id	sid	tid	token	lemma	pos
ejde-792	1	1	electronic	electronic	ADJ
ejde-792	1	2	journal	journal	NOUN
ejde-792	1	3	of	of	ADP
ejde-792	1	4	differential	differential	ADJ
ejde-792	1	5	equations	equation	NOUN
ejde-792	1	6	,	,	PUNCT
ejde-792	1	7	vol	vol	NOUN
ejde-792	1	8	.	.	NOUN
ejde-792	1	9	2024	2024	NUM
ejde-792	1	10	(	(	PUNCT
ejde-792	1	11	2024	2024	NUM
ejde-792	1	12	)	)	PUNCT
ejde-792	1	13	,	,	PUNCT
ejde-792	1	14	no	no	INTJ
ejde-792	1	15	.	.	NOUN
ejde-792	1	16	48	48	NUM
ejde-792	1	17	,	,	PUNCT
ejde-792	1	18	pp	pp	PROPN
ejde-792	1	19	.	.	PUNCT
ejde-792	2	1	1–10	1–10	PROPN
ejde-792	2	2	.	.	PUNCT
ejde-792	3	1	issn	issn	PROPN
ejde-792	3	2	:	:	PUNCT
ejde-792	3	3	1072	1072	NUM
ejde-792	3	4	-	-	SYM
ejde-792	3	5	6691	6691	NUM
ejde-792	3	6	.	.	PUNCT
ejde-792	4	1	url	url	PROPN
ejde-792	4	2	:	:	PUNCT
ejde-792	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-792	4	4	,	,	PUNCT
ejde-792	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-792	4	6	doi	doi	PROPN
ejde-792	4	7	:	:	PUNCT
ejde-792	4	8	10.58997	10.58997	NUM
ejde-792	4	9	/	/	SYM
ejde-792	4	10	ejde.2024.48	ejde.2024.48	ADJ
ejde-792	4	11	local	local	ADJ
ejde-792	4	12	bifurcation	bifurcation	NOUN
ejde-792	4	13	structure	structure	NOUN
ejde-792	4	14	and	and	CCONJ
ejde-792	4	15	stability	stability	NOUN
ejde-792	4	16	of	of	ADP
ejde-792	4	17	the	the	DET
ejde-792	4	18	mean	mean	ADJ
ejde-792	4	19	curvature	curvature	NOUN
ejde-792	4	20	equation	equation	NOUN
ejde-792	4	21	in	in	ADP
ejde-792	4	22	the	the	DET
ejde-792	4	23	static	static	ADJ
ejde-792	4	24	spacetime	spacetime	PROPN
ejde-792	4	25	siyu	siyu	PROPN
ejde-792	4	26	gao	gao	PROPN
ejde-792	4	27	,	,	PUNCT
ejde-792	4	28	qingbo	qingbo	PROPN
ejde-792	4	29	liu	liu	PROPN
ejde-792	4	30	,	,	PUNCT
ejde-792	4	31	yingxin	yingxin	PROPN
ejde-792	4	32	sun	sun	PROPN
ejde-792	4	33	abstract	abstract	NOUN
ejde-792	4	34	.	.	PUNCT
ejde-792	5	1	we	we	PRON
ejde-792	5	2	consider	consider	VERB
ejde-792	5	3	the	the	DET
ejde-792	5	4	curvature	curvature	NOUN
ejde-792	5	5	equation	equation	NOUN
ejde-792	5	6	in	in	ADP
ejde-792	5	7	the	the	DET
ejde-792	5	8	static	static	ADJ
ejde-792	5	9	spacetime	spacetime	NOUN
ejde-792	5	10	,	,	PUNCT
ejde-792	5	11	div	div	X
ejde-792	5	12	(	(	PUNCT
ejde-792	5	13	f(x)∇u√	f(x)∇u√	X
ejde-792	5	14	1−	1−	NUM
ejde-792	5	15	f2(x)|∇u|2	f2(x)|∇u|2	PROPN
ejde-792	5	16	)	)	PUNCT
ejde-792	6	1	+	+	CCONJ
ejde-792	6	2	∇u∇f(x)√	∇u∇f(x)√	NOUN
ejde-792	6	3	1−	1−	NUM
ejde-792	6	4	f2(x)|∇u|2	f2(x)|∇u|2	NOUN
ejde-792	6	5	=	=	SYM
ejde-792	6	6	λnh	λnh	PROPN
ejde-792	6	7	in	in	ADP
ejde-792	6	8	ω	ω	PROPN
ejde-792	6	9	,	,	PUNCT
ejde-792	6	10	where	where	SCONJ
ejde-792	6	11	ω	ω	PROPN
ejde-792	6	12	is	be	AUX
ejde-792	6	13	a	a	DET
ejde-792	6	14	bounded	bounded	ADJ
ejde-792	6	15	domain	domain	NOUN
ejde-792	6	16	in	in	ADP
ejde-792	6	17	rn	rn	PROPN
ejde-792	6	18	,	,	PUNCT
ejde-792	6	19	n	n	PRON
ejde-792	6	20	≥	≥	NUM
ejde-792	6	21	1	1	NUM
ejde-792	6	22	;	;	PUNCT
ejde-792	6	23	the	the	DET
ejde-792	6	24	function	function	NOUN
ejde-792	6	25	h	h	NOUN
ejde-792	6	26	gives	give	VERB
ejde-792	6	27	the	the	DET
ejde-792	6	28	mean	mean	ADJ
ejde-792	6	29	curvature	curvature	NOUN
ejde-792	6	30	.	.	PUNCT
ejde-792	7	1	we	we	PRON
ejde-792	7	2	investigate	investigate	VERB
ejde-792	7	3	the	the	DET
ejde-792	7	4	local	local	ADJ
ejde-792	7	5	bifurcation	bifurcation	NOUN
ejde-792	7	6	structure	structure	NOUN
ejde-792	7	7	and	and	CCONJ
ejde-792	7	8	stability	stability	NOUN
ejde-792	7	9	of	of	ADP
ejde-792	7	10	the	the	DET
ejde-792	7	11	solutions	solution	NOUN
ejde-792	7	12	to	to	ADP
ejde-792	7	13	this	this	DET
ejde-792	7	14	equation	equation	NOUN
ejde-792	7	15	.	.	PUNCT
ejde-792	8	1	1	1	X
ejde-792	8	2	.	.	X
ejde-792	8	3	introduction	introduction	NOUN
ejde-792	8	4	and	and	CCONJ
ejde-792	8	5	main	main	ADJ
ejde-792	8	6	results	result	NOUN
ejde-792	8	7	we	we	PRON
ejde-792	8	8	consider	consider	VERB
ejde-792	8	9	a	a	DET
ejde-792	8	10	domain	domain	NOUN
ejde-792	8	11	ω	ω	PROPN
ejde-792	8	12	⊆	⊆	NUM
ejde-792	8	13	rn	rn	PROPN
ejde-792	8	14	,	,	PUNCT
ejde-792	8	15	where	where	SCONJ
ejde-792	8	16	n	n	PRON
ejde-792	8	17	is	be	AUX
ejde-792	8	18	greater	great	ADJ
ejde-792	8	19	than	than	ADP
ejde-792	8	20	or	or	CCONJ
ejde-792	8	21	equal	equal	ADJ
ejde-792	8	22	to	to	ADP
ejde-792	8	23	1	1	NUM
ejde-792	8	24	.	.	PUNCT
ejde-792	9	1	let	let	VERB
ejde-792	9	2	f	f	PRON
ejde-792	9	3	be	be	AUX
ejde-792	9	4	a	a	DET
ejde-792	9	5	smooth	smooth	ADJ
ejde-792	9	6	positive	positive	ADJ
ejde-792	9	7	function	function	NOUN
ejde-792	9	8	on	on	ADP
ejde-792	9	9	ω	ω	PROPN
ejde-792	9	10	.	.	PUNCT
ejde-792	10	1	consider	consider	VERB
ejde-792	10	2	the	the	DET
ejde-792	10	3	n+1	n+1	ADJ
ejde-792	10	4	-	-	ADJ
ejde-792	10	5	dimensional	dimensional	ADJ
ejde-792	10	6	product	product	NOUN
ejde-792	10	7	manifold	manifold	ADJ
ejde-792	10	8	m	m	NOUN
ejde-792	10	9	=	=	VERB
ejde-792	11	1	i	i	PRON
ejde-792	11	2	×	×	PROPN
ejde-792	11	3	ω	ω	INTJ
ejde-792	11	4	equipped	equip	VERB
ejde-792	11	5	with	with	ADP
ejde-792	11	6	the	the	DET
ejde-792	11	7	lorentzian	lorentzian	ADJ
ejde-792	11	8	metric	metric	ADJ
ejde-792	11	9	g	g	PROPN
ejde-792	11	10	=	=	SYM
ejde-792	11	11	−f2(x	−f2(x	PROPN
ejde-792	11	12	)	)	PUNCT
ejde-792	12	1	dt2	dt2	PROPN
ejde-792	12	2	+	+	CCONJ
ejde-792	12	3	dx2	dx2	PROPN
ejde-792	12	4	.	.	PUNCT
ejde-792	13	1	in	in	ADP
ejde-792	13	2	[	[	X
ejde-792	13	3	22	22	NUM
ejde-792	13	4	,	,	PUNCT
ejde-792	13	5	lemma	lemma	PROPN
ejde-792	13	6	12.37	12.37	NUM
ejde-792	13	7	]	]	PUNCT
ejde-792	13	8	,	,	PUNCT
ejde-792	13	9	it	it	PRON
ejde-792	13	10	was	be	AUX
ejde-792	13	11	established	establish	VERB
ejde-792	13	12	that	that	SCONJ
ejde-792	13	13	m	m	PROPN
ejde-792	13	14	is	be	AUX
ejde-792	13	15	static	static	ADJ
ejde-792	13	16	with	with	ADP
ejde-792	13	17	respect	respect	NOUN
ejde-792	13	18	to	to	ADP
ejde-792	13	19	∂t	∂t	PROPN
ejde-792	13	20	/	/	SYM
ejde-792	13	21	f	f	PROPN
ejde-792	13	22	.	.	PUNCT
ejde-792	14	1	for	for	ADP
ejde-792	14	2	each	each	DET
ejde-792	14	3	u	u	PROPN
ejde-792	14	4	∈	∈	PROPN
ejde-792	14	5	c2(ω	c2(ω	PROPN
ejde-792	14	6	)	)	PUNCT
ejde-792	14	7	,	,	PUNCT
ejde-792	14	8	let	let	VERB
ejde-792	14	9	m	m	VERB
ejde-792	14	10	=	=	PRON
ejde-792	14	11	{	{	PUNCT
ejde-792	14	12	(	(	PUNCT
ejde-792	14	13	x	x	NOUN
ejde-792	14	14	,	,	PUNCT
ejde-792	14	15	u	u	NOUN
ejde-792	14	16	)	)	PUNCT
ejde-792	14	17	:	:	PUNCT
ejde-792	14	18	x	x	X
ejde-792	14	19	∈	∈	PROPN
ejde-792	14	20	ω	ω	PROPN
ejde-792	14	21	,	,	PUNCT
ejde-792	14	22	u	u	PROPN
ejde-792	14	23	∈	∈	PROPN
ejde-792	14	24	c2(ω	c2(ω	PROPN
ejde-792	14	25	)	)	PUNCT
ejde-792	14	26	}	}	PUNCT
ejde-792	14	27	.	.	PUNCT
ejde-792	15	1	a	a	DET
ejde-792	15	2	spacetime	spacetime	NOUN
ejde-792	15	3	m	m	VERB
ejde-792	15	4	is	be	AUX
ejde-792	15	5	termed	term	VERB
ejde-792	15	6	static	static	ADJ
ejde-792	15	7	in	in	ADP
ejde-792	15	8	relation	relation	NOUN
ejde-792	15	9	to	to	ADP
ejde-792	15	10	an	an	DET
ejde-792	15	11	observer	observer	NOUN
ejde-792	15	12	field	field	NOUN
ejde-792	15	13	q	q	PROPN
ejde-792	16	1	if	if	SCONJ
ejde-792	16	2	q	q	NOUN
ejde-792	16	3	is	be	AUX
ejde-792	16	4	irrotational	irrotational	ADJ
ejde-792	16	5	and	and	CCONJ
ejde-792	16	6	if	if	SCONJ
ejde-792	16	7	there	there	PRON
ejde-792	16	8	exists	exist	VERB
ejde-792	16	9	a	a	DET
ejde-792	16	10	smooth	smooth	ADJ
ejde-792	16	11	positive	positive	ADJ
ejde-792	16	12	function	function	NOUN
ejde-792	16	13	such	such	ADJ
ejde-792	16	14	that	that	SCONJ
ejde-792	16	15	fq	fq	PROPN
ejde-792	16	16	is	be	AUX
ejde-792	16	17	a	a	DET
ejde-792	16	18	killing	kill	VERB
ejde-792	16	19	vector	vector	NOUN
ejde-792	16	20	field	field	NOUN
ejde-792	16	21	.	.	PUNCT
ejde-792	17	1	then	then	ADV
ejde-792	17	2	,	,	PUNCT
ejde-792	17	3	(	(	PUNCT
ejde-792	17	4	m	m	NOUN
ejde-792	17	5	,	,	PUNCT
ejde-792	17	6	g	g	NOUN
ejde-792	17	7	)	)	PUNCT
ejde-792	17	8	=	=	SYM
ejde-792	17	9	u	u	NOUN
ejde-792	17	10	represents	represent	VERB
ejde-792	17	11	an	an	DET
ejde-792	17	12	n	n	CCONJ
ejde-792	17	13	-dimensional	-dimensional	ADJ
ejde-792	17	14	hypersurface	hypersurface	NOUN
ejde-792	17	15	in	in	ADP
ejde-792	17	16	m	m	PROPN
ejde-792	17	17	at	at	ADP
ejde-792	17	18	time	time	NOUN
ejde-792	17	19	t	t	PROPN
ejde-792	17	20	,	,	PUNCT
ejde-792	17	21	which	which	PRON
ejde-792	17	22	can	can	AUX
ejde-792	17	23	be	be	AUX
ejde-792	17	24	depicted	depict	VERB
ejde-792	17	25	by	by	ADP
ejde-792	17	26	the	the	DET
ejde-792	17	27	graph	graph	NOUN
ejde-792	17	28	of	of	ADP
ejde-792	17	29	t	t	PROPN
ejde-792	17	30	=	=	PUNCT
ejde-792	17	31	u.	u.	NOUN
ejde-792	17	32	u	u	NOUN
ejde-792	17	33	is	be	AUX
ejde-792	17	34	referred	refer	VERB
ejde-792	17	35	to	to	ADP
ejde-792	17	36	as	as	ADV
ejde-792	17	37	spacelike	spacelike	VERB
ejde-792	17	38	if	if	SCONJ
ejde-792	17	39	|∇u|	|∇u|	ADJ
ejde-792	17	40	<	<	X
ejde-792	17	41	1	1	NUM
ejde-792	17	42	/	/	SYM
ejde-792	17	43	f	f	PROPN
ejde-792	17	44	in	in	ADP
ejde-792	17	45	ω	ω	PROPN
ejde-792	17	46	(	(	PUNCT
ejde-792	17	47	see	see	VERB
ejde-792	17	48	[	[	X
ejde-792	17	49	20	20	NUM
ejde-792	17	50	]	]	NUM
ejde-792	17	51	)	)	PUNCT
ejde-792	17	52	.	.	PUNCT
ejde-792	18	1	we	we	PRON
ejde-792	18	2	define	define	VERB
ejde-792	18	3	u	u	NOUN
ejde-792	18	4	as	as	ADP
ejde-792	18	5	being	be	AUX
ejde-792	18	6	weakly	weakly	ADV
ejde-792	18	7	spacelike	spacelike	ADJ
ejde-792	18	8	if	if	SCONJ
ejde-792	18	9	|∇u|	|∇u|	ADJ
ejde-792	18	10	≤	≤	NUM
ejde-792	18	11	1	1	NUM
ejde-792	18	12	/	/	SYM
ejde-792	18	13	f	f	NOUN
ejde-792	18	14	,	,	PUNCT
ejde-792	18	15	i.e.	i.e.	X
ejde-792	18	16	,	,	PUNCT
ejde-792	18	17	if	if	SCONJ
ejde-792	18	18	it	it	PRON
ejde-792	18	19	is	be	AUX
ejde-792	18	20	in	in	ADP
ejde-792	18	21	ω	ω	PROPN
ejde-792	18	22	.	.	PUNCT
ejde-792	19	1	given	give	VERB
ejde-792	19	2	the	the	DET
ejde-792	19	3	mean	mean	ADJ
ejde-792	19	4	curvature	curvature	NOUN
ejde-792	19	5	h	h	NOUN
ejde-792	19	6	for	for	ADP
ejde-792	19	7	a	a	DET
ejde-792	19	8	spacelike	spacelike	ADJ
ejde-792	19	9	graph	graph	NOUN
ejde-792	19	10	u	u	PROPN
ejde-792	19	11	,	,	PUNCT
ejde-792	19	12	problem	problem	NOUN
ejde-792	19	13	(	(	PUNCT
ejde-792	19	14	1.1	1.1	NUM
ejde-792	19	15	)	)	PUNCT
ejde-792	19	16	has	have	VERB
ejde-792	19	17	implications	implication	NOUN
ejde-792	19	18	for	for	ADP
ejde-792	19	19	classical	classical	ADJ
ejde-792	19	20	relativity	relativity	NOUN
ejde-792	19	21	[	[	X
ejde-792	19	22	4	4	NUM
ejde-792	19	23	]	]	PUNCT
ejde-792	19	24	and	and	CCONJ
ejde-792	19	25	cosmology	cosmology	PROPN
ejde-792	19	26	research	research	NOUN
ejde-792	19	27	[	[	X
ejde-792	19	28	5	5	NUM
ejde-792	19	29	,	,	PUNCT
ejde-792	19	30	19	19	NUM
ejde-792	19	31	,	,	PUNCT
ejde-792	19	32	21	21	NUM
ejde-792	19	33	]	]	PUNCT
ejde-792	19	34	.	.	PUNCT
ejde-792	20	1	for	for	ADP
ejde-792	20	2	the	the	DET
ejde-792	20	3	case	case	NOUN
ejde-792	20	4	in	in	ADP
ejde-792	20	5	which	which	PRON
ejde-792	20	6	f	f	PROPN
ejde-792	20	7	is	be	AUX
ejde-792	20	8	constantly	constantly	ADV
ejde-792	20	9	equal	equal	ADJ
ejde-792	20	10	to	to	ADP
ejde-792	20	11	1	1	NUM
ejde-792	20	12	,	,	PUNCT
ejde-792	20	13	calabi	calabi	NOUN
ejde-792	21	1	[	[	X
ejde-792	21	2	9	9	NUM
ejde-792	21	3	]	]	PUNCT
ejde-792	21	4	explored	explore	VERB
ejde-792	21	5	the	the	DET
ejde-792	21	6	properties	property	NOUN
ejde-792	21	7	of	of	ADP
ejde-792	21	8	maximal	maximal	ADJ
ejde-792	21	9	surfaces	surface	NOUN
ejde-792	21	10	and	and	CCONJ
ejde-792	21	11	demonstrated	demonstrate	VERB
ejde-792	21	12	that	that	SCONJ
ejde-792	21	13	when	when	SCONJ
ejde-792	21	14	n	n	ADP
ejde-792	21	15	≤	≤	NUM
ejde-792	21	16	4	4	NUM
ejde-792	21	17	,	,	PUNCT
ejde-792	21	18	equation	equation	NOUN
ejde-792	21	19	(	(	PUNCT
ejde-792	21	20	1.2	1.2	NUM
ejde-792	21	21	)	)	PUNCT
ejde-792	21	22	allows	allow	VERB
ejde-792	21	23	only	only	ADV
ejde-792	21	24	linear	linear	ADJ
ejde-792	21	25	solutions	solution	NOUN
ejde-792	21	26	.	.	PUNCT
ejde-792	22	1	cheng	cheng	PROPN
ejde-792	22	2	and	and	CCONJ
ejde-792	22	3	yau	yau	PROPN
ejde-792	23	1	[	[	X
ejde-792	23	2	10	10	NUM
ejde-792	23	3	]	]	PUNCT
ejde-792	23	4	further	far	ADV
ejde-792	23	5	investigated	investigate	VERB
ejde-792	23	6	maximal	maximal	ADJ
ejde-792	23	7	surfaces	surface	NOUN
ejde-792	23	8	,	,	PUNCT
ejde-792	23	9	extending	extend	VERB
ejde-792	23	10	calabi	calabi	NOUN
ejde-792	23	11	’s	’s	PART
ejde-792	23	12	findings	finding	NOUN
ejde-792	23	13	to	to	ADP
ejde-792	23	14	all	all	DET
ejde-792	23	15	dimensions	dimension	NOUN
ejde-792	23	16	,	,	PUNCT
ejde-792	23	17	and	and	CCONJ
ejde-792	23	18	proposed	propose	VERB
ejde-792	23	19	the	the	DET
ejde-792	23	20	bernstein	bernstein	PROPN
ejde-792	23	21	theorem	theorem	PROPN
ejde-792	23	22	.	.	PROPN
ejde-792	24	1	for	for	ADP
ejde-792	24	2	cases	case	NOUN
ejde-792	24	3	in	in	ADP
ejde-792	24	4	which	which	PRON
ejde-792	24	5	f	f	PROPN
ejde-792	24	6	is	be	AUX
ejde-792	24	7	constantly	constantly	ADV
ejde-792	24	8	equal	equal	ADJ
ejde-792	24	9	to	to	ADP
ejde-792	24	10	1	1	NUM
ejde-792	24	11	,	,	PUNCT
ejde-792	24	12	treibergs	treibergs	X
ejde-792	24	13	[	[	X
ejde-792	24	14	24	24	NUM
ejde-792	24	15	]	]	PUNCT
ejde-792	24	16	provided	provide	VERB
ejde-792	24	17	significant	significant	ADJ
ejde-792	24	18	results	result	NOUN
ejde-792	24	19	for	for	ADP
ejde-792	24	20	entire	entire	ADJ
ejde-792	24	21	surfaces	surface	NOUN
ejde-792	24	22	with	with	ADP
ejde-792	24	23	a	a	DET
ejde-792	24	24	constant	constant	ADJ
ejde-792	24	25	mean	mean	NOUN
ejde-792	24	26	curvature	curvature	NOUN
ejde-792	24	27	.	.	PUNCT
ejde-792	25	1	for	for	ADP
ejde-792	25	2	cases	case	NOUN
ejde-792	25	3	in	in	ADP
ejde-792	25	4	which	which	PRON
ejde-792	25	5	f	f	PROPN
ejde-792	25	6	equals	equal	VERB
ejde-792	25	7	1	1	NUM
ejde-792	25	8	,	,	PUNCT
ejde-792	25	9	bartnik	bartnik	NOUN
ejde-792	25	10	and	and	CCONJ
ejde-792	25	11	simon	simon	PROPN
ejde-792	25	12	[	[	X
ejde-792	25	13	4	4	X
ejde-792	25	14	]	]	PUNCT
ejde-792	25	15	considered	consider	VERB
ejde-792	25	16	the	the	DET
ejde-792	25	17	dirichlet	dirichlet	PROPN
ejde-792	25	18	problem	problem	NOUN
ejde-792	25	19	for	for	ADP
ejde-792	25	20	equation	equation	NOUN
ejde-792	25	21	(	(	PUNCT
ejde-792	25	22	1.2	1.2	NUM
ejde-792	25	23	)	)	PUNCT
ejde-792	25	24	with	with	ADP
ejde-792	25	25	surfaces	surface	NOUN
ejde-792	25	26	of	of	ADP
ejde-792	25	27	bounded	bounded	ADJ
ejde-792	25	28	mean	mean	PROPN
ejde-792	25	29	curvature	curvature	NOUN
ejde-792	25	30	.	.	PUNCT
ejde-792	26	1	2020	2020	NUM
ejde-792	26	2	mathematics	mathematic	NOUN
ejde-792	26	3	subject	subject	ADJ
ejde-792	26	4	classification	classification	NOUN
ejde-792	26	5	.	.	PUNCT
ejde-792	27	1	35b32	35b32	NUM
ejde-792	27	2	,	,	PUNCT
ejde-792	27	3	35j93	35j93	NUM
ejde-792	27	4	,	,	PUNCT
ejde-792	27	5	35b35	35b35	NUM
ejde-792	27	6	.	.	PUNCT
ejde-792	28	1	key	key	ADJ
ejde-792	28	2	words	word	NOUN
ejde-792	28	3	and	and	CCONJ
ejde-792	28	4	phrases	phrase	NOUN
ejde-792	28	5	.	.	PUNCT
ejde-792	29	1	bifurcation	bifurcation	NOUN
ejde-792	29	2	;	;	PUNCT
ejde-792	29	3	mean	mean	VERB
ejde-792	29	4	curvature	curvature	NOUN
ejde-792	29	5	operator	operator	NOUN
ejde-792	29	6	;	;	PUNCT
ejde-792	29	7	stability	stability	NOUN
ejde-792	29	8	.	.	PUNCT
ejde-792	30	1	©	©	PROPN
ejde-792	30	2	2024	2024	NUM
ejde-792	30	3	.	.	PUNCT
ejde-792	31	1	this	this	DET
ejde-792	31	2	work	work	NOUN
ejde-792	31	3	is	be	AUX
ejde-792	31	4	licensed	license	VERB
ejde-792	31	5	under	under	ADP
ejde-792	31	6	a	a	DET
ejde-792	31	7	cc	cc	NOUN
ejde-792	31	8	by	by	ADP
ejde-792	31	9	4.0	4.0	NUM
ejde-792	31	10	license	license	NOUN
ejde-792	31	11	.	.	PUNCT
ejde-792	32	1	submitted	submit	VERB
ejde-792	32	2	july	july	PROPN
ejde-792	32	3	10	10	NUM
ejde-792	32	4	,	,	PUNCT
ejde-792	32	5	2024	2024	NUM
ejde-792	32	6	.	.	PUNCT
ejde-792	33	1	published	publish	VERB
ejde-792	33	2	august	august	PROPN
ejde-792	33	3	26	26	NUM
ejde-792	33	4	,	,	PUNCT
ejde-792	33	5	2024	2024	NUM
ejde-792	33	6	.	.	PUNCT
ejde-792	33	7	1	1	NUM
ejde-792	33	8	2	2	NUM
ejde-792	33	9	s.	s.	PROPN
ejde-792	33	10	gao	gao	PROPN
ejde-792	33	11	,	,	PUNCT
ejde-792	33	12	q.	q.	PROPN
ejde-792	33	13	liu	liu	PROPN
ejde-792	33	14	,	,	PUNCT
ejde-792	33	15	y.	y.	PROPN
ejde-792	33	16	sun	sun	PROPN
ejde-792	33	17	ejde-2024/48	ejde-2024/48	PROPN
ejde-792	33	18	the	the	DET
ejde-792	33	19	authors	author	NOUN
ejde-792	33	20	of	of	ADP
ejde-792	33	21	[	[	X
ejde-792	33	22	6	6	NUM
ejde-792	33	23	,	,	PUNCT
ejde-792	33	24	11	11	NUM
ejde-792	33	25	]	]	PUNCT
ejde-792	33	26	used	use	VERB
ejde-792	33	27	critical	critical	ADJ
ejde-792	33	28	point	point	NOUN
ejde-792	33	29	theory	theory	NOUN
ejde-792	33	30	and	and	CCONJ
ejde-792	33	31	topological	topological	ADJ
ejde-792	33	32	degree	degree	NOUN
ejde-792	33	33	arguments	argument	NOUN
ejde-792	33	34	to	to	PART
ejde-792	33	35	explore	explore	VERB
ejde-792	33	36	the	the	DET
ejde-792	33	37	nonexistence	nonexistence	NOUN
ejde-792	33	38	,	,	PUNCT
ejde-792	33	39	existence	existence	NOUN
ejde-792	33	40	,	,	PUNCT
ejde-792	33	41	and	and	CCONJ
ejde-792	33	42	multiplicity	multiplicity	NOUN
ejde-792	33	43	of	of	ADP
ejde-792	33	44	positive	positive	ADJ
ejde-792	33	45	solutions	solution	NOUN
ejde-792	33	46	for	for	ADP
ejde-792	33	47	f	f	PROPN
ejde-792	33	48	≡	≡	PROPN
ejde-792	33	49	1	1	NUM
ejde-792	33	50	in	in	ADP
ejde-792	33	51	bounded	bounded	ADJ
ejde-792	33	52	domains	domain	NOUN
ejde-792	33	53	.	.	PUNCT
ejde-792	34	1	in	in	ADP
ejde-792	34	2	[	[	X
ejde-792	34	3	13	13	NUM
ejde-792	34	4	]	]	PUNCT
ejde-792	34	5	,	,	PUNCT
ejde-792	34	6	the	the	DET
ejde-792	34	7	authors	author	NOUN
ejde-792	34	8	investigated	investigate	VERB
ejde-792	34	9	the	the	DET
ejde-792	34	10	nonexistence	nonexistence	NOUN
ejde-792	34	11	,	,	PUNCT
ejde-792	34	12	existence	existence	NOUN
ejde-792	34	13	,	,	PUNCT
ejde-792	34	14	and	and	CCONJ
ejde-792	34	15	multiplicity	multiplicity	NOUN
ejde-792	34	16	of	of	ADP
ejde-792	34	17	positive	positive	ADJ
ejde-792	34	18	radial	radial	ADJ
ejde-792	34	19	solutions	solution	NOUN
ejde-792	34	20	of	of	ADP
ejde-792	34	21	equation	equation	NOUN
ejde-792	34	22	(	(	PUNCT
ejde-792	34	23	1.2	1.2	NUM
ejde-792	34	24	)	)	PUNCT
ejde-792	34	25	with	with	ADP
ejde-792	34	26	nh	nh	PROPN
ejde-792	34	27	=	=	SYM
ejde-792	34	28	−λf(x	−λf(x	PROPN
ejde-792	34	29	,	,	PUNCT
ejde-792	34	30	s	s	PART
ejde-792	34	31	)	)	PUNCT
ejde-792	34	32	on	on	ADP
ejde-792	34	33	the	the	DET
ejde-792	34	34	unit	unit	NOUN
ejde-792	34	35	ball	ball	NOUN
ejde-792	34	36	via	via	ADP
ejde-792	34	37	the	the	DET
ejde-792	34	38	bifurcation	bifurcation	NOUN
ejde-792	34	39	method	method	NOUN
ejde-792	34	40	.	.	PUNCT
ejde-792	35	1	this	this	DET
ejde-792	35	2	work	work	NOUN
ejde-792	35	3	was	be	AUX
ejde-792	35	4	later	later	ADV
ejde-792	35	5	extended	extend	VERB
ejde-792	35	6	to	to	ADP
ejde-792	35	7	general	general	ADJ
ejde-792	35	8	domains	domain	NOUN
ejde-792	35	9	in	in	ADP
ejde-792	35	10	[	[	X
ejde-792	35	11	14	14	NUM
ejde-792	35	12	,	,	PUNCT
ejde-792	35	13	16	16	NUM
ejde-792	35	14	]	]	PUNCT
ejde-792	35	15	.	.	PUNCT
ejde-792	36	1	the	the	DET
