id	sid	tid	token	lemma	pos
ma-174	1	1	2023	2023	NUM
ma-174	1	2	ada	ada	PROPN
ma-174	1	3	academica	academica	PROPN
ma-174	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-174	1	5	.	.	PUNCT
ma-174	2	1	j.	j.	PROPN
ma-174	2	2	math	math	PROPN
ma-174	2	3	.	.	PUNCT
ma-174	3	1	anal	anal	ADJ
ma-174	3	2	.	.	PUNCT
ma-174	4	1	3	3	NUM
ma-174	4	2	(	(	PUNCT
ma-174	4	3	2023	2023	NUM
ma-174	4	4	)	)	PUNCT
ma-174	4	5	23doi	23doi	ADP
ma-174	4	6	:	:	PUNCT
ma-174	4	7	10.28924	10.28924	NUM
ma-174	4	8	/	/	SYM
ma-174	4	9	ada	ada	PROPN
ma-174	4	10	/	/	SYM
ma-174	4	11	ma.3.23	ma.3.23	PROPN
ma-174	4	12	strong	strong	ADJ
ma-174	4	13	continuity	continuity	NOUN
ma-174	4	14	of	of	ADP
ma-174	4	15	composition	composition	NOUN
ma-174	4	16	semigroups	semigroup	NOUN
ma-174	4	17	on	on	ADP
ma-174	4	18	the	the	DET
ma-174	4	19	generalized	generalized	ADJ
ma-174	4	20	bloch	bloch	NOUN
ma-174	4	21	spaces	space	NOUN
ma-174	4	22	of	of	ADP
ma-174	4	23	the	the	DET
ma-174	4	24	upper	upper	ADJ
ma-174	4	25	half	half	ADJ
ma-174	4	26	plane	plane	NOUN
ma-174	4	27	k.	k.	PROPN
ma-174	4	28	a.	a.	PROPN
ma-174	4	29	wandera1	wandera1	PROPN
ma-174	4	30	,	,	PUNCT
ma-174	4	31	j.	j.	PROPN
ma-174	4	32	o.	o.	PROPN
ma-174	4	33	bonyo2,∗	bonyo2,∗	PROPN
ma-174	4	34	,	,	PUNCT
ma-174	4	35	d.	d.	PROPN
ma-174	4	36	o.	o.	PROPN
ma-174	4	37	ambogo1	ambogo1	PROPN
ma-174	5	1	1department	1department	NUM
ma-174	5	2	of	of	ADP
ma-174	5	3	pure	pure	ADJ
ma-174	5	4	and	and	CCONJ
ma-174	5	5	applied	applied	ADJ
ma-174	5	6	mathematics	mathematic	NOUN
ma-174	5	7	,	,	PUNCT
ma-174	5	8	maseno	maseno	NOUN
ma-174	5	9	university	university	NOUN
ma-174	5	10	,	,	PUNCT
ma-174	5	11	p.o	p.o	PROPN
ma-174	5	12	.	.	PROPN
ma-174	5	13	box	box	PROPN
ma-174	5	14	333	333	NUM
ma-174	5	15	-	-	SYM
ma-174	5	16	40105	40105	NUM
ma-174	5	17	,	,	PUNCT
ma-174	5	18	maseno	maseno	VERB
ma-174	5	19	kenya	kenya	PROPN
ma-174	5	20	kwandera1@gmail.com	kwandera1@gmail.com	PROPN
ma-174	5	21	,	,	PUNCT
ma-174	5	22	ambogos@maseno.ac.ke	ambogos@maseno.ac.ke	PROPN
ma-174	5	23	2department	2department	NUM
ma-174	5	24	of	of	ADP
ma-174	5	25	mathematics	mathematic	NOUN
ma-174	5	26	,	,	PUNCT
ma-174	5	27	multimedia	multimedia	NOUN
ma-174	5	28	university	university	PROPN
ma-174	5	29	of	of	ADP
ma-174	5	30	kenya	kenya	PROPN
ma-174	5	31	,	,	PUNCT
ma-174	5	32	p.o	p.o	PROPN
ma-174	5	33	.	.	PROPN
ma-174	5	34	box	box	PROPN
ma-174	5	35	15653	15653	NUM
ma-174	5	36	-	-	SYM
ma-174	5	37	00503	00503	NUM
ma-174	5	38	,	,	PUNCT
ma-174	5	39	nairobi	nairobi	PROPN
ma-174	5	40	kenya	kenya	PROPN
ma-174	5	41	jbonyo@mmu.ac.ke	jbonyo@mmu.ac.ke	PROPN
ma-174	5	42	∗correspondence	∗correspondence	NOUN
ma-174	5	43	author	author	NOUN
ma-174	5	44	abstract	abstract	NOUN
ma-174	5	45	.	.	PUNCT
ma-174	6	1	we	we	PRON
ma-174	6	2	investigate	investigate	VERB
ma-174	6	3	strong	strong	ADJ
ma-174	6	4	continuity	continuity	NOUN
ma-174	6	5	of	of	ADP
ma-174	6	6	composition	composition	NOUN
ma-174	6	7	semigroups	semigroup	NOUN
ma-174	6	8	on	on	ADP
ma-174	6	9	the	the	DET
ma-174	6	10	generalized	generalized	ADJ
ma-174	6	11	blochspaces	blochspace	NOUN
ma-174	6	12	of	of	ADP
ma-174	6	13	the	the	DET
ma-174	6	14	upper	upper	ADJ
ma-174	6	15	half	half	ADJ
ma-174	6	16	plane	plane	NOUN
ma-174	6	17	.	.	PUNCT
ma-174	7	1	these	these	DET
ma-174	7	2	composition	composition	NOUN
ma-174	7	3	semigroups	semigroup	NOUN
ma-174	7	4	are	be	AUX
ma-174	7	5	induced	induce	VERB
ma-174	7	6	by	by	ADP
ma-174	7	7	automorphisms	automorphisms	PROPN
ma-174	7	8	ofthe	ofthe	VERB
ma-174	7	9	upper	upper	ADJ
ma-174	7	10	half	half	NOUN
ma-174	7	11	plane	plane	NOUN
ma-174	7	12	as	as	SCONJ
ma-174	7	13	classified	classify	VERB
ma-174	7	14	into	into	ADP
ma-174	7	15	three	three	NUM
ma-174	7	16	distinct	distinct	ADJ
ma-174	7	17	groups	group	NOUN
ma-174	7	18	in	in	ADP
ma-174	7	19	[	[	X
ma-174	7	20	3	3	NUM
ma-174	7	21	]	]	PUNCT
ma-174	7	22	.	.	PUNCT
ma-174	8	1	1	1	X
ma-174	8	2	.	.	X
ma-174	8	3	introduction	introduction	NOUN
ma-174	8	4	consider	consider	VERB
ma-174	8	5	h(ω	h(ω	PROPN
ma-174	8	6	)	)	PUNCT
ma-174	8	7	as	as	ADP
ma-174	8	8	the	the	DET
ma-174	8	9	fréchet	fréchet	NOUN
ma-174	8	10	space	space	NOUN
ma-174	8	11	of	of	ADP
ma-174	8	12	analytic	analytic	ADJ
ma-174	8	13	functions	function	NOUN
ma-174	8	14	f	f	NOUN
ma-174	8	15	:	:	PUNCT
ma-174	9	1	ω→	ω→	PUNCT
ma-174	9	2	c	c	NOUN
ma-174	9	3	endowed	endow	VERB
ma-174	9	4	with	with	ADP
ma-174	9	5	the	the	DET
ma-174	9	6	topologyof	topologyof	ADJ
ma-174	9	7	uniform	uniform	ADJ
ma-174	9	8	convergence	convergence	NOUN
ma-174	9	9	on	on	ADP
ma-174	9	10	compact	compact	ADJ
ma-174	9	11	subsets	subset	NOUN
ma-174	9	12	of	of	ADP
ma-174	9	13	ω	ω	PROPN
ma-174	9	14	.	.	PUNCT
ma-174	10	1	a	a	DET
ma-174	10	2	function	function	NOUN
ma-174	10	3	f	f	PROPN
ma-174	10	4	∈	∈	PROPN
ma-174	10	5	h(d	h(d	PROPN
ma-174	10	6	)	)	PUNCT
ma-174	10	7	is	be	AUX
ma-174	10	8	in	in	ADP
ma-174	10	9	the	the	DET
ma-174	10	10	bloch	bloch	PROPN
ma-174	10	11	space	space	NOUN
ma-174	10	12	of	of	ADP
ma-174	10	13	theunit	theunit	VERB
ma-174	10	14	disc	disc	PROPN
ma-174	10	15	b(d	b(d	PROPN
ma-174	10	16	)	)	PUNCT
ma-174	10	17	if	if	SCONJ
ma-174	10	18	‖f	‖f	ADP
ma-174	10	19	‖b1(d	‖b1(d	NOUN
ma-174	10	20	)	)	PUNCT
ma-174	10	21	:	:	PUNCT
ma-174	11	1	=	=	PUNCT
ma-174	11	2	sup	sup	NUM
ma-174	11	3	z∈d	z∈d	NOUN
ma-174	11	4	(	(	PUNCT
ma-174	11	5	1−	1−	NUM
ma-174	11	6	|z	|z	PROPN
ma-174	11	7	|2)|f	|2)|f	PROPN
ma-174	11	8	′(z)|	′(z)|	PROPN
ma-174	11	9	<	<	X
ma-174	11	10	∞	∞	NUM
ma-174	11	11	and	and	CCONJ
ma-174	11	12	in	in	ADP
ma-174	11	13	the	the	DET
ma-174	11	14	little	little	ADJ
ma-174	11	15	bloch	bloch	NOUN
ma-174	11	16	space	space	NOUN
ma-174	11	17	of	of	ADP
ma-174	11	18	the	the	DET
ma-174	11	19	unit	unit	NOUN
ma-174	11	20	disc	disc	VERB
ma-174	11	21	b0(d	b0(d	PROPN
ma-174	11	22	)	)	PUNCT
ma-174	12	1	if	if	SCONJ
ma-174	12	2	lim	lim	PROPN
ma-174	12	3	|z	|z	PROPN
ma-174	12	4	|−→1	|−→1	PROPN
ma-174	12	5	(	(	PUNCT
ma-174	12	6	1−	1−	NUM
ma-174	12	7	|z	|z	PROPN
ma-174	12	8	|2)|f	|2)|f	PROPN
ma-174	12	9	′(z)|	′(z)|	PROPN
ma-174	12	10	=	=	SYM
ma-174	12	11	0	0	X
ma-174	12	12	.	.	PUNCT
ma-174	13	1	for	for	ADP
ma-174	13	2	f	f	PROPN
ma-174	13	3	∈	∈	PROPN
ma-174	13	4	b(d	b(d	PROPN
ma-174	13	5	)	)	PUNCT
ma-174	13	6	,	,	PUNCT
ma-174	13	7	we	we	PRON
ma-174	13	8	define	define	VERB
ma-174	13	9	the	the	DET
ma-174	13	10	norm	norm	NOUN
ma-174	13	11	on	on	ADP
ma-174	13	12	b(d	b(d	PROPN
ma-174	13	13	)	)	PUNCT
ma-174	13	14	by	by	ADP
ma-174	13	15	‖f	‖f	ADP
ma-174	13	16	‖b(d	‖b(d	NOUN
ma-174	13	17	)	)	PUNCT
ma-174	13	18	:	:	PUNCT
ma-174	14	1	=	=	SYM
ma-174	14	2	|f	|f	PROPN
ma-174	14	3	(	(	PUNCT
ma-174	14	4	0)|+	0)|+	NUM
ma-174	14	5	‖f	‖f	ADJ
ma-174	14	6	‖b1(d),where	‖b1(d),where	CCONJ
ma-174	14	7	‖.‖b1(d	‖.‖b1(d	VERB
ma-174	14	8	)	)	PUNCT
ma-174	14	9	is	be	AUX
ma-174	14	10	a	a	DET
ma-174	14	11	seminorm	seminorm	NOUN
ma-174	14	12	on	on	ADP
ma-174	14	13	b(d).bloch	b(d).bloch	PROPN
ma-174	14	14	space	space	NOUN
ma-174	14	15	of	of	ADP
ma-174	14	16	the	the	DET
ma-174	14	17	upper	upper	ADJ
ma-174	14	18	half	half	ADJ
ma-174	14	19	plane	plane	NOUN
ma-174	14	20	b(u	b(u	PROPN
ma-174	14	21	)	)	PUNCT
ma-174	14	22	is	be	AUX
ma-174	14	23	a	a	DET
ma-174	14	24	set	set	NOUN
ma-174	14	25	of	of	ADP
ma-174	14	26	analytic	analytic	ADJ
ma-174	14	27	functions	function	NOUN
ma-174	14	28	f	f	PROPN
ma-174	14	29	∈	∈	PROPN
ma-174	14	30	h(u	h(u	PROPN
ma-174	14	31	)	)	PUNCT
ma-174	15	1	such	such	ADJ
ma-174	15	2	that	that	SCONJ
ma-174	15	3	‖f	‖f	ADJ
ma-174	15	4	‖b1(u	‖b1(u	NOUN
ma-174	15	5	)	)	PUNCT
ma-174	15	6	:	:	PUNCT
ma-174	15	7	=	=	SYM
ma-174	15	8	sup	sup	NOUN
ma-174	15	9	ω∈u	ω∈u	NOUN
ma-174	15	10	=(	=(	NOUN
ma-174	15	11	ω)|f	ω)|f	NOUN
ma-174	16	1	′(ω)|	′(ω)|	PROPN
ma-174	17	1	<	<	X
ma-174	17	2	∞.	∞.	PROPN
ma-174	17	3	for	for	ADP
ma-174	17	4	f	f	PROPN
ma-174	17	5	∈	∈	PROPN
ma-174	17	6	b(u	b(u	PROPN
ma-174	17	7	)	)	PUNCT
ma-174	17	8	,	,	PUNCT
ma-174	17	9	we	we	PRON
ma-174	17	10	define	define	VERB
ma-174	17	11	the	the	DET
ma-174	17	12	norm	norm	NOUN
ma-174	17	13	on	on	ADP
ma-174	17	14	b(u	b(u	PROPN
ma-174	17	15	)	)	PUNCT
ma-174	17	16	by	by	ADP
ma-174	17	17	‖f	‖f	PRON
ma-174	17	18	‖b(u	‖b(u	NOUN
ma-174	17	19	)	)	PUNCT
ma-174	17	20	:	:	PUNCT
ma-174	18	1	=	=	SYM
ma-174	18	2	|f	|f	PROPN
ma-174	18	3	(	(	PUNCT
ma-174	18	4	i)|+	i)|+	ADJ
ma-174	18	5	‖f	‖f	ADJ
ma-174	18	6	‖b1(u	‖b1(u	NOUN
ma-174	18	7	)	)	PUNCT
ma-174	18	8	,	,	PUNCT
ma-174	18	9	received	receive	VERB
ma-174	18	10	:	:	PUNCT
ma-174	18	11	5	5	NUM
ma-174	18	12	jun	jun	PROPN
ma-174	18	13	2023	2023	NUM
ma-174	18	14	.	.	PUNCT
ma-174	19	1	key	key	ADJ
ma-174	19	2	words	word	NOUN
ma-174	19	3	and	and	CCONJ
ma-174	19	4	phrases	phrase	NOUN
ma-174	19	5	.	.	PUNCT
ma-174	20	1	composition	composition	NOUN
ma-174	20	2	semigroup	semigroup	NOUN
ma-174	20	3	;	;	PUNCT
ma-174	20	4	analytic	analytic	ADJ
ma-174	20	5	functions	function	NOUN
ma-174	20	6	;	;	PUNCT
ma-174	20	7	self	self	NOUN
ma-174	20	8	analytic	analytic	ADJ
ma-174	20	9	maps	map	NOUN
ma-174	20	10	;	;	PUNCT
ma-174	20	11	bloch	bloch	PROPN
ma-174	20	12	spaces	space	VERB
ma-174	20	13	;	;	PUNCT
ma-174	20	14	unit	unit	NOUN
ma-174	20	15	disc	disc	NOUN
ma-174	20	16	;	;	PUNCT
ma-174	20	17	upperhalf	upperhalf	NOUN
ma-174	20	18	plane	plane	NOUN
ma-174	20	19	;	;	PUNCT
ma-174	20	20	strong	strong	ADJ
ma-174	20	21	continuity	continuity	NOUN
ma-174	20	22	;	;	PUNCT
ma-174	20	23	infinitesimal	infinitesimal	ADJ
ma-174	20	24	generator	generator	NOUN
ma-174	20	25	.	.	PUNCT
ma-174	21	1	1	1	NUM
ma-174	21	2	https://adac.ee	https://adac.ee	PROPN
ma-174	21	3	https://doi.org/10.28924/ada/ma.3.23	https://doi.org/10.28924/ada/ma.3.23	PROPN
ma-174	21	4	https://orcid.org/0000-0002-6442-4211	https://orcid.org/0000-0002-6442-4211	PROPN
ma-174	21	5	where	where	SCONJ
ma-174	21	6	‖.‖b1(u	‖.‖b1(u	NOUN
ma-174	21	7	)	)	PUNCT
ma-174	21	8	is	be	AUX
ma-174	21	9	a	a	DET
ma-174	21	10	seminorm	seminorm	NOUN
ma-174	21	11	on	on	ADP
ma-174	21	12	b(u).let	b(u).let	ADJ
ma-174	21	13	α	α	PROPN
ma-174	21	14	>	>	X
ma-174	21	15	0	0	PUNCT
ma-174	21	16	be	be	AUX
ma-174	21	17	a	a	DET
ma-174	21	18	real	real	ADJ
ma-174	21	19	number	number	NOUN
ma-174	21	20	,	,	PUNCT
ma-174	21	21	we	we	PRON
ma-174	21	22	define	define	VERB
ma-174	21	23	the	the	DET
ma-174	21	24	generalized	generalized	ADJ
ma-174	21	25	bloch	bloch	NOUN
