id	sid	tid	token	lemma	pos
ejpam-460	1	1	5_460_zhichun.dvi	5_460_zhichun.dvi	NUM
ejpam-460	1	2	european	european	ADJ
ejpam-460	1	3	journal	journal	NOUN
ejpam-460	1	4	of	of	ADP
ejpam-460	1	5	pure	pure	ADJ
ejpam-460	1	6	and	and	CCONJ
ejpam-460	1	7	applied	apply	VERB
ejpam-460	1	8	mathematics	mathematic	NOUN
ejpam-460	1	9	vol	vol	NOUN
ejpam-460	1	10	.	.	PROPN
ejpam-460	2	1	2	2	NUM
ejpam-460	2	2	,	,	PUNCT
ejpam-460	2	3	no	no	INTJ
ejpam-460	2	4	.	.	NOUN
ejpam-460	2	5	4	4	NUM
ejpam-460	2	6	,	,	PUNCT
ejpam-460	2	7	2009	2009	NUM
ejpam-460	2	8	,	,	PUNCT
ejpam-460	2	9	(	(	PUNCT
ejpam-460	2	10	532	532	NUM
ejpam-460	2	11	-	-	NUM
ejpam-460	2	12	543	543	NUM
ejpam-460	2	13	)	)	PUNCT
ejpam-460	2	14	issn	issn	PROPN
ejpam-460	2	15	1307	1307	NUM
ejpam-460	2	16	-	-	SYM
ejpam-460	2	17	5543	5543	NUM
ejpam-460	2	18	–	–	PUNCT
ejpam-460	2	19	www.ejpam.com	www.ejpam.com	X
ejpam-460	2	20	on	on	ADP
ejpam-460	2	21	the	the	DET
ejpam-460	2	22	solutions	solution	NOUN
ejpam-460	2	23	for	for	ADP
ejpam-460	2	24	grötzsch	grötzsch	ADJ
ejpam-460	2	25	annulus	annulus	PROPN
ejpam-460	2	26	extremal	extremal	PROPN
ejpam-460	2	27	problem	problem	NOUN
ejpam-460	2	28	xie	xie	PROPN
ejpam-460	2	29	zhichun∗	zhichun∗	PROPN
ejpam-460	2	30	and	and	CCONJ
ejpam-460	2	31	huang	huang	PROPN
ejpam-460	2	32	xinzhong	xinzhong	PROPN
ejpam-460	2	33	school	school	PROPN
ejpam-460	2	34	of	of	ADP
ejpam-460	2	35	mathematical	mathematical	ADJ
ejpam-460	2	36	science	science	NOUN
ejpam-460	2	37	,	,	PUNCT
ejpam-460	2	38	huaqiao	huaqiao	PROPN
ejpam-460	2	39	university	university	PROPN
ejpam-460	2	40	,	,	PUNCT
ejpam-460	2	41	quanzhou	quanzhou	PROPN
ejpam-460	2	42	,	,	PUNCT
ejpam-460	2	43	fujian	fujian	PROPN
ejpam-460	2	44	,	,	PUNCT
ejpam-460	2	45	362021	362021	NUM
ejpam-460	2	46	,	,	PUNCT
ejpam-460	3	1	p.	p.	PROPN
ejpam-460	3	2	r.	r.	PROPN
ejpam-460	3	3	c.	c.	PROPN
ejpam-460	3	4	abstract	abstract	PROPN
ejpam-460	3	5	.	.	PUNCT
ejpam-460	4	1	using	use	VERB
ejpam-460	4	2	the	the	DET
ejpam-460	4	3	properties	property	NOUN
ejpam-460	4	4	of	of	ADP
ejpam-460	4	5	planar	planar	ADJ
ejpam-460	4	6	quasiconformal	quasiconformal	ADJ
ejpam-460	4	7	mappings	mapping	NOUN
ejpam-460	4	8	,	,	PUNCT
ejpam-460	4	9	we	we	PRON
ejpam-460	4	10	obtain	obtain	VERB
ejpam-460	4	11	the	the	DET
ejpam-460	4	12	solutions	solution	NOUN
ejpam-460	4	13	for	for	ADP
ejpam-460	4	14	the	the	DET
ejpam-460	4	15	grötzsch	grötzsch	PROPN
ejpam-460	4	16	extremal	extremal	ADJ
ejpam-460	4	17	problem	problem	NOUN
ejpam-460	4	18	on	on	ADP
ejpam-460	4	19	the	the	DET
ejpam-460	4	20	annulus	annulus	NOUN
ejpam-460	4	21	.	.	PUNCT
ejpam-460	5	1	we	we	PRON
ejpam-460	5	2	also	also	ADV
ejpam-460	5	3	point	point	VERB
ejpam-460	5	4	out	out	ADP
ejpam-460	5	5	the	the	DET
ejpam-460	5	6	shortcoming	shortcoming	NOUN
ejpam-460	5	7	used	use	VERB
ejpam-460	5	8	in	in	ADP
ejpam-460	5	9	[	[	X
ejpam-460	5	10	3	3	X
ejpam-460	5	11	]	]	PUNCT
ejpam-460	5	12	for	for	ADP
ejpam-460	5	13	solving	solve	VERB
ejpam-460	5	14	this	this	DET
ejpam-460	5	15	extremal	extremal	ADJ
ejpam-460	5	16	problem	problem	NOUN
ejpam-460	5	17	.	.	PUNCT
ejpam-460	6	1	moreover	moreover	ADV
ejpam-460	6	2	,	,	PUNCT
ejpam-460	6	3	by	by	ADP
ejpam-460	6	4	the	the	DET
ejpam-460	6	5	hyperbolic	hyperbolic	ADJ
ejpam-460	6	6	area	area	NOUN
ejpam-460	6	7	distortion	distortion	NOUN
ejpam-460	6	8	and	and	CCONJ
ejpam-460	6	9	the	the	DET
ejpam-460	6	10	property	property	NOUN
ejpam-460	6	11	of	of	ADP
ejpam-460	6	12	domain	domain	NOUN
ejpam-460	6	13	module	module	NOUN
ejpam-460	6	14	,	,	PUNCT
ejpam-460	6	15	one	one	NUM
ejpam-460	6	16	criterion	criterion	NOUN
ejpam-460	6	17	for	for	ADP
ejpam-460	6	18	the	the	DET
ejpam-460	6	19	solution	solution	NOUN
ejpam-460	6	20	of	of	ADP
ejpam-460	6	21	the	the	DET
ejpam-460	6	22	grötzsch	grötzsch	ADJ
ejpam-460	6	23	annulus	annulus	PROPN
ejpam-460	6	24	extremal	extremal	ADJ
ejpam-460	6	25	problem	problem	NOUN
ejpam-460	6	26	is	be	AUX
ejpam-460	6	27	given	give	VERB
ejpam-460	6	28	under	under	ADP
ejpam-460	6	29	some	some	DET
ejpam-460	6	30	conditions	condition	NOUN
ejpam-460	6	31	.	.	PUNCT
ejpam-460	7	1	2000	2000	NUM
ejpam-460	7	2	mathematics	mathematic	NOUN
ejpam-460	7	3	subject	subject	NOUN
ejpam-460	7	4	classifications	classification	NOUN
ejpam-460	7	5	:	:	PUNCT
ejpam-460	7	6	primary	primary	NOUN
ejpam-460	7	7	30c62	30c62	NUM
ejpam-460	7	8	,	,	PUNCT
ejpam-460	7	9	secondary	secondary	ADJ
ejpam-460	7	10	30c55	30c55	NUM
ejpam-460	7	11	key	key	ADJ
ejpam-460	7	12	words	word	NOUN
ejpam-460	7	13	and	and	CCONJ
ejpam-460	7	14	phrases	phrase	NOUN
ejpam-460	7	15	:	:	PUNCT
ejpam-460	7	16	grötzsch	grötzsch	VERB
ejpam-460	7	17	extremal	extremal	ADJ
ejpam-460	7	18	problem	problem	NOUN
ejpam-460	7	19	,	,	PUNCT
ejpam-460	7	20	quasiconformal	quasiconformal	ADJ
ejpam-460	7	21	mapping	mapping	NOUN
ejpam-460	7	22	,	,	PUNCT
ejpam-460	7	23	hyperbolic	hyperbolic	ADJ
ejpam-460	7	24	area	area	NOUN
ejpam-460	7	25	distortion	distortion	NOUN
ejpam-460	7	26	1	1	NUM
ejpam-460	7	27	.	.	PUNCT
ejpam-460	8	1	introduction	introduction	NOUN
ejpam-460	8	2	let	let	VERB
ejpam-460	8	3	ω	ω	X
ejpam-460	8	4	,	,	PUNCT
ejpam-460	8	5	ω′	ω′	X
ejpam-460	8	6	⊂	⊂	PROPN
ejpam-460	8	7	c	c	AUX
ejpam-460	8	8	be	be	AUX
ejpam-460	8	9	planar	planar	ADJ
ejpam-460	8	10	domains	domain	NOUN
ejpam-460	8	11	,	,	PUNCT
ejpam-460	8	12	a	a	DET
ejpam-460	8	13	sense	sense	NOUN
ejpam-460	8	14	-	-	PUNCT
ejpam-460	8	15	preserving	preserve	VERB
ejpam-460	8	16	homeomorphism	homeomorphism	ADJ
ejpam-460	8	17	f	f	X
ejpam-460	8	18	:	:	PUNCT
ejpam-460	8	19	ω→	ω→	X
ejpam-460	8	20	ω′	ω′	PROPN
ejpam-460	8	21	is	be	AUX
ejpam-460	8	22	said	say	VERB
ejpam-460	8	23	to	to	PART
ejpam-460	8	24	be	be	AUX
ejpam-460	8	25	k−quasiconformal	k−quasiconformal	NOUN
ejpam-460	8	26	mapping	mapping	NOUN
ejpam-460	8	27	in	in	ADP
ejpam-460	8	28	ω	ω	PROPN
ejpam-460	8	29	,	,	PUNCT
ejpam-460	8	30	if	if	SCONJ
ejpam-460	8	31	it	it	PRON
ejpam-460	8	32	satisfies	satisfy	VERB
ejpam-460	8	33	:	:	PUNCT
ejpam-460	8	34	(	(	PUNCT
ejpam-460	8	35	1	1	X
ejpam-460	8	36	)	)	PUNCT
ejpam-460	8	37	f	f	PROPN
ejpam-460	8	38	is	be	AUX
ejpam-460	8	39	acl	acl	PROPN
ejpam-460	8	40	in	in	ADP
ejpam-460	8	41	ω	ω	PROPN
ejpam-460	8	42	;	;	PUNCT
ejpam-460	8	43	(	(	PUNCT
ejpam-460	8	44	2	2	X
ejpam-460	8	45	)	)	PUNCT
ejpam-460	8	46	∗corresponding	∗corresponde	VERB
ejpam-460	8	47	author	author	NOUN
ejpam-460	8	48	.	.	PUNCT
ejpam-460	9	1	email	email	NOUN
ejpam-460	9	2	addresses	address	NOUN
ejpam-460	9	3	:	:	PUNCT
ejpam-460	9	4	xiezhi	xiezhi	PROPN
ejpam-460	9	5	hun�hqu.edu	hun�hqu.edu	PROPN
ejpam-460	9	6	.	.	PUNCT
ejpam-460	10	1	n	n	PROPN
ejpam-460	10	2	(	(	PUNCT
ejpam-460	10	3	x.	x.	NOUN
ejpam-460	10	4	zhichun	zhichun	PROPN
ejpam-460	10	5	)	)	PUNCT
ejpam-460	10	6	,	,	PUNCT
ejpam-460	10	7	huangxz�hqu.edu	huangxz�hqu.edu	PROPN
ejpam-460	10	8	.	.	PUNCT
ejpam-460	11	1	n	n	PROPN
ejpam-460	11	2	(	(	PUNCT
ejpam-460	11	3	h.	h.	PROPN
ejpam-460	11	4	xinzhong	xinzhong	PROPN
ejpam-460	11	5	)	)	PUNCT
ejpam-460	11	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-460	12	1	532	532	NUM
ejpam-460	12	2	c	c	X
ejpam-460	12	3	©	©	PROPN
ejpam-460	12	4	2009	2009	NUM
ejpam-460	12	5	ejpam	ejpam	NOUN
ejpam-460	12	6	all	all	DET
ejpam-460	12	7	rights	right	NOUN
ejpam-460	12	8	reserved	reserve	VERB
ejpam-460	12	9	.	.	PUNCT
ejpam-460	13	1	x.	x.	NOUN
ejpam-460	13	2	zhichun	zhichun	PROPN
ejpam-460	13	3	and	and	CCONJ
ejpam-460	13	4	h.	h.	PROPN
ejpam-460	13	5	xinzhong	xinzhong	PROPN
ejpam-460	13	6	/	/	SYM
ejpam-460	13	7	eur	eur	PROPN
ejpam-460	13	8	.	.	PUNCT
ejpam-460	14	1	j.	j.	PROPN
ejpam-460	14	2	pure	pure	PROPN
ejpam-460	14	3	appl	appl	PROPN
ejpam-460	14	4	.	.	PROPN
ejpam-460	14	5	math	math	PROPN
ejpam-460	14	6	,	,	PUNCT
ejpam-460	14	7	2	2	NUM
ejpam-460	14	8	(	(	PUNCT
ejpam-460	14	9	2009	2009	NUM
ejpam-460	14	10	)	)	PUNCT
ejpam-460	14	11	,	,	PUNCT
ejpam-460	14	12	(	(	PUNCT
ejpam-460	14	13	532	532	NUM
ejpam-460	14	14	-	-	NUM
ejpam-460	14	15	543	543	NUM
ejpam-460	14	16	)	)	PUNCT
ejpam-460	14	17	533	533	NUM
ejpam-460	14	18	fz̄(z	fz̄(z	NOUN
ejpam-460	14	19	)	)	PUNCT
ejpam-460	14	20	=	=	SYM
ejpam-460	14	21	µ(z	µ(z	PROPN
ejpam-460	14	22	)	)	PUNCT
ejpam-460	14	23	fz(z	fz(z	NUM
ejpam-460	14	24	)	)	PUNCT
ejpam-460	15	1	a.e	a.e	PROPN
ejpam-460	15	2	.	.	PROPN
ejpam-460	15	3	,	,	PUNCT
ejpam-460	15	4	z	z	PROPN
ejpam-460	15	5	∈	∈	PROPN
ejpam-460	15	6	ω	ω	PROPN
ejpam-460	15	7	,	,	PUNCT
ejpam-460	15	8	where	where	SCONJ
ejpam-460	15	9	ess	ess	PROPN
ejpam-460	15	10	sup	sup	PROPN
ejpam-460	15	11	z∈ω	z∈ω	NOUN
ejpam-460	15	12	|µ(z)|=	|µ(z)|=	PROPN
ejpam-460	15	13	k	k	X
ejpam-460	15	14	<	<	X
ejpam-460	15	15	1	1	NUM
ejpam-460	15	16	,	,	PUNCT
ejpam-460	15	17	k	k	X
ejpam-460	15	18	=	=	PUNCT
ejpam-460	16	1	1+k	1+k	NUM
ejpam-460	16	2	1−k	1−k	NUM
ejpam-460	16	3	.	.	PUNCT
ejpam-460	17	1	suppose	suppose	VERB
ejpam-460	17	2	that	that	SCONJ
ejpam-460	17	3	f	f	PROPN
ejpam-460	17	4	(	(	PUNCT
ejpam-460	17	5	z	z	NOUN
ejpam-460	17	6	)	)	PUNCT
ejpam-460	17	7	is	be	AUX
ejpam-460	17	8	a	a	DET
ejpam-460	17	9	quasiconformal	quasiconformal	ADJ
ejpam-460	17	10	mapping	mapping	NOUN
ejpam-460	17	11	in	in	ADP
ejpam-460	17	12	ω	ω	PROPN
ejpam-460	17	13	,	,	PUNCT
ejpam-460	17	14	let	let	VERB
ejpam-460	17	15	d	d	PROPN
ejpam-460	17	16	f	f	X
ejpam-460	17	17	(	(	PUNCT
ejpam-460	17	18	z	z	NOUN
ejpam-460	17	19	)	)	PUNCT
ejpam-460	17	20	=	=	PUNCT
ejpam-460	18	1	|	|	ADV
ejpam-460	18	2	fz|+|	fz|+|	ADP
ejpam-460	18	3	fz̄	fz̄	ADJ
ejpam-460	18	4	|	|	ADV
ejpam-460	18	5	|	|	ADV
ejpam-460	18	6	fz|−|	fz|−|	PROPN
ejpam-460	18	7	fz̄	fz̄	ADJ
ejpam-460	18	8	|	|	ADV
ejpam-460	18	9	.	.	PUNCT
ejpam-460	19	1	the	the	DET
ejpam-460	19	2	grötzsch	grötzsch	PROPN
ejpam-460	19	3	extremal	extremal	ADJ
ejpam-460	19	4	problem	problem	NOUN
ejpam-460	19	5	is	be	AUX
ejpam-460	19	6	to	to	PART
ejpam-460	19	7	find	find	VERB
ejpam-460	19	8	a	a	DET
ejpam-460	19	9	quasiconformal	quasiconformal	ADJ
ejpam-460	19	10	mapping	mapping	NOUN
ejpam-460	19	11	f0(z	f0(z	NUM
ejpam-460	19	12	)	)	PUNCT
ejpam-460	19	13	among	among	ADP
ejpam-460	19	14	all	all	DET
ejpam-460	19	15	quasiconformal	quasiconformal	ADJ
ejpam-460	19	16	mappings	mapping	NOUN
ejpam-460	19	17	f	f	X
ejpam-460	19	18	(	(	PUNCT
ejpam-460	19	19	z	z	NOUN
ejpam-460	19	20	)	)	PUNCT
ejpam-460	19	21	in	in	ADP
ejpam-460	19	22	ω	ω	PROPN
ejpam-460	19	23	,	,	PUNCT
ejpam-460	19	24	such	such	ADJ
ejpam-460	19	25	that	that	SCONJ
ejpam-460	19	26	ess	ess	PROPN
ejpam-460	19	27	sup	sup	NOUN
ejpam-460	19	28	z∈ω	z∈ω	PROPN
ejpam-460	19	29	d	d	X
ejpam-460	19	30	f0	f0	PROPN
ejpam-460	19	31	(	(	PUNCT
ejpam-460	19	32	z	z	NOUN
ejpam-460	19	33	)	)	PUNCT
ejpam-460	19	34	=	=	SYM
ejpam-460	19	35	inf	inf	NOUN
ejpam-460	19	36	f	f	PROPN
ejpam-460	19	37	sup	sup	NOUN
ejpam-460	19	38	z	z	PROPN
ejpam-460	19	39	d	d	X
ejpam-460	19	40	f	f	X
ejpam-460	19	41	(	(	PUNCT
ejpam-460	19	42	z	z	NOUN
ejpam-460	19	43	)	)	PUNCT
ejpam-460	19	44	.	.	PUNCT
ejpam-460	20	1	(	(	PUNCT
ejpam-460	20	2	1	1	X
ejpam-460	20	3	)	)	PUNCT
ejpam-460	20	4	it	it	PRON
ejpam-460	20	5	is	be	AUX
ejpam-460	20	6	known	know	VERB
ejpam-460	20	7	by	by	ADP
ejpam-460	20	8	[	[	X
ejpam-460	20	9	1	1	X
ejpam-460	20	10	]	]	PUNCT
ejpam-460	20	11	that	that	SCONJ
ejpam-460	20	12	the	the	DET
ejpam-460	20	13	solution	solution	NOUN
ejpam-460	20	14	of	of	ADP
ejpam-460	20	15	the	the	DET
ejpam-460	20	16	grötzsch	grötzsch	ADJ
ejpam-460	20	17	extremal	extremal	ADJ
ejpam-460	20	18	problem	problem	NOUN
ejpam-460	20	19	from	from	ADP
ejpam-460	20	20	square	square	ADJ
ejpam-460	20	21	to	to	ADP
ejpam-460	20	22	rectangle	rectangle	NOUN
ejpam-460	20	23	is	be	AUX
ejpam-460	20	24	an	an	DET
ejpam-460	20	25	affine	affine	NOUN
ejpam-460	20	26	mapping	mapping	NOUN
ejpam-460	20	27	,	,	PUNCT
ejpam-460	20	28	and	and	CCONJ
ejpam-460	20	29	the	the	DET
ejpam-460	20	30	grötzsch	grötzsch	ADJ
ejpam-460	20	31	extremal	extremal	ADJ
ejpam-460	20	32	problem	problem	NOUN
ejpam-460	20	33	from	from	ADP
ejpam-460	20	34	annulus	annulus	NOUN
ejpam-460	20	35	{	{	PUNCT
ejpam-460	20	36	z|r	z|r	NOUN
ejpam-460	20	37	≤	≤	NUM
ejpam-460	20	38	|z|	|z|	VERB
ejpam-460	20	39	≤	≤	NOUN
ejpam-460	20	40	1	1	NUM
ejpam-460	20	41	}	}	PUNCT
ejpam-460	20	42	onto	onto	ADP
ejpam-460	20	43	{	{	PUNCT
ejpam-460	20	44	z|r	z|r	NOUN
ejpam-460	20	45	≤	≤	NUM
ejpam-460	20	46	|z|	|z|	VERB
ejpam-460	20	47	≤	≤	NOUN
ejpam-460	20	48	1	1	NUM
ejpam-460	20	49	}	}	PUNCT
ejpam-460	20	50	was	be	AUX
ejpam-460	20	51	investigated	investigate	VERB
ejpam-460	20	52	in	in	ADP
ejpam-460	20	53	[	[	X
ejpam-460	20	54	2	2	NUM
ejpam-460	20	55	]	]	PUNCT
ejpam-460	20	56	and	and	CCONJ
ejpam-460	20	57	[	[	X
ejpam-460	20	58	3	3	NUM
ejpam-460	20	59	]	]	PUNCT
ejpam-460	20	60	.	.	PUNCT
ejpam-460	21	1	the	the	DET
ejpam-460	21	2	following	following	ADJ
ejpam-460	21	3	result	result	NOUN
ejpam-460	21	4	was	be	AUX
ejpam-460	21	5	proved	prove	VERB
ejpam-460	21	6	in	in	ADP
ejpam-460	21	7	[	[	X
ejpam-460	21	8	2	2	NUM
ejpam-460	21	9	]	]	PUNCT
ejpam-460	21	10	.	.	PUNCT
ejpam-460	22	1	theorem	theorem	NOUN
ejpam-460	22	2	1	1	NUM
ejpam-460	22	3	.	.	PUNCT
ejpam-460	23	1	if	if	SCONJ
ejpam-460	23	2	f	f	PROPN
ejpam-460	23	3	(	(	PUNCT
ejpam-460	23	4	z	z	NOUN
ejpam-460	23	5	)	)	PUNCT
ejpam-460	23	6	be	be	AUX
ejpam-460	23	7	a	a	DET
ejpam-460	23	8	quasiconformal	quasiconformal	ADJ
ejpam-460	23	9	homeomorphism	homeomorphism	NOUN
ejpam-460	23	10	in	in	ADP
ejpam-460	23	11	the	the	DET
ejpam-460	23	12	unit	unit	NOUN
ejpam-460	23	13	disk	disk	NOUN
ejpam-460	23	14	onto	onto	ADP
ejpam-460	23	15	itself	itself	PRON
ejpam-460	23	16	such	such	ADJ
ejpam-460	23	17	that	that	SCONJ
ejpam-460	23	18	f	f	PROPN
ejpam-460	23	19	(	(	PUNCT
ejpam-460	23	20	0	0	NUM
ejpam-460	23	21	)	)	PUNCT
ejpam-460	23	22	=	=	SYM
ejpam-460	23	23	0	0	NUM
ejpam-460	23	24	,	,	PUNCT
ejpam-460	23	25	lim	lim	PROPN
ejpam-460	23	26	z→0	z→0	PROPN
ejpam-460	23	27	|	|	PROPN
ejpam-460	23	28	f	f	PROPN
ejpam-460	24	1	(	(	PUNCT
ejpam-460	24	2	z)|	z)|	INTJ
ejpam-460	24	3	|z|	|z|	VERB
ejpam-460	24	4	1	1	NUM
ejpam-460	24	5	k	k	NOUN
ejpam-460	24	6	=	=	SYM
ejpam-460	24	7	1	1	X
ejpam-460	24	8	.	.	PUNCT
ejpam-460	25	1	then	then	ADV
ejpam-460	25	2	the	the	DET
ejpam-460	25	3	solution	solution	NOUN
ejpam-460	25	4	for	for	ADP
ejpam-460	25	5	the	the	DET
