id	sid	tid	token	lemma	pos
ma-160	1	1	2023	2023	NUM
ma-160	1	2	ada	ada	PROPN
ma-160	1	3	academica	academica	PROPN
ma-160	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-160	1	5	.	.	PUNCT
ma-160	2	1	j.	j.	PROPN
ma-160	2	2	math	math	PROPN
ma-160	2	3	.	.	PUNCT
ma-160	3	1	anal	anal	ADJ
ma-160	3	2	.	.	PUNCT
ma-160	4	1	3	3	NUM
ma-160	4	2	(	(	PUNCT
ma-160	4	3	2023	2023	NUM
ma-160	4	4	)	)	PUNCT
ma-160	4	5	17doi	17doi	NUM
ma-160	4	6	:	:	PUNCT
ma-160	4	7	10.28924	10.28924	NUM
ma-160	4	8	/	/	SYM
ma-160	4	9	ada	ada	PROPN
ma-160	4	10	/	/	SYM
ma-160	4	11	ma.3.17	ma.3.17	NOUN
ma-160	4	12	on	on	ADP
ma-160	4	13	certain	certain	ADJ
ma-160	4	14	properties	property	NOUN
ma-160	4	15	of	of	ADP
ma-160	4	16	a	a	DET
ma-160	4	17	degenerate	degenerate	ADJ
ma-160	4	18	sigmoid	sigmoid	NOUN
ma-160	4	19	function	function	NOUN
ma-160	4	20	thomas	thomas	PROPN
ma-160	4	21	awinba	awinba	PROPN
ma-160	4	22	akugre1,∗	akugre1,∗	PROPN
ma-160	4	23	,	,	PUNCT
ma-160	4	24	kwara	kwara	PROPN
ma-160	4	25	nantomah2	nantomah2	PROPN
ma-160	4	26	,	,	PUNCT
ma-160	4	27	mohammed	mohammed	PROPN
ma-160	4	28	muniru	muniru	PROPN
ma-160	4	29	iddrisu3	iddrisu3	ADJ
ma-160	4	30	1department	1department	NUM
ma-160	4	31	of	of	ADP
ma-160	4	32	mathematics	mathematic	NOUN
ma-160	4	33	,	,	PUNCT
ma-160	4	34	school	school	NOUN
ma-160	4	35	of	of	ADP
ma-160	4	36	mathematical	mathematical	ADJ
ma-160	4	37	sciences	sciences	PROPN
ma-160	4	38	,	,	PUNCT
ma-160	4	39	c.	c.	PROPN
ma-160	4	40	k.	k.	PROPN
ma-160	4	41	tedam	tedam	PROPN
ma-160	4	42	university	university	PROPN
ma-160	4	43	of	of	ADP
ma-160	4	44	technology	technology	NOUN
ma-160	4	45	and	and	CCONJ
ma-160	4	46	applied	apply	VERB
ma-160	4	47	sciences	science	NOUN
ma-160	4	48	,	,	PUNCT
ma-160	4	49	p.	p.	PROPN
ma-160	4	50	o.	o.	PROPN
ma-160	4	51	box	box	PROPN
ma-160	4	52	24	24	NUM
ma-160	4	53	,	,	PUNCT
ma-160	4	54	navrongo	navrongo	NOUN
ma-160	4	55	,	,	PUNCT
ma-160	4	56	upper	upper	ADJ
ma-160	4	57	-	-	PUNCT
ma-160	4	58	east	east	NOUN
ma-160	4	59	region	region	NOUN
ma-160	4	60	,	,	PUNCT
ma-160	4	61	ghana	ghana	PROPN
ma-160	4	62	takugre.stu@cktutas.edu.gh	takugre.stu@cktutas.edu.gh	NOUN
ma-160	4	63	2department	2department	NUM
ma-160	4	64	of	of	ADP
ma-160	4	65	mathematics	mathematic	NOUN
ma-160	4	66	,	,	PUNCT
ma-160	4	67	school	school	NOUN
ma-160	4	68	of	of	ADP
ma-160	4	69	mathematical	mathematical	ADJ
ma-160	4	70	sciences	sciences	PROPN
ma-160	4	71	,	,	PUNCT
ma-160	4	72	c.	c.	PROPN
ma-160	4	73	k.	k.	PROPN
ma-160	4	74	tedam	tedam	PROPN
ma-160	4	75	university	university	PROPN
ma-160	4	76	of	of	ADP
ma-160	4	77	technology	technology	NOUN
ma-160	4	78	and	and	CCONJ
ma-160	4	79	applied	apply	VERB
ma-160	4	80	sciences	science	NOUN
ma-160	4	81	,	,	PUNCT
ma-160	4	82	p.	p.	PROPN
ma-160	4	83	o.	o.	PROPN
ma-160	4	84	box	box	PROPN
ma-160	4	85	24	24	NUM
ma-160	4	86	,	,	PUNCT
ma-160	4	87	navrongo	navrongo	NOUN
ma-160	4	88	,	,	PUNCT
ma-160	4	89	upper	upper	ADJ
ma-160	4	90	-	-	PUNCT
ma-160	4	91	east	east	NOUN
ma-160	4	92	region	region	NOUN
ma-160	4	93	,	,	PUNCT
ma-160	4	94	ghana	ghana	PROPN
ma-160	4	95	knantomah@cktutas.edu.gh	knantomah@cktutas.edu.gh	PROPN
ma-160	4	96	3department	3department	NUM
ma-160	4	97	of	of	ADP
ma-160	4	98	mathematics	mathematic	NOUN
ma-160	4	99	,	,	PUNCT
ma-160	4	100	school	school	NOUN
ma-160	4	101	of	of	ADP
ma-160	4	102	mathematical	mathematical	ADJ
ma-160	4	103	sciences	sciences	PROPN
ma-160	4	104	,	,	PUNCT
ma-160	4	105	c.	c.	PROPN
ma-160	4	106	k.	k.	PROPN
ma-160	4	107	tedam	tedam	PROPN
ma-160	4	108	university	university	PROPN
ma-160	4	109	of	of	ADP
ma-160	4	110	technology	technology	NOUN
ma-160	4	111	and	and	CCONJ
ma-160	4	112	applied	apply	VERB
ma-160	4	113	sciences	science	NOUN
ma-160	4	114	,	,	PUNCT
ma-160	4	115	p.	p.	PROPN
ma-160	4	116	o.	o.	PROPN
ma-160	4	117	box	box	PROPN
ma-160	4	118	24	24	NUM
ma-160	4	119	,	,	PUNCT
ma-160	4	120	navrongo	navrongo	NOUN
ma-160	4	121	,	,	PUNCT
ma-160	4	122	upper	upper	ADJ
ma-160	4	123	-	-	PUNCT
ma-160	4	124	east	east	NOUN
ma-160	4	125	region	region	NOUN
ma-160	4	126	,	,	PUNCT
ma-160	4	127	ghana	ghana	PROPN
ma-160	4	128	middrisu@cktutas.edu.gh	middrisu@cktutas.edu.gh	NOUN
ma-160	4	129	∗correspondence	∗correspondence	NOUN
ma-160	4	130	:	:	PUNCT
ma-160	4	131	takugre.stu@cktutas.edu.gh	takugre.stu@cktutas.edu.gh	NOUN
ma-160	4	132	abstract	abstract	NOUN
ma-160	4	133	.	.	PUNCT
ma-160	5	1	in	in	ADP
ma-160	5	2	this	this	DET
ma-160	5	3	paper	paper	NOUN
ma-160	5	4	,	,	PUNCT
ma-160	5	5	we	we	PRON
ma-160	5	6	introduce	introduce	VERB
ma-160	5	7	a	a	DET
ma-160	5	8	degenerate	degenerate	ADJ
ma-160	5	9	sigmoid	sigmoid	NOUN
ma-160	5	10	function	function	NOUN
ma-160	5	11	.	.	PUNCT
ma-160	6	1	by	by	ADP
ma-160	6	2	employing	employ	VERB
ma-160	6	3	analytical	analytical	ADJ
ma-160	6	4	tech	tech	NOUN
ma-160	6	5	-	-	PUNCT
ma-160	6	6	niques	nique	NOUN
ma-160	6	7	,	,	PUNCT
ma-160	6	8	we	we	PRON
ma-160	6	9	present	present	VERB
ma-160	6	10	some	some	DET
ma-160	6	11	properties	property	NOUN
ma-160	6	12	such	such	ADJ
ma-160	6	13	as	as	ADP
ma-160	6	14	logarithmic	logarithmic	ADJ
ma-160	6	15	concavity	concavity	NOUN
ma-160	6	16	,	,	PUNCT
ma-160	6	17	monotonicity	monotonicity	NOUN
ma-160	6	18	and	and	CCONJ
ma-160	6	19	inequalities	inequality	NOUN
ma-160	6	20	ofthe	ofthe	VERB
ma-160	6	21	new	new	ADJ
ma-160	6	22	function	function	NOUN
ma-160	6	23	.	.	PUNCT
ma-160	7	1	1	1	X
ma-160	7	2	.	.	X
ma-160	7	3	introduction	introduction	NOUN
ma-160	7	4	it	it	PRON
ma-160	7	5	is	be	AUX
ma-160	7	6	known	know	VERB
ma-160	7	7	that	that	SCONJ
ma-160	7	8	,	,	PUNCT
ma-160	7	9	what	what	PRON
ma-160	7	10	is	be	AUX
ma-160	7	11	currently	currently	ADV
ma-160	7	12	referred	refer	VERB
ma-160	7	13	to	to	ADP
ma-160	7	14	as	as	ADP
ma-160	7	15	the	the	DET
ma-160	7	16	logistic	logistic	ADJ
ma-160	7	17	equation	equation	NOUN
ma-160	7	18	or	or	CCONJ
ma-160	7	19	the	the	DET
ma-160	7	20	s	s	ADV
ma-160	7	21	-	-	PUNCT
ma-160	7	22	shaped	shape	VERB
ma-160	7	23	curve	curve	NOUN
ma-160	7	24	wasfirst	wasfirst	NOUN
ma-160	7	25	introduced	introduce	VERB
ma-160	7	26	by	by	ADP
ma-160	7	27	verhulst	verhulst	NOUN
ma-160	7	28	(	(	PUNCT
ma-160	7	29	see	see	VERB
ma-160	7	30	[	[	X
ma-160	7	31	17	17	NUM
ma-160	7	32	]	]	NUM
ma-160	7	33	)	)	PUNCT
ma-160	7	34	.	.	PUNCT
ma-160	8	1	it	it	PRON
ma-160	8	2	maps	map	VERB
ma-160	8	3	a	a	DET
ma-160	8	4	very	very	ADV
ma-160	8	5	large	large	ADJ
ma-160	8	6	input	input	NOUN
ma-160	8	7	domain	domain	NOUN
ma-160	8	8	to	to	ADP
ma-160	8	9	a	a	DET
ma-160	8	10	small	small	ADJ
ma-160	8	11	range	range	NOUN
ma-160	8	12	of	of	ADP
ma-160	8	13	outputof	outputof	ADP
ma-160	8	14	real	real	ADJ
ma-160	8	15	numbers	number	NOUN
ma-160	8	16	between	between	ADP
ma-160	8	17	0	0	NUM
ma-160	8	18	and	and	CCONJ
ma-160	8	19	1	1	NUM
ma-160	8	20	.	.	PUNCT
ma-160	9	1	it	it	PRON
ma-160	9	2	is	be	AUX
ma-160	9	3	a	a	DET
ma-160	9	4	one	one	NUM
ma-160	9	5	toone	toone	NOUN
ma-160	9	6	functioon	functioon	NOUN
ma-160	9	7	and	and	CCONJ
ma-160	9	8	increases	increase	VERB
ma-160	9	9	monotonically(see	monotonically(see	NOUN
ma-160	9	10	[	[	X
ma-160	9	11	8	8	NUM
ma-160	9	12	]	]	NUM
ma-160	9	13	)	)	PUNCT
ma-160	9	14	.	.	PUNCT
ma-160	10	1	the	the	DET
ma-160	10	2	sigmoid	sigmoid	NOUN
ma-160	10	3	function	function	NOUN
ma-160	10	4	,	,	PUNCT
ma-160	10	5	also	also	ADV
ma-160	10	6	known	know	VERB
ma-160	10	7	in	in	ADP
ma-160	10	8	the	the	DET
ma-160	10	9	literature	literature	NOUN
ma-160	10	10	as	as	SCONJ
ma-160	10	11	the	the	DET
ma-160	10	12	sigmoidal	sigmoidal	NOUN
ma-160	10	13	curve	curve	NOUN
ma-160	10	14	or	or	CCONJ
ma-160	10	15	standardlogistic	standardlogistic	ADJ
ma-160	10	16	function	function	NOUN
ma-160	10	17	is	be	AUX
ma-160	10	18	defined	define	VERB
ma-160	10	19	as	as	ADP
ma-160	10	20	(	(	PUNCT
ma-160	10	21	see	see	VERB
ma-160	10	22	[	[	X
ma-160	10	23	13	13	NUM
ma-160	10	24	]	]	NUM
ma-160	10	25	)	)	PUNCT
ma-160	10	26	,	,	PUNCT
ma-160	10	27	s	s	X
ma-160	10	28	(	(	PUNCT
ma-160	10	29	t	t	NOUN
ma-160	10	30	)	)	PUNCT
ma-160	10	31	=	=	SYM
ma-160	10	32	et	et	NOUN
ma-160	10	33	1	1	NUM
ma-160	10	34	+	+	NUM
ma-160	10	35	et	et	NOUN
ma-160	10	36	=	=	SYM
ma-160	10	37	1	1	NUM
ma-160	10	38	1	1	NUM
ma-160	10	39	+	+	NUM
ma-160	10	40	e−t	e−t	NOUN
ma-160	10	41	,	,	PUNCT
ma-160	10	42	t	t	PROPN
ma-160	10	43	∈	∈	PROPN
ma-160	10	44	(	(	PUNCT
ma-160	10	45	−∞,∞	−∞,∞	NOUN
ma-160	10	46	)	)	PUNCT
ma-160	10	47	,	,	PUNCT
ma-160	10	48	(	(	PUNCT
ma-160	10	49	1	1	X
ma-160	10	50	)	)	PUNCT
ma-160	10	51	=	=	SYM
ma-160	11	1	1	1	NUM
ma-160	11	2	2	2	NUM
ma-160	11	3	+	+	CCONJ
ma-160	11	4	1	1	NUM
ma-160	11	5	2	2	NUM
ma-160	11	6	tanh	tanh	NOUN
ma-160	11	7	(	(	PUNCT
ma-160	11	8	t	t	PROPN
ma-160	11	9	2	2	NUM
ma-160	11	10	)	)	PUNCT
ma-160	11	11	,	,	PUNCT
ma-160	11	12	t	t	PROPN
ma-160	11	13	∈	∈	PROPN
ma-160	11	14	(	(	PUNCT
ma-160	11	15	−∞,∞	−∞,∞	NOUN
ma-160	11	16	)	)	PUNCT
ma-160	11	17	.	.	PUNCT
ma-160	12	1	(	(	PUNCT
ma-160	12	2	2	2	X
ma-160	12	3	)	)	PUNCT
ma-160	12	4	it	it	PRON
ma-160	12	5	has	have	VERB
ma-160	12	6	the	the	DET
ma-160	12	7	following	following	NOUN
ma-160	12	8	as	as	SCONJ
ma-160	12	9	its	its	PRON
ma-160	12	10	first	first	ADJ
ma-160	12	11	and	and	CCONJ
ma-160	12	12	second	second	ADJ
ma-160	12	13	derivatives	derivative	NOUN
ma-160	12	14	received	receive	VERB
ma-160	12	15	:	:	PUNCT
ma-160	12	16	27	27	NUM
ma-160	12	17	feb	feb	NOUN
ma-160	12	18	2023	2023	NUM
ma-160	12	19	.	.	PUNCT
ma-160	13	1	key	key	ADJ
ma-160	13	2	words	word	NOUN
ma-160	13	3	and	and	CCONJ
ma-160	13	4	phrases	phrase	NOUN
ma-160	13	5	.	.	PUNCT
ma-160	14	1	degenerate	degenerate	ADJ
ma-160	14	2	sigmoid	sigmoid	NOUN
ma-160	14	3	function	function	NOUN
ma-160	14	4	;	;	PUNCT
ma-160	14	5	logarithmically	logarithmically	ADV
ma-160	14	6	concave	concave	VERB
ma-160	14	7	;	;	PUNCT
ma-160	14	8	inequality.1	inequality.1	PROPN
ma-160	14	9	https://adac.ee	https://adac.ee	PROPN
ma-160	14	10	https://doi.org/10.28924/ada/ma.3.17	https://doi.org/10.28924/ada/ma.3.17	PROPN
ma-160	14	11	https://orcid.org/0009-0005-1387-377x	https://orcid.org/0009-0005-1387-377x	PROPN
ma-160	14	12	https://orcid.org/0000-0003-0911-9537	https://orcid.org/0000-0003-0911-9537	PROPN
ma-160	14	13	https://orcid.org/0000-0001-7628-8168	https://orcid.org/0000-0001-7628-8168	PROPN
ma-160	14	14	eur	eur	PROPN
ma-160	14	15	.	.	PUNCT
ma-160	15	1	j.	j.	PROPN
ma-160	15	2	math	math	PROPN
ma-160	15	3	.	.	PUNCT
ma-160	16	1	anal	anal	PROPN
ma-160	16	2	.	.	PUNCT
ma-160	17	1	10.28924	10.28924	NUM
ma-160	17	2	/	/	SYM
ma-160	17	3	ada	ada	PROPN
ma-160	17	4	/	/	SYM
ma-160	17	5	ma.3.17	ma.3.17	PROPN
ma-160	17	6	2	2	NUM
ma-160	17	7	s	s	PART
ma-160	17	8	′	′	NOUN
ma-160	17	9	(	(	PUNCT
ma-160	17	10	t	t	NOUN
ma-160	17	11	)	)	PUNCT
ma-160	17	12	=	=	SYM
ma-160	17	13	et	et	NOUN
ma-160	17	14	(	(	PUNCT
ma-160	17	15	1	1	NUM
ma-160	17	16	+	+	CCONJ
ma-160	17	17	et)2	et)2	NOUN
ma-160	17	18	=	=	SYM
ma-160	17	19	s	s	X
ma-160	17	20	(	(	PUNCT
ma-160	17	21	t	t	PROPN
ma-160	17	22	)	)	PUNCT
ma-160	17	23	(	(	PUNCT
ma-160	17	24	1−	1−	NUM
ma-160	17	25	s	s	X
ma-160	17	26	(	(	PUNCT
ma-160	17	27	t	t	PROPN
ma-160	17	28	)	)	PUNCT
ma-160	17	29	)	)	PUNCT
ma-160	17	30	,	,	PUNCT
ma-160	17	31	(	(	PUNCT
ma-160	17	32	3	3	X
ma-160	17	33	)	)	PUNCT
ma-160	17	34	s	s	VERB
ma-160	17	35	′′	′′	PROPN
ma-160	17	36	(	(	PUNCT
ma-160	17	37	t	t	PROPN
ma-160	17	38	)	)	PUNCT
ma-160	17	39	=	=	SYM
ma-160	17	40	et	et	NOUN
ma-160	17	41	(	(	PUNCT
ma-160	17	42	1−	1−	NUM
ma-160	17	43	et	et	NOUN
ma-160	17	44	)	)	PUNCT
ma-160	17	45	(	(	PUNCT
ma-160	17	46	1	1	X
ma-160	17	47	+	+	CCONJ
ma-160	17	48	et)3	et)3	NOUN
ma-160	17	49	=	=	SYM
ma-160	17	50	s	s	X
ma-160	17	51	(	(	PUNCT
ma-160	17	52	t	t	PROPN
ma-160	17	53	)	)	PUNCT
ma-160	17	54	(	(	PUNCT
ma-160	17	55	1−	1−	NUM
ma-160	17	56	s	s	X
ma-160	17	57	(	(	PUNCT
ma-160	17	58	t	t	PROPN
ma-160	17	59	)	)	PUNCT
ma-160	17	60	)	)	PUNCT
ma-160	17	61	(	(	PUNCT
ma-160	17	62	1−	1−	NUM
ma-160	17	63	2s	2s	NUM
ma-160	17	64	(	(	PUNCT
ma-160	17	65	t	t	PROPN
ma-160	17	66	)	)	PUNCT
ma-160	17	67	)	)	PUNCT
ma-160	17	68	,	,	PUNCT
ma-160	17	69	(	(	PUNCT
ma-160	17	70	4	4	X
ma-160	17	71	)	)	PUNCT
ma-160	17	72	for	for	ADP
ma-160	17	73	all	all	DET
ma-160	17	74	t	t	NOUN
ma-160	17	75	∈	∈	PROPN
ma-160	17	76	(	(	PUNCT
ma-160	17	77	−∞,∞	−∞,∞	NOUN
ma-160	17	78	)	)	PUNCT
ma-160	17	79	.the	.the	PRON
ma-160	17	80	sigmoid	sigmoid	NOUN
ma-160	17	81	function	function	NOUN
ma-160	17	82	is	be	AUX
ma-160	17	83	used	use	VERB
ma-160	17	84	in	in	ADP
ma-160	17	85	a	a	DET
ma-160	17	86	wide	wide	ADJ
ma-160	17	87	range	range	NOUN
ma-160	17	88	of	of	ADP
ma-160	17	89	scientific	scientific	ADJ
ma-160	17	90	disciplines	discipline	NOUN
ma-160	17	91	,	,	PUNCT
ma-160	17	92	including	include	VERB
ma-160	17	93	probability	probability	NOUN
ma-160	17	94	andstatistics	andstatistic	NOUN
ma-160	17	95	,	,	PUNCT
ma-160	17	96	biology	biology	NOUN
ma-160	17	97	,	,	PUNCT
ma-160	17	98	demography	demography	NOUN
ma-160	17	99	,	,	PUNCT
ma-160	17	100	machine	machine	NOUN
ma-160	17	101	learning	learning	NOUN
ma-160	17	102	,	,	PUNCT
ma-160	17	103	population	population	NOUN
ma-160	17	104	dynamics	dynamic	NOUN
ma-160	17	105	,	,	PUNCT
ma-160	17	106	ecology	ecology	NOUN
ma-160	17	107	,	,	PUNCT
ma-160	17	108	and	and	CCONJ
ma-160	17	109	mathematicalpsychology(see	mathematicalpsychology(see	VERB
ma-160	18	1	[	[	X
ma-160	18	2	7	7	NUM
ma-160	18	3	]	]	PUNCT
ma-160	18	4	,	,	PUNCT
ma-160	19	1	[	[	X
ma-160	19	2	16	16	NUM
ma-160	19	3	]	]	PUNCT
ma-160	19	4	)	)	PUNCT
ma-160	19	5	.	.	PUNCT
ma-160	20	1	in	in	ADP
ma-160	20	2	the	the	DET
ma-160	20	3	business	business	NOUN
ma-160	20	4	sector	sector	NOUN
ma-160	20	5	,	,	PUNCT
ma-160	20	6	the	the	DET
ma-160	20	7	sigmoid	sigmoid	NOUN
ma-160	20	8	function	function	NOUN
ma-160	20	9	has	have	AUX
ma-160	20	10	been	be	AUX
ma-160	20	11	utilized	utilize	VERB
ma-160	20	12	to	to	PART
ma-160	20	13	analyzeperformance	analyzeperformance	VERB
ma-160	20	14	growth	growth	NOUN
ma-160	20	15	in	in	ADP