ma-174	21	26	space	space	NOUN
ma-174	21	27	of	of	ADP
ma-174	21	28	the	the	DET
ma-174	21	29	unit	unit	NOUN
ma-174	21	30	disc	disc	NOUN
ma-174	21	31	,	,	PUNCT
ma-174	21	32	bα(d	bα(d	NOUN
ma-174	21	33	)	)	PUNCT
ma-174	21	34	asthe	asthe	ADJ
ma-174	21	35	space	space	NOUN
ma-174	21	36	of	of	ADP
ma-174	21	37	all	all	DET
ma-174	21	38	functions	function	NOUN
ma-174	21	39	f	f	PROPN
ma-174	21	40	∈	∈	PROPN
ma-174	21	41	h(d	h(d	PROPN
ma-174	21	42	)	)	PUNCT
ma-174	21	43	such	such	ADJ
ma-174	21	44	that	that	SCONJ
ma-174	21	45	‖f	‖f	ADP
ma-174	21	46	‖bα1	‖bα1	PROPN
ma-174	21	47	(	(	PUNCT
ma-174	21	48	d	d	NOUN
ma-174	21	49	)	)	PUNCT
ma-174	21	50	:	:	PUNCT
ma-174	22	1	=	=	PUNCT
ma-174	22	2	sup	sup	NUM
ma-174	22	3	z∈d	z∈d	NOUN
ma-174	22	4	(	(	PUNCT
ma-174	22	5	1−	1−	NUM
ma-174	22	6	|z	|z	PROPN
ma-174	22	7	|2	|2	NUM
ma-174	22	8	)	)	PUNCT
ma-174	23	1	α	α	PROPN
ma-174	23	2	|f	|f	PROPN
ma-174	23	3	′(z)|	′(z)|	PROPN
ma-174	24	1	<	<	X
ma-174	24	2	∞.	∞.	PROPN
ma-174	24	3	for	for	ADP
ma-174	24	4	f	f	PROPN
ma-174	24	5	∈	∈	PROPN
ma-174	24	6	bα(d	bα(d	NOUN
ma-174	24	7	)	)	PUNCT
ma-174	24	8	,	,	PUNCT
ma-174	24	9	we	we	PRON
ma-174	24	10	define	define	VERB
ma-174	24	11	the	the	DET
ma-174	24	12	norm	norm	NOUN
ma-174	24	13	on	on	ADP
ma-174	24	14	bα(d	bα(d	NOUN
ma-174	24	15	)	)	PUNCT
ma-174	24	16	by	by	ADP
ma-174	24	17	‖f	‖f	ADP
ma-174	24	18	‖bα(d	‖bα(d	X
ma-174	24	19	)	)	PUNCT
ma-174	25	1	:	:	PUNCT
ma-174	25	2	=	=	SYM
ma-174	25	3	|f	|f	PROPN
ma-174	25	4	(	(	PUNCT
ma-174	25	5	0)|+	0)|+	NUM
ma-174	25	6	‖f	‖f	PRON
ma-174	25	7	‖bα1	‖bα1	PROPN
ma-174	25	8	(	(	PUNCT
ma-174	25	9	d	d	NOUN
ma-174	25	10	)	)	PUNCT
ma-174	25	11	.	.	PUNCT
ma-174	26	1	(	(	PUNCT
ma-174	26	2	1	1	X
ma-174	26	3	)	)	PUNCT
ma-174	26	4	we	we	PRON
ma-174	26	5	also	also	ADV
ma-174	26	6	define	define	VERB
ma-174	26	7	the	the	DET
ma-174	26	8	corresponding	corresponding	ADJ
ma-174	26	9	generalized	generalize	VERB
ma-174	26	10	little	little	ADJ
ma-174	26	11	bloch	bloch	NOUN
ma-174	26	12	space	space	NOUN
ma-174	26	13	of	of	ADP
ma-174	26	14	the	the	DET
ma-174	26	15	unit	unit	NOUN
ma-174	26	16	disc	disc	NOUN
ma-174	26	17	as	as	ADP
ma-174	26	18	the	the	DET
ma-174	26	19	space	space	NOUN
ma-174	26	20	of	of	ADP
ma-174	26	21	allfunctions	allfunction	NOUN
ma-174	26	22	f	f	PROPN
ma-174	26	23	∈	∈	PROPN
ma-174	26	24	h(d	h(d	PROPN
ma-174	26	25	)	)	PUNCT
ma-174	26	26	for	for	ADP
ma-174	26	27	which	which	PRON
ma-174	26	28	lim	lim	PROPN
ma-174	26	29	|z	|z	PROPN
ma-174	26	30	|→1	|→1	PROPN
ma-174	26	31	(	(	PUNCT
ma-174	26	32	1−	1−	NUM
ma-174	26	33	|z	|z	PROPN
ma-174	26	34	|2	|2	NUM
ma-174	26	35	)	)	PUNCT
ma-174	27	1	α	α	PROPN
ma-174	27	2	|f	|f	PROPN
ma-174	27	3	′(z)|	′(z)|	NOUN
ma-174	27	4	=	=	PUNCT
ma-174	27	5	0	0	NUM
ma-174	27	6	,	,	PUNCT
ma-174	27	7	with	with	ADP
ma-174	27	8	the	the	DET
ma-174	27	9	same	same	ADJ
ma-174	27	10	norm	norm	NOUN
ma-174	27	11	given	give	VERB
ma-174	27	12	by	by	ADP
ma-174	27	13	(	(	PUNCT
ma-174	27	14	1	1	NUM
ma-174	27	15	)	)	PUNCT
ma-174	27	16	.	.	PUNCT
ma-174	28	1	here	here	ADV
ma-174	28	2	,	,	PUNCT
ma-174	28	3	bα(d	bα(d	NOUN
ma-174	28	4	)	)	PUNCT
ma-174	28	5	and	and	CCONJ
ma-174	28	6	bα	bα	NOUN
ma-174	28	7	◦	◦	NOUN
ma-174	28	8	(	(	PUNCT
ma-174	28	9	d	d	X
ma-174	28	10	)	)	PUNCT
ma-174	28	11	are	be	AUX
ma-174	28	12	both	both	PRON
ma-174	28	13	banach	banach	NOUN
ma-174	28	14	spaces	space	NOUN
ma-174	28	15	with	with	ADP
ma-174	28	16	respectto	respectto	ADJ
ma-174	28	17	the	the	DET
ma-174	28	18	norm	norm	NOUN
ma-174	28	19	‖.‖bα(d	‖.‖bα(d	NUM
ma-174	28	20	)	)	PUNCT
ma-174	28	21	.	.	PUNCT
ma-174	29	1	the	the	DET
ma-174	29	2	generalized	generalize	VERB
ma-174	29	3	little	little	ADJ
ma-174	29	4	bloch	bloch	NOUN
ma-174	29	5	space	space	NOUN
ma-174	29	6	of	of	ADP
ma-174	29	7	the	the	DET
ma-174	29	8	unit	unit	NOUN
ma-174	29	9	disc	disc	NOUN
ma-174	29	10	,	,	PUNCT
ma-174	29	11	bα	bα	NOUN
ma-174	29	12	◦	◦	NOUN
ma-174	29	13	(	(	PUNCT
ma-174	29	14	d	d	X
ma-174	29	15	)	)	PUNCT
ma-174	29	16	is	be	AUX
ma-174	29	17	the	the	DET
ma-174	29	18	closure	closure	NOUN
ma-174	29	19	ofthe	ofthe	NOUN
ma-174	29	20	set	set	NOUN
ma-174	29	21	of	of	ADP
ma-174	29	22	polynomials	polynomial	NOUN
ma-174	29	23	in	in	ADP
ma-174	29	24	the	the	DET
ma-174	29	25	norm	norm	NOUN
ma-174	29	26	topology	topology	NOUN
ma-174	29	27	of	of	ADP
ma-174	29	28	bα(d	bα(d	NOUN
ma-174	29	29	)	)	PUNCT
ma-174	29	30	.	.	PUNCT
ma-174	30	1	for	for	SCONJ
ma-174	30	2	more	more	ADJ
ma-174	30	3	details	detail	NOUN
ma-174	30	4	see	see	VERB
ma-174	30	5	[	[	X
ma-174	30	6	17	17	NUM
ma-174	30	7	,	,	PUNCT
ma-174	30	8	18	18	NUM
ma-174	30	9	]	]	PUNCT
ma-174	30	10	.	.	PUNCT
ma-174	31	1	the	the	DET
ma-174	31	2	space	space	PROPN
ma-174	31	3	b(d	b(d	PROPN
ma-174	31	4	)	)	PUNCT
ma-174	31	5	has	have	AUX
ma-174	31	6	been	be	AUX
ma-174	31	7	studied	study	VERB
ma-174	31	8	by	by	ADP
ma-174	31	9	many	many	ADJ
ma-174	31	10	authors	author	NOUN
ma-174	31	11	because	because	SCONJ
ma-174	31	12	of	of	ADP
ma-174	31	13	its	its	PRON
ma-174	31	14	intrinsic	intrinsic	ADJ
ma-174	31	15	interest	interest	NOUN
ma-174	31	16	since	since	SCONJ
ma-174	31	17	its	its	PRON
ma-174	31	18	introduction[1	introduction[1	PROPN
ma-174	31	19	,	,	PUNCT
ma-174	31	20	4	4	NUM
ma-174	31	21	,	,	PUNCT
ma-174	31	22	8	8	NUM
ma-174	31	23	,	,	PUNCT
ma-174	31	24	10	10	NUM
ma-174	31	25	,	,	PUNCT
ma-174	31	26	13	13	NUM
ma-174	31	27	,	,	PUNCT
ma-174	31	28	14	14	NUM
ma-174	31	29	,	,	PUNCT
ma-174	31	30	18	18	NUM
ma-174	31	31	]	]	PUNCT
ma-174	31	32	.	.	PUNCT
ma-174	32	1	in	in	ADP
ma-174	32	2	[	[	X
ma-174	32	3	17	17	NUM
ma-174	32	4	]	]	PUNCT
ma-174	32	5	,	,	PUNCT
ma-174	32	6	the	the	DET
ma-174	32	7	generalized	generalized	ADJ
ma-174	32	8	bloch	bloch	NOUN
ma-174	32	9	spaces	space	NOUN
ma-174	32	10	of	of	ADP
ma-174	32	11	the	the	DET
ma-174	32	12	open	open	ADJ
ma-174	32	13	unit	unit	NOUN
ma-174	32	14	disc	disc	NOUN
ma-174	32	15	,	,	PUNCT
ma-174	32	16	bα(d	bα(d	NOUN
ma-174	32	17	)	)	PUNCT
ma-174	32	18	aredefined	aredefine	VERB
ma-174	32	19	and	and	CCONJ
ma-174	32	20	proved	prove	VERB
ma-174	32	21	to	to	PART
ma-174	32	22	be	be	AUX
ma-174	32	23	banach	banach	NOUN
ma-174	32	24	spaces	space	NOUN
ma-174	32	25	with	with	ADP
ma-174	32	26	respect	respect	NOUN
ma-174	32	27	to	to	ADP
ma-174	32	28	their	their	PRON
ma-174	32	29	norm	norm	NOUN
ma-174	32	30	.	.	PUNCT
ma-174	33	1	zhu	zhu	X
ma-174	34	1	[	[	X
ma-174	34	2	17	17	NUM
ma-174	34	3	]	]	PUNCT
ma-174	34	4	further	far	ADV
ma-174	34	5	establishedgeneralized	establishedgeneralize	VERB
ma-174	34	6	little	little	ADJ
ma-174	34	7	bloch	bloch	PROPN
ma-174	34	8	spaces	space	NOUN
ma-174	34	9	of	of	ADP
ma-174	34	10	the	the	DET
ma-174	34	11	unit	unit	NOUN
ma-174	34	12	disc	disc	VERB
ma-174	34	13	bα	bα	PROPN
ma-174	34	14	◦	◦	NOUN
ma-174	34	15	(	(	PUNCT
ma-174	34	16	d	d	NOUN
ma-174	34	17	)	)	PUNCT
ma-174	34	18	,	,	PUNCT
ma-174	34	19	as	as	SCONJ
ma-174	34	20	closed	closed	ADJ
ma-174	34	21	,	,	PUNCT
ma-174	34	22	separable	separable	ADJ
ma-174	34	23	subspaces	subspace	NOUN
ma-174	34	24	of	of	ADP
ma-174	34	25	bα(d).there	bα(d).there	PROPN
ma-174	34	26	is	be	AUX
ma-174	34	27	scanty	scanty	ADJ
ma-174	34	28	literature	literature	NOUN
ma-174	34	29	on	on	ADP
ma-174	34	30	the	the	DET
ma-174	34	31	properties	property	NOUN
ma-174	34	32	of	of	ADP
ma-174	34	33	the	the	DET
ma-174	34	34	generalized	generalized	ADJ
ma-174	34	35	bloch	bloch	NOUN
ma-174	34	36	spaces	space	NOUN
ma-174	34	37	of	of	ADP
ma-174	34	38	the	the	DET
ma-174	34	39	upper	upper	ADJ
ma-174	34	40	half	half	ADJ
ma-174	34	41	plane	plane	NOUN
ma-174	34	42	bα(u	bα(u	NOUN
ma-174	34	43	)	)	PUNCT
ma-174	34	44	,	,	PUNCT
ma-174	34	45	including	include	VERB
ma-174	34	46	whether	whether	SCONJ
ma-174	34	47	they	they	PRON
ma-174	34	48	are	be	AUX
ma-174	34	49	banach	banach	ADV
ma-174	34	50	spaces	space	NOUN
ma-174	34	51	.	.	PUNCT
ma-174	35	1	composition	composition	NOUN
ma-174	35	2	semigroups	semigroup	NOUN
ma-174	35	3	on	on	ADP
ma-174	35	4	bloch	bloch	PROPN
ma-174	35	5	spaces	space	VERB
ma-174	35	6	ofthe	ofthe	NOUN
ma-174	35	7	unit	unit	NOUN
ma-174	35	8	disc	disc	NOUN
ma-174	35	9	have	have	AUX
ma-174	35	10	been	be	AUX
ma-174	35	11	studied	study	VERB
ma-174	35	12	in	in	ADP
ma-174	35	13	literature	literature	NOUN
ma-174	35	14	,	,	PUNCT
ma-174	35	15	see	see	VERB
ma-174	35	16	for	for	ADP
ma-174	35	17	instance	instance	NOUN
ma-174	36	1	[	[	X
ma-174	36	2	2,11,12	2,11,12	NUM
ma-174	36	3	]	]	PUNCT
ma-174	36	4	and	and	CCONJ
ma-174	36	5	references	reference	NOUN
ma-174	36	6	therein	therein	ADV
ma-174	36	7	.	.	PUNCT
ma-174	37	1	onstrong	onstrong	NOUN
ma-174	37	2	continuity	continuity	NOUN
ma-174	37	3	of	of	ADP
ma-174	37	4	composition	composition	NOUN
ma-174	37	5	semigroups	semigroup	NOUN
ma-174	37	6	,	,	PUNCT
ma-174	37	7	siskakis	siskaki	NOUN
ma-174	38	1	[	[	X
ma-174	38	2	12	12	NUM
ma-174	38	3	]	]	PUNCT
ma-174	38	4	proved	prove	VERB
ma-174	38	5	that	that	SCONJ
ma-174	38	6	no	no	DET
ma-174	38	7	nontrivial	nontrivial	ADJ
ma-174	38	8	compositionsemigroups	compositionsemigroup	NOUN
ma-174	38	9	are	be	AUX
ma-174	38	10	strongly	strongly	ADV
ma-174	38	11	continuous	continuous	ADJ
ma-174	38	12	on	on	ADP
ma-174	38	13	the	the	DET
ma-174	38	14	bloch	bloch	PROPN
ma-174	38	15	space	space	NOUN
ma-174	38	16	of	of	ADP
ma-174	38	17	the	the	DET
ma-174	38	18	unit	unit	NOUN
ma-174	38	19	disc	disc	VERB
ma-174	38	20	b(d	b(d	PROPN
ma-174	38	21	)	)	PUNCT
ma-174	38	22	.	.	PUNCT
ma-174	39	1	the	the	DET
ma-174	39	2	correspondingstudy	correspondingstudy	NOUN
ma-174	39	3	of	of	ADP
ma-174	39	4	composition	composition	NOUN
ma-174	39	5	semigroups	semigroup	NOUN
ma-174	39	6	defined	define	VERB
ma-174	39	7	on	on	ADP
ma-174	39	8	the	the	DET
ma-174	39	9	bloch	bloch	PROPN
ma-174	39	10	spaces	space	NOUN
ma-174	39	11	of	of	ADP
ma-174	39	12	the	the	DET
ma-174	39	13	upper	upper	ADJ
ma-174	39	14	half	half	ADJ
ma-174	39	15	plane	plane	NOUN
ma-174	39	16	has	have	AUX
ma-174	39	17	not	not	PART
ma-174	39	18	yetbeen	yetbeen	VERB
ma-174	39	19	exhausted	exhaust	VERB
ma-174	39	20	.	.	PUNCT
ma-174	40	1	moreover	moreover	ADV
ma-174	40	2	,	,	PUNCT
ma-174	40	3	existing	exist	VERB
ma-174	40	4	works	work	NOUN
ma-174	40	5	on	on	ADP
ma-174	40	6	the	the	DET
ma-174	40	7	half	half	ADJ
ma-174	40	8	plane	plane	NOUN
ma-174	40	9	,	,	PUNCT
ma-174	40	10	see	see	VERB
ma-174	40	11	[	[	X
