id	sid	tid	token	lemma	pos
ma-225	1	1	2024	2024	NUM
ma-225	1	2	ada	ada	PROPN
ma-225	1	3	academica	academica	PROPN
ma-225	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-225	1	5	.	.	PUNCT
ma-225	2	1	j.	j.	PROPN
ma-225	2	2	math	math	PROPN
ma-225	2	3	.	.	PUNCT
ma-225	3	1	anal	anal	ADJ
ma-225	3	2	.	.	PUNCT
ma-225	4	1	4	4	NUM
ma-225	4	2	(	(	PUNCT
ma-225	4	3	2024	2024	NUM
ma-225	4	4	)	)	PUNCT
ma-225	4	5	19doi	19doi	NOUN
ma-225	4	6	:	:	PUNCT
ma-225	4	7	10.28924	10.28924	NUM
ma-225	4	8	/	/	SYM
ma-225	4	9	ada	ada	PROPN
ma-225	4	10	/	/	SYM
ma-225	4	11	ma.4.19	ma.4.19	PROPN
ma-225	4	12	convexity	convexity	NOUN
ma-225	4	13	properties	property	NOUN
ma-225	4	14	in	in	ADP
ma-225	4	15	non	non	ADJ
ma-225	4	16	-	-	ADJ
ma-225	4	17	newtonian	newtonian	ADJ
ma-225	4	18	calculus	calculus	NOUN
ma-225	4	19	and	and	CCONJ
ma-225	4	20	their	their	PRON
ma-225	4	21	applications	application	NOUN
ma-225	4	22	asambo	asambo	PROPN
ma-225	4	23	awini	awini	PROPN
ma-225	4	24	wilbert1,∗	wilbert1,∗	PROPN
ma-225	4	25	,	,	PUNCT
ma-225	4	26	mohammed	mohammed	PROPN
ma-225	4	27	muniru	muniru	PROPN
ma-225	4	28	iddrisu2	iddrisu2	PROPN
ma-225	4	29	,	,	PUNCT
ma-225	4	30	benedict	benedict	PROPN
ma-225	4	31	barnes3	barnes3	PUNCT
ma-225	5	1	1department	1department	NUM
ma-225	5	2	of	of	ADP
ma-225	5	3	mathematics	mathematic	NOUN
ma-225	5	4	,	,	PUNCT
ma-225	5	5	bongo	bongo	NOUN
ma-225	5	6	senior	senior	ADJ
ma-225	5	7	high	high	ADJ
ma-225	5	8	school	school	NOUN
ma-225	5	9	,	,	PUNCT
ma-225	5	10	box	box	PROPN
ma-225	5	11	7	7	NUM
ma-225	5	12	,	,	PUNCT
ma-225	5	13	bongo	bongo	NOUN
ma-225	5	14	district	district	NOUN
ma-225	5	15	,	,	PUNCT
ma-225	5	16	upper	upper	PROPN
ma-225	5	17	east	east	PROPN
ma-225	5	18	region	region	NOUN
ma-225	5	19	,	,	PUNCT
ma-225	5	20	ghana	ghana	PROPN
ma-225	5	21	awiniwilbert@gmail.com	awiniwilbert@gmail.com	PUNCT
ma-225	6	1	2department	2department	NUM
ma-225	6	2	of	of	ADP
ma-225	6	3	mathematics	mathematic	NOUN
ma-225	6	4	,	,	PUNCT
ma-225	6	5	faculty	faculty	NOUN
ma-225	6	6	of	of	ADP
ma-225	6	7	physical	physical	ADJ
ma-225	6	8	sciences	science	NOUN
ma-225	6	9	,	,	PUNCT
ma-225	6	10	university	university	NOUN
ma-225	6	11	for	for	ADP
ma-225	6	12	development	development	NOUN
ma-225	6	13	studies	study	NOUN
ma-225	6	14	,	,	PUNCT
ma-225	6	15	tamale	tamale	NOUN
ma-225	6	16	,	,	PUNCT
ma-225	6	17	ghana	ghana	PROPN
ma-225	6	18	mmuniru@uds.edu.gh	mmuniru@uds.edu.gh	PROPN
ma-225	6	19	3department	3department	PROPN
ma-225	6	20	of	of	ADP
ma-225	6	21	mathematics	mathematic	NOUN
ma-225	6	22	,	,	PUNCT
ma-225	6	23	faculty	faculty	NOUN
ma-225	6	24	of	of	ADP
ma-225	6	25	physical	physical	ADJ
ma-225	6	26	and	and	CCONJ
ma-225	6	27	computational	computational	ADJ
ma-225	6	28	sciences	science	NOUN
ma-225	6	29	,	,	PUNCT
ma-225	6	30	kwame	kwame	PROPN
ma-225	6	31	nkrumah	nkrumah	PROPN
ma-225	6	32	university	university	PROPN
ma-225	6	33	of	of	ADP
ma-225	6	34	science	science	NOUN
ma-225	6	35	and	and	CCONJ
ma-225	6	36	technology	technology	NOUN
ma-225	6	37	,	,	PUNCT
ma-225	6	38	kumasi	kumasi	PROPN
ma-225	6	39	,	,	PUNCT
ma-225	6	40	ghana	ghana	PROPN
ma-225	6	41	ewiekwamina@gmail.com	ewiekwamina@gmail.com	PUNCT
ma-225	7	1	∗correspondence	∗correspondence	NOUN
ma-225	7	2	:	:	PUNCT
ma-225	7	3	awiniwilbert@gmail.com	awiniwilbert@gmail.com	X
ma-225	8	1	abstract	abstract	ADJ
ma-225	8	2	.	.	PUNCT
ma-225	9	1	the	the	DET
ma-225	9	2	study	study	NOUN
ma-225	9	3	presented	present	VERB
ma-225	9	4	some	some	DET
ma-225	9	5	results	result	NOUN
ma-225	9	6	on	on	ADP
ma-225	9	7	convexity	convexity	NOUN
ma-225	9	8	properties	property	NOUN
ma-225	9	9	in	in	ADP
ma-225	9	10	non	non	ADJ
ma-225	9	11	-	-	ADJ
ma-225	9	12	newtonian	newtonian	ADJ
ma-225	9	13	calculus	calculus	NOUN
ma-225	9	14	.	.	PUNCT
ma-225	10	1	alsopresented	alsopresente	VERB
ma-225	10	2	is	be	AUX
ma-225	10	3	the	the	DET
ma-225	10	4	jensen	jensen	PROPN
ma-225	10	5	-	-	PUNCT
ma-225	10	6	steffensen	steffensen	PROPN
ma-225	10	7	inequality	inequality	NOUN
ma-225	10	8	in	in	ADP
ma-225	10	9	non	non	ADJ
ma-225	10	10	-	-	ADJ
ma-225	10	11	newtonian	newtonian	ADJ
ma-225	10	12	calculus	calculus	NOUN
ma-225	10	13	and	and	CCONJ
ma-225	10	14	some	some	DET
ma-225	10	15	applications	application	NOUN
ma-225	10	16	.	.	PUNCT
ma-225	11	1	theresearch	theresearch	NOUN
ma-225	11	2	was	be	AUX
ma-225	11	3	mainly	mainly	ADV
ma-225	11	4	on	on	ADP
ma-225	11	5	positive	positive	ADJ
ma-225	11	6	real	real	ADJ
ma-225	11	7	numbers	number	NOUN
ma-225	11	8	.	.	PUNCT
ma-225	12	1	1	1	X
ma-225	12	2	.	.	X
ma-225	12	3	introduction	introduction	NOUN
ma-225	12	4	classical	classical	ADJ
ma-225	12	5	calculus	calculus	NOUN
ma-225	12	6	was	be	AUX
ma-225	12	7	introduced	introduce	VERB
ma-225	12	8	by	by	ADP
ma-225	12	9	newton	newton	PROPN
ma-225	12	10	and	and	CCONJ
ma-225	12	11	leibnitz	leibnitz	PROPN
ma-225	12	12	which	which	PRON
ma-225	12	13	is	be	AUX
ma-225	12	14	applied	apply	VERB
ma-225	12	15	on	on	ADP
ma-225	12	16	our	our	PRON
ma-225	12	17	present	present	ADJ
ma-225	12	18	daymathematics	daymathematic	NOUN
ma-225	13	1	[	[	X
ma-225	13	2	1	1	NUM
ma-225	13	3	]	]	PUNCT
ma-225	13	4	.	.	PUNCT
ma-225	14	1	there	there	PRON
ma-225	14	2	are	be	VERB
ma-225	14	3	different	different	ADJ
ma-225	14	4	operations	operation	NOUN
ma-225	14	5	with	with	ADP
ma-225	14	6	respect	respect	NOUN
ma-225	14	7	to	to	ADP
ma-225	14	8	addition	addition	NOUN
ma-225	14	9	and	and	CCONJ
ma-225	14	10	subtraction	subtraction	NOUN
ma-225	14	11	of	of	ADP
ma-225	14	12	numbersunder	numbersunder	NOUN
ma-225	14	13	this	this	DET
ma-225	14	14	calculus	calculus	NOUN
ma-225	14	15	.	.	PUNCT
ma-225	15	1	however	however	ADV
ma-225	15	2	,	,	PUNCT
ma-225	15	3	grossman	grossman	NOUN
ma-225	15	4	and	and	CCONJ
ma-225	15	5	karts	kart	NOUN
ma-225	15	6	came	come	VERB
ma-225	15	7	out	out	ADP
ma-225	15	8	with	with	ADP
ma-225	15	9	another	another	DET
ma-225	15	10	calculus	calculus	NOUN
ma-225	15	11	known	know	VERB
ma-225	15	12	asnon	asnon	ADJ
ma-225	15	13	-	-	PUNCT
ma-225	15	14	newtonian	newtonian	ADJ
ma-225	15	15	calculus	calculus	NOUN
ma-225	15	16	in	in	ADP
ma-225	15	17	the	the	DET
ma-225	15	18	20th	20th	ADJ
ma-225	15	19	century	century	NOUN
ma-225	15	20	[	[	X
ma-225	15	21	2	2	NUM
ma-225	15	22	]	]	PUNCT
ma-225	15	23	.	.	PUNCT
ma-225	16	1	non	non	ADJ
ma-225	16	2	-	-	ADJ
ma-225	16	3	newtonian	newtonian	ADJ
ma-225	16	4	calculus	calculus	NOUN
ma-225	16	5	which	which	PRON
ma-225	16	6	is	be	AUX
ma-225	16	7	also	also	ADV
ma-225	16	8	calledmultiplicative	calledmultiplicative	ADJ
ma-225	16	9	calculus	calculus	NOUN
ma-225	16	10	is	be	AUX
ma-225	16	11	a	a	DET
ma-225	16	12	multiplicative	multiplicative	ADJ
ma-225	16	13	way	way	NOUN
ma-225	16	14	of	of	ADP
ma-225	16	15	generating	generate	VERB
ma-225	16	16	positive	positive	ADJ
ma-225	16	17	solutions	solution	NOUN
ma-225	16	18	to	to	PART
ma-225	16	19	mathematicalproblems	mathematicalproblems	VERB
ma-225	16	20	[	[	X
ma-225	16	21	3	3	NUM
ma-225	16	22	]	]	PUNCT
ma-225	16	23	.	.	PUNCT
ma-225	17	1	it	it	PRON
ma-225	17	2	is	be	AUX
ma-225	17	3	a	a	DET
ma-225	17	4	recent	recent	ADJ
ma-225	17	5	approach	approach	NOUN
ma-225	17	6	used	use	VERB
ma-225	17	7	to	to	PART
ma-225	17	8	solve	solve	VERB
ma-225	17	9	mathematical	mathematical	ADJ
ma-225	17	10	problems	problem	NOUN
ma-225	17	11	with	with	ADP
ma-225	17	12	positive	positive	ADJ
ma-225	17	13	realnumbers.it	realnumbers.it	PRON
ma-225	17	14	is	be	AUX
ma-225	17	15	evidently	evidently	ADV
ma-225	17	16	clear	clear	ADJ
ma-225	17	17	that	that	SCONJ
ma-225	17	18	addition	addition	NOUN
ma-225	17	19	is	be	AUX
ma-225	17	20	replaced	replace	VERB
ma-225	17	21	by	by	ADP
ma-225	17	22	multiplication	multiplication	NOUN
ma-225	17	23	in	in	ADP
ma-225	17	24	non	non	ADJ
ma-225	17	25	-	-	ADJ
ma-225	17	26	newtonian	newtonian	ADJ
ma-225	17	27	calculus	calculus	NOUN
ma-225	17	28	,	,	PUNCT
ma-225	17	29	andsubtraction	andsubtraction	NOUN
ma-225	17	30	by	by	ADP
ma-225	17	31	division	division	NOUN
ma-225	17	32	for	for	ADP
ma-225	17	33	example	example	NOUN
ma-225	17	34	,	,	PUNCT
ma-225	17	35	see	see	VERB
ma-225	17	36	authors	author	NOUN
ma-225	17	37	in	in	ADP
ma-225	17	38	[	[	X
ma-225	17	39	4	4	NUM
ma-225	17	40	,	,	PUNCT
ma-225	17	41	5	5	NUM
ma-225	17	42	]	]	PUNCT
ma-225	17	43	.	.	PUNCT
ma-225	18	1	this	this	DET
ma-225	18	2	result	result	NOUN
ma-225	18	3	has	have	AUX
ma-225	18	4	been	be	AUX
ma-225	18	5	supported	support	VERB
ma-225	18	6	by	by	ADP
ma-225	18	7	au	au	NOUN
ma-225	18	8	-	-	NOUN
ma-225	18	9	thors	thor	NOUN
ma-225	18	10	in	in	ADP
ma-225	18	11	[	[	X
ma-225	18	12	6	6	NUM
ma-225	18	13	]	]	PUNCT
ma-225	18	14	,	,	PUNCT
ma-225	18	15	when	when	SCONJ
ma-225	18	16	they	they	PRON
ma-225	18	17	introduced	introduce	VERB
ma-225	18	18	the	the	DET
ma-225	18	19	multiplicative	multiplicative	ADJ
ma-225	18	20	calculus	calculus	NOUN
ma-225	18	21	and	and	CCONJ
ma-225	18	22	its	its	PRON
ma-225	18	23	applications	application	NOUN
ma-225	18	24	,	,	PUNCT
ma-225	18	25	which	which	PRON
ma-225	18	26	has	have	AUX
ma-225	18	27	beenestablished	beenestablishe	VERB
ma-225	18	28	to	to	PART
ma-225	18	29	be	be	AUX
ma-225	18	30	applicable	applicable	ADJ
ma-225	18	31	to	to	ADP
ma-225	18	32	solving	solve	VERB
ma-225	18	33	mathematical	mathematical	ADJ
ma-225	18	34	problems	problem	NOUN
ma-225	18	35	[	[	X
ma-225	18	36	6	6	NUM
ma-225	18	37	]	]	PUNCT
ma-225	18	38	.	.	PUNCT
ma-225	19	1	non	non	ADJ
ma-225	19	2	-	-	ADJ
ma-225	19	3	newtonian	newtonian	ADJ
ma-225	19	4	calculus	calculus	NOUN
ma-225	19	5	hasbeen	hasbeen	VERB
ma-225	19	6	extended	extend	VERB
ma-225	19	7	in	in	ADP
ma-225	19	8	many	many	ADJ
ma-225	19	9	directions	direction	NOUN
ma-225	19	10	;	;	PUNCT
ma-225	19	11	fractional	fractional	ADJ
ma-225	19	12	derivative	derivative	ADJ
ma-225	19	13	,	,	PUNCT
ma-225	19	14	complex	complex	ADJ
ma-225	19	15	derivative	derivative	ADJ
ma-225	19	16	,	,	PUNCT
ma-225	19	17	integral	integral	ADJ
ma-225	19	18	transformations	transformation	NOUN
ma-225	19	19	,	,	PUNCT
ma-225	19	20	differential	differential	ADJ
ma-225	19	21	equations	equation	NOUN
ma-225	19	22	and	and	CCONJ
ma-225	19	23	applications	application	NOUN
ma-225	19	24	for	for	ADP
ma-225	19	25	science	science	NOUN
ma-225	19	26	and	and	CCONJ
ma-225	19	27	engineering	engineering	NOUN
ma-225	19	28	.	.	PUNCT
ma-225	20	1	received	receive	VERB
ma-225	20	2	:	:	PUNCT
ma-225	20	3	30	30	NUM
ma-225	20	4	jan	jan	PROPN
ma-225	20	5	2024	2024	NUM
ma-225	20	6	.	.	PUNCT
ma-225	21	1	key	key	ADJ
ma-225	21	2	words	word	NOUN
ma-225	21	3	and	and	CCONJ
ma-225	21	4	phrases	phrase	NOUN
ma-225	21	5	.	.	PUNCT
ma-225	22	1	non	non	ADJ
ma-225	22	2	-	-	ADJ
ma-225	22	3	newtonian	newtonian	ADJ
ma-225	22	4	calculus	calculus	NOUN
ma-225	22	5	,	,	PUNCT
ma-225	22	6	properties	property	NOUN
ma-225	22	7	,	,	PUNCT
ma-225	22	8	convexity	convexity	NOUN
ma-225	22	9	,	,	PUNCT
ma-225	22	10	jensen	jensen	PROPN
ma-225	22	11	-	-	PUNCT
ma-225	22	12	steffesen	steffesen	PROPN
ma-225	22	13	inequality.1	inequality.1	PROPN
ma-225	22	14	https://adac.ee	https://adac.ee	PROPN
ma-225	22	15	https://doi.org/10.28924/ada/ma.4.19	https://doi.org/10.28924/ada/ma.4.19	PRON
ma-225	22	16	https://orcid.org/0009-0004-4991-2067	https://orcid.org/0009-0004-4991-2067	NOUN
ma-225	22	17	https://orcid.org/0000-0001-7628-8168	https://orcid.org/0000-0001-7628-8168	PROPN
ma-225	22	18	https://orcid.org/0000-0002-0580-5655	https://orcid.org/0000-0002-0580-5655	PROPN
ma-225	22	19	the	the	DET
ma-225	22	20	authors	author	NOUN
ma-225	22	21	in	in	ADP
ma-225	22	22	[	[	X
ma-225	22	23	7	7	NUM
ma-225	22	24	]	]	PUNCT
ma-225	22	25	stated	state	VERB
ma-225	22	26	that	that	SCONJ
ma-225	22	27	,	,	PUNCT
ma-225	22	28	the	the	DET
ma-225	22	29	centre	centre	NOUN
ma-225	22	30	of	of	ADP
ma-225	22	31	all	all	DET
ma-225	22	32	analysis	analysis	NOUN
ma-225	22	33	in	in	ADP
ma-225	22	34	the	the	DET
ma-225	22	35	social	social	ADJ
ma-225	22	36	science	science	NOUN
ma-225	22	37	is	be	AUX
ma-225	22	38	the	the	DET
ma-225	22	39	derivative.they	derivative.they	PROPN
ma-225	22	40	were	be	AUX
ma-225	22	41	expecting	expect	VERB
ma-225	22	42	another	another	DET
ma-225	22	43	method	method	NOUN
ma-225	22	44	which	which	PRON
ma-225	22	45	may	may	AUX
ma-225	22	46	treat	treat	VERB
ma-225	22	47	realistic	realistic	ADJ
ma-225	22	48	growth	growth	NOUN
ma-225	22	49	phenomenon	phenomenon	NOUN
ma-225	22	50	better	well	ADV
ma-225	22	51	in	in	ADP
ma-225	22	52	oureconomy	oureconomy	ADV
ma-225	22	53	than	than	ADP
ma-225	22	54	the	the	DET
ma-225	22	55	ordinary	ordinary	ADJ
ma-225	22	56	approach	approach	NOUN
ma-225	22	57	.	.	PUNCT
ma-225	23	1	the	the	DET
ma-225	23	2	change	change	NOUN
ma-225	23	3	actually	actually	ADV
ma-225	23	4	came	come	VERB
ma-225	23	5	,	,	PUNCT
ma-225	23	6	which	which	PRON
ma-225	23	7	confirmed	confirm	VERB
ma-225	23	8	that	that	SCONJ
ma-225	23	9	variations	variation	NOUN
ma-225	23	10	aremore	aremore	ADV
ma-225	23	11	naturally	naturally	ADV
ma-225	23	12	measured	measure	VERB
ma-225	23	13	in	in	ADP
ma-225	23	14	ratios	ratio	NOUN
ma-225	23	15	than	than	ADP
ma-225	23	16	in	in	ADP
ma-225	23	17	differences	difference	NOUN
ma-225	23	18	[	[	X
ma-225	23	19	7	7	NUM
ma-225	23	20	]	]	PUNCT
ma-225	23	21	,	,	PUNCT
ma-225	23	22	until	until	ADP
ma-225	23	23	1972	1972	NUM
ma-225	23	24	that	that	SCONJ
ma-225	23	25	grossman	grossman	PROPN
ma-225	23	26	and	and	CCONJ
ma-225	23	27	katz	katz	PROPN
ma-225	23	28	cameout	cameout	PROPN
ma-225	23	29	with	with	ADP
ma-225	23	30	non	non	ADJ
ma-225	23	31	-	-	ADJ
ma-225	23	32	newtonian	newtonian	ADJ
ma-225	23	33	calculus	calculus	NOUN
ma-225	23	34	(	(	PUNCT
ma-225	23	35	see	see	VERB
ma-225	23	36	[	[	X
ma-225	23	37	2	2	NUM
ma-225	23	38	]	]	PUNCT
ma-225	23	39	)	)	PUNCT
ma-225	23	40	.	.	PUNCT
ma-225	24	1	in	in	ADP
ma-225	24	2	their	their	PRON
ma-225	24	3	work	work	NOUN
ma-225	24	4	,	,	PUNCT
ma-225	24	5	they	they	PRON
ma-225	24	6	also	also	ADV
ma-225	24	7	made	make	VERB
ma-225	24	8	it	it	PRON
ma-225	24	9	clearly	clearly	ADV
ma-225	24	10	that	that	PRON
ma-225	24	11	measuringgrowth	measuringgrowth	NOUN
ma-225	24	12	in	in	ADP
ma-225	24	13	ratios	ratio	NOUN
ma-225	24	14	gives	give	VERB
ma-225	24	15	a	a	DET
ma-225	24	16	better	well	ADJ
ma-225	24	17	variations	variation	NOUN
ma-225	24	18	than	than	ADP
ma-225	24	19	measuring	measure	VERB
ma-225	24	20	it	it	PRON
ma-225	24	21	in	in	ADP
ma-225	24	22	differences	difference	NOUN
ma-225	24	23	.	.	PUNCT
ma-225	25	1	non	non	ADJ
ma-225	25	2	-	-	ADJ
ma-225	25	3	newtonian	newtonian	ADJ
ma-225	25	4	calculusis	calculusis	NOUN
ma-225	25	5	essential	essential	ADJ
ma-225	25	6	in	in	ADP
ma-225	25	7	the	the	DET
ma-225	25	8	development	development	NOUN
ma-225	25	9	of	of	ADP
ma-225	25	10	our	our	PRON
ma-225	25	11	scientific	scientific	ADJ
ma-225	25	12	world	world	NOUN
ma-225	25	13	,	,	PUNCT
ma-225	25	14	which	which	PRON
ma-225	25	15	enhances	enhance	VERB
ma-225	25	16	production	production	NOUN
ma-225	25	17	and	and	CCONJ
ma-225	25	18	development.it	development.it	PRON
ma-225	25	19	is	be	AUX
ma-225	25	20	applicable	applicable	ADJ
ma-225	25	21	in	in	ADP
ma-225	25	22	various	various	ADJ
ma-225	25	23	ways	way	NOUN
ma-225	25	24	such	such	ADJ
ma-225	25	25	as	as	ADP
ma-225	25	26	finance	finance	NOUN
ma-225	25	27	(	(	PUNCT
ma-225	25	28	used	use	VERB
ma-225	25	29	in	in	ADP
ma-225	25	30	marketing	marketing	NOUN
ma-225	25	31	and	and	CCONJ
ma-225	25	32	determining	determine	VERB
ma-225	25	33	rates	rate	NOUN
ma-225	25	34	ofreturn	ofreturn	NOUN
ma-225	25	35	)	)	PUNCT
ma-225	25	36	,	,	PUNCT
ma-225	25	37	health	health	NOUN
ma-225	25	38	(	(	PUNCT
ma-225	25	39	used	use	VERB
ma-225	25	40	in	in	ADP
ma-225	25	41	tumor	tumor	NOUN
ma-225	25	42	therapy	therapy	NOUN
ma-225	25	43	and	and	CCONJ
ma-225	25	44	chemotherapy	chemotherapy	NOUN
ma-225	25	45	in	in	ADP
ma-225	25	46	medicine	medicine	NOUN
ma-225	25	47	,	,	PUNCT
ma-225	25	48	pathogen	pathogen	NOUN
ma-225	25	49	counts	count	NOUN
ma-225	25	50	in	in	ADP
ma-225	25	51	treatedwater	treatedwater	NOUN
ma-225	25	52	)	)	PUNCT
ma-225	25	53	,	,	PUNCT
ma-225	25	54	thermostatistics	thermostatistic	NOUN
ma-225	25	55	,	,	PUNCT
ma-225	25	56	quantum	quantum	ADJ
ma-225	25	57	theory	theory	NOUN
ma-225	25	58	,	,	PUNCT
ma-225	25	59	wave	wave	NOUN
ma-225	25	60	phenomenon	phenomenon	NOUN
ma-225	25	61	,	,	PUNCT
ma-225	25	62	pattern	pattern	NOUN
ma-225	25	63	recognition	recognition	NOUN
ma-225	25	64	in	in	ADP
ma-225	25	65	images	image	NOUN
ma-225	25	66	(	(	PUNCT
ma-225	25	67	eg	eg	NOUN
ma-225	25	68	.	.	PUNCT
ma-225	25	69	inbiomedicine	inbiomedicine	PROPN
ma-225	25	70	)	)	PUNCT
ma-225	25	71	,	,	PUNCT
ma-225	25	72	signal	signal	ADJ
ma-225	25	73	processing	processing	NOUN
ma-225	25	74	,	,	PUNCT
ma-225	25	75	biology	biology	NOUN
ma-225	25	76	-	-	PUNCT
ma-225	25	77	thus	thus	ADV
ma-225	25	78	the	the	DET
ma-225	25	79	rate	rate	NOUN
ma-225	25	80	at	at	ADP
ma-225	25	81	which	which	PRON
ma-225	25	82	growth	growth	NOUN
ma-225	25	83	increases	increase	VERB
ma-225	25	84	or	or	CCONJ
ma-225	25	85	decays.in	decays.in	PRON
ma-225	25	86	the	the	DET
ma-225	25	87	non	non	ADJ
ma-225	25	88	-	-	ADJ
ma-225	25	89	newtonian	newtonian	ADJ
ma-225	25	90	calculus	calculus	NOUN
ma-225	25	91	,	,	PUNCT
ma-225	25	92	ratios	ratio	NOUN
ma-225	25	93	are	be	AUX
ma-225	25	94	used	use	VERB
ma-225	25	95	in	in	ADP
ma-225	25	96	measuring	measure	VERB
ma-225	25	97	change	change	NOUN
ma-225	25	98	in	in	ADP
ma-225	25	99	values	value	NOUN
ma-225	25	100	whiles	while	NOUN
ma-225	25	101	in	in	ADP
ma-225	25	102	theclassical	theclassical	ADJ
ma-225	25	103	approach	approach	NOUN
ma-225	25	104	,	,	PUNCT
ma-225	25	105	differences	difference	NOUN
ma-225	25	106	are	be	AUX
ma-225	25	107	used	use	VERB
ma-225	25	108	in	in	ADP
ma-225	25	109	measuring	measure	VERB
ma-225	25	110	change	change	NOUN
ma-225	25	111	in	in	ADP
ma-225	25	112	values	value	NOUN
ma-225	25	113	.	.	PUNCT
ma-225	26	1	2	2	X
ma-225	26	2	.	.	X
ma-225	26	3	preliminaries	preliminary	NOUN
ma-225	26	4	in	in	ADP
ma-225	26	5	this	this	DET
ma-225	26	6	section	section	NOUN
ma-225	26	7	,	,	PUNCT
ma-225	26	8	we	we	PRON
ma-225	26	9	give	give	VERB
ma-225	26	10	an	an	DET
ma-225	26	11	overview	overview	NOUN
ma-225	26	12	of	of	ADP
ma-225	26	13	known	known	ADJ
ma-225	26	14	definitions	definition	NOUN
ma-225	26	15	and	and	CCONJ
ma-225	26	16	theories	theory	NOUN
ma-225	26	17	used	use	VERB
ma-225	26	18	in	in	ADP
ma-225	26	19	achieving	achieve	VERB
ma-225	26	20	ourresults	ourresult	NOUN
ma-225	26	21	.	.	PUNCT
ma-225	27	1	2.1	2.1	NUM
ma-225	27	2	.	.	PUNCT
ma-225	28	1	non	non	ADJ
ma-225	28	2	-	-	ADJ
ma-225	28	3	newtonian	newtonian	ADJ
ma-225	28	4	arithmetic	arithmetic	ADJ
ma-225	28	5	.	.	PUNCT
ma-225	29	1	a	a	DET
ma-225	29	2	system	system	NOUN
ma-225	29	3	that	that	PRON
ma-225	29	4	satisfies	satisfy	VERB
ma-225	29	5	the	the	DET
ma-225	29	6	basic	basic	ADJ
ma-225	29	7	assumptions	assumption	NOUN
ma-225	29	8	whose	whose	DET
ma-225	29	9	domain	domain	NOUN
ma-225	29	10	is	be	AUX
ma-225	29	11	asubset	asubset	VERB
ma-225	29	12	of	of	ADP
ma-225	29	13	r	r	NOUN
ma-225	29	14	is	be	AUX
ma-225	29	15	called	call	VERB
ma-225	29	16	arithmetic	arithmetic	ADJ
ma-225	29	17	.	.	PUNCT
ma-225	30	1	exactly	exactly	ADV
ma-225	30	2	one	one	NUM
ma-225	30	3	arithmetic	arithmetic	ADJ
ma-225	30	4	result	result	NOUN
ma-225	30	5	is	be	AUX
ma-225	30	6	produced	produce	VERB
ma-225	30	7	by	by	ADP
ma-225	30	8	a	a	DET
ma-225	30	9	generator	generator	NOUN
ma-225	30	10	,	,	PUNCT
ma-225	30	11	whichis	whichi	VERB
ma-225	30	12	a	a	DET
ma-225	30	13	one	one	NUM
ma-225	30	14	-	-	PUNCT
ma-225	30	15	to	to	ADP
ma-225	30	16	-	-	PUNCT
ma-225	30	17	one	one	NUM
ma-225	30	18	function	function	NOUN
ma-225	30	19	with	with	ADP
ma-225	30	20	a	a	DET
ma-225	30	21	range	range	NOUN
ma-225	30	22	of	of	ADP
ma-225	30	23	b	b	NOUN
ma-225	30	24	that	that	PRON
ma-225	30	25	is	be	AUX
ma-225	30	26	a	a	DET
ma-225	30	27	subset	subset	NOUN
ma-225	30	28	of	of	ADP
ma-225	30	29	the	the	DET
ma-225	30	30	domain	domain	NOUN
ma-225	30	31	r	r	NOUN
ma-225	31	1	[	[	X
ma-225	31	2	3,8	3,8	NUM
ma-225	31	3	]	]	PUNCT
ma-225	31	4	.	.	PUNCT
ma-225	32	1	the	the	DET
ma-225	32	2	fundamentalarithmetic	fundamentalarithmetic	ADJ
ma-225	32	3	operations	operation	NOUN
ma-225	32	4	are	be	AUX
ma-225	32	5	defined	define	VERB
ma-225	32	6	using	use	VERB
ma-225	32	7	the	the	DET
ma-225	32	8	generator	generator	NOUN
ma-225	32	9	as	as	SCONJ
ma-225	32	10	follows	follow	VERB
ma-225	32	11	[	[	X
ma-225	32	12	3	3	NUM
ma-225	32	13	,	,	PUNCT
ma-225	32	14	4]:addition	4]:addition	NUM
ma-225	32	15	,	,	PUNCT
ma-225	33	1	k+̇r	k+̇r	PROPN
ma-225	33	2	=	=	SYM
ma-225	33	3	α	α	X
ma-225	33	4	[	[	X
ma-225	33	5	α−1(k	α−1(k	X
ma-225	33	6	)	)	PUNCT
ma-225	33	7	+	+	NUM
ma-225	33	8	α−1(r)]subtraction	α−1(r)]subtraction	NOUN
ma-225	33	9	,	,	PUNCT