ma-160	20	16	manufacturing	manufacturing	NOUN
ma-160	20	17	and	and	CCONJ
ma-160	20	18	service	service	NOUN
ma-160	20	19	management	management	NOUN
ma-160	20	20	(	(	PUNCT
ma-160	20	21	see	see	VERB
ma-160	20	22	[	[	X
ma-160	20	23	10	10	NUM
ma-160	20	24	]	]	NUM
ma-160	20	25	)	)	PUNCT
ma-160	20	26	.	.	PUNCT
ma-160	21	1	at	at	ADP
ma-160	21	2	each	each	DET
ma-160	21	3	neuron	neuron	NOUN
ma-160	21	4	’s	’s	PART
ma-160	21	5	output	output	NOUN
ma-160	21	6	,	,	PUNCT
ma-160	21	7	the	the	DET
ma-160	21	8	function	function	NOUN
ma-160	21	9	serves	serve	VERB
ma-160	21	10	as	as	ADP
ma-160	21	11	an	an	DET
ma-160	21	12	activation	activation	NOUN
ma-160	21	13	function	function	NOUN
ma-160	21	14	in	in	ADP
ma-160	21	15	artificial	artificial	ADJ
ma-160	21	16	neural	neural	ADJ
ma-160	21	17	networks	network	NOUN
ma-160	21	18	(	(	PUNCT
ma-160	21	19	see	see	VERB
ma-160	21	20	[	[	X
ma-160	21	21	12	12	NUM
ma-160	21	22	]	]	PUNCT
ma-160	21	23	,	,	PUNCT
ma-160	21	24	[	[	X
ma-160	21	25	18	18	NUM
ma-160	21	26	]	]	PUNCT
ma-160	21	27	,	,	PUNCT
ma-160	21	28	[	[	X
ma-160	21	29	15	15	NUM
ma-160	21	30	]	]	PUNCT
ma-160	21	31	)	)	PUNCT
ma-160	21	32	andthe	andthe	ADJ
ma-160	21	33	references	reference	NOUN
ma-160	21	34	therein.in	therein.in	VERB
ma-160	21	35	addition	addition	NOUN
ma-160	21	36	,	,	PUNCT
ma-160	21	37	the	the	DET
ma-160	21	38	function	function	NOUN
ma-160	21	39	is	be	AUX
ma-160	21	40	used	use	VERB
ma-160	21	41	in	in	ADP
ma-160	21	42	medicine	medicine	NOUN
ma-160	21	43	to	to	PART
ma-160	21	44	research	research	VERB
ma-160	21	45	pharmacokinetic	pharmacokinetic	ADJ
ma-160	21	46	responses	response	NOUN
ma-160	21	47	and	and	CCONJ
ma-160	21	48	mimictumor	mimictumor	ADJ
ma-160	21	49	development	development	NOUN
ma-160	21	50	(	(	PUNCT
ma-160	21	51	see	see	VERB
ma-160	21	52	[	[	X
ma-160	21	53	11	11	NUM
ma-160	21	54	]	]	NUM
ma-160	21	55	)	)	PUNCT
ma-160	21	56	.	.	PUNCT
ma-160	22	1	in	in	ADP
ma-160	22	2	[	[	X
ma-160	22	3	5	5	NUM
ma-160	22	4	]	]	PUNCT
ma-160	22	5	,	,	PUNCT
ma-160	22	6	the	the	DET
ma-160	22	7	site	site	NOUN
ma-160	22	8	index	index	NOUN
ma-160	22	9	of	of	ADP
ma-160	22	10	unmanaged	unmanaged	ADJ
ma-160	22	11	loblolly	loblolly	ADV
ma-160	22	12	and	and	CCONJ
ma-160	22	13	slash	slash	VERB
ma-160	22	14	pine	pine	NOUN
ma-160	22	15	plantationsin	plantationsin	PROPN
ma-160	22	16	east	east	PROPN
ma-160	22	17	texas	texas	PROPN
ma-160	22	18	is	be	AUX
ma-160	22	19	predicted	predict	VERB
ma-160	22	20	using	use	VERB
ma-160	22	21	a	a	DET
ma-160	22	22	generic	generic	ADJ
ma-160	22	23	variant	variant	NOUN
ma-160	22	24	of	of	ADP
ma-160	22	25	the	the	DET
ma-160	22	26	sigmoid	sigmoid	NOUN
ma-160	22	27	function	function	NOUN
ma-160	22	28	.	.	PUNCT
ma-160	23	1	it	it	PRON
ma-160	23	2	is	be	AUX
ma-160	23	3	also	also	ADV
ma-160	23	4	used	use	VERB
ma-160	23	5	incomputer	incomputer	NOUN
ma-160	23	6	graphics	graphic	NOUN
ma-160	23	7	and	and	CCONJ
ma-160	23	8	image	image	NOUN
ma-160	23	9	processing	processing	NOUN
ma-160	23	10	to	to	PART
ma-160	23	11	improve	improve	VERB
ma-160	23	12	picture	picture	NOUN
ma-160	23	13	contrast	contrast	NOUN
ma-160	23	14	(	(	PUNCT
ma-160	23	15	see	see	VERB
ma-160	23	16	[	[	X
ma-160	23	17	4	4	NUM
ma-160	23	18	]	]	PUNCT
ma-160	23	19	,	,	PUNCT
ma-160	24	1	[	[	X
ma-160	24	2	9	9	NUM
ma-160	24	3	]	]	PUNCT
ma-160	24	4	,	,	PUNCT
ma-160	24	5	[	[	X
ma-160	24	6	6	6	NUM
ma-160	24	7	]	]	PUNCT
ma-160	24	8	)	)	PUNCT
ma-160	24	9	.	.	PUNCT
ma-160	25	1	it	it	PRON
ma-160	25	2	is	be	AUX
ma-160	25	3	clearfrom	clearfrom	VERB
ma-160	25	4	the	the	DET
ma-160	25	5	above	above	ADJ
ma-160	25	6	applications	application	NOUN
ma-160	25	7	of	of	ADP
ma-160	25	8	the	the	DET
ma-160	25	9	sigmoid	sigmoid	NOUN
ma-160	25	10	function	function	NOUN
ma-160	25	11	that	that	PRON
ma-160	25	12	,	,	PUNCT
ma-160	25	13	further	further	ADJ
ma-160	25	14	research	research	NOUN
ma-160	25	15	needs	need	VERB
ma-160	25	16	to	to	PART
ma-160	25	17	be	be	AUX
ma-160	25	18	conductedon	conductedon	ADJ
ma-160	25	19	this	this	DET
ma-160	25	20	very	very	ADV
ma-160	25	21	important	important	ADJ
ma-160	25	22	function	function	NOUN
ma-160	25	23	to	to	PART
ma-160	25	24	unearth	unearth	VERB
ma-160	25	25	more	more	ADJ
ma-160	25	26	of	of	ADP
ma-160	25	27	its	its	PRON
ma-160	25	28	properties	property	NOUN
ma-160	25	29	and	and	CCONJ
ma-160	25	30	potential	potential	ADJ
ma-160	25	31	applications	application	NOUN
ma-160	25	32	.	.	PUNCT
ma-160	26	1	recently	recently	ADV
ma-160	26	2	,	,	PUNCT
ma-160	26	3	in	in	ADP
ma-160	26	4	[	[	X
ma-160	26	5	13	13	NUM
ma-160	26	6	]	]	PUNCT
ma-160	26	7	,	,	PUNCT
ma-160	26	8	the	the	DET
ma-160	26	9	author	author	NOUN
ma-160	26	10	studied	study	VERB
ma-160	26	11	properties	property	NOUN
ma-160	26	12	such	such	ADJ
ma-160	26	13	as	as	ADP
ma-160	26	14	super	super	ADJ
ma-160	26	15	multiplicativity	multiplicativity	NOUN
ma-160	26	16	,	,	PUNCT
ma-160	26	17	subadditivity	subadditivity	NOUN
ma-160	26	18	,	,	PUNCT
ma-160	26	19	convexity	convexity	NOUN
ma-160	26	20	and	and	CCONJ
ma-160	26	21	inequalities	inequality	NOUN
ma-160	26	22	of	of	ADP
ma-160	26	23	the	the	DET
ma-160	26	24	sigmoid	sigmoid	NOUN
ma-160	26	25	function	function	NOUN
ma-160	26	26	.	.	PUNCT
ma-160	27	1	in	in	ADP
ma-160	27	2	this	this	DET
ma-160	27	3	paper	paper	NOUN
ma-160	27	4	,	,	PUNCT
ma-160	27	5	a	a	DET
ma-160	27	6	degenerate	degenerate	ADJ
ma-160	27	7	sigmoid	sigmoid	NOUN
ma-160	27	8	function	function	NOUN
ma-160	27	9	is	be	AUX
ma-160	27	10	introduced	introduce	VERB
ma-160	27	11	and	and	CCONJ
ma-160	27	12	properties	property	NOUN
ma-160	27	13	such	such	ADJ
ma-160	27	14	as	as	ADP
ma-160	27	15	logarithmicconcavity	logarithmicconcavity	NOUN
ma-160	27	16	,	,	PUNCT
ma-160	27	17	monotonicity	monotonicity	NOUN
ma-160	27	18	and	and	CCONJ
ma-160	27	19	inequlities	inequlitie	NOUN
ma-160	27	20	involving	involve	VERB
ma-160	27	21	the	the	DET
ma-160	27	22	function	function	NOUN
ma-160	27	23	are	be	AUX
ma-160	27	24	provided	provide	VERB
ma-160	27	25	.	.	PUNCT
ma-160	28	1	we	we	PRON
ma-160	28	2	start	start	VERB
ma-160	28	3	with	with	ADP
ma-160	28	4	thefollowing	thefollowing	ADJ
ma-160	28	5	definitions	definition	NOUN
ma-160	28	6	and	and	CCONJ
ma-160	28	7	lemmas	lemma	NOUN
ma-160	28	8	.	.	PUNCT
ma-160	29	1	2	2	X
ma-160	29	2	.	.	X
ma-160	30	1	some	some	DET
ma-160	30	2	definitions	definition	NOUN
ma-160	30	3	and	and	CCONJ
ma-160	30	4	lemmas	lemmas	ADJ
ma-160	30	5	definition	definition	NOUN
ma-160	30	6	2.1	2.1	NUM
ma-160	30	7	.	.	PUNCT
ma-160	31	1	[	[	X
ma-160	31	2	1	1	X
ma-160	31	3	]	]	PUNCT
ma-160	31	4	a	a	DET
ma-160	31	5	function	function	NOUN
ma-160	31	6	m	m	VERB
ma-160	31	7	:	:	PUNCT
ma-160	31	8	(	(	PUNCT
ma-160	31	9	0,∞)×(0,∞)→	0,∞)×(0,∞)→	NOUN
ma-160	31	10	(	(	PUNCT
ma-160	31	11	0,∞	0,∞	NOUN
ma-160	31	12	)	)	PUNCT
ma-160	31	13	is	be	AUX
ma-160	31	14	called	call	VERB
ma-160	31	15	a	a	DET
ma-160	31	16	mean	mean	ADJ
ma-160	31	17	function	function	NOUN
ma-160	31	18	if	if	SCONJ
ma-160	31	19	it	it	PRON
ma-160	31	20	satisfiesthe	satisfiesthe	VERB
ma-160	31	21	following.(1	following.(1	PROPN
ma-160	31	22	)	)	PUNCT
ma-160	31	23	m	m	VERB
ma-160	31	24	(	(	PUNCT
ma-160	31	25	r	r	NOUN
ma-160	31	26	,	,	PUNCT
ma-160	31	27	t	t	PROPN
ma-160	31	28	)	)	PUNCT
ma-160	32	1	=	=	SYM
ma-160	32	2	m	m	PROPN
ma-160	32	3	(	(	PUNCT
ma-160	32	4	t	t	PROPN
ma-160	32	5	,	,	PUNCT
ma-160	32	6	r	r	NOUN
ma-160	32	7	)	)	PUNCT
ma-160	32	8	,	,	PUNCT
ma-160	32	9	(	(	PUNCT
ma-160	32	10	2	2	X
ma-160	32	11	)	)	PUNCT
ma-160	32	12	m	m	PROPN
ma-160	32	13	(	(	PUNCT
ma-160	32	14	t	t	PROPN
ma-160	32	15	,	,	PUNCT
ma-160	32	16	t	t	PROPN
ma-160	32	17	)	)	PUNCT
ma-160	32	18	=	=	SYM
ma-160	32	19	t,(3	t,(3	PROPN
ma-160	32	20	)	)	PUNCT
ma-160	33	1	r	r	NOUN
ma-160	33	2	<	<	X
ma-160	33	3	m	m	X
ma-160	33	4	(	(	PUNCT
ma-160	33	5	r	r	NOUN
ma-160	33	6	,	,	PUNCT
ma-160	33	7	t	t	PROPN
ma-160	33	8	)	)	PUNCT
ma-160	33	9	<	<	X
ma-160	33	10	t	t	PROPN
ma-160	33	11	,	,	PUNCT
ma-160	33	12	for	for	ADP
ma-160	33	13	r	r	NOUN
ma-160	33	14	<	<	X
ma-160	33	15	t,(4	t,(4	PROPN
ma-160	33	16	)	)	PUNCT
ma-160	33	17	m	m	PROPN
ma-160	33	18	(	(	PUNCT
ma-160	33	19	ηr	ηr	NOUN
ma-160	33	20	,	,	PUNCT
ma-160	33	21	ηt	ηt	ADP
ma-160	33	22	)	)	PUNCT
ma-160	33	23	=	=	SYM
ma-160	33	24	ηm	ηm	X
ma-160	33	25	(	(	PUNCT
ma-160	33	26	r	r	NOUN
ma-160	33	27	,	,	PUNCT
ma-160	33	28	t	t	PROPN
ma-160	33	29	)	)	PUNCT
ma-160	33	30	,	,	PUNCT
ma-160	33	31	for	for	ADP
ma-160	33	32	η	η	PROPN
ma-160	33	33	>	>	X
ma-160	33	34	0	0	PROPN
ma-160	33	35	.	.	PUNCT
ma-160	34	1	there	there	PRON
ma-160	34	2	are	be	VERB
ma-160	34	3	many	many	ADJ
ma-160	34	4	well	well	ADV
ma-160	34	5	-	-	PUNCT
ma-160	34	6	known	know	VERB
ma-160	34	7	mean	mean	NOUN
ma-160	34	8	functions	function	NOUN
ma-160	34	9	in	in	ADP
ma-160	34	10	the	the	DET
ma-160	34	11	literature	literature	NOUN
ma-160	34	12	.	.	PUNCT
ma-160	35	1	amongst	amongst	ADP
ma-160	35	2	them	they	PRON
ma-160	35	3	are	be	AUX
ma-160	35	4	the	the	DET
ma-160	35	5	following.(1	following.(1	ADJ
ma-160	35	6	)	)	PUNCT
ma-160	35	7	arithmetic	arithmetic	ADJ
ma-160	35	8	mean	mean	NOUN
ma-160	35	9	:	:	PUNCT
ma-160	35	10	a	a	DET
ma-160	35	11	(	(	PUNCT
ma-160	35	12	r	r	NOUN
ma-160	35	13	,	,	PUNCT
ma-160	35	14	t	t	PROPN
ma-160	35	15	)	)	PUNCT
ma-160	35	16	=	=	PUNCT
ma-160	35	17	r+t	r+t	PROPN
ma-160	35	18	2	2	NUM
ma-160	35	19	,	,	PUNCT
ma-160	35	20	(	(	PUNCT
ma-160	35	21	2	2	X
ma-160	35	22	)	)	PUNCT
ma-160	35	23	geometric	geometric	ADJ
ma-160	35	24	mean	mean	NOUN
ma-160	35	25	:	:	PUNCT
ma-160	35	26	g	g	NOUN
ma-160	35	27	(	(	PUNCT
ma-160	35	28	r	r	NOUN
ma-160	35	29	,	,	PUNCT
ma-160	35	30	t	t	PROPN
ma-160	35	31	)	)	PUNCT
ma-160	35	32	=	=	SYM
ma-160	35	33	√	√	NUM
ma-160	35	34	r	r	NOUN
ma-160	35	35	t	t	PROPN
ma-160	35	36	,	,	PUNCT
ma-160	35	37	https://doi.org/10.28924/ada/ma.3.17	https://doi.org/10.28924/ada/ma.3.17	PROPN
ma-160	35	38	eur	eur	PROPN
ma-160	35	39	.	.	PUNCT
ma-160	36	1	j.	j.	PROPN
ma-160	36	2	math	math	PROPN
ma-160	36	3	.	.	PUNCT
ma-160	37	1	anal	anal	PROPN
ma-160	37	2	.	.	PUNCT
ma-160	38	1	10.28924	10.28924	NUM
ma-160	38	2	/	/	SYM
ma-160	38	3	ada	ada	PROPN
ma-160	38	4	/	/	SYM
ma-160	38	5	ma.3.17	ma.3.17	NOUN
ma-160	38	6	3(3	3(3	NUM
ma-160	38	7	)	)	PUNCT
ma-160	38	8	harmonic	harmonic	ADJ
ma-160	38	9	mean	mean	NOUN
ma-160	38	10	:	:	PUNCT
ma-160	39	1	h	h	NOUN
ma-160	39	2	(	(	PUNCT
ma-160	39	3	r	r	NOUN
ma-160	39	4	,	,	PUNCT
ma-160	39	5	t	t	PROPN
ma-160	39	6	)	)	PUNCT
ma-160	39	7	=	=	SYM
ma-160	40	1	1	1	NUM
ma-160	40	2	a	a	PRON
ma-160	40	3	(	(	PUNCT
ma-160	40	4	1r	1r	NUM
ma-160	40	5	,	,	PUNCT
ma-160	40	6	1	1	NUM
ma-160	40	7	t	t	NOUN
ma-160	40	8	)	)	PUNCT
ma-160	40	9	=	=	SYM
ma-160	41	1	2r	2r	NUM
ma-160	41	2	t	t	PROPN
ma-160	41	3	r+t	r+t	PROPN
ma-160	41	4	,	,	PUNCT
ma-160	41	5	(	(	PUNCT
ma-160	41	6	4	4	X
ma-160	41	7	)	)	PUNCT
ma-160	41	8	logarithmic	logarithmic	ADJ
ma-160	41	9	mean	mean	NOUN
ma-160	41	10	:	:	PUNCT
ma-160	42	1	l	l	NOUN
ma-160	42	2	(	(	PUNCT
ma-160	42	3	r	r	NOUN
ma-160	42	4	,	,	PUNCT
ma-160	42	5	t	t	PROPN
ma-160	42	6	)	)	PUNCT
ma-160	42	7	=	=	PRON
ma-160	42	8	r−t	r−t	PROPN
ma-160	42	9	ln	ln	NOUN
ma-160	42	10	r−ln	r−ln	PROPN
ma-160	42	11	t	t	PROPN
ma-160	42	12	,	,	PUNCT
ma-160	42	13	for	for	ADP
ma-160	42	14	r	r	NOUN
ma-160	42	15	6=	6=	PROPN
ma-160	42	16	t	t	NOUN
ma-160	42	17	and	and	CCONJ
ma-160	42	18	l	l	PROPN
ma-160	42	19	(	(	PUNCT
ma-160	42	20	t	t	PROPN
ma-160	42	21	,	,	PUNCT
ma-160	42	22	t	t	PROPN
ma-160	42	23	)	)	PUNCT
ma-160	42	24	=	=	SYM
ma-160	42	25	t,(5	t,(5	NOUN
ma-160	42	26	)	)	PUNCT
ma-160	42	27	identric	identric	NOUN
ma-160	42	28	mean	mean	VERB
ma-160	42	29	:	:	PUNCT
ma-160	43	1	i	i	PRON
ma-160	43	2	(	(	PUNCT
ma-160	43	3	r	r	NOUN
ma-160	43	4	,	,	PUNCT
ma-160	43	5	t	t	PROPN
ma-160	43	6	)	)	PUNCT
ma-160	43	7	=	=	SYM
ma-160	43	8	1	1	NUM
ma-160	43	9	e	e	X
ma-160	43	10	(	(	PUNCT
ma-160	43	11	r	r	NOUN
ma-160	43	12	r	r	PROPN
ma-160	43	13	tt	tt	PROPN
ma-160	43	14	)	)	PUNCT
ma-160	43	15	1	1	NUM
ma-160	43	16	r−t	r−t	NOUN
ma-160	43	17	,	,	PUNCT
ma-160	43	18	for	for	ADP
ma-160	43	19	r	r	NOUN
ma-160	43	20	6=	6=	PROPN
ma-160	43	21	t	t	PROPN
ma-160	43	22	and	and	CCONJ
ma-160	43	23	i	i	PRON
ma-160	43	24	(	(	PUNCT
ma-160	43	25	t	t	PROPN
ma-160	43	26	,	,	PUNCT
ma-160	43	27	t	t	PROPN
ma-160	43	28	)	)	PUNCT
ma-160	43	29	=	=	SYM
ma-160	44	1	t.	t.	NOUN
ma-160	44	2	definition	definition	NOUN
ma-160	44	3	2.2	2.2	NUM
ma-160	44	4	.	.	PUNCT
ma-160	45	1	[	[	X
ma-160	45	2	1	1	X
ma-160	45	3	]	]	PUNCT
ma-160	45	4	let	let	VERB
ma-160	45	5	g	g	NOUN
ma-160	45	6	:	:	PUNCT
ma-160	45	7	i	i	PROPN
ma-160	45	8	⊆	⊆	NUM
ma-160	45	9	(	(	PUNCT
ma-160	45	10	0,∞)→	0,∞)→	NOUN
ma-160	45	11	(	(	PUNCT
ma-160	45	12	0,∞	0,∞	NOUN
ma-160	45	13	)	)	PUNCT
ma-160	45	14	be	be	AUX
ma-160	45	15	a	a	DET
ma-160	45	16	continuous	continuous	ADJ
ma-160	45	17	function	function	NOUN
ma-160	45	18	and	and	CCONJ
ma-160	45	19	u	u	NOUN
ma-160	45	20	and	and	CCONJ
ma-160	45	21	v	v	NOUN
ma-160	45	22	be	be	AUX
ma-160	45	23	any	any	DET
ma-160	45	24	twomean	twomean	ADJ
ma-160	45	25	functions	function	NOUN
ma-160	45	26	.	.	PUNCT
ma-160	46	1	then	then	ADV
ma-160	46	2	,	,	PUNCT
ma-160	46	3	g	g	PROPN
ma-160	46	4	is	be	AUX
ma-160	46	5	said	say	VERB
ma-160	46	6	to	to	PART
ma-160	46	7	be	be	AUX
ma-160	46	8	uv	uv	ADP
ma-160	46	9	−convex	−convex	NOUN
ma-160	46	10	(	(	PUNCT
ma-160	46	11	uv	uv	NOUN
ma-160	46	12	−concave	−concave	NOUN
ma-160	46	13	)	)	PUNCT
ma-160	46	14	if	if	SCONJ
ma-160	46	15	g	g	PROPN
ma-160	46	16	(	(	PUNCT
ma-160	46	17	u	u	NOUN
ma-160	46	18	(	(	PUNCT
ma-160	46	19	r	r	PROPN
ma-160	46	20	,	,	PUNCT
ma-160	46	21	t	t	PROPN
ma-160	46	22	)	)	PUNCT
ma-160	46	23	)	)	PUNCT
ma-160	46	24	≤	≤	NOUN
ma-160	46	25	(	(	PUNCT
ma-160	46	26	≥	≥	NUM
ma-160	46	27	)	)	PUNCT
ma-160	46	28	v	v	NOUN
ma-160	46	29	(	(	PUNCT
ma-160	46	30	g	g	NOUN
ma-160	46	31	(	(	PUNCT
ma-160	46	32	r	r	NOUN
ma-160	46	33	)	)	PUNCT
ma-160	46	34	,	,	PUNCT
ma-160	46	35	g	g	PROPN
ma-160	46	36	(	(	PUNCT
ma-160	46	37	t	t	PROPN