ejde-792	36	2	author	author	NOUN
ejde-792	36	3	in	in	ADP
ejde-792	36	4	[	[	X
ejde-792	36	5	18	18	NUM
ejde-792	36	6	]	]	PUNCT
ejde-792	36	7	studied	study	VERB
ejde-792	36	8	the	the	DET
ejde-792	36	9	existence	existence	NOUN
ejde-792	36	10	and	and	CCONJ
ejde-792	36	11	uniqueness	uniqueness	NOUN
ejde-792	36	12	of	of	ADP
ejde-792	36	13	classical	classical	ADJ
ejde-792	36	14	solutions	solution	NOUN
ejde-792	36	15	,	,	PUNCT
ejde-792	36	16	the	the	DET
ejde-792	36	17	multiplicity	multiplicity	NOUN
ejde-792	36	18	of	of	ADP
ejde-792	36	19	strong	strong	ADJ
ejde-792	36	20	solutions	solution	NOUN
ejde-792	36	21	,	,	PUNCT
ejde-792	36	22	and	and	CCONJ
ejde-792	36	23	the	the	DET
ejde-792	36	24	symmetry	symmetry	NOUN
ejde-792	36	25	of	of	ADP
ejde-792	36	26	positive	positive	ADJ
ejde-792	36	27	solutions	solution	NOUN
ejde-792	36	28	.	.	PUNCT
ejde-792	37	1	the	the	DET
ejde-792	37	2	global	global	ADJ
ejde-792	37	3	structure	structure	NOUN
ejde-792	37	4	of	of	ADP
ejde-792	37	5	the	the	DET
ejde-792	37	6	positive	positive	ADJ
ejde-792	37	7	solutions	solution	NOUN
ejde-792	37	8	for	for	ADP
ejde-792	37	9	this	this	DET
ejde-792	37	10	problem	problem	NOUN
ejde-792	37	11	was	be	AUX
ejde-792	37	12	also	also	ADV
ejde-792	37	13	delineated	delineate	VERB
ejde-792	37	14	.	.	PUNCT
ejde-792	38	1	for	for	ADP
ejde-792	38	2	more	more	ADJ
ejde-792	38	3	research	research	NOUN
ejde-792	38	4	results	result	NOUN
ejde-792	38	5	on	on	ADP
ejde-792	38	6	the	the	DET
ejde-792	38	7	mean	mean	ADJ
ejde-792	38	8	curvature	curvature	NOUN
ejde-792	38	9	equation	equation	NOUN
ejde-792	38	10	,	,	PUNCT
ejde-792	38	11	see	see	VERB
ejde-792	38	12	references	reference	NOUN
ejde-792	38	13	[	[	X
ejde-792	38	14	7	7	NUM
ejde-792	38	15	,	,	PUNCT
ejde-792	38	16	8	8	NUM
ejde-792	38	17	,	,	PUNCT
ejde-792	38	18	17	17	NUM
ejde-792	38	19	,	,	PUNCT
ejde-792	38	20	15	15	NUM
ejde-792	38	21	]	]	PUNCT
ejde-792	38	22	and	and	CCONJ
ejde-792	38	23	their	their	PRON
ejde-792	38	24	cited	cite	VERB
ejde-792	38	25	literature	literature	NOUN
ejde-792	38	26	.	.	PUNCT
ejde-792	39	1	in	in	ADP
ejde-792	39	2	[	[	X
ejde-792	39	3	1	1	NUM
ejde-792	39	4	]	]	PUNCT
ejde-792	39	5	,	,	PUNCT
ejde-792	39	6	the	the	DET
ejde-792	39	7	stability	stability	NOUN
ejde-792	39	8	of	of	ADP
ejde-792	39	9	hypersurfaces	hypersurface	NOUN
ejde-792	39	10	with	with	ADP
ejde-792	39	11	a	a	DET
ejde-792	39	12	constant	constant	ADJ
ejde-792	39	13	mean	mean	NOUN
ejde-792	39	14	curvature	curvature	NOUN
ejde-792	39	15	was	be	AUX
ejde-792	39	16	studied	study	VERB
ejde-792	39	17	through	through	ADP
ejde-792	39	18	the	the	DET
ejde-792	39	19	calculus	calculus	NOUN
ejde-792	39	20	of	of	ADP
ejde-792	39	21	variations	variation	NOUN
ejde-792	39	22	.	.	PUNCT
ejde-792	40	1	the	the	DET
ejde-792	40	2	relationship	relationship	NOUN
ejde-792	40	3	between	between	ADP
ejde-792	40	4	stability	stability	NOUN
ejde-792	40	5	and	and	CCONJ
ejde-792	40	6	constant	constant	ADJ
ejde-792	40	7	mean	mean	ADJ
ejde-792	40	8	curvature	curvature	NOUN
ejde-792	40	9	was	be	AUX
ejde-792	40	10	presented	present	VERB
ejde-792	40	11	under	under	ADP
ejde-792	40	12	the	the	DET
ejde-792	40	13	condition	condition	NOUN
ejde-792	40	14	that	that	SCONJ
ejde-792	40	15	the	the	DET
ejde-792	40	16	hypersurface	hypersurface	NOUN
ejde-792	40	17	is	be	AUX
ejde-792	40	18	compact	compact	ADJ
ejde-792	40	19	.	.	PUNCT
ejde-792	41	1	in	in	ADP
ejde-792	41	2	[	[	X
ejde-792	41	3	2	2	NUM
ejde-792	41	4	]	]	PUNCT
ejde-792	41	5	,	,	PUNCT
ejde-792	41	6	the	the	DET
ejde-792	41	7	stability	stability	NOUN
ejde-792	41	8	of	of	ADP
ejde-792	41	9	hypersurfaces	hypersurface	NOUN
ejde-792	41	10	with	with	ADP
ejde-792	41	11	constant	constant	ADJ
ejde-792	41	12	mean	mean	ADJ
ejde-792	41	13	curvature	curvature	NOUN
ejde-792	41	14	in	in	ADP
ejde-792	41	15	riemannian	riemannian	ADJ
ejde-792	41	16	manifolds	manifold	NOUN
ejde-792	41	17	was	be	AUX
ejde-792	41	18	studied	study	VERB
ejde-792	41	19	.	.	PUNCT
ejde-792	42	1	barros	barros	PROPN
ejde-792	42	2	,	,	PUNCT
ejde-792	42	3	brasil	brasil	PROPN
ejde-792	42	4	,	,	PUNCT
ejde-792	42	5	and	and	CCONJ
ejde-792	42	6	caminha	caminha	VERB
ejde-792	42	7	[	[	X
ejde-792	42	8	3	3	NUM
ejde-792	42	9	]	]	PUNCT
ejde-792	42	10	investigated	investigate	VERB
ejde-792	42	11	stability	stability	NOUN
ejde-792	42	12	issues	issue	NOUN
ejde-792	42	13	concerning	concern	VERB
ejde-792	42	14	the	the	DET
ejde-792	42	15	generalized	generalized	ADJ
ejde-792	42	16	robertson	robertson	PROPN
ejde-792	42	17	-	-	PUNCT
ejde-792	42	18	walker	walker	PROPN
ejde-792	42	19	spacetime	spacetime	PROPN
ejde-792	42	20	.	.	PUNCT
ejde-792	43	1	in	in	ADP
ejde-792	43	2	this	this	DET
ejde-792	43	3	work	work	NOUN
ejde-792	43	4	,	,	PUNCT
ejde-792	43	5	we	we	PRON
ejde-792	43	6	investigate	investigate	VERB
ejde-792	43	7	the	the	DET
ejde-792	43	8	local	local	ADJ
ejde-792	43	9	bifurcation	bifurcation	NOUN
ejde-792	43	10	structure	structure	NOUN
ejde-792	43	11	and	and	CCONJ
ejde-792	43	12	stability	stability	NOUN
ejde-792	43	13	of	of	ADP
ejde-792	43	14	the	the	DET
ejde-792	43	15	mean	mean	ADJ
ejde-792	43	16	curvature	curvature	NOUN
ejde-792	43	17	equation	equation	NOUN
ejde-792	43	18	in	in	ADP
ejde-792	43	19	static	static	ADJ
ejde-792	43	20	the	the	DET
ejde-792	43	21	spacetime	spacetime	NOUN
ejde-792	43	22	.	.	PUNCT
ejde-792	44	1	we	we	PRON
ejde-792	44	2	consider	consider	VERB
ejde-792	44	3	the	the	DET
ejde-792	44	4	following	follow	VERB
ejde-792	44	5	0	0	NUM
ejde-792	44	6	-	-	PUNCT
ejde-792	44	7	dirichlet	dirichlet	PROPN
ejde-792	44	8	problem	problem	NOUN
ejde-792	44	9	involving	involve	VERB
ejde-792	44	10	the	the	DET
ejde-792	44	11	mean	mean	ADJ
ejde-792	44	12	curvature	curvature	NOUN
ejde-792	44	13	operator	operator	NOUN
ejde-792	44	14	in	in	ADP
ejde-792	44	15	minkowski	minkowski	ADJ
ejde-792	44	16	space	space	NOUN
ejde-792	44	17	:	:	PUNCT
ejde-792	44	18	−div	−div	NOUN
ejde-792	44	19	(	(	PUNCT
ejde-792	44	20	f2(x)∇u√	f2(x)∇u√	NOUN
ejde-792	44	21	1−	1−	NUM
ejde-792	44	22	f2(x)|∇u|2	f2(x)|∇u|2	PROPN
ejde-792	44	23	)	)	PUNCT
ejde-792	44	24	=	=	SYM
ejde-792	44	25	−λnf(x)h(x	−λnf(x)h(x	PROPN
ejde-792	44	26	,	,	PUNCT
ejde-792	44	27	u	u	NOUN
ejde-792	44	28	)	)	PUNCT
ejde-792	44	29	in	in	ADP
ejde-792	44	30	ω	ω	PROPN
ejde-792	44	31	,	,	PUNCT
ejde-792	44	32	u	u	NOUN
ejde-792	44	33	=	=	NOUN
ejde-792	44	34	0	0	NUM
ejde-792	44	35	on	on	ADP
ejde-792	44	36	∂ω	∂ω	PROPN
ejde-792	44	37	.	.	PUNCT
ejde-792	45	1	(	(	PUNCT
ejde-792	45	2	1.1	1.1	NUM
ejde-792	45	3	)	)	PUNCT
ejde-792	45	4	here	here	ADV
ejde-792	45	5	,	,	PUNCT
ejde-792	45	6	λ	λ	PROPN
ejde-792	45	7	is	be	AUX
ejde-792	45	8	a	a	DET
ejde-792	45	9	nonnegative	nonnegative	ADJ
ejde-792	45	10	parameter	parameter	NOUN
ejde-792	45	11	representing	represent	VERB
ejde-792	45	12	the	the	DET
ejde-792	45	13	strength	strength	NOUN
ejde-792	45	14	of	of	ADP
ejde-792	45	15	the	the	DET
ejde-792	45	16	mean	mean	ADJ
ejde-792	45	17	curvature	curvature	NOUN
ejde-792	45	18	function	function	NOUN
ejde-792	45	19	,	,	PUNCT
ejde-792	45	20	the	the	DET
ejde-792	45	21	real	real	ADV
ejde-792	45	22	-	-	PUNCT
ejde-792	45	23	valued	value	VERB
ejde-792	45	24	function	function	NOUN
ejde-792	45	25	h	h	NOUN
ejde-792	45	26	gives	give	VERB
ejde-792	45	27	the	the	DET
ejde-792	45	28	mean	mean	ADJ
ejde-792	45	29	curvature	curvature	NOUN
ejde-792	45	30	,	,	PUNCT
ejde-792	45	31	ω	ω	PROPN
ejde-792	45	32	is	be	AUX
ejde-792	45	33	a	a	DET
ejde-792	45	34	c2,α	c2,α	NOUN
ejde-792	45	35	bounded	bound	VERB
ejde-792	45	36	domain	domain	NOUN
ejde-792	45	37	in	in	ADP
ejde-792	45	38	rn	rn	PROPN
ejde-792	45	39	with	with	ADP
ejde-792	45	40	n	n	PRON
ejde-792	45	41	≥	≥	NOUN
ejde-792	45	42	1	1	NUM
ejde-792	45	43	for	for	ADP
ejde-792	45	44	some	some	DET
ejde-792	45	45	α	α	NOUN
ejde-792	45	46	>	>	X
ejde-792	45	47	0	0	NUM
ejde-792	45	48	,	,	PUNCT
ejde-792	45	49	and	and	CCONJ
ejde-792	45	50	f	f	PROPN
ejde-792	45	51	∈	∈	PROPN
ejde-792	46	1	c0,α(ω	c0,α(ω	PRON
ejde-792	46	2	×	×	NOUN
ejde-792	47	1	[	[	X
ejde-792	47	2	−d	−d	ADJ
ejde-792	47	3	,	,	PUNCT
ejde-792	47	4	d	d	X
ejde-792	47	5	]	]	X
ejde-792	47	6	)	)	PUNCT
ejde-792	47	7	,	,	PUNCT
ejde-792	47	8	where	where	SCONJ
ejde-792	47	9	d	d	NOUN
ejde-792	47	10	is	be	AUX
ejde-792	47	11	the	the	DET
ejde-792	47	12	diameter	diameter	NOUN
ejde-792	47	13	of	of	ADP
ejde-792	47	14	ω	ω	PROPN
ejde-792	47	15	.	.	PUNCT
ejde-792	48	1	using	use	VERB
ejde-792	48	2	the	the	DET
ejde-792	48	3	equation	equation	NOUN
ejde-792	48	4	div	div	X
ejde-792	48	5	(	(	PUNCT
ejde-792	48	6	f(x)∇u√	f(x)∇u√	X
ejde-792	48	7	1−	1−	NUM
ejde-792	48	8	f2(x)|∇u|2	f2(x)|∇u|2	PROPN
ejde-792	48	9	)	)	PUNCT
ejde-792	49	1	+	+	CCONJ
ejde-792	49	2	∇u∇f(x)√	∇u∇f(x)√	NOUN
ejde-792	49	3	1−	1−	NUM
ejde-792	49	4	f2(x)|∇u|2	f2(x)|∇u|2	PROPN
ejde-792	49	5	=	=	PROPN
ejde-792	49	6	nh	nh	PROPN
ejde-792	49	7	,	,	PUNCT
ejde-792	49	8	(	(	PUNCT
ejde-792	49	9	1.2	1.2	NUM
ejde-792	49	10	)	)	PUNCT
ejde-792	49	11	we	we	PRON
ejde-792	49	12	can	can	AUX
ejde-792	49	13	derive	derive	VERB
ejde-792	49	14	that	that	SCONJ
ejde-792	49	15	−	−	PROPN
ejde-792	49	16	div	div	X
ejde-792	49	17	(	(	PUNCT
ejde-792	49	18	f2(x)∇u√	f2(x)∇u√	NOUN
ejde-792	49	19	1−	1−	NUM
ejde-792	49	20	f2(x)|∇u|2	f2(x)|∇u|2	PROPN
ejde-792	49	21	)	)	PUNCT
ejde-792	50	1	=	=	SYM
ejde-792	50	2	−div	−div	NOUN
ejde-792	50	3	(	(	PUNCT
ejde-792	50	4	f(x	f(x	PROPN
ejde-792	50	5	)	)	PUNCT
ejde-792	50	6	·	·	PUNCT
ejde-792	50	7	f(x)∇u√	f(x)∇u√	X
ejde-792	50	8	1−	1−	NUM
ejde-792	50	9	f2(x)|∇u|2	f2(x)|∇u|2	NUM
ejde-792	50	10	)	)	PUNCT
ejde-792	50	11	=	=	SYM
ejde-792	50	12	−f(x	−f(x	PROPN
ejde-792	50	13	)	)	PUNCT
ejde-792	50	14	div	div	X
ejde-792	50	15	(	(	PUNCT
ejde-792	50	16	f(x)∇u√	f(x)∇u√	X
ejde-792	50	17	1−	1−	NUM
ejde-792	50	18	f2(x)|∇u|2	f2(x)|∇u|2	NUM
ejde-792	50	19	)	)	PUNCT
ejde-792	51	1	−	−	PROPN
ejde-792	51	2	f(x	f(x	PROPN
ejde-792	51	3	)	)	PUNCT
ejde-792	51	4	∇u∇f(x)√	∇u∇f(x)√	NOUN
ejde-792	51	5	1−	1−	NUM
ejde-792	51	6	f2(x)|∇u|2	f2(x)|∇u|2	PROPN
ejde-792	51	7	=	=	SYM
ejde-792	51	8	−nf(x)h	−nf(x)h	PROPN
ejde-792	51	9	.	.	PUNCT
ejde-792	52	1	from	from	ADP
ejde-792	52	2	[	[	X
ejde-792	52	3	18	18	NUM
ejde-792	52	4	]	]	PUNCT
ejde-792	52	5	,	,	PUNCT
ejde-792	52	6	we	we	PRON
ejde-792	52	7	have	have	VERB
ejde-792	52	8	div	div	X
ejde-792	52	9	(	(	PUNCT
ejde-792	52	10	f2(x)∇u√	f2(x)∇u√	NOUN
ejde-792	52	11	1−	1−	NUM
ejde-792	52	12	f2(x)|∇u|2	f2(x)|∇u|2	PROPN
ejde-792	52	13	)	)	PUNCT
ejde-792	53	1	=	=	SYM
ejde-792	54	1	nf(x)h	nf(x)h	PROPN
ejde-792	54	2	.	.	PUNCT
ejde-792	55	1	this	this	DET
ejde-792	55	2	equation	equation	NOUN
ejde-792	55	3	is	be	AUX
ejde-792	55	4	equivalent	equivalent	ADJ
ejde-792	55	5	to	to	ADP
ejde-792	55	6	(	(	PUNCT
ejde-792	55	7	1.2	1.2	NUM
ejde-792	55	8	)	)	PUNCT
ejde-792	55	9	.	.	PUNCT
ejde-792	56	1	next	next	ADV
ejde-792	56	2	,	,	PUNCT
ejde-792	56	3	we	we	PRON
ejde-792	56	4	present	present	VERB
ejde-792	56	5	the	the	DET
ejde-792	56	6	main	main	ADJ
ejde-792	56	7	theorem	theorem	NOUN
ejde-792	56	8	.	.	PUNCT
ejde-792	56	9	theorem	theorem	VERB
ejde-792	56	10	1.1	1.1	NUM
ejde-792	56	11	.	.	PUNCT
ejde-792	57	1	suppose	suppose	VERB
ejde-792	57	2	that	that	SCONJ
ejde-792	57	3	h	h	PROPN
ejde-792	57	4	is	be	AUX
ejde-792	57	5	c3	c3	NOUN
ejde-792	57	6	with	with	ADP
ejde-792	57	7	respect	respect	NOUN
ejde-792	57	8	to	to	ADP
ejde-792	57	9	its	its	PRON
ejde-792	57	10	second	second	ADJ
ejde-792	57	11	argument	argument	NOUN
ejde-792	57	12	and	and	CCONJ
ejde-792	57	13	that	that	SCONJ
ejde-792	57	14	h0	h0	NOUN
ejde-792	57	15	∈	∈	PROPN
ejde-792	57	16	(	(	PUNCT
ejde-792	57	17	0,+∞	0,+∞	NUM
ejde-792	57	18	)	)	PUNCT
ejde-792	57	19	.	.	PUNCT
ejde-792	58	1	then	then	ADV
ejde-792	58	2	,	,	PUNCT
ejde-792	58	3	all	all	DET
ejde-792	58	4	the	the	DET
ejde-792	58	5	solutions	solution	NOUN
ejde-792	58	6	of	of	ADP
ejde-792	58	7	problem	problem	NOUN
ejde-792	58	8	(	(	PUNCT
ejde-792	58	9	1.1	1.1	NUM
ejde-792	58	10	)	)	PUNCT
ejde-792	58	11	near	near	ADP
ejde-792	58	12	(	(	PUNCT
ejde-792	58	13	λ1	λ1	PROPN
ejde-792	58	14	/	/	SYM
ejde-792	58	15	h0	h0	PROPN
ejde-792	58	16	,	,	PUNCT
ejde-792	58	17	0	0	NUM
ejde-792	58	18	)	)	PUNCT
ejde-792	58	19	can	can	AUX
ejde-792	58	20	be	be	AUX
ejde-792	58	21	ejde-2024/48	ejde-2024/48	ADJ
ejde-792	58	22	local	local	ADJ
ejde-792	58	23	bifurcation	bifurcation	NOUN
ejde-792	58	24	structure	structure	NOUN
ejde-792	58	25	and	and	CCONJ
ejde-792	58	26	stability	stability	NOUN
ejde-792	58	27	3	3	NUM
ejde-792	58	28	expressed	express	VERB
ejde-792	58	29	as	as	ADP
ejde-792	58	30	(	(	PUNCT
ejde-792	58	31	λ(s	λ(s	PROPN
ejde-792	58	32	)	)	PUNCT
ejde-792	58	33	,	,	PUNCT
ejde-792	58	34	sφ1	sφ1	PROPN
ejde-792	58	35	+	+	X
ejde-792	58	36	sz(s	sz(s	NUM
ejde-792	58	37	)	)	PUNCT
ejde-792	58	38	)	)	PUNCT
ejde-792	58	39	for	for	ADP
ejde-792	58	40	s	s	PRON
ejde-792	58	41	in	in	ADP
ejde-792	58	42	an	an	DET
ejde-792	58	43	open	open	ADJ
ejde-792	58	44	interval	interval	NOUN
ejde-792	58	45	(	(	PUNCT
ejde-792	58	46	−δ	−δ	ADJ
ejde-792	58	47	,	,	PUNCT
ejde-792	58	48	δ	δ	PROPN
ejde-792	58	49	)	)	PUNCT
ejde-792	58	50	,	,	PUNCT
ejde-792	58	51	where	where	SCONJ
ejde-792	58	52	δ	δ	PROPN
ejde-792	58	53	>	>	X
ejde-792	58	54	0	0	PROPN
ejde-792	58	55	,	,	PUNCT
ejde-792	58	56	such	such	ADJ
ejde-792	58	57	that	that	DET
ejde-792	58	58	λ(0	λ(0	NOUN
ejde-792	58	59	)	)	PUNCT
ejde-792	59	1	=	=	PUNCT
ejde-792	59	2	λ1	λ1	ADJ
ejde-792	59	3	/	/	SYM
ejde-792	59	4	h0	h0	PROPN
ejde-792	59	5	and	and	CCONJ
ejde-792	59	6	λ′(0	λ′(0	NOUN
ejde-792	59	7	)	)	PUNCT
ejde-792	59	8	=	=	SYM
ejde-792	60	1	−	−	PROPN
ejde-792	60	2	−	−	PROPN
ejde-792	60	3	λ1	λ1	PROPN
ejde-792	60	4	h0	h0	PROPN
ejde-792	60	5	n	n	PROPN
ejde-792	60	6	∫	∫	PROPN
ejde-792	60	7	ω	ω	NUM
ejde-792	60	8	f(x)huu(x	f(x)huu(x	PROPN
ejde-792	60	9	,	,	PUNCT
ejde-792	60	10	0)φ	0)φ	NOUN
ejde-792	60	11	3	3	NUM
ejde-792	60	12	1	1	NUM
ejde-792	60	13	dx	dx	PROPN
ejde-792	60	14	2h0	2h0	NUM
ejde-792	60	15	∫	∫	PROPN
ejde-792	60	16	ω	ω	PROPN
ejde-792	60	17	f(x)φ2	f(x)φ2	PROPN
ejde-792	60	18	1	1	NUM
ejde-792	60	19	dx	dx	PROPN
ejde-792	60	20	.	.	PUNCT
ejde-792	61	1	here	here	ADV
ejde-792	61	2	,	,	PUNCT
ejde-792	61	3	z	z	NOUN
ejde-792	61	4	:	:	PUNCT
ejde-792	61	5	(	(	PUNCT
ejde-792	61	6	−δ	−δ	ADJ
ejde-792	61	7	,	,	PUNCT
ejde-792	61	8	δ	δ	PROPN
ejde-792	61	9	)	)	PUNCT
ejde-792	61	10	→	→	SYM
ejde-792	61	11	z	z	NOUN
ejde-792	61	12	is	be	AUX
ejde-792	61	13	a	a	DET
ejde-792	61	14	c2	c2	PROPN
ejde-792	61	15	function	function	NOUN
ejde-792	61	16	that	that	SCONJ
ejde-792	61	17	satisfies	satisfy	VERB
ejde-792	61	18	z(0	z(0	NOUN
ejde-792	61	19	)	)	PUNCT
ejde-792	61	20	=	=	SYM
ejde-792	62	1	0	0	X
ejde-792	62	2	.	.	PUNCT
ejde-792	63	1	additionally	additionally	ADV
ejde-792	63	2	,	,	PUNCT
ejde-792	63	3	if	if	SCONJ
ejde-792	63	4	huu(x	huu(x	PROPN
ejde-792	63	5	,	,	PUNCT
ejde-792	63	6	0	0	NUM
ejde-792	63	7	)	)	PUNCT
ejde-792	63	8	is	be	AUX
ejde-792	63	9	exactly	exactly	ADV
ejde-792	63	10	zero	zero	NUM
ejde-792	63	11	on	on	ADP
ejde-792	63	12	ω	ω	NUM
ejde-792	63	13	,	,	PUNCT
ejde-792	63	14	then	then	ADV
ejde-792	63	15	we	we	PRON
ejde-792	63	16	obtain	obtain	VERB
ejde-792	63	17	λ′′(0	λ′′(0	NOUN
ejde-792	63	18	)	)	PUNCT
ejde-792	64	1	=	=	SYM
ejde-792	65	1	−	−	PROPN
ejde-792	65	2	−	−	PROPN
ejde-792	65	3	λ1	λ1	PROPN
ejde-792	65	4	h0	h0	PROPN
ejde-792	65	5	n	n	PROPN
ejde-792	65	6	∫	∫	PROPN
ejde-792	65	7	ω	ω	NUM
ejde-792	65	8	f(x)huuu(x	f(x)huuu(x	PROPN
ejde-792	65	9	,	,	PUNCT
ejde-792	65	10	0)φ	0)φ	NOUN
ejde-792	65	11	4	4	NUM
ejde-792	65	12	1dx	1dx	NOUN
ejde-792	65	13	3h0	3h0	NUM
ejde-792	65	14	∫	∫	PROPN
ejde-792	65	15	ω	ω	PROPN
ejde-792	65	16	f(x)φ2	f(x)φ2	PROPN
ejde-792	65	17	1dx	1dx	NOUN
ejde-792	65	18	,	,	PUNCT
ejde-792	65	19	where	where	SCONJ
ejde-792	65	20	huuu(x	huuu(x	ADP
ejde-792	65	21	,	,	PUNCT
ejde-792	65	22	0	0	NUM
ejde-792	65	23	)	)	PUNCT
ejde-792	65	24	denotes	denote	VERB
ejde-792	65	25	the	the	DET
ejde-792	65	26	third	third	ADJ
ejde-792	65	27	derivative	derivative	NOUN
ejde-792	65	28	of	of	ADP
ejde-792	65	29	h	h	NOUN
ejde-792	65	30	with	with	ADP
ejde-792	65	31	respect	respect	NOUN
ejde-792	65	32	to	to	ADP
ejde-792	65	33	its	its	PRON
ejde-792	65	34	second	second	ADJ
ejde-792	65	35	variable	variable	NOUN
ejde-792	65	36	at	at	ADP
ejde-792	65	37	0	0	NUM
ejde-792	65	38	.	.	PUNCT
ejde-792	66	1	by	by	ADP
ejde-792	66	2	theorem	theorem	NOUN
ejde-792	66	3	1.1	1.1	NUM
ejde-792	66	4	,	,	PUNCT
ejde-792	66	5	we	we	PRON
ejde-792	66	6	can	can	AUX
ejde-792	66	7	deduce	deduce	VERB
ejde-792	66	8	the	the	DET
ejde-792	66	9	following	follow	VERB
ejde-792	66	10	stability	stability	NOUN
ejde-792	66	11	result	result	NOUN
ejde-792	66	12	.	.	PUNCT
ejde-792	67	1	theorem	theorem	VERB
ejde-792	67	2	1.2	1.2	NUM
ejde-792	67	3	.	.	PUNCT
ejde-792	68	1	let	let	VERB
ejde-792	68	2	λ1	λ1	PROPN
ejde-792	68	3	/	/	SYM
ejde-792	68	4	h0	h0	NOUN
ejde-792	68	5	be	be	AUX
ejde-792	68	6	a	a	DET
ejde-792	68	7	bifurcation	bifurcation	NOUN
ejde-792	68	8	point	point	NOUN
ejde-792	68	9	for	for	ADP
ejde-792	68	10	the	the	DET
ejde-792	68	11	equation	equation	NOUN
ejde-792	68	12	f	f	X
ejde-792	68	13	(	(	PUNCT
ejde-792	68	14	λ	λ	PROPN
ejde-792	68	15	,	,	PUNCT
ejde-792	68	16	u	u	NOUN
ejde-792	68	17	)	)	PUNCT
ejde-792	68	18	=	=	SYM
ejde-792	68	19	0	0	NUM
ejde-792	68	20	in	in	ADP
ejde-792	68	21	a	a	DET
ejde-792	68	22	banach	banach	NOUN
ejde-792	68	23	space	space	NOUN
ejde-792	68	24	x	x	PUNCT
ejde-792	68	25	,	,	PUNCT
ejde-792	68	26	and	and	CCONJ
ejde-792	68	27	assume	assume	VERB
ejde-792	68	28	that	that	SCONJ
ejde-792	68	29	0	0	NUM
ejde-792	68	30	is	be	AUX
ejde-792	68	31	a	a	DET
ejde-792	68	32	simple	simple	ADJ
ejde-792	68	33	eigenvalue	eigenvalue	NOUN
ejde-792	68	34	of	of	ADP
ejde-792	68	35	the	the	DET
ejde-792	68	36	linearized	linearize	VERB
ejde-792	68	37	operator	operator	NOUN
ejde-792	68	38	fu(λ1	fu(λ1	NOUN
ejde-792	68	39	/	/	SYM
ejde-792	68	40	h0	h0	PROPN
ejde-792	68	41	,	,	PUNCT
ejde-792	68	42	0	0	NUM
ejde-792	68	43	)	)	PUNCT
ejde-792	68	44	.	.	PUNCT
ejde-792	69	1	suppose	suppose	VERB
ejde-792	69	2	further	far	ADV
ejde-792	69	3	that	that	PRON
ejde-792	69	4	huu(x	huu(x	VERB
ejde-792	69	5	,	,	PUNCT
ejde-792	69	6	0	0	NUM
ejde-792	69	7	)	)	PUNCT
ejde-792	69	8	≡	≡	PROPN
ejde-792	69	9	0	0	NUM
ejde-792	70	1	in	in	ADP
ejde-792	70	2	ω	ω	PROPN
ejde-792	70	3	and	and	CCONJ
ejde-792	70	4	that	that	PRON
ejde-792	70	5	huuu(x	huuu(x	ADV
ejde-792	70	6	,	,	PUNCT
ejde-792	70	7	0	0	NUM
ejde-792	70	8	)	)	PUNCT
ejde-792	70	9	̸=	̸=	NOUN
ejde-792	70	10	0	0	NUM
ejde-792	70	11	in	in	ADP
ejde-792	70	12	ω	ω	PROPN
ejde-792	70	13	.	.	PUNCT
ejde-792	71	1	then	then	ADV
ejde-792	71	2	,	,	PUNCT
ejde-792	71	3	the	the	DET
ejde-792	71	4	stability	stability	NOUN
ejde-792	71	5	of	of	ADP
ejde-792	71	6	the	the	DET
ejde-792	71	7	solutions	solution	NOUN
ejde-792	71	8	u(s	u(s	ADJ
ejde-792	71	9	)	)	PUNCT
ejde-792	71	10	near	near	ADP
ejde-792	71	11	the	the	DET
ejde-792	71	12	bifurcation	bifurcation	NOUN
ejde-792	71	13	point	point	NOUN
