥	∥∥∥	PROPN
ejpam-6699	445	4	β	β	NOUN
ejpam-6699	445	5	≤	≤	NOUN
ejpam-6699	445	6	∣∣∣24tβ∣∣∣	∣∣∣24tβ∣∣∣	ADJ
ejpam-6699	445	7	ψ̃	ψ̃	PROPN
ejpam-6699	445	8	(	(	PUNCT
ejpam-6699	445	9	t	t	PROPN
ejpam-6699	445	10	2	2	NUM
ejpam-6699	445	11	t	t	NOUN
ejpam-6699	445	12	)	)	PUNCT
ejpam-6699	445	13	ϖ(ν1	ϖ(ν1	NOUN
ejpam-6699	445	14	,	,	PUNCT
ejpam-6699	445	15	.	.	PUNCT
ejpam-6699	445	16	.	.	PUNCT
ejpam-6699	445	17	.	.	PUNCT
ejpam-6699	446	1	,	,	PUNCT
ejpam-6699	446	2	νn−1	νn−1	PROPN
ejpam-6699	446	3	)	)	PUNCT
ejpam-6699	446	4	→	→	SYM
ejpam-6699	446	5	0	0	NUM
ejpam-6699	447	1	as	as	ADP
ejpam-6699	447	2	t	t	PROPN
ejpam-6699	447	3	→	→	SYM
ejpam-6699	447	4	∞	∞	NUM
ejpam-6699	447	5	by	by	ADP
ejpam-6699	447	6	assumption	assumption	NOUN
ejpam-6699	447	7	(	(	PUNCT
ejpam-6699	447	8	28	28	NUM
ejpam-6699	447	9	)	)	PUNCT
ejpam-6699	447	10	.	.	PUNCT
ejpam-6699	448	1	hence	hence	ADV
ejpam-6699	448	2	,	,	PUNCT
ejpam-6699	448	3	∥q4(t)−q′	∥q4(t)−q′	NOUN
ejpam-6699	448	4	4(t	4(t	NUM
ejpam-6699	448	5	)	)	PUNCT
ejpam-6699	448	6	,	,	PUNCT
ejpam-6699	448	7	ν1	ν1	NOUN
ejpam-6699	448	8	,	,	PUNCT
ejpam-6699	448	9	.	.	PUNCT
ejpam-6699	448	10	.	.	PUNCT
ejpam-6699	448	11	.	.	PUNCT
ejpam-6699	449	1	,	,	PUNCT
ejpam-6699	449	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	449	3	=	=	SYM
ejpam-6699	449	4	0	0	NUM
ejpam-6699	449	5	for	for	ADP
ejpam-6699	449	6	all	all	DET
ejpam-6699	449	7	t	t	NOUN
ejpam-6699	449	8	∈	∈	PROPN
ejpam-6699	449	9	e	e	NOUN
ejpam-6699	449	10	,	,	PUNCT
ejpam-6699	449	11	implying	imply	VERB
ejpam-6699	449	12	q4	q4	PROPN
ejpam-6699	449	13	=	=	SYM
ejpam-6699	449	14	q′	q′	NOUN
ejpam-6699	449	15	4	4	NUM
ejpam-6699	449	16	by	by	ADP
ejpam-6699	449	17	lemma	lemma	PROPN
ejpam-6699	449	18	2	2	NUM
ejpam-6699	449	19	.	.	NOUN
ejpam-6699	449	20	4	4	NUM
ejpam-6699	449	21	.	.	NOUN
ejpam-6699	449	22	consequences	consequence	NOUN
ejpam-6699	449	23	and	and	CCONJ
ejpam-6699	449	24	illustrative	illustrative	ADJ
ejpam-6699	449	25	example	example	NOUN
ejpam-6699	449	26	in	in	ADP
ejpam-6699	449	27	qp	qp	ADP
ejpam-6699	449	28	corollaries	corollary	NOUN
ejpam-6699	449	29	and	and	CCONJ
ejpam-6699	449	30	examples	example	NOUN
ejpam-6699	449	31	corollary	corollary	VERB
ejpam-6699	449	32	4.1	4.1	NUM
ejpam-6699	449	33	(	(	PUNCT
ejpam-6699	449	34	classical	classical	ADJ
ejpam-6699	449	35	non	non	ADJ
ejpam-6699	449	36	-	-	ADJ
ejpam-6699	449	37	archimedean	archimedean	ADJ
ejpam-6699	449	38	case	case	NOUN
ejpam-6699	449	39	)	)	PUNCT
ejpam-6699	449	40	let	let	VERB
ejpam-6699	449	41	e	e	NOUN
ejpam-6699	449	42	and	and	CCONJ
ejpam-6699	449	43	f	f	PROPN
ejpam-6699	449	44	be	be	AUX
ejpam-6699	449	45	non	non	ADJ
ejpam-6699	449	46	-	-	ADJ
ejpam-6699	449	47	archimedean	archimedean	ADJ
ejpam-6699	449	48	normed	normed	ADJ
ejpam-6699	449	49	spaces	space	NOUN
ejpam-6699	449	50	,	,	PUNCT
ejpam-6699	449	51	i.e.	i.e.	X
ejpam-6699	449	52	,	,	PUNCT
ejpam-6699	449	53	(	(	PUNCT
ejpam-6699	449	54	n	n	CCONJ
ejpam-6699	449	55	,	,	PUNCT
ejpam-6699	449	56	β)-normed	β)-normed	PUNCT
ejpam-6699	449	57	spaces	space	NOUN
ejpam-6699	449	58	with	with	ADP
ejpam-6699	449	59	n	n	NOUN
ejpam-6699	449	60	=	=	SYM
ejpam-6699	449	61	1	1	NUM
ejpam-6699	449	62	,	,	PUNCT
ejpam-6699	449	63	β	β	X
ejpam-6699	449	64	=	=	SYM
ejpam-6699	450	1	1	1	X
ejpam-6699	450	2	.	.	PUNCT
ejpam-6699	450	3	assume	assume	VERB
ejpam-6699	450	4	that	that	SCONJ
ejpam-6699	450	5	a	a	DET
ejpam-6699	450	6	mapping	mapping	NOUN
ejpam-6699	450	7	ϕ	ϕ	NOUN
ejpam-6699	450	8	:	:	PUNCT
ejpam-6699	450	9	e	e	X
ejpam-6699	450	10	→	→	SYM
ejpam-6699	450	11	f	f	PROPN
ejpam-6699	450	12	fulfills	fulfill	VERB
ejpam-6699	450	13	∥∆ϕ(t1	∥∆ϕ(t1	PROPN
ejpam-6699	450	14	,	,	PUNCT
ejpam-6699	450	15	t2	t2	NOUN
ejpam-6699	450	16	,	,	PUNCT
ejpam-6699	450	17	.	.	PUNCT
ejpam-6699	450	18	.	.	PUNCT
ejpam-6699	451	1	.	.	PUNCT
ejpam-6699	452	1	,	,	PUNCT
ejpam-6699	452	2	tr)∥	tr)∥	NOUN
ejpam-6699	452	3	≤	≤	NOUN
ejpam-6699	452	4	µ	µ	DET
ejpam-6699	452	5	r∑	r∑	NOUN
ejpam-6699	452	6	j=1	j=1	PROPN
ejpam-6699	452	7	∥tj∥s	∥tj∥s	PROPN
ejpam-6699	452	8	for	for	ADP
ejpam-6699	452	9	some	some	DET
ejpam-6699	452	10	constants	constant	NOUN
ejpam-6699	452	11	µ	µ	PRON
ejpam-6699	452	12	≥	≥	NOUN
ejpam-6699	452	13	0	0	NUM
ejpam-6699	452	14	,	,	PUNCT
ejpam-6699	452	15	s	s	VERB
ejpam-6699	452	16	>	>	X
ejpam-6699	452	17	1	1	NUM
ejpam-6699	452	18	,	,	PUNCT
ejpam-6699	452	19	and	and	CCONJ
ejpam-6699	452	20	all	all	DET
ejpam-6699	452	21	t1	t1	NOUN
ejpam-6699	452	22	,	,	PUNCT
ejpam-6699	452	23	t2	t2	NOUN
ejpam-6699	452	24	,	,	PUNCT
ejpam-6699	452	25	.	.	PUNCT
ejpam-6699	452	26	.	.	PUNCT
ejpam-6699	453	1	.	.	PUNCT
ejpam-6699	454	1	,	,	PUNCT
ejpam-6699	454	2	tr	tr	NOUN
ejpam-6699	454	3	∈	∈	PROPN
ejpam-6699	454	4	e.	e.	NOUN
ejpam-6699	454	5	then	then	ADV
ejpam-6699	454	6	there	there	PRON
ejpam-6699	454	7	is	be	VERB
ejpam-6699	454	8	a	a	DET
ejpam-6699	454	9	unique	unique	ADJ
ejpam-6699	454	10	quartic	quartic	ADJ
ejpam-6699	454	11	mapping	mapping	NOUN
ejpam-6699	454	12	q4	q4	NOUN
ejpam-6699	454	13	:	:	PUNCT
ejpam-6699	454	14	e	e	X
ejpam-6699	454	15	→	→	SYM
ejpam-6699	454	16	f	f	PROPN
ejpam-6699	454	17	fulfilling	fulfil	VERB
ejpam-6699	454	18	∥ϕ(t)−q4(t)∥	∥ϕ(t)−q4(t)∥	X
ejpam-6699	454	19	≤	≤	NOUN
ejpam-6699	454	20	µ|2−4|∥t∥s	µ|2−4|∥t∥s	NUM
ejpam-6699	454	21	,	,	PUNCT
ejpam-6699	454	22	for	for	ADP
ejpam-6699	454	23	every	every	DET
ejpam-6699	454	24	t	t	PROPN
ejpam-6699	454	25	∈	∈	PROPN
ejpam-6699	454	26	e.	e.	PROPN
ejpam-6699	454	27	corollary	corollary	PROPN
ejpam-6699	454	28	4.2	4.2	NUM
ejpam-6699	454	29	(	(	PUNCT
ejpam-6699	454	30	stability	stability	NOUN
ejpam-6699	454	31	in	in	ADP
ejpam-6699	454	32	β	β	ADJ
ejpam-6699	454	33	-	-	PUNCT
ejpam-6699	454	34	normed	norme	VERB
ejpam-6699	454	35	ultrametric	ultrametric	ADJ
ejpam-6699	454	36	spaces	space	NOUN
ejpam-6699	454	37	)	)	PUNCT
ejpam-6699	454	38	let	let	VERB
ejpam-6699	454	39	e	e	PRON
ejpam-6699	454	40	be	be	AUX
ejpam-6699	454	41	a	a	DET
ejpam-6699	454	42	β	β	NOUN
ejpam-6699	454	43	-	-	PUNCT
ejpam-6699	454	44	normed	normed	ADJ
ejpam-6699	454	45	space	space	NOUN
ejpam-6699	454	46	and	and	CCONJ
ejpam-6699	454	47	f	f	PROPN
ejpam-6699	454	48	a	a	DET
ejpam-6699	454	49	complete	complete	ADJ
ejpam-6699	454	50	non	non	ADJ
ejpam-6699	454	51	-	-	ADJ
ejpam-6699	454	52	archimedean	archimedean	ADJ
ejpam-6699	454	53	(	(	PUNCT
ejpam-6699	454	54	n	n	CCONJ
ejpam-6699	454	55	,	,	PUNCT
ejpam-6699	454	56	β)-normed	β)-normed	PUNCT
ejpam-6699	454	57	space	space	NOUN
ejpam-6699	454	58	with	with	ADP
ejpam-6699	454	59	0	0	NUM
ejpam-6699	454	60	<	<	X
ejpam-6699	454	61	β	β	X
ejpam-6699	454	62	<	<	X
ejpam-6699	454	63	1	1	NUM
ejpam-6699	454	64	.	.	PUNCT
ejpam-6699	455	1	if	if	SCONJ
ejpam-6699	455	2	a	a	DET
ejpam-6699	455	3	function	function	NOUN
ejpam-6699	455	4	ϕ	ϕ	NOUN
ejpam-6699	455	5	:	:	PUNCT
ejpam-6699	455	6	e	e	X
ejpam-6699	455	7	→	→	SYM
ejpam-6699	455	8	f	f	PROPN
ejpam-6699	455	9	satisfies	satisfie	NOUN
ejpam-6699	455	10	∥∆ϕ(t1	∥∆ϕ(t1	PROPN
ejpam-6699	455	11	,	,	PUNCT
ejpam-6699	455	12	t2	t2	NOUN