ma-174	40	12	7,13	7,13	NOUN
ma-174	40	13	]	]	PUNCT
ma-174	40	14	,	,	PUNCT
ma-174	40	15	have	have	VERB
ma-174	40	16	neither	neither	CCONJ
ma-174	40	17	exhausted	exhausted	ADJ
ma-174	40	18	theinvestigation	theinvestigation	NOUN
ma-174	40	19	of	of	ADP
ma-174	40	20	properties	property	NOUN
ma-174	40	21	of	of	ADP
ma-174	40	22	these	these	DET
ma-174	40	23	semigroups	semigroup	NOUN
ma-174	40	24	nor	nor	CCONJ
ma-174	40	25	considered	consider	VERB
ma-174	40	26	these	these	DET
ma-174	40	27	generalizations	generalization	NOUN
ma-174	40	28	.	.	PUNCT
ma-174	41	1	in	in	ADP
ma-174	41	2	this	this	DET
ma-174	41	3	papertherefore	papertherefore	NOUN
ma-174	41	4	,	,	PUNCT
ma-174	41	5	we	we	PRON
ma-174	41	6	investigate	investigate	VERB
ma-174	41	7	the	the	DET
ma-174	41	8	properties	property	NOUN
ma-174	41	9	of	of	ADP
ma-174	41	10	the	the	DET
ma-174	41	11	generalized	generalized	ADJ
ma-174	41	12	bloch	bloch	NOUN
ma-174	41	13	spaces	space	NOUN
ma-174	41	14	of	of	ADP
ma-174	41	15	the	the	DET
ma-174	41	16	upper	upper	ADJ
ma-174	41	17	half	half	ADJ
ma-174	41	18	plane	plane	NOUN
ma-174	41	19	asbanach	asbanach	NOUN
ma-174	41	20	spaces	space	VERB
ma-174	41	21	and	and	CCONJ
ma-174	41	22	extend	extend	VERB
ma-174	41	23	the	the	DET
ma-174	41	24	study	study	NOUN
ma-174	41	25	of	of	ADP
ma-174	41	26	semigroups	semigroup	NOUN
ma-174	41	27	of	of	ADP
ma-174	41	28	composition	composition	NOUN
ma-174	41	29	operators	operator	NOUN
ma-174	41	30	to	to	ADP
ma-174	41	31	the	the	DET
ma-174	41	32	setting	setting	NOUN
ma-174	41	33	of	of	ADP
ma-174	41	34	thegeneralized	thegeneralize	VERB
ma-174	41	35	bloch	bloch	PROPN
ma-174	41	36	spaces	space	NOUN
ma-174	41	37	of	of	ADP
ma-174	41	38	the	the	DET
ma-174	41	39	upper	upper	ADJ
ma-174	41	40	half	half	ADJ
ma-174	41	41	plane	plane	NOUN
ma-174	41	42	.	.	PUNCT
ma-174	42	1	2	2	X
ma-174	42	2	.	.	NUM
ma-174	42	3	preliminaries	preliminary	NOUN
ma-174	42	4	and	and	CCONJ
ma-174	42	5	definitions	definition	NOUN
ma-174	42	6	let	let	VERB
ma-174	42	7	c	c	NOUN
ma-174	42	8	be	be	AUX
ma-174	42	9	the	the	DET
ma-174	42	10	complex	complex	ADJ
ma-174	42	11	plane	plane	NOUN
ma-174	42	12	.	.	PUNCT
ma-174	43	1	the	the	DET
ma-174	43	2	set	set	NOUN
ma-174	43	3	d	d	NOUN
ma-174	43	4	:	:	PUNCT
ma-174	43	5	=	=	SYM
ma-174	43	6	{	{	PUNCT
ma-174	43	7	z	z	NOUN
ma-174	43	8	∈	∈	PROPN
ma-174	43	9	c	c	NOUN
ma-174	43	10	:	:	PUNCT
ma-174	43	11	|z	|z	PROPN
ma-174	44	1	|	|	ADV
ma-174	44	2	<	<	X
ma-174	44	3	1	1	NUM
ma-174	44	4	}	}	PUNCT
ma-174	44	5	is	be	AUX
ma-174	44	6	called	call	VERB
ma-174	44	7	the	the	DET
ma-174	44	8	open	open	ADJ
ma-174	44	9	unit	unit	NOUN
ma-174	44	10	disc.on	disc.on	VERB
ma-174	44	11	the	the	DET
ma-174	44	12	other	other	ADJ
ma-174	44	13	hand	hand	NOUN
ma-174	44	14	,	,	PUNCT
ma-174	44	15	the	the	DET
ma-174	44	16	set	set	ADJ
ma-174	44	17	u	u	NOUN
ma-174	44	18	:	:	PUNCT
ma-174	44	19	=	=	SYM
ma-174	44	20	{	{	PUNCT
ma-174	44	21	ω	ω	NUM
ma-174	44	22	∈	∈	PROPN
ma-174	44	23	c	c	NOUN
ma-174	44	24	:	:	PUNCT
ma-174	44	25	=(	=(	PROPN
ma-174	44	26	ω	ω	PROPN
ma-174	44	27	)	)	PUNCT
ma-174	44	28	>	>	X
ma-174	44	29	0	0	NUM
ma-174	44	30	}	}	PUNCT
ma-174	44	31	denotes	denote	VERB
ma-174	44	32	the	the	DET
ma-174	44	33	upper	upper	ADJ
ma-174	44	34	half	half	NOUN
ma-174	44	35	of	of	ADP
ma-174	44	36	the	the	DET
ma-174	44	37	complexplane	complexplane	NOUN
ma-174	44	38	c	c	NOUN
ma-174	44	39	,	,	PUNCT
ma-174	44	40	where	where	SCONJ
ma-174	44	41	=(	=(	PROPN
ma-174	44	42	ω	ω	NOUN
ma-174	44	43	)	)	PUNCT
ma-174	44	44	is	be	AUX
ma-174	44	45	the	the	DET
ma-174	44	46	imaginary	imaginary	ADJ
ma-174	44	47	part	part	NOUN
ma-174	44	48	of	of	ADP
ma-174	44	49	ω	ω	PROPN
ma-174	44	50	∈	∈	PROPN
ma-174	44	51	c.	c.	NOUN
ma-174	44	52	the	the	DET
ma-174	44	53	function	function	PROPN
ma-174	44	54	ψ(z)=	ψ(z)=	PROPN
ma-174	44	55	i(1+z	i(1+z	PROPN
ma-174	44	56	)	)	PUNCT
ma-174	44	57	1−z	1−z	NUM
ma-174	44	58	is	be	AUX
ma-174	44	59	referred	refer	VERB
ma-174	44	60	to	to	ADP
ma-174	44	61	asthe	asthe	DET
ma-174	44	62	cayley	cayley	ADJ
ma-174	44	63	transform	transform	NOUN
ma-174	44	64	and	and	CCONJ
ma-174	44	65	maps	map	VERB
ma-174	44	66	the	the	DET
ma-174	44	67	unit	unit	NOUN
ma-174	44	68	disc	disc	VERB
ma-174	44	69	d	d	PROPN
ma-174	44	70	conformally	conformally	ADV
ma-174	44	71	onto	onto	ADP
ma-174	44	72	the	the	DET
ma-174	44	73	upper	upper	ADJ
ma-174	44	74	half	half	ADJ
ma-174	44	75	-	-	PUNCT
ma-174	44	76	plane	plane	NOUN
ma-174	44	77	u	u	NOUN
ma-174	44	78	,	,	PUNCT
ma-174	44	79	with	with	ADP
ma-174	44	80	theinverse	theinverse	ADJ
ma-174	44	81	ψ−1(ω	ψ−1(ω	NOUN
ma-174	44	82	)	)	PUNCT
ma-174	44	83	=	=	PUNCT
ma-174	45	1	ω−i	ω−i	X
ma-174	45	2	ω+i	ω+i	NUM
ma-174	45	3	mapping	map	VERB
ma-174	45	4	the	the	DET
ma-174	45	5	upper	upper	ADJ
ma-174	45	6	half	half	ADJ
ma-174	45	7	plane	plane	NOUN
ma-174	45	8	u	u	NOUN
ma-174	45	9	,	,	PUNCT
ma-174	45	10	onto	onto	ADP
ma-174	45	11	the	the	DET
ma-174	45	12	unit	unit	NOUN
ma-174	45	13	disc	disc	NOUN
ma-174	45	14	,	,	PUNCT
ma-174	45	15	d.	d.	PROPN
ma-174	45	16	we	we	PRON
ma-174	45	17	refer	refer	VERB
ma-174	45	18	to	to	ADP
ma-174	45	19	[	[	X
ma-174	45	20	16	16	NUM
ma-174	45	21	]	]	X
ma-174	45	22	for2	for2	ADJ
ma-174	45	23	details	detail	NOUN
ma-174	45	24	.	.	PUNCT
ma-174	46	1	let	let	VERB
ma-174	46	2	α	α	PRON
ma-174	46	3	>	>	X
ma-174	46	4	0	0	PUNCT
ma-174	46	5	be	be	AUX
ma-174	46	6	a	a	DET
ma-174	46	7	real	real	ADJ
ma-174	46	8	number	number	NOUN
ma-174	46	9	.	.	PUNCT
ma-174	47	1	a	a	DET
ma-174	47	2	function	function	NOUN
ma-174	47	3	f	f	PROPN
ma-174	47	4	∈	∈	PROPN
ma-174	47	5	h(u	h(u	PROPN
ma-174	47	6	)	)	PUNCT
ma-174	47	7	belongs	belong	VERB
ma-174	47	8	to	to	ADP
ma-174	47	9	the	the	DET
ma-174	47	10	generalized	generalized	ADJ
ma-174	47	11	bloch	bloch	PROPN
ma-174	47	12	spaceof	spaceof	VERB
ma-174	47	13	the	the	DET
ma-174	47	14	upper	upper	ADJ
ma-174	47	15	half	half	ADJ
ma-174	47	16	plane	plane	NOUN
ma-174	47	17	,	,	PUNCT
ma-174	47	18	bα(u	bα(u	NOUN
ma-174	47	19	)	)	PUNCT
ma-174	47	20	if	if	SCONJ
ma-174	47	21	‖f	‖f	ADP
ma-174	47	22	‖bα1	‖bα1	PROPN
ma-174	47	23	(	(	PUNCT
ma-174	47	24	u	u	NOUN
ma-174	47	25	)	)	PUNCT
ma-174	47	26	:	:	PUNCT
ma-174	48	1	=	=	SYM
ma-174	48	2	sup	sup	NOUN
ma-174	48	3	ω∈u	ω∈u	NOUN
ma-174	48	4	=	=	SYM
ma-174	48	5	(	(	PUNCT
ma-174	48	6	ω)α	ω)α	X
ma-174	48	7	|f	|f	PRON
ma-174	49	1	′(ω)|	′(ω)|	X
ma-174	50	1	<	<	X
ma-174	50	2	∞	∞	PROPN
ma-174	50	3	with	with	ADP
ma-174	50	4	the	the	DET
ma-174	50	5	norm	norm	NOUN
ma-174	50	6	given	give	VERB
ma-174	50	7	by	by	ADP
ma-174	50	8	‖f	‖f	ADJ
ma-174	50	9	‖bα(u	‖bα(u	NUM
ma-174	50	10	)	)	PUNCT
ma-174	50	11	:	:	PUNCT
ma-174	50	12	=	=	SYM
ma-174	50	13	|f	|f	PROPN
ma-174	50	14	(	(	PUNCT
ma-174	50	15	i)|+	i)|+	ADJ
ma-174	50	16	‖f	‖f	VERB
ma-174	50	17	‖bα1	‖bα1	PROPN
ma-174	50	18	(	(	PUNCT
ma-174	50	19	u).the	u).the	PROPN
ma-174	50	20	corresponding	corresponding	ADJ
ma-174	50	21	generalized	generalize	VERB
ma-174	50	22	little	little	ADJ
ma-174	50	23	bloch	bloch	NOUN
ma-174	50	24	space	space	NOUN
ma-174	50	25	of	of	ADP
ma-174	50	26	the	the	DET
ma-174	50	27	upper	upper	ADJ
ma-174	50	28	half	half	ADJ
ma-174	50	29	plane	plane	NOUN
ma-174	50	30	,	,	PUNCT
ma-174	50	31	bα0	bα0	PROPN
ma-174	50	32	(	(	PUNCT
ma-174	50	33	u)is	u)is	PROPN
ma-174	50	34	defined	define	VERB
ma-174	50	35	as	as	ADP
ma-174	50	36	bα	bα	PROPN
ma-174	50	37	◦	◦	NOUN
ma-174	50	38	(	(	PUNCT
ma-174	50	39	u	u	NOUN
ma-174	50	40	)	)	PUNCT
ma-174	50	41	:	:	PUNCT
ma-174	51	1	=	=	X
ma-174	51	2	{	{	PUNCT
ma-174	51	3	f	f	PROPN
ma-174	51	4	∈	∈	PROPN
ma-174	51	5	h(u	h(u	PROPN
ma-174	51	6	)	)	PUNCT
ma-174	51	7	:	:	PUNCT
ma-174	52	1	lim	lim	PROPN
ma-174	52	2	=(	=(	PROPN
ma-174	52	3	ω)−→0	ω)−→0	PROPN
ma-174	52	4	=	=	SYM
ma-174	52	5	(	(	PUNCT
ma-174	52	6	ω)α	ω)α	X
ma-174	52	7	|f	|f	PRON
ma-174	53	1	′(ω)|	′(ω)|	PROPN
ma-174	53	2	=	=	SYM
ma-174	53	3	0	0	X
ma-174	53	4	}	}	PUNCT
ma-174	53	5	having	have	VERB
ma-174	53	6	the	the	DET
ma-174	53	7	same	same	ADJ
ma-174	53	8	norm	norm	NOUN
ma-174	53	9	as	as	ADP
ma-174	53	10	bα(u	bα(u	NOUN
ma-174	53	11	)	)	PUNCT
ma-174	53	12	.	.	PUNCT
ma-174	54	1	there	there	PRON
ma-174	54	2	is	be	VERB
ma-174	54	3	little	little	ADJ
ma-174	54	4	literature	literature	NOUN
ma-174	54	5	on	on	ADP
ma-174	54	6	the	the	DET
ma-174	54	7	properties	property	NOUN
ma-174	54	8	of	of	ADP
ma-174	54	9	the	the	DET
ma-174	54	10	generalized	generalized	ADJ
ma-174	54	11	blochspaces	blochspace	NOUN
ma-174	54	12	of	of	ADP
ma-174	54	13	the	the	DET
ma-174	54	14	upper	upper	ADJ
ma-174	54	15	half	half	ADJ
ma-174	54	16	plane	plane	NOUN
ma-174	54	17	as	as	ADP
ma-174	54	18	banach	banach	NOUN
ma-174	54	19	spaces	space	NOUN
ma-174	54	20	.	.	PUNCT
ma-174	55	1	let	let	VERB
ma-174	55	2	x	x	PRON
ma-174	55	3	be	be	AUX
ma-174	55	4	a	a	DET
ma-174	55	5	banach	banach	NOUN
ma-174	55	6	space	space	NOUN
ma-174	55	7	.	.	PUNCT
ma-174	56	1	a	a	DET
ma-174	56	2	one	one	NUM
ma-174	56	3	-	-	PUNCT
ma-174	56	4	parameterfamily	parameterfamily	ADV
ma-174	56	5	(	(	PUNCT
ma-174	56	6	tt)t≥0	tt)t≥0	NOUN
ma-174	56	7	is	be	AUX
ma-174	56	8	a	a	DET
ma-174	56	9	semigroup	semigroup	NOUN
ma-174	56	10	of	of	ADP
ma-174	56	11	bounded	bounded	ADJ
ma-174	56	12	linear	linear	PROPN
ma-174	56	13	operators	operator	NOUN
ma-174	56	14	on	on	ADP
ma-174	56	15	x	x	SYM
ma-174	56	16	,	,	PUNCT
ma-174	56	17	if(i	if(i	NUM
ma-174	56	18	)	)	PUNCT
ma-174	56	19	to	to	ADP
ma-174	56	20	=	=	SYM
ma-174	56	21	i	i	PROPN
ma-174	56	22	(	(	PUNCT
ma-174	56	23	identity	identity	NOUN
ma-174	56	24	operator	operator	NOUN
ma-174	56	25	on	on	ADP
ma-174	56	26	x	x	NOUN
ma-174	56	27	)	)	PUNCT
ma-174	56	28	,	,	PUNCT
ma-174	56	29	and(ii	and(ii	NUM
ma-174	56	30	)	)	PUNCT
ma-174	56	31	tt+s	tt+	NOUN
ma-174	56	32	=	=	PUNCT
ma-174	56	33	tt	tt	PART
ma-174	56	34	◦	◦	NOUN
ma-174	56	35	ts	ts	ADP
ma-174	56	36	for	for	ADP
ma-174	56	37	every	every	DET
ma-174	56	38	t	t	PROPN
ma-174	56	39	,	,	PUNCT
ma-174	56	40	s,≥	s,≥	PROPN
ma-174	56	41	0	0	PUNCT
ma-174	56	42	(	(	PUNCT
ma-174	56	43	semigroup	semigroup	PROPN
ma-174	56	44	property).a	property).a	NOUN
ma-174	56	45	semigroup	semigroup	NOUN
ma-174	56	46	(	(	PUNCT
ma-174	56	47	tt)t≥0	tt)t≥0	NOUN
ma-174	56	48	of	of	ADP
ma-174	56	49	bounded	bounded	ADJ
ma-174	56	50	linear	linear	PROPN
ma-174	56	51	operators	operator	NOUN
ma-174	56	52	on	on	ADP
ma-174	56	53	x	x	VERB