ejpam-460	25	6	grötzsch	grötzsch	ADJ
ejpam-460	25	7	annulus	annulus	PROPN
ejpam-460	25	8	extremal	extremal	ADJ
ejpam-460	25	9	problem	problem	NOUN
ejpam-460	25	10	is	be	AUX
ejpam-460	25	11	f	f	PROPN
ejpam-460	25	12	(	(	PUNCT
ejpam-460	25	13	z	z	NOUN
ejpam-460	25	14	)	)	PUNCT
ejpam-460	25	15	=	=	NOUN
ejpam-460	25	16	eiθz|z|	eiθz|z|	VERB
ejpam-460	25	17	1	1	NUM
ejpam-460	25	18	k	k	NOUN
ejpam-460	25	19	−1	−1	NOUN
ejpam-460	25	20	,	,	PUNCT
ejpam-460	25	21	where	where	SCONJ
ejpam-460	25	22	θ	θ	PROPN
ejpam-460	25	23	is	be	AUX
ejpam-460	25	24	a	a	DET
ejpam-460	25	25	real	real	ADJ
ejpam-460	25	26	number	number	NOUN
ejpam-460	25	27	.	.	PUNCT
ejpam-460	26	1	obviously	obviously	ADV
ejpam-460	26	2	,	,	PUNCT
ejpam-460	26	3	the	the	DET
ejpam-460	26	4	normalized	normalize	VERB
ejpam-460	26	5	condition	condition	NOUN
ejpam-460	26	6	f	f	X
ejpam-460	26	7	(	(	PUNCT
ejpam-460	26	8	0	0	NUM
ejpam-460	26	9	)	)	PUNCT
ejpam-460	26	10	=	=	SYM
ejpam-460	26	11	0	0	NUM
ejpam-460	26	12	is	be	AUX
ejpam-460	26	13	unnecessary	unnecessary	ADJ
ejpam-460	26	14	for	for	ADP
ejpam-460	26	15	solving	solve	VERB
ejpam-460	26	16	the	the	DET
ejpam-460	26	17	grötzsch	grötzsch	ADJ
ejpam-460	26	18	annulus	annulus	NOUN
ejpam-460	26	19	extremal	extremal	ADJ
ejpam-460	26	20	problem	problem	NOUN
ejpam-460	26	21	.	.	PUNCT
ejpam-460	27	1	by	by	ADP
ejpam-460	27	2	the	the	DET
ejpam-460	27	3	method	method	NOUN
ejpam-460	27	4	of	of	ADP
ejpam-460	27	5	extremal	extremal	ADJ
ejpam-460	27	6	length	length	NOUN
ejpam-460	27	7	,	,	PUNCT
ejpam-460	27	8	the	the	DET
ejpam-460	27	9	following	following	ADJ
ejpam-460	27	10	result	result	NOUN
ejpam-460	27	11	was	be	AUX
ejpam-460	27	12	proved	prove	VERB
ejpam-460	27	13	in	in	ADP
ejpam-460	27	14	[	[	X
ejpam-460	27	15	3	3	NUM
ejpam-460	27	16	]	]	PUNCT
ejpam-460	27	17	.	.	PUNCT
ejpam-460	28	1	lemma	lemma	PROPN
ejpam-460	28	2	1	1	X
ejpam-460	28	3	.	.	PUNCT
ejpam-460	29	1	if	if	SCONJ
ejpam-460	29	2	f	f	PROPN
ejpam-460	29	3	(	(	PUNCT
ejpam-460	29	4	z	z	NOUN
ejpam-460	29	5	)	)	PUNCT
ejpam-460	29	6	is	be	AUX
ejpam-460	29	7	a	a	DET
ejpam-460	29	8	k−quasiconformal	k−quasiconformal	PROPN
ejpam-460	29	9	homeomorphism	homeomorphism	NOUN
ejpam-460	29	10	from	from	ADP
ejpam-460	29	11	{	{	PUNCT
ejpam-460	29	12	z|r	z|r	NOUN
ejpam-460	29	13	≤	≤	NUM
ejpam-460	29	14	|z|	|z|	VERB
ejpam-460	29	15	≤	≤	NOUN
ejpam-460	29	16	1	1	NUM
ejpam-460	29	17	}	}	PUNCT
ejpam-460	29	18	to	to	ADP
ejpam-460	29	19	{	{	PUNCT
ejpam-460	29	20	z|r	z|r	NOUN
ejpam-460	29	21	1	1	NUM
ejpam-460	29	22	k	k	NOUN
ejpam-460	29	23	≤	≤	NUM
ejpam-460	29	24	|z|	|z|	VERB
ejpam-460	29	25	≤	≤	NUM
ejpam-460	29	26	1	1	NUM
ejpam-460	29	27	}	}	PUNCT
ejpam-460	29	28	,	,	PUNCT
ejpam-460	29	29	then	then	ADV
ejpam-460	29	30	f	f	X
ejpam-460	29	31	(	(	PUNCT
ejpam-460	29	32	z	z	NOUN
ejpam-460	29	33	)	)	PUNCT
ejpam-460	29	34	=	=	PRON
ejpam-460	29	35	λz|z|	λz|z|	VERB
ejpam-460	29	36	1	1	NUM
ejpam-460	29	37	k	k	NOUN
ejpam-460	29	38	−1	−1	NOUN
ejpam-460	29	39	,	,	PUNCT
ejpam-460	29	40	where	where	SCONJ
ejpam-460	29	41	λ	λ	PROPN
ejpam-460	29	42	is	be	AUX
ejpam-460	29	43	a	a	DET
ejpam-460	29	44	constant	constant	ADJ
ejpam-460	29	45	and	and	CCONJ
ejpam-460	29	46	|λ|=	|λ|=	ADJ
ejpam-460	29	47	1	1	NUM
ejpam-460	29	48	.	.	PUNCT
ejpam-460	30	1	we	we	PRON
ejpam-460	30	2	point	point	VERB
ejpam-460	30	3	out	out	ADP
ejpam-460	30	4	that	that	SCONJ
ejpam-460	30	5	lemma	lemma	PROPN
ejpam-460	30	6	1	1	NUM
ejpam-460	30	7	is	be	AUX
ejpam-460	30	8	not	not	PART
ejpam-460	30	9	true	true	ADJ
ejpam-460	30	10	.	.	PUNCT
ejpam-460	31	1	for	for	ADP
ejpam-460	31	2	example	example	NOUN
ejpam-460	31	3	,	,	PUNCT
ejpam-460	31	4	let	let	VERB
ejpam-460	31	5	g(z	g(z	ADJ
ejpam-460	31	6	)	)	PUNCT
ejpam-460	32	1	=	=	SYM
ejpam-460	32	2	r	r	NOUN
ejpam-460	32	3	1	1	NUM
ejpam-460	32	4	k	k	NOUN
ejpam-460	32	5	ei(θ+ksinθ	ei(θ+ksinθ	PROPN
ejpam-460	32	6	)	)	PUNCT
ejpam-460	32	7	,	,	PUNCT
ejpam-460	32	8	where	where	SCONJ
ejpam-460	32	9	z	z	NOUN
ejpam-460	32	10	=	=	SYM
ejpam-460	32	11	reiθ	reiθ	PROPN
ejpam-460	32	12	,	,	PUNCT
ejpam-460	32	13	k	k	PROPN
ejpam-460	32	14	=	=	PUNCT
ejpam-460	32	15	k−1	k−1	PROPN
ejpam-460	32	16	k+1	k+1	X
ejpam-460	32	17	,	,	PUNCT
ejpam-460	32	18	k	k	X
ejpam-460	32	19	≥	≥	NUM
ejpam-460	32	20	1	1	NUM
ejpam-460	32	21	,	,	PUNCT
ejpam-460	32	22	then	then	ADV
ejpam-460	32	23	g(z	g(z	PROPN
ejpam-460	32	24	)	)	PUNCT
ejpam-460	32	25	is	be	AUX
ejpam-460	32	26	a	a	DET
ejpam-460	32	27	2k2	2k2	NUM
ejpam-460	32	28	k+1	k+1	PRON
ejpam-460	32	29	−quasiconformal	−quasiconformal	ADJ
ejpam-460	32	30	homeomorphism	homeomorphism	PROPN
ejpam-460	32	31	from	from	ADP
ejpam-460	32	32	{	{	PUNCT
ejpam-460	32	33	z|r	z|r	NOUN
ejpam-460	32	34	≤	≤	NUM
ejpam-460	32	35	|z|	|z|	VERB
ejpam-460	32	36	≤	≤	NOUN
ejpam-460	32	37	1	1	NUM
ejpam-460	32	38	}	}	PUNCT
ejpam-460	32	39	onto	onto	ADP
ejpam-460	32	40	{	{	PUNCT
ejpam-460	32	41	z|r	z|r	NOUN
ejpam-460	32	42	1	1	NUM
ejpam-460	32	43	k	k	NOUN
ejpam-460	32	44	≤	≤	NUM
ejpam-460	32	45	|z|	|z|	NOUN
ejpam-460	32	46	≤	≤	NUM
ejpam-460	32	47	1	1	NUM
ejpam-460	32	48	}	}	PUNCT
ejpam-460	32	49	.	.	PUNCT
ejpam-460	33	1	therefore	therefore	ADV
ejpam-460	33	2	,	,	PUNCT
ejpam-460	33	3	lemma	lemma	PROPN
ejpam-460	33	4	1	1	NUM
ejpam-460	33	5	is	be	AUX
ejpam-460	33	6	meaningful	meaningful	ADJ
ejpam-460	33	7	only	only	ADV
ejpam-460	33	8	under	under	ADP
ejpam-460	33	9	x.	x.	PROPN
ejpam-460	33	10	zhichun	zhichun	PROPN
ejpam-460	33	11	and	and	CCONJ
ejpam-460	33	12	h.	h.	PROPN
ejpam-460	33	13	xinzhong	xinzhong	PROPN
ejpam-460	33	14	/	/	SYM
ejpam-460	33	15	eur	eur	PROPN
ejpam-460	33	16	.	.	PUNCT
ejpam-460	34	1	j.	j.	PROPN
ejpam-460	34	2	pure	pure	PROPN
ejpam-460	34	3	appl	appl	PROPN
ejpam-460	34	4	.	.	PROPN
ejpam-460	34	5	math	math	PROPN
ejpam-460	34	6	,	,	PUNCT
ejpam-460	34	7	2	2	NUM
ejpam-460	34	8	(	(	PUNCT
ejpam-460	34	9	2009	2009	NUM
ejpam-460	34	10	)	)	PUNCT
ejpam-460	34	11	,	,	PUNCT
ejpam-460	34	12	(	(	PUNCT
ejpam-460	34	13	532	532	NUM
ejpam-460	34	14	-	-	NUM
ejpam-460	34	15	543	543	NUM
ejpam-460	34	16	)	)	PUNCT
ejpam-460	34	17	534	534	NUM
ejpam-460	34	18	considering	consider	VERB
ejpam-460	34	19	the	the	DET
ejpam-460	34	20	grötzsch	grötzsch	PROPN
ejpam-460	34	21	extremal	extremal	ADJ
ejpam-460	34	22	problem	problem	NOUN
ejpam-460	34	23	on	on	ADP
ejpam-460	34	24	the	the	DET
ejpam-460	34	25	annulus	annulus	NOUN
ejpam-460	34	26	.	.	PUNCT
ejpam-460	35	1	on	on	ADP
ejpam-460	35	2	the	the	DET
ejpam-460	35	3	other	other	ADJ
ejpam-460	35	4	hand	hand	NOUN
ejpam-460	35	5	,	,	PUNCT
ejpam-460	35	6	[	[	X
ejpam-460	35	7	3	3	NUM
ejpam-460	35	8	]	]	PUNCT
ejpam-460	35	9	tried	try	VERB
ejpam-460	35	10	to	to	PART
ejpam-460	35	11	prove	prove	VERB
ejpam-460	35	12	the	the	DET
ejpam-460	35	13	solution	solution	NOUN
ejpam-460	35	14	of	of	ADP
ejpam-460	35	15	the	the	DET
ejpam-460	35	16	grö	grö	NOUN
ejpam-460	35	17	tzsch	tzsch	ADJ
ejpam-460	35	18	extremal	extremal	ADJ
ejpam-460	35	19	problem	problem	NOUN
ejpam-460	35	20	on	on	ADP
ejpam-460	35	21	the	the	DET
ejpam-460	35	22	annulus	annulus	NOUN
ejpam-460	35	23	by	by	ADP
ejpam-460	35	24	using	use	VERB
ejpam-460	35	25	the	the	DET
ejpam-460	35	26	method	method	NOUN
ejpam-460	35	27	of	of	ADP
ejpam-460	35	28	extremal	extremal	ADJ
ejpam-460	35	29	length	length	NOUN
ejpam-460	35	30	,	,	PUNCT
ejpam-460	35	31	the	the	DET
ejpam-460	35	32	property	property	NOUN
ejpam-460	35	33	of	of	ADP
ejpam-460	35	34	the	the	DET
ejpam-460	35	35	module	module	NOUN
ejpam-460	35	36	of	of	ADP
ejpam-460	35	37	ring	ring	NOUN
ejpam-460	35	38	domain	domain	NOUN
ejpam-460	35	39	and	and	CCONJ
ejpam-460	35	40	used	use	VERB
ejpam-460	35	41	the	the	DET
ejpam-460	35	42	following	following	ADJ
ejpam-460	35	43	equality	equality	NOUN
ejpam-460	35	44	1	1	NUM
ejpam-460	35	45	kmod(rr	kmod(rr	NOUN
ejpam-460	35	46	)	)	PUNCT
ejpam-460	35	47	=	=	SYM
ejpam-460	35	48	1	1	NUM
ejpam-460	35	49	mod(r	mod(r	NOUN
ejpam-460	35	50	r	r	NOUN
ejpam-460	35	51	1	1	NUM
ejpam-460	35	52	k	k	NOUN
ejpam-460	35	53	)	)	PUNCT
ejpam-460	35	54	.	.	PUNCT
ejpam-460	36	1	however	however	ADV
ejpam-460	36	2	,	,	PUNCT
ejpam-460	36	3	above	above	ADP
ejpam-460	36	4	equality	equality	NOUN
ejpam-460	36	5	is	be	AUX
ejpam-460	36	6	also	also	ADV
ejpam-460	36	7	not	not	PART
ejpam-460	36	8	true	true	ADJ
ejpam-460	36	9	,	,	PUNCT
ejpam-460	36	10	since	since	SCONJ
ejpam-460	36	11	we	we	PRON
ejpam-460	36	12	find	find	VERB
ejpam-460	36	13	that	that	SCONJ
ejpam-460	36	14	1	1	NUM
ejpam-460	36	15	kmod(rr	kmod(rr	NOUN
ejpam-460	36	16	)	)	PUNCT
ejpam-460	36	17	=	=	SYM
ejpam-460	36	18	1	1	NUM
ejpam-460	36	19	k	k	SYM
ejpam-460	36	20	1	1	NUM
ejpam-460	36	21	2π	2π	NOUN
ejpam-460	36	22	log	log	VERB
ejpam-460	36	23	1	1	NUM
ejpam-460	36	24	r	r	NOUN
ejpam-460	36	25	and	and	CCONJ
ejpam-460	36	26	1	1	NUM
ejpam-460	37	1	mod(r	mod(r	NOUN
ejpam-460	37	2	r	r	NOUN
ejpam-460	37	3	1	1	NUM
ejpam-460	37	4	k	k	NOUN
ejpam-460	37	5	)	)	PUNCT
ejpam-460	37	6	=	=	SYM
ejpam-460	38	1	1	1	NUM
ejpam-460	38	2	1	1	NUM
ejpam-460	38	3	k	k	SYM
ejpam-460	38	4	1	1	NUM
ejpam-460	38	5	2π	2π	NOUN
ejpam-460	38	6	log	log	VERB
ejpam-460	38	7	1	1	NUM
ejpam-460	38	8	r	r	NOUN
ejpam-460	38	9	.	.	PUNCT
ejpam-460	39	1	in	in	ADP
ejpam-460	39	2	this	this	DET
ejpam-460	39	3	paper	paper	NOUN
ejpam-460	39	4	,	,	PUNCT
ejpam-460	39	5	the	the	DET
ejpam-460	39	6	solution	solution	NOUN
ejpam-460	39	7	for	for	ADP
ejpam-460	39	8	the	the	DET
ejpam-460	39	9	grötzsch	grötzsch	PROPN
ejpam-460	39	10	extremal	extremal	ADJ
ejpam-460	39	11	problem	problem	NOUN
ejpam-460	39	12	on	on	ADP
ejpam-460	39	13	the	the	DET
ejpam-460	39	14	annulus	annulus	NOUN
ejpam-460	39	15	is	be	AUX
ejpam-460	39	16	solved	solve	VERB
ejpam-460	39	17	by	by	ADP
ejpam-460	39	18	using	use	VERB
ejpam-460	39	19	analytic	analytic	ADJ
ejpam-460	39	20	method	method	NOUN
ejpam-460	39	21	and	and	CCONJ
ejpam-460	39	22	the	the	DET
ejpam-460	39	23	extremal	extremal	ADJ
ejpam-460	39	24	condition	condition	NOUN
ejpam-460	39	25	.	.	PUNCT
ejpam-460	40	1	on	on	ADP
ejpam-460	40	2	the	the	DET
ejpam-460	40	3	other	other	ADJ
ejpam-460	40	4	hand	hand	NOUN
ejpam-460	40	5	,	,	PUNCT
ejpam-460	40	6	[	[	X
ejpam-460	40	7	3	3	X
ejpam-460	40	8	]	]	PUNCT
ejpam-460	40	9	also	also	ADV
ejpam-460	40	10	tried	try	VERB
ejpam-460	40	11	to	to	PART
ejpam-460	40	12	find	find	VERB
ejpam-460	40	13	the	the	DET
ejpam-460	40	14	solution	solution	NOUN
ejpam-460	40	15	of	of	ADP
ejpam-460	40	16	the	the	DET
ejpam-460	40	17	grötzsch	grötzsch	ADJ
ejpam-460	40	18	extremal	extremal	ADJ
ejpam-460	40	19	problem	problem	NOUN
ejpam-460	40	20	for	for	ADP
ejpam-460	40	21	area	area	NOUN
ejpam-460	40	22	distortion	distortion	NOUN
ejpam-460	40	23	problem	problem	NOUN
ejpam-460	40	24	.	.	PUNCT
ejpam-460	41	1	we	we	PRON
ejpam-460	41	2	also	also	ADV
ejpam-460	41	3	point	point	VERB
ejpam-460	41	4	out	out	ADP
ejpam-460	41	5	that	that	SCONJ
ejpam-460	41	6	the	the	DET
ejpam-460	41	7	proved	proved	ADJ
ejpam-460	41	8	result	result	NOUN
ejpam-460	41	9	is	be	AUX
ejpam-460	41	10	not	not	PART
ejpam-460	41	11	correct	correct	ADJ
ejpam-460	41	12	.	.	PUNCT
ejpam-460	42	1	at	at	ADP
ejpam-460	42	2	last	last	ADV
ejpam-460	42	3	,	,	PUNCT
ejpam-460	42	4	we	we	PRON
ejpam-460	42	5	obtain	obtain	VERB
ejpam-460	42	6	one	one	NUM
ejpam-460	42	7	criterion	criterion	NOUN
ejpam-460	42	8	for	for	ADP
ejpam-460	42	9	the	the	DET
ejpam-460	42	10	solution	solution	NOUN
ejpam-460	42	11	of	of	ADP
ejpam-460	42	12	the	the	DET
ejpam-460	42	13	grötzsch	grötzsch	ADJ
ejpam-460	42	14	annulus	annulus	NOUN
ejpam-460	42	15	extremal	extremal	ADJ
ejpam-460	42	16	problem	problem	NOUN
ejpam-460	42	17	considering	consider	VERB
ejpam-460	42	18	hyperbolic	hyperbolic	ADJ
ejpam-460	42	19	area	area	NOUN
ejpam-460	42	20	distortion	distortion	NOUN
ejpam-460	42	21	.	.	PUNCT
ejpam-460	43	1	2	2	X
ejpam-460	43	2	.	.	X
ejpam-460	43	3	main	main	ADJ
ejpam-460	43	4	results	result	NOUN
ejpam-460	43	5	and	and	CCONJ
ejpam-460	43	6	their	their	PRON
ejpam-460	43	7	proofs	proof	NOUN
ejpam-460	43	8	the	the	DET
ejpam-460	43	9	notations	notation	NOUN
ejpam-460	43	10	in	in	ADP
ejpam-460	43	11	this	this	DET
ejpam-460	43	12	paper	paper	NOUN
ejpam-460	43	13	are	be	AUX
ejpam-460	43	14	adopted	adopt	VERB
ejpam-460	43	15	as	as	ADP
ejpam-460	43	16	in	in	ADP
ejpam-460	43	17	[	[	X
ejpam-460	43	18	3	3	NUM
ejpam-460	43	19	]	]	PUNCT
ejpam-460	43	20	.	.	PUNCT
ejpam-460	44	1	let	let	VERB
ejpam-460	44	2	a(r1	a(r1	NOUN
ejpam-460	44	3	,	,	PUNCT
ejpam-460	44	4	r2	r2	PROPN
ejpam-460	44	5	)	)	PUNCT
ejpam-460	45	1	=	=	PRON
ejpam-460	45	2	{	{	PUNCT
ejpam-460	45	3	z|r1	z|r1	NOUN
ejpam-460	45	4	≤	≤	NUM
ejpam-460	45	5	|z|	|z|	VERB
ejpam-460	45	6	≤	≤	NUM
ejpam-460	45	7	r2	r2	NOUN
ejpam-460	45	8	}	}	PUNCT
ejpam-460	45	9	,	,	PUNCT
ejpam-460	45	10	a(r1	a(r1	ADV
ejpam-460	45	11	,	,	PUNCT
ejpam-460	45	12	r2;θ1,θ2	r2;θ1,θ2	PROPN
ejpam-460	45	13	)	)	PUNCT
ejpam-460	45	14	=	=	PRON
ejpam-460	46	1	{	{	PUNCT
ejpam-460	46	2	z|r1	z|r1	NOUN
ejpam-460	46	3	≤	≤	NUM
ejpam-460	46	4	|z|	|z|	VERB
ejpam-460	46	5	≤	≤	ADJ
ejpam-460	46	6	r2,θ1	r2,θ1	PROPN
ejpam-460	46	7	≤	≤	NUM
ejpam-460	46	8	argz	argz	PROPN
ejpam-460	46	9	≤	≤	PUNCT
ejpam-460	46	10	θ2	θ2	PROPN
ejpam-460	46	11	}	}	PUNCT
ejpam-460	46	12	.	.	PUNCT
ejpam-460	47	1	we	we	PRON
ejpam-460	47	2	will	will	AUX
ejpam-460	47	3	prove	prove	VERB
ejpam-460	47	4	the	the	DET
ejpam-460	47	5	following	follow	VERB
ejpam-460	47	6	result	result	NOUN
ejpam-460	47	7	.	.	PUNCT
ejpam-460	48	1	theorem	theorem	NOUN
ejpam-460	48	2	2	2	NUM
ejpam-460	48	3	.	.	PUNCT
ejpam-460	49	1	let	let	VERB
ejpam-460	49	2	q	q	PART
ejpam-460	49	3	be	be	AUX
ejpam-460	49	4	k	k	ADJ
ejpam-460	49	5	-	-	ADJ
ejpam-460	49	6	quasiconformal	quasiconformal	ADJ
ejpam-460	49	7	mappings	mapping	NOUN
ejpam-460	49	8	from	from	ADP
ejpam-460	49	9	{	{	PUNCT
ejpam-460	49	10	z|r1	z|r1	PROPN
ejpam-460	49	11	≤	≤	NUM
ejpam-460	49	12	|z|	|z|	NOUN
ejpam-460	49	13	≤	≤	NOUN
ejpam-460	49	14	1	1	NUM
ejpam-460	49	15	}	}	PUNCT
ejpam-460	49	16	to	to	ADP
ejpam-460	49	17	{	{	PUNCT
ejpam-460	49	18	z|r	z|r	NOUN
ejpam-460	49	19	1	1	NUM
ejpam-460	49	20	k	k	SYM
ejpam-460	49	21	1	1	NUM
ejpam-460	49	22	≤	≤	NOUN
ejpam-460	49	23	|z|	|z|	VERB
ejpam-460	49	24	≤	≤	NUM
ejpam-460	49	25	1	1	NUM
ejpam-460	49	26	}	}	PUNCT
