id	sid	tid	token	lemma	pos
ejpam-5103	1	1	european	european	PROPN
ejpam-5103	1	2	journal	journal	PROPN
ejpam-5103	1	3	of	of	ADP
ejpam-5103	1	4	pure	pure	ADJ
ejpam-5103	1	5	and	and	CCONJ
ejpam-5103	1	6	applied	apply	VERB
ejpam-5103	1	7	mathematics	mathematic	NOUN
ejpam-5103	1	8	vol	vol	NOUN
ejpam-5103	1	9	.	.	PROPN
ejpam-5103	2	1	17	17	NUM
ejpam-5103	2	2	,	,	PUNCT
ejpam-5103	2	3	no	no	INTJ
ejpam-5103	2	4	.	.	NOUN
ejpam-5103	2	5	2	2	NUM
ejpam-5103	2	6	,	,	PUNCT
ejpam-5103	2	7	2024	2024	NUM
ejpam-5103	2	8	,	,	PUNCT
ejpam-5103	2	9	860	860	NUM
ejpam-5103	2	10	-	-	SYM
ejpam-5103	2	11	869	869	NUM
ejpam-5103	2	12	issn	issn	PROPN
ejpam-5103	2	13	1307	1307	NUM
ejpam-5103	2	14	-	-	SYM
ejpam-5103	2	15	5543	5543	NUM
ejpam-5103	2	16	–	–	PUNCT
ejpam-5103	3	1	ejpam.com	ejpam.com	X
ejpam-5103	3	2	published	publish	VERB
ejpam-5103	3	3	by	by	ADP
ejpam-5103	3	4	new	new	PROPN
ejpam-5103	3	5	york	york	PROPN
ejpam-5103	3	6	business	business	PROPN
ejpam-5103	3	7	global	global	PROPN
ejpam-5103	3	8	strongly	strongly	ADV
ejpam-5103	3	9	geodesic	geodesic	ADJ
ejpam-5103	3	10	log	log	NOUN
ejpam-5103	3	11	-	-	PUNCT
ejpam-5103	3	12	preinvex	preinvex	NOUN
ejpam-5103	3	13	functions	function	NOUN
ejpam-5103	3	14	wedad	wedad	PROPN
ejpam-5103	3	15	saleh1,∗	saleh1,∗	PROPN
ejpam-5103	3	16	,	,	PUNCT
ejpam-5103	3	17	abdelghani	abdelghani	PROPN
ejpam-5103	3	18	lakhdari2	lakhdari2	PROPN
ejpam-5103	3	19	,	,	PUNCT
ejpam-5103	3	20	badreddine	badreddine	PROPN
ejpam-5103	3	21	meftah3	meftah3	PROPN
ejpam-5103	3	22	1department	1department	NUM
ejpam-5103	3	23	of	of	ADP
ejpam-5103	3	24	mathematics	mathematic	NOUN
ejpam-5103	3	25	,	,	PUNCT
ejpam-5103	3	26	taibah	taibah	PROPN
ejpam-5103	3	27	university	university	PROPN
ejpam-5103	3	28	,	,	PUNCT
ejpam-5103	3	29	almedina	almedina	NOUN
ejpam-5103	3	30	42353	42353	NUM
ejpam-5103	3	31	,	,	PUNCT
ejpam-5103	3	32	saudi	saudi	PROPN
ejpam-5103	3	33	arabia	arabia	PROPN
ejpam-5103	3	34	.	.	PUNCT
ejpam-5103	4	1	2department	2department	NUM
ejpam-5103	4	2	cpst	cpst	NOUN
ejpam-5103	4	3	,	,	PUNCT
ejpam-5103	4	4	national	national	ADJ
ejpam-5103	4	5	higher	high	ADJ
ejpam-5103	4	6	school	school	NOUN
ejpam-5103	4	7	of	of	ADP
ejpam-5103	4	8	technology	technology	NOUN
ejpam-5103	4	9	and	and	CCONJ
ejpam-5103	4	10	engineering	engineering	NOUN
ejpam-5103	4	11	,	,	PUNCT
ejpam-5103	4	12	annaba	annaba	PROPN
ejpam-5103	4	13	23005	23005	NUM
ejpam-5103	4	14	,	,	PUNCT
ejpam-5103	4	15	algeria	algeria	PROPN
ejpam-5103	4	16	.	.	PUNCT
ejpam-5103	5	1	3department	3department	NUM
ejpam-5103	5	2	of	of	ADP
ejpam-5103	5	3	mathematics	mathematic	NOUN
ejpam-5103	5	4	,	,	PUNCT
ejpam-5103	5	5	university	university	NOUN
ejpam-5103	5	6	8	8	NUM
ejpam-5103	5	7	may	may	PROPN
ejpam-5103	5	8	1945	1945	NUM
ejpam-5103	5	9	guelma	guelma	ADJ
ejpam-5103	5	10	,	,	PUNCT
ejpam-5103	5	11	algeria	algeria	PROPN
ejpam-5103	5	12	.	.	PUNCT
ejpam-5103	6	1	abstract	abstract	PROPN
ejpam-5103	6	2	.	.	PUNCT
ejpam-5103	7	1	in	in	ADP
ejpam-5103	7	2	this	this	DET
ejpam-5103	7	3	article	article	NOUN
ejpam-5103	7	4	,	,	PUNCT
ejpam-5103	7	5	we	we	PRON
ejpam-5103	7	6	delve	delve	VERB
ejpam-5103	7	7	into	into	ADP
ejpam-5103	7	8	the	the	DET
ejpam-5103	7	9	intriguing	intriguing	ADJ
ejpam-5103	7	10	concept	concept	NOUN
ejpam-5103	7	11	of	of	ADP
ejpam-5103	7	12	strongly	strongly	ADV
ejpam-5103	7	13	geodesic	geodesic	ADJ
ejpam-5103	7	14	log	log	NOUN
ejpam-5103	7	15	-	-	PUNCT
ejpam-5103	7	16	preinvex	preinvex	NOUN
ejpam-5103	7	17	functions	function	NOUN
ejpam-5103	7	18	in	in	ADP
ejpam-5103	7	19	riemannian	riemannian	ADJ
ejpam-5103	7	20	manifolds	manifold	NOUN
ejpam-5103	7	21	.	.	PUNCT
ejpam-5103	8	1	we	we	PRON
ejpam-5103	8	2	present	present	VERB
ejpam-5103	8	3	essential	essential	ADJ
ejpam-5103	8	4	preliminaries	preliminary	NOUN
ejpam-5103	8	5	and	and	CCONJ
ejpam-5103	8	6	fundamental	fundamental	ADJ
ejpam-5103	8	7	results	result	NOUN
ejpam-5103	8	8	that	that	PRON
ejpam-5103	8	9	shed	shed	VERB
ejpam-5103	8	10	light	light	NOUN
ejpam-5103	8	11	on	on	ADP
ejpam-5103	8	12	this	this	DET
ejpam-5103	8	13	specialized	specialized	ADJ
ejpam-5103	8	14	area	area	NOUN
ejpam-5103	8	15	of	of	ADP
ejpam-5103	8	16	study	study	NOUN
ejpam-5103	8	17	.	.	PUNCT
ejpam-5103	9	1	by	by	ADP
ejpam-5103	9	2	examining	examine	VERB
ejpam-5103	9	3	the	the	DET
ejpam-5103	9	4	properties	property	NOUN
ejpam-5103	9	5	and	and	CCONJ
ejpam-5103	9	6	implications	implication	NOUN
ejpam-5103	9	7	of	of	ADP
ejpam-5103	9	8	these	these	DET
ejpam-5103	9	9	functions	function	NOUN
ejpam-5103	9	10	,	,	PUNCT
ejpam-5103	9	11	we	we	PRON
ejpam-5103	9	12	aim	aim	VERB
ejpam-5103	9	13	to	to	PART
ejpam-5103	9	14	contribute	contribute	VERB
ejpam-5103	9	15	to	to	ADP
ejpam-5103	9	16	the	the	DET
ejpam-5103	9	17	growing	grow	VERB
ejpam-5103	9	18	body	body	NOUN
ejpam-5103	9	19	of	of	ADP
ejpam-5103	9	20	knowledge	knowledge	NOUN
ejpam-5103	9	21	in	in	ADP
ejpam-5103	9	22	convexity	convexity	NOUN
ejpam-5103	9	23	theory	theory	NOUN
ejpam-5103	9	24	within	within	ADP
ejpam-5103	9	25	the	the	DET
ejpam-5103	9	26	context	context	NOUN
ejpam-5103	9	27	of	of	ADP
ejpam-5103	9	28	riemannian	riemannian	ADJ
ejpam-5103	9	29	manifolds	manifold	NOUN
ejpam-5103	9	30	.	.	PUNCT
ejpam-5103	10	1	2020	2020	NUM
ejpam-5103	10	2	mathematics	mathematics	PROPN
ejpam-5103	10	3	subject	subject	NOUN
ejpam-5103	10	4	classifications	classification	NOUN
ejpam-5103	10	5	:	:	PUNCT
ejpam-5103	10	6	52a20	52a20	NUM
ejpam-5103	10	7	,	,	PUNCT
ejpam-5103	10	8	52a41	52a41	NUM
ejpam-5103	10	9	,	,	PUNCT
ejpam-5103	10	10	53c20	53c20	NUM
ejpam-5103	10	11	,	,	PUNCT
ejpam-5103	10	12	53c22	53c22	NUM
ejpam-5103	10	13	key	key	ADJ
ejpam-5103	10	14	words	word	NOUN
ejpam-5103	10	15	and	and	CCONJ
ejpam-5103	10	16	phrases	phrase	NOUN
ejpam-5103	10	17	:	:	PUNCT
ejpam-5103	10	18	log	log	NOUN
ejpam-5103	10	19	-	-	PUNCT
ejpam-5103	10	20	convex	convex	NOUN
ejpam-5103	10	21	functions	function	NOUN
ejpam-5103	10	22	,	,	PUNCT
ejpam-5103	10	23	geodesic	geodesic	ADJ
ejpam-5103	10	24	convex	convex	NOUN
ejpam-5103	10	25	functions	function	NOUN
ejpam-5103	10	26	,	,	PUNCT
ejpam-5103	10	27	geodesic	geodesic	ADJ
ejpam-5103	10	28	convex	convex	NOUN
ejpam-5103	10	29	sets	set	NOUN
ejpam-5103	10	30	,	,	PUNCT
ejpam-5103	10	31	riemannian	riemannian	ADJ
ejpam-5103	10	32	manifolds	manifold	NOUN
ejpam-5103	10	33	.	.	PUNCT
ejpam-5103	11	1	1	1	NUM
ejpam-5103	11	2	.	.	X
ejpam-5103	11	3	introduction	introduction	NOUN
ejpam-5103	11	4	and	and	CCONJ
ejpam-5103	11	5	preliminaries	preliminary	NOUN
ejpam-5103	11	6	let	let	VERB
ejpam-5103	11	7	ϑ	ϑ	PRON
ejpam-5103	11	8	⊆	⊆	NUM
ejpam-5103	11	9	r	r	NOUN
ejpam-5103	11	10	be	be	AUX
ejpam-5103	11	11	an	an	DET
ejpam-5103	11	12	interval	interval	NOUN
ejpam-5103	11	13	.	.	PUNCT
ejpam-5103	12	1	a	a	DET
ejpam-5103	12	2	function	function	NOUN
ejpam-5103	12	3	ξ	ξ	NOUN
ejpam-5103	12	4	:	:	PUNCT
ejpam-5103	12	5	ϑ	ϑ	X
ejpam-5103	12	6	−→	−→	NOUN
ejpam-5103	12	7	r	r	NOUN
ejpam-5103	12	8	is	be	AUX
ejpam-5103	12	9	said	say	VERB
ejpam-5103	12	10	to	to	PART
ejpam-5103	12	11	be	be	AUX
ejpam-5103	12	12	strongly	strongly	ADV
ejpam-5103	12	13	convex	convex	ADJ
ejpam-5103	12	14	with	with	ADP
ejpam-5103	12	15	modulus	modulus	ADJ
ejpam-5103	12	16	ε	ε	PROPN
ejpam-5103	12	17	>	>	X
ejpam-5103	12	18	0	0	PUNCT
ejpam-5103	13	1	if	if	SCONJ
ejpam-5103	13	2	ξ(ςu1+(1−ς)u2	ξ(ςu1+(1−ς)u2	NOUN
ejpam-5103	13	3	)	)	PUNCT
ejpam-5103	13	4	≤	≤	NOUN
ejpam-5103	13	5	ςξ(u1)+(1−ς)ξ(u2)−ες(1−ς)(u1−u2	ςξ(u1)+(1−ς)ξ(u2)−ες(1−ς)(u1−u2	NOUN
ejpam-5103	13	6	)	)	PUNCT
ejpam-5103	13	7	2	2	NUM
ejpam-5103	13	8	,	,	PUNCT
ejpam-5103	13	9	∀µ1	∀µ1	PROPN
ejpam-5103	13	10	,	,	PUNCT
ejpam-5103	13	11	µ2	µ2	PROPN
ejpam-5103	13	12	∈	∈	PROPN
ejpam-5103	13	13	ϑ	ϑ	NOUN
ejpam-5103	13	14	,	,	PUNCT
ejpam-5103	13	15	ς	ς	PROPN
ejpam-5103	13	16	∈	∈	PROPN
ejpam-5103	14	1	[	[	X
ejpam-5103	14	2	0	0	NUM
ejpam-5103	14	3	,	,	PUNCT
ejpam-5103	14	4	1	1	NUM
ejpam-5103	14	5	]	]	PUNCT
ejpam-5103	14	6	.	.	PUNCT
ejpam-5103	15	1	(	(	PUNCT
ejpam-5103	15	2	1	1	X
ejpam-5103	15	3	)	)	PUNCT
ejpam-5103	15	4	the	the	DET
ejpam-5103	15	5	concept	concept	NOUN
ejpam-5103	15	6	of	of	ADP
ejpam-5103	15	7	strongly	strongly	ADV
ejpam-5103	15	8	convex	convex	ADJ
ejpam-5103	15	9	functions	function	NOUN
ejpam-5103	15	10	,	,	PUNCT
ejpam-5103	15	11	initially	initially	ADV
ejpam-5103	15	12	introduced	introduce	VERB
ejpam-5103	15	13	by	by	ADP
ejpam-5103	15	14	polyak	polyak	NOUN
ejpam-5103	15	15	(	(	PUNCT
ejpam-5103	15	16	1966	1966	NUM
ejpam-5103	15	17	)	)	PUNCT
ejpam-5103	16	1	[	[	X
ejpam-5103	16	2	13	13	NUM
ejpam-5103	16	3	]	]	PUNCT
ejpam-5103	16	4	,	,	PUNCT
ejpam-5103	16	5	holds	hold	VERB
ejpam-5103	16	6	substantial	substantial	ADJ
ejpam-5103	16	7	relevance	relevance	NOUN
ejpam-5103	16	8	in	in	ADP
ejpam-5103	16	9	the	the	DET
ejpam-5103	16	10	fields	field	NOUN
ejpam-5103	16	11	of	of	ADP
ejpam-5103	16	12	optimization	optimization	NOUN
ejpam-5103	16	13	theory	theory	NOUN
ejpam-5103	16	14	and	and	CCONJ
ejpam-5103	16	15	mathematical	mathematical	ADJ
ejpam-5103	16	16	economics	economic	NOUN
ejpam-5103	16	17	.	.	PUNCT
ejpam-5103	17	1	an	an	DET
ejpam-5103	17	2	extensive	extensive	ADJ
ejpam-5103	17	3	exploration	exploration	NOUN
ejpam-5103	17	4	of	of	ADP
ejpam-5103	17	5	their	their	PRON
ejpam-5103	17	6	properties	property	NOUN
ejpam-5103	17	7	and	and	CCONJ
ejpam-5103	17	8	applications	application	NOUN
ejpam-5103	17	9	is	be	AUX
ejpam-5103	17	10	well	well	ADV
ejpam-5103	17	11	-	-	PUNCT
ejpam-5103	17	12	documented	document	VERB
ejpam-5103	17	13	across	across	ADP
ejpam-5103	17	14	various	various	ADJ
ejpam-5103	17	15	studies	study	NOUN
ejpam-5103	17	16	,	,	PUNCT
ejpam-5103	17	17	including	include	VERB
ejpam-5103	17	18	those	those	PRON
ejpam-5103	17	19	by	by	ADP
ejpam-5103	17	20	angulo	angulo	PROPN
ejpam-5103	17	21	et	et	PROPN
ejpam-5103	17	22	al	al	PROPN
ejpam-5103	17	23	.	.	PUNCT
ejpam-5103	18	1	[	[	X
ejpam-5103	18	2	1	1	NUM
ejpam-5103	18	3	]	]	PUNCT
ejpam-5103	18	4	,	,	PUNCT
ejpam-5103	18	5	awan	awan	PROPN
ejpam-5103	18	6	et	et	PROPN
ejpam-5103	18	7	al	al	PROPN
ejpam-5103	18	8	.	.	PUNCT
ejpam-5103	19	1	[	[	X
ejpam-5103	19	2	2	2	NUM
ejpam-5103	19	3	]	]	PUNCT
ejpam-5103	19	4	,	,	PUNCT
ejpam-5103	19	5	and	and	CCONJ
ejpam-5103	19	6	merentes	merente	VERB
ejpam-5103	19	7	et	et	PROPN
ejpam-5103	19	8	al	al	PROPN
ejpam-5103	19	9	.	.	PUNCT
ejpam-5103	20	1	[	[	X
ejpam-5103	20	2	6	6	NUM
ejpam-5103	20	3	]	]	PUNCT
ejpam-5103	20	4	,	,	PUNCT
ejpam-5103	20	5	among	among	ADP
ejpam-5103	20	6	others	other	NOUN
ejpam-5103	20	7	.	.	PUNCT
ejpam-5103	21	1	the	the	DET
ejpam-5103	21	2	notion	notion	NOUN
ejpam-5103	21	3	of	of	ADP
ejpam-5103	21	4	convexity	convexity	NOUN
ejpam-5103	21	5	has	have	AUX
ejpam-5103	21	6	been	be	AUX
ejpam-5103	21	7	expanded	expand	VERB
ejpam-5103	21	8	to	to	PART
ejpam-5103	21	9	encompass	encompass	VERB
ejpam-5103	21	10	strong	strong	ADJ
ejpam-5103	21	11	convexity	convexity	NOUN
ejpam-5103	21	12	of	of	ADP
ejpam-5103	21	13	order	order	NOUN
ejpam-5103	21	14	n	n	CCONJ
ejpam-5103	21	15	on	on	ADP
ejpam-5103	21	16	rn	rn	PROPN
ejpam-5103	21	17	,	,	PUNCT
ejpam-5103	21	18	as	as	SCONJ
ejpam-5103	21	19	defined	define	VERB
ejpam-5103	21	20	by	by	ADP
ejpam-5103	21	21	lin	lin	PROPN
ejpam-5103	21	22	et	et	PROPN
ejpam-5103	21	23	al	al	PROPN
ejpam-5103	21	24	.	.	PUNCT
ejpam-5103	22	1	[	[	X
ejpam-5103	22	2	5	5	NUM
ejpam-5103	22	3	]	]	PUNCT
ejpam-5103	22	4	:	:	PUNCT
ejpam-5103	22	5	a	a	DET
ejpam-5103	22	6	function	function	NOUN
ejpam-5103	22	7	ξ	ξ	NOUN
ejpam-5103	22	8	,	,	PUNCT
ejpam-5103	22	9	defined	define	VERB
ejpam-5103	22	10	on	on	ADP
ejpam-5103	22	11	a	a	DET
ejpam-5103	22	12	subset	subset	NOUN
ejpam-5103	22	13	ϑ	ϑ	X
ejpam-5103	22	14	of	of	ADP
ejpam-5103	22	15	the	the	DET
ejpam-5103	22	16	real	real	ADJ
ejpam-5103	22	17	numbers	number	NOUN
ejpam-5103	22	18	(	(	PUNCT
ejpam-5103	22	19	r	r	NOUN
ejpam-5103	22	20	)	)	PUNCT
ejpam-5103	22	21	,	,	PUNCT
ejpam-5103	22	22	is	be	AUX
ejpam-5103	22	23	termed	term	VERB
ejpam-5103	22	24	strongly	strongly	ADV
ejpam-5103	22	25	convex	convex	ADJ
ejpam-5103	22	26	of	of	ADP
ejpam-5103	22	27	order	order	NOUN
ejpam-5103	22	28	n	n	NOUN
ejpam-5103	22	29	if	if	SCONJ
ejpam-5103	22	30	it	it	PRON
ejpam-5103	22	31	satisfies	satisfy	VERB
ejpam-5103	22	32	the	the	DET
ejpam-5103	22	33	following	follow	VERB
ejpam-5103	22	34	condition	condition	NOUN
ejpam-5103	22	35	for	for	ADP
ejpam-5103	22	36	all	all	DET
ejpam-5103	22	37	µ1	µ1	NOUN
ejpam-5103	22	38	and	and	CCONJ
ejpam-5103	22	39	µ2	µ2	PROPN
ejpam-5103	22	40	within	within	ADP
ejpam-5103	22	41	ϑ	ϑ	PROPN
ejpam-5103	22	42	and	and	CCONJ
ejpam-5103	22	43	for	for	ADP
ejpam-5103	22	44	ς	ς	PROPN
ejpam-5103	22	45	in	in	ADP
ejpam-5103	22	46	the	the	DET
ejpam-5103	22	47	range	range	NOUN
ejpam-5103	23	1	[	[	X
ejpam-5103	23	2	0	0	NUM
ejpam-5103	23	3	,	,	PUNCT
ejpam-5103	23	4	1	1	NUM
ejpam-5103	23	5	]	]	PUNCT
ejpam-5103	23	6	:	:	PUNCT
ejpam-5103	23	7	there	there	PRON
ejpam-5103	23	8	exists	exist	VERB
ejpam-5103	23	9	a	a	DET
ejpam-5103	23	10	positive	positive	ADJ
ejpam-5103	23	11	constant	constant	ADJ
ejpam-5103	23	12	ε	ε	PROPN
ejpam-5103	23	13	>	>	X
ejpam-5103	23	14	0	0	NUM
ejpam-5103	23	15	such	such	ADJ
ejpam-5103	23	16	that	that	SCONJ
ejpam-5103	23	17	:	:	PUNCT
ejpam-5103	23	18	∗corresponding	∗corresponde	VERB
ejpam-5103	23	19	author	author	NOUN
ejpam-5103	23	20	.	.	PUNCT
ejpam-5103	24	1	doi	doi	NOUN
ejpam-5103	24	2	:	:	PUNCT
ejpam-5103	24	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5103	https://doi.org/10.29020/nybg.ejpam.v17i2.5103	NOUN
ejpam-5103	24	4	email	email	NOUN
ejpam-5103	24	5	addresses	address	NOUN
ejpam-5103	24	6	:	:	PUNCT
ejpam-5103	24	7	wlehabi@taibahu.edu.sa	wlehabi@taibahu.edu.sa	PROPN
ejpam-5103	24	8	(	(	PUNCT
ejpam-5103	24	9	w.	w.	PROPN
ejpam-5103	24	10	saleh	saleh	PROPN
ejpam-5103	24	11	)	)	PUNCT
ejpam-5103	24	12	,	,	PUNCT
ejpam-5103	24	13	a.lakhdari@ensti-annaba.dz	a.lakhdari@ensti-annaba.dz	NOUN
ejpam-5103	24	14	(	(	PUNCT
ejpam-5103	24	15	a.	a.	NOUN
ejpam-5103	24	16	lakhdari	lakhdari	PROPN
ejpam-5103	24	17	)	)	PUNCT
ejpam-5103	24	18	,	,	PUNCT
ejpam-5103	24	19	badrimeftah@yahoo.fr	badrimeftah@yahoo.fr	PROPN
ejpam-5103	24	20	(	(	PUNCT
ejpam-5103	24	21	b.	b.	PROPN
ejpam-5103	24	22	meftah	meftah	PROPN
ejpam-5103	24	23	)	)	PUNCT
ejpam-5103	24	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5103	24	25	860	860	NUM
ejpam-5103	25	1	©	©	PROPN
ejpam-5103	25	2	2024	2024	NUM
ejpam-5103	25	3	ejpam	ejpam	NOUN
ejpam-5103	25	4	all	all	DET
ejpam-5103	25	5	rights	right	NOUN
ejpam-5103	25	6	reserved	reserve	VERB
ejpam-5103	25	7	.	.	PUNCT
ejpam-5103	26	1	w.	w.	PROPN
ejpam-5103	26	2	saleh	saleh	PROPN
ejpam-5103	26	3	,	,	PUNCT
ejpam-5103	26	4	a.	a.	NOUN
ejpam-5103	26	5	lakhdari	lakhdari	PROPN
ejpam-5103	26	6	,	,	PUNCT
ejpam-5103	26	7	b.	b.	PROPN
ejpam-5103	26	8	meftah	meftah	PROPN
ejpam-5103	26	9	/	/	SYM
ejpam-5103	26	10	eur	eur	PROPN
ejpam-5103	26	11	.	.	PUNCT
ejpam-5103	27	1	j.	j.	PROPN
ejpam-5103	27	2	pure	pure	PROPN
ejpam-5103	27	3	appl	appl	PROPN
ejpam-5103	27	4	.	.	PROPN
ejpam-5103	27	5	math	math	PROPN
ejpam-5103	27	6	,	,	PUNCT
ejpam-5103	27	7	17	17	NUM
ejpam-5103	27	8	(	(	PUNCT
ejpam-5103	27	9	2	2	NUM
ejpam-5103	27	10	)	)	PUNCT
ejpam-5103	27	11	(	(	PUNCT
ejpam-5103	27	12	2024	2024	NUM
ejpam-5103	27	13	)	)	PUNCT
ejpam-5103	27	14	,	,	PUNCT
ejpam-5103	27	15	860	860	NUM
ejpam-5103	27	16	-	-	SYM
ejpam-5103	27	17	869	869	NUM
ejpam-5103	27	18	861	861	NUM
ejpam-5103	27	19	ξ(ςµ1	ξ(ςµ1	NOUN
ejpam-5103	27	20	+	+	CCONJ
ejpam-5103	27	21	(	(	PUNCT
ejpam-5103	27	22	1−	1−	NUM
ejpam-5103	27	23	ς)µ2	ς)µ2	PROPN
ejpam-5103	27	24	)	)	PUNCT
ejpam-5103	27	25	≤	≤	NUM
ejpam-5103	27	26	ςξ(µ1	ςξ(µ1	NOUN
ejpam-5103	27	27	)	)	PUNCT
ejpam-5103	28	1	+	+	CCONJ
ejpam-5103	28	2	(	(	PUNCT
ejpam-5103	28	3	1−	1−	NUM
ejpam-5103	28	4	ς)ξ(µ2)−	ς)ξ(µ2)−	NUM
ejpam-5103	28	5	ες(1−	ες(1−	PROPN
ejpam-5103	28	6	ς)∥µ1	ς)∥µ1	NOUN
ejpam-5103	28	7	−	−	NOUN
ejpam-5103	28	8	µ2∥n	µ2∥n	NOUN
ejpam-5103	28	9	.	.	PUNCT
ejpam-5103	29	1	this	this	DET
ejpam-5103	29	2	condition	condition	NOUN
ejpam-5103	29	3	characterizes	characterize	VERB
ejpam-5103	29	4	the	the	DET
ejpam-5103	29	5	strong	strong	ADJ
ejpam-5103	29	6	convexity	convexity	NOUN
ejpam-5103	29	7	property	property	NOUN
ejpam-5103	29	8	of	of	ADP
ejpam-5103	29	9	the	the	DET
ejpam-5103	29	10	function	function	NOUN
ejpam-5103	29	11	ξ	ξ	PROPN
ejpam-5103	29	12	in	in	ADP
ejpam-5103	29	13	the	the	DET
ejpam-5103	29	14	context	context	NOUN
ejpam-5103	29	15	of	of	ADP
ejpam-5103	29	16	mathematical	mathematical	ADJ
ejpam-5103	29	17	analysis	analysis	NOUN
ejpam-5103	29	18	.	.	PUNCT
ejpam-5103	30	1	additional	additional	ADJ
ejpam-5103	30	2	applications	application	NOUN
ejpam-5103	30	3	,	,	PUNCT
ejpam-5103	30	4	numerical	numerical	ADJ
ejpam-5103	30	5	techniques	technique	NOUN
ejpam-5103	30	6	,	,	PUNCT
ejpam-5103	30	7	and	and	CCONJ
ejpam-5103	30	8	variational	variational	ADJ
ejpam-5103	30	9	-	-	PUNCT
ejpam-5103	30	10	like	like	ADJ
ejpam-5103	30	11	inequalities	inequality	NOUN
ejpam-5103	30	12	for	for	ADP
ejpam-5103	30	13	convex	convex	NOUN
ejpam-5103	30	14	functions	function	NOUN
ejpam-5103	30	15	are	be	AUX
ejpam-5103	30	16	discussed	discuss	VERB
ejpam-5103	30	17	in	in	ADP
ejpam-5103	30	18	[	[	X
ejpam-5103	30	19	10	10	NUM
ejpam-5103	30	20	,	,	PUNCT
ejpam-5103	30	21	11	11	NUM
ejpam-5103	30	22	]	]	PUNCT
ejpam-5103	30	23	.	.	PUNCT
ejpam-5103	31	1	it	it	PRON
ejpam-5103	31	2	is	be	AUX
ejpam-5103	31	3	noteworthy	noteworthy	ADJ
ejpam-5103	31	4	that	that	SCONJ
ejpam-5103	31	5	log	log	NOUN
ejpam-5103	31	6	-	-	PUNCT
ejpam-5103	31	7	convex	convex	NOUN
ejpam-5103	31	8	functions	function	NOUN
ejpam-5103	31	9	,	,	PUNCT
ejpam-5103	31	10	as	as	SCONJ
ejpam-5103	31	11	opposed	oppose	VERB
ejpam-5103	31	12	to	to	PART
ejpam-5103	31	13	convex	convex	NOUN
ejpam-5103	31	14	functions	function	NOUN
ejpam-5103	31	15	,	,	PUNCT
ejpam-5103	31	16	have	have	AUX
ejpam-5103	31	17	been	be	AUX
ejpam-5103	31	18	shown	show	VERB
ejpam-5103	31	19	to	to	PART
ejpam-5103	31	20	yield	yield	VERB
ejpam-5103	31	21	more	more	ADV
ejpam-5103	31	22	precise	precise	ADJ
ejpam-5103	31	23	results	result	NOUN
ejpam-5103	31	24	and	and	CCONJ
ejpam-5103	31	25	inequalities	inequality	NOUN
ejpam-5103	31	26	.	.	PUNCT
ejpam-5103	32	1	numerous	numerous	ADJ
ejpam-5103	32	2	aspects	aspect	NOUN
ejpam-5103	32	3	related	relate	VERB
ejpam-5103	32	4	to	to	ADP
