id	sid	tid	token	lemma	pos
cana-4508	1	1	communications	communication	NOUN
cana-4508	1	2	on	on	ADP
cana-4508	1	3	applied	apply	VERB
cana-4508	1	4	nonlinear	nonlinear	ADJ
cana-4508	1	5	analysis	analysis	NOUN
cana-4508	1	6	issn	issn	NOUN
cana-4508	1	7	:	:	PUNCT
cana-4508	1	8	1074	1074	NUM
cana-4508	1	9	-	-	PUNCT
cana-4508	1	10	133x	133x	NUM
cana-4508	1	11	vol	vol	NOUN
cana-4508	1	12	32	32	NUM
cana-4508	1	13	no	no	NOUN
cana-4508	1	14	.	.	PUNCT
cana-4508	2	1	9s	9s	NUM
cana-4508	2	2	(	(	PUNCT
cana-4508	2	3	2025	2025	NUM
cana-4508	2	4	)	)	PUNCT
cana-4508	2	5	2232	2232	NUM
cana-4508	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	3	1	a	a	DET
cana-4508	3	2	new	new	ADJ
cana-4508	3	3	insight	insight	NOUN
cana-4508	3	4	with	with	ADP
cana-4508	3	5	trigonometric	trigonometric	ADJ
cana-4508	3	6	coefficients	coefficient	NOUN
cana-4508	3	7	of	of	ADP
cana-4508	3	8	additive	additive	ADJ
cana-4508	3	9	-	-	PUNCT
cana-4508	3	10	quadratic	quadratic	ADJ
cana-4508	3	11	functional	functional	ADJ
cana-4508	3	12	equations	equation	NOUN
cana-4508	3	13	and	and	CCONJ
cana-4508	3	14	its	its	PRON
cana-4508	3	15	stability	stability	NOUN
cana-4508	3	16	analysis	analysis	NOUN
cana-4508	3	17	v.	v.	ADP
cana-4508	3	18	vijayan	vijayan	PROPN
cana-4508	3	19	𝟏	𝟏	PROPN
cana-4508	4	1	p.	p.	NOUN
cana-4508	4	2	agilan	agilan	PROPN
cana-4508	5	1	𝟐∗	𝟐∗	ADP
cana-4508	5	2	,	,	PUNCT
cana-4508	5	3	m.	m.	PROPN
cana-4508	5	4	sophia	sophia	PROPN
cana-4508	5	5	𝟑	𝟑	PROPN
cana-4508	5	6	,	,	PUNCT
cana-4508	5	7	n.	n.	PROPN
cana-4508	5	8	maheshkumar	maheshkumar	PROPN
cana-4508	5	9	𝟒	𝟒	PROPN
cana-4508	5	10	1department	1department	NUM
cana-4508	5	11	of	of	ADP
cana-4508	5	12	electronics	electronic	NOUN
cana-4508	5	13	and	and	CCONJ
cana-4508	5	14	instrumentation	instrumentation	NOUN
cana-4508	5	15	engineering	engineering	NOUN
cana-4508	5	16	,	,	PUNCT
cana-4508	5	17	st.joseph	st.joseph	X
cana-4508	5	18	’s	’s	PART
cana-4508	5	19	college	college	NOUN
cana-4508	5	20	of	of	ADP
cana-4508	5	21	engineering	engineering	PROPN
cana-4508	5	22	,	,	PUNCT
cana-4508	5	23	omr	omr	PROPN
cana-4508	5	24	,	,	PUNCT
cana-4508	5	25	chennai	chennai	VERB
cana-4508	5	26	600	600	NUM
cana-4508	5	27	119	119	NUM
cana-4508	5	28	,	,	PUNCT
cana-4508	5	29	tamilnadu	tamilnadu	NOUN
cana-4508	5	30	,	,	PUNCT
cana-4508	5	31	india	india	PROPN
cana-4508	5	32	.	.	PUNCT
cana-4508	6	1	e-mail:vinvpn@gmail.com	e-mail:vinvpn@gmail.com	X
cana-4508	7	1	2department	2department	NUM
cana-4508	7	2	of	of	ADP
cana-4508	7	3	mathematics	mathematic	NOUN
cana-4508	7	4	,	,	PUNCT
cana-4508	7	5	st.joseph	st.joseph	X
cana-4508	7	6	’s	’s	PART
cana-4508	7	7	college	college	NOUN
cana-4508	7	8	of	of	ADP
cana-4508	7	9	engineering	engineering	PROPN
cana-4508	7	10	,	,	PUNCT
cana-4508	7	11	omr	omr	PROPN
cana-4508	7	12	,	,	PUNCT
cana-4508	7	13	chennai	chennai	VERB
cana-4508	7	14	600	600	NUM
cana-4508	7	15	119	119	NUM
cana-4508	7	16	,	,	PUNCT
cana-4508	7	17	tamilnadu	tamilnadu	ADJ
cana-4508	7	18	,	,	PUNCT
cana-4508	7	19	india	india	PROPN
cana-4508	7	20	.	.	PUNCT
cana-4508	8	1	e	e	X
cana-4508	8	2	-	-	NOUN
cana-4508	8	3	mail	mail	NOUN
cana-4508	8	4	:	:	PUNCT
cana-4508	9	1	agilram@gmail.com	agilram@gmail.com	X
cana-4508	9	2	3department	3department	NUM
cana-4508	9	3	of	of	ADP
cana-4508	9	4	mathematics	mathematic	NOUN
cana-4508	9	5	,	,	PUNCT
cana-4508	9	6	simats	simat	NOUN
cana-4508	9	7	engineering	engineering	PROPN
cana-4508	9	8	,	,	PUNCT
cana-4508	9	9	saveetha	saveetha	PROPN
cana-4508	9	10	nagar	nagar	PROPN
cana-4508	9	11	,	,	PUNCT
cana-4508	9	12	thandalam	thandalam	PROPN
cana-4508	9	13	,	,	PUNCT
cana-4508	9	14	kanchipuram	kanchipuram	PROPN
cana-4508	9	15	-	-	PUNCT
cana-4508	9	16	chennai	chennai	PROPN
cana-4508	9	17	rd	rd	PROPN
cana-4508	9	18	,	,	PUNCT
cana-4508	9	19	chennai	chennai	PROPN
cana-4508	9	20	602105	602105	NUM
cana-4508	9	21	,	,	PUNCT
cana-4508	9	22	tamil	tamil	PROPN
cana-4508	9	23	nadu	nadu	PROPN
cana-4508	9	24	,	,	PUNCT
cana-4508	9	25	india	india	PROPN
cana-4508	9	26	.	.	PUNCT
cana-4508	10	1	e	e	X
cana-4508	10	2	-	-	NOUN
cana-4508	10	3	mail	mail	NOUN
cana-4508	10	4	:	:	PUNCT
cana-4508	10	5	sophia.raj2005@gmail.com	sophia.raj2005@gmail.com	PROPN
cana-4508	10	6	.	.	PUNCT
cana-4508	11	1	4department	4department	NUM
cana-4508	11	2	of	of	ADP
cana-4508	11	3	science	science	NOUN
cana-4508	11	4	and	and	CCONJ
cana-4508	11	5	humanities	humanity	NOUN
cana-4508	11	6	,	,	PUNCT
cana-4508	11	7	faculty	faculty	NOUN
cana-4508	11	8	of	of	ADP
cana-4508	11	9	engineering	engineering	PROPN
cana-4508	11	10	,	,	PUNCT
cana-4508	11	11	karpagam	karpagam	PROPN
cana-4508	11	12	academy	academy	PROPN
cana-4508	11	13	of	of	ADP
cana-4508	11	14	higher	high	ADJ
cana-4508	11	15	education	education	NOUN
cana-4508	11	16	,	,	PUNCT
cana-4508	11	17	coimbatore-641021	coimbatore-641021	ADJ
cana-4508	11	18	,	,	PUNCT
cana-4508	11	19	tamilnadu	tamilnadu	NOUN
cana-4508	11	20	,	,	PUNCT
cana-4508	11	21	india	india	PROPN
cana-4508	11	22	.	.	PUNCT
cana-4508	12	1	e	e	X
cana-4508	12	2	-	-	NOUN
cana-4508	12	3	mail	mail	NOUN
cana-4508	12	4	:	:	PUNCT
cana-4508	12	5	maheshkumar.natarajan@kahedu.edu.in	maheshkumar.natarajan@kahedu.edu.in	ADJ
cana-4508	12	6	.	.	PUNCT
cana-4508	12	7	∗corresponding	∗corresponde	VERB
cana-4508	12	8	author	author	NOUN
cana-4508	12	9	:	:	PUNCT
cana-4508	12	10	agilram@gmail.com	agilram@gmail.com	X
cana-4508	12	11	article	article	PROPN
cana-4508	12	12	history	history	NOUN
cana-4508	12	13	:	:	PUNCT
cana-4508	12	14	received	receive	VERB
cana-4508	12	15	:	:	PUNCT
cana-4508	12	16	12	12	NUM
cana-4508	12	17	-	-	SYM
cana-4508	12	18	01	01	NUM
cana-4508	12	19	-	-	PUNCT
cana-4508	12	20	2025	2025	NUM
cana-4508	12	21	revised	revise	VERB
cana-4508	12	22	:	:	PUNCT
cana-4508	12	23	15	15	NUM
cana-4508	12	24	-	-	NUM
cana-4508	12	25	02	02	NUM
cana-4508	12	26	-	-	PUNCT
cana-4508	12	27	2025	2025	NUM
cana-4508	12	28	accepted	accept	VERB
cana-4508	12	29	:	:	PUNCT
cana-4508	12	30	01	01	NUM
cana-4508	12	31	-	-	SYM
cana-4508	12	32	03	03	NUM
cana-4508	12	33	-	-	PUNCT
cana-4508	12	34	2025	2025	NUM
cana-4508	12	35	abstract	abstract	NOUN
cana-4508	12	36	:	:	PUNCT
cana-4508	12	37	this	this	DET
cana-4508	12	38	study	study	NOUN
cana-4508	12	39	introduces	introduce	VERB
cana-4508	12	40	a	a	DET
cana-4508	12	41	novel	novel	ADJ
cana-4508	12	42	framework	framework	NOUN
cana-4508	12	43	for	for	ADP
cana-4508	12	44	analyzing	analyze	VERB
cana-4508	12	45	the	the	DET
cana-4508	12	46	ulam	ulam	NOUN
cana-4508	12	47	-	-	PUNCT
cana-4508	12	48	hyers	hyer	NOUN
cana-4508	12	49	stability	stability	NOUN
cana-4508	12	50	of	of	ADP
cana-4508	12	51	mixed	mixed	ADJ
cana-4508	12	52	-	-	PUNCT
cana-4508	12	53	type	type	NOUN
cana-4508	12	54	additive	additive	ADJ
cana-4508	12	55	-	-	PUNCT
cana-4508	12	56	quadratic	quadratic	ADJ
cana-4508	12	57	functional	functional	ADJ
cana-4508	12	58	equations	equation	NOUN
cana-4508	12	59	with	with	ADP
cana-4508	12	60	trigonometric	trigonometric	ADJ
cana-4508	12	61	constant	constant	ADJ
cana-4508	12	62	coefficients	coefficient	NOUN
cana-4508	12	63	in	in	ADP
cana-4508	12	64	banach	banach	NOUN
cana-4508	12	65	spaces	space	NOUN
cana-4508	12	66	.	.	PUNCT
cana-4508	13	1	employing	employ	VERB
cana-4508	13	2	advanced	advanced	ADJ
cana-4508	13	3	analytical	analytical	ADJ
cana-4508	13	4	techniques	technique	NOUN
cana-4508	13	5	and	and	CCONJ
cana-4508	13	6	leveraging	leverage	VERB
cana-4508	13	7	the	the	DET
cana-4508	13	8	unique	unique	ADJ
cana-4508	13	9	properties	property	NOUN
cana-4508	13	10	of	of	ADP
cana-4508	13	11	trigonometric	trigonometric	ADJ
cana-4508	13	12	functions	function	NOUN
cana-4508	13	13	,	,	PUNCT
cana-4508	13	14	we	we	PRON
cana-4508	13	15	derive	derive	VERB
cana-4508	13	16	sufficient	sufficient	ADJ
cana-4508	13	17	conditions	condition	NOUN
cana-4508	13	18	for	for	ADP
cana-4508	13	19	the	the	DET
cana-4508	13	20	stability	stability	NOUN
cana-4508	13	21	of	of	ADP
cana-4508	13	22	these	these	DET
cana-4508	13	23	equations	equation	NOUN
cana-4508	13	24	.	.	PUNCT
cana-4508	14	1	the	the	DET
cana-4508	14	2	intricate	intricate	ADJ
cana-4508	14	3	relationship	relationship	NOUN
cana-4508	14	4	between	between	ADP
cana-4508	14	5	additive	additive	ADJ
cana-4508	14	6	and	and	CCONJ
cana-4508	14	7	quadratic	quadratic	ADJ
cana-4508	14	8	components	component	NOUN
cana-4508	14	9	is	be	AUX
cana-4508	14	10	rigorously	rigorously	ADV
cana-4508	14	11	examined	examine	VERB
cana-4508	14	12	,	,	PUNCT
cana-4508	14	13	emphasizing	emphasize	VERB
cana-4508	14	14	the	the	DET
cana-4508	14	15	pivotal	pivotal	ADJ
cana-4508	14	16	role	role	NOUN
cana-4508	14	17	of	of	ADP
cana-4508	14	18	trigonometric	trigonometric	ADJ
cana-4508	14	19	coefficients	coefficient	NOUN
cana-4508	14	20	in	in	ADP
cana-4508	14	21	influencing	influence	VERB
cana-4508	14	22	stability	stability	NOUN
cana-4508	14	23	behavior	behavior	NOUN
cana-4508	14	24	.	.	PUNCT
cana-4508	15	1	our	our	PRON
cana-4508	15	2	results	result	NOUN
cana-4508	15	3	provide	provide	VERB
cana-4508	15	4	fresh	fresh	ADJ
cana-4508	15	5	insights	insight	NOUN
cana-4508	15	6	into	into	ADP
cana-4508	15	7	the	the	DET
cana-4508	15	8	structural	structural	ADJ
cana-4508	15	9	stability	stability	NOUN
cana-4508	15	10	of	of	ADP
cana-4508	15	11	functional	functional	ADJ
cana-4508	15	12	equations	equation	NOUN
cana-4508	15	13	and	and	CCONJ
cana-4508	15	14	broaden	broaden	VERB
cana-4508	15	15	the	the	DET
cana-4508	15	16	scope	scope	NOUN
cana-4508	15	17	of	of	ADP
cana-4508	15	18	existing	exist	VERB
cana-4508	15	19	stability	stability	NOUN
cana-4508	15	20	theories	theory	NOUN
cana-4508	15	21	.	.	PUNCT
cana-4508	16	1	this	this	DET
cana-4508	16	2	work	work	NOUN
cana-4508	16	3	lays	lay	VERB
cana-4508	16	4	the	the	DET
cana-4508	16	5	groundwork	groundwork	NOUN
cana-4508	16	6	for	for	ADP
cana-4508	16	7	future	future	ADJ
cana-4508	16	8	research	research	NOUN
cana-4508	16	9	on	on	ADP
cana-4508	16	10	mixed	mixed	ADJ
cana-4508	16	11	-	-	PUNCT
cana-4508	16	12	type	type	NOUN
cana-4508	16	13	functional	functional	ADJ
cana-4508	16	14	equations	equation	NOUN
cana-4508	16	15	in	in	ADP
cana-4508	16	16	both	both	CCONJ
cana-4508	16	17	theoretical	theoretical	ADJ
cana-4508	16	18	and	and	CCONJ
cana-4508	16	19	applied	apply	VERB
cana-4508	16	20	mathematical	mathematical	ADJ
cana-4508	16	21	contexts	contexts	NOUN
cana-4508	16	22	keywords	keyword	NOUN
cana-4508	16	23	:	:	PUNCT
cana-4508	16	24	additive	additive	ADJ
cana-4508	16	25	,	,	PUNCT
cana-4508	16	26	quadratic	quadratic	ADJ
cana-4508	16	27	functional	functional	ADJ
cana-4508	16	28	equations	equation	NOUN
cana-4508	16	29	,	,	PUNCT
cana-4508	16	30	generalized	generalize	VERB
cana-4508	16	31	hyers	hyer	NOUN
cana-4508	16	32	ulam	ulam	PROPN
cana-4508	16	33	rassias	rassias	PROPN
cana-4508	16	34	stability	stability	NOUN
cana-4508	16	35	1	1	NUM
cana-4508	16	36	.	.	PUNCT
cana-4508	17	1	introduction	introduction	NOUN
cana-4508	17	2	the	the	DET
cana-4508	17	3	study	study	NOUN
cana-4508	17	4	of	of	ADP
cana-4508	17	5	functional	functional	ADJ
cana-4508	17	6	equations	equation	NOUN
cana-4508	17	7	and	and	CCONJ
cana-4508	17	8	their	their	PRON
cana-4508	17	9	stability	stability	NOUN
cana-4508	17	10	has	have	AUX
cana-4508	17	11	been	be	AUX
cana-4508	17	12	a	a	DET
cana-4508	17	13	fundamental	fundamental	ADJ
cana-4508	17	14	aspect	aspect	NOUN
cana-4508	17	15	of	of	ADP
cana-4508	17	16	mathematical	mathematical	ADJ
cana-4508	17	17	analysis	analysis	NOUN
cana-4508	17	18	for	for	ADP
cana-4508	17	19	decades	decade	NOUN
cana-4508	17	20	.	.	PUNCT
cana-4508	18	1	the	the	DET
cana-4508	18	2	concept	concept	NOUN
cana-4508	18	3	of	of	ADP
cana-4508	18	4	stability	stability	NOUN
cana-4508	18	5	in	in	ADP
cana-4508	18	6	functional	functional	ADJ
cana-4508	18	7	equations	equation	NOUN
cana-4508	18	8	originated	originate	VERB
cana-4508	18	9	with	with	ADP
cana-4508	18	10	stanisław	stanisław	ADJ
cana-4508	18	11	ulam	ulam	PROPN
cana-4508	18	12	in	in	ADP
cana-4508	18	13	1940	1940	NUM
cana-4508	18	14	[	[	X
cana-4508	18	15	1	1	NUM
cana-4508	18	16	]	]	PUNCT
cana-4508	18	17	,	,	PUNCT
cana-4508	18	18	who	who	PRON
cana-4508	18	19	posed	pose	VERB
cana-4508	18	20	the	the	DET
cana-4508	18	21	question	question	NOUN
cana-4508	18	22	of	of	ADP
cana-4508	18	23	whether	whether	SCONJ
cana-4508	18	24	an	an	DET
cana-4508	18	25	approximate	approximate	ADJ
cana-4508	18	26	solution	solution	NOUN
cana-4508	18	27	to	to	ADP
cana-4508	18	28	a	a	DET
cana-4508	18	29	functional	functional	ADJ
cana-4508	18	30	equation	equation	NOUN
cana-4508	18	31	could	could	AUX
cana-4508	18	32	be	be	AUX
cana-4508	18	33	approximated	approximate	VERB
cana-4508	18	34	by	by	ADP
cana-4508	18	35	an	an	DET
cana-4508	18	36	exact	exact	ADJ
cana-4508	18	37	solution	solution	NOUN
cana-4508	18	38	.	.	PUNCT
cana-4508	19	1	in	in	ADP
cana-4508	19	2	1941	1941	NUM
cana-4508	19	3	,	,	PUNCT
cana-4508	19	4	donald	donald	PROPN
cana-4508	19	5	hyers	hyer	NOUN
cana-4508	19	6	[	[	X
cana-4508	19	7	2	2	X
cana-4508	19	8	]	]	PUNCT
cana-4508	19	9	provided	provide	VERB
cana-4508	19	10	the	the	DET
cana-4508	19	11	first	first	ADJ
cana-4508	19	12	affirmative	affirmative	ADJ
cana-4508	19	13	answer	answer	NOUN
cana-4508	19	14	to	to	ADP
cana-4508	19	15	ulam	ulam	PROPN
cana-4508	19	16	’s	’s	PART
cana-4508	19	17	question	question	NOUN
cana-4508	19	18	,	,	PUNCT
cana-4508	19	19	establishing	establish	VERB
cana-4508	19	20	the	the	DET
cana-4508	19	21	stability	stability	NOUN
cana-4508	19	22	of	of	ADP
cana-4508	19	23	linear	linear	ADJ
cana-4508	19	24	functional	functional	ADJ
cana-4508	19	25	equations	equation	NOUN
cana-4508	19	26	.	.	PUNCT
cana-4508	20	1	this	this	DET
cana-4508	20	2	foundational	foundational	ADJ
cana-4508	20	3	result	result	NOUN
cana-4508	20	4	,	,	PUNCT
cana-4508	20	5	now	now	ADV
cana-4508	20	6	known	know	VERB
cana-4508	20	7	as	as	ADP
cana-4508	20	8	ulam	ulam	NOUN
cana-4508	20	9	-	-	PUNCT
cana-4508	20	10	hyers	hyer	NOUN
cana-4508	20	11	stability	stability	NOUN
cana-4508	20	12	,	,	PUNCT
cana-4508	20	13	has	have	AUX
cana-4508	20	14	since	since	ADV
cana-4508	20	15	been	be	AUX
cana-4508	20	16	generalized	generalize	VERB
cana-4508	20	17	to	to	ADP
cana-4508	20	18	a	a	DET
cana-4508	20	19	wide	wide	ADJ
cana-4508	20	20	range	range	NOUN
cana-4508	20	21	of	of	ADP
cana-4508	20	22	functional	functional	ADJ
cana-4508	20	23	equations	equation	NOUN
cana-4508	20	24	[	[	X
cana-4508	20	25	3	3	NUM
cana-4508	20	26	,	,	PUNCT
cana-4508	20	27	4	4	NUM
cana-4508	20	28	,	,	PUNCT
cana-4508	20	29	5	5	NUM
cana-4508	20	30	]	]	PUNCT
cana-4508	20	31	,	,	PUNCT
cana-4508	20	32	including	include	VERB
cana-4508	20	33	quadratic	quadratic	ADJ
cana-4508	20	34	,	,	PUNCT
cana-4508	20	35	cubic	cubic	ADJ
cana-4508	20	36	,	,	PUNCT
cana-4508	20	37	and	and	CCONJ
cana-4508	20	38	mixed	mixed	ADJ
cana-4508	20	39	-	-	PUNCT
cana-4508	20	40	type	type	NOUN
cana-4508	20	41	equations	equation	NOUN
cana-4508	20	42	.	.	PUNCT
cana-4508	21	1	mixed	mix	VERB
cana-4508	21	2	-	-	PUNCT
cana-4508	21	3	type	type	NOUN
cana-4508	21	4	functional	functional	ADJ
cana-4508	21	5	equations	equation	NOUN
cana-4508	21	6	,	,	PUNCT
cana-4508	21	7	which	which	PRON
cana-4508	21	8	integrate	integrate	VERB
cana-4508	21	9	distinct	distinct	ADJ
cana-4508	21	10	mathematical	mathematical	ADJ
cana-4508	21	11	structures	structure	NOUN
cana-4508	21	12	such	such	ADJ
cana-4508	21	13	as	as	ADP
cana-4508	21	14	additive	additive	ADJ
cana-4508	21	15	and	and	CCONJ
cana-4508	21	16	quadratic	quadratic	ADJ
cana-4508	21	17	components	component	NOUN
cana-4508	21	18	,	,	PUNCT
cana-4508	21	19	have	have	AUX
cana-4508	21	20	garnered	garner	VERB
cana-4508	21	21	significant	significant	ADJ
cana-4508	21	22	attention	attention	NOUN
cana-4508	21	23	due	due	ADP
cana-4508	21	24	to	to	ADP
cana-4508	21	25	their	their	PRON
cana-4508	21	26	applications	application	NOUN
cana-4508	21	27	in	in	ADP
cana-4508	21	28	fields	field	NOUN
cana-4508	21	29	like	like	ADP
cana-4508	21	30	physics	physics	NOUN
cana-4508	21	31	,	,	PUNCT
cana-4508	21	32	economics	economic	NOUN
cana-4508	21	33	,	,	PUNCT
cana-4508	21	34	and	and	CCONJ
cana-4508	21	35	engineering	engineering	NOUN
cana-4508	21	36	.	.	PUNCT
cana-4508	22	1	the	the	DET
cana-4508	22	2	inclusion	inclusion	NOUN
cana-4508	22	3	of	of	ADP
cana-4508	22	4	trigonometric	trigonometric	ADJ
cana-4508	22	5	coefficients	coefficient	NOUN
cana-4508	22	6	introduces	introduce	VERB
cana-4508	22	7	additional	additional	ADJ
cana-4508	22	8	complexity	complexity	NOUN
cana-4508	22	9	and	and	CCONJ
cana-4508	22	10	depth	depth	NOUN
cana-4508	22	11	to	to	ADP
cana-4508	22	12	the	the	DET
cana-4508	22	13	analysis	analysis	NOUN
cana-4508	22	14	,	,	PUNCT
cana-4508	22	15	as	as	SCONJ
cana-4508	22	16	the	the	DET
cana-4508	22	17	inherent	inherent	ADJ
cana-4508	22	18	periodicity	periodicity	NOUN
cana-4508	22	19	and	and	CCONJ
cana-4508	22	20	symmetry	symmetry	NOUN
cana-4508	22	21	of	of	ADP
cana-4508	22	22	trigonometric	trigonometric	ADJ
cana-4508	22	23	functions	function	NOUN
cana-4508	22	24	play	play	VERB
cana-4508	22	25	a	a	DET
cana-4508	22	26	crucial	crucial	ADJ
cana-4508	22	27	role	role	NOUN
cana-4508	22	28	in	in	ADP
cana-4508	22	29	shaping	shape	VERB
cana-4508	22	30	stability	stability	NOUN
cana-4508	22	31	properties	property	NOUN
cana-4508	22	32	.	.	PUNCT
cana-4508	23	1	despite	despite	SCONJ
cana-4508	23	2	their	their	PRON
cana-4508	23	3	theoretical	theoretical	ADJ
cana-4508	23	4	and	and	CCONJ
cana-4508	23	5	practical	practical	ADJ
cana-4508	23	6	significance	significance	NOUN
cana-4508	23	7	,	,	PUNCT
cana-4508	23	8	the	the	DET
cana-4508	23	9	stability	stability	NOUN
cana-4508	23	10	of	of	ADP
cana-4508	23	11	mixed	mixed	ADJ
cana-4508	23	12	-	-	PUNCT
cana-4508	23	13	type	type	NOUN
cana-4508	23	14	functional	functional	ADJ
cana-4508	23	15	equations	equation	NOUN
cana-4508	23	16	with	with	ADP
cana-4508	23	17	trigonometric	trigonometric	ADJ
cana-4508	23	18	coefficients	coefficient	NOUN
cana-4508	23	19	remains	remain	VERB
cana-4508	23	20	a	a	DET
cana-4508	23	21	relatively	relatively	ADV
cana-4508	23	22	underexplored	underexplored	ADJ
cana-4508	23	23	area	area	NOUN
cana-4508	23	24	of	of	ADP
cana-4508	23	25	research	research	NOUN
cana-4508	23	26	[	[	X
cana-4508	23	27	6	6	NUM
cana-4508	23	28	,	,	PUNCT
cana-4508	23	29	7	7	NUM
cana-4508	23	30	,	,	PUNCT
cana-4508	23	31	8	8	NUM
cana-4508	23	32	]	]	PUNCT
cana-4508	23	33	.	.	PUNCT
cana-4508	24	1	communications	communication	NOUN
cana-4508	24	2	on	on	ADP
cana-4508	24	3	applied	apply	VERB
cana-4508	24	4	nonlinear	nonlinear	ADJ
cana-4508	24	5	analysis	analysis	NOUN
cana-4508	24	6	issn	issn	NOUN
cana-4508	24	7	:	:	PUNCT
cana-4508	24	8	1074	1074	NUM
cana-4508	24	9	-	-	PUNCT
cana-4508	24	10	133x	133x	NUM
cana-4508	24	11	vol	vol	NOUN
cana-4508	24	12	32	32	NUM
cana-4508	24	13	no	no	NOUN
cana-4508	24	14	.	.	PUNCT
cana-4508	25	1	9s	9s	NUM
cana-4508	25	2	(	(	PUNCT
cana-4508	25	3	2025	2025	NUM
cana-4508	25	4	)	)	PUNCT
cana-4508	25	5	2233	2233	NUM
cana-4508	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	26	1	the	the	DET
cana-4508	26	2	study	study	NOUN
cana-4508	26	3	of	of	ADP
cana-4508	26	4	ulam	ulam	PROPN
cana-4508	26	5	-	-	PUNCT
cana-4508	26	6	hyers	hyer	NOUN
cana-4508	26	7	stability	stability	NOUN
cana-4508	26	8	has	have	AUX
cana-4508	26	9	seen	see	VERB
cana-4508	26	10	significant	significant	ADJ
cana-4508	26	11	advancements	advancement	NOUN
cana-4508	26	12	,	,	PUNCT
cana-4508	26	13	emerging	emerge	VERB
cana-4508	26	14	as	as	ADP
cana-4508	26	15	a	a	DET
cana-4508	26	16	crucial	crucial	ADJ
cana-4508	26	17	area	area	NOUN
cana-4508	26	18	of	of	ADP
cana-4508	26	19	research	research	NOUN
cana-4508	26	20	in	in	ADP
cana-4508	26	21	functional	functional	ADJ
cana-4508	26	22	analysis	analysis	NOUN
cana-4508	26	23	and	and	CCONJ
cana-4508	26	24	its	its	PRON
cana-4508	26	25	applications	application	NOUN
cana-4508	26	26	[	[	X
cana-4508	26	27	9	9	NUM
cana-4508	26	28	,	,	PUNCT
cana-4508	26	29	10	10	NUM
cana-4508	26	30	,	,	PUNCT
cana-4508	26	31	11	11	NUM
cana-4508	26	32	,	,	PUNCT
cana-4508	26	33	12	12	NUM
cana-4508	26	34	,	,	PUNCT
cana-4508	26	35	13	13	NUM
cana-4508	26	36	]	]	PUNCT
cana-4508	26	37	.	.	PUNCT
cana-4508	27	1	functional	functional	ADJ
cana-4508	27	2	equations	equation	NOUN
cana-4508	27	3	with	with	ADP
cana-4508	27	4	mixed	mixed	ADJ
cana-4508	27	5	structures	structure	NOUN
cana-4508	27	6	,	,	PUNCT
cana-4508	27	7	such	such	ADJ
cana-4508	27	8	as	as	ADP
cana-4508	27	9	additive	additive	ADJ
cana-4508	27	10	-	-	PUNCT
cana-4508	27	11	quadratic	quadratic	ADJ
cana-4508	27	12	forms	form	NOUN
cana-4508	27	13	,	,	PUNCT
cana-4508	27	14	present	present	ADJ
cana-4508	27	15	unique	unique	ADJ
cana-4508	27	16	challenges	challenge	NOUN
cana-4508	27	17	and	and	CCONJ
cana-4508	27	18	opportunities	opportunity	NOUN
cana-4508	27	19	for	for	ADP
cana-4508	27	20	mathematical	mathematical	ADJ
cana-4508	27	21	investigation	investigation	NOUN
cana-4508	27	22	.	.	PUNCT
cana-4508	28	1	these	these	DET
cana-4508	28	2	equations	equation	NOUN
cana-4508	28	3	naturally	naturally	ADV
cana-4508	28	4	arise	arise	VERB
cana-4508	28	5	in	in	ADP
cana-4508	28	6	various	various	ADJ
cana-4508	28	7	fields	field	NOUN
cana-4508	28	8	,	,	PUNCT
cana-4508	28	9	modeling	model	VERB
cana-4508	28	10	systems	system	NOUN
cana-4508	28	11	where	where	SCONJ
cana-4508	28	12	linear	linear	NOUN
cana-4508	28	13	and	and	CCONJ
cana-4508	28	14	nonlinear	nonlinear	ADJ
cana-4508	28	15	behaviors	behavior	NOUN
cana-4508	28	16	interact	interact	VERB
cana-4508	28	17	.	.	PUNCT
cana-4508	29	1	this	this	DET
cana-4508	29	2	paper	paper	NOUN
cana-4508	29	3	focuses	focus	VERB
cana-4508	29	4	on	on	ADP
cana-4508	29	5	a	a	DET
cana-4508	29	6	novel	novel	ADJ
cana-4508	29	7	class	class	NOUN
cana-4508	29	8	of	of	ADP
cana-4508	29	9	mixed	mixed	ADJ
cana-4508	29	10	-	-	PUNCT
cana-4508	29	11	type	type	NOUN
cana-4508	29	12	additivequadratic	additivequadratic	ADJ
cana-4508	29	13	functional	functional	ADJ
cana-4508	29	14	equations	equation	NOUN
cana-4508	29	15	featuring	feature	VERB
cana-4508	29	16	trigonometric	trigonometric	ADJ
cana-4508	29	17	constant	constant	ADJ
cana-4508	29	18	coefficients	coefficient	NOUN
