id	sid	tid	token	lemma	pos
ejpam-6609	1	1	european	european	PROPN
ejpam-6609	1	2	journal	journal	PROPN
ejpam-6609	1	3	of	of	ADP
ejpam-6609	1	4	pure	pure	ADJ
ejpam-6609	1	5	and	and	CCONJ
ejpam-6609	1	6	applied	applied	ADJ
ejpam-6609	1	7	mathematics	mathematic	NOUN
ejpam-6609	1	8	2025	2025	NUM
ejpam-6609	1	9	,	,	PUNCT
ejpam-6609	1	10	vol	vol	NOUN
ejpam-6609	1	11	.	.	PROPN
ejpam-6609	1	12	18	18	NUM
ejpam-6609	1	13	,	,	PUNCT
ejpam-6609	1	14	issue	issue	NOUN
ejpam-6609	1	15	3	3	NUM
ejpam-6609	1	16	,	,	PUNCT
ejpam-6609	1	17	article	article	NOUN
ejpam-6609	1	18	number	number	NOUN
ejpam-6609	1	19	6609	6609	NUM
ejpam-6609	1	20	issn	issn	VERB
ejpam-6609	1	21	1307	1307	NUM
ejpam-6609	1	22	-	-	SYM
ejpam-6609	1	23	5543	5543	NUM
ejpam-6609	1	24	–	–	PUNCT
ejpam-6609	1	25	ejpam.com	ejpam.com	X
ejpam-6609	1	26	published	publish	VERB
ejpam-6609	1	27	by	by	ADP
ejpam-6609	1	28	new	new	PROPN
ejpam-6609	1	29	york	york	PROPN
ejpam-6609	1	30	business	business	PROPN
ejpam-6609	1	31	global	global	ADJ
ejpam-6609	1	32	triple	triple	ADJ
ejpam-6609	1	33	fixed	fix	VERB
ejpam-6609	1	34	-	-	PUNCT
ejpam-6609	1	35	point	point	NOUN
ejpam-6609	1	36	theorems	theorem	NOUN
ejpam-6609	1	37	:	:	PUNCT
ejpam-6609	1	38	generalized	generalize	VERB
ejpam-6609	1	39	analysis	analysis	NOUN
ejpam-6609	1	40	with	with	ADP
ejpam-6609	1	41	ai	ai	NOUN
ejpam-6609	1	42	/	/	SYM
ejpam-6609	1	43	crypto	crypto	NOUN
ejpam-6609	1	44	applications	application	NOUN
ejpam-6609	1	45	maha	maha	PROPN
ejpam-6609	1	46	noorwali1	noorwali1	PROPN
ejpam-6609	1	47	,	,	PUNCT
ejpam-6609	1	48	haitham	haitham	PROPN
ejpam-6609	1	49	qawaqneh2	qawaqneh2	PROPN
ejpam-6609	1	50	,	,	PUNCT
ejpam-6609	1	51	muhammad	muhammad	PROPN
ejpam-6609	1	52	tahir3	tahir3	PROPN
ejpam-6609	1	53	,	,	PUNCT
ejpam-6609	1	54	arif	arif	PROPN
ejpam-6609	1	55	mehmood3	mehmood3	PROPN
ejpam-6609	1	56	,	,	PUNCT
ejpam-6609	1	57	alaa	alaa	PROPN
ejpam-6609	1	58	m.	m.	PROPN
ejpam-6609	1	59	abd	abd	PROPN
ejpam-6609	1	60	el	el	PROPN
ejpam-6609	1	61	-	-	PROPN
ejpam-6609	1	62	latif4,∗	latif4,∗	PROPN
ejpam-6609	1	63	,	,	PUNCT
ejpam-6609	1	64	khaled	khaled	PROPN
ejpam-6609	1	65	a.	a.	NOUN
ejpam-6609	1	66	aldwoah5,∗	aldwoah5,∗	PROPN
ejpam-6609	1	67	,	,	PUNCT
ejpam-6609	1	68	cris	cris	PROPN
ejpam-6609	1	69	l.	l.	PROPN
ejpam-6609	1	70	armada6	armada6	PROPN
ejpam-6609	1	71	,	,	PUNCT
ejpam-6609	1	72	nurijam	nurijam	PROPN
ejpam-6609	1	73	hanna	hanna	PROPN
ejpam-6609	1	74	m.	m.	PROPN
ejpam-6609	1	75	mohammad7	mohammad7	PROPN
ejpam-6609	1	76	,	,	PUNCT
ejpam-6609	1	77	jamil	jamil	PROPN
ejpam-6609	1	78	j.	j.	PROPN
ejpam-6609	1	79	hamja7	hamja7	PROPN
ejpam-6609	1	80	1	1	NUM
ejpam-6609	1	81	department	department	NOUN
ejpam-6609	1	82	of	of	ADP
ejpam-6609	1	83	mathematics	mathematic	NOUN
ejpam-6609	1	84	,	,	PUNCT
ejpam-6609	1	85	king	king	PROPN
ejpam-6609	1	86	abdulaziz	abdulaziz	PROPN
ejpam-6609	1	87	university	university	PROPN
ejpam-6609	1	88	,	,	PUNCT
ejpam-6609	1	89	jeddah	jeddah	PROPN
ejpam-6609	1	90	,	,	PUNCT
ejpam-6609	1	91	saudi	saudi	PROPN
ejpam-6609	1	92	arabia	arabia	PROPN
ejpam-6609	1	93	2	2	NUM
ejpam-6609	1	94	al	al	PROPN
ejpam-6609	1	95	-	-	PUNCT
ejpam-6609	1	96	zaytoonah	zaytoonah	PROPN
ejpam-6609	1	97	university	university	PROPN
ejpam-6609	1	98	of	of	ADP
ejpam-6609	1	99	jordan	jordan	PROPN
ejpam-6609	1	100	,	,	PUNCT
ejpam-6609	1	101	amman	amman	PROPN
ejpam-6609	1	102	11733	11733	NUM
ejpam-6609	1	103	,	,	PUNCT
ejpam-6609	1	104	jordan	jordan	PROPN
ejpam-6609	1	105	3	3	NUM
ejpam-6609	1	106	department	department	PROPN
ejpam-6609	1	107	of	of	ADP
ejpam-6609	1	108	mathematics	mathematics	PROPN
ejpam-6609	1	109	,	,	PUNCT
ejpam-6609	1	110	institute	institute	PROPN
ejpam-6609	1	111	of	of	ADP
ejpam-6609	1	112	numerical	numerical	PROPN
ejpam-6609	1	113	sciences	sciences	PROPN
ejpam-6609	1	114	,	,	PUNCT
ejpam-6609	1	115	gomal	gomal	ADJ
ejpam-6609	1	116	university	university	NOUN
ejpam-6609	1	117	,	,	PUNCT
ejpam-6609	1	118	dera	dera	PROPN
ejpam-6609	1	119	ismail	ismail	PROPN
ejpam-6609	1	120	khan	khan	PROPN
ejpam-6609	1	121	29050	29050	NUM
ejpam-6609	1	122	,	,	PUNCT
ejpam-6609	1	123	kpk	kpk	PROPN
ejpam-6609	1	124	,	,	PUNCT
ejpam-6609	1	125	pakistan	pakistan	PROPN
ejpam-6609	1	126	4	4	NUM
ejpam-6609	1	127	department	department	NOUN
ejpam-6609	1	128	of	of	ADP
ejpam-6609	1	129	mathematics	mathematic	NOUN
ejpam-6609	1	130	,	,	PUNCT
ejpam-6609	1	131	college	college	NOUN
ejpam-6609	1	132	of	of	ADP
ejpam-6609	1	133	science	science	NOUN
ejpam-6609	1	134	,	,	PUNCT
ejpam-6609	1	135	northern	northern	ADJ
ejpam-6609	1	136	border	border	NOUN
ejpam-6609	1	137	university	university	NOUN
ejpam-6609	1	138	,	,	PUNCT
ejpam-6609	1	139	arar	arar	NOUN
ejpam-6609	1	140	91431	91431	NUM
ejpam-6609	1	141	,	,	PUNCT
ejpam-6609	1	142	saudi	saudi	PROPN
ejpam-6609	1	143	arabia	arabia	PROPN
ejpam-6609	1	144	5	5	NUM
ejpam-6609	1	145	department	department	NOUN
ejpam-6609	1	146	of	of	ADP
ejpam-6609	1	147	mathematics	mathematic	NOUN
ejpam-6609	1	148	,	,	PUNCT
ejpam-6609	1	149	faculty	faculty	NOUN
ejpam-6609	1	150	of	of	ADP
ejpam-6609	1	151	science	science	NOUN
ejpam-6609	1	152	,	,	PUNCT
ejpam-6609	1	153	islamic	islamic	PROPN
ejpam-6609	1	154	university	university	PROPN
ejpam-6609	1	155	of	of	ADP
ejpam-6609	1	156	madinah	madinah	PROPN
ejpam-6609	1	157	,	,	PUNCT
ejpam-6609	1	158	medinah	medinah	PROPN
ejpam-6609	1	159	,	,	PUNCT
ejpam-6609	1	160	saudi	saudi	PROPN
ejpam-6609	1	161	arabia	arabia	PROPN
ejpam-6609	1	162	6	6	NUM
ejpam-6609	1	163	vietnam	vietnam	PROPN
ejpam-6609	1	164	national	national	PROPN
ejpam-6609	1	165	university	university	PROPN
ejpam-6609	1	166	ho	ho	PROPN
ejpam-6609	1	167	chi	chi	PROPN
ejpam-6609	1	168	minh	minh	PROPN
ejpam-6609	1	169	city	city	PROPN
ejpam-6609	1	170	,	,	PUNCT
ejpam-6609	1	171	linh	linh	PROPN
ejpam-6609	1	172	trung	trung	VERB
ejpam-6609	1	173	and	and	CCONJ
ejpam-6609	1	174	department	department	NOUN
ejpam-6609	1	175	of	of	ADP
ejpam-6609	1	176	applied	apply	VERB
ejpam-6609	1	177	mathematics	mathematic	NOUN
ejpam-6609	1	178	,	,	PUNCT
ejpam-6609	1	179	faculty	faculty	NOUN
ejpam-6609	1	180	of	of	ADP
ejpam-6609	1	181	applied	apply	VERB
ejpam-6609	1	182	science	science	NOUN
ejpam-6609	1	183	,	,	PUNCT
ejpam-6609	1	184	ho	ho	PROPN
ejpam-6609	1	185	chi	chi	PROPN
ejpam-6609	1	186	minh	minh	PROPN
ejpam-6609	1	187	city	city	PROPN
ejpam-6609	1	188	university	university	PROPN
ejpam-6609	1	189	of	of	ADP
ejpam-6609	1	190	technology	technology	NOUN
ejpam-6609	1	191	(	(	PUNCT
ejpam-6609	1	192	hcmut	hcmut	ADJ
ejpam-6609	1	193	)	)	PUNCT
ejpam-6609	1	194	,	,	PUNCT
ejpam-6609	1	195	268	268	NUM
ejpam-6609	1	196	,	,	PUNCT
ejpam-6609	1	197	ly	ly	ADP
ejpam-6609	1	198	thuong	thuong	PROPN
ejpam-6609	1	199	kiet	kiet	PROPN
ejpam-6609	1	200	,	,	PUNCT
ejpam-6609	1	201	district	district	NOUN
ejpam-6609	1	202	10	10	NUM
ejpam-6609	1	203	,	,	PUNCT
ejpam-6609	1	204	ward	ward	NOUN
ejpam-6609	1	205	14	14	NUM
ejpam-6609	1	206	,	,	PUNCT
ejpam-6609	1	207	ho	ho	PROPN
ejpam-6609	1	208	chi	chi	PROPN
ejpam-6609	1	209	minh	minh	PROPN
ejpam-6609	1	210	city	city	PROPN
ejpam-6609	1	211	,	,	PUNCT
ejpam-6609	1	212	vietnam	vietnam	PROPN
ejpam-6609	1	213	7	7	NUM
ejpam-6609	1	214	department	department	NOUN
ejpam-6609	1	215	of	of	ADP
ejpam-6609	1	216	mathematics	mathematic	NOUN
ejpam-6609	1	217	,	,	PUNCT
ejpam-6609	1	218	college	college	NOUN
ejpam-6609	1	219	of	of	ADP
ejpam-6609	1	220	arts	art	NOUN
ejpam-6609	1	221	and	and	CCONJ
ejpam-6609	1	222	sciences	science	NOUN
ejpam-6609	1	223	,	,	PUNCT
ejpam-6609	1	224	mindanao	mindanao	PROPN
ejpam-6609	1	225	state	state	PROPN
ejpam-6609	1	226	university	university	PROPN
ejpam-6609	1	227	tawitawi	tawitawi	PROPN
ejpam-6609	1	228	college	college	PROPN
ejpam-6609	1	229	of	of	ADP
ejpam-6609	1	230	technology	technology	NOUN
ejpam-6609	1	231	and	and	CCONJ
ejpam-6609	1	232	oceanography	oceanography	NOUN
ejpam-6609	1	233	,	,	PUNCT
ejpam-6609	1	234	7500	7500	NUM
ejpam-6609	1	235	philippines	philippine	NOUN
ejpam-6609	1	236	abstract	abstract	ADJ
ejpam-6609	1	237	.	.	PUNCT
ejpam-6609	2	1	this	this	DET
ejpam-6609	2	2	study	study	NOUN
ejpam-6609	2	3	uses	use	VERB
ejpam-6609	2	4	important	important	ADJ
ejpam-6609	2	5	findings	finding	NOUN
ejpam-6609	2	6	from	from	ADP
ejpam-6609	2	7	hilbert	hilbert	NOUN
ejpam-6609	2	8	space	space	NOUN
ejpam-6609	2	9	(	(	PUNCT
ejpam-6609	2	10	hs	hs	NOUN
ejpam-6609	2	11	)	)	PUNCT
ejpam-6609	2	12	theory	theory	NOUN
ejpam-6609	2	13	to	to	PART
ejpam-6609	2	14	suggest	suggest	VERB
ejpam-6609	2	15	new	new	ADJ
ejpam-6609	2	16	fixedpoint	fixedpoint	NOUN
ejpam-6609	2	17	theorems	theorem	NOUN
ejpam-6609	2	18	for	for	ADP
ejpam-6609	2	19	three	three	NUM
ejpam-6609	2	20	self	self	NOUN
ejpam-6609	2	21	-	-	PUNCT
ejpam-6609	2	22	mappings	mapping	NOUN
ejpam-6609	2	23	(	(	PUNCT
ejpam-6609	2	24	3	3	NUM
ejpam-6609	2	25	-	-	PUNCT
ejpam-6609	2	26	sms	sms	NOUN
ejpam-6609	2	27	)	)	PUNCT
ejpam-6609	2	28	that	that	PRON
ejpam-6609	2	29	apply	apply	VERB
ejpam-6609	2	30	to	to	ADP
ejpam-6609	2	31	generalized	generalized	ADJ
ejpam-6609	2	32	inner	inner	ADJ
ejpam-6609	2	33	product	product	NOUN
ejpam-6609	2	34	spaces	space	NOUN
ejpam-6609	2	35	(	(	PUNCT
ejpam-6609	2	36	gips	gips	PROPN
ejpam-6609	2	37	)	)	PUNCT
ejpam-6609	2	38	.	.	PUNCT
ejpam-6609	3	1	we	we	PRON
ejpam-6609	3	2	made	make	VERB
ejpam-6609	3	3	it	it	PRON
ejpam-6609	3	4	apparent	apparent	ADJ
ejpam-6609	3	5	when	when	SCONJ
ejpam-6609	3	6	common	common	ADJ
ejpam-6609	3	7	fixed	fix	VERB
ejpam-6609	3	8	points	point	NOUN
ejpam-6609	3	9	(	(	PUNCT
ejpam-6609	3	10	cfps	cfps	NOUN
ejpam-6609	3	11	)	)	PUNCT
ejpam-6609	3	12	can	can	AUX
ejpam-6609	3	13	exist	exist	VERB
ejpam-6609	3	14	and	and	CCONJ
ejpam-6609	3	15	be	be	AUX
ejpam-6609	3	16	unique	unique	ADJ
ejpam-6609	3	17	,	,	PUNCT
ejpam-6609	3	18	even	even	ADV
ejpam-6609	3	19	when	when	SCONJ
ejpam-6609	3	20	the	the	DET
ejpam-6609	3	21	contraction	contraction	NOUN
ejpam-6609	3	22	criteria	criterion	NOUN
ejpam-6609	3	23	are	be	AUX
ejpam-6609	3	24	n’t	not	PART
ejpam-6609	3	25	as	as	ADV
ejpam-6609	3	26	strict	strict	ADJ
ejpam-6609	3	27	.	.	PUNCT
ejpam-6609	4	1	this	this	DET
ejpam-6609	4	2	improvement	improvement	NOUN
ejpam-6609	4	3	makes	make	VERB
ejpam-6609	4	4	the	the	DET
ejpam-6609	4	5	theory	theory	NOUN
ejpam-6609	4	6	better	well	ADV
ejpam-6609	4	7	for	for	ADP
ejpam-6609	4	8	systems	system	NOUN
ejpam-6609	4	9	with	with	ADP
ejpam-6609	4	10	more	more	ADJ
ejpam-6609	4	11	than	than	ADP
ejpam-6609	4	12	one	one	NUM
ejpam-6609	4	13	operator	operator	NOUN
ejpam-6609	4	14	.	.	PUNCT
ejpam-6609	5	1	our	our	PRON
ejpam-6609	5	2	results	result	NOUN
ejpam-6609	5	3	are	be	AUX
ejpam-6609	5	4	important	important	ADJ
ejpam-6609	5	5	in	in	ADP
ejpam-6609	5	6	two	two	NUM
ejpam-6609	5	7	main	main	ADJ
ejpam-6609	5	8	areas	area	NOUN
ejpam-6609	5	9	:	:	PUNCT
ejpam-6609	5	10	(	(	PUNCT
ejpam-6609	5	11	1	1	X
ejpam-6609	5	12	)	)	PUNCT
ejpam-6609	5	13	artificial	artificial	ADJ
ejpam-6609	5	14	intelligence	intelligence	NOUN
ejpam-6609	5	15	(	(	PUNCT
ejpam-6609	5	16	ai	ai	NOUN
ejpam-6609	5	17	)	)	PUNCT
ejpam-6609	5	18	,	,	PUNCT
ejpam-6609	5	19	where	where	SCONJ
ejpam-6609	5	20	we	we	PRON
ejpam-6609	5	21	use	use	VERB
ejpam-6609	5	22	fixed	fix	VERB
ejpam-6609	5	23	-	-	PUNCT
ejpam-6609	5	24	point	point	NOUN
ejpam-6609	5	25	analysis	analysis	NOUN
ejpam-6609	5	26	to	to	PART
ejpam-6609	5	27	make	make	VERB
ejpam-6609	5	28	sure	sure	ADJ
ejpam-6609	5	29	that	that	SCONJ
ejpam-6609	5	30	deep	deep	ADJ
ejpam-6609	5	31	equilibrium	equilibrium	NOUN
ejpam-6609	5	32	models	model	NOUN
ejpam-6609	5	33	(	(	PUNCT
ejpam-6609	5	34	dems	dem	NOUN
ejpam-6609	5	35	)	)	PUNCT
ejpam-6609	5	36	will	will	AUX
ejpam-6609	5	37	work	work	VERB
ejpam-6609	5	38	correctly	correctly	ADV
ejpam-6609	5	39	,	,	PUNCT
ejpam-6609	5	40	and	and	CCONJ
ejpam-6609	5	41	(	(	PUNCT
ejpam-6609	5	42	2	2	X
ejpam-6609	5	43	)	)	PUNCT
ejpam-6609	5	44	cryptography	cryptography	NOUN
ejpam-6609	5	45	(	(	PUNCT
ejpam-6609	5	46	crypto	crypto	NOUN
ejpam-6609	5	47	)	)	PUNCT
ejpam-6609	5	48	,	,	PUNCT
ejpam-6609	5	49	where	where	SCONJ
ejpam-6609	5	50	we	we	PRON
ejpam-6609	5	51	use	use	VERB
ejpam-6609	5	52	the	the	DET
ejpam-6609	5	53	mathematical	mathematical	ADJ
ejpam-6609	5	54	properties	property	NOUN
ejpam-6609	5	55	of	of	ADP
ejpam-6609	5	56	inner	inner	ADJ
ejpam-6609	5	57	product	product	NOUN
ejpam-6609	5	58	spaces	space	VERB
ejpam-6609	5	59	to	to	PART
ejpam-6609	5	60	make	make	VERB
ejpam-6609	5	61	new	new	ADJ
ejpam-6609	5	62	lattice	lattice	NOUN
ejpam-6609	5	63	-	-	PUNCT
ejpam-6609	5	64	based	base	VERB
ejpam-6609	5	65	designs	design	NOUN
ejpam-6609	5	66	for	for	ADP
ejpam-6609	5	67	post	post	ADJ
ejpam-6609	5	68	-	-	ADJ
ejpam-6609	5	69	quantum	quantum	ADJ
ejpam-6609	5	70	protocols	protocol	NOUN
ejpam-6609	5	71	(	(	PUNCT
ejpam-6609	5	72	pqps	pqps	ADJ
ejpam-6609	5	73	)	)	PUNCT
ejpam-6609	5	74	.	.	PUNCT
ejpam-6609	6	1	real	real	ADJ
ejpam-6609	6	2	-	-	PUNCT
ejpam-6609	6	3	world	world	NOUN
ejpam-6609	6	4	examples	example	NOUN
ejpam-6609	6	5	from	from	ADP
ejpam-6609	6	6	nonlinear	nonlinear	ADJ
ejpam-6609	6	7	analysis	analysis	NOUN
ejpam-6609	6	8	and	and	CCONJ
ejpam-6609	6	9	computational	computational	ADJ
ejpam-6609	6	10	proof	proof	NOUN
ejpam-6609	6	11	of	of	ADP
ejpam-6609	6	12	the	the	DET
ejpam-6609	6	13	presented	present	VERB
ejpam-6609	6	14	theorems	theorem	NOUN
ejpam-6609	6	15	support	support	VERB
ejpam-6609	6	16	the	the	DET
ejpam-6609	6	17	theoretical	theoretical	ADJ
ejpam-6609	6	18	contributions	contribution	NOUN
ejpam-6609	6	19	.	.	PUNCT
ejpam-6609	7	1	this	this	DET
ejpam-6609	7	2	work	work	NOUN
ejpam-6609	7	3	combines	combine	VERB
ejpam-6609	7	4	advanced	advanced	ADJ
ejpam-6609	7	5	functional	functional	ADJ
ejpam-6609	7	6	analysis	analysis	NOUN
ejpam-6609	7	7	with	with	ADP
ejpam-6609	7	8	modern	modern	ADJ
ejpam-6609	7	9	problems	problem	NOUN
ejpam-6609	7	10	in	in	ADP
ejpam-6609	7	11	machine	machine	NOUN
ejpam-6609	7	12	learning	learning	NOUN
ejpam-6609	7	13	(	(	PUNCT
ejpam-6609	7	14	ml	ml	NOUN
ejpam-6609	7	15	)	)	PUNCT
ejpam-6609	7	16	and	and	CCONJ
ejpam-6609	7	17	cybersecurity	cybersecurity	NOUN
ejpam-6609	7	18	,	,	PUNCT
ejpam-6609	7	19	giving	give	VERB
ejpam-6609	7	20	it	it	PRON
ejpam-6609	7	21	both	both	DET
ejpam-6609	7	22	mathematical	mathematical	ADJ
ejpam-6609	7	23	depth	depth	NOUN
ejpam-6609	7	24	and	and	CCONJ
ejpam-6609	7	25	real	real	ADJ
ejpam-6609	7	26	-	-	PUNCT
ejpam-6609	7	27	world	world	NOUN
ejpam-6609	7	28	usefulness	usefulness	NOUN
ejpam-6609	7	29	.	.	PUNCT
ejpam-6609	8	1	2020	2020	NUM
ejpam-6609	8	2	mathematics	mathematic	NOUN
ejpam-6609	8	3	subject	subject	NOUN
ejpam-6609	8	4	classifications	classification	NOUN
ejpam-6609	8	5	:	:	PUNCT
ejpam-6609	8	6	54a05	54a05	NUM
ejpam-6609	8	7	key	key	ADJ
ejpam-6609	8	8	words	word	NOUN
ejpam-6609	8	9	and	and	CCONJ
ejpam-6609	8	10	phrases	phrase	NOUN
ejpam-6609	8	11	:	:	PUNCT
ejpam-6609	8	12	three	three	NUM
ejpam-6609	8	13	self	self	NOUN
ejpam-6609	8	14	mappings	mapping	NOUN
ejpam-6609	8	15	,	,	PUNCT
ejpam-6609	8	16	generalized	generalize	VERB
ejpam-6609	8	17	inner	inner	ADJ
ejpam-6609	8	18	product	product	NOUN
ejpam-6609	8	19	spaces	space	NOUN
ejpam-6609	8	20	,	,	PUNCT
ejpam-6609	8	21	common	common	ADJ
ejpam-6609	8	22	fixed	fix	VERB
ejpam-6609	8	23	points	point	NOUN
ejpam-6609	8	24	,	,	PUNCT
ejpam-6609	8	25	deep	deep	ADJ
ejpam-6609	8	26	equilibrium	equilibrium	NOUN
ejpam-6609	8	27	models	model	NOUN
ejpam-6609	8	28	,	,	PUNCT
ejpam-6609	8	29	post	post	VERB
ejpam-6609	8	30	quantum	quantum	ADJ
ejpam-6609	8	31	protocols	protocol	NOUN
ejpam-6609	8	32	1	1	NUM
ejpam-6609	8	33	.	.	PUNCT
ejpam-6609	9	1	introduction	introduction	NOUN
ejpam-6609	9	2	fixed	fix	VERB
ejpam-6609	9	3	point	point	NOUN
ejpam-6609	9	4	theory	theory	NOUN
ejpam-6609	9	5	(	(	PUNCT
ejpam-6609	9	6	fpt	fpt	PROPN
ejpam-6609	9	7	)	)	PUNCT
ejpam-6609	9	8	has	have	AUX
ejpam-6609	9	9	grown	grow	VERB
ejpam-6609	9	10	considerably	considerably	ADV
ejpam-6609	9	11	since	since	SCONJ
ejpam-6609	9	12	it	it	PRON
ejpam-6609	9	13	started	start	VERB
ejpam-6609	9	14	with	with	ADP
ejpam-6609	9	15	the	the	DET
ejpam-6609	9	16	basic	basic	ADJ
ejpam-6609	9	17	contraction	contraction	NOUN
ejpam-6609	9	18	principle	principle	NOUN
ejpam-6609	9	19	,	,	PUNCT
ejpam-6609	9	20	which	which	PRON
ejpam-6609	9	21	indicated	indicate	VERB
ejpam-6609	9	22	that	that	SCONJ
ejpam-6609	9	23	fixed	fix	VERB
ejpam-6609	9	24	points	point	NOUN
ejpam-6609	9	25	exist	exist	VERB
ejpam-6609	9	26	in	in	ADP
ejpam-6609	9	27	complete	complete	ADJ
ejpam-6609	9	28	metric	metric	ADJ
ejpam-6609	9	29	spaces	space	NOUN
ejpam-6609	9	30	[	[	X
ejpam-6609	9	31	1	1	NUM
ejpam-6609	9	32	]	]	PUNCT
ejpam-6609	9	33	and	and	CCONJ
ejpam-6609	9	34	that	that	SCONJ
ejpam-6609	9	35	each	each	DET
ejpam-6609	9	36	one	one	NOUN
ejpam-6609	9	37	is	be	AUX
ejpam-6609	9	38	unique	unique	ADJ
ejpam-6609	9	39	for	for	ADP
ejpam-6609	9	40	single	single	ADJ
ejpam-6609	9	41	-	-	PUNCT
ejpam-6609	9	42	valued	value	VERB
ejpam-6609	9	43	mappings	mapping	NOUN
ejpam-6609	10	1	[	[	X
ejpam-6609	10	2	2	2	NUM
ejpam-6609	10	3	]	]	PUNCT
ejpam-6609	10	4	.	.	PUNCT
ejpam-6609	11	1	the	the	DET
ejpam-6609	11	2	field	field	NOUN
ejpam-6609	11	3	has	have	AUX
ejpam-6609	11	4	significantly	significantly	ADV
ejpam-6609	11	5	expanded	expand	VERB
ejpam-6609	11	6	to	to	PART
ejpam-6609	11	7	encompass	encompass	VERB
ejpam-6609	11	8	∗corresponding	∗corresponde	VERB
ejpam-6609	11	9	author	author	NOUN
ejpam-6609	11	10	.	.	PUNCT
ejpam-6609	12	1	doi	doi	NOUN
ejpam-6609	12	2	:	:	PUNCT
ejpam-6609	12	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6609	https://doi.org/10.29020/nybg.ejpam.v18i3.6609	NOUN
ejpam-6609	12	4	email	email	NOUN
ejpam-6609	12	5	addresses	address	NOUN
ejpam-6609	12	6	:	:	PUNCT
ejpam-6609	12	7	mnorwali@kau.edu.sa	mnorwali@kau.edu.sa	PROPN
ejpam-6609	12	8	(	(	PUNCT
ejpam-6609	12	9	m.	m.	PROPN
ejpam-6609	12	10	noorwali	noorwali	PROPN
ejpam-6609	12	11	)	)	PUNCT
ejpam-6609	12	12	,	,	PUNCT
ejpam-6609	12	13	h.alqawaqneh@zuj.edu.jo	h.alqawaqneh@zuj.edu.jo	PROPN
ejpam-6609	12	14	(	(	PUNCT
ejpam-6609	12	15	h.	h.	PROPN
ejpam-6609	12	16	alqawaqneh	alqawaqneh	PROPN
ejpam-6609	12	17	)	)	PUNCT
ejpam-6609	12	18	,	,	PUNCT
ejpam-6609	12	19	tahirkhanbaloch30@gmail.com	tahirkhanbaloch30@gmail.com	X
ejpam-6609	12	20	(	(	PUNCT
ejpam-6609	12	21	m.	m.	PROPN
ejpam-6609	12	22	tahir	tahir	PROPN
ejpam-6609	12	23	)	)	PUNCT
ejpam-6609	12	24	,	,	PUNCT
ejpam-6609	12	25	mehdaniyal@gmail.com	mehdaniyal@gmail.com	X
ejpam-6609	12	26	(	(	PUNCT
ejpam-6609	12	27	a.	a.	PROPN
ejpam-6609	12	28	mehmood	mehmood	PROPN
ejpam-6609	12	29	)	)	PUNCT
ejpam-6609	12	30	,	,	PUNCT
ejpam-6609	12	31	alaa.ali@nbu.edu.sa	alaa.ali@nbu.edu.sa	PROPN
ejpam-6609	12	32	(	(	PUNCT
ejpam-6609	12	33	a.	a.	NOUN
ejpam-6609	12	34	m.	m.	PROPN
ejpam-6609	12	35	abd	abd	PROPN
ejpam-6609	12	36	el	el	PROPN
ejpam-6609	12	37	-	-	PROPN
ejpam-6609	12	38	latif	latif	PROPN
ejpam-6609	12	39	)	)	PUNCT
ejpam-6609	12	40	,	,	PUNCT
ejpam-6609	12	41	aldwoah@yahoo.com	aldwoah@yahoo.com	X
ejpam-6609	12	42	(	(	PUNCT
ejpam-6609	12	43	k.	k.	PROPN
ejpam-6609	12	44	a.	a.	PROPN
ejpam-6609	12	45	aldwoah	aldwoah	PROPN
ejpam-6609	12	46	)	)	PUNCT
ejpam-6609	12	47	,	,	PUNCT
ejpam-6609	12	48	cris.armada@hcmut.edu.vn	cris.armada@hcmut.edu.vn	X
ejpam-6609	12	49	(	(	PUNCT
ejpam-6609	12	50	c.	c.	PROPN
ejpam-6609	12	51	l.	l.	PROPN
ejpam-6609	12	52	armada	armada	PROPN
ejpam-6609	12	53	)	)	PUNCT
ejpam-6609	12	54	,	,	PUNCT
ejpam-6609	12	55	jamilhamja@msutawi-tawi.edu.ph	jamilhamja@msutawi-tawi.edu.ph	PROPN
ejpam-6609	12	56	(	(	PUNCT
ejpam-6609	12	57	j.	j.	PROPN
ejpam-6609	12	58	j.	j.	PROPN
ejpam-6609	12	59	hamja	hamja	PROPN
ejpam-6609	12	60	)	)	PUNCT
ejpam-6609	12	61	,	,	PUNCT
ejpam-6609	12	62	hannamohammad@msutawi-tawi.edu.ph	hannamohammad@msutawi-tawi.edu.ph	PROPN
ejpam-6609	12	63	(	(	PUNCT
ejpam-6609	12	64	n.	n.	PROPN
ejpam-6609	12	65	h.	h.	PROPN
ejpam-6609	12	66	m.	m.	PROPN
ejpam-6609	12	67	mohammad	mohammad	PROPN
ejpam-6609	12	68	)	)	PUNCT
ejpam-6609	12	69	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6609	13	1	1	1	NUM
ejpam-6609	13	2	copyright	copyright	NOUN
ejpam-6609	13	3	:	:	PUNCT
ejpam-6609	13	4	©	©	PROPN
ejpam-6609	13	5	2025	2025	NUM
ejpam-6609	13	6	the	the	DET
ejpam-6609	13	7	author(s	author(s	NOUN
ejpam-6609	13	8	)	)	PUNCT
ejpam-6609	13	9	.	.	PUNCT
ejpam-6609	14	1	(	(	PUNCT
ejpam-6609	14	2	cc	cc	NOUN
ejpam-6609	14	3	by	by	ADP
ejpam-6609	14	4	-	-	PUNCT
ejpam-6609	14	5	nc	nc	PROPN
ejpam-6609	14	6	4.0	4.0	NUM
ejpam-6609	14	7	)	)	PUNCT
ejpam-6609	14	8	m.	m.	NOUN
ejpam-6609	14	9	noorwali	noorwali	PROPN
ejpam-6609	14	10	et	et	PROPN
ejpam-6609	14	11	al	al	PROPN
ejpam-6609	14	12	.	.	PUNCT
ejpam-6609	14	13	/	/	SYM
ejpam-6609	14	14	eur	eur	PROPN
ejpam-6609	14	15	.	.	PUNCT
ejpam-6609	15	1	j.	j.	PROPN
ejpam-6609	15	2	pure	pure	PROPN
ejpam-6609	15	3	appl	appl	PROPN
ejpam-6609	15	4	.	.	PROPN
ejpam-6609	15	5	math	math	PROPN
ejpam-6609	15	6	,	,	PUNCT
ejpam-6609	15	7	18	18	NUM
ejpam-6609	15	8	(	(	PUNCT
ejpam-6609	15	9	3	3	NUM
ejpam-6609	15	10	)	)	PUNCT
ejpam-6609	15	11	(	(	PUNCT
ejpam-6609	15	12	2025	2025	NUM
ejpam-6609	15	13	)	)	PUNCT
ejpam-6609	15	14	,	,	PUNCT
ejpam-6609	15	15	6609	6609	NUM
ejpam-6609	15	16	2	2	NUM
ejpam-6609	15	17	of	of	ADP
ejpam-6609	15	18	17	17	NUM
ejpam-6609	15	19	various	various	ADJ
ejpam-6609	15	20	generalizations	generalization	NOUN
ejpam-6609	15	21	and	and	CCONJ
ejpam-6609	15	22	expansions	expansion	NOUN
ejpam-6609	15	23	.	.	PUNCT
ejpam-6609	16	1	some	some	PRON
ejpam-6609	16	2	of	of	ADP
ejpam-6609	16	3	these	these	DET
ejpam-6609	16	4	studies	study	NOUN
ejpam-6609	16	5	explore	explore	VERB
ejpam-6609	16	6	mappings	mapping	NOUN
ejpam-6609	16	7	that	that	PRON
ejpam-6609	16	8	follow	follow	VERB
ejpam-6609	16	9	different	different	ADJ
ejpam-6609	16	10	contraction	contraction	NOUN
ejpam-6609	16	11	rules	rule	NOUN
ejpam-6609	16	12	,	,	PUNCT
ejpam-6609	16	13	identify	identify	VERB
ejpam-6609	16	14	fixed	fix	VERB
ejpam-6609	16	15	points	point	NOUN
ejpam-6609	16	16	for	for	ADP
ejpam-6609	16	17	multi	multi	ADJ
ejpam-6609	16	18	-	-	ADJ
ejpam-6609	16	19	valued	value	VERB
ejpam-6609	16	20	operators	operator	NOUN
ejpam-6609	16	21	without	without	ADP
ejpam-6609	16	22	requiring	require	VERB
ejpam-6609	16	23	compactness	compactness	NOUN
ejpam-6609	16	24	[	[	X
ejpam-6609	16	25	3	3	NUM
ejpam-6609	16	26	]	]	PUNCT
ejpam-6609	16	27	,	,	PUNCT
ejpam-6609	16	28	and	and	CCONJ
ejpam-6609	16	29	apply	apply	VERB
ejpam-6609	16	30	traditional	traditional	ADJ
ejpam-6609	16	31	results	result	NOUN
ejpam-6609	16	32	in	in	ADP
ejpam-6609	16	33	more	more	ADJ
ejpam-6609	16	34	complex	complex	ADJ
ejpam-6609	16	35	environments	environment	NOUN
ejpam-6609	16	36	like	like	ADP
ejpam-6609	16	37	cone	cone	NOUN
ejpam-6609	16	38	metric	metric	ADJ
ejpam-6609	16	39	spaces	space	NOUN
ejpam-6609	16	40	[	[	X
ejpam-6609	16	41	4	4	X
ejpam-6609	16	42	]	]	PUNCT
ejpam-6609	16	43	and	and	CCONJ
ejpam-6609	16	44	modular	modular	ADJ
ejpam-6609	16	45	metric	metric	ADJ
ejpam-6609	16	46	spaces	space	NOUN
ejpam-6609	16	47	[	[	X
ejpam-6609	16	48	5	5	NUM
ejpam-6609	16	49	]	]	PUNCT
ejpam-6609	16	50	.	.	PUNCT
ejpam-6609	17	1	the	the	DET
ejpam-6609	17	2	emergence	emergence	NOUN
ejpam-6609	17	3	of	of	ADP
ejpam-6609	17	4	generalized	generalized	ADJ
ejpam-6609	17	5	metric	metric	ADJ
ejpam-6609	17	6	spaces	space	NOUN
ejpam-6609	17	7	(	(	PUNCT
ejpam-6609	17	8	gmspaces	gmspace	NOUN
ejpam-6609	17	9	)	)	PUNCT
ejpam-6609	17	10	represented	represent	VERB
ejpam-6609	17	11	a	a	DET
ejpam-6609	17	12	significant	significant	ADJ
ejpam-6609	17	13	advancement	advancement	NOUN
ejpam-6609	17	14	in	in	ADP
ejpam-6609	17	15	the	the	DET
ejpam-6609	17	16	discipline	discipline	NOUN
ejpam-6609	17	17	[	[	X
ejpam-6609	17	18	6	6	NUM
ejpam-6609	17	19	]	]	PUNCT
ejpam-6609	17	20	.	.	PUNCT
ejpam-6609	18	1	it	it	PRON
ejpam-6609	18	2	brought	bring	VERB
ejpam-6609	18	3	in	in	ADP
ejpam-6609	18	4	new	new	ADJ
ejpam-6609	18	5	types	type	NOUN
ejpam-6609	18	6	of	of	ADP
ejpam-6609	18	7	single	single	ADV
ejpam-6609	18	8	-	-	PUNCT
ejpam-6609	18	9	valued	value	VERB
ejpam-6609	18	10	contractive	contractive	ADJ
ejpam-6609	18	11	mappings	mapping	NOUN
ejpam-6609	19	1	[	[	X
ejpam-6609	19	2	7	7	NUM
ejpam-6609	19	3	]	]	PUNCT
ejpam-6609	19	4	,	,	PUNCT
ejpam-6609	19	5	showed	show	VERB
ejpam-6609	19	6	more	more	ADJ
ejpam-6609	19	7	ways	way	NOUN
ejpam-6609	19	8	to	to	PART
ejpam-6609	19	9	find	find	VERB
ejpam-6609	19	10	solutions	solution	NOUN
ejpam-6609	19	11	in	in	ADP
ejpam-6609	19	12	unusual	unusual	ADJ
ejpam-6609	19	13	metric	metric	ADJ
ejpam-6609	19	14	situations	situation	NOUN
ejpam-6609	19	15	,	,	PUNCT
ejpam-6609	19	16	and	and	CCONJ
ejpam-6609	19	17	allowed	allow	VERB
ejpam-6609	19	18	the	the	DET
ejpam-6609	19	19	study	study	NOUN
ejpam-6609	19	20	of	of	ADP
ejpam-6609	19	21	mappings	mapping	NOUN
ejpam-6609	19	22	that	that	PRON
ejpam-6609	19	23	do	do	AUX
ejpam-6609	19	24	not	not	PART
ejpam-6609	19	25	follow	follow	VERB
ejpam-6609	19	26	the	the	DET
ejpam-6609	19	27	usual	usual	ADJ
ejpam-6609	19	28	order	order	NOUN
ejpam-6609	19	29	.	.	PUNCT
ejpam-6609	20	1	more	more	ADJ
ejpam-6609	20	2	research	research	NOUN
ejpam-6609	20	3	has	have	AUX
ejpam-6609	20	4	been	be	AUX
ejpam-6609	20	5	done	do	VERB
ejpam-6609	20	6	on	on	ADP
ejpam-6609	20	7	fixed	fix	VERB
ejpam-6609	20	8	points	point	NOUN
ejpam-6609	20	9	related	relate	VERB
ejpam-6609	20	10	to	to	ADP
ejpam-6609	20	11	repeating	repeat	VERB
ejpam-6609	20	12	patterns	pattern	NOUN
ejpam-6609	20	13	[	[	X
ejpam-6609	20	14	8	8	NUM
ejpam-6609	20	15	]	]	PUNCT
ejpam-6609	20	16	,	,	PUNCT
ejpam-6609	20	17	new	new	ADJ
ejpam-6609	20	18	ways	way	NOUN
ejpam-6609	20	19	to	to	PART
ejpam-6609	20	20	organize	organize	VERB
ejpam-6609	20	21	things	thing	NOUN
ejpam-6609	20	22	[	[	X
ejpam-6609	20	23	9	9	NUM
ejpam-6609	20	24	]	]	PUNCT
ejpam-6609	20	25	,	,	PUNCT
ejpam-6609	20	26	linked	link	VERB
ejpam-6609	20	27	mappings	mapping	NOUN
ejpam-6609	20	28	[	[	X
ejpam-6609	20	29	6	6	NUM
ejpam-6609	20	30	]	]	PUNCT
ejpam-6609	20	31	,	,	PUNCT
ejpam-6609	20	32	and	and	CCONJ
ejpam-6609	20	33	more	more	ADV
ejpam-6609	20	34	complicated	complicated	ADJ
ejpam-6609	20	35	structures	structure	NOUN
ejpam-6609	20	36	like	like	ADP
ejpam-6609	20	37	partial	partial	ADJ
ejpam-6609	20	38	metric	metric	ADJ
ejpam-6609	20	39	spaces	space	NOUN
ejpam-6609	20	40	[	[	X
ejpam-6609	20	41	10	10	NUM
ejpam-6609	20	42	]	]	PUNCT
ejpam-6609	20	43	and	and	CCONJ
ejpam-6609	20	44	set	set	NOUN
ejpam-6609	20	45	-	-	PUNCT
ejpam-6609	20	46	valued	value	VERB
ejpam-6609	20	47	operators	operator	NOUN
ejpam-6609	20	48	[	[	X
ejpam-6609	20	49	11	11	NUM
ejpam-6609	20	50	]	]	PUNCT
ejpam-6609	20	51	.	.	PUNCT
ejpam-6609	21	1	recent	recent	ADJ
ejpam-6609	21	2	progress	progress	NOUN
ejpam-6609	21	3	has	have	AUX
ejpam-6609	21	4	built	build	VERB
ejpam-6609	21	5	on	on	ADP
ejpam-6609	21	6	traditional	traditional	ADJ
ejpam-6609	21	7	findings	finding	NOUN
ejpam-6609	21	8	and	and	CCONJ
ejpam-6609	21	9	made	make	VERB
ejpam-6609	21	10	them	they	PRON
ejpam-6609	21	11	more	more	ADV
ejpam-6609	21	12	useful	useful	ADJ
ejpam-6609	21	13	in	in	ADP
ejpam-6609	21	14	nonlinear	nonlinear	ADJ
ejpam-6609	21	15	analysis	analysis	NOUN
ejpam-6609	21	16	by	by	ADP
ejpam-6609	21	17	creating	create	VERB
ejpam-6609	21	18	new	new	ADJ
ejpam-6609	21	19	fixed	fix	VERB
ejpam-6609	21	20	-	-	PUNCT
ejpam-6609	21	21	point	point	NOUN
ejpam-6609	21	22	theorems	theorem	NOUN
ejpam-6609	21	23	for	for	ADP
ejpam-6609	21	24	3	3	NUM
ejpam-6609	21	25	-	-	PUNCT
ejpam-6609	21	26	sms	sms	NOUN
ejpam-6609	21	27	in	in	ADP
ejpam-6609	21	28	gips	gips	PROPN
ejpam-6609	22	1	[	[	X
ejpam-6609	22	2	12	12	NUM
ejpam-6609	22	3	–	–	PUNCT
ejpam-6609	22	4	14	14	NUM
ejpam-6609	22	5	]	]	PUNCT
ejpam-6609	22	6	.	.	PUNCT
ejpam-6609	23	1	gips	gips	PROPN
ejpam-6609	23	2	can	can	AUX
ejpam-6609	23	3	provide	provide	VERB
ejpam-6609	23	4	cfps	cfps	NOUN
ejpam-6609	23	5	without	without	ADP
ejpam-6609	23	6	needing	need	VERB
ejpam-6609	23	7	strict	strict	ADJ
ejpam-6609	23	8	iteration	iteration	NOUN
ejpam-6609	23	9	rules	rule	NOUN
ejpam-6609	23	10	,	,	PUNCT
ejpam-6609	23	11	which	which	PRON
ejpam-6609	23	12	makes	make	VERB
ejpam-6609	23	13	fixed	fix	VERB
ejpam-6609	23	14	-	-	PUNCT
ejpam-6609	23	15	point	point	NOUN
ejpam-6609	23	16	theorems	theorem	NOUN
ejpam-6609	23	17	applicable	applicable	ADJ
ejpam-6609	23	18	to	to	ADP
ejpam-6609	23	19	a	a	DET
ejpam-6609	23	20	wider	wide	ADJ
ejpam-6609	23	21	variety	variety	NOUN
ejpam-6609	23	22	of	of	ADP
ejpam-6609	23	23	nonlinear	nonlinear	ADJ
ejpam-6609	23	24	problems	problem	NOUN
ejpam-6609	23	25	[	[	X
ejpam-6609	23	26	15	15	NUM
ejpam-6609	23	27	]	]	PUNCT
ejpam-6609	23	28	.	.	PUNCT
ejpam-6609	24	1	stability	stability	NOUN
ejpam-6609	24	2	in	in	ADP
ejpam-6609	24	3	dems	dems	PROPN
ejpam-6609	24	4	in	in	ADP
ejpam-6609	24	5	ai	ai	PROPN
ejpam-6609	24	6	[	[	X
ejpam-6609	24	7	16	16	NUM
ejpam-6609	24	8	]	]	PUNCT
ejpam-6609	24	9	and	and	CCONJ
ejpam-6609	24	10	validating	validate	VERB
ejpam-6609	24	11	iterative	iterative	ADJ
ejpam-6609	24	12	protocols	protocol	NOUN
ejpam-6609	24	13	in	in	ADP
ejpam-6609	24	14	cryptography	cryptography	NOUN
ejpam-6609	24	15	could	could	AUX
ejpam-6609	24	16	be	be	AUX
ejpam-6609	24	17	enhanced	enhance	VERB
ejpam-6609	24	18	by	by	ADP
ejpam-6609	24	19	using	use	VERB
ejpam-6609	24	20	the	the	DET
ejpam-6609	24	21	new	new	ADJ
ejpam-6609	24	22	results	result	NOUN
ejpam-6609	24	23	on	on	ADP
ejpam-6609	24	24	3	3	NUM
ejpam-6609	24	25	-	-	PUNCT
ejpam-6609	24	26	sms	sms	NOUN
ejpam-6609	24	27	in	in	ADP
ejpam-6609	24	28	gips	gips	PROPN
ejpam-6609	24	29	[	[	X
ejpam-6609	24	30	17	17	NUM
ejpam-6609	24	31	]	]	PUNCT
ejpam-6609	24	32	.	.	PUNCT
ejpam-6609	25	1	fpt	fpt	PROPN
ejpam-6609	25	2	plays	play	VERB
ejpam-6609	25	3	a	a	DET
ejpam-6609	25	4	highly	highly	ADV
ejpam-6609	25	5	significant	significant	ADJ
ejpam-6609	25	6	role	role	NOUN
ejpam-6609	25	7	in	in	ADP
ejpam-6609	25	8	analyzing	analyze	VERB
ejpam-6609	25	9	convergence	convergence	NOUN
ejpam-6609	25	10	and	and	CCONJ
ejpam-6609	25	11	accuracy	accuracy	NOUN
ejpam-6609	25	12	in	in	ADP
ejpam-6609	25	13	lattice	lattice	NOUN
ejpam-6609	25	14	-	-	PUNCT
ejpam-6609	25	15	based	base	VERB
ejpam-6609	25	16	designs	design	NOUN
ejpam-6609	25	17	for	for	ADP
ejpam-6609	25	18	pqps	pqps	ADJ
ejpam-6609	25	19	[	[	X
ejpam-6609	25	20	18	18	NUM
ejpam-6609	25	21	]	]	PUNCT
ejpam-6609	25	22	.	.	PUNCT
ejpam-6609	26	1	describe	describe	VERB
ejpam-6609	26	2	many	many	ADJ
ejpam-6609	26	3	improvements	improvement	NOUN
ejpam-6609	26	4	have	have	AUX
ejpam-6609	26	5	been	be	AUX
ejpam-6609	26	6	made	make	VERB
ejpam-6609	26	7	in	in	ADP
ejpam-6609	26	8	fixed	fix	VERB
ejpam-6609	26	9	point	point	NOUN
ejpam-6609	26	10	theory	theory	NOUN
ejpam-6609	26	11	for	for	ADP
ejpam-6609	26	12	different	different	ADJ
ejpam-6609	26	13	metric	metric	ADJ
ejpam-6609	26	14	spaces	space	NOUN
ejpam-6609	26	15	,	,	PUNCT
ejpam-6609	26	16	but	but	CCONJ
ejpam-6609	26	17	there	there	PRON
ejpam-6609	26	18	has	have	AUX
ejpam-6609	26	19	been	be	AUX
ejpam-6609	26	20	less	less	ADJ
ejpam-6609	26	21	study	study	NOUN
ejpam-6609	26	22	on	on	ADP
ejpam-6609	26	23	systems	system	NOUN
ejpam-6609	26	24	with	with	ADP
ejpam-6609	26	25	multiple	multiple	ADJ
ejpam-6609	26	26	operators	operator	NOUN
ejpam-6609	26	27	in	in	ADP
ejpam-6609	26	28	gips	gips	PROPN
ejpam-6609	27	1	[	[	X
ejpam-6609	27	2	19	19	NUM
ejpam-6609	27	3	]	]	PUNCT
ejpam-6609	27	4	,	,	PUNCT
ejpam-6609	27	5	which	which	PRON
ejpam-6609	27	6	are	be	AUX
ejpam-6609	27	7	larger	large	ADJ
ejpam-6609	27	8	types	type	NOUN
ejpam-6609	27	9	of	of	ADP
ejpam-6609	27	10	hs	hs	PROPN
ejpam-6609	27	11	and	and	CCONJ
ejpam-6609	27	12	gm	gm	PROPN