ejpam-5103	32	5	exponentially	exponentially	ADV
ejpam-5103	32	6	preinvex	preinvex	NOUN
ejpam-5103	32	7	functions	function	NOUN
ejpam-5103	32	8	and	and	CCONJ
ejpam-5103	32	9	their	their	PRON
ejpam-5103	32	10	variants	variant	NOUN
ejpam-5103	32	11	are	be	AUX
ejpam-5103	32	12	introduced	introduce	VERB
ejpam-5103	32	13	in	in	ADP
ejpam-5103	32	14	[	[	X
ejpam-5103	32	15	8	8	NUM
ejpam-5103	32	16	,	,	PUNCT
ejpam-5103	32	17	9	9	NUM
ejpam-5103	32	18	]	]	PUNCT
ejpam-5103	32	19	.	.	PUNCT
ejpam-5103	33	1	let	let	AUX
ejpam-5103	33	2	(	(	PUNCT
ejpam-5103	33	3	n	n	X
ejpam-5103	33	4	,	,	PUNCT
ejpam-5103	33	5	ξ	ξ	X
ejpam-5103	33	6	)	)	PUNCT
ejpam-5103	33	7	be	be	VERB
ejpam-5103	33	8	a	a	DET
ejpam-5103	33	9	complete	complete	ADJ
ejpam-5103	33	10	m	m	ADJ
ejpam-5103	33	11	-	-	ADJ
ejpam-5103	33	12	dimensional	dimensional	ADJ
ejpam-5103	33	13	riemannian	riemannian	ADJ
ejpam-5103	33	14	manifold	manifold	NOUN
ejpam-5103	33	15	equipped	equip	VERB
ejpam-5103	33	16	with	with	ADP
ejpam-5103	33	17	a	a	DET
ejpam-5103	33	18	riemannian	riemannian	ADJ
ejpam-5103	33	19	connection	connection	NOUN
ejpam-5103	33	20	∇.	∇.	PRON
ejpam-5103	33	21	consider	consider	VERB
ejpam-5103	33	22	a	a	DET
ejpam-5103	33	23	piecewise	piecewise	NOUN
ejpam-5103	33	24	c1	c1	NOUN
ejpam-5103	33	25	path	path	NOUN
ejpam-5103	33	26	γ	γ	X
ejpam-5103	33	27	:	:	PUNCT
ejpam-5103	34	1	[	[	X
ejpam-5103	34	2	µ1	µ1	ADJ
ejpam-5103	34	3	,	,	PUNCT
ejpam-5103	34	4	µ2	µ2	PROPN
ejpam-5103	34	5	]	]	PUNCT
ejpam-5103	34	6	−→	−→	NOUN
ejpam-5103	34	7	ϑ	ϑ	X
ejpam-5103	34	8	connecting	connect	VERB
ejpam-5103	34	9	a1	a1	NOUN
ejpam-5103	34	10	to	to	ADP
ejpam-5103	34	11	a2	a2	PROPN
ejpam-5103	34	12	,	,	PUNCT
ejpam-5103	34	13	where	where	SCONJ
ejpam-5103	34	14	γ(µ1	γ(µ1	NOUN
ejpam-5103	34	15	)	)	PUNCT
ejpam-5103	34	16	=	=	SYM
ejpam-5103	34	17	a2	a2	PROPN
ejpam-5103	34	18	and	and	CCONJ
ejpam-5103	34	19	γ(µ2	γ(µ2	PRON
ejpam-5103	34	20	)	)	PUNCT
ejpam-5103	34	21	=	=	SYM
ejpam-5103	34	22	a1	a1	NOUN
ejpam-5103	34	23	.	.	PUNCT
ejpam-5103	35	1	the	the	DET
ejpam-5103	35	2	length	length	NOUN
ejpam-5103	35	3	of	of	ADP
ejpam-5103	35	4	γ	γ	PROPN
ejpam-5103	35	5	is	be	AUX
ejpam-5103	35	6	defined	define	VERB
ejpam-5103	35	7	as	as	ADP
ejpam-5103	35	8	:	:	PUNCT
ejpam-5103	35	9	l(γ	l(γ	PROPN
ejpam-5103	35	10	)	)	PUNCT
ejpam-5103	36	1	=	=	SYM
ejpam-5103	36	2	∫	∫	PROPN
ejpam-5103	36	3	u2	u2	PROPN
ejpam-5103	36	4	u1	u1	PROPN
ejpam-5103	36	5	∥γ́(λ)∥γ(λ)dλ	∥γ́(λ)∥γ(λ)dλ	PROPN
ejpam-5103	36	6	.	.	PUNCT
ejpam-5103	37	1	for	for	ADP
ejpam-5103	37	2	any	any	DET
ejpam-5103	37	3	two	two	NUM
ejpam-5103	37	4	points	point	NOUN
ejpam-5103	37	5	a1	a1	NOUN
ejpam-5103	37	6	and	and	CCONJ
ejpam-5103	37	7	a2	a2	PROPN
ejpam-5103	37	8	in	in	ADP
ejpam-5103	37	9	n	n	PROPN
ejpam-5103	37	10	,	,	PUNCT
ejpam-5103	37	11	we	we	PRON
ejpam-5103	37	12	introduce	introduce	VERB
ejpam-5103	37	13	the	the	DET
ejpam-5103	37	14	following	follow	VERB
ejpam-5103	37	15	metric	metric	NOUN
ejpam-5103	37	16	:	:	PUNCT
ejpam-5103	37	17	d(a1	d(a1	NOUN
ejpam-5103	37	18	,	,	PUNCT
ejpam-5103	37	19	a2	a2	PROPN
ejpam-5103	37	20	)	)	PUNCT
ejpam-5103	37	21	=	=	SYM
ejpam-5103	37	22	inf	inf	NOUN
ejpam-5103	37	23	{	{	PUNCT
ejpam-5103	37	24	l(γ	l(γ	PROPN
ejpam-5103	37	25	)	)	PUNCT
ejpam-5103	37	26	:	:	PUNCT
ejpam-5103	38	1	γ	γ	X
ejpam-5103	38	2	is	be	AUX
ejpam-5103	38	3	a	a	DET
ejpam-5103	38	4	piecewise	piecewise	NOUN
ejpam-5103	38	5	c1	c1	NOUN
ejpam-5103	38	6	path	path	NOUN
ejpam-5103	38	7	connecting	connect	VERB
ejpam-5103	38	8	a1	a1	NOUN
ejpam-5103	38	9	to	to	ADP
ejpam-5103	38	10	a2	a2	PROPN
ejpam-5103	38	11	}	}	PUNCT
ejpam-5103	38	12	.	.	PUNCT
ejpam-5103	39	1	this	this	DET
ejpam-5103	39	2	metric	metric	ADJ
ejpam-5103	39	3	d	d	PROPN
ejpam-5103	39	4	induces	induce	VERB
ejpam-5103	39	5	the	the	DET
ejpam-5103	39	6	original	original	ADJ
ejpam-5103	39	7	topology	topology	NOUN
ejpam-5103	39	8	on	on	ADP
ejpam-5103	39	9	n	n	PROPN
ejpam-5103	39	10	.	.	PUNCT
ejpam-5103	40	1	in	in	ADP
ejpam-5103	40	2	every	every	DET
ejpam-5103	40	3	riemannian	riemannian	ADJ
ejpam-5103	40	4	manifold	manifold	NOUN
ejpam-5103	40	5	,	,	PUNCT
ejpam-5103	40	6	there	there	PRON
ejpam-5103	40	7	exists	exist	VERB
ejpam-5103	40	8	a	a	DET
ejpam-5103	40	9	uniquely	uniquely	ADV
ejpam-5103	40	10	determined	determined	ADJ
ejpam-5103	40	11	riemannian	riemannian	ADJ
ejpam-5103	40	12	connection	connection	NOUN
ejpam-5103	40	13	known	know	VERB
ejpam-5103	40	14	as	as	ADP
ejpam-5103	40	15	the	the	DET
ejpam-5103	40	16	levi	levi	PROPN
ejpam-5103	40	17	-	-	PUNCT
ejpam-5103	40	18	civita	civita	PROPN
ejpam-5103	40	19	connection	connection	NOUN
ejpam-5103	40	20	,	,	PUNCT
ejpam-5103	40	21	denoted	denote	VERB
ejpam-5103	40	22	by	by	ADP
ejpam-5103	40	23	∇xy	∇xy	PROPN
ejpam-5103	40	24	,	,	PUNCT
ejpam-5103	40	25	for	for	ADP
ejpam-5103	40	26	any	any	DET
ejpam-5103	40	27	vector	vector	NOUN
ejpam-5103	40	28	fields	field	NOUN
ejpam-5103	40	29	x	x	PUNCT
ejpam-5103	40	30	and	and	CCONJ
ejpam-5103	40	31	y	y	PROPN
ejpam-5103	40	32	in	in	ADP
ejpam-5103	40	33	ϑ.	ϑ.	NOUN
ejpam-5103	40	34	furthermore	furthermore	ADV
ejpam-5103	40	35	,	,	PUNCT
ejpam-5103	40	36	a	a	DET
ejpam-5103	40	37	smooth	smooth	ADJ
ejpam-5103	40	38	path	path	NOUN
ejpam-5103	40	39	γ	γ	NOUN
ejpam-5103	40	40	is	be	AUX
ejpam-5103	40	41	considered	consider	VERB
ejpam-5103	40	42	a	a	DET
ejpam-5103	40	43	geodesic	geodesic	NOUN
ejpam-5103	40	44	if	if	SCONJ
ejpam-5103	40	45	and	and	CCONJ
ejpam-5103	40	46	only	only	ADV
ejpam-5103	40	47	if	if	SCONJ
ejpam-5103	40	48	its	its	PRON
ejpam-5103	40	49	tangent	tangent	NOUN
ejpam-5103	40	50	vector	vector	NOUN
ejpam-5103	40	51	is	be	AUX
ejpam-5103	40	52	a	a	DET
ejpam-5103	40	53	parallel	parallel	ADJ
ejpam-5103	40	54	vector	vector	NOUN
ejpam-5103	40	55	field	field	NOUN
ejpam-5103	40	56	along	along	ADP
ejpam-5103	40	57	the	the	DET
ejpam-5103	40	58	path	path	NOUN
ejpam-5103	40	59	γ	γ	X
ejpam-5103	40	60	,	,	PUNCT
ejpam-5103	40	61	i.e.	i.e.	X
ejpam-5103	40	62	,	,	PUNCT
ejpam-5103	40	63	γ	γ	X
ejpam-5103	40	64	satisfies	satisfy	VERB
ejpam-5103	40	65	the	the	DET
ejpam-5103	40	66	equation	equation	NOUN
ejpam-5103	40	67	∇γ′γ′	∇γ′γ′	NOUN
ejpam-5103	40	68	=	=	PUNCT
ejpam-5103	40	69	0	0	X
ejpam-5103	40	70	.	.	PUNCT
ejpam-5103	41	1	any	any	DET
ejpam-5103	41	2	path	path	NOUN
ejpam-5103	41	3	γ	γ	NOUN
ejpam-5103	41	4	that	that	PRON
ejpam-5103	41	5	connects	connect	VERB
ejpam-5103	41	6	µ1	µ1	PROPN
ejpam-5103	41	7	and	and	CCONJ
ejpam-5103	41	8	µ2	µ2	PROPN
ejpam-5103	41	9	in	in	ADP
ejpam-5103	41	10	n	n	CCONJ
ejpam-5103	41	11	such	such	ADJ
ejpam-5103	41	12	that	that	SCONJ
ejpam-5103	41	13	l(γ	l(γ	PROPN
ejpam-5103	41	14	)	)	PUNCT
ejpam-5103	42	1	=	=	SYM
ejpam-5103	42	2	d(µ1	d(µ1	NOUN
ejpam-5103	42	3	,	,	PUNCT
ejpam-5103	42	4	µ2	µ2	PROPN
ejpam-5103	42	5	)	)	PUNCT
ejpam-5103	42	6	is	be	AUX
ejpam-5103	42	7	a	a	DET
ejpam-5103	42	8	geodesic	geodesic	NOUN
ejpam-5103	42	9	and	and	CCONJ
ejpam-5103	42	10	is	be	AUX
ejpam-5103	42	11	referred	refer	VERB
ejpam-5103	42	12	to	to	ADP
ejpam-5103	42	13	as	as	ADP
ejpam-5103	42	14	a	a	DET
ejpam-5103	42	15	minimal	minimal	ADJ
ejpam-5103	42	16	geodesic	geodesic	NOUN
ejpam-5103	42	17	.	.	PUNCT
ejpam-5103	43	1	let	let	VERB
ejpam-5103	43	2	n	n	PRON
ejpam-5103	43	3	be	be	AUX
ejpam-5103	43	4	a	a	DET
ejpam-5103	43	5	c∞	c∞	PROPN
ejpam-5103	43	6	complete	complete	ADJ
ejpam-5103	43	7	n	n	CCONJ
ejpam-5103	43	8	-	-	PUNCT
ejpam-5103	43	9	dimensional	dimensional	ADJ
ejpam-5103	43	10	riemannian	riemannian	NOUN
ejpam-5103	43	11	manifold	manifold	NOUN
ejpam-5103	43	12	with	with	ADP
ejpam-5103	43	13	metric	metric	ADJ
ejpam-5103	43	14	g	g	PROPN
ejpam-5103	43	15	and	and	CCONJ
ejpam-5103	43	16	levicivita	levicivita	PROPN
ejpam-5103	43	17	connection∇.	connection∇.	PROPN
ejpam-5103	43	18	additionally	additionally	ADV
ejpam-5103	43	19	,	,	PUNCT
ejpam-5103	43	20	consider	consider	VERB
ejpam-5103	43	21	the	the	DET
ejpam-5103	43	22	points	point	NOUN
ejpam-5103	43	23	µ1	µ1	PROPN
ejpam-5103	43	24	and	and	CCONJ
ejpam-5103	43	25	µ2	µ2	PROPN
ejpam-5103	43	26	inn	inn	PROPN
ejpam-5103	43	27	,	,	PUNCT
ejpam-5103	43	28	and	and	CCONJ
ejpam-5103	43	29	let	let	VERB
ejpam-5103	43	30	γ	γ	X
ejpam-5103	43	31	:	:	PUNCT
ejpam-5103	43	32	[	[	X
ejpam-5103	43	33	0	0	NUM
ejpam-5103	43	34	,	,	PUNCT
ejpam-5103	43	35	1	1	NUM
ejpam-5103	43	36	]	]	X
ejpam-5103	43	37	−→	−→	NOUN
ejpam-5103	43	38	n	n	AUX
ejpam-5103	43	39	be	be	AUX
ejpam-5103	43	40	a	a	DET
ejpam-5103	43	41	geodesic	geodesic	NOUN
ejpam-5103	43	42	connecting	connect	VERB
ejpam-5103	43	43	µ1	µ1	PROPN
ejpam-5103	43	44	and	and	CCONJ
ejpam-5103	43	45	µ2	µ2	PROPN
ejpam-5103	43	46	,	,	PUNCT
ejpam-5103	43	47	i.e.	i.e.	X
ejpam-5103	43	48	,	,	PUNCT
ejpam-5103	43	49	γµ1,µ2(0	γµ1,µ2(0	ADJ
ejpam-5103	43	50	)	)	PUNCT
ejpam-5103	43	51	=	=	VERB
ejpam-5103	44	1	µ2	µ2	NOUN
ejpam-5103	44	2	and	and	CCONJ
ejpam-5103	44	3	γµ1,µ2(1	γµ1,µ2(1	NOUN
ejpam-5103	44	4	)	)	PUNCT
ejpam-5103	44	5	=	=	SYM
ejpam-5103	44	6	µ1	µ1	PROPN
ejpam-5103	44	7	.	.	PUNCT
ejpam-5103	45	1	definition	definition	NOUN
ejpam-5103	45	2	1	1	NUM
ejpam-5103	45	3	.	.	PUNCT
ejpam-5103	46	1	[	[	X
ejpam-5103	46	2	3	3	NUM
ejpam-5103	46	3	]	]	PUNCT
ejpam-5103	46	4	.	.	PUNCT
ejpam-5103	47	1	let	let	VERB
ejpam-5103	47	2	a	a	DET
ejpam-5103	47	3	set	set	NOUN
ejpam-5103	47	4	ϑ	ϑ	AUX
ejpam-5103	47	5	⊂	⊂	X
ejpam-5103	47	6	n	n	AUX
ejpam-5103	47	7	be	be	AUX
ejpam-5103	47	8	geodesic	geodesic	ADJ
ejpam-5103	47	9	invex	invex	NOUN
ejpam-5103	47	10	w.r.t	w.r.t	PROPN
ejpam-5103	47	11	.	.	PUNCT
ejpam-5103	48	1	η	η	PROPN
ejpam-5103	48	2	:	:	PUNCT
ejpam-5103	48	3	n	n	PROPN
ejpam-5103	48	4	×	×	NOUN
ejpam-5103	48	5	n	n	CCONJ
ejpam-5103	48	6	−→	−→	ADJ
ejpam-5103	48	7	tn	tn	NOUN
ejpam-5103	48	8	.	.	PUNCT
ejpam-5103	49	1	a	a	DET
ejpam-5103	49	2	function	function	NOUN
ejpam-5103	49	3	ξ	ξ	NOUN
ejpam-5103	49	4	:	:	PUNCT
ejpam-5103	49	5	ϑ	ϑ	X
ejpam-5103	49	6	−→	−→	NOUN
ejpam-5103	49	7	ris	ris	PROPN
ejpam-5103	49	8	said	say	VERB
ejpam-5103	49	9	to	to	PART
ejpam-5103	49	10	be	be	AUX
ejpam-5103	49	11	geodesic	geodesic	ADJ
ejpam-5103	49	12	preinvex	preinvex	NOUN
ejpam-5103	50	1	w.r.t.η	w.r.t.η	ADP
ejpam-5103	50	2	iff	iff	PROPN
ejpam-5103	50	3	ξ(γµ1,µ2	ξ(γµ1,µ2	NUM
ejpam-5103	50	4	)	)	PUNCT
ejpam-5103	50	5	≤	≤	NOUN
ejpam-5103	50	6	(	(	PUNCT
ejpam-5103	50	7	1−	1−	NUM
ejpam-5103	50	8	ς)ξ(µ1	ς)ξ(µ1	NOUN
ejpam-5103	50	9	)	)	PUNCT
ejpam-5103	51	1	+	+	CCONJ
ejpam-5103	51	2	ςξ(µ2	ςξ(µ2	NUM
ejpam-5103	51	3	)	)	PUNCT
ejpam-5103	51	4	,	,	PUNCT
ejpam-5103	51	5	∀µ1	∀µ1	PROPN
ejpam-5103	51	6	,	,	PUNCT
ejpam-5103	51	7	µ2	µ2	PROPN
ejpam-5103	51	8	∈	∈	PROPN
ejpam-5103	51	9	ϑ	ϑ	NOUN
ejpam-5103	51	10	,	,	PUNCT
ejpam-5103	51	11	ς	ς	PROPN
ejpam-5103	51	12	∈	∈	PROPN
ejpam-5103	52	1	[	[	X
ejpam-5103	52	2	0	0	NUM
ejpam-5103	52	3	,	,	PUNCT
ejpam-5103	52	4	1	1	NUM
ejpam-5103	52	5	]	]	PUNCT
ejpam-5103	52	6	.	.	PUNCT
ejpam-5103	53	1	the	the	DET
ejpam-5103	53	2	strongly	strongly	ADV
ejpam-5103	53	3	geodesic	geodesic	ADJ
ejpam-5103	53	4	convexity	convexity	NOUN
ejpam-5103	53	5	of	of	ADP
ejpam-5103	53	6	order	order	NOUN
ejpam-5103	53	7	n	n	NOUN
ejpam-5103	53	8	on	on	ADP
ejpam-5103	53	9	a	a	DET
ejpam-5103	53	10	riemannian	riemannian	ADJ
ejpam-5103	53	11	manifold	manifold	NOUN
ejpam-5103	53	12	is	be	AUX
ejpam-5103	53	13	elaborated	elaborate	VERB
ejpam-5103	53	14	in	in	ADP
ejpam-5103	53	15	[	[	X
ejpam-5103	53	16	4	4	NUM
ejpam-5103	53	17	]	]	PUNCT
ejpam-5103	53	18	.	.	PUNCT
ejpam-5103	54	1	definition	definition	NOUN
ejpam-5103	54	2	2	2	NUM
ejpam-5103	54	3	.	.	PUNCT
ejpam-5103	54	4	suppose	suppose	VERB
ejpam-5103	54	5	ϑ	ϑ	X
ejpam-5103	54	6	⊆	⊆	NUM
ejpam-5103	54	7	n	n	PRON
ejpam-5103	54	8	is	be	AUX
ejpam-5103	54	9	a	a	DET
ejpam-5103	54	10	geodesically	geodesically	ADV
ejpam-5103	54	11	convex	convex	NOUN
ejpam-5103	54	12	subset	subset	NOUN
ejpam-5103	54	13	of	of	ADP
ejpam-5103	54	14	n	n	PROPN
ejpam-5103	54	15	.	.	PUNCT
ejpam-5103	55	1	a	a	DET
ejpam-5103	55	2	function	function	NOUN
ejpam-5103	55	3	ξ	ξ	NOUN
ejpam-5103	55	4	:	:	PUNCT
ejpam-5103	55	5	ϑ	ϑ	X
ejpam-5103	55	6	−→	−→	NOUN
ejpam-5103	55	7	r	r	NOUN
ejpam-5103	55	8	is	be	AUX
ejpam-5103	55	9	termed	term	VERB
ejpam-5103	55	10	strongly	strongly	ADV
ejpam-5103	55	11	geodesically	geodesically	ADV
ejpam-5103	55	12	convex	convex	NOUN
ejpam-5103	55	13	of	of	ADP
ejpam-5103	55	14	order	order	NOUN
ejpam-5103	55	15	n	n	CCONJ
ejpam-5103	55	16	>	>	X
ejpam-5103	55	17	0	0	PUNCT
ejpam-5103	56	1	on	on	ADP
ejpam-5103	56	2	ϑ	ϑ	PRON
ejpam-5103	56	3	if	if	SCONJ
ejpam-5103	56	4	there	there	PRON
ejpam-5103	56	5	exists	exist	VERB
ejpam-5103	56	6	a	a	DET
ejpam-5103	56	7	positive	positive	ADJ
ejpam-5103	56	8	constant	constant	ADJ
ejpam-5103	56	9	ε	ε	PROPN
ejpam-5103	56	10	>	>	X
ejpam-5103	56	11	0	0	NUM
ejpam-5103	56	12	such	such	ADJ
ejpam-5103	56	13	that	that	PRON
ejpam-5103	56	14	for	for	ADP
ejpam-5103	56	15	all	all	DET
ejpam-5103	56	16	µ1	µ1	NOUN
ejpam-5103	56	17	and	and	CCONJ
ejpam-5103	56	18	µ2	µ2	PROPN
ejpam-5103	56	19	in	in	ADP
ejpam-5103	56	20	ϑ	ϑ	PROPN
ejpam-5103	56	21	and	and	CCONJ
ejpam-5103	56	22	for	for	ADP
ejpam-5103	56	23	t	t	PROPN
ejpam-5103	56	24	in	in	ADP
ejpam-5103	56	25	the	the	DET
ejpam-5103	56	26	interval	interval	NOUN
ejpam-5103	56	27	[	[	X
ejpam-5103	56	28	0	0	NUM
ejpam-5103	56	29	,	,	PUNCT
ejpam-5103	56	30	1	1	NUM
ejpam-5103	56	31	]	]	PUNCT
ejpam-5103	56	32	,	,	PUNCT
ejpam-5103	56	33	the	the	DET
ejpam-5103	56	34	following	follow	VERB
ejpam-5103	56	35	inequality	inequality	NOUN
ejpam-5103	56	36	holds	hold	VERB
ejpam-5103	56	37	:	:	PUNCT
ejpam-5103	56	38	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	56	39	)	)	PUNCT
ejpam-5103	56	40	)	)	PUNCT
ejpam-5103	56	41	≤	≤	NUM
ejpam-5103	56	42	ςξ(µ1	ςξ(µ1	NOUN
ejpam-5103	56	43	)	)	PUNCT
ejpam-5103	57	1	+	+	CCONJ
ejpam-5103	57	2	(	(	PUNCT
ejpam-5103	57	3	1−	1−	NUM
ejpam-5103	57	4	ς)ξ(µ2)−	ς)ξ(µ2)−	NUM
ejpam-5103	57	5	ες(1−	ες(1−	NUM
ejpam-5103	57	6	ς)∥γ′µ1,µ2	ς)∥γ′µ1,µ2	NOUN
ejpam-5103	57	7	(	(	PUNCT
ejpam-5103	57	8	ς)∥n	ς)∥n	NUM
ejpam-5103	57	9	.	.	PUNCT
ejpam-5103	58	1	w.	w.	PROPN
ejpam-5103	58	2	saleh	saleh	PROPN
ejpam-5103	58	3	,	,	PUNCT
ejpam-5103	58	4	a.	a.	NOUN
ejpam-5103	58	5	lakhdari	lakhdari	PROPN
ejpam-5103	58	6	,	,	PUNCT
ejpam-5103	58	7	b.	b.	PROPN
ejpam-5103	58	8	meftah	meftah	PROPN
ejpam-5103	58	9	/	/	SYM
ejpam-5103	58	10	eur	eur	PROPN
ejpam-5103	58	11	.	.	PUNCT
ejpam-5103	59	1	j.	j.	PROPN
ejpam-5103	59	2	pure	pure	PROPN
ejpam-5103	59	3	appl	appl	PROPN
ejpam-5103	59	4	.	.	PROPN
ejpam-5103	59	5	math	math	PROPN
ejpam-5103	59	6	,	,	PUNCT
ejpam-5103	59	7	17	17	NUM
ejpam-5103	59	8	(	(	PUNCT
ejpam-5103	59	9	2	2	NUM
ejpam-5103	59	10	)	)	PUNCT
ejpam-5103	59	11	(	(	PUNCT
ejpam-5103	59	12	2024	2024	NUM
ejpam-5103	59	13	)	)	PUNCT
ejpam-5103	59	14	,	,	PUNCT
ejpam-5103	59	15	860	860	NUM
ejpam-5103	59	16	-	-	SYM
ejpam-5103	59	17	869	869	NUM
ejpam-5103	59	18	862	862	NUM
ejpam-5103	59	19	pini	pini	NOUN
ejpam-5103	60	1	[	[	X
ejpam-5103	60	2	12	12	NUM
ejpam-5103	60	3	]	]	PUNCT
ejpam-5103	60	4	conducted	conduct	VERB
ejpam-5103	60	5	an	an	DET
ejpam-5103	60	6	investigation	investigation	NOUN
ejpam-5103	60	7	into	into	ADP
ejpam-5103	60	8	various	various	ADJ
ejpam-5103	60	9	properties	property	NOUN
ejpam-5103	60	10	of	of	ADP
ejpam-5103	60	11	invex	invex	NOUN
ejpam-5103	60	12	functions	function	NOUN
ejpam-5103	60	13	on	on	ADP
ejpam-5103	60	14	riemannian	riemannian	ADJ
ejpam-5103	60	15	manifolds	manifold	NOUN
ejpam-5103	60	16	,	,	PUNCT
ejpam-5103	60	17	while	while	SCONJ
ejpam-5103	60	18	mititelu	mititelu	NOUN
ejpam-5103	60	19	[	[	X
ejpam-5103	60	20	7	7	NUM
ejpam-5103	60	21	]	]	PUNCT
ejpam-5103	60	22	explored	explore	VERB
ejpam-5103	60	23	its	its	PRON
ejpam-5103	60	24	generalization	generalization	NOUN
ejpam-5103	60	25	.	.	PUNCT
ejpam-5103	61	1	noor	noor	PROPN
ejpam-5103	61	2	and	and	CCONJ
ejpam-5103	61	3	noor	noor	PROPN
ejpam-5103	62	1	[	[	X
ejpam-5103	62	2	9	9	X
ejpam-5103	62	3	]	]	PUNCT
ejpam-5103	62	4	introduced	introduce	VERB
ejpam-5103	62	5	a	a	DET
ejpam-5103	62	6	novel	novel	ADJ
ejpam-5103	62	7	concept	concept	NOUN
ejpam-5103	62	8	known	know	VERB
ejpam-5103	62	9	as	as	ADP
ejpam-5103	62	10	exponentially	exponentially	ADV
ejpam-5103	62	11	preinvex	preinvex	NOUN
ejpam-5103	62	12	functions	function	NOUN
ejpam-5103	62	13	.	.	PUNCT
ejpam-5103	63	1	subsequently	subsequently	ADV
ejpam-5103	63	2	,	,	PUNCT
ejpam-5103	63	3	a	a	DET
ejpam-5103	63	4	multitude	multitude	NOUN
ejpam-5103	63	5	of	of	ADP
ejpam-5103	63	6	papers	paper	NOUN
ejpam-5103	63	7	have	have	AUX
ejpam-5103	63	8	emerged	emerge	VERB
ejpam-5103	63	9	in	in	ADP
ejpam-5103	63	10	the	the	DET
ejpam-5103	63	11	literature	literature	NOUN
ejpam-5103	63	12	,	,	PUNCT
ejpam-5103	63	13	delving	delve	VERB
ejpam-5103	63	14	into	into	ADP
ejpam-5103	63	15	the	the	DET
ejpam-5103	63	16	realm	realm	NOUN
ejpam-5103	63	17	of	of	ADP
ejpam-5103	63	18	(	(	PUNCT
ejpam-5103	63	19	generalized	generalized	ADJ
ejpam-5103	63	20	)	)	PUNCT
ejpam-5103	63	21	convexity	convexity	NOUN
ejpam-5103	63	22	on	on	ADP
ejpam-5103	63	23	riemannian	riemannian	ADJ
ejpam-5103	63	24	manifolds	manifold	NOUN
ejpam-5103	63	25	,	,	PUNCT
ejpam-5103	63	26	we	we	PRON
ejpam-5103	63	27	refer	refer	VERB
ejpam-5103	63	28	readers	reader	NOUN
ejpam-5103	63	29	to	to	ADP
ejpam-5103	63	30	[	[	X
ejpam-5103	63	31	14–16	14–16	NUM
ejpam-5103	63	32	]	]	PUNCT
ejpam-5103	63	33	.	.	PUNCT
ejpam-5103	64	1	in	in	ADP
ejpam-5103	64	2	this	this	DET
ejpam-5103	64	3	article	article	NOUN
ejpam-5103	64	4	,	,	PUNCT
ejpam-5103	64	5	we	we	PRON
ejpam-5103	64	6	present	present	VERB
ejpam-5103	64	7	some	some	DET
ejpam-5103	64	8	introductory	introductory	ADJ
ejpam-5103	64	9	concepts	concept	NOUN
ejpam-5103	64	10	and	and	CCONJ
ejpam-5103	64	11	fundamental	fundamental	ADJ
ejpam-5103	64	12	results	result	NOUN
ejpam-5103	64	13	pertaining	pertain	VERB
ejpam-5103	64	14	to	to	PART
ejpam-5103	64	15	strongly	strongly	ADV
ejpam-5103	64	16	geodesic	geodesic	VERB
ejpam-5103	64	17	log	log	NOUN
ejpam-5103	64	18	-	-	PUNCT
ejpam-5103	64	19	preinvex	preinvex	NOUN
ejpam-5103	64	20	functions	function	NOUN
ejpam-5103	64	21	in	in	ADP
ejpam-5103	64	22	riemannian	riemannian	ADJ
ejpam-5103	64	23	manifolds	manifold	NOUN
ejpam-5103	64	24	.	.	PUNCT
ejpam-5103	65	1	2	2	NUM
ejpam-5103	65	2	.	.	X
ejpam-5103	65	3	main	main	ADJ
ejpam-5103	65	4	results	result	NOUN
ejpam-5103	65	5	in	in	ADP
ejpam-5103	65	6	this	this	DET
ejpam-5103	65	7	article	article	NOUN