ma-160	46	38	)	)	PUNCT
ma-160	46	39	)	)	PUNCT
ma-160	46	40	,	,	PUNCT
ma-160	46	41	for	for	ADP
ma-160	46	42	all	all	DET
ma-160	46	43	r	r	NOUN
ma-160	46	44	,	,	PUNCT
ma-160	46	45	t	t	PROPN
ma-160	46	46	∈	∈	PROPN
ma-160	46	47	i.	i.	PROPN
ma-160	46	48	lemma	lemma	PROPN
ma-160	46	49	2.3	2.3	NUM
ma-160	46	50	.	.	PUNCT
ma-160	47	1	[	[	X
ma-160	47	2	1	1	X
ma-160	47	3	]	]	PUNCT
ma-160	47	4	let	let	VERB
ma-160	47	5	f	f	NOUN
ma-160	47	6	:	:	PUNCT
ma-160	47	7	i	i	PRON
ma-160	47	8	⊆	⊆	NUM
ma-160	47	9	(	(	PUNCT
ma-160	47	10	0,∞)→	0,∞)→	NOUN
ma-160	47	11	(	(	PUNCT
ma-160	47	12	0,∞	0,∞	NOUN
ma-160	47	13	)	)	PUNCT
ma-160	47	14	be	be	AUX
ma-160	47	15	a	a	DET
ma-160	47	16	differentiable	differentiable	ADJ
ma-160	47	17	function	function	NOUN
ma-160	47	18	.	.	PUNCT
ma-160	48	1	then(1	then(1	PROPN
ma-160	48	2	)	)	PUNCT
ma-160	49	1	f	f	PROPN
ma-160	49	2	is	be	AUX
ma-160	49	3	ag	ag	PROPN
ma-160	49	4	-	-	PUNCT
ma-160	49	5	convex(or	convex(or	NOUN
ma-160	49	6	concave	concave	NOUN
ma-160	49	7	)	)	PUNCT
ma-160	49	8	if	if	SCONJ
ma-160	49	9	and	and	CCONJ
ma-160	49	10	only	only	ADV
ma-160	49	11	if	if	SCONJ
ma-160	49	12	f	f	PROPN
ma-160	49	13	′	′	NUM
ma-160	49	14	(	(	PUNCT
ma-160	49	15	t	t	PROPN
ma-160	49	16	)	)	PUNCT
ma-160	49	17	f	f	PROPN
ma-160	49	18	(	(	PUNCT
ma-160	49	19	t	t	PROPN
ma-160	49	20	)	)	PUNCT
ma-160	49	21	is	be	AUX
ma-160	49	22	increasing(or	increasing(or	ADP
ma-160	49	23	decreasing	decrease	VERB
ma-160	49	24	)	)	PUNCT
ma-160	49	25	for	for	ADP
ma-160	49	26	all	all	DET
ma-160	49	27	t	t	NOUN
ma-160	49	28	∈	∈	PRON
ma-160	49	29	i	i	PRON
ma-160	49	30	.(2	.(2	PUNCT
ma-160	49	31	)	)	PUNCT
ma-160	50	1	f	f	PROPN
ma-160	50	2	is	be	AUX
ma-160	50	3	ah	ah	INTJ
ma-160	50	4	-	-	PUNCT
ma-160	50	5	convex	convex	NOUN
ma-160	50	6	(	(	PUNCT
ma-160	50	7	or	or	CCONJ
ma-160	50	8	concave	concave	VERB
ma-160	50	9	)	)	PUNCT
ma-160	50	10	if	if	SCONJ
ma-160	51	1	and	and	CCONJ
ma-160	51	2	only	only	ADV
ma-160	51	3	if	if	SCONJ
ma-160	51	4	f	f	PROPN
ma-160	51	5	′	′	NUM
ma-160	51	6	(	(	PUNCT
ma-160	51	7	t	t	PROPN
ma-160	51	8	)	)	PUNCT
ma-160	51	9	f	f	PROPN
ma-160	51	10	(	(	PUNCT
ma-160	51	11	t)2	t)2	PROPN
ma-160	51	12	is	be	AUX
ma-160	51	13	increasing(or	increasing(or	ADP
ma-160	51	14	decreasing	decrease	VERB
ma-160	51	15	)	)	PUNCT
ma-160	51	16	for	for	ADP
ma-160	51	17	all	all	DET
ma-160	51	18	t	t	NOUN
ma-160	51	19	∈	∈	PRON
ma-160	52	1	i	i	PRON
ma-160	52	2	.	.	PUNCT
ma-160	53	1	lemma	lemma	PROPN
ma-160	53	2	2.4	2.4	NUM
ma-160	53	3	.	.	PUNCT
ma-160	54	1	[	[	X
ma-160	54	2	2	2	X
ma-160	54	3	]	]	PUNCT
ma-160	54	4	let	let	VERB
ma-160	54	5	f	f	NOUN
ma-160	54	6	:	:	PUNCT
ma-160	54	7	i	i	PRON
ma-160	54	8	⊆	⊆	NUM
ma-160	54	9	(	(	PUNCT
ma-160	54	10	b,∞)→	b,∞)→	PROPN
ma-160	54	11	(	(	PUNCT
ma-160	54	12	−∞,∞	−∞,∞	NOUN
ma-160	54	13	)	)	PUNCT
ma-160	54	14	with	with	ADP
ma-160	54	15	b	b	PROPN
ma-160	54	16	≥	≥	NOUN
ma-160	54	17	0	0	NUM
ma-160	54	18	.	.	PUNCT
ma-160	55	1	if	if	SCONJ
ma-160	55	2	the	the	DET
ma-160	55	3	function	function	NOUN
ma-160	55	4	defined	define	VERB
ma-160	55	5	by	by	ADP
ma-160	55	6	g	g	PROPN
ma-160	55	7	(	(	PUNCT
ma-160	55	8	t	t	PROPN
ma-160	55	9	)	)	PUNCT
ma-160	55	10	=	=	SYM
ma-160	55	11	f	f	PROPN
ma-160	55	12	(	(	PUNCT
ma-160	55	13	t)−1	t)−1	NOUN
ma-160	55	14	t	t	PROPN
ma-160	55	15	is	be	AUX
ma-160	55	16	increasing	increase	VERB
ma-160	55	17	on	on	ADP
ma-160	55	18	(	(	PUNCT
ma-160	55	19	b,∞	b,∞	NOUN
ma-160	55	20	)	)	PUNCT
ma-160	55	21	,	,	PUNCT
ma-160	55	22	then	then	ADV
ma-160	55	23	the	the	DET
ma-160	55	24	function	function	NOUN
ma-160	55	25	h	h	NOUN
ma-160	55	26	(	(	PUNCT
ma-160	55	27	t	t	NOUN
ma-160	55	28	)	)	PUNCT
ma-160	55	29	=	=	SYM
ma-160	56	1	f	f	PROPN
ma-160	56	2	(	(	PUNCT
ma-160	56	3	t2	t2	PROPN
ma-160	56	4	)	)	PUNCT
ma-160	56	5	satisfies	satisfy	VERB
ma-160	56	6	the	the	DET
ma-160	56	7	grumbaum	grumbaum	ADJ
ma-160	56	8	-	-	PUNCT
ma-160	56	9	type	type	NOUN
ma-160	56	10	inequality	inequality	NOUN
ma-160	56	11	1	1	NUM
ma-160	56	12	+	+	NUM
ma-160	56	13	h	h	PROPN
ma-160	56	14	(	(	PUNCT
ma-160	56	15	z2	z2	PROPN
ma-160	56	16	)	)	PUNCT
ma-160	56	17	≥	≥	PROPN
ma-160	56	18	h	h	NOUN
ma-160	56	19	(	(	PUNCT
ma-160	56	20	r2	r2	PROPN
ma-160	56	21	)	)	PUNCT
ma-160	57	1	+	+	CCONJ
ma-160	57	2	h	h	NOUN
ma-160	57	3	(	(	PUNCT
ma-160	57	4	t2	t2	PROPN
ma-160	57	5	)	)	PUNCT
ma-160	57	6	,	,	PUNCT
ma-160	57	7	(	(	PUNCT
ma-160	57	8	5	5	X
ma-160	57	9	)	)	PUNCT
ma-160	57	10	where	where	SCONJ
ma-160	57	11	r	r	NOUN
ma-160	57	12	,	,	PUNCT
ma-160	57	13	t	t	PROPN
ma-160	57	14	≥	≥	PROPN
ma-160	57	15	b	b	PROPN
ma-160	57	16	and	and	CCONJ
ma-160	57	17	z2	z2	PROPN
ma-160	57	18	=	=	SYM
ma-160	57	19	r2	r2	PROPN
ma-160	57	20	+	+	CCONJ
ma-160	57	21	t2	t2	NOUN
ma-160	57	22	.	.	PUNCT
ma-160	58	1	if	if	SCONJ
ma-160	58	2	g	g	PROPN
ma-160	58	3	is	be	AUX
ma-160	58	4	decreasing	decrease	VERB
ma-160	58	5	,	,	PUNCT
ma-160	58	6	then	then	ADV
ma-160	58	7	the	the	DET
ma-160	58	8	inequality	inequality	NOUN
ma-160	58	9	(	(	PUNCT
ma-160	58	10	5	5	NUM
ma-160	58	11	)	)	PUNCT
ma-160	58	12	is	be	AUX
ma-160	58	13	reversed	reverse	VERB
ma-160	58	14	.	.	PUNCT
ma-160	59	1	3	3	X
ma-160	59	2	.	.	X
ma-160	59	3	main	main	ADJ
ma-160	59	4	results	result	NOUN
ma-160	59	5	definition	definition	NOUN
ma-160	59	6	3.1	3.1	NUM
ma-160	59	7	.	.	PUNCT
ma-160	60	1	the	the	DET
ma-160	60	2	degenerate	degenerate	ADJ
ma-160	60	3	sigmoid	sigmoid	NOUN
ma-160	60	4	function	function	NOUN
ma-160	60	5	is	be	AUX
ma-160	60	6	defined	define	VERB
ma-160	60	7	for	for	ADP
ma-160	60	8	λ	λ	PROPN
ma-160	60	9	∈	∈	PROPN
ma-160	60	10	(	(	PUNCT
ma-160	60	11	0,∞	0,∞	NOUN
ma-160	60	12	)	)	PUNCT
ma-160	60	13	and	and	CCONJ
ma-160	60	14	t	t	PROPN
ma-160	60	15	∈	∈	PROPN
ma-160	60	16	(	(	PUNCT
ma-160	60	17	−∞,∞	−∞,∞	NOUN
ma-160	60	18	)	)	PUNCT
ma-160	60	19	as	as	ADP
ma-160	60	20	sλ	sλ	NOUN
ma-160	60	21	(	(	PUNCT
ma-160	60	22	t	t	NOUN
ma-160	60	23	)	)	PUNCT
ma-160	60	24	=	=	PUNCT
ma-160	61	1	(	(	PUNCT
ma-160	61	2	1	1	NUM
ma-160	61	3	+	+	CCONJ
ma-160	61	4	λt	λt	X
ma-160	61	5	)	)	PUNCT
ma-160	61	6	1	1	NUM
ma-160	61	7	λ	λ	SYM
ma-160	61	8	1	1	NUM
ma-160	61	9	+	+	CCONJ
ma-160	61	10	(	(	PUNCT
ma-160	61	11	1	1	NUM
ma-160	61	12	+	+	CCONJ
ma-160	61	13	λt	λt	X
ma-160	61	14	)	)	PUNCT
ma-160	61	15	1	1	NUM
ma-160	61	16	λ	λ	X
ma-160	61	17	(	(	PUNCT
ma-160	61	18	6	6	NUM
ma-160	61	19	)	)	PUNCT
ma-160	61	20	=	=	SYM
ma-160	61	21	1	1	NUM
ma-160	61	22	1	1	NUM
ma-160	61	23	+	+	CCONJ
ma-160	61	24	(	(	PUNCT
ma-160	61	25	1	1	NUM
ma-160	61	26	+	+	CCONJ
ma-160	61	27	λt)−	λt)−	PROPN
ma-160	61	28	1	1	NUM
ma-160	61	29	λ	λ	X
ma-160	61	30	(	(	PUNCT
ma-160	61	31	7	7	NUM
ma-160	61	32	)	)	PUNCT
ma-160	61	33	=	=	SYM
ma-160	61	34	1	1	NUM
ma-160	61	35	2	2	NUM
ma-160	61	36	+	+	CCONJ
ma-160	61	37	1	1	NUM
ma-160	61	38	2	2	NUM
ma-160	61	39	tanhλ	tanhλ	NOUN
ma-160	61	40	(	(	PUNCT
ma-160	61	41	t	t	PROPN
ma-160	61	42	2	2	NUM
ma-160	61	43	)	)	PUNCT
ma-160	61	44	.	.	PUNCT
ma-160	62	1	(	(	PUNCT
ma-160	62	2	8)	8)	NUM
ma-160	62	3	it	it	PRON
ma-160	62	4	is	be	AUX
ma-160	62	5	clear	clear	ADJ
ma-160	62	6	that	that	SCONJ
ma-160	62	7	,	,	PUNCT
ma-160	62	8	taking	take	VERB
ma-160	62	9	the	the	DET
ma-160	62	10	limit	limit	NOUN
ma-160	62	11	of	of	ADP
ma-160	62	12	sλ	sλ	NOUN
ma-160	62	13	(	(	PUNCT
ma-160	62	14	t	t	NOUN
ma-160	62	15	)	)	PUNCT
ma-160	62	16	as	as	ADP
ma-160	62	17	λ→	λ→	PROPN
ma-160	62	18	0	0	NUM
ma-160	62	19	,	,	PUNCT
ma-160	62	20	then	then	ADV
ma-160	62	21	sλ	sλ	NOUN
ma-160	62	22	(	(	PUNCT
ma-160	62	23	t)→	t)→	PROPN
ma-160	62	24	s	s	X
ma-160	62	25	(	(	PUNCT
ma-160	62	26	t	t	PROPN
ma-160	62	27	)	)	PUNCT
ma-160	62	28	.the	.the	PUNCT
ma-160	63	1	first	first	ADJ
ma-160	63	2	derivative	derivative	NOUN
ma-160	63	3	of	of	ADP
ma-160	63	4	the	the	DET
ma-160	63	5	degenerate	degenerate	ADJ
ma-160	63	6	sigmoid	sigmoid	NOUN
ma-160	63	7	function	function	NOUN
ma-160	63	8	is	be	AUX
ma-160	63	9	given	give	VERB
ma-160	63	10	as	as	ADP
ma-160	63	11	s	s	NOUN
ma-160	63	12	′	′	NUM
ma-160	63	13	λ	λ	PROPN
ma-160	63	14	(	(	PUNCT
ma-160	63	15	t	t	PROPN
ma-160	63	16	)	)	PUNCT
ma-160	63	17	=	=	PUNCT
ma-160	64	1	(	(	PUNCT
ma-160	64	2	1	1	NUM
ma-160	64	3	+	+	CCONJ
ma-160	64	4	λt	λt	X
ma-160	64	5	)	)	PUNCT
ma-160	64	6	1	1	NUM
ma-160	64	7	λ	λ	SYM
ma-160	64	8	−1	−1	NOUN
ma-160	64	9	[	[	PUNCT
ma-160	64	10	1	1	NUM
ma-160	64	11	+	+	CCONJ
ma-160	64	12	(	(	PUNCT
ma-160	64	13	1	1	NUM
ma-160	64	14	+	+	CCONJ
ma-160	64	15	λt	λt	X
ma-160	64	16	)	)	PUNCT
ma-160	64	17	1	1	NUM
ma-160	64	18	λ	λ	NOUN
ma-160	64	19	]	]	X
ma-160	64	20	2	2	NUM
ma-160	64	21	>	>	SYM
ma-160	64	22	0	0	NUM
ma-160	64	23	,	,	PUNCT
ma-160	64	24	(	(	PUNCT
ma-160	64	25	9	9	NUM
ma-160	64	26	)	)	PUNCT
ma-160	64	27	for	for	ADP
ma-160	64	28	all	all	DET
ma-160	64	29	t	t	NOUN
ma-160	64	30	∈	∈	PROPN
ma-160	64	31	(	(	PUNCT
ma-160	64	32	−∞,∞	−∞,∞	NOUN
ma-160	64	33	)	)	PUNCT
ma-160	64	34	and	and	CCONJ
ma-160	64	35	λ	λ	X
ma-160	64	36	∈	∈	PROPN
ma-160	64	37	(	(	PUNCT
ma-160	64	38	0,∞	0,∞	NOUN
ma-160	64	39	)	)	PUNCT
ma-160	64	40	.	.	PUNCT
ma-160	65	1	the	the	DET
ma-160	65	2	degenerate	degenerate	ADJ
ma-160	65	3	sigmoid	sigmoid	NOUN
ma-160	65	4	function	function	NOUN
ma-160	65	5	satisfies	satisfy	VERB
ma-160	65	6	the	the	DET
ma-160	65	7	following	follow	VERB
ma-160	65	8	identities	identity	NOUN
ma-160	65	9	.	.	PUNCT
ma-160	66	1	https://doi.org/10.28924/ada/ma.3.17	https://doi.org/10.28924/ada/ma.3.17	PROPN
ma-160	66	2	eur	eur	PROPN
ma-160	66	3	.	.	PUNCT
ma-160	67	1	j.	j.	PROPN
ma-160	67	2	math	math	PROPN
ma-160	67	3	.	.	PUNCT
ma-160	68	1	anal	anal	PROPN
ma-160	68	2	.	.	PUNCT
ma-160	69	1	10.28924	10.28924	NUM
ma-160	69	2	/	/	SYM
ma-160	69	3	ada	ada	PROPN
ma-160	69	4	/	/	SYM
ma-160	69	5	ma.3.17	ma.3.17	ADJ
ma-160	69	6	4	4	NUM
ma-160	69	7	sλ	sλ	NOUN
ma-160	69	8	(	(	PUNCT
ma-160	69	9	t	t	NOUN
ma-160	69	10	)	)	PUNCT
ma-160	69	11	+	+	CCONJ
ma-160	69	12	sλ	sλ	NOUN
ma-160	69	13	(	(	PUNCT
ma-160	69	14	−t	−t	NOUN
ma-160	69	15	)	)	PUNCT
ma-160	69	16	=	=	SYM
ma-160	69	17	1	1	NUM
ma-160	69	18	,	,	PUNCT
ma-160	69	19	(	(	PUNCT
ma-160	69	20	10	10	NUM
ma-160	69	21	)	)	PUNCT
ma-160	69	22	sλ	sλ	NOUN
ma-160	69	23	(	(	PUNCT
ma-160	69	24	t)sλ	t)sλ	PROPN
ma-160	69	25	(	(	PUNCT
ma-160	69	26	−t	−t	PROPN
ma-160	69	27	)	)	PUNCT
ma-160	69	28	=	=	PUNCT
ma-160	70	1	(	(	PUNCT
ma-160	70	2	1	1	NUM
ma-160	70	3	+	+	CCONJ
ma-160	70	4	λt)s	λt)s	NUM
ma-160	70	5	′	′	NUM
ma-160	70	6	λ	λ	X
ma-160	70	7	(	(	PUNCT
ma-160	70	8	t	t	PROPN
ma-160	70	9	)	)	PUNCT
ma-160	70	10	,	,	PUNCT
ma-160	70	11	(	(	PUNCT
ma-160	70	12	11	11	NUM
ma-160	70	13	)	)	PUNCT
ma-160	70	14	s	s	PART
ma-160	70	15	′	′	NUM
ma-160	70	16	λ	λ	PROPN
ma-160	70	17	(	(	PUNCT
ma-160	70	18	t	t	PROPN
ma-160	70	19	)	)	PUNCT
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ma-160	70	24	(	(	PUNCT
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ma-160	70	29	12	12	X
ma-160	70	30	)	)	PUNCT
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ma-160	70	33	sλ	sλ	NOUN
ma-160	70	34	(	(	PUNCT
ma-160	70	35	t	t	NOUN
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ma-160	70	37	=	=	SYM
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ma-160	70	39	,	,	PUNCT
ma-160	70	40	(	(	PUNCT
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ma-160	70	42	)	)	PUNCT
ma-160	70	43	lim	lim	NOUN
ma-160	70	44	t→0	t→0	X
ma-160	70	45	sλ	sλ	NOUN
ma-160	70	46	(	(	PUNCT
ma-160	70	47	t	t	NOUN
ma-160	70	48	)	)	PUNCT
ma-160	70	49	=	=	SYM
ma-160	70	50	1	1	NUM
ma-160	70	51	2	2	NUM
ma-160	70	52	,	,	PUNCT
ma-160	70	53	(	(	PUNCT
ma-160	70	54	14	14	NUM
ma-160	70	55	)	)	PUNCT
ma-160	70	56	lim	lim	NOUN
ma-160	70	57	t→0	t→0	PUNCT
ma-160	70	58	s	s	PART
ma-160	70	59	′	′	NUM
ma-160	70	60	λ	λ	X
ma-160	70	61	(	(	PUNCT
ma-160	70	62	t	t	PROPN
ma-160	70	63	)	)	PUNCT
ma-160	70	64	=	=	SYM
ma-160	70	65	1	1	NUM
ma-160	70	66	4	4	NUM
ma-160	70	67	,	,	PUNCT
ma-160	70	68	(	(	PUNCT
ma-160	70	69	15	15	X
ma-160	70	70	)	)	PUNCT
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ma-160	70	72	t→∞	t→∞	PRON
ma-160	70	73	s	s	PART
ma-160	70	74	′	′	NUM
ma-160	71	1	λ	λ	PROPN
ma-160	71	2	(	(	PUNCT
ma-160	71	3	t	t	PROPN
ma-160	71	4	)	)	PUNCT
ma-160	71	5	=	=	NOUN
ma-160	71	6	0	0	X
ma-160	71	7	.	.	PUNCT
ma-160	72	1	(	(	PUNCT
ma-160	72	2	16	16	NUM
ma-160	72	3	)	)	PUNCT
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ma-160	72	5	3.2	3.2	NUM
ma-160	72	6	.	.	PUNCT
ma-160	73	1	the	the	DET
ma-160	73	2	function	function	NOUN
ma-160	73	3	sλ	sλ	NOUN
ma-160	73	4	(	(	PUNCT
ma-160	73	5	t	t	NOUN
ma-160	73	6	)	)	PUNCT
ma-160	73	7	is	be	AUX
ma-160	73	8	ag	ag	PROPN
ma-160	73	9	-	-	ADJ
ma-160	73	10	concave	concave	NOUN
ma-160	73	11	on	on	ADP
ma-160	73	12	(	(	PUNCT
ma-160	73	13	0,∞	0,∞	NOUN
ma-160	73	14	)	)	PUNCT
ma-160	73	15	.	.	PUNCT
ma-160	74	1	in	in	ADP
ma-160	74	2	other	other	ADJ
ma-160	74	3	words	word	NOUN
ma-160	74	4	,	,	PUNCT
ma-160	74	5	for	for	ADP
ma-160	74	6	all	all	DET
ma-160	74	7	r	r	NOUN
ma-160	74	8	,	,	PUNCT
ma-160	74	9	t	t	PROPN
ma-160	74	10	,	,	PUNCT
ma-160	74	11	λ	λ	PROPN
ma-160	74	12	∈	∈	PROPN
ma-160	74	13	(	(	PUNCT
ma-160	74	14	0,∞	0,∞	NOUN
ma-160	74	15	)	)	PUNCT
ma-160	74	16	,	,	PUNCT
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ma-160	74	18	inequality	inequality	NOUN
ma-160	74	19	sλ	sλ	NOUN
ma-160	74	20	(	(	PUNCT
ma-160	74	21	r	r	NOUN
ma-160	74	22	+	+	PROPN
ma-160	74	23	t	t	PROPN
ma-160	74	24	2	2	NUM
ma-160	74	25	)	)	PUNCT
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ma-160	75	1	[	[	X
ma-160	75	2	sλ	sλ	X
ma-160	75	3	(	(	PUNCT
ma-160	75	4	r)sλ	r)sλ	PROPN
ma-160	75	5	(	(	PUNCT
ma-160	75	6	t	t	PROPN
ma-160	75	7	)	)	PUNCT
ma-160	75	8	]	]	PUNCT
ma-160	75	9	1	1	NUM
ma-160	75	10	2	2	NUM
ma-160	75	11	(	(	PUNCT
ma-160	75	12	17	17	NUM
ma-160	75	13	)	)	PUNCT
ma-160	75	14	is	be	AUX
ma-160	75	15	satisfied	satisfied	ADJ
ma-160	75	16	.	.	PUNCT
ma-160	76	1	proof	proof	NOUN
ma-160	76	2	.	.	PUNCT
ma-160	77	1	we	we	PRON
ma-160	77	2	have	have	VERB
ma-160	77	3	s	s	NUM
ma-160	77	4	′	′	NUM
ma-160	77	5	λ	λ	X
ma-160	77	6	(	(	PUNCT