ejpam-460	49	27	,	,	PUNCT
ejpam-460	49	28	and	and	CCONJ
ejpam-460	49	29	if	if	SCONJ
ejpam-460	49	30	f	f	PROPN
ejpam-460	49	31	∈q	∈q	NOUN
ejpam-460	49	32	satisfies	satisfy	VERB
ejpam-460	49	33	ess	ess	NOUN
ejpam-460	49	34	sup	sup	NOUN
ejpam-460	49	35	z	z	PROPN
ejpam-460	49	36	d	d	X
ejpam-460	49	37	f	f	X
ejpam-460	49	38	(	(	PUNCT
ejpam-460	49	39	z	z	NOUN
ejpam-460	49	40	)	)	PUNCT
ejpam-460	49	41	=	=	SYM
ejpam-460	49	42	inf	inf	NOUN
ejpam-460	49	43	g∈q	g∈q	NOUN
ejpam-460	49	44	sup	sup	NOUN
ejpam-460	49	45	z	z	NOUN
ejpam-460	49	46	dg(z	dg(z	NUM
ejpam-460	49	47	)	)	PUNCT
ejpam-460	50	1	,	,	PUNCT
ejpam-460	50	2	then	then	ADV
ejpam-460	50	3	f	f	X
ejpam-460	50	4	(	(	PUNCT
ejpam-460	50	5	z	z	NOUN
ejpam-460	50	6	)	)	PUNCT
ejpam-460	50	7	=	=	SYM
ejpam-460	51	1	r	r	NOUN
ejpam-460	51	2	1	1	NUM
ejpam-460	51	3	k	k	NOUN
ejpam-460	51	4	ei(θ+α	ei(θ+α	PROPN
ejpam-460	51	5	)	)	PUNCT
ejpam-460	51	6	,	,	PUNCT
ejpam-460	51	7	z	z	X
ejpam-460	52	1	=	=	PUNCT
ejpam-460	52	2	reiθ	reiθ	PROPN
ejpam-460	52	3	,	,	PUNCT
ejpam-460	52	4	where	where	SCONJ
ejpam-460	52	5	α	α	NOUN
ejpam-460	52	6	is	be	AUX
ejpam-460	52	7	a	a	DET
ejpam-460	52	8	constant	constant	ADJ
ejpam-460	52	9	.	.	PUNCT
ejpam-460	53	1	x.	x.	NOUN
ejpam-460	53	2	zhichun	zhichun	PROPN
ejpam-460	53	3	and	and	CCONJ
ejpam-460	53	4	h.	h.	PROPN
ejpam-460	53	5	xinzhong	xinzhong	PROPN
ejpam-460	53	6	/	/	SYM
ejpam-460	53	7	eur	eur	PROPN
ejpam-460	53	8	.	.	PUNCT
ejpam-460	54	1	j.	j.	PROPN
ejpam-460	54	2	pure	pure	PROPN
ejpam-460	54	3	appl	appl	PROPN
ejpam-460	54	4	.	.	PROPN
ejpam-460	54	5	math	math	PROPN
ejpam-460	54	6	,	,	PUNCT
ejpam-460	54	7	2	2	NUM
ejpam-460	54	8	(	(	PUNCT
ejpam-460	54	9	2009	2009	NUM
ejpam-460	54	10	)	)	PUNCT
ejpam-460	54	11	,	,	PUNCT
ejpam-460	54	12	(	(	PUNCT
ejpam-460	54	13	532	532	NUM
ejpam-460	54	14	-	-	NUM
ejpam-460	54	15	543	543	NUM
ejpam-460	54	16	)	)	PUNCT
ejpam-460	54	17	535	535	NUM
ejpam-460	54	18	proof	proof	NOUN
ejpam-460	54	19	.	.	PUNCT
ejpam-460	55	1	for	for	ADP
ejpam-460	55	2	any	any	DET
ejpam-460	55	3	r1	r1	NOUN
ejpam-460	55	4	,	,	PUNCT
ejpam-460	55	5	r1	r1	PROPN
ejpam-460	55	6	<	<	X
ejpam-460	55	7	r1	r1	PROPN
ejpam-460	55	8	<	<	X
ejpam-460	55	9	1	1	NUM
ejpam-460	55	10	,	,	PUNCT
ejpam-460	55	11	let	let	VERB
ejpam-460	55	12	a=	a=	VERB
ejpam-460	55	13	{	{	PUNCT
ejpam-460	55	14	z|r1	z|r1	ADJ
ejpam-460	55	15	≤	≤	NUM
ejpam-460	55	16	|z|	|z|	VERB
ejpam-460	55	17	≤	≤	NUM
ejpam-460	55	18	r1	r1	NOUN
ejpam-460	55	19	}	}	PUNCT
ejpam-460	55	20	,	,	PUNCT
ejpam-460	55	21	b	b	X
ejpam-460	55	22	=	=	PRON
ejpam-460	55	23	{	{	PUNCT
ejpam-460	55	24	z|r1	z|r1	PROPN
ejpam-460	55	25	≤	≤	NUM
ejpam-460	55	26	|z|	|z|	NOUN
ejpam-460	55	27	≤	≤	NUM
ejpam-460	55	28	1	1	NUM
ejpam-460	55	29	}	}	PUNCT
ejpam-460	55	30	,	,	PUNCT
ejpam-460	55	31	by	by	ADP
ejpam-460	55	32	the	the	DET
ejpam-460	55	33	property	property	NOUN
ejpam-460	55	34	of	of	ADP
ejpam-460	55	35	the	the	DET
ejpam-460	55	36	module	module	NOUN
ejpam-460	55	37	of	of	ADP
ejpam-460	55	38	ring	ring	NOUN
ejpam-460	55	39	domain	domain	NOUN
ejpam-460	55	40	[	[	X
ejpam-460	55	41	4	4	NUM
ejpam-460	55	42	]	]	PUNCT
ejpam-460	55	43	,	,	PUNCT
ejpam-460	55	44	we	we	PRON
ejpam-460	55	45	have	have	VERB
ejpam-460	55	46	1	1	NUM
ejpam-460	55	47	k	k	SYM
ejpam-460	55	48	1	1	NUM
ejpam-460	55	49	2π	2π	NOUN
ejpam-460	55	50	log	log	VERB
ejpam-460	55	51	1	1	NUM
ejpam-460	55	52	r1	r1	NOUN
ejpam-460	55	53	=	=	SYM
ejpam-460	55	54	1	1	NUM
ejpam-460	55	55	k	k	X
ejpam-460	55	56	(	(	PUNCT
ejpam-460	55	57	mod(a	mod(a	PROPN
ejpam-460	55	58	)	)	PUNCT
ejpam-460	55	59	+	+	NOUN
ejpam-460	55	60	mod(b	mod(b	X
ejpam-460	55	61	)	)	PUNCT
ejpam-460	55	62	)	)	PUNCT
ejpam-460	55	63	≤	≤	NUM
ejpam-460	56	1	mod	mod	X
ejpam-460	56	2	(	(	PUNCT
ejpam-460	56	3	f	f	PROPN
ejpam-460	56	4	(	(	PUNCT
ejpam-460	56	5	a	a	NOUN
ejpam-460	56	6	)	)	PUNCT
ejpam-460	56	7	)	)	PUNCT
ejpam-460	57	1	+	+	NOUN
ejpam-460	57	2	mod	mod	X
ejpam-460	57	3	(	(	PUNCT
ejpam-460	57	4	f	f	PROPN
ejpam-460	57	5	(	(	PUNCT
ejpam-460	57	6	b	b	NOUN
ejpam-460	57	7	)	)	PUNCT
ejpam-460	57	8	)	)	PUNCT
ejpam-460	57	9	≤mod(a(r	≤mod(a(r	PROPN
ejpam-460	57	10	1	1	NUM
ejpam-460	57	11	k	k	PROPN
ejpam-460	57	12	1	1	NUM
ejpam-460	57	13	,	,	PUNCT
ejpam-460	57	14	1	1	NUM
ejpam-460	57	15	)	)	PUNCT
ejpam-460	57	16	)	)	PUNCT
ejpam-460	58	1	=	=	SYM
ejpam-460	58	2	1	1	NUM
ejpam-460	58	3	k	k	SYM
ejpam-460	58	4	1	1	NUM
ejpam-460	58	5	2π	2π	NOUN
ejpam-460	58	6	log	log	VERB
ejpam-460	58	7	1	1	NUM
ejpam-460	58	8	r1	r1	NOUN
ejpam-460	58	9	,	,	PUNCT
ejpam-460	58	10	thus	thus	ADV
ejpam-460	58	11	,	,	PUNCT
ejpam-460	58	12	mod	mod	PROPN
ejpam-460	58	13	(	(	PUNCT
ejpam-460	58	14	f	f	PROPN
ejpam-460	58	15	(	(	PUNCT
ejpam-460	58	16	a	a	NOUN
ejpam-460	58	17	)	)	PUNCT
ejpam-460	58	18	)	)	PUNCT
ejpam-460	59	1	+	+	NOUN
ejpam-460	59	2	mod	mod	X
ejpam-460	59	3	(	(	PUNCT
ejpam-460	59	4	f	f	PROPN
ejpam-460	59	5	(	(	PUNCT
ejpam-460	59	6	b	b	NOUN
ejpam-460	59	7	)	)	PUNCT
ejpam-460	59	8	)	)	PUNCT
ejpam-460	60	1	=	=	SYM
ejpam-460	60	2	mod(a(r	mod(a(r	NOUN
ejpam-460	60	3	1	1	NUM
ejpam-460	60	4	k	k	NOUN
ejpam-460	60	5	1	1	NUM
ejpam-460	60	6	,	,	PUNCT
ejpam-460	60	7	1	1	NUM
ejpam-460	60	8	)	)	PUNCT
ejpam-460	60	9	)	)	PUNCT
ejpam-460	60	10	=	=	SYM
ejpam-460	60	11	1	1	NUM
ejpam-460	60	12	k	k	SYM
ejpam-460	60	13	1	1	NUM
ejpam-460	60	14	2π	2π	NOUN
ejpam-460	60	15	log	log	VERB
ejpam-460	60	16	1	1	NUM
ejpam-460	60	17	r1	r1	NOUN
ejpam-460	60	18	.	.	PUNCT
ejpam-460	61	1	by	by	ADP
ejpam-460	61	2	lemma	lemma	PROPN
ejpam-460	61	3	1.3	1.3	NUM
ejpam-460	61	4	in	in	ADP
ejpam-460	61	5	[	[	X
ejpam-460	61	6	3	3	NUM
ejpam-460	61	7	]	]	PUNCT
ejpam-460	61	8	,	,	PUNCT
ejpam-460	61	9	we	we	PRON
ejpam-460	61	10	see	see	VERB
ejpam-460	61	11	that	that	SCONJ
ejpam-460	61	12	f	f	PROPN
ejpam-460	61	13	(	(	PUNCT
ejpam-460	61	14	a	a	NOUN
ejpam-460	61	15	)	)	PUNCT
ejpam-460	61	16	and	and	CCONJ
ejpam-460	61	17	f	f	PROPN
ejpam-460	61	18	(	(	PUNCT
ejpam-460	61	19	b	b	NOUN
ejpam-460	61	20	)	)	PUNCT
ejpam-460	61	21	are	be	AUX
ejpam-460	61	22	concentric	concentric	ADJ
ejpam-460	61	23	annulus	annulus	NOUN
ejpam-460	61	24	,	,	PUNCT
ejpam-460	61	25	we	we	PRON
ejpam-460	61	26	have	have	VERB
ejpam-460	61	27	|	|	ADV
ejpam-460	61	28	f	f	X
ejpam-460	61	29	(	(	PUNCT
ejpam-460	61	30	r1eiθ	r1eiθ	PROPN
ejpam-460	61	31	)	)	PUNCT
ejpam-460	61	32	|=	|=	NOUN
ejpam-460	61	33	r′	r′	NUM
ejpam-460	61	34	,	,	PUNCT
ejpam-460	61	35	0≤	0≤	NUM
ejpam-460	61	36	θ	θ	NOUN
ejpam-460	61	37	<	<	X
ejpam-460	62	1	2π	2π	NOUN
ejpam-460	62	2	.	.	PUNCT
ejpam-460	63	1	by	by	ADP
ejpam-460	63	2	the	the	DET
ejpam-460	63	3	quasi	quasi	ADJ
ejpam-460	63	4	-	-	ADJ
ejpam-460	63	5	invariant	invariant	ADJ
ejpam-460	63	6	property	property	NOUN
ejpam-460	63	7	of	of	ADP
ejpam-460	63	8	module	module	NOUN
ejpam-460	63	9	,	,	PUNCT
ejpam-460	63	10	we	we	PRON
ejpam-460	63	11	have	have	VERB
ejpam-460	63	12			PROPN
ejpam-460	63	13			PRON
ejpam-460	63	14			ADJ
ejpam-460	63	15	mod	mod	PROPN
ejpam-460	63	16	(	(	PUNCT
ejpam-460	63	17	f	f	PROPN
ejpam-460	63	18	(	(	PUNCT
ejpam-460	63	19	a	a	NOUN
ejpam-460	63	20	)	)	PUNCT
ejpam-460	63	21	)	)	PUNCT
ejpam-460	63	22	≥	≥	NOUN
ejpam-460	64	1	1	1	NUM
ejpam-460	64	2	k	k	PROPN
ejpam-460	64	3	mod(a	mod(a	PROPN
ejpam-460	64	4	)	)	PUNCT
ejpam-460	64	5	,	,	PUNCT
ejpam-460	64	6	mod	mod	PROPN
ejpam-460	64	7	(	(	PUNCT
ejpam-460	64	8	f	f	PROPN
ejpam-460	64	9	(	(	PUNCT
ejpam-460	64	10	b	b	NOUN
ejpam-460	64	11	)	)	PUNCT
ejpam-460	64	12	)	)	PUNCT
ejpam-460	64	13	≥	≥	NOUN
ejpam-460	64	14	1	1	NUM
ejpam-460	64	15	k	k	X
ejpam-460	64	16	mod(b	mod(b	PROPN
ejpam-460	64	17	)	)	PUNCT
ejpam-460	64	18	,	,	PUNCT
ejpam-460	64	19	that	that	PRON
ejpam-460	64	20	is	be	AUX
ejpam-460	64	21			PROPN
ejpam-460	64	22			ADJ
ejpam-460	64	23			PROPN
ejpam-460	64	24	r′	r′	PROPN
ejpam-460	64	25	≥	≥	PROPN
ejpam-460	64	26	r	r	NOUN
ejpam-460	64	27	1	1	NUM
ejpam-460	64	28	k	k	NOUN
ejpam-460	64	29	1	1	NUM
ejpam-460	64	30	,	,	PUNCT
ejpam-460	64	31	r′	r′	PROPN
ejpam-460	64	32	≤	≤	NUM
ejpam-460	64	33	r	r	NOUN
ejpam-460	64	34	1	1	NUM
ejpam-460	64	35	k	k	NOUN
ejpam-460	64	36	1	1	NUM
ejpam-460	64	37	.	.	PUNCT
ejpam-460	65	1	thus	thus	ADV
ejpam-460	65	2	,	,	PUNCT
ejpam-460	65	3	r′	r′	PROPN
ejpam-460	65	4	=	=	SYM
ejpam-460	65	5	r	r	NOUN
ejpam-460	65	6	1	1	NUM
ejpam-460	65	7	k	k	NOUN
ejpam-460	65	8	1	1	NUM
ejpam-460	65	9	.	.	PUNCT
ejpam-460	66	1	therefore	therefore	ADV
ejpam-460	66	2	,	,	PUNCT
ejpam-460	66	3	we	we	PRON
ejpam-460	66	4	obtain	obtain	VERB
ejpam-460	66	5	that	that	SCONJ
ejpam-460	66	6	f	f	PROPN
ejpam-460	66	7	(	(	PUNCT
ejpam-460	66	8	z	z	NOUN
ejpam-460	66	9	)	)	PUNCT
ejpam-460	66	10	=	=	SYM
ejpam-460	66	11	r	r	NOUN
ejpam-460	66	12	1	1	NUM
ejpam-460	66	13	k	k	PROPN
ejpam-460	66	14	eiϕ(r	eiϕ(r	PROPN
ejpam-460	66	15	,	,	PUNCT
ejpam-460	66	16	θ	θ	PROPN
ejpam-460	66	17	)	)	PUNCT
ejpam-460	66	18	,	,	PUNCT
ejpam-460	66	19	z	z	X
ejpam-460	66	20	=	=	PUNCT
ejpam-460	66	21	reiθ	reiθ	PROPN
ejpam-460	66	22	.	.	PUNCT
ejpam-460	67	1	next	next	ADV
ejpam-460	67	2	,	,	PUNCT
ejpam-460	67	3	we	we	PRON
ejpam-460	67	4	will	will	AUX
ejpam-460	67	5	prove	prove	VERB
ejpam-460	67	6	that	that	SCONJ
ejpam-460	67	7	ϕ(r	ϕ(r	PROPN
ejpam-460	67	8	,	,	PUNCT
ejpam-460	67	9	θ	θ	PROPN
ejpam-460	67	10	)	)	PUNCT
ejpam-460	67	11	depends	depend	VERB
ejpam-460	67	12	only	only	ADV
ejpam-460	67	13	on	on	ADP
ejpam-460	67	14	θ	θ	PROPN
ejpam-460	67	15	.	.	PUNCT
ejpam-460	68	1	by	by	ADP
ejpam-460	68	2	calculation	calculation	NOUN
ejpam-460	68	3	,	,	PUNCT
ejpam-460	68	4	we	we	PRON
ejpam-460	68	5	have	have	AUX
ejpam-460	68	6	fz̄	fz̄	VERB
ejpam-460	68	7	=	=	NOUN
ejpam-460	68	8	1	1	NUM
ejpam-460	68	9	2	2	NUM
ejpam-460	68	10	r	r	NOUN
ejpam-460	68	11	1	1	NUM
ejpam-460	68	12	k	k	NOUN
ejpam-460	68	13	−1ei(ϕ+θ	−1ei(ϕ+θ	PROPN
ejpam-460	68	14	)	)	PUNCT
ejpam-460	68	15	(	(	PUNCT
ejpam-460	68	16	1	1	NUM
ejpam-460	68	17	k	k	NOUN
ejpam-460	68	18	+	+	NOUN
ejpam-460	68	19	irϕr	irϕr	ADJ
ejpam-460	68	20	−ϕθ	−ϕθ	PROPN
ejpam-460	68	21	)	)	PUNCT
ejpam-460	68	22	,	,	PUNCT
ejpam-460	68	23	fz	fz	VERB
ejpam-460	69	1	=	=	NOUN
ejpam-460	69	2	1	1	NUM
ejpam-460	69	3	2	2	NUM
ejpam-460	69	4	r	r	NOUN
ejpam-460	69	5	1	1	NUM
ejpam-460	69	6	k	k	NOUN
ejpam-460	69	7	−1ei(ϕ−θ	−1ei(ϕ−θ	NOUN
ejpam-460	69	8	)	)	PUNCT
ejpam-460	69	9	(	(	PUNCT
ejpam-460	69	10	1	1	NUM
ejpam-460	69	11	k	k	NOUN
ejpam-460	69	12	+	+	NOUN
ejpam-460	69	13	irϕr	irϕr	ADJ
ejpam-460	69	14	+	+	NOUN
ejpam-460	69	15	ϕθ	ϕθ	NOUN
ejpam-460	69	16	)	)	PUNCT
ejpam-460	69	17	.	.	PUNCT
ejpam-460	70	1	obviously	obviously	ADV
ejpam-460	70	2	,	,	PUNCT
ejpam-460	70	3	w	w	PROPN
ejpam-460	70	4	=	=	NOUN
ejpam-460	70	5	λz|z|	λz|z|	VERB
ejpam-460	70	6	1	1	NUM
ejpam-460	70	7	k	k	NOUN
ejpam-460	70	8	−1	−1	NOUN
ejpam-460	70	9	is	be	AUX
ejpam-460	70	10	a	a	DET
ejpam-460	70	11	k	k	PROPN
ejpam-460	70	12	−	−	PROPN
ejpam-460	70	13	q.c	q.c	PROPN
ejpam-460	70	14	.	.	PROPN
ejpam-460	70	15	from	from	ADP
ejpam-460	70	16	{	{	PUNCT
ejpam-460	70	17	z|r1	z|r1	PROPN
ejpam-460	70	18	≤	≤	NUM
ejpam-460	70	19	|z|	|z|	NOUN
ejpam-460	70	20	≤	≤	NOUN
ejpam-460	70	21	1	1	NUM
ejpam-460	70	22	}	}	PUNCT
ejpam-460	70	23	onto	onto	ADP
ejpam-460	70	24	{	{	PUNCT
ejpam-460	70	25	z|r	z|r	NOUN
ejpam-460	70	26	1	1	NUM
ejpam-460	70	27	k	k	SYM
ejpam-460	70	28	1	1	NUM
ejpam-460	70	29	≤	≤	NOUN
ejpam-460	70	30	|z|	|z|	VERB
ejpam-460	70	31	≤	≤	NUM
ejpam-460	70	32	1	1	NUM
ejpam-460	70	33	}	}	PUNCT
ejpam-460	70	34	.	.	PUNCT
ejpam-460	71	1	if	if	SCONJ
ejpam-460	71	2	f	f	PROPN
ejpam-460	71	3	(	(	PUNCT
ejpam-460	71	4	z	z	NOUN
ejpam-460	71	5	)	)	PUNCT
ejpam-460	71	6	is	be	AUX
ejpam-460	71	7	an	an	DET
ejpam-460	71	8	extremal	extremal	ADJ
ejpam-460	71	9	k	k	PROPN
ejpam-460	71	10	−	−	PROPN
ejpam-460	71	11	q.c	q.c	PROPN
ejpam-460	71	12	.	.	PROPN
ejpam-460	71	13	,	,	PUNCT
ejpam-460	71	14	which	which	PRON
ejpam-460	71	15	satisfies	satisfy	VERB
ejpam-460	71	16	the	the	DET
ejpam-460	71	17	conditions	condition	NOUN
ejpam-460	71	18	of	of	ADP
ejpam-460	71	19	theorem	theorem	NOUN
ejpam-460	71	20	2	2	NUM
ejpam-460	71	21	,	,	PUNCT
ejpam-460	71	22	we	we	PRON
ejpam-460	71	23	have	have	VERB
ejpam-460	71	24	|	|	ADV
ejpam-460	71	25	fz̄	fz̄	ADJ
ejpam-460	71	26	fz	fz	VERB
ejpam-460	71	27	|=	|=	NOUN
ejpam-460	72	1	|	|	ADV
ejpam-460	72	2	1	1	NUM
ejpam-460	72	3	k	k	NOUN
ejpam-460	72	4	+	+	CCONJ
ejpam-460	72	5	irϕr	irϕr	ADJ
ejpam-460	72	6	−ϕθ	−ϕθ	PROPN
ejpam-460	72	7	1	1	NUM
ejpam-460	72	8	k	k	NOUN
ejpam-460	72	9	+	+	CCONJ
ejpam-460	72	10	irϕr	irϕr	ADJ
ejpam-460	72	11	+	+	PROPN
ejpam-460	72	12	ϕθ	ϕθ	NOUN
ejpam-460	72	13	|	|	ADV
ejpam-460	72	14	≤	≤	ADV
ejpam-460	73	1	k	k	NOUN
ejpam-460	73	2	−	−	PROPN
ejpam-460	74	1	1	1	NUM
ejpam-460	75	1	k	k	NOUN
ejpam-460	76	1	+	+	NOUN
ejpam-460	76	2	1	1	NUM
ejpam-460	76	3	,	,	PUNCT
ejpam-460	76	4	x.	x.	NOUN
ejpam-460	76	5	zhichun	zhichun	PROPN
ejpam-460	76	6	and	and	CCONJ
ejpam-460	76	7	h.	h.	PROPN
ejpam-460	76	8	xinzhong	xinzhong	PROPN
ejpam-460	76	9	/	/	SYM
ejpam-460	76	10	eur	eur	PROPN
ejpam-460	76	11	.	.	PUNCT
ejpam-460	77	1	j.	j.	PROPN
ejpam-460	77	2	pure	pure	PROPN
ejpam-460	77	3	appl	appl	PROPN
ejpam-460	77	4	.	.	PROPN
ejpam-460	77	5	math	math	PROPN
ejpam-460	77	6	,	,	PUNCT
ejpam-460	77	7	2	2	NUM
ejpam-460	77	8	(	(	PUNCT
ejpam-460	77	9	2009	2009	NUM
ejpam-460	77	10	)	)	PUNCT
ejpam-460	77	11	,	,	PUNCT
ejpam-460	77	12	(	(	PUNCT
ejpam-460	77	13	532	532	NUM
ejpam-460	77	14	-	-	NUM
ejpam-460	77	15	543	543	NUM
ejpam-460	77	16	)	)	PUNCT
ejpam-460	77	17	536	536	NUM
ejpam-460	77	18	thus	thus	ADV
ejpam-460	77	19	,	,	PUNCT
ejpam-460	77	20	k2ϕ2	k2ϕ2	PROPN
ejpam-460	77	21	θ	θ	X
ejpam-460	77	22	−	−	PROPN
ejpam-460	77	23	(	(	PUNCT
ejpam-460	77	24	k2	k2	PROPN
ejpam-460	77	25	+	+	CCONJ
ejpam-460	77	26	1)ϕθ	1)ϕθ	PROPN
ejpam-460	77	27	+	+	CCONJ
ejpam-460	77	28	1	1	NUM
ejpam-460	77	29	+	+	NUM
ejpam-460	77	30	k2r2ϕ2	k2r2ϕ2	PROPN
ejpam-460	77	31	r	r	NOUN
ejpam-460	77	32	≤	≤	NUM
ejpam-460	77	33	0	0	NUM
ejpam-460	77	34	,	,	PUNCT
ejpam-460	77	35	(	(	PUNCT
ejpam-460	77	36	2	2	X
ejpam-460	77	37	)	)	PUNCT
ejpam-460	77	38	if	if	SCONJ
ejpam-460	77	39	ϕr	ϕr	ADP
ejpam-460	77	40	6=	6=	ADP
ejpam-460	77	41	0	0	NUM
ejpam-460	77	42	,	,	PUNCT
ejpam-460	77	43	by	by	ADP
ejpam-460	77	44	the	the	DET
ejpam-460	77	45	inequality	inequality	NOUN
ejpam-460	77	46	(	(	PUNCT