ejpam-6699	455	13	,	,	PUNCT
ejpam-6699	455	14	.	.	PUNCT
ejpam-6699	455	15	.	.	PUNCT
ejpam-6699	455	16	.	.	PUNCT
ejpam-6699	456	1	,	,	PUNCT
ejpam-6699	456	2	tr	tr	VERB
ejpam-6699	456	3	)	)	PUNCT
ejpam-6699	456	4	,	,	PUNCT
ejpam-6699	456	5	ν1	ν1	NOUN
ejpam-6699	456	6	,	,	PUNCT
ejpam-6699	456	7	.	.	PUNCT
ejpam-6699	456	8	.	.	PUNCT
ejpam-6699	457	1	.	.	PUNCT
ejpam-6699	458	1	,	,	PUNCT
ejpam-6699	458	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	458	3	≤	≤	NOUN
ejpam-6699	458	4	µ	µ	PRON
ejpam-6699	458	5	r∑	r∑	NOUN
ejpam-6699	458	6	j=1	j=1	NOUN
ejpam-6699	458	7	∥tj∥sβ	∥tj∥sβ	VERB
ejpam-6699	458	8	for	for	ADP
ejpam-6699	458	9	all	all	DET
ejpam-6699	458	10	tj	tj	NOUN
ejpam-6699	458	11	∈	∈	PROPN
ejpam-6699	458	12	e	e	NOUN
ejpam-6699	458	13	,	,	PUNCT
ejpam-6699	458	14	νk	νk	NOUN
ejpam-6699	458	15	∈	∈	PROPN
ejpam-6699	458	16	f	f	NOUN
ejpam-6699	458	17	,	,	PUNCT
ejpam-6699	458	18	then	then	ADV
ejpam-6699	458	19	there	there	PRON
ejpam-6699	458	20	is	be	VERB
ejpam-6699	458	21	a	a	DET
ejpam-6699	458	22	unique	unique	ADJ
ejpam-6699	458	23	quartic	quartic	ADJ
ejpam-6699	458	24	function	function	NOUN
ejpam-6699	458	25	q4	q4	NOUN
ejpam-6699	458	26	:	:	PUNCT
ejpam-6699	458	27	e	e	X
ejpam-6699	458	28	→	→	SYM
ejpam-6699	458	29	f	f	PROPN
ejpam-6699	458	30	fulfilling	fulfil	VERB
ejpam-6699	458	31	∥ϕ(t)−q4(t	∥ϕ(t)−q4(t	NOUN
ejpam-6699	458	32	)	)	PUNCT
ejpam-6699	458	33	,	,	PUNCT
ejpam-6699	458	34	ν1	ν1	NOUN
ejpam-6699	458	35	,	,	PUNCT
ejpam-6699	458	36	.	.	PUNCT
ejpam-6699	458	37	.	.	PUNCT
ejpam-6699	459	1	.	.	PUNCT
ejpam-6699	460	1	,	,	PUNCT
ejpam-6699	460	2	νn−1∥β	νn−1∥β	NOUN
ejpam-6699	460	3	≤	≤	NUM
ejpam-6699	460	4	µ|2−4β|∥t∥sβ	µ|2−4β|∥t∥sβ	PUNCT
ejpam-6699	460	5	.	.	PUNCT
ejpam-6699	460	6	example	example	NOUN
ejpam-6699	460	7	4.3	4.3	NUM
ejpam-6699	460	8	(	(	PUNCT
ejpam-6699	460	9	mapping	mapping	NOUN
ejpam-6699	460	10	on	on	ADP
ejpam-6699	460	11	a	a	DET
ejpam-6699	460	12	qp	qp	NOUN
ejpam-6699	460	13	space	space	NOUN
ejpam-6699	460	14	)	)	PUNCT
ejpam-6699	460	15	let	let	VERB
ejpam-6699	460	16	p	p	PRON
ejpam-6699	460	17	>	>	X
ejpam-6699	460	18	3	3	NUM
ejpam-6699	460	19	be	be	AUX
ejpam-6699	460	20	a	a	DET
ejpam-6699	460	21	prime	prime	ADJ
ejpam-6699	460	22	number	number	NOUN
ejpam-6699	460	23	and	and	CCONJ
ejpam-6699	460	24	e	e	NOUN
ejpam-6699	460	25	=	=	SYM
ejpam-6699	460	26	qp	qp	PROPN
ejpam-6699	460	27	,	,	PUNCT
ejpam-6699	460	28	the	the	DET
ejpam-6699	460	29	field	field	NOUN
ejpam-6699	460	30	of	of	ADP
ejpam-6699	460	31	p	p	NOUN
ejpam-6699	460	32	-	-	PUNCT
ejpam-6699	460	33	adic	adic	ADJ
ejpam-6699	460	34	numbers	number	NOUN
ejpam-6699	460	35	.	.	PUNCT
ejpam-6699	461	1	define	define	VERB
ejpam-6699	461	2	the	the	DET
ejpam-6699	461	3	function	function	NOUN
ejpam-6699	461	4	ϕ	ϕ	NOUN
ejpam-6699	461	5	:	:	PUNCT
ejpam-6699	461	6	qp	qp	PROPN
ejpam-6699	461	7	→	→	SYM
ejpam-6699	461	8	qp	qp	NOUN
ejpam-6699	461	9	by	by	ADP
ejpam-6699	461	10	ϕ(t	ϕ(t	NUM
ejpam-6699	461	11	)	)	PUNCT
ejpam-6699	462	1	=	=	SYM
ejpam-6699	462	2	t4	t4	PROPN
ejpam-6699	462	3	+	+	CCONJ
ejpam-6699	462	4	ϵ(t	ϵ(t	NOUN
ejpam-6699	462	5	)	)	PUNCT
ejpam-6699	462	6	,	,	PUNCT
ejpam-6699	462	7	where	where	SCONJ
ejpam-6699	462	8	|ϵ(t)|p	|ϵ(t)|p	NOUN
ejpam-6699	462	9	≤	≤	ADJ
ejpam-6699	462	10	δ|t|sp	δ|t|sp	NOUN
ejpam-6699	462	11	for	for	ADP
ejpam-6699	462	12	some	some	DET
ejpam-6699	462	13	δ	δ	PROPN
ejpam-6699	462	14	>	>	X
ejpam-6699	462	15	0	0	PUNCT
ejpam-6699	462	16	and	and	CCONJ
ejpam-6699	462	17	s	s	X
ejpam-6699	462	18	>	>	X
ejpam-6699	462	19	4	4	NUM
ejpam-6699	462	20	.	.	PUNCT
ejpam-6699	463	1	then	then	ADV
ejpam-6699	463	2	ϕ	ϕ	X
ejpam-6699	463	3	satisfies	satisfy	VERB
ejpam-6699	463	4	the	the	DET
ejpam-6699	463	5	condition	condition	NOUN
ejpam-6699	463	6	of	of	ADP
ejpam-6699	463	7	theorem	theorem	NOUN
ejpam-6699	463	8	2	2	NUM
ejpam-6699	463	9	for	for	ADP
ejpam-6699	463	10	suitable	suitable	ADJ
ejpam-6699	463	11	µ	µ	NOUN
ejpam-6699	463	12	,	,	PUNCT
ejpam-6699	463	13	and	and	CCONJ
ejpam-6699	463	14	there	there	PRON
ejpam-6699	463	15	is	be	VERB
ejpam-6699	463	16	only	only	ADV
ejpam-6699	463	17	one	one	NUM
ejpam-6699	463	18	quartic	quartic	ADJ
ejpam-6699	463	19	function	function	NOUN
ejpam-6699	463	20	q4(t	q4(t	PROPN
ejpam-6699	463	21	)	)	PUNCT
ejpam-6699	463	22	=	=	PUNCT
ejpam-6699	464	1	t4	t4	PROPN
ejpam-6699	464	2	fulfilling	fulfil	VERB
ejpam-6699	464	3	∥ϕ(t)−q4(t)∥β	∥ϕ(t)−q4(t)∥β	SYM
ejpam-6699	464	4	≤	≤	X
ejpam-6699	464	5	µ∥t∥s	µ∥t∥s	NOUN
ejpam-6699	464	6	.	.	PUNCT
ejpam-6699	465	1	s.	s.	PROPN
ejpam-6699	465	2	gowri	gowri	PROPN
ejpam-6699	465	3	et	et	PROPN
ejpam-6699	465	4	al	al	PROPN
ejpam-6699	465	5	.	.	PUNCT
ejpam-6699	465	6	/	/	SYM
ejpam-6699	465	7	eur	eur	PROPN
ejpam-6699	465	8	.	.	PUNCT
ejpam-6699	466	1	j.	j.	PROPN
ejpam-6699	466	2	pure	pure	PROPN
ejpam-6699	466	3	appl	appl	PROPN
ejpam-6699	466	4	.	.	PROPN
ejpam-6699	466	5	math	math	PROPN
ejpam-6699	466	6	,	,	PUNCT
ejpam-6699	466	7	18	18	NUM
ejpam-6699	466	8	(	(	PUNCT
ejpam-6699	466	9	4	4	NUM
ejpam-6699	466	10	)	)	PUNCT
ejpam-6699	466	11	(	(	PUNCT
ejpam-6699	466	12	2025	2025	NUM
ejpam-6699	466	13	)	)	PUNCT
ejpam-6699	466	14	,	,	PUNCT
ejpam-6699	466	15	6699	6699	NUM
ejpam-6699	466	16	12	12	NUM
ejpam-6699	466	17	of	of	ADP
ejpam-6699	466	18	14	14	NUM
ejpam-6699	466	19	5	5	NUM
ejpam-6699	466	20	.	.	PUNCT
ejpam-6699	467	1	conclusion	conclusion	NOUN
ejpam-6699	467	2	this	this	DET
ejpam-6699	467	3	work	work	NOUN
ejpam-6699	467	4	examines	examine	VERB
ejpam-6699	467	5	the	the	DET
ejpam-6699	467	6	hyers	hyers	PROPN
ejpam-6699	467	7	-	-	PUNCT
ejpam-6699	467	8	ulam	ulam	ADJ
ejpam-6699	467	9	stability	stability	NOUN
ejpam-6699	467	10	of	of	ADP
ejpam-6699	467	11	a	a	DET
ejpam-6699	467	12	generalized	generalize	VERB
ejpam-6699	467	13	quartic	quartic	ADJ
ejpam-6699	467	14	functional	functional	ADJ
ejpam-6699	467	15	equation	equation	NOUN
ejpam-6699	467	16	within	within	ADP
ejpam-6699	467	17	non	non	ADJ
ejpam-6699	467	18	-	-	ADJ
ejpam-6699	467	19	archimedean	archimedean	ADJ
ejpam-6699	467	20	(	(	PUNCT
ejpam-6699	467	21	n	n	CCONJ
ejpam-6699	467	22	,	,	PUNCT
ejpam-6699	467	23	β)-normed	β)-normed	PUNCT
ejpam-6699	467	24	spaces	space	NOUN
ejpam-6699	467	25	.	.	PUNCT
ejpam-6699	468	1	these	these	DET
ejpam-6699	468	2	spaces	space	NOUN
ejpam-6699	468	3	,	,	PUNCT
ejpam-6699	468	4	which	which	PRON
ejpam-6699	468	5	generalize	generalize	VERB
ejpam-6699	468	6	traditional	traditional	ADJ
ejpam-6699	468	7	normed	normed	ADJ
ejpam-6699	468	8	and	and	CCONJ
ejpam-6699	468	9	ultrametric	ultrametric	ADJ
ejpam-6699	468	10	structures	structure	NOUN
ejpam-6699	468	11	,	,	PUNCT
ejpam-6699	468	12	offer	offer	VERB
ejpam-6699	468	13	a	a	DET
ejpam-6699	468	14	comprehensive	comprehensive	ADJ
ejpam-6699	468	15	framework	framework	NOUN
ejpam-6699	468	16	for	for	ADP
ejpam-6699	468	17	examining	examine	VERB
ejpam-6699	468	18	the	the	DET
ejpam-6699	468	19	behaviour	behaviour	NOUN
ejpam-6699	468	20	of	of	ADP
ejpam-6699	468	21	functional	functional	ADJ
