id	sid	tid	token	lemma	pos
ejde-196	1	1	special	special	ADJ
ejde-196	1	2	issue	issue	NOUN
ejde-196	1	3	in	in	ADP
ejde-196	1	4	honor	honor	NOUN
ejde-196	1	5	of	of	ADP
ejde-196	1	6	john	john	PROPN
ejde-196	1	7	w.	w.	PROPN
ejde-196	1	8	neuberger	neuberger	PROPN
ejde-196	1	9	electronic	electronic	PROPN
ejde-196	1	10	journal	journal	PROPN
ejde-196	1	11	of	of	ADP
ejde-196	1	12	differential	differential	ADJ
ejde-196	1	13	equations	equation	NOUN
ejde-196	1	14	,	,	PUNCT
ejde-196	1	15	special	special	ADJ
ejde-196	1	16	issue	issue	NOUN
ejde-196	1	17	02	02	NUM
ejde-196	1	18	(	(	PUNCT
ejde-196	1	19	2023	2023	NUM
ejde-196	1	20	)	)	PUNCT
ejde-196	1	21	,	,	PUNCT
ejde-196	1	22	pp	pp	ADP
ejde-196	1	23	.	.	PUNCT
ejde-196	2	1	175–192	175–192	X
ejde-196	2	2	.	.	PUNCT
ejde-196	3	1	issn	issn	PROPN
ejde-196	3	2	:	:	PUNCT
ejde-196	3	3	1072	1072	NUM
ejde-196	3	4	-	-	SYM
ejde-196	3	5	6691	6691	NUM
ejde-196	3	6	.	.	PUNCT
ejde-196	4	1	url	url	PROPN
ejde-196	4	2	:	:	PUNCT
ejde-196	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-196	4	4	or	or	CCONJ
ejde-196	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-196	4	6	local	local	ADJ
ejde-196	4	7	min	min	ADJ
ejde-196	4	8	-	-	ADJ
ejde-196	4	9	orthogonal	orthogonal	ADJ
ejde-196	4	10	principle	principle	NOUN
ejde-196	4	11	and	and	CCONJ
ejde-196	4	12	its	its	PRON
ejde-196	4	13	applications	application	NOUN
ejde-196	4	14	for	for	ADP
ejde-196	4	15	solving	solve	VERB
ejde-196	4	16	multiple	multiple	ADJ
ejde-196	4	17	solution	solution	NOUN
ejde-196	4	18	problems	problem	NOUN
ejde-196	4	19	meiqin	meiqin	PROPN
ejde-196	4	20	li	li	PROPN
ejde-196	4	21	,	,	PUNCT
ejde-196	4	22	bingbing	bingbing	PROPN
ejde-196	4	23	ji	ji	PROPN
ejde-196	4	24	,	,	PUNCT
ejde-196	4	25	jianxin	jianxin	PROPN
ejde-196	4	26	zhou	zhou	PROPN
ejde-196	4	27	in	in	ADP
ejde-196	4	28	memory	memory	NOUN
ejde-196	4	29	of	of	ADP
ejde-196	4	30	prof	prof	NOUN
ejde-196	4	31	.	.	PUNCT
ejde-196	5	1	john	john	PROPN
ejde-196	5	2	w.	w.	PROPN
ejde-196	5	3	neuberger	neuberger	PROPN
ejde-196	5	4	abstract	abstract	PROPN
ejde-196	5	5	.	.	PUNCT
ejde-196	6	1	in	in	ADP
ejde-196	6	2	this	this	DET
ejde-196	6	3	article	article	NOUN
ejde-196	6	4	we	we	PRON
ejde-196	6	5	establish	establish	VERB
ejde-196	6	6	a	a	DET
ejde-196	6	7	double	double	ADJ
ejde-196	6	8	-	-	PUNCT
ejde-196	6	9	orthogonal	orthogonal	ADJ
ejde-196	6	10	principle	principle	NOUN
ejde-196	6	11	,	,	PUNCT
ejde-196	6	12	and	and	CCONJ
ejde-196	6	13	a	a	DET
ejde-196	6	14	local	local	ADJ
ejde-196	6	15	min	min	ADJ
ejde-196	6	16	-	-	ADJ
ejde-196	6	17	orthogonal	orthogonal	ADJ
ejde-196	6	18	method	method	NOUN
ejde-196	6	19	with	with	ADP
ejde-196	6	20	its	its	PRON
ejde-196	6	21	step	step	NOUN
ejde-196	6	22	size	size	NOUN
ejde-196	6	23	rule	rule	NOUN
ejde-196	6	24	,	,	PUNCT
ejde-196	6	25	and	and	CCONJ
ejde-196	6	26	its	its	PRON
ejde-196	6	27	convergence	convergence	NOUN
ejde-196	6	28	under	under	ADP
ejde-196	6	29	assumptions	assumption	NOUN
ejde-196	6	30	more	more	ADV
ejde-196	6	31	general	general	ADJ
ejde-196	6	32	than	than	ADP
ejde-196	6	33	those	those	PRON
ejde-196	6	34	in	in	ADP
ejde-196	6	35	its	its	PRON
ejde-196	6	36	previous	previous	ADJ
ejde-196	6	37	versions	version	NOUN
ejde-196	6	38	.	.	PUNCT
ejde-196	7	1	with	with	ADP
ejde-196	7	2	such	such	DET
ejde-196	7	3	a	a	DET
ejde-196	7	4	general	general	ADJ
ejde-196	7	5	framework	framework	NOUN
ejde-196	7	6	,	,	PUNCT
ejde-196	7	7	we	we	PRON
ejde-196	7	8	justify	justify	VERB
ejde-196	7	9	mathematically	mathematically	ADV
ejde-196	7	10	the	the	DET
ejde-196	7	11	two	two	NUM
ejde-196	7	12	new	new	ADJ
ejde-196	7	13	algorithms	algorithm	NOUN
ejde-196	7	14	proposed	propose	VERB
ejde-196	7	15	for	for	ADP
ejde-196	7	16	solving	solve	VERB
ejde-196	7	17	w	w	NOUN
ejde-196	7	18	-	-	PUNCT
ejde-196	7	19	type	type	NOUN
ejde-196	7	20	problems	problem	NOUN
ejde-196	7	21	.	.	PUNCT
ejde-196	8	1	numerical	numerical	ADJ
ejde-196	8	2	examples	example	NOUN
ejde-196	8	3	for	for	ADP
ejde-196	8	4	finding	find	VERB
ejde-196	8	5	multiple	multiple	ADJ
ejde-196	8	6	solutions	solution	NOUN
ejde-196	8	7	to	to	ADP
ejde-196	8	8	w	w	NOUN
ejde-196	8	9	-	-	PUNCT
ejde-196	8	10	type	type	NOUN
ejde-196	8	11	and	and	CCONJ
ejde-196	8	12	to	to	AUX
ejde-196	8	13	mixed	mixed	ADJ
ejde-196	8	14	m	m	PROPN
ejde-196	8	15	-	-	PUNCT
ejde-196	8	16	w	w	NOUN
ejde-196	8	17	-	-	PUNCT
ejde-196	8	18	type	type	NOUN
ejde-196	8	19	problems	problem	NOUN
ejde-196	8	20	illustrate	illustrate	VERB
ejde-196	8	21	the	the	DET
ejde-196	8	22	flexibility	flexibility	NOUN
ejde-196	8	23	of	of	ADP
ejde-196	8	24	this	this	DET
ejde-196	8	25	method	method	NOUN
ejde-196	8	26	.	.	PUNCT
ejde-196	9	1	1	1	X
ejde-196	9	2	.	.	X
ejde-196	9	3	introduction	introduction	NOUN
ejde-196	9	4	consider	consider	VERB
ejde-196	9	5	the	the	DET
ejde-196	9	6	semilinear	semilinear	ADJ
ejde-196	9	7	elliptic	elliptic	ADJ
ejde-196	9	8	equation	equation	NOUN
ejde-196	9	9	−∆u(x)−	−∆u(x)−	PROPN
ejde-196	9	10	λu(x	λu(x	VERB
ejde-196	9	11	)	)	PUNCT
ejde-196	10	1	+	+	CCONJ
ejde-196	10	2	κf	κf	ADP
ejde-196	10	3	(	(	PUNCT
ejde-196	10	4	x	x	NOUN
ejde-196	10	5	,	,	PUNCT
ejde-196	10	6	u(x	u(x	NOUN
ejde-196	10	7	)	)	PUNCT
ejde-196	10	8	)	)	PUNCT
ejde-196	10	9	=	=	PUNCT
ejde-196	10	10	0	0	NUM
ejde-196	10	11	,	,	PUNCT
ejde-196	10	12	x	x	X
ejde-196	10	13	∈	∈	PROPN
ejde-196	10	14	ω	ω	PROPN
ejde-196	10	15	,	,	PUNCT
ejde-196	10	16	(	(	PUNCT
ejde-196	10	17	1.1	1.1	NUM
ejde-196	10	18	)	)	PUNCT
ejde-196	10	19	satisfying	satisfy	VERB
ejde-196	10	20	zero	zero	NUM
ejde-196	10	21	dirichlet	dirichlet	PROPN
ejde-196	10	22	or	or	CCONJ
ejde-196	10	23	neumann	neumann	PROPN
ejde-196	10	24	boundary	boundary	ADJ
ejde-196	10	25	conditions	condition	NOUN
ejde-196	10	26	,	,	PUNCT
ejde-196	10	27	where	where	SCONJ
ejde-196	10	28	ω	ω	PROPN
ejde-196	10	29	⊂	⊂	PROPN
ejde-196	10	30	rn	rn	PROPN
ejde-196	10	31	is	be	AUX
ejde-196	10	32	an	an	DET
ejde-196	10	33	open	open	ADJ
ejde-196	10	34	bounded	bounded	ADJ
ejde-196	10	35	domain	domain	NOUN
ejde-196	10	36	,	,	PUNCT
ejde-196	10	37	λ	λ	PROPN
ejde-196	10	38	and	and	CCONJ
ejde-196	10	39	κ	κ	PROPN
ejde-196	10	40	are	be	AUX
ejde-196	10	41	physical	physical	ADJ
ejde-196	10	42	parameters	parameter	NOUN
ejde-196	10	43	,	,	PUNCT
ejde-196	10	44	and	and	CCONJ
ejde-196	10	45	f	f	X
ejde-196	10	46	:	:	PUNCT
ejde-196	10	47	ω	ω	NUM
ejde-196	10	48	×	×	NOUN
ejde-196	10	49	r	r	NOUN
ejde-196	10	50	→	→	SYM
ejde-196	10	51	r	r	NOUN
ejde-196	10	52	satisfies	satisfy	VERB
ejde-196	10	53	certain	certain	ADJ
ejde-196	10	54	growth	growth	NOUN
ejde-196	10	55	and	and	CCONJ
ejde-196	10	56	regularity	regularity	NOUN
ejde-196	10	57	conditions	condition	NOUN
ejde-196	10	58	[	[	X
ejde-196	10	59	6	6	NUM
ejde-196	10	60	]	]	PUNCT
ejde-196	10	61	.	.	PUNCT
ejde-196	11	1	denoting	denote	VERB
ejde-196	11	2	∂	∂	ADJ
ejde-196	11	3	∂uf(x	∂uf(x	PROPN
ejde-196	11	4	,	,	PUNCT
ejde-196	11	5	u	u	NOUN
ejde-196	11	6	)	)	PUNCT
ejde-196	11	7	=	=	SYM
ejde-196	11	8	f	f	X
ejde-196	11	9	(	(	PUNCT
ejde-196	11	10	x	x	NOUN
ejde-196	11	11	,	,	PUNCT
ejde-196	11	12	u	u	NOUN
ejde-196	11	13	)	)	PUNCT
ejde-196	11	14	,	,	PUNCT
ejde-196	11	15	its	its	PRON
ejde-196	11	16	variational	variational	ADJ
ejde-196	11	17	functional	functional	NOUN
ejde-196	11	18	is	be	AUX
ejde-196	11	19	j(u	j(u	PROPN
ejde-196	11	20	)	)	PUNCT
ejde-196	12	1	=	=	SYM
ejde-196	12	2	∫	∫	PROPN
ejde-196	12	3	ω	ω	PROPN
ejde-196	12	4	[	[	PUNCT
ejde-196	12	5	1	1	NUM
ejde-196	12	6	2	2	NUM
ejde-196	12	7	(	(	PUNCT
ejde-196	12	8	|∇u(x)|2	|∇u(x)|2	X
ejde-196	12	9	−	−	PROPN
ejde-196	12	10	λu2(x	λu2(x	PROPN
ejde-196	12	11	)	)	PUNCT
ejde-196	12	12	)	)	PUNCT
ejde-196	13	1	+	+	CCONJ
ejde-196	13	2	κf(x	κf(x	NOUN
ejde-196	13	3	,	,	PUNCT
ejde-196	13	4	u(x))]dx	u(x))]dx	PROPN
ejde-196	13	5	.	.	PUNCT
ejde-196	13	6	(	(	PUNCT
ejde-196	13	7	1.2	1.2	NUM
ejde-196	13	8	)	)	PUNCT
ejde-196	13	9	for	for	ADP
ejde-196	13	10	a	a	DET
ejde-196	13	11	function	function	NOUN
ejde-196	13	12	j	j	PROPN
ejde-196	13	13	∈	∈	PROPN
ejde-196	13	14	c1(h	c1(h	PROPN
ejde-196	13	15	,	,	PUNCT
ejde-196	13	16	r	r	NOUN
ejde-196	13	17	)	)	PUNCT
ejde-196	13	18	where	where	SCONJ
ejde-196	13	19	h	h	NOUN
ejde-196	13	20	is	be	AUX
ejde-196	13	21	a	a	DET
ejde-196	13	22	hilbert	hilbert	NOUN
ejde-196	13	23	space	space	NOUN
ejde-196	13	24	,	,	PUNCT
ejde-196	13	25	a	a	DET
ejde-196	13	26	point	point	NOUN
ejde-196	13	27	u∗	u∗	NOUN
ejde-196	13	28	∈	∈	PROPN
ejde-196	13	29	h	h	NOUN
ejde-196	13	30	is	be	AUX
ejde-196	13	31	called	call	VERB
ejde-196	13	32	a	a	DET
ejde-196	13	33	critical	critical	ADJ
ejde-196	13	34	point	point	NOUN
ejde-196	13	35	of	of	ADP
ejde-196	13	36	j	j	PROPN
ejde-196	13	37	if	if	SCONJ
ejde-196	13	38	its	its	PRON
ejde-196	13	39	frechet	frechet	NOUN
ejde-196	13	40	derivative	derivative	PROPN
ejde-196	13	41	j	j	PROPN
ejde-196	13	42	′(u∗	′(u∗	ADV
ejde-196	13	43	)	)	PUNCT
ejde-196	13	44	=	=	SYM
ejde-196	13	45	0	0	NUM
ejde-196	13	46	;	;	PUNCT
ejde-196	13	47	a	a	DET
ejde-196	13	48	critical	critical	ADJ
ejde-196	13	49	point	point	NOUN
ejde-196	13	50	u∗	u∗	NOUN
ejde-196	13	51	is	be	AUX
ejde-196	13	52	called	call	VERB
ejde-196	13	53	a	a	DET
ejde-196	13	54	k	k	NOUN
ejde-196	13	55	-	-	NOUN
ejde-196	13	56	saddle	saddle	NOUN
ejde-196	13	57	if	if	SCONJ
ejde-196	13	58	it	it	PRON
ejde-196	13	59	is	be	AUX
ejde-196	13	60	a	a	DET
ejde-196	13	61	local	local	ADJ
ejde-196	13	62	maximum	maximum	ADJ
ejde-196	13	63	point	point	NOUN
ejde-196	13	64	of	of	ADP
ejde-196	13	65	j	j	PROPN
ejde-196	13	66	in	in	ADP
ejde-196	13	67	a	a	DET
ejde-196	13	68	k	k	ADJ
ejde-196	13	69	-	-	ADJ
ejde-196	13	70	dimensional	dimensional	ADJ
ejde-196	13	71	subspace	subspace	NOUN
ejde-196	13	72	and	and	CCONJ
ejde-196	13	73	a	a	DET
ejde-196	13	74	local	local	ADJ
ejde-196	13	75	minimum	minimum	NOUN
ejde-196	13	76	point	point	NOUN
ejde-196	13	77	in	in	ADP
ejde-196	13	78	the	the	DET
ejde-196	13	79	corresponding	corresponding	ADJ
ejde-196	13	80	k	k	PROPN
ejde-196	13	81	-	-	ADJ
ejde-196	13	82	co	co	ADJ
ejde-196	13	83	-	-	ADJ
ejde-196	13	84	dimensional	dimensional	ADJ
ejde-196	13	85	subspace	subspace	NOUN
ejde-196	13	86	.	.	PUNCT
ejde-196	14	1	such	such	ADJ
ejde-196	14	2	index	index	NOUN
ejde-196	14	3	k	k	PROPN
ejde-196	14	4	can	can	AUX
ejde-196	14	5	be	be	AUX
ejde-196	14	6	used	use	VERB
ejde-196	14	7	to	to	PART
ejde-196	14	8	measure	measure	VERB
ejde-196	14	9	the	the	DET
ejde-196	14	10	instability	instability	NOUN
ejde-196	14	11	of	of	ADP
ejde-196	14	12	the	the	DET
ejde-196	14	13	critical	critical	ADJ
ejde-196	14	14	point	point	NOUN
ejde-196	14	15	u∗.	u∗.	PUNCT
ejde-196	14	16	thus	thus	ADV
ejde-196	14	17	a	a	DET
ejde-196	14	18	0saddle	0saddle	NUM
ejde-196	14	19	is	be	AUX
ejde-196	14	20	a	a	DET
ejde-196	14	21	local	local	ADJ
ejde-196	14	22	minimum	minimum	NOUN
ejde-196	14	23	point	point	NOUN
ejde-196	14	24	of	of	ADP
ejde-196	14	25	j	j	PROPN
ejde-196	14	26	and	and	CCONJ
ejde-196	14	27	corresponds	correspond	VERB
ejde-196	14	28	to	to	ADP
ejde-196	14	29	a	a	DET
ejde-196	14	30	stable	stable	ADJ
ejde-196	14	31	local	local	ADJ
ejde-196	14	32	equilibrium	equilibrium	NOUN
ejde-196	14	33	state	state	NOUN
ejde-196	14	34	in	in	ADP
ejde-196	14	35	a	a	DET
ejde-196	14	36	physical	physical	ADJ
ejde-196	14	37	system	system	NOUN
ejde-196	14	38	;	;	PUNCT
ejde-196	14	39	while	while	SCONJ
ejde-196	14	40	k	k	NOUN
ejde-196	14	41	-	-	PUNCT
ejde-196	14	42	saddles	saddle	NOUN
ejde-196	14	43	with	with	ADP
ejde-196	14	44	k	k	PROPN
ejde-196	14	45	≥	≥	NUM
ejde-196	14	46	1	1	NUM
ejde-196	14	47	correspond	correspond	VERB
ejde-196	14	48	to	to	ADP
ejde-196	14	49	unstable	unstable	ADJ
ejde-196	14	50	local	local	ADJ
ejde-196	14	51	equilibria	equilibrium	NOUN
ejde-196	14	52	or	or	CCONJ
ejde-196	14	53	excited	excited	ADJ
ejde-196	14	54	states	state	NOUN
ejde-196	14	55	.	.	PUNCT
ejde-196	15	1	when	when	SCONJ
ejde-196	15	2	j	j	PROPN
ejde-196	15	3	is	be	AUX
ejde-196	15	4	c2	c2	PROPN
ejde-196	15	5	and	and	CCONJ
ejde-196	15	6	u∗	u∗	ADV
ejde-196	15	7	is	be	AUX
ejde-196	15	8	a	a	DET
ejde-196	15	9	critical	critical	ADJ
ejde-196	15	10	point	point	NOUN
ejde-196	15	11	,	,	PUNCT
ejde-196	15	12	we	we	PRON
ejde-196	15	13	denote	denote	VERB
ejde-196	15	14	the	the	DET
ejde-196	15	15	spectral	spectral	ADJ
ejde-196	15	16	decomposition	decomposition	NOUN
ejde-196	15	17	of	of	ADP
ejde-196	15	18	j	j	PROPN
ejde-196	15	19	′′(u∗	′′(u∗	PROPN
ejde-196	15	20	)	)	PUNCT
ejde-196	15	21	by	by	ADP
ejde-196	15	22	h	h	NOUN
ejde-196	15	23	=	=	SYM
ejde-196	15	24	h−	h−	PROPN
ejde-196	15	25	⊕h0	⊕h0	NUM
ejde-196	15	26	⊕h+	⊕h+	PROPN
ejde-196	15	27	where	where	SCONJ
ejde-196	15	28	h−	h−	PROPN
ejde-196	15	29	,	,	PUNCT
ejde-196	15	30	h0	h0	PROPN
ejde-196	15	31	,	,	PUNCT
ejde-196	15	32	h+	h+	X
ejde-196	15	33	are	be	AUX
ejde-196	15	34	respectively	respectively	ADV
ejde-196	15	35	the	the	DET
ejde-196	15	36	maximum	maximum	ADJ
ejde-196	15	37	negative	negative	ADJ
ejde-196	15	38	,	,	PUNCT
ejde-196	15	39	the	the	DET
ejde-196	15	40	null	null	NOUN
ejde-196	15	41	,	,	PUNCT
ejde-196	15	42	and	and	CCONJ
ejde-196	15	43	the	the	DET
ejde-196	15	44	maximum	maximum	ADJ
ejde-196	15	45	positive	positive	ADJ
ejde-196	15	46	subspaces	subspace	NOUN
ejde-196	15	47	of	of	ADP
ejde-196	15	48	j	j	PROPN
ejde-196	15	49	′′(u∗	′′(u∗	PROPN
ejde-196	15	50	)	)	PUNCT
ejde-196	15	51	with	with	ADP
ejde-196	15	52	dim(h0	dim(h0	NOUN
ejde-196	15	53	)	)	PUNCT
ejde-196	15	54	<	<	X
ejde-196	15	55	∞	∞	PROPN
ejde-196	15	56	,	,	PUNCT
ejde-196	15	57	and	and	CCONJ
ejde-196	15	58	mi(u∗	mi(u∗	NUM
ejde-196	15	59	)	)	PUNCT
ejde-196	16	1	=	=	SYM
ejde-196	16	2	dim(h−	dim(h−	X
ejde-196	16	3	)	)	PUNCT
ejde-196	16	4	is	be	AUX
ejde-196	16	5	called	call	VERB
ejde-196	16	6	the	the	DET
ejde-196	16	7	morse	morse	ADJ
ejde-196	16	8	index	index	NOUN
ejde-196	16	9	2020	2020	NUM
ejde-196	16	10	mathematics	mathematic	NOUN
ejde-196	16	11	subject	subject	ADJ
ejde-196	16	12	classification	classification	NOUN
ejde-196	16	13	.	.	PUNCT
ejde-196	17	1	35b38	35b38	NUM
ejde-196	17	2	,	,	PUNCT
ejde-196	17	3	58e05	58e05	NUM
ejde-196	17	4	,	,	PUNCT
ejde-196	17	5	65n20	65n20	NUM
ejde-196	17	6	.	.	PUNCT
ejde-196	18	1	key	key	ADJ
ejde-196	18	2	words	word	NOUN
ejde-196	18	3	and	and	CCONJ
ejde-196	18	4	phrases	phrase	NOUN
ejde-196	18	5	.	.	PUNCT
ejde-196	19	1	multiple	multiple	ADJ
ejde-196	19	2	solution	solution	NOUN
ejde-196	19	3	;	;	PUNCT
ejde-196	19	4	numerical	numerical	ADJ
ejde-196	19	5	algorithms	algorithm	NOUN
ejde-196	19	6	;	;	PUNCT
ejde-196	19	7	convergence	convergence	NOUN
ejde-196	19	8	analysis	analysis	NOUN
ejde-196	19	9	.	.	PUNCT
ejde-196	20	1	©	©	ADP
ejde-196	20	2	2023	2023	NUM
ejde-196	20	3	this	this	DET
ejde-196	20	4	work	work	NOUN
ejde-196	20	5	is	be	AUX
ejde-196	20	6	licensed	license	VERB
ejde-196	20	7	under	under	ADP
ejde-196	20	8	a	a	DET
ejde-196	20	9	cc	cc	NOUN
ejde-196	20	10	by	by	ADP
ejde-196	20	11	4.0	4.0	NUM
ejde-196	20	12	license	license	NOUN
ejde-196	20	13	.	.	PUNCT
ejde-196	21	1	published	publish	VERB
ejde-196	21	2	march	march	PROPN
ejde-196	21	3	27	27	NUM
ejde-196	21	4	,	,	PUNCT
ejde-196	21	5	2023	2023	NUM
ejde-196	21	6	.	.	PUNCT
ejde-196	22	1	175	175	NUM
ejde-196	22	2	176	176	NUM
ejde-196	22	3	m.	m.	NOUN
ejde-196	22	4	li	li	PROPN
ejde-196	22	5	,	,	PUNCT
ejde-196	22	6	b.	b.	PROPN
ejde-196	22	7	ji	ji	PROPN
ejde-196	22	8	,	,	PUNCT
ejde-196	22	9	j.	j.	PROPN
ejde-196	22	10	zhou	zhou	PROPN
ejde-196	22	11	ejde	ejde	PROPN
ejde-196	22	12	/	/	SYM
ejde-196	22	13	si/02	si/02	PROPN
ejde-196	22	14	of	of	ADP
ejde-196	22	15	u∗.	u∗.	PROPN
ejde-196	22	16	a	a	DET
ejde-196	22	17	critical	critical	ADJ
ejde-196	22	18	point	point	NOUN
ejde-196	22	19	u∗	u∗	ADV
ejde-196	22	20	is	be	AUX
ejde-196	22	21	non	non	ADJ
ejde-196	22	22	-	-	ADJ
ejde-196	22	23	degenerate	degenerate	ADJ
ejde-196	22	24	if	if	SCONJ
ejde-196	22	25	h0	h0	NOUN
ejde-196	22	26	=	=	SYM
ejde-196	22	27	{	{	PUNCT
ejde-196	22	28	0	0	NUM
ejde-196	22	29	}	}	PUNCT
ejde-196	22	30	and	and	CCONJ
ejde-196	22	31	degenerate	degenerate	ADJ
ejde-196	22	32	otherwise	otherwise	ADV
ejde-196	22	33	.	.	PUNCT
ejde-196	23	1	a	a	DET
ejde-196	23	2	non	non	ADJ
ejde-196	23	3	-	-	ADJ
ejde-196	23	4	degenerate	degenerate	ADJ
ejde-196	23	5	critical	critical	ADJ
ejde-196	23	6	point	point	NOUN
ejde-196	23	7	u∗	u∗	ADV
ejde-196	23	8	with	with	ADP
ejde-196	23	9	mi(u∗	mi(u∗	NUM
ejde-196	23	10	)	)	PUNCT
ejde-196	24	1	=	=	SYM
ejde-196	25	1	k	k	PROPN
ejde-196	25	2	is	be	AUX
ejde-196	25	3	a	a	DET
ejde-196	25	4	k	k	NOUN
ejde-196	25	5	-	-	NOUN
ejde-196	25	6	saddle	saddle	NOUN
ejde-196	25	7	.	.	PUNCT
ejde-196	26	1	let	let	VERB
ejde-196	26	2	λ1	λ1	PROPN
ejde-196	26	3	<	<	X
ejde-196	26	4	λ2	λ2	NOUN
ejde-196	26	5	<	<	X
ejde-196	26	6	.	.	PUNCT
ejde-196	26	7	.	.	PUNCT
ejde-196	27	1	.	.	PUNCT
ejde-196	28	1	be	be	AUX
ejde-196	28	2	the	the	DET
ejde-196	28	3	eigenvalues	eigenvalue	NOUN
ejde-196	28	4	of	of	ADP
ejde-196	28	5	−∆	−∆	NOUN
ejde-196	28	6	with	with	ADP
ejde-196	28	7	zero	zero	NUM
ejde-196	28	8	boundary	boundary	ADJ
ejde-196	28	9	condition	condition	NOUN
ejde-196	28	10	,	,	PUNCT
ejde-196	28	11	and	and	CCONJ
ejde-196	28	12	let	let	VERB
ejde-196	28	13	v1	v1	NOUN
ejde-196	28	14	,	,	PUNCT
ejde-196	28	15	v2	v2	NOUN
ejde-196	28	16	,	,	PUNCT
ejde-196	28	17	.	.	PUNCT
ejde-196	28	18	.	.	PUNCT
ejde-196	29	1	.	.	PUNCT
ejde-196	30	1	be	be	AUX
ejde-196	30	2	the	the	DET
ejde-196	30	3	corresponding	corresponding	ADJ
ejde-196	30	4	eigenfunctions	eigenfunction	NOUN
ejde-196	30	5	.	.	PUNCT
ejde-196	31	1	for	for	ADP
ejde-196	31	2	simplicity	simplicity	NOUN
ejde-196	31	3	,	,	PUNCT
ejde-196	31	4	we	we	PRON
ejde-196	31	5	first	first	ADV
ejde-196	31	6	consider	consider	VERB
ejde-196	31	7	f	f	PROPN
ejde-196	31	8	(	(	PUNCT
ejde-196	31	9	x	x	NOUN
ejde-196	31	10	,	,	PUNCT
ejde-196	31	11	u(x	u(x	NOUN
ejde-196	31	12	)	)	PUNCT
ejde-196	31	13	)	)	PUNCT
ejde-196	32	1	=	=	SYM
ejde-196	32	2	|u(x)|p−1u(x	|u(x)|p−1u(x	NOUN
ejde-196	32	3	)	)	PUNCT
ejde-196	32	4	with	with	ADP
ejde-196	32	5	p	p	PROPN
ejde-196	32	6	>	>	X
ejde-196	32	7	1	1	NUM
ejde-196	32	8	.	.	PUNCT
ejde-196	33	1	equation	equation	NOUN
ejde-196	33	2	(	(	PUNCT
ejde-196	33	3	1.1	1.1	NUM
ejde-196	33	4	)	)	PUNCT
ejde-196	33	5	is	be	AUX
ejde-196	33	6	called	call	VERB
ejde-196	33	7	focusing	focus	VERB
ejde-196	33	8	(	(	PUNCT
ejde-196	33	9	m	m	NOUN
ejde-196	33	10	-	-	NOUN
ejde-196	33	11	type	type	NOUN
ejde-196	33	12	)	)	PUNCT
ejde-196	33	13	if	if	SCONJ
ejde-196	33	14	κ	κ	PRON
ejde-196	33	15	<	<	X
ejde-196	33	16	0	0	PUNCT
ejde-196	34	1	and	and	CCONJ
ejde-196	34	2	defocusing	defocuse	VERB
ejde-196	34	3	(	(	PUNCT
ejde-196	34	4	w	w	NOUN
ejde-196	34	5	-	-	PUNCT
ejde-196	34	6	type	type	NOUN
ejde-196	34	7	)	)	PUNCT
ejde-196	34	8	if	if	SCONJ
ejde-196	34	9	κ	κ	X
ejde-196	34	10	>	>	X
ejde-196	34	11	0	0	NUM
ejde-196	34	12	.	.	PUNCT
ejde-196	35	1	it	it	PRON
ejde-196	35	2	is	be	AUX
ejde-196	35	3	known	know	VERB
ejde-196	35	4	that	that	SCONJ
ejde-196	35	5	when	when	SCONJ
ejde-196	35	6	κ	κ	X
ejde-196	35	7	>	>	X
ejde-196	35	8	0	0	PUNCT
ejde-196	35	9	and	and	CCONJ
ejde-196	35	10	λk	λk	X
ejde-196	35	11	<	<	X
ejde-196	35	12	λ	λ	X
ejde-196	35	13	<	<	X
ejde-196	35	14	λk+1	λk+1	PROPN
ejde-196	35	15	,	,	PUNCT
ejde-196	35	16	then	then	ADV
ejde-196	35	17	0	0	NUM
ejde-196	35	18	is	be	AUX
ejde-196	35	19	the	the	DET
ejde-196	35	20	only	only	ADJ
ejde-196	35	21	index	index	NOUN
ejde-196	35	22	k	k	NOUN
ejde-196	35	23	-	-	NOUN
ejde-196	35	24	saddle	saddle	ADJ
ejde-196	35	25	,	,	PUNCT
ejde-196	35	26	all	all	DET
ejde-196	35	27	nontrivial	nontrivial	ADJ
ejde-196	35	28	saddles	saddle	NOUN
ejde-196	35	29	have	have	VERB
ejde-196	35	30	index	index	NOUN
ejde-196	35	31	greater	great	ADJ
ejde-196	35	32	than	than	ADP
ejde-196	35	33	k	k	PROPN
ejde-196	35	34	,	,	PUNCT
ejde-196	35	35	and	and	CCONJ
ejde-196	35	36	for	for	ADP
ejde-196	35	37	all	all	DET
ejde-196	35	38	u	u	NOUN
ejde-196	35	39	∈	∈	PROPN
ejde-196	35	40	[	[	X
ejde-196	35	41	v1	v1	NOUN
ejde-196	35	42	,	,	PUNCT
ejde-196	35	43	.	.	PUNCT
ejde-196	35	44	.	.	PUNCT
ejde-196	36	1	.	.	PUNCT
ejde-196	37	1	,	,	PUNCT
ejde-196	37	2	vk]⊥	vk]⊥	VERB
ejde-196	37	3	,	,	PUNCT
ejde-196	37	4	there	there	PRON
ejde-196	37	5	is	be	VERB
ejde-196	37	6	tu	tu	PROPN
ejde-196	37	7	>	>	X
ejde-196	37	8	0	0	NUM
ejde-196	38	1	such	such	ADJ
ejde-196	38	2	that	that	SCONJ
ejde-196	38	3	tu	tu	PROPN
ejde-196	38	4	=	=	PUNCT
ejde-196	38	5	arg	arg	PROPN
ejde-196	38	6	maxt>0	maxt>0	NOUN
ejde-196	38	7	j(tu	j(tu	PROPN
ejde-196	38	8	)	)	PUNCT
ejde-196	38	9	;	;	PUNCT
ejde-196	38	10	when	when	SCONJ
ejde-196	38	11	κ	κ	X
ejde-196	38	12	<	<	X
ejde-196	38	13	0	0	PUNCT
ejde-196	38	14	and	and	CCONJ
ejde-196	38	15	λk	λk	X
ejde-196	38	16	<	<	X
ejde-196	38	17	λ	λ	X
ejde-196	38	18	<	<	X
ejde-196	38	19	λk+1	λk+1	PROPN
ejde-196	38	20	,	,	PUNCT
ejde-196	38	21	then	then	ADV
ejde-196	38	22	0	0	NUM
ejde-196	38	23	is	be	AUX
ejde-196	38	24	the	the	DET
ejde-196	38	25	only	only	ADJ
ejde-196	38	26	index	index	NOUN
ejde-196	38	27	k	k	NOUN
ejde-196	38	28	-	-	NOUN
ejde-196	38	29	saddle	saddle	ADJ
ejde-196	38	30	,	,	PUNCT
ejde-196	38	31	all	all	DET
ejde-196	38	32	nontrivial	nontrivial	ADJ
ejde-196	38	33	saddles	saddle	NOUN
ejde-196	38	34	have	have	VERB
ejde-196	38	35	index	index	NOUN
ejde-196	38	36	less	less	ADJ
ejde-196	38	37	than	than	ADP
ejde-196	38	38	k	k	PROPN
ejde-196	38	39	and	and	CCONJ
ejde-196	38	40	for	for	ADP
ejde-196	38	41	all	all	DET
ejde-196	38	42	u	u	NOUN
ejde-196	38	43	∈	∈	PROPN
ejde-196	38	44	[	[	X
ejde-196	38	45	v1	v1	NOUN
ejde-196	38	46	,	,	PUNCT
ejde-196	38	47	.	.	PUNCT
ejde-196	38	48	.	.	PUNCT
ejde-196	39	1	.	.	PUNCT
ejde-196	40	1	,	,	PUNCT
ejde-196	40	2	vk	vk	ADP
ejde-196	40	3	]	]	PUNCT
ejde-196	40	4	,	,	PUNCT
ejde-196	40	5	there	there	PRON
ejde-196	40	6	is	be	VERB
ejde-196	40	7	tu	tu	PROPN
ejde-196	40	8	>	>	X
ejde-196	40	9	0	0	NUM
ejde-196	41	1	such	such	ADJ
ejde-196	41	2	that	that	SCONJ
ejde-196	41	3	tu	tu	PROPN
ejde-196	41	4	=	=	PUNCT
ejde-196	41	5	arg	arg	NOUN
ejde-196	41	6	mint>0	mint>0	NOUN
ejde-196	41	7	j(tu	j(tu	PROPN
ejde-196	41	8	)	)	PUNCT
ejde-196	41	9	.	.	PUNCT
ejde-196	42	1	these	these	DET
ejde-196	42	2	two	two	NUM
ejde-196	42	3	types	type	NOUN
ejde-196	42	4	of	of	ADP
ejde-196	42	5	problems	problem	NOUN
ejde-196	42	6	are	be	AUX
ejde-196	42	7	very	very	ADV
ejde-196	42	8	different	different	ADJ
ejde-196	42	9	in	in	ADP
ejde-196	42	10	physical	physical	ADJ
ejde-196	42	11	nature	nature	NOUN
ejde-196	42	12	and	and	CCONJ
ejde-196	42	13	in	in	ADP
ejde-196	42	14	mathematical	mathematical	ADJ
ejde-196	42	15	structure	structure	NOUN
ejde-196	42	16	as	as	ADV
ejde-196	42	17	well	well	ADV
ejde-196	42	18	,	,	PUNCT
ejde-196	42	19	see	see	VERB
ejde-196	42	20	figure	figure	NOUN
ejde-196	42	21	1	1	NUM
ejde-196	42	22	.	.	PUNCT
ejde-196	43	1	in	in	ADP
ejde-196	43	2	the	the	DET
ejde-196	43	3	literature	literature	NOUN
ejde-196	43	4	,	,	PUNCT
ejde-196	43	5	these	these	DET
ejde-196	43	6	problems	problem	NOUN
ejde-196	43	7	have	have	VERB
ejde-196	43	8	to	to	PART
ejde-196	43	9	be	be	AUX
ejde-196	43	10	solved	solve	VERB
ejde-196	43	11	by	by	ADP
ejde-196	43	12	two	two	NUM
ejde-196	43	13	very	very	ADV
ejde-196	43	14	different	different	ADJ
ejde-196	43	15	types	type	NOUN
ejde-196	43	16	of	of	ADP
ejde-196	43	17	variational	variational	ADJ
ejde-196	43	18	methods	method	NOUN
ejde-196	43	19	.	.	PUNCT
ejde-196	44	1	k−saddle	k−saddle	PROPN
ejde-196	44	2	k−saddle	k−saddle	NOUN
ejde-196	44	3	∩-shape	∩-shape	NOUN
ejde-196	44	4	in	in	ADP
ejde-196	44	5	[	[	X
ejde-196	44	6	v1	v1	NOUN
ejde-196	44	7	,	,	PUNCT
ejde-196	44	8	.	.	PUNCT
ejde-196	44	9	.	.	PUNCT
ejde-196	45	1	.	.	PUNCT
ejde-196	46	1	,	,	PUNCT
ejde-196	46	2	vk	vk	ADP
ejde-196	46	3	]	]	PUNCT
ejde-196	46	4	,	,	PUNCT
ejde-196	46	5	m	m	NOUN
ejde-196	46	6	-	-	NOUN
ejde-196	46	7	shape	shape	NOUN
ejde-196	46	8	in	in	ADP
ejde-196	46	9	[	[	X
ejde-196	46	10	v1	v1	NOUN
ejde-196	46	11	,	,	PUNCT
ejde-196	46	12	.	.	PUNCT
ejde-196	46	13	.	.	PUNCT
ejde-196	47	1	.	.	PUNCT
ejde-196	48	1	,	,	PUNCT
ejde-196	48	2	vk]⊥	vk]⊥	VERB
ejde-196	48	3	;	;	PUNCT
ejde-196	48	4	∪-shape	∪-shape	VERB
ejde-196	48	5	in	in	ADP
ejde-196	48	6	[	[	X
ejde-196	48	7	v1	v1	NOUN
ejde-196	48	8	,	,	PUNCT
ejde-196	48	9	.	.	PUNCT
ejde-196	48	10	.	.	PUNCT
ejde-196	48	11	.	.	PUNCT
ejde-196	49	1	,	,	PUNCT
ejde-196	49	2	vk]⊥	vk]⊥	VERB
ejde-196	49	3	,	,	PUNCT
ejde-196	49	4	w	w	NOUN
ejde-196	49	5	-	-	PUNCT
ejde-196	49	6	shape	shape	NOUN
ejde-196	49	7	in	in	ADP
ejde-196	49	8	[	[	X
ejde-196	49	9	v1	v1	NOUN
ejde-196	49	10	,	,	PUNCT
ejde-196	49	11	.	.	PUNCT
ejde-196	49	12	.	.	PUNCT
ejde-196	50	1	.	.	PUNCT
ejde-196	51	1	,	,	PUNCT
ejde-196	51	2	vk	vk	ADP
ejde-196	51	3	]	]	PUNCT
ejde-196	51	4	.	.	PUNCT
ejde-196	52	1	figure	figure	NOUN
ejde-196	52	2	1	1	NUM
ejde-196	52	3	.	.	PUNCT
ejde-196	52	4	typical	typical	ADJ
ejde-196	52	5	functional	functional	ADJ
ejde-196	52	6	profiles	profile	NOUN
ejde-196	52	7	of	of	ADP
ejde-196	52	8	m	m	NOUN
ejde-196	52	9	-	-	NOUN
ejde-196	52	10	type	type	NOUN
ejde-196	52	11	(	(	PUNCT
ejde-196	52	12	left	left	ADJ
ejde-196	52	13	)	)	PUNCT
ejde-196	52	14	vs.	vs.	ADP
ejde-196	52	15	w	w	NOUN
ejde-196	52	16	-	-	PUNCT
ejde-196	52	17	type	type	NOUN
ejde-196	52	18	(	(	PUNCT
ejde-196	52	19	right	right	NOUN
ejde-196	52	20	)	)	PUNCT
ejde-196	52	21	.	.	PUNCT
ejde-196	53	1	because	because	SCONJ
ejde-196	53	2	of	of	ADP
ejde-196	53	3	the	the	DET
ejde-196	53	4	difference	difference	NOUN
ejde-196	53	5	in	in	ADP
ejde-196	53	6	space	space	NOUN
ejde-196	53	7	dimensions	dimension	NOUN
ejde-196	53	8	,	,	PUNCT
ejde-196	53	9	they	they	PRON
ejde-196	53	10	are	be	AUX
ejde-196	53	11	not	not	PART
ejde-196	53	12	upside	upside	ADV
ejde-196	53	13	-	-	PUNCT
ejde-196	53	14	down	down	NOUN
ejde-196	53	15	to	to	ADP
ejde-196	53	16	each	each	DET
ejde-196	53	17	other	other	ADJ
ejde-196	53	18	.	.	PUNCT
ejde-196	54	1	the	the	DET
ejde-196	54	2	case	case	NOUN
ejde-196	54	3	will	will	AUX
ejde-196	54	4	be	be	AUX
ejde-196	54	5	much	much	ADV
ejde-196	54	6	more	more	ADV
ejde-196	54	7	complex	complex	ADJ
ejde-196	54	8	if	if	SCONJ
ejde-196	54	9	f	f	PROPN
ejde-196	54	10	(	(	PUNCT
ejde-196	54	11	x	x	NOUN
ejde-196	54	12	,	,	PUNCT
ejde-196	54	13	u	u	NOUN
ejde-196	54	14	)	)	PUNCT
ejde-196	54	15	contains	contain	VERB
ejde-196	54	16	both	both	PRON
ejde-196	54	17	convex	convex	NOUN
ejde-196	54	18	and	and	CCONJ
ejde-196	54	19	concave	concave	VERB
ejde-196	54	20	nonlinear	nonlinear	ADJ
ejde-196	54	21	terms	term	NOUN
ejde-196	54	22	and	and	CCONJ
ejde-196	54	23	becomes	become	VERB
ejde-196	54	24	a	a	DET
ejde-196	54	25	mixed	mixed	ADJ
ejde-196	54	26	m	m	PROPN
ejde-196	54	27	-	-	PUNCT
ejde-196	54	28	w	w	NOUN
ejde-196	54	29	-	-	PUNCT
ejde-196	54	30	type	type	NOUN
ejde-196	54	31	problem	problem	NOUN
ejde-196	54	32	,	,	PUNCT
ejde-196	54	33	see	see	VERB
ejde-196	54	34	figure	figure	NOUN
ejde-196	54	35	2	2	NUM
ejde-196	54	36	,	,	PUNCT
ejde-196	54	37	f	f	PROPN
ejde-196	54	38	(	(	PUNCT
ejde-196	54	39	x	x	NOUN
ejde-196	54	40	,	,	PUNCT
ejde-196	54	41	u	u	NOUN
ejde-196	54	42	)	)	PUNCT
ejde-196	54	43	=	=	SYM
ejde-196	54	44	a(x)|u(x)|q−1u(x	a(x)|u(x)|q−1u(x	PROPN
ejde-196	54	45	)	)	PUNCT
ejde-196	55	1	+	+	CCONJ
ejde-196	55	2	b(x)|u(x)|p−1u(x	b(x)|u(x)|p−1u(x	NOUN
ejde-196	55	3	)	)	PUNCT
ejde-196	55	4	.	.	PUNCT
ejde-196	56	1	(	(	PUNCT
ejde-196	56	2	1.3	1.3	NUM
ejde-196	56	3	)	)	PUNCT
ejde-196	56	4	see	see	VERB
ejde-196	56	5	[	[	X
ejde-196	56	6	1	1	NUM
ejde-196	56	7	,	,	PUNCT
ejde-196	56	8	2	2	NUM
ejde-196	56	9	,	,	PUNCT
ejde-196	56	10	11	11	NUM
ejde-196	56	11	]	]	PUNCT
ejde-196	56	12	where	where	SCONJ
ejde-196	56	13	0	0	PUNCT
ejde-196	56	14	<	<	X
ejde-196	56	15	q	q	X
ejde-196	56	16	<	<	X
ejde-196	56	17	1	1	NUM
ejde-196	56	18	<	<	X
ejde-196	56	19	p	p	X
ejde-196	56	20	<	<	X
ejde-196	56	21	2∗	2∗	PROPN
ejde-196	56	22	,	,	PUNCT
ejde-196	56	23	2∗	2∗	NUM
ejde-196	57	1	=	=	SYM
ejde-196	57	2	n+2	n+2	NUM
ejde-196	57	3	n−2	n−2	PROPN
ejde-196	57	4	if	if	SCONJ
ejde-196	57	5	n	n	PRON
ejde-196	57	6	≥	≥	VERB
ejde-196	57	7	3	3	NUM
ejde-196	57	8	or	or	CCONJ
ejde-196	57	9	2∗	2∗	NUM
ejde-196	58	1	=	=	SYM
ejde-196	58	2	∞	∞	NOUN
ejde-196	58	3	if	if	SCONJ
ejde-196	58	4	n	n	NOUN
ejde-196	58	5	=	=	SYM
ejde-196	58	6	1	1	NUM
ejde-196	58	7	,	,	PUNCT
ejde-196	58	8	2	2	NUM
ejde-196	58	9	for	for	ADP
ejde-196	58	10	some	some	DET
ejde-196	58	11	given	give	VERB
ejde-196	58	12	nonnegative	nonnegative	ADJ
ejde-196	58	13	functions	function	NOUN
ejde-196	58	14	a(x	a(x	NOUN
ejde-196	58	15	)	)	PUNCT
ejde-196	58	16	and	and	CCONJ
ejde-196	58	17	b(x	b(x	NOUN
ejde-196	58	18	)	)	PUNCT
ejde-196	58	19	.	.	PUNCT
ejde-196	59	1	locally	locally	ADV
ejde-196	59	2	m−type	m−type	X
ejde-196	59	3	locally	locally	ADV
ejde-196	59	4	w−type	w−type	NOUN
ejde-196	59	5	figure	figure	NOUN
ejde-196	59	6	2	2	NUM
ejde-196	59	7	.	.	PUNCT
ejde-196	59	8	energy	energy	NOUN
ejde-196	59	9	profile	profile	NOUN
ejde-196	59	10	consists	consist	VERB
ejde-196	59	11	of	of	ADP
ejde-196	59	12	locally	locally	ADV
ejde-196	59	13	m	m	NOUN
ejde-196	59	14	and/or	and/or	CCONJ
ejde-196	59	15	w	w	NOUN
ejde-196	59	16	parts	part	NOUN
ejde-196	59	17	.	.	PUNCT
ejde-196	60	1	to	to	PART
ejde-196	60	2	solve	solve	VERB
ejde-196	60	3	m	m	NOUN
ejde-196	60	4	-	-	PUNCT
ejde-196	60	5	type	type	NOUN
ejde-196	60	6	problems	problem	NOUN
ejde-196	60	7	for	for	ADP
ejde-196	60	8	multiple	multiple	ADJ
ejde-196	60	9	solutions	solution	NOUN
ejde-196	60	10	,	,	PUNCT
ejde-196	60	11	a	a	DET
ejde-196	60	12	local	local	ADJ
ejde-196	60	13	min	min	ADJ
ejde-196	60	14	-	-	ADJ
ejde-196	60	15	max	max	PROPN
ejde-196	60	16	method	method	NOUN
ejde-196	60	17	(	(	PUNCT
ejde-196	60	18	lmm	lmm	PROPN
ejde-196	60	19	)	)	PUNCT
ejde-196	60	20	was	be	AUX
ejde-196	60	21	developed	develop	VERB
ejde-196	60	22	in	in	ADP
ejde-196	60	23	[	[	X
ejde-196	60	24	4	4	NUM
ejde-196	60	25	]	]	PUNCT
ejde-196	60	26	.	.	PUNCT
ejde-196	61	1	theoretically	theoretically	ADV
ejde-196	61	2	it	it	PRON
ejde-196	61	3	was	be	AUX
ejde-196	61	4	extended	extend	VERB
ejde-196	61	5	to	to	ADP
ejde-196	61	6	a	a	DET
ejde-196	61	7	local	local	ADJ
ejde-196	61	8	min	min	ADJ
ejde-196	61	9	-	-	ADJ
ejde-196	61	10	orthogonal	orthogonal	ADJ
ejde-196	61	11	method	method	NOUN
ejde-196	61	12	in	in	ADP
ejde-196	61	13	[	[	X
ejde-196	61	14	16	16	NUM
ejde-196	61	15	]	]	PUNCT
ejde-196	61	16	.	.	PUNCT
ejde-196	62	1	but	but	CCONJ
ejde-196	62	2	it	it	PRON
ejde-196	62	3	has	have	AUX
ejde-196	62	4	not	not	PART
ejde-196	62	5	been	be	AUX
ejde-196	62	6	used	use	VERB
ejde-196	62	7	to	to	PART
ejde-196	62	8	solve	solve	VERB
ejde-196	62	9	the	the	DET
ejde-196	62	10	w	w	NOUN
ejde-196	62	11	-	-	PUNCT
ejde-196	62	12	type	type	NOUN
ejde-196	62	13	or	or	CCONJ
ejde-196	62	14	other	other	ADJ
ejde-196	62	15	type	type	NOUN
ejde-196	62	16	problems	problem	NOUN
ejde-196	62	17	.	.	PUNCT
ejde-196	63	1	in	in	ADP
ejde-196	63	2	this	this	DET
ejde-196	63	3	paper	paper	NOUN
ejde-196	63	4	,	,	PUNCT
ejde-196	63	5	we	we	PRON
ejde-196	63	6	generalize	generalize	VERB
ejde-196	63	7	the	the	DET
ejde-196	63	8	local	local	ADJ
ejde-196	63	9	min	min	ADJ
ejde-196	63	10	-	-	ADJ
ejde-196	63	11	orthogonal	orthogonal	ADJ
ejde-196	63	12	method	method	NOUN
ejde-196	63	13	and	and	CCONJ
ejde-196	63	14	explore	explore	VERB
ejde-196	63	15	its	its	PRON
ejde-196	63	16	flexibilities	flexibility	NOUN
ejde-196	63	17	to	to	PART
ejde-196	63	18	solve	solve	VERB
ejde-196	63	19	the	the	DET
ejde-196	63	20	m	m	NOUN
ejde-196	63	21	-	-	NOUN
ejde-196	63	22	type	type	NOUN
ejde-196	63	23	,	,	PUNCT
ejde-196	63	24	the	the	DET
ejde-196	63	25	w	w	NOUN
ejde-196	63	26	-	-	PUNCT
ejde-196	63	27	type	type	NOUN
ejde-196	63	28	and	and	CCONJ
ejde-196	63	29	even	even	ADV
ejde-196	63	30	mixed	mixed	ADJ
ejde-196	63	31	m	m	PROPN
ejde-196	63	32	-	-	PUNCT
ejde-196	63	33	w	w	NOUN
ejde-196	63	34	-	-	PUNCT
ejde-196	63	35	type	type	NOUN
ejde-196	63	36	problems	problem	NOUN
ejde-196	63	37	.	.	PUNCT
ejde-196	64	1	ejde-2023	ejde-2023	ADJ
ejde-196	64	2	/	/	SYM
ejde-196	64	3	si/02	si/02	ADJ
ejde-196	64	4	short	short	ADJ
ejde-196	64	5	title	title	NOUN
ejde-196	64	6	177	177	NUM
ejde-196	64	7	let	let	VERB
ejde-196	64	8	l	l	NOUN
ejde-196	64	9	be	be	AUX
ejde-196	64	10	a	a	DET
ejde-196	64	11	closed	closed	ADJ
ejde-196	64	12	subset	subset	NOUN
ejde-196	64	13	of	of	ADP
ejde-196	64	14	h.	h.	PROPN
ejde-196	64	15	we	we	PRON
ejde-196	64	16	denote	denote	VERB
ejde-196	64	17	sl⊥	sl⊥	PROPN
ejde-196	64	18	=	=	PUNCT
ejde-196	64	19	{	{	PUNCT
ejde-196	64	20	v	v	NUM
ejde-196	64	21	∈	∈	NOUN
ejde-196	64	22	h	h	NOUN
ejde-196	64	23	:	:	PUNCT
ejde-196	65	1	‖v‖	‖v‖	ADJ
ejde-196	65	2	=	=	SYM
ejde-196	65	3	1	1	NUM
ejde-196	65	4	,	,	PUNCT
ejde-196	65	5	v⊥l	v⊥l	NOUN
ejde-196	65	6	}	}	PUNCT
ejde-196	65	7	,	,	PUNCT
ejde-196	65	8	and	and	CCONJ
ejde-196	65	9	for	for	ADP
ejde-196	65	10	each	each	DET
ejde-196	65	11	v	v	NOUN
ejde-196	65	12	∈	∈	NOUN
ejde-196	65	13	sl⊥	sl⊥	NOUN
ejde-196	65	14	,	,	PUNCT
ejde-196	65	15	let	let	VERB
ejde-196	65	16	[	[	X
ejde-196	65	17	v	v	ADP
ejde-196	65	18	,	,	PUNCT
ejde-196	65	19	l	l	NOUN
ejde-196	65	20	]	]	X
ejde-196	65	21	=	=	SYM
ejde-196	65	22	span{v	span{v	ADJ
ejde-196	65	23	,	,	PUNCT
ejde-196	65	24	l	l	NOUN
ejde-196	65	25	}	}	PUNCT
ejde-196	65	26	.	.	PUNCT
ejde-196	66	1	let	let	VERB
ejde-196	66	2	v∗	v∗	NOUN
ejde-196	66	3	=	=	NOUN
ejde-196	66	4	arg	arg	NOUN
ejde-196	66	5	min	min	PROPN
ejde-196	66	6	v∈s	v∈s	ADJ
ejde-196	66	7	l⊥	l⊥	NOUN
ejde-196	66	8	j(p(v	j(p(v	NOUN
ejde-196	66	9	)	)	PUNCT
ejde-196	66	10	)	)	PUNCT
ejde-196	66	11	,	,	PUNCT
ejde-196	67	1	where	where	SCONJ
ejde-196	67	2	for	for	ADP
ejde-196	67	3	the	the	DET
ejde-196	67	4	local	local	ADJ
ejde-196	67	5	min	min	ADJ
ejde-196	67	6	-	-	ADJ
ejde-196	67	7	max	max	PROPN
ejde-196	67	8	method	method	NOUN
ejde-196	67	9	(	(	PUNCT
ejde-196	67	10	lmm	lmm	PROPN
ejde-196	67	11	)	)	PUNCT
ejde-196	67	12	[	[	X
ejde-196	67	13	4	4	NUM
ejde-196	67	14	,	,	PUNCT
ejde-196	67	15	2001	2001	NUM
ejde-196	67	16	]	]	PUNCT
ejde-196	67	17	,	,	PUNCT
ejde-196	67	18	p(v	p(v	NOUN
ejde-196	67	19	)	)	PUNCT
ejde-196	67	20	=	=	SYM
ejde-196	67	21	arg	arg	NOUN
ejde-196	67	22	maxu∈[v	maxu∈[v	X
ejde-196	67	23	,	,	PUNCT
ejde-196	67	24	l	l	NOUN
ejde-196	67	25	]	]	X
ejde-196	67	26	j(u	j(u	PROPN
ejde-196	67	27	)	)	PUNCT
ejde-196	67	28	and	and	CCONJ
ejde-196	67	29	for	for	ADP
ejde-196	67	30	the	the	DET
ejde-196	67	31	local	local	ADJ
ejde-196	67	32	min	min	ADJ
ejde-196	67	33	-	-	ADJ
ejde-196	67	34	orthogonal	orthogonal	ADJ
ejde-196	67	35	method	method	NOUN
ejde-196	67	36	[	[	X
ejde-196	67	37	16	16	NUM
ejde-196	67	38	]	]	PUNCT
ejde-196	67	39	,	,	PUNCT
ejde-196	67	40	p(v	p(v	NOUN
ejde-196	67	41	)	)	PUNCT
ejde-196	67	42	∈	∈	PROPN
ejde-196	68	1	[	[	X
ejde-196	68	2	v	v	NOUN
ejde-196	68	3	,	,	PUNCT
ejde-196	68	4	l	l	NOUN
ejde-196	68	5	]	]	X
ejde-196	68	6	such	such	ADJ
ejde-196	68	7	that	that	SCONJ
ejde-196	68	8	j	j	PROPN
ejde-196	68	9	′(p(v))⊥[v	′(p(v))⊥[v	PROPN
ejde-196	68	10	,	,	PUNCT
ejde-196	68	11	l	l	NOUN
ejde-196	68	12	]	]	PUNCT
ejde-196	68	13	.	.	PUNCT
ejde-196	69	1	then	then	ADV
ejde-196	69	2	u∗	u∗	ADV
ejde-196	69	3	=	=	SYM
ejde-196	69	4	p(v∗	p(v∗	X
ejde-196	69	5	)	)	PUNCT
ejde-196	69	6	is	be	AUX
ejde-196	69	7	a	a	DET
ejde-196	69	8	critical	critical	ADJ
ejde-196	69	9	point	point	NOUN
ejde-196	69	10	of	of	ADP
ejde-196	69	11	j	j	PROPN
ejde-196	69	12	.	.	PUNCT
ejde-196	70	1	since	since	SCONJ
ejde-196	70	2	when	when	SCONJ
ejde-196	70	3	p(v	p(v	NOUN
ejde-196	70	4	)	)	PUNCT
ejde-196	70	5	∈	∈	PROPN
ejde-196	71	1	[	[	X
ejde-196	71	2	v	v	NOUN
ejde-196	71	3	,	,	PUNCT
ejde-196	71	4	l	l	NOUN
ejde-196	71	5	]	]	X
ejde-196	71	6	is	be	AUX
ejde-196	71	7	a	a	DET
ejde-196	71	8	local	local	ADJ
ejde-196	71	9	maximum	maximum	ADJ
ejde-196	71	10	point	point	NOUN
ejde-196	71	11	of	of	ADP
ejde-196	71	12	j	j	PROPN
ejde-196	71	13	in	in	ADP
ejde-196	71	14	[	[	X
ejde-196	71	15	v	v	NOUN
ejde-196	71	16	,	,	PUNCT
ejde-196	71	17	l	l	NOUN
ejde-196	71	18	]	]	X
ejde-196	71	19	,	,	PUNCT
ejde-196	71	20	we	we	PRON
ejde-196	71	21	have	have	VERB
ejde-196	71	22	j	j	PROPN
ejde-196	71	23	′(p(v))⊥[v	′(p(v))⊥[v	PROPN
ejde-196	71	24	,	,	PUNCT
ejde-196	71	25	l	l	NOUN
ejde-196	71	26	]	]	PUNCT
ejde-196	71	27	and	and	CCONJ
ejde-196	71	28	the	the	DET
ejde-196	71	29	local	local	ADJ
ejde-196	71	30	min	min	ADJ
ejde-196	71	31	-	-	ADJ
ejde-196	71	32	orthogonal	orthogonal	ADJ
ejde-196	71	33	method	method	NOUN
ejde-196	71	34	becomes	become	VERB
ejde-196	71	35	lmm	lmm	PROPN
ejde-196	71	36	which	which	PRON