ejpam-5103	65	8	,	,	PUNCT
ejpam-5103	65	9	we	we	PRON
ejpam-5103	65	10	introduce	introduce	VERB
ejpam-5103	65	11	an	an	DET
ejpam-5103	65	12	innovative	innovative	ADJ
ejpam-5103	65	13	concept	concept	NOUN
ejpam-5103	65	14	of	of	ADP
ejpam-5103	65	15	generalized	generalized	ADJ
ejpam-5103	65	16	convexity	convexity	NOUN
ejpam-5103	65	17	in	in	ADP
ejpam-5103	65	18	the	the	DET
ejpam-5103	65	19	context	context	NOUN
ejpam-5103	65	20	of	of	ADP
ejpam-5103	65	21	riemannian	riemannian	ADJ
ejpam-5103	65	22	manifolds	manifold	NOUN
ejpam-5103	65	23	.	.	PUNCT
ejpam-5103	66	1	specifically	specifically	ADV
ejpam-5103	66	2	,	,	PUNCT
ejpam-5103	66	3	we	we	PRON
ejpam-5103	66	4	define	define	VERB
ejpam-5103	66	5	the	the	DET
ejpam-5103	66	6	concept	concept	NOUN
ejpam-5103	66	7	of	of	ADP
ejpam-5103	66	8	strongly	strongly	ADV
ejpam-5103	66	9	geodesic	geodesic	ADJ
ejpam-5103	66	10	log	log	NOUN
ejpam-5103	66	11	-	-	PUNCT
ejpam-5103	66	12	preinvex	preinvex	NOUN
ejpam-5103	66	13	functions	function	NOUN
ejpam-5103	66	14	,	,	PUNCT
ejpam-5103	66	15	which	which	PRON
ejpam-5103	66	16	serves	serve	VERB
ejpam-5103	66	17	as	as	ADP
ejpam-5103	66	18	a	a	DET
ejpam-5103	66	19	comprehensive	comprehensive	ADJ
ejpam-5103	66	20	generalization	generalization	NOUN
ejpam-5103	66	21	and	and	CCONJ
ejpam-5103	66	22	extension	extension	NOUN
ejpam-5103	66	23	of	of	ADP
ejpam-5103	66	24	various	various	ADJ
ejpam-5103	66	25	previously	previously	ADV
ejpam-5103	66	26	introduced	introduce	VERB
ejpam-5103	66	27	notions	notion	NOUN
ejpam-5103	66	28	of	of	ADP
ejpam-5103	66	29	generalized	generalized	ADJ
ejpam-5103	66	30	convexity	convexity	NOUN
ejpam-5103	66	31	in	in	ADP
ejpam-5103	66	32	the	the	DET
ejpam-5103	66	33	existing	exist	VERB
ejpam-5103	66	34	literature	literature	NOUN
ejpam-5103	66	35	.	.	PUNCT
ejpam-5103	67	1	definition	definition	NOUN
ejpam-5103	67	2	3	3	NUM
ejpam-5103	67	3	.	.	PUNCT
ejpam-5103	68	1	a	a	DET
ejpam-5103	68	2	function	function	NOUN
ejpam-5103	68	3	ξ	ξ	NOUN
ejpam-5103	68	4	:	:	PUNCT
ejpam-5103	68	5	ϑ	ϑ	X
ejpam-5103	68	6	−→	−→	ADJ
ejpam-5103	68	7	r∗	r∗	NOUN
ejpam-5103	68	8	+	+	CCONJ
ejpam-5103	68	9	is	be	AUX
ejpam-5103	68	10	considered	consider	VERB
ejpam-5103	68	11	strongly	strongly	ADV
ejpam-5103	68	12	geodesic	geodesic	ADJ
ejpam-5103	68	13	log	log	NOUN
ejpam-5103	68	14	-	-	PUNCT
ejpam-5103	68	15	preinvex	preinvex	NOUN
ejpam-5103	68	16	w.r.t	w.r.t	NOUN
ejpam-5103	68	17	a	a	DET
ejpam-5103	68	18	bifunction	bifunction	NOUN
ejpam-5103	68	19	η	η	NOUN
ejpam-5103	68	20	if	if	SCONJ
ejpam-5103	68	21	there	there	PRON
ejpam-5103	68	22	exists	exist	VERB
ejpam-5103	68	23	a	a	DET
ejpam-5103	68	24	constant	constant	ADJ
ejpam-5103	68	25	ε	ε	PROPN
ejpam-5103	68	26	≥	≥	NOUN
ejpam-5103	68	27	0	0	NUM
ejpam-5103	68	28	such	such	ADJ
ejpam-5103	68	29	that	that	PRON
ejpam-5103	68	30	:	:	PUNCT
ejpam-5103	68	31	log	log	VERB
ejpam-5103	68	32	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	68	33	)	)	PUNCT
ejpam-5103	68	34	)	)	PUNCT
ejpam-5103	68	35	≤	≤	NOUN
ejpam-5103	68	36	(	(	PUNCT
ejpam-5103	68	37	1−	1−	NUM
ejpam-5103	68	38	ς	ς	NOUN
ejpam-5103	68	39	)	)	PUNCT
ejpam-5103	68	40	log	log	NOUN
ejpam-5103	68	41	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	68	42	)	)	PUNCT
ejpam-5103	69	1	+	+	CCONJ
ejpam-5103	69	2	ς	ς	PROPN
ejpam-5103	69	3	log	log	NOUN
ejpam-5103	69	4	ξ(µ2)−	ξ(µ2)−	VERB
ejpam-5103	69	5	ες(1−	ες(1−	NOUN
ejpam-5103	69	6	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	69	7	,	,	PUNCT
ejpam-5103	69	8	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	69	9	(	(	PUNCT
ejpam-5103	69	10	2	2	NUM
ejpam-5103	69	11	)	)	PUNCT
ejpam-5103	69	12	for	for	ADP
ejpam-5103	69	13	all	all	DET
ejpam-5103	69	14	µ1	µ1	PROPN
ejpam-5103	69	15	,	,	PUNCT
ejpam-5103	69	16	µ2	µ2	PROPN
ejpam-5103	69	17	∈	∈	PROPN
ejpam-5103	69	18	ϑ	ϑ	X
ejpam-5103	69	19	and	and	CCONJ
ejpam-5103	69	20	ς	ς	PROPN
ejpam-5103	69	21	∈	∈	PROPN
ejpam-5103	70	1	[	[	X
ejpam-5103	70	2	0	0	NUM
ejpam-5103	70	3	,	,	PUNCT
ejpam-5103	70	4	1	1	NUM
ejpam-5103	70	5	]	]	PUNCT
ejpam-5103	70	6	.	.	PUNCT
ejpam-5103	71	1	definition	definition	NOUN
ejpam-5103	71	2	4	4	NUM
ejpam-5103	71	3	.	.	PUNCT
ejpam-5103	72	1	a	a	DET
ejpam-5103	72	2	function	function	NOUN
ejpam-5103	72	3	ξ	ξ	NOUN
ejpam-5103	72	4	:	:	PUNCT
ejpam-5103	72	5	ϑ	ϑ	X
ejpam-5103	72	6	−→	−→	ADJ
ejpam-5103	72	7	r∗	r∗	NOUN
ejpam-5103	72	8	+	+	CCONJ
ejpam-5103	72	9	is	be	AUX
ejpam-5103	72	10	considered	consider	VERB
ejpam-5103	72	11	to	to	PART
ejpam-5103	72	12	be	be	AUX
ejpam-5103	72	13	strongly	strongly	ADV
ejpam-5103	72	14	geodesic	geodesic	ADJ
ejpam-5103	72	15	log	log	NOUN
ejpam-5103	72	16	-	-	PUNCT
ejpam-5103	72	17	quasi	quasi	NOUN
ejpam-5103	72	18	preinvex	preinvex	NOUN
ejpam-5103	72	19	w.r.t	w.r.t	VERB
ejpam-5103	72	20	the	the	DET
ejpam-5103	72	21	bifunction	bifunction	NOUN
ejpam-5103	72	22	η	η	PROPN
ejpam-5103	72	23	if	if	SCONJ
ejpam-5103	72	24	:	:	PUNCT
ejpam-5103	72	25	log	log	VERB
ejpam-5103	72	26	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	72	27	)	)	PUNCT
ejpam-5103	72	28	)	)	PUNCT
ejpam-5103	72	29	≤	≤	NUM
ejpam-5103	72	30	max	max	PROPN
ejpam-5103	72	31	{	{	PUNCT
ejpam-5103	72	32	log	log	PROPN
ejpam-5103	72	33	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	72	34	)	)	PUNCT
ejpam-5103	72	35	,	,	PUNCT
ejpam-5103	72	36	log	log	VERB
ejpam-5103	72	37	ξ(µ2)}−ες(1−ς)∥η(µ2	ξ(µ2)}−ες(1−ς)∥η(µ2	NOUN
ejpam-5103	72	38	,	,	PUNCT
ejpam-5103	72	39	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	72	40	,	,	PUNCT
ejpam-5103	72	41	∀µ1	∀µ1	PROPN
ejpam-5103	72	42	,	,	PUNCT
ejpam-5103	72	43	µ2	µ2	PROPN
ejpam-5103	72	44	∈	∈	PROPN
ejpam-5103	72	45	ϑ	ϑ	NOUN
ejpam-5103	72	46	,	,	PUNCT
ejpam-5103	72	47	ς	ς	PROPN
ejpam-5103	72	48	∈	∈	PROPN
ejpam-5103	73	1	[	[	X
ejpam-5103	73	2	0	0	NUM
ejpam-5103	73	3	,	,	PUNCT
ejpam-5103	73	4	1	1	NUM
ejpam-5103	73	5	]	]	PUNCT
ejpam-5103	73	6	.	.	PUNCT
ejpam-5103	74	1	definition	definition	NOUN
ejpam-5103	74	2	5	5	NUM
ejpam-5103	74	3	.	.	PUNCT
ejpam-5103	75	1	a	a	DET
ejpam-5103	75	2	function	function	NOUN
ejpam-5103	75	3	ξ	ξ	NOUN
ejpam-5103	75	4	:	:	PUNCT
ejpam-5103	75	5	ϑ	ϑ	X
ejpam-5103	75	6	−→	−→	ADJ
ejpam-5103	75	7	r∗	r∗	NOUN
ejpam-5103	75	8	+	+	CCONJ
ejpam-5103	75	9	is	be	AUX
ejpam-5103	75	10	said	say	VERB
ejpam-5103	75	11	to	to	PART
ejpam-5103	75	12	be	be	AUX
ejpam-5103	75	13	first	first	ADJ
ejpam-5103	75	14	kind	kind	ADV
ejpam-5103	75	15	of	of	ADV
ejpam-5103	75	16	strongly	strongly	ADV
ejpam-5103	75	17	geodesic	geodesic	ADJ
ejpam-5103	75	18	logpreinvex	logpreinvex	NOUN
ejpam-5103	75	19	w.r.t	w.r.t	NOUN
ejpam-5103	75	20	.	.	PUNCT
ejpam-5103	76	1	any	any	DET
ejpam-5103	76	2	arbitrary	arbitrary	ADJ
ejpam-5103	76	3	bifunction	bifunction	NOUN
ejpam-5103	76	4	η	η	PROPN
ejpam-5103	76	5	,	,	PUNCT
ejpam-5103	76	6	if	if	SCONJ
ejpam-5103	76	7	log	log	VERB
ejpam-5103	76	8	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	76	9	)	)	PUNCT
ejpam-5103	76	10	)	)	PUNCT
ejpam-5103	76	11	≤	≤	NOUN
ejpam-5103	76	12	(	(	PUNCT
ejpam-5103	76	13	log	log	NOUN
ejpam-5103	76	14	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	76	15	)	)	PUNCT
ejpam-5103	76	16	)	)	PUNCT
ejpam-5103	76	17	1−ς(log	1−ς(log	NUM
ejpam-5103	76	18	ξ(µ2	ξ(µ2	NUM
ejpam-5103	76	19	)	)	PUNCT
ejpam-5103	76	20	)	)	PUNCT
ejpam-5103	77	1	t	t	NOUN
ejpam-5103	78	1	−	−	NUM
ejpam-5103	78	2	ες(1−	ες(1−	NUM
ejpam-5103	78	3	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	78	4	,	,	PUNCT
ejpam-5103	78	5	µ1)(ς)∥2,∀µ1	µ1)(ς)∥2,∀µ1	NOUN
ejpam-5103	78	6	,	,	PUNCT
ejpam-5103	78	7	µ2	µ2	PROPN
ejpam-5103	78	8	∈	∈	PROPN
ejpam-5103	78	9	ϑ	ϑ	NOUN
ejpam-5103	78	10	,	,	PUNCT
ejpam-5103	78	11	ς	ς	PROPN
ejpam-5103	78	12	∈	∈	PROPN
ejpam-5103	79	1	[	[	X
ejpam-5103	79	2	0	0	NUM
ejpam-5103	79	3	,	,	PUNCT
ejpam-5103	79	4	1	1	NUM
ejpam-5103	79	5	]	]	PUNCT
ejpam-5103	79	6	.	.	PUNCT
ejpam-5103	80	1	from	from	ADP
ejpam-5103	80	2	the	the	DET
ejpam-5103	80	3	above	above	ADJ
ejpam-5103	80	4	defintions	defintion	NOUN
ejpam-5103	80	5	,	,	PUNCT
ejpam-5103	80	6	we	we	PRON
ejpam-5103	80	7	have	have	VERB
ejpam-5103	80	8	(	(	PUNCT
ejpam-5103	80	9	i	i	NOUN
ejpam-5103	80	10	)	)	PUNCT
ejpam-5103	80	11	log	log	VERB
ejpam-5103	80	12	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	80	13	)	)	PUNCT
ejpam-5103	80	14	)	)	PUNCT
ejpam-5103	80	15	≤	≤	NOUN
ejpam-5103	80	16	(	(	PUNCT
ejpam-5103	80	17	log	log	NOUN
ejpam-5103	80	18	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	80	19	)	)	PUNCT
ejpam-5103	80	20	)	)	PUNCT
ejpam-5103	80	21	1−ς(log	1−ς(log	NUM
ejpam-5103	80	22	ξ(µ2	ξ(µ2	NUM
ejpam-5103	80	23	)	)	PUNCT
ejpam-5103	80	24	)	)	PUNCT
ejpam-5103	81	1	t	t	NOUN
ejpam-5103	81	2	−	−	NUM
ejpam-5103	81	3	ες(1−	ες(1−	NUM
ejpam-5103	81	4	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	81	5	,	,	PUNCT
ejpam-5103	81	6	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	81	7	≤	≤	PROPN
ejpam-5103	81	8	(	(	PUNCT
ejpam-5103	81	9	1−	1−	NUM
ejpam-5103	81	10	ς	ς	NOUN
ejpam-5103	81	11	)	)	PUNCT
ejpam-5103	81	12	log	log	NOUN
ejpam-5103	81	13	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	81	14	)	)	PUNCT
ejpam-5103	82	1	+	+	CCONJ
ejpam-5103	82	2	ς	ς	PROPN
ejpam-5103	82	3	log	log	NOUN
ejpam-5103	82	4	ξ(µ2)−	ξ(µ2)−	VERB
ejpam-5103	82	5	ες(1−	ες(1−	NOUN
ejpam-5103	82	6	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	82	7	,	,	PUNCT
ejpam-5103	82	8	µ1)(ς)∥2,∀	µ1)(ς)∥2,∀	PUNCT
ejpam-5103	82	9	≤	≤	NUM
ejpam-5103	82	10	max	max	PROPN
ejpam-5103	82	11	{	{	PUNCT
ejpam-5103	82	12	log	log	PROPN
ejpam-5103	82	13	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	82	14	)	)	PUNCT
ejpam-5103	82	15	,	,	PUNCT
ejpam-5103	82	16	log	log	PROPN
ejpam-5103	82	17	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	82	18	)	)	PUNCT
ejpam-5103	82	19	}	}	PUNCT
ejpam-5103	83	1	−	−	PROPN
ejpam-5103	83	2	ες(1−	ες(1−	NUM
ejpam-5103	83	3	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	83	4	,	,	PUNCT
ejpam-5103	83	5	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	83	6	,	,	PUNCT
ejpam-5103	83	7	∀µ1	∀µ1	PROPN
ejpam-5103	83	8	,	,	PUNCT
ejpam-5103	83	9	µ2	µ2	PROPN
ejpam-5103	83	10	∈	∈	PROPN
ejpam-5103	83	11	ϑ	ϑ	NOUN
ejpam-5103	83	12	,	,	PUNCT
ejpam-5103	83	13	ς	ς	PROPN
ejpam-5103	83	14	∈	∈	PROPN
ejpam-5103	84	1	[	[	X
ejpam-5103	84	2	0	0	NUM
ejpam-5103	84	3	,	,	PUNCT
ejpam-5103	84	4	1	1	NUM
ejpam-5103	84	5	]	]	PUNCT
ejpam-5103	84	6	.	.	PUNCT
ejpam-5103	85	1	this	this	PRON
ejpam-5103	85	2	demonstrates	demonstrate	VERB
ejpam-5103	85	3	that	that	SCONJ
ejpam-5103	85	4	every	every	DET
ejpam-5103	85	5	first	first	ADJ
ejpam-5103	85	6	kind	kind	NOUN
ejpam-5103	85	7	of	of	ADP
ejpam-5103	85	8	strongly	strongly	ADV
ejpam-5103	85	9	geodesic	geodesic	ADJ
ejpam-5103	85	10	log	log	NOUN
ejpam-5103	85	11	-	-	PUNCT
ejpam-5103	85	12	preinvex	preinvex	NOUN
ejpam-5103	85	13	function	function	NOUN
ejpam-5103	85	14	is	be	AUX
ejpam-5103	85	15	indeed	indeed	ADV
ejpam-5103	85	16	a	a	DET
ejpam-5103	85	17	strongly	strongly	ADV
ejpam-5103	85	18	geodesic	geodesic	ADJ
ejpam-5103	85	19	log	log	NOUN
ejpam-5103	85	20	-	-	PUNCT
ejpam-5103	85	21	preinvex	preinvex	NOUN
ejpam-5103	85	22	function	function	NOUN
ejpam-5103	85	23	,	,	PUNCT
ejpam-5103	85	24	and	and	CCONJ
ejpam-5103	85	25	a	a	DET
ejpam-5103	85	26	strongly	strongly	ADV
ejpam-5103	85	27	geodesic	geodesic	ADJ
ejpam-5103	85	28	log	log	NOUN
ejpam-5103	85	29	-	-	PUNCT
ejpam-5103	85	30	preinvex	preinvex	NOUN
ejpam-5103	85	31	function	function	NOUN
ejpam-5103	85	32	can	can	AUX
ejpam-5103	85	33	be	be	AUX
ejpam-5103	85	34	considered	consider	VERB
ejpam-5103	85	35	a	a	DET
ejpam-5103	85	36	strongly	strongly	ADV
ejpam-5103	85	37	geodesic	geodesic	ADJ
ejpam-5103	85	38	log	log	NOUN
ejpam-5103	85	39	-	-	PUNCT
ejpam-5103	85	40	quasipreinvex	quasipreinvex	NOUN
ejpam-5103	85	41	function	function	NOUN
ejpam-5103	85	42	.	.	PUNCT
ejpam-5103	86	1	however	however	ADV
ejpam-5103	86	2	,	,	PUNCT
ejpam-5103	86	3	it	it	PRON
ejpam-5103	86	4	’s	’	VERB
ejpam-5103	86	5	important	important	ADJ
ejpam-5103	86	6	to	to	PART
ejpam-5103	86	7	note	note	VERB
ejpam-5103	86	8	that	that	SCONJ
ejpam-5103	86	9	the	the	DET
ejpam-5103	86	10	converse	converse	NOUN
ejpam-5103	86	11	is	be	AUX
ejpam-5103	86	12	not	not	PART
ejpam-5103	86	13	necessarily	necessarily	ADV
ejpam-5103	86	14	true	true	ADJ
ejpam-5103	86	15	.	.	PUNCT
ejpam-5103	87	1	w.	w.	PROPN
ejpam-5103	87	2	saleh	saleh	PROPN
ejpam-5103	87	3	,	,	PUNCT
ejpam-5103	87	4	a.	a.	NOUN
ejpam-5103	87	5	lakhdari	lakhdari	PROPN
ejpam-5103	87	6	,	,	PUNCT
ejpam-5103	87	7	b.	b.	PROPN
ejpam-5103	87	8	meftah	meftah	PROPN
ejpam-5103	87	9	/	/	SYM
ejpam-5103	87	10	eur	eur	PROPN
ejpam-5103	87	11	.	.	PUNCT
ejpam-5103	88	1	j.	j.	PROPN
ejpam-5103	88	2	pure	pure	PROPN
ejpam-5103	88	3	appl	appl	PROPN
ejpam-5103	88	4	.	.	PROPN
ejpam-5103	88	5	math	math	PROPN
ejpam-5103	88	6	,	,	PUNCT
ejpam-5103	88	7	17	17	NUM
ejpam-5103	88	8	(	(	PUNCT
ejpam-5103	88	9	2	2	NUM
ejpam-5103	88	10	)	)	PUNCT
ejpam-5103	88	11	(	(	PUNCT
ejpam-5103	88	12	2024	2024	NUM
ejpam-5103	88	13	)	)	PUNCT
ejpam-5103	88	14	,	,	PUNCT
ejpam-5103	88	15	860	860	NUM
ejpam-5103	88	16	-	-	SYM
ejpam-5103	88	17	869	869	NUM
ejpam-5103	88	18	863	863	NUM
ejpam-5103	88	19	(	(	PUNCT
ejpam-5103	88	20	ii	ii	NOUN
ejpam-5103	88	21	)	)	PUNCT
ejpam-5103	88	22	if	if	SCONJ
ejpam-5103	88	23	ε	ε	PROPN
ejpam-5103	88	24	=	=	SYM
ejpam-5103	88	25	0	0	PROPN
ejpam-5103	88	26	,	,	PUNCT
ejpam-5103	88	27	then	then	ADV
ejpam-5103	88	28	(	(	PUNCT
ejpam-5103	88	29	a	a	X
ejpam-5103	88	30	)	)	PUNCT
ejpam-5103	88	31	strongly	strongly	ADV
ejpam-5103	88	32	geodesic	geodesic	ADJ
ejpam-5103	88	33	log	log	NOUN
ejpam-5103	88	34	-	-	PUNCT
ejpam-5103	88	35	preinvex	preinvex	NOUN
ejpam-5103	88	36	function	function	NOUN
ejpam-5103	88	37	is	be	AUX
ejpam-5103	88	38	called	call	VERB
ejpam-5103	88	39	geodesic	geodesic	ADJ
ejpam-5103	88	40	log	log	NOUN
ejpam-5103	88	41	-	-	PUNCT
ejpam-5103	88	42	preinvex	preinvex	NOUN
ejpam-5103	88	43	function	function	NOUN
ejpam-5103	88	44	.	.	PUNCT
ejpam-5103	89	1	(	(	PUNCT
ejpam-5103	89	2	b	b	X
ejpam-5103	89	3	)	)	PUNCT
ejpam-5103	89	4	strongly	strongly	ADV
ejpam-5103	89	5	geodeic	geodeic	ADJ
ejpam-5103	89	6	log	log	NOUN
ejpam-5103	89	7	-	-	PUNCT
ejpam-5103	89	8	quasi	quasi	NOUN
ejpam-5103	89	9	preinvex	preinvex	NOUN
ejpam-5103	89	10	is	be	AUX
ejpam-5103	89	11	called	call	VERB
ejpam-5103	89	12	geodesic	geodesic	ADJ
ejpam-5103	89	13	log	log	NOUN
ejpam-5103	89	14	-	-	PUNCT
ejpam-5103	89	15	quasi	quasi	NOUN
ejpam-5103	89	16	preinvex	preinvex	NOUN
ejpam-5103	89	17	.	.	PUNCT
ejpam-5103	90	1	(	(	PUNCT
ejpam-5103	90	2	c	c	X
ejpam-5103	90	3	)	)	PUNCT
ejpam-5103	90	4	the	the	DET
ejpam-5103	90	5	first	first	ADJ
ejpam-5103	90	6	kind	kind	NOUN
ejpam-5103	90	7	of	of	ADP
ejpam-5103	90	8	strongly	strongly	ADV
ejpam-5103	90	9	geodesic	geodesic	ADJ
ejpam-5103	90	10	log	log	NOUN
ejpam-5103	90	11	-	-	PUNCT
ejpam-5103	90	12	preinvex	preinvex	NOUN
ejpam-5103	90	13	is	be	AUX
ejpam-5103	90	14	called	call	VERB
ejpam-5103	90	15	the	the	DET
ejpam-5103	90	16	first	first	ADJ
ejpam-5103	90	17	kind	kind	NOUN
ejpam-5103	90	18	of	of	ADP
ejpam-5103	90	19	geodesic	geodesic	ADJ
ejpam-5103	90	20	log	log	NOUN
ejpam-5103	90	21	-	-	PUNCT
ejpam-5103	90	22	preinvex	preinvex	NOUN
ejpam-5103	90	23	.	.	PUNCT
ejpam-5103	91	1	(	(	PUNCT
ejpam-5103	91	2	iii	iii	X
ejpam-5103	91	3	)	)	PUNCT
ejpam-5103	91	4	if	if	SCONJ
ejpam-5103	91	5	ς	ς	PROPN
ejpam-5103	91	6	=	=	SYM
ejpam-5103	91	7	1	1	NUM
ejpam-5103	91	8	,	,	PUNCT
ejpam-5103	91	9	then	then	ADV
ejpam-5103	91	10	definitions	definition	VERB
ejpam-5103	91	11	3	3	NUM
ejpam-5103	91	12	and	and	CCONJ
ejpam-5103	91	13	5	5	NUM
ejpam-5103	91	14	will	will	AUX
ejpam-5103	91	15	be	be	AUX
ejpam-5103	91	16	become	become	VERB
ejpam-5103	91	17	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	91	18	)	)	PUNCT
ejpam-5103	91	19	≤	≤	NOUN
ejpam-5103	91	20	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	91	21	)	)	PUNCT
ejpam-5103	91	22	,	,	PUNCT
ejpam-5103	91	23	∀µ1	∀µ1	PROPN
ejpam-5103	91	24	,	,	PUNCT
ejpam-5103	91	25	µ2	µ2	PROPN
ejpam-5103	91	26	∈	∈	PROPN
ejpam-5103	91	27	ϑ.	ϑ.	NOUN
ejpam-5103	91	28	definition	definition	NOUN
ejpam-5103	91	29	6	6	NUM
ejpam-5103	91	30	.	.	PUNCT
ejpam-5103	92	1	a	a	DET
ejpam-5103	92	2	function	function	NOUN
ejpam-5103	92	3	ξ	ξ	NOUN
ejpam-5103	92	4	:	:	PUNCT
ejpam-5103	92	5	ϑ	ϑ	X
ejpam-5103	92	6	−→	−→	ADJ
ejpam-5103	92	7	r∗	r∗	NOUN
ejpam-5103	92	8	+	+	CCONJ
ejpam-5103	92	9	is	be	AUX
ejpam-5103	92	10	said	say	VERB
ejpam-5103	92	11	to	to	PART
ejpam-5103	92	12	be	be	AUX
ejpam-5103	92	13	a	a	DET
ejpam-5103	92	14	strongly	strongly	ADV
ejpam-5103	92	15	affine	affine	ADJ
ejpam-5103	92	16	geodesic	geodesic	ADJ
ejpam-5103	92	17	log	log	NOUN
ejpam-5103	92	18	-	-	PUNCT
ejpam-5103	92	19	preinvex	preinvex	NOUN
ejpam-5103	92	20	w.r.t	w.r.t	NOUN
ejpam-5103	92	21	.	.	PUNCT
ejpam-5103	93	1	the	the	DET
ejpam-5103	93	2	bifunction	bifunction	PROPN
ejpam-5103	93	3	η	η	PROPN
ejpam-5103	93	4	,	,	PUNCT
ejpam-5103	93	5	if	if	SCONJ
ejpam-5103	93	6	there	there	PRON
ejpam-5103	93	7	exists	exist	VERB
ejpam-5103	93	8	a	a	DET
ejpam-5103	93	9	constant	constant	ADJ
ejpam-5103	93	10	ε	ε	PROPN
ejpam-5103	93	11	≥	≥	NUM
ejpam-5103	93	12	0	0	NUM
ejpam-5103	93	13	,	,	PUNCT
ejpam-5103	93	14	such	such	ADJ
ejpam-5103	93	15	that	that	PRON
ejpam-5103	93	16	log	log	NOUN
ejpam-5103	93	17	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	93	18	)	)	PUNCT
ejpam-5103	93	19	)	)	PUNCT
ejpam-5103	94	1	=	=	PUNCT
ejpam-5103	94	2	(	(	PUNCT
ejpam-5103	94	3	1−	1−	NUM
ejpam-5103	94	4	ς	ς	NOUN
ejpam-5103	94	5	)	)	PUNCT
ejpam-5103	94	6	log	log	NOUN
ejpam-5103	94	7	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	94	8	)	)	PUNCT
ejpam-5103	94	9	+	+	CCONJ
ejpam-5103	95	1	ς	ς	PROPN
ejpam-5103	95	2	log	log	NOUN
ejpam-5103	95	3	ξ(µ2)−	ξ(µ2)−	VERB
ejpam-5103	95	4	ες(1−	ες(1−	NOUN
ejpam-5103	95	5	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	95	6	,	,	PUNCT
ejpam-5103	95	7	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	95	8	,	,	PUNCT
ejpam-5103	95	9	(	(	PUNCT
ejpam-5103	95	10	3	3	X
ejpam-5103	95	11	)	)	PUNCT
ejpam-5103	95	12	∀µ1	∀µ1	PROPN
ejpam-5103	95	13	,	,	PUNCT
ejpam-5103	95	14	µ2	µ2	PROPN
ejpam-5103	95	15	∈	∈	PROPN
ejpam-5103	95	16	ϑ	ϑ	X
ejpam-5103	95	17	and	and	CCONJ
ejpam-5103	95	18	ς	ς	PROPN
ejpam-5103	95	19	∈	∈	PROPN
ejpam-5103	96	1	[	[	X
ejpam-5103	96	2	0	0	NUM
ejpam-5103	96	3	,	,	PUNCT