ejpam-6609	27	13	-	-	PUNCT
ejpam-6609	27	14	spaces	space	NOUN
ejpam-6609	27	15	[	[	X
ejpam-6609	27	16	20	20	NUM
ejpam-6609	27	17	]	]	PUNCT
ejpam-6609	27	18	.	.	PUNCT
ejpam-6609	28	1	the	the	DET
ejpam-6609	28	2	theory	theory	NOUN
ejpam-6609	28	3	of	of	ADP
ejpam-6609	28	4	fixed	fix	VERB
ejpam-6609	28	5	points	point	NOUN
ejpam-6609	28	6	has	have	AUX
ejpam-6609	28	7	recently	recently	ADV
ejpam-6609	28	8	led	lead	VERB
ejpam-6609	28	9	to	to	ADP
ejpam-6609	28	10	improvements	improvement	NOUN
ejpam-6609	28	11	in	in	ADP
ejpam-6609	28	12	the	the	DET
ejpam-6609	28	13	mathematical	mathematical	ADJ
ejpam-6609	28	14	modelling	modelling	NOUN
ejpam-6609	28	15	of	of	ADP
ejpam-6609	28	16	intelligent	intelligent	ADJ
ejpam-6609	28	17	systems	system	NOUN
ejpam-6609	28	18	.	.	PUNCT
ejpam-6609	29	1	in	in	ADP
ejpam-6609	29	2	[	[	X
ejpam-6609	29	3	21	21	NUM
ejpam-6609	29	4	]	]	PUNCT
ejpam-6609	29	5	,	,	PUNCT
ejpam-6609	29	6	researchers	researcher	NOUN
ejpam-6609	29	7	presented	present	VERB
ejpam-6609	29	8	fractional	fractional	ADJ
ejpam-6609	29	9	analytic	analytic	ADJ
ejpam-6609	29	10	solutions	solution	NOUN
ejpam-6609	29	11	to	to	ADP
ejpam-6609	29	12	real	real	ADJ
ejpam-6609	29	13	-	-	PUNCT
ejpam-6609	29	14	world	world	NOUN
ejpam-6609	29	15	problems	problem	NOUN
ejpam-6609	29	16	such	such	ADJ
ejpam-6609	29	17	as	as	ADP
ejpam-6609	29	18	aibased	aibased	ADJ
ejpam-6609	29	19	decision	decision	NOUN
ejpam-6609	29	20	systems	system	NOUN
ejpam-6609	29	21	,	,	PUNCT
ejpam-6609	29	22	while	while	SCONJ
ejpam-6609	29	23	generalised	generalised	ADJ
ejpam-6609	29	24	results	result	NOUN
ejpam-6609	29	25	in	in	ADP
ejpam-6609	29	26	partial	partial	ADJ
ejpam-6609	29	27	b	b	NOUN
ejpam-6609	29	28	-	-	ADJ
ejpam-6609	29	29	metric	metric	ADJ
ejpam-6609	29	30	spaces	space	NOUN
ejpam-6609	29	31	provided	provide	VERB
ejpam-6609	29	32	theoretical	theoretical	ADJ
ejpam-6609	29	33	contributions	contribution	NOUN
ejpam-6609	29	34	to	to	ADP
ejpam-6609	29	35	convergence	convergence	NOUN
ejpam-6609	29	36	in	in	ADP
ejpam-6609	29	37	uncertain	uncertain	ADJ
ejpam-6609	29	38	settings	setting	NOUN
ejpam-6609	29	39	[	[	X
ejpam-6609	29	40	22	22	NUM
ejpam-6609	29	41	]	]	PUNCT
ejpam-6609	29	42	.	.	PUNCT
ejpam-6609	30	1	in	in	ADP
ejpam-6609	30	2	[	[	X
ejpam-6609	30	3	23	23	NUM
ejpam-6609	30	4	]	]	PUNCT
ejpam-6609	30	5	,	,	PUNCT
ejpam-6609	30	6	it	it	PRON
ejpam-6609	30	7	was	be	AUX
ejpam-6609	30	8	shown	show	VERB
ejpam-6609	30	9	how	how	SCONJ
ejpam-6609	30	10	to	to	PART
ejpam-6609	30	11	use	use	VERB
ejpam-6609	30	12	new	new	ADJ
ejpam-6609	30	13	forms	form	NOUN
ejpam-6609	30	14	of	of	ADP
ejpam-6609	30	15	functions	function	NOUN
ejpam-6609	30	16	in	in	ADP
ejpam-6609	30	17	metric	metric	ADJ
ejpam-6609	30	18	spaces	space	NOUN
ejpam-6609	30	19	,	,	PUNCT
ejpam-6609	30	20	which	which	PRON
ejpam-6609	30	21	may	may	AUX
ejpam-6609	30	22	be	be	AUX
ejpam-6609	30	23	useful	useful	ADJ
ejpam-6609	30	24	in	in	ADP
ejpam-6609	30	25	testing	test	VERB
ejpam-6609	30	26	cryptographic	cryptographic	ADJ
ejpam-6609	30	27	protocols	protocol	NOUN
ejpam-6609	30	28	.	.	PUNCT
ejpam-6609	31	1	the	the	DET
ejpam-6609	31	2	applied	applied	ADJ
ejpam-6609	31	3	side	side	NOUN
ejpam-6609	31	4	was	be	AUX
ejpam-6609	31	5	served	serve	VERB
ejpam-6609	31	6	by	by	ADP
ejpam-6609	31	7	the	the	DET
ejpam-6609	31	8	work	work	NOUN
ejpam-6609	31	9	on	on	ADP
ejpam-6609	31	10	iot	iot	NOUN
ejpam-6609	31	11	-	-	PUNCT
ejpam-6609	31	12	based	base	VERB
ejpam-6609	31	13	learning	learning	NOUN
ejpam-6609	31	14	systems	system	NOUN
ejpam-6609	31	15	relying	rely	VERB
ejpam-6609	31	16	on	on	ADP
ejpam-6609	31	17	soft	soft	ADJ
ejpam-6609	31	18	computing	computing	NOUN
ejpam-6609	31	19	in	in	ADP
ejpam-6609	31	20	[	[	X
ejpam-6609	31	21	24	24	NUM
ejpam-6609	31	22	]	]	PUNCT
ejpam-6609	31	23	and	and	CCONJ
ejpam-6609	31	24	the	the	DET
ejpam-6609	31	25	development	development	NOUN
ejpam-6609	31	26	of	of	ADP
ejpam-6609	31	27	advanced	advanced	ADJ
ejpam-6609	31	28	control	control	NOUN
ejpam-6609	31	29	systems	system	NOUN
ejpam-6609	31	30	,	,	PUNCT
ejpam-6609	31	31	an	an	DET
ejpam-6609	31	32	example	example	NOUN
ejpam-6609	31	33	of	of	ADP
ejpam-6609	31	34	which	which	PRON
ejpam-6609	31	35	is	be	AUX
ejpam-6609	31	36	,	,	PUNCT
ejpam-6609	31	37	instead	instead	ADV
ejpam-6609	31	38	of	of	ADP
ejpam-6609	31	39	usual	usual	ADJ
ejpam-6609	31	40	pid	pid	NOUN
ejpam-6609	31	41	controllers	controller	NOUN
ejpam-6609	31	42	,	,	PUNCT
ejpam-6609	31	43	fractional	fractional	ADJ
ejpam-6609	31	44	-	-	PUNCT
ejpam-6609	31	45	order	order	NOUN
ejpam-6609	31	46	ones	one	NOUN
ejpam-6609	31	47	(	(	PUNCT
ejpam-6609	31	48	related	relate	VERB
ejpam-6609	31	49	to	to	ADP
ejpam-6609	31	50	robotics	robotic	NOUN
ejpam-6609	31	51	and	and	CCONJ
ejpam-6609	31	52	security	security	NOUN
ejpam-6609	31	53	automation	automation	NOUN
ejpam-6609	31	54	)	)	PUNCT
ejpam-6609	31	55	in	in	ADP
ejpam-6609	31	56	[	[	X
ejpam-6609	31	57	25	25	NUM
ejpam-6609	31	58	]	]	PUNCT
ejpam-6609	31	59	.	.	PUNCT
ejpam-6609	32	1	this	this	DET
ejpam-6609	32	2	paper	paper	NOUN
ejpam-6609	32	3	fills	fill	VERB
ejpam-6609	32	4	this	this	DET
ejpam-6609	32	5	gap	gap	NOUN
ejpam-6609	32	6	by	by	ADP
ejpam-6609	32	7	introducing	introduce	VERB
ejpam-6609	32	8	new	new	ADJ
ejpam-6609	32	9	fixed	fix	VERB
ejpam-6609	32	10	-	-	PUNCT
ejpam-6609	32	11	point	point	NOUN
ejpam-6609	32	12	theorems	theorem	NOUN
ejpam-6609	32	13	specifically	specifically	ADV
ejpam-6609	32	14	designed	design	VERB
ejpam-6609	32	15	for	for	ADP
ejpam-6609	32	16	3	3	NUM
ejpam-6609	32	17	-	-	PUNCT
ejpam-6609	32	18	sms	sms	NOUN
ejpam-6609	32	19	within	within	ADP
ejpam-6609	32	20	the	the	DET
ejpam-6609	32	21	gips	gip	NOUN
ejpam-6609	32	22	.	.	PUNCT
ejpam-6609	33	1	the	the	DET
ejpam-6609	33	2	primary	primary	ADJ
ejpam-6609	33	3	interest	interest	NOUN
ejpam-6609	33	4	is	be	AUX
ejpam-6609	33	5	to	to	PART
ejpam-6609	33	6	see	see	VERB
ejpam-6609	33	7	what	what	PRON
ejpam-6609	33	8	happens	happen	VERB
ejpam-6609	33	9	when	when	SCONJ
ejpam-6609	33	10	the	the	DET
ejpam-6609	33	11	conditions	condition	NOUN
ejpam-6609	33	12	that	that	PRON
ejpam-6609	33	13	make	make	VERB
ejpam-6609	33	14	the	the	DET
ejpam-6609	33	15	mapping	mapping	NOUN
ejpam-6609	33	16	contract	contract	NOUN
ejpam-6609	33	17	are	be	AUX
ejpam-6609	33	18	less	less	ADV
ejpam-6609	33	19	strict	strict	ADJ
ejpam-6609	33	20	.	.	PUNCT
ejpam-6609	34	1	because	because	SCONJ
ejpam-6609	34	2	of	of	ADP
ejpam-6609	34	3	this	this	PRON
ejpam-6609	34	4	,	,	PUNCT
ejpam-6609	34	5	progress	progress	NOUN
ejpam-6609	34	6	in	in	ADP
ejpam-6609	34	7	math	math	NOUN
ejpam-6609	34	8	and	and	CCONJ
ejpam-6609	34	9	practical	practical	ADJ
ejpam-6609	34	10	fields	field	NOUN
ejpam-6609	34	11	can	can	AUX
ejpam-6609	34	12	be	be	AUX
ejpam-6609	34	13	expected	expect	VERB
ejpam-6609	34	14	in	in	ADP
ejpam-6609	34	15	ml	ml	NOUN
ejpam-6609	34	16	and	and	CCONJ
ejpam-6609	34	17	cryptography	cryptography	NOUN
ejpam-6609	34	18	.	.	PUNCT
ejpam-6609	35	1	(	(	PUNCT
ejpam-6609	35	2	i	i	NOUN
ejpam-6609	35	3	)	)	PUNCT
ejpam-6609	35	4	theoretical	theoretical	ADJ
ejpam-6609	35	5	foundations	foundation	NOUN
ejpam-6609	35	6	:	:	PUNCT
ejpam-6609	35	7	for	for	ADP
ejpam-6609	35	8	3	3	NUM
ejpam-6609	35	9	-	-	PUNCT
ejpam-6609	35	10	sms	sms	NOUN
ejpam-6609	35	11	[	[	X
ejpam-6609	35	12	12	12	NUM
ejpam-6609	35	13	]	]	PUNCT
ejpam-6609	35	14	,	,	PUNCT
ejpam-6609	35	15	we	we	PRON
ejpam-6609	35	16	show	show	VERB
ejpam-6609	35	17	new	new	ADJ
ejpam-6609	35	18	cfp	cfp	PROPN
ejpam-6609	35	19	theorems	theorem	NOUN
ejpam-6609	35	20	under	under	ADP
ejpam-6609	35	21	less	less	ADV
ejpam-6609	35	22	strict	strict	ADJ
ejpam-6609	35	23	contraction	contraction	NOUN
ejpam-6609	35	24	conditions	condition	NOUN
ejpam-6609	35	25	,	,	PUNCT
ejpam-6609	35	26	which	which	PRON
ejpam-6609	35	27	significantly	significantly	ADV
ejpam-6609	35	28	broadens	broaden	VERB
ejpam-6609	35	29	and	and	CCONJ
ejpam-6609	35	30	combines	combine	VERB
ejpam-6609	35	31	earlier	early	ADJ
ejpam-6609	35	32	results	result	NOUN
ejpam-6609	35	33	found	find	VERB
ejpam-6609	35	34	in	in	ADP
ejpam-6609	35	35	various	various	ADJ
ejpam-6609	35	36	types	type	NOUN
ejpam-6609	35	37	of	of	ADP
ejpam-6609	35	38	metric	metric	ADJ
ejpam-6609	35	39	spaces	space	NOUN
ejpam-6609	35	40	.	.	PUNCT
ejpam-6609	36	1	these	these	DET
ejpam-6609	36	2	theorems	theorem	NOUN
ejpam-6609	36	3	offer	offer	VERB
ejpam-6609	36	4	a	a	DET
ejpam-6609	36	5	powerful	powerful	ADJ
ejpam-6609	36	6	and	and	CCONJ
ejpam-6609	36	7	flexible	flexible	ADJ
ejpam-6609	36	8	way	way	NOUN
ejpam-6609	36	9	to	to	PART
ejpam-6609	36	10	tackle	tackle	VERB
ejpam-6609	36	11	complicated	complicated	ADJ
ejpam-6609	36	12	problems	problem	NOUN
ejpam-6609	36	13	in	in	ADP
ejpam-6609	36	14	functional	functional	ADJ
ejpam-6609	36	15	analysis	analysis	NOUN
ejpam-6609	36	16	by	by	ADP
ejpam-6609	36	17	using	use	VERB
ejpam-6609	36	18	the	the	DET
ejpam-6609	36	19	mathematical	mathematical	ADJ
ejpam-6609	36	20	properties	property	NOUN
ejpam-6609	36	21	found	find	VERB
ejpam-6609	36	22	in	in	ADP
ejpam-6609	36	23	gips	gips	PROPN
ejpam-6609	36	24	.	.	PUNCT
ejpam-6609	37	1	(	(	PUNCT
ejpam-6609	37	2	ii	ii	NOUN
ejpam-6609	37	3	)	)	PUNCT
ejpam-6609	37	4	applications	application	NOUN
ejpam-6609	37	5	to	to	ADP
ejpam-6609	37	6	artificial	artificial	ADJ
ejpam-6609	37	7	intelligence	intelligence	NOUN
ejpam-6609	37	8	:	:	PUNCT
ejpam-6609	37	9	for	for	ADP
ejpam-6609	37	10	dems	dem	NOUN
ejpam-6609	37	11	[	[	X
ejpam-6609	37	12	16	16	NUM
ejpam-6609	37	13	]	]	PUNCT
ejpam-6609	37	14	,	,	PUNCT
ejpam-6609	37	15	which	which	PRON
ejpam-6609	37	16	are	be	AUX
ejpam-6609	37	17	a	a	DET
ejpam-6609	37	18	type	type	NOUN
ejpam-6609	37	19	of	of	ADP
ejpam-6609	37	20	neural	neural	ADJ
ejpam-6609	37	21	network	network	NOUN
ejpam-6609	37	22	that	that	PRON
ejpam-6609	37	23	requires	require	VERB
ejpam-6609	37	24	strong	strong	ADJ
ejpam-6609	37	25	theoretical	theoretical	ADJ
ejpam-6609	37	26	support	support	NOUN
ejpam-6609	37	27	for	for	ADP
ejpam-6609	37	28	stable	stable	ADJ
ejpam-6609	37	29	training	training	NOUN
ejpam-6609	37	30	,	,	PUNCT
ejpam-6609	37	31	we	we	PRON
ejpam-6609	37	32	demonstrate	demonstrate	VERB
ejpam-6609	37	33	how	how	SCONJ
ejpam-6609	37	34	the	the	DET
ejpam-6609	37	35	proposed	propose	VERB
ejpam-6609	37	36	cfp	cfp	PROPN
ejpam-6609	37	37	results	result	NOUN
ejpam-6609	37	38	ensure	ensure	VERB
ejpam-6609	37	39	that	that	SCONJ
ejpam-6609	37	40	the	the	DET
ejpam-6609	37	41	training	training	NOUN
ejpam-6609	37	42	process	process	NOUN
ejpam-6609	37	43	will	will	AUX
ejpam-6609	37	44	reach	reach	VERB
ejpam-6609	37	45	a	a	DET
ejpam-6609	37	46	solution	solution	NOUN
ejpam-6609	37	47	.	.	PUNCT
ejpam-6609	38	1	our	our	PRON
ejpam-6609	38	2	method	method	NOUN
ejpam-6609	38	3	extends	extend	VERB
ejpam-6609	38	4	the	the	DET
ejpam-6609	38	5	scope	scope	NOUN
ejpam-6609	38	6	of	of	ADP
ejpam-6609	38	7	conventional	conventional	ADJ
ejpam-6609	38	8	fixed	fix	VERB
ejpam-6609	38	9	-	-	PUNCT
ejpam-6609	38	10	point	point	NOUN
ejpam-6609	38	11	convergence	convergence	NOUN
ejpam-6609	38	12	analysis	analysis	NOUN
ejpam-6609	38	13	by	by	ADP
ejpam-6609	38	14	using	use	VERB
ejpam-6609	38	15	the	the	DET
ejpam-6609	38	16	framework	framework	NOUN
ejpam-6609	38	17	of	of	ADP
ejpam-6609	38	18	multi	multi	ADJ
ejpam-6609	38	19	-	-	NOUN
ejpam-6609	38	20	operator	operator	NOUN
ejpam-6609	38	21	fixed	fix	VERB
ejpam-6609	38	22	-	-	PUNCT
ejpam-6609	38	23	point	point	NOUN
ejpam-6609	38	24	theory	theory	NOUN
ejpam-6609	38	25	to	to	PART
ejpam-6609	38	26	study	study	VERB
ejpam-6609	38	27	complex	complex	ADJ
ejpam-6609	38	28	,	,	PUNCT
ejpam-6609	38	29	multi	multi	ADJ
ejpam-6609	38	30	-	-	ADJ
ejpam-6609	38	31	layered	layered	ADJ
ejpam-6609	38	32	,	,	PUNCT
ejpam-6609	38	33	feedback	feedback	NOUN
ejpam-6609	38	34	-	-	PUNCT
ejpam-6609	38	35	driven	drive	VERB
ejpam-6609	38	36	ai	ai	NOUN
ejpam-6609	38	37	systems	system	NOUN
ejpam-6609	38	38	.	.	PUNCT
ejpam-6609	39	1	(	(	PUNCT
ejpam-6609	39	2	iii	iii	X
ejpam-6609	39	3	)	)	PUNCT
ejpam-6609	39	4	cryptographic	cryptographic	ADJ
ejpam-6609	39	5	innovations	innovation	NOUN
ejpam-6609	39	6	:	:	PUNCT
ejpam-6609	39	7	we	we	PRON
ejpam-6609	39	8	introduce	introduce	VERB
ejpam-6609	39	9	a	a	DET
ejpam-6609	39	10	new	new	ADJ
ejpam-6609	39	11	type	type	NOUN
ejpam-6609	39	12	of	of	ADP
ejpam-6609	39	13	pqps	pqps	NOUN
ejpam-6609	39	14	that	that	PRON
ejpam-6609	39	15	use	use	VERB
ejpam-6609	39	16	lattices	lattice	NOUN
ejpam-6609	39	17	[	[	X
ejpam-6609	39	18	18	18	NUM
ejpam-6609	39	19	]	]	PUNCT
ejpam-6609	39	20	,	,	PUNCT
ejpam-6609	39	21	and	and	CCONJ
ejpam-6609	39	22	their	their	PRON
ejpam-6609	39	23	security	security	NOUN
ejpam-6609	39	24	relies	rely	VERB
ejpam-6609	39	25	on	on	ADP
ejpam-6609	39	26	repeating	repeat	VERB
ejpam-6609	39	27	three	three	NUM
ejpam-6609	39	28	specific	specific	ADJ
ejpam-6609	39	29	functions	function	NOUN
ejpam-6609	39	30	in	in	ADP
ejpam-6609	39	31	a	a	DET
ejpam-6609	39	32	special	special	ADJ
ejpam-6609	39	33	kind	kind	NOUN
ejpam-6609	39	34	of	of	ADP
ejpam-6609	39	35	mathematical	mathematical	ADJ
ejpam-6609	39	36	space	space	NOUN
ejpam-6609	39	37	called	call	VERB
ejpam-6609	39	38	gips	gips	PROPN
ejpam-6609	39	39	.	.	PUNCT
ejpam-6609	40	1	these	these	DET
ejpam-6609	40	2	protocols	protocol	NOUN
ejpam-6609	40	3	offer	offer	VERB
ejpam-6609	40	4	new	new	ADJ
ejpam-6609	40	5	methods	method	NOUN
ejpam-6609	40	6	to	to	PART
ejpam-6609	40	7	create	create	VERB
ejpam-6609	40	8	secure	secure	ADJ
ejpam-6609	40	9	systems	system	NOUN
ejpam-6609	40	10	that	that	PRON
ejpam-6609	40	11	can	can	AUX
ejpam-6609	40	12	withstand	withstand	VERB
ejpam-6609	40	13	quantum	quantum	NOUN
ejpam-6609	40	14	attacks	attack	NOUN
ejpam-6609	40	15	by	by	ADP
ejpam-6609	40	16	using	use	VERB
ejpam-6609	40	17	the	the	DET
ejpam-6609	40	18	relationship	relationship	NOUN
ejpam-6609	40	19	between	between	ADP
ejpam-6609	40	20	inner	inner	ADJ
ejpam-6609	40	21	product	product	NOUN
ejpam-6609	40	22	geometry	geometry	NOUN
ejpam-6609	40	23	and	and	CCONJ
ejpam-6609	40	24	contractive	contractive	ADJ
ejpam-6609	40	25	iteration	iteration	NOUN
ejpam-6609	40	26	.	.	PUNCT
ejpam-6609	41	1	in	in	ADP
ejpam-6609	41	2	this	this	DET
ejpam-6609	41	3	work	work	NOUN
ejpam-6609	41	4	,	,	PUNCT
ejpam-6609	41	5	traditional	traditional	ADJ
ejpam-6609	41	6	ideas	idea	NOUN
ejpam-6609	41	7	from	from	ADP
ejpam-6609	41	8	ordered	order	VERB
ejpam-6609	41	9	and	and	CCONJ
ejpam-6609	41	10	cone	cone	NOUN
ejpam-6609	41	11	metric	metric	ADJ
ejpam-6609	41	12	spaces	space	NOUN
ejpam-6609	41	13	are	be	AUX
ejpam-6609	41	14	extended	extend	VERB
ejpam-6609	41	15	into	into	ADP
ejpam-6609	41	16	a	a	DET
ejpam-6609	41	17	modern	modern	ADJ
ejpam-6609	41	18	cryptographic	cryptographic	ADJ
ejpam-6609	41	19	framework	framework	NOUN
ejpam-6609	41	20	.	.	PUNCT
ejpam-6609	42	1	as	as	ADP
ejpam-6609	42	2	a	a	DET
ejpam-6609	42	3	result	result	NOUN
ejpam-6609	42	4	,	,	PUNCT
ejpam-6609	42	5	this	this	DET
ejpam-6609	42	6	work	work	NOUN
ejpam-6609	42	7	makes	make	VERB
ejpam-6609	42	8	four	four	NUM
ejpam-6609	42	9	significant	significant	ADJ
ejpam-6609	42	10	contributions	contribution	NOUN
ejpam-6609	42	11	to	to	ADP
ejpam-6609	42	12	the	the	DET
ejpam-6609	42	13	literature	literature	NOUN
ejpam-6609	42	14	on	on	ADP
ejpam-6609	42	15	fixed	fix	VERB
ejpam-6609	42	16	point	point	NOUN
ejpam-6609	42	17	theory	theory	NOUN
ejpam-6609	42	18	:	:	PUNCT
ejpam-6609	42	19	in	in	ADP
ejpam-6609	42	20	gips	gips	PROPN
ejpam-6609	42	21	,	,	PUNCT
ejpam-6609	42	22	it	it	PRON
ejpam-6609	42	23	(	(	PUNCT
ejpam-6609	42	24	i	i	NOUN
ejpam-6609	42	25	)	)	PUNCT
ejpam-6609	42	26	establishes	establish	VERB
ejpam-6609	42	27	new	new	ADJ
ejpam-6609	42	28	cfp	cfp	NOUN
ejpam-6609	42	29	theorems	theorem	NOUN
ejpam-6609	42	30	for	for	ADP
ejpam-6609	42	31	3	3	NUM
ejpam-6609	42	32	-	-	PUNCT
ejpam-6609	42	33	sms	sm	NOUN
ejpam-6609	42	34	;	;	PUNCT
ejpam-6609	42	35	(	(	PUNCT
ejpam-6609	42	36	ii	ii	NOUN
ejpam-6609	42	37	)	)	PUNCT
ejpam-6609	42	38	uses	use	VERB
ejpam-6609	42	39	these	these	DET
ejpam-6609	42	40	findings	finding	NOUN
ejpam-6609	42	41	to	to	PART
ejpam-6609	42	42	m.	m.	PROPN
ejpam-6609	42	43	noorwali	noorwali	PROPN
ejpam-6609	42	44	et	et	PROPN
ejpam-6609	42	45	al	al	PROPN
ejpam-6609	42	46	.	.	PUNCT
ejpam-6609	42	47	/	/	SYM
ejpam-6609	42	48	eur	eur	PROPN
ejpam-6609	42	49	.	.	PUNCT
ejpam-6609	43	1	j.	j.	PROPN
ejpam-6609	43	2	pure	pure	PROPN
ejpam-6609	43	3	appl	appl	PROPN
ejpam-6609	43	4	.	.	PROPN
ejpam-6609	43	5	math	math	PROPN
ejpam-6609	43	6	,	,	PUNCT
ejpam-6609	43	7	18	18	NUM
ejpam-6609	43	8	(	(	PUNCT
ejpam-6609	43	9	3	3	NUM
ejpam-6609	43	10	)	)	PUNCT
ejpam-6609	43	11	(	(	PUNCT
ejpam-6609	43	12	2025	2025	NUM
ejpam-6609	43	13	)	)	PUNCT
ejpam-6609	43	14	,	,	PUNCT
ejpam-6609	43	15	6609	6609	NUM
ejpam-6609	43	16	3	3	NUM
ejpam-6609	43	17	of	of	ADP
ejpam-6609	43	18	17	17	NUM
ejpam-6609	43	19	examine	examine	NOUN
ejpam-6609	43	20	convergence	convergence	NOUN
ejpam-6609	43	21	in	in	ADP
ejpam-6609	43	22	deep	deep	ADJ
ejpam-6609	43	23	learning	learning	NOUN
ejpam-6609	43	24	models	model	NOUN
ejpam-6609	43	25	,	,	PUNCT
ejpam-6609	43	26	especially	especially	ADV
ejpam-6609	43	27	dqms	dqms	ADJ
ejpam-6609	43	28	;	;	PUNCT
ejpam-6609	43	29	(	(	PUNCT
ejpam-6609	43	30	iii	iii	X
ejpam-6609	43	31	)	)	PUNCT
ejpam-6609	43	32	presents	present	VERB
ejpam-6609	43	33	new	new	ADJ
ejpam-6609	43	34	pqp	pqp	NOUN
ejpam-6609	43	35	schemes	scheme	NOUN
ejpam-6609	43	36	based	base	VERB
ejpam-6609	43	37	on	on	ADP
ejpam-6609	43	38	fixed	fix	VERB
ejpam-6609	43	39	point	point	NOUN
ejpam-6609	43	40	iterations	iteration	NOUN
ejpam-6609	43	41	;	;	PUNCT
ejpam-6609	43	42	and	and	CCONJ
ejpam-6609	43	43	(	(	PUNCT
ejpam-6609	43	44	iv	iv	X
ejpam-6609	43	45	)	)	PUNCT
ejpam-6609	43	46	provides	provide	VERB
ejpam-6609	43	47	computational	computational	ADJ
ejpam-6609	43	48	experiments	experiment	NOUN
ejpam-6609	43	49	and	and	CCONJ
ejpam-6609	43	50	simulations	simulation	NOUN
ejpam-6609	43	51	to	to	PART
ejpam-6609	43	52	support	support	VERB
ejpam-6609	43	53	the	the	DET
ejpam-6609	43	54	theoretical	theoretical	ADJ
ejpam-6609	43	55	framework	framework	NOUN
ejpam-6609	43	56	.	.	PUNCT
ejpam-6609	44	1	problem	problem	NOUN
ejpam-6609	44	2	statement	statement	NOUN
ejpam-6609	44	3	despite	despite	SCONJ
ejpam-6609	44	4	major	major	ADJ
ejpam-6609	44	5	advancements	advancement	NOUN
ejpam-6609	44	6	in	in	ADP
ejpam-6609	44	7	fixed	fix	VERB
ejpam-6609	44	8	point	point	NOUN
ejpam-6609	44	9	theory	theory	NOUN
ejpam-6609	44	10	,	,	PUNCT
ejpam-6609	44	11	critical	critical	ADJ
ejpam-6609	44	12	gaps	gap	NOUN
ejpam-6609	44	13	still	still	ADV
ejpam-6609	44	14	exist	exist	VERB
ejpam-6609	44	15	in	in	ADP
ejpam-6609	44	16	the	the	DET
ejpam-6609	44	17	study	study	NOUN
ejpam-6609	44	18	of	of	ADP
ejpam-6609	44	19	cfps	cfps	NOUN
ejpam-6609	44	20	for	for	ADP
ejpam-6609	44	21	systems	system	NOUN
ejpam-6609	44	22	of	of	ADP
ejpam-6609	44	23	3	3	NUM
ejpam-6609	44	24	-	-	PUNCT
ejpam-6609	44	25	sms	sms	NOUN
ejpam-6609	44	26	in	in	ADP
ejpam-6609	44	27	gips	gips	PROPN
ejpam-6609	45	1	[	[	X
ejpam-6609	45	2	12	12	NUM
ejpam-6609	45	3	]	]	PUNCT
ejpam-6609	45	4	.	.	PUNCT
ejpam-6609	46	1	the	the	DET
ejpam-6609	46	2	theoretical	theoretical	ADJ
ejpam-6609	46	3	foundation	foundation	NOUN
ejpam-6609	46	4	for	for	ADP
ejpam-6609	46	5	3	3	NUM
ejpam-6609	46	6	-	-	PUNCT
ejpam-6609	46	7	sms	sms	NOUN
ejpam-6609	46	8	in	in	ADP
ejpam-6609	46	9	gips	gips	PROPN
ejpam-6609	46	10	is	be	AUX
ejpam-6609	46	11	still	still	ADV
ejpam-6609	46	12	lacking	lack	VERB
ejpam-6609	46	13	,	,	PUNCT
ejpam-6609	46	14	despite	despite	SCONJ
ejpam-6609	46	15	the	the	DET
ejpam-6609	46	16	fact	fact	NOUN
ejpam-6609	46	17	that	that	SCONJ
ejpam-6609	46	18	single	single	ADJ
ejpam-6609	46	19	-	-	PUNCT
ejpam-6609	46	20	mapping	mapping	NOUN
ejpam-6609	46	21	scenarios	scenario	NOUN
ejpam-6609	46	22	in	in	ADP
ejpam-6609	46	23	metric	metric	ADJ
ejpam-6609	46	24	spaces	space	NOUN
ejpam-6609	46	25	have	have	AUX
ejpam-6609	46	26	been	be	AUX
ejpam-6609	46	27	extensively	extensively	ADV
ejpam-6609	46	28	covered	cover	VERB
ejpam-6609	46	29	by	by	ADP
ejpam-6609	46	30	classical	classical	ADJ
ejpam-6609	46	31	results	result	NOUN
ejpam-6609	46	32	,	,	PUNCT
ejpam-6609	46	33	and	and	CCONJ
ejpam-6609	46	34	recent	recent	ADJ
ejpam-6609	46	35	advancements	advancement	NOUN
ejpam-6609	46	36	have	have	AUX
ejpam-6609	46	37	expanded	expand	VERB
ejpam-6609	46	38	these	these	DET
ejpam-6609	46	39	concepts	concept	NOUN
ejpam-6609	46	40	to	to	ADP
ejpam-6609	46	41	pairwise	pairwise	NOUN
ejpam-6609	46	42	mappings	mapping	NOUN
ejpam-6609	46	43	.	.	PUNCT
ejpam-6609	47	1	the	the	DET
ejpam-6609	47	2	three	three	NUM
ejpam-6609	47	3	main	main	ADJ
ejpam-6609	47	4	drawbacks	drawback	NOUN
ejpam-6609	47	5	of	of	ADP
ejpam-6609	47	6	current	current	ADJ
ejpam-6609	47	7	contraction	contraction	NOUN
ejpam-6609	47	8	-	-	PUNCT
ejpam-6609	47	9	type	type	NOUN
ejpam-6609	47	10	results	result	NOUN
ejpam-6609	47	11	are	be	AUX
ejpam-6609	47	12	as	as	SCONJ
ejpam-6609	47	13	follows	follow	VERB
ejpam-6609	47	14	:	:	PUNCT
ejpam-6609	47	15	(	(	PUNCT
ejpam-6609	47	16	1	1	X
ejpam-6609	47	17	)	)	PUNCT
ejpam-6609	47	18	they	they	PRON
ejpam-6609	47	19	mostly	mostly	ADV
ejpam-6609	47	20	concentrate	concentrate	VERB
ejpam-6609	47	21	on	on	ADP
ejpam-6609	47	22	classical	classical	ADJ
ejpam-6609	47	23	metric	metric	ADJ
ejpam-6609	47	24	spaces	space	NOUN
ejpam-6609	47	25	instead	instead	ADV
ejpam-6609	47	26	of	of	ADP
ejpam-6609	47	27	taking	take	VERB
ejpam-6609	47	28	advantage	advantage	NOUN
ejpam-6609	47	29	of	of	ADP
ejpam-6609	47	30	the	the	DET
ejpam-6609	47	31	more	more	ADV
ejpam-6609	47	32	complex	complex	ADJ
ejpam-6609	47	33	algebraic	algebraic	ADJ
ejpam-6609	47	34	and	and	CCONJ
ejpam-6609	47	35	geometric	geometric	ADJ
ejpam-6609	47	36	structure	structure	NOUN
ejpam-6609	47	37	of	of	ADP
ejpam-6609	47	38	gips	gip	NOUN
ejpam-6609	47	39	;	;	PUNCT
ejpam-6609	47	40	(	(	PUNCT
ejpam-6609	47	41	2	2	X
ejpam-6609	47	42	)	)	PUNCT
ejpam-6609	47	43	existing	exist	VERB
ejpam-6609	47	44	uniqueness	uniqueness	NOUN
ejpam-6609	47	45	proofs	proof	NOUN
ejpam-6609	47	46	for	for	ADP
ejpam-6609	47	47	cfps	cfps	NOUN
ejpam-6609	47	48	frequently	frequently	ADV
ejpam-6609	47	49	use	use	VERB
ejpam-6609	47	50	excessively	excessively	ADV
ejpam-6609	47	51	restrictive	restrictive	ADJ
ejpam-6609	47	52	contraction	contraction	NOUN
ejpam-6609	47	53	conditions	condition	NOUN
ejpam-6609	47	54	;	;	PUNCT
ejpam-6609	47	55	and	and	CCONJ
ejpam-6609	47	56	(	(	PUNCT
ejpam-6609	47	57	3	3	X
ejpam-6609	47	58	)	)	PUNCT
ejpam-6609	47	59	there	there	PRON
ejpam-6609	47	60	are	be	VERB
ejpam-6609	47	61	notably	notably	ADV
ejpam-6609	47	62	few	few	ADJ
ejpam-6609	47	63	applications	application	NOUN
ejpam-6609	47	64	to	to	ADP
ejpam-6609	47	65	contemporary	contemporary	ADJ
ejpam-6609	47	66	computational	computational	ADJ
ejpam-6609	47	67	domains	domain	NOUN
ejpam-6609	47	68	like	like	ADP
ejpam-6609	47	69	cryptography	cryptography	NOUN
ejpam-6609	47	70	and	and	CCONJ
ejpam-6609	47	71	ai	ai	VERB
ejpam-6609	47	72	.	.	PUNCT
ejpam-6609	48	1	to	to	PART
ejpam-6609	48	2	overcome	overcome	VERB
ejpam-6609	48	3	these	these	DET
ejpam-6609	48	4	limitations	limitation	NOUN
ejpam-6609	48	5	,	,	PUNCT
ejpam-6609	48	6	this	this	DET
ejpam-6609	48	7	work	work	NOUN
ejpam-6609	48	8	establishes	establish	VERB
ejpam-6609	48	9	new	new	ADJ
ejpam-6609	48	10	cfp	cfp	NOUN
ejpam-6609	48	11	theorems	theorem	NOUN
ejpam-6609	48	12	for	for	ADP
ejpam-6609	48	13	3	3	NUM
ejpam-6609	48	14	-	-	PUNCT
ejpam-6609	48	15	sms	sms	NOUN
ejpam-6609	48	16	in	in	ADP
ejpam-6609	48	17	gips	gips	PROPN
ejpam-6609	48	18	(	(	PUNCT
ejpam-6609	48	19	i	i	NOUN
ejpam-6609	48	20	)	)	PUNCT
ejpam-6609	48	21	relax	relax	VERB
ejpam-6609	48	22	conventional	conventional	ADJ
ejpam-6609	48	23	contraction	contraction	NOUN
ejpam-6609	48	24	assumptions	assumption	NOUN
ejpam-6609	48	25	while	while	SCONJ
ejpam-6609	48	26	preserving	preserve	VERB
ejpam-6609	48	27	uniqueness	uniqueness	NOUN
ejpam-6609	48	28	guarantees	guarantee	NOUN
ejpam-6609	48	29	,	,	PUNCT
ejpam-6609	48	30	(	(	PUNCT
ejpam-6609	48	31	ii	ii	NOUN
ejpam-6609	48	32	)	)	PUNCT
ejpam-6609	48	33	make	make	VERB
ejpam-6609	48	34	use	use	NOUN
ejpam-6609	48	35	of	of	ADP
ejpam-6609	48	36	gips	gips	PROPN
ejpam-6609	48	37	’s	’s	PART
ejpam-6609	48	38	vector	vector	NOUN
ejpam-6609	48	39	space	space	NOUN
ejpam-6609	48	40	and	and	CCONJ
ejpam-6609	48	41	inner	inner	ADJ
ejpam-6609	48	42	product	product	NOUN
ejpam-6609	48	43	properties	property	NOUN
ejpam-6609	48	44	to	to	PART
ejpam-6609	48	45	enable	enable	VERB
ejpam-6609	48	46	new	new	ADJ
ejpam-6609	48	47	applications	application	NOUN
ejpam-6609	48	48	,	,	PUNCT
ejpam-6609	48	49	and	and	CCONJ
ejpam-6609	48	50	(	(	PUNCT
ejpam-6609	48	51	iii	iii	X
ejpam-6609	48	52	)	)	PUNCT
ejpam-6609	48	53	offer	offer	VERB
ejpam-6609	48	54	tangible	tangible	ADJ
ejpam-6609	48	55	implementations	implementation	NOUN
ejpam-6609	48	56	in	in	ADP
ejpam-6609	48	57	two	two	NUM
ejpam-6609	48	58	modern	modern	ADJ
ejpam-6609	48	59	domains	domain	NOUN
ejpam-6609	48	60	:	:	PUNCT
ejpam-6609	48	61	ensuring	ensure	VERB
ejpam-6609	48	62	convergence	convergence	NOUN
ejpam-6609	48	63	in	in	ADP
ejpam-6609	48	64	ml	ml	ADP
ejpam-6609	48	65	dems	dem	NOUN
ejpam-6609	48	66	and	and	CCONJ
ejpam-6609	48	67	building	build	VERB
ejpam-6609	48	68	secure	secure	ADJ
ejpam-6609	48	69	pqps	pqps	ADJ
ejpam-6609	48	70	.	.	PUNCT
ejpam-6609	49	1	through	through	ADP
ejpam-6609	49	2	theoretical	theoretical	ADJ
ejpam-6609	49	3	advances	advance	NOUN
ejpam-6609	49	4	in	in	ADP
ejpam-6609	49	5	multi	multi	ADJ
ejpam-6609	49	6	-	-	ADJ
ejpam-6609	49	7	operator	operator	NOUN
ejpam-6609	49	8	systems	system	NOUN
ejpam-6609	49	9	(	(	PUNCT
ejpam-6609	49	10	sections	section	NOUN
ejpam-6609	49	11	2	2	NUM
ejpam-6609	49	12	-	-	SYM
ejpam-6609	49	13	3	3	NUM
ejpam-6609	49	14	)	)	PUNCT
ejpam-6609	49	15	and	and	CCONJ
ejpam-6609	49	16	demonstrable	demonstrable	ADJ
ejpam-6609	49	17	applications	application	NOUN
ejpam-6609	49	18	in	in	ADP
ejpam-6609	49	19	ai	ai	PRON
ejpam-6609	49	20	optimization	optimization	NOUN
ejpam-6609	49	21	(	(	PUNCT
ejpam-6609	49	22	section	section	NOUN
ejpam-6609	49	23	4	4	NUM
ejpam-6609	49	24	)	)	PUNCT
ejpam-6609	49	25	and	and	CCONJ
ejpam-6609	49	26	lattice	lattice	NOUN
ejpam-6609	49	27	-	-	PUNCT
ejpam-6609	49	28	based	base	VERB
ejpam-6609	49	29	cryptography	cryptography	NOUN
ejpam-6609	49	30	,	,	PUNCT
ejpam-6609	49	31	this	this	DET
ejpam-6609	49	32	method	method	NOUN
ejpam-6609	49	33	bridges	bridge	VERB
ejpam-6609	49	34	the	the	DET
ejpam-6609	49	35	gap	gap	NOUN
ejpam-6609	49	36	between	between	ADP
ejpam-6609	49	37	abstract	abstract	ADJ
ejpam-6609	49	38	fixed	fix	VERB
ejpam-6609	49	39	point	point	NOUN
ejpam-6609	49	40	theory	theory	NOUN
ejpam-6609	49	41	and	and	CCONJ
ejpam-6609	49	42	real	real	ADJ
ejpam-6609	49	43	-	-	PUNCT
ejpam-6609	49	44	world	world	NOUN
ejpam-6609	49	45	computational	computational	ADJ
ejpam-6609	49	46	challenges	challenge	NOUN
ejpam-6609	49	47	.	.	PUNCT
ejpam-6609	50	1	organization	organization	NOUN
ejpam-6609	50	2	we	we	PRON
ejpam-6609	50	3	structure	structure	VERB
ejpam-6609	50	4	the	the	DET
ejpam-6609	50	5	paper	paper	NOUN
ejpam-6609	50	6	as	as	SCONJ
ejpam-6609	50	7	follows	follow	VERB
ejpam-6609	50	8	:	:	PUNCT
ejpam-6609	50	9	the	the	DET
ejpam-6609	50	10	basic	basic	ADJ
ejpam-6609	50	11	foundation	foundation	NOUN
ejpam-6609	50	12	of	of	ADP
ejpam-6609	50	13	gips	gips	PROPN
ejpam-6609	50	14	is	be	AUX
ejpam-6609	50	15	established	establish	VERB
ejpam-6609	50	16	in	in	ADP
ejpam-6609	50	17	section	section	NOUN
ejpam-6609	50	18	2	2	NUM
ejpam-6609	50	19	,	,	PUNCT
ejpam-6609	50	20	which	which	PRON
ejpam-6609	50	21	also	also	ADV
ejpam-6609	50	22	goes	go	VERB
ejpam-6609	50	23	over	over	ADP
ejpam-6609	50	24	key	key	ADJ
ejpam-6609	50	25	fpt	fpt	PROPN
ejpam-6609	50	26	ideas	idea	NOUN
ejpam-6609	50	27	.	.	PUNCT
ejpam-6609	51	1	our	our	PRON
ejpam-6609	51	2	main	main	ADJ
ejpam-6609	51	3	theoretical	theoretical	ADJ
ejpam-6609	51	4	findings	finding	NOUN
ejpam-6609	51	5	are	be	AUX
ejpam-6609	51	6	shared	share	VERB
ejpam-6609	51	7	in	in	ADP
ejpam-6609	51	8	section	section	NOUN
ejpam-6609	51	9	3	3	NUM
ejpam-6609	51	10	,	,	PUNCT
ejpam-6609	51	11	which	which	PRON
ejpam-6609	51	12	also	also	ADV
ejpam-6609	51	13	explores	explore	VERB
ejpam-6609	51	14	how	how	SCONJ
ejpam-6609	51	15	these	these	DET
ejpam-6609	51	16	findings	finding	NOUN
ejpam-6609	51	17	can	can	AUX
ejpam-6609	51	18	be	be	AUX
ejpam-6609	51	19	used	use	VERB
ejpam-6609	51	20	in	in	ADP
ejpam-6609	51	21	ai	ai	NOUN
ejpam-6609	51	22	and	and	CCONJ
ejpam-6609	51	23	demonstrates	demonstrate	VERB
ejpam-6609	51	24	that	that	SCONJ
ejpam-6609	51	25	our	our	PRON
ejpam-6609	51	26	results	result	NOUN
ejpam-6609	51	27	ensure	ensure	VERB
ejpam-6609	51	28	convergence	convergence	NOUN
ejpam-6609	51	29	for	for	ADP
ejpam-6609	51	30	dems	dem	NOUN
ejpam-6609	51	31	.	.	PUNCT
ejpam-6609	52	1	these	these	PRON
ejpam-6609	52	2	include	include	VERB
ejpam-6609	52	3	novel	novel	ADJ
ejpam-6609	52	4	fixed	fix	VERB
ejpam-6609	52	5	-	-	PUNCT
ejpam-6609	52	6	point	point	NOUN
ejpam-6609	52	7	theorems	theorem	NOUN
ejpam-6609	52	8	for	for	ADP
ejpam-6609	52	9	3	3	NUM
ejpam-6609	52	10	-	-	PUNCT
ejpam-6609	52	11	sms	sms	NOUN
ejpam-6609	52	12	under	under	ADP
ejpam-6609	52	13	weakened	weakened	ADJ
ejpam-6609	52	14	contraction	contraction	NOUN
ejpam-6609	52	15	conditions	condition	NOUN
ejpam-6609	52	16	.	.	PUNCT
ejpam-6609	53	1	comparative	comparative	ADJ
ejpam-6609	53	2	analysis	analysis	NOUN
ejpam-6609	53	3	is	be	AUX
ejpam-6609	53	4	implemented	implement	VERB
ejpam-6609	53	5	in	in	ADP
ejpam-6609	53	6	section	section	NOUN
ejpam-6609	53	7	4	4	NUM
ejpam-6609	53	8	.	.	PUNCT
ejpam-6609	53	9	section	section	NOUN
ejpam-6609	53	10	5	5	NUM
ejpam-6609	53	11	wraps	wrap	NOUN
ejpam-6609	53	12	up	up	ADP
ejpam-6609	53	13	with	with	ADP
ejpam-6609	53	14	conclusions	conclusion	NOUN
ejpam-6609	53	15	and	and	CCONJ
ejpam-6609	53	16	suggestions	suggestion	NOUN
ejpam-6609	53	17	for	for	ADP
ejpam-6609	53	18	future	future	ADJ
ejpam-6609	53	19	research	research	NOUN
ejpam-6609	53	20	,	,	PUNCT
ejpam-6609	53	21	highlighting	highlight	VERB
ejpam-6609	53	22	the	the	DET
ejpam-6609	53	23	increasing	increase	VERB
ejpam-6609	53	24	importance	importance	NOUN
ejpam-6609	53	25	of	of	ADP
ejpam-6609	53	26	gips	gips	PROPN
ejpam-6609	53	27	in	in	ADP
ejpam-6609	53	28	both	both	DET
ejpam-6609	53	29	computer	computer	NOUN
ejpam-6609	53	30	applications	application	NOUN
ejpam-6609	53	31	and	and	CCONJ
ejpam-6609	53	32	theoretical	theoretical	ADJ
ejpam-6609	53	33	math	math	NOUN
ejpam-6609	53	34	.	.	PUNCT
ejpam-6609	54	1	2	2	X
ejpam-6609	54	2	.	.	X
ejpam-6609	54	3	preliminaries	preliminary	NOUN
ejpam-6609	54	4	this	this	DET
ejpam-6609	54	5	section	section	NOUN
ejpam-6609	54	6	focuses	focus	VERB
ejpam-6609	54	7	on	on	ADP
ejpam-6609	54	8	the	the	DET
ejpam-6609	54	9	essential	essential	ADJ
ejpam-6609	54	10	definitions	definition	NOUN
ejpam-6609	54	11	needed	need	VERB
ejpam-6609	54	12	for	for	ADP
ejpam-6609	54	13	the	the	DET
ejpam-6609	54	14	sections	section	NOUN
ejpam-6609	54	15	that	that	PRON
ejpam-6609	54	16	follow	follow	VERB
ejpam-6609	54	17	.	.	PUNCT
ejpam-6609	55	1	2.1	2.1	NUM
ejpam-6609	55	2	.	.	PUNCT
ejpam-6609	56	1	generalized	generalize	VERB
ejpam-6609	56	2	inner	inner	ADJ
ejpam-6609	56	3	product	product	NOUN
ejpam-6609	56	4	spaces	space	VERB
ejpam-6609	56	5	definition	definition	NOUN
ejpam-6609	56	6	1	1	NUM
ejpam-6609	56	7	.	.	PUNCT
ejpam-6609	57	1	[	[	X
ejpam-6609	57	2	13	13	NUM
ejpam-6609	57	3	]	]	PUNCT