ejpam-6699	468	22	equations	equation	NOUN
ejpam-6699	468	23	under	under	ADP
ejpam-6699	468	24	perturbations	perturbation	NOUN
ejpam-6699	468	25	.	.	PUNCT
ejpam-6699	469	1	theorem	theorem	ADJ
ejpam-6699	469	2	2	2	NUM
ejpam-6699	469	3	and	and	CCONJ
ejpam-6699	469	4	theorem	theorem	VERB
ejpam-6699	469	5	3	3	NUM
ejpam-6699	469	6	examined	examine	VERB
ejpam-6699	469	7	stability	stability	NOUN
ejpam-6699	469	8	in	in	ADP
ejpam-6699	469	9	the	the	DET
ejpam-6699	469	10	context	context	NOUN
ejpam-6699	469	11	of	of	ADP
ejpam-6699	469	12	a	a	DET
ejpam-6699	469	13	non	non	ADJ
ejpam-6699	469	14	-	-	ADJ
ejpam-6699	469	15	archimedean	archimedean	ADJ
ejpam-6699	469	16	β1	β1	PROPN
ejpam-6699	469	17	-	-	PUNCT
ejpam-6699	469	18	normed	norme	VERB
ejpam-6699	469	19	space	space	NOUN
ejpam-6699	469	20	as	as	ADP
ejpam-6699	469	21	the	the	DET
ejpam-6699	469	22	domain	domain	NOUN
ejpam-6699	469	23	and	and	CCONJ
ejpam-6699	469	24	a	a	DET
ejpam-6699	469	25	full	full	ADJ
ejpam-6699	469	26	non	non	ADJ
ejpam-6699	469	27	-	-	ADJ
ejpam-6699	469	28	archimedean	archimedean	ADJ
ejpam-6699	469	29	(	(	PUNCT
ejpam-6699	469	30	n	n	CCONJ
ejpam-6699	469	31	,	,	PUNCT
ejpam-6699	469	32	β)-normed	β)-normed	PUNCT
ejpam-6699	469	33	space	space	NOUN
ejpam-6699	469	34	as	as	ADP
ejpam-6699	469	35	the	the	DET
ejpam-6699	469	36	codomain	codomain	NOUN
ejpam-6699	469	37	.	.	PUNCT
ejpam-6699	470	1	theorem	theorem	VERB
ejpam-6699	470	2	4	4	NUM
ejpam-6699	470	3	and	and	CCONJ
ejpam-6699	470	4	theorem	theorem	VERB
ejpam-6699	470	5	5	5	NUM
ejpam-6699	470	6	broadened	broaden	VERB
ejpam-6699	470	7	these	these	DET
ejpam-6699	470	8	findings	finding	NOUN
ejpam-6699	470	9	to	to	PART
ejpam-6699	470	10	encompass	encompass	VERB
ejpam-6699	470	11	more	more	ADJ
ejpam-6699	470	12	complex	complex	ADJ
ejpam-6699	470	13	control	control	NOUN
ejpam-6699	470	14	functions	function	NOUN
ejpam-6699	470	15	,	,	PUNCT
ejpam-6699	470	16	hence	hence	ADV
ejpam-6699	470	17	permitting	permit	VERB
ejpam-6699	470	18	enhanced	enhanced	ADJ
ejpam-6699	470	19	flexibility	flexibility	NOUN
ejpam-6699	470	20	in	in	ADP
ejpam-6699	470	21	the	the	DET
ejpam-6699	470	22	assumptions	assumption	NOUN
ejpam-6699	470	23	regarding	regard	VERB
ejpam-6699	470	24	perturbations	perturbation	NOUN
ejpam-6699	470	25	.	.	PUNCT
ejpam-6699	471	1	our	our	PRON
ejpam-6699	471	2	results	result	NOUN
ejpam-6699	471	3	establish	establish	VERB
ejpam-6699	471	4	the	the	DET
ejpam-6699	471	5	existence	existence	NOUN
ejpam-6699	471	6	and	and	CCONJ
ejpam-6699	471	7	uniqueness	uniqueness	NOUN
ejpam-6699	471	8	of	of	ADP
ejpam-6699	471	9	quartic	quartic	ADJ
ejpam-6699	471	10	mappings	mapping	NOUN
ejpam-6699	471	11	that	that	PRON
ejpam-6699	471	12	resemble	resemble	VERB
ejpam-6699	471	13	the	the	DET
ejpam-6699	471	14	original	original	ADJ
ejpam-6699	471	15	functional	functional	ADJ
ejpam-6699	471	16	equation	equation	NOUN
ejpam-6699	471	17	,	,	PUNCT
ejpam-6699	471	18	therefore	therefore	ADV
ejpam-6699	471	19	validating	validate	VERB
ejpam-6699	471	20	its	its	PRON
ejpam-6699	471	21	ulam	ulam	ADJ
ejpam-6699	471	22	-	-	PUNCT
ejpam-6699	471	23	type	type	NOUN
ejpam-6699	471	24	stability	stability	NOUN
ejpam-6699	471	25	in	in	ADP
ejpam-6699	471	26	this	this	DET
ejpam-6699	471	27	extended	extended	ADJ
ejpam-6699	471	28	non	non	ADJ
ejpam-6699	471	29	-	-	ADJ
ejpam-6699	471	30	archimedean	archimedean	ADJ
ejpam-6699	471	31	context	context	NOUN
ejpam-6699	471	32	.	.	PUNCT
ejpam-6699	472	1	6	6	X
ejpam-6699	472	2	.	.	X
ejpam-6699	472	3	conflict	conflict	NOUN
ejpam-6699	472	4	of	of	ADP
ejpam-6699	472	5	interest	interest	NOUN
ejpam-6699	472	6	the	the	DET
ejpam-6699	472	7	authors	author	NOUN
ejpam-6699	472	8	declare	declare	VERB
ejpam-6699	472	9	that	that	SCONJ
ejpam-6699	472	10	they	they	PRON
ejpam-6699	472	11	have	have	VERB
ejpam-6699	472	12	no	no	DET
ejpam-6699	472	13	competing	compete	VERB
ejpam-6699	472	14	interests	interest	NOUN
ejpam-6699	472	15	.	.	PUNCT
ejpam-6699	473	1	availability	availability	NOUN
ejpam-6699	473	2	of	of	ADP
ejpam-6699	473	3	data	datum	NOUN
ejpam-6699	473	4	and	and	CCONJ
ejpam-6699	473	5	materials	material	NOUN
ejpam-6699	473	6	not	not	PART
ejpam-6699	473	7	applicable	applicable	ADJ
ejpam-6699	473	8	.	.	PUNCT
ejpam-6699	474	1	acknowledgements	acknowledgement	NOUN
ejpam-6699	474	2	this	this	DET
ejpam-6699	474	3	research	research	NOUN
ejpam-6699	474	4	was	be	AUX
ejpam-6699	474	5	supported	support	VERB
ejpam-6699	474	6	by	by	ADP
ejpam-6699	474	7	university	university	NOUN
ejpam-6699	474	8	of	of	ADP
ejpam-6699	474	9	phayao	phayao	NOUN
ejpam-6699	474	10	and	and	CCONJ
ejpam-6699	474	11	thailand	thailand	PROPN
ejpam-6699	474	12	science	science	PROPN
ejpam-6699	474	13	research	research	PROPN
ejpam-6699	474	14	and	and	CCONJ
ejpam-6699	474	15	innovation	innovation	NOUN
ejpam-6699	474	16	fund	fund	NOUN
ejpam-6699	474	17	(	(	PUNCT
ejpam-6699	474	18	fundamental	fundamental	ADJ
ejpam-6699	474	19	fund	fund	NOUN
ejpam-6699	474	20	2026	2026	NUM
ejpam-6699	474	21	,	,	PUNCT
ejpam-6699	474	22	grant	grant	VERB
ejpam-6699	474	23	no	no	NOUN
ejpam-6699	474	24	.	.	PUNCT
ejpam-6699	474	25	xxxx/2568	xxxx/2568	PROPN
ejpam-6699	474	26	)	)	PUNCT
ejpam-6699	474	27	.	.	PUNCT
ejpam-6699	475	1	authors	author	NOUN
ejpam-6699	475	2	’	'	PUNCT
ejpam-6699	475	3	contributions	contribution	NOUN
ejpam-6699	475	4	the	the	DET
ejpam-6699	475	5	authors	author	NOUN
ejpam-6699	475	6	equally	equally	ADV
ejpam-6699	475	7	conceived	conceive	VERB
ejpam-6699	475	8	of	of	ADP
ejpam-6699	475	9	the	the	DET
ejpam-6699	475	10	study	study	NOUN
ejpam-6699	475	11	,	,	PUNCT
ejpam-6699	475	12	participated	participate	VERB
ejpam-6699	475	13	in	in	ADP
ejpam-6699	475	14	its	its	PRON
ejpam-6699	475	15	design	design	NOUN
ejpam-6699	475	16	and	and	CCONJ
ejpam-6699	475	17	coordination	coordination	NOUN
ejpam-6699	475	18	,	,	PUNCT
ejpam-6699	475	19	drafted	draft	VERB
ejpam-6699	475	20	the	the	DET
ejpam-6699	475	21	manuscript	manuscript	NOUN
ejpam-6699	475	22	,	,	PUNCT
ejpam-6699	475	23	participated	participate	VERB
ejpam-6699	475	24	in	in	ADP
ejpam-6699	475	25	the	the	DET
ejpam-6699	475	26	sequence	sequence	NOUN
ejpam-6699	475	27	alignment	alignment	NOUN
ejpam-6699	475	28	,	,	PUNCT
ejpam-6699	475	29	and	and	CCONJ
ejpam-6699	475	30	read	read	VERB
ejpam-6699	475	31	and	and	CCONJ
ejpam-6699	475	32	approved	approve	VERB
ejpam-6699	475	33	the	the	DET
ejpam-6699	475	34	final	final	ADJ
ejpam-6699	475	35	manuscript	manuscript	NOUN
ejpam-6699	475	36	.	.	PUNCT
ejpam-6699	476	1	references	reference	NOUN
ejpam-6699	476	2	[	[	X
ejpam-6699	476	3	1	1	X
ejpam-6699	476	4	]	]	PUNCT
ejpam-6699	476	5	s.	s.	PROPN
ejpam-6699	476	6	m.	m.	PROPN
ejpam-6699	476	7	ulam	ulam	PROPN
ejpam-6699	476	8	.	.	PUNCT
ejpam-6699	477	1	a	a	DET
ejpam-6699	477	2	collection	collection	NOUN
ejpam-6699	477	3	of	of	ADP
ejpam-6699	477	4	mathematical	mathematical	ADJ
ejpam-6699	477	5	problems	problem	NOUN
ejpam-6699	477	6	.	.	PUNCT
ejpam-6699	478	1	interscience	interscience	NOUN