ejde-792	71	14	(	(	PUNCT
ejde-792	71	15	λ1	λ1	PROPN
ejde-792	71	16	/	/	SYM
ejde-792	71	17	h0	h0	PROPN
ejde-792	71	18	,	,	PUNCT
ejde-792	71	19	0	0	NUM
ejde-792	71	20	)	)	PUNCT
ejde-792	71	21	is	be	AUX
ejde-792	71	22	determined	determine	VERB
ejde-792	71	23	as	as	SCONJ
ejde-792	71	24	follows	follow	VERB
ejde-792	71	25	:	:	PUNCT
ejde-792	71	26	(	(	PUNCT
ejde-792	71	27	1	1	X
ejde-792	71	28	)	)	PUNCT
ejde-792	71	29	if	if	SCONJ
ejde-792	71	30	huuu(x	huuu(x	PROPN
ejde-792	71	31	,	,	PUNCT
ejde-792	71	32	0	0	NUM
ejde-792	71	33	)	)	PUNCT
ejde-792	71	34	>	>	X
ejde-792	71	35	0	0	PUNCT
ejde-792	72	1	in	in	ADP
ejde-792	72	2	ω	ω	PROPN
ejde-792	72	3	,	,	PUNCT
ejde-792	72	4	then	then	ADV
ejde-792	72	5	the	the	DET
ejde-792	72	6	solutions	solution	NOUN
ejde-792	72	7	are	be	AUX
ejde-792	72	8	asymptotically	asymptotically	ADV
ejde-792	72	9	linearly	linearly	ADV
ejde-792	72	10	stable	stable	ADJ
ejde-792	72	11	,	,	PUNCT
ejde-792	72	12	i.e.	i.e.	X
ejde-792	72	13	,	,	PUNCT
ejde-792	72	14	λ′′(0	λ′′(0	NOUN
ejde-792	72	15	)	)	PUNCT
ejde-792	72	16	>	>	X
ejde-792	73	1	0	0	X
ejde-792	73	2	.	.	PUNCT
ejde-792	74	1	(	(	PUNCT
ejde-792	74	2	2	2	X
ejde-792	74	3	)	)	PUNCT
ejde-792	74	4	if	if	SCONJ
ejde-792	74	5	huuu(x	huuu(x	PROPN
ejde-792	74	6	,	,	PUNCT
ejde-792	74	7	0	0	NUM
ejde-792	74	8	)	)	PUNCT
ejde-792	74	9	<	<	X
ejde-792	74	10	0	0	PUNCT
ejde-792	75	1	in	in	ADP
ejde-792	75	2	ω	ω	PROPN
ejde-792	75	3	,	,	PUNCT
ejde-792	75	4	then	then	ADV
ejde-792	75	5	the	the	DET
ejde-792	75	6	solutions	solution	NOUN
ejde-792	75	7	are	be	AUX
ejde-792	75	8	asymptotically	asymptotically	ADV
ejde-792	75	9	linearly	linearly	ADV
ejde-792	75	10	unstable	unstable	ADJ
ejde-792	75	11	,	,	PUNCT
ejde-792	75	12	i.e.	i.e.	X
ejde-792	75	13	,	,	PUNCT
ejde-792	75	14	λ′′(0	λ′′(0	NOUN
ejde-792	75	15	)	)	PUNCT
ejde-792	75	16	<	<	X
ejde-792	75	17	0	0	X
ejde-792	75	18	.	.	PUNCT
ejde-792	76	1	this	this	DET
ejde-792	76	2	article	article	NOUN
ejde-792	76	3	is	be	AUX
ejde-792	76	4	organized	organize	VERB
ejde-792	76	5	as	as	SCONJ
ejde-792	76	6	follows	follow	VERB
ejde-792	76	7	.	.	PUNCT
ejde-792	77	1	section	section	NOUN
ejde-792	77	2	2	2	NUM
ejde-792	77	3	discusses	discuss	VERB
ejde-792	77	4	the	the	DET
ejde-792	77	5	local	local	ADJ
ejde-792	77	6	bifurcation	bifurcation	NOUN
ejde-792	77	7	structure	structure	NOUN
ejde-792	77	8	of	of	ADP
ejde-792	77	9	the	the	DET
ejde-792	77	10	solution	solution	NOUN
ejde-792	77	11	set	set	VERB
ejde-792	77	12	of	of	ADP
ejde-792	77	13	equation	equation	NOUN
ejde-792	77	14	(	(	PUNCT
ejde-792	77	15	1.1	1.1	NUM
ejde-792	77	16	)	)	PUNCT
ejde-792	77	17	.	.	PUNCT
ejde-792	78	1	section	section	NOUN
ejde-792	78	2	3	3	NUM
ejde-792	78	3	presents	present	VERB
ejde-792	78	4	the	the	DET
ejde-792	78	5	stability	stability	NOUN
ejde-792	78	6	results	result	VERB
ejde-792	78	7	near	near	ADP
ejde-792	78	8	the	the	DET
ejde-792	78	9	bifurcation	bifurcation	NOUN
ejde-792	78	10	point	point	NOUN
ejde-792	78	11	.	.	PUNCT
ejde-792	79	1	2	2	X
ejde-792	79	2	.	.	X
ejde-792	79	3	local	local	ADJ
ejde-792	79	4	bifurcation	bifurcation	NOUN
ejde-792	79	5	structure	structure	NOUN
ejde-792	79	6	in	in	ADP
ejde-792	79	7	this	this	DET
ejde-792	79	8	section	section	NOUN
ejde-792	79	9	,	,	PUNCT
ejde-792	79	10	we	we	PRON
ejde-792	79	11	provide	provide	VERB
ejde-792	79	12	the	the	DET
ejde-792	79	13	proof	proof	NOUN
ejde-792	79	14	of	of	ADP
ejde-792	79	15	theorem	theorem	ADJ
ejde-792	79	16	1.1	1.1	NUM
ejde-792	79	17	.	.	PUNCT
ejde-792	80	1	consider	consider	VERB
ejde-792	80	2	the	the	DET
ejde-792	80	3	set	set	NOUN
ejde-792	80	4	x	x	PUNCT
ejde-792	80	5	defined	define	VERB
ejde-792	80	6	as	as	ADP
ejde-792	80	7	x	x	X
ejde-792	80	8	=	=	PRON
ejde-792	80	9	{	{	PUNCT
ejde-792	80	10	u	u	NOUN
ejde-792	80	11	∈	∈	PROPN
ejde-792	80	12	c1(ω	c1(ω	PROPN
ejde-792	80	13	)	)	PUNCT
ejde-792	80	14	:	:	PUNCT
ejde-792	81	1	u	u	NOUN
ejde-792	81	2	=	=	NOUN
ejde-792	81	3	0	0	NUM
ejde-792	81	4	on	on	ADP
ejde-792	81	5	∂ω	∂ω	ADJ
ejde-792	81	6	}	}	PUNCT
ejde-792	81	7	with	with	ADP
ejde-792	81	8	the	the	DET
ejde-792	81	9	norm	norm	NOUN
ejde-792	81	10	∥u∥	∥u∥	NOUN
ejde-792	81	11	:	:	PUNCT
ejde-792	81	12	=	=	SYM
ejde-792	81	13	∥f(x)∇u∥∞.	∥f(x)∇u∥∞.	NOUN
ejde-792	81	14	let	let	VERB
ejde-792	81	15	φ1	φ1	PRON
ejde-792	81	16	be	be	AUX
ejde-792	81	17	a	a	DET
ejde-792	81	18	positive	positive	ADJ
ejde-792	81	19	eigenfunction	eigenfunction	NOUN
ejde-792	81	20	corresponding	correspond	VERB
ejde-792	81	21	to	to	ADP
ejde-792	81	22	λ1	λ1	PROPN
ejde-792	81	23	with	with	ADP
ejde-792	81	24	∥φ1∥	∥φ1∥	PROPN
ejde-792	81	25	=	=	SYM
ejde-792	82	1	1	1	X
ejde-792	82	2	.	.	PUNCT
ejde-792	82	3	let	let	VERB
ejde-792	82	4	x0	x0	PROPN
ejde-792	82	5	be	be	AUX
ejde-792	82	6	a	a	DET
ejde-792	82	7	closed	closed	ADJ
ejde-792	82	8	subspace	subspace	NOUN
ejde-792	82	9	of	of	ADP
ejde-792	82	10	x	x	SYM
ejde-792	82	11	such	such	ADJ
ejde-792	82	12	that	that	SCONJ
ejde-792	82	13	x	x	X
ejde-792	83	1	=	=	SYM
ejde-792	83	2	x1	x1	NUM
ejde-792	83	3	⊕x0	⊕x0	NOUN
ejde-792	83	4	,	,	PUNCT
ejde-792	83	5	where	where	SCONJ
ejde-792	83	6	x1	x1	PROPN
ejde-792	83	7	=	=	SYM
ejde-792	83	8	span{φ1	span{φ1	ADJ
ejde-792	83	9	}	}	PUNCT
ejde-792	83	10	.	.	PUNCT
ejde-792	84	1	by	by	ADP
ejde-792	84	2	applying	apply	VERB
ejde-792	84	3	the	the	DET
ejde-792	84	4	hahn	hahn	NOUN
ejde-792	84	5	–	–	PUNCT
ejde-792	84	6	banach	banach	NOUN
ejde-792	84	7	theorem	theorem	VERB
ejde-792	84	8	,	,	PUNCT
ejde-792	84	9	we	we	PRON
ejde-792	84	10	can	can	AUX
ejde-792	84	11	find	find	VERB
ejde-792	84	12	a	a	DET
ejde-792	84	13	linear	linear	ADJ
ejde-792	84	14	continuous	continuous	ADJ
ejde-792	84	15	functional	functional	ADJ
ejde-792	84	16	l	l	NOUN
ejde-792	84	17	∈	∈	PROPN
ejde-792	84	18	x∗	x∗	PROPN
ejde-792	84	19	satisfying	satisfy	VERB
ejde-792	84	20	l(φ1	l(φ1	PROPN
ejde-792	84	21	)	)	PUNCT
ejde-792	84	22	=	=	SYM
ejde-792	84	23	1	1	NUM
ejde-792	84	24	and	and	CCONJ
ejde-792	84	25	x0	x0	NUM
ejde-792	84	26	=	=	PRON
ejde-792	84	27	{	{	PUNCT
ejde-792	84	28	u	u	NOUN
ejde-792	84	29	∈	∈	PROPN
ejde-792	84	30	x	x	X
ejde-792	84	31	:	:	PUNCT
ejde-792	84	32	l(u	l(u	PROPN
ejde-792	84	33	)	)	PUNCT
ejde-792	84	34	=	=	PUNCT
ejde-792	85	1	0	0	NUM
ejde-792	85	2	}	}	PUNCT
ejde-792	85	3	.	.	PUNCT
ejde-792	86	1	proof	proof	NOUN
ejde-792	86	2	of	of	ADP
ejde-792	86	3	theorem	theorem	ADJ
ejde-792	86	4	1.1	1.1	NUM
ejde-792	86	5	.	.	PUNCT
ejde-792	87	1	define	define	VERB
ejde-792	87	2	x	x	PUNCT
ejde-792	87	3	=	=	PRON
ejde-792	87	4	{	{	PUNCT
ejde-792	87	5	u	u	NOUN
ejde-792	87	6	∈	∈	PROPN
ejde-792	87	7	c2(ω	c2(ω	PROPN
ejde-792	87	8	)	)	PUNCT
ejde-792	87	9	:	:	PUNCT
ejde-792	88	1	u	u	NOUN
ejde-792	88	2	=	=	NOUN
ejde-792	88	3	0	0	NUM
ejde-792	88	4	,	,	PUNCT
ejde-792	88	5	on	on	ADP
ejde-792	88	6	∂ω	∂ω	ADJ
ejde-792	88	7	}	}	PUNCT
ejde-792	88	8	,	,	PUNCT
ejde-792	88	9	y	y	PROPN
ejde-792	88	10	=	=	SYM
ejde-792	88	11	c(ω	c(ω	PROPN
ejde-792	88	12	)	)	PUNCT
ejde-792	88	13	.	.	PUNCT
ejde-792	89	1	consider	consider	VERB
ejde-792	89	2	the	the	DET
ejde-792	89	3	function	function	NOUN
ejde-792	89	4	defined	define	VERB
ejde-792	89	5	by	by	ADP
ejde-792	89	6	f	f	PROPN
ejde-792	89	7	(	(	PUNCT
ejde-792	89	8	λ	λ	PROPN
ejde-792	89	9	,	,	PUNCT
ejde-792	89	10	u	u	NOUN
ejde-792	89	11	)	)	PUNCT
ejde-792	89	12	=	=	SYM
ejde-792	89	13	div	div	X
ejde-792	89	14	(	(	PUNCT
ejde-792	89	15	f2(x)∇u√	f2(x)∇u√	NOUN
ejde-792	89	16	1−	1−	NUM
ejde-792	89	17	f2(x)|∇u|2	f2(x)|∇u|2	PROPN
ejde-792	89	18	)	)	PUNCT
ejde-792	89	19	−	−	PROPN
ejde-792	90	1	λnf(x)h(x	λnf(x)h(x	PROPN
ejde-792	90	2	,	,	PUNCT
ejde-792	90	3	u	u	NOUN
ejde-792	90	4	)	)	PUNCT
ejde-792	90	5	.	.	PUNCT
ejde-792	91	1	since	since	SCONJ
ejde-792	91	2	h0	h0	PROPN
ejde-792	91	3	∈	∈	PROPN
ejde-792	91	4	(	(	PUNCT
ejde-792	91	5	0,+∞	0,+∞	NUM
ejde-792	91	6	)	)	PUNCT
ejde-792	91	7	and	and	CCONJ
ejde-792	91	8	h(x	h(x	PROPN
ejde-792	91	9	,	,	PUNCT
ejde-792	91	10	0	0	NUM
ejde-792	91	11	)	)	PUNCT
ejde-792	91	12	=	=	SYM
ejde-792	91	13	0	0	NUM
ejde-792	91	14	holds	hold	VERB
ejde-792	91	15	for	for	ADP
ejde-792	91	16	any	any	DET
ejde-792	91	17	x	x	SYM
ejde-792	91	18	∈	∈	PROPN
ejde-792	91	19	ω	ω	NOUN
ejde-792	91	20	,	,	PUNCT
ejde-792	91	21	we	we	PRON
ejde-792	91	22	have	have	VERB
ejde-792	91	23	f	f	PROPN
ejde-792	91	24	(	(	PUNCT
ejde-792	91	25	λ	λ	PROPN
ejde-792	91	26	,	,	PUNCT
ejde-792	91	27	0	0	NUM
ejde-792	91	28	)	)	PUNCT
ejde-792	91	29	=	=	VERB
ejde-792	92	1	div	div	X
ejde-792	92	2	(	(	PUNCT
ejde-792	92	3	f2(x)∇0√	f2(x)∇0√	PROPN
ejde-792	92	4	1−	1−	NUM
ejde-792	92	5	f2(x)|∇0|2	f2(x)|∇0|2	X
ejde-792	92	6	)	)	PUNCT
ejde-792	92	7	−	−	PROPN
ejde-792	93	1	λnf(x)h(x	λnf(x)h(x	PROPN
ejde-792	93	2	,	,	PUNCT
ejde-792	93	3	0	0	NUM
ejde-792	93	4	)	)	PUNCT
ejde-792	93	5	=	=	SYM
ejde-792	93	6	0	0	NUM
ejde-792	93	7	4	4	NUM
ejde-792	93	8	s.	s.	PROPN
ejde-792	93	9	gao	gao	PROPN
ejde-792	93	10	,	,	PUNCT
ejde-792	93	11	q.	q.	PROPN
ejde-792	93	12	liu	liu	PROPN
ejde-792	93	13	,	,	PUNCT
ejde-792	93	14	y.	y.	PROPN
ejde-792	93	15	sun	sun	PROPN
ejde-792	93	16	ejde-2024/48	ejde-2024/48	PROPN
ejde-792	93	17	(	(	PUNCT
ejde-792	93	18	a	a	X
ejde-792	93	19	)	)	PUNCT
ejde-792	93	20	transcritical	transcritical	ADJ
ejde-792	93	21	bifurcation	bifurcation	NOUN
ejde-792	93	22	(	(	PUNCT
ejde-792	93	23	b	b	NOUN
ejde-792	93	24	)	)	PUNCT
ejde-792	93	25	supercritical	supercritical	ADJ
ejde-792	93	26	pitchfork	pitchfork	NOUN
ejde-792	93	27	(	(	PUNCT
ejde-792	93	28	c	c	NOUN
ejde-792	93	29	)	)	PUNCT
ejde-792	93	30	subcritical	subcritical	ADJ
ejde-792	93	31	pitchfork	pitchfork	NOUN
ejde-792	93	32	figure	figure	NOUN
ejde-792	93	33	1	1	NUM
ejde-792	93	34	.	.	PUNCT
ejde-792	94	1	bifurcation	bifurcation	NOUN
ejde-792	94	2	diagrams	diagram	NOUN
ejde-792	94	3	of	of	ADP
ejde-792	94	4	theorem	theorem	ADJ
ejde-792	94	5	1.1	1.1	NUM
ejde-792	94	6	for	for	ADP
ejde-792	94	7	any	any	DET
ejde-792	94	8	λ	λ	NOUN
ejde-792	94	9	.	.	PUNCT
ejde-792	95	1	the	the	DET
ejde-792	95	2	partial	partial	ADJ
ejde-792	95	3	derivative	derivative	NOUN
ejde-792	95	4	of	of	ADP
ejde-792	95	5	f	f	PROPN
ejde-792	95	6	(	(	PUNCT
ejde-792	95	7	λ	λ	PROPN
ejde-792	95	8	,	,	PUNCT
ejde-792	95	9	u	u	NOUN
ejde-792	95	10	)	)	PUNCT
ejde-792	95	11	with	with	ADP
ejde-792	95	12	respect	respect	NOUN
ejde-792	95	13	to	to	ADP
ejde-792	95	14	λ	λ	PROPN
ejde-792	95	15	is	be	AUX
ejde-792	95	16	fλ(λ	fλ(λ	NUM
ejde-792	95	17	,	,	PUNCT
ejde-792	95	18	u	u	NOUN
ejde-792	95	19	)	)	PUNCT
ejde-792	95	20	=	=	SYM
ejde-792	95	21	−nf(x)h(x	−nf(x)h(x	PROPN
ejde-792	95	22	,	,	PUNCT
ejde-792	95	23	u	u	NOUN
ejde-792	95	24	)	)	PUNCT
ejde-792	95	25	.	.	PUNCT
ejde-792	96	1	according	accord	VERB
ejde-792	96	2	to	to	ADP
ejde-792	96	3	[	[	X
ejde-792	96	4	18	18	NUM
ejde-792	96	5	]	]	PUNCT
ejde-792	96	6	,	,	PUNCT
ejde-792	96	7	we	we	PRON
ejde-792	96	8	have	have	VERB
ejde-792	96	9	lim	lim	NOUN
ejde-792	96	10	t→0	t→0	PROPN
ejde-792	96	11	+	+	SYM
ejde-792	96	12	nh(x	nh(x	NUM
ejde-792	96	13	,	,	PUNCT
ejde-792	96	14	t	t	PROPN
ejde-792	96	15	)	)	PUNCT
ejde-792	96	16	t	t	NOUN
ejde-792	96	17	=	=	PUNCT
ejde-792	97	1	−h0	−h0	VERB
ejde-792	97	2	.	.	PUNCT
ejde-792	98	1	this	this	PRON
ejde-792	98	2	holds	hold	VERB
ejde-792	98	3	because	because	SCONJ
ejde-792	98	4	h	h	NOUN
ejde-792	98	5	is	be	AUX
ejde-792	98	6	a	a	DET
ejde-792	98	7	function	function	NOUN
ejde-792	98	8	with	with	ADP
ejde-792	98	9	third	third	ADJ
ejde-792	98	10	-	-	PUNCT
ejde-792	98	11	order	order	NOUN
ejde-792	98	12	continuous	continuous	ADJ
ejde-792	98	13	derivatives	derivative	NOUN
ejde-792	98	14	with	with	ADP
ejde-792	98	15	respect	respect	NOUN
ejde-792	98	16	to	to	ADP
ejde-792	98	17	its	its	PRON
ejde-792	98	18	second	second	ADJ
ejde-792	98	19	variable	variable	NOUN
ejde-792	98	20	,	,	PUNCT
ejde-792	98	21	and	and	CCONJ
ejde-792	98	22	f	f	PROPN
ejde-792	98	23	is	be	AUX
ejde-792	98	24	c3	c3	NOUN
ejde-792	98	25	with	with	ADP
ejde-792	98	26	respect	respect	NOUN
ejde-792	98	27	to	to	ADP
ejde-792	98	28	u	u	NOUN
ejde-792	98	29	in	in	ADP
ejde-792	98	30	some	some	DET
ejde-792	98	31	small	small	ADJ
ejde-792	98	32	neighborhood	neighborhood	NOUN
ejde-792	98	33	v	v	ADP
ejde-792	98	34	⊂	⊂	NOUN
ejde-792	98	35	x	x	X
ejde-792	98	36	of	of	ADP
ejde-792	98	37	0	0	NUM
ejde-792	98	38	.	.	PUNCT
ejde-792	99	1	by	by	ADP
ejde-792	99	2	calculation	calculation	NOUN
ejde-792	99	3	,	,	PUNCT
ejde-792	99	4	we	we	PRON
ejde-792	99	5	obtain	obtain	VERB
ejde-792	99	6	fu(λ	fu(λ	NUM
ejde-792	99	7	,	,	PUNCT
ejde-792	99	8	0)[φ1	0)[φ1	PROPN
ejde-792	99	9	]	]	X
ejde-792	99	10	=	=	PUNCT
ejde-792	99	11	div(f2(x)∇φ1	div(f2(x)∇φ1	X
ejde-792	99	12	)	)	PUNCT
ejde-792	99	13	+	+	CCONJ
ejde-792	99	14	λf(x)h0φ1	λf(x)h0φ1	X
ejde-792	99	15	,	,	PUNCT
ejde-792	99	16	fu	fu	ADJ
ejde-792	99	17	(	(	PUNCT
ejde-792	99	18	λ1	λ1	PROPN
ejde-792	99	19	h0	h0	PROPN
ejde-792	99	20	,	,	PUNCT
ejde-792	99	21	0)[φ	0)[φ	PROPN
ejde-792	99	22	]	]	X
ejde-792	99	23	=	=	SYM
ejde-792	99	24	div(f2(x)∇φ	div(f2(x)∇φ	NOUN
ejde-792	99	25	)	)	PUNCT
ejde-792	99	26	+	+	CCONJ
ejde-792	99	27	λ1	λ1	PROPN
ejde-792	99	28	h0	h0	PROPN
ejde-792	99	29	f(x)h0φ	f(x)h0φ	PROPN
ejde-792	99	30	,	,	PUNCT
ejde-792	99	31	where	where	SCONJ
ejde-792	99	32	φ	φ	PROPN
ejde-792	99	33	∈	∈	PROPN
ejde-792	99	34	x	x	X
ejde-792	99	35	.	.	PUNCT
ejde-792	100	1	the	the	DET
ejde-792	100	2	function	function	NOUN
ejde-792	100	3	φ1	φ1	PROPN
ejde-792	100	4	is	be	AUX
ejde-792	100	5	a	a	DET
ejde-792	100	6	positive	positive	ADJ
ejde-792	100	7	eigenfunction	eigenfunction	NOUN
ejde-792	100	8	corresponding	correspond	VERB
ejde-792	100	9	to	to	ADP
ejde-792	100	10	the	the	DET
ejde-792	100	11	principal	principal	ADJ
ejde-792	100	12	eigenvalue	eigenvalue	NOUN
ejde-792	100	13	λ1	λ1	PROPN
ejde-792	100	14	of	of	ADP
ejde-792	100	15	the	the	DET
ejde-792	100	16	linearized	linearize	VERB
ejde-792	100	17	problem	problem	NOUN
ejde-792	100	18	associated	associate	VERB
ejde-792	100	19	with	with	ADP
ejde-792	100	20	equation	equation	NOUN
ejde-792	100	21	(	(	PUNCT
ejde-792	100	22	1.1	1.1	NUM
ejde-792	100	23	)	)	PUNCT
ejde-792	100	24	.	.	PUNCT
ejde-792	101	1	specifically	specifically	ADV
ejde-792	101	2	,	,	PUNCT
ejde-792	101	3	φ1	φ1	PROPN
ejde-792	101	4	is	be	AUX
ejde-792	101	5	a	a	DET
ejde-792	101	6	solution	solution	NOUN
ejde-792	101	7	of	of	ADP
ejde-792	101	8	fu(λ1	fu(λ1	PROPN
ejde-792	101	9	/	/	SYM
ejde-792	101	10	h0	h0	PROPN
ejde-792	101	11	,	,	PUNCT
ejde-792	101	12	0)[φ	0)[φ	PROPN
ejde-792	101	13	]	]	X
ejde-792	101	14	=	=	SYM
ejde-792	101	15	0	0	X
ejde-792	101	16	.	.	PUNCT
ejde-792	102	1	then	then	ADV
ejde-792	102	2	,	,	PUNCT
ejde-792	102	3	we	we	PRON
ejde-792	102	4	have	have	VERB
ejde-792	102	5	div(f2(x)∇φ1	div(f2(x)∇φ1	VERB
ejde-792	102	6	)	)	PUNCT
ejde-792	103	1	+	+	CCONJ
ejde-792	104	1	λ1	λ1	ADJ
ejde-792	104	2	h0	h0	NOUN
ejde-792	104	3	f(x)h0φ1	f(x)h0φ1	NOUN
ejde-792	104	4	=	=	PUNCT
ejde-792	104	5	0	0	NUM
ejde-792	104	6	.	.	PUNCT
ejde-792	105	1	ejde-2024/48	ejde-2024/48	NOUN
ejde-792	105	2	local	local	ADJ
ejde-792	105	3	bifurcation	bifurcation	NOUN
ejde-792	105	4	structure	structure	NOUN
ejde-792	105	5	and	and	CCONJ
ejde-792	105	6	stability	stability	NOUN
ejde-792	105	7	5	5	NUM
ejde-792	105	8	since	since	SCONJ
ejde-792	105	9	φ1	φ1	PROPN
ejde-792	105	10	is	be	AUX
ejde-792	105	11	a	a	DET
ejde-792	105	12	nontrivial	nontrivial	ADJ
ejde-792	105	13	solution	solution	NOUN
ejde-792	105	14	,	,	PUNCT
ejde-792	105	15	it	it	PRON
ejde-792	105	16	follows	follow	VERB
ejde-792	105	17	that	that	SCONJ
ejde-792	105	18	φ1	φ1	PROPN
ejde-792	105	19	̸=	̸=	PROPN
ejde-792	105	20	0	0	NUM
ejde-792	105	21	and	and	CCONJ
ejde-792	105	22	φ2	φ2	PROPN
ejde-792	105	23	1	1	NUM
ejde-792	105	24	≥	≥	NOUN
ejde-792	105	25	0	0	NUM
ejde-792	105	26	.	.	PUNCT
ejde-792	106	1	hence	hence	ADV
ejde-792	106	2	,	,	PUNCT
ejde-792	106	3	the	the	DET
ejde-792	106	4	integral	integral	ADJ
ejde-792	106	5	∫	∫	PROPN
ejde-792	106	6	ω	ω	PROPN
ejde-792	106	7	φ2	φ2	PROPN
ejde-792	106	8	1	1	NUM
ejde-792	106	9	dx	dx	PROPN
ejde-792	106	10	>	>	X
ejde-792	106	11	0	0	PROPN
ejde-792	106	12	.	.	PUNCT
ejde-792	107	1	thus	thus	ADV
ejde-792	107	2	,	,	PUNCT
ejde-792	107	3	the	the	DET
ejde-792	107	4	kernel	kernel	PROPN
ejde-792	107	5	space	space	NOUN
ejde-792	107	6	is	be	AUX
ejde-792	107	7	n	n	DET
ejde-792	107	8	(	(	PUNCT
ejde-792	107	9	fu	fu	PROPN
ejde-792	107	10	(	(	PUNCT
ejde-792	107	11	λ1	λ1	PROPN
ejde-792	107	12	h0	h0	PROPN
ejde-792	107	13	,	,	PUNCT
ejde-792	107	14	0	0	NUM
ejde-792	107	15	)	)	PUNCT
ejde-792	107	16	)	)	PUNCT
ejde-792	108	1	=	=	SYM
ejde-792	108	2	span{φ1	span{φ1	ADJ
ejde-792	108	3	}	}	PUNCT
ejde-792	108	4	.	.	PUNCT
ejde-792	109	1	the	the	DET
ejde-792	109	2	codimension	codimension	NOUN
ejde-792	109	3	of	of	ADP
ejde-792	109	4	the	the	DET
ejde-792	109	5	image	image	NOUN
ejde-792	109	6	space	space	NOUN
ejde-792	109	7	is	be	AUX
ejde-792	109	8	r	r	NOUN
ejde-792	109	9	(	(	PUNCT
ejde-792	109	10	fu	fu	PROPN
ejde-792	109	11	(	(	PUNCT
ejde-792	109	12	λ1	λ1	PROPN
ejde-792	109	13	h0	h0	PROPN
ejde-792	109	14	,	,	PUNCT
ejde-792	109	15	0	0	NUM
ejde-792	109	16	)	)	PUNCT
ejde-792	109	17	)	)	PUNCT
ejde-792	110	1	=	=	PRON
ejde-792	110	2	{	{	PUNCT
ejde-792	110	3	v	v	NUM
ejde-792	110	4	∈	∈	X
ejde-792	110	5	y	y	NOUN
ejde-792	110	6	:	:	PUNCT
ejde-792	111	1	∫	∫	PROPN
ejde-792	111	2	ω	ω	NUM
ejde-792	111	3	vφ1	vφ1	NOUN
ejde-792	111	4	dx	dx	PROPN
ejde-792	111	5	=	=	SYM
ejde-792	111	6	0	0	NUM
ejde-792	111	7	}	}	PUNCT
ejde-792	111	8	.	.	PUNCT
ejde-792	112	1	therefore	therefore	ADV
ejde-792	112	2	,	,	PUNCT
ejde-792	112	3	dimn	dimn	PROPN
ejde-792	112	4	(	(	PUNCT
ejde-792	112	5	fu	fu	PROPN
ejde-792	112	6	(	(	PUNCT
ejde-792	112	7	λ1	λ1	PROPN
ejde-792	112	8	h0	h0	PROPN
ejde-792	112	9	,	,	PUNCT
ejde-792	112	10	0	0	NUM
ejde-792	112	11	)	)	PUNCT
ejde-792	112	12	)	)	PUNCT
ejde-792	113	1	=	=	SYM
ejde-792	113	2	codimr	codimr	NOUN
ejde-792	113	3	(	(	PUNCT
ejde-792	113	4	fu	fu	PROPN
ejde-792	113	5	(	(	PUNCT
ejde-792	113	6	λ1	λ1	PROPN
ejde-792	113	7	h0	h0	PROPN
ejde-792	113	8	,	,	PUNCT
ejde-792	113	9	0	0	NUM
ejde-792	113	10	)	)	PUNCT
ejde-792	113	11	)	)	PUNCT
ejde-792	113	12	=	=	PUNCT
ejde-792	114	1	1	1	X
ejde-792	114	2	.	.	PUNCT
ejde-792	114	3	(	(	PUNCT
ejde-792	114	4	2.1	2.1	NUM
ejde-792	114	5	)	)	PUNCT
ejde-792	114	6	clearly	clearly	ADV
ejde-792	114	7	,	,	PUNCT
ejde-792	114	8	f	f	PROPN
ejde-792	114	9	is	be	AUX
ejde-792	114	10	c1	c1	NOUN
ejde-792	114	11	with	with	ADP
ejde-792	114	12	respect	respect	NOUN
ejde-792	114	13	to	to	ADP
ejde-792	114	14	λ	λ	NOUN
ejde-792	114	15	,	,	PUNCT
ejde-792	114	16	and	and	CCONJ
ejde-792	114	17	fλu	fλu	NOUN
ejde-792	114	18	exists	exist	VERB
ejde-792	114	19	and	and	CCONJ
ejde-792	114	20	remains	remain	VERB
ejde-792	114	21	continuous	continuous	ADJ
ejde-792	114	22	in	in	ADP
ejde-792	114	23	a	a	DET
ejde-792	114	24	small	small	ADJ
ejde-792	114	25	neighborhood	neighborhood	NOUN
ejde-792	114	26	of	of	ADP
ejde-792	114	27	(	(	PUNCT
ejde-792	114	28	λ1	λ1	PROPN
ejde-792	114	29	/	/	SYM
ejde-792	114	30	h0	h0	PROPN
ejde-792	114	31	,	,	PUNCT