cana-4508	29	19	.	.	PUNCT
cana-4508	30	1	the	the	DET
cana-4508	30	2	incorporation	incorporation	NOUN
cana-4508	30	3	of	of	ADP
cana-4508	30	4	trigonometric	trigonometric	ADJ
cana-4508	30	5	terms	term	NOUN
cana-4508	30	6	introduces	introduce	VERB
cana-4508	30	7	distinctive	distinctive	ADJ
cana-4508	30	8	properties	property	NOUN
cana-4508	30	9	,	,	PUNCT
cana-4508	30	10	making	make	VERB
cana-4508	30	11	the	the	DET
cana-4508	30	12	stability	stability	NOUN
cana-4508	30	13	analysis	analysis	NOUN
cana-4508	30	14	both	both	CCONJ
cana-4508	30	15	complex	complex	ADJ
cana-4508	30	16	and	and	CCONJ
cana-4508	30	17	fascinating	fascinating	ADJ
cana-4508	30	18	.	.	PUNCT
cana-4508	31	1	trigonometric	trigonometric	ADJ
cana-4508	31	2	coefficients	coefficient	NOUN
cana-4508	31	3	are	be	AUX
cana-4508	31	4	not	not	PART
cana-4508	31	5	only	only	ADV
cana-4508	31	6	mathematically	mathematically	ADV
cana-4508	31	7	significant	significant	ADJ
cana-4508	31	8	but	but	CCONJ
cana-4508	31	9	also	also	ADV
cana-4508	31	10	hold	hold	VERB
cana-4508	31	11	practical	practical	ADJ
cana-4508	31	12	relevance	relevance	NOUN
cana-4508	31	13	in	in	ADP
cana-4508	31	14	modeling	model	VERB
cana-4508	31	15	periodic	periodic	ADJ
cana-4508	31	16	and	and	CCONJ
cana-4508	31	17	oscillatory	oscillatory	ADJ
cana-4508	31	18	phenomena	phenomenon	NOUN
cana-4508	31	19	,	,	PUNCT
cana-4508	31	20	such	such	ADJ
cana-4508	31	21	as	as	ADP
cana-4508	31	22	wave	wave	NOUN
cana-4508	31	23	functions	function	NOUN
cana-4508	31	24	and	and	CCONJ
cana-4508	31	25	signal	signal	ADJ
cana-4508	31	26	processing	processing	NOUN
cana-4508	31	27	.	.	PUNCT
cana-4508	32	1	the	the	DET
cana-4508	32	2	primary	primary	ADJ
cana-4508	32	3	objective	objective	NOUN
cana-4508	32	4	of	of	ADP
cana-4508	32	5	this	this	DET
cana-4508	32	6	study	study	NOUN
cana-4508	32	7	is	be	AUX
cana-4508	32	8	to	to	PART
cana-4508	32	9	establish	establish	VERB
cana-4508	32	10	the	the	DET
cana-4508	32	11	ulam	ulam	NOUN
cana-4508	32	12	-	-	PUNCT
cana-4508	32	13	hyers	hyer	NOUN
cana-4508	32	14	stability	stability	NOUN
cana-4508	32	15	of	of	ADP
cana-4508	32	16	these	these	DET
cana-4508	32	17	functional	functional	ADJ
cana-4508	32	18	equations	equation	NOUN
cana-4508	32	19	within	within	ADP
cana-4508	32	20	the	the	DET
cana-4508	32	21	framework	framework	NOUN
cana-4508	32	22	of	of	ADP
cana-4508	32	23	banach	banach	NOUN
cana-4508	32	24	spaces	space	NOUN
cana-4508	32	25	.	.	PUNCT
cana-4508	33	1	by	by	ADP
cana-4508	33	2	employing	employ	VERB
cana-4508	33	3	advanced	advanced	ADJ
cana-4508	33	4	techniques	technique	NOUN
cana-4508	33	5	in	in	ADP
cana-4508	33	6	functional	functional	ADJ
cana-4508	33	7	analysis	analysis	NOUN
cana-4508	33	8	and	and	CCONJ
cana-4508	33	9	leveraging	leverage	VERB
cana-4508	33	10	the	the	DET
cana-4508	33	11	inherent	inherent	ADJ
cana-4508	33	12	properties	property	NOUN
cana-4508	33	13	of	of	ADP
cana-4508	33	14	trigonometric	trigonometric	ADJ
cana-4508	33	15	coefficients	coefficient	NOUN
cana-4508	33	16	,	,	PUNCT
cana-4508	33	17	this	this	DET
cana-4508	33	18	work	work	NOUN
cana-4508	33	19	offers	offer	VERB
cana-4508	33	20	new	new	ADJ
cana-4508	33	21	insights	insight	NOUN
cana-4508	33	22	into	into	ADP
cana-4508	33	23	the	the	DET
cana-4508	33	24	stability	stability	NOUN
cana-4508	33	25	of	of	ADP
cana-4508	33	26	such	such	ADJ
cana-4508	33	27	equations	equation	NOUN
cana-4508	33	28	.	.	PUNCT
cana-4508	34	1	furthermore	furthermore	ADV
cana-4508	34	2	,	,	PUNCT
cana-4508	34	3	this	this	DET
cana-4508	34	4	research	research	NOUN
cana-4508	34	5	enhances	enhance	VERB
cana-4508	34	6	the	the	DET
cana-4508	34	7	broader	broad	ADJ
cana-4508	34	8	understanding	understanding	NOUN
cana-4508	34	9	of	of	ADP
cana-4508	34	10	how	how	SCONJ
cana-4508	34	11	trigonometric	trigonometric	ADJ
cana-4508	34	12	factors	factor	NOUN
cana-4508	34	13	influence	influence	VERB
cana-4508	34	14	the	the	DET
cana-4508	34	15	stability	stability	NOUN
cana-4508	34	16	of	of	ADP
cana-4508	34	17	mixed	mixed	ADJ
cana-4508	34	18	-	-	PUNCT
cana-4508	34	19	type	type	NOUN
cana-4508	34	20	functional	functional	ADJ
cana-4508	34	21	equations	equation	NOUN
cana-4508	34	22	,	,	PUNCT
cana-4508	34	23	paving	pave	VERB
cana-4508	34	24	the	the	DET
cana-4508	34	25	way	way	NOUN
cana-4508	34	26	for	for	ADP
cana-4508	34	27	further	further	ADJ
cana-4508	34	28	theoretical	theoretical	ADJ
cana-4508	34	29	developments	development	NOUN
cana-4508	34	30	and	and	CCONJ
cana-4508	34	31	practical	practical	ADJ
cana-4508	34	32	applications	application	NOUN
cana-4508	34	33	.	.	PUNCT
cana-4508	35	1	recently	recently	ADV
cana-4508	35	2	,	,	PUNCT
cana-4508	35	3	agilan	agilan	PROPN
cana-4508	35	4	et	et	PROPN
cana-4508	35	5	al	al	PROPN
cana-4508	35	6	.	.	PROPN
cana-4508	35	7	have	have	AUX
cana-4508	35	8	explored	explore	VERB
cana-4508	35	9	stability	stability	NOUN
cana-4508	35	10	results	result	NOUN
cana-4508	35	11	for	for	ADP
cana-4508	35	12	various	various	ADJ
cana-4508	35	13	additive	additive	ADJ
cana-4508	35	14	functional	functional	ADJ
cana-4508	35	15	equations	equation	NOUN
cana-4508	35	16	across	across	ADP
cana-4508	35	17	different	different	ADJ
cana-4508	35	18	normed	normed	ADJ
cana-4508	35	19	spaces	space	NOUN
cana-4508	35	20	,	,	PUNCT
cana-4508	35	21	as	as	SCONJ
cana-4508	35	22	evidenced	evidence	VERB
cana-4508	35	23	in	in	ADP
cana-4508	35	24	[	[	X
cana-4508	35	25	14	14	NUM
cana-4508	35	26	,	,	PUNCT
cana-4508	35	27	15	15	NUM
cana-4508	35	28	,	,	PUNCT
cana-4508	35	29	16	16	NUM
cana-4508	35	30	,	,	PUNCT
cana-4508	35	31	17	17	NUM
cana-4508	35	32	,	,	PUNCT
cana-4508	35	33	18	18	NUM
cana-4508	35	34	,	,	PUNCT
cana-4508	35	35	19	19	NUM
cana-4508	35	36	,	,	PUNCT
cana-4508	35	37	20	20	NUM
cana-4508	35	38	,	,	PUNCT
cana-4508	35	39	21,22	21,22	NOUN
cana-4508	35	40	]	]	X
cana-4508	35	41	.	.	PUNCT
cana-4508	36	1	through	through	ADP
cana-4508	36	2	these	these	DET
cana-4508	36	3	motivations	motivation	NOUN
cana-4508	36	4	,	,	PUNCT
cana-4508	36	5	the	the	DET
cana-4508	36	6	paper	paper	NOUN
cana-4508	36	7	aims	aim	VERB
cana-4508	36	8	to	to	PART
cana-4508	36	9	deepen	deepen	VERB
cana-4508	36	10	the	the	DET
cana-4508	36	11	understanding	understanding	NOUN
cana-4508	36	12	of	of	ADP
cana-4508	36	13	the	the	DET
cana-4508	36	14	stability	stability	NOUN
cana-4508	36	15	properties	property	NOUN
cana-4508	36	16	of	of	ADP
cana-4508	36	17	functional	functional	ADJ
cana-4508	36	18	equations	equation	NOUN
cana-4508	36	19	in	in	ADP
cana-4508	36	20	banach	banach	NOUN
cana-4508	36	21	spaces	space	NOUN
cana-4508	36	22	and	and	CCONJ
cana-4508	36	23	to	to	PART
cana-4508	36	24	demonstrate	demonstrate	VERB
cana-4508	36	25	the	the	DET
cana-4508	36	26	effectiveness	effectiveness	NOUN
cana-4508	36	27	of	of	ADP
cana-4508	36	28	combining	combine	VERB
cana-4508	36	29	direct	direct	ADJ
cana-4508	36	30	and	and	CCONJ
cana-4508	36	31	fixed	fix	VERB
cana-4508	36	32	point	point	NOUN
cana-4508	36	33	methods	method	NOUN
cana-4508	36	34	in	in	ADP
cana-4508	36	35	such	such	ADJ
cana-4508	36	36	analyses	analysis	NOUN
cana-4508	36	37	.	.	PUNCT
cana-4508	37	1	authors	author	NOUN
cana-4508	37	2	have	have	AUX
cana-4508	37	3	proved	prove	VERB
cana-4508	37	4	the	the	DET
cana-4508	37	5	generalized	generalize	VERB
cana-4508	37	6	ulam	ulam	PROPN
cana-4508	37	7	hyers	hyer	NOUN
cana-4508	37	8	stability	stability	NOUN
cana-4508	37	9	of	of	ADP
cana-4508	37	10	a	a	DET
cana-4508	37	11	mixed	mixed	ADJ
cana-4508	37	12	type	type	NOUN
cana-4508	37	13	general	general	ADJ
cana-4508	37	14	additive	additive	ADJ
cana-4508	37	15	quadratic	quadratic	ADJ
cana-4508	37	16	functional	functional	ADJ
cana-4508	37	17	equation	equation	NOUN
cana-4508	37	18	𝒬1	𝒬1	NOUN
cana-4508	37	19	(	(	PUNCT
cana-4508	37	20	𝒳𝑐𝑜𝑠𝑒𝑐	𝒳𝑐𝑜𝑠𝑒𝑐	PROPN
cana-4508	37	21	ℒ	ℒ	NOUN
cana-4508	37	22	(	(	PUNCT
cana-4508	37	23	𝜋	𝜋	NOUN
cana-4508	37	24	4	4	NUM
cana-4508	37	25	)	)	PUNCT
cana-4508	37	26	+	+	CCONJ
cana-4508	38	1	2𝒴𝑠𝑒𝑐ℒ	2𝒴𝑠𝑒𝑐ℒ	NUM
cana-4508	38	2	(	(	PUNCT
cana-4508	38	3	𝜋	𝜋	NOUN
cana-4508	38	4	4	4	NUM
cana-4508	38	5	)	)	PUNCT
cana-4508	38	6	)	)	PUNCT
cana-4508	39	1	+	+	CCONJ
cana-4508	39	2	𝒬1	𝒬1	NOUN
cana-4508	39	3	(	(	PUNCT
cana-4508	39	4	𝒴𝑠𝑒𝑐	𝒴𝑠𝑒𝑐	PROPN
cana-4508	39	5	ℒ	ℒ	NOUN
cana-4508	39	6	(	(	PUNCT
cana-4508	39	7	𝜋	𝜋	NOUN
cana-4508	39	8	4	4	NUM
cana-4508	39	9	)	)	PUNCT
cana-4508	39	10	−	−	NOUN
cana-4508	39	11	𝒳𝑐𝑜𝑠𝑒𝑐ℒ	𝒳𝑐𝑜𝑠𝑒𝑐ℒ	PUNCT
cana-4508	39	12	(	(	PUNCT
cana-4508	39	13	𝜋	𝜋	NOUN
cana-4508	39	14	4	4	NUM
cana-4508	39	15	)	)	PUNCT
cana-4508	39	16	)	)	PUNCT
cana-4508	40	1	+	+	CCONJ
cana-4508	40	2	𝒬1	𝒬1	NOUN
cana-4508	40	3	(	(	PUNCT
cana-4508	40	4	𝒳𝑐𝑜𝑠𝑒𝑐	𝒳𝑐𝑜𝑠𝑒𝑐	PROPN
cana-4508	40	5	ℒ	ℒ	NOUN
cana-4508	40	6	(	(	PUNCT
cana-4508	40	7	𝜋	𝜋	NOUN
cana-4508	40	8	4	4	NUM
cana-4508	40	9	)	)	PUNCT
cana-4508	40	10	−	−	PROPN
cana-4508	40	11	𝒴𝑠𝑒𝑐ℒ	𝒴𝑠𝑒𝑐ℒ	PROPN
cana-4508	40	12	(	(	PUNCT
cana-4508	40	13	𝜋	𝜋	NOUN
cana-4508	40	14	4	4	NUM
cana-4508	40	15	)	)	PUNCT
cana-4508	40	16	)	)	PUNCT
cana-4508	41	1	=	=	PRON
cana-4508	41	2	(	(	PUNCT
cana-4508	41	3	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	ADV
cana-4508	41	4	(	(	PUNCT
cana-4508	41	5	𝜋	𝜋	NOUN
cana-4508	41	6	4	4	NUM
cana-4508	41	7	)	)	PUNCT
cana-4508	41	8	+3𝑐𝑜𝑠𝑒𝑐2ℒ	+3𝑐𝑜𝑠𝑒𝑐2ℒ	NOUN
cana-4508	41	9	(	(	PUNCT
cana-4508	41	10	𝜋	𝜋	NOUN
cana-4508	41	11	4	4	NUM
cana-4508	41	12	)	)	PUNCT
cana-4508	41	13	2	2	NUM
cana-4508	41	14	)	)	PUNCT
cana-4508	41	15	𝒬1(𝒳	𝒬1(𝒳	NUM
cana-4508	41	16	)	)	PUNCT
cana-4508	42	1	+	+	CCONJ
cana-4508	42	2	(	(	PUNCT
cana-4508	42	3	3𝑐𝑜𝑠𝑒𝑐2ℒ	3𝑐𝑜𝑠𝑒𝑐2ℒ	NUM
cana-4508	42	4	(	(	PUNCT
cana-4508	42	5	𝜋	𝜋	NOUN
cana-4508	42	6	4	4	NUM
cana-4508	42	7	)	)	PUNCT
cana-4508	42	8	−𝑐𝑜𝑠𝑒𝑐ℒ	−𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	42	9	(	(	PUNCT
cana-4508	42	10	𝜋	𝜋	NOUN
cana-4508	42	11	4	4	NUM
cana-4508	42	12	)	)	PUNCT
cana-4508	42	13	2	2	NUM
cana-4508	42	14	)	)	PUNCT
cana-4508	42	15	𝒬1(−𝒳	𝒬1(−𝒳	X
cana-4508	42	16	)	)	PUNCT
cana-4508	42	17	+	+	CCONJ
cana-4508	42	18	(	(	PUNCT
cana-4508	42	19	𝑠𝑒𝑐ℒ	𝑠𝑒𝑐ℒ	ADV
cana-4508	42	20	(	(	PUNCT
cana-4508	42	21	𝜋	𝜋	NOUN
cana-4508	42	22	4	4	NUM
cana-4508	42	23	)	)	PUNCT
cana-4508	42	24	+	+	CCONJ
cana-4508	42	25	3𝑠𝑒𝑐2ℒ	3𝑠𝑒𝑐2ℒ	NUM
cana-4508	42	26	(	(	PUNCT
cana-4508	42	27	𝜋	𝜋	NOUN
cana-4508	42	28	4	4	NUM
cana-4508	42	29	)	)	PUNCT
cana-4508	42	30	)	)	PUNCT
cana-4508	42	31	𝒬1(𝒴	𝒬1(𝒴	PROPN
cana-4508	42	32	)	)	PUNCT
cana-4508	42	33	+	+	CCONJ
cana-4508	42	34	(	(	PUNCT
cana-4508	42	35	3𝑠𝑒𝑐	3𝑠𝑒𝑐	ADJ
cana-4508	42	36	2ℒ	2ℒ	NOUN
cana-4508	42	37	(	(	PUNCT
cana-4508	42	38	𝜋	𝜋	NOUN
cana-4508	42	39	4	4	NUM
cana-4508	42	40	)	)	PUNCT
cana-4508	42	41	−	−	PROPN
cana-4508	42	42	𝑠𝑒𝑐ℒ	𝑠𝑒𝑐ℒ	ADV
cana-4508	42	43	(	(	PUNCT
cana-4508	42	44	𝜋	𝜋	NOUN
cana-4508	42	45	4	4	NUM
cana-4508	42	46	)	)	PUNCT
cana-4508	42	47	)	)	PUNCT
cana-4508	42	48	𝒬1(−𝒴	𝒬1(−𝒴	PROPN
cana-4508	42	49	)	)	PUNCT
cana-4508	42	50	(	(	PUNCT
cana-4508	42	51	1	1	X
cana-4508	42	52	)	)	PUNCT
cana-4508	42	53	in	in	ADP
cana-4508	42	54	banach	banach	NOUN
cana-4508	42	55	spaces	space	NOUN
cana-4508	42	56	.	.	PUNCT
cana-4508	43	1	let	let	VERB
cana-4508	43	2	us	we	PRON
cana-4508	43	3	consider	consider	VERB
cana-4508	43	4	𝒳	𝒳	PRON
cana-4508	43	5	and	and	CCONJ
cana-4508	43	6	𝒴	𝒴	NOUN
cana-4508	43	7	to	to	PART
cana-4508	43	8	be	be	AUX
cana-4508	43	9	a	a	DET
cana-4508	43	10	normed	normed	ADJ
cana-4508	43	11	space	space	NOUN
cana-4508	43	12	and	and	CCONJ
cana-4508	43	13	a	a	DET
cana-4508	43	14	banach	banach	NOUN
cana-4508	43	15	space	space	NOUN
cana-4508	43	16	,	,	PUNCT
cana-4508	43	17	respectively	respectively	ADV
cana-4508	43	18	.	.	PUNCT
cana-4508	44	1	define	define	VERB
cana-4508	44	2	a	a	DET
cana-4508	44	3	mapping	mapping	NOUN
cana-4508	44	4	𝒬:ℋ	𝒬:ℋ	PUNCT
cana-4508	44	5	→	→	SYM
cana-4508	44	6	ℐ	ℐ	ADV
cana-4508	44	7	by	by	ADP
cana-4508	44	8	𝒬(𝒳	𝒬(𝒳	NOUN
cana-4508	44	9	,	,	PUNCT
cana-4508	44	10	𝒴	𝒴	PROPN
cana-4508	44	11	)	)	PUNCT
cana-4508	44	12	=	=	NUM
cana-4508	44	13	𝒬1	𝒬1	NOUN
cana-4508	44	14	(	(	PUNCT
cana-4508	44	15	𝒳𝑐𝑜𝑠𝑒𝑐	𝒳𝑐𝑜𝑠𝑒𝑐	PROPN
cana-4508	44	16	ℒ	ℒ	NOUN
cana-4508	44	17	(	(	PUNCT
cana-4508	44	18	𝜋	𝜋	NOUN
cana-4508	44	19	4	4	NUM
cana-4508	44	20	)	)	PUNCT
cana-4508	45	1	+	+	CCONJ
cana-4508	45	2	2𝒴𝑠𝑒𝑐ℒ	2𝒴𝑠𝑒𝑐ℒ	NUM
cana-4508	45	3	(	(	PUNCT
cana-4508	45	4	𝜋	𝜋	NOUN
cana-4508	45	5	4	4	NUM
cana-4508	45	6	)	)	PUNCT
cana-4508	45	7	)	)	PUNCT
cana-4508	46	1	+	+	CCONJ
cana-4508	46	2	𝒬1	𝒬1	NOUN
cana-4508	46	3	(	(	PUNCT
cana-4508	46	4	𝒴𝑠𝑒𝑐	𝒴𝑠𝑒𝑐	PROPN
cana-4508	46	5	ℒ	ℒ	NOUN
cana-4508	46	6	(	(	PUNCT
cana-4508	46	7	𝜋	𝜋	NOUN
cana-4508	46	8	4	4	NUM
cana-4508	46	9	)	)	PUNCT
cana-4508	46	10	−	−	NOUN
cana-4508	46	11	𝒳𝑐𝑜𝑠𝑒𝑐ℒ	𝒳𝑐𝑜𝑠𝑒𝑐ℒ	PUNCT
cana-4508	46	12	(	(	PUNCT
cana-4508	46	13	𝜋	𝜋	NOUN
cana-4508	46	14	4	4	NUM
cana-4508	46	15	)	)	PUNCT
cana-4508	46	16	)	)	PUNCT
cana-4508	47	1	+	+	VERB
cana-4508	47	2	𝒬1	𝒬1	NOUN
cana-4508	47	3	(	(	PUNCT
cana-4508	47	4	𝒳𝑐𝑜𝑠𝑒𝑐	𝒳𝑐𝑜𝑠𝑒𝑐	PROPN
cana-4508	47	5	ℒ	ℒ	NOUN
cana-4508	47	6	(	(	PUNCT
cana-4508	47	7	𝜋	𝜋	NOUN
cana-4508	47	8	4	4	NUM
cana-4508	47	9	)	)	PUNCT
cana-4508	47	10	−	−	PROPN
cana-4508	47	11	𝒴𝑠𝑒𝑐ℒ	𝒴𝑠𝑒𝑐ℒ	PROPN
cana-4508	47	12	(	(	PUNCT
cana-4508	47	13	𝜋	𝜋	NOUN
cana-4508	47	14	4	4	NUM
cana-4508	47	15	)	)	PUNCT
cana-4508	47	16	)	)	PUNCT
cana-4508	48	1	−	−	PROPN
cana-4508	49	1	(	(	PUNCT
cana-4508	49	2	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	49	3	(	(	PUNCT
cana-4508	49	4	𝜋	𝜋	NOUN
cana-4508	49	5	4	4	NUM
cana-4508	49	6	)	)	PUNCT
cana-4508	49	7	+3𝑐𝑜𝑠𝑒𝑐2ℒ	+3𝑐𝑜𝑠𝑒𝑐2ℒ	NOUN
cana-4508	49	8	(	(	PUNCT
cana-4508	49	9	𝜋	𝜋	NOUN
cana-4508	49	10	4	4	NUM
cana-4508	49	11	)	)	PUNCT
cana-4508	49	12	2	2	NUM
cana-4508	49	13	)	)	PUNCT
cana-4508	49	14	𝒬1(𝒳	𝒬1(𝒳	NOUN
cana-4508	49	15	)	)	PUNCT
cana-4508	49	16	−	−	PROPN
cana-4508	50	1	(	(	PUNCT
cana-4508	50	2	3𝑐𝑜𝑠𝑒𝑐2ℒ	3𝑐𝑜𝑠𝑒𝑐2ℒ	NUM
cana-4508	50	3	(	(	PUNCT
cana-4508	50	4	𝜋	𝜋	NOUN
cana-4508	50	5	4	4	NUM
cana-4508	50	6	)	)	PUNCT
cana-4508	50	7	−𝑐𝑜𝑠𝑒𝑐ℒ	−𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	50	8	(	(	PUNCT
cana-4508	50	9	𝜋	𝜋	NOUN
cana-4508	50	10	4	4	NUM
cana-4508	50	11	)	)	PUNCT
cana-4508	50	12	2	2	NUM
cana-4508	50	13	)	)	PUNCT
cana-4508	50	14	𝒬1(−𝒳	𝒬1(−𝒳	X
cana-4508	50	15	)	)	PUNCT
cana-4508	50	16	−	−	PROPN
cana-4508	50	17	(	(	PUNCT
cana-4508	50	18	𝑠𝑒𝑐ℒ	𝑠𝑒𝑐ℒ	ADV
cana-4508	50	19	(	(	PUNCT
cana-4508	50	20	𝜋	𝜋	NOUN
cana-4508	50	21	4	4	NUM
cana-4508	50	22	)	)	PUNCT
cana-4508	50	23	+	+	CCONJ
cana-4508	51	1	3𝑠𝑒𝑐2ℒ	3𝑠𝑒𝑐2ℒ	NUM
cana-4508	51	2	(	(	PUNCT
cana-4508	51	3	𝜋	𝜋	NOUN
cana-4508	51	4	4	4	NUM
cana-4508	51	5	)	)	PUNCT
cana-4508	51	6	)	)	PUNCT
cana-4508	52	1	𝒬1(𝒴	𝒬1(𝒴	PROPN
cana-4508	52	2	)	)	PUNCT
cana-4508	52	3	−	−	PROPN
cana-4508	53	1	(	(	PUNCT
cana-4508	53	2	3𝑠𝑒𝑐	3𝑠𝑒𝑐	ADJ
cana-4508	53	3	2ℒ	2ℒ	NOUN
cana-4508	53	4	(	(	PUNCT
cana-4508	53	5	𝜋	𝜋	NOUN
cana-4508	53	6	4	4	NUM
cana-4508	53	7	)	)	PUNCT
cana-4508	53	8	−	−	PROPN
cana-4508	53	9	𝑠𝑒𝑐ℒ	𝑠𝑒𝑐ℒ	ADV
cana-4508	54	1	(	(	PUNCT
cana-4508	54	2	𝜋	𝜋	NOUN
cana-4508	54	3	4	4	NUM
cana-4508	54	4	)	)	PUNCT
cana-4508	54	5	)	)	PUNCT
cana-4508	55	1	𝒬1(−𝒴	𝒬1(−𝒴	X
cana-4508	55	2	)	)	PUNCT
cana-4508	55	3	for	for	ADP
cana-4508	55	4	all	all	DET
cana-4508	55	5	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	55	6	∈	∈	PROPN
cana-4508	55	7	ℋ.	ℋ.	PROPN
cana-4508	55	8	communications	communication	NOUN
cana-4508	55	9	on	on	ADP
cana-4508	55	10	applied	apply	VERB
cana-4508	55	11	nonlinear	nonlinear	ADJ
cana-4508	55	12	analysis	analysis	NOUN
cana-4508	55	13	issn	issn	NOUN
cana-4508	55	14	:	:	PUNCT
cana-4508	55	15	1074	1074	NUM
cana-4508	55	16	-	-	PUNCT
cana-4508	55	17	133x	133x	NUM
cana-4508	55	18	vol	vol	NOUN
cana-4508	55	19	32	32	NUM
cana-4508	55	20	no	no	NOUN
cana-4508	55	21	.	.	PUNCT
cana-4508	56	1	9s	9s	NUM
cana-4508	56	2	(	(	PUNCT
cana-4508	56	3	2025	2025	NUM
cana-4508	56	4	)	)	PUNCT
cana-4508	56	5	2234	2234	NUM
cana-4508	56	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	56	7	2	2	X
cana-4508	56	8	.	.	PUNCT
cana-4508	56	9	stability	stability	NOUN
cana-4508	56	10	results	result	NOUN
cana-4508	56	11	:	:	PUNCT
cana-4508	56	12	odd	odd	ADJ
cana-4508	56	13	case	case	NOUN
cana-4508	56	14	theorem	theorem	VERB
cana-4508	56	15	2.1	2.1	NUM
cana-4508	56	16	let	let	VERB
cana-4508	56	17	𝒯:𝒳2	𝒯:𝒳2	NOUN
cana-4508	56	18	→	→	PUNCT
cana-4508	57	1	[	[	X
cana-4508	57	2	0,∞	0,∞	X
cana-4508	57	3	)	)	PUNCT
cana-4508	57	4	be	be	VERB
cana-4508	57	5	a	a	DET
cana-4508	57	6	function	function	NOUN
cana-4508	57	7	such	such	ADJ
cana-4508	57	8	that	that	SCONJ
cana-4508	57	9	∑∞𝒰=0	∑∞𝒰=0	PROPN
cana-4508	57	10	𝒯((cosecℒ	𝒯((cosecℒ	PROPN
cana-4508	57	11	(	(	PUNCT
cana-4508	57	12	π	π	PROPN
cana-4508	57	13	4	4	NUM
cana-4508	57	14	)	)	PUNCT
cana-4508	57	15	)	)	PUNCT
cana-4508	57	16	𝒰	𝒰	NOUN
cana-4508	57	17	𝒳,(cosecℒ	𝒳,(cosecℒ	PUNCT
cana-4508	57	18	(	(	PUNCT
cana-4508	57	19	π	π	NOUN
cana-4508	57	20	4	4	NUM
cana-4508	57	21	)	)	PUNCT
cana-4508	57	22	)	)	PUNCT
cana-4508	57	23	𝒰	𝒰	PROPN
cana-4508	57	24	𝒴	𝒴	PROPN
cana-4508	57	25	)	)	PUNCT
cana-4508	57	26	(	(	PUNCT
cana-4508	57	27	cosecℒ	cosecℒ	PROPN
cana-4508	57	28	(	(	PUNCT
cana-4508	57	29	π	π	PROPN
cana-4508	57	30	4	4	NUM
cana-4508	57	31	)	)	PUNCT
cana-4508	57	32	)	)	PUNCT
cana-4508	58	1	𝒰	𝒰	PROPN
cana-4508	58	2	converges	converge	VERB
cana-4508	58	3	in	in	ADP
cana-4508	58	4	ℛ	ℛ	PROPN
cana-4508	58	5	and	and	CCONJ
cana-4508	58	6	lim	lim	PROPN
cana-4508	58	7	𝒰→∞	𝒰→∞	PROPN
cana-4508	58	8	𝒯((cosecℒ	𝒯((cosecℒ	PROPN
cana-4508	58	9	(	(	PUNCT
cana-4508	58	10	π	π	PROPN
cana-4508	58	11	4	4	NUM
cana-4508	58	12	)	)	PUNCT
cana-4508	58	13	)	)	PUNCT
cana-4508	58	14	𝒰	𝒰	NOUN
cana-4508	58	15	𝒳,(cosecℒ	𝒳,(cosecℒ	PUNCT
cana-4508	58	16	(	(	PUNCT
cana-4508	58	17	π	π	NOUN
cana-4508	58	18	4	4	NUM
cana-4508	58	19	)	)	PUNCT
cana-4508	58	20	)	)	PUNCT
cana-4508	59	1	𝒰	𝒰	PROPN
cana-4508	59	2	𝒴	𝒴	PROPN
cana-4508	59	3	)	)	PUNCT
cana-4508	59	4	(	(	PUNCT
cana-4508	59	5	cosecℒ	cosecℒ	PROPN
cana-4508	59	6	(	(	PUNCT
cana-4508	59	7	π	π	PROPN
cana-4508	59	8	4	4	NUM
cana-4508	59	9	)	)	PUNCT
cana-4508	59	10	)	)	PUNCT
cana-4508	60	1	𝒰	𝒰	NOUN
cana-4508	60	2	=	=	SYM
cana-4508	60	3	0	0	PUNCT
cana-4508	60	4	(	(	PUNCT
cana-4508	60	5	1	1	NUM
cana-4508	60	6	)	)	PUNCT
cana-4508	60	7	for	for	ADP
cana-4508	60	8	all	all	DET
cana-4508	60	9	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	60	10	∈	∈	PROPN
cana-4508	60	11	ℋ.	ℋ.	PROPN
cana-4508	60	12	let	let	VERB
cana-4508	60	13	𝒬𝑎:ℋ	𝒬𝑎:ℋ	PRON
cana-4508	60	14	→	→	PUNCT
cana-4508	60	15	ℐ	ℐ	PRON
cana-4508	60	16	be	be	VERB
cana-4508	60	17	an	an	DET
cana-4508	60	18	odd	odd	ADJ
cana-4508	60	19	function	function	NOUN
cana-4508	60	20	satisfying	satisfy	VERB
cana-4508	60	21	the	the	DET
cana-4508	60	22	inequality	inequality	NOUN
cana-4508	60	23	‖𝒬𝑎(𝒳	‖𝒬𝑎(𝒳	NUM
cana-4508	60	24	,	,	PUNCT
cana-4508	60	25	𝒴)‖	𝒴)‖	VERB
cana-4508	60	26	≤	≤	NOUN
cana-4508	60	27	𝒯(𝒳,𝒴	𝒯(𝒳,𝒴	NUM
cana-4508	60	28	)	)	PUNCT
cana-4508	60	29	(	(	PUNCT
cana-4508	60	30	2	2	X
cana-4508	60	31	)	)	PUNCT
cana-4508	60	32	for	for	ADP
cana-4508	60	33	all	all	DET
cana-4508	60	34	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	60	35	∈	∈	PROPN
cana-4508	60	36	ℋ.	ℋ.	PROPN
cana-4508	60	37	then	then	ADV
cana-4508	60	38	there	there	PRON
cana-4508	60	39	exists	exist	VERB
cana-4508	60	40	a	a	DET
cana-4508	60	41	unique	unique	ADJ
cana-4508	60	42	additive	additive	ADJ
cana-4508	60	43	mapping	mapping	NOUN
cana-4508	60	44	𝐴:ℋ	𝐴:ℋ	PUNCT
cana-4508	61	1	→	→	PUNCT
cana-4508	61	2	ℐ	ℐ	PRON
cana-4508	61	3	such	such	ADJ
cana-4508	61	4	that	that	SCONJ
cana-4508	61	5	‖𝒬𝑎(𝒳	‖𝒬𝑎(𝒳	NUM
cana-4508	61	6	)	)	PUNCT
cana-4508	61	7	−	−	NOUN
cana-4508	61	8	𝐴(𝒳)‖	𝐴(𝒳)‖	NOUN
cana-4508	61	9	≤	≤	ADJ
cana-4508	61	10	1	1	NUM
cana-4508	61	11	(	(	PUNCT
cana-4508	61	12	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	61	13	(	(	PUNCT
cana-4508	61	14	𝜋	𝜋	NOUN
cana-4508	61	15	4	4	NUM
cana-4508	61	16	)	)	PUNCT
cana-4508	61	17	)	)	PUNCT
cana-4508	61	18	∑∞ℋ=0	∑∞ℋ=0	PUNCT
cana-4508	61	19	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	61	20	(	(	PUNCT
cana-4508	61	21	𝜋	𝜋	NOUN
cana-4508	61	22	4	4	NUM
cana-4508	61	23	)	)	PUNCT
cana-4508	61	24	)	)	PUNCT
cana-4508	62	1	ℋ	ℋ	NOUN
cana-4508	62	2	𝒳,0	𝒳,0	NUM
cana-4508	62	3	)	)	PUNCT
cana-4508	62	4	(	(	PUNCT
cana-4508	62	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	62	6	(	(	PUNCT
cana-4508	62	7	𝜋	𝜋	NOUN
cana-4508	62	8	4	4	NUM
cana-4508	62	9	)	)	PUNCT
cana-4508	62	10	)	)	PUNCT
cana-4508	63	1	ℋ	ℋ	PROPN
cana-4508	63	2	(	(	PUNCT
cana-4508	63	3	3	3	NUM
cana-4508	63	4	)	)	PUNCT
cana-4508	63	5	for	for	ADP
cana-4508	63	6	all	all	DET
cana-4508	63	7	𝒳	𝒳	PROPN
cana-4508	63	8	∈	∈	PROPN
cana-4508	63	9	ℋ.	ℋ.	PROPN
cana-4508	63	10	the	the	DET
cana-4508	63	11	mapping	mapping	NOUN
cana-4508	63	12	𝐴(𝒳	𝐴(𝒳	NOUN
cana-4508	63	13	)	)	PUNCT
cana-4508	63	14	is	be	AUX
cana-4508	63	15	defined	define	VERB
cana-4508	63	16	by	by	ADP
cana-4508	63	17	𝐴(𝒳	𝐴(𝒳	PROPN
cana-4508	63	18	)	)	PUNCT
cana-4508	64	1	=	=	VERB
cana-4508	64	2	lim	lim	PROPN
cana-4508	64	3	𝒰→∞	𝒰→∞	NOUN
cana-4508	65	1	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	65	2	ℒ	ℒ	X
cana-4508	65	3	(	(	PUNCT
cana-4508	65	4	𝜋	𝜋	NOUN
cana-4508	65	5	4	4	NUM
cana-4508	65	6	)	)	PUNCT
cana-4508	65	7	)	)	PUNCT
cana-4508	66	1	𝒰	𝒰	PROPN
cana-4508	66	2	𝒳	𝒳	PROPN
cana-4508	66	3	)	)	PUNCT
cana-4508	66	4	(	(	PUNCT
cana-4508	66	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	66	6	(	(	PUNCT
cana-4508	66	7	𝜋	𝜋	NOUN
cana-4508	66	8	4	4	NUM
cana-4508	66	9	)	)	PUNCT
cana-4508	66	10	)	)	PUNCT
cana-4508	67	1	𝒰	𝒰	NOUN
cana-4508	67	2	(	(	PUNCT
cana-4508	67	3	4	4	NUM
cana-4508	67	4	)	)	PUNCT
cana-4508	67	5	for	for	ADP
cana-4508	67	6	all	all	DET
cana-4508	67	7	𝒳	𝒳	PROPN
cana-4508	67	8	∈	∈	PROPN
cana-4508	67	9	ℋ.	ℋ.	PROPN
cana-4508	67	10	proof	proof	NOUN
cana-4508	67	11	.	.	PUNCT
cana-4508	68	1	replacing	replace	VERB
cana-4508	68	2	(	(	PUNCT
cana-4508	68	3	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	68	4	)	)	PUNCT
cana-4508	68	5	by	by	ADP
cana-4508	68	6	(	(	PUNCT
cana-4508	68	7	𝒳	𝒳	PROPN
cana-4508	68	8	,	,	PUNCT
cana-4508	68	9	0	0	NUM
cana-4508	68	10	)	)	PUNCT
cana-4508	68	11	in	in	ADP
cana-4508	68	12	(	(	PUNCT
cana-4508	68	13	2	2	NUM
cana-4508	68	14	)	)	PUNCT
cana-4508	68	15	and	and	CCONJ
cana-4508	68	16	using	use	VERB
cana-4508	68	17	oddness	oddness	NOUN
cana-4508	68	18	of	of	ADP
cana-4508	68	19	𝒬𝑎	𝒬𝑎	ADJ
cana-4508	68	20	,	,	PUNCT