ma-174	56	54	is	be	AUX
ma-174	56	55	strongly	strongly	ADV
ma-174	56	56	continuous	continuous	ADJ
ma-174	56	57	if	if	SCONJ
ma-174	56	58	lim	lim	PROPN
ma-174	56	59	t→0	t→0	AUX
ma-174	56	60	+	+	CCONJ
ma-174	56	61	‖ttx	‖ttx	PROPN
ma-174	56	62	−	−	PROPN
ma-174	56	63	x‖	x‖	PROPN
ma-174	57	1	=	=	NOUN
ma-174	57	2	0	0	PROPN
ma-174	57	3	for	for	ADP
ma-174	57	4	all	all	DET
ma-174	57	5	x	x	SYM
ma-174	57	6	∈	∈	NOUN
ma-174	57	7	x.	x.	NOUN
ma-174	58	1	the	the	DET
ma-174	58	2	infinitesimal	infinitesimal	ADJ
ma-174	58	3	generator	generator	NOUN
ma-174	58	4	denoted	denote	VERB
ma-174	58	5	by	by	ADP
ma-174	58	6	γ	γ	NOUN
ma-174	58	7	of	of	ADP
ma-174	58	8	(	(	PUNCT
ma-174	58	9	tt)t≥0	tt)t≥0	NOUN
ma-174	58	10	is	be	AUX
ma-174	58	11	defined	define	VERB
ma-174	58	12	by	by	ADP
ma-174	58	13	γx	γx	NOUN
ma-174	58	14	:	:	PUNCT
ma-174	58	15	=	=	PUNCT
ma-174	58	16	lim	lim	PROPN
ma-174	58	17	t→0	t→0	PROPN
ma-174	58	18	+	+	PROPN
ma-174	58	19	ttx	ttx	PROPN
ma-174	58	20	−	−	PROPN
ma-174	58	21	x	x	SYM
ma-174	58	22	t	t	PROPN
ma-174	58	23	=	=	SYM
ma-174	58	24	∂	∂	NUM
ma-174	58	25	∂t	∂t	PROPN
ma-174	58	26	(	(	PUNCT
ma-174	58	27	ttx	ttx	PROPN
ma-174	58	28	)	)	PUNCT
ma-174	58	29	∣∣∣∣	∣∣∣∣	PROPN
ma-174	58	30	t=0	t=0	PROPN
ma-174	58	31	for	for	ADP
ma-174	58	32	each	each	DET
ma-174	58	33	x	x	SYM
ma-174	58	34	∈	∈	PROPN
ma-174	58	35	dom(γ	dom(γ	PROPN
ma-174	58	36	)	)	PUNCT
ma-174	58	37	,	,	PUNCT
ma-174	58	38	where	where	SCONJ
ma-174	58	39	dom(γ	dom(γ	NOUN
ma-174	58	40	)	)	PUNCT
ma-174	58	41	denotes	denote	VERB
ma-174	58	42	the	the	DET
ma-174	58	43	domain	domain	NOUN
ma-174	58	44	of	of	ADP
ma-174	58	45	γ	γ	PROPN
ma-174	58	46	given	give	VERB
ma-174	58	47	by	by	ADP
ma-174	58	48	dom(γ	dom(γ	PROPN
ma-174	58	49	)	)	PUNCT
ma-174	58	50	=	=	PRON
ma-174	58	51	{	{	PUNCT
ma-174	58	52	x	x	PUNCT
ma-174	58	53	∈	∈	NOUN
ma-174	58	54	x	x	X
ma-174	58	55	:	:	PUNCT
ma-174	58	56	lim	lim	PROPN
ma-174	58	57	t→0	t→0	PROPN
ma-174	58	58	+	+	PROPN
ma-174	58	59	ttx	ttx	PROPN
ma-174	59	1	−	−	PROPN
ma-174	59	2	x	x	SYM
ma-174	59	3	t	t	PROPN
ma-174	59	4	exists	exist	VERB
ma-174	59	5	}	}	PUNCT
ma-174	59	6	.	.	PUNCT
ma-174	60	1	we	we	PRON
ma-174	60	2	define	define	VERB
ma-174	60	3	a	a	DET
ma-174	60	4	group	group	NOUN
ma-174	60	5	of	of	ADP
ma-174	60	6	bounded	bounded	ADJ
ma-174	60	7	linear	linear	PROPN
ma-174	60	8	operators	operator	NOUN
ma-174	60	9	as	as	ADP
ma-174	60	10	(	(	PUNCT
ma-174	60	11	tt)t∈r	tt)t∈r	X
ma-174	60	12	=	=	SYM
ma-174	60	13	tt	tt	VERB
ma-174	60	14	,	,	PUNCT
ma-174	60	15	t	t	PROPN
ma-174	60	16	≥	≥	NUM
ma-174	60	17	0	0	NUM
ma-174	60	18	,	,	PUNCT
ma-174	60	19	t−t	t−t	PROPN
ma-174	60	20	,	,	PUNCT
ma-174	60	21	t	t	PROPN
ma-174	60	22	≥	≥	NUM
ma-174	60	23	0	0	NUM
ma-174	60	24	.	.	PUNCT
ma-174	61	1	if	if	SCONJ
ma-174	61	2	both	both	PRON
ma-174	61	3	(	(	PUNCT
ma-174	61	4	tt)t≥0	tt)t≥0	NOUN
ma-174	61	5	and	and	CCONJ
ma-174	61	6	(	(	PUNCT
ma-174	61	7	t−t)t≥0	t−t)t≥0	NOUN
ma-174	61	8	are	be	AUX
ma-174	61	9	semigroups	semigroup	NOUN
ma-174	61	10	on	on	ADP
ma-174	61	11	x	x	X
ma-174	61	12	.	.	PUNCT
ma-174	62	1	for	for	ADP
ma-174	62	2	more	more	ADJ
ma-174	62	3	details	detail	NOUN
ma-174	62	4	see	see	VERB
ma-174	62	5	[	[	X
ma-174	62	6	5,6,9	5,6,9	NUM
ma-174	62	7	]	]	PUNCT
ma-174	62	8	.	.	PUNCT
ma-174	63	1	suppose	suppose	VERB
ma-174	63	2	ϕ	ϕ	X
ma-174	63	3	:	:	PUNCT
ma-174	63	4	ω→	ω→	NUM
ma-174	63	5	ωis	ωis	PROPN
ma-174	63	6	a	a	DET
ma-174	63	7	self	self	NOUN
ma-174	63	8	analytic	analytic	ADJ
ma-174	63	9	map	map	NOUN
ma-174	63	10	.	.	PUNCT
ma-174	64	1	the	the	DET
ma-174	64	2	composition	composition	NOUN
ma-174	64	3	operator	operator	NOUN
ma-174	64	4	induced	induce	VERB
ma-174	64	5	by	by	ADP
ma-174	64	6	ϕ	ϕ	NOUN
ma-174	64	7	on	on	ADP
ma-174	64	8	h(ω	h(ω	PROPN
ma-174	64	9	)	)	PUNCT
ma-174	64	10	is	be	AUX
ma-174	64	11	defined	define	VERB
ma-174	64	12	as	as	ADP
ma-174	64	13	cϕ(f	cϕ(f	X
ma-174	64	14	)	)	PUNCT
ma-174	65	1	=	=	PUNCT
ma-174	66	1	f	f	X
ma-174	66	2	o	o	X
ma-174	66	3	ϕ	ϕ	PROPN
ma-174	66	4	,	,	PUNCT
ma-174	66	5	for	for	ADP
ma-174	66	6	all	all	DET
ma-174	66	7	f	f	PROPN
ma-174	66	8	∈	∈	PROPN
ma-174	66	9	h(ω	h(ω	PROPN
ma-174	66	10	)	)	PUNCT
ma-174	66	11	.	.	PUNCT
ma-174	67	1	on	on	ADP
ma-174	67	2	the	the	DET
ma-174	67	3	other	other	ADJ
ma-174	67	4	hand	hand	NOUN
ma-174	67	5	,	,	PUNCT
ma-174	67	6	given	give	VERB
ma-174	67	7	t	t	PROPN
ma-174	67	8	≥	≥	NOUN
ma-174	67	9	0	0	NUM
ma-174	67	10	we	we	PRON
ma-174	67	11	define	define	VERB
ma-174	67	12	a	a	DET
ma-174	67	13	semigroup	semigroup	NOUN
ma-174	67	14	as	as	ADP
ma-174	67	15	a	a	DET
ma-174	67	16	family	family	NOUN
ma-174	67	17	(	(	PUNCT
ma-174	67	18	ϕt)t≥0	ϕt)t≥0	NOUN
ma-174	67	19	ofself	ofself	PRON
ma-174	67	20	analytic	analytic	ADJ
ma-174	67	21	maps	map	NOUN
ma-174	67	22	on	on	ADP
ma-174	67	23	ω	ω	NUM
ma-174	67	24	satisfying	satisfy	VERB
ma-174	67	25	the	the	DET
ma-174	67	26	following	follow	VERB
ma-174	67	27	properties(i	properties(i	NOUN
ma-174	67	28	)	)	PUNCT
ma-174	67	29	ϕ0(z	ϕ0(z	PROPN
ma-174	67	30	)	)	PUNCT
ma-174	68	1	=	=	SYM
ma-174	68	2	z	z	NOUN
ma-174	68	3	(	(	PUNCT
ma-174	68	4	identity	identity	NOUN
ma-174	68	5	map	map	NOUN
ma-174	68	6	on	on	ADP
ma-174	68	7	ω).(ii	ω).(ii	NOUN
ma-174	68	8	)	)	PUNCT
ma-174	68	9	ϕt+s	ϕt+s	PUNCT
ma-174	69	1	=	=	PUNCT
ma-174	69	2	ϕt	ϕt	ADV
ma-174	69	3	◦	◦	VERB
ma-174	69	4	ϕs	ϕs	INTJ
ma-174	69	5	,	,	PUNCT
ma-174	69	6	∀	∀	X
ma-174	69	7	t	t	PROPN
ma-174	69	8	,	,	PUNCT
ma-174	69	9	s	s	PART
ma-174	69	10	≥	≥	NOUN
ma-174	69	11	0	0	NUM
ma-174	69	12	(	(	PUNCT
ma-174	69	13	semigroup	semigroup	PROPN
ma-174	69	14	property).(iii	property).(iii	PROPN
ma-174	69	15	)	)	PUNCT
ma-174	69	16	ϕt	ϕt	PROPN
ma-174	70	1	→	→	PUNCT
ma-174	70	2	ϕ0	ϕ0	NOUN
ma-174	70	3	uniformly	uniformly	ADV
ma-174	70	4	on	on	ADP
ma-174	70	5	compact	compact	ADJ
ma-174	70	6	subsets	subset	NOUN
ma-174	70	7	of	of	ADP
ma-174	70	8	ω	ω	PROPN
ma-174	70	9	as	as	ADP
ma-174	70	10	t	t	PROPN
ma-174	70	11	→	→	SYM
ma-174	70	12	0.3	0.3	NUM
ma-174	70	13	composition	composition	NOUN
ma-174	70	14	semigroup	semigroup	NOUN
ma-174	70	15	induced	induce	VERB
ma-174	70	16	by	by	ADP
ma-174	70	17	ϕt	ϕt	PROPN
ma-174	70	18	on	on	ADP
ma-174	70	19	h(ω	h(ω	PROPN
ma-174	70	20	)	)	PUNCT
ma-174	70	21	is	be	AUX
ma-174	70	22	defined	define	VERB
ma-174	70	23	as	as	ADP
ma-174	70	24	cϕt	cϕt	PROPN
ma-174	70	25	(	(	PUNCT
ma-174	70	26	f	f	PROPN
ma-174	70	27	)	)	PUNCT
ma-174	71	1	=	=	PUNCT
ma-174	72	1	f	f	X
ma-174	73	1	o	o	INTJ
ma-174	73	2	ϕt	ϕt	INTJ
ma-174	73	3	,	,	PUNCT
ma-174	73	4	for	for	ADP
ma-174	73	5	all	all	DET
ma-174	73	6	f	f	PROPN
ma-174	73	7	∈	∈	PROPN
ma-174	73	8	h(ω	h(ω	PROPN
ma-174	73	9	)	)	PUNCT
ma-174	73	10	.	.	PUNCT
ma-174	74	1	3	3	X
ma-174	74	2	.	.	NUM
ma-174	74	3	generalized	generalize	VERB
ma-174	74	4	bloch	bloch	PROPN
ma-174	74	5	spaces	space	NOUN
ma-174	74	6	of	of	ADP
ma-174	74	7	the	the	DET
ma-174	74	8	upper	upper	ADJ
ma-174	74	9	half	half	ADJ
ma-174	74	10	plane	plane	NOUN
ma-174	74	11	in	in	ADP
ma-174	74	12	this	this	DET
ma-174	74	13	section	section	NOUN
ma-174	74	14	,	,	PUNCT
ma-174	74	15	we	we	PRON
ma-174	74	16	study	study	VERB
ma-174	74	17	properties	property	NOUN
ma-174	74	18	of	of	ADP
ma-174	74	19	the	the	DET
ma-174	74	20	generalized	generalized	ADJ
ma-174	74	21	bloch	bloch	NOUN
ma-174	74	22	spaces	space	VERB
ma-174	74	23	as	as	ADP
ma-174	74	24	banach	banach	NOUN
ma-174	74	25	spaces	space	VERB
ma-174	74	26	.	.	PUNCT
ma-174	75	1	we	we	PRON
ma-174	75	2	alsorelate	alsorelate	VERB
ma-174	75	3	functions	function	NOUN
ma-174	75	4	in	in	ADP
ma-174	75	5	the	the	DET
ma-174	75	6	generalized	generalized	ADJ
ma-174	75	7	bloch	bloch	NOUN
ma-174	75	8	space	space	NOUN
ma-174	75	9	of	of	ADP
ma-174	75	10	the	the	DET
ma-174	75	11	upper	upper	ADJ
ma-174	75	12	half	half	ADJ
ma-174	75	13	plane	plane	NOUN
ma-174	75	14	u	u	NOUN
ma-174	75	15	to	to	ADP
ma-174	75	16	their	their	PRON
ma-174	75	17	counterparts	counterpart	NOUN
ma-174	75	18	inthe	inthe	DET
ma-174	75	19	unit	unit	NOUN
ma-174	75	20	disc	disc	VERB
ma-174	75	21	d.	d.	PROPN
ma-174	75	22	following	follow	VERB
ma-174	75	23	[	[	X
ma-174	75	24	17,18	17,18	X
ma-174	75	25	]	]	X
ma-174	75	26	,	,	PUNCT
ma-174	75	27	it	it	PRON
ma-174	75	28	’s	’	VERB
ma-174	75	29	well	well	ADV
ma-174	75	30	known	know	VERB
ma-174	75	31	that	that	SCONJ
ma-174	75	32	bα(d	bα(d	NOUN
ma-174	75	33	)	)	PUNCT
ma-174	75	34	and	and	CCONJ
ma-174	75	35	bα0	bα0	PROPN
ma-174	75	36	(	(	PUNCT
ma-174	75	37	d	d	NOUN
ma-174	75	38	)	)	PUNCT
ma-174	75	39	are	be	AUX
ma-174	75	40	banach	banach	NOUN
ma-174	75	41	spaces	space	NOUN
ma-174	75	42	withrespect	withrespect	ADJ
ma-174	75	43	to	to	ADP
ma-174	75	44	the	the	DET
ma-174	75	45	norm	norm	NOUN
ma-174	75	46	‖.‖bα(d	‖.‖bα(d	NUM
ma-174	75	47	)	)	PUNCT
ma-174	75	48	.	.	PUNCT
ma-174	76	1	moreover	moreover	ADV
ma-174	76	2	the	the	DET
ma-174	76	3	set	set	NOUN
ma-174	76	4	of	of	ADP
ma-174	76	5	analytic	analytic	ADJ
ma-174	76	6	polynomials	polynomial	NOUN
ma-174	76	7	c[z	c[z	PART
ma-174	76	8	]	]	PUNCT
ma-174	76	9	:	:	PUNCT
ma-174	76	10	=	=	X
ma-174	76	11	{	{	PUNCT
ma-174	76	12	∞∑	∞∑	NUM
ma-174	76	13	n=0	n=0	NUM
ma-174	76	14	an	an	DET
ma-174	76	15	z	z	NOUN
ma-174	76	16	n	n	NOUN
ma-174	76	17	:	:	PUNCT
ma-174	76	18	z	z	X
ma-174	76	19	∈	∈	PROPN
ma-174	76	20	c	c	X
ma-174	76	21	}	}	PUNCT
ma-174	76	22	is	be	AUX
ma-174	76	23	dense	dense	ADJ
ma-174	76	24	in	in	ADP
ma-174	76	25	bα0	bα0	PROPN
ma-174	76	26	(	(	PUNCT
ma-174	76	27	d	d	NOUN
ma-174	76	28	)	)	PUNCT
ma-174	76	29	.	.	PUNCT
ma-174	77	1	these	these	DET
ma-174	77	2	results	result	NOUN
ma-174	77	3	are	be	AUX
ma-174	77	4	not	not	PART
ma-174	77	5	explicitly	explicitly	ADV
ma-174	77	6	clear	clear	ADJ
ma-174	77	7	from	from	ADP
ma-174	77	8	the	the	DET
ma-174	77	9	literature	literature	NOUN
ma-174	77	10	in	in	ADP
ma-174	77	11	the	the	DET
ma-174	77	12	setting	setting	NOUN
ma-174	77	13	of	of	ADP
ma-174	77	14	theupper	theupper	NOUN
ma-174	77	15	half	half	ADJ
ma-174	77	16	plane	plane	NOUN
ma-174	77	17	u.in	u.in	PROPN
ma-174	77	18	the	the	DET
ma-174	77	19	following	follow	VERB
ma-174	77	20	theorem	theorem	VERB
ma-174	77	21	,	,	PUNCT
ma-174	77	22	we	we	PRON
ma-174	77	23	establish	establish	VERB
ma-174	77	24	the	the	DET
ma-174	77	25	completeness	completeness	NOUN
ma-174	77	26	of	of	ADP
ma-174	77	27	bα(u	bα(u	NOUN
ma-174	77	28	)	)	PUNCT
ma-174	77	29	with	with	ADP