ejde-792	114	32	0	0	NUM
ejde-792	114	33	)	)	PUNCT
ejde-792	114	34	.	.	PUNCT
ejde-792	115	1	from	from	ADP
ejde-792	115	2	the	the	DET
ejde-792	115	3	calculations	calculation	NOUN
ejde-792	115	4	,	,	PUNCT
ejde-792	115	5	we	we	PRON
ejde-792	115	6	obtain	obtain	VERB
ejde-792	115	7	fλu	fλu	NOUN
ejde-792	115	8	(	(	PUNCT
ejde-792	115	9	λ1	λ1	PROPN
ejde-792	115	10	h0	h0	PROPN
ejde-792	115	11	,	,	PUNCT
ejde-792	115	12	0)[φ1	0)[φ1	PROPN
ejde-792	115	13	]	]	X
ejde-792	115	14	=	=	SYM
ejde-792	115	15	fh0φ1	fh0φ1	PROPN
ejde-792	115	16	,	,	PUNCT
ejde-792	115	17	(	(	PUNCT
ejde-792	115	18	2.2	2.2	NUM
ejde-792	115	19	)	)	PUNCT
ejde-792	115	20	fuu	fuu	PROPN
ejde-792	115	21	(	(	PUNCT
ejde-792	115	22	λ1	λ1	PROPN
ejde-792	115	23	h0	h0	PROPN
ejde-792	115	24	,	,	PUNCT
ejde-792	115	25	0)[φ1	0)[φ1	PROPN
ejde-792	115	26	]	]	X
ejde-792	115	27	2	2	NUM
ejde-792	115	28	=	=	SYM
ejde-792	115	29	−	−	PROPN
ejde-792	115	30	λ1	λ1	PROPN
ejde-792	115	31	h0	h0	PROPN
ejde-792	115	32	nfhuu(x	nfhuu(x	PROPN
ejde-792	115	33	,	,	PUNCT
ejde-792	115	34	0)φ	0)φ	NOUN
ejde-792	115	35	2	2	NUM
ejde-792	115	36	1	1	NUM
ejde-792	115	37	and	and	CCONJ
ejde-792	115	38	fuuu	fuuu	PROPN
ejde-792	115	39	(	(	PUNCT
ejde-792	115	40	λ1	λ1	PROPN
ejde-792	115	41	h0	h0	NOUN
ejde-792	115	42	,	,	PUNCT
ejde-792	115	43	0)[φ1	0)[φ1	PROPN
ejde-792	115	44	]	]	X
ejde-792	115	45	3	3	NUM
ejde-792	115	46	=	=	SYM
ejde-792	115	47	−	−	PROPN
ejde-792	115	48	λ1	λ1	PROPN
ejde-792	115	49	h0	h0	PROPN
ejde-792	115	50	nfhuuu(x	nfhuuu(x	PROPN
ejde-792	115	51	,	,	PUNCT
ejde-792	115	52	0)φ	0)φ	NOUN
ejde-792	115	53	3	3	NUM
ejde-792	115	54	1	1	NUM
ejde-792	115	55	.	.	PUNCT
ejde-792	115	56	combining	combine	VERB
ejde-792	115	57	this	this	PRON
ejde-792	115	58	with	with	ADP
ejde-792	115	59	h0	h0	PROPN
ejde-792	115	60	>	>	X
ejde-792	115	61	0	0	PROPN
ejde-792	115	62	,	,	PUNCT
ejde-792	115	63	we	we	PRON
ejde-792	115	64	obtain	obtain	VERB
ejde-792	115	65	h0	h0	PROPN
ejde-792	115	66	∫	∫	PROPN
ejde-792	115	67	ω	ω	PROPN
ejde-792	115	68	φ2	φ2	PROPN
ejde-792	115	69	1	1	NUM
ejde-792	115	70	dx	dx	PROPN
ejde-792	115	71	̸=	̸=	PROPN
ejde-792	115	72	0	0	NUM
ejde-792	115	73	.	.	PUNCT
ejde-792	116	1	thus	thus	ADV
ejde-792	116	2	,	,	PUNCT
ejde-792	116	3	∫	∫	PROPN
ejde-792	116	4	ω	ω	X
ejde-792	116	5	fh0φ	fh0φ	X
ejde-792	116	6	2	2	NUM
ejde-792	116	7	1	1	NUM
ejde-792	116	8	dx	dx	ADP
ejde-792	116	9	̸=	̸=	PROPN
ejde-792	116	10	0	0	NUM
ejde-792	116	11	.	.	PUNCT
ejde-792	117	1	this	this	PRON
ejde-792	117	2	leads	lead	VERB
ejde-792	117	3	to	to	ADP
ejde-792	117	4	the	the	DET
ejde-792	117	5	conclusion	conclusion	NOUN
ejde-792	117	6	that	that	SCONJ
ejde-792	117	7	fλu(λ1	fλu(λ1	PROPN
ejde-792	117	8	/	/	SYM
ejde-792	117	9	h0	h0	PROPN
ejde-792	117	10	,	,	PUNCT
ejde-792	117	11	0)[φ1	0)[φ1	PROPN
ejde-792	117	12	]	]	X
ejde-792	117	13	̸∈	̸∈	PROPN
ejde-792	117	14	r(fu(λ1	r(fu(λ1	PROPN
ejde-792	117	15	/	/	SYM
ejde-792	117	16	h0	h0	PROPN
ejde-792	117	17	,	,	PUNCT
ejde-792	117	18	0	0	NUM
ejde-792	117	19	)	)	PUNCT
ejde-792	117	20	)	)	PUNCT
ejde-792	117	21	.	.	PUNCT
ejde-792	118	1	(	(	PUNCT
ejde-792	118	2	2.3	2.3	NUM
ejde-792	118	3	)	)	PUNCT
ejde-792	118	4	by	by	ADP
ejde-792	118	5	applying	apply	VERB
ejde-792	118	6	[	[	X
ejde-792	118	7	16	16	NUM
ejde-792	118	8	]	]	PUNCT
ejde-792	118	9	,	,	PUNCT
ejde-792	118	10	we	we	PRON
ejde-792	118	11	deduce	deduce	VERB
ejde-792	118	12	that	that	SCONJ
ejde-792	118	13	all	all	DET
ejde-792	118	14	the	the	DET
ejde-792	118	15	solutions	solution	NOUN
ejde-792	118	16	near	near	ADP
ejde-792	118	17	(	(	PUNCT
ejde-792	118	18	λ1	λ1	PROPN
ejde-792	118	19	/	/	SYM
ejde-792	118	20	h0	h0	PROPN
ejde-792	118	21	,	,	PUNCT
ejde-792	118	22	0	0	NUM
ejde-792	118	23	)	)	PUNCT
ejde-792	118	24	for	for	ADP
ejde-792	118	25	problem	problem	NOUN
ejde-792	118	26	(	(	PUNCT
ejde-792	118	27	1.1	1.1	NUM
ejde-792	118	28	)	)	PUNCT
ejde-792	118	29	can	can	AUX
ejde-792	118	30	be	be	AUX
ejde-792	118	31	expressed	express	VERB
ejde-792	118	32	as	as	ADP
ejde-792	118	33	(	(	PUNCT
ejde-792	118	34	λ(s	λ(s	PROPN
ejde-792	118	35	)	)	PUNCT
ejde-792	118	36	,	,	PUNCT
ejde-792	118	37	sφ1+sz(s	sφ1+sz(s	PROPN
ejde-792	118	38	)	)	PUNCT
ejde-792	118	39	)	)	PUNCT
ejde-792	118	40	,	,	PUNCT
ejde-792	118	41	where	where	SCONJ
ejde-792	118	42	s	s	NOUN
ejde-792	118	43	belongs	belong	VERB
ejde-792	118	44	to	to	ADP
ejde-792	118	45	the	the	DET
ejde-792	118	46	interval	interval	NOUN
ejde-792	118	47	(	(	PUNCT
ejde-792	118	48	−δ	−δ	ADJ
ejde-792	118	49	,	,	PUNCT
ejde-792	118	50	δ	δ	PROPN
ejde-792	118	51	)	)	PUNCT
ejde-792	118	52	for	for	ADP
ejde-792	118	53	some	some	DET
ejde-792	118	54	positive	positive	ADJ
ejde-792	118	55	value	value	NOUN
ejde-792	118	56	of	of	ADP
ejde-792	118	57	δ	δ	PROPN
ejde-792	118	58	,	,	PUNCT
ejde-792	118	59	and	and	CCONJ
ejde-792	118	60	that	that	SCONJ
ejde-792	118	61	they	they	PRON
ejde-792	118	62	satisfy	satisfy	VERB
ejde-792	118	63	the	the	DET
ejde-792	118	64	conditions	condition	NOUN
ejde-792	118	65	λ(0	λ(0	NOUN
ejde-792	118	66	)	)	PUNCT
ejde-792	118	67	=	=	SYM
ejde-792	118	68	λ1	λ1	PROPN
ejde-792	118	69	/	/	SYM
ejde-792	118	70	h0	h0	NOUN
ejde-792	118	71	and	and	CCONJ
ejde-792	118	72	z(0	z(0	PROPN
ejde-792	118	73	)	)	PUNCT
ejde-792	118	74	=	=	SYM
ejde-792	119	1	0	0	X
ejde-792	119	2	.	.	PUNCT
ejde-792	120	1	we	we	PRON
ejde-792	120	2	rescale	rescale	VERB
ejde-792	120	3	φ1	φ1	PROPN
ejde-792	121	1	so	so	SCONJ
ejde-792	121	2	that	that	SCONJ
ejde-792	121	3	∫	∫	PROPN
ejde-792	121	4	ω	ω	NUM
ejde-792	121	5	φ2	φ2	PROPN
ejde-792	121	6	1	1	NUM
ejde-792	121	7	dx	dx	PROPN
ejde-792	121	8	=	=	SYM
ejde-792	121	9	1	1	X
ejde-792	121	10	.	.	PUNCT
ejde-792	121	11	(	(	PUNCT
ejde-792	121	12	2.4	2.4	NUM
ejde-792	121	13	)	)	PUNCT
ejde-792	121	14	then	then	ADV
ejde-792	121	15	,	,	PUNCT
ejde-792	121	16	we	we	PRON
ejde-792	121	17	define	define	VERB
ejde-792	121	18	the	the	DET
ejde-792	121	19	linear	linear	ADJ
ejde-792	121	20	functional	functional	ADJ
ejde-792	121	21	l(u	l(u	PROPN
ejde-792	121	22	)	)	PUNCT
ejde-792	122	1	=	=	SYM
ejde-792	122	2	∫	∫	PROPN
ejde-792	122	3	ω	ω	NUM
ejde-792	122	4	uφ1	uφ1	ADJ
ejde-792	122	5	dx	dx	PROPN
ejde-792	122	6	(	(	PUNCT
ejde-792	122	7	2.5	2.5	NUM
ejde-792	122	8	)	)	PUNCT
ejde-792	122	9	and	and	CCONJ
ejde-792	122	10	n	n	CCONJ
ejde-792	122	11	(	(	PUNCT
ejde-792	122	12	l	l	NOUN
ejde-792	122	13	)	)	PUNCT
ejde-792	122	14	=	=	SYM
ejde-792	122	15	r(fu	r(fu	PROPN
ejde-792	122	16	(	(	PUNCT
ejde-792	122	17	λ1	λ1	PROPN
ejde-792	122	18	h0	h0	PROPN
ejde-792	122	19	,	,	PUNCT
ejde-792	122	20	0	0	NUM
ejde-792	122	21	)	)	PUNCT
ejde-792	122	22	)	)	PUNCT
ejde-792	122	23	.	.	PUNCT
ejde-792	123	1	furthermore	furthermore	ADV
ejde-792	123	2	,	,	PUNCT
ejde-792	123	3	by	by	ADP
ejde-792	123	4	employing	employ	VERB
ejde-792	123	5	formula	formula	NOUN
ejde-792	123	6	(	(	PUNCT
ejde-792	123	7	4.5	4.5	NUM
ejde-792	123	8	)	)	PUNCT
ejde-792	123	9	derived	derive	VERB
ejde-792	123	10	in	in	ADP
ejde-792	123	11	[	[	X
ejde-792	123	12	23	23	NUM
ejde-792	123	13	]	]	PUNCT
ejde-792	123	14	,	,	PUNCT
ejde-792	123	15	we	we	PRON
ejde-792	123	16	can	can	AUX
ejde-792	123	17	deduce	deduce	VERB
ejde-792	123	18	that	that	PRON
ejde-792	123	19	λ′(0	λ′(0	ADJ
ejde-792	123	20	)	)	PUNCT
ejde-792	123	21	=	=	SYM
ejde-792	123	22	−	−	PROPN
ejde-792	123	23	⟨l	⟨l	NOUN
ejde-792	123	24	,	,	PUNCT
ejde-792	123	25	fuu	fuu	PROPN
ejde-792	123	26	(	(	PUNCT
ejde-792	123	27	λ1	λ1	PROPN
ejde-792	123	28	h0	h0	PROPN
ejde-792	123	29	,	,	PUNCT
ejde-792	123	30	0)[φ1	0)[φ1	PROPN
ejde-792	123	31	]	]	X
ejde-792	123	32	2⟩	2⟩	NUM
ejde-792	123	33	2⟨l	2⟨l	NUM
ejde-792	123	34	,	,	PUNCT
ejde-792	123	35	fλu	fλu	NOUN
ejde-792	123	36	(	(	PUNCT
ejde-792	123	37	λ1	λ1	PROPN
ejde-792	123	38	h0	h0	PROPN
ejde-792	123	39	,	,	PUNCT
ejde-792	123	40	0)[φ1]⟩	0)[φ1]⟩	PROPN
ejde-792	123	41	.	.	PUNCT
ejde-792	124	1	6	6	NUM
ejde-792	124	2	s.	s.	PROPN
ejde-792	124	3	gao	gao	PROPN
ejde-792	124	4	,	,	PUNCT
ejde-792	124	5	q.	q.	PROPN
ejde-792	124	6	liu	liu	PROPN
ejde-792	124	7	,	,	PUNCT
ejde-792	124	8	y.	y.	PROPN
ejde-792	124	9	sun	sun	PROPN
ejde-792	124	10	ejde-2024/48	ejde-2024/48	PROPN
ejde-792	124	11	subsequently	subsequently	ADV
ejde-792	124	12	,	,	PUNCT
ejde-792	124	13	⟨l	⟨l	PROPN
ejde-792	124	14	,	,	PUNCT
ejde-792	124	15	fuu	fuu	PROPN
ejde-792	124	16	(	(	PUNCT
ejde-792	124	17	λ1	λ1	PROPN
ejde-792	124	18	h0	h0	PROPN
ejde-792	124	19	,	,	PUNCT
ejde-792	124	20	0)[φ1	0)[φ1	PROPN
ejde-792	124	21	]	]	X
ejde-792	124	22	2⟩	2⟩	NUM
ejde-792	124	23	=	=	SYM
ejde-792	124	24	∫	∫	PROPN
ejde-792	124	25	ω	ω	PROPN
ejde-792	124	26	fuu	fuu	PROPN
ejde-792	124	27	(	(	PUNCT
ejde-792	124	28	λ1	λ1	PROPN
ejde-792	124	29	h0	h0	PROPN
ejde-792	124	30	,	,	PUNCT
ejde-792	124	31	0)[φ1	0)[φ1	PROPN
ejde-792	124	32	]	]	X
ejde-792	124	33	2φ1	2φ1	NUM
ejde-792	124	34	dx	dx	PROPN
ejde-792	124	35	=	=	SYM
ejde-792	124	36	−	−	PROPN
ejde-792	124	37	λ1	λ1	PROPN
ejde-792	124	38	h0	h0	PROPN
ejde-792	124	39	n	n	PROPN
ejde-792	124	40	∫	∫	PROPN
ejde-792	124	41	ω	ω	NUM
ejde-792	124	42	f(x)huu(x	f(x)huu(x	PROPN
ejde-792	124	43	,	,	PUNCT
ejde-792	124	44	0)φ	0)φ	NOUN
ejde-792	124	45	3	3	NUM
ejde-792	124	46	1	1	NUM
ejde-792	124	47	dx	dx	PROPN
ejde-792	124	48	(	(	PUNCT
ejde-792	124	49	2.6	2.6	NUM
ejde-792	124	50	)	)	PUNCT
ejde-792	124	51	and	and	CCONJ
ejde-792	124	52	2⟨l	2⟨l	NUM
ejde-792	124	53	,	,	PUNCT
ejde-792	124	54	fλu	fλu	NOUN
ejde-792	124	55	(	(	PUNCT
ejde-792	124	56	λ1	λ1	PROPN
ejde-792	124	57	h0	h0	PROPN
ejde-792	124	58	,	,	PUNCT
ejde-792	124	59	0)[φ1]⟩	0)[φ1]⟩	NOUN
ejde-792	124	60	=	=	SYM
ejde-792	124	61	2	2	NUM
ejde-792	124	62	∫	∫	PROPN
ejde-792	124	63	ω	ω	PROPN
ejde-792	124	64	fλu	fλu	PROPN
ejde-792	124	65	(	(	PUNCT
ejde-792	124	66	λ1	λ1	PROPN
ejde-792	124	67	h0	h0	PROPN
ejde-792	124	68	,	,	PUNCT
ejde-792	124	69	0)[φ1]φ1	0)[φ1]φ1	NUM
ejde-792	124	70	dx	dx	PROPN
ejde-792	125	1	=	=	SYM
ejde-792	126	1	2h0	2h0	NUM
ejde-792	126	2	∫	∫	PROPN
ejde-792	126	3	ω	ω	PROPN
ejde-792	126	4	f(x)φ2	f(x)φ2	PROPN
ejde-792	126	5	1	1	NUM
ejde-792	126	6	dx	dx	PROPN
ejde-792	126	7	.	.	PUNCT
ejde-792	127	1	(	(	PUNCT
ejde-792	127	2	2.7	2.7	NUM
ejde-792	127	3	)	)	PUNCT
ejde-792	127	4	from	from	ADP
ejde-792	127	5	(	(	PUNCT
ejde-792	127	6	2.6	2.6	NUM
ejde-792	127	7	)	)	PUNCT
ejde-792	127	8	and	and	CCONJ
ejde-792	127	9	(	(	PUNCT
ejde-792	127	10	2.7	2.7	NUM
ejde-792	127	11	)	)	PUNCT
ejde-792	127	12	,	,	PUNCT
ejde-792	127	13	we	we	PRON
ejde-792	127	14	obtain	obtain	VERB
ejde-792	127	15	λ′(0	λ′(0	NOUN
ejde-792	127	16	)	)	PUNCT
ejde-792	127	17	=	=	SYM
ejde-792	127	18	−	−	PROPN
ejde-792	127	19	⟨l	⟨l	NOUN
ejde-792	127	20	,	,	PUNCT
ejde-792	127	21	fuu	fuu	PROPN
ejde-792	127	22	(	(	PUNCT
ejde-792	127	23	λ1	λ1	PROPN
ejde-792	127	24	h0	h0	PROPN
ejde-792	127	25	,	,	PUNCT
ejde-792	127	26	0)[φ1	0)[φ1	PROPN
ejde-792	127	27	]	]	X
ejde-792	127	28	2⟩	2⟩	NUM
ejde-792	127	29	2⟨l	2⟨l	NUM
ejde-792	127	30	,	,	PUNCT
ejde-792	127	31	fλu	fλu	NOUN
ejde-792	127	32	(	(	PUNCT
ejde-792	127	33	λ1	λ1	PROPN
ejde-792	127	34	h0	h0	PROPN
ejde-792	127	35	,	,	PUNCT
ejde-792	127	36	0)[φ1]⟩	0)[φ1]⟩	NOUN
ejde-792	128	1	=	=	SYM
ejde-792	129	1	−	−	PROPN
ejde-792	129	2	−	−	PROPN
ejde-792	129	3	λ1	λ1	PROPN
ejde-792	129	4	h0	h0	PROPN
ejde-792	129	5	n	n	PROPN
ejde-792	129	6	∫	∫	PROPN
ejde-792	129	7	ω	ω	NUM
ejde-792	129	8	f(x)huu(x	f(x)huu(x	PROPN
ejde-792	129	9	,	,	PUNCT
ejde-792	129	10	0)φ	0)φ	NOUN
ejde-792	129	11	3	3	NUM
ejde-792	129	12	1	1	NUM
ejde-792	129	13	dx	dx	PROPN
ejde-792	129	14	2h0	2h0	NUM
ejde-792	129	15	∫	∫	PROPN
ejde-792	129	16	ω	ω	PROPN
ejde-792	129	17	f(x)φ2	f(x)φ2	PROPN
ejde-792	129	18	1	1	NUM
ejde-792	129	19	dx	dx	PROPN
ejde-792	129	20	.	.	PUNCT
ejde-792	130	1	if	if	SCONJ
ejde-792	130	2	huu(x	huu(x	NOUN
ejde-792	130	3	,	,	PUNCT
ejde-792	130	4	0	0	NUM
ejde-792	130	5	)	)	PUNCT
ejde-792	130	6	≡	≡	PROPN
ejde-792	130	7	0	0	NUM
ejde-792	131	1	in	in	ADP
ejde-792	131	2	ω	ω	NUM
ejde-792	131	3	,	,	PUNCT
ejde-792	131	4	using	use	VERB
ejde-792	131	5	[	[	X
ejde-792	131	6	23	23	NUM
ejde-792	131	7	,	,	PUNCT
ejde-792	131	8	(	(	PUNCT
ejde-792	131	9	4.6	4.6	NUM
ejde-792	131	10	)	)	PUNCT
ejde-792	131	11	]	]	PUNCT
ejde-792	131	12	,	,	PUNCT
ejde-792	131	13	we	we	PRON
ejde-792	131	14	deduce	deduce	VERB
ejde-792	131	15	that	that	SCONJ
ejde-792	131	16	λ′′(0	λ′′(0	NOUN
ejde-792	131	17	)	)	PUNCT
ejde-792	132	1	=	=	SYM
ejde-792	132	2	−	−	NOUN
ejde-792	132	3	⟨l	⟨l	NOUN
ejde-792	132	4	,	,	PUNCT
ejde-792	132	5	fuuu	fuuu	NOUN
ejde-792	132	6	(	(	PUNCT
ejde-792	132	7	λ1	λ1	PROPN
ejde-792	132	8	h0	h0	PROPN
ejde-792	132	9	,	,	PUNCT
ejde-792	132	10	0)[φ1	0)[φ1	PROPN
ejde-792	132	11	]	]	X
ejde-792	132	12	3⟩	3⟩	PROPN
ejde-792	132	13	3⟨l	3⟨l	NUM
ejde-792	132	14	,	,	PUNCT
ejde-792	132	15	fλu	fλu	PROPN
ejde-792	132	16	(	(	PUNCT
ejde-792	132	17	λ1	λ1	PROPN
ejde-792	132	18	h0	h0	PROPN
ejde-792	132	19	,	,	PUNCT
ejde-792	132	20	0)[φ1]⟩	0)[φ1]⟩	PROPN
ejde-792	132	21	,	,	PUNCT
ejde-792	132	22	where	where	SCONJ
ejde-792	132	23	⟨l	⟨l	NOUN
ejde-792	132	24	,	,	PUNCT
ejde-792	132	25	fuuu	fuuu	NOUN
ejde-792	132	26	(	(	PUNCT
ejde-792	132	27	λ1	λ1	PROPN
ejde-792	132	28	h0	h0	PROPN
ejde-792	132	29	,	,	PUNCT
ejde-792	132	30	0)[φ1	0)[φ1	PROPN
ejde-792	132	31	]	]	X
ejde-792	133	1	3⟩	3⟩	PROPN
ejde-792	133	2	=	=	SYM
ejde-792	134	1	∫	∫	PROPN
ejde-792	134	2	ω	ω	NUM
ejde-792	134	3	fuuu	fuuu	PROPN
ejde-792	134	4	(	(	PUNCT
ejde-792	134	5	λ1	λ1	PROPN
ejde-792	134	6	h0	h0	NOUN
ejde-792	134	7	,	,	PUNCT
ejde-792	134	8	0)[φ1	0)[φ1	PROPN
ejde-792	134	9	]	]	X
ejde-792	134	10	3φ1	3φ1	NUM
ejde-792	134	11	dx	dx	PROPN
ejde-792	134	12	=	=	SYM
ejde-792	134	13	−	−	PROPN
ejde-792	134	14	λ1	λ1	PROPN
ejde-792	134	15	h0	h0	PROPN
ejde-792	134	16	n	n	PROPN
ejde-792	134	17	∫	∫	PROPN
ejde-792	134	18	ω	ω	NUM
ejde-792	134	19	f(x)huuu(x	f(x)huuu(x	PROPN
ejde-792	134	20	,	,	PUNCT
ejde-792	134	21	0)φ	0)φ	NOUN
ejde-792	134	22	4	4	NUM
ejde-792	134	23	1	1	NUM
ejde-792	134	24	dx	dx	NOUN
ejde-792	134	25	and	and	CCONJ
ejde-792	134	26	3⟨l	3⟨l	NUM
ejde-792	134	27	,	,	PUNCT
ejde-792	134	28	fλu	fλu	NOUN
ejde-792	134	29	(	(	PUNCT
ejde-792	134	30	λ1	λ1	PROPN
ejde-792	134	31	h0	h0	PROPN
ejde-792	134	32	,	,	PUNCT
ejde-792	134	33	0)[φ1]⟩	0)[φ1]⟩	NOUN
ejde-792	134	34	=	=	SYM
ejde-792	134	35	3	3	NUM
ejde-792	134	36	∫	∫	PROPN
ejde-792	134	37	ω	ω	PROPN
ejde-792	134	38	fλu	fλu	PROPN
ejde-792	134	39	(	(	PUNCT
ejde-792	134	40	λ1	λ1	PROPN
ejde-792	134	41	h0	h0	PROPN
ejde-792	134	42	,	,	PUNCT
ejde-792	134	43	0)[φ1]φ1	0)[φ1]φ1	NUM
ejde-792	134	44	dx	dx	PROPN
ejde-792	135	1	=	=	SYM
ejde-792	135	2	3h0	3h0	NUM
ejde-792	135	3	∫	∫	PROPN
ejde-792	135	4	ω	ω	PROPN
ejde-792	135	5	f(x)φ2	f(x)φ2	PROPN
ejde-792	135	6	1	1	NUM
ejde-792	135	7	dx	dx	PROPN
ejde-792	135	8	.	.	PUNCT
ejde-792	136	1	thus	thus	ADV
ejde-792	136	2	,	,	PUNCT
ejde-792	136	3	λ′′(0	λ′′(0	NOUN
ejde-792	136	4	)	)	PUNCT
ejde-792	136	5	=	=	SYM
ejde-792	136	6	−	−	NOUN
ejde-792	136	7	⟨l	⟨l	NOUN
ejde-792	136	8	,	,	PUNCT
ejde-792	136	9	fuuu	fuuu	NOUN
ejde-792	136	10	(	(	PUNCT
ejde-792	136	11	λ1	λ1	PROPN
ejde-792	136	12	h0	h0	PROPN
ejde-792	136	13	,	,	PUNCT
ejde-792	136	14	0)[φ1	0)[φ1	PROPN
ejde-792	136	15	]	]	X
ejde-792	136	16	3⟩	3⟩	PROPN
ejde-792	136	17	3⟨l	3⟨l	NUM
ejde-792	136	18	,	,	PUNCT
ejde-792	136	19	fλu	fλu	PROPN
ejde-792	136	20	(	(	PUNCT
ejde-792	136	21	λ1	λ1	PROPN
ejde-792	136	22	h0	h0	PROPN
ejde-792	136	23	,	,	PUNCT
ejde-792	136	24	0)[φ1]⟩	0)[φ1]⟩	NOUN
ejde-792	136	25	=	=	SYM
ejde-792	137	1	−	−	PROPN
ejde-792	137	2	−	−	PROPN
ejde-792	137	3	λ1	λ1	PROPN
ejde-792	137	4	h0	h0	PROPN
ejde-792	137	5	n	n	PROPN
ejde-792	137	6	∫	∫	PROPN
ejde-792	137	7	ω	ω	NUM
ejde-792	137	8	f(x)huuu(x	f(x)huuu(x	PROPN
ejde-792	137	9	,	,	PUNCT
ejde-792	137	10	0)φ	0)φ	NOUN
ejde-792	137	11	4	4	NUM
ejde-792	137	12	1	1	NUM
ejde-792	137	13	dx	dx	PROPN
ejde-792	137	14	3h0	3h0	NUM
ejde-792	137	15	∫	∫	PROPN
ejde-792	137	16	ω	ω	PROPN
ejde-792	137	17	f(x)φ2	f(x)φ2	PROPN
ejde-792	137	18	1	1	NUM
ejde-792	137	19	dx	dx	PROPN
ejde-792	137	20	.	.	PUNCT
ejde-792	138	1	(	(	PUNCT
ejde-792	138	2	2.8	2.8	NUM
ejde-792	138	3	)	)	PUNCT
ejde-792	138	4	we	we	PRON
ejde-792	138	5	observe	observe	VERB
ejde-792	138	6	that	that	SCONJ
ejde-792	138	7	λ′(0	λ′(0	ADJ
ejde-792	138	8	)	)	PUNCT
ejde-792	138	9	̸=	̸=	PROPN
ejde-792	138	10	0	0	NUM
ejde-792	138	11	if	if	SCONJ
ejde-792	138	12	huu(x	huu(x	PROPN
ejde-792	138	13	,	,	PUNCT
ejde-792	138	14	0	0	NUM
ejde-792	138	15	)	)	PUNCT
ejde-792	138	16	̸=	̸=	NOUN
ejde-792	138	17	0	0	NUM
ejde-792	138	18	in	in	ADP
ejde-792	138	19	ω	ω	PROPN
ejde-792	138	20	.	.	PUNCT
ejde-792	139	1	this	this	PRON
ejde-792	139	2	indicates	indicate	VERB
ejde-792	139	3	the	the	DET
ejde-792	139	4	occurrence	occurrence	NOUN
ejde-792	139	5	of	of	ADP
ejde-792	139	6	a	a	DET
ejde-792	139	7	transcritical	transcritical	ADJ
ejde-792	139	8	bifurcation	bifurcation	NOUN
ejde-792	139	9	,	,	PUNCT
ejde-792	139	10	characterized	characterize	VERB
ejde-792	139	11	by	by	ADP
ejde-792	139	12	λ′(0	λ′(0	NOUN
ejde-792	139	13	)	)	PUNCT
ejde-792	139	14	̸=	̸=	PROPN
ejde-792	139	15	0	0	NUM
ejde-792	139	16	(	(	PUNCT
ejde-792	139	17	see	see	VERB
ejde-792	139	18	figure	figure	NOUN
ejde-792	139	19	1	1	NUM
ejde-792	139	20	)	)	PUNCT
ejde-792	139	21	.	.	PUNCT
ejde-792	140	1	if	if	SCONJ
ejde-792	140	2	huu(x	huu(x	NOUN
ejde-792	140	3	,	,	PUNCT
ejde-792	140	4	0	0	NUM
ejde-792	140	5	)	)	PUNCT
ejde-792	140	6	≡	≡	PROPN
ejde-792	140	7	0	0	PUNCT
ejde-792	141	1	in	in	ADP
ejde-792	141	2	ω	ω	PROPN
ejde-792	141	3	but	but	CCONJ
ejde-792	141	4	huuu(x	huuu(x	PROPN
ejde-792	141	5	,	,	PUNCT
ejde-792	141	6	0	0	NUM
ejde-792	141	7	)	)	PUNCT
ejde-792	141	8	̸=	̸=	NOUN
ejde-792	141	9	0	0	NUM
ejde-792	141	10	in	in	ADP
ejde-792	141	11	ω	ω	NUM
ejde-792	141	12	,	,	PUNCT
ejde-792	141	13	it	it	PRON
ejde-792	141	14	follows	follow	VERB
ejde-792	141	15	that	that	SCONJ
ejde-792	141	16	λ′(0	λ′(0	ADJ
ejde-792	141	17	)	)	PUNCT
ejde-792	141	18	=	=	SYM
ejde-792	141	19	0	0	NUM
ejde-792	141	20	and	and	CCONJ
ejde-792	141	21	λ′′(0	λ′′(0	NOUN
ejde-792	141	22	)	)	PUNCT
ejde-792	141	23	̸=	̸=	PROPN
ejde-792	141	24	0	0	NUM
ejde-792	141	25	,	,	PUNCT
ejde-792	141	26	implying	imply	VERB
ejde-792	141	27	a	a	DET
ejde-792	141	28	pitchfork	pitchfork	NOUN
ejde-792	141	29	bifurcation	bifurcation	NOUN
ejde-792	141	30	,	,	PUNCT
ejde-792	141	31	characterized	characterize	VERB
ejde-792	141	32	by	by	ADP
ejde-792	141	33	λ′′(0	λ′′(0	NOUN
ejde-792	141	34	)	)	PUNCT
ejde-792	141	35	̸=	̸=	PROPN
ejde-792	141	36	0	0	NUM
ejde-792	141	37	.	.	PUNCT
ejde-792	142	1	specifically	specifically	ADV
ejde-792	142	2	,	,	PUNCT
ejde-792	142	3	if	if	SCONJ
ejde-792	142	4	λ′′(0	λ′′(0	NOUN
ejde-792	142	5	)	)	PUNCT
ejde-792	142	6	>	>	X
ejde-792	142	7	0	0	NUM
ejde-792	142	8	,	,	PUNCT
ejde-792	142	9	a	a	DET
ejde-792	142	10	supercritical	supercritical	ADJ
ejde-792	142	11	pitchfork	pitchfork	NOUN
ejde-792	142	12	bifurcation	bifurcation	NOUN
ejde-792	142	13	occurs	occur	VERB
ejde-792	142	14	.	.	PUNCT
ejde-792	143	1	if	if	SCONJ
ejde-792	143	2	λ′′(0	λ′′(0	NOUN
ejde-792	143	3	)	)	PUNCT
ejde-792	143	4	<	<	X
