id	sid	tid	token	lemma	pos
cana-1549	1	1	communications	communication	NOUN
cana-1549	1	2	on	on	ADP
cana-1549	1	3	applied	apply	VERB
cana-1549	1	4	nonlinear	nonlinear	ADJ
cana-1549	1	5	analysis	analysis	NOUN
cana-1549	1	6	issn	issn	NOUN
cana-1549	1	7	:	:	PUNCT
cana-1549	1	8	1074	1074	NUM
cana-1549	1	9	-	-	PUNCT
cana-1549	1	10	133x	133x	NUM
cana-1549	1	11	vol	vol	NOUN
cana-1549	1	12	31	31	NUM
cana-1549	1	13	no	no	NOUN
cana-1549	1	14	.	.	PUNCT
cana-1549	2	1	8s	8s	PROPN
cana-1549	2	2	(	(	PUNCT
cana-1549	2	3	2024	2024	NUM
cana-1549	2	4	)	)	PUNCT
cana-1549	2	5	565	565	NUM
cana-1549	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	2	7	on	on	ADP
cana-1549	2	8	discrete	discrete	ADJ
cana-1549	2	9	harmonic	harmonic	ADJ
cana-1549	2	10	distributions	distribution	NOUN
cana-1549	2	11	related	relate	VERB
cana-1549	2	12	with	with	ADP
cana-1549	2	13	harmonic	harmonic	ADJ
cana-1549	2	14	mean	mean	ADJ
cana-1549	2	15	random	random	ADJ
cana-1549	2	16	variable	variable	ADJ
cana-1549	2	17	prasanta	prasanta	NOUN
cana-1549	2	18	kumar	kumar	PROPN
cana-1549	2	19	das	das	PROPN
cana-1549	2	20	school	school	NOUN
cana-1549	2	21	of	of	ADP
cana-1549	2	22	applied	apply	VERB
cana-1549	2	23	sciences	science	NOUN
cana-1549	2	24	,	,	PUNCT
cana-1549	2	25	kiit	kiit	PROPN
cana-1549	2	26	deemed	deem	VERB
cana-1549	2	27	to	to	PART
cana-1549	2	28	be	be	AUX
cana-1549	2	29	university	university	NOUN
cana-1549	2	30	,	,	PUNCT
cana-1549	2	31	bhubaneswar-751024	bhubaneswar-751024	PROPN
cana-1549	2	32	(	(	PUNCT
cana-1549	2	33	india	india	PROPN
cana-1549	2	34	)	)	PUNCT
cana-1549	2	35	pkdasfma@kiit.ac.in	pkdasfma@kiit.ac.in	NOUN
cana-1549	2	36	/	/	SYM
cana-1549	2	37	dasprasantkumar@yahoo.co.in	dasprasantkumar@yahoo.co.in	NOUN
cana-1549	2	38	article	article	NOUN
cana-1549	2	39	history	history	NOUN
cana-1549	2	40	:	:	PUNCT
cana-1549	2	41	received	receive	VERB
cana-1549	2	42	:	:	PUNCT
cana-1549	2	43	20	20	NUM
cana-1549	2	44	-	-	SYM
cana-1549	2	45	05	05	NUM
cana-1549	2	46	-	-	PUNCT
cana-1549	2	47	2024	2024	NUM
cana-1549	2	48	revised	revise	VERB
cana-1549	2	49	:	:	PUNCT
cana-1549	2	50	03	03	NUM
cana-1549	2	51	-	-	PUNCT
cana-1549	2	52	07	07	NUM
cana-1549	2	53	-	-	PUNCT
cana-1549	2	54	2024	2024	NUM
cana-1549	2	55	accepted	accept	VERB
cana-1549	2	56	:	:	PUNCT
cana-1549	2	57	21	21	NUM
cana-1549	2	58	-	-	SYM
cana-1549	2	59	07	07	NUM
cana-1549	2	60	-	-	PUNCT
cana-1549	2	61	2024	2024	NUM
cana-1549	2	62	abstract	abstract	NOUN
cana-1549	2	63	:	:	PUNCT
cana-1549	2	64	the	the	DET
cana-1549	2	65	main	main	ADJ
cana-1549	2	66	purpose	purpose	NOUN
cana-1549	2	67	of	of	ADP
cana-1549	2	68	this	this	DET
cana-1549	2	69	paper	paper	NOUN
cana-1549	2	70	is	be	AUX
cana-1549	2	71	to	to	PART
cana-1549	2	72	introduce	introduce	VERB
cana-1549	2	73	the	the	DET
cana-1549	2	74	concept	concept	NOUN
cana-1549	2	75	of	of	ADP
cana-1549	2	76	harmonically	harmonically	ADV
cana-1549	2	77	distributed	distribute	VERB
cana-1549	2	78	discrete	discrete	ADJ
cana-1549	2	79	random	random	ADJ
cana-1549	2	80	variable	variable	NOUN
cana-1549	2	81	.	.	PUNCT
cana-1549	3	1	we	we	PRON
cana-1549	3	2	define	define	VERB
cana-1549	3	3	probability	probability	NOUN
cana-1549	3	4	mass	mass	NOUN
cana-1549	3	5	function	function	NOUN
cana-1549	3	6	such	such	ADJ
cana-1549	3	7	as	as	ADP
cana-1549	3	8	harmonic	harmonic	ADJ
cana-1549	3	9	mass	mass	NOUN
cana-1549	3	10	function	function	NOUN
cana-1549	3	11	,	,	PUNCT
cana-1549	3	12	exponential	exponential	ADJ
cana-1549	3	13	harmonic	harmonic	ADJ
cana-1549	3	14	mass	mass	NOUN
cana-1549	3	15	function	function	NOUN
cana-1549	3	16	,	,	PUNCT
cana-1549	3	17	natural	natural	ADJ
cana-1549	3	18	logarithmic	logarithmic	ADJ
cana-1549	3	19	harmonic	harmonic	ADJ
cana-1549	3	20	mass	mass	NOUN
cana-1549	3	21	function	function	NOUN
cana-1549	3	22	and	and	CCONJ
cana-1549	3	23	their	their	PRON
cana-1549	3	24	associated	associate	VERB
cana-1549	3	25	cumulative	cumulative	ADJ
cana-1549	3	26	distribution	distribution	NOUN
cana-1549	3	27	functions	function	NOUN
cana-1549	3	28	.	.	PUNCT
cana-1549	4	1	existence	existence	NOUN
cana-1549	4	2	the	the	DET
cana-1549	4	3	distribution	distribution	NOUN
cana-1549	4	4	is	be	AUX
cana-1549	4	5	shown	show	VERB
cana-1549	4	6	with	with	ADP
cana-1549	4	7	some	some	DET
cana-1549	4	8	examples	example	NOUN
cana-1549	4	9	.	.	PUNCT
cana-1549	5	1	finally	finally	ADV
cana-1549	5	2	harmonic	harmonic	VERB
cana-1549	5	3	mass	mass	NOUN
cana-1549	5	4	function	function	NOUN
cana-1549	5	5	is	be	AUX
cana-1549	5	6	used	use	VERB
cana-1549	5	7	to	to	PART
cana-1549	5	8	solve	solve	VERB
cana-1549	5	9	a	a	DET
cana-1549	5	10	run	run	ADJ
cana-1549	5	11	-	-	PUNCT
cana-1549	5	12	time	time	NOUN
cana-1549	5	13	problem	problem	NOUN
cana-1549	5	14	and	and	CCONJ
cana-1549	5	15	a	a	DET
cana-1549	5	16	capacitance	capacitance	NOUN
cana-1549	5	17	problem	problem	NOUN
cana-1549	5	18	as	as	ADP
cana-1549	5	19	an	an	DET
cana-1549	5	20	application	application	NOUN
cana-1549	5	21	of	of	ADP
cana-1549	5	22	the	the	DET
cana-1549	5	23	distribution	distribution	NOUN
cana-1549	5	24	in	in	ADP
cana-1549	5	25	the	the	DET
cana-1549	5	26	field	field	NOUN
cana-1549	5	27	of	of	ADP
cana-1549	5	28	electrical	electrical	ADJ
cana-1549	5	29	engineering	engineering	NOUN
cana-1549	5	30	.	.	PUNCT
cana-1549	6	1	keywords	keyword	NOUN
cana-1549	6	2	:	:	PUNCT
cana-1549	6	3	harmonic	harmonic	ADJ
cana-1549	6	4	mean	mean	VERB
cana-1549	6	5	random	random	ADJ
cana-1549	6	6	variable	variable	NOUN
cana-1549	6	7	,	,	PUNCT
cana-1549	6	8	harmonic	harmonic	ADJ
cana-1549	6	9	mass	mass	NOUN
cana-1549	6	10	function	function	NOUN
cana-1549	6	11	,	,	PUNCT
cana-1549	6	12	exponential	exponential	ADJ
cana-1549	6	13	harmonic	harmonic	ADJ
cana-1549	6	14	mass	mass	NOUN
cana-1549	6	15	function	function	NOUN
cana-1549	6	16	,	,	PUNCT
cana-1549	6	17	natural	natural	ADJ
cana-1549	6	18	logarithmic	logarithmic	ADJ
cana-1549	6	19	harmonic	harmonic	ADJ
cana-1549	6	20	mass	mass	NOUN
cana-1549	6	21	function	function	NOUN
cana-1549	6	22	.	.	PUNCT
cana-1549	7	1	ams	am	NOUN
cana-1549	7	2	2010	2010	NUM
cana-1549	7	3	subject	subject	ADJ
cana-1549	7	4	classification	classification	NOUN
cana-1549	7	5	:	:	PUNCT
cana-1549	7	6	62exx	62exx	ADP
cana-1549	8	1	1	1	NUM
cana-1549	8	2	.	.	PUNCT
cana-1549	9	1	introduction	introduction	NOUN
cana-1549	9	2	and	and	CCONJ
cana-1549	9	3	preliminary	preliminary	ADJ
cana-1549	9	4	in	in	ADP
cana-1549	9	5	the	the	DET
cana-1549	9	6	theory	theory	NOUN
cana-1549	9	7	of	of	ADP
cana-1549	9	8	probability	probability	NOUN
cana-1549	9	9	and	and	CCONJ
cana-1549	9	10	statistics	statistic	NOUN
cana-1549	9	11	,	,	PUNCT
cana-1549	9	12	the	the	DET
cana-1549	9	13	continuous	continuous	ADJ
cana-1549	9	14	harmonic	harmonic	ADJ
cana-1549	9	15	distribution	distribution	NOUN
cana-1549	9	16	or	or	CCONJ
cana-1549	9	17	harmonic	harmonic	ADJ
cana-1549	9	18	law	law	NOUN
cana-1549	9	19	was	be	AUX
cana-1549	9	20	studied	study	VERB
cana-1549	9	21	by	by	ADP
cana-1549	9	22	etienne	etienne	PROPN
cana-1549	9	23	halphen	halphen	ADV
cana-1549	9	24	(	(	PUNCT
cana-1549	9	25	1941	1941	NUM
cana-1549	9	26	)	)	PUNCT
cana-1549	9	27	as	as	ADP
cana-1549	9	28	a	a	DET
cana-1549	9	29	special	special	ADJ
cana-1549	9	30	case	case	NOUN
cana-1549	9	31	of	of	ADP
cana-1549	9	32	generalized	generalized	ADJ
cana-1549	9	33	inverse	inverse	NOUN
cana-1549	9	34	gaussian	gaussian	ADJ
cana-1549	9	35	distribution	distribution	NOUN
cana-1549	9	36	family	family	NOUN
cana-1549	9	37	with	with	ADP
cana-1549	9	38	𝛾	𝛾	PROPN
cana-1549	9	39	=	=	SYM
cana-1549	9	40	0	0	NUM
cana-1549	9	41	.	.	PUNCT
cana-1549	10	1	the	the	DET
cana-1549	10	2	geometry	geometry	NOUN
cana-1549	10	3	of	of	ADP
cana-1549	10	4	statistical	statistical	ADJ
cana-1549	10	5	notions	notion	NOUN
cana-1549	10	6	or	or	CCONJ
cana-1549	10	7	the	the	DET
cana-1549	10	8	connections	connection	NOUN
cana-1549	10	9	between	between	ADP
cana-1549	10	10	geometry	geometry	NOUN
cana-1549	10	11	and	and	CCONJ
cana-1549	10	12	statistics	statistic	NOUN
cana-1549	10	13	has	have	AUX
cana-1549	10	14	existed	exist	VERB
cana-1549	10	15	from	from	ADP
cana-1549	10	16	the	the	DET
cana-1549	10	17	beginning	beginning	NOUN
cana-1549	10	18	of	of	ADP
cana-1549	10	19	both	both	DET
cana-1549	10	20	disciplines	discipline	NOUN
cana-1549	10	21	(	(	PUNCT
cana-1549	10	22	adler	adler	PROPN
cana-1549	10	23	c.	c.	PROPN
cana-1549	10	24	f.	f.	PROPN
cana-1549	10	25	1958	1958	NUM
cana-1549	11	1	[	[	X
cana-1549	11	2	1	1	NUM
cana-1549	11	3	]	]	PUNCT
cana-1549	11	4	,	,	PUNCT
cana-1549	11	5	and	and	CCONJ
cana-1549	11	6	fisher	fisher	PROPN
cana-1549	11	7	,	,	PUNCT
cana-1549	11	8	j.	j.	PROPN
cana-1549	11	9	b	b	PROPN
cana-1549	11	10	,	,	PUNCT
cana-1549	11	11	1978	1978	NUM
cana-1549	11	12	[	[	X
cana-1549	11	13	4	4	NUM
cana-1549	11	14	]	]	PUNCT
cana-1549	11	15	)	)	PUNCT
cana-1549	11	16	.	.	PUNCT
cana-1549	12	1	the	the	DET
cana-1549	12	2	connection	connection	NOUN
cana-1549	12	3	between	between	ADP
cana-1549	12	4	arithmetic	arithmetic	ADJ
cana-1549	12	5	,	,	PUNCT
cana-1549	12	6	geometric	geometric	ADJ
cana-1549	12	7	,	,	PUNCT
cana-1549	12	8	and	and	CCONJ
cana-1549	12	9	harmonic	harmonic	ADJ
cana-1549	12	10	mean	mean	NOUN
cana-1549	12	11	can	can	AUX
cana-1549	12	12	be	be	AUX
cana-1549	12	13	studied	study	VERB
cana-1549	12	14	by	by	ADP
cana-1549	12	15	the	the	DET
cana-1549	12	16	same	same	ADJ
cana-1549	12	17	approach	approach	NOUN
cana-1549	12	18	of	of	ADP
cana-1549	12	19	geometry	geometry	NOUN
cana-1549	12	20	of	of	ADP
cana-1549	12	21	statistics	statistic	NOUN
cana-1549	12	22	.	.	PUNCT
cana-1549	13	1	you	you	PRON
cana-1549	13	2	may	may	AUX
cana-1549	13	3	observe	observe	VERB
cana-1549	13	4	that	that	SCONJ
cana-1549	13	5	some	some	DET
cana-1549	13	6	other	other	ADJ
cana-1549	13	7	investigations	investigation	NOUN
cana-1549	13	8	are	be	AUX
cana-1549	13	9	based	base	VERB
cana-1549	13	10	on	on	ADP
cana-1549	13	11	the	the	DET
cana-1549	13	12	statistical	statistical	ADJ
cana-1549	13	13	properties	property	NOUN
cana-1549	13	14	of	of	ADP
cana-1549	13	15	geometric	geometric	ADJ
cana-1549	13	16	notions	notion	NOUN
cana-1549	13	17	(	(	PUNCT
cana-1549	14	1	ahangar	ahangar	PROPN
cana-1549	14	2	r.	r.	PROPN
cana-1549	14	3	r.	r.	PROPN
cana-1549	14	4	2010	2010	NUM
cana-1549	15	1	[	[	X
cana-1549	15	2	2	2	NUM
cana-1549	15	3	]	]	PUNCT
cana-1549	15	4	,	,	PUNCT
cana-1549	15	5	hilbert	hilbert	PROPN
cana-1549	15	6	,	,	PUNCT
cana-1549	15	7	d.	d.	PROPN
cana-1549	15	8	1902	1902	NUM
cana-1549	15	9	[	[	X
cana-1549	15	10	5	5	NUM
cana-1549	15	11	]	]	PUNCT
cana-1549	15	12	,	,	PUNCT
cana-1549	15	13	pearson	pearson	PROPN
cana-1549	15	14	,	,	PUNCT
cana-1549	15	15	r.	r.	PROPN
cana-1549	15	16	2011	2011	NUM
cana-1549	15	17	[	[	X
cana-1549	15	18	10	10	NUM
cana-1549	15	19	]	]	PUNCT
cana-1549	15	20	,	,	PUNCT
cana-1549	15	21	saville	saville	PROPN
cana-1549	15	22	,	,	PUNCT
cana-1549	15	23	j.	j.	PROPN
cana-1549	15	24	d.	d.	PROPN
cana-1549	15	25	,	,	PUNCT
cana-1549	15	26	and	and	CCONJ
cana-1549	15	27	wood	wood	PROPN
cana-1549	15	28	r.	r.	PROPN
cana-1549	15	29	g.	g.	PROPN
cana-1549	15	30	,	,	PUNCT
cana-1549	15	31	(	(	PUNCT
cana-1549	15	32	1977	1977	NUM
cana-1549	15	33	)	)	PUNCT
cana-1549	16	1	[	[	X
cana-1549	16	2	11	11	NUM
cana-1549	16	3	]	]	PUNCT
cana-1549	16	4	.	.	PUNCT
cana-1549	17	1	arithmetic	arithmetic	ADJ
cana-1549	17	2	mean	mean	NOUN
cana-1549	17	3	as	as	ADP
cana-1549	17	4	a	a	DET
cana-1549	17	5	great	great	ADJ
cana-1549	17	6	mode	mode	NOUN
cana-1549	17	7	of	of	ADP
cana-1549	17	8	investigation	investigation	NOUN
cana-1549	17	9	in	in	ADP
cana-1549	17	10	statistics	statistic	NOUN
cana-1549	17	11	is	be	AUX
cana-1549	17	12	used	use	VERB
cana-1549	17	13	in	in	ADP
cana-1549	17	14	many	many	ADJ
cana-1549	17	15	natural	natural	ADJ
cana-1549	17	16	phenomena	phenomenon	NOUN
cana-1549	17	17	,	,	PUNCT
cana-1549	17	18	but	but	CCONJ
cana-1549	17	19	many	many	ADJ
cana-1549	17	20	researches	research	NOUN
cana-1549	17	21	prefer	prefer	VERB
cana-1549	17	22	to	to	PART
cana-1549	17	23	use	use	VERB
cana-1549	17	24	either	either	CCONJ
cana-1549	17	25	geometric	geometric	ADJ
cana-1549	17	26	or	or	CCONJ
cana-1549	17	27	harmonic	harmonic	ADJ
cana-1549	17	28	mean	mean	NOUN
cana-1549	17	29	in	in	ADP
cana-1549	17	30	their	their	PRON
cana-1549	17	31	investigations	investigation	NOUN
cana-1549	17	32	,	,	PUNCT
cana-1549	17	33	(	(	PUNCT
cana-1549	17	34	maccluer	maccluer	NOUN
cana-1549	17	35	,	,	PUNCT
cana-1549	17	36	c.	c.	PROPN
cana-1549	17	37	r.	r.	PROPN
cana-1549	17	38	2000	2000	NUM
cana-1549	18	1	[	[	X
cana-1549	18	2	7	7	NUM
cana-1549	18	3	]	]	NUM
cana-1549	18	4	)	)	PUNCT
cana-1549	18	5	.	.	PUNCT
cana-1549	19	1	the	the	DET
cana-1549	19	2	arithmetic	arithmetic	ADJ
cana-1549	19	3	mean	mean	NOUN
cana-1549	19	4	and	and	CCONJ
cana-1549	19	5	the	the	DET
cana-1549	19	6	median	median	NOUN
cana-1549	19	7	may	may	AUX
cana-1549	19	8	be	be	AUX
cana-1549	19	9	the	the	DET
cana-1549	19	10	most	most	ADV
cana-1549	19	11	popular	popular	ADJ
cana-1549	19	12	and	and	CCONJ
cana-1549	19	13	convenient	convenient	ADJ
cana-1549	19	14	measures	measure	NOUN
cana-1549	19	15	that	that	PRON
cana-1549	19	16	financial	financial	ADJ
cana-1549	19	17	analysts	analyst	NOUN
cana-1549	19	18	use	use	VERB
cana-1549	19	19	for	for	ADP
cana-1549	19	20	their	their	PRON
cana-1549	19	21	valuations	valuation	NOUN
cana-1549	19	22	.	.	PUNCT
cana-1549	20	1	it	it	PRON
cana-1549	20	2	is	be	AUX
cana-1549	20	3	believed	believe	VERB
cana-1549	20	4	by	by	ADP
cana-1549	20	5	many	many	ADJ
cana-1549	20	6	financial	financial	ADJ
cana-1549	20	7	organizations	organization	NOUN
cana-1549	20	8	and	and	CCONJ
cana-1549	20	9	bankers	banker	NOUN
cana-1549	20	10	that	that	PRON
cana-1549	20	11	the	the	DET
cana-1549	20	12	harmonic	harmonic	NOUN
cana-1549	20	13	mean	mean	VERB
cana-1549	20	14	provides	provide	VERB
cana-1549	20	15	better	well	ADJ
cana-1549	20	16	information	information	NOUN
cana-1549	20	17	for	for	ADP
cana-1549	20	18	reasonable	reasonable	ADJ
cana-1549	20	19	measure	measure	NOUN
cana-1549	20	20	in	in	ADP
cana-1549	20	21	investment	investment	NOUN
cana-1549	20	22	strategy	strategy	NOUN
cana-1549	20	23	(	(	PUNCT
cana-1549	20	24	mathews	mathews	NOUN
cana-1549	20	25	and	and	CCONJ
cana-1549	20	26	gilbert2006	gilbert2006	PROPN
cana-1549	20	27	,	,	PUNCT
cana-1549	20	28	meyer	meyer	PROPN
cana-1549	20	29	d.	d.	PROPN
cana-1549	20	30	,	,	PUNCT
cana-1549	20	31	1970	1970	NUM
cana-1549	21	1	[	[	X
cana-1549	21	2	8	8	NUM
cana-1549	21	3	]	]	NUM
cana-1549	21	4	)	)	PUNCT
cana-1549	21	5	.	.	PUNCT
cana-1549	22	1	in	in	ADP
cana-1549	22	2	slowly	slowly	ADV
cana-1549	22	3	-	-	PUNCT
cana-1549	22	4	decaying	decay	VERB
cana-1549	22	5	distributions	distribution	NOUN
cana-1549	22	6	,	,	PUNCT
cana-1549	22	7	the	the	DET
cana-1549	22	8	harmonic	harmonic	ADJ
cana-1549	22	9	mean	mean	NOUN
cana-1549	22	10	often	often	ADV
cana-1549	22	11	turns	turn	VERB
cana-1549	22	12	out	out	ADP
cana-1549	22	13	to	to	PART
cana-1549	22	14	be	be	AUX
cana-1549	22	15	a	a	DET
cana-1549	22	16	much	much	ADV
cana-1549	22	17	better	well	ADJ
cana-1549	22	18	characterization	characterization	NOUN
cana-1549	22	19	than	than	ADP
cana-1549	22	20	the	the	DET
cana-1549	22	21	arithmetic	arithmetic	ADJ
cana-1549	22	22	mean	mean	NOUN