ma-225	33	10	k−̇r	k−̇r	PROPN
ma-225	33	11	=	=	PUNCT
ma-225	33	12	α	α	PROPN
ma-225	34	1	[	[	X
ma-225	34	2	α−1(k)−	α−1(k)−	NOUN
ma-225	34	3	α−1(r)]multiplication	α−1(r)]multiplication	NOUN
ma-225	34	4	,	,	PUNCT
ma-225	34	5	k×̇r	k×̇r	PROPN
ma-225	34	6	=	=	SYM
ma-225	34	7	α	α	PROPN
ma-225	34	8	[	[	X
ma-225	34	9	α−1(k)×	α−1(k)×	NOUN
ma-225	34	10	α−1(r)]division	α−1(r)]division	NOUN
ma-225	34	11	,	,	PUNCT
ma-225	34	12	k/̇r	k/̇r	PROPN
ma-225	34	13	=	=	PUNCT
ma-225	34	14	α	α	PROPN
ma-225	35	1	[	[	X
ma-225	35	2	α−1(k)/α−1(r)]when	α−1(k)/α−1(r)]when	ADV
ma-225	35	3	we	we	PRON
ma-225	35	4	take	take	VERB
ma-225	35	5	α	α	NOUN
ma-225	35	6	-	-	PUNCT
ma-225	35	7	generator	generator	NOUN
ma-225	35	8	as	as	ADP
ma-225	35	9	α(k	α(k	NOUN
ma-225	35	10	)	)	PUNCT
ma-225	36	1	=	=	SYM
ma-225	36	2	ek	ek	X
ma-225	36	3	,	,	PUNCT
ma-225	36	4	α−1(k	α−1(k	PROPN
ma-225	36	5	)	)	PUNCT
ma-225	36	6	=	=	SYM
ma-225	36	7	ln(k	ln(k	NUM
ma-225	36	8	)	)	PUNCT
ma-225	36	9	and	and	CCONJ
ma-225	36	10	k	k	X
ma-225	36	11	=	=	PUNCT
ma-225	36	12	r+	r+	PROPN
ma-225	36	13	,	,	PUNCT
ma-225	36	14	then	then	ADV
ma-225	36	15	α	α	PROPN
ma-225	36	16	arithmetic	arithmetic	PROPN
ma-225	36	17	reducesto	reducesto	PROPN
ma-225	36	18	non	non	ADJ
ma-225	36	19	-	-	ADJ
ma-225	36	20	newtonian	newtonian	ADJ
ma-225	36	21	arithmetic	arithmetic	ADJ
ma-225	36	22	as	as	SCONJ
ma-225	36	23	follows	follow	VERB
ma-225	36	24	:	:	PUNCT
ma-225	36	25	non	non	ADJ
ma-225	36	26	-	-	ADJ
ma-225	36	27	newtonian	newtonian	ADJ
ma-225	36	28	addition	addition	NOUN
ma-225	36	29	,	,	PUNCT
ma-225	36	30	k+̇r	k+̇r	PROPN
ma-225	36	31	=	=	SYM
ma-225	36	32	α	α	PROPN
ma-225	36	33	[	[	PUNCT
ma-225	36	34	α−1(k	α−1(k	PROPN
ma-225	36	35	)	)	PUNCT
ma-225	36	36	+	+	NUM
ma-225	36	37	α−1(r	α−1(r	NOUN
ma-225	36	38	)	)	PUNCT
ma-225	36	39	]	]	PUNCT
ma-225	37	1	=	=	PUNCT
ma-225	37	2	e(ln(k)+ln(r	e(ln(k)+ln(r	NOUN
ma-225	37	3	)	)	PUNCT
ma-225	37	4	)	)	PUNCT
ma-225	38	1	=	=	PUNCT
ma-225	39	1	k	k	X
ma-225	39	2	·	·	PUNCT
ma-225	39	3	r	r	NOUN
ma-225	39	4	(	(	PUNCT
ma-225	39	5	1	1	NUM
ma-225	39	6	)	)	PUNCT
ma-225	39	7	non	non	ADJ
ma-225	39	8	-	-	ADJ
ma-225	39	9	newtonian	newtonian	ADJ
ma-225	39	10	subtraction	subtraction	NOUN
ma-225	39	11	,	,	PUNCT
ma-225	39	12	k−̇r	k−̇r	PROPN
ma-225	39	13	=	=	PUNCT
ma-225	39	14	α	α	PROPN
ma-225	39	15	[	[	PUNCT
ma-225	39	16	α−1(k)−	α−1(k)−	PROPN
ma-225	39	17	α−1(r	α−1(r	NOUN
ma-225	39	18	)	)	PUNCT
ma-225	39	19	]	]	PUNCT
ma-225	40	1	=	=	PUNCT
ma-225	40	2	e(ln(k)−ln(r	e(ln(k)−ln(r	NOUN
ma-225	40	3	)	)	PUNCT
ma-225	40	4	)	)	PUNCT
ma-225	41	1	=	=	PUNCT
ma-225	42	1	k	k	X
ma-225	42	2	/	/	SYM
ma-225	42	3	r	r	NOUN
ma-225	42	4	(	(	PUNCT
ma-225	42	5	2	2	NUM
ma-225	42	6	)	)	PUNCT
ma-225	42	7	non	non	ADJ
ma-225	42	8	-	-	ADJ
ma-225	42	9	newtonian	newtonian	ADJ
ma-225	42	10	multiplication	multiplication	NOUN
ma-225	42	11	,	,	PUNCT
ma-225	42	12	k×̇r	k×̇r	PROPN
ma-225	42	13	=	=	SYM
ma-225	42	14	α	α	PROPN
ma-225	42	15	[	[	PUNCT
ma-225	42	16	α−1(k)×	α−1(k)×	NOUN
ma-225	42	17	α−1(r	α−1(r	NOUN
ma-225	42	18	)	)	PUNCT
ma-225	42	19	]	]	PUNCT
ma-225	43	1	=	=	PUNCT
ma-225	43	2	e(ln(k)×ln(r	e(ln(k)×ln(r	NUM
ma-225	43	3	)	)	PUNCT
ma-225	43	4	)	)	PUNCT
ma-225	44	1	=	=	SYM
ma-225	44	2	k	k	X
ma-225	44	3	ln(r	ln(r	NOUN
ma-225	44	4	)	)	PUNCT
ma-225	44	5	(	(	PUNCT
ma-225	44	6	3)2	3)2	NUM
ma-225	44	7	non	non	ADJ
ma-225	44	8	-	-	ADJ
ma-225	44	9	newtonian	newtonian	ADJ
ma-225	44	10	division	division	NOUN
ma-225	44	11	,	,	PUNCT
ma-225	44	12	k/̇r	k/̇r	PROPN
ma-225	44	13	=	=	PUNCT
ma-225	44	14	α	α	PROPN
ma-225	44	15	[	[	PUNCT
ma-225	44	16	α−1(k)/α−1(r	α−1(k)/α−1(r	NOUN
ma-225	44	17	)	)	PUNCT
ma-225	44	18	]	]	PUNCT
ma-225	45	1	=	=	SYM
ma-225	45	2	e(ln(k)/	e(ln(k)/	PROPN
ma-225	45	3	ln(r	ln(r	NOUN
ma-225	45	4	)	)	PUNCT
ma-225	45	5	)	)	PUNCT
ma-225	46	1	=	=	PUNCT
ma-225	46	2	k	k	NOUN
ma-225	46	3	1	1	NUM
ma-225	46	4	ln(r	ln(r	NOUN
ma-225	46	5	)	)	PUNCT
ma-225	46	6	(	(	PUNCT
ma-225	46	7	4	4	X
ma-225	46	8	)	)	PUNCT
ma-225	47	1	[	[	X
ma-225	47	2	1	1	NUM
ma-225	47	3	,	,	PUNCT
ma-225	47	4	4	4	NUM
ma-225	47	5	,	,	PUNCT
ma-225	47	6	8	8	NUM
ma-225	47	7	,	,	PUNCT
ma-225	47	8	9].the	9].the	PRON
ma-225	47	9	above	above	ADP
ma-225	47	10	non	non	ADJ
ma-225	47	11	-	-	ADJ
ma-225	47	12	newtonian	newtonian	ADJ
ma-225	47	13	arithmetic	arithmetic	NOUN
ma-225	47	14	are	be	AUX
ma-225	47	15	widely	widely	ADV
ma-225	47	16	accepted	accept	VERB
ma-225	47	17	in	in	ADP
ma-225	47	18	non	non	ADJ
ma-225	47	19	-	-	ADJ
ma-225	47	20	newtonian	newtonian	ADJ
ma-225	47	21	calculus.when	calculus.when	NOUN
ma-225	47	22	considering	consider	VERB
ma-225	47	23	n	n	CCONJ
ma-225	47	24	positive	positive	ADJ
ma-225	47	25	real	real	ADJ
ma-225	47	26	numbers	number	NOUN
ma-225	47	27	x1	x1	PROPN
ma-225	47	28	,	,	PUNCT
ma-225	47	29	x2	x2	PROPN
ma-225	47	30	,	,	PUNCT
ma-225	47	31	...	...	PUNCT
ma-225	47	32	,	,	PUNCT
ma-225	47	33	xn	xn	PROPN
ma-225	47	34	then	then	ADV
ma-225	47	35	the	the	DET
ma-225	47	36	α	α	NOUN
ma-225	47	37	-	-	ADJ
ma-225	47	38	arithmetic	arithmetic	ADJ
ma-225	47	39	mean	mean	NOUN
ma-225	47	40	is	be	AUX
ma-225	47	41	given	give	VERB
ma-225	47	42	as	as	ADP
ma-225	47	43	[	[	X
ma-225	47	44	10	10	NUM
ma-225	47	45	]	]	PUNCT
ma-225	47	46	:	:	PUNCT
ma-225	48	1	aα	aα	NOUN
ma-225	48	2	=	=	PUNCT
ma-225	49	1	∑n	∑n	PROPN
ma-225	49	2	i=1	i=1	X
ma-225	49	3	xi	xi	X
ma-225	49	4	/̇n	/̇n	PUNCT
ma-225	50	1	=	=	PUNCT
ma-225	50	2	∑n	∑n	PROPN
ma-225	51	1	i=1	i=1	PROPN
ma-225	51	2	α	α	PROPN
ma-225	51	3	[	[	PUNCT
ma-225	51	4	α−1(xi	α−1(xi	PROPN
ma-225	51	5	)	)	PUNCT
ma-225	51	6	n	n	NOUN
ma-225	51	7	]	]	PUNCT
ma-225	51	8	aα	aα	NOUN
ma-225	52	1	=	=	SYM
ma-225	52	2	α	α	PROPN
ma-225	52	3	[	[	PUNCT
ma-225	52	4	α−1(x1)+α−1(x2)+	α−1(x1)+α−1(x2)+	NUM
ma-225	52	5	...	...	PUNCT
ma-225	52	6	+α−1(xn	+α−1(xn	PROPN
ma-225	52	7	)	)	PUNCT
ma-225	52	8	n	n	CCONJ
ma-225	52	9	]	]	PUNCT
ma-225	52	10	considering	consider	VERB
ma-225	52	11	α	α	PROPN
ma-225	52	12	=	=	SYM
ma-225	52	13	exp	exp	NOUN
ma-225	52	14	,	,	PUNCT
ma-225	52	15	we	we	PRON
ma-225	52	16	have	have	VERB
ma-225	52	17	aexp	aexp	NOUN
ma-225	52	18	=	=	PUNCT
ma-225	53	1	[	[	X
ma-225	53	2	∏n	∏n	X
ma-225	53	3	i=1	i=1	X
ma-225	53	4	xi	xi	X
ma-225	53	5	]	]	PUNCT
ma-225	53	6	1	1	NUM
ma-225	53	7	n	n	NOUN
ma-225	53	8	=	=	SYM
ma-225	53	9	(	(	PUNCT
ma-225	53	10	x1	x1	NUM
ma-225	53	11	×	×	PROPN
ma-225	53	12	x2	x2	PROPN
ma-225	53	13	...	...	PUNCT
ma-225	53	14	xn	xn	X
ma-225	53	15	)	)	PUNCT
ma-225	53	16	1	1	NUM
ma-225	53	17	nalso	nalso	ADV
ma-225	53	18	considering	consider	VERB
ma-225	53	19	x1	x1	PROPN
ma-225	53	20	,	,	PUNCT
ma-225	53	21	x2	x2	PROPN
ma-225	53	22	,	,	PUNCT
ma-225	53	23	...	...	PUNCT
ma-225	53	24	,	,	PUNCT
ma-225	53	25	xn	xn	PROPN
ma-225	53	26	∈	∈	PROPN
ma-225	53	27	r+	r+	NOUN
ma-225	53	28	and	and	CCONJ
ma-225	53	29	gα	gα	VERB
ma-225	53	30	to	to	PART
ma-225	53	31	be	be	AUX
ma-225	53	32	the	the	DET
ma-225	53	33	α	α	NOUN
ma-225	53	34	-	-	ADJ
ma-225	53	35	geometric	geometric	ADJ
ma-225	53	36	mean	mean	NOUN
ma-225	53	37	then	then	ADV
ma-225	53	38	:	:	PUNCT
ma-225	53	39	gα	gα	ADP
ma-225	53	40	=	=	PUNCT
ma-225	54	1	[	[	X
ma-225	54	2	∏n	∏n	X
ma-225	54	3	i=1	i=1	X
ma-225	54	4	xi	xi	X
ma-225	54	5	]	]	PUNCT
ma-225	54	6	1	1	NUM
ma-225	54	7	n	n	NOUN
ma-225	54	8	=	=	SYM
ma-225	54	9	α	α	PROPN
ma-225	55	1	[	[	X
ma-225	55	2	∏n	∏n	PROPN
ma-225	55	3	i=1	i=1	PROPN
ma-225	55	4	α	α	PROPN
ma-225	55	5	−1(xi	−1(xi	NOUN
ma-225	55	6	)	)	PUNCT
ma-225	55	7	]	]	PUNCT
ma-225	56	1	1	1	NUM
ma-225	56	2	n	n	ADV
ma-225	56	3	gα	gα	ADP
ma-225	56	4	=	=	PUNCT
ma-225	56	5	α	α	X
ma-225	56	6	[	[	PUNCT
ma-225	56	7	(	(	PUNCT
ma-225	56	8	α−1(x1)×	α−1(x1)×	NUM
ma-225	56	9	α−1(x2)	α−1(x2)	NOUN
ma-225	56	10	...	...	PUNCT
ma-225	56	11	α−1(xn	α−1(xn	NOUN
ma-225	56	12	)	)	PUNCT
ma-225	56	13	)	)	PUNCT
ma-225	57	1	1	1	NUM
ma-225	57	2	n	n	NOUN
ma-225	57	3	]	]	X
ma-225	57	4	.in	.in	PUNCT
ma-225	57	5	a	a	DET
ma-225	57	6	similar	similar	ADJ
ma-225	57	7	way	way	NOUN
ma-225	57	8	,	,	PUNCT
ma-225	57	9	we	we	PRON
ma-225	57	10	take	take	VERB
ma-225	57	11	α	α	NOUN
ma-225	57	12	=	=	SYM
ma-225	57	13	exponent	exponent	NOUN
ma-225	57	14	,	,	PUNCT
ma-225	57	15	then	then	ADV
ma-225	57	16	the	the	DET
ma-225	57	17	α	α	NOUN
ma-225	57	18	-	-	ADJ
ma-225	57	19	geometric	geometric	ADJ
ma-225	57	20	mean	mean	NOUN
ma-225	57	21	can	can	AUX
ma-225	57	22	be	be	AUX
ma-225	57	23	interpreted	interpret	VERB
ma-225	57	24	as	as	ADP
ma-225	57	25	[	[	X
ma-225	57	26	10	10	NUM
ma-225	57	27	]	]	NOUN
ma-225	57	28	:	:	PUNCT
ma-225	57	29	gexp	gexp	NOUN
ma-225	57	30	=	=	PUNCT
ma-225	58	1	[	[	PUNCT
ma-225	58	2	(	(	PUNCT
ma-225	58	3	ln	ln	NOUN
ma-225	58	4	x1	x1	NUM
ma-225	58	5	×	×	NOUN
ma-225	58	6	ln	ln	ADJ
ma-225	58	7	x2	x2	PROPN
ma-225	58	8	...	...	PUNCT
ma-225	58	9	ln	ln	PROPN
ma-225	58	10	xn	xn	X
ma-225	58	11	)	)	PUNCT
ma-225	58	12	1	1	NUM
ma-225	58	13	n	n	NOUN
ma-225	58	14	]	]	PUNCT
ma-225	58	15	,	,	PUNCT
ma-225	58	16	(	(	PUNCT
ma-225	58	17	xn	xn	PROPN
ma-225	58	18	>	>	X
ma-225	58	19	1	1	NUM
ma-225	58	20	)	)	PUNCT
ma-225	58	21	.	.	PUNCT
ma-225	59	1	definition	definition	NOUN
ma-225	59	2	2.1	2.1	NUM
ma-225	59	3	.	.	PUNCT
ma-225	60	1	[	[	X
ma-225	60	2	11	11	NUM
ma-225	60	3	]	]	PUNCT
ma-225	60	4	a	a	DET
ma-225	60	5	set	set	NOUN
ma-225	60	6	c	c	NOUN
ma-225	61	1	=	=	PUNCT
ma-225	62	1	[	[	X
ma-225	62	2	a1	a1	NOUN
ma-225	62	3	,	,	PUNCT
ma-225	62	4	b1	b1	NOUN
ma-225	62	5	]	]	PUNCT
ma-225	62	6	⊆	⊆	NUM
ma-225	62	7	r	r	NOUN
ma-225	62	8	is	be	AUX
ma-225	62	9	said	say	VERB
ma-225	62	10	to	to	PART
ma-225	62	11	be	be	AUX
ma-225	62	12	convex	convex	ADJ
ma-225	62	13	if	if	SCONJ
ma-225	62	14	x	x	NOUN
ma-225	62	15	,	,	PUNCT
ma-225	62	16	y	y	PROPN
ma-225	62	17	∈	∈	PROPN
ma-225	62	18	c	c	NOUN
ma-225	62	19	,	,	PUNCT
ma-225	62	20	and	and	CCONJ
ma-225	62	21	qx	qx	PROPN
ma-225	63	1	+	+	CCONJ
ma-225	63	2	(	(	PUNCT
ma-225	63	3	1−	1−	NUM
ma-225	63	4	q)y	q)y	X
ma-225	63	5	∈	∈	PROPN
ma-225	63	6	c	c	NOUN
ma-225	63	7	,	,	PUNCT
ma-225	63	8	(	(	PUNCT
ma-225	63	9	5	5	NUM
ma-225	63	10	)	)	PUNCT
ma-225	63	11	for	for	ADP
ma-225	63	12	q	q	PROPN
ma-225	63	13	∈	∈	PROPN
ma-225	64	1	[	[	X
ma-225	64	2	0	0	NUM
ma-225	64	3	,	,	PUNCT
ma-225	64	4	1	1	NUM
ma-225	64	5	]	]	PUNCT
ma-225	64	6	.	.	PUNCT
ma-225	65	1	definition	definition	NOUN
ma-225	65	2	2.2	2.2	NUM
ma-225	65	3	.	.	PUNCT
ma-225	66	1	[	[	X
ma-225	66	2	11	11	NUM
ma-225	66	3	]	]	PUNCT
ma-225	66	4	let	let	VERB
ma-225	66	5	φ	φ	PROPN
ma-225	66	6	be	be	AUX
ma-225	66	7	defined	define	VERB
ma-225	66	8	on	on	ADP
ma-225	66	9	a	a	DET
ma-225	66	10	real	real	ADJ
ma-225	66	11	interval	interval	NOUN
ma-225	66	12	m	m	NOUN
ma-225	66	13	.	.	PUNCT
ma-225	67	1	the	the	DET
ma-225	67	2	function	function	NOUN
ma-225	67	3	φ	φ	PROPN
ma-225	67	4	is	be	AUX
ma-225	67	5	convex	convex	ADJ
ma-225	67	6	if	if	SCONJ
ma-225	67	7	:	:	PUNCT
ma-225	67	8	φ(λ1x	φ(λ1x	VERB
ma-225	67	9	+	+	CCONJ
ma-225	67	10	λ2y	λ2y	PROPN
ma-225	67	11	)	)	PUNCT
ma-225	67	12	≤	≤	NOUN
ma-225	67	13	λ1φ(x	λ1φ(x	PROPN
ma-225	67	14	)	)	PUNCT
ma-225	68	1	+	+	CCONJ
ma-225	69	1	λ2φ(y	λ2φ(y	PROPN
ma-225	69	2	)	)	PUNCT
ma-225	69	3	,	,	PUNCT
ma-225	69	4	(	(	PUNCT
ma-225	69	5	6	6	X
ma-225	69	6	)	)	PUNCT
ma-225	69	7	φ(λ1x	φ(λ1x	VERB
ma-225	69	8	+	+	CCONJ
ma-225	69	9	(	(	PUNCT
ma-225	69	10	1−	1−	NUM
ma-225	69	11	λ1)y	λ1)y	NOUN
ma-225	69	12	)	)	PUNCT
ma-225	69	13	≤	≤	NUM
ma-225	69	14	λ1φ(x	λ1φ(x	PROPN
ma-225	69	15	)	)	PUNCT
ma-225	70	1	+	+	CCONJ
ma-225	70	2	(	(	PUNCT
ma-225	70	3	1−	1−	NUM
ma-225	70	4	λ1)φ(y	λ1)φ(y	NOUN
ma-225	70	5	)	)	PUNCT
ma-225	70	6	,	,	PUNCT
ma-225	70	7	(	(	PUNCT
ma-225	70	8	7	7	X
ma-225	70	9	)	)	PUNCT
ma-225	70	10	where	where	SCONJ
ma-225	70	11	λ1	λ1	ADJ
ma-225	70	12	+	+	NUM
ma-225	70	13	λ2	λ2	NOUN
ma-225	70	14	=	=	SYM
ma-225	70	15	1	1	NUM
ma-225	70	16	,	,	PUNCT
ma-225	70	17	∀	∀	X
ma-225	70	18	x	x	NOUN
ma-225	70	19	,	,	PUNCT
ma-225	70	20	y	y	PROPN
ma-225	70	21	∈	∈	PROPN
ma-225	70	22	m	m	PROPN
ma-225	70	23	and	and	CCONJ
ma-225	70	24	λ1	λ1	ADJ
ma-225	70	25	,	,	PUNCT
ma-225	70	26	λ2	λ2	PROPN
ma-225	70	27	∈	∈	PROPN
ma-225	71	1	[	[	X
ma-225	71	2	0	0	NUM
ma-225	71	3	,	,	PUNCT
ma-225	71	4	1	1	NUM
ma-225	71	5	]	]	PUNCT
ma-225	71	6	.	.	PUNCT
ma-225	72	1	definition	definition	NOUN
ma-225	72	2	2.3	2.3	NUM
ma-225	72	3	.	.	PUNCT
ma-225	73	1	[	[	X
ma-225	73	2	12	12	NUM
ma-225	73	3	]	]	PUNCT
ma-225	73	4	let	let	VERB
ma-225	73	5	φ	φ	PROPN
ma-225	73	6	be	be	AUX
ma-225	73	7	convex	convex	ADJ
ma-225	73	8	function	function	NOUN
ma-225	73	9	and	and	CCONJ
ma-225	73	10	u1	u1	NOUN
ma-225	73	11	,	,	PUNCT
ma-225	73	12	v1	v1	NOUN
ma-225	73	13	∈	∈	NOUN
ma-225	73	14	r	r	NOUN
ma-225	73	15	,	,	PUNCT
ma-225	73	16	then	then	ADV
ma-225	73	17	φ	φ	PROPN
ma-225	73	18	(	(	PUNCT
ma-225	73	19	u1	u1	PROPN
ma-225	73	20	+	+	CCONJ
ma-225	73	21	v1	v1	PROPN
ma-225	73	22	2	2	NUM
ma-225	73	23	)	)	PUNCT
ma-225	73	24	≤	≤	NUM
ma-225	73	25	φ(u1	φ(u1	NOUN
ma-225	73	26	)	)	PUNCT
ma-225	73	27	+	+	SYM
ma-225	73	28	φ(v1	φ(v1	NOUN
ma-225	73	29	)	)	PUNCT
ma-225	73	30	2	2	NUM
ma-225	73	31	.	.	PUNCT
ma-225	74	1	(	(	PUNCT
ma-225	74	2	8)	8)	NUM
ma-225	74	3	proposition	proposition	NOUN
ma-225	74	4	2.1	2.1	NUM
ma-225	74	5	.	.	PUNCT
ma-225	75	1	[	[	X
ma-225	75	2	13	13	NUM
ma-225	75	3	]	]	PUNCT
ma-225	75	4	let	let	VERB
ma-225	75	5	x1	x1	PROPN
ma-225	75	6	≤	≤	NUM
ma-225	75	7	y1	y1	NOUN
ma-225	75	8	,	,	PUNCT
ma-225	75	9	x2	x2	PROPN
ma-225	75	10	≤	≤	NUM
ma-225	75	11	y2	y2	NOUN
ma-225	75	12	,	,	PUNCT
ma-225	75	13	and	and	CCONJ
ma-225	76	1	φw	φw	NOUN
ma-225	76	2	be	be	VERB
ma-225	76	3	a	a	DET
ma-225	76	4	convex	convex	NOUN
ma-225	76	5	function	function	NOUN
ma-225	76	6	on	on	ADP
ma-225	76	7	an	an	DET
ma-225	76	8	interval	interval	NOUN
ma-225	76	9	of	of	ADP
ma-225	76	10	real	real	ADJ
ma-225	76	11	positive	positive	ADJ
ma-225	76	12	values	value	NOUN
ma-225	76	13	,	,	PUNCT
ma-225	76	14	then	then	ADV
ma-225	76	15	φw	φw	PROPN
ma-225	76	16	(	(	PUNCT
ma-225	76	17	y2	y2	PROPN
ma-225	76	18	−	−	PROPN
ma-225	76	19	y1	y1	PROPN
ma-225	76	20	)	)	PUNCT
ma-225	76	21	x2	x2	NOUN
ma-225	77	1	−	−	PROPN
ma-225	78	1	x1	x1	PROPN
ma-225	78	2	≤	≤	PUNCT
ma-225	78	3	φw	φw	NOUN
ma-225	78	4	(	(	PUNCT
ma-225	78	5	y2)−	y2)−	PROPN
ma-225	78	6	φw	φw	PROPN
ma-225	78	7	(	(	PUNCT
ma-225	78	8	y1	y1	PROPN
ma-225	78	9	)	)	PUNCT
ma-225	78	10	x2	x2	NOUN
ma-225	78	11	−	−	PROPN
ma-225	79	1	x1	x1	INTJ
ma-225	79	2	.	.	PUNCT
ma-225	80	1	(	(	PUNCT
ma-225	80	2	9	9	X
ma-225	80	3	)	)	PUNCT
ma-225	80	4	theorem	theorem	VERB
ma-225	80	5	2.1	2.1	NUM
ma-225	80	6	(	(	PUNCT
ma-225	80	7	jensen	jensen	PROPN
ma-225	80	8	-	-	PUNCT
ma-225	80	9	steffensen	steffensen	PROPN
ma-225	80	10	inequality	inequality	NOUN
ma-225	80	11	)	)	PUNCT
ma-225	80	12	.	.	PUNCT
ma-225	81	1	[	[	X
ma-225	81	2	14	14	NUM
ma-225	81	3	]	]	PUNCT
ma-225	81	4	let	let	VERB
ma-225	81	5	φ	φ	PROPN
ma-225	81	6	be	be	AUX
ma-225	81	7	a	a	DET
ma-225	81	8	convex	convex	NOUN
ma-225	81	9	function	function	NOUN
ma-225	81	10	defined	define	VERB
ma-225	81	11	on	on	ADP
ma-225	81	12	an	an	DET
ma-225	81	13	interval	interval	NOUN
ma-225	81	14	of	of	ADP
ma-225	81	15	the	the	DET
ma-225	81	16	real	real	ADJ
ma-225	81	17	line	line	NOUN
ma-225	81	18	and	and	CCONJ
ma-225	81	19	let	let	VERB
ma-225	81	20	xi	xi	INTJ
ma-225	81	21	,	,	PUNCT
ma-225	82	1	pi	pi	PROPN
ma-225	82	2	∈	∈	PROPN
ma-225	82	3	r	r	NOUN
ma-225	82	4	,	,	PUNCT
ma-225	82	5	i	i	NOUN
ma-225	82	6	=	=	NOUN
ma-225	82	7	1	1	NUM
ma-225	82	8	,	,	PUNCT
ma-225	82	9	...	...	PUNCT
ma-225	82	10	m.	m.	NOUN
ma-225	82	11	if	if	SCONJ
ma-225	82	12	x1	x1	PROPN
ma-225	82	13	,	,	PUNCT
ma-225	82	14	...	...	PUNCT
ma-225	82	15	,	,	PUNCT
ma-225	82	16	xm	xm	PROPN
ma-225	82	17	and	and	CCONJ
ma-225	82	18	p1	p1	PROPN
ma-225	82	19	,	,	PUNCT
ma-225	82	20	...	...	PUNCT
ma-225	82	21	,	,	PUNCT
ma-225	82	22	pm	pm	NOUN
ma-225	82	23	,	,	PUNCT
ma-225	82	24	pm	pm	VERB
ma-225	82	25	>	>	X
ma-225	82	26	0	0	PROPN
ma-225	82	27	,	,	PUNCT
ma-225	82	28	then	then	ADV
ma-225	82	29	φ	φ	X
ma-225	82	30	(	(	PUNCT
ma-225	82	31	1	1	NUM
ma-225	82	32	pm	pm	NOUN
ma-225	82	33	m∑	m∑	NOUN
ma-225	82	34	i=1	i=1	ADP
ma-225	82	35	pixi	pixi	NOUN
ma-225	82	36	)	)	PUNCT
ma-225	82	37	≤	≤	NUM
ma-225	82	38	1	1	NUM
ma-225	82	39	pm	pm	NOUN
ma-225	82	40	m∑	m∑	VERB
ma-225	82	41	i=1	i=1	PRON
ma-225	82	42	piφ(xi	piφ(xi	NUM
ma-225	82	43	)	)	PUNCT
ma-225	82	44	.	.	PUNCT
ma-225	83	1	(	(	PUNCT
ma-225	83	2	10	10	NUM
ma-225	83	3	)	)	SYM
ma-225	83	4	3	3	NUM
ma-225	83	5	theorem	theorem	VERB
ma-225	83	6	2.2	2.2	NUM
ma-225	83	7	.	.	PUNCT
ma-225	84	1	[	[	X
ma-225	84	2	14	14	NUM
ma-225	84	3	]	]	PUNCT
ma-225	84	4	let	let	VERB
ma-225	84	5	φ	φ	PROPN
ma-225	84	6	be	be	AUX
ma-225	84	7	a	a	DET
ma-225	84	8	convex	convex	NOUN
ma-225	84	9	function	function	NOUN
ma-225	84	10	on	on	ADP
ma-225	84	11	an	an	DET
ma-225	84	12	interval	interval	NOUN
ma-225	84	13	s	s	PART
ma-225	84	14	=	=	PUNCT
ma-225	85	1	[	[	X
ma-225	85	2	a1	a1	NOUN
ma-225	85	3	,	,	PUNCT
ma-225	85	4	b1	b1	NOUN
ma-225	85	5	]	]	PUNCT
ma-225	85	6	⊂	⊂	X
ma-225	85	7	r+	r+	X
ma-225	85	8	,	,	PUNCT
ma-225	85	9	where	where	SCONJ
ma-225	85	10	a1	a1	NOUN
ma-225	85	11	<	<	X
ma-225	85	12	b1	b1	PROPN
ma-225	85	13	.	.	PUNCT
ma-225	86	1	let	let	VERB
ma-225	86	2	x	x	PUNCT
ma-225	86	3	=	=	SYM
ma-225	86	4	(	(	PUNCT
ma-225	86	5	x1	x1	PROPN
ma-225	86	6	,	,	PUNCT
ma-225	86	7	x2	x2	PROPN
ma-225	86	8	,	,	PUNCT
ma-225	86	9	...	...	PUNCT
ma-225	86	10	,	,	PUNCT
ma-225	86	11	xn	xn	PROPN
ma-225	86	12	)	)	PUNCT
ma-225	86	13	and	and	CCONJ
ma-225	86	14	p	p	NOUN
ma-225	86	15	=	=	SYM
ma-225	86	16	(	(	PUNCT
ma-225	86	17	p1	p1	PROPN
ma-225	86	18	,	,	PUNCT
ma-225	86	19	p2	p2	NOUN
ma-225	86	20	,	,	PUNCT
ma-225	86	21	...	...	PUNCT
ma-225	86	22	,	,	PUNCT
ma-225	86	23	pn	pn	PROPN
ma-225	86	24	)	)	PUNCT
ma-225	86	25	,	,	PUNCT
ma-225	86	26	then	then	ADV
ma-225	86	27	φk(a1	φk(a1	X
ma-225	86	28	+	+	CCONJ
ma-225	86	29	b1	b1	NOUN
ma-225	86	30	−	−	PROPN
ma-225	86	31	1	1	NUM
ma-225	86	32	pn	pn	PROPN
ma-225	86	33	n∑	n∑	PROPN
ma-225	86	34	i=1	i=1	PROPN
ma-225	86	35	pixi	pixi	NOUN
ma-225	86	36	)	)	PUNCT
ma-225	86	37	≤	≤	NOUN
ma-225	86	38	φk(a1	φk(a1	NOUN
ma-225	86	39	)	)	PUNCT
ma-225	86	40	+	+	NUM
ma-225	86	41	φk(b1)−	φk(b1)−	NOUN
ma-225	86	42	1	1	NUM
ma-225	86	43	pn	pn	PROPN
ma-225	86	44	n∑	n∑	PROPN
ma-225	86	45	i=1	i=1	PROPN
ma-225	86	46	piφk(xi	piφk(xi	ADJ
ma-225	86	47	)	)	PUNCT
ma-225	86	48	.	.	PUNCT
ma-225	87	1	(	(	PUNCT
ma-225	87	2	11	11	X
ma-225	87	3	)	)	PUNCT
ma-225	87	4	let	let	VERB
ma-225	87	5	pn	pn	NOUN
ma-225	87	6	=	=	PUNCT
ma-225	87	7	∑n	∑n	PROPN
ma-225	87	8	i=1	i=1	PROPN
ma-225	87	9	pi	pi	NOUN
ma-225	87	10	=	=	SYM
ma-225	87	11	1	1	NUM
ma-225	87	12	,	,	PUNCT
ma-225	87	13	then	then	ADV
ma-225	87	14	φk(a1	φk(a1	X
ma-225	87	15	+	+	CCONJ
ma-225	87	16	b1	b1	NOUN
ma-225	88	1	−	−	PROPN
ma-225	88	2	n∑	n∑	NOUN
ma-225	88	3	i=1	i=1	PROPN
ma-225	88	4	pixi	pixi	NOUN
ma-225	88	5	)	)	PUNCT
ma-225	89	1	≤	≤	NOUN
ma-225	89	2	φk(a1	φk(a1	NOUN
ma-225	89	3	)	)	PUNCT
ma-225	90	1	+	+	CCONJ
ma-225	90	2	φk(b1)−	φk(b1)−	NOUN
ma-225	90	3	n∑	n∑	PROPN
ma-225	90	4	i=1	i=1	PROPN
ma-225	90	5	piφk(xi	piφk(xi	ADJ
ma-225	90	6	)	)	PUNCT
ma-225	90	7	.	.	PUNCT
ma-225	91	1	(	(	PUNCT
ma-225	91	2	12	12	NUM
ma-225	91	3	)	)	PUNCT
ma-225	91	4	definition	definition	NOUN
ma-225	91	5	2.4	2.4	NUM
ma-225	91	6	.	.	PUNCT
ma-225	92	1	[	[	X
ma-225	92	2	15	15	NUM
ma-225	92	3	]	]	X
ma-225	92	4	the	the	DET
ma-225	92	5	definition	definition	NOUN
ma-225	92	6	of	of	ADP
ma-225	92	7	a	a	DET
ma-225	92	8	p	p	NOUN
ma-225	92	9	-	-	PUNCT
ma-225	92	10	convex	convex	NOUN
ma-225	92	11	set	set	NOUN
ma-225	92	12	for	for	ADP
ma-225	92	13	an	an	DET
ma-225	92	14	interval	interval	NOUN
ma-225	92	15	c	c	NOUN
ma-225	92	16	is	be	AUX
ma-225	92	17	(	(	PUNCT
ma-225	92	18	qup	qup	NOUN
ma-225	92	19	+	+	CCONJ
ma-225	92	20	(	(	PUNCT
ma-225	92	21	1−	1−	NUM
ma-225	92	22	q)vp	q)vp	PROPN
ma-225	92	23	)	)	PUNCT
ma-225	92	24	1	1	NUM
ma-225	92	25	p	p	NOUN
ma-225	92	26	∈	∈	PROPN
ma-225	92	27	c	c	X
ma-225	92	28	,	,	PUNCT
ma-225	92	29	(	(	PUNCT
ma-225	92	30	13	13	NUM
ma-225	92	31	)	)	PUNCT
ma-225	92	32	for	for	ADP
ma-225	92	33	all	all	DET
ma-225	92	34	u	u	NOUN
ma-225	92	35	,	,	PUNCT
ma-225	92	36	v	v	NOUN
ma-225	92	37	∈	∈	PROPN
ma-225	92	38	c	c	NOUN
ma-225	92	39	and	and	CCONJ
ma-225	92	40	q	q	NOUN
ma-225	92	41	∈	∈	PROPN
ma-225	93	1	[	[	X
ma-225	93	2	0	0	NUM
ma-225	93	3	,	,	PUNCT
ma-225	93	4	1	1	NUM
ma-225	93	5	]	]	PUNCT
ma-225	93	6	.	.	PUNCT
ma-225	94	1	definition	definition	NOUN
ma-225	94	2	2.5	2.5	NUM
ma-225	94	3	.	.	PUNCT
ma-225	95	1	[	[	X
ma-225	95	2	12	12	NUM
ma-225	95	3	,	,	PUNCT
ma-225	95	4	15	15	NUM
ma-225	95	5	,	,	PUNCT
ma-225	95	6	16	16	NUM
ma-225	95	7	]	]	PUNCT
ma-225	95	8	let	let	VERB
ma-225	95	9	c	c	NOUN
ma-225	95	10	=	=	PUNCT
ma-225	96	1	[	[	X
ma-225	96	2	a1	a1	NOUN
ma-225	96	3	,	,	PUNCT
ma-225	96	4	b1	b1	NOUN
ma-225	96	5	]	]	PUNCT
ma-225	96	6	be	be	VERB
ma-225	96	7	an	an	DET
ma-225	96	8	interval	interval	NOUN
ma-225	96	9	on	on	ADP
ma-225	96	10	real	real	ADJ