ejde-792	143	5	0	0	PROPN
ejde-792	143	6	,	,	PUNCT
ejde-792	143	7	a	a	DET
ejde-792	143	8	subcritical	subcritical	ADJ
ejde-792	143	9	pitchfork	pitchfork	NOUN
ejde-792	143	10	bifurcation	bifurcation	NOUN
ejde-792	143	11	occurs	occur	VERB
ejde-792	143	12	(	(	PUNCT
ejde-792	143	13	see	see	VERB
ejde-792	143	14	figure	figure	NOUN
ejde-792	143	15	1	1	NUM
ejde-792	143	16	)	)	PUNCT
ejde-792	143	17	.	.	PUNCT
ejde-792	144	1	hence	hence	ADV
ejde-792	144	2	,	,	PUNCT
ejde-792	144	3	the	the	DET
ejde-792	144	4	desired	desire	VERB
ejde-792	144	5	conclusions	conclusion	NOUN
ejde-792	144	6	are	be	AUX
ejde-792	144	7	obtained	obtain	VERB
ejde-792	144	8	.	.	PUNCT
ejde-792	145	1	□	□	PUNCT
ejde-792	145	2	ejde-2024/48	ejde-2024/48	NOUN
ejde-792	145	3	local	local	ADJ
ejde-792	145	4	bifurcation	bifurcation	NOUN
ejde-792	145	5	structure	structure	NOUN
ejde-792	145	6	and	and	CCONJ
ejde-792	145	7	stability	stability	NOUN
ejde-792	145	8	7	7	NUM
ejde-792	145	9	3	3	NUM
ejde-792	145	10	.	.	PUNCT
ejde-792	146	1	stability	stability	NOUN
ejde-792	146	2	properties	property	NOUN
ejde-792	146	3	in	in	ADP
ejde-792	146	4	this	this	DET
ejde-792	146	5	section	section	NOUN
ejde-792	146	6	,	,	PUNCT
ejde-792	146	7	we	we	PRON
ejde-792	146	8	provide	provide	VERB
ejde-792	146	9	the	the	DET
ejde-792	146	10	formal	formal	ADJ
ejde-792	146	11	stability	stability	NOUN
ejde-792	146	12	results	result	VERB
ejde-792	146	13	near	near	ADP
ejde-792	146	14	the	the	DET
ejde-792	146	15	bifurcation	bifurcation	NOUN
ejde-792	146	16	point	point	NOUN
ejde-792	146	17	.	.	PUNCT
ejde-792	147	1	the	the	DET
ejde-792	147	2	stability	stability	NOUN
ejde-792	147	3	properties	property	NOUN
ejde-792	147	4	are	be	AUX
ejde-792	147	5	obtained	obtain	VERB
ejde-792	147	6	via	via	ADP
ejde-792	147	7	the	the	DET
ejde-792	147	8	exchange	exchange	NOUN
ejde-792	147	9	of	of	ADP
ejde-792	147	10	stability	stability	NOUN
ejde-792	147	11	theorem	theorem	VERB
ejde-792	147	12	presented	present	VERB
ejde-792	147	13	in	in	ADP
ejde-792	147	14	[	[	X
ejde-792	147	15	12	12	NUM
ejde-792	147	16	]	]	PUNCT
ejde-792	147	17	,	,	PUNCT
ejde-792	147	18	which	which	PRON
ejde-792	147	19	is	be	AUX
ejde-792	147	20	our	our	PRON
ejde-792	147	21	fundamental	fundamental	ADJ
ejde-792	147	22	tool	tool	NOUN
ejde-792	147	23	.	.	PUNCT
ejde-792	148	1	theorem	theorem	VERB
ejde-792	148	2	3.1	3.1	NUM
ejde-792	148	3	(	(	PUNCT
ejde-792	148	4	crandall	crandall	NOUN
ejde-792	148	5	-	-	PUNCT
ejde-792	148	6	rabinowitz	rabinowitz	NOUN
ejde-792	148	7	)	)	PUNCT
ejde-792	148	8	.	.	PUNCT
ejde-792	149	1	let	let	VERB
ejde-792	149	2	x	x	PRON
ejde-792	149	3	and	and	CCONJ
ejde-792	149	4	y	y	PROPN
ejde-792	149	5	be	be	AUX
ejde-792	149	6	real	real	ADJ
ejde-792	149	7	banach	banach	NOUN
ejde-792	149	8	spaces	space	NOUN
ejde-792	149	9	,	,	PUNCT
ejde-792	149	10	and	and	CCONJ
ejde-792	149	11	let	let	VERB
ejde-792	149	12	k	k	NOUN
ejde-792	149	13	:	:	PUNCT
ejde-792	149	14	x	x	X
ejde-792	149	15	→	→	SYM
ejde-792	149	16	y	y	X
ejde-792	149	17	be	be	AUX
ejde-792	149	18	two	two	NUM
ejde-792	149	19	bounded	bounded	ADJ
ejde-792	149	20	linear	linear	PROPN
ejde-792	149	21	operators	operator	NOUN
ejde-792	149	22	.	.	PUNCT
ejde-792	150	1	assume	assume	VERB
ejde-792	150	2	that	that	SCONJ
ejde-792	150	3	f	f	X
ejde-792	150	4	:	:	PUNCT
ejde-792	150	5	r	r	NOUN
ejde-792	150	6	×	×	NOUN
ejde-792	150	7	x	x	INTJ
ejde-792	150	8	→	→	SYM
ejde-792	150	9	y	y	PROPN
ejde-792	150	10	is	be	AUX
ejde-792	150	11	c2	c2	PROPN
ejde-792	150	12	near	near	ADV
ejde-792	150	13	(	(	PUNCT
ejde-792	150	14	λ∗	λ∗	PROPN
ejde-792	150	15	,	,	PUNCT
ejde-792	150	16	0	0	NUM
ejde-792	150	17	)	)	PUNCT
ejde-792	150	18	∈	∈	NOUN
ejde-792	150	19	r	r	NOUN
ejde-792	150	20	×x	×x	NOUN
ejde-792	150	21	with	with	ADP
ejde-792	150	22	f	f	PROPN
ejde-792	150	23	(	(	PUNCT
ejde-792	150	24	λ	λ	PROPN
ejde-792	150	25	,	,	PUNCT
ejde-792	150	26	0	0	NUM
ejde-792	150	27	)	)	PUNCT
ejde-792	150	28	=	=	SYM
ejde-792	150	29	0	0	NUM
ejde-792	150	30	for	for	ADP
ejde-792	150	31	a	a	DET
ejde-792	150	32	sufficiently	sufficiently	ADV
ejde-792	150	33	small	small	ADJ
ejde-792	150	34	|λ∗	|λ∗	PROPN
ejde-792	150	35	−	−	PROPN
ejde-792	150	36	λ|	λ|	PROPN
ejde-792	150	37	.	.	PUNCT
ejde-792	151	1	let	let	VERB
ejde-792	151	2	t	t	NOUN
ejde-792	151	3	=	=	SYM
ejde-792	151	4	fu(λ∗	fu(λ∗	PROPN
ejde-792	151	5	,	,	PUNCT
ejde-792	151	6	0	0	NUM
ejde-792	151	7	)	)	PUNCT
ejde-792	151	8	.	.	PUNCT
ejde-792	152	1	if	if	SCONJ
ejde-792	152	2	β	β	PROPN
ejde-792	152	3	=	=	SYM
ejde-792	152	4	0	0	NUM
ejde-792	152	5	is	be	AUX
ejde-792	152	6	a	a	DET
ejde-792	152	7	fλu(λ∗	fλu(λ∗	NOUN
ejde-792	152	8	,	,	PUNCT
ejde-792	152	9	0)-simple	0)-simple	NUM
ejde-792	152	10	eigenvalue	eigenvalue	PROPN
ejde-792	152	11	of	of	ADP
ejde-792	152	12	operator	operator	NOUN
ejde-792	152	13	t	t	PROPN
ejde-792	152	14	and	and	CCONJ
ejde-792	152	15	a	a	DET
ejde-792	152	16	k	k	ADJ
ejde-792	152	17	-	-	ADJ
ejde-792	152	18	simple	simple	ADJ
ejde-792	152	19	eigenvalue	eigenvalue	NOUN
ejde-792	152	20	of	of	ADP
ejde-792	152	21	t	t	PROPN
ejde-792	152	22	,	,	PUNCT
ejde-792	152	23	then	then	ADV
ejde-792	152	24	there	there	PRON
ejde-792	152	25	locally	locally	ADV
ejde-792	152	26	exists	exist	VERB
ejde-792	152	27	a	a	DET
ejde-792	152	28	curve	curve	NOUN
ejde-792	152	29	(	(	PUNCT
ejde-792	152	30	λ(s	λ(s	PROPN
ejde-792	152	31	)	)	PUNCT
ejde-792	152	32	,	,	PUNCT
ejde-792	152	33	u(s	u(s	PROPN
ejde-792	152	34	)	)	PUNCT
ejde-792	152	35	)	)	PUNCT
ejde-792	153	1	∈	∈	PROPN
ejde-792	153	2	r	r	NOUN
ejde-792	153	3	×	×	NOUN
ejde-792	153	4	x	x	PUNCT
ejde-792	153	5	such	such	ADJ
ejde-792	153	6	that	that	SCONJ
ejde-792	153	7	(	(	PUNCT
ejde-792	153	8	λ(0	λ(0	NOUN
ejde-792	153	9	)	)	PUNCT
ejde-792	153	10	,	,	PUNCT
ejde-792	153	11	u(0	u(0	NOUN
ejde-792	153	12	)	)	PUNCT
ejde-792	153	13	)	)	PUNCT
ejde-792	154	1	=	=	SYM
ejde-792	154	2	(	(	PUNCT
ejde-792	154	3	λ∗	λ∗	PROPN
ejde-792	154	4	,	,	PUNCT
ejde-792	154	5	0	0	NUM
ejde-792	154	6	)	)	PUNCT
ejde-792	154	7	and	and	CCONJ
ejde-792	154	8	f	f	X
ejde-792	154	9	(	(	PUNCT
ejde-792	154	10	λ(s	λ(s	PROPN
ejde-792	154	11	)	)	PUNCT
ejde-792	154	12	,	,	PUNCT
ejde-792	154	13	u(s	u(s	PROPN
ejde-792	154	14	)	)	PUNCT
ejde-792	154	15	)	)	PUNCT
ejde-792	155	1	=	=	PUNCT
ejde-792	155	2	0	0	X
ejde-792	155	3	.	.	PUNCT
ejde-792	156	1	moreover	moreover	ADV
ejde-792	156	2	,	,	PUNCT
ejde-792	156	3	if	if	SCONJ
ejde-792	156	4	f	f	PROPN
ejde-792	156	5	(	(	PUNCT
ejde-792	156	6	λ	λ	PROPN
ejde-792	156	7	,	,	PUNCT
ejde-792	156	8	u	u	NOUN
ejde-792	156	9	)	)	PUNCT
ejde-792	156	10	=	=	SYM
ejde-792	156	11	0	0	NUM
ejde-792	156	12	with	with	ADP
ejde-792	156	13	u	u	NOUN
ejde-792	156	14	̸=	̸=	PROPN
ejde-792	156	15	0	0	PUNCT
ejde-792	156	16	and	and	CCONJ
ejde-792	156	17	(	(	PUNCT
ejde-792	156	18	λ	λ	PROPN
ejde-792	156	19	,	,	PUNCT
ejde-792	156	20	u	u	NOUN
ejde-792	156	21	)	)	PUNCT
ejde-792	156	22	near	near	ADP
ejde-792	156	23	(	(	PUNCT
ejde-792	156	24	λ∗	λ∗	PROPN
ejde-792	156	25	,	,	PUNCT
ejde-792	156	26	0	0	NUM
ejde-792	156	27	)	)	PUNCT
ejde-792	156	28	,	,	PUNCT
ejde-792	156	29	then	then	ADV
ejde-792	156	30	(	(	PUNCT
ejde-792	156	31	λ	λ	NOUN
ejde-792	156	32	,	,	PUNCT
ejde-792	156	33	u	u	NOUN
ejde-792	156	34	)	)	PUNCT
ejde-792	156	35	=	=	SYM
ejde-792	156	36	(	(	PUNCT
ejde-792	156	37	λ(s	λ(s	PROPN
ejde-792	156	38	)	)	PUNCT
ejde-792	156	39	,	,	PUNCT
ejde-792	156	40	u(s	u(s	PROPN
ejde-792	156	41	)	)	PUNCT
ejde-792	156	42	)	)	PUNCT
ejde-792	157	1	for	for	ADP
ejde-792	157	2	some	some	PRON
ejde-792	157	3	s	s	VERB
ejde-792	157	4	̸=	̸=	PROPN
ejde-792	157	5	0	0	NUM
ejde-792	157	6	.	.	PUNCT
ejde-792	158	1	furthermore	furthermore	ADV
ejde-792	158	2	,	,	PUNCT
ejde-792	158	3	there	there	PRON
ejde-792	158	4	are	be	VERB
ejde-792	158	5	eigenvalues	eigenvalue	NOUN
ejde-792	158	6	β(s	β(	NOUN
ejde-792	158	7	)	)	PUNCT
ejde-792	158	8	and	and	CCONJ
ejde-792	158	9	βtriv(λ	βtriv(λ	X
ejde-792	158	10	)	)	PUNCT
ejde-792	158	11	∈	∈	PROPN
ejde-792	158	12	r	r	NOUN
ejde-792	158	13	with	with	ADP
ejde-792	158	14	eigenvectors	eigenvector	NOUN
ejde-792	158	15	φ(s	φ(	VERB
ejde-792	158	16	)	)	PUNCT
ejde-792	158	17	and	and	CCONJ
ejde-792	158	18	φtriv(λ	φtriv(λ	NOUN
ejde-792	158	19	)	)	PUNCT
ejde-792	158	20	∈	∈	PROPN
ejde-792	158	21	x	x	PUNCT
ejde-792	158	22	such	such	ADJ
ejde-792	158	23	that	that	SCONJ
ejde-792	158	24	fu(λ(s	fu(λ(	NOUN
ejde-792	158	25	)	)	PUNCT
ejde-792	158	26	,	,	PUNCT
ejde-792	158	27	u(s))φ(s	u(s))φ(s	NOUN
ejde-792	158	28	)	)	PUNCT
ejde-792	158	29	=	=	PUNCT
ejde-792	158	30	β(s)kφ(s	β(s)kφ(s	PUNCT
ejde-792	158	31	)	)	PUNCT
ejde-792	158	32	,	,	PUNCT
ejde-792	158	33	fu(λ	fu(λ	PROPN
ejde-792	158	34	,	,	PUNCT
ejde-792	158	35	0)φtriv(λ	0)φtriv(λ	NUM
ejde-792	158	36	)	)	PUNCT
ejde-792	158	37	=	=	SYM
ejde-792	158	38	βtriv(λ)kφtriv(λ	βtriv(λ)kφtriv(λ	NOUN
ejde-792	158	39	)	)	PUNCT
ejde-792	158	40	,	,	PUNCT
ejde-792	158	41	with	with	ADP
ejde-792	158	42	β(0	β(0	NOUN
ejde-792	158	43	)	)	PUNCT
ejde-792	158	44	=	=	PUNCT
ejde-792	158	45	βtriv(λ∗	βtriv(λ∗	NOUN
ejde-792	158	46	)	)	PUNCT
ejde-792	158	47	=	=	SYM
ejde-792	158	48	0	0	NUM
ejde-792	158	49	,	,	PUNCT
ejde-792	158	50	φ(0	φ(0	ADJ
ejde-792	158	51	)	)	PUNCT
ejde-792	158	52	=	=	SYM
ejde-792	158	53	φtriv(λ∗	φtriv(λ∗	PRON
ejde-792	158	54	)	)	PUNCT
ejde-792	159	1	=	=	PUNCT
ejde-792	159	2	φ∗.	φ∗.	NOUN
ejde-792	159	3	each	each	DET
ejde-792	159	4	curve	curve	NOUN
ejde-792	159	5	is	be	AUX
ejde-792	159	6	c1	c1	NOUN
ejde-792	159	7	if	if	SCONJ
ejde-792	159	8	f	f	PROPN
ejde-792	159	9	is	be	AUX
ejde-792	159	10	c2	c2	PROPN
ejde-792	159	11	.	.	PUNCT
ejde-792	160	1	then	then	ADV
ejde-792	160	2	,	,	PUNCT
ejde-792	160	3	dβtriv(λ	dβtriv(λ	PROPN
ejde-792	160	4	)	)	PUNCT
ejde-792	160	5	dλ	dλ	NOUN
ejde-792	160	6	|λ	|λ	NOUN
ejde-792	160	7	=	=	SYM
ejde-792	160	8	λ∗	λ∗	NOUN
ejde-792	160	9	̸=	̸=	PROPN
ejde-792	160	10	0	0	NUM
ejde-792	160	11	,	,	PUNCT
ejde-792	160	12	lim	lim	PROPN
ejde-792	160	13	s→0,β(s)̸=0	s→0,β(s)̸=0	PROPN
ejde-792	160	14	sλ′(s	sλ′(s	PROPN
ejde-792	160	15	)	)	PUNCT
ejde-792	160	16	β(s	β(	NOUN
ejde-792	160	17	)	)	PUNCT
ejde-792	161	1	=	=	PUNCT
ejde-792	162	1	−	−	PROPN
ejde-792	162	2	1	1	NUM
ejde-792	162	3	β′	β′	NUM
ejde-792	162	4	triv(λ∗	triv(λ∗	NOUN
ejde-792	162	5	)	)	PUNCT
ejde-792	162	6	.	.	PUNCT
ejde-792	163	1	by	by	ADP
ejde-792	163	2	theorem	theorem	NOUN
ejde-792	163	3	3.1	3.1	NUM
ejde-792	163	4	,	,	PUNCT
ejde-792	163	5	we	we	PRON
ejde-792	163	6	obtain	obtain	VERB
ejde-792	163	7	the	the	DET
ejde-792	163	8	following	follow	VERB
ejde-792	163	9	formula	formula	NOUN
ejde-792	163	10	,	,	PUNCT
ejde-792	163	11	which	which	PRON
ejde-792	163	12	is	be	AUX
ejde-792	163	13	convenient	convenient	ADJ
ejde-792	163	14	to	to	PART
ejde-792	163	15	be	be	AUX
ejde-792	163	16	used	use	VERB
ejde-792	163	17	.	.	PUNCT
ejde-792	164	1	proposition	proposition	NOUN
ejde-792	164	2	3.2	3.2	NUM
ejde-792	164	3	.	.	PUNCT
ejde-792	165	1	under	under	ADP
ejde-792	165	2	the	the	DET
ejde-792	165	3	assumption	assumption	NOUN
ejde-792	165	4	of	of	ADP
ejde-792	165	5	theorem	theorem	NOUN
ejde-792	165	6	3.1	3.1	NUM
ejde-792	165	7	,	,	PUNCT
ejde-792	165	8	we	we	PRON
ejde-792	165	9	have	have	VERB
ejde-792	165	10	that	that	PRON
ejde-792	165	11	lim	lim	PROPN
ejde-792	165	12	s→0,β(s)̸=0	s→0,β(s)̸=0	PROPN
ejde-792	165	13	sλ′(s	sλ′(s	PROPN
ejde-792	165	14	)	)	PUNCT
ejde-792	165	15	β(s	β(	NOUN
ejde-792	165	16	)	)	PUNCT
ejde-792	166	1	l(fλu	l(fλu	PROPN
ejde-792	166	2	(	(	PUNCT
ejde-792	166	3	λ1	λ1	PROPN
ejde-792	166	4	h0	h0	PROPN
ejde-792	166	5	,	,	PUNCT
ejde-792	166	6	0)φ1	0)φ1	PROPN
ejde-792	166	7	)	)	PUNCT
ejde-792	166	8	l(kφ1	l(kφ1	PROPN
ejde-792	166	9	)	)	PUNCT
ejde-792	166	10	=	=	SYM
ejde-792	166	11	−1	−1	NOUN
ejde-792	166	12	,	,	PUNCT
ejde-792	166	13	where	where	SCONJ
ejde-792	166	14	l	l	PROPN
ejde-792	166	15	∈	∈	NOUN
ejde-792	166	16	x∗	x∗	PROPN
ejde-792	166	17	satisfies	satisfy	VERB
ejde-792	166	18	n	n	PRON
ejde-792	166	19	(	(	PUNCT
ejde-792	166	20	l	l	NOUN
ejde-792	166	21	)	)	PUNCT
ejde-792	166	22	=	=	SYM
ejde-792	166	23	r(fu(λ1	r(fu(λ1	NOUN
ejde-792	166	24	/	/	SYM
ejde-792	166	25	h0	h0	PROPN
ejde-792	166	26	,	,	PUNCT
ejde-792	166	27	0	0	NUM
ejde-792	166	28	)	)	PUNCT
ejde-792	166	29	)	)	PUNCT
ejde-792	166	30	,	,	PUNCT
ejde-792	166	31	with	with	ADP
ejde-792	166	32	x∗	x∗	PROPN
ejde-792	166	33	being	be	AUX
ejde-792	166	34	the	the	DET
ejde-792	166	35	dual	dual	ADJ
ejde-792	166	36	space	space	NOUN
ejde-792	166	37	of	of	ADP
ejde-792	166	38	x.	x.	PROPN
ejde-792	166	39	in	in	ADP
ejde-792	166	40	particular	particular	ADJ
ejde-792	166	41	,	,	PUNCT
ejde-792	166	42	if	if	SCONJ
ejde-792	166	43	k	k	PROPN
ejde-792	166	44	=	=	SYM
ejde-792	166	45	fλu(λ1	fλu(λ1	PROPN
ejde-792	166	46	/	/	SYM
ejde-792	166	47	h0	h0	PROPN
ejde-792	166	48	,	,	PUNCT
ejde-792	166	49	0	0	NUM
ejde-792	166	50	)	)	PUNCT
ejde-792	166	51	,	,	PUNCT
ejde-792	166	52	then	then	ADV
ejde-792	166	53	lim	lim	PROPN
ejde-792	166	54	s→0,β(s)̸=0	s→0,β(s)̸=0	PROPN
ejde-792	166	55	sλ′(s	sλ′(s	PROPN
ejde-792	166	56	)	)	PUNCT
ejde-792	166	57	β(s	β(	NOUN
ejde-792	166	58	)	)	PUNCT
ejde-792	166	59	=	=	SYM
ejde-792	166	60	−1	−1	NOUN
ejde-792	166	61	,	,	PUNCT
ejde-792	166	62	and	and	CCONJ
ejde-792	166	63	β′	β′	NUM
ejde-792	166	64	triv(λ1	triv(λ1	NOUN
ejde-792	166	65	/	/	SYM
ejde-792	166	66	h0	h0	NOUN
ejde-792	166	67	)	)	PUNCT
ejde-792	166	68	=	=	SYM
ejde-792	167	1	1	1	X
ejde-792	167	2	.	.	PUNCT
ejde-792	167	3	proof	proof	NOUN
ejde-792	167	4	.	.	PUNCT
ejde-792	168	1	by	by	ADP
ejde-792	168	2	differentiating	differentiate	VERB
ejde-792	168	3	fu(λ	fu(λ	PROPN
ejde-792	168	4	,	,	PUNCT
ejde-792	168	5	0)φtriv(λ	0)φtriv(λ	NUM
ejde-792	168	6	)	)	PUNCT
ejde-792	168	7	=	=	SYM
ejde-792	168	8	βtriv(λ)kφtriv(λ	βtriv(λ)kφtriv(λ	NOUN
ejde-792	168	9	)	)	PUNCT
ejde-792	168	10	,	,	PUNCT
ejde-792	168	11	we	we	PRON
ejde-792	168	12	have	have	VERB
ejde-792	168	13	fλu(λ	fλu(λ	PROPN
ejde-792	168	14	,	,	PUNCT
ejde-792	168	15	0)φtriv(λ	0)φtriv(λ	PROPN
ejde-792	168	16	)	)	PUNCT
ejde-792	169	1	+	+	NUM
ejde-792	169	2	fu(λ	fu(λ	NUM
ejde-792	169	3	,	,	PUNCT
ejde-792	169	4	0)φ	0)φ	X
ejde-792	169	5	′	′	NUM
ejde-792	170	1	triv(λ	triv(λ	NOUN
ejde-792	170	2	)	)	PUNCT
ejde-792	170	3	=	=	PUNCT
ejde-792	171	1	βtriv(λ)kφ′	βtriv(λ)kφ′	NUM
ejde-792	171	2	triv(λ	triv(λ	NOUN
ejde-792	171	3	)	)	PUNCT
ejde-792	172	1	+	+	NUM
ejde-792	172	2	β′	β′	NUM
ejde-792	172	3	triv(λ)kφtriv(λ	triv(λ)kφtriv(λ	NOUN
ejde-792	172	4	)	)	PUNCT
ejde-792	172	5	.	.	PUNCT
ejde-792	173	1	taking	take	VERB
ejde-792	173	2	λ	λ	PROPN
ejde-792	173	3	=	=	SYM
ejde-792	173	4	λ1	λ1	PROPN
ejde-792	173	5	/	/	SYM
ejde-792	173	6	h0	h0	PROPN
ejde-792	173	7	,	,	PUNCT
ejde-792	173	8	we	we	PRON
ejde-792	173	9	can	can	AUX
ejde-792	173	10	obtain	obtain	VERB
ejde-792	173	11	fλu	fλu	NOUN
ejde-792	173	12	(	(	PUNCT
ejde-792	173	13	λ1	λ1	PROPN
ejde-792	173	14	h0	h0	NOUN
ejde-792	173	15	,	,	PUNCT
ejde-792	173	16	0	0	NUM
ejde-792	173	17	)	)	PUNCT
ejde-792	173	18	φ1	φ1	NOUN
ejde-792	173	19	+	+	CCONJ
ejde-792	173	20	fu	fu	NOUN
ejde-792	173	21	(	(	PUNCT
ejde-792	173	22	λ1	λ1	PROPN
ejde-792	173	23	h0	h0	PROPN
ejde-792	173	24	,	,	PUNCT
ejde-792	173	25	0	0	NUM
ejde-792	173	26	)	)	PUNCT
ejde-792	173	27	φ′	φ′	NUM
ejde-792	173	28	triv	triv	NOUN
ejde-792	173	29	(	(	PUNCT
ejde-792	173	30	λ1	λ1	PROPN
ejde-792	173	31	h0	h0	NOUN
ejde-792	173	32	)	)	PUNCT
ejde-792	173	33	=	=	PUNCT
ejde-792	174	1	β′	β′	NUM
ejde-792	174	2	triv	triv	VERB
ejde-792	174	3	(	(	PUNCT
ejde-792	174	4	λ1	λ1	PROPN
ejde-792	174	5	h0	h0	PROPN
ejde-792	174	6	)	)	PUNCT
ejde-792	174	7	kφ1	kφ1	PROPN
ejde-792	174	8	.	.	PUNCT
ejde-792	175	1	(	(	PUNCT
ejde-792	175	2	3.1	3.1	NUM
ejde-792	175	3	)	)	PUNCT
ejde-792	175	4	since	since	SCONJ
ejde-792	175	5	β	β	X
ejde-792	175	6	=	=	SYM
ejde-792	175	7	0	0	NUM
ejde-792	175	8	is	be	AUX
ejde-792	175	9	an	an	DET
ejde-792	175	10	fλu(λ1	fλu(λ1	PROPN
ejde-792	175	11	/	/	SYM
ejde-792	175	12	h0	h0	PROPN
ejde-792	175	13	,	,	PUNCT
ejde-792	175	14	0)-simple	0)-simple	PROPN
ejde-792	175	15	eigenvalue	eigenvalue	PROPN
ejde-792	175	16	of	of	ADP
ejde-792	175	17	operator	operator	NOUN
ejde-792	175	18	fu(λ1	fu(λ1	PROPN
ejde-792	175	19	/	/	SYM
ejde-792	175	20	h0	h0	PROPN
ejde-792	175	21	,	,	PUNCT
ejde-792	175	22	0	0	NUM
ejde-792	175	23	)	)	PUNCT
ejde-792	175	24	,	,	PUNCT
ejde-792	175	25	we	we	PRON
ejde-792	175	26	have	have	VERB
ejde-792	175	27	fλu	fλu	NOUN
ejde-792	175	28	(	(	PUNCT
ejde-792	175	29	λ1	λ1	PROPN
ejde-792	175	30	h0	h0	NOUN
ejde-792	175	31	,	,	PUNCT
ejde-792	175	32	0	0	NUM
ejde-792	175	33	)	)	PUNCT
ejde-792	175	34	φ1	φ1	PROPN
ejde-792	175	35	̸∈	̸∈	PROPN
ejde-792	175	36	r	r	PROPN
ejde-792	175	37	(	(	PUNCT
ejde-792	175	38	fu	fu	NOUN
ejde-792	175	39	(	(	PUNCT
ejde-792	175	40	λ1	λ1	PROPN
ejde-792	175	41	h0	h0	NOUN
ejde-792	175	42	,	,	PUNCT
ejde-792	175	43	0	0	NUM
ejde-792	175	44	)	)	PUNCT
ejde-792	175	45	)	)	PUNCT
ejde-792	175	46	.	.	PUNCT
ejde-792	176	1	8	8	NUM
ejde-792	176	2	s.	s.	PROPN
ejde-792	176	3	gao	gao	PROPN
ejde-792	176	4	,	,	PUNCT
ejde-792	176	5	q.	q.	PROPN
ejde-792	176	6	liu	liu	PROPN
ejde-792	176	7	,	,	PUNCT
ejde-792	176	8	y.	y.	PROPN
ejde-792	176	9	sun	sun	PROPN
ejde-792	176	10	ejde-2024/48	ejde-2024/48	NOUN
ejde-792	176	11	by	by	ADP
ejde-792	176	12	taking	take	VERB
ejde-792	176	13	l	l	NOUN
ejde-792	176	14	on	on	ADP
ejde-792	176	15	both	both	DET
ejde-792	176	16	sides	side	NOUN
ejde-792	176	17	of	of	ADP
ejde-792	176	18	equation	equation	NOUN
ejde-792	176	19	(	(	PUNCT
ejde-792	176	20	3.1	3.1	NUM
ejde-792	176	21	)	)	PUNCT
ejde-792	176	22	and	and	CCONJ
ejde-792	176	23	using	use	VERB
ejde-792	176	24	the	the	DET
ejde-792	176	25	fact	fact	NOUN
ejde-792	176	26	that	that	SCONJ
ejde-792	176	27	n	n	PROPN
ejde-792	176	28	(	(	PUNCT
ejde-792	176	29	l	l	NOUN
ejde-792	176	30	)	)	PUNCT
ejde-792	176	31	=	=	SYM
ejde-792	176	32	r(fu	r(fu	PROPN
ejde-792	176	33	(	(	PUNCT
ejde-792	176	34	λ1	λ1	PROPN
ejde-792	176	35	h0	h0	PROPN
ejde-792	176	36	,	,	PUNCT
ejde-792	176	37	0	0	NUM
ejde-792	176	38	)	)	PUNCT
ejde-792	176	39	)	)	PUNCT
ejde-792	176	40	,	,	PUNCT
ejde-792	176	41	we	we	PRON
ejde-792	176	42	obtain	obtain	VERB
ejde-792	176	43	that	that	DET
ejde-792	176	44	l	l	NOUN
ejde-792	176	45	(	(	PUNCT
ejde-792	176	46	fλu	fλu	NOUN
ejde-792	176	47	(	(	PUNCT
ejde-792	176	48	λ1	λ1	PROPN
ejde-792	176	49	h0	h0	NOUN
ejde-792	176	50	,	,	PUNCT
ejde-792	176	51	0	0	NUM
ejde-792	176	52	)	)	PUNCT
ejde-792	176	53	φ1	φ1	NOUN
ejde-792	176	54	)	)	PUNCT
ejde-792	176	55	=	=	PUNCT
ejde-792	176	56	β′	β′	NUM
ejde-792	176	57	triv	triv	VERB
ejde-792	176	58	(	(	PUNCT
ejde-792	176	59	λ1	λ1	PROPN
ejde-792	176	60	h0	h0	PROPN
ejde-792	176	61	)	)	PUNCT
ejde-792	176	62	l(kφ1	l(kφ1	PROPN
ejde-792	176	63	)	)	PUNCT
ejde-792	176	64	.	.	PUNCT
ejde-792	177	1	it	it	PRON
ejde-792	177	2	can	can	AUX
ejde-792	177	3	be	be	AUX
ejde-792	177	4	deduced	deduce	VERB
ejde-792	177	5	that	that	SCONJ
ejde-792	177	6	l(kφ1	l(kφ1	ADP
ejde-792	177	7	)	)	PUNCT
ejde-792	177	8	̸=	̸=	PROPN
ejde-792	177	9	0	0	NUM
ejde-792	177	10	;	;	PUNCT