ejpam-6699	478	2	,	,	PUNCT
ejpam-6699	478	3	new	new	PROPN
ejpam-6699	478	4	york	york	PROPN
ejpam-6699	478	5	,	,	PUNCT
ejpam-6699	478	6	ny	ny	PROPN
ejpam-6699	478	7	,	,	PUNCT
ejpam-6699	478	8	usa	usa	PROPN
ejpam-6699	478	9	,	,	PUNCT
ejpam-6699	478	10	1960	1960	NUM
ejpam-6699	478	11	.	.	PUNCT
ejpam-6699	479	1	[	[	X
ejpam-6699	479	2	2	2	X
ejpam-6699	479	3	]	]	PUNCT
ejpam-6699	479	4	d.	d.	PROPN
ejpam-6699	479	5	h.	h.	PROPN
ejpam-6699	479	6	hyers	hyers	PROPN
ejpam-6699	479	7	.	.	PUNCT
ejpam-6699	480	1	on	on	ADP
ejpam-6699	480	2	the	the	DET
ejpam-6699	480	3	stability	stability	NOUN
ejpam-6699	480	4	of	of	ADP
ejpam-6699	480	5	the	the	DET
ejpam-6699	480	6	linear	linear	ADJ
ejpam-6699	480	7	functional	functional	ADJ
ejpam-6699	480	8	equation	equation	NOUN
ejpam-6699	480	9	.	.	PUNCT
ejpam-6699	481	1	proceedings	proceeding	NOUN
ejpam-6699	481	2	of	of	ADP
ejpam-6699	481	3	the	the	DET
ejpam-6699	481	4	national	national	PROPN
ejpam-6699	481	5	academy	academy	PROPN
ejpam-6699	481	6	of	of	ADP
ejpam-6699	481	7	sciences	sciences	PROPN
ejpam-6699	481	8	of	of	ADP
ejpam-6699	481	9	the	the	DET
ejpam-6699	481	10	usa	usa	PROPN
ejpam-6699	481	11	,	,	PUNCT
ejpam-6699	481	12	27:222–224	27:222–224	NUM
ejpam-6699	481	13	,	,	PUNCT
ejpam-6699	481	14	1941	1941	NUM
ejpam-6699	481	15	.	.	PUNCT
ejpam-6699	482	1	[	[	X
ejpam-6699	482	2	3	3	X
ejpam-6699	482	3	]	]	X
ejpam-6699	482	4	s.	s.	PROPN
ejpam-6699	482	5	czerwik	czerwik	PROPN
ejpam-6699	482	6	.	.	PUNCT
ejpam-6699	483	1	functional	functional	ADJ
ejpam-6699	483	2	equations	equation	NOUN
ejpam-6699	483	3	and	and	CCONJ
ejpam-6699	483	4	inequalities	inequality	NOUN
ejpam-6699	483	5	in	in	ADP
ejpam-6699	483	6	several	several	ADJ
ejpam-6699	483	7	variables	variable	NOUN
ejpam-6699	483	8	.	.	PUNCT
ejpam-6699	484	1	world	world	NOUN
ejpam-6699	484	2	scientific	scientific	PROPN
ejpam-6699	484	3	,	,	PUNCT
ejpam-6699	484	4	singapore	singapore	PROPN
ejpam-6699	484	5	,	,	PUNCT
ejpam-6699	484	6	2002	2002	NUM
ejpam-6699	484	7	.	.	PUNCT
ejpam-6699	485	1	s.	s.	PROPN
ejpam-6699	485	2	gowri	gowri	PROPN
ejpam-6699	485	3	et	et	PROPN
ejpam-6699	485	4	al	al	PROPN
ejpam-6699	485	5	.	.	PUNCT
ejpam-6699	485	6	/	/	SYM
ejpam-6699	485	7	eur	eur	PROPN
ejpam-6699	485	8	.	.	PUNCT
ejpam-6699	486	1	j.	j.	PROPN
ejpam-6699	486	2	pure	pure	PROPN
ejpam-6699	486	3	appl	appl	PROPN
ejpam-6699	486	4	.	.	PROPN
ejpam-6699	486	5	math	math	PROPN
ejpam-6699	486	6	,	,	PUNCT
ejpam-6699	486	7	18	18	NUM
ejpam-6699	486	8	(	(	PUNCT
ejpam-6699	486	9	4	4	NUM
ejpam-6699	486	10	)	)	PUNCT
ejpam-6699	486	11	(	(	PUNCT
ejpam-6699	486	12	2025	2025	NUM
ejpam-6699	486	13	)	)	PUNCT
ejpam-6699	486	14	,	,	PUNCT
ejpam-6699	486	15	6699	6699	NUM
ejpam-6699	486	16	13	13	NUM
ejpam-6699	486	17	of	of	ADP
ejpam-6699	486	18	14	14	NUM
ejpam-6699	486	19	[	[	SYM
ejpam-6699	486	20	4	4	NUM
ejpam-6699	486	21	]	]	PUNCT
ejpam-6699	486	22	s.	s.	PROPN
ejpam-6699	486	23	m.	m.	PROPN
ejpam-6699	486	24	jung	jung	PROPN
ejpam-6699	486	25	.	.	PUNCT
ejpam-6699	487	1	hyers	hyer	NOUN
ejpam-6699	487	2	–	–	PUNCT
ejpam-6699	487	3	ulam	ulam	X
ejpam-6699	487	4	–	–	PUNCT
ejpam-6699	487	5	rassias	rassia	NOUN
ejpam-6699	487	6	stability	stability	NOUN
ejpam-6699	487	7	of	of	ADP
ejpam-6699	487	8	functional	functional	ADJ
ejpam-6699	487	9	equations	equation	NOUN
ejpam-6699	487	10	in	in	ADP
ejpam-6699	487	11	nonlinear	nonlinear	ADJ
ejpam-6699	487	12	analysis	analysis	NOUN
ejpam-6699	487	13	.	.	PUNCT
ejpam-6699	488	1	springer	springer	NOUN
ejpam-6699	488	2	,	,	PUNCT
ejpam-6699	488	3	new	new	PROPN
ejpam-6699	488	4	york	york	PROPN
ejpam-6699	488	5	,	,	PUNCT
ejpam-6699	488	6	ny	ny	PROPN
ejpam-6699	488	7	,	,	PUNCT
ejpam-6699	488	8	usa	usa	PROPN
ejpam-6699	488	9	,	,	PUNCT
ejpam-6699	488	10	2011	2011	NUM
ejpam-6699	488	11	.	.	PUNCT
ejpam-6699	489	1	[	[	X
ejpam-6699	489	2	5	5	X
ejpam-6699	489	3	]	]	PUNCT
ejpam-6699	489	4	h.	h.	PROPN
ejpam-6699	489	5	azadi	azadi	PROPN
ejpam-6699	489	6	kenary	kenary	PROPN
ejpam-6699	489	7	,	,	PUNCT
ejpam-6699	489	8	h.	h.	PROPN
ejpam-6699	489	9	rezaei	rezaei	PROPN
ejpam-6699	489	10	,	,	PUNCT
ejpam-6699	489	11	m.	m.	NOUN
ejpam-6699	489	12	sharifzadeh	sharifzadeh	PROPN
ejpam-6699	489	13	,	,	PUNCT
ejpam-6699	489	14	d.	d.	PROPN
ejpam-6699	489	15	y.	y.	PROPN
ejpam-6699	489	16	shin	shin	PROPN
ejpam-6699	489	17	,	,	PUNCT
ejpam-6699	489	18	and	and	CCONJ
ejpam-6699	489	19	j.	j.	PROPN
ejpam-6699	489	20	r.	r.	PROPN
ejpam-6699	489	21	lee	lee	PROPN
ejpam-6699	489	22	.	.	PUNCT
ejpam-6699	490	1	nonarchimedean	nonarchimedean	PROPN
ejpam-6699	490	2	hyers	hyers	PROPN
ejpam-6699	490	3	-	-	PUNCT
ejpam-6699	490	4	ulam	ulam	ADJ
ejpam-6699	490	5	-	-	PUNCT
ejpam-6699	490	6	rassias	rassias	PROPN
ejpam-6699	490	7	stability	stability	NOUN
ejpam-6699	490	8	of	of	ADP
ejpam-6699	490	9	m	m	NOUN
ejpam-6699	490	10	-	-	ADJ
ejpam-6699	490	11	variable	variable	ADJ
ejpam-6699	490	12	functional	functional	ADJ
ejpam-6699	490	13	equation	equation	NOUN
ejpam-6699	490	14	.	.	PUNCT
ejpam-6699	491	1	advances	advance	NOUN
ejpam-6699	491	2	in	in	ADP
ejpam-6699	491	3	difference	difference	NOUN
ejpam-6699	491	4	equations	equation	NOUN
ejpam-6699	491	5	,	,	PUNCT
ejpam-6699	491	6	pages	page	NOUN
ejpam-6699	491	7	1–17	1–17	PROPN
ejpam-6699	491	8	,	,	PUNCT
ejpam-6699	491	9	2012	2012	NUM
ejpam-6699	491	10	.	.	PUNCT
ejpam-6699	492	1	[	[	X
ejpam-6699	492	2	6	6	NUM
ejpam-6699	492	3	]	]	X
ejpam-6699	492	4	y.	y.	PROPN
ejpam-6699	492	5	s.	s.	PROPN
ejpam-6699	492	6	lee	lee	PROPN
ejpam-6699	492	7	and	and	CCONJ
ejpam-6699	492	8	s.	s.	PROPN
ejpam-6699	492	9	y.	y.	PROPN
ejpam-6699	492	10	chung	chung	PROPN
ejpam-6699	492	11	.	.	PUNCT
ejpam-6699	493	1	stability	stability	NOUN
ejpam-6699	493	2	of	of	ADP
ejpam-6699	493	3	quartic	quartic	ADJ
ejpam-6699	493	4	functional	functional	ADJ
ejpam-6699	493	5	equations	equation	NOUN
ejpam-6699	493	6	in	in	ADP
ejpam-6699	493	7	the	the	DET
ejpam-6699	493	8	spaces	space	NOUN
ejpam-6699	493	9	of	of	ADP
ejpam-6699	493	10	generalized	generalized	ADJ
ejpam-6699	493	11	functions	function	NOUN
ejpam-6699	493	12	.	.	PUNCT
ejpam-6699	494	1	advances	advance	NOUN
ejpam-6699	494	2	in	in	ADP
ejpam-6699	494	3	difference	difference	NOUN
ejpam-6699	494	4	equations	equation	NOUN
ejpam-6699	494	5	,	,	PUNCT
ejpam-6699	494	6	pages	page	NOUN
ejpam-6699	494	7	1–16	1–16	PROPN
ejpam-6699	494	8	,	,	PUNCT
ejpam-6699	494	9	2009	2009	NUM
ejpam-6699	494	10	.	.	PUNCT
ejpam-6699	495	1	[	[	X
ejpam-6699	495	2	7	7	X
ejpam-6699	495	3	]	]	X
ejpam-6699	495	4	v.	v.	X
ejpam-6699	495	5	radu	radu	PROPN
ejpam-6699	495	6	.	.	PUNCT
ejpam-6699	496	1	the	the	DET
ejpam-6699	496	2	fixed	fixed	ADJ
ejpam-6699	496	3	point	point	NOUN
ejpam-6699	496	4	alternative	alternative	NOUN
ejpam-6699	496	5	and	and	CCONJ
ejpam-6699	496	6	the	the	DET
ejpam-6699	496	7	stability	stability	NOUN
ejpam-6699	496	8	of	of	ADP
ejpam-6699	496	9	functional	functional	ADJ
ejpam-6699	496	10	equations	equation	NOUN
ejpam-6699	496	11	.	.	PUNCT
ejpam-6699	497	1	fixed	fix	VERB
ejpam-6699	497	2	point	point	NOUN