ma-225	96	11	numbers	number	NOUN
ma-225	96	12	r.	r.	VERB
ma-225	96	13	an	an	DET
ma-225	96	14	expression	expression	NOUN
ma-225	96	15	φ	φ	X
ma-225	96	16	:	:	PUNCT
ma-225	97	1	c	c	X
ma-225	97	2	=	=	PUNCT
ma-225	98	1	[	[	X
ma-225	98	2	a1	a1	NOUN
ma-225	98	3	,	,	PUNCT
ma-225	98	4	b1	b1	NOUN
ma-225	98	5	]	]	PUNCT
ma-225	98	6	7→	7→	NUM
ma-225	98	7	r	r	NOUN
ma-225	98	8	is	be	AUX
ma-225	98	9	p	p	NOUN
ma-225	98	10	-	-	PUNCT
ma-225	98	11	convex	convex	NOUN
ma-225	98	12	if	if	SCONJ
ma-225	98	13	φ(qup	φ(qup	NOUN
ma-225	98	14	+	+	CCONJ
ma-225	98	15	(	(	PUNCT
ma-225	98	16	1−	1−	NUM
ma-225	98	17	q)vp	q)vp	PROPN
ma-225	98	18	)	)	PUNCT
ma-225	98	19	1	1	NUM
ma-225	98	20	p	p	NOUN
ma-225	98	21	≤	≤	NOUN
ma-225	98	22	qφ(u	qφ(u	PROPN
ma-225	98	23	)	)	PUNCT
ma-225	99	1	+	+	CCONJ
ma-225	99	2	(	(	PUNCT
ma-225	99	3	1−	1−	NUM
ma-225	99	4	q)φ(v	q)φ(v	ADV
ma-225	99	5	)	)	PUNCT
ma-225	99	6	,	,	PUNCT
ma-225	99	7	(	(	PUNCT
ma-225	99	8	14	14	NUM
ma-225	99	9	)	)	PUNCT
ma-225	99	10	∀	∀	NOUN
ma-225	99	11	u	u	NOUN
ma-225	99	12	,	,	PUNCT
ma-225	99	13	v	v	NOUN
ma-225	99	14	∈	∈	PROPN
ma-225	99	15	c	c	NOUN
ma-225	99	16	and	and	CCONJ
ma-225	99	17	q	q	NOUN
ma-225	99	18	∈	∈	PROPN
ma-225	100	1	[	[	X
ma-225	100	2	0	0	NUM
ma-225	100	3	,	,	PUNCT
ma-225	100	4	1	1	NUM
ma-225	100	5	]	]	PUNCT
ma-225	100	6	.	.	PUNCT
ma-225	101	1	definition	definition	NOUN
ma-225	101	2	2.6	2.6	NUM
ma-225	101	3	.	.	PUNCT
ma-225	102	1	[	[	X
ma-225	102	2	15	15	NUM
ma-225	102	3	]	]	PUNCT
ma-225	102	4	let	let	VERB
ma-225	102	5	φ	φ	PROPN
ma-225	102	6	be	be	AUX
ma-225	102	7	p	p	NOUN
ma-225	102	8	-	-	PUNCT
ma-225	102	9	convex	convex	NOUN
ma-225	102	10	function	function	NOUN
ma-225	102	11	and	and	CCONJ
ma-225	102	12	u	u	NOUN
ma-225	102	13	,	,	PUNCT
ma-225	102	14	v	v	PROPN
ma-225	102	15	∈	∈	PROPN
ma-225	102	16	r+	r+	NOUN
ma-225	102	17	,	,	PUNCT
ma-225	102	18	then	then	ADV
ma-225	102	19	φ	φ	PROPN
ma-225	102	20	(	(	PUNCT
ma-225	102	21	up	up	ADP
ma-225	102	22	+	+	CCONJ
ma-225	102	23	vp	vp	PROPN
ma-225	102	24	2	2	NUM
ma-225	102	25	)	)	PUNCT
ma-225	102	26	1	1	NUM
ma-225	102	27	p	p	NOUN
ma-225	102	28	≤	≤	NUM
ma-225	102	29	φ(u	φ(u	NOUN
ma-225	102	30	)	)	PUNCT
ma-225	102	31	+	+	CCONJ
ma-225	102	32	φ(v	φ(v	NOUN
ma-225	102	33	)	)	PUNCT
ma-225	102	34	2	2	NUM
ma-225	102	35	.	.	PUNCT
ma-225	103	1	(	(	PUNCT
ma-225	103	2	15	15	NUM
ma-225	103	3	)	)	PUNCT
ma-225	103	4	definition	definition	NOUN
ma-225	103	5	2.7	2.7	NUM
ma-225	103	6	.	.	PUNCT
ma-225	104	1	[	[	X
ma-225	104	2	15	15	NUM
ma-225	104	3	]	]	PUNCT
ma-225	104	4	suppose	suppose	VERB
ma-225	104	5	the	the	DET
ma-225	104	6	function	function	NOUN
ma-225	104	7	φ	φ	NOUN
ma-225	104	8	:	:	PUNCT
ma-225	105	1	c	c	X
ma-225	105	2	=	=	PUNCT
ma-225	106	1	[	[	X
ma-225	106	2	x	x	X
ma-225	106	3	,	,	PUNCT
ma-225	106	4	y	y	PROPN
ma-225	106	5	]	]	PUNCT
ma-225	106	6	7→	7→	NUM
ma-225	106	7	r+	r+	NOUN
ma-225	106	8	is	be	AUX
ma-225	106	9	strongly	strongly	ADV
ma-225	106	10	convex	convex	ADJ
ma-225	106	11	and	and	CCONJ
ma-225	106	12	β	β	X
ma-225	106	13	≥	≥	NUM
ma-225	106	14	1	1	NUM
ma-225	106	15	;	;	PUNCT
ma-225	106	16	then	then	ADV
ma-225	106	17	φ(qa	φ(qa	X
ma-225	106	18	+	+	CCONJ
ma-225	106	19	(	(	PUNCT
ma-225	106	20	1−	1−	NUM
ma-225	106	21	q)b	q)b	NOUN
ma-225	106	22	)	)	PUNCT
ma-225	106	23	≤	≤	NOUN
ma-225	106	24	qφ(a	qφ(a	PUNCT
ma-225	106	25	)	)	PUNCT
ma-225	107	1	+	+	CCONJ
ma-225	107	2	(	(	PUNCT
ma-225	107	3	1−	1−	NUM
ma-225	107	4	q)φ(b)−	q)φ(b)−	NOUN
ma-225	107	5	βq(1−	βq(1−	PROPN
ma-225	107	6	q)(b	q)(b	ADJ
ma-225	107	7	−	−	PROPN
ma-225	107	8	a)2	a)2	PROPN
ma-225	107	9	(	(	PUNCT
ma-225	107	10	16	16	NUM
ma-225	107	11	)	)	PUNCT
ma-225	107	12	for	for	ADP
ma-225	107	13	all	all	DET
ma-225	107	14	a	a	DET
ma-225	107	15	,	,	PUNCT
ma-225	107	16	b	b	X
ma-225	107	17	∈	∈	PROPN
ma-225	107	18	c	c	NOUN
ma-225	107	19	and	and	CCONJ
ma-225	107	20	q	q	NOUN
ma-225	107	21	∈	∈	PROPN
ma-225	108	1	[	[	X
ma-225	108	2	0	0	NUM
ma-225	108	3	,	,	PUNCT
ma-225	108	4	1	1	NUM
ma-225	108	5	]	]	PUNCT
ma-225	108	6	.	.	PUNCT
ma-225	109	1	definition	definition	NOUN
ma-225	109	2	2.8	2.8	NUM
ma-225	109	3	.	.	PUNCT
ma-225	110	1	[	[	X
ma-225	110	2	15	15	NUM
ma-225	110	3	]	]	X
ma-225	110	4	a	a	DET
ma-225	110	5	function	function	NOUN
ma-225	110	6	φ	φ	NOUN
ma-225	110	7	:	:	PUNCT
ma-225	111	1	c	c	X
ma-225	111	2	=	=	PUNCT
ma-225	112	1	[	[	X
ma-225	112	2	x	x	X
ma-225	112	3	,	,	PUNCT
ma-225	112	4	y	y	PROPN
ma-225	112	5	]	]	PUNCT
ma-225	112	6	7→	7→	NUM
ma-225	112	7	r	r	NOUN
ma-225	112	8	is	be	AUX
ma-225	112	9	strongly	strongly	ADV
ma-225	112	10	p	p	ADJ
ma-225	112	11	-	-	PUNCT
ma-225	112	12	convex	convex	NOUN
ma-225	112	13	function	function	NOUN
ma-225	112	14	,	,	PUNCT
ma-225	112	15	if	if	SCONJ
ma-225	112	16	φ(qap	φ(qap	PROPN
ma-225	112	17	+	+	CCONJ
ma-225	112	18	(	(	PUNCT
ma-225	112	19	1−	1−	NUM
ma-225	112	20	q)bp	q)bp	PROPN
ma-225	112	21	)	)	PUNCT
ma-225	112	22	1	1	NUM
ma-225	112	23	p	p	NOUN
ma-225	112	24	≤	≤	NOUN
ma-225	112	25	qφ(a	qφ(a	PUNCT
ma-225	112	26	)	)	PUNCT
ma-225	113	1	+	+	CCONJ
ma-225	113	2	(	(	PUNCT
ma-225	113	3	1−	1−	NUM
ma-225	113	4	q)φ(b)−	q)φ(b)−	PROPN
ma-225	113	5	βq(1−	βq(1−	PROPN
ma-225	113	6	q)(bp	q)(bp	PROPN
ma-225	113	7	−	−	PROPN
ma-225	113	8	ap)2	ap)2	PROPN
ma-225	113	9	,	,	PUNCT
ma-225	113	10	(	(	PUNCT
ma-225	113	11	17	17	NUM
ma-225	113	12	)	)	PUNCT
ma-225	113	13	for	for	ADP
ma-225	113	14	all	all	DET
ma-225	113	15	a	a	PRON
ma-225	113	16	,	,	PUNCT
ma-225	113	17	b	b	X
ma-225	113	18	∈	∈	PROPN
ma-225	113	19	c	c	NOUN
ma-225	113	20	and	and	CCONJ
ma-225	113	21	q	q	NOUN
ma-225	113	22	∈	∈	PROPN
ma-225	114	1	[	[	X
ma-225	114	2	0	0	NUM
ma-225	114	3	,	,	PUNCT
ma-225	114	4	1	1	NUM
ma-225	114	5	]	]	PUNCT
ma-225	114	6	.	.	PUNCT
ma-225	115	1	definition	definition	NOUN
ma-225	115	2	2.9	2.9	NUM
ma-225	115	3	.	.	PUNCT
ma-225	116	1	[	[	X
ma-225	116	2	15	15	NUM
ma-225	116	3	]	]	PUNCT
ma-225	116	4	in	in	ADP
ma-225	116	5	the	the	DET
ma-225	116	6	event	event	NOUN
ma-225	116	7	that	that	SCONJ
ma-225	116	8	an	an	DET
ma-225	116	9	interval	interval	NOUN
ma-225	116	10	c	c	NOUN
ma-225	116	11	is	be	AUX
ma-225	116	12	a	a	DET
ma-225	116	13	harmonic	harmonic	ADJ
ma-225	116	14	convex	convex	NOUN
ma-225	116	15	set	set	NOUN
ma-225	116	16	,	,	PUNCT
ma-225	116	17	then	then	ADV
ma-225	116	18	(	(	PUNCT
ma-225	116	19	uv	uv	NOUN
ma-225	116	20	qu	qu	PROPN
ma-225	116	21	+	+	X
ma-225	116	22	(	(	PUNCT
ma-225	116	23	1−	1−	NUM
ma-225	116	24	q)v	q)v	X
ma-225	116	25	)	)	PUNCT
ma-225	117	1	∈	∈	PROPN
ma-225	117	2	c	c	NOUN
ma-225	117	3	,	,	PUNCT
ma-225	117	4	(	(	PUNCT
ma-225	117	5	18	18	NUM
ma-225	117	6	)	)	PUNCT
ma-225	117	7	for	for	ADP
ma-225	117	8	all	all	DET
ma-225	117	9	u	u	NOUN
ma-225	117	10	,	,	PUNCT
ma-225	117	11	v	v	NOUN
ma-225	117	12	∈	∈	PROPN
ma-225	117	13	c	c	NOUN
ma-225	117	14	and	and	CCONJ
ma-225	117	15	q	q	NOUN
ma-225	117	16	∈	∈	PROPN
ma-225	118	1	[	[	X
ma-225	118	2	0	0	NUM
ma-225	118	3	,	,	PUNCT
ma-225	118	4	1	1	NUM
ma-225	118	5	]	]	PUNCT
ma-225	118	6	.	.	PUNCT
ma-225	119	1	definition	definition	NOUN
ma-225	119	2	2.10	2.10	NUM
ma-225	119	3	.	.	PUNCT
ma-225	120	1	[	[	X
ma-225	120	2	15	15	NUM
ma-225	120	3	]	]	PUNCT
ma-225	120	4	let	let	VERB
ma-225	120	5	the	the	DET
ma-225	120	6	function	function	NOUN
ma-225	120	7	φ	φ	NOUN
ma-225	120	8	:	:	PUNCT
ma-225	121	1	c	c	X
ma-225	121	2	=	=	PUNCT
ma-225	122	1	[	[	X
ma-225	122	2	a1	a1	NOUN
ma-225	122	3	,	,	PUNCT
ma-225	122	4	b1	b1	NOUN
ma-225	122	5	]	]	PUNCT
ma-225	122	6	⊆	⊆	NUM
ma-225	122	7	r+	r+	NOUN
ma-225	122	8	and	and	CCONJ
ma-225	122	9	c	c	NOUN
ma-225	122	10	=	=	PUNCT
ma-225	123	1	[	[	X
ma-225	123	2	a1	a1	NOUN
ma-225	123	3	,	,	PUNCT
ma-225	123	4	b1	b1	NOUN
ma-225	123	5	]	]	PUNCT
ma-225	123	6	be	be	VERB
ma-225	123	7	on	on	ADP
ma-225	123	8	an	an	DET
ma-225	123	9	interval	interval	NOUN
ma-225	123	10	on	on	ADP
ma-225	123	11	set	set	ADJ
ma-225	123	12	r+	r+	NOUN
ma-225	123	13	without	without	ADP
ma-225	123	14	zero	zero	NUM
ma-225	123	15	,	,	PUNCT
ma-225	123	16	then	then	ADV
ma-225	123	17	φ	φ	PROPN
ma-225	123	18	(	(	PUNCT
ma-225	123	19	uv	uv	NOUN
ma-225	123	20	qu	qu	PROPN
ma-225	123	21	+	+	X
ma-225	123	22	(	(	PUNCT
ma-225	123	23	1−	1−	NUM
ma-225	123	24	q)v	q)v	X
ma-225	123	25	)	)	PUNCT
ma-225	123	26	≤	≤	NUM
ma-225	123	27	(	(	PUNCT
ma-225	123	28	1−	1−	NUM
ma-225	123	29	q)φ(u	q)φ(u	NOUN
ma-225	123	30	)	)	PUNCT
ma-225	123	31	+	+	NUM
ma-225	123	32	qφ(v	qφ(v	NOUN
ma-225	123	33	)	)	PUNCT
ma-225	123	34	.	.	PUNCT
ma-225	124	1	(	(	PUNCT
ma-225	124	2	19	19	NUM
ma-225	124	3	)	)	SYM
ma-225	124	4	4	4	NUM
ma-225	124	5	definition	definition	NOUN
ma-225	124	6	2.11	2.11	NUM
ma-225	124	7	.	.	PUNCT
ma-225	125	1	[	[	X
ma-225	125	2	15	15	NUM
ma-225	125	3	]	]	PUNCT
ma-225	125	4	let	let	VERB
ma-225	125	5	c	c	NOUN
ma-225	125	6	=	=	PUNCT
ma-225	126	1	[	[	X
ma-225	126	2	a1	a1	NOUN
ma-225	126	3	,	,	PUNCT
ma-225	126	4	b1	b1	NOUN
ma-225	126	5	]	]	PUNCT
ma-225	126	6	represent	represent	VERB
ma-225	126	7	an	an	DET
ma-225	126	8	interval	interval	NOUN
ma-225	126	9	on	on	ADP
ma-225	126	10	the	the	DET
ma-225	126	11	p	p	NOUN
ma-225	126	12	-	-	PUNCT
ma-225	126	13	harmonic	harmonic	ADJ
ma-225	126	14	convex	convex	NOUN
ma-225	126	15	set	set	VERB
ma-225	126	16	r	r	NOUN
ma-225	126	17	without	without	ADP
ma-225	126	18	zero	zero	NUM
ma-225	126	19	.	.	PUNCT
ma-225	127	1	if	if	SCONJ
ma-225	127	2	a	a	DET
ma-225	127	3	function	function	NOUN
ma-225	127	4	φ	φ	NOUN
ma-225	127	5	:	:	PUNCT
ma-225	127	6	c	c	X
ma-225	127	7	=	=	PUNCT
ma-225	128	1	[	[	X
ma-225	128	2	a1	a1	NOUN
ma-225	128	3	,	,	PUNCT
ma-225	128	4	b1	b1	NOUN
ma-225	128	5	]	]	PUNCT
ma-225	128	6	⊆	⊆	NUM
ma-225	128	7	r	r	NOUN
ma-225	128	8	is	be	AUX
ma-225	128	9	p	p	NOUN
ma-225	128	10	-	-	PUNCT
ma-225	128	11	harmonic	harmonic	ADJ
ma-225	128	12	convex	convex	NOUN
ma-225	128	13	,	,	PUNCT
ma-225	128	14	it	it	PRON
ma-225	128	15	does	do	AUX
ma-225	128	16	not	not	PART
ma-225	128	17	include	include	VERB
ma-225	128	18	zero	zero	NUM
ma-225	128	19	if	if	SCONJ
ma-225	128	20	φ	φ	PROPN
ma-225	128	21	(	(	PUNCT
ma-225	128	22	upvp	upvp	ADJ
ma-225	128	23	qup	qup	NOUN
ma-225	128	24	+	+	CCONJ
ma-225	128	25	(	(	PUNCT
ma-225	128	26	1−	1−	NUM
ma-225	128	27	q)vp	q)vp	PROPN
ma-225	128	28	)	)	PUNCT
ma-225	128	29	1	1	NUM
ma-225	128	30	p	p	NOUN
ma-225	128	31	≤	≤	NOUN
ma-225	128	32	(	(	PUNCT
ma-225	128	33	1−	1−	NUM
ma-225	128	34	q)φ(u	q)φ(u	NOUN
ma-225	128	35	)	)	PUNCT
ma-225	128	36	+	+	NUM
ma-225	128	37	qφ(v	qφ(v	NOUN
ma-225	128	38	)	)	PUNCT
ma-225	128	39	,	,	PUNCT
ma-225	128	40	(	(	PUNCT
ma-225	128	41	20	20	NUM
ma-225	128	42	)	)	PUNCT
ma-225	128	43	for	for	ADP
ma-225	128	44	all	all	DET
ma-225	128	45	u	u	NOUN
ma-225	128	46	,	,	PUNCT
ma-225	128	47	v	v	NOUN
ma-225	128	48	∈	∈	PROPN
ma-225	128	49	c	c	NOUN
ma-225	128	50	and	and	CCONJ
ma-225	128	51	q	q	NOUN
ma-225	128	52	∈	∈	PROPN
ma-225	129	1	[	[	X
ma-225	129	2	0	0	NUM
ma-225	129	3	,	,	PUNCT
ma-225	129	4	1	1	NUM
ma-225	129	5	]	]	PUNCT
ma-225	129	6	.	.	PUNCT
ma-225	130	1	in	in	ADP
ma-225	130	2	1984	1984	NUM
ma-225	130	3	,	,	PUNCT
ma-225	130	4	g.	g.	PROPN
ma-225	130	5	toader	toader	PROPN
ma-225	130	6	defines	define	VERB
ma-225	130	7	m	m	ADJ
ma-225	130	8	-	-	ADJ
ma-225	130	9	convex	convex	ADJ
ma-225	130	10	function	function	NOUN
ma-225	130	11	as	as	SCONJ
ma-225	130	12	follows	follow	VERB
ma-225	130	13	[	[	X
ma-225	130	14	17	17	NUM
ma-225	130	15	]	]	NUM
ma-225	130	16	:	:	PUNCT
ma-225	130	17	definition	definition	NOUN
ma-225	130	18	2.12	2.12	NUM
ma-225	130	19	.	.	PUNCT
ma-225	131	1	let	let	VERB
ma-225	131	2	the	the	DET
ma-225	131	3	function	function	NOUN
ma-225	131	4	φ	φ	NOUN
ma-225	131	5	be	be	AUX
ma-225	131	6	a	a	DET
ma-225	131	7	real	real	ADJ
ma-225	131	8	r+	r+	NOUN
ma-225	131	9	on	on	ADP
ma-225	131	10	[	[	X
ma-225	131	11	u	u	NOUN
ma-225	131	12	,	,	PUNCT
ma-225	131	13	v	v	NOUN
ma-225	131	14	]	]	PUNCT
ma-225	131	15	and	and	CCONJ
ma-225	131	16	m	m	PROPN
ma-225	131	17	∈	∈	PROPN
ma-225	132	1	[	[	X
ma-225	132	2	0	0	NUM
ma-225	132	3	,	,	PUNCT
ma-225	132	4	1	1	NUM
ma-225	132	5	]	]	PUNCT
ma-225	132	6	,	,	PUNCT
ma-225	132	7	then	then	ADV
ma-225	132	8	m	m	NOUN
ma-225	132	9	-	-	ADJ
ma-225	132	10	convex	convex	ADJ
ma-225	132	11	function	function	NOUN
ma-225	132	12	is	be	AUX
ma-225	132	13	given	give	VERB
ma-225	132	14	as	as	ADP
ma-225	132	15	;	;	PUNCT
ma-225	132	16	φ	φ	X
ma-225	132	17	[	[	PUNCT
ma-225	132	18	qx1	qx1	PROPN
ma-225	132	19	+	+	ADJ
ma-225	132	20	m(1−	m(1−	ADJ
ma-225	132	21	q)y1	q)y1	PROPN
ma-225	132	22	]	]	PUNCT
ma-225	132	23	≤	≤	NUM
ma-225	132	24	qφ(x1	qφ(x1	NOUN
ma-225	132	25	)	)	PUNCT
ma-225	133	1	+	+	ADJ
ma-225	133	2	m(1−	m(1−	PROPN
ma-225	133	3	q)φ(y1	q)φ(y1	NOUN
ma-225	133	4	)	)	PUNCT
ma-225	133	5	,	,	PUNCT
ma-225	133	6	(	(	PUNCT
ma-225	133	7	21	21	NUM
ma-225	133	8	)	)	PUNCT
ma-225	133	9	for	for	ADP
ma-225	133	10	all	all	DET
ma-225	133	11	x1	x1	PROPN
ma-225	133	12	,	,	PUNCT
ma-225	133	13	y1	y1	PROPN
ma-225	133	14	∈	∈	PROPN
ma-225	134	1	[	[	X
ma-225	134	2	u	u	NOUN
ma-225	134	3	,	,	PUNCT
ma-225	134	4	v	v	NOUN
ma-225	134	5	]	]	PUNCT
ma-225	134	6	and	and	CCONJ
ma-225	134	7	q	q	ADJ
ma-225	134	8	∈	∈	PROPN
ma-225	135	1	[	[	X
ma-225	135	2	0	0	NUM
ma-225	135	3	,	,	PUNCT
ma-225	135	4	1	1	NUM
ma-225	135	5	]	]	PUNCT
ma-225	135	6	.	.	PUNCT
ma-225	136	1	also	also	ADV
ma-225	136	2	,	,	PUNCT
ma-225	136	3	φ	φ	PROPN
ma-225	136	4	is	be	AUX
ma-225	136	5	m	m	NOUN
ma-225	136	6	-	-	ADJ
ma-225	136	7	concave	concave	ADJ
ma-225	136	8	if	if	SCONJ
ma-225	136	9	−φ	−φ	NOUN
ma-225	136	10	is	be	AUX
ma-225	136	11	m−convex	m−convex	PROPN
ma-225	136	12	.	.	PROPN
ma-225	136	13	definition	definition	NOUN
ma-225	136	14	2.13	2.13	NUM
ma-225	136	15	.	.	PUNCT
ma-225	137	1	let	let	VERB
ma-225	137	2	the	the	DET
ma-225	137	3	function	function	NOUN
ma-225	137	4	φ	φ	PROPN
ma-225	137	5	be	be	AUX
ma-225	137	6	a	a	DET
ma-225	137	7	positive	positive	ADJ
ma-225	137	8	real	real	ADJ
ma-225	137	9	value	value	NOUN
ma-225	137	10	on	on	ADP
ma-225	137	11	s	s	NOUN
ma-225	137	12	=	=	PUNCT
ma-225	138	1	[	[	X
ma-225	138	2	u	u	NOUN
ma-225	138	3	,	,	PUNCT
ma-225	138	4	v	v	ADP
ma-225	138	5	]	]	PUNCT
ma-225	138	6	,	,	PUNCT
ma-225	138	7	then	then	ADV
ma-225	138	8	c	c	X
ma-225	138	9	-	-	PUNCT
ma-225	138	10	convex	convex	NOUN
ma-225	138	11	is	be	AUX
ma-225	138	12	represented	represent	VERB
ma-225	138	13	by	by	ADP
ma-225	138	14	luenberger	luenberger	NOUN
ma-225	138	15	(	(	PUNCT
ma-225	138	16	1969	1969	NUM
ma-225	138	17	)	)	PUNCT
ma-225	138	18	as	as	ADP
ma-225	138	19	;	;	PUNCT
ma-225	138	20	φ	φ	PROPN
ma-225	138	21	[	[	PUNCT
ma-225	138	22	(	(	PUNCT
ma-225	138	23	1−	1−	NUM
ma-225	138	24	q)x1	q)x1	NOUN
ma-225	138	25	+	+	CCONJ
ma-225	138	26	qy1	qy1	PROPN
ma-225	138	27	]	]	PUNCT
ma-225	138	28	≤	≤	NUM
ma-225	138	29	c(1−	c(1−	PROPN
ma-225	138	30	q)φ(x1	q)φ(x1	NOUN
ma-225	138	31	)	)	PUNCT
ma-225	139	1	+	+	NUM
ma-225	139	2	qφ(y1	qφ(y1	NOUN
ma-225	139	3	)	)	PUNCT
ma-225	139	4	,	,	PUNCT
ma-225	139	5	(	(	PUNCT
ma-225	139	6	22	22	NUM
ma-225	139	7	)	)	PUNCT
ma-225	139	8	where	where	SCONJ
ma-225	139	9	c	c	PROPN
ma-225	139	10	∈	∈	PROPN
ma-225	140	1	[	[	X
ma-225	140	2	0	0	NUM
ma-225	140	3	,	,	PUNCT
ma-225	140	4	1	1	NUM
ma-225	140	5	]	]	PUNCT
ma-225	140	6	,	,	PUNCT
ma-225	140	7	for	for	ADP
ma-225	140	8	all	all	DET
ma-225	140	9	x1	x1	PROPN
ma-225	140	10	,	,	PUNCT
ma-225	140	11	y1	y1	PROPN
ma-225	140	12	∈	∈	PROPN
ma-225	140	13	s	s	X
ma-225	140	14	and	and	CCONJ
ma-225	140	15	q	q	NOUN
ma-225	140	16	∈	∈	PROPN
ma-225	141	1	[	[	X
ma-225	141	2	0	0	NUM
ma-225	141	3	,	,	PUNCT
ma-225	141	4	1	1	NUM
ma-225	141	5	]	]	PUNCT
ma-225	141	6	.	.	PUNCT
ma-225	142	1	3	3	X
ma-225	142	2	.	.	NOUN
ma-225	142	3	results	result	NOUN
ma-225	142	4	and	and	CCONJ
ma-225	142	5	discussions	discussion	NOUN
ma-225	142	6	in	in	ADP
ma-225	142	7	this	this	DET
ma-225	142	8	section	section	NOUN
ma-225	142	9	,	,	PUNCT
ma-225	142	10	all	all	DET
ma-225	142	11	the	the	DET
ma-225	142	12	results	result	NOUN
ma-225	142	13	are	be	AUX
ma-225	142	14	presented	present	VERB
ma-225	142	15	in	in	ADP
ma-225	142	16	non	non	ADJ
ma-225	142	17	-	-	ADJ
ma-225	142	18	newtonian	newtonian	ADJ
ma-225	142	19	form	form	NOUN
ma-225	142	20	.	.	PUNCT
ma-225	143	1	definition	definition	NOUN
ma-225	143	2	3.1	3.1	NUM
ma-225	143	3	.	.	PUNCT
ma-225	144	1	let	let	VERB
ma-225	144	2	x	x	PRON
ma-225	144	3	,	,	PUNCT
ma-225	144	4	y	y	PROPN
ma-225	144	5	∈	∈	PROPN
ma-225	144	6	c	c	PROPN
ma-225	144	7	and	and	CCONJ
ma-225	144	8	c	c	NOUN
ma-225	144	9	=	=	PUNCT
ma-225	145	1	[	[	X
ma-225	145	2	a1	a1	NOUN
ma-225	145	3	,	,	PUNCT
ma-225	145	4	b1	b1	NOUN
ma-225	145	5	]	]	PUNCT
ma-225	145	6	⊆	⊆	NUM
ma-225	145	7	r+	r+	NOUN
ma-225	145	8	be	be	AUX
ma-225	145	9	a	a	DET
ma-225	145	10	set	set	NOUN
ma-225	145	11	.	.	PUNCT
ma-225	146	1	c	c	NOUN
ma-225	146	2	must	must	AUX
ma-225	146	3	be	be	AUX
ma-225	146	4	convex	convex	ADJ
ma-225	146	5	if	if	SCONJ
ma-225	146	6	x	x	NOUN
ma-225	146	7	ln(t	ln(t	PUNCT
ma-225	146	8	)	)	PUNCT
ma-225	146	9	·	·	PUNCT
ma-225	147	1	y	y	X
ma-225	147	2	ln	ln	ADJ
ma-225	147	3	(	(	PUNCT
ma-225	147	4	1	1	NUM
ma-225	147	5	t	t	NOUN
ma-225	147	6	)	)	PUNCT
ma-225	147	7	∈	∈	PROPN
ma-225	147	8	c	c	NOUN
ma-225	147	9	,	,	PUNCT
ma-225	147	10	(	(	PUNCT
ma-225	147	11	23	23	NUM
ma-225	147	12	)	)	PUNCT
ma-225	147	13	for	for	ADP
ma-225	147	14	t	t	PROPN
ma-225	147	15	∈	∈	PROPN
ma-225	148	1	[	[	X
ma-225	148	2	1	1	NUM
ma-225	148	3	,	,	PUNCT
ma-225	148	4	e	e	NOUN
ma-225	148	5	]	]	PUNCT
ma-225	148	6	.	.	PUNCT
ma-225	149	1	lemma	lemma	PROPN
ma-225	149	2	3.1	3.1	NUM
ma-225	149	3	.	.	PUNCT
ma-225	150	1	let	let	VERB
ma-225	150	2	,	,	PUNCT
ma-225	150	3	x1	x1	PROPN
ma-225	150	4	,	,	PUNCT
ma-225	150	5	y1	y1	PROPN
ma-225	150	6	∈	∈	PROPN
ma-225	150	7	r+	r+	NOUN
ma-225	150	8	and	and	CCONJ
ma-225	150	9	φ1	φ1	NOUN
ma-225	150	10	be	be	AUX
ma-225	150	11	a	a	DET
ma-225	150	12	convex	convex	NOUN
ma-225	150	13	function	function	NOUN
ma-225	150	14	.	.	PUNCT
ma-225	151	1	then	then	ADV
ma-225	151	2	φ1	φ1	PROPN
ma-225	151	3	[	[	PUNCT
ma-225	151	4	x	x	NOUN
ma-225	151	5	ln(t1	ln(t1	NOUN
ma-225	151	6	)	)	PUNCT
ma-225	151	7	1	1	NUM
ma-225	151	8	·	·	PUNCT
ma-225	151	9	y	y	X
ma-225	151	10	ln	ln	ADJ
ma-225	151	11	(	(	PUNCT
ma-225	151	12	1	1	NUM
ma-225	151	13	t1	t1	NOUN
ma-225	151	14	)	)	PUNCT
ma-225	151	15	1	1	NUM
ma-225	151	16	]	]	PUNCT
ma-225	151	17	≤	≤	NUM
ma-225	151	18	φ1(x1)ln(t1	φ1(x1)ln(t1	NOUN
ma-225	151	19	)	)	PUNCT
ma-225	151	20	·	·	PUNCT
ma-225	152	1	φ1(y1	φ1(y1	ADP
ma-225	152	2	)	)	PUNCT
ma-225	152	3	ln	ln	ADJ
ma-225	152	4	(	(	PUNCT
ma-225	152	5	1	1	NUM
ma-225	152	6	t1	t1	NOUN
ma-225	152	7	)	)	PUNCT
ma-225	152	8	,	,	PUNCT
ma-225	152	9	(	(	PUNCT
ma-225	152	10	24	24	NUM
ma-225	152	11	)	)	PUNCT
ma-225	152	12	for	for	ADP
ma-225	152	13	t1	t1	PROPN
ma-225	152	14	∈	∈	PROPN
ma-225	153	1	[	[	X
ma-225	153	2	1	1	NUM
ma-225	153	3	,	,	PUNCT
ma-225	153	4	e	e	NOUN
ma-225	153	5	]	]	X
ma-225	153	6	.	.	PUNCT
ma-225	154	1	5	5	NUM
ma-225	154	2	proof	proof	NOUN
ma-225	154	3	.	.	PUNCT
ma-225	155	1	using	use	VERB
ma-225	155	2	equation	equation	NOUN
ma-225	155	3	(	(	PUNCT
ma-225	155	4	3	3	NUM
ma-225	155	5	)	)	PUNCT
ma-225	155	6	,	,	PUNCT
ma-225	155	7	(	(	PUNCT
ma-225	155	8	1	1	X
ma-225	155	9	)	)	PUNCT
ma-225	155	10	and	and	CCONJ
ma-225	155	11	by	by	ADP
ma-225	155	12	convexity	convexity	NOUN
ma-225	155	13	,	,	PUNCT
ma-225	155	14	we	we	PRON
ma-225	155	15	have	have	VERB
ma-225	155	16	φ1	φ1	NOUN
ma-225	155	17	[	[	PUNCT
ma-225	155	18	x	x	NOUN
ma-225	155	19	ln(t1	ln(t1	NOUN
ma-225	155	20	)	)	PUNCT
ma-225	155	21	1	1	NUM
ma-225	155	22	·	·	PUNCT
ma-225	155	23	y	y	X
ma-225	155	24	ln	ln	ADJ
ma-225	155	25	(	(	PUNCT
ma-225	155	26	1	1	NUM
ma-225	155	27	t1	t1	NOUN
ma-225	155	28	)	)	PUNCT
ma-225	155	29	1	1	NUM
ma-225	155	30	]	]	PUNCT
ma-225	155	31	=	=	X
ma-225	155	32	φ1	φ1	PROPN
ma-225	155	33	[	[	PUNCT
ma-225	155	34	x1×̇t1+̇y1×̇	x1×̇t1+̇y1×̇	NOUN
ma-225	155	35	(	(	PUNCT
ma-225	155	36	1	1	NUM
ma-225	155	37	t1	t1	NOUN
ma-225	155	38	)	)	PUNCT
ma-225	155	39	]	]	PUNCT
ma-225	156	1	≤φ1(x1)×̇t1+̇φ1(y1)×̇	≤φ1(x1)×̇t1+̇φ1(y1)×̇	PROPN
ma-225	156	2	(	(	PUNCT
ma-225	156	3	1	1	NUM
ma-225	156	4	t1	t1	NOUN
ma-225	156	5	)	)	PUNCT
ma-225	157	1	≤α	≤α	NOUN
ma-225	157	2	[	[	PUNCT
ma-225	157	3	α−1φ1(x1)×̇α−1(t1)+̇α−1φ1(y1)×̇α−1	α−1φ1(x1)×̇α−1(t1)+̇α−1φ1(y1)×̇α−1	PRON
ma-225	157	4	(	(	PUNCT
ma-225	157	5	1	1	NUM
ma-225	157	6	t1	t1	NOUN
ma-225	157	7	)	)	PUNCT
ma-225	157	8	]	]	PUNCT
ma-225	158	1	≤α	≤α	NOUN
ma-225	158	2	[	[	PUNCT
ma-225	158	3	lnφ1(x1)×	lnφ1(x1)×	NOUN
ma-225	158	4	ln(t1	ln(t1	NOUN
ma-225	158	5	)	)	PUNCT
ma-225	159	1	+	+	CCONJ
ma-225	159	2	lnφ1(y1)×	lnφ1(y1)×	PROPN
ma-225	159	3	ln	ln	ADJ
ma-225	159	4	(	(	PUNCT
ma-225	159	5	1	1	NUM
ma-225	159	6	t1	t1	NOUN
ma-225	159	7	)	)	PUNCT
ma-225	159	8	]	]	PUNCT
ma-225	160	1	≤e	≤e	PROPN
ma-225	160	2	[	[	PUNCT
ma-225	160	3	lnφ1(x1	lnφ1(x1	PROPN
ma-225	160	4	)	)	PUNCT
ma-225	160	5	ln(t1)+lnφ1(y1	ln(t1)+lnφ1(y1	PROPN
ma-225	160	6	)	)	PUNCT
ma-225	160	7	ln	ln	PROPN