ejde-196	71	37	was	be	AUX
ejde-196	71	38	first	first	ADV
ejde-196	71	39	developed	develop	VERB
ejde-196	71	40	in	in	ADP
ejde-196	71	41	[	[	X
ejde-196	71	42	4	4	NUM
ejde-196	71	43	]	]	PUNCT
ejde-196	71	44	and	and	CCONJ
ejde-196	71	45	successfully	successfully	ADV
ejde-196	71	46	used	use	VERB
ejde-196	71	47	to	to	PART
ejde-196	71	48	solve	solve	VERB
ejde-196	71	49	many	many	ADJ
ejde-196	71	50	m	m	ADJ
ejde-196	71	51	-	-	PUNCT
ejde-196	71	52	type	type	NOUN
ejde-196	71	53	problems	problem	NOUN
ejde-196	71	54	.	.	PUNCT
ejde-196	72	1	definition	definition	NOUN
ejde-196	72	2	1.1	1.1	NUM
ejde-196	72	3	(	(	PUNCT
ejde-196	72	4	[	[	X
ejde-196	72	5	16	16	NUM
ejde-196	72	6	]	]	PUNCT
ejde-196	72	7	)	)	PUNCT
ejde-196	72	8	.	.	PUNCT
ejde-196	73	1	the	the	DET
ejde-196	73	2	set	set	NOUN
ejde-196	73	3	-	-	PUNCT
ejde-196	73	4	valued	value	VERB
ejde-196	73	5	mapping	mapping	NOUN
ejde-196	73	6	p	p	X
ejde-196	73	7	:	:	PUNCT
ejde-196	73	8	sl⊥	sl⊥	PROPN
ejde-196	73	9	→	→	SYM
ejde-196	73	10	2h	2h	NUM
ejde-196	73	11	is	be	AUX
ejde-196	73	12	called	call	VERB
ejde-196	73	13	the	the	DET
ejde-196	73	14	lorthogonal	lorthogonal	ADJ
ejde-196	73	15	mapping	mapping	NOUN
ejde-196	73	16	if	if	SCONJ
ejde-196	73	17	p	p	PROPN
ejde-196	73	18	(	(	PUNCT
ejde-196	73	19	v	v	NOUN
ejde-196	73	20	)	)	PUNCT
ejde-196	73	21	:	:	PUNCT
ejde-196	74	1	=	=	SYM
ejde-196	74	2	{	{	PUNCT
ejde-196	74	3	u	u	NOUN
ejde-196	74	4	∈	∈	PROPN
ejde-196	75	1	[	[	X
ejde-196	75	2	l	l	NOUN
ejde-196	75	3	,	,	PUNCT
ejde-196	75	4	v	v	NOUN
ejde-196	75	5	]	]	X
ejde-196	75	6	:	:	PUNCT
ejde-196	75	7	j	j	PROPN
ejde-196	75	8	′(u)⊥[l	′(u)⊥[l	PROPN
ejde-196	75	9	,	,	PUNCT
ejde-196	75	10	v	v	NOUN
ejde-196	75	11	]	]	X
ejde-196	75	12	}	}	PUNCT
ejde-196	75	13	for	for	ADP
ejde-196	75	14	each	each	DET
ejde-196	75	15	v	v	NOUN
ejde-196	75	16	∈	∈	PROPN
ejde-196	75	17	sl⊥	sl⊥	PROPN
ejde-196	75	18	.	.	PUNCT
ejde-196	76	1	a	a	DET
ejde-196	76	2	function	function	NOUN
ejde-196	76	3	p	p	NOUN
ejde-196	76	4	:	:	PUNCT
ejde-196	76	5	sl⊥	sl⊥	PROPN
ejde-196	76	6	→	→	SYM
ejde-196	76	7	h	h	NOUN
ejde-196	76	8	is	be	AUX
ejde-196	76	9	called	call	VERB
ejde-196	76	10	an	an	DET
ejde-196	76	11	l	l	ADJ
ejde-196	76	12	-	-	ADJ
ejde-196	76	13	orthogonal	orthogonal	ADJ
ejde-196	76	14	selection	selection	NOUN
ejde-196	76	15	if	if	SCONJ
ejde-196	76	16	p(v	p(v	NOUN
ejde-196	76	17	)	)	PUNCT
ejde-196	76	18	∈	∈	PROPN
ejde-196	76	19	p	p	X
ejde-196	76	20	(	(	PUNCT
ejde-196	76	21	v	v	NOUN
ejde-196	76	22	)	)	PUNCT
ejde-196	76	23	∀v	∀v	PROPN
ejde-196	76	24	∈	∈	PROPN
ejde-196	76	25	sl⊥	sl⊥	PROPN
ejde-196	76	26	.	.	PUNCT
ejde-196	77	1	if	if	SCONJ
ejde-196	77	2	p	p	NOUN
ejde-196	77	3	is	be	AUX
ejde-196	77	4	locally	locally	ADV
ejde-196	77	5	defined	define	VERB
ejde-196	77	6	then	then	ADV
ejde-196	77	7	p	p	NOUN
ejde-196	77	8	is	be	AUX
ejde-196	77	9	called	call	VERB
ejde-196	77	10	a	a	DET
ejde-196	77	11	local	local	ADJ
ejde-196	77	12	l	l	ADJ
ejde-196	77	13	-	-	ADJ
ejde-196	77	14	orthogonal	orthogonal	ADJ
ejde-196	77	15	selection	selection	NOUN
ejde-196	77	16	.	.	PUNCT
ejde-196	78	1	lemma	lemma	PROPN
ejde-196	78	2	1.2	1.2	NUM
ejde-196	78	3	(	(	PUNCT
ejde-196	78	4	[	[	X
ejde-196	78	5	16	16	NUM
ejde-196	78	6	]	]	PUNCT
ejde-196	78	7	)	)	PUNCT
ejde-196	78	8	.	.	PUNCT
ejde-196	79	1	if	if	SCONJ
ejde-196	79	2	j	j	PROPN
ejde-196	79	3	is	be	AUX
ejde-196	79	4	c1	c1	PROPN
ejde-196	79	5	,	,	PUNCT
ejde-196	79	6	then	then	ADV
ejde-196	79	7	g	g	PROPN
ejde-196	79	8	=	=	SYM
ejde-196	79	9	{	{	PUNCT
ejde-196	79	10	(	(	PUNCT
ejde-196	79	11	u	u	NOUN
ejde-196	79	12	,	,	PUNCT
ejde-196	79	13	v	v	NOUN
ejde-196	79	14	)	)	PUNCT
ejde-196	79	15	:	:	PUNCT
ejde-196	79	16	v	v	X
ejde-196	79	17	∈	∈	NOUN
ejde-196	79	18	sl⊥	sl⊥	PROPN
ejde-196	79	19	,	,	PUNCT
ejde-196	79	20	u	u	PROPN
ejde-196	79	21	∈	∈	PROPN
ejde-196	79	22	p	p	X
ejde-196	79	23	(	(	PUNCT
ejde-196	79	24	v	v	NOUN
ejde-196	79	25	)	)	PUNCT
ejde-196	79	26	}	}	PUNCT
ejde-196	79	27	is	be	AUX
ejde-196	79	28	closed	close	VERB
ejde-196	79	29	.	.	PUNCT
ejde-196	80	1	since	since	SCONJ
ejde-196	80	2	p(v	p(v	NOUN
ejde-196	80	3	)	)	PUNCT
ejde-196	80	4	=	=	SYM
ejde-196	80	5	tv	tv	NOUN
ejde-196	80	6	+	+	CCONJ
ejde-196	80	7	vl	vl	NOUN
ejde-196	80	8	for	for	ADP
ejde-196	80	9	some	some	DET
ejde-196	80	10	vl	vl	PROPN
ejde-196	80	11	∈	∈	PROPN
ejde-196	80	12	l	l	NOUN
ejde-196	80	13	,	,	PUNCT
ejde-196	80	14	it	it	PRON
ejde-196	80	15	is	be	AUX
ejde-196	80	16	clear	clear	ADJ
ejde-196	80	17	that	that	SCONJ
ejde-196	80	18	if	if	SCONJ
ejde-196	80	19	p	p	NOUN
ejde-196	80	20	is	be	AUX
ejde-196	80	21	continuous	continuous	ADJ
ejde-196	80	22	then	then	ADV
ejde-196	80	23	p(v∗	p(v∗	X
ejde-196	80	24	)	)	PUNCT
ejde-196	80	25	is	be	AUX
ejde-196	80	26	a	a	DET
ejde-196	80	27	critical	critical	ADJ
ejde-196	80	28	point	point	NOUN
ejde-196	80	29	of	of	ADP
ejde-196	80	30	j	j	PROPN
ejde-196	81	1	if	if	SCONJ
ejde-196	81	2	and	and	CCONJ
ejde-196	81	3	only	only	ADV
ejde-196	81	4	if	if	SCONJ
ejde-196	81	5	there	there	PRON
ejde-196	81	6	is	be	VERB
ejde-196	81	7	a	a	DET
ejde-196	81	8	neighborhood	neighborhood	NOUN
ejde-196	81	9	n	n	CCONJ
ejde-196	81	10	(	(	PUNCT
ejde-196	81	11	v∗	v∗	NOUN
ejde-196	81	12	)	)	PUNCT
ejde-196	81	13	of	of	ADP
ejde-196	81	14	v∗	v∗	NOUN
ejde-196	81	15	in	in	ADP
ejde-196	81	16	l⊥	l⊥	NOUN
ejde-196	81	17	such	such	ADJ
ejde-196	81	18	that	that	SCONJ
ejde-196	81	19	j	j	PROPN
ejde-196	81	20	′(p(v∗))⊥p(v	′(p(v∗))⊥p(v	PROPN
ejde-196	81	21	)	)	PUNCT
ejde-196	81	22	∀v	∀v	NOUN
ejde-196	81	23	∈	∈	PROPN
ejde-196	81	24	n	n	CCONJ
ejde-196	81	25	(	(	PUNCT
ejde-196	81	26	v∗	v∗	ADJ
ejde-196	81	27	)	)	PUNCT
ejde-196	81	28	∩	∩	NOUN
ejde-196	81	29	sl⊥	sl⊥	PROPN
ejde-196	81	30	.	.	PUNCT
ejde-196	82	1	(	(	PUNCT
ejde-196	82	2	1.4	1.4	NUM
ejde-196	82	3	)	)	PUNCT
ejde-196	82	4	we	we	PRON
ejde-196	82	5	call	call	VERB
ejde-196	82	6	(	(	PUNCT
ejde-196	82	7	1.4	1.4	NUM
ejde-196	82	8	)	)	PUNCT
ejde-196	82	9	a	a	DET
ejde-196	82	10	double	double	ADJ
ejde-196	82	11	-	-	PUNCT
ejde-196	82	12	orthogonal	orthogonal	ADJ
ejde-196	82	13	principle	principle	NOUN
ejde-196	82	14	for	for	ADP
ejde-196	82	15	critical	critical	ADJ
ejde-196	82	16	points	point	NOUN
ejde-196	82	17	.	.	PUNCT
ejde-196	83	1	an	an	DET
ejde-196	83	2	important	important	ADJ
ejde-196	83	3	feature	feature	NOUN
ejde-196	83	4	of	of	ADP
ejde-196	83	5	this	this	DET
ejde-196	83	6	principle	principle	NOUN
ejde-196	83	7	is	be	AUX
ejde-196	83	8	that	that	SCONJ
ejde-196	83	9	it	it	PRON
ejde-196	83	10	involves	involve	VERB
ejde-196	83	11	only	only	ADV
ejde-196	83	12	j	j	PROPN
ejde-196	83	13	′	′	INTJ
ejde-196	83	14	not	not	PART
ejde-196	83	15	the	the	DET
ejde-196	83	16	functional	functional	ADJ
ejde-196	83	17	j	j	PROPN
ejde-196	83	18	.	.	PUNCT
ejde-196	84	1	it	it	PRON
ejde-196	84	2	implies	imply	VERB
ejde-196	84	3	that	that	SCONJ
ejde-196	84	4	this	this	DET
ejde-196	84	5	principle	principle	NOUN
ejde-196	84	6	can	can	AUX
ejde-196	84	7	treat	treat	VERB
ejde-196	84	8	nonvariational	nonvariational	ADJ
ejde-196	84	9	multiple	multiple	ADJ
ejde-196	84	10	solution	solution	NOUN
ejde-196	84	11	problems	problem	NOUN
ejde-196	84	12	where	where	SCONJ
ejde-196	84	13	j	j	PROPN
ejde-196	84	14	′(v	′(v	PROPN
ejde-196	84	15	)	)	PUNCT
ejde-196	85	1	=	=	SYM
ejde-196	85	2	0	0	NUM
ejde-196	85	3	is	be	AUX
ejde-196	85	4	replaced	replace	VERB
ejde-196	85	5	by	by	ADP
ejde-196	85	6	a(v	a(v	NOUN
ejde-196	85	7	)	)	PUNCT
ejde-196	85	8	=	=	SYM
ejde-196	85	9	0	0	NUM
ejde-196	85	10	for	for	ADP
ejde-196	85	11	a	a	DET
ejde-196	85	12	nonlinear	nonlinear	ADJ
ejde-196	85	13	operator	operator	NOUN
ejde-196	85	14	a.	a.	NOUN
ejde-196	85	15	for	for	ADP
ejde-196	85	16	example	example	NOUN
ejde-196	85	17	,	,	PUNCT
ejde-196	85	18	it	it	PRON
ejde-196	85	19	is	be	AUX
ejde-196	85	20	applied	apply	VERB
ejde-196	85	21	to	to	PART
ejde-196	85	22	design	design	VERB
ejde-196	85	23	algorithms	algorithm	NOUN
ejde-196	85	24	to	to	PART
ejde-196	85	25	find	find	VERB
ejde-196	85	26	nonvariational	nonvariational	ADJ
ejde-196	85	27	multiple	multiple	ADJ
ejde-196	85	28	solutions	solution	NOUN
ejde-196	85	29	,	,	PUNCT
ejde-196	85	30	see	see	VERB
ejde-196	85	31	[	[	X
ejde-196	85	32	5	5	NUM
ejde-196	85	33	,	,	PUNCT
ejde-196	85	34	13	13	NUM
ejde-196	85	35	]	]	PUNCT
ejde-196	85	36	.	.	PUNCT
ejde-196	86	1	in	in	ADP
ejde-196	86	2	this	this	DET
ejde-196	86	3	paper	paper	NOUN
ejde-196	86	4	,	,	PUNCT
ejde-196	86	5	we	we	PRON
ejde-196	86	6	focus	focus	VERB
ejde-196	86	7	on	on	ADP
ejde-196	86	8	solving	solve	VERB
ejde-196	86	9	variational	variational	ADJ
ejde-196	86	10	multiple	multiple	ADJ
ejde-196	86	11	solution	solution	NOUN
ejde-196	86	12	problems	problem	NOUN
ejde-196	86	13	with	with	ADP
ejde-196	86	14	different	different	ADJ
ejde-196	86	15	mathematical	mathematical	ADJ
ejde-196	86	16	structures	structure	NOUN
ejde-196	86	17	by	by	ADP
ejde-196	86	18	exploring	explore	VERB
ejde-196	86	19	variations	variation	NOUN
ejde-196	86	20	of	of	ADP
ejde-196	86	21	this	this	DET
ejde-196	86	22	principle	principle	NOUN
ejde-196	86	23	.	.	PUNCT
ejde-196	87	1	in	in	ADP
ejde-196	87	2	numerical	numerical	ADJ
ejde-196	87	3	implementation	implementation	NOUN
ejde-196	87	4	,	,	PUNCT
ejde-196	87	5	the	the	DET
ejde-196	87	6	outer	outer	ADJ
ejde-196	87	7	-	-	PUNCT
ejde-196	87	8	layer	layer	NOUN
ejde-196	87	9	is	be	AUX
ejde-196	87	10	an	an	DET
ejde-196	87	11	infinite	infinite	ADJ
ejde-196	87	12	-	-	PUNCT
ejde-196	87	13	dimensional	dimensional	ADJ
ejde-196	87	14	problem	problem	NOUN
ejde-196	87	15	and	and	CCONJ
ejde-196	87	16	the	the	DET
ejde-196	87	17	innerlayer	innerlayer	NOUN
ejde-196	87	18	consists	consist	VERB
ejde-196	87	19	of	of	ADP
ejde-196	87	20	finite	finite	ADJ
ejde-196	87	21	-	-	ADJ
ejde-196	87	22	dimensional	dimensional	ADJ
ejde-196	87	23	sub	sub	NOUN
ejde-196	87	24	-	-	NOUN
ejde-196	87	25	problems	problem	NOUN
ejde-196	87	26	.	.	PUNCT
ejde-196	88	1	when	when	SCONJ
ejde-196	88	2	the	the	DET
ejde-196	88	3	outer	outer	ADJ
ejde-196	88	4	-	-	PUNCT
ejde-196	88	5	layer	layer	NOUN
ejde-196	88	6	orthogonal	orthogonal	ADJ
ejde-196	88	7	condition	condition	NOUN
ejde-196	88	8	in	in	ADP
ejde-196	88	9	(	(	PUNCT
ejde-196	88	10	1.4	1.4	NUM
ejde-196	88	11	)	)	PUNCT
ejde-196	88	12	is	be	AUX
ejde-196	88	13	achieved	achieve	VERB
ejde-196	88	14	by	by	ADP
ejde-196	88	15	a	a	DET
ejde-196	88	16	minimization	minimization	NOUN
ejde-196	88	17	process	process	NOUN
ejde-196	88	18	.	.	PUNCT
ejde-196	89	1	it	it	PRON
ejde-196	89	2	becomes	become	VERB
ejde-196	89	3	a	a	DET
ejde-196	89	4	local	local	ADJ
ejde-196	89	5	minorthogonal	minorthogonal	ADJ
ejde-196	89	6	principle	principle	NOUN
ejde-196	89	7	for	for	ADP
ejde-196	89	8	characterizing	characterize	VERB
ejde-196	89	9	variational	variational	ADJ
ejde-196	89	10	critical	critical	ADJ
ejde-196	89	11	points	point	NOUN
ejde-196	89	12	min	min	PROPN
ejde-196	89	13	v∈s	v∈s	ADJ
ejde-196	89	14	l⊥	l⊥	NOUN
ejde-196	89	15	j(p(v	j(p(v	NOUN
ejde-196	89	16	)	)	PUNCT
ejde-196	89	17	)	)	PUNCT
ejde-196	89	18	.	.	PUNCT
ejde-196	90	1	(	(	PUNCT
ejde-196	90	2	1.5	1.5	NUM
ejde-196	90	3	)	)	PUNCT
ejde-196	90	4	this	this	DET
ejde-196	90	5	result	result	NOUN
ejde-196	90	6	is	be	AUX
ejde-196	90	7	first	first	ADV
ejde-196	90	8	announced	announce	VERB
ejde-196	90	9	in	in	ADP
ejde-196	90	10	[	[	X
ejde-196	90	11	4	4	NUM
ejde-196	90	12	]	]	PUNCT
ejde-196	90	13	and	and	CCONJ
ejde-196	90	14	then	then	ADV
ejde-196	90	15	published	publish	VERB
ejde-196	90	16	in	in	ADP
ejde-196	90	17	[	[	X
ejde-196	90	18	16	16	NUM
ejde-196	90	19	]	]	PUNCT
ejde-196	90	20	.	.	PUNCT
ejde-196	91	1	as	as	ADP
ejde-196	91	2	one	one	NUM
ejde-196	91	3	of	of	ADP
ejde-196	91	4	the	the	DET
ejde-196	91	5	most	most	ADV
ejde-196	91	6	used	used	ADJ
ejde-196	91	7	mathematical	mathematical	ADJ
ejde-196	91	8	frameworks	framework	NOUN
ejde-196	91	9	in	in	ADP
ejde-196	91	10	critical	critical	ADJ
ejde-196	91	11	point	point	NOUN
ejde-196	91	12	theory	theory	NOUN
ejde-196	91	13	,	,	PUNCT
ejde-196	91	14	the	the	DET
ejde-196	91	15	ljusternik	ljusternik	NOUN
ejde-196	91	16	-	-	PUNCT
ejde-196	91	17	schnirelman	schnirelman	NOUN
ejde-196	91	18	principle	principle	NOUN
ejde-196	91	19	(	(	PUNCT
ejde-196	91	20	lsp	lsp	PROPN
ejde-196	91	21	)	)	PUNCT
ejde-196	91	22	[	[	X
ejde-196	91	23	15	15	NUM
ejde-196	91	24	]	]	PUNCT
ejde-196	91	25	characterizes	characterize	VERB
ejde-196	91	26	a	a	DET
ejde-196	91	27	critical	critical	ADJ
ejde-196	91	28	point	point	NOUN
ejde-196	91	29	as	as	ADP
ejde-196	91	30	a	a	DET
ejde-196	91	31	solution	solution	NOUN
ejde-196	91	32	to	to	ADP
ejde-196	91	33	the	the	DET
ejde-196	91	34	min	min	PROPN
ejde-196	91	35	-	-	ADJ
ejde-196	91	36	max	max	PROPN
ejde-196	91	37	problem	problem	NOUN
ejde-196	91	38	min	min	PROPN
ejde-196	91	39	a∈a	a∈a	PROPN
ejde-196	91	40	max	max	PROPN
ejde-196	91	41	u∈a	u∈a	PROPN
ejde-196	91	42	j(u	j(u	PROPN
ejde-196	91	43	)	)	PUNCT
ejde-196	91	44	(	(	PUNCT
ejde-196	91	45	1.6	1.6	NUM
ejde-196	91	46	)	)	PUNCT
ejde-196	91	47	where	where	SCONJ
ejde-196	91	48	a	a	PRON
ejde-196	91	49	is	be	AUX
ejde-196	91	50	a	a	DET
ejde-196	91	51	collection	collection	NOUN
ejde-196	91	52	of	of	ADP
ejde-196	91	53	certain	certain	ADJ
ejde-196	91	54	compact	compact	ADJ
ejde-196	91	55	sets	set	NOUN
ejde-196	91	56	a	a	PRON
ejde-196	91	57	,	,	PUNCT
ejde-196	91	58	e.g.	e.g.	ADV
ejde-196	91	59	,	,	PUNCT
ejde-196	91	60	a	a	DET
ejde-196	91	61	k	k	ADJ
ejde-196	91	62	-	-	ADJ
ejde-196	91	63	dimensional	dimensional	ADJ
ejde-196	91	64	simplex	simplex	NOUN
ejde-196	91	65	,	,	PUNCT
ejde-196	91	66	the	the	DET
ejde-196	91	67	min	min	PROPN
ejde-196	91	68	and	and	CCONJ
ejde-196	91	69	max	max	PROPN
ejde-196	91	70	are	be	AUX
ejde-196	91	71	all	all	ADV
ejde-196	91	72	in	in	ADP
ejde-196	91	73	the	the	DET
ejde-196	91	74	global	global	ADJ
ejde-196	91	75	sense	sense	NOUN
ejde-196	91	76	,	,	PUNCT
ejde-196	91	77	therefore	therefore	ADV
ejde-196	91	78	lsp	lsp	PROPN
ejde-196	91	79	is	be	AUX
ejde-196	91	80	not	not	PART
ejde-196	91	81	for	for	ADP
ejde-196	91	82	algorithm	algorithm	NOUN
ejde-196	91	83	implementation	implementation	NOUN
ejde-196	91	84	.	.	PUNCT
ejde-196	92	1	let	let	VERB
ejde-196	92	2	us	we	PRON
ejde-196	92	3	compare	compare	VERB
ejde-196	92	4	our	our	PRON
ejde-196	92	5	double	double	ADJ
ejde-196	92	6	-	-	PUNCT
ejde-196	92	7	orthogonal	orthogonal	ADJ
ejde-196	92	8	principle	principle	NOUN
ejde-196	92	9	to	to	ADP
ejde-196	92	10	lsp	lsp	PROPN
ejde-196	92	11	,	,	PUNCT
ejde-196	92	12	one	one	PRON
ejde-196	92	13	can	can	AUX
ejde-196	92	14	see	see	VERB
ejde-196	92	15	that	that	SCONJ
ejde-196	92	16	the	the	DET
ejde-196	92	17	former	former	ADJ
ejde-196	92	18	is	be	AUX
ejde-196	92	19	more	more	ADV
ejde-196	92	20	general	general	ADJ
ejde-196	92	21	,	,	PUNCT
ejde-196	92	22	where	where	SCONJ
ejde-196	92	23	(	(	PUNCT
ejde-196	92	24	1	1	X
ejde-196	92	25	)	)	PUNCT
ejde-196	92	26	the	the	DET
ejde-196	92	27	two	two	NUM
ejde-196	92	28	global	global	ADJ
ejde-196	92	29	min	min	PROPN
ejde-196	92	30	and	and	CCONJ
ejde-196	92	31	max	max	PROPN
ejde-196	92	32	in	in	ADP
ejde-196	92	33	lsp	lsp	PROPN
ejde-196	92	34	are	be	AUX
ejde-196	92	35	replaced	replace	VERB
ejde-196	92	36	by	by	ADP
ejde-196	92	37	two	two	NUM
ejde-196	92	38	⊥-conditions	⊥-condition	NOUN
ejde-196	92	39	which	which	PRON
ejde-196	92	40	can	can	AUX
ejde-196	92	41	be	be	AUX
ejde-196	92	42	satisfied	satisfied	ADJ
ejde-196	92	43	,	,	PUNCT
ejde-196	92	44	e.g.	e.g.	ADV
ejde-196	92	45	,	,	PUNCT
ejde-196	92	46	by	by	ADP
ejde-196	92	47	local	local	ADJ
ejde-196	92	48	min	min	PROPN
ejde-196	92	49	and/or	and/or	CCONJ
ejde-196	92	50	max	max	PROPN
ejde-196	92	51	process	process	NOUN
ejde-196	92	52	,	,	PUNCT
ejde-196	92	53	and	and	CCONJ
ejde-196	92	54	(	(	PUNCT
ejde-196	92	55	2	2	X
ejde-196	92	56	)	)	PUNCT
ejde-196	92	57	compact	compact	ADJ
ejde-196	92	58	sets	set	NOUN
ejde-196	92	59	in	in	ADP
ejde-196	92	60	lsp	lsp	PROPN
ejde-196	92	61	are	be	AUX
ejde-196	92	62	generalized	generalize	VERB
ejde-196	92	63	to	to	ADP
ejde-196	92	64	closed	closed	ADJ
ejde-196	92	65	subspaces	subspace	NOUN
ejde-196	92	66	which	which	PRON
ejde-196	92	67	can	can	AUX
ejde-196	92	68	be	be	AUX
ejde-196	92	69	either	either	CCONJ
ejde-196	92	70	finite	finite	ADJ
ejde-196	92	71	dimensional	dimensional	ADJ
ejde-196	92	72	or	or	CCONJ
ejde-196	92	73	infinite	infinite	ADJ
ejde-196	92	74	dimensional	dimensional	ADJ
ejde-196	92	75	.	.	PUNCT
ejde-196	93	1	this	this	PRON
ejde-196	93	2	represents	represent	VERB
ejde-196	93	3	a	a	DET
ejde-196	93	4	significant	significant	ADJ
ejde-196	93	5	advantage	advantage	NOUN
ejde-196	93	6	of	of	ADP
ejde-196	93	7	our	our	PRON
ejde-196	93	8	double	double	ADJ
ejde-196	93	9	-	-	PUNCT
ejde-196	93	10	orthogonal	orthogonal	ADJ
ejde-196	93	11	principle	principle	NOUN
ejde-196	93	12	over	over	ADP
ejde-196	93	13	lsp	lsp	PROPN
ejde-196	93	14	,	,	PUNCT
ejde-196	93	15	in	in	ADP
ejde-196	93	16	particular	particular	ADJ
ejde-196	93	17	,	,	PUNCT
ejde-196	93	18	in	in	ADP
ejde-196	93	19	algorithm	algorithm	NOUN
ejde-196	93	20	implementation	implementation	NOUN
ejde-196	93	21	.	.	PUNCT
ejde-196	94	1	178	178	NUM
ejde-196	94	2	m.	m.	NOUN
ejde-196	94	3	li	li	PROPN
ejde-196	94	4	,	,	PUNCT
ejde-196	94	5	b.	b.	PROPN
ejde-196	94	6	ji	ji	PROPN
ejde-196	94	7	,	,	PUNCT
ejde-196	94	8	j.	j.	PROPN
ejde-196	94	9	zhou	zhou	PROPN
ejde-196	94	10	ejde	ejde	PROPN
ejde-196	94	11	/	/	SYM
ejde-196	94	12	si/02	si/02	PROPN
ejde-196	94	13	in	in	ADP
ejde-196	94	14	1960	1960	NUM
ejde-196	94	15	,	,	PUNCT
ejde-196	94	16	the	the	DET
ejde-196	94	17	nehari	nehari	NOUN
ejde-196	94	18	manifold	manifold	VERB
ejde-196	94	19	[	[	X
ejde-196	94	20	7	7	NUM
ejde-196	94	21	]	]	SYM
ejde-196	94	22	n	n	NOUN
ejde-196	94	23	=	=	PUNCT
ejde-196	94	24	{	{	PUNCT
ejde-196	94	25	tuu	tuu	VERB
ejde-196	94	26	6=	6=	ADP
ejde-196	94	27	0	0	NUM
ejde-196	94	28	:	:	PUNCT
ejde-196	94	29	u	u	PROPN
ejde-196	94	30	∈	∈	PROPN
ejde-196	94	31	h	h	NOUN
ejde-196	94	32	,	,	PUNCT
ejde-196	94	33	‖u‖	‖u‖	PROPN
ejde-196	94	34	=	=	SYM
ejde-196	94	35	1	1	NUM
ejde-196	94	36	,	,	PUNCT
ejde-196	94	37	j	j	PROPN
ejde-196	94	38	′(tuu)⊥u	′(tuu)⊥u	PROPN
ejde-196	94	39	}	}	PUNCT
ejde-196	94	40	(	(	PUNCT
ejde-196	94	41	1.7	1.7	NUM
ejde-196	94	42	)	)	PUNCT
ejde-196	94	43	was	be	AUX
ejde-196	94	44	introduced	introduce	VERB
ejde-196	94	45	and	and	CCONJ
ejde-196	94	46	widely	widely	ADV
ejde-196	94	47	applied	apply	VERB
ejde-196	94	48	later	later	ADV
ejde-196	94	49	on	on	ADV
ejde-196	94	50	in	in	ADP
ejde-196	94	51	the	the	DET
ejde-196	94	52	literature	literature	NOUN
ejde-196	94	53	to	to	PART
ejde-196	94	54	prove	prove	VERB
ejde-196	94	55	the	the	DET
ejde-196	94	56	existence	existence	NOUN
ejde-196	94	57	of	of	ADP
ejde-196	94	58	1	1	NUM
ejde-196	94	59	-	-	PUNCT
ejde-196	94	60	saddle	saddle	NOUN
ejde-196	94	61	of	of	ADP
ejde-196	94	62	various	various	ADJ
ejde-196	94	63	nonlinear	nonlinear	ADJ
ejde-196	94	64	problems	problem	NOUN
ejde-196	94	65	through	through	ADP
ejde-196	94	66	minu∈n	minu∈n	PROPN
ejde-196	94	67	j(u	j(u	PROPN
ejde-196	94	68	)	)	PUNCT
ejde-196	94	69	.	.	PUNCT
ejde-196	95	1	in	in	ADP
ejde-196	95	2	2005	2005	NUM
ejde-196	95	3	and	and	CCONJ
ejde-196	95	4	after	after	ADP
ejde-196	95	5	[	[	X
ejde-196	95	6	8	8	NUM
ejde-196	95	7	,	,	PUNCT
ejde-196	95	8	9	9	NUM
ejde-196	95	9	,	,	PUNCT
ejde-196	95	10	12	12	NUM
ejde-196	95	11	]	]	X
ejde-196	95	12	]	]	X
ejde-196	95	13	,	,	PUNCT
ejde-196	95	14	a	a	DET
ejde-196	95	15	generalized	generalize	VERB
ejde-196	95	16	nehari	nehari	NOUN
ejde-196	95	17	manifold	manifold	ADJ
ejde-196	95	18	nl	nl	PROPN
ejde-196	95	19	=	=	PUNCT
ejde-196	95	20	{	{	PUNCT
ejde-196	95	21	u	u	NOUN
ejde-196	95	22	∈	∈	PROPN
ejde-196	95	23	[	[	X
ejde-196	95	24	v	v	NOUN
ejde-196	95	25	,	,	PUNCT
ejde-196	95	26	l	l	NOUN
ejde-196	95	27	]	]	X
ejde-196	95	28	:	:	PUNCT
ejde-196	95	29	u	u	PROPN
ejde-196	95	30	6=	6=	PROPN
ejde-196	95	31	0	0	NUM
ejde-196	95	32	,	,	PUNCT
ejde-196	95	33	v	v	NOUN
ejde-196	95	34	∈	∈	NOUN
ejde-196	95	35	sl⊥	sl⊥	PROPN
ejde-196	95	36	,	,	PUNCT
ejde-196	95	37	j	j	PROPN
ejde-196	95	38	′(u)⊥[v	′(u)⊥[v	PROPN
ejde-196	95	39	,	,	PUNCT
ejde-196	95	40	l	l	NOUN
ejde-196	95	41	]	]	X
ejde-196	95	42	}	}	PUNCT
ejde-196	95	43	was	be	AUX
ejde-196	95	44	proposed	propose	VERB
ejde-196	95	45	with	with	ADP
ejde-196	95	46	l	l	NOUN
ejde-196	95	47	=	=	PUNCT
ejde-196	96	1	[	[	X
ejde-196	96	2	v1	v1	NOUN
ejde-196	96	3	,	,	PUNCT
ejde-196	96	4	.	.	PUNCT
ejde-196	96	5	.	.	PUNCT
ejde-196	97	1	.	.	PUNCT
ejde-196	98	1	,	,	PUNCT
ejde-196	98	2	vk	vk	ADP
ejde-196	98	3	]	]	PUNCT
ejde-196	98	4	where	where	SCONJ
ejde-196	98	5	v1	v1	NOUN
ejde-196	98	6	,	,	PUNCT
ejde-196	98	7	.	.	PUNCT
ejde-196	98	8	.	.	PUNCT
ejde-196	99	1	.	.	PUNCT
ejde-196	100	1	,	,	PUNCT
ejde-196	100	2	vk	vk	NOUN
ejde-196	100	3	are	be	AUX
ejde-196	100	4	the	the	DET
ejde-196	100	5	first	first	ADJ
ejde-196	100	6	k	k	PROPN
ejde-196	100	7	eigenfunctions	eigenfunction	NOUN
ejde-196	100	8	of	of	ADP
ejde-196	100	9	−∆	−∆	NOUN
ejde-196	100	10	and	and	CCONJ
ejde-196	100	11	applied	apply	VERB
ejde-196	100	12	to	to	PART
ejde-196	100	13	prove	prove	VERB
ejde-196	100	14	the	the	DET
ejde-196	100	15	existence	existence	NOUN
ejde-196	100	16	of	of	ADP
ejde-196	100	17	a	a	DET
ejde-196	100	18	k	k	NOUN
ejde-196	100	19	-	-	NOUN
ejde-196	100	20	saddle	saddle	NOUN
ejde-196	100	21	of	of	ADP
ejde-196	100	22	semilinear	semilinear	PROPN
ejde-196	100	23	elliptic	elliptic	ADJ
ejde-196	100	24	pdes	pde	NOUN
ejde-196	100	25	through	through	ADP
ejde-196	100	26	minu∈nl	minu∈nl	NOUN
ejde-196	100	27	j(u	j(u	PROPN
ejde-196	100	28	)	)	PUNCT
ejde-196	100	29	.	.	PUNCT
ejde-196	101	1	it	it	PRON
ejde-196	101	2	is	be	AUX
ejde-196	101	3	also	also	ADV
ejde-196	101	4	proved	prove	VERB
ejde-196	101	5	that	that	SCONJ
ejde-196	101	6	such	such	DET
ejde-196	101	7	a	a	DET
ejde-196	101	8	manifold	manifold	ADJ
ejde-196	101	9	nl	nl	NOUN
ejde-196	101	10	is	be	AUX
ejde-196	101	11	c1	c1	NOUN
ejde-196	101	12	.	.	PUNCT
ejde-196	102	1	it	it	PRON
ejde-196	102	2	is	be	AUX
ejde-196	102	3	clear	clear	ADJ
ejde-196	102	4	that	that	SCONJ
ejde-196	102	5	the	the	DET
ejde-196	102	6	generalized	generalize	VERB
ejde-196	102	7	nehari	nehari	NOUN
ejde-196	102	8	method	method	NOUN
ejde-196	102	9	is	be	AUX
ejde-196	102	10	just	just	ADV
ejde-196	102	11	a	a	DET
ejde-196	102	12	special	special	ADJ
ejde-196	102	13	case	case	NOUN
ejde-196	102	14	of	of	ADP
ejde-196	102	15	our	our	PRON
ejde-196	102	16	local	local	ADJ
ejde-196	102	17	minorthogonal	minorthogonal	ADJ
ejde-196	102	18	method	method	NOUN
ejde-196	102	19	.	.	PUNCT
ejde-196	103	1	actually	actually	ADV
ejde-196	103	2	for	for	ADP
ejde-196	103	3	the	the	DET
ejde-196	103	4	double	double	ADJ
ejde-196	103	5	-	-	PUNCT
ejde-196	103	6	orthogonal	orthogonal	ADJ
ejde-196	103	7	principle	principle	NOUN
ejde-196	103	8	,	,	PUNCT
ejde-196	103	9	there	there	PRON
ejde-196	103	10	are	be	VERB
ejde-196	103	11	many	many	ADJ
ejde-196	103	12	different	different	ADJ
ejde-196	103	13	ways	way	NOUN
ejde-196	103	14	to	to	PART
ejde-196	103	15	acheive	acheive	VERB
ejde-196	103	16	the	the	DET
ejde-196	103	17	inner	inner	ADJ
ejde-196	103	18	-	-	PUNCT
ejde-196	103	19	layer	layer	NOUN
ejde-196	103	20	orthogonal	orthogonal	ADJ
ejde-196	103	21	condition	condition	NOUN
ejde-196	103	22	.	.	PUNCT
ejde-196	104	1	in	in	ADP
ejde-196	104	2	this	this	DET
ejde-196	104	3	paper	paper	NOUN
ejde-196	104	4	,	,	PUNCT
ejde-196	104	5	we	we	PRON
ejde-196	104	6	first	first	ADV
ejde-196	104	7	prove	prove	VERB
ejde-196	104	8	this	this	DET
ejde-196	104	9	principle	principle	NOUN
ejde-196	104	10	with	with	ADP
ejde-196	104	11	a	a	DET
ejde-196	104	12	weaker	weak	ADJ
ejde-196	104	13	continuity	continuity	NOUN
ejde-196	104	14	condition	condition	NOUN
ejde-196	104	15	,	,	PUNCT
ejde-196	104	16	then	then	ADV
ejde-196	104	17	establish	establish	VERB
ejde-196	104	18	two	two	NUM
ejde-196	104	19	new	new	ADJ
ejde-196	104	20	variations	variation	NOUN
ejde-196	104	21	to	to	PART
ejde-196	104	22	show	show	VERB
ejde-196	104	23	how	how	SCONJ
ejde-196	104	24	this	this	DET
ejde-196	104	25	principle	principle	NOUN
ejde-196	104	26	can	can	AUX
ejde-196	104	27	be	be	AUX
ejde-196	104	28	modified	modify	VERB
ejde-196	104	29	to	to	PART
ejde-196	104	30	find	find	VERB
ejde-196	104	31	multiple	multiple	ADJ
ejde-196	104	32	solutions	solution	NOUN
ejde-196	104	33	to	to	ADP
ejde-196	104	34	w	w	NOUN
ejde-196	104	35	-	-	PUNCT
ejde-196	104	36	type	type	NOUN
ejde-196	104	37	and	and	CCONJ
ejde-196	104	38	even	even	ADV
ejde-196	104	39	mixed	mixed	ADJ
ejde-196	104	40	m	m	PROPN
ejde-196	104	41	-	-	PUNCT
ejde-196	104	42	w	w	NOUN
ejde-196	104	43	-	-	PUNCT
ejde-196	104	44	type	type	NOUN
ejde-196	104	45	problems	problem	NOUN
ejde-196	104	46	.	.	PUNCT
ejde-196	105	1	2	2	X
ejde-196	105	2	.	.	X
ejde-196	105	3	a	a	DET
ejde-196	105	4	new	new	ADJ
ejde-196	105	5	local	local	ADJ
ejde-196	105	6	min	min	ADJ
ejde-196	105	7	-	-	ADJ
ejde-196	105	8	orthogonal	orthogonal	ADJ
ejde-196	105	9	method	method	NOUN
ejde-196	105	10	for	for	ADP
ejde-196	105	11	analysis	analysis	NOUN
ejde-196	105	12	purpose	purpose	NOUN
ejde-196	105	13	,	,	PUNCT
ejde-196	105	14	we	we	PRON
ejde-196	105	15	first	first	ADV
ejde-196	105	16	extend	extend	VERB
ejde-196	105	17	the	the	DET
ejde-196	105	18	domain	domain	NOUN
ejde-196	105	19	of	of	ADP
ejde-196	105	20	p	p	X
ejde-196	105	21	to	to	PART
ejde-196	105	22	s+	s+	ADV
ejde-196	105	23	l⊥	l⊥	PROPN
ejde-196	105	24	=	=	SYM
ejde-196	105	25	{	{	PUNCT
ejde-196	105	26	v	v	NUM
ejde-196	105	27	∈	∈	PROPN
ejde-196	105	28	l⊥	l⊥	NOUN
ejde-196	105	29	,	,	PUNCT
ejde-196	105	30	‖v‖	‖v‖	PROPN
ejde-196	105	31	≈	≈	PROPN
ejde-196	105	32	1	1	NUM
ejde-196	105	33	}	}	PUNCT
ejde-196	105	34	where	where	SCONJ
ejde-196	105	35	by	by	ADP
ejde-196	105	36	‖v‖	‖v‖	PROPN
ejde-196	105	37	≈	≈	PROPN
ejde-196	105	38	1	1	NUM
ejde-196	105	39	we	we	PRON
ejde-196	105	40	mean	mean	VERB
ejde-196	106	1	1	1	NUM
ejde-196	106	2	−	−	PROPN
ejde-196	106	3	δ	δ	X
ejde-196	106	4	<	<	X
ejde-196	106	5	‖v‖	‖v‖	X
ejde-196	106	6	<	<	X
ejde-196	106	7	1	1	NUM
ejde-196	106	8	+	+	CCONJ
ejde-196	106	9	δ	δ	PROPN
ejde-196	106	10	for	for	ADP
ejde-196	106	11	a	a	DET
ejde-196	106	12	given	give	VERB
ejde-196	106	13	δ	δ	PROPN
ejde-196	106	14	>	>	X
ejde-196	106	15	0	0	PROPN
ejde-196	106	16	.	.	PUNCT
ejde-196	107	1	while	while	SCONJ
ejde-196	107	2	in	in	ADP
ejde-196	107	3	our	our	PRON
ejde-196	107	4	algorithm	algorithm	NOUN
ejde-196	107	5	computation	computation	NOUN
ejde-196	107	6	and	and	CCONJ
ejde-196	107	7	convergence	convergence	NOUN
ejde-196	107	8	analysis	analysis	NOUN
ejde-196	107	9	we	we	PRON
ejde-196	107	10	will	will	AUX
ejde-196	107	11	evaluate	evaluate	VERB
ejde-196	107	12	p	p	NOUN
ejde-196	107	13	only	only	ADV
ejde-196	107	14	at	at	ADP
ejde-196	107	15	v	v	NOUN
ejde-196	107	16	or	or	CCONJ
ejde-196	107	17	v+sw	v+sw	NOUN
ejde-196	108	1	‖v+sw‖	‖v+sw‖	PROPN
ejde-196	108	2	where	where	SCONJ
ejde-196	108	3	v	v	NOUN
ejde-196	108	4	,	,	PUNCT
ejde-196	108	5	w	w	PROPN
ejde-196	108	6	∈	∈	PROPN
ejde-196	108	7	l⊥	l⊥	NOUN
ejde-196	108	8	,	,	PUNCT
ejde-196	108	9	‖v‖	‖v‖	PROPN
ejde-196	108	10	=	=	SYM
ejde-196	108	11	1	1	NUM
ejde-196	108	12	for	for	ADP
ejde-196	108	13	small	small	ADJ
ejde-196	108	14	s.	s.	PROPN
ejde-196	108	15	thus	thus	ADV
ejde-196	108	16	the	the	DET
ejde-196	108	17	closed	closed	ADJ
ejde-196	108	18	set	set	VERB
ejde-196	108	19	sl⊥	sl⊥	PROPN
ejde-196	108	20	will	will	AUX
ejde-196	108	21	be	be	AUX
ejde-196	108	22	enough	enough	ADJ
ejde-196	108	23	for	for	SCONJ
ejde-196	108	24	us	we	PRON
ejde-196	108	25	to	to	PART
ejde-196	108	26	define	define	VERB
ejde-196	108	27	ldlc	ldlc	NOUN
ejde-196	108	28	of	of	ADP
ejde-196	108	29	p.	p.	NOUN
ejde-196	108	30	definition	definition	NOUN
ejde-196	108	31	2.1	2.1	NUM
ejde-196	108	32	.	.	PUNCT
ejde-196	109	1	for	for	ADP
ejde-196	109	2	j	j	PROPN
ejde-196	109	3	∈	∈	PROPN
ejde-196	109	4	c1(h	c1(h	PROPN
ejde-196	109	5	,	,	PUNCT
ejde-196	109	6	r	r	NOUN
ejde-196	109	7	)	)	PUNCT
ejde-196	109	8	,	,	PUNCT
ejde-196	109	9	a	a	DET
ejde-196	109	10	set	set	NOUN
ejde-196	109	11	-	-	PUNCT
ejde-196	109	12	valued	value	VERB
ejde-196	109	13	mapping	mapping	NOUN
ejde-196	109	14	p	p	X
ejde-196	109	15	:	:	PUNCT
ejde-196	109	16	s+	s+	ADV
ejde-196	109	17	l⊥	l⊥	PROPN
ejde-196	109	18	→	→	SYM
ejde-196	109	19	h2	h2	PROPN
ejde-196	109	20	is	be	AUX
ejde-196	109	21	called	call	VERB
ejde-196	109	22	the	the	DET
ejde-196	109	23	l	l	ADJ
ejde-196	109	24	-	-	ADJ
ejde-196	109	25	orthogonal	orthogonal	ADJ
ejde-196	109	26	mapping	mapping	NOUN
ejde-196	109	27	if	if	SCONJ
ejde-196	109	28	for	for	ADP
ejde-196	109	29	each	each	DET
ejde-196	109	30	v	v	NOUN
ejde-196	109	31	∈	∈	PROPN
ejde-196	109	32	s+	s+	PUNCT
ejde-196	109	33	l⊥	l⊥	PROPN
ejde-196	109	34	p	p	X
ejde-196	109	35	(	(	PUNCT
ejde-196	109	36	v	v	NOUN
ejde-196	109	37	)	)	PUNCT
ejde-196	109	38	=	=	PUNCT
ejde-196	110	1	{	{	PUNCT
ejde-196	110	2	u	u	X
ejde-196	110	3	∈	∈	PROPN
ejde-196	110	4	[	[	X
ejde-196	110	5	v	v	NOUN
ejde-196	110	6	,	,	PUNCT
ejde-196	110	7	l	l	NOUN
ejde-196	110	8	]	]	X
ejde-196	110	9	:	:	PUNCT
ejde-196	110	10	j	j	PROPN
ejde-196	110	11	′(u)⊥[v	′(u)⊥[v	PROPN
ejde-196	110	12	,	,	PUNCT
ejde-196	110	13	l	l	NOUN
ejde-196	110	14	]	]	X
ejde-196	110	15	}	}	PUNCT
ejde-196	110	16	.	.	PUNCT
ejde-196	111	1	an	an	DET
ejde-196	111	2	l	l	ADJ
ejde-196	111	3	-	-	ADJ
ejde-196	111	4	orthogonal	orthogonal	ADJ
ejde-196	111	5	selection	selection	NOUN
ejde-196	111	6	p	p	NOUN
ejde-196	111	7	:	:	PUNCT
ejde-196	111	8	s+	s+	ADV
ejde-196	111	9	l⊥	l⊥	PROPN
ejde-196	111	10	→	→	SYM
ejde-196	111	11	h	h	NOUN
ejde-196	111	12	is	be	AUX
ejde-196	111	13	a	a	DET
ejde-196	111	14	mapping	mapping	NOUN
ejde-196	111	15	such	such	ADJ
ejde-196	111	16	that	that	DET
ejde-196	111	17	p(v	p(v	NOUN
ejde-196	111	18	)	)	PUNCT
ejde-196	111	19	∈	∈	PROPN
ejde-196	111	20	p	p	NOUN
ejde-196	111	21	(	(	PUNCT
ejde-196	111	22	v),∀v	v),∀v	PROPN
ejde-196	111	23	∈	∈	PROPN
ejde-196	111	24	s+	s+	PUNCT
ejde-196	111	25	l⊥	l⊥	PROPN
ejde-196	111	26	.	.	PUNCT
ejde-196	112	1	if	if	SCONJ
ejde-196	112	2	p	p	NOUN
ejde-196	112	3	is	be	AUX
ejde-196	112	4	locally	locally	ADV
ejde-196	112	5	defined	define	VERB
ejde-196	112	6	near	near	ADP
ejde-196	112	7	v	v	X
ejde-196	112	8	∈	∈	NOUN
ejde-196	112	9	s+	s+	PUNCT
ejde-196	112	10	l⊥	l⊥	PROPN
ejde-196	112	11	,	,	PUNCT
ejde-196	112	12	then	then	ADV
ejde-196	112	13	p	p	PROPN
ejde-196	112	14	is	be	AUX
ejde-196	112	15	called	call	VERB
ejde-196	112	16	a	a	DET
ejde-196	112	17	local	local	ADJ
ejde-196	112	18	l	l	ADJ
ejde-196	112	19	-	-	ADJ
ejde-196	112	20	orthogonal	orthogonal	ADJ
ejde-196	112	21	selection	selection	NOUN
ejde-196	112	22	near	near	ADP
ejde-196	112	23	v.	v.	ADP
ejde-196	112	24	definition	definition	NOUN
ejde-196	112	25	2.2	2.2	NUM
ejde-196	112	26	.	.	PUNCT
ejde-196	113	1	a	a	DET
ejde-196	113	2	local	local	ADJ
ejde-196	113	3	l	l	ADJ
ejde-196	113	4	-	-	ADJ
ejde-196	113	5	orthogonal	orthogonal	ADJ
ejde-196	113	6	selection	selection	NOUN
ejde-196	113	7	p	p	NOUN
ejde-196	113	8	is	be	AUX
ejde-196	113	9	said	say	VERB
ejde-196	113	10	to	to	PART
ejde-196	113	11	be	be	AUX
ejde-196	113	12	locally	locally	ADV
ejde-196	113	13	directional	directional	ADJ
ejde-196	113	14	lipschitz	lipschitz	NOUN
ejde-196	113	15	continuous	continuous	ADJ
ejde-196	113	16	(	(	PUNCT
ejde-196	113	17	ldlc	ldlc	NOUN
ejde-196	113	18	)	)	PUNCT
ejde-196	113	19	at	at	ADP
ejde-196	113	20	v	v	NUM
ejde-196	113	21	∈	∈	NOUN
ejde-196	113	22	sl⊥	sl⊥	PROPN
ejde-196	113	23	in	in	ADP
ejde-196	113	24	w	w	PROPN
ejde-196	113	25	∈	∈	PROPN
ejde-196	113	26	l⊥	l⊥	NOUN
ejde-196	113	27	,	,	PUNCT
ejde-196	113	28	if	if	SCONJ
ejde-196	113	29	there	there	PRON
ejde-196	113	30	is	be	VERB
ejde-196	113	31	a	a	DET
ejde-196	113	32	constant	constant	ADJ
ejde-196	113	33	`	`	PUNCT
ejde-196	113	34	0	0	NUM
ejde-196	113	35	depending	depend	VERB
ejde-196	113	36	on	on	ADP
ejde-196	113	37	v	v	NUM
ejde-196	113	38	and	and	CCONJ
ejde-196	113	39	w	w	NOUN
ejde-196	113	40	,	,	PUNCT
ejde-196	113	41	such	such	ADJ
ejde-196	113	42	that	that	SCONJ
ejde-196	113	43	for	for	ADP
ejde-196	113	44	all	all	PRON
ejde-196	113	45	s	s	PROPN
ejde-196	113	46	>	>	X
ejde-196	113	47	0	0	PUNCT
ejde-196	113	48	small	small	ADJ
ejde-196	113	49	,	,	PUNCT
ejde-196	113	50	it	it	PRON
ejde-196	113	51	holds	hold	VERB
ejde-196	113	52	‖p(v	‖p(v	PUNCT
ejde-196	113	53	+	+	CCONJ
ejde-196	113	54	sw)−	sw)−	ADJ
ejde-196	113	55	p(v)‖	p(v)‖	ADJ
ejde-196	113	56	≤	≤	PROPN
ejde-196	113	57	`	`	PUNCT
ejde-196	113	58	0s‖w‖	0s‖w‖	NUM
ejde-196	113	59	,	,	PUNCT
ejde-196	113	60	where	where	SCONJ
ejde-196	113	61	the	the	DET
ejde-196	113	62	term	term	NOUN
ejde-196	113	63	‖w‖	‖w‖	PROPN
ejde-196	113	64	can	can	AUX
ejde-196	113	65	be	be	AUX
ejde-196	113	66	removed	remove	VERB
ejde-196	113	67	since	since	SCONJ
ejde-196	113	68	`	`	PUNCT
ejde-196	113	69	0	0	NUM
ejde-196	113	70	depends	depend	VERB
ejde-196	113	71	on	on	ADP
ejde-196	113	72	w.	w.	PROPN
ejde-196	113	73	we	we	PRON
ejde-196	113	74	say	say	VERB
ejde-196	113	75	that	that	SCONJ
ejde-196	113	76	p	p	NOUN
ejde-196	113	77	is	be	AUX
ejde-196	113	78	ldlc	ldlc	NOUN
ejde-196	113	79	at	at	ADP
ejde-196	113	80	v	v	NOUN
ejde-196	113	81	∈	∈	NOUN
ejde-196	113	82	sl⊥	sl⊥	PROPN
ejde-196	113	83	if	if	SCONJ
ejde-196	113	84	it	it	PRON
ejde-196	113	85	is	be	AUX
ejde-196	113	86	ldlc	ldlc	NOUN
ejde-196	113	87	at	at	ADP
ejde-196	113	88	v	v	NOUN
ejde-196	113	89	∈	∈	NOUN
ejde-196	113	90	sl⊥	sl⊥	PROPN
ejde-196	113	91	in	in	ADP
ejde-196	113	92	each	each	DET
ejde-196	113	93	w	w	PROPN
ejde-196	113	94	∈	∈	PROPN
ejde-196	113	95	l⊥.	l⊥.	NOUN
ejde-196	113	96	we	we	PRON
ejde-196	113	97	say	say	VERB
ejde-196	113	98	that	that	SCONJ
ejde-196	113	99	p	p	NOUN
ejde-196	113	100	is	be	AUX
ejde-196	113	101	ldlc	ldlc	NOUN
ejde-196	113	102	if	if	SCONJ
ejde-196	113	103	it	it	PRON
ejde-196	113	104	is	be	AUX
ejde-196	113	105	ldlc	ldlc	NOUN
ejde-196	113	106	at	at	ADP
ejde-196	113	107	each	each	DET
ejde-196	113	108	v	v	NOUN
ejde-196	113	109	∈	∈	PROPN
ejde-196	113	110	sl⊥	sl⊥	PROPN
ejde-196	113	111	.	.	PUNCT
ejde-196	114	1	note	note	VERB
ejde-196	114	2	that	that	SCONJ
ejde-196	114	3	when	when	SCONJ
ejde-196	114	4	v	v	X
ejde-196	114	5	∈	∈	NOUN
ejde-196	114	6	sl⊥	sl⊥	PROPN
ejde-196	114	7	,	,	PUNCT
ejde-196	114	8	w	w	PROPN
ejde-196	114	9	∈	∈	PROPN
ejde-196	114	10	l⊥	l⊥	NOUN
ejde-196	114	11	are	be	AUX
ejde-196	114	12	given	give	VERB
ejde-196	114	13	,	,	PUNCT
ejde-196	114	14	s	s	PART
ejde-196	114	15	>	>	X
ejde-196	114	16	0	0	NUM
ejde-196	114	17	is	be	AUX
ejde-196	114	18	small	small	ADJ
ejde-196	114	19	,	,	PUNCT
ejde-196	114	20	we	we	PRON
ejde-196	114	21	have	have	VERB
ejde-196	114	22	v+sw	v+sw	NOUN
ejde-196	114	23	∈	∈	PROPN
ejde-196	114	24	s+	s+	PUNCT
ejde-196	114	25	l⊥	l⊥	PROPN
ejde-196	114	26	;	;	PUNCT
ejde-196	114	27	for	for	ADP
ejde-196	114	28	v(s	v(s	NOUN
ejde-196	114	29	)	)	PUNCT
ejde-196	115	1	=	=	SYM
ejde-196	115	2	v+sw	v+sw	NOUN
ejde-196	115	3	‖v+sw‖	‖v+sw‖	PROPN
ejde-196	115	4	,	,	PUNCT
ejde-196	115	5	we	we	PRON
ejde-196	115	6	have	have	VERB
ejde-196	115	7	‖v(s)‖	‖v(s)‖	NOUN
ejde-196	115	8	=	=	SYM
ejde-196	115	9	1	1	NUM
ejde-196	115	10	and	and	CCONJ
ejde-196	115	11	[	[	X
ejde-196	115	12	v(s	v(s	PROPN
ejde-196	115	13	)	)	PUNCT
ejde-196	115	14	,	,	PUNCT
ejde-196	115	15	l	l	NOUN
ejde-196	115	16	]	]	X
ejde-196	115	17	=	=	PUNCT
ejde-196	116	1	[	[	X
ejde-196	116	2	v	v	X
ejde-196	116	3	+	+	CCONJ
ejde-196	116	4	sw	sw	PROPN
ejde-196	116	5	,	,	PUNCT
ejde-196	116	6	l	l	NOUN
ejde-196	116	7	]	]	PUNCT
ejde-196	116	8	and	and	CCONJ
ejde-196	116	9	then	then	ADV
ejde-196	116	10	p(v(s	p(v(s	PROPN
ejde-196	116	11	)	)	PUNCT
ejde-196	116	12	)	)	PUNCT
ejde-196	116	13	=	=	PUNCT
ejde-196	116	14	tsv(s	tsv(s	PROPN
ejde-196	116	15	)	)	PUNCT
ejde-196	117	1	+	+	PUNCT
ejde-196	117	2	ul(s	ul(	NOUN
ejde-196	117	3	)	)	PUNCT
ejde-196	117	4	=	=	SYM
ejde-196	118	1	ts	ts	ADP
ejde-196	118	2	‖v	‖v	PROPN
ejde-196	119	1	+	+	CCONJ
ejde-196	119	2	sw‖	sw‖	PROPN
ejde-196	119	3	(	(	PUNCT
ejde-196	119	4	v	v	NOUN
ejde-196	119	5	+	+	CCONJ
ejde-196	119	6	sw	sw	NOUN
ejde-196	119	7	)	)	PUNCT
ejde-196	119	8	+	+	SYM
ejde-196	119	9	ul(s	ul(s	X
ejde-196	119	10	)	)	PUNCT
ejde-196	119	11	=	=	SYM
ejde-196	120	1	p(v	p(v	NOUN
ejde-196	120	2	+	+	CCONJ
ejde-196	120	3	sw	sw	PROPN
ejde-196	120	4	)	)	PUNCT
ejde-196	120	5	,	,	PUNCT
ejde-196	120	6	ejde-2023	ejde-2023	ADJ
ejde-196	120	7	/	/	SYM
ejde-196	120	8	si/02	si/02	ADJ