cana-4508	68	21	we	we	PRON
cana-4508	68	22	get	get	VERB
cana-4508	68	23	‖𝒬𝑎(𝒳	‖𝒬𝑎(𝒳	NUM
cana-4508	68	24	)	)	PUNCT
cana-4508	68	25	−	−	NOUN
cana-4508	69	1	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	69	2	ℒ	ℒ	X
cana-4508	69	3	(	(	PUNCT
cana-4508	69	4	𝜋	𝜋	NOUN
cana-4508	69	5	4	4	NUM
cana-4508	69	6	)	)	PUNCT
cana-4508	69	7	)	)	PUNCT
cana-4508	69	8	𝒳	𝒳	PROPN
cana-4508	69	9	)	)	PUNCT
cana-4508	69	10	(	(	PUNCT
cana-4508	69	11	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	69	12	(	(	PUNCT
cana-4508	69	13	𝜋	𝜋	NOUN
cana-4508	69	14	4	4	NUM
cana-4508	69	15	)	)	PUNCT
cana-4508	69	16	)	)	PUNCT
cana-4508	69	17	‖	‖	PROPN
cana-4508	69	18	≤	≤	ADV
cana-4508	69	19	1	1	NUM
cana-4508	69	20	(	(	PUNCT
cana-4508	69	21	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	69	22	(	(	PUNCT
cana-4508	69	23	𝜋	𝜋	NOUN
cana-4508	69	24	4	4	NUM
cana-4508	69	25	)	)	PUNCT
cana-4508	69	26	)	)	PUNCT
cana-4508	69	27	𝒯(𝒳	𝒯(𝒳	NOUN
cana-4508	69	28	,	,	PUNCT
cana-4508	69	29	0	0	NUM
cana-4508	69	30	)	)	PUNCT
cana-4508	69	31	(	(	PUNCT
cana-4508	69	32	5	5	NUM
cana-4508	69	33	)	)	PUNCT
cana-4508	69	34	for	for	ADP
cana-4508	69	35	all𝒳	all𝒳	SYM
cana-4508	69	36	∈	∈	PROPN
cana-4508	69	37	ℋ.	ℋ.	PROPN
cana-4508	69	38	now	now	ADV
cana-4508	69	39	replacing	replace	VERB
cana-4508	69	40	𝒳	𝒳	PRON
cana-4508	69	41	by	by	ADP
cana-4508	69	42	(	(	PUNCT
cana-4508	69	43	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	69	44	(	(	PUNCT
cana-4508	69	45	𝜋	𝜋	NOUN
cana-4508	69	46	4	4	NUM
cana-4508	69	47	)	)	PUNCT
cana-4508	69	48	𝒳	𝒳	PROPN
cana-4508	69	49	)	)	PUNCT
cana-4508	69	50	and	and	CCONJ
cana-4508	69	51	dividing	divide	VERB
cana-4508	69	52	by	by	ADP
cana-4508	69	53	(	(	PUNCT
cana-4508	69	54	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	69	55	(	(	PUNCT
cana-4508	69	56	𝜋	𝜋	NOUN
cana-4508	69	57	4	4	NUM
cana-4508	69	58	)	)	PUNCT
cana-4508	69	59	)	)	PUNCT
cana-4508	69	60	in	in	ADP
cana-4508	69	61	(	(	PUNCT
cana-4508	69	62	5	5	NUM
cana-4508	69	63	)	)	PUNCT
cana-4508	69	64	,	,	PUNCT
cana-4508	69	65	we	we	PRON
cana-4508	69	66	obtain	obtain	VERB
cana-4508	69	67	‖	‖	ADJ
cana-4508	69	68	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	69	69	ℒ	ℒ	NOUN
cana-4508	69	70	(	(	PUNCT
cana-4508	69	71	𝜋	𝜋	NOUN
cana-4508	69	72	4	4	NUM
cana-4508	69	73	)	)	PUNCT
cana-4508	69	74	)	)	PUNCT
cana-4508	69	75	𝒳	𝒳	PROPN
cana-4508	69	76	)	)	PUNCT
cana-4508	70	1	(	(	PUNCT
cana-4508	70	2	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	70	3	(	(	PUNCT
cana-4508	70	4	𝜋	𝜋	NOUN
cana-4508	70	5	4	4	NUM
cana-4508	70	6	)	)	PUNCT
cana-4508	70	7	)	)	PUNCT
cana-4508	70	8	−	−	ADP
cana-4508	70	9	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	70	10	ℒ	ℒ	X
cana-4508	70	11	(	(	PUNCT
cana-4508	70	12	𝜋	𝜋	NOUN
cana-4508	70	13	4	4	NUM
cana-4508	70	14	)	)	PUNCT
cana-4508	70	15	)	)	PUNCT
cana-4508	70	16	2	2	NUM
cana-4508	70	17	𝒳	𝒳	PROPN
cana-4508	70	18	)	)	PUNCT
cana-4508	70	19	(	(	PUNCT
cana-4508	70	20	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	70	21	(	(	PUNCT
cana-4508	70	22	𝜋	𝜋	NOUN
cana-4508	70	23	4	4	NUM
cana-4508	70	24	)	)	PUNCT
cana-4508	70	25	)	)	PUNCT
cana-4508	70	26	2	2	NUM
cana-4508	70	27	‖	‖	PROPN
cana-4508	70	28	≤	≤	NOUN
cana-4508	70	29	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	70	30	(	(	PUNCT
cana-4508	70	31	𝜋	𝜋	NOUN
cana-4508	70	32	4	4	NUM
cana-4508	70	33	)	)	PUNCT
cana-4508	70	34	)	)	PUNCT
cana-4508	70	35	𝒳,0	𝒳,0	X
cana-4508	70	36	)	)	PUNCT
cana-4508	70	37	(	(	PUNCT
cana-4508	70	38	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	70	39	(	(	PUNCT
cana-4508	70	40	𝜋	𝜋	NOUN
cana-4508	70	41	4	4	NUM
cana-4508	70	42	)	)	PUNCT
cana-4508	70	43	)	)	PUNCT
cana-4508	70	44	2	2	NUM
cana-4508	70	45	(	(	PUNCT
cana-4508	70	46	6	6	NUM
cana-4508	70	47	)	)	PUNCT
cana-4508	70	48	for	for	ADP
cana-4508	70	49	all𝒳	all𝒳	SYM
cana-4508	70	50	∈	∈	PROPN
cana-4508	70	51	ℋ.	ℋ.	PROPN
cana-4508	70	52	it	it	PRON
cana-4508	70	53	follows	follow	VERB
cana-4508	70	54	from	from	ADP
cana-4508	70	55	(	(	PUNCT
cana-4508	70	56	5	5	NUM
cana-4508	70	57	)	)	PUNCT
cana-4508	70	58	and	and	CCONJ
cana-4508	70	59	(	(	PUNCT
cana-4508	70	60	6	6	NUM
cana-4508	70	61	)	)	PUNCT
cana-4508	71	1	that	that	SCONJ
cana-4508	71	2	‖𝒬𝑎(𝒳	‖𝒬𝑎(𝒳	NUM
cana-4508	71	3	)	)	PUNCT
cana-4508	71	4	−	−	NOUN
cana-4508	71	5	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	71	6	ℒ	ℒ	X
cana-4508	71	7	(	(	PUNCT
cana-4508	71	8	𝜋	𝜋	NOUN
cana-4508	71	9	4	4	NUM
cana-4508	71	10	)	)	PUNCT
cana-4508	71	11	)	)	PUNCT
cana-4508	71	12	2	2	NUM
cana-4508	71	13	𝒳	𝒳	PROPN
cana-4508	71	14	)	)	PUNCT
cana-4508	71	15	(	(	PUNCT
cana-4508	71	16	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	71	17	(	(	PUNCT
cana-4508	71	18	𝜋	𝜋	NOUN
cana-4508	71	19	4	4	NUM
cana-4508	71	20	)	)	PUNCT
cana-4508	71	21	)	)	PUNCT
cana-4508	71	22	2	2	NUM
cana-4508	71	23	‖	‖	PROPN
cana-4508	71	24	≤	≤	NOUN
cana-4508	71	25	‖𝒬𝑎(𝒳	‖𝒬𝑎(𝒳	NUM
cana-4508	71	26	)	)	PUNCT
cana-4508	71	27	−	−	NOUN
cana-4508	71	28	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	71	29	ℒ	ℒ	X
cana-4508	71	30	(	(	PUNCT
cana-4508	71	31	𝜋	𝜋	NOUN
cana-4508	71	32	4	4	NUM
cana-4508	71	33	)	)	PUNCT
cana-4508	71	34	)	)	PUNCT
cana-4508	71	35	𝒳	𝒳	PROPN
cana-4508	71	36	)	)	PUNCT
cana-4508	71	37	(	(	PUNCT
cana-4508	71	38	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	71	39	(	(	PUNCT
cana-4508	71	40	𝜋	𝜋	NOUN
cana-4508	71	41	4	4	NUM
cana-4508	71	42	)	)	PUNCT
cana-4508	71	43	)	)	PUNCT
cana-4508	72	1	‖	‖	PROPN
cana-4508	72	2	(	(	PUNCT
cana-4508	72	3	7	7	NUM
cana-4508	72	4	)	)	PUNCT
cana-4508	72	5	+	+	CCONJ
cana-4508	72	6	‖	‖	ADJ
cana-4508	72	7	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	72	8	ℒ	ℒ	NOUN
cana-4508	72	9	(	(	PUNCT
cana-4508	72	10	𝜋	𝜋	NOUN
cana-4508	72	11	4	4	NUM
cana-4508	72	12	)	)	PUNCT
cana-4508	72	13	)	)	PUNCT
cana-4508	72	14	𝒳	𝒳	PROPN
cana-4508	72	15	)	)	PUNCT
cana-4508	72	16	(	(	PUNCT
cana-4508	72	17	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	72	18	(	(	PUNCT
cana-4508	72	19	𝜋	𝜋	NOUN
cana-4508	72	20	4	4	NUM
cana-4508	72	21	)	)	PUNCT
cana-4508	72	22	)	)	PUNCT
cana-4508	73	1	−	−	ADP
cana-4508	73	2	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	73	3	ℒ	ℒ	X
cana-4508	73	4	(	(	PUNCT
cana-4508	73	5	𝜋	𝜋	NOUN
cana-4508	73	6	4	4	NUM
cana-4508	73	7	)	)	PUNCT
cana-4508	73	8	)	)	PUNCT
cana-4508	73	9	2	2	NUM
cana-4508	73	10	𝒳	𝒳	PROPN
cana-4508	73	11	)	)	PUNCT
cana-4508	73	12	(	(	PUNCT
cana-4508	73	13	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	73	14	(	(	PUNCT
cana-4508	73	15	𝜋	𝜋	NOUN
cana-4508	73	16	4	4	NUM
cana-4508	73	17	)	)	PUNCT
cana-4508	73	18	)	)	PUNCT
cana-4508	73	19	2	2	NUM
cana-4508	73	20	‖	‖	PROPN
cana-4508	73	21	≤	≤	NUM
cana-4508	73	22	1	1	NUM
cana-4508	73	23	(	(	PUNCT
cana-4508	73	24	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	73	25	(	(	PUNCT
cana-4508	73	26	𝜋	𝜋	NOUN
cana-4508	73	27	4	4	NUM
cana-4508	73	28	)	)	PUNCT
cana-4508	73	29	)	)	PUNCT
cana-4508	74	1	[	[	X
cana-4508	74	2	𝒯(𝒳	𝒯(𝒳	NOUN
cana-4508	74	3	,	,	PUNCT
cana-4508	74	4	0	0	NUM
cana-4508	74	5	)	)	PUNCT
cana-4508	74	6	+	+	CCONJ
cana-4508	74	7	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	74	8	(	(	PUNCT
cana-4508	74	9	𝜋	𝜋	NOUN
cana-4508	74	10	4	4	NUM
cana-4508	74	11	)	)	PUNCT
cana-4508	74	12	)	)	PUNCT
cana-4508	74	13	𝒳,0	𝒳,0	X
cana-4508	74	14	)	)	PUNCT
cana-4508	74	15	(	(	PUNCT
cana-4508	74	16	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	74	17	(	(	PUNCT
cana-4508	74	18	𝜋	𝜋	NOUN
cana-4508	74	19	4	4	NUM
cana-4508	74	20	)	)	PUNCT
cana-4508	74	21	)	)	PUNCT
cana-4508	74	22	]	]	PUNCT
cana-4508	74	23	(	(	PUNCT
cana-4508	74	24	8)	8)	NUM
cana-4508	74	25	for	for	ADP
cana-4508	74	26	all𝒳	all𝒳	VERB
cana-4508	74	27	∈	∈	PROPN
cana-4508	74	28	ℋ.	ℋ.	PROPN
cana-4508	74	29	in	in	ADP
cana-4508	74	30	general	general	ADJ
cana-4508	74	31	for	for	ADP
cana-4508	74	32	any	any	DET
cana-4508	74	33	positive	positive	ADJ
cana-4508	74	34	integer	integer	NOUN
cana-4508	74	35	𝑁	𝑁	PROPN
cana-4508	74	36	,	,	PUNCT
cana-4508	74	37	we	we	PRON
cana-4508	74	38	get	get	VERB
cana-4508	74	39	‖𝒬𝑎(𝒳	‖𝒬𝑎(𝒳	NUM
cana-4508	74	40	)	)	PUNCT
cana-4508	74	41	−	−	NOUN
cana-4508	75	1	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	75	2	ℒ	ℒ	X
cana-4508	75	3	(	(	PUNCT
cana-4508	75	4	𝜋	𝜋	NOUN
cana-4508	75	5	4	4	NUM
cana-4508	75	6	)	)	PUNCT
cana-4508	75	7	)	)	PUNCT
cana-4508	76	1	𝑁	𝑁	PROPN
cana-4508	76	2	𝒳	𝒳	PROPN
cana-4508	76	3	)	)	PUNCT
cana-4508	76	4	(	(	PUNCT
cana-4508	76	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	76	6	(	(	PUNCT
cana-4508	76	7	𝜋	𝜋	NOUN
cana-4508	76	8	4	4	NUM
cana-4508	76	9	)	)	PUNCT
cana-4508	76	10	)	)	PUNCT
cana-4508	77	1	𝑁	𝑁	PROPN
cana-4508	77	2	‖	‖	PROPN
cana-4508	77	3	≤	≤	ADV
cana-4508	77	4	1	1	NUM
cana-4508	77	5	(	(	PUNCT
cana-4508	77	6	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	77	7	(	(	PUNCT
cana-4508	77	8	𝜋	𝜋	NOUN
cana-4508	77	9	4	4	NUM
cana-4508	77	10	)	)	PUNCT
cana-4508	77	11	)	)	PUNCT
cana-4508	77	12	∑𝒰−1ℋ=0	∑𝒰−1ℋ=0	VERB
cana-4508	77	13	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	77	14	(	(	PUNCT
cana-4508	77	15	𝜋	𝜋	NOUN
cana-4508	77	16	4	4	NUM
cana-4508	77	17	)	)	PUNCT
cana-4508	77	18	)	)	PUNCT
cana-4508	78	1	ℋ	ℋ	NOUN
cana-4508	78	2	𝒳,0	𝒳,0	NUM
cana-4508	78	3	)	)	PUNCT
cana-4508	78	4	(	(	PUNCT
cana-4508	78	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	78	6	(	(	PUNCT
cana-4508	78	7	𝜋	𝜋	NOUN
cana-4508	78	8	4	4	NUM
cana-4508	78	9	)	)	PUNCT
cana-4508	78	10	)	)	PUNCT
cana-4508	79	1	ℋ	ℋ	NOUN
cana-4508	79	2	communications	communication	NOUN
cana-4508	79	3	on	on	ADP
cana-4508	79	4	applied	apply	VERB
cana-4508	79	5	nonlinear	nonlinear	ADJ
cana-4508	79	6	analysis	analysis	NOUN
cana-4508	79	7	issn	issn	NOUN
cana-4508	79	8	:	:	PUNCT
cana-4508	79	9	1074	1074	NUM
cana-4508	79	10	-	-	PUNCT
cana-4508	79	11	133x	133x	NUM
cana-4508	79	12	vol	vol	NOUN
cana-4508	79	13	32	32	NUM
cana-4508	79	14	no	no	NOUN
cana-4508	79	15	.	.	PUNCT
cana-4508	80	1	9s	9s	NUM
cana-4508	80	2	(	(	PUNCT
cana-4508	80	3	2025	2025	NUM
cana-4508	80	4	)	)	PUNCT
cana-4508	80	5	2235	2235	NUM
cana-4508	80	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	80	7	≤	≤	ADV
cana-4508	80	8	1	1	NUM
cana-4508	80	9	(	(	PUNCT
cana-4508	80	10	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	80	11	(	(	PUNCT
cana-4508	80	12	𝜋	𝜋	NOUN
cana-4508	80	13	4	4	NUM
cana-4508	80	14	)	)	PUNCT
cana-4508	80	15	)	)	PUNCT
cana-4508	81	1	∑∞ℋ=0	∑∞ℋ=0	PUNCT
cana-4508	81	2	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	81	3	(	(	PUNCT
cana-4508	81	4	𝜋	𝜋	NOUN
cana-4508	81	5	4	4	NUM
cana-4508	81	6	)	)	PUNCT
cana-4508	81	7	)	)	PUNCT
cana-4508	82	1	ℋ	ℋ	NOUN
cana-4508	82	2	𝒳,0	𝒳,0	NUM
cana-4508	82	3	)	)	PUNCT
cana-4508	82	4	(	(	PUNCT
cana-4508	82	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	82	6	(	(	PUNCT
cana-4508	82	7	𝜋	𝜋	NOUN
cana-4508	82	8	4	4	NUM
cana-4508	82	9	)	)	PUNCT
cana-4508	82	10	)	)	PUNCT
cana-4508	83	1	ℋ	ℋ	PROPN
cana-4508	83	2	(	(	PUNCT
cana-4508	83	3	9	9	NUM
cana-4508	83	4	)	)	PUNCT
cana-4508	83	5	for	for	ADP
cana-4508	83	6	all𝒳	all𝒳	SYM
cana-4508	83	7	∈	∈	PROPN
cana-4508	83	8	ℋ.	ℋ.	PROPN
cana-4508	83	9	in	in	ADP
cana-4508	83	10	order	order	NOUN
cana-4508	83	11	to	to	PART
cana-4508	83	12	prove	prove	VERB
cana-4508	83	13	the	the	DET
cana-4508	83	14	convergence	convergence	NOUN
cana-4508	83	15	of	of	ADP
cana-4508	83	16	the	the	DET
cana-4508	83	17	sequence	sequence	NOUN
cana-4508	83	18	{	{	PUNCT
cana-4508	83	19	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	83	20	ℒ	ℒ	X
cana-4508	83	21	(	(	PUNCT
cana-4508	83	22	𝜋	𝜋	NOUN
cana-4508	83	23	4	4	NUM
cana-4508	83	24	)	)	PUNCT
cana-4508	83	25	)	)	PUNCT
cana-4508	84	1	𝑁	𝑁	PROPN
cana-4508	84	2	𝒳	𝒳	PROPN
cana-4508	84	3	)	)	PUNCT
cana-4508	84	4	(	(	PUNCT
cana-4508	84	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	84	6	(	(	PUNCT
cana-4508	84	7	𝜋	𝜋	NOUN
cana-4508	84	8	4	4	NUM
cana-4508	84	9	)	)	PUNCT
cana-4508	84	10	)	)	PUNCT
cana-4508	85	1	𝑁	𝑁	PROPN
cana-4508	85	2	}	}	PUNCT
cana-4508	85	3	,	,	PUNCT
cana-4508	85	4	replace	replace	VERB
cana-4508	85	5	𝒳	𝒳	PRON
cana-4508	85	6	by	by	ADP
cana-4508	85	7	(	(	PUNCT
cana-4508	85	8	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	85	9	(	(	PUNCT
cana-4508	85	10	𝜋	𝜋	NOUN
cana-4508	85	11	4	4	NUM
cana-4508	85	12	)	)	PUNCT
cana-4508	85	13	)	)	PUNCT
cana-4508	85	14	𝑀	𝑀	PROPN
cana-4508	85	15	𝒳	𝒳	PROPN
cana-4508	85	16	and	and	CCONJ
cana-4508	85	17	divide	divide	VERB
cana-4508	85	18	by	by	ADP
cana-4508	85	19	(	(	PUNCT
cana-4508	85	20	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	85	21	(	(	PUNCT
cana-4508	85	22	𝜋	𝜋	NOUN
cana-4508	85	23	4	4	NUM
cana-4508	85	24	)	)	PUNCT
cana-4508	85	25	)	)	PUNCT
cana-4508	85	26	𝑀	𝑀	PROPN
cana-4508	85	27	in	in	ADP
cana-4508	85	28	(	(	PUNCT
cana-4508	85	29	1	1	NUM
cana-4508	85	30	)	)	PUNCT
cana-4508	85	31	,	,	PUNCT
cana-4508	85	32	for	for	ADP
cana-4508	85	33	any	any	DET
cana-4508	85	34	𝑀,𝑁	𝑀,𝑁	NOUN
cana-4508	85	35	>	>	X
cana-4508	85	36	0	0	PUNCT
cana-4508	85	37	,	,	PUNCT
cana-4508	85	38	to	to	PART
cana-4508	85	39	deduce	deduce	VERB
cana-4508	85	40	‖	‖	ADJ
cana-4508	85	41	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	85	42	ℒ	ℒ	NOUN
cana-4508	85	43	(	(	PUNCT
cana-4508	85	44	𝜋	𝜋	NOUN
cana-4508	85	45	4	4	NUM
cana-4508	85	46	)	)	PUNCT
cana-4508	85	47	)	)	PUNCT
cana-4508	85	48	𝑀	𝑀	PROPN
cana-4508	85	49	𝒳	𝒳	PROPN
cana-4508	85	50	)	)	PUNCT
cana-4508	85	51	(	(	PUNCT
cana-4508	85	52	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	85	53	(	(	PUNCT
cana-4508	85	54	𝜋	𝜋	NOUN
cana-4508	85	55	4	4	NUM
cana-4508	85	56	)	)	PUNCT
cana-4508	85	57	)	)	PUNCT
cana-4508	86	1	𝑀	𝑀	PROPN
cana-4508	86	2	−	−	PROPN
cana-4508	86	3	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	86	4	ℒ	ℒ	X
cana-4508	86	5	(	(	PUNCT
cana-4508	86	6	𝜋	𝜋	NOUN
cana-4508	86	7	4	4	NUM
cana-4508	86	8	)	)	PUNCT
cana-4508	86	9	)	)	PUNCT
cana-4508	86	10	𝑁+𝑀	𝑁+𝑀	X
cana-4508	86	11	𝒳	𝒳	PROPN
cana-4508	86	12	)	)	PUNCT
cana-4508	86	13	(	(	PUNCT
cana-4508	86	14	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	86	15	(	(	PUNCT
cana-4508	86	16	𝜋	𝜋	NOUN
cana-4508	86	17	4	4	NUM
cana-4508	86	18	)	)	PUNCT
cana-4508	86	19	)	)	PUNCT
cana-4508	86	20	(	(	PUNCT
cana-4508	86	21	𝑁+𝑀	𝑁+𝑀	X
cana-4508	86	22	)	)	PUNCT
cana-4508	86	23	‖	‖	PROPN
cana-4508	87	1	=	=	SYM
cana-4508	87	2	1	1	NUM
cana-4508	87	3	(	(	PUNCT
cana-4508	87	4	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	87	5	(	(	PUNCT
cana-4508	87	6	𝜋	𝜋	NOUN
cana-4508	87	7	4	4	NUM
cana-4508	87	8	)	)	PUNCT
cana-4508	87	9	)	)	PUNCT
cana-4508	88	1	𝑀	𝑀	PROPN
cana-4508	88	2	‖𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	‖𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	NOUN
cana-4508	88	3	ℒ	ℒ	PROPN
cana-4508	88	4	(	(	PUNCT
cana-4508	88	5	𝜋	𝜋	NOUN
cana-4508	88	6	4	4	NUM
cana-4508	88	7	)	)	PUNCT
cana-4508	88	8	)	)	PUNCT
cana-4508	88	9	𝑀	𝑀	PROPN
cana-4508	88	10	𝒳	𝒳	PROPN
cana-4508	88	11	)	)	PUNCT
cana-4508	88	12	−	−	NUM
cana-4508	89	1	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	89	2	ℒ	ℒ	X
cana-4508	89	3	(	(	PUNCT
cana-4508	89	4	𝜋	𝜋	NOUN
cana-4508	89	5	4	4	NUM
cana-4508	89	6	)	)	PUNCT
cana-4508	89	7	)	)	PUNCT
cana-4508	90	1	𝑁	𝑁	PROPN
cana-4508	90	2	⋅(𝑐𝑜𝑠𝑒𝑐ℒ	⋅(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	90	3	(	(	PUNCT
cana-4508	90	4	𝜋	𝜋	NOUN
cana-4508	90	5	4	4	NUM
cana-4508	90	6	)	)	PUNCT
cana-4508	90	7	)	)	PUNCT
cana-4508	90	8	𝑀	𝑀	PROPN
cana-4508	90	9	𝒳	𝒳	PROPN
cana-4508	90	10	)	)	PUNCT
cana-4508	90	11	(	(	PUNCT
cana-4508	90	12	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	90	13	(	(	PUNCT
cana-4508	90	14	𝜋	𝜋	NOUN
cana-4508	90	15	4	4	NUM
cana-4508	90	16	)	)	PUNCT
cana-4508	90	17	)	)	PUNCT
cana-4508	91	1	𝑁	𝑁	PROPN
cana-4508	91	2	‖	‖	PROPN
cana-4508	91	3	≤	≤	ADV
cana-4508	91	4	1	1	NUM
cana-4508	91	5	(	(	PUNCT
cana-4508	91	6	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	91	7	(	(	PUNCT
cana-4508	91	8	𝜋	𝜋	NOUN
cana-4508	91	9	4	4	NUM
cana-4508	91	10	)	)	PUNCT
cana-4508	91	11	)	)	PUNCT
cana-4508	91	12	∑𝒰−1ℋ=0	∑𝒰−1ℋ=0	VERB
cana-4508	91	13	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	91	14	(	(	PUNCT
cana-4508	91	15	𝜋	𝜋	NOUN
cana-4508	91	16	4	4	NUM
cana-4508	91	17	)	)	PUNCT
cana-4508	91	18	)	)	PUNCT
cana-4508	91	19	ℋ+𝑀	ℋ+𝑀	NOUN
cana-4508	91	20	𝒳,0	𝒳,0	NUM
cana-4508	91	21	)	)	PUNCT
cana-4508	91	22	(	(	PUNCT
cana-4508	91	23	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	91	24	(	(	PUNCT
cana-4508	91	25	𝜋	𝜋	NOUN
cana-4508	91	26	4	4	NUM
cana-4508	91	27	)	)	PUNCT
cana-4508	91	28	)	)	PUNCT
cana-4508	91	29	ℋ+𝑀	ℋ+𝑀	PUNCT
cana-4508	92	1	≤	≤	ADV
cana-4508	92	2	1	1	NUM
cana-4508	92	3	(	(	PUNCT
cana-4508	92	4	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	92	5	(	(	PUNCT
cana-4508	92	6	𝜋	𝜋	NOUN
cana-4508	92	7	4	4	NUM
cana-4508	92	8	)	)	PUNCT
cana-4508	92	9	)	)	PUNCT
cana-4508	92	10	∑∞ℋ=0	∑∞ℋ=0	PUNCT
cana-4508	93	1	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	93	2	(	(	PUNCT
cana-4508	93	3	𝜋	𝜋	NOUN
cana-4508	93	4	4	4	NUM
cana-4508	93	5	)	)	PUNCT
cana-4508	93	6	)	)	PUNCT
cana-4508	93	7	ℋ+𝑀	ℋ+𝑀	NOUN
cana-4508	93	8	𝒳,0	𝒳,0	NUM
cana-4508	93	9	)	)	PUNCT
cana-4508	93	10	(	(	PUNCT
cana-4508	93	11	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	93	12	(	(	PUNCT
cana-4508	93	13	𝜋	𝜋	NOUN
cana-4508	93	14	4	4	NUM
cana-4508	93	15	)	)	PUNCT
cana-4508	93	16	)	)	PUNCT
cana-4508	93	17	ℋ+𝑀	ℋ+𝑀	PUNCT
cana-4508	94	1	→	→	SYM
cana-4508	94	2	0	0	NUM
cana-4508	94	3	𝑎𝑠	𝑎𝑠	PROPN
cana-4508	94	4	𝑀	𝑀	PROPN
cana-4508	94	5	→	→	PUNCT
cana-4508	94	6	∞	∞	PROPN
cana-4508	94	7	for	for	ADP
cana-4508	94	8	all	all	DET
cana-4508	94	9	𝒳	𝒳	PROPN
cana-4508	94	10	∈	∈	PROPN
cana-4508	94	11	ℋ.	ℋ.	PROPN
cana-4508	94	12	hence	hence	ADV
cana-4508	94	13	the	the	DET
cana-4508	94	14	sequence	sequence	NOUN
cana-4508	94	15	{	{	PUNCT
cana-4508	94	16	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	94	17	ℒ	ℒ	X
cana-4508	94	18	(	(	PUNCT
cana-4508	94	19	𝜋	𝜋	NOUN
cana-4508	94	20	4	4	NUM
cana-4508	94	21	)	)	PUNCT
cana-4508	94	22	)	)	PUNCT
cana-4508	95	1	𝑁	𝑁	PROPN
cana-4508	95	2	𝒳	𝒳	PROPN
cana-4508	95	3	)	)	PUNCT
cana-4508	95	4	(	(	PUNCT
cana-4508	95	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	95	6	(	(	PUNCT
cana-4508	95	7	𝜋	𝜋	NOUN
cana-4508	95	8	4	4	NUM
cana-4508	95	9	)	)	PUNCT
cana-4508	95	10	)	)	PUNCT
cana-4508	96	1	𝑁	𝑁	PROPN
cana-4508	96	2	}	}	PUNCT
cana-4508	96	3	is	be	AUX
cana-4508	96	4	a	a	DET
cana-4508	96	5	cauchy	cauchy	ADJ
cana-4508	96	6	sequence	sequence	NOUN
cana-4508	96	7	.	.	PUNCT
cana-4508	97	1	since	since	SCONJ
cana-4508	97	2	ℐ	ℐ	PRON
cana-4508	97	3	is	be	AUX
cana-4508	97	4	complete	complete	ADJ
cana-4508	97	5	,	,	PUNCT
cana-4508	97	6	there	there	PRON
cana-4508	97	7	exists	exist	VERB
cana-4508	97	8	a	a	DET
cana-4508	97	9	mapping	mapping	NOUN
cana-4508	97	10	𝐴:ℋ	𝐴:ℋ	PUNCT
cana-4508	98	1	→	→	PUNCT
cana-4508	98	2	ℐ	ℐ	PRON
cana-4508	98	3	such	such	ADJ
cana-4508	98	4	that	that	DET
cana-4508	98	5	𝐴(𝒳	𝐴(𝒳	NOUN
cana-4508	98	6	)	)	PUNCT
cana-4508	99	1	=	=	SYM
cana-4508	99	2	lim	lim	PROPN
cana-4508	99	3	𝒰→∞	𝒰→∞	NOUN
cana-4508	99	4	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	99	5	ℒ	ℒ	X
cana-4508	99	6	(	(	PUNCT
cana-4508	99	7	𝜋	𝜋	NOUN
cana-4508	99	8	4	4	NUM
cana-4508	99	9	)	)	PUNCT
cana-4508	99	10	)	)	PUNCT
cana-4508	100	1	𝑁	𝑁	PROPN
cana-4508	100	2	𝒳	𝒳	PROPN
cana-4508	100	3	)	)	PUNCT
cana-4508	100	4	(	(	PUNCT
cana-4508	100	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	100	6	(	(	PUNCT
cana-4508	100	7	𝜋	𝜋	NOUN
cana-4508	100	8	4	4	NUM
cana-4508	100	9	)	)	PUNCT
cana-4508	100	10	)	)	PUNCT
cana-4508	101	1	𝑁	𝑁	ADP
cana-4508	101	2	∀	∀	NOUN
cana-4508	101	3	𝒳	𝒳	NOUN
cana-4508	101	4	∈	∈	NOUN
cana-4508	101	5	ℋ.	ℋ.	PROPN
cana-4508	101	6	letting	let	VERB
cana-4508	101	7	𝑁	𝑁	PROPN
cana-4508	101	8	→	→	SYM
cana-4508	101	9	∞	∞	NUM
cana-4508	101	10	in	in	ADP
cana-4508	101	11	(	(	PUNCT
cana-4508	101	12	1	1	X
cana-4508	101	13	)	)	PUNCT
cana-4508	101	14	we	we	PRON
cana-4508	101	15	see	see	VERB
cana-4508	101	16	that	that	SCONJ
cana-4508	101	17	(	(	PUNCT
cana-4508	101	18	3	3	X
cana-4508	101	19	)	)	PUNCT
cana-4508	101	20	holds	hold	VERB
cana-4508	101	21	for	for	ADP
cana-4508	101	22	all𝒳	all𝒳	SYM
cana-4508	101	23	∈	∈	PROPN
cana-4508	101	24	ℋ.	ℋ.	PROPN
cana-4508	101	25	to	to	PART
cana-4508	101	26	prove	prove	VERB
cana-4508	101	27	𝐴	𝐴	PROPN
cana-4508	101	28	satisfies	satisfie	NOUN
cana-4508	101	29	(	(	PUNCT
cana-4508	101	30	1	1	NUM
cana-4508	101	31	)	)	PUNCT
cana-4508	101	32	,	,	PUNCT
cana-4508	101	33	replacing	replace	VERB
cana-4508	101	34	(	(	PUNCT
cana-4508	101	35	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	101	36	)	)	PUNCT
cana-4508	101	37	by	by	ADP
cana-4508	101	38	(	(	PUNCT
cana-4508	101	39	(	(	PUNCT
cana-4508	101	40	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	101	41	(	(	PUNCT
cana-4508	101	42	𝜋	𝜋	NOUN
cana-4508	101	43	4	4	NUM
cana-4508	101	44	)	)	PUNCT
cana-4508	101	45	)	)	PUNCT
cana-4508	102	1	𝑁	𝑁	PROPN
cana-4508	102	2	𝒳	𝒳	PROPN
cana-4508	102	3	,	,	PUNCT
cana-4508	102	4	(	(	PUNCT
cana-4508	102	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	102	6	(	(	PUNCT
cana-4508	102	7	𝜋	𝜋	NOUN
cana-4508	102	8	4	4	NUM
cana-4508	102	9	)	)	PUNCT
cana-4508	102	10	)	)	PUNCT
cana-4508	103	1	𝑁	𝑁	PROPN
cana-4508	103	2	𝒴	𝒴	PROPN
cana-4508	103	3	)	)	PUNCT
cana-4508	103	4	and	and	CCONJ
cana-4508	103	5	dividing	divide	VERB
cana-4508	103	6	by	by	ADP
cana-4508	103	7	(	(	PUNCT
cana-4508	103	8	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	103	9	(	(	PUNCT
cana-4508	103	10	𝜋	𝜋	NOUN
cana-4508	103	11	4	4	NUM
cana-4508	103	12	)	)	PUNCT
cana-4508	103	13	)	)	PUNCT
cana-4508	104	1	𝑁	𝑁	NOUN
cana-4508	104	2	in	in	ADP
cana-4508	104	3	(	(	PUNCT
cana-4508	104	4	2	2	NUM
cana-4508	104	5	)	)	PUNCT
cana-4508	104	6	,	,	PUNCT
cana-4508	104	7	we	we	PRON
cana-4508	104	8	obtain	obtain	VERB
cana-4508	104	9	1	1	NUM
cana-4508	104	10	(	(	PUNCT
cana-4508	104	11	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	ADV
cana-4508	104	12	(	(	PUNCT
cana-4508	104	13	𝜋	𝜋	NOUN
cana-4508	104	14	4	4	NUM
cana-4508	104	15	)	)	PUNCT
cana-4508	104	16	)	)	PUNCT
cana-4508	105	1	𝑁	𝑁	ADJ
cana-4508	105	2	∥	∥	X
cana-4508	105	3	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	105	4	ℒ	ℒ	NOUN
cana-4508	105	5	(	(	PUNCT
cana-4508	105	6	𝜋	𝜋	NOUN
cana-4508	105	7	4	4	NUM
cana-4508	105	8	)	)	PUNCT