ma-160	77	7	t	t	NOUN
ma-160	77	8	)	)	PUNCT
ma-160	77	9	sλ	sλ	NOUN
ma-160	77	10	(	(	PUNCT
ma-160	77	11	t	t	NOUN
ma-160	77	12	)	)	PUNCT
ma-160	78	1	=	=	NOUN
ma-160	78	2			NOUN
ma-160	78	3	(	(	PUNCT
ma-160	78	4	1	1	NUM
ma-160	78	5	+	+	CCONJ
ma-160	78	6	λt	λt	X
ma-160	78	7	)	)	PUNCT
ma-160	78	8	1	1	NUM
ma-160	78	9	λ	λ	SYM
ma-160	78	10	−1	−1	NOUN
ma-160	78	11	[	[	PUNCT
ma-160	78	12	1	1	NUM
ma-160	78	13	+	+	CCONJ
ma-160	78	14	(	(	PUNCT
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ma-160	78	16	+	+	CCONJ
ma-160	78	17	λt	λt	X
ma-160	78	18	)	)	PUNCT
ma-160	78	19	1	1	NUM
ma-160	78	20	λ	λ	NOUN
ma-160	78	21	]	]	X
ma-160	78	22	2	2	NUM
ma-160	78	23	(1	(1	NOUN
ma-160	78	24	+	+	CCONJ
ma-160	78	25	(	(	PUNCT
ma-160	78	26	1	1	NUM
ma-160	78	27	+	+	CCONJ
ma-160	78	28	λt	λt	X
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ma-160	78	30	1	1	NUM
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ma-160	78	34	+	+	CCONJ
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ma-160	78	36	)	)	PUNCT
ma-160	78	37	1	1	NUM
ma-160	78	38	λ	λ	NOUN
ma-160	78	39	)	)	PUNCT
ma-160	78	40	=	=	SYM
ma-160	78	41	1	1	NUM
ma-160	78	42	(	(	PUNCT
ma-160	78	43	1	1	NUM
ma-160	78	44	+	+	CCONJ
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ma-160	78	47	+	+	CCONJ
ma-160	78	48	(	(	PUNCT
ma-160	78	49	1	1	NUM
ma-160	78	50	+	+	CCONJ
ma-160	78	51	λt	λt	X
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ma-160	78	53	1	1	NUM
ma-160	78	54	λ	λ	NOUN
ma-160	78	55	+1	+1	NOUN
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ma-160	78	57	(	(	PUNCT
ma-160	78	58	s	s	VERB
ma-160	78	59	′	′	NUM
ma-160	78	60	λ	λ	X
ma-160	78	61	(	(	PUNCT
ma-160	78	62	t	t	NOUN
ma-160	78	63	)	)	PUNCT
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ma-160	78	65	(	(	PUNCT
ma-160	78	66	t	t	NOUN
ma-160	78	67	)	)	PUNCT
ma-160	78	68	)	)	PUNCT
ma-160	78	69	′	′	NUM
ma-160	79	1	=	=	PUNCT
ma-160	79	2	−	−	PROPN
ma-160	79	3	λ+	λ+	PUNCT
ma-160	79	4	(	(	PUNCT
ma-160	79	5	1	1	NUM
ma-160	79	6	+	+	NUM
ma-160	79	7	λ	λ	NOUN
ma-160	79	8	)	)	PUNCT
ma-160	79	9	(	(	PUNCT
ma-160	79	10	1	1	NUM
ma-160	79	11	+	+	CCONJ
ma-160	79	12	λt	λt	X
ma-160	79	13	)	)	PUNCT
ma-160	79	14	1	1	NUM
ma-160	79	15	λ	λ	NOUN
ma-160	79	16	[	[	PUNCT
ma-160	79	17	(	(	PUNCT
ma-160	79	18	1	1	NUM
ma-160	79	19	+	+	CCONJ
ma-160	79	20	λt	λt	X
ma-160	79	21	)	)	PUNCT
ma-160	79	22	+	+	CCONJ
ma-160	79	23	(	(	PUNCT
ma-160	79	24	1	1	NUM
ma-160	79	25	+	+	CCONJ
ma-160	79	26	λt	λt	X
ma-160	79	27	)	)	PUNCT
ma-160	79	28	1	1	NUM
ma-160	80	1	λ	λ	NOUN
ma-160	80	2	+1	+1	X
ma-160	80	3	]	]	X
ma-160	80	4	2	2	NUM
ma-160	80	5	<	<	X
ma-160	80	6	0	0	NUM
ma-160	80	7	,	,	PUNCT
ma-160	80	8	(	(	PUNCT
ma-160	80	9	18	18	NUM
ma-160	80	10	)	)	PUNCT
ma-160	80	11	which	which	PRON
ma-160	80	12	imlplies	imlplie	VERB
ma-160	80	13	that	that	PRON
ma-160	80	14	s	s	VERB
ma-160	80	15	′	′	NUM
ma-160	80	16	λ(t	λ(t	NOUN
ma-160	80	17	)	)	PUNCT
ma-160	80	18	sλ(t	sλ(t	NOUN
ma-160	80	19	)	)	PUNCT
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ma-160	80	21	decreasing	decrease	VERB
ma-160	80	22	on	on	ADP
ma-160	80	23	(	(	PUNCT
ma-160	80	24	0,∞	0,∞	NOUN
ma-160	80	25	)	)	PUNCT
ma-160	80	26	.	.	PUNCT
ma-160	81	1	hence	hence	ADV
ma-160	81	2	,	,	PUNCT
ma-160	81	3	by	by	ADP
ma-160	81	4	lemma	lemma	PROPN
ma-160	81	5	2.3(1	2.3(1	PROPN
ma-160	81	6	)	)	PUNCT
ma-160	81	7	,	,	PUNCT
ma-160	81	8	we	we	PRON
ma-160	81	9	obtain	obtain	VERB
ma-160	81	10	the	the	DET
ma-160	81	11	desiredresult	desiredresult	NOUN
ma-160	81	12	(	(	PUNCT
ma-160	81	13	17	17	NUM
ma-160	81	14	)	)	PUNCT
ma-160	81	15	.	.	PUNCT
ma-160	82	1	�	�	PROPN
ma-160	82	2	theorem	theorem	VERB
ma-160	82	3	3.3	3.3	NUM
ma-160	82	4	.	.	PUNCT
ma-160	83	1	the	the	DET
ma-160	83	2	function	function	NOUN
ma-160	83	3	sλ	sλ	NOUN
ma-160	83	4	(	(	PUNCT
ma-160	83	5	t	t	NOUN
ma-160	83	6	)	)	PUNCT
ma-160	83	7	is	be	AUX
ma-160	83	8	ah	ah	INTJ
ma-160	83	9	-	-	PUNCT
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ma-160	83	11	on	on	ADP
ma-160	83	12	(	(	PUNCT
ma-160	83	13	0,∞	0,∞	NOUN
ma-160	83	14	)	)	PUNCT
ma-160	83	15	.	.	PUNCT
ma-160	84	1	in	in	ADP
ma-160	84	2	other	other	ADJ
ma-160	84	3	words	word	NOUN
ma-160	84	4	,	,	PUNCT
ma-160	84	5	for	for	ADP
ma-160	84	6	all	all	DET
ma-160	84	7	r	r	NOUN
ma-160	84	8	,	,	PUNCT
ma-160	84	9	t	t	PROPN
ma-160	84	10	,	,	PUNCT
ma-160	84	11	λ	λ	PROPN
ma-160	84	12	∈	∈	PROPN
ma-160	84	13	(	(	PUNCT
ma-160	84	14	0,∞	0,∞	NOUN
ma-160	84	15	)	)	PUNCT
ma-160	84	16	,	,	PUNCT
ma-160	84	17	the	the	DET
ma-160	84	18	inequality	inequality	NOUN
ma-160	84	19	sλ	sλ	NOUN
ma-160	84	20	(	(	PUNCT
ma-160	84	21	r	r	NOUN
ma-160	84	22	+	+	PROPN
ma-160	84	23	t	t	PROPN
ma-160	84	24	2	2	NUM
ma-160	84	25	)	)	PUNCT
ma-160	84	26	≥	≥	NOUN
ma-160	84	27	2sλ	2sλ	NOUN
ma-160	84	28	(	(	PUNCT
ma-160	84	29	r)sλ	r)sλ	PROPN
ma-160	84	30	(	(	PUNCT
ma-160	84	31	t	t	PROPN
ma-160	84	32	)	)	PUNCT
ma-160	84	33	sλ	sλ	NOUN
ma-160	84	34	(	(	PUNCT
ma-160	84	35	r	r	NOUN
ma-160	84	36	)	)	PUNCT
ma-160	84	37	+	+	CCONJ
ma-160	84	38	sλ	sλ	X
ma-160	84	39	(	(	PUNCT
ma-160	84	40	t	t	NOUN
ma-160	84	41	)	)	PUNCT
ma-160	84	42	(	(	PUNCT
ma-160	84	43	19	19	NUM
ma-160	84	44	)	)	PUNCT
ma-160	84	45	is	be	AUX
ma-160	84	46	valid	valid	ADJ
ma-160	84	47	.	.	PUNCT
ma-160	85	1	https://doi.org/10.28924/ada/ma.3.17	https://doi.org/10.28924/ada/ma.3.17	PROPN
ma-160	85	2	eur	eur	PROPN
ma-160	85	3	.	.	PUNCT
ma-160	86	1	j.	j.	PROPN
ma-160	86	2	math	math	PROPN
ma-160	86	3	.	.	PUNCT
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ma-160	87	2	.	.	PUNCT
ma-160	88	1	10.28924	10.28924	NUM
ma-160	88	2	/	/	SYM
ma-160	88	3	ada	ada	PROPN
ma-160	88	4	/	/	SYM
ma-160	88	5	ma.3.17	ma.3.17	ADJ
ma-160	88	6	5	5	NUM
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ma-160	88	8	.	.	PUNCT
ma-160	89	1	now	now	ADV
ma-160	89	2	we	we	PRON
ma-160	89	3	have	have	VERB
ma-160	89	4	s	s	NUM
ma-160	90	1	′	′	NUM
ma-160	90	2	λ	λ	X
ma-160	90	3	(	(	PUNCT
ma-160	90	4	t	t	NOUN
ma-160	90	5	)	)	PUNCT
ma-160	90	6	sλ	sλ	NOUN
ma-160	90	7	(	(	PUNCT
ma-160	90	8	t)2	t)2	NOUN
ma-160	90	9	=	=	SYM
ma-160	90	10			X
ma-160	90	11	(	(	PUNCT
ma-160	90	12	1	1	NUM
ma-160	90	13	+	+	CCONJ
ma-160	90	14	λt	λt	X
ma-160	90	15	)	)	PUNCT
ma-160	90	16	1	1	NUM
ma-160	90	17	λ	λ	SYM
ma-160	90	18	−1	−1	NOUN
ma-160	90	19	[	[	PUNCT
ma-160	90	20	1	1	NUM
ma-160	90	21	+	+	CCONJ
ma-160	90	22	(	(	PUNCT
ma-160	90	23	1	1	NUM
ma-160	90	24	+	+	CCONJ
ma-160	90	25	λt	λt	X
ma-160	90	26	)	)	PUNCT
ma-160	90	27	1	1	NUM
ma-160	90	28	λ	λ	NOUN
ma-160	90	29	]	]	X
ma-160	90	30	2	2	NUM
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ma-160	90	32			X
ma-160	90	33	[	[	PUNCT
ma-160	90	34	1	1	NUM
ma-160	90	35	+	+	CCONJ
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ma-160	90	38	+	+	CCONJ
ma-160	90	39	λt	λt	X
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ma-160	90	41	1	1	NUM
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ma-160	90	44	2	2	NUM
ma-160	90	45	(	(	PUNCT
ma-160	90	46	1	1	NUM
ma-160	90	47	+	+	CCONJ
ma-160	90	48	λt	λt	X
ma-160	90	49	)	)	PUNCT
ma-160	90	50	2	2	NUM
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ma-160	90	53	=	=	NOUN
ma-160	90	54	1	1	NUM
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ma-160	90	57	+	+	CCONJ
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ma-160	90	62	+	+	CCONJ
ma-160	90	63	λt	λt	X
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ma-160	90	65	1	1	NUM
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ma-160	90	67	=	=	SYM
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ma-160	90	70	1	1	NUM
ma-160	90	71	+	+	CCONJ
ma-160	90	72	λt	λt	X
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ma-160	90	74	1	1	NUM
ma-160	90	75	λ	λ	NOUN
ma-160	90	76	+1	+1	NOUN
ma-160	91	1	and	and	CCONJ
ma-160	91	2	(	(	PUNCT
ma-160	91	3	s	s	AUX
ma-160	91	4	′	′	NUM
ma-160	91	5	λ	λ	X
ma-160	91	6	(	(	PUNCT
ma-160	91	7	t	t	NOUN
ma-160	91	8	)	)	PUNCT
ma-160	91	9	sλ	sλ	NOUN
ma-160	91	10	(	(	PUNCT
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ma-160	91	13	′	′	NUM
ma-160	92	1	=	=	PUNCT
ma-160	92	2	−	−	PROPN
ma-160	92	3	(	(	PUNCT
ma-160	92	4	1	1	NUM
ma-160	92	5	+	+	NUM
ma-160	92	6	λ	λ	NOUN
ma-160	92	7	)	)	PUNCT
ma-160	92	8	(	(	PUNCT
ma-160	92	9	1	1	NUM
ma-160	92	10	+	+	CCONJ
ma-160	92	11	λt	λt	X
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ma-160	92	13	1	1	NUM
ma-160	92	14	λ	λ	NOUN
ma-160	92	15	(	(	PUNCT
ma-160	92	16	1	1	NUM
ma-160	92	17	+	+	CCONJ
ma-160	92	18	λt	λt	X
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ma-160	92	20	2	2	NUM
ma-160	92	21	λ	λ	NOUN
ma-160	92	22	+2	+2	X
ma-160	92	23	<	<	X
ma-160	92	24	0	0	X
ma-160	92	25	.	.	PUNCT
ma-160	92	26	by	by	ADP
ma-160	92	27	lemma	lemma	PROPN
ma-160	92	28	2.3(2	2.3(2	NUM
ma-160	92	29	)	)	PUNCT
ma-160	92	30	,	,	PUNCT
ma-160	92	31	we	we	PRON
ma-160	92	32	conclude	conclude	VERB
ma-160	92	33	that	that	DET
ma-160	92	34	sλ	sλ	NOUN
ma-160	92	35	(	(	PUNCT
ma-160	92	36	t	t	NOUN
ma-160	92	37	)	)	PUNCT
ma-160	92	38	is	be	AUX
ma-160	92	39	ah	ah	INTJ
ma-160	92	40	-	-	PUNCT
ma-160	92	41	concave	concave	NOUN
ma-160	92	42	on	on	ADP
ma-160	92	43	(	(	PUNCT
ma-160	92	44	0,∞	0,∞	NUM
ma-160	92	45	)	)	PUNCT
ma-160	92	46	.	.	PUNCT
ma-160	93	1	this	this	PRON
ma-160	93	2	implies	imply	VERB
ma-160	93	3	inequality(19	inequality(19	NOUN
ma-160	93	4	)	)	PUNCT
ma-160	93	5	.	.	PUNCT
ma-160	94	1	�	�	PROPN
ma-160	94	2	theorem	theorem	VERB
ma-160	94	3	3.4	3.4	NUM
ma-160	94	4	.	.	PUNCT
ma-160	95	1	the	the	DET
ma-160	95	2	function	function	NOUN
ma-160	95	3	sλ	sλ	NOUN
ma-160	95	4	(	(	PUNCT
ma-160	95	5	t	t	PROPN
ma-160	95	6	)	)	PUNCT
ma-160	95	7	,	,	PUNCT
ma-160	95	8	for	for	ADP
ma-160	95	9	r	r	NOUN
ma-160	95	10	,	,	PUNCT
ma-160	95	11	t	t	PROPN
ma-160	95	12	,	,	PUNCT
ma-160	95	13	λ	λ	PROPN
ma-160	95	14	∈	∈	PROPN
ma-160	95	15	(	(	PUNCT
ma-160	95	16	0,∞	0,∞	NOUN
ma-160	95	17	)	)	PUNCT
ma-160	95	18	and	and	CCONJ
ma-160	95	19	z2	z2	PROPN
ma-160	95	20	=	=	SYM
ma-160	95	21	r2+t2	r2+t2	PROPN
ma-160	95	22	,	,	PUNCT
ma-160	95	23	satisfies	satisfy	VERB
ma-160	95	24	the	the	DET
ma-160	95	25	grunbaum	grunbaum	NOUN
ma-160	95	26	-	-	PUNCT
ma-160	95	27	type	type	NOUN
ma-160	95	28	inequality	inequality	NOUN
ma-160	95	29	1	1	NUM
ma-160	95	30	+	+	CCONJ
ma-160	95	31	sλ	sλ	NOUN
ma-160	95	32	(	(	PUNCT
ma-160	95	33	z2	z2	PROPN
ma-160	95	34	)	)	PUNCT
ma-160	95	35	≥	≥	NOUN
ma-160	95	36	sλ	sλ	NOUN
ma-160	95	37	(	(	PUNCT
ma-160	95	38	r2	r2	PROPN
ma-160	95	39	)	)	PUNCT
ma-160	96	1	+	+	CCONJ
ma-160	96	2	sλ	sλ	NOUN
ma-160	96	3	(	(	PUNCT
ma-160	96	4	t2	t2	PROPN
ma-160	96	5	)	)	PUNCT
ma-160	96	6	.	.	PUNCT
ma-160	97	1	(	(	PUNCT
ma-160	97	2	20	20	X
ma-160	97	3	)	)	PUNCT
ma-160	97	4	proof	proof	NOUN
ma-160	97	5	.	.	PUNCT
ma-160	98	1	let	let	VERB
ma-160	98	2	h	h	NOUN
ma-160	98	3	(	(	PUNCT
ma-160	98	4	t	t	PROPN
ma-160	98	5	)	)	PUNCT
ma-160	98	6	be	be	AUX
ma-160	98	7	defined	define	VERB
ma-160	98	8	for	for	ADP
ma-160	98	9	t	t	PROPN
ma-160	98	10	,	,	PUNCT
ma-160	99	1	λ	λ	PROPN
ma-160	99	2	∈	∈	PROPN
ma-160	99	3	(	(	PUNCT
ma-160	99	4	0,∞	0,∞	NOUN
ma-160	99	5	)	)	PUNCT
ma-160	99	6	as	as	ADP
ma-160	99	7	h	h	PROPN
ma-160	99	8	(	(	PUNCT
ma-160	99	9	t	t	PROPN
ma-160	99	10	)	)	PUNCT
ma-160	99	11	=	=	PRON
ma-160	99	12	sλ(t)−1	sλ(t)−1	PROPN
ma-160	99	13	t	t	PROPN
ma-160	99	14	.	.	PUNCT
ma-160	100	1	this	this	PRON
ma-160	100	2	implies	imply	VERB
ma-160	100	3	h	h	PROPN
ma-160	100	4	(	(	PUNCT
ma-160	100	5	t	t	PROPN
ma-160	100	6	)	)	PUNCT
ma-160	100	7	=	=	PUNCT
ma-160	100	8	(	(	PUNCT
ma-160	100	9	1+λt	1+λt	NUM
ma-160	100	10	)	)	PUNCT
ma-160	100	11	1	1	NUM
ma-160	100	12	λ	λ	NOUN
ma-160	100	13	1+(1+λt	1+(1+λt	NOUN
ma-160	100	14	)	)	PUNCT
ma-160	100	15	1	1	NUM
ma-160	100	16	λ	λ	NOUN
ma-160	100	17	−	−	PROPN
ma-160	100	18	1	1	NUM
ma-160	100	19	t	t	NOUN
ma-160	100	20	=	=	NOUN
ma-160	100	21	−	−	PROPN
ma-160	100	22	1	1	NUM
ma-160	100	23	t	t	NOUN
ma-160	100	24	+	+	X
ma-160	100	25	t	t	PROPN
ma-160	100	26	(	(	PUNCT
ma-160	100	27	1	1	NUM
ma-160	100	28	+	+	CCONJ
ma-160	100	29	λt	λt	X
ma-160	100	30	)	)	PUNCT
ma-160	100	31	1	1	NUM
ma-160	100	32	λ	λ	NOUN
ma-160	100	33	.	.	PUNCT
ma-160	101	1	differentiating	differentiate	VERB
ma-160	101	2	h	h	NOUN
ma-160	101	3	(	(	PUNCT
ma-160	101	4	t	t	PROPN
ma-160	101	5	)	)	PUNCT
ma-160	101	6	,	,	PUNCT
ma-160	101	7	we	we	PRON
ma-160	101	8	have	have	VERB
ma-160	101	9	h	h	NOUN
ma-160	101	10	′	′	NUM
ma-160	101	11	(	(	PUNCT
ma-160	101	12	t	t	NOUN
ma-160	101	13	)	)	PUNCT
ma-160	101	14	=	=	SYM
ma-160	102	1	1	1	NUM
ma-160	102	2	+	+	CCONJ
ma-160	102	3	(	(	PUNCT
ma-160	102	4	1	1	NUM
ma-160	102	5	+	+	CCONJ
ma-160	102	6	λt	λt	X
ma-160	102	7	)	)	PUNCT
ma-160	102	8	1	1	NUM
ma-160	102	9	λ	λ	NOUN
ma-160	102	10	+	+	X
ma-160	102	11	t	t	PROPN
ma-160	102	12	(	(	PUNCT
ma-160	102	13	1	1	NUM
ma-160	102	14	+	+	CCONJ
ma-160	102	15	λt	λt	X
ma-160	102	16	)	)	PUNCT
ma-160	102	17	1	1	NUM
ma-160	102	18	λ	λ	SYM
ma-160	102	19	−1	−1	NOUN
ma-160	102	20	[	[	PUNCT
ma-160	102	21	t	t	NOUN
ma-160	102	22	+	+	X
ma-160	102	23	t	t	PROPN
ma-160	102	24	(	(	PUNCT
ma-160	102	25	1	1	NUM
ma-160	102	26	+	+	CCONJ
ma-160	102	27	λt	λt	X
ma-160	102	28	)	)	PUNCT
ma-160	102	29	1	1	NUM
ma-160	102	30	λ	λ	NOUN
ma-160	102	31	]	]	X
ma-160	102	32	2	2	NUM
ma-160	102	33	>	>	SYM
ma-160	102	34	0	0	NUM
ma-160	102	35	,	,	PUNCT
ma-160	102	36	which	which	PRON
ma-160	102	37	implies	imply	VERB
ma-160	102	38	that	that	SCONJ
ma-160	102	39	h	h	NOUN
ma-160	102	40	(	(	PUNCT
ma-160	102	41	t	t	PROPN
ma-160	102	42	)	)	PUNCT
ma-160	102	43	is	be	AUX
ma-160	102	44	increasing	increase	VERB
ma-160	102	45	.	.	PUNCT
ma-160	103	1	by	by	ADP
ma-160	103	2	applying	apply	VERB
ma-160	103	3	lemma	lemma	PROPN
ma-160	103	4	2.4	2.4	NUM
ma-160	103	5	,	,	PUNCT
ma-160	103	6	we	we	PRON
ma-160	103	7	obtain	obtain	VERB
ma-160	103	8	the	the	DET
ma-160	103	9	desired	desire	VERB
ma-160	103	10	result	result	NOUN
ma-160	103	11	(	(	PUNCT
ma-160	103	12	20	20	NUM
ma-160	103	13	)	)	PUNCT
ma-160	103	14	.	.	PUNCT
ma-160	104	1	�	�	PROPN