cana-1549	22	23	,	,	PUNCT
cana-1549	22	24	which	which	PRON
cana-1549	22	25	is	be	AUX
cana-1549	22	26	a	a	DET
cana-1549	22	27	reciprocal	reciprocal	ADJ
cana-1549	22	28	transformation	transformation	NOUN
cana-1549	22	29	generally	generally	ADV
cana-1549	22	30	not	not	PART
cana-1549	22	31	even	even	ADV
cana-1549	22	32	well	well	ADV
cana-1549	22	33	-	-	PUNCT
cana-1549	22	34	defined	define	VERB
cana-1549	22	35	theoretically	theoretically	ADV
cana-1549	22	36	for	for	ADP
cana-1549	22	37	these	these	DET
cana-1549	22	38	distributions	distribution	NOUN
cana-1549	22	39	(	(	PUNCT
cana-1549	22	40	pearson	pearson	PROPN
cana-1549	22	41	,	,	PUNCT
cana-1549	22	42	r.	r.	PROPN
cana-1549	22	43	2011	2011	NUM
cana-1549	23	1	[	[	X
cana-1549	23	2	10	10	NUM
cana-1549	23	3	]	]	NUM
cana-1549	23	4	)	)	PUNCT
cana-1549	23	5	.	.	PUNCT
cana-1549	24	1	much	much	ADJ
cana-1549	24	2	research	research	NOUN
cana-1549	24	3	has	have	AUX
cana-1549	24	4	been	be	AUX
cana-1549	24	5	done	do	VERB
cana-1549	24	6	and	and	CCONJ
cana-1549	24	7	computational	computational	ADJ
cana-1549	24	8	tools	tool	NOUN
cana-1549	24	9	designed	design	VERB
cana-1549	24	10	these	these	DET
cana-1549	24	11	days	day	NOUN
cana-1549	24	12	that	that	PRON
cana-1549	24	13	are	be	AUX
cana-1549	24	14	equipped	equip	VERB
cana-1549	24	15	to	to	PART
cana-1549	24	16	change	change	VERB
cana-1549	24	17	the	the	DET
cana-1549	24	18	mode	mode	NOUN
cana-1549	24	19	of	of	ADP
cana-1549	24	20	computation	computation	NOUN
cana-1549	24	21	in	in	ADP
cana-1549	24	22	either	either	CCONJ
cana-1549	24	23	arithmetic	arithmetic	ADJ
cana-1549	24	24	,	,	PUNCT
cana-1549	24	25	geometric	geometric	ADJ
cana-1549	24	26	,	,	PUNCT
cana-1549	24	27	or	or	CCONJ
cana-1549	24	28	harmonic	harmonic	ADJ
cana-1549	24	29	sense	sense	NOUN
cana-1549	24	30	.	.	PUNCT
cana-1549	25	1	the	the	DET
cana-1549	25	2	continuous	continuous	ADJ
cana-1549	25	3	harmonic	harmonic	ADJ
cana-1549	25	4	density	density	NOUN
cana-1549	25	5	function	function	NOUN
cana-1549	25	6	,	,	PUNCT
cana-1549	25	7	transformation	transformation	NOUN
cana-1549	25	8	(	(	PUNCT
cana-1549	25	9	horizontal	horizontal	ADJ
cana-1549	25	10	shift	shift	NOUN
cana-1549	25	11	)	)	PUNCT
cana-1549	25	12	of	of	ADP
cana-1549	25	13	harmonic	harmonic	ADJ
cana-1549	25	14	density	density	NOUN
cana-1549	25	15	function	function	NOUN
cana-1549	25	16	,	,	PUNCT
cana-1549	25	17	harmonic	harmonic	ADJ
cana-1549	25	18	density	density	NOUN
cana-1549	25	19	with	with	ADP
cana-1549	25	20	stretch	stretch	NOUN
cana-1549	25	21	or	or	CCONJ
cana-1549	25	22	contraction	contraction	NOUN
cana-1549	25	23	,	,	PUNCT
cana-1549	25	24	and	and	CCONJ
cana-1549	25	25	general	general	ADJ
cana-1549	25	26	harmonic	harmonic	ADJ
cana-1549	25	27	function	function	NOUN
cana-1549	25	28	are	be	AUX
cana-1549	25	29	studied	study	VERB
cana-1549	25	30	by	by	ADP
cana-1549	25	31	(	(	PUNCT
cana-1549	25	32	ahangar	ahangar	PROPN
cana-1549	25	33	r.	r.	PROPN
cana-1549	25	34	2013	2013	NUM
cana-1549	25	35	communications	communication	NOUN
cana-1549	25	36	on	on	ADP
cana-1549	25	37	applied	apply	VERB
cana-1549	25	38	nonlinear	nonlinear	ADJ
cana-1549	25	39	analysis	analysis	NOUN
cana-1549	25	40	issn	issn	NOUN
cana-1549	25	41	:	:	PUNCT
cana-1549	25	42	1074	1074	NUM
cana-1549	25	43	-	-	PUNCT
cana-1549	25	44	133x	133x	NUM
cana-1549	25	45	vol	vol	NOUN
cana-1549	25	46	31	31	NUM
cana-1549	25	47	no	no	NOUN
cana-1549	25	48	.	.	PUNCT
cana-1549	26	1	8s	8s	PROPN
cana-1549	26	2	(	(	PUNCT
cana-1549	26	3	2024	2024	NUM
cana-1549	26	4	)	)	PUNCT
cana-1549	26	5	566	566	NUM
cana-1549	26	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	27	1	[	[	X
cana-1549	27	2	3	3	NUM
cana-1549	27	3	]	]	PUNCT
cana-1549	27	4	.	.	PUNCT
cana-1549	28	1	later	later	ADV
cana-1549	28	2	he	he	PRON
cana-1549	28	3	developed	develop	VERB
cana-1549	28	4	the	the	DET
cana-1549	28	5	algorithm	algorithm	NOUN
cana-1549	28	6	of	of	ADP
cana-1549	28	7	continuous	continuous	ADJ
cana-1549	28	8	harmonic	harmonic	ADJ
cana-1549	28	9	modelling	modelling	NOUN
cana-1549	28	10	and	and	CCONJ
cana-1549	28	11	continuous	continuous	ADJ
cana-1549	28	12	harmonic	harmonic	ADJ
cana-1549	28	13	regression	regression	NOUN
cana-1549	28	14	.	.	PUNCT
cana-1549	29	1	the	the	DET
cana-1549	29	2	total	total	ADJ
cana-1549	29	3	capacitor	capacitor	NOUN
cana-1549	29	4	(	(	PUNCT
cana-1549	29	5	𝐶𝑇	𝐶𝑇	PROPN
cana-1549	29	6	)	)	PUNCT
cana-1549	29	7	in	in	ADP
cana-1549	29	8	an	an	DET
cana-1549	29	9	series	series	NOUN
cana-1549	29	10	circuit	circuit	NOUN
cana-1549	29	11	of	of	ADP
cana-1549	29	12	𝑛	𝑛	DET
cana-1549	29	13	capacitance	capacitance	NOUN
cana-1549	29	14	𝐶𝑖	𝐶𝑖	PROPN
cana-1549	29	15	,	,	PUNCT
cana-1549	29	16	𝑖	𝑖	PROPN
cana-1549	29	17	=	=	SYM
cana-1549	29	18	1,2,⋯	1,2,⋯	PROPN
cana-1549	29	19	,	,	PUNCT
cana-1549	29	20	𝑛	𝑛	PRON
cana-1549	29	21	has	have	VERB
cana-1549	29	22	a	a	DET
cana-1549	29	23	harmonic	harmonic	ADJ
cana-1549	29	24	relation	relation	NOUN
cana-1549	29	25	and	and	CCONJ
cana-1549	29	26	the	the	DET
cana-1549	29	27	total	total	ADJ
cana-1549	29	28	resistance	resistance	NOUN
cana-1549	29	29	(	(	PUNCT
cana-1549	29	30	𝑅𝑇	𝑅𝑇	PROPN
cana-1549	29	31	)	)	PUNCT
cana-1549	29	32	in	in	ADP
cana-1549	29	33	an	an	DET
cana-1549	29	34	parallel	parallel	ADJ
cana-1549	29	35	circuit	circuit	NOUN
cana-1549	29	36	of	of	ADP
cana-1549	29	37	𝑛	𝑛	DET
cana-1549	29	38	resistance	resistance	NOUN
cana-1549	29	39	𝑅𝑖	𝑅𝑖	PROPN
cana-1549	29	40	,	,	PUNCT
cana-1549	29	41	𝑖	𝑖	PROPN
cana-1549	29	42	=	=	SYM
cana-1549	29	43	1,2,⋯	1,2,⋯	PROPN
cana-1549	29	44	,	,	PUNCT
cana-1549	29	45	𝑛	𝑛	PRON
cana-1549	29	46	has	have	VERB
cana-1549	29	47	a	a	DET
cana-1549	29	48	harmonic	harmonic	ADJ
cana-1549	29	49	relation	relation	NOUN
cana-1549	29	50	defined	define	VERB
cana-1549	29	51	by	by	ADP
cana-1549	29	52	circuits	circuit	NOUN
cana-1549	29	53	𝑅𝑇	𝑅𝑇	PROPN
cana-1549	29	54	𝐶𝑇	𝐶𝑇	PROPN
cana-1549	29	55	series	series	NOUN
cana-1549	29	56	𝑅1	𝑅1	NOUN
cana-1549	29	57	+	+	ADP
cana-1549	29	58	𝑅2	𝑅2	NOUN
cana-1549	29	59	+	+	ADJ
cana-1549	29	60	⋯+	⋯+	NOUN
cana-1549	29	61	𝑅𝑛	𝑅𝑛	NOUN
cana-1549	29	62	(	(	PUNCT
cana-1549	29	63	1	1	NUM
cana-1549	29	64	𝐶1	𝐶1	NOUN
cana-1549	30	1	+	+	NUM
cana-1549	30	2	1	1	NUM
cana-1549	30	3	𝐶2	𝐶2	NOUN
cana-1549	30	4	+	+	NOUN
cana-1549	30	5	⋯+	⋯+	NOUN
cana-1549	30	6	1	1	NUM
cana-1549	30	7	𝐶𝑛	𝐶𝑛	NOUN
cana-1549	30	8	)	)	PUNCT
cana-1549	30	9	−1	−1	NOUN
cana-1549	30	10	parallel	parallel	NOUN
cana-1549	30	11	(	(	PUNCT
cana-1549	30	12	1	1	NUM
cana-1549	30	13	𝑅1	𝑅1	NOUN
cana-1549	30	14	+	+	CCONJ
cana-1549	30	15	1	1	NUM
cana-1549	30	16	𝑅2	𝑅2	NOUN
cana-1549	30	17	+	+	ADJ
cana-1549	30	18	⋯+	⋯+	NOUN
cana-1549	30	19	1	1	NUM
cana-1549	30	20	𝑅𝑛	𝑅𝑛	PROPN
cana-1549	30	21	)	)	PUNCT
cana-1549	30	22	−1	−1	NOUN
cana-1549	30	23	𝐶1	𝐶1	NOUN
cana-1549	30	24	+	+	CCONJ
cana-1549	30	25	𝐶2	𝐶2	ADJ
cana-1549	30	26	+	+	NOUN
cana-1549	30	27	⋯+	⋯+	NOUN
cana-1549	30	28	𝐶𝑛	𝐶𝑛	VERB
cana-1549	30	29	in	in	ADP
cana-1549	30	30	order	order	NOUN
cana-1549	30	31	to	to	PART
cana-1549	30	32	reduce	reduce	VERB
cana-1549	30	33	the	the	DET
cana-1549	30	34	voltage	voltage	NOUN
cana-1549	30	35	drop	drop	NOUN
cana-1549	30	36	in	in	ADP
cana-1549	30	37	the	the	DET
cana-1549	30	38	series	series	NOUN
cana-1549	30	39	circuit	circuit	NOUN
cana-1549	30	40	,	,	PUNCT
cana-1549	30	41	we	we	PRON
cana-1549	30	42	need	need	VERB
cana-1549	30	43	to	to	PART
cana-1549	30	44	reduce	reduce	VERB
cana-1549	30	45	the	the	DET
cana-1549	30	46	𝐶𝑇	𝐶𝑇	PROPN
cana-1549	30	47	and	and	CCONJ
cana-1549	30	48	𝑅𝑇	𝑅𝑇	PROPN
cana-1549	30	49	in	in	ADP
cana-1549	30	50	parallel	parallel	ADJ
cana-1549	30	51	circuit	circuit	NOUN
cana-1549	30	52	.	.	PUNCT
cana-1549	31	1	according	accord	VERB
cana-1549	31	2	to	to	ADP
cana-1549	31	3	recent	recent	ADJ
cana-1549	31	4	study	study	NOUN
cana-1549	31	5	the	the	DET
cana-1549	31	6	drops	drop	NOUN
cana-1549	31	7	𝐶𝑇	𝐶𝑇	PROPN
cana-1549	31	8	or	or	CCONJ
cana-1549	31	9	𝑅𝑇	𝑅𝑇	PROPN
cana-1549	31	10	in	in	ADP
cana-1549	31	11	the	the	DET
cana-1549	31	12	line	line	NOUN
cana-1549	31	13	is	be	AUX
cana-1549	31	14	admissible	admissible	ADJ
cana-1549	31	15	upto	upto	NOUN
cana-1549	31	16	5	5	NUM
cana-1549	31	17	%	%	NOUN
cana-1549	31	18	either	either	CCONJ
cana-1549	31	19	in	in	ADP
cana-1549	31	20	series	series	NOUN
cana-1549	31	21	or	or	CCONJ
cana-1549	31	22	parallel	parallel	NOUN
cana-1549	31	23	respectively	respectively	ADV
cana-1549	31	24	.	.	PUNCT
cana-1549	32	1	therefore	therefore	ADV
cana-1549	32	2	a	a	DET
cana-1549	32	3	problem	problem	NOUN
cana-1549	32	4	is	be	AUX
cana-1549	32	5	there	there	ADV
cana-1549	32	6	to	to	PART
cana-1549	32	7	find	find	VERB
cana-1549	32	8	the	the	DET
cana-1549	32	9	probability	probability	NOUN
cana-1549	32	10	of	of	ADP
cana-1549	32	11	joining	join	VERB
cana-1549	32	12	of	of	ADP
cana-1549	32	13	(	(	PUNCT
cana-1549	32	14	𝑘	𝑘	PROPN
cana-1549	32	15	+	+	PROPN
cana-1549	32	16	1)𝑡ℎ	1)𝑡ℎ	NUM
cana-1549	32	17	capacitor	capacitor	NOUN
cana-1549	32	18	after	after	SCONJ
cana-1549	32	19	𝑘𝑡ℎ	𝑘𝑡ℎ	VERB
cana-1549	32	20	capacitors	capacitor	NOUN
cana-1549	32	21	in	in	ADP
cana-1549	32	22	the	the	DET
cana-1549	32	23	series	series	NOUN
cana-1549	32	24	line	line	NOUN
cana-1549	32	25	should	should	AUX
cana-1549	32	26	n’t	not	PART
cana-1549	32	27	exceed	exceed	VERB
cana-1549	32	28	0.05	0.05	NUM
cana-1549	32	29	.	.	PUNCT
cana-1549	33	1	this	this	DET
cana-1549	33	2	concept	concept	NOUN
cana-1549	33	3	encourages	encourage	VERB
cana-1549	33	4	us	we	PRON
cana-1549	33	5	to	to	PART
cana-1549	33	6	develop	develop	VERB
cana-1549	33	7	the	the	DET
cana-1549	33	8	discrete	discrete	ADJ
cana-1549	33	9	harmonic	harmonic	ADJ
cana-1549	33	10	distribution	distribution	NOUN
cana-1549	33	11	.	.	PUNCT
cana-1549	34	1	to	to	ADP
cana-1549	34	2	the	the	DET
cana-1549	34	3	best	good	ADJ
cana-1549	34	4	of	of	ADP
cana-1549	34	5	the	the	DET
cana-1549	34	6	author	author	NOUN
cana-1549	34	7	’s	’s	PART
cana-1549	34	8	knowledge	knowledge	NOUN
cana-1549	34	9	,	,	PUNCT
cana-1549	34	10	no	no	DET
cana-1549	34	11	direct	direct	ADJ
cana-1549	34	12	investigation	investigation	NOUN
cana-1549	34	13	exists	exist	VERB
cana-1549	34	14	to	to	PART
cana-1549	34	15	introduce	introduce	VERB
cana-1549	34	16	discrete	discrete	ADJ
cana-1549	34	17	harmonic	harmonic	NOUN
cana-1549	34	18	probability	probability	NOUN
cana-1549	34	19	mass	mass	NOUN
cana-1549	34	20	functions	function	NOUN
cana-1549	34	21	,	,	PUNCT
cana-1549	34	22	moments	moment	NOUN
cana-1549	34	23	,	,	PUNCT
cana-1549	34	24	central	central	ADJ
cana-1549	34	25	moments	moment	NOUN
cana-1549	34	26	and	and	CCONJ
cana-1549	34	27	its	its	PRON
cana-1549	34	28	applications	application	NOUN
cana-1549	34	29	.	.	PUNCT
cana-1549	35	1	in	in	ADP
cana-1549	35	2	this	this	DET
cana-1549	35	3	paper	paper	NOUN
cana-1549	35	4	we	we	PRON
cana-1549	35	5	develop	develop	VERB
cana-1549	35	6	the	the	DET
cana-1549	35	7	random	random	ADJ
cana-1549	35	8	variable	variable	NOUN
cana-1549	35	9	for	for	ADP
cana-1549	35	10	some	some	DET
cana-1549	35	11	types	type	NOUN
cana-1549	35	12	of	of	ADP
cana-1549	35	13	discrete	discrete	ADJ
cana-1549	35	14	harmonic	harmonic	ADJ
cana-1549	35	15	distribution	distribution	NOUN
cana-1549	35	16	and	and	CCONJ
cana-1549	35	17	their	their	PRON
cana-1549	35	18	cumulative	cumulative	ADJ
cana-1549	35	19	distribution	distribution	NOUN
cana-1549	35	20	mass	mass	NOUN
cana-1549	35	21	functions	function	NOUN
cana-1549	35	22	.	.	PUNCT
cana-1549	36	1	2	2	X
cana-1549	36	2	.	.	X
cana-1549	36	3	harmonic	harmonic	ADJ
cana-1549	36	4	mean	mean	PROPN
cana-1549	36	5	rv	rv	PROPN
cana-1549	36	6	(	(	PUNCT
cana-1549	36	7	hmrv	hmrv	PROPN
cana-1549	36	8	)	)	PUNCT
cana-1549	36	9	and	and	CCONJ
cana-1549	36	10	discrete	discrete	ADJ
cana-1549	36	11	harmonic	harmonic	ADJ
cana-1549	36	12	distribution	distribution	NOUN
cana-1549	36	13	for	for	ADP
cana-1549	36	14	any	any	DET
cana-1549	36	15	real	real	ADJ
cana-1549	36	16	(	(	PUNCT
cana-1549	36	17	𝑛	𝑛	NOUN
cana-1549	36	18	+	+	NOUN
cana-1549	36	19	1	1	NUM
cana-1549	36	20	)	)	PUNCT
cana-1549	36	21	real	real	ADJ
cana-1549	36	22	values	value	NOUN
cana-1549	36	23	𝑎0	𝑎0	PROPN
cana-1549	36	24	,	,	PUNCT
cana-1549	36	25	𝑎1	𝑎1	PROPN
cana-1549	36	26	,	,	PUNCT
cana-1549	36	27	⋯	⋯	PROPN
cana-1549	36	28	,	,	PUNCT
cana-1549	36	29	𝑎𝑛	𝑎𝑛	PROPN
cana-1549	36	30	,	,	PUNCT
cana-1549	36	31	harmonic	harmonic	ADJ
cana-1549	36	32	convexity	convexity	NOUN
cana-1549	36	33	of	of	ADP
cana-1549	36	34	𝐴	𝐴	PROPN
cana-1549	36	35	=	=	SYM
cana-1549	36	36	{	{	PUNCT
cana-1549	36	37	𝑎0	𝑎0	PROPN
cana-1549	36	38	,	,	PUNCT
cana-1549	36	39	𝑎1	𝑎1	PROPN
cana-1549	36	40	,	,	PUNCT
cana-1549	36	41	⋯	⋯	PROPN
cana-1549	36	42	,	,	PUNCT
cana-1549	36	43	𝑎𝑛	𝑎𝑛	PRON
cana-1549	36	44	}	}	PUNCT
cana-1549	36	45	is	be	AUX
cana-1549	36	46	𝐻𝐶(𝐴	𝐻𝐶(𝐴	ADJ
cana-1549	36	47	)	)	PUNCT
cana-1549	37	1	=	=	PUNCT
cana-1549	38	1	[	[	X
cana-1549	38	2	∑	∑	PROPN
cana-1549	38	3	𝑖=0	𝑖=0	PROPN
cana-1549	38	4	𝑛	𝑛	PROPN
cana-1549	38	5	(	(	PUNCT
cana-1549	38	6	𝛼𝑖/𝑎𝑖	𝛼𝑖/𝑎𝑖	NUM
cana-1549	38	7	)	)	PUNCT
cana-1549	38	8	]	]	PUNCT
cana-1549	38	9	−1	−1	NOUN
cana-1549	38	10	=	=	SYM
cana-1549	38	11	𝑎0𝑎1⋯𝑎𝑛	𝑎0𝑎1⋯𝑎𝑛	PROPN
cana-1549	38	12	∑	∑	PROPN
cana-1549	38	13	𝑖=0	𝑖=0	PROPN
cana-1549	38	14	𝑛	𝑛	PROPN
cana-1549	38	15	𝛼𝑖𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑛	𝛼𝑖𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑛	NOUN
cana-1549	38	16	satisfying	satisfy	VERB
cana-1549	38	17	∑	∑	PROPN
cana-1549	38	18	𝑖=0	𝑖=0	PROPN
cana-1549	38	19	𝑛	𝑛	PRON
cana-1549	38	20	𝛼𝑖	𝛼𝑖	NOUN
cana-1549	38	21	=	=	SYM
cana-1549	38	22	1	1	X
cana-1549	38	23	.	.	PUNCT
cana-1549	38	24	taking	take	VERB
cana-1549	38	25	𝛼𝑖	𝛼𝑖	ADP
cana-1549	38	26	=	=	SYM
cana-1549	38	27	1	1	NUM
cana-1549	38	28	𝑛+1	𝑛+1	PROPN
cana-1549	38	29	,	,	PUNCT
cana-1549	38	30	we	we	PRON
cana-1549	38	31	have	have	AUX
cana-1549	38	32	𝐻𝐶(𝐴	𝐻𝐶(𝐴	VERB
cana-1549	38	33	)	)	PUNCT
cana-1549	38	34	=	=	PUNCT
cana-1549	39	1	[	[	X
cana-1549	39	2	∑	∑	PROPN
cana-1549	39	3	𝑖=0	𝑖=0	PROPN
cana-1549	39	4	𝑛	𝑛	PROPN
cana-1549	39	5	(	(	PUNCT
cana-1549	39	6	𝛼𝑖/𝑎𝑖	𝛼𝑖/𝑎𝑖	NUM
cana-1549	39	7	)	)	PUNCT
cana-1549	39	8	]	]	PUNCT
cana-1549	39	9	−1	−1	NOUN
cana-1549	40	1	=	=	SYM
cana-1549	41	1	[	[	X
cana-1549	41	2	∑	∑	PROPN
cana-1549	41	3	𝑖=0	𝑖=0	PROPN
cana-1549	41	4	𝑛	𝑛	PROPN
cana-1549	41	5	(	(	PUNCT
cana-1549	41	6	1	1	NUM
cana-1549	41	7	(	(	PUNCT
cana-1549	41	8	𝑛	𝑛	PROPN
cana-1549	41	9	+	+	NUM
cana-1549	41	10	1)𝑎𝑖	1)𝑎𝑖	PROPN
cana-1549	41	11	]	]	PUNCT
cana-1549	41	12	−1	−1	NOUN
cana-1549	41	13	=	=	SYM
cana-1549	41	14	(	(	PUNCT
cana-1549	41	15	𝑛	𝑛	PROPN
cana-1549	41	16	+	+	NOUN
cana-1549	41	17	1	1	NUM
cana-1549	41	18	)	)	PUNCT
cana-1549	42	1	𝑎0𝑎1⋯𝑎𝑛	𝑎0𝑎1⋯𝑎𝑛	NOUN
cana-1549	42	2	∑	∑	PROPN
cana-1549	42	3	𝑖=0	𝑖=0	PROPN
cana-1549	42	4	𝑛	𝑛	PRON
cana-1549	42	5	𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑛	𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑛	NOUN
cana-1549	42	6	.	.	PUNCT
cana-1549	43	1	let	let	VERB
cana-1549	43	2	𝑋	𝑋	NOUN
cana-1549	43	3	be	be	AUX
cana-1549	43	4	the	the	DET
cana-1549	43	5	harmonic	harmonic	ADJ
cana-1549	43	6	mean	mean	ADJ
cana-1549	43	7	random	random	ADJ
cana-1549	43	8	variable	variable	NOUN
cana-1549	43	9	consists	consist	VERB
cana-1549	43	10	of	of	ADP
cana-1549	43	11	successive	successive	ADJ
cana-1549	43	12	harmonic	harmonic	ADJ