cana-4508	105	9	)	)	PUNCT
cana-4508	106	1	𝑁	𝑁	PROPN
cana-4508	106	2	𝒳	𝒳	PROPN
cana-4508	106	3	,	,	PUNCT
cana-4508	106	4	(	(	PUNCT
cana-4508	106	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	106	6	(	(	PUNCT
cana-4508	106	7	𝜋	𝜋	NOUN
cana-4508	106	8	4	4	NUM
cana-4508	106	9	)	)	PUNCT
cana-4508	106	10	)	)	PUNCT
cana-4508	107	1	𝑁	𝑁	PROPN
cana-4508	107	2	𝒴	𝒴	PROPN
cana-4508	107	3	)	)	PUNCT
cana-4508	107	4	∥	∥	PUNCT
cana-4508	107	5	≤	≤	NUM
cana-4508	107	6	1	1	NUM
cana-4508	107	7	(	(	PUNCT
cana-4508	107	8	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	107	9	(	(	PUNCT
cana-4508	107	10	𝜋	𝜋	NOUN
cana-4508	107	11	4	4	NUM
cana-4508	107	12	)	)	PUNCT
cana-4508	107	13	)	)	PUNCT
cana-4508	107	14	𝑁𝒯((𝑐𝑜𝑠𝑒𝑐	𝑁𝒯((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	107	15	ℒ	ℒ	X
cana-4508	107	16	(	(	PUNCT
cana-4508	107	17	𝜋	𝜋	NOUN
cana-4508	107	18	4	4	NUM
cana-4508	107	19	)	)	PUNCT
cana-4508	107	20	)	)	PUNCT
cana-4508	108	1	𝑁	𝑁	PROPN
cana-4508	108	2	𝒳	𝒳	PROPN
cana-4508	108	3	,	,	PUNCT
cana-4508	108	4	(	(	PUNCT
cana-4508	108	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	108	6	(	(	PUNCT
cana-4508	108	7	𝜋	𝜋	NOUN
cana-4508	108	8	4	4	NUM
cana-4508	108	9	)	)	PUNCT
cana-4508	108	10	)	)	PUNCT
cana-4508	109	1	𝑁	𝑁	PROPN
cana-4508	109	2	𝒴	𝒴	PROPN
cana-4508	109	3	)	)	PUNCT
cana-4508	109	4	for	for	ADP
cana-4508	109	5	all	all	DET
cana-4508	109	6	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	109	7	∈	∈	NOUN
cana-4508	109	8	ℋ.	ℋ.	PROPN
cana-4508	109	9	letting	let	VERB
cana-4508	109	10	𝑁	𝑁	PROPN
cana-4508	109	11	→	→	SYM
cana-4508	109	12	∞	∞	PROPN
cana-4508	109	13	in	in	ADP
cana-4508	109	14	the	the	DET
cana-4508	109	15	above	above	ADJ
cana-4508	109	16	inequality	inequality	NOUN
cana-4508	109	17	and	and	CCONJ
cana-4508	109	18	using	use	VERB
cana-4508	109	19	the	the	DET
cana-4508	109	20	definition	definition	NOUN
cana-4508	109	21	of	of	ADP
cana-4508	109	22	𝐴(𝒳	𝐴(𝒳	PROPN
cana-4508	109	23	)	)	PUNCT
cana-4508	109	24	,	,	PUNCT
cana-4508	109	25	we	we	PRON
cana-4508	109	26	see	see	VERB
cana-4508	109	27	that	that	DET
cana-4508	109	28	𝒬𝑎(𝒳,𝒴	𝒬𝑎(𝒳,𝒴	NOUN
cana-4508	109	29	)	)	PUNCT
cana-4508	109	30	=	=	SYM
cana-4508	110	1	0	0	X
cana-4508	110	2	.	.	PUNCT
cana-4508	111	1	hence	hence	ADV
cana-4508	111	2	𝐴	𝐴	PROPN
cana-4508	111	3	satisfies	satisfie	NOUN
cana-4508	111	4	(	(	PUNCT
cana-4508	111	5	1	1	NUM
cana-4508	111	6	)	)	PUNCT
cana-4508	111	7	for	for	ADP
cana-4508	111	8	all	all	DET
cana-4508	111	9	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	111	10	∈	∈	PROPN
cana-4508	111	11	ℋ.	ℋ.	PROPN
cana-4508	111	12	to	to	PART
cana-4508	111	13	show	show	VERB
cana-4508	111	14	𝐴	𝐴	PROPN
cana-4508	111	15	is	be	AUX
cana-4508	111	16	unique	unique	ADJ
cana-4508	111	17	,	,	PUNCT
cana-4508	111	18	let	let	VERB
cana-4508	111	19	𝐵(𝒳	𝐵(𝒳	NOUN
cana-4508	111	20	)	)	PUNCT
cana-4508	111	21	be	be	AUX
cana-4508	111	22	another	another	DET
cana-4508	111	23	additive	additive	ADJ
cana-4508	111	24	mapping	mapping	NOUN
cana-4508	111	25	satisfying	satisfying	ADJ
cana-4508	111	26	(	(	PUNCT
cana-4508	111	27	1	1	NUM
cana-4508	111	28	)	)	PUNCT
cana-4508	111	29	and	and	CCONJ
cana-4508	111	30	(	(	PUNCT
cana-4508	111	31	3	3	NUM
cana-4508	111	32	)	)	PUNCT
cana-4508	111	33	,	,	PUNCT
cana-4508	111	34	then	then	ADV
cana-4508	111	35	communications	communication	NOUN
cana-4508	111	36	on	on	ADP
cana-4508	111	37	applied	apply	VERB
cana-4508	111	38	nonlinear	nonlinear	ADJ
cana-4508	111	39	analysis	analysis	NOUN
cana-4508	111	40	issn	issn	NOUN
cana-4508	111	41	:	:	PUNCT
cana-4508	111	42	1074	1074	NUM
cana-4508	111	43	-	-	PUNCT
cana-4508	111	44	133x	133x	NUM
cana-4508	111	45	vol	vol	NOUN
cana-4508	111	46	32	32	NUM
cana-4508	112	1	no	no	NOUN
cana-4508	112	2	.	.	PUNCT
cana-4508	113	1	9s	9s	NUM
cana-4508	113	2	(	(	PUNCT
cana-4508	113	3	2025	2025	NUM
cana-4508	113	4	)	)	PUNCT
cana-4508	113	5	2236	2236	NUM
cana-4508	113	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	113	7	‖a(𝒳	‖a(𝒳	PROPN
cana-4508	113	8	)	)	PUNCT
cana-4508	113	9	−	−	NOUN
cana-4508	113	10	𝐵(𝒳)‖	𝐵(𝒳)‖	NOUN
cana-4508	113	11	=	=	SYM
cana-4508	113	12	1	1	NUM
cana-4508	113	13	(	(	PUNCT
cana-4508	113	14	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	113	15	(	(	PUNCT
cana-4508	113	16	𝜋	𝜋	NOUN
cana-4508	113	17	4	4	NUM
cana-4508	113	18	)	)	PUNCT
cana-4508	113	19	)	)	PUNCT
cana-4508	113	20	𝑁‖𝐴((𝑐𝑜𝑠𝑒𝑐	𝑁‖𝐴((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	113	21	ℒ	ℒ	NOUN
cana-4508	113	22	(	(	PUNCT
cana-4508	113	23	𝜋	𝜋	NOUN
cana-4508	113	24	4	4	NUM
cana-4508	113	25	)	)	PUNCT
cana-4508	113	26	)	)	PUNCT
cana-4508	114	1	𝑁	𝑁	PROPN
cana-4508	114	2	𝒳	𝒳	PROPN
cana-4508	114	3	)	)	PUNCT
cana-4508	114	4	−	−	PROPN
cana-4508	114	5	𝐵((𝑐𝑜𝑠𝑒𝑐ℒ	𝐵((𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	114	6	(	(	PUNCT
cana-4508	114	7	𝜋	𝜋	NOUN
cana-4508	114	8	4	4	NUM
cana-4508	114	9	)	)	PUNCT
cana-4508	114	10	)	)	PUNCT
cana-4508	115	1	𝑁	𝑁	PROPN
cana-4508	115	2	𝒳)‖	𝒳)‖	VERB
cana-4508	115	3	≤	≤	NUM
cana-4508	115	4	1	1	NUM
cana-4508	115	5	(	(	PUNCT
cana-4508	115	6	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	115	7	(	(	PUNCT
cana-4508	115	8	𝜋	𝜋	NOUN
cana-4508	115	9	4	4	NUM
cana-4508	115	10	)	)	PUNCT
cana-4508	115	11	)	)	PUNCT
cana-4508	116	1	𝑁	𝑁	NOUN
cana-4508	116	2	{	{	PUNCT
cana-4508	116	3	‖𝐴((𝑐𝑜𝑠𝑒𝑐	‖𝐴((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	116	4	ℒ	ℒ	NOUN
cana-4508	116	5	(	(	PUNCT
cana-4508	116	6	𝜋	𝜋	NOUN
cana-4508	116	7	4	4	NUM
cana-4508	116	8	)	)	PUNCT
cana-4508	116	9	)	)	PUNCT
cana-4508	117	1	𝑁	𝑁	PROPN
cana-4508	117	2	𝒳	𝒳	PROPN
cana-4508	117	3	)	)	PUNCT
cana-4508	117	4	−	−	NOUN
cana-4508	117	5	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	117	6	ℒ	ℒ	NOUN
cana-4508	117	7	(	(	PUNCT
cana-4508	117	8	𝜋	𝜋	NOUN
cana-4508	117	9	4	4	NUM
cana-4508	117	10	)	)	PUNCT
cana-4508	117	11	)	)	PUNCT
cana-4508	118	1	𝑁	𝑁	PROPN
cana-4508	118	2	𝒳)‖	𝒳)‖	NOUN
cana-4508	118	3	+	+	CCONJ
cana-4508	118	4	‖𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	‖𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	118	5	ℒ	ℒ	NOUN
cana-4508	118	6	(	(	PUNCT
cana-4508	118	7	𝜋	𝜋	NOUN
cana-4508	118	8	4	4	NUM
cana-4508	118	9	)	)	PUNCT
cana-4508	118	10	)	)	PUNCT
cana-4508	119	1	𝑁	𝑁	PROPN
cana-4508	119	2	𝒳	𝒳	PROPN
cana-4508	119	3	)	)	PUNCT
cana-4508	119	4	−	−	PROPN
cana-4508	119	5	𝐵((𝑐𝑜𝑠𝑒𝑐ℒ	𝐵((𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	119	6	(	(	PUNCT
cana-4508	119	7	𝜋	𝜋	NOUN
cana-4508	119	8	4	4	NUM
cana-4508	119	9	)	)	PUNCT
cana-4508	119	10	)	)	PUNCT
cana-4508	120	1	𝑁	𝑁	PROPN
cana-4508	120	2	𝒳)‖	𝒳)‖	NOUN
cana-4508	120	3	}	}	PUNCT
cana-4508	120	4	≤	≤	NOUN
cana-4508	120	5	2	2	NUM
cana-4508	120	6	(	(	PUNCT
cana-4508	120	7	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	120	8	(	(	PUNCT
cana-4508	120	9	𝜋	𝜋	NOUN
cana-4508	120	10	4	4	NUM
cana-4508	120	11	)	)	PUNCT
cana-4508	120	12	)	)	PUNCT
cana-4508	121	1	∑∞ℋ=0	∑∞ℋ=0	PUNCT
cana-4508	121	2	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	121	3	(	(	PUNCT
cana-4508	121	4	𝜋	𝜋	NOUN
cana-4508	121	5	4	4	NUM
cana-4508	121	6	)	)	PUNCT
cana-4508	121	7	)	)	PUNCT
cana-4508	121	8	ℋ+𝑁	ℋ+𝑁	PROPN
cana-4508	121	9	𝒳,0	𝒳,0	NUM
cana-4508	121	10	)	)	PUNCT
cana-4508	121	11	(	(	PUNCT
cana-4508	121	12	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	121	13	(	(	PUNCT
cana-4508	121	14	𝜋	𝜋	NOUN
cana-4508	121	15	4	4	NUM
cana-4508	121	16	)	)	PUNCT
cana-4508	121	17	)	)	PUNCT
cana-4508	121	18	(	(	PUNCT
cana-4508	121	19	ℋ+𝑁	ℋ+𝑁	PROPN
cana-4508	121	20	)	)	PUNCT
cana-4508	121	21	→	→	SYM
cana-4508	121	22	0	0	NUM
cana-4508	121	23	𝑎𝑠	𝑎𝑠	ADP
cana-4508	121	24	𝑁	𝑁	PROPN
cana-4508	121	25	→	→	SYM
cana-4508	121	26	∞	∞	PROPN
cana-4508	121	27	for	for	ADP
cana-4508	121	28	all	all	DET
cana-4508	121	29	𝒳	𝒳	PROPN
cana-4508	121	30	∈	∈	PROPN
cana-4508	121	31	ℋ.	ℋ.	PROPN
cana-4508	121	32	hence	hence	ADV
cana-4508	121	33	𝐴	𝐴	PROPN
cana-4508	121	34	is	be	AUX
cana-4508	121	35	unique	unique	ADJ
cana-4508	121	36	.	.	PUNCT
cana-4508	122	1	corollary	corollary	ADJ
cana-4508	122	2	2.2	2.2	NUM
cana-4508	122	3	let	let	VERB
cana-4508	122	4	𝒯	𝒯	PROPN
cana-4508	122	5	and	and	CCONJ
cana-4508	122	6	p	p	NOUN
cana-4508	122	7	be	be	VERB
cana-4508	122	8	non	non	ADJ
cana-4508	122	9	negative	negative	ADJ
cana-4508	122	10	real	real	ADJ
cana-4508	122	11	numbers	number	NOUN
cana-4508	122	12	.	.	PUNCT
cana-4508	123	1	let	let	VERB
cana-4508	123	2	an	an	DET
cana-4508	123	3	odd	odd	ADJ
cana-4508	123	4	function	function	NOUN
cana-4508	123	5	𝒬a:ℋ	𝒬a:ℋ	PUNCT
cana-4508	124	1	→	→	SYM
cana-4508	124	2	ℐ	ℐ	PRON
cana-4508	124	3	satisfy	satisfy	VERB
cana-4508	124	4	the	the	DET
cana-4508	124	5	inequality	inequality	NOUN
cana-4508	124	6	‖𝒬a(𝒳	‖𝒬a(𝒳	PROPN
cana-4508	124	7	,	,	PUNCT
cana-4508	124	8	𝒴)‖	𝒴)‖	X
cana-4508	124	9	≤	≤	NOUN
cana-4508	124	10	{	{	PUNCT
cana-4508	124	11	𝒯	𝒯	PROPN
cana-4508	124	12	,	,	PUNCT
cana-4508	124	13	𝒯{||𝒳||p	𝒯{||𝒳||p	PUNCT
cana-4508	124	14	+	+	NUM
cana-4508	124	15	||𝒴||p	||𝒴||p	NOUN
cana-4508	124	16	}	}	PUNCT
cana-4508	124	17	,	,	PUNCT
cana-4508	124	18	p	p	PROPN
cana-4508	124	19	≠	≠	PROPN
cana-4508	124	20	1	1	NUM
cana-4508	124	21	;	;	PUNCT
cana-4508	124	22	𝒯{||𝒳||p||𝒴||p	𝒯{||𝒳||p||𝒴||p	X
cana-4508	124	23	+	+	CCONJ
cana-4508	124	24	{	{	PUNCT
cana-4508	124	25	||𝒳||2p	||𝒳||2p	PROPN
cana-4508	124	26	+	+	CCONJ
cana-4508	124	27	||𝒴||2p	||𝒴||2p	NOUN
cana-4508	124	28	}	}	PUNCT
cana-4508	124	29	}	}	PUNCT
cana-4508	124	30	,	,	PUNCT
cana-4508	124	31	p	p	PROPN
cana-4508	124	32	≠	≠	PROPN
cana-4508	124	33	1	1	NUM
cana-4508	124	34	2	2	NUM
cana-4508	124	35	;	;	PUNCT
cana-4508	124	36	(	(	PUNCT
cana-4508	124	37	10	10	NUM
cana-4508	124	38	)	)	PUNCT
cana-4508	124	39	for	for	ADP
cana-4508	124	40	all	all	DET
cana-4508	124	41	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	124	42	∈	∈	PROPN
cana-4508	124	43	ℋ.	ℋ.	PROPN
cana-4508	124	44	then	then	ADV
cana-4508	124	45	there	there	PRON
cana-4508	124	46	exists	exist	VERB
cana-4508	124	47	a	a	DET
cana-4508	124	48	unique	unique	ADJ
cana-4508	124	49	additive	additive	ADJ
cana-4508	124	50	function	function	NOUN
cana-4508	124	51	a:ℋ	a:ℋ	NUM
cana-4508	124	52	→	→	SYM
cana-4508	124	53	ℐ	ℐ	PRON
cana-4508	124	54	such	such	ADJ
cana-4508	124	55	that	that	DET
cana-4508	124	56	‖𝒬1(𝒳	‖𝒬1(𝒳	PROPN
cana-4508	124	57	)	)	PUNCT
cana-4508	124	58	−	−	PROPN
cana-4508	125	1	a(𝒳)‖	a(𝒳)‖	NOUN
cana-4508	125	2	≤	≤	X
cana-4508	125	3	{	{	PUNCT
cana-4508	125	4	𝒯	𝒯	PROPN
cana-4508	125	5	(	(	PUNCT
cana-4508	125	6	cosecℒ	cosecℒ	PROPN
cana-4508	125	7	(	(	PUNCT
cana-4508	125	8	π	π	PROPN
cana-4508	125	9	4	4	NUM
cana-4508	125	10	)	)	PUNCT
cana-4508	125	11	)	)	PUNCT
cana-4508	125	12	−1	−1	NOUN
cana-4508	125	13	,	,	PUNCT
cana-4508	125	14	𝒯||𝒳||p	𝒯||𝒳||p	X
cana-4508	125	15	|(cosecℒ	|(cosecℒ	NOUN
cana-4508	125	16	(	(	PUNCT
cana-4508	125	17	π	π	PROPN
cana-4508	125	18	4	4	NUM
cana-4508	125	19	)	)	PUNCT
cana-4508	125	20	)	)	PUNCT
cana-4508	126	1	−(cosecℒ	−(cosecℒ	PROPN
cana-4508	126	2	(	(	PUNCT
cana-4508	126	3	π	π	PROPN
cana-4508	126	4	4	4	NUM
cana-4508	126	5	)	)	PUNCT
cana-4508	126	6	)	)	PUNCT
cana-4508	127	1	p	p	NOUN
cana-4508	128	1	|	|	ADV
cana-4508	128	2	,	,	PUNCT
cana-4508	128	3	𝒯||𝒳||2p	𝒯||𝒳||2p	X
cana-4508	128	4	|(cosecℒ	|(cosecℒ	NOUN
cana-4508	128	5	(	(	PUNCT
cana-4508	128	6	π	π	PROPN
cana-4508	128	7	4	4	NUM
cana-4508	128	8	)	)	PUNCT
cana-4508	128	9	)	)	PUNCT
cana-4508	129	1	−(cosecℒ	−(cosecℒ	PROPN
cana-4508	129	2	(	(	PUNCT
cana-4508	129	3	π	π	PROPN
cana-4508	129	4	4	4	NUM
cana-4508	129	5	)	)	PUNCT
cana-4508	129	6	)	)	PUNCT
cana-4508	129	7	2p	2p	NUM
cana-4508	130	1	|	|	ADV
cana-4508	130	2	,	,	PUNCT
cana-4508	130	3	(	(	PUNCT
cana-4508	130	4	11	11	NUM
cana-4508	130	5	)	)	PUNCT
cana-4508	130	6	for	for	ADP
cana-4508	130	7	all	all	DET
cana-4508	130	8	𝒳	𝒳	PROPN
cana-4508	130	9	∈	∈	PROPN
cana-4508	130	10	ℋ	ℋ	PROPN
cana-4508	130	11	3.stability	3.stability	PROPN
cana-4508	130	12	results	result	NOUN
cana-4508	130	13	:	:	PUNCT
cana-4508	130	14	even	even	ADV
cana-4508	130	15	case	case	NOUN
cana-4508	130	16	theorem	theorem	VERB
cana-4508	130	17	3.1	3.1	NUM
cana-4508	130	18	let	let	VERB
cana-4508	130	19	𝒯	𝒯	PROPN
cana-4508	130	20	:	:	PUNCT
cana-4508	130	21	x2	x2	PROPN
cana-4508	130	22	→	→	PUNCT
cana-4508	131	1	[	[	X
cana-4508	131	2	0,∞	0,∞	X
cana-4508	131	3	)	)	PUNCT
cana-4508	131	4	be	be	VERB
cana-4508	131	5	a	a	DET
cana-4508	131	6	function	function	NOUN
cana-4508	131	7	such	such	ADJ
cana-4508	131	8	that	that	SCONJ
cana-4508	131	9	∑∞𝒰=0	∑∞𝒰=0	PROPN
cana-4508	131	10	𝒯((cosecℒ	𝒯((cosecℒ	PROPN
cana-4508	131	11	(	(	PUNCT
cana-4508	131	12	π	π	PROPN
cana-4508	131	13	4	4	NUM
cana-4508	131	14	)	)	PUNCT
cana-4508	131	15	)	)	PUNCT
cana-4508	131	16	𝒰	𝒰	NOUN
cana-4508	131	17	𝒳,(cosecℒ	𝒳,(cosecℒ	PUNCT
cana-4508	131	18	(	(	PUNCT
cana-4508	131	19	π	π	NOUN
cana-4508	131	20	4	4	NUM
cana-4508	131	21	)	)	PUNCT
cana-4508	131	22	)	)	PUNCT
cana-4508	131	23	𝒰	𝒰	PROPN
cana-4508	131	24	𝒴	𝒴	PROPN
cana-4508	131	25	)	)	PUNCT
cana-4508	131	26	(	(	PUNCT
cana-4508	131	27	cosecℒ	cosecℒ	PROPN
cana-4508	131	28	(	(	PUNCT
cana-4508	131	29	π	π	PROPN
cana-4508	131	30	4	4	NUM
cana-4508	131	31	)	)	PUNCT
cana-4508	131	32	)	)	PUNCT
cana-4508	132	1	2n	2n	NUM
cana-4508	132	2	converges	converge	NOUN
cana-4508	132	3	in	in	ADP
cana-4508	132	4	ℛ	ℛ	PROPN
cana-4508	132	5	and	and	CCONJ
cana-4508	132	6	lim	lim	PROPN
cana-4508	132	7	𝒰→∞	𝒰→∞	PROPN
cana-4508	132	8	𝒯((cosecℒ	𝒯((cosecℒ	PROPN
cana-4508	132	9	(	(	PUNCT
cana-4508	132	10	π	π	PROPN
cana-4508	132	11	4	4	NUM
cana-4508	132	12	)	)	PUNCT
cana-4508	132	13	)	)	PUNCT
cana-4508	132	14	𝒰	𝒰	NOUN
cana-4508	132	15	𝒳,(cosecℒ	𝒳,(cosecℒ	PUNCT
cana-4508	132	16	(	(	PUNCT
cana-4508	132	17	π	π	NOUN
cana-4508	132	18	4	4	NUM
cana-4508	132	19	)	)	PUNCT
cana-4508	132	20	)	)	PUNCT
cana-4508	133	1	𝒰	𝒰	PROPN
cana-4508	133	2	𝒴	𝒴	PROPN
cana-4508	133	3	)	)	PUNCT
cana-4508	133	4	(	(	PUNCT
cana-4508	133	5	cosecℒ	cosecℒ	PROPN
cana-4508	133	6	(	(	PUNCT
cana-4508	133	7	π	π	PROPN
cana-4508	133	8	4	4	NUM
cana-4508	133	9	)	)	PUNCT
cana-4508	133	10	)	)	PUNCT
cana-4508	133	11	2n	2n	NUM
cana-4508	134	1	=	=	SYM
cana-4508	134	2	0	0	PUNCT
cana-4508	134	3	(	(	PUNCT
cana-4508	134	4	12	12	NUM
cana-4508	134	5	)	)	PUNCT
cana-4508	134	6	for	for	ADP
cana-4508	134	7	all	all	DET
cana-4508	134	8	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	134	9	∈	∈	PROPN
cana-4508	134	10	ℋ.	ℋ.	PROPN
cana-4508	134	11	let	let	VERB
cana-4508	134	12	𝒬𝑞:ℋ	𝒬𝑞:ℋ	PROPN
cana-4508	134	13	→	→	SYM
cana-4508	134	14	ℐ	ℐ	PRON
cana-4508	134	15	be	be	AUX
cana-4508	134	16	an	an	DET
cana-4508	134	17	even	even	ADJ
cana-4508	134	18	function	function	NOUN
cana-4508	134	19	satisfying	satisfy	VERB
cana-4508	134	20	the	the	DET
cana-4508	134	21	inequality	inequality	NOUN
cana-4508	134	22	‖𝑄(𝒳,𝒴)‖	‖𝑄(𝒳,𝒴)‖	NOUN
cana-4508	134	23	≤	≤	NUM
cana-4508	134	24	𝒯(𝒳,𝒴	𝒯(𝒳,𝒴	PROPN
cana-4508	134	25	)	)	PUNCT
cana-4508	134	26	(	(	PUNCT
cana-4508	134	27	13	13	NUM
cana-4508	134	28	)	)	PUNCT
cana-4508	134	29	for	for	ADP
cana-4508	134	30	all	all	DET
cana-4508	134	31	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	134	32	∈	∈	PROPN
cana-4508	134	33	ℋ.	ℋ.	PROPN
cana-4508	134	34	then	then	ADV
cana-4508	134	35	there	there	PRON
cana-4508	134	36	exists	exist	VERB
cana-4508	134	37	a	a	DET
cana-4508	134	38	unique	unique	ADJ
cana-4508	134	39	quadratic	quadratic	ADJ
cana-4508	134	40	mapping	mapping	NOUN
cana-4508	134	41	𝑄:ℋ	𝑄:ℋ	PUNCT
cana-4508	134	42	→	→	SYM
cana-4508	134	43	ℐ	ℐ	PRON
cana-4508	134	44	such	such	ADJ
cana-4508	134	45	that	that	SCONJ
cana-4508	134	46	communications	communication	NOUN
cana-4508	134	47	on	on	ADP
cana-4508	134	48	applied	apply	VERB
cana-4508	134	49	nonlinear	nonlinear	ADJ
cana-4508	134	50	analysis	analysis	NOUN
cana-4508	134	51	issn	issn	NOUN
cana-4508	134	52	:	:	PUNCT
cana-4508	134	53	1074	1074	NUM
cana-4508	134	54	-	-	PUNCT
cana-4508	134	55	133x	133x	NUM
cana-4508	134	56	vol	vol	NOUN
cana-4508	134	57	32	32	NUM
cana-4508	134	58	no	no	NOUN
cana-4508	134	59	.	.	PUNCT
cana-4508	135	1	9s	9s	NUM
cana-4508	135	2	(	(	PUNCT
cana-4508	135	3	2025	2025	NUM
cana-4508	135	4	)	)	PUNCT
cana-4508	135	5	2237	2237	NUM
cana-4508	135	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	135	7	‖𝒬𝑞(𝒳	‖𝒬𝑞(𝒳	NOUN
cana-4508	135	8	)	)	PUNCT
cana-4508	135	9	−	−	PROPN
cana-4508	135	10	𝑄(𝒳)‖	𝑄(𝒳)‖	NOUN
cana-4508	135	11	≤	≤	NUM
cana-4508	135	12	1	1	NUM
cana-4508	135	13	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	135	14	(	(	PUNCT
cana-4508	135	15	𝜋	𝜋	NOUN
cana-4508	135	16	4	4	NUM
cana-4508	135	17	)	)	PUNCT
cana-4508	135	18	)	)	PUNCT
cana-4508	136	1	2	2	NUM
cana-4508	136	2	∑	∑	SYM
cana-4508	136	3	∞	∞	NUM
cana-4508	136	4	ℋ=0	ℋ=0	PROPN
cana-4508	136	5	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	136	6	(	(	PUNCT
cana-4508	136	7	𝜋	𝜋	NOUN
cana-4508	136	8	4	4	NUM
cana-4508	136	9	)	)	PUNCT
cana-4508	136	10	)	)	PUNCT
cana-4508	137	1	ℋ	ℋ	NOUN
cana-4508	137	2	𝒳,0	𝒳,0	NUM
cana-4508	137	3	)	)	PUNCT
cana-4508	137	4	(	(	PUNCT
cana-4508	137	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	137	6	(	(	PUNCT
cana-4508	137	7	𝜋	𝜋	NOUN
cana-4508	137	8	4	4	NUM
cana-4508	137	9	)	)	PUNCT
cana-4508	137	10	)	)	PUNCT
cana-4508	137	11	2ℋ	2ℋ	NOUN
cana-4508	137	12	(	(	PUNCT
cana-4508	137	13	14	14	NUM
cana-4508	137	14	)	)	PUNCT
cana-4508	137	15	for	for	ADP
cana-4508	137	16	all	all	DET
cana-4508	137	17	𝒳	𝒳	PROPN
cana-4508	137	18	∈	∈	PROPN
cana-4508	137	19	ℋ.	ℋ.	PROPN
cana-4508	137	20	the	the	DET
cana-4508	137	21	mapping	mapping	NOUN
cana-4508	137	22	𝐴(𝒳	𝐴(𝒳	NOUN
cana-4508	137	23	)	)	PUNCT
cana-4508	137	24	is	be	AUX
cana-4508	137	25	defined	define	VERB
cana-4508	137	26	by	by	ADP
cana-4508	137	27	𝑄(𝒳	𝑄(𝒳	NOUN
cana-4508	137	28	)	)	PUNCT
cana-4508	137	29	=	=	SYM
cana-4508	137	30	lim	lim	PROPN
cana-4508	137	31	𝒰→∞	𝒰→∞	NOUN
cana-4508	137	32	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	137	33	ℒ	ℒ	X
cana-4508	137	34	(	(	PUNCT
cana-4508	137	35	𝜋	𝜋	NOUN
cana-4508	137	36	4	4	NUM
cana-4508	137	37	)	)	PUNCT
cana-4508	137	38	)	)	PUNCT
cana-4508	138	1	𝒰	𝒰	PROPN
cana-4508	138	2	𝒳	𝒳	PROPN
cana-4508	138	3	)	)	PUNCT
cana-4508	138	4	(	(	PUNCT
cana-4508	138	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	138	6	(	(	PUNCT
cana-4508	138	7	𝜋	𝜋	NOUN
cana-4508	138	8	4	4	NUM
cana-4508	138	9	)	)	PUNCT
cana-4508	138	10	)	)	PUNCT
cana-4508	138	11	2𝑁	2𝑁	NOUN
cana-4508	138	12	(	(	PUNCT
cana-4508	138	13	15	15	NUM
cana-4508	138	14	)	)	PUNCT
cana-4508	138	15	for	for	ADP
cana-4508	138	16	all	all	DET
cana-4508	138	17	𝒳	𝒳	PROPN
cana-4508	138	18	∈	∈	PROPN
cana-4508	138	19	ℋ.	ℋ.	PROPN
cana-4508	138	20	proof	proof	NOUN
cana-4508	138	21	.	.	PUNCT
cana-4508	139	1	replacing	replace	VERB
cana-4508	139	2	(	(	PUNCT
cana-4508	139	3	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	139	4	)	)	PUNCT
cana-4508	139	5	by	by	ADP
cana-4508	139	6	(	(	PUNCT
cana-4508	139	7	𝒳	𝒳	PROPN
cana-4508	139	8	,	,	PUNCT
cana-4508	139	9	0	0	NUM
cana-4508	139	10	)	)	PUNCT
cana-4508	139	11	in	in	ADP
cana-4508	139	12	(	(	PUNCT
cana-4508	139	13	13	13	NUM
cana-4508	139	14	)	)	PUNCT
cana-4508	139	15	and	and	CCONJ
cana-4508	139	16	using	use	VERB
cana-4508	139	17	evenness	evenness	NOUN
cana-4508	139	18	of	of	ADP
cana-4508	139	19	𝒬𝑞	𝒬𝑞	PROPN
cana-4508	139	20	,	,	PUNCT
cana-4508	139	21	we	we	PRON
cana-4508	139	22	get	get	VERB
cana-4508	139	23	‖𝒬𝑞(𝒳	‖𝒬𝑞(𝒳	NOUN
cana-4508	139	24	)	)	PUNCT
cana-4508	139	25	−	−	ADP
cana-4508	139	26	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	139	27	ℒ	ℒ	NOUN
cana-4508	139	28	(	(	PUNCT
cana-4508	139	29	𝜋	𝜋	NOUN
cana-4508	139	30	4	4	NUM
cana-4508	139	31	)	)	PUNCT
cana-4508	139	32	)	)	PUNCT
cana-4508	140	1	𝒳	𝒳	PROPN
cana-4508	140	2	)	)	PUNCT
cana-4508	140	3	(	(	PUNCT
cana-4508	140	4	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	140	5	(	(	PUNCT
cana-4508	140	6	𝜋	𝜋	NOUN
cana-4508	140	7	4	4	NUM
cana-4508	140	8	)	)	PUNCT
cana-4508	140	9	)	)	PUNCT
cana-4508	140	10	2	2	NUM
cana-4508	140	11	‖	‖	PROPN
cana-4508	140	12	≤	≤	NUM
cana-4508	140	13	1	1	NUM
cana-4508	140	14	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	140	15	(	(	PUNCT
cana-4508	140	16	𝜋	𝜋	NOUN
cana-4508	140	17	4	4	NUM
cana-4508	140	18	)	)	PUNCT
cana-4508	140	19	)	)	PUNCT
cana-4508	140	20	2𝒯(𝒳	2𝒯(𝒳	NOUN
cana-4508	140	21	,	,	PUNCT
cana-4508	140	22	0	0	NUM
cana-4508	140	23	)	)	PUNCT
cana-4508	140	24	(	(	PUNCT
cana-4508	140	25	16	16	NUM
cana-4508	140	26	)	)	PUNCT
cana-4508	140	27	for	for	ADP
cana-4508	140	28	all𝒳	all𝒳	SYM
cana-4508	140	29	∈	∈	PROPN
cana-4508	140	30	ℋ.	ℋ.	PROPN
cana-4508	140	31	now	now	ADV
cana-4508	140	32	replacing	replace	VERB
cana-4508	140	33	𝒳	𝒳	PRON
cana-4508	140	34	by	by	ADP
cana-4508	140	35	(	(	PUNCT
cana-4508	140	36	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	140	37	(	(	PUNCT
cana-4508	140	38	𝜋	𝜋	NOUN
cana-4508	140	39	4	4	NUM
cana-4508	140	40	)	)	PUNCT
cana-4508	140	41	)	)	PUNCT
cana-4508	140	42	𝒳	𝒳	PRON
cana-4508	140	43	and	and	CCONJ
cana-4508	140	44	dividing	divide	VERB
cana-4508	140	45	by	by	ADP
cana-4508	140	46	(	(	PUNCT
cana-4508	140	47	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	140	48	(	(	PUNCT
cana-4508	140	49	𝜋	𝜋	NOUN
cana-4508	140	50	4	4	NUM
cana-4508	140	51	)	)	PUNCT
cana-4508	140	52	)	)	PUNCT
cana-4508	140	53	2	2	NUM
cana-4508	140	54	in	in	ADP
cana-4508	140	55	(	(	PUNCT
cana-4508	140	56	16	16	NUM
cana-4508	140	57	)	)	PUNCT
cana-4508	140	58	,	,	PUNCT
cana-4508	140	59	we	we	PRON
cana-4508	140	60	obtain	obtain	VERB
cana-4508	140	61	‖	‖	ADJ
cana-4508	141	1	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	141	2	ℒ	ℒ	NOUN
cana-4508	141	3	(	(	PUNCT
cana-4508	141	4	𝜋	𝜋	NOUN
cana-4508	141	5	4	4	NUM
cana-4508	141	6	)	)	PUNCT
cana-4508	141	7	)	)	PUNCT
cana-4508	141	8	𝒳	𝒳	PROPN
cana-4508	141	9	)	)	PUNCT
cana-4508	141	10	(	(	PUNCT
cana-4508	141	11	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	141	12	(	(	PUNCT
cana-4508	141	13	𝜋	𝜋	NOUN
cana-4508	141	14	4	4	NUM
cana-4508	141	15	)	)	PUNCT
cana-4508	141	16	)	)	PUNCT
cana-4508	141	17	2	2	NUM
cana-4508	141	18	−	−	NOUN
cana-4508	141	19	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	141	20	ℒ	ℒ	NOUN
cana-4508	141	21	(	(	PUNCT
cana-4508	141	22	𝜋	𝜋	NOUN
cana-4508	141	23	4	4	NUM
cana-4508	141	24	)	)	PUNCT
cana-4508	141	25	)	)	PUNCT
cana-4508	141	26	2	2	NUM
cana-4508	141	27	𝒳	𝒳	PROPN