ma-174	77	30	respect	respect	NOUN
ma-174	77	31	to	to	ADP
ma-174	77	32	the	the	DET
ma-174	77	33	norm	norm	NOUN
ma-174	77	34	‖.‖bα(u	‖.‖bα(u	NUM
ma-174	77	35	)	)	PUNCT
ma-174	77	36	.	.	PUNCT
ma-174	78	1	theorem	theorem	VERB
ma-174	78	2	3.1	3.1	NUM
ma-174	78	3	.	.	PUNCT
ma-174	78	4	bα(u	bα(u	PROPN
ma-174	78	5	)	)	PUNCT
ma-174	79	1	is	be	AUX
ma-174	79	2	a	a	DET
ma-174	79	3	banach	banach	NOUN
ma-174	79	4	space	space	NOUN
ma-174	79	5	with	with	ADP
ma-174	79	6	respect	respect	NOUN
ma-174	79	7	to	to	ADP
ma-174	79	8	the	the	DET
ma-174	79	9	norm	norm	NOUN
ma-174	79	10	‖.‖bα(u	‖.‖bα(u	NUM
ma-174	79	11	)	)	PUNCT
ma-174	79	12	proof	proof	NOUN
ma-174	79	13	.	.	PUNCT
ma-174	80	1	it	it	PRON
ma-174	80	2	’s	’	VERB
ma-174	80	3	clear	clear	ADJ
ma-174	80	4	that	that	SCONJ
ma-174	80	5	(	(	PUNCT
ma-174	80	6	bα(u	bα(u	NOUN
ma-174	80	7	)	)	PUNCT
ma-174	80	8	,	,	PUNCT
ma-174	80	9	‖.‖bα(u	‖.‖bα(u	NUM
ma-174	80	10	)	)	PUNCT
ma-174	80	11	)	)	PUNCT
ma-174	80	12	is	be	AUX
ma-174	80	13	a	a	DET
ma-174	80	14	normed	normed	ADJ
ma-174	80	15	space	space	NOUN
ma-174	80	16	.	.	PUNCT
ma-174	81	1	now	now	ADV
ma-174	81	2	we	we	PRON
ma-174	81	3	prove	prove	VERB
ma-174	81	4	that	that	SCONJ
ma-174	81	5	the	the	DET
ma-174	81	6	space	space	NOUN
ma-174	81	7	bα(u)is	bα(u)is	ADJ
ma-174	81	8	complete	complete	ADJ
ma-174	81	9	in	in	ADP
ma-174	81	10	‖.‖bα(u	‖.‖bα(u	NUM
ma-174	81	11	)	)	PUNCT
ma-174	81	12	.	.	PUNCT
ma-174	82	1	let	let	VERB
ma-174	82	2	(	(	PUNCT
ma-174	82	3	fk)k	fk)k	PROPN
ma-174	82	4	denote	denote	VERB
ma-174	82	5	a	a	DET
ma-174	82	6	cauchy	cauchy	ADJ
ma-174	82	7	sequence	sequence	NOUN
ma-174	82	8	in	in	ADP
ma-174	82	9	bα(u	bα(u	NOUN
ma-174	82	10	)	)	PUNCT
ma-174	82	11	.	.	PUNCT
ma-174	83	1	for	for	ADP
ma-174	83	2	ε	ε	PROPN
ma-174	83	3	>	>	X
ma-174	83	4	0	0	PROPN
ma-174	83	5	,	,	PUNCT
ma-174	83	6	there	there	PRON
ma-174	83	7	exists	exist	VERB
ma-174	83	8	n	n	PRON
ma-174	83	9	∈	∈	PROPN
ma-174	83	10	n	n	PRON
ma-174	83	11	such	such	ADJ
ma-174	83	12	that	that	SCONJ
ma-174	83	13	‖fk	‖fk	PROPN
ma-174	83	14	−	−	PROPN
ma-174	83	15	fl‖bα(u	fl‖bα(u	NOUN
ma-174	83	16	)	)	PUNCT
ma-174	83	17	<	<	X
ma-174	83	18	ε	ε	PROPN
ma-174	83	19	,	,	PUNCT
ma-174	83	20	∀	∀	X
ma-174	83	21	k	k	NOUN
ma-174	83	22	,	,	PUNCT
ma-174	83	23	l	l	NOUN
ma-174	83	24	>	>	X
ma-174	83	25	n.	n.	NOUN
ma-174	83	26	hence	hence	ADV
ma-174	83	27	by	by	ADP
ma-174	83	28	the	the	DET
ma-174	83	29	definition	definition	NOUN
ma-174	83	30	of	of	ADP
ma-174	83	31	the	the	DET
ma-174	83	32	norm	norm	NOUN
ma-174	83	33	,	,	PUNCT
ma-174	83	34	we	we	PRON
ma-174	83	35	have	have	AUX
ma-174	83	36	forall	forall	NOUN
ma-174	83	37	∀	∀	X
ma-174	83	38	k	k	NOUN
ma-174	83	39	,	,	PUNCT
ma-174	83	40	l	l	NOUN
ma-174	83	41	>	>	X
ma-174	83	42	n	n	CCONJ
ma-174	83	43	,	,	PUNCT
ma-174	83	44	|fk(i)−	|fk(i)−	PROPN
ma-174	83	45	fl(i)|+	fl(i)|+	PROPN
ma-174	83	46	sup	sup	NOUN
ma-174	83	47	ω∈u	ω∈u	NOUN
ma-174	83	48	=	=	SYM
ma-174	83	49	(	(	PUNCT
ma-174	83	50	ω)α	ω)α	X
ma-174	83	51	|f	|f	PRON
ma-174	83	52	′k(ω)−	′k(ω)−	NOUN
ma-174	83	53	f	f	PROPN
ma-174	83	54	′l	′l	PROPN
ma-174	83	55	(	(	PUNCT
ma-174	83	56	ω)|	ω)|	ADV
ma-174	83	57	<	<	X
ma-174	83	58	ε	ε	PROPN
ma-174	83	59	,	,	PUNCT
ma-174	83	60	which	which	PRON
ma-174	83	61	means	mean	VERB
ma-174	83	62	that	that	SCONJ
ma-174	83	63	|fk(i)−	|fk(i)−	NOUN
ma-174	83	64	fl(i)|	fl(i)|	X
ma-174	83	65	<	<	X
ma-174	83	66	ε	ε	PROPN
ma-174	83	67	and	and	CCONJ
ma-174	83	68	(=	(=	ADJ
ma-174	83	69	(	(	PUNCT
ma-174	83	70	ω))α	ω))α	NOUN
ma-174	83	71	|f	|f	PROPN
ma-174	83	72	′k(ω)−	′k(ω)−	NOUN
ma-174	83	73	f	f	PROPN
ma-174	83	74	′l	′l	PROPN
ma-174	83	75	(	(	PUNCT
ma-174	83	76	ω)|	ω)|	ADV
ma-174	83	77	<	<	X
ma-174	83	78	ε	ε	PROPN
ma-174	83	79	,	,	PUNCT
ma-174	83	80	for	for	ADP
ma-174	83	81	ω	ω	PROPN
ma-174	83	82	∈	∈	PROPN
ma-174	83	83	u.	u.	NOUN
ma-174	84	1	so	so	ADV
ma-174	84	2	,	,	PUNCT
ma-174	84	3	(	(	PUNCT
ma-174	84	4	fk(i))k∈n	fk(i))k∈n	ADV
ma-174	84	5	is	be	AUX
ma-174	84	6	cauchy	cauchy	ADJ
ma-174	84	7	in	in	ADP
ma-174	84	8	c.	c.	NOUN
ma-174	84	9	by	by	ADP
ma-174	84	10	the	the	DET
ma-174	84	11	completeness	completeness	NOUN
ma-174	84	12	of	of	ADP
ma-174	84	13	c	c	PROPN
ma-174	84	14	,	,	PUNCT
ma-174	84	15	(	(	PUNCT
ma-174	84	16	fk(i))k	fk(i))k	NOUN
ma-174	84	17	converges	converge	VERB
ma-174	84	18	to	to	ADP
ma-174	84	19	a	a	DET
ma-174	84	20	limit	limit	NOUN
ma-174	84	21	,	,	PUNCT
ma-174	84	22	say	say	VERB
ma-174	84	23	u0	u0	ADJ
ma-174	84	24	.	.	PUNCT
ma-174	85	1	similarly	similarly	ADV
ma-174	85	2	,	,	PUNCT
ma-174	85	3	(	(	PUNCT
ma-174	85	4	f	f	PROPN
ma-174	85	5	′k(ω	′k(ω	PROPN
ma-174	85	6	)	)	PUNCT
ma-174	85	7	)	)	PUNCT
ma-174	86	1	k∈n	k∈n	PROPN
ma-174	86	2	is	be	AUX
ma-174	86	3	cauchy	cauchy	ADJ
ma-174	86	4	in	in	ADP
ma-174	86	5	c	c	PROPN
ma-174	86	6	and	and	CCONJ
ma-174	86	7	therefore	therefore	ADV
ma-174	86	8	converges	converge	VERB
ma-174	86	9	to	to	ADP
ma-174	86	10	a	a	DET
ma-174	86	11	limit	limit	NOUN
ma-174	86	12	,	,	PUNCT
ma-174	86	13	say	say	VERB
ma-174	86	14	g.since	g.since	NOUN
ma-174	86	15	|f	|f	PROPN
ma-174	86	16	′k(ω)−	′k(ω)−	NOUN
ma-174	86	17	f	f	PROPN
ma-174	86	18	′l	′l	PROPN
ma-174	86	19	(	(	PUNCT
ma-174	86	20	ω)|	ω)|	ADV
ma-174	86	21	<	<	X
ma-174	86	22	ε	ε	PROPN
ma-174	86	23	=(	=(	NOUN
ma-174	86	24	ω)α	ω)α	PUNCT
ma-174	86	25	and	and	CCONJ
ma-174	86	26	f	f	NUM
ma-174	86	27	′k(ω)→	′k(ω)→	NOUN
ma-174	86	28	g	g	PRON
ma-174	86	29	uniformly	uniformly	ADV
ma-174	86	30	on	on	ADP
ma-174	86	31	compact	compact	ADJ
ma-174	86	32	subsets	subset	NOUN
ma-174	86	33	of	of	ADP
ma-174	86	34	u	u	NOUN
ma-174	86	35	,	,	PUNCT
ma-174	86	36	then	then	ADV
ma-174	86	37	g	g	PROPN
ma-174	86	38	∈	∈	PROPN
ma-174	86	39	h(u).now	h(u).now	ADV
ma-174	86	40	,	,	PUNCT
ma-174	86	41	take	take	VERB
ma-174	86	42	f	f	PRON
ma-174	86	43	such	such	ADJ
ma-174	86	44	that	that	SCONJ
ma-174	86	45	f	f	PROPN
ma-174	86	46	′(ω	′(ω	PROPN
ma-174	86	47	)	)	PUNCT
ma-174	86	48	=	=	PUNCT
ma-174	86	49	g(ω)∀ω	g(ω)∀ω	NOUN
ma-174	86	50	∈	∈	PROPN
ma-174	86	51	u	u	NOUN
ma-174	86	52	and	and	CCONJ
ma-174	86	53	f	f	PROPN
ma-174	86	54	(	(	PUNCT
ma-174	86	55	i	i	NOUN
ma-174	86	56	)	)	PUNCT
ma-174	86	57	=	=	PUNCT
ma-174	87	1	u0.thus	u0.thu	NOUN
ma-174	87	2	,	,	PUNCT
ma-174	87	3	∀	∀	X
ma-174	87	4	ε	ε	X
ma-174	87	5	>	>	X
ma-174	87	6	0	0	PROPN
ma-174	87	7	,	,	PUNCT
ma-174	87	8	∃n	∃n	ADP
ma-174	87	9	such	such	ADJ
ma-174	87	10	that	that	SCONJ
ma-174	87	11	∀k	∀k	NOUN
ma-174	87	12	,	,	PUNCT
ma-174	87	13	l	l	NOUN
ma-174	87	14	>	>	X
ma-174	87	15	n	n	PROPN
ma-174	87	16	,	,	PUNCT
ma-174	87	17	=	=	SYM
ma-174	87	18	(	(	PUNCT
ma-174	87	19	ω)α	ω)α	X
ma-174	87	20	|f	|f	PRON
ma-174	87	21	′k(ω)−	′k(ω)−	NOUN
ma-174	87	22	f	f	PROPN
ma-174	87	23	′l	′l	PROPN
ma-174	87	24	(	(	PUNCT
ma-174	87	25	ω)|	ω)|	ADV
ma-174	87	26	<	<	X
ma-174	87	27	ε	ε	PROPN
ma-174	87	28	,	,	PUNCT
ma-174	87	29	∀ω	∀ω	PROPN
ma-174	87	30	∈	∈	PROPN
ma-174	87	31	u.	u.	NOUN
ma-174	87	32	taking	take	VERB
ma-174	87	33	limits	limit	NOUN
ma-174	87	34	as	as	ADP
ma-174	87	35	l	l	PROPN
ma-174	87	36	→∞	→∞	PROPN
ma-174	87	37	,	,	PUNCT
ma-174	87	38	then	then	ADV
ma-174	87	39	∀k	∀k	NOUN
ma-174	87	40	>	>	X
ma-174	87	41	n	n	PROPN
ma-174	87	42	,	,	PUNCT
ma-174	87	43	=	=	SYM
ma-174	87	44	(	(	PUNCT
ma-174	87	45	ω)α	ω)α	X
ma-174	87	46	|f	|f	PROPN
ma-174	87	47	′k(ω)−	′k(ω)−	NOUN
ma-174	87	48	f	f	X
ma-174	87	49	′(ω)|	′(ω)|	X
ma-174	87	50	<	<	X
ma-174	87	51	ε	ε	PROPN
ma-174	87	52	,	,	PUNCT
ma-174	87	53	∀ω	∀ω	PUNCT
ma-174	87	54	∈	∈	PROPN
ma-174	87	55	u.	u.	NOUN
ma-174	87	56	it	it	PRON
ma-174	87	57	follows	follow	VERB
ma-174	87	58	that	that	SCONJ
ma-174	87	59	‖fk	‖fk	NUM
ma-174	87	60	−	−	PUNCT
ma-174	87	61	f	f	PROPN
ma-174	87	62	‖bα(u	‖bα(u	PROPN
ma-174	87	63	)	)	PUNCT
ma-174	88	1	=	=	X
ma-174	88	2	|fk(i)−	|fk(i)−	PROPN
ma-174	88	3	f	f	X
ma-174	88	4	(	(	PUNCT
ma-174	88	5	i)|+	i)|+	ADJ
ma-174	88	6	sup	sup	NOUN
ma-174	88	7	ω∈u	ω∈u	NOUN
ma-174	88	8	=	=	SYM
ma-174	88	9	(	(	PUNCT
ma-174	88	10	ω)α	ω)α	X
ma-174	88	11	|f	|f	PROPN
ma-174	88	12	′k(ω)−	′k(ω)−	NOUN
ma-174	89	1	f	f	X
ma-174	89	2	′(ω)|	′(ω)|	X
ma-174	89	3	<	<	X
ma-174	89	4	ε	ε	PROPN
ma-174	89	5	4	4	NUM
ma-174	89	6	and	and	CCONJ
ma-174	89	7	so	so	ADV
ma-174	89	8	‖fk	‖fk	NUM
ma-174	89	9	−	−	PROPN
ma-174	89	10	f	f	PROPN
ma-174	89	11	‖bα(u	‖bα(u	PROPN
ma-174	89	12	)	)	PUNCT
ma-174	89	13	→	→	SYM
ma-174	89	14	0	0	PUNCT
ma-174	89	15	as	as	ADP
ma-174	89	16	k	k	PROPN
ma-174	89	17	→∞.now	→∞.now	PROPN
ma-174	89	18	,	,	PUNCT
ma-174	89	19	it	it	PRON
ma-174	89	20	remains	remain	VERB
ma-174	89	21	to	to	PART
ma-174	89	22	show	show	VERB
ma-174	89	23	that	that	SCONJ
ma-174	89	24	f	f	PROPN
ma-174	89	25	∈	∈	PROPN
ma-174	89	26	bα(u	bα(u	NOUN
ma-174	89	27	)	)	PUNCT
ma-174	89	28	.	.	PUNCT
ma-174	90	1	we	we	PRON
ma-174	90	2	have	have	VERB
ma-174	90	3	=	=	SYM
ma-174	90	4	(	(	PUNCT
ma-174	90	5	ω)α	ω)α	X
ma-174	90	6	|f	|f	PRON
ma-174	91	1	′(ω)|	′(ω)|	PROPN
ma-174	91	2	=	=	PUNCT
ma-174	92	1	=	=	SYM
ma-174	92	2	(	(	PUNCT
ma-174	92	3	ω)α	ω)α	X
ma-174	92	4	|f	|f	PROPN
ma-174	92	5	′(ω)−	′(ω)−	PROPN
ma-174	92	6	f	f	PROPN
ma-174	92	7	′k(ω	′k(ω	PROPN
ma-174	92	8	)	)	PUNCT
ma-174	93	1	+	+	CCONJ
ma-174	93	2	f	f	X
ma-174	93	3	′k(ω)|	′k(ω)|	ADV
ma-174	93	4	≤	≤	ADV
ma-174	93	5	=	=	SYM
ma-174	93	6	(	(	PUNCT
ma-174	93	7	ω)α	ω)α	X
ma-174	93	8	|f	|f	PROPN
ma-174	94	1	′(ω)−	′(ω)−	PROPN
ma-174	94	2	f	f	PROPN
ma-174	95	1	′kω|+	′kω|+	PROPN
ma-174	95	2	=	=	PRON
ma-174	95	3	(	(	PUNCT
ma-174	95	4	ω)α	ω)α	X
ma-174	95	5	|f	|f	PROPN
ma-174	96	1	′k(ω)|	′k(ω)|	ADV
ma-174	96	2	<	<	X
ma-174	96	3	ε+	ε+	X
ma-174	96	4	=	=	SYM
ma-174	96	5	(	(	PUNCT
ma-174	96	6	ω)α	ω)α	X
ma-174	96	7	|f	|f	PROPN
ma-174	96	8	′k(ω)|	′k(ω)|	PROPN
ma-174	96	9	<	<	X
ma-174	96	10	∞	∞	PROPN
ma-174	96	11	since	since	SCONJ
ma-174	96	12	(	(	PUNCT
ma-174	96	13	fk)k	fk)k	PROPN
ma-174	96	14	⊂	⊂	PROPN
ma-174	96	15	bα(u).now	bα(u).now	PROPN
ma-174	96	16	,	,	PUNCT
ma-174	96	17	taking	take	VERB
ma-174	96	18	supremum	supremum	ADV
ma-174	96	19	over	over	ADP
ma-174	96	20	all	all	DET
ma-174	96	21	ω	ω	NUM
ma-174	96	22	∈	∈	PROPN
ma-174	96	23	u	u	NOUN
ma-174	96	24	in	in	ADP
ma-174	96	25	the	the	DET
ma-174	96	26	above	above	ADJ
ma-174	96	27	equation	equation	NOUN
ma-174	96	28	,	,	PUNCT
ma-174	96	29	we	we	PRON
ma-174	96	30	have	have	VERB
ma-174	96	31	that	that	DET
ma-174	96	32	sup	sup	NOUN
ma-174	96	33	ω∈u	ω∈u	NOUN
ma-174	96	34	=	=	SYM