ma-160	104	2	theorem	theorem	VERB
ma-160	104	3	3.5	3.5	NUM
ma-160	104	4	.	.	PUNCT
ma-160	105	1	for	for	ADP
ma-160	105	2	λ	λ	PROPN
ma-160	105	3	∈	∈	PROPN
ma-160	105	4	(	(	PUNCT
ma-160	105	5	0,∞	0,∞	NOUN
ma-160	105	6	)	)	PUNCT
ma-160	105	7	,	,	PUNCT
ma-160	105	8	the	the	DET
ma-160	105	9	function	function	NOUN
ma-160	105	10	sλ	sλ	NOUN
ma-160	105	11	(	(	PUNCT
ma-160	105	12	t	t	NOUN
ma-160	105	13	)	)	PUNCT
ma-160	105	14	satisfies	satisfy	VERB
ma-160	105	15	the	the	DET
ma-160	105	16	inequalities	inequality	NOUN
ma-160	105	17	s2λ	s2λ	X
ma-160	105	18	(	(	PUNCT
ma-160	105	19	r	r	NOUN
ma-160	105	20	+	+	PROPN
ma-160	105	21	t	t	PROPN
ma-160	105	22	)	)	PUNCT
ma-160	105	23	≥	≥	NOUN
ma-160	105	24	sλ	sλ	NOUN
ma-160	105	25	(	(	PUNCT
ma-160	105	26	r)sλ	r)sλ	PROPN
ma-160	105	27	(	(	PUNCT
ma-160	105	28	t	t	PROPN
ma-160	105	29	)	)	PUNCT
ma-160	105	30	,	,	PUNCT
ma-160	105	31	r	r	X
ma-160	105	32	,	,	PUNCT
ma-160	105	33	t	t	PROPN
ma-160	105	34	∈	∈	PROPN
ma-160	106	1	[	[	X
ma-160	106	2	0,∞	0,∞	NUM
ma-160	106	3	)	)	PUNCT
ma-160	106	4	(	(	PUNCT
ma-160	106	5	21	21	NUM
ma-160	106	6	)	)	PUNCT
ma-160	106	7	and	and	CCONJ
ma-160	106	8	s2λ	s2λ	X
ma-160	106	9	(	(	PUNCT
ma-160	106	10	r	r	NOUN
ma-160	106	11	+	+	NOUN
ma-160	106	12	t	t	NOUN
ma-160	106	13	)	)	PUNCT
ma-160	106	14	≤	≤	NUM
ma-160	106	15	sλ	sλ	NOUN
ma-160	106	16	(	(	PUNCT
ma-160	106	17	r)sλ	r)sλ	PROPN
ma-160	106	18	(	(	PUNCT
ma-160	106	19	t	t	PROPN
ma-160	106	20	)	)	PUNCT
ma-160	106	21	,	,	PUNCT
ma-160	106	22	r	r	X
ma-160	106	23	,	,	PUNCT
ma-160	106	24	t	t	PROPN
ma-160	106	25	∈	∈	PROPN
ma-160	106	26	(	(	PUNCT
ma-160	106	27	−∞	−∞	NOUN
ma-160	106	28	,	,	PUNCT
ma-160	106	29	0	0	NUM
ma-160	106	30	]	]	PUNCT
ma-160	106	31	.	.	PUNCT
ma-160	107	1	(	(	PUNCT
ma-160	107	2	22	22	X
ma-160	107	3	)	)	PUNCT
ma-160	107	4	equality	equality	NOUN
ma-160	107	5	holds	hold	VERB
ma-160	107	6	if	if	SCONJ
ma-160	107	7	r	r	NOUN
ma-160	107	8	=	=	SYM
ma-160	107	9	t	t	NOUN
ma-160	107	10	=	=	SYM
ma-160	107	11	0	0	PROPN
ma-160	107	12	.	.	PUNCT
ma-160	108	1	https://doi.org/10.28924/ada/ma.3.17	https://doi.org/10.28924/ada/ma.3.17	PROPN
ma-160	108	2	eur	eur	PROPN
ma-160	108	3	.	.	PUNCT
ma-160	109	1	j.	j.	PROPN
ma-160	109	2	math	math	PROPN
ma-160	109	3	.	.	PUNCT
ma-160	110	1	anal	anal	PROPN
ma-160	110	2	.	.	PUNCT
ma-160	111	1	10.28924	10.28924	NUM
ma-160	111	2	/	/	SYM
ma-160	111	3	ada	ada	PROPN
ma-160	111	4	/	/	SYM
ma-160	111	5	ma.3.17	ma.3.17	ADJ
ma-160	111	6	6	6	NUM
ma-160	111	7	proof	proof	NOUN
ma-160	111	8	.	.	PUNCT
ma-160	112	1	let	let	VERB
ma-160	112	2	r	r	NOUN
ma-160	112	3	,	,	PUNCT
ma-160	112	4	t	t	PROPN
ma-160	112	5	∈	∈	PROPN
ma-160	113	1	[	[	X
ma-160	113	2	0,∞	0,∞	NUM
ma-160	113	3	)	)	PUNCT
ma-160	113	4	and	and	CCONJ
ma-160	113	5	λ	λ	X
ma-160	113	6	∈	∈	PROPN
ma-160	113	7	(	(	PUNCT
ma-160	113	8	0,∞	0,∞	NOUN
ma-160	113	9	)	)	PUNCT
ma-160	113	10	.	.	PUNCT
ma-160	114	1	recall	recall	VERB
ma-160	114	2	that	that	DET
ma-160	114	3	sλ	sλ	NOUN
ma-160	114	4	(	(	PUNCT
ma-160	114	5	t	t	NOUN
ma-160	114	6	)	)	PUNCT
ma-160	114	7	is	be	AUX
ma-160	114	8	increasing	increase	VERB
ma-160	114	9	.	.	PUNCT
ma-160	115	1	thus	thus	ADV
ma-160	115	2	we	we	PRON
ma-160	115	3	have	have	VERB
ma-160	115	4	sλ	sλ	NOUN
ma-160	115	5	(	(	PUNCT
ma-160	115	6	r	r	NOUN
ma-160	115	7	+	+	NOUN
ma-160	115	8	t	t	PROPN
ma-160	115	9	)	)	PUNCT
ma-160	115	10	≥	≥	NOUN
ma-160	115	11	sλ	sλ	NOUN
ma-160	115	12	(	(	PUNCT
ma-160	115	13	r	r	NOUN
ma-160	115	14	)	)	PUNCT
ma-160	115	15	>	>	X
ma-160	115	16	0	0	NUM
ma-160	115	17	,	,	PUNCT
ma-160	115	18	(	(	PUNCT
ma-160	115	19	23	23	NUM
ma-160	115	20	)	)	PUNCT
ma-160	115	21	sλ	sλ	NOUN
ma-160	115	22	(	(	PUNCT
ma-160	115	23	r	r	NOUN
ma-160	115	24	+	+	NOUN
ma-160	115	25	t	t	PROPN
ma-160	115	26	)	)	PUNCT
ma-160	115	27	≥	≥	NOUN
ma-160	115	28	sλ	sλ	NOUN
ma-160	115	29	(	(	PUNCT
ma-160	115	30	t	t	NOUN
ma-160	115	31	)	)	PUNCT
ma-160	115	32	>	>	X
ma-160	115	33	0	0	NUM
ma-160	115	34	,	,	PUNCT
ma-160	115	35	(	(	PUNCT
ma-160	115	36	24	24	NUM
ma-160	115	37	)	)	PUNCT
ma-160	115	38	since	since	SCONJ
ma-160	115	39	r	r	NOUN
ma-160	115	40	+	+	PROPN
ma-160	115	41	t	t	PROPN
ma-160	115	42	≥	≥	NOUN
ma-160	115	43	r	r	NOUN
ma-160	115	44	and	and	CCONJ
ma-160	115	45	r	r	PROPN
ma-160	115	46	+	+	PROPN
ma-160	115	47	t	t	NOUN
ma-160	115	48	≥	≥	NOUN
ma-160	115	49	t.	t.	PROPN
ma-160	115	50	now	now	ADV
ma-160	115	51	by	by	ADP
ma-160	115	52	multiplying	multiply	VERB
ma-160	115	53	(	(	PUNCT
ma-160	115	54	23)and	23)and	NUM
ma-160	115	55	(	(	PUNCT
ma-160	115	56	24	24	NUM
ma-160	115	57	)	)	PUNCT
ma-160	115	58	,	,	PUNCT
ma-160	115	59	we	we	PRON
ma-160	115	60	obtain	obtain	VERB
ma-160	115	61	the	the	DET
ma-160	115	62	desired	desire	VERB
ma-160	115	63	result	result	NOUN
ma-160	115	64	(	(	PUNCT
ma-160	115	65	21).next	21).next	NUM
ma-160	115	66	,	,	PUNCT
ma-160	115	67	let	let	VERB
ma-160	115	68	r	r	NOUN
ma-160	115	69	,	,	PUNCT
ma-160	115	70	t	t	PROPN
ma-160	115	71	∈	∈	PROPN
ma-160	115	72	(	(	PUNCT
ma-160	115	73	−∞	−∞	NOUN
ma-160	115	74	,	,	PUNCT
ma-160	115	75	0	0	NUM
ma-160	115	76	]	]	PUNCT
ma-160	115	77	and	and	CCONJ
ma-160	115	78	λ	λ	X
ma-160	115	79	∈	∈	PROPN
ma-160	115	80	(	(	PUNCT
ma-160	115	81	0,∞	0,∞	NOUN
ma-160	115	82	)	)	PUNCT
ma-160	115	83	,	,	PUNCT
ma-160	115	84	we	we	PRON
ma-160	115	85	have	have	VERB
ma-160	115	86	0	0	NUM
ma-160	115	87	<	<	X
ma-160	115	88	sλ	sλ	NOUN
ma-160	115	89	(	(	PUNCT
ma-160	115	90	r	r	NOUN
ma-160	115	91	+	+	NOUN
ma-160	115	92	t	t	NOUN
ma-160	115	93	)	)	PUNCT
ma-160	115	94	≤	≤	NUM
ma-160	116	1	sλ	sλ	NOUN
ma-160	116	2	(	(	PUNCT
ma-160	116	3	r	r	NOUN
ma-160	116	4	)	)	PUNCT
ma-160	116	5	,	,	PUNCT
ma-160	116	6	(	(	PUNCT
ma-160	116	7	25	25	NUM
ma-160	116	8	)	)	PUNCT
ma-160	116	9	0	0	PUNCT
ma-160	117	1	<	<	X
ma-160	117	2	sλ	sλ	NOUN
ma-160	117	3	(	(	PUNCT
ma-160	117	4	r	r	NOUN
ma-160	117	5	+	+	NOUN
ma-160	117	6	t	t	NOUN
ma-160	117	7	)	)	PUNCT
ma-160	117	8	≤	≤	NUM
ma-160	117	9	sλ	sλ	NOUN
ma-160	117	10	(	(	PUNCT
ma-160	117	11	t	t	PROPN
ma-160	117	12	)	)	PUNCT
ma-160	117	13	,	,	PUNCT
ma-160	117	14	(	(	PUNCT
ma-160	117	15	26	26	NUM
ma-160	117	16	)	)	PUNCT
ma-160	117	17	since	since	SCONJ
ma-160	117	18	r	r	NOUN
ma-160	117	19	+	+	NUM
ma-160	117	20	t	t	NOUN
ma-160	117	21	≤	≤	NUM
ma-160	117	22	r	r	NOUN
ma-160	117	23	and	and	CCONJ
ma-160	117	24	r	r	NOUN
ma-160	117	25	+	+	PROPN
ma-160	117	26	t	t	NOUN
ma-160	117	27	≤	≤	NOUN
ma-160	117	28	t.	t.	NOUN
ma-160	117	29	by	by	ADP
ma-160	117	30	multiplying	multiply	VERB
ma-160	117	31	the	the	DET
ma-160	117	32	inequalities	inequality	NOUN
ma-160	117	33	(	(	PUNCT
ma-160	117	34	25	25	NUM
ma-160	117	35	)	)	PUNCT
ma-160	117	36	and	and	CCONJ
ma-160	117	37	(	(	PUNCT
ma-160	117	38	26	26	NUM
ma-160	117	39	)	)	PUNCT
ma-160	117	40	,	,	PUNCT
ma-160	117	41	we	we	PRON
ma-160	117	42	have	have	VERB
ma-160	117	43	the	the	DET
ma-160	117	44	desiredresult	desiredresult	NOUN
ma-160	117	45	.	.	PUNCT
ma-160	118	1	�	�	PROPN
ma-160	118	2	theorem	theorem	VERB
ma-160	118	3	3.6	3.6	NUM
ma-160	118	4	.	.	PUNCT
ma-160	119	1	the	the	DET
ma-160	119	2	function	function	NOUN
ma-160	119	3	sλ	sλ	NOUN
ma-160	119	4	(	(	PUNCT
ma-160	119	5	t	t	PROPN
ma-160	119	6	)	)	PUNCT
ma-160	119	7	,	,	PUNCT
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ma-160	119	9	λ	λ	PROPN
ma-160	119	10	∈	∈	PROPN
ma-160	119	11	(	(	PUNCT
ma-160	119	12	0,∞	0,∞	NOUN
ma-160	119	13	)	)	PUNCT
ma-160	119	14	,	,	PUNCT
ma-160	119	15	satisfies	satisfy	VERB
ma-160	119	16	the	the	DET
ma-160	119	17	inequalities	inequality	NOUN
ma-160	119	18	s2λ	s2λ	X
ma-160	119	19	(	(	PUNCT
ma-160	119	20	r	r	NOUN
ma-160	119	21	t	t	PROPN
ma-160	119	22	)	)	PUNCT
ma-160	119	23	≤	≤	NUM
ma-160	119	24	sλ	sλ	NOUN
ma-160	119	25	(	(	PUNCT
ma-160	119	26	r)sλ	r)sλ	PROPN
ma-160	119	27	(	(	PUNCT
ma-160	119	28	t	t	PROPN
ma-160	119	29	)	)	PUNCT
ma-160	119	30	,	,	PUNCT
ma-160	119	31	r	r	X
ma-160	119	32	,	,	PUNCT
ma-160	119	33	t	t	PROPN
ma-160	119	34	∈	∈	PROPN
ma-160	120	1	[	[	X
ma-160	120	2	0	0	NUM
ma-160	120	3	,	,	PUNCT
ma-160	120	4	1	1	NUM
ma-160	120	5	]	]	PUNCT
ma-160	120	6	(	(	PUNCT
ma-160	120	7	27	27	NUM
ma-160	120	8	)	)	PUNCT
ma-160	120	9	and	and	CCONJ
ma-160	120	10	s2λ	s2λ	X
ma-160	120	11	(	(	PUNCT
ma-160	120	12	r	r	NOUN
ma-160	120	13	t	t	PROPN
ma-160	120	14	)	)	PUNCT
ma-160	120	15	≥	≥	NOUN
ma-160	120	16	sλ	sλ	NOUN
ma-160	120	17	(	(	PUNCT
ma-160	120	18	r)sλ	r)sλ	PROPN
ma-160	120	19	(	(	PUNCT
ma-160	120	20	t	t	PROPN
ma-160	120	21	)	)	PUNCT
ma-160	120	22	,	,	PUNCT
ma-160	120	23	r	r	X
ma-160	120	24	,	,	PUNCT
ma-160	120	25	t	t	PROPN
ma-160	120	26	∈	∈	PROPN
ma-160	121	1	[	[	X
ma-160	121	2	1,∞	1,∞	NUM
ma-160	121	3	)	)	PUNCT
ma-160	121	4	.	.	PUNCT
ma-160	122	1	(	(	PUNCT
ma-160	122	2	28	28	X
ma-160	122	3	)	)	PUNCT
ma-160	122	4	equality	equality	NOUN
ma-160	122	5	holds	hold	VERB
ma-160	122	6	if	if	SCONJ
ma-160	122	7	r	r	NOUN
ma-160	122	8	=	=	SYM
ma-160	122	9	t	t	NOUN
ma-160	122	10	=	=	SYM
ma-160	122	11	1	1	X
ma-160	122	12	.	.	PUNCT
ma-160	123	1	proof	proof	NOUN
ma-160	123	2	.	.	PUNCT
ma-160	124	1	let	let	VERB
ma-160	124	2	r	r	NOUN
ma-160	124	3	,	,	PUNCT
ma-160	124	4	t	t	PROPN
ma-160	124	5	∈	∈	PROPN
ma-160	125	1	[	[	X
ma-160	125	2	0	0	NUM
ma-160	125	3	,	,	PUNCT
ma-160	125	4	1	1	NUM
ma-160	125	5	]	]	PUNCT
ma-160	125	6	and	and	CCONJ
ma-160	125	7	λ	λ	X
ma-160	125	8	∈	∈	PROPN
ma-160	125	9	(	(	PUNCT
ma-160	125	10	0,∞	0,∞	NOUN
ma-160	125	11	)	)	PUNCT
ma-160	125	12	.	.	PUNCT
ma-160	126	1	recall	recall	VERB
ma-160	126	2	that	that	DET
ma-160	126	3	sλ	sλ	NOUN
ma-160	126	4	(	(	PUNCT
ma-160	126	5	t	t	NOUN
ma-160	126	6	)	)	PUNCT
ma-160	126	7	is	be	AUX
ma-160	126	8	increasing	increase	VERB
ma-160	126	9	.	.	PUNCT
ma-160	127	1	thus	thus	ADV
ma-160	127	2	we	we	PRON
ma-160	127	3	have	have	VERB
ma-160	127	4	0	0	NUM
ma-160	127	5	<	<	X
ma-160	127	6	sλ	sλ	NOUN
ma-160	127	7	(	(	PUNCT
ma-160	127	8	r	r	NOUN
ma-160	127	9	t	t	PROPN
ma-160	127	10	)	)	PUNCT
ma-160	127	11	≤	≤	NUM
ma-160	127	12	sλ	sλ	NOUN
ma-160	127	13	(	(	PUNCT
ma-160	127	14	r	r	NOUN
ma-160	127	15	)	)	PUNCT
ma-160	127	16	,	,	PUNCT
ma-160	127	17	(	(	PUNCT
ma-160	127	18	29	29	NUM
ma-160	127	19	)	)	PUNCT
ma-160	127	20	0	0	PUNCT
ma-160	128	1	<	<	X
ma-160	128	2	sλ	sλ	NOUN
ma-160	128	3	(	(	PUNCT
ma-160	128	4	r	r	NOUN
ma-160	128	5	t	t	PROPN
ma-160	128	6	)	)	PUNCT
ma-160	128	7	≤	≤	NUM
ma-160	128	8	sλ	sλ	NOUN
ma-160	128	9	(	(	PUNCT
ma-160	128	10	t	t	PROPN
ma-160	128	11	)	)	PUNCT
ma-160	128	12	,	,	PUNCT
ma-160	128	13	(	(	PUNCT
ma-160	128	14	30	30	NUM
ma-160	128	15	)	)	PUNCT
ma-160	128	16	since	since	SCONJ
ma-160	128	17	r	r	NOUN
ma-160	128	18	t	t	NOUN
ma-160	128	19	≤	≤	NOUN
ma-160	128	20	r	r	NOUN
ma-160	128	21	and	and	CCONJ
ma-160	128	22	r	r	NOUN
ma-160	128	23	t	t	NOUN
ma-160	128	24	≤	≤	NOUN
ma-160	128	25	t.	t.	NOUN
ma-160	128	26	now	now	ADV
ma-160	128	27	by	by	ADP
ma-160	128	28	multiplying	multiply	VERB
ma-160	128	29	(	(	PUNCT
ma-160	128	30	29)and	29)and	NUM
ma-160	128	31	(	(	PUNCT
ma-160	128	32	30	30	NUM
ma-160	128	33	)	)	PUNCT
ma-160	128	34	,	,	PUNCT
ma-160	128	35	we	we	PRON
ma-160	128	36	obtain	obtain	VERB
ma-160	128	37	the	the	DET
ma-160	128	38	result	result	NOUN
ma-160	128	39	(	(	PUNCT
ma-160	128	40	27).next	27).next	NUM
ma-160	128	41	,	,	PUNCT
ma-160	128	42	let	let	VERB
ma-160	128	43	r	r	NOUN
ma-160	128	44	,	,	PUNCT
ma-160	128	45	t	t	PROPN
ma-160	128	46	∈	∈	PROPN
ma-160	129	1	[	[	X
ma-160	129	2	1,∞	1,∞	NUM
ma-160	129	3	,	,	PUNCT
ma-160	129	4	)	)	PUNCT
ma-160	129	5	and	and	CCONJ
ma-160	129	6	λ	λ	X
ma-160	129	7	∈	∈	PROPN
ma-160	129	8	(	(	PUNCT
ma-160	129	9	0,∞	0,∞	NOUN
ma-160	129	10	)	)	PUNCT
ma-160	130	1	,	,	PUNCT
ma-160	130	2	we	we	PRON
ma-160	130	3	have	have	VERB
ma-160	130	4	sλ	sλ	NOUN
ma-160	130	5	(	(	PUNCT
ma-160	130	6	r	r	NOUN
ma-160	130	7	t	t	PROPN
ma-160	130	8	)	)	PUNCT
ma-160	130	9	≥	≥	NOUN
ma-160	130	10	sλ	sλ	NOUN
ma-160	130	11	(	(	PUNCT
ma-160	130	12	r	r	NOUN
ma-160	130	13	)	)	PUNCT
ma-160	130	14	>	>	X
ma-160	130	15	0	0	NUM
ma-160	130	16	,	,	PUNCT
ma-160	130	17	(	(	PUNCT
ma-160	130	18	31	31	NUM
ma-160	130	19	)	)	PUNCT
ma-160	130	20	sλ	sλ	NOUN
ma-160	130	21	(	(	PUNCT
ma-160	130	22	r	r	NOUN
ma-160	130	23	t	t	PROPN
ma-160	130	24	)	)	PUNCT
ma-160	130	25	≥	≥	NOUN
ma-160	130	26	sλ	sλ	NOUN
ma-160	130	27	(	(	PUNCT
ma-160	130	28	t	t	NOUN
ma-160	130	29	)	)	PUNCT
ma-160	130	30	>	>	X
ma-160	130	31	0	0	NUM
ma-160	130	32	,	,	PUNCT
ma-160	130	33	(	(	PUNCT
ma-160	130	34	32	32	NUM
ma-160	130	35	)	)	PUNCT
ma-160	130	36	since	since	SCONJ
ma-160	130	37	r	r	PROPN
ma-160	130	38	t	t	PROPN
ma-160	130	39	≥	≥	NOUN
ma-160	130	40	r	r	NOUN
ma-160	130	41	and	and	CCONJ
ma-160	130	42	r	r	NOUN
ma-160	130	43	t	t	NOUN
ma-160	130	44	≥	≥	NOUN
ma-160	130	45	t.	t.	X
ma-160	130	46	by	by	ADP
ma-160	130	47	multiplying	multiply	VERB
ma-160	130	48	the	the	DET
ma-160	130	49	inequalities	inequality	NOUN
ma-160	130	50	(	(	PUNCT
ma-160	130	51	31	31	NUM
ma-160	130	52	)	)	PUNCT
ma-160	130	53	and	and	CCONJ
ma-160	130	54	(	(	PUNCT
ma-160	130	55	32	32	NUM
ma-160	130	56	)	)	PUNCT
ma-160	130	57	,	,	PUNCT
ma-160	130	58	the	the	DET
ma-160	130	59	desired	desire	VERB
ma-160	130	60	result	result	NOUN
ma-160	130	61	isobtained	isobtaine	VERB
ma-160	130	62	(	(	PUNCT
ma-160	130	63	28	28	NUM
ma-160	130	64	)	)	PUNCT
ma-160	130	65	.	.	PUNCT
ma-160	131	1	�	�	PROPN
ma-160	131	2	theorem	theorem	VERB
ma-160	131	3	3.7	3.7	NUM
ma-160	131	4	.	.	PUNCT
ma-160	132	1	for	for	ADP
ma-160	132	2	r	r	NOUN
ma-160	132	3	,	,	PUNCT
ma-160	132	4	t	t	PROPN
ma-160	132	5	∈	∈	PROPN
ma-160	132	6	(	(	PUNCT
ma-160	132	7	−∞,∞	−∞,∞	NOUN
ma-160	132	8	)	)	PUNCT
ma-160	132	9	and	and	CCONJ
ma-160	132	10	λ	λ	X
ma-160	132	11	∈	∈	PROPN
ma-160	132	12	(	(	PUNCT
ma-160	132	13	0,∞	0,∞	NOUN
ma-160	132	14	)	)	PUNCT
ma-160	132	15	,	,	PUNCT
ma-160	132	16	the	the	DET
ma-160	132	17	function	function	NOUN
ma-160	132	18	sλ	sλ	NOUN
ma-160	132	19	(	(	PUNCT
ma-160	132	20	t	t	NOUN
ma-160	132	21	)	)	PUNCT
ma-160	132	22	is	be	AUX
ma-160	132	23	logarithmically	logarithmically	ADV
ma-160	132	24	concave	concave	VERB