ejpam-6609	57	4	suppose	suppose	VERB
ejpam-6609	57	5	v	v	SCONJ
ejpam-6609	57	6	̸=	̸=	PROPN
ejpam-6609	57	7	∅	∅	NOUN
ejpam-6609	57	8	is	be	AUX
ejpam-6609	57	9	a	a	DET
ejpam-6609	57	10	vector	vector	NOUN
ejpam-6609	57	11	space	space	NOUN
ejpam-6609	57	12	over	over	ADP
ejpam-6609	57	13	a	a	DET
ejpam-6609	57	14	field	field	NOUN
ejpam-6609	57	15	f	f	NOUN
ejpam-6609	57	16	(	(	PUNCT
ejpam-6609	57	17	where	where	SCONJ
ejpam-6609	57	18	f	f	PROPN
ejpam-6609	57	19	is	be	AUX
ejpam-6609	57	20	either	either	PRON
ejpam-6609	57	21	r	r	NOUN
ejpam-6609	57	22	or	or	CCONJ
ejpam-6609	57	23	c	c	NOUN
ejpam-6609	57	24	)	)	PUNCT
ejpam-6609	57	25	,	,	PUNCT
ejpam-6609	57	26	and	and	CCONJ
ejpam-6609	57	27	define	define	VERB
ejpam-6609	57	28	⟨	⟨	NOUN
ejpam-6609	57	29	·	·	PUNCT
ejpam-6609	57	30	,	,	PUNCT
ejpam-6609	57	31	·	·	PUNCT
ejpam-6609	57	32	⟩	⟩	NOUN
ejpam-6609	57	33	:	:	PUNCT
ejpam-6609	57	34	v	v	NUM
ejpam-6609	57	35	×	×	NOUN
ejpam-6609	57	36	v	v	NOUN
ejpam-6609	57	37	→	→	SYM
ejpam-6609	57	38	f	f	PROPN
ejpam-6609	57	39	as	as	ADP
ejpam-6609	57	40	an	an	DET
ejpam-6609	57	41	inner	inner	ADJ
ejpam-6609	57	42	product	product	NOUN
ejpam-6609	57	43	if	if	SCONJ
ejpam-6609	57	44	and	and	CCONJ
ejpam-6609	57	45	only	only	ADV
ejpam-6609	57	46	if	if	SCONJ
ejpam-6609	57	47	the	the	DET
ejpam-6609	57	48	subsequent	subsequent	ADJ
ejpam-6609	57	49	axioms	axiom	NOUN
ejpam-6609	57	50	are	be	AUX
ejpam-6609	57	51	satisfied	satisfied	ADJ
ejpam-6609	57	52	:	:	PUNCT
ejpam-6609	57	53	i.	i.	PROPN
ejpam-6609	57	54	⟨v	⟨v	PROPN
ejpam-6609	57	55	,	,	PUNCT
ejpam-6609	57	56	v⟩	v⟩	X
ejpam-6609	57	57	≥	≥	NOUN
ejpam-6609	57	58	0	0	NUM
ejpam-6609	57	59	∀	∀	NOUN
ejpam-6609	57	60	v	v	ADP
ejpam-6609	57	61	∈	∈	NOUN
ejpam-6609	57	62	v	v	NOUN
ejpam-6609	57	63	and	and	CCONJ
ejpam-6609	57	64	⟨v	⟨v	NOUN
ejpam-6609	57	65	,	,	PUNCT
ejpam-6609	57	66	v⟩	v⟩	NOUN
ejpam-6609	57	67	=	=	SYM
ejpam-6609	57	68	0	0	PUNCT
ejpam-6609	58	1	iff	iff	PROPN
ejpam-6609	58	2	v	v	NOUN
ejpam-6609	58	3	=	=	SYM
ejpam-6609	58	4	0	0	PUNCT
ejpam-6609	59	1	(	(	PUNCT
ejpam-6609	59	2	positive	positive	ADJ
ejpam-6609	59	3	definiteness	definiteness	NOUN
ejpam-6609	59	4	)	)	PUNCT
ejpam-6609	59	5	,	,	PUNCT
ejpam-6609	59	6	ii	ii	PROPN
ejpam-6609	59	7	.	.	PROPN
ejpam-6609	59	8	⟨u	⟨u	PROPN
ejpam-6609	59	9	,	,	PUNCT
ejpam-6609	59	10	v⟩	v⟩	NOUN
ejpam-6609	59	11	=	=	SYM
ejpam-6609	59	12	⟨v	⟨v	PROPN
ejpam-6609	59	13	,	,	PUNCT
ejpam-6609	59	14	u⟩	u⟩	X
ejpam-6609	59	15	∀	∀	X
ejpam-6609	59	16	u	u	NOUN
ejpam-6609	59	17	,	,	PUNCT
ejpam-6609	59	18	v	v	NOUN
ejpam-6609	59	19	∈	∈	PROPN
ejpam-6609	59	20	v	v	NOUN
ejpam-6609	59	21	(	(	PUNCT
ejpam-6609	59	22	conjugate	conjugate	ADJ
ejpam-6609	59	23	symmetry	symmetry	NOUN
ejpam-6609	59	24	)	)	PUNCT
ejpam-6609	59	25	,	,	PUNCT
ejpam-6609	59	26	iii	iii	X
ejpam-6609	59	27	.	.	PUNCT
ejpam-6609	60	1	⟨αu	⟨αu	NOUN
ejpam-6609	61	1	+	+	CCONJ
ejpam-6609	61	2	βv	βv	PUNCT
ejpam-6609	61	3	,	,	PUNCT
ejpam-6609	61	4	w⟩	w⟩	NOUN
ejpam-6609	61	5	=	=	SYM
ejpam-6609	62	1	α⟨u	α⟨u	NUM
ejpam-6609	62	2	,	,	PUNCT
ejpam-6609	62	3	w⟩	w⟩	NOUN
ejpam-6609	62	4	+	+	CCONJ
ejpam-6609	62	5	β⟨v	β⟨v	NUM
ejpam-6609	62	6	,	,	PUNCT
ejpam-6609	62	7	w⟩	w⟩	NOUN
ejpam-6609	62	8	∀	∀	X
ejpam-6609	62	9	u	u	NOUN
ejpam-6609	62	10	,	,	PUNCT
ejpam-6609	62	11	v	v	NOUN
ejpam-6609	62	12	,	,	PUNCT
ejpam-6609	62	13	w	w	PROPN
ejpam-6609	62	14	∈	∈	PROPN
ejpam-6609	62	15	v	v	NOUN
ejpam-6609	62	16	and	and	CCONJ
ejpam-6609	62	17	∀	∀	NOUN
ejpam-6609	62	18	α	α	NOUN
ejpam-6609	62	19	,	,	PUNCT
ejpam-6609	62	20	β	β	X
ejpam-6609	62	21	∈	∈	PROPN
ejpam-6609	62	22	f	f	X
ejpam-6609	62	23	(	(	PUNCT
ejpam-6609	62	24	linearity	linearity	NOUN
ejpam-6609	62	25	in	in	ADP
ejpam-6609	62	26	the	the	DET
ejpam-6609	62	27	first	first	ADJ
ejpam-6609	62	28	argument	argument	NOUN
ejpam-6609	62	29	)	)	PUNCT
ejpam-6609	62	30	.	.	PUNCT
ejpam-6609	63	1	the	the	DET
ejpam-6609	63	2	pair	pair	NOUN
ejpam-6609	63	3	(	(	PUNCT
ejpam-6609	63	4	v	v	NOUN
ejpam-6609	63	5	,	,	PUNCT
ejpam-6609	63	6	⟨	⟨	NOUN
ejpam-6609	63	7	·	·	SYM
ejpam-6609	63	8	,	,	PUNCT
ejpam-6609	63	9	·	·	PUNCT
ejpam-6609	63	10	⟩	⟩	NOUN
ejpam-6609	63	11	)	)	PUNCT
ejpam-6609	63	12	is	be	AUX
ejpam-6609	63	13	called	call	VERB
ejpam-6609	63	14	an	an	DET
ejpam-6609	63	15	inner	inner	ADJ
ejpam-6609	63	16	product	product	NOUN
ejpam-6609	63	17	space	space	NOUN
ejpam-6609	63	18	.	.	PUNCT
ejpam-6609	64	1	the	the	DET
ejpam-6609	64	2	concepts	concept	NOUN
ejpam-6609	64	3	of	of	ADP
ejpam-6609	64	4	length	length	NOUN
ejpam-6609	64	5	,	,	PUNCT
ejpam-6609	64	6	angle	angle	NOUN
ejpam-6609	64	7	,	,	PUNCT
ejpam-6609	64	8	and	and	CCONJ
ejpam-6609	64	9	orthogonality	orthogonality	NOUN
ejpam-6609	64	10	from	from	ADP
ejpam-6609	64	11	geometry	geometry	NOUN
ejpam-6609	64	12	are	be	AUX
ejpam-6609	64	13	important	important	ADJ
ejpam-6609	64	14	because	because	SCONJ
ejpam-6609	64	15	they	they	PRON
ejpam-6609	64	16	can	can	AUX
ejpam-6609	64	17	be	be	AUX
ejpam-6609	64	18	applied	apply	VERB
ejpam-6609	64	19	to	to	ADP
ejpam-6609	64	20	abstract	abstract	ADJ
ejpam-6609	64	21	vector	vector	NOUN
ejpam-6609	64	22	spaces	space	NOUN
ejpam-6609	64	23	.	.	PUNCT
ejpam-6609	65	1	such	such	ADJ
ejpam-6609	65	2	hilbert	hilbert	NOUN
ejpam-6609	65	3	spaces	space	NOUN
ejpam-6609	65	4	help	help	VERB
ejpam-6609	65	5	to	to	PART
ejpam-6609	65	6	frame	frame	VERB
ejpam-6609	65	7	the	the	DET
ejpam-6609	65	8	study	study	NOUN
ejpam-6609	65	9	,	,	PUNCT
ejpam-6609	65	10	which	which	PRON
ejpam-6609	65	11	plays	play	VERB
ejpam-6609	65	12	key	key	ADJ
ejpam-6609	65	13	roles	role	NOUN
ejpam-6609	65	14	in	in	ADP
ejpam-6609	65	15	quantum	quantum	ADJ
ejpam-6609	65	16	mechanics	mechanic	NOUN
ejpam-6609	65	17	,	,	PUNCT
ejpam-6609	65	18	signal	signal	ADJ
ejpam-6609	65	19	processing	processing	NOUN
ejpam-6609	65	20	,	,	PUNCT
ejpam-6609	65	21	and	and	CCONJ
ejpam-6609	65	22	functional	functional	ADJ
ejpam-6609	65	23	analysis	analysis	NOUN
ejpam-6609	65	24	.	.	PUNCT
ejpam-6609	66	1	the	the	DET
ejpam-6609	66	2	inner	inner	ADJ
ejpam-6609	66	3	product	product	NOUN
ejpam-6609	66	4	structure	structure	NOUN
ejpam-6609	66	5	supports	support	VERB
ejpam-6609	66	6	the	the	DET
ejpam-6609	66	7	use	use	NOUN
ejpam-6609	66	8	of	of	ADP
ejpam-6609	66	9	projections	projection	NOUN
ejpam-6609	66	10	,	,	PUNCT
ejpam-6609	66	11	orthogonal	orthogonal	ADJ
ejpam-6609	66	12	decompositions	decomposition	NOUN
ejpam-6609	66	13	,	,	PUNCT
ejpam-6609	66	14	and	and	CCONJ
ejpam-6609	66	15	different	different	ADJ
ejpam-6609	66	16	ways	way	NOUN
ejpam-6609	66	17	of	of	ADP
ejpam-6609	66	18	approximating	approximate	VERB
ejpam-6609	66	19	results	result	NOUN
ejpam-6609	66	20	important	important	ADJ
ejpam-6609	66	21	in	in	ADP
ejpam-6609	66	22	applied	apply	VERB
ejpam-6609	66	23	mathematics	mathematic	NOUN
ejpam-6609	66	24	and	and	CCONJ
ejpam-6609	66	25	physics	physics	PROPN
ejpam-6609	66	26	.	.	PUNCT
ejpam-6609	67	1	m.	m.	PROPN
ejpam-6609	67	2	noorwali	noorwali	PROPN
ejpam-6609	67	3	et	et	PROPN
ejpam-6609	67	4	al	al	PROPN
ejpam-6609	67	5	.	.	PUNCT
ejpam-6609	67	6	/	/	SYM
ejpam-6609	67	7	eur	eur	PROPN
ejpam-6609	67	8	.	.	PUNCT
ejpam-6609	68	1	j.	j.	PROPN
ejpam-6609	68	2	pure	pure	PROPN
ejpam-6609	68	3	appl	appl	PROPN
ejpam-6609	68	4	.	.	PROPN
ejpam-6609	68	5	math	math	PROPN
ejpam-6609	68	6	,	,	PUNCT
ejpam-6609	68	7	18	18	NUM
ejpam-6609	68	8	(	(	PUNCT
ejpam-6609	68	9	3	3	NUM
ejpam-6609	68	10	)	)	PUNCT
ejpam-6609	68	11	(	(	PUNCT
ejpam-6609	68	12	2025	2025	NUM
ejpam-6609	68	13	)	)	PUNCT
ejpam-6609	68	14	,	,	PUNCT
ejpam-6609	68	15	6609	6609	NUM
ejpam-6609	68	16	4	4	NUM
ejpam-6609	68	17	of	of	ADP
ejpam-6609	68	18	17	17	NUM
ejpam-6609	68	19	definition	definition	NOUN
ejpam-6609	68	20	2	2	NUM
ejpam-6609	68	21	.	.	PUNCT
ejpam-6609	69	1	[	[	X
ejpam-6609	69	2	13	13	NUM
ejpam-6609	69	3	]	]	PUNCT
ejpam-6609	69	4	suppose	suppose	VERB
ejpam-6609	69	5	(	(	PUNCT
ejpam-6609	69	6	v	v	NOUN
ejpam-6609	69	7	,	,	PUNCT
ejpam-6609	69	8	⟨	⟨	NOUN
ejpam-6609	69	9	·	·	SYM
ejpam-6609	69	10	,	,	PUNCT
ejpam-6609	69	11	·	·	PUNCT
ejpam-6609	69	12	⟩	⟩	NOUN
ejpam-6609	69	13	)	)	PUNCT
ejpam-6609	69	14	is	be	AUX
ejpam-6609	69	15	an	an	DET
ejpam-6609	69	16	inner	inner	ADJ
ejpam-6609	69	17	product	product	NOUN
ejpam-6609	69	18	space	space	NOUN
ejpam-6609	69	19	with	with	ADP
ejpam-6609	69	20	induced	induced	ADJ
ejpam-6609	69	21	norm	norm	NOUN
ejpam-6609	69	22	∥v∥	∥v∥	NOUN
ejpam-6609	69	23	=	=	SYM
ejpam-6609	69	24	√	√	NUM
ejpam-6609	69	25	⟨v	⟨v	NUM
ejpam-6609	69	26	,	,	PUNCT
ejpam-6609	69	27	v⟩	v⟩	NOUN
ejpam-6609	69	28	and	and	CCONJ
ejpam-6609	69	29	{	{	PUNCT
ejpam-6609	69	30	vi	vi	X
ejpam-6609	69	31	}	}	PUNCT
ejpam-6609	69	32	be	be	AUX
ejpam-6609	69	33	a	a	DET
ejpam-6609	69	34	sequence	sequence	NOUN
ejpam-6609	69	35	in	in	ADP
ejpam-6609	69	36	v.	v.	ADP
ejpam-6609	69	37	then	then	ADV
ejpam-6609	69	38	,	,	PUNCT
ejpam-6609	69	39	i.	i.	PROPN
ejpam-6609	69	40	{	{	PUNCT
ejpam-6609	69	41	vi	vi	PROPN
ejpam-6609	69	42	}	}	PUNCT
ejpam-6609	69	43	is	be	AUX
ejpam-6609	69	44	called	call	VERB
ejpam-6609	69	45	a	a	DET
ejpam-6609	69	46	cauchy	cauchy	ADJ
ejpam-6609	69	47	sequence	sequence	NOUN
ejpam-6609	69	48	if	if	SCONJ
ejpam-6609	69	49	for	for	ADP
ejpam-6609	69	50	any	any	DET
ejpam-6609	69	51	ε	ε	PROPN
ejpam-6609	69	52	>	>	X
ejpam-6609	69	53	0	0	PROPN
ejpam-6609	69	54	,	,	PUNCT
ejpam-6609	69	55	∃	∃	PROPN
ejpam-6609	69	56	n0	n0	PROPN
ejpam-6609	69	57	∈	∈	PROPN
ejpam-6609	69	58	n	n	CCONJ
ejpam-6609	69	59	such	such	ADJ
ejpam-6609	69	60	that	that	DET
ejpam-6609	69	61	∥vi	∥vi	PROPN
ejpam-6609	70	1	−	−	PROPN
ejpam-6609	71	1	vj∥	vj∥	INTJ
ejpam-6609	71	2	<	<	X
ejpam-6609	71	3	ε	ε	PROPN
ejpam-6609	71	4	∀	∀	X
ejpam-6609	72	1	i	i	PROPN
ejpam-6609	72	2	,	,	PUNCT
ejpam-6609	72	3	j	j	PROPN
ejpam-6609	72	4	≥	≥	PROPN
ejpam-6609	72	5	n0	n0	PROPN
ejpam-6609	72	6	.	.	PUNCT
ejpam-6609	72	7	ii	ii	PROPN
ejpam-6609	72	8	.	.	PUNCT
ejpam-6609	72	9	{	{	PUNCT
ejpam-6609	72	10	vi	vi	NOUN
ejpam-6609	72	11	}	}	PUNCT
ejpam-6609	72	12	converges	converge	NOUN
ejpam-6609	72	13	to	to	ADP
ejpam-6609	72	14	an	an	DET
ejpam-6609	72	15	element	element	NOUN
ejpam-6609	72	16	v	v	ADP
ejpam-6609	72	17	∈	∈	PROPN
ejpam-6609	72	18	v	v	NOUN
ejpam-6609	72	19	if	if	SCONJ
ejpam-6609	72	20	∀	∀	NOUN
ejpam-6609	72	21	ε	ε	X
ejpam-6609	72	22	>	>	X
ejpam-6609	72	23	0	0	PROPN
ejpam-6609	72	24	,	,	PUNCT
ejpam-6609	72	25	∃	∃	PROPN
ejpam-6609	72	26	n0	n0	PROPN
ejpam-6609	72	27	∈	∈	PROPN
ejpam-6609	72	28	n	n	CCONJ
ejpam-6609	72	29	such	such	ADJ
ejpam-6609	72	30	that	that	SCONJ
ejpam-6609	72	31	∥v	∥v	PROPN
ejpam-6609	72	32	−	−	PROPN
ejpam-6609	73	1	vi∥	vi∥	NOUN
ejpam-6609	73	2	<	<	X
ejpam-6609	73	3	ε	ε	PROPN
ejpam-6609	73	4	whenever	whenever	SCONJ
ejpam-6609	73	5	i	i	PRON
ejpam-6609	73	6	≥	≥	VERB
ejpam-6609	73	7	n0	n0	PROPN
ejpam-6609	73	8	.	.	PUNCT
ejpam-6609	73	9	iii	iii	PROPN
ejpam-6609	73	10	.	.	PUNCT
ejpam-6609	74	1	(	(	PUNCT
ejpam-6609	74	2	v	v	NOUN
ejpam-6609	74	3	,	,	PUNCT
ejpam-6609	74	4	⟨	⟨	NOUN
ejpam-6609	74	5	·	·	SYM
ejpam-6609	74	6	,	,	PUNCT
ejpam-6609	74	7	·	·	PUNCT
ejpam-6609	74	8	⟩	⟩	NOUN
ejpam-6609	74	9	)	)	PUNCT
ejpam-6609	74	10	is	be	AUX
ejpam-6609	74	11	complete	complete	ADJ
ejpam-6609	74	12	if	if	SCONJ
ejpam-6609	74	13	every	every	DET
ejpam-6609	74	14	cauchy	cauchy	ADJ
ejpam-6609	74	15	sequence	sequence	NOUN
ejpam-6609	74	16	converges	converge	VERB
ejpam-6609	74	17	in	in	ADP
ejpam-6609	74	18	v.	v.	ADP
ejpam-6609	74	19	a	a	DET
ejpam-6609	74	20	complete	complete	ADJ
ejpam-6609	74	21	inner	inner	ADJ
ejpam-6609	74	22	product	product	NOUN
ejpam-6609	74	23	space	space	NOUN
ejpam-6609	74	24	is	be	AUX
ejpam-6609	74	25	called	call	VERB
ejpam-6609	74	26	a	a	DET
ejpam-6609	74	27	hs	hs	PROPN
ejpam-6609	74	28	.	.	PROPN
ejpam-6609	74	29	proposition	proposition	NOUN
ejpam-6609	74	30	1	1	NUM
ejpam-6609	74	31	.	.	PUNCT
ejpam-6609	75	1	[	[	X
ejpam-6609	75	2	13	13	NUM
ejpam-6609	75	3	]	]	PUNCT
ejpam-6609	75	4	suppose	suppose	VERB
ejpam-6609	75	5	(	(	PUNCT
ejpam-6609	75	6	v	v	NOUN
ejpam-6609	75	7	,	,	PUNCT
ejpam-6609	75	8	⟨	⟨	NOUN
ejpam-6609	75	9	·	·	SYM
ejpam-6609	75	10	,	,	PUNCT
ejpam-6609	75	11	·	·	PUNCT
ejpam-6609	75	12	⟩	⟩	NOUN
ejpam-6609	75	13	)	)	PUNCT
ejpam-6609	75	14	be	be	VERB
ejpam-6609	75	15	an	an	DET
ejpam-6609	75	16	inner	inner	ADJ
ejpam-6609	75	17	product	product	NOUN
ejpam-6609	75	18	space	space	NOUN
ejpam-6609	75	19	with	with	ADP
ejpam-6609	75	20	induced	induced	ADJ
ejpam-6609	75	21	norm	norm	NOUN
ejpam-6609	75	22	∥v∥	∥v∥	ADJ
ejpam-6609	75	23	=	=	SYM
ejpam-6609	75	24	√	√	NUM
ejpam-6609	75	25	⟨v	⟨v	NUM
ejpam-6609	75	26	,	,	PUNCT
ejpam-6609	75	27	v⟩.	v⟩.	VERB
ejpam-6609	75	28	then	then	ADV
ejpam-6609	75	29	for	for	ADP
ejpam-6609	75	30	any	any	DET
ejpam-6609	75	31	u	u	NOUN
ejpam-6609	75	32	,	,	PUNCT
ejpam-6609	75	33	v	v	NOUN
ejpam-6609	75	34	,	,	PUNCT
ejpam-6609	75	35	w	w	PROPN
ejpam-6609	75	36	,	,	PUNCT
ejpam-6609	75	37	x	x	SYM
ejpam-6609	75	38	∈	∈	PROPN
ejpam-6609	75	39	v	v	NOUN
ejpam-6609	75	40	,	,	PUNCT
ejpam-6609	75	41	the	the	DET
ejpam-6609	75	42	following	follow	VERB
ejpam-6609	75	43	hold	hold	NOUN
ejpam-6609	75	44	:	:	PUNCT
ejpam-6609	75	45	i.	i.	NOUN
ejpam-6609	75	46	if	if	SCONJ
ejpam-6609	75	47	∥u−	∥u−	PROPN
ejpam-6609	75	48	v∥	v∥	PROPN
ejpam-6609	75	49	=	=	SYM
ejpam-6609	75	50	0	0	NUM
ejpam-6609	75	51	,	,	PUNCT
ejpam-6609	75	52	then	then	ADV
ejpam-6609	75	53	u	u	NOUN
ejpam-6609	75	54	=	=	PROPN
ejpam-6609	75	55	v	v	PROPN
ejpam-6609	75	56	iff	iff	PROPN
ejpam-6609	75	57	ii	ii	PROPN
ejpam-6609	75	58	.	.	PUNCT
ejpam-6609	76	1	∥u−	∥u−	PROPN
ejpam-6609	76	2	v∥	v∥	PROPN
ejpam-6609	76	3	≤	≤	NUM
ejpam-6609	76	4	∥u−	∥u−	PROPN
ejpam-6609	76	5	w∥+	w∥+	PROPN
ejpam-6609	76	6	∥w	∥w	PROPN
ejpam-6609	76	7	−	−	PROPN
ejpam-6609	76	8	v∥	v∥	PROPN
ejpam-6609	76	9	iii	iii	NOUN
ejpam-6609	76	10	.	.	PUNCT
ejpam-6609	77	1	|∥u∥	|∥u∥	NUM
ejpam-6609	77	2	−	−	PROPN
ejpam-6609	77	3	∥v∥|	∥v∥|	PROPN
ejpam-6609	77	4	≤	≤	NUM
ejpam-6609	77	5	∥u−	∥u−	PROPN
ejpam-6609	77	6	v∥	v∥	PROPN
ejpam-6609	77	7	iv	iv	NUM
ejpam-6609	77	8	.	.	PUNCT
ejpam-6609	78	1	∥u+	∥u+	NOUN
ejpam-6609	78	2	v∥2	v∥2	NOUN
ejpam-6609	78	3	+	+	CCONJ
ejpam-6609	78	4	∥u−	∥u−	NUM
ejpam-6609	78	5	v∥2	v∥2	NOUN
ejpam-6609	78	6	=	=	PUNCT
ejpam-6609	78	7	2(∥u∥2	2(∥u∥2	NUM
ejpam-6609	78	8	+	+	CCONJ
ejpam-6609	78	9	∥v∥2	∥v∥2	X
ejpam-6609	78	10	)	)	PUNCT
ejpam-6609	78	11	v.	v.	ADP
ejpam-6609	78	12	|⟨u	|⟨u	NOUN
ejpam-6609	78	13	,	,	PUNCT
ejpam-6609	78	14	v⟩|	v⟩|	PROPN
ejpam-6609	78	15	≤	≤	PROPN
ejpam-6609	78	16	∥u∥∥v∥	∥u∥∥v∥	PROPN
ejpam-6609	78	17	vi	vi	PROPN
ejpam-6609	78	18	.	.	PUNCT
ejpam-6609	79	1	∥u+	∥u+	NOUN
ejpam-6609	79	2	v∥	v∥	PROPN
ejpam-6609	79	3	≤	≤	NUM
ejpam-6609	79	4	∥u∥+	∥u∥+	VERB
ejpam-6609	79	5	∥v∥	∥v∥	PROPN
ejpam-6609	80	1	vii	vii	PROPN
ejpam-6609	80	2	.	.	PUNCT
ejpam-6609	80	3	|∥u−	|∥u−	PROPN
ejpam-6609	80	4	w∥	w∥	NOUN
ejpam-6609	81	1	−	−	PROPN
ejpam-6609	81	2	∥v	∥v	PROPN
ejpam-6609	81	3	−	−	PROPN
ejpam-6609	81	4	w∥|	w∥|	VERB
ejpam-6609	81	5	≤	≤	NUM
ejpam-6609	81	6	∥u−	∥u−	PROPN
ejpam-6609	81	7	v∥	v∥	PROPN
ejpam-6609	81	8	viii	viii	NOUN
ejpam-6609	81	9	.	.	PUNCT
ejpam-6609	82	1	|⟨u	|⟨u	NOUN
ejpam-6609	82	2	,	,	PUNCT
ejpam-6609	82	3	v⟩	v⟩	VERB
ejpam-6609	82	4	−	−	PROPN
ejpam-6609	82	5	⟨u	⟨u	NOUN
ejpam-6609	82	6	,	,	PUNCT
ejpam-6609	82	7	w⟩|	w⟩|	PROPN
ejpam-6609	82	8	≤	≤	PROPN
ejpam-6609	82	9	∥u∥∥v	∥u∥∥v	PROPN
ejpam-6609	82	10	−	−	PROPN
ejpam-6609	82	11	w∥	w∥	NOUN
ejpam-6609	82	12	ix	ix	PROPN
ejpam-6609	82	13	.	.	PUNCT
ejpam-6609	83	1	∥u−	∥u−	NOUN
ejpam-6609	83	2	v∥2	v∥2	NOUN
ejpam-6609	83	3	=	=	PUNCT
ejpam-6609	83	4	∥u∥2	∥u∥2	NOUN
ejpam-6609	83	5	+	+	CCONJ
ejpam-6609	83	6	∥v∥2	∥v∥2	ADJ
ejpam-6609	83	7	−	−	PROPN
ejpam-6609	83	8	2ℜ⟨u	2ℜ⟨u	NOUN
ejpam-6609	83	9	,	,	PUNCT
ejpam-6609	83	10	v⟩	v⟩	NOUN
ejpam-6609	83	11	x.	x.	NOUN
ejpam-6609	84	1	if	if	SCONJ
ejpam-6609	84	2	⟨u	⟨u	NOUN
ejpam-6609	84	3	,	,	PUNCT
ejpam-6609	84	4	w⟩	w⟩	NOUN
ejpam-6609	84	5	=	=	SYM
ejpam-6609	84	6	⟨v	⟨v	PROPN
ejpam-6609	84	7	,	,	PUNCT
ejpam-6609	84	8	w⟩	w⟩	NOUN
ejpam-6609	84	9	for	for	ADP
ejpam-6609	84	10	all	all	DET
ejpam-6609	84	11	w	w	PROPN
ejpam-6609	84	12	∈	∈	PROPN
ejpam-6609	84	13	v	v	NOUN
ejpam-6609	84	14	,	,	PUNCT
ejpam-6609	84	15	then	then	ADV
ejpam-6609	84	16	u	u	NOUN
ejpam-6609	84	17	=	=	PROPN
ejpam-6609	84	18	v	v	ADP
ejpam-6609	84	19	corollary	corollary	NOUN
ejpam-6609	84	20	1	1	NUM
ejpam-6609	84	21	.	.	PUNCT
ejpam-6609	85	1	[	[	X
ejpam-6609	85	2	13	13	NUM
ejpam-6609	85	3	]	]	PUNCT
ejpam-6609	85	4	suppose	suppose	VERB
ejpam-6609	85	5	(	(	PUNCT
ejpam-6609	85	6	v	v	NOUN
ejpam-6609	85	7	,	,	PUNCT
ejpam-6609	85	8	⟨	⟨	NOUN
ejpam-6609	85	9	·	·	SYM
ejpam-6609	85	10	,	,	PUNCT
ejpam-6609	85	11	·	·	PUNCT
ejpam-6609	85	12	⟩	⟩	NOUN
ejpam-6609	85	13	)	)	PUNCT
ejpam-6609	85	14	is	be	AUX
ejpam-6609	85	15	an	an	DET
ejpam-6609	85	16	inner	inner	ADJ
ejpam-6609	85	17	product	product	NOUN
ejpam-6609	85	18	space	space	NOUN
ejpam-6609	85	19	.	.	PUNCT
ejpam-6609	86	1	then	then	ADV
ejpam-6609	86	2	,	,	PUNCT
ejpam-6609	86	3	for	for	ADP
ejpam-6609	86	4	any	any	DET
ejpam-6609	86	5	u	u	NOUN
ejpam-6609	86	6	,	,	PUNCT
ejpam-6609	86	7	v	v	NOUN
ejpam-6609	86	8	∈	∈	PROPN
ejpam-6609	86	9	v	v	NOUN
ejpam-6609	86	10	:	:	PUNCT
ejpam-6609	86	11	(	(	PUNCT
ejpam-6609	86	12	i	i	NOUN
ejpam-6609	86	13	)	)	PUNCT
ejpam-6609	86	14	(	(	PUNCT
ejpam-6609	86	15	angle	angle	NOUN
ejpam-6609	86	16	inequality	inequality	NOUN
ejpam-6609	86	17	)	)	PUNCT
ejpam-6609	86	18	the	the	DET
ejpam-6609	86	19	angle	angle	NOUN
ejpam-6609	86	20	θ	θ	PROPN
ejpam-6609	86	21	between	between	ADP
ejpam-6609	86	22	u	u	NOUN
ejpam-6609	86	23	and	and	CCONJ
ejpam-6609	86	24	v	v	NOUN
ejpam-6609	86	25	satisfies	satisfie	NOUN
ejpam-6609	86	26	:	:	PUNCT
ejpam-6609	86	27	cos	cos	PROPN
ejpam-6609	86	28	θ	θ	PROPN
ejpam-6609	86	29	=	=	SYM
ejpam-6609	86	30	|⟨u	|⟨u	NOUN
ejpam-6609	86	31	,	,	PUNCT
ejpam-6609	86	32	v⟩|	v⟩|	PROPN
ejpam-6609	86	33	∥u∥∥v∥	∥u∥∥v∥	ADJ
ejpam-6609	86	34	≤	≤	ADJ
ejpam-6609	86	35	1	1	NUM
ejpam-6609	86	36	.	.	PUNCT
ejpam-6609	86	37	(	(	PUNCT
ejpam-6609	86	38	ii	ii	NOUN
ejpam-6609	86	39	)	)	PUNCT
ejpam-6609	86	40	(	(	PUNCT
ejpam-6609	86	41	orthogonality	orthogonality	NOUN
ejpam-6609	86	42	condition	condition	NOUN
ejpam-6609	86	43	)	)	PUNCT
ejpam-6609	87	1	if	if	SCONJ
ejpam-6609	87	2	⟨u	⟨u	NOUN
ejpam-6609	87	3	,	,	PUNCT
ejpam-6609	87	4	v⟩	v⟩	NOUN
ejpam-6609	87	5	=	=	SYM
ejpam-6609	87	6	0	0	NUM
ejpam-6609	87	7	,	,	PUNCT
ejpam-6609	87	8	then	then	ADV
ejpam-6609	87	9	:	:	PUNCT
ejpam-6609	87	10	∥u+	∥u+	VERB
ejpam-6609	87	11	v∥2	v∥2	NOUN
ejpam-6609	87	12	=	=	PUNCT
ejpam-6609	87	13	∥u∥2	∥u∥2	NOUN
ejpam-6609	87	14	+	+	CCONJ
ejpam-6609	87	15	∥v∥2	∥v∥2	X
ejpam-6609	87	16	(	(	PUNCT
ejpam-6609	87	17	pythagorean	pythagorean	PROPN
ejpam-6609	87	18	theorem	theorem	PROPN
ejpam-6609	87	19	)	)	PUNCT
ejpam-6609	87	20	.	.	PUNCT
ejpam-6609	88	1	(	(	PUNCT
ejpam-6609	88	2	iii	iii	X
ejpam-6609	88	3	)	)	PUNCT
ejpam-6609	88	4	(	(	PUNCT
ejpam-6609	88	5	norm	norm	NOUN
ejpam-6609	88	6	uniformity	uniformity	NOUN
ejpam-6609	88	7	)	)	PUNCT
ejpam-6609	88	8	for	for	ADP
ejpam-6609	88	9	any	any	DET
ejpam-6609	88	10	scalar	scalar	ADJ
ejpam-6609	88	11	α	α	NOUN
ejpam-6609	88	12	∈	∈	ADJ
ejpam-6609	89	1	f	f	X
ejpam-6609	89	2	:	:	PUNCT
ejpam-6609	89	3	∥αu∥	∥αu∥	PROPN
ejpam-6609	89	4	=	=	SYM
ejpam-6609	89	5	|α|	|α|	PROPN
ejpam-6609	89	6	·	·	PUNCT
ejpam-6609	89	7	∥u∥.	∥u∥.	NOUN
ejpam-6609	89	8	remark	remark	NOUN
ejpam-6609	89	9	1	1	NUM
ejpam-6609	89	10	.	.	PUNCT
ejpam-6609	90	1	[	[	X
ejpam-6609	90	2	13	13	NUM
ejpam-6609	90	3	]	]	SYM
ejpam-6609	90	4	inner	inner	ADJ
ejpam-6609	90	5	product	product	NOUN
ejpam-6609	90	6	spaces	space	NOUN
ejpam-6609	90	7	carry	carry	VERB
ejpam-6609	90	8	into	into	ADP
ejpam-6609	90	9	their	their	PRON
ejpam-6609	90	10	own	own	ADJ
ejpam-6609	90	11	abstract	abstract	ADJ
ejpam-6609	90	12	forms	form	NOUN
ejpam-6609	90	13	some	some	PRON
ejpam-6609	90	14	of	of	ADP
ejpam-6609	90	15	the	the	DET
ejpam-6609	90	16	features	feature	NOUN
ejpam-6609	90	17	of	of	ADP
ejpam-6609	90	18	euclidean	euclidean	ADJ
ejpam-6609	90	19	geometry	geometry	NOUN
ejpam-6609	90	20	.	.	PUNCT
ejpam-6609	91	1	•	•	NUM
ejpam-6609	91	2	the	the	DET
ejpam-6609	91	3	cauchy	cauchy	PROPN
ejpam-6609	91	4	-	-	PUNCT
ejpam-6609	91	5	schwarz	schwarz	PROPN
ejpam-6609	91	6	inequality	inequality	NOUN
ejpam-6609	91	7	makes	make	VERB
ejpam-6609	91	8	sure	sure	ADJ
ejpam-6609	91	9	that	that	SCONJ
ejpam-6609	91	10	the	the	DET
ejpam-6609	91	11	“	"	PUNCT
ejpam-6609	91	12	angle	angle	NOUN
ejpam-6609	91	13	”	"	PUNCT
ejpam-6609	91	14	between	between	ADP
ejpam-6609	91	15	vectors	vector	NOUN
ejpam-6609	91	16	is	be	AUX
ejpam-6609	91	17	clear	clear	ADJ
ejpam-6609	91	18	.	.	PUNCT
ejpam-6609	92	1	•	•	NUM
ejpam-6609	92	2	the	the	DET
ejpam-6609	92	3	parallelogram	parallelogram	NOUN
ejpam-6609	92	4	law	law	NOUN
ejpam-6609	92	5	sets	set	VERB
ejpam-6609	92	6	inner	inner	ADJ
ejpam-6609	92	7	-	-	PUNCT
ejpam-6609	92	8	product	product	NOUN
ejpam-6609	92	9	-	-	PUNCT
ejpam-6609	92	10	induced	induce	VERB
ejpam-6609	92	11	norms	norm	NOUN
ejpam-6609	92	12	apart	apart	ADV
ejpam-6609	92	13	from	from	ADP
ejpam-6609	92	14	general	general	ADJ
ejpam-6609	92	15	norms	norm	NOUN
ejpam-6609	92	16	.	.	PUNCT
ejpam-6609	93	1	•	•	NUM
ejpam-6609	93	2	the	the	DET
ejpam-6609	93	3	polarization	polarization	NOUN
ejpam-6609	93	4	identity	identity	NOUN
ejpam-6609	93	5	lets	let	VERB
ejpam-6609	93	6	you	you	PRON
ejpam-6609	93	7	get	get	VERB
ejpam-6609	93	8	the	the	DET
ejpam-6609	93	9	inner	inner	ADJ
ejpam-6609	93	10	product	product	NOUN
ejpam-6609	93	11	back	back	ADV
ejpam-6609	93	12	from	from	ADP
ejpam-6609	93	13	the	the	DET
ejpam-6609	93	14	norm	norm	NOUN
ejpam-6609	93	15	,	,	PUNCT
ejpam-6609	93	16	which	which	PRON
ejpam-6609	93	17	shows	show	VERB
ejpam-6609	93	18	that	that	SCONJ
ejpam-6609	93	19	they	they	PRON
ejpam-6609	93	20	are	be	AUX
ejpam-6609	93	21	the	the	DET
ejpam-6609	93	22	same	same	ADJ
ejpam-6609	93	23	.	.	PUNCT
ejpam-6609	94	1	m.	m.	PROPN
ejpam-6609	94	2	noorwali	noorwali	PROPN
ejpam-6609	94	3	et	et	PROPN
ejpam-6609	94	4	al	al	PROPN
ejpam-6609	94	5	.	.	PUNCT
ejpam-6609	94	6	/	/	SYM
ejpam-6609	94	7	eur	eur	PROPN
ejpam-6609	94	8	.	.	PUNCT
ejpam-6609	95	1	j.	j.	PROPN
ejpam-6609	95	2	pure	pure	PROPN
ejpam-6609	95	3	appl	appl	PROPN
ejpam-6609	95	4	.	.	PROPN
ejpam-6609	95	5	math	math	PROPN
ejpam-6609	95	6	,	,	PUNCT
ejpam-6609	95	7	18	18	NUM
ejpam-6609	95	8	(	(	PUNCT
ejpam-6609	95	9	3	3	NUM
ejpam-6609	95	10	)	)	PUNCT
ejpam-6609	95	11	(	(	PUNCT
ejpam-6609	95	12	2025	2025	NUM
ejpam-6609	95	13	)	)	PUNCT
ejpam-6609	95	14	,	,	PUNCT
ejpam-6609	95	15	6609	6609	NUM
ejpam-6609	95	16	5	5	NUM
ejpam-6609	95	17	of	of	ADP
ejpam-6609	95	18	17	17	NUM
ejpam-6609	95	19	•	•	NOUN
ejpam-6609	95	20	inner	inner	ADJ
ejpam-6609	95	21	product	product	NOUN
ejpam-6609	95	22	spaces	space	NOUN
ejpam-6609	95	23	have	have	VERB
ejpam-6609	95	24	uniform	uniform	ADJ
ejpam-6609	95	25	convexity	convexity	NOUN
ejpam-6609	95	26	,	,	PUNCT
ejpam-6609	95	27	which	which	PRON
ejpam-6609	95	28	is	be	AUX
ejpam-6609	95	29	different	different	ADJ
ejpam-6609	95	30	from	from	ADP
ejpam-6609	95	31	regular	regular	ADJ
ejpam-6609	95	32	metric	metric	ADJ
ejpam-6609	95	33	spaces	space	NOUN
ejpam-6609	95	34	.	.	PUNCT
ejpam-6609	96	1	this	this	PRON
ejpam-6609	96	2	lets	let	VERB
ejpam-6609	96	3	them	they	PRON
ejpam-6609	96	4	be	be	AUX
ejpam-6609	96	5	broken	break	VERB
ejpam-6609	96	6	down	down	ADP
ejpam-6609	96	7	into	into	ADP
ejpam-6609	96	8	orthogonal	orthogonal	ADJ
ejpam-6609	96	9	parts	part	NOUN
ejpam-6609	96	10	,	,	PUNCT
ejpam-6609	96	11	like	like	ADP
ejpam-6609	96	12	projections	projection	NOUN
ejpam-6609	96	13	in	in	ADP
ejpam-6609	96	14	hs	hs	PROPN
ejpam-6609	96	15	.	.	PUNCT
ejpam-6609	96	16	lemma	lemma	PROPN
ejpam-6609	97	1	1	1	NUM
ejpam-6609	97	2	.	.	PUNCT
ejpam-6609	98	1	[	[	X
ejpam-6609	98	2	13	13	NUM
ejpam-6609	98	3	]	]	PUNCT
ejpam-6609	98	4	suppose	suppose	VERB
ejpam-6609	98	5	w	w	NOUN
ejpam-6609	98	6	be	be	AUX
ejpam-6609	98	7	a	a	DET
ejpam-6609	98	8	closed	closed	ADJ
ejpam-6609	98	9	subspace	subspace	NOUN
ejpam-6609	98	10	of	of	ADP
ejpam-6609	98	11	an	an	DET
ejpam-6609	98	12	inner	inner	ADJ
ejpam-6609	98	13	product	product	NOUN
ejpam-6609	98	14	space	space	NOUN
ejpam-6609	98	15	(	(	PUNCT
ejpam-6609	98	16	v	v	NOUN
ejpam-6609	98	17	,	,	PUNCT
ejpam-6609	98	18	⟨	⟨	NOUN
ejpam-6609	98	19	·	·	SYM
ejpam-6609	98	20	,	,	PUNCT
ejpam-6609	98	21	·	·	PUNCT
ejpam-6609	98	22	⟩	⟩	NOUN
ejpam-6609	98	23	)	)	PUNCT
ejpam-6609	98	24	.	.	PUNCT
ejpam-6609	99	1	for	for	ADP
ejpam-6609	99	2	any	any	DET
ejpam-6609	99	3	v	v	NUM
ejpam-6609	99	4	∈	∈	PROPN
ejpam-6609	99	5	v	v	NOUN
ejpam-6609	99	6	,	,	PUNCT
ejpam-6609	99	7	there	there	PRON
ejpam-6609	99	8	exists	exist	VERB
ejpam-6609	99	9	a	a	DET
ejpam-6609	99	10	unique	unique	ADJ
ejpam-6609	99	11	w∗	w∗	NOUN
ejpam-6609	99	12	∈	∈	PROPN
ejpam-6609	99	13	w	w	ADP
ejpam-6609	99	14	such	such	ADJ
ejpam-6609	99	15	that	that	SCONJ
ejpam-6609	99	16	:	:	PUNCT
ejpam-6609	99	17	∥v	∥v	PROPN
ejpam-6609	99	18	−	−	PROPN
ejpam-6609	99	19	w∗∥	w∗∥	PROPN
ejpam-6609	99	20	=	=	PUNCT
ejpam-6609	99	21	inf	inf	PROPN
ejpam-6609	99	22	w∈w	w∈w	VERB
ejpam-6609	99	23	∥v	∥v	PROPN
ejpam-6609	99	24	−	−	NOUN
ejpam-6609	99	25	w∥	w∥	NOUN
ejpam-6609	99	26	,	,	PUNCT
ejpam-6609	99	27	and	and	CCONJ
ejpam-6609	99	28	v	v	ADP
ejpam-6609	99	29	−	−	PROPN
ejpam-6609	99	30	w∗	w∗	NOUN
ejpam-6609	99	31	is	be	AUX
ejpam-6609	99	32	orthogonal	orthogonal	ADJ
ejpam-6609	99	33	to	to	ADP
ejpam-6609	99	34	w.	w.	PROPN
ejpam-6609	99	35	this	this	DET
ejpam-6609	99	36	w∗	w∗	NOUN
ejpam-6609	99	37	is	be	AUX
ejpam-6609	99	38	the	the	DET
ejpam-6609	99	39	orthogonal	orthogonal	ADJ
ejpam-6609	99	40	projection	projection	NOUN
ejpam-6609	99	41	of	of	ADP
ejpam-6609	99	42	v	v	NOUN
ejpam-6609	99	43	onto	onto	ADP
ejpam-6609	99	44	w.	w.	PROPN
ejpam-6609	99	45	3	3	NUM
ejpam-6609	99	46	.	.	PUNCT
ejpam-6609	99	47	main	main	ADJ
ejpam-6609	99	48	results	result	NOUN
ejpam-6609	99	49	this	this	DET
ejpam-6609	99	50	section	section	NOUN
ejpam-6609	99	51	gives	give	VERB
ejpam-6609	99	52	major	major	ADJ
ejpam-6609	99	53	theorems	theorem	NOUN
ejpam-6609	99	54	that	that	PRON
ejpam-6609	99	55	guarantee	guarantee	VERB
ejpam-6609	99	56	that	that	SCONJ
ejpam-6609	99	57	for	for	ADP
ejpam-6609	99	58	contractive	contractive	ADJ
ejpam-6609	99	59	mappings	mapping	NOUN
ejpam-6609	99	60	that	that	PRON
ejpam-6609	99	61	play	play	VERB
ejpam-6609	99	62	a	a	DET
ejpam-6609	99	63	particular	particular	ADJ
ejpam-6609	99	64	role	role	NOUN
ejpam-6609	99	65	in	in	ADP
ejpam-6609	99	66	gips	gips	PROPN
ejpam-6609	99	67	,	,	PUNCT
ejpam-6609	99	68	there	there	PRON
ejpam-6609	99	69	is	be	VERB
ejpam-6609	99	70	one	one	NUM
ejpam-6609	99	71	unique	unique	ADJ
ejpam-6609	99	72	fixed	fix	VERB
ejpam-6609	99	73	point	point	NOUN
ejpam-6609	99	74	.	.	PUNCT
ejpam-6609	100	1	by	by	ADP
ejpam-6609	100	2	applying	apply	VERB
ejpam-6609	100	3	these	these	DET
ejpam-6609	100	4	theorems	theorem	NOUN
ejpam-6609	100	5	,	,	PUNCT
ejpam-6609	100	6	we	we	PRON
ejpam-6609	100	7	establish	establish	VERB
ejpam-6609	100	8	a	a	DET
ejpam-6609	100	9	robust	robust	ADJ
ejpam-6609	100	10	convergence	convergence	NOUN
ejpam-6609	100	11	algorithm	algorithm	NOUN
ejpam-6609	100	12	,	,	PUNCT
ejpam-6609	100	13	which	which	PRON
ejpam-6609	100	14	we	we	PRON
ejpam-6609	100	15	then	then	ADV
ejpam-6609	100	16	apply	apply	VERB
ejpam-6609	100	17	to	to	ADP
ejpam-6609	100	18	areas	area	NOUN
ejpam-6609	100	19	such	such	ADJ
ejpam-6609	100	20	as	as	ADP
ejpam-6609	100	21	ai	ai	NOUN
ejpam-6609	100	22	and	and	CCONJ
ejpam-6609	100	23	cryptography	cryptography	NOUN
ejpam-6609	100	24	once	once	SCONJ
ejpam-6609	100	25	quantum	quantum	NOUN
ejpam-6609	100	26	computing	computing	NOUN
ejpam-6609	100	27	emerges	emerge	NOUN
ejpam-6609	100	28	.	.	PUNCT
ejpam-6609	101	1	we	we	PRON
ejpam-6609	101	2	use	use	VERB
ejpam-6609	101	3	some	some	DET
ejpam-6609	101	4	lemmas	lemma	NOUN
ejpam-6609	101	5	,	,	PUNCT
ejpam-6609	101	6	remarks	remark	NOUN
ejpam-6609	101	7	,	,	PUNCT
ejpam-6609	101	8	and	and	CCONJ
ejpam-6609	101	9	examples	example	NOUN
ejpam-6609	101	10	to	to	PART
ejpam-6609	101	11	support	support	VERB
ejpam-6609	101	12	the	the	DET
ejpam-6609	101	13	theoretical	theoretical	ADJ
ejpam-6609	101	14	conclusions	conclusion	NOUN
ejpam-6609	101	15	.	.	PUNCT
ejpam-6609	102	1	as	as	ADP
ejpam-6609	102	2	a	a	DET
ejpam-6609	102	3	result	result	NOUN
ejpam-6609	102	4	,	,	PUNCT
ejpam-6609	102	5	these	these	DET
ejpam-6609	102	6	findings	finding	NOUN
ejpam-6609	102	7	support	support	VERB
ejpam-6609	102	8	more	more	ADJ
ejpam-6609	102	9	studying	study	VERB
ejpam-6609	102	10	in	in	ADP
ejpam-6609	102	11	optimization	optimization	NOUN
ejpam-6609	102	12	,	,	PUNCT
ejpam-6609	102	13	control	control	NOUN
ejpam-6609	102	14	,	,	PUNCT
ejpam-6609	102	15	and	and	CCONJ
ejpam-6609	102	16	secure	secure	ADJ
ejpam-6609	102	17	computation	computation	NOUN
ejpam-6609	102	18	.	.	PUNCT
ejpam-6609	103	1	numerous	numerous	ADJ
ejpam-6609	103	2	examples	example	NOUN
ejpam-6609	103	3	and	and	CCONJ
ejpam-6609	103	4	illustrations	illustration	NOUN
ejpam-6609	103	5	elucidate	elucidate	VERB
ejpam-6609	103	6	the	the	DET
ejpam-6609	103	7	application	application	NOUN
ejpam-6609	103	8	of	of	ADP
ejpam-6609	103	9	our	our	PRON
ejpam-6609	103	10	main	main	ADJ
ejpam-6609	103	11	theorems	theorem	NOUN
ejpam-6609	103	12	and	and	CCONJ
ejpam-6609	103	13	facilitate	facilitate	VERB
ejpam-6609	103	14	their	their	PRON
ejpam-6609	103	15	demonstration	demonstration	NOUN
ejpam-6609	103	16	through	through	ADP
ejpam-6609	103	17	exemplification	exemplification	NOUN
ejpam-6609	103	18	.	.	PUNCT
ejpam-6609	104	1	theorem	theorem	NOUN
ejpam-6609	104	2	1	1	NUM
ejpam-6609	104	3	.	.	PUNCT
ejpam-6609	105	1	let	let	VERB
ejpam-6609	105	2	(	(	PUNCT
ejpam-6609	105	3	h	h	NOUN
ejpam-6609	105	4	,	,	PUNCT
ejpam-6609	105	5	⟨	⟨	NOUN
ejpam-6609	105	6	·	·	SYM
ejpam-6609	105	7	,	,	PUNCT
ejpam-6609	105	8	·	·	PUNCT
ejpam-6609	105	9	⟩	⟩	NOUN
ejpam-6609	105	10	)	)	PUNCT
ejpam-6609	105	11	is	be	AUX
ejpam-6609	105	12	a	a	DET
ejpam-6609	105	13	hs	hs	PROPN
ejpam-6609	105	14	and	and	CCONJ
ejpam-6609	105	15	t1	t1	PROPN
ejpam-6609	105	16	,	,	PUNCT
ejpam-6609	105	17	t2	t2	NOUN
ejpam-6609	105	18	,	,	PUNCT
ejpam-6609	105	19	t3	t3	NOUN
ejpam-6609	105	20	:	:	PUNCT
ejpam-6609	105	21	h	h	NOUN
ejpam-6609	105	22	→	→	SYM
ejpam-6609	105	23	h	h	NOUN
ejpam-6609	105	24	be	be	AUX
ejpam-6609	105	25	mappings	mapping	NOUN
ejpam-6609	105	26	satisfying	satisfying	ADJ
ejpam-6609	105	27	:	:	PUNCT
ejpam-6609	105	28	∥t1x−	∥t1x−	NUM
ejpam-6609	105	29	t2y	t2y	ADV
ejpam-6609	105	30	−	−	PROPN
ejpam-6609	105	31	t3z∥2	t3z∥2	VERB
ejpam-6609	105	32	≤	≤	PUNCT
ejpam-6609	106	1	α∥x−	α∥x−	CCONJ
ejpam-6609	106	2	y	y	PROPN
ejpam-6609	106	3	−	−	PROPN
ejpam-6609	106	4	z∥2	z∥2	NOUN
ejpam-6609	106	5	+	+	NUM
ejpam-6609	106	6	β	β	X
ejpam-6609	106	7	∥x−	∥x−	NUM
ejpam-6609	106	8	t2y∥2	t2y∥2	NOUN
ejpam-6609	106	9	·	·	PUNCT
ejpam-6609	107	1	∥x−	∥x−	NUM
ejpam-6609	107	2	t3z∥2	t3z∥2	VERB
ejpam-6609	107	3	·	·	PUNCT
ejpam-6609	107	4	∥y	∥y	ADJ
ejpam-6609	107	5	−	−	NOUN
ejpam-6609	107	6	t1x∥2	t1x∥2	NOUN
ejpam-6609	107	7	1	1	NUM
ejpam-6609	107	8	+	+	CCONJ
ejpam-6609	107	9	∥y	∥y	ADJ
ejpam-6609	107	10	−	−	PROPN
ejpam-6609	107	11	t1x∥2∥z	t1x∥2∥z	NOUN
ejpam-6609	107	12	−	−	NOUN
ejpam-6609	107	13	t2y∥2∥x−	t2y∥2∥x−	ADV
ejpam-6609	107	14	t3z∥2	t3z∥2	VERB
ejpam-6609	107	15	+	+	CCONJ
ejpam-6609	107	16	γ	γ	PROPN
ejpam-6609	107	17	∥y	∥y	PROPN
ejpam-6609	107	18	−	−	PROPN
ejpam-6609	107	19	t3z∥2	t3z∥2	NOUN
ejpam-6609	107	20	·	·	PUNCT
ejpam-6609	107	21	∥z	∥z	NOUN
ejpam-6609	107	22	−	−	PROPN
ejpam-6609	107	23	t1x∥2	t1x∥2	NOUN
ejpam-6609	107	24	·	·	PUNCT
ejpam-6609	107	25	∥z	∥z	NOUN
ejpam-6609	107	26	−	−	NOUN
ejpam-6609	107	27	t2y∥2	t2y∥2	NOUN
ejpam-6609	107	28	1	1	NUM