ejpam-5103	96	4	1	1	NUM
ejpam-5103	96	5	]	]	PUNCT
ejpam-5103	96	6	.	.	PUNCT
ejpam-5103	97	1	example	example	NOUN
ejpam-5103	98	1	1	1	NUM
ejpam-5103	98	2	.	.	X
ejpam-5103	98	3	assume	assume	VERB
ejpam-5103	98	4	that	that	SCONJ
ejpam-5103	98	5	map	map	VERB
ejpam-5103	98	6	η	η	NOUN
ejpam-5103	98	7	:	:	PUNCT
ejpam-5103	98	8	r×	r×	NOUN
ejpam-5103	98	9	r	r	NOUN
ejpam-5103	98	10	−→	−→	NOUN
ejpam-5103	98	11	r	r	NOUN
ejpam-5103	98	12	is	be	AUX
ejpam-5103	98	13	defined	define	VERB
ejpam-5103	98	14	as	as	ADP
ejpam-5103	98	15	η(µ1	η(µ1	NOUN
ejpam-5103	98	16	,	,	PUNCT
ejpam-5103	98	17	µ2	µ2	PROPN
ejpam-5103	98	18	)	)	PUNCT
ejpam-5103	98	19	=	=	PRON
ejpam-5103	98	20	{	{	PUNCT
ejpam-5103	98	21	0	0	NUM
ejpam-5103	98	22	ifµ1	ifµ1	PROPN
ejpam-5103	98	23	=	=	SYM
ejpam-5103	98	24	µ2	µ2	PROPN
ejpam-5103	98	25	,	,	PUNCT
ejpam-5103	98	26	1−mu1	1−mu1	NUM
ejpam-5103	98	27	ifµ1	ifµ1	NOUN
ejpam-5103	98	28	̸=	̸=	PROPN
ejpam-5103	98	29	µ2	µ2	NOUN
ejpam-5103	98	30	,	,	PUNCT
ejpam-5103	98	31	also	also	ADV
ejpam-5103	98	32	,	,	PUNCT
ejpam-5103	98	33	γµ1,µ2(ς	γµ1,µ2(ς	NOUN
ejpam-5103	98	34	)	)	PUNCT
ejpam-5103	99	1	=	=	PRON
ejpam-5103	99	2	{	{	PUNCT
ejpam-5103	99	3	µ2	µ2	PROPN
ejpam-5103	99	4	ifµ1	ifµ1	PROPN
ejpam-5103	99	5	=	=	SYM
ejpam-5103	99	6	µ2	µ2	PROPN
ejpam-5103	99	7	,	,	PUNCT
ejpam-5103	99	8	µ2	µ2	PROPN
ejpam-5103	99	9	+	+	CCONJ
ejpam-5103	99	10	ς(1−	ς(1−	PROPN
ejpam-5103	99	11	µ1	µ1	PROPN
ejpam-5103	99	12	)	)	PUNCT
ejpam-5103	99	13	ifµ1	ifµ1	NOUN
ejpam-5103	99	14	̸=	̸=	PROPN
ejpam-5103	99	15	µ2	µ2	PROPN
ejpam-5103	99	16	,	,	PUNCT
ejpam-5103	99	17	assume	assume	VERB
ejpam-5103	99	18	that	that	SCONJ
ejpam-5103	99	19	ξ	ξ	X
ejpam-5103	99	20	:	:	PUNCT
ejpam-5103	99	21	r+	r+	NOUN
ejpam-5103	99	22	−→	−→	ADJ
ejpam-5103	99	23	r	r	NOUN
ejpam-5103	99	24	,	,	PUNCT
ejpam-5103	99	25	where	where	SCONJ
ejpam-5103	99	26	ξ(µ	ξ(µ	PROPN
ejpam-5103	99	27	)	)	PUNCT
ejpam-5103	100	1	=	=	SYM
ejpam-5103	100	2	expµ	expµ	NOUN
ejpam-5103	100	3	,	,	PUNCT
ejpam-5103	100	4	then	then	ADV
ejpam-5103	100	5	ξ	ξ	PROPN
ejpam-5103	100	6	is	be	AUX
ejpam-5103	100	7	strongly	strongly	ADV
ejpam-5103	100	8	geodesic	geodesic	ADJ
ejpam-5103	100	9	log	log	NOUN
ejpam-5103	100	10	-	-	PUNCT
ejpam-5103	100	11	preinvex	preinvex	NOUN
ejpam-5103	100	12	w.r.t	w.r.t	NOUN
ejpam-5103	100	13	.	.	PUNCT
ejpam-5103	101	1	the	the	DET
ejpam-5103	101	2	bifunction	bifunction	PROPN
ejpam-5103	101	3	η	η	PROPN
ejpam-5103	101	4	.	.	PROPN
ejpam-5103	101	5	example	example	NOUN
ejpam-5103	101	6	2	2	NUM
ejpam-5103	101	7	.	.	X
ejpam-5103	102	1	in	in	ADP
ejpam-5103	102	2	this	this	DET
ejpam-5103	102	3	example	example	NOUN
ejpam-5103	102	4	,	,	PUNCT
ejpam-5103	102	5	we	we	PRON
ejpam-5103	102	6	give	give	VERB
ejpam-5103	102	7	some	some	DET
ejpam-5103	102	8	new	new	ADJ
ejpam-5103	102	9	parallegram	parallegram	NOUN
ejpam-5103	102	10	low	low	NOUN
ejpam-5103	102	11	of	of	ADP
ejpam-5103	102	12	uniformly	uniformly	ADJ
ejpam-5103	102	13	banach	banach	NOUN
ejpam-5103	102	14	spaces	space	NOUN
ejpam-5103	102	15	involving	involve	VERB
ejpam-5103	102	16	the	the	DET
ejpam-5103	102	17	notion	notion	NOUN
ejpam-5103	102	18	of	of	ADP
ejpam-5103	102	19	strongly	strongly	ADV
ejpam-5103	102	20	affine	affine	VERB
ejpam-5103	102	21	geodesic	geodesic	ADJ
ejpam-5103	102	22	log	log	NOUN
ejpam-5103	102	23	-	-	PUNCT
ejpam-5103	102	24	preinvex	preinvex	NOUN
ejpam-5103	102	25	w.r.t	w.r.t	NOUN
ejpam-5103	102	26	.	.	PUNCT
ejpam-5103	103	1	the	the	DET
ejpam-5103	103	2	bifunction	bifunction	NOUN
ejpam-5103	103	3	η	η	PROPN
ejpam-5103	103	4	assume	assume	VERB
ejpam-5103	103	5	that	that	SCONJ
ejpam-5103	103	6	η(µ1	η(µ1	NOUN
ejpam-5103	103	7	,	,	PUNCT
ejpam-5103	103	8	µ2	µ2	PROPN
ejpam-5103	103	9	)	)	PUNCT
ejpam-5103	103	10	=	=	PRON
ejpam-5103	103	11	{	{	PUNCT
ejpam-5103	103	12	0	0	NUM
ejpam-5103	103	13	ifµ1	ifµ1	PROPN
ejpam-5103	103	14	=	=	SYM
ejpam-5103	103	15	µ2	µ2	PROPN
ejpam-5103	103	16	,	,	PUNCT
ejpam-5103	103	17	mu1	mu1	X
ejpam-5103	103	18	−mu2	−mu2	PROPN
ejpam-5103	103	19	ifµ1	ifµ1	PROPN
ejpam-5103	103	20	̸=	̸=	PROPN
ejpam-5103	103	21	µ2	µ2	NOUN
ejpam-5103	103	22	,	,	PUNCT
ejpam-5103	103	23	also	also	ADV
ejpam-5103	103	24	,	,	PUNCT
ejpam-5103	103	25	γµ1,µ2(ς	γµ1,µ2(ς	NOUN
ejpam-5103	103	26	)	)	PUNCT
ejpam-5103	103	27	=	=	PRON
ejpam-5103	103	28	{	{	PUNCT
ejpam-5103	103	29	µ2	µ2	PROPN
ejpam-5103	103	30	ifµ1	ifµ1	PROPN
ejpam-5103	103	31	=	=	SYM
ejpam-5103	103	32	µ2	µ2	PROPN
ejpam-5103	103	33	,	,	PUNCT
ejpam-5103	103	34	µ2	µ2	PROPN
ejpam-5103	103	35	+	+	CCONJ
ejpam-5103	103	36	ς(µ1	ς(µ1	NOUN
ejpam-5103	103	37	−	−	PROPN
ejpam-5103	103	38	µ2	µ2	PROPN
ejpam-5103	103	39	)	)	PUNCT
ejpam-5103	103	40	ifµ1	ifµ1	PROPN
ejpam-5103	103	41	̸=	̸=	PROPN
ejpam-5103	103	42	µ2	µ2	PROPN
ejpam-5103	103	43	,	,	PUNCT
ejpam-5103	103	44	frome	frome	ADJ
ejpam-5103	103	45	equality	equality	NOUN
ejpam-5103	103	46	(	(	PUNCT
ejpam-5103	103	47	3	3	NUM
ejpam-5103	103	48	)	)	PUNCT
ejpam-5103	103	49	,	,	PUNCT
ejpam-5103	103	50	we	we	PRON
ejpam-5103	103	51	get	get	VERB
ejpam-5103	103	52	∥	∥	NUM
ejpam-5103	103	53	log	log	NOUN
ejpam-5103	103	54	ξ(µ2	ξ(µ2	NOUN
ejpam-5103	103	55	+	+	CCONJ
ejpam-5103	103	56	ς(µ1	ς(µ1	NOUN
ejpam-5103	103	57	−	−	PROPN
ejpam-5103	103	58	µ2))∥2	µ2))∥2	NOUN
ejpam-5103	103	59	=	=	SYM
ejpam-5103	103	60	(	(	PUNCT
ejpam-5103	103	61	1−	1−	NUM
ejpam-5103	103	62	ς)∥	ς)∥	NUM
ejpam-5103	103	63	log	log	NOUN
ejpam-5103	103	64	ξ(µ1)∥2	ξ(µ1)∥2	PROPN
ejpam-5103	103	65	+	+	CCONJ
ejpam-5103	103	66	ς∥	ς∥	ADV
ejpam-5103	103	67	log	log	VERB
ejpam-5103	103	68	ξ(µ2)∥2	ξ(µ2)∥2	ADJ
ejpam-5103	103	69	−	−	NOUN
ejpam-5103	103	70	ες(1−	ες(1−	NUM
ejpam-5103	103	71	ς)∥µ2	ς)∥µ2	NOUN
ejpam-5103	103	72	−	−	PROPN
ejpam-5103	103	73	µ1∥2	µ1∥2	PROPN
ejpam-5103	103	74	,	,	PUNCT
ejpam-5103	103	75	(	(	PUNCT
ejpam-5103	103	76	4	4	X
ejpam-5103	103	77	)	)	PUNCT
ejpam-5103	103	78	∀µ1	∀µ1	PROPN
ejpam-5103	103	79	,	,	PUNCT
ejpam-5103	103	80	µ2	µ2	PROPN
ejpam-5103	103	81	∈	∈	PROPN
ejpam-5103	103	82	ϑ	ϑ	X
ejpam-5103	103	83	and	and	CCONJ
ejpam-5103	103	84	ς	ς	PROPN
ejpam-5103	103	85	∈	∈	PROPN
ejpam-5103	104	1	[	[	X
ejpam-5103	104	2	0	0	NUM
ejpam-5103	104	3	,	,	PUNCT
ejpam-5103	104	4	1	1	NUM
ejpam-5103	104	5	]	]	PUNCT
ejpam-5103	104	6	.	.	PUNCT
ejpam-5103	105	1	taking	take	VERB
ejpam-5103	105	2	ς	ς	PROPN
ejpam-5103	105	3	=	=	SYM
ejpam-5103	105	4	1	1	NUM
ejpam-5103	105	5	2	2	NUM
ejpam-5103	105	6	in	in	ADP
ejpam-5103	105	7	(	(	PUNCT
ejpam-5103	105	8	4	4	NUM
ejpam-5103	105	9	)	)	PUNCT
ejpam-5103	105	10	,	,	PUNCT
ejpam-5103	105	11	we	we	PRON
ejpam-5103	105	12	get	get	VERB
ejpam-5103	105	13	∥	∥	NUM
ejpam-5103	105	14	log	log	NOUN
ejpam-5103	105	15	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	105	16	+	+	CCONJ
ejpam-5103	105	17	µ2	µ2	PROPN
ejpam-5103	105	18	2	2	NUM
ejpam-5103	105	19	)	)	PUNCT
ejpam-5103	105	20	∥2	∥2	NOUN
ejpam-5103	106	1	+	+	CCONJ
ejpam-5103	106	2	ε	ε	PROPN
ejpam-5103	106	3	4	4	NUM
ejpam-5103	106	4	∥µ2	∥µ2	ADP
ejpam-5103	106	5	−	−	PROPN
ejpam-5103	106	6	µ1∥2	µ1∥2	PROPN
ejpam-5103	106	7	=	=	SYM
ejpam-5103	106	8	1	1	NUM
ejpam-5103	106	9	2	2	NUM
ejpam-5103	106	10	(	(	PUNCT
ejpam-5103	106	11	∥	∥	X
ejpam-5103	106	12	log	log	VERB
ejpam-5103	106	13	ξ(µ1)∥2	ξ(µ1)∥2	NOUN
ejpam-5103	107	1	+	+	CCONJ
ejpam-5103	107	2	∥	∥	NUM
ejpam-5103	107	3	log	log	NOUN
ejpam-5103	107	4	ξ(µ2)∥2	ξ(µ2)∥2	PROPN
ejpam-5103	107	5	)	)	PUNCT
ejpam-5103	107	6	,	,	PUNCT
ejpam-5103	107	7	(	(	PUNCT
ejpam-5103	107	8	5	5	X
ejpam-5103	107	9	)	)	PUNCT
ejpam-5103	107	10	∀µ1	∀µ1	PROPN
ejpam-5103	107	11	,	,	PUNCT
ejpam-5103	107	12	µ2	µ2	PROPN
ejpam-5103	107	13	∈	∈	PROPN
ejpam-5103	107	14	ϑ.	ϑ.	NOUN
ejpam-5103	107	15	which	which	PRON
ejpam-5103	107	16	ξ	ξ	PROPN
ejpam-5103	107	17	is	be	AUX
ejpam-5103	107	18	called	call	VERB
ejpam-5103	107	19	the	the	DET
ejpam-5103	107	20	log	log	NOUN
ejpam-5103	107	21	-	-	PUNCT
ejpam-5103	107	22	paralleogram	paralleogram	NOUN
ejpam-5103	107	23	for	for	ADP
ejpam-5103	107	24	the	the	DET
ejpam-5103	107	25	inner	inner	ADJ
ejpam-5103	107	26	product	product	NOUN
ejpam-5103	107	27	spaces	space	VERB
ejpam-5103	107	28	.	.	PUNCT
ejpam-5103	108	1	by	by	ADP
ejpam-5103	108	2	putting	put	VERB
ejpam-5103	108	3	log	log	NOUN
ejpam-5103	108	4	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	108	5	)	)	PUNCT
ejpam-5103	108	6	=	=	SYM
ejpam-5103	108	7	∥µ1∥2	∥µ1∥2	NOUN
ejpam-5103	108	8	in	in	ADP
ejpam-5103	108	9	(	(	PUNCT
ejpam-5103	108	10	5	5	NUM
ejpam-5103	108	11	)	)	PUNCT
ejpam-5103	108	12	,	,	PUNCT
ejpam-5103	108	13	we	we	PRON
ejpam-5103	108	14	have	have	VERB
ejpam-5103	108	15	the	the	DET
ejpam-5103	108	16	paralleogram	paralleogram	NOUN
ejpam-5103	108	17	for	for	SCONJ
ejpam-5103	108	18	the	the	DET
ejpam-5103	108	19	inner	inner	ADJ
ejpam-5103	108	20	product	product	NOUN
ejpam-5103	108	21	spaces	space	VERB
ejpam-5103	108	22	.	.	PUNCT
ejpam-5103	109	1	w.	w.	PROPN
ejpam-5103	109	2	saleh	saleh	PROPN
ejpam-5103	109	3	,	,	PUNCT
ejpam-5103	109	4	a.	a.	NOUN
ejpam-5103	109	5	lakhdari	lakhdari	PROPN
ejpam-5103	109	6	,	,	PUNCT
ejpam-5103	109	7	b.	b.	PROPN
ejpam-5103	109	8	meftah	meftah	PROPN
ejpam-5103	109	9	/	/	SYM
ejpam-5103	109	10	eur	eur	PROPN
ejpam-5103	109	11	.	.	PUNCT
ejpam-5103	110	1	j.	j.	PROPN
ejpam-5103	110	2	pure	pure	PROPN
ejpam-5103	110	3	appl	appl	PROPN
ejpam-5103	110	4	.	.	PROPN
ejpam-5103	110	5	math	math	PROPN
ejpam-5103	110	6	,	,	PUNCT
ejpam-5103	110	7	17	17	NUM
ejpam-5103	110	8	(	(	PUNCT
ejpam-5103	110	9	2	2	NUM
ejpam-5103	110	10	)	)	PUNCT
ejpam-5103	110	11	(	(	PUNCT
ejpam-5103	110	12	2024	2024	NUM
ejpam-5103	110	13	)	)	PUNCT
ejpam-5103	110	14	,	,	PUNCT
ejpam-5103	110	15	860	860	NUM
ejpam-5103	110	16	-	-	SYM
ejpam-5103	110	17	869	869	NUM
ejpam-5103	110	18	864	864	NUM
ejpam-5103	110	19	3	3	NUM
ejpam-5103	110	20	.	.	PUNCT
ejpam-5103	111	1	some	some	DET
ejpam-5103	111	2	aspects	aspect	NOUN
ejpam-5103	111	3	of	of	ADP
ejpam-5103	111	4	geodesic	geodesic	ADJ
ejpam-5103	111	5	log	log	NOUN
ejpam-5103	111	6	-	-	PUNCT
ejpam-5103	111	7	preinvexity	preinvexity	NOUN
ejpam-5103	111	8	properties	property	NOUN
ejpam-5103	111	9	in	in	ADP
ejpam-5103	111	10	this	this	DET
ejpam-5103	111	11	section	section	NOUN
ejpam-5103	111	12	,	,	PUNCT
ejpam-5103	111	13	we	we	PRON
ejpam-5103	111	14	examine	examine	VERB
ejpam-5103	111	15	fundamental	fundamental	ADJ
ejpam-5103	111	16	properties	property	NOUN
ejpam-5103	111	17	of	of	ADP
ejpam-5103	111	18	geodesic	geodesic	ADJ
ejpam-5103	111	19	log	log	NOUN
ejpam-5103	111	20	-	-	PUNCT
ejpam-5103	111	21	preinvex	preinvex	NOUN
ejpam-5103	111	22	functions	function	NOUN
ejpam-5103	111	23	.	.	PUNCT
ejpam-5103	112	1	theorem	theorem	NOUN
ejpam-5103	112	2	1	1	NUM
ejpam-5103	112	3	.	.	PUNCT
ejpam-5103	113	1	if	if	SCONJ
ejpam-5103	113	2	ξ	ξ	PROPN
ejpam-5103	113	3	is	be	AUX
ejpam-5103	113	4	a	a	DET
ejpam-5103	113	5	strongly	strongly	ADV
ejpam-5103	113	6	geodesic	geodesic	ADJ
ejpam-5103	113	7	log	log	NOUN
ejpam-5103	113	8	-	-	PUNCT
ejpam-5103	113	9	preinvex	preinvex	NOUN
ejpam-5103	113	10	fun	fun	NOUN
ejpam-5103	113	11	ction	ction	NOUN
ejpam-5103	113	12	,	,	PUNCT
ejpam-5103	113	13	then	then	ADV
ejpam-5103	113	14	any	any	DET
ejpam-5103	113	15	point	point	NOUN
ejpam-5103	113	16	that	that	PRON
ejpam-5103	113	17	serves	serve	VERB
ejpam-5103	113	18	as	as	ADP
ejpam-5103	113	19	a	a	DET
ejpam-5103	113	20	local	local	ADJ
ejpam-5103	113	21	minimum	minimum	NOUN
ejpam-5103	113	22	is	be	AUX
ejpam-5103	113	23	also	also	ADV
ejpam-5103	113	24	considered	consider	VERB
ejpam-5103	113	25	a	a	DET
ejpam-5103	113	26	global	global	ADJ
ejpam-5103	113	27	minimum	minimum	NOUN
ejpam-5103	113	28	.	.	PUNCT
ejpam-5103	114	1	proof	proof	NOUN
ejpam-5103	114	2	.	.	PUNCT
ejpam-5103	115	1	considering	consider	VERB
ejpam-5103	115	2	that	that	SCONJ
ejpam-5103	115	3	the	the	DET
ejpam-5103	115	4	function	function	NOUN
ejpam-5103	115	5	ξ	ξ	PROPN
ejpam-5103	115	6	is	be	AUX
ejpam-5103	115	7	geodesic	geodesic	ADJ
ejpam-5103	115	8	log	log	NOUN
ejpam-5103	115	9	-	-	PUNCT
ejpam-5103	115	10	preinvex	preinvex	NOUN
ejpam-5103	115	11	and	and	CCONJ
ejpam-5103	115	12	possesses	possess	VERB
ejpam-5103	115	13	a	a	DET
ejpam-5103	115	14	local	local	ADJ
ejpam-5103	115	15	minimum	minimum	NOUN
ejpam-5103	115	16	at	at	ADP
ejpam-5103	115	17	µ1	µ1	PROPN
ejpam-5103	115	18	∈	∈	PROPN
ejpam-5103	115	19	ϑ.	ϑ.	NOUN
ejpam-5103	115	20	let	let	VERB
ejpam-5103	115	21	’s	’s	PRON
ejpam-5103	115	22	assume	assume	VERB
ejpam-5103	115	23	the	the	DET
ejpam-5103	115	24	contrary	contrary	NOUN
ejpam-5103	115	25	,	,	PUNCT
ejpam-5103	115	26	which	which	PRON
ejpam-5103	115	27	is	be	AUX
ejpam-5103	115	28	,	,	PUNCT
ejpam-5103	115	29	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	115	30	)	)	PUNCT
ejpam-5103	115	31	<	<	X
ejpam-5103	115	32	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	115	33	)	)	PUNCT
ejpam-5103	115	34	for	for	ADP
ejpam-5103	115	35	some	some	DET
ejpam-5103	115	36	µ2	µ2	PROPN
ejpam-5103	115	37	∈	∈	PROPN
ejpam-5103	115	38	ϑ.	ϑ.	NOUN
ejpam-5103	115	39	since	since	SCONJ
ejpam-5103	115	40	ξ	ξ	PROPN
ejpam-5103	115	41	is	be	AUX
ejpam-5103	115	42	a	a	DET
ejpam-5103	115	43	geodesic	geodesic	ADJ
ejpam-5103	115	44	log	log	NOUN
ejpam-5103	115	45	-	-	PUNCT
ejpam-5103	115	46	preinvex	preinvex	NOUN
ejpam-5103	115	47	function	function	NOUN
ejpam-5103	115	48	,	,	PUNCT
ejpam-5103	115	49	we	we	PRON
ejpam-5103	115	50	have	have	AUX
ejpam-5103	115	51	:	:	PUNCT
ejpam-5103	115	52	log	log	VERB
ejpam-5103	115	53	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	115	54	)	)	PUNCT
ejpam-5103	115	55	)	)	PUNCT
ejpam-5103	116	1	<	<	X
ejpam-5103	116	2	(	(	PUNCT
ejpam-5103	116	3	1−	1−	NUM
ejpam-5103	116	4	ς	ς	NOUN
ejpam-5103	116	5	)	)	PUNCT
ejpam-5103	116	6	log	log	NOUN
ejpam-5103	116	7	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	116	8	)	)	PUNCT
ejpam-5103	117	1	+	+	CCONJ
ejpam-5103	117	2	ς	ς	PROPN
ejpam-5103	117	3	log	log	NOUN
ejpam-5103	117	4	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	117	5	)	)	PUNCT
ejpam-5103	117	6	,	,	PUNCT
ejpam-5103	117	7	for	for	ADP
ejpam-5103	117	8	ς	ς	PROPN
ejpam-5103	117	9	∈	∈	PROPN
ejpam-5103	117	10	(	(	PUNCT
ejpam-5103	117	11	0	0	NUM
ejpam-5103	117	12	,	,	PUNCT
ejpam-5103	117	13	1	1	NUM
ejpam-5103	117	14	)	)	PUNCT
ejpam-5103	117	15	.	.	PUNCT
ejpam-5103	118	1	thus	thus	ADV
ejpam-5103	118	2	,	,	PUNCT
ejpam-5103	118	3	we	we	PRON
ejpam-5103	118	4	can	can	AUX
ejpam-5103	118	5	derive	derive	VERB
ejpam-5103	118	6	:	:	PUNCT
ejpam-5103	118	7	log	log	VERB
ejpam-5103	118	8	ξ(γµ1,µ2(ς))−	ξ(γµ1,µ2(ς))−	PROPN
ejpam-5103	118	9	log	log	NOUN
ejpam-5103	118	10	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	118	11	)	)	PUNCT
ejpam-5103	118	12	<	<	X
ejpam-5103	118	13	t	t	PROPN
ejpam-5103	118	14	(	(	PUNCT
ejpam-5103	118	15	log	log	NOUN
ejpam-5103	118	16	ξ(µ2)−	ξ(µ2)−	VERB
ejpam-5103	118	17	log	log	NOUN
ejpam-5103	118	18	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	118	19	)	)	PUNCT
ejpam-5103	118	20	)	)	PUNCT
ejpam-5103	119	1	<	<	X
ejpam-5103	119	2	0	0	NUM
ejpam-5103	119	3	,	,	PUNCT
ejpam-5103	119	4	(	(	PUNCT
ejpam-5103	119	5	6	6	NUM
ejpam-5103	119	6	)	)	PUNCT
ejpam-5103	119	7	from	from	ADP
ejpam-5103	119	8	which	which	PRON
ejpam-5103	119	9	it	it	PRON
ejpam-5103	119	10	follows	follow	VERB
ejpam-5103	119	11	that	that	PRON
ejpam-5103	119	12	:	:	PUNCT
ejpam-5103	119	13	log	log	VERB
ejpam-5103	119	14	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	119	15	)	)	PUNCT
ejpam-5103	119	16	)	)	PUNCT
ejpam-5103	120	1	<	<	X
ejpam-5103	120	2	log	log	NOUN
ejpam-5103	120	3	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	120	4	)	)	PUNCT
ejpam-5103	120	5	,	,	PUNCT
ejpam-5103	120	6	for	for	ADP
ejpam-5103	120	7	arbitrary	arbitrary	ADJ
ejpam-5103	120	8	small	small	ADJ
ejpam-5103	120	9	ς	ς	PROPN
ejpam-5103	120	10	>	>	X
ejpam-5103	120	11	0	0	NUM
ejpam-5103	120	12	.	.	PUNCT
ejpam-5103	121	1	this	this	PRON
ejpam-5103	121	2	contradicts	contradict	VERB
ejpam-5103	121	3	the	the	DET
ejpam-5103	121	4	fact	fact	NOUN
ejpam-5103	121	5	that	that	SCONJ
ejpam-5103	121	6	the	the	DET
ejpam-5103	121	7	function	function	NOUN
ejpam-5103	121	8	has	have	AUX
ejpam-5103	121	9	a	a	DET
ejpam-5103	121	10	minimum	minimum	NOUN
ejpam-5103	121	11	at	at	ADP
ejpam-5103	121	12	the	the	DET
ejpam-5103	121	13	point	point	NOUN
ejpam-5103	121	14	µ1	µ1	PROPN
ejpam-5103	121	15	.	.	PUNCT
ejpam-5103	122	1	theorem	theorem	VERB
ejpam-5103	122	2	2	2	NUM
ejpam-5103	122	3	.	.	PUNCT
ejpam-5103	122	4	a	a	DET
ejpam-5103	122	5	positive	positive	ADJ
ejpam-5103	122	6	function	function	NOUN
ejpam-5103	122	7	f	f	PROPN
ejpam-5103	122	8	is	be	AUX
ejpam-5103	122	9	considered	consider	VERB
ejpam-5103	122	10	to	to	PART
ejpam-5103	122	11	be	be	AUX
ejpam-5103	122	12	geodesic	geodesic	ADJ
ejpam-5103	122	13	log	log	NOUN
ejpam-5103	122	14	-	-	PUNCT
ejpam-5103	122	15	preinvex	preinvex	NOUN
ejpam-5103	122	16	if	if	SCONJ
ejpam-5103	122	17	and	and	CCONJ
ejpam-5103	122	18	only	only	ADV
ejpam-5103	122	19	if	if	SCONJ
ejpam-5103	122	20	the	the	DET
ejpam-5103	122	21	set	set	NOUN
ejpam-5103	122	22	epi(f	epi(f	NOUN
ejpam-5103	122	23	)	)	PUNCT
ejpam-5103	122	24	=	=	SYM
ejpam-5103	122	25	{	{	PUNCT
ejpam-5103	122	26	(	(	PUNCT
ejpam-5103	122	27	u	u	NOUN
ejpam-5103	122	28	,	,	PUNCT
ejpam-5103	122	29	v	v	NOUN
ejpam-5103	122	30	)	)	PUNCT
ejpam-5103	122	31	:	:	PUNCT
ejpam-5103	122	32	u	u	PROPN
ejpam-5103	122	33	∈	∈	PROPN
ejpam-5103	122	34	u	u	NOUN
ejpam-5103	122	35	,	,	PUNCT
ejpam-5103	122	36	logf(u	logf(u	ADJ
ejpam-5103	122	37	)	)	PUNCT
ejpam-5103	122	38	≤	≤	NOUN
ejpam-5103	122	39	v	v	NOUN
ejpam-5103	122	40	,	,	PUNCT
ejpam-5103	122	41	v	v	NOUN
ejpam-5103	122	42	∈	∈	NOUN
ejpam-5103	122	43	r	r	NOUN
ejpam-5103	122	44	}	}	PUNCT
ejpam-5103	122	45	is	be	AUX
ejpam-5103	122	46	a	a	DET
ejpam-5103	122	47	geodesic	geodesic	ADJ
ejpam-5103	122	48	invex	invex	NOUN
ejpam-5103	122	49	set	set	NOUN
ejpam-5103	122	50	.	.	PUNCT
ejpam-5103	123	1	proof	proof	NOUN
ejpam-5103	123	2	.	.	PUNCT
ejpam-5103	124	1	let	let	VERB
ejpam-5103	124	2	f	f	PRON
ejpam-5103	124	3	be	be	AUX
ejpam-5103	124	4	a	a	DET
ejpam-5103	124	5	geodesic	geodesic	ADJ
ejpam-5103	124	6	log	log	NOUN
ejpam-5103	124	7	-	-	PUNCT
ejpam-5103	124	8	preinvex	preinvex	NOUN
ejpam-5103	124	9	function	function	NOUN
ejpam-5103	124	10	.	.	PUNCT
ejpam-5103	125	1	let	let	VERB
ejpam-5103	125	2	(	(	PUNCT
ejpam-5103	125	3	u1	u1	NOUN
ejpam-5103	125	4	,	,	PUNCT
ejpam-5103	125	5	v1	v1	NOUN
ejpam-5103	125	6	)	)	PUNCT