ma-160	132	25	.	.	PUNCT
ma-160	133	1	in	in	ADP
ma-160	133	2	other	other	ADJ
ma-160	133	3	words	word	NOUN
ma-160	133	4	,	,	PUNCT
ma-160	133	5	the	the	DET
ma-160	133	6	inequality	inequality	NOUN
ma-160	133	7	sλ	sλ	NOUN
ma-160	133	8	(	(	PUNCT
ma-160	133	9	r	r	NOUN
ma-160	133	10	a	a	PROPN
ma-160	133	11	+	+	NUM
ma-160	133	12	t	t	PROPN
ma-160	133	13	b	b	PROPN
ma-160	133	14	)	)	PUNCT
ma-160	133	15	≥	≥	NOUN
ma-160	133	16	[	[	X
ma-160	133	17	sλ	sλ	X
ma-160	133	18	(	(	PUNCT
ma-160	133	19	r	r	NOUN
ma-160	133	20	)	)	PUNCT
ma-160	133	21	]	]	PUNCT
ma-160	134	1	1	1	NUM
ma-160	134	2	a	a	PRON
ma-160	134	3	[	[	X
ma-160	134	4	sλ	sλ	NOUN
ma-160	134	5	(	(	PUNCT
ma-160	134	6	t	t	NOUN
ma-160	134	7	)	)	PUNCT
ma-160	134	8	]	]	PUNCT
ma-160	134	9	1	1	NUM
ma-160	134	10	b	b	X
ma-160	134	11	(	(	PUNCT
ma-160	134	12	33	33	NUM
ma-160	134	13	)	)	PUNCT
ma-160	134	14	is	be	AUX
ma-160	134	15	satisfied	satisfied	ADJ
ma-160	134	16	.	.	PUNCT
ma-160	135	1	where	where	SCONJ
ma-160	135	2	a	a	DET
ma-160	135	3	>	>	SYM
ma-160	135	4	1	1	NUM
ma-160	135	5	and	and	CCONJ
ma-160	135	6	1a	1a	NOUN
ma-160	135	7	+	+	CCONJ
ma-160	135	8	1	1	NUM
ma-160	135	9	b	b	X
ma-160	135	10	=	=	SYM
ma-160	135	11	1	1	PROPN
ma-160	135	12	.	.	PUNCT
ma-160	135	13	https://doi.org/10.28924/ada/ma.3.17	https://doi.org/10.28924/ada/ma.3.17	PROPN
ma-160	135	14	eur	eur	PROPN
ma-160	135	15	.	.	PUNCT
ma-160	136	1	j.	j.	PROPN
ma-160	136	2	math	math	PROPN
ma-160	136	3	.	.	PUNCT
ma-160	137	1	anal	anal	PROPN
ma-160	137	2	.	.	PUNCT
ma-160	138	1	10.28924	10.28924	NUM
ma-160	138	2	/	/	SYM
ma-160	138	3	ada	ada	PROPN
ma-160	138	4	/	/	SYM
ma-160	138	5	ma.3.17	ma.3.17	ADJ
ma-160	138	6	7	7	NUM
ma-160	138	7	proof	proof	NOUN
ma-160	138	8	.	.	PUNCT
ma-160	139	1	let	let	VERB
ma-160	139	2	q	q	NOUN
ma-160	139	3	(	(	PUNCT
ma-160	139	4	t	t	NOUN
ma-160	139	5	)	)	PUNCT
ma-160	139	6	=	=	SYM
ma-160	139	7	lnsλ	lnsλ	NOUN
ma-160	139	8	(	(	PUNCT
ma-160	139	9	t	t	PROPN
ma-160	139	10	)	)	PUNCT
ma-160	139	11	.	.	PUNCT
ma-160	140	1	then	then	ADV
ma-160	140	2	,	,	PUNCT
ma-160	140	3	q	q	NOUN
ma-160	140	4	′	′	NUM
ma-160	140	5	(	(	PUNCT
ma-160	140	6	t	t	NOUN
ma-160	140	7	)	)	PUNCT
ma-160	140	8	=	=	SYM
ma-160	141	1	s	s	PART
ma-160	142	1	′	′	NUM
ma-160	142	2	λ	λ	PROPN
ma-160	142	3	(	(	PUNCT
ma-160	142	4	t	t	NOUN
ma-160	142	5	)	)	PUNCT
ma-160	142	6	sλ	sλ	NOUN
ma-160	142	7	(	(	PUNCT
ma-160	142	8	t	t	NOUN
ma-160	142	9	)	)	PUNCT
ma-160	142	10	=	=	PUNCT
ma-160	142	11	(	(	PUNCT
ma-160	142	12	1+λt	1+λt	NUM
ma-160	142	13	)	)	PUNCT
ma-160	142	14	1	1	NUM
ma-160	142	15	λ	λ	SYM
ma-160	142	16	−1	−1	NOUN
ma-160	142	17	[	[	PUNCT
ma-160	142	18	1+(1+λt	1+(1+λt	NOUN
ma-160	142	19	)	)	PUNCT
ma-160	142	20	1	1	NUM
ma-160	142	21	λ	λ	NOUN
ma-160	142	22	]	]	X
ma-160	142	23	2	2	NUM
ma-160	142	24	(	(	PUNCT
ma-160	142	25	1+λt	1+λt	NUM
ma-160	142	26	)	)	PUNCT
ma-160	142	27	1	1	NUM
ma-160	142	28	λ	λ	NOUN
ma-160	142	29	1+(1+λt	1+(1+λt	NOUN
ma-160	142	30	)	)	PUNCT
ma-160	142	31	1	1	NUM
ma-160	142	32	λ	λ	NOUN
ma-160	142	33	=	=	PRON
ma-160	142	34			X
ma-160	142	35	(	(	PUNCT
ma-160	142	36	1	1	NUM
ma-160	142	37	+	+	CCONJ
ma-160	142	38	λt	λt	X
ma-160	142	39	)	)	PUNCT
ma-160	142	40	1	1	NUM
ma-160	142	41	λ	λ	NOUN
ma-160	142	42	(	(	PUNCT
ma-160	142	43	1	1	NUM
ma-160	142	44	+	+	CCONJ
ma-160	142	45	λt	λt	X
ma-160	142	46	)	)	PUNCT
ma-160	142	47	[	[	PUNCT
ma-160	142	48	1	1	NUM
ma-160	142	49	+	+	CCONJ
ma-160	142	50	(	(	PUNCT
ma-160	142	51	1	1	NUM
ma-160	142	52	+	+	CCONJ
ma-160	142	53	λt	λt	X
ma-160	142	54	)	)	PUNCT
ma-160	142	55	1	1	NUM
ma-160	142	56	λ	λ	NOUN
ma-160	142	57	]	]	X
ma-160	142	58	2	2	NUM
ma-160	142	59	(1	(1	NOUN
ma-160	142	60	+	+	CCONJ
ma-160	142	61	(	(	PUNCT
ma-160	142	62	1	1	NUM
ma-160	142	63	+	+	CCONJ
ma-160	142	64	λt	λt	X
ma-160	142	65	)	)	PUNCT
ma-160	142	66	1	1	NUM
ma-160	142	67	λ	λ	NOUN
ma-160	142	68	(	(	PUNCT
ma-160	142	69	1	1	NUM
ma-160	142	70	+	+	CCONJ
ma-160	142	71	λt	λt	X
ma-160	142	72	)	)	PUNCT
ma-160	142	73	1	1	NUM
ma-160	142	74	λ	λ	NOUN
ma-160	142	75	)	)	PUNCT
ma-160	142	76	=	=	SYM
ma-160	142	77	1	1	NUM
ma-160	142	78	(	(	PUNCT
ma-160	142	79	1	1	NUM
ma-160	142	80	+	+	CCONJ
ma-160	142	81	λt	λt	X
ma-160	142	82	)	)	PUNCT
ma-160	142	83	+	+	CCONJ
ma-160	142	84	(	(	PUNCT
ma-160	142	85	1	1	NUM
ma-160	142	86	+	+	CCONJ
ma-160	142	87	λt	λt	X
ma-160	142	88	)	)	PUNCT
ma-160	142	89	1	1	NUM
ma-160	142	90	λ	λ	NOUN
ma-160	142	91	+1	+1	INTJ
ma-160	142	92	.	.	PUNCT
ma-160	143	1	taking	take	VERB
ma-160	143	2	the	the	DET
ma-160	143	3	second	second	ADJ
ma-160	143	4	derivative	derivative	NOUN
ma-160	143	5	of	of	ADP
ma-160	143	6	q	q	PROPN
ma-160	143	7	(	(	PUNCT
ma-160	143	8	t	t	PROPN
ma-160	143	9	)	)	PUNCT
ma-160	143	10	,	,	PUNCT
ma-160	143	11	we	we	PRON
ma-160	143	12	have	have	VERB
ma-160	143	13	q	q	ADJ
ma-160	143	14	′′	′′	PROPN
ma-160	143	15	(	(	PUNCT
ma-160	143	16	t	t	PROPN
ma-160	143	17	)	)	PUNCT
ma-160	143	18	=	=	NOUN
ma-160	143	19	−	−	PROPN
ma-160	143	20	λ+	λ+	PUNCT
ma-160	143	21	(	(	PUNCT
ma-160	143	22	1	1	NUM
ma-160	143	23	+	+	NUM
ma-160	143	24	λ	λ	NOUN
ma-160	143	25	)	)	PUNCT
ma-160	143	26	(	(	PUNCT
ma-160	143	27	1	1	NUM
ma-160	143	28	+	+	CCONJ
ma-160	143	29	λt	λt	X
ma-160	143	30	)	)	PUNCT
ma-160	143	31	1	1	NUM
ma-160	143	32	λ	λ	NOUN
ma-160	143	33	[	[	PUNCT
ma-160	143	34	(	(	PUNCT
ma-160	143	35	1	1	NUM
ma-160	143	36	+	+	CCONJ
ma-160	143	37	λt	λt	X
ma-160	143	38	)	)	PUNCT
ma-160	143	39	+	+	CCONJ
ma-160	143	40	(	(	PUNCT
ma-160	143	41	1	1	NUM
ma-160	143	42	+	+	CCONJ
ma-160	143	43	λt	λt	X
ma-160	143	44	)	)	PUNCT
ma-160	143	45	1	1	NUM
ma-160	143	46	λ	λ	NOUN
ma-160	143	47	+1	+1	X
ma-160	143	48	]	]	X
ma-160	143	49	2	2	NUM
ma-160	143	50	<	<	X
ma-160	143	51	0	0	NUM
ma-160	143	52	,	,	PUNCT
ma-160	143	53	and	and	CCONJ
ma-160	143	54	this	this	PRON
ma-160	143	55	completes	complete	VERB
ma-160	143	56	the	the	DET
ma-160	143	57	proof	proof	NOUN
ma-160	143	58	.	.	PUNCT
ma-160	144	1	�	�	PROPN
ma-160	144	2	corollary	corollary	ADJ
ma-160	144	3	3.8	3.8	NUM
ma-160	144	4	.	.	PUNCT
ma-160	145	1	for	for	ADP
ma-160	145	2	λ	λ	PROPN
ma-160	145	3	∈	∈	PROPN
ma-160	145	4	(	(	PUNCT
ma-160	145	5	0,∞	0,∞	NOUN
ma-160	145	6	)	)	PUNCT
ma-160	145	7	and	and	CCONJ
ma-160	145	8	t	t	PROPN
ma-160	145	9	∈	∈	PROPN
ma-160	145	10	(	(	PUNCT
ma-160	145	11	−∞,∞	−∞,∞	NOUN
ma-160	145	12	)	)	PUNCT
ma-160	145	13	,	,	PUNCT
ma-160	145	14	the	the	DET
ma-160	145	15	inequalities	inequality	NOUN
ma-160	145	16	s	s	VERB
ma-160	145	17	′′	′′	PROPN
ma-160	145	18	λ	λ	PROPN
ma-160	145	19	(	(	PUNCT
ma-160	145	20	t)sλ	t)sλ	PROPN
ma-160	145	21	(	(	PUNCT
ma-160	145	22	t	t	PROPN
ma-160	145	23	)	)	PUNCT
ma-160	145	24	≤	≤	NOUN
ma-160	145	25	[	[	PUNCT
ma-160	145	26	s	s	NOUN
ma-160	145	27	′	′	NUM
ma-160	145	28	λ	λ	PROPN
ma-160	145	29	(	(	PUNCT
ma-160	145	30	t	t	PROPN
ma-160	145	31	)	)	PUNCT
ma-160	145	32	]	]	PUNCT
ma-160	145	33	2	2	NUM
ma-160	145	34	(	(	PUNCT
ma-160	145	35	34	34	NUM
ma-160	145	36	)	)	PUNCT
ma-160	145	37	and	and	CCONJ
ma-160	145	38	sλ	sλ	NOUN
ma-160	145	39	(	(	PUNCT
ma-160	145	40	1	1	NUM
ma-160	146	1	+	+	CCONJ
ma-160	146	2	u)sλ	u)sλ	PROPN
ma-160	146	3	(	(	PUNCT
ma-160	146	4	1−	1−	NUM
ma-160	146	5	u	u	NOUN
ma-160	146	6	)	)	PUNCT
ma-160	146	7	≤	≤	NOUN
ma-160	146	8	[	[	PUNCT
ma-160	146	9	(	(	PUNCT
ma-160	146	10	1	1	NUM
ma-160	146	11	+	+	NUM
ma-160	146	12	λ	λ	NOUN
ma-160	146	13	)	)	PUNCT
ma-160	146	14	1	1	NUM
ma-160	146	15	λ	λ	NOUN
ma-160	146	16	1	1	NUM
ma-160	146	17	+	+	CCONJ
ma-160	146	18	(	(	PUNCT
ma-160	146	19	1	1	NUM
ma-160	146	20	+	+	NUM
ma-160	146	21	λ	λ	NOUN
ma-160	146	22	)	)	PUNCT
ma-160	146	23	1	1	NUM
ma-160	146	24	λ	λ	NOUN
ma-160	146	25	]	]	X
ma-160	146	26	2	2	NUM
ma-160	146	27	(	(	PUNCT
ma-160	146	28	35	35	NUM
ma-160	146	29	)	)	PUNCT
ma-160	146	30	are	be	AUX
ma-160	146	31	valid	valid	ADJ
ma-160	146	32	.	.	PUNCT
ma-160	147	1	proof	proof	NOUN
ma-160	147	2	.	.	PUNCT
ma-160	148	1	since	since	SCONJ
ma-160	148	2	sλ	sλ	NOUN
ma-160	148	3	(	(	PUNCT
ma-160	148	4	t	t	NOUN
ma-160	148	5	)	)	PUNCT
ma-160	148	6	is	be	AUX
ma-160	148	7	logarithmically	logarithmically	ADV
ma-160	148	8	concave	concave	VERB
ma-160	148	9	,	,	PUNCT
ma-160	148	10	then	then	ADV
ma-160	148	11	[	[	X
ma-160	148	12	ln	ln	X
ma-160	148	13	(	(	PUNCT
ma-160	148	14	sλ	sλ	NOUN
ma-160	148	15	(	(	PUNCT
ma-160	148	16	t	t	NOUN
ma-160	148	17	)	)	PUNCT
ma-160	148	18	)	)	PUNCT
ma-160	148	19	]	]	PUNCT
ma-160	149	1	′′	′′	PROPN
ma-160	149	2	≤	≤	NOUN
ma-160	149	3	0	0	NUM
ma-160	149	4	,	,	PUNCT
ma-160	149	5	for	for	ADP
ma-160	149	6	all	all	DET
ma-160	149	7	t	t	NOUN
ma-160	149	8	∈	∈	PROPN
ma-160	149	9	(	(	PUNCT
ma-160	149	10	−∞,∞	−∞,∞	NOUN
ma-160	149	11	)	)	PUNCT
ma-160	149	12	and	and	CCONJ
ma-160	149	13	λ	λ	X
ma-160	149	14	∈	∈	PROPN
ma-160	149	15	(	(	PUNCT
ma-160	149	16	0,∞	0,∞	NOUN
ma-160	149	17	)	)	PUNCT
ma-160	149	18	.	.	PUNCT
ma-160	150	1	this	this	PRON
ma-160	150	2	implies	imply	VERB
ma-160	150	3	that	that	SCONJ
ma-160	150	4	,	,	PUNCT
ma-160	150	5	[	[	X
ma-160	150	6	ln	ln	X
ma-160	150	7	(	(	PUNCT
ma-160	150	8	sλ	sλ	NOUN
ma-160	150	9	(	(	PUNCT
ma-160	150	10	t	t	NOUN
ma-160	150	11	)	)	PUNCT
ma-160	150	12	)	)	PUNCT
ma-160	150	13	]	]	PUNCT
ma-160	151	1	′′	′′	PROPN
ma-160	151	2	=	=	PRON
ma-160	152	1	[	[	PUNCT
ma-160	152	2	s	s	NUM
ma-160	152	3	′	′	NUM
ma-160	152	4	λ	λ	PROPN
ma-160	152	5	(	(	PUNCT
ma-160	152	6	t	t	NOUN
ma-160	152	7	)	)	PUNCT
ma-160	152	8	sλ	sλ	NOUN
ma-160	152	9	(	(	PUNCT
ma-160	152	10	t	t	PROPN
ma-160	152	11	)	)	PUNCT
ma-160	152	12	]	]	PUNCT
ma-160	152	13	′	′	NUM
ma-160	152	14	=	=	PUNCT
ma-160	152	15	s	s	PART
ma-160	152	16	′′	′′	PROPN
ma-160	152	17	λ	λ	PROPN
ma-160	152	18	(	(	PUNCT
ma-160	152	19	t)sλ	t)sλ	PROPN
ma-160	152	20	(	(	PUNCT
ma-160	152	21	t)−	t)−	PROPN
ma-160	152	22	s′λ	s′λ	X
ma-160	152	23	(	(	PUNCT
ma-160	152	24	t)s	t)s	ADV
ma-160	152	25	′	′	NUM
ma-160	153	1	λ	λ	PROPN
ma-160	153	2	(	(	PUNCT
ma-160	153	3	t	t	PROPN
ma-160	153	4	)	)	PUNCT
ma-160	154	1	[	[	X
ma-160	154	2	sλ	sλ	X
ma-160	154	3	(	(	PUNCT
ma-160	154	4	t)]2	t)]2	NOUN
ma-160	154	5	=	=	SYM
ma-160	154	6	s	s	PART
ma-160	154	7	′′	′′	PROPN
ma-160	154	8	λ	λ	PROPN
ma-160	154	9	(	(	PUNCT
ma-160	154	10	t)sλ	t)sλ	PROPN
ma-160	154	11	(	(	PUNCT
ma-160	154	12	t)−	t)−	PROPN
ma-160	154	13	[	[	PUNCT
ma-160	154	14	s	s	NOUN
ma-160	154	15	′	′	NUM
ma-160	154	16	λ	λ	PROPN
ma-160	154	17	(	(	PUNCT
ma-160	154	18	t	t	PROPN
ma-160	154	19	)	)	PUNCT
ma-160	154	20	]	]	PUNCT
ma-160	154	21	2	2	NUM
ma-160	154	22	[	[	X
ma-160	154	23	sλ	sλ	NOUN
ma-160	154	24	(	(	PUNCT
ma-160	154	25	t)]2	t)]2	NOUN
ma-160	154	26	≤	≤	ADJ
ma-160	154	27	0	0	NUM
ma-160	154	28	.	.	PUNCT
ma-160	155	1	hence	hence	ADV
ma-160	155	2	,	,	PUNCT
ma-160	155	3	s′′λ	s′′λ	PROPN
ma-160	155	4	(	(	PUNCT
ma-160	155	5	t)sλ	t)sλ	PROPN
ma-160	155	6	(	(	PUNCT
ma-160	155	7	t)−	t)−	PROPN
ma-160	155	8	[	[	PUNCT
ma-160	155	9	s	s	NOUN
ma-160	155	10	′	′	NUM
ma-160	155	11	λ	λ	PROPN
ma-160	155	12	(	(	PUNCT
ma-160	155	13	t	t	PROPN
ma-160	155	14	)	)	PUNCT
ma-160	155	15	]	]	PUNCT
ma-160	155	16	2	2	NUM
ma-160	155	17	≤	≤	NUM
ma-160	155	18	0	0	NUM
ma-160	155	19	,	,	PUNCT
ma-160	155	20	which	which	PRON
ma-160	155	21	yields	yield	VERB
ma-160	155	22	equation	equation	NOUN
ma-160	155	23	(	(	PUNCT
ma-160	155	24	34	34	NUM
ma-160	155	25	)	)	PUNCT
ma-160	155	26	.	.	PUNCT
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ma-160	156	2	eur	eur	PROPN
ma-160	156	3	.	.	PUNCT
ma-160	157	1	j.	j.	PROPN
ma-160	157	2	math	math	PROPN
ma-160	157	3	.	.	PUNCT
ma-160	158	1	anal	anal	PROPN
ma-160	158	2	.	.	PUNCT
ma-160	159	1	10.28924	10.28924	NUM
ma-160	159	2	/	/	SYM
ma-160	159	3	ada	ada	PROPN
ma-160	159	4	/	/	SYM
ma-160	159	5	ma.3.17	ma.3.17	PROPN
ma-160	159	6	8next	8next	NUM
ma-160	159	7	,	,	PUNCT
ma-160	159	8	let	let	VERB
ma-160	159	9	a	a	DET
ma-160	159	10	=	=	SYM
ma-160	159	11	b	b	NOUN
ma-160	159	12	=	=	SYM
ma-160	159	13	2	2	NUM
ma-160	159	14	,	,	PUNCT
ma-160	159	15	t	t	NOUN
ma-160	159	16	=	=	SYM
ma-160	160	1	1	1	NUM
ma-160	160	2	+	+	NUM
ma-160	160	3	u	u	NOUN
ma-160	160	4	and	and	CCONJ
ma-160	160	5	r	r	NOUN
ma-160	160	6	=	=	SYM
ma-160	160	7	1−	1−	NUM
ma-160	160	8	u	u	NOUN
ma-160	160	9	in	in	ADP
ma-160	160	10	equation	equation	NOUN
ma-160	160	11	(	(	PUNCT
ma-160	160	12	33	33	NUM
ma-160	160	13	)	)	PUNCT
ma-160	160	14	.	.	PUNCT
ma-160	161	1	we	we	PRON
ma-160	161	2	have	have	VERB
ma-160	161	3	sλ	sλ	NOUN
ma-160	161	4	(	(	PUNCT
ma-160	161	5	1	1	NUM
ma-160	161	6	+	+	NUM
ma-160	161	7	u	u	NOUN
ma-160	161	8	2	2	NUM
ma-160	161	9	+	+	SYM
ma-160	161	10	1−	1−	NUM
ma-160	161	11	u	u	NOUN
ma-160	161	12	2	2	NUM
ma-160	161	13	)	)	PUNCT
ma-160	161	14	≥	≥	NOUN
ma-160	162	1	[	[	X
ma-160	162	2	sλ	sλ	NOUN
ma-160	162	3	(	(	PUNCT
ma-160	162	4	1	1	NUM
ma-160	162	5	+	+	NUM
ma-160	162	6	u	u	NOUN
ma-160	162	7	)	)	PUNCT
ma-160	162	8	]	]	PUNCT
ma-160	162	9	1	1	NUM
ma-160	162	10	2	2	NUM
ma-160	162	11	[	[	X
ma-160	162	12	sλ	sλ	NOUN
ma-160	162	13	(	(	PUNCT
ma-160	162	14	1−	1−	NUM
ma-160	162	15	u	u	NOUN
ma-160	162	16	)	)	PUNCT
ma-160	162	17	]	]	PUNCT
ma-160	162	18	1	1	NUM
ma-160	162	19	2	2	NUM
ma-160	162	20	sλ	sλ	NOUN
ma-160	162	21	(	(	PUNCT
ma-160	162	22	1	1	NUM
ma-160	162	23	)	)	PUNCT
ma-160	162	24	≥	≥	NOUN
ma-160	162	25	(	(	PUNCT
ma-160	162	26	[	[	X
ma-160	162	27	sλ	sλ	X
ma-160	162	28	(	(	PUNCT
ma-160	162	29	1	1	NUM
ma-160	162	30	+	+	NUM
ma-160	162	31	u	u	NOUN
ma-160	162	32	)	)	PUNCT
ma-160	162	33	]	]	PUNCT
ma-160	163	1	[	[	X
ma-160	163	2	sλ	sλ	NOUN
ma-160	163	3	(	(	PUNCT
ma-160	163	4	1−	1−	NUM
ma-160	163	5	u	u	NOUN
ma-160	163	6	)	)	PUNCT
ma-160	163	7	]	]	PUNCT
ma-160	163	8	)	)	PUNCT
ma-160	163	9	1	1	NUM
ma-160	163	10	2	2	NUM
ma-160	163	11	[	[	PUNCT
ma-160	163	12	(	(	PUNCT
ma-160	163	13	1	1	NUM
ma-160	163	14	+	+	NUM
ma-160	163	15	λ	λ	NOUN
ma-160	163	16	)	)	PUNCT
ma-160	163	17	1	1	NUM
ma-160	163	18	λ	λ	NOUN
ma-160	163	19	1	1	NUM
ma-160	163	20	+	+	CCONJ
ma-160	163	21	(	(	PUNCT
ma-160	163	22	1	1	NUM
ma-160	163	23	+	+	NUM
ma-160	163	24	λ	λ	NOUN
ma-160	163	25	)	)	PUNCT
ma-160	163	26	1	1	NUM
ma-160	163	27	λ	λ	NOUN
ma-160	163	28	]	]	X
ma-160	163	29	2	2	NUM
