id	sid	tid	token	lemma	pos
cana-395	1	1	communications	communication	NOUN
cana-395	1	2	on	on	ADP
cana-395	1	3	applied	apply	VERB
cana-395	1	4	nonlinear	nonlinear	ADJ
cana-395	1	5	analysis	analysis	NOUN
cana-395	1	6	issn	issn	NOUN
cana-395	1	7	:	:	PUNCT
cana-395	1	8	1074	1074	NUM
cana-395	1	9	-	-	PUNCT
cana-395	1	10	133x	133x	NUM
cana-395	1	11	vol	vol	NOUN
cana-395	1	12	31	31	NUM
cana-395	1	13	no	no	NOUN
cana-395	1	14	.	.	NOUN
cana-395	1	15	1	1	NUM
cana-395	1	16	(	(	PUNCT
cana-395	1	17	2024	2024	NUM
cana-395	1	18	)	)	PUNCT
cana-395	1	19	200	200	NUM
cana-395	1	20	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-395	1	21	fixed	fix	VERB
cana-395	1	22	point	point	NOUN
cana-395	1	23	results	result	NOUN
cana-395	1	24	under	under	ADP
cana-395	1	25	hausdorff	hausdorff	NOUN
cana-395	1	26	distance	distance	NOUN
cana-395	1	27	in	in	ADP
cana-395	1	28	the	the	DET
cana-395	1	29	fractal	fractal	ADJ
cana-395	1	30	spaces	space	NOUN
cana-395	1	31	rita	rita	PROPN
cana-395	1	32	pal1	pal1	PROPN
cana-395	1	33	,	,	PUNCT
cana-395	1	34	j.	j.	PROPN
cana-395	1	35	leo	leo	PROPN
cana-395	1	36	amalraj2	amalraj2	PROPN
cana-395	1	37	,	,	PUNCT
cana-395	1	38	v.	v.	PROPN
cana-395	1	39	venkata	venkata	PROPN
cana-395	1	40	kumar3	kumar3	PROPN
cana-395	1	41	,	,	PUNCT
cana-395	1	42	g.	g.	PROPN
cana-395	1	43	venkat	venkat	PROPN
cana-395	1	44	narayanan4	narayanan4	PROPN
cana-395	1	45	1department	1department	NUM
cana-395	1	46	of	of	ADP
cana-395	1	47	applied	apply	VERB
cana-395	1	48	mathematics	mathematics	PROPN
cana-395	1	49	bhilai	bhilai	PROPN
cana-395	1	50	institute	institute	PROPN
cana-395	1	51	of	of	ADP
cana-395	1	52	technology	technology	PROPN
cana-395	1	53	,	,	PUNCT
cana-395	1	54	bhilai	bhilai	PROPN
cana-395	1	55	,	,	PUNCT
cana-395	1	56	india	india	PROPN
cana-395	1	57	email	email	NOUN
cana-395	1	58	i	i	PROPN
cana-395	1	59	d	d	PROPN
cana-395	1	60	:	:	PUNCT
cana-395	1	61	ritapal001@gmail.com	ritapal001@gmail.com	X
cana-395	2	1	2department	2department	NUM
cana-395	2	2	of	of	ADP
cana-395	2	3	science	science	NOUN
cana-395	2	4	and	and	CCONJ
cana-395	2	5	humanities	humanity	NOUN
cana-395	2	6	rmk	rmk	PROPN
cana-395	2	7	college	college	PROPN
cana-395	2	8	of	of	ADP
cana-395	2	9	engineering	engineering	NOUN
cana-395	2	10	and	and	CCONJ
cana-395	2	11	technology	technology	NOUN
cana-395	2	12	puduvoyal	puduvoyal	NOUN
cana-395	2	13	,	,	PUNCT
cana-395	2	14	thiruvallur	thiruvallur	NOUN
cana-395	2	15	district	district	NOUN
cana-395	2	16	,	,	PUNCT
cana-395	2	17	tamilnadu	tamilnadu	NOUN
cana-395	2	18	,	,	PUNCT
cana-395	2	19	india	india	PROPN
cana-395	2	20	email	email	NOUN
cana-395	3	1	i	i	PROPN
cana-395	3	2	d	d	PROPN
cana-395	3	3	:	:	PUNCT
cana-395	3	4	leoamalraj@rmkcet.ac.in	leoamalraj@rmkcet.ac.in	PROPN
cana-395	3	5	3	3	NUM
cana-395	3	6	department	department	NOUN
cana-395	3	7	of	of	ADP
cana-395	3	8	mathematics	mathematic	NOUN
cana-395	3	9	,	,	PUNCT
cana-395	3	10	aditya	aditya	PROPN
cana-395	3	11	engineering	engineering	PROPN
cana-395	3	12	college	college	PROPN
cana-395	3	13	,	,	PUNCT
cana-395	3	14	surampalem	surampalem	NOUN
cana-395	3	15	,	,	PUNCT
cana-395	3	16	india	india	PROPN
cana-395	3	17	,	,	PUNCT
cana-395	3	18	email	email	NOUN
cana-395	4	1	i	i	PROPN
cana-395	4	2	d	d	PROPN
cana-395	4	3	:	:	PUNCT
cana-395	5	1	venkatakumar.v@aec.edu.in	venkatakumar.v@aec.edu.in	X
cana-395	5	2	4department	4department	NUM
cana-395	5	3	of	of	ADP
cana-395	5	4	mathematics	mathematics	PROPN
cana-395	5	5	st	st	PROPN
cana-395	5	6	.	.	PROPN
cana-395	5	7	joseph	joseph	PROPN
cana-395	5	8	’s	’s	PART
cana-395	5	9	college	college	PROPN
cana-395	5	10	of	of	ADP
cana-395	5	11	engineering	engineering	PROPN
cana-395	5	12	,	,	PUNCT
cana-395	5	13	omr	omr	PROPN
cana-395	5	14	,	,	PUNCT
cana-395	5	15	chennai	chennai	PROPN
cana-395	5	16	district	district	PROPN
cana-395	5	17	kanchipuram	kanchipuram	PROPN
cana-395	5	18	,	,	PUNCT
cana-395	5	19	tamil	tamil	PROPN
cana-395	5	20	nadu	nadu	PROPN
cana-395	5	21	,	,	PUNCT
cana-395	5	22	india	india	PROPN
cana-395	5	23	email	email	NOUN
cana-395	6	1	i	i	PROPN
cana-395	6	2	d	d	PROPN
cana-395	6	3	:	:	PUNCT
cana-395	7	1	gvenkatnarayanan@gmail.com	gvenkatnarayanan@gmail.com	X
cana-395	7	2	article	article	NOUN
cana-395	7	3	history	history	NOUN
cana-395	7	4	:	:	PUNCT
cana-395	7	5	received	receive	VERB
cana-395	7	6	:	:	PUNCT
cana-395	7	7	14	14	NUM
cana-395	7	8	-	-	SYM
cana-395	7	9	10	10	NUM
cana-395	7	10	-	-	PUNCT
cana-395	7	11	2023	2023	NUM
cana-395	7	12	revised	revise	VERB
cana-395	7	13	:	:	PUNCT
cana-395	7	14	28	28	NUM
cana-395	7	15	-	-	SYM
cana-395	7	16	11	11	NUM
cana-395	7	17	-	-	SYM
cana-395	7	18	2023	2023	NUM
cana-395	7	19	accepted	accept	VERB
cana-395	7	20	:	:	PUNCT
cana-395	7	21	18	18	NUM
cana-395	7	22	-	-	SYM
cana-395	7	23	12	12	NUM
cana-395	7	24	-	-	PUNCT
cana-395	7	25	2023	2023	NUM
cana-395	7	26	abstract	abstract	NOUN
cana-395	7	27	:	:	PUNCT
cana-395	7	28	we	we	PRON
cana-395	7	29	establish	establish	VERB
cana-395	7	30	a	a	DET
cana-395	7	31	novel	novel	ADJ
cana-395	7	32	notion	notion	NOUN
cana-395	7	33	of	of	ADP
cana-395	7	34	hausdorff	hausdorff	NOUN
cana-395	7	35	distance	distance	NOUN
cana-395	7	36	and	and	CCONJ
cana-395	7	37	explore	explore	VERB
cana-395	7	38	some	some	PRON
cana-395	7	39	of	of	ADP
cana-395	7	40	its	its	PRON
cana-395	7	41	topological	topological	ADJ
cana-395	7	42	characteristics	characteristic	NOUN
cana-395	7	43	by	by	ADP
cana-395	7	44	using	use	VERB
cana-395	7	45	an	an	DET
cana-395	7	46	extended	extended	ADJ
cana-395	7	47	modular	modular	ADJ
cana-395	7	48	metric	metric	NOUN
cana-395	7	49	.	.	PUNCT
cana-395	8	1	we	we	PRON
cana-395	8	2	prove	prove	VERB
cana-395	8	3	a	a	DET
cana-395	8	4	fixed	fix	VERB
cana-395	8	5	point	point	NOUN
cana-395	8	6	theorem	theorem	VERB
cana-395	8	7	on	on	ADP
cana-395	8	8	generalized	generalized	ADJ
cana-395	8	9	modular	modular	ADJ
cana-395	8	10	fractal	fractal	ADJ
cana-395	8	11	space	space	NOUN
cana-395	8	12	from	from	ADP
cana-395	8	13	the	the	DET
cana-395	8	14	concept	concept	NOUN
cana-395	8	15	of	of	ADP
cana-395	8	16	iterated	iterated	ADJ
cana-395	8	17	function	function	NOUN
cana-395	8	18	system	system	NOUN
cana-395	8	19	(	(	PUNCT
cana-395	8	20	ifs	ifs	PROPN
cana-395	8	21	)	)	PUNCT
cana-395	8	22	and	and	CCONJ
cana-395	8	23	contraction	contraction	NOUN
cana-395	8	24	.	.	PUNCT
cana-395	9	1	keywords	keyword	NOUN
cana-395	9	2	:	:	PUNCT
cana-395	9	3	fixed	fix	VERB
cana-395	9	4	points	point	NOUN
cana-395	9	5	;	;	PUNCT
cana-395	9	6	hausdorff	hausdorff	NOUN
cana-395	9	7	;	;	PUNCT
cana-395	9	8	fractal	fractal	ADJ
cana-395	9	9	space	space	NOUN
cana-395	9	10	;	;	PUNCT
cana-395	9	11	contraction	contraction	NOUN
cana-395	9	12	;	;	PUNCT
cana-395	9	13	ifs	ifs	PROPN
cana-395	9	14	;	;	PUNCT
cana-395	9	15	modular	modular	ADJ
cana-395	9	16	metric	metric	ADJ
cana-395	9	17	spaces	space	NOUN
cana-395	9	18	.	.	PUNCT
cana-395	10	1	2000	2000	NUM
cana-395	10	2	mathematics	mathematic	NOUN
cana-395	10	3	subject	subject	ADJ
cana-395	10	4	classification	classification	NOUN
cana-395	10	5	:	:	PUNCT
cana-395	10	6	47h10	47h10	NUM
cana-395	10	7	,	,	PUNCT
cana-395	10	8	54h25	54h25	NUM
cana-395	10	9	.	.	PUNCT
cana-395	11	1	1	1	X
cana-395	11	2	.	.	X
cana-395	11	3	introduction	introduction	NOUN
cana-395	11	4	the	the	DET
cana-395	11	5	metric	metric	ADJ
cana-395	11	6	modular	modular	ADJ
cana-395	11	7	space	space	NOUN
cana-395	11	8	structure	structure	NOUN
cana-395	11	9	was	be	AUX
cana-395	11	10	modified	modify	VERB
cana-395	11	11	by	by	ADP
cana-395	11	12	chistyakov	chistyakov	PROPN
cana-395	11	13	(	(	PUNCT
cana-395	11	14	chistyakov	chistyakov	NOUN
cana-395	11	15	,	,	PUNCT
cana-395	11	16	2008	2008	NUM
cana-395	11	17	;	;	PUNCT
cana-395	11	18	cho	cho	PROPN
cana-395	11	19	,	,	PUNCT
cana-395	11	20	saadati	saadati	NOUN
cana-395	11	21	and	and	CCONJ
cana-395	11	22	sadeghi	sadeghi	NOUN
cana-395	11	23	,	,	PUNCT
cana-395	11	24	2012	2012	NUM
cana-395	11	25	)	)	PUNCT
cana-395	11	26	,	,	PUNCT
cana-395	11	27	in	in	ADP
cana-395	11	28	an	an	DET
cana-395	11	29	insightful	insightful	ADJ
cana-395	11	30	way	way	NOUN
cana-395	11	31	and	and	CCONJ
cana-395	11	32	proposed	propose	VERB
cana-395	11	33	the	the	DET
cana-395	11	34	hausdorff	hausdorff	NOUN
cana-395	11	35	topology	topology	NOUN
cana-395	11	36	on	on	ADP
cana-395	11	37	it	it	PRON
cana-395	11	38	,	,	PUNCT
cana-395	11	39	is	be	AUX
cana-395	11	40	extremely	extremely	ADV
cana-395	11	41	well	well	ADV
cana-395	11	42	-	-	PUNCT
cana-395	11	43	liked	like	VERB
cana-395	11	44	in	in	ADP
cana-395	11	45	modern	modern	ADJ
cana-395	11	46	study	study	NOUN
cana-395	11	47	.	.	PUNCT
cana-395	12	1	now	now	ADV
cana-395	12	2	,	,	PUNCT
cana-395	12	3	using	use	VERB
cana-395	12	4	nonempty	nonempty	ADJ
cana-395	12	5	compact	compact	ADJ
cana-395	12	6	subsets	subset	NOUN
cana-395	12	7	,	,	PUNCT
cana-395	12	8	we	we	PRON
cana-395	12	9	investigate	investigate	VERB
cana-395	12	10	the	the	DET
cana-395	12	11	hausdorff	hausdorff	NOUN
cana-395	12	12	distance	distance	NOUN
cana-395	12	13	for	for	ADP
cana-395	12	14	a	a	DET
cana-395	12	15	certain	certain	ADJ
cana-395	12	16	(	(	PUNCT
cana-395	12	17	gmms	gmms	NOUN
cana-395	12	18	)	)	PUNCT
cana-395	12	19	.	.	PUNCT
cana-395	13	1	in	in	ADP
cana-395	13	2	order	order	NOUN
cana-395	13	3	to	to	PART
cana-395	13	4	demonstrate	demonstrate	VERB
cana-395	13	5	an	an	DET
cana-395	13	6	intriguing	intriguing	ADJ
cana-395	13	7	(	(	PUNCT
cana-395	13	8	fpt	fpt	PROPN
cana-395	13	9	)	)	PUNCT
cana-395	13	10	fixed	fix	VERB
cana-395	13	11	point	point	NOUN
cana-395	13	12	theorem	theorem	VERB
cana-395	13	13	,	,	PUNCT
cana-395	13	14	on	on	ADP
cana-395	13	15	a	a	DET
cana-395	13	16	generalised	generalise	VERB
cana-395	13	17	metric	metric	ADJ
cana-395	13	18	modular	modular	ADJ
cana-395	13	19	space	space	NOUN
cana-395	13	20	,	,	PUNCT
cana-395	13	21	we	we	PRON
cana-395	13	22	apply	apply	VERB
cana-395	13	23	the	the	DET
cana-395	13	24	iterated	iterated	ADJ
cana-395	13	25	function	function	NOUN
cana-395	13	26	system	system	NOUN
cana-395	13	27	(	(	PUNCT
cana-395	13	28	ifs	ifs	PROPN
cana-395	13	29	)	)	PUNCT
cana-395	13	30	and	and	CCONJ
cana-395	13	31	idea	idea	NOUN
cana-395	13	32	of	of	ADP
cana-395	13	33	contraction	contraction	NOUN
cana-395	13	34	together	together	ADV
cana-395	13	35	(	(	PUNCT
cana-395	13	36	abdou	abdou	PROPN
cana-395	13	37	,	,	PUNCT
cana-395	13	38	2016	2016	NUM
cana-395	13	39	;	;	PUNCT
cana-395	13	40	abdou	abdou	PROPN
cana-395	13	41	,	,	PUNCT
cana-395	13	42	2020	2020	NUM
cana-395	13	43	;	;	PUNCT
cana-395	13	44	chistyakov	chistyakov	NOUN
cana-395	13	45	,	,	PUNCT
cana-395	13	46	2008	2008	NUM
cana-395	13	47	;	;	PUNCT
cana-395	13	48	chistyakov	chistyakov	NOUN
cana-395	13	49	,	,	PUNCT
cana-395	13	50	2010	2010	NUM
cana-395	13	51	;	;	PUNCT
cana-395	13	52	cho	cho	PROPN
cana-395	13	53	et	et	PROPN
cana-395	13	54	al	al	PROPN
cana-395	13	55	.	.	PROPN
cana-395	13	56	,	,	PUNCT
cana-395	13	57	2012	2012	NUM
cana-395	13	58	;	;	PUNCT
cana-395	13	59	ege	ege	ADJ
cana-395	13	60	,	,	PUNCT
cana-395	13	61	park	park	NOUN
cana-395	13	62	and	and	CCONJ
cana-395	13	63	ansari	ansari	ADJ
cana-395	13	64	,	,	PUNCT
cana-395	13	65	2020	2020	NUM
cana-395	13	66	)	)	PUNCT
cana-395	13	67	.	.	PUNCT
cana-395	14	1	hutchinson	hutchinson	PROPN
cana-395	14	2	studied	study	VERB
cana-395	14	3	iterated	iterated	ADJ
cana-395	14	4	function	function	NOUN
cana-395	14	5	system	system	NOUN
cana-395	14	6	(	(	PUNCT
cana-395	14	7	ifs	ifs	PROPN
cana-395	14	8	)	)	PUNCT
cana-395	14	9	and	and	CCONJ
cana-395	14	10	thought	think	VERB
cana-395	14	11	about	about	ADP
cana-395	14	12	the	the	DET
cana-395	14	13	idea	idea	NOUN
cana-395	14	14	of	of	ADP
cana-395	14	15	fractal	fractal	ADJ
cana-395	14	16	theory	theory	NOUN
cana-395	14	17	(	(	PUNCT
cana-395	14	18	hutchinson	hutchinson	PROPN
cana-395	14	19	,	,	PUNCT
cana-395	14	20	1981	1981	NUM
cana-395	14	21	)	)	PUNCT
cana-395	14	22	.	.	PUNCT
cana-395	15	1	by	by	ADP
cana-395	15	2	ri	ri	PROPN
cana-395	15	3	(	(	PUNCT
cana-395	15	4	ri	ri	PROPN
cana-395	15	5	,	,	PUNCT
cana-395	15	6	2016),barnsley	2016),barnsley	NUM
cana-395	15	7	(	(	PUNCT
cana-395	15	8	good	good	ADJ
cana-395	15	9	,	,	PUNCT
cana-395	15	10	1990	1990	NUM
cana-395	15	11	)	)	PUNCT
cana-395	15	12	,	,	PUNCT
cana-395	15	13	bisht	bisht	PROPN
cana-395	15	14	(	(	PUNCT
cana-395	15	15	bisht	bisht	ADJ
cana-395	15	16	,	,	PUNCT
cana-395	15	17	2018	2018	NUM
cana-395	15	18	)	)	PUNCT
cana-395	15	19	,	,	PUNCT
cana-395	15	20	and	and	CCONJ
cana-395	15	21	imdad	imdad	PROPN
cana-395	15	22	(	(	PUNCT
cana-395	15	23	imdad	imdad	PROPN
cana-395	15	24	,	,	PUNCT
cana-395	15	25	alfaqih	alfaqih	VERB
cana-395	15	26	and	and	CCONJ
cana-395	15	27	khan	khan	PROPN
cana-395	15	28	,	,	PUNCT
cana-395	15	29	2018	2018	NUM
cana-395	15	30	)	)	PUNCT
cana-395	15	31	,	,	PUNCT
cana-395	15	32	this	this	DET
cana-395	15	33	topic	topic	NOUN
cana-395	15	34	was	be	AUX
cana-395	15	35	generalized	generalize	VERB
cana-395	15	36	.	.	PUNCT
cana-395	16	1	a	a	DET
cana-395	16	2	singular	singular	ADJ
cana-395	16	3	nonempty	nonempty	X
cana-395	16	4	compact	compact	ADJ
cana-395	16	5	set	set	VERB
cana-395	16	6	f	f	PROPN
cana-395	16	7	and	and	CCONJ
cana-395	16	8	f	f	PROPN
cana-395	16	9	=	=	PUNCT
cana-395	17	1	⋃	⋃	PROPN
cana-395	17	2	𝑄𝑚	𝑄𝑚	PROPN
cana-395	17	3	𝑖=1	𝑖=1	PUNCT
cana-395	18	1	i	i	PRON
cana-395	18	2	(	(	PUNCT
cana-395	18	3	f	f	X
cana-395	18	4	)	)	PUNCT
cana-395	18	5	of	of	ADP
cana-395	18	6	the	the	DET
cana-395	18	7	complete	complete	ADJ
cana-395	18	8	gmm	gmm	NOUN
cana-395	18	9	space	space	NOUN
cana-395	18	10	(	(	PUNCT
cana-395	18	11	l	l	NOUN
cana-395	18	12	,	,	PUNCT
cana-395	18	13	t	t	PROPN
cana-395	18	14	)	)	PUNCT
cana-395	18	15	for	for	ADP
cana-395	18	16	a	a	DET
cana-395	18	17	gmmifs	gmmif	NOUN
cana-395	18	18	,	,	PUNCT
cana-395	18	19	then	then	ADV
cana-395	18	20	a	a	DET
cana-395	18	21	fractal	fractal	ADJ
cana-395	18	22	set	set	NOUN
cana-395	18	23	f	f	PROPN
cana-395	18	24	is	be	AUX
cana-395	18	25	known	know	VERB
cana-395	18	26	as	as	ADP
cana-395	18	27	the	the	DET
cana-395	18	28	attractor	attractor	NOUN
cana-395	18	29	of	of	ADP
cana-395	18	30	the	the	DET
cana-395	18	31	relevant	relevant	ADJ
cana-395	18	32	generalized	generalized	ADJ
cana-395	18	33	modular	modular	ADJ
cana-395	18	34	metric	metric	ADJ
cana-395	18	35	iterated	iterate	VERB
cana-395	18	36	function	function	NOUN
cana-395	18	37	system	system	NOUN
cana-395	18	38	.	.	PUNCT
cana-395	19	1	the	the	DET
cana-395	19	2	associated	associated	ADJ