ejpam-5103	125	7	,	,	PUNCT
ejpam-5103	125	8	(	(	PUNCT
ejpam-5103	125	9	u2	u2	NOUN
ejpam-5103	125	10	,	,	PUNCT
ejpam-5103	125	11	v2	v2	PROPN
ejpam-5103	125	12	)	)	PUNCT
ejpam-5103	125	13	∈	∈	PROPN
ejpam-5103	126	1	epi(f	epi(f	NOUN
ejpam-5103	126	2	)	)	PUNCT
ejpam-5103	126	3	.	.	PUNCT
ejpam-5103	127	1	then	then	ADV
ejpam-5103	127	2	logf(u1	logf(u1	NOUN
ejpam-5103	127	3	)	)	PUNCT
ejpam-5103	127	4	≤	≤	NUM
ejpam-5103	127	5	v1	v1	NOUN
ejpam-5103	127	6	and	and	CCONJ
ejpam-5103	127	7	logf(u2	logf(u2	ADJ
ejpam-5103	127	8	)	)	PUNCT
ejpam-5103	127	9	≤	≤	NUM
ejpam-5103	127	10	v2	v2	NOUN
ejpam-5103	127	11	.	.	PUNCT
ejpam-5103	128	1	thus	thus	ADV
ejpam-5103	128	2	logf(γu1,u2(ς	logf(γu1,u2(ς	NOUN
ejpam-5103	128	3	)	)	PUNCT
ejpam-5103	128	4	)	)	PUNCT
ejpam-5103	128	5	≤	≤	NOUN
ejpam-5103	128	6	(	(	PUNCT
ejpam-5103	128	7	1−	1−	NUM
ejpam-5103	128	8	ς)logf(u1	ς)logf(u1	NOUN
ejpam-5103	128	9	)	)	PUNCT
ejpam-5103	129	1	+	+	NUM
ejpam-5103	129	2	ςlogf(u2	ςlogf(u2	X
ejpam-5103	129	3	)	)	PUNCT
ejpam-5103	129	4	≤	≤	NOUN
ejpam-5103	129	5	(	(	PUNCT
ejpam-5103	129	6	1−	1−	NUM
ejpam-5103	129	7	ς)v1	ς)v1	PROPN
ejpam-5103	129	8	+	+	SYM
ejpam-5103	129	9	ςv2.∀ς	ςv2.∀ς	NOUN
ejpam-5103	129	10	∈	∈	PROPN
ejpam-5103	130	1	[	[	X
ejpam-5103	130	2	0	0	NUM
ejpam-5103	130	3	,	,	PUNCT
ejpam-5103	130	4	1	1	NUM
ejpam-5103	130	5	]	]	PUNCT
ejpam-5103	130	6	.	.	PUNCT
ejpam-5103	131	1	that	that	PRON
ejpam-5103	131	2	means	mean	VERB
ejpam-5103	131	3	(	(	PUNCT
ejpam-5103	131	4	γu1,u2(ς	γu1,u2(ς	NOUN
ejpam-5103	131	5	)	)	PUNCT
ejpam-5103	131	6	,	,	PUNCT
ejpam-5103	131	7	(	(	PUNCT
ejpam-5103	131	8	1−	1−	NUM
ejpam-5103	131	9	ς)v1	ς)v1	PROPN
ejpam-5103	131	10	+	+	CCONJ
ejpam-5103	131	11	ςv2	ςv2	NOUN
ejpam-5103	131	12	)	)	PUNCT
ejpam-5103	131	13	∈	∈	PROPN
ejpam-5103	131	14	epi(f	epi(f	NOUN
ejpam-5103	131	15	)	)	PUNCT
ejpam-5103	131	16	.	.	PUNCT
ejpam-5103	132	1	this	this	DET
ejpam-5103	132	2	epi(f	epi(f	NOUN
ejpam-5103	132	3	)	)	PUNCT
ejpam-5103	132	4	is	be	AUX
ejpam-5103	132	5	geodesic	geodesic	ADJ
ejpam-5103	132	6	invex	invex	NOUN
ejpam-5103	132	7	set	set	NOUN
ejpam-5103	132	8	.	.	PUNCT
ejpam-5103	133	1	conversely	conversely	ADV
ejpam-5103	133	2	,	,	PUNCT
ejpam-5103	133	3	suppose	suppose	VERB
ejpam-5103	133	4	epi(f	epi(f	NOUN
ejpam-5103	133	5	)	)	PUNCT
ejpam-5103	133	6	is	be	AUX
ejpam-5103	133	7	a	a	DET
ejpam-5103	133	8	geodesic	geodesic	ADJ
ejpam-5103	133	9	invex	invex	NOUN
ejpam-5103	133	10	set	set	NOUN
ejpam-5103	133	11	.	.	PUNCT
ejpam-5103	134	1	let	let	VERB
ejpam-5103	134	2	u	u	NOUN
ejpam-5103	134	3	,	,	PUNCT
ejpam-5103	134	4	v	v	PROPN
ejpam-5103	134	5	∈	∈	PROPN
ejpam-5103	134	6	u	u	NOUN
ejpam-5103	134	7	.	.	PUNCT
ejpam-5103	135	1	then	then	ADV
ejpam-5103	135	2	,	,	PUNCT
ejpam-5103	135	3	we	we	PRON
ejpam-5103	135	4	have	have	VERB
ejpam-5103	135	5	(	(	PUNCT
ejpam-5103	135	6	u	u	NOUN
ejpam-5103	135	7	,	,	PUNCT
ejpam-5103	135	8	logf(u	logf(u	NOUN
ejpam-5103	135	9	)	)	PUNCT
ejpam-5103	135	10	)	)	PUNCT
ejpam-5103	136	1	∈	∈	PROPN
ejpam-5103	136	2	epi(f	epi(f	NOUN
ejpam-5103	136	3	)	)	PUNCT
ejpam-5103	136	4	and	and	CCONJ
ejpam-5103	136	5	(	(	PUNCT
ejpam-5103	136	6	v	v	NOUN
ejpam-5103	136	7	,	,	PUNCT
ejpam-5103	136	8	logf(v	logf(v	ADJ
ejpam-5103	136	9	)	)	PUNCT
ejpam-5103	136	10	)	)	PUNCT
ejpam-5103	137	1	∈	∈	PROPN
ejpam-5103	137	2	epi(f	epi(f	NOUN
ejpam-5103	137	3	)	)	PUNCT
ejpam-5103	137	4	.	.	PUNCT
ejpam-5103	138	1	since	since	SCONJ
ejpam-5103	138	2	epi(f	epi(f	NOUN
ejpam-5103	138	3	)	)	PUNCT
ejpam-5103	138	4	is	be	AUX
ejpam-5103	138	5	a	a	DET
ejpam-5103	138	6	geodesic	geodesic	ADJ
ejpam-5103	138	7	invex	invex	NOUN
ejpam-5103	138	8	set	set	NOUN
ejpam-5103	138	9	,	,	PUNCT
ejpam-5103	138	10	it	it	PRON
ejpam-5103	138	11	follows	follow	VERB
ejpam-5103	138	12	that	that	SCONJ
ejpam-5103	138	13	:	:	PUNCT
ejpam-5103	138	14	(	(	PUNCT
ejpam-5103	138	15	γu	γu	INTJ
ejpam-5103	138	16	,	,	PUNCT
ejpam-5103	138	17	v(ς	v(ς	NOUN
ejpam-5103	138	18	)	)	PUNCT
ejpam-5103	138	19	,	,	PUNCT
ejpam-5103	138	20	(	(	PUNCT
ejpam-5103	138	21	1−	1−	NUM
ejpam-5103	138	22	ς	ς	NOUN
ejpam-5103	138	23	)	)	PUNCT
ejpam-5103	138	24	logf(u	logf(u	NOUN
ejpam-5103	138	25	)	)	PUNCT
ejpam-5103	139	1	+	+	CCONJ
ejpam-5103	139	2	ς	ς	PROPN
ejpam-5103	139	3	logf(v	logf(v	ADJ
ejpam-5103	139	4	)	)	PUNCT
ejpam-5103	139	5	)	)	PUNCT
ejpam-5103	140	1	∈	∈	PROPN
ejpam-5103	140	2	epi(f	epi(f	NOUN
ejpam-5103	140	3	)	)	PUNCT
ejpam-5103	140	4	,	,	PUNCT
ejpam-5103	140	5	which	which	PRON
ejpam-5103	140	6	implies	imply	VERB
ejpam-5103	140	7	:	:	PUNCT
ejpam-5103	140	8	logf(γu	logf(γu	ADJ
ejpam-5103	140	9	,	,	PUNCT
ejpam-5103	140	10	v(ς	v(ς	NOUN
ejpam-5103	140	11	)	)	PUNCT
ejpam-5103	140	12	)	)	PUNCT
ejpam-5103	140	13	≤	≤	NOUN
ejpam-5103	140	14	(	(	PUNCT
ejpam-5103	140	15	1−	1−	NUM
ejpam-5103	140	16	ς	ς	NOUN
ejpam-5103	140	17	)	)	PUNCT
ejpam-5103	140	18	logf(u	logf(u	NOUN
ejpam-5103	140	19	)	)	PUNCT
ejpam-5103	141	1	+	+	CCONJ
ejpam-5103	141	2	ς	ς	PROPN
ejpam-5103	141	3	logf(v	logf(v	NUM
ejpam-5103	141	4	)	)	PUNCT
ejpam-5103	141	5	.	.	PUNCT
ejpam-5103	142	1	this	this	PRON
ejpam-5103	142	2	demonstrates	demonstrate	VERB
ejpam-5103	142	3	that	that	SCONJ
ejpam-5103	142	4	f	f	PROPN
ejpam-5103	142	5	is	be	AUX
ejpam-5103	142	6	a	a	DET
ejpam-5103	142	7	geodesic	geodesic	ADJ
ejpam-5103	142	8	log	log	NOUN
ejpam-5103	142	9	-	-	PUNCT
ejpam-5103	142	10	preinvex	preinvex	NOUN
ejpam-5103	142	11	function	function	NOUN
ejpam-5103	142	12	.	.	PUNCT
ejpam-5103	143	1	w.	w.	PROPN
ejpam-5103	143	2	saleh	saleh	PROPN
ejpam-5103	143	3	,	,	PUNCT
ejpam-5103	143	4	a.	a.	NOUN
ejpam-5103	143	5	lakhdari	lakhdari	PROPN
ejpam-5103	143	6	,	,	PUNCT
ejpam-5103	143	7	b.	b.	PROPN
ejpam-5103	143	8	meftah	meftah	PROPN
ejpam-5103	143	9	/	/	SYM
ejpam-5103	143	10	eur	eur	PROPN
ejpam-5103	143	11	.	.	PUNCT
ejpam-5103	144	1	j.	j.	PROPN
ejpam-5103	144	2	pure	pure	PROPN
ejpam-5103	144	3	appl	appl	PROPN
ejpam-5103	144	4	.	.	PROPN
ejpam-5103	144	5	math	math	PROPN
ejpam-5103	144	6	,	,	PUNCT
ejpam-5103	144	7	17	17	NUM
ejpam-5103	144	8	(	(	PUNCT
ejpam-5103	144	9	2	2	NUM
ejpam-5103	144	10	)	)	PUNCT
ejpam-5103	144	11	(	(	PUNCT
ejpam-5103	144	12	2024	2024	NUM
ejpam-5103	144	13	)	)	PUNCT
ejpam-5103	144	14	,	,	PUNCT
ejpam-5103	144	15	860	860	NUM
ejpam-5103	144	16	-	-	SYM
ejpam-5103	144	17	869	869	NUM
ejpam-5103	144	18	865	865	NUM
ejpam-5103	144	19	4	4	NUM
ejpam-5103	144	20	.	.	PUNCT
ejpam-5103	145	1	proporties	proportie	NOUN
ejpam-5103	145	2	of	of	ADP
ejpam-5103	145	3	strongly	strongly	ADV
ejpam-5103	145	4	geodesic	geodesic	ADJ
ejpam-5103	145	5	log	log	NOUN
ejpam-5103	145	6	-	-	PUNCT
ejpam-5103	145	7	preinvex	preinvex	NOUN
ejpam-5103	145	8	functions	function	NOUN
ejpam-5103	145	9	in	in	ADP
ejpam-5103	145	10	this	this	DET
ejpam-5103	145	11	section	section	NOUN
ejpam-5103	145	12	,	,	PUNCT
ejpam-5103	145	13	we	we	PRON
ejpam-5103	145	14	delve	delve	VERB
ejpam-5103	145	15	into	into	ADP
ejpam-5103	145	16	fundamental	fundamental	ADJ
ejpam-5103	145	17	properties	property	NOUN
ejpam-5103	145	18	of	of	ADP
ejpam-5103	145	19	functions	function	NOUN
ejpam-5103	145	20	exhibiting	exhibit	VERB
ejpam-5103	145	21	strongly	strongly	ADV
ejpam-5103	145	22	geodesic	geodesic	ADJ
ejpam-5103	145	23	log	log	NOUN
ejpam-5103	145	24	-	-	PUNCT
ejpam-5103	145	25	preinvexity	preinvexity	NOUN
ejpam-5103	145	26	.	.	PUNCT
ejpam-5103	146	1	theorem	theorem	NOUN
ejpam-5103	146	2	3	3	NUM
ejpam-5103	146	3	.	.	PUNCT
ejpam-5103	146	4	assume	assume	VERB
ejpam-5103	146	5	that	that	SCONJ
ejpam-5103	146	6	the	the	DET
ejpam-5103	146	7	function	function	NOUN
ejpam-5103	146	8	ξ	ξ	PROPN
ejpam-5103	146	9	is	be	AUX
ejpam-5103	146	10	differentiable	differentiable	ADJ
ejpam-5103	146	11	on	on	ADP
ejpam-5103	146	12	ϑ	ϑ	PRON
ejpam-5103	146	13	◦	◦	NOUN
ejpam-5103	146	14	.	.	PUNCT
ejpam-5103	147	1	if	if	SCONJ
ejpam-5103	147	2	ξ	ξ	PROPN
ejpam-5103	147	3	is	be	AUX
ejpam-5103	147	4	geodesic	geodesic	ADJ
ejpam-5103	147	5	logpreinvex	logpreinvex	NOUN
ejpam-5103	147	6	,	,	PUNCT
ejpam-5103	147	7	then	then	ADV
ejpam-5103	147	8	log	log	VERB
ejpam-5103	147	9	ξ(µ2)−	ξ(µ2)−	NOUN
ejpam-5103	147	10	log	log	NOUN
ejpam-5103	147	11	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	147	12	)	)	PUNCT
ejpam-5103	147	13	≥	≥	NOUN
ejpam-5103	147	14	dξµ1η(µ2	dξµ1η(µ2	PROPN
ejpam-5103	147	15	,	,	PUNCT
ejpam-5103	147	16	µ1	µ1	NOUN
ejpam-5103	147	17	)	)	PUNCT
ejpam-5103	147	18	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	147	19	)	)	PUNCT
ejpam-5103	148	1	+	+	CCONJ
ejpam-5103	148	2	ε∥η(µ2	ε∥η(µ2	ADJ
ejpam-5103	148	3	,	,	PUNCT
ejpam-5103	148	4	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	148	5	(	(	PUNCT
ejpam-5103	148	6	7	7	X
ejpam-5103	148	7	)	)	PUNCT
ejpam-5103	148	8	proof	proof	NOUN
ejpam-5103	148	9	.	.	PUNCT
ejpam-5103	149	1	assume	assume	VERB
ejpam-5103	149	2	that	that	SCONJ
ejpam-5103	149	3	the	the	DET
ejpam-5103	149	4	function	function	NOUN
ejpam-5103	149	5	ξ	ξ	PROPN
ejpam-5103	149	6	is	be	AUX
ejpam-5103	149	7	strongly	strongly	ADV
ejpam-5103	149	8	geodesic	geodesic	ADJ
ejpam-5103	149	9	log	log	NOUN
ejpam-5103	149	10	-	-	PUNCT
ejpam-5103	149	11	preinvex	preinvex	NOUN
ejpam-5103	149	12	,	,	PUNCT
ejpam-5103	149	13	one	one	NUM
ejpam-5103	149	14	has	have	VERB
ejpam-5103	149	15	that	that	PRON
ejpam-5103	149	16	log	log	VERB
ejpam-5103	149	17	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	149	18	)	)	PUNCT
ejpam-5103	149	19	)	)	PUNCT
ejpam-5103	149	20	≤	≤	NOUN
ejpam-5103	149	21	(	(	PUNCT
ejpam-5103	149	22	1−	1−	NUM
ejpam-5103	149	23	ς	ς	NOUN
ejpam-5103	149	24	)	)	PUNCT
ejpam-5103	149	25	log	log	NOUN
ejpam-5103	149	26	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	149	27	)	)	PUNCT
ejpam-5103	150	1	+	+	CCONJ
ejpam-5103	150	2	ς	ς	PROPN
ejpam-5103	150	3	log	log	NOUN
ejpam-5103	150	4	ξ(µ2)−	ξ(µ2)−	VERB
ejpam-5103	150	5	ες(1−	ες(1−	NOUN
ejpam-5103	150	6	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	150	7	,	,	PUNCT
ejpam-5103	150	8	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	150	9	,	,	PUNCT
ejpam-5103	150	10	hence	hence	ADV
ejpam-5103	150	11	log	log	VERB
ejpam-5103	150	12	ξ(µ2)−	ξ(µ2)−	NOUN
ejpam-5103	150	13	log	log	NOUN
ejpam-5103	150	14	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	150	15	)	)	PUNCT
ejpam-5103	150	16	≥	≥	NOUN
ejpam-5103	150	17	log	log	VERB
ejpam-5103	150	18	ξ(γµ1,µ2(ς))−	ξ(γµ1,µ2(ς))−	PROPN
ejpam-5103	150	19	log	log	NOUN
ejpam-5103	150	20	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	150	21	)	)	PUNCT
ejpam-5103	150	22	t	t	PROPN
ejpam-5103	150	23	+	+	CCONJ
ejpam-5103	150	24	ε∥η(µ2	ε∥η(µ2	ADJ
ejpam-5103	150	25	,	,	PUNCT
ejpam-5103	150	26	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	150	27	.	.	PUNCT
ejpam-5103	150	28	upon	upon	SCONJ
ejpam-5103	150	29	approaching	approach	VERB
ejpam-5103	150	30	the	the	DET
ejpam-5103	150	31	limit	limit	NOUN
ejpam-5103	150	32	as	as	SCONJ
ejpam-5103	150	33	ς	ς	PROPN
ejpam-5103	150	34	tends	tend	VERB
ejpam-5103	150	35	towards	towards	ADP
ejpam-5103	150	36	zero	zero	NUM
ejpam-5103	150	37	in	in	ADP
ejpam-5103	150	38	the	the	DET
ejpam-5103	150	39	preceding	precede	VERB
ejpam-5103	150	40	inequality	inequality	NOUN
ejpam-5103	150	41	,	,	PUNCT
ejpam-5103	150	42	we	we	PRON
ejpam-5103	150	43	obtain	obtain	VERB
ejpam-5103	150	44	:	:	PUNCT
ejpam-5103	150	45	log	log	NOUN
ejpam-5103	150	46	ξ(µ2)−	ξ(µ2)−	NOUN
ejpam-5103	150	47	log	log	NOUN
ejpam-5103	150	48	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	150	49	)	)	PUNCT
ejpam-5103	150	50	≥	≥	NOUN
ejpam-5103	150	51	dµ1ξγµ1,µ2	dµ1ξγµ1,µ2	PROPN
ejpam-5103	150	52	ξ(u	ξ(u	NOUN
ejpam-5103	150	53	)	)	PUNCT
ejpam-5103	151	1	+	+	CCONJ
ejpam-5103	151	2	ε∥η(µ2	ε∥η(µ2	ADJ
ejpam-5103	151	3	,	,	PUNCT
ejpam-5103	151	4	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	151	5	.	.	PUNCT
ejpam-5103	152	1	which	which	PRON
ejpam-5103	152	2	(	(	PUNCT
ejpam-5103	152	3	7	7	NUM
ejpam-5103	152	4	)	)	PUNCT
ejpam-5103	152	5	,	,	PUNCT
ejpam-5103	152	6	the	the	DET
ejpam-5103	152	7	required	require	VERB
ejpam-5103	152	8	result	result	NOUN
ejpam-5103	152	9	.	.	PUNCT
ejpam-5103	153	1	remark	remark	NOUN
ejpam-5103	153	2	1	1	NUM
ejpam-5103	153	3	.	.	PUNCT
ejpam-5103	154	1	form	form	NOUN
ejpam-5103	154	2	(	(	PUNCT
ejpam-5103	154	3	7	7	NUM
ejpam-5103	154	4	)	)	PUNCT
ejpam-5103	154	5	,	,	PUNCT
ejpam-5103	154	6	we	we	PRON
ejpam-5103	154	7	have	have	AUX
ejpam-5103	154	8	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	154	9	)	)	PUNCT
ejpam-5103	154	10	≥	≥	NOUN
ejpam-5103	154	11	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	154	12	)	)	PUNCT
ejpam-5103	154	13	exp	exp	NOUN
ejpam-5103	154	14	{	{	PUNCT
ejpam-5103	154	15	dµ1ξγµ1,µ2	dµ1ξγµ1,µ2	PROPN
ejpam-5103	154	16	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	154	17	)	)	PUNCT
ejpam-5103	155	1	+	+	CCONJ
ejpam-5103	155	2	ε∥η(µ2	ε∥η(µ2	ADJ
ejpam-5103	155	3	,	,	PUNCT
ejpam-5103	155	4	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	155	5	}	}	PUNCT
ejpam-5103	155	6	,	,	PUNCT
ejpam-5103	155	7	∀µ1	∀µ1	PROPN
ejpam-5103	155	8	,	,	PUNCT
ejpam-5103	155	9	µ2	µ2	PROPN
ejpam-5103	155	10	∈	∈	PROPN
ejpam-5103	155	11	ϑ.	ϑ.	NOUN
ejpam-5103	155	12	by	by	ADP
ejpam-5103	155	13	interchanging	interchange	VERB
ejpam-5103	155	14	the	the	DET
ejpam-5103	155	15	roles	role	NOUN
ejpam-5103	155	16	of	of	ADP
ejpam-5103	155	17	µ1	µ1	NOUN
ejpam-5103	155	18	and	and	CCONJ
ejpam-5103	155	19	µ2	µ2	PROPN
ejpam-5103	155	20	in	in	ADP
ejpam-5103	155	21	the	the	DET
ejpam-5103	155	22	inequality	inequality	NOUN
ejpam-5103	155	23	above	above	ADV
ejpam-5103	155	24	,	,	PUNCT
ejpam-5103	155	25	we	we	PRON
ejpam-5103	155	26	also	also	ADV
ejpam-5103	155	27	obtain	obtain	VERB
ejpam-5103	155	28	:	:	PUNCT
ejpam-5103	155	29	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	155	30	)	)	PUNCT
ejpam-5103	155	31	≥	≥	PROPN
ejpam-5103	155	32	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	155	33	)	)	PUNCT
ejpam-5103	155	34	exp	exp	NOUN
ejpam-5103	155	35	{	{	PUNCT
ejpam-5103	155	36	dµ2ξγµ1,µ2	dµ2ξγµ1,µ2	PROPN
ejpam-5103	155	37	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	155	38	)	)	PUNCT
ejpam-5103	156	1	+	+	CCONJ
ejpam-5103	156	2	ε∥η(µ2	ε∥η(µ2	ADJ
ejpam-5103	156	3	,	,	PUNCT
ejpam-5103	156	4	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	156	5	}	}	PUNCT
ejpam-5103	156	6	,	,	PUNCT
ejpam-5103	156	7	∀µ1	∀µ1	PROPN
ejpam-5103	156	8	,	,	PUNCT
ejpam-5103	156	9	µ2	µ2	PROPN
ejpam-5103	156	10	∈	∈	PROPN
ejpam-5103	156	11	ϑ.	ϑ.	NOUN
ejpam-5103	156	12	therefore	therefore	ADV
ejpam-5103	156	13	,	,	PUNCT
ejpam-5103	156	14	we	we	PRON
ejpam-5103	156	15	can	can	AUX
ejpam-5103	156	16	deduce	deduce	VERB
ejpam-5103	156	17	the	the	DET
ejpam-5103	156	18	subsequent	subsequent	ADJ
ejpam-5103	156	19	inequality	inequality	NOUN
ejpam-5103	156	20	:	:	PUNCT
ejpam-5103	156	21	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	156	22	)	)	PUNCT
ejpam-5103	156	23	+	+	NUM
ejpam-5103	156	24	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	156	25	)	)	PUNCT
ejpam-5103	156	26	≥	≥	PROPN
ejpam-5103	156	27	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	156	28	)	)	PUNCT
ejpam-5103	156	29	exp	exp	NOUN
ejpam-5103	156	30	{	{	PUNCT
ejpam-5103	156	31	dµ2ξγµ1,µ2	dµ2ξγµ1,µ2	PROPN
ejpam-5103	156	32	ξ(µ2	ξ(µ2	ADJ
ejpam-5103	156	33	)	)	PUNCT
ejpam-5103	157	1	+	+	CCONJ
ejpam-5103	157	2	ε∥η(µ2	ε∥η(µ2	ADJ
ejpam-5103	157	3	,	,	PUNCT
ejpam-5103	157	4	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	157	5	}	}	PUNCT
ejpam-5103	157	6	+	+	NOUN
ejpam-5103	157	7	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	157	8	)	)	PUNCT
ejpam-5103	157	9	exp	exp	NOUN
ejpam-5103	157	10	{	{	PUNCT
ejpam-5103	157	11	dµ1ξγµ1,µ2	dµ1ξγµ1,µ2	PROPN
ejpam-5103	157	12	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	157	13	)	)	PUNCT
ejpam-5103	158	1	+	+	CCONJ
ejpam-5103	158	2	ε∥η(µ2	ε∥η(µ2	ADJ
ejpam-5103	158	3	,	,	PUNCT
ejpam-5103	158	4	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	158	5	}	}	PUNCT
ejpam-5103	158	6	,	,	PUNCT
ejpam-5103	158	7	∀µ1	∀µ1	PROPN
ejpam-5103	158	8	,	,	PUNCT
ejpam-5103	158	9	µ2	µ2	PROPN
ejpam-5103	158	10	∈	∈	PROPN
ejpam-5103	158	11	ϑ.	ϑ.	NOUN
ejpam-5103	158	12	the	the	DET
ejpam-5103	158	13	last	last	ADJ
ejpam-5103	158	14	theorem	theorem	NOUN
ejpam-5103	158	15	allows	allow	VERB
ejpam-5103	158	16	us	we	PRON
ejpam-5103	158	17	to	to	PART
ejpam-5103	158	18	introduce	introduce	VERB
ejpam-5103	158	19	the	the	DET
ejpam-5103	158	20	concept	concept	NOUN
ejpam-5103	158	21	of	of	ADP
ejpam-5103	158	22	geodesic	geodesic	ADJ
ejpam-5103	158	23	log	log	NOUN
ejpam-5103	158	24	-	-	PUNCT
ejpam-5103	158	25	monotone	monotone	NOUN
ejpam-5103	158	26	operators	operator	NOUN
ejpam-5103	158	27	,	,	PUNCT
ejpam-5103	158	28	which	which	PRON
ejpam-5103	158	29	appears	appear	VERB
ejpam-5103	158	30	to	to	PART
ejpam-5103	158	31	be	be	AUX
ejpam-5103	158	32	a	a	DET
ejpam-5103	158	33	novel	novel	ADJ
ejpam-5103	158	34	addition	addition	NOUN
ejpam-5103	158	35	to	to	ADP
ejpam-5103	158	36	the	the	DET
ejpam-5103	158	37	field	field	NOUN
ejpam-5103	158	38	.	.	PUNCT
ejpam-5103	159	1	definition	definition	NOUN
ejpam-5103	159	2	7	7	NUM
ejpam-5103	159	3	.	.	PUNCT
ejpam-5103	160	1	(	(	PUNCT
ejpam-5103	160	2	i	i	NOUN
ejpam-5103	160	3	)	)	PUNCT
ejpam-5103	160	4	the	the	DET
ejpam-5103	160	5	differential	differential	NOUN
ejpam-5103	161	1	f	f	PROPN
ejpam-5103	161	2	′	′	NOUN
ejpam-5103	161	3	is	be	AUX
ejpam-5103	161	4	considered	consider	VERB
ejpam-5103	161	5	to	to	PART
ejpam-5103	161	6	be	be	AUX
ejpam-5103	161	7	strongly	strongly	ADV
ejpam-5103	161	8	geodesic	geodesic	ADJ
ejpam-5103	161	9	log	log	NOUN
ejpam-5103	161	10	-	-	PUNCT
ejpam-5103	161	11	monotone	monotone	NOUN
ejpam-5103	161	12	,	,	PUNCT
ejpam-5103	161	13	if	if	SCONJ
ejpam-5103	161	14	dfµ1η(µ2	dfµ1η(µ2	ADJ
ejpam-5103	161	15	,	,	PUNCT
ejpam-5103	161	16	µ1	µ1	NOUN
ejpam-5103	161	17	)	)	PUNCT
ejpam-5103	161	18	f(µ1	f(µ1	NOUN
ejpam-5103	161	19	)	)	PUNCT
ejpam-5103	162	1	+	+	CCONJ
ejpam-5103	162	2	dfµ2η(µ1	dfµ2η(µ1	ADJ
ejpam-5103	162	3	,	,	PUNCT
ejpam-5103	162	4	µ2	µ2	ADJ
ejpam-5103	162	5	)	)	PUNCT
ejpam-5103	162	6	f(v	f(v	PROPN
ejpam-5103	162	7	)	)	PUNCT
ejpam-5103	162	8	≤	≤	PROPN
ejpam-5103	162	9	−ε	−ε	PROPN
ejpam-5103	162	10	{	{	PUNCT
ejpam-5103	162	11	∥η(µ2	∥η(µ2	PROPN
ejpam-5103	162	12	,	,	PUNCT
ejpam-5103	162	13	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	162	14	+	+	NUM
ejpam-5103	162	15	∥η(µ1	∥η(µ1	NOUN
ejpam-5103	162	16	,	,	PUNCT