ma-225	160	8	(	(	PUNCT
ma-225	160	9	1	1	NUM
ma-225	160	10	t1	t1	NOUN
ma-225	160	11	)	)	PUNCT
ma-225	160	12	]	]	PUNCT
ma-225	161	1	≤(e	≤(e	PROPN
ma-225	161	2	lnφ1(x1))ln(t1	lnφ1(x1))ln(t1	PROPN
ma-225	161	3	)	)	PUNCT
ma-225	161	4	·	·	PUNCT
ma-225	162	1	(	(	PUNCT
ma-225	162	2	e	e	X
ma-225	162	3	lnφ1(y1))ln	lnφ1(y1))ln	PROPN
ma-225	162	4	(	(	PUNCT
ma-225	162	5	1	1	NUM
ma-225	162	6	t1	t1	NOUN
ma-225	162	7	)	)	PUNCT
ma-225	162	8	φ1	φ1	NOUN
ma-225	162	9	[	[	PUNCT
ma-225	162	10	x	x	SYM
ma-225	162	11	ln(t1	ln(t1	NOUN
ma-225	162	12	)	)	PUNCT
ma-225	162	13	1	1	NUM
ma-225	162	14	·	·	PUNCT
ma-225	162	15	y	y	X
ma-225	162	16	ln	ln	ADJ
ma-225	162	17	(	(	PUNCT
ma-225	162	18	1	1	NUM
ma-225	162	19	t1	t1	NOUN
ma-225	162	20	)	)	PUNCT
ma-225	162	21	1	1	NUM
ma-225	162	22	]	]	PUNCT
ma-225	162	23	≤φ1(x1)ln(t1	≤φ1(x1)ln(t1	X
ma-225	162	24	)	)	PUNCT
ma-225	162	25	·	·	PUNCT
ma-225	162	26	φ1(y1	φ1(y1	ADP
ma-225	162	27	)	)	PUNCT
ma-225	162	28	ln	ln	ADJ
ma-225	162	29	(	(	PUNCT
ma-225	162	30	1	1	NUM
ma-225	162	31	t1	t1	NOUN
ma-225	162	32	)	)	PUNCT
ma-225	162	33	,	,	PUNCT
ma-225	162	34	as	as	SCONJ
ma-225	162	35	required	require	VERB
ma-225	162	36	.	.	PUNCT
ma-225	163	1	�	�	PROPN
ma-225	163	2	lemma	lemma	PROPN
ma-225	163	3	3.2	3.2	NUM
ma-225	163	4	.	.	PUNCT
ma-225	164	1	let	let	VERB
ma-225	164	2	a1	a1	NOUN
ma-225	164	3	,	,	PUNCT
ma-225	164	4	b1	b1	NOUN
ma-225	164	5	∈	∈	PROPN
ma-225	164	6	r+	r+	NOUN
ma-225	164	7	and	and	CCONJ
ma-225	164	8	φ1	φ1	NOUN
ma-225	164	9	be	be	AUX
ma-225	164	10	a	a	DET
ma-225	164	11	convex	convex	NOUN
ma-225	164	12	function	function	NOUN
ma-225	164	13	,	,	PUNCT
ma-225	164	14	we	we	PRON
ma-225	164	15	have	have	VERB
ma-225	164	16	φ1	φ1	NOUN
ma-225	164	17	[	[	PUNCT
ma-225	164	18	(	(	PUNCT
ma-225	164	19	b1	b1	NOUN
ma-225	164	20	)	)	PUNCT
ma-225	164	21	(	(	PUNCT
ma-225	164	22	a1	a1	PROPN
ma-225	164	23	)	)	PUNCT
ma-225	164	24	]	]	PUNCT
ma-225	164	25	≤	≤	NUM
ma-225	164	26	φ1(b1	φ1(b1	NOUN
ma-225	164	27	)	)	PUNCT
ma-225	164	28	φ1(a1	φ1(a1	NUM
ma-225	164	29	)	)	PUNCT
ma-225	164	30	.	.	PUNCT
ma-225	165	1	(	(	PUNCT
ma-225	165	2	25	25	NUM
ma-225	165	3	)	)	PUNCT
ma-225	165	4	proof	proof	NOUN
ma-225	165	5	.	.	PUNCT
ma-225	166	1	using	use	VERB
ma-225	166	2	equation	equation	NOUN
ma-225	166	3	(	(	PUNCT
ma-225	166	4	2	2	NUM
ma-225	166	5	)	)	PUNCT
ma-225	166	6	and	and	CCONJ
ma-225	166	7	by	by	ADP
ma-225	166	8	convexity	convexity	NOUN
ma-225	166	9	,	,	PUNCT
ma-225	166	10	we	we	PRON
ma-225	166	11	have	have	VERB
ma-225	166	12	φ1	φ1	NOUN
ma-225	166	13	[	[	PUNCT
ma-225	166	14	(	(	PUNCT
ma-225	166	15	b1	b1	NOUN
ma-225	166	16	)	)	PUNCT
ma-225	166	17	(	(	PUNCT
ma-225	166	18	a1	a1	PROPN
ma-225	166	19	)	)	PUNCT
ma-225	166	20	]	]	PUNCT
ma-225	167	1	=	=	X
ma-225	167	2	φ1	φ1	PROPN
ma-225	167	3	[	[	PUNCT
ma-225	167	4	(	(	PUNCT
ma-225	167	5	b1)−̇(a1	b1)−̇(a1	NUM
ma-225	167	6	)	)	PUNCT
ma-225	167	7	]	]	PUNCT
ma-225	167	8	≤φ1(b1)−̇φ1(a1	≤φ1(b1)−̇φ1(a1	NOUN
ma-225	167	9	)	)	PUNCT
ma-225	167	10	≤α	≤α	NOUN
ma-225	167	11	[	[	PUNCT
ma-225	167	12	α−1(φ1(b1))−̇α−1(φ1(a1	α−1(φ1(b1))−̇α−1(φ1(a1	NOUN
ma-225	167	13	)	)	PUNCT
ma-225	167	14	)	)	PUNCT
ma-225	167	15	]	]	PUNCT
ma-225	167	16	≤e[lnφ1(b1)−lnφ1(a1	≤e[lnφ1(b1)−lnφ1(a1	PROPN
ma-225	167	17	)	)	PUNCT
ma-225	167	18	]	]	PUNCT
ma-225	167	19	≤	≤	NUM
ma-225	167	20	e	e	X
ma-225	167	21	lnφ1(b1	lnφ1(b1	X
ma-225	167	22	)	)	PUNCT
ma-225	167	23	e	e	X
ma-225	167	24	lnφ1(a1	lnφ1(a1	PROPN
ma-225	167	25	)	)	PUNCT
ma-225	167	26	≤	≤	NUM
ma-225	167	27	φ1(b1	φ1(b1	NOUN
ma-225	167	28	)	)	PUNCT
ma-225	167	29	φ1(a1	φ1(a1	NUM
ma-225	167	30	)	)	PUNCT
ma-225	167	31	φ1	φ1	NOUN
ma-225	167	32	[	[	PUNCT
ma-225	167	33	(	(	PUNCT
ma-225	167	34	b1	b1	NOUN
ma-225	167	35	)	)	PUNCT
ma-225	167	36	(	(	PUNCT
ma-225	167	37	a1	a1	PROPN
ma-225	167	38	)	)	PUNCT
ma-225	167	39	]	]	PUNCT
ma-225	167	40	≤	≤	NUM
ma-225	167	41	φ1(b1	φ1(b1	NOUN
ma-225	167	42	)	)	PUNCT
ma-225	167	43	φ1(a1	φ1(a1	NUM
ma-225	167	44	)	)	PUNCT
ma-225	167	45	,	,	PUNCT
ma-225	167	46	as	as	SCONJ
ma-225	167	47	required	require	VERB
ma-225	167	48	.	.	PUNCT
ma-225	168	1	�	�	PROPN
ma-225	168	2	lemma	lemma	PROPN
ma-225	168	3	3.3	3.3	NUM
ma-225	168	4	.	.	PUNCT
ma-225	169	1	let	let	VERB
ma-225	169	2	φ1	φ1	PROPN
ma-225	169	3	be	be	AUX
ma-225	169	4	convex	convex	ADJ
ma-225	169	5	and	and	CCONJ
ma-225	169	6	u1	u1	NOUN
ma-225	169	7	,	,	PUNCT
ma-225	169	8	v1	v1	PROPN
ma-225	169	9	∈	∈	PROPN
ma-225	169	10	r+	r+	NOUN
ma-225	169	11	.	.	PUNCT
ma-225	170	1	then	then	ADV
ma-225	170	2	φ1	φ1	PROPN
ma-225	170	3	[	[	PUNCT
ma-225	170	4	u1	u1	NOUN
ma-225	170	5	·	·	PUNCT
ma-225	170	6	v1	v1	NOUN
ma-225	170	7	2	2	NUM
ma-225	170	8	]	]	PUNCT
ma-225	170	9	≤	≤	NUM
ma-225	170	10	φ1(u1	φ1(u1	NOUN
ma-225	170	11	)	)	PUNCT
ma-225	170	12	·	·	PUNCT
ma-225	170	13	φ1(v1	φ1(v1	VERB
ma-225	170	14	)	)	PUNCT
ma-225	170	15	2	2	NUM
ma-225	170	16	.	.	PUNCT
ma-225	171	1	(	(	PUNCT
ma-225	171	2	26	26	NUM
ma-225	171	3	)	)	PUNCT
ma-225	171	4	6	6	NUM
ma-225	171	5	proof	proof	NOUN
ma-225	171	6	.	.	PUNCT
ma-225	172	1	using	use	VERB
ma-225	172	2	equation	equation	NOUN
ma-225	172	3	(	(	PUNCT
ma-225	172	4	1	1	NUM
ma-225	172	5	)	)	PUNCT
ma-225	172	6	and	and	CCONJ
ma-225	172	7	by	by	ADP
ma-225	172	8	convexity	convexity	NOUN
ma-225	172	9	,	,	PUNCT
ma-225	172	10	we	we	PRON
ma-225	172	11	have	have	VERB
ma-225	172	12	φ1	φ1	NOUN
ma-225	172	13	[	[	PUNCT
ma-225	172	14	u1	u1	NOUN
ma-225	172	15	·	·	PUNCT
ma-225	172	16	v1	v1	NOUN
ma-225	172	17	2	2	NUM
ma-225	172	18	]	]	PUNCT
ma-225	173	1	=	=	X
ma-225	173	2	φ1	φ1	NOUN
ma-225	173	3	[	[	PUNCT
ma-225	173	4	1	1	NUM
ma-225	173	5	2	2	NUM
ma-225	173	6	(	(	PUNCT
ma-225	173	7	u1+̇v1	u1+̇v1	ADJ
ma-225	173	8	)	)	PUNCT
ma-225	173	9	]	]	PUNCT
ma-225	173	10	≤	≤	NUM
ma-225	173	11	1	1	NUM
ma-225	173	12	2	2	NUM
ma-225	173	13	[	[	X
ma-225	173	14	(	(	PUNCT
ma-225	173	15	φ1(u1)+̇φ1(v1	φ1(u1)+̇φ1(v1	NOUN
ma-225	173	16	)	)	PUNCT
ma-225	173	17	)	)	PUNCT
ma-225	173	18	]	]	PUNCT
ma-225	174	1	≤	≤	NUM
ma-225	174	2	1	1	NUM
ma-225	174	3	2	2	NUM
ma-225	174	4	[	[	PUNCT
ma-225	174	5	α	α	X
ma-225	174	6	(	(	PUNCT
ma-225	174	7	α−1φ1(u1)+̇α	α−1φ1(u1)+̇α	PROPN
ma-225	174	8	−1φ1(v1	−1φ1(v1	PROPN
ma-225	174	9	)	)	PUNCT
ma-225	174	10	)	)	PUNCT
ma-225	174	11	]	]	PUNCT
ma-225	175	1	≤	≤	NUM
ma-225	175	2	1	1	NUM
ma-225	175	3	2	2	NUM
ma-225	175	4	[	[	PUNCT
ma-225	175	5	e(lnφ1(u1)+lnφ1(v1	e(lnφ1(u1)+lnφ1(v1	NOUN
ma-225	175	6	)	)	PUNCT
ma-225	175	7	)	)	PUNCT
ma-225	175	8	]	]	PUNCT
ma-225	176	1	≤	≤	NUM
ma-225	176	2	1	1	NUM
ma-225	176	3	2	2	NUM
ma-225	176	4	[	[	PUNCT
ma-225	176	5	e	e	X
ma-225	176	6	lnφ1(u1	lnφ1(u1	NOUN
ma-225	176	7	)	)	PUNCT
ma-225	176	8	·	·	PUNCT
ma-225	177	1	e	e	X
ma-225	177	2	lnφ1(v1	lnφ1(v1	PROPN
ma-225	177	3	)	)	PUNCT
ma-225	177	4	]	]	PUNCT
ma-225	178	1	≤	≤	NUM
ma-225	178	2	1	1	NUM
ma-225	178	3	2	2	NUM
ma-225	178	4	[	[	PUNCT
ma-225	178	5	φ1(u1	φ1(u1	NOUN
ma-225	178	6	)	)	PUNCT
ma-225	178	7	·	·	PUNCT
ma-225	179	1	φ1(v1	φ1(v1	NOUN
ma-225	179	2	)	)	PUNCT
ma-225	179	3	]	]	PUNCT
ma-225	179	4	φ1	φ1	PROPN
ma-225	179	5	[	[	PUNCT
ma-225	179	6	u1	u1	NOUN
ma-225	179	7	·	·	PUNCT
ma-225	179	8	v1	v1	NOUN
ma-225	179	9	2	2	NUM
ma-225	179	10	]	]	PUNCT
ma-225	179	11	≤	≤	NUM
ma-225	179	12	φ1(u1	φ1(u1	NOUN
ma-225	179	13	)	)	PUNCT
ma-225	179	14	·	·	PUNCT
ma-225	180	1	φ1(v1	φ1(v1	VERB
ma-225	180	2	)	)	PUNCT
ma-225	180	3	2	2	NUM
ma-225	180	4	,	,	PUNCT
ma-225	180	5	as	as	SCONJ
ma-225	180	6	required	require	VERB
ma-225	180	7	.	.	PUNCT
ma-225	181	1	�	�	PROPN
ma-225	181	2	lemma	lemma	PROPN
ma-225	181	3	3.4	3.4	NUM
ma-225	181	4	.	.	PUNCT
ma-225	181	5	consider	consider	VERB
ma-225	181	6	an	an	DET
ma-225	181	7	increasing	increase	VERB
ma-225	181	8	function	function	NOUN
ma-225	181	9	φ	φ	NOUN
ma-225	181	10	.	.	PUNCT
ma-225	182	1	if	if	SCONJ
ma-225	182	2	φ(vy	φ(vy	ADJ
ma-225	182	3	)	)	PUNCT
ma-225	182	4	≥	≥	NOUN
ma-225	182	5	φ(vx	φ(vx	NOUN
ma-225	182	6	)	)	PUNCT
ma-225	182	7	and	and	CCONJ
ma-225	182	8	vy	vy	ADP
ma-225	182	9	≥	≥	NUM
ma-225	182	10	vx	vx	PROPN
ma-225	182	11	,	,	PUNCT
ma-225	182	12	then	then	ADV
ma-225	182	13	φ(vx	φ(vx	NOUN
ma-225	182	14	)	)	PUNCT
ma-225	182	15	ln(vx	ln(vx	NOUN
ma-225	182	16	)	)	PUNCT
ma-225	182	17	·	·	PUNCT
ma-225	183	1	φ(vy	φ(vy	ADJ
ma-225	183	2	)	)	PUNCT
ma-225	183	3	ln(vy	ln(vy	PROPN
ma-225	183	4	)	)	PUNCT
ma-225	183	5	φ(vy	φ(vy	ADJ
ma-225	183	6	)	)	PUNCT
ma-225	183	7	ln(vx	ln(vx	NOUN
ma-225	183	8	)	)	PUNCT
ma-225	183	9	·	·	PUNCT
ma-225	184	1	φ(vx)ln(vy	φ(vx)ln(vy	NOUN
ma-225	184	2	)	)	PUNCT
ma-225	184	3	≤	≤	NUM
ma-225	184	4	1	1	NUM
ma-225	184	5	,	,	PUNCT
ma-225	184	6	(	(	PUNCT
ma-225	184	7	27	27	NUM
ma-225	184	8	)	)	PUNCT
ma-225	184	9	proof	proof	NOUN
ma-225	184	10	.	.	PUNCT
ma-225	185	1	using	use	VERB
ma-225	185	2	equation	equation	NOUN
ma-225	185	3	(	(	PUNCT
ma-225	185	4	3	3	NUM
ma-225	185	5	)	)	PUNCT
ma-225	185	6	and	and	CCONJ
ma-225	185	7	(	(	PUNCT
ma-225	185	8	1	1	NUM
ma-225	185	9	)	)	PUNCT
ma-225	185	10	,	,	PUNCT
ma-225	185	11	we	we	PRON
ma-225	185	12	have	have	VERB
ma-225	185	13	φ(vx	φ(vx	NOUN
ma-225	185	14	)	)	PUNCT
ma-225	185	15	ln(vx	ln(vx	NOUN
ma-225	185	16	)	)	PUNCT
ma-225	185	17	·	·	PUNCT
ma-225	185	18	φ(vy	φ(vy	ADJ
ma-225	185	19	)	)	PUNCT
ma-225	185	20	ln(vy	ln(vy	NOUN
ma-225	185	21	)	)	PUNCT
ma-225	186	1	=	=	PUNCT
ma-225	186	2	φ(vx)×̇vx	φ(vx)×̇vx	X
ma-225	186	3	+	+	NOUN
ma-225	186	4	̇φ(vy	̇φ(vy	NOUN
ma-225	186	5	)	)	PUNCT
ma-225	186	6	×̇vy	×̇vy	PROPN
ma-225	186	7	=	=	SYM
ma-225	186	8	vx	vx	NOUN
ma-225	186	9	×̇φ(vx)+̇vy	×̇φ(vx)+̇vy	ADJ
ma-225	186	10	×̇φ(vy	×̇φ(vy	ADJ
ma-225	186	11	)	)	PUNCT
ma-225	186	12	≤vx	≤vx	NOUN
ma-225	186	13	×̇φ(vy	×̇φ(vy	NOUN
ma-225	186	14	)	)	PUNCT
ma-225	186	15	+	+	VERB
ma-225	186	16	̇vy	̇vy	ADJ
ma-225	186	17	×̇φ(vx	×̇φ(vx	NOUN
ma-225	186	18	)	)	PUNCT
ma-225	186	19	≤α	≤α	NOUN
ma-225	186	20	[	[	PUNCT
ma-225	186	21	α−1(vx)×̇α−1φ(vy	α−1(vx)×̇α−1φ(vy	NOUN
ma-225	186	22	)	)	PUNCT
ma-225	186	23	+	+	ADJ
ma-225	186	24	̇α−1(vy	̇α−1(vy	ADJ
ma-225	186	25	)	)	PUNCT
ma-225	186	26	×̇α−1φ(vx	×̇α−1φ(vx	NOUN
ma-225	186	27	)	)	PUNCT
ma-225	186	28	]	]	PUNCT
ma-225	187	1	≤α	≤α	NOUN
ma-225	187	2	[	[	PUNCT
ma-225	187	3	ln(vx)×	ln(vx)×	PROPN
ma-225	187	4	lnφ(vy	lnφ(vy	PRON
ma-225	187	5	)	)	PUNCT
ma-225	187	6	+	+	CCONJ
ma-225	187	7	ln(vy	ln(vy	ADJ
ma-225	187	8	)	)	PUNCT
ma-225	187	9	×	×	NOUN
ma-225	187	10	lnφ(vx	lnφ(vx	NOUN
ma-225	187	11	)	)	PUNCT
ma-225	187	12	]	]	PUNCT
ma-225	187	13	≤e[ln(vx	≤e[ln(vx	NOUN
ma-225	187	14	)	)	PUNCT
ma-225	187	15	×lnφ(vy	×lnφ(vy	NOUN
ma-225	187	16	)	)	PUNCT
ma-225	188	1	+	+	ADP
ma-225	188	2	ln(vy	ln(vy	NOUN
ma-225	188	3	)	)	PUNCT
ma-225	188	4	×lnφ(vx	×lnφ(vx	X
ma-225	188	5	)	)	PUNCT
ma-225	188	6	]	]	PUNCT
ma-225	188	7	≤e	≤e	VERB
ma-225	188	8	ln(vx	ln(vx	NOUN
ma-225	188	9	)	)	PUNCT
ma-225	188	10	lnφ(vy	lnφ(vy	NOUN
ma-225	188	11	)	)	PUNCT
ma-225	188	12	·	·	PUNCT
ma-225	189	1	e	e	X
ma-225	189	2	ln(vy	ln(vy	NOUN
ma-225	189	3	)	)	PUNCT
ma-225	189	4	lnφ(vx	lnφ(vx	NOUN
ma-225	189	5	)	)	PUNCT
ma-225	189	6	≤	≤	NUM
ma-225	189	7	(	(	PUNCT
ma-225	189	8	e	e	X
ma-225	189	9	lnφ(vy	lnφ(vy	ADV
ma-225	189	10	)	)	PUNCT
ma-225	189	11	)	)	PUNCT
ma-225	190	1	ln(vx	ln(vx	NOUN
ma-225	190	2	)	)	PUNCT
ma-225	190	3	·	·	PUNCT
ma-225	190	4	(	(	PUNCT
ma-225	190	5	e	e	X
ma-225	190	6	lnφ(vx	lnφ(vx	X
ma-225	190	7	)	)	PUNCT
ma-225	190	8	)	)	PUNCT
ma-225	191	1	ln(vy	ln(vy	NOUN
ma-225	191	2	)	)	PUNCT
ma-225	191	3	φ(vx	φ(vx	NOUN
ma-225	191	4	)	)	PUNCT
ma-225	191	5	ln(vx	ln(vx	NOUN
ma-225	191	6	)	)	PUNCT
ma-225	191	7	·	·	PUNCT
ma-225	191	8	φ(vy	φ(vy	ADJ
ma-225	191	9	)	)	PUNCT
ma-225	191	10	ln(vy	ln(vy	NOUN
ma-225	191	11	)	)	PUNCT
ma-225	191	12	≤φ(vy	≤φ(vy	NOUN
ma-225	191	13	)	)	PUNCT
ma-225	191	14	ln(vx	ln(vx	PROPN
ma-225	191	15	)	)	PUNCT
ma-225	191	16	·	·	PUNCT
ma-225	192	1	φ(vx)ln(vy	φ(vx)ln(vy	NOUN
ma-225	192	2	)	)	PUNCT
ma-225	192	3	,	,	PUNCT
ma-225	192	4	as	as	SCONJ
ma-225	192	5	required	require	VERB
ma-225	192	6	.	.	PUNCT
ma-225	193	1	�	�	PROPN
ma-225	193	2	lemma	lemma	PROPN
ma-225	193	3	3.5	3.5	NUM
ma-225	193	4	.	.	PUNCT
ma-225	194	1	consider	consider	VERB
ma-225	194	2	a	a	DET
ma-225	194	3	decreasing	decrease	VERB
ma-225	194	4	function	function	NOUN
ma-225	194	5	φ	φ	NOUN
ma-225	194	6	.	.	PUNCT
ma-225	195	1	if	if	SCONJ
ma-225	195	2	φ(vx	φ(vx	NOUN
ma-225	195	3	)	)	PUNCT
ma-225	195	4	≥	≥	NOUN
ma-225	195	5	φ(vy	φ(vy	ADV
ma-225	195	6	)	)	PUNCT
ma-225	195	7	and	and	CCONJ
ma-225	195	8	vx	vx	X
ma-225	195	9	≥	≥	PROPN
ma-225	195	10	vy	vy	NOUN
ma-225	195	11	,	,	PUNCT
ma-225	195	12	then	then	ADV
ma-225	195	13	φ(vx	φ(vx	NOUN
ma-225	195	14	)	)	PUNCT
ma-225	195	15	ln(vx	ln(vx	NOUN
ma-225	195	16	)	)	PUNCT
ma-225	195	17	·	·	PUNCT
ma-225	195	18	φ(vy	φ(vy	ADJ
ma-225	195	19	)	)	PUNCT
ma-225	195	20	ln(vy	ln(vy	PROPN
ma-225	195	21	)	)	PUNCT
ma-225	195	22	φ(vy	φ(vy	ADJ
ma-225	195	23	)	)	PUNCT
ma-225	195	24	ln(vx	ln(vx	NOUN
ma-225	195	25	)	)	PUNCT
ma-225	195	26	·	·	PUNCT
ma-225	196	1	φ(vx)ln(vy	φ(vx)ln(vy	NOUN
ma-225	196	2	)	)	PUNCT
ma-225	196	3	≥	≥	NOUN
ma-225	196	4	1	1	NUM
ma-225	196	5	,	,	PUNCT
ma-225	196	6	(	(	PUNCT
ma-225	196	7	28	28	NUM
ma-225	196	8	)	)	PUNCT
ma-225	196	9	or	or	CCONJ
ma-225	196	10	φ(vx	φ(vx	NOUN
ma-225	196	11	)	)	PUNCT
ma-225	196	12	ln(vx	ln(vx	NOUN
ma-225	196	13	)	)	PUNCT
ma-225	196	14	·	·	PUNCT
ma-225	196	15	φ(vy	φ(vy	ADJ
ma-225	196	16	)	)	PUNCT
ma-225	196	17	ln(vy	ln(vy	PROPN
ma-225	196	18	)	)	PUNCT
ma-225	196	19	≥	≥	NOUN
ma-225	196	20	φ(vy	φ(vy	ADJ
ma-225	196	21	)	)	PUNCT
ma-225	196	22	ln(vx	ln(vx	PROPN
ma-225	196	23	)	)	PUNCT
ma-225	196	24	·	·	PUNCT
ma-225	197	1	φ(vx)ln(vy	φ(vx)ln(vy	NOUN
ma-225	197	2	)	)	PUNCT
ma-225	197	3	.	.	PUNCT
ma-225	198	1	(	(	PUNCT
ma-225	198	2	29)7	29)7	NUM
ma-225	198	3	the	the	DET
ma-225	198	4	proof	proof	NOUN
ma-225	198	5	is	be	AUX
ma-225	198	6	similar	similar	ADJ
ma-225	198	7	to	to	ADP
ma-225	198	8	inequality	inequality	NOUN
ma-225	198	9	(	(	PUNCT
ma-225	198	10	25	25	NUM
ma-225	198	11	)	)	PUNCT
ma-225	198	12	,	,	PUNCT
ma-225	198	13	with	with	SCONJ
ma-225	198	14	inequality	inequality	NOUN
ma-225	198	15	sign	sign	NOUN
ma-225	198	16	reversed	reverse	VERB
ma-225	198	17	.	.	PUNCT
ma-225	199	1	theorem	theorem	VERB
ma-225	199	2	3.1	3.1	NUM
ma-225	199	3	.	.	PUNCT
ma-225	200	1	let	let	VERB
ma-225	200	2	vi	vi	NOUN
ma-225	200	3	,	,	PUNCT
ma-225	200	4	zi	zi	PROPN
ma-225	200	5	∈	∈	PROPN
ma-225	200	6	r+	r+	ADV
ma-225	200	7	,	,	PUNCT
ma-225	200	8	i	i	PRON
ma-225	200	9	=	=	NOUN
ma-225	200	10	1	1	NUM
ma-225	200	11	,	,	PUNCT
ma-225	200	12	...	...	PUNCT
ma-225	200	13	,	,	PUNCT
ma-225	200	14	k	k	X
ma-225	200	15	,	,	PUNCT
ma-225	200	16	and	and	CCONJ
ma-225	200	17	let	let	VERB
ma-225	200	18	φ	φ	PROPN
ma-225	200	19	be	be	AUX
ma-225	200	20	a	a	DET
ma-225	200	21	convex	convex	NOUN
ma-225	200	22	function	function	NOUN
ma-225	200	23	defined	define	VERB
ma-225	200	24	on	on	ADP
ma-225	200	25	a	a	DET
ma-225	200	26	range	range	NOUN
ma-225	200	27	of	of	ADP
ma-225	200	28	the	the	DET
ma-225	200	29	real	real	ADJ
ma-225	200	30	line	line	NOUN
ma-225	200	31	.	.	PUNCT
ma-225	201	1	if	if	SCONJ
ma-225	201	2	bk	bk	VERB
ma-225	201	3	>	>	X
ma-225	201	4	0	0	NUM
ma-225	201	5	,	,	PUNCT
ma-225	201	6	then	then	ADV
ma-225	201	7	φ	φ	PROPN
ma-225	201	8			PROPN
ma-225	201	9	(	(	PUNCT
ma-225	201	10	1	1	NUM
ma-225	201	11	bk	bk	NOUN
ma-225	201	12	)	)	PUNCT
ma-225	201	13	ln∏k	ln∏k	NOUN
ma-225	201	14	i=1(vi	i=1(vi	X
ma-225	201	15	)	)	PUNCT
ma-225	201	16	ln(zi	ln(zi	PROPN
ma-225	201	17	)	)	PUNCT
ma-225	201	18			NOUN
ma-225	201	19	≤	≤	NUM
ma-225	201	20	(	(	PUNCT
ma-225	201	21	1	1	NUM
ma-225	201	22	bk	bk	NOUN
ma-225	201	23	)	)	PUNCT
ma-225	201	24	ln∏k	ln∏k	NOUN
ma-225	201	25	i=1	i=1	ADP
ma-225	201	26	φ(vi	φ(vi	PROPN
ma-225	201	27	)	)	PUNCT
ma-225	201	28	ln(zi	ln(zi	PROPN
ma-225	201	29	)	)	PUNCT
ma-225	201	30	.	.	PUNCT
ma-225	202	1	(	(	PUNCT
ma-225	202	2	30	30	X
ma-225	202	3	)	)	PUNCT
ma-225	202	4	proof	proof	NOUN
ma-225	202	5	.	.	PUNCT
ma-225	203	1	using	use	VERB
ma-225	203	2	equation	equation	NOUN
ma-225	203	3	(	(	PUNCT
ma-225	203	4	3	3	NUM
ma-225	203	5	)	)	PUNCT
ma-225	203	6	and	and	CCONJ
ma-225	203	7	by	by	ADP
ma-225	203	8	convexity	convexity	NOUN
ma-225	203	9	,	,	PUNCT
ma-225	203	10	we	we	PRON
ma-225	203	11	have	have	VERB
ma-225	203	12	φ	φ	NUM
ma-225	203	13			PROPN
ma-225	203	14	(	(	PUNCT
ma-225	203	15	1	1	NUM
ma-225	203	16	bk	bk	NOUN
ma-225	203	17	)	)	PUNCT
ma-225	203	18	ln∏k	ln∏k	NOUN
ma-225	203	19	i=1(vi	i=1(vi	X
ma-225	203	20	)	)	PUNCT
ma-225	203	21	ln(zi	ln(zi	PROPN
ma-225	203	22	)	)	PUNCT
ma-225	203	23			NOUN
ma-225	203	24	=	=	SYM
ma-225	203	25	φ	φ	NOUN
ma-225	203	26	1	1	NUM
ma-225	203	27	bk	bk	NOUN
ma-225	203	28	×̇	×̇	VERB
ma-225	203	29	k∏	k∏	PROPN
ma-225	203	30	i=1	i=1	PROPN
ma-225	203	31	(	(	PUNCT
ma-225	203	32	zi)×̇(vi	zi)×̇(vi	NOUN
ma-225	203	33	)	)	PUNCT
ma-225	203	34			NOUN
ma-225	203	35	≤	≤	ADV
ma-225	204	1	1	1	NUM
ma-225	204	2	bk	bk	NOUN
ma-225	204	3	×̇	×̇	VERB
ma-225	204	4	k∏	k∏	PROPN
ma-225	204	5	i=1	i=1	PROPN
ma-225	204	6	(	(	PUNCT
ma-225	204	7	zi)×̇φ(vi	zi)×̇φ(vi	PROPN
ma-225	204	8	)	)	PUNCT
ma-225	204	9	≤α	≤α	PROPN
ma-225	204	10	α−1	α−1	PROPN
ma-225	204	11	(	(	PUNCT
ma-225	204	12	1	1	NUM
ma-225	204	13	bk	bk	NOUN
ma-225	204	14	)	)	PUNCT
ma-225	204	15	×̇α−1	×̇α−1	VERB
ma-225	204	16	k∏	k∏	PROPN
ma-225	204	17	i=1	i=1	PROPN
ma-225	204	18	(	(	PUNCT
ma-225	204	19	zi)×̇α−1φ(vi	zi)×̇α−1φ(vi	NOUN
ma-225	204	20	)	)	PUNCT
ma-225	205	1			PROPN
ma-225	205	2	≤e	≤e	VERB
ma-225	205	3	(	(	PUNCT
ma-225	205	4	ln	ln	ADJ
ma-225	205	5	(	(	PUNCT
ma-225	205	6	1	1	NUM
ma-225	205	7	bk	bk	NOUN
ma-225	205	8	)	)	PUNCT
ma-225	206	1	ln	ln	PROPN
ma-225	206	2	∏k	∏k	X
ma-225	206	3	i=1(zi	i=1(zi	X
ma-225	206	4	)	)	PUNCT
ma-225	206	5	×lnφ(vi	×lnφ(vi	X
ma-225	206	6	)	)	PUNCT
ma-225	206	7	)	)	PUNCT
ma-225	206	8	≤	≤	NOUN
ma-225	206	9	(	(	PUNCT
ma-225	206	10	e	e	X
ma-225	206	11	ln	ln	X
ma-225	206	12	(	(	PUNCT
ma-225	206	13	1	1	NUM
ma-225	206	14	bk	bk	NOUN
ma-225	206	15	)	)	PUNCT
ma-225	206	16	)	)	PUNCT
ma-225	207	1	ln∏k	ln∏k	NOUN
ma-225	207	2	i=1(zi	i=1(zi	NOUN
ma-225	207	3	)	)	PUNCT
ma-225	207	4	×lnφ(vi	×lnφ(vi	X
ma-225	207	5	)	)	PUNCT
ma-225	207	6	≤	≤	NUM
ma-225	207	7	(	(	PUNCT
ma-225	207	8	1	1	NUM
ma-225	207	9	bk	bk	NOUN
ma-225	207	10	)	)	PUNCT
ma-225	207	11	ln∏k	ln∏k	NOUN
ma-225	207	12	i=1(zi	i=1(zi	NOUN
ma-225	207	13	)	)	PUNCT
ma-225	207	14	×lnφ(vi	×lnφ(vi	X
ma-225	207	15	)	)	PUNCT
ma-225	208	1	φ	φ	PROPN
ma-225	208	2			PROPN
ma-225	208	3	(	(	PUNCT
ma-225	208	4	1	1	NUM
ma-225	208	5	bk	bk	NOUN
ma-225	208	6	)	)	PUNCT
ma-225	208	7	ln∏k	ln∏k	NOUN
ma-225	208	8	i=1(vi	i=1(vi	X
ma-225	208	9	)	)	PUNCT
ma-225	208	10	ln(zi	ln(zi	PROPN
ma-225	208	11	)	)	PUNCT
ma-225	208	12			NOUN
ma-225	208	13	=(	=(	NOUN
ma-225	208	14	1	1	NUM
ma-225	208	15	bk	bk	NOUN
ma-225	208	16	)	)	PUNCT
ma-225	208	17	ln∏k	ln∏k	NOUN
ma-225	208	18	i=1	i=1	ADP
ma-225	208	19	φ(vi	φ(vi	PROPN
ma-225	208	20	)	)	PUNCT
ma-225	208	21	ln(zi	ln(zi	PROPN
ma-225	208	22	)	)	PUNCT
ma-225	208	23	,	,	PUNCT
ma-225	208	24	as	as	SCONJ
ma-225	208	25	required	require	VERB
ma-225	208	26	.	.	PUNCT
ma-225	209	1	�	�	PROPN
ma-225	209	2	definition	definition	NOUN
ma-225	209	3	3.2	3.2	NUM
ma-225	209	4	.	.	PUNCT
ma-225	210	1	a	a	DET
ma-225	210	2	set	set	NOUN
ma-225	210	3	m	m	NOUN
ma-225	210	4	=	=	PUNCT
ma-225	211	1	[	[	X
ma-225	211	2	a1	a1	NOUN
ma-225	211	3	,	,	PUNCT
ma-225	211	4	b1	b1	NOUN
ma-225	211	5	]	]	PUNCT
ma-225	211	6	⊆	⊆	NUM
ma-225	211	7	r+	r+	NOUN
ma-225	211	8	is	be	AUX
ma-225	211	9	a	a	DET
ma-225	211	10	p	p	NOUN
ma-225	211	11	-	-	PUNCT
ma-225	211	12	convex	convex	NOUN
ma-225	211	13	set	set	NOUN
ma-225	211	14	,	,	PUNCT
ma-225	211	15	if	if	SCONJ
ma-225	211	16	[	[	PUNCT
ma-225	211	17	[	[	X
ma-225	211	18	(	(	PUNCT
ma-225	211	19	x1	x1	PROPN
ma-225	211	20	)	)	PUNCT
ma-225	211	21	p]ln(j	p]ln(j	NOUN
ma-225	211	22	)	)	PUNCT
ma-225	211	23	·	·	PUNCT
ma-225	212	1	[	[	X
ma-225	212	2	(	(	PUNCT
ma-225	212	3	y1)p]ln	y1)p]ln	PROPN
ma-225	212	4	(	(	PUNCT
ma-225	212	5	1	1	NUM
ma-225	212	6	j	j	PROPN
ma-225	212	7	)	)	PUNCT
ma-225	212	8	]	]	PUNCT
ma-225	213	1	1	1	NUM
ma-225	213	2	p	p	X
ma-225	213	3	∈	∈	PROPN
ma-225	213	4	m	m	PRON
ma-225	213	5	,	,	PUNCT
ma-225	213	6	(	(	PUNCT
ma-225	213	7	31	31	NUM
ma-225	213	8	)	)	PUNCT
ma-225	213	9	∀	∀	PUNCT
ma-225	214	1	x1	x1	ADJ
ma-225	214	2	,	,	PUNCT
ma-225	214	3	y1	y1	PROPN
ma-225	214	4	∈	∈	PROPN
ma-225	214	5	m	m	NOUN
ma-225	214	6	and	and	CCONJ
ma-225	214	7	j	j	PROPN
ma-225	214	8	∈	∈	PROPN
ma-225	215	1	[	[	X
ma-225	215	2	1	1	NUM
ma-225	215	3	,	,	PUNCT
ma-225	215	4	e	e	NOUN
ma-225	215	5	]	]	PUNCT
ma-225	215	6	.	.	PUNCT
ma-225	216	1	lemma	lemma	PROPN
ma-225	216	2	3.6	3.6	NUM
ma-225	216	3	.	.	PUNCT
ma-225	217	1	let	let	VERB
ma-225	217	2	x	x	PRON
ma-225	217	3	,	,	PUNCT
ma-225	217	4	y	y	PROPN
ma-225	217	5	∈	∈	PROPN
ma-225	217	6	m	m	VERB
ma-225	218	1	where	where	SCONJ
ma-225	218	2	m	m	VERB
ma-225	218	3	⊆	⊆	NUM
ma-225	218	4	r+	r+	NOUN
ma-225	218	5	and	and	CCONJ
ma-225	218	6	j	j	PROPN
