id	sid	tid	token	lemma	pos
cana-3163	1	1	communications	communication	NOUN
cana-3163	1	2	on	on	ADP
cana-3163	1	3	applied	apply	VERB
cana-3163	1	4	nonlinear	nonlinear	ADJ
cana-3163	1	5	analysis	analysis	NOUN
cana-3163	1	6	issn	issn	NOUN
cana-3163	1	7	:	:	PUNCT
cana-3163	1	8	1074	1074	NUM
cana-3163	1	9	-	-	PUNCT
cana-3163	1	10	133x	133x	NUM
cana-3163	1	11	vol	vol	NOUN
cana-3163	1	12	32	32	NUM
cana-3163	1	13	no	no	NOUN
cana-3163	1	14	.	.	PUNCT
cana-3163	2	1	5s	5s	NUM
cana-3163	2	2	(	(	PUNCT
cana-3163	2	3	2025	2025	NUM
cana-3163	2	4	)	)	PUNCT
cana-3163	2	5	511	511	NUM
cana-3163	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	2	7	classification	classification	NOUN
cana-3163	2	8	of	of	ADP
cana-3163	2	9	maps	map	NOUN
cana-3163	2	10	on	on	ADP
cana-3163	2	11	intervals	interval	NOUN
cana-3163	2	12	that	that	PRON
cana-3163	2	13	exhibit	exhibit	VERB
cana-3163	2	14	dense	dense	ADJ
cana-3163	2	15	periodic	periodic	ADJ
cana-3163	2	16	points	point	NOUN
cana-3163	2	17	*	*	PUNCT
cana-3163	2	18	suja	suja	PROPN
cana-3163	2	19	n	n	CCONJ
cana-3163	2	20	thomas,**p.b	thomas,**p.b	PROPN
cana-3163	2	21	vinod	vinod	PROPN
cana-3163	2	22	kumar	kumar	PROPN
cana-3163	2	23	rajagiri	rajagiri	PROPN
cana-3163	2	24	school	school	NOUN
cana-3163	2	25	of	of	ADP
cana-3163	2	26	engineering	engineering	NOUN
cana-3163	2	27	and	and	CCONJ
cana-3163	2	28	technology	technology	NOUN
cana-3163	2	29	,	,	PUNCT
cana-3163	2	30	kochi	kochi	PROPN
cana-3163	2	31	apj	apj	PROPN
cana-3163	2	32	abdul	abdul	PROPN
cana-3163	2	33	kalam	kalam	PROPN
cana-3163	2	34	technological	technological	PROPN
cana-3163	2	35	university	university	NOUN
cana-3163	2	36	,	,	PUNCT
cana-3163	2	37	kerala	kerala	PROPN
cana-3163	2	38	,	,	PUNCT
cana-3163	2	39	india	india	PROPN
cana-3163	2	40	.	.	PUNCT
cana-3163	3	1	*	*	PUNCT
cana-3163	3	2	sujanthomas72@gmail.com	sujanthomas72@gmail.com	PROPN
cana-3163	3	3	muthoot	muthoot	PROPN
cana-3163	3	4	institute	institute	PROPN
cana-3163	3	5	of	of	ADP
cana-3163	3	6	technology	technology	NOUN
cana-3163	3	7	and	and	CCONJ
cana-3163	3	8	science	science	NOUN
cana-3163	3	9	,	,	PUNCT
cana-3163	3	10	kochi	kochi	PROPN
cana-3163	3	11	apj	apj	PROPN
cana-3163	3	12	abdul	abdul	PROPN
cana-3163	3	13	kalam	kalam	PROPN
cana-3163	3	14	technological	technological	PROPN
cana-3163	3	15	university	university	NOUN
cana-3163	3	16	,	,	PUNCT
cana-3163	3	17	kerala	kerala	PROPN
cana-3163	3	18	,	,	PUNCT
cana-3163	3	19	india	india	PROPN
cana-3163	3	20	.	.	PUNCT
cana-3163	4	1	*	*	PUNCT
cana-3163	4	2	*	*	PUNCT
cana-3163	4	3	vinodkumarpb@mgits.ac.in	vinodkumarpb@mgits.ac.in	PRON
cana-3163	4	4	article	article	NOUN
cana-3163	4	5	history	history	NOUN
cana-3163	4	6	:	:	PUNCT
cana-3163	4	7	received	receive	VERB
cana-3163	4	8	:	:	PUNCT
cana-3163	4	9	14	14	NUM
cana-3163	4	10	-	-	SYM
cana-3163	4	11	10	10	NUM
cana-3163	4	12	-	-	PUNCT
cana-3163	4	13	2024	2024	NUM
cana-3163	4	14	revised	revise	VERB
cana-3163	4	15	:	:	PUNCT
cana-3163	4	16	28	28	NUM
cana-3163	4	17	-	-	SYM
cana-3163	4	18	11	11	NUM
cana-3163	4	19	-	-	PUNCT
cana-3163	4	20	2024	2024	NUM
cana-3163	4	21	accepted	accept	VERB
cana-3163	4	22	:	:	PUNCT
cana-3163	4	23	10	10	NUM
cana-3163	4	24	-	-	SYM
cana-3163	4	25	12	12	NUM
cana-3163	4	26	-	-	PUNCT
cana-3163	4	27	2024	2024	NUM
cana-3163	4	28	abstract	abstract	NOUN
cana-3163	4	29	:	:	PUNCT
cana-3163	4	30	in	in	ADP
cana-3163	4	31	this	this	DET
cana-3163	4	32	paper	paper	NOUN
cana-3163	4	33	,	,	PUNCT
cana-3163	4	34	continuous	continuous	ADJ
cana-3163	4	35	maps	map	NOUN
cana-3163	4	36	on	on	ADP
cana-3163	4	37	interval	interval	NOUN
cana-3163	4	38	with	with	ADP
cana-3163	4	39	dense	dense	ADJ
cana-3163	4	40	set	set	NOUN
cana-3163	4	41	of	of	ADP
cana-3163	4	42	periodic	periodic	ADJ
cana-3163	4	43	points	point	NOUN
cana-3163	4	44	such	such	ADJ
cana-3163	4	45	that	that	SCONJ
cana-3163	4	46	the	the	DET
cana-3163	4	47	periods	period	NOUN
cana-3163	4	48	being	be	AUX
cana-3163	4	49	set	set	VERB
cana-3163	4	50	of	of	ADP
cana-3163	4	51	natural	natural	ADJ
cana-3163	4	52	numbers	number	NOUN
cana-3163	4	53	are	be	AUX
cana-3163	4	54	studied.these	studied.these	ADJ
cana-3163	4	55	maps	map	NOUN
cana-3163	4	56	named	name	VERB
cana-3163	4	57	as	as	ADP
cana-3163	4	58	peri	peri	PROPN
cana-3163	4	59	odically	odically	ADV
cana-3163	4	60	rich	rich	ADJ
cana-3163	4	61	maps	map	NOUN
cana-3163	4	62	could	could	AUX
cana-3163	4	63	be	be	AUX
cana-3163	4	64	divided	divide	VERB
cana-3163	4	65	into	into	ADP
cana-3163	4	66	two	two	NUM
cana-3163	4	67	classes	class	NOUN
cana-3163	4	68	on	on	ADP
cana-3163	4	69	interval	interval	NOUN
cana-3163	4	70	,	,	PUNCT
cana-3163	4	71	such	such	ADJ
cana-3163	4	72	as	as	ADP
cana-3163	4	73	the	the	DET
cana-3163	4	74	maps	map	NOUN
cana-3163	4	75	with	with	ADP
cana-3163	4	76	period	period	NOUN
cana-3163	4	77	four	four	NUM
cana-3163	4	78	points	point	NOUN
cana-3163	4	79	having	having	AUX
cana-3163	4	80	increasing	increase	VERB
cana-3163	4	81	periodic	periodic	ADJ
cana-3163	4	82	orbit	orbit	NOUN
cana-3163	4	83	and	and	CCONJ
cana-3163	4	84	the	the	DET
cana-3163	4	85	maps	map	NOUN
cana-3163	4	86	with	with	ADP
cana-3163	4	87	two	two	NUM
cana-3163	4	88	fixed	fix	VERB
cana-3163	4	89	points	point	NOUN
cana-3163	4	90	such	such	ADJ
cana-3163	4	91	that	that	DET
cana-3163	4	92	image	image	NOUN
cana-3163	4	93	of	of	ADP
cana-3163	4	94	a	a	DET
cana-3163	4	95	subset	subset	NOUN
cana-3163	4	96	of	of	ADP
cana-3163	4	97	interval	interval	NOUN
cana-3163	4	98	does	do	AUX
cana-3163	4	99	not	not	PART
cana-3163	4	100	belong	belong	VERB
cana-3163	4	101	to	to	ADP
cana-3163	4	102	the	the	DET
cana-3163	4	103	same	same	ADJ
cana-3163	4	104	subset	subset	NOUN
cana-3163	4	105	.	.	PUNCT
cana-3163	5	1	notion	notion	NOUN
cana-3163	5	2	of	of	ADP
cana-3163	5	3	bigness	bigness	ADJ
cana-3163	5	4	of	of	ADP
cana-3163	5	5	periodically	periodically	ADV
cana-3163	5	6	rich	rich	ADJ
cana-3163	5	7	maps	map	NOUN
cana-3163	5	8	is	be	AUX
cana-3163	5	9	also	also	ADV
cana-3163	5	10	discussed	discuss	VERB
cana-3163	5	11	.	.	PUNCT
cana-3163	6	1	keywords	keyword	NOUN
cana-3163	6	2	:	:	PUNCT
cana-3163	6	3	set	set	NOUN
cana-3163	6	4	of	of	ADP
cana-3163	6	5	periods	period	NOUN
cana-3163	6	6	,	,	PUNCT
cana-3163	6	7	transitive	transitive	ADJ
cana-3163	6	8	maps	map	NOUN
cana-3163	6	9	,	,	PUNCT
cana-3163	6	10	dense	dense	ADJ
cana-3163	6	11	periodic	periodic	ADJ
cana-3163	6	12	points	point	NOUN
cana-3163	6	13	.	.	PUNCT
cana-3163	7	1	1	1	X
cana-3163	7	2	.	.	X
cana-3163	7	3	introduction	introduction	NOUN
cana-3163	7	4	let	let	VERB
cana-3163	7	5	x	x	PRON
cana-3163	7	6	be	be	AUX
cana-3163	7	7	a	a	DET
cana-3163	7	8	hausdorff	hausdorff	NOUN
cana-3163	7	9	topological	topological	ADJ
cana-3163	7	10	space	space	NOUN
cana-3163	7	11	and	and	CCONJ
cana-3163	7	12	f	f	NOUN
cana-3163	7	13	:	:	PUNCT
cana-3163	7	14	x	x	X
cana-3163	7	15	→	→	PUNCT
cana-3163	7	16	x	x	PUNCT
cana-3163	7	17	be	be	AUX
cana-3163	7	18	a	a	DET
cana-3163	7	19	continuous	continuous	ADJ
cana-3163	7	20	map	map	NOUN
cana-3163	7	21	.	.	PUNCT
cana-3163	8	1	x	x	X
cana-3163	8	2	∈	∈	NOUN
cana-3163	8	3	x	x	PUNCT
cana-3163	8	4	such	such	ADJ
cana-3163	8	5	that	that	SCONJ
cana-3163	8	6	f	f	PROPN
cana-3163	8	7	(	(	PUNCT
cana-3163	8	8	x	x	X
cana-3163	8	9	)	)	PUNCT
cana-3163	8	10	=	=	PUNCT
cana-3163	9	1	x	x	X
cana-3163	9	2	is	be	AUX
cana-3163	9	3	called	call	VERB
cana-3163	9	4	a	a	DET
cana-3163	9	5	fixed	fix	VERB
cana-3163	9	6	point	point	NOUN
cana-3163	9	7	of	of	ADP
cana-3163	9	8	f	f	PROPN
cana-3163	9	9	.	.	PUNCT
cana-3163	10	1	a	a	DET
cana-3163	10	2	point	point	NOUN
cana-3163	10	3	x	x	X
cana-3163	10	4	∈	∈	NOUN
cana-3163	10	5	x	x	PUNCT
cana-3163	10	6	is	be	AUX
cana-3163	10	7	called	call	VERB
cana-3163	10	8	a	a	DET
cana-3163	10	9	periodic	periodic	ADJ
cana-3163	10	10	point	point	NOUN
cana-3163	10	11	if	if	SCONJ
cana-3163	10	12	there	there	PRON
cana-3163	10	13	exists	exist	VERB
cana-3163	10	14	n	n	PRON
cana-3163	10	15	∈	∈	PROPN
cana-3163	10	16	z+	z+	NUM
cana-3163	10	17	such	such	ADJ
cana-3163	10	18	that	that	SCONJ
cana-3163	10	19	fn(x	fn(x	X
cana-3163	10	20	)	)	PUNCT
cana-3163	10	21	=	=	PUNCT
cana-3163	10	22	x.	x.	NOUN
cana-3163	10	23	the	the	DET
cana-3163	10	24	smallest	small	ADJ
cana-3163	10	25	n	n	NOUN
cana-3163	10	26	for	for	ADP
cana-3163	10	27	which	which	PRON
cana-3163	10	28	fn(x	fn(x	NUM
cana-3163	10	29	)	)	PUNCT
cana-3163	11	1	=	=	NOUN
cana-3163	11	2	x	x	X
cana-3163	11	3	is	be	AUX
cana-3163	11	4	called	call	VERB
cana-3163	11	5	the	the	DET
cana-3163	11	6	period	period	NOUN
cana-3163	11	7	of	of	ADP
cana-3163	11	8	f	f	PROPN
cana-3163	11	9	.	.	PUNCT
cana-3163	12	1	the	the	DET
cana-3163	12	2	set	set	NOUN
cana-3163	12	3	of	of	ADP
cana-3163	12	4	periodic	periodic	ADJ
cana-3163	12	5	points	point	NOUN
cana-3163	12	6	of	of	ADP
cana-3163	12	7	period	period	NOUN
cana-3163	12	8	n	n	X
cana-3163	12	9	is	be	AUX
cana-3163	12	10	denoted	denote	VERB
cana-3163	12	11	by	by	ADP
cana-3163	12	12	pern(f	pern(f	PROPN
cana-3163	12	13	)	)	PUNCT
cana-3163	12	14	and	and	CCONJ
cana-3163	12	15	the	the	DET
cana-3163	12	16	set	set	NOUN
cana-3163	12	17	of	of	ADP
cana-3163	12	18	periods	period	NOUN
cana-3163	12	19	of	of	ADP
cana-3163	12	20	f	f	PROPN
cana-3163	12	21	is	be	AUX
cana-3163	12	22	denoted	denote	VERB
cana-3163	12	23	by	by	ADP
cana-3163	12	24	p	p	PROPN
cana-3163	12	25	(	(	PUNCT
cana-3163	12	26	f	f	PROPN
cana-3163	12	27	)	)	PUNCT
cana-3163	12	28	,	,	PUNCT
cana-3163	12	29	i.e.	i.e.	X
cana-3163	12	30	,	,	PUNCT
cana-3163	12	31	pern(f	pern(f	NOUN
cana-3163	12	32	)	)	PUNCT
cana-3163	13	1	=	=	PUNCT
cana-3163	13	2	{	{	PUNCT
cana-3163	13	3	x	x	PUNCT
cana-3163	13	4	∈	∈	PROPN
cana-3163	13	5	x|fn(x	x|fn(x	PROPN
cana-3163	13	6	)	)	PUNCT
cana-3163	14	1	=	=	SYM
cana-3163	14	2	x	x	X
cana-3163	14	3	and	and	CCONJ
cana-3163	14	4	fm(x	fm(x	NOUN
cana-3163	14	5	)	)	PUNCT
cana-3163	14	6	/=	/=	PUNCT
cana-3163	15	1	x	x	X
cana-3163	15	2	,	,	PUNCT
cana-3163	15	3	∀m	∀m	PROPN
cana-3163	15	4	<	<	X
cana-3163	15	5	n	n	CCONJ
cana-3163	15	6	}	}	PUNCT
cana-3163	15	7	p	p	X
cana-3163	15	8	(	(	PUNCT
cana-3163	15	9	f	f	PROPN
cana-3163	15	10	)	)	PUNCT
cana-3163	15	11	=	=	PRON
cana-3163	15	12	{	{	PUNCT
cana-3163	15	13	n	n	PRON
cana-3163	15	14	∈	∈	PROPN
cana-3163	15	15	z+|pern(f	z+|pern(f	X
cana-3163	15	16	)	)	PUNCT
cana-3163	16	1	/=	/=	NOUN
cana-3163	16	2	∅	∅	NOUN
cana-3163	16	3	}	}	PUNCT
cana-3163	16	4	.	.	PUNCT
cana-3163	17	1	per(f	per(f	NOUN
cana-3163	17	2	)	)	PUNCT
cana-3163	18	1	=	=	SYM
cana-3163	18	2	s	s	PART
cana-3163	18	3	n∈z+	n∈z+	ADJ
cana-3163	18	4	pern(f	pern(f	NOUN
cana-3163	18	5	)	)	PUNCT
cana-3163	18	6	is	be	AUX
cana-3163	18	7	the	the	DET
cana-3163	18	8	set	set	NOUN
cana-3163	18	9	of	of	ADP
cana-3163	18	10	periodic	periodic	ADJ
cana-3163	18	11	points	point	NOUN
cana-3163	18	12	of	of	ADP
cana-3163	18	13	f	f	PROPN
cana-3163	18	14	.	.	PUNCT
cana-3163	19	1	the	the	DET
cana-3163	19	2	forward	forward	ADJ
cana-3163	19	3	orbit	orbit	NOUN
cana-3163	19	4	of	of	ADP
cana-3163	19	5	a	a	DET
cana-3163	19	6	point	point	NOUN
cana-3163	19	7	x	x	SYM
cana-3163	19	8	∈	∈	NOUN
cana-3163	19	9	x	x	PUNCT
cana-3163	19	10	under	under	ADP
cana-3163	19	11	the	the	DET
cana-3163	19	12	mapping	mapping	NOUN
cana-3163	19	13	f	f	NOUN
cana-3163	19	14	is	be	AUX
cana-3163	19	15	denoted	denote	VERB
cana-3163	19	16	by	by	ADP
cana-3163	19	17	of	of	ADP
cana-3163	19	18	(	(	PUNCT
cana-3163	19	19	x),i.e	x),i.e	PROPN
cana-3163	19	20	.	.	PUNCT
cana-3163	19	21	,	,	PUNCT
cana-3163	19	22	of	of	ADP
cana-3163	19	23	(	(	PUNCT
cana-3163	19	24	x	x	X
cana-3163	19	25	)	)	PUNCT
cana-3163	19	26	=	=	SYM
cana-3163	19	27	{	{	PUNCT
cana-3163	19	28	x	x	PROPN
cana-3163	19	29	,	,	PUNCT
cana-3163	19	30	f	f	PROPN
cana-3163	19	31	(	(	PUNCT
cana-3163	19	32	x	x	NOUN
cana-3163	19	33	)	)	PUNCT
cana-3163	19	34	,	,	PUNCT
cana-3163	19	35	.	.	PUNCT
cana-3163	19	36	.	.	PUNCT
cana-3163	20	1	.	.	PUNCT
cana-3163	21	1	,	,	PUNCT
cana-3163	21	2	fn(x	fn(x	X
cana-3163	21	3	)	)	PUNCT
cana-3163	21	4	,	,	PUNCT
cana-3163	21	5	.	.	PUNCT
cana-3163	21	6	.	.	PUNCT
cana-3163	21	7	.	.	PUNCT
cana-3163	22	1	}	}	PUNCT
cana-3163	22	2	.	.	PUNCT
cana-3163	23	1	if	if	SCONJ
cana-3163	23	2	x	x	SYM
cana-3163	23	3	∈	∈	PROPN
cana-3163	23	4	pern(f	pern(f	NOUN
cana-3163	23	5	)	)	PUNCT
cana-3163	23	6	then	then	ADV
cana-3163	23	7	{	{	PUNCT
cana-3163	23	8	x	x	X
cana-3163	23	9	,	,	PUNCT
cana-3163	23	10	f	f	PROPN
cana-3163	23	11	(	(	PUNCT
cana-3163	23	12	x	x	NOUN
cana-3163	23	13	)	)	PUNCT
cana-3163	23	14	,	,	PUNCT
cana-3163	23	15	.	.	PUNCT
cana-3163	23	16	.	.	PUNCT
cana-3163	23	17	.	.	PUNCT
cana-3163	23	18	,	,	PUNCT
cana-3163	23	19	fn−1(x	fn−1(x	NOUN
cana-3163	23	20	)	)	PUNCT
cana-3163	23	21	}	}	PUNCT
cana-3163	23	22	is	be	AUX
cana-3163	23	23	called	call	VERB
cana-3163	23	24	periodic	periodic	ADJ
cana-3163	23	25	orbit	orbit	NOUN
cana-3163	23	26	of	of	ADP
cana-3163	23	27	x	x	PRON
cana-3163	23	28	,	,	PUNCT
cana-3163	23	29	in	in	ADP
cana-3163	23	30	this	this	DET
cana-3163	23	31	case	case	NOUN
cana-3163	23	32	clearly	clearly	ADV
cana-3163	23	33	|of	|of	NUM
cana-3163	23	34	(	(	PUNCT
cana-3163	23	35	x)|	x)|	NOUN
cana-3163	23	36	=	=	SYM
cana-3163	23	37	n.	n.	NOUN
cana-3163	23	38	the	the	DET
cana-3163	23	39	periodic	periodic	ADJ
cana-3163	23	40	orbit	orbit	NOUN
cana-3163	23	41	of	of	ADP
cana-3163	23	42	a	a	DET
cana-3163	23	43	point	point	NOUN
cana-3163	23	44	x	x	PUNCT
cana-3163	23	45	in	in	ADP
cana-3163	23	46	i	i	PRON
cana-3163	23	47	=	=	PUNCT
cana-3163	24	1	[	[	X
cana-3163	24	2	0	0	NUM
cana-3163	24	3	,	,	PUNCT
cana-3163	24	4	1	1	NUM
cana-3163	24	5	]	]	PUNCT
cana-3163	24	6	is	be	AUX
cana-3163	24	7	said	say	VERB
cana-3163	24	8	to	to	PART
cana-3163	24	9	be	be	AUX
cana-3163	24	10	increasing	increase	VERB
cana-3163	24	11	if	if	SCONJ
cana-3163	24	12	x	x	PROPN
cana-3163	24	13	<	<	X
cana-3163	24	14	f	f	X
cana-3163	24	15	(	(	PUNCT
cana-3163	24	16	x	x	X
cana-3163	24	17	)	)	PUNCT
cana-3163	24	18	<	<	X
cana-3163	24	19	·	·	PUNCT
cana-3163	24	20	·	·	PUNCT
cana-3163	24	21	·	·	PUNCT
cana-3163	24	22	<	<	X
cana-3163	24	23	fn−1(x	fn−1(x	NOUN
cana-3163	24	24	)	)	PUNCT
cana-3163	24	25	.	.	PUNCT
cana-3163	25	1	a	a	DET
cana-3163	25	2	map	map	NOUN
cana-3163	25	3	f	f	NOUN
cana-3163	25	4	is	be	AUX
cana-3163	25	5	said	say	VERB
cana-3163	25	6	to	to	PART
cana-3163	25	7	be	be	AUX
cana-3163	25	8	transitive	transitive	ADJ
cana-3163	25	9	,	,	PUNCT
cana-3163	25	10	if	if	SCONJ
cana-3163	25	11	for	for	ADP
cana-3163	25	12	every	every	DET
cana-3163	25	13	pair	pair	NOUN
cana-3163	25	14	of	of	ADP
cana-3163	25	15	non	non	ADJ
cana-3163	25	16	-	-	ADJ
cana-3163	25	17	empty	empty	ADJ
cana-3163	25	18	open	open	ADJ
cana-3163	25	19	subsets	subset	NOUN
cana-3163	25	20	u	u	NOUN
cana-3163	25	21	and	and	CCONJ
cana-3163	25	22	v	v	NOUN
cana-3163	25	23	in	in	ADP
cana-3163	25	24	x	x	NOUN
cana-3163	25	25	,	,	PUNCT
cana-3163	25	26	there	there	PRON
cana-3163	25	27	exists	exist	VERB
cana-3163	25	28	a	a	DET
cana-3163	25	29	positive	positive	ADJ
cana-3163	25	30	integer	integer	NOUN
cana-3163	25	31	n	n	CCONJ
cana-3163	25	32	such	such	ADJ
cana-3163	25	33	that	that	DET
cana-3163	25	34	fn(u	fn(u	NOUN
cana-3163	25	35	)	)	PUNCT
cana-3163	26	1	∩	∩	PROPN
cana-3163	26	2	v	v	ADP
cana-3163	26	3	/=	/=	PROPN
cana-3163	26	4	φ	φ	X
cana-3163	26	5	.	.	PUNCT
cana-3163	27	1	in	in	ADP
cana-3163	27	2	general	general	ADJ
cana-3163	27	3	,	,	PUNCT
cana-3163	27	4	the	the	DET
cana-3163	27	5	set	set	NOUN
cana-3163	27	6	of	of	ADP
cana-3163	27	7	periodic	periodic	ADJ
cana-3163	27	8	points	point	NOUN
cana-3163	27	9	per(f	per(f	NOUN
cana-3163	27	10	)	)	PUNCT
cana-3163	27	11	can	can	AUX
cana-3163	27	12	be	be	AUX
cana-3163	27	13	empty	empty	ADJ
cana-3163	27	14	,	,	PUNCT
cana-3163	27	15	a	a	DET
cana-3163	27	16	finite	finite	NOUN
cana-3163	27	17	set	set	NOUN
cana-3163	27	18	,	,	PUNCT
cana-3163	27	19	a	a	DET
cana-3163	27	20	countable	countable	ADJ
cana-3163	27	21	set	set	NOUN
cana-3163	27	22	or	or	CCONJ
cana-3163	27	23	an	an	DET
cana-3163	27	24	uncountable	uncountable	ADJ
cana-3163	27	25	set	set	NOUN
cana-3163	27	26	.	.	PUNCT
cana-3163	28	1	per(f	per(f	NOUN
cana-3163	28	2	)	)	PUNCT
cana-3163	28	3	can	can	AUX
cana-3163	28	4	be	be	AUX
cana-3163	28	5	even	even	ADV
cana-3163	28	6	rich	rich	ADJ
cana-3163	28	7	so	so	SCONJ
cana-3163	28	8	that	that	DET
cana-3163	28	9	per(f	per(f	NOUN
cana-3163	28	10	)	)	PUNCT
cana-3163	28	11	=	=	SYM
cana-3163	29	1	x	x	NOUN
cana-3163	29	2	,	,	PUNCT
cana-3163	29	3	which	which	PRON
cana-3163	29	4	is	be	AUX
cana-3163	29	5	one	one	NUM
cana-3163	29	6	of	of	ADP
cana-3163	29	7	the	the	DET
cana-3163	29	8	condition	condition	NOUN
cana-3163	29	9	for	for	SCONJ
cana-3163	29	10	f	f	PROPN
cana-3163	29	11	to	to	PART
cana-3163	29	12	be	be	AUX
cana-3163	29	13	chaotic	chaotic	ADJ
cana-3163	29	14	in	in	ADP
cana-3163	29	15	the	the	DET
cana-3163	29	16	sense	sense	NOUN
cana-3163	29	17	of	of	ADP
cana-3163	29	18	devaney	devaney	PROPN
cana-3163	30	1	[	[	X
cana-3163	30	2	5	5	NUM
cana-3163	30	3	]	]	PUNCT
cana-3163	30	4	.	.	PUNCT
cana-3163	31	1	if	if	SCONJ
cana-3163	31	2	f	f	PROPN
cana-3163	31	3	is	be	AUX
cana-3163	31	4	transitive	transitive	ADJ
cana-3163	31	5	on	on	ADP
cana-3163	31	6	i	i	PRON
cana-3163	31	7	then	then	ADV
cana-3163	31	8	communications	communication	VERB
cana-3163	31	9	on	on	ADP
cana-3163	31	10	applied	apply	VERB
cana-3163	31	11	nonlinear	nonlinear	ADJ
cana-3163	31	12	analysis	analysis	NOUN
cana-3163	31	13	issn	issn	NOUN
cana-3163	31	14	:	:	PUNCT
cana-3163	31	15	1074	1074	NUM
cana-3163	31	16	-	-	PUNCT
cana-3163	31	17	133x	133x	NUM
cana-3163	31	18	vol	vol	NOUN
cana-3163	31	19	32	32	NUM
cana-3163	31	20	no	no	NOUN
cana-3163	31	21	.	.	PUNCT
cana-3163	32	1	5s	5s	NUM
cana-3163	32	2	(	(	PUNCT
cana-3163	32	3	2025	2025	NUM
cana-3163	32	4	)	)	PUNCT
cana-3163	32	5	512	512	NUM
cana-3163	32	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	32	7	per(f	per(f	NOUN
cana-3163	32	8	)	)	PUNCT
cana-3163	32	9	=	=	PUNCT
cana-3163	33	1	i	i	PRON
cana-3163	34	1	[	[	X
cana-3163	34	2	12	12	NUM
cana-3163	34	3	]	]	PUNCT
cana-3163	34	4	.	.	PUNCT
cana-3163	35	1	a	a	DET
cana-3163	35	2	point	point	NOUN
cana-3163	35	3	x	x	X
cana-3163	35	4	∈	∈	NOUN
cana-3163	35	5	i	i	PRON
cana-3163	35	6	is	be	AUX
cana-3163	35	7	recurrent	recurrent	ADJ
cana-3163	35	8	if	if	SCONJ
cana-3163	35	9	for	for	ADP
cana-3163	35	10	each	each	DET
cana-3163	35	11	neighborhood	neighborhood	NOUN
cana-3163	35	12	u	u	NOUN
cana-3163	35	13	of	of	ADP
cana-3163	35	14	x	x	SYM
cana-3163	35	15	there	there	PRON
cana-3163	35	16	exists	exist	VERB
cana-3163	35	17	n	n	PRON
cana-3163	35	18	∈	∈	PROPN
cana-3163	35	19	z+	z+	NUM
cana-3163	35	20	such	such	ADJ
cana-3163	35	21	that	that	SCONJ
cana-3163	35	22	fn(x	fn(x	X
cana-3163	35	23	)	)	PUNCT
cana-3163	35	24	∈	∈	PROPN
cana-3163	35	25	u	u	NOUN
cana-3163	35	26	;	;	PUNCT
cana-3163	35	27	the	the	DET
cana-3163	35	28	set	set	NOUN
cana-3163	35	29	of	of	ADP
cana-3163	35	30	recurrent	recurrent	ADJ
cana-3163	35	31	points	point	NOUN
cana-3163	35	32	is	be	AUX
cana-3163	35	33	denoted	denote	VERB
cana-3163	35	34	by	by	ADP
cana-3163	35	35	rec(f	rec(f	PROPN
cana-3163	35	36	)	)	PUNCT
cana-3163	35	37	.	.	PUNCT
cana-3163	36	1	it	it	PRON
cana-3163	36	2	is	be	AUX
cana-3163	36	3	proved	prove	VERB
cana-3163	36	4	in	in	ADP
cana-3163	36	5	[	[	X
cana-3163	36	6	3	3	X
cana-3163	36	7	]	]	PUNCT
cana-3163	36	8	that	that	SCONJ
cana-3163	36	9	for	for	ADP
cana-3163	36	10	map	map	NOUN
cana-3163	36	11	f	f	PROPN
cana-3163	36	12	on	on	ADP
cana-3163	36	13	i	i	PRON
cana-3163	36	14	,	,	PUNCT
cana-3163	36	15	per(f	per(f	NOUN
cana-3163	36	16	)	)	PUNCT
cana-3163	36	17	=	=	SYM
cana-3163	36	18	rec(f	rec(f	PROPN
cana-3163	36	19	)	)	PUNCT
cana-3163	36	20	.	.	PUNCT
cana-3163	37	1	for	for	ADP
cana-3163	37	2	maps	map	NOUN
cana-3163	37	3	on	on	ADP
cana-3163	37	4	i	i	PRON
cana-3163	37	5	,	,	PUNCT
cana-3163	37	6	per1(f	per1(f	ADJ