cana-4508	141	28	)	)	PUNCT
cana-4508	141	29	(	(	PUNCT
cana-4508	141	30	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	141	31	(	(	PUNCT
cana-4508	141	32	𝜋	𝜋	NOUN
cana-4508	141	33	4	4	NUM
cana-4508	141	34	)	)	PUNCT
cana-4508	141	35	)	)	PUNCT
cana-4508	141	36	4	4	NUM
cana-4508	141	37	‖	‖	PROPN
cana-4508	141	38	≤	≤	NOUN
cana-4508	141	39	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	141	40	(	(	PUNCT
cana-4508	141	41	𝜋	𝜋	NOUN
cana-4508	141	42	4	4	NUM
cana-4508	141	43	)	)	PUNCT
cana-4508	141	44	)	)	PUNCT
cana-4508	141	45	𝒳,0	𝒳,0	NUM
cana-4508	141	46	)	)	PUNCT
cana-4508	141	47	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	141	48	(	(	PUNCT
cana-4508	141	49	𝜋	𝜋	NOUN
cana-4508	141	50	4	4	NUM
cana-4508	141	51	)	)	PUNCT
cana-4508	141	52	)	)	PUNCT
cana-4508	141	53	4	4	NUM
cana-4508	141	54	(	(	PUNCT
cana-4508	141	55	17	17	NUM
cana-4508	141	56	)	)	PUNCT
cana-4508	141	57	for	for	ADP
cana-4508	141	58	all𝒳	all𝒳	SYM
cana-4508	141	59	∈	∈	PROPN
cana-4508	141	60	ℋ.	ℋ.	PROPN
cana-4508	141	61	it	it	PRON
cana-4508	141	62	follows	follow	VERB
cana-4508	141	63	from	from	ADP
cana-4508	141	64	(	(	PUNCT
cana-4508	141	65	16	16	NUM
cana-4508	141	66	)	)	PUNCT
cana-4508	141	67	and	and	CCONJ
cana-4508	141	68	(	(	PUNCT
cana-4508	141	69	17	17	NUM
cana-4508	141	70	)	)	PUNCT
cana-4508	141	71	that	that	SCONJ
cana-4508	141	72	‖𝒬𝑞(𝒳	‖𝒬𝑞(𝒳	VERB
cana-4508	141	73	)	)	PUNCT
cana-4508	141	74	−	−	ADP
cana-4508	141	75	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	141	76	ℒ	ℒ	NOUN
cana-4508	141	77	(	(	PUNCT
cana-4508	141	78	𝜋	𝜋	NOUN
cana-4508	141	79	4	4	NUM
cana-4508	141	80	)	)	PUNCT
cana-4508	141	81	)	)	PUNCT
cana-4508	141	82	2	2	NUM
cana-4508	141	83	𝒳	𝒳	PROPN
cana-4508	141	84	)	)	PUNCT
cana-4508	142	1	(	(	PUNCT
cana-4508	142	2	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	142	3	(	(	PUNCT
cana-4508	142	4	𝜋	𝜋	NOUN
cana-4508	142	5	4	4	NUM
cana-4508	142	6	)	)	PUNCT
cana-4508	142	7	)	)	PUNCT
cana-4508	142	8	4	4	NUM
cana-4508	142	9	‖	‖	PROPN
cana-4508	142	10	≤	≤	NOUN
cana-4508	142	11	‖𝒬𝑞(𝒳	‖𝒬𝑞(𝒳	NUM
cana-4508	142	12	)	)	PUNCT
cana-4508	142	13	−	−	NOUN
cana-4508	142	14	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	142	15	ℒ	ℒ	NOUN
cana-4508	142	16	(	(	PUNCT
cana-4508	142	17	𝜋	𝜋	NOUN
cana-4508	142	18	4	4	NUM
cana-4508	142	19	)	)	PUNCT
cana-4508	142	20	)	)	PUNCT
cana-4508	142	21	𝒳	𝒳	PROPN
cana-4508	142	22	)	)	PUNCT
cana-4508	143	1	(	(	PUNCT
cana-4508	143	2	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	143	3	(	(	PUNCT
cana-4508	143	4	𝜋	𝜋	NOUN
cana-4508	143	5	4	4	NUM
cana-4508	143	6	)	)	PUNCT
cana-4508	143	7	)	)	PUNCT
cana-4508	143	8	2	2	NUM
cana-4508	143	9	‖	‖	PROPN
cana-4508	143	10	+	+	CCONJ
cana-4508	143	11	‖	‖	ADJ
cana-4508	143	12	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	143	13	ℒ	ℒ	NOUN
cana-4508	143	14	(	(	PUNCT
cana-4508	143	15	𝜋	𝜋	NOUN
cana-4508	143	16	4	4	NUM
cana-4508	143	17	)	)	PUNCT
cana-4508	143	18	)	)	PUNCT
cana-4508	143	19	𝒳	𝒳	PROPN
cana-4508	143	20	)	)	PUNCT
cana-4508	143	21	(	(	PUNCT
cana-4508	143	22	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	143	23	(	(	PUNCT
cana-4508	143	24	𝜋	𝜋	NOUN
cana-4508	143	25	4	4	NUM
cana-4508	143	26	)	)	PUNCT
cana-4508	143	27	)	)	PUNCT
cana-4508	143	28	2	2	NUM
cana-4508	143	29	−	−	NOUN
cana-4508	143	30	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	143	31	ℒ	ℒ	NOUN
cana-4508	143	32	(	(	PUNCT
cana-4508	143	33	𝜋	𝜋	NOUN
cana-4508	143	34	4	4	NUM
cana-4508	143	35	)	)	PUNCT
cana-4508	143	36	)	)	PUNCT
cana-4508	143	37	2	2	NUM
cana-4508	143	38	𝒳	𝒳	PROPN
cana-4508	143	39	)	)	PUNCT
cana-4508	143	40	(	(	PUNCT
cana-4508	143	41	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	143	42	(	(	PUNCT
cana-4508	143	43	𝜋	𝜋	NOUN
cana-4508	143	44	4	4	NUM
cana-4508	143	45	)	)	PUNCT
cana-4508	143	46	)	)	PUNCT
cana-4508	143	47	4	4	NUM
cana-4508	143	48	‖	‖	PROPN
cana-4508	143	49	≤	≤	NUM
cana-4508	143	50	1	1	NUM
cana-4508	143	51	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	143	52	(	(	PUNCT
cana-4508	143	53	𝜋	𝜋	NOUN
cana-4508	143	54	4	4	NUM
cana-4508	143	55	)	)	PUNCT
cana-4508	143	56	)	)	PUNCT
cana-4508	143	57	2	2	NUM
cana-4508	144	1	[	[	X
cana-4508	144	2	𝒯(𝒳	𝒯(𝒳	NOUN
cana-4508	144	3	,	,	PUNCT
cana-4508	144	4	0	0	NUM
cana-4508	144	5	)	)	PUNCT
cana-4508	144	6	+	+	CCONJ
cana-4508	144	7	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	144	8	(	(	PUNCT
cana-4508	144	9	𝜋	𝜋	NOUN
cana-4508	144	10	4	4	NUM
cana-4508	144	11	)	)	PUNCT
cana-4508	144	12	)	)	PUNCT
cana-4508	144	13	𝒳,0	𝒳,0	X
cana-4508	144	14	)	)	PUNCT
cana-4508	144	15	(	(	PUNCT
cana-4508	144	16	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	144	17	(	(	PUNCT
cana-4508	144	18	𝜋	𝜋	NOUN
cana-4508	144	19	4	4	NUM
cana-4508	144	20	)	)	PUNCT
cana-4508	144	21	)	)	PUNCT
cana-4508	144	22	2	2	NUM
cana-4508	144	23	]	]	PUNCT
cana-4508	144	24	(	(	PUNCT
cana-4508	144	25	18	18	NUM
cana-4508	144	26	)	)	PUNCT
cana-4508	144	27	for	for	ADP
cana-4508	144	28	all𝒳	all𝒳	SYM
cana-4508	144	29	∈	∈	PROPN
cana-4508	144	30	ℋ.	ℋ.	PROPN
cana-4508	144	31	in	in	ADP
cana-4508	144	32	general	general	ADJ
cana-4508	144	33	for	for	ADP
cana-4508	144	34	any	any	DET
cana-4508	144	35	positive	positive	ADJ
cana-4508	144	36	integer	integer	NOUN
cana-4508	144	37	𝑁	𝑁	PROPN
cana-4508	144	38	,	,	PUNCT
cana-4508	144	39	we	we	PRON
cana-4508	144	40	get	get	VERB
cana-4508	144	41	‖𝒬𝑞(𝒳	‖𝒬𝑞(𝒳	NOUN
cana-4508	144	42	)	)	PUNCT
cana-4508	144	43	−	−	ADP
cana-4508	144	44	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	144	45	ℒ	ℒ	NOUN
cana-4508	144	46	(	(	PUNCT
cana-4508	144	47	𝜋	𝜋	NOUN
cana-4508	144	48	4	4	NUM
cana-4508	144	49	)	)	PUNCT
cana-4508	144	50	)	)	PUNCT
cana-4508	145	1	𝑁	𝑁	PROPN
cana-4508	145	2	𝒳	𝒳	PROPN
cana-4508	145	3	)	)	PUNCT
cana-4508	145	4	(	(	PUNCT
cana-4508	145	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	145	6	(	(	PUNCT
cana-4508	145	7	𝜋	𝜋	NOUN
cana-4508	145	8	4	4	NUM
cana-4508	145	9	)	)	PUNCT
cana-4508	145	10	)	)	PUNCT
cana-4508	146	1	2𝑁	2𝑁	NOUN
cana-4508	146	2	‖	‖	PROPN
cana-4508	146	3	≤	≤	PROPN
cana-4508	146	4	1	1	NUM
cana-4508	146	5	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	146	6	(	(	PUNCT
cana-4508	146	7	𝜋	𝜋	NOUN
cana-4508	146	8	4	4	NUM
cana-4508	146	9	)	)	PUNCT
cana-4508	146	10	)	)	PUNCT
cana-4508	147	1	2∑	2∑	NOUN
cana-4508	148	1	𝒰−1	𝒰−1	X
cana-4508	148	2	ℋ=0	ℋ=0	NUM
cana-4508	148	3	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	148	4	(	(	PUNCT
cana-4508	148	5	𝜋	𝜋	NOUN
cana-4508	148	6	4	4	NUM
cana-4508	148	7	)	)	PUNCT
cana-4508	148	8	)	)	PUNCT
cana-4508	149	1	ℋ	ℋ	NOUN
cana-4508	149	2	𝒳,0	𝒳,0	NUM
cana-4508	149	3	)	)	PUNCT
cana-4508	149	4	(	(	PUNCT
cana-4508	149	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	149	6	(	(	PUNCT
cana-4508	149	7	𝜋	𝜋	NOUN
cana-4508	149	8	4	4	NUM
cana-4508	149	9	)	)	PUNCT
cana-4508	149	10	)	)	PUNCT
cana-4508	150	1	2ℋ	2ℋ	NOUN
cana-4508	150	2	≤	≤	NUM
cana-4508	150	3	1	1	NUM
cana-4508	150	4	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	150	5	(	(	PUNCT
cana-4508	150	6	𝜋	𝜋	NOUN
cana-4508	150	7	4	4	NUM
cana-4508	150	8	)	)	PUNCT
cana-4508	150	9	)	)	PUNCT
cana-4508	151	1	2∑	2∑	NUM
cana-4508	152	1	∞	∞	NUM
cana-4508	152	2	ℋ=0	ℋ=0	PROPN
cana-4508	153	1	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	153	2	(	(	PUNCT
cana-4508	153	3	𝜋	𝜋	NOUN
cana-4508	153	4	4	4	NUM
cana-4508	153	5	)	)	PUNCT
cana-4508	153	6	)	)	PUNCT
cana-4508	154	1	ℋ	ℋ	NOUN
cana-4508	154	2	𝒳,0	𝒳,0	NUM
cana-4508	154	3	)	)	PUNCT
cana-4508	154	4	(	(	PUNCT
cana-4508	154	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	154	6	(	(	PUNCT
cana-4508	154	7	𝜋	𝜋	NOUN
cana-4508	154	8	4	4	NUM
cana-4508	154	9	)	)	PUNCT
cana-4508	154	10	)	)	PUNCT
cana-4508	154	11	2ℋ	2ℋ	NOUN
cana-4508	154	12	(	(	PUNCT
cana-4508	154	13	19	19	NUM
cana-4508	154	14	)	)	PUNCT
cana-4508	154	15	for	for	ADP
cana-4508	154	16	all𝒳	all𝒳	SYM
cana-4508	154	17	∈	∈	PROPN
cana-4508	154	18	ℋ.	ℋ.	PROPN
cana-4508	154	19	in	in	ADP
cana-4508	154	20	order	order	NOUN
cana-4508	154	21	to	to	PART
cana-4508	154	22	prove	prove	VERB
cana-4508	154	23	the	the	DET
cana-4508	154	24	convergence	convergence	NOUN
cana-4508	154	25	of	of	ADP
cana-4508	154	26	the	the	DET
cana-4508	154	27	sequence	sequence	NOUN
cana-4508	154	28	communications	communication	NOUN
cana-4508	154	29	on	on	ADP
cana-4508	154	30	applied	apply	VERB
cana-4508	154	31	nonlinear	nonlinear	ADJ
cana-4508	154	32	analysis	analysis	NOUN
cana-4508	154	33	issn	issn	NOUN
cana-4508	154	34	:	:	PUNCT
cana-4508	154	35	1074	1074	NUM
cana-4508	154	36	-	-	PUNCT
cana-4508	154	37	133x	133x	NUM
cana-4508	154	38	vol	vol	NOUN
cana-4508	154	39	32	32	NUM
cana-4508	154	40	no	no	NOUN
cana-4508	154	41	.	.	PUNCT
cana-4508	155	1	9s	9s	NUM
cana-4508	155	2	(	(	PUNCT
cana-4508	155	3	2025	2025	NUM
cana-4508	155	4	)	)	PUNCT
cana-4508	155	5	2238	2238	NUM
cana-4508	156	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	156	2	{	{	PUNCT
cana-4508	156	3	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	156	4	ℒ	ℒ	X
cana-4508	156	5	(	(	PUNCT
cana-4508	156	6	𝜋	𝜋	NOUN
cana-4508	156	7	4	4	NUM
cana-4508	156	8	)	)	PUNCT
cana-4508	156	9	)	)	PUNCT
cana-4508	157	1	𝑁	𝑁	PROPN
cana-4508	157	2	𝒳	𝒳	PROPN
cana-4508	157	3	)	)	PUNCT
cana-4508	157	4	(	(	PUNCT
cana-4508	157	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	157	6	(	(	PUNCT
cana-4508	157	7	𝜋	𝜋	NOUN
cana-4508	157	8	4	4	NUM
cana-4508	157	9	)	)	PUNCT
cana-4508	157	10	)	)	PUNCT
cana-4508	157	11	2𝑁	2𝑁	NOUN
cana-4508	157	12	}	}	PUNCT
cana-4508	157	13	,	,	PUNCT
cana-4508	157	14	replace	replace	VERB
cana-4508	157	15	𝒳	𝒳	PRON
cana-4508	157	16	by	by	ADP
cana-4508	157	17	(	(	PUNCT
cana-4508	157	18	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	157	19	(	(	PUNCT
cana-4508	157	20	𝜋	𝜋	NOUN
cana-4508	157	21	4	4	NUM
cana-4508	157	22	)	)	PUNCT
cana-4508	157	23	)	)	PUNCT
cana-4508	157	24	2𝑀	2𝑀	PROPN
cana-4508	157	25	𝒳	𝒳	PROPN
cana-4508	157	26	and	and	CCONJ
cana-4508	157	27	divide	divide	VERB
cana-4508	157	28	by	by	ADP
cana-4508	157	29	(	(	PUNCT
cana-4508	157	30	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	157	31	(	(	PUNCT
cana-4508	157	32	𝜋	𝜋	NOUN
cana-4508	157	33	4	4	NUM
cana-4508	157	34	)	)	PUNCT
cana-4508	157	35	)	)	PUNCT
cana-4508	157	36	2𝑀	2𝑀	INTJ
cana-4508	157	37	in	in	ADP
cana-4508	157	38	(	(	PUNCT
cana-4508	157	39	1	1	NUM
cana-4508	157	40	)	)	PUNCT
cana-4508	157	41	,	,	PUNCT
cana-4508	157	42	for	for	ADP
cana-4508	157	43	any	any	DET
cana-4508	157	44	𝑀,𝑁	𝑀,𝑁	NOUN
cana-4508	157	45	>	>	X
cana-4508	157	46	0	0	PUNCT
cana-4508	157	47	,	,	PUNCT
cana-4508	157	48	to	to	PART
cana-4508	157	49	deduce	deduce	VERB
cana-4508	157	50	‖	‖	ADJ
cana-4508	157	51	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	157	52	ℒ	ℒ	NOUN
cana-4508	157	53	(	(	PUNCT
cana-4508	157	54	𝜋	𝜋	NOUN
cana-4508	157	55	4	4	NUM
cana-4508	157	56	)	)	PUNCT
cana-4508	157	57	)	)	PUNCT
cana-4508	158	1	𝑀	𝑀	PROPN
cana-4508	158	2	𝑥	𝑥	NOUN
cana-4508	158	3	)	)	PUNCT
cana-4508	158	4	(	(	PUNCT
cana-4508	158	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	158	6	(	(	PUNCT
cana-4508	158	7	𝜋	𝜋	NOUN
cana-4508	158	8	4	4	NUM
cana-4508	158	9	)	)	PUNCT
cana-4508	158	10	)	)	PUNCT
cana-4508	158	11	2𝑀	2𝑀	INTJ
cana-4508	158	12	−	−	PROPN
cana-4508	158	13	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	158	14	ℒ	ℒ	NOUN
cana-4508	158	15	(	(	PUNCT
cana-4508	158	16	𝜋	𝜋	NOUN
cana-4508	158	17	4	4	NUM
cana-4508	158	18	)	)	PUNCT
cana-4508	158	19	)	)	PUNCT
cana-4508	158	20	𝑁+𝑀	𝑁+𝑀	X
cana-4508	159	1	𝒳	𝒳	PROPN
cana-4508	159	2	)	)	PUNCT
cana-4508	159	3	(	(	PUNCT
cana-4508	159	4	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	159	5	(	(	PUNCT
cana-4508	159	6	𝜋	𝜋	NOUN
cana-4508	159	7	4	4	NUM
cana-4508	159	8	)	)	PUNCT
cana-4508	159	9	)	)	PUNCT
cana-4508	160	1	2(𝑁+𝑀	2(𝑁+𝑀	NUM
cana-4508	160	2	)	)	PUNCT
cana-4508	160	3	‖	‖	PROPN
cana-4508	160	4	=	=	SYM
cana-4508	160	5	1	1	NUM
cana-4508	160	6	(	(	PUNCT
cana-4508	160	7	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	160	8	(	(	PUNCT
cana-4508	160	9	𝜋	𝜋	NOUN
cana-4508	160	10	4	4	NUM
cana-4508	160	11	)	)	PUNCT
cana-4508	160	12	)	)	PUNCT
cana-4508	161	1	2𝑀	2𝑀	PROPN
cana-4508	161	2	‖𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	‖𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	161	3	ℒ	ℒ	NOUN
cana-4508	161	4	(	(	PUNCT
cana-4508	161	5	𝜋	𝜋	NOUN
cana-4508	161	6	4	4	NUM
cana-4508	161	7	)	)	PUNCT
cana-4508	161	8	)	)	PUNCT
cana-4508	161	9	𝑀	𝑀	PROPN
cana-4508	161	10	𝒳	𝒳	PROPN
cana-4508	161	11	)	)	PUNCT
cana-4508	161	12	−	−	NUM
cana-4508	162	1	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	𝒬𝑎((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	162	2	ℒ	ℒ	X
cana-4508	162	3	(	(	PUNCT
cana-4508	162	4	𝜋	𝜋	NOUN
cana-4508	162	5	4	4	NUM
cana-4508	162	6	)	)	PUNCT
cana-4508	162	7	)	)	PUNCT
cana-4508	163	1	𝑁	𝑁	PROPN
cana-4508	163	2	⋅(𝑐𝑜𝑠𝑒𝑐ℒ	⋅(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	163	3	(	(	PUNCT
cana-4508	163	4	𝜋	𝜋	NOUN
cana-4508	163	5	4	4	NUM
cana-4508	163	6	)	)	PUNCT
cana-4508	163	7	)	)	PUNCT
cana-4508	163	8	𝑀	𝑀	PROPN
cana-4508	163	9	𝒳	𝒳	PROPN
cana-4508	163	10	)	)	PUNCT
cana-4508	163	11	(	(	PUNCT
cana-4508	163	12	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	163	13	(	(	PUNCT
cana-4508	163	14	𝜋	𝜋	NOUN
cana-4508	163	15	4	4	NUM
cana-4508	163	16	)	)	PUNCT
cana-4508	163	17	)	)	PUNCT
cana-4508	164	1	2𝑁	2𝑁	NOUN
cana-4508	164	2	‖	‖	PROPN
cana-4508	164	3	≤	≤	PROPN
cana-4508	164	4	1	1	NUM
cana-4508	164	5	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	164	6	(	(	PUNCT
cana-4508	164	7	𝜋	𝜋	NOUN
cana-4508	164	8	4	4	NUM
cana-4508	164	9	)	)	PUNCT
cana-4508	164	10	)	)	PUNCT
cana-4508	165	1	2∑	2∑	NOUN
cana-4508	166	1	𝒰−1	𝒰−1	X
cana-4508	166	2	ℋ=0	ℋ=0	NUM
cana-4508	166	3	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	166	4	(	(	PUNCT
cana-4508	166	5	𝜋	𝜋	NOUN
cana-4508	166	6	4	4	NUM
cana-4508	166	7	)	)	PUNCT
cana-4508	166	8	)	)	PUNCT
cana-4508	166	9	ℋ+𝑀	ℋ+𝑀	NOUN
cana-4508	166	10	𝒳,0	𝒳,0	NUM
cana-4508	166	11	)	)	PUNCT
cana-4508	166	12	(	(	PUNCT
cana-4508	166	13	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	166	14	(	(	PUNCT
cana-4508	166	15	𝜋	𝜋	NOUN
cana-4508	166	16	4	4	NUM
cana-4508	166	17	)	)	PUNCT
cana-4508	166	18	)	)	PUNCT
cana-4508	166	19	2(ℋ+𝑀	2(ℋ+𝑀	NUM
cana-4508	166	20	)	)	PUNCT
cana-4508	166	21	≤	≤	NUM
cana-4508	166	22	1	1	NUM
cana-4508	166	23	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	166	24	(	(	PUNCT
cana-4508	166	25	𝜋	𝜋	NOUN
cana-4508	166	26	4	4	NUM
cana-4508	166	27	)	)	PUNCT
cana-4508	166	28	)	)	PUNCT
cana-4508	167	1	2∑	2∑	NUM
cana-4508	168	1	∞	∞	NUM
cana-4508	168	2	ℋ=0	ℋ=0	PROPN
cana-4508	169	1	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	169	2	(	(	PUNCT
cana-4508	169	3	𝜋	𝜋	NOUN
cana-4508	169	4	4	4	NUM
cana-4508	169	5	)	)	PUNCT
cana-4508	169	6	)	)	PUNCT
cana-4508	169	7	ℋ+𝑀	ℋ+𝑀	NOUN
cana-4508	169	8	𝒳,0	𝒳,0	NUM
cana-4508	169	9	)	)	PUNCT
cana-4508	169	10	(	(	PUNCT
cana-4508	169	11	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	169	12	(	(	PUNCT
cana-4508	169	13	𝜋	𝜋	NOUN
cana-4508	169	14	4	4	NUM
cana-4508	169	15	)	)	PUNCT
cana-4508	169	16	)	)	PUNCT
cana-4508	169	17	2(ℋ+𝑀	2(ℋ+𝑀	NUM
cana-4508	169	18	)	)	PUNCT
cana-4508	169	19	→	→	SYM
cana-4508	169	20	0	0	NUM
cana-4508	169	21	𝑎𝑠	𝑎𝑠	PROPN
cana-4508	169	22	𝑀	𝑀	PROPN
cana-4508	169	23	→	→	PUNCT
cana-4508	169	24	∞	∞	PROPN
cana-4508	169	25	for	for	ADP
cana-4508	169	26	all	all	DET
cana-4508	169	27	𝒳	𝒳	PROPN
cana-4508	169	28	∈	∈	PROPN
cana-4508	169	29	ℋ.	ℋ.	PROPN
cana-4508	169	30	hence	hence	ADV
cana-4508	169	31	the	the	DET
cana-4508	169	32	sequence	sequence	NOUN
cana-4508	169	33	{	{	PUNCT
cana-4508	169	34	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	169	35	ℒ	ℒ	X
cana-4508	169	36	(	(	PUNCT
cana-4508	169	37	𝜋	𝜋	NOUN
cana-4508	169	38	4	4	NUM
cana-4508	169	39	)	)	PUNCT
cana-4508	169	40	)	)	PUNCT
cana-4508	170	1	𝑁	𝑁	PROPN
cana-4508	170	2	𝒳	𝒳	PROPN
cana-4508	170	3	)	)	PUNCT
cana-4508	170	4	(	(	PUNCT
cana-4508	170	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	170	6	(	(	PUNCT
cana-4508	170	7	𝜋	𝜋	NOUN
cana-4508	170	8	4	4	NUM
cana-4508	170	9	)	)	PUNCT
cana-4508	170	10	)	)	PUNCT
cana-4508	170	11	2𝑁	2𝑁	NOUN
cana-4508	170	12	}	}	PUNCT
cana-4508	170	13	is	be	AUX
cana-4508	170	14	a	a	DET
cana-4508	170	15	cauchy	cauchy	ADJ
cana-4508	170	16	sequence	sequence	NOUN
cana-4508	170	17	.	.	PUNCT
cana-4508	171	1	since	since	SCONJ
cana-4508	171	2	ℐ	ℐ	PRON
cana-4508	171	3	is	be	AUX
cana-4508	171	4	complete	complete	ADJ
cana-4508	171	5	,	,	PUNCT
cana-4508	171	6	there	there	PRON
cana-4508	171	7	exists	exist	VERB
cana-4508	171	8	a	a	DET
cana-4508	171	9	mapping	mapping	NOUN
cana-4508	171	10	𝑄:ℋ	𝑄:ℋ	PUNCT
cana-4508	171	11	→	→	SYM
cana-4508	171	12	ℐ	ℐ	PRON
cana-4508	171	13	such	such	ADJ
cana-4508	171	14	that	that	SCONJ
cana-4508	171	15	𝒬𝑞(𝒳	𝒬𝑞(𝒳	ADJ
cana-4508	171	16	)	)	PUNCT
cana-4508	171	17	=	=	SYM
cana-4508	171	18	lim	lim	PROPN
cana-4508	171	19	𝒰→∞	𝒰→∞	NOUN
cana-4508	171	20	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	171	21	ℒ	ℒ	X
cana-4508	171	22	(	(	PUNCT
cana-4508	171	23	𝜋	𝜋	NOUN
cana-4508	171	24	4	4	NUM
cana-4508	171	25	)	)	PUNCT
cana-4508	171	26	)	)	PUNCT
cana-4508	172	1	𝑁	𝑁	PROPN
cana-4508	172	2	𝒳	𝒳	PROPN
cana-4508	172	3	)	)	PUNCT
cana-4508	172	4	(	(	PUNCT
cana-4508	172	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	172	6	(	(	PUNCT
cana-4508	172	7	𝜋	𝜋	NOUN
cana-4508	172	8	4	4	NUM
cana-4508	172	9	)	)	PUNCT
cana-4508	172	10	)	)	PUNCT
cana-4508	172	11	2𝑁	2𝑁	NOUN
cana-4508	172	12	∀	∀	PUNCT
cana-4508	172	13	𝒳	𝒳	ADP
cana-4508	172	14	∈	∈	PROPN
cana-4508	172	15	ℋ.	ℋ.	PROPN
cana-4508	172	16	letting	let	VERB
cana-4508	172	17	𝑁	𝑁	PROPN
cana-4508	172	18	→	→	SYM
cana-4508	172	19	∞	∞	NUM
cana-4508	172	20	in	in	ADP
cana-4508	172	21	(	(	PUNCT
cana-4508	172	22	1	1	X
cana-4508	172	23	)	)	PUNCT
cana-4508	172	24	we	we	PRON
cana-4508	172	25	see	see	VERB
cana-4508	172	26	that	that	SCONJ
cana-4508	172	27	(	(	PUNCT
cana-4508	172	28	14	14	NUM
cana-4508	172	29	)	)	PUNCT
cana-4508	172	30	holds	hold	VERB
cana-4508	172	31	for	for	ADP
cana-4508	172	32	all𝒳	all𝒳	SYM
cana-4508	172	33	∈	∈	PROPN
cana-4508	172	34	ℋ.	ℋ.	PROPN
cana-4508	172	35	to	to	PART
cana-4508	172	36	prove	prove	VERB
cana-4508	172	37	𝑄	𝑄	PRON
cana-4508	172	38	satisfies	satisfie	NOUN
cana-4508	172	39	(	(	PUNCT
cana-4508	172	40	1	1	NUM
cana-4508	172	41	)	)	PUNCT
cana-4508	172	42	,	,	PUNCT
cana-4508	172	43	replacing	replace	VERB
cana-4508	172	44	(	(	PUNCT
cana-4508	172	45	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	172	46	)	)	PUNCT
cana-4508	172	47	by	by	ADP
cana-4508	172	48	(	(	PUNCT
cana-4508	172	49	(	(	PUNCT
cana-4508	172	50	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	172	51	(	(	PUNCT
cana-4508	172	52	𝜋	𝜋	NOUN
cana-4508	172	53	4	4	NUM
cana-4508	172	54	)	)	PUNCT
cana-4508	172	55	)	)	PUNCT
cana-4508	173	1	𝑁	𝑁	PROPN
cana-4508	173	2	𝒳	𝒳	PROPN
cana-4508	173	3	,	,	PUNCT
cana-4508	173	4	(	(	PUNCT
cana-4508	173	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	173	6	(	(	PUNCT
cana-4508	173	7	𝜋	𝜋	NOUN
cana-4508	173	8	4	4	NUM
cana-4508	173	9	)	)	PUNCT
cana-4508	173	10	)	)	PUNCT
cana-4508	174	1	𝑁	𝑁	PROPN
cana-4508	174	2	𝒴	𝒴	PROPN
cana-4508	174	3	)	)	PUNCT
cana-4508	174	4	and	and	CCONJ
cana-4508	174	5	dividing	divide	VERB
cana-4508	174	6	by	by	ADP
cana-4508	174	7	(	(	PUNCT
cana-4508	174	8	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	174	9	(	(	PUNCT
cana-4508	174	10	𝜋	𝜋	NOUN
cana-4508	174	11	4	4	NUM
cana-4508	174	12	)	)	PUNCT
cana-4508	174	13	)	)	PUNCT
cana-4508	174	14	2𝑁	2𝑁	NOUN
cana-4508	174	15	in	in	ADP
cana-4508	174	16	(	(	PUNCT
cana-4508	174	17	13	13	NUM
cana-4508	174	18	)	)	PUNCT
cana-4508	174	19	,	,	PUNCT
cana-4508	174	20	we	we	PRON
cana-4508	174	21	obtain	obtain	VERB
cana-4508	174	22	1	1	NUM
cana-4508	174	23	(	(	PUNCT
cana-4508	174	24	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	ADV
cana-4508	174	25	(	(	PUNCT
cana-4508	174	26	𝜋	𝜋	NOUN
cana-4508	174	27	4	4	NUM
cana-4508	174	28	)	)	PUNCT
cana-4508	174	29	)	)	PUNCT
cana-4508	174	30	2𝑁	2𝑁	NOUN
cana-4508	174	31	∥	∥	PUNCT
cana-4508	174	32	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	174	33	ℒ	ℒ	NOUN
cana-4508	174	34	(	(	PUNCT
cana-4508	174	35	𝜋	𝜋	NOUN
cana-4508	174	36	4	4	NUM
cana-4508	174	37	)	)	PUNCT
cana-4508	174	38	)	)	PUNCT
cana-4508	175	1	𝑁	𝑁	PROPN
cana-4508	175	2	𝒳	𝒳	PROPN
cana-4508	175	3	,	,	PUNCT
cana-4508	175	4	(	(	PUNCT
cana-4508	175	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	175	6	(	(	PUNCT
cana-4508	175	7	𝜋	𝜋	NOUN
cana-4508	175	8	4	4	NUM
cana-4508	175	9	)	)	PUNCT
cana-4508	175	10	)	)	PUNCT
cana-4508	176	1	𝑁	𝑁	PROPN
cana-4508	176	2	𝒴	𝒴	PROPN
cana-4508	176	3	)	)	PUNCT
cana-4508	176	4	∥	∥	PUNCT
cana-4508	176	5	≤	≤	NUM
cana-4508	176	6	1	1	NUM
cana-4508	176	7	(	(	PUNCT
cana-4508	176	8	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	176	9	(	(	PUNCT
cana-4508	176	10	𝜋	𝜋	NOUN
cana-4508	176	11	4	4	NUM
cana-4508	176	12	)	)	PUNCT
cana-4508	176	13	)	)	PUNCT
cana-4508	176	14	2𝑁𝒯((𝑐𝑜𝑠𝑒𝑐	2𝑁𝒯((𝑐𝑜𝑠𝑒𝑐	NUM
cana-4508	176	15	ℒ	ℒ	NOUN
cana-4508	176	16	(	(	PUNCT
cana-4508	176	17	𝜋	𝜋	NOUN
cana-4508	176	18	4	4	NUM
cana-4508	176	19	)	)	PUNCT
cana-4508	176	20	)	)	PUNCT
cana-4508	177	1	𝑁	𝑁	PROPN
cana-4508	177	2	𝒳	𝒳	PROPN
cana-4508	177	3	,	,	PUNCT
cana-4508	177	4	(	(	PUNCT
cana-4508	177	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	177	6	(	(	PUNCT
cana-4508	177	7	𝜋	𝜋	NOUN
cana-4508	177	8	4	4	NUM
cana-4508	177	9	)	)	PUNCT
cana-4508	177	10	)	)	PUNCT
cana-4508	178	1	𝑁	𝑁	PROPN
cana-4508	178	2	𝒴	𝒴	PROPN
cana-4508	178	3	)	)	PUNCT
cana-4508	178	4	for	for	ADP
cana-4508	178	5	all	all	DET
cana-4508	178	6	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	178	7	∈	∈	NOUN
cana-4508	178	8	ℋ.	ℋ.	PROPN
cana-4508	178	9	letting	let	VERB
cana-4508	178	10	𝑁	𝑁	PROPN
cana-4508	178	11	→	→	SYM
cana-4508	178	12	∞	∞	PROPN
cana-4508	178	13	in	in	ADP
cana-4508	178	14	the	the	DET
cana-4508	178	15	above	above	ADJ
cana-4508	178	16	inequality	inequality	NOUN
cana-4508	178	17	and	and	CCONJ
cana-4508	178	18	using	use	VERB
cana-4508	178	19	the	the	DET
cana-4508	178	20	definition	definition	NOUN
cana-4508	178	21	of	of	ADP
cana-4508	178	22	𝑄(𝒳	𝑄(𝒳	NOUN
cana-4508	178	23	)	)	PUNCT
cana-4508	178	24	,	,	PUNCT
cana-4508	178	25	we	we	PRON
cana-4508	178	26	see	see	VERB