ma-225	218	7	∈	∈	PROPN
ma-225	219	1	[	[	X
ma-225	219	2	1	1	NUM
ma-225	219	3	,	,	PUNCT
ma-225	219	4	e	e	NOUN
ma-225	219	5	]	]	X
ma-225	219	6	.	.	PUNCT
ma-225	220	1	for	for	ADP
ma-225	220	2	a	a	DET
ma-225	220	3	p	p	ADJ
ma-225	220	4	-	-	PUNCT
ma-225	220	5	convex	convex	NOUN
ma-225	220	6	function	function	NOUN
ma-225	220	7	φ	φ	NOUN
ma-225	220	8	,	,	PUNCT
ma-225	220	9	we	we	PRON
ma-225	220	10	have	have	VERB
ma-225	220	11	φ	φ	NOUN
ma-225	220	12	[	[	PUNCT
ma-225	220	13	[	[	X
ma-225	220	14	(	(	PUNCT
ma-225	220	15	x1	x1	PROPN
ma-225	220	16	)	)	PUNCT
ma-225	220	17	p]ln(j	p]ln(j	NOUN
ma-225	220	18	)	)	PUNCT
ma-225	220	19	·	·	PUNCT
ma-225	221	1	[	[	X
ma-225	221	2	(	(	PUNCT
ma-225	221	3	y1)p]ln	y1)p]ln	PROPN
ma-225	221	4	(	(	PUNCT
ma-225	221	5	1	1	NUM
ma-225	221	6	j	j	PROPN
ma-225	221	7	)	)	PUNCT
ma-225	221	8	]	]	PUNCT
ma-225	221	9	1	1	NUM
ma-225	221	10	p	p	NOUN
ma-225	221	11	≤	≤	PROPN
ma-225	221	12	[	[	PUNCT
ma-225	221	13	φ(x1	φ(x1	NOUN
ma-225	221	14	)	)	PUNCT
ma-225	221	15	]	]	PUNCT
ma-225	221	16	ln(j	ln(j	X
ma-225	221	17	)	)	PUNCT
ma-225	221	18	·	·	PUNCT
ma-225	222	1	[	[	X
ma-225	222	2	φ(y1)]ln	φ(y1)]ln	ADJ
ma-225	222	3	(	(	PUNCT
ma-225	222	4	1j	1j	NUM
ma-225	222	5	)	)	PUNCT
ma-225	222	6	.	.	PUNCT
ma-225	223	1	(	(	PUNCT
ma-225	223	2	32)8	32)8	NUM
ma-225	223	3	proof	proof	NOUN
ma-225	223	4	.	.	PUNCT
ma-225	224	1	using	use	VERB
ma-225	224	2	equation	equation	NOUN
ma-225	224	3	(	(	PUNCT
ma-225	224	4	3	3	NUM
ma-225	224	5	)	)	PUNCT
ma-225	224	6	,	,	PUNCT
ma-225	224	7	(	(	PUNCT
ma-225	224	8	1	1	X
ma-225	224	9	)	)	PUNCT
ma-225	224	10	and	and	CCONJ
ma-225	224	11	by	by	ADP
ma-225	224	12	p	p	NOUN
ma-225	224	13	-	-	PUNCT
ma-225	224	14	convexity	convexity	NOUN
ma-225	224	15	,	,	PUNCT
ma-225	224	16	we	we	PRON
ma-225	224	17	have	have	VERB
ma-225	224	18	φ	φ	NOUN
ma-225	224	19	[	[	PUNCT
ma-225	224	20	[	[	X
ma-225	224	21	(	(	PUNCT
ma-225	224	22	x1	x1	PROPN
ma-225	224	23	)	)	PUNCT
ma-225	224	24	p]ln(j	p]ln(j	NOUN
ma-225	224	25	)	)	PUNCT
ma-225	224	26	·	·	PUNCT
ma-225	225	1	[	[	X
ma-225	225	2	(	(	PUNCT
ma-225	225	3	y1)p]ln	y1)p]ln	PROPN
ma-225	225	4	(	(	PUNCT
ma-225	225	5	1	1	NUM
ma-225	225	6	j	j	PROPN
ma-225	225	7	)	)	PUNCT
ma-225	225	8	]	]	PUNCT
ma-225	226	1	1	1	NUM
ma-225	226	2	p	p	X
ma-225	226	3	=	=	PRON
ma-225	226	4	φ	φ	X
ma-225	226	5	[	[	PUNCT
ma-225	226	6	(	(	PUNCT
ma-225	226	7	x1	x1	PROPN
ma-225	226	8	)	)	PUNCT
ma-225	226	9	p×̇j+̇(y1)p×̇	p×̇j+̇(y1)p×̇	PROPN
ma-225	226	10	(	(	PUNCT
ma-225	226	11	1	1	NUM
ma-225	226	12	j	j	NOUN
ma-225	226	13	)	)	PUNCT
ma-225	226	14	]	]	PUNCT
ma-225	226	15	1	1	NUM
ma-225	226	16	p	p	X
ma-225	226	17	≤φ(x1)×̇j+̇φ(y1)×̇	≤φ(x1)×̇j+̇φ(y1)×̇	PROPN
ma-225	226	18	(	(	PUNCT
ma-225	226	19	1	1	NUM
ma-225	226	20	j	j	PROPN
ma-225	226	21	)	)	PUNCT
ma-225	226	22	≤	≤	NOUN
ma-225	226	23	[	[	PUNCT
ma-225	226	24	α[α−1φ(x1)×̇α−1(j)+̇α−1φ(y1)×̇α−1	α[α−1φ(x1)×̇α−1(j)+̇α−1φ(y1)×̇α−1	NOUN
ma-225	226	25	(	(	PUNCT
ma-225	226	26	1	1	NUM
ma-225	226	27	j	j	NOUN
ma-225	226	28	)	)	PUNCT
ma-225	226	29	]	]	PUNCT
ma-225	226	30	]	]	PUNCT
ma-225	226	31	≤	≤	X
ma-225	226	32	[	[	PUNCT
ma-225	226	33	α[lnφ(x1)×̇	α[lnφ(x1)×̇	NUM
ma-225	226	34	ln(j)+̇	ln(j)+̇	PROPN
ma-225	226	35	lnφ(y1)×̇	lnφ(y1)×̇	PROPN
ma-225	226	36	ln	ln	PROPN
ma-225	226	37	(	(	PUNCT
ma-225	226	38	1	1	NUM
ma-225	226	39	j	j	NOUN
ma-225	226	40	)	)	PUNCT
ma-225	226	41	]	]	PUNCT
ma-225	226	42	]	]	PUNCT
ma-225	227	1	≤e	≤e	PROPN
ma-225	227	2	[	[	PUNCT
ma-225	227	3	lnφ(x1	lnφ(x1	PROPN
ma-225	227	4	)	)	PUNCT
ma-225	227	5	ln(j)+lnφ(y1	ln(j)+lnφ(y1	PROPN
ma-225	227	6	)	)	PUNCT
ma-225	227	7	ln	ln	ADJ
ma-225	227	8	(	(	PUNCT
ma-225	227	9	1	1	NUM
ma-225	227	10	j	j	NOUN
ma-225	227	11	)	)	PUNCT
ma-225	227	12	]	]	PUNCT
ma-225	228	1	≤e	≤e	VERB
ma-225	228	2	lnφ(x1	lnφ(x1	PROPN
ma-225	228	3	)	)	PUNCT
ma-225	228	4	ln(j	ln(j	X
ma-225	228	5	)	)	PUNCT
ma-225	228	6	·	·	PUNCT
ma-225	229	1	e	e	X
ma-225	229	2	lnφ(y1	lnφ(y1	PROPN
ma-225	229	3	)	)	PUNCT
ma-225	229	4	ln	ln	ADJ
ma-225	229	5	(	(	PUNCT
ma-225	229	6	1	1	NUM
ma-225	229	7	j	j	PROPN
ma-225	229	8	)	)	PUNCT
ma-225	229	9	≤	≤	NOUN
ma-225	229	10	(	(	PUNCT
ma-225	229	11	e	e	NOUN
ma-225	229	12	lnφ(x1	lnφ(x1	PROPN
ma-225	229	13	)	)	PUNCT
ma-225	229	14	)	)	PUNCT
ma-225	229	15	ln(j	ln(j	X
ma-225	229	16	)	)	PUNCT
ma-225	229	17	·	·	PUNCT
ma-225	230	1	(	(	PUNCT
ma-225	230	2	e	e	X
ma-225	230	3	lnφ(y1	lnφ(y1	PROPN
ma-225	230	4	)	)	PUNCT
ma-225	230	5	)	)	PUNCT
ma-225	231	1	ln	ln	X
ma-225	231	2	(	(	PUNCT
ma-225	231	3	1	1	NUM
ma-225	231	4	j	j	PROPN
ma-225	231	5	)	)	PUNCT
ma-225	231	6	≤	≤	NOUN
ma-225	231	7	[	[	PUNCT
ma-225	231	8	φ(x1	φ(x1	NOUN
ma-225	231	9	)	)	PUNCT
ma-225	231	10	]	]	PUNCT
ma-225	231	11	ln(j	ln(j	X
ma-225	231	12	)	)	PUNCT
ma-225	231	13	·	·	PUNCT
ma-225	232	1	[	[	X
ma-225	232	2	φ(y1)]ln	φ(y1)]ln	ADJ
ma-225	232	3	(	(	PUNCT
ma-225	232	4	1j	1j	NUM
ma-225	232	5	)	)	PUNCT
ma-225	232	6	φ	φ	PROPN
ma-225	232	7	[	[	PUNCT
ma-225	232	8	[	[	X
ma-225	232	9	(	(	PUNCT
ma-225	232	10	x1	x1	PROPN
ma-225	232	11	)	)	PUNCT
ma-225	232	12	p]ln(j	p]ln(j	NOUN
ma-225	232	13	)	)	PUNCT
ma-225	232	14	·	·	PUNCT
ma-225	233	1	[	[	X
ma-225	233	2	(	(	PUNCT
ma-225	233	3	y1)p]ln	y1)p]ln	PROPN
ma-225	233	4	(	(	PUNCT
ma-225	233	5	1	1	NUM
ma-225	233	6	j	j	PROPN
ma-225	233	7	)	)	PUNCT
ma-225	233	8	]	]	PUNCT
ma-225	233	9	1	1	NUM
ma-225	233	10	p	p	NOUN
ma-225	233	11	≤	≤	PROPN
ma-225	233	12	[	[	PUNCT
ma-225	233	13	φ(x1	φ(x1	NOUN
ma-225	233	14	)	)	PUNCT
ma-225	233	15	]	]	PUNCT
ma-225	233	16	ln(j	ln(j	X
ma-225	233	17	)	)	PUNCT
ma-225	233	18	·	·	PUNCT
ma-225	234	1	[	[	X
ma-225	234	2	φ(y1)]ln	φ(y1)]ln	ADJ
ma-225	234	3	(	(	PUNCT
ma-225	234	4	1j	1j	NUM
ma-225	234	5	)	)	PUNCT
ma-225	234	6	,	,	PUNCT
ma-225	234	7	as	as	SCONJ
ma-225	234	8	required	require	VERB
ma-225	234	9	.	.	PUNCT
ma-225	235	1	�	�	PROPN
ma-225	235	2	remark	remark	VERB
ma-225	235	3	1	1	NUM
ma-225	235	4	.	.	PUNCT
ma-225	236	1	the	the	DET
ma-225	236	2	inequality	inequality	NOUN
ma-225	236	3	(	(	PUNCT
ma-225	236	4	24	24	NUM
ma-225	236	5	)	)	PUNCT
ma-225	236	6	is	be	AUX
ma-225	236	7	obtained	obtain	VERB
ma-225	236	8	when	when	SCONJ
ma-225	236	9	p	p	PROPN
ma-225	236	10	=	=	NOUN
ma-225	236	11	1	1	X
ma-225	236	12	.	.	PUNCT
ma-225	237	1	lemma	lemma	PROPN
ma-225	237	2	3.7	3.7	NUM
ma-225	237	3	.	.	PUNCT
ma-225	238	1	let	let	VERB
ma-225	238	2	φ	φ	PROPN
ma-225	238	3	be	be	AUX
ma-225	238	4	convex	convex	ADJ
ma-225	238	5	and	and	CCONJ
ma-225	238	6	x1	x1	NUM
ma-225	238	7	,	,	PUNCT
ma-225	238	8	y1	y1	PROPN
ma-225	238	9	∈	∈	PROPN
ma-225	238	10	r+	r+	X
ma-225	238	11	.	.	PUNCT
ma-225	239	1	for	for	ADP
ma-225	239	2	a	a	DET
ma-225	239	3	p	p	ADJ
ma-225	239	4	-	-	PUNCT
ma-225	239	5	convex	convex	NOUN
ma-225	239	6	function	function	NOUN
ma-225	239	7	,	,	PUNCT
ma-225	239	8	we	we	PRON
ma-225	239	9	have	have	VERB
ma-225	239	10	φ	φ	PROPN
ma-225	239	11	[	[	PUNCT
ma-225	239	12	(	(	PUNCT
ma-225	239	13	x1	x1	PROPN
ma-225	239	14	)	)	PUNCT
ma-225	239	15	p	p	X
ma-225	239	16	·	·	PUNCT
ma-225	239	17	(	(	PUNCT
ma-225	239	18	y1)p	y1)p	NOUN
ma-225	239	19	2	2	NUM
ma-225	239	20	]	]	SYM
ma-225	239	21	1	1	NUM
ma-225	239	22	p	p	NOUN
ma-225	239	23	≤	≤	NUM
ma-225	239	24	φ(x1	φ(x1	NOUN
ma-225	239	25	)	)	PUNCT
ma-225	239	26	·	·	PUNCT
ma-225	239	27	φ(y1	φ(y1	ADJ
ma-225	239	28	)	)	PUNCT
ma-225	239	29	2	2	NUM
ma-225	239	30	.	.	PUNCT
ma-225	240	1	(	(	PUNCT
ma-225	240	2	33	33	NUM
ma-225	240	3	)	)	PUNCT
ma-225	240	4	proof	proof	NOUN
ma-225	240	5	.	.	PUNCT
ma-225	241	1	using	use	VERB
ma-225	241	2	equation	equation	NOUN
ma-225	241	3	(	(	PUNCT
ma-225	241	4	1	1	NUM
ma-225	241	5	)	)	PUNCT
ma-225	241	6	and	and	CCONJ
ma-225	241	7	by	by	ADP
ma-225	241	8	p	p	NOUN
ma-225	241	9	-	-	PUNCT
ma-225	241	10	convexity	convexity	NOUN
ma-225	241	11	:	:	PUNCT
ma-225	241	12	φ	φ	PROPN
ma-225	241	13	[	[	PUNCT
ma-225	241	14	(	(	PUNCT
ma-225	241	15	x1	x1	PROPN
ma-225	241	16	)	)	PUNCT
ma-225	241	17	p	p	X
ma-225	241	18	·	·	PUNCT
ma-225	241	19	(	(	PUNCT
ma-225	241	20	y1)p	y1)p	NOUN
ma-225	241	21	2	2	NUM
ma-225	241	22	]	]	SYM
ma-225	241	23	1	1	NUM
ma-225	241	24	p	p	NOUN
ma-225	241	25	=	=	PRON
ma-225	241	26	φ	φ	X
ma-225	241	27	[	[	PUNCT
ma-225	241	28	1	1	NUM
ma-225	241	29	2	2	NUM
ma-225	241	30	(	(	PUNCT
ma-225	241	31	(	(	PUNCT
ma-225	241	32	x1	x1	ADJ
ma-225	241	33	)	)	PUNCT
ma-225	241	34	p+̇(y1	p+̇(y1	NOUN
ma-225	241	35	)	)	PUNCT
ma-225	241	36	p	p	NOUN
ma-225	241	37	)	)	PUNCT
ma-225	241	38	]	]	PUNCT
ma-225	241	39	1	1	NUM
ma-225	241	40	p	p	NOUN
ma-225	241	41	≤	≤	NUM
ma-225	241	42	1	1	NUM
ma-225	241	43	2	2	NUM
ma-225	241	44	[	[	PUNCT
ma-225	241	45	φ(x1)+̇φ(y1	φ(x1)+̇φ(y1	PROPN
ma-225	241	46	)	)	PUNCT
ma-225	241	47	]	]	PUNCT
ma-225	242	1	≤	≤	NUM
ma-225	242	2	1	1	NUM
ma-225	242	3	2	2	NUM
ma-225	242	4	[	[	PUNCT
ma-225	242	5	α	α	X
ma-225	242	6	(	(	PUNCT
ma-225	242	7	α−1φ(x1)+̇α	α−1φ(x1)+̇α	PROPN
ma-225	242	8	−1φ(y1	−1φ(y1	NOUN
ma-225	242	9	)	)	PUNCT
ma-225	242	10	)	)	PUNCT
ma-225	242	11	]	]	PUNCT
ma-225	243	1	≤	≤	NUM
ma-225	243	2	1	1	NUM
ma-225	243	3	2	2	NUM
ma-225	243	4	[	[	PUNCT
ma-225	243	5	α	α	X
ma-225	243	6	(	(	PUNCT
ma-225	243	7	lnφ(x1	lnφ(x1	PROPN
ma-225	243	8	)	)	PUNCT
ma-225	243	9	+	+	SYM
ma-225	243	10	lnφ(y1	lnφ(y1	NOUN
ma-225	243	11	)	)	PUNCT
ma-225	243	12	)	)	PUNCT
ma-225	243	13	]	]	PUNCT
ma-225	244	1	≤	≤	NUM
ma-225	244	2	1	1	NUM
ma-225	244	3	2	2	NUM
ma-225	244	4	[	[	PUNCT
ma-225	244	5	e(lnφ(x1)+lnφ(y1	e(lnφ(x1)+lnφ(y1	ADJ
ma-225	244	6	)	)	PUNCT
ma-225	244	7	)	)	PUNCT
ma-225	244	8	]	]	PUNCT
ma-225	245	1	≤	≤	NUM
ma-225	245	2	1	1	NUM
ma-225	245	3	2	2	NUM
ma-225	245	4	[	[	PUNCT
ma-225	245	5	e	e	X
ma-225	245	6	lnφ(x1	lnφ(x1	PROPN
ma-225	245	7	)	)	PUNCT
ma-225	245	8	·	·	PUNCT
ma-225	246	1	e	e	X
ma-225	246	2	lnφ(y1	lnφ(y1	PROPN
ma-225	246	3	)	)	PUNCT
ma-225	246	4	]	]	PUNCT
ma-225	247	1	≤	≤	NUM
ma-225	247	2	1	1	NUM
ma-225	247	3	2	2	NUM
ma-225	247	4	[	[	PUNCT
ma-225	247	5	φ(x1	φ(x1	NOUN
ma-225	247	6	)	)	PUNCT
ma-225	247	7	·	·	PUNCT
ma-225	247	8	φ(y1	φ(y1	NUM
ma-225	247	9	)	)	PUNCT
ma-225	247	10	]	]	PUNCT
ma-225	248	1	φ	φ	PROPN
ma-225	248	2	[	[	PUNCT
ma-225	248	3	(	(	PUNCT
ma-225	248	4	x1	x1	PROPN
ma-225	248	5	)	)	PUNCT
ma-225	248	6	p	p	X
ma-225	248	7	·	·	PUNCT
ma-225	248	8	(	(	PUNCT
ma-225	248	9	y1)p	y1)p	NOUN
ma-225	248	10	2	2	NUM
ma-225	248	11	]	]	SYM
ma-225	248	12	1	1	NUM
ma-225	248	13	p	p	NOUN
ma-225	248	14	≤	≤	NUM
ma-225	248	15	φ(x1	φ(x1	NOUN
ma-225	248	16	)	)	PUNCT
ma-225	248	17	·	·	PUNCT
ma-225	248	18	φ(y1	φ(y1	CCONJ
ma-225	248	19	)	)	PUNCT
ma-225	248	20	2	2	NUM
ma-225	248	21	,	,	PUNCT
ma-225	248	22	9	9	NUM
ma-225	248	23	as	as	SCONJ
ma-225	248	24	required	require	VERB
ma-225	248	25	.	.	PUNCT
ma-225	249	1	�	�	PROPN
ma-225	249	2	remark	remark	VERB
ma-225	249	3	2	2	NUM
ma-225	249	4	.	.	PUNCT
ma-225	250	1	the	the	DET
ma-225	250	2	inequality	inequality	NOUN
ma-225	250	3	(	(	PUNCT
ma-225	250	4	26	26	NUM
ma-225	250	5	)	)	PUNCT
ma-225	250	6	is	be	AUX
ma-225	250	7	obtained	obtain	VERB
ma-225	250	8	when	when	SCONJ
ma-225	250	9	p	p	PROPN
ma-225	250	10	=	=	NOUN
ma-225	250	11	1	1	X
ma-225	250	12	.	.	PUNCT
ma-225	250	13	definition	definition	NOUN
ma-225	250	14	3.3	3.3	NUM
ma-225	250	15	.	.	PUNCT
ma-225	251	1	a	a	DET
ma-225	251	2	harmonic	harmonic	ADJ
ma-225	251	3	convex	convex	NOUN
ma-225	251	4	set	set	NOUN
ma-225	251	5	of	of	ADP
ma-225	251	6	an	an	DET
ma-225	251	7	interval	interval	NOUN
ma-225	251	8	m	m	VERB
ma-225	251	9	is	be	AUX
ma-225	251	10	described	describe	VERB
ma-225	251	11	,	,	PUNCT
ma-225	251	12	if	if	SCONJ
ma-225	251	13	x	x	X
ma-225	251	14	ln(y	ln(y	PUNCT
ma-225	251	15	)	)	PUNCT
ma-225	251	16	x	x	PUNCT
ma-225	251	17	ln(j	ln(j	X
ma-225	251	18	)	)	PUNCT
ma-225	251	19	·	·	PUNCT
ma-225	252	1	y	y	X
ma-225	252	2	ln	ln	ADJ
ma-225	252	3	(	(	PUNCT
ma-225	252	4	1	1	NUM
ma-225	252	5	j	j	PROPN
ma-225	252	6	)	)	PUNCT
ma-225	252	7	∈	∈	PROPN
ma-225	252	8	m	m	PROPN
ma-225	252	9	,	,	PUNCT
ma-225	252	10	(	(	PUNCT
ma-225	252	11	34	34	NUM
ma-225	252	12	)	)	PUNCT
ma-225	252	13	∀	∀	X
ma-225	253	1	x	x	NOUN
ma-225	253	2	,	,	PUNCT
ma-225	253	3	y	y	PROPN
ma-225	253	4	∈	∈	PROPN
ma-225	253	5	m	m	VERB
ma-225	253	6	and	and	CCONJ
ma-225	253	7	j	j	PROPN
ma-225	253	8	∈	∈	PROPN
ma-225	254	1	[	[	X
ma-225	254	2	1	1	NUM
ma-225	254	3	,	,	PUNCT
ma-225	254	4	e	e	NOUN
ma-225	254	5	]	]	PUNCT
ma-225	254	6	.	.	PUNCT
ma-225	255	1	lemma	lemma	PROPN
ma-225	255	2	3.8	3.8	NUM
ma-225	255	3	.	.	PUNCT
ma-225	256	1	let	let	VERB
ma-225	256	2	the	the	DET
ma-225	256	3	set	set	NOUN
ma-225	256	4	w	w	NOUN
ma-225	256	5	⊆	⊆	NUM
ma-225	256	6	r+	r+	NOUN
ma-225	256	7	be	be	AUX
ma-225	256	8	a	a	DET
ma-225	256	9	harmonic	harmonic	ADJ
ma-225	256	10	set	set	NOUN
ma-225	256	11	.	.	PUNCT
ma-225	257	1	for	for	ADP
ma-225	257	2	harmonic	harmonic	ADJ
ma-225	257	3	convex	convex	NOUN
ma-225	257	4	function	function	NOUN
ma-225	257	5	φ	φ	PROPN
ma-225	257	6	,	,	PUNCT
ma-225	257	7	we	we	PRON
ma-225	257	8	have	have	VERB
ma-225	257	9	φ	φ	VERB
ma-225	257	10			NUM
ma-225	257	11	x	x	SYM
ma-225	257	12	ln(y1	ln(y1	X
ma-225	257	13	)	)	PUNCT
ma-225	257	14	1	1	NUM
ma-225	257	15	x	x	SYM
ma-225	257	16	ln(q1	ln(q1	NOUN
ma-225	257	17	)	)	PUNCT
ma-225	257	18	·	·	PUNCT
ma-225	258	1	y	y	X
ma-225	258	2	ln	ln	ADJ
ma-225	258	3	(	(	PUNCT
ma-225	258	4	1	1	NUM
ma-225	258	5	q1	q1	NOUN
ma-225	258	6	)	)	PUNCT
ma-225	258	7	1	1	NUM
ma-225	258	8			NUM
ma-225	258	9	≤	≤	NUM
ma-225	258	10	φ(x1	φ(x1	NOUN
ma-225	258	11	)	)	PUNCT
ma-225	258	12	lnφ(y1	lnφ(y1	PROPN
ma-225	258	13	)	)	PUNCT
ma-225	258	14	φ(x1)ln(q1	φ(x1)ln(q1	NOUN
ma-225	258	15	)	)	PUNCT
ma-225	258	16	·	·	PUNCT
ma-225	258	17	φ(y1	φ(y1	NOUN
ma-225	258	18	)	)	PUNCT
ma-225	258	19	ln	ln	ADJ
ma-225	258	20	(	(	PUNCT
ma-225	258	21	1	1	NUM
ma-225	258	22	q1	q1	NOUN
ma-225	258	23	)	)	PUNCT
ma-225	258	24	,	,	PUNCT
ma-225	258	25	(	(	PUNCT
ma-225	258	26	35	35	NUM
ma-225	258	27	)	)	PUNCT
ma-225	258	28	∀	∀	PUNCT
ma-225	259	1	x1	x1	ADJ
ma-225	259	2	,	,	PUNCT
ma-225	259	3	y1	y1	PROPN
ma-225	259	4	∈	∈	PROPN
ma-225	259	5	w	w	PROPN
ma-225	259	6	and	and	CCONJ
ma-225	259	7	q1	q1	PROPN
ma-225	259	8	∈	∈	PROPN
ma-225	259	9	[	[	X
ma-225	259	10	1	1	NUM
ma-225	259	11	,	,	PUNCT
ma-225	259	12	e	e	NOUN
ma-225	259	13	]	]	PUNCT
ma-225	259	14	.	.	PUNCT
ma-225	260	1	proof	proof	NOUN
ma-225	260	2	.	.	PUNCT
ma-225	261	1	using	use	VERB
ma-225	261	2	equation	equation	NOUN
ma-225	261	3	(	(	PUNCT
ma-225	261	4	3	3	NUM
ma-225	261	5	)	)	PUNCT
ma-225	261	6	,	,	PUNCT
ma-225	261	7	(	(	PUNCT
ma-225	261	8	1	1	X
ma-225	261	9	)	)	PUNCT
ma-225	261	10	and	and	CCONJ
ma-225	261	11	by	by	ADP
ma-225	261	12	convexity	convexity	NOUN
ma-225	261	13	,	,	PUNCT
ma-225	261	14	we	we	PRON
ma-225	261	15	have	have	VERB
ma-225	261	16	φ	φ	VERB
ma-225	261	17			NUM
ma-225	261	18	x	x	SYM
ma-225	261	19	ln(y1	ln(y1	X
ma-225	261	20	)	)	PUNCT
ma-225	261	21	1	1	NUM
ma-225	261	22	x	x	SYM
ma-225	261	23	ln(q1	ln(q1	NOUN
ma-225	261	24	)	)	PUNCT
ma-225	261	25	1	1	NUM
ma-225	261	26	·	·	PUNCT
ma-225	261	27	y	y	X
ma-225	261	28	ln	ln	ADJ
ma-225	261	29	(	(	PUNCT
ma-225	261	30	1	1	NUM
ma-225	261	31	q1	q1	NOUN
ma-225	261	32	)	)	PUNCT
ma-225	261	33	1	1	NUM
ma-225	261	34			NUM
ma-225	261	35	=	=	SYM
ma-225	261	36	φ	φ	NOUN
ma-225	262	1	[	[	X
ma-225	262	2	x1×̇y1−̇(q1×̇x1+̇	x1×̇y1−̇(q1×̇x1+̇	PROPN
ma-225	262	3	(	(	PUNCT
ma-225	262	4	1	1	NUM
ma-225	262	5	q1	q1	NOUN
ma-225	262	6	)	)	PUNCT
ma-225	262	7	×̇y1	×̇y1	X
ma-225	262	8	)	)	PUNCT
ma-225	262	9	]	]	PUNCT
ma-225	263	1	≤φ(x1)×̇φ(y1)−̇(q1×̇φ(x1)+̇	≤φ(x1)×̇φ(y1)−̇(q1×̇φ(x1)+̇	NOUN
ma-225	263	2	(	(	PUNCT
ma-225	263	3	1	1	NUM
ma-225	263	4	q1	q1	NOUN
ma-225	263	5	)	)	PUNCT
ma-225	263	6	×̇φ(y1	×̇φ(y1	NOUN
ma-225	263	7	)	)	PUNCT
ma-225	263	8	)	)	PUNCT
ma-225	264	1	≤α	≤α	NOUN
ma-225	264	2	[	[	PUNCT
ma-225	264	3	α−1φ(x1)×̇α−1φ(y1)−̇(α−1(q1)×̇α−1φ(x1)+̇α−1	α−1φ(x1)×̇α−1φ(y1)−̇(α−1(q1)×̇α−1φ(x1)+̇α−1	X
ma-225	264	4	(	(	PUNCT
ma-225	264	5	1	1	NUM
ma-225	264	6	q1	q1	PROPN
ma-225	264	7	)	)	PUNCT
ma-225	264	8	×̇α−1φ(y1	×̇α−1φ(y1	NOUN
ma-225	264	9	)	)	PUNCT
ma-225	264	10	)	)	PUNCT
ma-225	264	11	]	]	PUNCT
ma-225	265	1	≤α	≤α	NOUN
ma-225	265	2	[	[	PUNCT
ma-225	265	3	lnφ(x1)×	lnφ(x1)×	PROPN
ma-225	265	4	lnφ(y1)−	lnφ(y1)−	PROPN
ma-225	265	5	(	(	PUNCT
ma-225	265	6	ln(q1)×	ln(q1)×	ADP
ma-225	265	7	lnφ(x1	lnφ(x1	PROPN
ma-225	265	8	)	)	PUNCT
ma-225	265	9	+	+	X
ma-225	265	10	ln	ln	ADJ
ma-225	265	11	(	(	PUNCT
ma-225	265	12	1	1	NUM
ma-225	265	13	q1	q1	NOUN
ma-225	265	14	)	)	PUNCT
ma-225	265	15	×	×	PROPN
ma-225	265	16	lnφ(y1	lnφ(y1	NOUN
ma-225	265	17	)	)	PUNCT
ma-225	265	18	)	)	PUNCT
ma-225	265	19	]	]	PUNCT
ma-225	266	1	≤e	≤e	VERB
ma-225	266	2	[	[	PUNCT
ma-225	266	3	lnφ(x1)×lnφ(y1)−(ln(q1)×lnφ(x1)+ln	lnφ(x1)×lnφ(y1)−(ln(q1)×lnφ(x1)+ln	PROPN
ma-225	266	4	(	(	PUNCT
ma-225	266	5	1q1	1q1	NUM
ma-225	266	6	)	)	PUNCT
ma-225	266	7	×lnφ(y1	×lnφ(y1	PROPN
ma-225	266	8	)	)	PUNCT
ma-225	266	9	)	)	PUNCT
ma-225	266	10	]	]	PUNCT
ma-225	267	1	≤e	≤e	PROPN
ma-225	267	2	[	[	PUNCT
ma-225	267	3	lnφ(x1)×lnφ(y1)−(lnφ(x1)×ln(q1)+lnφ(y1)×ln	lnφ(x1)×lnφ(y1)−(lnφ(x1)×ln(q1)+lnφ(y1)×ln	PROPN
ma-225	267	4	(	(	PUNCT
ma-225	267	5	1q1	1q1	NUM
ma-225	267	6	)	)	PUNCT
ma-225	267	7	)	)	PUNCT
ma-225	267	8	]	]	PUNCT
ma-225	268	1	≤	≤	NUM
ma-225	268	2	(	(	PUNCT
ma-225	268	3	e	e	NOUN
ma-225	268	4	lnφ(x1))lnφ(y1	lnφ(x1))lnφ(y1	X
ma-225	268	5	)	)	PUNCT
ma-225	268	6	(	(	PUNCT
ma-225	268	7	e	e	PROPN
ma-225	268	8	lnφ(x1))ln(q1	lnφ(x1))ln(q1	PROPN
ma-225	268	9	)	)	PUNCT
ma-225	268	10	·	·	PUNCT
ma-225	269	1	(	(	PUNCT
ma-225	269	2	e	e	X
ma-225	269	3	lnφ(y1))ln	lnφ(y1))ln	PROPN
ma-225	269	4	(	(	PUNCT
ma-225	269	5	1	1	NUM
ma-225	269	6	q1	q1	PROPN
ma-225	269	7	)	)	PUNCT
ma-225	269	8	φ	φ	PROPN
ma-225	269	9			PROPN
ma-225	269	10	x	x	SYM
ma-225	269	11	ln(y1	ln(y1	X
ma-225	269	12	)	)	PUNCT
ma-225	269	13	1	1	NUM
ma-225	269	14	x	x	SYM
ma-225	269	15	ln(q1	ln(q1	NOUN
ma-225	269	16	)	)	PUNCT
ma-225	269	17	1	1	NUM
ma-225	269	18	·	·	PUNCT
ma-225	269	19	y	y	X
ma-225	269	20	ln	ln	ADJ
ma-225	269	21	(	(	PUNCT
ma-225	269	22	1	1	NUM
ma-225	269	23	q1	q1	NOUN
ma-225	269	24	)	)	PUNCT
ma-225	269	25	1	1	NUM
ma-225	269	26			NUM
ma-225	269	27	≤	≤	NUM
ma-225	269	28	φ(x1	φ(x1	NOUN
ma-225	269	29	)	)	PUNCT
ma-225	269	30	lnφ(y1	lnφ(y1	PROPN
ma-225	269	31	)	)	PUNCT
ma-225	269	32	φx	φx	PROPN
ma-225	269	33	ln(q1	ln(q1	PROPN
ma-225	269	34	)	)	PUNCT
ma-225	269	35	1	1	NUM
ma-225	269	36	·	·	PUNCT
ma-225	269	37	φ(y1	φ(y1	NOUN
ma-225	269	38	)	)	PUNCT
ma-225	269	39	ln	ln	ADJ
ma-225	269	40	(	(	PUNCT
ma-225	269	41	1	1	NUM
ma-225	269	42	q1	q1	NOUN
ma-225	269	43	)	)	PUNCT
ma-225	269	44	,	,	PUNCT
ma-225	269	45	proved	prove	VERB
ma-225	269	46	.	.	PUNCT
ma-225	270	1	�	�	PROPN
ma-225	270	2	definition	definition	NOUN
ma-225	270	3	3.4	3.4	NUM
ma-225	270	4	.	.	PUNCT
ma-225	271	1	let	let	VERB
ma-225	271	2	w	w	NOUN
ma-225	271	3	be	be	AUX
ma-225	271	4	a	a	DET
ma-225	271	5	subset	subset	NOUN
ma-225	271	6	on	on	ADP
ma-225	271	7	r+	r+	X
ma-225	271	8	,	,	PUNCT
ma-225	271	9	then	then	ADV
ma-225	271	10	w	w	PROPN
ma-225	271	11	is	be	AUX
ma-225	271	12	p	p	ADJ
ma-225	271	13	-	-	PUNCT
ma-225	271	14	harmonic	harmonic	ADJ
ma-225	271	15	convex	convex	NOUN
ma-225	271	16	set	set	VERB
ma-225	271	17	if	if	NOUN
ma-225	271	18	[	[	X
ma-225	271	19	xp]ln(y	xp]ln(y	X
ma-225	271	20	)	)	PUNCT
ma-225	271	21	p	p	X
ma-225	272	1	[	[	X
ma-225	272	2	xp]ln(j	xp]ln(j	X
ma-225	272	3	)	)	PUNCT
ma-225	272	4	·	·	PUNCT
ma-225	273	1	[	[	X
ma-225	273	2	yp]ln	yp]ln	NOUN
ma-225	273	3	(	(	PUNCT
ma-225	273	4	1	1	NUM
ma-225	273	5	j	j	NOUN
ma-225	273	6	)	)	PUNCT
ma-225	273	7			VERB
ma-225	273	8	1p	1p	NUM
ma-225	273	9	∈	∈	PROPN
ma-225	273	10	w	w	PROPN
ma-225	273	11	,	,	PUNCT
ma-225	273	12	(	(	PUNCT
ma-225	273	13	36	36	NUM
ma-225	273	14	)	)	PUNCT
ma-225	273	15	∀	∀	X
ma-225	274	1	x	x	NOUN
ma-225	274	2	,	,	PUNCT
ma-225	274	3	y	y	PROPN
ma-225	274	4	∈	∈	PROPN
ma-225	274	5	w	w	PROPN
ma-225	274	6	and	and	CCONJ
ma-225	274	7	j	j	PROPN
ma-225	274	8	∈	∈	PROPN
ma-225	275	1	[	[	X
ma-225	275	2	1	1	NUM
ma-225	275	3	,	,	PUNCT
ma-225	275	4	e	e	NOUN
ma-225	275	5	]	]	X
ma-225	275	6	.	.	PUNCT
ma-225	276	1	10	10	NUM
ma-225	276	2	lemma	lemma	PROPN
ma-225	276	3	3.9	3.9	NUM
ma-225	276	4	.	.	PUNCT
ma-225	276	5	consider	consider	VERB
ma-225	276	6	the	the	DET
ma-225	276	7	p	p	NOUN
ma-225	276	8	-	-	PUNCT
ma-225	276	9	harmonic	harmonic	ADJ
ma-225	276	10	convex	convex	NOUN
ma-225	276	11	set	set	VERB
ma-225	276	12	m	m	VERB
ma-225	276	13	=	=	PUNCT
ma-225	277	1	[	[	X
ma-225	277	2	a	a	X
ma-225	277	3	,	,	PUNCT
ma-225	277	4	b	b	NOUN
ma-225	277	5	]	]	X
ma-225	277	6	⊆	⊆	NUM
ma-225	277	7	r+	r+	NOUN
ma-225	277	8	.	.	PUNCT
ma-225	278	1	if	if	SCONJ
ma-225	278	2	φ	φ	PROPN
ma-225	278	3	is	be	AUX
ma-225	278	4	p	p	ADJ
ma-225	278	5	-	-	PUNCT
ma-225	278	6	harmonic	harmonic	ADJ
ma-225	278	7	convex	convex	NOUN
ma-225	278	8	function	function	NOUN
ma-225	278	9	,	,	PUNCT
ma-225	278	10	we	we	PRON
ma-225	278	11	have	have	VERB
ma-225	278	12	φ	φ	NOUN
ma-225	278	13			PROPN
ma-225	278	14	[	[	X
ma-225	278	15	(	(	PUNCT