ejpam-6609	107	29	+	+	CCONJ
ejpam-6609	107	30	∥y	∥y	ADJ
ejpam-6609	107	31	−	−	PROPN
ejpam-6609	107	32	t1x∥2∥z	t1x∥2∥z	NOUN
ejpam-6609	107	33	−	−	NOUN
ejpam-6609	107	34	t2y∥2∥x−	t2y∥2∥x−	ADV
ejpam-6609	107	35	t3z∥2	t3z∥2	X
ejpam-6609	107	36	(	(	PUNCT
ejpam-6609	107	37	1	1	NUM
ejpam-6609	107	38	)	)	PUNCT
ejpam-6609	107	39	for	for	ADP
ejpam-6609	107	40	all	all	DET
ejpam-6609	107	41	x	x	NOUN
ejpam-6609	107	42	,	,	PUNCT
ejpam-6609	107	43	y	y	PROPN
ejpam-6609	107	44	,	,	PUNCT
ejpam-6609	107	45	z	z	PROPN
ejpam-6609	107	46	∈	∈	PROPN
ejpam-6609	107	47	h	h	NOUN
ejpam-6609	107	48	and	and	CCONJ
ejpam-6609	107	49	α	α	NOUN
ejpam-6609	107	50	,	,	PUNCT
ejpam-6609	107	51	β	β	X
ejpam-6609	107	52	,	,	PUNCT
ejpam-6609	107	53	γ	γ	X
ejpam-6609	107	54	≥	≥	X
ejpam-6609	107	55	0	0	NUM
ejpam-6609	107	56	with	with	ADP
ejpam-6609	107	57	α+	α+	X
ejpam-6609	107	58	β	β	NOUN
ejpam-6609	107	59	+	+	X
ejpam-6609	107	60	γ	γ	X
ejpam-6609	107	61	<	<	X
ejpam-6609	107	62	1	1	NUM
ejpam-6609	107	63	.	.	PUNCT
ejpam-6609	108	1	then	then	ADV
ejpam-6609	108	2	there	there	PRON
ejpam-6609	108	3	exists	exist	VERB
ejpam-6609	108	4	a	a	DET
ejpam-6609	108	5	unique	unique	ADJ
ejpam-6609	108	6	common	common	ADJ
ejpam-6609	108	7	fixed	fix	VERB
ejpam-6609	108	8	point	point	NOUN
ejpam-6609	108	9	u	u	PROPN
ejpam-6609	108	10	∈	∈	PROPN
ejpam-6609	108	11	h	h	NOUN
ejpam-6609	108	12	such	such	ADJ
ejpam-6609	108	13	that	that	DET
ejpam-6609	108	14	t1u	t1u	NOUN
ejpam-6609	108	15	=	=	PUNCT
ejpam-6609	108	16	t2u	t2u	VERB
ejpam-6609	108	17	=	=	PUNCT
ejpam-6609	108	18	t3u	t3u	NOUN
ejpam-6609	108	19	=	=	PUNCT
ejpam-6609	108	20	u.	u.	NOUN
ejpam-6609	108	21	proof	proof	NOUN
ejpam-6609	108	22	.	.	PUNCT
ejpam-6609	109	1	(	(	PUNCT
ejpam-6609	109	2	1	1	X
ejpam-6609	109	3	)	)	PUNCT
ejpam-6609	109	4	construction	construction	NOUN
ejpam-6609	109	5	of	of	ADP
ejpam-6609	109	6	iterative	iterative	NOUN
ejpam-6609	109	7	sequence	sequence	NOUN
ejpam-6609	109	8	give	give	VERB
ejpam-6609	109	9	a	a	DET
ejpam-6609	109	10	definition	definition	NOUN
ejpam-6609	109	11	of	of	ADP
ejpam-6609	109	12	the	the	DET
ejpam-6609	109	13	sequence	sequence	NOUN
ejpam-6609	109	14	{	{	PUNCT
ejpam-6609	109	15	xn	xn	VERB
ejpam-6609	109	16	}	}	PUNCT
ejpam-6609	109	17	by	by	ADP
ejpam-6609	109	18	x3n+1	x3n+1	PROPN
ejpam-6609	109	19	=	=	SYM
ejpam-6609	109	20	t1x3n	t1x3n	NUM
ejpam-6609	109	21	;	;	PUNCT
ejpam-6609	109	22	x3n+2	x3n+2	PUNCT
ejpam-6609	109	23	=	=	SYM
ejpam-6609	109	24	t2x3n+1	t2x3n+1	PROPN
ejpam-6609	109	25	;	;	PUNCT
ejpam-6609	109	26	x3n+3	x3n+3	PUNCT
ejpam-6609	110	1	=	=	SYM
ejpam-6609	111	1	t3x3n+2	t3x3n+2	NUM
ejpam-6609	111	2	.	.	PUNCT
ejpam-6609	112	1	(	(	PUNCT
ejpam-6609	112	2	2	2	X
ejpam-6609	112	3	)	)	PUNCT
ejpam-6609	112	4	contraction	contraction	NOUN
ejpam-6609	112	5	estimates	estimate	NOUN
ejpam-6609	112	6	from	from	ADP
ejpam-6609	112	7	eq	eq	PROPN
ejpam-6609	112	8	.	.	PUNCT
ejpam-6609	113	1	(	(	PUNCT
ejpam-6609	113	2	1	1	X
ejpam-6609	113	3	)	)	PUNCT
ejpam-6609	113	4	with	with	ADP
ejpam-6609	113	5	x	x	PROPN
ejpam-6609	113	6	=	=	SYM
ejpam-6609	113	7	x3n	x3n	PROPN
ejpam-6609	113	8	,	,	PUNCT
ejpam-6609	113	9	y	y	PROPN
ejpam-6609	113	10	=	=	SYM
ejpam-6609	113	11	x3n+1	x3n+1	PROPN
ejpam-6609	113	12	,	,	PUNCT
ejpam-6609	113	13	z	z	NOUN
ejpam-6609	113	14	=	=	SYM
ejpam-6609	113	15	x3n+2	x3n+2	PROPN
ejpam-6609	113	16	:	:	PUNCT
ejpam-6609	114	1	∥x3n+1	∥x3n+1	NOUN
ejpam-6609	114	2	−	−	PROPN
ejpam-6609	114	3	x3n+2	x3n+2	PUNCT
ejpam-6609	115	1	−	−	PROPN
ejpam-6609	115	2	x3n+3∥2	x3n+3∥2	PROPN
ejpam-6609	116	1	≤	≤	PROPN
ejpam-6609	117	1	α∥x3n	α∥x3n	PROPN
ejpam-6609	117	2	−	−	PROPN
ejpam-6609	117	3	x3n+1	x3n+1	PUNCT
ejpam-6609	117	4	−	−	PROPN
ejpam-6609	118	1	x3n+2∥2	x3n+2∥2	PUNCT
ejpam-6609	119	1	+	+	PUNCT
ejpam-6609	119	2	β	β	X
ejpam-6609	119	3	∥x3n	∥x3n	X
ejpam-6609	119	4	−	−	NUM
ejpam-6609	119	5	x3n+2∥2∥x3n	x3n+2∥2∥x3n	PUNCT
ejpam-6609	119	6	−	−	PUNCT
ejpam-6609	119	7	x3n+3∥2∥x3n+1	x3n+3∥2∥x3n+1	PUNCT
ejpam-6609	120	1	−	−	PROPN
ejpam-6609	120	2	x3n+1∥2	x3n+1∥2	PROPN
ejpam-6609	120	3	1	1	NUM
ejpam-6609	120	4	+	+	CCONJ
ejpam-6609	120	5	·	·	PUNCT
ejpam-6609	120	6	·	·	PUNCT
ejpam-6609	120	7	·	·	PUNCT
ejpam-6609	121	1	+	+	CCONJ
ejpam-6609	121	2	γ	γ	X
ejpam-6609	121	3	∥x3n+1	∥x3n+1	PROPN
ejpam-6609	121	4	−	−	PROPN
ejpam-6609	121	5	x3n+3∥2∥x3n+2	x3n+3∥2∥x3n+2	PROPN
ejpam-6609	122	1	−	−	PROPN
ejpam-6609	122	2	x3n+1∥2∥x3n+2	x3n+1∥2∥x3n+2	PUNCT
ejpam-6609	123	1	−	−	PROPN
ejpam-6609	124	1	x3n+2∥2	x3n+2∥2	NOUN
ejpam-6609	124	2	1	1	NUM
ejpam-6609	124	3	+	+	CCONJ
ejpam-6609	124	4	·	·	PUNCT
ejpam-6609	124	5	·	·	PUNCT
ejpam-6609	124	6	·	·	PUNCT
ejpam-6609	125	1	≤	≤	NUM
ejpam-6609	126	1	α∥x3n	α∥x3n	PROPN
ejpam-6609	126	2	−	−	PROPN
ejpam-6609	126	3	x3n+1	x3n+1	PROPN
ejpam-6609	127	1	−	−	PROPN
ejpam-6609	127	2	x3n+2∥2	x3n+2∥2	PRON
ejpam-6609	128	1	similarly	similarly	ADV
ejpam-6609	128	2	,	,	PUNCT
ejpam-6609	128	3	we	we	PRON
ejpam-6609	128	4	obtain	obtain	VERB
ejpam-6609	128	5	:	:	PUNCT
ejpam-6609	128	6	∥x3n+2	∥x3n+2	NUM
ejpam-6609	128	7	−	−	PUNCT
ejpam-6609	128	8	x3n+3	x3n+3	PUNCT
ejpam-6609	129	1	−	−	NOUN
ejpam-6609	130	1	x3n+4∥2	x3n+4∥2	DET
ejpam-6609	130	2	≤	≤	NUM
ejpam-6609	130	3	α∥x3n+1	α∥x3n+1	NOUN
ejpam-6609	130	4	−	−	PUNCT
ejpam-6609	130	5	x3n+2	x3n+2	PUNCT
ejpam-6609	131	1	−	−	PROPN
ejpam-6609	132	1	x3n+3∥2	x3n+3∥2	PROPN
ejpam-6609	133	1	∥x3n+3	∥x3n+3	NOUN
ejpam-6609	133	2	−	−	PROPN
ejpam-6609	134	1	x3n+4	x3n+4	X
ejpam-6609	134	2	−	−	PROPN
ejpam-6609	135	1	x3n+5∥2	x3n+5∥2	PROPN
ejpam-6609	135	2	≤	≤	PROPN
ejpam-6609	135	3	α∥x3n+2	α∥x3n+2	PROPN
ejpam-6609	135	4	−	−	PROPN
ejpam-6609	135	5	x3n+3	x3n+3	PUNCT
ejpam-6609	136	1	−	−	PROPN
ejpam-6609	137	1	x3n+4∥2	x3n+4∥2	PROPN
ejpam-6609	137	2	m.	m.	NOUN
ejpam-6609	137	3	noorwali	noorwali	PROPN
ejpam-6609	137	4	et	et	PROPN
ejpam-6609	137	5	al	al	PROPN
ejpam-6609	137	6	.	.	PUNCT
ejpam-6609	137	7	/	/	SYM
ejpam-6609	137	8	eur	eur	PROPN
ejpam-6609	137	9	.	.	PUNCT
ejpam-6609	138	1	j.	j.	PROPN
ejpam-6609	138	2	pure	pure	PROPN
ejpam-6609	138	3	appl	appl	PROPN
ejpam-6609	138	4	.	.	PROPN
ejpam-6609	138	5	math	math	PROPN
ejpam-6609	138	6	,	,	PUNCT
ejpam-6609	138	7	18	18	NUM
ejpam-6609	138	8	(	(	PUNCT
ejpam-6609	138	9	3	3	NUM
ejpam-6609	138	10	)	)	PUNCT
ejpam-6609	138	11	(	(	PUNCT
ejpam-6609	138	12	2025	2025	NUM
ejpam-6609	138	13	)	)	PUNCT
ejpam-6609	138	14	,	,	PUNCT
ejpam-6609	138	15	6609	6609	NUM
ejpam-6609	138	16	6	6	NUM
ejpam-6609	138	17	of	of	ADP
ejpam-6609	138	18	17	17	NUM
ejpam-6609	138	19	(	(	PUNCT
ejpam-6609	138	20	3	3	NUM
ejpam-6609	138	21	)	)	PUNCT
ejpam-6609	138	22	convergence	convergence	NOUN
ejpam-6609	138	23	analysis	analysis	NOUN
ejpam-6609	138	24	by	by	ADP
ejpam-6609	138	25	induction	induction	NOUN
ejpam-6609	138	26	:	:	PUNCT
ejpam-6609	138	27	∥xn+1	∥xn+1	VERB
ejpam-6609	138	28	−	−	PROPN
ejpam-6609	139	1	xn+2	xn+2	NUM
ejpam-6609	140	1	−	−	PROPN
ejpam-6609	140	2	xn+3∥	xn+3∥	NOUN
ejpam-6609	140	3	≤	≤	NOUN
ejpam-6609	140	4	αn/3∥x0	αn/3∥x0	NOUN
ejpam-6609	140	5	−	−	NOUN
ejpam-6609	141	1	x1	x1	NOUN
ejpam-6609	141	2	−	−	PROPN
ejpam-6609	141	3	x2∥	x2∥	NOUN
ejpam-6609	141	4	thus	thus	ADV
ejpam-6609	141	5	{	{	PUNCT
ejpam-6609	141	6	xn	xn	X
ejpam-6609	141	7	}	}	PUNCT
ejpam-6609	141	8	is	be	AUX
ejpam-6609	141	9	cauchy	cauchy	PROPN
ejpam-6609	141	10	.	.	PUNCT
ejpam-6609	142	1	since	since	SCONJ
ejpam-6609	142	2	h	h	NOUN
ejpam-6609	142	3	is	be	AUX
ejpam-6609	142	4	complete	complete	ADJ
ejpam-6609	142	5	,	,	PUNCT
ejpam-6609	142	6	∃u	∃u	PROPN
ejpam-6609	142	7	∈	∈	PROPN
ejpam-6609	142	8	h	h	NOUN
ejpam-6609	142	9	such	such	ADJ
ejpam-6609	142	10	that	that	PRON
ejpam-6609	142	11	xn	xn	PROPN
ejpam-6609	143	1	→	→	SYM
ejpam-6609	143	2	u.	u.	PROPN
ejpam-6609	143	3	(	(	PUNCT
ejpam-6609	143	4	4	4	NUM
ejpam-6609	143	5	)	)	PUNCT
ejpam-6609	143	6	fixed	fix	VERB
ejpam-6609	143	7	point	point	NOUN
ejpam-6609	143	8	verification	verification	NOUN
ejpam-6609	143	9	for	for	ADP
ejpam-6609	143	10	t1	t1	NOUN
ejpam-6609	143	11	:	:	PUNCT
ejpam-6609	143	12	∥t1u−	∥t1u−	DET
ejpam-6609	143	13	u∥2	u∥2	NOUN
ejpam-6609	143	14	=	=	PROPN
ejpam-6609	143	15	lim	lim	PROPN
ejpam-6609	143	16	∥t1u−	∥t1u−	NOUN
ejpam-6609	143	17	x3n+2	x3n+2	PROPN
ejpam-6609	143	18	−	−	PROPN
ejpam-6609	144	1	x3n+3∥2	x3n+3∥2	PROPN
ejpam-6609	144	2	≤	≤	PROPN
ejpam-6609	144	3	α∥u−	α∥u−	PROPN
ejpam-6609	144	4	u−	u−	PROPN
ejpam-6609	144	5	u∥2	u∥2	NOUN
ejpam-6609	144	6	+	+	CCONJ
ejpam-6609	145	1	(	(	PUNCT
ejpam-6609	145	2	higher	high	ADJ
ejpam-6609	145	3	order	order	NOUN
ejpam-6609	145	4	terms	term	NOUN
ejpam-6609	145	5	)	)	PUNCT
ejpam-6609	145	6	=	=	SYM
ejpam-6609	145	7	0	0	NUM
ejpam-6609	145	8	similarly	similarly	ADV
ejpam-6609	145	9	t2u	t2u	PUNCT
ejpam-6609	145	10	=	=	PUNCT
ejpam-6609	145	11	t3u	t3u	NOUN
ejpam-6609	145	12	=	=	SYM
ejpam-6609	145	13	u.	u.	PROPN
ejpam-6609	145	14	(	(	PUNCT
ejpam-6609	145	15	5	5	NUM
ejpam-6609	145	16	)	)	PUNCT
ejpam-6609	145	17	uniqueness	uniqueness	NOUN
ejpam-6609	145	18	suppose	suppose	VERB
ejpam-6609	145	19	v	v	NOUN
ejpam-6609	145	20	is	be	AUX
ejpam-6609	145	21	another	another	DET
ejpam-6609	145	22	fixed	fix	VERB
ejpam-6609	145	23	point	point	NOUN
ejpam-6609	145	24	:	:	PUNCT
ejpam-6609	145	25	∥u−	∥u−	NUM
ejpam-6609	145	26	v∥2	v∥2	NOUN
ejpam-6609	145	27	=	=	PUNCT
ejpam-6609	145	28	∥t1u−	∥t1u−	PRON
ejpam-6609	145	29	t2v	t2v	PROPN
ejpam-6609	145	30	−	−	NOUN
ejpam-6609	145	31	t3v∥2	t3v∥2	NOUN
ejpam-6609	145	32	≤	≤	NOUN
ejpam-6609	145	33	(	(	PUNCT
ejpam-6609	145	34	α+	α+	PRON
ejpam-6609	145	35	2β	2β	NOUN
ejpam-6609	145	36	+	+	CCONJ
ejpam-6609	145	37	2γ)∥u−	2γ)∥u−	NUM
ejpam-6609	145	38	v∥2	v∥2	NOUN
ejpam-6609	145	39	since	since	SCONJ
ejpam-6609	145	40	α+	α+	PRON
ejpam-6609	145	41	2β	2β	NOUN
ejpam-6609	145	42	+	+	CCONJ
ejpam-6609	145	43	2γ	2γ	NOUN
ejpam-6609	145	44	<	<	X
ejpam-6609	145	45	1	1	NUM
ejpam-6609	145	46	,	,	PUNCT
ejpam-6609	145	47	u	u	NOUN
ejpam-6609	145	48	=	=	PROPN
ejpam-6609	145	49	v.	v.	ADP
ejpam-6609	145	50	the	the	DET
ejpam-6609	145	51	iterative	iterative	NOUN
ejpam-6609	145	52	strategy	strategy	NOUN
ejpam-6609	145	53	in	in	ADP
ejpam-6609	145	54	algorithm	algorithm	NOUN
ejpam-6609	145	55	1	1	NUM
ejpam-6609	145	56	provides	provide	VERB
ejpam-6609	145	57	a	a	DET
ejpam-6609	145	58	structured	structured	ADJ
ejpam-6609	145	59	method	method	NOUN
ejpam-6609	145	60	for	for	ADP
ejpam-6609	145	61	finding	find	VERB
ejpam-6609	145	62	the	the	DET
ejpam-6609	145	63	common	common	ADJ
ejpam-6609	145	64	fixed	fix	VERB
ejpam-6609	145	65	point	point	NOUN
ejpam-6609	145	66	of	of	ADP
ejpam-6609	145	67	three	three	NUM
ejpam-6609	145	68	contractive	contractive	ADJ
ejpam-6609	145	69	mappings	mapping	NOUN
ejpam-6609	145	70	by	by	ADP
ejpam-6609	145	71	utilizing	utilize	VERB
ejpam-6609	145	72	each	each	DET
ejpam-6609	145	73	mapping	mapping	NOUN
ejpam-6609	145	74	in	in	ADP
ejpam-6609	145	75	a	a	DET
ejpam-6609	145	76	cycle	cycle	NOUN
ejpam-6609	145	77	.	.	PUNCT
ejpam-6609	146	1	algorithm	algorithm	NOUN
ejpam-6609	146	2	1	1	NUM
ejpam-6609	146	3	iterative	iterative	NOUN
ejpam-6609	146	4	approximation	approximation	NOUN
ejpam-6609	146	5	of	of	ADP
ejpam-6609	146	6	common	common	ADJ
ejpam-6609	146	7	fixed	fix	VERB
ejpam-6609	146	8	points	point	NOUN
ejpam-6609	146	9	for	for	ADP
ejpam-6609	146	10	three	three	NUM
ejpam-6609	146	11	mappings	mapping	NOUN
ejpam-6609	146	12	require	require	VERB
ejpam-6609	146	13	:	:	PUNCT
ejpam-6609	146	14	mappings	mapping	NOUN
ejpam-6609	146	15	t1	t1	NOUN
ejpam-6609	146	16	,	,	PUNCT
ejpam-6609	146	17	t2	t2	NOUN
ejpam-6609	146	18	,	,	PUNCT
ejpam-6609	146	19	t3	t3	NOUN
ejpam-6609	146	20	:	:	PUNCT
ejpam-6609	146	21	h	h	NOUN
ejpam-6609	146	22	→	→	SYM
ejpam-6609	146	23	h	h	NOUN
ejpam-6609	146	24	initial	initial	ADJ
ejpam-6609	146	25	guess	guess	NOUN
ejpam-6609	147	1	x0	x0	PROPN
ejpam-6609	147	2	∈	∈	PROPN
ejpam-6609	147	3	h	h	NOUN
ejpam-6609	147	4	tolerance	tolerance	NOUN
ejpam-6609	147	5	ϵ	ϵ	X
ejpam-6609	147	6	>	>	X
ejpam-6609	147	7	0	0	NUM
ejpam-6609	148	1	maximum	maximum	ADJ
ejpam-6609	148	2	iterations	iteration	NOUN
ejpam-6609	149	1	n	n	PRON
ejpam-6609	149	2	ensure	ensure	VERB
ejpam-6609	149	3	:	:	PUNCT
ejpam-6609	149	4	approximate	approximate	ADJ
ejpam-6609	149	5	common	common	ADJ
ejpam-6609	149	6	fixed	fix	VERB
ejpam-6609	149	7	point	point	NOUN
ejpam-6609	149	8	u	u	PROPN
ejpam-6609	149	9	≈	≈	PROPN
ejpam-6609	149	10	t1u	t1u	NOUN
ejpam-6609	149	11	=	=	PUNCT
ejpam-6609	149	12	t2u	t2u	VERB
ejpam-6609	150	1	=	=	PUNCT
ejpam-6609	150	2	t3u	t3u	NOUN
ejpam-6609	150	3	1	1	NUM
ejpam-6609	150	4	:	:	PUNCT
ejpam-6609	150	5	n←	n←	X
ejpam-6609	150	6	0	0	NUM
ejpam-6609	150	7	2	2	NUM
ejpam-6609	150	8	:	:	PUNCT
ejpam-6609	150	9	while	while	SCONJ
ejpam-6609	150	10	n	n	CCONJ
ejpam-6609	150	11	<	<	X
ejpam-6609	150	12	n	n	PROPN
ejpam-6609	150	13	and	and	CCONJ
ejpam-6609	150	14	∥xn+1	∥xn+1	VERB
ejpam-6609	150	15	−	−	PROPN
ejpam-6609	151	1	xn∥	xn∥	PROPN
ejpam-6609	151	2	≥	≥	PROPN
ejpam-6609	151	3	ϵ	ϵ	AUX
ejpam-6609	151	4	do	do	VERB
ejpam-6609	151	5	3	3	NUM
ejpam-6609	151	6	:	:	PUNCT
ejpam-6609	151	7	x3n+1	x3n+1	PROPN
ejpam-6609	151	8	←	←	PROPN
ejpam-6609	151	9	t1(x3n	t1(x3n	PROPN
ejpam-6609	151	10	)	)	PUNCT
ejpam-6609	151	11	4	4	NUM
ejpam-6609	151	12	:	:	PUNCT
ejpam-6609	151	13	x3n+2	x3n+2	PROPN
ejpam-6609	151	14	←	←	PROPN
ejpam-6609	151	15	t2(x3n+1	t2(x3n+1	PROPN
ejpam-6609	151	16	)	)	PUNCT
ejpam-6609	151	17	5	5	NUM
ejpam-6609	151	18	:	:	PUNCT
ejpam-6609	151	19	x3n+3	x3n+3	PROPN
ejpam-6609	151	20	←	←	PROPN
ejpam-6609	151	21	t3(x3n+2	t3(x3n+2	PROPN
ejpam-6609	151	22	)	)	PUNCT
ejpam-6609	151	23	6	6	NUM
ejpam-6609	151	24	:	:	PUNCT
ejpam-6609	151	25	if	if	SCONJ
ejpam-6609	151	26	∥x3n+3	∥x3n+3	VERB
ejpam-6609	151	27	−	−	NOUN
ejpam-6609	152	1	x3n∥	x3n∥	X
ejpam-6609	152	2	<	<	X
ejpam-6609	153	1	ϵ	ϵ	X
ejpam-6609	153	2	then	then	ADV
ejpam-6609	153	3	7	7	NUM
ejpam-6609	153	4	:	:	PUNCT
ejpam-6609	153	5	break	break	NOUN
ejpam-6609	153	6	8	8	NUM
ejpam-6609	153	7	:	:	PUNCT
ejpam-6609	153	8	end	end	VERB
ejpam-6609	153	9	if	if	SCONJ
ejpam-6609	153	10	9	9	NUM
ejpam-6609	153	11	:	:	PUNCT
ejpam-6609	153	12	n←	n←	NUM
ejpam-6609	153	13	n+	n+	X
ejpam-6609	153	14	1	1	NUM
ejpam-6609	153	15	10	10	NUM
ejpam-6609	153	16	:	:	PUNCT
ejpam-6609	153	17	end	end	VERB
ejpam-6609	153	18	while	while	SCONJ
ejpam-6609	153	19	11	11	NUM
ejpam-6609	153	20	:	:	PUNCT
ejpam-6609	153	21	if	if	SCONJ
ejpam-6609	153	22	n	n	NOUN
ejpam-6609	153	23	=	=	SYM
ejpam-6609	153	24	n	n	NOUN
ejpam-6609	153	25	then	then	ADV
ejpam-6609	153	26	12	12	NUM
ejpam-6609	153	27	:	:	PUNCT
ejpam-6609	153	28	return	return	VERB
ejpam-6609	153	29	“	"	PUNCT
ejpam-6609	153	30	maximum	maximum	ADJ
ejpam-6609	153	31	iterations	iteration	NOUN
ejpam-6609	153	32	reached	reach	VERB
ejpam-6609	153	33	”	"	PUNCT
ejpam-6609	153	34	13	13	NUM
ejpam-6609	153	35	:	:	PUNCT
ejpam-6609	153	36	else	else	ADV
ejpam-6609	153	37	14	14	NUM
ejpam-6609	153	38	:	:	PUNCT
ejpam-6609	153	39	return	return	VERB
ejpam-6609	153	40	u←	u←	PUNCT
ejpam-6609	153	41	x3n+3	x3n+3	PROPN
ejpam-6609	154	1	15	15	NUM
ejpam-6609	154	2	:	:	PUNCT
ejpam-6609	154	3	end	end	VERB
ejpam-6609	154	4	if	if	SCONJ
ejpam-6609	154	5	•	•	NOUN
ejpam-6609	154	6	convergence	convergence	NOUN
ejpam-6609	154	7	:	:	PUNCT
ejpam-6609	154	8	by	by	ADP
ejpam-6609	154	9	theorem	theorem	NOUN
ejpam-6609	154	10	1	1	NUM
ejpam-6609	154	11	,	,	PUNCT
ejpam-6609	154	12	the	the	DET
ejpam-6609	154	13	sequence	sequence	NOUN
ejpam-6609	154	14	{	{	PUNCT
ejpam-6609	154	15	xn	xn	NOUN
ejpam-6609	154	16	}	}	PUNCT
ejpam-6609	154	17	converges	converge	VERB
ejpam-6609	154	18	linearly	linearly	ADV
ejpam-6609	154	19	to	to	ADP
ejpam-6609	154	20	the	the	DET
ejpam-6609	154	21	one	one	NOUN
ejpam-6609	154	22	and	and	CCONJ
ejpam-6609	154	23	only	only	ADV
ejpam-6609	154	24	fixed	fix	VERB
ejpam-6609	154	25	point	point	NOUN
ejpam-6609	154	26	u.	u.	NOUN
ejpam-6609	154	27	•	•	NUM
ejpam-6609	154	28	complexity	complexity	NOUN
ejpam-6609	154	29	:	:	PUNCT
ejpam-6609	154	30	for	for	ADP
ejpam-6609	154	31	each	each	DET
ejpam-6609	154	32	iteration	iteration	NOUN
ejpam-6609	154	33	,	,	PUNCT
ejpam-6609	154	34	it	it	PRON
ejpam-6609	154	35	’s	’	VERB
ejpam-6609	154	36	necessary	necessary	ADJ
ejpam-6609	154	37	to	to	PART
ejpam-6609	154	38	look	look	VERB
ejpam-6609	154	39	at	at	ADP
ejpam-6609	154	40	each	each	DET
ejpam-6609	154	41	mapping	mapping	NOUN
ejpam-6609	154	42	t1	t1	NOUN
ejpam-6609	154	43	,	,	PUNCT
ejpam-6609	154	44	t2	t2	NOUN
ejpam-6609	154	45	,	,	PUNCT
ejpam-6609	154	46	t3	t3	PROPN
ejpam-6609	154	47	.	.	PUNCT
ejpam-6609	155	1	algorithms	algorithms	PROPN
ejpam-6609	155	2	2	2	NUM
ejpam-6609	155	3	-	-	SYM
ejpam-6609	155	4	3	3	NUM
ejpam-6609	155	5	achieve	achieve	VERB
ejpam-6609	155	6	the	the	DET
ejpam-6609	155	7	convergence	convergence	NOUN
ejpam-6609	155	8	demonstrated	demonstrate	VERB
ejpam-6609	155	9	in	in	ADP
ejpam-6609	155	10	examples	example	NOUN
ejpam-6609	155	11	1	1	NUM
ejpam-6609	155	12	and	and	CCONJ
ejpam-6609	155	13	2	2	NUM
ejpam-6609	155	14	,	,	PUNCT
ejpam-6609	155	15	which	which	PRON
ejpam-6609	155	16	rely	rely	VERB
ejpam-6609	155	17	on	on	ADP
ejpam-6609	155	18	lemma	lemma	PROPN
ejpam-6609	155	19	2	2	NUM
ejpam-6609	155	20	and	and	CCONJ
ejpam-6609	155	21	corollary	corollary	ADJ
ejpam-6609	155	22	2	2	NUM
ejpam-6609	155	23	.	.	PUNCT
ejpam-6609	155	24	theoretical	theoretical	ADJ
ejpam-6609	155	25	support	support	NOUN
ejpam-6609	155	26	for	for	ADP
ejpam-6609	155	27	all	all	DET
ejpam-6609	155	28	parts	part	NOUN
ejpam-6609	155	29	comes	come	VERB
ejpam-6609	155	30	from	from	ADP
ejpam-6609	155	31	theorem	theorem	ADJ
ejpam-6609	155	32	1	1	NUM
ejpam-6609	155	33	,	,	PUNCT
ejpam-6609	155	34	and	and	CCONJ
ejpam-6609	155	35	figures	figure	NOUN
ejpam-6609	155	36	1	1	NUM
ejpam-6609	155	37	and	and	CCONJ
ejpam-6609	155	38	2	2	NUM
ejpam-6609	155	39	show	show	VERB
ejpam-6609	155	40	how	how	SCONJ
ejpam-6609	155	41	the	the	DET
ejpam-6609	155	42	fixed	fix	VERB
ejpam-6609	155	43	-	-	PUNCT
ejpam-6609	155	44	point	point	NOUN
ejpam-6609	155	45	convergence	convergence	NOUN
ejpam-6609	155	46	works	work	VERB
ejpam-6609	155	47	in	in	ADP
ejpam-6609	155	48	both	both	DET
ejpam-6609	155	49	cases	case	NOUN
ejpam-6609	155	50	.	.	PUNCT
ejpam-6609	156	1	m.	m.	NOUN
ejpam-6609	156	2	noorwali	noorwali	PROPN
ejpam-6609	156	3	et	et	PROPN
ejpam-6609	156	4	al	al	PROPN
ejpam-6609	156	5	.	.	PUNCT
ejpam-6609	156	6	/	/	SYM
ejpam-6609	156	7	eur	eur	PROPN
ejpam-6609	156	8	.	.	PUNCT
ejpam-6609	157	1	j.	j.	PROPN
ejpam-6609	157	2	pure	pure	PROPN
ejpam-6609	157	3	appl	appl	PROPN
ejpam-6609	157	4	.	.	PROPN
ejpam-6609	157	5	math	math	PROPN
ejpam-6609	157	6	,	,	PUNCT
ejpam-6609	157	7	18	18	NUM
ejpam-6609	157	8	(	(	PUNCT
ejpam-6609	157	9	3	3	NUM
ejpam-6609	157	10	)	)	PUNCT
ejpam-6609	157	11	(	(	PUNCT
ejpam-6609	157	12	2025	2025	NUM
ejpam-6609	157	13	)	)	PUNCT
ejpam-6609	157	14	,	,	PUNCT
ejpam-6609	157	15	6609	6609	NUM
ejpam-6609	157	16	7	7	NUM
ejpam-6609	157	17	of	of	ADP
ejpam-6609	157	18	17	17	NUM
ejpam-6609	157	19	example	example	NOUN
ejpam-6609	157	20	1	1	NUM
ejpam-6609	157	21	.	.	X
ejpam-6609	158	1	for	for	ADP
ejpam-6609	158	2	reproducing	reproduce	VERB
ejpam-6609	158	3	kernel	kernel	PROPN
ejpam-6609	158	4	hilbert	hilbert	PROPN
ejpam-6609	158	5	space	space	NOUN
ejpam-6609	158	6	(	(	PUNCT
ejpam-6609	158	7	rkhs	rkh	NOUN
ejpam-6609	158	8	)	)	PUNCT
ejpam-6609	158	9	hk	hk	NOUN
ejpam-6609	158	10	with	with	ADP
ejpam-6609	158	11	kernel	kernel	PROPN
ejpam-6609	158	12	k(x	k(x	PROPN
ejpam-6609	158	13	,	,	PUNCT
ejpam-6609	158	14	y	y	NOUN
ejpam-6609	158	15	)	)	PUNCT
ejpam-6609	158	16	=	=	PUNCT
ejpam-6609	158	17	⟨ϕ(x	⟨ϕ(x	NOUN
ejpam-6609	158	18	)	)	PUNCT
ejpam-6609	158	19	,	,	PUNCT
ejpam-6609	158	20	ϕ(y)⟩	ϕ(y)⟩	PROPN
ejpam-6609	158	21	,	,	PUNCT
ejpam-6609	158	22	the	the	DET
ejpam-6609	158	23	theorem	theorem	NOUN
ejpam-6609	158	24	guarantees	guarantee	NOUN
ejpam-6609	158	25	convergence	convergence	NOUN
ejpam-6609	158	26	of	of	ADP
ejpam-6609	158	27	:	:	PUNCT
ejpam-6609	158	28	ϕn+1	ϕn+1	X
ejpam-6609	158	29	=	=	PUNCT
ejpam-6609	158	30	t1ϕn	t1ϕn	PUNCT
ejpam-6609	159	1	+	+	NUM
ejpam-6609	159	2	t2ϕn	t2ϕn	PUNCT
ejpam-6609	160	1	+	+	CCONJ
ejpam-6609	160	2	t3ϕn	t3ϕn	NOUN
ejpam-6609	160	3	3	3	NUM
ejpam-6609	160	4	where	where	SCONJ
ejpam-6609	160	5	ti	ti	PRON
ejpam-6609	160	6	are	be	AUX
ejpam-6609	160	7	nonlinear	nonlinear	ADJ
ejpam-6609	160	8	operators	operator	NOUN
ejpam-6609	160	9	in	in	ADP
ejpam-6609	160	10	feature	feature	NOUN
ejpam-6609	160	11	space	space	NOUN
ejpam-6609	160	12	.	.	PUNCT
ejpam-6609	161	1	algorithm	algorithm	NOUN
ejpam-6609	161	2	2	2	NUM
ejpam-6609	161	3	describes	describe	VERB
ejpam-6609	161	4	the	the	DET
ejpam-6609	161	5	corresponding	corresponding	ADJ
ejpam-6609	161	6	computational	computational	ADJ
ejpam-6609	161	7	technique	technique	NOUN
ejpam-6609	161	8	.	.	PUNCT
ejpam-6609	162	1	algorithm	algorithm	NOUN
ejpam-6609	162	2	2	2	NUM
ejpam-6609	162	3	kernel	kernel	NOUN
ejpam-6609	162	4	-	-	PUNCT
ejpam-6609	162	5	based	base	VERB
ejpam-6609	162	6	fixed	fix	VERB
ejpam-6609	162	7	point	point	NOUN
ejpam-6609	162	8	iteration	iteration	NOUN
ejpam-6609	162	9	for	for	ADP
ejpam-6609	162	10	rkhs	rkh	NOUN
ejpam-6609	162	11	require	require	VERB
ejpam-6609	162	12	:	:	PUNCT
ejpam-6609	163	1	rkhs	rkh	VERB
ejpam-6609	163	2	hk	hk	PROPN
ejpam-6609	163	3	with	with	ADP
ejpam-6609	163	4	kernel	kernel	PROPN
ejpam-6609	163	5	k	k	PROPN
ejpam-6609	163	6	(	(	PUNCT
ejpam-6609	163	7	·	·	PUNCT
ejpam-6609	163	8	,	,	PUNCT
ejpam-6609	163	9	·	·	PUNCT
ejpam-6609	163	10	)	)	PUNCT
ejpam-6609	163	11	nonlinear	nonlinear	PROPN
ejpam-6609	163	12	operators	operator	NOUN
ejpam-6609	163	13	t1	t1	VERB
ejpam-6609	163	14	,	,	PUNCT
ejpam-6609	163	15	t2	t2	NOUN
ejpam-6609	163	16	,	,	PUNCT
ejpam-6609	163	17	t3	t3	PROPN
ejpam-6609	163	18	:	:	PUNCT
ejpam-6609	163	19	hk	hk	PROPN
ejpam-6609	163	20	→	→	SYM
ejpam-6609	163	21	hk	hk	PROPN
ejpam-6609	163	22	initial	initial	ADJ
ejpam-6609	163	23	feature	feature	NOUN
ejpam-6609	163	24	map	map	NOUN
ejpam-6609	163	25	ϕ0	ϕ0	NOUN
ejpam-6609	163	26	∈	∈	PROPN
ejpam-6609	163	27	hk	hk	NOUN
ejpam-6609	163	28	tolerance	tolerance	NOUN
ejpam-6609	163	29	ϵ	ϵ	X
ejpam-6609	163	30	>	>	X
ejpam-6609	163	31	0	0	NUM
ejpam-6609	164	1	maximum	maximum	ADJ
ejpam-6609	164	2	iterations	iteration	NOUN
ejpam-6609	164	3	n	n	PRON
ejpam-6609	164	4	ensure	ensure	VERB
ejpam-6609	164	5	:	:	PUNCT
ejpam-6609	164	6	approximate	approximate	ADJ
ejpam-6609	164	7	fixed	fix	VERB
ejpam-6609	164	8	point	point	NOUN
ejpam-6609	164	9	ϕ∗	ϕ∗	ADV
ejpam-6609	164	10	satisfying	satisfy	VERB
ejpam-6609	164	11	1	1	NUM
ejpam-6609	164	12	3	3	NUM
ejpam-6609	164	13	∑3	∑3	PROPN
ejpam-6609	164	14	i=1	i=1	PROPN
ejpam-6609	164	15	tiϕ	tiϕ	NOUN
ejpam-6609	164	16	∗	∗	NOUN
ejpam-6609	164	17	=	=	PUNCT
ejpam-6609	165	1	ϕ∗	ϕ∗	NOUN
ejpam-6609	165	2	1	1	NUM
ejpam-6609	165	3	:	:	PUNCT
ejpam-6609	165	4	n←	n←	X
ejpam-6609	165	5	0	0	NUM
ejpam-6609	165	6	2	2	NUM
ejpam-6609	165	7	:	:	PUNCT
ejpam-6609	165	8	while	while	SCONJ
ejpam-6609	165	9	n	n	CCONJ
ejpam-6609	165	10	<	<	X
ejpam-6609	165	11	n	n	NOUN
ejpam-6609	165	12	and	and	CCONJ
ejpam-6609	165	13	∥ϕn+1	∥ϕn+1	ADP
ejpam-6609	165	14	−	−	NOUN
ejpam-6609	165	15	ϕn∥hk	ϕn∥hk	PROPN
ejpam-6609	165	16	≥	≥	PRON
ejpam-6609	165	17	ϵ	ϵ	AUX
ejpam-6609	165	18	do	do	VERB
ejpam-6609	165	19	3	3	NUM
ejpam-6609	165	20	:	:	PUNCT
ejpam-6609	165	21	compute	compute	ADJ
ejpam-6609	165	22	kernel	kernel	PROPN
ejpam-6609	165	23	evaluations	evaluation	NOUN
ejpam-6609	165	24	:	:	PUNCT
ejpam-6609	166	1	4	4	NUM
ejpam-6609	166	2	:	:	PUNCT
ejpam-6609	166	3	kn,1	kn,1	PROPN
ejpam-6609	166	4	←	←	PROPN
ejpam-6609	166	5	k	k	X
ejpam-6609	166	6	(	(	PUNCT
ejpam-6609	166	7	·	·	PUNCT
ejpam-6609	166	8	,	,	PUNCT
ejpam-6609	166	9	t1ϕn	t1ϕn	ADJ
ejpam-6609	166	10	)	)	PUNCT
ejpam-6609	166	11	5	5	NUM
ejpam-6609	166	12	:	:	PUNCT
ejpam-6609	166	13	kn,2	kn,2	PROPN
ejpam-6609	166	14	←	←	PROPN
ejpam-6609	166	15	k	k	X
ejpam-6609	166	16	(	(	PUNCT
ejpam-6609	166	17	·	·	PUNCT
ejpam-6609	166	18	,	,	PUNCT
ejpam-6609	166	19	t2ϕn	t2ϕn	NUM
ejpam-6609	166	20	)	)	PUNCT
ejpam-6609	166	21	6	6	NUM
ejpam-6609	166	22	:	:	PUNCT
ejpam-6609	166	23	kn,3	kn,3	PROPN
ejpam-6609	166	24	←	←	PROPN
ejpam-6609	166	25	k	k	PROPN
ejpam-6609	166	26	(	(	PUNCT
ejpam-6609	166	27	·	·	PUNCT
ejpam-6609	166	28	,	,	PUNCT
ejpam-6609	166	29	t3ϕn	t3ϕn	NOUN
ejpam-6609	166	30	)	)	PUNCT
ejpam-6609	166	31	7	7	NUM
ejpam-6609	166	32	:	:	PUNCT
ejpam-6609	166	33	update	update	VERB
ejpam-6609	166	34	feature	feature	NOUN
ejpam-6609	166	35	map	map	NOUN
ejpam-6609	166	36	:	:	PUNCT
ejpam-6609	166	37	8	8	NUM
ejpam-6609	166	38	:	:	SYM
ejpam-6609	166	39	ϕn+1	ϕn+1	NOUN
ejpam-6609	166	40	←	←	PROPN
ejpam-6609	166	41	1	1	NUM
ejpam-6609	166	42	3(kn,1	3(kn,1	NUM
ejpam-6609	166	43	+	+	CCONJ
ejpam-6609	166	44	kn,2	kn,2	PROPN
ejpam-6609	166	45	+	+	CCONJ
ejpam-6609	166	46	kn,3	kn,3	PROPN
ejpam-6609	166	47	)	)	PUNCT
ejpam-6609	166	48	9	9	NUM
ejpam-6609	166	49	:	:	PUNCT
ejpam-6609	166	50	if	if	SCONJ
ejpam-6609	166	51	∥ϕn+1	∥ϕn+1	ADP
ejpam-6609	166	52	−	−	PRON
ejpam-6609	166	53	ϕn∥hk	ϕn∥hk	PROPN
ejpam-6609	166	54	<	<	X
ejpam-6609	166	55	ϵ	ϵ	X
ejpam-6609	166	56	then	then	ADV
ejpam-6609	166	57	10	10	NUM
ejpam-6609	166	58	:	:	PUNCT
ejpam-6609	166	59	break	break	NOUN
ejpam-6609	166	60	11	11	NUM
ejpam-6609	166	61	:	:	PUNCT
ejpam-6609	166	62	end	end	VERB
ejpam-6609	166	63	if	if	SCONJ
ejpam-6609	166	64	12	12	NUM
ejpam-6609	166	65	:	:	PUNCT
ejpam-6609	166	66	n←	n←	NUM
ejpam-6609	166	67	n+	n+	PRON
ejpam-6609	166	68	1	1	NUM
ejpam-6609	166	69	13	13	NUM
ejpam-6609	166	70	:	:	PUNCT
ejpam-6609	166	71	end	end	VERB
ejpam-6609	166	72	while	while	SCONJ
ejpam-6609	166	73	14	14	NUM
ejpam-6609	166	74	:	:	PUNCT
ejpam-6609	166	75	if	if	SCONJ
ejpam-6609	166	76	n	n	NOUN
ejpam-6609	166	77	=	=	SYM
ejpam-6609	166	78	n	n	NOUN
ejpam-6609	166	79	then	then	ADV
ejpam-6609	166	80	15	15	NUM
ejpam-6609	166	81	:	:	PUNCT
ejpam-6609	166	82	return	return	VERB
ejpam-6609	166	83	“	"	PUNCT
ejpam-6609	166	84	maximum	maximum	ADJ
ejpam-6609	166	85	iterations	iteration	NOUN
ejpam-6609	166	86	reached	reach	VERB
ejpam-6609	166	87	”	"	PUNCT
ejpam-6609	166	88	16	16	NUM
ejpam-6609	166	89	:	:	PUNCT
ejpam-6609	166	90	else	else	ADV
ejpam-6609	166	91	17	17	NUM
ejpam-6609	166	92	:	:	PUNCT
ejpam-6609	166	93	return	return	VERB
ejpam-6609	166	94	ϕ∗	ϕ∗	ADV
ejpam-6609	166	95	←	←	PROPN
ejpam-6609	166	96	ϕn+1	ϕn+1	PROPN
ejpam-6609	166	97	18	18	NUM
ejpam-6609	166	98	:	:	PUNCT
ejpam-6609	166	99	end	end	VERB
ejpam-6609	166	100	if	if	SCONJ
ejpam-6609	166	101	example	example	NOUN
ejpam-6609	167	1	2	2	X
ejpam-6609	167	2	.	.	PUNCT
ejpam-6609	167	3	the	the	DET
ejpam-6609	167	4	lattice	lattice	PROPN
ejpam-6609	167	5	l	l	PROPN
ejpam-6609	167	6	⊂	⊂	PROPN
ejpam-6609	167	7	rn	rn	PROPN
ejpam-6609	167	8	has	have	VERB
ejpam-6609	167	9	a	a	DET
ejpam-6609	167	10	basis	basis	NOUN
ejpam-6609	167	11	b	b	NOUN
ejpam-6609	167	12	=	=	SYM
ejpam-6609	167	13	{	{	PUNCT
ejpam-6609	167	14	b1	b1	PROPN
ejpam-6609	167	15	,	,	PUNCT
ejpam-6609	167	16	...	...	PUNCT
ejpam-6609	167	17	,	,	PUNCT
ejpam-6609	167	18	bn	bn	ADJ
ejpam-6609	167	19	}	}	PUNCT
ejpam-6609	167	20	.	.	PUNCT
ejpam-6609	168	1	theorem	theorem	ADJ
ejpam-6609	168	2	1	1	NUM
ejpam-6609	168	3	ensures	ensure	VERB
ejpam-6609	168	4	convergence	convergence	NOUN
ejpam-6609	168	5	of	of	ADP
ejpam-6609	168	6	the	the	DET
ejpam-6609	168	7	decoding	decode	VERB
ejpam-6609	168	8	process	process	NOUN
ejpam-6609	168	9	using	use	VERB
ejpam-6609	168	10	fixed	fix	VERB
ejpam-6609	168	11	-	-	PUNCT
ejpam-6609	168	12	point	point	NOUN
ejpam-6609	168	13	iteration	iteration	NOUN
ejpam-6609	168	14	.	.	PUNCT
ejpam-6609	169	1	the	the	DET
ejpam-6609	169	2	entire	entire	ADJ
ejpam-6609	169	3	technique	technique	NOUN
ejpam-6609	169	4	is	be	AUX
ejpam-6609	169	5	described	describe	VERB
ejpam-6609	169	6	in	in	ADP
ejpam-6609	169	7	algorithm	algorithm	NOUN
ejpam-6609	169	8	3	3	NUM
ejpam-6609	169	9	.	.	PUNCT
ejpam-6609	169	10	m.	m.	PROPN
ejpam-6609	169	11	noorwali	noorwali	PROPN
ejpam-6609	169	12	et	et	PROPN
ejpam-6609	169	13	al	al	PROPN
ejpam-6609	169	14	.	.	PUNCT
ejpam-6609	169	15	/	/	SYM
ejpam-6609	169	16	eur	eur	PROPN
ejpam-6609	169	17	.	.	PUNCT
ejpam-6609	170	1	j.	j.	PROPN
ejpam-6609	170	2	pure	pure	PROPN
ejpam-6609	170	3	appl	appl	PROPN
ejpam-6609	170	4	.	.	PROPN
ejpam-6609	170	5	math	math	PROPN
ejpam-6609	170	6	,	,	PUNCT
ejpam-6609	170	7	18	18	NUM
ejpam-6609	170	8	(	(	PUNCT
ejpam-6609	170	9	3	3	NUM
ejpam-6609	170	10	)	)	PUNCT
ejpam-6609	170	11	(	(	PUNCT
ejpam-6609	170	12	2025	2025	NUM
ejpam-6609	170	13	)	)	PUNCT
ejpam-6609	170	14	,	,	PUNCT
ejpam-6609	170	15	6609	6609	NUM
ejpam-6609	170	16	8	8	NUM
ejpam-6609	170	17	of	of	ADP
ejpam-6609	170	18	17	17	NUM
ejpam-6609	170	19	algorithm	algorithm	NOUN
ejpam-6609	170	20	3	3	NUM
ejpam-6609	170	21	fixed	fix	VERB
ejpam-6609	170	22	-	-	PUNCT
ejpam-6609	170	23	point	point	NOUN
ejpam-6609	170	24	lattice	lattice	NOUN
ejpam-6609	170	25	decoding	decode	VERB
ejpam-6609	170	26	require	require	NOUN
ejpam-6609	170	27	:	:	PUNCT
ejpam-6609	170	28	lattice	lattice	PROPN
ejpam-6609	170	29	l	l	PROPN
ejpam-6609	170	30	⊂	⊂	PROPN
ejpam-6609	170	31	rn	rn	PROPN
ejpam-6609	170	32	with	with	ADP
ejpam-6609	170	33	basis	basis	NOUN
ejpam-6609	170	34	b	b	NOUN
ejpam-6609	170	35	target	target	NOUN
ejpam-6609	170	36	vector	vector	NOUN
ejpam-6609	170	37	t	t	PROPN
ejpam-6609	170	38	∈	∈	PROPN
ejpam-6609	170	39	rn	rn	PROPN
ejpam-6609	170	40	tolerance	tolerance	NOUN
ejpam-6609	170	41	ϵ	ϵ	X
ejpam-6609	170	42	>	>	X
ejpam-6609	170	43	0	0	PUNCT
ejpam-6609	170	44	ensure	ensure	VERB
ejpam-6609	170	45	:	:	PUNCT
ejpam-6609	170	46	approximate	approximate	ADJ
ejpam-6609	170	47	closest	close	ADJ
ejpam-6609	170	48	vector	vector	NOUN
ejpam-6609	170	49	x∗	x∗	PROPN
ejpam-6609	170	50	∈	∈	PROPN
ejpam-6609	170	51	l	l	NOUN
ejpam-6609	170	52	1	1	X
ejpam-6609	170	53	:	:	PUNCT
ejpam-6609	170	54	initialize	initialize	VERB
ejpam-6609	170	55	x0	x0	PROPN
ejpam-6609	170	56	←	←	PROPN
ejpam-6609	170	57	t	t	PROPN
ejpam-6609	170	58	2	2	NUM
ejpam-6609	170	59	:	:	PUNCT
ejpam-6609	170	60	n←	n←	X
ejpam-6609	170	61	0	0	SYM
ejpam-6609	171	1	3	3	NUM
ejpam-6609	171	2	:	:	PUNCT
ejpam-6609	171	3	repeat	repeat	VERB
ejpam-6609	171	4	4	4	NUM
ejpam-6609	171	5	:	:	PUNCT
ejpam-6609	171	6	x3n+1	x3n+1	PROPN
ejpam-6609	171	7	←	←	PROPN
ejpam-6609	171	8	projl(x3n	projl(x3n	PROPN
ejpam-6609	171	9	)	)	PUNCT
ejpam-6609	171	10	5	5	NUM
ejpam-6609	171	11	:	:	PUNCT
ejpam-6609	171	12	x3n+2	x3n+2	PROPN
ejpam-6609	171	13	←	←	PROPN
ejpam-6609	171	14	roundb(x3n+1	roundb(x3n+1	PROPN
ejpam-6609	171	15	)	)	PUNCT
ejpam-6609	171	16	6	6	NUM
ejpam-6609	171	17	:	:	PUNCT
ejpam-6609	171	18	x3n+3	x3n+3	PROPN
ejpam-6609	172	1	←	←	PROPN
ejpam-6609	172	2	x3n+2	x3n+2	PROPN
ejpam-6609	172	3	mod	mod	PROPN
ejpam-6609	172	4	b	b	PROPN
ejpam-6609	172	5	7	7	NUM
ejpam-6609	172	6	:	:	PUNCT
ejpam-6609	172	7	n←	n←	NUM
ejpam-6609	172	8	n+	n+	PUNCT
ejpam-6609	172	9	1	1	NUM
ejpam-6609	172	10	8	8	NUM
ejpam-6609	172	11	:	:	PUNCT
ejpam-6609	172	12	until	until	ADP
ejpam-6609	172	13	∥x3n	∥x3n	PROPN
ejpam-6609	172	14	−	−	NOUN
ejpam-6609	172	15	x3n−3∥	x3n−3∥	PUNCT
ejpam-6609	173	1	<	<	X
ejpam-6609	173	2	ϵ	ϵ	X
ejpam-6609	173	3	9	9	NUM
ejpam-6609	173	4	:	:	PUNCT
ejpam-6609	173	5	return	return	VERB
ejpam-6609	173	6	x∗	x∗	PROPN
ejpam-6609	173	7	←	←	PROPN
ejpam-6609	173	8	x3n	x3n	PROPN
ejpam-6609	173	9	remark	remark	PROPN
ejpam-6609	173	10	2	2	NUM
ejpam-6609	173	11	.	.	PUNCT
ejpam-6609	174	1	the	the	DET
ejpam-6609	174	2	three	three	NUM
ejpam-6609	174	3	operations	operation	NOUN
ejpam-6609	174	4	(	(	PUNCT
ejpam-6609	174	5	projection	projection	NOUN
ejpam-6609	174	6	,	,	PUNCT
ejpam-6609	174	7	rounding	rounding	NOUN
ejpam-6609	174	8	,	,	PUNCT
ejpam-6609	174	9	and	and	CCONJ
ejpam-6609	174	10	modulo	modulo	PROPN
ejpam-6609	174	11	reduction	reduction	NOUN