cana-395	19	3	attractor	attractor	NOUN
cana-395	19	4	generalized	generalize	VERB
cana-395	19	5	modular	modular	ADJ
cana-395	19	6	metric	metric	ADJ
cana-395	19	7	iterated	iterate	VERB
cana-395	19	8	function	function	NOUN
cana-395	19	9	system	system	NOUN
cana-395	19	10	in	in	ADP
cana-395	19	11	this	this	DET
cana-395	19	12	context	context	NOUN
cana-395	19	13	is	be	AUX
cana-395	19	14	referred	refer	VERB
cana-395	19	15	to	to	ADP
cana-395	19	16	as	as	ADP
cana-395	19	17	generalized	generalize	VERB
cana-395	19	18	modular	modular	ADJ
cana-395	19	19	metric	metric	ADJ
cana-395	19	20	fractal	fractal	ADJ
cana-395	19	21	space	space	NOUN
cana-395	19	22	.	.	PUNCT
cana-395	20	1	communications	communication	NOUN
cana-395	20	2	on	on	ADP
cana-395	20	3	applied	apply	VERB
cana-395	20	4	nonlinear	nonlinear	ADJ
cana-395	20	5	analysis	analysis	NOUN
cana-395	20	6	issn	issn	NOUN
cana-395	20	7	:	:	PUNCT
cana-395	20	8	1074	1074	NUM
cana-395	20	9	-	-	PUNCT
cana-395	20	10	133x	133x	NUM
cana-395	20	11	vol	vol	NOUN
cana-395	20	12	31	31	NUM
cana-395	20	13	no	no	NOUN
cana-395	20	14	.	.	NOUN
cana-395	20	15	1	1	NUM
cana-395	20	16	(	(	PUNCT
cana-395	20	17	2024	2024	NUM
cana-395	20	18	)	)	PUNCT
cana-395	20	19	201	201	NUM
cana-395	20	20	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-395	20	21	2	2	NUM
cana-395	20	22	preliminiers	preliminier	NOUN
cana-395	20	23	now	now	ADV
cana-395	20	24	let	let	VERB
cana-395	20	25	's	us	PRON
cana-395	20	26	review	review	VERB
cana-395	20	27	some	some	DET
cana-395	20	28	ideas	idea	NOUN
cana-395	20	29	and	and	CCONJ
cana-395	20	30	fundamental	fundamental	ADJ
cana-395	20	31	principles	principle	NOUN
cana-395	20	32	.	.	PUNCT
cana-395	21	1	here	here	ADV
cana-395	21	2	,	,	PUNCT
cana-395	21	3	we	we	PRON
cana-395	21	4	let	let	VERB
cana-395	21	5	p	p	NOUN
cana-395	21	6	=	=	PUNCT
cana-395	22	1	[	[	X
cana-395	22	2	0	0	NUM
cana-395	22	3	,	,	PUNCT
cana-395	22	4	1	1	NUM
cana-395	22	5	]	]	PUNCT
cana-395	22	6	,	,	PUNCT
cana-395	22	7	p0	p0	NOUN
cana-395	22	8	=	=	SYM
cana-395	22	9	(	(	PUNCT
cana-395	22	10	0	0	NUM
cana-395	22	11	,	,	PUNCT
cana-395	22	12	1	1	NUM
cana-395	22	13	)	)	PUNCT
cana-395	22	14	,	,	PUNCT
cana-395	22	15	q	q	NOUN
cana-395	23	1	=	=	PUNCT
cana-395	24	1	[	[	X
cana-395	24	2	0	0	NUM
cana-395	24	3	,	,	PUNCT
cana-395	24	4	∞	∞	PROPN
cana-395	24	5	)	)	PUNCT
cana-395	24	6	,	,	PUNCT
cana-395	24	7	q0	q0	PROPN
cana-395	24	8	=	=	SYM
cana-395	24	9	(	(	PUNCT
cana-395	24	10	0	0	NUM
cana-395	24	11	,	,	PUNCT
cana-395	24	12	∞	∞	PROPN
cana-395	24	13	)	)	PUNCT
cana-395	24	14	and	and	CCONJ
cana-395	24	15	a	a	DET
cana-395	24	16	set	set	ADJ
cana-395	24	17	l	l	NOUN
cana-395	24	18	≠	≠	PROPN
cana-395	24	19	ϕ.	ϕ.	ADJ
cana-395	24	20	definition	definition	NOUN
cana-395	24	21	2.1	2.1	NUM
cana-395	24	22	(	(	PUNCT
cana-395	24	23	azadifa	azadifa	NOUN
cana-395	24	24	,	,	PUNCT
cana-395	24	25	maramaei	maramaei	NOUN
cana-395	24	26	and	and	CCONJ
cana-395	24	27	sadeghi	sadeghi	NOUN
cana-395	24	28	,	,	PUNCT
cana-395	24	29	2013	2013	NUM
cana-395	24	30	)	)	PUNCT
cana-395	24	31	a	a	DET
cana-395	24	32	function	function	NOUN
cana-395	24	33	t	t	NOUN
cana-395	24	34	:	:	PUNCT
cana-395	24	35	l	l	NOUN
cana-395	24	36	×	×	NOUN
cana-395	24	37	l	l	NOUN
cana-395	24	38	×	×	NOUN
cana-395	24	39	l×	l×	PROPN
cana-395	24	40	q0	q0	PROPN
cana-395	24	41	→	→	X
cana-395	24	42	q	q	X
cana-395	24	43	is	be	AUX
cana-395	24	44	referred	refer	VERB
cana-395	24	45	to	to	ADP
cana-395	24	46	as	as	SCONJ
cana-395	24	47	generalized	generalize	VERB
cana-395	24	48	metric	metric	ADJ
cana-395	24	49	modular	modular	NOUN
cana-395	24	50	(	(	PUNCT
cana-395	24	51	gmm	gmm	NOUN
cana-395	24	52	)	)	PUNCT
cana-395	24	53	on	on	ADP
cana-395	24	54	l	l	PROPN
cana-395	24	55	a	a	DET
cana-395	24	56	no	no	DET
cana-395	24	57	-empty	-empty	NOUN
cana-395	24	58	set	set	VERB
cana-395	24	59	,	,	PUNCT
cana-395	24	60	if	if	SCONJ
cana-395	24	61	it	it	PRON
cana-395	24	62	follows	follow	VERB
cana-395	24	63	the	the	DET
cana-395	24	64	axioms	axiom	NOUN
cana-395	24	65	listed	list	VERB
cana-395	24	66	below	below	ADP
cana-395	24	67	:	:	PUNCT
cana-395	24	68	(	(	PUNCT
cana-395	24	69	gmm-1	gmm-1	X
cana-395	24	70	)	)	PUNCT
cana-395	24	71	t	t	PROPN
cana-395	24	72	ρ	ρ	PROPN
cana-395	24	73	(	(	PUNCT
cana-395	24	74	l	l	NOUN
cana-395	24	75	,	,	PUNCT
cana-395	24	76	l	l	NOUN
cana-395	24	77	,	,	PUNCT
cana-395	24	78	n	n	CCONJ
cana-395	24	79	)	)	PUNCT
cana-395	24	80	∈	∈	PROPN
cana-395	24	81	q0	q0	PROPN
cana-395	24	82	for	for	ADP
cana-395	24	83	all	all	DET
cana-395	24	84	l	l	NOUN
cana-395	24	85	,	,	PUNCT
cana-395	24	86	n	n	PROPN
cana-395	24	87	∈	∈	PROPN
cana-395	24	88	l	l	NOUN
cana-395	24	89	and	and	CCONJ
cana-395	24	90	ρ	ρ	PROPN
cana-395	24	91	∈	∈	PROPN
cana-395	24	92	q0	q0	PROPN
cana-395	24	93	with	with	ADP
cana-395	24	94	l	l	PROPN
cana-395	24	95	≠	≠	PROPN
cana-395	24	96	n.	n.	NOUN
cana-395	24	97	(	(	PUNCT
cana-395	24	98	gmm-2	gmm-2	NOUN
cana-395	24	99	)	)	PUNCT
cana-395	24	100	t	t	PROPN
cana-395	24	101	ρ	ρ	PROPN
cana-395	24	102	(	(	PUNCT
cana-395	24	103	l	l	PROPN
cana-395	24	104	,	,	PUNCT
cana-395	24	105	m	m	PROPN
cana-395	24	106	,	,	PUNCT
cana-395	24	107	n	n	CCONJ
cana-395	24	108	)	)	PUNCT
cana-395	24	109	=	=	SYM
cana-395	24	110	0	0	NUM
cana-395	24	111	,	,	PUNCT
cana-395	24	112	∀ρ	∀ρ	NOUN
cana-395	24	113	∈	∈	PROPN
cana-395	24	114	q0	q0	VERB
cana-395	24	115	if	if	SCONJ
cana-395	24	116	l	l	PROPN
cana-395	24	117	=	=	PUNCT
cana-395	24	118	m	m	VERB
cana-395	24	119	=	=	VERB
cana-395	24	120	n.	n.	NOUN
cana-395	24	121	(	(	PUNCT
cana-395	24	122	gmm-3	gmm-3	NOUN
cana-395	24	123	)	)	PUNCT
cana-395	24	124	t	t	PROPN
cana-395	24	125	ρ	ρ	PROPN
cana-395	24	126	(	(	PUNCT
cana-395	24	127	l	l	NOUN
cana-395	24	128	,	,	PUNCT
cana-395	24	129	l	l	NOUN
cana-395	24	130	,	,	PUNCT
cana-395	24	131	n	n	CCONJ
cana-395	24	132	)	)	PUNCT
cana-395	24	133	≤	≤	NOUN
cana-395	24	134	t	t	PROPN
cana-395	24	135	ρ	ρ	X
cana-395	24	136	(	(	PUNCT
cana-395	24	137	l	l	PROPN
cana-395	24	138	,	,	PUNCT
cana-395	24	139	m	m	PROPN
cana-395	24	140	,	,	PUNCT
cana-395	24	141	n	n	CCONJ
cana-395	24	142	)	)	PUNCT
cana-395	24	143	,	,	PUNCT
cana-395	24	144	∀ρ	∀ρ	X
cana-395	24	145	∈	∈	PROPN
cana-395	24	146	q0	q0	VERB
cana-395	24	147	if	if	SCONJ
cana-395	24	148	m	m	VERB
cana-395	24	149	≠	≠	PROPN
cana-395	24	150	n.	n.	NOUN
cana-395	24	151	(	(	PUNCT
cana-395	24	152	gmm-4	gmm-4	NUM
cana-395	24	153	)	)	PUNCT
cana-395	24	154	t	t	PROPN
cana-395	24	155	ρ	ρ	PROPN
cana-395	24	156	(	(	PUNCT
cana-395	24	157	l	l	PROPN
cana-395	24	158	,	,	PUNCT
cana-395	24	159	m	m	PROPN
cana-395	24	160	,	,	PUNCT
cana-395	24	161	n	n	CCONJ
cana-395	24	162	)	)	PUNCT
cana-395	24	163	=	=	SYM
cana-395	24	164	t	t	PROPN
cana-395	24	165	ρ	ρ	X
cana-395	24	166	(	(	PUNCT
cana-395	24	167	l	l	NOUN
cana-395	24	168	,	,	PUNCT
cana-395	24	169	n	n	CCONJ
cana-395	24	170	,	,	PUNCT
cana-395	24	171	m	m	NOUN
cana-395	24	172	)	)	PUNCT
cana-395	25	1	=	=	SYM
cana-395	25	2	t	t	PROPN
cana-395	25	3	ρ	ρ	PROPN
cana-395	25	4	(	(	PUNCT
cana-395	25	5	n	n	CCONJ
cana-395	25	6	,	,	PUNCT
cana-395	25	7	l	l	NOUN
cana-395	25	8	,	,	PUNCT
cana-395	25	9	m	m	NOUN
cana-395	25	10	)	)	PUNCT
cana-395	25	11	and	and	CCONJ
cana-395	25	12	so	so	ADV
cana-395	25	13	on	on	ADV
cana-395	25	14	.	.	PUNCT
cana-395	26	1	(	(	PUNCT
cana-395	26	2	gmm-5	gmm-5	NOUN
cana-395	26	3	)	)	PUNCT
cana-395	26	4	tρ	tρ	X
cana-395	27	1	+	+	PROPN
cana-395	27	2	δ	δ	PROPN
cana-395	27	3	(	(	PUNCT
cana-395	27	4	l	l	PROPN
cana-395	27	5	,	,	PUNCT
cana-395	27	6	m	m	PROPN
cana-395	27	7	,	,	PUNCT
cana-395	27	8	n	n	CCONJ
cana-395	27	9	)	)	PUNCT
cana-395	27	10	≤	≤	NOUN
cana-395	27	11	tρ	tρ	X
cana-395	27	12	(	(	PUNCT
cana-395	27	13	l	l	PROPN
cana-395	27	14	,	,	PUNCT
cana-395	27	15	v	v	NOUN
cana-395	27	16	,	,	PUNCT
cana-395	27	17	v	v	NOUN
cana-395	27	18	)	)	PUNCT
cana-395	27	19	+	+	CCONJ
cana-395	27	20	tδ	tδ	PROPN
cana-395	27	21	(	(	PUNCT
cana-395	27	22	v	v	NOUN
cana-395	27	23	,	,	PUNCT
cana-395	27	24	m	m	PROPN
cana-395	27	25	,	,	PUNCT
cana-395	27	26	v	v	NOUN
cana-395	27	27	)	)	PUNCT
cana-395	27	28	for	for	ADP
cana-395	27	29	all	all	DET
cana-395	27	30	ρ	ρ	NOUN
cana-395	27	31	,	,	PUNCT
cana-395	27	32	δ∈	δ∈	PROPN
cana-395	27	33	q0	q0	PROPN
cana-395	27	34	.	.	PUNCT
cana-395	28	1	then	then	ADV
cana-395	28	2	,	,	PUNCT
cana-395	28	3	(	(	PUNCT
cana-395	28	4	l	l	NOUN
cana-395	28	5	,	,	PUNCT
cana-395	28	6	t	t	PROPN
cana-395	28	7	)	)	PUNCT
cana-395	28	8	is	be	AUX
cana-395	28	9	referred	refer	VERB
cana-395	28	10	to	to	ADP
cana-395	28	11	as	as	ADP
cana-395	28	12	a	a	DET
cana-395	28	13	generalized	generalized	ADJ
cana-395	28	14	modular	modular	ADJ
cana-395	28	15	metric	metric	NOUN
cana-395	28	16	on	on	ADP
cana-395	28	17	l	l	PROPN
cana-395	28	18	.	.	PUNCT
cana-395	29	1	definition	definition	NOUN
cana-395	29	2	2.2	2.2	NUM
cana-395	29	3	(	(	PUNCT
cana-395	29	4	azadifa	azadifa	NOUN
cana-395	29	5	et	et	NOUN
cana-395	29	6	al	al	PROPN
cana-395	29	7	.	.	PROPN
cana-395	29	8	,	,	PUNCT
cana-395	29	9	2013	2013	NUM
cana-395	29	10	)	)	PUNCT
cana-395	29	11	let	let	VERB
cana-395	29	12	us	we	PRON
cana-395	29	13	set	set	VERB
cana-395	29	14	l0	l0	PROPN
cana-395	29	15	∈	∈	PROPN
cana-395	29	16	l	l	NOUN
cana-395	29	17	and	and	CCONJ
cana-395	29	18	lt	lt	NOUN
cana-395	29	19	=	=	X
cana-395	29	20	{	{	PUNCT
cana-395	29	21	m	m	PROPN
cana-395	29	22	∈	∈	PROPN
cana-395	29	23	l	l	NOUN
cana-395	29	24	;	;	PUNCT
cana-395	29	25	lim	lim	PROPN
cana-395	29	26	ρ	ρ	PROPN
cana-395	29	27	→0	→0	PUNCT
cana-395	29	28	t	t	PROPN
cana-395	29	29	ρ	ρ	PROPN
cana-395	29	30	(	(	PUNCT
cana-395	29	31	l0	l0	PROPN
cana-395	29	32	,	,	PUNCT
cana-395	29	33	m	m	PROPN
cana-395	29	34	,	,	PUNCT
cana-395	29	35	n)=0	n)=0	VERB
cana-395	29	36	for	for	ADP
cana-395	29	37	some	some	DET
cana-395	29	38	n	n	PRON
cana-395	29	39	∈	∈	NOUN
cana-395	29	40	l	l	NOUN
cana-395	29	41	}	}	PUNCT
cana-395	29	42	.	.	PUNCT
cana-395	30	1	the	the	DET
cana-395	30	2	set	set	NOUN
cana-395	30	3	lt	lt	NOUN
cana-395	30	4	is	be	AUX
cana-395	30	5	known	know	VERB
cana-395	30	6	as	as	ADP
cana-395	30	7	a	a	DET
cana-395	30	8	modular	modular	ADJ
cana-395	30	9	set	set	NOUN
cana-395	30	10	.	.	PUNCT
cana-395	31	1	definition	definition	NOUN
cana-395	31	2	2.3	2.3	NUM
cana-395	31	3	(	(	PUNCT
cana-395	31	4	azadifa	azadifa	NOUN
cana-395	31	5	et	et	NOUN
cana-395	31	6	al	al	PROPN
cana-395	31	7	.	.	PROPN
cana-395	31	8	,	,	PUNCT
cana-395	31	9	2013	2013	NUM
cana-395	31	10	)	)	PUNCT
cana-395	31	11	assume	assume	VERB
cana-395	31	12	that	that	SCONJ
cana-395	31	13	(	(	PUNCT
cana-395	31	14	l	l	NOUN
cana-395	31	15	,	,	PUNCT
cana-395	31	16	t	t	PROPN
cana-395	31	17	)	)	PUNCT
cana-395	31	18	be	be	AUX
cana-395	31	19	a	a	DET
cana-395	31	20	generalised	generalise	VERB
cana-395	31	21	modular	modular	ADJ
cana-395	31	22	metric	metric	NOUN
cana-395	31	23	(	(	PUNCT
cana-395	31	24	gmm	gmm	NOUN
cana-395	31	25	)	)	PUNCT
cana-395	31	26	space	space	NOUN
cana-395	31	27	.	.	PUNCT
cana-395	32	1	then	then	ADV
cana-395	32	2	,	,	PUNCT
cana-395	32	3	for	for	ADP
cana-395	32	4	l0	l0	PROPN
cana-395	32	5	∈	∈	PROPN
cana-395	32	6	lt	lt	NOUN
cana-395	32	7	and	and	CCONJ
cana-395	32	8	c	c	X
cana-395	32	9	>	>	X
cana-395	32	10	0	0	PROPN
cana-395	32	11	,	,	PUNCT
cana-395	32	12	the	the	DET
cana-395	32	13	t	t	PROPN
cana-395	32	14	-	-	PUNCT
cana-395	32	15	ball	ball	NOUN
cana-395	32	16	with	with	ADP
cana-395	32	17	radius	radius	NOUN
cana-395	32	18	c	c	PROPN
cana-395	32	19	and	and	CCONJ
cana-395	32	20	center	center	PROPN
cana-395	32	21	l0	l0	PROPN
cana-395	32	22	is	be	AUX
cana-395	32	23	bt	bt	PROPN
cana-395	32	24	(	(	PUNCT
cana-395	32	25	l0	l0	PROPN
cana-395	32	26	,	,	PUNCT
cana-395	32	27	c	c	NOUN
cana-395	32	28	)	)	PUNCT
cana-395	32	29	=	=	PRON
cana-395	33	1	{	{	PUNCT
cana-395	33	2	m	m	VERB
cana-395	33	3	∈	∈	NOUN
cana-395	33	4	lt	lt	NOUN
cana-395	33	5	:	:	PUNCT
cana-395	33	6	t	t	PROPN
cana-395	33	7	ρ	ρ	PROPN
cana-395	33	8	(	(	PUNCT
cana-395	33	9	l0	l0	PROPN
cana-395	33	10	,	,	PUNCT
cana-395	33	11	m	m	PROPN
cana-395	33	12	,	,	PUNCT
cana-395	33	13	m	m	NOUN
cana-395	33	14	)	)	PUNCT
cana-395	33	15	<	<	X
cana-395	33	16	c	c	X
cana-395	33	17	}	}	PUNCT
cana-395	33	18	,	,	PUNCT
cana-395	33	19	∀	∀	X
cana-395	33	20	ρ	ρ	NOUN
cana-395	33	21	>	>	X
cana-395	33	22	0	0	PROPN
cana-395	33	23	.	.	PUNCT
cana-395	34	1	proposition	proposition	NOUN
cana-395	34	2	2.3.1	2.3.1	NUM
cana-395	34	3	(	(	PUNCT
cana-395	34	4	azadifa	azadifa	NOUN
cana-395	34	5	et	et	NOUN
cana-395	34	6	al	al	PROPN
cana-395	34	7	.	.	PROPN
cana-395	34	8	,	,	PUNCT
cana-395	34	9	2013	2013	NUM
cana-395	34	10	)	)	PUNCT
cana-395	34	11	assume	assume	VERB
cana-395	34	12	that	that	SCONJ
cana-395	34	13	(	(	PUNCT
cana-395	34	14	l	l	NOUN
cana-395	34	15	,	,	PUNCT
cana-395	34	16	t	t	PROPN
cana-395	34	17	)	)	PUNCT
cana-395	34	18	be	be	AUX
cana-395	34	19	a	a	DET
cana-395	34	20	generalised	generalise	VERB
cana-395	34	21	modular	modular	ADJ
cana-395	34	22	metric	metric	NOUN
cana-395	34	23	(	(	PUNCT
cana-395	34	24	gmm	gmm	NOUN
cana-395	34	25	)	)	PUNCT
cana-395	34	26	space	space	NOUN
cana-395	34	27	.	.	PUNCT
cana-395	35	1	then	then	ADV
cana-395	35	2	for	for	ADP
cana-395	35	3	l0	l0	PROPN
cana-395	35	4	∈	∈	PROPN
cana-395	35	5	lt	lt	NOUN
cana-395	35	6	and	and	CCONJ
cana-395	35	7	c	c	X
cana-395	35	8	>	>	X
cana-395	35	9	0	0	NUM
cana-395	35	10	,	,	PUNCT
cana-395	35	11	(	(	PUNCT
cana-395	35	12	i	i	NOUN
cana-395	35	13	)	)	PUNCT
cana-395	35	14	if	if	SCONJ
cana-395	35	15	tρ	tρ	X
cana-395	35	16	(	(	PUNCT
cana-395	35	17	l0	l0	PROPN
cana-395	35	18	,	,	PUNCT
cana-395	35	19	l	l	NOUN
cana-395	35	20	,	,	PUNCT
cana-395	35	21	m	m	NOUN
cana-395	35	22	)	)	PUNCT
cana-395	35	23	<	<	X
cana-395	35	24	c	c	NOUN
cana-395	35	25	,	,	PUNCT
cana-395	35	26	∀	∀	X
cana-395	35	27	ρ	ρ	NOUN
cana-395	35	28	>	>	X
cana-395	35	29	0	0	NUM
cana-395	35	30	,	,	PUNCT
cana-395	35	31	then	then	ADV
cana-395	35	32	l	l	NOUN
cana-395	35	33	,	,	PUNCT
cana-395	35	34	m	m	VERB
cana-395	35	35	∈	∈	PROPN
cana-395	35	36	bt	bt	PROPN
cana-395	35	37	(	(	PUNCT
cana-395	35	38	l0	l0	PROPN
cana-395	35	39	,	,	PUNCT
cana-395	35	40	c	c	NOUN
cana-395	35	41	)	)	PUNCT
cana-395	35	42	.	.	PUNCT
cana-395	36	1	(	(	PUNCT
cana-395	36	2	ii	ii	NOUN
cana-395	36	3	)	)	PUNCT
cana-395	36	4	if	if	SCONJ
cana-395	36	5	m	m	VERB
cana-395	36	6	∈	∈	PROPN
cana-395	36	7	bt	bt	PROPN
cana-395	36	8	(	(	PUNCT
cana-395	36	9	l0	l0	PROPN
cana-395	36	10	,	,	PUNCT
cana-395	36	11	c	c	NOUN
cana-395	36	12	)	)	PUNCT
cana-395	36	13	:	:	PUNCT
cana-395	36	14	bt	bt	PROPN
cana-395	36	15	(	(	PUNCT
cana-395	36	16	m	m	PROPN