cana-4508	178	27	that	that	PRON
cana-4508	178	28	𝒬𝑞(𝒳,𝒴	𝒬𝑞(𝒳,𝒴	PUNCT
cana-4508	178	29	)	)	PUNCT
cana-4508	179	1	=	=	SYM
cana-4508	179	2	0	0	X
cana-4508	179	3	.	.	PUNCT
cana-4508	179	4	communications	communication	NOUN
cana-4508	179	5	on	on	ADP
cana-4508	179	6	applied	apply	VERB
cana-4508	179	7	nonlinear	nonlinear	ADJ
cana-4508	179	8	analysis	analysis	NOUN
cana-4508	179	9	issn	issn	NOUN
cana-4508	179	10	:	:	PUNCT
cana-4508	179	11	1074	1074	NUM
cana-4508	179	12	-	-	PUNCT
cana-4508	179	13	133x	133x	NUM
cana-4508	179	14	vol	vol	NOUN
cana-4508	179	15	32	32	NUM
cana-4508	179	16	no	no	NOUN
cana-4508	179	17	.	.	PUNCT
cana-4508	180	1	9s	9s	NUM
cana-4508	180	2	(	(	PUNCT
cana-4508	180	3	2025	2025	NUM
cana-4508	180	4	)	)	PUNCT
cana-4508	180	5	2239	2239	NUM
cana-4508	180	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	180	7	hence	hence	ADV
cana-4508	180	8	𝑄	𝑄	PROPN
cana-4508	180	9	satisfies	satisfie	NOUN
cana-4508	180	10	(	(	PUNCT
cana-4508	180	11	1	1	NUM
cana-4508	180	12	)	)	PUNCT
cana-4508	180	13	for	for	ADP
cana-4508	180	14	all	all	DET
cana-4508	180	15	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	180	16	∈	∈	PROPN
cana-4508	180	17	ℋ.	ℋ.	PROPN
cana-4508	180	18	to	to	PART
cana-4508	180	19	show	show	VERB
cana-4508	180	20	𝑄	𝑄	PRON
cana-4508	180	21	is	be	AUX
cana-4508	180	22	unique	unique	ADJ
cana-4508	180	23	,	,	PUNCT
cana-4508	180	24	let	let	VERB
cana-4508	180	25	𝐵(𝒳	𝐵(𝒳	NOUN
cana-4508	180	26	)	)	PUNCT
cana-4508	180	27	be	be	AUX
cana-4508	180	28	another	another	DET
cana-4508	180	29	quadratic	quadratic	ADJ
cana-4508	180	30	mapping	mapping	NOUN
cana-4508	180	31	satisfying	satisfying	ADJ
cana-4508	180	32	(	(	PUNCT
cana-4508	180	33	1	1	NUM
cana-4508	180	34	)	)	PUNCT
cana-4508	180	35	and	and	CCONJ
cana-4508	180	36	(	(	PUNCT
cana-4508	180	37	14	14	NUM
cana-4508	180	38	)	)	PUNCT
cana-4508	180	39	,	,	PUNCT
cana-4508	180	40	then	then	ADV
cana-4508	180	41	‖𝐴(𝒳	‖𝐴(𝒳	NUM
cana-4508	180	42	)	)	PUNCT
cana-4508	181	1	−	−	NUM
cana-4508	181	2	𝐵(𝒳)‖	𝐵(𝒳)‖	NOUN
cana-4508	181	3	=	=	SYM
cana-4508	181	4	1	1	NUM
cana-4508	181	5	(	(	PUNCT
cana-4508	181	6	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	181	7	(	(	PUNCT
cana-4508	181	8	𝜋	𝜋	NOUN
cana-4508	181	9	4	4	NUM
cana-4508	181	10	)	)	PUNCT
cana-4508	181	11	)	)	PUNCT
cana-4508	181	12	2𝑁‖𝐴((𝑐𝑜𝑠𝑒𝑐	2𝑁‖𝐴((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	181	13	ℒ	ℒ	NOUN
cana-4508	181	14	(	(	PUNCT
cana-4508	181	15	𝜋	𝜋	NOUN
cana-4508	181	16	4	4	NUM
cana-4508	181	17	)	)	PUNCT
cana-4508	181	18	)	)	PUNCT
cana-4508	182	1	𝑁	𝑁	PROPN
cana-4508	182	2	𝒳	𝒳	PROPN
cana-4508	182	3	)	)	PUNCT
cana-4508	182	4	−	−	PROPN
cana-4508	182	5	𝐵((𝑐𝑜𝑠𝑒𝑐ℒ	𝐵((𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	182	6	(	(	PUNCT
cana-4508	182	7	𝜋	𝜋	NOUN
cana-4508	182	8	4	4	NUM
cana-4508	182	9	)	)	PUNCT
cana-4508	182	10	)	)	PUNCT
cana-4508	183	1	𝑁	𝑁	PROPN
cana-4508	183	2	𝒳)‖	𝒳)‖	VERB
cana-4508	183	3	≤	≤	NUM
cana-4508	183	4	1	1	NUM
cana-4508	183	5	(	(	PUNCT
cana-4508	183	6	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	183	7	(	(	PUNCT
cana-4508	183	8	𝜋	𝜋	NOUN
cana-4508	183	9	4	4	NUM
cana-4508	183	10	)	)	PUNCT
cana-4508	183	11	)	)	PUNCT
cana-4508	183	12	2𝑁	2𝑁	NOUN
cana-4508	183	13	{	{	PUNCT
cana-4508	183	14	‖𝐴((𝑐𝑜𝑠𝑒𝑐	‖𝐴((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	183	15	ℒ	ℒ	NOUN
cana-4508	183	16	(	(	PUNCT
cana-4508	183	17	𝜋	𝜋	NOUN
cana-4508	183	18	4	4	NUM
cana-4508	183	19	)	)	PUNCT
cana-4508	183	20	)	)	PUNCT
cana-4508	184	1	𝑁	𝑁	PROPN
cana-4508	184	2	𝒳	𝒳	PROPN
cana-4508	184	3	)	)	PUNCT
cana-4508	184	4	−	−	PUNCT
cana-4508	184	5	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	VERB
cana-4508	184	6	ℒ	ℒ	NOUN
cana-4508	184	7	(	(	PUNCT
cana-4508	184	8	𝜋	𝜋	NOUN
cana-4508	184	9	4	4	NUM
cana-4508	184	10	)	)	PUNCT
cana-4508	184	11	)	)	PUNCT
cana-4508	185	1	𝑁	𝑁	PROPN
cana-4508	185	2	𝒳)‖	𝒳)‖	NOUN
cana-4508	185	3	+	+	CCONJ
cana-4508	185	4	‖𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	‖𝒬𝑞((𝑐𝑜𝑠𝑒𝑐	ADJ
cana-4508	185	5	ℒ	ℒ	NOUN
cana-4508	185	6	(	(	PUNCT
cana-4508	185	7	𝜋	𝜋	NOUN
cana-4508	185	8	4	4	NUM
cana-4508	185	9	)	)	PUNCT
cana-4508	185	10	)	)	PUNCT
cana-4508	186	1	𝑁	𝑁	PROPN
cana-4508	186	2	𝒳	𝒳	PROPN
cana-4508	186	3	)	)	PUNCT
cana-4508	186	4	−	−	PROPN
cana-4508	186	5	𝐵((𝑐𝑜𝑠𝑒𝑐ℒ	𝐵((𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	186	6	(	(	PUNCT
cana-4508	186	7	𝜋	𝜋	NOUN
cana-4508	186	8	4	4	NUM
cana-4508	186	9	)	)	PUNCT
cana-4508	186	10	)	)	PUNCT
cana-4508	187	1	𝑁	𝑁	PROPN
cana-4508	187	2	𝒳)‖	𝒳)‖	NOUN
cana-4508	187	3	}	}	PUNCT
cana-4508	187	4	≤	≤	NUM
cana-4508	187	5	2	2	NUM
cana-4508	187	6	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	187	7	(	(	PUNCT
cana-4508	187	8	𝜋	𝜋	NOUN
cana-4508	187	9	4	4	NUM
cana-4508	187	10	)	)	PUNCT
cana-4508	187	11	)	)	PUNCT
cana-4508	188	1	2∑	2∑	NUM
cana-4508	189	1	∞	∞	NUM
cana-4508	189	2	ℋ=0	ℋ=0	PROPN
cana-4508	190	1	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	190	2	(	(	PUNCT
cana-4508	190	3	𝜋	𝜋	NOUN
cana-4508	190	4	4	4	NUM
cana-4508	190	5	)	)	PUNCT
cana-4508	190	6	)	)	PUNCT
cana-4508	190	7	ℋ+𝑁	ℋ+𝑁	PROPN
cana-4508	190	8	𝒳,0	𝒳,0	NUM
cana-4508	190	9	)	)	PUNCT
cana-4508	190	10	(	(	PUNCT
cana-4508	190	11	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	190	12	(	(	PUNCT
cana-4508	190	13	𝜋	𝜋	NOUN
cana-4508	190	14	4	4	NUM
cana-4508	190	15	)	)	PUNCT
cana-4508	190	16	)	)	PUNCT
cana-4508	190	17	2(ℋ+𝑁	2(ℋ+𝑁	NUM
cana-4508	190	18	)	)	PUNCT
cana-4508	190	19	→	→	SYM
cana-4508	190	20	0	0	NUM
cana-4508	190	21	𝑎𝑠	𝑎𝑠	ADP
cana-4508	190	22	𝑁	𝑁	PROPN
cana-4508	190	23	→	→	SYM
cana-4508	190	24	∞	∞	PROPN
cana-4508	190	25	for	for	ADP
cana-4508	190	26	all	all	DET
cana-4508	190	27	𝒳	𝒳	PROPN
cana-4508	190	28	∈	∈	NOUN
cana-4508	190	29	ℋ.	ℋ.	NOUN
cana-4508	190	30	hence	hence	ADV
cana-4508	190	31	𝑄	𝑄	PROPN
cana-4508	190	32	is	be	AUX
cana-4508	190	33	unique	unique	ADJ
cana-4508	190	34	.	.	PUNCT
cana-4508	191	1	corollary	corollary	ADJ
cana-4508	191	2	3.2	3.2	NUM
cana-4508	191	3	let	let	VERB
cana-4508	191	4	𝒯	𝒯	PROPN
cana-4508	191	5	and	and	CCONJ
cana-4508	191	6	p	p	NOUN
cana-4508	191	7	be	be	VERB
cana-4508	191	8	non	non	ADJ
cana-4508	191	9	negative	negative	ADJ
cana-4508	191	10	real	real	ADJ
cana-4508	191	11	numbers	number	NOUN
cana-4508	191	12	.	.	PUNCT
cana-4508	192	1	let	let	VERB
cana-4508	192	2	an	an	DET
cana-4508	192	3	even	even	ADV
cana-4508	192	4	function	function	NOUN
cana-4508	192	5	𝒬q:ℋ	𝒬q:ℋ	PUNCT
cana-4508	193	1	→	→	X
cana-4508	193	2	ℐ	ℐ	PRON
cana-4508	193	3	satisfy	satisfy	VERB
cana-4508	193	4	the	the	DET
cana-4508	193	5	inequality	inequality	NOUN
cana-4508	193	6	‖𝑄(𝒳,𝒴)‖	‖𝑄(𝒳,𝒴)‖	NOUN
cana-4508	193	7	≤	≤	NOUN
cana-4508	193	8	{	{	PUNCT
cana-4508	193	9	𝒯	𝒯	PROPN
cana-4508	193	10	,	,	PUNCT
cana-4508	193	11	𝒯{||𝒳||p	𝒯{||𝒳||p	PUNCT
cana-4508	193	12	+	+	NUM
cana-4508	193	13	||𝒴||p	||𝒴||p	NOUN
cana-4508	193	14	}	}	PUNCT
cana-4508	193	15	,	,	PUNCT
cana-4508	193	16	p	p	PROPN
cana-4508	193	17	≠	≠	PROPN
cana-4508	193	18	2	2	NUM
cana-4508	193	19	;	;	PUNCT
cana-4508	193	20	𝒯{||𝒳||p||𝒴||p	𝒯{||𝒳||p||𝒴||p	X
cana-4508	193	21	+	+	CCONJ
cana-4508	193	22	{	{	PUNCT
cana-4508	193	23	||𝒳||2p	||𝒳||2p	PROPN
cana-4508	193	24	+	+	CCONJ
cana-4508	193	25	||𝒴||2p	||𝒴||2p	NOUN
cana-4508	193	26	}	}	PUNCT
cana-4508	193	27	}	}	PUNCT
cana-4508	193	28	,	,	PUNCT
cana-4508	193	29	p	p	PROPN
cana-4508	193	30	≠	≠	PROPN
cana-4508	193	31	1	1	NUM
cana-4508	193	32	;	;	PUNCT
cana-4508	193	33	(	(	PUNCT
cana-4508	193	34	20	20	NUM
cana-4508	193	35	)	)	PUNCT
cana-4508	193	36	for	for	ADP
cana-4508	193	37	all	all	DET
cana-4508	193	38	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	193	39	∈	∈	PROPN
cana-4508	193	40	ℋ.	ℋ.	PROPN
cana-4508	193	41	then	then	ADV
cana-4508	193	42	there	there	PRON
cana-4508	193	43	exists	exist	VERB
cana-4508	193	44	a	a	DET
cana-4508	193	45	unique	unique	ADJ
cana-4508	193	46	quadratic	quadratic	ADJ
cana-4508	193	47	function	function	NOUN
cana-4508	193	48	q:ℋ	q:ℋ	PROPN
cana-4508	193	49	→	→	SYM
cana-4508	193	50	ℐ	ℐ	PRON
cana-4508	193	51	such	such	ADJ
cana-4508	193	52	that	that	DET
cana-4508	193	53	‖𝒬1(𝒳	‖𝒬1(𝒳	PROPN
cana-4508	193	54	)	)	PUNCT
cana-4508	193	55	−	−	PROPN
cana-4508	194	1	q(𝒳)‖	q(𝒳)‖	VERB
cana-4508	194	2	≤	≤	NOUN
cana-4508	194	3	{	{	PUNCT
cana-4508	194	4	𝒯	𝒯	PROPN
cana-4508	194	5	3((cosecℒ	3((cosecℒ	PROPN
cana-4508	194	6	(	(	PUNCT
cana-4508	194	7	π	π	PROPN
cana-4508	194	8	4	4	NUM
cana-4508	194	9	)	)	PUNCT
cana-4508	194	10	)	)	PUNCT
cana-4508	194	11	2	2	NUM
cana-4508	194	12	−1	−1	NOUN
cana-4508	194	13	)	)	PUNCT
cana-4508	194	14	,	,	PUNCT
cana-4508	194	15	𝒯||𝒳||p	𝒯||𝒳||p	PROPN
cana-4508	194	16	3|(cosecℒ	3|(cosecℒ	NUM
cana-4508	194	17	(	(	PUNCT
cana-4508	194	18	π	π	PROPN
cana-4508	194	19	4	4	NUM
cana-4508	194	20	)	)	PUNCT
cana-4508	194	21	)	)	PUNCT
cana-4508	195	1	2	2	NUM
cana-4508	195	2	−(cosecℒ	−(cosecℒ	PROPN
cana-4508	195	3	(	(	PUNCT
cana-4508	195	4	π	π	PROPN
cana-4508	195	5	4	4	NUM
cana-4508	195	6	)	)	PUNCT
cana-4508	195	7	)	)	PUNCT
cana-4508	196	1	p	p	NOUN
cana-4508	197	1	|	|	ADV
cana-4508	197	2	,	,	PUNCT
cana-4508	197	3	𝒯||𝒳||2p	𝒯||𝒳||2p	PROPN
cana-4508	197	4	3|(cosecℒ	3|(cosecℒ	NUM
cana-4508	197	5	(	(	PUNCT
cana-4508	197	6	π	π	PROPN
cana-4508	197	7	4	4	NUM
cana-4508	197	8	)	)	PUNCT
cana-4508	197	9	)	)	PUNCT
cana-4508	198	1	2	2	NUM
cana-4508	198	2	−(cosecℒ	−(cosecℒ	PROPN
cana-4508	198	3	(	(	PUNCT
cana-4508	198	4	π	π	PROPN
cana-4508	198	5	4	4	NUM
cana-4508	198	6	)	)	PUNCT
cana-4508	198	7	)	)	PUNCT
cana-4508	198	8	2p	2p	NUM
cana-4508	199	1	|	|	ADV
cana-4508	199	2	,	,	PUNCT
cana-4508	199	3	(	(	PUNCT
cana-4508	199	4	21	21	NUM
cana-4508	199	5	)	)	PUNCT
cana-4508	199	6	for	for	ADP
cana-4508	199	7	all	all	PRON
cana-4508	199	8	𝒳	𝒳	PROPN
cana-4508	199	9	∈	∈	PROPN
cana-4508	199	10	ℋ.	ℋ.	PROPN
cana-4508	199	11	4	4	NUM
cana-4508	199	12	stability	stability	NOUN
cana-4508	199	13	results	result	NOUN
cana-4508	199	14	:	:	PUNCT
cana-4508	199	15	mixed	mixed	ADJ
cana-4508	199	16	case	case	NOUN
cana-4508	199	17	theorem	theorem	VERB
cana-4508	199	18	4.1	4.1	NUM
cana-4508	199	19	let	let	VERB
cana-4508	199	20	𝒯	𝒯	PROPN
cana-4508	199	21	:	:	PUNCT
cana-4508	199	22	x2	x2	PROPN
cana-4508	199	23	→	→	PUNCT
cana-4508	200	1	[	[	X
cana-4508	200	2	0,∞	0,∞	X
cana-4508	200	3	)	)	PUNCT
cana-4508	200	4	be	be	VERB
cana-4508	200	5	a	a	DET
cana-4508	200	6	function	function	NOUN
cana-4508	200	7	satisfying	satisfy	VERB
cana-4508	200	8	(	(	PUNCT
cana-4508	200	9	2.1	2.1	NUM
cana-4508	200	10	)	)	PUNCT
cana-4508	200	11	and	and	CCONJ
cana-4508	200	12	(	(	PUNCT
cana-4508	200	13	3.1	3.1	NUM
cana-4508	200	14	)	)	PUNCT
cana-4508	200	15	for	for	ADP
cana-4508	200	16	all	all	DET
cana-4508	200	17	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	200	18	∈	∈	NOUN
cana-4508	200	19	ℋ.	ℋ.	PROPN
cana-4508	200	20	let	let	VERB
cana-4508	200	21	𝒬:ℋ	𝒬:ℋ	PUNCT
cana-4508	200	22	→	→	SYM
cana-4508	200	23	ℐ	ℐ	PRON
cana-4508	200	24	be	be	AUX
cana-4508	200	25	a	a	DET
cana-4508	200	26	function	function	NOUN
cana-4508	200	27	satisfying	satisfy	VERB
cana-4508	200	28	the	the	DET
cana-4508	200	29	inequality	inequality	NOUN
cana-4508	200	30	‖𝒬(𝒳	‖𝒬(𝒳	PROPN
cana-4508	200	31	,	,	PUNCT
cana-4508	200	32	𝒴)‖	𝒴)‖	VERB
cana-4508	200	33	≤	≤	NOUN
cana-4508	200	34	𝒯(𝒳,𝒴	𝒯(𝒳,𝒴	NUM
cana-4508	200	35	)	)	PUNCT
cana-4508	200	36	(	(	PUNCT
cana-4508	200	37	22	22	NUM
cana-4508	200	38	)	)	PUNCT
cana-4508	200	39	for	for	ADP
cana-4508	200	40	all	all	DET
cana-4508	200	41	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	200	42	∈	∈	PROPN
cana-4508	200	43	ℋ.	ℋ.	PROPN
cana-4508	200	44	then	then	ADV
cana-4508	200	45	there	there	PRON
cana-4508	200	46	exists	exist	VERB
cana-4508	200	47	a	a	DET
cana-4508	200	48	unique	unique	ADJ
cana-4508	200	49	additive	additive	ADJ
cana-4508	200	50	mapping	mapping	NOUN
cana-4508	200	51	a:ℋ	a:ℋ	NUM
cana-4508	200	52	→	→	SYM
cana-4508	200	53	ℐ	ℐ	PRON
cana-4508	200	54	and	and	CCONJ
cana-4508	200	55	a	a	DET
cana-4508	200	56	unique	unique	ADJ
cana-4508	200	57	quadratic	quadratic	ADJ
cana-4508	200	58	mapping	mapping	NOUN
cana-4508	200	59	q:ℋ	q:ℋ	NOUN
cana-4508	200	60	→	→	SYM
cana-4508	200	61	ℐ	ℐ	PRON
cana-4508	200	62	such	such	ADJ
cana-4508	200	63	that	that	SCONJ
cana-4508	200	64	communications	communication	NOUN
cana-4508	200	65	on	on	ADP
cana-4508	200	66	applied	apply	VERB
cana-4508	200	67	nonlinear	nonlinear	ADJ
cana-4508	200	68	analysis	analysis	NOUN
cana-4508	200	69	issn	issn	NOUN
cana-4508	200	70	:	:	PUNCT
cana-4508	200	71	1074	1074	NUM
cana-4508	200	72	-	-	PUNCT
cana-4508	200	73	133x	133x	NUM
cana-4508	200	74	vol	vol	NOUN
cana-4508	200	75	32	32	NUM
cana-4508	200	76	no	no	NOUN
cana-4508	200	77	.	.	PUNCT
cana-4508	201	1	9s	9s	NUM
cana-4508	201	2	(	(	PUNCT
cana-4508	201	3	2025	2025	NUM
cana-4508	201	4	)	)	PUNCT
cana-4508	201	5	2240	2240	NUM
cana-4508	201	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	201	7	‖𝒬1(𝒳	‖𝒬1(𝒳	PROPN
cana-4508	201	8	)	)	PUNCT
cana-4508	201	9	−	−	PROPN
cana-4508	201	10	𝐴(𝒳	𝐴(𝒳	NOUN
cana-4508	201	11	)	)	PUNCT
cana-4508	201	12	−	−	PROPN
cana-4508	201	13	𝑄(𝒳)‖	𝑄(𝒳)‖	NOUN
cana-4508	201	14	≤	≤	NUM
cana-4508	201	15	1	1	NUM
cana-4508	201	16	2	2	NUM
cana-4508	201	17	[	[	PUNCT
cana-4508	201	18	1	1	NUM
cana-4508	201	19	(	(	PUNCT
cana-4508	201	20	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	201	21	(	(	PUNCT
cana-4508	201	22	𝜋	𝜋	NOUN
cana-4508	201	23	4	4	NUM
cana-4508	201	24	)	)	PUNCT
cana-4508	201	25	)	)	PUNCT
cana-4508	201	26	∑	∑	PUNCT
cana-4508	202	1	∞	∞	X
cana-4508	202	2	ℋ=0	ℋ=0	PROPN
cana-4508	202	3	(	(	PUNCT
cana-4508	202	4	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	202	5	(	(	PUNCT
cana-4508	202	6	𝜋	𝜋	NOUN
cana-4508	202	7	4	4	NUM
cana-4508	202	8	)	)	PUNCT
cana-4508	202	9	)	)	PUNCT
cana-4508	202	10	ℋ	ℋ	PROPN
cana-4508	202	11	𝒳	𝒳	PROPN
cana-4508	202	12	,	,	PUNCT
cana-4508	202	13	0	0	PUNCT
cana-4508	203	1	(	(	PUNCT
cana-4508	203	2	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	203	3	(	(	PUNCT
cana-4508	203	4	𝜋	𝜋	NOUN
cana-4508	203	5	4	4	NUM
cana-4508	203	6	)	)	PUNCT
cana-4508	203	7	)	)	PUNCT
cana-4508	203	8	ℋ	ℋ	PROPN
cana-4508	203	9	+	+	CCONJ
cana-4508	203	10	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	203	11	(	(	PUNCT
cana-4508	203	12	𝜋	𝜋	NOUN
cana-4508	203	13	4	4	NUM
cana-4508	203	14	)	)	PUNCT
cana-4508	203	15	)	)	PUNCT
cana-4508	203	16	ℋ	ℋ	PROPN
cana-4508	203	17	𝒳	𝒳	PROPN
cana-4508	203	18	,	,	PUNCT
cana-4508	203	19	0	0	NUM
cana-4508	203	20	)	)	PUNCT
cana-4508	203	21	(	(	PUNCT
cana-4508	203	22	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	203	23	(	(	PUNCT
cana-4508	203	24	𝜋	𝜋	NOUN
cana-4508	203	25	4	4	NUM
cana-4508	203	26	)	)	PUNCT
cana-4508	203	27	)	)	PUNCT
cana-4508	204	1	ℋ	ℋ	NOUN
cana-4508	204	2	)	)	PUNCT
cana-4508	205	1	+	+	CCONJ
cana-4508	205	2	1	1	NUM
cana-4508	205	3	3(𝑐𝑜𝑠𝑒𝑐ℒ	3(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	205	4	(	(	PUNCT
cana-4508	205	5	𝜋	𝜋	NOUN
cana-4508	205	6	4	4	NUM
cana-4508	205	7	)	)	PUNCT
cana-4508	205	8	)	)	PUNCT
cana-4508	206	1	2∑	2∑	NUM
cana-4508	206	2	∞	∞	NUM
cana-4508	206	3	ℋ=0	ℋ=0	PROPN
cana-4508	206	4	(	(	PUNCT
cana-4508	206	5	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	206	6	(	(	PUNCT
cana-4508	206	7	𝜋	𝜋	NOUN
cana-4508	206	8	4	4	NUM
cana-4508	206	9	)	)	PUNCT
cana-4508	206	10	)	)	PUNCT
cana-4508	207	1	ℋ	ℋ	NOUN
cana-4508	207	2	𝒳,0	𝒳,0	NUM
cana-4508	207	3	)	)	PUNCT
cana-4508	207	4	(	(	PUNCT
cana-4508	207	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	207	6	(	(	PUNCT
cana-4508	207	7	𝜋	𝜋	NOUN
cana-4508	207	8	4	4	NUM
cana-4508	207	9	)	)	PUNCT
cana-4508	207	10	)	)	PUNCT
cana-4508	208	1	2ℋ	2ℋ	NOUN
cana-4508	208	2	+	+	CCONJ
cana-4508	208	3	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	208	4	(	(	PUNCT
cana-4508	208	5	𝜋	𝜋	NOUN
cana-4508	208	6	4	4	NUM
cana-4508	208	7	)	)	PUNCT
cana-4508	208	8	)	)	PUNCT
cana-4508	209	1	ℋ	ℋ	NOUN
cana-4508	209	2	𝒳,0	𝒳,0	NUM
cana-4508	209	3	)	)	PUNCT
cana-4508	209	4	(	(	PUNCT
cana-4508	209	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	209	6	(	(	PUNCT
cana-4508	209	7	𝜋	𝜋	NOUN
cana-4508	209	8	4	4	NUM
cana-4508	209	9	)	)	PUNCT
cana-4508	209	10	)	)	PUNCT
cana-4508	209	11	2ℋ	2ℋ	NOUN
cana-4508	209	12	)	)	PUNCT
cana-4508	209	13	]	]	PUNCT
cana-4508	209	14	(	(	PUNCT
cana-4508	209	15	23	23	NUM
cana-4508	209	16	)	)	PUNCT
cana-4508	209	17	for	for	ADP
cana-4508	209	18	all	all	DET
cana-4508	209	19	𝒳	𝒳	PROPN
cana-4508	209	20	∈	∈	PROPN
cana-4508	209	21	ℋ.	ℋ.	PROPN
cana-4508	209	22	the	the	DET
cana-4508	209	23	mapping	mapping	NOUN
cana-4508	209	24	𝐴(𝒳	𝐴(𝒳	NOUN
cana-4508	209	25	)	)	PUNCT
cana-4508	209	26	and	and	CCONJ
cana-4508	209	27	𝑄(𝒳	𝑄(𝒳	NOUN
cana-4508	209	28	)	)	PUNCT
cana-4508	209	29	are	be	AUX
cana-4508	209	30	defined	define	VERB
cana-4508	209	31	in	in	ADP
cana-4508	209	32	(	(	PUNCT
cana-4508	209	33	4	4	NUM
cana-4508	209	34	)	)	PUNCT
cana-4508	209	35	and	and	CCONJ
cana-4508	209	36	(	(	PUNCT
cana-4508	209	37	15	15	NUM
cana-4508	209	38	)	)	PUNCT
cana-4508	209	39	respectively	respectively	ADV
cana-4508	209	40	for	for	ADP
cana-4508	209	41	all	all	DET
cana-4508	209	42	𝒳	𝒳	PROPN
cana-4508	209	43	∈	∈	PROPN
cana-4508	209	44	ℋ.	ℋ.	PROPN
cana-4508	209	45	proof	proof	NOUN
cana-4508	209	46	.	.	PUNCT
cana-4508	210	1	let	let	VERB
cana-4508	210	2	𝒬𝑜(𝒳	𝒬𝑜(𝒳	PRON
cana-4508	210	3	)	)	PUNCT
cana-4508	210	4	=	=	SYM
cana-4508	210	5	𝒬𝑎(𝒳)−𝒬𝑎(−𝒳	𝒬𝑎(𝒳)−𝒬𝑎(−𝒳	X
cana-4508	210	6	)	)	PUNCT
cana-4508	210	7	2	2	NUM
cana-4508	210	8	for	for	ADP
cana-4508	210	9	all𝒳	all𝒳	SYM
cana-4508	210	10	∈	∈	PROPN
cana-4508	210	11	ℋ.	ℋ.	PROPN
cana-4508	210	12	then	then	ADV
cana-4508	210	13	𝒬𝑜(0	𝒬𝑜(0	NOUN
cana-4508	210	14	)	)	PUNCT
cana-4508	210	15	=	=	SYM
cana-4508	210	16	0	0	NUM
cana-4508	210	17	and	and	CCONJ
cana-4508	210	18	𝒬𝑜(−𝒳	𝒬𝑜(−𝒳	PRON
cana-4508	210	19	)	)	PUNCT
cana-4508	211	1	=	=	SYM
cana-4508	211	2	−𝒬𝑜(𝒳	−𝒬𝑜(𝒳	NOUN
cana-4508	211	3	)	)	PUNCT
cana-4508	211	4	for	for	ADP
cana-4508	211	5	all𝒳	all𝒳	SYM
cana-4508	211	6	∈	∈	PROPN
cana-4508	211	7	ℋ.	ℋ.	PROPN
cana-4508	211	8	hence	hence	ADV
cana-4508	211	9	‖𝒬𝑜(𝒳,𝒴)‖	‖𝒬𝑜(𝒳,𝒴)‖	NOUN
cana-4508	211	10	≤	≤	NUM
cana-4508	211	11	𝒯(𝒳,𝒴	𝒯(𝒳,𝒴	NUM
cana-4508	211	12	)	)	PUNCT
cana-4508	211	13	2	2	NUM
cana-4508	212	1	+	+	CCONJ
cana-4508	212	2	𝒯(−𝒳,−𝒴	𝒯(−𝒳,−𝒴	ADJ
cana-4508	212	3	)	)	PUNCT
cana-4508	212	4	2	2	NUM
cana-4508	212	5	(	(	PUNCT
cana-4508	212	6	24	24	NUM
cana-4508	212	7	)	)	PUNCT
cana-4508	212	8	for	for	ADP
cana-4508	212	9	all	all	DET
cana-4508	212	10	𝒳,𝒳	𝒳,𝒳	NOUN
cana-4508	212	11	∈	∈	PROPN
cana-4508	212	12	ℋ.	ℋ.	PROPN
cana-4508	212	13	by	by	ADP
cana-4508	212	14	theorem	theorem	NOUN
cana-4508	212	15	2.1	2.1	NUM
cana-4508	212	16	,	,	PUNCT
cana-4508	212	17	we	we	PRON
cana-4508	212	18	have	have	VERB
cana-4508	212	19	‖𝒬𝑜(𝒳	‖𝒬𝑜(𝒳	NOUN
cana-4508	212	20	)	)	PUNCT
cana-4508	212	21	−	−	PROPN
cana-4508	213	1	𝐴(𝒳)‖	𝐴(𝒳)‖	NOUN
cana-4508	214	1	≤	≤	NOUN
cana-4508	214	2	1	1	NUM
cana-4508	214	3	2(𝑐𝑜𝑠𝑒𝑐ℒ	2(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	214	4	(	(	PUNCT
cana-4508	214	5	𝜋	𝜋	NOUN
cana-4508	214	6	4	4	NUM
cana-4508	214	7	)	)	PUNCT
cana-4508	214	8	)	)	PUNCT
cana-4508	214	9	∑∞ℋ=0	∑∞ℋ=0	PUNCT
cana-4508	215	1	(	(	PUNCT
cana-4508	215	2	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	215	3	(	(	PUNCT
cana-4508	215	4	𝜋	𝜋	NOUN
cana-4508	215	5	4	4	NUM
cana-4508	215	6	)	)	PUNCT
cana-4508	215	7	)	)	PUNCT
cana-4508	216	1	ℋ	ℋ	NOUN
cana-4508	216	2	𝒳,0	𝒳,0	NUM
cana-4508	216	3	)	)	PUNCT
cana-4508	216	4	(	(	PUNCT
cana-4508	216	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	216	6	(	(	PUNCT
cana-4508	216	7	𝜋	𝜋	NOUN
cana-4508	216	8	4	4	NUM
cana-4508	216	9	)	)	PUNCT
cana-4508	216	10	)	)	PUNCT
cana-4508	217	1	ℋ	ℋ	PROPN
cana-4508	217	2	+	+	CCONJ
cana-4508	217	3	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	217	4	(	(	PUNCT
cana-4508	217	5	𝜋	𝜋	NOUN
cana-4508	217	6	4	4	NUM
cana-4508	217	7	)	)	PUNCT
cana-4508	217	8	)	)	PUNCT
cana-4508	218	1	ℋ	ℋ	NOUN
cana-4508	218	2	𝒳,0	𝒳,0	NUM
cana-4508	218	3	)	)	PUNCT
cana-4508	218	4	(	(	PUNCT
cana-4508	218	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	218	6	(	(	PUNCT
cana-4508	218	7	𝜋	𝜋	NOUN
cana-4508	218	8	4	4	NUM
cana-4508	218	9	)	)	PUNCT
cana-4508	218	10	)	)	PUNCT
cana-4508	219	1	ℋ	ℋ	NOUN
cana-4508	219	2	)	)	PUNCT
cana-4508	219	3	(	(	PUNCT
cana-4508	219	4	25	25	NUM
cana-4508	219	5	)	)	PUNCT
cana-4508	219	6	for	for	ADP
cana-4508	219	7	all	all	DET
cana-4508	219	8	𝒳	𝒳	PROPN
cana-4508	219	9	∈	∈	PROPN
cana-4508	219	10	ℋ.	ℋ.	PROPN
cana-4508	219	11	also	also	ADV
cana-4508	219	12	,	,	PUNCT
cana-4508	219	13	let	let	VERB
cana-4508	219	14	𝒬𝑒(𝒳	𝒬𝑒(𝒳	VERB
cana-4508	219	15	)	)	PUNCT
cana-4508	219	16	=	=	SYM
cana-4508	219	17	𝒬𝑞(𝒳)+𝒬𝑞(−𝒳	𝒬𝑞(𝒳)+𝒬𝑞(−𝒳	NOUN
cana-4508	219	18	)	)	PUNCT
cana-4508	219	19	2	2	NUM
cana-4508	219	20	for	for	ADP
cana-4508	219	21	all𝒳	all𝒳	SYM
cana-4508	219	22	∈	∈	PROPN
cana-4508	219	23	ℋ.	ℋ.	PROPN
cana-4508	219	24	then	then	ADV
cana-4508	219	25	𝒬𝑒(0	𝒬𝑒(0	ADJ
cana-4508	219	26	)	)	PUNCT
cana-4508	220	1	=	=	SYM
cana-4508	220	2	0	0	NUM
cana-4508	220	3	and	and	CCONJ
cana-4508	220	4	𝒬𝑒(−𝒳	𝒬𝑒(−𝒳	PUNCT
cana-4508	220	5	)	)	PUNCT
cana-4508	220	6	=	=	SYM
cana-4508	220	7	𝒬𝑒(𝒳	𝒬𝑒(𝒳	NOUN
cana-4508	220	8	)	)	PUNCT
cana-4508	220	9	for	for	ADP
cana-4508	220	10	all𝒳	all𝒳	SYM
cana-4508	220	11	∈	∈	PROPN
cana-4508	220	12	ℋ.	ℋ.	PROPN
cana-4508	220	13	hence	hence	ADV
cana-4508	220	14	‖𝒬𝑒(𝒳,𝒴)‖	‖𝒬𝑒(𝒳,𝒴)‖	PROPN
cana-4508	220	15	≤	≤	NUM
cana-4508	220	16	𝒯(𝒳,𝒴	𝒯(𝒳,𝒴	NUM
cana-4508	220	17	)	)	PUNCT
cana-4508	220	18	2	2	NUM
cana-4508	220	19	+	+	CCONJ
cana-4508	220	20	𝒯(−𝒳,−𝒴	𝒯(−𝒳,−𝒴	ADJ
cana-4508	220	21	)	)	PUNCT
cana-4508	220	22	2	2	NUM
cana-4508	220	23	(	(	PUNCT
cana-4508	220	24	26	26	NUM
cana-4508	220	25	)	)	PUNCT
cana-4508	220	26	for	for	ADP
cana-4508	220	27	all	all	DET
cana-4508	220	28	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	220	29	∈	∈	PROPN
cana-4508	220	30	ℋ.	ℋ.	PROPN
cana-4508	220	31	by	by	ADP
cana-4508	220	32	theorem	theorem	NOUN
cana-4508	220	33	3.1	3.1	NUM
cana-4508	220	34	,	,	PUNCT
cana-4508	220	35	we	we	PRON
cana-4508	220	36	have	have	VERB
cana-4508	220	37	‖𝒬𝑒(𝒳	‖𝒬𝑒(𝒳	NUM
cana-4508	220	38	)	)	PUNCT
cana-4508	220	39	−	−	PROPN
cana-4508	220	40	𝑄(𝒳)‖	𝑄(𝒳)‖	NOUN
cana-4508	220	41	≤	≤	NOUN
cana-4508	220	42	1	1	NUM
cana-4508	220	43	6(𝑐𝑜𝑠𝑒𝑐ℒ	6(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	220	44	(	(	PUNCT