cana-3163	37	7	)	)	PUNCT
cana-3163	37	8	∪per2(f	∪per2(f	NOUN
cana-3163	37	9	)	)	PUNCT
cana-3163	37	10	is	be	AUX
cana-3163	37	11	closed	close	VERB
cana-3163	37	12	[	[	PUNCT
cana-3163	37	13	4	4	NUM
cana-3163	37	14	]	]	PUNCT
cana-3163	37	15	.	.	PUNCT
cana-3163	38	1	considering	consider	VERB
cana-3163	38	2	the	the	DET
cana-3163	38	3	set	set	NOUN
cana-3163	38	4	of	of	ADP
cana-3163	38	5	periods	period	NOUN
cana-3163	38	6	,	,	PUNCT
cana-3163	38	7	p	p	X
cana-3163	38	8	(	(	PUNCT
cana-3163	38	9	f	f	PROPN
cana-3163	38	10	)	)	PUNCT
cana-3163	38	11	can	can	AUX
cana-3163	38	12	also	also	ADV
cana-3163	38	13	be	be	AUX
cana-3163	38	14	empty	empty	ADJ
cana-3163	38	15	,	,	PUNCT
cana-3163	38	16	a	a	DET
cana-3163	38	17	finite	finite	NOUN
cana-3163	38	18	or	or	CCONJ
cana-3163	38	19	an	an	DET
cana-3163	38	20	infinite	infinite	ADJ
cana-3163	38	21	subset	subset	NOUN
cana-3163	38	22	of	of	ADP
cana-3163	38	23	z+	z+	NUM
cana-3163	38	24	or	or	CCONJ
cana-3163	38	25	the	the	DET
cana-3163	38	26	whole	whole	ADJ
cana-3163	38	27	z+	z+	NUM
cana-3163	38	28	.	.	PUNCT
cana-3163	39	1	sharkovskii	sharkovskii	PROPN
cana-3163	39	2	has	have	AUX
cana-3163	39	3	proved	prove	VERB
cana-3163	39	4	that	that	SCONJ
cana-3163	39	5	for	for	ADP
cana-3163	39	6	maps	map	NOUN
cana-3163	39	7	on	on	ADP
cana-3163	39	8	r	r	NOUN
cana-3163	39	9	or	or	CCONJ
cana-3163	39	10	i	i	PRON
cana-3163	39	11	,	,	PUNCT
cana-3163	39	12	if	if	SCONJ
cana-3163	39	13	n	n	PRON
cana-3163	39	14	∈	∈	NOUN
cana-3163	39	15	p	p	X
cana-3163	39	16	(	(	PUNCT
cana-3163	39	17	f	f	PROPN
cana-3163	39	18	)	)	PUNCT
cana-3163	39	19	then	then	ADV
cana-3163	39	20	m	m	VERB
cana-3163	39	21	∈	∈	PROPN
cana-3163	39	22	p	p	X
cana-3163	39	23	(	(	PUNCT
cana-3163	39	24	f	f	PROPN
cana-3163	39	25	)	)	PUNCT
cana-3163	39	26	for	for	ADP
cana-3163	39	27	every	every	DET
cana-3163	39	28	m	m	NOUN
cana-3163	39	29	which	which	PRON
cana-3163	39	30	follows	follow	VERB
cana-3163	39	31	n	n	PRON
cana-3163	39	32	in	in	ADP
cana-3163	39	33	the	the	DET
cana-3163	39	34	sharkovskii	sharkovskii	PROPN
cana-3163	39	35	ordering	order	VERB
cana-3163	39	36	3	3	NUM
cana-3163	39	37	≺	≺	NOUN
cana-3163	39	38	5	5	NUM
cana-3163	39	39	≺	≺	NOUN
cana-3163	39	40	7	7	NUM
cana-3163	39	41	≺	≺	NOUN
cana-3163	39	42	9	9	NUM
cana-3163	39	43	≺	≺	NOUN
cana-3163	39	44	...	...	PUNCT
cana-3163	39	45	≺	≺	VERB
cana-3163	39	46	2.3	2.3	NUM
cana-3163	39	47	≺	≺	NOUN
cana-3163	39	48	2.5	2.5	NUM
cana-3163	39	49	≺	≺	NOUN
cana-3163	39	50	2.7	2.7	NUM
cana-3163	39	51	≺	≺	NOUN
cana-3163	39	52	...	...	PUNCT
cana-3163	39	53	22.3	22.3	NUM
cana-3163	39	54	≺	≺	NOUN
cana-3163	39	55	22.5	22.5	NUM
cana-3163	39	56	≺	≺	NOUN
cana-3163	39	57	...	...	PUNCT
cana-3163	39	58	23.3	23.3	NUM
cana-3163	39	59	≺	≺	NOUN
cana-3163	39	60	23.5	23.5	NUM
cana-3163	39	61	≺	≺	NOUN
cana-3163	39	62	23.7	23.7	NUM
cana-3163	39	63	...	...	PUNCT
cana-3163	39	64	≺	≺	NOUN
cana-3163	39	65	23	23	NUM
cana-3163	39	66	≺	≺	NOUN
cana-3163	39	67	22	22	NUM
cana-3163	39	68	≺	≺	NOUN
cana-3163	39	69	2	2	NUM
cana-3163	39	70	≺	≺	NOUN
cana-3163	39	71	1	1	NUM
cana-3163	39	72	.	.	PUNCT
cana-3163	40	1	[	[	X
cana-3163	40	2	10	10	NUM
cana-3163	40	3	]	]	PUNCT
cana-3163	40	4	at	at	ADP
cana-3163	40	5	this	this	DET
cana-3163	40	6	juncture	juncture	NOUN
cana-3163	40	7	a	a	DET
cana-3163	40	8	natural	natural	ADJ
cana-3163	40	9	question	question	NOUN
cana-3163	40	10	arises	arise	VERB
cana-3163	40	11	.	.	PUNCT
cana-3163	41	1	is	be	AUX
cana-3163	41	2	there	there	PRON
cana-3163	41	3	any	any	DET
cana-3163	41	4	relation	relation	NOUN
cana-3163	41	5	between	between	ADP
cana-3163	41	6	per	per	ADP
cana-3163	41	7	(	(	PUNCT
cana-3163	41	8	f	f	PROPN
cana-3163	41	9	)	)	PUNCT
cana-3163	41	10	and	and	CCONJ
cana-3163	41	11	p	p	X
cana-3163	41	12	(	(	PUNCT
cana-3163	41	13	f	f	PROPN
cana-3163	41	14	)	)	PUNCT
cana-3163	41	15	?	?	PUNCT
cana-3163	42	1	the	the	DET
cana-3163	42	2	question	question	NOUN
cana-3163	42	3	prompts	prompt	VERB
cana-3163	42	4	to	to	PART
cana-3163	42	5	think	think	VERB
cana-3163	42	6	about	about	ADP
cana-3163	42	7	the	the	DET
cana-3163	42	8	’	'	PUNCT
cana-3163	42	9	bigness	bigness	NOUN
cana-3163	42	10	’	'	PUNCT
cana-3163	42	11	of	of	ADP
cana-3163	42	12	set	set	NOUN
cana-3163	42	13	of	of	ADP
cana-3163	42	14	periodic	periodic	ADJ
cana-3163	42	15	points	point	NOUN
cana-3163	42	16	.	.	PUNCT
cana-3163	43	1	more	more	ADV
cana-3163	43	2	specifically	specifically	ADV
cana-3163	43	3	,	,	PUNCT
cana-3163	43	4	the	the	DET
cana-3163	43	5	‘	'	PUNCT
cana-3163	43	6	bigness	bigness	NOUN
cana-3163	43	7	’	'	PUNCT
cana-3163	43	8	of	of	ADP
cana-3163	43	9	set	set	NOUN
cana-3163	43	10	of	of	ADP
cana-3163	43	11	periodic	periodic	ADJ
cana-3163	43	12	points	point	NOUN
cana-3163	43	13	is	be	AUX
cana-3163	43	14	treated	treat	VERB
cana-3163	43	15	in	in	ADP
cana-3163	43	16	two	two	NUM
cana-3163	43	17	ways	way	NOUN
cana-3163	43	18	,	,	PUNCT
cana-3163	43	19	(	(	PUNCT
cana-3163	43	20	1	1	X
cana-3163	43	21	)	)	PUNCT
cana-3163	43	22	perf	perf	NOUN
cana-3163	43	23	=	=	PUNCT
cana-3163	44	1	x	x	X
cana-3163	44	2	and	and	CCONJ
cana-3163	44	3	(	(	PUNCT
cana-3163	44	4	2	2	X
cana-3163	44	5	)	)	PUNCT
cana-3163	44	6	p	p	NOUN
cana-3163	44	7	(	(	PUNCT
cana-3163	44	8	f	f	PROPN
cana-3163	44	9	)	)	PUNCT
cana-3163	44	10	=	=	PUNCT
cana-3163	44	11	z+	z+	X
cana-3163	44	12	.	.	PUNCT
cana-3163	44	13	to	to	PART
cana-3163	44	14	understand	understand	VERB
cana-3163	44	15	,	,	PUNCT
cana-3163	44	16	some	some	DET
cana-3163	44	17	examples	example	NOUN
cana-3163	44	18	are	be	AUX
cana-3163	44	19	listed	list	VERB
cana-3163	44	20	below	below	ADV
cana-3163	44	21	.	.	PUNCT
cana-3163	45	1	example	example	NOUN
cana-3163	46	1	1	1	NUM
cana-3163	46	2	.	.	PUNCT
cana-3163	46	3	a	a	DET
cana-3163	46	4	map	map	NOUN
cana-3163	46	5	for	for	ADP
cana-3163	46	6	which	which	PRON
cana-3163	46	7	per(f	per(f	NOUN
cana-3163	46	8	)	)	PUNCT
cana-3163	46	9	=	=	PUNCT
cana-3163	47	1	x	x	PROPN
cana-3163	47	2	and	and	CCONJ
cana-3163	47	3	p	p	X
cana-3163	47	4	(	(	PUNCT
cana-3163	47	5	f	f	PROPN
cana-3163	47	6	)	)	PUNCT
cana-3163	47	7	/=	/=	PROPN
cana-3163	48	1	z+	z+	NUM
cana-3163	48	2	let	let	VERB
cana-3163	48	3	f	f	PROPN
cana-3163	48	4	:	:	PUNCT
cana-3163	48	5	i	i	PRON
cana-3163	48	6	—	—	PUNCT
cana-3163	48	7	→	→	PUNCT
cana-3163	48	8	i	i	PRON
cana-3163	48	9	be	be	AUX
cana-3163	48	10	defined	define	VERB
cana-3163	48	11	as	as	ADP
cana-3163	48	12	2	2	NUM
cana-3163	48	13	x	x	SYM
cana-3163	48	14	+	+	NUM
cana-3163	48	15	1/2	1/2	NUM
cana-3163	48	16	,	,	PUNCT
cana-3163	48	17	0	0	NUM
cana-3163	48	18	≤	≤	NUM
cana-3163	48	19	x	x	SYM
cana-3163	48	20	≤	≤	NUM
cana-3163	48	21	1/4	1/4	NUM
cana-3163	48	22	f1(x	f1(x	NUM
cana-3163	48	23	)	)	PUNCT
cana-3163	48	24	=	=	PUNCT
cana-3163	49	1	2x	2x	NOUN
cana-3163	50	1	+	+	SYM
cana-3163	50	2	3/2	3/2	NUM
cana-3163	50	3	,	,	PUNCT
cana-3163	50	4	1/4	1/4	NUM
cana-3163	50	5	<	<	X
cana-3163	50	6	x	x	SYM
cana-3163	50	7	≤	≤	NUM
cana-3163	50	8	3/4	3/4	NUM
cana-3163	50	9	2x	2x	NUM
cana-3163	50	10	3/2	3/2	NUM
cana-3163	50	11	,	,	PUNCT
cana-3163	50	12	3/4	3/4	NUM
cana-3163	50	13	<	<	X
cana-3163	50	14	x	x	SYM
cana-3163	50	15	≤	≤	ADJ
cana-3163	50	16	1	1	NUM
cana-3163	50	17	figure	figure	NOUN
cana-3163	50	18	1	1	NUM
cana-3163	50	19	.	.	PUNCT
cana-3163	51	1	graph	graph	NOUN
cana-3163	51	2	of	of	ADP
cana-3163	51	3	f1	f1	PROPN
cana-3163	51	4	communications	communication	NOUN
cana-3163	51	5	on	on	ADP
cana-3163	51	6	applied	apply	VERB
cana-3163	51	7	nonlinear	nonlinear	ADJ
cana-3163	51	8	analysis	analysis	NOUN
cana-3163	51	9	issn	issn	NOUN
cana-3163	51	10	:	:	PUNCT
cana-3163	51	11	1074	1074	NUM
cana-3163	51	12	-	-	PUNCT
cana-3163	51	13	133x	133x	NUM
cana-3163	51	14	vol	vol	NOUN
cana-3163	51	15	32	32	NUM
cana-3163	51	16	no	no	NOUN
cana-3163	51	17	.	.	PUNCT
cana-3163	52	1	5s	5s	NUM
cana-3163	52	2	(	(	PUNCT
cana-3163	52	3	2025	2025	NUM
cana-3163	52	4	)	)	PUNCT
cana-3163	52	5	513	513	NUM
cana-3163	52	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-3163	52	7	1	1	NUM
cana-3163	52	8	1	1	NUM
cana-3163	52	9	1	1	NUM
cana-3163	52	10	1	1	NUM
cana-3163	52	11	fixed	fix	VERB
cana-3163	52	12	point	point	NOUN
cana-3163	52	13	of	of	ADP
cana-3163	52	14	f1	f1	NOUN
cana-3163	52	15	is	be	AUX
cana-3163	52	16	at	at	ADP
cana-3163	52	17	x	x	X
cana-3163	52	18	=	=	SYM
cana-3163	52	19	1/2	1/2	NUM
cana-3163	52	20	.	.	PUNCT
cana-3163	53	1	f1([0	f1([0	NOUN
cana-3163	53	2	,	,	PUNCT
cana-3163	53	3	1/2	1/2	NUM
cana-3163	53	4	]	]	PUNCT
cana-3163	53	5	)	)	PUNCT
cana-3163	54	1	=	=	PUNCT
cana-3163	55	1	[	[	X
cana-3163	55	2	1/2	1/2	NUM
cana-3163	55	3	,	,	PUNCT
cana-3163	55	4	1	1	NUM
cana-3163	55	5	]	]	PUNCT
cana-3163	55	6	and	and	CCONJ
cana-3163	55	7	f1([1/2	f1([1/2	PROPN
cana-3163	55	8	,	,	PUNCT
cana-3163	55	9	1	1	NUM
cana-3163	55	10	]	]	PUNCT
cana-3163	55	11	)	)	PUNCT
cana-3163	56	1	=	=	PUNCT
cana-3163	57	1	[	[	X
cana-3163	57	2	0	0	NUM
cana-3163	57	3	,	,	PUNCT
cana-3163	57	4	1/2	1/2	NUM
cana-3163	57	5	]	]	X
cana-3163	57	6	f	f	PROPN
cana-3163	57	7	2([0	2([0	PROPN
cana-3163	57	8	,	,	PUNCT
cana-3163	57	9	1/2	1/2	NUM
cana-3163	57	10	]	]	PUNCT
cana-3163	57	11	)	)	PUNCT
cana-3163	57	12	=	=	SYM
cana-3163	57	13	f1(f1([0	f1(f1([0	PROPN
cana-3163	57	14	,	,	PUNCT
cana-3163	57	15	1/2	1/2	NUM
cana-3163	57	16	]	]	PUNCT
cana-3163	57	17	)	)	PUNCT
cana-3163	57	18	)	)	PUNCT
cana-3163	58	1	=	=	SYM
cana-3163	58	2	f1([1/2	f1([1/2	PROPN
cana-3163	58	3	,	,	PUNCT
cana-3163	58	4	1	1	NUM
cana-3163	58	5	]	]	PUNCT
cana-3163	58	6	)	)	PUNCT
cana-3163	59	1	=	=	PUNCT
cana-3163	60	1	[	[	X
cana-3163	60	2	0	0	NUM
cana-3163	60	3	,	,	PUNCT
cana-3163	60	4	1/2	1/2	NUM
cana-3163	60	5	]	]	PUNCT
cana-3163	60	6	f	f	PROPN
cana-3163	60	7	2([1/2	2([1/2	NUM
cana-3163	60	8	,	,	PUNCT
cana-3163	60	9	1	1	NUM
cana-3163	60	10	]	]	PUNCT
cana-3163	60	11	)	)	PUNCT
cana-3163	60	12	=	=	SYM
cana-3163	60	13	f1(f1([1/2	f1(f1([1/2	PROPN
cana-3163	60	14	,	,	PUNCT
cana-3163	60	15	1	1	NUM
cana-3163	60	16	]	]	NUM
cana-3163	60	17	)	)	PUNCT
cana-3163	60	18	)	)	PUNCT
cana-3163	61	1	=	=	SYM
cana-3163	61	2	f1([0	f1([0	NOUN
cana-3163	61	3	,	,	PUNCT
cana-3163	61	4	1/2	1/2	NUM
cana-3163	61	5	]	]	PUNCT
cana-3163	61	6	)	)	PUNCT
cana-3163	61	7	=	=	PUNCT
cana-3163	62	1	[	[	X
cana-3163	62	2	1/2	1/2	NUM
cana-3163	62	3	,	,	PUNCT
cana-3163	62	4	1	1	NUM
cana-3163	62	5	]	]	SYM
cana-3163	62	6	f	f	PROPN
cana-3163	62	7	3([0	3([0	NUM
cana-3163	62	8	,	,	PUNCT
cana-3163	62	9	1/2	1/2	NUM
cana-3163	62	10	]	]	PUNCT
cana-3163	62	11	)	)	PUNCT
cana-3163	62	12	=	=	SYM
cana-3163	63	1	f1(f	f1(f	PROPN
cana-3163	63	2	2([0	2([0	NUM
cana-3163	63	3	,	,	PUNCT
cana-3163	63	4	1/2	1/2	NUM
cana-3163	63	5	]	]	PUNCT
cana-3163	63	6	)	)	PUNCT
cana-3163	63	7	)	)	PUNCT
cana-3163	64	1	=	=	SYM
cana-3163	64	2	f1([0	f1([0	NOUN
cana-3163	64	3	,	,	PUNCT
cana-3163	64	4	1/2	1/2	NUM
cana-3163	64	5	]	]	PUNCT
cana-3163	64	6	)	)	PUNCT
cana-3163	64	7	=	=	PUNCT
cana-3163	65	1	[	[	X
cana-3163	65	2	1/2	1/2	NUM
cana-3163	65	3	,	,	PUNCT
cana-3163	65	4	1	1	NUM
cana-3163	65	5	]	]	SYM
cana-3163	65	6	1	1	NUM
cana-3163	65	7	1	1	NUM
cana-3163	65	8	f	f	NOUN
cana-3163	65	9	3([1/2	3([1/2	NUM
cana-3163	65	10	,	,	PUNCT
cana-3163	65	11	1	1	NUM
cana-3163	65	12	]	]	PUNCT
cana-3163	65	13	)	)	PUNCT
cana-3163	66	1	=	=	SYM
cana-3163	67	1	f1(f	f1(f	PROPN
cana-3163	67	2	2([1/2	2([1/2	NUM
cana-3163	67	3	,	,	PUNCT
cana-3163	67	4	1	1	NUM
cana-3163	67	5	]	]	NUM
cana-3163	67	6	)	)	PUNCT
cana-3163	67	7	)	)	PUNCT
cana-3163	68	1	=	=	SYM
cana-3163	68	2	f1([1/2	f1([1/2	PROPN
cana-3163	68	3	,	,	PUNCT
cana-3163	68	4	1	1	NUM
cana-3163	68	5	]	]	PUNCT
cana-3163	68	6	)	)	PUNCT
cana-3163	69	1	=	=	PUNCT
cana-3163	70	1	[	[	X
cana-3163	70	2	0	0	NUM
cana-3163	70	3	,	,	PUNCT
cana-3163	70	4	1/2	1/2	NUM
cana-3163	70	5	]	]	SYM
cana-3163	70	6	1	1	NUM
cana-3163	70	7	1	1	NUM
cana-3163	70	8	so	so	ADV
cana-3163	70	9	,	,	PUNCT
cana-3163	70	10	f	f	PROPN
cana-3163	70	11	3(x	3(x	NUM
cana-3163	70	12	)	)	PUNCT
cana-3163	70	13	/=	/=	NOUN
cana-3163	70	14	x	x	PUNCT
cana-3163	71	1	for	for	ADP
cana-3163	71	2	all	all	DET
cana-3163	71	3	x	x	SYM
cana-3163	71	4	∈	∈	PROPN
cana-3163	71	5	[	[	X
cana-3163	71	6	0	0	NUM
cana-3163	71	7	,	,	PUNCT
cana-3163	71	8	1	1	NUM
cana-3163	71	9	]	]	PUNCT
cana-3163	71	10	except	except	SCONJ
cana-3163	71	11	for	for	ADP
cana-3163	71	12	x	x	SYM
cana-3163	71	13	=	=	SYM
cana-3163	71	14	1/2	1/2	NUM
cana-3163	71	15	.	.	PUNCT
cana-3163	72	1	similarly	similarly	ADV
cana-3163	72	2	,	,	PUNCT
cana-3163	72	3	if	if	SCONJ
cana-3163	72	4	n	n	PRON
cana-3163	72	5	is	be	AUX
cana-3163	72	6	odd	odd	ADJ
cana-3163	72	7	,	,	PUNCT
cana-3163	72	8	fn(x	fn(x	PRON
cana-3163	72	9	)	)	PUNCT
cana-3163	72	10	/=	/=	PUNCT
cana-3163	73	1	x	x	SYM
cana-3163	73	2	,	,	PUNCT
cana-3163	73	3	∀x	∀x	X
cana-3163	73	4	∈	∈	PROPN
cana-3163	74	1	[	[	X
cana-3163	74	2	0	0	NUM
cana-3163	74	3	,	,	PUNCT
cana-3163	74	4	1	1	NUM
cana-3163	74	5	]	]	PUNCT
cana-3163	74	6	except	except	SCONJ
cana-3163	74	7	for	for	ADP
cana-3163	74	8	x	x	SYM
cana-3163	74	9	=	=	SYM
cana-3163	74	10	1/2	1/2	NUM
cana-3163	74	11	.	.	PUNCT
cana-3163	75	1	if	if	SCONJ
cana-3163	75	2	n	n	NOUN
cana-3163	75	3	is	be	AUX
cana-3163	75	4	even	even	ADV
cana-3163	75	5	,	,	PUNCT
cana-3163	75	6	fn([0	fn([0	NOUN
cana-3163	75	7	,	,	PUNCT
cana-3163	75	8	1/2	1/2	NUM
cana-3163	75	9	]	]	PUNCT
cana-3163	75	10	)	)	PUNCT
cana-3163	76	1	=	=	PUNCT
cana-3163	77	1	[	[	X
cana-3163	77	2	0	0	NUM
cana-3163	77	3	,	,	PUNCT
cana-3163	77	4	1/2	1/2	NUM
cana-3163	77	5	]	]	PUNCT
cana-3163	77	6	and	and	CCONJ
cana-3163	77	7	fn([1/2	fn([1/2	PROPN
cana-3163	77	8	,	,	PUNCT
cana-3163	77	9	1	1	NUM
cana-3163	77	10	]	]	PUNCT
cana-3163	77	11	)	)	PUNCT
cana-3163	78	1	=	=	PUNCT
cana-3163	79	1	[	[	X
cana-3163	79	2	1/2	1/2	NUM
cana-3163	79	3	,	,	PUNCT
cana-3163	79	4	1	1	NUM
cana-3163	79	5	]	]	PUNCT
cana-3163	79	6	.	.	PUNCT
cana-3163	80	1	i.e.	i.e.	X
cana-3163	80	2	,	,	PUNCT
cana-3163	80	3	fn	fn	PROPN
cana-3163	80	4	has	have	AUX
cana-3163	80	5	fixed	fix	VERB
cana-3163	80	6	points	point	NOUN
cana-3163	80	7	1	1	NUM
cana-3163	80	8	1	1	NUM
cana-3163	80	9	1	1	NUM
cana-3163	80	10	in	in	ADP
cana-3163	80	11	[	[	X
cana-3163	80	12	0	0	NUM
cana-3163	80	13	,	,	PUNCT
cana-3163	80	14	1	1	NUM
cana-3163	80	15	]	]	PUNCT
cana-3163	80	16	for	for	ADP
cana-3163	80	17	even	even	ADV
cana-3163	80	18	n.	n.	NOUN
cana-3163	80	19	so	so	SCONJ
cana-3163	80	20	there	there	PRON
cana-3163	80	21	are	be	VERB
cana-3163	80	22	points	point	NOUN
cana-3163	80	23	with	with	ADP
cana-3163	80	24	even	even	ADV
cana-3163	80	25	periods	period	NOUN
cana-3163	80	26	in	in	ADP
cana-3163	80	27	[	[	X
cana-3163	80	28	0	0	NUM
cana-3163	80	29	,	,	PUNCT
cana-3163	80	30	1	1	NUM
cana-3163	80	31	]	]	PUNCT
cana-3163	80	32	.	.	PUNCT
cana-3163	81	1	therefore	therefore	ADV
cana-3163	81	2	p	p	X
cana-3163	81	3	(	(	PUNCT
cana-3163	81	4	f1	f1	NOUN
cana-3163	81	5	)	)	PUNCT
cana-3163	81	6	=	=	PUNCT
cana-3163	81	7	z+\{3	z+\{3	NOUN
cana-3163	81	8	,	,	PUNCT
cana-3163	81	9	5	5	NUM
cana-3163	81	10	,	,	PUNCT
cana-3163	81	11	7	7	NUM
cana-3163	81	12	,	,	PUNCT
cana-3163	81	13	.	.	PUNCT
cana-3163	81	14	.	.	PUNCT
cana-3163	81	15	.	.	PUNCT
cana-3163	81	16	}	}	PUNCT
cana-3163	81	17	.	.	PUNCT
cana-3163	82	1	on	on	ADP
cana-3163	82	2	the	the	DET
cana-3163	82	3	other	other	ADJ
cana-3163	82	4	hand	hand	NOUN
cana-3163	82	5	,	,	PUNCT
cana-3163	82	6	if	if	SCONJ
cana-3163	82	7	x	x	PRON
cana-3163	82	8	is	be	AUX
cana-3163	82	9	any	any	DET
cana-3163	82	10	irrational	irrational	ADJ
cana-3163	82	11	number	number	NOUN
cana-3163	82	12	in	in	ADP
cana-3163	82	13	[	[	X
cana-3163	82	14	0	0	NUM
cana-3163	82	15	,	,	PUNCT
cana-3163	82	16	1	1	NUM
cana-3163	82	17	]	]	PUNCT
cana-3163	82	18	then	then	ADV
cana-3163	82	19	of1	of1	INTJ
cana-3163	82	20	(	(	PUNCT
cana-3163	82	21	x	x	NOUN
cana-3163	82	22	)	)	PUNCT
cana-3163	82	23	=	=	SYM
cana-3163	82	24	i.	i.	NOUN
cana-3163	82	25	so	so	ADV
cana-3163	82	26	f1	f1	PROPN
cana-3163	82	27	is	be	AUX
cana-3163	82	28	transitive	transitive	ADJ
cana-3163	82	29	on	on	ADP
cana-3163	82	30	i	i	PRON
cana-3163	82	31	[	[	X
cana-3163	82	32	11	11	NUM
cana-3163	82	33	]	]	PUNCT
cana-3163	82	34	.	.	PUNCT
cana-3163	83	1	hence	hence	ADV
cana-3163	83	2	per(f1	per(f1	PROPN
cana-3163	83	3	)	)	PUNCT
cana-3163	83	4	=	=	SYM
cana-3163	84	1	i	i	PRON
cana-3163	85	1	[	[	X
cana-3163	85	2	12	12	NUM
cana-3163	85	3	]	]	PUNCT
cana-3163	85	4	.	.	PUNCT
cana-3163	85	5	example	example	NOUN
cana-3163	85	6	2	2	NUM
cana-3163	85	7	.	.	X
cana-3163	86	1	consider	consider	VERB
cana-3163	86	2	a	a	DET
cana-3163	86	3	map	map	NOUN
cana-3163	86	4	with	with	ADP
cana-3163	86	5	p	p	PROPN
cana-3163	86	6	(	(	PUNCT
cana-3163	86	7	f	f	PROPN
cana-3163	86	8	)	)	PUNCT
cana-3163	86	9	=	=	SYM
cana-3163	86	10	z+	z+	NUM
cana-3163	86	11	and	and	CCONJ
cana-3163	86	12	per(f	per(f	NOUN
cana-3163	86	13	)	)	PUNCT
cana-3163	87	1	/=	/=	NOUN
cana-3163	87	2	x.	x.	NOUN
cana-3163	87	3	let	let	VERB
cana-3163	87	4	f2	f2	PRON
cana-3163	87	5	:	:	PUNCT
cana-3163	87	6	i	i	PRON
cana-3163	87	7	→	→	PUNCT
cana-3163	87	8	i	i	PRON
cana-3163	87	9	be	be	AUX
cana-3163	87	10	defined	define	VERB
cana-3163	87	11	as	as	ADP
cana-3163	87	12	figure	figure	NOUN
cana-3163	87	13	2	2	NUM
cana-3163	87	14	.	.	PUNCT
cana-3163	88	1	graph	graph	NOUN
cana-3163	88	2	of	of	ADP
cana-3163	88	3	f2	f2	PROPN
cana-3163	88	4	3x	3x	NUM
cana-3163	88	5	,	,	PUNCT
cana-3163	88	6	0	0	NUM
cana-3163	88	7	≤	≤	NUM
cana-3163	88	8	x	x	SYM
cana-3163	88	9	≤	≤	NOUN
cana-3163	88	10	1/3	1/3	NUM
cana-3163	88	11	f2(x	f2(x	NOUN
cana-3163	88	12	)	)	PUNCT
cana-3163	88	13	=	=	SYM
cana-3163	88	14	1	1	NUM
cana-3163	88	15	,	,	PUNCT
cana-3163	88	16	1/3	1/3	PRON
cana-3163	88	17	<	<	X
cana-3163	88	18	x	x	SYM
cana-3163	88	19	≤	≤	NOUN
cana-3163	88	20	2/3	2/3	NUM
cana-3163	88	21	33x	33x	NOUN
cana-3163	88	22	,	,	PUNCT
cana-3163	88	23	2/3	2/3	NUM
cana-3163	88	24	<	<	X
cana-3163	88	25	x	x	SYM
cana-3163	88	26	≤	≤	NUM
cana-3163	88	27	1	1	NUM
cana-3163	88	28	of2	of2	PROPN
cana-3163	88	29	(	(	PUNCT
cana-3163	88	30	9/28	9/28	NUM
cana-3163	88	31	)	)	PUNCT
cana-3163	88	32	=	=	PRON
cana-3163	88	33	{	{	PUNCT
cana-3163	88	34	9/28	9/28	NUM
cana-3163	88	35	,	,	PUNCT
cana-3163	88	36	27/28	27/28	NUM
cana-3163	88	37	,	,	PUNCT
cana-3163	88	38	3/28	3/28	NUM
cana-3163	88	39	}	}	PUNCT
cana-3163	88	40	.	.	PUNCT
cana-3163	89	1	communications	communication	NOUN
cana-3163	89	2	on	on	ADP
cana-3163	89	3	applied	apply	VERB
cana-3163	89	4	nonlinear	nonlinear	ADJ
cana-3163	89	5	analysis	analysis	NOUN
cana-3163	89	6	issn	issn	NOUN
cana-3163	89	7	:	:	PUNCT
cana-3163	89	8	1074	1074	NUM
cana-3163	89	9	-	-	PUNCT
cana-3163	89	10	133x	133x	NUM
cana-3163	89	11	vol	vol	NOUN
cana-3163	89	12	32	32	NUM
cana-3163	89	13	no	no	NOUN
cana-3163	89	14	.	.	PUNCT
cana-3163	90	1	5s	5s	NUM
cana-3163	90	2	(	(	PUNCT
cana-3163	90	3	2025	2025	NUM
cana-3163	90	4	)	)	PUNCT
cana-3163	90	5	514	514	NUM
cana-3163	91	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	91	2	2	2	NUM
cana-3163	91	3	4	4	NUM
cana-3163	91	4	i.e.	i.e.	X
cana-3163	91	5	,	,	PUNCT
cana-3163	91	6	9/28	9/28	NUM
cana-3163	91	7	is	be	AUX
cana-3163	91	8	a	a	DET
cana-3163	91	9	periodic	periodic	ADJ
cana-3163	91	10	point	point	NOUN
cana-3163	91	11	of	of	ADP
cana-3163	91	12	period	period	NOUN
cana-3163	91	13	3	3	NUM
cana-3163	91	14	.	.	PUNCT
cana-3163	91	15	hence	hence	ADV
cana-3163	91	16	by	by	ADP
cana-3163	91	17	proposition	proposition	NOUN
cana-3163	91	18	1	1	NUM
cana-3163	91	19	,	,	PUNCT
cana-3163	91	20	p	p	X
cana-3163	91	21	(	(	PUNCT
cana-3163	91	22	f2	f2	PROPN
cana-3163	91	23	)	)	PUNCT
cana-3163	91	24	=	=	SYM
cana-3163	91	25	z+	z+	X
cana-3163	91	26	.	.	PUNCT
cana-3163	92	1	now	now	ADV
cana-3163	92	2	consider	consider	VERB
cana-3163	92	3	open	open	ADJ
cana-3163	92	4	sets	set	NOUN
cana-3163	92	5	u	u	NOUN
cana-3163	92	6	and	and	CCONJ
cana-3163	92	7	v	v	NOUN
cana-3163	92	8	in	in	ADP
cana-3163	92	9	(	(	PUNCT
cana-3163	92	10	1/3	1/3	NUM
cana-3163	92	11	,	,	PUNCT