cana-1549	43	13	mean	mean	NOUN
cana-1549	43	14	values	value	NOUN
cana-1549	43	15	of	of	ADP
cana-1549	43	16	𝑎𝑖	𝑎𝑖	X
cana-1549	43	17	’s	’s	ADV
cana-1549	43	18	,	,	PUNCT
cana-1549	43	19	i.e.	i.e.	X
cana-1549	43	20	,	,	PUNCT
cana-1549	43	21	𝑋	𝑋	PROPN
cana-1549	43	22	=	=	SYM
cana-1549	43	23	{	{	PUNCT
cana-1549	43	24	𝑥0	𝑥0	PROPN
cana-1549	43	25	,	,	PUNCT
cana-1549	43	26	𝑥1	𝑥1	NOUN
cana-1549	43	27	,	,	PUNCT
cana-1549	43	28	⋯	⋯	PROPN
cana-1549	43	29	,	,	PUNCT
cana-1549	43	30	𝑥𝑛	𝑥𝑛	VERB
cana-1549	43	31	}	}	PUNCT
cana-1549	43	32	where	where	SCONJ
cana-1549	43	33	𝑥𝑘	𝑥𝑘	X
cana-1549	43	34	=	=	PUNCT
cana-1549	44	1	[	[	X
cana-1549	44	2	∑	∑	PROPN
cana-1549	44	3	𝑖=0	𝑖=0	PROPN
cana-1549	44	4	𝑘	𝑘	PROPN
cana-1549	44	5	(	(	PUNCT
cana-1549	44	6	1/𝑎𝑖	1/𝑎𝑖	NUM
cana-1549	44	7	)	)	PUNCT
cana-1549	44	8	]	]	PUNCT
cana-1549	44	9	−1	−1	NOUN
cana-1549	44	10	=	=	SYM
cana-1549	44	11	𝑎0𝑎1⋯𝑎𝑘	𝑎0𝑎1⋯𝑎𝑘	PROPN
cana-1549	44	12	∑	∑	PUNCT
cana-1549	44	13	𝑖=0	𝑖=0	PROPN
cana-1549	44	14	𝑘	𝑘	PRON
cana-1549	44	15	𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘	𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘	NOUN
cana-1549	44	16	(	(	PUNCT
cana-1549	44	17	2.1	2.1	NUM
cana-1549	44	18	)	)	PUNCT
cana-1549	44	19	for	for	ADP
cana-1549	44	20	𝑘	𝑘	PROPN
cana-1549	44	21	=	=	SYM
cana-1549	44	22	0,1,⋯	0,1,⋯	NUM
cana-1549	44	23	,	,	PUNCT
cana-1549	44	24	𝑛	𝑛	PROPN
cana-1549	44	25	,	,	PUNCT
cana-1549	44	26	then	then	ADV
cana-1549	44	27	we	we	PRON
cana-1549	44	28	have	have	VERB
cana-1549	44	29	communications	communication	NOUN
cana-1549	44	30	on	on	ADP
cana-1549	44	31	applied	apply	VERB
cana-1549	44	32	nonlinear	nonlinear	ADJ
cana-1549	44	33	analysis	analysis	NOUN
cana-1549	44	34	issn	issn	NOUN
cana-1549	44	35	:	:	PUNCT
cana-1549	44	36	1074	1074	NUM
cana-1549	44	37	-	-	PUNCT
cana-1549	44	38	133x	133x	NUM
cana-1549	44	39	vol	vol	NOUN
cana-1549	44	40	31	31	NUM
cana-1549	44	41	no	no	NOUN
cana-1549	44	42	.	.	PUNCT
cana-1549	45	1	8s	8s	PROPN
cana-1549	45	2	(	(	PUNCT
cana-1549	45	3	2024	2024	NUM
cana-1549	45	4	)	)	PUNCT
cana-1549	45	5	567	567	NUM
cana-1549	45	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	45	7	𝑥0	𝑥0	NOUN
cana-1549	45	8	=	=	SYM
cana-1549	45	9	𝑎0	𝑎0	PROPN
cana-1549	45	10	,	,	PUNCT
cana-1549	45	11	𝑥1	𝑥1	NOUN
cana-1549	45	12	=	=	SYM
cana-1549	45	13	𝑎0𝑎1	𝑎0𝑎1	PROPN
cana-1549	45	14	𝑎0	𝑎0	PROPN
cana-1549	45	15	+	+	CCONJ
cana-1549	45	16	𝑎1	𝑎1	ADV
cana-1549	45	17	,	,	PUNCT
cana-1549	45	18	𝑥2	𝑥2	NOUN
cana-1549	45	19	=	=	SYM
cana-1549	45	20	𝑎0𝑎1𝑎2	𝑎0𝑎1𝑎2	PROPN
cana-1549	45	21	𝑎1𝑎2	𝑎1𝑎2	PROPN
cana-1549	46	1	+	+	CCONJ
cana-1549	46	2	𝑎0𝑎2	𝑎0𝑎2	ADJ
cana-1549	46	3	+	+	X
cana-1549	46	4	𝑎0𝑎1	𝑎0𝑎1	X
cana-1549	46	5	,	,	PUNCT
cana-1549	46	6	⋯	⋯	PROPN
cana-1549	46	7	,	,	PUNCT
cana-1549	46	8	𝑥𝑛	𝑥𝑛	PROPN
cana-1549	46	9	=	=	SYM
cana-1549	46	10	𝑎0𝑎1⋯𝑎𝑛	𝑎0𝑎1⋯𝑎𝑛	PROPN
cana-1549	46	11	𝑎1𝑎2⋯𝑎𝑛	𝑎1𝑎2⋯𝑎𝑛	PROPN
cana-1549	46	12	+	+	CCONJ
cana-1549	46	13	𝑎0𝑎2⋯𝑎𝑛	𝑎0𝑎2⋯𝑎𝑛	PROPN
cana-1549	46	14	+	+	CCONJ
cana-1549	46	15	𝑎0𝑎2⋯𝑎𝑛−1	𝑎0𝑎2⋯𝑎𝑛−1	PROPN
cana-1549	46	16	.	.	PUNCT
cana-1549	47	1	consider	consider	VERB
cana-1549	47	2	𝑝(𝑥	𝑝(𝑥	NOUN
cana-1549	47	3	)	)	PUNCT
cana-1549	47	4	=	=	SYM
cana-1549	47	5	𝐶	𝐶	PROPN
cana-1549	47	6	𝑥	𝑥	PROPN
cana-1549	47	7	such	such	ADJ
cana-1549	47	8	that	that	SCONJ
cana-1549	47	9	∑	∑	PROPN
cana-1549	47	10	𝑖=0	𝑖=0	PROPN
cana-1549	47	11	𝑛	𝑛	PRON
cana-1549	47	12	𝑝(𝑥𝑖	𝑝(𝑥𝑖	PROPN
cana-1549	47	13	)	)	PUNCT
cana-1549	47	14	=	=	SYM
cana-1549	47	15	1	1	NUM
cana-1549	47	16	for	for	ADP
cana-1549	47	17	some	some	DET
cana-1549	47	18	𝐶	𝐶	PROPN
cana-1549	47	19	>	>	X
cana-1549	47	20	0	0	PROPN
cana-1549	47	21	.	.	PUNCT
cana-1549	47	22	assume	assume	VERB
cana-1549	47	23	that	that	SCONJ
cana-1549	47	24	𝐹(𝑥0	𝐹(𝑥0	PROPN
cana-1549	47	25	)	)	PUNCT
cana-1549	47	26	=	=	SYM
cana-1549	47	27	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	47	28	)	)	PUNCT
cana-1549	48	1	=	=	SYM
cana-1549	48	2	𝐶	𝐶	PROPN
cana-1549	48	3	𝑎0	𝑎0	ADV
cana-1549	48	4	𝐹(𝑥1	𝐹(𝑥1	ADV
cana-1549	48	5	)	)	PUNCT
cana-1549	48	6	=	=	SYM
cana-1549	48	7	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	48	8	)	)	PUNCT
cana-1549	48	9	+	+	SYM
cana-1549	48	10	𝑝(𝑥1	𝑝(𝑥1	X
cana-1549	48	11	)	)	PUNCT
cana-1549	48	12	=	=	SYM
cana-1549	48	13	𝐶	𝐶	PROPN
cana-1549	48	14	[	[	PUNCT
cana-1549	48	15	1	1	NUM
cana-1549	48	16	𝑎0	𝑎0	PROPN
cana-1549	48	17	+	+	CCONJ
cana-1549	48	18	𝑎0	𝑎0	PROPN
cana-1549	49	1	+	+	CCONJ
cana-1549	49	2	𝑎1	𝑎1	ADV
cana-1549	49	3	𝑎0𝑎1	𝑎0𝑎1	X
cana-1549	49	4	]	]	PUNCT
cana-1549	49	5	=	=	PUNCT
cana-1549	49	6	𝐶	𝐶	PROPN
cana-1549	49	7	𝑎0	𝑎0	PROPN
cana-1549	49	8	+	+	CCONJ
cana-1549	49	9	2𝑎1	2𝑎1	NUM
cana-1549	49	10	𝑎0𝑎1	𝑎0𝑎1	X
cana-1549	49	11	=	=	SYM
cana-1549	49	12	𝐶	𝐶	PROPN
cana-1549	49	13	[	[	PUNCT
cana-1549	49	14	1	1	NUM
cana-1549	49	15	𝑎1	𝑎1	NOUN
cana-1549	49	16	+	+	CCONJ
cana-1549	49	17	2	2	NUM
cana-1549	49	18	𝑎0	𝑎0	PROPN
cana-1549	49	19	]	]	PUNCT
cana-1549	49	20	𝐹(𝑥2	𝐹(𝑥2	NUM
cana-1549	49	21	)	)	PUNCT
cana-1549	49	22	=	=	SYM
cana-1549	49	23	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	49	24	)	)	PUNCT
cana-1549	49	25	+	+	SYM
cana-1549	49	26	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	49	27	)	)	PUNCT
cana-1549	49	28	+	+	CCONJ
cana-1549	49	29	𝑝(𝑥2	𝑝(𝑥2	NOUN
cana-1549	49	30	)	)	PUNCT
cana-1549	50	1	=	=	SYM
cana-1549	50	2	𝐶	𝐶	PROPN
cana-1549	50	3	[	[	PUNCT
cana-1549	50	4	1	1	NUM
cana-1549	50	5	𝑎0	𝑎0	PROPN
cana-1549	50	6	+	+	CCONJ
cana-1549	50	7	𝑎0	𝑎0	PROPN
cana-1549	50	8	+	+	NOUN
cana-1549	50	9	𝑎1	𝑎1	ADV
cana-1549	50	10	𝑎0𝑎1	𝑎0𝑎1	PUNCT
cana-1549	51	1	+	+	CCONJ
cana-1549	51	2	𝑎1𝑎2	𝑎1𝑎2	PROPN
cana-1549	52	1	+	+	CCONJ
cana-1549	52	2	𝑎0𝑎2	𝑎0𝑎2	ADJ
cana-1549	52	3	+	+	CCONJ
cana-1549	52	4	𝑎0𝑎1	𝑎0𝑎1	NOUN
cana-1549	52	5	𝑎0𝑎1𝑎2	𝑎0𝑎1𝑎2	X
cana-1549	52	6	]	]	PUNCT
cana-1549	52	7	=	=	SYM
cana-1549	52	8	𝐶	𝐶	PROPN
cana-1549	52	9	𝑎0𝑎1	𝑎0𝑎1	PROPN
cana-1549	52	10	+	+	CCONJ
cana-1549	52	11	2𝑎0𝑎2	2𝑎0𝑎2	NUM
cana-1549	52	12	+	+	CCONJ
cana-1549	52	13	3𝑎1𝑎2	3𝑎1𝑎2	NUM
cana-1549	52	14	𝑎0𝑎1𝑎2	𝑎0𝑎1𝑎2	PROPN
cana-1549	52	15	=	=	SYM
cana-1549	52	16	𝐶	𝐶	PROPN
cana-1549	52	17	[	[	PUNCT
cana-1549	52	18	1	1	NUM
cana-1549	52	19	𝑎2	𝑎2	NOUN
cana-1549	52	20	+	+	CCONJ
cana-1549	52	21	2	2	NUM
cana-1549	52	22	𝑎1	𝑎1	NOUN
cana-1549	52	23	+	+	CCONJ
cana-1549	52	24	3	3	NUM
cana-1549	52	25	𝑎0	𝑎0	PROPN
cana-1549	52	26	]	]	PUNCT
cana-1549	52	27	.	.	PUNCT
cana-1549	53	1	in	in	ADP
cana-1549	53	2	general	general	ADJ
cana-1549	53	3	,	,	PUNCT
cana-1549	53	4	for	for	ADP
cana-1549	53	5	𝑘	𝑘	PROPN
cana-1549	53	6	=	=	SYM
cana-1549	53	7	0,1,⋯	0,1,⋯	NUM
cana-1549	53	8	,	,	PUNCT
cana-1549	53	9	𝑛	𝑛	PROPN
cana-1549	53	10	,	,	PUNCT
cana-1549	53	11	we	we	PRON
cana-1549	53	12	have	have	VERB
cana-1549	53	13	𝐹(𝑥𝑘	𝐹(𝑥𝑘	ADJ
cana-1549	53	14	)	)	PUNCT
cana-1549	53	15	=	=	SYM
cana-1549	53	16	𝐶	𝐶	PROPN
cana-1549	53	17	[	[	PUNCT
cana-1549	53	18	1	1	NUM
cana-1549	53	19	𝑎𝑘	𝑎𝑘	NOUN
cana-1549	53	20	+	+	NOUN
cana-1549	53	21	2	2	NUM
cana-1549	53	22	𝑎𝑘−1	𝑎𝑘−1	PROPN
cana-1549	53	23	+	+	CCONJ
cana-1549	53	24	3	3	NUM
cana-1549	53	25	𝑎𝑘−2	𝑎𝑘−2	NOUN
cana-1549	53	26	+	+	NOUN
cana-1549	53	27	⋯+	⋯+	NOUN
cana-1549	53	28	𝑘	𝑘	DET
cana-1549	53	29	−	−	PROPN
cana-1549	53	30	1	1	NUM
cana-1549	53	31	𝑎2	𝑎2	NOUN
cana-1549	54	1	+	+	CCONJ
cana-1549	55	1	𝑘	𝑘	X
cana-1549	55	2	𝑎1	𝑎1	NOUN
cana-1549	56	1	+	+	CCONJ
cana-1549	56	2	𝑘	𝑘	PROPN
cana-1549	56	3	+	+	ADJ
cana-1549	56	4	1	1	NUM
cana-1549	56	5	𝑎0	𝑎0	PROPN
cana-1549	56	6	]	]	PUNCT
cana-1549	56	7	=	=	PUNCT
cana-1549	56	8	𝐶∑	𝐶∑	PROPN
cana-1549	56	9	𝑘	𝑘	PRON
cana-1549	56	10	𝑖=0	𝑖=0	PROPN
cana-1549	56	11	𝑖	𝑖	PROPN
cana-1549	57	1	+	+	NOUN
cana-1549	57	2	1	1	NUM
cana-1549	57	3	𝑎𝑘−𝑖	𝑎𝑘−𝑖	NOUN
cana-1549	57	4	.	.	PUNCT
cana-1549	58	1	now	now	ADV
cana-1549	58	2	1	1	NUM
cana-1549	58	3	=	=	SYM
cana-1549	58	4	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	58	5	)	)	PUNCT
cana-1549	59	1	+	+	SYM
cana-1549	59	2	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	59	3	)	)	PUNCT
cana-1549	59	4	+	+	CCONJ
cana-1549	59	5	𝑝(𝑥2	𝑝(𝑥2	NOUN
cana-1549	59	6	)	)	PUNCT
cana-1549	60	1	+	+	NUM
cana-1549	60	2	⋯+	⋯+	NUM
cana-1549	60	3	𝑝(𝑥𝑛	𝑝(𝑥𝑛	NUM
cana-1549	60	4	)	)	PUNCT
cana-1549	60	5	=	=	SYM
cana-1549	60	6	𝐶	𝐶	PROPN
cana-1549	60	7	[	[	PUNCT
cana-1549	60	8	1	1	NUM
cana-1549	60	9	𝑎0	𝑎0	PROPN
cana-1549	60	10	+	+	CCONJ
cana-1549	60	11	𝑎0	𝑎0	PROPN
cana-1549	61	1	+	+	NOUN
cana-1549	61	2	𝑎1	𝑎1	ADP
cana-1549	61	3	𝑎0𝑎1	𝑎0𝑎1	PUNCT
cana-1549	62	1	+	+	PUNCT
cana-1549	62	2	𝑎1𝑎2⋯𝑎𝑛	𝑎1𝑎2⋯𝑎𝑛	X
cana-1549	62	3	+	+	CCONJ
cana-1549	62	4	𝑎0𝑎2⋯𝑎𝑛	𝑎0𝑎2⋯𝑎𝑛	PROPN
cana-1549	62	5	+	+	CCONJ
cana-1549	62	6	𝑎0𝑎2⋯𝑎𝑛−1	𝑎0𝑎2⋯𝑎𝑛−1	PROPN
cana-1549	62	7	𝑎0𝑎1⋯𝑎𝑛	𝑎0𝑎1⋯𝑎𝑛	NOUN
cana-1549	62	8	]	]	PUNCT
cana-1549	62	9	=	=	SYM
cana-1549	62	10	𝐶	𝐶	PROPN
cana-1549	62	11	[	[	PUNCT
cana-1549	62	12	1	1	NUM
cana-1549	62	13	𝑎𝑛	𝑎𝑛	PROPN
cana-1549	62	14	+	+	CCONJ
cana-1549	62	15	2	2	NUM
cana-1549	62	16	𝑎𝑛−1	𝑎𝑛−1	NOUN
cana-1549	62	17	+	+	CCONJ
cana-1549	62	18	3	3	NUM
cana-1549	62	19	𝑎𝑛−2	𝑎𝑛−2	PROPN
cana-1549	62	20	+	+	NOUN
cana-1549	62	21	⋯+	⋯+	NOUN
cana-1549	62	22	𝑛	𝑛	DET
cana-1549	62	23	−	−	NUM
cana-1549	62	24	1	1	NUM
cana-1549	62	25	𝑎2	𝑎2	NOUN
cana-1549	62	26	+	+	CCONJ
cana-1549	62	27	𝑛	𝑛	PROPN
cana-1549	62	28	𝑎1	𝑎1	NOUN
cana-1549	62	29	+	+	CCONJ
cana-1549	62	30	𝑛	𝑛	PRON
cana-1549	62	31	+	+	SYM
cana-1549	62	32	1	1	NUM
cana-1549	62	33	𝑎0	𝑎0	PROPN
cana-1549	62	34	]	]	PUNCT
cana-1549	62	35	=	=	PUNCT
cana-1549	62	36	𝐶∑	𝐶∑	PROPN
cana-1549	62	37	𝑛	𝑛	PROPN
cana-1549	62	38	𝑖=0	𝑖=0	PROPN
cana-1549	62	39	𝑖	𝑖	PROPN
cana-1549	63	1	+	+	NOUN
cana-1549	63	2	1	1	NUM
cana-1549	63	3	𝑎𝑛−𝑖	𝑎𝑛−𝑖	ADJ
cana-1549	63	4	⇒	⇒	NOUN
cana-1549	63	5	𝐶	𝐶	PROPN
cana-1549	63	6	=	=	SYM
cana-1549	63	7	(	(	PUNCT
cana-1549	63	8	∑	∑	PROPN
cana-1549	63	9	𝑖=0	𝑖=0	PROPN
cana-1549	63	10	𝑛	𝑛	ADP
cana-1549	63	11	𝑖	𝑖	SYM
cana-1549	64	1	+	+	NOUN
cana-1549	64	2	1	1	NUM
cana-1549	64	3	𝑎𝑛−𝑖	𝑎𝑛−𝑖	NOUN
cana-1549	64	4	)	)	PUNCT
cana-1549	64	5	−1	−1	NOUN
cana-1549	64	6	(	(	PUNCT
cana-1549	64	7	2.2	2.2	NUM
cana-1549	64	8	)	)	PUNCT
cana-1549	64	9	.	.	PUNCT
cana-1549	65	1	hence	hence	ADV
cana-1549	65	2	𝑝(𝑥	𝑝(𝑥	NUM
cana-1549	65	3	)	)	PUNCT
cana-1549	66	1	=	=	PRON
cana-1549	66	2	(	(	PUNCT
cana-1549	66	3	∑	∑	PROPN
cana-1549	66	4	𝑖=0	𝑖=0	PROPN
cana-1549	66	5	𝑛	𝑛	ADP
cana-1549	66	6	𝑖	𝑖	SYM
cana-1549	67	1	+	+	NOUN
cana-1549	67	2	1	1	NUM
cana-1549	67	3	𝑎𝑛−𝑖	𝑎𝑛−𝑖	ADJ
cana-1549	67	4	)	)	PUNCT
cana-1549	67	5	−1	−1	NOUN
cana-1549	67	6	1	1	NUM
cana-1549	67	7	𝑥	𝑥	NOUN
cana-1549	67	8	,	,	PUNCT
cana-1549	67	9	𝑥	𝑥	NOUN
cana-1549	67	10	=	=	SYM
cana-1549	67	11	𝑥0	𝑥0	PROPN
cana-1549	67	12	,	,	PUNCT
cana-1549	67	13	𝑥1	𝑥1	NOUN
cana-1549	67	14	,	,	PUNCT
cana-1549	67	15	⋯	⋯	PROPN
cana-1549	67	16	,	,	PUNCT
cana-1549	67	17	𝑥𝑛.	𝑥𝑛.	X
cana-1549	67	18	(	(	PUNCT
cana-1549	67	19	2.3	2.3	NUM
cana-1549	67	20	)	)	PUNCT
cana-1549	67	21	is	be	AUX
cana-1549	67	22	a	a	DET
cana-1549	67	23	special	special	ADJ
cana-1549	67	24	type	type	NOUN
cana-1549	67	25	of	of	ADP
cana-1549	67	26	harmonic	harmonic	ADJ
cana-1549	67	27	pmf	pmf	NOUN
cana-1549	67	28	in	in	ADP
cana-1549	67	29	the	the	DET
cana-1549	67	30	harmonic	harmonic	ADJ
cana-1549	67	31	set	set	NOUN
cana-1549	67	32	.	.	PUNCT
cana-1549	68	1	example	example	NOUN
cana-1549	68	2	2.1	2.1	NUM
cana-1549	68	3	if	if	SCONJ
cana-1549	68	4	a	a	DET
cana-1549	68	5	runner	runner	NOUN
cana-1549	68	6	runs	run	VERB
cana-1549	68	7	as	as	ADP
cana-1549	68	8	per	per	ADP
cana-1549	68	9	the	the	DET
cana-1549	68	10	time	time	NOUN
cana-1549	68	11	(	(	PUNCT
cana-1549	68	12	in	in	ADP
cana-1549	68	13	seconds	second	NOUN
cana-1549	68	14	)	)	PUNCT
cana-1549	68	15	given	give	VERB
cana-1549	68	16	in	in	ADP
cana-1549	68	17	the	the	DET
cana-1549	68	18	time	time	NOUN
cana-1549	68	19	space	space	NOUN
cana-1549	68	20	𝑇	𝑇	PROPN
cana-1549	68	21	=	=	SYM
cana-1549	68	22	{	{	PUNCT
cana-1549	68	23	𝑡0	𝑡0	PROPN
cana-1549	68	24	,	,	PUNCT
cana-1549	68	25	𝑡1	𝑡1	NOUN
cana-1549	68	26	,	,	PUNCT
cana-1549	68	27	⋯	⋯	PROPN
cana-1549	68	28	,	,	PUNCT
cana-1549	68	29	𝑡𝑛	𝑡𝑛	NOUN
cana-1549	68	30	}	}	PUNCT
cana-1549	68	31	,	,	PUNCT
cana-1549	68	32	𝑡𝑖	𝑡𝑖	NOUN
cana-1549	68	33	≠	≠	PROPN
cana-1549	68	34	0	0	NUM
cana-1549	68	35	with	with	ADP
cana-1549	68	36	relative	relative	ADJ
cana-1549	68	37	speed	speed	NOUN
cana-1549	68	38	has	have	VERB
cana-1549	68	39	a	a	DET
cana-1549	68	40	harmonic	harmonic	ADJ
cana-1549	68	41	random	random	ADJ
cana-1549	68	42	variable	variable	NOUN
cana-1549	68	43	𝑋𝑇	𝑋𝑇	NOUN
cana-1549	68	44	=	=	SYM
cana-1549	68	45	{	{	PUNCT
cana-1549	68	46	𝑥	𝑥	NOUN
cana-1549	68	47	:	:	PUNCT
cana-1549	68	48	𝑥	𝑥	NOUN
cana-1549	68	49	=	=	SYM
cana-1549	68	50	𝑥(𝑡	𝑥(𝑡	PROPN
cana-1549	68	51	)	)	PUNCT
cana-1549	68	52	,	,	PUNCT
cana-1549	68	53	𝑡	𝑡	PROPN
cana-1549	68	54	∈	∈	PROPN
cana-1549	68	55	𝑇	𝑇	PROPN
cana-1549	68	56	}	}	PUNCT
cana-1549	68	57	=	=	SYM
cana-1549	68	58	{	{	PUNCT
cana-1549	68	59	𝑥0	𝑥0	NOUN
cana-1549	68	60	,	,	PUNCT
cana-1549	68	61	𝑥1	𝑥1	NOUN
cana-1549	68	62	,	,	PUNCT
cana-1549	68	63	⋯	⋯	PROPN
cana-1549	68	64	,	,	PUNCT
cana-1549	68	65	𝑥𝑛	𝑥𝑛	AUX
cana-1549	68	66	}	}	PUNCT
cana-1549	68	67	obtained	obtain	VERB
cana-1549	68	68	by	by	ADP
cana-1549	68	69	equation	equation	NOUN
cana-1549	68	70	(	(	PUNCT
cana-1549	68	71	2.1	2.1	NUM
cana-1549	68	72	)	)	PUNCT
cana-1549	68	73	.	.	PUNCT
cana-1549	69	1	the	the	DET
cana-1549	69	2	runner	runner	NOUN
cana-1549	69	3	has	have	VERB
cana-1549	69	4	a	a	DET
cana-1549	69	5	relative	relative	ADJ
cana-1549	69	6	distance	distance	NOUN
cana-1549	69	7	covered	cover	VERB
cana-1549	69	8	in	in	ADP
cana-1549	69	9	time	time	NOUN
cana-1549	69	10	𝑡	𝑡	PROPN
cana-1549	69	11	∈	∈	PROPN
cana-1549	69	12	𝐴	𝐴	PROPN
cana-1549	69	13	communications	communication	NOUN
cana-1549	69	14	on	on	ADP
cana-1549	69	15	applied	apply	VERB
cana-1549	69	16	nonlinear	nonlinear	ADJ
cana-1549	69	17	analysis	analysis	NOUN
cana-1549	69	18	issn	issn	NOUN
cana-1549	69	19	:	:	PUNCT
cana-1549	69	20	1074	1074	NUM
cana-1549	69	21	-	-	PUNCT
cana-1549	69	22	133x	133x	NUM
cana-1549	69	23	vol	vol	NOUN
cana-1549	69	24	31	31	NUM
cana-1549	69	25	no	no	NOUN
cana-1549	69	26	.	.	PUNCT
cana-1549	70	1	8s	8s	PROPN
cana-1549	70	2	(	(	PUNCT
cana-1549	70	3	2024	2024	NUM
cana-1549	70	4	)	)	PUNCT
cana-1549	70	5	568	568	NUM
cana-1549	70	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	70	7	has	have	VERB
cana-1549	70	8	the	the	DET
cana-1549	70	9	harmonic	harmonic	ADJ
cana-1549	70	10	pmf	pmf	NOUN
cana-1549	70	11	measured	measure	VERB
cana-1549	70	12	by	by	ADP
cana-1549	70	13	(	(	PUNCT
cana-1549	70	14	2.3	2.3	NUM
cana-1549	70	15	)	)	PUNCT
cana-1549	70	16	.	.	PUNCT
cana-1549	71	1	thus	thus	ADV
cana-1549	71	2	for	for	ADP
cana-1549	71	3	𝑎	𝑎	PROPN
cana-1549	71	4	=	=	SYM
cana-1549	71	5	2,3,5	2,3,5	NUM
cana-1549	71	6	,	,	PUNCT
cana-1549	71	7	the	the	DET
cana-1549	71	8	equation	equation	NOUN
cana-1549	71	9	(	(	PUNCT
cana-1549	71	10	2.1	2.1	NUM
cana-1549	71	11	)	)	PUNCT
cana-1549	71	12	gives	give	VERB
cana-1549	71	13	harmonic	harmonic	PROPN
cana-1549	71	14	rv	rv	PROPN
cana-1549	71	15	𝑋𝑇	𝑋𝑇	PROPN
cana-1549	71	16	with	with	ADP
cana-1549	71	17	images	image	NOUN
cana-1549	71	18	𝑥	𝑥	X
cana-1549	71	19	=	=	SYM
cana-1549	71	20	2	2	NUM
cana-1549	71	21	,	,	PUNCT
cana-1549	71	22	6	6	NUM
cana-1549	71	23	5	5	NUM
cana-1549	71	24	,	,	PUNCT
cana-1549	71	25	30	30	NUM
cana-1549	71	26	31	31	NUM
cana-1549	71	27	.	.	PUNCT
cana-1549	72	1	since	since	SCONJ
cana-1549	72	2	equation	equation	NOUN
cana-1549	72	3	(	(	PUNCT