ejde-196	120	9	short	short	ADJ
ejde-196	120	10	title	title	NOUN
ejde-196	120	11	179	179	NUM
ejde-196	120	12	with	with	ADP
ejde-196	120	13	ul(s	ul(	NOUN
ejde-196	120	14	)	)	PUNCT
ejde-196	120	15	∈	∈	PROPN
ejde-196	120	16	l.	l.	NOUN
ejde-196	120	17	it	it	PRON
ejde-196	120	18	implies	imply	VERB
ejde-196	120	19	that	that	SCONJ
ejde-196	120	20	if	if	SCONJ
ejde-196	120	21	p	p	NOUN
ejde-196	120	22	is	be	AUX
ejde-196	120	23	ldlc	ldlc	NOUN
ejde-196	120	24	at	at	ADP
ejde-196	120	25	v	v	NOUN
ejde-196	120	26	∈	∈	NOUN
ejde-196	120	27	sl⊥	sl⊥	PROPN
ejde-196	120	28	in	in	ADP
ejde-196	120	29	w	w	PROPN
ejde-196	120	30	∈	∈	PROPN
ejde-196	120	31	l⊥	l⊥	NOUN
ejde-196	120	32	,	,	PUNCT
ejde-196	120	33	there	there	PRON
ejde-196	120	34	is	be	VERB
ejde-196	120	35	`	`	PUNCT
ejde-196	120	36	0	0	PUNCT
ejde-196	120	37	>	>	X
ejde-196	120	38	0	0	NUM
ejde-196	120	39	such	such	ADJ
ejde-196	120	40	that	that	SCONJ
ejde-196	120	41	when	when	SCONJ
ejde-196	120	42	s	s	VERB
ejde-196	120	43	>	>	X
ejde-196	120	44	0	0	PUNCT
ejde-196	120	45	small	small	ADJ
ejde-196	120	46	,	,	PUNCT
ejde-196	120	47	we	we	PRON
ejde-196	120	48	have	have	VERB
ejde-196	120	49	‖p(v(s))−	‖p(v(s))−	NUM
ejde-196	121	1	p(v)‖	p(v)‖	NOUN
ejde-196	121	2	=	=	PUNCT
ejde-196	121	3	‖p(v	‖p(v	PUNCT
ejde-196	121	4	+	+	CCONJ
ejde-196	121	5	sw)−	sw)−	ADJ
ejde-196	121	6	p(v)‖	p(v)‖	ADJ
ejde-196	121	7	≤	≤	PROPN
ejde-196	121	8	`	`	PUNCT
ejde-196	121	9	0|s|‖w‖	0|s|‖w‖	NUM
ejde-196	121	10	=	=	SYM
ejde-196	121	11	o(s	o(s	PROPN
ejde-196	121	12	)	)	PUNCT
ejde-196	121	13	.	.	PUNCT
ejde-196	122	1	(	(	PUNCT
ejde-196	122	2	2.1	2.1	NUM
ejde-196	122	3	)	)	PUNCT
ejde-196	122	4	it	it	PRON
ejde-196	122	5	is	be	AUX
ejde-196	122	6	clear	clear	ADJ
ejde-196	122	7	that	that	SCONJ
ejde-196	122	8	if	if	SCONJ
ejde-196	122	9	the	the	DET
ejde-196	122	10	constant	constant	ADJ
ejde-196	122	11	`	`	PUNCT
ejde-196	122	12	0	0	NUM
ejde-196	122	13	is	be	AUX
ejde-196	122	14	independent	independent	ADJ
ejde-196	122	15	of	of	ADP
ejde-196	122	16	w	w	PROPN
ejde-196	122	17	,	,	PUNCT
ejde-196	122	18	p	p	PRON
ejde-196	122	19	becomes	becomes	AUX
ejde-196	122	20	locally	locally	ADV
ejde-196	122	21	lipschitz	lipschitz	VERB
ejde-196	122	22	continuous	continuous	ADJ
ejde-196	122	23	at	at	ADP
ejde-196	122	24	v	v	NOUN
ejde-196	122	25	in	in	ADP
ejde-196	122	26	all	all	DET
ejde-196	122	27	directions	direction	NOUN
ejde-196	122	28	in	in	ADP
ejde-196	122	29	l⊥.	l⊥.	NOUN
ejde-196	122	30	if	if	SCONJ
ejde-196	122	31	the	the	DET
ejde-196	122	32	gateaux	gateaux	ADV
ejde-196	122	33	derivative	derivative	ADJ
ejde-196	122	34	δp(v;w	δp(v;w	PROPN
ejde-196	122	35	)	)	PUNCT
ejde-196	122	36	of	of	ADP
ejde-196	122	37	p	p	NOUN
ejde-196	122	38	exists	exist	VERB
ejde-196	122	39	at	at	ADP
ejde-196	122	40	v	v	NOUN
ejde-196	122	41	in	in	ADP
ejde-196	122	42	w	w	PROPN
ejde-196	122	43	∈	∈	PROPN
ejde-196	122	44	l⊥	l⊥	PROPN
ejde-196	122	45	and	and	CCONJ
ejde-196	122	46	lim	lim	PROPN
ejde-196	122	47	s→0	s→0	PROPN
ejde-196	123	1	1	1	NUM
ejde-196	123	2	s	s	PART
ejde-196	123	3	(	(	PUNCT
ejde-196	123	4	p(v	p(v	NOUN
ejde-196	123	5	+	+	CCONJ
ejde-196	123	6	sw)−	sw)−	ADJ
ejde-196	123	7	p(v	p(v	NOUN
ejde-196	123	8	)	)	PUNCT
ejde-196	123	9	)	)	PUNCT
ejde-196	124	1	=	=	SYM
ejde-196	124	2	δp(v;w	δp(v;w	PROPN
ejde-196	124	3	)	)	PUNCT
ejde-196	124	4	,	,	PUNCT
ejde-196	124	5	where	where	SCONJ
ejde-196	124	6	δp(v;αw	δp(v;αw	NOUN
ejde-196	124	7	)	)	PUNCT
ejde-196	124	8	=	=	SYM
ejde-196	124	9	αδp(v;w	αδp(v;w	NOUN
ejde-196	124	10	)	)	PUNCT
ejde-196	124	11	for	for	ADP
ejde-196	124	12	any	any	DET
ejde-196	124	13	scalar	scalar	ADJ
ejde-196	124	14	α	α	NOUN
ejde-196	124	15	,	,	PUNCT
ejde-196	124	16	but	but	CCONJ
ejde-196	124	17	not	not	PART
ejde-196	124	18	necessarily	necessarily	ADV
ejde-196	124	19	linear	linear	VERB
ejde-196	124	20	in	in	ADP
ejde-196	124	21	w.	w.	PROPN
ejde-196	124	22	denote	denote	VERB
ejde-196	124	23	`	`	PUNCT
ejde-196	124	24	0	0	NUM
ejde-196	124	25	=	=	SYM
ejde-196	124	26	max{1	max{1	NOUN
ejde-196	124	27	,	,	PUNCT
ejde-196	124	28	2	2	NUM
ejde-196	124	29	‖w‖	‖w‖	PROPN
ejde-196	124	30	|δp(v;w)|	|δp(v;w)|	VERB
ejde-196	124	31	}	}	PUNCT
ejde-196	124	32	>	>	X
ejde-196	124	33	0	0	NUM
ejde-196	124	34	,	,	PUNCT
ejde-196	124	35	then	then	ADV
ejde-196	124	36	there	there	PRON
ejde-196	124	37	is	be	VERB
ejde-196	124	38	s0	s0	PROPN
ejde-196	124	39	>	>	PUNCT
ejde-196	124	40	0	0	PROPN
ejde-196	124	41	,	,	PUNCT
ejde-196	124	42	such	such	ADJ
ejde-196	124	43	that	that	SCONJ
ejde-196	124	44	when	when	SCONJ
ejde-196	124	45	s0	s0	PROPN
ejde-196	124	46	>	>	X
ejde-196	124	47	|s|	|s|	PROPN
ejde-196	124	48	>	>	SYM
ejde-196	124	49	0	0	NUM
ejde-196	124	50	,	,	PUNCT
ejde-196	124	51	it	it	PRON
ejde-196	124	52	holds	hold	VERB
ejde-196	124	53	‖p(v	‖p(v	PUNCT
ejde-196	124	54	+	+	CCONJ
ejde-196	124	55	sw)−	sw)−	ADJ
ejde-196	124	56	p(v)‖	p(v)‖	INTJ
ejde-196	124	57	<	<	X
ejde-196	124	58	`	`	PUNCT
ejde-196	124	59	0|s|‖w‖	0|s|‖w‖	NOUN
ejde-196	124	60	,	,	PUNCT
ejde-196	124	61	or	or	CCONJ
ejde-196	124	62	p	p	NOUN
ejde-196	124	63	is	be	AUX
ejde-196	124	64	ldlc	ldlc	NOUN
ejde-196	124	65	at	at	ADP
ejde-196	124	66	v	v	NOUN
ejde-196	124	67	in	in	ADP
ejde-196	124	68	w.	w.	NOUN
ejde-196	124	69	we	we	PRON
ejde-196	124	70	conclude	conclude	VERB
ejde-196	124	71	that	that	SCONJ
ejde-196	124	72	the	the	DET
ejde-196	124	73	gateaux	gateaux	ADV
ejde-196	124	74	-	-	PUNCT
ejde-196	124	75	differential	differential	NOUN
ejde-196	124	76	of	of	ADP
ejde-196	124	77	p	p	NOUN
ejde-196	124	78	at	at	ADP
ejde-196	124	79	v	v	NUM
ejde-196	124	80	in	in	ADP
ejde-196	124	81	w	w	PROPN
ejde-196	124	82	∈	∈	PROPN
ejde-196	124	83	l⊥	l⊥	NOUN
ejde-196	124	84	implies	imply	VERB
ejde-196	124	85	ldlc	ldlc	NOUN
ejde-196	124	86	at	at	ADP
ejde-196	124	87	v	v	NOUN
ejde-196	124	88	in	in	ADP
ejde-196	124	89	w	w	PROPN
ejde-196	124	90	,	,	PUNCT
ejde-196	124	91	then	then	ADV
ejde-196	124	92	the	the	DET
ejde-196	124	93	directional	directional	ADJ
ejde-196	124	94	continuity	continuity	NOUN
ejde-196	124	95	at	at	ADP
ejde-196	124	96	v	v	NOUN
ejde-196	124	97	in	in	ADP
ejde-196	124	98	w	w	PROPN
ejde-196	124	99	but	but	CCONJ
ejde-196	124	100	not	not	PART
ejde-196	124	101	the	the	DET
ejde-196	124	102	continuity	continuity	NOUN
ejde-196	124	103	at	at	ADP
ejde-196	124	104	v.	v.	ADP
ejde-196	124	105	the	the	DET
ejde-196	124	106	directional	directional	ADJ
ejde-196	124	107	continuity	continuity	NOUN
ejde-196	124	108	is	be	AUX
ejde-196	124	109	clearly	clearly	ADV
ejde-196	124	110	weaker	weak	ADJ
ejde-196	124	111	than	than	ADP
ejde-196	124	112	the	the	DET
ejde-196	124	113	weakcontinuity	weakcontinuity	NOUN
ejde-196	124	114	since	since	SCONJ
ejde-196	124	115	the	the	DET
ejde-196	124	116	latter	latter	ADJ
ejde-196	124	117	implies	imply	VERB
ejde-196	124	118	the	the	DET
ejde-196	124	119	continuity	continuity	NOUN
ejde-196	124	120	in	in	ADP
ejde-196	124	121	any	any	DET
ejde-196	124	122	finite	finite	ADJ
ejde-196	124	123	-	-	ADJ
ejde-196	124	124	dimensional	dimensional	ADJ
ejde-196	124	125	space	space	NOUN
ejde-196	124	126	.	.	PUNCT
ejde-196	125	1	lemma	lemma	PROPN
ejde-196	125	2	2.3	2.3	NUM
ejde-196	125	3	(	(	PUNCT
ejde-196	125	4	stepsize	stepsize	VERB
ejde-196	125	5	rule	rule	NOUN
ejde-196	125	6	)	)	PUNCT
ejde-196	125	7	.	.	PUNCT
ejde-196	126	1	let	let	VERB
ejde-196	126	2	p	p	PRON
ejde-196	126	3	be	be	AUX
ejde-196	126	4	a	a	DET
ejde-196	126	5	local	local	ADJ
ejde-196	126	6	l	l	ADJ
ejde-196	126	7	-	-	ADJ
ejde-196	126	8	orthogonal	orthogonal	ADJ
ejde-196	126	9	selection	selection	NOUN
ejde-196	126	10	of	of	ADP
ejde-196	126	11	j	j	PROPN
ejde-196	126	12	in	in	ADP
ejde-196	126	13	s+	s+	AUX
ejde-196	126	14	l⊥	l⊥	VERB
ejde-196	126	15	such	such	ADJ
ejde-196	126	16	that	that	SCONJ
ejde-196	126	17	p	p	NOUN
ejde-196	126	18	is	be	AUX
ejde-196	126	19	ldlc	ldlc	NOUN
ejde-196	126	20	at	at	ADP
ejde-196	126	21	v	v	NOUN
ejde-196	126	22	∈	∈	NOUN
ejde-196	126	23	sl⊥	sl⊥	PROPN
ejde-196	126	24	and	and	CCONJ
ejde-196	126	25	dis(p(v	dis(p(v	PROPN
ejde-196	126	26	)	)	PUNCT
ejde-196	126	27	,	,	PUNCT
ejde-196	126	28	l	l	NOUN
ejde-196	126	29	)	)	PUNCT
ejde-196	126	30	>	>	X
ejde-196	127	1	0	0	X
ejde-196	127	2	.	.	PUNCT
ejde-196	128	1	if	if	SCONJ
ejde-196	128	2	d	d	PROPN
ejde-196	128	3	=	=	PUNCT
ejde-196	128	4	−j	−j	NOUN
ejde-196	128	5	′(p(v	′(p(v	NOUN
ejde-196	128	6	)	)	PUNCT
ejde-196	128	7	)	)	PUNCT
ejde-196	129	1	6=	6=	ADP
ejde-196	129	2	0	0	NUM
ejde-196	129	3	,	,	PUNCT
ejde-196	129	4	set	set	VERB
ejde-196	129	5	w	w	NOUN
ejde-196	129	6	=	=	PUNCT
ejde-196	129	7	d	d	PROPN
ejde-196	129	8	/	/	SYM
ejde-196	129	9	c	c	NOUN
ejde-196	129	10	where	where	SCONJ
ejde-196	129	11	c	c	NOUN
ejde-196	129	12	=	=	SYM
ejde-196	129	13	max{1	max{1	PROPN
ejde-196	129	14	,	,	PUNCT
ejde-196	129	15	‖d‖	‖d‖	PROPN
ejde-196	129	16	}	}	PUNCT
ejde-196	129	17	and	and	CCONJ
ejde-196	129	18	v(s	v(s	NOUN
ejde-196	129	19	)	)	PUNCT
ejde-196	130	1	=	=	SYM
ejde-196	130	2	v+sw	v+sw	NOUN
ejde-196	130	3	‖v+sw‖	‖v+sw‖	PROPN
ejde-196	130	4	,	,	PUNCT
ejde-196	130	5	then	then	ADV
ejde-196	130	6	there	there	PRON
ejde-196	130	7	is	be	VERB
ejde-196	130	8	s0	s0	PROPN
ejde-196	130	9	>	>	X
ejde-196	130	10	0	0	NUM
ejde-196	130	11	such	such	ADJ
ejde-196	130	12	that	that	SCONJ
ejde-196	130	13	when	when	SCONJ
ejde-196	130	14	s0	s0	PROPN
ejde-196	130	15	>	>	X
ejde-196	130	16	s	s	PROPN
ejde-196	130	17	>	>	X
ejde-196	130	18	0	0	NUM
ejde-196	130	19	,	,	PUNCT
ejde-196	130	20	we	we	PRON
ejde-196	130	21	have	have	VERB
ejde-196	130	22	a	a	DET
ejde-196	130	23	stepsize	stepsize	NOUN
ejde-196	130	24	rule	rule	NOUN
ejde-196	130	25	j(p(v(s)))−	j(p(v(s)))−	PROPN
ejde-196	130	26	j(p(v	j(p(v	NOUN
ejde-196	130	27	)	)	PUNCT
ejde-196	130	28	)	)	PUNCT
ejde-196	130	29	<	<	X
ejde-196	130	30	−1	−1	NOUN
ejde-196	130	31	4	4	NUM
ejde-196	130	32	tvs‖d‖2	tvs‖d‖2	NUM
ejde-196	130	33	/	/	SYM
ejde-196	130	34	c.	c.	NOUN
ejde-196	130	35	(	(	PUNCT
ejde-196	130	36	2.2	2.2	NUM
ejde-196	130	37	)	)	PUNCT
ejde-196	130	38	furthermore	furthermore	ADV
ejde-196	130	39	,	,	PUNCT
ejde-196	130	40	if	if	SCONJ
ejde-196	130	41	p(vk	p(vk	PROPN
ejde-196	130	42	)	)	PUNCT
ejde-196	130	43	→	→	SYM
ejde-196	130	44	p(v	p(v	NOUN
ejde-196	130	45	)	)	PUNCT
ejde-196	130	46	as	as	ADP
ejde-196	130	47	vk	vk	PROPN
ejde-196	130	48	→	→	SYM
ejde-196	130	49	v	v	NOUN
ejde-196	130	50	,	,	PUNCT
ejde-196	130	51	then	then	ADV
ejde-196	130	52	there	there	PRON
ejde-196	130	53	exists	exist	VERB
ejde-196	130	54	n	n	PROPN
ejde-196	130	55	>	>	X
ejde-196	130	56	0	0	NUM
ejde-196	130	57	such	such	ADJ
ejde-196	130	58	that	that	SCONJ
ejde-196	130	59	when	when	SCONJ
ejde-196	130	60	k	k	PROPN
ejde-196	130	61	>	>	X
ejde-196	130	62	n	n	PROPN
ejde-196	130	63	,	,	PUNCT
ejde-196	130	64	we	we	PRON
ejde-196	130	65	have	have	VERB
ejde-196	130	66	a	a	DET
ejde-196	130	67	uniform	uniform	ADJ
ejde-196	130	68	stepsize	stepsize	NOUN
ejde-196	130	69	rule	rule	NOUN
ejde-196	130	70	j(p(vk))−	j(p(vk))−	X
ejde-196	130	71	j(p(v	j(p(v	NOUN
ejde-196	130	72	)	)	PUNCT
ejde-196	130	73	)	)	PUNCT
ejde-196	131	1	<	<	X
ejde-196	131	2	−tv	−tv	NOUN
ejde-196	131	3	4	4	NUM
ejde-196	131	4	‖j	‖j	ADP
ejde-196	131	5	′(p(v))‖2	′(p(v))‖2	ADV
ejde-196	131	6	/	/	SYM
ejde-196	131	7	c.	c.	NOUN
ejde-196	131	8	(	(	PUNCT
ejde-196	131	9	2.3	2.3	NUM
ejde-196	131	10	)	)	PUNCT
ejde-196	131	11	proof	proof	NOUN
ejde-196	131	12	.	.	PUNCT
ejde-196	132	1	denote	denote	PROPN
ejde-196	132	2	p(v	p(v	NOUN
ejde-196	132	3	)	)	PUNCT
ejde-196	132	4	=	=	SYM
ejde-196	132	5	tvv+uv	tvv+uv	PROPN
ejde-196	132	6	,	,	PUNCT
ejde-196	132	7	p(v(s	p(v(s	NOUN
ejde-196	132	8	)	)	PUNCT
ejde-196	132	9	)	)	PUNCT
ejde-196	133	1	=	=	SYM
ejde-196	133	2	tsv(s)+ul(s	tsv(s)+ul(s	X
ejde-196	133	3	)	)	PUNCT
ejde-196	133	4	for	for	ADP
ejde-196	133	5	tv	tv	NOUN
ejde-196	133	6	,	,	PUNCT
ejde-196	133	7	ts	ts	ADP
ejde-196	133	8	>	>	X
ejde-196	133	9	0	0	PUNCT
ejde-196	134	1	and	and	CCONJ
ejde-196	134	2	uv	uv	NOUN
ejde-196	134	3	,	,	PUNCT
ejde-196	134	4	ul(s	ul(s	ADJ
ejde-196	134	5	)	)	PUNCT
ejde-196	134	6	∈	∈	PROPN
ejde-196	134	7	l.	l.	NOUN
ejde-196	134	8	then	then	ADV
ejde-196	134	9	there	there	PRON
ejde-196	134	10	is	be	VERB
ejde-196	134	11	s0	s0	PROPN
ejde-196	134	12	>	>	X
ejde-196	134	13	0	0	NUM
ejde-196	134	14	such	such	ADJ
ejde-196	134	15	that	that	SCONJ
ejde-196	134	16	when	when	SCONJ
ejde-196	134	17	s0	s0	PROPN
ejde-196	134	18	>	>	X
ejde-196	134	19	s	s	PROPN
ejde-196	134	20	>	>	X
ejde-196	134	21	0	0	NUM
ejde-196	134	22	,	,	PUNCT
ejde-196	134	23	we	we	PRON
ejde-196	134	24	have	have	VERB
ejde-196	134	25	j(p(v(s)))−	j(p(v(s)))−	PROPN
ejde-196	134	26	j(p(v	j(p(v	NOUN
ejde-196	134	27	)	)	PUNCT
ejde-196	134	28	)	)	PUNCT
ejde-196	135	1	=	=	PUNCT
ejde-196	135	2	〈	〈	PROPN
ejde-196	135	3	j	j	PROPN
ejde-196	135	4	′(p(v	′(p(v	NOUN
ejde-196	135	5	)	)	PUNCT
ejde-196	135	6	)	)	PUNCT
ejde-196	135	7	,	,	PUNCT
ejde-196	135	8	p(v(s))−	p(v(s))−	ADP
ejde-196	135	9	p(v)〉+	p(v)〉+	PRON
ejde-196	135	10	o(‖p(v(s))−	o(‖p(v(s))−	NUM
ejde-196	135	11	p(v)‖	p(v)‖	NUM
ejde-196	135	12	)	)	PUNCT
ejde-196	135	13	=	=	SYM
ejde-196	135	14	tss	tss	PROPN
ejde-196	135	15	‖v	‖v	NOUN
ejde-196	136	1	+	+	CCONJ
ejde-196	136	2	sw‖	sw‖	PROPN
ejde-196	136	3	〈	〈	PROPN
ejde-196	136	4	j	j	PROPN
ejde-196	136	5	′(p(v	′(p(v	NOUN
ejde-196	136	6	)	)	PUNCT
ejde-196	136	7	)	)	PUNCT
ejde-196	136	8	,	,	PUNCT
ejde-196	136	9	w〉+	w〉+	PROPN
ejde-196	136	10	o(‖p(v	o(‖p(v	NOUN
ejde-196	136	11	+	+	CCONJ
ejde-196	136	12	sw)−	sw)−	ADJ
ejde-196	136	13	p(v)‖	p(v)‖	NUM
ejde-196	136	14	)	)	PUNCT
ejde-196	136	15	=	=	SYM
ejde-196	136	16	−	−	PROPN
ejde-196	136	17	tss	tss	NOUN
ejde-196	136	18	‖v	‖v	NOUN
ejde-196	137	1	+	+	CCONJ
ejde-196	137	2	sw‖	sw‖	PROPN
ejde-196	137	3	‖d‖2	‖d‖2	PROPN
ejde-196	137	4	/	/	SYM
ejde-196	137	5	c	c	NOUN
ejde-196	137	6	+	+	PUNCT
ejde-196	137	7	o(s	o(s	PROPN
ejde-196	137	8	)	)	PUNCT
ejde-196	137	9	<	<	X
ejde-196	137	10	−1	−1	NOUN
ejde-196	137	11	4	4	NUM
ejde-196	137	12	tvs‖d‖2	tvs‖d‖2	NUM
ejde-196	137	13	/	/	SYM
ejde-196	137	14	c	c	NOUN
ejde-196	137	15	,	,	PUNCT
ejde-196	137	16	(	(	PUNCT
ejde-196	137	17	2.4	2.4	NUM
ejde-196	137	18	)	)	PUNCT
ejde-196	137	19	where	where	SCONJ
ejde-196	137	20	we	we	PRON
ejde-196	137	21	have	have	AUX
ejde-196	137	22	used	use	VERB
ejde-196	137	23	the	the	DET
ejde-196	137	24	properties	property	NOUN
ejde-196	137	25	j	j	PROPN
ejde-196	137	26	′(p(v))⊥[v	′(p(v))⊥[v	PROPN
ejde-196	137	27	,	,	PUNCT
ejde-196	137	28	l	l	NOUN
ejde-196	137	29	]	]	PUNCT
ejde-196	137	30	and	and	CCONJ
ejde-196	137	31	‖p(v(s	‖p(v(s	PROPN
ejde-196	137	32	)	)	PUNCT
ejde-196	137	33	)	)	PUNCT
ejde-196	138	1	−	−	PROPN
ejde-196	138	2	p(v)‖	p(v)‖	PROPN
ejde-196	138	3	=	=	SYM
ejde-196	138	4	o(s	o(s	PROPN
ejde-196	138	5	)	)	PUNCT
ejde-196	138	6	in	in	ADP
ejde-196	138	7	(	(	PUNCT
ejde-196	138	8	2.1	2.1	NUM
ejde-196	138	9	)	)	PUNCT
ejde-196	138	10	.	.	PUNCT
ejde-196	139	1	the	the	DET
ejde-196	139	2	uniform	uniform	ADJ
ejde-196	139	3	stepsize	stepsize	NOUN
ejde-196	139	4	rule	rule	NOUN
ejde-196	139	5	follows	follow	VERB
ejde-196	139	6	from	from	ADP
ejde-196	139	7	the	the	DET
ejde-196	139	8	fact	fact	NOUN
ejde-196	139	9	that	that	SCONJ
ejde-196	139	10	j	j	PROPN
ejde-196	139	11	is	be	AUX
ejde-196	139	12	c1	c1	NOUN
ejde-196	139	13	,	,	PUNCT
ejde-196	139	14	p(vk	p(vk	PROPN
ejde-196	139	15	)	)	PUNCT
ejde-196	139	16	→	→	SYM
ejde-196	139	17	p(v	p(v	NOUN
ejde-196	139	18	)	)	PUNCT
ejde-196	139	19	leads	lead	VERB
ejde-196	139	20	to	to	ADP
ejde-196	139	21	j	j	PROPN
ejde-196	140	1	′(p(vk))→	′(p(vk))→	PROPN
ejde-196	140	2	j	j	PROPN
ejde-196	140	3	′(p(v	′(p(v	NOUN
ejde-196	140	4	)	)	PUNCT
ejde-196	140	5	)	)	PUNCT
ejde-196	141	1	as	as	SCONJ
ejde-196	141	2	vk	vk	PROPN
ejde-196	141	3	→	→	SYM
ejde-196	141	4	v.	v.	PROPN
ejde-196	141	5	�	�	PROPN
ejde-196	141	6	theorem	theorem	VERB
ejde-196	141	7	2.4	2.4	NUM
ejde-196	141	8	(	(	PUNCT
ejde-196	141	9	local	local	ADJ
ejde-196	141	10	min	min	ADJ
ejde-196	141	11	-	-	ADJ
ejde-196	141	12	orthogonal	orthogonal	ADJ
ejde-196	141	13	characterization	characterization	NOUN
ejde-196	141	14	)	)	PUNCT
ejde-196	141	15	.	.	PUNCT
ejde-196	142	1	let	let	VERB
ejde-196	142	2	p	p	PRON
ejde-196	142	3	be	be	AUX
ejde-196	142	4	an	an	DET
ejde-196	142	5	l	l	ADJ
ejde-196	142	6	-	-	ADJ
ejde-196	142	7	orthogonal	orthogonal	ADJ
ejde-196	142	8	selection	selection	NOUN
ejde-196	142	9	of	of	ADP
ejde-196	142	10	j	j	PROPN
ejde-196	142	11	in	in	ADP
ejde-196	142	12	s+	s+	AUX
ejde-196	142	13	l⊥	l⊥	VERB
ejde-196	142	14	such	such	ADJ
ejde-196	142	15	that	that	SCONJ
ejde-196	142	16	(	(	PUNCT
ejde-196	142	17	1	1	X
ejde-196	142	18	)	)	PUNCT
ejde-196	142	19	p	p	NOUN
ejde-196	142	20	is	be	AUX
ejde-196	142	21	ldlc	ldlc	NOUN
ejde-196	142	22	at	at	ADP
ejde-196	142	23	v	v	NOUN
ejde-196	142	24	∈	∈	NOUN
ejde-196	142	25	sl⊥	sl⊥	PROPN
ejde-196	142	26	and	and	CCONJ
ejde-196	142	27	dis(p(v	dis(p(v	PROPN
ejde-196	142	28	)	)	PUNCT
ejde-196	142	29	,	,	PUNCT
ejde-196	142	30	l	l	NOUN
ejde-196	142	31	)	)	PUNCT
ejde-196	142	32	>	>	X
ejde-196	142	33	0	0	NUM
ejde-196	142	34	,	,	PUNCT
ejde-196	142	35	(	(	PUNCT
ejde-196	142	36	2	2	X
ejde-196	142	37	)	)	PUNCT
ejde-196	142	38	v	v	NOUN
ejde-196	142	39	=	=	NOUN
ejde-196	142	40	arg	arg	NOUN
ejde-196	142	41	local	local	ADJ
ejde-196	142	42	minu∈s	minu∈s	PROPN
ejde-196	142	43	l⊥	l⊥	NOUN
ejde-196	142	44	j(p(u	j(p(u	NOUN
ejde-196	142	45	)	)	PUNCT
ejde-196	142	46	)	)	PUNCT
ejde-196	142	47	.	.	PUNCT
ejde-196	143	1	then	then	ADV
ejde-196	143	2	p(v	p(v	NOUN
ejde-196	143	3	)	)	PUNCT
ejde-196	143	4	is	be	AUX
ejde-196	143	5	a	a	DET
ejde-196	143	6	critical	critical	ADJ
ejde-196	143	7	point	point	NOUN
ejde-196	143	8	of	of	ADP
ejde-196	143	9	j	j	PROPN
ejde-196	143	10	.	.	PUNCT
ejde-196	144	1	180	180	NUM
ejde-196	144	2	m.	m.	PROPN
ejde-196	144	3	li	li	PROPN
ejde-196	144	4	,	,	PUNCT
ejde-196	144	5	b.	b.	PROPN
ejde-196	144	6	ji	ji	PROPN
ejde-196	144	7	,	,	PUNCT
ejde-196	144	8	j.	j.	PROPN
ejde-196	144	9	zhou	zhou	PROPN
ejde-196	144	10	ejde	ejde	PROPN
ejde-196	144	11	/	/	SYM
ejde-196	144	12	si/02	si/02	ADJ
ejde-196	144	13	proof	proof	NOUN
ejde-196	144	14	.	.	PUNCT
ejde-196	145	1	if	if	SCONJ
ejde-196	145	2	d	d	PROPN
ejde-196	145	3	=	=	PUNCT
ejde-196	145	4	−j	−j	NOUN
ejde-196	145	5	′(p(v	′(p(v	NOUN
ejde-196	145	6	)	)	PUNCT
ejde-196	145	7	)	)	PUNCT
ejde-196	146	1	6=	6=	ADP
ejde-196	146	2	0	0	NUM
ejde-196	146	3	,	,	PUNCT
ejde-196	146	4	set	set	VERB
ejde-196	146	5	w	w	NOUN
ejde-196	146	6	=	=	PUNCT
ejde-196	146	7	d	d	PROPN
ejde-196	146	8	/	/	SYM
ejde-196	146	9	c	c	NOUN
ejde-196	146	10	where	where	SCONJ
ejde-196	146	11	c	c	NOUN
ejde-196	146	12	=	=	SYM
ejde-196	146	13	max{1	max{1	PROPN
ejde-196	146	14	,	,	PUNCT
ejde-196	146	15	‖d‖	‖d‖	PROPN
ejde-196	146	16	}	}	PUNCT
ejde-196	146	17	and	and	CCONJ
ejde-196	146	18	v(s	v(s	NOUN
ejde-196	146	19	)	)	PUNCT
ejde-196	147	1	=	=	SYM
ejde-196	147	2	v+sw	v+sw	NOUN
ejde-196	147	3	‖v+sw‖	‖v+sw‖	PROPN
ejde-196	147	4	∈	∈	PROPN
ejde-196	147	5	sl⊥	sl⊥	PROPN
ejde-196	147	6	,	,	PUNCT
ejde-196	147	7	then	then	ADV
ejde-196	147	8	by	by	ADP
ejde-196	147	9	the	the	DET
ejde-196	147	10	step	step	NOUN
ejde-196	147	11	size	size	NOUN
ejde-196	147	12	rule	rule	NOUN
ejde-196	147	13	(	(	PUNCT
ejde-196	147	14	2.2	2.2	NUM
ejde-196	147	15	)	)	PUNCT
ejde-196	147	16	,	,	PUNCT
ejde-196	147	17	as	as	SCONJ
ejde-196	147	18	s	s	PROPN
ejde-196	147	19	>	>	X
ejde-196	147	20	0	0	PUNCT
ejde-196	147	21	sufficiently	sufficiently	ADV
ejde-196	147	22	small	small	ADJ
ejde-196	147	23	,	,	PUNCT
ejde-196	147	24	we	we	PRON
ejde-196	147	25	have	have	VERB
ejde-196	147	26	j(p(v(s)))−	j(p(v(s)))−	PROPN
ejde-196	147	27	j(p(v	j(p(v	NOUN
ejde-196	147	28	)	)	PUNCT
ejde-196	147	29	)	)	PUNCT
ejde-196	148	1	<	<	X
ejde-196	148	2	−1	−1	NOUN
ejde-196	148	3	4	4	NUM
ejde-196	148	4	tvs‖d‖2	tvs‖d‖2	NUM
ejde-196	148	5	/	/	SYM
ejde-196	148	6	c.	c.	NOUN
ejde-196	148	7	it	it	PRON
ejde-196	148	8	violates	violate	VERB
ejde-196	148	9	assumption	assumption	NOUN
ejde-196	148	10	(	(	PUNCT
ejde-196	148	11	2	2	NUM
ejde-196	148	12	)	)	PUNCT
ejde-196	148	13	.	.	PUNCT
ejde-196	149	1	�	�	PROPN
ejde-196	149	2	3	3	NUM
ejde-196	149	3	.	.	PUNCT
ejde-196	150	1	a	a	DET
ejde-196	150	2	numerical	numerical	ADJ
ejde-196	150	3	local	local	ADJ
ejde-196	150	4	min	min	ADJ
ejde-196	150	5	-	-	ADJ
ejde-196	150	6	orthogonal	orthogonal	ADJ
ejde-196	150	7	algorithm	algorithm	NOUN
ejde-196	150	8	the	the	DET
ejde-196	150	9	local	local	ADJ
ejde-196	150	10	min	min	ADJ
ejde-196	150	11	-	-	ADJ
ejde-196	150	12	orthogonal	orthogonal	ADJ
ejde-196	150	13	characterization	characterization	NOUN
ejde-196	150	14	of	of	ADP
ejde-196	150	15	a	a	DET
ejde-196	150	16	critical	critical	ADJ
ejde-196	150	17	point	point	NOUN
ejde-196	150	18	,	,	PUNCT
ejde-196	150	19	theorem	theorem	VERB
ejde-196	150	20	2.4	2.4	NUM
ejde-196	150	21	suggests	suggest	VERB
ejde-196	150	22	a	a	DET
ejde-196	150	23	local	local	ADJ
ejde-196	150	24	min	min	ADJ
ejde-196	150	25	-	-	ADJ
ejde-196	150	26	orthogonal	orthogonal	ADJ
ejde-196	150	27	method	method	NOUN
ejde-196	150	28	with	with	ADP
ejde-196	150	29	the	the	DET
ejde-196	150	30	stepsize	stepsize	NOUN
ejde-196	150	31	rule	rule	NOUN
ejde-196	150	32	(	(	PUNCT
ejde-196	150	33	2.2	2.2	NUM
ejde-196	150	34	)	)	PUNCT
ejde-196	150	35	if	if	SCONJ
ejde-196	150	36	the	the	DET
ejde-196	150	37	min	min	NOUN
ejde-196	150	38	is	be	AUX
ejde-196	150	39	approximated	approximate	VERB
ejde-196	150	40	by	by	ADP
ejde-196	150	41	using	use	VERB
ejde-196	150	42	a	a	DET
ejde-196	150	43	steepest	steep	ADJ
ejde-196	150	44	decent	decent	ADJ
ejde-196	150	45	method	method	NOUN
ejde-196	150	46	.	.	PUNCT
ejde-196	151	1	we	we	PRON
ejde-196	151	2	present	present	VERB
ejde-196	151	3	the	the	DET
ejde-196	151	4	steps	step	NOUN
ejde-196	151	5	below	below	ADV
ejde-196	151	6	:	:	PUNCT
ejde-196	151	7	step	step	NOUN
ejde-196	151	8	1	1	NUM
ejde-196	151	9	:	:	PUNCT
ejde-196	151	10	given	give	VERB
ejde-196	151	11	ε	ε	PROPN
ejde-196	151	12	>	>	X
ejde-196	151	13	0	0	PROPN
ejde-196	151	14	,	,	PUNCT
ejde-196	151	15	λ	λ	X
ejde-196	151	16	>	>	X
ejde-196	151	17	0	0	PUNCT
ejde-196	151	18	and	and	CCONJ
ejde-196	151	19	n	n	PRON
ejde-196	151	20	previously	previously	ADV
ejde-196	151	21	found	find	VERB
ejde-196	151	22	critical	critical	ADJ
ejde-196	151	23	points	point	NOUN
ejde-196	151	24	w1	w1	NOUN
ejde-196	151	25	,	,	PUNCT
ejde-196	151	26	w2	w2	NOUN
ejde-196	151	27	,	,	PUNCT
ejde-196	151	28	.	.	PUNCT
ejde-196	151	29	.	.	PUNCT
ejde-196	152	1	.	.	PUNCT
ejde-196	153	1	,	,	PUNCT
ejde-196	153	2	wn	wn	PROPN
ejde-196	153	3	of	of	ADP
ejde-196	153	4	j	j	PROPN
ejde-196	153	5	,	,	PUNCT
ejde-196	153	6	of	of	ADP
ejde-196	153	7	which	which	PRON
ejde-196	153	8	wn	wn	PROPN
ejde-196	153	9	has	have	VERB
ejde-196	153	10	the	the	DET
ejde-196	153	11	highest	high	ADJ
ejde-196	153	12	critical	critical	ADJ
ejde-196	153	13	value	value	NOUN
ejde-196	153	14	.	.	PUNCT
ejde-196	154	1	set	set	VERB
ejde-196	154	2	l	l	NOUN
ejde-196	154	3	=	=	SYM
ejde-196	154	4	span{w1	span{w1	PROPN
ejde-196	154	5	,	,	PUNCT
ejde-196	154	6	w2	w2	NOUN
ejde-196	154	7	,	,	PUNCT
ejde-196	154	8	.	.	PUNCT
ejde-196	154	9	.	.	PUNCT
ejde-196	155	1	.	.	PUNCT
ejde-196	156	1	,	,	PUNCT
ejde-196	156	2	wn	wn	PROPN
ejde-196	156	3	}	}	PUNCT
ejde-196	156	4	.	.	PUNCT
ejde-196	157	1	let	let	VERB
ejde-196	157	2	v1	v1	NOUN
ejde-196	157	3	∈	∈	PROPN
ejde-196	157	4	sl⊥	sl⊥	PROPN
ejde-196	157	5	be	be	AUX
ejde-196	157	6	an	an	DET
ejde-196	157	7	ascent	ascent	ADJ
ejde-196	157	8	direction	direction	NOUN
ejde-196	157	9	at	at	ADP
ejde-196	157	10	wn	wn	PROPN
ejde-196	157	11	.	.	PUNCT
ejde-196	158	1	let	let	VERB
ejde-196	158	2	t00	t00	NOUN
ejde-196	158	3	=	=	SYM
ejde-196	158	4	1	1	NUM
ejde-196	158	5	,	,	PUNCT
ejde-196	159	1	v0	v0	NOUN
ejde-196	159	2	l	l	NOUN
ejde-196	159	3	=	=	SYM
ejde-196	159	4	wn	wn	PROPN
ejde-196	159	5	and	and	CCONJ
ejde-196	159	6	set	set	VERB
ejde-196	159	7	k	k	PROPN
ejde-196	159	8	=	=	PUNCT
ejde-196	159	9	0	0	X
ejde-196	159	10	.	.	PUNCT
ejde-196	160	1	step	step	NOUN
ejde-196	160	2	2	2	NUM
ejde-196	160	3	:	:	PUNCT
ejde-196	160	4	using	use	VERB
ejde-196	160	5	the	the	DET
ejde-196	160	6	initial	initial	ADJ
ejde-196	160	7	guess	guess	NOUN
ejde-196	160	8	u	u	NOUN
ejde-196	160	9	=	=	PROPN
ejde-196	160	10	tk0v	tk0v	PROPN
ejde-196	160	11	k	k	PROPN
ejde-196	161	1	+	+	CCONJ
ejde-196	161	2	vkl	vkl	NOUN
ejde-196	161	3	,	,	PUNCT
ejde-196	161	4	solve	solve	VERB
ejde-196	161	5	tk0	tk0	NOUN
ejde-196	161	6	,	,	PUNCT
ejde-196	161	7	t	t	PROPN
ejde-196	161	8	k	k	PROPN
ejde-196	161	9	1	1	NUM
ejde-196	161	10	,	,	PUNCT
ejde-196	161	11	.	.	PUNCT
ejde-196	161	12	.	.	PUNCT
ejde-196	162	1	.	.	PUNCT
ejde-196	163	1	,	,	PUNCT
ejde-196	163	2	t	t	PROPN
ejde-196	163	3	k	k	PROPN
ejde-196	163	4	n	n	PROPN
ejde-196	163	5	from	from	ADP
ejde-196	163	6	〈	〈	PROPN
ejde-196	163	7	j	j	PROPN
ejde-196	163	8	′(t0vk0	′(t0vk0	NOUN
ejde-196	163	9	+	+	CCONJ
ejde-196	164	1	t1w1	t1w1	X
ejde-196	164	2	+	+	X
ejde-196	164	3	·	·	PUNCT
ejde-196	164	4	·	·	PUNCT
ejde-196	164	5	·	·	PUNCT
ejde-196	164	6	+	+	CCONJ
ejde-196	164	7	tnwn	tnwn	NOUN
ejde-196	164	8	)	)	PUNCT
ejde-196	164	9	,	,	PUNCT
ejde-196	164	10	vk	vk	VERB
ejde-196	164	11	〉	〉	NOUN
ejde-196	164	12	=	=	SYM
ejde-196	164	13	0	0	NUM
ejde-196	164	14	,	,	PUNCT
ejde-196	164	15	〈	〈	PROPN
ejde-196	164	16	j	j	PROPN
ejde-196	164	17	′(t0vk0	′(t0vk0	VERB
ejde-196	164	18	+	+	CCONJ
ejde-196	164	19	t1w1	t1w1	X
ejde-196	164	20	+	+	X
ejde-196	164	21	·	·	PUNCT
ejde-196	164	22	·	·	PUNCT
ejde-196	164	23	·	·	PUNCT
ejde-196	164	24	+	+	NUM
ejde-196	164	25	tvwn	tvwn	NOUN
ejde-196	164	26	)	)	PUNCT
ejde-196	164	27	,	,	PUNCT
ejde-196	164	28	wj	wj	PROPN
ejde-196	164	29	〉	〉	PROPN
ejde-196	164	30	=	=	SYM
ejde-196	164	31	0	0	NUM
ejde-196	164	32	,	,	PUNCT
ejde-196	164	33	for	for	ADP
ejde-196	164	34	j	j	PROPN
ejde-196	164	35	=	=	SYM
ejde-196	164	36	1	1	PROPN
ejde-196	164	37	,	,	PUNCT
ejde-196	164	38	.	.	PUNCT
ejde-196	164	39	.	.	PUNCT
ejde-196	165	1	.	.	PUNCT
ejde-196	166	1	,	,	PUNCT
ejde-196	166	2	n.	n.	PROPN
ejde-196	166	3	denote	denote	VERB
ejde-196	166	4	uk	uk	PROPN
ejde-196	166	5	≡	≡	PROPN
ejde-196	166	6	p(vk	p(vk	PROPN
ejde-196	166	7	)	)	PUNCT
ejde-196	166	8	=	=	SYM
ejde-196	167	1	tk0v	tk0v	PROPN
ejde-196	167	2	k	k	NOUN
ejde-196	167	3	0	0	PUNCT
ejde-196	168	1	+	+	CCONJ
ejde-196	168	2	vkl	vkl	NOUN
ejde-196	168	3	=	=	SYM
ejde-196	169	1	tk0v	tk0v	PROPN
ejde-196	169	2	k	k	NOUN
ejde-196	169	3	0	0	PUNCT
ejde-196	170	1	+	+	CCONJ
ejde-196	171	1	vk1w1	vk1w1	AUX
ejde-196	171	2	+	+	NUM
ejde-196	171	3	·	·	PUNCT
ejde-196	171	4	·	·	PUNCT
ejde-196	171	5	·	·	PUNCT
ejde-196	171	6	tknwn	tknwn	ADJ
ejde-196	171	7	.	.	PUNCT
ejde-196	172	1	step	step	NOUN
ejde-196	172	2	3	3	NUM
ejde-196	172	3	:	:	PUNCT
ejde-196	172	4	compute	compute	VERB
ejde-196	172	5	the	the	DET
ejde-196	172	6	steepest	steep	ADJ
ejde-196	172	7	descent	descent	NOUN
ejde-196	172	8	vector	vector	NOUN
ejde-196	172	9	dk	dk	PROPN
ejde-196	172	10	=	=	PROPN
ejde-196	172	11	−j	−j	NOUN
ejde-196	172	12	′(uk	′(uk	PROPN
ejde-196	172	13	)	)	PUNCT
ejde-196	172	14	.	.	PUNCT
ejde-196	173	1	step	step	NOUN
ejde-196	173	2	4	4	NUM
ejde-196	173	3	:	:	PUNCT
ejde-196	173	4	if	if	SCONJ
ejde-196	173	5	‖dk‖	‖dk‖	VERB
ejde-196	173	6	≤	≤	NUM
ejde-196	173	7	ε	ε	PROPN
ejde-196	173	8	then	then	ADV
ejde-196	173	9	output	output	PROPN
ejde-196	173	10	wn+1	wn+1	PROPN
ejde-196	173	11	=	=	SYM
ejde-196	173	12	uk	uk	PROPN
ejde-196	173	13	,	,	PUNCT
ejde-196	173	14	stop	stop	NOUN
ejde-196	173	15	;	;	PUNCT
ejde-196	173	16	else	else	ADV
ejde-196	173	17	goto	goto	ADJ
ejde-196	173	18	step	step	NOUN
ejde-196	173	19	5	5	NUM
ejde-196	173	20	.	.	PUNCT
ejde-196	173	21	step	step	NOUN
ejde-196	173	22	5	5	NUM
ejde-196	173	23	:	:	PUNCT
ejde-196	173	24	set	set	VERB
ejde-196	173	25	vk(s	vk(s	PRON
ejde-196	173	26	)	)	PUNCT
ejde-196	174	1	=	=	SYM
ejde-196	174	2	vk+sdk	vk+sdk	NOUN
ejde-196	174	3	‖vk+sdk‖	‖vk+sdk‖	NUM
ejde-196	174	4	and	and	CCONJ
ejde-196	174	5	find	find	VERB
ejde-196	174	6	sk	sk	NOUN
ejde-196	174	7	=	=	PUNCT
ejde-196	174	8	max	max	PROPN
ejde-196	174	9	{	{	PUNCT
ejde-196	174	10	λ	λ	PROPN
ejde-196	174	11	2	2	NUM
ejde-196	174	12	m	m	VERB
ejde-196	174	13	:	:	PUNCT
ejde-196	174	14	m	m	VERB
ejde-196	174	15	∈	∈	PROPN
ejde-196	174	16	n	n	CCONJ
ejde-196	174	17	,	,	PUNCT
ejde-196	174	18	2	2	NUM
ejde-196	174	19	m	m	NOUN
ejde-196	174	20	>	>	X
ejde-196	174	21	‖dk‖	‖dk‖	PROPN
ejde-196	174	22	,	,	PUNCT
ejde-196	174	23	j(p(vk	j(p(vk	PROPN
ejde-196	174	24	(	(	PUNCT
ejde-196	174	25	λ	λ	PROPN
ejde-196	174	26	2	2	NUM
ejde-196	174	27	m	m	NOUN
ejde-196	174	28	)	)	PUNCT
ejde-196	174	29	)	)	PUNCT
ejde-196	174	30	)	)	PUNCT
ejde-196	175	1	−	−	ADP
ejde-196	175	2	j(wk	j(wk	NUM
ejde-196	175	3	)	)	PUNCT
ejde-196	175	4	≤	≤	NUM
ejde-196	175	5	−	−	PROPN
ejde-196	175	6	t	t	PROPN
ejde-196	175	7	k	k	NOUN
ejde-196	175	8	0	0	NUM
ejde-196	175	9	2	2	NUM
ejde-196	175	10	‖dk‖	‖dk‖	VERB
ejde-196	175	11	‖vk	‖vk	NUM
ejde-196	175	12	(	(	PUNCT
ejde-196	175	13	λ	λ	PROPN
ejde-196	175	14	2	2	NUM
ejde-196	175	15	m	m	NOUN
ejde-196	175	16	)	)	PUNCT
ejde-196	175	17	−	−	PROPN
ejde-196	176	1	vk‖	vk‖	PROPN
ejde-196	176	2	}	}	PUNCT
ejde-196	176	3	.	.	PUNCT
ejde-196	177	1	initial	initial	ADJ
ejde-196	177	2	guess	guess	NOUN
ejde-196	177	3	u	u	NOUN
ejde-196	177	4	=	=	PROPN
ejde-196	177	5	tk0v	tk0v	PROPN
ejde-196	177	6	k	k	PROPN
ejde-196	177	7	(	(	PUNCT
ejde-196	177	8	λ	λ	PROPN
ejde-196	177	9	2	2	NUM
ejde-196	177	10	m	m	NOUN
ejde-196	177	11	)	)	PUNCT
ejde-196	178	1	+	+	NOUN
ejde-196	178	2	vkl	vkl	NOUN
ejde-196	178	3	is	be	AUX
ejde-196	178	4	used	use	VERB
ejde-196	178	5	to	to	PART
ejde-196	178	6	find	find	VERB
ejde-196	178	7	p(vk	p(vk	PROPN
ejde-196	178	8	(	(	PUNCT
ejde-196	178	9	λ	λ	PROPN
ejde-196	178	10	2	2	NUM
ejde-196	178	11	m	m	NOUN
ejde-196	178	12	)	)	PUNCT
ejde-196	178	13	)	)	PUNCT
ejde-196	178	14	in	in	ADP
ejde-196	178	15	{	{	PUNCT
ejde-196	178	16	l	l	NOUN
ejde-196	178	17	,	,	PUNCT
ejde-196	178	18	vk	vk	PROPN
ejde-196	178	19	(	(	PUNCT
ejde-196	178	20	λ	λ	PROPN
ejde-196	178	21	2	2	NUM
ejde-196	178	22	m	m	NOUN
ejde-196	178	23	)	)	PUNCT
ejde-196	178	24	}	}	PUNCT
ejde-196	178	25	\l	\l	VERB
ejde-196	178	26	as	as	ADP
ejde-196	178	27	similar	similar	ADJ
ejde-196	178	28	in	in	ADP
ejde-196	178	29	step	step	NOUN
ejde-196	178	30	2	2	NUM
ejde-196	178	31	and	and	CCONJ
ejde-196	178	32	where	where	SCONJ
ejde-196	178	33	tk0	tk0	NOUN
ejde-196	178	34	and	and	CCONJ
ejde-196	178	35	vkl	vkl	NOUN
ejde-196	178	36	are	be	AUX
ejde-196	178	37	found	find	VERB
ejde-196	178	38	in	in	ADP
ejde-196	178	39	step	step	NOUN
ejde-196	178	40	2	2	NUM
ejde-196	178	41	.	.	NOUN
ejde-196	178	42	step	step	NOUN
ejde-196	178	43	6	6	NUM
ejde-196	178	44	:	:	PUNCT
ejde-196	178	45	set	set	VERB
ejde-196	178	46	vk+1	vk+1	ADJ
ejde-196	178	47	=	=	SYM
ejde-196	178	48	vk(sk	vk(sk	NOUN
ejde-196	178	49	)	)	PUNCT
ejde-196	178	50	and	and	CCONJ
ejde-196	178	51	update	update	VERB
ejde-196	178	52	k	k	NOUN
ejde-196	179	1	=	=	PUNCT
ejde-196	179	2	k	k	PROPN
ejde-196	180	1	+	+	CCONJ
ejde-196	180	2	1	1	NUM
ejde-196	180	3	then	then	ADV
ejde-196	180	4	goto	goto	ADJ
ejde-196	180	5	step	step	NOUN
ejde-196	180	6	2	2	NUM
ejde-196	180	7	.	.	PUNCT
ejde-196	180	8	remark	remark	PROPN
ejde-196	180	9	3.1	3.1	NUM
ejde-196	180	10	.	.	PUNCT
ejde-196	181	1	(	(	PUNCT
ejde-196	181	2	1	1	X
ejde-196	181	3	)	)	PUNCT
ejde-196	181	4	the	the	DET
ejde-196	181	5	algorithm	algorithm	NOUN
ejde-196	181	6	starts	start	VERB
ejde-196	181	7	with	with	ADP
ejde-196	181	8	n	n	NOUN
ejde-196	181	9	=	=	SYM
ejde-196	181	10	0	0	NUM
ejde-196	181	11	,	,	PUNCT
ejde-196	181	12	l	l	NOUN
ejde-196	181	13	=	=	PUNCT
ejde-196	181	14	{	{	PUNCT
ejde-196	181	15	0	0	NUM
ejde-196	181	16	}	}	PUNCT
ejde-196	181	17	to	to	PART
ejde-196	181	18	find	find	VERB
ejde-196	181	19	w1	w1	NOUN
ejde-196	181	20	,	,	PUNCT
ejde-196	181	21	then	then	ADV
ejde-196	181	22	n	n	NOUN
ejde-196	181	23	=	=	SYM
ejde-196	181	24	1	1	NUM
ejde-196	181	25	,	,	PUNCT
ejde-196	181	26	l	l	NOUN
ejde-196	181	27	=	=	SYM
ejde-196	181	28	{	{	PUNCT
ejde-196	181	29	w1	w1	NOUN
ejde-196	181	30	}	}	PUNCT
ejde-196	181	31	to	to	PART
ejde-196	181	32	find	find	VERB
ejde-196	181	33	w2	w2	NOUN
ejde-196	181	34	,	,	PUNCT
ejde-196	181	35	etc	etc	X
ejde-196	181	36	.	.	X
ejde-196	181	37	to	to	PART
ejde-196	181	38	stay	stay	VERB
ejde-196	181	39	away	away	ADV
ejde-196	181	40	from	from	ADP
ejde-196	181	41	previously	previously	ADV
ejde-196	181	42	found	find	VERB
ejde-196	181	43	solutions	solution	NOUN
ejde-196	181	44	contained	contain	VERB
ejde-196	181	45	in	in	ADP
ejde-196	181	46	l	l	NOUN
ejde-196	181	47	,	,	PUNCT
ejde-196	181	48	we	we	PRON
ejde-196	181	49	should	should	AUX
ejde-196	181	50	choose	choose	VERB
ejde-196	181	51	an	an	DET
ejde-196	181	52	initial	initial	ADJ
ejde-196	181	53	guess	guess	NOUN
ejde-196	181	54	v0	v0	NOUN
ejde-196	181	55	which	which	PRON
ejde-196	181	56	is	be	AUX
ejde-196	181	57	at	at	ADP
ejde-196	181	58	least	least	ADJ
ejde-196	181	59	nearly	nearly	ADV
ejde-196	181	60	orthogonal	orthogonal	ADJ
ejde-196	181	61	to	to	ADP
ejde-196	181	62	l.	l.	PROPN
ejde-196	181	63	(	(	PUNCT
ejde-196	181	64	2	2	NUM
ejde-196	181	65	)	)	PUNCT
ejde-196	181	66	the	the	DET
ejde-196	181	67	finite	finite	ADJ
ejde-196	181	68	-	-	ADJ
ejde-196	181	69	dimensional	dimensional	ADJ
ejde-196	181	70	sub	sub	NOUN
ejde-196	181	71	-	-	NOUN
ejde-196	181	72	problem	problem	NOUN
ejde-196	181	73	for	for	ADP
ejde-196	181	74	finding	find	VERB
ejde-196	181	75	p(vk	p(vk	PROPN
ejde-196	181	76	)	)	PUNCT
ejde-196	181	77	in	in	ADP
ejde-196	181	78	step	step	NOUN
ejde-196	181	79	2	2	NUM
ejde-196	181	80	or	or	CCONJ
ejde-196	181	81	p(vk	p(vk	PROPN
ejde-196	181	82	(	(	PUNCT
ejde-196	181	83	λ	λ	PROPN
ejde-196	181	84	2	2	NUM
ejde-196	181	85	m	m	NOUN
ejde-196	181	86	)	)	PUNCT
ejde-196	181	87	)	)	PUNCT
ejde-196	182	1	in	in	ADP
ejde-196	182	2	step	step	NOUN
ejde-196	182	3	5	5	NUM
ejde-196	182	4	can	can	AUX
ejde-196	182	5	be	be	AUX
ejde-196	182	6	solved	solve	VERB
ejde-196	182	7	,	,	PUNCT
ejde-196	182	8	for	for	ADP
ejde-196	182	9	example	example	NOUN
ejde-196	182	10	,	,	PUNCT
ejde-196	182	11	by	by	ADP
ejde-196	182	12	matlab	matlab	PROPN
ejde-196	182	13	subroutine	subroutine	PROPN
ejde-196	182	14	fsolve	fsolve	NOUN
ejde-196	182	15	,	,	PUNCT
ejde-196	182	16	where	where	SCONJ
ejde-196	182	17	tk0	tk0	NOUN
ejde-196	182	18	>	>	X
ejde-196	182	19	0	0	NUM
ejde-196	182	20	is	be	AUX
ejde-196	182	21	determined	determine	VERB
ejde-196	182	22	by	by	ADP
ejde-196	182	23	the	the	DET
ejde-196	182	24	problem	problem	NOUN
ejde-196	182	25	structure	structure	NOUN
ejde-196	182	26	.	.	PUNCT
ejde-196	183	1	when	when	SCONJ
ejde-196	183	2	tk0	tk0	VERB
ejde-196	183	3	=	=	SYM
ejde-196	183	4	0	0	PROPN
ejde-196	183	5	,	,	PUNCT
ejde-196	183	6	the	the	DET
ejde-196	183	7	algorithm	algorithm	NOUN
ejde-196	183	8	fails	fail	VERB
ejde-196	183	9	to	to	PART
ejde-196	183	10	find	find	VERB
ejde-196	183	11	a	a	DET
ejde-196	183	12	new	new	ADJ
ejde-196	183	13	critical	critical	ADJ
ejde-196	183	14	point	point	NOUN
ejde-196	183	15	.	.	PUNCT
ejde-196	184	1	(	(	PUNCT
ejde-196	184	2	3	3	X
ejde-196	184	3	)	)	PUNCT
ejde-196	184	4	in	in	ADP
ejde-196	184	5	step	step	NOUN
ejde-196	184	6	5	5	NUM
ejde-196	184	7	,	,	PUNCT
ejde-196	184	8	the	the	DET
ejde-196	184	9	step	step	NOUN
ejde-196	184	10	size	size	NOUN
ejde-196	184	11	rule	rule	NOUN
ejde-196	184	12	has	have	AUX
ejde-196	184	13	been	be	AUX
ejde-196	184	14	equivalently	equivalently	ADV
ejde-196	184	15	modified	modify	VERB
ejde-196	184	16	due	due	ADP
ejde-196	184	17	to	to	ADP
ejde-196	184	18	the	the	DET
ejde-196	184	19	inequalities	inequality	NOUN
ejde-196	184	20	s‖d‖	s‖d‖	PROPN
ejde-196	184	21	‖v	‖v	NOUN
ejde-196	184	22	+	+	CCONJ
ejde-196	184	23	sd‖	sd‖	NOUN
ejde-196	184	24	≤	≤	NUM
ejde-196	185	1	‖v(s)−	‖v(s)−	PROPN
ejde-196	185	2	v‖	v‖	NOUN
ejde-196	185	3	≤	≤	NUM
ejde-196	185	4	√	√	NUM
ejde-196	185	5	2s‖d‖	2s‖d‖	NUM
ejde-196	185	6	‖v	‖v	NOUN
ejde-196	185	7	+	+	CCONJ
ejde-196	185	8	sd‖	sd‖	ADV
ejde-196	185	9	.	.	PUNCT
ejde-196	186	1	also	also	ADV
ejde-196	186	2	using	use	VERB
ejde-196	186	3	the	the	DET
ejde-196	186	4	designated	designate	VERB
ejde-196	186	5	initial	initial	ADJ
ejde-196	186	6	guess	guess	NOUN
ejde-196	186	7	as	as	SCONJ
ejde-196	186	8	in	in	ADP
ejde-196	186	9	step	step	NOUN
ejde-196	186	10	5	5	NUM
ejde-196	186	11	is	be	AUX
ejde-196	186	12	very	very	ADV
ejde-196	186	13	important	important	ADJ
ejde-196	186	14	to	to	PART
ejde-196	186	15	continuously	continuously	ADV