cana-3163	92	12	2/3	2/3	NUM
cana-3163	92	13	)	)	PUNCT
cana-3163	92	14	.	.	PUNCT
cana-3163	93	1	fn(u	fn(u	PUNCT
cana-3163	93	2	)	)	PUNCT
cana-3163	94	1	=	=	SYM
cana-3163	94	2	fn(v	fn(v	X
cana-3163	94	3	)	)	PUNCT
cana-3163	94	4	=	=	PUNCT
cana-3163	94	5	{	{	PUNCT
cana-3163	94	6	0	0	NUM
cana-3163	94	7	}	}	PUNCT
cana-3163	94	8	,	,	PUNCT
cana-3163	94	9	∀	∀	NOUN
cana-3163	94	10	n	n	PRON
cana-3163	94	11	∈	∈	PROPN
cana-3163	94	12	z+	z+	NUM
cana-3163	94	13	.	.	PUNCT
cana-3163	95	1	so	so	ADV
cana-3163	95	2	,	,	PUNCT
cana-3163	95	3	2	2	NUM
cana-3163	95	4	2	2	NUM
cana-3163	95	5	there	there	PRON
cana-3163	95	6	does	do	AUX
cana-3163	95	7	not	not	PART
cana-3163	95	8	exists	exist	VERB
cana-3163	95	9	n	n	PRON
cana-3163	95	10	∈	∈	PROPN
cana-3163	95	11	z+	z+	NUM
cana-3163	95	12	such	such	ADJ
cana-3163	95	13	that	that	DET
cana-3163	95	14	fn(u	fn(u	NOUN
cana-3163	95	15	)	)	PUNCT
cana-3163	96	1	∩	∩	PROPN
cana-3163	96	2	v	v	ADP
cana-3163	96	3	/=	/=	PROPN
cana-3163	96	4	∅.	∅.	VERB
cana-3163	96	5	therefore	therefore	ADV
cana-3163	96	6	per(f2	per(f2	NOUN
cana-3163	96	7	)	)	PUNCT
cana-3163	97	1	/=	/=	PROPN
cana-3163	97	2	i.	i.	PROPN
cana-3163	97	3	example	example	NOUN
cana-3163	98	1	3	3	NUM
cana-3163	98	2	.	.	PUNCT
cana-3163	98	3	tent	tent	NOUN
cana-3163	98	4	map	map	NOUN
cana-3163	98	5	defined	define	VERB
cana-3163	98	6	on	on	ADP
cana-3163	98	7	i.	i.	PROPN
cana-3163	98	8	f3	f3	PROPN
cana-3163	98	9	(	(	PUNCT
cana-3163	98	10	x	x	X
cana-3163	98	11	)	)	PUNCT
cana-3163	98	12	=	=	SYM
cana-3163	98	13	2x	2x	NUM
cana-3163	98	14	,	,	PUNCT
cana-3163	98	15	0	0	NUM
cana-3163	98	16	≤	≤	NUM
cana-3163	98	17	x	x	SYM
cana-3163	98	18	≤	≤	NUM
cana-3163	98	19	1/2	1/2	NUM
cana-3163	98	20	2(1	2(1	NUM
cana-3163	98	21	—	—	PUNCT
cana-3163	98	22	x	x	X
cana-3163	98	23	)	)	PUNCT
cana-3163	98	24	,	,	PUNCT
cana-3163	98	25	1/2	1/2	NUM
cana-3163	98	26	<	<	X
cana-3163	98	27	x	x	SYM
cana-3163	98	28	≤	≤	NUM
cana-3163	98	29	1	1	NUM
cana-3163	98	30	p	p	NOUN
cana-3163	98	31	(	(	PUNCT
cana-3163	98	32	f3	f3	ADJ
cana-3163	98	33	)	)	PUNCT
cana-3163	98	34	=	=	SYM
cana-3163	98	35	z+	z+	NUM
cana-3163	98	36	and	and	CCONJ
cana-3163	98	37	per(f3	per(f3	ADJ
cana-3163	98	38	)	)	PUNCT
cana-3163	99	1	=	=	SYM
cana-3163	99	2	i	i	PRON
cana-3163	100	1	[	[	X
cana-3163	100	2	9	9	NUM
cana-3163	100	3	]	]	PUNCT
cana-3163	100	4	.	.	PUNCT
cana-3163	101	1	example	example	NOUN
cana-3163	101	2	4	4	NUM
cana-3163	101	3	.	.	X
cana-3163	102	1	consider	consider	VERB
cana-3163	102	2	f4	f4	NOUN
cana-3163	102	3	defined	define	VERB
cana-3163	102	4	on	on	ADP
cana-3163	102	5	i	i	PRON
cana-3163	102	6	by	by	ADP
cana-3163	102	7	f4(x	f4(x	NUM
cana-3163	103	1	)	)	PUNCT
cana-3163	103	2	=	=	SYM
cana-3163	103	3	x2	x2	PROPN
cana-3163	103	4	.	.	PUNCT
cana-3163	104	1	fixed	fix	VERB
cana-3163	104	2	points	point	NOUN
cana-3163	104	3	are	be	AUX
cana-3163	104	4	0	0	NUM
cana-3163	104	5	,	,	PUNCT
cana-3163	104	6	1	1	NUM
cana-3163	104	7	and	and	CCONJ
cana-3163	104	8	fn(x	fn(x	NOUN
cana-3163	104	9	)	)	PUNCT
cana-3163	105	1	→	→	SYM
cana-3163	105	2	0	0	NUM
cana-3163	105	3	,	,	PUNCT
cana-3163	105	4	∀x	∀x	X
cana-3163	105	5	∈	∈	PROPN
cana-3163	105	6	(	(	PUNCT
cana-3163	105	7	0	0	NUM
cana-3163	105	8	,	,	PUNCT
cana-3163	105	9	1	1	NUM
cana-3163	105	10	)	)	PUNCT
cana-3163	105	11	.	.	PUNCT
cana-3163	106	1	so	so	ADV
cana-3163	106	2	no	no	DET
cana-3163	106	3	periodic	periodic	ADJ
cana-3163	106	4	points	point	NOUN
cana-3163	106	5	other	other	ADJ
cana-3163	106	6	than	than	ADP
cana-3163	106	7	fixed	fix	VERB
cana-3163	106	8	points	point	NOUN
cana-3163	106	9	.	.	PUNCT
cana-3163	107	1	here	here	ADV
cana-3163	107	2	neither	neither	DET
cana-3163	107	3	per(f4	per(f4	NOUN
cana-3163	107	4	)	)	PUNCT
cana-3163	107	5	/=	/=	PROPN
cana-3163	108	1	i	i	PRON
cana-3163	108	2	nor	nor	CCONJ
cana-3163	108	3	p	p	X
cana-3163	108	4	(	(	PUNCT
cana-3163	108	5	f4	f4	PROPN
cana-3163	108	6	)	)	PUNCT
cana-3163	108	7	/=	/=	NOUN
cana-3163	108	8	z+	z+	PROPN
cana-3163	108	9	.	.	PUNCT
cana-3163	109	1	figure	figure	NOUN
cana-3163	109	2	3	3	NUM
cana-3163	109	3	.	.	PUNCT
cana-3163	110	1	graph	graph	NOUN
cana-3163	110	2	of	of	ADP
cana-3163	110	3	f3	f3	ADJ
cana-3163	110	4	figure	figure	NOUN
cana-3163	110	5	4	4	NUM
cana-3163	110	6	.	.	PUNCT
cana-3163	110	7	graph	graph	NOUN
cana-3163	110	8	of	of	ADP
cana-3163	110	9	f4	f4	ADJ
cana-3163	110	10	example	example	NOUN
cana-3163	110	11	5	5	NUM
cana-3163	110	12	.	.	PUNCT
cana-3163	111	1	let	let	VERB
cana-3163	111	2	sı	sı	NOUN
cana-3163	111	3	=	=	PUNCT
cana-3163	111	4	{	{	PUNCT
cana-3163	111	5	z	z	NOUN
cana-3163	111	6	∈	∈	PROPN
cana-3163	111	7	c	c	X
cana-3163	111	8	|ı	|ı	X
cana-3163	111	9	z	z	NOUN
cana-3163	111	10	ı=	ı=	NOUN
cana-3163	111	11	1	1	NUM
cana-3163	111	12	}	}	PUNCT
cana-3163	111	13	and	and	CCONJ
cana-3163	111	14	f5	f5	NOUN
cana-3163	111	15	:	:	PUNCT
cana-3163	111	16	sı	sı	NOUN
cana-3163	111	17	—	—	PUNCT
cana-3163	111	18	→	→	SYM
cana-3163	111	19	sı	sı	NOUN
cana-3163	111	20	be	be	AUX
cana-3163	111	21	defined	define	VERB
cana-3163	111	22	as	as	ADP
cana-3163	111	23	f5(z	f5(z	NOUN
cana-3163	111	24	)	)	PUNCT
cana-3163	111	25	=	=	SYM
cana-3163	111	26	z2	z2	PROPN
cana-3163	111	27	.	.	PUNCT
cana-3163	112	1	per(f5	per(f5	NOUN
cana-3163	112	2	)	)	PUNCT
cana-3163	112	3	=	=	NOUN
cana-3163	112	4	sı	sı	NOUN
cana-3163	112	5	and	and	CCONJ
cana-3163	112	6	|	|	ADV
cana-3163	112	7	pern(f5	pern(f5	ADJ
cana-3163	112	8	)	)	PUNCT
cana-3163	113	1	|=	|=	X
cana-3163	113	2	2n	2n	NUM
cana-3163	113	3	—	—	PUNCT
cana-3163	113	4	1	1	NUM
cana-3163	113	5	,	,	PUNCT
cana-3163	113	6	n	n	PRON
cana-3163	113	7	≥	≥	NOUN
cana-3163	113	8	1	1	NUM
cana-3163	113	9	[	[	X
cana-3163	113	10	6	6	NUM
cana-3163	113	11	]	]	PUNCT
cana-3163	113	12	example	example	NOUN
cana-3163	113	13	6	6	NUM
cana-3163	113	14	.	.	PUNCT
cana-3163	114	1	let	let	VERB
cana-3163	114	2	f6	f6	PRON
cana-3163	114	3	:	:	PUNCT
cana-3163	114	4	r	r	X
cana-3163	114	5	—	—	PUNCT
cana-3163	114	6	→	→	PUNCT
cana-3163	114	7	r	r	NOUN
cana-3163	114	8	be	be	AUX
cana-3163	114	9	defined	define	VERB
cana-3163	114	10	as	as	ADP
cana-3163	114	11	f	f	PROPN
cana-3163	114	12	(	(	PUNCT
cana-3163	114	13	x	x	NOUN
cana-3163	114	14	)	)	PUNCT
cana-3163	115	1	=	=	SYM
cana-3163	115	2	x2−7	x2−7	PROPN
cana-3163	115	3	.	.	PUNCT
cana-3163	116	1	6	6	NUM
cana-3163	116	2	2	2	NUM
cana-3163	116	3	communications	communication	NOUN
cana-3163	116	4	on	on	ADP
cana-3163	116	5	applied	apply	VERB
cana-3163	116	6	nonlinear	nonlinear	ADJ
cana-3163	116	7	analysis	analysis	NOUN
cana-3163	116	8	issn	issn	NOUN
cana-3163	116	9	:	:	PUNCT
cana-3163	116	10	1074	1074	NUM
cana-3163	116	11	-	-	PUNCT
cana-3163	116	12	133x	133x	NUM
cana-3163	116	13	vol	vol	NOUN
cana-3163	116	14	32	32	NUM
cana-3163	116	15	no	no	NOUN
cana-3163	116	16	.	.	PUNCT
cana-3163	117	1	5s	5s	NUM
cana-3163	117	2	(	(	PUNCT
cana-3163	117	3	2025	2025	NUM
cana-3163	117	4	)	)	PUNCT
cana-3163	117	5	515	515	NUM
cana-3163	117	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	117	7	figure	figure	NOUN
cana-3163	117	8	5	5	NUM
cana-3163	117	9	.	.	PUNCT
cana-3163	117	10	graph	graph	NOUN
cana-3163	117	11	of	of	ADP
cana-3163	117	12	f6	f6	PROPN
cana-3163	117	13	per3(f6	per3(f6	PROPN
cana-3163	117	14	)	)	PUNCT
cana-3163	118	1	/=	/=	NOUN
cana-3163	118	2	∅	∅	NOUN
cana-3163	118	3	and	and	CCONJ
cana-3163	118	4	therefore	therefore	ADV
cana-3163	118	5	p	p	X
cana-3163	118	6	(	(	PUNCT
cana-3163	118	7	f6	f6	PROPN
cana-3163	118	8	)	)	PUNCT
cana-3163	118	9	=	=	PUNCT
cana-3163	118	10	z+	z+	NUM
cana-3163	119	1	[	[	X
cana-3163	119	2	10	10	NUM
cana-3163	119	3	]	]	PUNCT
cana-3163	119	4	.	.	PUNCT
cana-3163	120	1	also	also	ADV
cana-3163	120	2	per(f6	per(f6	PRON
cana-3163	120	3	)	)	PUNCT
cana-3163	120	4	∩	∩	ADJ
cana-3163	120	5	q	q	NOUN
cana-3163	120	6	=	=	NOUN
cana-3163	120	7	∅	∅	NOUN
cana-3163	120	8	and	and	CCONJ
cana-3163	120	9	per(f6	per(f6	NOUN
cana-3163	120	10	)	)	PUNCT
cana-3163	121	1	=	=	PUNCT
cana-3163	121	2	r.	r.	PROPN
cana-3163	121	3	example	example	NOUN
cana-3163	121	4	7	7	X
cana-3163	121	5	.	.	PUNCT
cana-3163	122	1	let	let	VERB
cana-3163	122	2	f7	f7	PROPN
cana-3163	123	1	:	:	PUNCT
cana-3163	123	2	c	c	X
cana-3163	123	3	—	—	PUNCT
cana-3163	123	4	→	→	PUNCT
cana-3163	123	5	c	c	X
cana-3163	123	6	be	be	AUX
cana-3163	123	7	defined	define	VERB
cana-3163	123	8	as	as	ADP
cana-3163	123	9	f7(z	f7(z	NOUN
cana-3163	123	10	)	)	PUNCT
cana-3163	123	11	=	=	SYM
cana-3163	123	12	z2	z2	PROPN
cana-3163	123	13	—	—	PUNCT
cana-3163	123	14	z.	z.	PROPN
cana-3163	123	15	p	p	PROPN
cana-3163	123	16	(	(	PUNCT
cana-3163	123	17	f7	f7	PROPN
cana-3163	123	18	)	)	PUNCT
cana-3163	123	19	=	=	PUNCT
cana-3163	123	20	z+\{2	z+\{2	X
cana-3163	123	21	}	}	PUNCT
cana-3163	124	1	[	[	X
cana-3163	124	2	2	2	NUM
cana-3163	124	3	]	]	PUNCT
cana-3163	124	4	.	.	PUNCT
cana-3163	125	1	here	here	ADV
cana-3163	125	2	per(f7	per(f7	ADJ
cana-3163	125	3	)	)	PUNCT
cana-3163	125	4	/=	/=	PROPN
cana-3163	125	5	c.	c.	NOUN
cana-3163	125	6	from	from	ADP
cana-3163	125	7	the	the	DET
cana-3163	125	8	examples	example	NOUN
cana-3163	125	9	given	give	VERB
cana-3163	125	10	above	above	ADV
cana-3163	125	11	,	,	PUNCT
cana-3163	125	12	it	it	PRON
cana-3163	125	13	is	be	AUX
cana-3163	125	14	observed	observe	VERB
cana-3163	125	15	that	that	SCONJ
cana-3163	125	16	there	there	PRON
cana-3163	125	17	is	be	VERB
cana-3163	125	18	no	no	DET
cana-3163	125	19	implication	implication	NOUN
cana-3163	125	20	between	between	ADP
cana-3163	125	21	p	p	PROPN
cana-3163	125	22	(	(	PUNCT
cana-3163	125	23	f	f	PROPN
cana-3163	125	24	)	)	PUNCT
cana-3163	125	25	=	=	SYM
cana-3163	125	26	z+	z+	NUM
cana-3163	125	27	and	and	CCONJ
cana-3163	125	28	per(f	per(f	NOUN
cana-3163	125	29	)	)	PUNCT
cana-3163	125	30	=	=	PUNCT
cana-3163	126	1	x.	x.	NOUN
cana-3163	126	2	definition	definition	NOUN
cana-3163	126	3	1	1	NUM
cana-3163	126	4	.	.	PUNCT
cana-3163	127	1	let	let	VERB
cana-3163	127	2	f	f	PRON
cana-3163	127	3	be	be	AUX
cana-3163	127	4	a	a	DET
cana-3163	127	5	continuous	continuous	ADJ
cana-3163	127	6	map	map	NOUN
cana-3163	127	7	defined	define	VERB
cana-3163	127	8	on	on	ADP
cana-3163	127	9	a	a	DET
cana-3163	127	10	topological	topological	ADJ
cana-3163	127	11	space	space	NOUN
cana-3163	127	12	x.	x.	NOUN
cana-3163	128	1	we	we	PRON
cana-3163	128	2	say	say	VERB
cana-3163	128	3	that	that	SCONJ
cana-3163	128	4	f	f	PROPN
cana-3163	128	5	is	be	AUX
cana-3163	128	6	periodically	periodically	ADV
cana-3163	128	7	rich	rich	ADJ
cana-3163	128	8	if	if	SCONJ
cana-3163	128	9	f	f	PROPN
cana-3163	128	10	has	have	VERB
cana-3163	128	11	both	both	CCONJ
cana-3163	128	12	a	a	DET
cana-3163	128	13	dense	dense	ADJ
cana-3163	128	14	set	set	NOUN
cana-3163	128	15	of	of	ADP
cana-3163	128	16	periodic	periodic	ADJ
cana-3163	128	17	points	point	NOUN
cana-3163	128	18	and	and	CCONJ
cana-3163	128	19	the	the	DET
cana-3163	128	20	set	set	NOUN
cana-3163	128	21	of	of	ADP
cana-3163	128	22	periods	period	NOUN
cana-3163	128	23	is	be	AUX
cana-3163	128	24	equal	equal	ADJ
cana-3163	128	25	to	to	ADP
cana-3163	128	26	z+	z+	NOUN
cana-3163	128	27	.	.	PUNCT
cana-3163	129	1	i.e.	i.e.	X
cana-3163	129	2	,	,	PUNCT
cana-3163	129	3	f	f	PROPN
cana-3163	129	4	is	be	AUX
cana-3163	129	5	periodically	periodically	ADV
cana-3163	129	6	rich	rich	ADJ
cana-3163	129	7	if	if	SCONJ
cana-3163	129	8	(	(	PUNCT
cana-3163	129	9	1	1	X
cana-3163	129	10	)	)	PUNCT
cana-3163	129	11	per(f	per(f	NOUN
cana-3163	129	12	)	)	PUNCT
cana-3163	129	13	=	=	SYM
cana-3163	130	1	x	x	X
cana-3163	130	2	and	and	CCONJ
cana-3163	130	3	(	(	PUNCT
cana-3163	130	4	2	2	X
cana-3163	130	5	)	)	PUNCT
cana-3163	130	6	p	p	NOUN
cana-3163	130	7	(	(	PUNCT
cana-3163	130	8	f	f	PROPN
cana-3163	130	9	)	)	PUNCT
cana-3163	130	10	=	=	PUNCT
cana-3163	131	1	z+	z+	X
cana-3163	131	2	.	.	PUNCT
cana-3163	132	1	let	let	AUX
cana-3163	132	2	pr(x	pr(x	X
cana-3163	132	3	)	)	PUNCT
cana-3163	132	4	be	be	AUX
cana-3163	132	5	the	the	DET
cana-3163	132	6	set	set	NOUN
cana-3163	132	7	of	of	ADP
cana-3163	132	8	continuous	continuous	ADJ
cana-3163	132	9	maps	map	NOUN
cana-3163	132	10	on	on	ADP
cana-3163	132	11	x	x	PUNCT
cana-3163	132	12	which	which	PRON
cana-3163	132	13	are	be	AUX
cana-3163	132	14	periodically	periodically	ADV
cana-3163	132	15	rich	rich	ADJ
cana-3163	132	16	.	.	PUNCT
cana-3163	133	1	if	if	SCONJ
cana-3163	133	2	pr(x	pr(x	VERB
cana-3163	133	3	)	)	PUNCT
cana-3163	133	4	/=	/=	NOUN
cana-3163	133	5	∅	∅	NOUN
cana-3163	133	6	,	,	PUNCT
cana-3163	133	7	we	we	PRON
cana-3163	133	8	say	say	VERB
cana-3163	133	9	that	that	SCONJ
cana-3163	133	10	x	x	PRON
cana-3163	133	11	admits	admit	VERB
cana-3163	133	12	periodically	periodically	ADV
cana-3163	133	13	rich	rich	ADJ
cana-3163	133	14	maps	map	NOUN
cana-3163	133	15	.	.	PUNCT
cana-3163	134	1	for	for	ADP
cana-3163	134	2	example	example	NOUN
cana-3163	134	3	interval	interval	NOUN
cana-3163	134	4	i	i	PRON
cana-3163	134	5	admits	admit	VERB
cana-3163	134	6	periodically	periodically	ADV
cana-3163	134	7	rich	rich	ADJ
cana-3163	134	8	maps	map	NOUN
cana-3163	134	9	since	since	SCONJ
cana-3163	134	10	tent	tent	NOUN
cana-3163	134	11	map	map	NOUN
cana-3163	134	12	satisfies	satisfy	VERB
cana-3163	134	13	both	both	CCONJ
cana-3163	134	14	the	the	DET
cana-3163	134	15	conditions	condition	NOUN
cana-3163	134	16	of	of	ADP
cana-3163	134	17	definition	definition	NOUN
cana-3163	134	18	.	.	PUNCT
cana-3163	135	1	2	2	X
cana-3163	135	2	.	.	X
cana-3163	135	3	periodically	periodically	ADV
cana-3163	135	4	rich	rich	ADJ
cana-3163	135	5	maps	map	NOUN
cana-3163	135	6	on	on	ADP
cana-3163	135	7	interval	interval	NOUN
cana-3163	135	8	the	the	DET
cana-3163	135	9	following	follow	VERB
cana-3163	135	10	proposition	proposition	NOUN
cana-3163	135	11	is	be	AUX
cana-3163	135	12	a	a	DET
cana-3163	135	13	corollary	corollary	NOUN
cana-3163	135	14	to	to	ADP
cana-3163	135	15	sharkovskii	sharkovskii	PROPN
cana-3163	135	16	’s	’s	PART
cana-3163	135	17	result	result	NOUN
cana-3163	135	18	.	.	PUNCT
cana-3163	136	1	proposition	proposition	NOUN
cana-3163	136	2	1	1	NUM
cana-3163	136	3	.	.	PUNCT
cana-3163	136	4	for	for	ADP
cana-3163	136	5	continuous	continuous	ADJ
cana-3163	136	6	maps	map	NOUN
cana-3163	137	1	f	f	NOUN
cana-3163	137	2	:	:	PUNCT
cana-3163	137	3	i	i	PRON
cana-3163	137	4	→	→	PUNCT
cana-3163	137	5	i	i	PRON
cana-3163	137	6	if	if	SCONJ
cana-3163	137	7	per3(f	per3(f	X
cana-3163	137	8	)	)	PUNCT
cana-3163	137	9	/=	/=	PRON
cana-3163	137	10	∅	∅	NOUN
cana-3163	137	11	then	then	ADV
cana-3163	137	12	p	p	X
cana-3163	137	13	(	(	PUNCT
cana-3163	137	14	f	f	PROPN
cana-3163	137	15	)	)	PUNCT
cana-3163	137	16	=	=	SYM
cana-3163	137	17	z+	z+	X
cana-3163	137	18	.	.	PUNCT
cana-3163	138	1	proposition	proposition	NOUN
cana-3163	138	2	2	2	NUM
cana-3163	138	3	.	.	PUNCT
cana-3163	139	1	let	let	VERB
cana-3163	139	2	f	f	NOUN
cana-3163	139	3	:	:	PUNCT
cana-3163	139	4	i	i	PRON
cana-3163	139	5	—	—	PUNCT
cana-3163	139	6	→	→	PUNCT
cana-3163	139	7	i	i	PRON
cana-3163	139	8	be	be	VERB
cana-3163	139	9	a	a	DET
cana-3163	139	10	continuous	continuous	ADJ
cana-3163	139	11	map	map	NOUN
cana-3163	139	12	.	.	PUNCT
cana-3163	140	1	if	if	SCONJ
cana-3163	140	2	f	f	PROPN
cana-3163	140	3	is	be	AUX
cana-3163	140	4	transitive	transitive	ADJ
cana-3163	140	5	and	and	CCONJ
cana-3163	140	6	per3(f	per3(f	NOUN
cana-3163	140	7	)	)	PUNCT
cana-3163	141	1	/=	/=	PRON
cana-3163	141	2	∅	∅	NOUN
cana-3163	141	3	then	then	ADV
cana-3163	141	4	f	f	PROPN
cana-3163	141	5	∈	∈	PROPN
cana-3163	141	6	pr(i	pr(i	NOUN
cana-3163	141	7	)	)	PUNCT
cana-3163	141	8	.	.	PUNCT
cana-3163	142	1	proof	proof	NOUN
cana-3163	142	2	.	.	PUNCT
cana-3163	143	1	if	if	SCONJ
cana-3163	143	2	f	f	PROPN
cana-3163	143	3	is	be	AUX
cana-3163	143	4	transitive	transitive	ADJ
cana-3163	143	5	then	then	ADV
cana-3163	143	6	per(f	per(f	NOUN
cana-3163	143	7	)	)	PUNCT
cana-3163	144	1	=	=	PUNCT
cana-3163	145	1	i	i	PRON
cana-3163	146	1	[	[	X
cana-3163	146	2	12	12	NUM
cana-3163	146	3	]	]	PUNCT
cana-3163	146	4	and	and	CCONJ
cana-3163	146	5	per3(f	per3(f	X
cana-3163	146	6	)	)	PUNCT
cana-3163	147	1	/=	/=	PROPN
cana-3163	147	2	∅	∅	NOUN
cana-3163	147	3	implies	imply	VERB
cana-3163	147	4	p	p	X
cana-3163	147	5	(	(	PUNCT
cana-3163	147	6	f	f	NOUN
cana-3163	147	7	)	)	PUNCT
cana-3163	147	8	=	=	SYM
cana-3163	148	1	z+	z+	X
cana-3163	148	2	by	by	ADP
cana-3163	148	3	proposition	proposition	NOUN
cana-3163	148	4	communications	communication	NOUN
cana-3163	148	5	on	on	ADP
cana-3163	148	6	applied	apply	VERB
cana-3163	148	7	nonlinear	nonlinear	ADJ
cana-3163	148	8	analysis	analysis	NOUN
cana-3163	148	9	issn	issn	NOUN
cana-3163	148	10	:	:	PUNCT
cana-3163	148	11	1074	1074	NUM
cana-3163	148	12	-	-	PUNCT
cana-3163	148	13	133x	133x	NUM
cana-3163	148	14	vol	vol	NOUN
cana-3163	148	15	32	32	NUM
cana-3163	148	16	no	no	NOUN
cana-3163	148	17	.	.	PUNCT
cana-3163	149	1	5s	5s	NUM
cana-3163	149	2	(	(	PUNCT
cana-3163	149	3	2025	2025	NUM
cana-3163	149	4	)	)	PUNCT
cana-3163	149	5	516	516	NUM
cana-3163	149	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	149	7	1	1	NUM
cana-3163	149	8	.	.	PUNCT
cana-3163	150	1	hence	hence	ADV
cana-3163	150	2	f	f	PROPN
cana-3163	150	3	∈	∈	PROPN
cana-3163	150	4	pr(i	pr(i	NOUN
cana-3163	150	5	)	)	PUNCT
cana-3163	150	6	.	.	PUNCT
cana-3163	151	1	proposition	proposition	NOUN
cana-3163	151	2	3	3	X
cana-3163	151	3	.	.	PUNCT
cana-3163	152	1	let	let	VERB
cana-3163	152	2	f	f	NOUN
cana-3163	152	3	:	:	PUNCT
cana-3163	152	4	i	i	PRON
cana-3163	152	5	—	—	PUNCT
cana-3163	152	6	→	→	PUNCT
cana-3163	152	7	i	i	PRON
cana-3163	152	8	be	be	VERB
cana-3163	152	9	a	a	DET
cana-3163	152	10	transitive	transitive	ADJ
cana-3163	152	11	map	map	NOUN
cana-3163	152	12	.	.	PUNCT
cana-3163	153	1	if	if	SCONJ
cana-3163	153	2	there	there	PRON
cana-3163	153	3	exists	exist	VERB
cana-3163	153	4	x	x	X
cana-3163	153	5	∈	∈	PROPN
cana-3163	153	6	per4(f	per4(f	PUNCT
cana-3163	153	7	)	)	PUNCT
cana-3163	153	8	with	with	ADP
cana-3163	153	9	increasing	increase	VERB
cana-3163	153	10	orbit	orbit	NOUN
cana-3163	153	11	then	then	ADV
cana-3163	153	12	f	f	PROPN
cana-3163	153	13	∈	∈	PROPN
cana-3163	153	14	pr(i	pr(i	NOUN
cana-3163	153	15	)	)	PUNCT
cana-3163	153	16	.	.	PUNCT
cana-3163	154	1	since	since	SCONJ
cana-3163	154	2	f	f	PROPN
cana-3163	154	3	is	be	AUX
cana-3163	154	4	transitive	transitive	ADJ
cana-3163	154	5	,	,	PUNCT
cana-3163	154	6	per(f	per(f	NOUN
cana-3163	154	7	)	)	PUNCT
cana-3163	155	1	=	=	PUNCT
cana-3163	155	2	i	i	PRON
cana-3163	156	1	[	[	X
cana-3163	156	2	12	12	NUM
cana-3163	156	3	]	]	PUNCT
cana-3163	156	4	.	.	PUNCT
cana-3163	157	1	we	we	PRON
cana-3163	157	2	use	use	VERB
cana-3163	157	3	the	the	DET
cana-3163	157	4	following	follow	VERB
cana-3163	157	5	lemmas	lemma	NOUN
cana-3163	157	6	from	from	ADP
cana-3163	157	7	the	the	DET
cana-3163	157	8	literature	literature	NOUN
cana-3163	157	9	[	[	X
cana-3163	157	10	1	1	X
cana-3163	157	11	]	]	PUNCT
cana-3163	157	12	to	to	PART
cana-3163	157	13	show	show	VERB
cana-3163	157	14	that	that	PRON
cana-3163	157	15	per3(f	per3(f	NOUN
cana-3163	157	16	)	)	PUNCT
cana-3163	157	17	/=	/=	PROPN
cana-3163	158	1	φ	φ	PROPN
cana-3163	158	2	.	.	PUNCT
cana-3163	159	1	lemma	lemma	PROPN
cana-3163	159	2	1	1	X
cana-3163	159	3	.	.	PUNCT
cana-3163	160	1	let	let	VERB
cana-3163	160	2	f	f	NOUN
cana-3163	160	3	:	:	PUNCT
cana-3163	160	4	i	i	PRON
cana-3163	160	5	→	→	PUNCT
cana-3163	160	6	i	i	PRON
cana-3163	160	7	be	be	VERB
cana-3163	160	8	a	a	DET
cana-3163	160	9	continuous	continuous	ADJ
cana-3163	160	10	map	map	NOUN
cana-3163	160	11	and	and	CCONJ
cana-3163	160	12	let	let	VERB
cana-3163	160	13	j	j	PROPN
cana-3163	160	14	,	,	PUNCT
cana-3163	160	15	k	k	PROPN
cana-3163	161	1	c	c	VERB
cana-3163	161	2	i	i	PRON
cana-3163	161	3	be	be	VERB
cana-3163	161	4	closed	close	VERB
cana-3163	161	5	intervals	interval	NOUN
cana-3163	161	6	with	with	ADP
cana-3163	161	7	f	f	PROPN
cana-3163	161	8	(	(	PUNCT
cana-3163	161	9	j	j	PROPN
cana-3163	161	10	)	)	PUNCT