ejpam-6699	497	3	theory	theory	NOUN
ejpam-6699	497	4	,	,	PUNCT
ejpam-6699	497	5	4:91–96	4:91–96	PROPN
ejpam-6699	497	6	,	,	PUNCT
ejpam-6699	497	7	2003	2003	NUM
ejpam-6699	497	8	.	.	PUNCT
ejpam-6699	498	1	[	[	X
ejpam-6699	498	2	8	8	NUM
ejpam-6699	498	3	]	]	SYM
ejpam-6699	498	4	th	th	X
ejpam-6699	498	5	.	.	PUNCT
ejpam-6699	498	6	m.	m.	NOUN
ejpam-6699	498	7	rassias	rassias	PROPN
ejpam-6699	498	8	.	.	PUNCT
ejpam-6699	499	1	on	on	ADP
ejpam-6699	499	2	the	the	DET
ejpam-6699	499	3	stability	stability	NOUN
ejpam-6699	499	4	of	of	ADP
ejpam-6699	499	5	the	the	DET
ejpam-6699	499	6	linear	linear	ADJ
ejpam-6699	499	7	mapping	mapping	NOUN
ejpam-6699	499	8	in	in	ADP
ejpam-6699	499	9	banach	banach	NOUN
ejpam-6699	499	10	spaces	space	NOUN
ejpam-6699	499	11	.	.	PUNCT
ejpam-6699	500	1	proceedings	proceeding	NOUN
ejpam-6699	500	2	of	of	ADP
ejpam-6699	500	3	the	the	DET
ejpam-6699	500	4	american	american	PROPN
ejpam-6699	500	5	mathematical	mathematical	PROPN
ejpam-6699	500	6	society	society	NOUN
ejpam-6699	500	7	,	,	PUNCT
ejpam-6699	500	8	72:297–300	72:297–300	PROPN
ejpam-6699	500	9	,	,	PUNCT
ejpam-6699	500	10	1978	1978	NUM
ejpam-6699	500	11	.	.	PUNCT
ejpam-6699	501	1	[	[	X
ejpam-6699	501	2	9	9	NUM
ejpam-6699	501	3	]	]	X
ejpam-6699	501	4	n.	n.	NOUN
ejpam-6699	501	5	uthirasamy	uthirasamy	PROPN
ejpam-6699	501	6	,	,	PUNCT
ejpam-6699	501	7	k.	k.	PROPN
ejpam-6699	501	8	tamilvanan	tamilvanan	PROPN
ejpam-6699	501	9	,	,	PUNCT
ejpam-6699	501	10	h.	h.	PROPN
ejpam-6699	501	11	k.	k.	PROPN
ejpam-6699	501	12	nashine	nashine	PROPN
ejpam-6699	501	13	,	,	PUNCT
ejpam-6699	501	14	and	and	CCONJ
ejpam-6699	501	15	r.	r.	PROPN
ejpam-6699	501	16	george	george	PROPN
ejpam-6699	501	17	.	.	PUNCT
ejpam-6699	501	18	solution	solution	NOUN
ejpam-6699	501	19	and	and	CCONJ
ejpam-6699	501	20	stability	stability	NOUN
ejpam-6699	501	21	of	of	ADP
ejpam-6699	501	22	quartic	quartic	ADJ
ejpam-6699	501	23	functional	functional	ADJ
ejpam-6699	501	24	equations	equation	NOUN
ejpam-6699	501	25	in	in	ADP
ejpam-6699	501	26	modular	modular	ADJ
ejpam-6699	501	27	spaces	space	NOUN
ejpam-6699	501	28	by	by	ADP
ejpam-6699	501	29	using	use	VERB
ejpam-6699	501	30	fatou	fatou	NOUN
ejpam-6699	501	31	property	property	NOUN
ejpam-6699	501	32	.	.	PUNCT
ejpam-6699	502	1	journal	journal	PROPN
ejpam-6699	502	2	of	of	ADP
ejpam-6699	502	3	function	function	NOUN
ejpam-6699	502	4	spaces	space	NOUN
ejpam-6699	502	5	,	,	PUNCT
ejpam-6699	502	6	1:1–9	1:1–9	NUM
ejpam-6699	502	7	,	,	PUNCT
ejpam-6699	502	8	2022	2022	NUM
ejpam-6699	502	9	.	.	PUNCT
ejpam-6699	503	1	[	[	X
ejpam-6699	503	2	10	10	NUM
ejpam-6699	503	3	]	]	X
ejpam-6699	503	4	n.	n.	NOUN
ejpam-6699	503	5	koblitz	koblitz	PROPN
ejpam-6699	503	6	.	.	PUNCT
ejpam-6699	504	1	p	p	X
ejpam-6699	504	2	-	-	PUNCT
ejpam-6699	504	3	adic	adic	ADJ
ejpam-6699	504	4	numbers	number	NOUN
ejpam-6699	504	5	,	,	PUNCT
ejpam-6699	504	6	p	p	ADJ
ejpam-6699	504	7	-	-	PUNCT
ejpam-6699	504	8	adic	adic	ADJ
ejpam-6699	504	9	analysis	analysis	NOUN
ejpam-6699	504	10	,	,	PUNCT
ejpam-6699	504	11	and	and	CCONJ
ejpam-6699	504	12	zeta	zeta	NOUN
ejpam-6699	504	13	-	-	PUNCT
ejpam-6699	504	14	functions	function	NOUN
ejpam-6699	504	15	.	.	PUNCT
ejpam-6699	505	1	springer	springer	NOUN
ejpam-6699	505	2	,	,	PUNCT
ejpam-6699	505	3	new	new	PROPN
ejpam-6699	505	4	york	york	PROPN
ejpam-6699	505	5	,	,	PUNCT
ejpam-6699	505	6	ny	ny	PROPN
ejpam-6699	505	7	,	,	PUNCT
ejpam-6699	505	8	usa	usa	PROPN
ejpam-6699	505	9	,	,	PUNCT
ejpam-6699	505	10	1984	1984	NUM
ejpam-6699	505	11	.	.	PUNCT
ejpam-6699	506	1	[	[	X
ejpam-6699	506	2	11	11	NUM
ejpam-6699	506	3	]	]	PUNCT
ejpam-6699	506	4	w.	w.	PROPN
ejpam-6699	506	5	h.	h.	PROPN
ejpam-6699	506	6	schikhof	schikhof	PROPN
ejpam-6699	506	7	.	.	PUNCT
ejpam-6699	507	1	ultrametric	ultrametric	ADJ
ejpam-6699	507	2	calculus	calculus	NOUN
ejpam-6699	507	3	:	:	PUNCT
ejpam-6699	507	4	an	an	DET
ejpam-6699	507	5	introduction	introduction	NOUN
ejpam-6699	507	6	to	to	ADP
ejpam-6699	507	7	p	p	NOUN
ejpam-6699	507	8	-	-	PUNCT
ejpam-6699	507	9	adic	adic	ADJ
ejpam-6699	507	10	analysis	analysis	NOUN
ejpam-6699	507	11	.	.	PUNCT
ejpam-6699	508	1	cambridge	cambridge	PROPN
ejpam-6699	508	2	university	university	PROPN
ejpam-6699	508	3	press	press	PROPN
ejpam-6699	508	4	,	,	PUNCT
ejpam-6699	508	5	cambridge	cambridge	PROPN
ejpam-6699	508	6	,	,	PUNCT
ejpam-6699	508	7	uk	uk	PROPN
ejpam-6699	508	8	,	,	PUNCT
ejpam-6699	508	9	1984	1984	NUM
ejpam-6699	508	10	.	.	PUNCT
ejpam-6699	509	1	[	[	X
ejpam-6699	509	2	12	12	NUM
ejpam-6699	509	3	]	]	X
ejpam-6699	509	4	s.	s.	PROPN
ejpam-6699	509	5	bosch	bosch	PROPN
ejpam-6699	509	6	,	,	PUNCT
ejpam-6699	509	7	u.	u.	NOUN
ejpam-6699	509	8	güntzer	güntzer	NOUN
ejpam-6699	509	9	,	,	PUNCT
ejpam-6699	509	10	and	and	CCONJ
ejpam-6699	509	11	r.	r.	PROPN
ejpam-6699	509	12	remmert	remmert	PROPN
ejpam-6699	509	13	.	.	PUNCT
ejpam-6699	510	1	non	non	ADJ
ejpam-6699	510	2	-	-	ADJ
ejpam-6699	510	3	archimedean	archimedean	ADJ
ejpam-6699	510	4	analysis	analysis	NOUN
ejpam-6699	510	5	.	.	PUNCT
ejpam-6699	511	1	springer	springer	NOUN
ejpam-6699	511	2	,	,	PUNCT
ejpam-6699	511	3	berlin	berlin	PROPN
ejpam-6699	511	4	/	/	SYM
ejpam-6699	511	5	heidelberg	heidelberg	PROPN
ejpam-6699	511	6	,	,	PUNCT
ejpam-6699	511	7	germany	germany	PROPN
ejpam-6699	511	8	,	,	PUNCT
ejpam-6699	511	9	1984	1984	NUM
ejpam-6699	511	10	.	.	PUNCT
ejpam-6699	512	1	[	[	X
ejpam-6699	512	2	13	13	NUM
ejpam-6699	512	3	]	]	PUNCT
ejpam-6699	512	4	f.	f.	PROPN
ejpam-6699	512	5	q.	q.	PROPN
ejpam-6699	512	6	gouvêa	gouvêa	PROPN
ejpam-6699	512	7	.	.	PUNCT
ejpam-6699	513	1	p	p	X
ejpam-6699	513	2	-	-	PUNCT
ejpam-6699	513	3	adic	adic	ADJ
ejpam-6699	513	4	numbers	number	NOUN
ejpam-6699	513	5	:	:	PUNCT
ejpam-6699	513	6	an	an	DET
ejpam-6699	513	7	introduction	introduction	NOUN
ejpam-6699	513	8	.	.	PUNCT
ejpam-6699	514	1	springer	springer	NOUN
ejpam-6699	514	2	,	,	PUNCT
ejpam-6699	514	3	berlin	berlin	PROPN
ejpam-6699	514	4	/	/	SYM
ejpam-6699	514	5	heidelberg	heidelberg	PROPN
ejpam-6699	514	6	,	,	PUNCT
ejpam-6699	514	7	germany	germany	PROPN
ejpam-6699	514	8	,	,	PUNCT
ejpam-6699	514	9	2nd	2nd	PROPN
ejpam-6699	514	10	edition	edition	NOUN
ejpam-6699	514	11	,	,	PUNCT
ejpam-6699	514	12	1997	1997	NUM
ejpam-6699	514	13	.	.	PUNCT
ejpam-6699	515	1	[	[	X
ejpam-6699	515	2	14	14	NUM
ejpam-6699	515	3	]	]	PUNCT
ejpam-6699	515	4	a.	a.	PROPN
ejpam-6699	515	5	f.	f.	PROPN
ejpam-6699	515	6	monna	monna	PROPN
ejpam-6699	515	7	.	.	PUNCT
ejpam-6699	516	1	dirichlet	dirichlet	PROPN
ejpam-6699	516	2	’s	’s	PART
ejpam-6699	516	3	principle	principle	NOUN
ejpam-6699	516	4	:	:	PUNCT
ejpam-6699	516	5	a	a	DET
ejpam-6699	516	6	mathematical	mathematical	ADJ
ejpam-6699	516	7	comedy	comedy	NOUN
ejpam-6699	516	8	of	of	ADP
ejpam-6699	516	9	errors	error	NOUN
ejpam-6699	516	10	and	and	CCONJ
ejpam-6699	516	11	its	its	PRON
ejpam-6699	516	12	influence	influence	NOUN
ejpam-6699	516	13	on	on	ADP
ejpam-6699	516	14	the	the	DET
ejpam-6699	516	15	development	development	NOUN
ejpam-6699	516	16	of	of	ADP