cana-395	36	17	,	,	PUNCT
cana-395	36	18	δ	δ	PROPN
cana-395	36	19	)	)	PUNCT
cana-395	36	20	⊆	⊆	NUM
cana-395	36	21	bt	bt	NOUN
cana-395	36	22	(	(	PUNCT
cana-395	36	23	l0	l0	PROPN
cana-395	36	24	,	,	PUNCT
cana-395	36	25	c	c	NOUN
cana-395	36	26	)	)	PUNCT
cana-395	36	27	and	and	CCONJ
cana-395	36	28	δ	δ	PROPN
cana-395	36	29	>	>	PROPN
cana-395	36	30	0	0	PROPN
cana-395	36	31	.	.	PUNCT
cana-395	37	1	definition	definition	NOUN
cana-395	37	2	2.4	2.4	NUM
cana-395	37	3	(	(	PUNCT
cana-395	37	4	azadifa	azadifa	NOUN
cana-395	37	5	et	et	NOUN
cana-395	37	6	al	al	PROPN
cana-395	37	7	.	.	PROPN
cana-395	37	8	,	,	PUNCT
cana-395	37	9	2013	2013	NUM
cana-395	37	10	)	)	PUNCT
cana-395	37	11	assume	assume	VERB
cana-395	37	12	that	that	SCONJ
cana-395	37	13	(	(	PUNCT
cana-395	37	14	l	l	NOUN
cana-395	37	15	,	,	PUNCT
cana-395	37	16	t	t	PROPN
cana-395	37	17	)	)	PUNCT
cana-395	37	18	is	be	AUX
cana-395	37	19	a	a	DET
cana-395	37	20	generalized	generalized	ADJ
cana-395	37	21	(	(	PUNCT
cana-395	37	22	gmms	gmms	NOUN
cana-395	37	23	)	)	PUNCT
cana-395	37	24	modular	modular	ADJ
cana-395	37	25	metric	metric	ADJ
cana-395	37	26	space	space	NOUN
cana-395	37	27	.	.	PUNCT
cana-395	38	1	sequence	sequence	NOUN
cana-395	38	2	{	{	PUNCT
cana-395	38	3	ln	ln	ADJ
cana-395	38	4	}	}	PUNCT
cana-395	38	5	⊆	⊆	NUM
cana-395	38	6	l	l	NOUN
cana-395	38	7	and	and	CCONJ
cana-395	38	8	tl	tl	PROPN
cana-395	38	9	is	be	AUX
cana-395	38	10	tconvergent	tconvergent	NOUN
cana-395	38	11	to	to	ADP
cana-395	38	12	l	l	NOUN
cana-395	38	13	if	if	SCONJ
cana-395	38	14	it	it	PRON
cana-395	38	15	converges	converge	VERB
cana-395	38	16	to	to	ADP
cana-395	38	17	l	l	PROPN
cana-395	38	18	of	of	ADP
cana-395	38	19	τ	τ	PROPN
cana-395	38	20	(	(	PUNCT
cana-395	38	21	t	t	PROPN
cana-395	38	22	ρ	ρ	PROPN
cana-395	38	23	)	)	PUNCT
cana-395	38	24	,	,	PUNCT
cana-395	38	25	∀	∀	NUM
cana-395	38	26	n∈	n∈	NOUN
cana-395	38	27	n	n	NOUN
cana-395	38	28	.	.	PUNCT
cana-395	39	1	proposition	proposition	NOUN
cana-395	39	2	2.4.1	2.4.1	NUM
cana-395	39	3	(	(	PUNCT
cana-395	39	4	azadifa	azadifa	NOUN
cana-395	39	5	et	et	NOUN
cana-395	39	6	al	al	PROPN
cana-395	39	7	.	.	PROPN
cana-395	39	8	,	,	PUNCT
cana-395	39	9	2013	2013	NUM
cana-395	39	10	)	)	PUNCT
cana-395	39	11	assume	assume	VERB
cana-395	39	12	that	that	SCONJ
cana-395	39	13	(	(	PUNCT
cana-395	39	14	l	l	NOUN
cana-395	39	15	,	,	PUNCT
cana-395	39	16	t	t	PROPN
cana-395	39	17	)	)	PUNCT
cana-395	39	18	is	be	AUX
cana-395	39	19	a	a	DET
cana-395	39	20	generalized	generalized	ADJ
cana-395	39	21	(	(	PUNCT
cana-395	39	22	gmms	gmms	NOUN
cana-395	39	23	)	)	PUNCT
cana-395	39	24	modular	modular	ADJ
cana-395	39	25	metric	metric	ADJ
cana-395	39	26	space	space	NOUN
cana-395	39	27	and	and	CCONJ
cana-395	39	28	sequence	sequence	NOUN
cana-395	39	29	{	{	PUNCT
cana-395	39	30	ln	ln	ADJ
cana-395	39	31	}	}	PUNCT
cana-395	39	32	⊆	⊆	NUM
cana-395	39	33	lt	lt	NOUN
cana-395	39	34	,	,	PUNCT
cana-395	39	35	∀	∀	NOUN
cana-395	39	36	n	n	PRON
cana-395	39	37	∈	∈	PROPN
cana-395	39	38	n.	n.	NOUN
cana-395	39	39	then	then	ADV
cana-395	39	40	the	the	DET
cana-395	39	41	followings	following	NOUN
cana-395	39	42	are	be	AUX
cana-395	39	43	satisfied	satisfied	ADJ
cana-395	39	44	:	:	PUNCT
cana-395	39	45	(	(	PUNCT
cana-395	39	46	1	1	X
cana-395	39	47	)	)	PUNCT
cana-395	39	48	sequence{ln	sequence{ln	NOUN
cana-395	39	49	}	}	PUNCT
cana-395	39	50	is	be	AUX
cana-395	39	51	t	t	NOUN
cana-395	39	52	-	-	PUNCT
cana-395	39	53	convergent	convergent	NOUN
cana-395	39	54	to	to	ADP
cana-395	39	55	l.	l.	PROPN
cana-395	39	56	(	(	PUNCT
cana-395	39	57	2	2	NUM
cana-395	39	58	)	)	PUNCT
cana-395	39	59	σ	σ	PROPN
cana-395	39	60	ρ	ρ	PROPN
cana-395	39	61	t	t	PROPN
cana-395	39	62	(	(	PUNCT
cana-395	39	63	ln	ln	PROPN
cana-395	39	64	,	,	PUNCT
cana-395	39	65	l	l	NOUN
cana-395	39	66	)	)	PUNCT
cana-395	40	1	→	→	SYM
cana-395	40	2	0	0	NUM
cana-395	40	3	when	when	SCONJ
cana-395	40	4	n	n	X
cana-395	40	5	→	→	SYM
cana-395	40	6	∞	∞	PROPN
cana-395	40	7	,	,	PUNCT
cana-395	40	8	(	(	PUNCT
cana-395	40	9	3	3	X
cana-395	40	10	)	)	PUNCT
cana-395	40	11	t	t	NOUN
cana-395	40	12	ρ	ρ	PROPN
cana-395	40	13	(	(	PUNCT
cana-395	40	14	ln	ln	ADJ
cana-395	40	15	,	,	PUNCT
cana-395	40	16	ln	ln	ADJ
cana-395	40	17	,	,	PUNCT
cana-395	40	18	l	l	NOUN
cana-395	40	19	)	)	PUNCT
cana-395	40	20	→	→	SYM
cana-395	40	21	0	0	NUM
cana-395	41	1	when	when	SCONJ
cana-395	41	2	n	n	X
cana-395	41	3	→∞	→∞	PROPN
cana-395	41	4	for	for	ADP
cana-395	41	5	all	all	DET
cana-395	41	6	ρ	ρ	PROPN
cana-395	41	7	>	>	X
cana-395	41	8	0	0	NUM
cana-395	41	9	;	;	PUNCT
cana-395	41	10	(	(	PUNCT
cana-395	41	11	4	4	X
cana-395	41	12	)	)	PUNCT
cana-395	41	13	t	t	NOUN
cana-395	41	14	ρ	ρ	PROPN
cana-395	41	15	(	(	PUNCT
cana-395	41	16	ln	ln	PROPN
cana-395	41	17	,	,	PUNCT
cana-395	41	18	l	l	NOUN
cana-395	41	19	,	,	PUNCT
cana-395	41	20	l	l	NOUN
cana-395	41	21	)	)	PUNCT
cana-395	41	22	→	→	SYM
cana-395	41	23	0	0	NUM
cana-395	41	24	as	as	ADP
cana-395	41	25	n	n	X
cana-395	41	26	→∞	→∞	PROPN
cana-395	41	27	for	for	ADP
cana-395	41	28	all	all	DET
cana-395	41	29	ρ	ρ	PROPN
cana-395	41	30	>	>	X
cana-395	41	31	0	0	NUM
cana-395	41	32	;	;	PUNCT
cana-395	41	33	(	(	PUNCT
cana-395	41	34	5	5	X
cana-395	41	35	)	)	PUNCT
cana-395	41	36	t	t	NOUN
cana-395	41	37	ρ	ρ	PROPN
cana-395	41	38	(	(	PUNCT
cana-395	41	39	lm	lm	INTJ
cana-395	41	40	,	,	PUNCT
cana-395	41	41	ln	ln	ADJ
cana-395	41	42	,	,	PUNCT
cana-395	41	43	l	l	NOUN
cana-395	41	44	)	)	PUNCT
cana-395	41	45	→	→	SYM
cana-395	41	46	0	0	NUM
cana-395	41	47	when	when	SCONJ
cana-395	41	48	m	m	PROPN
cana-395	41	49	,	,	PUNCT
cana-395	41	50	n	n	PROPN
cana-395	41	51	→∞	→∞	NUM
cana-395	41	52	,	,	PUNCT
cana-395	41	53	∀	∀	X
cana-395	41	54	ρ	ρ	NOUN
cana-395	41	55	>	>	X
cana-395	41	56	0	0	PROPN
cana-395	41	57	.	.	PUNCT
cana-395	42	1	definition	definition	NOUN
cana-395	42	2	2.5	2.5	NUM
cana-395	42	3	(	(	PUNCT
cana-395	42	4	azadifa	azadifa	NOUN
cana-395	42	5	et	et	NOUN
cana-395	42	6	al	al	PROPN
cana-395	42	7	.	.	PROPN
cana-395	42	8	,	,	PUNCT
cana-395	42	9	2013	2013	NUM
cana-395	42	10	)	)	PUNCT
cana-395	42	11	assume	assume	VERB
cana-395	42	12	that	that	SCONJ
cana-395	42	13	(	(	PUNCT
cana-395	42	14	l	l	NOUN
cana-395	42	15	,	,	PUNCT
cana-395	42	16	t	t	PROPN
cana-395	42	17	)	)	PUNCT
cana-395	42	18	is	be	AUX
cana-395	42	19	a	a	DET
cana-395	42	20	generalized	generalized	ADJ
cana-395	42	21	(	(	PUNCT
cana-395	42	22	gmms	gmms	NOUN
cana-395	42	23	)	)	PUNCT
cana-395	42	24	modular	modular	ADJ
cana-395	42	25	metric	metric	ADJ
cana-395	42	26	space	space	NOUN
cana-395	42	27	.	.	PUNCT
cana-395	43	1	then	then	ADV
cana-395	43	2	sequence	sequence	NOUN
cana-395	43	3	{	{	PUNCT
cana-395	43	4	ln	ln	ADJ
cana-395	43	5	}	}	PUNCT
cana-395	43	6	⊆	⊆	NUM
cana-395	43	7	lt	lt	NOUN
cana-395	43	8	,	,	PUNCT
cana-395	43	9	is	be	AUX
cana-395	43	10	called	call	VERB
cana-395	43	11	tcauchy	tcauchy	NOUN
cana-395	43	12	sequence	sequence	NOUN
cana-395	43	13	if	if	SCONJ
cana-395	43	14	,	,	PUNCT
cana-395	43	15	nε	nε	PROPN
cana-395	43	16	∈	∈	PROPN
cana-395	43	17	n	n	CCONJ
cana-395	43	18	:	:	PUNCT
cana-395	43	19	t	t	PROPN
cana-395	43	20	ρ	ρ	PROPN
cana-395	43	21	(	(	PUNCT
cana-395	43	22	ln	ln	ADJ
cana-395	43	23	,	,	PUNCT
cana-395	43	24	lm	lm	INTJ
cana-395	43	25	,	,	PUNCT
cana-395	43	26	lq	lq	PROPN
cana-395	43	27	)	)	PUNCT
cana-395	43	28	<	<	X
cana-395	43	29	ε	ε	PROPN
cana-395	43	30	,	,	PUNCT
cana-395	43	31	∀	∀	X
cana-395	43	32	communications	communication	NOUN
cana-395	43	33	on	on	ADP
cana-395	43	34	applied	apply	VERB
cana-395	43	35	nonlinear	nonlinear	ADJ
cana-395	43	36	analysis	analysis	NOUN
cana-395	43	37	issn	issn	NOUN
cana-395	43	38	:	:	PUNCT
cana-395	43	39	1074	1074	NUM
cana-395	43	40	-	-	PUNCT
cana-395	43	41	133x	133x	NUM
cana-395	43	42	vol	vol	NOUN
cana-395	43	43	31	31	NUM
cana-395	43	44	no	no	NOUN
cana-395	43	45	.	.	NOUN
cana-395	43	46	1	1	NUM
cana-395	43	47	(	(	PUNCT
cana-395	43	48	2024	2024	NUM
cana-395	43	49	)	)	PUNCT
cana-395	43	50	202	202	NUM
cana-395	43	51	https://internationalpubls.com	https://internationalpubls.com	X
cana-395	43	52	,	,	PUNCT
cana-395	43	53	n	n	CCONJ
cana-395	43	54	,	,	PUNCT
cana-395	43	55	m	m	PROPN
cana-395	43	56	,	,	PUNCT
cana-395	43	57	q	q	PUNCT
cana-395	43	58	≥	≥	NOUN
cana-395	43	59	nε	nε	ADJ
cana-395	43	60	and	and	CCONJ
cana-395	43	61	for	for	ADP
cana-395	43	62	every	every	DET
cana-395	43	63	ε	ε	PROPN
cana-395	43	64	,	,	PUNCT
cana-395	43	65	ρ	ρ	PROPN
cana-395	43	66	>	>	X
cana-395	43	67	0	0	PROPN
cana-395	43	68	.	.	PUNCT
cana-395	44	1	if	if	SCONJ
cana-395	44	2	every	every	DET
cana-395	44	3	tcauchy	tcauchy	NOUN
cana-395	44	4	sequence	sequence	NOUN
cana-395	44	5	in	in	ADP
cana-395	44	6	a	a	DET
cana-395	44	7	gmm	gmm	NOUN
cana-395	44	8	-	-	PUNCT
cana-395	44	9	space	space	NOUN
cana-395	44	10	l	l	NOUN
cana-395	44	11	is	be	AUX
cana-395	44	12	a	a	DET
cana-395	44	13	tconvergent	tconvergent	NOUN
cana-395	44	14	sequence	sequence	NOUN
cana-395	44	15	in	in	ADP
cana-395	44	16	that	that	DET
cana-395	44	17	space	space	NOUN
cana-395	44	18	,	,	PUNCT
cana-395	44	19	the	the	DET
cana-395	44	20	space	space	NOUN
cana-395	44	21	is	be	AUX
cana-395	44	22	said	say	VERB
cana-395	44	23	to	to	PART
cana-395	44	24	be	be	AUX
cana-395	44	25	"	"	PUNCT
cana-395	44	26	t	t	NOUN
cana-395	44	27	-	-	PUNCT
cana-395	44	28	complete	complete	ADJ
cana-395	44	29	.	.	PUNCT
cana-395	44	30	"	"	PUNCT
cana-395	45	1	proposition	proposition	NOUN
cana-395	45	2	2.5.1	2.5.1	NUM
cana-395	45	3	(	(	PUNCT
cana-395	45	4	azadifa	azadifa	NOUN
cana-395	45	5	et	et	NOUN
cana-395	45	6	al	al	PROPN
cana-395	45	7	.	.	PROPN
cana-395	45	8	,	,	PUNCT
cana-395	45	9	2013	2013	NUM
cana-395	45	10	)	)	PUNCT
cana-395	45	11	assume	assume	VERB
cana-395	45	12	that	that	SCONJ
cana-395	45	13	(	(	PUNCT
cana-395	45	14	l	l	NOUN
cana-395	45	15	,	,	PUNCT
cana-395	45	16	t	t	PROPN
cana-395	45	17	)	)	PUNCT
cana-395	45	18	is	be	AUX
cana-395	45	19	a	a	DET
cana-395	45	20	generalized(gmms	generalized(gmms	NOUN
cana-395	45	21	)	)	PUNCT
cana-395	45	22	modular	modular	ADJ
cana-395	45	23	metric	metric	ADJ
cana-395	45	24	space	space	NOUN
cana-395	45	25	and	and	CCONJ
cana-395	45	26	sequence	sequence	NOUN
cana-395	45	27	{	{	PUNCT
cana-395	45	28	ln	ln	ADJ
cana-395	45	29	}	}	PUNCT
cana-395	45	30	⊆	⊆	NUM
cana-395	45	31	lt	lt	NOUN
cana-395	45	32	for	for	ADP
cana-395	45	33	all	all	DET
cana-395	45	34	n∈n	n∈n	NOUN
cana-395	45	35	.	.	PUNCT
cana-395	46	1	then	then	ADV
cana-395	46	2	the	the	DET
cana-395	46	3	followings	following	NOUN
cana-395	46	4	are	be	AUX
cana-395	46	5	equivalent	equivalent	ADJ
cana-395	46	6	:	:	PUNCT
cana-395	46	7	(	(	PUNCT
cana-395	46	8	1	1	X
cana-395	46	9	)	)	PUNCT
cana-395	46	10	sequence	sequence	NOUN
cana-395	46	11	{	{	PUNCT
cana-395	46	12	ln	ln	ADJ
cana-395	46	13	}	}	PUNCT
cana-395	46	14	is	be	AUX
cana-395	46	15	a	a	DET
cana-395	46	16	t	t	PROPN
cana-395	46	17	-	-	PUNCT
cana-395	46	18	cauchy	cauchy	NOUN
cana-395	46	19	sequence	sequence	NOUN
cana-395	46	20	.	.	PUNCT
cana-395	47	1	(	(	PUNCT
cana-395	47	2	2	2	X
cana-395	47	3	)	)	PUNCT
cana-395	47	4	we	we	PRON
cana-395	47	5	can	can	AUX
cana-395	47	6	locate	locate	VERB
cana-395	47	7	nε	nε	PROPN
cana-395	47	8	∈	∈	PROPN
cana-395	47	9	n	n	CCONJ
cana-395	47	10	:	:	PUNCT
cana-395	47	11	t	t	PROPN
cana-395	47	12	ρ	ρ	PROPN
cana-395	47	13	(	(	PUNCT
cana-395	47	14	ln	ln	ADJ
cana-395	47	15	,	,	PUNCT
cana-395	47	16	lm	lm	INTJ
cana-395	47	17	,	,	PUNCT
cana-395	47	18	lm	lm	PROPN
cana-395	47	19	)	)	PUNCT
cana-395	47	20	<	<	X
cana-395	47	21	ε	ε	PROPN
cana-395	47	22	,	,	PUNCT
cana-395	47	23	for	for	ADP
cana-395	47	24	each	each	DET
cana-395	47	25	ε	ε	PROPN
cana-395	47	26	>	>	X
cana-395	47	27	0	0	PROPN
cana-395	47	28	,	,	PUNCT
cana-395	47	29	ρ	ρ	PROPN
cana-395	47	30	>	>	X
cana-395	47	31	0	0	NUM
cana-395	47	32	,	,	PUNCT
cana-395	47	33	for	for	ADP
cana-395	47	34	every	every	DET
cana-395	47	35	n	n	CCONJ
cana-395	47	36	,	,	PUNCT
cana-395	47	37	m	m	VERB
cana-395	47	38	≥	≥	NOUN
cana-395	47	39	nε	nε	ADJ
cana-395	47	40	.	.	PUNCT
cana-395	48	1	(	(	PUNCT
cana-395	48	2	3	3	X
cana-395	48	3	)	)	PUNCT
cana-395	48	4	sequence{ln	sequence{ln	NOUN
cana-395	48	5	}	}	PUNCT
cana-395	48	6	is	be	AUX
cana-395	48	7	a	a	DET
cana-395	48	8	cauchy	cauchy	ADJ
cana-395	48	9	sequence	sequence	NOUN
cana-395	48	10	.	.	PUNCT
cana-395	49	1	proposition	proposition	NOUN
cana-395	49	2	2.5.2	2.5.2	NUM
cana-395	49	3	(	(	PUNCT
cana-395	49	4	azadifa	azadifa	NOUN
cana-395	49	5	et	et	NOUN
cana-395	49	6	al	al	PROPN
cana-395	49	7	.	.	PROPN
cana-395	49	8	,	,	PUNCT
cana-395	49	9	2013	2013	NUM
cana-395	49	10	)	)	PUNCT
cana-395	49	11	assume	assume	VERB
cana-395	49	12	that	that	SCONJ
cana-395	49	13	(	(	PUNCT
cana-395	49	14	l	l	NOUN
cana-395	49	15	,	,	PUNCT
cana-395	49	16	t	t	PROPN
cana-395	49	17	)	)	PUNCT
cana-395	49	18	is	be	AUX
cana-395	49	19	a	a	DET
cana-395	49	20	(	(	PUNCT
cana-395	49	21	gmm	gmm	NOUN
cana-395	49	22	)	)	PUNCT
cana-395	49	23	space	space	NOUN
cana-395	49	24	.	.	PUNCT
cana-395	50	1	then	then	ADV
cana-395	50	2	for	for	ADP
cana-395	50	3	l	l	NOUN
cana-395	50	4	×	×	PROPN
cana-395	50	5	l	l	NOUN
cana-395	50	6	×	×	NOUN
cana-395	50	7	l	l	NOUN
cana-395	50	8	×	×	NOUN
cana-395	50	9	q0	q0	NOUN
cana-395	50	10	,	,	PUNCT
cana-395	50	11	t	t	PROPN
cana-395	50	12	is	be	AUX
cana-395	50	13	a	a	DET
cana-395	50	14	continuous	continuous	ADJ
cana-395	50	15	function	function	NOUN
cana-395	50	16	.	.	PUNCT
cana-395	51	1	let	let	VERB
cana-395	51	2	us	we	PRON
cana-395	51	3	assume	assume	VERB
cana-395	51	4	that	that	SCONJ
cana-395	51	5	a	a	DET
cana-395	51	6	gmm	gmm	NOUN
cana-395	51	7	-	-	PUNCT
cana-395	51	8	space	space	NOUN
cana-395	51	9	(	(	PUNCT
cana-395	51	10	l	l	NOUN
cana-395	51	11	,	,	PUNCT
cana-395	51	12	t	t	PROPN
cana-395	51	13	)	)	PUNCT
cana-395	51	14	has	have	VERB
cana-395	51	15	two	two	NUM
cana-395	51	16	(	(	PUNCT
cana-395	51	17	nonempty	nonempty	NOUN
cana-395	51	18	)	)	PUNCT
cana-395	51	19	subsets	subset	NOUN
cana-395	51	20	,	,	PUNCT
cana-395	51	21	t	t	PROPN
cana-395	51	22	and	and	CCONJ
cana-395	51	23	w.	w.	PROPN
cana-395	51	24	t	t	PROPN
cana-395	51	25	ρ	ρ	PROPN
cana-395	51	26	(	(	PUNCT
cana-395	51	27	l	l	PROPN
cana-395	51	28	,	,	PUNCT
cana-395	51	29	t	t	PROPN
cana-395	51	30	,	,	PUNCT
cana-395	51	31	w	w	PROPN
cana-395	51	32	)	)	PUNCT
cana-395	51	33	=	=	PUNCT
cana-395	51	34	inf{t	inf{t	NOUN
cana-395	51	35	ρ	ρ	NOUN
cana-395	51	36	(	(	PUNCT
cana-395	51	37	l	l	PROPN
cana-395	51	38	,	,	PUNCT
cana-395	51	39	t	t	PROPN
cana-395	51	40	,	,	PUNCT
cana-395	51	41	w	w	PROPN
cana-395	51	42	):	):	PUNCT