cana-3163	161	11	m	m	VERB
cana-3163	161	12	k	k	ADJ
cana-3163	161	13	,	,	PUNCT
cana-3163	161	14	then	then	ADV
cana-3163	161	15	there	there	PRON
cana-3163	161	16	exists	exist	VERB
cana-3163	161	17	a	a	DET
cana-3163	161	18	closed	closed	ADJ
cana-3163	161	19	interval	interval	NOUN
cana-3163	161	20	h	h	NOUN
cana-3163	161	21	c	c	PROPN
cana-3163	161	22	j	j	PROPN
cana-3163	161	23	with	with	ADP
cana-3163	161	24	f	f	PROPN
cana-3163	161	25	(	(	PUNCT
cana-3163	161	26	h	h	NOUN
cana-3163	161	27	)	)	PUNCT
cana-3163	162	1	=	=	PUNCT
cana-3163	162	2	k.	k.	PROPN
cana-3163	162	3	lemma	lemma	PROPN
cana-3163	163	1	2	2	X
cana-3163	163	2	.	.	PUNCT
cana-3163	163	3	let	let	VERB
cana-3163	163	4	f	f	NOUN
cana-3163	163	5	:	:	PUNCT
cana-3163	163	6	i	i	PRON
cana-3163	163	7	→	→	PUNCT
cana-3163	163	8	i	i	PRON
cana-3163	163	9	be	be	VERB
cana-3163	163	10	a	a	DET
cana-3163	163	11	continuous	continuous	ADJ
cana-3163	163	12	map	map	NOUN
cana-3163	163	13	.	.	PUNCT
cana-3163	164	1	suppose	suppose	VERB
cana-3163	164	2	h	h	NOUN
cana-3163	164	3	and	and	CCONJ
cana-3163	164	4	k	k	PROPN
cana-3163	164	5	are	be	AUX
cana-3163	164	6	closed	close	VERB
cana-3163	164	7	intervals	interval	NOUN
cana-3163	164	8	with	with	ADP
cana-3163	164	9	h	h	NOUN
cana-3163	164	10	c	c	NOUN
cana-3163	165	1	k	k	PROPN
cana-3163	165	2	c	c	PROPN
cana-3163	166	1	i	i	PRON
cana-3163	166	2	and	and	CCONJ
cana-3163	166	3	f	f	PROPN
cana-3163	166	4	(	(	PUNCT
cana-3163	166	5	h	h	NOUN
cana-3163	166	6	)	)	PUNCT
cana-3163	166	7	=	=	SYM
cana-3163	167	1	k	k	NOUN
cana-3163	167	2	,	,	PUNCT
cana-3163	167	3	then	then	ADV
cana-3163	167	4	f	f	PROPN
cana-3163	167	5	has	have	VERB
cana-3163	167	6	a	a	DET
cana-3163	167	7	fixed	fix	VERB
cana-3163	167	8	point	point	NOUN
cana-3163	167	9	in	in	ADP
cana-3163	167	10	h.	h.	PROPN
cana-3163	167	11	proof	proof	NOUN
cana-3163	167	12	of	of	ADP
cana-3163	167	13	proposition	proposition	NOUN
cana-3163	167	14	3	3	NUM
cana-3163	167	15	:	:	PUNCT
cana-3163	167	16	let	let	VERB
cana-3163	167	17	x	x	X
cana-3163	167	18	∈	∈	PROPN
cana-3163	167	19	per4(f	per4(f	PART
cana-3163	167	20	)	)	PUNCT
cana-3163	167	21	with	with	ADP
cana-3163	167	22	increasing	increase	VERB
cana-3163	167	23	orbit	orbit	NOUN
cana-3163	167	24	,	,	PUNCT
cana-3163	167	25	i.e.	i.e.	X
cana-3163	167	26	,	,	PUNCT
cana-3163	167	27	x	x	X
cana-3163	167	28	<	<	X
cana-3163	167	29	f	f	X
cana-3163	167	30	(	(	PUNCT
cana-3163	167	31	x	x	X
cana-3163	167	32	)	)	PUNCT
cana-3163	167	33	<	<	X
cana-3163	167	34	f	f	PROPN
cana-3163	167	35	2(x	2(x	NUM
cana-3163	167	36	)	)	PUNCT
cana-3163	167	37	<	<	X
cana-3163	167	38	f	f	PROPN
cana-3163	167	39	3(x	3(x	NUM
cana-3163	167	40	)	)	PUNCT
cana-3163	167	41	.	.	PUNCT
cana-3163	168	1	f	f	X
cana-3163	169	1	(	(	PUNCT
cana-3163	169	2	[	[	X
cana-3163	169	3	x	x	X
cana-3163	169	4	,	,	PUNCT
cana-3163	169	5	f	f	PROPN
cana-3163	169	6	(	(	PUNCT
cana-3163	169	7	x	x	NOUN
cana-3163	169	8	)	)	PUNCT
cana-3163	169	9	]	]	PUNCT
cana-3163	169	10	)	)	PUNCT
cana-3163	169	11	m	m	VERB
cana-3163	170	1	[	[	X
cana-3163	170	2	f	f	X
cana-3163	170	3	(	(	PUNCT
cana-3163	170	4	x	x	NOUN
cana-3163	170	5	)	)	PUNCT
cana-3163	170	6	,	,	PUNCT
cana-3163	170	7	f	f	PROPN
cana-3163	170	8	2(x	2(x	NUM
cana-3163	170	9	)	)	PUNCT
cana-3163	170	10	]	]	PUNCT
cana-3163	170	11	f	f	X
cana-3163	170	12	(	(	PUNCT
cana-3163	170	13	[	[	X
cana-3163	170	14	f	f	X
cana-3163	170	15	(	(	PUNCT
cana-3163	170	16	x	x	NOUN
cana-3163	170	17	)	)	PUNCT
cana-3163	170	18	,	,	PUNCT
cana-3163	170	19	f	f	PROPN
cana-3163	170	20	2(x	2(x	NUM
cana-3163	170	21	)	)	PUNCT
cana-3163	170	22	]	]	PUNCT
cana-3163	170	23	)	)	PUNCT
cana-3163	170	24	m	m	VERB
cana-3163	171	1	[	[	X
cana-3163	171	2	f	f	PROPN
cana-3163	171	3	2(x	2(x	NUM
cana-3163	171	4	)	)	PUNCT
cana-3163	171	5	,	,	PUNCT
cana-3163	171	6	f	f	PROPN
cana-3163	171	7	3(x	3(x	NUM
cana-3163	171	8	)	)	PUNCT
cana-3163	171	9	]	]	PUNCT
cana-3163	172	1	f	f	X
cana-3163	172	2	(	(	PUNCT
cana-3163	172	3	[	[	X
cana-3163	172	4	f	f	PROPN
cana-3163	172	5	2(x	2(x	NUM
cana-3163	172	6	)	)	PUNCT
cana-3163	172	7	,	,	PUNCT
cana-3163	172	8	f	f	PROPN
cana-3163	172	9	3(x	3(x	NUM
cana-3163	172	10	)	)	PUNCT
cana-3163	172	11	]	]	PUNCT
cana-3163	172	12	)	)	PUNCT
cana-3163	172	13	m	m	VERB
cana-3163	173	1	[	[	X
cana-3163	173	2	x	x	X
cana-3163	173	3	,	,	PUNCT
cana-3163	173	4	f	f	PROPN
cana-3163	173	5	(	(	PUNCT
cana-3163	173	6	x	x	NOUN
cana-3163	173	7	)	)	PUNCT
cana-3163	173	8	]	]	PUNCT
cana-3163	173	9	so	so	ADV
cana-3163	173	10	by	by	ADP
cana-3163	173	11	lemma	lemma	PROPN
cana-3163	173	12	2	2	NUM
cana-3163	173	13	,	,	PUNCT
cana-3163	173	14	there	there	PRON
cana-3163	173	15	exists	exist	VERB
cana-3163	173	16	f1	f1	NOUN
cana-3163	173	17	c	c	PROPN
cana-3163	174	1	[	[	X
cana-3163	174	2	x	x	X
cana-3163	174	3	,	,	PUNCT
cana-3163	174	4	f	f	PROPN
cana-3163	174	5	(	(	PUNCT
cana-3163	174	6	x	x	NOUN
cana-3163	174	7	)	)	PUNCT
cana-3163	174	8	]	]	PUNCT
cana-3163	174	9	such	such	ADJ
cana-3163	174	10	that	that	SCONJ
cana-3163	174	11	f	f	PROPN
cana-3163	174	12	(	(	PUNCT
cana-3163	174	13	f1	f1	PROPN
cana-3163	174	14	)	)	PUNCT
cana-3163	174	15	=	=	PUNCT
cana-3163	175	1	[	[	X
cana-3163	175	2	f	f	X
cana-3163	175	3	(	(	PUNCT
cana-3163	175	4	x	x	NOUN
cana-3163	175	5	)	)	PUNCT
cana-3163	175	6	,	,	PUNCT
cana-3163	175	7	f	f	PROPN
cana-3163	175	8	2(x	2(x	NUM
cana-3163	175	9	)	)	PUNCT
cana-3163	175	10	]	]	PUNCT
cana-3163	176	1	f2	f2	PROPN
cana-3163	176	2	c	c	PROPN
cana-3163	177	1	[	[	X
cana-3163	177	2	f	f	X
cana-3163	177	3	(	(	PUNCT
cana-3163	177	4	x	x	NOUN
cana-3163	177	5	)	)	PUNCT
cana-3163	177	6	,	,	PUNCT
cana-3163	177	7	f	f	PROPN
cana-3163	177	8	2(x	2(x	NUM
cana-3163	177	9	)	)	PUNCT
cana-3163	177	10	]	]	PUNCT
cana-3163	177	11	such	such	ADJ
cana-3163	177	12	that	that	SCONJ
cana-3163	177	13	f	f	PROPN
cana-3163	177	14	(	(	PUNCT
cana-3163	177	15	f2	f2	PROPN
cana-3163	177	16	)	)	PUNCT
cana-3163	177	17	=	=	PUNCT
cana-3163	178	1	[	[	X
cana-3163	178	2	f	f	PROPN
cana-3163	178	3	2(x	2(x	NUM
cana-3163	178	4	)	)	PUNCT
cana-3163	178	5	,	,	PUNCT
cana-3163	178	6	f	f	PROPN
cana-3163	178	7	3(x	3(x	NUM
cana-3163	178	8	)	)	PUNCT
cana-3163	178	9	]	]	PUNCT
cana-3163	179	1	f3	f3	PROPN
cana-3163	179	2	c	c	PROPN
cana-3163	180	1	[	[	X
cana-3163	180	2	f	f	PROPN
cana-3163	180	3	2(x	2(x	NUM
cana-3163	180	4	)	)	PUNCT
cana-3163	180	5	,	,	PUNCT
cana-3163	180	6	f	f	PROPN
cana-3163	180	7	3(x	3(x	NUM
cana-3163	180	8	)	)	PUNCT
cana-3163	180	9	]	]	PUNCT
cana-3163	180	10	such	such	ADJ
cana-3163	180	11	that	that	SCONJ
cana-3163	180	12	f	f	PROPN
cana-3163	180	13	(	(	PUNCT
cana-3163	180	14	f3	f3	ADJ
cana-3163	180	15	)	)	PUNCT
cana-3163	180	16	=	=	PUNCT
cana-3163	181	1	[	[	X
cana-3163	181	2	x	x	X
cana-3163	181	3	,	,	PUNCT
cana-3163	181	4	f	f	PROPN
cana-3163	181	5	(	(	PUNCT
cana-3163	181	6	x	x	NOUN
cana-3163	181	7	)	)	PUNCT
cana-3163	181	8	]	]	PUNCT
cana-3163	181	9	.	.	PUNCT
cana-3163	182	1	now	now	ADV
cana-3163	182	2	,	,	PUNCT
cana-3163	182	3	f	f	PROPN
cana-3163	182	4	2(f3	2(f3	NUM
cana-3163	182	5	)	)	PUNCT
cana-3163	183	1	=	=	SYM
cana-3163	183	2	f	f	PROPN
cana-3163	183	3	(	(	PUNCT
cana-3163	183	4	f1	f1	PROPN
cana-3163	183	5	)	)	PUNCT
cana-3163	183	6	;	;	PUNCT
cana-3163	183	7	f	f	PROPN
cana-3163	183	8	2(f1	2(f1	NUM
cana-3163	183	9	)	)	PUNCT
cana-3163	183	10	=	=	SYM
cana-3163	184	1	f	f	PROPN
cana-3163	184	2	(	(	PUNCT
cana-3163	184	3	f2	f2	PROPN
cana-3163	184	4	)	)	PUNCT
cana-3163	184	5	;	;	PUNCT
cana-3163	184	6	f	f	PROPN
cana-3163	184	7	2(f2	2(f2	PROPN
cana-3163	184	8	)	)	PUNCT
cana-3163	184	9	=	=	PUNCT
cana-3163	185	1	[	[	X
cana-3163	185	2	x	x	X
cana-3163	185	3	,	,	PUNCT
cana-3163	185	4	f	f	PROPN
cana-3163	185	5	(	(	PUNCT
cana-3163	185	6	x)].therefore	x)].therefore	PROPN
cana-3163	186	1	[	[	X
cana-3163	186	2	x	x	X
cana-3163	186	3	,	,	PUNCT
cana-3163	186	4	f	f	PROPN
cana-3163	186	5	(	(	PUNCT
cana-3163	186	6	x	x	NOUN
cana-3163	186	7	)	)	PUNCT
cana-3163	186	8	]	]	PUNCT
cana-3163	187	1	=	=	PUNCT
cana-3163	187	2	f	f	PROPN
cana-3163	187	3	3(f1	3(f1	PROPN
cana-3163	187	4	)	)	PUNCT
cana-3163	187	5	.	.	PUNCT
cana-3163	188	1	so	so	ADV
cana-3163	188	2	by	by	ADP
cana-3163	188	3	lemma	lemma	PROPN
cana-3163	188	4	2	2	NUM
cana-3163	188	5	,	,	PUNCT
cana-3163	188	6	f	f	PROPN
cana-3163	188	7	3	3	NUM
cana-3163	188	8	has	have	VERB
cana-3163	188	9	a	a	DET
cana-3163	188	10	fixed	fix	VERB
cana-3163	188	11	point	point	NOUN
cana-3163	188	12	in	in	ADP
cana-3163	188	13	f1	f1	NOUN
cana-3163	188	14	,	,	PUNCT
cana-3163	188	15	say	say	VERB
cana-3163	188	16	y.	y.	NOUN
cana-3163	188	17	now	now	ADV
cana-3163	188	18	y	y	PROPN
cana-3163	188	19	/∈	/∈	PUNCT
cana-3163	188	20	per2(f	per2(f	ADJ
cana-3163	188	21	)	)	PUNCT
cana-3163	188	22	,	,	PUNCT
cana-3163	188	23	since	since	SCONJ
cana-3163	188	24	f	f	PROPN
cana-3163	188	25	2(y	2(y	NUM
cana-3163	188	26	)	)	PUNCT
cana-3163	188	27	∈	∈	PROPN
cana-3163	189	1	[	[	X
cana-3163	189	2	f	f	X
cana-3163	189	3	(	(	PUNCT
cana-3163	189	4	x	x	NOUN
cana-3163	189	5	)	)	PUNCT
cana-3163	189	6	,	,	PUNCT
cana-3163	189	7	f	f	PROPN
cana-3163	189	8	2(x	2(x	NUM
cana-3163	189	9	)	)	PUNCT
cana-3163	189	10	]	]	PUNCT
cana-3163	189	11	.	.	PUNCT
cana-3163	190	1	therefore	therefore	ADV
cana-3163	190	2	y	y	PROPN
cana-3163	190	3	∈	∈	PROPN
cana-3163	190	4	per3(f	per3(f	NOUN
cana-3163	190	5	)	)	PUNCT
cana-3163	190	6	.	.	PUNCT
cana-3163	191	1	so	so	ADV
cana-3163	191	2	per3(f	per3(f	X
cana-3163	191	3	)	)	PUNCT
cana-3163	192	1	/=	/=	PROPN
cana-3163	192	2	∅.	∅.	ADP
cana-3163	192	3	now	now	ADV
cana-3163	192	4	,	,	PUNCT
cana-3163	192	5	by	by	ADP
cana-3163	192	6	proposition	proposition	NOUN
cana-3163	192	7	2	2	NUM
cana-3163	192	8	,	,	PUNCT
cana-3163	192	9	f	f	PROPN
cana-3163	192	10	∈	∈	PROPN
cana-3163	192	11	pr(i	pr(i	NOUN
cana-3163	192	12	)	)	PUNCT
cana-3163	192	13	.	.	PUNCT
cana-3163	193	1	proposition	proposition	NOUN
cana-3163	193	2	4	4	NUM
cana-3163	193	3	.	.	PUNCT
cana-3163	194	1	let	let	VERB
cana-3163	194	2	f	f	PRON
cana-3163	194	3	be	be	AUX
cana-3163	194	4	a	a	DET
cana-3163	194	5	continuous	continuous	ADJ
cana-3163	194	6	map	map	NOUN
cana-3163	194	7	such	such	ADJ
cana-3163	194	8	that	that	SCONJ
cana-3163	194	9	(	(	PUNCT
cana-3163	194	10	i	i	NOUN
cana-3163	194	11	)	)	PUNCT
cana-3163	194	12	f	f	PROPN
cana-3163	194	13	(	(	PUNCT
cana-3163	194	14	x	x	X
cana-3163	194	15	)	)	PUNCT
cana-3163	194	16	=	=	PUNCT
cana-3163	195	1	x	x	PUNCT
cana-3163	195	2	for	for	ADP
cana-3163	195	3	some	some	DET
cana-3163	195	4	x	x	SYM
cana-3163	195	5	∈	∈	PROPN
cana-3163	195	6	(	(	PUNCT
cana-3163	195	7	0	0	NUM
cana-3163	195	8	,	,	PUNCT
cana-3163	195	9	1	1	NUM
cana-3163	195	10	)	)	PUNCT
cana-3163	195	11	(	(	PUNCT
cana-3163	195	12	ii	ii	NOUN
cana-3163	195	13	)	)	PUNCT
cana-3163	195	14	f	f	PROPN
cana-3163	195	15	(	(	PUNCT
cana-3163	195	16	1	1	NUM
cana-3163	195	17	)	)	PUNCT
cana-3163	195	18	=	=	SYM
cana-3163	195	19	1	1	NUM
cana-3163	195	20	communications	communication	NOUN
cana-3163	195	21	on	on	ADP
cana-3163	195	22	applied	apply	VERB
cana-3163	195	23	nonlinear	nonlinear	ADJ
cana-3163	195	24	analysis	analysis	NOUN
cana-3163	195	25	issn	issn	NOUN
cana-3163	195	26	:	:	PUNCT
cana-3163	195	27	1074	1074	NUM
cana-3163	195	28	-	-	PUNCT
cana-3163	195	29	133x	133x	NUM
cana-3163	195	30	vol	vol	NOUN
cana-3163	195	31	32	32	NUM
cana-3163	195	32	no	no	NOUN
cana-3163	195	33	.	.	PUNCT
cana-3163	196	1	5s	5s	NUM
cana-3163	196	2	(	(	PUNCT
cana-3163	196	3	2025	2025	NUM
cana-3163	196	4	)	)	PUNCT
cana-3163	197	1	517	517	NUM
cana-3163	197	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	197	3	(	(	PUNCT
cana-3163	197	4	iii	iii	NOUN
cana-3163	197	5	)	)	PUNCT
cana-3163	197	6	for	for	ADP
cana-3163	197	7	every	every	DET
cana-3163	197	8	j	j	PROPN
cana-3163	197	9	c	c	PROPN
cana-3163	197	10	i	i	PROPN
cana-3163	197	11	,	,	PUNCT
cana-3163	197	12	f	f	PROPN
cana-3163	197	13	(	(	PUNCT
cana-3163	197	14	j	j	PROPN
cana-3163	197	15	)	)	PUNCT
cana-3163	197	16	/⊆	/⊆	PUNCT
cana-3163	198	1	j	j	PROPN
cana-3163	198	2	(	(	PUNCT
cana-3163	198	3	iv	iv	X
cana-3163	198	4	)	)	PUNCT
cana-3163	198	5	f	f	PROPN
cana-3163	198	6	is	be	AUX
cana-3163	198	7	transitive	transitive	ADJ
cana-3163	198	8	.	.	PUNCT
cana-3163	199	1	then	then	ADV
cana-3163	199	2	f	f	PROPN
cana-3163	199	3	∈	∈	PROPN
cana-3163	199	4	pr(i	pr(i	NOUN
cana-3163	199	5	)	)	PUNCT
cana-3163	199	6	.	.	PUNCT
cana-3163	200	1	before	before	ADP
cana-3163	200	2	proving	prove	VERB
cana-3163	200	3	proposition	proposition	NOUN
cana-3163	200	4	,	,	PUNCT
cana-3163	200	5	we	we	PRON
cana-3163	200	6	prove	prove	VERB
cana-3163	200	7	a	a	DET
cana-3163	200	8	lemma	lemma	PROPN
cana-3163	200	9	.	.	PUNCT
cana-3163	201	1	lemma	lemma	PROPN
cana-3163	201	2	3	3	X
cana-3163	201	3	.	.	PUNCT
cana-3163	202	1	let	let	VERB
cana-3163	202	2	f	f	NOUN
cana-3163	202	3	:	:	PUNCT
cana-3163	202	4	i	i	PRON
cana-3163	202	5	→	→	PUNCT
cana-3163	202	6	i	i	PRON
cana-3163	202	7	be	be	VERB
cana-3163	202	8	a	a	DET
cana-3163	202	9	map	map	NOUN
cana-3163	202	10	and	and	CCONJ
cana-3163	202	11	there	there	PRON
cana-3163	202	12	exists	exist	VERB
cana-3163	202	13	a	a	DET
cana-3163	202	14	<	<	X
cana-3163	202	15	b	b	X
cana-3163	202	16	<	<	X
cana-3163	202	17	c	c	X
cana-3163	202	18	in	in	ADP
cana-3163	202	19	(	(	PUNCT
cana-3163	202	20	0	0	NUM
cana-3163	202	21	,	,	PUNCT
cana-3163	202	22	1	1	NUM
cana-3163	202	23	)	)	PUNCT
cana-3163	202	24	such	such	ADJ
cana-3163	202	25	that	that	SCONJ
cana-3163	202	26	(	(	PUNCT
cana-3163	202	27	i	i	NOUN
cana-3163	202	28	)	)	PUNCT
cana-3163	202	29	f	f	PROPN
cana-3163	202	30	(	(	PUNCT
cana-3163	202	31	a	a	NOUN
cana-3163	202	32	)	)	PUNCT
cana-3163	202	33	=	=	SYM
cana-3163	202	34	a	a	DET
cana-3163	202	35	(	(	PUNCT
cana-3163	202	36	ii	ii	NOUN
cana-3163	202	37	)	)	PUNCT
cana-3163	202	38	f	f	PROPN
cana-3163	202	39	(	(	PUNCT
cana-3163	202	40	c	c	NOUN
cana-3163	202	41	)	)	PUNCT
cana-3163	202	42	≤	≤	NOUN
cana-3163	202	43	a	a	DET
cana-3163	202	44	(	(	PUNCT
cana-3163	202	45	iii	iii	NOUN
cana-3163	202	46	)	)	PUNCT
cana-3163	202	47	f	f	NOUN
cana-3163	202	48	(	(	PUNCT
cana-3163	202	49	b	b	NOUN
cana-3163	202	50	)	)	PUNCT
cana-3163	202	51	≥	≥	NOUN
cana-3163	203	1	c	c	NOUN
cana-3163	203	2	then	then	ADV
cana-3163	203	3	p	p	X
cana-3163	203	4	(	(	PUNCT
cana-3163	203	5	f	f	PROPN
cana-3163	203	6	)	)	PUNCT
cana-3163	203	7	=	=	SYM
cana-3163	203	8	z+	z+	X
cana-3163	203	9	.	.	PUNCT
cana-3163	204	1	proof	proof	NOUN
cana-3163	204	2	.	.	PUNCT
cana-3163	205	1	let	let	VERB
cana-3163	205	2	a	a	DET
cana-3163	205	3	<	<	X
cana-3163	205	4	b	b	X
cana-3163	205	5	<	<	X
cana-3163	205	6	c	c	PROPN
cana-3163	205	7	and	and	CCONJ
cana-3163	205	8	f	f	PROPN
cana-3163	205	9	(	(	PUNCT
cana-3163	205	10	a	a	X
cana-3163	205	11	)	)	PUNCT
cana-3163	205	12	=	=	SYM
cana-3163	205	13	a.	a.	NOUN
cana-3163	205	14	let	let	VERB
cana-3163	205	15	f	f	PROPN
cana-3163	205	16	(	(	PUNCT
cana-3163	205	17	b	b	NOUN
cana-3163	205	18	)	)	PUNCT
cana-3163	205	19	=	=	SYM
cana-3163	205	20	br	br	PROPN
cana-3163	205	21	,	,	PUNCT
cana-3163	205	22	f	f	PROPN
cana-3163	205	23	(	(	PUNCT
cana-3163	205	24	c	c	NOUN
cana-3163	205	25	)	)	PUNCT
cana-3163	205	26	=	=	SYM
cana-3163	206	1	cr	cr	PROPN
cana-3163	206	2	.	.	PUNCT
cana-3163	207	1	for	for	ADP
cana-3163	207	2	given	give	VERB
cana-3163	207	3	conditions	condition	NOUN
cana-3163	207	4	,	,	PUNCT
cana-3163	207	5	cr	cr	ADP
cana-3163	207	6	≤	≤	NOUN
cana-3163	207	7	a	a	PRON
cana-3163	207	8	and	and	CCONJ
cana-3163	207	9	c	c	NOUN
cana-3163	207	10	≤	≤	NUM
cana-3163	207	11	br	br	NOUN
cana-3163	207	12	.	.	PUNCT
cana-3163	208	1	so	so	ADV
cana-3163	208	2	c	c	PROPN
cana-3163	208	3	∈	∈	PROPN
cana-3163	209	1	[	[	X
cana-3163	209	2	cr	cr	X
cana-3163	209	3	,	,	PUNCT
cana-3163	209	4	br	br	X
cana-3163	209	5	]	]	PUNCT
cana-3163	209	6	.	.	PUNCT
cana-3163	210	1	by	by	ADP
cana-3163	210	2	intermediate	intermediate	ADJ
cana-3163	210	3	value	value	NOUN
cana-3163	210	4	theorem	theorem	VERB
cana-3163	210	5	,	,	PUNCT
cana-3163	210	6	there	there	PRON
cana-3163	210	7	exists	exist	VERB
cana-3163	210	8	x	x	X
cana-3163	210	9	∈	∈	PROPN
cana-3163	210	10	[	[	X
cana-3163	210	11	b	b	X
cana-3163	210	12	,	,	PUNCT
cana-3163	210	13	c	c	X
cana-3163	210	14	]	]	X
cana-3163	210	15	such	such	ADJ
cana-3163	210	16	that	that	SCONJ
cana-3163	210	17	f	f	PROPN
cana-3163	210	18	(	(	PUNCT
cana-3163	210	19	x	x	X
cana-3163	210	20	)	)	PUNCT
cana-3163	210	21	=	=	SYM
cana-3163	211	1	c.	c.	NOUN
cana-3163	211	2	now	now	ADV
cana-3163	211	3	since	since	SCONJ
cana-3163	211	4	x	x	PROPN
cana-3163	211	5	∈	∈	PROPN
cana-3163	211	6	(	(	PUNCT
cana-3163	211	7	a	a	DET
cana-3163	211	8	,	,	PUNCT
cana-3163	211	9	br	br	NOUN
cana-3163	211	10	)	)	PUNCT
cana-3163	211	11	,	,	PUNCT
cana-3163	211	12	f	f	PROPN
cana-3163	211	13	(	(	PUNCT
cana-3163	211	14	a	a	NOUN
cana-3163	211	15	)	)	PUNCT
cana-3163	211	16	=	=	SYM
cana-3163	211	17	a	a	PRON
cana-3163	211	18	,	,	PUNCT
cana-3163	211	19	and	and	CCONJ
cana-3163	211	20	f	f	PROPN
cana-3163	211	21	(	(	PUNCT
cana-3163	211	22	b	b	NOUN
cana-3163	211	23	)	)	PUNCT
cana-3163	211	24	=	=	PUNCT
cana-3163	211	25	br	br	NOUN
cana-3163	211	26	by	by	ADP
cana-3163	211	27	intermediate	intermediate	ADJ
cana-3163	211	28	value	value	NOUN
cana-3163	211	29	theorem	theorem	VERB
cana-3163	211	30	,	,	PUNCT
cana-3163	211	31	there	there	PRON
cana-3163	211	32	exists	exist	VERB
cana-3163	211	33	y	y	PROPN
cana-3163	211	34	∈	∈	PROPN
cana-3163	211	35	(	(	PUNCT
cana-3163	211	36	a	a	DET
cana-3163	211	37	,	,	PUNCT
cana-3163	211	38	b	b	NOUN
cana-3163	211	39	)	)	PUNCT
cana-3163	211	40	with	with	ADP
cana-3163	211	41	f	f	PROPN
cana-3163	211	42	(	(	PUNCT
cana-3163	211	43	y	y	NOUN
cana-3163	211	44	)	)	PUNCT
cana-3163	211	45	=	=	PUNCT
cana-3163	212	1	x.	x.	NOUN
cana-3163	212	2	therefore	therefore	ADV
cana-3163	212	3	cr	cr	X
cana-3163	212	4	=	=	SYM
cana-3163	212	5	f	f	PROPN
cana-3163	212	6	(	(	PUNCT
cana-3163	212	7	c	c	NOUN
cana-3163	212	8	)	)	PUNCT
cana-3163	212	9	<	<	X
cana-3163	213	1	y	y	X
cana-3163	213	2	<	<	X
cana-3163	213	3	x	x	PUNCT
cana-3163	213	4	=	=	SYM
cana-3163	213	5	f	f	X
cana-3163	213	6	(	(	PUNCT
cana-3163	213	7	y	y	NOUN
cana-3163	213	8	)	)	PUNCT
cana-3163	213	9	<	<	X
cana-3163	213	10	c	c	X
cana-3163	213	11	=	=	SYM
cana-3163	213	12	f	f	PROPN
cana-3163	213	13	(	(	PUNCT
cana-3163	213	14	x	x	NOUN
cana-3163	213	15	)	)	PUNCT
cana-3163	213	16	.	.	PUNCT
cana-3163	214	1	by	by	ADP
cana-3163	214	2	standard	standard	ADJ
cana-3163	214	3	result	result	NOUN
cana-3163	214	4	[	[	X
cana-3163	214	5	8	8	NUM
cana-3163	214	6	]	]	PUNCT
cana-3163	214	7	,	,	PUNCT
cana-3163	214	8	p	p	X
cana-3163	214	9	(	(	PUNCT
cana-3163	214	10	f	f	NOUN
cana-3163	214	11	)	)	PUNCT
cana-3163	214	12	=	=	SYM
cana-3163	215	1	z+	z+	NUM
cana-3163	215	2	proof	proof	NOUN
cana-3163	215	3	of	of	ADP
cana-3163	215	4	proposition	proposition	NOUN
cana-3163	215	5	4	4	NUM
cana-3163	215	6	:	:	PUNCT
cana-3163	215	7	by	by	ADP
cana-3163	215	8	condition	condition	NOUN
cana-3163	215	9	(	(	PUNCT
cana-3163	215	10	i	i	NOUN
cana-3163	215	11	)	)	PUNCT
cana-3163	215	12	,	,	PUNCT
cana-3163	215	13	there	there	PRON
cana-3163	215	14	exists	exist	VERB
cana-3163	215	15	a	a	DET
cana-3163	215	16	∈	∈	PROPN
cana-3163	215	17	(	(	PUNCT
cana-3163	215	18	0	0	NUM
cana-3163	215	19	,	,	PUNCT
cana-3163	215	20	1	1	NUM
cana-3163	215	21	)	)	PUNCT
cana-3163	215	22	such	such	ADJ
cana-3163	215	23	that	that	SCONJ
cana-3163	215	24	f	f	PROPN
cana-3163	215	25	(	(	PUNCT
cana-3163	215	26	a	a	PROPN
cana-3163	215	27	)	)	PUNCT
cana-3163	215	28	=	=	SYM
cana-3163	215	29	a	a	PRON
cana-3163	215	30	and	and	CCONJ
cana-3163	215	31	figure	figure	VERB
cana-3163	215	32	6	6	NUM
cana-3163	215	33	.	.	PUNCT
cana-3163	215	34	case	case	NOUN
cana-3163	215	35	1	1	NUM
cana-3163	215	36	of	of	ADP
cana-3163	215	37	proposition	proposition	NOUN
cana-3163	215	38	4	4	NUM
cana-3163	215	39	by	by	ADP
cana-3163	215	40	condition	condition	NOUN
cana-3163	215	41	(	(	PUNCT
cana-3163	215	42	ii	ii	NOUN