ejde-792	177	11	consequently	consequently	ADV
ejde-792	177	12	,	,	PUNCT
ejde-792	177	13	β′	β′	NUM
ejde-792	177	14	triv	triv	VERB
ejde-792	177	15	(	(	PUNCT
ejde-792	177	16	λ1	λ1	PROPN
ejde-792	177	17	h0	h0	NOUN
ejde-792	177	18	)	)	PUNCT
ejde-792	178	1	=	=	X
ejde-792	178	2	l	l	NOUN
ejde-792	178	3	(	(	PUNCT
ejde-792	178	4	fλu	fλu	NOUN
ejde-792	178	5	(	(	PUNCT
ejde-792	178	6	λ1	λ1	PROPN
ejde-792	178	7	h0	h0	NOUN
ejde-792	178	8	,	,	PUNCT
ejde-792	178	9	0	0	NUM
ejde-792	178	10	)	)	PUNCT
ejde-792	178	11	φ1	φ1	NOUN
ejde-792	178	12	)	)	PUNCT
ejde-792	178	13	l(kφ1	l(kφ1	PROPN
ejde-792	178	14	)	)	PUNCT
ejde-792	178	15	,	,	PUNCT
ejde-792	178	16	which	which	PRON
ejde-792	178	17	yields	yield	VERB
ejde-792	178	18	the	the	DET
ejde-792	178	19	desired	desire	VERB
ejde-792	178	20	formula	formula	NOUN
ejde-792	178	21	.	.	PUNCT
ejde-792	179	1	□	□	PUNCT
ejde-792	179	2	before	before	ADP
ejde-792	179	3	providing	provide	VERB
ejde-792	179	4	the	the	DET
ejde-792	179	5	stability	stability	NOUN
ejde-792	179	6	result	result	NOUN
ejde-792	179	7	(	(	PUNCT
ejde-792	179	8	in	in	ADP
ejde-792	179	9	the	the	DET
ejde-792	179	10	linearized	linearize	VERB
ejde-792	179	11	sense	sense	NOUN
ejde-792	179	12	)	)	PUNCT
ejde-792	179	13	for	for	ADP
ejde-792	179	14	(	(	PUNCT
ejde-792	179	15	1.1	1.1	NUM
ejde-792	179	16	)	)	PUNCT
ejde-792	179	17	near	near	ADP
ejde-792	179	18	the	the	DET
ejde-792	179	19	bifurcation	bifurcation	NOUN
ejde-792	179	20	point	point	NOUN
ejde-792	179	21	,	,	PUNCT
ejde-792	179	22	we	we	PRON
ejde-792	179	23	review	review	VERB
ejde-792	179	24	the	the	DET
ejde-792	179	25	concept	concept	NOUN
ejde-792	179	26	of	of	ADP
ejde-792	179	27	stability	stability	NOUN
ejde-792	179	28	.	.	PUNCT
ejde-792	180	1	the	the	DET
ejde-792	180	2	operator	operator	NOUN
ejde-792	180	3	equation	equation	NOUN
ejde-792	180	4	f	f	X
ejde-792	180	5	(	(	PUNCT
ejde-792	180	6	λ	λ	PROPN
ejde-792	180	7	,	,	PUNCT
ejde-792	180	8	x	x	X
ejde-792	180	9	)	)	PUNCT
ejde-792	180	10	=	=	SYM
ejde-792	180	11	0	0	NUM
ejde-792	180	12	represents	represent	VERB
ejde-792	180	13	the	the	DET
ejde-792	180	14	equilibrium	equilibrium	NOUN
ejde-792	180	15	form	form	NOUN
ejde-792	180	16	of	of	ADP
ejde-792	180	17	the	the	DET
ejde-792	180	18	evolution	evolution	NOUN
ejde-792	180	19	equation	equation	NOUN
ejde-792	180	20	dx	dx	PROPN
ejde-792	180	21	dt	dt	NOUN
ejde-792	181	1	=	=	SYM
ejde-792	181	2	f	f	PROPN
ejde-792	181	3	(	(	PUNCT
ejde-792	181	4	λ	λ	PROPN
ejde-792	181	5	,	,	PUNCT
ejde-792	181	6	x	x	NOUN
ejde-792	181	7	)	)	PUNCT
ejde-792	181	8	.	.	PUNCT
ejde-792	182	1	(	(	PUNCT
ejde-792	182	2	3.2	3.2	NUM
ejde-792	182	3	)	)	PUNCT
ejde-792	182	4	suppose	suppose	VERB
ejde-792	182	5	that	that	SCONJ
ejde-792	182	6	f	f	PROPN
ejde-792	182	7	(	(	PUNCT
ejde-792	182	8	λ0	λ0	NOUN
ejde-792	182	9	,	,	PUNCT
ejde-792	182	10	x0	x0	NUM
ejde-792	182	11	)	)	PUNCT
ejde-792	183	1	=	=	SYM
ejde-792	183	2	0	0	X
ejde-792	183	3	.	.	PUNCT
ejde-792	184	1	if	if	SCONJ
ejde-792	184	2	all	all	DET
ejde-792	184	3	the	the	DET
ejde-792	184	4	eigenvalues	eigenvalue	NOUN
ejde-792	184	5	of	of	ADP
ejde-792	184	6	fx(λ0	fx(λ0	NOUN
ejde-792	184	7	,	,	PUNCT
ejde-792	184	8	x0	x0	PROPN
ejde-792	184	9	)	)	PUNCT
ejde-792	184	10	are	be	AUX
ejde-792	184	11	negative	negative	ADJ
ejde-792	184	12	,	,	PUNCT
ejde-792	184	13	then	then	ADV
ejde-792	184	14	x0	x0	PROPN
ejde-792	184	15	is	be	AUX
ejde-792	184	16	called	call	VERB
ejde-792	184	17	an	an	DET
ejde-792	184	18	asymptotically	asymptotically	ADV
ejde-792	184	19	linearly	linearly	ADV
ejde-792	184	20	stable	stable	ADJ
ejde-792	184	21	solution	solution	NOUN
ejde-792	184	22	of	of	ADP
ejde-792	184	23	(	(	PUNCT
ejde-792	184	24	3.2	3.2	NUM
ejde-792	184	25	)	)	PUNCT
ejde-792	184	26	.	.	PUNCT
ejde-792	185	1	on	on	ADP
ejde-792	185	2	the	the	DET
ejde-792	185	3	other	other	ADJ
ejde-792	185	4	hand	hand	NOUN
ejde-792	185	5	,	,	PUNCT
ejde-792	185	6	if	if	SCONJ
ejde-792	185	7	a	a	DET
ejde-792	185	8	positive	positive	ADJ
ejde-792	185	9	eigenvalue	eigenvalue	NOUN
ejde-792	185	10	of	of	ADP
ejde-792	185	11	fx(λ0	fx(λ0	NOUN
ejde-792	185	12	,	,	PUNCT
ejde-792	185	13	x0	x0	PROPN
ejde-792	185	14	)	)	PUNCT
ejde-792	185	15	exists	exist	VERB
ejde-792	185	16	,	,	PUNCT
ejde-792	185	17	then	then	ADV
ejde-792	185	18	x0	x0	PROPN
ejde-792	185	19	is	be	AUX
ejde-792	185	20	called	call	VERB
ejde-792	185	21	an	an	DET
ejde-792	185	22	unstable	unstable	ADJ
ejde-792	185	23	solution	solution	NOUN
ejde-792	185	24	of	of	ADP
ejde-792	185	25	(	(	PUNCT
ejde-792	185	26	3.2	3.2	NUM
ejde-792	185	27	)	)	PUNCT
ejde-792	185	28	.	.	PUNCT
ejde-792	186	1	proof	proof	NOUN
ejde-792	186	2	of	of	ADP
ejde-792	186	3	theorem	theorem	ADJ
ejde-792	186	4	1.2	1.2	NUM
ejde-792	186	5	.	.	PUNCT
ejde-792	187	1	let	let	VERB
ejde-792	187	2	x	x	PRON
ejde-792	187	3	be	be	AUX
ejde-792	187	4	a	a	DET
ejde-792	187	5	banach	banach	NOUN
ejde-792	187	6	space	space	NOUN
ejde-792	187	7	.	.	PUNCT
ejde-792	188	1	from	from	ADP
ejde-792	188	2	equations	equation	NOUN
ejde-792	188	3	(	(	PUNCT
ejde-792	188	4	2.1	2.1	NUM
ejde-792	188	5	)	)	PUNCT
ejde-792	188	6	and	and	CCONJ
ejde-792	188	7	(	(	PUNCT
ejde-792	188	8	2.3	2.3	NUM
ejde-792	188	9	)	)	PUNCT
ejde-792	188	10	,	,	PUNCT
ejde-792	188	11	it	it	PRON
ejde-792	188	12	can	can	AUX
ejde-792	188	13	be	be	AUX
ejde-792	188	14	inferred	infer	VERB
ejde-792	188	15	that	that	SCONJ
ejde-792	188	16	0	0	NUM
ejde-792	188	17	is	be	AUX
ejde-792	188	18	a	a	DET
ejde-792	188	19	simple	simple	ADJ
ejde-792	188	20	eigenvalue	eigenvalue	NOUN
ejde-792	188	21	of	of	ADP
ejde-792	188	22	fu(λ1	fu(λ1	PROPN
ejde-792	188	23	/	/	SYM
ejde-792	188	24	h0	h0	PROPN
ejde-792	188	25	,	,	PUNCT
ejde-792	188	26	0	0	NUM
ejde-792	188	27	)	)	PUNCT
ejde-792	188	28	:	:	PUNCT
ejde-792	189	1	=	=	PROPN
ejde-792	189	2	t	t	PROPN
ejde-792	189	3	.	.	PUNCT
ejde-792	190	1	let	let	VERB
ejde-792	190	2	φ1	φ1	PROPN
ejde-792	190	3	represent	represent	VERB
ejde-792	190	4	the	the	DET
ejde-792	190	5	eigenfunction	eigenfunction	NOUN
ejde-792	190	6	corresponding	correspond	VERB
ejde-792	190	7	to	to	ADP
ejde-792	190	8	the	the	DET
ejde-792	190	9	eigenvalue	eigenvalue	NOUN
ejde-792	190	10	0	0	PUNCT
ejde-792	190	11	with	with	ADP
ejde-792	190	12	∥φ1∥	∥φ1∥	PROPN
ejde-792	190	13	=	=	SYM
ejde-792	191	1	1	1	X
ejde-792	191	2	.	.	X
ejde-792	192	1	we	we	PRON
ejde-792	192	2	denote	denote	VERB
ejde-792	192	3	t	t	PROPN
ejde-792	192	4	=	=	SYM
ejde-792	192	5	fu(λ1	fu(λ1	NOUN
ejde-792	192	6	/	/	SYM
ejde-792	192	7	h0	h0	PROPN
ejde-792	192	8	,	,	PUNCT
ejde-792	192	9	0	0	NUM
ejde-792	192	10	)	)	PUNCT
ejde-792	192	11	and	and	CCONJ
ejde-792	192	12	k	k	PROPN
ejde-792	192	13	=	=	PUNCT
ejde-792	192	14	fλu(λ1	fλu(λ1	PROPN
ejde-792	192	15	/	/	SYM
ejde-792	192	16	h0	h0	PROPN
ejde-792	192	17	,	,	PUNCT
ejde-792	192	18	0	0	NUM
ejde-792	192	19	)	)	PUNCT
ejde-792	192	20	.	.	PUNCT
ejde-792	193	1	according	accord	VERB
ejde-792	193	2	to	to	ADP
ejde-792	193	3	theorem	theorem	ADJ
ejde-792	193	4	3.1	3.1	NUM
ejde-792	193	5	,	,	PUNCT
ejde-792	193	6	we	we	PRON
ejde-792	193	7	have	have	VERB
ejde-792	193	8	k[φ1	k[φ1	X
ejde-792	193	9	]	]	X
ejde-792	193	10	=	=	SYM
ejde-792	193	11	fλ(λ	fλ(λ	NUM
ejde-792	193	12	,	,	PUNCT
ejde-792	193	13	u	u	NOUN
ejde-792	193	14	)	)	PUNCT
ejde-792	193	15	∣∣	∣∣	NUM
ejde-792	193	16	(	(	PUNCT
ejde-792	193	17	λ	λ	INTJ
ejde-792	193	18	,	,	PUNCT
ejde-792	193	19	u)=(λ1	u)=(λ1	NOUN
ejde-792	193	20	/	/	SYM
ejde-792	193	21	h0,0	h0,0	NOUN
ejde-792	193	22	)	)	PUNCT
ejde-792	194	1	[	[	X
ejde-792	194	2	φ1	φ1	X
ejde-792	194	3	]	]	PUNCT
ejde-792	194	4	.	.	PUNCT
ejde-792	195	1	thus	thus	ADV
ejde-792	195	2	,	,	PUNCT
ejde-792	195	3	we	we	PRON
ejde-792	195	4	obtain	obtain	VERB
ejde-792	195	5	k[φ1	k[φ1	ADP
ejde-792	195	6	]	]	PUNCT
ejde-792	195	7	=	=	SYM
ejde-792	195	8	−nf(x)h(x	−nf(x)h(x	PROPN
ejde-792	195	9	,	,	PUNCT
ejde-792	195	10	0)φ1	0)φ1	NOUN
ejde-792	195	11	.	.	PUNCT
ejde-792	196	1	next	next	ADJ
ejde-792	196	2	,	,	PUNCT
ejde-792	196	3	we	we	PRON
ejde-792	196	4	need	need	VERB
ejde-792	196	5	to	to	PART
ejde-792	196	6	verify	verify	VERB
ejde-792	196	7	whether	whether	SCONJ
ejde-792	196	8	0	0	NUM
ejde-792	196	9	is	be	AUX
ejde-792	196	10	a	a	DET
ejde-792	196	11	simple	simple	ADJ
ejde-792	196	12	eigenvalue	eigenvalue	NOUN
ejde-792	196	13	of	of	ADP
ejde-792	196	14	k.	k.	PROPN
ejde-792	196	15	consider	consider	VERB
ejde-792	196	16	the	the	DET
ejde-792	196	17	eigenvalue	eigenvalue	NOUN
ejde-792	196	18	problem	problem	NOUN
ejde-792	196	19	k[φ1	k[φ1	NOUN
ejde-792	196	20	]	]	X
ejde-792	197	1	=	=	SYM
ejde-792	197	2	−nf(x)h(x	−nf(x)h(x	PROPN
ejde-792	197	3	,	,	PUNCT
ejde-792	197	4	0)φ1	0)φ1	X
ejde-792	197	5	=	=	SYM
ejde-792	197	6	0	0	X
ejde-792	197	7	.	.	PUNCT
ejde-792	198	1	this	this	PRON
ejde-792	198	2	implies	imply	VERB
ejde-792	198	3	that	that	SCONJ
ejde-792	198	4	φ1	φ1	PROPN
ejde-792	198	5	satisfies	satisfy	VERB
ejde-792	198	6	f(x)h(x	f(x)h(x	PRON
ejde-792	198	7	,	,	PUNCT
ejde-792	198	8	0)φ1	0)φ1	NOUN
ejde-792	198	9	=	=	SYM
ejde-792	198	10	0	0	NUM
ejde-792	198	11	.	.	PUNCT
ejde-792	199	1	for	for	ADP
ejde-792	199	2	nonzero	nonzero	PROPN
ejde-792	199	3	φ1	φ1	PROPN
ejde-792	199	4	,	,	PUNCT
ejde-792	199	5	this	this	PRON
ejde-792	199	6	can	can	AUX
ejde-792	199	7	only	only	ADV
ejde-792	199	8	hold	hold	VERB
ejde-792	199	9	if	if	SCONJ
ejde-792	199	10	f(x)h(x	f(x)h(x	NOUN
ejde-792	199	11	,	,	PUNCT
ejde-792	199	12	0	0	NUM
ejde-792	199	13	)	)	PUNCT
ejde-792	199	14	=	=	SYM
ejde-792	200	1	0	0	X
ejde-792	200	2	.	.	PUNCT
ejde-792	201	1	however	however	ADV
ejde-792	201	2	,	,	PUNCT
ejde-792	201	3	since	since	SCONJ
ejde-792	201	4	f(x	f(x	PROPN
ejde-792	201	5	)	)	PUNCT
ejde-792	201	6	̸=	̸=	PROPN
ejde-792	201	7	0	0	NUM
ejde-792	201	8	and	and	CCONJ
ejde-792	201	9	h(x	h(x	PROPN
ejde-792	201	10	,	,	PUNCT
ejde-792	201	11	0	0	NUM
ejde-792	201	12	)	)	PUNCT
ejde-792	201	13	̸=	̸=	NOUN
ejde-792	201	14	0	0	NUM
ejde-792	201	15	,	,	PUNCT
ejde-792	201	16	φ1	φ1	PROPN
ejde-792	201	17	must	must	AUX
ejde-792	201	18	be	be	AUX
ejde-792	201	19	zero	zero	NUM
ejde-792	201	20	,	,	PUNCT
ejde-792	201	21	which	which	PRON
ejde-792	201	22	contradicts	contradict	VERB
ejde-792	201	23	our	our	PRON
ejde-792	201	24	assumption	assumption	NOUN
ejde-792	201	25	.	.	PUNCT
ejde-792	202	1	this	this	PRON
ejde-792	202	2	indicates	indicate	VERB
ejde-792	202	3	that	that	SCONJ
ejde-792	202	4	k[φ1	k[φ1	NOUN
ejde-792	202	5	]	]	PUNCT
ejde-792	202	6	/∈	/∈	PUNCT
ejde-792	203	1	r(t	r(t	NOUN
ejde-792	203	2	)	)	PUNCT
ejde-792	203	3	.	.	PUNCT
ejde-792	204	1	therefore	therefore	ADV
ejde-792	204	2	,	,	PUNCT
ejde-792	204	3	0	0	NUM
ejde-792	204	4	is	be	AUX
ejde-792	204	5	a	a	DET
ejde-792	204	6	k	k	ADJ
ejde-792	204	7	-	-	ADJ
ejde-792	204	8	simple	simple	ADJ
ejde-792	204	9	eigenvalue	eigenvalue	NOUN
ejde-792	204	10	of	of	ADP
ejde-792	204	11	fu(λ1	fu(λ1	PROPN
ejde-792	204	12	/	/	SYM
ejde-792	204	13	h0	h0	PROPN
ejde-792	204	14	,	,	PUNCT
ejde-792	204	15	0	0	NUM
ejde-792	204	16	)	)	PUNCT
ejde-792	204	17	.	.	PUNCT
ejde-792	205	1	according	accord	VERB
ejde-792	205	2	to	to	ADP
ejde-792	205	3	theorem	theorem	ADJ
ejde-792	205	4	3.1	3.1	NUM
ejde-792	205	5	,	,	PUNCT
ejde-792	205	6	there	there	PRON
ejde-792	205	7	exist	exist	VERB
ejde-792	205	8	eigenvalues	eigenvalue	NOUN
ejde-792	205	9	β(s	β(	NOUN
ejde-792	205	10	)	)	PUNCT
ejde-792	205	11	and	and	CCONJ
ejde-792	205	12	βtriv(λ	βtriv(λ	X
ejde-792	205	13	)	)	PUNCT
ejde-792	205	14	∈	∈	PROPN
ejde-792	205	15	r	r	NOUN
ejde-792	205	16	and	and	CCONJ
ejde-792	205	17	eigenvectors	eigenvector	NOUN
ejde-792	205	18	ϕ1(s	ϕ1(	NOUN
ejde-792	205	19	)	)	PUNCT
ejde-792	205	20	and	and	CCONJ
ejde-792	205	21	ϕtriv(λ	ϕtriv(λ	NUM
ejde-792	205	22	)	)	PUNCT
ejde-792	205	23	in	in	ADP
ejde-792	205	24	the	the	DET
ejde-792	205	25	vector	vector	NOUN
ejde-792	205	26	space	space	NOUN
ejde-792	205	27	x	x	PUNCT
ejde-792	205	28	such	such	ADJ
ejde-792	205	29	that	that	SCONJ
ejde-792	205	30	fu(λ(s	fu(λ(	NOUN
ejde-792	205	31	)	)	PUNCT
ejde-792	205	32	,	,	PUNCT
ejde-792	205	33	u(s))ϕ1(s	u(s))ϕ1(s	ADP
ejde-792	205	34	)	)	PUNCT
ejde-792	205	35	=	=	SYM
ejde-792	205	36	β(s)fλu(λ1	β(s)fλu(λ1	NOUN
ejde-792	205	37	/	/	SYM
ejde-792	205	38	h0	h0	PROPN
ejde-792	205	39	,	,	PUNCT
ejde-792	205	40	0)ϕ1(s	0)ϕ1(s	NOUN
ejde-792	205	41	)	)	PUNCT
ejde-792	205	42	,	,	PUNCT
ejde-792	205	43	fu(λ	fu(λ	PROPN
ejde-792	205	44	,	,	PUNCT
ejde-792	205	45	0)ϕtriv(λ	0)ϕtriv(λ	NUM
ejde-792	205	46	)	)	PUNCT
ejde-792	205	47	=	=	SYM
ejde-792	206	1	βtriv(λ)fλu(λ1	βtriv(λ)fλu(λ1	PROPN
ejde-792	206	2	/	/	SYM
ejde-792	206	3	h0	h0	PROPN
ejde-792	206	4	,	,	PUNCT
ejde-792	206	5	0)ϕtriv(λ	0)ϕtriv(λ	NUM
ejde-792	206	6	)	)	PUNCT
ejde-792	206	7	with	with	ADP
ejde-792	206	8	β(0	β(0	NOUN
ejde-792	206	9	)	)	PUNCT
ejde-792	206	10	=	=	SYM
ejde-792	206	11	βtriv(λ1	βtriv(λ1	NOUN
ejde-792	206	12	/	/	SYM
ejde-792	206	13	h0	h0	PROPN
ejde-792	206	14	)	)	PUNCT
ejde-792	206	15	=	=	SYM
ejde-792	206	16	0	0	NUM
ejde-792	206	17	,	,	PUNCT
ejde-792	206	18	ϕ1(0	ϕ1(0	PROPN
ejde-792	206	19	)	)	PUNCT
ejde-792	207	1	=	=	SYM
ejde-792	207	2	ϕtriv(λ1	ϕtriv(λ1	NOUN
ejde-792	207	3	/	/	SYM
ejde-792	207	4	h0	h0	PROPN
ejde-792	207	5	)	)	PUNCT
ejde-792	207	6	=	=	SYM
ejde-792	207	7	φ1	φ1	PROPN
ejde-792	207	8	.	.	PUNCT
ejde-792	208	1	each	each	DET
ejde-792	208	2	curve	curve	NOUN
ejde-792	208	3	is	be	AUX
ejde-792	208	4	c1	c1	NOUN
ejde-792	208	5	if	if	SCONJ
ejde-792	208	6	f	f	PROPN
ejde-792	208	7	belongs	belong	VERB
ejde-792	208	8	to	to	ADP
ejde-792	208	9	c2	c2	PROPN
ejde-792	208	10	;	;	PUNCT
ejde-792	208	11	then	then	ADV
ejde-792	208	12	,	,	PUNCT
ejde-792	208	13	dβtriv(λ	dβtriv(λ	PROPN
ejde-792	208	14	)	)	PUNCT
ejde-792	208	15	dλ	dλ	NOUN
ejde-792	208	16	∣∣	∣∣	X
ejde-792	208	17	λ	λ	X
ejde-792	208	18	=	=	SYM
ejde-792	208	19	λ1	λ1	ADJ
ejde-792	208	20	/	/	SYM
ejde-792	208	21	h0	h0	NOUN
ejde-792	208	22	̸=	̸=	PROPN
ejde-792	208	23	0	0	NUM
ejde-792	208	24	,	,	PUNCT
ejde-792	208	25	lim	lim	PROPN
ejde-792	208	26	s→0,β(s	s→0,β(	VERB
ejde-792	208	27	)	)	PUNCT
ejde-792	208	28	̸=0	̸=0	ADJ
ejde-792	208	29	sλ′(s	sλ′(s	PROPN
ejde-792	208	30	)	)	PUNCT
ejde-792	208	31	β(s	β(	NOUN
ejde-792	208	32	)	)	PUNCT
ejde-792	209	1	=	=	PUNCT
ejde-792	209	2	−	−	PROPN
ejde-792	209	3	1	1	NUM
ejde-792	209	4	β′	β′	NUM
ejde-792	209	5	triv(λ1	triv(λ1	NOUN
ejde-792	209	6	/	/	SYM
ejde-792	209	7	h0	h0	NOUN
ejde-792	209	8	)	)	PUNCT
ejde-792	209	9	.	.	PUNCT
ejde-792	210	1	ejde-2024/48	ejde-2024/48	NOUN
ejde-792	210	2	local	local	ADJ
ejde-792	210	3	bifurcation	bifurcation	NOUN
ejde-792	210	4	structure	structure	NOUN
ejde-792	210	5	and	and	CCONJ
ejde-792	210	6	stability	stability	NOUN
ejde-792	210	7	9	9	NUM
ejde-792	210	8	by	by	ADP
ejde-792	210	9	(	(	PUNCT
ejde-792	210	10	2.2	2.2	NUM
ejde-792	210	11	)	)	PUNCT
ejde-792	210	12	and	and	CCONJ
ejde-792	210	13	(	(	PUNCT
ejde-792	210	14	2.5	2.5	NUM
ejde-792	210	15	)	)	PUNCT
ejde-792	210	16	,	,	PUNCT
ejde-792	210	17	we	we	PRON
ejde-792	210	18	obtain	obtain	VERB
ejde-792	210	19	l	l	NOUN
ejde-792	210	20	(	(	PUNCT
ejde-792	210	21	fλu	fλu	NOUN
ejde-792	210	22	(	(	PUNCT
ejde-792	210	23	λ1	λ1	PROPN
ejde-792	210	24	h0	h0	NOUN
ejde-792	210	25	,	,	PUNCT
ejde-792	210	26	0	0	NUM
ejde-792	210	27	)	)	PUNCT
ejde-792	210	28	φ1	φ1	NOUN
ejde-792	210	29	)	)	PUNCT
ejde-792	211	1	=	=	SYM
ejde-792	211	2	∫	∫	PROPN
ejde-792	211	3	ω	ω	NUM
ejde-792	211	4	fλu	fλu	NOUN
ejde-792	211	5	(	(	PUNCT
ejde-792	211	6	λ1	λ1	PROPN
ejde-792	211	7	h0	h0	PROPN
ejde-792	211	8	,	,	PUNCT
ejde-792	211	9	0	0	NUM
ejde-792	211	10	)	)	PUNCT
ejde-792	211	11	φ1	φ1	NOUN
ejde-792	211	12	·	·	PUNCT
ejde-792	211	13	φ1	φ1	PROPN
ejde-792	211	14	dx	dx	PROPN
ejde-792	211	15	.	.	PROPN
ejde-792	212	1	from	from	ADP
ejde-792	212	2	(	(	PUNCT
ejde-792	212	3	2.4	2.4	NUM
ejde-792	212	4	)	)	PUNCT
ejde-792	212	5	,	,	PUNCT
ejde-792	212	6	we	we	PRON
ejde-792	212	7	have	have	VERB
ejde-792	212	8	l	l	NOUN
ejde-792	212	9	(	(	PUNCT
ejde-792	212	10	fλu	fλu	NOUN
ejde-792	212	11	(	(	PUNCT
ejde-792	212	12	λ1	λ1	PROPN
ejde-792	212	13	h0	h0	NOUN
ejde-792	212	14	,	,	PUNCT
ejde-792	212	15	0	0	NUM
ejde-792	212	16	)	)	PUNCT
ejde-792	212	17	φ1	φ1	NOUN
ejde-792	212	18	)	)	PUNCT
ejde-792	212	19	=	=	PUNCT
ejde-792	213	1	fh0	fh0	PROPN
ejde-792	213	2	∫	∫	PROPN
ejde-792	213	3	ω	ω	PROPN
ejde-792	213	4	φ2	φ2	PROPN
ejde-792	213	5	1	1	NUM
ejde-792	213	6	dx	dx	PROPN
ejde-792	213	7	=	=	SYM
ejde-792	213	8	fh0	fh0	X
ejde-792	213	9	·	·	PUNCT
ejde-792	213	10	1	1	NUM
ejde-792	213	11	=	=	SYM
ejde-792	213	12	fh0	fh0	X
ejde-792	213	13	>	>	X
ejde-792	213	14	0	0	X
ejde-792	213	15	.	.	PUNCT
ejde-792	214	1	this	this	PRON
ejde-792	214	2	suggests	suggest	VERB
ejde-792	214	3	that	that	SCONJ
ejde-792	214	4	if	if	SCONJ
ejde-792	214	5	β(s	β(	NOUN
ejde-792	214	6	)	)	PUNCT
ejde-792	214	7	>	>	X
ejde-792	214	8	0	0	PUNCT
ejde-792	214	9	(	(	PUNCT
ejde-792	214	10	β(s	β(	NOUN
ejde-792	214	11	)	)	PUNCT
ejde-792	214	12	<	<	X
ejde-792	214	13	0	0	NUM
ejde-792	214	14	)	)	PUNCT
ejde-792	214	15	,	,	PUNCT
ejde-792	214	16	u(s	u(s	PROPN
ejde-792	214	17	)	)	PUNCT
ejde-792	214	18	is	be	AUX
ejde-792	214	19	(	(	PUNCT
ejde-792	214	20	formally	formally	ADV
ejde-792	214	21	)	)	PUNCT
ejde-792	214	22	unstable	unstable	ADJ
ejde-792	214	23	(	(	PUNCT
ejde-792	214	24	stable	stable	ADJ
ejde-792	214	25	)	)	PUNCT
ejde-792	214	26	.	.	PUNCT
ejde-792	215	1	by	by	ADP
ejde-792	215	2	proposition	proposition	NOUN
ejde-792	215	3	3.2	3.2	NUM
ejde-792	215	4	,	,	PUNCT
ejde-792	215	5	we	we	PRON
ejde-792	215	6	can	can	AUX
ejde-792	215	7	deduce	deduce	VERB
ejde-792	215	8	that	that	PRON
ejde-792	215	9	lim	lim	PROPN
ejde-792	215	10	s→0	s→0	PROPN
ejde-792	215	11	sλ′(s	sλ′(s	PROPN
ejde-792	215	12	)	)	PUNCT
ejde-792	215	13	β(s	β(	NOUN
ejde-792	215	14	)	)	PUNCT
ejde-792	215	15	=	=	PUNCT
ejde-792	215	16	−1	−1	NOUN
ejde-792	215	17	.	.	PUNCT
ejde-792	216	1	if	if	SCONJ
ejde-792	216	2	huu(x	huu(x	NOUN
ejde-792	216	3	,	,	PUNCT
ejde-792	216	4	0	0	NUM
ejde-792	216	5	)	)	PUNCT
ejde-792	216	6	≡	≡	PROPN
ejde-792	216	7	0	0	PUNCT
ejde-792	217	1	in	in	ADP
ejde-792	217	2	ω	ω	PROPN
ejde-792	217	3	but	but	CCONJ
ejde-792	217	4	huuu(x	huuu(x	PROPN
ejde-792	217	5	,	,	PUNCT
ejde-792	217	6	0	0	NUM
ejde-792	217	7	)	)	PUNCT
ejde-792	217	8	̸=	̸=	NOUN
ejde-792	217	9	0	0	NUM
ejde-792	217	10	in	in	ADP
ejde-792	217	11	ω	ω	NUM
ejde-792	217	12	,	,	PUNCT
ejde-792	217	13	it	it	PRON
ejde-792	217	14	has	have	AUX
ejde-792	217	15	been	be	AUX
ejde-792	217	16	demonstrated	demonstrate	VERB
ejde-792	217	17	that	that	SCONJ
ejde-792	217	18	λ′(0	λ′(0	VERB
ejde-792	217	19	)	)	PUNCT
ejde-792	217	20	=	=	SYM
ejde-792	218	1	0	0	X
ejde-792	218	2	.	.	PUNCT
ejde-792	219	1	we	we	PRON
ejde-792	219	2	can	can	AUX
ejde-792	219	3	write	write	VERB
ejde-792	219	4	λ′(s	λ′(s	X
ejde-792	219	5	)	)	PUNCT
ejde-792	219	6	=	=	SYM
ejde-792	219	7	sλ′′(0	sλ′′(0	NOUN
ejde-792	219	8	)	)	PUNCT
ejde-792	220	1	+	+	NOUN
ejde-792	220	2	o(s2	o(s2	NOUN
ejde-792	220	3	)	)	PUNCT
ejde-792	220	4	.	.	PUNCT
ejde-792	221	1	thus	thus	ADV
ejde-792	221	2	,	,	PUNCT
ejde-792	221	3	we	we	PRON
ejde-792	221	4	see	see	VERB