ma-225	278	16	x1	x1	ADJ
ma-225	278	17	)	)	PUNCT
ma-225	278	18	p]ln(y1	p]ln(y1	PROPN
ma-225	278	19	)	)	PUNCT
ma-225	278	20	p	p	X
ma-225	279	1	[	[	X
ma-225	279	2	(	(	PUNCT
ma-225	279	3	x1)p]ln(j1	x1)p]ln(j1	X
ma-225	279	4	)	)	PUNCT
ma-225	279	5	·	·	PUNCT
ma-225	280	1	[	[	X
ma-225	280	2	(	(	PUNCT
ma-225	280	3	y1)p	y1)p	NOUN
ma-225	280	4	]	]	X
ma-225	280	5	ln	ln	ADJ
ma-225	280	6	(	(	PUNCT
ma-225	280	7	1	1	NUM
ma-225	280	8	j1	j1	NOUN
ma-225	280	9	)	)	PUNCT
ma-225	280	10			NOUN
ma-225	280	11	1p	1p	VERB
ma-225	280	12	≤	≤	NOUN
ma-225	280	13	[	[	X
ma-225	280	14	φ(x1	φ(x1	NOUN
ma-225	280	15	)	)	PUNCT
ma-225	280	16	]	]	PUNCT
ma-225	281	1	lnφ(y1	lnφ(y1	NOUN
ma-225	281	2	)	)	PUNCT
ma-225	281	3	[	[	X
ma-225	281	4	φ(x1)]ln(j	φ(x1)]ln(j	X
ma-225	281	5	)	)	PUNCT
ma-225	281	6	·	·	PUNCT
ma-225	282	1	[	[	X
ma-225	282	2	φ(y1	φ(y1	NOUN
ma-225	282	3	)	)	PUNCT
ma-225	282	4	]	]	PUNCT
ma-225	283	1	ln	ln	X
ma-225	283	2	(	(	PUNCT
ma-225	283	3	1	1	NUM
ma-225	283	4	j1	j1	PROPN
ma-225	283	5	)	)	PUNCT
ma-225	283	6	,	,	PUNCT
ma-225	283	7	(	(	PUNCT
ma-225	283	8	37	37	NUM
ma-225	283	9	)	)	PUNCT
ma-225	283	10	∀	∀	PUNCT
ma-225	284	1	x1	x1	ADJ
ma-225	284	2	,	,	PUNCT
ma-225	284	3	y1	y1	PROPN
ma-225	284	4	∈	∈	PROPN
ma-225	284	5	m	m	NOUN
ma-225	284	6	and	and	CCONJ
ma-225	284	7	j1	j1	PROPN
ma-225	284	8	∈	∈	PROPN
ma-225	285	1	[	[	X
ma-225	285	2	1	1	NUM
ma-225	285	3	,	,	PUNCT
ma-225	285	4	e	e	NOUN
ma-225	285	5	]	]	PUNCT
ma-225	285	6	.	.	PUNCT
ma-225	286	1	proof	proof	NOUN
ma-225	286	2	.	.	PUNCT
ma-225	287	1	using	use	VERB
ma-225	287	2	equation	equation	NOUN
ma-225	287	3	(	(	PUNCT
ma-225	287	4	3	3	NUM
ma-225	287	5	)	)	PUNCT
ma-225	287	6	,	,	PUNCT
ma-225	287	7	(	(	PUNCT
ma-225	287	8	2	2	X
ma-225	287	9	)	)	PUNCT
ma-225	287	10	and	and	CCONJ
ma-225	287	11	by	by	ADP
ma-225	287	12	p	p	NOUN
ma-225	287	13	-	-	PUNCT
ma-225	287	14	convexity	convexity	NOUN
ma-225	287	15	,	,	PUNCT
ma-225	287	16	we	we	PRON
ma-225	287	17	have	have	VERB
ma-225	287	18	φ	φ	NOUN
ma-225	287	19			PROPN
ma-225	287	20	[	[	X
ma-225	287	21	(	(	PUNCT
ma-225	287	22	x1	x1	ADJ
ma-225	287	23	)	)	PUNCT
ma-225	287	24	p]ln(y1	p]ln(y1	PROPN
ma-225	287	25	)	)	PUNCT
ma-225	287	26	p	p	X
ma-225	288	1	[	[	X
ma-225	288	2	(	(	PUNCT
ma-225	288	3	x1)p]ln(j1	x1)p]ln(j1	X
ma-225	288	4	)	)	PUNCT
ma-225	288	5	·	·	PUNCT
ma-225	289	1	[	[	X
ma-225	289	2	(	(	PUNCT
ma-225	289	3	y1)p	y1)p	NOUN
ma-225	289	4	]	]	X
ma-225	289	5	ln	ln	ADJ
ma-225	289	6	(	(	PUNCT
ma-225	289	7	1	1	NUM
ma-225	289	8	j1	j1	NOUN
ma-225	289	9	)	)	PUNCT
ma-225	289	10			NOUN
ma-225	289	11	1p	1p	NUM
ma-225	289	12	=	=	SYM
ma-225	289	13	φ	φ	NOUN
ma-225	289	14	[	[	X
ma-225	289	15	(	(	PUNCT
ma-225	289	16	x1)p×̇(y1)p−̇(j1×̇(x1)p+̇	x1)p×̇(y1)p−̇(j1×̇(x1)p+̇	ADJ
ma-225	289	17	(	(	PUNCT
ma-225	289	18	1	1	NUM
ma-225	289	19	j1	j1	NOUN
ma-225	289	20	)	)	PUNCT
ma-225	289	21	×̇(y1)p	×̇(y1)p	NOUN
ma-225	289	22	)	)	PUNCT
ma-225	289	23	]	]	PUNCT
ma-225	290	1	1	1	NUM
ma-225	290	2	p	p	PRON
ma-225	290	3	≤φ(x1)×̇φ(y1)−̇(j1×̇φ(x1)+̇	≤φ(x1)×̇φ(y1)−̇(j1×̇φ(x1)+̇	ADJ
ma-225	290	4	(	(	PUNCT
ma-225	290	5	1	1	NUM
ma-225	290	6	j1	j1	PROPN
ma-225	290	7	)	)	PUNCT
ma-225	290	8	×̇φ(y1	×̇φ(y1	NOUN
ma-225	290	9	)	)	PUNCT
ma-225	290	10	)	)	PUNCT
ma-225	291	1	≤α	≤α	NOUN
ma-225	291	2	[	[	PUNCT
ma-225	291	3	lnφ(x1)×	lnφ(x1)×	PROPN
ma-225	291	4	lnφ(y1)−	lnφ(y1)−	PROPN
ma-225	291	5	(	(	PUNCT
ma-225	291	6	ln(j1)×	ln(j1)×	PROPN
ma-225	291	7	lnφ(x1	lnφ(x1	PROPN
ma-225	291	8	)	)	PUNCT
ma-225	292	1	+	+	X
ma-225	292	2	ln	ln	ADJ
ma-225	292	3	(	(	PUNCT
ma-225	292	4	1	1	NUM
ma-225	292	5	j1	j1	PROPN
ma-225	292	6	)	)	PUNCT
ma-225	292	7	×	×	PROPN
ma-225	292	8	lnφ(y1	lnφ(y1	PROPN
ma-225	292	9	)	)	PUNCT
ma-225	292	10	]	]	PUNCT
ma-225	293	1	≤e	≤e	NOUN
ma-225	293	2	[	[	PUNCT
ma-225	293	3	lnφ(x1)×lnφ(y1)−(ln(j1)×lnφ(x1)+ln	lnφ(x1)×lnφ(y1)−(ln(j1)×lnφ(x1)+ln	PROPN
ma-225	293	4	(	(	PUNCT
ma-225	293	5	1j1	1j1	NUM
ma-225	293	6	)	)	PUNCT
ma-225	293	7	×lnφ(y1	×lnφ(y1	PROPN
ma-225	293	8	)	)	PUNCT
ma-225	293	9	]	]	PUNCT
ma-225	294	1	≤e	≤e	NOUN
ma-225	294	2	[	[	PUNCT
ma-225	294	3	lnφ(x1)×lnφ(y1)−(lnφ(x1)×ln(j1)+lnφ(y1)×ln	lnφ(x1)×lnφ(y1)−(lnφ(x1)×ln(j1)+lnφ(y1)×ln	PROPN
ma-225	294	4	(	(	PUNCT
ma-225	294	5	1j1	1j1	NUM
ma-225	294	6	)	)	PUNCT
ma-225	294	7	]	]	PUNCT
ma-225	294	8	≤	≤	NUM
ma-225	294	9	e[lnφ(x1	e[lnφ(x1	PROPN
ma-225	294	10	)	)	PUNCT
ma-225	294	11	lnφ(y1	lnφ(y1	PROPN
ma-225	294	12	)	)	PUNCT
ma-225	294	13	]	]	X
ma-225	295	1	e[lnφ(x1	e[lnφ(x1	ADJ
ma-225	295	2	)	)	PUNCT
ma-225	295	3	ln(j1	ln(j1	X
ma-225	295	4	)	)	PUNCT
ma-225	295	5	]	]	PUNCT
ma-225	295	6	·	·	PUNCT
ma-225	295	7	e[lnφ(y1	e[lnφ(y1	NUM
ma-225	295	8	)	)	PUNCT
ma-225	295	9	ln	ln	ADJ
ma-225	295	10	(	(	PUNCT
ma-225	295	11	1	1	NUM
ma-225	295	12	j1	j1	NOUN
ma-225	295	13	)	)	PUNCT
ma-225	295	14	]	]	PUNCT
ma-225	296	1	≤	≤	NUM
ma-225	296	2	(	(	PUNCT
ma-225	296	3	φ(x1	φ(x1	NOUN
ma-225	296	4	)	)	PUNCT
ma-225	296	5	)	)	PUNCT
ma-225	297	1	lnφ(y1	lnφ(y1	NOUN
ma-225	297	2	)	)	PUNCT
ma-225	297	3	(	(	PUNCT
ma-225	297	4	φ(x1))ln(j1	φ(x1))ln(j1	X
ma-225	297	5	)	)	PUNCT
ma-225	297	6	·	·	PUNCT
ma-225	297	7	(	(	PUNCT
ma-225	297	8	φ(y1	φ(y1	NOUN
ma-225	297	9	)	)	PUNCT
ma-225	297	10	)	)	PUNCT
ma-225	298	1	ln	ln	ADJ
ma-225	298	2	(	(	PUNCT
ma-225	298	3	1	1	NUM
ma-225	298	4	j1	j1	PROPN
ma-225	298	5	)	)	PUNCT
ma-225	298	6	φ	φ	PROPN
ma-225	298	7			PROPN
ma-225	298	8	[	[	X
ma-225	298	9	(	(	PUNCT
ma-225	298	10	x1	x1	ADJ
ma-225	298	11	)	)	PUNCT
ma-225	298	12	p]ln(y1	p]ln(y1	PROPN
ma-225	298	13	)	)	PUNCT
ma-225	298	14	p	p	X
ma-225	298	15	[	[	X
ma-225	298	16	(	(	PUNCT
ma-225	298	17	x1)p]ln(j1	x1)p]ln(j1	X
ma-225	298	18	)	)	PUNCT
ma-225	298	19	·	·	PUNCT
ma-225	299	1	[	[	X
ma-225	299	2	(	(	PUNCT
ma-225	299	3	y1)p	y1)p	NOUN
ma-225	299	4	]	]	X
ma-225	299	5	ln	ln	ADJ
ma-225	299	6	(	(	PUNCT
ma-225	299	7	1	1	NUM
ma-225	299	8	j1	j1	NOUN
ma-225	299	9	)	)	PUNCT
ma-225	299	10			NOUN
ma-225	299	11	1p	1p	VERB
ma-225	299	12	≤	≤	NOUN
ma-225	299	13	[	[	X
ma-225	299	14	φ(x1	φ(x1	NOUN
ma-225	299	15	)	)	PUNCT
ma-225	299	16	]	]	PUNCT
ma-225	300	1	lnφ(y1	lnφ(y1	NOUN
ma-225	300	2	)	)	PUNCT
ma-225	300	3	[	[	X
ma-225	300	4	φ(x1)]ln(j1	φ(x1)]ln(j1	NOUN
ma-225	300	5	)	)	PUNCT
ma-225	300	6	·	·	PUNCT
ma-225	301	1	[	[	X
ma-225	301	2	φ(y1	φ(y1	NOUN
ma-225	301	3	)	)	PUNCT
ma-225	301	4	]	]	PUNCT
ma-225	302	1	ln	ln	X
ma-225	302	2	(	(	PUNCT
ma-225	302	3	1	1	NUM
ma-225	302	4	j1	j1	PROPN
ma-225	302	5	)	)	PUNCT
ma-225	302	6	,	,	PUNCT
ma-225	302	7	as	as	SCONJ
ma-225	302	8	required	require	VERB
ma-225	302	9	.	.	PUNCT
ma-225	303	1	�	�	PROPN
ma-225	303	2	remark	remark	VERB
ma-225	303	3	3	3	NUM
ma-225	303	4	.	.	PUNCT
ma-225	304	1	when	when	SCONJ
ma-225	304	2	p	p	NOUN
ma-225	304	3	=	=	NOUN
ma-225	304	4	1	1	NUM
ma-225	304	5	,	,	PUNCT
ma-225	304	6	the	the	DET
ma-225	304	7	inequality	inequality	NOUN
ma-225	304	8	(	(	PUNCT
ma-225	304	9	35	35	NUM
ma-225	304	10	)	)	PUNCT
ma-225	304	11	is	be	AUX
ma-225	304	12	obtained	obtain	VERB
ma-225	304	13	.	.	PUNCT
ma-225	305	1	definition	definition	NOUN
ma-225	305	2	3.5	3.5	NUM
ma-225	305	3	.	.	PUNCT
ma-225	306	1	m	m	PROPN
ma-225	306	2	is	be	AUX
ma-225	306	3	referred	refer	VERB
ma-225	306	4	to	to	ADP
ma-225	306	5	as	as	SCONJ
ma-225	306	6	p	p	PROPN
ma-225	306	7	-	-	PUNCT
ma-225	306	8	jensen	jensen	PROPN
ma-225	306	9	-	-	PUNCT
ma-225	306	10	steffensen	steffensen	PROPN
ma-225	306	11	’s	’s	PART
ma-225	306	12	set	set	VERB
ma-225	306	13	if	if	SCONJ
ma-225	306	14	,	,	PUNCT
ma-225	306	15	it	it	PRON
ma-225	306	16	is	be	AUX
ma-225	306	17	a	a	DET
ma-225	306	18	subset	subset	NOUN
ma-225	306	19	of	of	ADP
ma-225	306	20	r+	r+	NOUN
ma-225	306	21	,	,	PUNCT
ma-225	306	22	assuming	assume	VERB
ma-225	306	23	that	that	PROPN
ma-225	306	24	(	(	PUNCT
ma-225	306	25	1	1	NUM
ma-225	306	26	bn	bn	NOUN
ma-225	306	27	)	)	PUNCT
ma-225	306	28	ln∏n	ln∏n	PROPN
ma-225	306	29	i=1((xi	i=1((xi	ADJ
ma-225	306	30	)	)	PUNCT
ma-225	306	31	p)ln(zi	p)ln(zi	ADJ
ma-225	306	32	)	)	PUNCT
ma-225	306	33			NOUN
ma-225	306	34	1p	1p	NUM
ma-225	306	35	∈	∈	NOUN
ma-225	306	36	m.	m.	NOUN
ma-225	306	37	(	(	PUNCT
ma-225	306	38	38	38	NUM
ma-225	306	39	)	)	PUNCT
ma-225	306	40	lemma	lemma	PROPN
ma-225	306	41	3.10	3.10	NUM
ma-225	306	42	.	.	PUNCT
ma-225	307	1	let	let	VERB
ma-225	307	2	m	m	PRON
ma-225	307	3	⊆	⊆	NUM
ma-225	307	4	r+	r+	NOUN
ma-225	307	5	be	be	AUX
ma-225	307	6	p	p	PROPN
ma-225	307	7	-	-	PUNCT
ma-225	307	8	jensen	jensen	PROPN
ma-225	307	9	-	-	PUNCT
ma-225	307	10	steffensen	steffensen	NOUN
ma-225	307	11	set	set	NOUN
ma-225	307	12	.	.	PUNCT
ma-225	308	1	for	for	ADP
ma-225	308	2	p	p	PROPN
ma-225	308	3	-	-	PUNCT
ma-225	308	4	jensen	jensen	PROPN
ma-225	308	5	-	-	PUNCT
ma-225	308	6	steffensen	steffensen	PROPN
ma-225	308	7	’s	’s	PART
ma-225	308	8	inequality	inequality	NOUN
ma-225	308	9	,	,	PUNCT
ma-225	308	10	we	we	PRON
ma-225	308	11	have	have	VERB
ma-225	308	12	φ	φ	NUM
ma-225	308	13			PROPN
ma-225	308	14	(	(	PUNCT
ma-225	308	15	1	1	NUM
ma-225	308	16	bn	bn	NOUN
ma-225	308	17	)	)	PUNCT
ma-225	308	18	ln∏n	ln∏n	PROPN
ma-225	308	19	r=1((xi	r=1((xi	PROPN
ma-225	308	20	)	)	PUNCT
ma-225	308	21	p)ln(zi	p)ln(zi	ADJ
ma-225	308	22	)	)	PUNCT
ma-225	308	23			NOUN
ma-225	308	24	1p	1p	VERB
ma-225	308	25	≤	≤	NOUN
ma-225	308	26	(	(	PUNCT
ma-225	308	27	1	1	NUM
ma-225	308	28	bn	bn	NOUN
ma-225	308	29	)	)	PUNCT
ma-225	308	30	ln∏n	ln∏n	NOUN
ma-225	309	1	r=1	r=1	NOUN
ma-225	309	2	φ(xi	φ(xi	X
ma-225	309	3	)	)	PUNCT
ma-225	309	4	ln(zi	ln(zi	PROPN
ma-225	309	5	)	)	PUNCT
ma-225	309	6	.	.	PUNCT
ma-225	310	1	(	(	PUNCT
ma-225	310	2	39	39	NUM
ma-225	310	3	)	)	PUNCT
ma-225	310	4	11	11	NUM
ma-225	310	5	proof	proof	NOUN
ma-225	310	6	.	.	PUNCT
ma-225	311	1	using	use	VERB
ma-225	311	2	equation	equation	NOUN
ma-225	311	3	(	(	PUNCT
ma-225	311	4	3	3	NUM
ma-225	311	5	)	)	PUNCT
ma-225	311	6	and	and	CCONJ
ma-225	311	7	by	by	ADP
ma-225	311	8	p	p	NOUN
ma-225	311	9	-	-	PUNCT
ma-225	311	10	convexity	convexity	NOUN
ma-225	311	11	,	,	PUNCT
ma-225	311	12	we	we	PRON
ma-225	311	13	have	have	VERB
ma-225	311	14	φ	φ	NUM
ma-225	311	15			PROPN
ma-225	311	16	(	(	PUNCT
ma-225	311	17	1	1	NUM
ma-225	311	18	bn	bn	NOUN
ma-225	311	19	)	)	PUNCT
ma-225	311	20	ln∏n	ln∏n	PROPN
ma-225	311	21	r=1((xi	r=1((xi	PROPN
ma-225	311	22	)	)	PUNCT
ma-225	311	23	p)ln(zi	p)ln(zi	ADJ
ma-225	311	24	)	)	PUNCT
ma-225	311	25			NOUN
ma-225	311	26	1p	1p	NUM
ma-225	311	27	=	=	SYM
ma-225	311	28	φ	φ	NOUN
ma-225	311	29	1	1	NUM
ma-225	311	30	bn	bn	NOUN
ma-225	311	31	×̇	×̇	PROPN
ma-225	311	32	n∏	n∏	PROPN
ma-225	311	33	r=1	r=1	PROPN
ma-225	311	34	(	(	PUNCT
ma-225	311	35	zi)×̇(xi)p	zi)×̇(xi)p	X
ma-225	311	36			NOUN
ma-225	311	37	1p	1p	VERB
ma-225	311	38	≤	≤	NUM
ma-225	312	1	1	1	NUM
ma-225	312	2	bn	bn	NUM
ma-225	312	3	×̇	×̇	PROPN
ma-225	312	4	n∏	n∏	PROPN
ma-225	312	5	r=1	r=1	PROPN
ma-225	312	6	(	(	PUNCT
ma-225	312	7	zi)×̇φ(xi	zi)×̇φ(xi	NOUN
ma-225	312	8	)	)	PUNCT
ma-225	312	9	≤α	≤α	NOUN
ma-225	312	10	α−1	α−1	PROPN
ma-225	312	11	(	(	PUNCT
ma-225	312	12	1	1	NUM
ma-225	312	13	bn	bn	NOUN
ma-225	312	14	)	)	PUNCT
ma-225	312	15	×̇α−1	×̇α−1	NOUN
ma-225	312	16	n∏	n∏	PROPN
ma-225	312	17	r=1	r=1	PROPN
ma-225	312	18	(	(	PUNCT
ma-225	312	19	zi)×̇α−1φ(xi	zi)×̇α−1φ(xi	PRON
ma-225	312	20	)	)	PUNCT
ma-225	313	1			PROPN
ma-225	313	2	≤e	≤e	VERB
ma-225	313	3	(	(	PUNCT
ma-225	313	4	ln	ln	ADJ
ma-225	313	5	(	(	PUNCT
ma-225	313	6	1	1	NUM
ma-225	313	7	bn	bn	NOUN
ma-225	313	8	)	)	PUNCT
ma-225	313	9	ln	ln	PROPN
ma-225	313	10	∏n	∏n	ADJ
ma-225	313	11	r=1(zi	r=1(zi	X
ma-225	313	12	)	)	PUNCT
ma-225	313	13	×lnφ(xi	×lnφ(xi	X
ma-225	313	14	)	)	PUNCT
ma-225	313	15	)	)	PUNCT
ma-225	314	1	≤	≤	NOUN
ma-225	314	2	(	(	PUNCT
ma-225	314	3	e	e	X
ma-225	314	4	ln	ln	X
ma-225	314	5	(	(	PUNCT
ma-225	314	6	1	1	NUM
ma-225	314	7	bn	bn	NUM
ma-225	314	8	)	)	PUNCT
ma-225	314	9	)	)	PUNCT
ma-225	314	10	ln∏n	ln∏n	NUM
ma-225	314	11	r=1(zi	r=1(zi	NOUN
ma-225	314	12	)	)	PUNCT
ma-225	314	13	×lnφ(xi	×lnφ(xi	X
ma-225	314	14	)	)	PUNCT
ma-225	315	1	≤	≤	NUM
ma-225	315	2	(	(	PUNCT
ma-225	315	3	1	1	NUM
ma-225	315	4	bn	bn	NOUN
ma-225	315	5	)	)	PUNCT
ma-225	315	6	ln∏n	ln∏n	NOUN
ma-225	315	7	r=1(zi	r=1(zi	NOUN
ma-225	315	8	)	)	PUNCT
ma-225	315	9	×lnφ(xi	×lnφ(xi	X
ma-225	315	10	)	)	PUNCT
ma-225	316	1	φ	φ	PROPN
ma-225	316	2			PROPN
ma-225	316	3	(	(	PUNCT
ma-225	316	4	1	1	NUM
ma-225	316	5	bn	bn	NOUN
ma-225	316	6	)	)	PUNCT
ma-225	316	7	ln∏n	ln∏n	PROPN
ma-225	316	8	r=1((xi	r=1((xi	ADJ
ma-225	316	9	)	)	PUNCT
ma-225	316	10	)	)	PUNCT
ma-225	317	1	ln(zi	ln(zi	PROPN
ma-225	317	2	)	)	PUNCT
ma-225	317	3			NOUN
ma-225	317	4	1p	1p	ADJ
ma-225	317	5	≤	≤	NOUN
ma-225	317	6	(	(	PUNCT
ma-225	317	7	1	1	NUM
ma-225	317	8	bn	bn	NOUN
ma-225	317	9	)	)	PUNCT
ma-225	317	10	ln∏n	ln∏n	NOUN
ma-225	318	1	r=1	r=1	NOUN
ma-225	318	2	φ(xi	φ(xi	X
ma-225	318	3	)	)	PUNCT
ma-225	318	4	ln(zi	ln(zi	PROPN
ma-225	318	5	)	)	PUNCT
ma-225	318	6	,	,	PUNCT
ma-225	318	7	as	as	SCONJ
ma-225	318	8	required	require	VERB
ma-225	318	9	.	.	PUNCT
ma-225	319	1	�	�	PROPN
ma-225	319	2	definition	definition	NOUN
ma-225	319	3	3.6	3.6	NUM
ma-225	319	4	.	.	PUNCT
ma-225	320	1	let	let	VERB
ma-225	320	2	the	the	DET
ma-225	320	3	set	set	NOUN
ma-225	320	4	w	w	NOUN
ma-225	320	5	be	be	AUX
ma-225	320	6	a	a	DET
ma-225	320	7	subset	subset	NOUN
ma-225	320	8	on	on	ADP
ma-225	320	9	r+	r+	X
ma-225	320	10	,	,	PUNCT
ma-225	320	11	then	then	ADV
ma-225	320	12	w	w	NOUN
ma-225	320	13	is	be	AUX
ma-225	320	14	strongly	strongly	ADV
ma-225	320	15	convex	convex	ADJ
ma-225	320	16	set	set	VERB
ma-225	320	17	if	if	SCONJ
ma-225	320	18	[	[	X
ma-225	320	19	[	[	PUNCT
ma-225	320	20	µln(j	µln(j	PROPN
ma-225	320	21	)	)	PUNCT
ma-225	320	22	]	]	X
ma-225	320	23	ln	ln	X
ma-225	320	24	(	(	PUNCT
ma-225	320	25	1	1	NUM
ma-225	320	26	j	j	NOUN
ma-225	320	27	)	)	PUNCT
ma-225	321	1	]	]	PUNCT
ma-225	322	1	ln	ln	X
ma-225	322	2	(	(	PUNCT
ma-225	322	3	y	y	NOUN
ma-225	322	4	x	x	PROPN
ma-225	322	5	)	)	PUNCT
ma-225	322	6	2	2	NUM
ma-225	322	7	∈	∈	PROPN
ma-225	322	8	w	w	NOUN
ma-225	322	9	,	,	PUNCT
ma-225	322	10	(	(	PUNCT
ma-225	322	11	40	40	NUM
ma-225	322	12	)	)	PUNCT
ma-225	322	13	where	where	SCONJ
ma-225	322	14	µ	µ	PRON
ma-225	322	15	≥	≥	NOUN
ma-225	322	16	1	1	NUM
ma-225	322	17	,	,	PUNCT
ma-225	322	18	∀	∀	X
ma-225	322	19	x	x	NOUN
ma-225	322	20	,	,	PUNCT
ma-225	322	21	y	y	PROPN
ma-225	322	22	∈	∈	PROPN
ma-225	322	23	w	w	PROPN
ma-225	322	24	and	and	CCONJ
ma-225	322	25	j	j	PROPN
ma-225	322	26	∈	∈	PROPN
ma-225	323	1	[	[	X
ma-225	323	2	1	1	NUM
ma-225	323	3	,	,	PUNCT
ma-225	323	4	e	e	NOUN
ma-225	323	5	]	]	PUNCT
ma-225	323	6	.	.	PUNCT
ma-225	324	1	lemma	lemma	PROPN
ma-225	324	2	3.11	3.11	NUM
ma-225	324	3	.	.	PUNCT
ma-225	325	1	consider	consider	VERB
ma-225	325	2	a	a	DET
ma-225	325	3	convex	convex	NOUN
ma-225	325	4	set	set	VERB
ma-225	325	5	w	w	NOUN
ma-225	325	6	⊆	⊆	NUM
ma-225	325	7	r+	r+	NOUN
ma-225	325	8	.	.	PUNCT
ma-225	326	1	for	for	ADP
ma-225	326	2	a	a	DET
ma-225	326	3	strongly	strongly	ADV
ma-225	326	4	convex	convex	ADJ
ma-225	326	5	function	function	NOUN
ma-225	326	6	φ	φ	PROPN
ma-225	326	7	,	,	PUNCT
ma-225	326	8	we	we	PRON
ma-225	326	9	have	have	VERB
ma-225	326	10	φ	φ	PROPN
ma-225	326	11	[	[	PUNCT
ma-225	326	12	x	x	SYM
ma-225	326	13	ln(q	ln(q	NOUN
ma-225	326	14	)	)	PUNCT
ma-225	326	15	·	·	PUNCT
ma-225	327	1	y	y	X
ma-225	327	2	ln	ln	ADJ
ma-225	327	3	(	(	PUNCT
ma-225	327	4	1	1	NUM
ma-225	327	5	q	q	NOUN
ma-225	327	6	)	)	PUNCT
ma-225	327	7	]	]	PUNCT
ma-225	327	8	≤	≤	NUM
ma-225	327	9	φ(x)ln(q	φ(x)ln(q	NOUN
ma-225	327	10	)	)	PUNCT
ma-225	327	11	·	·	PUNCT
ma-225	328	1	φ(y)ln	φ(y)ln	X
ma-225	328	2	(	(	PUNCT
ma-225	328	3	1	1	NUM
ma-225	328	4	q	q	NOUN
ma-225	328	5	)	)	PUNCT
ma-225	329	1	[	[	X
ma-225	329	2	[	[	X
ma-225	329	3	µln(q	µln(q	NOUN
ma-225	329	4	)	)	PUNCT
ma-225	329	5	]	]	X
ma-225	330	1	ln	ln	X
ma-225	330	2	(	(	PUNCT
ma-225	330	3	1	1	NUM
ma-225	330	4	q	q	NOUN
ma-225	330	5	)	)	PUNCT
ma-225	330	6	]	]	PUNCT
ma-225	331	1	ln	ln	X
ma-225	331	2	(	(	PUNCT
ma-225	331	3	y	y	NOUN
ma-225	331	4	x	x	PROPN
ma-225	331	5	)	)	PUNCT
ma-225	331	6	2	2	NUM
ma-225	331	7	,	,	PUNCT
ma-225	331	8	(	(	PUNCT
ma-225	331	9	41	41	NUM
ma-225	331	10	)	)	PUNCT
ma-225	331	11	where	where	SCONJ
ma-225	331	12	µ	µ	PRON
ma-225	331	13	≥	≥	NOUN
ma-225	331	14	1	1	NUM
ma-225	331	15	,	,	PUNCT
ma-225	331	16	∀	∀	X
ma-225	331	17	x	x	NOUN
ma-225	331	18	,	,	PUNCT
ma-225	331	19	y	y	PROPN
ma-225	331	20	∈	∈	PROPN
ma-225	331	21	w	w	PROPN
ma-225	331	22	and	and	CCONJ
ma-225	331	23	q	q	NOUN
ma-225	331	24	∈	∈	PROPN
ma-225	332	1	[	[	X
ma-225	332	2	1	1	NUM
ma-225	332	3	,	,	PUNCT
ma-225	332	4	e	e	NOUN
ma-225	332	5	]	]	X
ma-225	332	6	.	.	PUNCT
ma-225	333	1	12	12	NUM
ma-225	333	2	proof	proof	NOUN
ma-225	333	3	.	.	PUNCT
ma-225	334	1	using	use	VERB
ma-225	334	2	equation	equation	NOUN
ma-225	334	3	(	(	PUNCT
ma-225	334	4	3	3	NUM
ma-225	334	5	)	)	PUNCT
ma-225	334	6	,	,	PUNCT
ma-225	334	7	(	(	PUNCT
ma-225	334	8	1	1	X
ma-225	334	9	)	)	PUNCT
ma-225	334	10	and	and	CCONJ
ma-225	334	11	by	by	ADP
ma-225	334	12	convexity	convexity	NOUN
ma-225	334	13	,	,	PUNCT
ma-225	334	14	we	we	PRON
ma-225	334	15	have	have	VERB
ma-225	334	16	φ	φ	PROPN
ma-225	334	17	[	[	PUNCT
ma-225	334	18	x	x	SYM
ma-225	334	19	ln(q	ln(q	NOUN
ma-225	334	20	)	)	PUNCT
ma-225	334	21	·	·	PUNCT
ma-225	335	1	y	y	X
ma-225	335	2	ln	ln	ADJ
ma-225	335	3	(	(	PUNCT
ma-225	335	4	1	1	NUM
ma-225	335	5	q	q	NOUN
ma-225	335	6	)	)	PUNCT
ma-225	335	7	]	]	PUNCT
ma-225	336	1	=	=	X
ma-225	336	2	φ	φ	X
ma-225	336	3	[	[	PUNCT
ma-225	336	4	x×̇q+̇y×̇	x×̇q+̇y×̇	PROPN
ma-225	336	5	(	(	PUNCT
ma-225	336	6	1	1	NUM
ma-225	336	7	q	q	NOUN
ma-225	336	8	)	)	PUNCT
ma-225	336	9	]	]	PUNCT
ma-225	336	10	≤q×̇φ(x)+̇	≤q×̇φ(x)+̇	PROPN
ma-225	336	11	(	(	PUNCT
ma-225	336	12	1	1	NUM
ma-225	336	13	q	q	NOUN
ma-225	336	14	)	)	PUNCT
ma-225	336	15	×̇φ(y)−̇µ×̇j×̇	×̇φ(y)−̇µ×̇j×̇	PROPN
ma-225	336	16	(	(	PUNCT
ma-225	336	17	1	1	NUM
ma-225	336	18	q	q	NOUN
ma-225	336	19	)	)	PUNCT
ma-225	336	20	×̇	×̇	PROPN
ma-225	336	21	(	(	PUNCT
ma-225	336	22	y	y	NOUN
ma-225	336	23	x	x	PROPN
ma-225	336	24	)	)	PUNCT
ma-225	336	25	2	2	NUM
ma-225	336	26	≤α	≤α	NOUN
ma-225	336	27	[	[	PUNCT
ma-225	336	28	α−1q×̇α−1φ(x)+̇α−1	α−1q×̇α−1φ(x)+̇α−1	NUM
ma-225	336	29	(	(	PUNCT
ma-225	336	30	1	1	NUM
ma-225	336	31	q	q	NOUN
ma-225	336	32	)	)	PUNCT
ma-225	336	33	×̇α−1φ(y)−̇α−1(µ)×̇α−1(q)×̇α−1	×̇α−1φ(y)−̇α−1(µ)×̇α−1(q)×̇α−1	NOUN
ma-225	336	34	(	(	PUNCT
ma-225	336	35	1	1	NUM
ma-225	336	36	q	q	NOUN
ma-225	336	37	)	)	PUNCT
ma-225	336	38	×̇	×̇	PROPN
ma-225	336	39	(	(	PUNCT
ma-225	336	40	y	y	NOUN
ma-225	336	41	x	x	PROPN
ma-225	336	42	)	)	PUNCT
ma-225	336	43	2	2	NUM
ma-225	336	44	]	]	PUNCT
ma-225	336	45	≤	≤	NUM
ma-225	336	46	e	e	X
ma-225	336	47	ln(q	ln(q	NOUN
ma-225	336	48	)	)	PUNCT
ma-225	336	49	lnφ(x	lnφ(x	PROPN
ma-225	336	50	)	)	PUNCT
ma-225	336	51	·	·	PUNCT
ma-225	337	1	e	e	X
ma-225	337	2	ln	ln	ADJ
ma-225	337	3	(	(	PUNCT
ma-225	337	4	1	1	NUM
ma-225	337	5	q	q	NOUN
ma-225	337	6	)	)	PUNCT
ma-225	337	7	lnφ(y	lnφ(y	NOUN
ma-225	337	8	)	)	PUNCT
ma-225	337	9	e	e	NOUN
ma-225	337	10	ln(µ	ln(µ	X
ma-225	337	11	)	)	PUNCT
ma-225	337	12	ln(q	ln(q	PUNCT
ma-225	337	13	)	)	PUNCT
ma-225	337	14	ln	ln	ADJ
ma-225	337	15	(	(	PUNCT
ma-225	337	16	1	1	NUM
ma-225	337	17	q	q	NOUN
ma-225	337	18	)	)	PUNCT
ma-225	338	1	ln	ln	PROPN
ma-225	338	2	(	(	PUNCT
ma-225	338	3	y	y	NOUN
ma-225	338	4	x	x	PROPN
ma-225	338	5	)	)	PUNCT
ma-225	338	6	2	2	NUM
ma-225	338	7	)	)	PUNCT
ma-225	338	8	≤	≤	NOUN
ma-225	338	9	(	(	PUNCT
ma-225	338	10	e	e	NOUN
ma-225	338	11	lnφ(x	lnφ(x	PROPN
ma-225	338	12	)	)	PUNCT
ma-225	338	13	)	)	PUNCT
ma-225	338	14	ln(q	ln(q	X
ma-225	338	15	)	)	PUNCT
ma-225	338	16	·	·	PUNCT
ma-225	338	17	(	(	PUNCT
ma-225	338	18	e	e	NOUN
ma-225	338	19	lnφ(y	lnφ(y	PROPN
ma-225	338	20	)	)	PUNCT
ma-225	338	21	)	)	PUNCT
ma-225	339	1	ln	ln	X
ma-225	339	2	(	(	PUNCT
ma-225	339	3	1	1	NUM
ma-225	339	4	q	q	NOUN
ma-225	339	5	)	)	PUNCT
ma-225	339	6	(	(	PUNCT
ma-225	339	7	e	e	NOUN
ma-225	339	8	ln(µ	ln(µ	NOUN
ma-225	339	9	)	)	PUNCT
ma-225	339	10	)	)	PUNCT
ma-225	339	11	ln(q	ln(q	X
ma-225	339	12	)	)	PUNCT
ma-225	339	13	ln	ln	ADJ
ma-225	339	14	(	(	PUNCT
ma-225	339	15	1	1	NUM
ma-225	339	16	q	q	NOUN
ma-225	339	17	)	)	PUNCT
ma-225	339	18	ln	ln	PROPN
ma-225	339	19	(	(	PUNCT
ma-225	339	20	y	y	NOUN
ma-225	339	21	x	x	PROPN
ma-225	339	22	)	)	PUNCT
ma-225	339	23	2	2	NUM
ma-225	339	24	φ	φ	NOUN
ma-225	339	25	[	[	PUNCT
ma-225	339	26	x	x	SYM
ma-225	339	27	ln(q	ln(q	NOUN
ma-225	339	28	)	)	PUNCT
ma-225	339	29	·	·	PUNCT
ma-225	340	1	y	y	X
ma-225	340	2	ln	ln	ADJ
ma-225	340	3	(	(	PUNCT
ma-225	340	4	1	1	NUM
ma-225	340	5	q	q	NOUN
ma-225	340	6	)	)	PUNCT
ma-225	340	7	]	]	PUNCT
ma-225	340	8	≤	≤	NUM
ma-225	340	9	φ(x)ln(q	φ(x)ln(q	NOUN
ma-225	340	10	)	)	PUNCT
ma-225	340	11	·	·	PUNCT
ma-225	341	1	φ(y)ln	φ(y)ln	X
ma-225	341	2	(	(	PUNCT
ma-225	341	3	1	1	NUM
ma-225	341	4	q	q	NOUN
ma-225	341	5	)	)	PUNCT
ma-225	342	1	[	[	X
ma-225	342	2	[	[	X
ma-225	342	3	µln(q	µln(q	NOUN
ma-225	342	4	)	)	PUNCT
ma-225	342	5	]	]	X