ejpam-460	77	47	2	2	NUM
ejpam-460	77	48	)	)	PUNCT
ejpam-460	77	49	,	,	PUNCT
ejpam-460	77	50	we	we	PRON
ejpam-460	77	51	have	have	VERB
ejpam-460	77	52	k2ϕ2	k2ϕ2	X
ejpam-460	77	53	θ	θ	X
ejpam-460	77	54	−	−	PROPN
ejpam-460	77	55	(	(	PUNCT
ejpam-460	77	56	k2	k2	PROPN
ejpam-460	77	57	+	+	CCONJ
ejpam-460	77	58	1)ϕθ	1)ϕθ	PROPN
ejpam-460	77	59	+	+	CCONJ
ejpam-460	77	60	1	1	NUM
ejpam-460	77	61	≤−k2r2ϕ2	≤−k2r2ϕ2	NOUN
ejpam-460	77	62	r	r	NOUN
ejpam-460	77	63	<	<	X
ejpam-460	77	64	0	0	NUM
ejpam-460	77	65	,	,	PUNCT
ejpam-460	77	66	that	that	ADV
ejpam-460	77	67	is	is	ADV
ejpam-460	77	68	(	(	PUNCT
ejpam-460	77	69	k2ϕθ	k2ϕθ	X
ejpam-460	77	70	−	−	PROPN
ejpam-460	78	1	1)(ϕθ	1)(ϕθ	NUM
ejpam-460	79	1	−	−	NOUN
ejpam-460	79	2	1	1	NUM
ejpam-460	79	3	)	)	PUNCT
ejpam-460	79	4	<	<	X
ejpam-460	79	5	0	0	NUM
ejpam-460	79	6	,	,	PUNCT
ejpam-460	79	7	so	so	SCONJ
ejpam-460	79	8	we	we	PRON
ejpam-460	79	9	obtain	obtain	VERB
ejpam-460	79	10	that	that	PRON
ejpam-460	79	11	1	1	NUM
ejpam-460	79	12	k2	k2	NOUN
ejpam-460	79	13	<	<	X
ejpam-460	79	14	ϕθ	ϕθ	NOUN
ejpam-460	79	15	<	<	X
ejpam-460	79	16	1	1	NUM
ejpam-460	79	17	.	.	PUNCT
ejpam-460	80	1	for	for	ADP
ejpam-460	80	2	any	any	DET
ejpam-460	80	3	r	r	NOUN
ejpam-460	80	4	∈	∈	PROPN
ejpam-460	80	5	(	(	PUNCT
ejpam-460	80	6	0	0	NUM
ejpam-460	80	7	,	,	PUNCT
ejpam-460	80	8	1	1	NUM
ejpam-460	80	9	)	)	PUNCT
ejpam-460	80	10	,	,	PUNCT
ejpam-460	80	11	we	we	PRON
ejpam-460	80	12	have	have	VERB
ejpam-460	80	13	2π	2π	NOUN
ejpam-460	80	14	=	=	SYM
ejpam-460	81	1	ϕ(r	ϕ(r	PROPN
ejpam-460	81	2	,	,	PUNCT
ejpam-460	81	3	2π)−ϕ(r	2π)−ϕ(r	NUM
ejpam-460	81	4	,	,	PUNCT
ejpam-460	81	5	0	0	NUM
ejpam-460	81	6	)	)	PUNCT
ejpam-460	81	7	=	=	SYM
ejpam-460	82	1	∫	∫	PROPN
ejpam-460	82	2	2π	2π	NOUN
ejpam-460	82	3	0	0	NUM
ejpam-460	83	1	ϕθ	ϕθ	NOUN
ejpam-460	83	2	(	(	PUNCT
ejpam-460	83	3	r	r	NOUN
ejpam-460	83	4	,	,	PUNCT
ejpam-460	83	5	θ	θ	PROPN
ejpam-460	83	6	)	)	PUNCT
ejpam-460	84	1	dθ	dθ	PROPN
ejpam-460	84	2	<	<	X
ejpam-460	84	3	∫	∫	PROPN
ejpam-460	84	4	2π	2π	PROPN
ejpam-460	84	5	0	0	NUM
ejpam-460	85	1	dθ	dθ	PROPN
ejpam-460	85	2	=	=	PUNCT
ejpam-460	85	3	2π	2π	NOUN
ejpam-460	85	4	.	.	PUNCT
ejpam-460	86	1	this	this	PRON
ejpam-460	86	2	is	be	AUX
ejpam-460	86	3	impossible	impossible	ADJ
ejpam-460	86	4	.	.	PUNCT
ejpam-460	87	1	so	so	ADV
ejpam-460	87	2	we	we	PRON
ejpam-460	87	3	have	have	AUX
ejpam-460	87	4	proved	prove	VERB
ejpam-460	87	5	that	that	PRON
ejpam-460	87	6	ϕr	ϕr	NOUN
ejpam-460	87	7	=	=	NOUN
ejpam-460	87	8	0	0	NUM
ejpam-460	87	9	.	.	PUNCT
ejpam-460	88	1	therefore	therefore	ADV
ejpam-460	88	2	,	,	PUNCT
ejpam-460	88	3	we	we	PRON
ejpam-460	88	4	may	may	AUX
ejpam-460	88	5	assume	assume	VERB
ejpam-460	88	6	that	that	SCONJ
ejpam-460	88	7	f	f	PROPN
ejpam-460	88	8	(	(	PUNCT
ejpam-460	88	9	z	z	NOUN
ejpam-460	88	10	)	)	PUNCT
ejpam-460	89	1	=	=	SYM
ejpam-460	89	2	r	r	NOUN
ejpam-460	89	3	1	1	NUM
ejpam-460	89	4	k	k	NOUN
ejpam-460	89	5	eiψ(θ	eiψ(θ	PROPN
ejpam-460	89	6	)	)	PUNCT
ejpam-460	89	7	.	.	PUNCT
ejpam-460	90	1	at	at	ADP
ejpam-460	90	2	last	last	ADV
ejpam-460	90	3	,	,	PUNCT
ejpam-460	90	4	we	we	PRON
ejpam-460	90	5	will	will	AUX
ejpam-460	90	6	prove	prove	VERB
ejpam-460	90	7	that	that	SCONJ
ejpam-460	90	8	ψ(θ	ψ(θ	PUNCT
ejpam-460	90	9	)	)	PUNCT
ejpam-460	91	1	=	=	SYM
ejpam-460	91	2	θ	θ	PROPN
ejpam-460	91	3	+	+	NUM
ejpam-460	91	4	α	α	NOUN
ejpam-460	91	5	,	,	PUNCT
ejpam-460	91	6	where	where	SCONJ
ejpam-460	91	7	α	α	NOUN
ejpam-460	91	8	is	be	AUX
ejpam-460	91	9	a	a	DET
ejpam-460	91	10	constant	constant	ADJ
ejpam-460	91	11	.	.	PUNCT
ejpam-460	92	1	for	for	ADP
ejpam-460	92	2	any	any	DET
ejpam-460	92	3	θ1,θ2	θ1,θ2	PROPN
ejpam-460	92	4	,	,	PUNCT
ejpam-460	92	5	0	0	NUM
ejpam-460	92	6	≤	≤	NUM
ejpam-460	92	7	θ1	θ1	NOUN
ejpam-460	92	8	<	<	X
ejpam-460	92	9	θ2	θ2	ADV
ejpam-460	92	10	≤	≤	ADJ
ejpam-460	92	11	2π	2π	NOUN
ejpam-460	92	12	,	,	PUNCT
ejpam-460	92	13	let	let	VERB
ejpam-460	92	14	a1	a1	NOUN
ejpam-460	92	15	=	=	VERB
ejpam-460	92	16	a(r1	a(r1	PROPN
ejpam-460	92	17	,	,	PUNCT
ejpam-460	92	18	1;θ1,θ2	1;θ1,θ2	NUM
ejpam-460	92	19	)	)	PUNCT
ejpam-460	92	20	,	,	PUNCT
ejpam-460	92	21	a2	a2	PROPN
ejpam-460	92	22	=	=	PUNCT
ejpam-460	93	1	a(r1	a(r1	ADV
ejpam-460	93	2	,	,	PUNCT
ejpam-460	93	3	1;θ2,θ1	1;θ2,θ1	PROPN
ejpam-460	93	4	+	+	ADJ
ejpam-460	93	5	2π	2π	NOUN
ejpam-460	93	6	)	)	PUNCT
ejpam-460	93	7	,	,	PUNCT
ejpam-460	93	8	then	then	ADV
ejpam-460	93	9	f	f	PROPN
ejpam-460	93	10	(	(	PUNCT
ejpam-460	93	11	a1	a1	PROPN
ejpam-460	93	12	)	)	PUNCT
ejpam-460	93	13	=	=	PUNCT
ejpam-460	94	1	a(r	a(r	PROPN
ejpam-460	94	2	1	1	NUM
ejpam-460	94	3	k	k	PROPN
ejpam-460	94	4	1	1	NUM
ejpam-460	94	5	,	,	PUNCT
ejpam-460	94	6	1;ψ(θ1),ψ(θ2	1;ψ(θ1),ψ(θ2	NUM
ejpam-460	94	7	)	)	PUNCT
ejpam-460	94	8	)	)	PUNCT
ejpam-460	94	9	,	,	PUNCT
ejpam-460	94	10	f	f	PROPN
ejpam-460	94	11	(	(	PUNCT
ejpam-460	94	12	a2	a2	PROPN
ejpam-460	94	13	)	)	PUNCT
ejpam-460	94	14	=	=	PUNCT
ejpam-460	95	1	a(r	a(r	PROPN
ejpam-460	95	2	1	1	NUM
ejpam-460	95	3	k	k	PROPN
ejpam-460	95	4	1	1	NUM
ejpam-460	95	5	,	,	PUNCT
ejpam-460	95	6	1;ψ(θ2),ψ(θ1	1;ψ(θ2),ψ(θ1	NUM
ejpam-460	95	7	)	)	PUNCT
ejpam-460	95	8	+	+	CCONJ
ejpam-460	95	9	2π	2π	NOUN
ejpam-460	95	10	)	)	PUNCT
ejpam-460	95	11	.	.	PUNCT
ejpam-460	96	1	by	by	ADP
ejpam-460	96	2	the	the	DET
ejpam-460	96	3	quasi	quasi	ADJ
ejpam-460	96	4	-	-	ADJ
ejpam-460	96	5	invariant	invariant	ADJ
ejpam-460	96	6	property	property	NOUN
ejpam-460	96	7	of	of	ADP
ejpam-460	96	8	module	module	NOUN
ejpam-460	96	9	,	,	PUNCT
ejpam-460	96	10	we	we	PRON
ejpam-460	96	11	have	have	VERB
ejpam-460	96	12			PROPN
ejpam-460	96	13			PRON
ejpam-460	96	14			NOUN
ejpam-460	96	15	1	1	NUM
ejpam-460	96	16	k	k	NOUN
ejpam-460	96	17	mod(a1)≤mod	mod(a1)≤mod	PROPN
ejpam-460	96	18	(	(	PUNCT
ejpam-460	96	19	f	f	PROPN
ejpam-460	96	20	(	(	PUNCT
ejpam-460	96	21	a1	a1	PROPN
ejpam-460	96	22	)	)	PUNCT
ejpam-460	96	23	)	)	PUNCT
ejpam-460	96	24	,	,	PUNCT
ejpam-460	96	25	1	1	NUM
ejpam-460	96	26	k	k	X
ejpam-460	96	27	mod(a2)≤mod	mod(a2)≤mod	PROPN
ejpam-460	96	28	(	(	PUNCT
ejpam-460	96	29	f	f	PROPN
ejpam-460	96	30	(	(	PUNCT
ejpam-460	96	31	a2	a2	PROPN
ejpam-460	96	32	)	)	PUNCT
ejpam-460	96	33	)	)	PUNCT
ejpam-460	96	34	,	,	PUNCT
ejpam-460	96	35	x.	x.	NOUN
ejpam-460	96	36	zhichun	zhichun	PROPN
ejpam-460	96	37	and	and	CCONJ
ejpam-460	96	38	h.	h.	PROPN
ejpam-460	96	39	xinzhong	xinzhong	PROPN
ejpam-460	96	40	/	/	SYM
ejpam-460	96	41	eur	eur	PROPN
ejpam-460	96	42	.	.	PUNCT
ejpam-460	97	1	j.	j.	PROPN
ejpam-460	97	2	pure	pure	PROPN
ejpam-460	97	3	appl	appl	PROPN
ejpam-460	97	4	.	.	PROPN
ejpam-460	97	5	math	math	PROPN
ejpam-460	97	6	,	,	PUNCT
ejpam-460	97	7	2	2	NUM
ejpam-460	97	8	(	(	PUNCT
ejpam-460	97	9	2009	2009	NUM
ejpam-460	97	10	)	)	PUNCT
ejpam-460	97	11	,	,	PUNCT
ejpam-460	97	12	(	(	PUNCT
ejpam-460	97	13	532	532	NUM
ejpam-460	97	14	-	-	NUM
ejpam-460	97	15	543	543	NUM
ejpam-460	97	16	)	)	PUNCT
ejpam-460	97	17	537	537	NUM
ejpam-460	97	18	that	that	PRON
ejpam-460	97	19	is	be	AUX
ejpam-460	97	20			PROPN
ejpam-460	97	21			NOUN
ejpam-460	97	22			ADJ
ejpam-460	97	23	ψ(θ2)−ψ(θ1)≤	ψ(θ2)−ψ(θ1)≤	NUM
ejpam-460	97	24	θ2−	θ2−	NOUN
ejpam-460	97	25	θ1	θ1	NOUN
ejpam-460	97	26	,	,	PUNCT
ejpam-460	97	27	ψ(θ2)−ψ(θ1)≥	ψ(θ2)−ψ(θ1)≥	PUNCT
ejpam-460	97	28	θ2−	θ2−	NOUN
ejpam-460	97	29	θ1	θ1	NOUN
ejpam-460	97	30	,	,	PUNCT
ejpam-460	97	31	therefore	therefore	ADV
ejpam-460	97	32	,	,	PUNCT
ejpam-460	97	33	ψ(θ2)−ψ(θ1	ψ(θ2)−ψ(θ1	X
ejpam-460	97	34	)	)	PUNCT
ejpam-460	98	1	=	=	PUNCT
ejpam-460	98	2	θ2	θ2	PROPN
ejpam-460	98	3	−	−	PROPN
ejpam-460	98	4	θ1	θ1	NOUN
ejpam-460	98	5	.	.	PUNCT
ejpam-460	99	1	for	for	ADP
ejpam-460	99	2	any	any	DET
ejpam-460	99	3	θ	θ	NOUN
ejpam-460	99	4	,	,	PUNCT
ejpam-460	99	5	0≤	0≤	NUM
ejpam-460	99	6	θ	θ	NOUN
ejpam-460	99	7	≤	≤	ADJ
ejpam-460	99	8	2π	2π	NOUN
ejpam-460	99	9	,	,	PUNCT
ejpam-460	99	10	since	since	SCONJ
ejpam-460	99	11	ψ′(θ	ψ′(θ	PRON
ejpam-460	99	12	)	)	PUNCT
ejpam-460	100	1	=	=	SYM
ejpam-460	100	2	lim	lim	PROPN
ejpam-460	100	3	h→0	h→0	ADV
ejpam-460	100	4	ψ(θ	ψ(θ	PROPN
ejpam-460	100	5	+	+	NUM
ejpam-460	100	6	h)−ψ(θ	h)−ψ(θ	X
ejpam-460	100	7	)	)	PUNCT
ejpam-460	100	8	h	h	NOUN
ejpam-460	101	1	=	=	SYM
ejpam-460	101	2	lim	lim	PROPN
ejpam-460	101	3	h→0	h→0	ADV
ejpam-460	102	1	(	(	PUNCT
ejpam-460	102	2	θ	θ	PROPN
ejpam-460	102	3	+	+	CCONJ
ejpam-460	102	4	h)−	h)−	PROPN
ejpam-460	102	5	θ	θ	PROPN
ejpam-460	102	6	h	h	NOUN
ejpam-460	102	7	=	=	SYM
ejpam-460	102	8	1	1	NUM
ejpam-460	102	9	,	,	PUNCT
ejpam-460	102	10	then	then	ADV
ejpam-460	102	11	ψ(θ	ψ(θ	PUNCT
ejpam-460	102	12	)	)	PUNCT
ejpam-460	103	1	=	=	PUNCT
ejpam-460	103	2	θ	θ	X
ejpam-460	104	1	+	+	NOUN
ejpam-460	104	2	α	α	X
ejpam-460	104	3	,	,	PUNCT
ejpam-460	104	4	α	α	PROPN
ejpam-460	104	5	is	be	AUX
ejpam-460	104	6	a	a	DET
ejpam-460	104	7	constant	constant	ADJ
ejpam-460	104	8	,	,	PUNCT
ejpam-460	104	9	thus	thus	ADV
ejpam-460	104	10	,	,	PUNCT
ejpam-460	104	11	we	we	PRON
ejpam-460	104	12	obtain	obtain	VERB
ejpam-460	104	13	that	that	SCONJ
ejpam-460	104	14	f	f	PROPN
ejpam-460	104	15	(	(	PUNCT
ejpam-460	104	16	z	z	NOUN
ejpam-460	104	17	)	)	PUNCT
ejpam-460	105	1	=	=	SYM
ejpam-460	105	2	r	r	NOUN
ejpam-460	105	3	1	1	NUM
ejpam-460	105	4	k	k	NOUN
ejpam-460	105	5	ei(θ+α	ei(θ+α	PROPN
ejpam-460	105	6	)	)	PUNCT
ejpam-460	105	7	.	.	PUNCT
ejpam-460	106	1	the	the	DET
ejpam-460	106	2	theorem	theorem	NOUN
ejpam-460	106	3	is	be	AUX
ejpam-460	106	4	proved	prove	VERB
ejpam-460	106	5	.	.	PUNCT
ejpam-460	107	1	by	by	ADP
ejpam-460	107	2	theorem	theorem	NOUN
ejpam-460	107	3	2	2	NUM
ejpam-460	107	4	,	,	PUNCT
ejpam-460	107	5	the	the	DET
ejpam-460	107	6	following	following	ADJ
ejpam-460	107	7	result	result	NOUN
ejpam-460	107	8	can	can	AUX
ejpam-460	107	9	be	be	AUX
ejpam-460	107	10	proved	prove	VERB
ejpam-460	107	11	.	.	PUNCT
ejpam-460	108	1	corollary	corollary	ADJ
ejpam-460	108	2	1	1	NUM
ejpam-460	108	3	.	.	PUNCT
ejpam-460	109	1	let	let	VERB
ejpam-460	109	2	q	q	PART
ejpam-460	109	3	be	be	AUX
ejpam-460	109	4	k−quasiconformal	k−quasiconformal	NOUN
ejpam-460	109	5	mappings	mapping	NOUN
ejpam-460	109	6	from	from	ADP
ejpam-460	109	7	{	{	PUNCT
ejpam-460	109	8	z|r1	z|r1	PROPN
ejpam-460	109	9	≤	≤	NUM
ejpam-460	109	10	|z|	|z|	NOUN
ejpam-460	109	11	≤	≤	NOUN
ejpam-460	109	12	1	1	NUM
ejpam-460	109	13	}	}	PUNCT
ejpam-460	109	14	to	to	ADP
ejpam-460	109	15	{	{	PUNCT
ejpam-460	109	16	z||rk	z||rk	VERB
ejpam-460	109	17	1	1	NUM
ejpam-460	109	18	≤	≤	NOUN
ejpam-460	109	19	|z|	|z|	VERB
ejpam-460	109	20	≤	≤	NUM
ejpam-460	109	21	1	1	NUM
ejpam-460	109	22	}	}	PUNCT
ejpam-460	109	23	,	,	PUNCT
ejpam-460	109	24	and	and	CCONJ
ejpam-460	109	25	if	if	SCONJ
ejpam-460	109	26	f	f	PROPN
ejpam-460	109	27	∈q	∈q	NOUN
ejpam-460	109	28	satisfies	satisfy	VERB
ejpam-460	109	29	ess	ess	NOUN
ejpam-460	109	30	sup	sup	NOUN
ejpam-460	109	31	z	z	PROPN
ejpam-460	109	32	d	d	X
ejpam-460	109	33	f	f	X
ejpam-460	109	34	(	(	PUNCT
ejpam-460	109	35	z	z	NOUN
ejpam-460	109	36	)	)	PUNCT
ejpam-460	109	37	=	=	SYM
ejpam-460	109	38	inf	inf	NOUN
ejpam-460	109	39	g∈q	g∈q	NOUN
ejpam-460	109	40	sup	sup	NOUN
ejpam-460	109	41	z	z	NOUN
ejpam-460	109	42	dg(z	dg(z	NUM
ejpam-460	109	43	)	)	PUNCT
ejpam-460	109	44	,	,	PUNCT
ejpam-460	109	45	then	then	ADV
ejpam-460	109	46	f	f	X
ejpam-460	109	47	(	(	PUNCT
ejpam-460	109	48	z	z	NOUN
ejpam-460	109	49	)	)	PUNCT
ejpam-460	110	1	=	=	PRON
ejpam-460	110	2	rk	rk	NOUN
ejpam-460	110	3	ei(θ+α	ei(θ+α	PROPN
ejpam-460	110	4	)	)	PUNCT
ejpam-460	110	5	,	,	PUNCT
ejpam-460	110	6	z	z	X
ejpam-460	110	7	=	=	PUNCT
ejpam-460	110	8	reiθ	reiθ	PROPN
ejpam-460	110	9	,	,	PUNCT
ejpam-460	110	10	where	where	SCONJ
ejpam-460	110	11	α	α	NOUN
ejpam-460	110	12	is	be	AUX
ejpam-460	110	13	a	a	DET
ejpam-460	110	14	constant	constant	ADJ
ejpam-460	110	15	.	.	PUNCT
ejpam-460	111	1	on	on	ADP
ejpam-460	111	2	the	the	DET
ejpam-460	111	3	other	other	ADJ
ejpam-460	111	4	hand	hand	NOUN
ejpam-460	111	5	,	,	PUNCT
ejpam-460	111	6	[	[	X
ejpam-460	111	7	3	3	X
ejpam-460	111	8	]	]	PUNCT
ejpam-460	111	9	also	also	ADV
ejpam-460	111	10	considered	consider	VERB
ejpam-460	111	11	the	the	DET
ejpam-460	111	12	grötzsch	grötzsch	ADJ
ejpam-460	111	13	extremal	extremal	ADJ
ejpam-460	111	14	problem	problem	NOUN
ejpam-460	111	15	on	on	ADP
ejpam-460	111	16	annulus	annulus	NOUN
ejpam-460	111	17	under	under	ADP
ejpam-460	111	18	some	some	DET
ejpam-460	111	19	restriction	restriction	NOUN
ejpam-460	111	20	of	of	ADP
ejpam-460	111	21	area	area	NOUN
ejpam-460	111	22	distortion	distortion	NOUN
ejpam-460	111	23	,	,	PUNCT
ejpam-460	111	24	and	and	CCONJ
ejpam-460	111	25	proved	prove	VERB
ejpam-460	111	26	the	the	DET
ejpam-460	111	27	following	follow	VERB
ejpam-460	111	28	result	result	NOUN
ejpam-460	111	29	.	.	PUNCT
ejpam-460	112	1	x.	x.	NOUN
ejpam-460	112	2	zhichun	zhichun	PROPN
ejpam-460	112	3	and	and	CCONJ
ejpam-460	112	4	h.	h.	PROPN
ejpam-460	112	5	xinzhong	xinzhong	PROPN
ejpam-460	112	6	/	/	SYM
ejpam-460	112	7	eur	eur	PROPN
ejpam-460	112	8	.	.	PUNCT
ejpam-460	113	1	j.	j.	PROPN
ejpam-460	113	2	pure	pure	PROPN
ejpam-460	113	3	appl	appl	PROPN
ejpam-460	113	4	.	.	PROPN
ejpam-460	113	5	math	math	PROPN
ejpam-460	113	6	,	,	PUNCT
ejpam-460	113	7	2	2	NUM
ejpam-460	113	8	(	(	PUNCT
ejpam-460	113	9	2009	2009	NUM
ejpam-460	113	10	)	)	PUNCT
ejpam-460	113	11	,	,	PUNCT
ejpam-460	113	12	(	(	PUNCT
ejpam-460	113	13	532	532	NUM
ejpam-460	113	14	-	-	NUM
ejpam-460	113	15	543	543	NUM
ejpam-460	113	16	)	)	PUNCT
ejpam-460	113	17	538	538	NUM
ejpam-460	113	18	theorem	theorem	NOUN
ejpam-460	113	19	3	3	NUM
ejpam-460	113	20	.	.	PUNCT
ejpam-460	114	1	let	let	VERB
ejpam-460	114	2	f	f	NOUN
ejpam-460	114	3	:	:	PUNCT
ejpam-460	114	4	∆=	∆=	VERB
ejpam-460	114	5	{	{	PUNCT
ejpam-460	114	6	z||z|	z||z|	PROPN
ejpam-460	114	7	<	<	X
ejpam-460	114	8	1	1	NUM
ejpam-460	114	9	}	}	PUNCT
ejpam-460	114	10	→∆	→∆	VERB
ejpam-460	114	11	be	be	AUX
ejpam-460	114	12	a	a	DET
ejpam-460	114	13	k−quasiconformal	k−quasiconformal	NOUN
ejpam-460	114	14	mapping	mapping	NOUN
ejpam-460	114	15	and	and	CCONJ
ejpam-460	114	16	f	f	PROPN
ejpam-460	114	17	(	(	PUNCT
ejpam-460	114	18	∆	∆	X
ejpam-460	114	19	)	)	PUNCT
ejpam-460	114	20	=	=	SYM
ejpam-460	114	21	∆	∆	PROPN
ejpam-460	114	22	,	,	PUNCT
ejpam-460	114	23	f	f	X
ejpam-460	114	24	(	(	PUNCT
ejpam-460	114	25	0	0	NUM
ejpam-460	114	26	)	)	PUNCT