ejde-792	221	5	that	that	SCONJ
ejde-792	221	6	lim	lim	PROPN
ejde-792	221	7	s→0	s→0	AUX
ejde-792	221	8	s2λ′′(0	s2λ′′(0	PUNCT
ejde-792	221	9	)	)	PUNCT
ejde-792	222	1	+	+	NOUN
ejde-792	222	2	o(s3	o(s3	NOUN
ejde-792	222	3	)	)	PUNCT
ejde-792	222	4	β(s	β(	NOUN
ejde-792	222	5	)	)	PUNCT
ejde-792	222	6	=	=	PUNCT
ejde-792	223	1	−1	−1	NOUN
ejde-792	223	2	.	.	PUNCT
ejde-792	224	1	furthermore	furthermore	ADV
ejde-792	224	2	,	,	PUNCT
ejde-792	224	3	it	it	PRON
ejde-792	224	4	can	can	AUX
ejde-792	224	5	be	be	AUX
ejde-792	224	6	observed	observe	VERB
ejde-792	224	7	that	that	SCONJ
ejde-792	224	8	lim	lim	PROPN
ejde-792	224	9	s→0	s→0	PUNCT
ejde-792	224	10	λ′′(0	λ′′(0	NUM
ejde-792	224	11	)	)	PUNCT
ejde-792	224	12	β(s	β(	NOUN
ejde-792	224	13	)	)	PUNCT
ejde-792	224	14	s2	s2	NOUN
ejde-792	224	15	=	=	PUNCT
ejde-792	224	16	−1	−1	NOUN
ejde-792	224	17	.	.	PUNCT
ejde-792	225	1	we	we	PRON
ejde-792	225	2	therefore	therefore	ADV
ejde-792	225	3	have	have	VERB
ejde-792	225	4	lim	lim	PROPN
ejde-792	225	5	s→0	s→0	PROPN
ejde-792	225	6	β(s	β(s	PROPN
ejde-792	225	7	)	)	PUNCT
ejde-792	225	8	s2	s2	NOUN
ejde-792	225	9	=	=	PUNCT
ejde-792	225	10	−λ′′(0	−λ′′(0	PROPN
ejde-792	225	11	)	)	PUNCT
ejde-792	225	12	.	.	PUNCT
ejde-792	226	1	(	(	PUNCT
ejde-792	226	2	3.3	3.3	NUM
ejde-792	226	3	)	)	PUNCT
ejde-792	226	4	if	if	SCONJ
ejde-792	226	5	λ′′(0	λ′′(0	NOUN
ejde-792	226	6	)	)	PUNCT
ejde-792	226	7	>	>	X
ejde-792	226	8	0	0	NUM
ejde-792	226	9	,	,	PUNCT
ejde-792	226	10	we	we	PRON
ejde-792	226	11	can	can	AUX
ejde-792	226	12	conclude	conclude	VERB
ejde-792	226	13	from	from	ADP
ejde-792	226	14	(	(	PUNCT
ejde-792	226	15	3.3	3.3	NUM
ejde-792	226	16	)	)	PUNCT
ejde-792	226	17	that	that	SCONJ
ejde-792	226	18	for	for	ADP
ejde-792	226	19	small	small	ADJ
ejde-792	226	20	values	value	NOUN
ejde-792	226	21	of	of	ADP
ejde-792	226	22	|s|	|s|	PROPN
ejde-792	226	23	,	,	PUNCT
ejde-792	226	24	β(s	β(	NOUN
ejde-792	226	25	)	)	PUNCT
ejde-792	226	26	is	be	AUX
ejde-792	226	27	negative	negative	ADJ
ejde-792	226	28	.	.	PUNCT
ejde-792	227	1	on	on	ADP
ejde-792	227	2	the	the	DET
ejde-792	227	3	other	other	ADJ
ejde-792	227	4	hand	hand	NOUN
ejde-792	227	5	,	,	PUNCT
ejde-792	227	6	if	if	SCONJ
ejde-792	227	7	λ′′(0	λ′′(0	NOUN
ejde-792	227	8	)	)	PUNCT
ejde-792	227	9	<	<	X
ejde-792	227	10	0	0	NUM
ejde-792	227	11	,	,	PUNCT
ejde-792	227	12	we	we	PRON
ejde-792	227	13	can	can	AUX
ejde-792	227	14	conclude	conclude	VERB
ejde-792	227	15	from	from	ADP
ejde-792	227	16	(	(	PUNCT
ejde-792	227	17	3.3	3.3	NUM
ejde-792	227	18	)	)	PUNCT
ejde-792	227	19	that	that	SCONJ
ejde-792	227	20	for	for	ADP
ejde-792	227	21	small	small	ADJ
ejde-792	227	22	values	value	NOUN
ejde-792	227	23	of	of	ADP
ejde-792	227	24	|s|	|s|	PROPN
ejde-792	227	25	,	,	PUNCT
ejde-792	227	26	β(s	β(	NOUN
ejde-792	227	27	)	)	PUNCT
ejde-792	227	28	is	be	AUX
ejde-792	227	29	positive	positive	ADJ
ejde-792	227	30	.	.	PUNCT
ejde-792	228	1	therefore	therefore	ADV
ejde-792	228	2	,	,	PUNCT
ejde-792	228	3	when	when	SCONJ
ejde-792	228	4	huuu(x	huuu(x	ADP
ejde-792	228	5	,	,	PUNCT
ejde-792	228	6	0	0	NUM
ejde-792	228	7	)	)	PUNCT
ejde-792	228	8	>	>	X
ejde-792	228	9	0	0	PUNCT
ejde-792	229	1	in	in	ADP
ejde-792	229	2	ω	ω	NUM
ejde-792	229	3	,	,	PUNCT
ejde-792	229	4	we	we	PRON
ejde-792	229	5	have	have	VERB
ejde-792	229	6	λ′′(0	λ′′(0	NOUN
ejde-792	229	7	)	)	PUNCT
ejde-792	229	8	>	>	X
ejde-792	229	9	0	0	PUNCT
ejde-792	229	10	along	along	ADP
ejde-792	229	11	the	the	DET
ejde-792	229	12	nontrivial	nontrivial	NOUN
ejde-792	229	13	bifurcating	bifurcate	VERB
ejde-792	229	14	curve	curve	NOUN
ejde-792	229	15	passing	pass	VERB
ejde-792	229	16	through	through	ADP
ejde-792	229	17	(	(	PUNCT
ejde-792	229	18	λ1	λ1	PROPN
ejde-792	229	19	/	/	SYM
ejde-792	229	20	h0	h0	PROPN
ejde-792	229	21	,	,	PUNCT
ejde-792	229	22	0	0	NUM
ejde-792	229	23	)	)	PUNCT
ejde-792	229	24	,	,	PUNCT
ejde-792	229	25	indicating	indicate	VERB
ejde-792	229	26	asymptotic	asymptotic	ADJ
ejde-792	229	27	linear	linear	ADJ
ejde-792	229	28	stability	stability	NOUN
ejde-792	229	29	of	of	ADP
ejde-792	229	30	the	the	DET
ejde-792	229	31	nontrivial	nontrivial	ADJ
ejde-792	229	32	solutions	solution	NOUN
ejde-792	229	33	.	.	PUNCT
ejde-792	230	1	similarly	similarly	ADV
ejde-792	230	2	,	,	PUNCT
ejde-792	230	3	when	when	SCONJ
ejde-792	230	4	huuu(x	huuu(x	ADP
ejde-792	230	5	,	,	PUNCT
ejde-792	230	6	0	0	NUM
ejde-792	230	7	)	)	PUNCT
ejde-792	230	8	<	<	X
ejde-792	230	9	0	0	PUNCT
ejde-792	230	10	in	in	ADP
ejde-792	230	11	ω	ω	PROPN
ejde-792	230	12	,	,	PUNCT
ejde-792	230	13	we	we	PRON
ejde-792	230	14	have	have	VERB
ejde-792	230	15	λ′′(0	λ′′(0	NOUN
ejde-792	230	16	)	)	PUNCT
ejde-792	230	17	<	<	X
ejde-792	230	18	0	0	PUNCT
ejde-792	230	19	along	along	ADP
ejde-792	230	20	the	the	DET
ejde-792	230	21	nontrivial	nontrivial	NOUN
ejde-792	230	22	bifurcating	bifurcate	VERB
ejde-792	230	23	curve	curve	NOUN
ejde-792	230	24	passing	pass	VERB
ejde-792	230	25	through	through	ADP
ejde-792	230	26	(	(	PUNCT
ejde-792	230	27	λ1	λ1	PROPN
ejde-792	230	28	/	/	SYM
ejde-792	230	29	h0	h0	PROPN
ejde-792	230	30	,	,	PUNCT
ejde-792	230	31	0	0	NUM
ejde-792	230	32	)	)	PUNCT
ejde-792	230	33	,	,	PUNCT
ejde-792	230	34	indicating	indicate	VERB
ejde-792	230	35	the	the	DET
ejde-792	230	36	asymptotic	asymptotic	ADJ
ejde-792	230	37	linear	linear	NOUN
ejde-792	230	38	instability	instability	NOUN
ejde-792	230	39	of	of	ADP
ejde-792	230	40	the	the	DET
ejde-792	230	41	nontrivial	nontrivial	ADJ
ejde-792	230	42	solutions	solution	NOUN
ejde-792	230	43	.	.	PUNCT
ejde-792	231	1	□	□	PUNCT
ejde-792	231	2	acknwledgments	acknwledgment	NOUN
ejde-792	231	3	.	.	PUNCT
ejde-792	232	1	we	we	PRON
ejde-792	232	2	would	would	AUX
ejde-792	232	3	like	like	VERB
ejde-792	232	4	to	to	PART
ejde-792	232	5	express	express	VERB
ejde-792	232	6	our	our	PRON
ejde-792	232	7	deepest	deep	ADJ
ejde-792	232	8	gratitude	gratitude	NOUN
ejde-792	232	9	to	to	ADP
ejde-792	232	10	our	our	PRON
ejde-792	232	11	doctoral	doctoral	ADJ
ejde-792	232	12	adviser	adviser	NOUN
ejde-792	232	13	,	,	PUNCT
ejde-792	232	14	professor	professor	NOUN
ejde-792	232	15	guowei	guowei	PROPN
ejde-792	232	16	dai	dai	PROPN
ejde-792	232	17	,	,	PUNCT
ejde-792	232	18	for	for	ADP
ejde-792	232	19	his	his	PRON
ejde-792	232	20	invaluable	invaluable	ADJ
ejde-792	232	21	guidance	guidance	NOUN
ejde-792	232	22	and	and	CCONJ
ejde-792	232	23	support	support	NOUN
ejde-792	232	24	in	in	ADP
ejde-792	232	25	the	the	DET
ejde-792	232	26	selection	selection	NOUN
ejde-792	232	27	of	of	ADP
ejde-792	232	28	our	our	PRON
ejde-792	232	29	research	research	NOUN
ejde-792	232	30	topic	topic	NOUN
ejde-792	232	31	,	,	PUNCT
ejde-792	232	32	as	as	ADV
ejde-792	232	33	well	well	ADV
ejde-792	232	34	as	as	ADP
ejde-792	232	35	through	through	ADP
ejde-792	232	36	the	the	DET
ejde-792	232	37	writing	writing	NOUN
ejde-792	232	38	and	and	CCONJ
ejde-792	232	39	research	research	NOUN
ejde-792	232	40	process	process	NOUN
ejde-792	232	41	.	.	PUNCT
ejde-792	233	1	his	his	PRON
ejde-792	233	2	contributions	contribution	NOUN
ejde-792	233	3	have	have	AUX
ejde-792	233	4	been	be	AUX
ejde-792	233	5	instrumental	instrumental	ADJ
ejde-792	233	6	in	in	ADP
ejde-792	233	7	the	the	DET
ejde-792	233	8	successful	successful	ADJ
ejde-792	233	9	completion	completion	NOUN
ejde-792	233	10	of	of	ADP
ejde-792	233	11	this	this	DET
ejde-792	233	12	work	work	NOUN
ejde-792	233	13	.	.	PUNCT
ejde-792	234	1	this	this	DET
ejde-792	234	2	research	research	NOUN
ejde-792	234	3	was	be	AUX
ejde-792	234	4	supported	support	VERB
ejde-792	234	5	by	by	ADP
ejde-792	234	6	the	the	DET
ejde-792	234	7	nnsf	nnsf	PROPN
ejde-792	234	8	of	of	ADP
ejde-792	234	9	china	china	PROPN
ejde-792	234	10	(	(	PUNCT
ejde-792	234	11	no	no	INTJ
ejde-792	234	12	.	.	NOUN
ejde-792	234	13	12371110	12371110	NUM
ejde-792	234	14	)	)	PUNCT
ejde-792	234	15	.	.	PUNCT
ejde-792	235	1	references	reference	NOUN
ejde-792	235	2	[	[	X
ejde-792	235	3	1	1	X
ejde-792	235	4	]	]	PUNCT
ejde-792	235	5	j.	j.	PROPN
ejde-792	235	6	l.	l.	PROPN
ejde-792	235	7	barbosa	barbosa	PROPN
ejde-792	235	8	,	,	PUNCT
ejde-792	235	9	m.	m.	PROPN
ejde-792	235	10	d.	d.	PROPN
ejde-792	235	11	carmo	carmo	PROPN
ejde-792	235	12	;	;	PUNCT
ejde-792	235	13	stability	stability	NOUN
ejde-792	235	14	of	of	ADP
ejde-792	235	15	hypersurfaces	hypersurface	NOUN
ejde-792	235	16	with	with	ADP
ejde-792	235	17	constant	constant	ADJ
ejde-792	235	18	mean	mean	NOUN
ejde-792	235	19	curvature	curvature	NOUN
ejde-792	235	20	,	,	PUNCT
ejde-792	235	21	math	math	NOUN
ejde-792	235	22	.	.	PUNCT
ejde-792	236	1	z.	z.	PROPN
ejde-792	236	2	,	,	PUNCT
ejde-792	236	3	185	185	NUM
ejde-792	236	4	(	(	PUNCT
ejde-792	236	5	1984	1984	NUM
ejde-792	236	6	)	)	PUNCT
ejde-792	236	7	,	,	PUNCT
ejde-792	236	8	339–353	339–353	NUM
ejde-792	236	9	.	.	PUNCT
ejde-792	237	1	[	[	X
ejde-792	237	2	2	2	X
ejde-792	237	3	]	]	PUNCT
ejde-792	237	4	j.	j.	PROPN
ejde-792	237	5	l.	l.	PROPN
ejde-792	237	6	barbosa	barbosa	PROPN
ejde-792	237	7	,	,	PUNCT
ejde-792	237	8	m.	m.	PROPN
ejde-792	237	9	d.	d.	PROPN
ejde-792	237	10	carmo	carmo	PROPN
ejde-792	237	11	,	,	PUNCT
ejde-792	237	12	j.	j.	PROPN
ejde-792	237	13	eschenburg	eschenburg	PROPN
ejde-792	237	14	;	;	PUNCT
ejde-792	237	15	stability	stability	NOUN
ejde-792	237	16	of	of	ADP
ejde-792	237	17	hypersurfaces	hypersurface	NOUN
ejde-792	237	18	of	of	ADP
ejde-792	237	19	constant	constant	ADJ
ejde-792	237	20	mean	mean	ADJ
ejde-792	237	21	curvature	curvature	NOUN
ejde-792	237	22	in	in	ADP
ejde-792	237	23	riemannian	riemannian	ADJ
ejde-792	237	24	manifolds	manifold	NOUN
ejde-792	237	25	,	,	PUNCT
ejde-792	237	26	math	math	NOUN
ejde-792	237	27	.	.	PUNCT
ejde-792	238	1	z.	z.	PROPN
ejde-792	238	2	,	,	PUNCT
ejde-792	238	3	197	197	NUM
ejde-792	238	4	(	(	PUNCT
ejde-792	238	5	1988	1988	NUM
ejde-792	238	6	)	)	PUNCT
ejde-792	238	7	,	,	PUNCT
ejde-792	238	8	123–138	123–138	NUM
ejde-792	238	9	.	.	PUNCT
ejde-792	239	1	[	[	X
ejde-792	239	2	3	3	NUM
ejde-792	239	3	]	]	X
ejde-792	239	4	a.	a.	NOUN
ejde-792	239	5	barros	barros	PROPN
ejde-792	239	6	,	,	PUNCT
ejde-792	239	7	a.	a.	PROPN
ejde-792	239	8	brasil	brasil	PROPN
ejde-792	239	9	,	,	PUNCT
ejde-792	239	10	a.	a.	NOUN
ejde-792	239	11	caminha	caminha	NOUN
ejde-792	239	12	;	;	PUNCT
ejde-792	239	13	stability	stability	NOUN
ejde-792	239	14	of	of	ADP
ejde-792	239	15	spacelike	spacelike	ADJ
ejde-792	239	16	hypersurfaces	hypersurface	NOUN
ejde-792	239	17	in	in	ADP
ejde-792	239	18	foliated	foliated	ADJ
ejde-792	239	19	spacetimes	spacetime	NOUN
ejde-792	239	20	,	,	PUNCT
ejde-792	239	21	differential	differential	ADJ
ejde-792	239	22	geom	geom	NOUN
ejde-792	239	23	.	.	PUNCT
ejde-792	240	1	appl	appl	PROPN
ejde-792	240	2	.	.	PROPN
ejde-792	240	3	,	,	PUNCT
ejde-792	240	4	26	26	NUM
ejde-792	240	5	(	(	PUNCT
ejde-792	240	6	2008	2008	NUM
ejde-792	240	7	)	)	PUNCT
ejde-792	240	8	,	,	PUNCT
ejde-792	240	9	357	357	NUM
ejde-792	240	10	-	-	SYM
ejde-792	240	11	365	365	NUM
ejde-792	240	12	.	.	PUNCT
ejde-792	241	1	[	[	X
ejde-792	241	2	4	4	NUM
ejde-792	241	3	]	]	X
ejde-792	241	4	r.	r.	PROPN
ejde-792	241	5	bartnik	bartnik	PROPN
ejde-792	241	6	,	,	PUNCT
ejde-792	241	7	l.	l.	PROPN
ejde-792	241	8	simon	simon	PROPN
ejde-792	241	9	;	;	PUNCT
ejde-792	241	10	spacelike	spacelike	PROPN
ejde-792	241	11	hypersurfaces	hypersurface	NOUN
ejde-792	241	12	with	with	ADP
ejde-792	241	13	prescribed	prescribe	VERB
ejde-792	241	14	boundary	boundary	ADJ
ejde-792	241	15	values	value	NOUN
ejde-792	241	16	and	and	CCONJ
ejde-792	241	17	mean	mean	ADJ
ejde-792	241	18	curvature	curvature	NOUN
ejde-792	241	19	,	,	PUNCT
ejde-792	241	20	comm	comm	NOUN
ejde-792	241	21	.	.	PUNCT
ejde-792	241	22	math	math	NOUN
ejde-792	241	23	.	.	PUNCT
ejde-792	242	1	phys	phy	NOUN
ejde-792	242	2	.	.	PUNCT
ejde-792	243	1	87	87	NUM
ejde-792	243	2	(	(	PUNCT
ejde-792	243	3	1982	1982	NUM
ejde-792	243	4	-	-	SYM
ejde-792	243	5	1983	1983	NUM
ejde-792	243	6	)	)	PUNCT
ejde-792	243	7	,	,	PUNCT
ejde-792	243	8	131–152	131–152	NUM
ejde-792	243	9	.	.	PUNCT
ejde-792	244	1	10	10	NUM
ejde-792	244	2	s.	s.	PROPN
ejde-792	244	3	gao	gao	PROPN
ejde-792	244	4	,	,	PUNCT
ejde-792	244	5	q.	q.	PROPN
ejde-792	244	6	liu	liu	PROPN
ejde-792	244	7	,	,	PUNCT
ejde-792	244	8	y.	y.	PROPN
ejde-792	244	9	sun	sun	PROPN
ejde-792	244	10	ejde-2024/48	ejde-2024/48	NOUN
ejde-792	244	11	[	[	X
ejde-792	244	12	5	5	X
ejde-792	244	13	]	]	X
ejde-792	244	14	c.	c.	PROPN
ejde-792	244	15	bereanu	bereanu	PROPN
ejde-792	244	16	,	,	PUNCT
ejde-792	244	17	d.	d.	PROPN
ejde-792	244	18	de	de	PROPN
ejde-792	244	19	la	la	PROPN
ejde-792	244	20	fuente	fuente	PROPN
ejde-792	244	21	,	,	PUNCT
ejde-792	244	22	a.	a.	PROPN
ejde-792	244	23	romero	romero	PROPN
ejde-792	244	24	,	,	PUNCT
ejde-792	244	25	p.j	p.j	PROPN
ejde-792	244	26	.	.	PROPN
ejde-792	244	27	torres	torre	NOUN
ejde-792	244	28	;	;	PUNCT
ejde-792	244	29	existence	existence	NOUN
ejde-792	244	30	and	and	CCONJ
ejde-792	244	31	multiplicity	multiplicity	NOUN
ejde-792	244	32	of	of	ADP
ejde-792	244	33	entire	entire	ADJ
ejde-792	244	34	radial	radial	ADJ
ejde-792	244	35	space	space	NOUN
ejde-792	244	36	like	like	ADP
ejde-792	244	37	graphs	graph	NOUN
ejde-792	244	38	with	with	ADP
ejde-792	244	39	prescribed	prescribed	ADJ
ejde-792	244	40	mean	mean	NOUN
ejde-792	244	41	curvature	curvature	NOUN
ejde-792	244	42	function	function	NOUN
ejde-792	244	43	in	in	ADP
ejde-792	244	44	certain	certain	ADJ
ejde-792	244	45	friedmannlemâıtre	friedmannlemâıtre	PROPN
ejde-792	244	46	-	-	PUNCT
ejde-792	244	47	robertson	robertson	PROPN
ejde-792	244	48	-	-	PUNCT
ejde-792	244	49	walker	walker	PROPN
ejde-792	244	50	space	space	PROPN
ejde-792	244	51	times	times	PROPN
ejde-792	244	52	,	,	PUNCT
ejde-792	244	53	commun	commun	PROPN
ejde-792	244	54	.	.	PUNCT
ejde-792	244	55	contemp	contemp	PROPN
ejde-792	244	56	.	.	PUNCT
ejde-792	245	1	math	math	NOUN
ejde-792	245	2	.	.	PUNCT
ejde-792	245	3	,	,	PUNCT
ejde-792	245	4	19	19	NUM
ejde-792	245	5	(	(	PUNCT
ejde-792	245	6	2017	2017	NUM
ejde-792	245	7	)	)	PUNCT
ejde-792	245	8	,	,	PUNCT
ejde-792	245	9	1–18	1–18	NUM
ejde-792	245	10	.	.	PUNCT
ejde-792	246	1	[	[	X
ejde-792	246	2	6	6	NUM
ejde-792	246	3	]	]	X
ejde-792	246	4	c.	c.	PROPN
ejde-792	246	5	bereanu	bereanu	PROPN
ejde-792	246	6	,	,	PUNCT
ejde-792	246	7	p.	p.	PROPN
ejde-792	246	8	jebelean	jebelean	PROPN
ejde-792	246	9	,	,	PUNCT
ejde-792	246	10	j.	j.	PROPN
ejde-792	246	11	mawhin	mawhin	PROPN
ejde-792	246	12	;	;	PUNCT
ejde-792	246	13	the	the	DET
ejde-792	246	14	dirichlet	dirichlet	PROPN
ejde-792	246	15	problem	problem	NOUN
ejde-792	246	16	with	with	ADP
ejde-792	246	17	mean	mean	ADJ
ejde-792	246	18	curvature	curvature	NOUN
ejde-792	246	19	operator	operator	NOUN
ejde-792	246	20	in	in	ADP
ejde-792	246	21	minkowski	minkowski	ADJ
ejde-792	246	22	space	space	NOUN
ejde-792	246	23	-	-	PUNCT
ejde-792	246	24	a	a	DET
ejde-792	246	25	variational	variational	ADJ
ejde-792	246	26	approach	approach	NOUN
ejde-792	246	27	,	,	PUNCT
ejde-792	246	28	adv	adv	PROPN
ejde-792	246	29	.	.	PUNCT
ejde-792	246	30	nonlinear	nonlinear	ADJ
ejde-792	246	31	stud	stud	NOUN
ejde-792	246	32	.	.	PUNCT
ejde-792	247	1	14	14	NUM
ejde-792	247	2	(	(	PUNCT
ejde-792	247	3	2014	2014	NUM
ejde-792	247	4	)	)	PUNCT
ejde-792	247	5	,	,	PUNCT
ejde-792	247	6	315–326	315–326	NUM
ejde-792	247	7	.	.	PUNCT
ejde-792	248	1	[	[	X
ejde-792	248	2	7	7	X
ejde-792	248	3	]	]	X
ejde-792	248	4	c.	c.	PROPN
ejde-792	248	5	bereanu	bereanu	PROPN
ejde-792	248	6	,	,	PUNCT
ejde-792	248	7	p.	p.	PROPN
ejde-792	248	8	jebelean	jebelean	PROPN
ejde-792	248	9	,	,	PUNCT
ejde-792	248	10	p.	p.	PROPN
ejde-792	248	11	j.	j.	PROPN
ejde-792	248	12	torres	torres	PROPN
ejde-792	248	13	;	;	PUNCT
ejde-792	248	14	positive	positive	ADJ
ejde-792	248	15	radial	radial	ADJ
ejde-792	248	16	solutions	solution	NOUN
ejde-792	248	17	for	for	ADP
ejde-792	248	18	dirichlet	dirichlet	PROPN
ejde-792	248	19	problems	problem	NOUN
ejde-792	248	20	with	with	ADP
ejde-792	248	21	mean	mean	ADJ
ejde-792	248	22	curvature	curvature	NOUN
ejde-792	248	23	operators	operator	NOUN
ejde-792	248	24	in	in	ADP
ejde-792	248	25	minkowski	minkowski	ADJ
ejde-792	248	26	space	space	NOUN
ejde-792	248	27	,	,	PUNCT
ejde-792	248	28	j.	j.	PROPN
ejde-792	248	29	funct	funct	PROPN
ejde-792	248	30	.	.	PUNCT
ejde-792	249	1	anal	anal	PROPN
ejde-792	249	2	.	.	PROPN
ejde-792	249	3	,	,	PUNCT
ejde-792	249	4	264	264	NUM
ejde-792	249	5	(	(	PUNCT
ejde-792	249	6	2013	2013	NUM
ejde-792	249	7	)	)	PUNCT
ejde-792	249	8	,	,	PUNCT
ejde-792	249	9	270–287	270–287	NUM
ejde-792	249	10	.	.	PUNCT
ejde-792	250	1	[	[	X
ejde-792	250	2	8	8	NUM
ejde-792	250	3	]	]	X
ejde-792	250	4	c.	c.	PROPN
ejde-792	250	5	bereanu	bereanu	PROPN
ejde-792	250	6	,	,	PUNCT
ejde-792	250	7	p.	p.	PROPN
ejde-792	250	8	jebelean	jebelean	PROPN
ejde-792	250	9	,	,	PUNCT
ejde-792	250	10	p.	p.	PROPN
ejde-792	250	11	j.	j.	PROPN
ejde-792	250	12	torres	torres	PROPN
ejde-792	250	13	;	;	PUNCT
ejde-792	250	14	multiple	multiple	ADJ
ejde-792	250	15	positive	positive	ADJ
ejde-792	250	16	radial	radial	ADJ
ejde-792	250	17	solutions	solution	NOUN
ejde-792	250	18	for	for	ADP
ejde-792	250	19	a	a	DET
ejde-792	250	20	dirichlet	dirichlet	PROPN
ejde-792	250	21	problem	problem	NOUN
ejde-792	250	22	involving	involve	VERB
ejde-792	250	23	the	the	DET
ejde-792	250	24	mean	mean	ADJ
ejde-792	250	25	curvature	curvature	NOUN
ejde-792	250	26	operator	operator	NOUN
ejde-792	250	27	in	in	ADP
ejde-792	250	28	minkowski	minkowski	ADJ
ejde-792	250	29	space	space	NOUN
ejde-792	250	30	,	,	PUNCT
ejde-792	250	31	j.	j.	PROPN
ejde-792	250	32	funct	funct	PROPN
ejde-792	250	33	.	.	PUNCT
ejde-792	251	1	anal	anal	PROPN
ejde-792	251	2	.	.	PUNCT
ejde-792	252	1	265	265	NUM
ejde-792	252	2	(	(	PUNCT
ejde-792	252	3	2013	2013	NUM
ejde-792	252	4	)	)	PUNCT
ejde-792	252	5	,	,	PUNCT
ejde-792	252	6	644	644	NUM
ejde-792	252	7	–	–	PUNCT
ejde-792	252	8	659	659	NUM
ejde-792	252	9	.	.	PUNCT
ejde-792	253	1	[	[	X
ejde-792	253	2	9	9	NUM
ejde-792	253	3	]	]	PUNCT
ejde-792	253	4	e.	e.	PROPN
ejde-792	253	5	calabi	calabi	PROPN
ejde-792	253	6	;	;	PUNCT
ejde-792	253	7	examples	example	NOUN
ejde-792	253	8	of	of	ADP
ejde-792	253	9	berstein	berstein	ADJ
ejde-792	253	10	problems	problem	NOUN
ejde-792	253	11	for	for	ADP
ejde-792	253	12	some	some	DET
ejde-792	253	13	nonlinear	nonlinear	ADJ
ejde-792	253	14	equations	equation	NOUN
ejde-792	253	15	,	,	PUNCT
ejde-792	253	16	proc	proc	NOUN
ejde-792	253	17	.	.	PUNCT
ejde-792	254	1	sym	sym	NOUN
ejde-792	254	2	.	.	PUNCT
ejde-792	255	1	global	global	ADJ
ejde-792	255	2	analysis	analysis	NOUN
ejde-792	255	3	,	,	PUNCT
ejde-792	255	4	univ	univ	PROPN
ejde-792	255	5	.	.	PROPN
ejde-792	255	6	of	of	ADP
ejde-792	255	7	calif	calif	PROPN
ejde-792	255	8	.	.	PROPN
ejde-792	255	9	,	,	PUNCT
ejde-792	255	10	berkeley	berkeley	PROPN
ejde-792	255	11	,	,	PUNCT
ejde-792	255	12	1968	1968	NUM
ejde-792	255	13	.	.	PUNCT
ejde-792	256	1	[	[	X
ejde-792	256	2	10	10	NUM
ejde-792	256	3	]	]	PUNCT
ejde-792	256	4	s.-y	s.-y	NOUN