ma-174	96	35	(	(	PUNCT
ma-174	96	36	ω)α	ω)α	X
ma-174	96	37	|f	|f	PRON
ma-174	97	1	′(ω)|	′(ω)|	PROPN
ma-174	98	1	<	<	X
ma-174	98	2	∞	∞	X
ma-174	98	3	which	which	PRON
ma-174	98	4	implies	imply	VERB
ma-174	98	5	that	that	SCONJ
ma-174	98	6	f	f	PROPN
ma-174	98	7	∈	∈	PROPN
ma-174	98	8	bα(u	bα(u	NOUN
ma-174	98	9	)	)	PUNCT
ma-174	98	10	,	,	PUNCT
ma-174	98	11	as	as	SCONJ
ma-174	98	12	desired	desire	VERB
ma-174	98	13	.	.	PUNCT
ma-174	99	1	�	�	PROPN
ma-174	99	2	as	as	ADP
ma-174	99	3	an	an	DET
ma-174	99	4	immediate	immediate	ADJ
ma-174	99	5	consequence	consequence	NOUN
ma-174	99	6	,	,	PUNCT
ma-174	99	7	we	we	PRON
ma-174	99	8	have	have	VERB
ma-174	99	9	corollary	corollary	ADJ
ma-174	99	10	3.2	3.2	NUM
ma-174	99	11	.	.	PUNCT
ma-174	100	1	b(u	b(u	PROPN
ma-174	100	2	)	)	PUNCT
ma-174	100	3	is	be	AUX
ma-174	100	4	a	a	DET
ma-174	100	5	banach	banach	NOUN
ma-174	100	6	space	space	NOUN
ma-174	100	7	with	with	ADP
ma-174	100	8	respect	respect	NOUN
ma-174	100	9	to	to	ADP
ma-174	100	10	the	the	DET
ma-174	100	11	norm	norm	NOUN
ma-174	100	12	‖	‖	PROPN
ma-174	100	13	.	.	PUNCT
ma-174	101	1	‖b(u	‖b(u	X
ma-174	101	2	)	)	PUNCT
ma-174	101	3	proof	proof	NOUN
ma-174	101	4	.	.	PUNCT
ma-174	101	5	follows	follow	VERB
ma-174	101	6	immediately	immediately	ADV
ma-174	101	7	by	by	ADP
ma-174	101	8	taking	take	VERB
ma-174	101	9	α	α	NOUN
ma-174	101	10	=	=	SYM
ma-174	101	11	1	1	NUM
ma-174	101	12	in	in	ADP
ma-174	101	13	theorem	theorem	ADJ
ma-174	101	14	3.1	3.1	NUM
ma-174	101	15	.	.	PUNCT
ma-174	101	16	�	�	PROPN
ma-174	101	17	under	under	ADP
ma-174	101	18	the	the	DET
ma-174	101	19	norm	norm	NOUN
ma-174	101	20	‖	‖	PROPN
ma-174	101	21	.	.	PUNCT
ma-174	102	1	‖bα(u	‖bα(u	PROPN
ma-174	102	2	)	)	PUNCT
ma-174	103	1	,	,	PUNCT
ma-174	103	2	the	the	DET
ma-174	103	3	space	space	NOUN
ma-174	103	4	bα0	bα0	NOUN
ma-174	103	5	(	(	PUNCT
ma-174	103	6	u	u	NOUN
ma-174	103	7	)	)	PUNCT
ma-174	103	8	also	also	ADV
ma-174	103	9	becomes	become	VERB
ma-174	103	10	a	a	DET
ma-174	103	11	banach	banach	NOUN
ma-174	103	12	space	space	NOUN
ma-174	103	13	as	as	ADP
ma-174	103	14	in	in	ADP
ma-174	103	15	the	the	DET
ma-174	103	16	followingtheorem	followingtheorem	ADJ
ma-174	103	17	,	,	PUNCT
ma-174	103	18	theorem	theorem	VERB
ma-174	103	19	3.3	3.3	NUM
ma-174	103	20	.	.	PUNCT
ma-174	104	1	bα0	bα0	PROPN
ma-174	104	2	(	(	PUNCT
ma-174	104	3	u	u	NOUN
ma-174	104	4	)	)	PUNCT
ma-174	104	5	is	be	AUX
ma-174	104	6	a	a	DET
ma-174	104	7	banach	banach	NOUN
ma-174	104	8	space	space	NOUN
ma-174	104	9	with	with	ADP
ma-174	104	10	respect	respect	NOUN
ma-174	104	11	to	to	ADP
ma-174	104	12	the	the	DET
ma-174	104	13	norm	norm	NOUN
ma-174	104	14	‖	‖	PROPN
ma-174	104	15	.	.	PUNCT
ma-174	105	1	‖bα(u	‖bα(u	PROPN
ma-174	105	2	)	)	PUNCT
ma-174	105	3	.	.	PUNCT
ma-174	106	1	proof	proof	NOUN
ma-174	106	2	.	.	PUNCT
ma-174	107	1	following	follow	VERB
ma-174	107	2	theorem	theorem	VERB
ma-174	107	3	3.1	3.1	NUM
ma-174	107	4	,	,	PUNCT
ma-174	107	5	we	we	PRON
ma-174	107	6	need	need	VERB
ma-174	107	7	to	to	PART
ma-174	107	8	show	show	VERB
ma-174	107	9	that	that	SCONJ
ma-174	107	10	every	every	DET
ma-174	107	11	sequence	sequence	NOUN
ma-174	107	12	in	in	ADP
ma-174	107	13	bα0	bα0	PROPN
ma-174	107	14	(	(	PUNCT
ma-174	107	15	u	u	NOUN
ma-174	107	16	)	)	PUNCT
ma-174	107	17	convergent	convergent	NOUN
ma-174	107	18	in	in	ADP
ma-174	107	19	bα(u)has	bα(u)ha	VERB
ma-174	107	20	its	its	PRON
ma-174	107	21	limit	limit	NOUN
ma-174	107	22	in	in	ADP
ma-174	107	23	bα0	bα0	PROPN
ma-174	107	24	(	(	PUNCT
ma-174	107	25	u).let	u).let	PROPN
ma-174	107	26	(	(	PUNCT
ma-174	107	27	fn	fn	NOUN
ma-174	107	28	)	)	PUNCT
ma-174	107	29	⊂	⊂	PROPN
ma-174	107	30	bα0	bα0	PROPN
ma-174	107	31	(	(	PUNCT
ma-174	107	32	u	u	NOUN
ma-174	107	33	)	)	PUNCT
ma-174	107	34	and	and	CCONJ
ma-174	107	35	g	g	PROPN
ma-174	107	36	∈	∈	PROPN
ma-174	107	37	bα(u	bα(u	NOUN
ma-174	107	38	)	)	PUNCT
ma-174	107	39	be	be	AUX
ma-174	107	40	such	such	ADJ
ma-174	107	41	that	that	PRON
ma-174	107	42	fn	fn	NOUN
ma-174	107	43	→	→	SYM
ma-174	107	44	g	g	NOUN
ma-174	107	45	as	as	ADP
ma-174	107	46	n	n	PROPN
ma-174	107	47	→	→	SYM
ma-174	107	48	∞.	∞.	PROPN
ma-174	107	49	we	we	PRON
ma-174	107	50	need	need	VERB
ma-174	107	51	to	to	PART
ma-174	107	52	prove	prove	VERB
ma-174	107	53	that	that	SCONJ
ma-174	107	54	g	g	PROPN
ma-174	107	55	∈	∈	PROPN
ma-174	107	56	bα0	bα0	PROPN
ma-174	107	57	(	(	PUNCT
ma-174	107	58	u	u	NOUN
ma-174	107	59	)	)	PUNCT
ma-174	107	60	.	.	PUNCT
ma-174	108	1	since	since	SCONJ
ma-174	108	2	fn	fn	NOUN
ma-174	108	3	,	,	PUNCT
ma-174	108	4	g	g	PROPN
ma-174	108	5	are	be	AUX
ma-174	108	6	holomorphic	holomorphic	ADJ
ma-174	108	7	on	on	ADP
ma-174	108	8	compact	compact	ADJ
ma-174	108	9	subsets	subset	NOUN
ma-174	108	10	of	of	ADP
ma-174	108	11	u	u	NOUN
ma-174	108	12	,	,	PUNCT
ma-174	108	13	and	and	CCONJ
ma-174	108	14	fn	fn	NOUN
ma-174	108	15	→	→	SYM
ma-174	108	16	g	g	PROPN
ma-174	108	17	,	,	PUNCT
ma-174	108	18	we	we	PRON
ma-174	108	19	have	have	VERB
ma-174	108	20	f	f	PROPN
ma-174	108	21	′n	′n	PROPN
ma-174	108	22	→	→	SYM
ma-174	108	23	g′uniformly	g′uniformly	PROPN
ma-174	108	24	.	.	PUNCT
ma-174	109	1	now	now	ADV
ma-174	109	2	that	that	SCONJ
ma-174	109	3	fn	fn	PROPN
ma-174	109	4	⊂	⊂	PROPN
ma-174	109	5	bα0	bα0	PROPN
ma-174	109	6	(	(	PUNCT
ma-174	109	7	u	u	NOUN
ma-174	109	8	)	)	PUNCT
ma-174	109	9	,	,	PUNCT
ma-174	109	10	we	we	PRON
ma-174	109	11	have	have	VERB
ma-174	109	12	lim	lim	PROPN
ma-174	109	13	=(	=(	PROPN
ma-174	109	14	ω)→0	ω)→0	NOUN
ma-174	109	15	(=	(=	NOUN
ma-174	109	16	(	(	PUNCT
ma-174	109	17	ω))α	ω))α	NOUN
ma-174	109	18	|f	|f	PROPN
ma-174	110	1	′n(ω)|	′n(ω)|	PROPN
ma-174	110	2	=	=	SYM
ma-174	110	3	0,∀	0,∀	NUM
ma-174	110	4	n.	n.	NOUN
ma-174	110	5	(	(	PUNCT
ma-174	110	6	2	2	NUM
ma-174	110	7	)	)	PUNCT
ma-174	110	8	since	since	SCONJ
ma-174	110	9	limn→∞	limn→∞	PROPN
ma-174	110	10	f	f	X
ma-174	110	11	′	′	NUM
ma-174	110	12	n	n	PROPN
ma-174	110	13	=	=	SYM
ma-174	110	14	g′	g′	NOUN
ma-174	110	15	,	,	PUNCT
ma-174	110	16	we	we	PRON
ma-174	110	17	have	have	VERB
ma-174	110	18	lim	lim	PROPN
ma-174	110	19	=(	=(	PROPN
ma-174	110	20	ω)→0	ω)→0	NOUN
ma-174	110	21	(=	(=	NOUN
ma-174	110	22	(	(	PUNCT
ma-174	110	23	ω))α	ω))α	NOUN
ma-174	110	24	|g′(ω)|	|g′(ω)|	PROPN
ma-174	111	1	=	=	SYM
ma-174	111	2	lim	lim	PROPN
ma-174	111	3	=(	=(	NOUN
ma-174	111	4	ω)→0	ω)→0	NOUN
ma-174	111	5	(=	(=	NOUN
ma-174	111	6	(	(	PUNCT
ma-174	111	7	ω))α	ω))α	NOUN
ma-174	111	8	|	|	PROPN
ma-174	111	9	lim	lim	PROPN
ma-174	111	10	n→∞	n→∞	PROPN
ma-174	111	11	f	f	PROPN
ma-174	111	12	′n(ω)|	′n(ω)|	PROPN
ma-174	111	13	which	which	PRON
ma-174	111	14	is	be	AUX
ma-174	111	15	equivalent	equivalent	ADJ
ma-174	111	16	to	to	ADP
ma-174	111	17	lim	lim	PROPN
ma-174	111	18	=(	=(	PROPN
ma-174	111	19	ω)→0	ω)→0	NOUN
ma-174	111	20	(=	(=	NOUN
ma-174	111	21	(	(	PUNCT
ma-174	111	22	ω))α	ω))α	NOUN
ma-174	111	23	|g′(ω)|	|g′(ω)|	PROPN
ma-174	111	24	=	=	SYM
ma-174	111	25	lim	lim	PROPN
ma-174	111	26	n→∞	n→∞	X
ma-174	112	1	(	(	PUNCT
ma-174	112	2	lim	lim	PROPN
ma-174	112	3	=(	=(	NOUN
ma-174	112	4	ω)→0	ω)→0	NOUN
ma-174	112	5	(=	(=	NOUN
ma-174	112	6	(	(	PUNCT
ma-174	112	7	ω))α	ω))α	NOUN
ma-174	112	8	|f	|f	PROPN
ma-174	112	9	′n(ω)|	′n(ω)|	PROPN
ma-174	112	10	)	)	PUNCT
ma-174	112	11	.	.	PUNCT
ma-174	113	1	following	follow	VERB
ma-174	113	2	equation	equation	NOUN
ma-174	113	3	(	(	PUNCT
ma-174	113	4	2	2	NUM
ma-174	113	5	)	)	PUNCT
ma-174	113	6	,	,	PUNCT
ma-174	113	7	we	we	PRON
ma-174	113	8	see	see	VERB
ma-174	113	9	that	that	SCONJ
ma-174	113	10	lim	lim	PROPN
ma-174	113	11	=(	=(	PROPN
ma-174	113	12	ω)→0	ω)→0	NOUN
ma-174	113	13	(=	(=	NOUN
ma-174	113	14	(	(	PUNCT
ma-174	113	15	ω))α	ω))α	NOUN
ma-174	113	16	|g′(ω)|	|g′(ω)|	PROPN
ma-174	113	17	=	=	SYM
ma-174	113	18	0	0	X
ma-174	113	19	.	.	PUNCT
ma-174	114	1	so	so	ADV
ma-174	114	2	,	,	PUNCT
ma-174	114	3	g	g	PROPN
ma-174	114	4	∈	∈	PROPN
ma-174	114	5	bα0	bα0	PROPN
ma-174	114	6	(	(	PUNCT
ma-174	114	7	u	u	NOUN
ma-174	114	8	)	)	PUNCT
ma-174	114	9	,	,	PUNCT
ma-174	114	10	completing	complete	VERB
ma-174	114	11	the	the	DET
ma-174	114	12	proof	proof	NOUN
ma-174	114	13	.	.	PUNCT
ma-174	115	1	�	�	NOUN
ma-174	115	2	5	5	NUM
ma-174	115	3	as	as	ADP
ma-174	115	4	a	a	DET
ma-174	115	5	consequence	consequence	NOUN
ma-174	115	6	,	,	PUNCT
ma-174	115	7	we	we	PRON
ma-174	115	8	have	have	VERB
ma-174	115	9	the	the	DET
ma-174	115	10	following	follow	VERB
ma-174	115	11	,	,	PUNCT
ma-174	115	12	corollary	corollary	ADJ
ma-174	115	13	3.4	3.4	NUM
ma-174	115	14	.	.	PUNCT
ma-174	116	1	b0(u	b0(u	X
ma-174	116	2	)	)	PUNCT
ma-174	117	1	is	be	AUX
ma-174	117	2	a	a	DET
ma-174	117	3	banach	banach	NOUN
ma-174	117	4	space	space	NOUN
ma-174	117	5	with	with	ADP
ma-174	117	6	respect	respect	NOUN
ma-174	117	7	to	to	ADP
ma-174	117	8	the	the	DET
ma-174	117	9	norm	norm	NOUN
ma-174	117	10	‖.‖b(u	‖.‖b(u	VERB
ma-174	117	11	)	)	PUNCT
ma-174	117	12	proof	proof	NOUN
ma-174	117	13	.	.	PUNCT
ma-174	118	1	follows	follow	VERB
ma-174	118	2	immediately	immediately	ADV
ma-174	118	3	by	by	ADP
ma-174	118	4	taking	take	VERB
ma-174	118	5	α	α	NOUN
ma-174	118	6	=	=	SYM
ma-174	118	7	1	1	NUM
ma-174	118	8	in	in	ADP
ma-174	118	9	theorem	theorem	ADJ
ma-174	118	10	3.3	3.3	NUM
ma-174	118	11	.	.	PUNCT
ma-174	119	1	�	�	PROPN
ma-174	119	2	in	in	ADP
ma-174	119	3	the	the	DET
ma-174	119	4	next	next	ADJ
ma-174	119	5	results	result	NOUN
ma-174	119	6	,	,	PUNCT
ma-174	119	7	we	we	PRON
ma-174	119	8	generate	generate	VERB
ma-174	119	9	a	a	DET
ma-174	119	10	relationship	relationship	NOUN
ma-174	119	11	between	between	ADP
ma-174	119	12	functions	function	NOUN
ma-174	119	13	in	in	ADP
ma-174	119	14	the	the	DET
ma-174	119	15	generalized	generalized	ADJ
ma-174	119	16	bloch	bloch	PROPN
ma-174	119	17	spaceof	spaceof	VERB
ma-174	120	1	the	the	DET
ma-174	120	2	upper	upper	ADJ
ma-174	120	3	half	half	ADJ
ma-174	120	4	plane	plane	NOUN
ma-174	120	5	u	u	NOUN
ma-174	120	6	and	and	CCONJ
ma-174	120	7	their	their	PRON
ma-174	120	8	counterparts	counterpart	NOUN
ma-174	120	9	in	in	ADP
ma-174	120	10	the	the	DET
ma-174	120	11	unit	unit	NOUN
ma-174	120	12	disc	disc	VERB
ma-174	120	13	d	d	PROPN
ma-174	120	14	proposition	proposition	NOUN
ma-174	120	15	3.5	3.5	NUM
ma-174	120	16	.	.	PUNCT
ma-174	121	1	let	let	VERB
ma-174	121	2	f	f	PROPN
ma-174	121	3	∈	∈	PROPN
ma-174	121	4	bα(u	bα(u	NOUN
ma-174	121	5	)	)	PUNCT