ejpam-460	114	27	=	=	NOUN
ejpam-460	115	1	0	0	X
ejpam-460	115	2	.	.	X
ejpam-460	116	1	for	for	ADP
ejpam-460	116	2	any	any	DET
ejpam-460	116	3	∆r	∆r	NOUN
ejpam-460	116	4	=	=	PUNCT
ejpam-460	116	5	{	{	PUNCT
ejpam-460	116	6	z||z|	z||z|	PROPN
ejpam-460	116	7	<	<	X
ejpam-460	116	8	r	r	NOUN
ejpam-460	116	9	}	}	PUNCT
ejpam-460	116	10	⊂∆	⊂∆	NOUN
ejpam-460	116	11	,	,	PUNCT
ejpam-460	116	12	if	if	SCONJ
ejpam-460	116	13	area	area	NOUN
ejpam-460	116	14	(	(	PUNCT
ejpam-460	116	15	f	f	PROPN
ejpam-460	116	16	(	(	PUNCT
ejpam-460	116	17	∆r	∆r	NOUN
ejpam-460	116	18	)	)	PUNCT
ejpam-460	116	19	)	)	PUNCT
ejpam-460	116	20	area(∆r	area(∆r	NOUN
ejpam-460	116	21	)	)	PUNCT
ejpam-460	116	22	1	1	NUM
ejpam-460	116	23	k	k	NOUN
ejpam-460	116	24	≥	≥	NUM
ejpam-460	116	25	π1−	π1−	NOUN
ejpam-460	116	26	1	1	NUM
ejpam-460	116	27	k	k	NOUN
ejpam-460	116	28	,	,	PUNCT
ejpam-460	116	29	then	then	ADV
ejpam-460	116	30	,	,	PUNCT
ejpam-460	116	31	f	f	PROPN
ejpam-460	116	32	(	(	PUNCT
ejpam-460	116	33	z	z	NOUN
ejpam-460	116	34	)	)	PUNCT
ejpam-460	116	35	=	=	PRON
ejpam-460	116	36	λz|z|	λz|z|	VERB
ejpam-460	116	37	1	1	NUM
ejpam-460	116	38	k	k	NOUN
ejpam-460	116	39	−1	−1	NOUN
ejpam-460	116	40	for	for	ADP
ejpam-460	116	41	any	any	DET
ejpam-460	116	42	z	z	PROPN
ejpam-460	116	43	∈	∈	PROPN
ejpam-460	116	44	rr	rr	NOUN
ejpam-460	116	45	=	=	SYM
ejpam-460	117	1	{	{	PUNCT
ejpam-460	117	2	z|r	z|r	NOUN
ejpam-460	117	3	<	<	X
ejpam-460	117	4	|z|	|z|	PROPN
ejpam-460	117	5	<	<	X
ejpam-460	117	6	1	1	NUM
ejpam-460	117	7	}	}	PUNCT
ejpam-460	117	8	,	,	PUNCT
ejpam-460	117	9	where	where	SCONJ
ejpam-460	117	10	λ	λ	PROPN
ejpam-460	117	11	is	be	AUX
ejpam-460	117	12	a	a	DET
ejpam-460	117	13	constant	constant	ADJ
ejpam-460	117	14	with	with	ADP
ejpam-460	117	15	|λ|=	|λ|=	NOUN
ejpam-460	117	16	1	1	NUM
ejpam-460	117	17	.	.	PUNCT
ejpam-460	118	1	we	we	PRON
ejpam-460	118	2	will	will	AUX
ejpam-460	118	3	point	point	VERB
ejpam-460	118	4	out	out	ADP
ejpam-460	118	5	that	that	SCONJ
ejpam-460	118	6	theorem	theorem	NOUN
ejpam-460	118	7	3	3	NUM
ejpam-460	118	8	is	be	AUX
ejpam-460	118	9	not	not	PART
ejpam-460	118	10	correct	correct	ADJ
ejpam-460	118	11	.	.	PUNCT
ejpam-460	119	1	for	for	ADP
ejpam-460	119	2	example	example	NOUN
ejpam-460	119	3	,	,	PUNCT
ejpam-460	119	4	if	if	SCONJ
ejpam-460	119	5	we	we	PRON
ejpam-460	119	6	again	again	ADV
ejpam-460	119	7	take	take	VERB
ejpam-460	119	8	g(z	g(z	ADJ
ejpam-460	119	9	)	)	PUNCT
ejpam-460	120	1	=	=	SYM
ejpam-460	120	2	r	r	NOUN
ejpam-460	120	3	1	1	NUM
ejpam-460	120	4	k	k	NOUN
ejpam-460	120	5	ei(θ+ksinθ	ei(θ+ksinθ	PROPN
ejpam-460	120	6	)	)	PUNCT
ejpam-460	120	7	,	,	PUNCT
ejpam-460	120	8	z	z	X
ejpam-460	120	9	=	=	PUNCT
ejpam-460	120	10	reiθ	reiθ	PROPN
ejpam-460	120	11	,	,	PUNCT
ejpam-460	120	12	it	it	PRON
ejpam-460	120	13	is	be	AUX
ejpam-460	120	14	easy	easy	ADJ
ejpam-460	120	15	to	to	PART
ejpam-460	120	16	see	see	VERB
ejpam-460	120	17	that	that	DET
ejpam-460	120	18	area(g(∆r	area(g(∆r	PROPN
ejpam-460	120	19	)	)	PUNCT
ejpam-460	120	20	)	)	PUNCT
ejpam-460	120	21	area(∆r	area(∆r	X
ejpam-460	120	22	)	)	PUNCT
ejpam-460	121	1	1	1	NUM
ejpam-460	121	2	k	k	X
ejpam-460	121	3	=	=	PUNCT
ejpam-460	121	4	π1−	π1−	NOUN
ejpam-460	121	5	1	1	NUM
ejpam-460	121	6	k	k	NOUN
ejpam-460	121	7	,	,	PUNCT
ejpam-460	121	8	but	but	CCONJ
ejpam-460	121	9	g(z	g(z	PROPN
ejpam-460	121	10	)	)	PUNCT
ejpam-460	121	11	is	be	AUX
ejpam-460	121	12	not	not	PART
ejpam-460	121	13	the	the	DET
ejpam-460	121	14	required	require	VERB
ejpam-460	121	15	form	form	NOUN
ejpam-460	121	16	,	,	PUNCT
ejpam-460	121	17	thus	thus	ADV
ejpam-460	121	18	theorem	theorem	VERB
ejpam-460	121	19	3	3	NUM
ejpam-460	121	20	is	be	AUX
ejpam-460	121	21	fault	fault	NOUN
ejpam-460	121	22	.	.	PUNCT
ejpam-460	122	1	next	next	ADV
ejpam-460	122	2	,	,	PUNCT
ejpam-460	122	3	by	by	ADP
ejpam-460	122	4	considering	consider	VERB
ejpam-460	122	5	the	the	DET
ejpam-460	122	6	relationship	relationship	NOUN
ejpam-460	122	7	between	between	ADP
ejpam-460	122	8	the	the	DET
ejpam-460	122	9	hyperbolic	hyperbolic	ADJ
ejpam-460	122	10	area	area	NOUN
ejpam-460	122	11	of	of	ADP
ejpam-460	122	12	ring	ring	NOUN
ejpam-460	122	13	domain	domain	NOUN
ejpam-460	122	14	and	and	CCONJ
ejpam-460	122	15	its	its	PRON
ejpam-460	122	16	module	module	NOUN
ejpam-460	122	17	,	,	PUNCT
ejpam-460	122	18	we	we	PRON
ejpam-460	122	19	can	can	AUX
ejpam-460	122	20	obtain	obtain	VERB
ejpam-460	122	21	one	one	NUM
ejpam-460	122	22	criterion	criterion	NOUN
ejpam-460	122	23	for	for	ADP
ejpam-460	122	24	extremal	extremal	ADJ
ejpam-460	122	25	mapping	mapping	NOUN
ejpam-460	122	26	under	under	ADP
ejpam-460	122	27	the	the	DET
ejpam-460	122	28	hyperbolic	hyperbolic	ADJ
ejpam-460	122	29	area	area	NOUN
ejpam-460	122	30	distortion	distortion	NOUN
ejpam-460	122	31	condition	condition	NOUN
ejpam-460	122	32	.	.	PUNCT
ejpam-460	123	1	suppose	suppose	VERB
ejpam-460	123	2	that	that	SCONJ
ejpam-460	123	3	e	e	PROPN
ejpam-460	123	4	⊂∆	⊂∆	NOUN
ejpam-460	123	5	is	be	AUX
ejpam-460	123	6	a	a	DET
ejpam-460	123	7	measurable	measurable	ADJ
ejpam-460	123	8	subset	subset	NOUN
ejpam-460	123	9	,	,	PUNCT
ejpam-460	123	10	let	let	VERB
ejpam-460	123	11	|e|hyp	|e|hyp	PUNCT
ejpam-460	123	12	=	=	SYM
ejpam-460	123	13	∫∫	∫∫	ADV
ejpam-460	123	14	e	e	ADP
ejpam-460	123	15	1	1	NUM
ejpam-460	123	16	(	(	PUNCT
ejpam-460	123	17	1−	1−	NUM
ejpam-460	123	18	|z|2)2	|z|2)2	PROPN
ejpam-460	123	19	|dz|2	|dz|2	PROPN
ejpam-460	123	20	,	,	PUNCT
ejpam-460	123	21	be	be	AUX
ejpam-460	123	22	the	the	DET
ejpam-460	123	23	hyperbolic	hyperbolic	ADJ
ejpam-460	123	24	area	area	NOUN
ejpam-460	123	25	of	of	ADP
ejpam-460	123	26	e	e	PROPN
ejpam-460	123	27	,	,	PUNCT
ejpam-460	123	28	and	and	CCONJ
ejpam-460	123	29	area(e	area(e	NOUN
ejpam-460	123	30	)	)	PUNCT
ejpam-460	123	31	be	be	AUX
ejpam-460	123	32	its	its	PRON
ejpam-460	123	33	euclidean	euclidean	ADJ
ejpam-460	123	34	area	area	NOUN
ejpam-460	123	35	.	.	PUNCT
ejpam-460	124	1	we	we	PRON
ejpam-460	124	2	need	need	VERB
ejpam-460	124	3	the	the	DET
ejpam-460	124	4	following	follow	VERB
ejpam-460	124	5	result	result	NOUN
ejpam-460	124	6	made	make	VERB
ejpam-460	124	7	in	in	ADP
ejpam-460	124	8	[	[	X
ejpam-460	124	9	5	5	NUM
ejpam-460	124	10	]	]	PUNCT
ejpam-460	124	11	.	.	PUNCT
ejpam-460	125	1	lemma	lemma	PROPN
ejpam-460	125	2	2	2	X
ejpam-460	125	3	.	.	PUNCT
ejpam-460	126	1	let	let	VERB
ejpam-460	126	2	r	r	PRON
ejpam-460	126	3	be	be	AUX
ejpam-460	126	4	a	a	DET
ejpam-460	126	5	ring	ring	NOUN
ejpam-460	126	6	domain	domain	NOUN
ejpam-460	126	7	bounded	bound	VERB
ejpam-460	126	8	by	by	ADP
ejpam-460	126	9	two	two	NUM
ejpam-460	126	10	mutually	mutually	ADV
ejpam-460	126	11	disjoint	disjoint	VERB
ejpam-460	126	12	closed	closed	ADJ
ejpam-460	126	13	curves	curve	NOUN
ejpam-460	126	14	b0	b0	NOUN
ejpam-460	126	15	and	and	CCONJ
ejpam-460	126	16	b1	b1	NOUN
ejpam-460	126	17	.	.	PUNCT
ejpam-460	127	1	for	for	ADP
ejpam-460	127	2	any	any	DET
ejpam-460	127	3	arbitrary	arbitrary	ADJ
ejpam-460	127	4	concentric	concentric	ADJ
ejpam-460	127	5	ring	ring	NOUN
ejpam-460	127	6	r∗	r∗	PROPN
ejpam-460	127	7	bounded	bound	VERB
ejpam-460	127	8	by	by	ADP
ejpam-460	127	9	concentric	concentric	ADJ
ejpam-460	127	10	circles	circle	NOUN
ejpam-460	127	11	b∗	b∗	ADJ
ejpam-460	127	12	0	0	PUNCT
ejpam-460	127	13	and	and	CCONJ
ejpam-460	127	14	b∗	b∗	ADJ
ejpam-460	127	15	1	1	NUM
ejpam-460	127	16	,	,	PUNCT
ejpam-460	127	17	if	if	SCONJ
ejpam-460	127	18	area(r	area(r	NOUN
ejpam-460	127	19	)	)	PUNCT
ejpam-460	127	20	=	=	SYM
ejpam-460	127	21	area(r∗	area(r∗	NOUN
ejpam-460	127	22	)	)	PUNCT
ejpam-460	127	23	,	,	PUNCT
ejpam-460	127	24	then	then	ADV
ejpam-460	127	25	mod(r)≤mod(r∗	mod(r)≤mod(r∗	PROPN
ejpam-460	127	26	)	)	PUNCT
ejpam-460	127	27	.	.	PUNCT
ejpam-460	128	1	lemma	lemma	PROPN
ejpam-460	129	1	3	3	X
ejpam-460	129	2	.	.	PUNCT
ejpam-460	130	1	if	if	SCONJ
ejpam-460	130	2	f	f	PROPN
ejpam-460	130	3	(	(	PUNCT
ejpam-460	130	4	x	x	X
ejpam-460	130	5	)	)	PUNCT
ejpam-460	130	6	is	be	AUX
ejpam-460	130	7	a	a	DET
ejpam-460	130	8	positive	positive	ADJ
ejpam-460	130	9	continuous	continuous	ADJ
ejpam-460	130	10	function	function	NOUN
ejpam-460	130	11	in	in	ADP
ejpam-460	130	12	[	[	X
ejpam-460	130	13	0	0	NUM
ejpam-460	130	14	,	,	PUNCT
ejpam-460	130	15	2π	2π	NOUN
ejpam-460	130	16	]	]	PUNCT
ejpam-460	130	17	,	,	PUNCT
ejpam-460	130	18	then	then	ADV
ejpam-460	130	19	∫	∫	PROPN
ejpam-460	130	20	2π	2π	NOUN
ejpam-460	130	21	0	0	NUM
ejpam-460	130	22	1	1	NUM
ejpam-460	130	23	f	f	NOUN
ejpam-460	130	24	(	(	PUNCT
ejpam-460	130	25	x	x	X
ejpam-460	130	26	)	)	PUNCT
ejpam-460	130	27	d	d	NOUN
ejpam-460	130	28	x	x	SYM
ejpam-460	130	29	≥	≥	NUM
ejpam-460	130	30	4π2	4π2	NUM
ejpam-460	130	31	∫	∫	PROPN
ejpam-460	130	32	2π	2π	PROPN
ejpam-460	130	33	0	0	NUM
ejpam-460	131	1	f	f	X
ejpam-460	131	2	(	(	PUNCT
ejpam-460	131	3	x	x	X
ejpam-460	131	4	)	)	PUNCT
ejpam-460	131	5	d	d	NOUN
ejpam-460	131	6	x	x	X
ejpam-460	131	7	,	,	PUNCT
ejpam-460	131	8	with	with	ADP
ejpam-460	131	9	equality	equality	NOUN
ejpam-460	131	10	if	if	SCONJ
ejpam-460	131	11	and	and	CCONJ
ejpam-460	131	12	only	only	ADV
ejpam-460	131	13	if	if	SCONJ
ejpam-460	131	14	f	f	PROPN
ejpam-460	131	15	(	(	PUNCT
ejpam-460	131	16	x)≡	x)≡	PROPN
ejpam-460	131	17	c	c	PROPN
ejpam-460	131	18	>	>	X
ejpam-460	131	19	0	0	X
ejpam-460	131	20	.	.	PUNCT
ejpam-460	131	21	x.	x.	PROPN
ejpam-460	131	22	zhichun	zhichun	PROPN
ejpam-460	131	23	and	and	CCONJ
ejpam-460	131	24	h.	h.	PROPN
ejpam-460	131	25	xinzhong	xinzhong	PROPN
ejpam-460	131	26	/	/	SYM
ejpam-460	131	27	eur	eur	PROPN
ejpam-460	131	28	.	.	PUNCT
ejpam-460	132	1	j.	j.	PROPN
ejpam-460	132	2	pure	pure	PROPN
ejpam-460	132	3	appl	appl	PROPN
ejpam-460	132	4	.	.	PROPN
ejpam-460	132	5	math	math	PROPN
ejpam-460	132	6	,	,	PUNCT
ejpam-460	132	7	2	2	NUM
ejpam-460	132	8	(	(	PUNCT
ejpam-460	132	9	2009	2009	NUM
ejpam-460	132	10	)	)	PUNCT
ejpam-460	132	11	,	,	PUNCT
ejpam-460	132	12	(	(	PUNCT
ejpam-460	132	13	532	532	NUM
ejpam-460	132	14	-	-	NUM
ejpam-460	132	15	543	543	NUM
ejpam-460	132	16	)	)	PUNCT
ejpam-460	132	17	539	539	NUM
ejpam-460	132	18	proof	proof	NOUN
ejpam-460	132	19	.	.	PUNCT
ejpam-460	133	1	if	if	SCONJ
ejpam-460	133	2	f	f	PROPN
ejpam-460	133	3	(	(	PUNCT
ejpam-460	133	4	x	x	X
ejpam-460	133	5	)	)	PUNCT
ejpam-460	133	6	∈	∈	PROPN
ejpam-460	133	7	c[0	c[0	PROPN
ejpam-460	133	8	,	,	PUNCT
ejpam-460	133	9	2π	2π	NOUN
ejpam-460	133	10	]	]	PUNCT
ejpam-460	133	11	and	and	CCONJ
ejpam-460	133	12	f	f	PROPN
ejpam-460	133	13	(	(	PUNCT
ejpam-460	133	14	x	x	X
ejpam-460	133	15	)	)	PUNCT
ejpam-460	133	16	>	>	X
ejpam-460	133	17	0	0	NUM
ejpam-460	133	18	,	,	PUNCT
ejpam-460	133	19	we	we	PRON
ejpam-460	133	20	have	have	VERB
ejpam-460	133	21	[	[	PUNCT
ejpam-460	133	22	∫	∫	PROPN
ejpam-460	133	23	2π	2π	PROPN
ejpam-460	133	24	0	0	PUNCT
ejpam-460	134	1	p	p	X
ejpam-460	134	2	f	f	X
ejpam-460	134	3	(	(	PUNCT
ejpam-460	134	4	x	x	NOUN
ejpam-460	134	5	)	)	PUNCT
ejpam-460	134	6	·	·	PUNCT
ejpam-460	134	7	1	1	NUM
ejpam-460	135	1	p	p	NOUN
ejpam-460	135	2	f	f	X
ejpam-460	135	3	(	(	PUNCT
ejpam-460	135	4	x	x	X
ejpam-460	135	5	)	)	PUNCT
ejpam-460	135	6	d	d	NOUN
ejpam-460	135	7	x]2	x]2	PUNCT
ejpam-460	135	8	≤	≤	NUM
ejpam-460	135	9	∫	∫	PROPN
ejpam-460	135	10	2π	2π	PROPN
ejpam-460	135	11	0	0	NUM
ejpam-460	136	1	f	f	X
ejpam-460	136	2	(	(	PUNCT
ejpam-460	136	3	x	x	X
ejpam-460	136	4	)	)	PUNCT
ejpam-460	136	5	d	d	NOUN
ejpam-460	136	6	x	x	SYM
ejpam-460	136	7	·	·	PUNCT
ejpam-460	136	8	∫	∫	PROPN
ejpam-460	136	9	2π	2π	NOUN
ejpam-460	136	10	0	0	NUM
ejpam-460	136	11	1	1	NUM
ejpam-460	136	12	f	f	NOUN
ejpam-460	136	13	(	(	PUNCT
ejpam-460	136	14	x	x	X
ejpam-460	136	15	)	)	PUNCT
ejpam-460	136	16	d	d	NOUN
ejpam-460	136	17	x	x	SYM
ejpam-460	136	18	,	,	PUNCT
ejpam-460	136	19	thus	thus	ADV
ejpam-460	136	20	4π2	4π2	NUM
ejpam-460	136	21	≤	≤	ADV
ejpam-460	136	22	2π	2π	PROPN
ejpam-460	136	23	∫	∫	NOUN
ejpam-460	136	24	0	0	NUM
ejpam-460	137	1	f	f	PROPN
ejpam-460	137	2	(	(	PUNCT
ejpam-460	137	3	x)d	x)d	PUNCT
ejpam-460	137	4	x	x	SYM
ejpam-460	137	5	·	·	PUNCT
ejpam-460	137	6	2π	2π	NUM
ejpam-460	137	7	∫	∫	NOUN
ejpam-460	137	8	0	0	NUM
ejpam-460	137	9	1	1	NUM
ejpam-460	137	10	f	f	PROPN
ejpam-460	137	11	(	(	PUNCT
ejpam-460	137	12	x	x	X
ejpam-460	137	13	)	)	PUNCT
ejpam-460	137	14	d	d	NOUN
ejpam-460	137	15	x	x	SYM
ejpam-460	137	16	,	,	PUNCT
ejpam-460	137	17	that	that	ADV
ejpam-460	137	18	is	is	ADV
ejpam-460	137	19	,	,	PUNCT
ejpam-460	137	20	∫	∫	PROPN
ejpam-460	137	21	2π	2π	NOUN
ejpam-460	137	22	0	0	NUM
ejpam-460	137	23	1	1	NUM
ejpam-460	137	24	f	f	NOUN
ejpam-460	137	25	(	(	PUNCT
ejpam-460	137	26	x	x	X
ejpam-460	137	27	)	)	PUNCT
ejpam-460	137	28	d	d	NOUN
ejpam-460	137	29	x	x	SYM
ejpam-460	137	30	≥	≥	NUM
ejpam-460	137	31	4π2	4π2	NUM
ejpam-460	137	32	∫	∫	PROPN
ejpam-460	137	33	2π	2π	PROPN
ejpam-460	137	34	0	0	NUM
ejpam-460	138	1	f	f	X
ejpam-460	138	2	(	(	PUNCT
ejpam-460	138	3	x	x	X
ejpam-460	138	4	)	)	PUNCT
ejpam-460	138	5	d	d	NOUN
ejpam-460	138	6	x	x	X
ejpam-460	138	7	.	.	PUNCT
ejpam-460	139	1	obviously	obviously	ADV
ejpam-460	139	2	,	,	PUNCT
ejpam-460	139	3	with	with	ADP
ejpam-460	139	4	equality	equality	NOUN
ejpam-460	139	5	if	if	SCONJ
ejpam-460	139	6	and	and	CCONJ
ejpam-460	139	7	only	only	ADV
ejpam-460	139	8	if	if	SCONJ
ejpam-460	139	9	f	f	PROPN
ejpam-460	139	10	(	(	PUNCT
ejpam-460	139	11	x)≡	x)≡	PROPN
ejpam-460	140	1	c	c	PROPN
ejpam-460	140	2	>	>	X
ejpam-460	141	1	0	0	X
ejpam-460	141	2	.	.	PUNCT
ejpam-460	142	1	the	the	DET
ejpam-460	142	2	proof	proof	NOUN
ejpam-460	142	3	of	of	ADP
ejpam-460	142	4	lemma	lemma	PROPN
ejpam-460	142	5	3	3	NUM
ejpam-460	142	6	is	be	AUX
ejpam-460	142	7	finished	finish	VERB
ejpam-460	142	8	.	.	PUNCT
ejpam-460	143	1	next	next	ADV
ejpam-460	143	2	we	we	PRON
ejpam-460	143	3	will	will	AUX
ejpam-460	143	4	prove	prove	VERB
ejpam-460	143	5	the	the	DET
ejpam-460	143	6	following	follow	VERB
ejpam-460	143	7	theorem	theorem	ADJ
ejpam-460	143	8	4	4	NUM
ejpam-460	143	9	.	.	PUNCT
ejpam-460	143	10	suppose	suppose	VERB
ejpam-460	143	11	that	that	SCONJ
ejpam-460	143	12	s1	s1	PROPN
ejpam-460	143	13	=	=	PUNCT
ejpam-460	143	14	a(r1	a(r1	ADV
ejpam-460	143	15	,	,	PUNCT
ejpam-460	143	16	r2	r2	PROPN
ejpam-460	143	17	)	)	PUNCT
ejpam-460	143	18	,	,	PUNCT
ejpam-460	143	19	0	0	PUNCT
ejpam-460	143	20	<	<	X