ejpam-6699	516	17	analysis	analysis	NOUN
ejpam-6699	516	18	.	.	PUNCT
ejpam-6699	517	1	reidel	reidel	PROPN
ejpam-6699	517	2	,	,	PUNCT
ejpam-6699	517	3	dordrecht	dordrecht	PROPN
ejpam-6699	517	4	,	,	PUNCT
ejpam-6699	517	5	the	the	DET
ejpam-6699	517	6	netherlands	netherlands	PROPN
ejpam-6699	517	7	,	,	PUNCT
ejpam-6699	517	8	1975	1975	NUM
ejpam-6699	517	9	.	.	PUNCT
ejpam-6699	518	1	[	[	X
ejpam-6699	518	2	15	15	NUM
ejpam-6699	518	3	]	]	X
ejpam-6699	518	4	n.	n.	NOUN
ejpam-6699	518	5	alessa	alessa	PROPN
ejpam-6699	518	6	,	,	PUNCT
ejpam-6699	518	7	k.	k.	PROPN
ejpam-6699	518	8	tamilvanan	tamilvanan	PROPN
ejpam-6699	518	9	,	,	PUNCT
ejpam-6699	518	10	k.	k.	PROPN
ejpam-6699	518	11	loganathan	loganathan	PROPN
ejpam-6699	518	12	,	,	PUNCT
ejpam-6699	518	13	and	and	CCONJ
ejpam-6699	518	14	k.	k.	PROPN
ejpam-6699	518	15	k.	k.	PROPN
ejpam-6699	518	16	selvi	selvi	PROPN
ejpam-6699	518	17	.	.	PUNCT
ejpam-6699	519	1	hyers	hyer	NOUN
ejpam-6699	519	2	–	–	PUNCT
ejpam-6699	519	3	ulam	ulam	PROPN
ejpam-6699	519	4	stability	stability	NOUN
ejpam-6699	519	5	of	of	ADP
ejpam-6699	519	6	functional	functional	ADJ
ejpam-6699	519	7	equation	equation	NOUN
ejpam-6699	519	8	deriving	derive	VERB
ejpam-6699	519	9	from	from	ADP
ejpam-6699	519	10	quadratic	quadratic	ADJ
ejpam-6699	519	11	mapping	mapping	NOUN
ejpam-6699	519	12	in	in	ADP
ejpam-6699	519	13	non	non	ADJ
ejpam-6699	519	14	-	-	ADJ
ejpam-6699	519	15	archimedean	archimedean	ADJ
ejpam-6699	519	16	(	(	PUNCT
ejpam-6699	519	17	n	n	CCONJ
ejpam-6699	519	18	,	,	PUNCT
ejpam-6699	519	19	β)normed	β)normed	ADJ
ejpam-6699	519	20	spaces	space	NOUN
ejpam-6699	519	21	.	.	PUNCT
ejpam-6699	520	1	journal	journal	NOUN
ejpam-6699	520	2	of	of	ADP
ejpam-6699	520	3	function	function	NOUN
ejpam-6699	520	4	spaces	space	NOUN
ejpam-6699	520	5	,	,	PUNCT
ejpam-6699	520	6	2021:9953214	2021:9953214	NUM
ejpam-6699	520	7	,	,	PUNCT
ejpam-6699	520	8	2021	2021	NUM
ejpam-6699	520	9	.	.	PUNCT
ejpam-6699	521	1	[	[	X
ejpam-6699	521	2	16	16	NUM
ejpam-6699	521	3	]	]	PUNCT
ejpam-6699	521	4	a.	a.	NOUN
ejpam-6699	521	5	pasupathi	pasupathi	PROPN
ejpam-6699	521	6	,	,	PUNCT
ejpam-6699	521	7	j.	j.	PROPN
ejpam-6699	521	8	konsalraj	konsalraj	PROPN
ejpam-6699	521	9	,	,	PUNCT
ejpam-6699	521	10	n.	n.	PROPN
ejpam-6699	521	11	fatima	fatima	PROPN
ejpam-6699	521	12	,	,	PUNCT
ejpam-6699	521	13	v.	v.	ADP
ejpam-6699	521	14	velusamy	velusamy	PROPN
ejpam-6699	521	15	,	,	PUNCT
ejpam-6699	521	16	n.	n.	PROPN
ejpam-6699	521	17	mlaiki	mlaiki	PROPN
ejpam-6699	521	18	,	,	PUNCT
ejpam-6699	521	19	and	and	CCONJ
ejpam-6699	521	20	n.	n.	NOUN
ejpam-6699	521	21	souayah	souayah	NOUN
ejpam-6699	521	22	.	.	PUNCT
ejpam-6699	522	1	direct	direct	ADJ
ejpam-6699	522	2	and	and	CCONJ
ejpam-6699	522	3	fixed	fix	VERB
ejpam-6699	522	4	-	-	PUNCT
ejpam-6699	522	5	point	point	NOUN
ejpam-6699	522	6	stability	stability	NOUN
ejpam-6699	522	7	–	–	PUNCT
ejpam-6699	522	8	instability	instability	NOUN
ejpam-6699	522	9	of	of	ADP
ejpam-6699	522	10	additive	additive	ADJ
ejpam-6699	522	11	functional	functional	ADJ
ejpam-6699	522	12	equation	equation	NOUN
ejpam-6699	522	13	in	in	ADP
ejpam-6699	522	14	banach	banach	NOUN
ejpam-6699	522	15	and	and	CCONJ
ejpam-6699	522	16	quasi	quasi	ADJ
ejpam-6699	522	17	-	-	ADJ
ejpam-6699	522	18	beta	beta	ADJ
ejpam-6699	522	19	normed	norme	VERB
ejpam-6699	522	20	spaces	space	NOUN
ejpam-6699	522	21	.	.	PUNCT
ejpam-6699	522	22	symmetry	symmetry	NOUN
ejpam-6699	522	23	,	,	PUNCT
ejpam-6699	522	24	8:1700	8:1700	NUM
ejpam-6699	522	25	,	,	PUNCT
ejpam-6699	522	26	2022	2022	NUM
ejpam-6699	522	27	.	.	PUNCT
ejpam-6699	523	1	[	[	X
ejpam-6699	523	2	17	17	NUM
ejpam-6699	523	3	]	]	PUNCT
ejpam-6699	523	4	r.	r.	PROPN
ejpam-6699	523	5	k.	k.	PROPN
ejpam-6699	523	6	sharma	sharma	PROPN
ejpam-6699	523	7	and	and	CCONJ
ejpam-6699	523	8	s.	s.	PROPN
ejpam-6699	523	9	chandok	chandok	PROPN
ejpam-6699	523	10	.	.	PUNCT
ejpam-6699	524	1	quartic	quartic	ADJ
ejpam-6699	524	2	functional	functional	ADJ
ejpam-6699	524	3	equation	equation	NOUN
ejpam-6699	524	4	:	:	PUNCT
ejpam-6699	524	5	ulam	ulam	ADJ
ejpam-6699	524	6	-	-	PUNCT
ejpam-6699	524	7	type	type	NOUN
ejpam-6699	524	8	stability	stability	NOUN
ejpam-6699	524	9	in	in	ADP
ejpam-6699	524	10	(	(	PUNCT
ejpam-6699	524	11	β	β	X
ejpam-6699	524	12	,	,	PUNCT
ejpam-6699	524	13	p)-banach	p)-banach	PUNCT
ejpam-6699	524	14	space	space	NOUN
ejpam-6699	524	15	and	and	CCONJ
ejpam-6699	524	16	non	non	ADJ
ejpam-6699	524	17	-	-	ADJ
ejpam-6699	524	18	archimedean	archimedean	ADJ
ejpam-6699	524	19	β	β	X
ejpam-6699	524	20	-	-	ADJ
ejpam-6699	524	21	normed	normed	ADJ
ejpam-6699	524	22	space	space	NOUN
ejpam-6699	524	23	.	.	PUNCT
ejpam-6699	525	1	journal	journal	PROPN
ejpam-6699	525	2	of	of	ADP
ejpam-6699	525	3	mathematics	mathematic	NOUN
ejpam-6699	525	4	,	,	PUNCT
ejpam-6699	525	5	1:9908530	1:9908530	NUM
ejpam-6699	525	6	,	,	PUNCT
ejpam-6699	525	7	2022	2022	NUM
ejpam-6699	525	8	.	.	PUNCT
ejpam-6699	526	1	[	[	X
ejpam-6699	526	2	18	18	NUM
ejpam-6699	526	3	]	]	X
ejpam-6699	526	4	j.	j.	PROPN
ejpam-6699	526	5	aczél	aczél	PROPN
ejpam-6699	526	6	.	.	PUNCT
ejpam-6699	527	1	lectures	lecture	NOUN
ejpam-6699	527	2	on	on	ADP
ejpam-6699	527	3	functional	functional	ADJ
ejpam-6699	527	4	equations	equation	NOUN
ejpam-6699	527	5	and	and	CCONJ
ejpam-6699	527	6	their	their	PRON
ejpam-6699	527	7	applications	application	NOUN
ejpam-6699	527	8	.	.	PUNCT
ejpam-6699	528	1	academic	academic	ADJ
ejpam-6699	528	2	press	press	NOUN
ejpam-6699	528	3	,	,	PUNCT
ejpam-6699	528	4	new	new	PROPN
ejpam-6699	528	5	york	york	PROPN
ejpam-6699	528	6	,	,	PUNCT
ejpam-6699	528	7	ny	ny	PROPN
ejpam-6699	528	8	,	,	PUNCT
ejpam-6699	528	9	usa	usa	PROPN
ejpam-6699	528	10	,	,	PUNCT
ejpam-6699	528	11	1966	1966	NUM
ejpam-6699	528	12	.	.	PUNCT
ejpam-6699	529	1	[	[	X
ejpam-6699	529	2	19	19	NUM
ejpam-6699	529	3	]	]	X
ejpam-6699	529	4	y.	y.	PROPN
ejpam-6699	529	5	almalki	almalki	PROPN
ejpam-6699	529	6	,	,	PUNCT
ejpam-6699	529	7	b.	b.	PROPN
ejpam-6699	529	8	radhakrishnan	radhakrishnan	PROPN
ejpam-6699	529	9	,	,	PUNCT
ejpam-6699	529	10	u.	u.	PROPN
ejpam-6699	529	11	jayaraman	jayaraman	PROPN
ejpam-6699	529	12	,	,	PUNCT
ejpam-6699	529	13	and	and	CCONJ
ejpam-6699	529	14	k.	k.	PROPN
ejpam-6699	529	15	tamilvanan	tamilvanan	PROPN
ejpam-6699	529	16	.	.	PUNCT
ejpam-6699	530	1	some	some	DET
ejpam-6699	530	2	common	common	ADJ
ejpam-6699	530	3	fixed	fix	VERB
ejpam-6699	530	4	point	point	NOUN
ejpam-6699	530	5	results	result	NOUN
ejpam-6699	530	6	in	in	ADP
ejpam-6699	530	7	modular	modular	ADJ
ejpam-6699	530	8	ultrametric	ultrametric	ADJ
ejpam-6699	530	9	space	space	NOUN
ejpam-6699	530	10	using	use	VERB
ejpam-6699	530	11	various	various	ADJ
ejpam-6699	530	12	contractions	contraction	NOUN
ejpam-6699	530	13	and	and	CCONJ
ejpam-6699	530	14	their	their	PRON
ejpam-6699	530	15	application	application	NOUN
ejpam-6699	530	16	to	to	ADP
ejpam-6699	530	17	well	well	ADV
ejpam-6699	530	18	-	-	PUNCT
ejpam-6699	530	19	posedness	posedness	NOUN
ejpam-6699	530	20	.	.	PUNCT
ejpam-6699	531	1	mathematics	mathematic	NOUN