cana-1549	72	4	2.2	2.2	NUM
cana-1549	72	5	)	)	PUNCT
cana-1549	72	6	gives	give	VERB
cana-1549	72	7	𝐶	𝐶	PROPN
cana-1549	72	8	=	=	PROPN
cana-1549	72	9	30	30	NUM
cana-1549	72	10	71	71	NUM
cana-1549	72	11	,	,	PUNCT
cana-1549	72	12	we	we	PRON
cana-1549	72	13	have	have	VERB
cana-1549	72	14	the	the	DET
cana-1549	72	15	harmonic	harmonic	ADJ
cana-1549	72	16	pmf	pmf	NOUN
cana-1549	72	17	𝑝(𝑥	𝑝(𝑥	PROPN
cana-1549	72	18	)	)	PUNCT
cana-1549	72	19	=	=	SYM
cana-1549	72	20	30	30	NUM
cana-1549	72	21	71𝑥	71𝑥	NOUN
cana-1549	72	22	,	,	PUNCT
cana-1549	72	23	𝑥	𝑥	NOUN
cana-1549	72	24	=	=	SYM
cana-1549	72	25	2	2	NUM
cana-1549	72	26	,	,	PUNCT
cana-1549	72	27	6	6	NUM
cana-1549	72	28	5	5	NUM
cana-1549	72	29	,	,	PUNCT
cana-1549	72	30	30	30	NUM
cana-1549	72	31	31	31	NUM
cana-1549	72	32	obtained	obtain	VERB
cana-1549	72	33	by	by	ADP
cana-1549	72	34	(	(	PUNCT
cana-1549	72	35	2.3	2.3	NUM
cana-1549	72	36	)	)	PUNCT
cana-1549	72	37	satisfies	satisfie	NOUN
cana-1549	72	38	∑	∑	PUNCT
cana-1549	72	39	𝑖=0	𝑖=0	PROPN
cana-1549	72	40	𝑛	𝑛	PRON
cana-1549	72	41	𝑝(𝑥𝑖	𝑝(𝑥𝑖	PROPN
cana-1549	72	42	)	)	PUNCT
cana-1549	72	43	=	=	SYM
cana-1549	73	1	1	1	X
cana-1549	73	2	.	.	X
cana-1549	73	3	hence	hence	ADV
cana-1549	73	4	the	the	DET
cana-1549	73	5	random	random	ADJ
cana-1549	73	6	variable	variable	ADJ
cana-1549	73	7	𝑋𝑇	𝑋𝑇	PROPN
cana-1549	73	8	has	have	VERB
cana-1549	73	9	the	the	DET
cana-1549	73	10	pmf	pmf	PROPN
cana-1549	73	11	values	value	NOUN
cana-1549	73	12	are	be	AUX
cana-1549	73	13	𝑝(2	𝑝(2	NOUN
cana-1549	73	14	)	)	PUNCT
cana-1549	73	15	=	=	PUNCT
cana-1549	73	16	15	15	NUM
cana-1549	73	17	71	71	NUM
cana-1549	73	18	,	,	PUNCT
cana-1549	73	19	𝑝(6/5	𝑝(6/5	NUM
cana-1549	73	20	)	)	PUNCT
cana-1549	73	21	=	=	NUM
cana-1549	73	22	25	25	NUM
cana-1549	73	23	71	71	NUM
cana-1549	73	24	,	,	PUNCT
cana-1549	73	25	and	and	CCONJ
cana-1549	73	26	𝑝(30/31	𝑝(30/31	PROPN
cana-1549	73	27	)	)	PUNCT
cana-1549	73	28	=	=	PUNCT
cana-1549	74	1	31	31	NUM
cana-1549	74	2	71	71	NUM
cana-1549	74	3	and	and	CCONJ
cana-1549	74	4	cdf	cdf	PROPN
cana-1549	74	5	values	value	NOUN
cana-1549	74	6	are	be	AUX
cana-1549	74	7	𝐹(2	𝐹(2	NUM
cana-1549	74	8	)	)	PUNCT
cana-1549	75	1	=	=	SYM
cana-1549	75	2	15	15	NUM
cana-1549	75	3	71	71	NUM
cana-1549	75	4	,	,	PUNCT
cana-1549	75	5	𝐹(6/5	𝐹(6/5	PROPN
cana-1549	75	6	)	)	PUNCT
cana-1549	75	7	=	=	NUM
cana-1549	76	1	40	40	NUM
cana-1549	76	2	71	71	NUM
cana-1549	76	3	,	,	PUNCT
cana-1549	76	4	and	and	CCONJ
cana-1549	76	5	𝐹(30/31	𝐹(30/31	NUM
cana-1549	76	6	)	)	PUNCT
cana-1549	76	7	=	=	SYM
cana-1549	77	1	1	1	X
cana-1549	77	2	.	.	PUNCT
cana-1549	77	3	example	example	NOUN
cana-1549	77	4	2.2	2.2	NUM
cana-1549	77	5	in	in	ADP
cana-1549	77	6	a	a	DET
cana-1549	77	7	series	series	NOUN
cana-1549	77	8	with	with	ADP
cana-1549	77	9	three	three	NUM
cana-1549	77	10	capacitors	capacitor	NOUN
cana-1549	77	11	𝐶1	𝐶1	VERB
cana-1549	77	12	=	=	SYM
cana-1549	77	13	2𝜇f	2𝜇f	NOUN
cana-1549	77	14	,	,	PUNCT
cana-1549	77	15	𝐶2	𝐶2	ADJ
cana-1549	77	16	=	=	PUNCT
cana-1549	77	17	4𝜇f	4𝜇f	NOUN
cana-1549	77	18	and	and	CCONJ
cana-1549	77	19	𝐶3	𝐶3	PROPN
cana-1549	77	20	=	=	SYM
cana-1549	77	21	5𝜇f	5𝜇f	NOUN
cana-1549	77	22	are	be	AUX
cana-1549	77	23	connected	connect	VERB
cana-1549	77	24	,	,	PUNCT
cana-1549	77	25	then	then	ADV
cana-1549	77	26	rv	rv	PROPN
cana-1549	77	27	𝑋𝐶	𝑋𝐶	PROPN
cana-1549	77	28	measures	measure	VERB
cana-1549	77	29	total	total	ADJ
cana-1549	77	30	capacitance	capacitance	NOUN
cana-1549	77	31	after	after	ADP
cana-1549	77	32	connecting	connect	VERB
cana-1549	77	33	to	to	ADP
cana-1549	77	34	the	the	DET
cana-1549	77	35	next	next	ADJ
cana-1549	77	36	capacitor	capacitor	NOUN
cana-1549	77	37	,	,	PUNCT
cana-1549	77	38	then	then	ADV
cana-1549	77	39	𝑋𝐶	𝑋𝐶	PROPN
cana-1549	77	40	=	=	SYM
cana-1549	77	41	{	{	PUNCT
cana-1549	77	42	𝑥1	𝑥1	NOUN
cana-1549	77	43	,	,	PUNCT
cana-1549	77	44	𝑥2	𝑥2	NOUN
cana-1549	77	45	,	,	PUNCT
cana-1549	77	46	𝑥3	𝑥3	NOUN
cana-1549	77	47	}	}	PUNCT
cana-1549	78	1	where	where	SCONJ
cana-1549	78	2	𝑥1	𝑥1	NOUN
cana-1549	78	3	=	=	SYM
cana-1549	78	4	2	2	NUM
cana-1549	78	5	,	,	PUNCT
cana-1549	78	6	𝑥2	𝑥2	NOUN
cana-1549	78	7	=	=	SYM
cana-1549	78	8	(	(	PUNCT
cana-1549	78	9	1	1	NUM
cana-1549	78	10	2	2	NUM
cana-1549	78	11	+	+	CCONJ
cana-1549	78	12	1	1	NUM
cana-1549	78	13	4	4	NUM
cana-1549	78	14	)	)	PUNCT
cana-1549	78	15	−1	−1	NOUN
cana-1549	78	16	=	=	SYM
cana-1549	78	17	4/3	4/3	NUM
cana-1549	78	18	and	and	CCONJ
cana-1549	78	19	𝑥3	𝑥3	NOUN
cana-1549	78	20	=	=	SYM
cana-1549	78	21	(	(	PUNCT
cana-1549	78	22	1	1	NUM
cana-1549	78	23	2	2	NUM
cana-1549	78	24	+	+	CCONJ
cana-1549	78	25	1	1	NUM
cana-1549	78	26	4	4	NUM
cana-1549	78	27	+	+	CCONJ
cana-1549	78	28	1	1	NUM
cana-1549	78	29	5	5	NUM
cana-1549	78	30	)	)	PUNCT
cana-1549	78	31	−1	−1	NOUN
cana-1549	78	32	=	=	SYM
cana-1549	78	33	20/19	20/19	NUM
cana-1549	78	34	.	.	PUNCT
cana-1549	79	1	the	the	DET
cana-1549	79	2	harmonic	harmonic	ADJ
cana-1549	79	3	pmf	pmf	NOUN
cana-1549	79	4	of	of	ADP
cana-1549	79	5	𝑋𝐶	𝑋𝐶	PROPN
cana-1549	79	6	is	be	AUX
cana-1549	79	7	𝑝(𝑥	𝑝(𝑥	NUM
cana-1549	79	8	)	)	PUNCT
cana-1549	79	9	=	=	SYM
cana-1549	79	10	5	5	NUM
cana-1549	79	11	11𝑥	11𝑥	X
cana-1549	79	12	for	for	ADP
cana-1549	79	13	𝑥	𝑥	NOUN
cana-1549	79	14	=	=	SYM
cana-1549	79	15	2	2	NUM
cana-1549	79	16	,	,	PUNCT
cana-1549	79	17	4/3	4/3	NUM
cana-1549	79	18	,	,	PUNCT
cana-1549	79	19	20/19	20/19	NUM
cana-1549	79	20	,	,	PUNCT
cana-1549	79	21	i.e.	i.e.	X
cana-1549	79	22	,	,	PUNCT
cana-1549	79	23	𝑝(2	𝑝(2	PROPN
cana-1549	79	24	)	)	PUNCT
cana-1549	79	25	=	=	SYM
cana-1549	79	26	5	5	NUM
cana-1549	79	27	22	22	NUM
cana-1549	79	28	,	,	PUNCT
cana-1549	79	29	𝑝(4/3	𝑝(4/3	ADJ
cana-1549	79	30	)	)	PUNCT
cana-1549	79	31	=	=	SYM
cana-1549	79	32	15/44	15/44	NUM
cana-1549	79	33	and	and	CCONJ
cana-1549	79	34	𝑝(20/19	𝑝(20/19	NUM
cana-1549	79	35	)	)	PUNCT
cana-1549	80	1	=	=	SYM
cana-1549	80	2	19/44	19/44	NUM
cana-1549	80	3	and	and	CCONJ
cana-1549	80	4	the	the	DET
cana-1549	80	5	cdf	cdf	PROPN
cana-1549	80	6	of	of	ADP
cana-1549	80	7	𝑋𝐶	𝑋𝐶	PROPN
cana-1549	80	8	are	be	AUX
cana-1549	80	9	𝐹(2	𝐹(2	NUM
cana-1549	80	10	)	)	PUNCT
cana-1549	81	1	=	=	SYM
cana-1549	81	2	5	5	NUM
cana-1549	81	3	22	22	NUM
cana-1549	81	4	,	,	PUNCT
cana-1549	81	5	𝐹(4/3	𝐹(4/3	PROPN
cana-1549	81	6	)	)	PUNCT
cana-1549	81	7	=	=	SYM
cana-1549	82	1	25/44	25/44	NUM
cana-1549	82	2	and	and	CCONJ
cana-1549	82	3	𝐹(20/19	𝐹(20/19	NUM
cana-1549	82	4	)	)	PUNCT
cana-1549	82	5	=	=	SYM
cana-1549	83	1	1	1	NUM
cana-1549	83	2	.	.	NOUN
cana-1549	83	3	3	3	NUM
cana-1549	83	4	.	.	NOUN
cana-1549	83	5	discrete	discrete	ADJ
cana-1549	83	6	exponential	exponential	ADJ
cana-1549	83	7	harmonic	harmonic	ADJ
cana-1549	83	8	distribution	distribution	NOUN
cana-1549	83	9	let	let	VERB
cana-1549	83	10	ℤ𝑚	ℤ𝑚	PRON
cana-1549	83	11	𝑛	𝑛	PART
cana-1549	83	12	be	be	AUX
cana-1549	83	13	the	the	DET
cana-1549	83	14	set	set	NOUN
cana-1549	83	15	of	of	ADP
cana-1549	83	16	integers	integer	NOUN
cana-1549	83	17	lie	lie	VERB
cana-1549	83	18	in	in	ADP
cana-1549	83	19	between	between	ADP
cana-1549	83	20	𝑚	𝑚	PROPN
cana-1549	83	21	and	and	CCONJ
cana-1549	83	22	𝑛	𝑛	PRON
cana-1549	83	23	for	for	ADP
cana-1549	83	24	0	0	NUM
cana-1549	83	25	≤	≤	NUM
cana-1549	83	26	𝑚	𝑚	ADP
cana-1549	83	27	≤	≤	NUM
cana-1549	83	28	𝑛	𝑛	DET
cana-1549	83	29	−	−	PROPN
cana-1549	83	30	1	1	NUM
cana-1549	83	31	and	and	CCONJ
cana-1549	83	32	𝐾	𝐾	PROPN
cana-1549	83	33	=	=	SYM
cana-1549	83	34	{	{	PUNCT
cana-1549	83	35	𝑥	𝑥	PROPN
cana-1549	83	36	≠	≠	PROPN
cana-1549	83	37	0	0	NUM
cana-1549	83	38	:	:	PUNCT
cana-1549	83	39	𝑥	𝑥	X
cana-1549	83	40	=	=	SYM
cana-1549	83	41	𝑘	𝑘	PRON
cana-1549	83	42	exp	exp	NOUN
cana-1549	83	43	(	(	PUNCT
cana-1549	83	44	1	1	NUM
cana-1549	83	45	𝑘−1	𝑘−1	PROPN
cana-1549	83	46	)	)	PUNCT
cana-1549	83	47	,	,	PUNCT
cana-1549	83	48	𝑘	𝑘	DET
cana-1549	83	49	≥	≥	NOUN
cana-1549	83	50	1	1	NUM
cana-1549	83	51	}	}	PUNCT
cana-1549	83	52	satisfying	satisfy	VERB
cana-1549	83	53	𝑥𝑘.	𝑥𝑘.	ADV
cana-1549	83	54	let	let	VERB
cana-1549	83	55	𝑋	𝑋	NOUN
cana-1549	83	56	be	be	AUX
cana-1549	83	57	a	a	DET
cana-1549	83	58	discrete	discrete	ADJ
cana-1549	83	59	random	random	ADJ
cana-1549	83	60	variable	variable	NOUN
cana-1549	83	61	defined	define	VERB
cana-1549	83	62	by	by	ADP
cana-1549	83	63	𝑋	𝑋	PROPN
cana-1549	83	64	=	=	SYM
cana-1549	83	65	{	{	PUNCT
cana-1549	83	66	𝑧𝑥	𝑧𝑥	NOUN
cana-1549	83	67	∈	∈	PROPN
cana-1549	83	68	ℝ	ℝ	PROPN
cana-1549	83	69	:	:	PUNCT
cana-1549	83	70	𝑥	𝑥	X
cana-1549	83	71	=	=	SYM
cana-1549	83	72	0,1,⋯	0,1,⋯	PROPN
cana-1549	83	73	,	,	PUNCT
cana-1549	83	74	𝑛	𝑛	PROPN
cana-1549	83	75	}	}	PUNCT
cana-1549	83	76	and	and	CCONJ
cana-1549	83	77	the	the	DET
cana-1549	83	78	discrete	discrete	ADJ
cana-1549	83	79	harmonic	harmonic	ADJ
cana-1549	83	80	probability	probability	NOUN
cana-1549	83	81	mass	mass	NOUN
cana-1549	83	82	function	function	NOUN
cana-1549	83	83	(	(	PUNCT
cana-1549	83	84	dhpmf	dhpmf	NOUN
cana-1549	83	85	)	)	PUNCT
cana-1549	83	86	be	be	AUX
cana-1549	83	87	𝑝(𝑥	𝑝(𝑥	NUM
cana-1549	83	88	;	;	PUNCT
cana-1549	83	89	𝑧	𝑧	X
cana-1549	83	90	)	)	PUNCT
cana-1549	83	91	=	=	SYM
cana-1549	83	92	𝐶	𝐶	PROPN
cana-1549	83	93	(	(	PUNCT
cana-1549	83	94	𝑛	𝑛	PROPN
cana-1549	83	95	+	+	X
cana-1549	83	96	1)𝑧𝑥	1)𝑧𝑥	NUM
cana-1549	83	97	,	,	PUNCT
cana-1549	83	98	𝑥	𝑥	PROPN
cana-1549	83	99	=	=	NUM
cana-1549	83	100	0,1,⋯	0,1,⋯	NUM
cana-1549	83	101	,	,	PUNCT
cana-1549	83	102	𝑛	𝑛	PROPN
cana-1549	83	103	,	,	PUNCT
cana-1549	83	104	and	and	CCONJ
cana-1549	83	105	,	,	PUNCT
cana-1549	83	106	0	0	NUM
cana-1549	83	107	otherwise	otherwise	ADV
cana-1549	83	108	,	,	PUNCT
cana-1549	83	109	for	for	ADP
cana-1549	83	110	some	some	DET
cana-1549	83	111	𝐶	𝐶	PROPN
cana-1549	83	112	>	>	X
cana-1549	83	113	0	0	PUNCT
cana-1549	84	1	satisfying	satisfy	VERB
cana-1549	84	2	0	0	NUM
cana-1549	84	3	≤	≤	NOUN
cana-1549	84	4	𝑝(𝑥	𝑝(𝑥	NOUN
cana-1549	84	5	;	;	PUNCT
cana-1549	84	6	𝑧	𝑧	X
cana-1549	84	7	)	)	PUNCT
cana-1549	84	8	≤	≤	NUM
cana-1549	84	9	1	1	NUM
cana-1549	84	10	and	and	CCONJ
cana-1549	84	11	∑𝑛𝑥=0	∑𝑛𝑥=0	NUM
cana-1549	84	12	𝑝(𝑥	𝑝(𝑥	NUM
cana-1549	84	13	;	;	PUNCT
cana-1549	84	14	𝑎	𝑎	X
cana-1549	84	15	)	)	PUNCT
cana-1549	84	16	=	=	SYM
cana-1549	84	17	1	1	X
cana-1549	84	18	.	.	PUNCT
cana-1549	85	1	since	since	SCONJ
cana-1549	85	2	0	0	NUM
cana-1549	85	3	≠	≠	PROPN
cana-1549	85	4	𝑥	𝑥	DET
cana-1549	85	5	∈	∈	PROPN
cana-1549	85	6	𝐾	𝐾	PROPN
cana-1549	85	7	,	,	PUNCT
cana-1549	85	8	we	we	PRON
cana-1549	85	9	obtain	obtain	VERB
cana-1549	85	10	𝑥	𝑥	PRON
cana-1549	85	11	=	=	SYM
cana-1549	85	12	𝑘exp	𝑘exp	NOUN
cana-1549	85	13	(	(	PUNCT
cana-1549	85	14	1	1	NUM
cana-1549	85	15	𝑘−1	𝑘−1	PROPN
cana-1549	85	16	)	)	PUNCT
cana-1549	85	17	,	,	PUNCT
cana-1549	85	18	i.e.	i.e.	X
cana-1549	85	19	,	,	PUNCT
cana-1549	85	20	𝑥𝑘	𝑥𝑘	X
cana-1549	85	21	=	=	PUNCT
cana-1549	85	22	𝑘𝑥	𝑘𝑥	PROPN
cana-1549	85	23	for	for	ADP
cana-1549	85	24	some	some	DET
cana-1549	85	25	𝑘	𝑘	PROPN
cana-1549	85	26	>	>	X
cana-1549	85	27	0	0	PUNCT
cana-1549	86	1	and	and	CCONJ
cana-1549	86	2	𝑥	𝑥	AUX
cana-1549	86	3	>	>	X
cana-1549	86	4	0	0	X
cana-1549	86	5	.	.	PUNCT
cana-1549	86	6	letting	let	VERB
cana-1549	86	7	𝑧𝑥	𝑧𝑥	NOUN
cana-1549	86	8	=	=	PUNCT
cana-1549	86	9	𝑎	𝑎	NOUN
cana-1549	86	10	𝑥	𝑥	NOUN
cana-1549	86	11	for	for	ADP
cana-1549	86	12	some	some	DET
cana-1549	86	13	parameter	parameter	NOUN
cana-1549	86	14	𝑎	𝑎	X
cana-1549	86	15	>	>	X
cana-1549	86	16	0	0	NUM
cana-1549	86	17	,	,	PUNCT
cana-1549	86	18	𝑥	𝑥	NOUN
cana-1549	86	19	=	=	NOUN
cana-1549	86	20	0,1,2,⋯	0,1,2,⋯	NUM
cana-1549	86	21	,	,	PUNCT
cana-1549	86	22	𝑛	𝑛	PROPN
cana-1549	86	23	,	,	PUNCT
cana-1549	86	24	we	we	PRON
cana-1549	86	25	obtain	obtain	VERB
cana-1549	86	26	𝑝(𝑥	𝑝(𝑥	NUM
cana-1549	86	27	;	;	PUNCT
cana-1549	86	28	𝑎	𝑎	X
cana-1549	86	29	,	,	PUNCT
cana-1549	86	30	𝑛	𝑛	ADJ
cana-1549	86	31	)	)	PUNCT
cana-1549	86	32	=	=	SYM
cana-1549	86	33	𝐶	𝐶	PROPN
cana-1549	86	34	(	(	PUNCT
cana-1549	86	35	𝑛	𝑛	PROPN
cana-1549	86	36	+	+	X
cana-1549	86	37	1)𝑎𝑥	1)𝑎𝑥	PROPN
cana-1549	86	38	,	,	PUNCT
cana-1549	86	39	𝑥	𝑥	PROPN
cana-1549	86	40	=	=	NUM
cana-1549	86	41	0,1,⋯	0,1,⋯	NUM
cana-1549	86	42	,	,	PUNCT
cana-1549	86	43	𝑛	𝑛	PROPN
cana-1549	86	44	,	,	PUNCT
cana-1549	86	45	and	and	CCONJ
cana-1549	86	46	,	,	PUNCT
cana-1549	86	47	0	0	NUM
cana-1549	86	48	otherwise	otherwise	ADV
cana-1549	86	49	,	,	PUNCT
cana-1549	86	50	considering	consider	VERB
cana-1549	86	51	it	it	PRON
cana-1549	86	52	as	as	ADP
cana-1549	86	53	a	a	DET
cana-1549	86	54	legitimate	legitimate	ADJ
cana-1549	86	55	pmf	pmf	NOUN
cana-1549	86	56	,	,	PUNCT
cana-1549	86	57	we	we	PRON
cana-1549	86	58	have	have	VERB
cana-1549	86	59	unity	unity	NOUN
cana-1549	86	60	property	property	NOUN
cana-1549	86	61	,	,	PUNCT
cana-1549	86	62	i.e.	i.e.	X
cana-1549	86	63	,	,	PUNCT
cana-1549	86	64	1	1	NUM
cana-1549	86	65	=	=	NOUN
cana-1549	86	66	∑	∑	PART
cana-1549	86	67	𝑛	𝑛	PROPN
cana-1549	86	68	𝑥=0	𝑥=0	X
cana-1549	86	69	𝑝(𝑥	𝑝(𝑥	NUM
cana-1549	86	70	;	;	PUNCT
cana-1549	86	71	𝑎	𝑎	X
cana-1549	86	72	,	,	PUNCT
cana-1549	86	73	𝑛	𝑛	ADJ
cana-1549	86	74	)	)	PUNCT
cana-1549	86	75	=	=	NOUN
cana-1549	86	76	∑	∑	ADP
cana-1549	86	77	𝑛	𝑛	PRON
cana-1549	86	78	𝑥=0	𝑥=0	PUNCT
cana-1549	86	79	𝐶	𝐶	PROPN
cana-1549	86	80	(	(	PUNCT
cana-1549	86	81	𝑛	𝑛	PROPN
cana-1549	86	82	+	+	NUM
cana-1549	86	83	1)𝑎𝑥	1)𝑎𝑥	PROPN
cana-1549	86	84	=	=	SYM
cana-1549	86	85	𝐶	𝐶	PROPN
cana-1549	86	86	(	(	PUNCT
cana-1549	86	87	𝑛	𝑛	PROPN
cana-1549	86	88	+	+	NOUN
cana-1549	86	89	1	1	NUM
cana-1549	86	90	)	)	PUNCT
cana-1549	86	91	(	(	PUNCT
cana-1549	86	92	1	1	NUM
cana-1549	86	93	+	+	NUM
cana-1549	86	94	1	1	NUM
cana-1549	86	95	𝑎	𝑎	X
cana-1549	86	96	+	+	NUM
cana-1549	86	97	1	1	NUM
cana-1549	86	98	𝑎2	𝑎2	NOUN
cana-1549	86	99	+	+	PROPN
cana-1549	86	100	⋯+	⋯+	PROPN
cana-1549	86	101	1	1	NUM
cana-1549	86	102	𝑎𝑛−1	𝑎𝑛−1	NOUN
cana-1549	86	103	+	+	CCONJ
cana-1549	86	104	1	1	NUM
cana-1549	86	105	𝑎𝑛	𝑎𝑛	NOUN
cana-1549	86	106	)	)	PUNCT
cana-1549	86	107	=	=	SYM
cana-1549	87	1	𝐶𝑎−𝑛	𝐶𝑎−𝑛	ADJ
cana-1549	87	2	(	(	PUNCT
cana-1549	87	3	𝑛	𝑛	PROPN
cana-1549	87	4	+	+	NOUN
cana-1549	87	5	1	1	NUM
cana-1549	87	6	)	)	PUNCT
cana-1549	87	7	(	(	PUNCT
cana-1549	87	8	1	1	NUM
cana-1549	87	9	+	+	CCONJ
cana-1549	87	10	𝑎	𝑎	PRON
cana-1549	87	11	+	+	NOUN
cana-1549	87	12	𝑎2	𝑎2	NOUN
cana-1549	87	13	+	+	NOUN
cana-1549	87	14	⋯+	⋯+	NOUN
cana-1549	87	15	𝑎𝑛	𝑎𝑛	NOUN
cana-1549	87	16	)	)	PUNCT
cana-1549	87	17	communications	communication	NOUN
cana-1549	87	18	on	on	ADP
cana-1549	87	19	applied	apply	VERB
cana-1549	87	20	nonlinear	nonlinear	ADJ
cana-1549	87	21	analysis	analysis	NOUN
cana-1549	87	22	issn	issn	NOUN
cana-1549	87	23	:	:	PUNCT
cana-1549	87	24	1074	1074	NUM
cana-1549	87	25	-	-	PUNCT
cana-1549	87	26	133x	133x	NUM
cana-1549	87	27	vol	vol	NOUN
cana-1549	87	28	31	31	NUM
cana-1549	87	29	no	no	NOUN
cana-1549	87	30	.	.	PUNCT
cana-1549	88	1	8s	8s	PROPN
cana-1549	88	2	(	(	PUNCT
cana-1549	88	3	2024	2024	NUM
cana-1549	88	4	)	)	PUNCT
cana-1549	88	5	569	569	NUM
cana-1549	88	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	88	7	implying	imply	VERB
cana-1549	88	8	𝐶	𝐶	PROPN
cana-1549	88	9	=	=	SYM
cana-1549	88	10	(	(	PUNCT
cana-1549	88	11	𝑛	𝑛	PROPN
cana-1549	88	12	+	+	NOUN
cana-1549	88	13	1	1	NUM
cana-1549	88	14	)	)	PUNCT
cana-1549	88	15	(	(	PUNCT
cana-1549	88	16	𝑎−1	𝑎−1	PROPN
cana-1549	88	17	𝑎−𝑎−𝑛	𝑎−𝑎−𝑛	ADJ
cana-1549	88	18	)	)	PUNCT
cana-1549	88	19	for	for	ADP
cana-1549	88	20	𝑎	𝑎	PROPN
cana-1549	88	21	>	>	SYM
cana-1549	88	22	1	1	NUM
cana-1549	88	23	.	.	PUNCT