ejde-196	186	16	trace	trace	VERB
ejde-196	186	17	a	a	DET
ejde-196	186	18	solution	solution	NOUN
ejde-196	186	19	branch	branch	NOUN
ejde-196	186	20	and	and	CCONJ
ejde-196	186	21	to	to	PART
ejde-196	186	22	keep	keep	VERB
ejde-196	186	23	the	the	DET
ejde-196	186	24	algorithm	algorithm	NOUN
ejde-196	186	25	stable	stable	ADJ
ejde-196	186	26	;	;	PUNCT
ejde-196	186	27	(	(	PUNCT
ejde-196	186	28	4	4	X
ejde-196	186	29	)	)	PUNCT
ejde-196	186	30	for	for	ADP
ejde-196	186	31	an	an	DET
ejde-196	186	32	initial	initial	ADJ
ejde-196	186	33	guess	guess	NOUN
ejde-196	186	34	u0	u0	ADJ
ejde-196	186	35	,	,	PUNCT
ejde-196	186	36	when	when	SCONJ
ejde-196	186	37	the	the	DET
ejde-196	186	38	nehari	nehari	NOUN
ejde-196	186	39	manifold	manifold	VERB
ejde-196	186	40	n	n	CCONJ
ejde-196	186	41	defined	define	VERB
ejde-196	186	42	in	in	ADP
ejde-196	186	43	[	[	X
ejde-196	186	44	7	7	NUM
ejde-196	186	45	]	]	PUNCT
ejde-196	186	46	has	have	VERB
ejde-196	186	47	only	only	ADV
ejde-196	186	48	one	one	NUM
ejde-196	186	49	branch	branch	NOUN
ejde-196	186	50	,	,	PUNCT
ejde-196	186	51	we	we	PRON
ejde-196	186	52	can	can	AUX
ejde-196	186	53	simply	simply	ADV
ejde-196	186	54	find	find	VERB
ejde-196	186	55	t0	t0	PROPN
ejde-196	186	56	>	>	X
ejde-196	186	57	0	0	NUM
ejde-196	187	1	such	such	ADJ
ejde-196	187	2	that	that	SCONJ
ejde-196	187	3	t0u0	t0u0	PROPN
ejde-196	187	4	∈	∈	PROPN
ejde-196	187	5	n	n	ADV
ejde-196	187	6	and	and	CCONJ
ejde-196	187	7	use	use	VERB
ejde-196	187	8	t0u0	t0u0	PROPN
ejde-196	187	9	as	as	ADP
ejde-196	187	10	an	an	DET
ejde-196	187	11	initial	initial	ADJ
ejde-196	187	12	guess	guess	NOUN
ejde-196	187	13	;	;	PUNCT
ejde-196	187	14	when	when	SCONJ
ejde-196	187	15	n	n	PRON
ejde-196	187	16	contains	contain	VERB
ejde-196	187	17	multiple	multiple	ADJ
ejde-196	187	18	branches	branch	NOUN
ejde-196	187	19	,	,	PUNCT
ejde-196	187	20	with	with	ADP
ejde-196	187	21	extra	extra	ADJ
ejde-196	187	22	information	information	NOUN
ejde-196	187	23	on	on	ADP
ejde-196	187	24	the	the	DET
ejde-196	187	25	ejde-2023	ejde-2023	NOUN
ejde-196	187	26	/	/	SYM
ejde-196	187	27	si/02	si/02	ADJ
ejde-196	187	28	short	short	ADJ
ejde-196	187	29	title	title	NOUN
ejde-196	187	30	181	181	NUM
ejde-196	187	31	branches	branch	NOUN
ejde-196	187	32	such	such	ADJ
ejde-196	187	33	as	as	ADP
ejde-196	187	34	the	the	DET
ejde-196	187	35	sign	sign	NOUN
ejde-196	187	36	of	of	ADP
ejde-196	187	37	j	j	PROPN
ejde-196	187	38	-	-	PUNCT
ejde-196	187	39	values	value	NOUN
ejde-196	187	40	,	,	PUNCT
ejde-196	187	41	we	we	PRON
ejde-196	187	42	can	can	AUX
ejde-196	187	43	find	find	VERB
ejde-196	187	44	a	a	DET
ejde-196	187	45	proper	proper	ADJ
ejde-196	187	46	value	value	NOUN
ejde-196	187	47	t0	t0	PROPN
ejde-196	187	48	>	>	X
ejde-196	187	49	0	0	NUM
ejde-196	188	1	such	such	ADJ
ejde-196	188	2	that	that	SCONJ
ejde-196	188	3	t0u0	t0u0	PROPN
ejde-196	188	4	belongs	belong	VERB
ejde-196	188	5	to	to	ADP
ejde-196	188	6	a	a	DET
ejde-196	188	7	designated	designate	VERB
ejde-196	188	8	branch	branch	NOUN
ejde-196	188	9	.	.	PUNCT
ejde-196	189	1	4	4	X
ejde-196	189	2	.	.	X
ejde-196	189	3	algorithm	algorithm	NOUN
ejde-196	189	4	convergence	convergence	NOUN
ejde-196	189	5	analysis	analysis	NOUN
ejde-196	189	6	we	we	PRON
ejde-196	189	7	assume	assume	VERB
ejde-196	189	8	that	that	SCONJ
ejde-196	189	9	j	j	PROPN
ejde-196	189	10	satisfies	satisfy	VERB
ejde-196	189	11	the	the	DET
ejde-196	189	12	palais	palais	PROPN
ejde-196	189	13	-	-	PUNCT
ejde-196	189	14	smale	smale	ADJ
ejde-196	189	15	(	(	PUNCT
ejde-196	189	16	ps	ps	NOUN
ejde-196	189	17	)	)	PUNCT
ejde-196	189	18	condition	condition	NOUN
ejde-196	189	19	,	,	PUNCT
ejde-196	189	20	i.e.	i.e.	X
ejde-196	189	21	,	,	PUNCT
ejde-196	189	22	any	any	DET
ejde-196	189	23	sequence	sequence	NOUN
ejde-196	189	24	{	{	PUNCT
ejde-196	189	25	uk	uk	PROPN
ejde-196	189	26	}	}	PUNCT
ejde-196	189	27	⊂	⊂	PROPN
ejde-196	189	28	h	h	NOUN
ejde-196	189	29	with	with	ADP
ejde-196	189	30	{	{	PUNCT
ejde-196	189	31	j(uk	j(uk	PROPN
ejde-196	189	32	)	)	PUNCT
ejde-196	189	33	}	}	PUNCT
ejde-196	189	34	bounded	bound	VERB
ejde-196	189	35	and	and	CCONJ
ejde-196	189	36	j	j	PROPN
ejde-196	189	37	′(uk)→	′(uk)→	NUM
ejde-196	189	38	0	0	NUM
ejde-196	189	39	has	have	VERB
ejde-196	189	40	a	a	DET
ejde-196	189	41	convergent	convergent	NOUN
ejde-196	189	42	subsequence	subsequence	NOUN
ejde-196	189	43	.	.	PUNCT
ejde-196	190	1	let	let	AUX
ejde-196	190	2	{	{	PUNCT
ejde-196	190	3	wk	wk	NOUN
ejde-196	190	4	}	}	PUNCT
ejde-196	190	5	=	=	SYM
ejde-196	190	6	{	{	PUNCT
ejde-196	190	7	p(vk	p(vk	PROPN
ejde-196	190	8	)	)	PUNCT
ejde-196	190	9	}	}	PUNCT
ejde-196	190	10	be	be	VERB
ejde-196	190	11	the	the	DET
ejde-196	190	12	sequence	sequence	NOUN
ejde-196	190	13	generated	generate	VERB
ejde-196	190	14	by	by	ADP
ejde-196	190	15	the	the	DET
ejde-196	190	16	local	local	ADJ
ejde-196	190	17	min	min	ADJ
ejde-196	190	18	-	-	ADJ
ejde-196	190	19	orthogonal	orthogonal	ADJ
ejde-196	190	20	method	method	NOUN
ejde-196	190	21	with	with	ADP
ejde-196	190	22	ε	ε	PROPN
ejde-196	190	23	=	=	SYM
ejde-196	190	24	0	0	PROPN
ejde-196	190	25	.	.	PUNCT
ejde-196	191	1	next	next	ADV
ejde-196	191	2	we	we	PRON
ejde-196	191	3	present	present	VERB
ejde-196	191	4	an	an	DET
ejde-196	191	5	improved	improved	ADJ
ejde-196	191	6	convergence	convergence	NOUN
ejde-196	191	7	result	result	NOUN
ejde-196	191	8	[	[	X
ejde-196	191	9	16	16	NUM
ejde-196	191	10	,	,	PUNCT
ejde-196	191	11	18	18	NUM
ejde-196	191	12	]	]	PUNCT
ejde-196	191	13	by	by	ADP
ejde-196	191	14	modifying	modify	VERB
ejde-196	191	15	the	the	DET
ejde-196	191	16	proof	proof	NOUN
ejde-196	191	17	of	of	ADP
ejde-196	191	18	[	[	X
ejde-196	191	19	18	18	NUM
ejde-196	191	20	,	,	PUNCT
ejde-196	191	21	theorem	theorem	VERB
ejde-196	191	22	2.4	2.4	NUM
ejde-196	191	23	]	]	PUNCT
ejde-196	191	24	.	.	PUNCT
ejde-196	192	1	theorem	theorem	VERB
ejde-196	192	2	4.1	4.1	NUM
ejde-196	192	3	.	.	PUNCT
ejde-196	193	1	if	if	SCONJ
ejde-196	193	2	p	p	NOUN
ejde-196	193	3	is	be	AUX
ejde-196	193	4	defined	define	VERB
ejde-196	193	5	in	in	ADP
ejde-196	193	6	s+	s+	ADV
ejde-196	193	7	l⊥	l⊥	NOUN
ejde-196	193	8	,	,	PUNCT
ejde-196	193	9	and	and	CCONJ
ejde-196	193	10	continuous	continuous	ADJ
ejde-196	193	11	and	and	CCONJ
ejde-196	193	12	ldlc	ldlc	NOUN
ejde-196	193	13	on	on	ADP
ejde-196	193	14	sl⊥	sl⊥	PROPN
ejde-196	193	15	,	,	PUNCT
ejde-196	193	16	d(l	d(l	ADV
ejde-196	193	17	,	,	PUNCT
ejde-196	193	18	wk	wk	NOUN
ejde-196	193	19	)	)	PUNCT
ejde-196	193	20	>	>	X
ejde-196	194	1	α>0	α>0	NOUN
ejde-196	194	2	and	and	CCONJ
ejde-196	194	3	infv∈s	infv∈s	PROPN
ejde-196	194	4	l⊥	l⊥	PROPN
ejde-196	194	5	j(p(v))>−∞	j(p(v))>−∞	NOUN
ejde-196	194	6	,	,	PUNCT
ejde-196	194	7	then	then	ADV
ejde-196	194	8	(	(	PUNCT
ejde-196	194	9	a	a	X
ejde-196	194	10	)	)	PUNCT
ejde-196	194	11	skdk	skdk	NOUN
ejde-196	194	12	→	→	X
ejde-196	194	13	0	0	NUM
ejde-196	194	14	;	;	PUNCT
ejde-196	194	15	(	(	PUNCT
ejde-196	194	16	b	b	X
ejde-196	194	17	)	)	PUNCT
ejde-196	194	18	there	there	PRON
ejde-196	194	19	is	be	VERB
ejde-196	194	20	{	{	PUNCT
ejde-196	194	21	vki	vki	NOUN
ejde-196	194	22	}	}	PUNCT
ejde-196	194	23	⊂	⊂	PROPN
ejde-196	194	24	{	{	PUNCT
ejde-196	194	25	vk	vk	PROPN
ejde-196	194	26	}	}	PUNCT
ejde-196	194	27	such	such	ADJ
ejde-196	194	28	that	that	DET
ejde-196	194	29	vki	vki	NOUN
ejde-196	194	30	→	→	SYM
ejde-196	194	31	v∗	v∗	NOUN
ejde-196	194	32	with	with	ADP
ejde-196	194	33	w∗	w∗	NOUN
ejde-196	194	34	=	=	SYM
ejde-196	194	35	p(v∗	p(v∗	NOUN
ejde-196	194	36	)	)	PUNCT
ejde-196	194	37	,	,	PUNCT
ejde-196	194	38	j	j	PROPN
ejde-196	194	39	′(w∗	′(w∗	X
ejde-196	194	40	)	)	PUNCT
ejde-196	194	41	=	=	SYM
ejde-196	194	42	0	0	NUM
ejde-196	194	43	;	;	PUNCT
ejde-196	194	44	(	(	PUNCT
ejde-196	194	45	c	c	X
ejde-196	194	46	)	)	PUNCT
ejde-196	194	47	if	if	SCONJ
ejde-196	194	48	w∗	w∗	NOUN
ejde-196	194	49	is	be	AUX
ejde-196	194	50	isolated	isolate	VERB
ejde-196	194	51	then	then	ADV
ejde-196	194	52	vk	vk	INTJ
ejde-196	194	53	→	→	SYM
ejde-196	194	54	v∗.	v∗.	NOUN
ejde-196	194	55	proof	proof	NOUN
ejde-196	194	56	.	.	PUNCT
ejde-196	195	1	by	by	ADP
ejde-196	195	2	the	the	DET
ejde-196	195	3	step	step	NOUN
ejde-196	195	4	size	size	NOUN
ejde-196	195	5	rule	rule	NOUN
ejde-196	195	6	(	(	PUNCT
ejde-196	195	7	2.2	2.2	NUM
ejde-196	195	8	)	)	PUNCT
ejde-196	195	9	,	,	PUNCT
ejde-196	195	10	√	√	PROPN
ejde-196	195	11	1+(sk‖dk‖)2√	1+(sk‖dk‖)2√	NUM
ejde-196	195	12	2	2	NUM
ejde-196	195	13	>	>	SYM
ejde-196	195	14	1/2	1/2	NUM
ejde-196	195	15	and	and	CCONJ
ejde-196	195	16	the	the	DET
ejde-196	195	17	inequality	inequality	NOUN
ejde-196	195	18	sk‖dk‖√	sk‖dk‖√	NOUN
ejde-196	195	19	1	1	NUM
ejde-196	195	20	+	+	CCONJ
ejde-196	195	21	(	(	PUNCT
ejde-196	195	22	sk‖dk‖)2	sk‖dk‖)2	X
ejde-196	195	23	≤	≤	NUM
ejde-196	195	24	‖vk+1	‖vk+1	PUNCT
ejde-196	195	25	−	−	PROPN
ejde-196	195	26	vk‖	vk‖	PROPN
ejde-196	195	27	≤	≤	PROPN
ejde-196	195	28	√	√	NUM
ejde-196	195	29	2sk‖dk‖√	2sk‖dk‖√	NUM
ejde-196	195	30	1	1	NUM
ejde-196	196	1	+	+	CCONJ
ejde-196	196	2	(	(	PUNCT
ejde-196	196	3	sk‖dk‖)2	sk‖dk‖)2	X
ejde-196	196	4	,	,	PUNCT
ejde-196	196	5	(	(	PUNCT
ejde-196	196	6	4.1	4.1	NUM
ejde-196	196	7	)	)	PUNCT
ejde-196	196	8	we	we	PRON
ejde-196	196	9	obtain	obtain	VERB
ejde-196	196	10	j(wk+1)−	j(wk+1)−	ADJ
ejde-196	196	11	j(wk	j(wk	NOUN
ejde-196	196	12	)	)	PUNCT
ejde-196	196	13	≤	≤	NUM
ejde-196	196	14	−1	−1	NOUN
ejde-196	196	15	4	4	NUM
ejde-196	196	16	|tk0	|tk0	PROPN
ejde-196	196	17	|sk‖dk‖2ck	|sk‖dk‖2ck	NOUN
ejde-196	196	18	≤	≤	NUM
ejde-196	196	19	−1	−1	NOUN
ejde-196	196	20	8	8	NUM
ejde-196	196	21	|tk0	|tk0	PROPN
ejde-196	196	22	|‖j	|‖j	NOUN
ejde-196	196	23	′(wk)‖‖vk+1	′(wk)‖‖vk+1	NOUN
ejde-196	196	24	−	−	X
ejde-196	196	25	vk‖.	vk‖.	ADP
ejde-196	196	26	adding	add	VERB
ejde-196	196	27	it	it	PRON
ejde-196	196	28	up	up	ADP
ejde-196	196	29	for	for	ADP
ejde-196	196	30	all	all	PRON
ejde-196	196	31	k	k	NOUN
ejde-196	196	32	=	=	SYM
ejde-196	196	33	1	1	NUM
ejde-196	196	34	,	,	PUNCT
ejde-196	196	35	2	2	NUM
ejde-196	196	36	,	,	PUNCT
ejde-196	196	37	.	.	PUNCT
ejde-196	196	38	.	.	PUNCT
ejde-196	196	39	.	.	PUNCT
ejde-196	197	1	and	and	CCONJ
ejde-196	197	2	noting	note	VERB
ejde-196	197	3	|tk0	|tk0	PROPN
ejde-196	197	4	|	|	ADV
ejde-196	197	5	>	>	X
ejde-196	197	6	α	α	X
ejde-196	197	7	>	>	X
ejde-196	197	8	0	0	NUM
ejde-196	197	9	,	,	PUNCT
ejde-196	197	10	j(wk	j(wk	NUM
ejde-196	197	11	)	)	PUNCT
ejde-196	197	12	>	>	X
ejde-196	198	1	−m	−m	PROPN
ejde-196	198	2	>	>	X
ejde-196	198	3	−∞	−∞	NOUN
ejde-196	198	4	,	,	PUNCT
ejde-196	198	5	we	we	PRON
ejde-196	198	6	obtain	obtain	VERB
ejde-196	198	7	−∞	−∞	ADP
ejde-196	198	8	<	<	X
ejde-196	198	9	lim	lim	PROPN
ejde-196	198	10	k→∞	k→∞	PROPN
ejde-196	198	11	j(wk)−	j(wk)−	PROPN
ejde-196	198	12	j(w1	j(w1	PROPN
ejde-196	198	13	)	)	PUNCT
ejde-196	198	14	≤	≤	NUM
ejde-196	198	15	−1	−1	NOUN
ejde-196	198	16	4	4	NUM
ejde-196	198	17	∞∑	∞∑	NUM
ejde-196	198	18	k=1	k=1	ADP
ejde-196	198	19	|tk0	|tk0	PROPN
ejde-196	198	20	|sk‖dk‖2ck	|sk‖dk‖2ck	NOUN
ejde-196	198	21	≤	≤	ADJ
ejde-196	198	22	−α	−α	NOUN
ejde-196	198	23	4	4	NUM
ejde-196	198	24	∞∑	∞∑	NUM
ejde-196	198	25	k=1	k=1	PUNCT
ejde-196	199	1	sk‖dk‖2ck	sk‖dk‖2ck	PROPN
ejde-196	199	2	<	<	X
ejde-196	199	3	−	−	PROPN
ejde-196	199	4	α	α	NOUN
ejde-196	199	5	8	8	NUM
ejde-196	199	6	∞∑	∞∑	NOUN
ejde-196	199	7	k=1	k=1	PUNCT
ejde-196	199	8	‖j	‖j	ADV
ejde-196	199	9	′(wk)‖‖vk+1	′(wk)‖‖vk+1	VERB
ejde-196	199	10	−	−	PROPN
ejde-196	199	11	vk‖.	vk‖.	PROPN
ejde-196	199	12	(	(	PUNCT
ejde-196	199	13	4.2	4.2	NUM
ejde-196	199	14	)	)	PUNCT
ejde-196	199	15	thus	thus	ADV
ejde-196	199	16	sk‖dk‖2ck	sk‖dk‖2ck	NOUN
ejde-196	199	17	=	=	SYM
ejde-196	199	18	sk‖j	sk‖j	NUM
ejde-196	199	19	′(wk)‖2	′(wk)‖2	NOUN
ejde-196	199	20	/	/	SYM
ejde-196	199	21	ck	ck	NOUN
ejde-196	199	22	→	→	SYM
ejde-196	199	23	0	0	NUM
ejde-196	199	24	.	.	PUNCT
ejde-196	200	1	then	then	ADV
ejde-196	200	2	ck	ck	NOUN
ejde-196	200	3	=	=	SYM
ejde-196	200	4	max{1	max{1	NOUN
ejde-196	200	5	,	,	PUNCT
ejde-196	200	6	‖j	‖j	SYM
ejde-196	200	7	′(wk)‖	′(wk)‖	NOUN
ejde-196	200	8	}	}	PUNCT
ejde-196	200	9	and	and	CCONJ
ejde-196	200	10	0	0	NUM
ejde-196	200	11	<	<	X
ejde-196	200	12	sk	sk	X
ejde-196	200	13	<	<	X
ejde-196	200	14	λ	λ	X
ejde-196	200	15	lead	lead	NOUN
ejde-196	200	16	to	to	PART
ejde-196	200	17	sk‖dk‖	sk‖dk‖	VERB
ejde-196	200	18	→	→	SYM
ejde-196	200	19	0	0	PUNCT
ejde-196	200	20	as	as	ADP
ejde-196	200	21	in	in	ADP
ejde-196	200	22	(	(	PUNCT
ejde-196	200	23	a	a	NOUN
ejde-196	200	24	)	)	PUNCT
ejde-196	200	25	.	.	PUNCT
ejde-196	201	1	next	next	ADJ
ejde-196	201	2	to	to	PART
ejde-196	201	3	prove	prove	VERB
ejde-196	201	4	(	(	PUNCT
ejde-196	201	5	b	b	NOUN
ejde-196	201	6	)	)	PUNCT
ejde-196	201	7	,	,	PUNCT
ejde-196	201	8	there	there	PRON
ejde-196	201	9	are	be	VERB
ejde-196	201	10	totally	totally	ADV
ejde-196	201	11	two	two	NUM
ejde-196	201	12	cases	case	NOUN
ejde-196	201	13	,	,	PUNCT
ejde-196	201	14	either	either	CCONJ
ejde-196	201	15	(	(	PUNCT
ejde-196	201	16	1	1	X
ejde-196	201	17	)	)	PUNCT
ejde-196	201	18	‖dk‖	‖dk‖	VERB
ejde-196	201	19	>	>	PUNCT
ejde-196	201	20	η	η	X
ejde-196	201	21	>	>	X
ejde-196	201	22	0	0	PROPN
ejde-196	202	1	for	for	ADP
ejde-196	202	2	k	k	PROPN
ejde-196	202	3	=	=	SYM
ejde-196	202	4	1	1	NUM
ejde-196	202	5	,	,	PUNCT
ejde-196	202	6	2	2	NUM
ejde-196	202	7	,	,	PUNCT
ejde-196	202	8	.	.	PUNCT
ejde-196	202	9	.	.	PUNCT
ejde-196	202	10	.	.	PUNCT
ejde-196	203	1	for	for	ADP
ejde-196	203	2	some	some	PRON
ejde-196	203	3	1	1	NUM
ejde-196	203	4	2	2	NUM
ejde-196	203	5	>	>	X
ejde-196	203	6	η	η	X
ejde-196	203	7	>	>	X
ejde-196	203	8	0	0	PROPN
ejde-196	203	9	,	,	PUNCT
ejde-196	203	10	or	or	CCONJ
ejde-196	203	11	(	(	PUNCT
ejde-196	203	12	2	2	X
ejde-196	203	13	)	)	PUNCT
ejde-196	203	14	there	there	PRON
ejde-196	203	15	is	be	VERB
ejde-196	203	16	a	a	DET
ejde-196	203	17	subsequence	subsequence	NOUN
ejde-196	203	18	{	{	PUNCT
ejde-196	203	19	dki	dki	NOUN
ejde-196	203	20	}	}	PUNCT
ejde-196	203	21	⊂	⊂	PROPN
ejde-196	203	22	{	{	PUNCT
ejde-196	203	23	dk	dk	X
ejde-196	203	24	}	}	PUNCT
ejde-196	203	25	such	such	ADJ
ejde-196	203	26	that	that	DET
ejde-196	203	27	dki	dki	NOUN
ejde-196	203	28	→	→	PUNCT
ejde-196	203	29	0	0	X
ejde-196	203	30	.	.	PUNCT
ejde-196	204	1	in	in	ADP
ejde-196	204	2	case	case	NOUN
ejde-196	204	3	(	(	PUNCT
ejde-196	204	4	1	1	NUM
ejde-196	204	5	)	)	PUNCT
ejde-196	204	6	,	,	PUNCT
ejde-196	204	7	η	η	X
ejde-196	204	8	<	<	X
ejde-196	204	9	‖dk‖	‖dk‖	VERB
ejde-196	204	10	≤	≤	NUM
ejde-196	204	11	1	1	NUM
ejde-196	204	12	,	,	PUNCT
ejde-196	204	13	then	then	ADV
ejde-196	204	14	(	(	PUNCT
ejde-196	204	15	4.2	4.2	NUM
ejde-196	204	16	)	)	PUNCT
ejde-196	204	17	becomes	become	VERB
ejde-196	204	18	−∞	−∞	ADP
ejde-196	204	19	<	<	X
ejde-196	204	20	lim	lim	PROPN
ejde-196	204	21	k→∞	k→∞	PROPN
ejde-196	204	22	j(wk)−	j(wk)−	PROPN
ejde-196	204	23	j(w1	j(w1	PROPN
ejde-196	204	24	)	)	PUNCT
ejde-196	204	25	≤	≤	NUM
ejde-196	204	26	−α	−α	NOUN
ejde-196	204	27	8	8	NUM
ejde-196	204	28	∞∑	∞∑	NOUN
ejde-196	204	29	k=1	k=1	PUNCT
ejde-196	204	30	‖j	‖j	ADV
ejde-196	205	1	′(wk)‖‖vk+1	′(wk)‖‖vk+1	VERB
ejde-196	205	2	−	−	PROPN
ejde-196	205	3	vk‖	vk‖	PROPN
ejde-196	205	4	≤	≤	NUM
ejde-196	205	5	−αη	−αη	NOUN
ejde-196	205	6	8	8	NUM
ejde-196	205	7	∞∑	∞∑	NUM
ejde-196	205	8	k=1	k=1	PUNCT
ejde-196	205	9	‖vk+1	‖vk+1	PUNCT
ejde-196	206	1	−	−	PROPN
ejde-196	206	2	vk‖	vk‖	PROPN
ejde-196	206	3	,	,	PUNCT
ejde-196	206	4	(	(	PUNCT
ejde-196	206	5	4.3	4.3	NUM
ejde-196	206	6	)	)	PUNCT
ejde-196	206	7	i.e.	i.e.	X
ejde-196	206	8	,	,	PUNCT
ejde-196	206	9	{	{	PUNCT
ejde-196	206	10	vk	vk	NOUN
ejde-196	206	11	}	}	PUNCT
ejde-196	206	12	⊂	⊂	PRON
ejde-196	206	13	sl⊥	sl⊥	PROPN
ejde-196	206	14	is	be	AUX
ejde-196	206	15	a	a	DET
ejde-196	206	16	cauchy	cauchy	ADJ
ejde-196	206	17	sequence	sequence	NOUN
ejde-196	206	18	in	in	ADP
ejde-196	206	19	the	the	DET
ejde-196	206	20	hilbert	hilbert	PROPN
ejde-196	206	21	space	space	NOUN
ejde-196	206	22	h.	h.	PROPN
ejde-196	207	1	thus	thus	ADV
ejde-196	207	2	there	there	PRON
ejde-196	207	3	exists	exist	VERB
ejde-196	207	4	v∗	v∗	PROPN
ejde-196	207	5	∈	∈	PROPN
ejde-196	207	6	sl⊥	sl⊥	PROPN
ejde-196	207	7	such	such	ADJ
ejde-196	207	8	that	that	PRON
ejde-196	207	9	vk	vk	NOUN
ejde-196	207	10	→	→	SYM
ejde-196	207	11	v∗	v∗	PROPN
ejde-196	207	12	as	as	ADP
ejde-196	207	13	k	k	PROPN
ejde-196	207	14	→∞.	→∞.	PROPN
ejde-196	207	15	since	since	SCONJ
ejde-196	207	16	p	p	NOUN
ejde-196	207	17	is	be	AUX
ejde-196	207	18	continuous	continuous	ADJ
ejde-196	207	19	and	and	CCONJ
ejde-196	207	20	j	j	PROPN
ejde-196	207	21	is	be	AUX
ejde-196	207	22	c1	c1	PROPN
ejde-196	207	23	,	,	PUNCT
ejde-196	207	24	we	we	PRON
ejde-196	207	25	have	have	VERB
ejde-196	207	26	w∗	w∗	NOUN
ejde-196	207	27	=	=	SYM
ejde-196	207	28	p(v∗	p(v∗	NOUN
ejde-196	207	29	)	)	PUNCT
ejde-196	207	30	=	=	SYM
ejde-196	208	1	limk→∞	limk→∞	ADJ
ejde-196	208	2	p(vk	p(vk	PROPN
ejde-196	208	3	)	)	PUNCT
ejde-196	208	4	and	and	CCONJ
ejde-196	208	5	‖j	‖j	PRON
ejde-196	208	6	′(w∗)‖	′(w∗)‖	X
ejde-196	208	7	≥	≥	PROPN
ejde-196	208	8	η	η	PROPN
ejde-196	208	9	.	.	PROPN
ejde-196	208	10	hence	hence	ADV
ejde-196	208	11	w∗	w∗	PROPN
ejde-196	208	12	is	be	AUX
ejde-196	208	13	not	not	PART
ejde-196	208	14	a	a	DET
ejde-196	208	15	critical	critical	ADJ
ejde-196	208	16	point	point	NOUN
ejde-196	208	17	.	.	PUNCT
ejde-196	209	1	then	then	ADV
ejde-196	209	2	skdk	skdk	PROPN
ejde-196	209	3	→	→	SYM
ejde-196	209	4	0	0	NUM
ejde-196	209	5	and	and	CCONJ
ejde-196	209	6	‖dk‖	‖dk‖	VERB
ejde-196	209	7	>	>	PUNCT
ejde-196	209	8	η	η	PROPN
ejde-196	209	9	>	>	X
ejde-196	209	10	0	0	NUM
ejde-196	210	1	imply	imply	VERB
ejde-196	210	2	sk	sk	INTJ
ejde-196	210	3	→	→	SYM
ejde-196	210	4	0	0	NUM
ejde-196	210	5	,	,	PUNCT
ejde-196	210	6	which	which	PRON
ejde-196	210	7	contradicts	contradict	VERB
ejde-196	210	8	the	the	DET
ejde-196	210	9	uniform	uniform	ADJ
ejde-196	210	10	step	step	NOUN
ejde-196	210	11	size	size	NOUN
ejde-196	210	12	rule	rule	NOUN
ejde-196	210	13	,	,	PUNCT
ejde-196	210	14	lemma	lemma	PROPN
ejde-196	210	15	2.3	2.3	NUM
ejde-196	210	16	.	.	PUNCT
ejde-196	211	1	thus	thus	ADV
ejde-196	211	2	case	case	NOUN
ejde-196	211	3	(	(	PUNCT
ejde-196	211	4	2	2	X
ejde-196	211	5	)	)	PUNCT
ejde-196	211	6	must	must	AUX
ejde-196	211	7	hold	hold	VERB
ejde-196	211	8	,	,	PUNCT
ejde-196	211	9	i.e.	i.e.	X
ejde-196	211	10	,	,	PUNCT
ejde-196	211	11	there	there	PRON
ejde-196	211	12	is	be	VERB
ejde-196	211	13	a	a	DET
ejde-196	211	14	subsequence	subsequence	NOUN
ejde-196	211	15	182	182	NUM
ejde-196	211	16	m.	m.	NOUN
ejde-196	211	17	li	li	PROPN
ejde-196	211	18	,	,	PUNCT
ejde-196	211	19	b.	b.	PROPN
ejde-196	212	1	ji	ji	PROPN
ejde-196	212	2	,	,	PUNCT
ejde-196	212	3	j.	j.	PROPN
ejde-196	212	4	zhou	zhou	PROPN
ejde-196	212	5	ejde	ejde	PROPN
ejde-196	212	6	/	/	SYM
ejde-196	212	7	si/02	si/02	PROPN
ejde-196	212	8	{	{	PUNCT
ejde-196	212	9	dki	dki	PROPN
ejde-196	212	10	}	}	PUNCT
ejde-196	212	11	⊂	⊂	PROPN
ejde-196	212	12	{	{	PUNCT
ejde-196	212	13	dk	dk	X
ejde-196	212	14	}	}	PUNCT
ejde-196	212	15	such	such	ADJ
ejde-196	212	16	that	that	DET
ejde-196	212	17	dki	dki	NOUN
ejde-196	212	18	→	→	PUNCT
ejde-196	212	19	0	0	X
ejde-196	212	20	.	.	PUNCT
ejde-196	213	1	since	since	SCONJ
ejde-196	213	2	{	{	PUNCT
ejde-196	213	3	j(wki	j(wki	NOUN
ejde-196	213	4	)	)	PUNCT
ejde-196	213	5	}	}	PUNCT
ejde-196	213	6	is	be	AUX
ejde-196	213	7	bounded	bound	VERB
ejde-196	213	8	,	,	PUNCT
ejde-196	213	9	by	by	ADP
ejde-196	213	10	the	the	DET
ejde-196	213	11	ps	ps	NOUN
ejde-196	213	12	condition	condition	NOUN
ejde-196	213	13	,	,	PUNCT
ejde-196	213	14	there	there	PRON
ejde-196	213	15	is	be	VERB
ejde-196	213	16	a	a	DET
ejde-196	213	17	subsequence	subsequence	NOUN
ejde-196	213	18	denoted	denote	VERB
ejde-196	213	19	by	by	ADP
ejde-196	213	20	wki	wki	PROPN
ejde-196	213	21	=	=	PUNCT
ejde-196	213	22	p(vki	p(vki	NOUN
ejde-196	213	23	)	)	PUNCT
ejde-196	213	24	=	=	VERB
ejde-196	214	1	tki0	tki0	NOUN
ejde-196	214	2	v	v	INTJ
ejde-196	214	3	ki	ki	PROPN
ejde-196	215	1	+	+	CCONJ
ejde-196	215	2	wkil	wkil	ADJ
ejde-196	215	3	again	again	ADV
ejde-196	215	4	for	for	ADP
ejde-196	215	5	some	some	DET
ejde-196	215	6	wkil	wkil	ADJ
ejde-196	215	7	∈	∈	PROPN
ejde-196	215	8	l	l	NOUN
ejde-196	215	9	,	,	PUNCT
ejde-196	215	10	such	such	ADJ
ejde-196	215	11	that	that	SCONJ
ejde-196	215	12	wki	wki	PROPN
ejde-196	215	13	→	→	SYM
ejde-196	215	14	w∗	w∗	NOUN
ejde-196	215	15	=	=	SYM
ejde-196	215	16	w∗⊥	w∗⊥	X
ejde-196	215	17	+	+	CCONJ
ejde-196	215	18	w∗l	w∗l	NOUN
ejde-196	215	19	with	with	ADP
ejde-196	215	20	w∗⊥	w∗⊥	PROPN
ejde-196	215	21	∈	∈	PROPN
ejde-196	215	22	l⊥	l⊥	NOUN
ejde-196	215	23	,	,	PUNCT
ejde-196	215	24	w∗l	w∗l	NOUN
ejde-196	215	25	∈	∈	PROPN
ejde-196	215	26	l.	l.	NOUN
ejde-196	215	27	by	by	ADP
ejde-196	215	28	the	the	DET
ejde-196	215	29	condition	condition	NOUN
ejde-196	215	30	|tk0	|tk0	PROPN
ejde-196	216	1	|	|	ADV
ejde-196	216	2	>	>	X
ejde-196	216	3	α	α	X
ejde-196	216	4	>	>	X
ejde-196	216	5	0	0	NUM
ejde-196	216	6	,	,	PUNCT
ejde-196	216	7	we	we	PRON
ejde-196	216	8	must	must	AUX
ejde-196	216	9	have	have	VERB
ejde-196	216	10	tki0	tki0	NOUN
ejde-196	216	11	v	v	ADP
ejde-196	216	12	ki	ki	PROPN
ejde-196	216	13	→	→	PUNCT
ejde-196	216	14	w∗⊥	w∗⊥	ADV
ejde-196	216	15	,	,	PUNCT
ejde-196	216	16	w	w	PROPN
ejde-196	216	17	ki	ki	PROPN
ejde-196	216	18	l	l	PROPN
ejde-196	216	19	→	→	SYM
ejde-196	216	20	w∗l	w∗l	NUM
ejde-196	216	21	and	and	CCONJ
ejde-196	216	22	vki	vki	NOUN
ejde-196	216	23	→	→	SYM
ejde-196	216	24	v∗	v∗	NOUN
ejde-196	216	25	for	for	ADP
ejde-196	216	26	some	some	DET
ejde-196	216	27	v∗	v∗	ADJ
ejde-196	216	28	∈	∈	PROPN
ejde-196	216	29	sl⊥	sl⊥	PROPN
ejde-196	216	30	.	.	PUNCT
ejde-196	217	1	thus	thus	ADV
ejde-196	217	2	tki0	tki0	X
ejde-196	217	3	→	→	PUNCT
ejde-196	217	4	t∗0	t∗0	NOUN
ejde-196	217	5	for	for	ADP
ejde-196	217	6	some	some	DET
ejde-196	217	7	|t∗0|	|t∗0|	NOUN
ejde-196	217	8	≥	≥	NOUN
ejde-196	217	9	α	α	X
ejde-196	217	10	>	>	X
ejde-196	217	11	0	0	PROPN
ejde-196	217	12	.	.	PUNCT
ejde-196	218	1	the	the	DET
ejde-196	218	2	continuity	continuity	NOUN
ejde-196	218	3	of	of	ADP
ejde-196	218	4	p	p	NOUN
ejde-196	218	5	then	then	ADV
ejde-196	218	6	leads	lead	VERB
ejde-196	218	7	to	to	ADP
ejde-196	218	8	w∗	w∗	NOUN
ejde-196	218	9	=	=	SYM
ejde-196	218	10	p(v∗	p(v∗	NOUN
ejde-196	218	11	)	)	PUNCT
ejde-196	218	12	.	.	PUNCT
ejde-196	219	1	we	we	PRON
ejde-196	219	2	have	have	AUX
ejde-196	219	3	proved	prove	VERB
ejde-196	219	4	that	that	SCONJ
ejde-196	219	5	p	p	NOUN
ejde-196	219	6	is	be	AUX
ejde-196	219	7	a	a	DET
ejde-196	219	8	homeomorphism	homeomorphism	PROPN
ejde-196	219	9	and	and	CCONJ
ejde-196	219	10	j	j	PROPN
ejde-196	219	11	′(w∗	′(w∗	PROPN
ejde-196	219	12	)	)	PUNCT
ejde-196	219	13	=	=	PUNCT
ejde-196	219	14	limi→∞	limi→∞	PROPN
ejde-196	219	15	j	j	PROPN
ejde-196	219	16	′(wki	′(wki	PROPN
ejde-196	219	17	)	)	PUNCT
ejde-196	219	18	=	=	PUNCT
ejde-196	219	19	limi→∞	limi→∞	PROPN
ejde-196	219	20	dki	dki	NOUN
ejde-196	219	21	=	=	PUNCT
ejde-196	220	1	0	0	PROPN
ejde-196	220	2	.	.	PUNCT
ejde-196	221	1	so	so	ADV
ejde-196	221	2	(	(	PUNCT
ejde-196	221	3	b	b	X
ejde-196	221	4	)	)	PUNCT
ejde-196	221	5	is	be	AUX
ejde-196	221	6	proved	prove	VERB
ejde-196	221	7	.	.	PUNCT
ejde-196	222	1	finally	finally	ADV
ejde-196	222	2	to	to	PART
ejde-196	222	3	prove	prove	VERB
ejde-196	222	4	(	(	PUNCT
ejde-196	222	5	c	c	NOUN
ejde-196	222	6	)	)	PUNCT
ejde-196	222	7	vk	vk	PROPN
ejde-196	222	8	→	→	SYM
ejde-196	222	9	v∗	v∗	PROPN
ejde-196	222	10	if	if	SCONJ
ejde-196	222	11	w∗	w∗	NOUN
ejde-196	222	12	=	=	SYM
ejde-196	222	13	p(v∗	p(v∗	PRON
ejde-196	222	14	)	)	PUNCT
ejde-196	222	15	is	be	AUX
ejde-196	222	16	an	an	DET
ejde-196	222	17	isolated	isolated	ADJ
ejde-196	222	18	saddle	saddle	NOUN
ejde-196	222	19	,	,	PUNCT
ejde-196	222	20	since	since	SCONJ
ejde-196	222	21	p	p	NOUN
ejde-196	222	22	is	be	AUX
ejde-196	222	23	a	a	DET
ejde-196	222	24	homeomorphism	homeomorphism	NOUN
ejde-196	222	25	,	,	PUNCT
ejde-196	222	26	this	this	PRON
ejde-196	222	27	means	mean	VERB
ejde-196	222	28	that	that	SCONJ
ejde-196	222	29	there	there	PRON
ejde-196	222	30	is	be	VERB
ejde-196	222	31	δ	δ	PROPN
ejde-196	222	32	>	>	X
ejde-196	222	33	0	0	NUM
ejde-196	222	34	such	such	ADJ
ejde-196	222	35	that	that	SCONJ
ejde-196	222	36	there	there	PRON
ejde-196	222	37	is	be	VERB
ejde-196	222	38	no	no	DET
ejde-196	222	39	point	point	NOUN
ejde-196	223	1	v′	v′	NOUN
ejde-196	223	2	∈	∈	NOUN
ejde-196	223	3	sl⊥	sl⊥	PROPN
ejde-196	223	4	satisfying	satisfy	VERB
ejde-196	223	5	‖v∗	‖v∗	ADP
ejde-196	223	6	−	−	PROPN
ejde-196	223	7	v′‖	v′‖	ADP
ejde-196	223	8	<	<	X
ejde-196	223	9	δ	δ	PROPN
ejde-196	223	10	and	and	CCONJ
ejde-196	223	11	j	j	PROPN
ejde-196	223	12	′(p(v′	′(p(v′	PROPN
ejde-196	223	13	)	)	PUNCT
ejde-196	223	14	)	)	PUNCT
ejde-196	224	1	=	=	PUNCT
ejde-196	224	2	0	0	X
ejde-196	224	3	.	.	PUNCT
ejde-196	225	1	let	let	VERB
ejde-196	225	2	i	i	PRON
ejde-196	225	3	⊂	⊂	PRON
ejde-196	225	4	n	n	NOUN
ejde-196	225	5	=	=	PRON
ejde-196	225	6	{	{	PUNCT
ejde-196	225	7	1	1	NUM
ejde-196	225	8	,	,	PUNCT
ejde-196	225	9	2	2	NUM
ejde-196	225	10	,	,	PUNCT
ejde-196	225	11	.	.	PUNCT
ejde-196	225	12	.	.	PUNCT
ejde-196	225	13	.	.	PUNCT
ejde-196	226	1	}	}	PUNCT
ejde-196	226	2	.	.	PUNCT
ejde-196	227	1	we	we	PRON
ejde-196	227	2	call	call	VERB
ejde-196	227	3	∑	∑	ADV
ejde-196	227	4	i∈i	i∈i	ADJ
ejde-196	227	5	‖vi+1	‖vi+1	PUNCT
ejde-196	228	1	−	−	NOUN
ejde-196	228	2	vi‖	vi‖	NOUN
ejde-196	228	3	the	the	DET
ejde-196	228	4	total	total	ADJ
ejde-196	228	5	distance	distance	NOUN
ejde-196	228	6	traveled	travel	VERB
ejde-196	228	7	by	by	ADP
ejde-196	228	8	the	the	DET
ejde-196	228	9	subsequence	subsequence	NOUN
ejde-196	228	10	{	{	PUNCT
ejde-196	228	11	vi}i∈i	vi}i∈i	NOUN
ejde-196	228	12	.	.	PUNCT
ejde-196	229	1	for	for	ADP
ejde-196	229	2	any	any	DET
ejde-196	229	3	η	η	PROPN
ejde-196	229	4	>	>	X
ejde-196	229	5	0	0	PROPN
ejde-196	229	6	,	,	PUNCT
ejde-196	229	7	let	let	VERB
ejde-196	229	8	i	i	PRON
ejde-196	229	9	∈	∈	VERB
ejde-196	229	10	i	i	PRON
ejde-196	229	11	⊂	⊂	PROPN
ejde-196	229	12	n	n	PRON
ejde-196	229	13	denote	denote	VERB
ejde-196	229	14	the	the	DET
ejde-196	229	15	whole	whole	ADJ
ejde-196	229	16	index	index	NOUN
ejde-196	229	17	set	set	VERB
ejde-196	229	18	in	in	ADP
ejde-196	229	19	n	n	CCONJ
ejde-196	229	20	with	with	ADP
ejde-196	229	21	‖di‖	‖di‖	PROPN
ejde-196	229	22	>	>	X
ejde-196	229	23	η	η	PROPN
ejde-196	229	24	.	.	PROPN
ejde-196	230	1	then	then	ADV
ejde-196	230	2	(	(	PUNCT
ejde-196	230	3	4.2	4.2	NUM
ejde-196	230	4	)	)	PUNCT
ejde-196	230	5	leads	lead	VERB
ejde-196	230	6	to	to	ADP
ejde-196	230	7	−∞	−∞	X
ejde-196	230	8	<	<	X
ejde-196	230	9	lim	lim	PROPN
ejde-196	230	10	i→∞	i→∞	NUM
ejde-196	230	11	j(wi)−	j(wi)−	NOUN
ejde-196	230	12	j(w1	j(w1	NOUN
ejde-196	230	13	)	)	PUNCT
ejde-196	230	14	≤	≤	NUM
ejde-196	230	15	−α	−α	NOUN
ejde-196	230	16	8	8	NUM
ejde-196	230	17	∑	∑	ADP
ejde-196	230	18	i∈i	i∈i	ADV
ejde-196	230	19	‖j	‖j	ADV
ejde-196	231	1	′(wi)‖‖vi+1	′(wi)‖‖vi+1	NUM
ejde-196	232	1	−	−	PROPN
ejde-196	232	2	vi‖	vi‖	PROPN
ejde-196	232	3	<	<	X
ejde-196	232	4	−αη	−αη	PROPN
ejde-196	232	5	8	8	NUM
ejde-196	232	6	∑	∑	ADP
ejde-196	232	7	i∈i	i∈i	ADJ
ejde-196	232	8	‖vi+1	‖vi+1	PUNCT
ejde-196	233	1	−	−	PROPN
ejde-196	233	2	vi‖	vi‖	PROPN
ejde-196	233	3	,	,	PUNCT
ejde-196	233	4	(	(	PUNCT
ejde-196	233	5	4.4	4.4	NUM
ejde-196	233	6	)	)	PUNCT
ejde-196	233	7	i.e.	i.e.	X
ejde-196	233	8	,	,	PUNCT
ejde-196	233	9	the	the	DET
ejde-196	233	10	total	total	ADJ
ejde-196	233	11	distance	distance	NOUN
ejde-196	233	12	traveled	travel	VERB
ejde-196	233	13	by	by	ADP
ejde-196	233	14	{	{	PUNCT
ejde-196	233	15	vi}i∈i	vi}i∈i	INTJ
ejde-196	233	16	is	be	AUX
ejde-196	233	17	finite	finite	PROPN
ejde-196	233	18	.	.	PUNCT
ejde-196	234	1	note	note	VERB
ejde-196	234	2	that	that	SCONJ
ejde-196	234	3	by	by	ADP
ejde-196	234	4	(	(	PUNCT
ejde-196	234	5	a	a	X
ejde-196	234	6	)	)	PUNCT
ejde-196	234	7	,	,	PUNCT
ejde-196	234	8	we	we	PRON
ejde-196	234	9	have	have	VERB
ejde-196	234	10	‖vk+1	‖vk+1	PUNCT
ejde-196	235	1	−	−	NUM
ejde-196	235	2	vk‖	vk‖	PROPN
ejde-196	235	3	≤	≤	PROPN
ejde-196	235	4	√	√	NUM
ejde-196	235	5	2sk‖dk‖√	2sk‖dk‖√	NUM
ejde-196	235	6	1	1	NUM
ejde-196	236	1	+	+	CCONJ
ejde-196	236	2	(	(	PUNCT
ejde-196	236	3	sk‖dk‖)2	sk‖dk‖)2	X
ejde-196	236	4	→	→	SYM
ejde-196	236	5	0	0	NUM
ejde-196	236	6	.	.	PUNCT
ejde-196	237	1	(	(	PUNCT
ejde-196	237	2	4.5	4.5	NUM
ejde-196	237	3	)	)	PUNCT
ejde-196	237	4	suppose	suppose	VERB
ejde-196	237	5	there	there	PRON
ejde-196	237	6	is	be	VERB
ejde-196	237	7	δ3	δ3	PROPN
ejde-196	237	8	>	>	X
ejde-196	237	9	0	0	NUM
ejde-196	237	10	such	such	ADJ
ejde-196	237	11	that	that	SCONJ
ejde-196	237	12	there	there	PRON
ejde-196	237	13	are	be	VERB
ejde-196	237	14	infinitely	infinitely	ADV
ejde-196	237	15	many	many	ADJ
ejde-196	237	16	points	point	NOUN
ejde-196	237	17	v	v	NOUN
ejde-196	237	18	in	in	ADP
ejde-196	237	19	{	{	PUNCT
ejde-196	237	20	vk	vk	INTJ
ejde-196	237	21	}	}	PUNCT
ejde-196	237	22	with	with	ADP
ejde-196	237	23	‖v	‖v	PROPN
ejde-196	237	24	−	−	PROPN
ejde-196	237	25	v∗‖	v∗‖	PROPN
ejde-196	237	26	>	>	X
ejde-196	237	27	δ3	δ3	PROPN
ejde-196	237	28	.	.	PUNCT
ejde-196	238	1	by	by	ADP
ejde-196	238	2	the	the	DET
ejde-196	238	3	inequality	inequality	NOUN
ejde-196	238	4	(	(	PUNCT
ejde-196	238	5	4.5	4.5	NUM
ejde-196	238	6	)	)	PUNCT
ejde-196	238	7	,	,	PUNCT
ejde-196	238	8	for	for	ADP
ejde-196	238	9	any	any	DET
ejde-196	238	10	0	0	NUM
ejde-196	238	11	<	<	X
ejde-196	238	12	δ1	δ1	NOUN
ejde-196	238	13	<	<	X
ejde-196	238	14	δ2	δ2	PROPN
ejde-196	238	15	<	<	X
ejde-196	238	16	δ3	δ3	PROPN
ejde-196	238	17	,	,	PUNCT
ejde-196	238	18	there	there	PRON
ejde-196	238	19	is	be	VERB
ejde-196	238	20	k	k	PROPN
ejde-196	238	21	>	>	X
ejde-196	238	22	0	0	NUM
ejde-196	239	1	such	such	ADJ
ejde-196	239	2	that	that	SCONJ
ejde-196	239	3	when	when	SCONJ
ejde-196	239	4	k	k	PROPN
ejde-196	239	5	>	>	X
ejde-196	239	6	k	k	PROPN
ejde-196	239	7	,	,	PUNCT
ejde-196	239	8	‖vk+1−	‖vk+1−	PROPN
ejde-196	239	9	vk‖	vk‖	NOUN
ejde-196	239	10	<	<	X
ejde-196	239	11	1	1	NUM
ejde-196	239	12	2	2	NUM
ejde-196	239	13	(	(	PUNCT
ejde-196	239	14	δ2−	δ2−	NOUN
ejde-196	239	15	δ1	δ1	NOUN
ejde-196	239	16	)	)	PUNCT
ejde-196	239	17	.	.	PUNCT
ejde-196	240	1	since	since	SCONJ
ejde-196	240	2	v∗	v∗	PROPN
ejde-196	240	3	is	be	AUX
ejde-196	240	4	a	a	DET
ejde-196	240	5	limit	limit	NOUN
ejde-196	240	6	point	point	NOUN
ejde-196	240	7	of	of	ADP
ejde-196	240	8	{	{	PUNCT
ejde-196	240	9	vk	vk	PROPN
ejde-196	240	10	}	}	PUNCT
ejde-196	240	11	,	,	PUNCT
ejde-196	240	12	there	there	PRON
ejde-196	240	13	are	be	VERB
ejde-196	240	14	infinitely	infinitely	ADV
ejde-196	240	15	many	many	ADJ
ejde-196	240	16	points	point	NOUN
ejde-196	240	17	v	v	ADP
ejde-196	240	18	∈	∈	PROPN
ejde-196	240	19	{	{	PUNCT
ejde-196	240	20	vk	vk	NOUN
ejde-196	240	21	}	}	PUNCT
ejde-196	240	22	such	such	ADJ
ejde-196	240	23	that	that	SCONJ
ejde-196	240	24	0	0	NUM
ejde-196	240	25	<	<	X
ejde-196	240	26	‖v	‖v	PROPN
ejde-196	240	27	−	−	PROPN
ejde-196	240	28	v∗‖	v∗‖	PROPN
ejde-196	240	29	<	<	X
ejde-196	240	30	δ1	δ1	NOUN
ejde-196	240	31	and	and	CCONJ
ejde-196	240	32	there	there	PRON
ejde-196	240	33	are	be	VERB
ejde-196	240	34	also	also	ADV
ejde-196	240	35	infinitely	infinitely	ADV
ejde-196	240	36	many	many	ADJ
ejde-196	240	37	points	point	NOUN
ejde-196	240	38	{	{	PUNCT
ejde-196	240	39	vki	vki	NOUN
ejde-196	240	40	}	}	PUNCT
ejde-196	240	41	⊂	⊂	PROPN
ejde-196	240	42	{	{	PUNCT
ejde-196	240	43	vk	vk	PROPN
ejde-196	240	44	}	}	PUNCT
ejde-196	240	45	such	such	ADJ
ejde-196	240	46	that	that	SCONJ
ejde-196	240	47	0	0	NUM
ejde-196	240	48	<	<	X
ejde-196	240	49	δ1	δ1	NOUN
ejde-196	240	50	<	<	X
ejde-196	240	51	‖vki	‖vki	NOUN
ejde-196	240	52	−	−	PROPN
ejde-196	240	53	v∗‖	v∗‖	PROPN
ejde-196	240	54	<	<	X
ejde-196	240	55	δ2	δ2	PROPN
ejde-196	240	56	,	,	PUNCT
ejde-196	240	57	i.e.	i.e.	X
ejde-196	240	58	,	,	PUNCT
ejde-196	240	59	the	the	DET
ejde-196	240	60	sequence	sequence	NOUN
ejde-196	240	61	{	{	PUNCT
ejde-196	240	62	vk	vk	PROPN
ejde-196	240	63	}	}	PUNCT
ejde-196	240	64	enters	enter	VERB
ejde-196	240	65	the	the	DET
ejde-196	240	66	three	three	NUM
ejde-196	240	67	ring	ring	NOUN
ejde-196	240	68	regions	region	NOUN
ejde-196	240	69	centered	center	VERB
ejde-196	240	70	at	at	ADP
ejde-196	240	71	v∗	v∗	NOUN
ejde-196	240	72	and	and	CCONJ
ejde-196	240	73	defined	define	VERB
ejde-196	240	74	by	by	ADP
ejde-196	240	75	0	0	NUM
ejde-196	240	76	<	<	X
ejde-196	240	77	δ1	δ1	NOUN
ejde-196	240	78	<	<	X
ejde-196	240	79	δ2	δ2	PROPN
ejde-196	240	80	<	<	X
ejde-196	240	81	δ3	δ3	PROPN
ejde-196	240	82	infinitely	infinitely	ADV
ejde-196	240	83	many	many	ADJ
ejde-196	240	84	times	time	NOUN
ejde-196	240	85	.	.	PUNCT
ejde-196	241	1	thus	thus	ADV
ejde-196	241	2	the	the	DET
ejde-196	241	3	total	total	ADJ
ejde-196	241	4	distance	distance	NOUN
ejde-196	241	5	traveled	travel	VERB
ejde-196	241	6	by	by	ADP
ejde-196	241	7	such	such	ADJ
ejde-196	241	8	{	{	PUNCT
ejde-196	241	9	vki	vki	NOUN
ejde-196	241	10	}	}	PUNCT
ejde-196	241	11	has	have	VERB
ejde-196	241	12	to	to	PART
ejde-196	241	13	be	be	AUX
ejde-196	241	14	infinite	infinite	ADJ
ejde-196	241	15	.	.	PUNCT
ejde-196	242	1	however	however	ADV
ejde-196	242	2	by	by	ADP
ejde-196	242	3	(	(	PUNCT
ejde-196	242	4	4.4	4.4	NUM
ejde-196	242	5	)	)	PUNCT
ejde-196	242	6	,	,	PUNCT
ejde-196	242	7	for	for	ADP
ejde-196	242	8	any	any	DET
ejde-196	242	9	η	η	PROPN
ejde-196	242	10	>	>	X
ejde-196	242	11	0	0	PROPN
ejde-196	242	12	,	,	PUNCT
ejde-196	242	13	the	the	DET
ejde-196	242	14	total	total	ADJ
ejde-196	242	15	distance	distance	NOUN
ejde-196	242	16	traveled	travel	VERB
ejde-196	242	17	by	by	ADP
ejde-196	242	18	all	all	DET
ejde-196	242	19	the	the	DET
ejde-196	242	20	points	point	NOUN
ejde-196	242	21	vk	vk	VERB
ejde-196	242	22	′	′	NUM
ejde-196	243	1	i	i	PRON
ejde-196	243	2	∈	∈	PROPN
ejde-196	243	3	{	{	PUNCT
ejde-196	243	4	vk	vk	NOUN
ejde-196	243	5	}	}	PUNCT
ejde-196	243	6	with	with	ADP
ejde-196	243	7	‖j	‖j	PRON
ejde-196	243	8	′(p(vk′i))‖	′(p(vk′i))‖	X
ejde-196	243	9	>	>	X
ejde-196	243	10	η	η	PROPN
ejde-196	243	11	is	be	AUX
ejde-196	243	12	finite	finite	ADJ
ejde-196	243	13	.	.	PUNCT
ejde-196	244	1	thus	thus	ADV
ejde-196	244	2	there	there	PRON
ejde-196	244	3	must	must	AUX
ejde-196	244	4	be	be	AUX
ejde-196	244	5	infinitely	infinitely	ADV
ejde-196	244	6	many	many	ADJ
ejde-196	244	7	points	point	NOUN
ejde-196	244	8	{	{	PUNCT
ejde-196	244	9	vki′	vki′	NUM
ejde-196	244	10	}	}	PUNCT
ejde-196	244	11	⊂	⊂	PROPN
ejde-196	244	12	{	{	PUNCT
ejde-196	244	13	vki	vki	NOUN
ejde-196	244	14	}	}	PUNCT
ejde-196	244	15	such	such	ADJ
ejde-196	244	16	that	that	SCONJ
ejde-196	244	17	j	j	PROPN
ejde-196	244	18	′(p(vki′	′(p(vki′	ADV
ejde-196	244	19	)	)	PUNCT
ejde-196	244	20	)	)	PUNCT
ejde-196	245	1	→	→	SYM
ejde-196	245	2	0	0	X
ejde-196	245	3	.	.	PUNCT
ejde-196	245	4	by	by	ADP
ejde-196	245	5	the	the	DET
ejde-196	245	6	ps	ps	NOUN
ejde-196	245	7	condition	condition	NOUN
ejde-196	245	8	,	,	PUNCT
ejde-196	245	9	there	there	PRON
ejde-196	245	10	is	be	VERB
ejde-196	245	11	a	a	DET
ejde-196	245	12	subsequence	subsequence	NOUN
ejde-196	245	13	,	,	PUNCT
ejde-196	245	14	denoted	denote	VERB
ejde-196	245	15	by	by	ADP
ejde-196	245	16	{	{	PUNCT
ejde-196	245	17	wki′	wki′	NOUN
ejde-196	245	18	}	}	PUNCT
ejde-196	245	19	=	=	SYM
ejde-196	245	20	{	{	PUNCT
ejde-196	245	21	p(vki′	p(vki′	NOUN
ejde-196	245	22	)	)	PUNCT
ejde-196	245	23	}	}	PUNCT
ejde-196	245	24	again	again	ADV
ejde-196	245	25	,	,	PUNCT
ejde-196	245	26	such	such	ADJ
ejde-196	245	27	that	that	SCONJ
ejde-196	245	28	wki′	wki′	PROPN
ejde-196	245	29	→	→	SYM
ejde-196	245	30	w′	w′	X
ejde-196	245	31	=	=	PUNCT
ejde-196	246	1	t′v′	t′v′	PROPN
ejde-196	246	2	+	+	CCONJ
ejde-196	246	3	w′l	w′l	NOUN
ejde-196	246	4	for	for	ADP
ejde-196	246	5	some	some	DET
ejde-196	246	6	t′	t′	NUM
ejde-196	246	7	>	>	SYM
ejde-196	246	8	0	0	NUM
ejde-196	246	9	,	,	PUNCT
ejde-196	246	10	v′	v′	NOUN
ejde-196	246	11	∈	∈	NOUN
ejde-196	246	12	sl⊥	sl⊥	PROPN
ejde-196	246	13	,	,	PUNCT
ejde-196	246	14	w′l	w′l	NOUN
ejde-196	246	15	∈	∈	PROPN
ejde-196	246	16	l	l	NOUN
ejde-196	246	17	with	with	ADP
ejde-196	246	18	j	j	PROPN
ejde-196	246	19	′(w′	′(w′	PUNCT
ejde-196	246	20	)	)	PUNCT
ejde-196	246	21	=	=	SYM
ejde-196	246	22	0	0	NUM
ejde-196	246	23	and	and	CCONJ
ejde-196	246	24	δ1	δ1	VERB
ejde-196	246	25	≤	≤	NUM
ejde-196	246	26	‖v∗	‖v∗	ADP
ejde-196	246	27	−	−	PROPN
ejde-196	246	28	v′‖	v′‖	ADP
ejde-196	246	29	≤	≤	ADV
ejde-196	246	30	δ2	δ2	VERB
ejde-196	246	31	.	.	PUNCT
ejde-196	247	1	since	since	SCONJ