ejpam-6699	531	2	,	,	PUNCT
ejpam-6699	531	3	11(19):1–18	11(19):1–18	NUM
ejpam-6699	531	4	,	,	PUNCT
ejpam-6699	531	5	2023	2023	NUM
ejpam-6699	531	6	.	.	PUNCT
ejpam-6699	532	1	[	[	X
ejpam-6699	532	2	20	20	NUM
ejpam-6699	532	3	]	]	PUNCT
ejpam-6699	532	4	j.	j.	PROPN
ejpam-6699	532	5	c.	c.	PROPN
ejpam-6699	532	6	bae	bae	PROPN
ejpam-6699	532	7	and	and	CCONJ
ejpam-6699	532	8	c.	c.	PROPN
ejpam-6699	532	9	g.	g.	PROPN
ejpam-6699	532	10	park	park	PROPN
ejpam-6699	532	11	.	.	PUNCT
ejpam-6699	533	1	stability	stability	NOUN
ejpam-6699	533	2	of	of	ADP
ejpam-6699	533	3	a	a	DET
ejpam-6699	533	4	functional	functional	ADJ
ejpam-6699	533	5	equation	equation	NOUN
ejpam-6699	533	6	associated	associate	VERB
ejpam-6699	533	7	with	with	ADP
ejpam-6699	533	8	quartic	quartic	ADJ
ejpam-6699	533	9	mapping	mapping	NOUN
ejpam-6699	533	10	.	.	PUNCT
ejpam-6699	534	1	journal	journal	PROPN
ejpam-6699	534	2	of	of	ADP
ejpam-6699	534	3	mathematical	mathematical	ADJ
ejpam-6699	534	4	inequalities	inequality	NOUN
ejpam-6699	534	5	,	,	PUNCT
ejpam-6699	534	6	9:103–110	9:103–110	NUM
ejpam-6699	534	7	,	,	PUNCT
ejpam-6699	534	8	2004	2004	NUM
ejpam-6699	534	9	.	.	PUNCT
ejpam-6699	535	1	[	[	X
ejpam-6699	535	2	21	21	NUM
ejpam-6699	535	3	]	]	X
ejpam-6699	535	4	e.	e.	PROPN
ejpam-6699	535	5	castillo	castillo	PROPN
ejpam-6699	535	6	.	.	PUNCT
ejpam-6699	536	1	functional	functional	ADJ
ejpam-6699	536	2	equations	equation	NOUN
ejpam-6699	536	3	and	and	CCONJ
ejpam-6699	536	4	modeling	modeling	NOUN
ejpam-6699	536	5	in	in	ADP
ejpam-6699	536	6	mathematical	mathematical	ADJ
ejpam-6699	536	7	physics	physics	NOUN
ejpam-6699	536	8	.	.	PUNCT
ejpam-6699	537	1	springer	springer	PROPN
ejpam-6699	537	2	,	,	PUNCT
ejpam-6699	537	3	cham	cham	PROPN
ejpam-6699	537	4	,	,	PUNCT
ejpam-6699	537	5	switzerland	switzerland	PROPN
ejpam-6699	537	6	,	,	PUNCT
ejpam-6699	537	7	2021	2021	NUM
ejpam-6699	537	8	.	.	PUNCT
ejpam-6699	538	1	[	[	X
ejpam-6699	538	2	22	22	NUM
ejpam-6699	538	3	]	]	PUNCT
ejpam-6699	538	4	s.	s.	PROPN
ejpam-6699	538	5	s.	s.	PROPN
ejpam-6699	538	6	dragomir	dragomir	PROPN
ejpam-6699	538	7	.	.	PUNCT
ejpam-6699	539	1	some	some	DET
ejpam-6699	539	2	new	new	ADJ
ejpam-6699	539	3	applications	application	NOUN
ejpam-6699	539	4	of	of	ADP
ejpam-6699	539	5	the	the	DET
ejpam-6699	539	6	jensen	jensen	PROPN
ejpam-6699	539	7	functional	functional	PROPN
ejpam-6699	539	8	in	in	ADP
ejpam-6699	539	9	information	information	NOUN
ejpam-6699	539	10	theory	theory	NOUN
ejpam-6699	539	11	.	.	PUNCT
ejpam-6699	540	1	s.	s.	PROPN
ejpam-6699	540	2	gowri	gowri	PROPN
ejpam-6699	540	3	et	et	PROPN
ejpam-6699	540	4	al	al	PROPN
ejpam-6699	540	5	.	.	PUNCT
ejpam-6699	540	6	/	/	SYM
ejpam-6699	540	7	eur	eur	PROPN
ejpam-6699	540	8	.	.	PUNCT
ejpam-6699	541	1	j.	j.	PROPN
ejpam-6699	541	2	pure	pure	PROPN
ejpam-6699	541	3	appl	appl	PROPN
ejpam-6699	541	4	.	.	PROPN
ejpam-6699	541	5	math	math	PROPN
ejpam-6699	541	6	,	,	PUNCT
ejpam-6699	541	7	18	18	NUM
ejpam-6699	541	8	(	(	PUNCT
ejpam-6699	541	9	4	4	NUM
ejpam-6699	541	10	)	)	PUNCT
ejpam-6699	541	11	(	(	PUNCT
ejpam-6699	541	12	2025	2025	NUM
ejpam-6699	541	13	)	)	PUNCT
ejpam-6699	541	14	,	,	PUNCT
ejpam-6699	541	15	6699	6699	NUM
ejpam-6699	541	16	14	14	NUM
ejpam-6699	541	17	of	of	ADP
ejpam-6699	541	18	14	14	NUM
ejpam-6699	541	19	tamkang	tamkang	PROPN
ejpam-6699	541	20	journal	journal	NOUN
ejpam-6699	541	21	of	of	ADP
ejpam-6699	541	22	mathematics	mathematic	NOUN
ejpam-6699	541	23	,	,	PUNCT
ejpam-6699	541	24	31:223–234	31:223–234	NUM
ejpam-6699	541	25	,	,	PUNCT
ejpam-6699	541	26	2000	2000	NUM
ejpam-6699	541	27	.	.	PUNCT
ejpam-6699	542	1	[	[	X
ejpam-6699	542	2	23	23	NUM
ejpam-6699	542	3	]	]	PUNCT
ejpam-6699	542	4	m.	m.	NOUN
ejpam-6699	542	5	s.	s.	PROPN
ejpam-6699	542	6	khan	khan	PROPN
ejpam-6699	542	7	and	and	CCONJ
ejpam-6699	542	8	m.	m.	PROPN
ejpam-6699	542	9	imdad	imdad	PROPN
ejpam-6699	542	10	.	.	PUNCT
ejpam-6699	543	1	generalizations	generalization	NOUN
ejpam-6699	543	2	of	of	ADP
ejpam-6699	543	3	banach	banach	NOUN
ejpam-6699	543	4	and	and	CCONJ
ejpam-6699	543	5	kannan	kannan	PROPN
ejpam-6699	543	6	mappings	mapping	NOUN
ejpam-6699	543	7	in	in	ADP
ejpam-6699	543	8	non	non	ADJ
ejpam-6699	543	9	-	-	ADJ
ejpam-6699	543	10	archimedean	archimedean	ADJ
ejpam-6699	543	11	spaces	space	NOUN
ejpam-6699	543	12	.	.	PUNCT
ejpam-6699	544	1	journal	journal	NOUN
ejpam-6699	544	2	of	of	ADP
ejpam-6699	544	3	fixed	fix	VERB
ejpam-6699	544	4	point	point	NOUN
ejpam-6699	544	5	theory	theory	NOUN
ejpam-6699	544	6	and	and	CCONJ
ejpam-6699	544	7	applications	application	NOUN
ejpam-6699	544	8	,	,	PUNCT
ejpam-6699	544	9	23:1–12	23:1–12	NUM
ejpam-6699	544	10	,	,	PUNCT
ejpam-6699	544	11	2021	2021	NUM
ejpam-6699	544	12	.	.	PUNCT
ejpam-6699	545	1	[	[	X
ejpam-6699	545	2	24	24	NUM
ejpam-6699	545	3	]	]	X
ejpam-6699	545	4	d.	d.	PROPN
ejpam-6699	545	5	miheţ.	miheţ.	PROPN
ejpam-6699	545	6	stability	stability	PROPN
ejpam-6699	545	7	results	result	VERB
ejpam-6699	545	8	in	in	ADP
ejpam-6699	545	9	fuzzy	fuzzy	ADJ
ejpam-6699	545	10	normed	normed	ADJ
ejpam-6699	545	11	spaces	space	NOUN
ejpam-6699	545	12	.	.	PUNCT
ejpam-6699	546	1	journal	journal	PROPN
ejpam-6699	546	2	of	of	ADP
ejpam-6699	546	3	inequalities	inequality	NOUN
ejpam-6699	546	4	in	in	ADP
ejpam-6699	546	5	pure	pure	ADJ
ejpam-6699	546	6	and	and	CCONJ
ejpam-6699	546	7	applied	applied	ADJ
ejpam-6699	546	8	mathematics	mathematic	NOUN
ejpam-6699	546	9	,	,	PUNCT
ejpam-6699	546	10	9:1–6	9:1–6	NUM
ejpam-6699	546	11	,	,	PUNCT
ejpam-6699	546	12	2008	2008	NUM
ejpam-6699	546	13	.	.	PUNCT
ejpam-6699	547	1	[	[	X
ejpam-6699	547	2	25	25	NUM
ejpam-6699	547	3	]	]	X
ejpam-6699	547	4	b.	b.	PROPN
ejpam-6699	547	5	radhakrishnan	radhakrishnan	PROPN
ejpam-6699	547	6	and	and	CCONJ
ejpam-6699	547	7	u.	u.	PROPN
ejpam-6699	547	8	jayaraman	jayaraman	PROPN
ejpam-6699	547	9	.	.	PUNCT
ejpam-6699	548	1	fixed	fix	VERB
ejpam-6699	548	2	point	point	NOUN
ejpam-6699	548	3	results	result	NOUN
ejpam-6699	548	4	in	in	ADP
ejpam-6699	548	5	partially	partially	ADV
ejpam-6699	548	6	ordered	order	VERB
ejpam-6699	548	7	ultrametric	ultrametric	ADJ
ejpam-6699	548	8	space	space	NOUN
ejpam-6699	548	9	via	via	ADP
ejpam-6699	548	10	p	p	ADJ
ejpam-6699	548	11	-	-	PUNCT
ejpam-6699	548	12	adic	adic	ADJ
ejpam-6699	548	13	distance	distance	NOUN
ejpam-6699	548	14	.	.	PUNCT
ejpam-6699	549	1	iaeng	iaeng	PROPN
ejpam-6699	549	2	international	international	PROPN
ejpam-6699	549	3	journal	journal	PROPN
ejpam-6699	549	4	of	of	ADP
ejpam-6699	549	5	applied	apply	VERB
ejpam-6699	549	6	mathematics	mathematic	NOUN
ejpam-6699	549	7	,	,	PUNCT
ejpam-6699	549	8	53(3):1–7	53(3):1–7	NUM
ejpam-6699	549	9	,	,	PUNCT
ejpam-6699	549	10	2022	2022	NUM
ejpam-6699	549	11	.	.	PUNCT
ejpam-6699	550	1	[	[	X
ejpam-6699	550	2	26	26	NUM
ejpam-6699	550	3	]	]	X
ejpam-6699	550	4	b.	b.	PROPN
ejpam-6699	550	5	radhakrishnan	radhakrishnan	PROPN
ejpam-6699	550	6	,	,	PUNCT
ejpam-6699	550	7	u.	u.	PROPN
ejpam-6699	550	8	jayaraman	jayaraman	PROPN
ejpam-6699	550	9	,	,	PUNCT