cana-395	51	43	t	t	PROPN
cana-395	51	44	∈	∈	PROPN
cana-395	51	45	t	t	PROPN
cana-395	51	46	,	,	PUNCT
cana-395	51	47	w	w	PROPN
cana-395	51	48	∈	∈	PROPN
cana-395	51	49	w	w	PROPN
cana-395	51	50	}	}	PUNCT
cana-395	51	51	for	for	ADP
cana-395	51	52	l	l	NOUN
cana-395	51	53	∈	∈	PROPN
cana-395	51	54	l	l	NOUN
cana-395	51	55	and	and	CCONJ
cana-395	51	56	ρ	ρ	NOUN
cana-395	51	57	>	>	X
cana-395	51	58	0	0	NUM
cana-395	51	59	,	,	PUNCT
cana-395	51	60	proposition	proposition	NOUN
cana-395	51	61	2.5.3	2.5.3	NUM
cana-395	51	62	(	(	PUNCT
cana-395	51	63	azadifa	azadifa	NOUN
cana-395	51	64	et	et	PROPN
cana-395	51	65	al	al	PROPN
cana-395	51	66	.	.	PROPN
cana-395	51	67	,	,	PUNCT
cana-395	51	68	2013	2013	NUM
cana-395	51	69	)	)	PUNCT
cana-395	51	70	assume	assume	VERB
cana-395	51	71	that	that	SCONJ
cana-395	51	72	(	(	PUNCT
cana-395	51	73	l	l	NOUN
cana-395	51	74	,	,	PUNCT
cana-395	51	75	t	t	PROPN
cana-395	51	76	)	)	PUNCT
cana-395	51	77	is	be	AUX
cana-395	51	78	a	a	DET
cana-395	51	79	generalized	generalize	VERB
cana-395	51	80	modular	modular	ADJ
cana-395	51	81	metric	metric	ADJ
cana-395	51	82	space	space	NOUN
cana-395	51	83	.	.	PUNCT
cana-395	52	1	for	for	ADP
cana-395	52	2	each	each	DET
cana-395	52	3	m	m	NOUN
cana-395	52	4	,	,	PUNCT
cana-395	52	5	n	n	CCONJ
cana-395	52	6	,	,	PUNCT
cana-395	52	7	p	p	PROPN
cana-395	52	8	∈	∈	PROPN
cana-395	52	9	h0(l	h0(l	PROPN
cana-395	52	10	)	)	PUNCT
cana-395	52	11	,	,	PUNCT
cana-395	52	12	the	the	DET
cana-395	52	13	function	function	NOUN
cana-395	52	14	δ	δ	PROPN
cana-395	52	15	|→	|→	PROPN
cana-395	52	16	supm∈m	supm∈m	PROPN
cana-395	52	17	tρ	tρ	X
cana-395	52	18	(	(	PUNCT
cana-395	52	19	m	m	PROPN
cana-395	52	20	,	,	PUNCT
cana-395	52	21	n	n	CCONJ
cana-395	52	22	,	,	PUNCT
cana-395	52	23	p	p	NOUN
cana-395	52	24	)	)	PUNCT
cana-395	52	25	is	be	AUX
cana-395	52	26	continuous	continuous	ADJ
cana-395	52	27	on	on	ADP
cana-395	52	28	q0	q0	ADJ
cana-395	52	29	proposition	proposition	NOUN
cana-395	52	30	2.5.4	2.5.4	NUM
cana-395	52	31	(	(	PUNCT
cana-395	52	32	alihajimohammad	alihajimohammad	NOUN
cana-395	52	33	and	and	CCONJ
cana-395	52	34	saadati	saadati	NOUN
cana-395	52	35	,	,	PUNCT
cana-395	52	36	2021	2021	NUM
cana-395	52	37	)	)	PUNCT
cana-395	53	1	assume	assume	VERB
cana-395	53	2	that	that	SCONJ
cana-395	53	3	(	(	PUNCT
cana-395	53	4	l	l	NOUN
cana-395	53	5	,	,	PUNCT
cana-395	53	6	t	t	PROPN
cana-395	53	7	)	)	PUNCT
cana-395	53	8	is	be	AUX
cana-395	53	9	a	a	DET
cana-395	53	10	generalized	generalized	ADJ
cana-395	53	11	(	(	PUNCT
cana-395	53	12	gmms	gmms	NOUN
cana-395	53	13	)	)	PUNCT
cana-395	53	14	modular	modular	ADJ
cana-395	53	15	metric	metric	ADJ
cana-395	53	16	space	space	NOUN
cana-395	53	17	.	.	PUNCT
cana-395	54	1	suppose	suppose	VERB
cana-395	54	2	sequence	sequence	NOUN
cana-395	54	3	{	{	PUNCT
cana-395	54	4	ln}⊆	ln}⊆	PROPN
cana-395	54	5	l	l	NOUN
cana-395	54	6	:	:	PUNCT
cana-395	54	7	tφn	tφn	PROPN
cana-395	54	8	(	(	PUNCT
cana-395	54	9	ρ	ρ	PROPN
cana-395	54	10	)	)	PUNCT
cana-395	54	11	(	(	PUNCT
cana-395	54	12	l	l	NOUN
cana-395	54	13	n	n	X
cana-395	54	14	,	,	PUNCT
cana-395	54	15	l	l	PROPN
cana-395	54	16	n+1	n+1	PROPN
cana-395	54	17	,	,	PUNCT
cana-395	54	18	l	l	PROPN
cana-395	54	19	n+1	n+1	NOUN
cana-395	54	20	)	)	PUNCT
cana-395	54	21	≤	≤	NOUN
cana-395	54	22	t	t	PROPN
cana-395	54	23	ρ	ρ	X
cana-395	54	24	(	(	PUNCT
cana-395	54	25	l	l	NOUN
cana-395	54	26	0	0	NUM
cana-395	54	27	,	,	PUNCT
cana-395	54	28	l	l	NOUN
cana-395	54	29	1	1	NUM
cana-395	54	30	,	,	PUNCT
cana-395	54	31	l	l	NOUN
cana-395	54	32	1	1	X
cana-395	54	33	)	)	PUNCT
cana-395	54	34	for	for	ADP
cana-395	54	35	all	all	DET
cana-395	54	36	ρ	ρ	NOUN
cana-395	54	37	∈	∈	PROPN
cana-395	54	38	q0	q0	NOUN
cana-395	54	39	.	.	PUNCT
cana-395	55	1	then{ln	then{ln	X
cana-395	55	2	}	}	PUNCT
cana-395	55	3	is	be	AUX
cana-395	55	4	a	a	DET
cana-395	55	5	t	t	PROPN
cana-395	55	6	-	-	PUNCT
cana-395	55	7	cauchy	cauchy	NOUN
cana-395	55	8	sequence	sequence	NOUN
cana-395	55	9	.	.	PUNCT
cana-395	56	1	proposition	proposition	NOUN
cana-395	56	2	2.5.5	2.5.5	NUM
cana-395	56	3	(	(	PUNCT
cana-395	56	4	alihajimohammad	alihajimohammad	NOUN
cana-395	56	5	and	and	CCONJ
cana-395	56	6	saadati	saadati	NOUN
cana-395	56	7	,	,	PUNCT
cana-395	56	8	2021	2021	NUM
cana-395	56	9	)	)	PUNCT
cana-395	56	10	let	let	VERB
cana-395	56	11	(	(	PUNCT
cana-395	56	12	l	l	NOUN
cana-395	56	13	,	,	PUNCT
cana-395	56	14	t	t	PROPN
cana-395	56	15	)	)	PUNCT
cana-395	56	16	is	be	AUX
cana-395	56	17	a	a	DET
cana-395	56	18	generalized	generalized	ADJ
cana-395	56	19	(	(	PUNCT
cana-395	56	20	gmm	gmm	NOUN
cana-395	56	21	)	)	PUNCT
cana-395	56	22	modular	modular	ADJ
cana-395	56	23	metric	metric	ADJ
cana-395	56	24	space	space	NOUN
cana-395	56	25	.	.	PUNCT
cana-395	57	1	if	if	SCONJ
cana-395	57	2	t	t	PROPN
cana-395	57	3	ρ	ρ	X
cana-395	57	4	(	(	PUNCT
cana-395	57	5	l	l	PROPN
cana-395	57	6	,	,	PUNCT
cana-395	57	7	m	m	PROPN
cana-395	57	8	,	,	PUNCT
cana-395	57	9	n	n	CCONJ
cana-395	57	10	)	)	PUNCT
cana-395	57	11	=	=	SYM
cana-395	58	1	c	c	NOUN
cana-395	58	2	for	for	ADP
cana-395	58	3	all	all	DET
cana-395	58	4	l	l	NOUN
cana-395	58	5	,	,	PUNCT
cana-395	58	6	m	m	PROPN
cana-395	58	7	,	,	PUNCT
cana-395	58	8	n	n	PROPN
cana-395	58	9	∈	∈	PROPN
cana-395	58	10	l	l	NOUN
cana-395	58	11	and	and	CCONJ
cana-395	58	12	ρ	ρ	PROPN
cana-395	58	13	∈	∈	PROPN
cana-395	58	14	q0	q0	NOUN
cana-395	58	15	,	,	PUNCT
cana-395	58	16	then	then	ADV
cana-395	58	17	c	c	NOUN
cana-395	58	18	=	=	SYM
cana-395	58	19	0	0	X
cana-395	58	20	.	.	PUNCT
cana-395	59	1	lemma	lemma	PROPN
cana-395	59	2	2.6	2.6	NUM
cana-395	59	3	let	let	VERB
cana-395	59	4	(	(	PUNCT
cana-395	59	5	l	l	NOUN
cana-395	59	6	,	,	PUNCT
cana-395	59	7	t	t	PROPN
cana-395	59	8	)	)	PUNCT
cana-395	59	9	is	be	AUX
cana-395	59	10	a	a	DET
cana-395	59	11	(	(	PUNCT
cana-395	59	12	gmm	gmm	NOUN
cana-395	59	13	)	)	PUNCT
cana-395	59	14	space	space	NOUN
cana-395	59	15	.	.	PUNCT
cana-395	60	1	then	then	ADV
cana-395	60	2	,	,	PUNCT
cana-395	60	3	for	for	ADP
cana-395	60	4	each	each	DET
cana-395	60	5	l	l	NOUN
cana-395	60	6	∈	∈	PROPN
cana-395	60	7	l	l	NOUN
cana-395	60	8	,	,	PUNCT
cana-395	60	9	m	m	PROPN
cana-395	60	10	,	,	PUNCT
cana-395	60	11	n	n	PRON
cana-395	60	12	∈	∈	PROPN
cana-395	60	13	h0(l	h0(l	PROPN
cana-395	60	14	)	)	PUNCT
cana-395	60	15	and	and	CCONJ
cana-395	60	16	ρ	ρ	PROPN
cana-395	60	17	∈	∈	PROPN
cana-395	60	18	q0	q0	NOUN
cana-395	60	19	,	,	PUNCT
cana-395	60	20	there	there	PRON
cana-395	60	21	are	be	VERB
cana-395	60	22	m0	m0	PROPN
cana-395	60	23	∈	∈	PROPN
cana-395	60	24	m	m	PROPN
cana-395	60	25	,	,	PUNCT
cana-395	60	26	n0	n0	PROPN
cana-395	60	27	∈	∈	PROPN
cana-395	60	28	n	n	PRON
cana-395	60	29	such	such	ADJ
cana-395	60	30	that	that	SCONJ
cana-395	60	31	t(l	t(l	PROPN
cana-395	60	32	,	,	PUNCT
cana-395	60	33	m	m	INTJ
cana-395	60	34	,	,	PUNCT
cana-395	60	35	n)=	n)=	PROPN
cana-395	60	36	t	t	PROPN
cana-395	60	37	ρ	ρ	PROPN
cana-395	60	38	(	(	PUNCT
cana-395	60	39	l	l	NOUN
cana-395	60	40	,	,	PUNCT
cana-395	60	41	m0,n0	m0,n0	PROPN
cana-395	60	42	)	)	PUNCT
cana-395	60	43	.	.	PUNCT
cana-395	61	1	proof	proof	NOUN
cana-395	61	2	let	let	VERB
cana-395	61	3	l	l	PROPN
cana-395	61	4	∈	∈	PROPN
cana-395	61	5	l	l	NOUN
cana-395	61	6	,	,	PUNCT
cana-395	61	7	m	m	PROPN
cana-395	61	8	,	,	PUNCT
cana-395	61	9	n	n	PRON
cana-395	61	10	∈	∈	PROPN
cana-395	61	11	h0(l	h0(l	PROPN
cana-395	61	12	)	)	PUNCT
cana-395	61	13	and	and	CCONJ
cana-395	61	14	ρ	ρ	NOUN
cana-395	61	15	>	>	X
cana-395	61	16	0	0	NUM
cana-395	61	17	.	.	PUNCT
cana-395	62	1	by	by	ADP
cana-395	62	2	proposition	proposition	NOUN
cana-395	62	3	2.5.2	2.5.2	NUM
cana-395	62	4	.	.	PUNCT
cana-395	63	1	the	the	DET
cana-395	63	2	functions	function	NOUN
cana-395	63	3	t	t	PROPN
cana-395	63	4	,	,	PUNCT
cana-395	63	5	u	u	PROPN
cana-395	63	6	|→	|→	PROPN
cana-395	63	7	t	t	PROPN
cana-395	63	8	ρ	ρ	X
cana-395	63	9	(	(	PUNCT
cana-395	63	10	l	l	PROPN
cana-395	63	11	,	,	PUNCT
cana-395	63	12	m	m	PROPN
cana-395	63	13	,	,	PUNCT
cana-395	63	14	n	n	CCONJ
cana-395	63	15	)	)	PUNCT
cana-395	63	16	are	be	AUX
cana-395	63	17	continuous	continuous	ADJ
cana-395	63	18	.	.	PUNCT
cana-395	64	1	thus	thus	ADV
cana-395	64	2	,	,	PUNCT
cana-395	64	3	by	by	ADP
cana-395	64	4	compactness	compactness	NOUN
cana-395	64	5	of	of	ADP
cana-395	64	6	m	m	PROPN
cana-395	64	7	and	and	CCONJ
cana-395	64	8	n	n	CCONJ
cana-395	64	9	,	,	PUNCT
cana-395	64	10	∃	∃	PROPN
cana-395	64	11	m0	m0	PROPN
cana-395	64	12	∈	∈	PROPN
cana-395	64	13	m	m	PROPN
cana-395	64	14	,	,	PUNCT
cana-395	64	15	n0	n0	PROPN
cana-395	64	16	∈	∈	PROPN
cana-395	64	17	n	n	CCONJ
cana-395	64	18	:	:	PUNCT
cana-395	64	19	inf	inf	PROPN
cana-395	64	20	t	t	PROPN
cana-395	64	21	ρ	ρ	PROPN
cana-395	64	22	(	(	PUNCT
cana-395	64	23	l	l	PROPN
cana-395	64	24	,	,	PUNCT
cana-395	64	25	m	m	PROPN
cana-395	64	26	,	,	PUNCT
cana-395	64	27	n)=	n)=	PROPN
cana-395	64	28	t	t	PROPN
cana-395	64	29	ρ	ρ	PROPN
cana-395	64	30	(	(	PUNCT
cana-395	64	31	l	l	NOUN
cana-395	64	32	,	,	PUNCT
cana-395	64	33	m0	m0	NOUN
cana-395	64	34	,	,	PUNCT
cana-395	64	35	n0	n0	NUM
cana-395	64	36	)	)	PUNCT
cana-395	64	37	,	,	PUNCT
cana-395	64	38	for	for	ADP
cana-395	64	39	all	all	DET
cana-395	64	40	m	m	NOUN
cana-395	64	41	∈m	∈m	NOUN
cana-395	64	42	,	,	PUNCT
cana-395	64	43	n	n	PRON
cana-395	64	44	∈	∈	PROPN
cana-395	64	45	n	n	CCONJ
cana-395	64	46	lemma	lemma	PROPN
cana-395	64	47	2.7	2.7	NUM
cana-395	64	48	assume	assume	VERB
cana-395	64	49	that	that	SCONJ
cana-395	64	50	(	(	PUNCT
cana-395	64	51	l	l	NOUN
cana-395	64	52	,	,	PUNCT
cana-395	64	53	t	t	PROPN
cana-395	64	54	)	)	PUNCT
cana-395	64	55	is	be	AUX
cana-395	64	56	a	a	DET
cana-395	64	57	generalized	generalized	ADJ
cana-395	64	58	(	(	PUNCT
cana-395	64	59	gmms	gmms	NOUN
cana-395	64	60	)	)	PUNCT
cana-395	64	61	modular	modular	ADJ
cana-395	64	62	metric	metric	ADJ
cana-395	64	63	space	space	NOUN
cana-395	64	64	.then	.then	NOUN
cana-395	64	65	,	,	PUNCT
cana-395	64	66	for	for	ADP
cana-395	64	67	every	every	DET
cana-395	64	68	m	m	NOUN
cana-395	64	69	∈	∈	NOUN
cana-395	64	70	h0(l	h0(l	PROPN
cana-395	64	71	)	)	PUNCT
cana-395	64	72	,	,	PUNCT
cana-395	64	73	n	n	CCONJ
cana-395	64	74	,	,	PUNCT
cana-395	64	75	p	p	PRON
cana-395	64	76	∈	∈	PROPN
cana-395	64	77	f0(l	f0(l	NOUN
cana-395	64	78	)	)	PUNCT
cana-395	64	79	and	and	CCONJ
cana-395	64	80	ρ	ρ	PROPN
cana-395	64	81	∈	∈	PROPN
cana-395	64	82	q0	q0	NOUN
cana-395	64	83	we	we	PRON
cana-395	64	84	can	can	AUX
cana-395	64	85	find	find	VERB
cana-395	64	86	m0	m0	NOUN
cana-395	64	87	∈	∈	PROPN
cana-395	64	88	m	m	VERB
cana-395	64	89	such	such	ADJ
cana-395	64	90	that	that	DET
cana-395	64	91	sup	sup	NOUN
cana-395	64	92	t	t	PROPN
cana-395	64	93	ρ	ρ	PROPN
cana-395	64	94	(	(	PUNCT
cana-395	64	95	m	m	PROPN
cana-395	64	96	,	,	PUNCT
cana-395	64	97	n	n	CCONJ
cana-395	64	98	,	,	PUNCT
cana-395	64	99	p	p	NOUN
cana-395	64	100	)	)	PUNCT
cana-395	64	101	=	=	SYM
cana-395	64	102	t	t	PROPN
cana-395	64	103	ρ	ρ	PROPN
cana-395	64	104	(	(	PUNCT
cana-395	64	105	m0	m0	PROPN
cana-395	64	106	,	,	PUNCT
cana-395	64	107	n	n	CCONJ
cana-395	64	108	,	,	PUNCT
cana-395	64	109	p	p	NOUN
cana-395	64	110	)	)	PUNCT
cana-395	64	111	.	.	PUNCT
cana-395	65	1	proof	proof	NOUN
cana-395	65	2	put	put	VERB
cana-395	65	3	δ	δ	X
cana-395	65	4	=	=	SYM
cana-395	65	5	supt∈t	supt∈t	PROPN
cana-395	65	6	t	t	PROPN
cana-395	65	7	ρ	ρ	PROPN
cana-395	65	8	(	(	PUNCT
cana-395	65	9	m	m	PROPN
cana-395	65	10	,	,	PUNCT
cana-395	65	11	n	n	CCONJ
cana-395	65	12	,	,	PUNCT
cana-395	65	13	p	p	NOUN
cana-395	65	14	)	)	PUNCT
cana-395	65	15	.	.	PUNCT
cana-395	66	1	then	then	ADV
cana-395	66	2	we	we	PRON
cana-395	66	3	get	get	VERB
cana-395	66	4	a	a	DET
cana-395	66	5	sequence	sequence	NOUN
cana-395	66	6	(	(	PUNCT
cana-395	66	7	mn)n	mn)n	PROPN
cana-395	66	8	in	in	ADP
cana-395	66	9	m	m	PROPN
cana-395	66	10	:	:	PUNCT
cana-395	66	11	δ	δ	X
cana-395	66	12	–	–	PUNCT
cana-395	66	13	1	1	NUM
cana-395	66	14	𝑛	𝑛	PRON
cana-395	66	15	<	<	X
cana-395	66	16	t	t	PROPN
cana-395	66	17	ρ	ρ	PROPN
cana-395	66	18	(	(	PUNCT
cana-395	66	19	mn	mn	PROPN
cana-395	66	20	,	,	PUNCT
cana-395	66	21	n	n	CCONJ
cana-395	66	22	,	,	PUNCT
cana-395	66	23	p	p	NOUN
cana-395	66	24	)	)	PUNCT
cana-395	66	25	in	in	ADP
cana-395	66	26	which	which	PRON
cana-395	66	27	n	n	NUM
cana-395	66	28	∈	∈	PROPN
cana-395	66	29	n.	n.	NOUN
cana-395	66	30	from	from	ADP
cana-395	66	31	m	m	PROPN
cana-395	66	32	∈	∈	PROPN
cana-395	66	33	h0(l	h0(l	PROPN
cana-395	66	34	)	)	PUNCT
cana-395	66	35	,	,	PUNCT
cana-395	66	36	a	a	DET
cana-395	66	37	subsequence	subsequence	NOUN
cana-395	66	38	(	(	PUNCT
cana-395	66	39	tnk	tnk	PROPN
cana-395	66	40	)	)	PUNCT
cana-395	67	1	k	k	PROPN
cana-395	67	2	of	of	ADP
cana-395	67	3	(	(	PUNCT
cana-395	67	4	mn)n	mn)n	PROPN
cana-395	67	5	and	and	CCONJ
cana-395	67	6	m0	m0	PROPN
cana-395	67	7	∈	∈	PROPN
cana-395	67	8	m	m	VERB
cana-395	67	9	:	:	PUNCT
cana-395	67	10	mnk	mnk	PROPN
cana-395	67	11	→	→	SYM
cana-395	67	12	m0	m0	PROPN
cana-395	67	13	in	in	ADP
cana-395	67	14	(	(	PUNCT
cana-395	67	15	l	l	NOUN
cana-395	67	16	,	,	PUNCT
cana-395	67	17	t	t	PROPN
cana-395	67	18	)	)	PUNCT
cana-395	67	19	.	.	PUNCT
cana-395	68	1	select	select	VERB
cana-395	68	2	n	n	PRON
cana-395	68	3	∈	∈	PROPN
cana-395	68	4	n	n	CCONJ
cana-395	68	5	,	,	PUNCT
cana-395	68	6	p∈	p∈	PROPN
cana-395	68	7	p	p	NOUN
cana-395	68	8	.	.	PUNCT
cana-395	69	1	from	from	ADP
cana-395	69	2	the	the	DET
cana-395	69	3	proposition	proposition	NOUN
cana-395	69	4	2.5.2	2.5.2	NUM
cana-395	69	5	,	,	PUNCT
cana-395	69	6	we	we	PRON
cana-395	69	7	get	get	VERB
cana-395	69	8	lim	lim	PROPN
cana-395	69	9	𝑘	𝑘	PRON
cana-395	69	10	𝑇	𝑇	PROPN
cana-395	69	11	ρ	ρ	PROPN
cana-395	69	12	(	(	PUNCT
cana-395	69	13	mnk	mnk	PROPN
cana-395	69	14	,	,	PUNCT
cana-395	69	15	n	n	CCONJ
cana-395	69	16	,	,	PUNCT
cana-395	69	17	p)=	p)=	PROPN
cana-395	69	18	t	t	PROPN
cana-395	69	19	ρ	ρ	PROPN
cana-395	69	20	(	(	PUNCT
cana-395	69	21	m0	m0	PROPN
cana-395	69	22	,	,	PUNCT
cana-395	69	23	n	n	CCONJ
cana-395	69	24	,	,	PUNCT
cana-395	69	25	p	p	NOUN
cana-395	69	26	)	)	PUNCT
cana-395	69	27	.	.	PUNCT
cana-395	70	1	since	since	SCONJ
cana-395	70	2	,	,	PUNCT
cana-395	70	3	for	for	ADP
cana-395	70	4	each	each	DET
cana-395	70	5	k	k	PROPN
cana-395	70	6	∈	∈	PROPN
cana-395	70	7	n	n	CCONJ
cana-395	70	8	,	,	PUNCT
cana-395	70	9	δ	δ	PROPN
cana-395	70	10	1	1	NUM
cana-395	70	11	𝑛𝑘	𝑛𝑘	ADP
cana-395	70	12	<	<	X
cana-395	70	13	t	t	PROPN
cana-395	70	14	ρ	ρ	PROPN
cana-395	70	15	(	(	PUNCT
cana-395	70	16	mnk	mnk	PROPN
cana-395	70	17	,	,	PUNCT
cana-395	70	18	n	n	CCONJ
cana-395	70	19	,	,	PUNCT
cana-395	70	20	p	p	NOUN