cana-1549	88	24	thus	thus	ADV
cana-1549	88	25	fixed	fix	VERB
cana-1549	88	26	fixed	fixed	ADJ
cana-1549	88	27	parameters	parameter	NOUN
cana-1549	88	28	𝑛	𝑛	PROPN
cana-1549	88	29	=	=	SYM
cana-1549	88	30	2,3,⋯	2,3,⋯	NUM
cana-1549	88	31	and	and	CCONJ
cana-1549	88	32	𝑎	𝑎	NOUN
cana-1549	88	33	=	=	NOUN
cana-1549	88	34	2,3,⋯	2,3,⋯	NUM
cana-1549	88	35	,	,	PUNCT
cana-1549	88	36	the	the	DET
cana-1549	88	37	dhpmf	dhpmf	NOUN
cana-1549	88	38	is	be	AUX
cana-1549	88	39	obtained	obtain	VERB
cana-1549	88	40	as	as	ADP
cana-1549	88	41	𝑑ℎ(𝑥	𝑑ℎ(𝑥	NOUN
cana-1549	88	42	;	;	PUNCT
cana-1549	88	43	𝑎	𝑎	X
cana-1549	88	44	,	,	PUNCT
cana-1549	88	45	𝑛	𝑛	ADJ
cana-1549	88	46	)	)	PUNCT
cana-1549	89	1	=	=	SYM
cana-1549	89	2	𝑃(𝑋	𝑃(𝑋	PROPN
cana-1549	89	3	=	=	SYM
cana-1549	89	4	𝑥	𝑥	PROPN
cana-1549	89	5	)	)	PUNCT
cana-1549	89	6	=	=	NOUN
cana-1549	90	1	(	(	PUNCT
cana-1549	90	2	𝑎	𝑎	X
cana-1549	90	3	−	−	NUM
cana-1549	90	4	1	1	NUM
cana-1549	90	5	𝑎	𝑎	PRON
cana-1549	90	6	−	−	NOUN
cana-1549	90	7	𝑎−𝑛	𝑎−𝑛	NUM
cana-1549	90	8	)	)	PUNCT
cana-1549	90	9	1	1	NUM
cana-1549	90	10	𝑎𝑥	𝑎𝑥	NOUN
cana-1549	90	11	,	,	PUNCT
cana-1549	90	12	𝑥	𝑥	PROPN
cana-1549	90	13	=	=	SYM
cana-1549	90	14	0,1,⋯	0,1,⋯	NUM
cana-1549	90	15	,	,	PUNCT
cana-1549	90	16	𝑛	𝑛	NOUN
cana-1549	90	17	;	;	PUNCT
cana-1549	90	18	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1549	90	19	0	0	NUM
cana-1549	91	1	otherwise	otherwise	ADV
cana-1549	91	2	and	and	CCONJ
cana-1549	91	3	the	the	DET
cana-1549	91	4	corresponding	correspond	VERB
cana-1549	91	5	discrete	discrete	ADJ
cana-1549	91	6	harmonic	harmonic	ADJ
cana-1549	91	7	probability	probability	NOUN
cana-1549	91	8	distribution	distribution	NOUN
cana-1549	91	9	function	function	NOUN
cana-1549	91	10	(	(	PUNCT
cana-1549	91	11	dhpdf	dhpdf	PROPN
cana-1549	91	12	)	)	PUNCT
cana-1549	91	13	is	be	AUX
cana-1549	91	14	given	give	VERB
cana-1549	91	15	by	by	ADP
cana-1549	91	16	𝐷𝐻(𝑥	𝐷𝐻(𝑥	NOUN
cana-1549	91	17	;	;	PUNCT
cana-1549	91	18	𝑎	𝑎	X
cana-1549	91	19	,	,	PUNCT
cana-1549	91	20	𝑛	𝑛	ADJ
cana-1549	91	21	)	)	PUNCT
cana-1549	91	22	=	=	X
cana-1549	91	23	𝑃(𝑋	𝑃(𝑋	PROPN
cana-1549	91	24	≤	≤	NOUN
cana-1549	91	25	𝑥	𝑥	X
cana-1549	91	26	)	)	PUNCT
cana-1549	92	1	=	=	NOUN
cana-1549	92	2	∑	∑	PART
cana-1549	92	3	𝑛	𝑛	PROPN
cana-1549	92	4	𝑥=0	𝑥=0	PUNCT
cana-1549	92	5	𝑑ℎ(𝑦	𝑑ℎ(𝑦	NUM
cana-1549	92	6	;	;	PUNCT
cana-1549	92	7	𝑎	𝑎	X
cana-1549	92	8	)	)	PUNCT
cana-1549	92	9	=	=	SYM
cana-1549	92	10	(	(	PUNCT
cana-1549	92	11	𝑎	𝑎	X
cana-1549	92	12	−	−	NUM
cana-1549	92	13	1	1	NUM
cana-1549	92	14	𝑎	𝑎	PRON
cana-1549	92	15	−	−	NOUN
cana-1549	92	16	𝑎−𝑛	𝑎−𝑛	NUM
cana-1549	92	17	)	)	PUNCT
cana-1549	92	18	∑	∑	ADP
cana-1549	92	19	𝑥	𝑥	PART
cana-1549	92	20	𝑦=0	𝑦=0	ADP
cana-1549	92	21	1	1	NUM
cana-1549	92	22	𝑎𝑦	𝑎𝑦	NOUN
cana-1549	92	23	,	,	PUNCT
cana-1549	92	24	𝑥	𝑥	PROPN
cana-1549	92	25	≤	≤	NUM
cana-1549	92	26	𝑛	𝑛	NOUN
cana-1549	92	27	=	=	PUNCT
cana-1549	92	28	(	(	PUNCT
cana-1549	92	29	𝑎	𝑎	X
cana-1549	92	30	−	−	NUM
cana-1549	92	31	1	1	NUM
cana-1549	92	32	𝑎	𝑎	PRON
cana-1549	92	33	−	−	NOUN
cana-1549	92	34	𝑎−𝑛	𝑎−𝑛	NUM
cana-1549	92	35	)	)	PUNCT
cana-1549	92	36	𝑎−𝑥	𝑎−𝑥	PROPN
cana-1549	92	37	(	(	PUNCT
cana-1549	92	38	𝑎𝑥+1	𝑎𝑥+1	NOUN
cana-1549	92	39	−	−	PROPN
cana-1549	92	40	1	1	NUM
cana-1549	92	41	𝑎	𝑎	DET
cana-1549	92	42	−	−	NOUN
cana-1549	92	43	1	1	NUM
cana-1549	92	44	)	)	PUNCT
cana-1549	92	45	=	=	PUNCT
cana-1549	93	1	𝑎	𝑎	DET
cana-1549	93	2	−	−	NOUN
cana-1549	93	3	𝑎−𝑥	𝑎−𝑥	NOUN
cana-1549	93	4	𝑎	𝑎	NOUN
cana-1549	93	5	−	−	NOUN
cana-1549	93	6	𝑎−𝑛	𝑎−𝑛	VERB
cana-1549	93	7	for	for	ADP
cana-1549	93	8	𝑥	𝑥	NOUN
cana-1549	93	9	=	=	SYM
cana-1549	93	10	0,1,⋯	0,1,⋯	PUNCT
cana-1549	93	11	,	,	PUNCT
cana-1549	93	12	𝑛.	𝑛.	NOUN
cana-1549	93	13	we	we	PRON
cana-1549	93	14	define	define	VERB
cana-1549	93	15	the	the	DET
cana-1549	93	16	discrete	discrete	ADJ
cana-1549	93	17	complementary	complementary	ADJ
cana-1549	93	18	harmonic	harmonic	ADJ
cana-1549	93	19	probability	probability	NOUN
cana-1549	93	20	distribution	distribution	NOUN
cana-1549	93	21	function	function	NOUN
cana-1549	93	22	(	(	PUNCT
cana-1549	93	23	dchpdf	dchpdf	NOUN
cana-1549	93	24	)	)	PUNCT
cana-1549	93	25	as	as	SCONJ
cana-1549	93	26	𝐶𝐻(𝑥	𝐶𝐻(𝑥	ADJ
cana-1549	93	27	;	;	PUNCT
cana-1549	93	28	𝑎	𝑎	X
cana-1549	93	29	,	,	PUNCT
cana-1549	93	30	𝑛	𝑛	ADJ
cana-1549	93	31	)	)	PUNCT
cana-1549	93	32	=	=	SYM
cana-1549	93	33	𝑃(𝑋	𝑃(𝑋	PROPN
cana-1549	93	34	≥	≥	NOUN
cana-1549	93	35	𝑥	𝑥	NOUN
cana-1549	93	36	)	)	PUNCT
cana-1549	93	37	=	=	SYM
cana-1549	93	38	1	1	NUM
cana-1549	93	39	−	−	PROPN
cana-1549	93	40	𝑃(𝑋	𝑃(𝑋	PROPN
cana-1549	93	41	≤	≤	VERB
cana-1549	93	42	𝑥	𝑥	PRON
cana-1549	93	43	−	−	NUM
cana-1549	93	44	1	1	NUM
cana-1549	93	45	)	)	PUNCT
cana-1549	93	46	=	=	SYM
cana-1549	94	1	1	1	NUM
cana-1549	94	2	−	−	NUM
cana-1549	94	3	𝑎	𝑎	PRON
cana-1549	94	4	−	−	NOUN
cana-1549	94	5	𝑎−(𝑥−1	𝑎−(𝑥−1	NOUN
cana-1549	94	6	)	)	PUNCT
cana-1549	94	7	𝑎	𝑎	PRON
cana-1549	94	8	−	−	NOUN
cana-1549	94	9	𝑎−𝑛	𝑎−𝑛	PROPN
cana-1549	94	10	=	=	SYM
cana-1549	94	11	𝑎−(𝑥−1	𝑎−(𝑥−1	NOUN
cana-1549	94	12	)	)	PUNCT
cana-1549	94	13	−	−	NOUN
cana-1549	95	1	𝑎−𝑛	𝑎−𝑛	NUM
cana-1549	95	2	𝑎	𝑎	PRON
cana-1549	95	3	−	−	NOUN
cana-1549	95	4	𝑎−𝑛	𝑎−𝑛	NUM
cana-1549	95	5	for	for	ADP
cana-1549	95	6	𝑥	𝑥	NOUN
cana-1549	95	7	=	=	SYM
cana-1549	95	8	0,1,⋯	0,1,⋯	PUNCT
cana-1549	95	9	,	,	PUNCT
cana-1549	95	10	𝑛.	𝑛.	NOUN
cana-1549	95	11	thus	thus	ADV
cana-1549	95	12	for	for	ADP
cana-1549	95	13	𝑎	𝑎	PROPN
cana-1549	95	14	=	=	SYM
cana-1549	95	15	2	2	NUM
cana-1549	95	16	,	,	PUNCT
cana-1549	95	17	the	the	DET
cana-1549	95	18	dhpmf	dhpmf	NOUN
cana-1549	95	19	is	be	AUX
cana-1549	95	20	𝑑ℎ(𝑥	𝑑ℎ(𝑥	VERB
cana-1549	95	21	;	;	PUNCT
cana-1549	95	22	𝑎	𝑎	X
cana-1549	95	23	)	)	PUNCT
cana-1549	95	24	=	=	SYM
cana-1549	95	25	(	(	PUNCT
cana-1549	95	26	1	1	NUM
cana-1549	95	27	2	2	NUM
cana-1549	95	28	−	−	PROPN
cana-1549	95	29	2−𝑛	2−𝑛	NUM
cana-1549	95	30	)	)	PUNCT
cana-1549	95	31	1	1	NUM
cana-1549	95	32	2𝑥	2𝑥	NOUN
cana-1549	95	33	,	,	PUNCT
cana-1549	95	34	𝑥	𝑥	NOUN
cana-1549	95	35	=	=	NOUN
cana-1549	95	36	0,1,2,⋯𝑛	0,1,2,⋯𝑛	NOUN
cana-1549	95	37	;	;	PUNCT
cana-1549	95	38	and	and	CCONJ
cana-1549	95	39	0	0	NUM
cana-1549	95	40	otherwise	otherwise	ADV
cana-1549	95	41	and	and	CCONJ
cana-1549	95	42	the	the	DET
cana-1549	95	43	dhpdf	dhpdf	NOUN
cana-1549	95	44	is	be	AUX
cana-1549	95	45	𝐷𝐻(𝑥	𝐷𝐻(𝑥	X
cana-1549	95	46	;	;	PUNCT
cana-1549	95	47	2	2	NUM
cana-1549	95	48	,	,	PUNCT
cana-1549	95	49	𝑛	𝑛	ADJ
cana-1549	95	50	)	)	PUNCT
cana-1549	95	51	=	=	SYM
cana-1549	95	52	2	2	NUM
cana-1549	95	53	−	−	NUM
cana-1549	95	54	2−𝑥	2−𝑥	NUM
cana-1549	95	55	2	2	NUM
cana-1549	95	56	−	−	PROPN
cana-1549	95	57	2−𝑛	2−𝑛	NUM
cana-1549	95	58	,	,	PUNCT
cana-1549	95	59	𝑥	𝑥	NOUN
cana-1549	95	60	=	=	SYM
cana-1549	95	61	0,1	0,1	NUM
cana-1549	95	62	,	,	PUNCT
cana-1549	95	63	2,⋯	2,⋯	NUM
cana-1549	95	64	,	,	PUNCT
cana-1549	95	65	𝑛.	𝑛.	NOUN
cana-1549	95	66	for	for	ADP
cana-1549	95	67	𝑎	𝑎	PROPN
cana-1549	95	68	=	=	SYM
cana-1549	95	69	3	3	NUM
cana-1549	95	70	,	,	PUNCT
cana-1549	95	71	the	the	DET
cana-1549	95	72	dhpmf	dhpmf	NOUN
cana-1549	95	73	is	be	AUX
cana-1549	95	74	𝑑ℎ(𝑥	𝑑ℎ(𝑥	VERB
cana-1549	95	75	;	;	PUNCT
cana-1549	95	76	3,3	3,3	NUM
cana-1549	95	77	)	)	PUNCT
cana-1549	95	78	=	=	NOUN
cana-1549	95	79	(	(	PUNCT
cana-1549	95	80	2	2	NUM
cana-1549	95	81	3	3	NUM
cana-1549	95	82	−	−	PROPN
cana-1549	95	83	3−𝑛	3−𝑛	NUM
cana-1549	95	84	)	)	PUNCT
cana-1549	95	85	1	1	NUM
cana-1549	95	86	3𝑥	3𝑥	NUM
cana-1549	95	87	,	,	PUNCT
cana-1549	95	88	𝑥	𝑥	NOUN
cana-1549	95	89	=	=	NOUN
cana-1549	95	90	0,1,2,⋯	0,1,2,⋯	NUM
cana-1549	95	91	,	,	PUNCT
cana-1549	95	92	𝑛	𝑛	NOUN
cana-1549	95	93	;	;	PUNCT
cana-1549	95	94	and	and	CCONJ
cana-1549	95	95	0	0	NUM
cana-1549	96	1	otherwise	otherwise	ADV
cana-1549	96	2	and	and	CCONJ
cana-1549	96	3	the	the	DET
cana-1549	96	4	dhpdf	dhpdf	NOUN
cana-1549	96	5	is	be	AUX
cana-1549	96	6	𝐷𝐻(𝑥	𝐷𝐻(𝑥	X
cana-1549	96	7	;	;	PUNCT
cana-1549	96	8	3	3	NUM
cana-1549	96	9	,	,	PUNCT
cana-1549	96	10	𝑛	𝑛	ADJ
cana-1549	96	11	)	)	PUNCT
cana-1549	96	12	=	=	SYM
cana-1549	96	13	3	3	NUM
cana-1549	96	14	−	−	PROPN
cana-1549	96	15	3−𝑥	3−𝑥	NUM
cana-1549	96	16	3	3	NUM
cana-1549	96	17	−	−	PROPN
cana-1549	96	18	3−𝑛	3−𝑛	NUM
cana-1549	96	19	.	.	PUNCT
cana-1549	97	1	𝑥	𝑥	NOUN
cana-1549	98	1	=	=	NOUN
cana-1549	98	2	0,1,2,⋯	0,1,2,⋯	NOUN
cana-1549	98	3	,	,	PUNCT
cana-1549	98	4	𝑛.	𝑛.	NOUN
cana-1549	98	5	the	the	DET
cana-1549	98	6	accompanying	accompany	VERB
cana-1549	98	7	table	table	NOUN
cana-1549	98	8	is	be	AUX
cana-1549	98	9	given	give	VERB
cana-1549	98	10	for	for	ADP
cana-1549	98	11	𝑎	𝑎	PROPN
cana-1549	98	12	=	=	SYM
cana-1549	98	13	2	2	NUM
cana-1549	98	14	and	and	CCONJ
cana-1549	98	15	𝑛	𝑛	ADJ
cana-1549	98	16	=	=	SYM
cana-1549	98	17	3	3	NUM
cana-1549	98	18	:	:	PUNCT
cana-1549	98	19	𝑎	𝑎	NOUN
cana-1549	98	20	=	=	SYM
cana-1549	98	21	2	2	NUM
cana-1549	98	22	,	,	PUNCT
cana-1549	98	23	𝑛	𝑛	NOUN
cana-1549	98	24	=	=	SYM
cana-1549	98	25	3	3	NUM
cana-1549	98	26	𝑥	𝑥	NOUN
cana-1549	98	27	=	=	SYM
cana-1549	98	28	0	0	PUNCT
cana-1549	99	1	𝑥	𝑥	NOUN
cana-1549	99	2	=	=	SYM
cana-1549	99	3	1	1	NUM
cana-1549	99	4	𝑥	𝑥	NOUN
cana-1549	99	5	=	=	SYM
cana-1549	99	6	2	2	NUM
cana-1549	99	7	𝑥	𝑥	NOUN
cana-1549	99	8	=	=	SYM
cana-1549	99	9	3	3	NUM
cana-1549	99	10	𝑑ℎ(𝑥	𝑑ℎ(𝑥	X
cana-1549	99	11	;	;	PUNCT
cana-1549	99	12	2,3	2,3	NUM
cana-1549	99	13	)	)	PUNCT
cana-1549	99	14	0.5333	0.5333	NUM
cana-1549	99	15	0.2667	0.2667	NUM
cana-1549	99	16	0.1333	0.1333	NUM
cana-1549	99	17	0.0667	0.0667	NUM
cana-1549	99	18	𝐷𝐻(𝑥	𝐷𝐻(𝑥	NOUN
cana-1549	99	19	;	;	PUNCT
cana-1549	99	20	2,3	2,3	NUM
cana-1549	99	21	)	)	PUNCT
cana-1549	99	22	0.5333	0.5333	NUM
cana-1549	99	23	0.8000	0.8000	NUM
cana-1549	99	24	0.9333	0.9333	NUM
cana-1549	99	25	1.0000	1.0000	NUM
cana-1549	99	26	𝑎	𝑎	PRON
cana-1549	99	27	=	=	SYM
cana-1549	99	28	2	2	NUM
cana-1549	99	29	,	,	PUNCT
cana-1549	99	30	𝑛	𝑛	NOUN
cana-1549	99	31	=	=	SYM
cana-1549	99	32	3	3	NUM
cana-1549	99	33	𝑥	𝑥	NOUN
cana-1549	99	34	=	=	SYM
cana-1549	99	35	0	0	PUNCT
cana-1549	99	36	𝑥	𝑥	NOUN
cana-1549	99	37	=	=	SYM
cana-1549	99	38	1	1	NUM
cana-1549	99	39	𝑥	𝑥	NOUN
cana-1549	99	40	=	=	SYM
cana-1549	99	41	2	2	NUM
cana-1549	99	42	𝑥	𝑥	NOUN
cana-1549	99	43	=	=	SYM
cana-1549	99	44	3	3	NUM
cana-1549	99	45	𝑑ℎ(𝑥	𝑑ℎ(𝑥	X
cana-1549	99	46	;	;	PUNCT
cana-1549	99	47	3,3	3,3	NUM
cana-1549	99	48	)	)	PUNCT
cana-1549	99	49	0.6750	0.6750	NUM
cana-1549	99	50	0.2250	0.2250	NUM
cana-1549	99	51	0.0750	0.0750	NUM
cana-1549	99	52	0.0250	0.0250	NUM
cana-1549	99	53	𝐷𝐻(𝑥	𝐷𝐻(𝑥	NOUN
cana-1549	99	54	;	;	PUNCT
cana-1549	99	55	3,3	3,3	NUM
cana-1549	99	56	)	)	PUNCT
cana-1549	99	57	0.6750	0.6750	NUM
cana-1549	99	58	0.9000	0.9000	NUM
cana-1549	99	59	0.9750	0.9750	NUM
cana-1549	99	60	1.0000	1.0000	NUM
cana-1549	99	61	communications	communication	NOUN
cana-1549	99	62	on	on	ADP
cana-1549	99	63	applied	apply	VERB
cana-1549	99	64	nonlinear	nonlinear	ADJ
cana-1549	99	65	analysis	analysis	NOUN
cana-1549	99	66	issn	issn	NOUN
cana-1549	99	67	:	:	PUNCT
cana-1549	99	68	1074	1074	NUM
cana-1549	99	69	-	-	PUNCT
cana-1549	99	70	133x	133x	NUM
cana-1549	99	71	vol	vol	NOUN
cana-1549	99	72	31	31	NUM
cana-1549	99	73	no	no	NOUN
cana-1549	99	74	.	.	PUNCT
cana-1549	100	1	8s	8s	PROPN
cana-1549	100	2	(	(	PUNCT
cana-1549	100	3	2024	2024	NUM
cana-1549	100	4	)	)	PUNCT
cana-1549	100	5	570	570	NUM
cana-1549	100	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	100	7	the	the	DET
cana-1549	100	8	accompanying	accompany	VERB
cana-1549	100	9	table	table	NOUN
cana-1549	100	10	is	be	AUX
cana-1549	100	11	given	give	VERB
cana-1549	100	12	for	for	ADP
cana-1549	100	13	𝑎	𝑎	PROPN
cana-1549	100	14	=	=	SYM
cana-1549	100	15	𝑒	𝑒	NOUN
cana-1549	100	16	and	and	CCONJ
cana-1549	100	17	𝑛	𝑛	PRON
cana-1549	100	18	=	=	NOUN
cana-1549	101	1	1,2,3	1,2,3	NUM
cana-1549	101	2	:	:	PUNCT
cana-1549	101	3	𝑎	𝑎	PROPN
cana-1549	101	4	=	=	SYM
cana-1549	101	5	𝑒	𝑒	NOUN
cana-1549	101	6	,	,	PUNCT
cana-1549	101	7	𝑛	𝑛	NOUN
cana-1549	101	8	=	=	SYM
cana-1549	101	9	1	1	NUM
cana-1549	101	10	𝑥	𝑥	NOUN
cana-1549	101	11	=	=	SYM
cana-1549	101	12	0	0	PUNCT
cana-1549	101	13	𝑥	𝑥	NOUN
cana-1549	101	14	=	=	NOUN
cana-1549	101	15	1	1	NUM
cana-1549	101	16	𝑑ℎ(𝑥	𝑑ℎ(𝑥	X
cana-1549	101	17	;	;	PUNCT
cana-1549	101	18	𝑒	𝑒	X
cana-1549	101	19	,	,	PUNCT
cana-1549	101	20	1	1	X
cana-1549	101	21	)	)	PUNCT
cana-1549	101	22	0.7311	0.7311	NUM
cana-1549	101	23	0.2689	0.2689	NUM
cana-1549	101	24	𝐷𝐻(𝑥	𝐷𝐻(𝑥	NOUN
cana-1549	101	25	;	;	PUNCT
cana-1549	101	26	𝑒	𝑒	X
cana-1549	101	27	,	,	PUNCT
cana-1549	101	28	1	1	X
cana-1549	101	29	)	)	PUNCT
cana-1549	101	30	0.7311	0.7311	NUM
cana-1549	101	31	1.0000	1.0000	NUM
cana-1549	101	32	𝑎	𝑎	X
cana-1549	101	33	=	=	SYM
cana-1549	101	34	𝑒	𝑒	NOUN
cana-1549	101	35	,	,	PUNCT
cana-1549	101	36	𝑛	𝑛	NOUN
cana-1549	101	37	=	=	SYM
cana-1549	101	38	2	2	NUM
cana-1549	101	39	𝑥	𝑥	NOUN
cana-1549	101	40	=	=	SYM
cana-1549	101	41	0	0	PUNCT
cana-1549	102	1	𝑥	𝑥	NOUN
cana-1549	102	2	=	=	SYM
cana-1549	102	3	1	1	NUM
cana-1549	102	4	𝑥	𝑥	NOUN
cana-1549	102	5	=	=	SYM
cana-1549	102	6	2	2	NUM
cana-1549	102	7	𝑑ℎ(𝑥	𝑑ℎ(𝑥	X
cana-1549	102	8	;	;	PUNCT
cana-1549	102	9	𝑒	𝑒	X
cana-1549	102	10	,	,	PUNCT
cana-1549	102	11	2	2	X
cana-1549	102	12	)	)	PUNCT
cana-1549	102	13	0.6652	0.6652	NUM
cana-1549	102	14	0.2447	0.2447	NUM
cana-1549	102	15	0.0009	0.0009	NUM
cana-1549	102	16	𝐷𝐻(𝑥	𝐷𝐻(𝑥	X
cana-1549	102	17	;	;	PUNCT
cana-1549	102	18	𝑒	𝑒	X
cana-1549	102	19	,	,	PUNCT
cana-1549	102	20	2	2	X
cana-1549	102	21	)	)	PUNCT
cana-1549	102	22	0.6652	0.6652	NUM
cana-1549	102	23	0.9100	0.9100	NUM
cana-1549	102	24	1.0000	1.0000	NUM
cana-1549	102	25	𝑎	𝑎	X
cana-1549	102	26	=	=	SYM
cana-1549	102	27	𝑒	𝑒	NOUN
cana-1549	102	28	,	,	PUNCT
cana-1549	102	29	𝑛	𝑛	PROPN
cana-1549	102	30	=	=	SYM
cana-1549	102	31	3	3	NUM
cana-1549	102	32	𝑥	𝑥	NOUN
cana-1549	102	33	=	=	SYM
cana-1549	102	34	0	0	PUNCT
cana-1549	103	1	𝑥	𝑥	NOUN
cana-1549	103	2	=	=	SYM
cana-1549	103	3	1	1	NUM
cana-1549	103	4	𝑥	𝑥	NOUN
cana-1549	103	5	=	=	SYM
cana-1549	103	6	2	2	NUM
cana-1549	103	7	𝑥	𝑥	NOUN
cana-1549	103	8	=	=	SYM
cana-1549	103	9	3	3	NUM
cana-1549	103	10	𝑑ℎ(𝑥	𝑑ℎ(𝑥	X
cana-1549	103	11	;	;	PUNCT
cana-1549	103	12	𝑒	𝑒	X
cana-1549	103	13	,	,	PUNCT
cana-1549	103	14	3	3	X
cana-1549	103	15	)	)	PUNCT
cana-1549	103	16	0.6439	0.6439	NUM
cana-1549	104	1	0.2369	0.2369	NUM
cana-1549	104	2	0.0871	0.0871	NUM
cana-1549	104	3	0.0321	0.0321	NUM
cana-1549	104	4	𝐷𝐻(𝑥	𝐷𝐻(𝑥	NOUN
cana-1549	104	5	;	;	PUNCT
cana-1549	104	6	𝑒	𝑒	X
cana-1549	104	7	,	,	PUNCT
cana-1549	104	8	3	3	X
cana-1549	104	9	)	)	PUNCT
cana-1549	104	10	0.6439	0.6439	NUM
cana-1549	104	11	0.8808	0.8808	NUM
cana-1549	104	12	0.9679	0.9679	NUM
cana-1549	104	13	1.0000	1.0000	NUM
cana-1549	104	14	4	4	NUM
cana-1549	104	15	.	.	NOUN
cana-1549	104	16	discrete	discrete	ADJ
cana-1549	104	17	natural	natural	ADJ
cana-1549	104	18	logarithmic	logarithmic	ADJ
cana-1549	104	19	harmonic	harmonic	ADJ
cana-1549	104	20	distribution	distribution	NOUN
cana-1549	104	21	for	for	ADP
cana-1549	104	22	any	any	DET
cana-1549	104	23	real	real	ADJ
cana-1549	104	24	𝑥	𝑥	X
cana-1549	104	25	>	>	X
cana-1549	104	26	1	1	NUM
cana-1549	104	27	,	,	PUNCT
cana-1549	104	28	we	we	PRON
cana-1549	104	29	have	have	AUX
cana-1549	104	30	ln𝑥	ln𝑥	NOUN
cana-1549	104	31	>	>	X
cana-1549	104	32	0	0	X
cana-1549	104	33	.	.	PUNCT
cana-1549	105	1	let	let	VERB
cana-1549	105	2	𝑎0	𝑎0	VERB
cana-1549	105	3	≥	≥	VERB
cana-1549	105	4	𝑎𝑘	𝑎𝑘	VERB
cana-1549	105	5	for	for	ADP