ma-160	163	30	≥	≥	NOUN
ma-160	163	31	sλ	sλ	NOUN
ma-160	163	32	(	(	PUNCT
ma-160	163	33	1	1	NUM
ma-160	163	34	+	+	CCONJ
ma-160	163	35	u)sλ	u)sλ	PROPN
ma-160	163	36	(	(	PUNCT
ma-160	163	37	1−	1−	NUM
ma-160	163	38	u	u	NOUN
ma-160	163	39	)	)	PUNCT
ma-160	163	40	,	,	PUNCT
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ma-160	163	42	in	in	ADP
ma-160	163	43	equation	equation	NOUN
ma-160	163	44	(	(	PUNCT
ma-160	163	45	35	35	NUM
ma-160	163	46	)	)	PUNCT
ma-160	163	47	.	.	PUNCT
ma-160	164	1	this	this	PRON
ma-160	164	2	concludes	conclude	VERB
ma-160	164	3	the	the	DET
ma-160	164	4	proof	proof	NOUN
ma-160	164	5	.	.	PUNCT
ma-160	165	1	�	�	PROPN
ma-160	165	2	theorem	theorem	VERB
ma-160	165	3	3.9	3.9	NUM
ma-160	165	4	.	.	PUNCT
ma-160	165	5	for	for	ADP
ma-160	165	6	t	t	PROPN
ma-160	165	7	,	,	PUNCT
ma-160	165	8	λ	λ	PROPN
ma-160	165	9	∈	∈	PROPN
ma-160	165	10	(	(	PUNCT
ma-160	165	11	0,∞	0,∞	NOUN
ma-160	165	12	)	)	PUNCT
ma-160	165	13	,	,	PUNCT
ma-160	165	14	the	the	DET
ma-160	165	15	function	function	NOUN
ma-160	165	16	sλ	sλ	NOUN
ma-160	165	17	(	(	PUNCT
ma-160	165	18	t	t	NOUN
ma-160	165	19	)	)	PUNCT
ma-160	165	20	satisfies	satisfy	VERB
ma-160	165	21	the	the	DET
ma-160	165	22	inequality	inequality	NOUN
ma-160	165	23	1	1	NUM
ma-160	165	24	<	<	X
ma-160	165	25	sλ	sλ	NOUN
ma-160	165	26	(	(	PUNCT
ma-160	165	27	t	t	NOUN
ma-160	165	28	+	+	CCONJ
ma-160	165	29	1	1	X
ma-160	165	30	)	)	PUNCT
ma-160	165	31	sλ	sλ	NOUN
ma-160	165	32	(	(	PUNCT
ma-160	165	33	t	t	NOUN
ma-160	165	34	)	)	PUNCT
ma-160	165	35	<	<	X
ma-160	165	36	2	2	NUM
ma-160	165	37	(	(	PUNCT
ma-160	165	38	1	1	NUM
ma-160	165	39	+	+	NUM
ma-160	165	40	λ	λ	NOUN
ma-160	165	41	)	)	PUNCT
ma-160	165	42	1	1	NUM
ma-160	165	43	λ	λ	NOUN
ma-160	165	44	1	1	NUM
ma-160	165	45	+	+	CCONJ
ma-160	165	46	(	(	PUNCT
ma-160	165	47	1	1	NUM
ma-160	165	48	+	+	NUM
ma-160	165	49	λ	λ	NOUN
ma-160	165	50	)	)	PUNCT
ma-160	165	51	1	1	NUM
ma-160	165	52	λ	λ	NOUN
ma-160	165	53	.	.	PUNCT
ma-160	166	1	(	(	PUNCT
ma-160	166	2	36	36	NUM
ma-160	166	3	)	)	PUNCT
ma-160	166	4	proof	proof	NOUN
ma-160	166	5	.	.	PUNCT
ma-160	167	1	recall	recall	NOUN
ma-160	167	2	from	from	ADP
ma-160	167	3	equation	equation	NOUN
ma-160	167	4	(	(	PUNCT
ma-160	167	5	18	18	NUM
ma-160	167	6	)	)	PUNCT
ma-160	167	7	,	,	PUNCT
ma-160	167	8	that	that	SCONJ
ma-160	167	9	(	(	PUNCT
ma-160	167	10	s	s	VERB
ma-160	167	11	′	′	NUM
ma-160	167	12	λ	λ	X
ma-160	167	13	(	(	PUNCT
ma-160	167	14	t	t	NOUN
ma-160	167	15	)	)	PUNCT
ma-160	167	16	sλ	sλ	NOUN
ma-160	167	17	(	(	PUNCT
ma-160	167	18	t	t	NOUN
ma-160	167	19	)	)	PUNCT
ma-160	167	20	)	)	PUNCT
ma-160	167	21	′	′	NUM
ma-160	168	1	=	=	PUNCT
ma-160	168	2	−	−	PROPN
ma-160	168	3	λ+	λ+	PUNCT
ma-160	168	4	(	(	PUNCT
ma-160	168	5	1	1	NUM
ma-160	168	6	+	+	NUM
ma-160	168	7	λ	λ	NOUN
ma-160	168	8	)	)	PUNCT
ma-160	168	9	(	(	PUNCT
ma-160	168	10	1	1	NUM
ma-160	168	11	+	+	CCONJ
ma-160	168	12	λt	λt	X
ma-160	168	13	)	)	PUNCT
ma-160	168	14	1	1	NUM
ma-160	168	15	λ	λ	NOUN
ma-160	168	16	[	[	PUNCT
ma-160	168	17	(	(	PUNCT
ma-160	168	18	1	1	NUM
ma-160	168	19	+	+	CCONJ
ma-160	168	20	λt	λt	X
ma-160	168	21	)	)	PUNCT
ma-160	168	22	+	+	CCONJ
ma-160	168	23	(	(	PUNCT
ma-160	168	24	1	1	NUM
ma-160	168	25	+	+	CCONJ
ma-160	168	26	λt	λt	X
ma-160	168	27	)	)	PUNCT
ma-160	168	28	1	1	NUM
ma-160	169	1	λ	λ	NOUN
ma-160	169	2	+1	+1	X
ma-160	169	3	]	]	X
ma-160	169	4	2	2	NUM
ma-160	169	5	<	<	X
ma-160	169	6	0	0	NUM
ma-160	169	7	,	,	PUNCT
ma-160	169	8	for	for	ADP
ma-160	169	9	all	all	DET
ma-160	169	10	t	t	PROPN
ma-160	169	11	,	,	PUNCT
ma-160	169	12	λ	λ	PROPN
ma-160	169	13	∈	∈	PROPN
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ma-160	169	15	0,∞	0,∞	NOUN
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ma-160	170	8	λ(t	λ(t	NOUN
ma-160	170	9	)	)	PUNCT
ma-160	170	10	sλ(t	sλ(t	NOUN
ma-160	170	11	)	)	PUNCT
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ma-160	171	10	)	)	PUNCT
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ma-160	172	3	1	1	NUM
ma-160	172	4	+	+	NUM
ma-160	172	5	λ	λ	X
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ma-160	172	7	t	t	PROPN
ma-160	172	8	+	+	NOUN
ma-160	172	9	1	1	NUM
ma-160	172	10	)	)	PUNCT
ma-160	172	11	]	]	PUNCT
ma-160	172	12	1	1	NUM
ma-160	172	13	λ	λ	SYM
ma-160	172	14	1	1	NUM
ma-160	172	15	+	+	CCONJ
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ma-160	172	17	1	1	NUM
ma-160	172	18	+	+	NUM
ma-160	172	19	λ	λ	X
ma-160	172	20	(	(	PUNCT
ma-160	172	21	t	t	PROPN
ma-160	172	22	+	+	NOUN
ma-160	172	23	1	1	NUM
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ma-160	172	47	=	=	PUNCT
ma-160	173	1	[	[	X
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ma-160	173	3	+	+	NUM
ma-160	173	4	λ	λ	X
ma-160	173	5	(	(	PUNCT
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ma-160	173	7	+	+	NOUN
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ma-160	173	9	)	)	PUNCT
ma-160	173	10	]	]	PUNCT
ma-160	173	11	1	1	NUM
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ma-160	173	16	+	+	CCONJ
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ma-160	173	19	1	1	NUM
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ma-160	174	1	[	[	X
ma-160	174	2	1	1	NUM
ma-160	174	3	+	+	NUM
ma-160	174	4	λ	λ	X
ma-160	174	5	(	(	PUNCT
ma-160	174	6	t	t	PROPN
ma-160	174	7	+	+	NOUN
ma-160	174	8	1	1	NUM
ma-160	174	9	)	)	PUNCT
ma-160	174	10	]	]	PUNCT
ma-160	174	11	1	1	NUM
ma-160	174	12	λ	λ	X
ma-160	174	13	(	(	PUNCT
ma-160	174	14	1	1	NUM
ma-160	174	15	+	+	CCONJ
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ma-160	174	17	)	)	PUNCT
ma-160	174	18	1	1	NUM
ma-160	174	19	λ	λ	NOUN
ma-160	174	20	+	+	CCONJ
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ma-160	174	22	1	1	NUM
ma-160	174	23	+	+	CCONJ
ma-160	174	24	λt	λt	X
ma-160	174	25	)	)	PUNCT
ma-160	174	26	1	1	NUM
ma-160	174	27	λ	λ	NOUN
ma-160	174	28	[	[	X
ma-160	174	29	1	1	NUM
ma-160	174	30	+	+	NUM
ma-160	174	31	λ	λ	X
ma-160	174	32	(	(	PUNCT
ma-160	174	33	t	t	PROPN
ma-160	174	34	+	+	NOUN
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ma-160	174	38	1	1	NUM
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ma-160	174	42	(	(	PUNCT
ma-160	174	43	t	t	PROPN
ma-160	174	44	)	)	PUNCT
ma-160	174	45	=	=	PUNCT
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ma-160	175	2	(	(	PUNCT
ma-160	175	3	t	t	NOUN
ma-160	175	4	)	)	PUNCT
ma-160	175	5	=	=	SYM
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ma-160	176	2	(	(	PUNCT
ma-160	176	3	t	t	PROPN
ma-160	176	4	+	+	PROPN
ma-160	176	5	1)−	1)−	PROPN
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ma-160	176	7	(	(	PUNCT
ma-160	176	8	t	t	PROPN
ma-160	176	9	)	)	PUNCT
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ma-160	177	4	′	′	NUM
ma-160	177	5	(	(	PUNCT
ma-160	177	6	t	t	NOUN
ma-160	177	7	)	)	PUNCT
ma-160	177	8	=	=	SYM
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ma-160	177	10	′	′	NUM
ma-160	177	11	λ	λ	X
ma-160	177	12	(	(	PUNCT
ma-160	177	13	t	t	PROPN
ma-160	177	14	+	+	CCONJ
ma-160	177	15	1	1	X
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ma-160	177	18	(	(	PUNCT
ma-160	177	19	t	t	NOUN
ma-160	177	20	+	+	CCONJ
ma-160	177	21	1	1	NUM
ma-160	177	22	)	)	PUNCT
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ma-160	177	25	′	′	NUM
ma-160	177	26	λ	λ	PROPN
ma-160	177	27	(	(	PUNCT
ma-160	177	28	t	t	NOUN
ma-160	177	29	)	)	PUNCT
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ma-160	177	31	(	(	PUNCT
ma-160	177	32	t	t	NOUN
ma-160	177	33	)	)	PUNCT
ma-160	177	34	<	<	X
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ma-160	177	36	,	,	PUNCT
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ma-160	177	39	′	′	NUM
ma-160	177	40	λ(t	λ(t	NOUN
ma-160	177	41	)	)	PUNCT
ma-160	177	42	sλ(t	sλ(t	NOUN
ma-160	177	43	)	)	PUNCT
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ma-160	177	45	decreasing	decrease	VERB
ma-160	177	46	.	.	PUNCT
ma-160	178	1	this	this	PRON
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ma-160	178	3	ω	ω	PROPN
ma-160	178	4	(	(	PUNCT
ma-160	178	5	t	t	PROPN
ma-160	178	6	)	)	PUNCT
ma-160	178	7	and	and	CCONJ
ma-160	178	8	consequently	consequently	ADV
ma-160	178	9	p	p	X
ma-160	178	10	(	(	PUNCT
ma-160	178	11	t	t	NOUN
ma-160	178	12	)	)	PUNCT
ma-160	178	13	are	be	AUX
ma-160	178	14	decreasing	decrease	VERB
ma-160	178	15	.	.	PUNCT
ma-160	179	1	hence	hence	ADV
ma-160	179	2	,	,	PUNCT
ma-160	179	3	forall	forall	PROPN
ma-160	179	4	t	t	PROPN
ma-160	179	5	,	,	PUNCT
ma-160	179	6	λ	λ	PROPN
ma-160	179	7	∈	∈	PROPN
ma-160	179	8	(	(	PUNCT
ma-160	179	9	0,∞	0,∞	NOUN
ma-160	179	10	)	)	PUNCT
ma-160	179	11	,	,	PUNCT
ma-160	179	12	we	we	PRON
ma-160	179	13	have	have	VERB
ma-160	179	14	1	1	NUM
ma-160	179	15	=	=	SYM
ma-160	179	16	lim	lim	PROPN
ma-160	179	17	t→∞	t→∞	ADP
ma-160	179	18	p	p	X
ma-160	179	19	(	(	PUNCT
ma-160	179	20	t	t	PROPN
ma-160	179	21	)	)	PUNCT
ma-160	179	22	<	<	X
ma-160	180	1	p	p	X
ma-160	180	2	(	(	PUNCT
ma-160	180	3	t	t	PROPN
ma-160	180	4	)	)	PUNCT
ma-160	180	5	<	<	X
ma-160	180	6	lim	lim	PROPN
ma-160	180	7	t→0	t→0	PROPN
ma-160	180	8	p	p	PROPN
ma-160	180	9	(	(	PUNCT
ma-160	180	10	t	t	NOUN
ma-160	180	11	)	)	PUNCT
ma-160	180	12	=	=	SYM
ma-160	180	13	2	2	NUM
ma-160	180	14	(	(	PUNCT
ma-160	180	15	1	1	NUM
ma-160	180	16	+	+	NUM
ma-160	180	17	λ	λ	NOUN
ma-160	180	18	)	)	PUNCT
ma-160	180	19	1	1	NUM
ma-160	180	20	λ	λ	NOUN
ma-160	180	21	1	1	NUM
ma-160	180	22	+	+	CCONJ
ma-160	180	23	(	(	PUNCT
ma-160	180	24	1	1	NUM
ma-160	180	25	+	+	NUM
ma-160	180	26	λ	λ	NOUN
ma-160	180	27	)	)	PUNCT
ma-160	180	28	1	1	NUM
ma-160	180	29	λ	λ	NOUN
ma-160	180	30	,	,	PUNCT
ma-160	180	31	which	which	PRON
ma-160	180	32	yields	yield	VERB
ma-160	180	33	the	the	DET
ma-160	180	34	desired	desire	VERB
ma-160	180	35	result	result	NOUN
ma-160	180	36	(	(	PUNCT
ma-160	180	37	36	36	NUM
ma-160	180	38	)	)	PUNCT
ma-160	180	39	.	.	PUNCT
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ma-160	181	4	.	.	PUNCT
ma-160	182	1	j.	j.	PROPN
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ma-160	182	3	.	.	PUNCT
ma-160	183	1	anal	anal	PROPN
ma-160	183	2	.	.	PUNCT
ma-160	184	1	10.28924	10.28924	NUM
ma-160	184	2	/	/	SYM
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ma-160	184	7	.	.	PUNCT
ma-160	185	1	conclusion	conclusion	NOUN
ma-160	185	2	we	we	PRON
ma-160	185	3	have	have	AUX
ma-160	185	4	introduced	introduce	VERB
ma-160	185	5	a	a	DET
ma-160	185	6	degenerate	degenerate	ADJ
ma-160	185	7	sigmoid	sigmoid	NOUN
ma-160	185	8	function	function	NOUN
ma-160	185	9	.	.	PUNCT
ma-160	186	1	properties	property	NOUN
ma-160	186	2	such	such	ADJ
ma-160	186	3	as	as	ADP
ma-160	186	4	concavity	concavity	NOUN
ma-160	186	5	,	,	PUNCT
ma-160	186	6	monotonicity	monotonicity	NOUN
ma-160	186	7	andinequalities	andinequalitie	NOUN
ma-160	186	8	involving	involve	VERB
ma-160	186	9	the	the	DET
ma-160	186	10	new	new	ADJ
ma-160	186	11	function	function	NOUN
ma-160	186	12	have	have	AUX
ma-160	186	13	been	be	AUX
ma-160	186	14	established	establish	VERB
ma-160	186	15	.	.	PUNCT
ma-160	187	1	these	these	DET
ma-160	187	2	established	establish	VERB
ma-160	187	3	properties	property	NOUN
ma-160	187	4	canbe	canbe	VERB
ma-160	187	5	applied	apply	VERB
ma-160	187	6	in	in	ADP
ma-160	187	7	several	several	ADJ
ma-160	187	8	areas	area	NOUN
ma-160	187	9	of	of	ADP
ma-160	187	10	mathematics	mathematic	NOUN
ma-160	187	11	.	.	PUNCT
ma-160	188	1	5	5	X
ma-160	188	2	.	.	X
ma-160	188	3	conflicts	conflict	NOUN
ma-160	188	4	of	of	ADP
ma-160	188	5	interest	interest	NOUN
ma-160	188	6	the	the	DET
ma-160	188	7	corresponding	corresponding	ADJ
ma-160	188	8	author	author	NOUN
ma-160	188	9	affirms	affirm	VERB
ma-160	188	10	on	on	ADP
ma-160	188	11	behalf	behalf	NOUN
ma-160	188	12	of	of	ADP
ma-160	188	13	all	all	DET
ma-160	188	14	authors	author	NOUN
ma-160	188	15	that	that	SCONJ
ma-160	188	16	there	there	PRON
ma-160	188	17	is	be	VERB
ma-160	188	18	no	no	DET
ma-160	188	19	conflict	conflict	NOUN
ma-160	188	20	of	of	ADP
ma-160	188	21	interest	interest	NOUN
ma-160	188	22	for	for	ADP
ma-160	188	23	thepublication	thepublication	NOUN
ma-160	188	24	of	of	ADP
ma-160	188	25	this	this	DET
ma-160	188	26	research	research	NOUN
ma-160	188	27	.	.	PUNCT
ma-160	189	1	references	reference	NOUN
ma-160	189	2	[	[	X
ma-160	189	3	1	1	NUM
ma-160	189	4	]	]	X
ma-160	189	5	g.d	g.d	PROPN
ma-160	189	6	.	.	PROPN
ma-160	189	7	anderson	anderson	PROPN
ma-160	189	8	,	,	PUNCT
ma-160	189	9	m.k	m.k	PROPN
ma-160	189	10	.	.	PROPN
ma-160	189	11	vamanamurthy	vamanamurthy	PROPN
ma-160	189	12	and	and	CCONJ
ma-160	189	13	m.	m.	NOUN
ma-160	189	14	vuorinen	vuorinen	PROPN
ma-160	189	15	,	,	PUNCT
ma-160	189	16	generalized	generalized	ADJ
ma-160	189	17	convexity	convexity	NOUN
ma-160	189	18	and	and	CCONJ
ma-160	189	19	inequalities	inequality	NOUN
ma-160	189	20	,	,	PUNCT
ma-160	189	21	j.	j.	PROPN
ma-160	189	22	math	math	PROPN
ma-160	189	23	.	.	PUNCT
ma-160	190	1	anal	anal	PROPN
ma-160	190	2	.	.	PUNCT
ma-160	191	1	appl.335	appl.335	NOUN
ma-160	191	2	(	(	PUNCT
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ma-160	191	4	)	)	PUNCT
ma-160	191	5	1294	1294	NUM
ma-160	191	6	-	-	SYM
ma-160	191	7	1308	1308	NUM
ma-160	191	8	.	.	PUNCT
ma-160	192	1	https://doi.org/10.1016/j.jmaa.2007.02.016.[2	https://doi.org/10.1016/j.jmaa.2007.02.016.[2	PROPN
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ma-160	192	7	-	-	PUNCT
ma-160	192	8	type	type	NOUN
ma-160	192	9	inequalities	inequality	NOUN
ma-160	192	10	for	for	ADP
ma-160	192	11	special	special	ADJ
ma-160	192	12	functions	function	NOUN
ma-160	192	13	,	,	PUNCT
ma-160	192	14	j.	j.	PROPN
ma-160	192	15	ineq	ineq	PROPN
ma-160	192	16	.	.	PUNCT
ma-160	193	1	pure	pure	ADJ
ma-160	193	2	appl	appl	PROPN
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ma-160	193	4	math	math	NOUN
ma-160	193	5	.	.	PUNCT
ma-160	194	1	7	7	NUM
ma-160	194	2	(	(	PUNCT
ma-160	194	3	2006	2006	NUM
ma-160	194	4	)	)	PUNCT
ma-160	194	5	175	175	NUM
ma-160	194	6	.	.	PUNCT
ma-160	195	1	https://www	https://www	PROPN