ejpam-460	143	21	r1	r1	PROPN
ejpam-460	143	22	<	<	X
ejpam-460	143	23	r2	r2	PROPN
ejpam-460	143	24	<	<	X
ejpam-460	143	25	1	1	NUM
ejpam-460	143	26	,	,	PUNCT
ejpam-460	143	27	and	and	CCONJ
ejpam-460	143	28	s2	s2	PROPN
ejpam-460	143	29	⊂∆	⊂∆	NOUN
ejpam-460	143	30	is	be	AUX
ejpam-460	143	31	a	a	DET
ejpam-460	143	32	ring	ring	NOUN
ejpam-460	143	33	domain	domain	NOUN
ejpam-460	143	34	bounded	bound	VERB
ejpam-460	143	35	by	by	ADP
ejpam-460	143	36	two	two	NUM
ejpam-460	143	37	mutually	mutually	ADV
ejpam-460	143	38	disjoint	disjoint	VERB
ejpam-460	143	39	closed	close	VERB
ejpam-460	143	40	curves	curve	NOUN
ejpam-460	143	41	c1	c1	NOUN
ejpam-460	143	42	and	and	CCONJ
ejpam-460	143	43	c2	c2	PROPN
ejpam-460	143	44	,	,	PUNCT
ejpam-460	143	45	where	where	SCONJ
ejpam-460	143	46	c1	c1	PROPN
ejpam-460	143	47	=	=	PUNCT
ejpam-460	143	48	{	{	PUNCT
ejpam-460	143	49	z||z|	z||z|	PROPN
ejpam-460	143	50	=	=	SYM
ejpam-460	143	51	r1	r1	PROPN
ejpam-460	143	52	}	}	PUNCT
ejpam-460	143	53	and	and	CCONJ
ejpam-460	143	54	c2	c2	PROPN
ejpam-460	143	55	=	=	SYM
ejpam-460	143	56	{	{	PUNCT
ejpam-460	143	57	z|z	z|z	NUM
ejpam-460	143	58	=	=	SYM
ejpam-460	143	59	f	f	PROPN
ejpam-460	143	60	(	(	PUNCT
ejpam-460	143	61	θ	θ	PROPN
ejpam-460	143	62	)	)	PUNCT
ejpam-460	143	63	eiθ	eiθ	PROPN
ejpam-460	143	64	}	}	PUNCT
ejpam-460	143	65	,	,	PUNCT
ejpam-460	143	66	where	where	SCONJ
ejpam-460	143	67	f	f	PROPN
ejpam-460	143	68	(	(	PUNCT
ejpam-460	143	69	θ	θ	PROPN
ejpam-460	143	70	)	)	PUNCT
ejpam-460	143	71	is	be	AUX
ejpam-460	143	72	a	a	DET
ejpam-460	143	73	continuous	continuous	ADJ
ejpam-460	143	74	function	function	NOUN
ejpam-460	143	75	in	in	ADP
ejpam-460	143	76	[	[	X
ejpam-460	143	77	0	0	NUM
ejpam-460	143	78	,	,	PUNCT
ejpam-460	143	79	2π	2π	NOUN
ejpam-460	143	80	]	]	PUNCT
ejpam-460	143	81	and	and	CCONJ
ejpam-460	143	82	f	f	X
ejpam-460	143	83	(	(	PUNCT
ejpam-460	143	84	0	0	NUM
ejpam-460	143	85	)	)	PUNCT
ejpam-460	143	86	=	=	SYM
ejpam-460	143	87	f	f	X
ejpam-460	143	88	(	(	PUNCT
ejpam-460	143	89	2π	2π	NOUN
ejpam-460	143	90	)	)	PUNCT
ejpam-460	143	91	,	,	PUNCT
ejpam-460	143	92	are	be	AUX
ejpam-460	143	93	the	the	DET
ejpam-460	143	94	inner	inner	ADJ
ejpam-460	143	95	and	and	CCONJ
ejpam-460	143	96	outer	outer	ADJ
ejpam-460	143	97	boundaries	boundary	NOUN
ejpam-460	143	98	of	of	ADP
ejpam-460	143	99	s2	s2	PROPN
ejpam-460	143	100	.	.	PUNCT
ejpam-460	144	1	if	if	SCONJ
ejpam-460	144	2	|s1|hyp	|s1|hyp	PROPN
ejpam-460	144	3	≥	≥	PROPN
ejpam-460	144	4	|s2|hyp	|s2|hyp	PROPN
ejpam-460	144	5	,	,	PUNCT
ejpam-460	144	6	then	then	ADV
ejpam-460	144	7	mod(s1	mod(s1	VERB
ejpam-460	144	8	)	)	PUNCT
ejpam-460	144	9	≥mod(s2	≥mod(s2	PROPN
ejpam-460	144	10	)	)	PUNCT
ejpam-460	144	11	.	.	PUNCT
ejpam-460	145	1	with	with	ADP
ejpam-460	145	2	equality	equality	NOUN
ejpam-460	145	3	if	if	SCONJ
ejpam-460	145	4	and	and	CCONJ
ejpam-460	145	5	only	only	ADV
ejpam-460	145	6	if	if	SCONJ
ejpam-460	145	7	s2	s2	PROPN
ejpam-460	145	8	=	=	SYM
ejpam-460	145	9	s1	s1	PROPN
ejpam-460	145	10	=	=	PUNCT
ejpam-460	145	11	a(r1	a(r1	ADV
ejpam-460	145	12	,	,	PUNCT
ejpam-460	145	13	r2	r2	PROPN
ejpam-460	145	14	)	)	PUNCT
ejpam-460	145	15	.	.	PUNCT
ejpam-460	146	1	proof	proof	NOUN
ejpam-460	146	2	.	.	PUNCT
ejpam-460	147	1	since	since	SCONJ
ejpam-460	147	2	|s1|hyp	|s1|hyp	PROPN
ejpam-460	147	3	=	=	SYM
ejpam-460	147	4	∫∫	∫∫	PROPN
ejpam-460	147	5	s1	s1	NOUN
ejpam-460	147	6	1	1	NUM
ejpam-460	147	7	(	(	PUNCT
ejpam-460	147	8	1−	1−	NUM
ejpam-460	147	9	|z|2)2	|z|2)2	NOUN
ejpam-460	147	10	|dz|2	|dz|2	PUNCT
ejpam-460	147	11	=	=	SYM
ejpam-460	147	12	∫	∫	PROPN
ejpam-460	147	13	2π	2π	PROPN
ejpam-460	147	14	0	0	NUM
ejpam-460	147	15	∫	∫	PROPN
ejpam-460	147	16	r2	r2	PROPN
ejpam-460	147	17	r1	r1	PROPN
ejpam-460	147	18	r	r	PROPN
ejpam-460	147	19	(	(	PUNCT
ejpam-460	147	20	1−	1−	NUM
ejpam-460	147	21	r2)2	r2)2	NOUN
ejpam-460	147	22	drdθ	drdθ	NOUN
ejpam-460	147	23	=	=	PUNCT
ejpam-460	147	24	π	π	PROPN
ejpam-460	147	25	(	(	PUNCT
ejpam-460	147	26	1	1	NUM
ejpam-460	147	27	1−	1−	NUM
ejpam-460	147	28	r2	r2	NOUN
ejpam-460	147	29	2	2	NUM
ejpam-460	147	30	−	−	PROPN
ejpam-460	147	31	1	1	NUM
ejpam-460	147	32	1−	1−	NUM
ejpam-460	147	33	r2	r2	PROPN
ejpam-460	147	34	1	1	NUM
ejpam-460	147	35	)	)	PUNCT
ejpam-460	147	36	,	,	PUNCT
ejpam-460	147	37	x.	x.	NOUN
ejpam-460	147	38	zhichun	zhichun	PROPN
ejpam-460	147	39	and	and	CCONJ
ejpam-460	147	40	h.	h.	PROPN
ejpam-460	147	41	xinzhong	xinzhong	PROPN
ejpam-460	147	42	/	/	SYM
ejpam-460	147	43	eur	eur	PROPN
ejpam-460	147	44	.	.	PUNCT
ejpam-460	148	1	j.	j.	PROPN
ejpam-460	148	2	pure	pure	PROPN
ejpam-460	148	3	appl	appl	PROPN
ejpam-460	148	4	.	.	PROPN
ejpam-460	148	5	math	math	PROPN
ejpam-460	148	6	,	,	PUNCT
ejpam-460	148	7	2	2	NUM
ejpam-460	148	8	(	(	PUNCT
ejpam-460	148	9	2009	2009	NUM
ejpam-460	148	10	)	)	PUNCT
ejpam-460	148	11	,	,	PUNCT
ejpam-460	148	12	(	(	PUNCT
ejpam-460	148	13	532	532	NUM
ejpam-460	148	14	-	-	NUM
ejpam-460	148	15	543	543	NUM
ejpam-460	148	16	)	)	PUNCT
ejpam-460	148	17	540	540	NUM
ejpam-460	148	18	|s2|hyp	|s2|hyp	NOUN
ejpam-460	148	19	=	=	PUNCT
ejpam-460	149	1	∫∫	∫∫	PROPN
ejpam-460	149	2	s2	s2	NOUN
ejpam-460	149	3	1	1	NUM
ejpam-460	149	4	(	(	PUNCT
ejpam-460	149	5	1−	1−	NUM
ejpam-460	149	6	|z|2)2	|z|2)2	NOUN
ejpam-460	149	7	|dz|2	|dz|2	PUNCT
ejpam-460	149	8	=	=	SYM
ejpam-460	149	9	∫	∫	PROPN
ejpam-460	149	10	2π	2π	PROPN
ejpam-460	149	11	0	0	NUM
ejpam-460	150	1	∫	∫	PROPN
ejpam-460	150	2	f	f	PROPN
ejpam-460	150	3	(	(	PUNCT
ejpam-460	150	4	θ	θ	PROPN
ejpam-460	150	5	)	)	PUNCT
ejpam-460	150	6	r1	r1	PROPN
ejpam-460	150	7	r	r	NOUN
ejpam-460	150	8	(	(	PUNCT
ejpam-460	150	9	1−	1−	NUM
ejpam-460	150	10	r2)2	r2)2	NOUN
ejpam-460	150	11	drdθ	drdθ	NOUN
ejpam-460	150	12	=	=	SYM
ejpam-460	150	13	1	1	NUM
ejpam-460	150	14	2	2	NUM
ejpam-460	150	15	∫	∫	NOUN
ejpam-460	150	16	2π	2π	NOUN
ejpam-460	150	17	0	0	PUNCT
ejpam-460	150	18	(	(	PUNCT
ejpam-460	150	19	1	1	NUM
ejpam-460	150	20	1−	1−	NUM
ejpam-460	150	21	f	f	NOUN
ejpam-460	150	22	2(θ	2(θ	NUM
ejpam-460	150	23	)	)	PUNCT
ejpam-460	150	24	−	−	ADP
ejpam-460	150	25	1	1	NUM
ejpam-460	150	26	1−	1−	NUM
ejpam-460	150	27	r2	r2	PROPN
ejpam-460	150	28	1	1	NUM
ejpam-460	150	29	)	)	PUNCT
ejpam-460	150	30	dθ	dθ	NOUN
ejpam-460	150	31	,	,	PUNCT
ejpam-460	150	32	by	by	ADP
ejpam-460	150	33	the	the	DET
ejpam-460	150	34	condition	condition	NOUN
ejpam-460	150	35	|s1|hyp	|s1|hyp	PROPN
ejpam-460	150	36	≥	≥	PROPN
ejpam-460	150	37	|s2|hyp	|s2|hyp	PROPN
ejpam-460	150	38	,	,	PUNCT
ejpam-460	150	39	we	we	PRON
ejpam-460	150	40	have	have	VERB
ejpam-460	150	41	π	π	PROPN
ejpam-460	150	42	1−	1−	NUM
ejpam-460	150	43	r2	r2	PROPN
ejpam-460	150	44	2	2	NUM
ejpam-460	150	45	≥	≥	NOUN
ejpam-460	150	46	1	1	NUM
ejpam-460	150	47	2	2	NUM
ejpam-460	150	48	∫	∫	NOUN
ejpam-460	150	49	2π	2π	NOUN
ejpam-460	150	50	0	0	NUM
ejpam-460	150	51	1	1	NUM
ejpam-460	150	52	1−	1−	NUM
ejpam-460	150	53	f	f	NOUN
ejpam-460	150	54	2(θ	2(θ	NUM
ejpam-460	150	55	)	)	PUNCT
ejpam-460	151	1	dθ	dθ	PROPN
ejpam-460	151	2	.	.	PUNCT
ejpam-460	152	1	by	by	ADP
ejpam-460	152	2	lemma	lemma	PROPN
ejpam-460	152	3	3	3	NUM
ejpam-460	152	4	,	,	PUNCT
ejpam-460	152	5	we	we	PRON
ejpam-460	152	6	obtain	obtain	VERB
ejpam-460	152	7	π	π	PROPN
ejpam-460	152	8	1	1	NUM
ejpam-460	152	9	1−	1−	NUM
ejpam-460	152	10	r2	r2	PROPN
ejpam-460	152	11	2	2	NUM
ejpam-460	152	12	≥	≥	NOUN
ejpam-460	152	13	1	1	NUM
ejpam-460	152	14	2	2	NUM
ejpam-460	152	15	∫	∫	NOUN
ejpam-460	152	16	2π	2π	NOUN
ejpam-460	152	17	0	0	NUM
ejpam-460	152	18	1	1	NUM
ejpam-460	152	19	1−	1−	NUM
ejpam-460	152	20	f	f	NOUN
ejpam-460	152	21	2(θ	2(θ	NUM
ejpam-460	152	22	)	)	PUNCT
ejpam-460	153	1	dθ	dθ	PROPN
ejpam-460	153	2	≥	≥	NUM
ejpam-460	153	3	1	1	NUM
ejpam-460	153	4	2	2	NUM
ejpam-460	153	5	4π2	4π2	NUM
ejpam-460	153	6	∫	∫	PROPN
ejpam-460	153	7	2π	2π	PROPN
ejpam-460	153	8	0	0	NUM
ejpam-460	154	1	1−	1−	NUM
ejpam-460	154	2	f	f	NOUN
ejpam-460	154	3	2(θ	2(θ	NUM
ejpam-460	154	4	)	)	PUNCT
ejpam-460	155	1	dθ	dθ	PROPN
ejpam-460	155	2	,	,	PUNCT
ejpam-460	155	3	thus	thus	ADV
ejpam-460	155	4	,	,	PUNCT
ejpam-460	155	5	1	1	NUM
ejpam-460	155	6	2π	2π	NUM
ejpam-460	155	7	1	1	NUM
ejpam-460	155	8	1−	1−	NUM
ejpam-460	155	9	r2	r2	PROPN
ejpam-460	155	10	2	2	NUM
ejpam-460	155	11	≥	≥	NOUN
ejpam-460	155	12	1	1	NUM
ejpam-460	155	13	∫	∫	PROPN
ejpam-460	155	14	2π	2π	PROPN
ejpam-460	155	15	0	0	NUM
ejpam-460	155	16	1−	1−	NUM
ejpam-460	155	17	f	f	NOUN
ejpam-460	155	18	2(θ	2(θ	NUM
ejpam-460	155	19	)	)	PUNCT
ejpam-460	155	20	dθ	dθ	PROPN
ejpam-460	155	21	,	,	PUNCT
ejpam-460	155	22	by	by	ADP
ejpam-460	155	23	calculation	calculation	NOUN
ejpam-460	155	24	,	,	PUNCT
ejpam-460	155	25	we	we	PRON
ejpam-460	155	26	get	get	VERB
ejpam-460	155	27	area(s1	area(s1	ADJ
ejpam-460	155	28	∪∆r1	∪∆r1	PROPN
ejpam-460	155	29	)	)	PUNCT
ejpam-460	155	30	≥	≥	NOUN
ejpam-460	155	31	1	1	NUM
ejpam-460	155	32	2	2	NUM
ejpam-460	155	33	∫	∫	NOUN
ejpam-460	155	34	2π	2π	NOUN
ejpam-460	155	35	0	0	PUNCT
ejpam-460	156	1	f	f	X
ejpam-460	156	2	(	(	PUNCT
ejpam-460	156	3	θ	θ	NOUN
ejpam-460	156	4	)	)	PUNCT
ejpam-460	156	5	2	2	NUM
ejpam-460	156	6	dθ	dθ	NOUN
ejpam-460	156	7	=	=	PROPN
ejpam-460	156	8	area(s2	area(s2	PROPN
ejpam-460	156	9	∪∆r1	∪∆r1	PROPN
ejpam-460	156	10	)	)	PUNCT
ejpam-460	156	11	,	,	PUNCT
ejpam-460	156	12	thus	thus	ADV
ejpam-460	156	13	area(s1	area(s1	ADJ
ejpam-460	156	14	)	)	PUNCT
ejpam-460	156	15	≥	≥	NOUN
ejpam-460	156	16	area(s2	area(s2	NUM
ejpam-460	156	17	)	)	PUNCT
ejpam-460	156	18	,	,	PUNCT
ejpam-460	156	19	owing	owe	VERB
ejpam-460	156	20	to	to	ADP
ejpam-460	156	21	lemma	lemma	PROPN
ejpam-460	156	22	2	2	NUM
ejpam-460	156	23	,	,	PUNCT
ejpam-460	156	24	we	we	PRON
ejpam-460	156	25	obtain	obtain	VERB
ejpam-460	156	26	mod(s1	mod(s1	NOUN
ejpam-460	156	27	)	)	PUNCT
ejpam-460	156	28	≥mod(s2	≥mod(s2	PROPN
ejpam-460	156	29	)	)	PUNCT
ejpam-460	156	30	.	.	PUNCT
ejpam-460	157	1	next	next	ADV
ejpam-460	157	2	,	,	PUNCT
ejpam-460	157	3	we	we	PRON
ejpam-460	157	4	will	will	AUX
ejpam-460	157	5	discuss	discuss	VERB
ejpam-460	157	6	the	the	DET
ejpam-460	157	7	equality	equality	NOUN
ejpam-460	157	8	.	.	PUNCT
ejpam-460	158	1	if	if	SCONJ
ejpam-460	158	2	mod(s1	mod(s1	VERB
ejpam-460	158	3	)	)	PUNCT
ejpam-460	158	4	=	=	PUNCT
ejpam-460	159	1	mod(s2	mod(s2	PROPN
ejpam-460	159	2	)	)	PUNCT
ejpam-460	159	3	,	,	PUNCT
ejpam-460	159	4	we	we	PRON
ejpam-460	159	5	conclude	conclude	VERB
ejpam-460	159	6	that	that	SCONJ
ejpam-460	159	7	area(s1	area(s1	ADV
ejpam-460	159	8	)	)	PUNCT
ejpam-460	159	9	=	=	SYM
ejpam-460	159	10	area(s2	area(s2	X
ejpam-460	159	11	)	)	PUNCT
ejpam-460	159	12	.	.	PUNCT
ejpam-460	160	1	otherwise	otherwise	ADV
ejpam-460	160	2	,	,	PUNCT
ejpam-460	160	3	if	if	SCONJ
ejpam-460	160	4	area(s1	area(s1	ADJ
ejpam-460	160	5	)	)	PUNCT
ejpam-460	160	6	>	>	X
ejpam-460	161	1	area(s2	area(s2	PROPN
ejpam-460	161	2	)	)	PUNCT
ejpam-460	162	1	,	,	PUNCT
ejpam-460	162	2	then	then	ADV
ejpam-460	162	3	there	there	PRON
ejpam-460	162	4	exists	exist	VERB
ejpam-460	162	5	an	an	DET
ejpam-460	162	6	a(r1	a(r1	NOUN
ejpam-460	162	7	,	,	PUNCT
ejpam-460	162	8	r	r	NOUN
ejpam-460	162	9	′	′	NOUN
ejpam-460	162	10	)	)	PUNCT
ejpam-460	163	1	⊂	⊂	PROPN
ejpam-460	163	2	s1	s1	NOUN
ejpam-460	163	3	=	=	PUNCT
ejpam-460	163	4	a(r1	a(r1	ADV
ejpam-460	163	5	,	,	PUNCT
ejpam-460	163	6	r2	r2	PROPN
ejpam-460	163	7	)	)	PUNCT
ejpam-460	163	8	,	,	PUNCT
ejpam-460	164	1	r	r	NOUN
ejpam-460	164	2	′	′	NOUN
ejpam-460	164	3	<	<	X
ejpam-460	164	4	r2	r2	PROPN
ejpam-460	164	5	,	,	PUNCT
ejpam-460	164	6	such	such	ADJ
ejpam-460	164	7	that	that	SCONJ
ejpam-460	164	8	area(a(r1	area(a(r1	VERB
ejpam-460	164	9	,	,	PUNCT
ejpam-460	164	10	r	r	NOUN
ejpam-460	164	11	′	′	NUM
ejpam-460	164	12	)	)	PUNCT
ejpam-460	164	13	)	)	PUNCT
ejpam-460	165	1	=	=	PUNCT
ejpam-460	165	2	area(s2	area(s2	X
ejpam-460	165	3	)	)	PUNCT
ejpam-460	165	4	.	.	PUNCT
ejpam-460	166	1	by	by	ADP
ejpam-460	166	2	lemma	lemma	PROPN
ejpam-460	166	3	2	2	NUM
ejpam-460	166	4	,	,	PUNCT
ejpam-460	166	5	we	we	PRON
ejpam-460	166	6	get	get	VERB
ejpam-460	166	7	mod(s1)>mod(a(r1	mod(s1)>mod(a(r1	NOUN
ejpam-460	166	8	,	,	PUNCT
ejpam-460	166	9	r	r	NOUN
ejpam-460	166	10	′))≥mod(s2	′))≥mod(s2	NOUN
ejpam-460	166	11	)	)	PUNCT
ejpam-460	166	12	,	,	PUNCT
ejpam-460	166	13	x.	x.	NOUN
ejpam-460	166	14	zhichun	zhichun	PROPN
ejpam-460	166	15	and	and	CCONJ
ejpam-460	166	16	h.	h.	PROPN
ejpam-460	166	17	xinzhong	xinzhong	PROPN
ejpam-460	166	18	/	/	SYM
ejpam-460	166	19	eur	eur	PROPN
ejpam-460	166	20	.	.	PUNCT
ejpam-460	167	1	j.	j.	PROPN
ejpam-460	167	2	pure	pure	PROPN
ejpam-460	167	3	appl	appl	PROPN
ejpam-460	167	4	.	.	PROPN
ejpam-460	167	5	math	math	PROPN
ejpam-460	167	6	,	,	PUNCT
ejpam-460	167	7	2	2	NUM
ejpam-460	167	8	(	(	PUNCT
ejpam-460	167	9	2009	2009	NUM
ejpam-460	167	10	)	)	PUNCT
ejpam-460	167	11	,	,	PUNCT
ejpam-460	167	12	(	(	PUNCT
ejpam-460	167	13	532	532	NUM
ejpam-460	167	14	-	-	NUM
ejpam-460	167	15	543	543	NUM
ejpam-460	167	16	)	)	PUNCT
ejpam-460	167	17	541	541	NUM
ejpam-460	167	18	this	this	PRON
ejpam-460	167	19	is	be	AUX
ejpam-460	167	20	a	a	DET
ejpam-460	167	21	contradiction	contradiction	NOUN
ejpam-460	167	22	with	with	ADP
ejpam-460	167	23	mod(s1	mod(s1	NOUN
ejpam-460	167	24	)	)	PUNCT
ejpam-460	167	25	=	=	PUNCT
ejpam-460	167	26	mod(s2	mod(s2	NOUN
ejpam-460	167	27	)	)	PUNCT
ejpam-460	167	28	.	.	PUNCT
ejpam-460	168	1	therefore	therefore	ADV
ejpam-460	168	2	,	,	PUNCT
ejpam-460	168	3	if	if	SCONJ
ejpam-460	168	4	mod(s1	mod(s1	VERB
ejpam-460	168	5	)	)	PUNCT
ejpam-460	168	6	=	=	NOUN
ejpam-460	168	7	mod(s2	mod(s2	X
ejpam-460	168	8	)	)	PUNCT
ejpam-460	168	9	,	,	PUNCT
ejpam-460	168	10	by	by	ADP
ejpam-460	168	11	lemma	lemma	PROPN
ejpam-460	168	12	3	3	NUM
ejpam-460	168	13	and	and	CCONJ
ejpam-460	168	14	the	the	DET
ejpam-460	168	15	proof	proof	NOUN
ejpam-460	168	16	above	above	ADV
ejpam-460	168	17	,	,	PUNCT
ejpam-460	168	18	we	we	PRON
ejpam-460	168	19	have	have	VERB
ejpam-460	168	20			PROPN
ejpam-460	168	21			PRON
ejpam-460	168	22			NOUN
ejpam-460	168	23	|s1|hyp	|s1|hyp	PROPN
ejpam-460	168	24	=	=	SYM
ejpam-460	168	25	|s2|hyp	|s2|hyp	PROPN
ejpam-460	168	26	,	,	PUNCT
ejpam-460	168	27	f	f	PROPN
ejpam-460	168	28	(	(	PUNCT
ejpam-460	168	29	θ	θ	PROPN
ejpam-460	168	30	)	)	PUNCT
ejpam-460	169	1	≡	≡	PROPN
ejpam-460	169	2	c	c	NOUN