ejde-196	247	2	0	0	NUM
ejde-196	247	3	<	<	X
ejde-196	247	4	δ1	δ1	NOUN
ejde-196	247	5	<	<	X
ejde-196	247	6	δ2	δ2	PROPN
ejde-196	247	7	can	can	AUX
ejde-196	247	8	be	be	AUX
ejde-196	247	9	any	any	DET
ejde-196	247	10	numbers	number	NOUN
ejde-196	247	11	less	less	ADJ
ejde-196	247	12	than	than	ADP
ejde-196	247	13	δ3	δ3	PROPN
ejde-196	247	14	,	,	PUNCT
ejde-196	247	15	this	this	PRON
ejde-196	247	16	contradicts	contradict	VERB
ejde-196	247	17	to	to	ADP
ejde-196	247	18	the	the	DET
ejde-196	247	19	assumption	assumption	NOUN
ejde-196	247	20	that	that	SCONJ
ejde-196	247	21	w∗	w∗	NOUN
ejde-196	247	22	=	=	SYM
ejde-196	247	23	p(v∗	p(v∗	PRON
ejde-196	247	24	)	)	PUNCT
ejde-196	247	25	is	be	AUX
ejde-196	247	26	an	an	DET
ejde-196	247	27	isolated	isolated	ADJ
ejde-196	247	28	critical	critical	ADJ
ejde-196	247	29	point	point	NOUN
ejde-196	247	30	.	.	PUNCT
ejde-196	248	1	thus	thus	ADV
ejde-196	248	2	for	for	ADP
ejde-196	248	3	any	any	DET
ejde-196	248	4	δ3	δ3	PROPN
ejde-196	248	5	>	>	X
ejde-196	248	6	0	0	NUM
ejde-196	248	7	,	,	PUNCT
ejde-196	248	8	there	there	PRON
ejde-196	248	9	can	can	AUX
ejde-196	248	10	be	be	AUX
ejde-196	248	11	at	at	ADP
ejde-196	248	12	most	most	ADJ
ejde-196	248	13	a	a	DET
ejde-196	248	14	finite	finite	ADJ
ejde-196	248	15	number	number	NOUN
ejde-196	248	16	of	of	ADP
ejde-196	248	17	points	point	NOUN
ejde-196	248	18	v	v	NOUN
ejde-196	248	19	in	in	ADP
ejde-196	248	20	{	{	PUNCT
ejde-196	248	21	vk	vk	INTJ
ejde-196	248	22	}	}	PUNCT
ejde-196	248	23	with	with	ADP
ejde-196	248	24	‖v∗	‖v∗	PROPN
ejde-196	248	25	−	−	PROPN
ejde-196	248	26	v‖	v‖	PROPN
ejde-196	248	27	>	>	X
ejde-196	248	28	δ3	δ3	PROPN
ejde-196	248	29	,	,	PUNCT
ejde-196	248	30	i.e.	i.e.	X
ejde-196	248	31	,	,	PUNCT
ejde-196	248	32	vk	vk	X
ejde-196	248	33	→	→	SYM
ejde-196	248	34	v∗.	v∗.	PROPN
ejde-196	248	35	(	(	PUNCT
ejde-196	248	36	c	c	X
ejde-196	248	37	)	)	PUNCT
ejde-196	248	38	is	be	AUX
ejde-196	248	39	proved	prove	VERB
ejde-196	248	40	.	.	PUNCT
ejde-196	249	1	this	this	PRON
ejde-196	249	2	completes	complete	VERB
ejde-196	249	3	the	the	DET
ejde-196	249	4	proof	proof	NOUN
ejde-196	249	5	of	of	ADP
ejde-196	249	6	the	the	DET
ejde-196	249	7	theorem	theorem	PROPN
ejde-196	249	8	.	.	PUNCT
ejde-196	249	9	�	�	PROPN
ejde-196	249	10	5	5	NUM
ejde-196	249	11	.	.	PUNCT
ejde-196	249	12	two	two	NUM
ejde-196	249	13	new	new	ADJ
ejde-196	249	14	variations	variation	NOUN
ejde-196	249	15	to	to	ADP
ejde-196	249	16	the	the	DET
ejde-196	249	17	min	min	ADJ
ejde-196	249	18	-	-	ADJ
ejde-196	249	19	orthogonal	orthogonal	ADJ
ejde-196	249	20	method	method	NOUN
ejde-196	249	21	the	the	DET
ejde-196	249	22	local	local	ADJ
ejde-196	249	23	min	min	ADJ
ejde-196	249	24	-	-	ADJ
ejde-196	249	25	orthogonal	orthogonal	ADJ
ejde-196	249	26	method	method	NOUN
ejde-196	249	27	is	be	AUX
ejde-196	249	28	first	first	ADV
ejde-196	249	29	introduced	introduce	VERB
ejde-196	249	30	in	in	ADP
ejde-196	249	31	[	[	X
ejde-196	249	32	16	16	NUM
ejde-196	249	33	]	]	PUNCT
ejde-196	249	34	as	as	ADP
ejde-196	249	35	a	a	DET
ejde-196	249	36	theoretical	theoretical	ADJ
ejde-196	249	37	extension	extension	NOUN
ejde-196	249	38	of	of	ADP
ejde-196	249	39	lmm	lmm	PROPN
ejde-196	249	40	for	for	ADP
ejde-196	249	41	solving	solve	VERB
ejde-196	249	42	m	m	NOUN
ejde-196	249	43	-	-	PUNCT
ejde-196	249	44	type	type	NOUN
ejde-196	249	45	multiple	multiple	ADJ
ejde-196	249	46	solution	solution	NOUN
ejde-196	249	47	problems	problem	NOUN
ejde-196	249	48	.	.	PUNCT
ejde-196	250	1	next	next	ADV
ejde-196	250	2	we	we	PRON
ejde-196	250	3	show	show	VERB
ejde-196	250	4	that	that	SCONJ
ejde-196	250	5	this	this	DET
ejde-196	250	6	method	method	NOUN
ejde-196	250	7	turns	turn	VERB
ejde-196	250	8	out	out	ADP
ejde-196	250	9	to	to	PART
ejde-196	250	10	be	be	AUX
ejde-196	250	11	quite	quite	ADV
ejde-196	250	12	flexible	flexible	ADJ
ejde-196	250	13	.	.	PUNCT
ejde-196	251	1	it	it	PRON
ejde-196	251	2	can	can	AUX
ejde-196	251	3	be	be	AUX
ejde-196	251	4	actually	actually	ADV
ejde-196	251	5	modified	modify	VERB
ejde-196	251	6	to	to	PART
ejde-196	251	7	solve	solve	VERB
ejde-196	251	8	w	w	NOUN
ejde-196	251	9	-	-	PUNCT
ejde-196	251	10	type	type	NOUN
ejde-196	251	11	or	or	CCONJ
ejde-196	251	12	even	even	ADV
ejde-196	251	13	mixed	mixed	ADJ
ejde-196	251	14	m	m	PROPN
ejde-196	251	15	-	-	PUNCT
ejde-196	251	16	w	w	NOUN
ejde-196	251	17	type	type	NOUN
ejde-196	251	18	problems	problem	NOUN
ejde-196	251	19	for	for	ADP
ejde-196	251	20	multiple	multiple	ADJ
ejde-196	251	21	solutions	solution	NOUN
ejde-196	251	22	.	.	PUNCT
ejde-196	252	1	first	first	ADV
ejde-196	252	2	we	we	PRON
ejde-196	252	3	observe	observe	VERB
ejde-196	252	4	closely	closely	ADV
ejde-196	252	5	the	the	DET
ejde-196	252	6	profile	profile	NOUN
ejde-196	252	7	of	of	ADP
ejde-196	252	8	a	a	DET
ejde-196	252	9	typical	typical	ADJ
ejde-196	252	10	w	w	NOUN
ejde-196	252	11	-	-	PUNCT
ejde-196	252	12	type	type	NOUN
ejde-196	252	13	functional	functional	ADJ
ejde-196	252	14	j	j	NOUN
ejde-196	252	15	in	in	ADP
ejde-196	252	16	figure	figure	NOUN
ejde-196	252	17	1	1	NUM
ejde-196	252	18	,	,	PUNCT
ejde-196	252	19	we	we	PRON
ejde-196	252	20	found	find	VERB
ejde-196	252	21	that	that	SCONJ
ejde-196	252	22	(	(	PUNCT
ejde-196	252	23	1	1	NUM
ejde-196	252	24	)	)	PUNCT
ejde-196	252	25	0	0	NUM
ejde-196	252	26	is	be	AUX
ejde-196	252	27	the	the	DET
ejde-196	252	28	only	only	ADJ
ejde-196	252	29	k	k	ADJ
ejde-196	252	30	-	-	ADJ
ejde-196	252	31	saddle	saddle	NOUN
ejde-196	252	32	and	and	CCONJ
ejde-196	252	33	all	all	DET
ejde-196	252	34	other	other	ADJ
ejde-196	252	35	critical	critical	ADJ
ejde-196	252	36	points	point	NOUN
ejde-196	252	37	must	must	AUX
ejde-196	252	38	have	have	VERB
ejde-196	252	39	saddle	saddle	NOUN
ejde-196	252	40	index	index	NOUN
ejde-196	252	41	less	less	ADJ
ejde-196	252	42	than	than	ADP
ejde-196	252	43	k	k	NOUN
ejde-196	252	44	;	;	PUNCT
ejde-196	252	45	ejde-2023	ejde-2023	ADJ
ejde-196	252	46	/	/	SYM
ejde-196	252	47	si/02	si/02	ADJ
ejde-196	252	48	short	short	ADJ
ejde-196	252	49	title	title	NOUN
ejde-196	252	50	183	183	NUM
ejde-196	252	51	(	(	PUNCT
ejde-196	252	52	2	2	NUM
ejde-196	252	53	)	)	PUNCT
ejde-196	252	54	for	for	SCONJ
ejde-196	252	55	any	any	DET
ejde-196	252	56	nontrivial	nontrivial	ADJ
ejde-196	252	57	solution	solution	NOUN
ejde-196	252	58	u∗	u∗	ADV
ejde-196	252	59	,	,	PUNCT
ejde-196	252	60	it	it	PRON
ejde-196	252	61	must	must	AUX
ejde-196	252	62	be	be	AUX
ejde-196	252	63	a	a	DET
ejde-196	252	64	local	local	ADJ
ejde-196	252	65	minimum	minimum	NOUN
ejde-196	252	66	point	point	NOUN
ejde-196	252	67	of	of	ADP
ejde-196	252	68	j	j	PROPN
ejde-196	252	69	along	along	ADP
ejde-196	252	70	its	its	PRON
ejde-196	252	71	direction	direction	NOUN
ejde-196	252	72	with	with	ADP
ejde-196	252	73	j(u∗	j(u∗	NOUN
ejde-196	252	74	)	)	PUNCT
ejde-196	252	75	<	<	X
ejde-196	253	1	j(0	j(0	PROPN
ejde-196	253	2	)	)	PUNCT
ejde-196	254	1	=	=	SYM
ejde-196	254	2	0	0	X
ejde-196	254	3	.	.	PUNCT
ejde-196	254	4	based	base	VERB
ejde-196	254	5	on	on	ADP
ejde-196	254	6	the	the	DET
ejde-196	254	7	variational	variational	ADJ
ejde-196	254	8	structures	structure	NOUN
ejde-196	254	9	we	we	PRON
ejde-196	254	10	observed	observe	VERB
ejde-196	254	11	,	,	PUNCT
ejde-196	254	12	we	we	PRON
ejde-196	254	13	propose	propose	VERB
ejde-196	254	14	two	two	NUM
ejde-196	254	15	new	new	ADJ
ejde-196	254	16	variations	variation	NOUN
ejde-196	254	17	of	of	ADP
ejde-196	254	18	the	the	DET
ejde-196	254	19	min	min	ADJ
ejde-196	254	20	-	-	ADJ
ejde-196	254	21	orthogonal	orthogonal	ADJ
ejde-196	254	22	method	method	NOUN
ejde-196	254	23	:	:	PUNCT
ejde-196	254	24	(	(	PUNCT
ejde-196	254	25	a1	a1	NOUN
ejde-196	254	26	)	)	PUNCT
ejde-196	254	27	a	a	DET
ejde-196	254	28	min	min	PROPN
ejde-196	254	29	-	-	PUNCT
ejde-196	254	30	min	min	ADJ
ejde-196	254	31	-	-	ADJ
ejde-196	254	32	max	max	NOUN
ejde-196	254	33	algorithm	algorithm	NOUN
ejde-196	254	34	.	.	PUNCT
ejde-196	255	1	(	(	PUNCT
ejde-196	255	2	this	this	DET
ejde-196	255	3	method	method	NOUN
ejde-196	255	4	was	be	AUX
ejde-196	255	5	proposed	propose	VERB
ejde-196	255	6	by	by	ADP
ejde-196	255	7	dr	dr	PROPN
ejde-196	255	8	.	.	PROPN
ejde-196	255	9	zhi	zhi	PROPN
ejde-196	255	10	-	-	PROPN
ejde-196	255	11	qiang	qiang	PROPN
ejde-196	255	12	wang	wang	PROPN
ejde-196	255	13	in	in	ADP
ejde-196	255	14	2012	2012	NUM
ejde-196	255	15	in	in	ADP
ejde-196	255	16	a	a	DET
ejde-196	255	17	discussion	discussion	NOUN
ejde-196	255	18	.	.	PUNCT
ejde-196	255	19	)	)	PUNCT
ejde-196	256	1	min	min	PROPN
ejde-196	256	2	v∈s	v∈s	ADJ
ejde-196	256	3	l⊥	l⊥	PROPN
ejde-196	256	4	min	min	PROPN
ejde-196	256	5	r>0	r>0	PROPN
ejde-196	256	6	max	max	PROPN
ejde-196	256	7	u∈[v	u∈[v	PROPN
ejde-196	256	8	,	,	PUNCT
ejde-196	256	9	l],‖u‖=r	l],‖u‖=r	PROPN
ejde-196	256	10	j(u	j(u	PROPN
ejde-196	256	11	)	)	PUNCT
ejde-196	256	12	.	.	PUNCT
ejde-196	257	1	(	(	PUNCT
ejde-196	257	2	a2	a2	PROPN
ejde-196	257	3	)	)	PUNCT
ejde-196	257	4	a	a	DET
ejde-196	257	5	min	min	PROPN
ejde-196	257	6	-	-	PUNCT
ejde-196	257	7	max	max	ADJ
ejde-196	257	8	-	-	PUNCT
ejde-196	257	9	min	min	NOUN
ejde-196	257	10	algorithm	algorithm	NOUN
ejde-196	257	11	.	.	PUNCT
ejde-196	258	1	min	min	PROPN
ejde-196	258	2	v∈s	v∈s	ADJ
ejde-196	258	3	l⊥	l⊥	NOUN
ejde-196	258	4	max	max	PROPN
ejde-196	258	5	u∈[v	u∈[v	PROPN
ejde-196	258	6	,	,	PUNCT
ejde-196	258	7	l],‖u‖≈1	l],‖u‖≈1	PROPN
ejde-196	258	8	min	min	PROPN
ejde-196	258	9	t>0	t>0	PROPN
ejde-196	258	10	j(tu	j(tu	PROPN
ejde-196	258	11	)	)	PUNCT
ejde-196	258	12	.	.	PUNCT
ejde-196	259	1	note	note	VERB
ejde-196	259	2	that	that	SCONJ
ejde-196	259	3	those	those	DET
ejde-196	259	4	two	two	NUM
ejde-196	259	5	algorithms	algorithm	NOUN
ejde-196	259	6	start	start	VERB
ejde-196	259	7	with	with	ADP
ejde-196	259	8	l	l	NOUN
ejde-196	259	9	=	=	PUNCT
ejde-196	259	10	{	{	PUNCT
ejde-196	259	11	0	0	NUM
ejde-196	259	12	}	}	PUNCT
ejde-196	259	13	.	.	PUNCT
ejde-196	260	1	in	in	ADP
ejde-196	260	2	this	this	DET
ejde-196	260	3	case	case	NOUN
ejde-196	260	4	,	,	PUNCT
ejde-196	260	5	the	the	DET
ejde-196	260	6	maxoperation	maxoperation	NOUN
ejde-196	260	7	in	in	ADP
ejde-196	260	8	the	the	DET
ejde-196	260	9	algorithms	algorithms	NOUN
ejde-196	260	10	is	be	AUX
ejde-196	260	11	void	void	ADJ
ejde-196	260	12	and	and	CCONJ
ejde-196	260	13	the	the	DET
ejde-196	260	14	algorithms	algorithm	NOUN
ejde-196	260	15	find	find	VERB
ejde-196	260	16	a	a	DET
ejde-196	260	17	local	local	ADJ
ejde-196	260	18	minimum	minimum	NOUN
ejde-196	260	19	point	point	NOUN
ejde-196	260	20	u∗0	u∗0	NOUN
ejde-196	260	21	of	of	ADP
ejde-196	260	22	j	j	PROPN
ejde-196	260	23	.	.	PUNCT
ejde-196	261	1	then	then	ADV
ejde-196	261	2	by	by	ADP
ejde-196	261	3	setting	set	VERB
ejde-196	261	4	l	l	NOUN
ejde-196	261	5	=	=	PUNCT
ejde-196	261	6	{	{	PUNCT
ejde-196	261	7	u∗0	u∗0	ADV
ejde-196	261	8	}	}	PUNCT
ejde-196	261	9	,	,	PUNCT
ejde-196	261	10	the	the	DET
ejde-196	261	11	algorithms	algorithm	NOUN
ejde-196	261	12	find	find	VERB
ejde-196	261	13	a	a	DET
ejde-196	261	14	1	1	NUM
ejde-196	261	15	-	-	PUNCT
ejde-196	261	16	saddle	saddle	NOUN
ejde-196	261	17	,	,	PUNCT
ejde-196	261	18	.	.	PUNCT
ejde-196	261	19	.	.	PUNCT
ejde-196	261	20	.	.	PUNCT
ejde-196	262	1	,	,	PUNCT
ejde-196	263	1	etc	etc	X
ejde-196	263	2	.	.	X
ejde-196	263	3	for	for	ADP
ejde-196	263	4	each	each	DET
ejde-196	263	5	v	v	NOUN
ejde-196	263	6	∈	∈	NOUN
ejde-196	263	7	sl⊥	sl⊥	PROPN
ejde-196	263	8	,	,	PUNCT
ejde-196	263	9	we	we	PRON
ejde-196	263	10	denote	denote	VERB
ejde-196	263	11	,	,	PUNCT
ejde-196	263	12	in	in	ADP
ejde-196	263	13	the	the	DET
ejde-196	263	14	min	min	PROPN
ejde-196	263	15	-	-	PUNCT
ejde-196	263	16	min	min	ADJ
ejde-196	263	17	-	-	ADJ
ejde-196	263	18	max	max	ADJ
ejde-196	263	19	algorithm	algorithm	NOUN
ejde-196	263	20	(	(	PUNCT
ejde-196	263	21	a1	a1	NOUN
ejde-196	263	22	)	)	PUNCT
ejde-196	263	23	,	,	PUNCT
ejde-196	263	24	p(v	p(v	NOUN
ejde-196	263	25	)	)	PUNCT
ejde-196	263	26	=	=	SYM
ejde-196	263	27	arg	arg	NOUN
ejde-196	263	28	min	min	PROPN
ejde-196	263	29	r>0	r>0	PROPN
ejde-196	263	30	max	max	PROPN
ejde-196	263	31	u∈[v	u∈[v	PROPN
ejde-196	263	32	,	,	PUNCT
ejde-196	263	33	l],‖u‖=r	l],‖u‖=r	PROPN
ejde-196	263	34	j(u	j(u	PROPN
ejde-196	263	35	)	)	PUNCT
ejde-196	263	36	(	(	PUNCT
ejde-196	263	37	5.1	5.1	NUM
ejde-196	263	38	)	)	PUNCT
ejde-196	263	39	and	and	CCONJ
ejde-196	263	40	in	in	ADP
ejde-196	263	41	the	the	DET
ejde-196	263	42	min	min	PROPN
ejde-196	263	43	-	-	PUNCT
ejde-196	263	44	max	max	ADJ
ejde-196	263	45	-	-	PUNCT
ejde-196	263	46	min	min	NOUN
ejde-196	263	47	algorithm	algorithm	NOUN
ejde-196	263	48	(	(	PUNCT
ejde-196	263	49	a2	a2	PROPN
ejde-196	263	50	)	)	PUNCT
ejde-196	263	51	,	,	PUNCT
ejde-196	263	52	p(v	p(v	NOUN
ejde-196	263	53	)	)	PUNCT
ejde-196	263	54	=	=	PUNCT
ejde-196	263	55	arg	arg	NOUN
ejde-196	263	56	max	max	PROPN
ejde-196	263	57	u∈[v	u∈[v	PROPN
ejde-196	263	58	,	,	PUNCT
ejde-196	263	59	l],‖u‖≈1	l],‖u‖≈1	PROPN
ejde-196	263	60	min	min	PROPN
ejde-196	263	61	t>0	t>0	PROPN
ejde-196	263	62	j(tu	j(tu	PROPN
ejde-196	263	63	)	)	PUNCT
ejde-196	263	64	,	,	PUNCT
ejde-196	263	65	(	(	PUNCT
ejde-196	263	66	5.2	5.2	NUM
ejde-196	263	67	)	)	PUNCT
ejde-196	263	68	then	then	ADV
ejde-196	263	69	the	the	DET
ejde-196	263	70	two	two	NUM
ejde-196	263	71	algorithms	algorithm	NOUN
ejde-196	263	72	can	can	AUX
ejde-196	263	73	be	be	AUX
ejde-196	263	74	expressed	express	VERB
ejde-196	263	75	as	as	ADP
ejde-196	263	76	minv∈s	minv∈s	PROPN
ejde-196	263	77	l⊥	l⊥	PROPN
ejde-196	263	78	j(p(v	j(p(v	NOUN
ejde-196	263	79	)	)	PUNCT
ejde-196	263	80	)	)	PUNCT
ejde-196	263	81	.	.	PUNCT
ejde-196	264	1	thus	thus	ADV
ejde-196	264	2	to	to	PART
ejde-196	264	3	establish	establish	VERB
ejde-196	264	4	their	their	PRON
ejde-196	264	5	mathematical	mathematical	ADJ
ejde-196	264	6	justifications	justification	NOUN
ejde-196	264	7	by	by	ADP
ejde-196	264	8	the	the	DET
ejde-196	264	9	local	local	ADJ
ejde-196	264	10	min	min	ADJ
ejde-196	264	11	-	-	ADJ
ejde-196	264	12	orthogonal	orthogonal	ADJ
ejde-196	264	13	principle	principle	NOUN
ejde-196	264	14	,	,	PUNCT
ejde-196	264	15	we	we	PRON
ejde-196	264	16	only	only	ADV
ejde-196	264	17	have	have	VERB
ejde-196	264	18	to	to	PART
ejde-196	264	19	show	show	VERB
ejde-196	264	20	j	j	PROPN
ejde-196	264	21	′(p(v))⊥[v	′(p(v))⊥[v	PROPN
ejde-196	264	22	,	,	PUNCT
ejde-196	264	23	l	l	NOUN
ejde-196	264	24	]	]	X
ejde-196	264	25	.	.	PUNCT
ejde-196	265	1	5.1	5.1	NUM
ejde-196	265	2	.	.	PUNCT
ejde-196	265	3	justification	justification	NOUN
ejde-196	265	4	of	of	ADP
ejde-196	265	5	the	the	DET
ejde-196	265	6	min	min	PROPN
ejde-196	265	7	-	-	PUNCT
ejde-196	265	8	min	min	ADJ
ejde-196	265	9	-	-	ADJ
ejde-196	265	10	max	max	NOUN
ejde-196	265	11	algorithm	algorithm	NOUN
ejde-196	265	12	.	.	PUNCT
ejde-196	266	1	in	in	ADP
ejde-196	266	2	the	the	DET
ejde-196	266	3	min	min	PROPN
ejde-196	266	4	-	-	PUNCT
ejde-196	266	5	min	min	ADJ
ejde-196	266	6	-	-	ADJ
ejde-196	266	7	max	max	ADJ
ejde-196	266	8	algorithm	algorithm	NOUN
ejde-196	266	9	(	(	PUNCT
ejde-196	266	10	a1	a1	NOUN
ejde-196	266	11	)	)	PUNCT
ejde-196	266	12	,	,	PUNCT
ejde-196	266	13	when	when	SCONJ
ejde-196	266	14	l	l	PROPN
ejde-196	266	15	is	be	AUX
ejde-196	266	16	k	k	ADJ
ejde-196	266	17	-	-	ADJ
ejde-196	266	18	dimensional	dimensional	ADJ
ejde-196	266	19	and	and	CCONJ
ejde-196	266	20	v	v	ADP
ejde-196	266	21	∈	∈	NOUN
ejde-196	266	22	sl⊥	sl⊥	PROPN
ejde-196	266	23	is	be	AUX
ejde-196	266	24	given	give	VERB
ejde-196	266	25	,	,	PUNCT
ejde-196	266	26	choose	choose	VERB
ejde-196	266	27	an	an	DET
ejde-196	266	28	orthonormal	orthonormal	ADJ
ejde-196	266	29	basis	basis	NOUN
ejde-196	266	30	in	in	ADP
ejde-196	266	31	[	[	X
ejde-196	266	32	v	v	NOUN
ejde-196	266	33	,	,	PUNCT
ejde-196	266	34	l	l	NOUN
ejde-196	266	35	]	]	X
ejde-196	266	36	.	.	PUNCT
ejde-196	267	1	for	for	ADP
ejde-196	267	2	each	each	DET
ejde-196	267	3	u	u	NOUN
ejde-196	267	4	∈	∈	PROPN
ejde-196	267	5	[	[	X
ejde-196	267	6	v	v	NOUN
ejde-196	267	7	,	,	PUNCT
ejde-196	267	8	l	l	NOUN
ejde-196	267	9	]	]	X
ejde-196	267	10	,	,	PUNCT
ejde-196	267	11	let	let	VERB
ejde-196	267	12	(	(	PUNCT
ejde-196	267	13	r	r	NOUN
ejde-196	267	14	,	,	PUNCT
ejde-196	267	15	θ	θ	NOUN
ejde-196	267	16	)	)	PUNCT
ejde-196	267	17	=	=	SYM
ejde-196	267	18	(	(	PUNCT
ejde-196	267	19	r	r	NOUN
ejde-196	267	20	,	,	PUNCT
ejde-196	267	21	θ1	θ1	NOUN
ejde-196	267	22	,	,	PUNCT
ejde-196	267	23	.	.	PUNCT
ejde-196	267	24	.	.	PUNCT
ejde-196	267	25	.	.	PUNCT
ejde-196	268	1	,	,	PUNCT
ejde-196	268	2	θk	θk	PROPN
ejde-196	268	3	)	)	PUNCT
ejde-196	268	4	be	be	VERB
ejde-196	268	5	the	the	DET
ejde-196	268	6	k+1dimensional	k+1dimensional	ADJ
ejde-196	268	7	spherical	spherical	ADJ
ejde-196	268	8	coordinates	coordinate	NOUN
ejde-196	268	9	of	of	ADP
ejde-196	268	10	u	u	NOUN
ejde-196	268	11	in	in	ADP
ejde-196	268	12	[	[	X
ejde-196	268	13	v	v	NOUN
ejde-196	268	14	,	,	PUNCT
ejde-196	268	15	l	l	NOUN
ejde-196	268	16	]	]	PUNCT
ejde-196	268	17	.	.	PUNCT
ejde-196	269	1	denote	denote	VERB
ejde-196	269	2	χ	χ	X
ejde-196	269	3	:	:	PUNCT
ejde-196	269	4	(	(	PUNCT
ejde-196	269	5	r	r	NOUN
ejde-196	269	6	,	,	PUNCT
ejde-196	269	7	θ	θ	NOUN
ejde-196	269	8	)	)	PUNCT
ejde-196	269	9	→	→	SYM
ejde-196	269	10	u	u	NOUN
ejde-196	269	11	=	=	SYM
ejde-196	269	12	χ(r	χ(r	PROPN
ejde-196	269	13	,	,	PUNCT
ejde-196	269	14	θ	θ	NOUN
ejde-196	269	15	)	)	PUNCT
ejde-196	269	16	the	the	DET
ejde-196	269	17	coordinate	coordinate	NOUN
ejde-196	269	18	transformation	transformation	NOUN
ejde-196	269	19	from	from	ADP
ejde-196	269	20	the	the	DET
ejde-196	269	21	spherical	spherical	ADJ
ejde-196	269	22	coordinates	coordinate	NOUN
ejde-196	269	23	to	to	ADP
ejde-196	269	24	[	[	X
ejde-196	269	25	v	v	ADP
ejde-196	269	26	,	,	PUNCT
ejde-196	269	27	l	l	NOUN
ejde-196	269	28	]	]	PUNCT
ejde-196	269	29	with	with	ADP
ejde-196	269	30	respect	respect	NOUN
ejde-196	269	31	to	to	ADP
ejde-196	269	32	the	the	DET
ejde-196	269	33	orthonormal	orthonormal	ADJ
ejde-196	269	34	basis	basis	NOUN
ejde-196	269	35	.	.	PUNCT
ejde-196	270	1	we	we	PRON
ejde-196	270	2	denote	denote	VERB
ejde-196	270	3	u(r	u(r	PROPN
ejde-196	270	4	,	,	PUNCT
ejde-196	270	5	θ	θ	X
ejde-196	270	6	)	)	PUNCT
ejde-196	270	7	the	the	DET
ejde-196	270	8	spherical	spherical	ADJ
ejde-196	270	9	coordinates	coordinate	NOUN
ejde-196	270	10	of	of	ADP
ejde-196	270	11	u.	u.	PROPN
ejde-196	270	12	since	since	SCONJ
ejde-196	270	13	p(v	p(v	NOUN
ejde-196	270	14	)	)	PUNCT
ejde-196	270	15	∈	∈	PROPN
ejde-196	271	1	[	[	X
ejde-196	271	2	v	v	NOUN
ejde-196	271	3	,	,	PUNCT
ejde-196	271	4	l	l	NOUN
ejde-196	271	5	]	]	X
ejde-196	271	6	,	,	PUNCT
ejde-196	271	7	to	to	PART
ejde-196	271	8	prove	prove	VERB
ejde-196	271	9	j	j	PROPN
ejde-196	271	10	′(p(v))⊥[v	′(p(v))⊥[v	PROPN
ejde-196	271	11	,	,	PUNCT
ejde-196	271	12	l	l	PROPN
ejde-196	271	13	]	]	X
ejde-196	271	14	,	,	PUNCT
ejde-196	271	15	we	we	PRON
ejde-196	271	16	consider	consider	VERB
ejde-196	271	17	js(r	js(r	NOUN
ejde-196	271	18	,	,	PUNCT
ejde-196	271	19	θ	θ	NOUN
ejde-196	271	20	)	)	PUNCT
ejde-196	271	21	=	=	SYM
ejde-196	271	22	j(χ(r	j(χ(r	PROPN
ejde-196	271	23	,	,	PUNCT
ejde-196	271	24	θ	θ	NOUN
ejde-196	271	25	)	)	PUNCT
ejde-196	271	26	)	)	PUNCT
ejde-196	272	1	=	=	SYM
ejde-196	272	2	j(u	j(u	PROPN
ejde-196	272	3	)	)	PUNCT
ejde-196	272	4	.	.	PUNCT
ejde-196	273	1	denote	denote	NOUN
ejde-196	273	2	,	,	PUNCT
ejde-196	273	3	for	for	ADP
ejde-196	273	4	each	each	DET
ejde-196	273	5	r	r	NOUN
ejde-196	273	6	>	>	X
ejde-196	273	7	0	0	NUM
ejde-196	273	8	,	,	PUNCT
ejde-196	273	9	θ(r	θ(r	X
ejde-196	273	10	)	)	PUNCT
ejde-196	274	1	=	=	PUNCT
ejde-196	274	2	arg	arg	NOUN
ejde-196	274	3	max	max	PROPN
ejde-196	274	4	θ	θ	PROPN
ejde-196	274	5	js(r	js(r	PROPN
ejde-196	274	6	,	,	PUNCT
ejde-196	274	7	θ	θ	NOUN
ejde-196	274	8	)	)	PUNCT
ejde-196	274	9	(	(	PUNCT
ejde-196	274	10	5.3	5.3	NUM
ejde-196	274	11	)	)	PUNCT
ejde-196	274	12	and	and	CCONJ
ejde-196	274	13	u	u	X
ejde-196	274	14	=	=	PROPN
ejde-196	274	15	p(v	p(v	NOUN
ejde-196	274	16	)	)	PUNCT
ejde-196	274	17	in	in	ADP
ejde-196	274	18	the	the	DET
ejde-196	274	19	spherical	spherical	ADJ
ejde-196	274	20	coordinates	coordinate	NOUN
ejde-196	274	21	u(rv	u(rv	NOUN
ejde-196	274	22	,	,	PUNCT
ejde-196	274	23	θ(rv	θ(rv	NUM
ejde-196	274	24	)	)	PUNCT
ejde-196	274	25	)	)	PUNCT
ejde-196	275	1	=	=	PUNCT
ejde-196	275	2	arg	arg	NOUN
ejde-196	275	3	min	min	PROPN
ejde-196	275	4	r>0	r>0	PROPN
ejde-196	275	5	js(r	js(r	PROPN
ejde-196	275	6	,	,	PUNCT
ejde-196	275	7	θ(r	θ(r	NOUN
ejde-196	275	8	)	)	PUNCT
ejde-196	275	9	)	)	PUNCT
ejde-196	275	10	.	.	PUNCT
ejde-196	276	1	(	(	PUNCT
ejde-196	276	2	5.4	5.4	NUM
ejde-196	276	3	)	)	PUNCT
ejde-196	276	4	assume	assume	VERB
ejde-196	276	5	θ(r	θ(r	X
ejde-196	276	6	)	)	PUNCT
ejde-196	276	7	is	be	AUX
ejde-196	276	8	continuous	continuous	ADJ
ejde-196	276	9	,	,	PUNCT
ejde-196	276	10	we	we	PRON
ejde-196	276	11	show	show	VERB
ejde-196	276	12	that	that	SCONJ
ejde-196	276	13	djs(rv	djs(rv	NOUN
ejde-196	276	14	,	,	PUNCT
ejde-196	276	15	θ(rv	θ(rv	NUM
ejde-196	276	16	)	)	PUNCT
ejde-196	276	17	)	)	PUNCT
ejde-196	277	1	=	=	SYM
ejde-196	277	2	(	(	PUNCT
ejde-196	277	3	jsr	jsr	PROPN
ejde-196	277	4	(	(	PUNCT
ejde-196	277	5	rv	rv	PROPN
ejde-196	277	6	,	,	PUNCT
ejde-196	277	7	θ(rv	θ(rv	NUM
ejde-196	277	8	)	)	PUNCT
ejde-196	277	9	)	)	PUNCT
ejde-196	277	10	,	,	PUNCT
ejde-196	278	1	j	j	PROPN
ejde-196	278	2	s	s	PROPN
ejde-196	278	3	θ	θ	PROPN
ejde-196	278	4	(	(	PUNCT
ejde-196	278	5	rv	rv	PROPN
ejde-196	278	6	,	,	PUNCT
ejde-196	278	7	θ(rv	θ(rv	NUM
ejde-196	278	8	)	)	PUNCT
ejde-196	278	9	)	)	PUNCT
ejde-196	278	10	)	)	PUNCT
ejde-196	279	1	=	=	PUNCT
ejde-196	279	2	(	(	PUNCT
ejde-196	279	3	0	0	NUM
ejde-196	279	4	,	,	PUNCT
ejde-196	279	5	0	0	NUM
ejde-196	279	6	,	,	PUNCT
ejde-196	279	7	.	.	PUNCT
ejde-196	279	8	.	.	PUNCT
ejde-196	280	1	.	.	PUNCT
ejde-196	281	1	,	,	PUNCT
ejde-196	281	2	0	0	NUM
ejde-196	281	3	)	)	PUNCT
ejde-196	281	4	.	.	PUNCT
ejde-196	282	1	by	by	ADP
ejde-196	282	2	(	(	PUNCT
ejde-196	282	3	5.3	5.3	NUM
ejde-196	282	4	)	)	PUNCT
ejde-196	282	5	we	we	PRON
ejde-196	282	6	have	have	AUX
ejde-196	282	7	jsθ	jsθ	AUX
ejde-196	282	8	(	(	PUNCT
ejde-196	282	9	r	r	NOUN
ejde-196	282	10	,	,	PUNCT
ejde-196	282	11	θ(r	θ(r	NOUN
ejde-196	282	12	)	)	PUNCT
ejde-196	282	13	)	)	PUNCT
ejde-196	282	14	=	=	PUNCT
ejde-196	282	15	(	(	PUNCT
ejde-196	282	16	jsθ1(r	jsθ1(r	NUM
ejde-196	282	17	,	,	PUNCT
ejde-196	282	18	θ(r	θ(r	NOUN
ejde-196	282	19	)	)	PUNCT
ejde-196	282	20	)	)	PUNCT
ejde-196	282	21	,	,	PUNCT
ejde-196	282	22	.	.	PUNCT
ejde-196	282	23	.	.	PUNCT
ejde-196	282	24	.	.	PUNCT
ejde-196	283	1	,	,	PUNCT
ejde-196	283	2	jsθk(r	jsθk(r	INTJ
ejde-196	283	3	,	,	PUNCT
ejde-196	283	4	θ(r	θ(r	NOUN
ejde-196	283	5	)	)	PUNCT
ejde-196	283	6	)	)	PUNCT
ejde-196	283	7	)	)	PUNCT
ejde-196	284	1	=	=	PUNCT
ejde-196	284	2	(	(	PUNCT
ejde-196	284	3	0	0	NUM
ejde-196	284	4	,	,	PUNCT
ejde-196	284	5	.	.	PUNCT
ejde-196	284	6	.	.	PUNCT
ejde-196	285	1	.	.	PUNCT
ejde-196	286	1	,	,	PUNCT
ejde-196	286	2	0	0	NUM
ejde-196	286	3	)	)	PUNCT
ejde-196	286	4	or	or	CCONJ
ejde-196	286	5	djs(r	djs(r	PROPN
ejde-196	286	6	,	,	PUNCT
ejde-196	286	7	θ(r	θ(r	NOUN
ejde-196	286	8	)	)	PUNCT
ejde-196	286	9	)	)	PUNCT
ejde-196	287	1	=	=	SYM
ejde-196	287	2	(	(	PUNCT
ejde-196	287	3	jsr	jsr	PROPN
ejde-196	287	4	(	(	PUNCT
ejde-196	287	5	r	r	NOUN
ejde-196	287	6	,	,	PUNCT
ejde-196	287	7	θ(r	θ(r	NOUN
ejde-196	287	8	)	)	PUNCT
ejde-196	287	9	)	)	PUNCT
ejde-196	287	10	,	,	PUNCT
ejde-196	287	11	jsθ	jsθ	PROPN
ejde-196	287	12	(	(	PUNCT
ejde-196	287	13	r	r	NOUN
ejde-196	287	14	,	,	PUNCT
ejde-196	287	15	θ(r	θ(r	NOUN
ejde-196	287	16	)	)	PUNCT
ejde-196	287	17	)	)	PUNCT
ejde-196	287	18	)	)	PUNCT
ejde-196	288	1	=	=	PRON
ejde-196	288	2	(	(	PUNCT
ejde-196	288	3	jsr	jsr	PROPN
ejde-196	288	4	(	(	PUNCT
ejde-196	288	5	r	r	NOUN
ejde-196	288	6	,	,	PUNCT
ejde-196	288	7	θ(r	θ(r	NOUN
ejde-196	288	8	)	)	PUNCT
ejde-196	288	9	)	)	PUNCT
ejde-196	288	10	,	,	PUNCT
ejde-196	288	11	0	0	NUM
ejde-196	288	12	,	,	PUNCT
ejde-196	288	13	.	.	PUNCT
ejde-196	288	14	.	.	PUNCT
ejde-196	288	15	.	.	PUNCT
ejde-196	289	1	,	,	PUNCT
ejde-196	289	2	0	0	NUM
ejde-196	289	3	)	)	PUNCT
ejde-196	289	4	.	.	PUNCT
ejde-196	290	1	suppose	suppose	VERB
ejde-196	291	1	jsr	jsr	PROPN
ejde-196	291	2	(	(	PUNCT
ejde-196	291	3	rv	rv	PROPN
ejde-196	291	4	,	,	PUNCT
ejde-196	291	5	θ(rv	θ(rv	NUM
ejde-196	291	6	)	)	PUNCT
ejde-196	291	7	)	)	PUNCT
ejde-196	291	8	6=	6=	ADP
ejde-196	291	9	0	0	X
ejde-196	291	10	.	.	X
ejde-196	291	11	denote	denote	NOUN
ejde-196	291	12	rv(s	rv(s	NOUN
ejde-196	291	13	)	)	PUNCT
ejde-196	292	1	=	=	PUNCT
ejde-196	292	2	rv	rv	PROPN
ejde-196	292	3	−	−	PROPN
ejde-196	292	4	sjsr	sjsr	ADV
ejde-196	292	5	(	(	PUNCT
ejde-196	292	6	rv	rv	NOUN
ejde-196	292	7	,	,	PUNCT
ejde-196	292	8	θ(rv	θ(rv	NUM
ejde-196	292	9	)	)	PUNCT
ejde-196	292	10	)	)	PUNCT
ejde-196	292	11	.	.	PUNCT
ejde-196	293	1	184	184	NUM
ejde-196	293	2	m.	m.	NOUN
ejde-196	293	3	li	li	PROPN
ejde-196	293	4	,	,	PUNCT
ejde-196	293	5	b.	b.	PROPN
ejde-196	293	6	ji	ji	PROPN
ejde-196	293	7	,	,	PUNCT
ejde-196	293	8	j.	j.	PROPN
ejde-196	293	9	zhou	zhou	PROPN
ejde-196	293	10	ejde	ejde	PROPN
ejde-196	293	11	/	/	SYM
ejde-196	293	12	si/02	si/02	PROPN
ejde-196	293	13	let	let	NOUN
ejde-196	293	14	(	(	PUNCT
ejde-196	293	15	rv(s	rv(s	NUM
ejde-196	293	16	)	)	PUNCT
ejde-196	293	17	,	,	PUNCT
ejde-196	293	18	θ(s	θ(s	NOUN
ejde-196	293	19	)	)	PUNCT
ejde-196	293	20	)	)	PUNCT
ejde-196	293	21	be	be	AUX
ejde-196	293	22	defined	define	VERB
ejde-196	293	23	in	in	ADP
ejde-196	293	24	(	(	PUNCT
ejde-196	293	25	5.3	5.3	NUM
ejde-196	293	26	)	)	PUNCT
ejde-196	293	27	where	where	SCONJ
ejde-196	293	28	θ(s	θ(s	NOUN
ejde-196	293	29	)	)	PUNCT
ejde-196	293	30	=	=	SYM
ejde-196	293	31	θv(rv(s	θv(rv(s	NOUN
ejde-196	293	32	)	)	PUNCT
ejde-196	293	33	)	)	PUNCT
ejde-196	293	34	.	.	PUNCT
ejde-196	294	1	for	for	ADP
ejde-196	294	2	each	each	DET
ejde-196	294	3	fixed	fix	VERB
ejde-196	294	4	s	s	PROPN
ejde-196	294	5	>	>	X
ejde-196	294	6	0	0	NUM
ejde-196	294	7	,	,	PUNCT
ejde-196	294	8	we	we	PRON
ejde-196	294	9	define	define	VERB
ejde-196	294	10	g(λ	g(λ	PROPN
ejde-196	294	11	)	)	PUNCT
ejde-196	295	1	=	=	PRON
ejde-196	295	2	js(rv	js(rv	VERB
ejde-196	295	3	−	−	PRON
ejde-196	295	4	λsjsr	λsjsr	NOUN
ejde-196	295	5	(	(	PUNCT
ejde-196	295	6	rv	rv	NOUN
ejde-196	295	7	,	,	PUNCT
ejde-196	295	8	θ(rv	θ(rv	NUM
ejde-196	295	9	)	)	PUNCT
ejde-196	295	10	)	)	PUNCT
ejde-196	295	11	,	,	PUNCT
ejde-196	295	12	θ(s	θ(s	NOUN
ejde-196	295	13	)	)	PUNCT
ejde-196	295	14	)	)	PUNCT
ejde-196	295	15	.	.	PUNCT
ejde-196	296	1	by	by	ADP
ejde-196	296	2	the	the	DET
ejde-196	296	3	mean	mean	ADJ
ejde-196	296	4	-	-	PUNCT
ejde-196	296	5	value	value	NOUN
ejde-196	296	6	theorem	theorem	NOUN
ejde-196	296	7	,	,	PUNCT
ejde-196	296	8	there	there	PRON
ejde-196	296	9	is	be	VERB
ejde-196	296	10	0	0	NUM
ejde-196	296	11	<	<	X
ejde-196	296	12	λs	λs	X
ejde-196	296	13	<	<	X
ejde-196	296	14	1	1	NUM
ejde-196	296	15	such	such	ADJ
ejde-196	296	16	that	that	DET
ejde-196	296	17	g′(λs	g′(λs	NOUN
ejde-196	296	18	)	)	PUNCT
ejde-196	296	19	=	=	SYM
ejde-196	296	20	g(1)−g(0	g(1)−g(0	NOUN
ejde-196	296	21	)	)	PUNCT
ejde-196	296	22	=	=	SYM
ejde-196	296	23	js(rv(s	js(rv(s	PROPN
ejde-196	296	24	)	)	PUNCT
ejde-196	296	25	,	,	PUNCT
ejde-196	296	26	θ(s))−js(rv	θ(s))−js(rv	ADJ
ejde-196	296	27	,	,	PUNCT
ejde-196	296	28	θ(s	θ(s	PROPN
ejde-196	296	29	)	)	PUNCT
ejde-196	296	30	)	)	PUNCT
ejde-196	296	31	≥	≥	NOUN
ejde-196	296	32	js(rv(s	js(rv(s	NOUN
ejde-196	296	33	)	)	PUNCT
ejde-196	296	34	,	,	PUNCT
ejde-196	296	35	θ(s))−js(rv	θ(s))−js(rv	ADJ
ejde-196	296	36	,	,	PUNCT
ejde-196	296	37	θ(rv	θ(rv	NUM
ejde-196	296	38	)	)	PUNCT
ejde-196	296	39	)	)	PUNCT
ejde-196	296	40	since	since	SCONJ
ejde-196	296	41	by	by	ADP
ejde-196	296	42	(	(	PUNCT
ejde-196	296	43	5.3	5.3	NUM
ejde-196	296	44	)	)	PUNCT
ejde-196	296	45	,	,	PUNCT
ejde-196	296	46	js(rv	js(rv	PROPN
ejde-196	296	47	,	,	PUNCT
ejde-196	296	48	θ(rv	θ(rv	NUM
ejde-196	296	49	)	)	PUNCT
ejde-196	296	50	)	)	PUNCT
ejde-196	296	51	≥	≥	NOUN
ejde-196	297	1	js(rv	js(rv	PROPN
ejde-196	297	2	,	,	PUNCT
ejde-196	297	3	θ(s	θ(s	PROPN
ejde-196	297	4	)	)	PUNCT
ejde-196	297	5	)	)	PUNCT
ejde-196	298	1	when	when	SCONJ
ejde-196	298	2	s	s	VERB
ejde-196	298	3	>	>	X
ejde-196	298	4	0	0	NUM
ejde-196	298	5	is	be	AUX
ejde-196	298	6	small	small	ADJ
ejde-196	298	7	.	.	PUNCT
ejde-196	299	1	on	on	ADP
ejde-196	299	2	the	the	DET
ejde-196	299	3	other	other	ADJ
ejde-196	299	4	hand	hand	NOUN
ejde-196	299	5	g′(λs	g′(λs	NOUN
ejde-196	299	6	)	)	PUNCT
ejde-196	299	7	=	=	PUNCT
ejde-196	300	1	〈	〈	PROPN
ejde-196	300	2	jsr	jsr	PROPN
ejde-196	300	3	(	(	PUNCT
ejde-196	300	4	rv	rv	PROPN
ejde-196	300	5	−	−	PROPN
ejde-196	301	1	λssjsr	λssjsr	ADJ
ejde-196	301	2	(	(	PUNCT
ejde-196	301	3	rv	rv	NOUN
ejde-196	301	4	,	,	PUNCT
ejde-196	301	5	θ(rv	θ(rv	NUM
ejde-196	301	6	)	)	PUNCT
ejde-196	301	7	)	)	PUNCT
ejde-196	301	8	,	,	PUNCT
ejde-196	301	9	θ(s)),−sjsr	θ(s)),−sjsr	NOUN
ejde-196	301	10	(	(	PUNCT
ejde-196	301	11	rv	rv	NOUN
ejde-196	301	12	,	,	PUNCT
ejde-196	301	13	θ(rv	θ(rv	NUM
ejde-196	301	14	)	)	PUNCT
ejde-196	301	15	)	)	PUNCT
ejde-196	301	16	〉	〉	PROPN
ejde-196	301	17	≤	≤	NOUN
ejde-196	301	18	−	−	PROPN
ejde-196	301	19	s	s	NOUN
ejde-196	301	20	4	4	NUM
ejde-196	301	21	‖jsr	‖jsr	NOUN
ejde-196	301	22	(	(	PUNCT
ejde-196	301	23	rv	rv	NOUN
ejde-196	301	24	,	,	PUNCT
ejde-196	301	25	θ(rv))‖2	θ(rv))‖2	PROPN
ejde-196	301	26	since	since	SCONJ
ejde-196	301	27	rv	rv	PROPN
ejde-196	301	28	−	−	PROPN
ejde-196	301	29	λssjsr	λssjsr	ADJ
ejde-196	301	30	(	(	PUNCT
ejde-196	301	31	rv	rv	NOUN
ejde-196	301	32	,	,	PUNCT
ejde-196	301	33	θ(rv))→	θ(rv))→	PROPN
ejde-196	301	34	rv	rv	PROPN
ejde-196	301	35	implies	imply	VERB
ejde-196	301	36	θ(s)→	θ(s)→	ADJ
ejde-196	301	37	θ(rv	θ(rv	NOUN
ejde-196	301	38	)	)	PUNCT
ejde-196	301	39	as	as	ADP
ejde-196	301	40	s→	s→	X
ejde-196	301	41	0	0	NUM
ejde-196	301	42	.	.	PUNCT
ejde-196	302	1	then	then	ADV
ejde-196	302	2	we	we	PRON
ejde-196	302	3	have	have	VERB
ejde-196	302	4	js(rv(s	js(rv(s	NOUN
ejde-196	302	5	)	)	PUNCT
ejde-196	302	6	,	,	PUNCT
ejde-196	302	7	θ(s))−	θ(s))−	NOUN
ejde-196	302	8	js(rv	js(rv	NOUN
ejde-196	302	9	,	,	PUNCT
ejde-196	302	10	θ(rv	θ(rv	NUM
ejde-196	302	11	)	)	PUNCT
ejde-196	302	12	)	)	PUNCT
ejde-196	302	13	<	<	X
ejde-196	303	1	−	−	PROPN
ejde-196	303	2	s	s	PART
ejde-196	303	3	4	4	NUM
ejde-196	303	4	‖jsr	‖jsr	NOUN
ejde-196	303	5	(	(	PUNCT
ejde-196	303	6	rv	rv	NOUN
ejde-196	303	7	,	,	PUNCT
ejde-196	303	8	θ(rv))‖2	θ(rv))‖2	PROPN
ejde-196	303	9	when	when	SCONJ
ejde-196	303	10	s	s	VERB
ejde-196	303	11	>	>	X
ejde-196	303	12	0	0	NUM
ejde-196	303	13	is	be	AUX
ejde-196	303	14	small	small	ADJ
ejde-196	303	15	.	.	PUNCT
ejde-196	304	1	this	this	PRON
ejde-196	304	2	contradicts	contradict	VERB
ejde-196	304	3	(	(	PUNCT
ejde-196	304	4	5.4	5.4	NUM
ejde-196	304	5	)	)	PUNCT
ejde-196	304	6	if	if	SCONJ
ejde-196	304	7	jsr	jsr	PROPN
ejde-196	304	8	(	(	PUNCT
ejde-196	304	9	rv	rv	PROPN
ejde-196	304	10	,	,	PUNCT
ejde-196	304	11	θ(rv	θ(rv	NUM
ejde-196	304	12	)	)	PUNCT
ejde-196	304	13	)	)	PUNCT
ejde-196	305	1	6=	6=	ADP
ejde-196	305	2	0	0	X
ejde-196	305	3	.	.	PUNCT
ejde-196	306	1	thus	thus	ADV
ejde-196	306	2	we	we	PRON
ejde-196	306	3	have	have	AUX
ejde-196	306	4	shown	show	VERB
ejde-196	306	5	djs(rv	djs(rv	NOUN
ejde-196	306	6	,	,	PUNCT
ejde-196	306	7	θ(rv	θ(rv	NUM
ejde-196	306	8	)	)	PUNCT
ejde-196	306	9	)	)	PUNCT
ejde-196	307	1	=	=	SYM
ejde-196	307	2	(	(	PUNCT
ejde-196	307	3	jsr	jsr	PROPN
ejde-196	307	4	(	(	PUNCT
ejde-196	307	5	rv	rv	PROPN
ejde-196	307	6	,	,	PUNCT
ejde-196	307	7	θ(rv	θ(rv	NUM
ejde-196	307	8	)	)	PUNCT
ejde-196	307	9	)	)	PUNCT
ejde-196	307	10	,	,	PUNCT
ejde-196	308	1	j	j	PROPN
ejde-196	308	2	s	s	PROPN
ejde-196	308	3	θ	θ	PROPN
ejde-196	308	4	(	(	PUNCT
ejde-196	308	5	rv	rv	PROPN
ejde-196	308	6	,	,	PUNCT
ejde-196	308	7	θ(rv	θ(rv	NUM
ejde-196	308	8	)	)	PUNCT
ejde-196	308	9	)	)	PUNCT
ejde-196	308	10	)	)	PUNCT
ejde-196	309	1	=	=	PUNCT
ejde-196	309	2	(	(	PUNCT
ejde-196	309	3	0	0	NUM
ejde-196	309	4	,	,	PUNCT
ejde-196	309	5	0	0	NUM
ejde-196	309	6	,	,	PUNCT
ejde-196	309	7	.	.	PUNCT
ejde-196	309	8	.	.	PUNCT
ejde-196	310	1	.	.	PUNCT
ejde-196	311	1	,	,	PUNCT
ejde-196	311	2	0	0	NUM
ejde-196	311	3	)	)	PUNCT
ejde-196	311	4	.	.	PUNCT
ejde-196	312	1	to	to	PART
ejde-196	312	2	show	show	VERB
ejde-196	312	3	j	j	PROPN
ejde-196	312	4	′(p(v))⊥[v	′(p(v))⊥[v	PROPN
ejde-196	312	5	,	,	PUNCT
ejde-196	312	6	l	l	PROPN
ejde-196	312	7	]	]	X
ejde-196	312	8	,	,	PUNCT
ejde-196	312	9	we	we	PRON
ejde-196	312	10	use	use	VERB
ejde-196	312	11	the	the	DET
ejde-196	312	12	same	same	ADJ
ejde-196	312	13	notation	notation	NOUN
ejde-196	312	14	as	as	ADP
ejde-196	312	15	before	before	ADV
ejde-196	312	16	.	.	PUNCT
ejde-196	313	1	for	for	ADP
ejde-196	313	2	u	u	NOUN
ejde-196	313	3	=	=	PROPN
ejde-196	313	4	p(v	p(v	PROPN
ejde-196	313	5	)	)	PUNCT
ejde-196	313	6	,	,	PUNCT
ejde-196	313	7	we	we	PRON
ejde-196	313	8	have	have	VERB
ejde-196	313	9	u	u	NOUN
ejde-196	313	10	=	=	SYM
ejde-196	313	11	χ(rv	χ(rv	NOUN
ejde-196	313	12	,	,	PUNCT
ejde-196	313	13	θ(rv	θ(rv	NUM
ejde-196	313	14	)	)	PUNCT
ejde-196	313	15	)	)	PUNCT
ejde-196	313	16	.	.	PUNCT
ejde-196	314	1	we	we	PRON
ejde-196	314	2	note	note	VERB
ejde-196	314	3	that	that	SCONJ
ejde-196	314	4	χ′r(rv	χ′r(rv	NOUN
ejde-196	314	5	,	,	PUNCT
ejde-196	314	6	θ(rv	θ(rv	NUM
ejde-196	314	7	)	)	PUNCT
ejde-196	314	8	)	)	PUNCT
ejde-196	314	9	is	be	AUX
ejde-196	314	10	the	the	DET
ejde-196	314	11	ray	ray	NOUN
ejde-196	314	12	-	-	PUNCT
ejde-196	314	13	direction	direction	NOUN
ejde-196	314	14	of	of	ADP
ejde-196	314	15	the	the	DET
ejde-196	314	16	point	point	NOUN
ejde-196	314	17	u	u	NOUN
ejde-196	314	18	and	and	CCONJ
ejde-196	314	19	χ′θ(rv	χ′θ(rv	NOUN
ejde-196	314	20	,	,	PUNCT
ejde-196	314	21	θ(rv	θ(rv	NUM
ejde-196	314	22	)	)	PUNCT
ejde-196	314	23	)	)	PUNCT
ejde-196	314	24	forms	form	VERB
ejde-196	314	25	a	a	DET
ejde-196	314	26	basis	basis	NOUN
ejde-196	314	27	for	for	ADP
ejde-196	314	28	the	the	DET
ejde-196	314	29	tangent	tangent	ADJ
ejde-196	314	30	space	space	NOUN
ejde-196	314	31	of	of	ADP
ejde-196	314	32	the	the	DET
ejde-196	314	33	k+1	k+1	ADV
ejde-196	314	34	-	-	ADJ
ejde-196	314	35	dimensional	dimensional	ADJ
ejde-196	314	36	sphere	sphere	NOUN
ejde-196	314	37	at	at	ADP
ejde-196	314	38	the	the	DET
ejde-196	314	39	point	point	NOUN
ejde-196	314	40	u.	u.	NOUN
ejde-196	314	41	by	by	ADP
ejde-196	314	42	the	the	DET
ejde-196	314	43	chain	chain	NOUN
ejde-196	314	44	rule	rule	NOUN
ejde-196	314	45	,	,	PUNCT
ejde-196	314	46	we	we	PRON
ejde-196	314	47	have	have	VERB
ejde-196	314	48	0	0	NUM
ejde-196	314	49	=	=	SYM
ejde-196	314	50	∂	∂	NOUN
ejde-196	314	51	∂r	∂r	PROPN
ejde-196	314	52	js(rv	js(rv	PROPN
ejde-196	314	53	,	,	PUNCT
ejde-196	314	54	θ(rv	θ(rv	NUM
ejde-196	314	55	)	)	PUNCT
ejde-196	314	56	)	)	PUNCT
ejde-196	315	1	=	=	SYM
ejde-196	315	2	∂	∂	NUM
ejde-196	316	1	∂r	∂r	PROPN
ejde-196	316	2	j(χ(r	j(χ(r	PROPN
ejde-196	316	3	,	,	PUNCT
ejde-196	316	4	θ))|(r	θ))|(r	ADV
ejde-196	316	5	,	,	PUNCT
ejde-196	316	6	θ)=(rv	θ)=(rv	NOUN
ejde-196	316	7	,	,	PUNCT
ejde-196	316	8	θ(rv	θ(rv	NUM
ejde-196	316	9	)	)	PUNCT
ejde-196	316	10	)	)	PUNCT
ejde-196	317	1	=	=	SYM
ejde-196	317	2	j	j	PROPN
ejde-196	317	3	′(p(v))χ′r(rv	′(p(v))χ′r(rv	PROPN
ejde-196	317	4	,	,	PUNCT
ejde-196	317	5	θ(rv	θ(rv	NUM
ejde-196	317	6	)	)	PUNCT
ejde-196	317	7	)	)	PUNCT
ejde-196	317	8	,	,	PUNCT
ejde-196	317	9	(	(	PUNCT
ejde-196	317	10	0	0	NUM
ejde-196	317	11	,	,	PUNCT
ejde-196	317	12	.	.	PUNCT
ejde-196	317	13	.	.	PUNCT
ejde-196	317	14	.	.	PUNCT
ejde-196	318	1	,	,	PUNCT
ejde-196	318	2	0	0	X
ejde-196	318	3	)	)	PUNCT
ejde-196	318	4	=	=	SYM
ejde-196	318	5	∂	∂	NUM
ejde-196	318	6	∂θ	∂θ	NOUN
ejde-196	318	7	js(rv	js(rv	PROPN
ejde-196	318	8	,	,	PUNCT
ejde-196	318	9	θ(rv	θ(rv	NUM
ejde-196	318	10	)	)	PUNCT
ejde-196	318	11	)	)	PUNCT
ejde-196	319	1	=	=	SYM
ejde-196	319	2	∂	∂	NUM
ejde-196	320	1	∂θ	∂θ	PROPN
ejde-196	320	2	j(χ(r	j(χ(r	PROPN
ejde-196	320	3	,	,	PUNCT
ejde-196	320	4	θ))|(r	θ))|(r	ADV
ejde-196	320	5	,	,	PUNCT
ejde-196	320	6	θ)=(rv	θ)=(rv	NOUN
ejde-196	320	7	,	,	PUNCT
ejde-196	320	8	θ(rv	θ(rv	NUM
ejde-196	320	9	)	)	PUNCT
ejde-196	320	10	)	)	PUNCT
ejde-196	321	1	=	=	SYM
ejde-196	321	2	j	j	PROPN
ejde-196	321	3	′(p(v))χ′θ(rv	′(p(v))χ′θ(rv	PROPN
ejde-196	321	4	,	,	PUNCT
ejde-196	321	5	θ(rv	θ(rv	NUM
ejde-196	321	6	)	)	PUNCT
ejde-196	321	7	)	)	PUNCT
ejde-196	321	8	,	,	PUNCT
ejde-196	321	9	i.e.	i.e.	X
ejde-196	321	10	,	,	PUNCT
ejde-196	321	11	in	in	ADP
ejde-196	321	12	the	the	DET