ejpam-6609	174	12	)	)	PUNCT
ejpam-6609	174	13	correspond	correspond	VERB
ejpam-6609	174	14	to	to	ADP
ejpam-6609	174	15	the	the	DET
ejpam-6609	174	16	mappings	mapping	NOUN
ejpam-6609	174	17	t1	t1	NOUN
ejpam-6609	174	18	,	,	PUNCT
ejpam-6609	174	19	t2	t2	NOUN
ejpam-6609	174	20	,	,	PUNCT
ejpam-6609	174	21	t3	t3	PROPN
ejpam-6609	174	22	in	in	ADP
ejpam-6609	174	23	theorem	theorem	NOUN
ejpam-6609	174	24	1	1	NUM
ejpam-6609	174	25	.	.	PUNCT
ejpam-6609	175	1	the	the	DET
ejpam-6609	175	2	contraction	contraction	NOUN
ejpam-6609	175	3	requirement	requirement	NOUN
ejpam-6609	175	4	can	can	AUX
ejpam-6609	175	5	be	be	AUX
ejpam-6609	175	6	satisfied	satisfied	ADJ
ejpam-6609	175	7	for	for	ADP
ejpam-6609	175	8	wellconditioned	wellconditione	VERB
ejpam-6609	175	9	bases	basis	NOUN
ejpam-6609	175	10	.	.	PUNCT
ejpam-6609	176	1	this	this	DET
ejpam-6609	176	2	result	result	NOUN
ejpam-6609	176	3	establishes	establish	VERB
ejpam-6609	176	4	the	the	DET
ejpam-6609	176	5	theoretical	theoretical	ADJ
ejpam-6609	176	6	framework	framework	NOUN
ejpam-6609	176	7	for	for	ADP
ejpam-6609	176	8	lattice	lattice	NOUN
ejpam-6609	176	9	decoding	decode	VERB
ejpam-6609	176	10	convergence	convergence	NOUN
ejpam-6609	176	11	.	.	PUNCT
ejpam-6609	177	1	the	the	DET
ejpam-6609	177	2	equation	equation	NOUN
ejpam-6609	177	3	α+	α+	X
ejpam-6609	177	4	β	β	X
ejpam-6609	177	5	+	+	X
ejpam-6609	177	6	γ	γ	X
ejpam-6609	177	7	<	<	X
ejpam-6609	177	8	1	1	NUM
ejpam-6609	177	9	ensures	ensure	VERB
ejpam-6609	177	10	that	that	SCONJ
ejpam-6609	177	11	the	the	DET
ejpam-6609	177	12	operator	operator	NOUN
ejpam-6609	177	13	t	t	PROPN
ejpam-6609	177	14	(	(	PUNCT
ejpam-6609	177	15	x	x	NOUN
ejpam-6609	177	16	)	)	PUNCT
ejpam-6609	177	17	=	=	SYM
ejpam-6609	178	1	t1x+	t1x+	NOUN
ejpam-6609	178	2	t2x+	t2x+	VERB
ejpam-6609	179	1	t3x	t3x	X
ejpam-6609	179	2	3	3	NUM
ejpam-6609	179	3	is	be	AUX
ejpam-6609	179	4	a	a	DET
ejpam-6609	179	5	contraction	contraction	NOUN
ejpam-6609	179	6	mapping	mapping	NOUN
ejpam-6609	179	7	.	.	PUNCT
ejpam-6609	180	1	lemma	lemma	PROPN
ejpam-6609	180	2	2	2	NUM
ejpam-6609	180	3	.	.	PUNCT
ejpam-6609	181	1	the	the	DET
ejpam-6609	181	2	contraction	contraction	NOUN
ejpam-6609	181	3	parameters	parameter	NOUN
ejpam-6609	181	4	meet	meet	VERB
ejpam-6609	181	5	the	the	DET
ejpam-6609	181	6	following	following	NOUN
ejpam-6609	181	7	:	:	PUNCT
ejpam-6609	181	8	α+	α+	X
ejpam-6609	181	9	β	β	NOUN
ejpam-6609	181	10	sup	sup	NOUN
ejpam-6609	181	11	x	x	PROPN
ejpam-6609	181	12	,	,	PUNCT
ejpam-6609	181	13	y	y	PROPN
ejpam-6609	181	14	∥x−	∥x−	PROPN
ejpam-6609	181	15	t2y∥2∥x−	t2y∥2∥x−	ADV
ejpam-6609	181	16	t3z∥2	t3z∥2	VERB
ejpam-6609	181	17	1	1	NUM
ejpam-6609	181	18	+	+	CCONJ
ejpam-6609	181	19	·	·	PUNCT
ejpam-6609	181	20	·	·	PUNCT
ejpam-6609	181	21	·	·	PUNCT
ejpam-6609	182	1	+	+	CCONJ
ejpam-6609	182	2	γ	γ	X
ejpam-6609	182	3	sup	sup	NOUN
ejpam-6609	182	4	x	x	NOUN
ejpam-6609	182	5	,	,	PUNCT
ejpam-6609	182	6	y	y	PROPN
ejpam-6609	182	7	∥y	∥y	PROPN
ejpam-6609	182	8	−	−	PROPN
ejpam-6609	182	9	t3z∥2∥z	t3z∥2∥z	NOUN
ejpam-6609	182	10	−	−	NOUN
ejpam-6609	182	11	t1x∥2	t1x∥2	NOUN
ejpam-6609	182	12	1	1	NUM
ejpam-6609	182	13	+	+	CCONJ
ejpam-6609	182	14	·	·	PUNCT
ejpam-6609	182	15	·	·	PUNCT
ejpam-6609	182	16	·	·	PUNCT
ejpam-6609	182	17	<	<	X
ejpam-6609	182	18	1	1	NUM
ejpam-6609	182	19	corollary	corollary	ADJ
ejpam-6609	182	20	2	2	NUM
ejpam-6609	182	21	(	(	PUNCT
ejpam-6609	182	22	special	special	ADJ
ejpam-6609	182	23	case	case	NOUN
ejpam-6609	182	24	for	for	ADP
ejpam-6609	182	25	two	two	NUM
ejpam-6609	182	26	operators	operator	NOUN
ejpam-6609	182	27	)	)	PUNCT
ejpam-6609	182	28	.	.	PUNCT
ejpam-6609	183	1	let	let	VERB
ejpam-6609	183	2	(	(	PUNCT
ejpam-6609	183	3	h	h	NOUN
ejpam-6609	183	4	,	,	PUNCT
ejpam-6609	183	5	⟨	⟨	NOUN
ejpam-6609	183	6	·	·	SYM
ejpam-6609	183	7	,	,	PUNCT
ejpam-6609	183	8	·	·	PUNCT
ejpam-6609	183	9	⟩	⟩	NOUN
ejpam-6609	183	10	)	)	PUNCT
ejpam-6609	183	11	be	be	VERB
ejpam-6609	183	12	a	a	DET
ejpam-6609	183	13	hilbert	hilbert	NOUN
ejpam-6609	183	14	space	space	NOUN
ejpam-6609	183	15	and	and	CCONJ
ejpam-6609	183	16	t1	t1	NOUN
ejpam-6609	183	17	,	,	PUNCT
ejpam-6609	183	18	t2	t2	NOUN
ejpam-6609	183	19	:	:	PUNCT
ejpam-6609	184	1	h	h	NOUN
ejpam-6609	184	2	→	→	SYM
ejpam-6609	184	3	h	h	NOUN
ejpam-6609	184	4	satisfy	satisfy	NOUN
ejpam-6609	184	5	:	:	PUNCT
ejpam-6609	184	6	∥t1x−	∥t1x−	NUM
ejpam-6609	184	7	t2y∥2	t2y∥2	NOUN
ejpam-6609	184	8	≤	≤	PUNCT
ejpam-6609	185	1	α∥x−	α∥x−	CCONJ
ejpam-6609	185	2	y∥2	y∥2	NOUN
ejpam-6609	185	3	+	+	CCONJ
ejpam-6609	185	4	β	β	X
ejpam-6609	185	5	∥x−	∥x−	NUM
ejpam-6609	185	6	t2y∥2	t2y∥2	NOUN
ejpam-6609	185	7	·	·	PUNCT
ejpam-6609	185	8	∥y	∥y	ADJ
ejpam-6609	185	9	−	−	NOUN
ejpam-6609	185	10	t1x∥2	t1x∥2	NOUN
ejpam-6609	185	11	1	1	NUM
ejpam-6609	185	12	+	+	CCONJ
ejpam-6609	185	13	∥y	∥y	ADJ
ejpam-6609	185	14	−	−	NOUN
ejpam-6609	185	15	t1x∥2∥x−	t1x∥2∥x−	NOUN
ejpam-6609	185	16	t2y∥2	t2y∥2	NOUN
ejpam-6609	185	17	for	for	ADP
ejpam-6609	185	18	all	all	DET
ejpam-6609	185	19	x	x	NOUN
ejpam-6609	185	20	,	,	PUNCT
ejpam-6609	185	21	y	y	PROPN
ejpam-6609	185	22	∈	∈	PROPN
ejpam-6609	185	23	h	h	NOUN
ejpam-6609	185	24	,	,	PUNCT
ejpam-6609	185	25	where	where	SCONJ
ejpam-6609	185	26	α	α	X
ejpam-6609	185	27	,	,	PUNCT
ejpam-6609	185	28	β	β	X
ejpam-6609	185	29	≥	≥	NOUN
ejpam-6609	185	30	0	0	NUM
ejpam-6609	185	31	with	with	ADP
ejpam-6609	185	32	α+	α+	X
ejpam-6609	185	33	β	β	X
ejpam-6609	185	34	<	<	X
ejpam-6609	185	35	1	1	NUM
ejpam-6609	185	36	.	.	PUNCT
ejpam-6609	186	1	then	then	ADV
ejpam-6609	186	2	there	there	PRON
ejpam-6609	186	3	exists	exist	VERB
ejpam-6609	186	4	a	a	DET
ejpam-6609	186	5	unique	unique	ADJ
ejpam-6609	186	6	common	common	ADJ
ejpam-6609	186	7	fixed	fix	VERB
ejpam-6609	186	8	point	point	NOUN
ejpam-6609	186	9	u	u	PROPN
ejpam-6609	186	10	∈	∈	PROPN
ejpam-6609	186	11	h	h	NOUN
ejpam-6609	186	12	such	such	ADJ
ejpam-6609	186	13	that	that	DET
ejpam-6609	186	14	t1u	t1u	NOUN
ejpam-6609	186	15	=	=	PUNCT
ejpam-6609	187	1	t2u	t2u	PUNCT
ejpam-6609	187	2	=	=	PUNCT
ejpam-6609	187	3	u.	u.	NOUN
ejpam-6609	187	4	figure	figure	NOUN
ejpam-6609	187	5	1	1	NUM
ejpam-6609	187	6	:	:	PUNCT
ejpam-6609	187	7	kernel	kernel	NOUN
ejpam-6609	187	8	method	method	PROPN
ejpam-6609	187	9	and	and	CCONJ
ejpam-6609	187	10	lattice	lattice	NOUN
ejpam-6609	187	11	decoding	decode	VERB
ejpam-6609	187	12	m.	m.	NOUN
ejpam-6609	187	13	noorwali	noorwali	PROPN
ejpam-6609	187	14	et	et	PROPN
ejpam-6609	187	15	al	al	PROPN
ejpam-6609	187	16	.	.	PUNCT
ejpam-6609	187	17	/	/	SYM
ejpam-6609	187	18	eur	eur	PROPN
ejpam-6609	187	19	.	.	PUNCT
ejpam-6609	188	1	j.	j.	PROPN
ejpam-6609	188	2	pure	pure	PROPN
ejpam-6609	188	3	appl	appl	PROPN
ejpam-6609	188	4	.	.	PROPN
ejpam-6609	188	5	math	math	PROPN
ejpam-6609	188	6	,	,	PUNCT
ejpam-6609	188	7	18	18	NUM
ejpam-6609	188	8	(	(	PUNCT
ejpam-6609	188	9	3	3	NUM
ejpam-6609	188	10	)	)	PUNCT
ejpam-6609	188	11	(	(	PUNCT
ejpam-6609	188	12	2025	2025	NUM
ejpam-6609	188	13	)	)	PUNCT
ejpam-6609	188	14	,	,	PUNCT
ejpam-6609	188	15	6609	6609	NUM
ejpam-6609	188	16	9	9	NUM
ejpam-6609	188	17	of	of	ADP
ejpam-6609	188	18	17	17	NUM
ejpam-6609	188	19	figure	figure	NOUN
ejpam-6609	188	20	2	2	NUM
ejpam-6609	188	21	:	:	PUNCT
ejpam-6609	188	22	contraction	contraction	NOUN
ejpam-6609	188	23	condition	condition	NOUN
ejpam-6609	188	24	verify	verify	NOUN
ejpam-6609	188	25	theorem	theorem	NOUN
ejpam-6609	188	26	2	2	X
ejpam-6609	188	27	.	.	PUNCT
ejpam-6609	189	1	let	let	AUX
ejpam-6609	189	2	(	(	PUNCT
ejpam-6609	189	3	h	h	NOUN
ejpam-6609	189	4	,	,	PUNCT
ejpam-6609	189	5	⟨	⟨	NOUN
ejpam-6609	189	6	·	·	SYM
ejpam-6609	189	7	,	,	PUNCT
ejpam-6609	189	8	·	·	PUNCT
ejpam-6609	189	9	⟩	⟩	NOUN
ejpam-6609	189	10	)	)	PUNCT
ejpam-6609	189	11	be	be	VERB
ejpam-6609	189	12	a	a	DET
ejpam-6609	189	13	hs	hs	PROPN
ejpam-6609	189	14	and	and	CCONJ
ejpam-6609	189	15	t1	t1	PROPN
ejpam-6609	189	16	,	,	PUNCT
ejpam-6609	189	17	t2	t2	NOUN
ejpam-6609	189	18	,	,	PUNCT
ejpam-6609	189	19	t3	t3	NOUN
ejpam-6609	189	20	:	:	PUNCT
ejpam-6609	189	21	h	h	NOUN
ejpam-6609	189	22	→	→	PUNCT
ejpam-6609	189	23	h	h	NOUN
ejpam-6609	189	24	be	be	AUX
ejpam-6609	189	25	operators	operator	NOUN
ejpam-6609	189	26	satisfying	satisfy	VERB
ejpam-6609	189	27	:	:	PUNCT
ejpam-6609	189	28	∥t1x−	∥t1x−	NUM
ejpam-6609	189	29	t2y	t2y	ADV
ejpam-6609	189	30	−	−	PROPN
ejpam-6609	189	31	t3z∥2	t3z∥2	VERB
ejpam-6609	189	32	≤	≤	PUNCT
ejpam-6609	190	1	α∥x−	α∥x−	CCONJ
ejpam-6609	190	2	y	y	PROPN
ejpam-6609	190	3	−	−	PROPN
ejpam-6609	190	4	z∥2	z∥2	NOUN
ejpam-6609	190	5	+	+	PUNCT
ejpam-6609	190	6	β	β	VERB
ejpam-6609	190	7	∥x−	∥x−	NUM
ejpam-6609	190	8	t1x∥2∥y	t1x∥2∥y	PROPN
ejpam-6609	190	9	−	−	PROPN
ejpam-6609	190	10	t2y∥2∥z	t2y∥2∥z	NOUN
ejpam-6609	190	11	−	−	PROPN
ejpam-6609	190	12	t3z∥2	t3z∥2	NOUN
ejpam-6609	190	13	1	1	NUM
ejpam-6609	190	14	+	+	CCONJ
ejpam-6609	190	15	∥y	∥y	ADJ
ejpam-6609	190	16	−	−	NOUN
ejpam-6609	190	17	t2y∥2∥t2y	t2y∥2∥t2y	ADP
ejpam-6609	190	18	−	−	PROPN
ejpam-6609	190	19	t3z∥2	t3z∥2	NOUN
ejpam-6609	190	20	(	(	PUNCT
ejpam-6609	190	21	2	2	NUM
ejpam-6609	190	22	)	)	PUNCT
ejpam-6609	190	23	for	for	ADP
ejpam-6609	190	24	all	all	DET
ejpam-6609	190	25	x	x	NOUN
ejpam-6609	190	26	,	,	PUNCT
ejpam-6609	190	27	y	y	PROPN
ejpam-6609	190	28	,	,	PUNCT
ejpam-6609	190	29	z	z	PROPN
ejpam-6609	190	30	∈	∈	PROPN
ejpam-6609	190	31	h	h	NOUN
ejpam-6609	190	32	,	,	PUNCT
ejpam-6609	190	33	where	where	SCONJ
ejpam-6609	190	34	α	α	X
ejpam-6609	190	35	,	,	PUNCT
ejpam-6609	190	36	β	β	X
ejpam-6609	190	37	≥	≥	NOUN
ejpam-6609	190	38	0	0	NUM
ejpam-6609	190	39	with	with	ADP
ejpam-6609	190	40	α	α	PROPN
ejpam-6609	190	41	+	+	X
ejpam-6609	190	42	β	β	X
ejpam-6609	190	43	<	<	X
ejpam-6609	190	44	1	1	NUM
ejpam-6609	190	45	.	.	PUNCT
ejpam-6609	190	46	then	then	ADV
ejpam-6609	190	47	there	there	PRON
ejpam-6609	190	48	exists	exist	VERB
ejpam-6609	190	49	a	a	DET
ejpam-6609	190	50	unique	unique	ADJ
ejpam-6609	190	51	cfp	cfp	NOUN
ejpam-6609	190	52	u	u	NOUN
ejpam-6609	190	53	∈	∈	PROPN
ejpam-6609	190	54	h	h	NOUN
ejpam-6609	190	55	so	so	SCONJ
ejpam-6609	190	56	that	that	DET
ejpam-6609	190	57	t1u	t1u	VERB
ejpam-6609	190	58	=	=	PUNCT
ejpam-6609	190	59	t2u	t2u	VERB
ejpam-6609	191	1	=	=	PUNCT
ejpam-6609	191	2	t3u	t3u	NOUN
ejpam-6609	191	3	=	=	PUNCT
ejpam-6609	191	4	u.	u.	NOUN
ejpam-6609	191	5	proof	proof	NOUN
ejpam-6609	191	6	.	.	PUNCT
ejpam-6609	192	1	(	(	PUNCT
ejpam-6609	192	2	1	1	X
ejpam-6609	192	3	)	)	PUNCT
ejpam-6609	192	4	iterative	iterative	NOUN
ejpam-6609	192	5	sequence	sequence	NOUN
ejpam-6609	192	6	construction	construction	NOUN
ejpam-6609	192	7	fix	fix	NOUN
ejpam-6609	192	8	x0	x0	PROPN
ejpam-6609	192	9	∈	∈	PROPN
ejpam-6609	192	10	h	h	NOUN
ejpam-6609	192	11	and	and	CCONJ
ejpam-6609	192	12	define	define	VERB
ejpam-6609	192	13	the	the	DET
ejpam-6609	192	14	sequence	sequence	NOUN
ejpam-6609	192	15	:	:	PUNCT
ejpam-6609	192	16	x3n+1	x3n+1	X
ejpam-6609	192	17	=	=	PUNCT
ejpam-6609	192	18	t1x3n	t1x3n	X
ejpam-6609	192	19	;	;	PUNCT
ejpam-6609	192	20	x3n+2	x3n+2	PUNCT
ejpam-6609	192	21	=	=	SYM
ejpam-6609	192	22	t2x3n+1	t2x3n+1	PROPN
ejpam-6609	192	23	;	;	PUNCT
ejpam-6609	192	24	x3n+3	x3n+3	PUNCT
ejpam-6609	193	1	=	=	SYM
ejpam-6609	193	2	t3x3n+2	t3x3n+2	NUM
ejpam-6609	193	3	.	.	PUNCT
ejpam-6609	194	1	(	(	PUNCT
ejpam-6609	194	2	2	2	X
ejpam-6609	194	3	)	)	PUNCT
ejpam-6609	194	4	contraction	contraction	NOUN
ejpam-6609	194	5	estimates	estimate	NOUN
ejpam-6609	194	6	applying	apply	VERB
ejpam-6609	194	7	eq	eq	ADP
ejpam-6609	194	8	.	.	PUNCT
ejpam-6609	195	1	(	(	PUNCT
ejpam-6609	195	2	2	2	NUM
ejpam-6609	195	3	)	)	PUNCT
ejpam-6609	195	4	with	with	ADP
ejpam-6609	195	5	x	x	PROPN
ejpam-6609	195	6	=	=	SYM
ejpam-6609	195	7	x3n	x3n	PROPN
ejpam-6609	195	8	,	,	PUNCT
ejpam-6609	195	9	y	y	PROPN
ejpam-6609	195	10	=	=	SYM
ejpam-6609	195	11	x3n+1	x3n+1	PROPN
ejpam-6609	195	12	,	,	PUNCT
ejpam-6609	195	13	z	z	NOUN
ejpam-6609	195	14	=	=	SYM
ejpam-6609	195	15	x3n+2	x3n+2	PROPN
ejpam-6609	195	16	:	:	PUNCT
ejpam-6609	196	1	∥x3n+1	∥x3n+1	NOUN
ejpam-6609	196	2	−	−	PROPN
ejpam-6609	196	3	x3n+2	x3n+2	PUNCT
ejpam-6609	197	1	−	−	PROPN
ejpam-6609	197	2	x3n+3∥2	x3n+3∥2	PROPN
ejpam-6609	198	1	≤	≤	PROPN
ejpam-6609	199	1	α∥x3n	α∥x3n	PROPN
ejpam-6609	199	2	−	−	PROPN
ejpam-6609	199	3	x3n+1	x3n+1	PUNCT
ejpam-6609	199	4	−	−	PROPN
ejpam-6609	200	1	x3n+2∥2	x3n+2∥2	PUNCT
ejpam-6609	201	1	+	+	PUNCT
ejpam-6609	201	2	β	β	X
ejpam-6609	201	3	∥x3n	∥x3n	X
ejpam-6609	201	4	−	−	PROPN
ejpam-6609	201	5	x3n+1∥2∥x3n+1	x3n+1∥2∥x3n+1	NUM
ejpam-6609	202	1	−	−	PROPN
ejpam-6609	203	1	x3n+2∥2∥x3n+2	x3n+2∥2∥x3n+2	PROPN
ejpam-6609	204	1	−	−	PROPN
ejpam-6609	204	2	x3n+3∥2	x3n+3∥2	PROPN
ejpam-6609	204	3	1	1	NUM
ejpam-6609	205	1	+	+	NUM
ejpam-6609	205	2	∥x3n+1	∥x3n+1	NOUN
ejpam-6609	205	3	−	−	PROPN
ejpam-6609	205	4	x3n+2∥2∥x3n+2	x3n+2∥2∥x3n+2	PROPN
ejpam-6609	206	1	−	−	PROPN
ejpam-6609	206	2	x3n+3∥2	x3n+3∥2	PROPN
ejpam-6609	206	3	using	use	VERB
ejpam-6609	206	4	the	the	DET
ejpam-6609	206	5	identity	identity	NOUN
ejpam-6609	206	6	∥a−	∥a−	X
ejpam-6609	206	7	b∥2	b∥2	NOUN
ejpam-6609	206	8	=	=	PUNCT
ejpam-6609	206	9	∥a∥2	∥a∥2	PROPN
ejpam-6609	206	10	+	+	CCONJ
ejpam-6609	206	11	∥b∥2	∥b∥2	PROPN
ejpam-6609	206	12	−	−	PROPN
ejpam-6609	206	13	2⟨a	2⟨a	NOUN
ejpam-6609	206	14	,	,	PUNCT
ejpam-6609	206	15	b⟩	b⟩	PUNCT
ejpam-6609	206	16	and	and	CCONJ
ejpam-6609	206	17	the	the	DET
ejpam-6609	206	18	cauchy	cauchy	PROPN
ejpam-6609	206	19	-	-	PUNCT
ejpam-6609	206	20	schwarz	schwarz	PROPN
ejpam-6609	206	21	inequality	inequality	NOUN
ejpam-6609	206	22	:	:	PUNCT
ejpam-6609	206	23	≤	≤	NUM
ejpam-6609	206	24	α(∥x3n∥2	α(∥x3n∥2	NOUN
ejpam-6609	207	1	+	+	CCONJ
ejpam-6609	207	2	∥x3n+1∥2	∥x3n+1∥2	ADJ
ejpam-6609	207	3	+	+	CCONJ
ejpam-6609	207	4	∥x3n+2∥2	∥x3n+2∥2	PROPN
ejpam-6609	207	5	−	−	PROPN
ejpam-6609	207	6	2⟨x3n	2⟨x3n	NUM
ejpam-6609	207	7	,	,	PUNCT
ejpam-6609	207	8	x3n+1⟩	x3n+1⟩	PROPN
ejpam-6609	207	9	−	−	PROPN
ejpam-6609	207	10	2⟨x3n	2⟨x3n	NUM
ejpam-6609	207	11	,	,	PUNCT
ejpam-6609	207	12	x3n+2⟩+	x3n+2⟩+	PROPN
ejpam-6609	207	13	2⟨x3n+1	2⟨x3n+1	NUM
ejpam-6609	207	14	,	,	PUNCT
ejpam-6609	207	15	x3n+2⟩	x3n+2⟩	PROPN
ejpam-6609	207	16	)	)	PUNCT
ejpam-6609	208	1	+	+	CCONJ
ejpam-6609	208	2	β	β	X
ejpam-6609	208	3	(	(	PUNCT
ejpam-6609	208	4	∥x3n∥2	∥x3n∥2	NOUN
ejpam-6609	208	5	+	+	CCONJ
ejpam-6609	208	6	∥x3n+1∥2	∥x3n+1∥2	ADJ
ejpam-6609	208	7	−	−	NOUN
ejpam-6609	208	8	2⟨x3n	2⟨x3n	NUM
ejpam-6609	208	9	,	,	PUNCT
ejpam-6609	208	10	x3n+1⟩)(∥x3n+1∥2	x3n+1⟩)(∥x3n+1∥2	PROPN
ejpam-6609	208	11	+	+	CCONJ
ejpam-6609	208	12	∥x3n+2∥2	∥x3n+2∥2	PROPN
ejpam-6609	208	13	−	−	PROPN
ejpam-6609	208	14	2⟨x3n+1	2⟨x3n+1	NUM
ejpam-6609	208	15	,	,	PUNCT
ejpam-6609	208	16	x3n+2⟩	x3n+2⟩	PROPN
ejpam-6609	208	17	)	)	PUNCT
ejpam-6609	208	18	1	1	NUM
ejpam-6609	209	1	+	+	CCONJ
ejpam-6609	209	2	(	(	PUNCT
ejpam-6609	209	3	∥x3n+1∥2	∥x3n+1∥2	ADJ
ejpam-6609	209	4	+	+	CCONJ
ejpam-6609	209	5	∥x3n+2∥2	∥x3n+2∥2	PROPN
ejpam-6609	209	6	−	−	PROPN
ejpam-6609	209	7	2⟨x3n+1	2⟨x3n+1	NUM
ejpam-6609	209	8	,	,	PUNCT
ejpam-6609	209	9	x3n+2⟩	x3n+2⟩	PROPN
ejpam-6609	209	10	)	)	PUNCT
ejpam-6609	209	11	·	·	PUNCT
ejpam-6609	210	1	(	(	PUNCT
ejpam-6609	210	2	∥x3n+2∥2	∥x3n+2∥2	PROPN
ejpam-6609	210	3	+	+	CCONJ
ejpam-6609	210	4	∥x3n+3∥2	∥x3n+3∥2	PROPN
ejpam-6609	210	5	−	−	PROPN
ejpam-6609	210	6	2⟨x3n+2	2⟨x3n+2	PROPN
ejpam-6609	210	7	,	,	PUNCT
ejpam-6609	210	8	x3n+3⟩	x3n+3⟩	PROPN
ejpam-6609	210	9	)	)	PUNCT
ejpam-6609	210	10	1	1	NUM
ejpam-6609	211	1	+	+	CCONJ
ejpam-6609	211	2	(	(	PUNCT
ejpam-6609	211	3	∥x3n+2∥2	∥x3n+2∥2	PROPN
ejpam-6609	211	4	+	+	CCONJ
ejpam-6609	211	5	∥x3n+3∥2	∥x3n+3∥2	PROPN
ejpam-6609	211	6	−	−	PROPN
ejpam-6609	211	7	2⟨x3n+2	2⟨x3n+2	PROPN
ejpam-6609	211	8	,	,	PUNCT
ejpam-6609	211	9	x3n+3⟩	x3n+3⟩	PROPN
ejpam-6609	211	10	)	)	PUNCT
ejpam-6609	211	11	after	after	ADP
ejpam-6609	211	12	simplifying	simplify	VERB
ejpam-6609	211	13	utilizing	utilize	VERB
ejpam-6609	211	14	the	the	DET
ejpam-6609	211	15	contraction	contraction	NOUN
ejpam-6609	211	16	property	property	NOUN
ejpam-6609	211	17	:	:	PUNCT
ejpam-6609	211	18	∥x3n+1	∥x3n+1	NOUN
ejpam-6609	211	19	−	−	PROPN
ejpam-6609	211	20	x3n+2	x3n+2	PUNCT
ejpam-6609	212	1	−	−	PROPN
ejpam-6609	212	2	x3n+3∥	x3n+3∥	PROPN
ejpam-6609	213	1	≤	≤	PROPN
ejpam-6609	213	2	η∥x3n	η∥x3n	PROPN
ejpam-6609	213	3	−	−	PROPN
ejpam-6609	213	4	x3n+1	x3n+1	PROPN
ejpam-6609	213	5	−	−	PROPN
ejpam-6609	213	6	x3n+2∥	x3n+2∥	PROPN
ejpam-6609	213	7	(	(	PUNCT
ejpam-6609	213	8	η	η	PROPN
ejpam-6609	213	9	=	=	PROPN
ejpam-6609	213	10	√	√	PROPN
ejpam-6609	213	11	α+	α+	X
ejpam-6609	213	12	β	β	X
ejpam-6609	213	13	<	<	X
ejpam-6609	213	14	1	1	NUM
ejpam-6609	213	15	)	)	PUNCT
ejpam-6609	213	16	(	(	PUNCT
ejpam-6609	213	17	3	3	X
ejpam-6609	213	18	)	)	PUNCT
ejpam-6609	213	19	m.	m.	NOUN
ejpam-6609	213	20	noorwali	noorwali	PROPN
ejpam-6609	213	21	et	et	PROPN
ejpam-6609	213	22	al	al	PROPN
ejpam-6609	213	23	.	.	PUNCT
ejpam-6609	213	24	/	/	SYM
ejpam-6609	213	25	eur	eur	PROPN
ejpam-6609	213	26	.	.	PUNCT
ejpam-6609	214	1	j.	j.	PROPN
ejpam-6609	214	2	pure	pure	PROPN
ejpam-6609	214	3	appl	appl	PROPN
ejpam-6609	214	4	.	.	PROPN
ejpam-6609	214	5	math	math	PROPN
ejpam-6609	214	6	,	,	PUNCT
ejpam-6609	214	7	18	18	NUM
ejpam-6609	214	8	(	(	PUNCT
ejpam-6609	214	9	3	3	NUM
ejpam-6609	214	10	)	)	PUNCT
ejpam-6609	214	11	(	(	PUNCT
ejpam-6609	214	12	2025	2025	NUM
ejpam-6609	214	13	)	)	PUNCT
ejpam-6609	214	14	,	,	PUNCT
ejpam-6609	214	15	6609	6609	NUM
ejpam-6609	214	16	10	10	NUM
ejpam-6609	214	17	of	of	ADP
ejpam-6609	214	18	17	17	NUM
ejpam-6609	214	19	(	(	PUNCT
ejpam-6609	214	20	3	3	NUM
ejpam-6609	214	21	)	)	PUNCT
ejpam-6609	214	22	convergence	convergence	NOUN
ejpam-6609	214	23	analysis	analysis	NOUN
ejpam-6609	214	24	by	by	ADP
ejpam-6609	214	25	induction	induction	NOUN
ejpam-6609	214	26	:	:	PUNCT
ejpam-6609	214	27	∥xn+1	∥xn+1	VERB
ejpam-6609	214	28	−	−	PROPN
ejpam-6609	215	1	xn+2	xn+2	NUM
ejpam-6609	216	1	−	−	PROPN
ejpam-6609	216	2	xn+3∥	xn+3∥	NOUN
ejpam-6609	216	3	≤	≤	NOUN
ejpam-6609	216	4	ηn∥x1	ηn∥x1	NOUN
ejpam-6609	216	5	−	−	PROPN
ejpam-6609	216	6	x2	x2	PROPN
ejpam-6609	217	1	−	−	PROPN
ejpam-6609	217	2	x3∥	x3∥	PROPN
ejpam-6609	217	3	≤	≤	ADV
ejpam-6609	217	4	ηn(∥x1∥+	ηn(∥x1∥+	PROPN
ejpam-6609	217	5	∥x2∥+	∥x2∥+	ADJ
ejpam-6609	217	6	∥x3∥)→	∥x3∥)→	NOUN
ejpam-6609	217	7	0	0	NUM
ejpam-6609	217	8	as	as	ADP
ejpam-6609	217	9	n→∞	n→∞	NOUN
ejpam-6609	217	10	the	the	DET
ejpam-6609	217	11	sequence	sequence	NOUN
ejpam-6609	217	12	is	be	AUX
ejpam-6609	217	13	cauchy	cauchy	ADJ
ejpam-6609	217	14	since	since	SCONJ
ejpam-6609	217	15	for	for	ADP
ejpam-6609	217	16	m	m	PROPN
ejpam-6609	217	17	>	>	X
ejpam-6609	217	18	n	n	CCONJ
ejpam-6609	217	19	:	:	PUNCT
ejpam-6609	217	20	∥xn	∥xn	NUM
ejpam-6609	218	1	−	−	PROPN
ejpam-6609	218	2	xm∥	xm∥	PROPN
ejpam-6609	218	3	≤	≤	NOUN
ejpam-6609	219	1	m−1∑	m−1∑	NUM
ejpam-6609	219	2	k	k	NOUN
ejpam-6609	219	3	=	=	PROPN
ejpam-6609	219	4	n	n	NOUN
ejpam-6609	219	5	∥xk	∥xk	NOUN
ejpam-6609	219	6	−	−	PROPN
ejpam-6609	219	7	xk+1∥	xk+1∥	PROPN
ejpam-6609	219	8	≤	≤	NOUN
ejpam-6609	219	9	m−1∑	m−1∑	NUM
ejpam-6609	219	10	k	k	NOUN
ejpam-6609	219	11	=	=	NOUN
ejpam-6609	219	12	n	n	NOUN
ejpam-6609	219	13	ηk∥x0	ηk∥x0	NOUN
ejpam-6609	219	14	−	−	NOUN
ejpam-6609	219	15	x1∥	x1∥	PROPN
ejpam-6609	219	16	≤	≤	PROPN
ejpam-6609	219	17	ηn	ηn	VERB
ejpam-6609	219	18	1−	1−	NUM
ejpam-6609	219	19	η	η	PROPN
ejpam-6609	219	20	∥x0	∥x0	X
ejpam-6609	219	21	−	−	PROPN
ejpam-6609	219	22	x1∥	x1∥	PROPN
ejpam-6609	219	23	→	→	SYM
ejpam-6609	219	24	0	0	NUM
ejpam-6609	219	25	by	by	ADP
ejpam-6609	219	26	completeness	completeness	NOUN
ejpam-6609	219	27	,	,	PUNCT
ejpam-6609	219	28	∃u	∃u	PROPN
ejpam-6609	219	29	∈	∈	PROPN
ejpam-6609	219	30	h	h	NOUN
ejpam-6609	219	31	with	with	ADP
ejpam-6609	219	32	xn	xn	PROPN
ejpam-6609	219	33	→	→	SYM
ejpam-6609	219	34	u.	u.	PROPN
ejpam-6609	219	35	(	(	PUNCT
ejpam-6609	219	36	4	4	NUM
ejpam-6609	219	37	)	)	PUNCT
ejpam-6609	219	38	fixed	fix	VERB
ejpam-6609	219	39	point	point	NOUN
ejpam-6609	219	40	verification	verification	NOUN
ejpam-6609	219	41	for	for	ADP
ejpam-6609	219	42	t1	t1	NOUN
ejpam-6609	219	43	:	:	PUNCT
ejpam-6609	219	44	∥t1u−	∥t1u−	PRON
ejpam-6609	219	45	u∥	u∥	X
ejpam-6609	220	1	=	=	X
ejpam-6609	220	2	lim	lim	PROPN
ejpam-6609	220	3	∥t1u−	∥t1u−	NOUN
ejpam-6609	220	4	x3n+2	x3n+2	PROPN
ejpam-6609	220	5	−	−	PROPN
ejpam-6609	220	6	x3n+3∥	x3n+3∥	PROPN
ejpam-6609	220	7	≤	≤	NUM
ejpam-6609	220	8	α∥u−	α∥u−	NOUN
ejpam-6609	220	9	u−	u−	PROPN
ejpam-6609	220	10	u∥+	u∥+	PROPN
ejpam-6609	220	11	β	β	PROPN
ejpam-6609	220	12	lim	lim	PROPN
ejpam-6609	220	13	∥u−	∥u−	PROPN
ejpam-6609	220	14	t1u∥2∥u−	t1u∥2∥u−	PRON
ejpam-6609	220	15	t2u∥2∥u−	t2u∥2∥u−	ADJ
ejpam-6609	220	16	t3u∥2	t3u∥2	NOUN
ejpam-6609	220	17	1	1	NUM
ejpam-6609	220	18	+	+	CCONJ
ejpam-6609	220	19	·	·	PUNCT
ejpam-6609	220	20	·	·	PUNCT
ejpam-6609	220	21	·	·	PUNCT
ejpam-6609	221	1	=	=	SYM
ejpam-6609	221	2	0	0	NUM
ejpam-6609	221	3	thus	thus	ADV
ejpam-6609	221	4	t1u	t1u	VERB
ejpam-6609	221	5	=	=	PUNCT
ejpam-6609	221	6	u.	u.	NOUN
ejpam-6609	221	7	similarly	similarly	ADV
ejpam-6609	221	8	t2u	t2u	PUNCT
ejpam-6609	222	1	=	=	PUNCT
ejpam-6609	223	1	t3u	t3u	NOUN
ejpam-6609	223	2	=	=	SYM
ejpam-6609	223	3	u.	u.	PROPN
ejpam-6609	223	4	(	(	PUNCT
ejpam-6609	223	5	5	5	NUM
ejpam-6609	223	6	)	)	PUNCT
ejpam-6609	223	7	uniqueness	uniqueness	NOUN
ejpam-6609	223	8	let	let	VERB
ejpam-6609	223	9	v	v	PART
ejpam-6609	223	10	be	be	AUX
ejpam-6609	223	11	another	another	DET
ejpam-6609	223	12	fixed	fix	VERB
ejpam-6609	223	13	point	point	NOUN
ejpam-6609	223	14	:	:	PUNCT
ejpam-6609	223	15	∥u−	∥u−	PROPN
ejpam-6609	223	16	v∥	v∥	NOUN
ejpam-6609	223	17	=	=	PUNCT
ejpam-6609	223	18	∥t1u−	∥t1u−	PART
ejpam-6609	223	19	t2v	t2v	NUM
ejpam-6609	223	20	−	−	ADP
ejpam-6609	223	21	t3v∥	t3v∥	X
ejpam-6609	223	22	≤	≤	NUM
ejpam-6609	223	23	(	(	PUNCT
ejpam-6609	223	24	α+	α+	NUM
ejpam-6609	223	25	2β)∥u−	2β)∥u−	NUM
ejpam-6609	223	26	v∥	v∥	NOUN
ejpam-6609	223	27	<	<	X
ejpam-6609	223	28	∥u−	∥u−	PROPN
ejpam-6609	223	29	v∥	v∥	PROPN
ejpam-6609	223	30	implies	imply	VERB
ejpam-6609	223	31	u	u	NOUN
ejpam-6609	223	32	=	=	PUNCT
ejpam-6609	223	33	v.	v.	ADP
ejpam-6609	223	34	algorithm	algorithm	NOUN
ejpam-6609	223	35	4	4	NUM
ejpam-6609	223	36	iteratively	iteratively	ADV
ejpam-6609	223	37	approximates	approximate	VERB
ejpam-6609	223	38	a	a	DET
ejpam-6609	223	39	cfp	cfp	NOUN
ejpam-6609	223	40	of	of	ADP
ejpam-6609	223	41	three	three	NUM
ejpam-6609	223	42	nonlinear	nonlinear	ADJ
ejpam-6609	223	43	operators	operator	NOUN
ejpam-6609	223	44	in	in	ADP
ejpam-6609	223	45	an	an	DET
ejpam-6609	223	46	hs	hs	PROPN
ejpam-6609	223	47	,	,	PUNCT
ejpam-6609	223	48	using	use	VERB
ejpam-6609	223	49	a	a	DET
ejpam-6609	223	50	contraction	contraction	NOUN
ejpam-6609	223	51	-	-	PUNCT
ejpam-6609	223	52	based	base	VERB
ejpam-6609	223	53	error	error	NOUN
ejpam-6609	223	54	metric	metric	NOUN
ejpam-6609	223	55	to	to	PART
ejpam-6609	223	56	monitor	monitor	VERB
ejpam-6609	223	57	convergence	convergence	NOUN
ejpam-6609	223	58	.	.	PUNCT
ejpam-6609	224	1	m.	m.	NOUN
ejpam-6609	224	2	noorwali	noorwali	PROPN
ejpam-6609	224	3	et	et	PROPN
ejpam-6609	224	4	al	al	PROPN
ejpam-6609	224	5	.	.	PUNCT
ejpam-6609	224	6	/	/	SYM
ejpam-6609	224	7	eur	eur	PROPN
ejpam-6609	224	8	.	.	PUNCT
ejpam-6609	225	1	j.	j.	PROPN
ejpam-6609	225	2	pure	pure	PROPN
ejpam-6609	225	3	appl	appl	PROPN
ejpam-6609	225	4	.	.	PROPN
ejpam-6609	225	5	math	math	PROPN
ejpam-6609	225	6	,	,	PUNCT
ejpam-6609	225	7	18	18	NUM
ejpam-6609	225	8	(	(	PUNCT
ejpam-6609	225	9	3	3	NUM
ejpam-6609	225	10	)	)	PUNCT
ejpam-6609	225	11	(	(	PUNCT
ejpam-6609	225	12	2025	2025	NUM
ejpam-6609	225	13	)	)	PUNCT
ejpam-6609	225	14	,	,	PUNCT
ejpam-6609	225	15	6609	6609	NUM
ejpam-6609	225	16	11	11	NUM
ejpam-6609	225	17	of	of	ADP
ejpam-6609	225	18	17	17	NUM
ejpam-6609	225	19	algorithm	algorithm	NOUN
ejpam-6609	225	20	4	4	NUM
ejpam-6609	225	21	iterative	iterative	NOUN
ejpam-6609	225	22	fixed	fix	VERB
ejpam-6609	225	23	point	point	NOUN
ejpam-6609	225	24	approximation	approximation	NOUN
ejpam-6609	225	25	for	for	ADP
ejpam-6609	225	26	three	three	NUM
ejpam-6609	225	27	operators	operator	NOUN
ejpam-6609	225	28	require	require	VERB
ejpam-6609	225	29	:	:	PUNCT
ejpam-6609	225	30	hs	hs	PROPN
ejpam-6609	225	31	h	h	PROPN
ejpam-6609	225	32	with	with	ADP
ejpam-6609	225	33	inner	inner	ADJ
ejpam-6609	225	34	product	product	NOUN
ejpam-6609	225	35	⟨	⟨	VERB
ejpam-6609	225	36	·	·	PUNCT
ejpam-6609	225	37	,	,	PUNCT
ejpam-6609	225	38	·	·	PUNCT
ejpam-6609	225	39	⟩	⟩	NOUN
ejpam-6609	225	40	operators	operator	NOUN
ejpam-6609	225	41	t1	t1	VERB
ejpam-6609	225	42	,	,	PUNCT
ejpam-6609	225	43	t2	t2	NOUN
ejpam-6609	225	44	,	,	PUNCT
ejpam-6609	225	45	t3	t3	NOUN
ejpam-6609	225	46	:	:	PUNCT
ejpam-6609	225	47	h	h	NOUN
ejpam-6609	225	48	→	→	SYM
ejpam-6609	225	49	h	h	NOUN
ejpam-6609	225	50	satisfying	satisfying	NOUN
ejpam-6609	225	51	theorem	theorem	NOUN
ejpam-6609	225	52	2	2	NUM
ejpam-6609	225	53	initial	initial	ADJ
ejpam-6609	225	54	point	point	NOUN
ejpam-6609	226	1	x0	x0	PROPN
ejpam-6609	226	2	∈	∈	PROPN
ejpam-6609	226	3	h	h	NOUN
ejpam-6609	226	4	tolerance	tolerance	NOUN
ejpam-6609	226	5	ϵ	ϵ	X
ejpam-6609	226	6	>	>	X
ejpam-6609	226	7	0	0	NUM
ejpam-6609	227	1	maximum	maximum	ADJ
ejpam-6609	227	2	iterations	iteration	NOUN
ejpam-6609	227	3	n	n	PRON
ejpam-6609	227	4	ensure	ensure	VERB
ejpam-6609	227	5	:	:	PUNCT
ejpam-6609	227	6	common	common	ADJ
ejpam-6609	227	7	fixed	fix	VERB
ejpam-6609	227	8	point	point	NOUN
ejpam-6609	227	9	u	u	NOUN
ejpam-6609	227	10	satisfying	satisfy	VERB
ejpam-6609	227	11	t1u	t1u	NOUN
ejpam-6609	227	12	=	=	PUNCT
ejpam-6609	227	13	t2u	t2u	VERB
ejpam-6609	228	1	=	=	PUNCT
ejpam-6609	228	2	t3u	t3u	NOUN
ejpam-6609	228	3	=	=	SYM
ejpam-6609	228	4	u	u	NOUN
ejpam-6609	228	5	1	1	NUM
ejpam-6609	228	6	:	:	PUNCT
ejpam-6609	228	7	n←	n←	X
ejpam-6609	228	8	0	0	NUM
ejpam-6609	228	9	2	2	NUM
ejpam-6609	228	10	:	:	PUNCT
ejpam-6609	228	11	err←∞	err←∞	PROPN
ejpam-6609	228	12	3	3	NUM
ejpam-6609	228	13	:	:	PUNCT
ejpam-6609	228	14	while	while	SCONJ
ejpam-6609	228	15	n	n	CCONJ
ejpam-6609	228	16	<	<	X
ejpam-6609	228	17	n	n	NOUN
ejpam-6609	228	18	and	and	CCONJ
ejpam-6609	228	19	err	err	VERB
ejpam-6609	228	20	≥	≥	NUM
ejpam-6609	228	21	ϵ	ϵ	AUX
ejpam-6609	228	22	do	do	AUX
ejpam-6609	228	23	4	4	NUM
ejpam-6609	228	24	:	:	PUNCT
ejpam-6609	228	25	x3n+1	x3n+1	PROPN
ejpam-6609	228	26	←	←	PROPN
ejpam-6609	228	27	t1(x3n	t1(x3n	PROPN
ejpam-6609	228	28	)	)	PUNCT
ejpam-6609	228	29	5	5	NUM
ejpam-6609	228	30	:	:	PUNCT
ejpam-6609	228	31	x3n+2	x3n+2	PROPN
ejpam-6609	228	32	←	←	PROPN
ejpam-6609	228	33	t2(x3n+1	t2(x3n+1	PROPN
ejpam-6609	228	34	)	)	PUNCT
ejpam-6609	228	35	6	6	NUM
ejpam-6609	228	36	:	:	PUNCT
ejpam-6609	228	37	x3n+3	x3n+3	PROPN
ejpam-6609	228	38	←	←	PROPN
ejpam-6609	228	39	t3(x3n+2	t3(x3n+2	PROPN
ejpam-6609	228	40	)	)	PUNCT
ejpam-6609	228	41	7	7	NUM
ejpam-6609	228	42	:	:	PUNCT
ejpam-6609	228	43	compute	compute	NOUN
ejpam-6609	228	44	contraction	contraction	NOUN
ejpam-6609	228	45	metric	metric	NOUN
ejpam-6609	228	46	:	:	PUNCT
ejpam-6609	228	47	8	8	NUM
ejpam-6609	228	48	:	:	PUNCT
ejpam-6609	228	49	err←	err←	PROPN
ejpam-6609	228	50	∥x3n+1	∥x3n+1	PROPN
ejpam-6609	228	51	−	−	PROPN
ejpam-6609	228	52	x3n+2	x3n+2	X
ejpam-6609	228	53	−	−	PROPN
ejpam-6609	228	54	x3n+3∥2	x3n+3∥2	PROPN
ejpam-6609	229	1	−	−	PROPN
ejpam-6609	229	2	α∥x3n	α∥x3n	PROPN
ejpam-6609	229	3	−	−	PROPN
ejpam-6609	229	4	x3n+1	x3n+1	PUNCT
ejpam-6609	230	1	−	−	PROPN
ejpam-6609	231	1	x3n+2∥2	x3n+2∥2	PROPN
ejpam-6609	231	2	9	9	NUM
ejpam-6609	231	3	:	:	PUNCT
ejpam-6609	231	4	−β	−β	PROPN
ejpam-6609	231	5	∥x3n−x3n+1∥2∥x3n+1−x3n+2∥2∥x3n+2−x3n+3∥2	∥x3n−x3n+1∥2∥x3n+1−x3n+2∥2∥x3n+2−x3n+3∥2	PROPN
ejpam-6609	231	6	1+∥x3n+1−x3n+2∥2∥x3n+2−x3n+3∥2	1+∥x3n+1−x3n+2∥2∥x3n+2−x3n+3∥2	NUM
ejpam-6609	231	7	10	10	NUM
ejpam-6609	231	8	:	:	PUNCT
ejpam-6609	231	9	if	if	SCONJ
ejpam-6609	231	10	∥x3n+3	∥x3n+3	VERB
ejpam-6609	231	11	−	−	NOUN
ejpam-6609	232	1	x3n∥	x3n∥	X
ejpam-6609	232	2	<	<	X
ejpam-6609	233	1	ϵ	ϵ	X
ejpam-6609	233	2	then	then	ADV
ejpam-6609	233	3	qa	qa	PROPN
ejpam-6609	233	4	11	11	NUM
ejpam-6609	233	5	:	:	PUNCT
ejpam-6609	233	6	break	break	NOUN
ejpam-6609	233	7	12	12	NUM
ejpam-6609	233	8	:	:	PUNCT
ejpam-6609	233	9	end	end	VERB
ejpam-6609	233	10	if	if	SCONJ
ejpam-6609	233	11	13	13	NUM
ejpam-6609	233	12	:	:	PUNCT
ejpam-6609	233	13	n←	n←	X
ejpam-6609	233	14	n+	n+	PUNCT
ejpam-6609	233	15	1	1	NUM
ejpam-6609	233	16	14	14	NUM
ejpam-6609	233	17	:	:	PUNCT
ejpam-6609	233	18	end	end	VERB
ejpam-6609	233	19	while	while	SCONJ
ejpam-6609	233	20	15	15	NUM
ejpam-6609	233	21	:	:	PUNCT
ejpam-6609	233	22	return	return	NOUN
ejpam-6609	233	23	u←	u←	PUNCT
ejpam-6609	233	24	x3n+3	x3n+3	PROPN
ejpam-6609	234	1	the	the	DET
ejpam-6609	234	2	fixed	fix	VERB
ejpam-6609	234	3	-	-	PUNCT
ejpam-6609	234	4	point	point	NOUN
ejpam-6609	234	5	framework	framework	NOUN
ejpam-6609	234	6	from	from	ADP
ejpam-6609	234	7	theorem	theorem	ADJ
ejpam-6609	234	8	2	2	NUM
ejpam-6609	234	9	has	have	VERB
ejpam-6609	234	10	many	many	ADJ
ejpam-6609	234	11	practical	practical	ADJ
ejpam-6609	234	12	applications	application	NOUN
ejpam-6609	234	13	and	and	CCONJ
ejpam-6609	234	14	specific	specific	ADJ
ejpam-6609	234	15	instances	instance	NOUN
ejpam-6609	234	16	.	.	PUNCT
ejpam-6609	235	1	lemma	lemma	PROPN
ejpam-6609	235	2	3	3	NUM
ejpam-6609	235	3	demonstrates	demonstrate	VERB
ejpam-6609	235	4	that	that	PRON
ejpam-6609	235	5	orthogonal	orthogonal	ADJ
ejpam-6609	235	6	projection	projection	NOUN
ejpam-6609	235	7	operators	operator	NOUN
ejpam-6609	235	8	easily	easily	ADV
ejpam-6609	235	9	satisfy	satisfy	VERB
ejpam-6609	235	10	the	the	DET
ejpam-6609	235	11	theorem	theorem	PROPN
ejpam-6609	235	12	’s	’s	PART
ejpam-6609	235	13	contraction	contraction	NOUN
ejpam-6609	235	14	requirement	requirement	NOUN
ejpam-6609	235	15	,	,	PUNCT
ejpam-6609	235	16	making	make	VERB
ejpam-6609	235	17	them	they	PRON
ejpam-6609	235	18	helpful	helpful	ADJ
ejpam-6609	235	19	for	for	ADP
ejpam-6609	235	20	problems	problem	NOUN
ejpam-6609	235	21	involving	involve	VERB
ejpam-6609	235	22	subspace	subspace	NOUN
ejpam-6609	235	23	optimization	optimization	NOUN
ejpam-6609	235	24	.	.	PUNCT
ejpam-6609	236	1	corollary	corollary	ADJ
ejpam-6609	236	2	3	3	NUM
ejpam-6609	236	3	goes	go	VERB
ejpam-6609	236	4	into	into	ADP
ejpam-6609	236	5	further	further	ADJ
ejpam-6609	236	6	detail	detail	NOUN
ejpam-6609	236	7	about	about	ADP
ejpam-6609	236	8	these	these	DET
ejpam-6609	236	9	concepts	concept	NOUN
ejpam-6609	236	10	by	by	ADP
ejpam-6609	236	11	giving	give	VERB
ejpam-6609	236	12	specific	specific	ADJ
ejpam-6609	236	13	logarithmic	logarithmic	ADJ
ejpam-6609	236	14	convergence	convergence	NOUN
ejpam-6609	236	15	rates	rate	NOUN
ejpam-6609	236	16	for	for	ADP
ejpam-6609	236	17	the	the	DET
ejpam-6609	236	18	symmetric	symmetric	ADJ
ejpam-6609	236	19	operator	operator	NOUN
ejpam-6609	236	20	situation	situation	NOUN
ejpam-6609	236	21	when	when	SCONJ
ejpam-6609	236	22	algorithm	algorithm	NOUN
ejpam-6609	236	23	4	4	NUM
ejpam-6609	236	24	is	be	AUX
ejpam-6609	236	25	used	use	VERB
ejpam-6609	236	26	.	.	PUNCT
ejpam-6609	237	1	two	two	NUM
ejpam-6609	237	2	important	important	ADJ
ejpam-6609	237	3	examples	example	NOUN
ejpam-6609	237	4	show	show	VERB
ejpam-6609	237	5	how	how	SCONJ
ejpam-6609	237	6	to	to	PART
ejpam-6609	237	7	put	put	VERB
ejpam-6609	237	8	these	these	DET
ejpam-6609	237	9	ideas	idea	NOUN
ejpam-6609	237	10	into	into	ADP
ejpam-6609	237	11	practice	practice	NOUN
ejpam-6609	237	12	:	:	PUNCT
ejpam-6609	237	13	example	example	NOUN
ejpam-6609	237	14	3	3	NUM
ejpam-6609	237	15	shows	show	VERB
ejpam-6609	237	16	how	how	SCONJ
ejpam-6609	237	17	the	the	DET
ejpam-6609	237	18	theorem	theorem	NOUN
ejpam-6609	237	19	can	can	AUX
ejpam-6609	237	20	be	be	AUX
ejpam-6609	237	21	used	use	VERB
ejpam-6609	237	22	in	in	ADP