cana-395	70	21	)	)	PUNCT
cana-395	70	22	,	,	PUNCT
cana-395	70	23	we	we	PRON
cana-395	70	24	get	get	VERB
cana-395	70	25	δ	δ	NOUN
cana-395	70	26	≤	≤	PROPN
cana-395	70	27	t	t	PROPN
cana-395	70	28	ρ	ρ	PROPN
cana-395	70	29	(	(	PUNCT
cana-395	70	30	m0	m0	PROPN
cana-395	70	31	,	,	PUNCT
cana-395	70	32	n	n	CCONJ
cana-395	70	33	,	,	PUNCT
cana-395	70	34	p	p	NOUN
cana-395	70	35	)	)	PUNCT
cana-395	70	36	.	.	PUNCT
cana-395	71	1	we	we	PRON
cana-395	71	2	conclude	conclude	VERB
cana-395	71	3	δ	δ	PROPN
cana-395	71	4	=	=	SYM
cana-395	71	5	t	t	PROPN
cana-395	71	6	ρ	ρ	PROPN
cana-395	71	7	(	(	PUNCT
cana-395	71	8	m0	m0	PROPN
cana-395	71	9	,	,	PUNCT
cana-395	71	10	n	n	CCONJ
cana-395	71	11	,	,	PUNCT
cana-395	71	12	p	p	NOUN
cana-395	71	13	)	)	PUNCT
cana-395	71	14	.	.	PUNCT
cana-395	72	1	definition	definition	NOUN
cana-395	72	2	2.8	2.8	NUM
cana-395	72	3	(	(	PUNCT
cana-395	72	4	alihajimohammad	alihajimohammad	NOUN
cana-395	72	5	and	and	CCONJ
cana-395	72	6	saadati	saadati	NOUN
cana-395	72	7	,	,	PUNCT
cana-395	72	8	2021	2021	NUM
cana-395	72	9	)	)	PUNCT
cana-395	72	10	let	let	VERB
cana-395	72	11	(	(	PUNCT
cana-395	72	12	l	l	NOUN
cana-395	72	13	,	,	PUNCT
cana-395	72	14	t	t	PROPN
cana-395	72	15	)	)	PUNCT
cana-395	72	16	is	be	AUX
cana-395	72	17	a	a	DET
cana-395	72	18	generalized	generalized	ADJ
cana-395	72	19	(	(	PUNCT
cana-395	72	20	gmm	gmm	NOUN
cana-395	72	21	)	)	PUNCT
cana-395	72	22	modular	modular	ADJ
cana-395	72	23	metric	metric	NOUN
cana-395	72	24	and	and	CCONJ
cana-395	72	25	q	q	NOUN
cana-395	72	26	:	:	PUNCT
cana-395	72	27	l	l	PUNCT
cana-395	72	28	→	→	PUNCT
cana-395	72	29	l	l	NOUN
cana-395	72	30	is	be	AUX
cana-395	72	31	called	call	VERB
cana-395	72	32	a	a	DET
cana-395	72	33	gmm	gmm	PROPN
cana-395	72	34	-	-	PUNCT
cana-395	72	35	φ	φ	VERB
cana-395	72	36	-	-	ADJ
cana-395	72	37	contractive	contractive	ADJ
cana-395	72	38	mapping	mapping	NOUN
cana-395	72	39	if	if	SCONJ
cana-395	72	40	tφ(ρ	tφ(ρ	NUM
cana-395	72	41	)	)	PUNCT
cana-395	72	42	(	(	PUNCT
cana-395	72	43	q(l	q(l	NOUN
cana-395	72	44	)	)	PUNCT
cana-395	72	45	,	,	PUNCT
cana-395	72	46	q(t	q(t	PROPN
cana-395	72	47	)	)	PUNCT
cana-395	72	48	,	,	PUNCT
cana-395	72	49	q(u	q(u	NOUN
cana-395	72	50	)	)	PUNCT
cana-395	72	51	)	)	PUNCT
cana-395	73	1	≤	≤	NOUN
cana-395	73	2	t	t	PROPN
cana-395	73	3	ρ	ρ	X
cana-395	73	4	(	(	PUNCT
cana-395	73	5	l	l	PROPN
cana-395	73	6	,	,	PUNCT
cana-395	73	7	m	m	PROPN
cana-395	73	8	,	,	PUNCT
cana-395	73	9	n	n	CCONJ
cana-395	73	10	)	)	PUNCT
cana-395	73	11	for	for	ADP
cana-395	73	12	every	every	DET
cana-395	73	13	l	l	NOUN
cana-395	73	14	,	,	PUNCT
cana-395	73	15	m	m	PROPN
cana-395	73	16	,	,	PUNCT
cana-395	73	17	n	n	PROPN
cana-395	73	18	∈	∈	PROPN
cana-395	73	19	s	s	PART
cana-395	73	20	and	and	CCONJ
cana-395	73	21	ρ	ρ	PROPN
cana-395	73	22	∈	∈	PROPN
cana-395	73	23	q0	q0	NOUN
cana-395	73	24	.	.	PUNCT
cana-395	74	1	communications	communication	NOUN
cana-395	74	2	on	on	ADP
cana-395	74	3	applied	apply	VERB
cana-395	74	4	nonlinear	nonlinear	ADJ
cana-395	74	5	analysis	analysis	NOUN
cana-395	74	6	issn	issn	NOUN
cana-395	74	7	:	:	PUNCT
cana-395	74	8	1074	1074	NUM
cana-395	74	9	-	-	PUNCT
cana-395	74	10	133x	133x	NUM
cana-395	74	11	vol	vol	NOUN
cana-395	74	12	31	31	NUM
cana-395	74	13	no	no	NOUN
cana-395	74	14	.	.	NOUN
cana-395	74	15	1	1	NUM
cana-395	74	16	(	(	PUNCT
cana-395	74	17	2024	2024	NUM
cana-395	74	18	)	)	PUNCT
cana-395	74	19	203	203	NUM
cana-395	74	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-395	74	21	definition	definition	NOUN
cana-395	74	22	2.9	2.9	NUM
cana-395	74	23	(	(	PUNCT
cana-395	74	24	alihajimohammad	alihajimohammad	NOUN
cana-395	74	25	and	and	CCONJ
cana-395	74	26	saadati	saadati	NOUN
cana-395	74	27	,	,	PUNCT
cana-395	74	28	2021	2021	NUM
cana-395	74	29	)	)	PUNCT
cana-395	74	30	a	a	DET
cana-395	74	31	generalized	generalize	VERB
cana-395	74	32	modular	modular	ADJ
cana-395	74	33	metric	metric	ADJ
cana-395	74	34	-φcontractions	-φcontraction	NOUN
cana-395	74	35	{	{	PUNCT
cana-395	74	36	q1	q1	NOUN
cana-395	74	37	,	,	PUNCT
cana-395	74	38	q2	q2	NOUN
cana-395	74	39	,	,	PUNCT
cana-395	74	40	...	...	PUNCT
cana-395	74	41	,	,	PUNCT
cana-395	74	42	qm	qm	INTJ
cana-395	74	43	,	,	PUNCT
cana-395	74	44	:	:	PUNCT
cana-395	74	45	m	m	VERB
cana-395	74	46	≥	≥	NOUN
cana-395	74	47	2	2	NUM
cana-395	74	48	}	}	PUNCT
cana-395	74	49	is	be	AUX
cana-395	74	50	a	a	DET
cana-395	74	51	finite	finite	NOUN
cana-395	74	52	set	set	VERB
cana-395	74	53	on	on	ADP
cana-395	74	54	a	a	DET
cana-395	74	55	complete	complete	ADJ
cana-395	74	56	gmm	gmm	NOUN
cana-395	74	57	-	-	PUNCT
cana-395	74	58	space	space	NOUN
cana-395	74	59	(	(	PUNCT
cana-395	74	60	l	l	NOUN
cana-395	74	61	,	,	PUNCT
cana-395	74	62	t	t	PROPN
cana-395	74	63	)	)	PUNCT
cana-395	74	64	is	be	AUX
cana-395	74	65	known	know	VERB
cana-395	74	66	as	as	ADP
cana-395	74	67	a	a	DET
cana-395	74	68	generalized	generalized	ADJ
cana-395	74	69	modular	modular	ADJ
cana-395	74	70	metric	metric	ADJ
cana-395	74	71	iterated	iterate	VERB
cana-395	74	72	function	function	NOUN
cana-395	74	73	system	system	NOUN
cana-395	74	74	.	.	PUNCT
cana-395	75	1	3	3	NUM
cana-395	75	2	main	main	ADJ
cana-395	75	3	results	result	NOUN
cana-395	75	4	:	:	PUNCT
cana-395	75	5	mathematical	mathematical	ADJ
cana-395	75	6	theorem	theorem	NOUN
cana-395	75	7	let	let	VERB
cana-395	75	8	(	(	PUNCT
cana-395	75	9	l	l	NOUN
cana-395	75	10	,	,	PUNCT
cana-395	75	11	t	t	PROPN
cana-395	75	12	)	)	PUNCT
cana-395	75	13	is	be	AUX
cana-395	75	14	a	a	DET
cana-395	75	15	generalized	generalized	ADJ
cana-395	75	16	(	(	PUNCT
cana-395	75	17	gmm	gmm	NOUN
cana-395	75	18	)	)	PUNCT
cana-395	75	19	modular	modular	ADJ
cana-395	75	20	metric	metric	ADJ
cana-395	75	21	space	space	NOUN
cana-395	75	22	and	and	CCONJ
cana-395	75	23	assume	assume	VERB
cana-395	75	24	the	the	DET
cana-395	75	25	followings	following	NOUN
cana-395	75	26	:	:	PUNCT
cana-395	75	27	f0(l	f0(l	NUM
cana-395	75	28	)	)	PUNCT
cana-395	75	29	=	=	PUNCT
cana-395	75	30	nonempty	nonempty	ADJ
cana-395	75	31	subsets	subset	NOUN
cana-395	75	32	of	of	ADP
cana-395	75	33	l	l	NOUN
cana-395	75	34	,	,	PUNCT
cana-395	75	35	g0(l	g0(l	PROPN
cana-395	75	36	)	)	PUNCT
cana-395	75	37	=	=	PUNCT
cana-395	75	38	nonempty	nonempty	ADJ
cana-395	75	39	finite	finite	ADJ
cana-395	75	40	subsets	subset	NOUN
cana-395	75	41	and	and	CCONJ
cana-395	75	42	h0(l	h0(l	NOUN
cana-395	75	43	)	)	PUNCT
cana-395	75	44	=	=	NOUN
cana-395	75	45	nonempty	nonempty	ADJ
cana-395	75	46	compact	compact	ADJ
cana-395	75	47	subset	subset	NOUN
cana-395	75	48	of	of	ADP
cana-395	75	49	l.	l.	PROPN
cana-395	75	50	then	then	ADV
cana-395	75	51	a	a	DET
cana-395	75	52	function	function	NOUN
cana-395	75	53	ht	ht	X
cana-395	75	54	on	on	ADP
cana-395	75	55	h0(l	h0(l	PROPN
cana-395	75	56	)	)	PUNCT
cana-395	75	57	×	×	NOUN
cana-395	75	58	h0(l	h0(l	PROPN
cana-395	75	59	)	)	PUNCT
cana-395	75	60	×	×	NOUN
cana-395	75	61	h0(l	h0(l	NOUN
cana-395	75	62	)	)	PUNCT
cana-395	75	63	×	×	NOUN
cana-395	75	64	q0	q0	PROPN
cana-395	75	65	is	be	AUX
cana-395	75	66	defined	define	VERB
cana-395	75	67	by	by	ADP
cana-395	75	68	ht	ht	PROPN
cana-395	75	69	(	(	PUNCT
cana-395	75	70	m	m	PROPN
cana-395	75	71	,	,	PUNCT
cana-395	75	72	n	n	CCONJ
cana-395	75	73	,	,	PUNCT
cana-395	75	74	p	p	X
cana-395	75	75	,	,	PUNCT
cana-395	75	76	ρ	ρ	NOUN
cana-395	75	77	)	)	PUNCT
cana-395	75	78	=	=	SYM
cana-395	75	79	max{supm∈mtρ(m	max{supm∈mtρ(m	PROPN
cana-395	75	80	,	,	PUNCT
cana-395	75	81	n	n	CCONJ
cana-395	75	82	,	,	PUNCT
cana-395	75	83	p	p	NOUN
cana-395	75	84	)	)	PUNCT
cana-395	75	85	,	,	PUNCT
cana-395	75	86	supn∈	supn∈	PROPN
cana-395	75	87	ntρ	ntρ	ADV
cana-395	75	88	(	(	PUNCT
cana-395	75	89	m	m	PROPN
cana-395	75	90	,	,	PUNCT
cana-395	75	91	n	n	CCONJ
cana-395	75	92	,	,	PUNCT
cana-395	75	93	p	p	NOUN
cana-395	75	94	)	)	PUNCT
cana-395	75	95	,	,	PUNCT
cana-395	75	96	supp∈p	supp∈p	PROPN
cana-395	75	97	t	t	PROPN
cana-395	75	98	ρ	ρ	PROPN
cana-395	75	99	(	(	PUNCT
cana-395	75	100	m	m	PROPN
cana-395	75	101	,	,	PUNCT
cana-395	75	102	n	n	CCONJ
cana-395	75	103	,	,	PUNCT
cana-395	75	104	p	p	NOUN
cana-395	75	105	)	)	PUNCT
cana-395	75	106	}	}	PUNCT
cana-395	75	107	for	for	ADP
cana-395	75	108	every	every	DET
cana-395	75	109	m	m	NOUN
cana-395	75	110	,	,	PUNCT
cana-395	75	111	n	n	CCONJ
cana-395	75	112	,	,	PUNCT
cana-395	75	113	p	p	PROPN
cana-395	75	114	∈	∈	PROPN
cana-395	75	115	h0(l	h0(l	PROPN
cana-395	75	116	)	)	PUNCT
cana-395	75	117	and	and	CCONJ
cana-395	75	118	ρ	ρ	PROPN
cana-395	75	119	∈	∈	PROPN
cana-395	75	120	q0	q0	NOUN
cana-395	75	121	.	.	PUNCT
cana-395	76	1	lemma	lemma	PROPN
cana-395	76	2	3.1	3.1	NUM
cana-395	76	3	let	let	VERB
cana-395	76	4	(	(	PUNCT
cana-395	76	5	l	l	NOUN
cana-395	76	6	,	,	PUNCT
cana-395	76	7	t	t	PROPN
cana-395	76	8	)	)	PUNCT
cana-395	76	9	is	be	AUX
cana-395	76	10	a	a	DET
cana-395	76	11	generalized	generalized	ADJ
cana-395	76	12	(	(	PUNCT
cana-395	76	13	gmm	gmm	NOUN
cana-395	76	14	)	)	PUNCT
cana-395	76	15	modular	modular	ADJ
cana-395	76	16	metric	metric	ADJ
cana-395	76	17	space	space	NOUN
cana-395	76	18	l	l	PROPN
cana-395	76	19	∈	∈	PROPN
cana-395	76	20	l	l	NOUN
cana-395	76	21	,	,	PUNCT
cana-395	76	22	m	m	PROPN
cana-395	76	23	,	,	PUNCT
cana-395	76	24	n	n	PRON
cana-395	76	25	∈	∈	PROPN
cana-395	76	26	h0(l	h0(l	PROPN
cana-395	76	27	)	)	PUNCT
cana-395	76	28	,	,	PUNCT
cana-395	76	29	p∈	p∈	PROPN
cana-395	76	30	f0(l	f0(l	NOUN
cana-395	76	31	)	)	PUNCT
cana-395	76	32	,	,	PUNCT
cana-395	76	33	and	and	CCONJ
cana-395	76	34	α	α	X
cana-395	76	35	,	,	PUNCT
cana-395	76	36	β	β	PROPN
cana-395	76	37	∈	∈	PROPN
cana-395	76	38	q0	q0	PROPN
cana-395	76	39	.	.	PUNCT
cana-395	77	1	then	then	ADV
cana-395	77	2	tα+β	tα+β	PROPN
cana-395	77	3	(	(	PUNCT
cana-395	77	4	l	l	NOUN
cana-395	77	5	,	,	PUNCT
cana-395	77	6	m	m	PROPN
cana-395	77	7	,	,	PUNCT
cana-395	77	8	p	p	NOUN
cana-395	77	9	)	)	PUNCT
cana-395	77	10	≤	≤	NOUN
cana-395	77	11	tα(l	tα(l	NOUN
cana-395	77	12	,	,	PUNCT
cana-395	77	13	n	n	CCONJ
cana-395	77	14	,	,	PUNCT
cana-395	77	15	n)+tβ	n)+tβ	PROPN
cana-395	77	16	(	(	PUNCT
cana-395	77	17	ul	ul	INTJ
cana-395	77	18	,	,	PUNCT
cana-395	77	19	m	m	VERB
cana-395	77	20	,	,	PUNCT
cana-395	77	21	p	p	NOUN
cana-395	77	22	)	)	PUNCT
cana-395	77	23	,	,	PUNCT
cana-395	77	24	where	where	SCONJ
cana-395	77	25	nl	nl	PROPN
cana-395	77	26	∈	∈	PROPN
cana-395	77	27	n	n	PRON
cana-395	77	28	satisfies	satisfie	NOUN
cana-395	77	29	tα(l	tα(l	NUM
cana-395	77	30	,	,	PUNCT
cana-395	77	31	n	n	CCONJ
cana-395	77	32	,	,	PUNCT
cana-395	77	33	n)=	n)=	NOUN
cana-395	77	34	tα(l	tα(l	NUM
cana-395	77	35	,	,	PUNCT
cana-395	77	36	nl	nl	NOUN
cana-395	77	37	,	,	PUNCT
cana-395	77	38	nl	nl	NOUN
cana-395	77	39	)	)	PUNCT
cana-395	77	40	.	.	PUNCT
cana-395	78	1	proof	proof	NOUN
cana-395	78	2	:	:	PUNCT
cana-395	78	3	using	use	VERB
cana-395	78	4	lemma	lemma	PROPN
cana-395	78	5	2.6	2.6	NUM
cana-395	78	6	,	,	PUNCT
cana-395	78	7	tα(l	tα(l	NUM
cana-395	78	8	,	,	PUNCT
cana-395	78	9	n	n	CCONJ
cana-395	78	10	,	,	PUNCT
cana-395	78	11	n	n	CCONJ
cana-395	78	12	)	)	PUNCT
cana-395	78	13	=	=	SYM
cana-395	78	14	tα(l	tα(l	NOUN
cana-395	78	15	,	,	PUNCT
cana-395	78	16	nl	nl	NOUN
cana-395	78	17	,	,	PUNCT
cana-395	78	18	nl	nl	PROPN
cana-395	78	19	)	)	PUNCT
cana-395	78	20	.	.	PUNCT
cana-395	79	1	for	for	ADP
cana-395	79	2	each	each	DET
cana-395	79	3	m	m	NOUN
cana-395	79	4	∈	∈	NOUN
cana-395	79	5	m	m	NOUN
cana-395	79	6	,	,	PUNCT
cana-395	79	7	p	p	PROPN
cana-395	79	8	∈	∈	PROPN
cana-395	79	9	p	p	NOUN
cana-395	79	10	,	,	PUNCT
cana-395	79	11	we	we	PRON
cana-395	79	12	have	have	VERB
cana-395	79	13	tα+β	tα+β	NOUN
cana-395	79	14	(	(	PUNCT
cana-395	79	15	l	l	NOUN
cana-395	79	16	,	,	PUNCT
cana-395	79	17	m	m	PROPN
cana-395	79	18	,	,	PUNCT
cana-395	79	19	p	p	NOUN
cana-395	79	20	)	)	PUNCT
cana-395	79	21	≤	≤	NUM
cana-395	79	22	tα+β	tα+β	NOUN
cana-395	79	23	(	(	PUNCT
cana-395	79	24	l	l	NOUN
cana-395	79	25	,	,	PUNCT
cana-395	79	26	m	m	PROPN
cana-395	79	27	,	,	PUNCT
cana-395	79	28	p	p	NOUN
cana-395	79	29	)	)	PUNCT
cana-395	79	30	≤	≤	NOUN
cana-395	79	31	tα(l	tα(l	NOUN
cana-395	79	32	,	,	PUNCT
cana-395	79	33	nl	nl	PROPN
cana-395	79	34	,	,	PUNCT
cana-395	79	35	nl)+	nl)+	ADV
cana-395	79	36	tβ(nl	tβ(nl	PROPN
cana-395	79	37	,	,	PUNCT
cana-395	79	38	m	m	PROPN
cana-395	79	39	,	,	PUNCT
cana-395	79	40	p	p	NOUN
cana-395	79	41	)	)	PUNCT
cana-395	79	42	.	.	PUNCT
cana-395	80	1	then	then	ADV
cana-395	80	2	tα+β	tα+β	PROPN
cana-395	80	3	(	(	PUNCT
cana-395	80	4	l	l	NOUN
cana-395	80	5	,	,	PUNCT
cana-395	80	6	m	m	VERB
cana-395	80	7	,	,	PUNCT
cana-395	80	8	p	p	NOUN
cana-395	80	9	)	)	PUNCT
cana-395	80	10	≤	≤	NOUN
cana-395	80	11	tα(l	tα(l	NOUN
cana-395	80	12	,	,	PUNCT
cana-395	80	13	n	n	CCONJ
cana-395	80	14	,	,	PUNCT
cana-395	80	15	n)+	n)+	PROPN
cana-395	80	16	tβ	tβ	PROPN
cana-395	80	17	(	(	PUNCT
cana-395	80	18	nl	nl	PROPN
cana-395	80	19	,	,	PUNCT
cana-395	80	20	m	m	PROPN
cana-395	80	21	,	,	PUNCT
cana-395	80	22	p	p	NOUN
cana-395	80	23	)	)	PUNCT
cana-395	80	24	theorem	theorem	VERB
cana-395	80	25	3.2	3.2	NUM
cana-395	80	26	let	let	VERB
cana-395	80	27	(	(	PUNCT
cana-395	80	28	l	l	NOUN
cana-395	80	29	,	,	PUNCT
cana-395	80	30	t	t	PROPN
cana-395	80	31	)	)	PUNCT
cana-395	80	32	be	be	AUX
cana-395	80	33	a	a	DET
cana-395	80	34	generalized	generalized	ADJ
cana-395	80	35	modular	modular	ADJ
cana-395	80	36	metric	metric	NOUN
cana-395	80	37	(	(	PUNCT
cana-395	80	38	gmm	gmm	NOUN
cana-395	80	39	)	)	PUNCT
cana-395	80	40	space	space	NOUN
cana-395	80	41	.	.	PUNCT
cana-395	81	1	then	then	ADV
cana-395	81	2	(	(	PUNCT
cana-395	81	3	h0(l	h0(l	NOUN
cana-395	81	4	)	)	PUNCT
cana-395	81	5	,	,	PUNCT
cana-395	81	6	ht	ht	PROPN
cana-395	81	7	)	)	PUNCT
cana-395	81	8	is	be	AUX
cana-395	81	9	a	a	DET
cana-395	81	10	generalized	generalized	ADJ
cana-395	81	11	(	(	PUNCT
cana-395	81	12	gmms	gmms	NOUN
cana-395	81	13	)	)	PUNCT
cana-395	81	14	modular	modular	ADJ
cana-395	81	15	metric	metric	ADJ
cana-395	81	16	space	space	NOUN
cana-395	81	17	.	.	PUNCT
cana-395	82	1	proof	proof	NOUN
cana-395	82	2	:	:	PUNCT
cana-395	82	3	let	let	VERB
cana-395	82	4	m	m	PRON
cana-395	82	5	,	,	PUNCT
cana-395	82	6	n	n	CCONJ
cana-395	82	7	,	,	PUNCT
cana-395	82	8	p	p	X
cana-395	82	9	,	,	PUNCT
cana-395	82	10	w	w	PROPN
cana-395	82	11	∈	∈	PROPN
cana-395	82	12	h0(l	h0(l	PROPN
cana-395	82	13	)	)	PUNCT
cana-395	82	14	and	and	CCONJ
cana-395	82	15	α	α	NOUN
cana-395	82	16	,	,	PUNCT
cana-395	82	17	β	β	PROPN
cana-395	82	18	∈	∈	NOUN
cana-395	82	19	q0	q0	PROPN
cana-395	82	20	.	.	PUNCT
cana-395	83	1	by	by	ADP
cana-395	83	2	lemma	lemma	PROPN