ejde-196	321	13	space	space	NOUN
ejde-196	322	1	[	[	X
ejde-196	322	2	v	v	NOUN
ejde-196	322	3	,	,	PUNCT
ejde-196	322	4	l	l	NOUN
ejde-196	322	5	]	]	X
ejde-196	322	6	,	,	PUNCT
ejde-196	322	7	j	j	PROPN
ejde-196	322	8	′(p(v	′(p(v	NOUN
ejde-196	322	9	)	)	PUNCT
ejde-196	322	10	)	)	PUNCT
ejde-196	322	11	is	be	AUX
ejde-196	322	12	orthogonal	orthogonal	ADJ
ejde-196	322	13	to	to	ADP
ejde-196	322	14	the	the	DET
ejde-196	322	15	vector	vector	NOUN
ejde-196	322	16	(	(	PUNCT
ejde-196	322	17	ray	ray	NOUN
ejde-196	322	18	)	)	PUNCT
ejde-196	322	19	u	u	NOUN
ejde-196	322	20	=	=	PROPN
ejde-196	322	21	p(v	p(v	PROPN
ejde-196	322	22	)	)	PUNCT
ejde-196	322	23	and	and	CCONJ
ejde-196	322	24	the	the	DET
ejde-196	322	25	tangent	tangent	ADJ
ejde-196	322	26	space	space	NOUN
ejde-196	322	27	of	of	ADP
ejde-196	322	28	the	the	DET
ejde-196	322	29	sphere	sphere	NOUN
ejde-196	322	30	at	at	ADP
ejde-196	322	31	u	u	NOUN
ejde-196	322	32	=	=	PROPN
ejde-196	322	33	p(v	p(v	PROPN
ejde-196	322	34	)	)	PUNCT
ejde-196	322	35	.	.	PUNCT
ejde-196	323	1	consequently	consequently	ADV
ejde-196	323	2	j	j	PROPN
ejde-196	323	3	′(p(v))⊥[v	′(p(v))⊥[v	PROPN
ejde-196	323	4	,	,	PUNCT
ejde-196	323	5	l	l	NOUN
ejde-196	323	6	]	]	X
ejde-196	323	7	.	.	PUNCT
ejde-196	324	1	we	we	PRON
ejde-196	324	2	have	have	AUX
ejde-196	324	3	theoretically	theoretically	ADV
ejde-196	324	4	verified	verify	VERB
ejde-196	324	5	that	that	SCONJ
ejde-196	324	6	the	the	DET
ejde-196	324	7	min	min	PROPN
ejde-196	324	8	-	-	PUNCT
ejde-196	324	9	min	min	ADJ
ejde-196	324	10	-	-	ADJ
ejde-196	324	11	max	max	ADJ
ejde-196	324	12	algorithm	algorithm	NOUN
ejde-196	324	13	(	(	PUNCT
ejde-196	324	14	a1	a1	NOUN
ejde-196	324	15	)	)	PUNCT
ejde-196	324	16	fits	fit	VERB
ejde-196	324	17	into	into	ADP
ejde-196	324	18	the	the	DET
ejde-196	324	19	local	local	ADJ
ejde-196	324	20	min	min	ADJ
ejde-196	324	21	-	-	ADJ
ejde-196	324	22	orthogonal	orthogonal	ADJ
ejde-196	324	23	principle	principle	NOUN
ejde-196	324	24	framework	framework	NOUN
ejde-196	324	25	by	by	ADP
ejde-196	324	26	introducing	introduce	VERB
ejde-196	324	27	a	a	DET
ejde-196	324	28	k+1	k+1	ADV
ejde-196	324	29	-	-	ADJ
ejde-196	324	30	dimensional	dimensional	ADJ
ejde-196	324	31	spherical	spherical	ADJ
ejde-196	324	32	coordinates	coordinate	NOUN
ejde-196	324	33	.	.	PUNCT
ejde-196	325	1	in	in	ADP
ejde-196	325	2	numerical	numerical	ADJ
ejde-196	325	3	implementation	implementation	NOUN
ejde-196	325	4	,	,	PUNCT
ejde-196	325	5	this	this	PRON
ejde-196	325	6	is	be	AUX
ejde-196	325	7	not	not	PART
ejde-196	325	8	necessary	necessary	ADJ
ejde-196	325	9	,	,	PUNCT
ejde-196	325	10	we	we	PRON
ejde-196	325	11	may	may	AUX
ejde-196	325	12	conveniently	conveniently	ADV
ejde-196	325	13	work	work	VERB
ejde-196	325	14	with	with	ADP
ejde-196	325	15	the	the	DET
ejde-196	325	16	original	original	ADJ
ejde-196	325	17	coordinates	coordinate	NOUN
ejde-196	325	18	system	system	NOUN
ejde-196	325	19	while	while	SCONJ
ejde-196	325	20	matching	match	VERB
ejde-196	325	21	the	the	DET
ejde-196	325	22	above	above	ADJ
ejde-196	325	23	analysis	analysis	NOUN
ejde-196	325	24	.	.	PUNCT
ejde-196	326	1	also	also	ADV
ejde-196	326	2	numerically	numerically	ADV
ejde-196	326	3	the	the	DET
ejde-196	326	4	two	two	NUM
ejde-196	326	5	mins	min	NOUN
ejde-196	326	6	can	can	AUX
ejde-196	326	7	be	be	AUX
ejde-196	326	8	put	put	VERB
ejde-196	326	9	into	into	ADP
ejde-196	326	10	one	one	NUM
ejde-196	326	11	to	to	PART
ejde-196	326	12	form	form	VERB
ejde-196	326	13	a	a	DET
ejde-196	326	14	two	two	NUM
ejde-196	326	15	-	-	PUNCT
ejde-196	326	16	level	level	NOUN
ejde-196	326	17	algorithm	algorithm	NOUN
ejde-196	326	18	.	.	PUNCT
ejde-196	327	1	a	a	DET
ejde-196	327	2	similar	similar	ADJ
ejde-196	327	3	method	method	NOUN
ejde-196	327	4	has	have	AUX
ejde-196	327	5	been	be	AUX
ejde-196	327	6	proposed	propose	VERB
ejde-196	327	7	independently	independently	ADV
ejde-196	327	8	in	in	ADP
ejde-196	327	9	[	[	X
ejde-196	327	10	14	14	NUM
ejde-196	327	11	]	]	PUNCT
ejde-196	327	12	in	in	ADP
ejde-196	327	13	the	the	DET
ejde-196	327	14	framework	framework	NOUN
ejde-196	327	15	of	of	ADP
ejde-196	327	16	lsp	lsp	PROPN
ejde-196	327	17	which	which	PRON
ejde-196	327	18	actually	actually	ADV
ejde-196	327	19	uses	use	VERB
ejde-196	327	20	global	global	ADJ
ejde-196	327	21	min	min	PROPN
ejde-196	327	22	and	and	CCONJ
ejde-196	327	23	max	max	PROPN
ejde-196	327	24	in	in	ADP
ejde-196	327	25	its	its	PRON
ejde-196	327	26	mathematical	mathematical	ADJ
ejde-196	327	27	formulation	formulation	NOUN
ejde-196	327	28	while	while	SCONJ
ejde-196	327	29	our	our	PRON
ejde-196	327	30	local	local	ADJ
ejde-196	327	31	min	min	ADJ
ejde-196	327	32	-	-	ADJ
ejde-196	327	33	orthogonal	orthogonal	ADJ
ejde-196	327	34	principle	principle	NOUN
ejde-196	327	35	uses	use	VERB
ejde-196	327	36	local	local	ADJ
ejde-196	327	37	min	min	PROPN
ejde-196	327	38	and	and	CCONJ
ejde-196	327	39	max	max	PROPN
ejde-196	327	40	in	in	ADP
ejde-196	327	41	its	its	PRON
ejde-196	327	42	mathematical	mathematical	ADJ
ejde-196	327	43	formulation	formulation	NOUN
ejde-196	327	44	as	as	SCONJ
ejde-196	327	45	showed	show	VERB
ejde-196	327	46	in	in	ADP
ejde-196	327	47	the	the	DET
ejde-196	327	48	previous	previous	ADJ
ejde-196	327	49	sections	section	NOUN
ejde-196	327	50	.	.	PUNCT
ejde-196	328	1	since	since	SCONJ
ejde-196	328	2	some	some	DET
ejde-196	328	3	numerical	numerical	ADJ
ejde-196	328	4	examples	example	NOUN
ejde-196	328	5	on	on	ADP
ejde-196	328	6	w	w	NOUN
ejde-196	328	7	-	-	PUNCT
ejde-196	328	8	type	type	NOUN
ejde-196	328	9	problems	problem	NOUN
ejde-196	328	10	are	be	AUX
ejde-196	328	11	already	already	ADV
ejde-196	328	12	carried	carry	VERB
ejde-196	328	13	out	out	ADP
ejde-196	328	14	in	in	ADP
ejde-196	328	15	[	[	X
ejde-196	328	16	14	14	NUM
ejde-196	328	17	]	]	PUNCT
ejde-196	328	18	,	,	PUNCT
ejde-196	328	19	we	we	PRON
ejde-196	328	20	will	will	AUX
ejde-196	328	21	not	not	PART
ejde-196	328	22	go	go	VERB
ejde-196	328	23	further	far	ADV
ejde-196	328	24	on	on	ADP
ejde-196	328	25	this	this	DET
ejde-196	328	26	algorithm	algorithm	NOUN
ejde-196	328	27	.	.	PUNCT
ejde-196	329	1	5.2	5.2	NUM
ejde-196	329	2	.	.	PUNCT
ejde-196	329	3	justification	justification	NOUN
ejde-196	329	4	of	of	ADP
ejde-196	329	5	the	the	DET
ejde-196	329	6	min	min	PROPN
ejde-196	329	7	-	-	PUNCT
ejde-196	329	8	max	max	PROPN
ejde-196	329	9	-	-	PUNCT
ejde-196	329	10	min	min	NOUN
ejde-196	329	11	algorithm	algorithm	NOUN
ejde-196	329	12	.	.	PUNCT
ejde-196	330	1	in	in	ADP
ejde-196	330	2	the	the	DET
ejde-196	330	3	min	min	PROPN
ejde-196	330	4	-	-	PUNCT
ejde-196	330	5	max	max	ADJ
ejde-196	330	6	-	-	PUNCT
ejde-196	330	7	min	min	NOUN
ejde-196	330	8	algorithm	algorithm	NOUN
ejde-196	330	9	(	(	PUNCT
ejde-196	330	10	a2	a2	PROPN
ejde-196	330	11	)	)	PUNCT
ejde-196	330	12	,	,	PUNCT
ejde-196	330	13	for	for	ADP
ejde-196	330	14	each	each	DET
ejde-196	330	15	v	v	NOUN
ejde-196	330	16	∈	∈	NOUN
ejde-196	330	17	sl⊥	sl⊥	PROPN
ejde-196	330	18	,	,	PUNCT
ejde-196	330	19	u	u	PROPN
ejde-196	330	20	∈	∈	PROPN
ejde-196	331	1	[	[	X
ejde-196	331	2	v	v	NOUN
ejde-196	331	3	,	,	PUNCT
ejde-196	331	4	l	l	NOUN
ejde-196	331	5	]	]	X
ejde-196	331	6	,	,	PUNCT
ejde-196	331	7	‖u‖	‖u‖	PROPN
ejde-196	331	8	≈	≈	PROPN
ejde-196	331	9	1	1	NUM
ejde-196	331	10	,	,	PUNCT
ejde-196	331	11	denote	denote	NOUN
ejde-196	331	12	p̄(u	p̄(u	NOUN
ejde-196	331	13	)	)	PUNCT
ejde-196	331	14	=	=	PUNCT
ejde-196	332	1	tuu	tuu	VERB
ejde-196	332	2	where	where	SCONJ
ejde-196	332	3	tu	tu	PROPN
ejde-196	332	4	=	=	PUNCT
ejde-196	332	5	arg	arg	NOUN
ejde-196	332	6	min	min	PROPN
ejde-196	332	7	t>0	t>0	PROPN
ejde-196	332	8	j(tu	j(tu	PROPN
ejde-196	332	9	)	)	PUNCT
ejde-196	332	10	(	(	PUNCT
ejde-196	332	11	5.5	5.5	NUM
ejde-196	332	12	)	)	PUNCT
ejde-196	332	13	and	and	CCONJ
ejde-196	332	14	p(v	p(v	NOUN
ejde-196	332	15	)	)	PUNCT
ejde-196	332	16	=	=	SYM
ejde-196	332	17	p̄(uv	p̄(uv	NOUN
ejde-196	332	18	)	)	PUNCT
ejde-196	332	19	=	=	SYM
ejde-196	333	1	tvuv	tvuv	NOUN
ejde-196	333	2	where	where	SCONJ
ejde-196	333	3	uv	uv	NOUN
ejde-196	333	4	=	=	PUNCT
ejde-196	333	5	arg	arg	NOUN
ejde-196	333	6	max	max	PROPN
ejde-196	333	7	u∈[v	u∈[v	PROPN
ejde-196	333	8	,	,	PUNCT
ejde-196	333	9	l],‖u‖≈1	l],‖u‖≈1	NOUN
ejde-196	333	10	j(p̄(u	j(p̄(u	PROPN
ejde-196	333	11	)	)	PUNCT
ejde-196	333	12	)	)	PUNCT
ejde-196	333	13	.	.	PUNCT
ejde-196	334	1	ejde-2023	ejde-2023	ADJ
ejde-196	334	2	/	/	SYM
ejde-196	334	3	si/02	si/02	ADJ
ejde-196	334	4	short	short	ADJ
ejde-196	334	5	title	title	NOUN
ejde-196	334	6	185	185	NUM
ejde-196	334	7	assume	assume	NOUN
ejde-196	334	8	p̄	p̄	PROPN
ejde-196	334	9	is	be	AUX
ejde-196	334	10	continuous	continuous	ADJ
ejde-196	334	11	and	and	CCONJ
ejde-196	334	12	suppose	suppose	VERB
ejde-196	334	13	the	the	DET
ejde-196	334	14	projection	projection	NOUN
ejde-196	334	15	w	w	PROPN
ejde-196	334	16	of	of	ADP
ejde-196	334	17	j	j	PROPN
ejde-196	334	18	′(p(v	′(p(v	NOUN
ejde-196	334	19	)	)	PUNCT
ejde-196	334	20	)	)	PUNCT
ejde-196	334	21	onto	onto	ADP
ejde-196	334	22	the	the	DET
ejde-196	334	23	subspace	subspace	NOUN
ejde-196	335	1	[	[	X
ejde-196	335	2	v	v	NOUN
ejde-196	335	3	,	,	PUNCT
ejde-196	335	4	l	l	NOUN
ejde-196	335	5	]	]	X
ejde-196	335	6	satisfies	satisfie	NOUN
ejde-196	335	7	w	w	PROPN
ejde-196	335	8	=	=	SYM
ejde-196	335	9	j	j	PROPN
ejde-196	335	10	′(p(v))[v	′(p(v))[v	PROPN
ejde-196	335	11	,	,	PUNCT
ejde-196	335	12	l	l	NOUN
ejde-196	335	13	]	]	PUNCT
ejde-196	335	14	6=	6=	ADP
ejde-196	335	15	0	0	NUM
ejde-196	335	16	.	.	PUNCT
ejde-196	336	1	for	for	ADP
ejde-196	336	2	s	s	PROPN
ejde-196	336	3	>	>	X
ejde-196	336	4	0	0	PUNCT
ejde-196	336	5	small	small	ADJ
ejde-196	336	6	,	,	PUNCT
ejde-196	336	7	we	we	PRON
ejde-196	336	8	let	let	VERB
ejde-196	336	9	uv(s	uv(s	PUNCT
ejde-196	336	10	)	)	PUNCT
ejde-196	337	1	=	=	SYM
ejde-196	337	2	uv	uv	NOUN
ejde-196	338	1	+	+	NUM
ejde-196	338	2	sw	sw	PROPN
ejde-196	338	3	‖uv	‖uv	PRON
ejde-196	338	4	+	+	CCONJ
ejde-196	338	5	sw‖	sw‖	PROPN
ejde-196	338	6	∈	∈	PROPN
ejde-196	338	7	s[v	s[v	NOUN
ejde-196	338	8	,	,	PUNCT
ejde-196	338	9	l	l	NOUN
ejde-196	338	10	]	]	X
ejde-196	338	11	(	(	PUNCT
ejde-196	338	12	5.6	5.6	NUM
ejde-196	338	13	)	)	PUNCT
ejde-196	338	14	and	and	CCONJ
ejde-196	338	15	p̄(uv(s	p̄(uv(s	NOUN
ejde-196	338	16	)	)	PUNCT
ejde-196	338	17	)	)	PUNCT
ejde-196	339	1	=	=	SYM
ejde-196	339	2	tsuv(s	tsuv(s	PROPN
ejde-196	339	3	)	)	PUNCT
ejde-196	339	4	=	=	PUNCT
ejde-196	339	5	ts	ts	PART
ejde-196	339	6	uv+sw	uv+sw	PROPN
ejde-196	339	7	‖uv+sw‖	‖uv+sw‖	NOUN
ejde-196	339	8	for	for	ADP
ejde-196	339	9	some	some	PRON
ejde-196	339	10	ts	ts	ADP
ejde-196	339	11	>	>	X
ejde-196	339	12	0	0	X
ejde-196	339	13	.	.	PUNCT
ejde-196	339	14	thus	thus	ADV
ejde-196	339	15	p̄(uv(s	p̄(uv(s	NOUN
ejde-196	339	16	)	)	PUNCT
ejde-196	339	17	)	)	PUNCT
ejde-196	340	1	→	→	SYM
ejde-196	340	2	p̄(uv	p̄(uv	NOUN
ejde-196	340	3	)	)	PUNCT
ejde-196	340	4	as	as	ADP
ejde-196	340	5	s→	s→	X
ejde-196	340	6	0	0	NUM
ejde-196	340	7	.	.	PUNCT
ejde-196	340	8	for	for	ADP
ejde-196	340	9	each	each	PRON
ejde-196	340	10	s	s	X
ejde-196	340	11	>	>	X
ejde-196	340	12	0	0	NUM
ejde-196	340	13	,	,	PUNCT
ejde-196	340	14	we	we	PRON
ejde-196	340	15	define	define	VERB
ejde-196	340	16	g(λ	g(λ	PROPN
ejde-196	340	17	)	)	PUNCT
ejde-196	341	1	=	=	SYM
ejde-196	341	2	j	j	PROPN
ejde-196	341	3	(	(	PUNCT
ejde-196	341	4	tsuv	tsuv	NOUN
ejde-196	341	5	‖uv	‖uv	PRON
ejde-196	341	6	+	+	CCONJ
ejde-196	341	7	sw‖	sw‖	NOUN
ejde-196	341	8	+	+	CCONJ
ejde-196	341	9	λ	λ	NOUN
ejde-196	341	10	tssw	tssw	NOUN
ejde-196	341	11	‖uv	‖uv	PRON
ejde-196	341	12	+	+	CCONJ
ejde-196	341	13	sw‖	sw‖	NOUN
ejde-196	341	14	)	)	PUNCT
ejde-196	341	15	.	.	PUNCT
ejde-196	342	1	by	by	ADP
ejde-196	342	2	the	the	DET
ejde-196	342	3	mean	mean	ADJ
ejde-196	342	4	-	-	PUNCT
ejde-196	342	5	value	value	NOUN
ejde-196	342	6	theorem	theorem	NOUN
ejde-196	342	7	,	,	PUNCT
ejde-196	342	8	there	there	PRON
ejde-196	342	9	is	be	VERB
ejde-196	342	10	0	0	NUM
ejde-196	342	11	<	<	X
ejde-196	342	12	λs	λs	X
ejde-196	342	13	<	<	X
ejde-196	342	14	1	1	NUM
ejde-196	342	15	such	such	ADJ
ejde-196	342	16	that	that	DET
ejde-196	342	17	g′(λs	g′(λs	NOUN
ejde-196	342	18	)	)	PUNCT
ejde-196	342	19	=	=	SYM
ejde-196	342	20	g(1)−	g(1)−	NOUN
ejde-196	342	21	g(0	g(0	PROPN
ejde-196	342	22	)	)	PUNCT
ejde-196	342	23	=	=	PUNCT
ejde-196	343	1	j(p̄(uv(s)))−	j(p̄(uv(s)))−	PROPN
ejde-196	343	2	j	j	PROPN
ejde-196	343	3	(	(	PUNCT
ejde-196	343	4	tsuv	tsuv	NOUN
ejde-196	343	5	‖uv	‖uv	PRON
ejde-196	343	6	+	+	CCONJ
ejde-196	343	7	sw‖	sw‖	NOUN
ejde-196	343	8	)	)	PUNCT
ejde-196	343	9	≤	≤	PUNCT
ejde-196	344	1	j(p̄(uv(s)))−	j(p̄(uv(s)))−	VERB
ejde-196	344	2	j(p̄(uv	j(p̄(uv	NOUN
ejde-196	344	3	)	)	PUNCT
ejde-196	344	4	)	)	PUNCT
ejde-196	344	5	,	,	PUNCT
ejde-196	344	6	since	since	SCONJ
ejde-196	344	7	ts	ts	ADP
ejde-196	344	8	→	→	SYM
ejde-196	344	9	tv‖uv‖	tv‖uv‖	PROPN
ejde-196	344	10	as	as	ADP
ejde-196	344	11	s→	s→	X
ejde-196	344	12	0	0	NUM
ejde-196	345	1	and	and	CCONJ
ejde-196	345	2	when	when	SCONJ
ejde-196	345	3	s	s	AUX
ejde-196	345	4	>	>	X
ejde-196	345	5	0	0	NUM
ejde-196	345	6	is	be	AUX
ejde-196	345	7	small	small	ADJ
ejde-196	345	8	,	,	PUNCT
ejde-196	345	9	we	we	PRON
ejde-196	345	10	have	have	VERB
ejde-196	345	11	j	j	PROPN
ejde-196	345	12	(	(	PUNCT
ejde-196	345	13	tsuv	tsuv	NOUN
ejde-196	345	14	‖uv+sw‖	‖uv+sw‖	PROPN
ejde-196	345	15	)	)	PUNCT
ejde-196	345	16	≥	≥	PROPN
ejde-196	345	17	j(p̄(uv	j(p̄(uv	NOUN
ejde-196	345	18	)	)	PUNCT
ejde-196	345	19	)	)	PUNCT
ejde-196	346	1	by	by	ADP
ejde-196	346	2	(	(	PUNCT
ejde-196	346	3	5.5	5.5	NUM
ejde-196	346	4	)	)	PUNCT
ejde-196	346	5	.	.	PUNCT
ejde-196	347	1	on	on	ADP
ejde-196	347	2	the	the	DET
ejde-196	347	3	other	other	ADJ
ejde-196	347	4	hand	hand	NOUN
ejde-196	347	5	when	when	SCONJ
ejde-196	347	6	s	s	VERB
ejde-196	347	7	>	>	X
ejde-196	347	8	0	0	NUM
ejde-196	347	9	is	be	AUX
ejde-196	347	10	small	small	ADJ
ejde-196	347	11	,	,	PUNCT
ejde-196	347	12	we	we	PRON
ejde-196	347	13	have	have	VERB
ejde-196	347	14	ts	ts	ADP
ejde-196	347	15	→	→	SYM
ejde-196	347	16	tv‖uv‖	tv‖uv‖	PROPN
ejde-196	347	17	and	and	CCONJ
ejde-196	347	18	then	then	ADV
ejde-196	347	19	g′(λs	g′(λs	PROPN
ejde-196	347	20	)	)	PUNCT
ejde-196	347	21	=	=	PUNCT
ejde-196	348	1	〈	〈	PROPN
ejde-196	348	2	j	j	PROPN
ejde-196	348	3	′	′	PROPN
ejde-196	348	4	(	(	PUNCT
ejde-196	348	5	tsuv	tsuv	NOUN
ejde-196	348	6	‖uv	‖uv	PRON
ejde-196	348	7	+	+	CCONJ
ejde-196	348	8	sw‖	sw‖	NOUN
ejde-196	349	1	+	+	CCONJ
ejde-196	349	2	λs	λs	ADP
ejde-196	349	3	tssw	tssw	NOUN
ejde-196	350	1	‖uv	‖uv	PRON
ejde-196	350	2	+	+	CCONJ
ejde-196	350	3	sw‖	sw‖	NOUN
ejde-196	350	4	)	)	PUNCT
ejde-196	350	5	,	,	PUNCT
ejde-196	350	6	tssw	tssw	NOUN
ejde-196	350	7	‖uv	‖uv	PRON
ejde-196	351	1	+	+	CCONJ
ejde-196	351	2	sw‖	sw‖	NUM
ejde-196	351	3	〉	〉	NOUN
ejde-196	351	4	>	>	SYM
ejde-196	351	5	1	1	NUM
ejde-196	351	6	4	4	NUM
ejde-196	351	7	tss‖w‖2	tss‖w‖2	NUM
ejde-196	351	8	,	,	PUNCT
ejde-196	351	9	where	where	SCONJ
ejde-196	351	10	the	the	DET
ejde-196	351	11	last	last	ADJ
ejde-196	351	12	inequality	inequality	NOUN
ejde-196	351	13	is	be	AUX
ejde-196	351	14	due	due	ADJ
ejde-196	351	15	to	to	ADP
ejde-196	351	16	the	the	DET
ejde-196	351	17	facts	fact	NOUN
ejde-196	351	18	that	that	SCONJ
ejde-196	351	19	as	as	ADP
ejde-196	351	20	s→	s→	X
ejde-196	351	21	0	0	NUM
ejde-196	351	22	,	,	PUNCT
ejde-196	351	23	j	j	PROPN
ejde-196	351	24	′	′	PROPN
ejde-196	351	25	(	(	PUNCT
ejde-196	351	26	tsuv	tsuv	NOUN
ejde-196	351	27	‖uv+sw‖+λs	‖uv+sw‖+λs	PROPN
ejde-196	351	28	tssw	tssw	NOUN
ejde-196	351	29	‖uv+sw‖	‖uv+sw‖	PROPN
ejde-196	351	30	)	)	PUNCT
ejde-196	351	31	→	→	SYM
ejde-196	351	32	j	j	PROPN
ejde-196	351	33	′(p̄(uv	′(p̄(uv	NOUN
ejde-196	351	34	)	)	PUNCT
ejde-196	351	35	)	)	PUNCT
ejde-196	351	36	and	and	CCONJ
ejde-196	351	37	〈	〈	PROPN
ejde-196	351	38	j	j	PROPN
ejde-196	351	39	′(p̄(uv	′(p̄(uv	NOUN
ejde-196	351	40	)	)	PUNCT
ejde-196	351	41	)	)	PUNCT
ejde-196	351	42	,	,	PUNCT
ejde-196	352	1	w	w	NOUN
ejde-196	352	2	〉	〉	NOUN
ejde-196	352	3	=	=	SYM
ejde-196	352	4	‖w‖2	‖w‖2	PROPN
ejde-196	352	5	>	>	X
ejde-196	352	6	0	0	X
ejde-196	352	7	.	.	PUNCT
ejde-196	353	1	consequently	consequently	ADV
ejde-196	353	2	as	as	SCONJ
ejde-196	353	3	s	s	PROPN
ejde-196	353	4	>	>	X
ejde-196	353	5	0	0	PUNCT
ejde-196	353	6	small	small	ADJ
ejde-196	353	7	,	,	PUNCT
ejde-196	353	8	we	we	PRON
ejde-196	353	9	have	have	AUX
ejde-196	353	10	j(p̄(uv(s)))−	j(p̄(uv(s)))−	VERB
ejde-196	353	11	j(p̄(uv	j(p̄(uv	NOUN
ejde-196	353	12	)	)	PUNCT
ejde-196	354	1	=	=	PUNCT
ejde-196	354	2	j(p̄(uv(s)))−	j(p̄(uv(s)))−	NOUN
ejde-196	354	3	j(p(v	j(p(v	NOUN
ejde-196	354	4	)	)	PUNCT
ejde-196	354	5	>	>	X
ejde-196	354	6	1	1	NUM
ejde-196	354	7	4	4	NUM
ejde-196	354	8	tss‖w‖2	tss‖w‖2	X
ejde-196	354	9	>	>	X
ejde-196	354	10	0	0	NUM
ejde-196	354	11	,	,	PUNCT
ejde-196	354	12	(	(	PUNCT
ejde-196	354	13	5.7	5.7	NUM
ejde-196	354	14	)	)	PUNCT
ejde-196	354	15	which	which	PRON
ejde-196	354	16	violates	violate	VERB
ejde-196	354	17	(	(	PUNCT
ejde-196	354	18	5.2	5.2	NUM
ejde-196	354	19	)	)	PUNCT
ejde-196	354	20	unless	unless	SCONJ
ejde-196	354	21	w	w	PROPN
ejde-196	354	22	=	=	SYM
ejde-196	354	23	j	j	PROPN
ejde-196	354	24	′(p(v))[v	′(p(v))[v	PROPN
ejde-196	354	25	,	,	PUNCT
ejde-196	354	26	l	l	NOUN
ejde-196	354	27	]	]	X
ejde-196	354	28	=	=	SYM
ejde-196	354	29	0	0	NUM
ejde-196	354	30	or	or	CCONJ
ejde-196	354	31	j	j	PROPN
ejde-196	354	32	′(p(v))⊥[v	′(p(v))⊥[v	PROPN
ejde-196	354	33	,	,	PUNCT
ejde-196	354	34	l	l	NOUN
ejde-196	354	35	]	]	X
ejde-196	354	36	.	.	PUNCT
ejde-196	355	1	thus	thus	ADV
ejde-196	355	2	we	we	PRON
ejde-196	355	3	have	have	AUX
ejde-196	355	4	proved	prove	VERB
ejde-196	355	5	that	that	SCONJ
ejde-196	355	6	the	the	DET
ejde-196	355	7	min	min	PROPN
ejde-196	355	8	-	-	PUNCT
ejde-196	355	9	max	max	ADJ
ejde-196	355	10	-	-	PUNCT
ejde-196	355	11	min	min	NOUN
ejde-196	355	12	algorithm	algorithm	NOUN
ejde-196	355	13	(	(	PUNCT
ejde-196	355	14	a2	a2	PROPN
ejde-196	355	15	)	)	PUNCT
ejde-196	355	16	fits	fit	VERB
ejde-196	355	17	into	into	ADP
ejde-196	355	18	the	the	DET
ejde-196	355	19	local	local	ADJ
ejde-196	355	20	min	min	ADJ
ejde-196	355	21	-	-	ADJ
ejde-196	355	22	orthogonal	orthogonal	ADJ
ejde-196	355	23	principle	principle	NOUN
ejde-196	355	24	framework	framework	NOUN
ejde-196	355	25	.	.	PUNCT
ejde-196	356	1	on	on	ADP
ejde-196	356	2	the	the	DET
ejde-196	356	3	other	other	ADJ
ejde-196	356	4	hand	hand	NOUN
ejde-196	356	5	,	,	PUNCT
ejde-196	356	6	denote	denote	VERB
ejde-196	356	7	f	f	PROPN
ejde-196	356	8	(	(	PUNCT
ejde-196	356	9	t	t	PROPN
ejde-196	356	10	,	,	PUNCT
ejde-196	356	11	u	u	NOUN
ejde-196	356	12	)	)	PUNCT
ejde-196	356	13	=	=	PUNCT
ejde-196	357	1	〈	〈	PROPN
ejde-196	357	2	j	j	PROPN
ejde-196	357	3	′(tu	′(tu	PROPN
ejde-196	357	4	)	)	PUNCT
ejde-196	357	5	,	,	PUNCT
ejde-196	357	6	u	u	NOUN
ejde-196	357	7	〉	〉	NOUN
ejde-196	357	8	=	=	SYM
ejde-196	357	9	0	0	NUM
ejde-196	357	10	,	,	PUNCT
ejde-196	357	11	when	when	SCONJ
ejde-196	357	12	f	f	PROPN
ejde-196	357	13	′t	′t	PROPN
ejde-196	357	14	=	=	SYM
ejde-196	357	15	〈	〈	PROPN
ejde-196	357	16	j	j	PROPN
ejde-196	357	17	′′(tuu)u	′′(tuu)u	PROPN
ejde-196	357	18	,	,	PUNCT
ejde-196	357	19	u	u	NOUN
ejde-196	357	20	〉	〉	PROPN
ejde-196	357	21	6=	6=	NUM
ejde-196	357	22	0	0	NUM
ejde-196	357	23	(	(	PUNCT
ejde-196	357	24	>	>	X
ejde-196	357	25	0	0	PUNCT
ejde-196	357	26	for	for	ADP
ejde-196	357	27	w	w	NOUN
ejde-196	357	28	-	-	PUNCT
ejde-196	357	29	type	type	NOUN
ejde-196	357	30	)	)	PUNCT
ejde-196	357	31	,	,	PUNCT
ejde-196	357	32	(	(	PUNCT
ejde-196	357	33	5.8	5.8	NUM
ejde-196	357	34	)	)	PUNCT
ejde-196	357	35	by	by	ADP
ejde-196	357	36	the	the	DET
ejde-196	357	37	implicit	implicit	ADJ
ejde-196	357	38	function	function	NOUN
ejde-196	357	39	theorem	theorem	VERB
ejde-196	357	40	,	,	PUNCT
ejde-196	357	41	t′(u	t′(u	PROPN
ejde-196	357	42	)	)	PUNCT
ejde-196	357	43	=	=	SYM
ejde-196	358	1	p̄′(u	p̄′(u	PROPN
ejde-196	358	2	)	)	PUNCT
ejde-196	358	3	is	be	AUX
ejde-196	358	4	actually	actually	ADV
ejde-196	358	5	locally	locally	ADV
ejde-196	358	6	c1	c1	PROPN
ejde-196	358	7	at	at	ADP
ejde-196	358	8	u.	u.	NOUN
ejde-196	358	9	for	for	ADP
ejde-196	358	10	many	many	ADJ
ejde-196	358	11	problems	problem	NOUN
ejde-196	358	12	in	in	ADP
ejde-196	358	13	application	application	NOUN
ejde-196	358	14	,	,	PUNCT
ejde-196	358	15	p̄(u	p̄(u	NOUN
ejde-196	358	16	)	)	PUNCT
ejde-196	358	17	=	=	PUNCT
ejde-196	359	1	tuu	tuu	ADV
ejde-196	359	2	has	have	VERB
ejde-196	359	3	an	an	DET
ejde-196	359	4	explicit	explicit	ADJ
ejde-196	359	5	expression	expression	NOUN
ejde-196	359	6	,	,	PUNCT
ejde-196	359	7	then	then	ADV
ejde-196	359	8	the	the	DET
ejde-196	359	9	algorithm	algorithm	NOUN
ejde-196	359	10	becomes	become	VERB
ejde-196	359	11	to	to	PART
ejde-196	359	12	solve	solve	VERB
ejde-196	359	13	minv∈s	minv∈s	PROPN
ejde-196	359	14	l⊥	l⊥	PROPN
ejde-196	359	15	maxu∈[v	maxu∈[v	NUM
ejde-196	359	16	,	,	PUNCT
ejde-196	359	17	l],‖u‖≈1	l],‖u‖≈1	NOUN
ejde-196	359	18	j(p̄(u	j(p̄(u	PROPN
ejde-196	359	19	)	)	PUNCT
ejde-196	359	20	)	)	PUNCT
ejde-196	359	21	,	,	PUNCT
ejde-196	359	22	a	a	DET
ejde-196	359	23	two	two	NUM
ejde-196	359	24	-	-	PUNCT
ejde-196	359	25	level	level	NOUN
ejde-196	359	26	local	local	ADJ
ejde-196	359	27	min	min	NOUN
ejde-196	359	28	-	-	ADJ
ejde-196	359	29	max	max	NOUN
ejde-196	359	30	algorithm	algorithm	NOUN
ejde-196	359	31	.	.	PUNCT
ejde-196	360	1	6	6	NUM
ejde-196	360	2	.	.	X
ejde-196	360	3	numerical	numerical	ADJ
ejde-196	360	4	examples	example	NOUN
ejde-196	360	5	since	since	SCONJ
ejde-196	360	6	the	the	DET
ejde-196	360	7	m	m	NOUN
ejde-196	360	8	-	-	PUNCT
ejde-196	360	9	type	type	NOUN
ejde-196	360	10	problems	problem	NOUN
ejde-196	360	11	have	have	AUX
ejde-196	360	12	been	be	AUX
ejde-196	360	13	successfully	successfully	ADV
ejde-196	360	14	solved	solve	VERB
ejde-196	360	15	before	before	ADV
ejde-196	360	16	,	,	PUNCT
ejde-196	360	17	in	in	ADP
ejde-196	360	18	this	this	DET
ejde-196	360	19	section	section	NOUN
ejde-196	360	20	,	,	PUNCT
ejde-196	360	21	we	we	PRON
ejde-196	360	22	present	present	VERB
ejde-196	360	23	numerical	numerical	ADJ
ejde-196	360	24	multiple	multiple	ADJ
ejde-196	360	25	solutions	solution	NOUN
ejde-196	360	26	to	to	ADP
ejde-196	360	27	a	a	DET
ejde-196	360	28	w	w	NOUN
ejde-196	360	29	-	-	PUNCT
ejde-196	360	30	type	type	NOUN
ejde-196	360	31	problem	problem	NOUN
ejde-196	360	32	by	by	ADP
ejde-196	360	33	the	the	DET
ejde-196	360	34	local	local	ADJ
ejde-196	360	35	min	min	ADJ
ejde-196	360	36	-	-	ADJ
ejde-196	360	37	maxmin	maxmin	ADJ
ejde-196	360	38	algorithm	algorithm	NOUN
ejde-196	360	39	(	(	PUNCT
ejde-196	360	40	a2	a2	PROPN
ejde-196	360	41	)	)	PUNCT
ejde-196	360	42	and	and	CCONJ
ejde-196	360	43	to	to	ADP
ejde-196	360	44	a	a	DET
ejde-196	360	45	mixed	mixed	ADJ
ejde-196	360	46	m	m	PROPN
ejde-196	360	47	-	-	PUNCT
ejde-196	360	48	w	w	NOUN
ejde-196	360	49	-	-	PUNCT
ejde-196	360	50	type	type	NOUN
ejde-196	360	51	problem	problem	NOUN
ejde-196	360	52	by	by	ADP
ejde-196	360	53	the	the	DET
ejde-196	360	54	local	local	ADJ
ejde-196	360	55	min	min	ADJ
ejde-196	360	56	-	-	ADJ
ejde-196	360	57	orthogonal	orthogonal	ADJ
ejde-196	360	58	algorithm	algorithm	NOUN
ejde-196	360	59	.	.	PUNCT
ejde-196	361	1	6.1	6.1	NUM
ejde-196	361	2	.	.	PUNCT
ejde-196	361	3	numerical	numerical	ADJ
ejde-196	361	4	examples	example	NOUN
ejde-196	361	5	by	by	ADP
ejde-196	361	6	the	the	DET
ejde-196	361	7	mini	mini	PROPN
ejde-196	361	8	-	-	PROPN
ejde-196	361	9	max	max	ADJ
ejde-196	361	10	-	-	PUNCT
ejde-196	361	11	min	min	NOUN
ejde-196	361	12	method	method	NOUN
ejde-196	361	13	.	.	PUNCT
ejde-196	362	1	a	a	DET
ejde-196	362	2	w	w	NOUN
ejde-196	362	3	-	-	PUNCT
ejde-196	362	4	type	type	NOUN
ejde-196	362	5	problem	problem	NOUN
ejde-196	362	6	.	.	PUNCT
ejde-196	363	1	in	in	ADP
ejde-196	363	2	(	(	PUNCT
ejde-196	363	3	1.1	1.1	NUM
ejde-196	363	4	)	)	PUNCT
ejde-196	363	5	,	,	PUNCT
ejde-196	363	6	we	we	PRON
ejde-196	363	7	set	set	VERB
ejde-196	363	8	κ	κ	NOUN
ejde-196	363	9	=	=	SYM
ejde-196	363	10	1	1	NUM
ejde-196	363	11	,	,	PUNCT
ejde-196	363	12	f	f	PROPN
ejde-196	363	13	(	(	PUNCT
ejde-196	363	14	x	x	NOUN
ejde-196	363	15	,	,	PUNCT
ejde-196	363	16	u(x	u(x	NOUN
ejde-196	363	17	)	)	PUNCT
ejde-196	363	18	)	)	PUNCT
ejde-196	364	1	=	=	SYM
ejde-196	364	2	|u(x)|p−1u(x	|u(x)|p−1u(x	NOUN
ejde-196	364	3	)	)	PUNCT
ejde-196	364	4	with	with	ADP
ejde-196	364	5	h	h	NOUN
ejde-196	364	6	=	=	PUNCT
ejde-196	364	7	h1	h1	PROPN
ejde-196	364	8	0	0	NUM
ejde-196	365	1	(	(	PUNCT
ejde-196	365	2	ω),ω	ω),ω	PROPN
ejde-196	365	3	=	=	SYM
ejde-196	365	4	(	(	PUNCT
ejde-196	365	5	0	0	NUM
ejde-196	365	6	,	,	PUNCT
ejde-196	365	7	1)2	1)2	NUM
ejde-196	365	8	⊂	⊂	PROPN
ejde-196	365	9	r2	r2	PROPN
ejde-196	365	10	and	and	CCONJ
ejde-196	365	11	p	p	NOUN
ejde-196	365	12	=	=	NOUN
ejde-196	365	13	3	3	X
ejde-196	365	14	.	.	PUNCT
ejde-196	366	1	the	the	DET
ejde-196	366	2	operator	operator	NOUN
ejde-196	366	3	−∆	−∆	NOUN
ejde-196	366	4	subject	subject	ADJ
ejde-196	366	5	to	to	ADP
ejde-196	366	6	zero	zero	NUM
ejde-196	366	7	dirichlet	dirichlet	PROPN
ejde-196	366	8	boundary	boundary	ADJ
ejde-196	366	9	condition	condition	NOUN
ejde-196	366	10	.	.	PUNCT
ejde-196	367	1	has	have	AUX
ejde-196	367	2	eigenvalues	eigenvalue	VERB
ejde-196	367	3	µi	µi	ADP
ejde-196	367	4	=	=	SYM
ejde-196	367	5	4.9348	4.9348	NUM
ejde-196	367	6	,	,	PUNCT
ejde-196	367	7	12.3370	12.3370	NUM
ejde-196	367	8	,	,	PUNCT
ejde-196	367	9	12.3370	12.3370	NUM
ejde-196	367	10	,	,	PUNCT
ejde-196	367	11	19.7392	19.7392	NUM
ejde-196	367	12	,	,	PUNCT
ejde-196	367	13	24.6740	24.6740	NUM
ejde-196	367	14	,	,	PUNCT
ejde-196	367	15	24.6740	24.6740	NUM
ejde-196	367	16	,	,	PUNCT
ejde-196	367	17	32.0762	32.0762	NUM
ejde-196	367	18	,	,	PUNCT
ejde-196	367	19	32.0762	32.0762	NUM
ejde-196	367	20	,	,	PUNCT
ejde-196	367	21	.	.	PUNCT
ejde-196	367	22	.	.	PUNCT
ejde-196	368	1	.	.	PUNCT
ejde-196	369	1	.	.	PUNCT
ejde-196	370	1	it	it	PRON
ejde-196	370	2	is	be	AUX
ejde-196	370	3	known	know	VERB
ejde-196	370	4	that	that	SCONJ
ejde-196	370	5	when	when	SCONJ
ejde-196	370	6	µn	µn	PROPN
ejde-196	370	7	<	<	X
ejde-196	370	8	λ	λ	X
ejde-196	370	9	<	<	X
ejde-196	370	10	µn+1	µn+1	PROPN
ejde-196	370	11	,	,	PUNCT
ejde-196	370	12	the	the	DET
ejde-196	370	13	problem	problem	NOUN
ejde-196	370	14	has	have	VERB
ejde-196	370	15	at	at	ADV
ejde-196	370	16	least	least	ADJ
ejde-196	370	17	n	n	PRON
ejde-196	370	18	pairs	pair	NOUN
ejde-196	370	19	of	of	ADP
ejde-196	370	20	solutions	solution	NOUN
ejde-196	370	21	.	.	PUNCT
ejde-196	371	1	the	the	DET
ejde-196	371	2	local	local	ADJ
ejde-196	371	3	min	min	PROPN
ejde-196	371	4	-	-	PUNCT
ejde-196	371	5	max	max	ADJ
ejde-196	371	6	-	-	PUNCT
ejde-196	371	7	min	min	NOUN
ejde-196	371	8	algorithm	algorithm	NOUN
ejde-196	371	9	(	(	PUNCT
ejde-196	371	10	a2	a2	PROPN
ejde-196	371	11	)	)	PUNCT
ejde-196	371	12	has	have	VERB
ejde-196	371	13	numerical	numerical	ADJ
ejde-196	371	14	solutions	solution	NOUN
ejde-196	371	15	u1	u1	PROPN
ejde-196	371	16	–	–	PUNCT
ejde-196	371	17	u10	u10	NOUN
ejde-196	371	18	shown	show	VERB
ejde-196	371	19	in	in	ADP
ejde-196	371	20	figs	fig	NOUN
ejde-196	371	21	.	.	PUNCT
ejde-196	372	1	3	3	NUM
ejde-196	372	2	-	-	SYM
ejde-196	372	3	4	4	NUM
ejde-196	372	4	,	,	PUNCT
ejde-196	372	5	and	and	CCONJ
ejde-196	372	6	their	their	PRON
ejde-196	372	7	numerical	numerical	ADJ
ejde-196	372	8	data	datum	NOUN
ejde-196	372	9	documented	document	VERB
ejde-196	372	10	in	in	ADP
ejde-196	372	11	table	table	NOUN
ejde-196	372	12	1	1	NUM
ejde-196	372	13	.	.	X
ejde-196	372	14	186	186	NUM
ejde-196	372	15	m.	m.	NOUN
ejde-196	372	16	li	li	PROPN
ejde-196	372	17	,	,	PUNCT
ejde-196	372	18	b.	b.	PROPN
ejde-196	372	19	ji	ji	PROPN
ejde-196	372	20	,	,	PUNCT
ejde-196	372	21	j.	j.	PROPN
ejde-196	372	22	zhou	zhou	PROPN
ejde-196	372	23	ejde	ejde	PROPN
ejde-196	372	24	/	/	SYM
ejde-196	372	25	si/02	si/02	PROPN
ejde-196	372	26	table	table	NOUN
ejde-196	372	27	1	1	NUM
ejde-196	372	28	.	.	PUNCT
ejde-196	372	29	w	w	NOUN
ejde-196	372	30	-	-	PUNCT
ejde-196	372	31	type	type	NOUN
ejde-196	372	32	solutions	solution	NOUN
ejde-196	372	33	nth	nth	NOUN
ejde-196	372	34	figure	figure	NOUN
ejde-196	372	35	mi	mi	PROPN
ejde-196	372	36	support	support	PROPN
ejde-196	372	37	j	j	PROPN
ejde-196	372	38	(	(	PUNCT
ejde-196	372	39	·	·	PUNCT
ejde-196	372	40	)	)	PUNCT
ejde-196	372	41	‖j	‖j	PUNCT
ejde-196	372	42	′(·)‖	′(·)‖	PUNCT
ejde-196	372	43	errmax	errmax	PROPN
ejde-196	372	44	nit	nit	PROPN
ejde-196	372	45	1	1	NUM
ejde-196	372	46	(	(	PUNCT
ejde-196	372	47	u1	u1	NOUN
ejde-196	372	48	)	)	PUNCT
ejde-196	372	49	0	0	NUM
ejde-196	373	1	na	na	ADP
ejde-196	373	2	-313.4176	-313.4176	PROPN
ejde-196	373	3	0.0002	0.0002	NUM
ejde-196	373	4	0.0044	0.0044	NUM
ejde-196	373	5	8	8	NUM
ejde-196	373	6	2	2	NUM
ejde-196	373	7	(	(	PUNCT
ejde-196	373	8	u2	u2	NOUN
ejde-196	373	9	)	)	PUNCT
ejde-196	373	10	1	1	NUM
ejde-196	373	11	1	1	NUM
ejde-196	373	12	-126.9724	-126.9724	NUM
ejde-196	373	13	0.0007	0.0007	NUM
ejde-196	373	14	0.0052	0.0052	NUM
ejde-196	373	15	10	10	NUM
ejde-196	373	16	3	3	NUM
ejde-196	373	17	(	(	PUNCT
ejde-196	373	18	u3	u3	PROPN
ejde-196	373	19	)	)	PUNCT
ejde-196	373	20	1	1	NUM
ejde-196	373	21	1	1	NUM
ejde-196	373	22	-126.9722	-126.9722	NUM
ejde-196	373	23	0.0004	0.0004	NUM
ejde-196	373	24	0.0053	0.0053	NUM
ejde-196	373	25	8	8	NUM
ejde-196	373	26	4	4	NUM
ejde-196	373	27	(	(	PUNCT
ejde-196	373	28	u4	u4	NOUN
ejde-196	373	29	)	)	PUNCT
ejde-196	373	30	1	1	NUM
ejde-196	373	31	1	1	NUM
ejde-196	373	32	-109.1923	-109.1923	NUM
ejde-196	373	33	0.0052	0.0052	NUM
ejde-196	373	34	0.0296	0.0296	NUM
ejde-196	373	35	9	9	NUM
ejde-196	373	36	5	5	NUM
ejde-196	373	37	(	(	PUNCT
ejde-196	373	38	u5	u5	PROPN
ejde-196	373	39	)	)	PUNCT
ejde-196	373	40	1	1	NUM
ejde-196	373	41	1	1	NUM
ejde-196	373	42	-109.1914	-109.1914	NUM
ejde-196	373	43	0.006	0.006	NUM
ejde-196	373	44	0.0719	0.0719	NUM
ejde-196	373	45	6	6	NUM
ejde-196	373	46	6	6	NUM
ejde-196	373	47	(	(	PUNCT
ejde-196	373	48	u6	u6	NOUN
ejde-196	373	49	)	)	PUNCT
ejde-196	373	50	2	2	NUM
ejde-196	374	1	[	[	X
ejde-196	374	2	1,2	1,2	NUM
ejde-196	374	3	]	]	PUNCT
ejde-196	374	4	-31.7034	-31.7034	NUM
ejde-196	374	5	0.001	0.001	NUM
ejde-196	374	6	0.0075	0.0075	NUM
ejde-196	374	7	7	7	NUM
ejde-196	374	8	7	7	NUM
ejde-196	374	9	(	(	PUNCT
ejde-196	374	10	u7	u7	PROPN
ejde-196	374	11	)	)	PUNCT
ejde-196	374	12	3	3	NUM
ejde-196	375	1	[	[	X
ejde-196	375	2	1,2,6	1,2,6	NUM
ejde-196	375	3	]	]	X
ejde-196	375	4	-5.1074	-5.1074	PROPN
ejde-196	375	5	0.0029	0.0029	NUM
ejde-196	375	6	0.0457	0.0457	NUM
ejde-196	375	7	5	5	NUM
ejde-196	375	8	8	8	NUM
ejde-196	375	9	(	(	PUNCT
ejde-196	375	10	u8	u8	PROPN
ejde-196	375	11	)	)	PUNCT
ejde-196	375	12	3	3	NUM
ejde-196	376	1	[	[	X
ejde-196	376	2	1,2,6	1,2,6	NUM
ejde-196	376	3	]	]	PUNCT
ejde-196	376	4	-5.1073	-5.1073	VERB
ejde-196	377	1	0.0014	0.0014	NUM
ejde-196	377	2	0.0304	0.0304	NUM
ejde-196	377	3	6	6	NUM
ejde-196	377	4	9	9	NUM
ejde-196	377	5	(	(	PUNCT
ejde-196	377	6	u9	u9	PROPN
ejde-196	377	7	)	)	PUNCT
ejde-196	377	8	3	3	NUM
ejde-196	378	1	[	[	X
ejde-196	378	2	1,2,4	1,2,4	NUM
ejde-196	378	3	]	]	X
ejde-196	378	4	-4.2985	-4.2985	NOUN
ejde-196	378	5	0.001	0.001	NUM
ejde-196	378	6	0.0215	0.0215	NUM
ejde-196	378	7	4	4	NUM
ejde-196	378	8	10	10	NUM
ejde-196	378	9	(	(	PUNCT
ejde-196	378	10	u10	u10	PROPN
ejde-196	378	11	)	)	PUNCT
ejde-196	378	12	3	3	NUM
ejde-196	379	1	[	[	X
ejde-196	379	2	1,2,4	1,2,4	X
ejde-196	379	3	]	]	X
ejde-196	379	4	-4.2607	-4.2607	VERB
ejde-196	379	5	0.0011	0.0011	NUM
ejde-196	379	6	0.0709	0.0709	NUM
ejde-196	379	7	5	5	NUM
ejde-196	379	8	(	(	PUNCT
ejde-196	379	9	u1	u1	NOUN
ejde-196	379	10	)	)	PUNCT
ejde-196	379	11	(	(	PUNCT
ejde-196	379	12	u2	u2	PROPN
ejde-196	379	13	)	)	PUNCT
ejde-196	379	14	(	(	PUNCT
ejde-196	379	15	u3	u3	PROPN
ejde-196	379	16	)	)	PUNCT
ejde-196	379	17	(	(	PUNCT
ejde-196	379	18	u4	u4	NOUN
ejde-196	379	19	)	)	PUNCT
ejde-196	379	20	figure	figure	NOUN
ejde-196	380	1	3	3	NUM
ejde-196	380	2	.	.	PUNCT
ejde-196	380	3	critical	critical	ADJ
ejde-196	380	4	points	point	NOUN
ejde-196	380	5	u1	u1	NOUN
ejde-196	380	6	–	–	PUNCT
ejde-196	380	7	u4	u4	PROPN
ejde-196	380	8	when	when	SCONJ
ejde-196	380	9	λ	λ	X
ejde-196	380	10	=	=	NOUN
ejde-196	380	11	28	28	NUM
ejde-196	380	12	as	as	SCONJ
ejde-196	380	13	stated	state	VERB
ejde-196	380	14	in	in	ADP
ejde-196	380	15	the	the	DET
ejde-196	380	16	last	last	ADJ
ejde-196	380	17	paragraph	paragraph	NOUN
ejde-196	380	18	of	of	ADP
ejde-196	380	19	section	section	NOUN
ejde-196	380	20	5	5	NUM
ejde-196	380	21	,	,	PUNCT
ejde-196	380	22	for	for	ADP
ejde-196	380	23	this	this	DET
ejde-196	380	24	problem	problem	NOUN
ejde-196	380	25	,	,	PUNCT
ejde-196	380	26	p̄(u	p̄(u	NOUN
ejde-196	380	27	)	)	PUNCT
ejde-196	380	28	has	have	VERB
ejde-196	380	29	an	an	DET
ejde-196	380	30	explicit	explicit	ADJ
ejde-196	380	31	expression	expression	NOUN
ejde-196	380	32	,	,	PUNCT
ejde-196	380	33	the	the	DET
ejde-196	380	34	local	local	ADJ
ejde-196	380	35	min	min	ADJ
ejde-196	380	36	-	-	ADJ
ejde-196	380	37	max	max	PROPN
ejde-196	380	38	method	method	NOUN
ejde-196	380	39	can	can	AUX
ejde-196	380	40	actually	actually	ADV
ejde-196	380	41	be	be	AUX
ejde-196	380	42	used	use	VERB
ejde-196	380	43	to	to	PART
ejde-196	380	44	find	find	VERB
ejde-196	380	45	multiple	multiple	ADJ
ejde-196	380	46	solutions	solution	NOUN
ejde-196	380	47	.	.	PUNCT
ejde-196	381	1	much	much	ADV
ejde-196	381	2	more	more	ADJ
ejde-196	381	3	numerical	numerical	ADJ
ejde-196	381	4	details	detail	NOUN
ejde-196	381	5	can	can	AUX
ejde-196	381	6	be	be	AUX
ejde-196	381	7	found	find	VERB
ejde-196	381	8	in	in	ADP
ejde-196	381	9	[	[	X
ejde-196	381	10	3	3	NUM
ejde-196	381	11	]	]	PUNCT
ejde-196	381	12	.	.	PUNCT
ejde-196	382	1	6.2	6.2	NUM
ejde-196	382	2	.	.	PUNCT
ejde-196	383	1	numerical	numerical	ADJ
ejde-196	383	2	examples	example	NOUN
ejde-196	383	3	by	by	ADP
ejde-196	383	4	the	the	DET
ejde-196	383	5	min	min	ADJ
ejde-196	383	6	-	-	ADJ
ejde-196	383	7	orthogonal	orthogonal	ADJ
ejde-196	383	8	method	method	NOUN
ejde-196	383	9	.	.	PUNCT
ejde-196	384	1	a	a	DET
ejde-196	384	2	mixed	mixed	ADJ
ejde-196	384	3	m	m	NOUN
ejde-196	384	4	-	-	PUNCT
ejde-196	384	5	wtype	wtype	NOUN
ejde-196	384	6	problem	problem	NOUN
ejde-196	384	7	.	.	PUNCT
ejde-196	385	1	consider	consider	VERB
ejde-196	385	2	numerically	numerically	ADV
ejde-196	385	3	solutions	solution	NOUN
ejde-196	385	4	to	to	ADP
ejde-196	385	5	the	the	DET
ejde-196	385	6	problem	problem	NOUN
ejde-196	385	7	mixed	mix	VERB
ejde-196	385	8	with	with	ADP
ejde-196	385	9	concave	concave	NOUN
ejde-196	385	10	and	and	CCONJ
ejde-196	385	11	convex	convex	NOUN
ejde-196	385	12	nonlinearities	nonlinearitie	NOUN
ejde-196	385	13	[	[	X
ejde-196	385	14	1	1	NUM
ejde-196	385	15	,	,	PUNCT
ejde-196	385	16	2	2	NUM
ejde-196	385	17	,	,	PUNCT
ejde-196	385	18	11	11	NUM
ejde-196	385	19	]	]	PUNCT
ejde-196	386	1	−∆u+	−∆u+	NOUN
ejde-196	387	1	λ(x)u−	λ(x)u−	INTJ
ejde-196	387	2	a|u(x)|q−1u(x)−	a|u(x)|q−1u(x)−	PROPN
ejde-196	387	3	b|u(x)|p−1u(x	b|u(x)|p−1u(x	PROPN
ejde-196	387	4	)	)	PUNCT
ejde-196	388	1	=	=	SYM
ejde-196	388	2	0	0	NUM
ejde-196	388	3	,	,	PUNCT
ejde-196	388	4	(	(	PUNCT
ejde-196	388	5	6.1	6.1	NUM
ejde-196	388	6	)	)	PUNCT
ejde-196	388	7	ejde-2023	ejde-2023	NOUN
ejde-196	388	8	/	/	SYM
ejde-196	388	9	si/02	si/02	ADJ
ejde-196	388	10	short	short	ADJ
ejde-196	388	11	title	title	NOUN
ejde-196	388	12	187	187	NUM
ejde-196	388	13	(	(	PUNCT
ejde-196	388	14	u5	u5	PROPN
ejde-196	388	15	)	)	PUNCT
ejde-196	388	16	(	(	PUNCT
ejde-196	388	17	u6	u6	PROPN
ejde-196	388	18	)	)	PUNCT
ejde-196	388	19	(	(	PUNCT
ejde-196	388	20	u7	u7	PROPN
ejde-196	388	21	)	)	PUNCT
ejde-196	388	22	(	(	PUNCT
ejde-196	388	23	u8	u8	PROPN
ejde-196	388	24	)	)	PUNCT
ejde-196	388	25	(	(	PUNCT
ejde-196	388	26	u9	u9	PROPN
ejde-196	388	27	)	)	PUNCT
ejde-196	388	28	(	(	PUNCT
ejde-196	388	29	u10	u10	PROPN
ejde-196	388	30	)	)	PUNCT
ejde-196	388	31	figure	figure	NOUN
ejde-196	388	32	4	4	NUM
ejde-196	388	33	.	.	PUNCT
ejde-196	388	34	critical	critical	ADJ
ejde-196	388	35	points	point	NOUN
ejde-196	388	36	u5	u5	PROPN
ejde-196	388	37	–	–	PUNCT
ejde-196	388	38	u10	u10	NOUN
ejde-196	388	39	when	when	SCONJ
ejde-196	388	40	λ	λ	X
ejde-196	388	41	=	=	NOUN
ejde-196	388	42	28	28	NUM
ejde-196	388	43	where	where	SCONJ
ejde-196	388	44	u	u	PROPN
ejde-196	388	45	∈	∈	NOUN
ejde-196	388	46	h	h	NOUN
ejde-196	388	47	=	=	PUNCT
ejde-196	388	48	h1	h1	PROPN
ejde-196	388	49	0	0	NUM
ejde-196	388	50	(	(	PUNCT