ma-174	121	6	and	and	CCONJ
ma-174	121	7	ψ	ψ	X
ma-174	121	8	be	be	AUX
ma-174	121	9	the	the	DET
ma-174	121	10	cayley	cayley	ADJ
ma-174	121	11	transform	transform	NOUN
ma-174	121	12	,	,	PUNCT
ma-174	121	13	then	then	ADV
ma-174	121	14	f	f	PROPN
ma-174	121	15	∈	∈	PROPN
ma-174	121	16	bα(u	bα(u	NOUN
ma-174	121	17	)	)	PUNCT
ma-174	122	1	if	if	SCONJ
ma-174	122	2	and	and	CCONJ
ma-174	122	3	only	only	ADV
ma-174	122	4	if	if	SCONJ
ma-174	122	5	f	f	PROPN
ma-174	122	6	◦	◦	NOUN
ma-174	122	7	ψ	ψ	X
ma-174	122	8	∈	∈	NOUN
ma-174	122	9	bα(d	bα(d	NOUN
ma-174	122	10	)	)	PUNCT
ma-174	122	11	proof	proof	NOUN
ma-174	122	12	.	.	PUNCT
ma-174	123	1	it	it	PRON
ma-174	123	2	suffices	suffice	VERB
ma-174	123	3	to	to	PART
ma-174	123	4	prove	prove	VERB
ma-174	123	5	that	that	SCONJ
ma-174	123	6	‖f	‖f	ADP
ma-174	123	7	‖bα1	‖bα1	PROPN
ma-174	123	8	(	(	PUNCT
ma-174	123	9	u	u	NOUN
ma-174	123	10	)	)	PUNCT
ma-174	123	11	<	<	X
ma-174	123	12	∞	∞	NUM
ma-174	123	13	if	if	SCONJ
ma-174	123	14	and	and	CCONJ
ma-174	123	15	only	only	ADV
ma-174	123	16	if	if	SCONJ
ma-174	123	17	‖f	‖f	DET
ma-174	123	18	◦	◦	NOUN
ma-174	123	19	ψ‖bα1	ψ‖bα1	X
ma-174	123	20	(	(	PUNCT
ma-174	123	21	d	d	X
ma-174	123	22	)	)	PUNCT
ma-174	123	23	<	<	X
ma-174	123	24	∞.	∞.	PROPN
ma-174	123	25	let	let	VERB
ma-174	123	26	f	f	PRON
ma-174	123	27	be	be	AUX
ma-174	123	28	a	a	DET
ma-174	123	29	functionin	functionin	ADJ
ma-174	123	30	bα(u	bα(u	NOUN
ma-174	123	31	)	)	PUNCT
ma-174	123	32	.	.	PUNCT
ma-174	124	1	then	then	ADV
ma-174	124	2	by	by	ADP
ma-174	124	3	definition	definition	NOUN
ma-174	124	4	,	,	PUNCT
ma-174	124	5	‖f	‖f	ADP
ma-174	124	6	‖bα1	‖bα1	PROPN
ma-174	124	7	(	(	PUNCT
ma-174	124	8	u	u	NOUN
ma-174	124	9	)	)	PUNCT
ma-174	124	10	=	=	PUNCT
ma-174	124	11	supω∈u=(ω)α|f	supω∈u=(ω)α|f	VERB
ma-174	124	12	′(ω)|	′(ω)|	PROPN
ma-174	124	13	<	<	X
ma-174	124	14	∞.	∞.	PROPN
ma-174	124	15	now	now	ADV
ma-174	124	16	,	,	PUNCT
ma-174	124	17	by	by	ADP
ma-174	124	18	changing	change	VERB
ma-174	124	19	variables	variable	NOUN
ma-174	124	20	,	,	PUNCT
ma-174	124	21	let	let	VERB
ma-174	124	22	ω	ω	NUM
ma-174	124	23	=	=	SYM
ma-174	124	24	ψ(z	ψ(z	PROPN
ma-174	124	25	)	)	PUNCT
ma-174	124	26	,	,	PUNCT
ma-174	124	27	where	where	SCONJ
ma-174	124	28	ψ	ψ	NOUN
ma-174	124	29	is	be	AUX
ma-174	124	30	the	the	DET
ma-174	124	31	cayley	cayley	ADJ
ma-174	124	32	transform	transform	NOUN
ma-174	124	33	.	.	PUNCT
ma-174	125	1	then	then	ADV
ma-174	125	2	=(	=(	PROPN
ma-174	125	3	ω	ω	PROPN
ma-174	125	4	)	)	PUNCT
ma-174	125	5	=	=	SYM
ma-174	125	6	ω	ω	NUM
ma-174	125	7	−	−	PROPN
ma-174	125	8	ω	ω	NUM
ma-174	125	9	2i	2i	NUM
ma-174	125	10	=	=	SYM
ma-174	125	11	ψ(z)−	ψ(z)−	PROPN
ma-174	125	12	ψ(z	ψ(z	PROPN
ma-174	125	13	)	)	PUNCT
ma-174	125	14	2i	2i	NOUN
ma-174	125	15	.	.	PUNCT
ma-174	126	1	using	use	VERB
ma-174	126	2	ψ(z	ψ(z	PROPN
ma-174	126	3	)	)	PUNCT
ma-174	126	4	=	=	SYM
ma-174	126	5	i(1+z	i(1+z	PROPN
ma-174	126	6	)	)	PUNCT
ma-174	126	7	1−z	1−z	NUM
ma-174	126	8	and	and	CCONJ
ma-174	126	9	ψ(z	ψ(z	PROPN
ma-174	126	10	)	)	PUNCT
ma-174	126	11	=	=	SYM
ma-174	126	12	−i(1+z	−i(1+z	PROPN
ma-174	126	13	)	)	PUNCT
ma-174	126	14	1−z	1−z	NUM
ma-174	126	15	,	,	PUNCT
ma-174	126	16	we	we	PRON
ma-174	126	17	have	have	VERB
ma-174	126	18	=(	=(	NOUN
ma-174	126	19	ω	ω	NUM
ma-174	126	20	)	)	PUNCT
ma-174	126	21	=	=	SYM
ma-174	126	22	i(1+z	i(1+z	PROPN
ma-174	126	23	)	)	PUNCT
ma-174	126	24	1−z	1−z	NUM
ma-174	127	1	−	−	PROPN
ma-174	127	2	−i(1+z	−i(1+z	PROPN
ma-174	127	3	)	)	PUNCT
ma-174	127	4	1−z	1−z	PROPN
ma-174	127	5	2i	2i	NUM
ma-174	127	6	=	=	SYM
ma-174	127	7	i(1	i(1	PROPN
ma-174	127	8	+	+	PROPN
ma-174	127	9	z)(1−	z)(1−	PROPN
ma-174	127	10	z	z	PROPN
ma-174	127	11	)	)	PUNCT
ma-174	128	1	+	+	CCONJ
ma-174	129	1	i(1	i(1	PROPN
ma-174	129	2	+	+	CCONJ
ma-174	129	3	z)(1−	z)(1−	PROPN
ma-174	129	4	z	z	PROPN
ma-174	129	5	)	)	PUNCT
ma-174	129	6	2i(1−	2i(1−	PROPN
ma-174	129	7	z)(1−	z)(1−	PROPN
ma-174	129	8	z	z	PROPN
ma-174	129	9	)	)	PUNCT
ma-174	129	10	=	=	NOUN
ma-174	129	11	i(2−	i(2−	PROPN
ma-174	129	12	2zz	2zz	ADV
ma-174	129	13	)	)	PUNCT
ma-174	129	14	2i(1−	2i(1−	NUM
ma-174	129	15	z)(1−	z)(1−	PROPN
ma-174	129	16	z	z	PROPN
ma-174	129	17	)	)	PUNCT
ma-174	129	18	=	=	SYM
ma-174	129	19	1−	1−	NUM
ma-174	129	20	|z	|z	PROPN
ma-174	129	21	|2	|2	NUM
ma-174	129	22	|1−	|1−	PROPN
ma-174	129	23	z	z	PROPN
ma-174	129	24	|2	|2	NUM
ma-174	129	25	.we	.we	PUNCT
ma-174	129	26	get	get	VERB
ma-174	129	27	the	the	DET
ma-174	129	28	absolute	absolute	ADJ
ma-174	129	29	of	of	ADP
ma-174	129	30	ψ′(z	ψ′(z	PROPN
ma-174	129	31	)	)	PUNCT
ma-174	130	1	=	=	SYM
ma-174	130	2	2i	2i	NOUN
ma-174	130	3	(	(	PUNCT
ma-174	130	4	1−z)2	1−z)2	NUM
ma-174	130	5	as	as	ADP
ma-174	130	6	|ψ′(z)|	|ψ′(z)|	PRON
ma-174	130	7	=	=	SYM
ma-174	130	8	2	2	NUM
ma-174	130	9	|1−	|1−	NOUN
ma-174	130	10	z	z	NOUN
ma-174	130	11	|2	|2	NUM
ma-174	130	12	.	.	PUNCT
ma-174	131	1	(	(	PUNCT
ma-174	131	2	3	3	X
ma-174	131	3	)	)	PUNCT
ma-174	131	4	now	now	ADV
ma-174	131	5	,	,	PUNCT
ma-174	131	6	by	by	ADP
ma-174	131	7	definition	definition	NOUN
ma-174	131	8	we	we	PRON
ma-174	131	9	have	have	VERB
ma-174	131	10	‖f	‖f	ADP
ma-174	131	11	‖bα1	‖bα1	PROPN
ma-174	131	12	(	(	PUNCT
ma-174	131	13	u	u	NOUN
ma-174	131	14	)	)	PUNCT
ma-174	131	15	=	=	SYM
ma-174	131	16	sup	sup	NOUN
ma-174	131	17	z∈d	z∈d	NUM
ma-174	131	18	(	(	PUNCT
ma-174	131	19	1−	1−	NUM
ma-174	131	20	|z	|z	PROPN
ma-174	131	21	|2	|2	NUM
ma-174	131	22	|1−	|1−	PROPN
ma-174	131	23	z	z	PROPN
ma-174	131	24	|2	|2	NUM
ma-174	131	25	)	)	PUNCT
ma-174	132	1	α	α	PROPN
ma-174	132	2	|f	|f	X
ma-174	132	3	′(ψ(z))|	′(ψ(z))|	PROPN
ma-174	132	4	.	.	PUNCT
ma-174	133	1	from	from	ADP
ma-174	133	2	equation	equation	NOUN
ma-174	133	3	(	(	PUNCT
ma-174	133	4	3	3	NUM
ma-174	133	5	)	)	PUNCT
ma-174	133	6	,	,	PUNCT
ma-174	133	7	we	we	PRON
ma-174	133	8	have	have	VERB
ma-174	133	9	|1−	|1−	PROPN
ma-174	133	10	z	z	NOUN
ma-174	133	11	|2	|2	NUM
ma-174	133	12	=	=	SYM
ma-174	133	13	2	2	NUM
ma-174	133	14	|ψ′(z)|	|ψ′(z)|	DET
ma-174	133	15	,	,	PUNCT
ma-174	133	16	therefore	therefore	ADV
ma-174	133	17	‖f	‖f	ADP
ma-174	133	18	‖bα1	‖bα1	PROPN
ma-174	133	19	(	(	PUNCT
ma-174	133	20	u	u	NOUN
ma-174	133	21	)	)	PUNCT
ma-174	133	22	=	=	SYM
ma-174	133	23	1	1	NUM
ma-174	133	24	2α	2α	NOUN
ma-174	133	25	sup	sup	NOUN
ma-174	133	26	z∈d	z∈d	NUM
ma-174	133	27	(	(	PUNCT
ma-174	133	28	1−	1−	NUM
ma-174	133	29	|z	|z	PROPN
ma-174	133	30	|2)α|ψ′(z)|α|f	|2)α|ψ′(z)|α|f	PROPN
ma-174	133	31	′(ψ(z))|	′(ψ(z))|	PROPN
ma-174	133	32	.	.	PROPN
ma-174	133	33	6	6	NUM
ma-174	133	34	since	since	SCONJ
ma-174	133	35	,	,	PUNCT
ma-174	133	36	(	(	PUNCT
ma-174	133	37	f	f	X
ma-174	133	38	◦	◦	NOUN
ma-174	133	39	ψ)′(z	ψ)′(z	PROPN
ma-174	133	40	)	)	PUNCT
ma-174	134	1	=	=	SYM
ma-174	134	2	f	f	PROPN
ma-174	134	3	′(ψ(z))ψ′(z	′(ψ(z))ψ′(z	PROPN
ma-174	134	4	)	)	PUNCT
ma-174	134	5	,	,	PUNCT
ma-174	134	6	we	we	PRON
ma-174	134	7	have	have	AUX
ma-174	134	8	|ψ′(z)|α|f	|ψ′(z)|α|f	VERB
ma-174	134	9	′(ψ(z))|	′(ψ(z))|	NOUN
ma-174	134	10	=	=	PUNCT
ma-174	134	11	|ψ′(z)(f	|ψ′(z)(f	NOUN
ma-174	134	12	◦	◦	VERB
ma-174	134	13	ψ)′(z)||ψ′(z)α−1|	ψ)′(z)||ψ′(z)α−1|	PROPN
ma-174	134	14	and	and	CCONJ
ma-174	134	15	hence	hence	ADV
ma-174	134	16	‖f	‖f	ADP
ma-174	134	17	‖bα1	‖bα1	PROPN
ma-174	134	18	(	(	PUNCT
ma-174	134	19	u	u	NOUN
ma-174	134	20	)	)	PUNCT
ma-174	134	21	=	=	SYM
ma-174	134	22	1	1	NUM
ma-174	134	23	2α	2α	NOUN
ma-174	134	24	sup	sup	NOUN
ma-174	134	25	z∈d	z∈d	NUM
ma-174	134	26	(	(	PUNCT
ma-174	134	27	1−	1−	NUM
ma-174	134	28	|z	|z	PROPN
ma-174	134	29	|2)α|ψ′(z)(f	|2)α|ψ′(z)(f	PUNCT
ma-174	134	30	◦	◦	VERB
ma-174	134	31	ψ)′(z)||ψ′(z)α−1|	ψ)′(z)||ψ′(z)α−1|	NOUN
ma-174	134	32	=	=	SYM
ma-174	134	33	1	1	NUM
ma-174	134	34	2α	2α	NOUN
ma-174	134	35	|ψ′(z)α−1|‖f	|ψ′(z)α−1|‖f	NOUN
ma-174	134	36	◦	◦	NOUN
ma-174	134	37	ψ‖bα1	ψ‖bα1	NOUN
ma-174	134	38	(	(	PUNCT
ma-174	134	39	d),which	d),which	PROPN
ma-174	134	40	is	be	AUX
ma-174	134	41	finite	finite	ADJ
ma-174	134	42	if	if	SCONJ
ma-174	134	43	and	and	CCONJ
ma-174	134	44	only	only	ADV
ma-174	134	45	if	if	SCONJ
ma-174	134	46	‖f	‖f	DET
ma-174	134	47	◦	◦	NOUN
ma-174	134	48	ψ‖bα1	ψ‖bα1	X
ma-174	134	49	(	(	PUNCT
ma-174	134	50	d	d	X
ma-174	134	51	)	)	PUNCT
ma-174	134	52	is	be	AUX
ma-174	134	53	finite	finite	ADJ
ma-174	134	54	.	.	PUNCT
ma-174	135	1	this	this	PRON
ma-174	135	2	completes	complete	VERB
ma-174	135	3	the	the	DET
ma-174	135	4	proof	proof	NOUN
ma-174	135	5	.	.	PUNCT
ma-174	136	1	�	�	PROPN
ma-174	136	2	an	an	DET
ma-174	136	3	immediate	immediate	ADJ
ma-174	136	4	consequence	consequence	NOUN
ma-174	136	5	is	be	AUX
ma-174	136	6	the	the	DET
ma-174	136	7	following	follow	VERB
ma-174	136	8	,	,	PUNCT
ma-174	136	9	corollary	corollary	ADJ
ma-174	136	10	3.6	3.6	NUM
ma-174	136	11	.	.	PUNCT
ma-174	137	1	let	let	VERB
ma-174	137	2	f	f	PROPN
ma-174	137	3	∈	∈	PROPN
ma-174	137	4	b(u	b(u	PROPN
ma-174	137	5	)	)	PUNCT
ma-174	137	6	and	and	CCONJ
ma-174	137	7	ψ	ψ	X
ma-174	137	8	be	be	AUX
ma-174	137	9	the	the	DET
ma-174	137	10	cayley	cayley	ADJ
ma-174	137	11	transform	transform	NOUN
ma-174	137	12	,	,	PUNCT
ma-174	137	13	then	then	ADV
ma-174	137	14	‖f	‖f	ADP
ma-174	137	15	‖b1(u	‖b1(u	X
ma-174	137	16	)	)	PUNCT
ma-174	137	17	=	=	SYM
ma-174	137	18	1	1	NUM
ma-174	137	19	2	2	NUM
ma-174	137	20	‖f	‖f	ADP
ma-174	137	21	◦	◦	NOUN
ma-174	137	22	ψ‖b1(d	ψ‖b1(d	NOUN
ma-174	137	23	)	)	PUNCT
ma-174	137	24	(	(	PUNCT
ma-174	137	25	4	4	X
ma-174	137	26	)	)	PUNCT
ma-174	137	27	in	in	ADP
ma-174	137	28	particular	particular	ADJ
ma-174	137	29	,	,	PUNCT
ma-174	137	30	a	a	DET
ma-174	137	31	function	function	NOUN
ma-174	137	32	f	f	PROPN
ma-174	137	33	∈	∈	PROPN
ma-174	137	34	b(u	b(u	PROPN
ma-174	137	35	)	)	PUNCT
ma-174	137	36	if	if	SCONJ
ma-174	137	37	and	and	CCONJ
ma-174	137	38	only	only	ADV
ma-174	137	39	if	if	SCONJ
ma-174	137	40	f	f	PROPN
ma-174	137	41	◦	◦	NOUN
ma-174	137	42	ψ	ψ	X
ma-174	137	43	∈	∈	PROPN
ma-174	137	44	b(d	b(d	PROPN
ma-174	137	45	)	)	PUNCT
ma-174	137	46	.	.	PUNCT
ma-174	138	1	proof	proof	NOUN
ma-174	138	2	.	.	PUNCT
ma-174	139	1	this	this	PRON
ma-174	139	2	follows	follow	VERB
ma-174	139	3	immediately	immediately	ADV
ma-174	139	4	from	from	ADP
ma-174	139	5	proposition	proposition	NOUN
ma-174	139	6	3.5	3.5	NUM
ma-174	139	7	by	by	ADP
ma-174	139	8	taking	take	VERB
ma-174	139	9	α	α	NOUN
ma-174	139	10	=	=	SYM
ma-174	139	11	1	1	X
ma-174	139	12	.	.	X
ma-174	139	13	�	�	PROPN