cana-1549	105	6	𝑘	𝑘	PROPN
cana-1549	105	7	=	=	SYM
cana-1549	105	8	1,2,⋯	1,2,⋯	PROPN
cana-1549	105	9	,	,	PUNCT
cana-1549	105	10	𝑛	𝑛	PRON
cana-1549	105	11	such	such	ADJ
cana-1549	105	12	that	that	SCONJ
cana-1549	105	13	the	the	DET
cana-1549	105	14	greatest	great	ADJ
cana-1549	105	15	integer	integer	NOUN
cana-1549	105	16	of	of	ADP
cana-1549	105	17	𝑎0	𝑎0	PROPN
cana-1549	105	18	is	be	AUX
cana-1549	105	19	at	at	ADP
cana-1549	105	20	least	least	ADJ
cana-1549	105	21	𝑛	𝑛	DET
cana-1549	105	22	+	+	NOUN
cana-1549	106	1	1	1	X
cana-1549	106	2	.	.	PUNCT
cana-1549	106	3	let	let	VERB
cana-1549	106	4	𝑋	𝑋	NOUN
cana-1549	106	5	be	be	AUX
cana-1549	106	6	the	the	DET
cana-1549	106	7	harmonic	harmonic	ADJ
cana-1549	106	8	mean	mean	NOUN
cana-1549	106	9	random	random	ADJ
cana-1549	106	10	variable	variable	NOUN
cana-1549	106	11	with	with	ADP
cana-1549	106	12	images	image	NOUN
cana-1549	106	13	𝑥𝑘	𝑥𝑘	NOUN
cana-1549	106	14	,	,	PUNCT
cana-1549	106	15	𝑘	𝑘	X
cana-1549	106	16	=	=	X
cana-1549	106	17	0,1,2,⋯	0,1,2,⋯	NOUN
cana-1549	106	18	,	,	PUNCT
cana-1549	107	1	𝑛	𝑛	PROPN
cana-1549	107	2	obtained	obtain	VERB
cana-1549	107	3	by	by	ADP
cana-1549	107	4	the	the	DET
cana-1549	107	5	relation	relation	NOUN
cana-1549	107	6	given	give	VERB
cana-1549	107	7	in	in	ADP
cana-1549	107	8	(	(	PUNCT
cana-1549	107	9	2.1	2.1	NUM
cana-1549	107	10	)	)	PUNCT
cana-1549	107	11	.	.	PUNCT
cana-1549	108	1	we	we	PRON
cana-1549	108	2	have	have	VERB
cana-1549	108	3	𝑥0	𝑥0	NOUN
cana-1549	108	4	=	=	SYM
cana-1549	108	5	𝑎0	𝑎0	PROPN
cana-1549	108	6	,	,	PUNCT
cana-1549	108	7	𝑥1	𝑥1	NOUN
cana-1549	108	8	=	=	SYM
cana-1549	108	9	𝑎0𝑎1	𝑎0𝑎1	PROPN
cana-1549	108	10	𝑎0	𝑎0	PROPN
cana-1549	108	11	+	+	CCONJ
cana-1549	108	12	𝑎1	𝑎1	ADV
cana-1549	108	13	,	,	PUNCT
cana-1549	108	14	𝑥2	𝑥2	NOUN
cana-1549	108	15	=	=	SYM
cana-1549	108	16	𝑎0𝑎1𝑎2	𝑎0𝑎1𝑎2	PROPN
cana-1549	108	17	𝑎1𝑎2	𝑎1𝑎2	PROPN
cana-1549	109	1	+	+	CCONJ
cana-1549	109	2	𝑎0𝑎2	𝑎0𝑎2	ADJ
cana-1549	109	3	+	+	X
cana-1549	109	4	𝑎0𝑎1	𝑎0𝑎1	X
cana-1549	109	5	,	,	PUNCT
cana-1549	109	6	⋯	⋯	PROPN
cana-1549	109	7	,	,	PUNCT
cana-1549	109	8	𝑥𝑛	𝑥𝑛	PROPN
cana-1549	109	9	=	=	SYM
cana-1549	109	10	𝑎0𝑎1⋯𝑎𝑛	𝑎0𝑎1⋯𝑎𝑛	PROPN
cana-1549	109	11	𝑎1𝑎2⋯𝑎𝑛	𝑎1𝑎2⋯𝑎𝑛	PROPN
cana-1549	109	12	+	+	CCONJ
cana-1549	109	13	𝑎0𝑎2⋯𝑎𝑛	𝑎0𝑎2⋯𝑎𝑛	PROPN
cana-1549	109	14	+	+	CCONJ
cana-1549	109	15	𝑎0𝑎2⋯𝑎𝑛−1	𝑎0𝑎2⋯𝑎𝑛−1	PROPN
cana-1549	109	16	,	,	PUNCT
cana-1549	109	17	so	so	ADV
cana-1549	109	18	𝑥𝑘	𝑥𝑘	X
cana-1549	109	19	>	>	X
cana-1549	109	20	1	1	NUM
cana-1549	109	21	for	for	ADP
cana-1549	109	22	each	each	DET
cana-1549	109	23	𝑘	𝑘	NOUN
cana-1549	109	24	as	as	ADP
cana-1549	109	25	[	[	X
cana-1549	109	26	𝑎0	𝑎0	PROPN
cana-1549	109	27	]	]	X
cana-1549	109	28	≥	≥	NOUN
cana-1549	109	29	𝑛	𝑛	PRON
cana-1549	110	1	+	+	NOUN
cana-1549	110	2	1	1	X
cana-1549	110	3	.	.	PUNCT
cana-1549	111	1	the	the	DET
cana-1549	111	2	logarithmic	logarithmic	ADJ
cana-1549	111	3	harmonic	harmonic	ADJ
cana-1549	111	4	pmf	pmf	NOUN
cana-1549	111	5	of	of	ADP
cana-1549	111	6	𝑋	𝑋	PROPN
cana-1549	111	7	is	be	AUX
cana-1549	111	8	defined	define	VERB
cana-1549	111	9	by	by	ADP
cana-1549	111	10	𝑝(𝑥	𝑝(𝑥	PROPN
cana-1549	111	11	;	;	PUNCT
cana-1549	111	12	𝑛	𝑛	X
cana-1549	111	13	)	)	PUNCT
cana-1549	111	14	=	=	PRON
cana-1549	111	15	{	{	PUNCT
cana-1549	111	16	𝐶	𝐶	PROPN
cana-1549	111	17	(	(	PUNCT
cana-1549	111	18	𝑛	𝑛	PROPN
cana-1549	111	19	+	+	NOUN
cana-1549	111	20	1	1	X
cana-1549	111	21	)	)	PUNCT
cana-1549	111	22	ln	ln	NOUN
cana-1549	112	1	𝑥	𝑥	X
cana-1549	112	2	,	,	PUNCT
cana-1549	112	3	if	if	SCONJ
cana-1549	112	4	𝑥	𝑥	PRON
cana-1549	112	5	=	=	SYM
cana-1549	112	6	𝑥0	𝑥0	PROPN
cana-1549	112	7	,	,	PUNCT
cana-1549	112	8	𝑥1	𝑥1	NOUN
cana-1549	112	9	,	,	PUNCT
cana-1549	112	10	⋯	⋯	PROPN
cana-1549	112	11	,	,	PUNCT
cana-1549	112	12	𝑥𝑛	𝑥𝑛	PROPN
cana-1549	112	13	;	;	PUNCT
cana-1549	112	14	0	0	NUM
cana-1549	112	15	,	,	PUNCT
cana-1549	112	16	otherwise	otherwise	ADV
cana-1549	112	17	,	,	PUNCT
cana-1549	112	18	for	for	ADP
cana-1549	112	19	some	some	DET
cana-1549	112	20	𝐶	𝐶	PROPN
cana-1549	112	21	>	>	X
cana-1549	112	22	0	0	PUNCT
cana-1549	113	1	satisfying	satisfy	VERB
cana-1549	113	2	0	0	NUM
cana-1549	113	3	≤	≤	NOUN
cana-1549	113	4	𝑝(𝑥	𝑝(𝑥	ADP
cana-1549	113	5	;	;	PUNCT
cana-1549	113	6	𝑛	𝑛	X
cana-1549	113	7	)	)	PUNCT
cana-1549	113	8	≤	≤	NUM
cana-1549	113	9	1	1	NUM
cana-1549	113	10	and	and	CCONJ
cana-1549	113	11	∑	∑	PUNCT
cana-1549	113	12	𝑥=0	𝑥=0	PUNCT
cana-1549	113	13	𝑛	𝑛	PROPN
cana-1549	113	14	𝑝(𝑥	𝑝(𝑥	NOUN
cana-1549	113	15	;	;	PUNCT
cana-1549	113	16	𝑛	𝑛	X
cana-1549	113	17	)	)	PUNCT
cana-1549	113	18	=	=	SYM
cana-1549	113	19	1	1	NUM
cana-1549	113	20	where	where	SCONJ
cana-1549	113	21	ln	ln	ADV
cana-1549	113	22	𝑥𝑘	𝑥𝑘	X
cana-1549	114	1	=	=	X
cana-1549	114	2	ln	ln	PROPN
cana-1549	114	3	(	(	PUNCT
cana-1549	114	4	𝑎0𝑎1⋯𝑎𝑛	𝑎0𝑎1⋯𝑎𝑛	PROPN
cana-1549	114	5	(	(	PUNCT
cana-1549	114	6	∑	∑	PROPN
cana-1549	114	7	𝑖=0	𝑖=0	PROPN
cana-1549	114	8	𝑘	𝑘	PRON
cana-1549	114	9	𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘	𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘	NOUN
cana-1549	114	10	)	)	PUNCT
cana-1549	114	11	)	)	PUNCT
cana-1549	115	1	=	=	NOUN
cana-1549	115	2	∑	∑	X
cana-1549	115	3	𝑘	𝑘	PRON
cana-1549	115	4	𝑖=0	𝑖=0	PROPN
cana-1549	115	5	ln	ln	NOUN
cana-1549	116	1	𝑎𝑖	𝑎𝑖	ADP
cana-1549	116	2	−	−	PROPN
cana-1549	116	3	ln	ln	ADJ
cana-1549	116	4	(	(	PUNCT
cana-1549	116	5	∑	∑	PROPN
cana-1549	116	6	𝑖=0	𝑖=0	PROPN
cana-1549	116	7	𝑘	𝑘	PRON
cana-1549	116	8	𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘	𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘	NOUN
cana-1549	116	9	)	)	PUNCT
cana-1549	116	10	.	.	PUNCT
cana-1549	117	1	for	for	ADP
cana-1549	117	2	for	for	ADP
cana-1549	117	3	𝑘	𝑘	NOUN
cana-1549	117	4	=	=	SYM
cana-1549	117	5	0,1,⋯	0,1,⋯	PUNCT
cana-1549	117	6	,	,	PUNCT
cana-1549	117	7	𝑛.	𝑛.	NOUN
cana-1549	117	8	1	1	NUM
cana-1549	117	9	=	=	SYM
cana-1549	117	10	∑	∑	PART
cana-1549	117	11	𝑛	𝑛	PRON
cana-1549	117	12	𝑘=0	𝑘=0	PROPN
cana-1549	117	13	𝑝(𝑦𝑘	𝑝(𝑦𝑘	NOUN
cana-1549	117	14	;	;	PUNCT
cana-1549	117	15	𝑛	𝑛	X
cana-1549	117	16	)	)	PUNCT
cana-1549	117	17	=	=	SYM
cana-1549	117	18	𝐶	𝐶	PROPN
cana-1549	117	19	(	(	PUNCT
cana-1549	117	20	𝑛	𝑛	PROPN
cana-1549	117	21	+	+	NOUN
cana-1549	117	22	1	1	NUM
cana-1549	117	23	)	)	PUNCT
cana-1549	117	24	∑	∑	ADP
cana-1549	117	25	𝑛	𝑛	PRON
cana-1549	117	26	𝑘=0	𝑘=0	ADP
cana-1549	117	27	1	1	NUM
cana-1549	117	28	ln	ln	NOUN
cana-1549	117	29	𝑥𝑘	𝑥𝑘	X
cana-1549	117	30	=	=	SYM
cana-1549	117	31	𝐶	𝐶	PROPN
cana-1549	117	32	(	(	PUNCT
cana-1549	117	33	𝑛	𝑛	PROPN
cana-1549	117	34	+	+	NOUN
cana-1549	117	35	1	1	NUM
cana-1549	117	36	)	)	PUNCT
cana-1549	117	37	[	[	PUNCT
cana-1549	117	38	1	1	NUM
cana-1549	117	39	ln	ln	ADJ
cana-1549	117	40	𝑥0	𝑥0	NOUN
cana-1549	117	41	+	+	CCONJ
cana-1549	117	42	1	1	NUM
cana-1549	117	43	ln	ln	NOUN
cana-1549	117	44	𝑥1	𝑥1	NOUN
cana-1549	117	45	+	+	CCONJ
cana-1549	117	46	1	1	NUM
cana-1549	117	47	ln	ln	ADJ
cana-1549	117	48	𝑥2	𝑥2	NOUN
cana-1549	117	49	+	+	NOUN
cana-1549	117	50	⋯+	⋯+	PROPN
cana-1549	117	51	1	1	NUM
cana-1549	117	52	ln	ln	NOUN
cana-1549	117	53	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-1549	117	54	+	+	CCONJ
cana-1549	117	55	1	1	NUM
cana-1549	117	56	ln	ln	NOUN
cana-1549	117	57	𝑥𝑛	𝑥𝑛	NOUN
cana-1549	117	58	]	]	PUNCT
cana-1549	117	59	communications	communication	NOUN
cana-1549	117	60	on	on	ADP
cana-1549	117	61	applied	apply	VERB
cana-1549	117	62	nonlinear	nonlinear	ADJ
cana-1549	117	63	analysis	analysis	NOUN
cana-1549	117	64	issn	issn	NOUN
cana-1549	117	65	:	:	PUNCT
cana-1549	117	66	1074	1074	NUM
cana-1549	117	67	-	-	PUNCT
cana-1549	117	68	133x	133x	NUM
cana-1549	117	69	vol	vol	NOUN
cana-1549	117	70	31	31	NUM
cana-1549	117	71	no	no	NOUN
cana-1549	117	72	.	.	PUNCT
cana-1549	118	1	8s	8s	PROPN
cana-1549	118	2	(	(	PUNCT
cana-1549	118	3	2024	2024	NUM
cana-1549	118	4	)	)	PUNCT
cana-1549	118	5	571	571	NUM
cana-1549	118	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	118	7	⇒	⇒	PROPN
cana-1549	118	8	𝐶	𝐶	PROPN
cana-1549	118	9	=	=	PUNCT
cana-1549	118	10	(	(	PUNCT
cana-1549	118	11	𝑛	𝑛	PROPN
cana-1549	118	12	+	+	NOUN
cana-1549	118	13	1	1	NUM
cana-1549	118	14	)	)	PUNCT
cana-1549	118	15	[	[	PUNCT
cana-1549	118	16	1	1	NUM
cana-1549	118	17	ln	ln	ADJ
cana-1549	118	18	𝑥0	𝑥0	NOUN
cana-1549	118	19	+	+	CCONJ
cana-1549	118	20	1	1	NUM
cana-1549	118	21	ln	ln	NOUN
cana-1549	118	22	𝑥1	𝑥1	NOUN
cana-1549	118	23	+	+	CCONJ
cana-1549	118	24	1	1	NUM
cana-1549	118	25	ln	ln	ADJ
cana-1549	118	26	𝑥2	𝑥2	NOUN
cana-1549	118	27	+	+	NOUN
cana-1549	118	28	⋯+	⋯+	PROPN
cana-1549	118	29	1	1	NUM
cana-1549	118	30	ln	ln	NOUN
cana-1549	118	31	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-1549	118	32	+	+	CCONJ
cana-1549	118	33	1	1	NUM
cana-1549	118	34	ln	ln	NOUN
cana-1549	118	35	𝑥𝑛	𝑥𝑛	NOUN
cana-1549	118	36	]	]	PUNCT
cana-1549	118	37	−1	−1	NOUN
cana-1549	118	38	.	.	PUNCT
cana-1549	119	1	thus	thus	ADV
cana-1549	119	2	the	the	DET
cana-1549	119	3	logarithmic	logarithmic	ADJ
cana-1549	119	4	harmonic	harmonic	ADJ
cana-1549	119	5	pmf	pmf	NOUN
cana-1549	119	6	of	of	ADP
cana-1549	119	7	𝑋	𝑋	PROPN
cana-1549	119	8	is	be	AUX
cana-1549	119	9	𝑝(𝑥	𝑝(𝑥	NUM
cana-1549	119	10	)	)	PUNCT
cana-1549	119	11	=	=	PUNCT
cana-1549	119	12	𝑙ℎ(𝑥	𝑙ℎ(𝑥	NOUN
cana-1549	119	13	;	;	PUNCT
cana-1549	119	14	𝑛	𝑛	X
cana-1549	119	15	)	)	PUNCT
cana-1549	119	16	=	=	NOUN
cana-1549	119	17	{	{	PUNCT
cana-1549	119	18	𝑘	𝑘	X
cana-1549	119	19	ln	ln	NOUN
cana-1549	119	20	𝑥	𝑥	X
cana-1549	119	21	,	,	PUNCT
cana-1549	119	22	𝑖𝑓	𝑖𝑓	ADP
cana-1549	119	23	𝑥	𝑥	PRON
cana-1549	119	24	=	=	SYM
cana-1549	119	25	𝑥0	𝑥0	PROPN
cana-1549	119	26	,	,	PUNCT
cana-1549	119	27	𝑥1	𝑥1	NOUN
cana-1549	119	28	,	,	PUNCT
cana-1549	119	29	⋯	⋯	PROPN
cana-1549	119	30	,	,	PUNCT
cana-1549	119	31	𝑥𝑛	𝑥𝑛	PROPN
cana-1549	119	32	;	;	PUNCT
cana-1549	119	33	0	0	NUM
cana-1549	119	34	,	,	PUNCT
cana-1549	119	35	otherwise	otherwise	ADV
cana-1549	119	36	,	,	PUNCT
cana-1549	119	37	where	where	SCONJ
cana-1549	119	38	𝑘	𝑘	X
cana-1549	119	39	=	=	PUNCT
cana-1549	119	40	[	[	PUNCT
cana-1549	119	41	1	1	NUM
cana-1549	119	42	ln	ln	ADJ
cana-1549	119	43	𝑥0	𝑥0	NOUN
cana-1549	119	44	+	+	CCONJ
cana-1549	119	45	1	1	NUM
cana-1549	119	46	ln	ln	NOUN
cana-1549	119	47	𝑥1	𝑥1	NOUN
cana-1549	119	48	+	+	CCONJ
cana-1549	119	49	1	1	NUM
cana-1549	119	50	ln	ln	ADJ
cana-1549	119	51	𝑥2	𝑥2	NOUN
cana-1549	119	52	+	+	NOUN
cana-1549	119	53	⋯+	⋯+	PROPN
cana-1549	119	54	1	1	NUM
cana-1549	119	55	ln	ln	NOUN
cana-1549	119	56	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-1549	119	57	+	+	CCONJ
cana-1549	119	58	1	1	NUM
cana-1549	119	59	ln	ln	NOUN
cana-1549	119	60	𝑥𝑛	𝑥𝑛	NOUN
cana-1549	119	61	]	]	PUNCT
cana-1549	119	62	−1	−1	NOUN
cana-1549	119	63	,	,	PUNCT
cana-1549	119	64	ln	ln	ADJ
cana-1549	119	65	𝑥0	𝑥0	NOUN
cana-1549	119	66	=	=	SYM
cana-1549	119	67	ln	ln	PROPN
cana-1549	119	68	𝑎0	𝑎0	PROPN
cana-1549	119	69	ln	ln	PROPN
cana-1549	119	70	𝑥1	𝑥1	NOUN
cana-1549	119	71	=	=	SYM
cana-1549	119	72	ln	ln	PROPN
cana-1549	119	73	𝑎0	𝑎0	PROPN
cana-1549	119	74	+	+	CCONJ
cana-1549	119	75	ln𝑎1	ln𝑎1	NOUN
cana-1549	119	76	−	−	PROPN
cana-1549	119	77	ln(𝑎0	ln(𝑎0	VERB
cana-1549	119	78	+	+	CCONJ
cana-1549	119	79	𝑎1	𝑎1	ADJ
cana-1549	119	80	)	)	PUNCT
cana-1549	119	81	ln	ln	ADJ
cana-1549	119	82	𝑥2	𝑥2	NOUN
cana-1549	119	83	=	=	SYM
cana-1549	119	84	ln	ln	NOUN
cana-1549	119	85	𝑎0	𝑎0	PROPN
cana-1549	120	1	+	+	CCONJ
cana-1549	120	2	ln	ln	ADJ
cana-1549	120	3	𝑎1	𝑎1	NOUN
cana-1549	120	4	+	+	CCONJ
cana-1549	120	5	ln	ln	ADJ
cana-1549	120	6	𝑎2	𝑎2	NOUN
cana-1549	120	7	−	−	PROPN
cana-1549	120	8	ln(𝑎0𝑎1	ln(𝑎0𝑎1	X
cana-1549	121	1	+	+	CCONJ
cana-1549	121	2	𝑎0𝑎2	𝑎0𝑎2	ADJ
cana-1549	121	3	+	+	CCONJ
cana-1549	121	4	𝑎1𝑎2	𝑎1𝑎2	X
cana-1549	121	5	)	)	PUNCT
cana-1549	121	6	and	and	CCONJ
cana-1549	121	7	so	so	ADV
cana-1549	121	8	on	on	ADV
cana-1549	121	9	,	,	PUNCT
cana-1549	121	10	ln	ln	ADJ
cana-1549	121	11	𝑥𝑛	𝑥𝑛	VERB
cana-1549	121	12	=	=	NOUN
cana-1549	121	13	∑	∑	PART
cana-1549	121	14	𝑛	𝑛	PROPN
cana-1549	121	15	𝑘=0	𝑘=0	ADP
cana-1549	121	16	ln	ln	ADV
cana-1549	121	17	𝑎𝑘	𝑎𝑘	ADP
cana-1549	121	18	−	−	PROPN
cana-1549	121	19	ln(∑	ln(∑	NOUN
cana-1549	121	20	𝑘=0	𝑘=0	VERB
cana-1549	121	21	𝑛	𝑛	DET
cana-1549	121	22	𝑎0𝑎1⋯𝑎𝑘−1𝑎𝑘+1⋯𝑎𝑛	𝑎0𝑎1⋯𝑎𝑘−1𝑎𝑘+1⋯𝑎𝑛	NOUN
cana-1549	121	23	)	)	PUNCT
cana-1549	121	24	.	.	PUNCT
cana-1549	122	1	provided	provide	VERB
cana-1549	122	2	𝑥𝑘	𝑥𝑘	NOUN
cana-1549	122	3	>	>	X
cana-1549	122	4	1	1	NUM
cana-1549	122	5	for	for	ADP
cana-1549	122	6	each	each	PRON
cana-1549	122	7	𝑘	𝑘	NOUN
cana-1549	122	8	=	=	PUNCT
cana-1549	122	9	0,1,2⋯	0,1,2⋯	PRON
cana-1549	122	10	and	and	CCONJ
cana-1549	122	11	cdf	cdf	PROPN
cana-1549	122	12	of	of	ADP
cana-1549	122	13	𝑋	𝑋	PROPN
cana-1549	122	14	is	be	AUX
cana-1549	122	15	𝐿𝐻(𝑥	𝐿𝐻(𝑥	ADJ
cana-1549	122	16	;	;	PUNCT
cana-1549	122	17	𝑛	𝑛	X
cana-1549	122	18	)	)	PUNCT
cana-1549	122	19	=	=	X
cana-1549	122	20	𝑃(𝑋	𝑃(𝑋	PROPN
cana-1549	122	21	≤	≤	NOUN
cana-1549	122	22	𝑥	𝑥	X
cana-1549	122	23	)	)	PUNCT
cana-1549	122	24	for	for	ADP
cana-1549	122	25	all	all	DET
cana-1549	122	26	𝑥	𝑥	PRON
cana-1549	122	27	=	=	SYM
cana-1549	122	28	𝑥0	𝑥0	PROPN
cana-1549	122	29	,	,	PUNCT
cana-1549	122	30	𝑥1	𝑥1	NOUN
cana-1549	122	31	,	,	PUNCT
cana-1549	122	32	⋯	⋯	PROPN
cana-1549	122	33	,	,	PUNCT
cana-1549	122	34	𝑥𝑛	𝑥𝑛	AUX
cana-1549	122	35	given	give	VERB
cana-1549	122	36	by	by	ADP
cana-1549	122	37	𝐿𝐻(𝑥0	𝐿𝐻(𝑥0	ADP
cana-1549	122	38	;	;	PUNCT
cana-1549	122	39	𝑛	𝑛	X
cana-1549	122	40	)	)	PUNCT
cana-1549	122	41	=	=	SYM
cana-1549	122	42	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	122	43	;	;	PUNCT
cana-1549	122	44	𝑛	𝑛	X
cana-1549	122	45	)	)	PUNCT
cana-1549	122	46	=	=	PUNCT
cana-1549	122	47	𝑘	𝑘	PRON
cana-1549	122	48	ln	ln	ADJ
cana-1549	122	49	𝑥0	𝑥0	NOUN
cana-1549	122	50	,	,	PUNCT
cana-1549	122	51	𝐿𝐻(𝑥1	𝐿𝐻(𝑥1	PROPN
cana-1549	122	52	;	;	PUNCT
cana-1549	122	53	𝑛	𝑛	X
cana-1549	122	54	)	)	PUNCT
cana-1549	122	55	=	=	SYM
cana-1549	122	56	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	122	57	;	;	PUNCT
cana-1549	122	58	𝑛	𝑛	X
cana-1549	122	59	)	)	PUNCT
cana-1549	122	60	+	+	X
cana-1549	122	61	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	122	62	;	;	PUNCT
cana-1549	122	63	𝑛	𝑛	X
cana-1549	122	64	)	)	PUNCT
cana-1549	122	65	=	=	SYM
cana-1549	122	66	𝑘	𝑘	X
cana-1549	122	67	[	[	PUNCT
cana-1549	122	68	1	1	NUM
cana-1549	122	69	ln	ln	ADJ
cana-1549	122	70	𝑥0	𝑥0	NOUN
cana-1549	122	71	+	+	CCONJ
cana-1549	122	72	1	1	NUM
cana-1549	122	73	ln	ln	NOUN
cana-1549	122	74	𝑥1	𝑥1	NOUN
cana-1549	122	75	]	]	PUNCT
cana-1549	122	76	𝐿𝐻(𝑥2	𝐿𝐻(𝑥2	PROPN
cana-1549	122	77	;	;	PUNCT
cana-1549	122	78	𝑛	𝑛	X
cana-1549	122	79	)	)	PUNCT
cana-1549	122	80	=	=	SYM
cana-1549	122	81	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	122	82	;	;	PUNCT
cana-1549	122	83	𝑛	𝑛	X
cana-1549	122	84	)	)	PUNCT
cana-1549	122	85	+	+	X
cana-1549	122	86	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	122	87	;	;	PUNCT
cana-1549	122	88	𝑛	𝑛	X
cana-1549	122	89	)	)	PUNCT
cana-1549	122	90	+	+	CCONJ
cana-1549	122	91	𝑝(𝑥2	𝑝(𝑥2	NOUN
cana-1549	122	92	;	;	PUNCT
cana-1549	122	93	𝑛	𝑛	X
cana-1549	122	94	)	)	PUNCT
cana-1549	123	1	=	=	SYM
cana-1549	123	2	𝑘	𝑘	X
cana-1549	123	3	[	[	PUNCT
cana-1549	123	4	1	1	NUM
cana-1549	123	5	ln	ln	ADJ
cana-1549	123	6	𝑥0	𝑥0	NOUN
cana-1549	123	7	+	+	CCONJ
cana-1549	123	8	1	1	NUM
cana-1549	123	9	ln	ln	NOUN
cana-1549	123	10	𝑥1	𝑥1	NOUN
cana-1549	123	11	+	+	CCONJ
cana-1549	123	12	1	1	NUM
cana-1549	123	13	ln	ln	ADJ
cana-1549	123	14	𝑥2	𝑥2	NOUN