cana-3163	215	43	)	)	PUNCT
cana-3163	215	44	,	,	PUNCT
cana-3163	215	45	f	f	PROPN
cana-3163	215	46	(	(	PUNCT
cana-3163	215	47	1	1	NUM
cana-3163	215	48	)	)	PUNCT
cana-3163	215	49	=	=	SYM
cana-3163	215	50	1	1	X
cana-3163	215	51	.	.	PUNCT
cana-3163	215	52	let	let	VERB
cana-3163	215	53	us	we	PRON
cana-3163	215	54	take	take	VERB
cana-3163	215	55	one	one	NUM
cana-3163	215	56	of	of	ADP
cana-3163	215	57	the	the	DET
cana-3163	215	58	two	two	NUM
cana-3163	215	59	possibilities	possibility	NOUN
cana-3163	215	60	f	f	X
cana-3163	215	61	(	(	PUNCT
cana-3163	215	62	0	0	NUM
cana-3163	215	63	)	)	PUNCT
cana-3163	215	64	/=	/=	NOUN
cana-3163	215	65	1	1	NUM
cana-3163	215	66	or	or	CCONJ
cana-3163	215	67	f	f	X
cana-3163	215	68	(	(	PUNCT
cana-3163	215	69	0	0	NUM
cana-3163	215	70	)	)	PUNCT
cana-3163	215	71	=	=	SYM
cana-3163	215	72	1	1	X
cana-3163	215	73	.	.	X
cana-3163	215	74	case	case	NOUN
cana-3163	215	75	1	1	NUM
cana-3163	215	76	:	:	PUNCT
cana-3163	215	77	if	if	SCONJ
cana-3163	215	78	f	f	PROPN
cana-3163	215	79	(	(	PUNCT
cana-3163	215	80	0	0	NUM
cana-3163	215	81	)	)	PUNCT
cana-3163	215	82	/=	/=	NOUN
cana-3163	216	1	1	1	X
cana-3163	216	2	.	.	PUNCT
cana-3163	216	3	let	let	VERB
cana-3163	216	4	m	m	VERB
cana-3163	216	5	=	=	VERB
cana-3163	216	6	sup{f	sup{f	PROPN
cana-3163	216	7	(	(	PUNCT
cana-3163	216	8	x)|x	x)|x	X
cana-3163	216	9	∈	∈	PROPN
cana-3163	216	10	(	(	PUNCT
cana-3163	216	11	0	0	NUM
cana-3163	216	12	,	,	PUNCT
cana-3163	216	13	a	a	PRON
cana-3163	216	14	]	]	X
cana-3163	216	15	}	}	PUNCT
cana-3163	216	16	.	.	PUNCT
cana-3163	217	1	clearly	clearly	ADV
cana-3163	217	2	m	m	VERB
cana-3163	217	3	>	>	X
cana-3163	217	4	a.	a.	NOUN
cana-3163	217	5	there	there	PRON
cana-3163	217	6	are	be	VERB
cana-3163	217	7	two	two	NUM
cana-3163	217	8	possibilities	possibility	NOUN
cana-3163	217	9	either	either	CCONJ
cana-3163	217	10	m	m	VERB
cana-3163	217	11	/=	/=	ADJ
cana-3163	217	12	1	1	NUM
cana-3163	217	13	or	or	CCONJ
cana-3163	217	14	m	m	VERB
cana-3163	217	15	=	=	ADJ
cana-3163	217	16	1	1	X
cana-3163	217	17	.	.	PUNCT
cana-3163	217	18	consider	consider	VERB
cana-3163	217	19	m	m	PRON
cana-3163	217	20	/=	/=	ADJ
cana-3163	217	21	1	1	NUM
cana-3163	217	22	.	.	PUNCT
cana-3163	218	1	let	let	VERB
cana-3163	218	2	b	b	NOUN
cana-3163	218	3	=	=	PUNCT
cana-3163	218	4	inf{x	inf{x	NOUN
cana-3163	218	5	∈	∈	PROPN
cana-3163	218	6	(	(	PUNCT
cana-3163	218	7	a	a	PRON
cana-3163	218	8	,	,	PUNCT
cana-3163	218	9	m]|f	m]|f	PROPN
cana-3163	218	10	(	(	PUNCT
cana-3163	218	11	x	x	X
cana-3163	218	12	)	)	PUNCT
cana-3163	218	13	>	>	X
cana-3163	218	14	m	m	PROPN
cana-3163	218	15	}	}	PUNCT
cana-3163	218	16	,	,	PUNCT
cana-3163	218	17	then	then	ADV
cana-3163	218	18	for	for	ADP
cana-3163	218	19	all	all	DET
cana-3163	218	20	x	x	SYM
cana-3163	218	21	∈	∈	PROPN
cana-3163	218	22	[	[	X
cana-3163	218	23	a	a	X
cana-3163	218	24	,	,	PUNCT
cana-3163	218	25	b	b	NOUN
cana-3163	218	26	]	]	X
cana-3163	218	27	,	,	PUNCT
cana-3163	218	28	f	f	PROPN
cana-3163	218	29	(	(	PUNCT
cana-3163	218	30	x	x	NOUN
cana-3163	218	31	)	)	PUNCT
cana-3163	218	32	≤	≤	NUM
cana-3163	218	33	m	m	NOUN
cana-3163	218	34	,	,	PUNCT
cana-3163	218	35	andf	andf	NOUN
cana-3163	218	36	(	(	PUNCT
cana-3163	218	37	b	b	NOUN
cana-3163	218	38	)	)	PUNCT
cana-3163	218	39	>	>	X
cana-3163	218	40	m.	m.	NOUN
cana-3163	218	41	let	let	VERB
cana-3163	218	42	ar	ar	NOUN
cana-3163	218	43	=	=	PUNCT
cana-3163	218	44	sup{x	sup{x	NOUN
cana-3163	218	45	∈	∈	PROPN
cana-3163	219	1	[	[	X
cana-3163	219	2	a	a	X
cana-3163	219	3	,	,	PUNCT
cana-3163	219	4	b]|f	b]|f	PROPN
cana-3163	219	5	(	(	PUNCT
cana-3163	219	6	x	x	X
cana-3163	219	7	)	)	PUNCT
cana-3163	219	8	=	=	SYM
cana-3163	219	9	x	x	X
cana-3163	219	10	}	}	PUNCT
cana-3163	219	11	,	,	PUNCT
cana-3163	219	12	ar	ar	NOUN
cana-3163	219	13	∈	∈	PROPN
cana-3163	219	14	[	[	X
cana-3163	219	15	a	a	DET
cana-3163	219	16	,	,	PUNCT
cana-3163	219	17	b	b	NOUN
cana-3163	219	18	)	)	PUNCT
cana-3163	219	19	.	.	PUNCT
cana-3163	220	1	now	now	ADV
cana-3163	220	2	f	f	X
cana-3163	220	3	(	(	PUNCT
cana-3163	220	4	x	x	X
cana-3163	220	5	)	)	PUNCT
cana-3163	220	6	>	>	PUNCT
cana-3163	220	7	x	x	PUNCT
cana-3163	220	8	for	for	ADP
cana-3163	220	9	all	all	DET
cana-3163	220	10	x	x	SYM
cana-3163	220	11	∈	∈	PROPN
cana-3163	220	12	(	(	PUNCT
cana-3163	220	13	ar	ar	PROPN
cana-3163	220	14	,	,	PUNCT
cana-3163	220	15	b	b	NOUN
cana-3163	220	16	]	]	X
cana-3163	220	17	and	and	CCONJ
cana-3163	220	18	f	f	X
cana-3163	220	19	(	(	PUNCT
cana-3163	220	20	[	[	X
cana-3163	220	21	ar	ar	NOUN
cana-3163	220	22	,	,	PUNCT
cana-3163	220	23	b	b	NOUN
cana-3163	220	24	]	]	X
cana-3163	220	25	)	)	PUNCT
cana-3163	221	1	=	=	PUNCT
cana-3163	222	1	[	[	X
cana-3163	222	2	ar	ar	PROPN
cana-3163	222	3	,	,	PUNCT
cana-3163	222	4	m	m	VERB
cana-3163	222	5	]	]	PUNCT
cana-3163	222	6	.	.	PUNCT
cana-3163	223	1	let	let	VERB
cana-3163	223	2	n	n	NOUN
cana-3163	223	3	=	=	VERB
cana-3163	223	4	min{f	min{f	X
cana-3163	223	5	(	(	PUNCT
cana-3163	223	6	x)|x	x)|x	PRON
cana-3163	223	7	∈	∈	PROPN
cana-3163	224	1	[	[	X
cana-3163	224	2	b	b	X
cana-3163	224	3	,	,	PUNCT
cana-3163	224	4	1	1	NUM
cana-3163	224	5	]	]	PUNCT
cana-3163	224	6	}	}	PUNCT
cana-3163	224	7	and	and	CCONJ
cana-3163	224	8	c	c	NOUN
cana-3163	224	9	=	=	PUNCT
cana-3163	224	10	inf{x	inf{x	NOUN
cana-3163	224	11	∈	∈	PROPN
cana-3163	225	1	[	[	X
cana-3163	225	2	b	b	NOUN
cana-3163	225	3	,	,	PUNCT
cana-3163	225	4	1]|f	1]|f	NUM
cana-3163	225	5	(	(	PUNCT
cana-3163	225	6	x	x	NOUN
cana-3163	225	7	)	)	PUNCT
cana-3163	225	8	=	=	SYM
cana-3163	225	9	n	n	CCONJ
cana-3163	225	10	}	}	PUNCT
cana-3163	225	11	.	.	PUNCT
cana-3163	226	1	now	now	ADV
cana-3163	226	2	ar	ar	VERB
cana-3163	226	3	>	>	X
cana-3163	226	4	f	f	PROPN
cana-3163	226	5	(	(	PUNCT
cana-3163	226	6	c	c	NOUN
cana-3163	226	7	)	)	PUNCT
cana-3163	226	8	=	=	VERB
cana-3163	226	9	n.	n.	NOUN
cana-3163	226	10	communications	communication	NOUN
cana-3163	226	11	on	on	ADP
cana-3163	226	12	applied	apply	VERB
cana-3163	226	13	nonlinear	nonlinear	ADJ
cana-3163	226	14	analysis	analysis	NOUN
cana-3163	226	15	issn	issn	NOUN
cana-3163	226	16	:	:	PUNCT
cana-3163	226	17	1074	1074	NUM
cana-3163	226	18	-	-	PUNCT
cana-3163	226	19	133x	133x	NUM
cana-3163	226	20	vol	vol	NOUN
cana-3163	226	21	32	32	NUM
cana-3163	226	22	no	no	NOUN
cana-3163	226	23	.	.	PUNCT
cana-3163	227	1	5s	5s	NUM
cana-3163	227	2	(	(	PUNCT
cana-3163	227	3	2025	2025	NUM
cana-3163	227	4	)	)	PUNCT
cana-3163	227	5	518	518	NUM
cana-3163	227	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	227	7	claim	claim	NOUN
cana-3163	227	8	:	:	PUNCT
cana-3163	227	9	there	there	PRON
cana-3163	227	10	exists	exist	VERB
cana-3163	227	11	d	d	PROPN
cana-3163	227	12	∈	∈	PROPN
cana-3163	227	13	(	(	PUNCT
cana-3163	227	14	ar	ar	PROPN
cana-3163	227	15	,	,	PUNCT
cana-3163	227	16	c	c	NOUN
cana-3163	227	17	)	)	PUNCT
cana-3163	227	18	such	such	ADJ
cana-3163	227	19	that	that	SCONJ
cana-3163	227	20	f	f	PROPN
cana-3163	227	21	(	(	PUNCT
cana-3163	227	22	d	d	PROPN
cana-3163	227	23	)	)	PUNCT
cana-3163	227	24	>	>	PUNCT
cana-3163	227	25	c.	c.	NOUN
cana-3163	227	26	if	if	SCONJ
cana-3163	227	27	c	c	PROPN
cana-3163	227	28	<	<	X
cana-3163	227	29	m	m	PROPN
cana-3163	227	30	,	,	PUNCT
cana-3163	227	31	take	take	VERB
cana-3163	227	32	d	d	X
cana-3163	227	33	=	=	X
cana-3163	227	34	b.	b.	PROPN
cana-3163	228	1	if	if	SCONJ
cana-3163	228	2	c	c	PROPN
cana-3163	228	3	≥	≥	X
cana-3163	228	4	m	m	VERB
cana-3163	228	5	,	,	PUNCT
cana-3163	228	6	there	there	PRON
cana-3163	228	7	exists	exist	VERB
cana-3163	228	8	y	y	PROPN
cana-3163	228	9	∈	∈	PROPN
cana-3163	228	10	(	(	PUNCT
cana-3163	228	11	b	b	NOUN
cana-3163	228	12	,	,	PUNCT
cana-3163	228	13	c	c	NOUN
cana-3163	228	14	)	)	PUNCT
cana-3163	228	15	such	such	ADJ
cana-3163	228	16	that	that	SCONJ
cana-3163	228	17	f	f	PROPN
cana-3163	228	18	(	(	PUNCT
cana-3163	228	19	y	y	PROPN
cana-3163	228	20	)	)	PUNCT
cana-3163	228	21	>	>	X
cana-3163	229	1	c	c	X
cana-3163	229	2	,	,	PUNCT
cana-3163	229	3	take	take	VERB
cana-3163	229	4	d	d	NOUN
cana-3163	229	5	=	=	SYM
cana-3163	229	6	y.	y.	NOUN
cana-3163	229	7	claim	claim	NOUN
cana-3163	229	8	is	be	AUX
cana-3163	229	9	proved	prove	VERB
cana-3163	229	10	.	.	PUNCT
cana-3163	230	1	so	so	ADV
cana-3163	230	2	we	we	PRON
cana-3163	230	3	have	have	AUX
cana-3163	230	4	,	,	PUNCT
cana-3163	230	5	ar	ar	VERB
cana-3163	230	6	<	<	X
cana-3163	230	7	d	d	X
cana-3163	230	8	<	<	X
cana-3163	230	9	c	c	PROPN
cana-3163	230	10	with	with	ADP
cana-3163	230	11	f	f	PROPN
cana-3163	230	12	(	(	PUNCT
cana-3163	230	13	ar	ar	PROPN
cana-3163	230	14	)	)	PUNCT
cana-3163	230	15	=	=	SYM
cana-3163	230	16	ar	ar	PROPN
cana-3163	230	17	,	,	PUNCT
cana-3163	230	18	f	f	PROPN
cana-3163	230	19	(	(	PUNCT
cana-3163	230	20	c	c	NOUN
cana-3163	230	21	)	)	PUNCT
cana-3163	231	1	≤	≤	NUM
cana-3163	232	1	f	f	X
cana-3163	232	2	(	(	PUNCT
cana-3163	232	3	ar	ar	NOUN
cana-3163	232	4	)	)	PUNCT
cana-3163	232	5	and	and	CCONJ
cana-3163	232	6	c	c	X
cana-3163	232	7	<	<	X
cana-3163	232	8	f	f	X
cana-3163	232	9	(	(	PUNCT
cana-3163	232	10	d	d	NOUN
cana-3163	232	11	)	)	PUNCT
cana-3163	232	12	.	.	PUNCT
cana-3163	233	1	by	by	ADP
cana-3163	233	2	lemma	lemma	PROPN
cana-3163	233	3	3	3	NUM
cana-3163	233	4	,	,	PUNCT
cana-3163	233	5	p	p	X
cana-3163	233	6	(	(	PUNCT
cana-3163	233	7	f	f	PROPN
cana-3163	233	8	)	)	PUNCT
cana-3163	233	9	=	=	PUNCT
cana-3163	233	10	z+	z+	X
cana-3163	233	11	.	.	PUNCT
cana-3163	233	12	now	now	ADV
cana-3163	233	13	consider	consider	VERB
cana-3163	233	14	the	the	DET
cana-3163	233	15	other	other	ADJ
cana-3163	233	16	case	case	NOUN
cana-3163	233	17	m	m	NOUN
cana-3163	233	18	=	=	NOUN
cana-3163	233	19	1	1	X
cana-3163	233	20	.	.	PUNCT
cana-3163	234	1	let	let	VERB
cana-3163	234	2	x	x	X
cana-3163	234	3	∈	∈	PROPN
cana-3163	235	1	[	[	X
cana-3163	235	2	0	0	NUM
cana-3163	235	3	,	,	PUNCT
cana-3163	235	4	a	a	PRON
cana-3163	235	5	]	]	X
cana-3163	235	6	such	such	ADJ
cana-3163	235	7	that	that	SCONJ
cana-3163	235	8	f	f	PROPN
cana-3163	235	9	(	(	PUNCT
cana-3163	235	10	x	x	X
cana-3163	235	11	)	)	PUNCT
cana-3163	235	12	=	=	PUNCT
cana-3163	235	13	m	m	NOUN
cana-3163	235	14	=	=	NOUN
cana-3163	235	15	1	1	X
cana-3163	235	16	.	.	PUNCT
cana-3163	235	17	let	let	VERB
cana-3163	235	18	n	n	NOUN
cana-3163	235	19	=	=	VERB
cana-3163	235	20	min{f	min{f	X
cana-3163	235	21	(	(	PUNCT
cana-3163	235	22	x)|x	x)|x	X
cana-3163	235	23	∈	∈	PROPN
cana-3163	236	1	[	[	X
cana-3163	236	2	a	a	X
cana-3163	236	3	,	,	PUNCT
cana-3163	236	4	1	1	NUM
cana-3163	236	5	]	]	PUNCT
cana-3163	236	6	}	}	PUNCT
cana-3163	236	7	and	and	CCONJ
cana-3163	236	8	let	let	VERB
cana-3163	236	9	figure	figure	VERB
cana-3163	236	10	7	7	NUM
cana-3163	236	11	.	.	PUNCT
cana-3163	237	1	when	when	SCONJ
cana-3163	237	2	m	m	VERB
cana-3163	237	3	=	=	SYM
cana-3163	237	4	1	1	NUM
cana-3163	237	5	,	,	PUNCT
cana-3163	237	6	n	n	PRON
cana-3163	237	7	≥	≥	NOUN
cana-3163	237	8	x	x	SYM
cana-3163	237	9	f	f	X
cana-3163	237	10	(	(	PUNCT
cana-3163	237	11	c	c	NOUN
cana-3163	237	12	)	)	PUNCT
cana-3163	237	13	=	=	SYM
cana-3163	238	1	n	n	CCONJ
cana-3163	238	2	,	,	PUNCT
cana-3163	238	3	c	c	PROPN
cana-3163	238	4	∈	∈	PROPN
cana-3163	239	1	[	[	X
cana-3163	239	2	a	a	X
cana-3163	239	3	,	,	PUNCT
cana-3163	239	4	1	1	NUM
cana-3163	239	5	]	]	PUNCT
cana-3163	239	6	.	.	PUNCT
cana-3163	240	1	either	either	CCONJ
cana-3163	240	2	n	n	ADV
cana-3163	240	3	/=	/=	NOUN
cana-3163	240	4	0	0	PUNCT
cana-3163	241	1	or	or	CCONJ
cana-3163	241	2	n	n	CCONJ
cana-3163	241	3	=	=	SYM
cana-3163	241	4	0	0	PROPN
cana-3163	241	5	.	.	PUNCT
cana-3163	242	1	if	if	SCONJ
cana-3163	242	2	n	n	ADV
cana-3163	242	3	/=	/=	NOUN
cana-3163	242	4	0	0	PUNCT
cana-3163	243	1	and	and	CCONJ
cana-3163	243	2	n	n	PRON
cana-3163	243	3	≥	≥	NOUN
cana-3163	243	4	x	x	NOUN
cana-3163	243	5	,	,	PUNCT
cana-3163	243	6	then	then	ADV
cana-3163	243	7	there	there	PRON
cana-3163	243	8	exists	exist	VERB
cana-3163	243	9	z	z	PROPN
cana-3163	243	10	∈	∈	PROPN
cana-3163	243	11	(	(	PUNCT
cana-3163	243	12	x	x	X
cana-3163	243	13	,	,	PUNCT
cana-3163	243	14	a	a	PRON
cana-3163	243	15	)	)	PUNCT
cana-3163	243	16	such	such	ADJ
cana-3163	243	17	that	that	SCONJ
cana-3163	243	18	f	f	PROPN
cana-3163	243	19	(	(	PUNCT
cana-3163	243	20	z	z	NOUN
cana-3163	243	21	)	)	PUNCT
cana-3163	243	22	≥	≥	NOUN
cana-3163	243	23	x.	x.	PUNCT
cana-3163	244	1	so	so	ADV
cana-3163	244	2	x	x	X
cana-3163	244	3	<	<	X
cana-3163	244	4	z	z	X
cana-3163	244	5	<	<	X
cana-3163	244	6	a	a	PRON
cana-3163	244	7	with	with	ADP
cana-3163	244	8	f	f	PROPN
cana-3163	244	9	(	(	PUNCT
cana-3163	244	10	x	x	NOUN
cana-3163	244	11	)	)	PUNCT
cana-3163	244	12	≥	≥	NOUN
cana-3163	244	13	f	f	PROPN
cana-3163	244	14	(	(	PUNCT
cana-3163	244	15	a	a	NOUN
cana-3163	244	16	)	)	PUNCT
cana-3163	244	17	and	and	CCONJ
cana-3163	244	18	x	x	SYM
cana-3163	244	19	≤	≤	ADJ
cana-3163	244	20	f	f	X
cana-3163	244	21	(	(	PUNCT
cana-3163	244	22	z	z	NOUN
cana-3163	244	23	)	)	PUNCT
cana-3163	244	24	.	.	PUNCT
cana-3163	245	1	by	by	ADP
cana-3163	245	2	lemma	lemma	PROPN
cana-3163	245	3	3	3	NUM
cana-3163	245	4	,	,	PUNCT
cana-3163	245	5	p	p	X
cana-3163	245	6	(	(	PUNCT
cana-3163	245	7	f	f	PROPN
cana-3163	245	8	)	)	PUNCT
cana-3163	245	9	=	=	PUNCT
cana-3163	245	10	z+	z+	X
cana-3163	245	11	.	.	PUNCT
cana-3163	246	1	if	if	SCONJ
cana-3163	246	2	n	n	ADV
cana-3163	246	3	/=	/=	NOUN
cana-3163	246	4	0	0	PUNCT
cana-3163	247	1	and	and	CCONJ
cana-3163	247	2	n	n	CCONJ
cana-3163	247	3	<	<	X
cana-3163	247	4	x	x	X
cana-3163	247	5	,	,	PUNCT
cana-3163	247	6	then	then	ADV
cana-3163	247	7	there	there	PRON
cana-3163	247	8	exists	exist	VERB
cana-3163	247	9	z	z	PROPN
cana-3163	247	10	∈	∈	PROPN
cana-3163	247	11	(	(	PUNCT
cana-3163	247	12	n	n	CCONJ
cana-3163	247	13	,	,	PUNCT
cana-3163	247	14	a	a	X
cana-3163	247	15	)	)	PUNCT
cana-3163	247	16	such	such	ADJ
cana-3163	247	17	that	that	SCONJ
cana-3163	247	18	f	f	PROPN
cana-3163	247	19	(	(	PUNCT
cana-3163	247	20	z	z	NOUN
cana-3163	247	21	)	)	PUNCT
cana-3163	247	22	>	>	X
cana-3163	248	1	n.	n.	NOUN
cana-3163	248	2	here	here	ADV
cana-3163	248	3	z	z	PROPN
cana-3163	248	4	∈	∈	PROPN
cana-3163	248	5	(	(	PUNCT
cana-3163	248	6	n	n	X
cana-3163	248	7	,	,	PUNCT
cana-3163	248	8	x	x	NOUN
cana-3163	248	9	)	)	PUNCT
cana-3163	248	10	∪	∪	ADV
cana-3163	248	11	(	(	PUNCT
cana-3163	248	12	x	x	X
cana-3163	248	13	,	,	PUNCT
cana-3163	248	14	a	a	PRON
cana-3163	248	15	)	)	PUNCT
cana-3163	248	16	.	.	PUNCT
cana-3163	249	1	if	if	SCONJ
cana-3163	249	2	z	z	PROPN
cana-3163	249	3	∈	∈	PROPN
cana-3163	249	4	(	(	PUNCT
cana-3163	249	5	n	n	X
cana-3163	249	6	,	,	PUNCT
cana-3163	249	7	x	x	X
cana-3163	249	8	)	)	PUNCT
cana-3163	249	9	then	then	ADV
cana-3163	249	10	z	z	X
cana-3163	249	11	<	<	X
cana-3163	249	12	x	x	X
cana-3163	249	13	<	<	X
cana-3163	249	14	c	c	PROPN
cana-3163	249	15	with	with	ADP
cana-3163	249	16	f	f	PROPN
cana-3163	249	17	(	(	PUNCT
cana-3163	249	18	z	z	NOUN
cana-3163	249	19	)	)	PUNCT
cana-3163	249	20	>	>	PUNCT
cana-3163	250	1	z	z	PROPN
cana-3163	250	2	and	and	CCONJ
cana-3163	250	3	f	f	PROPN
cana-3163	250	4	(	(	PUNCT
cana-3163	250	5	x	x	X
cana-3163	250	6	)	)	PUNCT
cana-3163	251	1	>	>	X
cana-3163	251	2	x.	x.	NOUN
cana-3163	251	3	define	define	VERB
cana-3163	251	4	g(y	g(y	NOUN
cana-3163	251	5	)	)	PUNCT
cana-3163	251	6	=	=	SYM
cana-3163	252	1	f	f	PROPN
cana-3163	252	2	(	(	PUNCT
cana-3163	252	3	y	y	NOUN
cana-3163	252	4	)	)	PUNCT
cana-3163	252	5	—	—	PUNCT
cana-3163	253	1	y.	y.	NOUN
cana-3163	253	2	by	by	ADP
cana-3163	253	3	intermediate	intermediate	ADJ
cana-3163	253	4	value	value	NOUN
cana-3163	253	5	theorem	theorem	VERB
cana-3163	253	6	,	,	PUNCT
cana-3163	253	7	there	there	PRON
cana-3163	253	8	exists	exist	VERB
cana-3163	253	9	s	s	PROPN
cana-3163	253	10	∈	∈	PROPN
cana-3163	253	11	(	(	PUNCT
cana-3163	253	12	z	z	NOUN
cana-3163	253	13	,	,	PUNCT
cana-3163	253	14	x	x	NOUN
cana-3163	253	15	)	)	PUNCT
cana-3163	253	16	such	such	ADJ
cana-3163	253	17	that	that	DET
cana-3163	253	18	g(s	g(s	NOUN
cana-3163	253	19	)	)	PUNCT
cana-3163	253	20	=	=	SYM
cana-3163	253	21	0	0	NUM
cana-3163	253	22	and	and	CCONJ
cana-3163	253	23	g(z	g(z	PROPN
cana-3163	253	24	)	)	PUNCT
cana-3163	253	25	<	<	X
cana-3163	253	26	0	0	NUM
cana-3163	253	27	,	,	PUNCT
cana-3163	253	28	g(x	g(x	NOUN
cana-3163	253	29	)	)	PUNCT
cana-3163	253	30	>	>	X
cana-3163	254	1	0	0	X
cana-3163	254	2	.	.	PUNCT
cana-3163	255	1	therefore	therefore	ADV
cana-3163	255	2	s	s	X
cana-3163	255	3	∈	∈	PROPN
cana-3163	255	4	(	(	PUNCT
cana-3163	255	5	z	z	NOUN
cana-3163	255	6	,	,	PUNCT
cana-3163	255	7	x	x	X
cana-3163	255	8	)	)	PUNCT
cana-3163	255	9	is	be	AUX
cana-3163	255	10	fixed	fix	VERB
cana-3163	255	11	for	for	ADP
cana-3163	255	12	f	f	PROPN
cana-3163	255	13	.	.	PUNCT
cana-3163	256	1	so	so	ADV
cana-3163	256	2	s	s	VERB
cana-3163	256	3	<	<	X
cana-3163	256	4	x	x	X
cana-3163	256	5	<	<	X
cana-3163	256	6	c	c	NOUN
cana-3163	256	7	with	with	ADP
cana-3163	256	8	fixed	fix	VERB
cana-3163	256	9	s	s	PROPN
cana-3163	256	10	,	,	PUNCT
cana-3163	256	11	f	f	PROPN
cana-3163	256	12	(	(	PUNCT
cana-3163	256	13	c	c	NOUN
cana-3163	256	14	)	)	PUNCT
cana-3163	256	15	≤	≤	NUM
cana-3163	257	1	f	f	X
cana-3163	257	2	(	(	PUNCT
cana-3163	257	3	s	s	X
cana-3163	257	4	)	)	PUNCT
cana-3163	257	5	and	and	CCONJ
cana-3163	257	6	c	c	NOUN
cana-3163	257	7	≤	≤	PROPN
cana-3163	257	8	f	f	X
cana-3163	257	9	(	(	PUNCT
cana-3163	257	10	x	x	NOUN
cana-3163	257	11	)	)	PUNCT
cana-3163	257	12	.	.	PUNCT
cana-3163	258	1	by	by	ADP
cana-3163	258	2	lemma	lemma	PROPN
cana-3163	258	3	3	3	NUM
cana-3163	258	4	,	,	PUNCT
cana-3163	258	5	p	p	X
cana-3163	258	6	(	(	PUNCT
cana-3163	258	7	f	f	PROPN
cana-3163	258	8	)	)	PUNCT
cana-3163	258	9	=	=	SYM
cana-3163	258	10	z+	z+	PROPN
cana-3163	258	11	.	.	PUNCT
cana-3163	258	12	figure	figure	NOUN
cana-3163	258	13	8	8	NUM
cana-3163	258	14	.	.	PUNCT
cana-3163	259	1	when	when	SCONJ
cana-3163	259	2	m	m	AUX
cana-3163	259	3	=	=	SYM
cana-3163	259	4	1	1	NUM
cana-3163	259	5	,	,	PUNCT
cana-3163	259	6	n	n	CCONJ
cana-3163	259	7	<	<	X
cana-3163	259	8	x	x	PUNCT
cana-3163	259	9	communications	communication	NOUN
cana-3163	259	10	on	on	ADP
cana-3163	259	11	applied	apply	VERB
cana-3163	259	12	nonlinear	nonlinear	ADJ
cana-3163	259	13	analysis	analysis	NOUN
cana-3163	259	14	issn	issn	NOUN
cana-3163	259	15	:	:	PUNCT
cana-3163	259	16	1074	1074	NUM
cana-3163	259	17	-	-	PUNCT
cana-3163	259	18	133x	133x	NUM
cana-3163	259	19	vol	vol	NOUN
cana-3163	259	20	32	32	NUM
cana-3163	259	21	no	no	NOUN
cana-3163	259	22	.	.	PUNCT
cana-3163	260	1	5s	5s	NUM
cana-3163	260	2	(	(	PUNCT
cana-3163	260	3	2025	2025	NUM
cana-3163	260	4	)	)	PUNCT
cana-3163	260	5	519	519	NUM
cana-3163	260	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-3163	260	7	case	case	NOUN
cana-3163	260	8	2	2	NUM
cana-3163	260	9	:	:	PUNCT
cana-3163	260	10	if	if	SCONJ
cana-3163	260	11	f	f	PROPN
cana-3163	260	12	(	(	PUNCT
cana-3163	260	13	0	0	NUM
cana-3163	260	14	)	)	PUNCT
cana-3163	260	15	=	=	SYM
cana-3163	260	16	1	1	NUM
cana-3163	260	17	let	let	VERB
cana-3163	260	18	c	c	PROPN
cana-3163	260	19	∈	∈	PROPN
cana-3163	260	20	(	(	PUNCT
cana-3163	260	21	0	0	NUM
cana-3163	260	22	,	,	PUNCT
cana-3163	260	23	1	1	NUM
cana-3163	260	24	)	)	PUNCT
cana-3163	260	25	such	such	ADJ
cana-3163	260	26	that	that	SCONJ
cana-3163	260	27	f	f	PROPN
cana-3163	260	28	(	(	PUNCT
cana-3163	260	29	c	c	X
cana-3163	260	30	)	)	PUNCT
cana-3163	260	31	=	=	SYM
cana-3163	260	32	0	0	X
cana-3163	260	33	.	.	PUNCT
cana-3163	261	1	if	if	SCONJ
cana-3163	261	2	such	such	ADJ
cana-3163	261	3	c	c	NOUN
cana-3163	261	4	does	do	AUX
cana-3163	261	5	n’t	not	PART
cana-3163	261	6	exist	exist	VERB
cana-3163	261	7	,	,	PUNCT
cana-3163	261	8	p	p	NOUN
cana-3163	261	9	=	=	PUNCT
cana-3163	261	10	inf{f	inf{f	NOUN
cana-3163	261	11	(	(	PUNCT
cana-3163	261	12	x)|x	x)|x	X
cana-3163	261	13	∈	∈	PROPN
cana-3163	261	14	i	i	PRON
cana-3163	261	15	}	}	PUNCT
cana-3163	261	16	>	>	X
cana-3163	262	1	0	0	X
cana-3163	262	2	.	.	PUNCT
cana-3163	263	1	hence	hence	ADV
cana-3163	263	2	[	[	X