ejpam-460	170	1	=	=	PUNCT
ejpam-460	170	2	r2	r2	PROPN
ejpam-460	170	3	.	.	PUNCT
ejpam-460	171	1	⇔	⇔	PROPN
ejpam-460	171	2	s1	s1	PROPN
ejpam-460	171	3	=	=	SYM
ejpam-460	171	4	s2	s2	PROPN
ejpam-460	171	5	.	.	PUNCT
ejpam-460	172	1	the	the	DET
ejpam-460	172	2	proof	proof	NOUN
ejpam-460	172	3	is	be	AUX
ejpam-460	172	4	complete	complete	ADJ
ejpam-460	172	5	.	.	PUNCT
ejpam-460	173	1	we	we	PRON
ejpam-460	173	2	have	have	AUX
ejpam-460	173	3	pointed	point	VERB
ejpam-460	173	4	out	out	ADP
ejpam-460	173	5	that	that	SCONJ
ejpam-460	173	6	the	the	DET
ejpam-460	173	7	area	area	NOUN
ejpam-460	173	8	distortion	distortion	NOUN
ejpam-460	173	9	used	use	VERB
ejpam-460	173	10	in	in	ADP
ejpam-460	173	11	[	[	X
ejpam-460	173	12	3	3	NUM
ejpam-460	173	13	,	,	PUNCT
ejpam-460	173	14	theorem	theorem	VERB
ejpam-460	173	15	2.1	2.1	NUM
ejpam-460	173	16	]	]	PUNCT
ejpam-460	173	17	could	could	AUX
ejpam-460	173	18	not	not	PART
ejpam-460	173	19	characterize	characterize	VERB
ejpam-460	173	20	extremal	extremal	ADJ
ejpam-460	173	21	quasiconformal	quasiconformal	ADJ
ejpam-460	173	22	mapping	mapping	NOUN
ejpam-460	173	23	.	.	PUNCT
ejpam-460	174	1	we	we	PRON
ejpam-460	174	2	will	will	AUX
ejpam-460	174	3	prove	prove	VERB
ejpam-460	174	4	the	the	DET
ejpam-460	174	5	following	follow	VERB
ejpam-460	174	6	theorem	theorem	NOUN
ejpam-460	174	7	5	5	NUM
ejpam-460	174	8	.	.	PUNCT
ejpam-460	175	1	let	let	VERB
ejpam-460	175	2	w	w	NOUN
ejpam-460	175	3	=	=	SYM
ejpam-460	175	4	f	f	X
ejpam-460	175	5	(	(	PUNCT
ejpam-460	175	6	z	z	NOUN
ejpam-460	175	7	)	)	PUNCT
ejpam-460	175	8	be	be	AUX
ejpam-460	175	9	a	a	DET
ejpam-460	175	10	quasiconformal	quasiconformal	ADJ
ejpam-460	175	11	mapping	mapping	NOUN
ejpam-460	175	12	from	from	ADP
ejpam-460	175	13	s	s	NOUN
ejpam-460	175	14	=	=	PUNCT
ejpam-460	175	15	{	{	PUNCT
ejpam-460	175	16	z|r1	z|r1	PROPN
ejpam-460	175	17	≤	≤	NUM
ejpam-460	175	18	|z|	|z|	VERB
ejpam-460	175	19	≤	≤	NUM
ejpam-460	175	20	r2	r2	NOUN
ejpam-460	175	21	,	,	PUNCT
ejpam-460	176	1	0	0	NUM
ejpam-460	176	2	<	<	X
ejpam-460	176	3	r1	r1	NOUN
ejpam-460	176	4	<	<	X
ejpam-460	176	5	r2	r2	PROPN
ejpam-460	176	6	<	<	X
ejpam-460	176	7	1	1	NUM
ejpam-460	176	8	}	}	PUNCT
ejpam-460	176	9	onto	onto	ADP
ejpam-460	176	10	s′	s′	NOUN
ejpam-460	176	11	,	,	PUNCT
ejpam-460	176	12	where	where	SCONJ
ejpam-460	176	13	s′	s′	ADJ
ejpam-460	176	14	⊂	⊂	PROPN
ejpam-460	176	15	d	d	X
ejpam-460	176	16	=	=	PUNCT
ejpam-460	176	17	{	{	PUNCT
ejpam-460	176	18	w||w|	w||w|	PROPN
ejpam-460	176	19	<	<	X
ejpam-460	176	20	1	1	NUM
ejpam-460	176	21	}	}	PUNCT
ejpam-460	176	22	is	be	AUX
ejpam-460	176	23	a	a	DET
ejpam-460	176	24	ring	ring	NOUN
ejpam-460	176	25	domain	domain	NOUN
ejpam-460	176	26	bounded	bound	VERB
ejpam-460	176	27	by	by	ADP
ejpam-460	176	28	two	two	NUM
ejpam-460	176	29	mutually	mutually	ADV
ejpam-460	176	30	disjoint	disjoint	VERB
ejpam-460	176	31	closed	closed	ADJ
ejpam-460	176	32	curves	curve	NOUN
ejpam-460	176	33	γ1	γ1	NOUN
ejpam-460	176	34	=	=	SYM
ejpam-460	176	35	{	{	PUNCT
ejpam-460	176	36	w||w|	w||w|	PROPN
ejpam-460	176	37	=	=	SYM
ejpam-460	176	38	r	r	NOUN
ejpam-460	176	39	1	1	NUM
ejpam-460	176	40	k	k	NOUN
ejpam-460	176	41	1	1	NUM
ejpam-460	176	42	}	}	PUNCT
ejpam-460	176	43	and	and	CCONJ
ejpam-460	176	44	γ2	γ2	PROPN
ejpam-460	176	45	=	=	SYM
ejpam-460	176	46	{	{	PUNCT
ejpam-460	176	47	w|w	w|w	NOUN
ejpam-460	176	48	=	=	SYM
ejpam-460	176	49	f	f	PROPN
ejpam-460	176	50	(	(	PUNCT
ejpam-460	176	51	r2eiθ	r2eiθ	PROPN
ejpam-460	176	52	)	)	PUNCT
ejpam-460	176	53	,	,	PUNCT
ejpam-460	176	54	0	0	NUM
ejpam-460	176	55	≤	≤	NUM
ejpam-460	176	56	θ	θ	NOUN
ejpam-460	176	57	≤	≤	ADJ
ejpam-460	176	58	2π	2π	NOUN
ejpam-460	176	59	}	}	PUNCT
ejpam-460	176	60	.	.	PUNCT
ejpam-460	177	1	if	if	SCONJ
ejpam-460	177	2	f	f	PROPN
ejpam-460	177	3	(	(	PUNCT
ejpam-460	177	4	z	z	NOUN
ejpam-460	177	5	)	)	PUNCT
ejpam-460	177	6	satisfies	satisfie	NOUN
ejpam-460	177	7	|	|	ADV
ejpam-460	177	8	f	f	PROPN
ejpam-460	177	9	(	(	PUNCT
ejpam-460	177	10	s)|hyp	s)|hyp	PROPN
ejpam-460	177	11	|s|hyp	|s|hyp	PROPN
ejpam-460	177	12	=	=	PRON
ejpam-460	177	13	(	(	PUNCT
ejpam-460	177	14	r	r	NOUN
ejpam-460	177	15	2	2	NUM
ejpam-460	177	16	k	k	NOUN
ejpam-460	177	17	2	2	NUM
ejpam-460	177	18	−	−	NOUN
ejpam-460	177	19	r	r	NOUN
ejpam-460	177	20	2	2	NUM
ejpam-460	177	21	k	k	NOUN
ejpam-460	177	22	1	1	NUM
ejpam-460	177	23	)	)	PUNCT
ejpam-460	177	24	(	(	PUNCT
ejpam-460	177	25	1−	1−	NUM
ejpam-460	177	26	r2	r2	PROPN
ejpam-460	177	27	2	2	NUM
ejpam-460	177	28	)	)	PUNCT
ejpam-460	177	29	(	(	PUNCT
ejpam-460	177	30	1−	1−	NUM
ejpam-460	177	31	r2	r2	PROPN
ejpam-460	177	32	1	1	NUM
ejpam-460	177	33	)	)	PUNCT
ejpam-460	177	34	(	(	PUNCT
ejpam-460	177	35	r2	r2	PROPN
ejpam-460	177	36	2−	2−	NUM
ejpam-460	177	37	r2	r2	NOUN
ejpam-460	177	38	1)(1−	1)(1−	NUM
ejpam-460	177	39	r	r	NOUN
ejpam-460	177	40	2	2	NUM
ejpam-460	177	41	k	k	NOUN
ejpam-460	177	42	2	2	NUM
ejpam-460	177	43	)	)	PUNCT
ejpam-460	177	44	(	(	PUNCT
ejpam-460	177	45	1−	1−	NUM
ejpam-460	177	46	r	r	NOUN
ejpam-460	177	47	2	2	NUM
ejpam-460	177	48	k	k	NOUN
ejpam-460	177	49	1	1	NUM
ejpam-460	177	50	)	)	PUNCT
ejpam-460	177	51	,	,	PUNCT
ejpam-460	177	52	and	and	CCONJ
ejpam-460	177	53	k	k	X
ejpam-460	177	54	[	[	X
ejpam-460	177	55	f	f	X
ejpam-460	177	56	]	]	PUNCT
ejpam-460	177	57	=	=	SYM
ejpam-460	177	58	k	k	NOUN
ejpam-460	177	59	,	,	PUNCT
ejpam-460	177	60	then	then	ADV
ejpam-460	177	61	f	f	X
ejpam-460	177	62	(	(	PUNCT
ejpam-460	177	63	z	z	NOUN
ejpam-460	177	64	)	)	PUNCT
ejpam-460	177	65	is	be	AUX
ejpam-460	177	66	an	an	DET
ejpam-460	177	67	extremal	extremal	ADJ
ejpam-460	177	68	quasiconformal	quasiconformal	ADJ
ejpam-460	177	69	mapping	mapping	NOUN
ejpam-460	177	70	from	from	ADP
ejpam-460	177	71	s	s	PRON
ejpam-460	177	72	onto	onto	ADP
ejpam-460	177	73	s′	s′	NUM
ejpam-460	177	74	,	,	PUNCT
ejpam-460	177	75	and	and	CCONJ
ejpam-460	177	76	f	f	PROPN
ejpam-460	177	77	(	(	PUNCT
ejpam-460	177	78	z	z	NOUN
ejpam-460	177	79	)	)	PUNCT
ejpam-460	178	1	=	=	SYM
ejpam-460	178	2	r	r	NOUN
ejpam-460	178	3	1	1	NUM
ejpam-460	178	4	k	k	NOUN
ejpam-460	178	5	ei(θ+α	ei(θ+α	PROPN
ejpam-460	178	6	)	)	PUNCT
ejpam-460	178	7	,	,	PUNCT
ejpam-460	178	8	z	z	X
ejpam-460	178	9	=	=	PUNCT
ejpam-460	178	10	reiθ	reiθ	PROPN
ejpam-460	178	11	,	,	PUNCT
ejpam-460	178	12	where	where	SCONJ
ejpam-460	178	13	α	α	NOUN
ejpam-460	178	14	is	be	AUX
ejpam-460	178	15	a	a	DET
ejpam-460	178	16	constant	constant	ADJ
ejpam-460	178	17	.	.	PUNCT
ejpam-460	179	1	proof	proof	NOUN
ejpam-460	179	2	.	.	PUNCT
ejpam-460	180	1	let	let	VERB
ejpam-460	180	2	s′′	s′′	PROPN
ejpam-460	180	3	=	=	SYM
ejpam-460	180	4	a(r	a(r	PROPN
ejpam-460	180	5	1	1	NUM
ejpam-460	180	6	k	k	PROPN
ejpam-460	180	7	1	1	NUM
ejpam-460	180	8	,	,	PUNCT
ejpam-460	180	9	r	r	NOUN
ejpam-460	180	10	1	1	NUM
ejpam-460	180	11	k	k	NOUN
ejpam-460	180	12	2	2	NUM
ejpam-460	180	13	)	)	PUNCT
ejpam-460	180	14	,	,	PUNCT
ejpam-460	180	15	by	by	ADP
ejpam-460	180	16	calculation	calculation	NOUN
ejpam-460	180	17	,	,	PUNCT
ejpam-460	180	18	we	we	PRON
ejpam-460	180	19	have	have	VERB
ejpam-460	180	20	|s|hyp	|s|hyp	PROPN
ejpam-460	180	21	=	=	SYM
ejpam-460	180	22	π	π	NOUN
ejpam-460	180	23	r2	r2	NOUN
ejpam-460	180	24	2	2	NUM
ejpam-460	180	25	−	−	PROPN
ejpam-460	180	26	r2	r2	PROPN
ejpam-460	180	27	1	1	NUM
ejpam-460	180	28	(	(	PUNCT
ejpam-460	180	29	1−	1−	NUM
ejpam-460	180	30	r2	r2	PROPN
ejpam-460	180	31	2)(1−	2)(1−	NUM
ejpam-460	180	32	r2	r2	PROPN
ejpam-460	180	33	1	1	NUM
ejpam-460	180	34	)	)	PUNCT
ejpam-460	180	35	,	,	PUNCT
ejpam-460	180	36	|s′′|hyp	|s′′|hyp	PROPN
ejpam-460	181	1	=	=	PUNCT
ejpam-460	181	2	π	π	NOUN
ejpam-460	181	3	r	r	NOUN
ejpam-460	181	4	2	2	NUM
ejpam-460	181	5	k	k	NOUN
ejpam-460	181	6	2	2	NUM
ejpam-460	181	7	−	−	NOUN
ejpam-460	181	8	r	r	NOUN
ejpam-460	181	9	2	2	NUM
ejpam-460	181	10	k	k	NOUN
ejpam-460	181	11	1	1	NUM
ejpam-460	181	12	(	(	PUNCT
ejpam-460	181	13	1−	1−	NUM
ejpam-460	181	14	r	r	NOUN
ejpam-460	181	15	2	2	NUM
ejpam-460	181	16	k	k	NOUN
ejpam-460	181	17	2	2	NUM
ejpam-460	181	18	)	)	PUNCT
ejpam-460	181	19	(	(	PUNCT
ejpam-460	181	20	1−	1−	NUM
ejpam-460	181	21	r	r	NOUN
ejpam-460	181	22	2	2	NUM
ejpam-460	181	23	k	k	NOUN
ejpam-460	181	24	1	1	NUM
ejpam-460	181	25	)	)	PUNCT
ejpam-460	181	26	,	,	PUNCT
ejpam-460	181	27	by	by	ADP
ejpam-460	181	28	the	the	DET
ejpam-460	181	29	hypothesis	hypothesis	NOUN
ejpam-460	181	30	that	that	PRON
ejpam-460	182	1	|	|	ADV
ejpam-460	182	2	f	f	X
ejpam-460	182	3	(	(	PUNCT
ejpam-460	182	4	s)|hyp	s)|hyp	PROPN
ejpam-460	182	5	|s|hyp	|s|hyp	PROPN
ejpam-460	182	6	=	=	PRON
ejpam-460	182	7	(	(	PUNCT
ejpam-460	182	8	r	r	NOUN
ejpam-460	182	9	2	2	NUM
ejpam-460	182	10	k	k	NOUN
ejpam-460	182	11	2	2	NUM
ejpam-460	182	12	−	−	NOUN
ejpam-460	182	13	r	r	NOUN
ejpam-460	182	14	2	2	NUM
ejpam-460	182	15	k	k	NOUN
ejpam-460	182	16	1	1	NUM
ejpam-460	182	17	)	)	PUNCT
ejpam-460	182	18	(	(	PUNCT
ejpam-460	182	19	1−	1−	NUM
ejpam-460	182	20	r2	r2	PROPN
ejpam-460	182	21	2	2	NUM
ejpam-460	182	22	)	)	PUNCT
ejpam-460	182	23	(	(	PUNCT
ejpam-460	182	24	1−	1−	NUM
ejpam-460	182	25	r2	r2	PROPN
ejpam-460	182	26	1	1	NUM
ejpam-460	182	27	)	)	PUNCT
ejpam-460	182	28	(	(	PUNCT
ejpam-460	182	29	r2	r2	PROPN
ejpam-460	182	30	2−	2−	NUM
ejpam-460	182	31	r2	r2	NOUN
ejpam-460	182	32	1)(1−	1)(1−	NUM
ejpam-460	182	33	r	r	NOUN
ejpam-460	182	34	2	2	NUM
ejpam-460	182	35	k	k	NOUN
ejpam-460	182	36	2	2	NUM
ejpam-460	182	37	)	)	PUNCT
ejpam-460	182	38	(	(	PUNCT
ejpam-460	182	39	1−	1−	NUM
ejpam-460	182	40	r	r	NOUN
ejpam-460	182	41	2	2	NUM
ejpam-460	182	42	k	k	NOUN
ejpam-460	182	43	1	1	NUM
ejpam-460	182	44	)	)	PUNCT
ejpam-460	182	45	,	,	PUNCT
ejpam-460	182	46	x.	x.	NOUN
ejpam-460	182	47	zhichun	zhichun	PROPN
ejpam-460	182	48	and	and	CCONJ
ejpam-460	182	49	h.	h.	PROPN
ejpam-460	182	50	xinzhong	xinzhong	PROPN
ejpam-460	182	51	/	/	SYM
ejpam-460	182	52	eur	eur	PROPN
ejpam-460	182	53	.	.	PUNCT
ejpam-460	183	1	j.	j.	PROPN
ejpam-460	183	2	pure	pure	PROPN
ejpam-460	183	3	appl	appl	PROPN
ejpam-460	183	4	.	.	PROPN
ejpam-460	183	5	math	math	PROPN
ejpam-460	183	6	,	,	PUNCT
ejpam-460	183	7	2	2	NUM
ejpam-460	183	8	(	(	PUNCT
ejpam-460	183	9	2009	2009	NUM
ejpam-460	183	10	)	)	PUNCT
ejpam-460	183	11	,	,	PUNCT
ejpam-460	183	12	(	(	PUNCT
ejpam-460	183	13	532	532	NUM
ejpam-460	183	14	-	-	NUM
ejpam-460	183	15	543	543	NUM
ejpam-460	183	16	)	)	PUNCT
ejpam-460	183	17	542	542	NUM
ejpam-460	183	18	we	we	PRON
ejpam-460	183	19	have	have	VERB
ejpam-460	183	20	|	|	ADV
ejpam-460	183	21	f	f	X
ejpam-460	183	22	(	(	PUNCT
ejpam-460	183	23	s)|hyp	s)|hyp	PROPN
ejpam-460	183	24	=	=	SYM
ejpam-460	183	25	|s|hyp	|s|hyp	PROPN
ejpam-460	184	1	(	(	PUNCT
ejpam-460	184	2	r	r	NOUN
ejpam-460	184	3	2	2	NUM
ejpam-460	184	4	k	k	NOUN
ejpam-460	184	5	2	2	NUM
ejpam-460	184	6	−	−	NOUN
ejpam-460	184	7	r	r	NOUN
ejpam-460	184	8	2	2	NUM
ejpam-460	184	9	k	k	NOUN
ejpam-460	184	10	1	1	NUM
ejpam-460	184	11	)	)	PUNCT
ejpam-460	184	12	(	(	PUNCT
ejpam-460	184	13	1−	1−	NUM
ejpam-460	184	14	r2	r2	PROPN
ejpam-460	184	15	2	2	NUM
ejpam-460	184	16	)	)	PUNCT
ejpam-460	184	17	(	(	PUNCT
ejpam-460	184	18	1−	1−	NUM
ejpam-460	184	19	r2	r2	PROPN
ejpam-460	184	20	1	1	NUM
ejpam-460	184	21	)	)	PUNCT
ejpam-460	184	22	(	(	PUNCT
ejpam-460	184	23	r2	r2	PROPN
ejpam-460	184	24	2	2	NUM
ejpam-460	184	25	−	−	PROPN
ejpam-460	184	26	r2	r2	NOUN
ejpam-460	184	27	1)(1−	1)(1−	NUM
ejpam-460	184	28	r	r	NOUN
ejpam-460	184	29	2	2	NUM
ejpam-460	184	30	k	k	NOUN
ejpam-460	184	31	2	2	NUM
ejpam-460	184	32	)	)	PUNCT
ejpam-460	184	33	(	(	PUNCT
ejpam-460	184	34	1−	1−	NUM
ejpam-460	184	35	r	r	NOUN
ejpam-460	184	36	2	2	NUM
ejpam-460	184	37	k	k	NOUN
ejpam-460	184	38	1	1	NUM
ejpam-460	184	39	)	)	PUNCT
ejpam-460	184	40	=	=	PUNCT
ejpam-460	185	1	π	π	X
ejpam-460	185	2	r2	r2	NOUN
ejpam-460	185	3	2	2	NUM
ejpam-460	185	4	−	−	PROPN
ejpam-460	185	5	r2	r2	PROPN
ejpam-460	185	6	1	1	NUM
ejpam-460	185	7	(	(	PUNCT
ejpam-460	185	8	1−	1−	NUM
ejpam-460	185	9	r2	r2	PROPN
ejpam-460	185	10	2)(1−	2)(1−	NUM
ejpam-460	185	11	r2	r2	PROPN
ejpam-460	185	12	1	1	NUM
ejpam-460	185	13	)	)	PUNCT
ejpam-460	185	14	·	·	PUNCT
ejpam-460	186	1	(	(	PUNCT
ejpam-460	186	2	r	r	NOUN
ejpam-460	186	3	2	2	NUM
ejpam-460	186	4	k	k	NOUN
ejpam-460	186	5	2	2	NUM
ejpam-460	186	6	−	−	NOUN
ejpam-460	186	7	r	r	NOUN
ejpam-460	186	8	2	2	NUM
ejpam-460	186	9	k	k	NOUN
ejpam-460	186	10	1	1	NUM
ejpam-460	186	11	)	)	PUNCT
ejpam-460	186	12	(	(	PUNCT
ejpam-460	186	13	1−	1−	NUM
ejpam-460	186	14	r2	r2	PROPN
ejpam-460	186	15	2	2	NUM
ejpam-460	186	16	)	)	PUNCT
ejpam-460	186	17	(	(	PUNCT
ejpam-460	186	18	1−	1−	NUM
ejpam-460	186	19	r2	r2	PROPN
ejpam-460	186	20	1	1	NUM
ejpam-460	186	21	)	)	PUNCT
ejpam-460	186	22	(	(	PUNCT
ejpam-460	186	23	r2	r2	PROPN
ejpam-460	186	24	2−	2−	NUM
ejpam-460	186	25	r2	r2	NOUN
ejpam-460	186	26	1)(1−	1)(1−	NUM
ejpam-460	186	27	r	r	NOUN
ejpam-460	186	28	2	2	NUM
ejpam-460	186	29	k	k	NOUN
ejpam-460	186	30	2	2	NUM
ejpam-460	186	31	)	)	PUNCT
ejpam-460	186	32	(	(	PUNCT
ejpam-460	186	33	1−	1−	NUM
ejpam-460	186	34	r	r	NOUN
ejpam-460	186	35	2	2	NUM
ejpam-460	186	36	k	k	NOUN
ejpam-460	186	37	1	1	NUM
ejpam-460	186	38	)	)	PUNCT
ejpam-460	186	39	=	=	PUNCT
ejpam-460	187	1	π	π	NOUN
ejpam-460	187	2	r	r	NOUN
ejpam-460	187	3	2	2	NUM
ejpam-460	187	4	k	k	NOUN
ejpam-460	187	5	2	2	NUM
ejpam-460	187	6	−	−	NOUN
ejpam-460	187	7	r	r	NOUN
ejpam-460	187	8	2	2	NUM
ejpam-460	187	9	k	k	NOUN
ejpam-460	187	10	1	1	NUM
ejpam-460	187	11	(	(	PUNCT
ejpam-460	187	12	1−	1−	NUM
ejpam-460	187	13	r	r	NOUN
ejpam-460	187	14	2	2	NUM
ejpam-460	187	15	k	k	NOUN
ejpam-460	187	16	2	2	NUM
ejpam-460	187	17	)	)	PUNCT
ejpam-460	187	18	(	(	PUNCT
ejpam-460	187	19	1−	1−	NUM
ejpam-460	187	20	r	r	NOUN
ejpam-460	187	21	2	2	NUM
ejpam-460	187	22	k	k	NOUN
ejpam-460	187	23	1	1	NUM
ejpam-460	187	24	)	)	PUNCT
ejpam-460	187	25	=	=	SYM
ejpam-460	187	26	|s′′|hyp	|s′′|hyp	PROPN
ejpam-460	187	27	.	.	PUNCT
ejpam-460	187	28	using	use	VERB
ejpam-460	187	29	theorem	theorem	NOUN
ejpam-460	187	30	4	4	NUM
ejpam-460	187	31	,	,	PUNCT
ejpam-460	187	32	we	we	PRON
ejpam-460	187	33	get	get	VERB
ejpam-460	187	34	mod(s′′	mod(s′′	PROPN
ejpam-460	187	35	)	)	PUNCT
ejpam-460	187	36	≥mod	≥mod	PROPN
ejpam-460	187	37	(	(	PUNCT
ejpam-460	187	38	f	f	X
ejpam-460	187	39	(	(	PUNCT
ejpam-460	187	40	s	s	NOUN
ejpam-460	187	41	)	)	PUNCT