ejpam-5103	162	17	µ2)(ς)∥2	µ2)(ς)∥2	PROPN
ejpam-5103	162	18	}	}	PUNCT
ejpam-5103	162	19	,	,	PUNCT
ejpam-5103	162	20	∀µ1	∀µ1	PROPN
ejpam-5103	162	21	,	,	PUNCT
ejpam-5103	162	22	µ2	µ2	PROPN
ejpam-5103	162	23	∈	∈	PROPN
ejpam-5103	162	24	ϑ.	ϑ.	PROPN
ejpam-5103	162	25	w.	w.	PROPN
ejpam-5103	162	26	saleh	saleh	PROPN
ejpam-5103	162	27	,	,	PUNCT
ejpam-5103	162	28	a.	a.	NOUN
ejpam-5103	162	29	lakhdari	lakhdari	PROPN
ejpam-5103	162	30	,	,	PUNCT
ejpam-5103	162	31	b.	b.	PROPN
ejpam-5103	162	32	meftah	meftah	PROPN
ejpam-5103	162	33	/	/	SYM
ejpam-5103	162	34	eur	eur	PROPN
ejpam-5103	162	35	.	.	PUNCT
ejpam-5103	163	1	j.	j.	PROPN
ejpam-5103	163	2	pure	pure	PROPN
ejpam-5103	163	3	appl	appl	PROPN
ejpam-5103	163	4	.	.	PROPN
ejpam-5103	163	5	math	math	PROPN
ejpam-5103	163	6	,	,	PUNCT
ejpam-5103	163	7	17	17	NUM
ejpam-5103	163	8	(	(	PUNCT
ejpam-5103	163	9	2	2	NUM
ejpam-5103	163	10	)	)	PUNCT
ejpam-5103	163	11	(	(	PUNCT
ejpam-5103	163	12	2024	2024	NUM
ejpam-5103	163	13	)	)	PUNCT
ejpam-5103	163	14	,	,	PUNCT
ejpam-5103	163	15	860	860	NUM
ejpam-5103	163	16	-	-	SYM
ejpam-5103	163	17	869	869	NUM
ejpam-5103	163	18	866	866	NUM
ejpam-5103	163	19	(	(	PUNCT
ejpam-5103	163	20	ii	ii	PROPN
ejpam-5103	163	21	)	)	PUNCT
ejpam-5103	163	22	the	the	DET
ejpam-5103	163	23	differential	differential	NOUN
ejpam-5103	163	24	f	f	PROPN
ejpam-5103	163	25	′	′	NOUN
ejpam-5103	163	26	is	be	AUX
ejpam-5103	163	27	considered	consider	VERB
ejpam-5103	163	28	to	to	PART
ejpam-5103	163	29	be	be	AUX
ejpam-5103	163	30	geodesic	geodesic	ADJ
ejpam-5103	163	31	log	log	NOUN
ejpam-5103	163	32	-	-	PUNCT
ejpam-5103	163	33	monotone	monotone	NOUN
ejpam-5103	163	34	,	,	PUNCT
ejpam-5103	163	35	if	if	SCONJ
ejpam-5103	163	36	dfµ1η(µ2	dfµ1η(µ2	ADJ
ejpam-5103	163	37	,	,	PUNCT
ejpam-5103	163	38	µ1	µ1	NOUN
ejpam-5103	163	39	)	)	PUNCT
ejpam-5103	163	40	f(µ1	f(µ1	NOUN
ejpam-5103	163	41	)	)	PUNCT
ejpam-5103	164	1	+	+	CCONJ
ejpam-5103	164	2	dfµ2η(µ1	dfµ2η(µ1	ADJ
ejpam-5103	164	3	,	,	PUNCT
ejpam-5103	164	4	µ2	µ2	ADJ
ejpam-5103	164	5	)	)	PUNCT
ejpam-5103	164	6	f(µ2	f(µ2	NOUN
ejpam-5103	164	7	)	)	PUNCT
ejpam-5103	164	8	≤	≤	NOUN
ejpam-5103	164	9	0	0	NUM
ejpam-5103	164	10	,	,	PUNCT
ejpam-5103	164	11	∀µ1	∀µ1	PROPN
ejpam-5103	164	12	,	,	PUNCT
ejpam-5103	164	13	µ2	µ2	PROPN
ejpam-5103	164	14	∈	∈	PROPN
ejpam-5103	164	15	ϑ.	ϑ.	NOUN
ejpam-5103	164	16	(	(	PUNCT
ejpam-5103	164	17	iii	iii	X
ejpam-5103	164	18	)	)	PUNCT
ejpam-5103	164	19	the	the	DET
ejpam-5103	164	20	differential	differential	NOUN
ejpam-5103	165	1	f	f	PROPN
ejpam-5103	165	2	′	′	NOUN
ejpam-5103	165	3	is	be	AUX
ejpam-5103	165	4	considered	consider	VERB
ejpam-5103	165	5	to	to	PART
ejpam-5103	165	6	be	be	AUX
ejpam-5103	165	7	geodesic	geodesic	ADJ
ejpam-5103	165	8	logpseudo	logpseudo	NOUN
ejpam-5103	165	9	-	-	ADJ
ejpam-5103	165	10	monotone	monotone	ADJ
ejpam-5103	165	11	,	,	PUNCT
ejpam-5103	165	12	if	if	SCONJ
ejpam-5103	165	13	dfµ1η(v	dfµ1η(v	NOUN
ejpam-5103	165	14	,	,	PUNCT
ejpam-5103	165	15	µ1	µ1	NOUN
ejpam-5103	165	16	)	)	PUNCT
ejpam-5103	165	17	f(µ1	f(µ1	NOUN
ejpam-5103	165	18	)	)	PUNCT
ejpam-5103	165	19	≥	≥	NOUN
ejpam-5103	165	20	0	0	NUM
ejpam-5103	166	1	=	=	NOUN
ejpam-5103	166	2	⇒	⇒	PRON
ejpam-5103	166	3	−dfvη(µ1	−dfvη(µ1	PROPN
ejpam-5103	166	4	,	,	PUNCT
ejpam-5103	166	5	v	v	NOUN
ejpam-5103	166	6	)	)	PUNCT
ejpam-5103	166	7	f(v	f(v	NOUN
ejpam-5103	166	8	)	)	PUNCT
ejpam-5103	166	9	≥	≥	NOUN
ejpam-5103	166	10	0∀µ1	0∀µ1	NOUN
ejpam-5103	166	11	,	,	PUNCT
ejpam-5103	166	12	v	v	NOUN
ejpam-5103	166	13	∈	∈	NOUN
ejpam-5103	166	14	ϑ.	ϑ.	NOUN
ejpam-5103	166	15	from	from	ADP
ejpam-5103	166	16	these	these	DET
ejpam-5103	166	17	definitions	definition	NOUN
ejpam-5103	166	18	,	,	PUNCT
ejpam-5103	166	19	we	we	PRON
ejpam-5103	166	20	can	can	AUX
ejpam-5103	166	21	deduce	deduce	VERB
ejpam-5103	166	22	that	that	PRON
ejpam-5103	166	23	strongly	strongly	ADV
ejpam-5103	166	24	geodesic	geodesic	ADJ
ejpam-5103	166	25	log	log	NOUN
ejpam-5103	166	26	-	-	PUNCT
ejpam-5103	166	27	monotonicity	monotonicity	NOUN
ejpam-5103	166	28	entails	entail	VERB
ejpam-5103	166	29	geodesic	geodesic	ADJ
ejpam-5103	166	30	log	log	NOUN
ejpam-5103	166	31	-	-	PUNCT
ejpam-5103	166	32	monotonicity	monotonicity	NOUN
ejpam-5103	166	33	,	,	PUNCT
ejpam-5103	166	34	which	which	PRON
ejpam-5103	166	35	,	,	PUNCT
ejpam-5103	166	36	in	in	ADP
ejpam-5103	166	37	turn	turn	NOUN
ejpam-5103	166	38	,	,	PUNCT
ejpam-5103	166	39	implies	imply	VERB
ejpam-5103	166	40	geodesic	geodesic	ADJ
ejpam-5103	166	41	log	log	NOUN
ejpam-5103	166	42	-	-	PUNCT
ejpam-5103	166	43	pseudo	pseudo	NOUN
ejpam-5103	166	44	-	-	NOUN
ejpam-5103	166	45	monotonicity	monotonicity	NOUN
ejpam-5103	166	46	.	.	PUNCT
ejpam-5103	167	1	it	it	PRON
ejpam-5103	167	2	’s	’	VERB
ejpam-5103	167	3	crucial	crucial	ADJ
ejpam-5103	167	4	to	to	PART
ejpam-5103	167	5	emphasize	emphasize	VERB
ejpam-5103	167	6	that	that	SCONJ
ejpam-5103	167	7	the	the	DET
ejpam-5103	167	8	reverse	reverse	NOUN
ejpam-5103	167	9	may	may	AUX
ejpam-5103	167	10	not	not	PART
ejpam-5103	167	11	always	always	ADV
ejpam-5103	167	12	hold	hold	VERB
ejpam-5103	167	13	true	true	ADJ
ejpam-5103	167	14	.	.	PUNCT
ejpam-5103	168	1	theorem	theorem	ADJ
ejpam-5103	168	2	4	4	NUM
ejpam-5103	168	3	.	.	PUNCT
ejpam-5103	169	1	let	let	VERB
ejpam-5103	169	2	f	f	PRON
ejpam-5103	169	3	be	be	AUX
ejpam-5103	169	4	differentiable	differentiable	ADJ
ejpam-5103	169	5	strongly	strongly	ADV
ejpam-5103	169	6	geodesic	geodesic	ADJ
ejpam-5103	169	7	log	log	NOUN
ejpam-5103	169	8	-	-	PUNCT
ejpam-5103	169	9	preinvex	preinvex	NOUN
ejpam-5103	169	10	function	function	NOUN
ejpam-5103	169	11	on	on	ADP
ejpam-5103	169	12	the	the	DET
ejpam-5103	169	13	geodesic	geodesic	ADJ
ejpam-5103	169	14	set	set	NOUN
ejpam-5103	169	15	.	.	PUNCT
ejpam-5103	170	1	if	if	SCONJ
ejpam-5103	170	2	(	(	PUNCT
ejpam-5103	170	3	7	7	X
ejpam-5103	170	4	)	)	PUNCT
ejpam-5103	170	5	holds	hold	VERB
ejpam-5103	170	6	,	,	PUNCT
ejpam-5103	170	7	then	then	ADV
ejpam-5103	170	8	f	f	NOUN
ejpam-5103	170	9	′	′	NUM
ejpam-5103	170	10	satisfies	satisfie	NOUN
ejpam-5103	170	11	dfµ1η(µ2	dfµ1η(µ2	PROPN
ejpam-5103	170	12	,	,	PUNCT
ejpam-5103	170	13	µ1	µ1	NOUN
ejpam-5103	170	14	)	)	PUNCT
ejpam-5103	170	15	f(µ1	f(µ1	NOUN
ejpam-5103	170	16	)	)	PUNCT
ejpam-5103	171	1	+	+	CCONJ
ejpam-5103	171	2	dfµ2η(µ1	dfµ2η(µ1	ADJ
ejpam-5103	171	3	,	,	PUNCT
ejpam-5103	171	4	v	v	NOUN
ejpam-5103	171	5	)	)	PUNCT
ejpam-5103	171	6	f(µ2	f(µ2	NOUN
ejpam-5103	171	7	)	)	PUNCT
ejpam-5103	171	8	≤	≤	NOUN
ejpam-5103	171	9	−ε	−ε	PROPN
ejpam-5103	171	10	{	{	PUNCT
ejpam-5103	171	11	∥η(µ2	∥η(µ2	PROPN
ejpam-5103	171	12	,	,	PUNCT
ejpam-5103	171	13	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	171	14	+	+	NUM
ejpam-5103	171	15	∥η(µ1	∥η(µ1	NOUN
ejpam-5103	171	16	,	,	PUNCT
ejpam-5103	171	17	µ2)(ς)∥2	µ2)(ς)∥2	PROPN
ejpam-5103	171	18	}	}	PUNCT
ejpam-5103	171	19	,	,	PUNCT
ejpam-5103	171	20	∀µ1	∀µ1	PROPN
ejpam-5103	171	21	,	,	PUNCT
ejpam-5103	171	22	µ2	µ2	PROPN
ejpam-5103	171	23	∈	∈	PROPN
ejpam-5103	171	24	ϑ.	ϑ.	NOUN
ejpam-5103	171	25	(	(	PUNCT
ejpam-5103	171	26	8)	8)	NUM
ejpam-5103	171	27	proof	proof	NOUN
ejpam-5103	171	28	.	.	PUNCT
ejpam-5103	172	1	log	log	VERB
ejpam-5103	172	2	f(µ2)−	f(µ2)−	ADV
ejpam-5103	172	3	log	log	VERB
ejpam-5103	172	4	f(µ1	f(µ1	NOUN
ejpam-5103	172	5	)	)	PUNCT
ejpam-5103	172	6	≥	≥	NOUN
ejpam-5103	173	1	dfµ1η(µ2	dfµ1η(µ2	PROPN
ejpam-5103	173	2	,	,	PUNCT
ejpam-5103	173	3	µ1	µ1	NOUN
ejpam-5103	173	4	)	)	PUNCT
ejpam-5103	173	5	f(µ1	f(µ1	NOUN
ejpam-5103	173	6	)	)	PUNCT
ejpam-5103	174	1	+	+	CCONJ
ejpam-5103	174	2	ε∥η(µ2	ε∥η(µ2	ADJ
ejpam-5103	174	3	,	,	PUNCT
ejpam-5103	174	4	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	174	5	.	.	PUNCT
ejpam-5103	175	1	(	(	PUNCT
ejpam-5103	175	2	9	9	NUM
ejpam-5103	175	3	)	)	PUNCT
ejpam-5103	175	4	by	by	ADP
ejpam-5103	175	5	swapping	swap	VERB
ejpam-5103	175	6	the	the	DET
ejpam-5103	175	7	roles	role	NOUN
ejpam-5103	175	8	of	of	ADP
ejpam-5103	175	9	µ1	µ1	NOUN
ejpam-5103	175	10	and	and	CCONJ
ejpam-5103	175	11	µ2	µ2	PROPN
ejpam-5103	175	12	in	in	ADP
ejpam-5103	175	13	inequality	inequality	NOUN
ejpam-5103	175	14	(	(	PUNCT
ejpam-5103	175	15	9	9	NUM
ejpam-5103	175	16	)	)	PUNCT
ejpam-5103	175	17	,	,	PUNCT
ejpam-5103	175	18	we	we	PRON
ejpam-5103	175	19	obtain	obtain	VERB
ejpam-5103	175	20	:	:	PUNCT
ejpam-5103	175	21	log	log	VERB
ejpam-5103	175	22	f(µ1)−	f(µ1)−	PROPN
ejpam-5103	175	23	log	log	NOUN
ejpam-5103	175	24	f(µ2	f(µ2	NOUN
ejpam-5103	175	25	)	)	PUNCT
ejpam-5103	175	26	≥	≥	NOUN
ejpam-5103	175	27	dfµ2η(µ1	dfµ2η(µ1	ADJ
ejpam-5103	175	28	,	,	PUNCT
ejpam-5103	175	29	µ2	µ2	PROPN
ejpam-5103	175	30	)	)	PUNCT
ejpam-5103	175	31	f(µ2	f(µ2	NOUN
ejpam-5103	175	32	)	)	PUNCT
ejpam-5103	175	33	+	+	SYM
ejpam-5103	175	34	ε∥η(µ1	ε∥η(µ1	PROPN
ejpam-5103	175	35	,	,	PUNCT
ejpam-5103	175	36	µ2)(ς)∥2	µ2)(ς)∥2	PROPN
ejpam-5103	175	37	.	.	PUNCT
ejpam-5103	176	1	(	(	PUNCT
ejpam-5103	176	2	10	10	NUM
ejpam-5103	176	3	)	)	PUNCT
ejpam-5103	176	4	adding	add	VERB
ejpam-5103	176	5	(	(	PUNCT
ejpam-5103	176	6	9	9	NUM
ejpam-5103	176	7	)	)	PUNCT
ejpam-5103	176	8	and	and	CCONJ
ejpam-5103	176	9	(	(	PUNCT
ejpam-5103	176	10	10	10	NUM
ejpam-5103	176	11	)	)	PUNCT
ejpam-5103	176	12	,	,	PUNCT
ejpam-5103	176	13	we	we	PRON
ejpam-5103	176	14	have	have	VERB
ejpam-5103	176	15	dfµ1η(µ2	dfµ1η(µ2	ADJ
ejpam-5103	176	16	,	,	PUNCT
ejpam-5103	176	17	µ1	µ1	NOUN
ejpam-5103	176	18	)	)	PUNCT
ejpam-5103	176	19	f(µ1	f(µ1	NOUN
ejpam-5103	176	20	)	)	PUNCT
ejpam-5103	177	1	+	+	CCONJ
ejpam-5103	177	2	dfµ2η(µ1	dfµ2η(µ1	ADJ
ejpam-5103	177	3	,	,	PUNCT
ejpam-5103	177	4	µ2	µ2	ADJ
ejpam-5103	177	5	)	)	PUNCT
ejpam-5103	177	6	f(µ2	f(µ2	NOUN
ejpam-5103	177	7	)	)	PUNCT
ejpam-5103	177	8	≤	≤	NOUN
ejpam-5103	177	9	−ε	−ε	PROPN
ejpam-5103	177	10	{	{	PUNCT
ejpam-5103	177	11	∥η(µ2	∥η(µ2	PROPN
ejpam-5103	177	12	,	,	PUNCT
ejpam-5103	177	13	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	177	14	+	+	NUM
ejpam-5103	177	15	∥η(µ1	∥η(µ1	NOUN
ejpam-5103	177	16	,	,	PUNCT
ejpam-5103	177	17	µ2)(ς)∥2	µ2)(ς)∥2	PROPN
ejpam-5103	177	18	}	}	PUNCT
ejpam-5103	177	19	,	,	PUNCT
ejpam-5103	177	20	∀µ1	∀µ1	PROPN
ejpam-5103	177	21	,	,	PUNCT
ejpam-5103	177	22	µ2	µ2	PROPN
ejpam-5103	177	23	∈	∈	PROPN
ejpam-5103	177	24	ϑ	ϑ	X
ejpam-5103	177	25	,	,	PUNCT
ejpam-5103	177	26	(	(	PUNCT
ejpam-5103	177	27	11	11	NUM
ejpam-5103	177	28	)	)	PUNCT
ejpam-5103	177	29	demonstrating	demonstrate	VERB
ejpam-5103	177	30	that	that	SCONJ
ejpam-5103	177	31	the	the	DET
ejpam-5103	177	32	derivative	derivative	ADJ
ejpam-5103	177	33	f	f	NOUN
ejpam-5103	177	34	′	′	NUM
ejpam-5103	177	35	is	be	AUX
ejpam-5103	177	36	strongly	strongly	ADV
ejpam-5103	177	37	geodesic	geodesic	ADJ
ejpam-5103	177	38	log	log	NOUN
ejpam-5103	177	39	-	-	PUNCT
ejpam-5103	177	40	monotone	monotone	NOUN
ejpam-5103	177	41	.	.	PUNCT
ejpam-5103	178	1	definition	definition	NOUN
ejpam-5103	178	2	8	8	NUM
ejpam-5103	178	3	.	.	PUNCT
ejpam-5103	179	1	the	the	DET
ejpam-5103	179	2	function	function	NOUN
ejpam-5103	179	3	f	f	PROPN
ejpam-5103	179	4	is	be	AUX
ejpam-5103	179	5	considered	consider	VERB
ejpam-5103	179	6	to	to	PART
ejpam-5103	179	7	be	be	AUX
ejpam-5103	179	8	sharply	sharply	ADV
ejpam-5103	179	9	geodesic	geodesic	ADJ
ejpam-5103	179	10	log	log	NOUN
ejpam-5103	179	11	-	-	PUNCT
ejpam-5103	179	12	pseudo	pseudo	NOUN
ejpam-5103	179	13	preinvex	preinvex	NOUN
ejpam-5103	179	14	,	,	PUNCT
ejpam-5103	179	15	if	if	SCONJ
ejpam-5103	179	16	dfµ1η(µ2	dfµ1η(µ2	ADJ
ejpam-5103	179	17	,	,	PUNCT
ejpam-5103	179	18	µ1	µ1	NOUN
ejpam-5103	179	19	)	)	PUNCT
ejpam-5103	179	20	f(µ1	f(µ1	NOUN
ejpam-5103	179	21	)	)	PUNCT
ejpam-5103	179	22	≥	≥	NOUN
ejpam-5103	179	23	0	0	NUM
ejpam-5103	180	1	=	=	NOUN
ejpam-5103	180	2	⇒	⇒	NOUN
ejpam-5103	180	3	f(µ2	f(µ2	NOUN
ejpam-5103	180	4	)	)	PUNCT
ejpam-5103	180	5	≥	≥	PRON
ejpam-5103	180	6	log	log	VERB
ejpam-5103	180	7	f(γµ1,µ2(ς)),∀µ1	f(γµ1,µ2(ς)),∀µ1	PROPN
ejpam-5103	180	8	,	,	PUNCT
ejpam-5103	180	9	µ2	µ2	PROPN
ejpam-5103	180	10	∈	∈	PROPN
ejpam-5103	180	11	ϑ	ϑ	NOUN
ejpam-5103	180	12	,	,	PUNCT
ejpam-5103	180	13	ς	ς	PROPN
ejpam-5103	180	14	∈	∈	PROPN
ejpam-5103	181	1	[	[	X
ejpam-5103	181	2	0	0	NUM
ejpam-5103	181	3	,	,	PUNCT
ejpam-5103	181	4	1	1	NUM
ejpam-5103	181	5	]	]	PUNCT
ejpam-5103	181	6	.	.	PUNCT
ejpam-5103	182	1	theorem	theorem	ADJ
ejpam-5103	182	2	5	5	NUM
ejpam-5103	182	3	.	.	PUNCT
ejpam-5103	182	4	assume	assume	VERB
ejpam-5103	182	5	that	that	SCONJ
ejpam-5103	182	6	f	f	PROPN
ejpam-5103	182	7	is	be	AUX
ejpam-5103	182	8	a	a	DET
ejpam-5103	182	9	sharply	sharply	ADV
ejpam-5103	182	10	geodesic	geodesic	ADJ
ejpam-5103	182	11	log	log	NOUN
ejpam-5103	182	12	-	-	PUNCT
ejpam-5103	182	13	pseudo	pseudo	NOUN
ejpam-5103	182	14	preinvex	preinvex	NOUN
ejpam-5103	182	15	function	function	NOUN
ejpam-5103	182	16	on	on	ADP
ejpam-5103	182	17	a.	a.	NOUN
ejpam-5103	182	18	then	then	ADV
ejpam-5103	182	19	dfµ2η(µ1	dfµ2η(µ1	ADJ
ejpam-5103	182	20	,	,	PUNCT
ejpam-5103	182	21	µ2	µ2	PROPN
ejpam-5103	182	22	)	)	PUNCT
ejpam-5103	182	23	f(µ2	f(µ2	NOUN
ejpam-5103	182	24	)	)	PUNCT
ejpam-5103	182	25	≥	≥	NOUN
ejpam-5103	182	26	0	0	NUM
ejpam-5103	182	27	,	,	PUNCT
ejpam-5103	182	28	∀µ1	∀µ1	PROPN
ejpam-5103	182	29	,	,	PUNCT
ejpam-5103	182	30	µ2	µ2	PROPN
ejpam-5103	182	31	∈	∈	PROPN
ejpam-5103	182	32	ϑ.	ϑ.	NOUN
ejpam-5103	182	33	proof	proof	NOUN
ejpam-5103	182	34	.	.	PUNCT
ejpam-5103	183	1	let	let	VERB
ejpam-5103	183	2	f	f	PRON
ejpam-5103	183	3	be	be	AUX
ejpam-5103	183	4	a	a	DET
ejpam-5103	183	5	sharply	sharply	ADV
ejpam-5103	183	6	geodesic	geodesic	ADJ
ejpam-5103	183	7	log	log	NOUN
ejpam-5103	183	8	-	-	PUNCT
ejpam-5103	183	9	pseudo	pseudo	NOUN
ejpam-5103	183	10	preinvex	preinvex	NOUN
ejpam-5103	183	11	function	function	NOUN
ejpam-5103	183	12	on	on	ADP
ejpam-5103	183	13	ϑ	ϑ	SYM
ejpam-5103	183	14	,	,	PUNCT
ejpam-5103	183	15	then	then	ADV
ejpam-5103	183	16	f(µ2	f(µ2	NOUN
ejpam-5103	183	17	)	)	PUNCT
ejpam-5103	183	18	≥	≥	NOUN
ejpam-5103	183	19	log	log	VERB
ejpam-5103	183	20	f(γµ1,µ2(ς	f(γµ1,µ2(ς	NOUN
ejpam-5103	183	21	)	)	PUNCT
ejpam-5103	183	22	)	)	PUNCT
ejpam-5103	183	23	,	,	PUNCT
ejpam-5103	184	1	∀µ1	∀µ1	PROPN
ejpam-5103	184	2	,	,	PUNCT
ejpam-5103	184	3	µ2	µ2	PROPN
ejpam-5103	184	4	∈	∈	PROPN
ejpam-5103	184	5	ϑ	ϑ	NOUN
ejpam-5103	184	6	,	,	PUNCT
ejpam-5103	184	7	ς	ς	PROPN
ejpam-5103	184	8	∈	∈	PROPN
ejpam-5103	184	9	[	[	X
ejpam-5103	184	10	0	0	NUM
ejpam-5103	184	11	,	,	PUNCT
ejpam-5103	184	12	1	1	NUM
ejpam-5103	184	13	]	]	PUNCT
ejpam-5103	184	14	.	.	PUNCT
ejpam-5103	185	1	hence	hence	ADV
ejpam-5103	185	2	,	,	PUNCT
ejpam-5103	185	3	by	by	ADP
ejpam-5103	185	4	taking	take	VERB
ejpam-5103	185	5	ς	ς	PROPN
ejpam-5103	185	6	−→	−→	NOUN
ejpam-5103	185	7	0	0	NUM
ejpam-5103	185	8	,	,	PUNCT
ejpam-5103	185	9	we	we	PRON
ejpam-5103	185	10	have	have	VERB
ejpam-5103	185	11	the	the	DET
ejpam-5103	185	12	result	result	NOUN
ejpam-5103	185	13	.	.	PUNCT
ejpam-5103	186	1	w.	w.	PROPN
ejpam-5103	186	2	saleh	saleh	PROPN
ejpam-5103	186	3	,	,	PUNCT
ejpam-5103	186	4	a.	a.	NOUN
ejpam-5103	186	5	lakhdari	lakhdari	PROPN
ejpam-5103	186	6	,	,	PUNCT
ejpam-5103	186	7	b.	b.	PROPN
ejpam-5103	186	8	meftah	meftah	PROPN
ejpam-5103	186	9	/	/	SYM
ejpam-5103	186	10	eur	eur	PROPN
ejpam-5103	186	11	.	.	PUNCT
ejpam-5103	187	1	j.	j.	PROPN
ejpam-5103	187	2	pure	pure	PROPN
ejpam-5103	187	3	appl	appl	PROPN
ejpam-5103	187	4	.	.	PROPN
ejpam-5103	187	5	math	math	PROPN
ejpam-5103	187	6	,	,	PUNCT
ejpam-5103	187	7	17	17	NUM
ejpam-5103	187	8	(	(	PUNCT
ejpam-5103	187	9	2	2	NUM
ejpam-5103	187	10	)	)	PUNCT
ejpam-5103	187	11	(	(	PUNCT
ejpam-5103	187	12	2024	2024	NUM
ejpam-5103	187	13	)	)	PUNCT
ejpam-5103	187	14	,	,	PUNCT
ejpam-5103	187	15	860	860	NUM
ejpam-5103	187	16	-	-	SYM
ejpam-5103	187	17	869	869	NUM
ejpam-5103	187	18	867	867	NUM
ejpam-5103	187	19	definition	definition	NOUN
ejpam-5103	187	20	9	9	NUM
ejpam-5103	187	21	.	.	PUNCT
ejpam-5103	188	1	a	a	DET
ejpam-5103	188	2	function	function	NOUN
ejpam-5103	188	3	ξ	ξ	PROPN
ejpam-5103	188	4	is	be	AUX
ejpam-5103	188	5	considered	consider	VERB
ejpam-5103	188	6	geodesic	geodesic	ADJ
ejpam-5103	188	7	log	log	NOUN
ejpam-5103	188	8	-	-	PUNCT
ejpam-5103	188	9	pseudo	pseudo	NOUN
ejpam-5103	188	10	preinvex	preinvex	NOUN
ejpam-5103	188	11	w.r.t	w.r.t	NOUN
ejpam-5103	188	12	.	.	PUNCT
ejpam-5103	189	1	a	a	DET
ejpam-5103	189	2	strictly	strictly	ADV
ejpam-5103	189	3	positive	positive	ADJ
ejpam-5103	189	4	bifunction	bifunction	NOUN
ejpam-5103	189	5	β	β	X
ejpam-5103	189	6	,	,	PUNCT
ejpam-5103	189	7	such	such	ADJ
ejpam-5103	189	8	that	that	DET
ejpam-5103	189	9	log	log	NOUN
ejpam-5103	189	10	ξ(µ2	ξ(µ2	VERB
ejpam-5103	189	11	)	)	PUNCT
ejpam-5103	189	12	<	<	X
ejpam-5103	189	13	log	log	NOUN
ejpam-5103	189	14	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	189	15	)	)	PUNCT
ejpam-5103	190	1	=	=	NOUN
ejpam-5103	190	2	⇒	⇒	NOUN
ejpam-5103	190	3	log	log	VERB
ejpam-5103	190	4	ξ(γµ1,µ2	ξ(γµ1,µ2	NUM
ejpam-5103	190	5	)	)	PUNCT
ejpam-5103	190	6	<	<	X
ejpam-5103	190	7	log	log	NOUN
ejpam-5103	190	8	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	190	9	)	)	PUNCT
ejpam-5103	191	1	+	+	PUNCT
ejpam-5103	191	2	ς(ς	ς(ς	PROPN
ejpam-5103	191	3	−	−	PROPN
ejpam-5103	191	4	1)β(µ2	1)β(µ2	NUM
ejpam-5103	191	5	,	,	PUNCT
ejpam-5103	191	6	µ1	µ1	PROPN
ejpam-5103	191	7	)	)	PUNCT
ejpam-5103	191	8	,	,	PUNCT
ejpam-5103	191	9	∀µ1	∀µ1	PROPN
ejpam-5103	191	10	,	,	PUNCT
ejpam-5103	191	11	µ2	µ2	PROPN
ejpam-5103	191	12	∈	∈	PROPN
ejpam-5103	191	13	ϑ	ϑ	X
ejpam-5103	191	14	and	and	CCONJ
ejpam-5103	191	15	ς	ς	PROPN
ejpam-5103	191	16	∈	∈	PROPN
ejpam-5103	192	1	[	[	X
ejpam-5103	192	2	0	0	NUM
ejpam-5103	192	3	,	,	PUNCT
ejpam-5103	192	4	1	1	NUM
ejpam-5103	192	5	]	]	PUNCT
ejpam-5103	192	6	.	.	PUNCT
ejpam-5103	193	1	theorem	theorem	VERB
ejpam-5103	193	2	6	6	NUM
ejpam-5103	193	3	.	.	PUNCT
ejpam-5103	194	1	if	if	SCONJ
ejpam-5103	194	2	ξ	ξ	PROPN
ejpam-5103	194	3	is	be	AUX
ejpam-5103	194	4	a	a	DET
ejpam-5103	194	5	strongly	strongly	ADV
ejpam-5103	194	6	geodesic	geodesic	ADJ
ejpam-5103	194	7	log	log	NOUN
ejpam-5103	194	8	-	-	PUNCT
ejpam-5103	194	9	preinvex	preinvex	NOUN
ejpam-5103	194	10	function	function	NOUN
ejpam-5103	194	11	such	such	DET
ejpam-5103	194	12	that	that	DET
ejpam-5103	194	13	log	log	NOUN
ejpam-5103	194	14	ξ(µ2	ξ(µ2	VERB
ejpam-5103	194	15	)	)	PUNCT
ejpam-5103	194	16	<	<	X
ejpam-5103	194	17	log	log	PROPN
ejpam-5103	194	18	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	194	19	)	)	PUNCT
ejpam-5103	194	20	,	,	PUNCT