ejde-792	256	5	.	.	PUNCT
ejde-792	257	1	cheng	cheng	PROPN
ejde-792	257	2	,	,	PUNCT
ejde-792	257	3	s.-t	s.-t	PROPN
ejde-792	257	4	.	.	PUNCT
ejde-792	258	1	yau	yau	PROPN
ejde-792	258	2	;	;	PUNCT
ejde-792	258	3	maximal	maximal	ADJ
ejde-792	258	4	spacelike	spacelike	ADJ
ejde-792	258	5	hypersurfaces	hypersurface	NOUN
ejde-792	258	6	in	in	ADP
ejde-792	258	7	the	the	DET
ejde-792	258	8	lorentz	lorentz	PROPN
ejde-792	258	9	-	-	PUNCT
ejde-792	258	10	minkowski	minkowski	ADJ
ejde-792	258	11	spaces	space	NOUN
ejde-792	258	12	,	,	PUNCT
ejde-792	258	13	ann	ann	PROPN
ejde-792	258	14	.	.	PROPN
ejde-792	258	15	of	of	ADP
ejde-792	258	16	math	math	NOUN
ejde-792	258	17	.	.	PUNCT
ejde-792	258	18	,	,	PUNCT
ejde-792	258	19	104	104	NUM
ejde-792	258	20	(	(	PUNCT
ejde-792	258	21	1976	1976	NUM
ejde-792	258	22	)	)	PUNCT
ejde-792	258	23	,	,	PUNCT
ejde-792	258	24	407–419	407–419	NUM
ejde-792	258	25	.	.	PUNCT
ejde-792	259	1	[	[	X
ejde-792	259	2	11	11	NUM
ejde-792	259	3	]	]	X
ejde-792	259	4	c.	c.	NOUN
ejde-792	259	5	corsato	corsato	PROPN
ejde-792	259	6	,	,	PUNCT
ejde-792	259	7	f.	f.	PROPN
ejde-792	259	8	obersnel	obersnel	PROPN
ejde-792	259	9	,	,	PUNCT
ejde-792	259	10	p.	p.	PROPN
ejde-792	259	11	omari	omari	PROPN
ejde-792	259	12	,	,	PUNCT
ejde-792	259	13	s.	s.	PROPN
ejde-792	259	14	rivetti	rivetti	PROPN
ejde-792	259	15	;	;	PUNCT
ejde-792	259	16	positive	positive	ADJ
ejde-792	259	17	solutions	solution	NOUN
ejde-792	259	18	of	of	ADP
ejde-792	259	19	the	the	DET
ejde-792	259	20	dirichlet	dirichlet	PROPN
ejde-792	259	21	problem	problem	NOUN
ejde-792	259	22	for	for	ADP
ejde-792	259	23	the	the	DET
ejde-792	259	24	prescribed	prescribed	ADJ
ejde-792	259	25	mean	mean	NOUN
ejde-792	259	26	curvature	curvature	NOUN
ejde-792	259	27	equation	equation	NOUN
ejde-792	259	28	in	in	ADP
ejde-792	259	29	minkowski	minkowski	ADJ
ejde-792	259	30	space	space	NOUN
ejde-792	259	31	,	,	PUNCT
ejde-792	259	32	j.	j.	PROPN
ejde-792	259	33	math	math	PROPN
ejde-792	259	34	.	.	PUNCT
ejde-792	260	1	anal	anal	PROPN
ejde-792	260	2	.	.	PUNCT
ejde-792	260	3	appl	appl	PROPN
ejde-792	260	4	.	.	PROPN
ejde-792	261	1	,	,	PUNCT
ejde-792	261	2	405	405	NUM
ejde-792	261	3	(	(	PUNCT
ejde-792	261	4	2013	2013	NUM
ejde-792	261	5	)	)	PUNCT
ejde-792	261	6	,	,	PUNCT
ejde-792	261	7	227–239	227–239	NUM
ejde-792	261	8	.	.	PUNCT
ejde-792	262	1	[	[	X
ejde-792	262	2	12	12	NUM
ejde-792	262	3	]	]	PUNCT
ejde-792	262	4	m.	m.	NOUN
ejde-792	262	5	g.	g.	PROPN
ejde-792	262	6	crandall	crandall	PROPN
ejde-792	262	7	,	,	PUNCT
ejde-792	262	8	p.	p.	PROPN
ejde-792	262	9	h.	h.	PROPN
ejde-792	262	10	rabinowitz	rabinowitz	PROPN
ejde-792	262	11	;	;	PUNCT
ejde-792	262	12	bifurcation	bifurcation	NOUN
ejde-792	262	13	,	,	PUNCT
ejde-792	262	14	perturbation	perturbation	NOUN
ejde-792	262	15	of	of	ADP
ejde-792	262	16	simple	simple	ADJ
ejde-792	262	17	eigenvalues	eigenvalue	NOUN
ejde-792	262	18	and	and	CCONJ
ejde-792	262	19	linearied	linearie	VERB
ejde-792	262	20	stability	stability	NOUN
ejde-792	262	21	,	,	PUNCT
ejde-792	262	22	arch	arch	NOUN
ejde-792	262	23	.	.	PUNCT
ejde-792	263	1	ration	ration	NOUN
ejde-792	263	2	.	.	PUNCT
ejde-792	264	1	mech	mech	PROPN
ejde-792	264	2	.	.	PUNCT
ejde-792	265	1	anal	anal	PROPN
ejde-792	265	2	.	.	PROPN
ejde-792	265	3	,	,	PUNCT
ejde-792	265	4	52	52	NUM
ejde-792	265	5	(	(	PUNCT
ejde-792	265	6	1973	1973	NUM
ejde-792	265	7	)	)	PUNCT
ejde-792	265	8	,	,	PUNCT
ejde-792	265	9	161–180	161–180	NUM
ejde-792	265	10	.	.	PUNCT
ejde-792	266	1	[	[	X
ejde-792	266	2	13	13	NUM
ejde-792	266	3	]	]	X
ejde-792	266	4	g.	g.	PROPN
ejde-792	266	5	dai	dai	PROPN
ejde-792	266	6	;	;	PUNCT
ejde-792	266	7	bifurcation	bifurcation	NOUN
ejde-792	266	8	and	and	CCONJ
ejde-792	266	9	positive	positive	ADJ
ejde-792	266	10	solutions	solution	NOUN
ejde-792	266	11	for	for	ADP
ejde-792	266	12	problem	problem	NOUN
ejde-792	266	13	with	with	ADP
ejde-792	266	14	mean	mean	ADJ
ejde-792	266	15	curvature	curvature	NOUN
ejde-792	266	16	operator	operator	NOUN
ejde-792	266	17	in	in	ADP
ejde-792	266	18	minkowski	minkowski	ADJ
ejde-792	266	19	space	space	NOUN
ejde-792	266	20	,	,	PUNCT
ejde-792	266	21	calc	calc	NOUN
ejde-792	266	22	.	.	PUNCT
ejde-792	267	1	var	var	PROPN
ejde-792	267	2	.	.	PUNCT
ejde-792	268	1	,	,	PUNCT
ejde-792	268	2	55	55	NUM
ejde-792	268	3	(	(	PUNCT
ejde-792	268	4	2016	2016	NUM
ejde-792	268	5	)	)	PUNCT
ejde-792	268	6	,	,	PUNCT
ejde-792	268	7	1–17	1–17	PROPN
ejde-792	268	8	.	.	PUNCT
ejde-792	269	1	[	[	X
ejde-792	269	2	14	14	NUM
ejde-792	269	3	]	]	X
ejde-792	269	4	g.	g.	PROPN
ejde-792	269	5	dai	dai	PROPN
ejde-792	269	6	;	;	PUNCT
ejde-792	269	7	global	global	ADJ
ejde-792	269	8	bifurcation	bifurcation	NOUN
ejde-792	269	9	for	for	ADP
ejde-792	269	10	problem	problem	NOUN
ejde-792	269	11	with	with	ADP
ejde-792	269	12	mean	mean	ADJ
ejde-792	269	13	curvature	curvature	NOUN
ejde-792	269	14	operator	operator	NOUN
ejde-792	269	15	on	on	ADP
ejde-792	269	16	general	general	ADJ
ejde-792	269	17	domain	domain	NOUN
ejde-792	269	18	,	,	PUNCT
ejde-792	269	19	nonlinear	nonlinear	NOUN
ejde-792	269	20	differ	differ	VERB
ejde-792	269	21	.	.	PUNCT
ejde-792	270	1	equ	equ	PROPN
ejde-792	270	2	.	.	PUNCT
ejde-792	270	3	appl	appl	PROPN
ejde-792	270	4	.	.	PROPN
ejde-792	270	5	,	,	PUNCT
ejde-792	270	6	24	24	NUM
ejde-792	270	7	(	(	PUNCT
ejde-792	270	8	2017	2017	NUM
ejde-792	270	9	)	)	PUNCT
ejde-792	270	10	,	,	PUNCT
ejde-792	270	11	30	30	NUM
ejde-792	270	12	.	.	PUNCT
ejde-792	271	1	[	[	X
ejde-792	271	2	15	15	NUM
ejde-792	271	3	]	]	X
ejde-792	271	4	g.	g.	PROPN
ejde-792	271	5	dai	dai	PROPN
ejde-792	271	6	;	;	PUNCT
ejde-792	271	7	global	global	ADJ
ejde-792	271	8	structure	structure	NOUN
ejde-792	271	9	of	of	ADP
ejde-792	271	10	one	one	NUM
ejde-792	271	11	-	-	PUNCT
ejde-792	271	12	sign	sign	NOUN
ejde-792	271	13	solutions	solution	NOUN
ejde-792	271	14	for	for	ADP
ejde-792	271	15	problem	problem	NOUN
ejde-792	271	16	with	with	ADP
ejde-792	271	17	mean	mean	ADJ
ejde-792	271	18	curvature	curvature	NOUN
ejde-792	271	19	operator	operator	NOUN
ejde-792	271	20	.	.	PUNCT
ejde-792	272	1	nonlinearity	nonlinearity	NOUN
ejde-792	272	2	,	,	PUNCT
ejde-792	272	3	31	31	NUM
ejde-792	272	4	(	(	PUNCT
ejde-792	272	5	2018	2018	NUM
ejde-792	272	6	)	)	PUNCT
ejde-792	272	7	,	,	PUNCT
ejde-792	272	8	5309–5328	5309–5328	NUM
ejde-792	272	9	.	.	PUNCT
ejde-792	273	1	[	[	X
ejde-792	273	2	16	16	NUM
ejde-792	273	3	]	]	X
ejde-792	273	4	g.	g.	PROPN
ejde-792	273	5	dai	dai	PROPN
ejde-792	273	6	;	;	PUNCT
ejde-792	273	7	bifurcation	bifurcation	NOUN
ejde-792	273	8	and	and	CCONJ
ejde-792	273	9	nonnegative	nonnegative	ADJ
ejde-792	273	10	solutions	solution	NOUN
ejde-792	273	11	for	for	ADP
ejde-792	273	12	problem	problem	NOUN
ejde-792	273	13	with	with	ADP
ejde-792	273	14	mean	mean	ADJ
ejde-792	273	15	curvature	curvature	NOUN
ejde-792	273	16	operator	operator	NOUN
ejde-792	273	17	on	on	ADP
ejde-792	273	18	general	general	ADJ
ejde-792	273	19	domain	domain	NOUN
ejde-792	273	20	,	,	PUNCT
ejde-792	273	21	indiana	indiana	PROPN
ejde-792	273	22	univ	univ	PROPN
ejde-792	273	23	.	.	PUNCT
ejde-792	273	24	math	math	PROPN
ejde-792	273	25	.	.	PUNCT
ejde-792	274	1	j.	j.	PROPN
ejde-792	274	2	,	,	PUNCT
ejde-792	274	3	67	67	NUM
ejde-792	274	4	(	(	PUNCT
ejde-792	274	5	2018	2018	NUM
ejde-792	274	6	)	)	PUNCT
ejde-792	274	7	,	,	PUNCT
ejde-792	274	8	2103–2121	2103–2121	NUM
ejde-792	274	9	.	.	PUNCT
ejde-792	275	1	[	[	X
ejde-792	275	2	17	17	NUM
ejde-792	275	3	]	]	X
ejde-792	275	4	g.	g.	PROPN
ejde-792	275	5	dai	dai	PROPN
ejde-792	275	6	,	,	PUNCT
ejde-792	275	7	z.	z.	PROPN
ejde-792	275	8	zhang	zhang	PROPN
ejde-792	275	9	;	;	PUNCT
ejde-792	275	10	spectrum	spectrum	NOUN
ejde-792	275	11	,	,	PUNCT
ejde-792	275	12	bifurcation	bifurcation	NOUN
ejde-792	275	13	and	and	CCONJ
ejde-792	275	14	hypersurfaces	hypersurface	NOUN
ejde-792	275	15	of	of	ADP
ejde-792	275	16	prescribed	prescribed	ADJ
ejde-792	275	17	k	k	PROPN
ejde-792	275	18	-	-	PUNCT
ejde-792	275	19	th	th	VERB
ejde-792	275	20	mean	mean	ADJ
ejde-792	275	21	curvature	curvature	NOUN
ejde-792	275	22	in	in	ADP
ejde-792	275	23	minkowski	minkowski	ADJ
ejde-792	275	24	space	space	NOUN
ejde-792	275	25	.	.	PUNCT
ejde-792	276	1	asymptot	asymptot	PROPN
ejde-792	276	2	.	.	PUNCT
ejde-792	277	1	anal	anal	PROPN
ejde-792	277	2	.	.	PUNCT
ejde-792	278	1	136	136	NUM
ejde-792	278	2	(	(	PUNCT
ejde-792	278	3	2024	2024	NUM
ejde-792	278	4	)	)	PUNCT
ejde-792	278	5	,	,	PUNCT
ejde-792	278	6	257–289	257–289	NUM
ejde-792	278	7	.	.	PUNCT
ejde-792	279	1	[	[	X
ejde-792	279	2	18	18	NUM
ejde-792	279	3	]	]	X
ejde-792	279	4	g.	g.	PROPN
ejde-792	279	5	dai	dai	PROPN
ejde-792	279	6	,	,	PUNCT
ejde-792	279	7	s.	s.	PROPN
ejde-792	279	8	gao	gao	PROPN
ejde-792	279	9	,	,	PUNCT
ejde-792	279	10	h.	h.	PROPN
ejde-792	279	11	luo	luo	PROPN
ejde-792	279	12	;	;	PUNCT
ejde-792	279	13	existence	existence	NOUN
ejde-792	279	14	and	and	CCONJ
ejde-792	279	15	regularity	regularity	NOUN
ejde-792	279	16	of	of	ADP
ejde-792	279	17	spacelike	spacelike	ADJ
ejde-792	279	18	hypersurfaces	hypersurface	NOUN
ejde-792	279	19	for	for	ADP
ejde-792	279	20	mean	mean	ADJ
ejde-792	279	21	curvature	curvature	NOUN
ejde-792	279	22	equation	equation	NOUN
ejde-792	279	23	in	in	ADP
ejde-792	279	24	the	the	DET
ejde-792	279	25	static	static	ADJ
ejde-792	279	26	spacetime	spacetime	NOUN
ejde-792	279	27	,	,	PUNCT
ejde-792	279	28	rocky	rocky	ADJ
ejde-792	279	29	mountain	mountain	NOUN
ejde-792	279	30	j.	j.	PROPN
ejde-792	279	31	math	math	PROPN
ejde-792	279	32	.	.	PUNCT
ejde-792	280	1	2024	2024	NUM
ejde-792	280	2	.	.	PUNCT
ejde-792	281	1	accepted	accept	VERB
ejde-792	281	2	for	for	ADP
ejde-792	281	3	publication	publication	NOUN
ejde-792	281	4	,	,	PUNCT
ejde-792	281	5	available	available	ADJ
ejde-792	281	6	at	at	ADP
ejde-792	281	7	https://api.semanticscholar.org/corpusid:270309258	https://api.semanticscholar.org/corpusid:270309258	PROPN
ejde-792	281	8	[	[	X
ejde-792	281	9	19	19	NUM
ejde-792	281	10	]	]	X
ejde-792	281	11	d.	d.	PROPN
ejde-792	281	12	de	de	PROPN
ejde-792	281	13	la	la	PROPN
ejde-792	281	14	fuente	fuente	PROPN
ejde-792	281	15	,	,	PUNCT
ejde-792	281	16	a.	a.	PROPN
ejde-792	281	17	romero	romero	PROPN
ejde-792	281	18	,	,	PUNCT
ejde-792	281	19	p.j	p.j	PROPN
ejde-792	281	20	.	.	PROPN
ejde-792	281	21	torres	torre	NOUN
ejde-792	281	22	;	;	PUNCT
ejde-792	281	23	radial	radial	ADJ
ejde-792	281	24	solutions	solution	NOUN
ejde-792	281	25	of	of	ADP
ejde-792	281	26	the	the	DET
ejde-792	281	27	dirichlet	dirichlet	PROPN
ejde-792	281	28	problem	problem	NOUN
ejde-792	281	29	for	for	ADP
ejde-792	281	30	the	the	DET
ejde-792	281	31	prescribed	prescribed	ADJ
ejde-792	281	32	mean	mean	NOUN
ejde-792	281	33	curvature	curvature	NOUN
ejde-792	281	34	equation	equation	NOUN
ejde-792	281	35	,	,	PUNCT
ejde-792	281	36	adv	adv	PROPN
ejde-792	281	37	.	.	PUNCT
ejde-792	281	38	nonlinear	nonlinear	ADJ
ejde-792	281	39	stud	stud	NOUN
ejde-792	281	40	.	.	PUNCT
ejde-792	282	1	15	15	NUM
ejde-792	282	2	(	(	PUNCT
ejde-792	282	3	2014	2014	NUM
ejde-792	282	4	)	)	PUNCT
ejde-792	282	5	,	,	PUNCT
ejde-792	282	6	171–181	171–181	NUM
ejde-792	282	7	.	.	PUNCT
ejde-792	283	1	[	[	X
ejde-792	283	2	20	20	NUM
ejde-792	283	3	]	]	PUNCT
ejde-792	283	4	d.	d.	PROPN
ejde-792	283	5	fuente	fuente	PROPN
ejde-792	283	6	,	,	PUNCT
ejde-792	283	7	a.	a.	PROPN
ejde-792	283	8	romero	romero	PROPN
ejde-792	283	9	,	,	PUNCT
ejde-792	283	10	p.	p.	PROPN
ejde-792	283	11	j.	j.	PROPN
ejde-792	283	12	torres	torres	PROPN
ejde-792	283	13	;	;	PUNCT
ejde-792	283	14	entire	entire	ADJ
ejde-792	283	15	spherically	spherically	NOUN
ejde-792	283	16	symmetric	symmetric	ADJ
ejde-792	283	17	spacelike	spacelike	PROPN
ejde-792	283	18	graphs	graph	NOUN
ejde-792	283	19	with	with	ADP
ejde-792	283	20	prescribed	prescribed	ADJ
ejde-792	283	21	mean	mean	NOUN
ejde-792	283	22	curvature	curvature	NOUN
ejde-792	283	23	function	function	NOUN
ejde-792	283	24	in	in	ADP
ejde-792	283	25	schwarzschild	schwarzschild	NOUN
ejde-792	283	26	and	and	CCONJ
ejde-792	283	27	reissner	reissner	NOUN
ejde-792	283	28	-	-	PUNCT
ejde-792	283	29	nordström	nordström	NOUN
ejde-792	283	30	spacetimes	spacetime	NOUN
ejde-792	283	31	,	,	PUNCT
ejde-792	283	32	class	class	NOUN
ejde-792	283	33	.	.	PUNCT
ejde-792	283	34	quantum	quantum	PROPN
ejde-792	283	35	grav	grav	PROPN
ejde-792	283	36	.	.	PROPN
ejde-792	283	37	,	,	PUNCT
ejde-792	283	38	32	32	NUM
ejde-792	283	39	(	(	PUNCT
ejde-792	283	40	2015	2015	NUM
ejde-792	283	41	)	)	PUNCT
ejde-792	283	42	,	,	PUNCT
ejde-792	283	43	035018	035018	NUM
ejde-792	283	44	(	(	PUNCT
ejde-792	283	45	17pp	17pp	NOUN
ejde-792	283	46	)	)	PUNCT
ejde-792	283	47	.	.	PUNCT
ejde-792	284	1	[	[	X
ejde-792	284	2	21	21	NUM
ejde-792	284	3	]	]	X
ejde-792	284	4	j.	j.	PROPN
ejde-792	284	5	mawhin	mawhin	PROPN
ejde-792	284	6	,	,	PUNCT
ejde-792	284	7	p.	p.	PROPN
ejde-792	284	8	j.	j.	PROPN
ejde-792	284	9	torres	torres	PROPN
ejde-792	284	10	;	;	PUNCT
ejde-792	284	11	prescribed	prescribed	ADJ
ejde-792	284	12	mean	mean	NOUN
ejde-792	284	13	curvature	curvature	NOUN
ejde-792	284	14	graphs	graph	NOUN
ejde-792	284	15	with	with	ADP
ejde-792	284	16	neumann	neumann	PROPN
ejde-792	284	17	boundary	boundary	ADJ
ejde-792	284	18	conditions	condition	NOUN
ejde-792	284	19	in	in	ADP
ejde-792	284	20	some	some	DET
ejde-792	284	21	flrw	flrw	ADJ
ejde-792	284	22	spacetimes	spacetime	NOUN
ejde-792	284	23	,	,	PUNCT
ejde-792	284	24	j.	j.	PROPN
ejde-792	284	25	differential	differential	PROPN
ejde-792	284	26	equations	equation	NOUN
ejde-792	284	27	261	261	NUM
ejde-792	284	28	(	(	PUNCT
ejde-792	284	29	2016	2016	NUM
ejde-792	284	30	)	)	PUNCT
ejde-792	284	31	,	,	PUNCT
ejde-792	284	32	7145–7156	7145–7156	NUM
ejde-792	284	33	.	.	PUNCT
ejde-792	285	1	[	[	X
ejde-792	285	2	22	22	NUM
ejde-792	285	3	]	]	X
ejde-792	285	4	b.	b.	PROPN
ejde-792	285	5	o’neill	o’neill	PROPN
ejde-792	285	6	;	;	PUNCT
ejde-792	285	7	semi	semi	ADJ
ejde-792	285	8	-	-	ADJ
ejde-792	285	9	riemannian	riemannian	ADJ
ejde-792	285	10	geometry	geometry	NOUN
ejde-792	285	11	,	,	PUNCT
ejde-792	285	12	academic	academic	ADJ
ejde-792	285	13	press	press	NOUN
ejde-792	285	14	,	,	PUNCT
ejde-792	285	15	1983	1983	NUM
ejde-792	285	16	.	.	PUNCT
ejde-792	286	1	[	[	X
ejde-792	286	2	23	23	NUM
ejde-792	286	3	]	]	PUNCT
ejde-792	286	4	j.	j.	PROPN
ejde-792	286	5	shi	shi	PROPN
ejde-792	286	6	;	;	PUNCT
ejde-792	286	7	persistence	persistence	NOUN
ejde-792	286	8	and	and	CCONJ
ejde-792	286	9	bifurcation	bifurcation	NOUN
ejde-792	286	10	of	of	ADP
ejde-792	286	11	degenerate	degenerate	ADJ
ejde-792	286	12	solutions	solution	NOUN
ejde-792	286	13	,	,	PUNCT
ejde-792	286	14	j.	j.	PROPN
ejde-792	286	15	funct	funct	PROPN
ejde-792	286	16	.	.	PUNCT
ejde-792	287	1	anal	anal	PROPN
ejde-792	287	2	.	.	PROPN
ejde-792	287	3	,	,	PUNCT
ejde-792	287	4	169	169	NUM
ejde-792	287	5	(	(	PUNCT
ejde-792	287	6	2	2	NUM
ejde-792	287	7	)	)	PUNCT
ejde-792	287	8	(	(	PUNCT
ejde-792	287	9	1999	1999	NUM
ejde-792	287	10	)	)	PUNCT
ejde-792	287	11	,	,	PUNCT
ejde-792	287	12	494–531	494–531	NUM
ejde-792	287	13	.	.	PUNCT
ejde-792	288	1	[	[	X
ejde-792	288	2	24	24	NUM
ejde-792	288	3	]	]	PUNCT
ejde-792	288	4	a.	a.	PROPN
ejde-792	288	5	e.	e.	PROPN
ejde-792	288	6	treibergs	treibergs	PROPN
ejde-792	288	7	;	;	PUNCT
ejde-792	288	8	entire	entire	ADJ
ejde-792	288	9	spacelike	spacelike	ADJ
ejde-792	288	10	hypersurfaces	hypersurface	NOUN
ejde-792	288	11	of	of	ADP
ejde-792	288	12	constant	constant	ADJ
ejde-792	288	13	mean	mean	ADJ
ejde-792	288	14	curvature	curvature	NOUN
ejde-792	288	15	in	in	ADP
ejde-792	288	16	minkowski	minkowski	ADJ
ejde-792	288	17	space	space	NOUN
ejde-792	288	18	,	,	PUNCT
ejde-792	288	19	invent	invent	NOUN
ejde-792	288	20	.	.	PUNCT
ejde-792	289	1	math	math	NOUN
ejde-792	289	2	.	.	PUNCT
ejde-792	290	1	,	,	PUNCT
ejde-792	290	2	66	66	NUM
ejde-792	290	3	(	(	PUNCT
ejde-792	290	4	1982	1982	NUM
ejde-792	290	5	)	)	PUNCT
ejde-792	290	6	,	,	PUNCT
ejde-792	290	7	39–56	39–56	NUM
ejde-792	290	8	.	.	PUNCT
ejde-792	291	1	siyu	siyu	PROPN
ejde-792	291	2	gao	gao	PROPN
ejde-792	291	3	(	(	PUNCT
ejde-792	291	4	corresponding	corresponding	ADJ
ejde-792	291	5	author	author	NOUN
ejde-792	291	6	)	)	PUNCT
ejde-792	291	7	school	school	NOUN
ejde-792	291	8	of	of	ADP
ejde-792	291	9	mathematical	mathematical	ADJ
ejde-792	291	10	sciences	sciences	PROPN
ejde-792	291	11	,	,	PUNCT
ejde-792	291	12	dalian	dalian	PROPN
ejde-792	291	13	university	university	PROPN
ejde-792	291	14	of	of	ADP
ejde-792	291	15	technology	technology	PROPN
ejde-792	291	16	,	,	PUNCT
ejde-792	291	17	dalian	dalian	PROPN
ejde-792	291	18	,	,	PUNCT
ejde-792	291	19	116024	116024	NUM
ejde-792	291	20	,	,	PUNCT
ejde-792	291	21	china	china	PROPN
ejde-792	291	22	email	email	NOUN
ejde-792	291	23	address	address	NOUN
ejde-792	291	24	:	:	PUNCT
ejde-792	291	25	gao15898107523@163.com	gao15898107523@163.com	ADP
ejde-792	291	26	qingbo	qingbo	NOUN
ejde-792	291	27	liu	liu	PROPN
ejde-792	291	28	school	school	PROPN
ejde-792	291	29	of	of	ADP
ejde-792	291	30	mathematical	mathematical	ADJ
ejde-792	291	31	sciences	sciences	PROPN
ejde-792	291	32	,	,	PUNCT
ejde-792	291	33	dalian	dalian	PROPN
ejde-792	291	34	university	university	PROPN
ejde-792	291	35	of	of	ADP
ejde-792	291	36	technology	technology	PROPN
ejde-792	291	37	,	,	PUNCT
ejde-792	291	38	dalian	dalian	PROPN
ejde-792	291	39	,	,	PUNCT
ejde-792	291	40	116024	116024	NUM
ejde-792	291	41	,	,	PUNCT
ejde-792	291	42	china	china	PROPN
ejde-792	291	43	email	email	PROPN
ejde-792	291	44	address	address	NOUN
ejde-792	291	45	:	:	PUNCT
ejde-792	291	46	liuqingbo@mail.dlut.edu.cn	liuqingbo@mail.dlut.edu.cn	PROPN
ejde-792	291	47	yingxin	yingxin	PROPN
ejde-792	291	48	sun	sun	PROPN
ejde-792	291	49	school	school	PROPN
ejde-792	291	50	of	of	ADP
ejde-792	291	51	mathematical	mathematical	ADJ
ejde-792	291	52	sciences	sciences	PROPN
ejde-792	291	53	,	,	PUNCT
ejde-792	291	54	dalian	dalian	PROPN
ejde-792	291	55	university	university	PROPN
ejde-792	291	56	of	of	ADP
ejde-792	291	57	technology	technology	PROPN
ejde-792	291	58	,	,	PUNCT
ejde-792	291	59	dalian	dalian	PROPN
ejde-792	291	60	,	,	PUNCT
ejde-792	291	61	116024	116024	NUM
ejde-792	291	62	,	,	PUNCT
ejde-792	291	63	china	china	PROPN
ejde-792	291	64	email	email	NOUN
ejde-792	291	65	address	address	NOUN
ejde-792	291	66	:	:	PUNCT
ejde-792	291	67	sunyingxin2023@mail.dlut.edu.cn	sunyingxin2023@mail.dlut.edu.cn	X
ejde-792	291	68	1	1	NUM
ejde-792	291	69	.	.	PUNCT
ejde-792	291	70	introduction	introduction	NOUN
ejde-792	291	71	and	and	CCONJ
ejde-792	291	72	main	main	ADJ
ejde-792	291	73	results	result	NOUN
ejde-792	291	74	2	2	NUM
ejde-792	291	75	.	.	PUNCT
ejde-792	291	76	local	local	ADJ
ejde-792	291	77	bifurcation	bifurcation	NOUN
ejde-792	291	78	structure	structure	NOUN
ejde-792	291	79	3	3	NUM
ejde-792	291	80	.	.	PUNCT
ejde-792	291	81	stability	stability	NOUN
ejde-792	291	82	properties	property	NOUN
ejde-792	291	83	acknwledgments	acknwledgment	NOUN
ejde-792	291	84	references	reference	NOUN