cana-3163	263	3	p	p	X
cana-3163	263	4	,	,	PUNCT
cana-3163	263	5	0	0	NUM
cana-3163	263	6	]	]	PUNCT
cana-3163	263	7	is	be	AUX
cana-3163	263	8	invariant	invariant	ADJ
cana-3163	263	9	under	under	ADP
cana-3163	263	10	f	f	PROPN
cana-3163	263	11	.	.	PUNCT
cana-3163	264	1	so	so	ADV
cana-3163	264	2	there	there	PRON
cana-3163	264	3	exists	exist	VERB
cana-3163	264	4	z	z	PROPN
cana-3163	264	5	∈	∈	PROPN
cana-3163	264	6	(	(	PUNCT
cana-3163	264	7	c	c	NOUN
cana-3163	264	8	,	,	PUNCT
cana-3163	264	9	1	1	NUM
cana-3163	264	10	)	)	PUNCT
cana-3163	265	1	such	such	ADJ
cana-3163	265	2	that	that	SCONJ
cana-3163	265	3	f	f	PROPN
cana-3163	265	4	(	(	PUNCT
cana-3163	265	5	z	z	NOUN
cana-3163	265	6	)	)	PUNCT
cana-3163	265	7	=	=	SYM
cana-3163	265	8	c.	c.	NOUN
cana-3163	265	9	therefore	therefore	ADV
cana-3163	265	10	p	p	PROPN
cana-3163	265	11	(	(	PUNCT
cana-3163	265	12	f	f	PROPN
cana-3163	265	13	)	)	PUNCT
cana-3163	265	14	=	=	SYM
cana-3163	265	15	z+	z+	X
cana-3163	265	16	.	.	PUNCT
cana-3163	266	1	by	by	ADP
cana-3163	266	2	condition	condition	NOUN
cana-3163	266	3	(	(	PUNCT
cana-3163	266	4	iv	iv	NOUN
cana-3163	266	5	)	)	PUNCT
cana-3163	266	6	,	,	PUNCT
cana-3163	266	7	per(f	per(f	NOUN
cana-3163	266	8	)	)	PUNCT
cana-3163	266	9	=	=	SYM
cana-3163	266	10	i.	i.	NOUN
cana-3163	266	11	therefore	therefore	ADV
cana-3163	266	12	f	f	PROPN
cana-3163	266	13	∈	∈	PROPN
cana-3163	266	14	pr(i	pr(i	NOUN
cana-3163	266	15	)	)	PUNCT
cana-3163	266	16	.	.	PUNCT
cana-3163	267	1	hence	hence	ADV
cana-3163	267	2	the	the	DET
cana-3163	267	3	proof	proof	NOUN
cana-3163	267	4	.	.	PUNCT
cana-3163	268	1	we	we	PRON
cana-3163	268	2	summarize	summarize	VERB
cana-3163	268	3	the	the	DET
cana-3163	268	4	above	above	ADJ
cana-3163	268	5	propositions	proposition	NOUN
cana-3163	268	6	3	3	NUM
cana-3163	268	7	and	and	CCONJ
cana-3163	268	8	4	4	NUM
cana-3163	268	9	as	as	SCONJ
cana-3163	268	10	given	give	VERB
cana-3163	268	11	below	below	ADP
cana-3163	268	12	proposition	proposition	NOUN
cana-3163	268	13	5	5	NUM
cana-3163	268	14	.	.	PUNCT
cana-3163	269	1	let	let	VERB
cana-3163	269	2	f	f	PRON
cana-3163	269	3	be	be	AUX
cana-3163	269	4	transitive	transitive	ADJ
cana-3163	269	5	map	map	NOUN
cana-3163	269	6	on	on	ADP
cana-3163	269	7	i.	i.	PROPN
cana-3163	269	8	then	then	ADV
cana-3163	270	1	f	f	PROPN
cana-3163	270	2	∈	∈	PROPN
cana-3163	270	3	pr(i	pr(i	NOUN
cana-3163	270	4	)	)	PUNCT
cana-3163	270	5	if	if	SCONJ
cana-3163	270	6	f	f	PROPN
cana-3163	270	7	belongs	belong	VERB
cana-3163	270	8	to	to	ADP
cana-3163	270	9	anyone	anyone	PRON
cana-3163	270	10	of	of	ADP
cana-3163	270	11	the	the	DET
cana-3163	270	12	following	follow	VERB
cana-3163	270	13	two	two	NUM
cana-3163	270	14	classes	class	NOUN
cana-3163	270	15	.	.	PUNCT
cana-3163	271	1	class	class	NOUN
cana-3163	271	2	1	1	NUM
cana-3163	271	3	.	.	PUNCT
cana-3163	272	1	there	there	PRON
cana-3163	272	2	exists	exist	VERB
cana-3163	272	3	x	x	X
cana-3163	272	4	∈	∈	PROPN
cana-3163	272	5	per4(f	per4(f	PUNCT
cana-3163	272	6	)	)	PUNCT
cana-3163	272	7	with	with	ADP
cana-3163	272	8	increasing	increase	VERB
cana-3163	272	9	periodic	periodic	ADJ
cana-3163	272	10	orbit	orbit	NOUN
cana-3163	272	11	.	.	PUNCT
cana-3163	273	1	class	class	NOUN
cana-3163	273	2	2	2	NUM
cana-3163	273	3	.	.	PUNCT
cana-3163	274	1	(	(	PUNCT
cana-3163	274	2	i	i	NOUN
cana-3163	274	3	)	)	PUNCT
cana-3163	274	4	f	f	PROPN
cana-3163	274	5	(	(	PUNCT
cana-3163	274	6	x	x	X
cana-3163	274	7	)	)	PUNCT
cana-3163	274	8	=	=	PUNCT
cana-3163	275	1	x	x	PUNCT
cana-3163	275	2	for	for	ADP
cana-3163	275	3	some	some	DET
cana-3163	275	4	x	x	SYM
cana-3163	275	5	∈	∈	PROPN
cana-3163	275	6	(	(	PUNCT
cana-3163	275	7	0	0	NUM
cana-3163	275	8	,	,	PUNCT
cana-3163	275	9	1	1	NUM
cana-3163	275	10	)	)	PUNCT
cana-3163	275	11	(	(	PUNCT
cana-3163	275	12	ii	ii	NOUN
cana-3163	275	13	)	)	PUNCT
cana-3163	275	14	f	f	PROPN
cana-3163	275	15	(	(	PUNCT
cana-3163	275	16	1	1	NUM
cana-3163	275	17	)	)	PUNCT
cana-3163	275	18	=	=	SYM
cana-3163	275	19	1	1	NUM
cana-3163	275	20	(	(	PUNCT
cana-3163	275	21	iii	iii	NOUN
cana-3163	275	22	)	)	PUNCT
cana-3163	275	23	for	for	ADP
cana-3163	275	24	every	every	DET
cana-3163	275	25	j	j	PROPN
cana-3163	275	26	c	c	PROPN
cana-3163	275	27	i	i	PROPN
cana-3163	275	28	,	,	PUNCT
cana-3163	275	29	f	f	PROPN
cana-3163	275	30	(	(	PUNCT
cana-3163	275	31	j	j	PROPN
cana-3163	275	32	)	)	PUNCT
cana-3163	275	33	/⊆	/⊆	PUNCT
cana-3163	276	1	j.	j.	PROPN
cana-3163	276	2	now	now	ADV
cana-3163	276	3	,	,	PUNCT
cana-3163	276	4	we	we	PRON
cana-3163	276	5	shall	shall	AUX
cana-3163	276	6	give	give	VERB
cana-3163	276	7	an	an	DET
cana-3163	276	8	example	example	NOUN
cana-3163	276	9	to	to	PART
cana-3163	276	10	show	show	VERB
cana-3163	276	11	that	that	SCONJ
cana-3163	276	12	transitivity	transitivity	NOUN
cana-3163	276	13	of	of	ADP
cana-3163	276	14	f	f	PROPN
cana-3163	276	15	is	be	AUX
cana-3163	276	16	not	not	PART
cana-3163	276	17	a	a	DET
cana-3163	276	18	necessary	necessary	ADJ
cana-3163	276	19	condition	condition	NOUN
cana-3163	276	20	for	for	SCONJ
cana-3163	276	21	f	f	PROPN
cana-3163	276	22	to	to	PART
cana-3163	276	23	be	be	AUX
cana-3163	276	24	in	in	ADP
cana-3163	276	25	pr(x	pr(x	NOUN
cana-3163	276	26	)	)	PUNCT
cana-3163	276	27	.	.	PUNCT
cana-3163	277	1	example	example	NOUN
cana-3163	278	1	8	8	NUM
cana-3163	278	2	.	.	PUNCT
cana-3163	279	1	let	let	VERB
cana-3163	279	2	f	f	NOUN
cana-3163	279	3	:	:	PUNCT
cana-3163	279	4	r	r	NOUN
cana-3163	279	5	→	→	SYM
cana-3163	279	6	r	r	NOUN
cana-3163	279	7	be	be	AUX
cana-3163	279	8	defined	define	VERB
cana-3163	279	9	as	as	ADP
cana-3163	279	10	3x	3x	NUM
cana-3163	279	11	,	,	PUNCT
cana-3163	279	12	0	0	NUM
cana-3163	279	13	≤	≤	NUM
cana-3163	279	14	x	x	PUNCT
cana-3163	279	15	≤	≤	NOUN
cana-3163	279	16	1/3	1/3	NUM
cana-3163	279	17	-3x	-3x	NOUN
cana-3163	279	18	+	+	CCONJ
cana-3163	279	19	2	2	NUM
cana-3163	279	20	,	,	PUNCT
cana-3163	279	21	1/3	1/3	NUM
cana-3163	279	22	≤	≤	NOUN
cana-3163	279	23	x	x	PUNCT
cana-3163	279	24	<	<	X
cana-3163	279	25	2/3	2/3	NUM
cana-3163	279	26	f	f	X
cana-3163	279	27	(	(	PUNCT
cana-3163	279	28	x	x	NOUN
cana-3163	279	29	)	)	PUNCT
cana-3163	279	30	=	=	SYM
cana-3163	279	31	3x	3x	NUM
cana-3163	279	32	—	—	PUNCT
cana-3163	279	33	2	2	NUM
cana-3163	279	34	,	,	PUNCT
cana-3163	279	35	2/3	2/3	NUM
cana-3163	279	36	<	<	X
cana-3163	279	37	x	x	SYM
cana-3163	279	38	≤	≤	NUM
cana-3163	279	39	1	1	NUM
cana-3163	279	40	f	f	NOUN
cana-3163	279	41	(	(	PUNCT
cana-3163	279	42	x	x	X
cana-3163	279	43	—	—	PUNCT
cana-3163	279	44	1	1	X
cana-3163	279	45	)	)	PUNCT
cana-3163	279	46	+	+	NUM
cana-3163	279	47	1	1	NUM
cana-3163	279	48	,	,	PUNCT
cana-3163	279	49	x	x	X
cana-3163	279	50	≥	≥	NUM
cana-3163	279	51	1	1	NUM
cana-3163	279	52	f	f	X
cana-3163	279	53	(	(	PUNCT
cana-3163	279	54	x	x	PROPN
cana-3163	279	55	+	+	NUM
cana-3163	279	56	1	1	NUM
cana-3163	279	57	)	)	PUNCT
cana-3163	279	58	—	—	PUNCT
cana-3163	279	59	1	1	NUM
cana-3163	279	60	,	,	PUNCT
cana-3163	279	61	x	x	SYM
cana-3163	279	62	≤	≤	ADV
cana-3163	279	63	0	0	NUM
cana-3163	279	64	communications	communication	NOUN
cana-3163	279	65	on	on	ADP
cana-3163	279	66	applied	apply	VERB
cana-3163	279	67	nonlinear	nonlinear	ADJ
cana-3163	279	68	analysis	analysis	NOUN
cana-3163	279	69	issn	issn	NOUN
cana-3163	279	70	:	:	PUNCT
cana-3163	279	71	1074	1074	NUM
cana-3163	279	72	-	-	PUNCT
cana-3163	279	73	133x	133x	NUM
cana-3163	279	74	vol	vol	NOUN
cana-3163	279	75	32	32	NUM
cana-3163	279	76	no	no	NOUN
cana-3163	279	77	.	.	PUNCT
cana-3163	280	1	5s	5s	NUM
cana-3163	280	2	(	(	PUNCT
cana-3163	280	3	2025	2025	NUM
cana-3163	280	4	)	)	PUNCT
cana-3163	280	5	520	520	NUM
cana-3163	280	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	280	7	figure	figure	NOUN
cana-3163	280	8	9	9	NUM
cana-3163	280	9	.	.	PUNCT
cana-3163	280	10	graph	graph	NOUN
cana-3163	280	11	of	of	ADP
cana-3163	280	12	f	f	PROPN
cana-3163	280	13	f|[n	f|[n	NOUN
cana-3163	280	14	,	,	PUNCT
cana-3163	280	15	n+1	n+1	PROPN
cana-3163	280	16	]	]	PUNCT
cana-3163	280	17	has	have	AUX
cana-3163	280	18	one	one	NUM
cana-3163	280	19	fixed	fix	VERB
cana-3163	280	20	point	point	NOUN
cana-3163	280	21	in	in	ADP
cana-3163	280	22	(	(	PUNCT
cana-3163	280	23	n	n	X
cana-3163	280	24	,	,	PUNCT
cana-3163	280	25	n	n	PROPN
cana-3163	280	26	+	+	NOUN
cana-3163	280	27	1	1	NUM
cana-3163	280	28	)	)	PUNCT
cana-3163	280	29	fi|[n	fi|[n	NOUN
cana-3163	280	30	,	,	PUNCT
cana-3163	280	31	n+1	n+1	X
cana-3163	280	32	]	]	PUNCT
cana-3163	280	33	has	have	VERB
cana-3163	280	34	3i	3i	NUM
cana-3163	280	35	—	—	PUNCT
cana-3163	280	36	2	2	NUM
cana-3163	280	37	fixed	fix	VERB
cana-3163	280	38	points	point	NOUN
cana-3163	280	39	in	in	ADP
cana-3163	280	40	(	(	PUNCT
cana-3163	280	41	n	n	X
cana-3163	280	42	,	,	PUNCT
cana-3163	280	43	n	n	PROPN
cana-3163	280	44	+	+	NOUN
cana-3163	280	45	1	1	NUM
cana-3163	280	46	)	)	PUNCT
cana-3163	280	47	.	.	PUNCT
cana-3163	281	1	if	if	SCONJ
cana-3163	281	2	x	x	PRON
cana-3163	281	3	and	and	CCONJ
cana-3163	281	4	y	y	PROPN
cana-3163	281	5	are	be	AUX
cana-3163	281	6	fixed	fix	VERB
cana-3163	281	7	points	point	NOUN
cana-3163	281	8	of	of	ADP
cana-3163	281	9	f	f	PROPN
cana-3163	281	10	i	i	PRON
cana-3163	281	11	in	in	ADP
cana-3163	281	12	(	(	PUNCT
cana-3163	281	13	n	n	X
cana-3163	281	14	,	,	PUNCT
cana-3163	281	15	n	n	PROPN
cana-3163	281	16	+	+	NOUN
cana-3163	281	17	1	1	NUM
cana-3163	281	18	)	)	PUNCT
cana-3163	281	19	,	,	PUNCT
cana-3163	281	20	then	then	ADV
cana-3163	281	21	|fi(x	|fi(x	NUM
cana-3163	281	22	)	)	PUNCT
cana-3163	281	23	—	—	PUNCT
cana-3163	282	1	fi(y)|	fi(y)|	AUX
cana-3163	282	2	<	<	X
cana-3163	282	3	(	(	PUNCT
cana-3163	282	4	1/3)i−1	1/3)i−1	NUM
cana-3163	282	5	.	.	PUNCT
cana-3163	282	6	therefore	therefore	ADV
cana-3163	282	7	per(f	per(f	NOUN
cana-3163	282	8	)	)	PUNCT
cana-3163	282	9	=	=	PUNCT
cana-3163	283	1	r.	r.	PROPN
cana-3163	283	2	hence	hence	ADV
cana-3163	283	3	f	f	PROPN
cana-3163	283	4	∈	∈	PROPN
cana-3163	283	5	pr(r	pr(r	NOUN
cana-3163	283	6	)	)	PUNCT
cana-3163	283	7	.	.	PUNCT
cana-3163	284	1	here	here	ADV
cana-3163	284	2	f	f	X
cana-3163	284	3	is	be	AUX
cana-3163	284	4	not	not	PART
cana-3163	284	5	transitive	transitive	ADJ
cana-3163	284	6	.	.	PUNCT
cana-3163	285	1	note	note	NOUN
cana-3163	285	2	:	:	PUNCT
cana-3163	285	3	let	let	VERB
cana-3163	285	4	c(i	c(i	NOUN
cana-3163	285	5	)	)	PUNCT
cana-3163	285	6	be	be	AUX
cana-3163	285	7	the	the	DET
cana-3163	285	8	space	space	NOUN
cana-3163	285	9	of	of	ADP
cana-3163	285	10	all	all	DET
cana-3163	285	11	continuous	continuous	ADJ
cana-3163	285	12	function	function	NOUN
cana-3163	285	13	defined	define	VERB
cana-3163	285	14	on	on	ADP
cana-3163	285	15	i	i	PRON
cana-3163	285	16	and	and	CCONJ
cana-3163	285	17	let	let	VERB
cana-3163	285	18	(	(	PUNCT
cana-3163	285	19	i	i	NOUN
cana-3163	285	20	)	)	PUNCT
cana-3163	285	21	t	t	PROPN
cana-3163	285	22	(	(	PUNCT
cana-3163	285	23	i	i	NOUN
cana-3163	285	24	)	)	PUNCT
cana-3163	285	25	=	=	PUNCT
cana-3163	286	1	{	{	PUNCT
cana-3163	286	2	f	f	PROPN
cana-3163	286	3	∈	∈	PROPN
cana-3163	286	4	c(i)|f	c(i)|f	PROPN
cana-3163	286	5	is	be	AUX
cana-3163	286	6	transitive	transitive	ADJ
cana-3163	286	7	}	}	PUNCT
cana-3163	286	8	figure	figure	NOUN
cana-3163	286	9	10	10	NUM
cana-3163	286	10	.	.	PUNCT
cana-3163	287	1	the	the	DET
cana-3163	287	2	relationship	relationship	NOUN
cana-3163	287	3	between	between	ADP
cana-3163	287	4	per(i),p	per(i),p	PROPN
cana-3163	287	5	(	(	PUNCT
cana-3163	287	6	i),t	i),t	PROPN
cana-3163	287	7	(	(	PUNCT
cana-3163	287	8	i),and	i),and	NOUN
cana-3163	287	9	pr(i	pr(i	NOUN
cana-3163	287	10	)	)	PUNCT
cana-3163	287	11	.	.	PUNCT
cana-3163	288	1	(	(	PUNCT
cana-3163	288	2	ii	ii	NOUN
cana-3163	288	3	)	)	PUNCT
cana-3163	288	4	per(i	per(i	PROPN
cana-3163	288	5	)	)	PUNCT
cana-3163	288	6	=	=	PRON
cana-3163	289	1	{	{	PUNCT
cana-3163	289	2	f	f	PROPN
cana-3163	289	3	∈	∈	PROPN
cana-3163	289	4	c(i)|per(f	c(i)|per(f	NOUN
cana-3163	289	5	)	)	PUNCT
cana-3163	290	1	=	=	PUNCT
cana-3163	290	2	i	i	PROPN
cana-3163	290	3	}	}	PUNCT
cana-3163	290	4	(	(	PUNCT
cana-3163	290	5	iii	iii	X
cana-3163	290	6	)	)	PUNCT
cana-3163	290	7	p	p	NOUN
cana-3163	290	8	(	(	PUNCT
cana-3163	290	9	i	i	NOUN
cana-3163	290	10	)	)	PUNCT
cana-3163	290	11	=	=	PUNCT
cana-3163	291	1	{	{	PUNCT
cana-3163	291	2	f	f	PROPN
cana-3163	291	3	∈	∈	PROPN
cana-3163	291	4	c(i)|p	c(i)|p	PROPN
cana-3163	291	5	(	(	PUNCT
cana-3163	291	6	f	f	PROPN
cana-3163	291	7	)	)	PUNCT
cana-3163	291	8	=	=	PUNCT
cana-3163	291	9	z+	z+	X
cana-3163	291	10	}	}	PUNCT
cana-3163	291	11	.	.	PUNCT
cana-3163	292	1	then	then	ADV
cana-3163	292	2	we	we	PRON
cana-3163	292	3	have	have	VERB
cana-3163	292	4	,	,	PUNCT
cana-3163	292	5	t	t	PROPN
cana-3163	292	6	(	(	PUNCT
cana-3163	292	7	i	i	NOUN
cana-3163	292	8	)	)	PUNCT
cana-3163	292	9	c	c	PROPN
cana-3163	292	10	per(i	per(i	PROPN
cana-3163	292	11	)	)	PUNCT
cana-3163	292	12	and	and	CCONJ
cana-3163	292	13	pr(i	pr(i	NOUN
cana-3163	292	14	)	)	PUNCT
cana-3163	292	15	=	=	SYM
cana-3163	292	16	per(i	per(i	PROPN
cana-3163	292	17	)	)	PUNCT
cana-3163	292	18	∩	∩	NOUN
cana-3163	293	1	p	p	X
cana-3163	293	2	(	(	PUNCT
cana-3163	293	3	i	i	NOUN
cana-3163	293	4	)	)	PUNCT
cana-3163	293	5	.	.	PUNCT
cana-3163	294	1	also	also	ADV
cana-3163	294	2	t	t	PROPN
cana-3163	294	3	(	(	PUNCT
cana-3163	294	4	i	i	NOUN
cana-3163	294	5	)	)	PUNCT
cana-3163	294	6	∩	∩	PROPN
cana-3163	294	7	p	p	X
cana-3163	294	8	(	(	PUNCT
cana-3163	294	9	i	i	NOUN
cana-3163	294	10	)	)	PUNCT
cana-3163	294	11	/=	/=	NOUN
cana-3163	294	12	∅	∅	NOUN
cana-3163	294	13	(	(	PUNCT
cana-3163	294	14	see	see	VERB
cana-3163	294	15	figure	figure	NOUN
cana-3163	294	16	10	10	NUM
cana-3163	294	17	)	)	PUNCT
cana-3163	294	18	.	.	PUNCT
cana-3163	295	1	communications	communication	NOUN
cana-3163	295	2	on	on	ADP
cana-3163	295	3	applied	apply	VERB
cana-3163	295	4	nonlinear	nonlinear	ADJ
cana-3163	295	5	analysis	analysis	NOUN
cana-3163	295	6	issn	issn	NOUN
cana-3163	295	7	:	:	PUNCT
cana-3163	295	8	1074	1074	NUM
cana-3163	295	9	-	-	PUNCT
cana-3163	295	10	133x	133x	NUM
cana-3163	295	11	vol	vol	NOUN
cana-3163	295	12	32	32	NUM
cana-3163	295	13	no	no	NOUN
cana-3163	295	14	.	.	PUNCT
cana-3163	296	1	5s	5s	NUM
cana-3163	296	2	(	(	PUNCT
cana-3163	296	3	2025	2025	NUM
cana-3163	296	4	)	)	PUNCT
cana-3163	296	5	521	521	NUM
cana-3163	296	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	296	7	2	2	NUM
cana-3163	296	8	4	4	NUM
cana-3163	296	9	2θ	2θ	NUM
cana-3163	296	10	it	it	PRON
cana-3163	296	11	is	be	AUX
cana-3163	296	12	proved	prove	VERB
cana-3163	296	13	that	that	SCONJ
cana-3163	296	14	t	t	PROPN
cana-3163	296	15	(	(	PUNCT
cana-3163	296	16	i	i	NOUN
cana-3163	296	17	)	)	PUNCT
cana-3163	296	18	∼=	∼=	PROPN
cana-3163	297	1	t	t	NOUN
cana-3163	297	2	(	(	PUNCT
cana-3163	297	3	i	i	NOUN
cana-3163	297	4	)	)	PUNCT
cana-3163	297	5	∼=	∼=	VERB
cana-3163	297	6	l2	l2	NOUN
cana-3163	297	7	[	[	X
cana-3163	297	8	7	7	NUM
cana-3163	297	9	]	]	PUNCT
cana-3163	297	10	.	.	PUNCT
cana-3163	298	1	now	now	ADV
cana-3163	298	2	,	,	PUNCT
cana-3163	298	3	we	we	PRON
cana-3163	298	4	shall	shall	AUX
cana-3163	298	5	prove	prove	VERB
cana-3163	298	6	the	the	DET
cana-3163	298	7	following	follow	VERB
cana-3163	298	8	proposition	proposition	NOUN
cana-3163	298	9	.	.	PUNCT
cana-3163	299	1	proposition	proposition	NOUN
cana-3163	299	2	6	6	NUM
cana-3163	299	3	.	.	PUNCT
cana-3163	300	1	p	p	X
cana-3163	300	2	(	(	PUNCT
cana-3163	300	3	i	i	NOUN
cana-3163	300	4	)	)	PUNCT
cana-3163	300	5	=	=	PUNCT
cana-3163	300	6	c(i	c(i	PROPN
cana-3163	300	7	)	)	PUNCT
cana-3163	300	8	.	.	PUNCT
cana-3163	301	1	proof	proof	NOUN
cana-3163	301	2	.	.	PUNCT
cana-3163	302	1	let	let	VERB
cana-3163	302	2	c	c	PRON
cana-3163	302	3	be	be	AUX
cana-3163	302	4	a	a	DET
cana-3163	302	5	fixed	fix	VERB
cana-3163	302	6	point	point	NOUN
cana-3163	302	7	of	of	ADP
cana-3163	302	8	f	f	PROPN
cana-3163	302	9	,	,	PUNCT
cana-3163	303	1	i.e.	i.e.	X
cana-3163	303	2	,	,	PUNCT
cana-3163	303	3	f	f	PROPN
cana-3163	303	4	(	(	PUNCT
cana-3163	303	5	c	c	NOUN
cana-3163	303	6	)	)	PUNCT
cana-3163	303	7	=	=	SYM
cana-3163	303	8	c.	c.	NOUN
cana-3163	303	9	given	give	VERB
cana-3163	303	10	an	an	DET
cana-3163	303	11	ϵ	ϵ	X
cana-3163	303	12	>	>	X
cana-3163	303	13	0	0	NUM
cana-3163	303	14	,	,	PUNCT
cana-3163	303	15	we	we	PRON
cana-3163	303	16	will	will	AUX
cana-3163	303	17	construct	construct	VERB
cana-3163	303	18	a	a	DET
cana-3163	303	19	g	g	PROPN
cana-3163	303	20	∈	∈	PROPN
cana-3163	303	21	p	p	X
cana-3163	303	22	(	(	PUNCT
cana-3163	303	23	i	i	NOUN
cana-3163	303	24	)	)	PUNCT
cana-3163	304	1	such	such	ADJ
cana-3163	304	2	that	that	SCONJ
cana-3163	304	3	p	p	X
cana-3163	304	4	(	(	PUNCT
cana-3163	304	5	g	g	NOUN
cana-3163	304	6	)	)	PUNCT
cana-3163	304	7	=	=	SYM
cana-3163	304	8	z+	z+	X
cana-3163	304	9	.	.	PUNCT
cana-3163	304	10	case	case	NOUN
cana-3163	304	11	1	1	NUM
cana-3163	304	12	:	:	PUNCT
cana-3163	304	13	c	c	NOUN
cana-3163	304	14	/=	/=	NOUN
cana-3163	305	1	1	1	X
cana-3163	305	2	.	.	PUNCT
cana-3163	305	3	since	since	SCONJ
cana-3163	305	4	f	f	PROPN
cana-3163	305	5	is	be	AUX
cana-3163	305	6	continuous	continuous	ADJ
cana-3163	305	7	at	at	ADP
cana-3163	305	8	c	c	PROPN
cana-3163	305	9	,	,	PUNCT
cana-3163	305	10	there	there	PRON
cana-3163	305	11	exists	exist	VERB
cana-3163	305	12	a	a	DET
cana-3163	305	13	δ	δ	PROPN
cana-3163	305	14	>	>	X
cana-3163	305	15	0	0	PUNCT
cana-3163	306	1	(	(	PUNCT
cana-3163	306	2	choose	choose	VERB
cana-3163	306	3	δ	δ	PROPN
cana-3163	306	4	<	<	X
cana-3163	306	5	є	є	PROPN
cana-3163	306	6	)	)	PUNCT
cana-3163	306	7	such	such	ADJ
cana-3163	306	8	that	that	SCONJ
cana-3163	306	9	|f	|f	PROPN
cana-3163	306	10	(	(	PUNCT
cana-3163	306	11	x	x	NOUN
cana-3163	306	12	)	)	PUNCT
cana-3163	306	13	—	—	PUNCT
cana-3163	306	14	c|	c|	VERB
cana-3163	306	15	<	<	X
cana-3163	306	16	ϵ	ϵ	X
cana-3163	306	17	,	,	PUNCT
cana-3163	306	18	for	for	ADP
cana-3163	306	19	all	all	DET
cana-3163	306	20	x	x	SYM
cana-3163	306	21	∈	∈	PROPN
cana-3163	307	1	[	[	X
cana-3163	307	2	c	c	NOUN
cana-3163	307	3	,	,	PUNCT
cana-3163	307	4	c	c	PROPN
cana-3163	307	5	+	+	CCONJ
cana-3163	307	6	δ	δ	X
cana-3163	307	7	]	]	X
cana-3163	307	8	.	.	PUNCT
cana-3163	308	1	2	2	NUM
cana-3163	308	2	let	let	VERB
cana-3163	308	3	θ	θ	NOUN
cana-3163	308	4	=	=	SYM
cana-3163	308	5	δ	δ	PROPN
cana-3163	308	6	,	,	PUNCT
cana-3163	308	7	a	a	DET
cana-3163	308	8	be	be	NOUN
cana-3163	308	9	[	[	X
cana-3163	308	10	c	c	NOUN
cana-3163	308	11	,	,	PUNCT
cana-3163	308	12	c	c	NOUN
cana-3163	308	13	+	+	CCONJ
cana-3163	308	14	2θ	2θ	NUM
cana-3163	308	15	]	]	PUNCT
cana-3163	308	16	and	and	CCONJ
cana-3163	308	17	b	b	X
cana-3163	308	18	be	be	VERB
cana-3163	308	19	[	[	X
cana-3163	308	20	c	c	NOUN
cana-3163	308	21	+	+	NUM
cana-3163	308	22	2θ	2θ	NUM
cana-3163	308	23	,	,	PUNCT
cana-3163	308	24	c	c	NOUN
cana-3163	308	25	+	+	X
cana-3163	308	26	3θ	3θ	NUM
cana-3163	308	27	]	]	PUNCT
cana-3163	308	28	.	.	PUNCT
cana-3163	309	1	define	define	VERB
cana-3163	309	2	g1	g1	PROPN
cana-3163	309	3	:	:	PUNCT
cana-3163	309	4	a	a	PRON
cana-3163	309	5	→	→	SYM
cana-3163	309	6	a	a	PRON
cana-3163	309	7	as	as	ADP
cana-3163	309	8	figure	figure	NOUN
cana-3163	309	9	11	11	NUM
cana-3163	309	10	.	.	PUNCT
cana-3163	310	1	graph	graph	NOUN
cana-3163	310	2	of	of	ADP
cana-3163	310	3	g1	g1	PROPN
cana-3163	310	4	g1	g1	PROPN
cana-3163	310	5	(	(	PUNCT
cana-3163	310	6	x	x	X
cana-3163	310	7	)	)	PUNCT
cana-3163	310	8	=	=	SYM
cana-3163	310	9	2x	2x	NUM
cana-3163	310	10	-	-	SYM
cana-3163	310	11	c	c	NOUN
cana-3163	310	12	,	,	PUNCT
cana-3163	310	13	x	x	SYM
cana-3163	310	14	∈	∈	PROPN
cana-3163	311	1	[	[	X
cana-3163	311	2	c	c	X
cana-3163	311	3	,	,	PUNCT
cana-3163	311	4	c	c	NOUN
cana-3163	311	5	+	+	CCONJ
cana-3163	311	6	θ	θ	PROPN
cana-3163	311	7	]	]	X
cana-3163	311	8	-2x	-2x	PROPN
cana-3163	312	1	+	+	CCONJ
cana-3163	312	2	3c	3c	NUM
cana-3163	312	3	+	+	NUM
cana-3163	312	4	4θ	4θ	NOUN
cana-3163	312	5	.	.	PUNCT
cana-3163	313	1	x	x	X
cana-3163	313	2	∈	∈	PROPN