cana-395	83	3	2.7	2.7	NUM
cana-395	83	4	,	,	PUNCT
cana-395	83	5	there	there	PRON
cana-395	83	6	exist	exist	VERB
cana-395	83	7	m0	m0	NOUN
cana-395	83	8	∈	∈	PROPN
cana-395	83	9	m	m	PROPN
cana-395	83	10	,	,	PUNCT
cana-395	83	11	n0	n0	PROPN
cana-395	83	12	∈	∈	PROPN
cana-395	83	13	n	n	CCONJ
cana-395	83	14	,	,	PUNCT
cana-395	83	15	and	and	CCONJ
cana-395	83	16	p0	p0	PROPN
cana-395	83	17	∈	∈	PROPN
cana-395	83	18	p	p	NOUN
cana-395	83	19	such	such	ADJ
cana-395	83	20	that	that	PRON
cana-395	83	21	:	:	PUNCT
cana-395	83	22	supm∈m	supm∈m	PROPN
cana-395	83	23	t	t	PROPN
cana-395	83	24	(	(	PUNCT
cana-395	83	25	m	m	PROPN
cana-395	83	26	,	,	PUNCT
cana-395	83	27	n	n	CCONJ
cana-395	83	28	,	,	PUNCT
cana-395	83	29	p	p	NOUN
cana-395	83	30	)	)	PUNCT
cana-395	83	31	=	=	SYM
cana-395	83	32	t	t	PROPN
cana-395	83	33	(	(	PUNCT
cana-395	83	34	m0	m0	PROPN
cana-395	83	35	,	,	PUNCT
cana-395	83	36	n	n	CCONJ
cana-395	83	37	,	,	PUNCT
cana-395	83	38	p	p	NOUN
cana-395	83	39	)	)	PUNCT
cana-395	83	40	,	,	PUNCT
cana-395	83	41	supn∈n	supn∈n	PROPN
cana-395	83	42	t(m	t(m	PROPN
cana-395	83	43	,	,	PUNCT
cana-395	83	44	n	n	CCONJ
cana-395	83	45	,	,	PUNCT
cana-395	83	46	p	p	NOUN
cana-395	83	47	)	)	PUNCT
cana-395	83	48	=	=	SYM
cana-395	83	49	t(m	t(m	PROPN
cana-395	83	50	,	,	PUNCT
cana-395	83	51	n0	n0	PROPN
cana-395	83	52	,	,	PUNCT
cana-395	83	53	p	p	NOUN
cana-395	83	54	)	)	PUNCT
cana-395	83	55	,	,	PUNCT
cana-395	83	56	and	and	CCONJ
cana-395	83	57	supp∈p	supp∈p	PROPN
cana-395	83	58	t(m	t(m	PROPN
cana-395	83	59	,	,	PUNCT
cana-395	83	60	p	p	X
cana-395	83	61	,	,	PUNCT
cana-395	83	62	p)=	p)=	NOUN
cana-395	83	63	t(m	t(m	PROPN
cana-395	83	64	,	,	PUNCT
cana-395	83	65	p	p	X
cana-395	83	66	,	,	PUNCT
cana-395	83	67	p0	p0	NOUN
cana-395	83	68	)	)	PUNCT
cana-395	83	69	.	.	PUNCT
cana-395	84	1	then	then	ADV
cana-395	84	2	ht	ht	INTJ
cana-395	84	3	(	(	PUNCT
cana-395	84	4	m	m	PROPN
cana-395	84	5	,	,	PUNCT
cana-395	84	6	n	n	CCONJ
cana-395	84	7	,	,	PUNCT
cana-395	84	8	p	p	X
cana-395	84	9	,	,	PUNCT
cana-395	84	10	α	α	NOUN
cana-395	84	11	)	)	PUNCT
cana-395	84	12	≥	≥	NOUN
cana-395	84	13	0	0	NUM
cana-395	84	14	.	.	PUNCT
cana-395	85	1	moreover	moreover	ADV
cana-395	85	2	,	,	PUNCT
cana-395	85	3	it	it	PRON
cana-395	85	4	is	be	AUX
cana-395	85	5	clear	clear	ADJ
cana-395	85	6	that	that	SCONJ
cana-395	85	7	m	m	VERB
cana-395	85	8	=	=	SYM
cana-395	86	1	n	n	PROPN
cana-395	86	2	=	=	SYM
cana-395	86	3	p	p	PROPN
cana-395	86	4	⇔	⇔	PROPN
cana-395	86	5	ht	ht	PROPN
cana-395	86	6	(	(	PUNCT
cana-395	86	7	m	m	PROPN
cana-395	86	8	,	,	PUNCT
cana-395	86	9	n	n	CCONJ
cana-395	86	10	,	,	PUNCT
cana-395	86	11	p	p	X
cana-395	86	12	,	,	PUNCT
cana-395	86	13	α)=	α)=	ADJ
cana-395	86	14	0	0	PUNCT
cana-395	87	1	then	then	ADV
cana-395	87	2	from	from	ADP
cana-395	87	3	lemma	lemma	PROPN
cana-395	87	4	3.1	3.1	NUM
cana-395	87	5	we	we	PRON
cana-395	87	6	have	have	AUX
cana-395	87	7	supm∈m	supm∈m	VERB
cana-395	87	8	tα+β	tα+β	PROPN
cana-395	87	9	(	(	PUNCT
cana-395	87	10	m	m	PROPN
cana-395	87	11	,	,	PUNCT
cana-395	87	12	n	n	CCONJ
cana-395	87	13	,	,	PUNCT
cana-395	87	14	w	w	NOUN
cana-395	87	15	)	)	PUNCT
cana-395	87	16	≤	≤	NOUN
cana-395	87	17	supm∈m	supm∈m	PROPN
cana-395	87	18	tα	tα	PROPN
cana-395	87	19	(	(	PUNCT
cana-395	87	20	m	m	PROPN
cana-395	87	21	,	,	PUNCT
cana-395	87	22	p	p	X
cana-395	87	23	,	,	PUNCT
cana-395	87	24	p	p	NOUN
cana-395	87	25	)	)	PUNCT
cana-395	88	1	+	+	CCONJ
cana-395	88	2	supm∈m	supm∈m	PROPN
cana-395	88	3	tβ	tβ	PROPN
cana-395	88	4	(	(	PUNCT
cana-395	88	5	pm	pm	NOUN
cana-395	88	6	,	,	PUNCT
cana-395	88	7	n	n	CCONJ
cana-395	88	8	,	,	PUNCT
cana-395	88	9	w	w	NOUN
cana-395	88	10	)	)	PUNCT
cana-395	88	11	since{pm	since{pm	NUM
cana-395	88	12	:	:	PUNCT
cana-395	88	13	m	m	VERB
cana-395	88	14	∈	∈	PROPN
cana-395	88	15	m	m	NOUN
cana-395	88	16	}	}	PUNCT
cana-395	88	17	⊆	⊆	NUM
cana-395	88	18	p	p	NOUN
cana-395	88	19	,	,	PUNCT
cana-395	88	20	supm∈m	supm∈m	PROPN
cana-395	88	21	tβ	tβ	PROPN
cana-395	88	22	(	(	PUNCT
cana-395	88	23	pm	pm	NOUN
cana-395	88	24	,	,	PUNCT
cana-395	88	25	n	n	CCONJ
cana-395	88	26	,	,	PUNCT
cana-395	88	27	w	w	NOUN
cana-395	88	28	)	)	PUNCT
cana-395	88	29	≤	≤	NOUN
cana-395	88	30	supp∈p	supp∈p	PROPN
cana-395	88	31	tβ(p	tβ(p	NOUN
cana-395	88	32	,	,	PUNCT
cana-395	88	33	n	n	CCONJ
cana-395	88	34	,	,	PUNCT
cana-395	88	35	w	w	NOUN
cana-395	88	36	)	)	PUNCT
cana-395	88	37	communications	communication	NOUN
cana-395	88	38	on	on	ADP
cana-395	88	39	applied	apply	VERB
cana-395	88	40	nonlinear	nonlinear	ADJ
cana-395	88	41	analysis	analysis	NOUN
cana-395	88	42	issn	issn	NOUN
cana-395	88	43	:	:	PUNCT
cana-395	88	44	1074	1074	NUM
cana-395	88	45	-	-	PUNCT
cana-395	88	46	133x	133x	NUM
cana-395	88	47	vol	vol	NOUN
cana-395	88	48	31	31	NUM
cana-395	88	49	no	no	NOUN
cana-395	88	50	.	.	NOUN
cana-395	88	51	1	1	NUM
cana-395	88	52	(	(	PUNCT
cana-395	88	53	2024	2024	NUM
cana-395	88	54	)	)	PUNCT
cana-395	88	55	204	204	NUM
cana-395	88	56	https://internationalpubls.com	https://internationalpubls.com	X
cana-395	88	57	supm∈m	supm∈m	PROPN
cana-395	88	58	tα	tα	PROPN
cana-395	89	1	+	+	PROPN
cana-395	89	2	β	β	X
cana-395	89	3	(	(	PUNCT
cana-395	89	4	m	m	PROPN
cana-395	89	5	,	,	PUNCT
cana-395	89	6	n	n	CCONJ
cana-395	89	7	,	,	PUNCT
cana-395	89	8	w	w	NOUN
cana-395	89	9	)	)	PUNCT
cana-395	89	10	≤	≤	NUM
cana-395	89	11	supm	supm	NOUN
cana-395	89	12	∈	∈	PROPN
cana-395	89	13	m	m	VERB
cana-395	89	14	tα	tα	NOUN
cana-395	89	15	(	(	PUNCT
cana-395	89	16	m	m	PROPN
cana-395	89	17	,	,	PUNCT
cana-395	89	18	p	p	X
cana-395	89	19	,	,	PUNCT
cana-395	89	20	p	p	NOUN
cana-395	89	21	)	)	PUNCT
cana-395	89	22	+	+	CCONJ
cana-395	89	23	sup	sup	NUM
cana-395	89	24	p∈p	p∈p	VERB
cana-395	90	1	tβ	tβ	INTJ
cana-395	90	2	(	(	PUNCT
cana-395	90	3	p	p	NOUN
cana-395	90	4	,	,	PUNCT
cana-395	90	5	n	n	CCONJ
cana-395	90	6	,	,	PUNCT
cana-395	90	7	w	w	NOUN
cana-395	90	8	)	)	PUNCT
cana-395	90	9	in	in	ADP
cana-395	90	10	the	the	DET
cana-395	90	11	same	same	ADJ
cana-395	90	12	way	way	NOUN
cana-395	90	13	,	,	PUNCT
cana-395	90	14	we	we	PRON
cana-395	90	15	obtain	obtain	VERB
cana-395	90	16	sup	sup	NOUN
cana-395	90	17	n∈n	n∈n	ADV
cana-395	90	18	tα+β	tα+β	NOUN
cana-395	90	19	(	(	PUNCT
cana-395	90	20	m	m	PROPN
cana-395	90	21	,	,	PUNCT
cana-395	90	22	n	n	CCONJ
cana-395	90	23	,	,	PUNCT
cana-395	90	24	w	w	NOUN
cana-395	90	25	)	)	PUNCT
cana-395	90	26	≤	≤	NUM
cana-395	90	27	sup	sup	NUM
cana-395	90	28	n∈n	n∈n	NOUN
cana-395	90	29	tα(n	tα(n	NOUN
cana-395	90	30	,	,	PUNCT
cana-395	90	31	p	p	X
cana-395	90	32	,	,	PUNCT
cana-395	90	33	p	p	NOUN
cana-395	90	34	)	)	PUNCT
cana-395	91	1	+	+	NUM
cana-395	91	2	sup	sup	NUM
cana-395	91	3	p∈p	p∈p	VERB
cana-395	92	1	tβ	tβ	INTJ
cana-395	92	2	(	(	PUNCT
cana-395	92	3	p	p	X
cana-395	92	4	,	,	PUNCT
cana-395	92	5	m	m	PROPN
cana-395	92	6	,	,	PUNCT
cana-395	92	7	w	w	PROPN
cana-395	92	8	)	)	PUNCT
cana-395	92	9	,	,	PUNCT
cana-395	92	10	sup	sup	NOUN
cana-395	92	11	w	w	PROPN
cana-395	92	12	∈	∈	PROPN
cana-395	92	13	w	w	NOUN
cana-395	92	14	tα+β	tα+β	NOUN
cana-395	92	15	(	(	PUNCT
cana-395	92	16	m	m	PROPN
cana-395	92	17	,	,	PUNCT
cana-395	92	18	n	n	CCONJ
cana-395	92	19	,	,	PUNCT
cana-395	92	20	w	w	NOUN
cana-395	92	21	)	)	PUNCT
cana-395	92	22	≤	≤	NOUN
cana-395	92	23	supw	supw	VERB
cana-395	93	1	∈	∈	PROPN
cana-395	93	2	w	w	PROPN
cana-395	93	3	tα(w	tα(w	NOUN
cana-395	93	4	,	,	PUNCT
cana-395	93	5	p	p	X
cana-395	93	6	,	,	PUNCT
cana-395	93	7	p	p	NOUN
cana-395	93	8	)	)	PUNCT
cana-395	93	9	+	+	NUM
cana-395	93	10	supp	supp	PROPN
cana-395	93	11	∈	∈	PROPN
cana-395	94	1	p	p	X
cana-395	94	2	tβ	tβ	PROPN
cana-395	95	1	(	(	PUNCT
cana-395	95	2	p	p	X
cana-395	95	3	,	,	PUNCT
cana-395	95	4	m	m	PROPN
cana-395	95	5	,	,	PUNCT
cana-395	95	6	w	w	PROPN
cana-395	95	7	)	)	PUNCT
cana-395	95	8	.	.	PUNCT
cana-395	96	1	therefore	therefore	ADV
cana-395	96	2	,	,	PUNCT
cana-395	96	3	it	it	PRON
cana-395	96	4	is	be	AUX
cana-395	96	5	obvious	obvious	ADJ
cana-395	96	6	to	to	PART
cana-395	96	7	conclude	conclude	VERB
cana-395	96	8	that	that	SCONJ
cana-395	96	9	ht	ht	PROPN
cana-395	96	10	(	(	PUNCT
cana-395	96	11	m	m	PROPN
cana-395	96	12	,	,	PUNCT
cana-395	96	13	n	n	CCONJ
cana-395	96	14	,	,	PUNCT
cana-395	96	15	w	w	PROPN
cana-395	96	16	,	,	PUNCT
cana-395	96	17	α	α	PROPN
cana-395	96	18	+	+	CCONJ
cana-395	96	19	β	β	X
cana-395	96	20	)	)	PUNCT
cana-395	96	21	≤	≤	NOUN
cana-395	97	1	ht	ht	PROPN
cana-395	98	1	(	(	PUNCT
cana-395	98	2	m	m	PROPN
cana-395	98	3	,	,	PUNCT
cana-395	98	4	p	p	X
cana-395	98	5	,	,	PUNCT
cana-395	98	6	p	p	X
cana-395	98	7	,	,	PUNCT
cana-395	98	8	α)+	α)+	NOUN
cana-395	98	9	ht	ht	PROPN
cana-395	98	10	(	(	PUNCT
cana-395	98	11	p	p	NOUN
cana-395	98	12	,	,	PUNCT
cana-395	98	13	n	n	CCONJ
cana-395	98	14	,	,	PUNCT
cana-395	98	15	w	w	PROPN
cana-395	98	16	,	,	PUNCT
cana-395	98	17	β	β	NOUN
cana-395	98	18	)	)	PUNCT
cana-395	98	19	.	.	PUNCT
cana-395	99	1	α	α	PROPN
cana-395	99	2	|→	|→	PROPN
cana-395	99	3	ht	ht	X
cana-395	99	4	(	(	PUNCT
cana-395	99	5	m	m	PROPN
cana-395	99	6	,	,	PUNCT
cana-395	99	7	n	n	CCONJ
cana-395	99	8	,	,	PUNCT
cana-395	99	9	p	p	X
cana-395	99	10	,	,	PUNCT
cana-395	99	11	α	α	X
cana-395	99	12	)	)	PUNCT
cana-395	99	13	is	be	AUX
cana-395	99	14	continuous	continuous	ADJ
cana-395	99	15	on	on	ADP
cana-395	99	16	q0	q0	PROPN
cana-395	99	17	,	,	PUNCT
cana-395	99	18	by	by	ADP
cana-395	99	19	the	the	DET
cana-395	99	20	proposition	proposition	NOUN
cana-395	99	21	2.5.3	2.5.3	NUM
cana-395	99	22	.	.	PUNCT
cana-395	100	1	then	then	ADV
cana-395	100	2	(	(	PUNCT
cana-395	100	3	h0(l	h0(l	NOUN
cana-395	100	4	)	)	PUNCT
cana-395	100	5	,	,	PUNCT
cana-395	100	6	ht	ht	PROPN
cana-395	100	7	)	)	PUNCT
cana-395	100	8	is	be	AUX
cana-395	100	9	a	a	DET
cana-395	100	10	generalized	generalized	ADJ
cana-395	100	11	modular	modular	ADJ
cana-395	100	12	metric	metric	ADJ
cana-395	100	13	space	space	NOUN
cana-395	100	14	.	.	PUNCT
cana-395	101	1	theorem	theorem	VERB
cana-395	101	2	3.3	3.3	NUM
cana-395	101	3	let	let	NOUN
cana-395	101	4	(	(	PUNCT
cana-395	101	5	l	l	NOUN
cana-395	101	6	,	,	PUNCT
cana-395	101	7	t	t	PROPN
cana-395	101	8	)	)	PUNCT
cana-395	101	9	is	be	AUX
cana-395	101	10	a	a	DET
cana-395	101	11	(	(	PUNCT
cana-395	101	12	gmm	gmm	NOUN
cana-395	101	13	)	)	PUNCT
cana-395	101	14	space	space	NOUN
cana-395	101	15	.a	.a	NOUN
cana-395	101	16	function	function	NOUN
cana-395	101	17	q	q	NOUN
cana-395	101	18	:	:	PUNCT
cana-395	101	19	l	l	X
cana-395	101	20	→	→	PUNCT
cana-395	101	21	l	l	NOUN
cana-395	101	22	is	be	AUX
cana-395	101	23	given	give	VERB
cana-395	101	24	by	by	ADP
cana-395	101	25	:	:	PUNCT
cana-395	101	26	tφ(ρ	tφ(ρ	NUM
cana-395	101	27	)	)	PUNCT
cana-395	101	28	(	(	PUNCT
cana-395	101	29	q	q	X
cana-395	101	30	(	(	PUNCT
cana-395	101	31	l	l	NOUN
cana-395	101	32	)	)	PUNCT
cana-395	101	33	,	,	PUNCT
cana-395	101	34	q(m	q(m	PROPN
cana-395	101	35	)	)	PUNCT
cana-395	101	36	,	,	PUNCT
cana-395	101	37	q(n	q(n	PROPN
cana-395	101	38	)	)	PUNCT
cana-395	101	39	≤	≤	NOUN
cana-395	101	40	t	t	PROPN
cana-395	101	41	ρ	ρ	X
cana-395	101	42	(	(	PUNCT
cana-395	101	43	l	l	PROPN
cana-395	101	44	,	,	PUNCT
cana-395	101	45	m	m	PROPN
cana-395	101	46	,	,	PUNCT
cana-395	101	47	n	n	CCONJ
cana-395	101	48	)	)	PUNCT
cana-395	101	49	,	,	PUNCT
cana-395	101	50	for	for	ADP
cana-395	101	51	all	all	DET
cana-395	101	52	ρ	ρ	NOUN
cana-395	101	53	∈	∈	PROPN
cana-395	101	54	q0	q0	NOUN
cana-395	101	55	and	and	CCONJ
cana-395	101	56	l	l	NOUN
cana-395	101	57	,	,	PUNCT
cana-395	101	58	m	m	PROPN
cana-395	101	59	,	,	PUNCT
cana-395	101	60	n	n	PROPN
cana-395	101	61	∈	∈	PROPN
cana-395	101	62	l.	l.	NOUN
cana-395	101	63	then	then	ADV
cana-395	101	64	the	the	DET
cana-395	101	65	sequence	sequence	NOUN
cana-395	101	66	𝑄𝑛(𝑙)𝑛=1	𝑄𝑛(𝑙)𝑛=1	X
cana-395	102	1	∞	∞	PROPN
cana-395	102	2	is	be	AUX
cana-395	102	3	generalized	generalize	VERB
cana-395	102	4	modular	modular	ADJ
cana-395	102	5	metric	metric	ADJ
cana-395	102	6	complete	complete	ADJ
cana-395	102	7	space	space	NOUN
cana-395	102	8	.	.	PUNCT
cana-395	103	1	proof	proof	NOUN
cana-395	103	2	let	let	VERB
cana-395	103	3	{	{	PUNCT
cana-395	103	4	ln	ln	ADJ
cana-395	103	5	:	:	PUNCT
cana-395	103	6	q	q	PROPN
cana-395	103	7	n(l)}𝑛=1	n(l)}𝑛=1	PROPN
cana-395	103	8	+	+	NOUN
cana-395	103	9	∞	∞	NUM
cana-395	103	10	}	}	PUNCT
cana-395	103	11	,	,	PUNCT
cana-395	103	12	{	{	PUNCT
cana-395	103	13	ln	ln	ADJ
cana-395	103	14	}	}	PUNCT
cana-395	103	15	is	be	AUX
cana-395	103	16	a	a	DET
cana-395	103	17	sequence	sequence	NOUN
cana-395	103	18	that	that	PRON
cana-395	103	19	complies	comply	VERB
cana-395	103	20	with	with	ADP
cana-395	103	21	the	the	DET
cana-395	103	22	requirements	requirement	NOUN
cana-395	103	23	of	of	ADP
cana-395	103	24	proposition	proposition	NOUN
cana-395	103	25	2.5.4	2.5.4	NUM
cana-395	103	26	t	t	NOUN
cana-395	103	27	ρ	ρ	X
cana-395	103	28	(	(	PUNCT
cana-395	103	29	l	l	NOUN
cana-395	103	30	,	,	PUNCT
cana-395	103	31	q(l	q(l	NOUN
cana-395	103	32	)	)	PUNCT
cana-395	103	33	,	,	PUNCT
cana-395	103	34	q(l	q(l	NOUN
cana-395	103	35	)	)	PUNCT
cana-395	103	36	≤	≤	NUM
cana-395	103	37	t	t	PROPN
cana-395	103	38	ρ	ρ	X
cana-395	103	39	(	(	PUNCT
cana-395	103	40	l	l	NOUN
cana-395	103	41	,	,	PUNCT
cana-395	103	42	q(l	q(l	NOUN
cana-395	103	43	)	)	PUNCT
cana-395	103	44	,	,	PUNCT
cana-395	103	45	q(l	q(l	NOUN
cana-395	103	46	)	)	PUNCT
cana-395	103	47	)	)	PUNCT
cana-395	104	1	(	(	PUNCT
cana-395	104	2	using	use	VERB
cana-395	104	3	the	the	DET
cana-395	104	4	induction	induction	NOUN
cana-395	104	5	)	)	PUNCT
cana-395	104	6	if	if	SCONJ
cana-395	104	7	tφn(ρ	tφn(ρ	NUM
cana-395	104	8	)	)	PUNCT
cana-395	104	9	(	(	PUNCT
cana-395	104	10	qn(l	qn(l	NUM
cana-395	104	11	)	)	PUNCT
cana-395	104	12	,	,	PUNCT
cana-395	104	13	qn+1(l	qn+1(l	ADP
cana-395	104	14	)	)	PUNCT
cana-395	104	15	,	,	PUNCT
cana-395	104	16	qn+1(l	qn+1(l	PROPN
cana-395	104	17	)	)	PUNCT
cana-395	104	18	)	)	PUNCT
cana-395	104	19	≤	≤	PROPN
cana-395	105	1	t	t	PROPN
cana-395	105	2	ρ	ρ	X
cana-395	105	3	(	(	PUNCT
cana-395	105	4	l	l	NOUN
cana-395	105	5	,	,	PUNCT
cana-395	105	6	q(l	q(l	NOUN
cana-395	105	7	)	)	PUNCT
cana-395	105	8	,	,	PUNCT
cana-395	105	9	q(l	q(l	NOUN
cana-395	105	10	)	)	PUNCT
cana-395	105	11	)	)	PUNCT
cana-395	105	12	then	then	ADV
cana-395	105	13	tφ	tφ	PROPN
cana-395	105	14	n+1	n+1	PROPN
cana-395	105	15	(	(	PUNCT
cana-395	105	16	ρ	ρ	PROPN
cana-395	105	17	)	)	PUNCT
cana-395	105	18	(	(	PUNCT
cana-395	105	19	qn+1(l	qn+1(l	NOUN
cana-395	105	20	)	)	PUNCT
cana-395	105	21	,	,	PUNCT
cana-395	105	22	qn+2(l	qn+2(l	NOUN
cana-395	105	23	)	)	PUNCT
cana-395	105	24	,	,	PUNCT
cana-395	105	25	qn+2(l	qn+2(l	NOUN