ma-225	343	1	ln	ln	X
ma-225	343	2	(	(	PUNCT
ma-225	343	3	1	1	NUM
ma-225	343	4	q	q	NOUN
ma-225	343	5	)	)	PUNCT
ma-225	343	6	]	]	PUNCT
ma-225	344	1	ln	ln	X
ma-225	344	2	(	(	PUNCT
ma-225	344	3	y	y	NOUN
ma-225	344	4	x	x	PROPN
ma-225	344	5	)	)	PUNCT
ma-225	344	6	2	2	NUM
ma-225	344	7	,	,	PUNCT
ma-225	344	8	proved	prove	VERB
ma-225	344	9	.	.	PUNCT
ma-225	345	1	�	�	PROPN
ma-225	345	2	lemma	lemma	PROPN
ma-225	345	3	3.12	3.12	NUM
ma-225	345	4	.	.	PUNCT
ma-225	346	1	consider	consider	VERB
ma-225	346	2	the	the	DET
ma-225	346	3	convex	convex	NOUN
ma-225	346	4	set	set	VERB
ma-225	346	5	w	w	NOUN
ma-225	346	6	=	=	PUNCT
ma-225	347	1	[	[	X
ma-225	347	2	a	a	X
ma-225	347	3	,	,	PUNCT
ma-225	347	4	b	b	NOUN
ma-225	347	5	]	]	X
ma-225	347	6	⊆	⊆	NUM
ma-225	347	7	r+	r+	NOUN
ma-225	347	8	.	.	PUNCT
ma-225	348	1	if	if	SCONJ
ma-225	348	2	φ	φ	PROPN
ma-225	348	3	is	be	AUX
ma-225	348	4	highly	highly	ADV
ma-225	348	5	p	p	ADJ
ma-225	348	6	-	-	PUNCT
ma-225	348	7	convex	convex	NOUN
ma-225	348	8	function	function	NOUN
ma-225	348	9	,	,	PUNCT
ma-225	348	10	we	we	PRON
ma-225	348	11	have	have	VERB
ma-225	348	12	φ	φ	PROPN
ma-225	348	13	[	[	PUNCT
ma-225	348	14	(	(	PUNCT
ma-225	348	15	xp)ln(j	xp)ln(j	PROPN
ma-225	348	16	)	)	PUNCT
ma-225	348	17	·	·	PUNCT
ma-225	348	18	(	(	PUNCT
ma-225	348	19	yp)ln	yp)ln	NOUN
ma-225	348	20	(	(	PUNCT
ma-225	348	21	1	1	NUM
ma-225	348	22	j	j	NOUN
ma-225	348	23	)	)	PUNCT
ma-225	348	24	]	]	PUNCT
ma-225	348	25	1	1	NUM
ma-225	348	26	p	p	NOUN
ma-225	348	27	≤	≤	NUM
ma-225	348	28	φ(x)ln(j	φ(x)ln(j	NOUN
ma-225	348	29	)	)	PUNCT
ma-225	348	30	·	·	PUNCT
ma-225	349	1	φ(y)ln	φ(y)ln	X
ma-225	349	2	(	(	PUNCT
ma-225	349	3	1	1	NUM
ma-225	349	4	j	j	NOUN
ma-225	349	5	)	)	PUNCT
ma-225	350	1	[	[	X
ma-225	350	2	[	[	PUNCT
ma-225	350	3	µln(j	µln(j	PROPN
ma-225	350	4	)	)	PUNCT
ma-225	350	5	]	]	X
ma-225	350	6	ln	ln	X
ma-225	350	7	(	(	PUNCT
ma-225	350	8	1	1	NUM
ma-225	350	9	j	j	NOUN
ma-225	350	10	)	)	PUNCT
ma-225	351	1	]	]	PUNCT
ma-225	351	2	ln	ln	X
ma-225	351	3	(	(	PUNCT
ma-225	351	4	y	y	NOUN
ma-225	351	5	x	x	PROPN
ma-225	351	6	)	)	PUNCT
ma-225	351	7	2	2	NUM
ma-225	351	8	,	,	PUNCT
ma-225	351	9	(	(	PUNCT
ma-225	351	10	42	42	NUM
ma-225	351	11	)	)	PUNCT
ma-225	351	12	where	where	SCONJ
ma-225	351	13	µ	µ	PRON
ma-225	351	14	≥	≥	NOUN
ma-225	351	15	1	1	NUM
ma-225	351	16	,	,	PUNCT
ma-225	351	17	∀	∀	X
ma-225	351	18	x	x	NOUN
ma-225	351	19	,	,	PUNCT
ma-225	351	20	y	y	PROPN
ma-225	351	21	∈	∈	PROPN
ma-225	351	22	w	w	PROPN
ma-225	351	23	and	and	CCONJ
ma-225	351	24	j	j	PROPN
ma-225	351	25	∈	∈	PROPN
ma-225	352	1	[	[	X
ma-225	352	2	1	1	NUM
ma-225	352	3	,	,	PUNCT
ma-225	352	4	e	e	NOUN
ma-225	352	5	]	]	PUNCT
ma-225	352	6	.	.	PUNCT
ma-225	353	1	remark	remark	PROPN
ma-225	353	2	4	4	NUM
ma-225	353	3	.	.	PUNCT
ma-225	354	1	when	when	SCONJ
ma-225	354	2	p	p	NOUN
ma-225	354	3	=	=	NOUN
ma-225	354	4	1	1	NUM
ma-225	354	5	,	,	PUNCT
ma-225	354	6	the	the	DET
ma-225	354	7	strongly	strongly	ADV
ma-225	354	8	p	p	NOUN
ma-225	354	9	-	-	PUNCT
ma-225	354	10	convex	convex	NOUN
ma-225	354	11	function	function	NOUN
ma-225	354	12	returns	return	NOUN
ma-225	354	13	to	to	PART
ma-225	354	14	strongly	strongly	ADV
ma-225	354	15	convex	convex	VERB
ma-225	354	16	function	function	NOUN
ma-225	354	17	.	.	PUNCT
ma-225	355	1	thus	thus	ADV
ma-225	355	2	the	the	DET
ma-225	355	3	inequality	inequality	NOUN
ma-225	355	4	(	(	PUNCT
ma-225	355	5	41	41	NUM
ma-225	355	6	)	)	PUNCT
ma-225	355	7	is	be	AUX
ma-225	355	8	obtained	obtain	VERB
ma-225	355	9	.	.	PUNCT
ma-225	356	1	proof	proof	NOUN
ma-225	356	2	.	.	PUNCT
ma-225	357	1	using	use	VERB
ma-225	357	2	equation	equation	NOUN
ma-225	357	3	(	(	PUNCT
ma-225	357	4	3	3	NUM
ma-225	357	5	)	)	PUNCT
ma-225	357	6	,	,	PUNCT
ma-225	357	7	(	(	PUNCT
ma-225	357	8	1	1	X
ma-225	357	9	)	)	PUNCT
ma-225	357	10	and	and	CCONJ
ma-225	357	11	by	by	ADP
ma-225	357	12	p	p	NOUN
ma-225	357	13	-	-	PUNCT
ma-225	357	14	convexity	convexity	NOUN
ma-225	357	15	,	,	PUNCT
ma-225	357	16	we	we	PRON
ma-225	357	17	have	have	VERB
ma-225	357	18	φ	φ	PROPN
ma-225	357	19	[	[	PUNCT
ma-225	357	20	(	(	PUNCT
ma-225	357	21	xp)ln(j	xp)ln(j	PROPN
ma-225	357	22	)	)	PUNCT
ma-225	357	23	·	·	PUNCT
ma-225	357	24	(	(	PUNCT
ma-225	357	25	yp)ln	yp)ln	NOUN
ma-225	357	26	(	(	PUNCT
ma-225	357	27	1	1	NUM
ma-225	357	28	j	j	NOUN
ma-225	357	29	)	)	PUNCT
ma-225	357	30	]	]	PUNCT
ma-225	358	1	1	1	NUM
ma-225	358	2	p	p	PRON
ma-225	358	3	≤j×̇φ(x)+̇	≤j×̇φ(x)+̇	PROPN
ma-225	358	4	(	(	PUNCT
ma-225	358	5	1	1	NUM
ma-225	358	6	j	j	NOUN
ma-225	358	7	)	)	PUNCT
ma-225	358	8	×̇φ(y)−̇µ×̇j×̇	×̇φ(y)−̇µ×̇j×̇	PROPN
ma-225	358	9	(	(	PUNCT
ma-225	358	10	1	1	NUM
ma-225	358	11	j	j	NOUN
ma-225	358	12	)	)	PUNCT
ma-225	358	13	×̇	×̇	PROPN
ma-225	358	14	(	(	PUNCT
ma-225	358	15	y	y	NOUN
ma-225	358	16	x	x	PROPN
ma-225	358	17	)	)	PUNCT
ma-225	358	18	2	2	NUM
ma-225	358	19	≤α	≤α	NOUN
ma-225	358	20	[	[	PUNCT
ma-225	358	21	α−1j×̇α−1φ(x)+̇α−1	α−1j×̇α−1φ(x)+̇α−1	NOUN
ma-225	358	22	(	(	PUNCT
ma-225	358	23	1	1	NUM
ma-225	358	24	j	j	PROPN
ma-225	358	25	)	)	PUNCT
ma-225	358	26	×̇α−1φ(y)−̇α−1(µ)×̇α−1(j)×̇α−1	×̇α−1φ(y)−̇α−1(µ)×̇α−1(j)×̇α−1	NOUN
ma-225	358	27	(	(	PUNCT
ma-225	358	28	1	1	NUM
ma-225	358	29	j	j	PROPN
ma-225	358	30	)	)	PUNCT
ma-225	358	31	×̇	×̇	PROPN
ma-225	358	32	(	(	PUNCT
ma-225	358	33	y	y	NOUN
ma-225	358	34	x	x	PROPN
ma-225	358	35	)	)	PUNCT
ma-225	358	36	2	2	NUM
ma-225	358	37	]	]	PUNCT
ma-225	358	38	≤e	≤e	NOUN
ma-225	358	39	(	(	PUNCT
ma-225	358	40	ln(j	ln(j	NOUN
ma-225	358	41	)	)	PUNCT
ma-225	358	42	lnφ(x)+ln	lnφ(x)+ln	NOUN
ma-225	358	43	(	(	PUNCT
ma-225	358	44	1	1	NUM
ma-225	358	45	j	j	NOUN
ma-225	358	46	)	)	PUNCT
ma-225	358	47	lnφ(y)−ln(µ	lnφ(y)−ln(µ	VERB
ma-225	358	48	)	)	PUNCT
ma-225	358	49	ln(j	ln(j	PROPN
ma-225	358	50	)	)	PUNCT
ma-225	358	51	ln	ln	ADJ
ma-225	358	52	(	(	PUNCT
ma-225	358	53	1	1	NUM
ma-225	358	54	j	j	PROPN
ma-225	358	55	)	)	PUNCT
ma-225	359	1	ln	ln	PROPN
ma-225	359	2	(	(	PUNCT
ma-225	359	3	y	y	NOUN
ma-225	359	4	x	x	PROPN
ma-225	359	5	)	)	PUNCT
ma-225	359	6	2	2	X
ma-225	359	7	)	)	PUNCT
ma-225	359	8	≤	≤	NUM
ma-225	359	9	e	e	X
ma-225	359	10	ln(j	ln(j	X
ma-225	359	11	)	)	PUNCT
ma-225	359	12	lnφ(x	lnφ(x	PROPN
ma-225	359	13	)	)	PUNCT
ma-225	359	14	·	·	PUNCT
ma-225	360	1	e	e	X
ma-225	360	2	ln	ln	ADJ
ma-225	360	3	(	(	PUNCT
ma-225	360	4	1	1	NUM
ma-225	360	5	j	j	NOUN
ma-225	360	6	)	)	PUNCT
ma-225	360	7	lnφ(y	lnφ(y	PROPN
ma-225	360	8	)	)	PUNCT
ma-225	360	9	e	e	NOUN
ma-225	360	10	ln(µ	ln(µ	X
ma-225	360	11	)	)	PUNCT
ma-225	360	12	ln(j	ln(j	NOUN
ma-225	360	13	)	)	PUNCT
ma-225	360	14	ln	ln	ADJ
ma-225	360	15	(	(	PUNCT
ma-225	360	16	1	1	NUM
ma-225	360	17	j	j	PROPN
ma-225	360	18	)	)	PUNCT
ma-225	361	1	ln	ln	PROPN
ma-225	361	2	(	(	PUNCT
ma-225	361	3	y	y	NOUN
ma-225	361	4	x	x	PROPN
ma-225	361	5	)	)	PUNCT
ma-225	361	6	2	2	NUM
ma-225	361	7	)	)	PUNCT
ma-225	361	8	≤	≤	NOUN
ma-225	361	9	(	(	PUNCT
ma-225	361	10	e	e	NOUN
ma-225	361	11	lnφ(x	lnφ(x	PROPN
ma-225	361	12	)	)	PUNCT
ma-225	361	13	)	)	PUNCT
ma-225	361	14	ln(j	ln(j	X
ma-225	361	15	)	)	PUNCT
ma-225	361	16	·	·	PUNCT
ma-225	362	1	(	(	PUNCT
ma-225	362	2	e	e	NOUN
ma-225	362	3	lnφ(y	lnφ(y	PROPN
ma-225	362	4	)	)	PUNCT
ma-225	362	5	)	)	PUNCT
ma-225	363	1	ln	ln	X
ma-225	363	2	(	(	PUNCT
ma-225	363	3	1	1	NUM
ma-225	363	4	j	j	NOUN
ma-225	363	5	)	)	PUNCT
ma-225	364	1	(	(	PUNCT
ma-225	364	2	e	e	NOUN
ma-225	364	3	ln(µ	ln(µ	NOUN
ma-225	364	4	)	)	PUNCT
ma-225	364	5	)	)	PUNCT
ma-225	364	6	ln(j	ln(j	X
ma-225	364	7	)	)	PUNCT
ma-225	365	1	ln	ln	ADJ
ma-225	365	2	(	(	PUNCT
ma-225	365	3	1	1	NUM
ma-225	365	4	j	j	PROPN
ma-225	365	5	)	)	PUNCT
ma-225	366	1	ln	ln	PROPN
ma-225	366	2	(	(	PUNCT
ma-225	366	3	y	y	NOUN
ma-225	366	4	x	x	PROPN
ma-225	366	5	)	)	PUNCT
ma-225	366	6	2	2	NUM
ma-225	366	7	13	13	NUM
ma-225	366	8	φ	φ	NOUN
ma-225	366	9	[	[	PUNCT
ma-225	366	10	(	(	PUNCT
ma-225	366	11	xp)ln(j	xp)ln(j	PROPN
ma-225	366	12	)	)	PUNCT
ma-225	366	13	·	·	PUNCT
ma-225	366	14	(	(	PUNCT
ma-225	366	15	yp)ln	yp)ln	NOUN
ma-225	366	16	(	(	PUNCT
ma-225	366	17	1	1	NUM
ma-225	366	18	j	j	NOUN
ma-225	366	19	)	)	PUNCT
ma-225	366	20	]	]	PUNCT
ma-225	366	21	1	1	NUM
ma-225	366	22	p	p	NOUN
ma-225	366	23	≤	≤	NUM
ma-225	366	24	φ(x)ln(j	φ(x)ln(j	NOUN
ma-225	366	25	)	)	PUNCT
ma-225	366	26	·	·	PUNCT
ma-225	366	27	φ(y)ln	φ(y)ln	X
ma-225	366	28	(	(	PUNCT
ma-225	366	29	1	1	NUM
ma-225	366	30	j	j	NOUN
ma-225	366	31	)	)	PUNCT
ma-225	367	1	[	[	X
ma-225	367	2	[	[	PUNCT
ma-225	367	3	µln(j	µln(j	PROPN
ma-225	367	4	)	)	PUNCT
ma-225	367	5	]	]	X
ma-225	367	6	ln	ln	X
ma-225	367	7	(	(	PUNCT
ma-225	367	8	1	1	NUM
ma-225	367	9	j	j	NOUN
ma-225	367	10	)	)	PUNCT
ma-225	368	1	]	]	PUNCT
ma-225	368	2	ln	ln	X
ma-225	368	3	(	(	PUNCT
ma-225	368	4	y	y	NOUN
ma-225	368	5	x	x	PROPN
ma-225	368	6	)	)	PUNCT
ma-225	368	7	2	2	NUM
ma-225	368	8	,	,	PUNCT
ma-225	368	9	proved	prove	VERB
ma-225	368	10	.	.	PUNCT
ma-225	369	1	�	�	PROPN
ma-225	369	2	definition	definition	NOUN
ma-225	369	3	3.7	3.7	NUM
ma-225	369	4	.	.	PUNCT
ma-225	370	1	let	let	VERB
ma-225	370	2	the	the	DET
ma-225	370	3	set	set	NOUN
ma-225	370	4	w	w	NOUN
ma-225	370	5	be	be	AUX
ma-225	370	6	a	a	DET
ma-225	370	7	subset	subset	NOUN
ma-225	370	8	on	on	ADP
ma-225	370	9	r+	r+	X
ma-225	370	10	,	,	PUNCT
ma-225	370	11	then	then	ADV
ma-225	370	12	w	w	PROPN
ma-225	370	13	is	be	AUX
ma-225	370	14	m	m	NOUN
ma-225	370	15	-	-	ADJ
ma-225	370	16	convex	convex	NOUN
ma-225	370	17	set	set	VERB
ma-225	370	18	if	if	SCONJ
ma-225	370	19	x	x	PRON
ma-225	370	20	ln(t	ln(t	PUNCT
ma-225	370	21	)	)	PUNCT
ma-225	370	22	·	·	PUNCT
ma-225	371	1	[	[	PUNCT
ma-225	371	2	(	(	PUNCT
ma-225	371	3	y)ln(m	y)ln(m	NOUN
ma-225	371	4	)	)	PUNCT
ma-225	371	5	]	]	X
ma-225	372	1	ln	ln	X
ma-225	372	2	(	(	PUNCT
ma-225	372	3	1	1	NUM
ma-225	372	4	t	t	NOUN
ma-225	372	5	)	)	PUNCT
ma-225	372	6	∈	∈	PROPN
ma-225	372	7	w	w	PROPN
ma-225	372	8	,	,	PUNCT
ma-225	372	9	(	(	PUNCT
ma-225	372	10	43	43	NUM
ma-225	372	11	)	)	PUNCT
ma-225	372	12	where	where	SCONJ
ma-225	372	13	m	m	VERB
ma-225	372	14	∈	∈	X
ma-225	373	1	[	[	X
ma-225	373	2	1	1	NUM
ma-225	373	3	,	,	PUNCT
ma-225	373	4	e	e	NOUN
ma-225	373	5	]	]	X
ma-225	373	6	,	,	PUNCT
ma-225	373	7	∀	∀	X
ma-225	373	8	x	x	NOUN
ma-225	373	9	,	,	PUNCT
ma-225	373	10	y	y	PROPN
ma-225	373	11	∈	∈	PROPN
ma-225	373	12	w	w	PROPN
ma-225	373	13	and	and	CCONJ
ma-225	373	14	j	j	PROPN
ma-225	373	15	∈	∈	PROPN
ma-225	374	1	[	[	X
ma-225	374	2	1	1	NUM
ma-225	374	3	,	,	PUNCT
ma-225	374	4	e	e	NOUN
ma-225	374	5	]	]	PUNCT
ma-225	374	6	.	.	PUNCT
ma-225	375	1	lemma	lemma	PROPN
ma-225	375	2	3.13	3.13	NUM
ma-225	375	3	.	.	PUNCT
ma-225	376	1	let	let	VERB
ma-225	376	2	the	the	DET
ma-225	376	3	function	function	NOUN
ma-225	376	4	φ1	φ1	PROPN
ma-225	376	5	be	be	AUX
ma-225	376	6	m	m	NOUN
ma-225	376	7	-	-	ADJ
ma-225	376	8	convex	convex	ADJ
ma-225	376	9	and	and	CCONJ
ma-225	376	10	m1	m1	PROPN
ma-225	376	11	∈	∈	PROPN
ma-225	377	1	[	[	X
ma-225	377	2	1	1	NUM
ma-225	377	3	,	,	PUNCT
ma-225	377	4	e	e	NOUN
ma-225	377	5	]	]	X
ma-225	377	6	,	,	PUNCT
ma-225	377	7	then	then	ADV
ma-225	377	8	φ1	φ1	PROPN
ma-225	377	9	[	[	PUNCT
ma-225	377	10	x	x	NOUN
ma-225	377	11	ln(t1	ln(t1	NOUN
ma-225	377	12	)	)	PUNCT
ma-225	377	13	1	1	NUM
ma-225	377	14	·	·	PUNCT
ma-225	377	15	[	[	PUNCT
ma-225	377	16	y	y	PROPN
ma-225	377	17	ln(m	ln(m	PROPN
ma-225	377	18	)	)	PUNCT
ma-225	377	19	1	1	NUM
ma-225	377	20	]	]	PUNCT
ma-225	377	21	ln	ln	ADJ
ma-225	377	22	(	(	PUNCT
ma-225	377	23	1	1	NUM
ma-225	377	24	t1	t1	NOUN
ma-225	377	25	)	)	PUNCT
ma-225	377	26	]	]	PUNCT
ma-225	377	27	≤	≤	NUM
ma-225	377	28	φ1(x1)ln(t1	φ1(x1)ln(t1	NOUN
ma-225	377	29	)	)	PUNCT
ma-225	377	30	·	·	PUNCT
ma-225	378	1	[	[	PUNCT
ma-225	378	2	φ1(y1	φ1(y1	ADP
ma-225	378	3	)	)	PUNCT
ma-225	378	4	ln(m	ln(m	NOUN
ma-225	378	5	)	)	PUNCT
ma-225	378	6	]	]	X
ma-225	379	1	ln	ln	X
ma-225	379	2	(	(	PUNCT
ma-225	379	3	1	1	NUM
ma-225	379	4	t1	t1	NOUN
ma-225	379	5	)	)	PUNCT
ma-225	379	6	,	,	PUNCT
ma-225	379	7	(	(	PUNCT
ma-225	379	8	44	44	NUM
ma-225	379	9	)	)	PUNCT
ma-225	379	10	for	for	ADP
ma-225	379	11	all	all	DET
ma-225	379	12	x1	x1	PROPN
ma-225	379	13	,	,	PUNCT
ma-225	379	14	y1	y1	PROPN
ma-225	379	15	∈	∈	PROPN
ma-225	379	16	r+	r+	NOUN
ma-225	379	17	and	and	CCONJ
ma-225	379	18	t	t	NOUN
ma-225	379	19	∈	∈	PROPN
ma-225	380	1	[	[	X
ma-225	380	2	1	1	NUM
ma-225	380	3	,	,	PUNCT
ma-225	380	4	e	e	NOUN
ma-225	380	5	]	]	PUNCT
ma-225	380	6	.	.	PUNCT
ma-225	381	1	proof	proof	NOUN
ma-225	381	2	.	.	PUNCT
ma-225	382	1	using	use	VERB
ma-225	382	2	equation	equation	NOUN
ma-225	382	3	(	(	PUNCT
ma-225	382	4	3	3	NUM
ma-225	382	5	)	)	PUNCT
ma-225	382	6	,	,	PUNCT
ma-225	382	7	(	(	PUNCT
ma-225	382	8	1	1	X
ma-225	382	9	)	)	PUNCT
ma-225	382	10	and	and	CCONJ
ma-225	382	11	by	by	ADP
ma-225	382	12	convexity	convexity	NOUN
ma-225	382	13	,	,	PUNCT
ma-225	382	14	we	we	PRON
ma-225	382	15	have	have	VERB
ma-225	382	16	φ1	φ1	NOUN
ma-225	382	17	[	[	PUNCT
ma-225	382	18	x	x	NOUN
ma-225	382	19	ln(t1	ln(t1	NOUN
ma-225	382	20	)	)	PUNCT
ma-225	382	21	1	1	NUM
ma-225	382	22	·	·	PUNCT
ma-225	382	23	[	[	PUNCT
ma-225	382	24	(	(	PUNCT
ma-225	382	25	y1	y1	INTJ
ma-225	382	26	)	)	PUNCT
ma-225	382	27	ln(m1	ln(m1	NOUN
ma-225	382	28	)	)	PUNCT
ma-225	382	29	]	]	PUNCT
ma-225	383	1	ln	ln	X
ma-225	383	2	(	(	PUNCT
ma-225	383	3	1	1	NUM
ma-225	383	4	t1	t1	NOUN
ma-225	383	5	)	)	PUNCT
ma-225	383	6	]	]	PUNCT
ma-225	384	1	=	=	X
ma-225	384	2	φ1	φ1	PROPN
ma-225	384	3	[	[	PUNCT
ma-225	384	4	x1×̇t+̇m1×̇(y1)×̇	x1×̇t+̇m1×̇(y1)×̇	PROPN
ma-225	384	5	(	(	PUNCT
ma-225	384	6	1	1	NUM
ma-225	384	7	t1	t1	NOUN
ma-225	384	8	)	)	PUNCT
ma-225	384	9	]	]	PUNCT
ma-225	384	10	≤φ1(x1)×̇t1+̇m1×̇φ1(y1)×̇	≤φ1(x1)×̇t1+̇m1×̇φ1(y1)×̇	PROPN
ma-225	384	11	(	(	PUNCT
ma-225	384	12	1	1	NUM
ma-225	384	13	t1	t1	NOUN
ma-225	384	14	)	)	PUNCT
ma-225	384	15	≤	≤	NOUN
ma-225	385	1	[	[	PUNCT
ma-225	385	2	α[α−1φ1(x1)×̇α−1(t1)+̇α−1(m1)×̇α−1φ1(y1)×̇α−1	α[α−1φ1(x1)×̇α−1(t1)+̇α−1(m1)×̇α−1φ1(y1)×̇α−1	NOUN
ma-225	385	3	(	(	PUNCT
ma-225	385	4	1	1	NUM
ma-225	385	5	t1	t1	NOUN
ma-225	385	6	)	)	PUNCT
ma-225	385	7	]	]	PUNCT
ma-225	385	8	]	]	PUNCT
ma-225	386	1	≤	≤	X
ma-225	386	2	[	[	PUNCT
ma-225	386	3	α[lnφ1(x1)×	α[lnφ1(x1)×	NUM
ma-225	386	4	ln(t1	ln(t1	NOUN
ma-225	386	5	)	)	PUNCT
ma-225	387	1	+	+	CCONJ
ma-225	388	1	ln(m1)×̇	ln(m1)×̇	PROPN
ma-225	388	2	lnφ1(y1)×	lnφ1(y1)×	PROPN
ma-225	388	3	ln	ln	ADJ
ma-225	388	4	(	(	PUNCT
ma-225	388	5	1	1	NUM
ma-225	388	6	t1	t1	NOUN
ma-225	388	7	)	)	PUNCT
ma-225	388	8	]	]	PUNCT
ma-225	388	9	]	]	PUNCT
ma-225	388	10	≤	≤	X
ma-225	388	11	[	[	PUNCT
ma-225	388	12	e	e	X
ma-225	388	13	[	[	X
ma-225	388	14	lnφ1(x1)×ln(t1)+ln(m1)×lnφ1(y1)×ln	lnφ1(x1)×ln(t1)+ln(m1)×lnφ1(y1)×ln	PROPN
ma-225	388	15	(	(	PUNCT
ma-225	388	16	1t1	1t1	NUM
ma-225	388	17	)	)	PUNCT
ma-225	388	18	]	]	PUNCT
ma-225	388	19	]	]	PUNCT
ma-225	389	1	≤	≤	X
ma-225	389	2	[	[	PUNCT
ma-225	389	3	e	e	X
ma-225	389	4	[	[	X
ma-225	389	5	lnφ1(x1	lnφ1(x1	PROPN
ma-225	389	6	)	)	PUNCT
ma-225	389	7	ln(t1)+lnm1	ln(t1)+lnm1	PROPN
ma-225	389	8	lnφ1(y1	lnφ1(y1	PROPN
ma-225	389	9	)	)	PUNCT
ma-225	389	10	ln	ln	ADJ
ma-225	389	11	(	(	PUNCT
ma-225	389	12	1	1	NUM
ma-225	389	13	t1	t1	NOUN
ma-225	389	14	)	)	PUNCT
ma-225	389	15	]	]	PUNCT
ma-225	389	16	]	]	PUNCT
ma-225	389	17	≤(e	≤(e	X
ma-225	389	18	lnφ1(x1))ln(t1	lnφ1(x1))ln(t1	PROPN
ma-225	389	19	)	)	PUNCT
ma-225	389	20	·	·	PUNCT
ma-225	389	21	(	(	PUNCT
ma-225	389	22	(	(	PUNCT
ma-225	389	23	e	e	NOUN
ma-225	389	24	lnφ1(y1))ln(m1))ln	lnφ1(y1))ln(m1))ln	X
ma-225	389	25	(	(	PUNCT
ma-225	389	26	1	1	NUM
ma-225	389	27	t1	t1	NOUN
ma-225	389	28	)	)	PUNCT
ma-225	389	29	φ1	φ1	NOUN
ma-225	389	30	[	[	PUNCT
ma-225	389	31	x	x	SYM
ma-225	389	32	ln(t1	ln(t1	NOUN
ma-225	389	33	)	)	PUNCT
ma-225	389	34	1	1	NUM
ma-225	389	35	·	·	PUNCT
ma-225	389	36	[	[	PUNCT
ma-225	389	37	(	(	PUNCT
ma-225	389	38	y1	y1	INTJ
ma-225	389	39	)	)	PUNCT
ma-225	389	40	ln(m1	ln(m1	NOUN
ma-225	389	41	)	)	PUNCT
ma-225	390	1	]	]	PUNCT
ma-225	390	2	ln	ln	X
ma-225	390	3	(	(	PUNCT
ma-225	390	4	1	1	NUM
ma-225	390	5	t	t	NOUN
ma-225	390	6	)	)	PUNCT
ma-225	390	7	]	]	PUNCT
ma-225	391	1	≤φ1(x1)ln(t1	≤φ1(x1)ln(t1	X
ma-225	391	2	)	)	PUNCT
ma-225	391	3	·	·	PUNCT
ma-225	392	1	[	[	PUNCT
ma-225	392	2	φ1(y1	φ1(y1	ADP
ma-225	392	3	)	)	PUNCT
ma-225	392	4	ln(m1	ln(m1	NOUN
ma-225	392	5	)	)	PUNCT
ma-225	393	1	]	]	PUNCT
ma-225	394	1	ln	ln	X
ma-225	394	2	(	(	PUNCT
ma-225	394	3	1	1	NUM
ma-225	394	4	t1	t1	NOUN
ma-225	394	5	)	)	PUNCT
ma-225	394	6	,	,	PUNCT
ma-225	394	7	as	as	SCONJ
ma-225	394	8	required	require	VERB
ma-225	394	9	.	.	PUNCT
ma-225	395	1	�	�	PROPN
ma-225	395	2	definition	definition	NOUN
ma-225	395	3	3.8	3.8	NUM
ma-225	395	4	.	.	PUNCT
ma-225	396	1	let	let	VERB
ma-225	396	2	w	w	NOUN
ma-225	396	3	⊆	⊆	NUM
ma-225	396	4	r+	r+	NOUN
ma-225	396	5	,	,	PUNCT
ma-225	396	6	then	then	ADV
ma-225	396	7	w	w	PROPN
ma-225	396	8	is	be	AUX
ma-225	396	9	said	say	VERB
ma-225	396	10	to	to	PART
ma-225	396	11	be	be	AUX
ma-225	396	12	c	c	NOUN
ma-225	396	13	-	-	NOUN
ma-225	396	14	convex	convex	NOUN
ma-225	396	15	set	set	VERB
ma-225	396	16	if	if	SCONJ
ma-225	396	17	x	x	X
ma-225	396	18	ln	ln	ADJ
ma-225	396	19	(	(	PUNCT
ma-225	396	20	1	1	NUM
ma-225	396	21	t	t	NOUN
ma-225	396	22	)	)	PUNCT
ma-225	396	23	1	1	NUM
ma-225	396	24	·	·	PUNCT
ma-225	396	25	y	y	PROPN
ma-225	396	26	ln(t)1	ln(t)1	PROPN
ma-225	396	27	∈	∈	PROPN
ma-225	396	28	w	w	PROPN
ma-225	396	29	,	,	PUNCT
ma-225	396	30	(	(	PUNCT
ma-225	396	31	45	45	NUM
ma-225	396	32	)	)	PUNCT
ma-225	396	33	where	where	SCONJ
ma-225	396	34	c	c	PROPN
ma-225	396	35	∈	∈	PROPN
ma-225	397	1	[	[	X
ma-225	397	2	1	1	NUM
ma-225	397	3	,	,	PUNCT
ma-225	397	4	e	e	NOUN
ma-225	397	5	]	]	X
ma-225	397	6	,	,	PUNCT
ma-225	397	7	∀	∀	X
ma-225	397	8	x1	x1	ADJ
ma-225	397	9	,	,	PUNCT
ma-225	397	10	y1	y1	PROPN
ma-225	397	11	∈	∈	PROPN
ma-225	397	12	w	w	NOUN
ma-225	397	13	and	and	CCONJ
ma-225	397	14	t	t	NOUN
ma-225	397	15	∈	∈	PROPN
ma-225	398	1	[	[	X
ma-225	398	2	1	1	NUM
ma-225	398	3	,	,	PUNCT
ma-225	398	4	e	e	NOUN
ma-225	398	5	]	]	X
ma-225	398	6	.	.	PUNCT
ma-225	399	1	14	14	NUM
ma-225	399	2	lemma	lemma	PROPN
ma-225	399	3	3.14	3.14	NUM
ma-225	399	4	.	.	PUNCT
ma-225	400	1	let	let	VERB
ma-225	400	2	the	the	DET
ma-225	400	3	function	function	NOUN
ma-225	400	4	φ1	φ1	NOUN
ma-225	400	5	be	be	AUX
ma-225	400	6	c	c	NOUN
ma-225	400	7	-	-	NOUN
ma-225	400	8	convex	convex	NOUN
ma-225	400	9	on	on	ADP
ma-225	400	10	w	w	NOUN
ma-225	400	11	=	=	PUNCT
ma-225	401	1	[	[	X
ma-225	401	2	a	a	PRON
ma-225	401	3	,	,	PUNCT
ma-225	401	4	a1	a1	NOUN
ma-225	401	5	]	]	PUNCT
ma-225	401	6	and	and	CCONJ
ma-225	401	7	c	c	NOUN
ma-225	401	8	∈	∈	PROPN
ma-225	402	1	[	[	X
ma-225	402	2	1	1	NUM
ma-225	402	3	,	,	PUNCT
ma-225	402	4	e	e	NOUN
ma-225	402	5	]	]	X
ma-225	402	6	,	,	PUNCT
ma-225	402	7	then	then	ADV
ma-225	402	8	φ1	φ1	PROPN
ma-225	402	9	[	[	PUNCT
ma-225	402	10	(	(	PUNCT
ma-225	402	11	x	x	SYM
ma-225	402	12	ln	ln	ADJ
ma-225	402	13	(	(	PUNCT
ma-225	402	14	1	1	NUM
ma-225	402	15	t1	t1	NOUN
ma-225	402	16	)	)	PUNCT
ma-225	402	17	1	1	NUM
ma-225	402	18	·	·	PUNCT
ma-225	402	19	y	y	SYM
ma-225	402	20	ln(t1)1	ln(t1)1	PROPN
ma-225	402	21	]	]	PUNCT
ma-225	402	22	≤	≤	NOUN
ma-225	402	23	[	[	PUNCT
ma-225	402	24	φ1(x1	φ1(x1	NOUN
ma-225	402	25	)	)	PUNCT
ma-225	402	26	ln(c	ln(c	PROPN
ma-225	402	27	)	)	PUNCT
ma-225	402	28	]	]	PUNCT
ma-225	403	1	ln	ln	X
ma-225	403	2	(	(	PUNCT
ma-225	403	3	1	1	NUM
ma-225	403	4	t1	t1	NOUN
ma-225	403	5	)	)	PUNCT
ma-225	403	6	·	·	PUNCT
ma-225	403	7	φ1(y1)ln(t1	φ1(y1)ln(t1	NOUN
ma-225	403	8	)	)	PUNCT
ma-225	403	9	,	,	PUNCT
ma-225	403	10	(	(	PUNCT
ma-225	403	11	46	46	NUM
ma-225	403	12	)	)	PUNCT
ma-225	403	13	∀	∀	PUNCT
ma-225	404	1	x1	x1	ADJ
ma-225	404	2	,	,	PUNCT
ma-225	404	3	y1	y1	PROPN
ma-225	404	4	∈	∈	PROPN
ma-225	404	5	w	w	NOUN
ma-225	404	6	and	and	CCONJ
ma-225	404	7	t1	t1	NOUN
ma-225	404	8	∈	∈	PROPN
ma-225	405	1	[	[	X
ma-225	405	2	1	1	NUM
ma-225	405	3	,	,	PUNCT
ma-225	405	4	e	e	NOUN
ma-225	405	5	]	]	PUNCT
ma-225	405	6	.	.	PUNCT
ma-225	406	1	proof	proof	NOUN
ma-225	406	2	.	.	PUNCT
ma-225	407	1	using	use	VERB
ma-225	407	2	equation	equation	NOUN
ma-225	407	3	(	(	PUNCT
ma-225	407	4	3	3	NUM
ma-225	407	5	)	)	PUNCT
ma-225	407	6	,	,	PUNCT
ma-225	407	7	(	(	PUNCT
ma-225	407	8	1	1	X
ma-225	407	9	)	)	PUNCT
ma-225	407	10	and	and	CCONJ
ma-225	407	11	by	by	ADP
ma-225	407	12	convexity	convexity	NOUN
ma-225	407	13	,	,	PUNCT
ma-225	407	14	we	we	PRON
ma-225	407	15	have	have	VERB
ma-225	407	16	φ1	φ1	NOUN
ma-225	407	17	[	[	PUNCT
ma-225	407	18	x	x	X
ma-225	407	19	ln	ln	ADJ
ma-225	407	20	(	(	PUNCT
ma-225	407	21	1	1	NUM
ma-225	407	22	t1	t1	NOUN
ma-225	407	23	)	)	PUNCT
ma-225	407	24	1	1	NUM
ma-225	407	25	·	·	PUNCT
ma-225	407	26	(	(	PUNCT
ma-225	407	27	y1)ln(t1	y1)ln(t1	PROPN
ma-225	407	28	)	)	PUNCT
ma-225	407	29	]	]	PUNCT
ma-225	408	1	=	=	X
ma-225	408	2	φ1	φ1	PROPN
ma-225	408	3	[	[	PUNCT
ma-225	408	4	x1×̇	x1×̇	PROPN
ma-225	408	5	(	(	PUNCT
ma-225	408	6	1	1	NUM
ma-225	408	7	t1	t1	NOUN
ma-225	408	8	)	)	PUNCT
ma-225	408	9	+	+	PUNCT