cana-3163	314	1	[	[	X
cana-3163	314	2	c	c	X
cana-3163	314	3	+	+	NUM
cana-3163	314	4	θ	θ	PROPN
cana-3163	314	5	,	,	PUNCT
cana-3163	314	6	c	c	NOUN
cana-3163	314	7	+	+	CCONJ
cana-3163	314	8	2θ	2θ	NUM
cana-3163	314	9	]	]	PUNCT
cana-3163	314	10	.	.	PUNCT
cana-3163	315	1	(	(	PUNCT
cana-3163	315	2	see	see	VERB
cana-3163	315	3	figure	figure	NOUN
cana-3163	315	4	11	11	NUM
cana-3163	315	5	)	)	PUNCT
cana-3163	315	6	g1	g1	NOUN
cana-3163	315	7	on	on	ADP
cana-3163	315	8	[	[	X
cana-3163	315	9	c	c	X
cana-3163	315	10	,	,	PUNCT
cana-3163	315	11	c	c	NOUN
cana-3163	315	12	+	+	CCONJ
cana-3163	315	13	2θ	2θ	NUM
cana-3163	315	14	]	]	PUNCT
cana-3163	315	15	is	be	AUX
cana-3163	315	16	topologically	topologically	ADV
cana-3163	315	17	conjugate	conjugate	ADJ
cana-3163	315	18	to	to	ADP
cana-3163	315	19	the	the	DET
cana-3163	315	20	tent	tent	NOUN
cana-3163	315	21	map	map	NOUN
cana-3163	315	22	on	on	ADP
cana-3163	315	23	[	[	X
cana-3163	315	24	0	0	NUM
cana-3163	315	25	,	,	PUNCT
cana-3163	315	26	1	1	NUM
cana-3163	315	27	]	]	PUNCT
cana-3163	315	28	defined	define	VERB
cana-3163	315	29	by	by	ADP
cana-3163	315	30	t	t	PROPN
cana-3163	315	31	(	(	PUNCT
cana-3163	315	32	x	x	NOUN
cana-3163	315	33	)	)	PUNCT
cana-3163	315	34	=	=	SYM
cana-3163	315	35	2x	2x	NUM
cana-3163	315	36	,	,	PUNCT
cana-3163	315	37	≤	≤	NUM
cana-3163	315	38	x	x	SYM
cana-3163	315	39	≤	≤	NUM
cana-3163	315	40	1/2	1/2	NUM
cana-3163	315	41	2(1	2(1	NUM
cana-3163	315	42	—	—	PUNCT
cana-3163	315	43	x	x	X
cana-3163	315	44	)	)	PUNCT
cana-3163	315	45	,	,	PUNCT
cana-3163	315	46	1/	1/	NUM
cana-3163	315	47	2	2	NUM
cana-3163	315	48	≤	≤	NUM
cana-3163	315	49	x	x	PUNCT
cana-3163	315	50	≤	≤	NUM
cana-3163	315	51	1	1	NUM
cana-3163	315	52	via	via	ADP
cana-3163	315	53	the	the	DET
cana-3163	315	54	homeomorphism	homeomorphism	PROPN
cana-3163	315	55	h	h	NOUN
cana-3163	315	56	:	:	PUNCT
cana-3163	316	1	[	[	X
cana-3163	316	2	c	c	X
cana-3163	316	3	,	,	PUNCT
cana-3163	316	4	c	c	NOUN
cana-3163	316	5	+	+	CCONJ
cana-3163	316	6	2θ	2θ	NUM
cana-3163	316	7	]	]	PUNCT
cana-3163	316	8	→	→	PUNCT
cana-3163	317	1	[	[	X
cana-3163	317	2	0	0	NUM
cana-3163	317	3	,	,	PUNCT
cana-3163	317	4	1	1	NUM
cana-3163	317	5	]	]	PUNCT
cana-3163	317	6	defined	define	VERB
cana-3163	317	7	as	as	ADP
cana-3163	317	8	h(x	h(x	PROPN
cana-3163	317	9	)	)	PUNCT
cana-3163	317	10	=	=	PUNCT
cana-3163	318	1	x−c	x−c	PROPN
cana-3163	318	2	.	.	PUNCT
cana-3163	319	1	therefore	therefore	ADV
cana-3163	319	2	per(g1	per(g1	NOUN
cana-3163	319	3	)	)	PUNCT
cana-3163	319	4	=	=	PUNCT
cana-3163	320	1	[	[	X
cana-3163	320	2	c	c	X
cana-3163	320	3	,	,	PUNCT
cana-3163	320	4	c	c	NOUN
cana-3163	320	5	+	+	CCONJ
cana-3163	320	6	2θ	2θ	NUM
cana-3163	320	7	]	]	PUNCT
cana-3163	320	8	.	.	PUNCT
cana-3163	321	1	communications	communication	NOUN
cana-3163	321	2	on	on	ADP
cana-3163	321	3	applied	apply	VERB
cana-3163	321	4	nonlinear	nonlinear	ADJ
cana-3163	321	5	analysis	analysis	NOUN
cana-3163	321	6	issn	issn	NOUN
cana-3163	321	7	:	:	PUNCT
cana-3163	321	8	1074	1074	NUM
cana-3163	321	9	-	-	PUNCT
cana-3163	321	10	133x	133x	NUM
cana-3163	321	11	vol	vol	NOUN
cana-3163	321	12	32	32	NUM
cana-3163	321	13	no	no	NOUN
cana-3163	321	14	.	.	PUNCT
cana-3163	322	1	5s	5s	NUM
cana-3163	322	2	(	(	PUNCT
cana-3163	322	3	2025	2025	NUM
cana-3163	322	4	)	)	PUNCT
cana-3163	322	5	522	522	NUM
cana-3163	322	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	322	7	2	2	NUM
cana-3163	322	8	θ	θ	NOUN
cana-3163	322	9	7	7	NUM
cana-3163	322	10	x∈a	x∈a	ADJ
cana-3163	322	11	x∈a	x∈a	NOUN
cana-3163	322	12	let	let	VERB
cana-3163	322	13	g2	g2	PROPN
cana-3163	322	14	:	:	PUNCT
cana-3163	323	1	b	b	X
cana-3163	323	2	→	→	SYM
cana-3163	323	3	[	[	X
cana-3163	323	4	0	0	NUM
cana-3163	323	5	,	,	PUNCT
cana-3163	323	6	1	1	NUM
cana-3163	323	7	]	]	PUNCT
cana-3163	323	8	be	be	AUX
cana-3163	323	9	defined	define	VERB
cana-3163	323	10	as	as	ADP
cana-3163	323	11	g	g	PROPN
cana-3163	323	12	(	(	PUNCT
cana-3163	323	13	x	x	NOUN
cana-3163	323	14	)	)	PUNCT
cana-3163	323	15	=	=	SYM
cana-3163	324	1	c	c	NOUN
cana-3163	324	2	+	+	CCONJ
cana-3163	324	3	(	(	PUNCT
cana-3163	324	4	x	x	X
cana-3163	324	5	—	—	PUNCT
cana-3163	324	6	c	c	NOUN
cana-3163	324	7	—	—	PUNCT
cana-3163	324	8	2θ	2θ	NUM
cana-3163	324	9	)	)	PUNCT
cana-3163	324	10	f	f	NOUN
cana-3163	324	11	(	(	PUNCT
cana-3163	324	12	c	c	NOUN
cana-3163	324	13	+	+	X
cana-3163	324	14	3θ	3θ	NUM
cana-3163	324	15	)	)	PUNCT
cana-3163	324	16	—	—	PUNCT
cana-3163	325	1	c	c	X
cana-3163	325	2	.	.	PUNCT
cana-3163	326	1	i.e.	i.e.	X
cana-3163	326	2	,	,	PUNCT
cana-3163	326	3	the	the	DET
cana-3163	326	4	graph	graph	NOUN
cana-3163	326	5	of	of	ADP
cana-3163	326	6	g2	g2	PROPN
cana-3163	326	7	is	be	AUX
cana-3163	326	8	a	a	DET
cana-3163	326	9	line	line	NOUN
cana-3163	326	10	segment	segment	NOUN
cana-3163	326	11	joining	join	VERB
cana-3163	326	12	(	(	PUNCT
cana-3163	326	13	c	c	NOUN
cana-3163	326	14	+	+	NOUN
cana-3163	326	15	2θ	2θ	NUM
cana-3163	326	16	,	,	PUNCT
cana-3163	326	17	c	c	NOUN
cana-3163	326	18	)	)	PUNCT
cana-3163	326	19	and	and	CCONJ
cana-3163	326	20	(	(	PUNCT
cana-3163	326	21	c	c	PROPN
cana-3163	326	22	+	+	SYM
cana-3163	326	23	3θ	3θ	NUM
cana-3163	326	24	,	,	PUNCT
cana-3163	326	25	f	f	PROPN
cana-3163	326	26	(	(	PUNCT
cana-3163	326	27	c	c	PROPN
cana-3163	326	28	+	+	X
cana-3163	326	29	3θ	3θ	NUM
cana-3163	326	30	)	)	PUNCT
cana-3163	326	31	)	)	PUNCT
cana-3163	327	1	(	(	PUNCT
cana-3163	327	2	see	see	VERB
cana-3163	327	3	figure	figure	NOUN
cana-3163	327	4	12	12	NUM
cana-3163	327	5	)	)	PUNCT
cana-3163	327	6	.	.	PUNCT
cana-3163	328	1	figure	figure	NOUN
cana-3163	328	2	12	12	NUM
cana-3163	328	3	.	.	PUNCT
cana-3163	329	1	graph	graph	NOUN
cana-3163	329	2	of	of	ADP
cana-3163	329	3	g2	g2	PROPN
cana-3163	329	4	figure	figure	NOUN
cana-3163	329	5	13	13	NUM
cana-3163	329	6	.	.	PUNCT
cana-3163	330	1	graph	graph	NOUN
cana-3163	330	2	of	of	ADP
cana-3163	330	3	g	g	NOUN
cana-3163	330	4	at	at	ADP
cana-3163	330	5	c	c	PROPN
cana-3163	330	6	/=	/=	PROPN
cana-3163	330	7	1	1	NUM
cana-3163	330	8	now	now	ADV
cana-3163	330	9	,	,	PUNCT
cana-3163	330	10	define	define	VERB
cana-3163	330	11	g	g	NOUN
cana-3163	330	12	:	:	PUNCT
cana-3163	331	1	[	[	X
cana-3163	331	2	0	0	NUM
cana-3163	331	3	,	,	PUNCT
cana-3163	331	4	1	1	NUM
cana-3163	331	5	]	]	PUNCT
cana-3163	331	6	→	→	PUNCT
cana-3163	331	7	[	[	X
cana-3163	331	8	0	0	NUM
cana-3163	331	9	,	,	PUNCT
cana-3163	331	10	1	1	NUM
cana-3163	331	11	]	]	PUNCT
cana-3163	331	12	as	as	ADP
cana-3163	331	13	f	f	PROPN
cana-3163	331	14	(	(	PUNCT
cana-3163	331	15	x	x	NOUN
cana-3163	331	16	)	)	PUNCT
cana-3163	331	17	,	,	PUNCT
cana-3163	331	18	x	x	PUNCT
cana-3163	331	19	∈	∈	PROPN
cana-3163	332	1	[	[	X
cana-3163	332	2	0	0	NUM
cana-3163	332	3	,	,	PUNCT
cana-3163	332	4	1]\(a	1]\(a	NUM
cana-3163	332	5	∪	∪	PROPN
cana-3163	332	6	b	b	NOUN
cana-3163	332	7	)	)	PUNCT
cana-3163	332	8	g(x	g(x	NOUN
cana-3163	332	9	)	)	PUNCT
cana-3163	332	10	=	=	SYM
cana-3163	332	11	g1(x	g1(x	NOUN
cana-3163	332	12	)	)	PUNCT
cana-3163	332	13	,	,	PUNCT
cana-3163	332	14	x	x	PUNCT
cana-3163	332	15	∈	∈	VERB
cana-3163	332	16	a	a	DET
cana-3163	332	17	g2(x	g2(x	NOUN
cana-3163	332	18	)	)	PUNCT
cana-3163	332	19	,	,	PUNCT
cana-3163	332	20	x	x	PUNCT
cana-3163	332	21	∈	∈	PROPN
cana-3163	332	22	b	b	X
cana-3163	332	23	(	(	PUNCT
cana-3163	332	24	see	see	VERB
cana-3163	332	25	figure	figure	NOUN
cana-3163	332	26	13	13	NUM
cana-3163	332	27	)	)	PUNCT
cana-3163	332	28	.	.	PUNCT
cana-3163	333	1	0	0	NUM
cana-3163	333	2	,	,	PUNCT
cana-3163	333	3	x	x	X
cana-3163	333	4	∈	∈	PROPN
cana-3163	334	1	[	[	X
cana-3163	334	2	0	0	NUM
cana-3163	334	3	,	,	PUNCT
cana-3163	334	4	1]\(a	1]\(a	NUM
cana-3163	334	5	∪	∪	PROPN
cana-3163	334	6	b	b	NOUN
cana-3163	334	7	)	)	PUNCT
cana-3163	334	8	|f	|f	PROPN
cana-3163	334	9	(	(	PUNCT
cana-3163	334	10	x	x	NOUN
cana-3163	334	11	)	)	PUNCT
cana-3163	334	12	—	—	PUNCT
cana-3163	334	13	g(x)|	g(x)|	ADP
cana-3163	334	14	=	=	SYM
cana-3163	334	15	f	f	X
cana-3163	334	16	(	(	PUNCT
cana-3163	334	17	x	x	NOUN
cana-3163	334	18	)	)	PUNCT
cana-3163	334	19	—	—	PUNCT
cana-3163	335	1	g1(x)|	g1(x)|	ADJ
cana-3163	335	2	,	,	PUNCT
cana-3163	335	3	x	x	SYM
cana-3163	335	4	∈	∈	PROPN
cana-3163	335	5	a	a	DET
cana-3163	335	6	(	(	PUNCT
cana-3163	335	7	x	x	NOUN
cana-3163	335	8	)	)	PUNCT
cana-3163	335	9	—	—	PUNCT
cana-3163	335	10	g2(x)|	g2(x)|	NOUN
cana-3163	335	11	,	,	PUNCT
cana-3163	335	12	x	x	PUNCT
cana-3163	335	13	∈	∈	NOUN
cana-3163	335	14	b	b	NOUN
cana-3163	335	15	hence	hence	ADV
cana-3163	335	16	f	f	PROPN
cana-3163	335	17	—	—	PUNCT
cana-3163	335	18	g	g	NOUN
cana-3163	335	19	=	=	PUNCT
cana-3163	335	20	max{max	max{max	PROPN
cana-3163	335	21	|f	|f	PROPN
cana-3163	335	22	(	(	PUNCT
cana-3163	335	23	x	x	NOUN
cana-3163	335	24	)	)	PUNCT
cana-3163	335	25	—	—	PUNCT
cana-3163	336	1	g1(x)|	g1(x)|	PROPN
cana-3163	336	2	,	,	PUNCT
cana-3163	336	3	max	max	PROPN
cana-3163	336	4	|f	|f	PROPN
cana-3163	336	5	(	(	PUNCT
cana-3163	336	6	x	x	NOUN
cana-3163	336	7	)	)	PUNCT
cana-3163	336	8	—	—	PUNCT
cana-3163	336	9	g2(x)|	g2(x)|	PROPN
cana-3163	336	10	}	}	PUNCT
cana-3163	336	11	x∈a	x∈a	VERB
cana-3163	336	12	x∈b	x∈b	NOUN
cana-3163	336	13	≤	≤	NUM
cana-3163	336	14	max	max	PROPN
cana-3163	336	15	max	max	PROPN
cana-3163	336	16	|f	|f	PROPN
cana-3163	336	17	(	(	PUNCT
cana-3163	336	18	x	x	NOUN
cana-3163	336	19	)	)	PUNCT
cana-3163	336	20	—	—	PUNCT
cana-3163	337	1	c|	c|	PROPN
cana-3163	337	2	+	+	CCONJ
cana-3163	337	3	max	max	PROPN
cana-3163	337	4	|c	|c	ADJ
cana-3163	337	5	—	—	PUNCT
cana-3163	337	6	g1(x)|	g1(x)|	PROPN
cana-3163	337	7	,	,	PUNCT
cana-3163	337	8	max	max	PROPN
cana-3163	337	9	|f	|f	PROPN
cana-3163	337	10	(	(	PUNCT
cana-3163	337	11	x	x	NOUN
cana-3163	337	12	)	)	PUNCT
cana-3163	337	13	—	—	PUNCT
cana-3163	338	1	c|	c|	PROPN
cana-3163	338	2	+	+	CCONJ
cana-3163	338	3	max	max	PROPN
cana-3163	338	4	|c	|c	PROPN
cana-3163	338	5	—	—	PUNCT
cana-3163	338	6	g2(x)|	g2(x)|	PROPN
cana-3163	338	7	let	let	VERB
cana-3163	338	8	x0	x0	PROPN
cana-3163	338	9	=	=	PUNCT
cana-3163	339	1	c	c	PROPN
cana-3163	339	2	+	+	CCONJ
cana-3163	339	3	4	4	NUM
cana-3163	339	4	θ	θ	NOUN
cana-3163	339	5	.	.	PUNCT
cana-3163	340	1	x∈b	x∈b	PROPN
cana-3163	340	2	<	<	X
cana-3163	340	3	max	max	PROPN
cana-3163	340	4	{	{	PUNCT
cana-3163	340	5	ϵ	ϵ	X
cana-3163	340	6	+	+	CCONJ
cana-3163	340	7	ϵ	ϵ	X
cana-3163	340	8	,	,	PUNCT
cana-3163	340	9	,	,	PUNCT
cana-3163	340	10	ϵ	ϵ	X
cana-3163	340	11	+	+	NOUN
cana-3163	340	12	2	2	NUM
cana-3163	340	13	4	4	NUM
cana-3163	340	14	2	2	NUM
cana-3163	340	15	x∈b	x∈b	NOUN
cana-3163	340	16	ϵ	ϵ	NOUN
cana-3163	340	17	}	}	PUNCT
cana-3163	340	18	=	=	SYM
cana-3163	340	19	max	max	PROPN
cana-3163	340	20	{	{	PUNCT
cana-3163	340	21	3ϵ	3ϵ	NOUN
cana-3163	340	22	2	2	NUM
cana-3163	340	23	4	4	NUM
cana-3163	340	24	,	,	PUNCT
cana-3163	340	25	ϵ	ϵ	NOUN
cana-3163	340	26	}	}	PUNCT
cana-3163	340	27	=	=	PUNCT
cana-3163	340	28	ϵ.	ϵ.	NOUN
cana-3163	340	29	communications	communication	NOUN
cana-3163	340	30	on	on	ADP
cana-3163	340	31	applied	apply	VERB
cana-3163	340	32	nonlinear	nonlinear	ADJ
cana-3163	340	33	analysis	analysis	NOUN
cana-3163	340	34	issn	issn	NOUN
cana-3163	340	35	:	:	PUNCT
cana-3163	340	36	1074	1074	NUM
cana-3163	340	37	-	-	PUNCT
cana-3163	340	38	133x	133x	NUM
cana-3163	340	39	vol	vol	NOUN
cana-3163	340	40	32	32	NUM
cana-3163	340	41	no	no	NOUN
cana-3163	340	42	.	.	PUNCT
cana-3163	341	1	5s	5s	NUM
cana-3163	341	2	(	(	PUNCT
cana-3163	341	3	2025	2025	NUM
cana-3163	341	4	)	)	PUNCT
cana-3163	341	5	523	523	NUM
cana-3163	341	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	341	7	therefore	therefore	ADV
cana-3163	341	8	x0	x0	PROPN
cana-3163	341	9	∈	∈	PROPN
cana-3163	341	10	per3(g	per3(g	NUM
cana-3163	341	11	)	)	PUNCT
cana-3163	341	12	.	.	PUNCT
cana-3163	342	1	hence	hence	ADV
cana-3163	342	2	p	p	X
cana-3163	342	3	(	(	PUNCT
cana-3163	342	4	g	g	NOUN
cana-3163	342	5	)	)	PUNCT
cana-3163	342	6	=	=	SYM
cana-3163	342	7	z+	z+	X
cana-3163	342	8	.	.	PUNCT
cana-3163	342	9	case	case	NOUN
cana-3163	342	10	2	2	NUM
cana-3163	342	11	:	:	PUNCT
cana-3163	342	12	if	if	SCONJ
cana-3163	342	13	c	c	PROPN
cana-3163	342	14	=	=	NOUN
cana-3163	342	15	1	1	X
cana-3163	342	16	.	.	PUNCT
cana-3163	342	17	let	let	VERB
cana-3163	342	18	a	a	DET
cana-3163	342	19	be	be	AUX
cana-3163	342	20	[	[	X
cana-3163	342	21	1	1	NUM
cana-3163	342	22	—	—	PUNCT
cana-3163	342	23	2θ	2θ	NUM
cana-3163	342	24	,	,	PUNCT
cana-3163	342	25	1	1	NUM
cana-3163	342	26	]	]	PUNCT
cana-3163	342	27	and	and	CCONJ
cana-3163	342	28	b	b	AUX
cana-3163	342	29	be	be	AUX
cana-3163	342	30	[	[	X
cana-3163	342	31	1	1	NUM
cana-3163	342	32	—	—	PUNCT
cana-3163	342	33	3θ	3θ	NUM
cana-3163	342	34	,	,	PUNCT
cana-3163	342	35	1	1	NUM
cana-3163	342	36	—	—	PUNCT
cana-3163	342	37	2θ	2θ	NUM
cana-3163	342	38	]	]	PUNCT
cana-3163	342	39	.	.	PUNCT
cana-3163	343	1	let	let	VERB
cana-3163	343	2	g1	g1	PROPN
cana-3163	343	3	:	:	PUNCT
cana-3163	343	4	a	a	DET
cana-3163	343	5	→	→	X
cana-3163	343	6	a	a	PRON
cana-3163	343	7	be	be	AUX
cana-3163	343	8	defined	define	VERB
cana-3163	343	9	as	as	ADP
cana-3163	343	10	g	g	PROPN
cana-3163	343	11	(	(	PUNCT
cana-3163	343	12	x	x	NOUN
cana-3163	343	13	)	)	PUNCT
cana-3163	343	14	=	=	SYM
cana-3163	343	15	2x-1	2x-1	NUM
cana-3163	343	16	,	,	PUNCT
cana-3163	344	1	x	x	PUNCT
cana-3163	344	2	∈	∈	PROPN
cana-3163	345	1	[	[	X
cana-3163	345	2	1	1	NUM
cana-3163	345	3	—	—	PUNCT
cana-3163	345	4	θ	θ	NOUN
cana-3163	345	5	,	,	PUNCT
cana-3163	345	6	1	1	NUM
cana-3163	345	7	]	]	SYM
cana-3163	345	8	3	3	NUM
cana-3163	345	9	—	—	PUNCT
cana-3163	345	10	2x	2x	NUM
cana-3163	345	11	—	—	PUNCT
cana-3163	345	12	4θ	4θ	NOUN
cana-3163	345	13	,	,	PUNCT
cana-3163	345	14	x	x	PUNCT
cana-3163	345	15	∈	∈	PROPN
cana-3163	346	1	[	[	X
cana-3163	346	2	1	1	NUM
cana-3163	346	3	—	—	PUNCT
cana-3163	346	4	2θ	2θ	NUM
cana-3163	346	5	,	,	PUNCT
cana-3163	346	6	1	1	NUM
cana-3163	346	7	—	—	SYM
cana-3163	346	8	θ	θ	NOUN
cana-3163	346	9	]	]	X
cana-3163	346	10	let	let	VERB
cana-3163	346	11	g2	g2	PROPN
cana-3163	346	12	:	:	PUNCT
cana-3163	346	13	b	b	X
cana-3163	346	14	→	→	SYM
cana-3163	346	15	[	[	X
cana-3163	346	16	0	0	NUM
cana-3163	346	17	,	,	PUNCT
cana-3163	346	18	1	1	NUM
cana-3163	346	19	]	]	PUNCT
cana-3163	346	20	be	be	AUX
cana-3163	346	21	defined	define	VERB
cana-3163	346	22	as	as	ADP
cana-3163	346	23	𝑔2(𝑥	𝑔2(𝑥	X
cana-3163	346	24	)	)	PUNCT
cana-3163	346	25	=	=	SYM
cana-3163	347	1	𝑓	𝑓	PROPN
cana-3163	347	2	(	(	PUNCT
cana-3163	347	3	1	1	NUM
cana-3163	347	4	—	—	PUNCT
cana-3163	347	5	3𝜃	3𝜃	NUM
cana-3163	347	6	)	)	PUNCT
cana-3163	348	1	+	+	CCONJ
cana-3163	348	2	(	(	PUNCT
cana-3163	348	3	𝑥	𝑥	X
cana-3163	348	4	—	—	PUNCT
cana-3163	348	5	1	1	NUM
cana-3163	348	6	+	+	SYM
cana-3163	348	7	3𝜃)(1	3𝜃)(1	NUM
cana-3163	348	8	—	—	PUNCT
cana-3163	348	9	𝑓	𝑓	X
cana-3163	348	10	(	(	PUNCT
cana-3163	348	11	1	1	NUM
cana-3163	348	12	—	—	PUNCT
cana-3163	348	13	3𝜃	3𝜃	NUM
cana-3163	348	14	)	)	PUNCT
cana-3163	348	15	)	)	PUNCT
cana-3163	349	1	𝜃	𝜃	PRON
cana-3163	349	2	let	let	VERB
cana-3163	349	3	g	g	NOUN
cana-3163	349	4	:	:	PUNCT
cana-3163	350	1	[	[	X
cana-3163	350	2	0	0	NUM
cana-3163	350	3	,	,	PUNCT
cana-3163	350	4	1	1	NUM
cana-3163	350	5	]	]	PUNCT
cana-3163	350	6	→	→	PUNCT
cana-3163	350	7	[	[	X
cana-3163	350	8	0	0	NUM
cana-3163	350	9	,	,	PUNCT
cana-3163	350	10	1	1	NUM
cana-3163	350	11	]	]	PUNCT
cana-3163	350	12	be	be	AUX
cana-3163	350	13	such	such	ADJ
cana-3163	350	14	that	that	SCONJ
cana-3163	350	15	f	f	PROPN
cana-3163	350	16	(	(	PUNCT
cana-3163	350	17	x	x	X
cana-3163	350	18	)	)	PUNCT
cana-3163	350	19	,	,	PUNCT
cana-3163	350	20	x	x	PUNCT
cana-3163	350	21	∈	∈	PROPN
cana-3163	351	1	[	[	X
cana-3163	351	2	0	0	NUM
cana-3163	351	3	,	,	PUNCT
cana-3163	351	4	1]\(a	1]\(a	NUM
cana-3163	351	5	∪	∪	PROPN
cana-3163	351	6	b	b	NOUN
cana-3163	351	7	)	)	PUNCT
cana-3163	351	8	g(x	g(x	NOUN
cana-3163	351	9	)	)	PUNCT
cana-3163	351	10	=	=	SYM
cana-3163	351	11	g2(x	g2(x	PROPN
cana-3163	351	12	)	)	PUNCT
cana-3163	351	13	,	,	PUNCT
cana-3163	351	14	x	x	PUNCT
cana-3163	351	15	∈	∈	NOUN
cana-3163	351	16	b	b	NOUN
cana-3163	351	17	g	g	PROPN
cana-3163	351	18	1	1	NUM
cana-3163	351	19	(	(	PUNCT
cana-3163	351	20	x	x	PROPN
cana-3163	351	21	)	)	PUNCT
cana-3163	351	22	,	,	PUNCT
cana-3163	351	23	x	x	PUNCT
cana-3163	351	24	∈	∈	PROPN
cana-3163	351	25	a	a	DET
cana-3163	351	26	(	(	PUNCT
cana-3163	351	27	see	see	VERB
cana-3163	351	28	figure	figure	NOUN
cana-3163	351	29	14	14	NUM
cana-3163	351	30	)	)	PUNCT
cana-3163	351	31	figure	figure	NOUN
cana-3163	351	32	14	14	NUM
cana-3163	351	33	.	.	PUNCT
cana-3163	352	1	graph	graph	NOUN
cana-3163	352	2	of	of	ADP
cana-3163	352	3	g	g	NOUN
cana-3163	352	4	at	at	ADP
cana-3163	352	5	c	c	NOUN
cana-3163	352	6	=	=	SYM
cana-3163	352	7	1	1	NUM
cana-3163	352	8	then	then	ADV
cana-3163	352	9	as	as	ADP
cana-3163	352	10	in	in	ADP
cana-3163	352	11	the	the	DET
cana-3163	352	12	earlier	early	ADJ
cana-3163	352	13	case	case	NOUN
cana-3163	352	14	,	,	PUNCT
cana-3163	352	15	given	give	VERB
cana-3163	352	16	ϵ	ϵ	ADP
cana-3163	352	17	>	>	X
cana-3163	352	18	0	0	PROPN
cana-3163	352	19	,	,	PUNCT
cana-3163	352	20	f	f	PROPN
cana-3163	352	21	—	—	PUNCT
cana-3163	352	22	g	g	X
cana-3163	352	23	<	<	X
cana-3163	352	24	ϵ.	ϵ.	NOUN
cana-3163	352	25	(	(	PUNCT
cana-3163	352	26	1	1	NUM
cana-3163	352	27	—	—	SYM
cana-3163	352	28	4/7θ	4/7θ	NUM
cana-3163	352	29	)	)	PUNCT
cana-3163	352	30	∈	∈	NOUN
cana-3163	352	31	per3(g	per3(g	NOUN
cana-3163	352	32	)	)	PUNCT
cana-3163	352	33	since	since	SCONJ
cana-3163	352	34	communications	communication	NOUN
cana-3163	352	35	on	on	ADP
cana-3163	352	36	applied	apply	VERB
cana-3163	352	37	nonlinear	nonlinear	ADJ
cana-3163	352	38	analysis	analysis	NOUN
cana-3163	352	39	issn	issn	NOUN
cana-3163	352	40	:	:	PUNCT
cana-3163	352	41	1074	1074	NUM
cana-3163	352	42	-	-	PUNCT
cana-3163	352	43	133x	133x	NUM
cana-3163	352	44	vol	vol	NOUN
cana-3163	352	45	32	32	NUM
cana-3163	352	46	no	no	NOUN
cana-3163	352	47	.	.	PUNCT
cana-3163	353	1	5s	5s	NUM
cana-3163	353	2	(	(	PUNCT
cana-3163	353	3	2025	2025	NUM
cana-3163	353	4	)	)	PUNCT
cana-3163	353	5	524	524	NUM
cana-3163	353	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	353	7	hence	hence	ADV
cana-3163	353	8	the	the	DET
cana-3163	353	9	proposition	proposition	NOUN
cana-3163	353	10	.	.	PUNCT
cana-3163	354	1	corollary	corollary	ADJ
cana-3163	354	2	1	1	NUM
cana-3163	354	3	.	.	PUNCT
cana-3163	355	1	let	let	VERB
cana-3163	355	2	f	f	PROPN
cana-3163	355	3	∈	∈	PROPN
cana-3163	355	4	c(i	c(i	PROPN
cana-3163	355	5	)	)	PUNCT
cana-3163	355	6	.	.	PUNCT
cana-3163	356	1	then	then	ADV
cana-3163	356	2	(	(	PUNCT
cana-3163	356	3	i	i	NOUN
cana-3163	356	4	)	)	PUNCT
cana-3163	356	5	given	give	VERB
cana-3163	356	6	ϵ	ϵ	PROPN
cana-3163	356	7	>	>	X
cana-3163	356	8	0	0	NUM
cana-3163	356	9	,	,	PUNCT
cana-3163	356	10	there	there	PRON
cana-3163	356	11	exists	exist	VERB
cana-3163	356	12	g	g	PROPN
cana-3163	356	13	∈	∈	PROPN
cana-3163	356	14	p	p	X
cana-3163	356	15	(	(	PUNCT
cana-3163	356	16	i	i	NOUN
cana-3163	356	17	)	)	PUNCT
cana-3163	356	18	such	such	ADJ
cana-3163	356	19	that	that	PRON
cana-3163	356	20	|f	|f	PROPN
cana-3163	356	21	—	—	PUNCT
cana-3163	356	22	g|	g|	X
cana-3163	356	23	<	<	X
cana-3163	356	24	ϵ	ϵ	X
cana-3163	356	25	and	and	CCONJ
cana-3163	356	26	g	g	PROPN
cana-3163	356	27	/	/	SYM
cana-3163	356	28	j	j	PROPN
cana-3163	356	29	∈	∈	PROPN
cana-3163	356	30	pr(j	pr(j	NOUN
cana-3163	356	31	)	)	PUNCT
cana-3163	356	32	for	for	ADP
cana-3163	356	33	some	some	DET
cana-3163	356	34	interval	interval	NOUN
cana-3163	356	35	j	j	PROPN
cana-3163	356	36	c	c	PROPN
cana-3163	356	37	i.	i.	PROPN
cana-3163	356	38	(	(	PUNCT
cana-3163	356	39	ii	ii	PROPN
cana-3163	356	40	)	)	PUNCT
cana-3163	356	41	{	{	PUNCT
cana-3163	356	42	g	g	NOUN
cana-3163	356	43	:	:	PUNCT
cana-3163	356	44	i	i	PROPN
cana-3163	356	45	→	→	SYM
cana-3163	356	46	i|	i|	PROPN
cana-3163	356	47	given	give	VERB
cana-3163	356	48	ϵ	ϵ	PROPN
cana-3163	356	49	>	>	X