ejde-196	388	51	ω	ω	NOUN
ejde-196	388	52	)	)	PUNCT
ejde-196	388	53	,	,	PUNCT
ejde-196	388	54	ω	ω	PROPN
ejde-196	388	55	⊂	⊂	PROPN
ejde-196	388	56	rn	rn	PROPN
ejde-196	388	57	is	be	AUX
ejde-196	388	58	open	open	ADJ
ejde-196	388	59	bounded	bound	VERB
ejde-196	388	60	,	,	PUNCT
ejde-196	388	61	0	0	PUNCT
ejde-196	388	62	<	<	X
ejde-196	388	63	q	q	X
ejde-196	388	64	<	<	X
ejde-196	388	65	1	1	NUM
ejde-196	388	66	<	<	X
ejde-196	388	67	p	p	X
ejde-196	388	68	<	<	X
ejde-196	388	69	2∗	2∗	PROPN
ejde-196	388	70	,	,	PUNCT
ejde-196	388	71	2∗	2∗	NUM
ejde-196	389	1	=	=	SYM
ejde-196	389	2	n+2	n+2	NUM
ejde-196	389	3	n−2	n−2	PROPN
ejde-196	389	4	if	if	SCONJ
ejde-196	389	5	n	n	PRON
ejde-196	389	6	≥	≥	NOUN
ejde-196	389	7	3	3	NUM
ejde-196	389	8	and	and	CCONJ
ejde-196	389	9	2∗	2∗	NUM
ejde-196	389	10	=	=	SYM
ejde-196	389	11	∞	∞	NOUN
ejde-196	389	12	if	if	SCONJ
ejde-196	389	13	n	n	NOUN
ejde-196	389	14	≤	≤	ADV
ejde-196	389	15	2	2	NUM
ejde-196	389	16	,	,	PUNCT
ejde-196	389	17	a	a	DET
ejde-196	389	18	and	and	CCONJ
ejde-196	389	19	b	b	NOUN
ejde-196	389	20	are	be	AUX
ejde-196	389	21	nonnegative	nonnegative	ADJ
ejde-196	389	22	functions	function	NOUN
ejde-196	389	23	on	on	ADP
ejde-196	389	24	ω	ω	PROPN
ejde-196	389	25	.	.	PUNCT
ejde-196	390	1	its	its	PRON
ejde-196	390	2	energy	energy	NOUN
ejde-196	390	3	functional	functional	NOUN
ejde-196	390	4	is	be	AUX
ejde-196	390	5	j(u	j(u	PROPN
ejde-196	390	6	)	)	PUNCT
ejde-196	391	1	=	=	SYM
ejde-196	392	1	∫	∫	PROPN
ejde-196	392	2	ω	ω	PROPN
ejde-196	393	1	[	[	X
ejde-196	393	2	1	1	NUM
ejde-196	393	3	2	2	NUM
ejde-196	393	4	|∇u(x)|2	|∇u(x)|2	NOUN
ejde-196	393	5	+	+	CCONJ
ejde-196	393	6	1	1	NUM
ejde-196	393	7	2	2	NUM
ejde-196	393	8	λ(x)u2(x)−	λ(x)u2(x)−	SYM
ejde-196	393	9	a(x	a(x	PROPN
ejde-196	393	10	)	)	PUNCT
ejde-196	393	11	q	q	NOUN
ejde-196	394	1	+	+	NUM
ejde-196	394	2	1	1	NUM
ejde-196	394	3	|u(x)|q+1	|u(x)|q+1	PROPN
ejde-196	394	4	−	−	PRON
ejde-196	394	5	b(x	b(x	PROPN
ejde-196	394	6	)	)	PUNCT
ejde-196	394	7	p+	p+	AUX
ejde-196	394	8	1	1	NUM
ejde-196	394	9	|u(x)|p+1	|u(x)|p+1	NOUN
ejde-196	394	10	]	]	X
ejde-196	394	11	dx	dx	PROPN
ejde-196	394	12	.	.	PUNCT
ejde-196	395	1	(	(	PUNCT
ejde-196	395	2	6.2	6.2	NUM
ejde-196	395	3	)	)	PUNCT
ejde-196	395	4	we	we	PRON
ejde-196	395	5	note	note	VERB
ejde-196	395	6	that	that	SCONJ
ejde-196	395	7	its	its	PRON
ejde-196	395	8	function	function	NOUN
ejde-196	395	9	profile	profile	NOUN
ejde-196	395	10	shows	show	VERB
ejde-196	395	11	a	a	DET
ejde-196	395	12	mixed	mixed	ADJ
ejde-196	395	13	locally	locally	ADV
ejde-196	395	14	m	m	NOUN
ejde-196	395	15	-	-	NOUN
ejde-196	395	16	type	type	NOUN
ejde-196	395	17	and	and	CCONJ
ejde-196	395	18	locally	locally	ADV
ejde-196	395	19	w	w	NOUN
ejde-196	395	20	-	-	PUNCT
ejde-196	395	21	type	type	NOUN
ejde-196	395	22	feature	feature	NOUN
ejde-196	395	23	,	,	PUNCT
ejde-196	395	24	see	see	VERB
ejde-196	395	25	figure	figure	NOUN
ejde-196	395	26	2	2	NUM
ejde-196	395	27	,	,	PUNCT
ejde-196	395	28	and	and	CCONJ
ejde-196	395	29	its	its	PRON
ejde-196	395	30	nehari	nehari	NOUN
ejde-196	395	31	manifold	manifold	NOUN
ejde-196	395	32	n	n	CCONJ
ejde-196	395	33	defined	define	VERB
ejde-196	395	34	in	in	ADP
ejde-196	395	35	(	(	PUNCT
ejde-196	395	36	1.7	1.7	NUM
ejde-196	395	37	)	)	PUNCT
ejde-196	395	38	consists	consist	VERB
ejde-196	395	39	of	of	ADP
ejde-196	395	40	two	two	NUM
ejde-196	395	41	branches	branch	NOUN
ejde-196	395	42	,	,	PUNCT
ejde-196	395	43	one	one	NUM
ejde-196	395	44	is	be	AUX
ejde-196	395	45	a	a	DET
ejde-196	395	46	local	local	ADJ
ejde-196	395	47	m	m	NOUN
ejde-196	395	48	-	-	NOUN
ejde-196	395	49	type	type	NOUN
ejde-196	395	50	where	where	SCONJ
ejde-196	395	51	j	j	NOUN
ejde-196	395	52	-	-	PUNCT
ejde-196	395	53	values	value	NOUN
ejde-196	395	54	are	be	AUX
ejde-196	395	55	positive	positive	ADJ
ejde-196	395	56	and	and	CCONJ
ejde-196	395	57	the	the	DET
ejde-196	395	58	other	other	ADJ
ejde-196	395	59	is	be	AUX
ejde-196	395	60	a	a	DET
ejde-196	395	61	local	local	ADJ
ejde-196	395	62	w	w	NOUN
ejde-196	395	63	-	-	PUNCT
ejde-196	395	64	type	type	NOUN
ejde-196	395	65	where	where	SCONJ
ejde-196	395	66	j	j	NOUN
ejde-196	395	67	-	-	PUNCT
ejde-196	395	68	values	value	NOUN
ejde-196	395	69	are	be	AUX
ejde-196	395	70	negative	negative	ADJ
ejde-196	395	71	.	.	PUNCT
ejde-196	396	1	usually	usually	ADV
ejde-196	396	2	two	two	NUM
ejde-196	396	3	different	different	ADJ
ejde-196	396	4	variational	variational	ADJ
ejde-196	396	5	methods	method	NOUN
ejde-196	396	6	,	,	PUNCT
ejde-196	396	7	e.g.	e.g.	ADV
ejde-196	396	8	,	,	PUNCT
ejde-196	396	9	the	the	DET
ejde-196	396	10	local	local	ADJ
ejde-196	396	11	min	min	ADJ
ejde-196	396	12	-	-	ADJ
ejde-196	396	13	max	max	PROPN
ejde-196	396	14	method	method	NOUN
ejde-196	396	15	for	for	ADP
ejde-196	396	16	the	the	DET
ejde-196	396	17	locally	locally	ADJ
ejde-196	396	18	m	m	NOUN
ejde-196	396	19	-	-	PUNCT
ejde-196	396	20	type	type	NOUN
ejde-196	396	21	branch	branch	NOUN
ejde-196	396	22	and	and	CCONJ
ejde-196	396	23	the	the	DET
ejde-196	396	24	min	min	PROPN
ejde-196	396	25	-	-	PUNCT
ejde-196	396	26	minmax	minmax	ADJ
ejde-196	396	27	algorithm	algorithm	NOUN
ejde-196	396	28	(	(	PUNCT
ejde-196	396	29	a1	a1	PROPN
ejde-196	396	30	)	)	PUNCT
ejde-196	396	31	for	for	ADP
ejde-196	396	32	the	the	DET
ejde-196	396	33	locally	locally	ADV
ejde-196	396	34	w	w	NOUN
ejde-196	396	35	-	-	PUNCT
ejde-196	396	36	type	type	NOUN
ejde-196	396	37	branch	branch	NOUN
ejde-196	396	38	have	have	VERB
ejde-196	396	39	to	to	PART
ejde-196	396	40	be	be	AUX
ejde-196	396	41	used	use	VERB
ejde-196	396	42	separately	separately	ADV
ejde-196	396	43	.	.	PUNCT
ejde-196	397	1	we	we	PRON
ejde-196	397	2	use	use	VERB
ejde-196	397	3	the	the	DET
ejde-196	397	4	min	min	ADJ
ejde-196	397	5	-	-	ADJ
ejde-196	397	6	orthogonal	orthogonal	ADJ
ejde-196	397	7	method	method	NOUN
ejde-196	397	8	described	describe	VERB
ejde-196	397	9	in	in	ADP
ejde-196	397	10	section	section	NOUN
ejde-196	397	11	3	3	NUM
ejde-196	397	12	,	,	PUNCT
ejde-196	397	13	since	since	SCONJ
ejde-196	397	14	the	the	DET
ejde-196	397	15	⊥-operation	⊥-operation	NOUN
ejde-196	397	16	does	do	AUX
ejde-196	397	17	not	not	PART
ejde-196	397	18	differentiate	differentiate	VERB
ejde-196	397	19	and	and	CCONJ
ejde-196	397	20	can	can	AUX
ejde-196	397	21	treat	treat	VERB
ejde-196	397	22	both	both	DET
ejde-196	397	23	types	type	NOUN
ejde-196	397	24	.	.	PUNCT
ejde-196	398	1	the	the	DET
ejde-196	398	2	difficulty	difficulty	NOUN
ejde-196	398	3	is	be	AUX
ejde-196	398	4	how	how	SCONJ
ejde-196	398	5	the	the	DET
ejde-196	398	6	algorithm	algorithm	NOUN
ejde-196	398	7	188	188	NUM
ejde-196	398	8	m.	m.	NOUN
ejde-196	398	9	li	li	PROPN
ejde-196	398	10	,	,	PUNCT
ejde-196	398	11	b.	b.	PROPN
ejde-196	398	12	ji	ji	PROPN
ejde-196	398	13	,	,	PUNCT
ejde-196	398	14	j.	j.	PROPN
ejde-196	398	15	zhou	zhou	PROPN
ejde-196	398	16	ejde	ejde	PROPN
ejde-196	398	17	/	/	SYM
ejde-196	398	18	si/02	si/02	PROPN
ejde-196	398	19	search	search	NOUN
ejde-196	398	20	will	will	AUX
ejde-196	398	21	consistently	consistently	ADV
ejde-196	398	22	stay	stay	VERB
ejde-196	398	23	on	on	ADP
ejde-196	398	24	the	the	DET
ejde-196	398	25	desired	desire	VERB
ejde-196	398	26	branch	branch	NOUN
ejde-196	398	27	.	.	PUNCT
ejde-196	399	1	to	to	PART
ejde-196	399	2	resolve	resolve	VERB
ejde-196	399	3	this	this	DET
ejde-196	399	4	difficulty	difficulty	NOUN
ejde-196	399	5	,	,	PUNCT
ejde-196	399	6	in	in	ADP
ejde-196	399	7	our	our	PRON
ejde-196	399	8	numerical	numerical	ADJ
ejde-196	399	9	computation	computation	NOUN
ejde-196	399	10	we	we	PRON
ejde-196	399	11	add	add	VERB
ejde-196	399	12	a	a	DET
ejde-196	399	13	constraint	constraint	NOUN
ejde-196	399	14	j	j	PROPN
ejde-196	399	15	>	>	PUNCT
ejde-196	399	16	0	0	PUNCT
ejde-196	399	17	or	or	CCONJ
ejde-196	399	18	j	j	ADJ
ejde-196	399	19	<	<	X
ejde-196	399	20	0	0	PUNCT
ejde-196	399	21	to	to	ADP
ejde-196	399	22	the	the	DET
ejde-196	399	23	⊥-operation	⊥-operation	NOUN
ejde-196	399	24	.	.	PUNCT
ejde-196	400	1	case	case	NOUN
ejde-196	400	2	1	1	X
ejde-196	400	3	.	.	PUNCT
ejde-196	401	1	we	we	PRON
ejde-196	401	2	set	set	VERB
ejde-196	401	3	λ(x	λ(x	PRON
ejde-196	401	4	)	)	PUNCT
ejde-196	402	1	=	=	SYM
ejde-196	402	2	0	0	NUM
ejde-196	402	3	,	,	PUNCT
ejde-196	402	4	a(x	a(x	PROPN
ejde-196	402	5	)	)	PUNCT
ejde-196	402	6	=	=	NOUN
ejde-196	402	7	1.4	1.4	NUM
ejde-196	402	8	,	,	PUNCT
ejde-196	402	9	b(x	b(x	NOUN
ejde-196	402	10	)	)	PUNCT
ejde-196	402	11	=	=	SYM
ejde-196	402	12	1	1	NUM
ejde-196	402	13	,	,	PUNCT
ejde-196	402	14	p	p	NOUN
ejde-196	402	15	=	=	NOUN
ejde-196	402	16	4	4	NUM
ejde-196	402	17	,	,	PUNCT
ejde-196	402	18	q	q	NOUN
ejde-196	402	19	=	=	NOUN
ejde-196	402	20	0.05	0.05	NUM
ejde-196	402	21	,	,	PUNCT
ejde-196	402	22	r	r	NOUN
ejde-196	402	23	=	=	SYM
ejde-196	402	24	0	0	NUM
ejde-196	402	25	,	,	PUNCT
ejde-196	402	26	and	and	CCONJ
ejde-196	402	27	ω	ω	X
ejde-196	402	28	=	=	SYM
ejde-196	402	29	(	(	PUNCT
ejde-196	402	30	−1	−1	NOUN
ejde-196	402	31	,	,	PUNCT
ejde-196	402	32	1)2	1)2	NUM
ejde-196	402	33	.	.	PUNCT
ejde-196	403	1	locally	locally	ADV
ejde-196	403	2	w	w	NOUN
ejde-196	403	3	-	-	PUNCT
ejde-196	403	4	type	type	NOUN
ejde-196	403	5	saddles	saddle	NOUN
ejde-196	403	6	with	with	ADP
ejde-196	403	7	j	j	PROPN
ejde-196	403	8	<	<	X
ejde-196	403	9	0	0	NUM
ejde-196	403	10	.	.	PUNCT
ejde-196	404	1	the	the	DET
ejde-196	404	2	algorithm	algorithm	NOUN
ejde-196	404	3	finds	find	VERB
ejde-196	404	4	the	the	DET
ejde-196	404	5	solutions	solution	NOUN
ejde-196	404	6	in	in	ADP
ejde-196	404	7	figure	figure	NOUN
ejde-196	404	8	5	5	NUM
ejde-196	404	9	and	and	CCONJ
ejde-196	404	10	generates	generate	VERB
ejde-196	404	11	the	the	DET
ejde-196	404	12	numerical	numerical	ADJ
ejde-196	404	13	data	datum	NOUN
ejde-196	404	14	in	in	ADP
ejde-196	404	15	table	table	NOUN
ejde-196	404	16	2	2	NUM
ejde-196	404	17	.	.	PUNCT
ejde-196	404	18	table	table	NOUN
ejde-196	404	19	2	2	NUM
ejde-196	404	20	.	.	PUNCT
ejde-196	404	21	numerical	numerical	ADJ
ejde-196	404	22	data	datum	NOUN
ejde-196	404	23	for	for	ADP
ejde-196	404	24	w	w	NOUN
ejde-196	404	25	-	-	PUNCT
ejde-196	404	26	type	type	NOUN
ejde-196	404	27	saddles	saddle	NOUN
ejde-196	404	28	‖d‖	‖d‖	SYM
ejde-196	404	29	ε	ε	PROPN
ejde-196	404	30	‖j	‖j	PROPN
ejde-196	405	1	′(·)‖∞	′(·)‖∞	PROPN
ejde-196	405	2	j	j	PROPN
ejde-196	405	3	(	(	PUNCT
ejde-196	405	4	·	·	PUNCT
ejde-196	405	5	)	)	PUNCT
ejde-196	406	1	nit	nit	NOUN
ejde-196	406	2	a	a	DET
ejde-196	406	3	local	local	ADJ
ejde-196	406	4	min	min	NOUN
ejde-196	406	5	0.0007	0.0007	NUM
ejde-196	406	6	1e-3	1e-3	NUM
ejde-196	406	7	0.0031	0.0031	NUM
ejde-196	406	8	-0.4314	-0.4314	NOUN
ejde-196	406	9	9	9	NUM
ejde-196	406	10	1	1	NUM
ejde-196	406	11	-	-	PUNCT
ejde-196	406	12	saddle	saddle	NOUN
ejde-196	406	13	1	1	NUM
ejde-196	406	14	0.0008	0.0008	NUM
ejde-196	406	15	1e-3	1e-3	NUM
ejde-196	406	16	1.7330	1.7330	NUM
ejde-196	406	17	-0.1589	-0.1589	NOUN
ejde-196	406	18	8	8	NUM
ejde-196	406	19	1	1	NUM
ejde-196	406	20	-	-	PUNCT
ejde-196	406	21	saddle	saddle	NOUN
ejde-196	406	22	2	2	NUM
ejde-196	406	23	0.0064	0.0064	NUM
ejde-196	406	24	7e-3	7e-3	NUM
ejde-196	406	25	1.9031	1.9031	NUM
ejde-196	406	26	-0.1445	-0.1445	NOUN
ejde-196	406	27	6	6	NUM
ejde-196	406	28	2	2	NUM
ejde-196	406	29	-	-	PUNCT
ejde-196	406	30	saddle	saddle	NOUN
ejde-196	406	31	1	1	NUM
ejde-196	406	32	0.0009	0.0009	NUM
ejde-196	406	33	1e-3	1e-3	NUM
ejde-196	406	34	1.6276	1.6276	NUM
ejde-196	406	35	-0.0930	-0.0930	NOUN
ejde-196	406	36	7	7	NUM
ejde-196	406	37	2	2	NUM
ejde-196	406	38	-	-	PUNCT
ejde-196	406	39	saddle	saddle	NOUN
ejde-196	406	40	2	2	NUM
ejde-196	406	41	0.0008	0.0008	NUM
ejde-196	406	42	1e-3	1e-3	NUM
ejde-196	406	43	1.6924	1.6924	NUM
ejde-196	406	44	-0.0671	-0.0671	NOUN
ejde-196	406	45	8	8	NUM
ejde-196	406	46	3	3	NUM
ejde-196	406	47	-	-	PUNCT
ejde-196	406	48	saddle	saddle	NOUN
ejde-196	406	49	0.0007	0.0007	NUM
ejde-196	406	50	1e-3	1e-3	NUM
ejde-196	406	51	1.7164	1.7164	NUM
ejde-196	406	52	-0.0756	-0.0756	NOUN
ejde-196	406	53	8	8	NUM
ejde-196	406	54	j	j	NOUN
ejde-196	406	55	=	=	SYM
ejde-196	406	56	−0.4314	−0.4314	PROPN
ejde-196	406	57	,	,	PUNCT
ejde-196	406	58	‖u‖∞	‖u‖∞	PROPN
ejde-196	406	59	=	=	SYM
ejde-196	406	60	0.3898	0.3898	NUM
ejde-196	406	61	at	at	ADP
ejde-196	406	62	(	(	PUNCT
ejde-196	406	63	−0.0005,−0.0032	−0.0005,−0.0032	NOUN
ejde-196	406	64	)	)	PUNCT
ejde-196	406	65	j	j	PROPN
ejde-196	406	66	=	=	SYM
ejde-196	406	67	−0.1589	−0.1589	PROPN
ejde-196	406	68	,	,	PUNCT
ejde-196	406	69	‖u‖∞	‖u‖∞	PROPN
ejde-196	406	70	=	=	SYM
ejde-196	406	71	0.1428	0.1428	NUM
ejde-196	406	72	at	at	ADP
ejde-196	406	73	(	(	PUNCT
ejde-196	406	74	0.5003	0.5003	NUM
ejde-196	406	75	,	,	PUNCT
ejde-196	406	76	0.0031	0.0031	NUM
ejde-196	406	77	)	)	PUNCT
ejde-196	406	78	j	j	PROPN
ejde-196	406	79	=	=	SYM
ejde-196	406	80	−0.1445	−0.1445	PROPN
ejde-196	406	81	,	,	PUNCT
ejde-196	407	1	‖u‖∞	‖u‖∞	PROPN
ejde-196	407	2	=	=	PUNCT
ejde-196	407	3	0.1497	0.1497	NUM
ejde-196	407	4	at	at	ADP
ejde-196	407	5	(	(	PUNCT
ejde-196	407	6	−0.3853,−0.3897	−0.3853,−0.3897	PROPN
ejde-196	407	7	)	)	PUNCT
ejde-196	407	8	(	(	PUNCT
ejde-196	407	9	a	a	X
ejde-196	407	10	)	)	PUNCT
ejde-196	407	11	a	a	DET
ejde-196	407	12	local	local	ADJ
ejde-196	407	13	min	min	NOUN
ejde-196	407	14	(	(	PUNCT
ejde-196	407	15	b	b	NOUN
ejde-196	407	16	)	)	PUNCT
ejde-196	407	17	1	1	NUM
ejde-196	407	18	-	-	NUM
ejde-196	407	19	saddle	saddle	NOUN
ejde-196	407	20	1	1	NUM
ejde-196	407	21	(	(	PUNCT
ejde-196	407	22	c	c	NOUN
ejde-196	407	23	)	)	PUNCT
ejde-196	407	24	1	1	NUM
ejde-196	407	25	-	-	NUM
ejde-196	407	26	saddle	saddle	NOUN
ejde-196	407	27	2	2	NUM
ejde-196	407	28	j	j	NOUN
ejde-196	407	29	=	=	SYM
ejde-196	407	30	−0.0930	−0.0930	PROPN
ejde-196	407	31	,	,	PUNCT
ejde-196	407	32	‖u‖∞	‖u‖∞	PROPN
ejde-196	407	33	=	=	SYM
ejde-196	407	34	0.0901	0.0901	NUM
ejde-196	407	35	at	at	ADP
ejde-196	407	36	(	(	PUNCT
ejde-196	407	37	−0.5012	−0.5012	NUM
ejde-196	407	38	,	,	PUNCT
ejde-196	407	39	0.4955	0.4955	NUM
ejde-196	407	40	)	)	PUNCT
ejde-196	407	41	,	,	PUNCT
ejde-196	407	42	sid	sid	PROPN
ejde-196	407	43	=	=	SYM
ejde-196	407	44	a	a	PROPN
ejde-196	407	45	,	,	PUNCT
ejde-196	407	46	b	b	PROPN
ejde-196	407	47	j	j	PROPN
ejde-196	407	48	=	=	SYM
ejde-196	407	49	−0.0671	−0.0671	PROPN
ejde-196	407	50	,	,	PUNCT
ejde-196	407	51	‖u‖∞	‖u‖∞	NUM
ejde-196	407	52	=	=	SYM
ejde-196	407	53	0.0714	0.0714	NUM
ejde-196	407	54	at	at	ADP
ejde-196	407	55	(	(	PUNCT
ejde-196	407	56	0.0018,−0.6103	0.0018,−0.6103	NOUN
ejde-196	407	57	)	)	PUNCT
ejde-196	407	58	,	,	PUNCT
ejde-196	407	59	sid=	sid=	PROPN
ejde-196	407	60	a	a	PROPN
ejde-196	407	61	,	,	PUNCT
ejde-196	407	62	d	d	X
ejde-196	407	63	j	j	PROPN
ejde-196	407	64	=	=	SYM
ejde-196	407	65	−0.0756	−0.0756	PROPN
ejde-196	407	66	,	,	PUNCT
ejde-196	407	67	‖u‖∞	‖u‖∞	NUM
ejde-196	407	68	=	=	NUM
ejde-196	407	69	0.0662	0.0662	NUM
ejde-196	407	70	at	at	ADP
ejde-196	407	71	(	(	PUNCT
ejde-196	407	72	0.6662	0.6662	NUM
ejde-196	407	73	,	,	PUNCT
ejde-196	407	74	0.0038	0.0038	NUM
ejde-196	407	75	)	)	PUNCT
ejde-196	407	76	,	,	PUNCT
ejde-196	407	77	sid	sid	PROPN
ejde-196	407	78	=	=	PUNCT
ejde-196	407	79	a	a	PROPN
ejde-196	407	80	,	,	PUNCT
ejde-196	407	81	b	b	NOUN
ejde-196	407	82	,	,	PUNCT
ejde-196	407	83	d	d	X
ejde-196	407	84	(	(	PUNCT
ejde-196	407	85	d	d	NOUN
ejde-196	407	86	)	)	PUNCT
ejde-196	407	87	2	2	NUM
ejde-196	407	88	-	-	NUM
ejde-196	407	89	saddle	saddle	NOUN
ejde-196	407	90	1	1	NUM
ejde-196	407	91	(	(	PUNCT
ejde-196	407	92	e	e	NOUN
ejde-196	407	93	)	)	PUNCT
ejde-196	407	94	2	2	NUM
ejde-196	407	95	-	-	NUM
ejde-196	407	96	saddle	saddle	NOUN
ejde-196	407	97	2	2	NUM
ejde-196	407	98	(	(	PUNCT
ejde-196	407	99	f	f	NOUN
ejde-196	407	100	)	)	PUNCT
ejde-196	407	101	3	3	NUM
ejde-196	407	102	-	-	PUNCT
ejde-196	407	103	saddle	saddle	NOUN
ejde-196	407	104	figure	figure	NOUN
ejde-196	407	105	5	5	NUM
ejde-196	407	106	.	.	PUNCT
ejde-196	408	1	locally	locally	ADV
ejde-196	408	2	w	w	NOUN
ejde-196	408	3	-	-	PUNCT
ejde-196	408	4	type	type	NOUN
ejde-196	408	5	saddles	saddle	NOUN
ejde-196	408	6	with	with	ADP
ejde-196	408	7	with	with	ADP
ejde-196	408	8	j	j	PROPN
ejde-196	408	9	<	<	X
ejde-196	408	10	0	0	PUNCT
ejde-196	409	1	locally	locally	ADV
ejde-196	409	2	m	m	NOUN
ejde-196	409	3	-	-	PUNCT
ejde-196	409	4	type	type	NOUN
ejde-196	409	5	saddles	saddle	NOUN
ejde-196	409	6	with	with	ADP
ejde-196	409	7	j	j	PROPN
ejde-196	409	8	>	>	X
ejde-196	409	9	0	0	NUM
ejde-196	410	1	the	the	DET
ejde-196	410	2	algorithm	algorithm	NOUN
ejde-196	410	3	finds	find	VERB
ejde-196	410	4	the	the	DET
ejde-196	410	5	solutions	solution	NOUN
ejde-196	410	6	in	in	ADP
ejde-196	410	7	figure	figure	NOUN
ejde-196	410	8	6	6	NUM
ejde-196	410	9	and	and	CCONJ
ejde-196	410	10	generates	generate	VERB
ejde-196	410	11	the	the	DET
ejde-196	410	12	numerical	numerical	ADJ
ejde-196	410	13	data	datum	NOUN
ejde-196	410	14	in	in	ADP
ejde-196	410	15	table	table	NOUN
ejde-196	410	16	3	3	NUM
ejde-196	410	17	.	.	PUNCT
ejde-196	410	18	case	case	NOUN
ejde-196	410	19	2	2	NUM
ejde-196	410	20	.	.	PUNCT
ejde-196	410	21	to	to	PART
ejde-196	410	22	investigate	investigate	VERB
ejde-196	410	23	possible	possible	ADJ
ejde-196	410	24	bifurcation	bifurcation	NOUN
ejde-196	410	25	phenomenon	phenomenon	NOUN
ejde-196	410	26	,	,	PUNCT
ejde-196	410	27	we	we	PRON
ejde-196	410	28	set	set	VERB
ejde-196	410	29	λ(x	λ(x	PRON
ejde-196	410	30	)	)	PUNCT
ejde-196	411	1	=	=	SYM
ejde-196	411	2	0	0	NUM
ejde-196	411	3	,	,	PUNCT
ejde-196	411	4	a(x	a(x	PROPN
ejde-196	411	5	)	)	PUNCT
ejde-196	411	6	=	=	NOUN
ejde-196	411	7	1.4	1.4	NUM
ejde-196	411	8	,	,	PUNCT
ejde-196	411	9	b(x	b(x	NOUN
ejde-196	411	10	)	)	PUNCT
ejde-196	411	11	=	=	SYM
ejde-196	411	12	1	1	NUM
ejde-196	411	13	,	,	PUNCT
ejde-196	411	14	p	p	NOUN
ejde-196	411	15	=	=	NOUN
ejde-196	411	16	4	4	NUM
ejde-196	411	17	,	,	PUNCT
ejde-196	411	18	q	q	NOUN
ejde-196	411	19	=	=	NOUN
ejde-196	411	20	0.05	0.05	NUM
ejde-196	411	21	,	,	PUNCT
ejde-196	411	22	r	r	NOUN
ejde-196	411	23	=	=	SYM
ejde-196	411	24	4	4	NUM
ejde-196	411	25	,	,	PUNCT
ejde-196	411	26	and	and	CCONJ
ejde-196	411	27	ω	ω	X
ejde-196	411	28	=	=	SYM
ejde-196	411	29	(	(	PUNCT
ejde-196	411	30	−1	−1	NOUN
ejde-196	411	31	,	,	PUNCT
ejde-196	411	32	1)2	1)2	NUM
ejde-196	411	33	.	.	PUNCT
ejde-196	412	1	ejde-2023	ejde-2023	ADJ
ejde-196	412	2	/	/	SYM
ejde-196	412	3	si/02	si/02	ADJ
ejde-196	412	4	short	short	ADJ
ejde-196	412	5	title	title	NOUN
ejde-196	412	6	189	189	NUM
ejde-196	412	7	table	table	NOUN
ejde-196	412	8	3	3	NUM
ejde-196	412	9	.	.	PUNCT
ejde-196	412	10	numerical	numerical	ADJ
ejde-196	412	11	data	datum	NOUN
ejde-196	412	12	for	for	ADP
ejde-196	412	13	locally	locally	ADV
ejde-196	412	14	m	m	NOUN
ejde-196	412	15	-	-	PUNCT
ejde-196	412	16	type	type	NOUN
ejde-196	412	17	saddles	saddle	NOUN
ejde-196	412	18	‖d‖	‖d‖	SYM
ejde-196	412	19	ε	ε	PROPN
ejde-196	412	20	‖j	‖j	PROPN
ejde-196	413	1	′(·)‖∞	′(·)‖∞	PROPN
ejde-196	413	2	j	j	PROPN
ejde-196	413	3	(	(	PUNCT
ejde-196	413	4	·	·	PUNCT
ejde-196	413	5	)	)	PUNCT
ejde-196	413	6	nit	nit	NOUN
ejde-196	414	1	1	1	NUM
ejde-196	414	2	-	-	NUM
ejde-196	414	3	saddle	saddle	NOUN
ejde-196	414	4	0.0008	0.0008	NUM
ejde-196	414	5	1e-3	1e-3	NUM
ejde-196	414	6	0.0281	0.0281	NUM
ejde-196	414	7	2.1680	2.1680	NUM
ejde-196	414	8	18	18	NUM
ejde-196	414	9	2	2	NUM
ejde-196	414	10	-	-	PUNCT
ejde-196	414	11	saddle	saddle	NOUN
ejde-196	414	12	1	1	NUM
ejde-196	414	13	0.0009	0.0009	NUM
ejde-196	414	14	1e-3	1e-3	PROPN
ejde-196	414	15	1.7601	1.7601	NUM
ejde-196	414	16	18.0417	18.0417	NUM
ejde-196	414	17	45	45	NUM
ejde-196	414	18	2	2	NUM
ejde-196	414	19	-	-	PUNCT
ejde-196	414	20	saddle	saddle	NOUN
ejde-196	414	21	2	2	NUM
ejde-196	414	22	0.0009	0.0009	NUM
ejde-196	414	23	1e-3	1e-3	NUM
ejde-196	414	24	0.0630	0.0630	NUM
ejde-196	414	25	19.5637	19.5637	NUM
ejde-196	414	26	31	31	NUM
ejde-196	414	27	2	2	NUM
ejde-196	414	28	-	-	PUNCT
ejde-196	414	29	saddle	saddle	NOUN
ejde-196	414	30	3	3	NUM
ejde-196	414	31	0.0009	0.0009	NUM
ejde-196	414	32	1e-3	1e-3	PROPN
ejde-196	414	33	0.0556	0.0556	NUM
ejde-196	414	34	19.5638	19.5638	NUM
ejde-196	414	35	31	31	NUM
ejde-196	414	36	3	3	NUM
ejde-196	414	37	-	-	PUNCT
ejde-196	414	38	saddle	saddle	NOUN
ejde-196	414	39	0.0029	0.0029	NUM
ejde-196	414	40	3e-3	3e-3	NUM
ejde-196	414	41	1.7163	1.7163	NUM
ejde-196	414	42	56.6547	56.6547	NUM
ejde-196	414	43	44	44	NUM
ejde-196	414	44	4	4	NUM
ejde-196	414	45	-	-	PUNCT
ejde-196	414	46	saddle	saddle	ADJ
ejde-196	414	47	1	1	NUM
ejde-196	414	48	0.0007	0.0007	NUM
ejde-196	414	49	1e-3	1e-3	NUM
ejde-196	414	50	0.9094	0.9094	NUM
ejde-196	414	51	53.2727	53.2727	NUM
ejde-196	414	52	36	36	NUM
ejde-196	414	53	4	4	NUM
ejde-196	414	54	-	-	NUM
ejde-196	414	55	saddle	saddle	NOUN
ejde-196	414	56	2	2	NUM
ejde-196	414	57	0.0036	0.0036	NUM
ejde-196	414	58	4e-3	4e-3	NUM
ejde-196	414	59	1.5387	1.5387	NUM
ejde-196	414	60	65.1722	65.1722	NUM
ejde-196	414	61	60	60	NUM
ejde-196	414	62	locally	locally	ADV
ejde-196	414	63	m	m	ADJ
ejde-196	414	64	-	-	PUNCT
ejde-196	414	65	type	type	NOUN
ejde-196	414	66	saddles	saddle	NOUN
ejde-196	414	67	with	with	ADP
ejde-196	414	68	j	j	PROPN
ejde-196	414	69	>	>	X
ejde-196	414	70	0	0	NUM
ejde-196	415	1	the	the	DET
ejde-196	415	2	algorithm	algorithm	NOUN
ejde-196	415	3	finds	find	VERB
ejde-196	415	4	the	the	DET
ejde-196	415	5	solutions	solution	NOUN
ejde-196	415	6	in	in	ADP
ejde-196	415	7	figure	figure	NOUN
ejde-196	415	8	7	7	NUM
ejde-196	415	9	and	and	CCONJ
ejde-196	415	10	generates	generate	VERB
ejde-196	415	11	the	the	DET
ejde-196	415	12	numerical	numerical	ADJ
ejde-196	415	13	data	datum	NOUN
ejde-196	415	14	in	in	ADP
ejde-196	415	15	table	table	NOUN
ejde-196	415	16	4	4	NUM
ejde-196	415	17	.	.	PUNCT
ejde-196	415	18	table	table	NOUN
ejde-196	415	19	4	4	NUM
ejde-196	415	20	.	.	PUNCT
ejde-196	416	1	numerical	numerical	ADJ
ejde-196	416	2	data	datum	NOUN
ejde-196	416	3	for	for	ADP
ejde-196	416	4	locally	locally	ADV
ejde-196	416	5	m	m	NOUN
ejde-196	416	6	-	-	PUNCT
ejde-196	416	7	type	type	NOUN
ejde-196	416	8	saddles	saddle	NOUN
ejde-196	416	9	‖d‖	‖d‖	SYM
ejde-196	416	10	ε	ε	PROPN
ejde-196	416	11	‖j	‖j	PROPN
ejde-196	417	1	′(·)‖∞	′(·)‖∞	PROPN
ejde-196	417	2	j	j	PROPN
ejde-196	417	3	(	(	PUNCT
ejde-196	417	4	·	·	PUNCT
ejde-196	417	5	)	)	PUNCT
ejde-196	417	6	nit	nit	NOUN
ejde-196	418	1	1	1	NUM
ejde-196	418	2	-	-	NUM
ejde-196	418	3	saddle	saddle	NOUN
ejde-196	418	4	0.0009	0.0009	NUM
ejde-196	418	5	1e-3	1e-3	NUM
ejde-196	418	6	0.0916	0.0916	NUM
ejde-196	418	7	17.6073	17.6073	NUM
ejde-196	418	8	98	98	NUM
ejde-196	418	9	2	2	NUM
ejde-196	418	10	-	-	PUNCT
ejde-196	418	11	saddle	saddle	NOUN
ejde-196	418	12	1	1	NUM
ejde-196	418	13	0.0009	0.0009	NUM
ejde-196	418	14	1e-3	1e-3	NUM
ejde-196	418	15	1.6902	1.6902	NUM
ejde-196	418	16	38.7233	38.7233	NUM
ejde-196	418	17	98	98	NUM
ejde-196	418	18	2	2	NUM
ejde-196	418	19	-	-	PUNCT
ejde-196	418	20	saddle	saddle	NOUN
ejde-196	418	21	2	2	NUM
ejde-196	418	22	0.0009	0.0009	NUM
ejde-196	418	23	1e-3	1e-3	NUM
ejde-196	418	24	1.6926	1.6926	NUM
ejde-196	418	25	39.8931	39.8931	NUM
ejde-196	418	26	93	93	NUM
ejde-196	418	27	2	2	NUM
ejde-196	418	28	-	-	PUNCT
ejde-196	418	29	saddle	saddle	NOUN
ejde-196	418	30	3	3	NUM
ejde-196	418	31	0.0010	0.0010	NUM
ejde-196	418	32	1e-3	1e-3	NUM
ejde-196	418	33	1.6629	1.6629	NUM
ejde-196	418	34	39.8933	39.8933	NUM
ejde-196	418	35	93	93	NUM
ejde-196	418	36	3	3	NUM
ejde-196	418	37	-	-	PUNCT
ejde-196	418	38	saddle	saddle	NOUN
ejde-196	418	39	0.0009	0.0009	NUM
ejde-196	418	40	1e-3	1e-3	NUM
ejde-196	418	41	0.9003	0.9003	NUM
ejde-196	418	42	56.6546	56.6546	NUM
ejde-196	418	43	171	171	NUM
ejde-196	418	44	4	4	NUM
ejde-196	418	45	-	-	NUM
ejde-196	418	46	saddle	saddle	NOUN
ejde-196	418	47	1	1	NUM
ejde-196	418	48	0.0010	0.0010	NUM
ejde-196	418	49	1e-3	1e-3	NUM
ejde-196	418	50	1.7205	1.7205	NUM
ejde-196	418	51	82.5416	82.5416	NUM
ejde-196	418	52	154	154	NUM
ejde-196	418	53	from	from	ADP
ejde-196	418	54	figure	figure	NOUN
ejde-196	418	55	7	7	NUM
ejde-196	418	56	(	(	PUNCT
ejde-196	418	57	a	a	NOUN
ejde-196	418	58	)	)	PUNCT
ejde-196	418	59	,	,	PUNCT
ejde-196	418	60	a	a	DET
ejde-196	418	61	symmetry	symmetry	NOUN
ejde-196	418	62	breaking	break	VERB
ejde-196	418	63	phenomenon	phenomenon	NOUN
ejde-196	418	64	can	can	AUX
ejde-196	418	65	be	be	AUX
ejde-196	418	66	clearly	clearly	ADV
ejde-196	418	67	observed	observe	VERB
ejde-196	418	68	.	.	PUNCT
ejde-196	419	1	to	to	PART
ejde-196	419	2	further	far	ADV
ejde-196	419	3	explore	explore	VERB
ejde-196	419	4	such	such	DET
ejde-196	419	5	a	a	DET
ejde-196	419	6	bifurcation	bifurcation	NOUN
ejde-196	419	7	process	process	NOUN
ejde-196	419	8	,	,	PUNCT
ejde-196	419	9	we	we	PRON
ejde-196	419	10	use	use	VERB
ejde-196	419	11	the	the	DET
ejde-196	419	12	same	same	ADJ
ejde-196	419	13	initial	initial	ADJ
ejde-196	419	14	guess	guess	NOUN
ejde-196	419	15	u0	u0	ADJ
ejde-196	419	16	,	,	PUNCT
ejde-196	419	17	the	the	DET
ejde-196	419	18	eigenfunction	eigenfunction	NOUN
ejde-196	419	19	of	of	ADP
ejde-196	419	20	−∆	−∆	NOUN
ejde-196	419	21	corresponding	correspond	VERB
ejde-196	419	22	to	to	ADP
ejde-196	419	23	the	the	DET
ejde-196	419	24	first	first	ADJ
ejde-196	419	25	eigenvalue	eigenvalue	PROPN
ejde-196	419	26	µ1	µ1	PROPN
ejde-196	419	27	.	.	PUNCT
ejde-196	420	1	thus	thus	ADV
ejde-196	420	2	u0	u0	PROPN
ejde-196	420	3	is	be	AUX
ejde-196	420	4	symmetric	symmetric	ADJ
ejde-196	420	5	about	about	ADP
ejde-196	420	6	the	the	DET
ejde-196	420	7	x	x	NOUN
ejde-196	420	8	-	-	NOUN
ejde-196	420	9	axis	axis	ADJ
ejde-196	420	10	,	,	PUNCT
ejde-196	420	11	the	the	DET
ejde-196	420	12	y	y	NOUN
ejde-196	420	13	-	-	PUNCT
ejde-196	420	14	axis	axis	NOUN
ejde-196	420	15	and	and	CCONJ
ejde-196	420	16	the	the	DET
ejde-196	420	17	lines	line	NOUN
ejde-196	420	18	x	x	PUNCT
ejde-196	420	19	=	=	PUNCT
ejde-196	420	20	y	y	PROPN
ejde-196	420	21	and	and	CCONJ
ejde-196	420	22	x	x	X
ejde-196	420	23	=	=	PRON
ejde-196	420	24	−y	−y	PROPN
ejde-196	420	25	.	.	PUNCT
ejde-196	421	1	the	the	DET
ejde-196	421	2	min	min	ADJ
ejde-196	421	3	-	-	ADJ
ejde-196	421	4	orthogonal	orthogonal	ADJ
ejde-196	421	5	method	method	NOUN
ejde-196	421	6	is	be	AUX
ejde-196	421	7	applied	apply	VERB
ejde-196	421	8	with	with	ADP
ejde-196	421	9	l	l	NOUN
ejde-196	421	10	=	=	PUNCT
ejde-196	421	11	{	{	PUNCT
ejde-196	421	12	0	0	NUM
ejde-196	421	13	}	}	PUNCT
ejde-196	421	14	for	for	ADP
ejde-196	421	15	the	the	DET
ejde-196	421	16	first	first	ADJ
ejde-196	421	17	nmo	nmo	PROPN
ejde-196	421	18	times	time	NOUN
ejde-196	421	19	and	and	CCONJ
ejde-196	421	20	then	then	ADV
ejde-196	421	21	it	it	PRON
ejde-196	421	22	is	be	AUX
ejde-196	421	23	followed	follow	VERB
ejde-196	421	24	by	by	ADP
ejde-196	421	25	a	a	DET
ejde-196	421	26	newton	newton	PROPN
ejde-196	421	27	method	method	NOUN
ejde-196	421	28	to	to	PART
ejde-196	421	29	speedup	speedup	VERB
ejde-196	421	30	local	local	ADJ
ejde-196	421	31	convergence	convergence	NOUN
ejde-196	421	32	.	.	PUNCT
ejde-196	422	1	since	since	SCONJ
ejde-196	422	2	the	the	DET
ejde-196	422	3	newton	newton	PROPN
ejde-196	422	4	method	method	NOUN
ejde-196	422	5	is	be	AUX
ejde-196	422	6	symmetry	symmetry	NOUN
ejde-196	422	7	invariant	invariant	ADJ
ejde-196	422	8	and	and	CCONJ
ejde-196	422	9	insensitive	insensitive	ADJ
ejde-196	422	10	to	to	ADP
ejde-196	422	11	numerical	numerical	ADJ
ejde-196	422	12	errors	error	NOUN
ejde-196	422	13	[	[	X
ejde-196	422	14	10	10	NUM
ejde-196	422	15	]	]	PUNCT
ejde-196	422	16	,	,	PUNCT
ejde-196	422	17	we	we	PRON
ejde-196	422	18	obtain	obtain	VERB
ejde-196	422	19	the	the	DET
ejde-196	422	20	following	follow	VERB
ejde-196	422	21	three	three	NUM
ejde-196	422	22	solutions	solution	NOUN
ejde-196	422	23	with	with	ADP
ejde-196	422	24	different	different	ADJ
ejde-196	422	25	symmetries	symmetry	NOUN
ejde-196	422	26	as	as	ADP
ejde-196	422	27	in	in	ADP
ejde-196	422	28	figure	figure	NOUN
ejde-196	422	29	8	8	NUM
ejde-196	422	30	,	,	PUNCT
ejde-196	422	31	a	a	DET
ejde-196	422	32	typical	typical	ADJ
ejde-196	422	33	symmetry	symmetry	NOUN
ejde-196	422	34	-	-	PUNCT
ejde-196	422	35	breaking	break	VERB
ejde-196	422	36	phenomenon	phenomenon	NOUN
ejde-196	422	37	due	due	ADP
ejde-196	422	38	to	to	ADP
ejde-196	422	39	a	a	DET
ejde-196	422	40	bifurcation	bifurcation	NOUN
ejde-196	422	41	process	process	NOUN
ejde-196	422	42	along	along	ADP
ejde-196	422	43	the	the	DET
ejde-196	422	44	parameter	parameter	PROPN
ejde-196	422	45	r.	r.	PROPN
ejde-196	422	46	since	since	SCONJ
ejde-196	422	47	in	in	ADP
ejde-196	422	48	this	this	DET
ejde-196	422	49	case	case	NOUN
ejde-196	422	50	,	,	PUNCT
ejde-196	422	51	the	the	DET
ejde-196	422	52	locally	locally	ADV
ejde-196	422	53	w	w	NOUN
ejde-196	422	54	-	-	PUNCT
ejde-196	422	55	type	type	NOUN
ejde-196	422	56	saddles	saddle	NOUN
ejde-196	422	57	did	do	AUX
ejde-196	422	58	not	not	PART
ejde-196	422	59	show	show	VERB
ejde-196	422	60	any	any	DET
ejde-196	422	61	bifurcation	bifurcation	NOUN
ejde-196	422	62	phenomenon	phenomenon	NOUN
ejde-196	422	63	and	and	CCONJ
ejde-196	422	64	their	their	PRON
ejde-196	422	65	solution	solution	NOUN
ejde-196	422	66	profiles	profile	NOUN
ejde-196	422	67	are	be	AUX
ejde-196	422	68	similar	similar	ADJ
ejde-196	422	69	to	to	ADP
ejde-196	422	70	those	those	PRON
ejde-196	422	71	in	in	ADP
ejde-196	422	72	figure	figure	NOUN
ejde-196	422	73	5	5	NUM
ejde-196	422	74	,	,	PUNCT
ejde-196	422	75	they	they	PRON
ejde-196	422	76	are	be	AUX
ejde-196	422	77	omitted	omit	VERB
ejde-196	422	78	here	here	ADV
ejde-196	422	79	.	.	PUNCT
ejde-196	423	1	it	it	PRON
ejde-196	423	2	is	be	AUX
ejde-196	423	3	interesting	interesting	ADJ
ejde-196	423	4	to	to	PART
ejde-196	423	5	compare	compare	VERB
ejde-196	423	6	the	the	DET
ejde-196	423	7	three	three	NUM
ejde-196	423	8	solutions	solution	NOUN
ejde-196	423	9	in	in	ADP
ejde-196	423	10	figure	figure	NOUN
ejde-196	423	11	8	8	NUM
ejde-196	423	12	with	with	ADP
ejde-196	423	13	the	the	DET
ejde-196	423	14	three	three	NUM
ejde-196	423	15	solutions	solution	NOUN
ejde-196	423	16	in	in	ADP
ejde-196	423	17	[	[	X
ejde-196	423	18	18	18	NUM
ejde-196	423	19	,	,	PUNCT
ejde-196	423	20	figure	figure	NOUN
ejde-196	423	21	9	9	NUM
ejde-196	423	22	]	]	PUNCT
ejde-196	423	23	,	,	PUNCT
ejde-196	423	24	where	where	SCONJ
ejde-196	423	25	a	a	DET
ejde-196	423	26	=	=	SYM
ejde-196	423	27	0	0	NUM
ejde-196	423	28	,	,	PUNCT
ejde-196	423	29	r	r	NOUN
ejde-196	423	30	=	=	SYM
ejde-196	423	31	2	2	NUM
ejde-196	423	32	and	and	CCONJ
ejde-196	423	33	nmo	nmo	NOUN
ejde-196	423	34	=	=	SYM
ejde-196	423	35	4	4	NUM
ejde-196	423	36	,	,	PUNCT
ejde-196	423	37	14	14	NUM
ejde-196	423	38	,	,	PUNCT
ejde-196	423	39	20	20	NUM
ejde-196	423	40	.	.	PUNCT
ejde-196	424	1	they	they	PRON
ejde-196	424	2	have	have	VERB
ejde-196	424	3	exactly	exactly	ADV
ejde-196	424	4	the	the	DET
ejde-196	424	5	same	same	ADJ
ejde-196	424	6	symmetries	symmetry	NOUN
ejde-196	424	7	in	in	ADP
ejde-196	424	8	order	order	NOUN
ejde-196	424	9	.	.	PUNCT
ejde-196	425	1	since	since	SCONJ
ejde-196	425	2	a	a	DET
ejde-196	425	3	=	=	SYM
ejde-196	425	4	0	0	NUM
ejde-196	425	5	,	,	PUNCT
ejde-196	425	6	the	the	DET
ejde-196	425	7	latter	latter	ADJ
ejde-196	425	8	is	be	AUX
ejde-196	425	9	an	an	DET
ejde-196	425	10	m	m	NOUN
ejde-196	425	11	-	-	PUNCT
ejde-196	425	12	type	type	NOUN
ejde-196	425	13	problem	problem	NOUN
ejde-196	425	14	.	.	PUNCT
ejde-196	426	1	the	the	DET
ejde-196	426	2	differences	difference	NOUN
ejde-196	426	3	in	in	ADP
ejde-196	426	4	nmo	nmo	PROPN
ejde-196	426	5	values	value	NOUN
ejde-196	426	6	imply	imply	VERB
ejde-196	426	7	that	that	SCONJ
ejde-196	426	8	the	the	DET
ejde-196	426	9	latter	latter	ADJ
ejde-196	426	10	is	be	AUX
ejde-196	426	11	much	much	ADV
ejde-196	426	12	easier	easy	ADJ
ejde-196	426	13	to	to	PART
ejde-196	426	14	bifurcate	bifurcate	VERB
ejde-196	426	15	to	to	ADP
ejde-196	426	16	an	an	DET
ejde-196	426	17	asymmetric	asymmetric	ADJ
ejde-196	426	18	solution	solution	NOUN
ejde-196	426	19	.	.	PUNCT
ejde-196	427	1	this	this	PRON
ejde-196	427	2	may	may	AUX
ejde-196	427	3	indicate	indicate	VERB
ejde-196	427	4	the	the	DET
ejde-196	427	5	influence	influence	NOUN
ejde-196	427	6	by	by	ADP
ejde-196	427	7	the	the	DET
ejde-196	427	8	concave	concave	ADJ
ejde-196	427	9	term	term	NOUN
ejde-196	427	10	−a|u(x)|q−1u(x	−a|u(x)|q−1u(x	NOUN
ejde-196	427	11	)	)	PUNCT
ejde-196	427	12	in	in	ADP
ejde-196	427	13	the	the	DET
ejde-196	427	14	problem	problem	NOUN
ejde-196	427	15	.	.	PUNCT
ejde-196	428	1	final	final	ADJ
ejde-196	428	2	remarks	remark	NOUN
ejde-196	428	3	.	.	PUNCT
ejde-196	429	1	from	from	ADP
ejde-196	429	2	a	a	DET
ejde-196	429	3	double	double	ADJ
ejde-196	429	4	-	-	PUNCT
ejde-196	429	5	orthogonal	orthogonal	ADJ
ejde-196	429	6	principle	principle	NOUN
ejde-196	429	7	,	,	PUNCT
ejde-196	429	8	which	which	PRON
ejde-196	429	9	does	do	AUX
ejde-196	429	10	not	not	PART
ejde-196	429	11	have	have	VERB
ejde-196	429	12	to	to	PART
ejde-196	429	13	have	have	VERB
ejde-196	429	14	a	a	DET
ejde-196	429	15	variational	variational	ADJ
ejde-196	429	16	structure	structure	NOUN
ejde-196	429	17	,	,	PUNCT
ejde-196	429	18	to	to	ADP
ejde-196	429	19	a	a	DET
ejde-196	429	20	local	local	ADJ
ejde-196	429	21	min	min	ADJ
ejde-196	429	22	-	-	ADJ
ejde-196	429	23	orthogonal	orthogonal	ADJ
ejde-196	429	24	principle	principle	NOUN
ejde-196	429	25	,	,	PUNCT
ejde-196	429	26	which	which	PRON
ejde-196	429	27	needs	need	VERB
ejde-196	429	28	only	only	ADV
ejde-196	429	29	a	a	DET
ejde-196	429	30	general	general	ADJ
ejde-196	429	31	variational	variational	ADJ
ejde-196	429	32	structure	structure	NOUN
ejde-196	429	33	,	,	PUNCT
ejde-196	429	34	to	to	ADP
ejde-196	429	35	its	its	PRON
ejde-196	429	36	numerical	numerical	ADJ
ejde-196	429	37	algorithm	algorithm	NOUN
ejde-196	429	38	,	,	PUNCT
ejde-196	429	39	we	we	PRON
ejde-196	429	40	established	establish	VERB
ejde-196	429	41	its	its	PRON
ejde-196	429	42	solution	solution	NOUN
ejde-196	429	43	characterization	characterization	NOUN
ejde-196	429	44	,	,	PUNCT
ejde-196	429	45	step	step	NOUN
ejde-196	429	46	size	size	NOUN
ejde-196	429	47	rule	rule	NOUN
ejde-196	429	48	and	and	CCONJ
ejde-196	429	49	convergence	convergence	NOUN
ejde-196	429	50	.	.	PUNCT
ejde-196	430	1	then	then	ADV
ejde-196	430	2	we	we	PRON
ejde-196	430	3	used	use	VERB
ejde-196	430	4	the	the	DET
ejde-196	430	5	local	local	ADJ
ejde-196	430	6	minorthogonal	minorthogonal	ADJ
ejde-196	430	7	principle	principle	NOUN
ejde-196	430	8	as	as	ADP
ejde-196	430	9	a	a	DET
ejde-196	430	10	general	general	ADJ
ejde-196	430	11	mathematical	mathematical	ADJ
ejde-196	430	12	framework	framework	NOUN
ejde-196	430	13	to	to	PART
ejde-196	430	14	justify	justify	VERB
ejde-196	430	15	two	two	NUM
ejde-196	430	16	algorithms	algorithm	NOUN
ejde-196	430	17	,	,	PUNCT
ejde-196	430	18	a	a	DET
ejde-196	430	19	min	min	PROPN
ejde-196	430	20	-	-	PUNCT
ejde-196	430	21	max	max	ADJ
ejde-196	430	22	-	-	PUNCT
ejde-196	430	23	min	min	NOUN
ejde-196	430	24	algorithm	algorithm	NOUN
ejde-196	430	25	and	and	CCONJ
ejde-196	430	26	a	a	DET
ejde-196	430	27	min	min	PROPN
ejde-196	430	28	-	-	PUNCT
ejde-196	430	29	min	min	ADJ
ejde-196	430	30	-	-	ADJ
ejde-196	430	31	max	max	NOUN
ejde-196	430	32	algorithm	algorithm	NOUN
ejde-196	430	33	,	,	PUNCT
ejde-196	430	34	used	use	VERB
ejde-196	430	35	to	to	PART
ejde-196	430	36	solve	solve	VERB
ejde-196	430	37	w	w	NOUN
ejde-196	430	38	-	-	PUNCT
ejde-196	430	39	type	type	NOUN
ejde-196	430	40	problems	problem	NOUN
ejde-196	430	41	for	for	ADP
ejde-196	430	42	multiple	multiple	ADJ
ejde-196	430	43	solutions	solution	NOUN
ejde-196	430	44	.	.	PUNCT
ejde-196	431	1	in	in	ADP
ejde-196	431	2	the	the	DET
ejde-196	431	3	final	final	ADJ
ejde-196	431	4	section	section	NOUN
ejde-196	431	5	,	,	PUNCT
ejde-196	431	6	to	to	PART
ejde-196	431	7	illustrate	illustrate	VERB
ejde-196	431	8	the	the	DET
ejde-196	431	9	flexibilities	flexibility	NOUN
ejde-196	431	10	of	of	ADP
ejde-196	431	11	190	190	NUM
ejde-196	431	12	m.	m.	NOUN
ejde-196	431	13	li	li	PROPN
ejde-196	431	14	,	,	PUNCT
ejde-196	431	15	b.	b.	PROPN
ejde-196	432	1	ji	ji	PROPN
ejde-196	432	2	,	,	PUNCT
ejde-196	432	3	j.	j.	PROPN
ejde-196	432	4	zhou	zhou	PROPN
ejde-196	432	5	ejde	ejde	PROPN
ejde-196	432	6	/	/	SYM
ejde-196	432	7	si/02	si/02	PROPN
ejde-196	432	8	j	j	NOUN
ejde-196	432	9	=	=	SYM
ejde-196	432	10	2.1680	2.1680	NUM
ejde-196	432	11	,	,	PUNCT
ejde-196	432	12	‖u‖∞	‖u‖∞	PROPN
ejde-196	432	13	=	=	PUNCT
ejde-196	432	14	2.1990	2.1990	NUM