ma-225	408	10	̇(y1)×̇t1	̇(y1)×̇t1	PUNCT
ma-225	408	11	]	]	X
ma-225	408	12	≤c×̇φ1(x1)×̇	≤c×̇φ1(x1)×̇	PROPN
ma-225	408	13	(	(	PUNCT
ma-225	408	14	1	1	NUM
ma-225	408	15	t1	t1	NOUN
ma-225	408	16	)	)	PUNCT
ma-225	408	17	+	+	VERB
ma-225	408	18	̇φ1(y1)×̇t1	̇φ1(y1)×̇t1	X
ma-225	408	19	≤	≤	NOUN
ma-225	408	20	[	[	PUNCT
ma-225	408	21	α[α−1(c)×̇α−1φ1(x1)×̇α−1	α[α−1(c)×̇α−1φ1(x1)×̇α−1	NOUN
ma-225	408	22	(	(	PUNCT
ma-225	408	23	1	1	NUM
ma-225	408	24	t1	t1	NOUN
ma-225	408	25	)	)	PUNCT
ma-225	408	26	+	+	NOUN
ma-225	408	27	̇α−1φ1(y1)×̇α−1(t1	̇α−1φ1(y1)×̇α−1(t1	X
ma-225	408	28	)	)	PUNCT
ma-225	408	29	]	]	PUNCT
ma-225	409	1	]	]	PUNCT
ma-225	409	2	≤	≤	X
ma-225	409	3	[	[	PUNCT
ma-225	409	4	α[ln(c)×	α[ln(c)×	PROPN
ma-225	409	5	lnφ1(x1)×	lnφ1(x1)×	PROPN
ma-225	409	6	ln	ln	PROPN
ma-225	409	7	(	(	PUNCT
ma-225	409	8	1	1	NUM
ma-225	409	9	t1	t1	NOUN
ma-225	409	10	)	)	PUNCT
ma-225	410	1	+	+	NUM
ma-225	410	2	lnφ1(y1)×	lnφ1(y1)×	PROPN
ma-225	410	3	ln(t1	ln(t1	NUM
ma-225	410	4	)	)	PUNCT
ma-225	410	5	]	]	PUNCT
ma-225	411	1	]	]	PUNCT
ma-225	411	2	≤	≤	X
ma-225	411	3	[	[	PUNCT
ma-225	411	4	e	e	X
ma-225	411	5	[	[	X
ma-225	411	6	lnφ1(x1)×ln(c)×ln	lnφ1(x1)×ln(c)×ln	PROPN
ma-225	411	7	(	(	PUNCT
ma-225	411	8	1t1	1t1	NUM
ma-225	411	9	)	)	PUNCT
ma-225	411	10	+	+	NOUN
ma-225	411	11	lnφ1(y1)×ln(t1	lnφ1(y1)×ln(t1	NOUN
ma-225	411	12	)	)	PUNCT
ma-225	411	13	]	]	PUNCT
ma-225	411	14	]	]	PUNCT
ma-225	412	1	≤	≤	X
ma-225	412	2	[	[	PUNCT
ma-225	412	3	e	e	X
ma-225	412	4	[	[	X
ma-225	412	5	lnφ1(x1	lnφ1(x1	PROPN
ma-225	412	6	)	)	PUNCT
ma-225	412	7	ln(c	ln(c	PROPN
ma-225	412	8	)	)	PUNCT
ma-225	413	1	ln	ln	ADJ
ma-225	413	2	(	(	PUNCT
ma-225	413	3	1	1	NUM
ma-225	413	4	t1	t1	NOUN
ma-225	413	5	)	)	PUNCT
ma-225	414	1	+	+	SYM
ma-225	414	2	lnφ1(y1	lnφ1(y1	PROPN
ma-225	414	3	)	)	PUNCT
ma-225	414	4	ln(t1	ln(t1	NOUN
ma-225	414	5	)	)	PUNCT
ma-225	414	6	]	]	PUNCT
ma-225	415	1	]	]	PUNCT
ma-225	415	2	≤	≤	X
ma-225	415	3	[	[	PUNCT
ma-225	415	4	(	(	PUNCT
ma-225	415	5	e	e	NOUN
ma-225	415	6	lnφ1(x1))ln(c	lnφ1(x1))ln(c	NOUN
ma-225	415	7	)	)	PUNCT
ma-225	415	8	]	]	PUNCT
ma-225	416	1	ln	ln	X
ma-225	416	2	(	(	PUNCT
ma-225	416	3	1	1	NUM
ma-225	416	4	t1	t1	NOUN
ma-225	416	5	)	)	PUNCT
ma-225	416	6	·	·	PUNCT
ma-225	417	1	(	(	PUNCT
ma-225	417	2	e	e	NOUN
ma-225	417	3	lnφ1(y1))ln(t1	lnφ1(y1))ln(t1	PROPN
ma-225	417	4	)	)	PUNCT
ma-225	417	5	φ1	φ1	NOUN
ma-225	417	6	[	[	PUNCT
ma-225	417	7	x	x	X
ma-225	417	8	ln	ln	ADJ
ma-225	417	9	(	(	PUNCT
ma-225	417	10	1	1	NUM
ma-225	417	11	t1	t1	NOUN
ma-225	417	12	)	)	PUNCT
ma-225	417	13	1	1	NUM
ma-225	417	14	·	·	PUNCT
ma-225	417	15	(	(	PUNCT
ma-225	417	16	y1)ln(t1	y1)ln(t1	PROPN
ma-225	417	17	)	)	PUNCT
ma-225	417	18	]	]	PUNCT
ma-225	418	1	≤	≤	NOUN
ma-225	418	2	[	[	PUNCT
ma-225	418	3	φ1(x1	φ1(x1	NOUN
ma-225	418	4	)	)	PUNCT
ma-225	418	5	ln(c	ln(c	PROPN
ma-225	418	6	)	)	PUNCT
ma-225	419	1	]	]	PUNCT
ma-225	419	2	ln	ln	X
ma-225	419	3	(	(	PUNCT
ma-225	419	4	1	1	NUM
ma-225	419	5	t1	t1	NOUN
ma-225	419	6	)	)	PUNCT
ma-225	420	1	·	·	PUNCT
ma-225	420	2	φ1(y1)ln(t1),as	φ1(y1)ln(t1),as	PRON
ma-225	420	3	required	require	VERB
ma-225	420	4	.	.	PUNCT
ma-225	421	1	�	�	PROPN
ma-225	421	2	4	4	NUM
ma-225	421	3	.	.	PUNCT
ma-225	421	4	conclusion	conclusion	NOUN
ma-225	421	5	in	in	ADP
ma-225	421	6	this	this	DET
ma-225	421	7	paper	paper	NOUN
ma-225	421	8	,	,	PUNCT
ma-225	421	9	some	some	DET
ma-225	421	10	classes	class	NOUN
ma-225	421	11	of	of	ADP
ma-225	421	12	convex	convex	NOUN
ma-225	421	13	functions	function	NOUN
ma-225	421	14	have	have	AUX
ma-225	421	15	been	be	AUX
ma-225	421	16	identified	identify	VERB
ma-225	421	17	and	and	CCONJ
ma-225	421	18	presented	present	VERB
ma-225	421	19	.	.	PUNCT
ma-225	422	1	the	the	DET
ma-225	422	2	pa	pa	PROPN
ma-225	422	3	-	-	PUNCT
ma-225	422	4	per	per	NOUN
ma-225	422	5	established	establish	VERB
ma-225	422	6	some	some	DET
ma-225	422	7	convexity	convexity	NOUN
ma-225	422	8	properties	property	NOUN
ma-225	422	9	and	and	CCONJ
ma-225	422	10	inequalities	inequality	NOUN
ma-225	422	11	in	in	ADP
ma-225	422	12	non	non	ADJ
ma-225	422	13	-	-	ADJ
ma-225	422	14	newtonian	newtonian	ADJ
ma-225	422	15	calculus	calculus	NOUN
ma-225	422	16	and	and	CCONJ
ma-225	422	17	theirapplications	theirapplication	NOUN
ma-225	422	18	.	.	PUNCT
ma-225	423	1	references	reference	NOUN
ma-225	423	2	[	[	X
ma-225	423	3	1	1	X
ma-225	423	4	]	]	PUNCT
ma-225	423	5	e.	e.	PROPN
ma-225	423	6	unluyol	unluyol	PROPN
ma-225	423	7	,	,	PUNCT
ma-225	423	8	s.	s.	PROPN
ma-225	423	9	salas	salas	PROPN
ma-225	423	10	,	,	PUNCT
ma-225	423	11	i̇.	i̇.	NOUN
ma-225	423	12	i̇scan	i̇scan	ADJ
ma-225	423	13	,	,	PUNCT
ma-225	423	14	convex	convex	NOUN
ma-225	423	15	functions	function	NOUN
ma-225	423	16	and	and	CCONJ
ma-225	423	17	some	some	DET
ma-225	423	18	inequalities	inequality	NOUN
ma-225	423	19	in	in	ADP
ma-225	423	20	terms	term	NOUN
ma-225	423	21	of	of	ADP
ma-225	423	22	the	the	DET
ma-225	423	23	non	non	ADJ
ma-225	423	24	-	-	ADJ
ma-225	423	25	newtonian	newtonian	ADJ
ma-225	423	26	calculus	calculus	NOUN
ma-225	423	27	,	,	PUNCT
ma-225	423	28	in	in	ADP
ma-225	423	29	:	:	PUNCT
ma-225	423	30	aip	aip	PROPN
ma-225	423	31	conference	conference	NOUN
ma-225	423	32	proceedings	proceeding	NOUN
ma-225	423	33	,	,	PUNCT
ma-225	423	34	vol	vol	NOUN
ma-225	423	35	.	.	PUNCT
ma-225	423	36	1833	1833	NUM
ma-225	423	37	,	,	PUNCT
ma-225	423	38	aip	aip	PROPN
ma-225	423	39	publishing	publishing	PROPN
ma-225	423	40	llc	llc	PROPN
ma-225	423	41	,	,	PUNCT
ma-225	423	42	2017	2017	NUM
ma-225	423	43	,	,	PUNCT
ma-225	423	44	p.	p.	NOUN
ma-225	423	45	020043.[2	020043.[2	NUM
ma-225	423	46	]	]	X
ma-225	423	47	a.	a.	PROPN
ma-225	423	48	e.	e.	PROPN
ma-225	423	49	bashirov	bashirov	PROPN
ma-225	423	50	,	,	PUNCT
ma-225	423	51	r.	r.	PROPN
ma-225	423	52	mustafa	mustafa	PROPN
ma-225	423	53	,	,	PUNCT
ma-225	423	54	on	on	ADP
ma-225	423	55	complex	complex	ADJ
ma-225	423	56	multiplicative	multiplicative	ADJ
ma-225	423	57	differentiation	differentiation	NOUN
ma-225	423	58	,	,	PUNCT
ma-225	423	59	twms	twms	PROPN
ma-225	423	60	journal	journal	NOUN
ma-225	423	61	of	of	ADP
ma-225	423	62	applied	apply	VERB
ma-225	423	63	and	and	CCONJ
ma-225	423	64	engineeringmathematics	engineeringmathematic	NOUN
ma-225	423	65	1	1	NUM
ma-225	423	66	(	(	PUNCT
ma-225	423	67	1	1	NUM
ma-225	423	68	)	)	PUNCT
ma-225	423	69	(	(	PUNCT
ma-225	423	70	2011	2011	NUM
ma-225	423	71	)	)	PUNCT
ma-225	423	72	75–85.[3	75–85.[3	PROPN
ma-225	423	73	]	]	X
ma-225	423	74	k.	k.	PROPN
ma-225	423	75	boruah	boruah	PROPN
ma-225	423	76	,	,	PUNCT
ma-225	423	77	b.	b.	PROPN
ma-225	423	78	hazarika	hazarika	NOUN
ma-225	423	79	,	,	PUNCT
ma-225	423	80	g	g	NOUN
ma-225	423	81	-	-	PUNCT
ma-225	423	82	calculus	calculus	NOUN
ma-225	423	83	,	,	PUNCT
ma-225	423	84	twms	twms	PROPN
ma-225	423	85	journal	journal	NOUN
ma-225	423	86	of	of	ADP
ma-225	423	87	applied	apply	VERB
ma-225	423	88	and	and	CCONJ
ma-225	423	89	engineering	engineering	NOUN
ma-225	423	90	mathematics	mathematic	NOUN
ma-225	423	91	8	8	NUM
ma-225	423	92	(	(	PUNCT
ma-225	423	93	1	1	NUM
ma-225	423	94	)	)	PUNCT
ma-225	423	95	(	(	PUNCT
ma-225	423	96	2018	2018	NUM
ma-225	423	97	)	)	PUNCT
ma-225	424	1	94–105.[4	94–105.[4	NUM
ma-225	424	2	]	]	X
ma-225	424	3	d.	d.	PROPN
ma-225	424	4	f.	f.	PROPN
ma-225	424	5	torres	torres	PROPN
ma-225	424	6	,	,	PUNCT
ma-225	424	7	on	on	ADP
ma-225	424	8	a	a	DET
ma-225	424	9	non	non	ADJ
ma-225	424	10	-	-	ADJ
ma-225	424	11	newtonian	newtonian	ADJ
ma-225	424	12	calculus	calculus	NOUN
ma-225	424	13	of	of	ADP
ma-225	424	14	variations	variation	NOUN
ma-225	424	15	,	,	PUNCT
ma-225	424	16	axioms	axiom	VERB
ma-225	424	17	10	10	NUM
ma-225	424	18	(	(	PUNCT
ma-225	424	19	3	3	NUM
ma-225	424	20	)	)	PUNCT
ma-225	424	21	(	(	PUNCT
ma-225	424	22	2021	2021	NUM
ma-225	424	23	)	)	PUNCT
ma-225	424	24	171.[5	171.[5	NUM
ma-225	424	25	]	]	PUNCT
ma-225	424	26	m.	m.	NOUN
ma-225	424	27	czachor	czachor	PROPN
ma-225	424	28	,	,	PUNCT
ma-225	424	29	non	non	ADJ
ma-225	424	30	-	-	ADJ
ma-225	424	31	newtonian	newtonian	ADJ
ma-225	424	32	mathematics	mathematic	NOUN
ma-225	424	33	instead	instead	ADV
ma-225	424	34	of	of	ADP
ma-225	424	35	non	non	ADJ
ma-225	424	36	-	-	ADJ
ma-225	424	37	newtonian	newtonian	ADJ
ma-225	424	38	physics	physics	NOUN
ma-225	424	39	:	:	PUNCT
ma-225	424	40	dark	dark	ADJ
ma-225	424	41	matter	matter	NOUN
ma-225	424	42	and	and	CCONJ
ma-225	424	43	dark	dark	ADJ
ma-225	424	44	energy	energy	NOUN
ma-225	424	45	from	from	ADP
ma-225	424	46	amismatch	amismatch	NOUN
ma-225	424	47	of	of	ADP
ma-225	424	48	arithmetics	arithmetic	NOUN
ma-225	424	49	,	,	PUNCT
ma-225	424	50	foundations	foundation	NOUN
ma-225	424	51	of	of	ADP
ma-225	424	52	science	science	NOUN
ma-225	424	53	26	26	NUM
ma-225	424	54	(	(	PUNCT
ma-225	424	55	2021	2021	NUM
ma-225	424	56	)	)	PUNCT
ma-225	424	57	75–95.[6	75–95.[6	NUM
ma-225	424	58	]	]	PUNCT
ma-225	424	59	a.	a.	PROPN
ma-225	424	60	e.	e.	PROPN
ma-225	424	61	bashirov	bashirov	PROPN
ma-225	424	62	,	,	PUNCT
ma-225	424	63	e.	e.	PROPN
ma-225	424	64	m.	m.	PROPN
ma-225	424	65	kurpınar	kurpınar	PROPN
ma-225	424	66	,	,	PUNCT
ma-225	424	67	a.	a.	NOUN
ma-225	424	68	özyapıcı	özyapıcı	PROPN
ma-225	424	69	,	,	PUNCT
ma-225	424	70	multiplicative	multiplicative	ADJ
ma-225	424	71	calculus	calculus	NOUN
ma-225	424	72	and	and	CCONJ
ma-225	424	73	its	its	PRON
ma-225	424	74	applications	application	NOUN
ma-225	424	75	,	,	PUNCT
ma-225	424	76	journal	journal	NOUN
ma-225	424	77	of	of	ADP
ma-225	424	78	mathematicalanalysis	mathematicalanalysis	NOUN
ma-225	424	79	and	and	CCONJ
ma-225	424	80	applications	application	NOUN
ma-225	424	81	337	337	NUM
ma-225	424	82	(	(	PUNCT
ma-225	424	83	1	1	NUM
ma-225	424	84	)	)	PUNCT
ma-225	424	85	(	(	PUNCT
ma-225	424	86	2008	2008	NUM
ma-225	424	87	)	)	PUNCT
ma-225	425	1	36–48.[7	36–48.[7	NUM
ma-225	425	2	]	]	X
ma-225	425	3	d.	d.	PROPN
ma-225	425	4	filip	filip	PROPN
ma-225	425	5	,	,	PUNCT
ma-225	425	6	c.	c.	PROPN
ma-225	425	7	piatecki	piatecki	PROPN
ma-225	425	8	,	,	PUNCT
ma-225	425	9	an	an	DET
ma-225	425	10	overview	overview	NOUN
ma-225	425	11	on	on	ADP
ma-225	425	12	the	the	DET
ma-225	425	13	non	non	ADJ
ma-225	425	14	-	-	ADJ
ma-225	425	15	newtonian	newtonian	ADJ
ma-225	425	16	calculus	calculus	NOUN
ma-225	425	17	and	and	CCONJ
ma-225	425	18	its	its	PRON
ma-225	425	19	potential	potential	ADJ
ma-225	425	20	applications	application	NOUN
ma-225	425	21	to	to	PART
ma-225	425	22	economics.15	economics.15	VERB
ma-225	425	23	[	[	X
ma-225	425	24	8	8	NUM
ma-225	425	25	]	]	PUNCT
ma-225	425	26	m.	m.	NOUN
ma-225	425	27	grossman	grossman	NOUN
ma-225	425	28	,	,	PUNCT
ma-225	425	29	bigeometric	bigeometric	ADJ
ma-225	425	30	calculus	calculus	NOUN
ma-225	425	31	:	:	PUNCT
ma-225	425	32	a	a	DET
ma-225	425	33	system	system	NOUN
ma-225	425	34	with	with	ADP
ma-225	425	35	a	a	DET
ma-225	425	36	scale	scale	NOUN
ma-225	425	37	-	-	PUNCT
ma-225	425	38	free	free	ADJ
ma-225	425	39	derivative	derivative	NOUN
ma-225	425	40	,	,	PUNCT
ma-225	425	41	archimedes	archimedes	PROPN
ma-225	425	42	foundation	foundation	PROPN
ma-225	425	43	,	,	PUNCT
ma-225	425	44	1983.[9	1983.[9	NUM
ma-225	425	45	]	]	PUNCT
ma-225	425	46	m.	m.	NOUN
ma-225	425	47	grossman	grossman	PROPN
ma-225	425	48	,	,	PUNCT
ma-225	425	49	r.	r.	PROPN
ma-225	425	50	katz	katz	PROPN
ma-225	425	51	,	,	PUNCT
ma-225	425	52	non	non	ADJ
ma-225	425	53	-	-	ADJ
ma-225	425	54	newtonian	newtonian	ADJ
ma-225	425	55	calculus	calculus	NOUN
ma-225	425	56	:	:	PUNCT
ma-225	425	57	a	a	DET
ma-225	425	58	self	self	NOUN
ma-225	425	59	-	-	PUNCT
ma-225	425	60	contained	contain	VERB
ma-225	425	61	,	,	PUNCT
ma-225	425	62	elementary	elementary	ADJ
ma-225	425	63	exposition	exposition	NOUN
ma-225	425	64	of	of	ADP
ma-225	425	65	the	the	DET
ma-225	425	66	authors	author	NOUN
ma-225	425	67	’	’	PART
ma-225	425	68	investi	investi	PROPN
ma-225	425	69	-	-	PUNCT
ma-225	425	70	gations	gation	NOUN
ma-225	425	71	...	...	PUNCT
ma-225	425	72	,	,	PUNCT
ma-225	425	73	non	non	ADJ
ma-225	425	74	-	-	ADJ
ma-225	425	75	newtonian	newtonian	ADJ
ma-225	425	76	calculus	calculus	NOUN
ma-225	425	77	,	,	PUNCT
ma-225	425	78	1972.[10	1972.[10	NUM
ma-225	425	79	]	]	X
ma-225	425	80	u.	u.	PROPN
ma-225	425	81	kadak	kadak	PROPN
ma-225	425	82	,	,	PUNCT
ma-225	425	83	y.	y.	PROPN
ma-225	425	84	gürefe	gürefe	PROPN
ma-225	425	85	,	,	PUNCT
ma-225	425	86	a	a	DET
ma-225	425	87	generalization	generalization	NOUN
ma-225	425	88	on	on	ADP
ma-225	425	89	weighted	weight	VERB
ma-225	425	90	means	mean	NOUN
ma-225	425	91	and	and	CCONJ
ma-225	425	92	convex	convex	NOUN
ma-225	425	93	functions	function	NOUN
ma-225	425	94	with	with	ADP
ma-225	425	95	respect	respect	NOUN
ma-225	425	96	to	to	ADP
ma-225	425	97	the	the	DET
ma-225	425	98	non	non	ADJ
ma-225	425	99	-	-	ADJ
ma-225	425	100	newtoniancalculus	newtoniancalculus	ADJ
ma-225	425	101	,	,	PUNCT
ma-225	425	102	international	international	ADJ
ma-225	425	103	journal	journal	NOUN
ma-225	425	104	of	of	ADP
ma-225	425	105	analysis	analysis	NOUN
ma-225	425	106	2016.[11	2016.[11	NUM
ma-225	425	107	]	]	X
ma-225	425	108	m.	m.	NOUN
ma-225	425	109	a.	a.	PROPN
ma-225	425	110	noor	noor	PROPN
ma-225	425	111	,	,	PUNCT
ma-225	425	112	k.	k.	PROPN
ma-225	425	113	i.	i.	PROPN
ma-225	425	114	noor	noor	PROPN
ma-225	425	115	,	,	PUNCT
ma-225	425	116	s.	s.	PROPN
ma-225	425	117	iftikhar	iftikhar	PROPN
ma-225	425	118	,	,	PUNCT
ma-225	425	119	hermite	hermite	PROPN
ma-225	425	120	-	-	PUNCT
ma-225	425	121	hadamard	hadamard	ADJ
ma-225	425	122	inequalities	inequality	NOUN
ma-225	425	123	for	for	ADP
ma-225	425	124	harmonic	harmonic	ADJ
ma-225	425	125	nonconvex	nonconvex	NOUN
ma-225	425	126	functions	function	NOUN
ma-225	425	127	,	,	PUNCT
ma-225	425	128	magnt	magnt	X
ma-225	425	129	res.rep	res.rep	NOUN
ma-225	425	130	4	4	NUM
ma-225	425	131	(	(	PUNCT
ma-225	425	132	2016	2016	NUM
ma-225	425	133	)	)	PUNCT
ma-225	425	134	24–40.[12	24–40.[12	NUM
ma-225	425	135	]	]	PUNCT
ma-225	425	136	s.	s.	PROPN
ma-225	425	137	s.	s.	PROPN
ma-225	425	138	dragomir	dragomir	PROPN
ma-225	425	139	,	,	PUNCT
ma-225	425	140	n	n	CCONJ
ma-225	425	141	-	-	PUNCT
ma-225	425	142	points	point	NOUN
ma-225	425	143	inequalities	inequality	NOUN
ma-225	425	144	of	of	ADP
ma-225	425	145	hermite	hermite	ADJ
ma-225	425	146	-	-	PUNCT
ma-225	425	147	hadamard	hadamard	ADJ
ma-225	425	148	type	type	NOUN
ma-225	425	149	for	for	ADP
ma-225	425	150	h	h	NOUN
ma-225	425	151	-	-	PUNCT
ma-225	425	152	convex	convex	NOUN
ma-225	425	153	functions	function	NOUN
ma-225	425	154	on	on	ADP
ma-225	425	155	linear	linear	ADJ
ma-225	425	156	spaces	space	NOUN
ma-225	425	157	,	,	PUNCT
ma-225	425	158	armenianjournal	armenianjournal	ADJ
ma-225	425	159	of	of	ADP
ma-225	425	160	mathematics	mathematic	NOUN
ma-225	425	161	8	8	NUM
ma-225	425	162	(	(	PUNCT
ma-225	425	163	1	1	NUM
ma-225	425	164	)	)	PUNCT
ma-225	425	165	(	(	PUNCT
ma-225	425	166	2016	2016	NUM
ma-225	425	167	)	)	PUNCT
ma-225	425	168	38–57.[13	38–57.[13	NUM
ma-225	425	169	]	]	X
ma-225	425	170	i.	i.	PROPN
ma-225	425	171	franjić	franjić	PROPN
ma-225	425	172	,	,	PUNCT
ma-225	425	173	s.	s.	PROPN
ma-225	425	174	khalid	khalid	PROPN
ma-225	425	175	,	,	PUNCT
ma-225	425	176	j.	j.	PROPN
ma-225	425	177	pečarić	pečarić	PROPN
ma-225	425	178	,	,	PUNCT
ma-225	425	179	on	on	ADP
ma-225	425	180	the	the	DET
ma-225	425	181	refinements	refinement	NOUN
ma-225	425	182	of	of	ADP
ma-225	425	183	the	the	DET
ma-225	425	184	jensen	jensen	PROPN
ma-225	425	185	-	-	PUNCT
ma-225	425	186	steffensen	steffensen	PROPN
ma-225	425	187	inequality	inequality	PROPN
ma-225	425	188	,	,	PUNCT
ma-225	425	189	journal	journal	NOUN
ma-225	425	190	of	of	ADP
ma-225	425	191	inequalities	inequality	NOUN
ma-225	425	192	andapplications	andapplication	NOUN
ma-225	425	193	2011	2011	NUM
ma-225	425	194	(	(	PUNCT
ma-225	425	195	1	1	NUM
ma-225	425	196	)	)	PUNCT
ma-225	425	197	(	(	PUNCT
ma-225	425	198	2011	2011	NUM
ma-225	425	199	)	)	PUNCT
ma-225	425	200	1–11.[14	1–11.[14	NUM
ma-225	425	201	]	]	PUNCT
ma-225	425	202	m.	m.	NOUN
ma-225	425	203	bakula	bakula	PROPN
ma-225	425	204	,	,	PUNCT
ma-225	425	205	m.	m.	NOUN
ma-225	425	206	matić	matić	PROPN
ma-225	425	207	,	,	PUNCT
ma-225	425	208	j.	j.	PROPN
ma-225	425	209	pečarić	pečarić	PROPN
ma-225	425	210	,	,	PUNCT
ma-225	425	211	generalizations	generalization	NOUN
ma-225	425	212	of	of	ADP
ma-225	425	213	the	the	DET
ma-225	425	214	jensen	jensen	PROPN
ma-225	425	215	-	-	PUNCT
ma-225	425	216	steffensen	steffensen	PROPN
ma-225	425	217	and	and	CCONJ
ma-225	425	218	related	related	ADJ
ma-225	425	219	inequalities	inequality	NOUN
ma-225	425	220	,	,	PUNCT
ma-225	425	221	open	open	ADJ
ma-225	425	222	mathematics7	mathematics7	NOUN
ma-225	425	223	(	(	PUNCT
ma-225	425	224	4	4	NUM
ma-225	425	225	)	)	PUNCT
ma-225	425	226	(	(	PUNCT
ma-225	425	227	2009	2009	NUM
ma-225	425	228	)	)	PUNCT
ma-225	425	229	787–803.[15	787–803.[15	PROPN
ma-225	425	230	]	]	X
ma-225	425	231	h.	h.	PROPN
ma-225	425	232	li	li	PROPN
ma-225	425	233	,	,	PUNCT
ma-225	425	234	m.	m.	PROPN
ma-225	425	235	s.	s.	PROPN
ma-225	425	236	saleem	saleem	PROPN
ma-225	425	237	,	,	PUNCT
ma-225	425	238	i.	i.	PROPN
ma-225	425	239	ahmed	ahmed	PROPN
ma-225	425	240	,	,	PUNCT
ma-225	425	241	k.	k.	PROPN
ma-225	425	242	n.	n.	PROPN
ma-225	425	243	aslam	aslam	PROPN
ma-225	425	244	,	,	PUNCT
ma-225	425	245	hermite	hermite	ADJ
ma-225	425	246	–	–	PUNCT
ma-225	425	247	hadamard	hadamard	ADJ
ma-225	425	248	and	and	CCONJ
ma-225	425	249	fejér	fejér	NOUN
ma-225	425	250	-	-	PUNCT
ma-225	425	251	type	type	NOUN
ma-225	425	252	inequalities	inequality	NOUN
ma-225	425	253	for	for	ADP
ma-225	425	254	strongly	strongly	ADV
ma-225	425	255	reciprocally(p	reciprocally(p	PROPN
ma-225	425	256	,	,	PUNCT
ma-225	425	257	h)-convex	h)-convex	NOUN
ma-225	425	258	functions	function	NOUN
ma-225	425	259	of	of	ADP
ma-225	425	260	higher	high	ADJ
ma-225	425	261	order	order	NOUN
ma-225	425	262	,	,	PUNCT
ma-225	425	263	journal	journal	NOUN
ma-225	425	264	of	of	ADP
ma-225	425	265	inequalities	inequality	NOUN
ma-225	425	266	and	and	CCONJ
ma-225	425	267	applications	application	NOUN
ma-225	425	268	2023	2023	NUM
ma-225	425	269	(	(	PUNCT
ma-225	425	270	1	1	NUM
ma-225	425	271	)	)	PUNCT
ma-225	425	272	(	(	PUNCT
ma-225	425	273	2023	2023	NUM
ma-225	425	274	)	)	PUNCT
ma-225	425	275	1–20.[16	1–20.[16	NUM
ma-225	425	276	]	]	X
ma-225	425	277	i.	i.	PROPN
ma-225	425	278	iscan	iscan	PROPN
ma-225	425	279	,	,	PUNCT
ma-225	425	280	ostrowski	ostrowski	ADJ
ma-225	425	281	type	type	NOUN
ma-225	425	282	inequalities	inequality	NOUN
ma-225	425	283	for	for	ADP
ma-225	425	284	p	p	NOUN
ma-225	425	285	-	-	PUNCT
ma-225	425	286	convex	convex	NOUN
ma-225	425	287	functions	function	NOUN
ma-225	425	288	,	,	PUNCT
ma-225	425	289	new	new	ADJ
ma-225	425	290	trends	trend	NOUN
ma-225	425	291	in	in	ADP
ma-225	425	292	mathematical	mathematical	ADJ
ma-225	425	293	sciences	science	NOUN
ma-225	425	294	4	4	NUM
ma-225	425	295	(	(	PUNCT
ma-225	425	296	3	3	NUM
ma-225	425	297	)	)	PUNCT
ma-225	425	298	(	(	PUNCT
ma-225	425	299	2016)140–150.[17	2016)140–150.[17	PROPN
ma-225	425	300	]	]	PUNCT
ma-225	425	301	j.	j.	PROPN
ma-225	425	302	n.	n.	PROPN
ma-225	425	303	valdés	valdés	PROPN
ma-225	425	304	,	,	PUNCT
ma-225	425	305	f.	f.	PROPN
ma-225	425	306	rabossi	rabossi	PROPN
ma-225	425	307	,	,	PUNCT
ma-225	425	308	a.	a.	PROPN
ma-225	425	309	d.	d.	PROPN
ma-225	425	310	samaniego	samaniego	PROPN
ma-225	425	311	,	,	PUNCT
ma-225	425	312	convex	convex	NOUN
ma-225	425	313	functions	function	NOUN
ma-225	425	314	:	:	PUNCT
ma-225	425	315	ariadne	ariadne	PROPN
ma-225	425	316	’s	’s	PART
ma-225	425	317	thread	thread	NOUN
ma-225	425	318	or	or	CCONJ
ma-225	425	319	charlotte	charlotte	PROPN
ma-225	425	320	’s	’s	PART
ma-225	425	321	spiderweb	spiderweb	PROPN
ma-225	425	322	,	,	PUNCT
ma-225	425	323	advancedmathematical	advancedmathematical	ADJ
ma-225	425	324	models	model	NOUN
ma-225	425	325	&	&	CCONJ
ma-225	425	326	applications	application	NOUN
ma-225	425	327	5	5	NUM
ma-225	425	328	(	(	PUNCT
ma-225	425	329	2	2	NUM
ma-225	425	330	)	)	PUNCT
ma-225	425	331	(	(	PUNCT
ma-225	425	332	2020	2020	NUM
ma-225	425	333	)	)	PUNCT
ma-225	425	334	176–191	176–191	NUM
ma-225	425	335	.	.	PUNCT
ma-225	426	1	16	16	NUM
ma-225	426	2	1	1	NUM
ma-225	426	3	.	.	PUNCT
ma-225	426	4	introduction	introduction	NOUN
ma-225	426	5	2	2	NUM
ma-225	426	6	.	.	PUNCT
ma-225	426	7	preliminaries	preliminary	NOUN
ma-225	426	8	2.1	2.1	NUM
ma-225	426	9	.	.	PUNCT
ma-225	427	1	non	non	ADJ
ma-225	427	2	-	-	ADJ
ma-225	427	3	newtonian	newtonian	ADJ
ma-225	427	4	arithmetic	arithmetic	ADJ
ma-225	427	5	3	3	NUM
ma-225	427	6	.	.	NOUN
ma-225	427	7	results	result	NOUN
ma-225	427	8	and	and	CCONJ
ma-225	427	9	discussions	discussion	NOUN
ma-225	427	10	4	4	NUM
ma-225	427	11	.	.	PUNCT
ma-225	428	1	conclusion	conclusion	NOUN
ma-225	428	2	references	reference	NOUN