cana-4508	220	45	𝜋	𝜋	NOUN
cana-4508	220	46	4	4	NUM
cana-4508	220	47	)	)	PUNCT
cana-4508	220	48	)	)	PUNCT
cana-4508	221	1	2∑	2∑	NUM
cana-4508	221	2	∞	∞	NUM
cana-4508	221	3	ℋ=0	ℋ=0	PROPN
cana-4508	221	4	(	(	PUNCT
cana-4508	221	5	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	221	6	(	(	PUNCT
cana-4508	221	7	𝜋	𝜋	NOUN
cana-4508	221	8	4	4	NUM
cana-4508	221	9	)	)	PUNCT
cana-4508	221	10	)	)	PUNCT
cana-4508	222	1	ℋ	ℋ	NOUN
cana-4508	222	2	𝒳,0	𝒳,0	NUM
cana-4508	222	3	)	)	PUNCT
cana-4508	222	4	(	(	PUNCT
cana-4508	222	5	(	(	PUNCT
cana-4508	222	6	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	222	7	(	(	PUNCT
cana-4508	222	8	𝜋	𝜋	NOUN
cana-4508	222	9	4	4	NUM
cana-4508	222	10	)	)	PUNCT
cana-4508	222	11	)	)	PUNCT
cana-4508	223	1	2ℋ	2ℋ	NOUN
cana-4508	223	2	+	+	CCONJ
cana-4508	223	3	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	223	4	(	(	PUNCT
cana-4508	223	5	𝜋	𝜋	NOUN
cana-4508	223	6	4	4	NUM
cana-4508	223	7	)	)	PUNCT
cana-4508	223	8	)	)	PUNCT
cana-4508	224	1	ℋ	ℋ	NOUN
cana-4508	224	2	𝒳,0	𝒳,0	NUM
cana-4508	224	3	)	)	PUNCT
cana-4508	224	4	(	(	PUNCT
cana-4508	224	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	224	6	(	(	PUNCT
cana-4508	224	7	𝜋	𝜋	NOUN
cana-4508	224	8	4	4	NUM
cana-4508	224	9	)	)	PUNCT
cana-4508	224	10	)	)	PUNCT
cana-4508	224	11	2ℋ	2ℋ	NOUN
cana-4508	224	12	)	)	PUNCT
cana-4508	224	13	(	(	PUNCT
cana-4508	224	14	27	27	NUM
cana-4508	224	15	)	)	PUNCT
cana-4508	224	16	for	for	ADP
cana-4508	224	17	all	all	DET
cana-4508	224	18	𝒳	𝒳	PROPN
cana-4508	224	19	∈	∈	PROPN
cana-4508	224	20	ℋ.	ℋ.	PROPN
cana-4508	224	21	define	define	VERB
cana-4508	224	22	𝒬(𝒳	𝒬(𝒳	NOUN
cana-4508	224	23	)	)	PUNCT
cana-4508	224	24	=	=	SYM
cana-4508	224	25	𝒬𝑒(𝒳	𝒬𝑒(𝒳	NOUN
cana-4508	224	26	)	)	PUNCT
cana-4508	224	27	+	+	PUNCT
cana-4508	224	28	𝒬𝑜(𝒳	𝒬𝑜(𝒳	X
cana-4508	224	29	)	)	PUNCT
cana-4508	224	30	(	(	PUNCT
cana-4508	224	31	28	28	NUM
cana-4508	224	32	)	)	PUNCT
cana-4508	224	33	for	for	ADP
cana-4508	224	34	all	all	PRON
cana-4508	224	35	𝒳	𝒳	PROPN
cana-4508	224	36	∈	∈	PROPN
cana-4508	224	37	ℋ.	ℋ.	PROPN
cana-4508	224	38	from	from	ADP
cana-4508	224	39	(	(	PUNCT
cana-4508	224	40	25),(27	25),(27	NUM
cana-4508	224	41	)	)	PUNCT
cana-4508	224	42	and	and	CCONJ
cana-4508	224	43	(	(	PUNCT
cana-4508	224	44	28	28	NUM
cana-4508	224	45	)	)	PUNCT
cana-4508	224	46	,	,	PUNCT
cana-4508	224	47	we	we	PRON
cana-4508	224	48	arrive	arrive	VERB
cana-4508	224	49	‖𝒬1(𝒳	‖𝒬1(𝒳	PROPN
cana-4508	224	50	)	)	PUNCT
cana-4508	225	1	−	−	PROPN
cana-4508	225	2	𝐴(𝒳	𝐴(𝒳	NOUN
cana-4508	225	3	)	)	PUNCT
cana-4508	225	4	−	−	PROPN
cana-4508	225	5	𝑄(𝒳)‖	𝑄(𝒳)‖	NOUN
cana-4508	225	6	=	=	SYM
cana-4508	225	7	‖𝒬𝑒(𝒳	‖𝒬𝑒(𝒳	PROPN
cana-4508	225	8	)	)	PUNCT
cana-4508	225	9	+	+	CCONJ
cana-4508	225	10	𝒬𝑜(𝒳	𝒬𝑜(𝒳	SYM
cana-4508	225	11	)	)	PUNCT
cana-4508	225	12	−	−	NOUN
cana-4508	225	13	𝐴(𝒳	𝐴(𝒳	NOUN
cana-4508	225	14	)	)	PUNCT
cana-4508	225	15	−	−	PROPN
cana-4508	226	1	𝑄(𝒳)‖	𝑄(𝒳)‖	NOUN
cana-4508	226	2	≤	≤	NUM
cana-4508	226	3	‖𝒬𝑜(𝒳	‖𝒬𝑜(𝒳	NOUN
cana-4508	226	4	)	)	PUNCT
cana-4508	226	5	−	−	PROPN
cana-4508	227	1	𝐴(𝒳)‖	𝐴(𝒳)‖	NOUN
cana-4508	227	2	+	+	CCONJ
cana-4508	227	3	‖𝒬𝑒(𝒳	‖𝒬𝑒(𝒳	NUM
cana-4508	227	4	)	)	PUNCT
cana-4508	227	5	−	−	PROPN
cana-4508	227	6	𝑄(𝒳)‖	𝑄(𝒳)‖	NOUN
cana-4508	227	7	≤	≤	NOUN
cana-4508	227	8	1	1	NUM
cana-4508	227	9	2(𝑐𝑜𝑠𝑒𝑐ℒ	2(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	227	10	(	(	PUNCT
cana-4508	227	11	𝜋	𝜋	NOUN
cana-4508	227	12	4	4	NUM
cana-4508	227	13	)	)	PUNCT
cana-4508	227	14	)	)	PUNCT
cana-4508	227	15	∑∞ℋ=0	∑∞ℋ=0	PUNCT
cana-4508	228	1	(	(	PUNCT
cana-4508	228	2	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	228	3	(	(	PUNCT
cana-4508	228	4	𝜋	𝜋	NOUN
cana-4508	228	5	4	4	NUM
cana-4508	228	6	)	)	PUNCT
cana-4508	228	7	)	)	PUNCT
cana-4508	229	1	ℋ	ℋ	NOUN
cana-4508	229	2	𝒳,0	𝒳,0	NUM
cana-4508	229	3	)	)	PUNCT
cana-4508	229	4	(	(	PUNCT
cana-4508	229	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	229	6	(	(	PUNCT
cana-4508	229	7	𝜋	𝜋	NOUN
cana-4508	229	8	4	4	NUM
cana-4508	229	9	)	)	PUNCT
cana-4508	229	10	)	)	PUNCT
cana-4508	230	1	ℋ	ℋ	PROPN
cana-4508	230	2	+	+	CCONJ
cana-4508	230	3	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	230	4	(	(	PUNCT
cana-4508	230	5	𝜋	𝜋	NOUN
cana-4508	230	6	4	4	NUM
cana-4508	230	7	)	)	PUNCT
cana-4508	230	8	)	)	PUNCT
cana-4508	231	1	ℋ	ℋ	NOUN
cana-4508	231	2	𝒳,0	𝒳,0	NUM
cana-4508	231	3	)	)	PUNCT
cana-4508	231	4	(	(	PUNCT
cana-4508	231	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	231	6	(	(	PUNCT
cana-4508	231	7	𝜋	𝜋	NOUN
cana-4508	231	8	4	4	NUM
cana-4508	231	9	)	)	PUNCT
cana-4508	231	10	)	)	PUNCT
cana-4508	231	11	ℋ	ℋ	NOUN
cana-4508	231	12	)	)	PUNCT
cana-4508	231	13	communications	communication	NOUN
cana-4508	231	14	on	on	ADP
cana-4508	231	15	applied	apply	VERB
cana-4508	231	16	nonlinear	nonlinear	ADJ
cana-4508	231	17	analysis	analysis	NOUN
cana-4508	231	18	issn	issn	NOUN
cana-4508	231	19	:	:	PUNCT
cana-4508	231	20	1074	1074	NUM
cana-4508	231	21	-	-	PUNCT
cana-4508	231	22	133x	133x	NUM
cana-4508	231	23	vol	vol	NOUN
cana-4508	231	24	32	32	NUM
cana-4508	231	25	no	no	NOUN
cana-4508	231	26	.	.	PUNCT
cana-4508	232	1	9s	9s	NUM
cana-4508	232	2	(	(	PUNCT
cana-4508	232	3	2025	2025	NUM
cana-4508	232	4	)	)	PUNCT
cana-4508	232	5	2241	2241	NUM
cana-4508	232	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	233	1	+	+	CCONJ
cana-4508	233	2	1	1	NUM
cana-4508	233	3	6(𝑐𝑜𝑠𝑒𝑐ℒ	6(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	233	4	(	(	PUNCT
cana-4508	233	5	𝜋	𝜋	NOUN
cana-4508	233	6	4	4	NUM
cana-4508	233	7	)	)	PUNCT
cana-4508	233	8	)	)	PUNCT
cana-4508	234	1	2∑	2∑	NUM
cana-4508	234	2	∞	∞	NUM
cana-4508	234	3	ℋ=0	ℋ=0	PROPN
cana-4508	234	4	(	(	PUNCT
cana-4508	234	5	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	𝒯((𝑐𝑜𝑠𝑒𝑐ℒ	VERB
cana-4508	234	6	(	(	PUNCT
cana-4508	234	7	𝜋	𝜋	NOUN
cana-4508	234	8	4	4	NUM
cana-4508	234	9	)	)	PUNCT
cana-4508	234	10	)	)	PUNCT
cana-4508	235	1	ℋ	ℋ	NOUN
cana-4508	235	2	𝒳,0	𝒳,0	NUM
cana-4508	235	3	)	)	PUNCT
cana-4508	235	4	(	(	PUNCT
cana-4508	235	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	235	6	(	(	PUNCT
cana-4508	235	7	𝜋	𝜋	NOUN
cana-4508	235	8	4	4	NUM
cana-4508	235	9	)	)	PUNCT
cana-4508	235	10	)	)	PUNCT
cana-4508	236	1	2ℋ	2ℋ	NOUN
cana-4508	236	2	+	+	CCONJ
cana-4508	236	3	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	𝒯(−(𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	236	4	(	(	PUNCT
cana-4508	236	5	𝜋	𝜋	NOUN
cana-4508	236	6	4	4	NUM
cana-4508	236	7	)	)	PUNCT
cana-4508	236	8	)	)	PUNCT
cana-4508	237	1	ℋ	ℋ	NOUN
cana-4508	237	2	𝒳,0	𝒳,0	NUM
cana-4508	237	3	)	)	PUNCT
cana-4508	237	4	(	(	PUNCT
cana-4508	237	5	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	237	6	(	(	PUNCT
cana-4508	237	7	𝜋	𝜋	NOUN
cana-4508	237	8	4	4	NUM
cana-4508	237	9	)	)	PUNCT
cana-4508	237	10	)	)	PUNCT
cana-4508	237	11	2ℋ	2ℋ	NOUN
cana-4508	237	12	)	)	PUNCT
cana-4508	237	13	for	for	ADP
cana-4508	237	14	all	all	DET
cana-4508	237	15	𝒳	𝒳	PROPN
cana-4508	237	16	∈	∈	PROPN
cana-4508	237	17	ℋ	ℋ	PROPN
cana-4508	237	18	corollary	corollary	ADJ
cana-4508	237	19	4.2	4.2	NUM
cana-4508	237	20	let	let	VERB
cana-4508	237	21	𝒯	𝒯	PROPN
cana-4508	237	22	and	and	CCONJ
cana-4508	237	23	p	p	NOUN
cana-4508	237	24	be	be	VERB
cana-4508	237	25	non	non	ADJ
cana-4508	237	26	negative	negative	ADJ
cana-4508	237	27	real	real	ADJ
cana-4508	237	28	numbers	number	NOUN
cana-4508	237	29	.	.	PUNCT
cana-4508	238	1	let	let	VERB
cana-4508	238	2	a	a	DET
cana-4508	238	3	function	function	NOUN
cana-4508	238	4	𝒬:ℋ	𝒬:ℋ	PUNCT
cana-4508	238	5	→	→	AUX
cana-4508	238	6	ℐ	ℐ	PRON
cana-4508	238	7	satisfy	satisfy	VERB
cana-4508	238	8	the	the	DET
cana-4508	238	9	inequality	inequality	PROPN
cana-4508	238	10	‖𝒬(𝒳	‖𝒬(𝒳	PROPN
cana-4508	238	11	,	,	PUNCT
cana-4508	238	12	𝒴)‖	𝒴)‖	ADP
cana-4508	238	13	≤	≤	NOUN
cana-4508	238	14	{	{	PUNCT
cana-4508	238	15	𝒯	𝒯	PROPN
cana-4508	238	16	,	,	PUNCT
cana-4508	238	17	𝒯{||𝒳||p	𝒯{||𝒳||p	PUNCT
cana-4508	238	18	+	+	NUM
cana-4508	238	19	||𝒴||p	||𝒴||p	NOUN
cana-4508	238	20	}	}	PUNCT
cana-4508	238	21	,	,	PUNCT
cana-4508	238	22	p	p	PROPN
cana-4508	238	23	≠	≠	PROPN
cana-4508	238	24	1,2	1,2	NUM
cana-4508	238	25	;	;	PUNCT
cana-4508	238	26	𝒯{||𝒳||p||𝒴||p	𝒯{||𝒳||p||𝒴||p	X
cana-4508	238	27	+	+	CCONJ
cana-4508	238	28	{	{	PUNCT
cana-4508	238	29	||𝒳||2p	||𝒳||2p	PROPN
cana-4508	238	30	+	+	CCONJ
cana-4508	238	31	||𝒴||2p	||𝒴||2p	NOUN
cana-4508	238	32	}	}	PUNCT
cana-4508	238	33	}	}	PUNCT
cana-4508	238	34	,	,	PUNCT
cana-4508	238	35	p	p	PROPN
cana-4508	238	36	≠	≠	PROPN
cana-4508	238	37	1	1	NUM
cana-4508	238	38	2	2	NUM
cana-4508	238	39	,	,	PUNCT
cana-4508	238	40	1	1	NUM
cana-4508	238	41	;	;	PUNCT
cana-4508	238	42	(	(	PUNCT
cana-4508	238	43	29	29	NUM
cana-4508	238	44	)	)	PUNCT
cana-4508	238	45	for	for	ADP
cana-4508	238	46	all	all	DET
cana-4508	238	47	𝒳,𝒴	𝒳,𝒴	NOUN
cana-4508	238	48	∈	∈	PROPN
cana-4508	238	49	ℋ.	ℋ.	PROPN
cana-4508	238	50	then	then	ADV
cana-4508	238	51	there	there	PRON
cana-4508	238	52	exists	exist	VERB
cana-4508	238	53	a	a	DET
cana-4508	238	54	unique	unique	ADJ
cana-4508	238	55	additive	additive	ADJ
cana-4508	238	56	function	function	NOUN
cana-4508	238	57	a:ℋ	a:ℋ	NUM
cana-4508	238	58	→	→	SYM
cana-4508	238	59	ℐ	ℐ	PRON
cana-4508	238	60	and	and	CCONJ
cana-4508	238	61	a	a	DET
cana-4508	238	62	unique	unique	ADJ
cana-4508	238	63	quadratic	quadratic	ADJ
cana-4508	238	64	function	function	NOUN
cana-4508	238	65	q:ℋ	q:ℋ	PROPN
cana-4508	238	66	→	→	SYM
cana-4508	238	67	ℐ	ℐ	PRON
cana-4508	238	68	such	such	ADJ
cana-4508	238	69	that	that	PRON
cana-4508	238	70	‖𝒬1(𝒳	‖𝒬1(𝒳	PROPN
cana-4508	238	71	)	)	PUNCT
cana-4508	238	72	−	−	PROPN
cana-4508	238	73	𝐴(𝒳	𝐴(𝒳	NOUN
cana-4508	238	74	)	)	PUNCT
cana-4508	238	75	−	−	ADP
cana-4508	239	1	𝒬(𝒳)‖	𝒬(𝒳)‖	NOUN
cana-4508	239	2	≤	≤	NOUN
cana-4508	239	3	{	{	PUNCT
cana-4508	239	4	(	(	PUNCT
cana-4508	239	5	𝒯	𝒯	PROPN
cana-4508	239	6	(	(	PUNCT
cana-4508	239	7	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	PROPN
cana-4508	239	8	(	(	PUNCT
cana-4508	239	9	𝜋	𝜋	NOUN
cana-4508	239	10	4	4	NUM
cana-4508	239	11	)	)	PUNCT
cana-4508	239	12	)	)	PUNCT
cana-4508	239	13	−1	−1	NOUN
cana-4508	239	14	)	)	PUNCT
cana-4508	240	1	+	+	CCONJ
cana-4508	240	2	(	(	PUNCT
cana-4508	240	3	𝒯	𝒯	PROPN
cana-4508	240	4	(	(	PUNCT
cana-4508	240	5	(	(	PUNCT
cana-4508	240	6	𝑐𝑜𝑠𝑒𝑐ℒ	𝑐𝑜𝑠𝑒𝑐ℒ	INTJ
cana-4508	240	7	(	(	PUNCT
cana-4508	240	8	𝜋	𝜋	NOUN
cana-4508	240	9	4	4	NUM
cana-4508	240	10	)	)	PUNCT
cana-4508	240	11	)	)	PUNCT
cana-4508	240	12	2	2	NUM
cana-4508	240	13	−1	−1	NOUN
cana-4508	240	14	)	)	PUNCT
cana-4508	240	15	)	)	PUNCT
cana-4508	240	16	,	,	PUNCT
cana-4508	240	17	[	[	PUNCT
cana-4508	240	18	1	1	NUM
cana-4508	240	19	|(𝑐𝑜𝑠𝑒𝑐ℒ	|(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	240	20	(	(	PUNCT
cana-4508	240	21	𝜋	𝜋	NOUN
cana-4508	240	22	4	4	NUM
cana-4508	240	23	)	)	PUNCT
cana-4508	240	24	)	)	PUNCT
cana-4508	240	25	−(𝑐𝑜𝑠𝑒𝑐ℒ	−(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	240	26	(	(	PUNCT
cana-4508	240	27	𝜋	𝜋	NOUN
cana-4508	240	28	4	4	NUM
cana-4508	240	29	)	)	PUNCT
cana-4508	240	30	)	)	PUNCT
cana-4508	240	31	𝑃	𝑃	VERB
cana-4508	240	32	|	|	ADV
cana-4508	241	1	+	+	CCONJ
cana-4508	242	1	1	1	NUM
cana-4508	242	2	3|(𝑐𝑜𝑠𝑒𝑐ℒ	3|(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	242	3	(	(	PUNCT
cana-4508	242	4	𝜋	𝜋	NOUN
cana-4508	242	5	4	4	NUM
cana-4508	242	6	)	)	PUNCT
cana-4508	242	7	)	)	PUNCT
cana-4508	242	8	2	2	NUM
cana-4508	242	9	−(𝑐𝑜𝑠𝑒𝑐ℒ	−(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	242	10	(	(	PUNCT
cana-4508	242	11	𝜋	𝜋	NOUN
cana-4508	242	12	4	4	NUM
cana-4508	242	13	)	)	PUNCT
cana-4508	242	14	)	)	PUNCT
cana-4508	243	1	𝑃	𝑃	VERB
cana-4508	243	2	|	|	ADV
cana-4508	243	3	]	]	PUNCT
cana-4508	243	4	||𝒳||𝑃	||𝒳||𝑃	NOUN
cana-4508	243	5	,	,	PUNCT
cana-4508	243	6	[	[	PUNCT
cana-4508	243	7	1	1	NUM
cana-4508	243	8	|(𝑐𝑜𝑠𝑒𝑐ℒ	|(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	243	9	(	(	PUNCT
cana-4508	243	10	𝜋	𝜋	NOUN
cana-4508	243	11	4	4	NUM
cana-4508	243	12	)	)	PUNCT
cana-4508	243	13	)	)	PUNCT
cana-4508	243	14	−(𝑐𝑜𝑠𝑒𝑐ℒ	−(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	243	15	(	(	PUNCT
cana-4508	243	16	𝜋	𝜋	NOUN
cana-4508	243	17	4	4	NUM
cana-4508	243	18	)	)	PUNCT
cana-4508	243	19	)	)	PUNCT
cana-4508	243	20	2𝑃	2𝑃	ADV
cana-4508	244	1	|	|	ADV
cana-4508	245	1	+	+	CCONJ
cana-4508	245	2	1	1	NUM
cana-4508	245	3	3|(𝑐𝑜𝑠𝑒𝑐ℒ	3|(𝑐𝑜𝑠𝑒𝑐ℒ	NUM
cana-4508	245	4	(	(	PUNCT
cana-4508	245	5	𝜋	𝜋	NOUN
cana-4508	245	6	4	4	NUM
cana-4508	245	7	)	)	PUNCT
cana-4508	245	8	)	)	PUNCT
cana-4508	245	9	2	2	NUM
cana-4508	245	10	−(𝑐𝑜𝑠𝑒𝑐ℒ	−(𝑐𝑜𝑠𝑒𝑐ℒ	NOUN
cana-4508	245	11	(	(	PUNCT
cana-4508	245	12	𝜋	𝜋	NOUN
cana-4508	245	13	4	4	NUM
cana-4508	245	14	)	)	PUNCT
cana-4508	245	15	)	)	PUNCT
cana-4508	245	16	2𝑃	2𝑃	ADV
cana-4508	246	1	|	|	ADV
cana-4508	246	2	]	]	PUNCT
cana-4508	246	3	||𝒳||2𝑃	||𝒳||2𝑃	PROPN
cana-4508	246	4	(	(	PUNCT
cana-4508	246	5	30	30	NUM
cana-4508	246	6	)	)	PUNCT
cana-4508	246	7	for	for	ADP
cana-4508	246	8	all	all	DET
cana-4508	246	9	𝒳	𝒳	PROPN
cana-4508	246	10	∈	∈	PROPN
cana-4508	246	11	ℋ	ℋ	PROPN
cana-4508	246	12	refrences	refrence	NOUN
cana-4508	246	13	[	[	X
cana-4508	246	14	1	1	X
cana-4508	246	15	]	]	X
cana-4508	246	16	s.m	s.m	PROPN
cana-4508	246	17	.	.	PROPN
cana-4508	246	18	ulam	ulam	PROPN
cana-4508	246	19	,	,	PUNCT
cana-4508	246	20	problems	problem	NOUN
cana-4508	246	21	in	in	ADP
cana-4508	246	22	modern	modern	ADJ
cana-4508	246	23	mathematics	mathematic	NOUN
cana-4508	246	24	,	,	PUNCT
cana-4508	246	25	science	science	NOUN
cana-4508	246	26	editions	edition	NOUN
cana-4508	246	27	,	,	PUNCT
cana-4508	246	28	wiley	wiley	NOUN
cana-4508	246	29	,	,	PUNCT
cana-4508	246	30	newyork	newyork	PROPN
cana-4508	246	31	,	,	PUNCT
cana-4508	246	32	1964	1964	NUM
cana-4508	246	33	(	(	PUNCT
cana-4508	246	34	chapter	chapter	NOUN
cana-4508	246	35	vi	vi	PROPN
cana-4508	246	36	,	,	PUNCT
cana-4508	246	37	some	some	DET
cana-4508	246	38	questions	question	NOUN
cana-4508	246	39	in	in	ADP
cana-4508	246	40	analysis	analysis	NOUN
cana-4508	246	41	:	:	PUNCT
cana-4508	246	42	1	1	NUM
cana-4508	246	43	,	,	PUNCT
cana-4508	246	44	stability	stability	NOUN
cana-4508	246	45	)	)	PUNCT
cana-4508	246	46	.	.	PUNCT
cana-4508	247	1	[	[	X
cana-4508	247	2	2	2	NUM
cana-4508	247	3	]	]	X
cana-4508	247	4	d.h	d.h	PROPN
cana-4508	247	5	.	.	PROPN
cana-4508	247	6	hyers	hyer	NOUN
cana-4508	247	7	,	,	PUNCT
cana-4508	247	8	on	on	ADP
cana-4508	247	9	the	the	DET
cana-4508	247	10	stability	stability	NOUN
cana-4508	247	11	of	of	ADP
cana-4508	247	12	the	the	DET
cana-4508	247	13	linear	linear	ADJ
cana-4508	247	14	functional	functional	ADJ
cana-4508	247	15	equation	equation	NOUN
cana-4508	247	16	,	,	PUNCT
cana-4508	247	17	proc.nat	proc.nat	PROPN
cana-4508	247	18	.	.	PUNCT
cana-4508	248	1	acad.sci	acad.sci	X
cana-4508	248	2	.	.	PUNCT
cana-4508	248	3	,u.s.a	,u.s.a	PROPN
cana-4508	248	4	.	.	PUNCT
cana-4508	248	5	,27	,27	PUNCT
cana-4508	248	6	(	(	PUNCT
cana-4508	248	7	1941	1941	NUM
cana-4508	248	8	)	)	PUNCT
cana-4508	248	9	222	222	NUM
cana-4508	248	10	-	-	SYM
cana-4508	248	11	224	224	NUM
cana-4508	248	12	.	.	PUNCT
cana-4508	249	1	[	[	X
cana-4508	249	2	3	3	X
cana-4508	249	3	]	]	PUNCT
cana-4508	249	4	t.	t.	PROPN
cana-4508	249	5	aoki	aoki	PROPN
cana-4508	249	6	,	,	PUNCT
cana-4508	249	7	on	on	ADP
cana-4508	249	8	the	the	DET
cana-4508	249	9	stability	stability	NOUN
cana-4508	249	10	of	of	ADP
cana-4508	249	11	the	the	DET
cana-4508	249	12	linear	linear	ADJ
cana-4508	249	13	transformation	transformation	NOUN
cana-4508	249	14	in	in	ADP
cana-4508	249	15	banach	banach	NOUN
cana-4508	249	16	spaces	space	NOUN
cana-4508	249	17	,	,	PUNCT
cana-4508	249	18	j.	j.	PROPN
cana-4508	249	19	math	math	PROPN
cana-4508	249	20	.	.	PUNCT
cana-4508	250	1	soc	soc	PROPN
cana-4508	250	2	.	.	PUNCT
cana-4508	251	1	japan	japan	PROPN
cana-4508	251	2	,	,	PUNCT
cana-4508	251	3	2	2	NUM
cana-4508	251	4	(	(	PUNCT
cana-4508	251	5	1950	1950	NUM
cana-4508	251	6	)	)	PUNCT
cana-4508	251	7	,	,	PUNCT
cana-4508	251	8	64	64	NUM
cana-4508	251	9	-	-	SYM
cana-4508	251	10	66	66	NUM
cana-4508	251	11	.	.	PUNCT
cana-4508	252	1	[	[	X
cana-4508	252	2	4	4	NUM
cana-4508	252	3	]	]	X
cana-4508	252	4	p.	p.	NOUN
cana-4508	252	5	găvrută	găvrută	NOUN
cana-4508	252	6	,	,	PUNCT
cana-4508	252	7	an	an	DET
cana-4508	252	8	answer	answer	NOUN
cana-4508	252	9	to	to	ADP
cana-4508	252	10	a	a	DET
cana-4508	252	11	question	question	NOUN
cana-4508	252	12	of	of	ADP
cana-4508	252	13	j.m.rassias	j.m.rassia	NOUN
cana-4508	252	14	concerning	concern	VERB
cana-4508	252	15	the	the	DET
cana-4508	252	16	stability	stability	NOUN
cana-4508	252	17	of	of	ADP
cana-4508	252	18	cauchy	cauchy	ADJ
cana-4508	252	19	functional	functional	ADJ
cana-4508	252	20	equation	equation	NOUN
cana-4508	252	21	,	,	PUNCT
cana-4508	252	22	advances	advance	NOUN
cana-4508	252	23	in	in	ADP
cana-4508	252	24	equations	equation	NOUN
cana-4508	252	25	and	and	CCONJ
cana-4508	252	26	inequalities	inequality	NOUN
cana-4508	252	27	,	,	PUNCT
cana-4508	252	28	hadronic	hadronic	ADJ
cana-4508	252	29	math	math	NOUN
cana-4508	252	30	.	.	PUNCT
cana-4508	253	1	ser	ser	PROPN
cana-4508	253	2	.	.	PROPN
cana-4508	253	3	,	,	PUNCT
cana-4508	253	4	(	(	PUNCT
cana-4508	253	5	1999	1999	NUM
cana-4508	253	6	)	)	PUNCT
cana-4508	253	7	,	,	PUNCT
cana-4508	253	8	67	67	NUM
cana-4508	253	9	-	-	SYM
cana-4508	253	10	71	71	NUM
cana-4508	253	11	.	.	PUNCT
cana-4508	254	1	[	[	X
cana-4508	254	2	5	5	NUM
cana-4508	254	3	]	]	PUNCT
cana-4508	254	4	p.	p.	NOUN
cana-4508	254	5	găvrută	găvrută	NOUN
cana-4508	254	6	,	,	PUNCT
cana-4508	254	7	on	on	ADP
cana-4508	254	8	a	a	DET
cana-4508	254	9	problem	problem	NOUN
cana-4508	254	10	of	of	ADP
cana-4508	254	11	g.	g.	PROPN
cana-4508	254	12	isac	isac	PROPN
cana-4508	254	13	and	and	CCONJ
cana-4508	254	14	th	th	PROPN
cana-4508	254	15	.	.	PUNCT
cana-4508	255	1	m.	m.	NOUN
cana-4508	255	2	rassias	rassias	PROPN
cana-4508	255	3	concerning	concern	VERB
cana-4508	255	4	the	the	DET
cana-4508	255	5	stability	stability	NOUN
cana-4508	255	6	of	of	ADP
cana-4508	255	7	mappings	mapping	NOUN
cana-4508	255	8	,	,	PUNCT
cana-4508	255	9	j.	j.	PROPN
cana-4508	255	10	math	math	PROPN
cana-4508	255	11	.	.	PUNCT
cana-4508	256	1	anal	anal	PROPN
cana-4508	256	2	.	.	PUNCT
cana-4508	257	1	appl	appl	PROPN
cana-4508	257	2	.	.	PUNCT
cana-4508	258	1	261	261	NUM
cana-4508	258	2	(	(	PUNCT
cana-4508	258	3	2001	2001	NUM
cana-4508	258	4	)	)	PUNCT
cana-4508	258	5	,	,	PUNCT
cana-4508	258	6	543	543	NUM
cana-4508	258	7	-	-	SYM
cana-4508	258	8	553	553	NUM
cana-4508	258	9	.	.	PUNCT
cana-4508	259	1	[	[	X
cana-4508	259	2	6	6	NUM
cana-4508	259	3	]	]	X
cana-4508	259	4	j.m	j.m	PROPN
cana-4508	259	5	.	.	PROPN
cana-4508	259	6	rassias	rassias	PROPN
cana-4508	259	7	,	,	PUNCT
cana-4508	259	8	on	on	ADP
cana-4508	259	9	approximately	approximately	ADV
cana-4508	259	10	of	of	ADP
cana-4508	259	11	approximately	approximately	ADV
cana-4508	259	12	linear	linear	ADJ
cana-4508	259	13	mappings	mapping	NOUN
cana-4508	259	14	by	by	ADP
cana-4508	259	15	linear	linear	PROPN
cana-4508	259	16	mappings	mapping	NOUN
cana-4508	259	17	,	,	PUNCT
cana-4508	259	18	j.	j.	PROPN
cana-4508	259	19	funct	funct	PROPN
cana-4508	259	20	.	.	PUNCT
cana-4508	260	1	anal	anal	PROPN
cana-4508	260	2	.	.	PUNCT
cana-4508	261	1	usa	usa	PROPN
cana-4508	261	2	,	,	PUNCT
cana-4508	261	3	46	46	NUM
cana-4508	261	4	,	,	PUNCT
cana-4508	261	5	(	(	PUNCT
cana-4508	261	6	1982	1982	NUM
cana-4508	261	7	)	)	PUNCT
cana-4508	261	8	126	126	NUM
cana-4508	261	9	-	-	SYM
cana-4508	261	10	130	130	NUM
cana-4508	261	11	.	.	PUNCT
cana-4508	262	1	communications	communication	NOUN
cana-4508	262	2	on	on	ADP
cana-4508	262	3	applied	apply	VERB
cana-4508	262	4	nonlinear	nonlinear	ADJ
cana-4508	262	5	analysis	analysis	NOUN
cana-4508	262	6	issn	issn	NOUN
cana-4508	262	7	:	:	PUNCT
cana-4508	262	8	1074	1074	NUM
cana-4508	262	9	-	-	PUNCT
cana-4508	262	10	133x	133x	NUM
cana-4508	262	11	vol	vol	NOUN
cana-4508	262	12	32	32	NUM
cana-4508	262	13	no	no	NOUN
cana-4508	262	14	.	.	PUNCT
cana-4508	263	1	9s	9s	NUM
cana-4508	263	2	(	(	PUNCT
cana-4508	263	3	2025	2025	NUM
cana-4508	263	4	)	)	PUNCT
cana-4508	263	5	2242	2242	NUM
cana-4508	263	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	264	1	[	[	X
cana-4508	264	2	7	7	X
cana-4508	264	3	]	]	X
cana-4508	264	4	j.m	j.m	PROPN
cana-4508	264	5	.	.	PROPN
cana-4508	264	6	rassias	rassias	PROPN
cana-4508	264	7	,	,	PUNCT
cana-4508	264	8	on	on	ADP
cana-4508	264	9	approximately	approximately	ADV
cana-4508	264	10	of	of	ADP
cana-4508	264	11	approximately	approximately	ADV
cana-4508	264	12	linear	linear	ADJ
cana-4508	264	13	mappings	mapping	NOUN
cana-4508	264	14	by	by	ADP
cana-4508	264	15	linear	linear	ADJ
cana-4508	264	16	mappings	mapping	NOUN
cana-4508	264	17	,	,	PUNCT
cana-4508	264	18	bull	bull	NOUN
cana-4508	264	19	.	.	PUNCT
cana-4508	265	1	sc	sc	PROPN
cana-4508	265	2	.	.	PROPN
cana-4508	265	3	math	math	PROPN
cana-4508	265	4	,	,	PUNCT
cana-4508	265	5	108	108	NUM
cana-4508	265	6	,	,	PUNCT
cana-4508	265	7	(	(	PUNCT
cana-4508	265	8	1984	1984	NUM
cana-4508	265	9	)	)	PUNCT
cana-4508	265	10	445	445	NUM
cana-4508	265	11	-	-	SYM
cana-4508	265	12	446	446	NUM
cana-4508	265	13	.	.	PUNCT
cana-4508	266	1	[	[	X
cana-4508	266	2	8	8	NUM
cana-4508	266	3	]	]	X
cana-4508	266	4	j.m	j.m	PROPN
cana-4508	266	5	.	.	PROPN
cana-4508	266	6	rassias	rassias	PROPN
cana-4508	266	7	,	,	PUNCT
cana-4508	266	8	k.w	k.w	PROPN
cana-4508	266	9	.	.	PROPN
cana-4508	266	10	jun	jun	PROPN
cana-4508	266	11	,	,	PUNCT
cana-4508	266	12	h.m	h.m	PROPN
cana-4508	266	13	.	.	PROPN
cana-4508	266	14	kim	kim	PROPN
cana-4508	266	15	,	,	PUNCT
cana-4508	266	16	approximate	approximate	ADJ
cana-4508	266	17	(	(	PUNCT
cana-4508	266	18	m	m	PROPN
cana-4508	266	19	,	,	PUNCT
cana-4508	266	20	n	n	CCONJ
cana-4508	266	21	)	)	PUNCT
cana-4508	266	22	−cauchy	−cauchy	ADJ
cana-4508	266	23	jensen	jensen	PROPN
cana-4508	266	24	additive	additive	ADJ
cana-4508	266	25	mappings	mapping	NOUN
cana-4508	266	26	in	in	ADP
cana-4508	266	27	c	c	PROPN
cana-4508	266	28	∗-algebras	∗-algebra	NOUN
cana-4508	266	29	,	,	PUNCT
cana-4508	266	30	acta	acta	PROPN
cana-4508	266	31	mathematica	mathematica	PROPN
cana-4508	266	32	sinica	sinica	PROPN
cana-4508	266	33	,	,	PUNCT
cana-4508	266	34	english	english	ADJ
cana-4508	266	35	series	series	NOUN
cana-4508	266	36	,	,	PUNCT