cana-1549	123	15	]	]	PUNCT
cana-1549	123	16	𝐿𝐻(𝑥3	𝐿𝐻(𝑥3	NOUN
cana-1549	123	17	;	;	PUNCT
cana-1549	123	18	𝑛	𝑛	X
cana-1549	123	19	)	)	PUNCT
cana-1549	124	1	=	=	SYM
cana-1549	124	2	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	124	3	;	;	PUNCT
cana-1549	124	4	𝑛	𝑛	X
cana-1549	124	5	)	)	PUNCT
cana-1549	124	6	+	+	X
cana-1549	124	7	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	124	8	;	;	PUNCT
cana-1549	124	9	𝑛	𝑛	X
cana-1549	124	10	)	)	PUNCT
cana-1549	124	11	+	+	CCONJ
cana-1549	124	12	𝑝(𝑥2	𝑝(𝑥2	NOUN
cana-1549	124	13	;	;	PUNCT
cana-1549	124	14	𝑛	𝑛	X
cana-1549	124	15	)	)	PUNCT
cana-1549	124	16	+	+	CCONJ
cana-1549	124	17	𝑝(𝑥3	𝑝(𝑥3	NOUN
cana-1549	124	18	;	;	PUNCT
cana-1549	124	19	𝑛	𝑛	X
cana-1549	124	20	)	)	PUNCT
cana-1549	124	21	=	=	SYM
cana-1549	124	22	𝑘	𝑘	X
cana-1549	124	23	[	[	PUNCT
cana-1549	124	24	1	1	NUM
cana-1549	124	25	ln	ln	ADJ
cana-1549	124	26	𝑥0	𝑥0	NOUN
cana-1549	124	27	+	+	CCONJ
cana-1549	124	28	1	1	NUM
cana-1549	124	29	ln	ln	NOUN
cana-1549	124	30	𝑥1	𝑥1	NOUN
cana-1549	124	31	+	+	CCONJ
cana-1549	124	32	1	1	NUM
cana-1549	124	33	ln	ln	ADJ
cana-1549	124	34	𝑥2	𝑥2	NOUN
cana-1549	124	35	+	+	CCONJ
cana-1549	124	36	1	1	NUM
cana-1549	124	37	ln	ln	NOUN
cana-1549	124	38	𝑥3	𝑥3	NOUN
cana-1549	124	39	]	]	PUNCT
cana-1549	124	40	and	and	CCONJ
cana-1549	124	41	so	so	ADV
cana-1549	124	42	on	on	ADV
cana-1549	124	43	.	.	PUNCT
cana-1549	125	1	example	example	NOUN
cana-1549	125	2	4.1	4.1	NUM
cana-1549	125	3	let	let	VERB
cana-1549	125	4	𝑛	𝑛	PRON
cana-1549	125	5	=	=	SYM
cana-1549	125	6	4	4	X
cana-1549	125	7	.	.	X
cana-1549	126	1	for	for	ADP
cana-1549	126	2	𝑎0	𝑎0	ADV
cana-1549	126	3	=	=	SYM
cana-1549	126	4	5	5	NUM
cana-1549	126	5	,	,	PUNCT
cana-1549	126	6	𝑎1	𝑎1	NOUN
cana-1549	126	7	=	=	NOUN
cana-1549	126	8	5.1	5.1	NUM
cana-1549	126	9	,	,	PUNCT
cana-1549	126	10	𝑎2	𝑎2	NOUN
cana-1549	126	11	=	=	SYM
cana-1549	126	12	5.2	5.2	NUM
cana-1549	126	13	,	,	PUNCT
cana-1549	126	14	𝑎3	𝑎3	PROPN
cana-1549	126	15	=	=	SYM
cana-1549	126	16	5.3	5.3	NUM
cana-1549	126	17	,	,	PUNCT
cana-1549	126	18	𝑎4	𝑎4	PROPN
cana-1549	126	19	=	=	SYM
cana-1549	126	20	5.4	5.4	NUM
cana-1549	126	21	,	,	PUNCT
cana-1549	126	22	we	we	PRON
cana-1549	126	23	have	have	VERB
cana-1549	126	24	𝑥0	𝑥0	NOUN
cana-1549	126	25	=	=	SYM
cana-1549	126	26	5	5	NUM
cana-1549	126	27	,	,	PUNCT
cana-1549	126	28	𝑥1	𝑥1	NOUN
cana-1549	126	29	=	=	SYM
cana-1549	126	30	2.52475	2.52475	NUM
cana-1549	126	31	,	,	PUNCT
cana-1549	126	32	𝑥2	𝑥2	NOUN
cana-1549	126	33	=	=	SYM
cana-1549	126	34	1.69956	1.69956	NUM
cana-1549	126	35	,	,	PUNCT
cana-1549	126	36	𝑥3	𝑥3	NOUN
cana-1549	126	37	=	=	SYM
cana-1549	126	38	1.28689	1.28689	NUM
cana-1549	126	39	,	,	PUNCT
cana-1549	126	40	𝑥4	𝑥4	NOUN
cana-1549	126	41	=	=	SYM
cana-1549	126	42	1.03923	1.03923	NUM
cana-1549	126	43	.	.	PUNCT
cana-1549	127	1	therefore	therefore	ADV
cana-1549	127	2	𝑙ℎ(𝑥	𝑙ℎ(𝑥	NUM
cana-1549	127	3	;	;	PUNCT
cana-1549	127	4	4	4	X
cana-1549	127	5	)	)	PUNCT
cana-1549	127	6	=	=	PRON
cana-1549	127	7	{	{	PUNCT
cana-1549	127	8	𝑘	𝑘	X
cana-1549	127	9	ln	ln	NOUN
cana-1549	127	10	𝑥	𝑥	X
cana-1549	127	11	,	,	PUNCT
cana-1549	127	12	𝑖𝑓	𝑖𝑓	ADP
cana-1549	127	13	𝑥	𝑥	PRON
cana-1549	127	14	=	=	SYM
cana-1549	127	15	𝑥0	𝑥0	PROPN
cana-1549	127	16	,	,	PUNCT
cana-1549	127	17	𝑥1	𝑥1	NOUN
cana-1549	127	18	,	,	PUNCT
cana-1549	127	19	⋯	⋯	PROPN
cana-1549	127	20	,	,	PUNCT
cana-1549	127	21	𝑥𝑛	𝑥𝑛	PROPN
cana-1549	127	22	;	;	PUNCT
cana-1549	127	23	0	0	NUM
cana-1549	127	24	,	,	PUNCT
cana-1549	127	25	otherwise	otherwise	ADV
cana-1549	127	26	,	,	PUNCT
cana-1549	127	27	where	where	SCONJ
cana-1549	127	28	communications	communication	NOUN
cana-1549	127	29	on	on	ADP
cana-1549	127	30	applied	apply	VERB
cana-1549	127	31	nonlinear	nonlinear	ADJ
cana-1549	127	32	analysis	analysis	NOUN
cana-1549	127	33	issn	issn	NOUN
cana-1549	127	34	:	:	PUNCT
cana-1549	127	35	1074	1074	NUM
cana-1549	127	36	-	-	PUNCT
cana-1549	127	37	133x	133x	NUM
cana-1549	127	38	vol	vol	NOUN
cana-1549	127	39	31	31	NUM
cana-1549	127	40	no	no	NOUN
cana-1549	127	41	.	.	PUNCT
cana-1549	128	1	8s	8s	PROPN
cana-1549	128	2	(	(	PUNCT
cana-1549	128	3	2024	2024	NUM
cana-1549	128	4	)	)	PUNCT
cana-1549	128	5	572	572	NUM
cana-1549	128	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	128	7	𝑘	𝑘	X
cana-1549	128	8	=	=	PUNCT
cana-1549	128	9	[	[	PUNCT
cana-1549	128	10	1	1	NUM
cana-1549	128	11	ln	ln	ADJ
cana-1549	128	12	𝑥0	𝑥0	NOUN
cana-1549	128	13	+	+	CCONJ
cana-1549	128	14	1	1	NUM
cana-1549	128	15	ln	ln	NOUN
cana-1549	128	16	𝑥1	𝑥1	NOUN
cana-1549	128	17	+	+	CCONJ
cana-1549	128	18	1	1	NUM
cana-1549	128	19	ln	ln	ADJ
cana-1549	128	20	𝑥2	𝑥2	NOUN
cana-1549	128	21	+	+	CCONJ
cana-1549	128	22	1	1	NUM
cana-1549	128	23	ln	ln	NOUN
cana-1549	128	24	𝑥3	𝑥3	NOUN
cana-1549	128	25	+	+	CCONJ
cana-1549	128	26	1	1	NUM
cana-1549	128	27	ln	ln	ADJ
cana-1549	128	28	𝑥4	𝑥4	NOUN
cana-1549	128	29	]	]	PUNCT
cana-1549	128	30	−1	−1	NOUN
cana-1549	128	31	=	=	SYM
cana-1549	128	32	0.0298	0.0298	NUM
cana-1549	128	33	which	which	PRON
cana-1549	128	34	is	be	AUX
cana-1549	128	35	approximated	approximate	VERB
cana-1549	128	36	to	to	ADP
cana-1549	128	37	4	4	NUM
cana-1549	128	38	significant	significant	ADJ
cana-1549	128	39	digit	digit	NOUN
cana-1549	128	40	form	form	NOUN
cana-1549	128	41	.	.	PUNCT
cana-1549	129	1	thus	thus	ADV
cana-1549	129	2	ln	ln	ADJ
cana-1549	129	3	𝑥0	𝑥0	NOUN
cana-1549	129	4	=	=	PUNCT
cana-1549	129	5	1.60944	1.60944	NUM
cana-1549	129	6	;	;	PUNCT
cana-1549	129	7	ln	ln	PROPN
cana-1549	129	8	𝑥1	𝑥1	NOUN
cana-1549	129	9	=	=	PROPN
cana-1549	129	10	0.92614	0.92614	NUM
cana-1549	129	11	;	;	PUNCT
cana-1549	129	12	ln	ln	ADJ
cana-1549	129	13	𝑥2	𝑥2	NOUN
cana-1549	129	14	=	=	NOUN
cana-1549	129	15	0.53037	0.53037	NUM
cana-1549	129	16	;	;	PUNCT
cana-1549	129	17	ln	ln	ADJ
cana-1549	129	18	𝑥3	𝑥3	NOUN
cana-1549	129	19	=	=	SYM
cana-1549	129	20	0.25223	0.25223	NUM
cana-1549	129	21	;	;	PUNCT
cana-1549	129	22	ln	ln	ADJ
cana-1549	129	23	𝑥4	𝑥4	NOUN
cana-1549	129	24	=	=	SYM
cana-1549	129	25	0.03848	0.03848	NUM
cana-1549	129	26	,	,	PUNCT
cana-1549	129	27	implying	imply	VERB
cana-1549	129	28	the	the	DET
cana-1549	129	29	pmf	pmf	NOUN
cana-1549	129	30	of	of	ADP
cana-1549	129	31	𝑋	𝑋	PROPN
cana-1549	129	32	is	be	AUX
cana-1549	129	33	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	129	34	;	;	PUNCT
cana-1549	129	35	5	5	X
cana-1549	129	36	)	)	PUNCT
cana-1549	129	37	=	=	PUNCT
cana-1549	129	38	𝑘	𝑘	PRON
cana-1549	129	39	ln	ln	ADJ
cana-1549	129	40	𝑥0	𝑥0	NOUN
cana-1549	129	41	=	=	SYM
cana-1549	129	42	0.01852	0.01852	NUM
cana-1549	129	43	;	;	PUNCT
cana-1549	129	44	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	129	45	;	;	PUNCT
cana-1549	129	46	5	5	X
cana-1549	129	47	)	)	PUNCT
cana-1549	129	48	=	=	PUNCT
cana-1549	129	49	𝑘	𝑘	PROPN
cana-1549	129	50	ln	ln	PROPN
cana-1549	129	51	𝑥1	𝑥1	PROPN
cana-1549	129	52	=	=	PROPN
cana-1549	129	53	0.03218	0.03218	NUM
cana-1549	129	54	;	;	PUNCT
cana-1549	129	55	𝑝(𝑥2	𝑝(𝑥2	PROPN
cana-1549	129	56	;	;	PUNCT
cana-1549	129	57	5	5	X
cana-1549	129	58	)	)	PUNCT
cana-1549	129	59	=	=	PUNCT
cana-1549	129	60	𝑘	𝑘	PRON
cana-1549	129	61	ln	ln	ADJ
cana-1549	129	62	𝑥2	𝑥2	NOUN
cana-1549	129	63	=	=	SYM
cana-1549	129	64	0.05619	0.05619	NUM
cana-1549	129	65	𝑝(𝑥3	𝑝(𝑥3	NOUN
cana-1549	129	66	;	;	PUNCT
cana-1549	129	67	5	5	X
cana-1549	129	68	)	)	PUNCT
cana-1549	129	69	=	=	PUNCT
cana-1549	129	70	𝑘	𝑘	PRON
cana-1549	129	71	ln	ln	ADJ
cana-1549	129	72	𝑥3	𝑥3	NOUN
cana-1549	129	73	=	=	SYM
cana-1549	129	74	0.11815	0.11815	NUM
cana-1549	129	75	;	;	PUNCT
cana-1549	129	76	𝑝(𝑥4	𝑝(𝑥4	NOUN
cana-1549	129	77	;	;	PUNCT
cana-1549	129	78	5	5	X
cana-1549	129	79	)	)	PUNCT
cana-1549	129	80	=	=	PUNCT
cana-1549	129	81	𝑘	𝑘	PRON
cana-1549	129	82	ln	ln	ADJ
cana-1549	129	83	𝑥4	𝑥4	NOUN
cana-1549	129	84	=	=	SYM
cana-1549	129	85	0.77443	0.77443	NUM
cana-1549	129	86	.	.	PUNCT
cana-1549	130	1	and	and	CCONJ
cana-1549	130	2	cdf	cdf	PROPN
cana-1549	130	3	of	of	ADP
cana-1549	130	4	𝑋	𝑋	PROPN
cana-1549	130	5	is	be	AUX
cana-1549	130	6	𝐿𝐻(𝑥	𝐿𝐻(𝑥	ADJ
cana-1549	130	7	;	;	PUNCT
cana-1549	130	8	𝑛	𝑛	X
cana-1549	130	9	)	)	PUNCT
cana-1549	131	1	=	=	X
cana-1549	131	2	𝑃(𝑋	𝑃(𝑋	PROPN
cana-1549	131	3	≤	≤	NOUN
cana-1549	131	4	𝑥	𝑥	X
cana-1549	131	5	)	)	PUNCT
cana-1549	131	6	for	for	ADP
cana-1549	131	7	all	all	DET
cana-1549	131	8	𝑥	𝑥	PRON
cana-1549	131	9	=	=	SYM
cana-1549	131	10	𝑥0	𝑥0	PROPN
cana-1549	131	11	,	,	PUNCT
cana-1549	131	12	𝑥1	𝑥1	NOUN
cana-1549	131	13	,	,	PUNCT
cana-1549	131	14	⋯	⋯	PROPN
cana-1549	131	15	,	,	PUNCT
cana-1549	131	16	𝑥𝑛	𝑥𝑛	AUX
cana-1549	131	17	given	give	VERB
cana-1549	131	18	as	as	SCONJ
cana-1549	131	19	follows	follow	NOUN
cana-1549	131	20	.	.	PUNCT
cana-1549	132	1	𝐿𝐻(𝑥0	𝐿𝐻(𝑥0	VERB
cana-1549	132	2	;	;	PUNCT
cana-1549	132	3	4	4	X
cana-1549	132	4	)	)	PUNCT
cana-1549	132	5	=	=	SYM
cana-1549	132	6	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	132	7	;	;	PUNCT
cana-1549	132	8	4	4	X
cana-1549	132	9	)	)	PUNCT
cana-1549	132	10	=	=	SYM
cana-1549	133	1	𝑘	𝑘	PRON
cana-1549	133	2	𝑙𝑛𝑥0	𝑙𝑛𝑥0	NOUN
cana-1549	134	1	=	=	SYM
cana-1549	134	2	0.01853	0.01853	NUM
cana-1549	134	3	,	,	PUNCT
cana-1549	134	4	𝐿𝐻(𝑥1	𝐿𝐻(𝑥1	X
cana-1549	134	5	;	;	PUNCT
cana-1549	134	6	4	4	X
cana-1549	134	7	)	)	PUNCT
cana-1549	134	8	=	=	SYM
cana-1549	134	9	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	134	10	;	;	PUNCT
cana-1549	134	11	4	4	X
cana-1549	134	12	)	)	PUNCT
cana-1549	134	13	+	+	X
cana-1549	134	14	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	134	15	;	;	PUNCT
cana-1549	134	16	4	4	X
cana-1549	134	17	)	)	PUNCT
cana-1549	134	18	=	=	SYM
cana-1549	135	1	𝑘	𝑘	X
cana-1549	135	2	[	[	PUNCT
cana-1549	135	3	1	1	NUM
cana-1549	135	4	ln	ln	ADJ
cana-1549	135	5	𝑥0	𝑥0	NOUN
cana-1549	135	6	+	+	CCONJ
cana-1549	135	7	1	1	NUM
cana-1549	135	8	ln	ln	NOUN
cana-1549	135	9	𝑥1	𝑥1	NOUN
cana-1549	135	10	]	]	PUNCT
cana-1549	135	11	=	=	PUNCT
cana-1549	135	12	0.05073	0.05073	NUM
cana-1549	135	13	𝐿𝐻(𝑥2	𝐿𝐻(𝑥2	NUM
cana-1549	135	14	;	;	PUNCT
cana-1549	135	15	4	4	X
cana-1549	135	16	)	)	PUNCT
cana-1549	135	17	=	=	SYM
cana-1549	135	18	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	135	19	;	;	PUNCT
cana-1549	135	20	4	4	X
cana-1549	135	21	)	)	PUNCT
cana-1549	135	22	+	+	X
cana-1549	135	23	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	135	24	;	;	PUNCT
cana-1549	135	25	4	4	X
cana-1549	135	26	)	)	PUNCT
cana-1549	135	27	+	+	CCONJ
cana-1549	135	28	𝑝(𝑥2	𝑝(𝑥2	NOUN
cana-1549	135	29	;	;	PUNCT
cana-1549	135	30	4	4	X
cana-1549	135	31	)	)	PUNCT
cana-1549	135	32	=	=	SYM
cana-1549	136	1	𝑘	𝑘	X
cana-1549	136	2	[	[	PUNCT
cana-1549	136	3	1	1	NUM
cana-1549	136	4	ln	ln	ADJ
cana-1549	136	5	𝑥0	𝑥0	NOUN
cana-1549	136	6	+	+	CCONJ
cana-1549	136	7	1	1	NUM
cana-1549	136	8	ln	ln	NOUN
cana-1549	136	9	𝑥1	𝑥1	NOUN
cana-1549	136	10	+	+	CCONJ
cana-1549	136	11	1	1	NUM
cana-1549	136	12	ln	ln	ADJ
cana-1549	136	13	𝑥2	𝑥2	NOUN
cana-1549	136	14	]	]	PUNCT
cana-1549	136	15	=	=	SYM
cana-1549	136	16	0.10695	0.10695	NUM
cana-1549	136	17	𝐿𝐻(𝑥3	𝐿𝐻(𝑥3	NOUN
cana-1549	136	18	;	;	PUNCT
cana-1549	136	19	4	4	X
cana-1549	136	20	)	)	PUNCT
cana-1549	136	21	=	=	SYM
cana-1549	136	22	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	136	23	;	;	PUNCT
cana-1549	136	24	4	4	X
cana-1549	136	25	)	)	PUNCT
cana-1549	136	26	+	+	X
cana-1549	136	27	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	136	28	;	;	PUNCT
cana-1549	136	29	4	4	X
cana-1549	136	30	)	)	PUNCT
cana-1549	136	31	+	+	CCONJ
cana-1549	136	32	𝑝(𝑥2	𝑝(𝑥2	NOUN
cana-1549	136	33	;	;	PUNCT
cana-1549	136	34	4	4	X
cana-1549	136	35	)	)	PUNCT
cana-1549	136	36	+	+	CCONJ
cana-1549	136	37	𝑝(𝑥3	𝑝(𝑥3	NOUN
cana-1549	136	38	;	;	PUNCT
cana-1549	136	39	4	4	X
cana-1549	136	40	)	)	PUNCT
cana-1549	136	41	=	=	SYM
cana-1549	137	1	𝑘	𝑘	X
cana-1549	137	2	[	[	PUNCT
cana-1549	137	3	1	1	NUM
cana-1549	137	4	ln	ln	ADJ
cana-1549	137	5	𝑥0	𝑥0	NOUN
cana-1549	137	6	+	+	CCONJ
cana-1549	137	7	1	1	NUM
cana-1549	137	8	ln	ln	NOUN
cana-1549	137	9	𝑥1	𝑥1	NOUN
cana-1549	137	10	+	+	CCONJ
cana-1549	137	11	1	1	NUM
cana-1549	137	12	ln	ln	ADJ
cana-1549	137	13	𝑥2	𝑥2	NOUN
cana-1549	137	14	+	+	CCONJ
cana-1549	137	15	1	1	NUM
cana-1549	137	16	ln	ln	NOUN
cana-1549	137	17	𝑥3	𝑥3	NOUN
cana-1549	137	18	]	]	PUNCT
cana-1549	137	19	=	=	SYM
cana-1549	137	20	0.22518	0.22518	NUM
cana-1549	137	21	𝐿𝐻(𝑥4	𝐿𝐻(𝑥4	NOUN
cana-1549	137	22	;	;	PUNCT
cana-1549	137	23	4	4	X
cana-1549	137	24	)	)	PUNCT
cana-1549	137	25	=	=	SYM
cana-1549	137	26	𝑝(𝑥0	𝑝(𝑥0	ADJ
cana-1549	137	27	;	;	PUNCT
cana-1549	137	28	4	4	X
cana-1549	137	29	)	)	PUNCT
cana-1549	137	30	+	+	X
cana-1549	137	31	𝑝(𝑥1	𝑝(𝑥1	PROPN
cana-1549	137	32	;	;	PUNCT
cana-1549	137	33	4	4	X
cana-1549	137	34	)	)	PUNCT
cana-1549	137	35	+	+	CCONJ
cana-1549	137	36	𝑝(𝑥2	𝑝(𝑥2	NOUN
cana-1549	137	37	;	;	PUNCT
cana-1549	137	38	4	4	X
cana-1549	137	39	)	)	PUNCT
cana-1549	137	40	+	+	CCONJ
cana-1549	137	41	𝑝(𝑥3	𝑝(𝑥3	NOUN
cana-1549	137	42	;	;	PUNCT
cana-1549	137	43	4	4	X
cana-1549	137	44	)	)	PUNCT
cana-1549	137	45	+	+	NUM
cana-1549	137	46	𝑝(𝑥4	𝑝(𝑥4	NOUN
cana-1549	137	47	;	;	PUNCT
cana-1549	137	48	4	4	X
cana-1549	137	49	)	)	PUNCT
cana-1549	137	50	=	=	SYM
cana-1549	138	1	𝑘	𝑘	X
cana-1549	138	2	[	[	PUNCT
cana-1549	138	3	1	1	NUM
cana-1549	138	4	ln	ln	ADJ
cana-1549	138	5	𝑥0	𝑥0	NOUN
cana-1549	138	6	+	+	CCONJ
cana-1549	138	7	1	1	NUM
cana-1549	138	8	ln	ln	NOUN
cana-1549	138	9	𝑥1	𝑥1	NOUN
cana-1549	138	10	+	+	CCONJ
cana-1549	138	11	1	1	NUM
cana-1549	138	12	ln	ln	ADJ
cana-1549	138	13	𝑥2	𝑥2	NOUN
cana-1549	138	14	+	+	CCONJ
cana-1549	138	15	1	1	NUM
cana-1549	138	16	ln	ln	NOUN
cana-1549	138	17	𝑥3	𝑥3	NOUN
cana-1549	138	18	+	+	CCONJ
cana-1549	138	19	1	1	NUM
cana-1549	138	20	ln	ln	ADJ
cana-1549	138	21	𝑥4	𝑥4	NOUN
cana-1549	138	22	]	]	PUNCT
cana-1549	138	23	=	=	SYM
cana-1549	138	24	1.00012	1.00012	NUM
cana-1549	138	25	.	.	PUNCT
cana-1549	139	1	the	the	DET
cana-1549	139	2	distribution	distribution	NOUN
cana-1549	139	3	value	value	NOUN
cana-1549	139	4	at	at	ADP
cana-1549	139	5	𝑥	𝑥	PRON
cana-1549	139	6	≥	≥	NOUN
cana-1549	139	7	𝑥4	𝑥4	NOUN
cana-1549	139	8	is	be	AUX
cana-1549	139	9	approximated	approximate	VERB
cana-1549	139	10	to	to	ADP
cana-1549	139	11	1	1	NUM
cana-1549	139	12	−	−	NOUN
cana-1549	139	13	ϵ	ϵ	ADP
cana-1549	139	14	where	where	SCONJ
cana-1549	139	15	ϵ	ϵ	X
cana-1549	139	16	=	=	SYM
cana-1549	139	17	0.00055	0.00055	NUM
cana-1549	139	18	because	because	SCONJ
cana-1549	139	19	of	of	ADP
cana-1549	139	20	truncational	truncational	ADJ
cana-1549	139	21	errors	error	NOUN
cana-1549	139	22	present	present	ADJ
cana-1549	139	23	in	in	ADP
cana-1549	139	24	𝑥𝑘	𝑥𝑘	PROPN
cana-1549	139	25	’s	’s	PART
cana-1549	139	26	and	and	CCONJ
cana-1549	139	27	pmf	pmf	PROPN
cana-1549	139	28	values	value	NOUN
cana-1549	139	29	in	in	ADP
cana-1549	139	30	𝑥𝑘	𝑥𝑘	NOUN
cana-1549	139	31	’s	’s	NOUN
cana-1549	139	32	.	.	PUNCT
cana-1549	140	1	future	future	ADJ
cana-1549	140	2	scope	scope	NOUN
cana-1549	140	3	our	our	PRON
cana-1549	140	4	next	next	ADJ
cana-1549	140	5	aim	aim	NOUN
cana-1549	140	6	to	to	PART
cana-1549	140	7	study	study	VERB
cana-1549	140	8	the	the	DET
cana-1549	140	9	mean	mean	ADJ
cana-1549	140	10	,	,	PUNCT
cana-1549	140	11	variance	variance	NOUN
cana-1549	140	12	,	,	PUNCT
cana-1549	140	13	moment	moment	NOUN
cana-1549	140	14	,	,	PUNCT
cana-1549	140	15	skewness	skewness	NOUN
cana-1549	140	16	,	,	PUNCT
cana-1549	140	17	kurtosis	kurtosis	NOUN
cana-1549	140	18	,	,	PUNCT