ejpam-6609	237	23	distributed	distribute	VERB
ejpam-6609	237	24	machine	machine	NOUN
ejpam-6609	237	25	learning	learning	NOUN
ejpam-6609	237	26	systems	system	NOUN
ejpam-6609	237	27	,	,	PUNCT
ejpam-6609	237	28	where	where	SCONJ
ejpam-6609	237	29	gradient	gradient	NOUN
ejpam-6609	237	30	operators	operator	NOUN
ejpam-6609	237	31	t1	t1	VERB
ejpam-6609	237	32	,	,	PUNCT
ejpam-6609	237	33	t2	t2	NOUN
ejpam-6609	237	34	,	,	PUNCT
ejpam-6609	237	35	t3	t3	PROPN
ejpam-6609	237	36	stand	stand	NOUN
ejpam-6609	237	37	for	for	ADP
ejpam-6609	237	38	parallel	parallel	ADJ
ejpam-6609	237	39	processing	processing	NOUN
ejpam-6609	237	40	units	unit	NOUN
ejpam-6609	237	41	.	.	PUNCT
ejpam-6609	237	42	example	example	NOUN
ejpam-6609	237	43	4	4	NUM
ejpam-6609	237	44	uses	use	VERB
ejpam-6609	237	45	the	the	DET
ejpam-6609	237	46	theorem	theorem	ADJ
ejpam-6609	237	47	2	2	NUM
ejpam-6609	237	48	construction	construction	NOUN
ejpam-6609	237	49	to	to	PART
ejpam-6609	237	50	make	make	VERB
ejpam-6609	237	51	cryptographic	cryptographic	ADJ
ejpam-6609	237	52	hash	hash	NOUN
ejpam-6609	237	53	functions	function	NOUN
ejpam-6609	237	54	.	.	PUNCT
ejpam-6609	238	1	algorithms	algorithm	NOUN
ejpam-6609	238	2	5	5	NUM
ejpam-6609	238	3	and	and	CCONJ
ejpam-6609	238	4	6	6	NUM
ejpam-6609	238	5	make	make	VERB
ejpam-6609	238	6	these	these	DET
ejpam-6609	238	7	applications	application	NOUN
ejpam-6609	238	8	work	work	VERB
ejpam-6609	238	9	by	by	ADP
ejpam-6609	238	10	giving	give	VERB
ejpam-6609	238	11	them	they	PRON
ejpam-6609	238	12	specific	specific	ADJ
ejpam-6609	238	13	ways	way	NOUN
ejpam-6609	238	14	to	to	PART
ejpam-6609	238	15	run	run	VERB
ejpam-6609	238	16	on	on	ADP
ejpam-6609	238	17	a	a	DET
ejpam-6609	238	18	computer	computer	NOUN
ejpam-6609	238	19	.	.	PUNCT
ejpam-6609	239	1	figure	figure	NOUN
ejpam-6609	239	2	3	3	NUM
ejpam-6609	239	3	shows	show	VERB
ejpam-6609	239	4	how	how	SCONJ
ejpam-6609	239	5	various	various	ADJ
ejpam-6609	239	6	techniques	technique	NOUN
ejpam-6609	239	7	converge	converge	VERB
ejpam-6609	239	8	by	by	ADP
ejpam-6609	239	9	showing	show	VERB
ejpam-6609	239	10	how	how	SCONJ
ejpam-6609	239	11	the	the	DET
ejpam-6609	239	12	contraction	contraction	NOUN
ejpam-6609	239	13	dynamics	dynamic	VERB
ejpam-6609	239	14	change	change	VERB
ejpam-6609	239	15	with	with	ADP
ejpam-6609	239	16	different	different	ADJ
ejpam-6609	239	17	parameter	parameter	NOUN
ejpam-6609	239	18	settings	setting	NOUN
ejpam-6609	239	19	.	.	PUNCT
ejpam-6609	240	1	when	when	SCONJ
ejpam-6609	240	2	taken	take	VERB
ejpam-6609	240	3	together	together	ADV
ejpam-6609	240	4	,	,	PUNCT
ejpam-6609	240	5	these	these	DET
ejpam-6609	240	6	parts	part	NOUN
ejpam-6609	240	7	provide	provide	VERB
ejpam-6609	240	8	a	a	DET
ejpam-6609	240	9	complete	complete	ADJ
ejpam-6609	240	10	structure	structure	NOUN
ejpam-6609	240	11	for	for	ADP
ejpam-6609	240	12	fixed	fix	VERB
ejpam-6609	240	13	-	-	PUNCT
ejpam-6609	240	14	point	point	NOUN
ejpam-6609	240	15	computation	computation	NOUN
ejpam-6609	240	16	in	in	ADP
ejpam-6609	240	17	hs	hs	PROPN
ejpam-6609	240	18	,	,	PUNCT
ejpam-6609	240	19	with	with	ADP
ejpam-6609	240	20	both	both	CCONJ
ejpam-6609	240	21	theoretical	theoretical	ADJ
ejpam-6609	240	22	guarantees	guarantee	NOUN
ejpam-6609	240	23	and	and	CCONJ
ejpam-6609	240	24	real	real	ADJ
ejpam-6609	240	25	-	-	PUNCT
ejpam-6609	240	26	world	world	NOUN
ejpam-6609	240	27	examples	example	NOUN
ejpam-6609	240	28	.	.	PUNCT
ejpam-6609	241	1	example	example	NOUN
ejpam-6609	241	2	3	3	X
ejpam-6609	241	3	.	.	X
ejpam-6609	242	1	consider	consider	VERB
ejpam-6609	242	2	a	a	DET
ejpam-6609	242	3	distributed	distribute	VERB
ejpam-6609	242	4	learning	learning	NOUN
ejpam-6609	242	5	system	system	NOUN
ejpam-6609	242	6	with	with	ADP
ejpam-6609	242	7	three	three	NUM
ejpam-6609	242	8	workers	worker	NOUN
ejpam-6609	242	9	calculating	calculate	VERB
ejpam-6609	242	10	partial	partial	ADJ
ejpam-6609	242	11	gradients	gradient	NOUN
ejpam-6609	242	12	t1	t1	NOUN
ejpam-6609	242	13	,	,	PUNCT
ejpam-6609	242	14	t2	t2	NOUN
ejpam-6609	242	15	,	,	PUNCT
ejpam-6609	242	16	t3	t3	PROPN
ejpam-6609	242	17	:	:	PUNCT
ejpam-6609	242	18	rd	rd	PROPN
ejpam-6609	242	19	→	→	SYM
ejpam-6609	242	20	rd	rd	PROPN
ejpam-6609	242	21	on	on	ADP
ejpam-6609	242	22	data	datum	NOUN
ejpam-6609	242	23	that	that	PRON
ejpam-6609	242	24	has	have	AUX
ejpam-6609	242	25	been	be	AUX
ejpam-6609	242	26	split	split	VERB
ejpam-6609	242	27	up	up	ADP
ejpam-6609	242	28	.	.	PUNCT
ejpam-6609	243	1	theorem	theorem	ADJ
ejpam-6609	243	2	2	2	NUM
ejpam-6609	243	3	shows	show	VERB
ejpam-6609	243	4	that	that	SCONJ
ejpam-6609	243	5	this	this	DET
ejpam-6609	243	6	system	system	NOUN
ejpam-6609	243	7	will	will	AUX
ejpam-6609	243	8	converge	converge	VERB
ejpam-6609	243	9	if	if	SCONJ
ejpam-6609	243	10	the	the	DET
ejpam-6609	243	11	learning	learning	NOUN
ejpam-6609	243	12	rate	rate	NOUN
ejpam-6609	243	13	is	be	AUX
ejpam-6609	243	14	set	set	VERB
ejpam-6609	243	15	correctly	correctly	ADV
ejpam-6609	243	16	.	.	PUNCT
ejpam-6609	244	1	algorithm	algorithm	NOUN
ejpam-6609	244	2	5	5	NUM
ejpam-6609	244	3	will	will	AUX
ejpam-6609	244	4	give	give	VERB
ejpam-6609	244	5	a	a	DET
ejpam-6609	244	6	full	full	ADJ
ejpam-6609	244	7	description	description	NOUN
ejpam-6609	244	8	of	of	ADP
ejpam-6609	244	9	the	the	DET
ejpam-6609	244	10	training	training	NOUN
ejpam-6609	244	11	process	process	NOUN
ejpam-6609	244	12	.	.	PUNCT
ejpam-6609	245	1	m.	m.	NOUN
ejpam-6609	245	2	noorwali	noorwali	PROPN
ejpam-6609	245	3	et	et	PROPN
ejpam-6609	245	4	al	al	PROPN
ejpam-6609	245	5	.	.	PUNCT
ejpam-6609	245	6	/	/	SYM
ejpam-6609	245	7	eur	eur	PROPN
ejpam-6609	245	8	.	.	PUNCT
ejpam-6609	246	1	j.	j.	PROPN
ejpam-6609	246	2	pure	pure	PROPN
ejpam-6609	246	3	appl	appl	PROPN
ejpam-6609	246	4	.	.	PROPN
ejpam-6609	246	5	math	math	PROPN
ejpam-6609	246	6	,	,	PUNCT
ejpam-6609	246	7	18	18	NUM
ejpam-6609	246	8	(	(	PUNCT
ejpam-6609	246	9	3	3	NUM
ejpam-6609	246	10	)	)	PUNCT
ejpam-6609	246	11	(	(	PUNCT
ejpam-6609	246	12	2025	2025	NUM
ejpam-6609	246	13	)	)	PUNCT
ejpam-6609	246	14	,	,	PUNCT
ejpam-6609	246	15	6609	6609	NUM
ejpam-6609	246	16	12	12	NUM
ejpam-6609	246	17	of	of	ADP
ejpam-6609	246	18	17	17	NUM
ejpam-6609	246	19	algorithm	algorithm	NOUN
ejpam-6609	246	20	5	5	NUM
ejpam-6609	246	21	distributed	distribute	VERB
ejpam-6609	246	22	gradient	gradient	ADJ
ejpam-6609	246	23	descent	descent	NOUN
ejpam-6609	246	24	protocol	protocol	NOUN
ejpam-6609	246	25	require	require	VERB
ejpam-6609	246	26	:	:	PUNCT
ejpam-6609	247	1	data	datum	NOUN
ejpam-6609	247	2	set	set	VERB
ejpam-6609	247	3	d	d	NOUN
ejpam-6609	247	4	=	=	SYM
ejpam-6609	247	5	d1	d1	PROPN
ejpam-6609	247	6	∪	∪	ADP
ejpam-6609	247	7	d2	d2	PROPN
ejpam-6609	247	8	∪	∪	NOUN
ejpam-6609	247	9	d3	d3	PROPN
ejpam-6609	247	10	initial	initial	ADJ
ejpam-6609	247	11	parameters	parameter	NOUN
ejpam-6609	248	1	θ0	θ0	PROPN
ejpam-6609	248	2	∈	∈	PROPN
ejpam-6609	248	3	rd	rd	NOUN
ejpam-6609	248	4	learning	learning	PROPN
ejpam-6609	248	5	rate	rate	PROPN
ejpam-6609	248	6	η	η	PROPN
ejpam-6609	248	7	>	>	X
ejpam-6609	248	8	0	0	NUM
ejpam-6609	249	1	maximum	maximum	ADJ
ejpam-6609	249	2	iterations	iteration	NOUN
ejpam-6609	249	3	k	k	NOUN
ejpam-6609	249	4	ensure	ensure	VERB
ejpam-6609	249	5	:	:	PUNCT
ejpam-6609	249	6	optimized	optimize	VERB
ejpam-6609	249	7	parameters	parameter	NOUN
ejpam-6609	249	8	θ∗	θ∗	VERB
ejpam-6609	249	9	1	1	NUM
ejpam-6609	249	10	:	:	PUNCT
ejpam-6609	249	11	partition	partition	NOUN
ejpam-6609	249	12	d	d	NOUN
ejpam-6609	249	13	into	into	ADP
ejpam-6609	249	14	d1,d2,d3	d1,d2,d3	NOUN
ejpam-6609	249	15	2	2	NUM
ejpam-6609	249	16	:	:	PUNCT
ejpam-6609	249	17	for	for	ADP
ejpam-6609	249	18	k	k	PROPN
ejpam-6609	249	19	=	=	SYM
ejpam-6609	249	20	0	0	PROPN
ejpam-6609	249	21	to	to	ADP
ejpam-6609	249	22	k	k	PROPN
ejpam-6609	249	23	−	−	PROPN
ejpam-6609	249	24	1	1	NUM
ejpam-6609	249	25	do	do	AUX
ejpam-6609	249	26	3	3	NUM
ejpam-6609	249	27	:	:	PUNCT
ejpam-6609	249	28	worker	worker	NOUN
ejpam-6609	249	29	1	1	NUM
ejpam-6609	249	30	computes	compute	VERB
ejpam-6609	249	31	t1(θk	t1(θk	NOUN
ejpam-6609	249	32	)	)	PUNCT
ejpam-6609	249	33	=	=	SYM
ejpam-6609	250	1	∇θl(θk	∇θl(θk	NOUN
ejpam-6609	250	2	,	,	PUNCT
ejpam-6609	250	3	d1	d1	NOUN
ejpam-6609	250	4	)	)	PUNCT
ejpam-6609	250	5	4	4	NUM
ejpam-6609	250	6	:	:	PUNCT
ejpam-6609	250	7	worker	worker	NOUN
ejpam-6609	250	8	2	2	NUM
ejpam-6609	250	9	computes	compute	NOUN
ejpam-6609	250	10	t2(θk	t2(θk	PROPN
ejpam-6609	250	11	)	)	PUNCT
ejpam-6609	250	12	=	=	SYM
ejpam-6609	251	1	∇θl(θk	∇θl(θk	NOUN
ejpam-6609	251	2	,	,	PUNCT
ejpam-6609	251	3	d2	d2	PROPN
ejpam-6609	251	4	)	)	PUNCT
ejpam-6609	251	5	5	5	NUM
ejpam-6609	251	6	:	:	PUNCT
ejpam-6609	251	7	worker	worker	NOUN
ejpam-6609	251	8	3	3	NUM
ejpam-6609	251	9	computes	compute	VERB
ejpam-6609	251	10	t3(θk	t3(θk	PRON
ejpam-6609	251	11	)	)	PUNCT
ejpam-6609	251	12	=	=	SYM
ejpam-6609	252	1	∇θl(θk	∇θl(θk	NOUN
ejpam-6609	252	2	,	,	PUNCT
ejpam-6609	252	3	d3	d3	PROPN
ejpam-6609	252	4	)	)	PUNCT
ejpam-6609	252	5	6	6	NUM
ejpam-6609	252	6	:	:	PUNCT
ejpam-6609	252	7	aggregate	aggregate	ADJ
ejpam-6609	252	8	updates	update	NOUN
ejpam-6609	252	9	:	:	PUNCT
ejpam-6609	252	10	7	7	NUM
ejpam-6609	252	11	:	:	PUNCT
ejpam-6609	252	12	θk+1	θk+1	NUM
ejpam-6609	252	13	←	←	PROPN
ejpam-6609	252	14	θk	θk	NOUN
ejpam-6609	252	15	−	−	PROPN
ejpam-6609	252	16	η(t1(θk	η(t1(θk	NOUN
ejpam-6609	252	17	)	)	PUNCT
ejpam-6609	252	18	+	+	CCONJ
ejpam-6609	252	19	t2(θk	t2(θk	PROPN
ejpam-6609	252	20	)	)	PUNCT
ejpam-6609	252	21	+	+	NUM
ejpam-6609	252	22	t3(θk	t3(θk	NUM
ejpam-6609	252	23	)	)	PUNCT
ejpam-6609	252	24	)	)	PUNCT
ejpam-6609	252	25	8	8	NUM
ejpam-6609	252	26	:	:	PUNCT
ejpam-6609	252	27	if	if	SCONJ
ejpam-6609	252	28	∥θk+1	∥θk+1	VERB
ejpam-6609	252	29	−	−	PROPN
ejpam-6609	252	30	θk∥2	θk∥2	PROPN
ejpam-6609	252	31	≥	≥	NUM
ejpam-6609	252	32	1	1	NUM
ejpam-6609	252	33	3∥θk	3∥θk	NUM
ejpam-6609	252	34	−	−	PROPN
ejpam-6609	252	35	θk−1∥2	θk−1∥2	NOUN
ejpam-6609	252	36	then	then	ADV
ejpam-6609	252	37	9	9	NUM
ejpam-6609	252	38	:	:	PUNCT
ejpam-6609	252	39	η	η	PROPN
ejpam-6609	252	40	←	←	PROPN
ejpam-6609	252	41	η/2	η/2	NUM
ejpam-6609	252	42	10	10	NUM
ejpam-6609	252	43	:	:	PUNCT
ejpam-6609	252	44	end	end	VERB
ejpam-6609	252	45	if	if	SCONJ
ejpam-6609	252	46	11	11	NUM
ejpam-6609	252	47	:	:	PUNCT
ejpam-6609	252	48	end	end	VERB
ejpam-6609	252	49	for	for	ADP
ejpam-6609	252	50	12	12	NUM
ejpam-6609	252	51	:	:	PUNCT
ejpam-6609	252	52	return	return	NOUN
ejpam-6609	252	53	θ∗	θ∗	PROPN
ejpam-6609	252	54	←	←	PROPN
ejpam-6609	252	55	θk	θk	NOUN
ejpam-6609	252	56	remark	remark	NOUN
ejpam-6609	252	57	3	3	NUM
ejpam-6609	252	58	.	.	PUNCT
ejpam-6609	253	1	it	it	PRON
ejpam-6609	253	2	’s	’	VERB
ejpam-6609	253	3	evident	evident	ADJ
ejpam-6609	253	4	how	how	SCONJ
ejpam-6609	253	5	theorem	theorem	ADJ
ejpam-6609	253	6	2	2	NUM
ejpam-6609	253	7	fits	fit	VERB
ejpam-6609	253	8	in	in	ADP
ejpam-6609	253	9	when	when	SCONJ
ejpam-6609	253	10	:	:	PUNCT
ejpam-6609	253	11	•	•	ADP
ejpam-6609	253	12	the	the	DET
ejpam-6609	253	13	gradient	gradient	NOUN
ejpam-6609	253	14	operators	operator	NOUN
ejpam-6609	253	15	ti	ti	NOUN
ejpam-6609	253	16	satisfy	satisfy	VERB
ejpam-6609	253	17	condition	condition	NOUN
ejpam-6609	253	18	(	(	PUNCT
ejpam-6609	253	19	2	2	NUM
ejpam-6609	253	20	)	)	PUNCT
ejpam-6609	253	21	with	with	ADP
ejpam-6609	253	22	α	α	NOUN
ejpam-6609	253	23	=	=	SYM
ejpam-6609	253	24	1	1	NUM
ejpam-6609	253	25	3	3	NUM
ejpam-6609	253	26	,	,	PUNCT
ejpam-6609	253	27	β	β	X
ejpam-6609	253	28	=	=	SYM
ejpam-6609	253	29	0	0	NUM
ejpam-6609	253	30	•	•	NOUN
ejpam-6609	253	31	the	the	DET
ejpam-6609	253	32	learning	learning	PROPN
ejpam-6609	253	33	rate	rate	PROPN
ejpam-6609	253	34	η	η	PROPN
ejpam-6609	253	35	ensures	ensure	VERB
ejpam-6609	253	36	α+	α+	PUNCT
ejpam-6609	253	37	β	β	NOUN
ejpam-6609	253	38	=	=	SYM
ejpam-6609	253	39	1	1	NUM
ejpam-6609	253	40	3	3	NUM
ejpam-6609	253	41	<	<	SYM
ejpam-6609	253	42	1	1	NUM
ejpam-6609	253	43	•	•	NUM
ejpam-6609	253	44	the	the	DET
ejpam-6609	253	45	fixed	fix	VERB
ejpam-6609	253	46	-	-	PUNCT
ejpam-6609	253	47	point	point	NOUN
ejpam-6609	253	48	iteration	iteration	NOUN
ejpam-6609	253	49	approach	approach	NOUN
ejpam-6609	253	50	is	be	AUX
ejpam-6609	253	51	used	use	VERB
ejpam-6609	253	52	to	to	PART
ejpam-6609	253	53	update	update	VERB
ejpam-6609	253	54	the	the	DET
ejpam-6609	253	55	parameters	parameter	NOUN
ejpam-6609	253	56	.	.	PUNCT
ejpam-6609	254	1	this	this	PRON
ejpam-6609	254	2	gives	give	VERB
ejpam-6609	254	3	theoretical	theoretical	ADJ
ejpam-6609	254	4	guarantees	guarantee	NOUN
ejpam-6609	254	5	that	that	PRON
ejpam-6609	254	6	dispersed	disperse	VERB
ejpam-6609	254	7	training	training	NOUN
ejpam-6609	254	8	will	will	AUX
ejpam-6609	254	9	converge	converge	VERB
ejpam-6609	254	10	.	.	PUNCT
ejpam-6609	254	11	example	example	NOUN
ejpam-6609	255	1	4	4	NUM
ejpam-6609	255	2	.	.	X
ejpam-6609	255	3	consider	consider	VERB
ejpam-6609	255	4	three	three	NUM
ejpam-6609	255	5	cryptographic	cryptographic	ADJ
ejpam-6609	255	6	hash	hash	NOUN
ejpam-6609	255	7	functions	function	NOUN
ejpam-6609	255	8	h1	h1	PROPN
ejpam-6609	255	9	,	,	PUNCT
ejpam-6609	255	10	h2	h2	PROPN
ejpam-6609	255	11	,	,	PUNCT
ejpam-6609	255	12	h3	h3	NOUN
ejpam-6609	255	13	:	:	PUNCT
ejpam-6609	255	14	{	{	PUNCT
ejpam-6609	255	15	0	0	NUM
ejpam-6609	255	16	,	,	PUNCT
ejpam-6609	255	17	1}∗	1}∗	PROPN
ejpam-6609	255	18	→	→	SYM
ejpam-6609	255	19	{	{	PUNCT
ejpam-6609	255	20	0	0	NUM
ejpam-6609	255	21	,	,	PUNCT
ejpam-6609	255	22	1}n	1}n	NUM
ejpam-6609	255	23	that	that	PRON
ejpam-6609	255	24	have	have	VERB
ejpam-6609	255	25	contraction	contraction	NOUN
ejpam-6609	255	26	features	feature	NOUN
ejpam-6609	255	27	.	.	PUNCT
ejpam-6609	256	1	the	the	DET
ejpam-6609	256	2	fixed	fix	VERB
ejpam-6609	256	3	-	-	PUNCT
ejpam-6609	256	4	point	point	NOUN
ejpam-6609	256	5	construction	construction	NOUN
ejpam-6609	256	6	converges	converge	NOUN
ejpam-6609	256	7	by	by	ADP
ejpam-6609	256	8	theorem	theorem	NOUN
ejpam-6609	256	9	2	2	NUM
ejpam-6609	256	10	when	when	SCONJ
ejpam-6609	256	11	each	each	DET
ejpam-6609	256	12	hi	hi	INTJ
ejpam-6609	256	13	meets	meet	VERB
ejpam-6609	256	14	the	the	DET
ejpam-6609	256	15	lipschitz	lipschitz	NOUN
ejpam-6609	256	16	requirement	requirement	NOUN
ejpam-6609	256	17	∥hi(x)−hi(y)∥	∥hi(x)−hi(y)∥	NOUN
ejpam-6609	256	18	≤	≤	NOUN
ejpam-6609	256	19	l∥x−	l∥x−	ADV
ejpam-6609	256	20	y∥	y∥	VERB
ejpam-6609	256	21	for	for	ADP
ejpam-6609	256	22	l	l	NOUN
ejpam-6609	256	23	<	<	X
ejpam-6609	256	24	1/	1/	NUM
ejpam-6609	256	25	√	√	NUM
ejpam-6609	256	26	3	3	NUM
ejpam-6609	256	27	.	.	PUNCT
ejpam-6609	256	28	algorithm	algorithm	PROPN
ejpam-6609	256	29	6	6	NUM
ejpam-6609	256	30	shows	show	VERB
ejpam-6609	256	31	how	how	SCONJ
ejpam-6609	256	32	to	to	PART
ejpam-6609	256	33	make	make	VERB
ejpam-6609	256	34	the	the	DET
ejpam-6609	256	35	full	full	ADJ
ejpam-6609	256	36	hash	hash	NOUN
ejpam-6609	256	37	.	.	PUNCT
ejpam-6609	257	1	algorithm	algorithm	NOUN
ejpam-6609	257	2	6	6	NUM
ejpam-6609	257	3	fixed	fix	VERB
ejpam-6609	257	4	-	-	PUNCT
ejpam-6609	257	5	point	point	NOUN
ejpam-6609	257	6	cryptographic	cryptographic	ADJ
ejpam-6609	257	7	hash	hash	NOUN
ejpam-6609	257	8	construction	construction	NOUN
ejpam-6609	257	9	require	require	NOUN
ejpam-6609	257	10	:	:	PUNCT
ejpam-6609	257	11	input	input	NOUN
ejpam-6609	257	12	message	message	NOUN
ejpam-6609	257	13	x	x	X
ejpam-6609	257	14	∈	∈	PROPN
ejpam-6609	257	15	{	{	PUNCT
ejpam-6609	257	16	0	0	NUM
ejpam-6609	257	17	,	,	PUNCT
ejpam-6609	257	18	1}∗	1}∗	ADJ
ejpam-6609	257	19	hash	hash	NOUN
ejpam-6609	257	20	functions	function	NOUN
ejpam-6609	257	21	h1	h1	PROPN
ejpam-6609	257	22	,	,	PUNCT
ejpam-6609	257	23	h2	h2	PROPN
ejpam-6609	257	24	,	,	PUNCT
ejpam-6609	257	25	h3	h3	NOUN
ejpam-6609	257	26	with	with	ADP
ejpam-6609	257	27	l	l	ADJ
ejpam-6609	257	28	-	-	ADJ
ejpam-6609	257	29	lipschitz	lipschitz	ADJ
ejpam-6609	257	30	constants	constant	NOUN
ejpam-6609	257	31	convergence	convergence	NOUN
ejpam-6609	257	32	threshold	threshold	NOUN
ejpam-6609	257	33	ϵ	ϵ	X
ejpam-6609	257	34	>	>	X
ejpam-6609	257	35	0	0	PUNCT
ejpam-6609	258	1	maximum	maximum	ADJ
ejpam-6609	258	2	iterations	iteration	NOUN
ejpam-6609	258	3	n	n	PRON
ejpam-6609	258	4	(	(	PUNCT
ejpam-6609	258	5	security	security	NOUN
ejpam-6609	258	6	parameter	parameter	NOUN
ejpam-6609	258	7	)	)	PUNCT
ejpam-6609	258	8	ensure	ensure	VERB
ejpam-6609	258	9	:	:	PUNCT
ejpam-6609	258	10	fixed	fix	VERB
ejpam-6609	258	11	-	-	PUNCT
ejpam-6609	258	12	point	point	NOUN
ejpam-6609	258	13	hash	hash	NOUN
ejpam-6609	258	14	digest	digest	NOUN
ejpam-6609	258	15	h∗	h∗	PROPN
ejpam-6609	258	16	∈	∈	PROPN
ejpam-6609	258	17	{	{	PUNCT
ejpam-6609	258	18	0	0	NUM
ejpam-6609	258	19	,	,	PUNCT
ejpam-6609	258	20	1}n	1}n	NUM
ejpam-6609	258	21	1	1	NUM
ejpam-6609	258	22	:	:	PUNCT
ejpam-6609	258	23	initialize	initialize	VERB
ejpam-6609	258	24	h0	h0	PROPN
ejpam-6609	258	25	←	←	PROPN
ejpam-6609	258	26	sha3	sha3	PROPN
ejpam-6609	258	27	-	-	PUNCT
ejpam-6609	258	28	512(x	512(x	NOUN
ejpam-6609	258	29	)	)	PUNCT
ejpam-6609	258	30	2	2	NUM
ejpam-6609	258	31	:	:	PUNCT
ejpam-6609	258	32	n←	n←	X
ejpam-6609	258	33	0	0	SYM
ejpam-6609	259	1	3	3	NUM
ejpam-6609	259	2	:	:	PUNCT
ejpam-6609	259	3	repeat	repeat	VERB
ejpam-6609	259	4	4	4	NUM
ejpam-6609	259	5	:	:	PUNCT
ejpam-6609	259	6	h3n+1	h3n+1	VERB
ejpam-6609	259	7	←	←	PROPN
ejpam-6609	259	8	h1(h3n	h1(h3n	PRON
ejpam-6609	259	9	)	)	PUNCT
ejpam-6609	259	10	5	5	NUM
ejpam-6609	259	11	:	:	PUNCT
ejpam-6609	259	12	h3n+2	h3n+2	PROPN
ejpam-6609	259	13	←	←	PROPN
ejpam-6609	259	14	h2(h3n+1	h2(h3n+1	PROPN
ejpam-6609	259	15	)	)	PUNCT
ejpam-6609	259	16	6	6	NUM
ejpam-6609	259	17	:	:	PUNCT
ejpam-6609	259	18	h3n+3	h3n+3	PROPN
ejpam-6609	259	19	←	←	PROPN
ejpam-6609	259	20	h3(h3n+2	h3(h3n+2	PROPN
ejpam-6609	259	21	)	)	PUNCT
ejpam-6609	259	22	7	7	NUM
ejpam-6609	259	23	:	:	PUNCT
ejpam-6609	259	24	n←	n←	X
ejpam-6609	259	25	n+	n+	PUNCT
ejpam-6609	259	26	1	1	NUM
ejpam-6609	259	27	8	8	NUM
ejpam-6609	259	28	:	:	PUNCT
ejpam-6609	259	29	until	until	ADP
ejpam-6609	259	30	∥h3n	∥h3n	PROPN
ejpam-6609	259	31	−	−	PROPN
ejpam-6609	259	32	h3n−3∥	h3n−3∥	X
ejpam-6609	259	33	<	<	X
ejpam-6609	259	34	ϵ	ϵ	X
ejpam-6609	259	35	or	or	CCONJ
ejpam-6609	259	36	n	n	PRON
ejpam-6609	259	37	≥	≥	NOUN
ejpam-6609	259	38	n	n	ADV
ejpam-6609	259	39	9	9	NUM
ejpam-6609	259	40	:	:	PUNCT
ejpam-6609	259	41	return	return	VERB
ejpam-6609	259	42	h∗	h∗	PROPN
ejpam-6609	259	43	←	←	PROPN
ejpam-6609	259	44	h3n	h3n	PROPN
ejpam-6609	259	45	remark	remark	VERB
ejpam-6609	259	46	4	4	NUM
ejpam-6609	259	47	.	.	PUNCT
ejpam-6609	259	48	for	for	ADP
ejpam-6609	259	49	theorem	theorem	NOUN
ejpam-6609	259	50	2	2	NUM
ejpam-6609	259	51	to	to	PART
ejpam-6609	259	52	be	be	AUX
ejpam-6609	259	53	useful	useful	ADJ
ejpam-6609	259	54	in	in	ADP
ejpam-6609	259	55	cryptography	cryptography	NOUN
ejpam-6609	259	56	,	,	PUNCT
ejpam-6609	259	57	it	it	PRON
ejpam-6609	259	58	needs	need	VERB
ejpam-6609	259	59	:	:	PUNCT
ejpam-6609	259	60	m.	m.	PROPN
ejpam-6609	259	61	noorwali	noorwali	PROPN
ejpam-6609	259	62	et	et	PROPN
ejpam-6609	259	63	al	al	PROPN
ejpam-6609	259	64	.	.	PUNCT
ejpam-6609	259	65	/	/	SYM
ejpam-6609	259	66	eur	eur	PROPN
ejpam-6609	259	67	.	.	PUNCT
ejpam-6609	260	1	j.	j.	PROPN
ejpam-6609	260	2	pure	pure	PROPN
ejpam-6609	260	3	appl	appl	PROPN
ejpam-6609	260	4	.	.	PROPN
ejpam-6609	260	5	math	math	PROPN
ejpam-6609	260	6	,	,	PUNCT
ejpam-6609	260	7	18	18	NUM
ejpam-6609	260	8	(	(	PUNCT
ejpam-6609	260	9	3	3	NUM
ejpam-6609	260	10	)	)	PUNCT
ejpam-6609	260	11	(	(	PUNCT
ejpam-6609	260	12	2025	2025	NUM
ejpam-6609	260	13	)	)	PUNCT
ejpam-6609	260	14	,	,	PUNCT
ejpam-6609	260	15	6609	6609	NUM
ejpam-6609	260	16	13	13	NUM
ejpam-6609	260	17	of	of	ADP
ejpam-6609	260	18	17	17	NUM
ejpam-6609	260	19	•	•	NOUN
ejpam-6609	260	20	form	form	NOUN
ejpam-6609	260	21	h3	h3	NOUN
ejpam-6609	260	22	◦	◦	NOUN
ejpam-6609	260	23	h2	h2	NOUN
ejpam-6609	260	24	◦	◦	NOUN
ejpam-6609	260	25	h1	h1	NOUN
ejpam-6609	260	26	makes	make	VERB
ejpam-6609	260	27	a	a	DET
ejpam-6609	260	28	contraction	contraction	NOUN
ejpam-6609	260	29	mapping	mapping	NOUN
ejpam-6609	260	30	.	.	PUNCT
ejpam-6609	261	1	•	•	NUM
ejpam-6609	261	2	the	the	DET
ejpam-6609	261	3	security	security	NOUN
ejpam-6609	261	4	parameter	parameter	NOUN
ejpam-6609	261	5	n	n	PRON
ejpam-6609	261	6	stops	stop	VERB
ejpam-6609	261	7	infinite	infinite	ADJ
ejpam-6609	261	8	loops	loop	NOUN
ejpam-6609	261	9	.	.	PUNCT
ejpam-6609	262	1	•	•	NUM
ejpam-6609	262	2	the	the	DET
ejpam-6609	262	3	fixed	fix	VERB
ejpam-6609	262	4	-	-	PUNCT
ejpam-6609	262	5	point	point	NOUN
ejpam-6609	262	6	h∗	h∗	NOUN
ejpam-6609	262	7	gets	get	VERB
ejpam-6609	262	8	its	its	PRON
ejpam-6609	262	9	collision	collision	NOUN
ejpam-6609	262	10	resistance	resistance	NOUN
ejpam-6609	262	11	from	from	ADP
ejpam-6609	262	12	the	the	DET
ejpam-6609	262	13	hashes	hash	NOUN
ejpam-6609	262	14	of	of	ADP
ejpam-6609	262	15	its	its	PRON
ejpam-6609	262	16	parts	part	NOUN
ejpam-6609	262	17	.	.	PUNCT
ejpam-6609	263	1	•	•	NUM
ejpam-6609	263	2	the	the	DET
ejpam-6609	263	3	constraint	constraint	NOUN
ejpam-6609	263	4	l	l	NOUN
ejpam-6609	263	5	<	<	X
ejpam-6609	263	6	1/	1/	NUM
ejpam-6609	263	7	√	√	NUM
ejpam-6609	263	8	3	3	NUM
ejpam-6609	263	9	makes	make	VERB
ejpam-6609	263	10	sure	sure	ADJ
ejpam-6609	263	11	that	that	SCONJ
ejpam-6609	263	12	the	the	DET
ejpam-6609	263	13	composite	composite	ADJ
ejpam-6609	263	14	mapping	mapping	NOUN
ejpam-6609	263	15	meets	meet	VERB
ejpam-6609	263	16	condition	condition	NOUN
ejpam-6609	263	17	(	(	PUNCT
ejpam-6609	263	18	2	2	NUM
ejpam-6609	263	19	)	)	PUNCT
ejpam-6609	263	20	.	.	PUNCT
ejpam-6609	264	1	this	this	DET
ejpam-6609	264	2	construction	construction	NOUN
ejpam-6609	264	3	makes	make	VERB
ejpam-6609	264	4	a	a	DET
ejpam-6609	264	5	hash	hash	NOUN
ejpam-6609	264	6	function	function	NOUN
ejpam-6609	264	7	that	that	PRON
ejpam-6609	264	8	is	be	AUX
ejpam-6609	264	9	demonstrably	demonstrably	ADV
ejpam-6609	264	10	convergent	convergent	ADJ
ejpam-6609	264	11	and	and	CCONJ
ejpam-6609	264	12	has	have	VERB
ejpam-6609	264	13	security	security	NOUN
ejpam-6609	264	14	properties	property	NOUN
ejpam-6609	264	15	that	that	PRON
ejpam-6609	264	16	are	be	AUX
ejpam-6609	264	17	not	not	PART
ejpam-6609	264	18	conventional	conventional	ADJ
ejpam-6609	264	19	.	.	PUNCT
ejpam-6609	265	1	corollary	corollary	ADJ
ejpam-6609	265	2	3	3	NUM
ejpam-6609	265	3	.	.	PUNCT
ejpam-6609	266	1	for	for	ADP
ejpam-6609	266	2	t1	t1	NOUN
ejpam-6609	266	3	=	=	SYM
ejpam-6609	266	4	t2	t2	PROPN
ejpam-6609	266	5	=	=	SYM
ejpam-6609	266	6	t3	t3	PROPN
ejpam-6609	266	7	=	=	SYM
ejpam-6609	266	8	t	t	PROPN
ejpam-6609	266	9	,	,	PUNCT
ejpam-6609	266	10	algorithm	algorithm	NOUN
ejpam-6609	266	11	4	4	NUM
ejpam-6609	266	12	converges	converge	NOUN
ejpam-6609	266	13	in	in	ADP
ejpam-6609	266	14	o(log	o(log	PROPN
ejpam-6609	266	15	1	1	NUM
ejpam-6609	266	16	ϵ	ϵ	NOUN
ejpam-6609	266	17	)	)	PUNCT
ejpam-6609	266	18	steps	step	NOUN
ejpam-6609	266	19	:	:	PUNCT
ejpam-6609	266	20	1	1	NUM
ejpam-6609	266	21	:	:	PUNCT
ejpam-6609	266	22	convergence	convergence	NOUN
ejpam-6609	266	23	rate	rate	NOUN
ejpam-6609	266	24	:	:	PUNCT
ejpam-6609	266	25	log	log	VERB
ejpam-6609	266	26	ϵ	ϵ	X
ejpam-6609	266	27	log(α+β	log(α+β	NOUN
ejpam-6609	266	28	)	)	PUNCT
ejpam-6609	266	29	lemma	lemma	PROPN
ejpam-6609	266	30	3	3	NUM
ejpam-6609	266	31	.	.	PUNCT
ejpam-6609	267	1	the	the	DET
ejpam-6609	267	2	orthogonal	orthogonal	ADJ
ejpam-6609	267	3	projection	projection	NOUN
ejpam-6609	267	4	ps	ps	PROPN
ejpam-6609	267	5	onto	onto	ADP
ejpam-6609	267	6	a	a	DET
ejpam-6609	267	7	subspace	subspace	NOUN
ejpam-6609	267	8	s	s	X
ejpam-6609	267	9	⊂	⊂	PROPN
ejpam-6609	267	10	h	h	PROPN
ejpam-6609	267	11	meets	meet	VERB
ejpam-6609	267	12	the	the	DET
ejpam-6609	267	13	following	following	ADJ
ejpam-6609	267	14	conditions	condition	NOUN
ejpam-6609	267	15	:	:	PUNCT
ejpam-6609	267	16	∥psx−	∥psx−	PROPN
ejpam-6609	267	17	psy	psy	PROPN
ejpam-6609	267	18	−	−	PROPN
ejpam-6609	267	19	psz∥2	psz∥2	VERB
ejpam-6609	267	20	≤	≤	PUNCT
ejpam-6609	268	1	∥x−	∥x−	PROPN
ejpam-6609	268	2	y	y	PROPN
ejpam-6609	268	3	−	−	PROPN
ejpam-6609	268	4	z∥2	z∥2	NOUN
ejpam-6609	268	5	figure	figure	NOUN
ejpam-6609	268	6	3	3	NUM
ejpam-6609	268	7	:	:	PUNCT
ejpam-6609	268	8	machine	machine	NOUN
ejpam-6609	268	9	learning	learn	VERB
ejpam-6609	268	10	interpretation	interpretation	NOUN
ejpam-6609	268	11	of	of	ADP
ejpam-6609	268	12	theorem	theorem	ADJ
ejpam-6609	268	13	2	2	NUM
ejpam-6609	268	14	4	4	NUM
ejpam-6609	268	15	.	.	PUNCT
ejpam-6609	268	16	comparative	comparative	ADJ
ejpam-6609	268	17	analysis	analysis	NOUN
ejpam-6609	268	18	as	as	SCONJ
ejpam-6609	268	19	you	you	PRON
ejpam-6609	268	20	see	see	VERB
ejpam-6609	268	21	in	in	ADP
ejpam-6609	268	22	figure	figure	NOUN
ejpam-6609	268	23	4	4	NUM
ejpam-6609	268	24	,	,	PUNCT
ejpam-6609	268	25	a	a	DET
ejpam-6609	268	26	normalized	normalize	VERB
ejpam-6609	268	27	bar	bar	NOUN
ejpam-6609	268	28	chart	chart	NOUN
ejpam-6609	268	29	shows	show	VERB
ejpam-6609	268	30	the	the	DET
ejpam-6609	268	31	relative	relative	ADJ
ejpam-6609	268	32	differences	difference	NOUN
ejpam-6609	268	33	in	in	ADP
ejpam-6609	268	34	the	the	DET
ejpam-6609	268	35	complexity	complexity	NOUN
ejpam-6609	268	36	and	and	CCONJ
ejpam-6609	268	37	flexibility	flexibility	NOUN
ejpam-6609	268	38	of	of	ADP
ejpam-6609	268	39	the	the	DET
ejpam-6609	268	40	two	two	NUM
ejpam-6609	268	41	theorems	theorem	NOUN
ejpam-6609	268	42	by	by	ADP
ejpam-6609	268	43	comparing	compare	VERB
ejpam-6609	268	44	several	several	ADJ
ejpam-6609	268	45	aspects	aspect	NOUN
ejpam-6609	268	46	.	.	PUNCT
ejpam-6609	269	1	the	the	DET
ejpam-6609	269	2	two	two	NUM
ejpam-6609	269	3	main	main	ADJ
ejpam-6609	269	4	theorems	theorem	NOUN
ejpam-6609	269	5	,	,	PUNCT
ejpam-6609	269	6	theorem	theorem	ADJ
ejpam-6609	269	7	1	1	NUM
ejpam-6609	269	8	and	and	CCONJ
ejpam-6609	269	9	theorem	theorem	VERB
ejpam-6609	269	10	2	2	NUM
ejpam-6609	269	11	,	,	PUNCT
ejpam-6609	269	12	are	be	AUX
ejpam-6609	269	13	alike	alike	ADV
ejpam-6609	269	14	in	in	ADP
ejpam-6609	269	15	some	some	DET
ejpam-6609	269	16	ways	way	NOUN
ejpam-6609	269	17	but	but	CCONJ
ejpam-6609	269	18	not	not	PART
ejpam-6609	269	19	in	in	ADP
ejpam-6609	269	20	others	other	NOUN
ejpam-6609	269	21	.	.	PUNCT
ejpam-6609	270	1	table	table	NOUN
ejpam-6609	270	2	1	1	NUM
ejpam-6609	270	3	:	:	PUNCT
ejpam-6609	270	4	comparison	comparison	NOUN
ejpam-6609	270	5	of	of	ADP
ejpam-6609	270	6	main	main	ADJ
ejpam-6609	270	7	theorems	theorem	NOUN
ejpam-6609	270	8	aspect	aspect	NOUN
ejpam-6609	270	9	theorem	theorem	VERB
ejpam-6609	270	10	1	1	NUM
ejpam-6609	270	11	theorem	theorem	ADJ
ejpam-6609	270	12	2	2	NUM
ejpam-6609	270	13	space	space	NOUN
ejpam-6609	270	14	general	general	PROPN
ejpam-6609	270	15	hs	hs	PROPN
ejpam-6609	270	16	general	general	PROPN
ejpam-6609	270	17	hs	hs	PROPN
ejpam-6609	270	18	mappings	mapping	NOUN
ejpam-6609	270	19	three	three	NUM
ejpam-6609	270	20	distinct	distinct	ADJ
ejpam-6609	270	21	operators	operator	NOUN
ejpam-6609	270	22	three	three	NUM
ejpam-6609	270	23	distinct	distinct	ADJ
ejpam-6609	270	24	operators	operator	NOUN
ejpam-6609	270	25	contraction	contraction	VERB
ejpam-6609	270	26	condition	condition	NOUN
ejpam-6609	270	27	three	three	NUM
ejpam-6609	270	28	-	-	PUNCT
ejpam-6609	270	29	term	term	NOUN
ejpam-6609	270	30	inequality	inequality	NOUN
ejpam-6609	270	31	two	two	NUM
ejpam-6609	270	32	-	-	PUNCT
ejpam-6609	270	33	term	term	NOUN
ejpam-6609	270	34	inequality	inequality	NOUN
ejpam-6609	270	35	parameters	parameter	NOUN
ejpam-6609	270	36	α	α	PRON
ejpam-6609	270	37	,	,	PUNCT
ejpam-6609	270	38	β	β	X
ejpam-6609	270	39	,	,	PUNCT
ejpam-6609	270	40	γ	γ	PROPN
ejpam-6609	270	41	α	α	PROPN
ejpam-6609	270	42	,	,	PUNCT
ejpam-6609	270	43	β	β	X
ejpam-6609	270	44	constraint	constraint	NOUN
ejpam-6609	270	45	α+	α+	X
ejpam-6609	270	46	β	β	NOUN
ejpam-6609	270	47	+	+	X
ejpam-6609	270	48	γ	γ	X
ejpam-6609	270	49	<	<	X
ejpam-6609	270	50	1	1	NUM
ejpam-6609	270	51	α+	α+	NOUN
ejpam-6609	270	52	β	β	X
ejpam-6609	270	53	<	<	X
ejpam-6609	270	54	1	1	NUM
ejpam-6609	270	55	convergence	convergence	NOUN
ejpam-6609	270	56	rate	rate	NOUN
ejpam-6609	270	57	o(αn/3	o(αn/3	NOUN
ejpam-6609	270	58	)	)	PUNCT
ejpam-6609	270	59	o((α+	o((α+	NOUN
ejpam-6609	270	60	β)n/3	β)n/3	NOUN
ejpam-6609	270	61	)	)	PUNCT
ejpam-6609	270	62	uniqueness	uniqueness	NOUN
ejpam-6609	270	63	proof	proof	NOUN
ejpam-6609	270	64	uses	use	VERB
ejpam-6609	270	65	α+	α+	PRON
ejpam-6609	270	66	2β	2β	NOUN
ejpam-6609	270	67	+	+	CCONJ
ejpam-6609	270	68	2γ	2γ	NOUN
ejpam-6609	270	69	<	<	X
ejpam-6609	270	70	1	1	NUM
ejpam-6609	270	71	uses	use	VERB
ejpam-6609	270	72	α+	α+	PUNCT
ejpam-6609	270	73	2β	2β	NOUN
ejpam-6609	270	74	<	<	X
ejpam-6609	270	75	1	1	NUM
ejpam-6609	270	76	m.	m.	NOUN
ejpam-6609	270	77	noorwali	noorwali	PROPN
ejpam-6609	270	78	et	et	PROPN
ejpam-6609	270	79	al	al	PROPN
ejpam-6609	270	80	.	.	PUNCT
ejpam-6609	270	81	/	/	SYM
ejpam-6609	270	82	eur	eur	PROPN
ejpam-6609	270	83	.	.	PUNCT
ejpam-6609	271	1	j.	j.	PROPN
ejpam-6609	271	2	pure	pure	PROPN
ejpam-6609	271	3	appl	appl	PROPN
ejpam-6609	271	4	.	.	PROPN
ejpam-6609	271	5	math	math	PROPN
ejpam-6609	271	6	,	,	PUNCT
ejpam-6609	271	7	18	18	NUM
ejpam-6609	271	8	(	(	PUNCT
ejpam-6609	271	9	3	3	NUM
ejpam-6609	271	10	)	)	PUNCT
ejpam-6609	271	11	(	(	PUNCT
ejpam-6609	271	12	2025	2025	NUM
ejpam-6609	271	13	)	)	PUNCT
ejpam-6609	271	14	,	,	PUNCT
ejpam-6609	271	15	6609	6609	NUM
ejpam-6609	271	16	14	14	NUM
ejpam-6609	271	17	of	of	ADP
ejpam-6609	271	18	17	17	NUM
ejpam-6609	271	19	figure	figure	NOUN
ejpam-6609	271	20	4	4	NUM
ejpam-6609	271	21	:	:	PUNCT
ejpam-6609	271	22	theoretical	theoretical	ADJ
ejpam-6609	271	23	comparison	comparison	NOUN
ejpam-6609	271	24	between	between	ADP
ejpam-6609	271	25	theorems	theorem	NOUN
ejpam-6609	271	26	•	•	NUM
ejpam-6609	271	27	structural	structural	ADJ
ejpam-6609	271	28	complexity	complexity	NOUN
ejpam-6609	271	29	:	:	PUNCT
ejpam-6609	271	30	theorem	theorem	NOUN
ejpam-6609	271	31	1	1	NUM
ejpam-6609	271	32	includes	include	VERB
ejpam-6609	271	33	a	a	DET
ejpam-6609	271	34	more	more	ADV
ejpam-6609	271	35	complicated	complicated	ADJ
ejpam-6609	271	36	contraction	contraction	NOUN
ejpam-6609	271	37	condition	condition	NOUN
ejpam-6609	271	38	with	with	ADP
ejpam-6609	271	39	extra	extra	ADJ
ejpam-6609	271	40	nonlinear	nonlinear	ADJ
ejpam-6609	271	41	terms	term	NOUN
ejpam-6609	271	42	(	(	PUNCT
ejpam-6609	271	43	parameter	parameter	NOUN
ejpam-6609	271	44	γ	γ	PROPN
ejpam-6609	271	45	)	)	PUNCT
ejpam-6609	271	46	,	,	PUNCT
ejpam-6609	271	47	while	while	SCONJ
ejpam-6609	271	48	theorem	theorem	VERB
ejpam-6609	271	49	2	2	NUM
ejpam-6609	271	50	offers	offer	VERB
ejpam-6609	271	51	a	a	DET
ejpam-6609	271	52	simpler	simple	ADJ
ejpam-6609	271	53	version	version	NOUN
ejpam-6609	271	54	that	that	PRON
ejpam-6609	271	55	might	might	AUX
ejpam-6609	271	56	be	be	AUX
ejpam-6609	271	57	easier	easy	ADJ
ejpam-6609	271	58	to	to	PART
ejpam-6609	271	59	use	use	VERB
ejpam-6609	271	60	in	in	ADP
ejpam-6609	271	61	real	real	ADJ
ejpam-6609	271	62	situations	situation	NOUN
ejpam-6609	271	63	.	.	PUNCT
ejpam-6609	272	1	•	•	NUM
ejpam-6609	272	2	parameter	parameter	NOUN
ejpam-6609	272	3	flexibility	flexibility	NOUN
ejpam-6609	272	4	:	:	PUNCT
ejpam-6609	272	5	the	the	DET
ejpam-6609	272	6	three	three	NUM
ejpam-6609	272	7	-	-	PUNCT
ejpam-6609	272	8	parameter	parameter	NOUN
ejpam-6609	272	9	version	version	NOUN
ejpam-6609	272	10	(	(	PUNCT
ejpam-6609	272	11	theorem	theorem	NOUN
ejpam-6609	272	12	1	1	NUM
ejpam-6609	272	13	)	)	PUNCT
ejpam-6609	272	14	offers	offer	VERB
ejpam-6609	272	15	greater	great	ADJ
ejpam-6609	272	16	modeling	modeling	NOUN
ejpam-6609	272	17	flexibility	flexibility	NOUN
ejpam-6609	272	18	for	for	ADP