cana-4508	266	37	vol	vol	NOUN
cana-4508	266	38	.	.	PROPN
cana-4508	266	39	27	27	NUM
cana-4508	266	40	,	,	PUNCT
cana-4508	266	41	no	no	INTJ
cana-4508	266	42	.	.	NOUN
cana-4508	266	43	10	10	NUM
cana-4508	266	44	,	,	PUNCT
cana-4508	266	45	(	(	PUNCT
cana-4508	266	46	2011	2011	NUM
cana-4508	266	47	)	)	PUNCT
cana-4508	266	48	,	,	PUNCT
cana-4508	266	49	1907	1907	NUM
cana-4508	266	50	-	-	SYM
cana-4508	266	51	1922	1922	NUM
cana-4508	266	52	.	.	PUNCT
cana-4508	267	1	[	[	X
cana-4508	267	2	9	9	NUM
cana-4508	267	3	]	]	SYM
cana-4508	267	4	rus	rus	NOUN
cana-4508	267	5	,	,	PUNCT
cana-4508	267	6	ioan	ioan	PROPN
cana-4508	267	7	a.	a.	PROPN
cana-4508	267	8	"	"	PUNCT
cana-4508	267	9	ulam	ulam	X
cana-4508	267	10	stability	stability	NOUN
cana-4508	267	11	of	of	ADP
cana-4508	267	12	ordinary	ordinary	ADJ
cana-4508	267	13	differential	differential	ADJ
cana-4508	267	14	equation	equation	NOUN
cana-4508	267	15	.	.	PUNCT
cana-4508	267	16	"	"	PUNCT
cana-4508	268	1	studia	studia	PROPN
cana-4508	268	2	universitatis	universitatis	PROPN
cana-4508	268	3	babesbolyai	babesbolyai	PROPN
cana-4508	268	4	,	,	PUNCT
cana-4508	268	5	mathematica	mathematica	PROPN
cana-4508	268	6	4	4	NUM
cana-4508	268	7	(	(	PUNCT
cana-4508	268	8	2009	2009	NUM
cana-4508	268	9	)	)	PUNCT
cana-4508	268	10	.	.	PUNCT
cana-4508	269	1	[	[	X
cana-4508	269	2	10	10	NUM
cana-4508	269	3	]	]	PUNCT
cana-4508	269	4	kumama	kumama	NOUN
cana-4508	269	5	,	,	PUNCT
cana-4508	269	6	poom	poom	NOUN
cana-4508	269	7	,	,	PUNCT
cana-4508	269	8	amjad	amjad	PROPN
cana-4508	269	9	ali	ali	PROPN
cana-4508	269	10	,	,	PUNCT
cana-4508	269	11	kamal	kamal	PROPN
cana-4508	269	12	shah	shah	PROPN
cana-4508	269	13	,	,	PUNCT
cana-4508	269	14	and	and	CCONJ
cana-4508	269	15	rahmat	rahmat	PROPN
cana-4508	269	16	ali	ali	PROPN
cana-4508	269	17	khan	khan	PROPN
cana-4508	269	18	.	.	PUNCT
cana-4508	270	1	"	"	PUNCT
cana-4508	270	2	existence	existence	NOUN
cana-4508	270	3	results	result	NOUN
cana-4508	270	4	and	and	CCONJ
cana-4508	270	5	hyers	hyer	NOUN
cana-4508	270	6	–	–	PUNCT
cana-4508	270	7	ulam	ulam	PROPN
cana-4508	270	8	stability	stability	NOUN
cana-4508	270	9	to	to	ADP
cana-4508	270	10	a	a	DET
cana-4508	270	11	class	class	NOUN
cana-4508	270	12	of	of	ADP
cana-4508	270	13	nonlinear	nonlinear	ADJ
cana-4508	270	14	arbitrary	arbitrary	ADJ
cana-4508	270	15	order	order	NOUN
cana-4508	270	16	differential	differential	NOUN
cana-4508	270	17	equations	equation	NOUN
cana-4508	270	18	.	.	PUNCT
cana-4508	270	19	"	"	PUNCT
cana-4508	271	1	j.	j.	PROPN
cana-4508	271	2	nonlinear	nonlinear	PROPN
cana-4508	271	3	sci	sci	PROPN
cana-4508	271	4	.	.	PUNCT
cana-4508	271	5	appl	appl	PROPN
cana-4508	271	6	10	10	NUM
cana-4508	271	7	,	,	PUNCT
cana-4508	271	8	no	no	INTJ
cana-4508	271	9	.	.	NOUN
cana-4508	271	10	6	6	NUM
cana-4508	271	11	(	(	PUNCT
cana-4508	271	12	2017	2017	NUM
cana-4508	271	13	):	):	PUNCT
cana-4508	271	14	2986	2986	NUM
cana-4508	271	15	-	-	SYM
cana-4508	271	16	2997	2997	NUM
cana-4508	271	17	.	.	PUNCT
cana-4508	272	1	[	[	X
cana-4508	272	2	11	11	NUM
cana-4508	272	3	]	]	X
cana-4508	272	4	agarwal	agarwal	PROPN
cana-4508	272	5	,	,	PUNCT
cana-4508	272	6	ravi	ravi	PROPN
cana-4508	272	7	p.	p.	PROPN
cana-4508	272	8	,	,	PUNCT
cana-4508	272	9	snezhana	snezhana	PROPN
cana-4508	272	10	hristova	hristova	PROPN
cana-4508	272	11	,	,	PUNCT
cana-4508	272	12	and	and	CCONJ
cana-4508	272	13	donal	donal	PROPN
cana-4508	272	14	o’regan	o’regan	PROPN
cana-4508	272	15	.	.	PUNCT
cana-4508	273	1	"	"	PUNCT
cana-4508	273	2	ulam	ulam	PROPN
cana-4508	273	3	stability	stability	NOUN
cana-4508	273	4	for	for	ADP
cana-4508	273	5	boundary	boundary	ADJ
cana-4508	273	6	value	value	NOUN
cana-4508	273	7	problems	problem	NOUN
cana-4508	273	8	of	of	ADP
cana-4508	273	9	differential	differential	ADJ
cana-4508	273	10	equations	equation	NOUN
cana-4508	273	11	—	—	PUNCT
cana-4508	273	12	main	main	ADJ
cana-4508	273	13	misunderstandings	misunderstanding	NOUN
cana-4508	273	14	and	and	CCONJ
cana-4508	273	15	how	how	SCONJ
cana-4508	273	16	to	to	PART
cana-4508	273	17	avoid	avoid	VERB
cana-4508	273	18	them	they	PRON
cana-4508	273	19	.	.	PUNCT
cana-4508	273	20	"	"	PUNCT
cana-4508	274	1	mathematics	mathematic	NOUN
cana-4508	274	2	12	12	NUM
cana-4508	274	3	,	,	PUNCT
cana-4508	274	4	no	no	INTJ
cana-4508	274	5	.	.	NOUN
cana-4508	274	6	11	11	NUM
cana-4508	274	7	(	(	PUNCT
cana-4508	274	8	2024	2024	NUM
cana-4508	274	9	):	):	PUNCT
cana-4508	274	10	1626	1626	NUM
cana-4508	274	11	.	.	PUNCT
cana-4508	275	1	[	[	X
cana-4508	275	2	12	12	NUM
cana-4508	275	3	]	]	PUNCT
cana-4508	275	4	baias	baia	NOUN
cana-4508	275	5	,	,	PUNCT
cana-4508	275	6	alina	alina	PROPN
cana-4508	275	7	ramona	ramona	PROPN
cana-4508	275	8	,	,	PUNCT
cana-4508	275	9	and	and	CCONJ
cana-4508	275	10	dorian	dorian	PROPN
cana-4508	275	11	popa	popa	NOUN
cana-4508	275	12	.	.	PUNCT
cana-4508	276	1	"	"	PUNCT
cana-4508	276	2	on	on	ADP
cana-4508	276	3	ulam	ulam	PROPN
cana-4508	276	4	stability	stability	NOUN
cana-4508	276	5	of	of	ADP
cana-4508	276	6	a	a	DET
cana-4508	276	7	linear	linear	ADJ
cana-4508	276	8	difference	difference	NOUN
cana-4508	276	9	equation	equation	NOUN
cana-4508	276	10	in	in	ADP
cana-4508	276	11	banach	banach	NOUN
cana-4508	276	12	spaces	space	NOUN
cana-4508	276	13	.	.	PUNCT
cana-4508	276	14	"	"	PUNCT
cana-4508	276	15	bulletin	bulletin	NOUN
cana-4508	276	16	of	of	ADP
cana-4508	276	17	the	the	DET
cana-4508	276	18	malaysian	malaysian	PROPN
cana-4508	276	19	mathematical	mathematical	PROPN
cana-4508	276	20	sciences	sciences	PROPN
cana-4508	276	21	society	society	NOUN
cana-4508	276	22	43	43	NUM
cana-4508	276	23	,	,	PUNCT
cana-4508	276	24	no	no	INTJ
cana-4508	276	25	.	.	NOUN
cana-4508	276	26	2	2	NUM
cana-4508	276	27	(	(	PUNCT
cana-4508	276	28	2020	2020	NUM
cana-4508	276	29	):	):	PUNCT
cana-4508	276	30	13571371	13571371	NUM
cana-4508	276	31	.	.	PUNCT
cana-4508	277	1	[	[	X
cana-4508	277	2	13	13	NUM
cana-4508	277	3	]	]	X
cana-4508	277	4	tripathy	tripathy	ADJ
cana-4508	277	5	,	,	PUNCT
cana-4508	277	6	a.k	a.k	PROPN
cana-4508	277	7	.	.	PROPN
cana-4508	277	8	,	,	PUNCT
cana-4508	277	9	2021	2021	NUM
cana-4508	277	10	.	.	PUNCT
cana-4508	278	1	hyers	hyer	NOUN
cana-4508	278	2	-	-	PUNCT
cana-4508	278	3	ulam	ulam	PROPN
cana-4508	278	4	stability	stability	NOUN
cana-4508	278	5	of	of	ADP
cana-4508	278	6	ordinary	ordinary	ADJ
cana-4508	278	7	differential	differential	ADJ
cana-4508	278	8	equations	equation	NOUN
cana-4508	278	9	.	.	PUNCT
cana-4508	279	1	chapman	chapman	NOUN
cana-4508	279	2	and	and	CCONJ
cana-4508	279	3	hall	hall	PROPN
cana-4508	279	4	/	/	SYM
cana-4508	279	5	crc	crc	NOUN
cana-4508	279	6	.	.	PUNCT
cana-4508	280	1	[	[	X
cana-4508	280	2	14	14	NUM
cana-4508	280	3	]	]	X
cana-4508	280	4	pasupathi	pasupathi	NOUN
cana-4508	280	5	,	,	PUNCT
cana-4508	280	6	a.	a.	NOUN
cana-4508	280	7	;	;	PUNCT
cana-4508	280	8	konsalraj	konsalraj	PROPN
cana-4508	280	9	,	,	PUNCT
cana-4508	280	10	j.	j.	PROPN
cana-4508	280	11	;	;	PUNCT
cana-4508	280	12	fatima	fatima	PROPN
cana-4508	280	13	,	,	PUNCT
cana-4508	280	14	n.	n.	NOUN
cana-4508	280	15	;	;	PUNCT
cana-4508	280	16	velusamy	velusamy	PROPN
cana-4508	280	17	,	,	PUNCT
cana-4508	280	18	v.	v.	PROPN
cana-4508	280	19	;	;	PUNCT
cana-4508	280	20	mlaiki	mlaiki	PROPN
cana-4508	280	21	,	,	PUNCT
cana-4508	280	22	n.	n.	NOUN
cana-4508	280	23	;	;	PUNCT
cana-4508	280	24	souayah	souayah	NOUN
cana-4508	280	25	,	,	PUNCT
cana-4508	280	26	n.	n.	NOUN
cana-4508	280	27	direct	direct	ADJ
cana-4508	280	28	and	and	CCONJ
cana-4508	280	29	fixedpoint	fixedpoint	NOUN
cana-4508	280	30	stability	stability	NOUN
cana-4508	280	31	–	–	PUNCT
cana-4508	280	32	instability	instability	NOUN
cana-4508	280	33	of	of	ADP
cana-4508	280	34	additive	additive	ADJ
cana-4508	280	35	functional	functional	ADJ
cana-4508	280	36	equation	equation	NOUN
cana-4508	280	37	in	in	ADP
cana-4508	280	38	banach	banach	NOUN
cana-4508	280	39	and	and	CCONJ
cana-4508	280	40	quasi	quasi	ADJ
cana-4508	280	41	-	-	ADJ
cana-4508	280	42	beta	beta	ADJ
cana-4508	280	43	normed	norme	VERB
cana-4508	280	44	spaces	space	NOUN
cana-4508	280	45	.	.	PUNCT
cana-4508	281	1	symmetry	symmetry	NOUN
cana-4508	281	2	2022	2022	NUM
cana-4508	281	3	,	,	PUNCT
cana-4508	281	4	14	14	NUM
cana-4508	281	5	,	,	PUNCT
cana-4508	281	6	1700	1700	NUM
cana-4508	281	7	.	.	PUNCT
cana-4508	282	1	[	[	X
cana-4508	282	2	15	15	NUM
cana-4508	282	3	]	]	X
cana-4508	282	4	agilan	agilan	ADJ
cana-4508	282	5	,	,	PUNCT
cana-4508	282	6	p.	p.	NOUN
cana-4508	282	7	;	;	PUNCT
cana-4508	282	8	julietraja	julietraja	PROPN
cana-4508	282	9	,	,	PUNCT
cana-4508	282	10	k.	k.	PROPN
cana-4508	282	11	;	;	PUNCT
cana-4508	282	12	mlaiki	mlaiki	PROPN
cana-4508	282	13	,	,	PUNCT
cana-4508	282	14	n.	n.	NOUN
cana-4508	282	15	;	;	PUNCT
cana-4508	282	16	mukheimer	mukheimer	NOUN
cana-4508	282	17	,	,	PUNCT
cana-4508	282	18	a.	a.	NOUN
cana-4508	282	19	intuitionistic	intuitionistic	ADJ
cana-4508	282	20	fuzzy	fuzzy	ADJ
cana-4508	282	21	stability	stability	NOUN
cana-4508	282	22	of	of	ADP
cana-4508	282	23	an	an	DET
cana-4508	282	24	euler	euler	NOUN
cana-4508	282	25	–	–	PUNCT
cana-4508	282	26	lagrange	lagrange	NOUN
cana-4508	282	27	symmetry	symmetry	NOUN
cana-4508	282	28	additive	additive	ADJ
cana-4508	282	29	functional	functional	ADJ
cana-4508	282	30	equation	equation	NOUN
cana-4508	282	31	via	via	ADP
cana-4508	282	32	direct	direct	ADJ
cana-4508	282	33	and	and	CCONJ
cana-4508	282	34	fixed	fix	VERB
cana-4508	282	35	point	point	NOUN
cana-4508	282	36	technique	technique	NOUN
cana-4508	282	37	(	(	PUNCT
cana-4508	282	38	fpt	fpt	PROPN
cana-4508	282	39	)	)	PUNCT
cana-4508	282	40	.	.	PUNCT
cana-4508	283	1	symmetry	symmetry	PROPN
cana-4508	283	2	2022	2022	NUM
cana-4508	283	3	,	,	PUNCT
cana-4508	283	4	14	14	NUM
cana-4508	283	5	,	,	PUNCT
cana-4508	283	6	2454	2454	NUM
cana-4508	283	7	.	.	PUNCT
cana-4508	284	1	[	[	X
cana-4508	284	2	16	16	NUM
cana-4508	284	3	]	]	SYM
cana-4508	284	4	agilan	agilan	ADJ
cana-4508	284	5	,	,	PUNCT
cana-4508	284	6	p.	p.	NOUN
cana-4508	284	7	;	;	PUNCT
cana-4508	284	8	almazah	almazah	PROPN
cana-4508	284	9	,	,	PUNCT
cana-4508	284	10	m.a.a	m.a.a	PROPN
cana-4508	284	11	.	.	PUNCT
cana-4508	284	12	;	;	PUNCT
cana-4508	284	13	julietraja	julietraja	PROPN
cana-4508	284	14	,	,	PUNCT
cana-4508	284	15	k.	k.	PROPN
cana-4508	284	16	;	;	PUNCT
cana-4508	284	17	alsinai	alsinai	PROPN
cana-4508	284	18	,	,	PUNCT
cana-4508	284	19	a.	a.	NOUN
cana-4508	284	20	classical	classical	NOUN
cana-4508	284	21	and	and	CCONJ
cana-4508	284	22	fixed	fix	VERB
cana-4508	284	23	point	point	NOUN
cana-4508	284	24	approach	approach	NOUN
cana-4508	284	25	to	to	ADP
cana-4508	284	26	the	the	DET
cana-4508	284	27	stability	stability	NOUN
cana-4508	284	28	analysis	analysis	NOUN
cana-4508	284	29	of	of	ADP
cana-4508	284	30	a	a	DET
cana-4508	284	31	bilateral	bilateral	ADJ
cana-4508	284	32	symmetric	symmetric	ADJ
cana-4508	284	33	additive	additive	ADJ
cana-4508	284	34	functional	functional	ADJ
cana-4508	284	35	equation	equation	NOUN
cana-4508	284	36	in	in	ADP
cana-4508	284	37	fuzzy	fuzzy	ADJ
cana-4508	284	38	and	and	CCONJ
cana-4508	284	39	random	random	ADJ
cana-4508	284	40	normed	normed	ADJ
cana-4508	284	41	spaces	space	NOUN
cana-4508	284	42	.	.	PUNCT
cana-4508	285	1	mathematics	mathematic	NOUN
cana-4508	285	2	2023	2023	NUM
cana-4508	285	3	,	,	PUNCT
cana-4508	285	4	11	11	NUM
cana-4508	285	5	,	,	PUNCT
cana-4508	285	6	681	681	NUM
cana-4508	285	7	.	.	PUNCT
cana-4508	286	1	[	[	X
cana-4508	286	2	17	17	NUM
cana-4508	286	3	]	]	SYM
cana-4508	286	4	agilan	agilan	ADJ
cana-4508	286	5	,	,	PUNCT
cana-4508	286	6	p.	p.	NOUN
cana-4508	286	7	;	;	PUNCT
cana-4508	286	8	julietraja	julietraja	PROPN
cana-4508	286	9	.	.	PUNCT
cana-4508	286	10	;	;	PUNCT
cana-4508	286	11	k	k	PROPN
cana-4508	286	12	almazah	almazah	PROPN
cana-4508	286	13	,	,	PUNCT
cana-4508	286	14	m.a.a	m.a.a	PROPN
cana-4508	286	15	.	.	PUNCT
cana-4508	286	16	;	;	PUNCT
cana-4508	287	1	alsinai	alsinai	PROPN
cana-4508	287	2	,	,	PUNCT
cana-4508	287	3	a.	a.	NOUN
cana-4508	287	4	stability	stability	NOUN
cana-4508	287	5	analysis	analysis	NOUN
cana-4508	287	6	of	of	ADP
cana-4508	287	7	a	a	DET
cana-4508	287	8	new	new	ADJ
cana-4508	287	9	class	class	NOUN
cana-4508	287	10	of	of	ADP
cana-4508	287	11	series	series	NOUN
cana-4508	287	12	type	type	NOUN
cana-4508	287	13	additive	additive	ADJ
cana-4508	287	14	functional	functional	ADJ
cana-4508	287	15	equation	equation	NOUN
cana-4508	287	16	in	in	ADP
cana-4508	287	17	banach	banach	NOUN
cana-4508	287	18	spaces	space	NOUN
cana-4508	287	19	:	:	PUNCT
cana-4508	287	20	direct	direct	ADJ
cana-4508	287	21	and	and	CCONJ
cana-4508	287	22	fixed	fix	VERB
cana-4508	287	23	point	point	NOUN
cana-4508	287	24	techniques	technique	NOUN
cana-4508	287	25	mathematics	mathematic	NOUN
cana-4508	287	26	2023	2023	NUM
cana-4508	287	27	,	,	PUNCT
cana-4508	287	28	11	11	NUM
cana-4508	287	29	,	,	PUNCT
cana-4508	287	30	887	887	NUM
cana-4508	287	31	.	.	PUNCT
cana-4508	287	32	doi.org/10.3390/math11040887	doi.org/10.3390/math11040887	VERB
cana-4508	287	33	.	.	PUNCT
cana-4508	288	1	[	[	X
cana-4508	288	2	18	18	NUM
cana-4508	288	3	]	]	PUNCT
cana-4508	288	4	aloqaily	aloqaily	ADV
cana-4508	288	5	,	,	PUNCT
cana-4508	288	6	ahmad	ahmad	PROPN
cana-4508	288	7	,	,	PUNCT
cana-4508	288	8	p.	p.	PROPN
cana-4508	288	9	agilan	agilan	PROPN
cana-4508	288	10	,	,	PUNCT
cana-4508	288	11	k.	k.	PROPN
cana-4508	288	12	julietraja	julietraja	PROPN
cana-4508	288	13	,	,	PUNCT
cana-4508	288	14	s.	s.	PROPN
cana-4508	288	15	annadurai	annadurai	PROPN
cana-4508	288	16	,	,	PUNCT
cana-4508	288	17	and	and	CCONJ
cana-4508	288	18	nabil	nabil	PROPN
cana-4508	288	19	mlaiki	mlaiki	PROPN
cana-4508	288	20	.	.	PUNCT
cana-4508	289	1	a	a	DET
cana-4508	289	2	novel	novel	ADJ
cana-4508	289	3	stability	stability	NOUN
cana-4508	289	4	analysis	analysis	NOUN
cana-4508	289	5	of	of	ADP
cana-4508	289	6	functional	functional	ADJ
cana-4508	289	7	equation	equation	NOUN
cana-4508	289	8	in	in	ADP
cana-4508	289	9	neutrosophic	neutrosophic	ADJ
cana-4508	289	10	normed	norme	VERB
cana-4508	289	11	spaces	space	NOUN
cana-4508	289	12	.	.	PUNCT
cana-4508	290	1	boundary	boundary	ADJ
cana-4508	290	2	value	value	NOUN
cana-4508	290	3	problems	problem	NOUN
cana-4508	290	4	,	,	PUNCT
cana-4508	290	5	2024	2024	NUM
cana-4508	290	6	,	,	PUNCT
cana-4508	290	7	no	no	INTJ
cana-4508	290	8	.	.	NOUN
cana-4508	290	9	1	1	NUM
cana-4508	290	10	(	(	PUNCT
cana-4508	290	11	2024	2024	NUM
cana-4508	290	12	)	)	PUNCT
cana-4508	290	13	,	,	PUNCT
cana-4508	290	14	47	47	NUM
cana-4508	290	15	.	.	PUNCT
cana-4508	291	1	[	[	X
cana-4508	291	2	19	19	NUM
cana-4508	291	3	]	]	SYM
cana-4508	291	4	agilan	agilan	ADJ
cana-4508	291	5	,	,	PUNCT
cana-4508	291	6	p.	p.	PROPN
cana-4508	291	7	,	,	PUNCT
cana-4508	291	8	julietraja	julietraja	PROPN
cana-4508	291	9	,	,	PUNCT
cana-4508	291	10	k.	k.	PROPN
cana-4508	291	11	,	,	PUNCT
cana-4508	291	12	kanimozhi	kanimozhi	PROPN
cana-4508	291	13	,	,	PUNCT
cana-4508	291	14	b.	b.	PROPN
cana-4508	291	15	and	and	CCONJ
cana-4508	291	16	alsinai	alsinai	PROPN
cana-4508	291	17	,	,	PUNCT
cana-4508	291	18	a.	a.	PROPN
cana-4508	291	19	,	,	PUNCT
cana-4508	291	20	hyers	hyer	NOUN
cana-4508	291	21	stability	stability	NOUN
cana-4508	291	22	of	of	ADP
cana-4508	291	23	aqc	aqc	PROPN
cana-4508	291	24	functional	functional	ADJ
cana-4508	291	25	equation	equation	NOUN
cana-4508	291	26	.	.	PUNCT
cana-4508	292	1	dynamics	dynamic	NOUN
cana-4508	292	2	of	of	ADP
cana-4508	292	3	continuous	continuous	ADJ
cana-4508	292	4	,	,	PUNCT
cana-4508	292	5	discrete	discrete	ADJ
cana-4508	292	6	and	and	CCONJ
cana-4508	292	7	impulsive	impulsive	ADJ
cana-4508	292	8	systems	system	NOUN
cana-4508	292	9	series	series	NOUN
cana-4508	292	10	b	b	NOUN
cana-4508	292	11	:	:	PUNCT
cana-4508	292	12	applications	application	NOUN
cana-4508	292	13	and	and	CCONJ
cana-4508	292	14	algorithms	algorithm	NOUN
cana-4508	292	15	,	,	PUNCT
cana-4508	292	16	2024	2024	NUM
cana-4508	292	17	,	,	PUNCT
cana-4508	292	18	31	31	NUM
cana-4508	292	19	,	,	PUNCT
cana-4508	292	20	63	63	NUM
cana-4508	292	21	-	-	SYM
cana-4508	292	22	75	75	NUM
cana-4508	292	23	.	.	PUNCT
cana-4508	293	1	[	[	X
cana-4508	293	2	20	20	NUM
cana-4508	293	3	]	]	SYM
cana-4508	293	4	agilan	agilan	ADJ
cana-4508	293	5	,	,	PUNCT
cana-4508	293	6	p.	p.	PROPN
cana-4508	293	7	,	,	PUNCT
cana-4508	293	8	julietraja	julietraja	PROPN
cana-4508	293	9	,	,	PUNCT
cana-4508	293	10	k	k	PROPN
cana-4508	293	11	,	,	PUNCT
cana-4508	293	12	sarah	sarah	PROPN
cana-4508	293	13	aljohani	aljohani	PROPN
cana-4508	293	14	,	,	PUNCT
cana-4508	293	15	nabil	nabil	PROPN
cana-4508	293	16	mlaiki	mlaiki	PROPN
cana-4508	293	17	,	,	PUNCT
cana-4508	293	18	generalised	generalise	VERB
cana-4508	293	19	ulam	ulam	PROPN
cana-4508	293	20	-	-	PUNCT
cana-4508	293	21	hyers	hyer	NOUN
cana-4508	293	22	stability	stability	NOUN
cana-4508	293	23	analysis	analysis	NOUN
cana-4508	293	24	for	for	ADP
cana-4508	293	25	system	system	NOUN
cana-4508	293	26	of	of	ADP
cana-4508	293	27	additive	additive	ADJ
cana-4508	293	28	functional	functional	ADJ
cana-4508	293	29	equation	equation	NOUN
cana-4508	293	30	in	in	ADP
cana-4508	293	31	fuzzy	fuzzy	ADJ
cana-4508	293	32	and	and	CCONJ
cana-4508	293	33	random	random	ADJ
cana-4508	293	34	normed	normed	ADJ
cana-4508	293	35	spaces	space	NOUN
cana-4508	293	36	:	:	PUNCT
cana-4508	293	37	direct	direct	ADJ
cana-4508	293	38	and	and	CCONJ
cana-4508	293	39	fixed	fix	VERB
cana-4508	293	40	point	point	NOUN
cana-4508	293	41	approach	approach	NOUN
cana-4508	293	42	.	.	PUNCT
cana-4508	294	1	int	int	NOUN
cana-4508	294	2	.	.	PUNCT
cana-4508	295	1	j.	j.	PROPN
cana-4508	295	2	anal	anal	PROPN
cana-4508	295	3	.	.	PUNCT
cana-4508	296	1	appl	appl	PROPN
cana-4508	296	2	.	.	PROPN
cana-4508	296	3	,	,	PUNCT
cana-4508	296	4	22	22	NUM
cana-4508	296	5	2024	2024	NUM
cana-4508	296	6	,	,	PUNCT
cana-4508	296	7	201	201	NUM
cana-4508	296	8	.	.	PUNCT
cana-4508	297	1	communications	communication	NOUN
cana-4508	297	2	on	on	ADP
cana-4508	297	3	applied	apply	VERB
cana-4508	297	4	nonlinear	nonlinear	ADJ
cana-4508	297	5	analysis	analysis	NOUN
cana-4508	297	6	issn	issn	NOUN
cana-4508	297	7	:	:	PUNCT
cana-4508	297	8	1074	1074	NUM
cana-4508	297	9	-	-	PUNCT
cana-4508	297	10	133x	133x	NUM
cana-4508	297	11	vol	vol	NOUN
cana-4508	297	12	32	32	NUM
cana-4508	297	13	no	no	NOUN
cana-4508	297	14	.	.	PUNCT
cana-4508	298	1	9s	9s	NUM
cana-4508	298	2	(	(	PUNCT
cana-4508	298	3	2025	2025	NUM
cana-4508	298	4	)	)	PUNCT
cana-4508	298	5	2243	2243	NUM
cana-4508	298	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4508	299	1	[	[	X
cana-4508	299	2	21	21	NUM
cana-4508	299	3	]	]	SYM
cana-4508	299	4	agilan.p	agilan.p	PROPN
cana-4508	299	5	,	,	PUNCT
cana-4508	299	6	vijayan.v	vijayan.v	PROPN
cana-4508	299	7	,	,	PUNCT
cana-4508	299	8	sophia.m	sophia.m	NUM
cana-4508	299	9	,	,	PUNCT
cana-4508	299	10	ganapathy.g	ganapathy.g	PROPN
cana-4508	299	11	.	.	PUNCT
cana-4508	299	12	,	,	PUNCT
cana-4508	299	13	exploring	explore	VERB
cana-4508	299	14	advanced	advanced	ADJ
cana-4508	299	15	stability	stability	NOUN
cana-4508	299	16	of	of	ADP
cana-4508	299	17	higher	high	ADJ
cana-4508	299	18	-	-	PUNCT
cana-4508	299	19	order	order	NOUN
cana-4508	299	20	functional	functional	ADJ
cana-4508	299	21	equations	equation	NOUN
cana-4508	299	22	in	in	ADP
cana-4508	299	23	neutrosophic	neutrosophic	ADJ
cana-4508	299	24	normed	norme	VERB
cana-4508	299	25	spaces	space	NOUN
cana-4508	299	26	via	via	ADP
cana-4508	299	27	hyers	hyer	NOUN
cana-4508	299	28	-	-	PUNCT
cana-4508	299	29	ulam	ulam	PROPN
cana-4508	299	30	methodologies	methodology	NOUN
cana-4508	299	31	.	.	PUNCT
cana-4508	300	1	communications	communication	NOUN
cana-4508	300	2	on	on	ADP
cana-4508	300	3	applied	apply	VERB
cana-4508	300	4	nonlinear	nonlinear	ADJ
cana-4508	300	5	analysis	analysis	NOUN
cana-4508	300	6	,	,	PUNCT
cana-4508	300	7	2025	2025	NUM
cana-4508	300	8	,	,	PUNCT
cana-4508	300	9	vol	vol	NOUN
cana-4508	300	10	32	32	NUM
cana-4508	300	11	no	no	NOUN
cana-4508	300	12	.	.	PUNCT
cana-4508	301	1	7s	7	NOUN
cana-4508	301	2	(	(	PUNCT
cana-4508	301	3	2025	2025	NUM
cana-4508	301	4	)	)	PUNCT
cana-4508	301	5	,	,	PUNCT
cana-4508	301	6	806	806	NUM
cana-4508	301	7	-	-	SYM
cana-4508	301	8	822	822	NUM
cana-4508	301	9	.	.	PUNCT
cana-4508	302	1	[	[	X
cana-4508	302	2	22]agilan.p	22]agilan.p	NUM
cana-4508	302	3	,	,	PUNCT
cana-4508	302	4	vijayan.v	vijayan.v	PROPN
cana-4508	302	5	,	,	PUNCT
cana-4508	302	6	sophia	sophia	PROPN
cana-4508	302	7	.	.	PUNCT
cana-4508	303	1	m	m	PROPN
cana-4508	303	2	,	,	PUNCT
cana-4508	303	3	banu	banu	PROPN
cana-4508	303	4	priya.v	priya.v	PROPN
cana-4508	303	5	.	.	PUNCT
cana-4508	303	6	,	,	PUNCT
cana-4508	303	7	investigating	investigate	VERB
cana-4508	303	8	a	a	DET
cana-4508	303	9	novel	novel	ADJ
cana-4508	303	10	stability	stability	NOUN
cana-4508	303	11	results	result	NOUN
cana-4508	303	12	of	of	ADP
cana-4508	303	13	generalized	generalized	ADJ
cana-4508	303	14	alternate	alternate	ADJ
cana-4508	303	15	cubic	cubic	ADJ
cana-4508	303	16	functional	functional	ADJ
cana-4508	303	17	equations	equation	NOUN
cana-4508	303	18	:	:	PUNCT
cana-4508	303	19	classical	classical	ADJ
cana-4508	303	20	method	method	NOUN
cana-4508	303	21	for	for	ADP
cana-4508	303	22	banach	banach	NOUN
cana-4508	303	23	spaces	space	NOUN
cana-4508	303	24	and	and	CCONJ
cana-4508	303	25	direct	direct	ADJ
cana-4508	303	26	-fixed	-fixe	VERB
cana-4508	303	27	point	point	NOUN
cana-4508	303	28	approaches	approach	NOUN
cana-4508	303	29	for	for	ADP
cana-4508	303	30	fuzzy	fuzzy	ADJ
cana-4508	303	31	normed	normed	ADJ
cana-4508	303	32	spaces	space	NOUN
cana-4508	303	33	,	,	PUNCT
cana-4508	303	34	communications	communication	NOUN
cana-4508	303	35	on	on	ADP
cana-4508	303	36	applied	apply	VERB
cana-4508	303	37	nonlinear	nonlinear	ADJ
cana-4508	303	38	analysis	analysis	NOUN
cana-4508	303	39	,	,	PUNCT
cana-4508	303	40	2025	2025	NUM
cana-4508	303	41	,	,	PUNCT
cana-4508	303	42	vol	vol	NOUN
cana-4508	303	43	32	32	NUM
cana-4508	304	1	no	no	NOUN
cana-4508	304	2	.	.	PUNCT
cana-4508	305	1	9s	9s	NUM
cana-4508	305	2	(	(	PUNCT
cana-4508	305	3	2025),475	2025),475	NOUN
cana-4508	305	4	-	-	NUM
cana-4508	305	5	492	492	NUM
cana-4508	305	6	.	.	PUNCT