cana-395	105	26	)	)	PUNCT
cana-395	105	27	)	)	PUNCT
cana-395	106	1	=	=	PUNCT
cana-395	107	1	tφ(φn	tφ(φn	PROPN
cana-395	107	2	(	(	PUNCT
cana-395	107	3	ρ	ρ	PROPN
cana-395	107	4	)	)	PUNCT
cana-395	107	5	)	)	PUNCT
cana-395	107	6	(	(	PUNCT
cana-395	107	7	q	q	X
cana-395	107	8	(	(	PUNCT
cana-395	107	9	qn(l	qn(l	NUM
cana-395	107	10	)	)	PUNCT
cana-395	107	11	,	,	PUNCT
cana-395	107	12	q	q	PROPN
cana-395	107	13	(	(	PUNCT
cana-395	107	14	qn+1(l	qn+1(l	NOUN
cana-395	107	15	)	)	PUNCT
cana-395	107	16	)	)	PUNCT
cana-395	107	17	)	)	PUNCT
cana-395	108	1	now	now	ADV
cana-395	108	2	we	we	PRON
cana-395	108	3	have	have	VERB
cana-395	108	4	q	q	PROPN
cana-395	108	5	(	(	PUNCT
cana-395	108	6	qn+1(l	qn+1(l	NOUN
cana-395	108	7	)	)	PUNCT
cana-395	108	8	)	)	PUNCT
cana-395	109	1	≤	≤	NUM
cana-395	109	2	tφn	tφn	INTJ
cana-395	109	3	(	(	PUNCT
cana-395	109	4	ρ	ρ	PROPN
cana-395	109	5	)	)	PUNCT
cana-395	109	6	(	(	PUNCT
cana-395	109	7	qn(l	qn(l	NUM
cana-395	109	8	)	)	PUNCT
cana-395	109	9	,	,	PUNCT
cana-395	109	10	qn+1(l	qn+1(l	ADP
cana-395	109	11	)	)	PUNCT
cana-395	109	12	,	,	PUNCT
cana-395	109	13	qn+1(l	qn+1(l	PROPN
cana-395	109	14	)	)	PUNCT
cana-395	109	15	)	)	PUNCT
cana-395	109	16	≤	≤	PROPN
cana-395	110	1	t	t	PROPN
cana-395	110	2	ρ	ρ	X
cana-395	110	3	(	(	PUNCT
cana-395	110	4	l	l	NOUN
cana-395	110	5	,	,	PUNCT
cana-395	110	6	q(l	q(l	NOUN
cana-395	110	7	)	)	PUNCT
cana-395	110	8	,	,	PUNCT
cana-395	110	9	q(l	q(l	NOUN
cana-395	110	10	)	)	PUNCT
cana-395	110	11	)	)	PUNCT
cana-395	110	12	therefore	therefore	ADV
cana-395	110	13	,	,	PUNCT
cana-395	110	14	tφn	tφn	PROPN
cana-395	110	15	(	(	PUNCT
cana-395	110	16	ρ	ρ	PROPN
cana-395	110	17	)	)	PUNCT
cana-395	110	18	(	(	PUNCT
cana-395	110	19	l	l	NOUN
cana-395	110	20	n	n	X
cana-395	110	21	,	,	PUNCT
cana-395	110	22	l	l	PROPN
cana-395	110	23	n+1	n+1	PROPN
cana-395	110	24	,	,	PUNCT
cana-395	110	25	l	l	PROPN
cana-395	110	26	n+1	n+1	NOUN
cana-395	110	27	)	)	PUNCT
cana-395	110	28	≤	≤	NOUN
cana-395	111	1	t	t	PROPN
cana-395	111	2	ρ	ρ	X
cana-395	111	3	(	(	PUNCT
cana-395	111	4	l	l	NOUN
cana-395	111	5	0	0	NUM
cana-395	111	6	,	,	PUNCT
cana-395	111	7	l	l	NOUN
cana-395	111	8	1	1	NUM
cana-395	111	9	,	,	PUNCT
cana-395	111	10	l	l	NOUN
cana-395	111	11	1	1	NUM
cana-395	111	12	)	)	PUNCT
cana-395	111	13	,	,	PUNCT
cana-395	111	14	hence	hence	ADV
cana-395	111	15	qn	qn	PROPN
cana-395	111	16	(	(	PUNCT
cana-395	111	17	l	l	NOUN
cana-395	111	18	)	)	PUNCT
cana-395	111	19	𝑛=1	𝑛=1	NOUN
cana-395	111	20	∞	∞	NOUN
cana-395	111	21	is	be	AUX
cana-395	111	22	generalized	generalize	VERB
cana-395	111	23	modular	modular	ADJ
cana-395	111	24	metric	metric	ADJ
cana-395	111	25	complete	complete	ADJ
cana-395	111	26	space	space	NOUN
cana-395	111	27	(	(	PUNCT
cana-395	111	28	gmmcs	gmmcs	NOUN
cana-395	111	29	)	)	PUNCT
cana-395	111	30	.	.	PUNCT
cana-395	112	1	theorem	theorem	VERB
cana-395	112	2	3.4	3.4	NUM
cana-395	112	3	suppose	suppose	VERB
cana-395	112	4	(	(	PUNCT
cana-395	112	5	l	l	NOUN
cana-395	112	6	,	,	PUNCT
cana-395	112	7	t	t	PROPN
cana-395	112	8	)	)	PUNCT
cana-395	112	9	is	be	AUX
cana-395	112	10	a	a	DET
cana-395	112	11	generalized	generalized	ADJ
cana-395	112	12	(	(	PUNCT
cana-395	112	13	gmm	gmm	NOUN
cana-395	112	14	)	)	PUNCT
cana-395	112	15	space	space	NOUN
cana-395	112	16	and	and	CCONJ
cana-395	112	17	map	map	VERB
cana-395	112	18	q	q	NOUN
cana-395	112	19	,	,	PUNCT
cana-395	112	20	gmm	gmm	NOUN
cana-395	112	21	-	-	PUNCT
cana-395	112	22	φcontractive	φcontractive	ADJ
cana-395	112	23	mapping	mapping	NOUN
cana-395	112	24	for	for	ADP
cana-395	112	25	all	all	DET
cana-395	112	26	ρ	ρ	NOUN
cana-395	112	27	∈	∈	PROPN
cana-395	112	28	q0	q0	NOUN
cana-395	112	29	and	and	CCONJ
cana-395	112	30	l	l	NOUN
cana-395	112	31	,	,	PUNCT
cana-395	112	32	m	m	PROPN
cana-395	112	33	,	,	PUNCT
cana-395	112	34	n	n	PROPN
cana-395	112	35	∈	∈	NOUN
cana-395	112	36	l	l	NOUN
cana-395	112	37	:	:	PUNCT
cana-395	112	38	tφ(ρ)(q(l	tφ(ρ)(q(l	NOUN
cana-395	112	39	)	)	PUNCT
cana-395	112	40	,	,	PUNCT
cana-395	112	41	q(t	q(t	PROPN
cana-395	112	42	)	)	PUNCT
cana-395	112	43	,	,	PUNCT
cana-395	112	44	q(u	q(u	NOUN
cana-395	112	45	)	)	PUNCT
cana-395	112	46	≤	≤	NOUN
cana-395	112	47	t	t	PROPN
cana-395	112	48	ρ	ρ	X
cana-395	112	49	(	(	PUNCT
cana-395	112	50	l	l	PROPN
cana-395	112	51	,	,	PUNCT
cana-395	112	52	m	m	PROPN
cana-395	112	53	,	,	PUNCT
cana-395	112	54	n	n	CCONJ
cana-395	112	55	)	)	PUNCT
cana-395	112	56	for	for	ADP
cana-395	112	57	every	every	DET
cana-395	112	58	l	l	NOUN
cana-395	112	59	,	,	PUNCT
cana-395	112	60	m	m	PROPN
cana-395	112	61	,	,	PUNCT
cana-395	112	62	n	n	PROPN
cana-395	112	63	∈	∈	PROPN
cana-395	112	64	l	l	NOUN
cana-395	112	65	and	and	CCONJ
cana-395	112	66	ρ	ρ	NUM
cana-395	112	67	∈	∈	PROPN
cana-395	112	68	q0.then	q0.then	ADV
cana-395	112	69	in	in	ADP
cana-395	112	70	l	l	NOUN
cana-395	112	71	,	,	PUNCT
cana-395	112	72	q	q	PUNCT
cana-395	112	73	posseses	possese	VERB
cana-395	112	74	a	a	DET
cana-395	112	75	unique	unique	ADJ
cana-395	112	76	fixed	fix	VERB
cana-395	112	77	point	point	NOUN
cana-395	112	78	.	.	PUNCT
cana-395	113	1	proof	proof	NOUN
cana-395	113	2	now	now	ADV
cana-395	113	3	from	from	ADP
cana-395	113	4	the	the	DET
cana-395	113	5	above	above	ADJ
cana-395	113	6	theorem	theorem	NOUN
cana-395	113	7	3.3	3.3	NUM
cana-395	113	8	,	,	PUNCT
cana-395	113	9	we	we	PRON
cana-395	113	10	get	get	VERB
cana-395	113	11	{	{	PUNCT
cana-395	113	12	ln	ln	ADJ
cana-395	113	13	:	:	PUNCT
cana-395	113	14	{	{	PUNCT
cana-395	113	15	qn(l)}}n=1	qn(l)}}n=1	PROPN
cana-395	113	16	+	+	NOUN
cana-395	113	17	∞	∞	NUM
cana-395	113	18	}	}	PUNCT
cana-395	113	19	is	be	AUX
cana-395	113	20	generalized(gmmcs	generalized(gmmcs	PROPN
cana-395	113	21	)	)	PUNCT
cana-395	113	22	modular	modular	ADJ
cana-395	113	23	metric	metric	ADJ
cana-395	113	24	complete	complete	ADJ
cana-395	113	25	space	space	NOUN
cana-395	113	26	,	,	PUNCT
cana-395	113	27	for	for	ADP
cana-395	113	28	each	each	DET
cana-395	113	29	l	l	NOUN
cana-395	113	30	∈	∈	PROPN
cana-395	113	31	l	l	NOUN
cana-395	113	32	and	and	CCONJ
cana-395	113	33	limn→∞	limn→∞	PROPN
cana-395	113	34	qn(l	qn(l	NUM
cana-395	113	35	)	)	PUNCT
cana-395	114	1	=	=	SYM
cana-395	114	2	x	x	SYM
cana-395	114	3	∈l	∈l	PROPN
cana-395	114	4	.	.	PUNCT
cana-395	115	1	letting	let	VERB
cana-395	115	2	l0	l0	NOUN
cana-395	115	3	=	=	PUNCT
cana-395	115	4	l	l	PROPN
cana-395	115	5	and	and	CCONJ
cana-395	115	6	ln	ln	NOUN
cana-395	115	7	=	=	NOUN
cana-395	115	8	qn(l	qn(l	X
cana-395	115	9	)	)	PUNCT
cana-395	115	10	for	for	ADP
cana-395	115	11	each	each	DET
cana-395	115	12	n	n	PRON
cana-395	115	13	≥	≥	NOUN
cana-395	115	14	1	1	NUM
cana-395	115	15	,	,	PUNCT
cana-395	115	16	since	since	SCONJ
cana-395	115	17	limn→∞	limn→∞	PROPN
cana-395	115	18	qn(l)=	qn(l)=	PROPN
cana-395	115	19	x	x	NOUN
cana-395	115	20	,	,	PUNCT
cana-395	115	21	communications	communication	NOUN
cana-395	115	22	on	on	ADP
cana-395	115	23	applied	apply	VERB
cana-395	115	24	nonlinear	nonlinear	ADJ
cana-395	115	25	analysis	analysis	NOUN
cana-395	115	26	issn	issn	NOUN
cana-395	115	27	:	:	PUNCT
cana-395	115	28	1074	1074	NUM
cana-395	115	29	-	-	PUNCT
cana-395	115	30	133x	133x	NUM
cana-395	115	31	vol	vol	NOUN
cana-395	115	32	31	31	NUM
cana-395	115	33	no	no	NOUN
cana-395	115	34	.	.	NOUN
cana-395	115	35	1	1	NUM
cana-395	115	36	(	(	PUNCT
cana-395	115	37	2024	2024	NUM
cana-395	115	38	)	)	PUNCT
cana-395	115	39	205	205	NUM
cana-395	115	40	https://internationalpubls.com	https://internationalpubls.com	X
cana-395	116	1	we	we	PRON
cana-395	116	2	have	have	VERB
cana-395	116	3	lim	lim	PROPN
cana-395	116	4	t	t	PROPN
cana-395	116	5	ρ	ρ	PROPN
cana-395	116	6	(	(	PUNCT
cana-395	116	7	ln	ln	ADJ
cana-395	116	8	,	,	PUNCT
cana-395	116	9	x	x	X
cana-395	116	10	,	,	PUNCT
cana-395	116	11	x)=0for	x)=0for	ADP
cana-395	116	12	each	each	DET
cana-395	116	13	ρ	ρ	PROPN
cana-395	116	14	∈	∈	PROPN
cana-395	116	15	q0	q0	NOUN
cana-395	116	16	.	.	PUNCT
cana-395	117	1	on	on	ADP
cana-395	117	2	the	the	DET
cana-395	117	3	other	other	ADJ
cana-395	117	4	hand	hand	NOUN
cana-395	117	5	,	,	PUNCT
cana-395	117	6	we	we	PRON
cana-395	117	7	recognize	recognize	VERB
cana-395	117	8	tφ(ρ)(q(x	tφ(ρ)(q(x	NOUN
cana-395	117	9	)	)	PUNCT
cana-395	117	10	,	,	PUNCT
cana-395	117	11	ln+1	ln+1	PROPN
cana-395	117	12	,	,	PUNCT
cana-395	117	13	ln+1	ln+1	ADJ
cana-395	117	14	)	)	PUNCT
cana-395	117	15	≤	≤	NOUN
cana-395	117	16	t	t	PROPN
cana-395	117	17	ρ	ρ	X
cana-395	117	18	(	(	PUNCT
cana-395	117	19	x	x	X
cana-395	117	20	,	,	PUNCT
cana-395	117	21	ln	ln	ADJ
cana-395	117	22	,	,	PUNCT
cana-395	117	23	ln	ln	ADJ
cana-395	117	24	)	)	PUNCT
cana-395	117	25	for	for	ADP
cana-395	117	26	each	each	DET
cana-395	117	27	n	n	PRON
cana-395	117	28	∈	∈	PROPN
cana-395	117	29	n	n	NOUN
cana-395	117	30	and	and	CCONJ
cana-395	117	31	each	each	DET
cana-395	117	32	ρ	ρ	PROPN
cana-395	117	33	>	>	X
cana-395	117	34	0	0	PROPN
cana-395	117	35	.	.	PUNCT
cana-395	118	1	then	then	ADV
cana-395	118	2	tφ(ρ)(q(x	tφ(ρ)(q(x	NOUN
cana-395	118	3	)	)	PUNCT
cana-395	118	4	,	,	PUNCT
cana-395	118	5	x	x	NOUN
cana-395	118	6	,	,	PUNCT
cana-395	118	7	x	x	X
cana-395	118	8	)	)	PUNCT
cana-395	118	9	=	=	SYM
cana-395	119	1	.𝑛→∞	.𝑛→∞	PROPN
cana-395	119	2	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-395	119	3	𝑇φ(ρ	𝑇φ(ρ	PROPN
cana-395	119	4	)	)	PUNCT
cana-395	119	5	(	(	PUNCT
cana-395	119	6	q(x	q(x	PROPN
cana-395	119	7	)	)	PUNCT
cana-395	119	8	,	,	PUNCT
cana-395	119	9	ln+1	ln+1	PROPN
cana-395	119	10	,	,	PUNCT
cana-395	119	11	ln+1	ln+1	ADJ
cana-395	119	12	)	)	PUNCT
cana-395	119	13	≤	≤	NOUN
cana-395	119	14	.𝑛→∞	.𝑛→∞	PUNCT
cana-395	120	1	𝑙𝑖𝑚	𝑙𝑖𝑚	PROPN
cana-395	120	2	t	t	PROPN
cana-395	120	3	ρ	ρ	X
cana-395	120	4	(	(	PUNCT
cana-395	120	5	x	x	X
cana-395	120	6	,	,	PUNCT
cana-395	120	7	ln	ln	ADJ
cana-395	120	8	,	,	PUNCT
cana-395	120	9	ln)=0,for	ln)=0,for	ADP
cana-395	120	10	each	each	DET
cana-395	120	11	ρ	ρ	PROPN
cana-395	120	12	>	>	X
cana-395	120	13	0	0	NUM
cana-395	120	14	.	.	PUNCT
cana-395	120	15	.	.	PUNCT
cana-395	121	1	=	=	PRON
cana-395	121	2	>	>	X
cana-395	121	3	x	x	X
cana-395	121	4	=	=	SYM
cana-395	121	5	q(x	q(x	PROPN
cana-395	121	6	)	)	PUNCT
cana-395	121	7	,	,	PUNCT
cana-395	121	8	now	now	ADV
cana-395	121	9	,	,	PUNCT
cana-395	121	10	to	to	PART
cana-395	121	11	prove	prove	VERB
cana-395	121	12	:	:	PUNCT
cana-395	121	13	uniqueness	uniqueness	NOUN
cana-395	121	14	let	let	VERB
cana-395	121	15	y	y	PROPN
cana-395	121	16	∈	∈	PROPN
cana-395	121	17	q	q	NOUN
cana-395	121	18	is	be	AUX
cana-395	121	19	another	another	DET
cana-395	121	20	point	point	NOUN
cana-395	121	21	,	,	PUNCT
cana-395	121	22	and	and	CCONJ
cana-395	121	23	ρ	ρ	PROPN
cana-395	121	24	∈	∈	PROPN
cana-395	121	25	q0	q0	PROPN
cana-395	121	26	t	t	PROPN
cana-395	121	27	ρ	ρ	X
cana-395	121	28	(	(	PUNCT
cana-395	121	29	x	x	X
cana-395	121	30	,	,	PUNCT
cana-395	121	31	x	x	NOUN
cana-395	121	32	,	,	PUNCT
cana-395	121	33	y	y	NOUN
cana-395	121	34	)	)	PUNCT
cana-395	122	1	=	=	SYM
cana-395	122	2	t	t	PROPN
cana-395	122	3	ρ	ρ	PROPN
cana-395	122	4	(	(	PUNCT
cana-395	122	5	q(x	q(x	PROPN
cana-395	122	6	)	)	PUNCT
cana-395	122	7	,	,	PUNCT
cana-395	122	8	q(x	q(x	PROPN
cana-395	122	9	)	)	PUNCT
cana-395	122	10	,	,	PUNCT
cana-395	122	11	q(y	q(y	PROPN
cana-395	122	12	)	)	PUNCT
cana-395	122	13	)	)	PUNCT
cana-395	122	14	≥	≥	NOUN
cana-395	122	15	tφ(ρ	tφ(ρ	NUM
cana-395	122	16	)	)	PUNCT
cana-395	122	17	(	(	PUNCT
cana-395	122	18	q(x	q(x	PROPN
cana-395	122	19	)	)	PUNCT
cana-395	122	20	,	,	PUNCT
cana-395	122	21	q(x	q(x	PROPN
cana-395	122	22	)	)	PUNCT
cana-395	122	23	,	,	PUNCT
cana-395	122	24	q(y	q(y	PROPN
cana-395	122	25	)	)	PUNCT
cana-395	122	26	)	)	PUNCT
cana-395	122	27	.	.	PUNCT
cana-395	123	1	since	since	SCONJ
cana-395	123	2	t	t	PROPN
cana-395	123	3	ρ	ρ	PROPN
cana-395	123	4	(	(	PUNCT
cana-395	123	5	l	l	PROPN
cana-395	123	6	,	,	PUNCT
cana-395	123	7	m	m	PROPN
cana-395	123	8	,	,	PUNCT
cana-395	123	9	m	m	VERB
cana-395	123	10	)	)	PUNCT
cana-395	123	11	is	be	AUX
cana-395	123	12	nonincreasing	nonincrease	VERB
cana-395	123	13	and	and	CCONJ
cana-395	123	14	φ(ρ	φ(ρ	NOUN
cana-395	123	15	)	)	PUNCT
cana-395	123	16	<	<	X
cana-395	123	17	ρ	ρ	PROPN
cana-395	123	18	,	,	PUNCT
cana-395	123	19	we	we	PRON
cana-395	123	20	have	have	VERB
cana-395	123	21	tφ(ρ	tφ(ρ	VERB
cana-395	123	22	)	)	PUNCT
cana-395	123	23	(	(	PUNCT
cana-395	123	24	q(x	q(x	PROPN
cana-395	123	25	)	)	PUNCT
cana-395	123	26	,	,	PUNCT
cana-395	123	27	q(x	q(x	PROPN
cana-395	123	28	)	)	PUNCT
cana-395	123	29	,	,	PUNCT
cana-395	123	30	q(y	q(y	PROPN
cana-395	123	31	)	)	PUNCT
cana-395	123	32	)	)	PUNCT
cana-395	123	33	≥	≥	PROPN
cana-395	123	34	t	t	PROPN
cana-395	123	35	ρ	ρ	PROPN
cana-395	123	36	(	(	PUNCT
cana-395	123	37	q(x	q(x	PROPN
cana-395	123	38	)	)	PUNCT
cana-395	123	39	,	,	PUNCT
cana-395	123	40	q(x	q(x	PROPN
cana-395	123	41	)	)	PUNCT
cana-395	123	42	,	,	PUNCT
cana-395	123	43	q(y	q(y	NOUN
cana-395	123	44	)	)	PUNCT
cana-395	123	45	)	)	PUNCT
cana-395	124	1	=	=	SYM
cana-395	124	2	tρ	tρ	NOUN
cana-395	124	3	(	(	PUNCT
cana-395	124	4	x	x	NOUN
cana-395	124	5	,	,	PUNCT
cana-395	124	6	x	x	PROPN
cana-395	124	7	,	,	PUNCT
cana-395	124	8	y	y	PROPN
cana-395	124	9	)	)	PUNCT
cana-395	124	10	.	.	PUNCT
cana-395	125	1	hence	hence	ADV
cana-395	125	2	t	t	PROPN
cana-395	125	3	ρ	ρ	X
cana-395	125	4	(	(	PUNCT
cana-395	125	5	x	x	X
cana-395	125	6	,	,	PUNCT
cana-395	125	7	x	x	NOUN
cana-395	125	8	,	,	PUNCT
cana-395	125	9	y)=	y)=	ADJ
cana-395	125	10	c	c	PROPN
cana-395	125	11	from	from	ADP
cana-395	125	12	proposition	proposition	NOUN
cana-395	125	13	2.5.5	2.5.5	NUM
cana-395	125	14	,	,	PUNCT
cana-395	125	15	we	we	PRON
cana-395	125	16	get	get	VERB
cana-395	125	17	c	c	NOUN
cana-395	125	18	=	=	SYM
cana-395	125	19	0	0	NUM
cana-395	125	20	.	.	PUNCT
cana-395	126	1	therefore	therefore	ADV
cana-395	126	2	,	,	PUNCT
cana-395	126	3	x	x	PUNCT
cana-395	126	4	=	=	PUNCT
cana-395	126	5	y.	y.	NOUN
cana-395	126	6	4	4	NUM
cana-395	126	7	.	.	PUNCT
cana-395	126	8	conclusions	conclusion	NOUN
cana-395	126	9	we	we	PRON
cana-395	126	10	have	have	AUX
cana-395	126	11	studied	study	VERB
cana-395	126	12	certain	certain	ADJ
cana-395	126	13	topological	topological	ADJ
cana-395	126	14	aspects	aspect	NOUN
cana-395	126	15	of	of	ADP
cana-395	126	16	the	the	DET
cana-395	126	17	hausdorff	hausdorff	NOUN
cana-395	126	18	distance	distance	NOUN
cana-395	126	19	on	on	ADP
cana-395	126	20	gmm	gmm	NOUN
cana-395	126	21	and	and	CCONJ
cana-395	126	22	defined	define	VERB
cana-395	126	23	a	a	DET
cana-395	126	24	(	(	PUNCT
cana-395	126	25	gmfs	gmfs	NOUN