cana-3163	356	50	0	0	PROPN
cana-3163	356	51	,	,	PUNCT
cana-3163	356	52	|f	|f	PROPN
cana-3163	356	53	—	—	PUNCT
cana-3163	356	54	g|	g|	X
cana-3163	356	55	<	<	X
cana-3163	356	56	ϵ	ϵ	X
cana-3163	356	57	and	and	CCONJ
cana-3163	356	58	g	g	PROPN
cana-3163	356	59	/	/	SYM
cana-3163	356	60	j	j	PROPN
cana-3163	356	61	∈	∈	PROPN
cana-3163	356	62	pr(j	pr(j	NOUN
cana-3163	356	63	)	)	PUNCT
cana-3163	356	64	}	}	PUNCT
cana-3163	356	65	=	=	SYM
cana-3163	356	66	|fix(f	|fix(f	NOUN
cana-3163	356	67	)	)	PUNCT
cana-3163	356	68	|	|	ADV
cana-3163	356	69	.	.	PUNCT
cana-3163	357	1	proof	proof	NOUN
cana-3163	357	2	.	.	PUNCT
cana-3163	358	1	(	(	PUNCT
cana-3163	358	2	i)take	i)take	NOUN
cana-3163	358	3	j	j	X
cana-3163	358	4	=	=	PUNCT
cana-3163	359	1	[	[	X
cana-3163	359	2	c	c	X
cana-3163	359	3	,	,	PUNCT
cana-3163	359	4	c+	c+	VERB
cana-3163	359	5	2θ	2θ	NUM
cana-3163	359	6	]	]	PUNCT
cana-3163	359	7	if	if	SCONJ
cana-3163	359	8	c	c	PROPN
cana-3163	359	9	/=	/=	NOUN
cana-3163	359	10	1	1	NUM
cana-3163	359	11	[	[	PUNCT
cana-3163	359	12	1	1	NUM
cana-3163	359	13	-	-	NUM
cana-3163	359	14	2θ	2θ	NUM
cana-3163	359	15	,	,	PUNCT
cana-3163	359	16	1	1	NUM
cana-3163	359	17	]	]	PUNCT
cana-3163	359	18	if	if	SCONJ
cana-3163	359	19	c	c	NOUN
cana-3163	359	20	=	=	NOUN
cana-3163	360	1	1	1	X
cana-3163	360	2	.	.	PUNCT
cana-3163	361	1	g	g	NOUN
cana-3163	361	2	/	/	SYM
cana-3163	361	3	j	j	PROPN
cana-3163	361	4	∼=	∼=	PROPN
cana-3163	361	5	t/[0,1	t/[0,1	NOUN
cana-3163	361	6	]	]	PUNCT
cana-3163	361	7	.	.	PUNCT
cana-3163	362	1	therefore	therefore	ADV
cana-3163	362	2	g	g	PROPN
cana-3163	362	3	∈	∈	PROPN
cana-3163	362	4	pr(j	pr(j	X
cana-3163	362	5	)	)	PUNCT
cana-3163	362	6	.	.	PUNCT
cana-3163	363	1	(	(	PUNCT
cana-3163	363	2	ii	ii	X
cana-3163	363	3	)	)	PUNCT
cana-3163	363	4	the	the	DET
cana-3163	363	5	cardinality	cardinality	NOUN
cana-3163	363	6	of	of	ADP
cana-3163	363	7	set	set	NOUN
cana-3163	363	8	of	of	ADP
cana-3163	363	9	all	all	DET
cana-3163	363	10	g	g	NOUN
cana-3163	363	11	’s	’s	NOUN
cana-3163	363	12	in	in	ADP
cana-3163	363	13	corollary	corollary	ADJ
cana-3163	363	14	(	(	PUNCT
cana-3163	363	15	i	i	NOUN
cana-3163	363	16	)	)	PUNCT
cana-3163	363	17	is	be	AUX
cana-3163	363	18	the	the	DET
cana-3163	363	19	number	number	NOUN
cana-3163	363	20	of	of	ADP
cana-3163	363	21	fixed	fix	VERB
cana-3163	363	22	points	point	NOUN
cana-3163	363	23	of	of	ADP
cana-3163	363	24	f	f	PROPN
cana-3163	363	25	.	.	PUNCT
cana-3163	364	1	because	because	SCONJ
cana-3163	364	2	at	at	ADP
cana-3163	364	3	each	each	DET
cana-3163	364	4	fixed	fix	VERB
cana-3163	364	5	points	point	NOUN
cana-3163	364	6	of	of	ADP
cana-3163	364	7	f	f	NOUN
cana-3163	364	8	we	we	PRON
cana-3163	364	9	can	can	AUX
cana-3163	364	10	construct	construct	VERB
cana-3163	364	11	g	g	NOUN
cana-3163	364	12	as	as	ADP
cana-3163	364	13	above	above	ADV
cana-3163	364	14	.	.	PUNCT
cana-3163	365	1	note	note	NOUN
cana-3163	365	2	:	:	PUNCT
cana-3163	365	3	if	if	SCONJ
cana-3163	365	4	h	h	NOUN
cana-3163	365	5	is	be	AUX
cana-3163	365	6	a	a	DET
cana-3163	365	7	homeomorphism	homeomorphism	NOUN
cana-3163	365	8	on	on	ADP
cana-3163	365	9	i	i	PRON
cana-3163	365	10	,	,	PUNCT
cana-3163	365	11	then	then	ADV
cana-3163	365	12	h	h	NOUN
cana-3163	365	13	∈/	∈/	PROPN
cana-3163	365	14	pr(i	pr(i	NOUN
cana-3163	365	15	)	)	PUNCT
cana-3163	365	16	as	as	ADP
cana-3163	365	17	per(h	per(h	PROPN
cana-3163	365	18	)	)	PUNCT
cana-3163	365	19	=	=	PUNCT
cana-3163	365	20	per1(h	per1(h	NOUN
cana-3163	365	21	)	)	PUNCT
cana-3163	365	22	∪	∪	ADP
cana-3163	365	23	per2(h	per2(h	NOUN
cana-3163	365	24	)	)	PUNCT
cana-3163	365	25	.	.	PUNCT
cana-3163	366	1	here	here	ADV
cana-3163	366	2	p	p	X
cana-3163	366	3	(	(	PUNCT
cana-3163	366	4	h	h	NOUN
cana-3163	366	5	)	)	PUNCT
cana-3163	366	6	=	=	SYM
cana-3163	366	7	{	{	PUNCT
cana-3163	366	8	1	1	NUM
cana-3163	366	9	,	,	PUNCT
cana-3163	366	10	2	2	NUM
cana-3163	366	11	}	}	PUNCT
cana-3163	366	12	.	.	PUNCT
cana-3163	367	1	3	3	X
cana-3163	367	2	.	.	X
cana-3163	367	3	conclusion	conclusion	NOUN
cana-3163	367	4	studies	study	NOUN
cana-3163	367	5	of	of	ADP
cana-3163	367	6	periodically	periodically	ADV
cana-3163	367	7	rich	rich	ADJ
cana-3163	367	8	maps	map	NOUN
cana-3163	367	9	on	on	ADP
cana-3163	367	10	various	various	ADJ
cana-3163	367	11	topological	topological	ADJ
cana-3163	367	12	spaces	space	NOUN
cana-3163	367	13	would	would	AUX
cana-3163	367	14	contribute	contribute	VERB
cana-3163	367	15	to	to	ADP
cana-3163	367	16	literature	literature	NOUN
cana-3163	367	17	of	of	ADP
cana-3163	367	18	the	the	DET
cana-3163	367	19	chaotic	chaotic	ADJ
cana-3163	367	20	dynamical	dynamical	ADJ
cana-3163	367	21	systems.the	systems.the	DET
cana-3163	367	22	set	set	NOUN
cana-3163	367	23	of	of	ADP
cana-3163	367	24	periodic	periodic	ADJ
cana-3163	367	25	points	point	NOUN
cana-3163	367	26	is	be	AUX
cana-3163	367	27	a	a	DET
cana-3163	367	28	critical	critical	ADJ
cana-3163	367	29	concept	concept	NOUN
cana-3163	367	30	in	in	ADP
cana-3163	367	31	studying	study	VERB
cana-3163	367	32	chaotic	chaotic	ADJ
cana-3163	367	33	maps	map	NOUN
cana-3163	367	34	as	as	SCONJ
cana-3163	367	35	it	it	PRON
cana-3163	367	36	captures	capture	VERB
cana-3163	367	37	the	the	DET
cana-3163	367	38	recurrence	recurrence	NOUN
cana-3163	367	39	and	and	CCONJ
cana-3163	367	40	regularity	regularity	NOUN
cana-3163	367	41	in	in	ADP
cana-3163	367	42	the	the	DET
cana-3163	367	43	otherwise	otherwise	ADV
cana-3163	367	44	seemingly	seemingly	ADV
cana-3163	367	45	erratic	erratic	ADJ
cana-3163	367	46	dynamics.the	dynamics.the	DET
cana-3163	367	47	set	set	NOUN
cana-3163	367	48	of	of	ADP
cana-3163	367	49	periods	period	NOUN
cana-3163	367	50	reflects	reflect	VERB
cana-3163	367	51	the	the	DET
cana-3163	367	52	variety	variety	NOUN
cana-3163	367	53	and	and	CCONJ
cana-3163	367	54	richness	richness	NOUN
cana-3163	367	55	of	of	ADP
cana-3163	367	56	periodic	periodic	ADJ
cana-3163	367	57	behavior	behavior	NOUN
cana-3163	367	58	in	in	ADP
cana-3163	367	59	chaotic	chaotic	ADJ
cana-3163	367	60	systems	system	NOUN
cana-3163	367	61	,	,	PUNCT
cana-3163	367	62	often	often	ADV
cana-3163	367	63	being	be	AUX
cana-3163	367	64	dense	dense	ADJ
cana-3163	367	65	in	in	ADP
cana-3163	367	66	natural	natural	ADJ
cana-3163	367	67	numbers	number	NOUN
cana-3163	367	68	,	,	PUNCT
cana-3163	367	69	indicating	indicate	VERB
cana-3163	367	70	that	that	SCONJ
cana-3163	367	71	the	the	DET
cana-3163	367	72	system	system	NOUN
cana-3163	367	73	exhibits	exhibit	VERB
cana-3163	367	74	periodic	periodic	ADJ
cana-3163	367	75	behavior	behavior	NOUN
cana-3163	367	76	at	at	ADP
cana-3163	367	77	every	every	DET
cana-3163	367	78	scale	scale	NOUN
cana-3163	367	79	.	.	PUNCT
cana-3163	368	1	acknowledgment	acknowledgment	NOUN
cana-3163	368	2	:	:	PUNCT
cana-3163	369	1	we	we	PRON
cana-3163	369	2	gratefully	gratefully	ADV
cana-3163	369	3	acknowledge	acknowledge	VERB
cana-3163	369	4	the	the	DET
cana-3163	369	5	administrative	administrative	ADJ
cana-3163	369	6	support	support	NOUN
cana-3163	369	7	,	,	PUNCT
cana-3163	369	8	facilities	facility	NOUN
cana-3163	369	9	and	and	CCONJ
cana-3163	369	10	opportunities	opportunity	NOUN
cana-3163	369	11	provided	provide	VERB
cana-3163	369	12	by	by	ADP
cana-3163	369	13	the	the	DET
cana-3163	369	14	management	management	NOUN
cana-3163	369	15	of	of	ADP
cana-3163	369	16	rajagiri	rajagiri	NOUN
cana-3163	369	17	school	school	NOUN
cana-3163	369	18	of	of	ADP
cana-3163	369	19	engineering	engineering	NOUN
cana-3163	369	20	&	&	CCONJ
cana-3163	369	21	technology	technology	NOUN
cana-3163	369	22	,	,	PUNCT
cana-3163	369	23	kochi	kochi	NOUN
cana-3163	369	24	and	and	CCONJ
cana-3163	369	25	ihrd	ihrd	NOUN
cana-3163	369	26	,	,	PUNCT
cana-3163	369	27	kerala	kerala	PROPN
cana-3163	369	28	.	.	PUNCT
cana-3163	370	1	references	reference	NOUN
cana-3163	370	2	[	[	X
cana-3163	370	3	1	1	NUM
cana-3163	370	4	]	]	PUNCT
cana-3163	370	5	luis	luis	PROPN
cana-3163	370	6	alseda	alseda	PROPN
cana-3163	370	7	,	,	PUNCT
cana-3163	370	8	jaume	jaume	PROPN
cana-3163	370	9	llibre	llibre	PROPN
cana-3163	370	10	,	,	PUNCT
cana-3163	370	11	and	and	CCONJ
cana-3163	370	12	michal	michal	PROPN
cana-3163	370	13	misiurewicz	misiurewicz	PROPN
cana-3163	370	14	.	.	PUNCT
cana-3163	371	1	combinatorial	combinatorial	ADJ
cana-3163	371	2	dynamics	dynamic	NOUN
cana-3163	371	3	and	and	CCONJ
cana-3163	371	4	entropy	entropy	NOUN
cana-3163	371	5	in	in	ADP
cana-3163	371	6	dimension	dimension	NOUN
cana-3163	371	7	one	one	NUM
cana-3163	371	8	,	,	PUNCT
cana-3163	371	9	volume	volume	NOUN
cana-3163	371	10	5	5	NUM
cana-3163	371	11	.	.	PUNCT
cana-3163	372	1	world	world	NOUN
cana-3163	372	2	scientific	scientific	ADJ
cana-3163	372	3	publishing	publishing	NOUN
cana-3163	372	4	company	company	NOUN
cana-3163	372	5	,	,	PUNCT
cana-3163	372	6	2000	2000	NUM
cana-3163	372	7	.	.	PUNCT
cana-3163	373	1	communications	communication	NOUN
cana-3163	373	2	on	on	ADP
cana-3163	373	3	applied	apply	VERB
cana-3163	373	4	nonlinear	nonlinear	ADJ
cana-3163	373	5	analysis	analysis	NOUN
cana-3163	373	6	issn	issn	NOUN
cana-3163	373	7	:	:	PUNCT
cana-3163	373	8	1074	1074	NUM
cana-3163	373	9	-	-	PUNCT
cana-3163	373	10	133x	133x	NUM
cana-3163	373	11	vol	vol	NOUN
cana-3163	373	12	32	32	NUM
cana-3163	373	13	no	no	NOUN
cana-3163	373	14	.	.	PUNCT
cana-3163	374	1	5s	5s	NUM
cana-3163	374	2	(	(	PUNCT
cana-3163	374	3	2025	2025	NUM
cana-3163	374	4	)	)	PUNCT
cana-3163	374	5	525	525	NUM
cana-3163	374	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3163	375	1	[	[	X
cana-3163	375	2	2	2	NUM
cana-3163	375	3	]	]	PUNCT
cana-3163	375	4	detlef	detlef	NOUN
cana-3163	375	5	bargmann	bargmann	PROPN
cana-3163	375	6	and	and	CCONJ
cana-3163	375	7	walter	walter	PROPN
cana-3163	375	8	bergweiler	bergweiler	NOUN
cana-3163	375	9	.	.	PUNCT
cana-3163	376	1	periodic	periodic	ADJ
cana-3163	376	2	points	point	NOUN
cana-3163	376	3	and	and	CCONJ
cana-3163	376	4	normal	normal	ADJ
cana-3163	376	5	families	family	NOUN
cana-3163	376	6	.	.	PUNCT
cana-3163	377	1	proceedings	proceeding	NOUN
cana-3163	377	2	of	of	ADP
cana-3163	377	3	the	the	DET
cana-3163	377	4	american	american	PROPN
cana-3163	377	5	mathematical	mathematical	PROPN
cana-3163	377	6	society	society	NOUN
cana-3163	377	7	,	,	PUNCT
cana-3163	377	8	129(10):2881–2888	129(10):2881–2888	NUM
cana-3163	377	9	,	,	PUNCT
cana-3163	377	10	2001	2001	NUM
cana-3163	377	11	.	.	PUNCT
cana-3163	378	1	[	[	X
cana-3163	378	2	3	3	X
cana-3163	378	3	]	]	X
cana-3163	378	4	ethan	ethan	PROPN
cana-3163	378	5	m	m	PROPN
cana-3163	378	6	coven	coven	NOUN
cana-3163	378	7	and	and	CCONJ
cana-3163	378	8	ga	ga	PROPN
cana-3163	378	9	hedlund	hedlund	PROPN
cana-3163	378	10	.	.	PUNCT
cana-3163	379	1	p¯	p¯	X
cana-3163	380	1	=	=	PUNCT
cana-3163	380	2	r̄	r̄	NOUN
cana-3163	380	3	for	for	ADP
cana-3163	380	4	maps	map	NOUN
cana-3163	380	5	of	of	ADP
cana-3163	380	6	the	the	DET
cana-3163	380	7	interval	interval	NOUN
cana-3163	380	8	,	,	PUNCT
cana-3163	380	9	1980	1980	NUM
cana-3163	380	10	.	.	PUNCT
cana-3163	381	1	[	[	X
cana-3163	381	2	4	4	NUM
cana-3163	381	3	]	]	SYM
cana-3163	381	4	jp	jp	NOUN
cana-3163	381	5	delahaye	delahaye	NOUN
cana-3163	381	6	.	.	PUNCT
cana-3163	382	1	the	the	DET
cana-3163	382	2	set	set	NOUN
cana-3163	382	3	of	of	ADP
cana-3163	382	4	periodic	periodic	ADJ
cana-3163	382	5	points	point	NOUN
cana-3163	382	6	.	.	PUNCT
cana-3163	383	1	the	the	DET
cana-3163	383	2	american	american	PROPN
cana-3163	383	3	mathematical	mathematical	PROPN
cana-3163	383	4	monthly	monthly	ADV
cana-3163	383	5	,	,	PUNCT
cana-3163	383	6	88(9):646–651	88(9):646–651	NUM
cana-3163	383	7	,	,	PUNCT
cana-3163	383	8	1981	1981	NUM
cana-3163	383	9	.	.	PUNCT
cana-3163	384	1	[	[	X
cana-3163	384	2	5	5	X
cana-3163	384	3	]	]	X
cana-3163	384	4	robert	robert	PROPN
cana-3163	384	5	l.	l.	PROPN
cana-3163	384	6	devaney	devaney	PROPN
cana-3163	384	7	.	.	PUNCT
cana-3163	385	1	dynamics	dynamic	NOUN
cana-3163	385	2	of	of	ADP
cana-3163	385	3	simple	simple	ADJ
cana-3163	385	4	maps	map	NOUN
cana-3163	385	5	.	.	PUNCT
cana-3163	386	1	chaos	chaos	NOUN
cana-3163	386	2	and	and	CCONJ
cana-3163	386	3	fractals	fractal	NOUN
cana-3163	386	4	.	.	PUNCT
cana-3163	387	1	in	in	ADP
cana-3163	387	2	proc	proc	NOUN
cana-3163	387	3	.	.	PUNCT
cana-3163	388	1	symposia	symposia	NOUN
cana-3163	388	2	.	.	PUNCT
cana-3163	389	1	appl	appl	PROPN
cana-3163	389	2	.	.	PROPN
cana-3163	389	3	math	math	PROPN
cana-3163	389	4	.	.	PUNCT
cana-3163	390	1	,	,	PUNCT
cana-3163	390	2	volume	volume	NOUN
cana-3163	390	3	39	39	NUM
cana-3163	390	4	of	of	ADP
cana-3163	390	5	ams	am	NOUN
cana-3163	390	6	short	short	ADJ
cana-3163	390	7	course	course	NOUN
cana-3163	390	8	lecture	lecture	NOUN
cana-3163	390	9	notes	note	NOUN
cana-3163	390	10	,	,	PUNCT
cana-3163	390	11	pages	page	NOUN
cana-3163	390	12	1–24	1–24	PROPN
cana-3163	390	13	,	,	PUNCT
cana-3163	390	14	amer	amer	PROPN
cana-3163	390	15	.	.	PROPN
cana-3163	390	16	math	math	PROPN
cana-3163	390	17	.	.	PUNCT
cana-3163	391	1	soc	soc	PROPN
cana-3163	391	2	.	.	PUNCT
cana-3163	391	3	,	,	PUNCT
cana-3163	391	4	providence	providence	NOUN
cana-3163	391	5	,	,	PUNCT
cana-3163	391	6	ri	ri	NOUN
cana-3163	391	7	,	,	PUNCT
cana-3163	391	8	1989	1989	NUM
cana-3163	391	9	.	.	PUNCT
cana-3163	392	1	[	[	X
cana-3163	392	2	6	6	NUM
cana-3163	392	3	]	]	X
cana-3163	392	4	graham	graham	PROPN
cana-3163	392	5	everest	everest	PROPN
cana-3163	392	6	,	,	PUNCT
cana-3163	392	7	alf	alf	PROPN
cana-3163	392	8	j	j	PROPN
cana-3163	392	9	van	van	PROPN
cana-3163	392	10	der	der	PROPN
cana-3163	392	11	poorten	poorten	VERB
cana-3163	392	12	,	,	PUNCT
cana-3163	392	13	yash	yash	PROPN
cana-3163	392	14	puri	puri	PROPN
cana-3163	392	15	,	,	PUNCT
cana-3163	392	16	and	and	CCONJ
cana-3163	392	17	thomas	thomas	PROPN
cana-3163	392	18	b	b	PROPN
cana-3163	392	19	ward	ward	PROPN
cana-3163	392	20	.	.	PUNCT
cana-3163	393	1	integer	integer	NOUN
cana-3163	393	2	sequences	sequence	NOUN
cana-3163	393	3	and	and	CCONJ
cana-3163	393	4	periodic	periodic	ADJ
cana-3163	393	5	points	point	NOUN
cana-3163	393	6	.	.	PUNCT
cana-3163	394	1	journal	journal	NOUN
cana-3163	394	2	of	of	ADP
cana-3163	394	3	integer	integer	NOUN
cana-3163	394	4	sequences	sequence	NOUN
cana-3163	395	1	[	[	X
cana-3163	395	2	electronic	electronic	ADJ
cana-3163	395	3	only	only	ADV
cana-3163	395	4	]	]	PUNCT
cana-3163	395	5	,	,	PUNCT
cana-3163	395	6	5(2):art–02	5(2):art–02	NUM
cana-3163	395	7	,	,	PUNCT
cana-3163	395	8	2002	2002	NUM
cana-3163	395	9	.	.	PUNCT
cana-3163	396	1	[	[	X
cana-3163	396	2	7	7	X
cana-3163	396	3	]	]	PUNCT
cana-3163	396	4	zhaorong	zhaorong	NOUN
cana-3163	396	5	he	he	PRON
cana-3163	396	6	,	,	PUNCT
cana-3163	396	7	j.	j.	PROPN
cana-3163	396	8	li	li	PROPN
cana-3163	396	9	,	,	PUNCT
cana-3163	396	10	and	and	CCONJ
cana-3163	396	11	zhongqiang	zhongqiang	PROPN
cana-3163	396	12	yang	yang	PROPN
cana-3163	396	13	.	.	PUNCT
cana-3163	397	1	the	the	DET
cana-3163	397	2	topological	topological	ADJ
cana-3163	397	3	structure	structure	NOUN
cana-3163	397	4	of	of	ADP
cana-3163	397	5	function	function	NOUN
cana-3163	397	6	space	space	NOUN
cana-3163	397	7	of	of	ADP
cana-3163	397	8	transitive	transitive	ADJ
cana-3163	397	9	maps	map	NOUN
cana-3163	397	10	.	.	PUNCT
cana-3163	398	1	topology	topology	NOUN
cana-3163	398	2	and	and	CCONJ
cana-3163	398	3	its	its	PRON
cana-3163	398	4	applications	application	NOUN
cana-3163	398	5	,	,	PUNCT
cana-3163	398	6	275:107009	275:107009	NUM
cana-3163	398	7	,	,	PUNCT
cana-3163	398	8	2020	2020	NUM
cana-3163	398	9	.	.	PUNCT
cana-3163	399	1	[	[	X
cana-3163	399	2	8	8	NUM
cana-3163	399	3	]	]	X
cana-3163	399	4	tien	tien	PROPN
cana-3163	399	5	-	-	PUNCT
cana-3163	399	6	yien	yien	PROPN
cana-3163	399	7	li	li	PROPN
cana-3163	399	8	and	and	CCONJ
cana-3163	399	9	james	james	PROPN
cana-3163	399	10	a	a	DET
cana-3163	399	11	yorke	yorke	PROPN
cana-3163	399	12	.	.	PUNCT
cana-3163	400	1	period	period	NOUN
cana-3163	400	2	three	three	NUM
cana-3163	400	3	implies	imply	VERB
cana-3163	400	4	chaos	chaos	NOUN
cana-3163	400	5	.	.	PUNCT
cana-3163	401	1	the	the	DET
cana-3163	401	2	american	american	PROPN
cana-3163	401	3	mathematical	mathematical	PROPN
cana-3163	401	4	monthly	monthly	ADV
cana-3163	401	5	,	,	PUNCT
cana-3163	401	6	82(10):985	82(10):985	NUM
cana-3163	401	7	–	–	PUNCT
cana-3163	401	8	992	992	NUM
cana-3163	401	9	,	,	PUNCT
cana-3163	401	10	1975	1975	NUM
cana-3163	401	11	.	.	PUNCT
cana-3163	402	1	[	[	X
cana-3163	402	2	9	9	NUM
cana-3163	402	3	]	]	X
cana-3163	402	4	sylvie	sylvie	NOUN
cana-3163	402	5	ruette	ruette	NOUN
cana-3163	402	6	.	.	PUNCT
cana-3163	403	1	chaos	chaos	NOUN
cana-3163	403	2	on	on	ADP
cana-3163	403	3	the	the	DET
cana-3163	403	4	interval	interval	NOUN
cana-3163	403	5	,	,	PUNCT
cana-3163	403	6	volume	volume	NOUN
cana-3163	403	7	67	67	NUM
cana-3163	403	8	.	.	PUNCT
cana-3163	404	1	american	american	PROPN
cana-3163	404	2	mathematical	mathematical	PROPN
cana-3163	404	3	soc	soc	PROPN
cana-3163	404	4	.	.	PUNCT
cana-3163	404	5	,	,	PUNCT
cana-3163	404	6	2017	2017	NUM
cana-3163	404	7	.	.	PUNCT
cana-3163	405	1	[	[	X
cana-3163	405	2	10	10	NUM
cana-3163	405	3	]	]	PUNCT
cana-3163	405	4	a.	a.	NOUN
cana-3163	405	5	n.	n.	PROPN
cana-3163	405	6	sharkovski˘ı	sharkovski˘ı	PROPN
cana-3163	405	7	.	.	PUNCT
cana-3163	405	8	coexistence	coexistence	NOUN
cana-3163	405	9	of	of	ADP
cana-3163	405	10	cycles	cycle	NOUN
cana-3163	405	11	of	of	ADP
cana-3163	405	12	a	a	DET
cana-3163	405	13	continuous	continuous	ADJ
cana-3163	405	14	map	map	NOUN
cana-3163	405	15	of	of	ADP
cana-3163	405	16	the	the	DET
cana-3163	405	17	line	line	NOUN
cana-3163	405	18	into	into	ADP
cana-3163	405	19	itself(english	itself(english	ADJ
cana-3163	405	20	version	version	NOUN
cana-3163	405	21	)	)	PUNCT
cana-3163	405	22	.	.	PUNCT
cana-3163	406	1	international	international	ADJ
cana-3163	406	2	journal	journal	PROPN
cana-3163	406	3	of	of	ADP
cana-3163	406	4	bifurcation	bifurcation	NOUN
cana-3163	406	5	and	and	CCONJ
cana-3163	406	6	chaos	chaos	NOUN
cana-3163	406	7	,	,	PUNCT
cana-3163	406	8	5(05):1263–1273	5(05):1263–1273	NUM
cana-3163	406	9	,	,	PUNCT
cana-3163	406	10	1995	1995	NUM
cana-3163	406	11	.	.	PUNCT
cana-3163	407	1	[	[	X
cana-3163	407	2	11	11	NUM
cana-3163	407	3	]	]	PUNCT
cana-3163	407	4	stephen	stephen	PROPN
cana-3163	407	5	silverman	silverman	PROPN
cana-3163	407	6	.	.	PUNCT
cana-3163	408	1	on	on	ADP
cana-3163	408	2	maps	map	NOUN
cana-3163	408	3	with	with	ADP
cana-3163	408	4	dense	dense	ADJ
cana-3163	408	5	orbits	orbit	NOUN
cana-3163	408	6	and	and	CCONJ
cana-3163	408	7	the	the	DET
cana-3163	408	8	definition	definition	NOUN
cana-3163	408	9	of	of	ADP
cana-3163	408	10	chaos	chaos	NOUN
cana-3163	408	11	.	.	PUNCT
cana-3163	409	1	the	the	DET
cana-3163	409	2	rocky	rocky	ADJ
cana-3163	409	3	mountain	mountain	NOUN
cana-3163	409	4	journal	journal	NOUN
cana-3163	409	5	of	of	ADP
cana-3163	409	6	mathematics	mathematic	NOUN
cana-3163	409	7	,	,	PUNCT
cana-3163	409	8	22(1):353–375	22(1):353–375	PROPN
cana-3163	409	9	,	,	PUNCT
cana-3163	409	10	1992	1992	NUM
cana-3163	409	11	.	.	PUNCT
cana-3163	410	1	[	[	X
cana-3163	410	2	12	12	NUM
cana-3163	410	3	]	]	PUNCT
cana-3163	410	4	michel	michel	PROPN
cana-3163	410	5	vellekoop	vellekoop	PROPN
cana-3163	410	6	and	and	CCONJ
cana-3163	410	7	raoul	raoul	PROPN
cana-3163	410	8	berglund	berglund	PROPN
cana-3163	410	9	.	.	PUNCT
cana-3163	411	1	on	on	ADP
cana-3163	411	2	intervals	interval	NOUN
cana-3163	411	3	,	,	PUNCT
cana-3163	411	4	transitivity	transitivity	NOUN
cana-3163	411	5	=	=	SYM
cana-3163	411	6	chaos	chaos	NOUN
cana-3163	411	7	.	.	PUNCT
cana-3163	412	1	the	the	DET
cana-3163	412	2	american	american	PROPN
cana-3163	412	3	mathematical	mathematical	PROPN
cana-3163	412	4	monthly	monthly	ADV
cana-3163	412	5	,	,	PUNCT
cana-3163	412	6	101(4):353–355	101(4):353–355	NUM
cana-3163	412	7	,	,	PUNCT
cana-3163	412	8	1994	1994	NUM
cana-3163	412	9	.	.	PUNCT