ejpam-5103	194	21	then	then	ADV
ejpam-5103	194	22	ξ	ξ	PROPN
ejpam-5103	194	23	is	be	AUX
ejpam-5103	194	24	strongly	strongly	ADV
ejpam-5103	194	25	geodesic	geodesic	ADJ
ejpam-5103	194	26	log	log	NOUN
ejpam-5103	194	27	-	-	PUNCT
ejpam-5103	194	28	pseudo	pseudo	NOUN
ejpam-5103	194	29	preinex	preinex	NOUN
ejpam-5103	194	30	.	.	PUNCT
ejpam-5103	195	1	proof	proof	NOUN
ejpam-5103	195	2	.	.	PUNCT
ejpam-5103	196	1	since	since	SCONJ
ejpam-5103	196	2	log	log	PROPN
ejpam-5103	196	3	ξ(µ2	ξ(µ2	NUM
ejpam-5103	196	4	)	)	PUNCT
ejpam-5103	196	5	<	<	X
ejpam-5103	196	6	log	log	NOUN
ejpam-5103	196	7	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	196	8	)	)	PUNCT
ejpam-5103	196	9	and	and	CCONJ
ejpam-5103	196	10	ξ	ξ	PROPN
ejpam-5103	196	11	is	be	AUX
ejpam-5103	196	12	a	a	DET
ejpam-5103	196	13	strongly	strongly	ADV
ejpam-5103	196	14	geodesic	geodesic	ADJ
ejpam-5103	196	15	log	log	NOUN
ejpam-5103	196	16	-	-	PUNCT
ejpam-5103	196	17	preinvex	preinvex	NOUN
ejpam-5103	196	18	function	function	NOUN
ejpam-5103	196	19	,	,	PUNCT
ejpam-5103	196	20	then	then	ADV
ejpam-5103	196	21	∀µ1	∀µ1	X
ejpam-5103	196	22	,	,	PUNCT
ejpam-5103	196	23	µ2	µ2	PROPN
ejpam-5103	196	24	∈	∈	PROPN
ejpam-5103	196	25	ϑ	ϑ	NOUN
ejpam-5103	196	26	,	,	PUNCT
ejpam-5103	196	27	ς	ς	PROPN
ejpam-5103	196	28	∈	∈	PROPN
ejpam-5103	197	1	[	[	X
ejpam-5103	197	2	0	0	NUM
ejpam-5103	197	3	,	,	PUNCT
ejpam-5103	197	4	1	1	NUM
ejpam-5103	197	5	]	]	PUNCT
ejpam-5103	197	6	,	,	PUNCT
ejpam-5103	197	7	we	we	PRON
ejpam-5103	197	8	have	have	AUX
ejpam-5103	197	9	log	log	VERB
ejpam-5103	197	10	ξ(γµ1,µ2(ς	ξ(γµ1,µ2(ς	NOUN
ejpam-5103	197	11	)	)	PUNCT
ejpam-5103	197	12	)	)	PUNCT
ejpam-5103	197	13	≤	≤	NUM
ejpam-5103	197	14	log	log	NOUN
ejpam-5103	197	15	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	197	16	)	)	PUNCT
ejpam-5103	198	1	+	+	CCONJ
ejpam-5103	198	2	ς	ς	PROPN
ejpam-5103	198	3	(	(	PUNCT
ejpam-5103	198	4	log	log	NOUN
ejpam-5103	198	5	ξ(µ2)−	ξ(µ2)−	NOUN
ejpam-5103	198	6	log	log	VERB
ejpam-5103	198	7	ξ(µ1))−	ξ(µ1))−	NOUN
ejpam-5103	198	8	ες(1−	ες(1−	NUM
ejpam-5103	198	9	ς)∥η(µ2	ς)∥η(µ2	NOUN
ejpam-5103	198	10	,	,	PUNCT
ejpam-5103	198	11	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	198	12	<	<	X
ejpam-5103	198	13	log	log	NOUN
ejpam-5103	198	14	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	198	15	)	)	PUNCT
ejpam-5103	199	1	+	+	CCONJ
ejpam-5103	199	2	ς(1−	ς(1−	PROPN
ejpam-5103	199	3	ς	ς	PROPN
ejpam-5103	199	4	)	)	PUNCT
ejpam-5103	199	5	(	(	PUNCT
ejpam-5103	199	6	log	log	NOUN
ejpam-5103	199	7	ξ(µ2)−	ξ(µ2)−	NOUN
ejpam-5103	199	8	log	log	VERB
ejpam-5103	199	9	ξ(µ1))−	ξ(µ1))−	NOUN
ejpam-5103	199	10	ες(1−	ες(1−	NUM
ejpam-5103	199	11	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	199	12	,	,	PUNCT
ejpam-5103	199	13	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	199	14	=	=	PUNCT
ejpam-5103	199	15	log	log	NOUN
ejpam-5103	199	16	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	199	17	)	)	PUNCT
ejpam-5103	200	1	+	+	CCONJ
ejpam-5103	200	2	ς(ς	ς(ς	NUM
ejpam-5103	200	3	−	−	NUM
ejpam-5103	200	4	1	1	NUM
ejpam-5103	200	5	)	)	PUNCT
ejpam-5103	200	6	(	(	PUNCT
ejpam-5103	200	7	log	log	NOUN
ejpam-5103	200	8	ξ(µ1)−	ξ(µ1)−	VERB
ejpam-5103	200	9	log	log	NOUN
ejpam-5103	200	10	ξ(µ2))−	ξ(µ2))−	NOUN
ejpam-5103	200	11	ες(1−	ες(1−	NUM
ejpam-5103	200	12	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	200	13	,	,	PUNCT
ejpam-5103	200	14	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	200	15	<	<	X
ejpam-5103	200	16	log	log	NOUN
ejpam-5103	200	17	ξ(µ1	ξ(µ1	NOUN
ejpam-5103	200	18	)	)	PUNCT
ejpam-5103	201	1	+	+	CCONJ
ejpam-5103	201	2	ς(1−	ς(1−	ADJ
ejpam-5103	201	3	ς)β(µ2	ς)β(µ2	NOUN
ejpam-5103	201	4	,	,	PUNCT
ejpam-5103	201	5	µ1)−	µ1)−	ADV
ejpam-5103	201	6	ες(1−	ες(1−	NOUN
ejpam-5103	201	7	ς)∥η(µ2	ς)∥η(µ2	NUM
ejpam-5103	201	8	,	,	PUNCT
ejpam-5103	201	9	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	201	10	,	,	PUNCT
ejpam-5103	201	11	where	where	SCONJ
ejpam-5103	201	12	β(µ2	β(µ2	ADV
ejpam-5103	201	13	,	,	PUNCT
ejpam-5103	201	14	µ1	µ1	PROPN
ejpam-5103	201	15	)	)	PUNCT
ejpam-5103	201	16	=	=	PUNCT
ejpam-5103	201	17	log	log	NOUN
ejpam-5103	201	18	ξ(µ1)−	ξ(µ1)−	ADJ
ejpam-5103	201	19	log	log	NOUN
ejpam-5103	201	20	ξ(v	ξ(v	NOUN
ejpam-5103	201	21	)	)	PUNCT
ejpam-5103	201	22	>	>	X
ejpam-5103	202	1	0	0	X
ejpam-5103	202	2	.	.	PUNCT
ejpam-5103	203	1	this	this	PRON
ejpam-5103	203	2	demonstrates	demonstrate	VERB
ejpam-5103	203	3	that	that	SCONJ
ejpam-5103	203	4	ξ	ξ	PROPN
ejpam-5103	203	5	is	be	AUX
ejpam-5103	203	6	a	a	DET
ejpam-5103	203	7	strongly	strongly	ADV
ejpam-5103	203	8	geodesic	geodesic	ADJ
ejpam-5103	203	9	log	log	NOUN
ejpam-5103	203	10	-	-	PUNCT
ejpam-5103	203	11	preinvex	preinvex	NOUN
ejpam-5103	203	12	function	function	NOUN
ejpam-5103	203	13	.	.	PUNCT
ejpam-5103	204	1	now	now	ADV
ejpam-5103	204	2	,	,	PUNCT
ejpam-5103	204	3	we	we	PRON
ejpam-5103	204	4	demonstrate	demonstrate	VERB
ejpam-5103	204	5	that	that	SCONJ
ejpam-5103	204	6	the	the	DET
ejpam-5103	204	7	subtraction	subtraction	NOUN
ejpam-5103	204	8	of	of	ADP
ejpam-5103	204	9	a	a	DET
ejpam-5103	204	10	strongly	strongly	ADV
ejpam-5103	204	11	geodesic	geodesic	ADJ
ejpam-5103	204	12	log	log	NOUN
ejpam-5103	204	13	-	-	PUNCT
ejpam-5103	204	14	preinvex	preinvex	NOUN
ejpam-5103	204	15	function	function	NOUN
ejpam-5103	204	16	and	and	CCONJ
ejpam-5103	204	17	an	an	DET
ejpam-5103	204	18	affine	affine	NOUN
ejpam-5103	204	19	geodesic	geodesic	NOUN
ejpam-5103	204	20	strongly	strongly	ADV
ejpam-5103	204	21	log	log	NOUN
ejpam-5103	204	22	-	-	PUNCT
ejpam-5103	204	23	preinvex	preinvex	NOUN
ejpam-5103	204	24	function	function	NOUN
ejpam-5103	204	25	results	result	NOUN
ejpam-5103	204	26	in	in	ADP
ejpam-5103	204	27	another	another	DET
ejpam-5103	204	28	geodesic	geodesic	ADJ
ejpam-5103	204	29	logpreinvex	logpreinvex	NOUN
ejpam-5103	204	30	function	function	NOUN
ejpam-5103	204	31	.	.	PUNCT
ejpam-5103	205	1	theorem	theorem	VERB
ejpam-5103	205	2	7	7	NUM
ejpam-5103	205	3	.	.	PUNCT
ejpam-5103	206	1	let	let	VERB
ejpam-5103	206	2	f	f	PRON
ejpam-5103	206	3	be	be	AUX
ejpam-5103	206	4	affine	affine	VERB
ejpam-5103	206	5	strongly	strongly	ADV
ejpam-5103	206	6	geodesic	geodesic	ADJ
ejpam-5103	206	7	log	log	NOUN
ejpam-5103	206	8	-	-	PUNCT
ejpam-5103	206	9	preinvex	preinvex	NOUN
ejpam-5103	206	10	function	function	NOUN
ejpam-5103	206	11	.	.	PUNCT
ejpam-5103	207	1	if	if	SCONJ
ejpam-5103	207	2	f	f	PROPN
ejpam-5103	207	3	is	be	AUX
ejpam-5103	207	4	a	a	DET
ejpam-5103	207	5	strongly	strongly	ADV
ejpam-5103	207	6	geodesic	geodesic	ADJ
ejpam-5103	207	7	log	log	NOUN
ejpam-5103	207	8	-	-	PUNCT
ejpam-5103	207	9	preinvex	preinvex	NOUN
ejpam-5103	207	10	,	,	PUNCT
ejpam-5103	207	11	then	then	ADV
ejpam-5103	207	12	f	f	PROPN
ejpam-5103	208	1	−	−	PROPN
ejpam-5103	208	2	f	f	PROPN
ejpam-5103	208	3	is	be	AUX
ejpam-5103	208	4	a	a	DET
ejpam-5103	208	5	geodesic	geodesic	ADJ
ejpam-5103	208	6	log	log	NOUN
ejpam-5103	208	7	-	-	PUNCT
ejpam-5103	208	8	preinvex	preinvex	NOUN
ejpam-5103	208	9	function	function	NOUN
ejpam-5103	208	10	.	.	PUNCT
ejpam-5103	209	1	proof	proof	NOUN
ejpam-5103	209	2	.	.	PUNCT
ejpam-5103	210	1	assume	assume	VERB
ejpam-5103	210	2	that	that	SCONJ
ejpam-5103	210	3	f	f	PROPN
ejpam-5103	210	4	is	be	AUX
ejpam-5103	210	5	an	an	DET
ejpam-5103	210	6	affine	affine	NOUN
ejpam-5103	210	7	strongly	strongly	ADV
ejpam-5103	210	8	geodesic	geodesic	ADJ
ejpam-5103	210	9	log	log	NOUN
ejpam-5103	210	10	-	-	PUNCT
ejpam-5103	210	11	preinvex	preinvex	NOUN
ejpam-5103	210	12	function.then	function.then	ADV
ejpam-5103	210	13	log	log	NOUN
ejpam-5103	210	14	f(γµ1,µ2(ς	f(γµ1,µ2(ς	NOUN
ejpam-5103	210	15	)	)	PUNCT
ejpam-5103	210	16	)	)	PUNCT
ejpam-5103	211	1	=	=	SYM
ejpam-5103	211	2	(	(	PUNCT
ejpam-5103	211	3	1−ς	1−ς	NUM
ejpam-5103	211	4	)	)	PUNCT
ejpam-5103	211	5	log	log	NOUN
ejpam-5103	211	6	f(µ1)+ς	f(µ1)+ς	NUM
ejpam-5103	211	7	log	log	VERB
ejpam-5103	211	8	f(µ2)−ες(1−ς)∥η(µ2	f(µ2)−ες(1−ς)∥η(µ2	NOUN
ejpam-5103	211	9	,	,	PUNCT
ejpam-5103	211	10	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	211	11	,	,	PUNCT
ejpam-5103	211	12	∀µ1	∀µ1	PROPN
ejpam-5103	211	13	,	,	PUNCT
ejpam-5103	211	14	µ2	µ2	PROPN
ejpam-5103	211	15	∈	∈	PROPN
ejpam-5103	211	16	ϑ	ϑ	NOUN
ejpam-5103	211	17	,	,	PUNCT
ejpam-5103	211	18	ς	ς	PROPN
ejpam-5103	211	19	∈	∈	PROPN
ejpam-5103	212	1	[	[	X
ejpam-5103	212	2	0	0	NUM
ejpam-5103	212	3	,	,	PUNCT
ejpam-5103	212	4	1	1	NUM
ejpam-5103	212	5	]	]	PUNCT
ejpam-5103	212	6	.	.	PUNCT
ejpam-5103	213	1	(	(	PUNCT
ejpam-5103	213	2	12	12	NUM
ejpam-5103	213	3	)	)	PUNCT
ejpam-5103	213	4	from	from	ADP
ejpam-5103	213	5	the	the	DET
ejpam-5103	213	6	strongly	strongly	ADV
ejpam-5103	213	7	geodesic	geodesic	ADJ
ejpam-5103	213	8	log	log	NOUN
ejpam-5103	213	9	-	-	PUNCT
ejpam-5103	213	10	preivexity	preivexity	NOUN
ejpam-5103	213	11	of	of	ADP
ejpam-5103	213	12	f	f	PROPN
ejpam-5103	213	13	,	,	PUNCT
ejpam-5103	213	14	we	we	PRON
ejpam-5103	213	15	have	have	VERB
ejpam-5103	213	16	logf(γµ1,µ2(ς	logf(γµ1,µ2(ς	NOUN
ejpam-5103	213	17	)	)	PUNCT
ejpam-5103	213	18	)	)	PUNCT
ejpam-5103	214	1	=	=	SYM
ejpam-5103	214	2	(	(	PUNCT
ejpam-5103	214	3	1−ς	1−ς	NUM
ejpam-5103	214	4	)	)	PUNCT
ejpam-5103	214	5	logf(µ1)+ς	logf(µ1)+ς	PROPN
ejpam-5103	214	6	logf(µ2)−ες(1−ς)∥η(µ2	logf(µ2)−ες(1−ς)∥η(µ2	PROPN
ejpam-5103	214	7	,	,	PUNCT
ejpam-5103	214	8	µ1)(ς)∥2,∀µ1	µ1)(ς)∥2,∀µ1	NOUN
ejpam-5103	214	9	,	,	PUNCT
ejpam-5103	214	10	µ2	µ2	PROPN
ejpam-5103	214	11	∈	∈	PROPN
ejpam-5103	214	12	ϑ	ϑ	NOUN
ejpam-5103	214	13	,	,	PUNCT
ejpam-5103	214	14	ς	ς	PROPN
ejpam-5103	214	15	∈	∈	PROPN
ejpam-5103	215	1	[	[	X
ejpam-5103	215	2	0	0	NUM
ejpam-5103	215	3	,	,	PUNCT
ejpam-5103	215	4	1	1	NUM
ejpam-5103	215	5	]	]	PUNCT
ejpam-5103	215	6	.	.	PUNCT
ejpam-5103	216	1	(	(	PUNCT
ejpam-5103	216	2	13	13	NUM
ejpam-5103	216	3	)	)	PUNCT
ejpam-5103	216	4	from	from	ADP
ejpam-5103	216	5	(	(	PUNCT
ejpam-5103	216	6	1	1	NUM
ejpam-5103	216	7	)	)	PUNCT
ejpam-5103	216	8	and	and	CCONJ
ejpam-5103	216	9	(	(	PUNCT
ejpam-5103	216	10	12	12	NUM
ejpam-5103	216	11	)	)	PUNCT
ejpam-5103	216	12	,	,	PUNCT
ejpam-5103	216	13	we	we	PRON
ejpam-5103	216	14	have	have	VERB
ejpam-5103	216	15	logf(γµ1,µ2(ς))−	logf(γµ1,µ2(ς))−	NOUN
ejpam-5103	216	16	log	log	VERB
ejpam-5103	216	17	f(γµ1,µ2(ς	f(γµ1,µ2(ς	NOUN
ejpam-5103	216	18	)	)	PUNCT
ejpam-5103	216	19	)	)	PUNCT
ejpam-5103	216	20	≤	≤	NOUN
ejpam-5103	216	21	(	(	PUNCT
ejpam-5103	216	22	1−	1−	NUM
ejpam-5103	216	23	ς)(logf(µ1)−	ς)(logf(µ1)−	PROPN
ejpam-5103	216	24	log(µ1	log(µ1	PROPN
ejpam-5103	216	25	)	)	PUNCT
ejpam-5103	216	26	)	)	PUNCT
ejpam-5103	217	1	+	+	CCONJ
ejpam-5103	217	2	ς(logf(µ2)−	ς(logf(µ2)−	VERB
ejpam-5103	217	3	log	log	NOUN
ejpam-5103	217	4	f(µ2	f(µ2	NOUN
ejpam-5103	217	5	)	)	PUNCT
ejpam-5103	217	6	)	)	PUNCT
ejpam-5103	217	7	.	.	PUNCT
ejpam-5103	218	1	(	(	PUNCT
ejpam-5103	218	2	14	14	NUM
ejpam-5103	218	3	)	)	PUNCT
ejpam-5103	218	4	logf(γµ1,µ2(ς))−log	logf(γµ1,µ2(ς))−log	NOUN
ejpam-5103	218	5	f(γµ1,µ2(ς	f(γµ1,µ2(ς	PROPN
ejpam-5103	218	6	)	)	PUNCT
ejpam-5103	218	7	)	)	PUNCT
ejpam-5103	219	1	≤	≤	NOUN
ejpam-5103	219	2	(	(	PUNCT
ejpam-5103	219	3	1−ς)(logf(µ1)−log	1−ς)(logf(µ1)−log	NUM
ejpam-5103	219	4	f(µ1))+ς(logf(µ2)−log	f(µ1))+ς(logf(µ2)−log	PROPN
ejpam-5103	219	5	f(µ2	f(µ2	PROPN
ejpam-5103	219	6	)	)	PUNCT
ejpam-5103	219	7	)	)	PUNCT
ejpam-5103	219	8	.	.	PUNCT
ejpam-5103	220	1	this	this	PRON
ejpam-5103	220	2	demonstrates	demonstrate	VERB
ejpam-5103	220	3	that	that	SCONJ
ejpam-5103	220	4	f	f	PROPN
ejpam-5103	221	1	−	−	PROPN
ejpam-5103	221	2	f	f	PROPN
ejpam-5103	221	3	is	be	AUX
ejpam-5103	221	4	a	a	DET
ejpam-5103	221	5	geodesic	geodesic	ADJ
ejpam-5103	221	6	log	log	NOUN
ejpam-5103	221	7	-	-	PUNCT
ejpam-5103	221	8	preinvex	preinvex	NOUN
ejpam-5103	221	9	function	function	NOUN
ejpam-5103	221	10	.	.	PUNCT
ejpam-5103	222	1	references	reference	NOUN
ejpam-5103	222	2	868	868	NUM
ejpam-5103	222	3	remark	remark	NOUN
ejpam-5103	222	4	2	2	NUM
ejpam-5103	222	5	.	.	PUNCT
ejpam-5103	223	1	we	we	PRON
ejpam-5103	223	2	observe	observe	VERB
ejpam-5103	223	3	that	that	SCONJ
ejpam-5103	223	4	if	if	SCONJ
ejpam-5103	223	5	a	a	DET
ejpam-5103	223	6	strictly	strictly	ADV
ejpam-5103	223	7	positive	positive	ADJ
ejpam-5103	223	8	function	function	NOUN
ejpam-5103	223	9	f	f	NOUN
ejpam-5103	223	10	is	be	AUX
ejpam-5103	223	11	strongly	strongly	ADV
ejpam-5103	223	12	geodesic	geodesic	ADJ
ejpam-5103	223	13	logpreinvex	logpreinvex	NOUN
ejpam-5103	223	14	,	,	PUNCT
ejpam-5103	223	15	then	then	ADV
ejpam-5103	223	16	the	the	DET
ejpam-5103	223	17	following	follow	VERB
ejpam-5103	223	18	inequality	inequality	NOUN
ejpam-5103	223	19	holds	hold	VERB
ejpam-5103	223	20	for	for	ADP
ejpam-5103	223	21	all	all	DET
ejpam-5103	223	22	µ1	µ1	PROPN
ejpam-5103	223	23	,	,	PUNCT
ejpam-5103	223	24	µ2	µ2	PROPN
ejpam-5103	223	25	∈	∈	PROPN
ejpam-5103	223	26	ϑ	ϑ	X
ejpam-5103	223	27	and	and	CCONJ
ejpam-5103	223	28	ς	ς	PROPN
ejpam-5103	223	29	∈	∈	PROPN
ejpam-5103	224	1	[	[	X
ejpam-5103	224	2	0	0	NUM
ejpam-5103	224	3	,	,	PUNCT
ejpam-5103	224	4	1	1	NUM
ejpam-5103	224	5	]	]	PUNCT
ejpam-5103	224	6	,	,	PUNCT
ejpam-5103	224	7	known	know	VERB
ejpam-5103	224	8	as	as	ADP
ejpam-5103	224	9	the	the	DET
ejpam-5103	224	10	wright	wright	PROPN
ejpam-5103	224	11	strongly	strongly	ADV
ejpam-5103	224	12	geodesic	geodesic	ADJ
ejpam-5103	224	13	log	log	NOUN
ejpam-5103	224	14	-	-	PUNCT
ejpam-5103	224	15	preinvex	preinvex	NOUN
ejpam-5103	224	16	function	function	NOUN
ejpam-5103	224	17	:	:	PUNCT
ejpam-5103	224	18	logf(γµ1,µ2(ς))+logf(γµ2,µ1(ς	logf(γµ1,µ2(ς))+logf(γµ2,µ1(ς	PROPN
ejpam-5103	224	19	)	)	PUNCT
ejpam-5103	224	20	)	)	PUNCT
ejpam-5103	224	21	≤	≤	NOUN
ejpam-5103	224	22	logf(µ1)+logf(µ2)−2ς(1−ς)∥η(µ2	logf(µ1)+logf(µ2)−2ς(1−ς)∥η(µ2	PROPN
ejpam-5103	224	23	,	,	PUNCT
ejpam-5103	224	24	µ1)(ς)∥2	µ1)(ς)∥2	PROPN
ejpam-5103	224	25	.	.	PUNCT
ejpam-5103	224	26	(	(	PUNCT
ejpam-5103	224	27	15	15	NUM
ejpam-5103	224	28	)	)	PUNCT
ejpam-5103	224	29	from	from	ADP
ejpam-5103	224	30	(	(	PUNCT
ejpam-5103	224	31	15	15	NUM
ejpam-5103	224	32	)	)	PUNCT
ejpam-5103	224	33	,	,	PUNCT
ejpam-5103	224	34	we	we	PRON
ejpam-5103	224	35	can	can	AUX
ejpam-5103	224	36	deduce	deduce	VERB
ejpam-5103	224	37	the	the	DET
ejpam-5103	224	38	following	following	NOUN
ejpam-5103	224	39	:	:	PUNCT
ejpam-5103	224	40	f(γµ1,µ2(ς))f(γµ2,µ1(ς	f(γµ1,µ2(ς))f(γµ2,µ1(ς	ADJ
ejpam-5103	224	41	)	)	PUNCT
ejpam-5103	224	42	)	)	PUNCT
ejpam-5103	225	1	=	=	SYM
ejpam-5103	225	2	logf(γµ1,µ2(ς	logf(γµ1,µ2(ς	NOUN
ejpam-5103	225	3	)	)	PUNCT
ejpam-5103	225	4	)	)	PUNCT
ejpam-5103	226	1	+	+	CCONJ
ejpam-5103	226	2	logf(γµ2,µ1(ς	logf(γµ2,µ1(ς	NOUN
ejpam-5103	226	3	)	)	PUNCT
ejpam-5103	226	4	)	)	PUNCT
ejpam-5103	227	1	≤	≤	NUM
ejpam-5103	227	2	logf(µ1	logf(µ1	NOUN
ejpam-5103	227	3	)	)	PUNCT
ejpam-5103	228	1	+	+	CCONJ
ejpam-5103	228	2	logf(µ2	logf(µ2	X
ejpam-5103	228	3	)	)	PUNCT
ejpam-5103	228	4	=	=	SYM
ejpam-5103	228	5	logf(µ1)f(µ2	logf(µ1)f(µ2	PROPN
ejpam-5103	228	6	)	)	PUNCT
ejpam-5103	228	7	,	,	PUNCT
ejpam-5103	228	8	∀µ1	∀µ1	PROPN
ejpam-5103	228	9	,	,	PUNCT
ejpam-5103	228	10	µ2	µ2	PROPN
ejpam-5103	228	11	∈	∈	PROPN
ejpam-5103	228	12	ϑ	ϑ	NOUN
ejpam-5103	228	13	,	,	PUNCT
ejpam-5103	228	14	ς	ς	PROPN
ejpam-5103	228	15	∈	∈	PROPN
ejpam-5103	229	1	[	[	X
ejpam-5103	229	2	0	0	NUM
ejpam-5103	229	3	,	,	PUNCT
ejpam-5103	229	4	1	1	NUM
ejpam-5103	229	5	]	]	PUNCT
ejpam-5103	229	6	.	.	PUNCT
ejpam-5103	230	1	this	this	PRON
ejpam-5103	230	2	implies	imply	VERB
ejpam-5103	230	3	that	that	PRON
ejpam-5103	230	4	f(γµ1,µ2(ς))f(γµ2,µ1(ς	f(γµ1,µ2(ς))f(γµ2,µ1(ς	PROPN
ejpam-5103	230	5	)	)	PUNCT
ejpam-5103	230	6	)	)	PUNCT
ejpam-5103	230	7	≤	≤	NOUN
ejpam-5103	230	8	f(µ1)f(µ2	f(µ1)f(µ2	NOUN
ejpam-5103	230	9	)	)	PUNCT
ejpam-5103	230	10	,	,	PUNCT
ejpam-5103	230	11	∀µ1	∀µ1	PROPN
ejpam-5103	230	12	,	,	PUNCT
ejpam-5103	230	13	µ2	µ2	PROPN
ejpam-5103	230	14	∈	∈	PROPN
ejpam-5103	230	15	ϑ	ϑ	NOUN
ejpam-5103	230	16	,	,	PUNCT
ejpam-5103	230	17	ς	ς	PROPN
ejpam-5103	230	18	∈	∈	PROPN
ejpam-5103	231	1	[	[	X
ejpam-5103	231	2	0	0	NUM
ejpam-5103	231	3	,	,	PUNCT
ejpam-5103	231	4	1	1	NUM
ejpam-5103	231	5	]	]	PUNCT
ejpam-5103	231	6	,	,	PUNCT
ejpam-5103	231	7	which	which	PRON
ejpam-5103	231	8	demonstrates	demonstrate	VERB
ejpam-5103	231	9	that	that	SCONJ
ejpam-5103	231	10	every	every	DET
ejpam-5103	231	11	strictly	strictly	ADV
ejpam-5103	231	12	positive	positive	ADJ
ejpam-5103	231	13	function	function	NOUN
ejpam-5103	231	14	f	f	PROPN
ejpam-5103	231	15	is	be	AUX
ejpam-5103	231	16	multiplicative	multiplicative	ADJ
ejpam-5103	231	17	wright	wright	PROPN
ejpam-5103	231	18	strongly	strongly	ADV
ejpam-5103	231	19	geodesic	geodesic	ADJ
ejpam-5103	231	20	log	log	NOUN
ejpam-5103	231	21	-	-	PUNCT
ejpam-5103	231	22	preinvex	preinvex	NOUN
ejpam-5103	231	23	.	.	PUNCT
ejpam-5103	232	1	funding	funding	NOUN
ejpam-5103	232	2	:	:	PUNCT
ejpam-5103	232	3	the	the	DET
ejpam-5103	232	4	authors	author	NOUN
ejpam-5103	232	5	confirm	confirm	VERB
ejpam-5103	232	6	that	that	SCONJ
ejpam-5103	232	7	they	they	PRON
ejpam-5103	232	8	did	do	AUX
ejpam-5103	232	9	not	not	PART
ejpam-5103	232	10	receive	receive	VERB
ejpam-5103	232	11	any	any	DET
ejpam-5103	232	12	financial	financial	ADJ
ejpam-5103	232	13	support	support	NOUN
ejpam-5103	232	14	for	for	ADP
ejpam-5103	232	15	this	this	DET
ejpam-5103	232	16	article	article	NOUN
ejpam-5103	232	17	.	.	PUNCT
ejpam-5103	233	1	conflict	conflict	NOUN
ejpam-5103	233	2	of	of	ADP
ejpam-5103	233	3	interest	interest	NOUN
ejpam-5103	233	4	:	:	PUNCT
ejpam-5103	233	5	the	the	DET
ejpam-5103	233	6	authors	author	NOUN
ejpam-5103	233	7	affirm	affirm	VERB
ejpam-5103	233	8	the	the	DET
ejpam-5103	233	9	absence	absence	NOUN
ejpam-5103	233	10	of	of	ADP
ejpam-5103	233	11	any	any	DET
ejpam-5103	233	12	conflicts	conflict	NOUN
ejpam-5103	233	13	of	of	ADP
ejpam-5103	233	14	interest	interest	NOUN
ejpam-5103	233	15	.	.	PUNCT
ejpam-5103	234	1	data	datum	NOUN
ejpam-5103	234	2	and	and	CCONJ
ejpam-5103	234	3	material	material	NOUN
ejpam-5103	234	4	availability	availability	NOUN
ejpam-5103	234	5	:	:	PUNCT
ejpam-5103	234	6	data	datum	NOUN
ejpam-5103	234	7	and	and	CCONJ
ejpam-5103	234	8	materials	material	NOUN
ejpam-5103	234	9	are	be	AUX
ejpam-5103	234	10	not	not	PART
ejpam-5103	234	11	applicable	applicable	ADJ
ejpam-5103	234	12	to	to	ADP
ejpam-5103	234	13	this	this	DET
ejpam-5103	234	14	study	study	NOUN
ejpam-5103	234	15	.	.	PUNCT
ejpam-5103	235	1	contributions	contribution	NOUN