ejpam-6609	272	19	complex	complex	ADJ
ejpam-6609	272	20	operator	operator	NOUN
ejpam-6609	272	21	interactions	interaction	NOUN
ejpam-6609	272	22	,	,	PUNCT
ejpam-6609	272	23	whereas	whereas	SCONJ
ejpam-6609	272	24	the	the	DET
ejpam-6609	272	25	two	two	NUM
ejpam-6609	272	26	-	-	PUNCT
ejpam-6609	272	27	parameter	parameter	NOUN
ejpam-6609	272	28	version	version	NOUN
ejpam-6609	272	29	(	(	PUNCT
ejpam-6609	272	30	theorem	theorem	NOUN
ejpam-6609	272	31	2	2	NUM
ejpam-6609	272	32	)	)	PUNCT
ejpam-6609	272	33	provides	provide	VERB
ejpam-6609	272	34	simpler	simple	ADJ
ejpam-6609	272	35	verification	verification	NOUN
ejpam-6609	272	36	conditions	condition	NOUN
ejpam-6609	272	37	.	.	PUNCT
ejpam-6609	273	1	•	•	NUM
ejpam-6609	273	2	convergence	convergence	NOUN
ejpam-6609	273	3	behavior	behavior	NOUN
ejpam-6609	273	4	:	:	PUNCT
ejpam-6609	273	5	linear	linear	ADJ
ejpam-6609	273	6	convergence	convergence	NOUN
ejpam-6609	273	7	is	be	AUX
ejpam-6609	273	8	promised	promise	VERB
ejpam-6609	273	9	by	by	ADP
ejpam-6609	273	10	both	both	DET
ejpam-6609	273	11	theorems	theorem	NOUN
ejpam-6609	273	12	,	,	PUNCT
ejpam-6609	273	13	yet	yet	CCONJ
ejpam-6609	273	14	the	the	DET
ejpam-6609	273	15	speed	speed	NOUN
ejpam-6609	273	16	of	of	ADP
ejpam-6609	273	17	convergence	convergence	NOUN
ejpam-6609	273	18	depends	depend	VERB
ejpam-6609	273	19	on	on	ADP
ejpam-6609	273	20	what	what	PRON
ejpam-6609	273	21	parameters	parameter	NOUN
ejpam-6609	273	22	are	be	AUX
ejpam-6609	273	23	used	use	VERB
ejpam-6609	273	24	together	together	ADV
ejpam-6609	273	25	.	.	PUNCT
ejpam-6609	274	1	γ	γ	PROPN
ejpam-6609	274	2	in	in	ADP
ejpam-6609	274	3	theorem	theorem	NOUN
ejpam-6609	274	4	1	1	NUM
ejpam-6609	274	5	makes	make	VERB
ejpam-6609	274	6	it	it	PRON
ejpam-6609	274	7	possible	possible	ADJ
ejpam-6609	274	8	to	to	PART
ejpam-6609	274	9	have	have	VERB
ejpam-6609	274	10	more	more	ADJ
ejpam-6609	274	11	exact	exact	ADJ
ejpam-6609	274	12	control	control	NOUN
ejpam-6609	274	13	over	over	ADP
ejpam-6609	274	14	complex	complex	ADJ
ejpam-6609	274	15	interactions	interaction	NOUN
ejpam-6609	274	16	.	.	PUNCT
ejpam-6609	275	1	the	the	DET
ejpam-6609	275	2	adaptability	adaptability	NOUN
ejpam-6609	275	3	of	of	ADP
ejpam-6609	275	4	theorem	theorem	NOUN
ejpam-6609	275	5	2	2	NUM
ejpam-6609	275	6	across	across	ADP
ejpam-6609	275	7	various	various	ADJ
ejpam-6609	275	8	domains	domain	NOUN
ejpam-6609	275	9	is	be	AUX
ejpam-6609	275	10	illustrated	illustrate	VERB
ejpam-6609	275	11	by	by	ADP
ejpam-6609	275	12	the	the	DET
ejpam-6609	275	13	applications	application	NOUN
ejpam-6609	275	14	in	in	ADP
ejpam-6609	275	15	examples	example	NOUN
ejpam-6609	275	16	3	3	NUM
ejpam-6609	275	17	and	and	CCONJ
ejpam-6609	275	18	4	4	NUM
ejpam-6609	275	19	.	.	PUNCT
ejpam-6609	276	1	the	the	DET
ejpam-6609	276	2	three	three	NUM
ejpam-6609	276	3	-	-	PUNCT
ejpam-6609	276	4	operator	operator	NOUN
ejpam-6609	276	5	fixed	fix	VERB
ejpam-6609	276	6	-	-	PUNCT
ejpam-6609	276	7	point	point	NOUN
ejpam-6609	276	8	framework	framework	NOUN
ejpam-6609	276	9	is	be	AUX
ejpam-6609	276	10	used	use	VERB
ejpam-6609	276	11	in	in	ADP
ejpam-6609	276	12	both	both	DET
ejpam-6609	276	13	algorithm	algorithm	NOUN
ejpam-6609	276	14	5	5	NUM
ejpam-6609	276	15	(	(	PUNCT
ejpam-6609	276	16	for	for	ADP
ejpam-6609	276	17	distributed	distributed	ADJ
ejpam-6609	276	18	learning	learning	NOUN
ejpam-6609	276	19	)	)	PUNCT
ejpam-6609	276	20	and	and	CCONJ
ejpam-6609	276	21	algorithm	algorithm	NOUN
ejpam-6609	276	22	6	6	NUM
ejpam-6609	276	23	(	(	PUNCT
ejpam-6609	276	24	for	for	ADP
ejpam-6609	276	25	cryptographic	cryptographic	ADJ
ejpam-6609	276	26	hashing	hashing	NOUN
ejpam-6609	276	27	)	)	PUNCT
ejpam-6609	276	28	,	,	PUNCT
ejpam-6609	276	29	but	but	CCONJ
ejpam-6609	276	30	each	each	PRON
ejpam-6609	276	31	is	be	AUX
ejpam-6609	276	32	adjusted	adjust	VERB
ejpam-6609	276	33	to	to	PART
ejpam-6609	276	34	meet	meet	VERB
ejpam-6609	276	35	specific	specific	ADJ
ejpam-6609	276	36	needs	need	NOUN
ejpam-6609	276	37	of	of	ADP
ejpam-6609	276	38	their	their	PRON
ejpam-6609	276	39	fields	field	NOUN
ejpam-6609	276	40	.	.	PUNCT
ejpam-6609	277	1	the	the	DET
ejpam-6609	277	2	operators	operator	NOUN
ejpam-6609	277	3	ti	ti	VERB
ejpam-6609	277	4	=	=	SYM
ejpam-6609	277	5	∇li	∇li	PROPN
ejpam-6609	277	6	in	in	ADP
ejpam-6609	277	7	distributed	distributed	ADJ
ejpam-6609	277	8	learning	learning	NOUN
ejpam-6609	277	9	rely	rely	VERB
ejpam-6609	277	10	on	on	ADP
ejpam-6609	277	11	the	the	DET
ejpam-6609	277	12	learning	learning	NOUN
ejpam-6609	277	13	rate	rate	NOUN
ejpam-6609	277	14	η	η	PROPN
ejpam-6609	277	15	to	to	PART
ejpam-6609	277	16	maintain	maintain	VERB
ejpam-6609	277	17	the	the	DET
ejpam-6609	277	18	contraction	contraction	NOUN
ejpam-6609	277	19	condition	condition	NOUN
ejpam-6609	277	20	,	,	PUNCT
ejpam-6609	277	21	and	and	CCONJ
ejpam-6609	277	22	gradient	gradient	NOUN
ejpam-6609	277	23	norms	norm	NOUN
ejpam-6609	277	24	are	be	AUX
ejpam-6609	277	25	used	use	VERB
ejpam-6609	277	26	to	to	PART
ejpam-6609	277	27	check	check	VERB
ejpam-6609	277	28	if	if	SCONJ
ejpam-6609	277	29	the	the	DET
ejpam-6609	277	30	process	process	NOUN
ejpam-6609	277	31	is	be	AUX
ejpam-6609	277	32	getting	get	VERB
ejpam-6609	277	33	closer	close	ADJ
ejpam-6609	277	34	to	to	ADP
ejpam-6609	277	35	a	a	DET
ejpam-6609	277	36	solution	solution	NOUN
ejpam-6609	277	37	.	.	PUNCT
ejpam-6609	278	1	on	on	ADP
ejpam-6609	278	2	the	the	DET
ejpam-6609	278	3	other	other	ADJ
ejpam-6609	278	4	hand	hand	NOUN
ejpam-6609	278	5	,	,	PUNCT
ejpam-6609	278	6	the	the	DET
ejpam-6609	278	7	cryptographic	cryptographic	ADJ
ejpam-6609	278	8	hashing	hashing	NOUN
ejpam-6609	278	9	application	application	NOUN
ejpam-6609	278	10	uses	use	VERB
ejpam-6609	278	11	hash	hash	NOUN
ejpam-6609	278	12	functions	function	NOUN
ejpam-6609	278	13	hi	hi	INTJ
ejpam-6609	278	14	that	that	PRON
ejpam-6609	278	15	satisfy	satisfy	NOUN
ejpam-6609	278	16	lipschitz	lipschitz	VERB
ejpam-6609	278	17	continuity	continuity	NOUN
ejpam-6609	278	18	with	with	ADP
ejpam-6609	278	19	constants	constant	NOUN
ejpam-6609	278	20	l	l	NOUN
ejpam-6609	278	21	<	<	X
ejpam-6609	278	22	1/	1/	NUM
ejpam-6609	278	23	√	√	NUM
ejpam-6609	278	24	3	3	NUM
ejpam-6609	278	25	.	.	PUNCT
ejpam-6609	278	26	iteration	iteration	NOUN
ejpam-6609	278	27	is	be	AUX
ejpam-6609	278	28	bounded	bound	VERB
ejpam-6609	278	29	by	by	ADP
ejpam-6609	278	30	a	a	DET
ejpam-6609	278	31	security	security	NOUN
ejpam-6609	278	32	parameter	parameter	NOUN
ejpam-6609	278	33	n	n	NOUN
ejpam-6609	278	34	,	,	PUNCT
ejpam-6609	278	35	and	and	CCONJ
ejpam-6609	278	36	convergence	convergence	NOUN
ejpam-6609	278	37	is	be	AUX
ejpam-6609	278	38	confirmed	confirm	VERB
ejpam-6609	278	39	by	by	ADP
ejpam-6609	278	40	comparing	compare	VERB
ejpam-6609	278	41	output	output	NOUN
ejpam-6609	278	42	differences	difference	NOUN
ejpam-6609	278	43	.	.	PUNCT
ejpam-6609	279	1	table	table	NOUN
ejpam-6609	279	2	2	2	NUM
ejpam-6609	279	3	provides	provide	VERB
ejpam-6609	279	4	a	a	DET
ejpam-6609	279	5	summary	summary	NOUN
ejpam-6609	279	6	of	of	ADP
ejpam-6609	279	7	these	these	DET
ejpam-6609	279	8	distinctions	distinction	NOUN
ejpam-6609	279	9	,	,	PUNCT
ejpam-6609	279	10	demonstrating	demonstrate	VERB
ejpam-6609	279	11	that	that	SCONJ
ejpam-6609	279	12	while	while	SCONJ
ejpam-6609	279	13	both	both	DET
ejpam-6609	279	14	algorithms	algorithm	NOUN
ejpam-6609	279	15	are	be	AUX
ejpam-6609	279	16	based	base	VERB
ejpam-6609	279	17	on	on	ADP
ejpam-6609	279	18	the	the	DET
ejpam-6609	279	19	same	same	ADJ
ejpam-6609	279	20	theoretical	theoretical	ADJ
ejpam-6609	279	21	outcome	outcome	NOUN
ejpam-6609	279	22	(	(	PUNCT
ejpam-6609	279	23	theorem	theorem	NOUN
ejpam-6609	279	24	2	2	NUM
ejpam-6609	279	25	)	)	PUNCT
ejpam-6609	279	26	,	,	PUNCT
ejpam-6609	279	27	their	their	PRON
ejpam-6609	279	28	approaches	approach	NOUN
ejpam-6609	279	29	to	to	ADP
ejpam-6609	279	30	implementation	implementation	NOUN
ejpam-6609	279	31	are	be	AUX
ejpam-6609	279	32	completely	completely	ADV
ejpam-6609	279	33	unique	unique	ADJ
ejpam-6609	279	34	.	.	PUNCT
ejpam-6609	280	1	both	both	DET
ejpam-6609	280	2	algorithms	algorithm	NOUN
ejpam-6609	280	3	show	show	VERB
ejpam-6609	280	4	that	that	SCONJ
ejpam-6609	280	5	they	they	PRON
ejpam-6609	280	6	get	get	VERB
ejpam-6609	280	7	closer	close	ADJ
ejpam-6609	280	8	to	to	ADP
ejpam-6609	280	9	a	a	DET
ejpam-6609	280	10	solution	solution	NOUN
ejpam-6609	280	11	,	,	PUNCT
ejpam-6609	280	12	as	as	SCONJ
ejpam-6609	280	13	seen	see	VERB
ejpam-6609	280	14	in	in	ADP
ejpam-6609	280	15	figure	figure	NOUN
ejpam-6609	280	16	5	5	NUM
ejpam-6609	280	17	,	,	PUNCT
ejpam-6609	280	18	but	but	CCONJ
ejpam-6609	280	19	they	they	PRON
ejpam-6609	280	20	do	do	VERB
ejpam-6609	280	21	so	so	ADV
ejpam-6609	280	22	at	at	ADP
ejpam-6609	280	23	different	different	ADJ
ejpam-6609	280	24	speeds	speed	NOUN
ejpam-6609	280	25	and	and	CCONJ
ejpam-6609	280	26	with	with	ADP
ejpam-6609	280	27	different	different	ADJ
ejpam-6609	280	28	patterns	pattern	NOUN
ejpam-6609	280	29	based	base	VERB
ejpam-6609	280	30	on	on	ADP
ejpam-6609	280	31	how	how	SCONJ
ejpam-6609	280	32	their	their	PRON
ejpam-6609	280	33	operators	operator	NOUN
ejpam-6609	280	34	are	be	AUX
ejpam-6609	280	35	defined	define	VERB
ejpam-6609	280	36	.	.	PUNCT
ejpam-6609	281	1	m.	m.	PROPN
ejpam-6609	281	2	noorwali	noorwali	PROPN
ejpam-6609	281	3	et	et	PROPN
ejpam-6609	281	4	al	al	PROPN
ejpam-6609	281	5	.	.	PUNCT
ejpam-6609	281	6	/	/	SYM
ejpam-6609	281	7	eur	eur	PROPN
ejpam-6609	281	8	.	.	PUNCT
ejpam-6609	282	1	j.	j.	PROPN
ejpam-6609	282	2	pure	pure	PROPN
ejpam-6609	282	3	appl	appl	PROPN
ejpam-6609	282	4	.	.	PROPN
ejpam-6609	282	5	math	math	PROPN
ejpam-6609	282	6	,	,	PUNCT
ejpam-6609	282	7	18	18	NUM
ejpam-6609	282	8	(	(	PUNCT
ejpam-6609	282	9	3	3	NUM
ejpam-6609	282	10	)	)	PUNCT
ejpam-6609	282	11	(	(	PUNCT
ejpam-6609	282	12	2025	2025	NUM
ejpam-6609	282	13	)	)	PUNCT
ejpam-6609	282	14	,	,	PUNCT
ejpam-6609	282	15	6609	6609	NUM
ejpam-6609	282	16	15	15	NUM
ejpam-6609	282	17	of	of	ADP
ejpam-6609	282	18	17	17	NUM
ejpam-6609	282	19	table	table	NOUN
ejpam-6609	282	20	2	2	NUM
ejpam-6609	282	21	:	:	PUNCT
ejpam-6609	282	22	comparison	comparison	NOUN
ejpam-6609	282	23	of	of	ADP
ejpam-6609	282	24	applications	application	NOUN
ejpam-6609	282	25	feature	feature	VERB
ejpam-6609	282	26	distributed	distribute	VERB
ejpam-6609	282	27	learning	learn	VERB
ejpam-6609	282	28	(	(	PUNCT
ejpam-6609	282	29	alg	alg	PROPN
ejpam-6609	282	30	.	.	PROPN
ejpam-6609	282	31	5	5	NUM
ejpam-6609	282	32	)	)	PUNCT
ejpam-6609	282	33	cryptographic	cryptographic	ADJ
ejpam-6609	282	34	hashing	hashing	NOUN
ejpam-6609	282	35	(	(	PUNCT
ejpam-6609	282	36	alg	alg	PROPN
ejpam-6609	282	37	.	.	PROPN
ejpam-6609	282	38	6	6	NUM
ejpam-6609	282	39	)	)	PUNCT
ejpam-6609	282	40	theorem	theorem	ADJ
ejpam-6609	282	41	used	use	VERB
ejpam-6609	282	42	theorem	theorem	ADJ
ejpam-6609	282	43	2	2	NUM
ejpam-6609	282	44	theorem	theorem	ADJ
ejpam-6609	282	45	2	2	NUM
ejpam-6609	282	46	operators	operator	NOUN
ejpam-6609	282	47	ti	ti	NOUN
ejpam-6609	283	1	=	=	NOUN
ejpam-6609	283	2	∇li	∇li	AUX
ejpam-6609	283	3	hi	hi	INTJ
ejpam-6609	283	4	(	(	PUNCT
ejpam-6609	283	5	hash	hash	NOUN
ejpam-6609	283	6	functions	function	NOUN
ejpam-6609	283	7	)	)	PUNCT
ejpam-6609	283	8	contraction	contraction	NOUN
ejpam-6609	283	9	source	source	NOUN
ejpam-6609	283	10	learning	learn	VERB
ejpam-6609	283	11	rate	rate	NOUN
ejpam-6609	283	12	η	η	PROPN
ejpam-6609	283	13	lipschitz	lipschitz	PROPN
ejpam-6609	283	14	constant	constant	ADJ
ejpam-6609	283	15	l	l	NOUN
ejpam-6609	283	16	<	<	X
ejpam-6609	283	17	1/	1/	NUM
ejpam-6609	283	18	√	√	SYM
ejpam-6609	283	19	3	3	NUM
ejpam-6609	283	20	convergence	convergence	NOUN
ejpam-6609	283	21	check	check	NOUN
ejpam-6609	283	22	∥∇li∥	∥∇li∥	PROPN
ejpam-6609	283	23	norm	norm	VERB
ejpam-6609	283	24	hash	hash	NOUN
ejpam-6609	283	25	output	output	NOUN
ejpam-6609	283	26	difference	difference	NOUN
ejpam-6609	283	27	key	key	NOUN
ejpam-6609	283	28	parameters	parameter	NOUN
ejpam-6609	283	29	α	α	NOUN
ejpam-6609	283	30	=	=	SYM
ejpam-6609	283	31	1	1	NUM
ejpam-6609	283	32	3	3	NUM
ejpam-6609	283	33	,	,	PUNCT
ejpam-6609	283	34	β	β	X
ejpam-6609	283	35	=	=	SYM
ejpam-6609	283	36	0	0	NUM
ejpam-6609	283	37	α	α	NOUN
ejpam-6609	283	38	=	=	SYM
ejpam-6609	283	39	l2	l2	NOUN
ejpam-6609	283	40	,	,	PUNCT
ejpam-6609	283	41	β	β	NOUN
ejpam-6609	283	42	=	=	SYM
ejpam-6609	283	43	0	0	NUM
ejpam-6609	283	44	stopping	stopping	NOUN
ejpam-6609	283	45	criterion	criterion	NOUN
ejpam-6609	283	46	adaptive	adaptive	ADJ
ejpam-6609	283	47	η	η	PROPN
ejpam-6609	283	48	threshold	threshold	NOUN
ejpam-6609	283	49	security	security	NOUN
ejpam-6609	283	50	parameter	parameter	NOUN
ejpam-6609	283	51	n	n	NUM
ejpam-6609	284	1	implementation	implementation	NOUN
ejpam-6609	284	2	parallel	parallel	NOUN
ejpam-6609	284	3	gradient	gradient	NOUN
ejpam-6609	284	4	updates	update	VERB
ejpam-6609	284	5	iterative	iterative	NOUN
ejpam-6609	284	6	hashing	hashing	NOUN
ejpam-6609	284	7	figure	figure	NOUN
ejpam-6609	284	8	5	5	NUM
ejpam-6609	284	9	:	:	PUNCT
ejpam-6609	284	10	comparison	comparison	NOUN
ejpam-6609	284	11	between	between	ADP
ejpam-6609	284	12	distributed	distribute	VERB
ejpam-6609	284	13	learning	learning	NOUN
ejpam-6609	284	14	and	and	CCONJ
ejpam-6609	284	15	cryptographic	cryptographic	ADJ
ejpam-6609	284	16	hashing	hashing	NOUN
ejpam-6609	284	17	5	5	NUM
ejpam-6609	284	18	.	.	PUNCT
ejpam-6609	285	1	conclusion	conclusion	PROPN
ejpam-6609	285	2	hilton	hilton	PROPN
ejpam-6609	285	3	’s	’s	PART
ejpam-6609	285	4	work	work	NOUN
ejpam-6609	285	5	shows	show	VERB
ejpam-6609	285	6	that	that	PRON
ejpam-6609	285	7	holds	hold	VERB
ejpam-6609	285	8	for	for	ADP
ejpam-6609	285	9	gips	gip	NOUN
ejpam-6609	285	10	,	,	PUNCT
ejpam-6609	285	11	but	but	CCONJ
ejpam-6609	285	12	it	it	PRON
ejpam-6609	285	13	was	be	AUX
ejpam-6609	285	14	not	not	PART
ejpam-6609	285	15	proven	prove	VERB
ejpam-6609	285	16	by	by	ADP
ejpam-6609	285	17	classical	classical	ADJ
ejpam-6609	285	18	duality	duality	NOUN
ejpam-6609	285	19	.	.	PUNCT
ejpam-6609	286	1	we	we	PRON
ejpam-6609	286	2	have	have	AUX
ejpam-6609	286	3	proven	prove	VERB
ejpam-6609	286	4	that	that	SCONJ
ejpam-6609	286	5	in	in	ADP
ejpam-6609	286	6	some	some	DET
ejpam-6609	286	7	parts	part	NOUN
ejpam-6609	286	8	of	of	ADP
ejpam-6609	286	9	the	the	DET
ejpam-6609	286	10	theory	theory	NOUN
ejpam-6609	286	11	,	,	PUNCT
ejpam-6609	286	12	there	there	PRON
ejpam-6609	286	13	can	can	AUX
ejpam-6609	286	14	be	be	AUX
ejpam-6609	286	15	just	just	ADV
ejpam-6609	286	16	one	one	NUM
ejpam-6609	286	17	set	set	NOUN
ejpam-6609	286	18	of	of	ADP
ejpam-6609	286	19	common	common	ADJ
ejpam-6609	286	20	fixed	fix	VERB
ejpam-6609	286	21	points	point	NOUN
ejpam-6609	286	22	,	,	PUNCT
ejpam-6609	286	23	and	and	CCONJ
ejpam-6609	286	24	they	they	PRON
ejpam-6609	286	25	still	still	ADV
ejpam-6609	286	26	appear	appear	VERB
ejpam-6609	286	27	with	with	ADP
ejpam-6609	286	28	less	less	ADV
ejpam-6609	286	29	strict	strict	ADJ
ejpam-6609	286	30	conditions	condition	NOUN
ejpam-6609	286	31	.	.	PUNCT
ejpam-6609	287	1	in	in	ADP
ejpam-6609	287	2	gips	gips	PROPN
ejpam-6609	287	3	,	,	PUNCT
ejpam-6609	287	4	we	we	PRON
ejpam-6609	287	5	step	step	VERB
ejpam-6609	287	6	further	far	ADV
ejpam-6609	287	7	than	than	ADP
ejpam-6609	287	8	the	the	DET
ejpam-6609	287	9	regular	regular	ADJ
ejpam-6609	287	10	fixed	fix	VERB
ejpam-6609	287	11	-	-	PUNCT
ejpam-6609	287	12	point	point	NOUN
ejpam-6609	287	13	theory	theory	NOUN
ejpam-6609	287	14	and	and	CCONJ
ejpam-6609	287	15	look	look	VERB
ejpam-6609	287	16	at	at	ADP
ejpam-6609	287	17	systems	system	NOUN
ejpam-6609	287	18	with	with	ADP
ejpam-6609	287	19	multiple	multiple	ADJ
ejpam-6609	287	20	operators	operator	NOUN
ejpam-6609	287	21	,	,	PUNCT
ejpam-6609	287	22	which	which	PRON
ejpam-6609	287	23	are	be	AUX
ejpam-6609	287	24	based	base	VERB
ejpam-6609	287	25	on	on	ADP
ejpam-6609	287	26	real	real	ADJ
ejpam-6609	287	27	cases	case	NOUN
ejpam-6609	287	28	in	in	ADP
ejpam-6609	287	29	nonlinear	nonlinear	ADJ
ejpam-6609	287	30	analysis	analysis	NOUN
ejpam-6609	287	31	.	.	PUNCT
ejpam-6609	288	1	every	every	DET
ejpam-6609	288	2	theorem	theorem	NOUN
ejpam-6609	288	3	we	we	PRON
ejpam-6609	288	4	discovered	discover	VERB
ejpam-6609	288	5	was	be	AUX
ejpam-6609	288	6	then	then	ADV
ejpam-6609	288	7	checked	check	VERB
ejpam-6609	288	8	by	by	ADP
ejpam-6609	288	9	software	software	NOUN
ejpam-6609	288	10	to	to	PART
ejpam-6609	288	11	develop	develop	VERB
ejpam-6609	288	12	stronger	strong	ADJ
ejpam-6609	288	13	certainty	certainty	NOUN
ejpam-6609	288	14	.	.	PUNCT
ejpam-6609	289	1	there	there	PRON
ejpam-6609	289	2	is	be	VERB
ejpam-6609	289	3	potential	potential	ADJ
ejpam-6609	289	4	for	for	ADP
ejpam-6609	289	5	two	two	NUM
ejpam-6609	289	6	key	key	ADJ
ejpam-6609	289	7	areas	area	NOUN
ejpam-6609	289	8	of	of	ADP
ejpam-6609	289	9	science	science	NOUN
ejpam-6609	289	10	to	to	PART
ejpam-6609	289	11	advance	advance	VERB
ejpam-6609	289	12	a	a	DET
ejpam-6609	289	13	lot	lot	NOUN
ejpam-6609	289	14	through	through	ADP
ejpam-6609	289	15	this	this	DET
ejpam-6609	289	16	research	research	NOUN
ejpam-6609	289	17	.	.	PUNCT
ejpam-6609	290	1	fixed	fix	VERB
ejpam-6609	290	2	-	-	PUNCT
ejpam-6609	290	3	point	point	NOUN
ejpam-6609	290	4	analysis	analysis	NOUN
ejpam-6609	290	5	in	in	ADP
ejpam-6609	290	6	ai	ai	INTJ
ejpam-6609	290	7	helped	help	VERB
ejpam-6609	290	8	us	we	PRON
ejpam-6609	290	9	correct	correct	VERB
ejpam-6609	290	10	dems	dems	PROPN
ejpam-6609	290	11	so	so	SCONJ
ejpam-6609	290	12	neural	neural	ADJ
ejpam-6609	290	13	networks	network	NOUN
ejpam-6609	290	14	performed	perform	VERB
ejpam-6609	290	15	well	well	ADV
ejpam-6609	290	16	.	.	PUNCT
ejpam-6609	291	1	noticing	notice	VERB
ejpam-6609	291	2	the	the	DET
ejpam-6609	291	3	role	role	NOUN
ejpam-6609	291	4	of	of	ADP
ejpam-6609	291	5	pqps	pqps	ADJ
ejpam-6609	291	6	in	in	ADP
ejpam-6609	291	7	cryptography	cryptography	NOUN
ejpam-6609	291	8	gave	give	VERB
ejpam-6609	291	9	us	we	PRON
ejpam-6609	291	10	our	our	PRON
ejpam-6609	291	11	idea	idea	NOUN
ejpam-6609	291	12	,	,	PUNCT
ejpam-6609	291	13	so	so	ADV
ejpam-6609	291	14	we	we	PRON
ejpam-6609	291	15	used	use	VERB
ejpam-6609	291	16	our	our	PRON
ejpam-6609	291	17	knowledge	knowledge	NOUN
ejpam-6609	291	18	of	of	ADP
ejpam-6609	291	19	inner	inner	ADJ
ejpam-6609	291	20	product	product	NOUN
ejpam-6609	291	21	spaces	space	VERB
ejpam-6609	291	22	to	to	PART
ejpam-6609	291	23	create	create	VERB
ejpam-6609	291	24	new	new	ADJ
ejpam-6609	291	25	systems	system	NOUN
ejpam-6609	291	26	for	for	ADP
ejpam-6609	291	27	cryptography	cryptography	NOUN
ejpam-6609	291	28	based	base	VERB
ejpam-6609	291	29	on	on	ADP
ejpam-6609	291	30	lattices	lattice	NOUN
ejpam-6609	291	31	.	.	PUNCT
ejpam-6609	292	1	future	future	ADJ
ejpam-6609	292	2	work	work	NOUN
ejpam-6609	292	3	generalization	generalization	NOUN
ejpam-6609	292	4	to	to	ADP
ejpam-6609	292	5	n	n	CCONJ
ejpam-6609	292	6	-	-	PUNCT
ejpam-6609	292	7	self	self	NOUN
ejpam-6609	292	8	-	-	PUNCT
ejpam-6609	292	9	mappings	mapping	NOUN
ejpam-6609	292	10	:	:	PUNCT
ejpam-6609	292	11	a	a	DET
ejpam-6609	292	12	first	first	ADJ
ejpam-6609	292	13	further	further	ADJ
ejpam-6609	292	14	step	step	NOUN
ejpam-6609	292	15	is	be	AUX
ejpam-6609	292	16	to	to	PART
ejpam-6609	292	17	generalize	generalize	VERB
ejpam-6609	292	18	the	the	DET
ejpam-6609	292	19	fixed	fix	VERB
ejpam-6609	292	20	-	-	PUNCT
ejpam-6609	292	21	point	point	NOUN
ejpam-6609	292	22	results	result	NOUN
ejpam-6609	292	23	to	to	ADP
ejpam-6609	292	24	situations	situation	NOUN
ejpam-6609	292	25	where	where	SCONJ
ejpam-6609	292	26	four	four	NUM
ejpam-6609	292	27	or	or	CCONJ
ejpam-6609	292	28	more	more	ADJ
ejpam-6609	292	29	self	self	NOUN
ejpam-6609	292	30	-	-	PUNCT
ejpam-6609	292	31	mappings	mapping	NOUN
ejpam-6609	292	32	satisfy	satisfy	NOUN
ejpam-6609	292	33	contractive	contractive	ADJ
ejpam-6609	292	34	-	-	PUNCT
ejpam-6609	292	35	type	type	NOUN
ejpam-6609	292	36	conditions	condition	NOUN
ejpam-6609	292	37	.	.	PUNCT
ejpam-6609	293	1	this	this	DET
ejpam-6609	293	2	process	process	NOUN
ejpam-6609	293	3	means	mean	VERB
ejpam-6609	293	4	coming	come	VERB
ejpam-6609	293	5	up	up	ADP
ejpam-6609	293	6	with	with	ADP
ejpam-6609	293	7	suitable	suitable	ADJ
ejpam-6609	293	8	repeated	repeat	VERB
ejpam-6609	293	9	sequences	sequence	NOUN
ejpam-6609	293	10	and	and	CCONJ
ejpam-6609	293	11	inequalities	inequality	NOUN
ejpam-6609	293	12	that	that	PRON
ejpam-6609	293	13	introduce	introduce	VERB
ejpam-6609	293	14	rules	rule	NOUN
ejpam-6609	293	15	for	for	ADP
ejpam-6609	293	16	higher	high	ADJ
ejpam-6609	293	17	-	-	PUNCT
ejpam-6609	293	18	order	order	NOUN
ejpam-6609	293	19	interactions	interaction	NOUN
ejpam-6609	293	20	among	among	ADP
ejpam-6609	293	21	several	several	ADJ
ejpam-6609	293	22	mappings	mapping	NOUN
ejpam-6609	293	23	.	.	PUNCT
ejpam-6609	294	1	although	although	SCONJ
ejpam-6609	294	2	these	these	DET
ejpam-6609	294	3	additions	addition	NOUN
ejpam-6609	294	4	to	to	ADP
ejpam-6609	294	5	the	the	DET
ejpam-6609	294	6	theory	theory	NOUN
ejpam-6609	294	7	bring	bring	VERB
ejpam-6609	294	8	fresh	fresh	ADJ
ejpam-6609	294	9	mathematical	mathematical	ADJ
ejpam-6609	294	10	difficulties	difficulty	NOUN
ejpam-6609	294	11	,	,	PUNCT
ejpam-6609	294	12	the	the	DET
ejpam-6609	294	13	framework	framework	NOUN
ejpam-6609	294	14	discussed	discuss	VERB
ejpam-6609	294	15	here	here	ADV
ejpam-6609	294	16	provides	provide	VERB
ejpam-6609	294	17	an	an	DET
ejpam-6609	294	18	excellent	excellent	ADJ
ejpam-6609	294	19	starting	starting	NOUN
ejpam-6609	294	20	point	point	NOUN
ejpam-6609	294	21	for	for	ADP
ejpam-6609	294	22	new	new	ADJ
ejpam-6609	294	23	work	work	NOUN
ejpam-6609	294	24	.	.	PUNCT
ejpam-6609	295	1	m.	m.	NOUN
ejpam-6609	295	2	noorwali	noorwali	PROPN
ejpam-6609	295	3	et	et	PROPN
ejpam-6609	295	4	al	al	PROPN
ejpam-6609	295	5	.	.	PUNCT
ejpam-6609	295	6	/	/	SYM
ejpam-6609	295	7	eur	eur	PROPN
ejpam-6609	295	8	.	.	PUNCT
ejpam-6609	296	1	j.	j.	PROPN
ejpam-6609	296	2	pure	pure	PROPN
ejpam-6609	296	3	appl	appl	PROPN
ejpam-6609	296	4	.	.	PROPN
ejpam-6609	296	5	math	math	PROPN
ejpam-6609	296	6	,	,	PUNCT
ejpam-6609	296	7	18	18	NUM
ejpam-6609	296	8	(	(	PUNCT
ejpam-6609	296	9	3	3	NUM
ejpam-6609	296	10	)	)	PUNCT
ejpam-6609	296	11	(	(	PUNCT
ejpam-6609	296	12	2025	2025	NUM
ejpam-6609	296	13	)	)	PUNCT
ejpam-6609	296	14	,	,	PUNCT
ejpam-6609	296	15	6609	6609	NUM
ejpam-6609	296	16	16	16	NUM
ejpam-6609	296	17	of	of	ADP
ejpam-6609	296	18	17	17	NUM
ejpam-6609	296	19	advanced	advanced	ADJ
ejpam-6609	296	20	convergence	convergence	NOUN
ejpam-6609	296	21	analysis	analysis	NOUN
ejpam-6609	296	22	:	:	PUNCT
ejpam-6609	296	23	an	an	DET
ejpam-6609	296	24	essential	essential	ADJ
ejpam-6609	296	25	task	task	NOUN
ejpam-6609	296	26	for	for	ADP
ejpam-6609	296	27	the	the	DET
ejpam-6609	296	28	future	future	NOUN
ejpam-6609	296	29	is	be	AUX
ejpam-6609	296	30	to	to	PART
ejpam-6609	296	31	improve	improve	VERB
ejpam-6609	296	32	the	the	DET
ejpam-6609	296	33	way	way	NOUN
ejpam-6609	296	34	iterative	iterative	NOUN
ejpam-6609	296	35	processes	process	NOUN
ejpam-6609	296	36	come	come	VERB
ejpam-6609	296	37	together	together	ADV
ejpam-6609	296	38	.	.	PUNCT
ejpam-6609	297	1	for	for	ADP
ejpam-6609	297	2	example	example	NOUN
ejpam-6609	297	3	,	,	PUNCT
ejpam-6609	297	4	this	this	PRON
ejpam-6609	297	5	means	mean	VERB
ejpam-6609	297	6	exploring	explore	VERB
ejpam-6609	297	7	wider	wide	ADJ
ejpam-6609	297	8	classes	class	NOUN
ejpam-6609	297	9	of	of	ADP
ejpam-6609	297	10	recurrence	recurrence	NOUN
ejpam-6609	297	11	relations	relation	NOUN
ejpam-6609	297	12	,	,	PUNCT
ejpam-6609	297	13	less	less	ADV
ejpam-6609	297	14	strict	strict	ADJ
ejpam-6609	297	15	convergence	convergence	NOUN
ejpam-6609	297	16	conditions	condition	NOUN
ejpam-6609	297	17	,	,	PUNCT
ejpam-6609	297	18	and	and	CCONJ
ejpam-6609	297	19	how	how	SCONJ
ejpam-6609	297	20	quickly	quickly	ADV
ejpam-6609	297	21	sequences	sequence	NOUN
ejpam-6609	297	22	converge	converge	VERB
ejpam-6609	297	23	with	with	ADP
ejpam-6609	297	24	multi	multi	ADJ
ejpam-6609	297	25	-	-	ADJ
ejpam-6609	297	26	operator	operator	NOUN
ejpam-6609	297	27	schemes	scheme	NOUN
ejpam-6609	297	28	.	.	PUNCT
ejpam-6609	298	1	applying	apply	VERB
ejpam-6609	298	2	these	these	DET
ejpam-6609	298	3	improvements	improvement	NOUN
ejpam-6609	298	4	will	will	AUX
ejpam-6609	298	5	add	add	VERB
ejpam-6609	298	6	much	much	ADJ
ejpam-6609	298	7	strength	strength	NOUN
ejpam-6609	298	8	to	to	ADP
ejpam-6609	298	9	how	how	SCONJ
ejpam-6609	298	10	computations	computation	NOUN
ejpam-6609	298	11	are	be	AUX
ejpam-6609	298	12	carried	carry	VERB
ejpam-6609	298	13	out	out	ADP
ejpam-6609	298	14	in	in	ADP
ejpam-6609	298	15	theory	theory	NOUN
ejpam-6609	298	16	and	and	CCONJ
ejpam-6609	298	17	in	in	ADP
ejpam-6609	298	18	practical	practical	ADJ
ejpam-6609	298	19	circumstances	circumstance	NOUN
ejpam-6609	298	20	.	.	PUNCT
ejpam-6609	299	1	applications	application	NOUN
ejpam-6609	299	2	in	in	ADP
ejpam-6609	299	3	ai	ai	PROPN
ejpam-6609	299	4	and	and	CCONJ
ejpam-6609	299	5	machine	machine	NOUN
ejpam-6609	299	6	learning	learning	NOUN
ejpam-6609	299	7	:	:	PUNCT
ejpam-6609	299	8	studies	study	NOUN
ejpam-6609	299	9	should	should	AUX
ejpam-6609	299	10	continue	continue	VERB
ejpam-6609	299	11	to	to	PART
ejpam-6609	299	12	research	research	VERB
ejpam-6609	299	13	how	how	SCONJ
ejpam-6609	299	14	fixed	fix	VERB
ejpam-6609	299	15	-	-	PUNCT
ejpam-6609	299	16	point	point	NOUN
ejpam-6609	299	17	theory	theory	NOUN
ejpam-6609	299	18	affects	affect	VERB
ejpam-6609	299	19	artificial	artificial	ADJ
ejpam-6609	299	20	intelligence	intelligence	NOUN
ejpam-6609	299	21	,	,	PUNCT
ejpam-6609	299	22	most	most	ADV
ejpam-6609	299	23	notably	notably	ADV
ejpam-6609	299	24	in	in	ADP
ejpam-6609	299	25	understanding	understanding	NOUN
ejpam-6609	299	26	and	and	CCONJ
ejpam-6609	299	27	designing	design	VERB
ejpam-6609	299	28	stable	stable	ADJ
ejpam-6609	299	29	learning	learning	NOUN
ejpam-6609	299	30	designs	design	NOUN
ejpam-6609	299	31	.	.	PUNCT
ejpam-6609	300	1	more	more	ADJ
ejpam-6609	300	2	research	research	NOUN
ejpam-6609	300	3	can	can	AUX
ejpam-6609	300	4	look	look	VERB
ejpam-6609	300	5	into	into	ADP
ejpam-6609	300	6	when	when	SCONJ
ejpam-6609	300	7	deep	deep	ADJ
ejpam-6609	300	8	equilibrium	equilibrium	NOUN
ejpam-6609	300	9	models	model	NOUN
ejpam-6609	300	10	come	come	VERB
ejpam-6609	300	11	together	together	ADV
ejpam-6609	300	12	,	,	PUNCT
ejpam-6609	300	13	examine	examine	VERB
ejpam-6609	300	14	how	how	SCONJ
ejpam-6609	300	15	stable	stable	ADJ
ejpam-6609	300	16	complicated	complicated	ADJ
ejpam-6609	300	17	neural	neural	ADJ
ejpam-6609	300	18	systems	system	NOUN
ejpam-6609	300	19	are	be	AUX
ejpam-6609	300	20	,	,	PUNCT
ejpam-6609	300	21	and	and	CCONJ
ejpam-6609	300	22	see	see	VERB
ejpam-6609	300	23	if	if	SCONJ
ejpam-6609	300	24	fixed	fix	VERB
ejpam-6609	300	25	-	-	PUNCT
ejpam-6609	300	26	point	point	NOUN
ejpam-6609	300	27	results	result	NOUN
ejpam-6609	300	28	can	can	AUX
ejpam-6609	300	29	help	help	VERB
ejpam-6609	300	30	in	in	ADP
ejpam-6609	300	31	learning	learn	VERB
ejpam-6609	300	32	algorithms	algorithm	NOUN
ejpam-6609	300	33	,	,	PUNCT
ejpam-6609	300	34	reinforcement	reinforcement	NOUN
ejpam-6609	300	35	learning	learning	NOUN
ejpam-6609	300	36	,	,	PUNCT
ejpam-6609	300	37	and	and	CCONJ
ejpam-6609	300	38	adversarial	adversarial	ADJ
ejpam-6609	300	39	networks	network	NOUN
ejpam-6609	300	40	.	.	PUNCT
ejpam-6609	301	1	cryptographic	cryptographic	ADJ
ejpam-6609	301	2	and	and	CCONJ
ejpam-6609	301	3	algebraic	algebraic	ADJ
ejpam-6609	301	4	extensions	extension	NOUN
ejpam-6609	301	5	:	:	PUNCT
ejpam-6609	301	6	based	base	VERB
ejpam-6609	301	7	on	on	ADP
ejpam-6609	301	8	our	our	PRON
ejpam-6609	301	9	theory	theory	NOUN
ejpam-6609	301	10	,	,	PUNCT
ejpam-6609	301	11	we	we	PRON
ejpam-6609	301	12	can	can	AUX
ejpam-6609	301	13	suggest	suggest	VERB
ejpam-6609	301	14	fresh	fresh	ADJ
ejpam-6609	301	15	plans	plan	NOUN
ejpam-6609	301	16	for	for	ADP
ejpam-6609	301	17	constructing	construct	VERB
ejpam-6609	301	18	secure	secure	ADJ
ejpam-6609	301	19	lattice	lattice	NOUN
ejpam-6609	301	20	-	-	PUNCT
ejpam-6609	301	21	based	base	VERB
ejpam-6609	301	22	post	post	ADJ
ejpam-6609	301	23	-	-	ADJ
ejpam-6609	301	24	quantum	quantum	ADJ
ejpam-6609	301	25	protocols	protocol	NOUN
ejpam-6609	301	26	in	in	ADP
ejpam-6609	301	27	the	the	DET
ejpam-6609	301	28	world	world	NOUN
ejpam-6609	301	29	of	of	ADP
ejpam-6609	301	30	cryptography	cryptography	NOUN
ejpam-6609	301	31	.	.	PUNCT
ejpam-6609	302	1	as	as	SCONJ
ejpam-6609	302	2	time	time	NOUN
ejpam-6609	302	3	goes	go	VERB
ejpam-6609	302	4	on	on	ADP
ejpam-6609	302	5	,	,	PUNCT
ejpam-6609	302	6	we	we	PRON
ejpam-6609	302	7	could	could	AUX
ejpam-6609	302	8	work	work	VERB
ejpam-6609	302	9	on	on	ADP
ejpam-6609	302	10	new	new	ADJ
ejpam-6609	302	11	types	type	NOUN
ejpam-6609	302	12	of	of	ADP
ejpam-6609	302	13	algebra	algebra	NOUN
ejpam-6609	302	14	for	for	ADP
ejpam-6609	302	15	hardness	hardness	NOUN
ejpam-6609	302	16	,	,	PUNCT
ejpam-6609	302	17	construct	construct	VERB
ejpam-6609	302	18	security	security	NOUN
ejpam-6609	302	19	for	for	ADP
ejpam-6609	302	20	postquantum	postquantum	NOUN
ejpam-6609	302	21	systems	system	NOUN
ejpam-6609	302	22	,	,	PUNCT
ejpam-6609	302	23	and	and	CCONJ
ejpam-6609	302	24	develop	develop	VERB
ejpam-6609	302	25	primitives	primitive	NOUN
ejpam-6609	302	26	built	build	VERB
ejpam-6609	302	27	from	from	ADP
ejpam-6609	302	28	how	how	SCONJ
ejpam-6609	302	29	operator	operator	NOUN
ejpam-6609	302	30	systems	system	NOUN
ejpam-6609	302	31	behave	behave	VERB
ejpam-6609	302	32	when	when	SCONJ
ejpam-6609	302	33	they	they	PRON
ejpam-6609	302	34	reach	reach	VERB
ejpam-6609	302	35	fixed	fix	VERB
ejpam-6609	302	36	points	point	NOUN
ejpam-6609	302	37	in	in	ADP
ejpam-6609	302	38	an	an	DET
ejpam-6609	302	39	inner	inner	ADJ
ejpam-6609	302	40	product	product	NOUN
ejpam-6609	302	41	space	space	NOUN
ejpam-6609	302	42	.	.	PUNCT
ejpam-6609	303	1	acknowledgements	acknowledgement	NOUN
ejpam-6609	303	2	the	the	DET
ejpam-6609	303	3	project	project	NOUN
ejpam-6609	303	4	was	be	AUX
ejpam-6609	303	5	funded	fund	VERB
ejpam-6609	303	6	by	by	ADP
ejpam-6609	303	7	the	the	DET
ejpam-6609	303	8	kau	kau	PROPN
ejpam-6609	303	9	endowment	endowment	PROPN
ejpam-6609	303	10	(	(	PUNCT
ejpam-6609	303	11	waqf	waqf	NOUN
ejpam-6609	303	12	)	)	PUNCT
ejpam-6609	303	13	at	at	ADP
ejpam-6609	303	14	king	king	PROPN
ejpam-6609	303	15	abdulaziz	abdulaziz	PROPN
ejpam-6609	303	16	university	university	PROPN
ejpam-6609	303	17	,	,	PUNCT
ejpam-6609	303	18	jeddah	jeddah	PROPN
ejpam-6609	303	19	,	,	PUNCT
ejpam-6609	303	20	saudi	saudi	PROPN
ejpam-6609	303	21	arabia	arabia	PROPN
ejpam-6609	303	22	.	.	PUNCT
ejpam-6609	304	1	the	the	DET
ejpam-6609	304	2	authors	author	NOUN
ejpam-6609	304	3	,	,	PUNCT
ejpam-6609	304	4	therefore	therefore	ADV
ejpam-6609	304	5	,	,	PUNCT
ejpam-6609	304	6	acknowledge	acknowledge	VERB
ejpam-6609	304	7	waqf	waqf	NOUN
ejpam-6609	304	8	and	and	CCONJ
ejpam-6609	304	9	the	the	DET
ejpam-6609	304	10	deanship	deanship	NOUN
ejpam-6609	304	11	of	of	ADP
ejpam-6609	304	12	scientific	scientific	ADJ
ejpam-6609	304	13	research	research	NOUN
ejpam-6609	304	14	(	(	PUNCT
ejpam-6609	304	15	dsr	dsr	PROPN
ejpam-6609	304	16	)	)	PUNCT
ejpam-6609	304	17	for	for	ADP
ejpam-6609	304	18	technical	technical	ADJ
ejpam-6609	304	19	and	and	CCONJ
ejpam-6609	304	20	financial	financial	ADJ
ejpam-6609	304	21	support	support	NOUN
ejpam-6609	304	22	.	.	PUNCT
ejpam-6609	305	1	declarations	declaration	NOUN
ejpam-6609	305	2	author	author	NOUN
ejpam-6609	305	3	contributions	contribution	NOUN
ejpam-6609	305	4	:	:	PUNCT