ejpam-460	187	42	)	)	PUNCT
ejpam-460	187	43	,	,	PUNCT
ejpam-460	187	44	and	and	CCONJ
ejpam-460	187	45	by	by	ADP
ejpam-460	187	46	the	the	DET
ejpam-460	187	47	quasiconformallity	quasiconformallity	NOUN
ejpam-460	187	48	of	of	ADP
ejpam-460	187	49	f	f	PROPN
ejpam-460	187	50	(	(	PUNCT
ejpam-460	187	51	z	z	NOUN
ejpam-460	187	52	)	)	PUNCT
ejpam-460	187	53	,	,	PUNCT
ejpam-460	187	54	we	we	PRON
ejpam-460	187	55	have	have	VERB
ejpam-460	187	56	mod	mod	PROPN
ejpam-460	187	57	(	(	PUNCT
ejpam-460	187	58	f	f	PROPN
ejpam-460	187	59	(	(	PUNCT
ejpam-460	187	60	s	s	NOUN
ejpam-460	187	61	)	)	PUNCT
ejpam-460	187	62	)	)	PUNCT
ejpam-460	187	63	≥	≥	NOUN
ejpam-460	188	1	1	1	NUM
ejpam-460	188	2	k	k	NOUN
ejpam-460	188	3	mod(s	mod(s	PROPN
ejpam-460	188	4	)	)	PUNCT
ejpam-460	188	5	,	,	PUNCT
ejpam-460	188	6	thus	thus	ADV
ejpam-460	188	7	,	,	PUNCT
ejpam-460	188	8	mod	mod	PROPN
ejpam-460	188	9	(	(	PUNCT
ejpam-460	188	10	f	f	PROPN
ejpam-460	188	11	(	(	PUNCT
ejpam-460	188	12	s	s	NOUN
ejpam-460	188	13	)	)	PUNCT
ejpam-460	188	14	)	)	PUNCT
ejpam-460	189	1	=	=	SYM
ejpam-460	189	2	1	1	NUM
ejpam-460	189	3	k	k	NOUN
ejpam-460	189	4	mod(s	mod(s	PROPN
ejpam-460	189	5	)	)	PUNCT
ejpam-460	189	6	=	=	SYM
ejpam-460	189	7	mod(s′′	mod(s′′	PROPN
ejpam-460	189	8	)	)	PUNCT
ejpam-460	189	9	,	,	PUNCT
ejpam-460	189	10	again	again	ADV
ejpam-460	189	11	by	by	ADP
ejpam-460	189	12	theorem	theorem	NOUN
ejpam-460	189	13	4	4	NUM
ejpam-460	189	14	,	,	PUNCT
ejpam-460	189	15	we	we	PRON
ejpam-460	189	16	obtain	obtain	VERB
ejpam-460	189	17	f	f	PROPN
ejpam-460	189	18	(	(	PUNCT
ejpam-460	189	19	s	s	X
ejpam-460	189	20	)	)	PUNCT
ejpam-460	189	21	=	=	SYM
ejpam-460	190	1	s′′	s′′	PROPN
ejpam-460	190	2	=	=	PUNCT
ejpam-460	190	3	a(r	a(r	PROPN
ejpam-460	190	4	1	1	NUM
ejpam-460	190	5	k	k	PROPN
ejpam-460	190	6	1	1	NUM
ejpam-460	190	7	,	,	PUNCT
ejpam-460	190	8	r	r	NOUN
ejpam-460	190	9	1	1	NUM
ejpam-460	190	10	k	k	NOUN
ejpam-460	190	11	2	2	NUM
ejpam-460	190	12	)	)	PUNCT
ejpam-460	190	13	.	.	PUNCT
ejpam-460	191	1	therefore	therefore	ADV
ejpam-460	191	2	,	,	PUNCT
ejpam-460	191	3	as	as	SCONJ
ejpam-460	191	4	it	it	PRON
ejpam-460	191	5	is	be	AUX
ejpam-460	191	6	proved	prove	VERB
ejpam-460	191	7	in	in	ADP
ejpam-460	191	8	theorem	theorem	NOUN
ejpam-460	191	9	2	2	NUM
ejpam-460	191	10	,	,	PUNCT
ejpam-460	191	11	we	we	PRON
ejpam-460	191	12	have	have	VERB
ejpam-460	191	13	f	f	PROPN
ejpam-460	191	14	(	(	PUNCT
ejpam-460	191	15	z	z	NOUN
ejpam-460	191	16	)	)	PUNCT
ejpam-460	191	17	=	=	SYM
ejpam-460	192	1	r	r	NOUN
ejpam-460	192	2	1	1	NUM
ejpam-460	192	3	k	k	NOUN
ejpam-460	192	4	ei(θ+α	ei(θ+α	PROPN
ejpam-460	192	5	)	)	PUNCT
ejpam-460	192	6	,	,	PUNCT
ejpam-460	192	7	z	z	X
ejpam-460	193	1	=	=	PUNCT
ejpam-460	193	2	reiθ	reiθ	PROPN
ejpam-460	193	3	,	,	PUNCT
ejpam-460	193	4	where	where	SCONJ
ejpam-460	193	5	α	α	NOUN
ejpam-460	193	6	is	be	AUX
ejpam-460	193	7	a	a	DET
ejpam-460	193	8	constant	constant	ADJ
ejpam-460	193	9	.	.	PUNCT
ejpam-460	194	1	using	use	VERB
ejpam-460	194	2	theorem	theorem	NOUN
ejpam-460	194	3	5	5	NUM
ejpam-460	194	4	,	,	PUNCT
ejpam-460	194	5	we	we	PRON
ejpam-460	194	6	have	have	VERB
ejpam-460	194	7	the	the	DET
ejpam-460	194	8	following	follow	VERB
ejpam-460	194	9	corollary	corollary	NOUN
ejpam-460	194	10	2	2	NUM
ejpam-460	194	11	.	.	PUNCT
ejpam-460	195	1	let	let	VERB
ejpam-460	195	2	w	w	NOUN
ejpam-460	195	3	=	=	SYM
ejpam-460	195	4	f	f	X
ejpam-460	195	5	(	(	PUNCT
ejpam-460	195	6	z	z	NOUN
ejpam-460	195	7	)	)	PUNCT
ejpam-460	195	8	be	be	AUX
ejpam-460	195	9	a	a	DET
ejpam-460	195	10	quasiconformal	quasiconformal	ADJ
ejpam-460	195	11	mapping	mapping	NOUN
ejpam-460	195	12	from	from	ADP
ejpam-460	195	13	s	s	PRON
ejpam-460	195	14	onto	onto	ADP
ejpam-460	195	15	s′	s′	ADJ
ejpam-460	195	16	=	=	PUNCT
ejpam-460	195	17	{	{	PUNCT
ejpam-460	195	18	w|rk	w|rk	NOUN
ejpam-460	195	19	1	1	NUM
ejpam-460	195	20	≤	≤	NOUN
ejpam-460	195	21	|w|	|w|	VERB
ejpam-460	195	22	≤	≤	NUM
ejpam-460	195	23	rk	rk	NOUN
ejpam-460	195	24	2	2	NUM
ejpam-460	195	25	,	,	PUNCT
ejpam-460	195	26	0	0	PUNCT
ejpam-460	195	27	<	<	X
ejpam-460	195	28	r1	r1	PROPN
ejpam-460	195	29	<	<	X
ejpam-460	195	30	r2	r2	PROPN
ejpam-460	195	31	<	<	X
ejpam-460	195	32	1	1	NUM
ejpam-460	195	33	}	}	PUNCT
ejpam-460	195	34	,	,	PUNCT
ejpam-460	195	35	where	where	SCONJ
ejpam-460	195	36	s	s	VERB
ejpam-460	195	37	⊂	⊂	PROPN
ejpam-460	195	38	d	d	X
ejpam-460	195	39	=	=	PUNCT
ejpam-460	195	40	{	{	PUNCT
ejpam-460	195	41	z||z|	z||z|	PROPN
ejpam-460	195	42	<	<	X
ejpam-460	195	43	1	1	NUM
ejpam-460	195	44	}	}	PUNCT
ejpam-460	195	45	is	be	AUX
ejpam-460	195	46	a	a	DET
ejpam-460	195	47	ring	ring	NOUN
ejpam-460	195	48	domain	domain	NOUN
ejpam-460	195	49	bounded	bound	VERB
ejpam-460	195	50	by	by	ADP
ejpam-460	195	51	two	two	NUM
ejpam-460	195	52	mutually	mutually	ADV
ejpam-460	195	53	disjoint	disjoint	VERB
ejpam-460	195	54	closed	closed	ADJ
ejpam-460	195	55	curves	curve	NOUN
ejpam-460	195	56	γ1	γ1	NOUN
ejpam-460	195	57	=	=	SYM
ejpam-460	195	58	{	{	PUNCT
ejpam-460	195	59	z||z|	z||z|	PROPN
ejpam-460	195	60	=	=	SYM
ejpam-460	195	61	r1	r1	PROPN
ejpam-460	195	62	}	}	PUNCT
ejpam-460	195	63	and	and	CCONJ
ejpam-460	195	64	γ2	γ2	PROPN
ejpam-460	195	65	=	=	SYM
ejpam-460	195	66	{	{	PUNCT
ejpam-460	195	67	z||	z||	PROPN
ejpam-460	195	68	f	f	PROPN
ejpam-460	195	69	(	(	PUNCT
ejpam-460	195	70	z)|	z)|	VERB
ejpam-460	195	71	=	=	PRON
ejpam-460	195	72	rk	rk	NOUN
ejpam-460	195	73	2	2	NUM
ejpam-460	195	74	}	}	PUNCT
ejpam-460	195	75	.	.	PUNCT
ejpam-460	196	1	if	if	SCONJ
ejpam-460	196	2	f	f	PROPN
ejpam-460	196	3	(	(	PUNCT
ejpam-460	196	4	z	z	NOUN
ejpam-460	196	5	)	)	PUNCT
ejpam-460	196	6	satisfies	satisfie	NOUN
ejpam-460	196	7	|s|hyp	|s|hyp	ADV
ejpam-460	196	8	|	|	ADV
ejpam-460	196	9	f	f	X
ejpam-460	196	10	(	(	PUNCT
ejpam-460	196	11	s)|hyp	s)|hyp	PROPN
ejpam-460	196	12	=	=	SYM
ejpam-460	196	13	(	(	PUNCT
ejpam-460	196	14	r2	r2	PROPN
ejpam-460	196	15	2	2	NUM
ejpam-460	196	16	−	−	PROPN
ejpam-460	197	1	r2	r2	PROPN
ejpam-460	198	1	1	1	NUM
ejpam-460	199	1	)	)	PUNCT
ejpam-460	199	2	(	(	PUNCT
ejpam-460	199	3	1−	1−	NUM
ejpam-460	199	4	r2k	r2k	NOUN
ejpam-460	199	5	2	2	NUM
ejpam-460	199	6	)	)	PUNCT
ejpam-460	199	7	(	(	PUNCT
ejpam-460	199	8	1−	1−	NUM
ejpam-460	199	9	r2k	r2k	NOUN
ejpam-460	199	10	1	1	NUM
ejpam-460	199	11	)	)	PUNCT
ejpam-460	199	12	(	(	PUNCT
ejpam-460	199	13	r2k	r2k	NOUN
ejpam-460	199	14	2	2	NUM
ejpam-460	199	15	−	−	NOUN
ejpam-460	199	16	r2k	r2k	NOUN
ejpam-460	199	17	1	1	NUM
ejpam-460	199	18	)	)	PUNCT
ejpam-460	199	19	(	(	PUNCT
ejpam-460	199	20	1−	1−	NUM
ejpam-460	199	21	r2	r2	PROPN
ejpam-460	199	22	2)(1−	2)(1−	NUM
ejpam-460	199	23	r2	r2	PROPN
ejpam-460	199	24	1	1	NUM
ejpam-460	199	25	)	)	PUNCT
ejpam-460	199	26	,	,	PUNCT
ejpam-460	199	27	references	reference	NOUN
ejpam-460	199	28	543	543	NUM
ejpam-460	199	29	and	and	CCONJ
ejpam-460	199	30	k	k	X
ejpam-460	200	1	[	[	X
ejpam-460	200	2	f	f	X
ejpam-460	200	3	]	]	PUNCT
ejpam-460	200	4	=	=	SYM
ejpam-460	200	5	k	k	NOUN
ejpam-460	200	6	,	,	PUNCT
ejpam-460	200	7	then	then	ADV
ejpam-460	200	8	f	f	X
ejpam-460	200	9	(	(	PUNCT
ejpam-460	200	10	z	z	NOUN
ejpam-460	200	11	)	)	PUNCT
ejpam-460	200	12	is	be	AUX
ejpam-460	200	13	an	an	DET
ejpam-460	200	14	extremal	extremal	ADJ
ejpam-460	200	15	quasiconformal	quasiconformal	ADJ
ejpam-460	200	16	mapping	mapping	NOUN
ejpam-460	200	17	from	from	ADP
ejpam-460	200	18	s	s	PRON
ejpam-460	200	19	onto	onto	ADP
ejpam-460	200	20	s′	s′	NUM
ejpam-460	200	21	,	,	PUNCT
ejpam-460	200	22	and	and	CCONJ
ejpam-460	200	23	f	f	PROPN
ejpam-460	200	24	(	(	PUNCT
ejpam-460	200	25	z	z	NOUN
ejpam-460	200	26	)	)	PUNCT
ejpam-460	201	1	=	=	PRON
ejpam-460	201	2	rk	rk	NOUN
ejpam-460	201	3	ei(θ+α	ei(θ+α	PROPN
ejpam-460	201	4	)	)	PUNCT
ejpam-460	201	5	,	,	PUNCT
ejpam-460	201	6	z	z	X
ejpam-460	201	7	=	=	PUNCT
ejpam-460	201	8	reiθ	reiθ	PROPN
ejpam-460	201	9	,	,	PUNCT
ejpam-460	201	10	where	where	SCONJ
ejpam-460	201	11	α	α	NOUN
ejpam-460	201	12	is	be	AUX
ejpam-460	201	13	a	a	DET
ejpam-460	201	14	constant	constant	ADJ
ejpam-460	201	15	.	.	PUNCT
ejpam-460	202	1	acknowledgements	acknowledgement	NOUN
ejpam-460	202	2	the	the	DET
ejpam-460	202	3	authors	author	NOUN
ejpam-460	202	4	would	would	AUX
ejpam-460	202	5	like	like	VERB
ejpam-460	202	6	to	to	PART
ejpam-460	202	7	thank	thank	VERB
ejpam-460	202	8	the	the	DET
ejpam-460	202	9	referee	referee	NOUN
ejpam-460	202	10	very	very	ADV
ejpam-460	202	11	much	much	ADV
ejpam-460	202	12	for	for	ADP
ejpam-460	202	13	his	his	PRON
ejpam-460	202	14	many	many	ADJ
ejpam-460	202	15	valuable	valuable	ADJ
ejpam-460	202	16	suggestions	suggestion	NOUN
ejpam-460	202	17	.	.	PUNCT
ejpam-460	203	1	this	this	DET
ejpam-460	203	2	project	project	NOUN
ejpam-460	203	3	was	be	AUX
ejpam-460	203	4	supported	support	VERB
ejpam-460	203	5	by	by	ADP
ejpam-460	203	6	the	the	DET
ejpam-460	203	7	natural	natural	ADJ
ejpam-460	203	8	science	science	PROPN
ejpam-460	203	9	foundation	foundation	NOUN
ejpam-460	203	10	of	of	ADP
ejpam-460	203	11	fujian	fujian	PROPN
ejpam-460	203	12	province	province	PROPN
ejpam-460	203	13	,	,	PUNCT
ejpam-460	203	14	pr	pr	PROPN
ejpam-460	203	15	china	china	PROPN
ejpam-460	203	16	(	(	PUNCT
ejpam-460	203	17	2008j0195	2008j0195	PROPN
ejpam-460	203	18	)	)	PUNCT
ejpam-460	203	19	.	.	PUNCT
ejpam-460	204	1	references	reference	NOUN
ejpam-460	204	2	[	[	X
ejpam-460	204	3	1	1	NUM
ejpam-460	204	4	]	]	PUNCT
ejpam-460	204	5	l.	l.	NOUN
ejpam-460	204	6	v.	v.	ADP
ejpam-460	204	7	ahlfors	ahlfors	PROPN
ejpam-460	204	8	,	,	PUNCT
ejpam-460	204	9	lectures	lecture	NOUN
ejpam-460	204	10	on	on	ADP
ejpam-460	204	11	quasiconformal	quasiconformal	ADJ
ejpam-460	204	12	mappings	mapping	NOUN
ejpam-460	204	13	,	,	PUNCT
ejpam-460	204	14	new	new	PROPN
ejpam-460	204	15	jersey	jersey	PROPN
ejpam-460	204	16	:	:	PUNCT
ejpam-460	204	17	d.	d.	PROPN
ejpam-460	204	18	van	van	PROPN
ejpam-460	204	19	nostrandreinhold	nostrandreinhold	PROPN
ejpam-460	204	20	company	company	PROPN
ejpam-460	204	21	,	,	PUNCT
ejpam-460	204	22	inc	inc	PROPN
ejpam-460	204	23	.	.	PROPN
ejpam-460	204	24	princeton	princeton	PROPN
ejpam-460	204	25	(	(	PUNCT
ejpam-460	204	26	1966	1966	NUM
ejpam-460	204	27	)	)	PUNCT
ejpam-460	204	28	.	.	PUNCT
ejpam-460	205	1	[	[	X
ejpam-460	205	2	2	2	NUM
ejpam-460	205	3	]	]	PUNCT
ejpam-460	205	4	zhu	zhu	PROPN
ejpam-460	205	5	huacheng	huacheng	PROPN
ejpam-460	205	6	,	,	PUNCT
ejpam-460	205	7	zhou	zhou	PROPN
ejpam-460	205	8	zemin	zemin	PROPN
ejpam-460	205	9	and	and	CCONJ
ejpam-460	205	10	he	he	PRON
ejpam-460	205	11	chengqi	chengqi	VERB
ejpam-460	205	12	,	,	PUNCT
ejpam-460	205	13	the	the	DET
ejpam-460	205	14	characterization	characterization	NOUN
ejpam-460	205	15	of	of	ADP
ejpam-460	205	16	grötzsch	grötzsch	PROPN
ejpam-460	205	17	’s	’s	PART
ejpam-460	205	18	problem	problem	NOUN
ejpam-460	205	19	in	in	ADP
ejpam-460	205	20	a	a	DET
ejpam-460	205	21	domain	domain	NOUN
ejpam-460	205	22	,	,	PUNCT
ejpam-460	205	23	j.	j.	PROPN
ejpam-460	205	24	of	of	ADP
ejpam-460	205	25	fudan	fudan	PROPN
ejpam-460	205	26	univ	univ	PROPN
ejpam-460	205	27	.	.	PROPN
ejpam-460	205	28	,	,	PUNCT
ejpam-460	205	29	38(2	38(2	NUM
ejpam-460	205	30	)	)	PUNCT
ejpam-460	205	31	(	(	PUNCT
ejpam-460	205	32	1999	1999	NUM
ejpam-460	205	33	)	)	PUNCT
ejpam-460	205	34	,	,	PUNCT
ejpam-460	205	35	205	205	NUM
ejpam-460	205	36	-	-	SYM
ejpam-460	205	37	207	207	NUM
ejpam-460	205	38	.	.	PUNCT
ejpam-460	206	1	[	[	X
ejpam-460	206	2	3	3	X
ejpam-460	206	3	]	]	X
ejpam-460	206	4	li	li	PROPN
ejpam-460	206	5	shulong	shulong	PROPN
ejpam-460	206	6	,	,	PUNCT
ejpam-460	206	7	liu	liu	PROPN
ejpam-460	206	8	lixin	lixin	PROPN
ejpam-460	206	9	and	and	CCONJ
ejpam-460	206	10	zeng	zeng	PROPN
ejpam-460	206	11	cuiping	cuiping	PROPN
ejpam-460	206	12	,	,	PUNCT
ejpam-460	206	13	the	the	DET
ejpam-460	206	14	schwarz	schwarz	PROPN
ejpam-460	206	15	type	type	NOUN
ejpam-460	206	16	theorems	theorem	NOUN
ejpam-460	206	17	for	for	ADP
ejpam-460	206	18	quasiconformal	quasiconformal	ADJ
ejpam-460	206	19	mappings	mapping	NOUN
ejpam-460	206	20	under	under	ADP
ejpam-460	206	21	area	area	NOUN
ejpam-460	206	22	distortion	distortion	NOUN
ejpam-460	206	23	conditions	condition	NOUN
ejpam-460	206	24	,	,	PUNCT
ejpam-460	206	25	chinese	chinese	PROPN
ejpam-460	206	26	ann	ann	PROPN
ejpam-460	206	27	.	.	PUNCT
ejpam-460	206	28	math	math	PROPN
ejpam-460	206	29	.	.	PUNCT
ejpam-460	206	30	,	,	PUNCT
ejpam-460	206	31	28a(1	28a(1	NUM
ejpam-460	206	32	)	)	PUNCT
ejpam-460	206	33	(	(	PUNCT
ejpam-460	206	34	2007	2007	NUM
ejpam-460	206	35	)	)	PUNCT
ejpam-460	206	36	,	,	PUNCT
ejpam-460	206	37	111120	111120	NUM
ejpam-460	206	38	.	.	PUNCT
ejpam-460	207	1	[	[	X
ejpam-460	207	2	4	4	NUM
ejpam-460	207	3	]	]	X
ejpam-460	207	4	li	li	PROPN
ejpam-460	207	5	zhong	zhong	PROPN
ejpam-460	207	6	,	,	PUNCT
ejpam-460	207	7	an	an	DET
ejpam-460	207	8	introduction	introduction	NOUN
ejpam-460	207	9	to	to	ADP
ejpam-460	207	10	complex	complex	ADJ
ejpam-460	207	11	analysis	analysis	NOUN
ejpam-460	207	12	,	,	PUNCT
ejpam-460	207	13	beijing	beijing	PROPN
ejpam-460	207	14	university	university	PROPN
ejpam-460	207	15	press	press	NOUN
ejpam-460	207	16	(	(	PUNCT
ejpam-460	207	17	2005	2005	NUM
ejpam-460	207	18	)	)	PUNCT
ejpam-460	208	1	[	[	X
ejpam-460	208	2	5	5	X
ejpam-460	208	3	]	]	PUNCT
ejpam-460	208	4	f.	f.	PROPN
ejpam-460	208	5	w.	w.	PROPN
ejpam-460	208	6	gehring	gehring	PROPN
ejpam-460	208	7	and	and	CCONJ
ejpam-460	208	8	väisälä	väisälä	PROPN
ejpam-460	208	9	jussi	jussi	PROPN
ejpam-460	208	10	,	,	PUNCT
ejpam-460	208	11	on	on	ADP
ejpam-460	208	12	the	the	DET
ejpam-460	208	13	geometric	geometric	ADJ
ejpam-460	208	14	definition	definition	NOUN
ejpam-460	208	15	for	for	ADP
ejpam-460	208	16	quasiconformal	quasiconformal	ADJ
ejpam-460	208	17	mappings	mapping	NOUN
ejpam-460	208	18	,	,	PUNCT
ejpam-460	208	19	comment	comment	NOUN
ejpam-460	208	20	.	.	PUNCT
ejpam-460	209	1	math	math	NOUN
ejpam-460	209	2	.	.	PUNCT
ejpam-460	210	1	helv	helv	PROPN
ejpam-460	210	2	.	.	PROPN
ejpam-460	210	3	,	,	PUNCT
ejpam-460	210	4	36	36	NUM
ejpam-460	210	5	(	(	PUNCT
ejpam-460	210	6	1962	1962	NUM
ejpam-460	210	7	)	)	PUNCT
ejpam-460	210	8	,	,	PUNCT
ejpam-460	210	9	19	19	NUM
ejpam-460	210	10	-	-	SYM
ejpam-460	210	11	32	32	NUM
ejpam-460	210	12	.	.	PUNCT