cana-395	126	26	)	)	PUNCT
cana-395	126	27	in	in	ADP
cana-395	126	28	the	the	DET
cana-395	126	29	sense	sense	NOUN
cana-395	126	30	of	of	ADP
cana-395	126	31	chistyakov	chistyakov	NOUN
cana-395	126	32	by	by	ADP
cana-395	126	33	iterated	iterated	ADJ
cana-395	126	34	function	function	NOUN
cana-395	126	35	system	system	NOUN
cana-395	126	36	.	.	PUNCT
cana-395	127	1	as	as	ADP
cana-395	127	2	anapplication	anapplication	NOUN
cana-395	127	3	some	some	DET
cana-395	127	4	concepts	concept	NOUN
cana-395	127	5	of	of	ADP
cana-395	127	6	fixed	fix	VERB
cana-395	127	7	point	point	NOUN
cana-395	127	8	have	have	AUX
cana-395	127	9	been	be	AUX
cana-395	127	10	implemented	implement	VERB
cana-395	127	11	in	in	ADP
cana-395	127	12	generalized	generalized	ADJ
cana-395	127	13	modular	modular	ADJ
cana-395	127	14	metric	metric	ADJ
cana-395	127	15	space	space	NOUN
cana-395	127	16	and	and	CCONJ
cana-395	127	17	generalized	generalize	VERB
cana-395	127	18	modular	modular	ADJ
cana-395	127	19	metric	metric	ADJ
cana-395	127	20	fractal	fractal	ADJ
cana-395	127	21	space	space	NOUN
cana-395	127	22	(	(	PUNCT
cana-395	127	23	gmmf	gmmf	NOUN
cana-395	127	24	-	-	PUNCT
cana-395	127	25	space	space	NOUN
cana-395	127	26	)	)	PUNCT
cana-395	127	27	.	.	PUNCT
cana-395	128	1	5	5	X
cana-395	128	2	.	.	X
cana-395	128	3	acknowledgement	acknowledgement	NOUN
cana-395	128	4	the	the	DET
cana-395	128	5	authors	author	NOUN
cana-395	128	6	are	be	AUX
cana-395	128	7	grateful	grateful	ADJ
cana-395	128	8	to	to	ADP
cana-395	128	9	the	the	DET
cana-395	128	10	knowledgeable	knowledgeable	ADJ
cana-395	128	11	referee	referee	NOUN
cana-395	128	12	for	for	ADP
cana-395	128	13	his	his	PRON
cana-395	128	14	insightful	insightful	ADJ
cana-395	128	15	observations	observation	NOUN
cana-395	128	16	and	and	CCONJ
cana-395	128	17	comments	comment	NOUN
cana-395	128	18	,	,	PUNCT
cana-395	128	19	which	which	PRON
cana-395	128	20	substantially	substantially	ADV
cana-395	128	21	assisted	assist	VERB
cana-395	128	22	us	we	PRON
cana-395	128	23	in	in	ADP
cana-395	128	24	significantly	significantly	ADV
cana-395	128	25	improving	improve	VERB
cana-395	128	26	the	the	DET
cana-395	128	27	manuscript	manuscript	NOUN
cana-395	128	28	.	.	PUNCT
cana-395	129	1	*	*	PUNCT
cana-395	129	2	references	reference	NOUN
cana-395	129	3	[	[	X
cana-395	129	4	1	1	NUM
cana-395	129	5	]	]	X
cana-395	129	6	abdou	abdou	PROPN
cana-395	129	7	,	,	PUNCT
cana-395	129	8	a.	a.	NOUN
cana-395	129	9	a.	a.	NOUN
cana-395	129	10	2016	2016	NUM
cana-395	129	11	.	.	PUNCT
cana-395	130	1	some	some	DET
cana-395	130	2	fixed	fix	VERB
cana-395	130	3	point	point	NOUN
cana-395	130	4	theorems	theorem	NOUN
cana-395	130	5	in	in	ADP
cana-395	130	6	modular	modular	ADJ
cana-395	130	7	metric	metric	ADJ
cana-395	130	8	spaces	space	NOUN
cana-395	130	9	,	,	PUNCT
cana-395	130	10	j.	j.	PROPN
cana-395	130	11	nonlinear	nonlinear	PROPN
cana-395	130	12	sci	sci	PROPN
cana-395	130	13	.	.	PUNCT
cana-395	130	14	appl	appl	PROPN
cana-395	130	15	9(6	9(6	PROPN
cana-395	130	16	):	):	PUNCT
cana-395	130	17	4381–4387	4381–4387	NOUN
cana-395	130	18	.	.	PUNCT
cana-395	131	1	[	[	X
cana-395	131	2	2	2	NUM
cana-395	131	3	]	]	X
cana-395	131	4	abdou	abdou	PROPN
cana-395	131	5	,	,	PUNCT
cana-395	131	6	a.	a.	NOUN
cana-395	131	7	a.	a.	NOUN
cana-395	131	8	2020	2020	NUM
cana-395	131	9	.	.	PUNCT
cana-395	131	10	fixed	fix	VERB
cana-395	131	11	points	point	NOUN
cana-395	131	12	of	of	ADP
cana-395	131	13	kannan	kannan	PROPN
cana-395	131	14	maps	map	NOUN
cana-395	131	15	in	in	ADP
cana-395	131	16	modular	modular	ADJ
cana-395	131	17	metric	metric	ADJ
cana-395	131	18	spaces	space	NOUN
cana-395	131	19	,	,	PUNCT
cana-395	131	20	aims	aim	VERB
cana-395	131	21	mathematics	mathematics	PROPN
cana-395	131	22	5(6	5(6	NUM
cana-395	131	23	):	):	PUNCT
cana-395	131	24	6395–6403	6395–6403	NOUN
cana-395	131	25	.	.	PUNCT
cana-395	132	1	[	[	X
cana-395	132	2	3	3	NUM
cana-395	132	3	]	]	PUNCT
cana-395	132	4	alihajimohammad	alihajimohammad	NOUN
cana-395	132	5	,	,	PUNCT
cana-395	132	6	a.	a.	NOUN
cana-395	132	7	and	and	CCONJ
cana-395	132	8	saadati	saadati	PROPN
cana-395	132	9	,	,	PUNCT
cana-395	132	10	r.	r.	PROPN
cana-395	132	11	2021	2021	NUM
cana-395	132	12	.	.	PUNCT
cana-395	133	1	generalized	generalize	VERB
cana-395	133	2	modular	modular	ADJ
cana-395	133	3	fractal	fractal	ADJ
cana-395	133	4	spaces	space	NOUN
cana-395	133	5	and	and	CCONJ
cana-395	133	6	fixed	fix	VERB
cana-395	133	7	point	point	NOUN
cana-395	133	8	theorems	theorem	NOUN
cana-395	133	9	,	,	PUNCT
cana-395	133	10	advances	advance	NOUN
cana-395	133	11	in	in	ADP
cana-395	133	12	difference	difference	NOUN
cana-395	133	13	equations	equation	NOUN
cana-395	133	14	2021(1	2021(1	NUM
cana-395	133	15	):	):	PUNCT
cana-395	133	16	1–10	1–10	NOUN
cana-395	133	17	.	.	PUNCT
cana-395	134	1	[	[	X
cana-395	134	2	4	4	NUM
cana-395	134	3	]	]	SYM
cana-395	134	4	azadifa	azadifa	PROPN
cana-395	134	5	,	,	PUNCT
cana-395	134	6	b.	b.	PROPN
cana-395	134	7	,	,	PUNCT
cana-395	134	8	maramaei	maramaei	NOUN
cana-395	134	9	,	,	PUNCT
cana-395	134	10	m.	m.	NOUN
cana-395	134	11	and	and	CCONJ
cana-395	134	12	sadeghi	sadeghi	PROPN
cana-395	134	13	,	,	PUNCT
cana-395	134	14	g.	g.	PROPN
cana-395	134	15	2013	2013	NUM
cana-395	134	16	.	.	PUNCT
cana-395	135	1	on	on	ADP
cana-395	135	2	the	the	DET
cana-395	135	3	modular	modular	ADJ
cana-395	135	4	g	g	NOUN
cana-395	135	5	-	-	PUNCT
cana-395	135	6	metric	metric	ADJ
cana-395	135	7	spaces	space	NOUN
cana-395	135	8	and	and	CCONJ
cana-395	135	9	fixed	fix	VERB
cana-395	135	10	point	point	NOUN
cana-395	135	11	theorems	theorem	NOUN
cana-395	135	12	.	.	PROPN
cana-395	135	13	,	,	PUNCT
cana-395	135	14	journal	journal	PROPN
cana-395	135	15	of	of	ADP
cana-395	135	16	nonlinear	nonlinear	PROPN
cana-395	135	17	sciences	sciences	PROPN
cana-395	135	18	&	&	CCONJ
cana-395	135	19	applications	application	NOUN
cana-395	135	20	(	(	PUNCT
cana-395	135	21	jnsa	jnsa	PROPN
cana-395	135	22	)	)	PUNCT
cana-395	135	23	6(4	6(4	NUM
cana-395	135	24	)	)	PUNCT
cana-395	135	25	.	.	PUNCT
cana-395	136	1	[	[	X
cana-395	136	2	5	5	NUM
cana-395	136	3	]	]	X
cana-395	136	4	bisht	bisht	PROPN
cana-395	136	5	,	,	PUNCT
cana-395	136	6	r.	r.	PROPN
cana-395	136	7	k.	k.	PROPN
cana-395	136	8	2018	2018	NUM
cana-395	136	9	.	.	PUNCT
cana-395	137	1	comment	comment	NOUN
cana-395	137	2	on	on	ADP
cana-395	137	3	:	:	PUNCT
cana-395	137	4	a	a	DET
cana-395	137	5	new	new	ADJ
cana-395	137	6	fixed	fix	VERB
cana-395	137	7	point	point	NOUN
cana-395	137	8	theorem	theorem	VERB
cana-395	137	9	in	in	ADP
cana-395	137	10	the	the	DET
cana-395	137	11	fractal	fractal	ADJ
cana-395	137	12	space	space	NOUN
cana-395	137	13	,	,	PUNCT
cana-395	137	14	indagationes	indagatione	VERB
cana-395	137	15	mathematicae	mathematicae	PROPN
cana-395	137	16	29(2	29(2	NOUN
cana-395	137	17	):	):	PUNCT
cana-395	137	18	819–823	819–823	NUM
cana-395	137	19	.	.	PUNCT
cana-395	138	1	communications	communication	NOUN
cana-395	138	2	on	on	ADP
cana-395	138	3	applied	apply	VERB
cana-395	138	4	nonlinear	nonlinear	ADJ
cana-395	138	5	analysis	analysis	NOUN
cana-395	138	6	issn	issn	NOUN
cana-395	138	7	:	:	PUNCT
cana-395	138	8	1074	1074	NUM
cana-395	138	9	-	-	PUNCT
cana-395	138	10	133x	133x	NUM
cana-395	138	11	vol	vol	NOUN
cana-395	138	12	31	31	NUM
cana-395	138	13	no	no	NOUN
cana-395	138	14	.	.	NOUN
cana-395	138	15	1	1	NUM
cana-395	138	16	(	(	PUNCT
cana-395	138	17	2024	2024	NUM
cana-395	138	18	)	)	PUNCT
cana-395	138	19	206	206	NUM
cana-395	138	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-395	139	1	[	[	X
cana-395	139	2	6	6	NUM
cana-395	139	3	]	]	SYM
cana-395	139	4	chistyakov	chistyakov	NOUN
cana-395	139	5	,	,	PUNCT
cana-395	139	6	v.	v.	ADP
cana-395	139	7	v.	v.	ADP
cana-395	139	8	2008	2008	NUM
cana-395	139	9	.	.	PUNCT
cana-395	140	1	modular	modular	ADJ
cana-395	140	2	metric	metric	ADJ
cana-395	140	3	spaces	space	NOUN
cana-395	140	4	generated	generate	VERB
cana-395	140	5	by	by	ADP
cana-395	140	6	f	f	NOUN
cana-395	140	7	-	-	PUNCT
cana-395	140	8	modulars	modular	NOUN
cana-395	140	9	,	,	PUNCT
cana-395	140	10	folia	folia	PROPN
cana-395	140	11	math	math	PROPN
cana-395	140	12	14(3	14(3	NUM
cana-395	140	13	)	)	PUNCT
cana-395	140	14	.	.	PUNCT
cana-395	141	1	[	[	X
cana-395	141	2	7	7	NUM
cana-395	141	3	]	]	SYM
cana-395	141	4	chistyakov	chistyakov	NOUN
cana-395	141	5	,	,	PUNCT
cana-395	141	6	v.	v.	PROPN
cana-395	141	7	v.	v.	ADP
cana-395	141	8	2010	2010	NUM
cana-395	141	9	.	.	PUNCT
cana-395	142	1	modular	modular	ADJ
cana-395	142	2	metric	metric	ADJ
cana-395	142	3	spaces	space	NOUN
cana-395	142	4	,	,	PUNCT
cana-395	142	5	i	i	PRON
cana-395	142	6	:	:	PUNCT
cana-395	142	7	basic	basic	ADJ
cana-395	142	8	concepts	concept	NOUN
cana-395	142	9	,	,	PUNCT
cana-395	142	10	nonlinear	nonlinear	ADJ
cana-395	142	11	analysis	analysis	NOUN
cana-395	142	12	:	:	PUNCT
cana-395	142	13	theory	theory	NOUN
cana-395	142	14	,	,	PUNCT
cana-395	142	15	methods	method	NOUN
cana-395	142	16	&	&	CCONJ
cana-395	142	17	applications	application	NOUN
cana-395	142	18	72(1	72(1	NOUN
cana-395	142	19	):	):	PUNCT
cana-395	142	20	1–14	1–14	PROPN
cana-395	142	21	.	.	PUNCT
cana-395	143	1	[	[	X
cana-395	143	2	8	8	NUM
cana-395	143	3	]	]	X
cana-395	143	4	cho	cho	PROPN
cana-395	143	5	,	,	PUNCT
cana-395	143	6	y.	y.	PROPN
cana-395	143	7	j.	j.	PROPN
cana-395	143	8	,	,	PUNCT
cana-395	143	9	saadati	saadati	PROPN
cana-395	143	10	,	,	PUNCT
cana-395	143	11	r.	r.	PROPN
cana-395	143	12	and	and	CCONJ
cana-395	143	13	sadeghi	sadeghi	PROPN
cana-395	143	14	,	,	PUNCT
cana-395	143	15	g.	g.	PROPN
cana-395	143	16	2012	2012	NUM
cana-395	143	17	.	.	PUNCT
cana-395	144	1	quasi	quasi	ADJ
cana-395	144	2	-	-	ADJ
cana-395	144	3	contractive	contractive	ADJ
cana-395	144	4	mappings	mapping	NOUN
cana-395	144	5	in	in	ADP
cana-395	144	6	modular	modular	ADJ
cana-395	144	7	metric	metric	ADJ
cana-395	144	8	spaces	space	NOUN
cana-395	144	9	,	,	PUNCT
cana-395	144	10	journal	journal	NOUN
cana-395	144	11	of	of	ADP
cana-395	144	12	applied	apply	VERB
cana-395	144	13	mathematics	mathematic	NOUN
cana-395	144	14	2012	2012	NUM
cana-395	144	15	.	.	PUNCT
cana-395	145	1	[	[	X
cana-395	145	2	9	9	NUM
cana-395	145	3	]	]	X
cana-395	145	4	ege	ege	PROPN
cana-395	145	5	,	,	PUNCT
cana-395	145	6	o.	o.	NOUN
cana-395	145	7	,	,	PUNCT
cana-395	145	8	park	park	NOUN
cana-395	145	9	,	,	PUNCT
cana-395	145	10	c.	c.	PROPN
cana-395	145	11	and	and	CCONJ
cana-395	145	12	ansari	ansari	ADJ
cana-395	145	13	,	,	PUNCT
cana-395	145	14	a.	a.	PROPN
cana-395	145	15	h.	h.	PROPN
cana-395	145	16	2020	2020	NUM
cana-395	145	17	.	.	PUNCT
cana-395	146	1	a	a	DET
cana-395	146	2	different	different	ADJ
cana-395	146	3	approach	approach	NOUN
cana-395	146	4	to	to	ADP
cana-395	146	5	complex	complex	PROPN
cana-395	146	6	valued	value	VERB
cana-395	146	7	gb	gb	ADP
cana-395	146	8	g	g	PROPN
cana-395	146	9	{	{	PUNCT
cana-395	146	10	b	b	NOUN
cana-395	146	11	}	}	PUNCT
cana-395	146	12	metric	metric	ADJ
cana-395	146	13	spaces	space	NOUN
cana-395	146	14	,	,	PUNCT
cana-395	146	15	advances	advance	NOUN
cana-395	146	16	in	in	ADP
cana-395	146	17	difference	difference	NOUN
cana-395	146	18	equations	equation	NOUN
cana-395	146	19	2020(1	2020(1	NUM
cana-395	146	20	):	):	PUNCT
cana-395	146	21	1–13	1–13	NOUN
cana-395	146	22	.	.	PUNCT
cana-395	147	1	[	[	X
cana-395	147	2	10	10	NUM
cana-395	147	3	]	]	X
cana-395	147	4	good	good	ADJ
cana-395	147	5	,	,	PUNCT
cana-395	147	6	i.	i.	NOUN
cana-395	147	7	1990	1990	NUM
cana-395	147	8	.	.	PUNCT
cana-395	148	1	fractals	fractal	NOUN
cana-395	148	2	everywhere	everywhere	ADV
cana-395	148	3	(	(	PUNCT
cana-395	148	4	michael	michael	PROPN
cana-395	148	5	barnsley	barnsley	PROPN
cana-395	148	6	)	)	PUNCT
cana-395	148	7	.	.	PUNCT
cana-395	149	1	[	[	X
cana-395	149	2	11	11	NUM
cana-395	149	3	]	]	X
cana-395	149	4	hutchinson	hutchinson	PROPN
cana-395	149	5	,	,	PUNCT
cana-395	149	6	j.	j.	PROPN
cana-395	149	7	e.	e.	PROPN
cana-395	149	8	1981	1981	NUM
cana-395	149	9	.	.	PUNCT
cana-395	150	1	fractals	fractal	NOUN
cana-395	150	2	and	and	CCONJ
cana-395	150	3	self	self	NOUN
cana-395	150	4	similarity	similarity	NOUN
cana-395	150	5	,	,	PUNCT
cana-395	150	6	indiana	indiana	PROPN
cana-395	150	7	university	university	PROPN
cana-395	150	8	mathematics	mathematics	PROPN
cana-395	150	9	journal	journal	PROPN
cana-395	150	10	30(5	30(5	NUM
cana-395	150	11	):	):	PUNCT
cana-395	150	12	713–747	713–747	NUM
cana-395	150	13	.	.	PUNCT
cana-395	151	1	[	[	X
cana-395	151	2	12	12	NUM
cana-395	151	3	]	]	X
cana-395	151	4	imdad	imdad	PROPN
cana-395	151	5	,	,	PUNCT
cana-395	151	6	m.	m.	NOUN
cana-395	151	7	,	,	PUNCT
cana-395	151	8	alfaqih	alfaqih	VERB
cana-395	151	9	,	,	PUNCT
cana-395	151	10	w.	w.	PROPN
cana-395	151	11	m.	m.	PROPN
cana-395	151	12	and	and	CCONJ
cana-395	151	13	khan	khan	PROPN
cana-395	151	14	,	,	PUNCT
cana-395	151	15	i.	i.	PROPN
cana-395	151	16	a.	a.	PROPN
cana-395	151	17	2018	2018	NUM
cana-395	151	18	.	.	PUNCT
cana-395	152	1	weak	weak	ADJ
cana-395	152	2	θ	θ	NOUN
cana-395	152	3	-	-	PUNCT
cana-395	152	4	contractions	contraction	NOUN
cana-395	152	5	and	and	CCONJ
cana-395	152	6	some	some	DET
cana-395	152	7	fixed	fix	VERB
cana-395	152	8	point	point	NOUN
cana-395	152	9	results	result	NOUN
cana-395	152	10	with	with	ADP
cana-395	152	11	applications	application	NOUN
cana-395	152	12	to	to	ADP
cana-395	152	13	fractal	fractal	ADJ
cana-395	152	14	theory	theory	NOUN
cana-395	152	15	,	,	PUNCT
cana-395	152	16	advances	advance	NOUN
cana-395	152	17	in	in	ADP
cana-395	152	18	difference	difference	NOUN
cana-395	152	19	equations	equation	NOUN
cana-395	152	20	2018(1	2018(1	NOUN
cana-395	152	21	):	):	PUNCT
cana-395	152	22	1	1	NUM
cana-395	152	23	–	–	PUNCT
cana-395	152	24	18	18	NUM
cana-395	152	25	.	.	PUNCT
cana-395	153	1	[	[	X
cana-395	153	2	13	13	NUM
cana-395	153	3	]	]	SYM
cana-395	153	4	ri	ri	PROPN
cana-395	153	5	,	,	PUNCT
cana-395	153	6	s.-i	s.-i	PROPN
cana-395	153	7	.	.	PROPN
cana-395	153	8	2016	2016	NUM
cana-395	153	9	.	.	PUNCT
cana-395	154	1	a	a	DET
cana-395	154	2	new	new	ADJ
cana-395	154	3	fixed	fix	VERB
cana-395	154	4	point	point	NOUN
cana-395	154	5	theorem	theorem	VERB
cana-395	154	6	in	in	ADP
cana-395	154	7	the	the	DET
cana-395	154	8	fractal	fractal	ADJ
cana-395	154	9	space	space	NOUN
cana-395	154	10	,	,	PUNCT
cana-395	154	11	indagationes	indagatione	NOUN
cana-395	154	12	mathematicae	mathematicae	NOUN
cana-395	154	13	27(1	27(1	NUM
cana-395	154	14	):	):	PUNCT
cana-395	154	15	85–93	85–93	NUM
cana-395	154	16	.	.	PUNCT