ejpam-6609	305	5	all	all	DET
ejpam-6609	305	6	authors	author	NOUN
ejpam-6609	305	7	equally	equally	ADV
ejpam-6609	305	8	contributed	contribute	VERB
ejpam-6609	305	9	.	.	PUNCT
ejpam-6609	306	1	conflicts	conflict	NOUN
ejpam-6609	306	2	of	of	ADP
ejpam-6609	306	3	interest	interest	NOUN
ejpam-6609	306	4	:	:	PUNCT
ejpam-6609	306	5	the	the	DET
ejpam-6609	306	6	authors	author	NOUN
ejpam-6609	306	7	declare	declare	VERB
ejpam-6609	306	8	no	no	DET
ejpam-6609	306	9	conflict	conflict	NOUN
ejpam-6609	306	10	of	of	ADP
ejpam-6609	306	11	interest	interest	NOUN
ejpam-6609	306	12	.	.	PUNCT
ejpam-6609	307	1	data	datum	NOUN
ejpam-6609	307	2	availability	availability	NOUN
ejpam-6609	307	3	:	:	PUNCT
ejpam-6609	307	4	all	all	DET
ejpam-6609	307	5	the	the	DET
ejpam-6609	307	6	data	datum	NOUN
ejpam-6609	307	7	is	be	AUX
ejpam-6609	307	8	provided	provide	VERB
ejpam-6609	307	9	in	in	ADP
ejpam-6609	307	10	the	the	DET
ejpam-6609	307	11	manuscript	manuscript	NOUN
ejpam-6609	307	12	.	.	PUNCT
ejpam-6609	308	1	references	reference	NOUN
ejpam-6609	308	2	[	[	X
ejpam-6609	308	3	1	1	X
ejpam-6609	308	4	]	]	PUNCT
ejpam-6609	308	5	e.	e.	PROPN
ejpam-6609	308	6	karapinar	karapinar	PROPN
ejpam-6609	308	7	and	and	CCONJ
ejpam-6609	308	8	r.	r.	PROPN
ejpam-6609	308	9	p.	p.	PROPN
ejpam-6609	308	10	agarwal	agarwal	PROPN
ejpam-6609	308	11	.	.	PUNCT
ejpam-6609	309	1	fixed	fix	VERB
ejpam-6609	309	2	point	point	NOUN
ejpam-6609	309	3	theory	theory	NOUN
ejpam-6609	309	4	in	in	ADP
ejpam-6609	309	5	generalized	generalized	ADJ
ejpam-6609	309	6	metric	metric	ADJ
ejpam-6609	309	7	spaces	space	NOUN
ejpam-6609	309	8	.	.	PUNCT
ejpam-6609	310	1	springer	springer	NOUN
ejpam-6609	310	2	,	,	PUNCT
ejpam-6609	310	3	2022	2022	NUM
ejpam-6609	310	4	.	.	PUNCT
ejpam-6609	311	1	[	[	X
ejpam-6609	311	2	2	2	X
ejpam-6609	311	3	]	]	PUNCT
ejpam-6609	311	4	v.	v.	PROPN
ejpam-6609	311	5	i.	i.	PROPN
ejpam-6609	311	6	istratescu	istratescu	PROPN
ejpam-6609	311	7	.	.	PUNCT
ejpam-6609	312	1	fixed	fix	VERB
ejpam-6609	312	2	point	point	NOUN
ejpam-6609	312	3	theory	theory	NOUN
ejpam-6609	312	4	:	:	PUNCT
ejpam-6609	312	5	an	an	DET
ejpam-6609	312	6	introduction	introduction	NOUN
ejpam-6609	312	7	,	,	PUNCT
ejpam-6609	312	8	volume	volume	NOUN
ejpam-6609	312	9	7	7	NUM
ejpam-6609	312	10	of	of	ADP
ejpam-6609	312	11	mathematics	mathematic	NOUN
ejpam-6609	312	12	and	and	CCONJ
ejpam-6609	312	13	its	its	PRON
ejpam-6609	312	14	applications	application	NOUN
ejpam-6609	312	15	.	.	PUNCT
ejpam-6609	313	1	springer	springer	NOUN
ejpam-6609	313	2	science	science	PROPN
ejpam-6609	313	3	&	&	CCONJ
ejpam-6609	313	4	business	business	NOUN
ejpam-6609	313	5	media	medium	NOUN
ejpam-6609	313	6	,	,	PUNCT
ejpam-6609	313	7	2001	2001	NUM
ejpam-6609	313	8	.	.	PUNCT
ejpam-6609	314	1	[	[	X
ejpam-6609	314	2	3	3	X
ejpam-6609	314	3	]	]	PUNCT
ejpam-6609	314	4	k.	k.	PROPN
ejpam-6609	314	5	ullah	ullah	PROPN
ejpam-6609	314	6	,	,	PUNCT
ejpam-6609	314	7	m.	m.	NOUN
ejpam-6609	314	8	s.	s.	PROPN
ejpam-6609	314	9	u.	u.	PROPN
ejpam-6609	314	10	khan	khan	PROPN
ejpam-6609	314	11	,	,	PUNCT
ejpam-6609	314	12	and	and	CCONJ
ejpam-6609	314	13	m.	m.	PROPN
ejpam-6609	314	14	de	de	PROPN
ejpam-6609	314	15	la	la	PROPN
ejpam-6609	314	16	sen	sen	PROPN
ejpam-6609	314	17	.	.	PROPN
ejpam-6609	314	18	fixed	fix	VERB
ejpam-6609	314	19	point	point	NOUN
ejpam-6609	314	20	results	result	NOUN
ejpam-6609	314	21	on	on	ADP
ejpam-6609	314	22	multi	multi	ADJ
ejpam-6609	314	23	-	-	ADJ
ejpam-6609	314	24	valued	value	VERB
ejpam-6609	314	25	generalized	generalize	VERB
ejpam-6609	314	26	(	(	PUNCT
ejpam-6609	314	27	α	α	NOUN
ejpam-6609	314	28	,	,	PUNCT
ejpam-6609	314	29	β)-nonexpansive	β)-nonexpansive	PUNCT
ejpam-6609	314	30	mappings	mapping	NOUN
ejpam-6609	314	31	in	in	ADP
ejpam-6609	314	32	banach	banach	NOUN
ejpam-6609	314	33	spaces	space	NOUN
ejpam-6609	314	34	.	.	PUNCT
ejpam-6609	315	1	algorithms	algorithm	NOUN
ejpam-6609	315	2	,	,	PUNCT
ejpam-6609	315	3	14(8):223	14(8):223	NUM
ejpam-6609	315	4	,	,	PUNCT
ejpam-6609	315	5	2021	2021	NUM
ejpam-6609	315	6	.	.	PUNCT
ejpam-6609	316	1	[	[	X
ejpam-6609	316	2	4	4	X
ejpam-6609	316	3	]	]	PUNCT
ejpam-6609	316	4	a.	a.	NOUN
ejpam-6609	316	5	abou	abou	PROPN
ejpam-6609	316	6	bakr	bakr	PROPN
ejpam-6609	316	7	and	and	CCONJ
ejpam-6609	316	8	s.	s.	PROPN
ejpam-6609	316	9	mohamed	mohamed	PROPN
ejpam-6609	316	10	.	.	PROPN
ejpam-6609	317	1	on	on	ADP
ejpam-6609	317	2	various	various	ADJ
ejpam-6609	317	3	types	type	NOUN
ejpam-6609	317	4	of	of	ADP
ejpam-6609	317	5	cone	cone	NOUN
ejpam-6609	317	6	metric	metric	ADJ
ejpam-6609	317	7	spaces	space	NOUN
ejpam-6609	317	8	and	and	CCONJ
ejpam-6609	317	9	some	some	DET
ejpam-6609	317	10	applications	application	NOUN
ejpam-6609	317	11	in	in	ADP
ejpam-6609	317	12	fixed	fix	VERB
ejpam-6609	317	13	point	point	NOUN
ejpam-6609	317	14	theory	theory	NOUN
ejpam-6609	317	15	.	.	PUNCT
ejpam-6609	318	1	international	international	ADJ
ejpam-6609	318	2	journal	journal	PROPN
ejpam-6609	318	3	of	of	ADP
ejpam-6609	318	4	nonlinear	nonlinear	ADJ
ejpam-6609	318	5	analysis	analysis	NOUN
ejpam-6609	318	6	and	and	CCONJ
ejpam-6609	318	7	applications	application	NOUN
ejpam-6609	318	8	,	,	PUNCT
ejpam-6609	318	9	14(1):163–184	14(1):163–184	NUM
ejpam-6609	318	10	,	,	PUNCT
ejpam-6609	318	11	2023	2023	NUM
ejpam-6609	318	12	.	.	PUNCT
ejpam-6609	319	1	[	[	X
ejpam-6609	319	2	5	5	X
ejpam-6609	319	3	]	]	PUNCT
ejpam-6609	319	4	s.	s.	PROPN
ejpam-6609	319	5	grailoo	grailoo	PROPN
ejpam-6609	319	6	,	,	PUNCT
ejpam-6609	319	7	a.	a.	PROPN
ejpam-6609	319	8	bodaghi	bodaghi	PROPN
ejpam-6609	319	9	,	,	PUNCT
ejpam-6609	319	10	and	and	CCONJ
ejpam-6609	319	11	a.	a.	NOUN
ejpam-6609	319	12	n.	n.	PROPN
ejpam-6609	319	13	motlagh	motlagh	PROPN
ejpam-6609	319	14	.	.	PUNCT
ejpam-6609	320	1	some	some	DET
ejpam-6609	320	2	results	result	NOUN
ejpam-6609	320	3	in	in	ADP
ejpam-6609	320	4	metric	metric	ADJ
ejpam-6609	320	5	modular	modular	ADJ
ejpam-6609	320	6	spaces	space	NOUN
ejpam-6609	320	7	.	.	PUNCT
ejpam-6609	321	1	international	international	ADJ
ejpam-6609	321	2	journal	journal	PROPN
ejpam-6609	321	3	of	of	ADP
ejpam-6609	321	4	nonlinear	nonlinear	ADJ
ejpam-6609	321	5	analysis	analysis	NOUN
ejpam-6609	321	6	and	and	CCONJ
ejpam-6609	321	7	applications	application	NOUN
ejpam-6609	321	8	,	,	PUNCT
ejpam-6609	321	9	13(2):983–988	13(2):983–988	NUM
ejpam-6609	321	10	,	,	PUNCT
ejpam-6609	321	11	2022	2022	NUM
ejpam-6609	321	12	.	.	PUNCT
ejpam-6609	322	1	[	[	X
ejpam-6609	322	2	6	6	NUM
ejpam-6609	322	3	]	]	PUNCT
ejpam-6609	322	4	m.	m.	NOUN
ejpam-6609	322	5	noorwali	noorwali	PROPN
ejpam-6609	322	6	,	,	PUNCT
ejpam-6609	322	7	r.	r.	PROPN
ejpam-6609	322	8	hatamleh	hatamleh	PROPN
ejpam-6609	322	9	,	,	PUNCT
ejpam-6609	322	10	a.	a.	NOUN
ejpam-6609	322	11	a.	a.	NOUN
ejpam-6609	322	12	abubaker	abubaker	PROPN
ejpam-6609	322	13	,	,	PUNCT
ejpam-6609	322	14	a.	a.	PROPN
ejpam-6609	322	15	al	al	PROPN
ejpam-6609	322	16	-	-	PUNCT
ejpam-6609	322	17	husban	husban	PROPN
ejpam-6609	322	18	,	,	PUNCT
ejpam-6609	322	19	j.	j.	PROPN
ejpam-6609	322	20	j.	j.	PROPN
ejpam-6609	322	21	hamja	hamja	PROPN
ejpam-6609	322	22	,	,	PUNCT
ejpam-6609	322	23	g.	g.	PROPN
ejpam-6609	322	24	nordo	nordo	PROPN
ejpam-6609	322	25	,	,	PUNCT
ejpam-6609	322	26	t.	t.	PROPN
ejpam-6609	322	27	fujita	fujita	PROPN
ejpam-6609	322	28	,	,	PUNCT
ejpam-6609	322	29	and	and	CCONJ
ejpam-6609	322	30	a.	a.	PROPN
ejpam-6609	322	31	m.	m.	PROPN
ejpam-6609	322	32	khattak	khattak	PROPN
ejpam-6609	322	33	.	.	PUNCT
ejpam-6609	322	34	characterization	characterization	NOUN
ejpam-6609	322	35	and	and	CCONJ
ejpam-6609	322	36	application	application	NOUN
ejpam-6609	322	37	of	of	ADP
ejpam-6609	322	38	fixed	fix	VERB
ejpam-6609	322	39	point	point	NOUN
ejpam-6609	322	40	theorems	theorem	NOUN
ejpam-6609	322	41	for	for	ADP
ejpam-6609	322	42	3	3	NUM
ejpam-6609	322	43	-	-	PUNCT
ejpam-6609	322	44	self	self	NOUN
ejpam-6609	322	45	mappings	mapping	NOUN
ejpam-6609	322	46	in	in	ADP
ejpam-6609	322	47	generalized	generalized	ADJ
ejpam-6609	322	48	metric	metric	ADJ
ejpam-6609	322	49	spaces	space	NOUN
ejpam-6609	322	50	.	.	PUNCT
ejpam-6609	323	1	european	european	ADJ
ejpam-6609	323	2	journal	journal	PROPN
ejpam-6609	323	3	of	of	ADP
ejpam-6609	323	4	pure	pure	ADJ
ejpam-6609	323	5	and	and	CCONJ
ejpam-6609	323	6	applied	applied	ADJ
ejpam-6609	323	7	mathematics	mathematic	NOUN
ejpam-6609	323	8	,	,	PUNCT
ejpam-6609	323	9	18(2):5838	18(2):5838	NUM
ejpam-6609	323	10	,	,	PUNCT
ejpam-6609	323	11	2025	2025	NUM
ejpam-6609	323	12	.	.	PUNCT
ejpam-6609	324	1	[	[	X
ejpam-6609	324	2	7	7	X
ejpam-6609	324	3	]	]	X
ejpam-6609	324	4	v.	v.	ADP
ejpam-6609	324	5	berinde	berinde	NOUN
ejpam-6609	324	6	and	and	CCONJ
ejpam-6609	324	7	m.	m.	NOUN
ejpam-6609	324	8	păcurar	păcurar	NOUN
ejpam-6609	324	9	.	.	PUNCT
ejpam-6609	325	1	recent	recent	ADJ
ejpam-6609	325	2	developments	development	NOUN
ejpam-6609	325	3	in	in	ADP
ejpam-6609	325	4	the	the	DET
ejpam-6609	325	5	fixed	fix	VERB
ejpam-6609	325	6	point	point	NOUN
ejpam-6609	325	7	theory	theory	NOUN
ejpam-6609	325	8	of	of	ADP
ejpam-6609	325	9	enriched	enrich	VERB
ejpam-6609	325	10	contractive	contractive	ADJ
ejpam-6609	325	11	mappings	mapping	NOUN
ejpam-6609	325	12	.	.	PUNCT
ejpam-6609	326	1	a	a	DET
ejpam-6609	326	2	survey	survey	NOUN
ejpam-6609	326	3	.	.	PUNCT
ejpam-6609	327	1	arxiv	arxiv	PROPN
ejpam-6609	327	2	preprint	preprint	PROPN
ejpam-6609	327	3	,	,	PUNCT
ejpam-6609	327	4	2024	2024	NUM
ejpam-6609	327	5	.	.	PUNCT
ejpam-6609	328	1	arxiv:2404.04928	arxiv:2404.04928	PROPN
ejpam-6609	328	2	.	.	PUNCT
ejpam-6609	329	1	[	[	X
ejpam-6609	329	2	8	8	X
ejpam-6609	329	3	]	]	PUNCT
ejpam-6609	329	4	t.	t.	PROPN
ejpam-6609	329	5	j.	j.	PROPN
ejpam-6609	329	6	piotrowski	piotrowski	PROPN
ejpam-6609	329	7	,	,	PUNCT
ejpam-6609	329	8	r.	r.	PROPN
ejpam-6609	329	9	l.	l.	PROPN
ejpam-6609	329	10	g.	g.	PROPN
ejpam-6609	329	11	cavalcante	cavalcante	PROPN
ejpam-6609	329	12	,	,	PUNCT
ejpam-6609	329	13	and	and	CCONJ
ejpam-6609	329	14	m.	m.	NOUN
ejpam-6609	329	15	gabor	gabor	PROPN
ejpam-6609	329	16	.	.	PUNCT
ejpam-6609	330	1	fixed	fix	VERB
ejpam-6609	330	2	points	point	NOUN
ejpam-6609	330	3	of	of	ADP
ejpam-6609	330	4	nonnegative	nonnegative	ADJ
ejpam-6609	330	5	neural	neural	ADJ
ejpam-6609	330	6	networks	network	NOUN
ejpam-6609	330	7	.	.	PUNCT
ejpam-6609	331	1	journal	journal	NOUN
ejpam-6609	331	2	of	of	ADP
ejpam-6609	331	3	machine	machine	NOUN
ejpam-6609	331	4	learning	learn	VERB
ejpam-6609	331	5	research	research	NOUN
ejpam-6609	331	6	,	,	PUNCT
ejpam-6609	331	7	25(139):1–40	25(139):1–40	PROPN
ejpam-6609	331	8	,	,	PUNCT
ejpam-6609	331	9	2024	2024	NUM
ejpam-6609	331	10	.	.	PUNCT
ejpam-6609	332	1	m.	m.	PROPN
ejpam-6609	332	2	noorwali	noorwali	PROPN
ejpam-6609	332	3	et	et	PROPN
ejpam-6609	332	4	al	al	PROPN
ejpam-6609	332	5	.	.	PUNCT
ejpam-6609	332	6	/	/	SYM
ejpam-6609	332	7	eur	eur	PROPN
ejpam-6609	332	8	.	.	PUNCT
ejpam-6609	333	1	j.	j.	PROPN
ejpam-6609	333	2	pure	pure	PROPN
ejpam-6609	333	3	appl	appl	PROPN
ejpam-6609	333	4	.	.	PROPN
ejpam-6609	333	5	math	math	PROPN
ejpam-6609	333	6	,	,	PUNCT
ejpam-6609	333	7	18	18	NUM
ejpam-6609	333	8	(	(	PUNCT
ejpam-6609	333	9	3	3	NUM
ejpam-6609	333	10	)	)	PUNCT
ejpam-6609	333	11	(	(	PUNCT
ejpam-6609	333	12	2025	2025	NUM
ejpam-6609	333	13	)	)	PUNCT
ejpam-6609	333	14	,	,	PUNCT
ejpam-6609	333	15	6609	6609	NUM
ejpam-6609	333	16	17	17	NUM
ejpam-6609	333	17	of	of	ADP
ejpam-6609	333	18	17	17	NUM
ejpam-6609	334	1	[	[	X
ejpam-6609	334	2	9	9	NUM
ejpam-6609	334	3	]	]	PUNCT
ejpam-6609	334	4	v.	v.	CCONJ
ejpam-6609	334	5	goudar	goudar	PROPN
ejpam-6609	334	6	,	,	PUNCT
ejpam-6609	334	7	b.	b.	PROPN
ejpam-6609	334	8	peysakhovich	peysakhovich	PROPN
ejpam-6609	334	9	,	,	PUNCT
ejpam-6609	334	10	d.	d.	PROPN
ejpam-6609	334	11	j.	j.	PROPN
ejpam-6609	334	12	freedman	freedman	PROPN
ejpam-6609	334	13	,	,	PUNCT
ejpam-6609	334	14	e.	e.	PROPN
ejpam-6609	334	15	a.	a.	PROPN
ejpam-6609	334	16	buffalo	buffalo	PROPN
ejpam-6609	334	17	,	,	PUNCT
ejpam-6609	334	18	and	and	CCONJ
ejpam-6609	334	19	x.-j	x.-j	PROPN
ejpam-6609	334	20	.	.	PUNCT
ejpam-6609	335	1	wang	wang	PROPN
ejpam-6609	335	2	.	.	PUNCT
ejpam-6609	336	1	schema	schema	VERB
ejpam-6609	336	2	formation	formation	NOUN
ejpam-6609	336	3	in	in	ADP
ejpam-6609	336	4	a	a	DET
ejpam-6609	336	5	neural	neural	ADJ
ejpam-6609	336	6	population	population	NOUN
ejpam-6609	336	7	subspace	subspace	NOUN
ejpam-6609	336	8	underlies	underlie	VERB
ejpam-6609	336	9	learning	learn	VERB
ejpam-6609	336	10	-	-	PUNCT
ejpam-6609	336	11	to	to	PART
ejpam-6609	336	12	-	-	PUNCT
ejpam-6609	336	13	learn	learn	VERB
ejpam-6609	336	14	in	in	ADP
ejpam-6609	336	15	flexible	flexible	ADJ
ejpam-6609	336	16	sensorimotor	sensorimotor	NOUN
ejpam-6609	336	17	problem	problem	NOUN
ejpam-6609	336	18	-	-	PUNCT
ejpam-6609	336	19	solving	solving	NOUN
ejpam-6609	336	20	.	.	PUNCT
ejpam-6609	337	1	nature	nature	NOUN
ejpam-6609	337	2	neuroscience	neuroscience	NOUN
ejpam-6609	337	3	,	,	PUNCT
ejpam-6609	337	4	26(5):879–890	26(5):879–890	NOUN
ejpam-6609	337	5	,	,	PUNCT
ejpam-6609	337	6	2023	2023	NUM
ejpam-6609	337	7	.	.	PUNCT
ejpam-6609	338	1	[	[	X
ejpam-6609	338	2	10	10	NUM
ejpam-6609	338	3	]	]	PUNCT
ejpam-6609	338	4	k.	k.	PROPN
ejpam-6609	338	5	abbas	abbas	PROPN
ejpam-6609	338	6	.	.	PUNCT
ejpam-6609	339	1	from	from	ADP
ejpam-6609	339	2	algorithms	algorithm	NOUN
ejpam-6609	339	3	to	to	ADP
ejpam-6609	339	4	hardware	hardware	NOUN
ejpam-6609	339	5	architectures	architecture	NOUN
ejpam-6609	339	6	.	.	PUNCT
ejpam-6609	340	1	springer	springer	NOUN
ejpam-6609	340	2	,	,	PUNCT
ejpam-6609	340	3	(	(	PUNCT
ejpam-6609	340	4	year	year	NOUN
ejpam-6609	340	5	)	)	PUNCT
ejpam-6609	340	6	.	.	PUNCT
ejpam-6609	341	1	[	[	X
ejpam-6609	341	2	11	11	NUM
ejpam-6609	341	3	]	]	PUNCT
ejpam-6609	341	4	e.	e.	PROPN
ejpam-6609	341	5	e.	e.	PROPN
ejpam-6609	341	6	berdysheva	berdysheva	PROPN
ejpam-6609	341	7	,	,	PUNCT
ejpam-6609	341	8	n.	n.	PROPN
ejpam-6609	341	9	dyn	dyn	PROPN
ejpam-6609	341	10	,	,	PUNCT
ejpam-6609	341	11	e.	e.	PROPN
ejpam-6609	341	12	farkhi	farkhi	PROPN
ejpam-6609	341	13	,	,	PUNCT
ejpam-6609	341	14	and	and	CCONJ
ejpam-6609	341	15	a.	a.	NOUN
ejpam-6609	341	16	mokhov	mokhov	NOUN
ejpam-6609	341	17	.	.	PUNCT
ejpam-6609	342	1	metric	metric	ADJ
ejpam-6609	342	2	approximation	approximation	NOUN
ejpam-6609	342	3	of	of	ADP
ejpam-6609	342	4	set	set	NOUN
ejpam-6609	342	5	-	-	PUNCT
ejpam-6609	342	6	valued	value	VERB
ejpam-6609	342	7	functions	function	NOUN
ejpam-6609	342	8	of	of	ADP
ejpam-6609	342	9	bounded	bounded	ADJ
ejpam-6609	342	10	variation	variation	NOUN
ejpam-6609	342	11	by	by	ADP
ejpam-6609	342	12	integral	integral	ADJ
ejpam-6609	342	13	operators	operator	NOUN
ejpam-6609	342	14	.	.	PUNCT
ejpam-6609	343	1	constructive	constructive	ADJ
ejpam-6609	343	2	approximation	approximation	NOUN
ejpam-6609	343	3	,	,	PUNCT
ejpam-6609	343	4	pages	page	NOUN
ejpam-6609	343	5	1–31	1–31	PROPN
ejpam-6609	343	6	,	,	PUNCT
ejpam-6609	343	7	2024	2024	NUM
ejpam-6609	343	8	.	.	PUNCT
ejpam-6609	344	1	[	[	X
ejpam-6609	344	2	12	12	NUM
ejpam-6609	344	3	]	]	PUNCT
ejpam-6609	345	1	s.	s.	PROPN
ejpam-6609	345	2	k.	k.	PROPN
ejpam-6609	345	3	datta	datta	PROPN
ejpam-6609	345	4	,	,	PUNCT
ejpam-6609	345	5	r.	r.	PROPN
ejpam-6609	345	6	sarka	sarka	PROPN
ejpam-6609	345	7	,	,	PUNCT
ejpam-6609	345	8	n.	n.	PROPN
ejpam-6609	345	9	biswas	biswas	PROPN
ejpam-6609	345	10	,	,	PUNCT
ejpam-6609	345	11	and	and	CCONJ
ejpam-6609	345	12	j.	j.	PROPN
ejpam-6609	345	13	saha	saha	PROPN
ejpam-6609	345	14	.	.	PUNCT
ejpam-6609	346	1	common	common	ADJ
ejpam-6609	346	2	fixed	fix	VERB
ejpam-6609	346	3	point	point	NOUN
ejpam-6609	346	4	theorems	theorem	NOUN
ejpam-6609	346	5	for	for	ADP
ejpam-6609	346	6	three	three	NUM
ejpam-6609	346	7	self	self	NOUN
ejpam-6609	346	8	mappings	mapping	NOUN
ejpam-6609	346	9	in	in	ADP
ejpam-6609	346	10	bicomplex	bicomplex	NOUN
ejpam-6609	346	11	valued	value	VERB
ejpam-6609	346	12	metric	metric	ADJ
ejpam-6609	346	13	spaces	space	NOUN
ejpam-6609	346	14	.	.	PUNCT
ejpam-6609	347	1	south	south	PROPN
ejpam-6609	347	2	east	east	PROPN
ejpam-6609	347	3	asian	asian	PROPN
ejpam-6609	347	4	journal	journal	PROPN
ejpam-6609	347	5	of	of	ADP
ejpam-6609	347	6	mathematics	mathematics	PROPN
ejpam-6609	347	7	&	&	CCONJ
ejpam-6609	347	8	mathematical	mathematical	PROPN
ejpam-6609	347	9	sciences	sciences	PROPN
ejpam-6609	347	10	,	,	PUNCT
ejpam-6609	347	11	17(1	17(1	NUM
ejpam-6609	347	12	)	)	PUNCT
ejpam-6609	347	13	,	,	PUNCT
ejpam-6609	347	14	2021	2021	NUM
ejpam-6609	347	15	.	.	PUNCT
ejpam-6609	348	1	[	[	X
ejpam-6609	348	2	13	13	NUM
ejpam-6609	348	3	]	]	PUNCT
ejpam-6609	348	4	m.	m.	NOUN
ejpam-6609	348	5	a.	a.	PROPN
ejpam-6609	348	6	dritschel	dritschel	PROPN
ejpam-6609	348	7	and	and	CCONJ
ejpam-6609	348	8	j.	j.	PROPN
ejpam-6609	348	9	rovnyak	rovnyak	PROPN
ejpam-6609	348	10	.	.	PUNCT
ejpam-6609	349	1	operators	operator	NOUN
ejpam-6609	349	2	on	on	ADP
ejpam-6609	349	3	indefinite	indefinite	ADJ
ejpam-6609	349	4	inner	inner	ADJ
ejpam-6609	349	5	product	product	NOUN
ejpam-6609	349	6	spaces	space	NOUN
ejpam-6609	349	7	.	.	PUNCT
ejpam-6609	350	1	lectures	lecture	NOUN
ejpam-6609	350	2	on	on	ADP
ejpam-6609	350	3	operator	operator	NOUN
ejpam-6609	350	4	theory	theory	NOUN
ejpam-6609	350	5	and	and	CCONJ
ejpam-6609	350	6	its	its	PRON
ejpam-6609	350	7	applications	application	NOUN
ejpam-6609	350	8	,	,	PUNCT
ejpam-6609	350	9	3:141–232	3:141–232	NUM
ejpam-6609	350	10	,	,	PUNCT
ejpam-6609	350	11	1996	1996	NUM
ejpam-6609	350	12	.	.	PUNCT
ejpam-6609	351	1	[	[	X
ejpam-6609	351	2	14	14	NUM
ejpam-6609	351	3	]	]	X
ejpam-6609	351	4	e.	e.	PROPN
ejpam-6609	351	5	prugovecki	prugovecki	PROPN
ejpam-6609	351	6	.	.	PUNCT
ejpam-6609	352	1	topologies	topology	NOUN
ejpam-6609	352	2	on	on	ADP
ejpam-6609	352	3	generalized	generalized	ADJ
ejpam-6609	352	4	inner	inner	ADJ
ejpam-6609	352	5	product	product	NOUN
ejpam-6609	352	6	spaces	space	NOUN
ejpam-6609	352	7	.	.	PUNCT
ejpam-6609	353	1	canadian	canadian	ADJ
ejpam-6609	353	2	journal	journal	PROPN
ejpam-6609	353	3	of	of	ADP
ejpam-6609	353	4	mathematics	mathematics	PROPN
ejpam-6609	353	5	,	,	PUNCT
ejpam-6609	353	6	21:158–169	21:158–169	PROPN
ejpam-6609	353	7	,	,	PUNCT
ejpam-6609	353	8	1969	1969	NUM
ejpam-6609	353	9	.	.	PUNCT
ejpam-6609	354	1	[	[	X
ejpam-6609	354	2	15	15	NUM
ejpam-6609	354	3	]	]	X
ejpam-6609	354	4	m.	m.	PROPN
ejpam-6609	354	5	e.	e.	PROPN
ejpam-6609	354	6	gordji	gordji	PROPN
ejpam-6609	354	7	and	and	CCONJ
ejpam-6609	354	8	h.	h.	PROPN
ejpam-6609	354	9	khodaei	khodaei	PROPN
ejpam-6609	354	10	.	.	PUNCT
ejpam-6609	355	1	the	the	DET
ejpam-6609	355	2	fixed	fix	VERB
ejpam-6609	355	3	point	point	NOUN
ejpam-6609	355	4	method	method	NOUN
ejpam-6609	355	5	for	for	ADP
ejpam-6609	355	6	fuzzy	fuzzy	ADJ
ejpam-6609	355	7	approximation	approximation	NOUN
ejpam-6609	355	8	of	of	ADP
ejpam-6609	355	9	a	a	DET
ejpam-6609	355	10	functional	functional	ADJ
ejpam-6609	355	11	equation	equation	NOUN
ejpam-6609	355	12	associated	associate	VERB
ejpam-6609	355	13	with	with	ADP
ejpam-6609	355	14	inner	inner	ADJ
ejpam-6609	355	15	product	product	NOUN
ejpam-6609	355	16	spaces	space	VERB
ejpam-6609	355	17	.	.	PUNCT
ejpam-6609	356	1	discrete	discrete	ADJ
ejpam-6609	356	2	dynamics	dynamic	NOUN
ejpam-6609	356	3	in	in	ADP
ejpam-6609	356	4	nature	nature	NOUN
ejpam-6609	356	5	and	and	CCONJ
ejpam-6609	356	6	society	society	NOUN
ejpam-6609	356	7	,	,	PUNCT
ejpam-6609	356	8	2010(1):140767	2010(1):140767	NOUN
ejpam-6609	356	9	,	,	PUNCT
ejpam-6609	356	10	2010	2010	NUM
ejpam-6609	356	11	.	.	PUNCT
ejpam-6609	357	1	[	[	X
ejpam-6609	357	2	16	16	NUM
ejpam-6609	357	3	]	]	X
ejpam-6609	357	4	h.	h.	PROPN
ejpam-6609	357	5	chu	chu	PROPN
ejpam-6609	357	6	,	,	PUNCT
ejpam-6609	357	7	s.	s.	PROPN
ejpam-6609	357	8	wei	wei	PROPN
ejpam-6609	357	9	,	,	PUNCT
ejpam-6609	357	10	t.	t.	PROPN
ejpam-6609	357	11	liu	liu	PROPN
ejpam-6609	357	12	,	,	PUNCT
ejpam-6609	357	13	y.	y.	PROPN
ejpam-6609	357	14	zhao	zhao	PROPN
ejpam-6609	357	15	,	,	PUNCT
ejpam-6609	357	16	and	and	CCONJ
ejpam-6609	357	17	y.	y.	PROPN
ejpam-6609	357	18	miyatake	miyatake	NOUN
ejpam-6609	357	19	.	.	PUNCT
ejpam-6609	358	1	lyapunov	lyapunov	NOUN
ejpam-6609	358	2	-	-	PUNCT
ejpam-6609	358	3	stable	stable	ADJ
ejpam-6609	358	4	deep	deep	ADJ
ejpam-6609	358	5	equilibrium	equilibrium	NOUN
ejpam-6609	358	6	models	model	NOUN
ejpam-6609	358	7	.	.	PUNCT
ejpam-6609	359	1	proceedings	proceeding	NOUN
ejpam-6609	359	2	of	of	ADP
ejpam-6609	359	3	the	the	DET
ejpam-6609	359	4	aaai	aaai	PROPN
ejpam-6609	359	5	conference	conference	NOUN
ejpam-6609	359	6	on	on	ADP
ejpam-6609	359	7	artificial	artificial	ADJ
ejpam-6609	359	8	intelligence	intelligence	NOUN
ejpam-6609	359	9	,	,	PUNCT
ejpam-6609	359	10	38(10):11615–11623	38(10):11615–11623	NUM
ejpam-6609	359	11	,	,	PUNCT
ejpam-6609	359	12	2024	2024	NUM
ejpam-6609	359	13	.	.	PUNCT
ejpam-6609	360	1	[	[	X
ejpam-6609	360	2	17	17	NUM
ejpam-6609	360	3	]	]	PUNCT
ejpam-6609	360	4	j.	j.	PROPN
ejpam-6609	360	5	r.	r.	PROPN
ejpam-6609	360	6	troncoso	troncoso	PROPN
ejpam-6609	360	7	-	-	PUNCT
ejpam-6609	360	8	pastoriza	pastoriza	NOUN
ejpam-6609	360	9	,	,	PUNCT
ejpam-6609	360	10	p.	p.	PROPN
ejpam-6609	360	11	comesaña	comesaña	PROPN
ejpam-6609	360	12	,	,	PUNCT
ejpam-6609	360	13	and	and	CCONJ
ejpam-6609	360	14	f.	f.	PROPN
ejpam-6609	360	15	pérez	pérez	PROPN
ejpam-6609	360	16	-	-	PUNCT
ejpam-6609	360	17	gonzález	gonzález	NOUN
ejpam-6609	360	18	.	.	PUNCT
ejpam-6609	360	19	secure	secure	VERB
ejpam-6609	360	20	direct	direct	ADJ
ejpam-6609	360	21	and	and	CCONJ
ejpam-6609	360	22	iterative	iterative	ADJ
ejpam-6609	360	23	protocols	protocol	NOUN
ejpam-6609	360	24	for	for	ADP
ejpam-6609	360	25	solving	solve	VERB
ejpam-6609	360	26	systems	system	NOUN
ejpam-6609	360	27	of	of	ADP
ejpam-6609	360	28	linear	linear	PROPN
ejpam-6609	360	29	equations	equation	NOUN
ejpam-6609	360	30	.	.	PUNCT
ejpam-6609	361	1	proceedings	proceeding	NOUN
ejpam-6609	361	2	of	of	ADP
ejpam-6609	361	3	1st	1st	ADJ
ejpam-6609	361	4	international	international	ADJ
ejpam-6609	361	5	workshop	workshop	NOUN
ejpam-6609	361	6	on	on	ADP
ejpam-6609	361	7	signal	signal	ADJ
ejpam-6609	361	8	processing	processing	NOUN
ejpam-6609	361	9	in	in	ADP
ejpam-6609	361	10	the	the	DET
ejpam-6609	361	11	encrypted	encrypt	VERB
ejpam-6609	361	12	domain	domain	NOUN
ejpam-6609	361	13	(	(	PUNCT
ejpam-6609	361	14	speed	speed	NOUN
ejpam-6609	361	15	)	)	PUNCT
ejpam-6609	361	16	,	,	PUNCT
ejpam-6609	361	17	pages	page	NOUN
ejpam-6609	361	18	122–141	122–141	NUM
ejpam-6609	361	19	,	,	PUNCT
ejpam-6609	361	20	2009	2009	NUM
ejpam-6609	361	21	.	.	PUNCT
ejpam-6609	362	1	[	[	X
ejpam-6609	362	2	18	18	NUM
ejpam-6609	362	3	]	]	X
ejpam-6609	362	4	c.	c.	PROPN
ejpam-6609	362	5	peikert	peikert	PROPN
ejpam-6609	362	6	et	et	PROPN
ejpam-6609	362	7	al	al	PROPN
ejpam-6609	362	8	.	.	PUNCT
ejpam-6609	363	1	a	a	DET
ejpam-6609	363	2	decade	decade	NOUN
ejpam-6609	363	3	of	of	ADP
ejpam-6609	363	4	lattice	lattice	ADJ
ejpam-6609	363	5	cryptography	cryptography	NOUN
ejpam-6609	363	6	.	.	PUNCT
ejpam-6609	364	1	foundations	foundation	NOUN
ejpam-6609	364	2	and	and	CCONJ
ejpam-6609	364	3	trends	trend	NOUN
ejpam-6609	364	4	in	in	ADP
ejpam-6609	364	5	theoretical	theoretical	ADJ
ejpam-6609	364	6	computer	computer	NOUN
ejpam-6609	364	7	science	science	NOUN
ejpam-6609	364	8	,	,	PUNCT
ejpam-6609	364	9	10(4):283–424	10(4):283–424	PROPN
ejpam-6609	364	10	,	,	PUNCT
ejpam-6609	364	11	2016	2016	NUM
ejpam-6609	364	12	.	.	PUNCT
ejpam-6609	365	1	[	[	X
ejpam-6609	365	2	19	19	NUM
ejpam-6609	365	3	]	]	PUNCT
ejpam-6609	365	4	m.	m.	NOUN
ejpam-6609	365	5	keçeci	keçeci	PROPN
ejpam-6609	365	6	.	.	PUNCT
ejpam-6609	366	1	understanding	understand	VERB
ejpam-6609	366	2	quantum	quantum	ADJ
ejpam-6609	366	3	mechanics	mechanic	NOUN
ejpam-6609	366	4	through	through	ADP
ejpam-6609	366	5	hilbert	hilbert	PROPN
ejpam-6609	366	6	spaces	space	NOUN
ejpam-6609	366	7	:	:	PUNCT
ejpam-6609	366	8	applications	application	NOUN
ejpam-6609	366	9	in	in	ADP
ejpam-6609	366	10	quantum	quantum	NOUN
ejpam-6609	366	11	computing	computing	NOUN
ejpam-6609	366	12	.	.	PUNCT
ejpam-6609	367	1	preprint	preprint	NOUN
ejpam-6609	367	2	,	,	PUNCT
ejpam-6609	367	3	2025	2025	NUM
ejpam-6609	367	4	.	.	PUNCT
ejpam-6609	368	1	[	[	X
ejpam-6609	368	2	20	20	NUM
ejpam-6609	368	3	]	]	PUNCT
ejpam-6609	368	4	p.	p.	PROPN
ejpam-6609	368	5	f.	f.	PROPN
ejpam-6609	369	1	dos	dos	PROPN
ejpam-6609	369	2	santos	santos	PROPN
ejpam-6609	369	3	,	,	PUNCT
ejpam-6609	369	4	c.	c.	PROPN
ejpam-6609	369	5	florentino	florentino	PROPN
ejpam-6609	369	6	,	,	PUNCT
ejpam-6609	369	7	and	and	CCONJ
ejpam-6609	369	8	j.	j.	PROPN
ejpam-6609	369	9	orts	orts	PROPN
ejpam-6609	369	10	.	.	PUNCT
ejpam-6609	370	1	characterizing	characterize	VERB
ejpam-6609	370	2	maximal	maximal	ADJ
ejpam-6609	370	3	varieties	variety	NOUN
ejpam-6609	370	4	via	via	ADP
ejpam-6609	370	5	bredon	bredon	PROPN
ejpam-6609	370	6	cohomology	cohomology	NOUN
ejpam-6609	370	7	.	.	PUNCT
ejpam-6609	371	1	arxiv	arxiv	PROPN
ejpam-6609	371	2	preprint	preprint	PROPN
ejpam-6609	371	3	,	,	PUNCT
ejpam-6609	371	4	2023	2023	NUM
ejpam-6609	371	5	.	.	PUNCT
ejpam-6609	372	1	arxiv:2310.17554	arxiv:2310.17554	PROPN
ejpam-6609	372	2	.	.	PUNCT
ejpam-6609	373	1	[	[	X
ejpam-6609	373	2	21	21	NUM
ejpam-6609	373	3	]	]	X
ejpam-6609	373	4	h.	h.	PROPN
ejpam-6609	373	5	qawaqneh	qawaqneh	PROPN
ejpam-6609	373	6	.	.	PUNCT
ejpam-6609	374	1	fractional	fractional	ADJ
ejpam-6609	374	2	analytic	analytic	ADJ
ejpam-6609	374	3	solutions	solution	NOUN
ejpam-6609	374	4	and	and	CCONJ
ejpam-6609	374	5	fixed	fix	VERB
ejpam-6609	374	6	point	point	NOUN
ejpam-6609	374	7	results	result	NOUN
ejpam-6609	374	8	with	with	ADP
ejpam-6609	374	9	some	some	DET
ejpam-6609	374	10	applications	application	NOUN
ejpam-6609	374	11	.	.	PUNCT
ejpam-6609	375	1	advances	advance	NOUN
ejpam-6609	375	2	in	in	ADP
ejpam-6609	375	3	fixed	fix	VERB
ejpam-6609	375	4	point	point	NOUN
ejpam-6609	375	5	theory	theory	NOUN
ejpam-6609	375	6	,	,	PUNCT
ejpam-6609	375	7	14(1	14(1	NUM
ejpam-6609	375	8	)	)	PUNCT
ejpam-6609	375	9	,	,	PUNCT
ejpam-6609	375	10	2024	2024	NUM
ejpam-6609	375	11	.	.	PUNCT
ejpam-6609	376	1	[	[	X
ejpam-6609	376	2	22	22	NUM
ejpam-6609	376	3	]	]	X
ejpam-6609	376	4	h.	h.	PROPN
ejpam-6609	376	5	qawaqneh	qawaqneh	PROPN
ejpam-6609	376	6	,	,	PUNCT
ejpam-6609	376	7	m.	m.	PROPN
ejpam-6609	376	8	s.	s.	PROPN
ejpam-6609	376	9	md	md	PROPN
ejpam-6609	376	10	.	.	PROPN
ejpam-6609	376	11	noorani	noorani	PROPN
ejpam-6609	376	12	,	,	PUNCT
ejpam-6609	376	13	h.	h.	PROPN
ejpam-6609	376	14	aydi	aydi	PROPN
ejpam-6609	376	15	,	,	PUNCT
ejpam-6609	376	16	a.	a.	NOUN
ejpam-6609	376	17	zraiqat	zraiqat	PROPN
ejpam-6609	376	18	,	,	PUNCT
ejpam-6609	376	19	and	and	CCONJ
ejpam-6609	376	20	a.	a.	NOUN
ejpam-6609	376	21	h.	h.	PROPN
ejpam-6609	376	22	ansari	ansari	PROPN
ejpam-6609	376	23	.	.	PUNCT
ejpam-6609	377	1	on	on	ADP
ejpam-6609	377	2	fixed	fix	VERB
ejpam-6609	377	3	point	point	NOUN
ejpam-6609	377	4	results	result	NOUN
ejpam-6609	377	5	in	in	ADP
ejpam-6609	377	6	partial	partial	ADJ
ejpam-6609	377	7	b	b	NOUN
ejpam-6609	377	8	-	-	PUNCT
ejpam-6609	377	9	metric	metric	ADJ
ejpam-6609	377	10	spaces	space	NOUN
ejpam-6609	377	11	.	.	PUNCT
ejpam-6609	378	1	journal	journal	NOUN
ejpam-6609	378	2	of	of	ADP
ejpam-6609	378	3	function	function	NOUN
ejpam-6609	378	4	spaces	space	NOUN
ejpam-6609	378	5	,	,	PUNCT
ejpam-6609	378	6	page	page	NOUN
ejpam-6609	378	7	8769190	8769190	NUM
ejpam-6609	378	8	,	,	PUNCT
ejpam-6609	378	9	2021	2021	NUM
ejpam-6609	378	10	.	.	PUNCT
ejpam-6609	379	1	[	[	X
ejpam-6609	379	2	23	23	NUM
ejpam-6609	379	3	]	]	X
ejpam-6609	379	4	h.	h.	PROPN
ejpam-6609	379	5	qawaqneh	qawaqneh	PROPN
ejpam-6609	379	6	.	.	PUNCT
ejpam-6609	380	1	new	new	ADJ
ejpam-6609	380	2	functions	function	NOUN
ejpam-6609	380	3	for	for	ADP
ejpam-6609	380	4	fixed	fix	VERB
ejpam-6609	380	5	point	point	NOUN
ejpam-6609	380	6	results	result	NOUN
ejpam-6609	380	7	in	in	ADP
ejpam-6609	380	8	metric	metric	ADJ
ejpam-6609	380	9	spaces	space	NOUN
ejpam-6609	380	10	with	with	ADP
ejpam-6609	380	11	some	some	DET
ejpam-6609	380	12	applications	application	NOUN
ejpam-6609	380	13	.	.	PUNCT
ejpam-6609	381	1	indian	indian	ADJ
ejpam-6609	381	2	journal	journal	PROPN
ejpam-6609	381	3	of	of	ADP
ejpam-6609	381	4	mathematics	mathematic	NOUN
ejpam-6609	381	5	,	,	PUNCT
ejpam-6609	381	6	66(1):55–84	66(1):55–84	NOUN
ejpam-6609	381	7	,	,	PUNCT
ejpam-6609	381	8	2024	2024	NUM
ejpam-6609	381	9	.	.	PUNCT
ejpam-6609	382	1	[	[	X
ejpam-6609	382	2	24	24	NUM
ejpam-6609	382	3	]	]	PUNCT
ejpam-6609	382	4	t.	t.	PROPN
ejpam-6609	382	5	kanan	kanan	PROPN
ejpam-6609	382	6	,	,	PUNCT
ejpam-6609	382	7	m.	m.	NOUN
ejpam-6609	382	8	elbes	elbes	PROPN
ejpam-6609	382	9	,	,	PUNCT
ejpam-6609	382	10	k.	k.	PROPN
ejpam-6609	382	11	abu	abu	PROPN
ejpam-6609	382	12	maria	maria	PROPN
ejpam-6609	382	13	,	,	PUNCT
ejpam-6609	382	14	and	and	CCONJ
ejpam-6609	382	15	m.	m.	NOUN
ejpam-6609	382	16	alia	alia	PROPN
ejpam-6609	382	17	.	.	PUNCT
ejpam-6609	383	1	exploring	explore	VERB
ejpam-6609	383	2	the	the	DET
ejpam-6609	383	3	potential	potential	NOUN
ejpam-6609	383	4	of	of	ADP
ejpam-6609	383	5	iot	iot	NOUN
ejpam-6609	383	6	-	-	PUNCT
ejpam-6609	383	7	based	base	VERB
ejpam-6609	383	8	learning	learning	NOUN
ejpam-6609	383	9	environments	environment	NOUN
ejpam-6609	383	10	in	in	ADP
ejpam-6609	383	11	education	education	NOUN
ejpam-6609	383	12	.	.	PUNCT
ejpam-6609	384	1	international	international	ADJ
ejpam-6609	384	2	journal	journal	NOUN
ejpam-6609	384	3	of	of	ADP
ejpam-6609	384	4	advances	advance	NOUN
ejpam-6609	384	5	in	in	ADP
ejpam-6609	384	6	soft	soft	ADJ
ejpam-6609	384	7	computing	computing	NOUN
ejpam-6609	384	8	and	and	CCONJ
ejpam-6609	384	9	its	its	PRON
ejpam-6609	384	10	applications	application	NOUN
ejpam-6609	384	11	,	,	PUNCT
ejpam-6609	384	12	15(2	15(2	NUM
ejpam-6609	384	13	)	)	PUNCT
ejpam-6609	384	14	,	,	PUNCT
ejpam-6609	384	15	2023	2023	NUM
ejpam-6609	384	16	.	.	PUNCT
ejpam-6609	385	1	[	[	X
ejpam-6609	385	2	25	25	NUM
ejpam-6609	385	3	]	]	PUNCT
ejpam-6609	385	4	i.	i.	PROPN
ejpam-6609	385	5	m.	m.	PROPN
ejpam-6609	385	6	batiha	batiha	PROPN
ejpam-6609	385	7	,	,	PUNCT
ejpam-6609	385	8	s.	s.	PROPN
ejpam-6609	385	9	a.	a.	PROPN
ejpam-6609	385	10	njadat	njadat	PROPN
ejpam-6609	385	11	,	,	PUNCT
ejpam-6609	385	12	r.	r.	PROPN
ejpam-6609	385	13	m.	m.	PROPN
ejpam-6609	385	14	batyha	batyha	PROPN
ejpam-6609	385	15	,	,	PUNCT
ejpam-6609	385	16	a.	a.	NOUN
ejpam-6609	385	17	zraiqat	zraiqat	PROPN
ejpam-6609	385	18	,	,	PUNCT
ejpam-6609	385	19	a.	a.	NOUN
ejpam-6609	385	20	dababneh	dababneh	PROPN
ejpam-6609	385	21	,	,	PUNCT
ejpam-6609	385	22	and	and	CCONJ
ejpam-6609	385	23	sh	sh	PROPN
ejpam-6609	385	24	.	.	PROPN
ejpam-6609	385	25	momani	momani	PROPN
ejpam-6609	385	26	.	.	PUNCT
ejpam-6609	386	1	design	design	NOUN
ejpam-6609	386	2	fractional	fractional	ADJ
ejpam-6609	386	3	-	-	PUNCT
ejpam-6609	386	4	order	order	NOUN
ejpam-6609	386	5	pid	pid	NOUN
ejpam-6609	386	6	controllers	controller	NOUN
ejpam-6609	386	7	for	for	ADP
ejpam-6609	386	8	single	single	ADJ
ejpam-6609	386	9	-	-	PUNCT
ejpam-6609	386	10	joint	joint	ADJ
ejpam-6609	386	11	robot	robot	NOUN
ejpam-6609	386	12	arm	arm	NOUN
ejpam-6609	386	13	model	model	NOUN
ejpam-6609	386	14	.	.	PUNCT
ejpam-6609	387	1	international	international	ADJ
ejpam-6609	387	2	journal	journal	NOUN
ejpam-6609	387	3	of	of	ADP
ejpam-6609	387	4	advances	advance	NOUN
ejpam-6609	387	5	in	in	ADP
ejpam-6609	387	6	soft	soft	ADJ
ejpam-6609	387	7	computing	computing	NOUN
ejpam-6609	387	8	and	and	CCONJ
ejpam-6609	387	9	its	its	PRON
ejpam-6609	387	10	applications	application	NOUN
ejpam-6609	387	11	,	,	PUNCT
ejpam-6609	387	12	14(2):96–114	14(2):96–114	NUM
ejpam-6609	387	13	,	,	PUNCT
ejpam-6609	387	14	2022	2022	NUM
ejpam-6609	387	15	.	.	PUNCT