ma-160	195	2	.	.	PUNCT
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ma-160	196	3	p.	p.	PROPN
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ma-160	196	10	riordan	riordan	PROPN
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ma-160	196	13	and	and	CCONJ
ma-160	196	14	exponential	exponential	PROPN
ma-160	196	15	riordan	riordan	PROPN
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ma-160	196	17	,	,	PUNCT
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ma-160	197	1	[	[	X
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ma-160	197	3	]	]	PUNCT
ma-160	197	4	.	.	PUNCT
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ma-160	198	3	b.	b.	PROPN
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ma-160	198	7	socha	socha	PROPN
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ma-160	198	12	of	of	ADP
ma-160	198	13	approximations	approximation	NOUN
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ma-160	198	16	s	s	NOUN
ma-160	198	17	-	-	PUNCT
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ma-160	198	19	functions	function	NOUN
ma-160	198	20	for	for	ADP
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ma-160	198	22	and	and	CCONJ
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ma-160	198	24	graphics	graphic	NOUN
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ma-160	198	27	image	image	NOUN
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ma-160	199	2	.	.	X
ma-160	200	1	2	2	NUM
ma-160	200	2	(	(	PUNCT
ma-160	200	3	2012	2012	NUM
ma-160	200	4	)	)	PUNCT
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ma-160	200	6	-	-	SYM
ma-160	200	7	28	28	NUM
ma-160	200	8	.	.	PUNCT
ma-160	201	1	https://doi.org/10.2478/	https://doi.org/10.2478/	PROPN
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ma-160	201	3	-	-	PUNCT
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ma-160	201	5	-	-	PUNCT
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ma-160	201	7	-	-	PUNCT
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ma-160	203	2	,	,	PUNCT
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ma-160	203	5	(	(	PUNCT
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ma-160	203	7	)	)	PUNCT
ma-160	203	8	,	,	PUNCT
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ma-160	203	23	.	.	PROPN
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ma-160	203	25	,	,	PUNCT
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ma-160	203	44	,	,	PUNCT
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ma-160	203	46	)	)	PUNCT
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ma-160	203	48	238	238	NUM
ma-160	203	49	-	-	SYM
ma-160	203	50	246	246	NUM
ma-160	203	51	.	.	PUNCT
ma-160	204	1	https://doi.org/	https://doi.org/	VERB
ma-160	204	2	10.22266	10.22266	NUM
ma-160	204	3	/	/	SYM
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ma-160	204	10	a	a	DET
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ma-160	204	14	for	for	ADP
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ma-160	204	17	networks	network	NOUN
ma-160	204	18	,	,	PUNCT
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ma-160	204	24	institutefor	institutefor	ADP
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ma-160	204	27	,	,	PUNCT
ma-160	204	28	(	(	PUNCT
ma-160	204	29	1993	1993	NUM
ma-160	204	30	)	)	PUNCT
ma-160	204	31	,	,	PUNCT
ma-160	204	32	8	8	NUM
ma-160	204	33	-	-	SYM
ma-160	204	34	93	93	NUM
ma-160	204	35	.	.	PUNCT
ma-160	205	1	http://hdl.handle.net/1903/5355.[8	http://hdl.handle.net/1903/5355.[8	PROPN
ma-160	205	2	]	]	PUNCT
ma-160	205	3	j.	j.	PROPN
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ma-160	205	5	,	,	PUNCT
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ma-160	205	10	ezeafulukwe	ezeafulukwe	PROPN
ma-160	205	11	,	,	PUNCT
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ma-160	205	13	sigmoid	sigmoid	NOUN
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ma-160	205	15	in	in	ADP
ma-160	205	16	univalent	univalent	ADJ
ma-160	205	17	function	function	NOUN
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ma-160	205	19	,	,	PUNCT
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ma-160	205	21	.	.	PUNCT
ma-160	206	1	j.	j.	PROPN
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ma-160	206	3	.	.	PUNCT
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ma-160	207	4	.	.	PROPN
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ma-160	207	6	(	(	PUNCT
ma-160	207	7	2013	2013	NUM
ma-160	207	8	)	)	PUNCT
ma-160	207	9	313	313	NUM
ma-160	207	10	-	-	SYM
ma-160	207	11	317	317	NUM
ma-160	207	12	.	.	PUNCT
ma-160	207	13	http://www.ascent-journals.com/ijmsea/vol7no5/33-joseph.pdf.[9	http://www.ascent-journals.com/ijmsea/vol7no5/33-joseph.pdf.[9	PROPN
ma-160	207	14	]	]	PUNCT
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ma-160	207	20	a	a	DET
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ma-160	207	22	approach	approach	NOUN
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ma-160	207	24	contrast	contrast	NOUN
ma-160	207	25	enhancement	enhancement	NOUN
ma-160	207	26	using	use	VERB
ma-160	207	27	sigmoid	sigmoid	NOUN
ma-160	207	28	function	function	NOUN
ma-160	207	29	,	,	PUNCT
ma-160	207	30	int	int	PROPN
ma-160	207	31	.	.	PUNCT
ma-160	208	1	arab	arab	PROPN
ma-160	208	2	j.	j.	PROPN
ma-160	208	3	inf.techn	inf.techn	PROPN
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ma-160	208	5	1	1	NUM
ma-160	208	6	(	(	PUNCT
ma-160	208	7	2004	2004	NUM
ma-160	208	8	)	)	PUNCT
ma-160	208	9	221	221	NUM
ma-160	208	10	-	-	SYM
ma-160	208	11	225	225	NUM
ma-160	208	12	.	.	PUNCT
ma-160	209	1	https://iajit.org/portal/pdf/vol.1,no.2/10-nagla.pdf[10	https://iajit.org/portal/pdf/vol.1,no.2/10-nagla.pdf[10	PROPN
ma-160	209	2	]	]	PUNCT
ma-160	209	3	t.	t.	PROPN
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ma-160	209	5	,	,	PUNCT
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ma-160	209	7	functions	function	NOUN
ma-160	209	8	in	in	ADP
ma-160	209	9	reliability	reliability	NOUN
ma-160	209	10	based	base	VERB
ma-160	209	11	management	management	NOUN
ma-160	209	12	,	,	PUNCT
ma-160	209	13	period	period	NOUN
ma-160	209	14	.	.	PUNCT
ma-160	210	1	polytechn	polytechn	PROPN
ma-160	210	2	.	.	PUNCT
ma-160	211	1	soc	soc	PROPN
ma-160	211	2	.	.	PUNCT
ma-160	212	1	manage	manage	VERB
ma-160	212	2	.	.	PUNCT
ma-160	213	1	sci	sci	PROPN
ma-160	213	2	.	.	PROPN
ma-160	213	3	2	2	NUM
ma-160	213	4	(	(	PUNCT
ma-160	213	5	2007	2007	NUM
ma-160	213	6	)	)	PUNCT
ma-160	213	7	,	,	PUNCT
ma-160	213	8	67	67	NUM
ma-160	213	9	-	-	SYM
ma-160	213	10	72	72	NUM
ma-160	213	11	.	.	PUNCT
ma-160	214	1	https://doi.org/10.3311/pp.so.2007-2.04.[11	https://doi.org/10.3311/pp.so.2007-2.04.[11	PROPN
ma-160	214	2	]	]	X
ma-160	215	1	n.	n.	NOUN
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ma-160	215	4	s.	s.	PROPN
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ma-160	215	7	on	on	ADP
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ma-160	215	9	hausdorff	hausdorff	NOUN
ma-160	215	10	distance	distance	NOUN
ma-160	215	11	between	between	ADP
ma-160	215	12	the	the	DET
ma-160	215	13	heaviside	heaviside	ADJ
ma-160	215	14	step	step	NOUN
ma-160	215	15	function	function	NOUN
ma-160	215	16	and	and	CCONJ
ma-160	215	17	verhulst	verhulst	ADP
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ma-160	215	19	,	,	PUNCT
ma-160	215	20	j.	j.	PROPN
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ma-160	215	22	.	.	PUNCT
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ma-160	216	2	.	.	PUNCT
ma-160	217	1	1	1	NUM
ma-160	217	2	(	(	PUNCT
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ma-160	217	4	)	)	PUNCT
ma-160	217	5	109	109	NUM
ma-160	217	6	-	-	SYM
ma-160	217	7	119	119	NUM
ma-160	217	8	.	.	PUNCT
ma-160	218	1	https://doi.org/10.1007/s10910-015-0552-0.[12	https://doi.org/10.1007/s10910-015-0552-0.[12	PROPN
ma-160	218	2	]	]	X
ma-160	218	3	a.	a.	PROPN
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ma-160	218	5	,	,	PUNCT
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ma-160	218	7	mehrotra	mehrotra	PROPN
ma-160	218	8	,	,	PUNCT
ma-160	218	9	c.k	c.k	PROPN
ma-160	218	10	.	.	PROPN
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ma-160	218	13	s.	s.	PROPN
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ma-160	218	15	,	,	PUNCT
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ma-160	218	24	applicationsto	applicationsto	NOUN
ma-160	218	25	neural	neural	ADJ
ma-160	218	26	networks	network	NOUN
ma-160	218	27	,	,	PUNCT
ma-160	218	28	neural	neural	ADJ
ma-160	218	29	networks	network	NOUN
ma-160	218	30	,	,	PUNCT
ma-160	218	31	5	5	NUM
ma-160	218	32	(	(	PUNCT
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ma-160	218	34	)	)	PUNCT
ma-160	218	35	,	,	PUNCT
ma-160	218	36	819	819	NUM
ma-160	218	37	-	-	SYM
ma-160	218	38	835	835	NUM
ma-160	218	39	.	.	PUNCT
ma-160	219	1	https://doi.org/10.1016/0893-6080(95)00107-7.[13	https://doi.org/10.1016/0893-6080(95)00107-7.[13	PROPN
ma-160	219	2	]	]	X
ma-160	219	3	k.	k.	PROPN
ma-160	219	4	nantomah	nantomah	PROPN
ma-160	219	5	,	,	PUNCT
ma-160	219	6	on	on	ADP
ma-160	219	7	some	some	DET
ma-160	219	8	properties	property	NOUN
ma-160	219	9	of	of	ADP
ma-160	219	10	the	the	DET
ma-160	219	11	sigmoid	sigmoid	NOUN
ma-160	219	12	function	function	NOUN
ma-160	219	13	,	,	PUNCT
ma-160	219	14	asia	asia	PROPN
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ma-160	219	16	.	.	PUNCT
ma-160	220	1	1	1	NUM
ma-160	220	2	(	(	PUNCT
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ma-160	220	4	)	)	PUNCT
ma-160	220	5	,	,	PUNCT
ma-160	220	6	79	79	NUM
ma-160	220	7	-	-	SYM
ma-160	220	8	90	90	NUM
ma-160	220	9	.	.	PUNCT
ma-160	221	1	http://www.asiamath	http://www.asiamath	NOUN
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ma-160	222	6	/	/	SYM
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ma-160	222	8	-	-	PUNCT
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ma-160	223	1	k.	k.	PROPN
ma-160	223	2	nantomah	nantomah	PROPN
ma-160	223	3	,	,	PUNCT
ma-160	223	4	c.a	c.a	PROPN
ma-160	223	5	.	.	PROPN
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ma-160	223	8	s.	s.	PROPN
ma-160	223	9	nasiru	nasiru	PROPN
ma-160	223	10	,	,	PUNCT
ma-160	223	11	on	on	ADP
ma-160	223	12	a	a	DET
ma-160	223	13	generalized	generalize	VERB
ma-160	223	14	sigmoid	sigmoid	NOUN
ma-160	223	15	function	function	NOUN
ma-160	223	16	and	and	CCONJ
ma-160	223	17	its	its	PRON
ma-160	223	18	properties	property	NOUN
ma-160	223	19	,	,	PUNCT
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ma-160	223	21	j.	j.	PROPN
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ma-160	224	1	appl.2020	appl.2020	CCONJ
ma-160	224	2	(	(	PUNCT
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ma-160	224	4	)	)	PUNCT
ma-160	224	5	,	,	PUNCT
ma-160	224	6	1	1	NUM
ma-160	224	7	-	-	SYM
ma-160	224	8	11	11	NUM
ma-160	224	9	.	.	PUNCT
ma-160	225	1	https://scienceasia.asia/files/528.pdf	https://scienceasia.asia/files/528.pdf	PROPN
ma-160	225	2	.	.	PUNCT
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ma-160	226	12	http://hdl.handle.net/1903/5355	http://hdl.handle.net/1903/5355	PROPN
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ma-160	226	14	https://iajit.org/portal/pdf/vol.1,no.2/10-nagla.pdf	https://iajit.org/portal/pdf/vol.1,no.2/10-nagla.pdf	PROPN
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ma-160	226	18	http://www.asiamath.org/issue1/vol3iss1/am-1904-4107.pdf	http://www.asiamath.org/issue1/vol3iss1/am-1904-4107.pdf	X
ma-160	226	19	http://www.asiamath.org/issue1/vol3iss1/am-1904-4107.pdf	http://www.asiamath.org/issue1/vol3iss1/am-1904-4107.pdf	VERB
ma-160	226	20	https://scienceasia.asia/files/528.pdf	https://scienceasia.asia/files/528.pdf	X
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ma-160	226	22	.	.	PUNCT
ma-160	227	1	j.	j.	PROPN
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ma-160	233	2	)	)	PUNCT
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ma-160	237	1	4	4	NUM
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ma-160	237	7	-	-	SYM
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ma-160	239	13	network	network	NOUN
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ma-160	239	17	,	,	PUNCT
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ma-160	239	19	systems	system	NOUN
ma-160	239	20	,	,	PUNCT
ma-160	239	21	(	(	PUNCT
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ma-160	239	23	)	)	PUNCT
ma-160	239	24	,	,	PUNCT
ma-160	239	25	1	1	NUM
ma-160	239	26	-	-	SYM
ma-160	239	27	19	19	NUM
ma-160	239	28	.	.	PUNCT
ma-160	239	29	https	https	NOUN
ma-160	239	30	:	:	PUNCT
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ma-160	240	2	/	/	SYM
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ma-160	241	2	.	.	PUNCT
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ma-160	242	5	https://doi.org/10.1007/bf02309004	https://doi.org/10.1007/bf02309004	X
ma-160	242	6	https://doi.org/10.5772/intechopen.80416	https://doi.org/10.5772/intechopen.80416	NOUN
ma-160	242	7	https://doi.org/10.5772/intechopen.80416	https://doi.org/10.5772/intechopen.80416	ADJ
ma-160	242	8	1	1	NUM
ma-160	242	9	.	.	PUNCT
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ma-160	242	11	2	2	NUM
ma-160	242	12	.	.	PUNCT
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ma-160	243	2	definitions	definition	NOUN
ma-160	243	3	and	and	CCONJ
ma-160	243	4	lemmas	lemmas	PROPN
ma-160	243	5	3	3	NUM
ma-160	243	6	.	.	NOUN
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ma-160	243	9	4	4	NUM
ma-160	243	10	.	.	PUNCT
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ma-160	243	12	5	5	NUM
ma-160	243	13	.	.	PUNCT
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ma-160	244	3	interest	interest	NOUN
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