ejpam-5103	235	2	of	of	ADP
ejpam-5103	235	3	authors	author	NOUN
ejpam-5103	235	4	:	:	PUNCT
ejpam-5103	235	5	every	every	DET
ejpam-5103	235	6	author	author	NOUN
ejpam-5103	235	7	played	play	VERB
ejpam-5103	235	8	a	a	DET
ejpam-5103	235	9	role	role	NOUN
ejpam-5103	235	10	in	in	ADP
ejpam-5103	235	11	the	the	DET
ejpam-5103	235	12	study	study	NOUN
ejpam-5103	235	13	by	by	ADP
ejpam-5103	235	14	contributing	contribute	VERB
ejpam-5103	235	15	to	to	ADP
ejpam-5103	235	16	its	its	PRON
ejpam-5103	235	17	conceptualization	conceptualization	NOUN
ejpam-5103	235	18	and	and	CCONJ
ejpam-5103	235	19	coordination	coordination	NOUN
ejpam-5103	235	20	,	,	PUNCT
ejpam-5103	235	21	drafting	draft	VERB
ejpam-5103	235	22	the	the	DET
ejpam-5103	235	23	manuscript	manuscript	NOUN
ejpam-5103	235	24	,	,	PUNCT
ejpam-5103	235	25	and	and	CCONJ
ejpam-5103	235	26	participating	participate	VERB
ejpam-5103	235	27	in	in	ADP
ejpam-5103	235	28	the	the	DET
ejpam-5103	235	29	review	review	NOUN
ejpam-5103	235	30	and	and	CCONJ
ejpam-5103	235	31	approval	approval	NOUN
ejpam-5103	235	32	of	of	ADP
ejpam-5103	235	33	the	the	DET
ejpam-5103	235	34	final	final	ADJ
ejpam-5103	235	35	version	version	NOUN
ejpam-5103	235	36	.	.	PUNCT
ejpam-5103	236	1	references	reference	NOUN
ejpam-5103	236	2	[	[	X
ejpam-5103	236	3	1	1	X
ejpam-5103	236	4	]	]	X
ejpam-5103	236	5	hiliana	hiliana	NOUN
ejpam-5103	236	6	angulo	angulo	PROPN
ejpam-5103	236	7	,	,	PUNCT
ejpam-5103	236	8	josé	josé	PROPN
ejpam-5103	236	9	giménez	giménez	PROPN
ejpam-5103	236	10	,	,	PUNCT
ejpam-5103	236	11	ana	ana	PROPN
ejpam-5103	236	12	milena	milena	PROPN
ejpam-5103	236	13	moros	moros	PROPN
ejpam-5103	236	14	,	,	PUNCT
ejpam-5103	236	15	and	and	CCONJ
ejpam-5103	236	16	kazimierz	kazimierz	PROPN
ejpam-5103	236	17	nikodem	nikodem	PROPN
ejpam-5103	236	18	.	.	PUNCT
ejpam-5103	237	1	on	on	ADP
ejpam-5103	237	2	strongly	strongly	ADV
ejpam-5103	237	3	h	h	ADJ
ejpam-5103	237	4	-	-	PUNCT
ejpam-5103	237	5	convex	convex	NOUN
ejpam-5103	237	6	functions	function	NOUN
ejpam-5103	237	7	.	.	PUNCT
ejpam-5103	238	1	annals	annal	NOUN
ejpam-5103	238	2	of	of	ADP
ejpam-5103	238	3	functional	functional	ADJ
ejpam-5103	238	4	analysis	analysis	NOUN
ejpam-5103	238	5	,	,	PUNCT
ejpam-5103	238	6	2(2):85–91	2(2):85–91	NUM
ejpam-5103	238	7	,	,	PUNCT
ejpam-5103	238	8	2011	2011	NUM
ejpam-5103	238	9	.	.	PUNCT
ejpam-5103	239	1	[	[	X
ejpam-5103	239	2	2	2	X
ejpam-5103	239	3	]	]	PUNCT
ejpam-5103	239	4	muhammad	muhammad	PROPN
ejpam-5103	239	5	uzair	uzair	PROPN
ejpam-5103	239	6	awan	awan	PROPN
ejpam-5103	239	7	,	,	PUNCT
ejpam-5103	239	8	muhammad	muhammad	PROPN
ejpam-5103	239	9	aslam	aslam	PROPN
ejpam-5103	239	10	noor	noor	PROPN
ejpam-5103	239	11	,	,	PUNCT
ejpam-5103	239	12	khalida	khalida	PROPN
ejpam-5103	239	13	inayat	inayat	PROPN
ejpam-5103	239	14	noor	noor	PROPN
ejpam-5103	239	15	,	,	PUNCT
ejpam-5103	239	16	and	and	CCONJ
ejpam-5103	239	17	farhat	farhat	PROPN
ejpam-5103	239	18	safdar	safdar	PROPN
ejpam-5103	239	19	.	.	PUNCT
ejpam-5103	240	1	on	on	ADP
ejpam-5103	240	2	strongly	strongly	ADV
ejpam-5103	240	3	generalized	generalize	VERB
ejpam-5103	240	4	convex	convex	NOUN
ejpam-5103	240	5	functions	function	NOUN
ejpam-5103	240	6	.	.	PUNCT
ejpam-5103	241	1	filomat	filomat	NOUN
ejpam-5103	241	2	,	,	PUNCT
ejpam-5103	241	3	31(18):5783–5790	31(18):5783–5790	NUM
ejpam-5103	241	4	,	,	PUNCT
ejpam-5103	241	5	2017	2017	NUM
ejpam-5103	241	6	.	.	PUNCT
ejpam-5103	242	1	[	[	X
ejpam-5103	242	2	3	3	X
ejpam-5103	242	3	]	]	PUNCT
ejpam-5103	242	4	a	a	DET
ejpam-5103	242	5	barani	barani	NOUN
ejpam-5103	242	6	and	and	CCONJ
ejpam-5103	242	7	mr	mr	PROPN
ejpam-5103	242	8	pouryayevali	pouryayevali	PROPN
ejpam-5103	242	9	.	.	PROPN
ejpam-5103	242	10	invex	invex	PROPN
ejpam-5103	242	11	sets	set	NOUN
ejpam-5103	242	12	and	and	CCONJ
ejpam-5103	242	13	preinvex	preinvex	NOUN
ejpam-5103	242	14	functions	function	NOUN
ejpam-5103	242	15	on	on	ADP
ejpam-5103	242	16	riemannian	riemannian	ADJ
ejpam-5103	242	17	manifolds	manifold	NOUN
ejpam-5103	242	18	.	.	PUNCT
ejpam-5103	243	1	journal	journal	PROPN
ejpam-5103	243	2	of	of	ADP
ejpam-5103	243	3	mathematical	mathematical	ADJ
ejpam-5103	243	4	analysis	analysis	NOUN
ejpam-5103	243	5	and	and	CCONJ
ejpam-5103	243	6	applications	application	NOUN
ejpam-5103	243	7	,	,	PUNCT
ejpam-5103	243	8	328(2):767–779	328(2):767–779	PROPN
ejpam-5103	243	9	,	,	PUNCT
ejpam-5103	243	10	2007	2007	NUM
ejpam-5103	243	11	.	.	PUNCT
ejpam-5103	244	1	[	[	X
ejpam-5103	244	2	4	4	NUM
ejpam-5103	244	3	]	]	PUNCT
ejpam-5103	244	4	akhlad	akhlad	PROPN
ejpam-5103	244	5	iqbal	iqbal	PROPN
ejpam-5103	244	6	and	and	CCONJ
ejpam-5103	244	7	izhar	izhar	PROPN
ejpam-5103	244	8	ahmad	ahmad	PROPN
ejpam-5103	244	9	.	.	PUNCT
ejpam-5103	245	1	strong	strong	ADJ
ejpam-5103	245	2	geodesic	geodesic	ADJ
ejpam-5103	245	3	convex	convex	NOUN
ejpam-5103	245	4	functions	function	NOUN
ejpam-5103	245	5	of	of	ADP
ejpam-5103	245	6	order	order	NOUN
ejpam-5103	245	7	m.	m.	NOUN
ejpam-5103	245	8	numerical	numerical	ADJ
ejpam-5103	245	9	functional	functional	ADJ
ejpam-5103	245	10	analysis	analysis	NOUN
ejpam-5103	245	11	and	and	CCONJ
ejpam-5103	245	12	optimization	optimization	NOUN
ejpam-5103	245	13	,	,	PUNCT
ejpam-5103	245	14	40(15):1840–1846	40(15):1840–1846	NUM
ejpam-5103	245	15	,	,	PUNCT
ejpam-5103	245	16	2019	2019	NUM
ejpam-5103	245	17	.	.	PUNCT
ejpam-5103	246	1	references	reference	NOUN
ejpam-5103	246	2	869	869	NUM
ejpam-5103	246	3	[	[	X
ejpam-5103	246	4	5	5	NUM
ejpam-5103	246	5	]	]	PUNCT
ejpam-5103	246	6	gui	gui	PROPN
ejpam-5103	246	7	-	-	PUNCT
ejpam-5103	246	8	hua	hua	PROPN
ejpam-5103	246	9	lin	lin	PROPN
ejpam-5103	246	10	and	and	CCONJ
ejpam-5103	246	11	masao	masao	PROPN
ejpam-5103	246	12	fukushima	fukushima	PROPN
ejpam-5103	246	13	.	.	PUNCT
ejpam-5103	247	1	some	some	DET
ejpam-5103	247	2	exact	exact	ADJ
ejpam-5103	247	3	penalty	penalty	NOUN
ejpam-5103	247	4	results	result	NOUN
ejpam-5103	247	5	for	for	ADP
ejpam-5103	247	6	nonlinear	nonlinear	ADJ
ejpam-5103	247	7	programs	program	NOUN
ejpam-5103	247	8	and	and	CCONJ
ejpam-5103	247	9	mathematical	mathematical	ADJ
ejpam-5103	247	10	programs	program	NOUN
ejpam-5103	247	11	with	with	ADP
ejpam-5103	247	12	equilibrium	equilibrium	NOUN
ejpam-5103	247	13	constraints	constraint	NOUN
ejpam-5103	247	14	.	.	PUNCT
ejpam-5103	248	1	journal	journal	NOUN
ejpam-5103	248	2	of	of	ADP
ejpam-5103	248	3	optimization	optimization	NOUN
ejpam-5103	248	4	theory	theory	NOUN
ejpam-5103	248	5	and	and	CCONJ
ejpam-5103	248	6	applications	application	NOUN
ejpam-5103	248	7	,	,	PUNCT
ejpam-5103	248	8	118:67–80	118:67–80	NUM
ejpam-5103	248	9	,	,	PUNCT
ejpam-5103	248	10	2003	2003	NUM
ejpam-5103	248	11	.	.	PUNCT
ejpam-5103	249	1	[	[	X
ejpam-5103	249	2	6	6	NUM
ejpam-5103	249	3	]	]	X
ejpam-5103	249	4	nelson	nelson	PROPN
ejpam-5103	249	5	merentes	merentes	PROPN
ejpam-5103	249	6	,	,	PUNCT
ejpam-5103	249	7	kazimierz	kazimierz	PROPN
ejpam-5103	249	8	nikodem	nikodem	PROPN
ejpam-5103	249	9	,	,	PUNCT
ejpam-5103	249	10	and	and	CCONJ
ejpam-5103	249	11	sergio	sergio	PROPN
ejpam-5103	249	12	rivas	rivas	PROPN
ejpam-5103	249	13	.	.	PROPN
ejpam-5103	249	14	remarks	remark	NOUN
ejpam-5103	249	15	on	on	ADP
ejpam-5103	249	16	strongly	strongly	ADV
ejpam-5103	249	17	wrightconvex	wrightconvex	PROPN
ejpam-5103	249	18	functions	function	NOUN
ejpam-5103	249	19	.	.	PUNCT
ejpam-5103	250	1	ann	ann	PROPN
ejpam-5103	250	2	.	.	PUNCT
ejpam-5103	250	3	polon	polon	PROPN
ejpam-5103	250	4	.	.	PUNCT
ejpam-5103	251	1	math	math	NOUN
ejpam-5103	251	2	,	,	PUNCT
ejpam-5103	251	3	102(3):271–278	102(3):271–278	NUM
ejpam-5103	251	4	,	,	PUNCT
ejpam-5103	251	5	2011	2011	NUM
ejpam-5103	251	6	.	.	PUNCT
ejpam-5103	252	1	[	[	X
ejpam-5103	252	2	7	7	X
ejpam-5103	252	3	]	]	PUNCT
ejpam-5103	252	4	stephan	stephan	PROPN
ejpam-5103	252	5	mititelu	mititelu	NOUN
ejpam-5103	252	6	.	.	PUNCT
ejpam-5103	253	1	generalized	generalize	VERB
ejpam-5103	253	2	invexity	invexity	NOUN
ejpam-5103	253	3	and	and	CCONJ
ejpam-5103	253	4	vector	vector	NOUN
ejpam-5103	253	5	optimization	optimization	NOUN
ejpam-5103	253	6	on	on	ADP
ejpam-5103	253	7	differentiable	differentiable	ADJ
ejpam-5103	253	8	manifolds	manifold	NOUN
ejpam-5103	253	9	.	.	PUNCT
ejpam-5103	253	10	differ	differ	VERB
ejpam-5103	253	11	.	.	PUNCT
ejpam-5103	254	1	geom	geom	PROPN
ejpam-5103	254	2	.	.	PUNCT
ejpam-5103	255	1	dyn	dyn	PROPN
ejpam-5103	255	2	.	.	PUNCT
ejpam-5103	256	1	syst	syst	PROPN
ejpam-5103	256	2	,	,	PUNCT
ejpam-5103	256	3	3(1):21–31	3(1):21–31	NUM
ejpam-5103	256	4	,	,	PUNCT
ejpam-5103	256	5	2001	2001	NUM
ejpam-5103	256	6	.	.	PUNCT
ejpam-5103	257	1	[	[	X
ejpam-5103	257	2	8	8	X
ejpam-5103	257	3	]	]	X
ejpam-5103	257	4	muhammad	muhammad	PROPN
ejpam-5103	257	5	aslam	aslam	PROPN
ejpam-5103	257	6	noor	noor	PROPN
ejpam-5103	257	7	and	and	CCONJ
ejpam-5103	257	8	khalida	khalida	PROPN
ejpam-5103	257	9	inayat	inayat	PROPN
ejpam-5103	257	10	noor	noor	PROPN
ejpam-5103	257	11	.	.	PUNCT
ejpam-5103	258	1	new	new	ADJ
ejpam-5103	258	2	classes	class	NOUN
ejpam-5103	258	3	of	of	ADP
ejpam-5103	258	4	strongly	strongly	ADV
ejpam-5103	258	5	exponentially	exponentially	ADV
ejpam-5103	258	6	preinvex	preinvex	NOUN
ejpam-5103	258	7	functions	function	NOUN
ejpam-5103	258	8	.	.	PUNCT
ejpam-5103	259	1	aims	aim	VERB
ejpam-5103	259	2	math	math	NOUN
ejpam-5103	259	3	,	,	PUNCT
ejpam-5103	259	4	4(6):1554–1568	4(6):1554–1568	NOUN
ejpam-5103	259	5	,	,	PUNCT
ejpam-5103	259	6	2019	2019	NUM
ejpam-5103	259	7	.	.	PUNCT
ejpam-5103	260	1	[	[	X
ejpam-5103	260	2	9	9	X
ejpam-5103	260	3	]	]	PUNCT
ejpam-5103	260	4	muhammad	muhammad	PROPN
ejpam-5103	260	5	aslam	aslam	PROPN
ejpam-5103	260	6	noor	noor	PROPN
ejpam-5103	260	7	and	and	CCONJ
ejpam-5103	260	8	khalida	khalida	PROPN
ejpam-5103	260	9	inayat	inayat	PROPN
ejpam-5103	260	10	noor	noor	PROPN
ejpam-5103	260	11	.	.	PUNCT
ejpam-5103	261	1	some	some	DET
ejpam-5103	261	2	properties	property	NOUN
ejpam-5103	261	3	of	of	ADP
ejpam-5103	261	4	exponentially	exponentially	ADV
ejpam-5103	261	5	preinvex	preinvex	NOUN
ejpam-5103	261	6	functions	function	NOUN
ejpam-5103	261	7	.	.	PUNCT
ejpam-5103	262	1	facta	facta	PROPN
ejpam-5103	262	2	universitat	universitat	PROPN
ejpam-5103	262	3	(	(	PUNCT
ejpam-5103	262	4	nis	nis	NOUN
ejpam-5103	262	5	)	)	PUNCT
ejpam-5103	262	6	.	.	PUNCT
ejpam-5103	263	1	ser.math	ser.math	PROPN
ejpam-5103	263	2	.	.	PUNCT
ejpam-5103	264	1	inform	inform	NOUN
ejpam-5103	264	2	,	,	PUNCT
ejpam-5103	264	3	34(5	34(5	PROPN
ejpam-5103	264	4	)	)	PUNCT
ejpam-5103	264	5	,	,	PUNCT
ejpam-5103	264	6	2019	2019	NUM
ejpam-5103	264	7	.	.	PUNCT
ejpam-5103	265	1	[	[	X
ejpam-5103	265	2	10	10	NUM
ejpam-5103	265	3	]	]	X
ejpam-5103	265	4	muhammad	muhammad	PROPN
ejpam-5103	265	5	aslam	aslam	PROPN
ejpam-5103	265	6	noor	noor	PROPN
ejpam-5103	265	7	,	,	PUNCT
ejpam-5103	265	8	khalida	khalida	PROPN
ejpam-5103	265	9	inayat	inayat	PROPN
ejpam-5103	265	10	noor	noor	PROPN
ejpam-5103	265	11	,	,	PUNCT
ejpam-5103	265	12	and	and	CCONJ
ejpam-5103	265	13	muhammad	muhammad	PROPN
ejpam-5103	265	14	uzair	uzair	PROPN
ejpam-5103	265	15	awan	awan	PROPN
ejpam-5103	265	16	.	.	PUNCT
ejpam-5103	266	1	hermite	hermite	PROPN
ejpam-5103	266	2	hadamard	hadamard	ADJ
ejpam-5103	266	3	inequalities	inequality	NOUN
ejpam-5103	266	4	for	for	ADP
ejpam-5103	266	5	modified	modified	ADJ
ejpam-5103	266	6	h	h	ADJ
ejpam-5103	266	7	-	-	PUNCT
ejpam-5103	266	8	convex	convex	NOUN
ejpam-5103	266	9	functions	function	NOUN
ejpam-5103	266	10	.	.	PUNCT
ejpam-5103	267	1	transylvanian	transylvanian	ADJ
ejpam-5103	267	2	journal	journal	NOUN
ejpam-5103	267	3	of	of	ADP
ejpam-5103	267	4	mathematics	mathematic	NOUN
ejpam-5103	267	5	and	and	CCONJ
ejpam-5103	267	6	mechanics	mechanic	NOUN
ejpam-5103	267	7	,	,	PUNCT
ejpam-5103	267	8	6:1–10	6:1–10	NUM
ejpam-5103	267	9	,	,	PUNCT
ejpam-5103	267	10	2014	2014	NUM
ejpam-5103	267	11	.	.	PUNCT
ejpam-5103	268	1	[	[	X
ejpam-5103	268	2	11	11	NUM
ejpam-5103	268	3	]	]	PUNCT
ejpam-5103	268	4	ammara	ammara	ADV
ejpam-5103	268	5	nosheen	nosheen	PROPN
ejpam-5103	268	6	,	,	PUNCT
ejpam-5103	268	7	sana	sana	PROPN
ejpam-5103	268	8	ijaz	ijaz	PROPN
ejpam-5103	268	9	,	,	PUNCT
ejpam-5103	268	10	khuram	khuram	PROPN
ejpam-5103	268	11	ali	ali	PROPN
ejpam-5103	268	12	khan	khan	PROPN
ejpam-5103	268	13	,	,	PUNCT
ejpam-5103	268	14	khalid	khalid	PROPN
ejpam-5103	268	15	mahmood	mahmood	PROPN
ejpam-5103	268	16	awan	awan	PROPN
ejpam-5103	268	17	,	,	PUNCT
ejpam-5103	268	18	marwan	marwan	PROPN
ejpam-5103	268	19	ali	ali	PROPN
ejpam-5103	268	20	albahar	albahar	PROPN
ejpam-5103	268	21	,	,	PUNCT
ejpam-5103	268	22	and	and	CCONJ
ejpam-5103	268	23	mohammed	mohammed	PROPN
ejpam-5103	268	24	thanoon	thanoon	NOUN
ejpam-5103	268	25	.	.	PUNCT
ejpam-5103	269	1	some	some	DET
ejpam-5103	269	2	q	q	ADJ
ejpam-5103	269	3	-	-	ADJ
ejpam-5103	269	4	symmetric	symmetric	ADJ
ejpam-5103	269	5	integral	integral	ADJ
ejpam-5103	269	6	inequalities	inequality	NOUN
ejpam-5103	269	7	involving	involve	VERB
ejpam-5103	269	8	s	s	NOUN
ejpam-5103	269	9	-	-	PUNCT
ejpam-5103	269	10	convex	convex	ADJ
ejpam-5103	269	11	functions	function	NOUN
ejpam-5103	269	12	.	.	PUNCT
ejpam-5103	270	1	symmetry	symmetry	NOUN
ejpam-5103	270	2	,	,	PUNCT
ejpam-5103	270	3	15(6):1169	15(6):1169	NUM
ejpam-5103	270	4	,	,	PUNCT
ejpam-5103	270	5	2023	2023	NUM
ejpam-5103	270	6	.	.	PUNCT
ejpam-5103	271	1	[	[	X
ejpam-5103	271	2	12	12	NUM
ejpam-5103	271	3	]	]	X
ejpam-5103	271	4	r	r	NOUN
ejpam-5103	271	5	pini	pini	NOUN
ejpam-5103	271	6	.	.	PUNCT
ejpam-5103	272	1	convexity	convexity	NOUN
ejpam-5103	272	2	along	along	ADP
ejpam-5103	272	3	curves	curve	NOUN
ejpam-5103	272	4	and	and	CCONJ
ejpam-5103	272	5	indunvexity	indunvexity	NOUN
ejpam-5103	272	6	.	.	PUNCT
ejpam-5103	273	1	optimization	optimization	NOUN
ejpam-5103	273	2	,	,	PUNCT
ejpam-5103	273	3	29(4):301–309	29(4):301–309	NUM
ejpam-5103	273	4	,	,	PUNCT
ejpam-5103	273	5	1994	1994	NUM
ejpam-5103	273	6	.	.	PUNCT
ejpam-5103	274	1	[	[	X
ejpam-5103	274	2	13	13	NUM
ejpam-5103	274	3	]	]	PUNCT
ejpam-5103	274	4	boris	boris	PROPN
ejpam-5103	274	5	teodorovich	teodorovich	PROPN
ejpam-5103	274	6	polyak	polyak	PROPN
ejpam-5103	274	7	.	.	PUNCT
ejpam-5103	275	1	existence	existence	NOUN
ejpam-5103	275	2	theorems	theorem	NOUN
ejpam-5103	275	3	and	and	CCONJ
ejpam-5103	275	4	convergence	convergence	NOUN
ejpam-5103	275	5	of	of	ADP
ejpam-5103	275	6	minimizing	minimize	VERB
ejpam-5103	275	7	sequences	sequence	NOUN
ejpam-5103	275	8	for	for	ADP
ejpam-5103	275	9	extremal	extremal	ADJ
ejpam-5103	275	10	problems	problem	NOUN
ejpam-5103	275	11	with	with	ADP
ejpam-5103	275	12	constraints	constraint	NOUN
ejpam-5103	275	13	.	.	PUNCT
ejpam-5103	276	1	in	in	ADP
ejpam-5103	276	2	doklady	doklady	PROPN
ejpam-5103	276	3	akademii	akademii	PROPN
ejpam-5103	276	4	nauk	nauk	PROPN
ejpam-5103	276	5	,	,	PUNCT
ejpam-5103	276	6	volume	volume	NOUN
ejpam-5103	276	7	166	166	NUM
ejpam-5103	276	8	,	,	PUNCT
ejpam-5103	276	9	pages	page	NOUN
ejpam-5103	276	10	287–290	287–290	NUM
ejpam-5103	276	11	.	.	PUNCT
ejpam-5103	277	1	russian	russian	ADJ
ejpam-5103	277	2	academy	academy	PROPN
ejpam-5103	277	3	of	of	ADP
ejpam-5103	277	4	sciences	sciences	PROPN
ejpam-5103	277	5	,	,	PUNCT
ejpam-5103	277	6	1966	1966	NUM
ejpam-5103	277	7	.	.	PUNCT
ejpam-5103	278	1	[	[	X
ejpam-5103	278	2	14	14	NUM
ejpam-5103	278	3	]	]	X
ejpam-5103	278	4	wedad	wedad	PROPN
ejpam-5103	278	5	saleh	saleh	PROPN
ejpam-5103	278	6	.	.	PUNCT
ejpam-5103	279	1	some	some	DET
ejpam-5103	279	2	properties	property	NOUN
ejpam-5103	279	3	of	of	ADP
ejpam-5103	279	4	geodesic	geodesic	NOUN
ejpam-5103	279	5	strongly	strongly	ADV
ejpam-5103	279	6	eb	eb	PROPN
ejpam-5103	279	7	-	-	PUNCT
ejpam-5103	279	8	vex	vex	NOUN
ejpam-5103	279	9	functions	function	NOUN
ejpam-5103	279	10	.	.	PUNCT
ejpam-5103	280	1	international	international	ADJ
ejpam-5103	280	2	journal	journal	NOUN
ejpam-5103	280	3	of	of	ADP
ejpam-5103	280	4	analysis	analysis	NOUN
ejpam-5103	280	5	and	and	CCONJ
ejpam-5103	280	6	applications	application	NOUN
ejpam-5103	280	7	,	,	PUNCT
ejpam-5103	280	8	17(3):388–395	17(3):388–395	PROPN
ejpam-5103	280	9	,	,	PUNCT
ejpam-5103	280	10	2019	2019	NUM
ejpam-5103	280	11	.	.	PUNCT
ejpam-5103	281	1	[	[	X
ejpam-5103	281	2	15	15	NUM
ejpam-5103	281	3	]	]	X
ejpam-5103	281	4	constantin	constantin	NOUN
ejpam-5103	281	5	udriste	udriste	NOUN
ejpam-5103	281	6	.	.	PUNCT
ejpam-5103	282	1	convex	convex	NOUN
ejpam-5103	282	2	functions	function	NOUN
ejpam-5103	282	3	and	and	CCONJ
ejpam-5103	282	4	optimization	optimization	NOUN
ejpam-5103	282	5	methods	method	NOUN
ejpam-5103	282	6	on	on	ADP
ejpam-5103	282	7	riemannian	riemannian	ADJ
ejpam-5103	282	8	manifolds	manifold	NOUN
ejpam-5103	282	9	,	,	PUNCT
ejpam-5103	282	10	volume	volume	NOUN
ejpam-5103	282	11	297	297	NUM
ejpam-5103	282	12	.	.	PUNCT
ejpam-5103	283	1	springer	springer	NOUN
ejpam-5103	283	2	science	science	PROPN
ejpam-5103	283	3	&	&	CCONJ
ejpam-5103	283	4	business	business	NOUN
ejpam-5103	283	5	media	medium	NOUN
ejpam-5103	283	6	,	,	PUNCT
ejpam-5103	283	7	2013	2013	NUM
ejpam-5103	283	8	.	.	PUNCT
ejpam-5103	284	1	[	[	X
ejpam-5103	284	2	16	16	NUM
ejpam-5103	284	3	]	]	SYM
ejpam-5103	284	4	taiyin	taiyin	PROPN
ejpam-5103	284	5	zhao	zhao	PROPN
ejpam-5103	284	6	,	,	PUNCT
ejpam-5103	284	7	muhammad	muhammad	PROPN
ejpam-5103	284	8	shoaib	shoaib	PROPN
ejpam-5103	284	9	saleem	saleem	PROPN
ejpam-5103	284	10	,	,	PUNCT
ejpam-5103	284	11	waqas	waqas	PROPN
ejpam-5103	284	12	nazeer	nazeer	PROPN
ejpam-5103	284	13	,	,	PUNCT
ejpam-5103	284	14	imran	imran	PROPN
ejpam-5103	284	15	bashir	bashir	PROPN
ejpam-5103	284	16	,	,	PUNCT
ejpam-5103	284	17	and	and	CCONJ
ejpam-5103	284	18	ijaz	ijaz	PROPN
ejpam-5103	284	19	hussain	hussain	PROPN
ejpam-5103	284	20	.	.	PUNCT
ejpam-5103	285	1	on	on	ADP
ejpam-5103	285	2	generalized	generalize	VERB
ejpam-5103	285	3	strongly	strongly	ADV
ejpam-5103	285	4	modified	modify	VERB
ejpam-5103	285	5	h	h	ADJ
ejpam-5103	285	6	-	-	PUNCT
ejpam-5103	285	7	convex	convex	NOUN
ejpam-5103	285	8	functions	function	NOUN
ejpam-5103	285	9	.	.	PUNCT
ejpam-5103	286	1	journal	journal	NOUN
ejpam-5103	286	2	of	of	ADP
ejpam-5103	286	3	inequalities	inequality	NOUN
ejpam-5103	286	4	and	and	CCONJ
ejpam-5103	286	5	applications	application	NOUN
ejpam-5103	286	6	,	,	PUNCT
ejpam-5103	286	7	2020:1–12	2020:1–12	NUM
ejpam-5103	286	8	,	,	PUNCT
ejpam-5103	286	9	2020	2020	NUM
ejpam-5103	286	10	.	.	PUNCT