ejpam-6699	550	10	s.	s.	PROPN
ejpam-6699	550	11	o.	o.	PROPN
ejpam-6699	550	12	hilali	hilali	PROPN
ejpam-6699	550	13	,	,	PUNCT
ejpam-6699	550	14	m.	m.	PROPN
ejpam-6699	550	15	kameswari	kameswari	PROPN
ejpam-6699	550	16	,	,	PUNCT
ejpam-6699	550	17	m.	m.	NOUN
ejpam-6699	550	18	alhagyan	alhagyan	PROPN
ejpam-6699	550	19	,	,	PUNCT
ejpam-6699	550	20	k.	k.	PROPN
ejpam-6699	550	21	tamilvanan	tamilvanan	PROPN
ejpam-6699	550	22	,	,	PUNCT
ejpam-6699	550	23	and	and	CCONJ
ejpam-6699	550	24	a.	a.	PROPN
ejpam-6699	550	25	gargouri	gargouri	PROPN
ejpam-6699	550	26	.	.	PUNCT
ejpam-6699	551	1	coincidence	coincidence	NOUN
ejpam-6699	551	2	point	point	NOUN
ejpam-6699	551	3	results	result	NOUN
ejpam-6699	551	4	for	for	ADP
ejpam-6699	551	5	self	self	NOUN
ejpam-6699	551	6	-	-	PUNCT
ejpam-6699	551	7	mapping	mapping	NOUN
ejpam-6699	551	8	with	with	ADP
ejpam-6699	551	9	extended	extended	ADJ
ejpam-6699	551	10	rational	rational	ADJ
ejpam-6699	551	11	contraction	contraction	NOUN
ejpam-6699	551	12	in	in	ADP
ejpam-6699	551	13	partially	partially	ADV
ejpam-6699	551	14	ordered	order	VERB
ejpam-6699	551	15	ultrametric	ultrametric	ADJ
ejpam-6699	551	16	spaces	space	NOUN
ejpam-6699	551	17	using	use	VERB
ejpam-6699	551	18	p	p	NOUN
ejpam-6699	551	19	-	-	PUNCT
ejpam-6699	551	20	adic	adic	ADJ
ejpam-6699	551	21	distance	distance	NOUN
ejpam-6699	551	22	.	.	PUNCT
ejpam-6699	552	1	journal	journal	NOUN
ejpam-6699	552	2	of	of	ADP
ejpam-6699	552	3	mathematics	mathematics	PROPN
ejpam-6699	552	4	,	,	PUNCT
ejpam-6699	552	5	2024(1):1–14	2024(1):1–14	PROPN
ejpam-6699	552	6	,	,	PUNCT
ejpam-6699	552	7	2024	2024	NUM
ejpam-6699	552	8	.	.	PUNCT
ejpam-6699	553	1	[	[	X
ejpam-6699	553	2	27	27	NUM
ejpam-6699	553	3	]	]	X
ejpam-6699	553	4	d.	d.	NOUN
ejpam-6699	553	5	shukla	shukla	PROPN
ejpam-6699	553	6	.	.	PUNCT
ejpam-6699	554	1	p	p	X
ejpam-6699	554	2	-	-	PUNCT
ejpam-6699	554	3	adic	adic	ADJ
ejpam-6699	554	4	fixed	fix	VERB
ejpam-6699	554	5	point	point	NOUN
ejpam-6699	554	6	theory	theory	NOUN
ejpam-6699	554	7	and	and	CCONJ
ejpam-6699	554	8	applications	application	NOUN
ejpam-6699	554	9	to	to	PART
ejpam-6699	554	10	nonlinear	nonlinear	VERB
ejpam-6699	554	11	integral	integral	ADJ
ejpam-6699	554	12	equations	equation	NOUN
ejpam-6699	554	13	.	.	PUNCT
ejpam-6699	555	1	mathematical	mathematical	ADJ
ejpam-6699	555	2	reports	report	NOUN
ejpam-6699	555	3	,	,	PUNCT
ejpam-6699	555	4	24:23–34	24:23–34	NUM
ejpam-6699	555	5	,	,	PUNCT
ejpam-6699	555	6	2022	2022	NUM
ejpam-6699	555	7	.	.	PUNCT
ejpam-6699	556	1	[	[	X
ejpam-6699	556	2	28	28	NUM
ejpam-6699	556	3	]	]	X
ejpam-6699	556	4	x.	x.	PROPN
ejpam-6699	556	5	yang	yang	PROPN
ejpam-6699	556	6	,	,	PUNCT
ejpam-6699	556	7	l.	l.	PROPN
ejpam-6699	556	8	chang	chang	PROPN
ejpam-6699	556	9	,	,	PUNCT
ejpam-6699	556	10	g.	g.	PROPN
ejpam-6699	556	11	liu	liu	PROPN
ejpam-6699	556	12	,	,	PUNCT
ejpam-6699	556	13	and	and	CCONJ
ejpam-6699	556	14	g.	g.	PROPN
ejpam-6699	556	15	shen	shen	PROPN
ejpam-6699	556	16	.	.	PUNCT
ejpam-6699	557	1	stability	stability	NOUN
ejpam-6699	557	2	of	of	ADP
ejpam-6699	557	3	functional	functional	ADJ
ejpam-6699	557	4	equation	equation	NOUN
ejpam-6699	557	5	in	in	ADP
ejpam-6699	557	6	(	(	PUNCT
ejpam-6699	557	7	n	n	CCONJ
ejpam-6699	557	8	,	,	PUNCT
ejpam-6699	557	9	β)normed	β)normed	ADJ
ejpam-6699	557	10	spaces	space	NOUN
ejpam-6699	557	11	.	.	PUNCT
ejpam-6699	558	1	journal	journal	PROPN
ejpam-6699	558	2	of	of	ADP
ejpam-6699	558	3	inequalities	inequality	NOUN
ejpam-6699	558	4	and	and	CCONJ
ejpam-6699	558	5	applications	application	NOUN
ejpam-6699	558	6	,	,	PUNCT
ejpam-6699	558	7	2015(112):1–18	2015(112):1–18	NUM
ejpam-6699	558	8	,	,	PUNCT
ejpam-6699	558	9	2015	2015	NUM
ejpam-6699	558	10	.	.	PUNCT
ejpam-6699	559	1	[	[	X
ejpam-6699	559	2	29	29	NUM
ejpam-6699	559	3	]	]	PUNCT
ejpam-6699	559	4	s.	s.	PROPN
ejpam-6699	559	5	pinelas	pinelas	PROPN
ejpam-6699	559	6	,	,	PUNCT
ejpam-6699	559	7	v.	v.	CCONJ
ejpam-6699	559	8	govindan	govindan	NOUN
ejpam-6699	559	9	,	,	PUNCT
ejpam-6699	559	10	and	and	CCONJ
ejpam-6699	559	11	k.	k.	PROPN
ejpam-6699	559	12	tamilvanan	tamilvanan	PROPN
ejpam-6699	559	13	.	.	PUNCT
ejpam-6699	560	1	stability	stability	NOUN
ejpam-6699	560	2	of	of	ADP
ejpam-6699	560	3	a	a	DET
ejpam-6699	560	4	quartic	quartic	ADJ
ejpam-6699	560	5	functional	functional	ADJ
ejpam-6699	560	6	equation	equation	NOUN
ejpam-6699	560	7	.	.	PUNCT
ejpam-6699	561	1	journal	journal	NOUN
ejpam-6699	561	2	of	of	ADP
ejpam-6699	561	3	fixed	fix	VERB
ejpam-6699	561	4	point	point	NOUN
ejpam-6699	561	5	theory	theory	NOUN
ejpam-6699	561	6	and	and	CCONJ
ejpam-6699	561	7	applications	application	NOUN
ejpam-6699	561	8	,	,	PUNCT
ejpam-6699	561	9	2018(20):1–10	2018(20):1–10	NUM
ejpam-6699	561	10	,	,	PUNCT
ejpam-6699	561	11	2018	2018	NUM
ejpam-6699	561	12	.	.	PUNCT
ejpam-6699	562	1	[	[	X
ejpam-6699	562	2	30	30	NUM
ejpam-6699	562	3	]	]	X
ejpam-6699	562	4	l.	l.	PROPN
ejpam-6699	562	5	i.	i.	PROPN
ejpam-6699	562	6	cădariu	cădariu	PROPN
ejpam-6699	562	7	and	and	CCONJ
ejpam-6699	562	8	v.	v.	ADP
ejpam-6699	562	9	radu	radu	PROPN
ejpam-6699	562	10	.	.	PUNCT
ejpam-6699	562	11	fixed	fix	VERB
ejpam-6699	562	12	points	point	NOUN
ejpam-6699	562	13	and	and	CCONJ
ejpam-6699	562	14	the	the	DET
ejpam-6699	562	15	stability	stability	NOUN
ejpam-6699	562	16	of	of	ADP
ejpam-6699	562	17	quadratic	quadratic	ADJ
ejpam-6699	562	18	functional	functional	ADJ
ejpam-6699	562	19	equations	equation	NOUN
ejpam-6699	562	20	.	.	PUNCT
ejpam-6699	563	1	analele	analele	PROPN
ejpam-6699	563	2	universităţii	universităţii	AUX
ejpam-6699	563	3	din	din	VERB
ejpam-6699	563	4	timişoara	timişoara	NOUN
ejpam-6699	563	5	,	,	PUNCT
ejpam-6699	563	6	seria	seria	PROPN
ejpam-6699	563	7	matematică-informatică	matematică-informatică	NOUN
ejpam-6699	563	8	,	,	PUNCT
ejpam-6699	563	9	41:25–48	41:25–48	PROPN
ejpam-6699	563	10	,	,	PUNCT
ejpam-6699	563	11	2003	2003	NUM
ejpam-6699	563	12	.	.	PUNCT
ejpam-6699	564	1	[	[	X
ejpam-6699	564	2	31	31	NUM
ejpam-6699	564	3	]	]	SYM
ejpam-6699	564	4	s.-m	s.-m	PROPN
ejpam-6699	564	5	.	.	PUNCT
ejpam-6699	564	6	jung	jung	PROPN
ejpam-6699	564	7	.	.	PUNCT
ejpam-6699	565	1	hyers	hyer	NOUN
ejpam-6699	565	2	-	-	PUNCT
ejpam-6699	565	3	ulam	ulam	NOUN
ejpam-6699	565	4	-	-	PUNCT
ejpam-6699	565	5	rassias	rassias	PROPN
ejpam-6699	565	6	stability	stability	NOUN
ejpam-6699	565	7	of	of	ADP
ejpam-6699	565	8	functional	functional	ADJ
ejpam-6699	565	9	equations	equation	NOUN
ejpam-6699	565	10	in	in	ADP
ejpam-6699	565	11	connection	connection	NOUN
ejpam-6699	565	12	with	with	ADP
ejpam-6699	565	13	classical	classical	ADJ
ejpam-6699	565	14	inequalities	inequality	NOUN
ejpam-6699	565	15	.	.	PUNCT
ejpam-6699	566	1	nonlinear	nonlinear	ADJ
ejpam-6699	566	2	functional	functional	ADJ
ejpam-6699	566	3	analysis	analysis	NOUN
ejpam-6699	566	4	and	and	CCONJ
ejpam-6699	566	5	applications	application	NOUN
ejpam-6699	566	6	,	,	PUNCT
ejpam-6699	566	7	8:123–164	8:123–164	NOUN
ejpam-6699	566	8	,	,	PUNCT
ejpam-6699	566	9	2003	2003	NUM
ejpam-6699	566	10	.	.	PUNCT