ejde-196	432	15	at	at	ADP
ejde-196	432	16	(	(	PUNCT
ejde-196	432	17	−0.0005,−0.0032	−0.0005,−0.0032	NOUN
ejde-196	432	18	)	)	PUNCT
ejde-196	432	19	j	j	PROPN
ejde-196	432	20	=	=	PUNCT
ejde-196	432	21	18.0417	18.0417	NUM
ejde-196	432	22	,	,	PUNCT
ejde-196	432	23	‖u‖∞	‖u‖∞	PROPN
ejde-196	432	24	=	=	SYM
ejde-196	432	25	3.3582	3.3582	NUM
ejde-196	432	26	at	at	ADP
ejde-196	432	27	(	(	PUNCT
ejde-196	432	28	0.3972	0.3972	NUM
ejde-196	432	29	,	,	PUNCT
ejde-196	432	30	0.3990	0.3990	NUM
ejde-196	432	31	)	)	PUNCT
ejde-196	432	32	j	j	PROPN
ejde-196	432	33	=	=	SYM
ejde-196	432	34	19.5637	19.5637	NUM
ejde-196	432	35	,	,	PUNCT
ejde-196	432	36	‖u‖∞	‖u‖∞	PROPN
ejde-196	432	37	=	=	SYM
ejde-196	432	38	3.4559	3.4559	NUM
ejde-196	432	39	at	at	ADP
ejde-196	432	40	(	(	PUNCT
ejde-196	432	41	0.5003	0.5003	NUM
ejde-196	432	42	,	,	PUNCT
ejde-196	432	43	0.0031	0.0031	NUM
ejde-196	432	44	)	)	PUNCT
ejde-196	432	45	(	(	PUNCT
ejde-196	432	46	a	a	X
ejde-196	432	47	)	)	PUNCT
ejde-196	432	48	1	1	NUM
ejde-196	432	49	-	-	NUM
ejde-196	432	50	saddle	saddle	NOUN
ejde-196	432	51	(	(	PUNCT
ejde-196	432	52	b	b	NOUN
ejde-196	432	53	)	)	PUNCT
ejde-196	432	54	2	2	NUM
ejde-196	432	55	-	-	NUM
ejde-196	432	56	saddle	saddle	NOUN
ejde-196	432	57	1	1	NUM
ejde-196	432	58	(	(	PUNCT
ejde-196	432	59	c	c	NOUN
ejde-196	432	60	)	)	PUNCT
ejde-196	432	61	2	2	NUM
ejde-196	432	62	-	-	NUM
ejde-196	432	63	saddle	saddle	NOUN
ejde-196	432	64	2	2	NUM
ejde-196	432	65	j	j	NOUN
ejde-196	432	66	=	=	SYM
ejde-196	432	67	19.5638	19.5638	NUM
ejde-196	432	68	,	,	PUNCT
ejde-196	432	69	‖u‖∞	‖u‖∞	PROPN
ejde-196	432	70	=	=	SYM
ejde-196	432	71	3.4559	3.4559	NUM
ejde-196	432	72	at	at	ADP
ejde-196	432	73	(	(	PUNCT
ejde-196	432	74	−0.0014,−0.5010	−0.0014,−0.5010	NOUN
ejde-196	432	75	)	)	PUNCT
ejde-196	432	76	j	j	PROPN
ejde-196	432	77	=	=	SYM
ejde-196	432	78	56.6547	56.6547	NUM
ejde-196	432	79	,	,	PUNCT
ejde-196	432	80	‖u‖∞	‖u‖∞	PROPN
ejde-196	432	81	=	=	SYM
ejde-196	432	82	4.6177	4.6177	NUM
ejde-196	432	83	at	at	ADP
ejde-196	432	84	(	(	PUNCT
ejde-196	432	85	−0.0005,−0.0032	−0.0005,−0.0032	NOUN
ejde-196	432	86	)	)	PUNCT
ejde-196	432	87	,	,	PUNCT
ejde-196	432	88	sid	sid	PROPN
ejde-196	432	89	=	=	PUNCT
ejde-196	432	90	a	a	PROPN
ejde-196	432	91	,	,	PUNCT
ejde-196	432	92	c	c	PROPN
ejde-196	432	93	j	j	PROPN
ejde-196	432	94	=	=	SYM
ejde-196	432	95	53.2727	53.2727	NUM
ejde-196	432	96	,	,	PUNCT
ejde-196	432	97	‖u‖∞	‖u‖∞	PROPN
ejde-196	432	98	=	=	SYM
ejde-196	432	99	3.9597	3.9597	NUM
ejde-196	432	100	at	at	ADP
ejde-196	432	101	(	(	PUNCT
ejde-196	432	102	0.4978	0.4978	NUM
ejde-196	432	103	,	,	PUNCT
ejde-196	432	104	0.5000	0.5000	NUM
ejde-196	432	105	)	)	PUNCT
ejde-196	432	106	,	,	PUNCT
ejde-196	432	107	sid	sid	PROPN
ejde-196	432	108	=	=	PUNCT
ejde-196	432	109	a	a	PROPN
ejde-196	432	110	,	,	PUNCT
ejde-196	432	111	c	c	NOUN
ejde-196	432	112	,	,	PUNCT
ejde-196	432	113	d	d	X
ejde-196	432	114	(	(	PUNCT
ejde-196	432	115	d	d	NOUN
ejde-196	432	116	)	)	PUNCT
ejde-196	432	117	2	2	NUM
ejde-196	432	118	-	-	NUM
ejde-196	432	119	saddle	saddle	NOUN
ejde-196	432	120	3	3	NUM
ejde-196	432	121	(	(	PUNCT
ejde-196	432	122	e	e	NOUN
ejde-196	432	123	)	)	PUNCT
ejde-196	432	124	3	3	NUM
ejde-196	432	125	-	-	NUM
ejde-196	432	126	saddle	saddle	NOUN
ejde-196	432	127	(	(	PUNCT
ejde-196	432	128	f	f	NOUN
ejde-196	432	129	)	)	PUNCT
ejde-196	432	130	4	4	NUM
ejde-196	432	131	-	-	NUM
ejde-196	432	132	saddle	saddle	ADJ
ejde-196	432	133	1	1	NUM
ejde-196	432	134	j	j	NOUN
ejde-196	432	135	=	=	SYM
ejde-196	432	136	65.1722	65.1722	NUM
ejde-196	432	137	,	,	PUNCT
ejde-196	432	138	‖u‖∞	‖u‖∞	PROPN
ejde-196	432	139	=	=	SYM
ejde-196	432	140	4.3352	4.3352	NUM
ejde-196	432	141	at	at	ADP
ejde-196	432	142	(	(	PUNCT
ejde-196	432	143	−0.6056	−0.6056	PROPN
ejde-196	432	144	,	,	PUNCT
ejde-196	432	145	0.0019	0.0019	NUM
ejde-196	432	146	)	)	PUNCT
ejde-196	432	147	,	,	PUNCT
ejde-196	432	148	sid	sid	PROPN
ejde-196	432	149	=	=	PUNCT
ejde-196	433	1	a	a	PROPN
ejde-196	433	2	,	,	PUNCT
ejde-196	433	3	b	b	NOUN
ejde-196	433	4	,	,	PUNCT
ejde-196	433	5	c	c	PROPN
ejde-196	433	6	4	4	NUM
ejde-196	433	7	-	-	NUM
ejde-196	433	8	saddle	saddle	ADJ
ejde-196	433	9	2	2	NUM
ejde-196	433	10	figure	figure	NOUN
ejde-196	433	11	6	6	NUM
ejde-196	433	12	.	.	PUNCT
ejde-196	434	1	locally	locally	ADV
ejde-196	434	2	m	m	NOUN
ejde-196	434	3	-	-	PUNCT
ejde-196	434	4	type	type	NOUN
ejde-196	434	5	saddles	saddle	NOUN
ejde-196	434	6	with	with	ADP
ejde-196	434	7	j	j	PROPN
ejde-196	434	8	>	>	X
ejde-196	434	9	0	0	NUM
ejde-196	435	1	the	the	DET
ejde-196	435	2	method	method	NOUN
ejde-196	435	3	developed	develop	VERB
ejde-196	435	4	in	in	ADP
ejde-196	435	5	the	the	DET
ejde-196	435	6	previous	previous	ADJ
ejde-196	435	7	sections	section	NOUN
ejde-196	435	8	,	,	PUNCT
ejde-196	435	9	we	we	PRON
ejde-196	435	10	present	present	VERB
ejde-196	435	11	some	some	DET
ejde-196	435	12	numerical	numerical	ADJ
ejde-196	435	13	examples	example	NOUN
ejde-196	435	14	for	for	ADP
ejde-196	435	15	solving	solve	VERB
ejde-196	435	16	w	w	NOUN
ejde-196	435	17	-	-	PUNCT
ejde-196	435	18	type	type	NOUN
ejde-196	435	19	problems	problem	NOUN
ejde-196	435	20	and	and	CCONJ
ejde-196	435	21	,	,	PUNCT
ejde-196	435	22	in	in	ADP
ejde-196	435	23	particular	particular	ADJ
ejde-196	435	24	,	,	PUNCT
ejde-196	435	25	for	for	ADP
ejde-196	435	26	mixed	mixed	ADJ
ejde-196	435	27	m	m	PROPN
ejde-196	435	28	-	-	PUNCT
ejde-196	435	29	w	w	NOUN
ejde-196	435	30	-	-	PUNCT
ejde-196	435	31	type	type	NOUN
ejde-196	435	32	problems	problem	NOUN
ejde-196	435	33	where	where	SCONJ
ejde-196	435	34	both	both	CCONJ
ejde-196	435	35	convex	convex	VERB
ejde-196	435	36	and	and	CCONJ
ejde-196	435	37	concave	concave	NOUN
ejde-196	435	38	nonlinearities	nonlinearitie	NOUN
ejde-196	435	39	are	be	AUX
ejde-196	435	40	present	present	ADJ
ejde-196	435	41	,	,	PUNCT
ejde-196	435	42	for	for	ADP
ejde-196	435	43	multiple	multiple	ADJ
ejde-196	435	44	solutions	solution	NOUN
ejde-196	435	45	.	.	PUNCT
ejde-196	436	1	as	as	ADP
ejde-196	436	2	a	a	DET
ejde-196	436	3	trade	trade	NOUN
ejde-196	436	4	-	-	PUNCT
ejde-196	436	5	off	off	NOUN
ejde-196	436	6	for	for	ADP
ejde-196	436	7	the	the	DET
ejde-196	436	8	generalization	generalization	NOUN
ejde-196	436	9	of	of	ADP
ejde-196	436	10	the	the	DET
ejde-196	436	11	method	method	NOUN
ejde-196	436	12	,	,	PUNCT
ejde-196	436	13	so	so	ADV
ejde-196	436	14	far	far	ADV
ejde-196	436	15	we	we	PRON
ejde-196	436	16	can	can	AUX
ejde-196	436	17	establish	establish	VERB
ejde-196	436	18	only	only	ADV
ejde-196	436	19	a	a	DET
ejde-196	436	20	one	one	NUM
ejde-196	436	21	side	side	NOUN
ejde-196	436	22	bound	bind	VERB
ejde-196	436	23	(	(	PUNCT
ejde-196	436	24	inequality	inequality	NOUN
ejde-196	436	25	)	)	PUNCT
ejde-196	436	26	estimate	estimate	NOUN
ejde-196	436	27	for	for	ADP
ejde-196	436	28	the	the	DET
ejde-196	436	29	morse	morse	ADJ
ejde-196	436	30	index	index	NOUN
ejde-196	436	31	of	of	ADP
ejde-196	436	32	a	a	DET
ejde-196	436	33	solution	solution	NOUN
ejde-196	436	34	found	find	VERB
ejde-196	436	35	by	by	ADP
ejde-196	436	36	the	the	DET
ejde-196	436	37	local	local	ADJ
ejde-196	436	38	min	min	ADJ
ejde-196	436	39	-	-	ADJ
ejde-196	436	40	orthogonal	orthogonal	ADJ
ejde-196	436	41	method	method	NOUN
ejde-196	436	42	[	[	X
ejde-196	436	43	3	3	NUM
ejde-196	436	44	]	]	PUNCT
ejde-196	436	45	,	,	PUNCT
ejde-196	436	46	not	not	PART
ejde-196	436	47	as	as	ADP
ejde-196	436	48	an	an	DET
ejde-196	436	49	equality	equality	NOUN
ejde-196	436	50	established	establish	VERB
ejde-196	436	51	by	by	ADP
ejde-196	436	52	the	the	DET
ejde-196	436	53	local	local	ADJ
ejde-196	436	54	min	min	ADJ
ejde-196	436	55	-	-	ADJ
ejde-196	436	56	max	max	PROPN
ejde-196	436	57	method	method	NOUN
ejde-196	436	58	[	[	X
ejde-196	436	59	17	17	NUM
ejde-196	436	60	]	]	PUNCT
ejde-196	436	61	.	.	PUNCT
ejde-196	437	1	ejde-2023	ejde-2023	ADJ
ejde-196	437	2	/	/	SYM
ejde-196	437	3	si/02	si/02	ADJ
ejde-196	437	4	short	short	ADJ
ejde-196	437	5	title	title	NOUN
ejde-196	437	6	191	191	NUM
ejde-196	437	7	j	j	NOUN
ejde-196	437	8	=	=	SYM
ejde-196	437	9	17.6073	17.6073	NUM
ejde-196	437	10	,	,	PUNCT
ejde-196	437	11	‖u‖∞	‖u‖∞	PROPN
ejde-196	437	12	=	=	PUNCT
ejde-196	437	13	4.6509	4.6509	NUM
ejde-196	437	14	at	at	ADP
ejde-196	437	15	(	(	PUNCT
ejde-196	437	16	0.6375	0.6375	NUM
ejde-196	437	17	,	,	PUNCT
ejde-196	437	18	0.6408	0.6408	NUM
ejde-196	437	19	)	)	PUNCT
ejde-196	437	20	j	j	PROPN
ejde-196	437	21	=	=	SYM
ejde-196	437	22	38.7233	38.7233	NUM
ejde-196	437	23	,	,	PUNCT
ejde-196	437	24	‖u‖∞	‖u‖∞	PROPN
ejde-196	437	25	=	=	PUNCT
ejde-196	437	26	4.7529	4.7529	NUM
ejde-196	437	27	at	at	ADP
ejde-196	437	28	(	(	PUNCT
ejde-196	437	29	−0.6686,−0.6668	−0.6686,−0.6668	NOUN
ejde-196	437	30	)	)	PUNCT
ejde-196	437	31	j	j	NOUN
ejde-196	437	32	=	=	PUNCT
ejde-196	437	33	39.8931	39.8931	NUM
ejde-196	437	34	,	,	PUNCT
ejde-196	437	35	‖u‖∞	‖u‖∞	PROPN
ejde-196	437	36	=	=	SYM
ejde-196	437	37	4.8060	4.8060	NUM
ejde-196	437	38	at	at	ADP
ejde-196	437	39	(	(	PUNCT
ejde-196	437	40	−0.6855	−0.6855	NOUN
ejde-196	437	41	,	,	PUNCT
ejde-196	437	42	0.6760	0.6760	NUM
ejde-196	437	43	)	)	PUNCT
ejde-196	437	44	(	(	PUNCT
ejde-196	437	45	a	a	X
ejde-196	437	46	)	)	PUNCT
ejde-196	437	47	1	1	NUM
ejde-196	437	48	-	-	NUM
ejde-196	437	49	saddle	saddle	NOUN
ejde-196	437	50	(	(	PUNCT
ejde-196	437	51	b	b	NOUN
ejde-196	437	52	)	)	PUNCT
ejde-196	437	53	2	2	NUM
ejde-196	437	54	-	-	NUM
ejde-196	437	55	saddle	saddle	NOUN
ejde-196	437	56	1	1	NUM
ejde-196	437	57	2	2	NUM
ejde-196	437	58	-	-	PUNCT
ejde-196	437	59	saddle	saddle	NOUN
ejde-196	437	60	2	2	NUM
ejde-196	437	61	j	j	NOUN
ejde-196	437	62	=	=	NOUN
ejde-196	437	63	39.8933	39.8933	NUM
ejde-196	437	64	,	,	PUNCT
ejde-196	437	65	‖u‖∞	‖u‖∞	PROPN
ejde-196	437	66	=	=	SYM
ejde-196	437	67	4.8022	4.8022	NUM
ejde-196	437	68	at	at	ADP
ejde-196	437	69	(	(	PUNCT
ejde-196	437	70	0.6838,−0.6831	0.6838,−0.6831	X
ejde-196	437	71	)	)	PUNCT
ejde-196	437	72	j	j	PROPN
ejde-196	437	73	=	=	PUNCT
ejde-196	437	74	56.6546	56.6546	NUM
ejde-196	437	75	,	,	PUNCT
ejde-196	437	76	‖u‖∞	‖u‖∞	PROPN
ejde-196	437	77	=	=	SYM
ejde-196	437	78	4.9724	4.9724	NUM
ejde-196	437	79	at	at	ADP
ejde-196	437	80	(	(	PUNCT
ejde-196	437	81	−0.7159,−0.7175	−0.7159,−0.7175	PROPN
ejde-196	437	82	)	)	PUNCT
ejde-196	437	83	,	,	PUNCT
ejde-196	437	84	sid	sid	PROPN
ejde-196	437	85	=	=	PUNCT
ejde-196	437	86	a	a	PROPN
ejde-196	437	87	,	,	PUNCT
ejde-196	437	88	c	c	PROPN
ejde-196	437	89	j	j	PROPN
ejde-196	437	90	=	=	SYM
ejde-196	437	91	82.5416	82.5416	NUM
ejde-196	437	92	,	,	PUNCT
ejde-196	437	93	‖u‖∞	‖u‖∞	PROPN
ejde-196	437	94	=	=	SYM
ejde-196	437	95	4.8665	4.8665	NUM
ejde-196	437	96	at	at	ADP
ejde-196	437	97	(	(	PUNCT
ejde-196	437	98	0.6910,−0.6946	0.6910,−0.6946	NUM
ejde-196	437	99	)	)	PUNCT
ejde-196	437	100	,	,	PUNCT
ejde-196	437	101	sid	sid	PROPN
ejde-196	437	102	=	=	PUNCT
ejde-196	437	103	a	a	PROPN
ejde-196	437	104	,	,	PUNCT
ejde-196	437	105	c	c	NOUN
ejde-196	437	106	,	,	PUNCT
ejde-196	437	107	d	d	X
ejde-196	437	108	(	(	PUNCT
ejde-196	437	109	d	d	NOUN
ejde-196	437	110	)	)	PUNCT
ejde-196	437	111	2	2	NUM
ejde-196	437	112	-	-	NUM
ejde-196	437	113	saddle	saddle	NOUN
ejde-196	437	114	3	3	NUM
ejde-196	437	115	(	(	PUNCT
ejde-196	437	116	e	e	NOUN
ejde-196	437	117	)	)	PUNCT
ejde-196	437	118	3	3	NUM
ejde-196	437	119	-	-	NUM
ejde-196	437	120	saddle	saddle	NOUN
ejde-196	437	121	(	(	PUNCT
ejde-196	437	122	f	f	NOUN
ejde-196	437	123	)	)	PUNCT
ejde-196	437	124	4	4	NUM
ejde-196	437	125	-	-	PUNCT
ejde-196	437	126	saddle	saddle	NOUN
ejde-196	437	127	figure	figure	NOUN
ejde-196	437	128	7	7	NUM
ejde-196	437	129	.	.	PUNCT
ejde-196	438	1	locally	locally	ADV
ejde-196	438	2	m	m	NOUN
ejde-196	438	3	-	-	PUNCT
ejde-196	438	4	type	type	NOUN
ejde-196	438	5	saddles	saddle	NOUN
ejde-196	438	6	with	with	ADP
ejde-196	438	7	j	j	PROPN
ejde-196	438	8	>	>	X
ejde-196	438	9	0	0	PUNCT
ejde-196	439	1	j	j	PROPN
ejde-196	439	2	=	=	NOUN
ejde-196	439	3	46.1140	46.1140	NUM
ejde-196	439	4	,	,	PUNCT
ejde-196	439	5	‖u‖∞	‖u‖∞	PROPN
ejde-196	439	6	=	=	SYM
ejde-196	439	7	4.5370	4.5370	NUM
ejde-196	439	8	at	at	ADP
ejde-196	439	9	(	(	PUNCT
ejde-196	439	10	0.0208,−0.0104	0.0208,−0.0104	NOUN
ejde-196	439	11	)	)	PUNCT
ejde-196	439	12	j	j	PROPN
ejde-196	439	13	=	=	SYM
ejde-196	439	14	29.4731	29.4731	NUM
ejde-196	439	15	,	,	PUNCT
ejde-196	439	16	‖u‖∞	‖u‖∞	PROPN
ejde-196	439	17	=	=	SYM
ejde-196	439	18	5.7561	5.7561	NUM
ejde-196	439	19	at	at	ADP
ejde-196	439	20	(	(	PUNCT
ejde-196	439	21	0.6510,−0.0052	0.6510,−0.0052	NOUN
ejde-196	439	22	)	)	PUNCT
ejde-196	439	23	j	j	PROPN
ejde-196	440	1	=	=	SYM
ejde-196	440	2	17.6390	17.6390	NUM
ejde-196	440	3	,	,	PUNCT
ejde-196	440	4	‖u‖∞	‖u‖∞	PROPN
ejde-196	440	5	=	=	SYM
ejde-196	440	6	4.6441	4.6441	NUM
ejde-196	440	7	at	at	ADP
ejde-196	440	8	(	(	PUNCT
ejde-196	440	9	0.6458,−0.6354	0.6458,−0.6354	NOUN
ejde-196	440	10	)	)	PUNCT
ejde-196	440	11	(	(	PUNCT
ejde-196	440	12	a)nmo	a)nmo	NOUN
ejde-196	440	13	=	=	SYM
ejde-196	440	14	15	15	NUM
ejde-196	440	15	nmo	nmo	NOUN
ejde-196	440	16	=	=	SYM
ejde-196	440	17	90	90	NUM
ejde-196	440	18	nmo	nmo	NOUN
ejde-196	440	19	=	=	SYM
ejde-196	440	20	110	110	NUM
ejde-196	440	21	figure	figure	NOUN
ejde-196	440	22	8	8	NUM
ejde-196	440	23	.	.	PUNCT
ejde-196	441	1	locally	locally	ADV
ejde-196	441	2	m	m	NOUN
ejde-196	441	3	-	-	ADJ
ejde-196	441	4	type	type	NOUN
ejde-196	441	5	positive	positive	ADJ
ejde-196	441	6	solutions	solution	NOUN
ejde-196	441	7	with	with	ADP
ejde-196	441	8	newton	newton	PROPN
ejde-196	441	9	’s	’s	PART
ejde-196	441	10	method	method	PROPN
ejde-196	441	11	192	192	NUM
ejde-196	441	12	m.	m.	NOUN
ejde-196	441	13	li	li	PROPN
ejde-196	441	14	,	,	PUNCT
ejde-196	441	15	b.	b.	PROPN
ejde-196	442	1	ji	ji	PROPN
ejde-196	442	2	,	,	PUNCT
ejde-196	442	3	j.	j.	PROPN
ejde-196	442	4	zhou	zhou	PROPN
ejde-196	442	5	ejde	ejde	PROPN
ejde-196	442	6	/	/	SYM
ejde-196	442	7	si/02	si/02	PROPN
ejde-196	442	8	references	reference	NOUN
ejde-196	442	9	[	[	X
ejde-196	442	10	1	1	NUM
ejde-196	442	11	]	]	PUNCT
ejde-196	442	12	a.	a.	NOUN
ejde-196	442	13	ambrosetti	ambrosetti	PROPN
ejde-196	442	14	,	,	PUNCT
ejde-196	442	15	h.	h.	PROPN
ejde-196	442	16	brezis	brezis	PROPN
ejde-196	442	17	;	;	PUNCT
ejde-196	442	18	combined	combined	ADJ
ejde-196	442	19	effects	effect	NOUN
ejde-196	442	20	of	of	ADP
ejde-196	442	21	concave	concave	NOUN
ejde-196	442	22	and	and	CCONJ
ejde-196	442	23	convex	convex	NOUN
ejde-196	442	24	nonlinearities	nonlinearitie	NOUN
ejde-196	442	25	in	in	ADP
ejde-196	442	26	some	some	DET
ejde-196	442	27	elliptic	elliptic	ADJ
ejde-196	442	28	problems	problem	NOUN
ejde-196	442	29	,	,	PUNCT
ejde-196	442	30	j.	j.	PROPN
ejde-196	442	31	func	func	PROPN
ejde-196	442	32	.	.	PUNCT
ejde-196	443	1	anal	anal	PROPN
ejde-196	443	2	.	.	PROPN
ejde-196	443	3	,	,	PUNCT
ejde-196	443	4	122(1994	122(1994	NUM
ejde-196	443	5	)	)	PUNCT
ejde-196	443	6	519	519	NUM
ejde-196	443	7	-	-	SYM
ejde-196	443	8	543	543	NUM
ejde-196	443	9	.	.	PUNCT
ejde-196	444	1	[	[	X
ejde-196	444	2	2	2	NUM
ejde-196	444	3	]	]	PUNCT
ejde-196	444	4	k.-j	k.-j	PROPN
ejde-196	444	5	.	.	PUNCT
ejde-196	445	1	chen	chen	PROPN
ejde-196	445	2	;	;	PUNCT
ejde-196	445	3	combined	combined	ADJ
ejde-196	445	4	effects	effect	NOUN
ejde-196	445	5	of	of	ADP
ejde-196	445	6	concave	concave	NOUN
ejde-196	445	7	and	and	CCONJ
ejde-196	445	8	convex	convex	NOUN
ejde-196	445	9	nonlinearities	nonlinearitie	NOUN
ejde-196	445	10	in	in	ADP
ejde-196	445	11	elliptic	elliptic	ADJ
ejde-196	445	12	equation	equation	NOUN
ejde-196	445	13	on	on	ADP
ejde-196	445	14	rn	rn	PROPN
ejde-196	445	15	,	,	PUNCT
ejde-196	445	16	jmaa	jmaa	PROPN
ejde-196	445	17	,	,	PUNCT
ejde-196	445	18	355(2009	355(2009	NUM
ejde-196	445	19	)	)	PUNCT
ejde-196	445	20	767	767	NUM
ejde-196	445	21	-	-	SYM
ejde-196	445	22	777	777	NUM
ejde-196	445	23	.	.	PUNCT
ejde-196	446	1	[	[	X
ejde-196	446	2	3	3	X
ejde-196	446	3	]	]	PUNCT
ejde-196	446	4	m.	m.	PROPN
ejde-196	446	5	q.	q.	PROPN
ejde-196	446	6	li	li	PROPN
ejde-196	446	7	;	;	PUNCT
ejde-196	446	8	finding	find	VERB
ejde-196	446	9	multiple	multiple	ADJ
ejde-196	446	10	saddle	saddle	NOUN
ejde-196	446	11	points	point	NOUN
ejde-196	446	12	for	for	ADP
ejde-196	446	13	defocused	defocused	ADJ
ejde-196	446	14	nonlinear	nonlinear	ADJ
ejde-196	446	15	problems	problem	NOUN
ejde-196	446	16	and	and	CCONJ
ejde-196	446	17	gdifferentiable	gdifferentiable	ADJ
ejde-196	446	18	functionals	functional	NOUN
ejde-196	446	19	,	,	PUNCT
ejde-196	447	1	ph.d	ph.d	PROPN
ejde-196	447	2	.	.	PUNCT
ejde-196	447	3	dissertation	dissertation	PROPN
ejde-196	447	4	,	,	PUNCT
ejde-196	447	5	texas	texas	PROPN
ejde-196	447	6	a&m	a&m	PROPN
ejde-196	447	7	university	university	PROPN
ejde-196	447	8	,	,	PUNCT
ejde-196	447	9	2017	2017	NUM
ejde-196	447	10	.	.	PUNCT
ejde-196	448	1	[	[	X
ejde-196	448	2	4	4	X
ejde-196	448	3	]	]	X
ejde-196	448	4	y.	y.	PROPN
ejde-196	448	5	li	li	PROPN
ejde-196	448	6	,	,	PUNCT
ejde-196	448	7	j.	j.	PROPN
ejde-196	448	8	zhou	zhou	PROPN
ejde-196	448	9	;	;	PUNCT
ejde-196	448	10	a	a	DET
ejde-196	448	11	minimax	minimax	NOUN
ejde-196	448	12	method	method	NOUN
ejde-196	448	13	for	for	ADP
ejde-196	448	14	finding	find	VERB
ejde-196	448	15	multiple	multiple	ADJ
ejde-196	448	16	critical	critical	ADJ
ejde-196	448	17	points	point	NOUN
ejde-196	448	18	and	and	CCONJ
ejde-196	448	19	its	its	PRON
ejde-196	448	20	applications	application	NOUN
ejde-196	448	21	to	to	ADP
ejde-196	448	22	nonlinear	nonlinear	ADJ
ejde-196	448	23	pdes	pde	NOUN
ejde-196	448	24	,	,	PUNCT
ejde-196	448	25	siam	siam	PROPN
ejde-196	448	26	sci	sci	PROPN
ejde-196	448	27	.	.	PUNCT
ejde-196	448	28	comp	comp	PROPN
ejde-196	448	29	.	.	PROPN
ejde-196	448	30	,	,	PUNCT
ejde-196	448	31	23(2001	23(2001	NUM
ejde-196	448	32	)	)	PUNCT
ejde-196	448	33	840	840	NUM
ejde-196	448	34	-	-	SYM
ejde-196	448	35	865	865	NUM
ejde-196	448	36	.	.	PUNCT
ejde-196	449	1	[	[	X
ejde-196	449	2	5	5	X
ejde-196	449	3	]	]	PUNCT
ejde-196	449	4	z.	z.	PROPN
ejde-196	449	5	x.	x.	PROPN
ejde-196	449	6	li	li	PROPN
ejde-196	449	7	,	,	PUNCT
ejde-196	449	8	z.-q	z.-q	PROPN
ejde-196	449	9	.	.	PUNCT
ejde-196	449	10	wang	wang	PROPN
ejde-196	449	11	,	,	PUNCT
ejde-196	449	12	j.	j.	PROPN
ejde-196	449	13	zhou	zhou	PROPN
ejde-196	449	14	;	;	PUNCT
ejde-196	449	15	a	a	DET
ejde-196	449	16	new	new	ADJ
ejde-196	449	17	augmented	augment	VERB
ejde-196	449	18	singular	singular	NOUN
ejde-196	449	19	transform	transform	NOUN
ejde-196	449	20	and	and	CCONJ
ejde-196	449	21	its	its	PRON
ejde-196	449	22	partial	partial	ADJ
ejde-196	449	23	newtoncorrection	newtoncorrection	NOUN
ejde-196	449	24	method	method	NOUN
ejde-196	449	25	for	for	ADP
ejde-196	449	26	finding	find	VERB
ejde-196	449	27	more	more	ADJ
ejde-196	449	28	solutions	solution	NOUN
ejde-196	449	29	,	,	PUNCT
ejde-196	449	30	j.	j.	PROPN
ejde-196	449	31	scientific	scientific	PROPN
ejde-196	449	32	computing	computing	NOUN
ejde-196	449	33	,	,	PUNCT
ejde-196	449	34	71(2017	71(2017	NUM
ejde-196	449	35	)	)	PUNCT
ejde-196	449	36	634	634	NUM
ejde-196	449	37	-	-	SYM
ejde-196	449	38	665	665	NUM
ejde-196	449	39	.	.	PUNCT
ejde-196	450	1	[	[	X
ejde-196	450	2	6	6	NUM
ejde-196	450	3	]	]	PUNCT
ejde-196	450	4	z.	z.	PROPN
ejde-196	450	5	liu	liu	PROPN
ejde-196	450	6	,	,	PUNCT
ejde-196	450	7	z.-q	z.-q	PROPN
ejde-196	450	8	.	.	PUNCT
ejde-196	450	9	wang	wang	PROPN
ejde-196	450	10	;	;	PUNCT
ejde-196	450	11	schrödinger	schrödinger	NOUN
ejde-196	450	12	equations	equation	NOUN
ejde-196	450	13	with	with	ADP
ejde-196	450	14	concave	concave	ADJ
ejde-196	450	15	and	and	CCONJ
ejde-196	450	16	convex	convex	NOUN
ejde-196	450	17	nonlinearities	nonlinearitie	NOUN
ejde-196	450	18	,	,	PUNCT
ejde-196	450	19	z.	z.	PROPN
ejde-196	450	20	angew	angew	PROPN
ejde-196	450	21	.	.	PUNCT
ejde-196	451	1	math	math	NOUN
ejde-196	451	2	.	.	PUNCT
ejde-196	452	1	phys	phy	NOUN
ejde-196	452	2	.	.	PUNCT
ejde-196	453	1	56(2005	56(2005	NUM
ejde-196	453	2	)	)	PUNCT
ejde-196	454	1	609–629	609–629	NUM
ejde-196	454	2	.	.	PUNCT
ejde-196	455	1	[	[	X
ejde-196	455	2	7	7	X
ejde-196	455	3	]	]	PUNCT
ejde-196	455	4	z.	z.	PROPN
ejde-196	455	5	nehari	nehari	PROPN
ejde-196	455	6	;	;	PUNCT
ejde-196	455	7	on	on	ADP
ejde-196	455	8	a	a	DET
ejde-196	455	9	class	class	NOUN
ejde-196	455	10	of	of	ADP
ejde-196	455	11	nonlinear	nonlinear	ADJ
ejde-196	455	12	second	second	ADJ
ejde-196	455	13	-	-	PUNCT
ejde-196	455	14	order	order	NOUN
ejde-196	455	15	differential	differential	ADJ
ejde-196	455	16	equations	equation	NOUN
ejde-196	455	17	,	,	PUNCT
ejde-196	455	18	trans	trans	PROPN
ejde-196	455	19	.	.	PROPN
ejde-196	455	20	amer	amer	PROPN
ejde-196	455	21	.	.	PUNCT
ejde-196	455	22	math	math	PROPN
ejde-196	455	23	.	.	PUNCT
ejde-196	456	1	soc	soc	PROPN
ejde-196	456	2	.	.	PUNCT
ejde-196	457	1	95(1960	95(1960	NUM
ejde-196	457	2	)	)	PUNCT
ejde-196	458	1	101–123	101–123	NUM
ejde-196	458	2	.	.	PUNCT
ejde-196	459	1	[	[	X
ejde-196	459	2	8	8	NUM
ejde-196	459	3	]	]	X
ejde-196	459	4	a.	a.	NOUN
ejde-196	459	5	pankov	pankov	NOUN
ejde-196	459	6	;	;	PUNCT
ejde-196	459	7	periodic	periodic	ADJ
ejde-196	459	8	nonlinear	nonlinear	ADJ
ejde-196	459	9	schrödinger	schrödinger	NOUN
ejde-196	459	10	equation	equation	NOUN
ejde-196	459	11	with	with	ADP
ejde-196	459	12	application	application	NOUN
ejde-196	459	13	to	to	ADP
ejde-196	459	14	photonic	photonic	ADJ
ejde-196	459	15	crystals	crystal	NOUN
ejde-196	459	16	,	,	PUNCT
ejde-196	459	17	milan	milan	PROPN
ejde-196	459	18	j.	j.	PROPN
ejde-196	459	19	of	of	ADP
ejde-196	459	20	mathematics	mathematics	PROPN
ejde-196	459	21	,	,	PUNCT
ejde-196	459	22	73(2005	73(2005	NUM
ejde-196	459	23	)	)	PUNCT
ejde-196	459	24	259–287	259–287	NUM
ejde-196	459	25	.	.	PUNCT
ejde-196	460	1	[	[	X
ejde-196	460	2	9	9	NUM
ejde-196	460	3	]	]	PUNCT
ejde-196	460	4	a.	a.	NOUN
ejde-196	460	5	pankov	pankov	NOUN
ejde-196	460	6	;	;	PUNCT
ejde-196	460	7	gap	gap	NOUN
ejde-196	460	8	solitons	soliton	NOUN
ejde-196	460	9	in	in	ADP
ejde-196	460	10	periodic	periodic	ADJ
ejde-196	460	11	discret	discret	ADJ
ejde-196	460	12	nonlinear	nonlinear	PROPN
ejde-196	460	13	schodinger	schodinger	NOUN
ejde-196	460	14	equations	equation	NOUN
ejde-196	460	15	,	,	PUNCT
ejde-196	460	16	nonlinearities	nonlinearitie	NOUN
ejde-196	460	17	,	,	PUNCT
ejde-196	460	18	1(2006	1(2006	NUM
ejde-196	460	19	)	)	PUNCT
ejde-196	460	20	19	19	NUM
ejde-196	460	21	-	-	SYM
ejde-196	460	22	27	27	NUM
ejde-196	460	23	.	.	PUNCT
ejde-196	461	1	[	[	X
ejde-196	461	2	10	10	NUM
ejde-196	461	3	]	]	X
ejde-196	461	4	z.-q	z.-q	PROPN
ejde-196	461	5	.	.	PUNCT
ejde-196	461	6	wang	wang	PROPN
ejde-196	461	7	,	,	PUNCT
ejde-196	461	8	j.	j.	PROPN
ejde-196	461	9	zhou	zhou	PROPN
ejde-196	461	10	;	;	PUNCT
ejde-196	461	11	a	a	DET
ejde-196	461	12	local	local	ADJ
ejde-196	461	13	minimax	minimax	NOUN
ejde-196	461	14	-	-	PUNCT
ejde-196	461	15	newton	newton	NOUN
ejde-196	461	16	method	method	NOUN
ejde-196	461	17	for	for	ADP
ejde-196	461	18	finding	find	VERB
ejde-196	461	19	critical	critical	ADJ
ejde-196	461	20	points	point	NOUN
ejde-196	461	21	with	with	ADP
ejde-196	461	22	symmetries	symmetry	NOUN
ejde-196	461	23	,	,	PUNCT
ejde-196	461	24	siam	siam	PROPN
ejde-196	461	25	j.	j.	PROPN
ejde-196	461	26	num	num	PROPN
ejde-196	461	27	.	.	PROPN
ejde-196	462	1	anal	anal	PROPN
ejde-196	462	2	.	.	PROPN
ejde-196	462	3	,	,	PUNCT
ejde-196	462	4	42(2004	42(2004	PROPN
ejde-196	462	5	)	)	PUNCT
ejde-196	462	6	,	,	PUNCT
ejde-196	462	7	1745	1745	NUM
ejde-196	462	8	-	-	SYM
ejde-196	462	9	1759	1759	NUM
ejde-196	462	10	.	.	PUNCT
ejde-196	463	1	[	[	X
ejde-196	463	2	11	11	NUM
ejde-196	463	3	]	]	PUNCT
ejde-196	463	4	y.-f	y.-f	PROPN
ejde-196	463	5	.	.	PUNCT
ejde-196	463	6	wu	wu	PROPN
ejde-196	463	7	;	;	PUNCT
ejde-196	463	8	on	on	ADP
ejde-196	463	9	semilinear	semilinear	PROPN
ejde-196	463	10	elliptic	elliptic	ADJ
ejde-196	463	11	equations	equation	NOUN
ejde-196	463	12	involving	involve	VERB
ejde-196	463	13	concave	concave	VERB
ejde-196	463	14	-	-	PUNCT
ejde-196	463	15	convex	convex	NOUN
ejde-196	463	16	nonlinearities	nonlinearitie	NOUN
ejde-196	463	17	and	and	CCONJ
ejde-196	463	18	signchanging	signchange	VERB
ejde-196	463	19	weight	weight	NOUN
ejde-196	463	20	function	function	NOUN
ejde-196	463	21	,	,	PUNCT
ejde-196	463	22	j.	j.	PROPN
ejde-196	463	23	math	math	PROPN
ejde-196	463	24	.	.	PUNCT
ejde-196	464	1	anal	anal	PROPN
ejde-196	464	2	.	.	PUNCT
ejde-196	464	3	appl	appl	PROPN
ejde-196	464	4	.	.	PROPN
ejde-196	464	5	,	,	PUNCT
ejde-196	464	6	318(2006	318(2006	NUM
ejde-196	464	7	)	)	PUNCT
ejde-196	464	8	253	253	NUM
ejde-196	464	9	-	-	SYM
ejde-196	464	10	270	270	NUM
ejde-196	464	11	.	.	PUNCT
ejde-196	465	1	[	[	X
ejde-196	465	2	12	12	NUM
ejde-196	465	3	]	]	PUNCT
ejde-196	465	4	a.	a.	NOUN
ejde-196	465	5	szulkin	szulkin	PROPN
ejde-196	465	6	,	,	PUNCT
ejde-196	465	7	t.	t.	PROPN
ejde-196	465	8	weth	weth	PROPN
ejde-196	465	9	;	;	PUNCT
ejde-196	465	10	the	the	DET
ejde-196	465	11	method	method	NOUN
ejde-196	465	12	of	of	ADP
ejde-196	465	13	nehari	nehari	PROPN
ejde-196	465	14	manifold	manifold	ADJ
ejde-196	465	15	,	,	PUNCT
ejde-196	465	16	handbook	handbook	NOUN
ejde-196	465	17	of	of	ADP
ejde-196	465	18	nonconvex	nonconvex	NOUN
ejde-196	465	19	analysis	analysis	NOUN
ejde-196	465	20	and	and	CCONJ
ejde-196	465	21	applications	application	NOUN
ejde-196	465	22	,	,	PUNCT
ejde-196	465	23	(	(	PUNCT
ejde-196	465	24	2010	2010	NUM
ejde-196	465	25	)	)	PUNCT
ejde-196	465	26	597	597	NUM
ejde-196	465	27	-	-	SYM
ejde-196	465	28	632	632	NUM
ejde-196	465	29	.	.	PUNCT
ejde-196	466	1	[	[	X
ejde-196	466	2	13	13	NUM
ejde-196	466	3	]	]	PUNCT
ejde-196	466	4	z.	z.	PROPN
ejde-196	466	5	q.	q.	PROPN
ejde-196	466	6	xie	xie	PROPN
ejde-196	466	7	,	,	PUNCT
ejde-196	466	8	w.	w.	PROPN
ejde-196	466	9	f.	f.	PROPN
ejde-196	466	10	yi	yi	PROPN
ejde-196	466	11	,	,	PUNCT
ejde-196	466	12	j.	j.	PROPN
ejde-196	466	13	zhou	zhou	PROPN
ejde-196	466	14	;	;	PUNCT
ejde-196	466	15	an	an	DET
ejde-196	466	16	augmented	augment	VERB
ejde-196	466	17	singular	singular	NOUN
ejde-196	466	18	transform	transform	NOUN
ejde-196	466	19	and	and	CCONJ
ejde-196	466	20	its	its	PRON
ejde-196	466	21	partial	partial	ADJ
ejde-196	466	22	newton	newton	NOUN
ejde-196	466	23	method	method	NOUN
ejde-196	466	24	for	for	ADP
ejde-196	466	25	finding	find	VERB
ejde-196	466	26	multiple	multiple	ADJ
ejde-196	466	27	solutions	solution	NOUN
ejde-196	466	28	,	,	PUNCT
ejde-196	466	29	jcam	jcam	NOUN
ejde-196	466	30	,	,	PUNCT
ejde-196	466	31	286(2015	286(2015	NUM
ejde-196	466	32	)	)	PUNCT
ejde-196	466	33	145	145	NUM
ejde-196	466	34	-	-	SYM
ejde-196	466	35	157	157	NUM
ejde-196	466	36	.	.	PUNCT
ejde-196	467	1	[	[	X
ejde-196	467	2	14	14	NUM
ejde-196	467	3	]	]	PUNCT
ejde-196	467	4	x.	x.	NOUN
ejde-196	467	5	yao	yao	PROPN
ejde-196	467	6	;	;	PUNCT
ejde-196	467	7	ljusternik	ljusternik	X
ejde-196	467	8	–	–	PUNCT
ejde-196	467	9	schnirelman	schnirelman	NOUN
ejde-196	467	10	minimax	minimax	NOUN
ejde-196	467	11	algorithms	algorithm	NOUN
ejde-196	467	12	and	and	CCONJ
ejde-196	467	13	an	an	DET
ejde-196	467	14	application	application	NOUN
ejde-196	467	15	for	for	ADP
ejde-196	467	16	finding	find	VERB
ejde-196	467	17	multiple	multiple	ADJ
ejde-196	467	18	negative	negative	ADJ
ejde-196	467	19	energy	energy	NOUN
ejde-196	467	20	solutions	solution	NOUN
ejde-196	467	21	of	of	ADP
ejde-196	467	22	semilinear	semilinear	PROPN
ejde-196	467	23	elliptic	elliptic	ADJ
ejde-196	467	24	dirichlet	dirichlet	PROPN
ejde-196	467	25	problem	problem	NOUN
ejde-196	467	26	involving	involve	VERB
ejde-196	467	27	concave	concave	NOUN
ejde-196	467	28	and	and	CCONJ
ejde-196	467	29	convex	convex	NOUN
ejde-196	467	30	nonlinearities	nonlinearitie	NOUN
ejde-196	467	31	:	:	PUNCT
ejde-196	467	32	part	part	NOUN
ejde-196	467	33	i.	i.	PROPN
ejde-196	467	34	algorithms	algorithms	PROPN
ejde-196	467	35	and	and	CCONJ
ejde-196	467	36	convergence	convergence	NOUN
ejde-196	467	37	,	,	PUNCT
ejde-196	467	38	j.	j.	PROPN
ejde-196	467	39	scientific	scientific	PROPN
ejde-196	467	40	computing	computing	NOUN
ejde-196	467	41	,	,	PUNCT
ejde-196	467	42	66(2016	66(2016	NUM
ejde-196	467	43	)	)	PUNCT
ejde-196	467	44	,	,	PUNCT
ejde-196	467	45	19	19	NUM
ejde-196	467	46	-	-	SYM
ejde-196	467	47	40	40	NUM
ejde-196	467	48	.	.	PUNCT
ejde-196	468	1	[	[	X
ejde-196	468	2	15	15	NUM
ejde-196	468	3	]	]	X
ejde-196	468	4	e.	e.	PROPN
ejde-196	468	5	zeidler	zeidler	PROPN
ejde-196	468	6	;	;	PUNCT
ejde-196	468	7	nonlinear	nonlinear	ADJ
ejde-196	468	8	functional	functional	ADJ
ejde-196	468	9	analysis	analysis	NOUN
ejde-196	468	10	and	and	CCONJ
ejde-196	468	11	its	its	PRON
ejde-196	468	12	applications	application	NOUN
ejde-196	468	13	i	i	PRON
ejde-196	468	14	and	and	CCONJ
ejde-196	468	15	iii	iii	PROPN
ejde-196	468	16	,	,	PUNCT
ejde-196	468	17	springer	springer	NOUN
ejde-196	468	18	-	-	PUNCT
ejde-196	468	19	verlag	verlag	PROPN
ejde-196	468	20	,	,	PUNCT
ejde-196	468	21	new	new	PROPN
ejde-196	468	22	york	york	PROPN
ejde-196	468	23	,	,	PUNCT
ejde-196	468	24	1985	1985	NUM
ejde-196	468	25	.	.	PUNCT
ejde-196	469	1	[	[	X
ejde-196	469	2	16	16	NUM
ejde-196	469	3	]	]	PUNCT
ejde-196	469	4	j.	j.	PROPN
ejde-196	469	5	zhou	zhou	PROPN
ejde-196	469	6	;	;	PUNCT
ejde-196	469	7	a	a	DET
ejde-196	469	8	local	local	ADJ
ejde-196	469	9	min	min	ADJ
ejde-196	469	10	-	-	ADJ
ejde-196	469	11	orthogonal	orthogonal	ADJ
ejde-196	469	12	method	method	NOUN
ejde-196	469	13	for	for	ADP
ejde-196	469	14	finding	find	VERB
ejde-196	469	15	multiple	multiple	ADJ
ejde-196	469	16	saddle	saddle	NOUN
ejde-196	469	17	points	point	NOUN
ejde-196	469	18	,	,	PUNCT
ejde-196	469	19	j.	j.	PROPN
ejde-196	469	20	math	math	PROPN
ejde-196	469	21	.	.	PUNCT
ejde-196	470	1	anal	anal	PROPN
ejde-196	470	2	.	.	PUNCT
ejde-196	471	1	appl	appl	PROPN
ejde-196	471	2	.	.	PROPN
ejde-196	471	3	,	,	PUNCT
ejde-196	471	4	291(2004	291(2004	NUM
ejde-196	471	5	)	)	PUNCT
ejde-196	471	6	66	66	NUM
ejde-196	471	7	-	-	SYM
ejde-196	471	8	81	81	NUM
ejde-196	471	9	.	.	PUNCT
ejde-196	472	1	[	[	X
ejde-196	472	2	17	17	NUM
ejde-196	472	3	]	]	PUNCT
ejde-196	472	4	j.	j.	PROPN
ejde-196	472	5	zhou	zhou	PROPN
ejde-196	472	6	;	;	PUNCT
ejde-196	472	7	instability	instability	NOUN
ejde-196	472	8	analysis	analysis	NOUN
ejde-196	472	9	of	of	ADP
ejde-196	472	10	saddle	saddle	ADJ
ejde-196	472	11	points	point	NOUN
ejde-196	472	12	by	by	ADP
ejde-196	472	13	a	a	DET
ejde-196	472	14	local	local	ADJ
ejde-196	472	15	minimax	minimax	NOUN
ejde-196	472	16	method	method	NOUN
ejde-196	472	17	,	,	PUNCT
ejde-196	472	18	math	math	NOUN
ejde-196	472	19	.	.	PUNCT
ejde-196	473	1	comp	comp	PROPN
ejde-196	473	2	.	.	PUNCT
ejde-196	473	3	,	,	PUNCT
ejde-196	473	4	74(2005	74(2005	NUM
ejde-196	473	5	)	)	PUNCT
ejde-196	473	6	,	,	PUNCT
ejde-196	473	7	1391	1391	NUM
ejde-196	473	8	-	-	SYM
ejde-196	473	9	1411	1411	NUM
ejde-196	473	10	.	.	PUNCT
ejde-196	474	1	[	[	X
ejde-196	474	2	18	18	NUM
ejde-196	474	3	]	]	PUNCT
ejde-196	474	4	j.	j.	PROPN
ejde-196	474	5	zhou	zhou	PROPN
ejde-196	474	6	;	;	PUNCT
ejde-196	474	7	solving	solve	VERB
ejde-196	474	8	multiple	multiple	ADJ
ejde-196	474	9	solution	solution	NOUN
ejde-196	474	10	problems	problem	NOUN
ejde-196	474	11	:	:	PUNCT
ejde-196	474	12	computational	computational	ADJ
ejde-196	474	13	methods	method	NOUN
ejde-196	474	14	and	and	CCONJ
ejde-196	474	15	theory	theory	NOUN
ejde-196	474	16	revisited	revisit	VERB
ejde-196	474	17	,	,	PUNCT
ejde-196	474	18	communication	communication	NOUN
ejde-196	474	19	appl	appl	NOUN
ejde-196	475	1	.	.	PUNCT
ejde-196	475	2	math	math	PROPN
ejde-196	475	3	.	.	PUNCT
ejde-196	476	1	&	&	CCONJ
ejde-196	476	2	comput	comput	PROPN
ejde-196	476	3	.	.	PUNCT
ejde-196	477	1	,	,	PUNCT
ejde-196	477	2	31(2017	31(2017	NUM
ejde-196	477	3	)	)	PUNCT
ejde-196	477	4	1	1	NUM
ejde-196	477	5	-	-	SYM
ejde-196	477	6	31	31	NUM
ejde-196	477	7	.	.	PUNCT
ejde-196	478	1	meiqin	meiqin	PROPN
ejde-196	478	2	li	li	PROPN
ejde-196	478	3	school	school	PROPN
ejde-196	478	4	of	of	ADP
ejde-196	478	5	engineering	engineering	NOUN
ejde-196	478	6	,	,	PUNCT
ejde-196	478	7	university	university	PROPN
ejde-196	478	8	of	of	ADP
ejde-196	478	9	virginia	virginia	PROPN
ejde-196	478	10	,	,	PUNCT
ejde-196	478	11	charlottesville	charlottesville	PROPN
ejde-196	478	12	,	,	PUNCT
ejde-196	478	13	va	va	PROPN
ejde-196	478	14	22904	22904	NUM
ejde-196	478	15	,	,	PUNCT
ejde-196	478	16	usa	usa	PROPN
ejde-196	478	17	email	email	NOUN
ejde-196	478	18	address	address	NOUN
ejde-196	478	19	:	:	PUNCT
ejde-196	479	1	ml2vq@virginia.edu	ml2vq@virginia.edu	NOUN
ejde-196	479	2	bingbing	bingbe	VERB
ejde-196	479	3	ji	ji	PROPN
ejde-196	479	4	microstrategy	microstrategy	PROPN
ejde-196	479	5	inc	inc	PROPN
ejde-196	479	6	.	.	PROPN
ejde-196	479	7	,	,	PUNCT
ejde-196	479	8	tysons	tysons	PROPN
ejde-196	479	9	corner	corner	PROPN
ejde-196	479	10	,	,	PUNCT
ejde-196	479	11	va	va	PROPN
ejde-196	479	12	22182	22182	NUM
ejde-196	479	13	usa	usa	PROPN
ejde-196	479	14	email	email	NOUN
ejde-196	479	15	address	address	NOUN
ejde-196	479	16	:	:	PUNCT
ejde-196	479	17	bingbing.tamu@gmail.com	bingbing.tamu@gmail.com	PROPN
ejde-196	479	18	jianxin	jianxin	PROPN
ejde-196	479	19	zhou	zhou	PROPN
ejde-196	479	20	texas	texas	PROPN
ejde-196	479	21	a&m	a&m	PROPN
ejde-196	479	22	university	university	PROPN
ejde-196	479	23	,	,	PUNCT
ejde-196	479	24	college	college	NOUN
ejde-196	479	25	station	station	NOUN
ejde-196	479	26	,	,	PUNCT
ejde-196	479	27	tx	tx	PROPN
ejde-196	479	28	77843	77843	NUM
ejde-196	479	29	,	,	PUNCT
ejde-196	479	30	usa	usa	PROPN
ejde-196	479	31	email	email	NOUN
ejde-196	479	32	address	address	NOUN
ejde-196	479	33	:	:	PUNCT
ejde-196	480	1	jzhou@math.tamu.edu	jzhou@math.tamu.edu	NUM
ejde-196	480	2	1	1	NUM
ejde-196	480	3	.	.	PUNCT
ejde-196	480	4	introduction	introduction	NOUN
ejde-196	480	5	2	2	NUM
ejde-196	480	6	.	.	PUNCT
ejde-196	481	1	a	a	DET
ejde-196	481	2	new	new	ADJ
ejde-196	481	3	local	local	ADJ
ejde-196	481	4	min	min	ADJ
ejde-196	481	5	-	-	ADJ
ejde-196	481	6	orthogonal	orthogonal	ADJ
ejde-196	481	7	method	method	NOUN
ejde-196	481	8	3	3	NUM
ejde-196	481	9	.	.	PUNCT
ejde-196	482	1	a	a	DET
ejde-196	482	2	numerical	numerical	ADJ
ejde-196	482	3	local	local	ADJ
ejde-196	482	4	min	min	ADJ
ejde-196	482	5	-	-	ADJ
ejde-196	482	6	orthogonal	orthogonal	ADJ
ejde-196	482	7	algorithm	algorithm	NOUN
ejde-196	482	8	4	4	NUM
ejde-196	482	9	.	.	PUNCT
ejde-196	482	10	algorithm	algorithm	NOUN
ejde-196	482	11	convergence	convergence	NOUN
ejde-196	482	12	analysis	analysis	NOUN
ejde-196	482	13	5	5	NUM
ejde-196	482	14	.	.	PUNCT
ejde-196	482	15	two	two	NUM
ejde-196	482	16	new	new	ADJ
ejde-196	482	17	variations	variation	NOUN
ejde-196	482	18	to	to	ADP
ejde-196	482	19	the	the	DET
ejde-196	482	20	min	min	ADJ
ejde-196	482	21	-	-	ADJ
ejde-196	482	22	orthogonal	orthogonal	ADJ
ejde-196	482	23	method	method	NOUN
ejde-196	482	24	5.1	5.1	NUM
ejde-196	482	25	.	.	PUNCT
ejde-196	482	26	justification	justification	NOUN
ejde-196	482	27	of	of	ADP
ejde-196	482	28	the	the	DET
ejde-196	482	29	min	min	PROPN
ejde-196	482	30	-	-	PUNCT
ejde-196	482	31	min	min	ADJ
ejde-196	482	32	-	-	ADJ
ejde-196	482	33	max	max	ADJ
ejde-196	482	34	algorithm	algorithm	PROPN
ejde-196	482	35	5.2	5.2	NUM
ejde-196	482	36	.	.	PUNCT
ejde-196	483	1	justification	justification	NOUN
ejde-196	483	2	of	of	ADP
ejde-196	483	3	the	the	DET
ejde-196	483	4	min	min	PROPN
ejde-196	483	5	-	-	PUNCT
ejde-196	483	6	max	max	PROPN
ejde-196	483	7	-	-	PUNCT
ejde-196	483	8	min	min	NOUN
ejde-196	483	9	algorithm	algorithm	PROPN
ejde-196	483	10	6	6	NUM
ejde-196	483	11	.	.	PUNCT
ejde-196	483	12	numerical	numerical	ADJ
ejde-196	483	13	examples	example	NOUN
ejde-196	483	14	6.1	6.1	NUM
ejde-196	483	15	.	.	PUNCT
ejde-196	484	1	numerical	numerical	ADJ
ejde-196	484	2	examples	example	NOUN
ejde-196	484	3	by	by	ADP
ejde-196	484	4	the	the	DET
ejde-196	484	5	mini	mini	PROPN
ejde-196	484	6	-	-	PROPN
ejde-196	484	7	max	max	ADJ
ejde-196	484	8	-	-	PUNCT
ejde-196	484	9	min	min	NOUN
ejde-196	484	10	method	method	PROPN
ejde-196	484	11	6.2	6.2	NUM
ejde-196	484	12	.	.	PUNCT
ejde-196	485	1	numerical	numerical	ADJ
ejde-196	485	2	examples	example	NOUN
ejde-196	485	3	by	by	ADP
ejde-196	485	4	the	the	DET
ejde-196	485	5	min	min	ADJ
ejde-196	485	6	-	-	ADJ
ejde-196	485	7	orthogonal	orthogonal	ADJ
ejde-196	485	8	method	method	NOUN
ejde-196	485	9	final	final	ADJ
ejde-196	485	10	remarks	remark	NOUN
ejde-196	485	11	references	reference	NOUN