cana-1549	140	19	moment	moment	NOUN
cana-1549	140	20	generating	generating	NOUN
cana-1549	140	21	functions	function	NOUN
cana-1549	140	22	of	of	ADP
cana-1549	140	23	the	the	DET
cana-1549	140	24	discrete	discrete	ADJ
cana-1549	140	25	harmonic	harmonic	NOUN
cana-1549	140	26	mean	mean	NOUN
cana-1549	140	27	distributed	distribute	VERB
cana-1549	140	28	random	random	ADJ
cana-1549	140	29	variables	variable	NOUN
cana-1549	140	30	with	with	ADP
cana-1549	140	31	the	the	DET
cana-1549	140	32	harmonic	harmonic	ADJ
cana-1549	140	33	pmf	pmf	NOUN
cana-1549	140	34	,	,	PUNCT
cana-1549	140	35	exponential	exponential	ADJ
cana-1549	140	36	harmonic	harmonic	ADJ
cana-1549	140	37	pmf	pmf	NOUN
cana-1549	140	38	,	,	PUNCT
cana-1549	140	39	natural	natural	ADJ
cana-1549	140	40	logarithmic	logarithmic	ADJ
cana-1549	140	41	harmonic	harmonic	ADJ
cana-1549	140	42	pmf	pmf	NOUN
cana-1549	140	43	in	in	ADP
cana-1549	140	44	future	future	NOUN
cana-1549	140	45	.	.	PUNCT
cana-1549	141	1	refrences	refrence	VERB
cana-1549	141	2	[	[	X
cana-1549	141	3	1	1	NUM
cana-1549	141	4	]	]	X
cana-1549	141	5	adler	adler	NOUN
cana-1549	141	6	,	,	PUNCT
cana-1549	141	7	c.	c.	PROPN
cana-1549	141	8	f.	f.	PROPN
cana-1549	141	9	,	,	PUNCT
cana-1549	141	10	(	(	PUNCT
cana-1549	141	11	claire	claire	PROPN
cana-1549	141	12	fisher	fisher	PROPN
cana-1549	141	13	adler	adler	PROPN
cana-1549	141	14	)	)	PUNCT
cana-1549	141	15	1958	1958	NUM
cana-1549	141	16	,	,	PUNCT
cana-1549	141	17	“	"	PUNCT
cana-1549	141	18	modern	modern	ADJ
cana-1549	141	19	geometry	geometry	NOUN
cana-1549	141	20	"	"	PUNCT
cana-1549	141	21	,	,	PUNCT
cana-1549	141	22	an	an	DET
cana-1549	141	23	integrated	integrated	ADJ
cana-1549	141	24	first	first	ADJ
cana-1549	141	25	course	course	NOUN
cana-1549	141	26	,	,	PUNCT
cana-1549	141	27	second	second	ADJ
cana-1549	141	28	edition	edition	NOUN
cana-1549	141	29	,	,	PUNCT
cana-1549	141	30	c.w	c.w	PROPN
cana-1549	141	31	.	.	PROPN
cana-1549	141	32	post	post	PROPN
cana-1549	141	33	college	college	NOUN
cana-1549	141	34	of	of	ADP
cana-1549	141	35	long	long	PROPN
cana-1549	141	36	island	island	PROPN
cana-1549	141	37	university	university	PROPN
cana-1549	141	38	,	,	PUNCT
cana-1549	141	39	mcgraw	mcgraw	PROPN
cana-1549	141	40	-	-	PUNCT
cana-1549	141	41	hill	hill	NOUN
cana-1549	141	42	book	book	NOUN
cana-1549	141	43	company	company	NOUN
cana-1549	141	44	.	.	PUNCT
cana-1549	142	1	[	[	X
cana-1549	142	2	2	2	NUM
cana-1549	142	3	]	]	SYM
cana-1549	142	4	ahangar	ahangar	NOUN
cana-1549	142	5	,	,	PUNCT
cana-1549	142	6	r.	r.	PROPN
cana-1549	142	7	r.	r.	PROPN
cana-1549	142	8	,	,	PUNCT
cana-1549	142	9	2010	2010	NUM
cana-1549	142	10	,	,	PUNCT
cana-1549	142	11	“	"	PUNCT
cana-1549	142	12	introduction	introduction	NOUN
cana-1549	142	13	to	to	ADP
cana-1549	142	14	statistical	statistical	ADJ
cana-1549	142	15	geometry	geometry	NOUN
cana-1549	142	16	"	"	PUNCT
cana-1549	142	17	,	,	PUNCT
cana-1549	142	18	proceedings	proceeding	NOUN
cana-1549	142	19	of	of	ADP
cana-1549	142	20	the	the	DET
cana-1549	142	21	international	international	ADJ
cana-1549	142	22	conference	conference	NOUN
cana-1549	142	23	on	on	ADP
cana-1549	142	24	scientific	scientific	ADJ
cana-1549	142	25	computing	computing	NOUN
cana-1549	142	26	,	,	PUNCT
cana-1549	142	27	editors	editor	NOUN
cana-1549	142	28	:	:	PUNCT
cana-1549	142	29	hamid	hamid	PROPN
cana-1549	142	30	r.	r.	PROPN
cana-1549	142	31	arabnia	arabnia	PROPN
cana-1549	142	32	,	,	PUNCT
cana-1549	142	33	george	george	PROPN
cana-1549	142	34	a.	a.	PROPN
cana-1549	142	35	gravvanis	gravvanis	PROPN
cana-1549	142	36	,	,	PUNCT
cana-1549	142	37	csrea	csrea	PROPN
cana-1549	142	38	press	press	PROPN
cana-1549	142	39	,	,	PUNCT
cana-1549	142	40	www.world	www.world	NOUN
cana-1549	142	41	-	-	PUNCT
cana-1549	142	42	academy	academy	NOUN
cana-1549	142	43	-	-	PUNCT
cana-1549	142	44	ofscience.org	ofscience.org	PROPN
cana-1549	142	45	,	,	PUNCT
cana-1549	142	46	scs	scs	NOUN
cana-1549	142	47	.	.	PUNCT
cana-1549	143	1	[	[	X
cana-1549	143	2	3	3	NUM
cana-1549	143	3	]	]	X
cana-1549	143	4	ahangar	ahangar	NOUN
cana-1549	143	5	,	,	PUNCT
cana-1549	143	6	r.	r.	PROPN
cana-1549	143	7	,	,	PUNCT
cana-1549	143	8	(	(	PUNCT
cana-1549	143	9	2013	2013	NUM
cana-1549	143	10	)	)	PUNCT
cana-1549	143	11	,	,	PUNCT
cana-1549	143	12	“	"	PUNCT
cana-1549	143	13	harmonic	harmonic	VERB
cana-1549	143	14	probability	probability	NOUN
cana-1549	143	15	density	density	NOUN
cana-1549	143	16	function	function	NOUN
cana-1549	143	17	,	,	PUNCT
cana-1549	143	18	modelling	modelling	NOUN
cana-1549	143	19	and	and	CCONJ
cana-1549	143	20	regression	regression	NOUN
cana-1549	143	21	"	"	PUNCT
cana-1549	143	22	,	,	PUNCT
cana-1549	143	23	european	european	PROPN
cana-1549	143	24	international	international	PROPN
cana-1549	143	25	journal	journal	PROPN
cana-1549	143	26	of	of	ADP
cana-1549	143	27	science	science	NOUN
cana-1549	143	28	and	and	CCONJ
cana-1549	143	29	technology	technology	NOUN
cana-1549	143	30	,	,	PUNCT
cana-1549	143	31	2(5	2(5	NUM
cana-1549	143	32	)	)	PUNCT
cana-1549	143	33	communications	communication	NOUN
cana-1549	143	34	on	on	ADP
cana-1549	143	35	applied	apply	VERB
cana-1549	143	36	nonlinear	nonlinear	ADJ
cana-1549	143	37	analysis	analysis	NOUN
cana-1549	143	38	issn	issn	NOUN
cana-1549	143	39	:	:	PUNCT
cana-1549	143	40	1074	1074	NUM
cana-1549	143	41	-	-	PUNCT
cana-1549	143	42	133x	133x	NUM
cana-1549	143	43	vol	vol	NOUN
cana-1549	143	44	31	31	NUM
cana-1549	143	45	no	no	NOUN
cana-1549	143	46	.	.	PUNCT
cana-1549	144	1	8s	8s	PROPN
cana-1549	144	2	(	(	PUNCT
cana-1549	144	3	2024	2024	NUM
cana-1549	144	4	)	)	PUNCT
cana-1549	144	5	573	573	NUM
cana-1549	144	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1549	145	1	[	[	X
cana-1549	145	2	4	4	NUM
cana-1549	145	3	]	]	X
cana-1549	145	4	fisher	fisher	PROPN
cana-1549	145	5	,	,	PUNCT
cana-1549	145	6	b.	b.	PROPN
cana-1549	145	7	j.	j.	PROPN
cana-1549	145	8	(	(	PUNCT
cana-1549	145	9	box	box	PROPN
cana-1549	145	10	,	,	PUNCT
cana-1549	145	11	john	john	PROPN
cana-1549	145	12	fisher	fisher	PROPN
cana-1549	145	13	)	)	PUNCT
cana-1549	145	14	,	,	PUNCT
cana-1549	145	15	“	"	PUNCT
cana-1549	145	16	r.	r.	PROPN
cana-1549	145	17	a.	a.	PROPN
cana-1549	145	18	fisher	fisher	PROPN
cana-1549	145	19	,	,	PUNCT
cana-1549	145	20	a	a	DET
cana-1549	145	21	life	life	NOUN
cana-1549	145	22	of	of	ADP
cana-1549	145	23	scientist	scientist	NOUN
cana-1549	145	24	"	"	PUNCT
cana-1549	145	25	,	,	PUNCT
cana-1549	145	26	new	new	PROPN
cana-1549	145	27	york	york	PROPN
cana-1549	145	28	,	,	PUNCT
cana-1549	145	29	wiley	wiley	NOUN
cana-1549	145	30	,	,	PUNCT
cana-1549	145	31	1978	1978	NUM
cana-1549	145	32	.	.	PUNCT
cana-1549	146	1	[	[	X
cana-1549	146	2	5	5	NUM
cana-1549	146	3	]	]	X
cana-1549	146	4	hilbert	hilbert	PROPN
cana-1549	146	5	,	,	PUNCT
cana-1549	146	6	d.	d.	PROPN
cana-1549	146	7	,	,	PUNCT
cana-1549	146	8	(	(	PUNCT
cana-1549	146	9	1902	1902	NUM
cana-1549	146	10	)	)	PUNCT
cana-1549	146	11	,	,	PUNCT
cana-1549	146	12	"	"	PUNCT
cana-1549	146	13	the	the	DET
cana-1549	146	14	foundations	foundation	NOUN
cana-1549	146	15	of	of	ADP
cana-1549	146	16	geometry	geometry	NOUN
cana-1549	146	17	"	"	PUNCT
cana-1549	146	18	,	,	PUNCT
cana-1549	146	19	chicago	chicago	PROPN
cana-1549	146	20	,	,	PUNCT
cana-1549	146	21	il	il	PROPN
cana-1549	146	22	:	:	PUNCT
cana-1549	146	23	the	the	DET
cana-1549	146	24	open	open	ADJ
cana-1549	146	25	court	court	NOUN
cana-1549	146	26	publishing	publishing	NOUN
cana-1549	146	27	company	company	NOUN
cana-1549	146	28	.	.	PUNCT
cana-1549	147	1	[	[	X
cana-1549	147	2	6	6	NUM
cana-1549	147	3	]	]	SYM
cana-1549	147	4	kapur	kapur	PROPN
cana-1549	147	5	,	,	PUNCT
cana-1549	147	6	d.	d.	PROPN
cana-1549	147	7	,	,	PUNCT
cana-1549	147	8	and	and	CCONJ
cana-1549	147	9	mundy	mundy	PROPN
cana-1549	147	10	,	,	PUNCT
cana-1549	147	11	j.	j.	PROPN
cana-1549	147	12	l.	l.	PROPN
cana-1549	147	13	,	,	PUNCT
cana-1549	147	14	(	(	PUNCT
cana-1549	147	15	1989	1989	NUM
cana-1549	147	16	)	)	PUNCT
cana-1549	147	17	,	,	PUNCT
cana-1549	147	18	geometric	geometric	ADJ
cana-1549	147	19	reasoning	reasoning	NOUN
cana-1549	147	20	"	"	PUNCT
cana-1549	147	21	,	,	PUNCT
cana-1549	147	22	a	a	DET
cana-1549	147	23	bradford	bradford	PROPN
cana-1549	147	24	book	book	NOUN
cana-1549	147	25	.	.	PUNCT
cana-1549	148	1	a	a	DET
cana-1549	148	2	collection	collection	NOUN
cana-1549	148	3	of	of	ADP
cana-1549	148	4	papers	paper	NOUN
cana-1549	148	5	presented	present	VERB
cana-1549	148	6	at	at	ADP
cana-1549	148	7	international	international	ADJ
cana-1549	148	8	workshop	workshop	NOUN
cana-1549	148	9	on	on	ADP
cana-1549	148	10	geometric	geometric	ADJ
cana-1549	148	11	reasoning	reasoning	NOUN
cana-1549	148	12	held	hold	VERB
cana-1549	148	13	at	at	ADP
cana-1549	148	14	oxford	oxford	PROPN
cana-1549	148	15	university	university	NOUN
cana-1549	148	16	on	on	ADP
cana-1549	148	17	1986	1986	NUM
cana-1549	148	18	.	.	PUNCT
cana-1549	149	1	[	[	X
cana-1549	149	2	7	7	NUM
cana-1549	149	3	]	]	X
cana-1549	149	4	maccluer	maccluer	NOUN
cana-1549	149	5	,	,	PUNCT
cana-1549	149	6	c.	c.	PROPN
cana-1549	149	7	r.	r.	PROPN
cana-1549	149	8	(	(	PUNCT
cana-1549	149	9	2000	2000	NUM
cana-1549	149	10	)	)	PUNCT
cana-1549	149	11	,	,	PUNCT
cana-1549	149	12	"	"	PUNCT
cana-1549	149	13	industrial	industrial	ADJ
cana-1549	149	14	mathematics	mathematic	NOUN
cana-1549	149	15	,	,	PUNCT
cana-1549	149	16	modeling	modeling	NOUN
cana-1549	149	17	in	in	ADP
cana-1549	149	18	industry	industry	NOUN
cana-1549	149	19	,	,	PUNCT
cana-1549	149	20	science	science	NOUN
cana-1549	149	21	,	,	PUNCT
cana-1549	149	22	and	and	CCONJ
cana-1549	149	23	government	government	NOUN
cana-1549	149	24	"	"	PUNCT
cana-1549	149	25	,	,	PUNCT
cana-1549	149	26	prentice	prentice	NOUN
cana-1549	149	27	hall	hall	NOUN
cana-1549	149	28	.	.	PUNCT
cana-1549	150	1	[	[	X
cana-1549	150	2	8	8	NUM
cana-1549	150	3	]	]	X
cana-1549	150	4	mathews	mathews	PROPN
cana-1549	150	5	and	and	CCONJ
cana-1549	150	6	gilbert	gilbert	PROPN
cana-1549	150	7	,	,	PUNCT
cana-1549	150	8	(	(	PUNCT
cana-1549	150	9	2006	2006	NUM
cana-1549	150	10	)	)	PUNCT
cana-1549	150	11	"	"	PUNCT
cana-1549	150	12	when	when	SCONJ
cana-1549	150	13	averaging	average	VERB
cana-1549	150	14	multiples	multiple	NOUN
cana-1549	150	15	,	,	PUNCT
cana-1549	150	16	apply	apply	VERB
cana-1549	150	17	the	the	DET
cana-1549	150	18	harmonic	harmonic	ADJ
cana-1549	150	19	mean	mean	NOUN
cana-1549	150	20	"	"	PUNCT
cana-1549	150	21	,	,	PUNCT
cana-1549	150	22	bvupdate	bvupdate	ADJ
cana-1549	150	23	:	:	PUNCT
cana-1549	150	24	a	a	DET
cana-1549	150	25	business	business	NOUN
cana-1549	150	26	valuation	valuation	NOUN
cana-1549	150	27	library	library	PROPN
cana-1549	150	28	publication	publication	NOUN
cana-1549	150	29	,	,	PUNCT
cana-1549	150	30	www.bvlibrary.com	www.bvlibrary.com	X
cana-1549	150	31	.	.	PUNCT
cana-1549	151	1	[	[	X
cana-1549	151	2	9	9	NUM
cana-1549	151	3	]	]	X
cana-1549	151	4	meyer	meyer	PROPN
cana-1549	151	5	d.	d.	PROPN
cana-1549	151	6	,	,	PUNCT
cana-1549	151	7	(	(	PUNCT
cana-1549	151	8	1970	1970	NUM
cana-1549	151	9	)	)	PUNCT
cana-1549	151	10	,	,	PUNCT
cana-1549	151	11	"	"	PUNCT
cana-1549	151	12	probability	probability	NOUN
cana-1549	151	13	and	and	CCONJ
cana-1549	151	14	statistics	statistic	NOUN
cana-1549	151	15	an	an	DET
cana-1549	151	16	undergraduate	undergraduate	ADJ
cana-1549	151	17	course	course	NOUN
cana-1549	151	18	"	"	PUNCT
cana-1549	151	19	,	,	PUNCT
cana-1549	151	20	w.a	w.a	PROPN
cana-1549	151	21	.	.	PROPN
cana-1549	151	22	benjamin	benjamin	PROPN
cana-1549	151	23	inc	inc	PROPN
cana-1549	151	24	.	.	PROPN
cana-1549	151	25	new	new	PROPN
cana-1549	151	26	york	york	PROPN
cana-1549	151	27	.	.	PUNCT
cana-1549	152	1	[	[	X
cana-1549	152	2	10	10	NUM
cana-1549	152	3	]	]	X
cana-1549	152	4	pearson	pearson	PROPN
cana-1549	152	5	,	,	PUNCT
cana-1549	152	6	r.	r.	PROPN
cana-1549	152	7	,	,	PUNCT
cana-1549	152	8	(	(	PUNCT
cana-1549	152	9	november	november	PROPN
cana-1549	152	10	11	11	NUM
cana-1549	152	11	,	,	PUNCT
cana-1549	152	12	2011	2011	NUM
cana-1549	152	13	)	)	PUNCT
cana-1549	152	14	,	,	PUNCT
cana-1549	152	15	"	"	PUNCT
cana-1549	152	16	harmonic	harmonic	ADJ
cana-1549	152	17	means	mean	NOUN
cana-1549	152	18	,	,	PUNCT
cana-1549	152	19	reciprocals	reciprocal	NOUN
cana-1549	152	20	,	,	PUNCT
cana-1549	152	21	and	and	CCONJ
cana-1549	152	22	ratios	ratio	NOUN
cana-1549	152	23	of	of	ADP
cana-1549	152	24	random	random	ADJ
cana-1549	152	25	variables	variable	NOUN
cana-1549	152	26	"	"	PUNCT
cana-1549	152	27	,	,	PUNCT
cana-1549	152	28	this	this	DET
cana-1549	152	29	article	article	NOUN
cana-1549	152	30	was	be	AUX
cana-1549	152	31	first	first	ADV
cana-1549	152	32	published	publish	VERB
cana-1549	152	33	on	on	ADP
cana-1549	152	34	exploringdatablogs	exploringdatablog	NOUN
cana-1549	152	35	,	,	PUNCT
cana-1549	152	36	and	and	CCONJ
cana-1549	152	37	later	later	ADV
cana-1549	152	38	contributed	contribute	VERB
cana-1549	152	39	to	to	ADP
cana-1549	152	40	r	r	NOUN
cana-1549	152	41	-	-	PUNCT
cana-1549	152	42	bloggers	blogger	NOUN
cana-1549	152	43	:	:	PUNCT
cana-1549	152	44	www.rbloggers.com	www.rbloggers.com	X
cana-1549	152	45	.	.	PUNCT
cana-1549	153	1	[	[	X
cana-1549	153	2	11	11	NUM
cana-1549	153	3	]	]	X
cana-1549	153	4	saville	saville	PROPN
cana-1549	153	5	j.	j.	PROPN
cana-1549	153	6	d.	d.	PROPN
cana-1549	153	7	,	,	PUNCT
cana-1549	153	8	and	and	CCONJ
cana-1549	153	9	wood	wood	NOUN
cana-1549	153	10	,	,	PUNCT
cana-1549	153	11	r.	r.	PROPN
cana-1549	153	12	g.	g.	PROPN
cana-1549	153	13	,	,	PUNCT
cana-1549	153	14	third	third	ADJ
cana-1549	153	15	print	print	NOUN
cana-1549	153	16	(	(	PUNCT
cana-1549	153	17	1997	1997	NUM
cana-1549	153	18	)	)	PUNCT
cana-1549	153	19	,	,	PUNCT
cana-1549	153	20	"	"	PUNCT
cana-1549	153	21	statistical	statistical	ADJ
cana-1549	153	22	methods	method	NOUN
cana-1549	153	23	:	:	PUNCT
cana-1549	153	24	the	the	DET
cana-1549	153	25	geometric	geometric	ADJ
cana-1549	153	26	approach	approach	NOUN
cana-1549	153	27	,	,	PUNCT
cana-1549	153	28	springer	springer	NOUN
cana-1549	153	29	.	.	PUNCT
cana-1549	154	1	[	[	X
cana-1549	154	2	12	12	NUM
cana-1549	154	3	]	]	PUNCT
cana-1549	154	4	tesi	tesi	PROPN
cana-1549	154	5	di	di	PROPN
cana-1549	154	6	,	,	PUNCT
cana-1549	154	7	l.	l.	PROPN
cana-1549	154	8	d.	d.	PROPN
cana-1549	154	9	,	,	PUNCT
cana-1549	154	10	(	(	PUNCT
cana-1549	154	11	2006	2006	NUM
cana-1549	154	12	-	-	SYM
cana-1549	154	13	2007	2007	NUM
cana-1549	154	14	)	)	PUNCT
cana-1549	154	15	,	,	PUNCT
cana-1549	154	16	"	"	PUNCT
cana-1549	154	17	simulation	simulation	NOUN
cana-1549	154	18	and	and	CCONJ
cana-1549	154	19	analysis	analysis	NOUN
cana-1549	154	20	of	of	ADP
cana-1549	154	21	chemical	chemical	ADJ
cana-1549	154	22	reactions	reaction	NOUN
cana-1549	154	23	using	use	VERB
cana-1549	154	24	stochastic	stochastic	ADJ
cana-1549	154	25	differential	differential	ADJ
cana-1549	154	26	equations	equation	NOUN
cana-1549	154	27	"	"	PUNCT
cana-1549	154	28	,	,	PUNCT
cana-1549	154	29	thesis	thesis	INTJ
cana-1549	154	30	,	,	PUNCT
cana-1549	154	31	universita	universita	PROPN
cana-1549	154	32	degli	degli	PROPN
cana-1549	154	33	studi	studi	PROPN
cana-1549	154	34	di	di	PROPN
cana-1549	154	35	torino	torino	PROPN
cana-1549	154	36	,	,	PUNCT
cana-1549	154	37	anno	anno	NOUN
cana-1549	154	38	academico	academico	NOUN
cana-1549	154	39	.	.	PUNCT