ma-174	139	14	4	4	NUM
ma-174	139	15	.	.	PUNCT
ma-174	139	16	composition	composition	NOUN
ma-174	139	17	semigroups	semigroup	NOUN
ma-174	139	18	on	on	ADP
ma-174	139	19	the	the	DET
ma-174	139	20	generalized	generalized	ADJ
ma-174	139	21	little	little	ADJ
ma-174	139	22	bloch	bloch	NOUN
ma-174	139	23	space	space	NOUN
ma-174	139	24	of	of	ADP
ma-174	139	25	the	the	DET
ma-174	139	26	upper	upper	ADJ
ma-174	139	27	half	half	ADJ
ma-174	139	28	plane	plane	NOUN
ma-174	139	29	in	in	ADP
ma-174	139	30	[	[	X
ma-174	139	31	3	3	NUM
ma-174	139	32	]	]	PUNCT
ma-174	139	33	,	,	PUNCT
ma-174	139	34	the	the	DET
ma-174	139	35	non	non	ADJ
ma-174	139	36	trivial	trivial	ADJ
ma-174	139	37	automorphisms	automorphism	NOUN
ma-174	139	38	of	of	ADP
ma-174	139	39	the	the	DET
ma-174	139	40	upper	upper	ADJ
ma-174	139	41	half	half	ADJ
ma-174	139	42	plane	plane	NOUN
ma-174	139	43	u	u	NOUN
ma-174	139	44	were	be	AUX
ma-174	139	45	classified	classify	VERB
ma-174	139	46	according	accord	VERB
ma-174	139	47	to	to	ADP
ma-174	139	48	thelocation	thelocation	NOUN
ma-174	139	49	of	of	ADP
ma-174	139	50	their	their	PRON
ma-174	139	51	fixed	fix	VERB
ma-174	139	52	points	point	NOUN
ma-174	139	53	into	into	ADP
ma-174	139	54	three	three	NUM
ma-174	139	55	distinct	distinct	ADJ
ma-174	139	56	classes	class	NOUN
ma-174	139	57	namely	namely	ADV
ma-174	139	58	;	;	PUNCT
ma-174	139	59	scaling	scaling	NOUN
ma-174	139	60	,	,	PUNCT
ma-174	139	61	translation	translation	NOUN
ma-174	139	62	and	and	CCONJ
ma-174	139	63	rotationgroups	rotationgroup	NOUN
ma-174	139	64	.	.	PUNCT
ma-174	140	1	in	in	ADP
ma-174	140	2	this	this	DET
ma-174	140	3	section	section	NOUN
ma-174	140	4	,	,	PUNCT
ma-174	140	5	we	we	PRON
ma-174	140	6	determine	determine	VERB
ma-174	140	7	composition	composition	NOUN
ma-174	140	8	semigroups	semigroup	NOUN
ma-174	140	9	induced	induce	VERB
ma-174	140	10	by	by	ADP
ma-174	140	11	these	these	DET
ma-174	140	12	automorphismgroups	automorphismgroup	NOUN
ma-174	140	13	of	of	ADP
ma-174	140	14	the	the	DET
ma-174	140	15	upper	upper	ADJ
ma-174	140	16	half	half	ADJ
ma-174	140	17	plane	plane	NOUN
ma-174	140	18	u	u	NOUN
ma-174	140	19	,	,	PUNCT
ma-174	140	20	on	on	ADP
ma-174	140	21	the	the	DET
ma-174	140	22	generalized	generalized	ADJ
ma-174	140	23	bloch	bloch	NOUN
ma-174	140	24	space	space	NOUN
ma-174	140	25	of	of	ADP
ma-174	140	26	the	the	DET
ma-174	140	27	upper	upper	ADJ
ma-174	140	28	half	half	ADJ
ma-174	140	29	plane	plane	NOUN
ma-174	140	30	bα(u	bα(u	NOUN
ma-174	140	31	)	)	PUNCT
ma-174	140	32	.	.	PUNCT
ma-174	141	1	wethen	wethen	PROPN
ma-174	141	2	employ	employ	VERB
ma-174	141	3	the	the	DET
ma-174	141	4	theory	theory	NOUN
ma-174	141	5	of	of	ADP
ma-174	141	6	linear	linear	PROPN
ma-174	141	7	operators	operator	NOUN
ma-174	141	8	on	on	ADP
ma-174	141	9	banach	banach	NOUN
ma-174	141	10	spaces	space	NOUN
ma-174	141	11	to	to	PART
ma-174	141	12	investigate	investigate	VERB
ma-174	141	13	the	the	DET
ma-174	141	14	semigroup	semigroup	ADJ
ma-174	141	15	propertiesof	propertiesof	NOUN
ma-174	141	16	the	the	DET
ma-174	141	17	induced	induced	ADJ
ma-174	141	18	composition	composition	NOUN
ma-174	141	19	semigroup	semigroup	NOUN
ma-174	141	20	.	.	PUNCT
ma-174	142	1	for	for	ADP
ma-174	142	2	any	any	DET
ma-174	142	3	given	give	VERB
ma-174	142	4	semigroup	semigroup	NOUN
ma-174	142	5	ϕt	ϕt	ADV
ma-174	142	6	,	,	PUNCT
ma-174	142	7	the	the	DET
ma-174	142	8	induced	induced	ADJ
ma-174	142	9	operator	operator	NOUN
ma-174	142	10	semigroup	semigroup	NOUN
ma-174	142	11	cϕt	cϕt	PROPN
ma-174	142	12	is	be	AUX
ma-174	142	13	known	know	VERB
ma-174	142	14	to	to	PART
ma-174	142	15	be	be	AUX
ma-174	142	16	strongly	strongly	ADV
ma-174	142	17	continuous	continuous	ADJ
ma-174	142	18	on	on	ADP
ma-174	142	19	the	the	DET
ma-174	142	20	little	little	ADJ
ma-174	142	21	bloch	bloch	PROPN
ma-174	142	22	space	space	NOUN
ma-174	142	23	.	.	PUNCT
ma-174	143	1	on	on	ADP
ma-174	143	2	the	the	DET
ma-174	143	3	other	other	ADJ
ma-174	143	4	hand	hand	NOUN
ma-174	143	5	,	,	PUNCT
ma-174	143	6	no	no	DET
ma-174	143	7	non	non	ADJ
ma-174	143	8	trivialcomposition	trivialcomposition	NOUN
ma-174	143	9	semigroup	semigroup	NOUN
ma-174	143	10	is	be	AUX
ma-174	143	11	strongly	strongly	ADV
ma-174	143	12	continuous	continuous	ADJ
ma-174	143	13	on	on	ADP
ma-174	143	14	the	the	DET
ma-174	143	15	bloch	bloch	PROPN
ma-174	143	16	space	space	NOUN
ma-174	143	17	,	,	PUNCT
ma-174	143	18	see	see	VERB
ma-174	143	19	[	[	X
ma-174	143	20	11	11	NUM
ma-174	143	21	]	]	PUNCT
ma-174	143	22	.	.	PUNCT
ma-174	144	1	therefore	therefore	ADV
ma-174	144	2	,	,	PUNCT
ma-174	144	3	we	we	PRON
ma-174	144	4	shalldetermine	shalldetermine	VERB
ma-174	144	5	the	the	DET
ma-174	144	6	composition	composition	NOUN
ma-174	144	7	semigroup	semigroup	NOUN
ma-174	144	8	induced	induce	VERB
ma-174	144	9	by	by	ADP
ma-174	144	10	these	these	DET
ma-174	144	11	automorphism	automorphism	NOUN
ma-174	144	12	groups	group	NOUN
ma-174	144	13	on	on	ADP
ma-174	144	14	the	the	DET
ma-174	144	15	generalizedlittle	generalizedlittle	PROPN
ma-174	144	16	bloch	bloch	PROPN
ma-174	144	17	space	space	NOUN
ma-174	144	18	of	of	ADP
ma-174	144	19	the	the	DET
ma-174	144	20	upper	upper	ADJ
ma-174	144	21	half	half	ADJ
ma-174	144	22	plane	plane	NOUN
ma-174	144	23	,	,	PUNCT
ma-174	144	24	bα0	bα0	PROPN
ma-174	144	25	(	(	PUNCT
ma-174	144	26	u	u	NOUN
ma-174	144	27	)	)	PUNCT
ma-174	144	28	.	.	PUNCT
ma-174	145	1	further	far	ADV
ma-174	145	2	,	,	PUNCT
ma-174	145	3	we	we	PRON
ma-174	145	4	show	show	VERB
ma-174	145	5	that	that	SCONJ
ma-174	145	6	composition	composition	NOUN
ma-174	145	7	semigroupsinduced	semigroupsinduce	VERB
ma-174	145	8	by	by	ADP
ma-174	145	9	scaling	scale	VERB
ma-174	145	10	and	and	CCONJ
ma-174	145	11	translation	translation	NOUN
ma-174	145	12	groups	group	NOUN
ma-174	145	13	are	be	AUX
ma-174	145	14	strongly	strongly	ADV
ma-174	145	15	continuous	continuous	ADJ
ma-174	145	16	on	on	ADP
ma-174	145	17	bα0	bα0	PROPN
ma-174	145	18	(	(	PUNCT
ma-174	145	19	u	u	NOUN
ma-174	145	20	)	)	PUNCT
ma-174	145	21	.	.	PUNCT
ma-174	146	1	we	we	PRON
ma-174	146	2	also	also	ADV
ma-174	146	3	establishstrong	establishstrong	VERB
ma-174	146	4	continuity	continuity	NOUN
ma-174	146	5	of	of	ADP
ma-174	146	6	composition	composition	NOUN
ma-174	146	7	semigroups	semigroup	NOUN
ma-174	146	8	induced	induce	VERB
ma-174	146	9	by	by	ADP
ma-174	146	10	rotation	rotation	NOUN
ma-174	146	11	group	group	NOUN
ma-174	146	12	on	on	ADP
ma-174	146	13	bα0	bα0	PROPN
ma-174	146	14	(	(	PUNCT
ma-174	146	15	d	d	NOUN
ma-174	146	16	)	)	PUNCT
ma-174	146	17	.	.	PUNCT
ma-174	147	1	the	the	DET
ma-174	147	2	infinitesimalgenerator	infinitesimalgenerator	NOUN
ma-174	147	3	is	be	AUX
ma-174	147	4	identified	identify	VERB
ma-174	147	5	and	and	CCONJ
ma-174	147	6	its	its	PRON
ma-174	147	7	domain	domain	NOUN
ma-174	147	8	stated	state	VERB
ma-174	147	9	.	.	PUNCT
ma-174	148	1	4.1	4.1	NUM
ma-174	148	2	.	.	PUNCT
ma-174	148	3	scaling	scale	VERB
ma-174	148	4	group	group	NOUN
ma-174	148	5	.	.	PUNCT
ma-174	149	1	the	the	DET
ma-174	149	2	automorphisms	automorphism	NOUN
ma-174	149	3	of	of	ADP
ma-174	149	4	this	this	DET
ma-174	149	5	group	group	NOUN
ma-174	149	6	are	be	AUX
ma-174	149	7	of	of	ADP
ma-174	149	8	the	the	DET
ma-174	149	9	form	form	NOUN
ma-174	149	10	ϕt(z	ϕt(z	NUM
ma-174	149	11	)	)	PUNCT
ma-174	150	1	=	=	SYM
ma-174	151	1	k	k	PROPN
ma-174	151	2	tz	tz	NOUN
ma-174	151	3	,	,	PUNCT
ma-174	151	4	where	where	SCONJ
ma-174	151	5	z	z	PROPN
ma-174	151	6	∈	∈	PROPN
ma-174	151	7	uand	uand	NOUN
ma-174	151	8	k	k	PROPN
ma-174	151	9	,	,	PUNCT
ma-174	151	10	t	t	PROPN
ma-174	151	11	∈	∈	PROPN
ma-174	151	12	r	r	NOUN
ma-174	151	13	with	with	ADP
ma-174	151	14	k	k	PROPN
ma-174	151	15	6=	6=	PROPN
ma-174	151	16	0	0	NUM
ma-174	151	17	.	.	PUNCT
ma-174	152	1	as	as	SCONJ
ma-174	152	2	noted	note	VERB
ma-174	152	3	in	in	ADP
ma-174	152	4	[	[	X
ma-174	152	5	3	3	NUM
ma-174	152	6	]	]	PUNCT
ma-174	152	7	,	,	PUNCT
ma-174	152	8	the	the	DET
ma-174	152	9	semigroup	semigroup	ADJ
ma-174	152	10	properties	property	NOUN
ma-174	152	11	of	of	ADP
ma-174	152	12	the	the	DET
ma-174	152	13	induced	induced	ADJ
ma-174	152	14	compositionoperators	compositionoperator	NOUN
ma-174	152	15	will	will	AUX
ma-174	152	16	differ	differ	VERB
ma-174	152	17	significantly	significantly	ADV
ma-174	152	18	depending	depend	VERB
ma-174	152	19	on	on	ADP
ma-174	152	20	whether	whether	SCONJ
ma-174	152	21	0	0	NUM
ma-174	152	22	<	<	X
ma-174	152	23	k	k	X
ma-174	152	24	<	<	X
ma-174	152	25	1	1	NUM
ma-174	152	26	or	or	CCONJ
ma-174	152	27	k	k	ADJ
ma-174	152	28	>	>	X
ma-174	152	29	1	1	X
ma-174	152	30	.	.	PUNCT
ma-174	152	31	thus	thus	ADV
ma-174	152	32	for	for	ADP
ma-174	152	33	0	0	NUM
ma-174	152	34	<	<	X
ma-174	152	35	k	k	X
ma-174	152	36	<	<	X
ma-174	152	37	1,we	1,we	NUM
ma-174	152	38	consider	consider	VERB
ma-174	152	39	without	without	ADP
ma-174	152	40	loss	loss	NOUN
ma-174	152	41	of	of	ADP
ma-174	152	42	generality	generality	NOUN
ma-174	152	43	,	,	PUNCT
ma-174	152	44	the	the	DET
ma-174	152	45	analytic	analytic	ADJ
ma-174	152	46	self	self	NOUN
ma-174	152	47	maps	map	NOUN
ma-174	152	48	ϕt	ϕt	ADV
ma-174	152	49	:	:	PUNCT
ma-174	152	50	u	u	NOUN
ma-174	152	51	−→	−→	ADJ
ma-174	152	52	u	u	NOUN
ma-174	152	53	of	of	ADP
ma-174	152	54	the	the	DET
ma-174	152	55	form	form	NOUN
ma-174	152	56	ϕt(z	ϕt(z	NUM
ma-174	152	57	)	)	PUNCT
ma-174	152	58	=	=	SYM
ma-174	152	59	e−tz	e−tz	VERB
ma-174	152	60	,	,	PUNCT
ma-174	152	61	z	z	PROPN
ma-174	152	62	∈	∈	PROPN
ma-174	152	63	u.	u.	NOUN
ma-174	152	64	(	(	PUNCT
ma-174	152	65	5	5	NUM
ma-174	152	66	)	)	PUNCT
ma-174	152	67	the	the	DET
ma-174	152	68	composition	composition	NOUN
ma-174	152	69	semigroup	semigroup	NOUN
ma-174	152	70	induced	induce	VERB
ma-174	152	71	by	by	ADP
ma-174	152	72	equation	equation	NOUN
ma-174	152	73	(	(	PUNCT
ma-174	152	74	5	5	NUM
ma-174	152	75	)	)	PUNCT
ma-174	152	76	on	on	ADP
ma-174	152	77	bα0	bα0	PROPN
ma-174	152	78	(	(	PUNCT
ma-174	152	79	u	u	NOUN
ma-174	152	80	)	)	PUNCT
ma-174	152	81	is	be	AUX
ma-174	152	82	given	give	VERB
ma-174	152	83	by	by	ADP
ma-174	152	84	cϕt	cϕt	PROPN
ma-174	152	85	f	f	PROPN
ma-174	152	86	(	(	PUNCT
ma-174	152	87	z	z	NOUN
ma-174	152	88	)	)	PUNCT
ma-174	152	89	=	=	PUNCT
ma-174	153	1	(	(	PUNCT
ma-174	153	2	f	f	X
ma-174	153	3	◦	◦	VERB
ma-174	153	4	ϕt	ϕt	PROPN
ma-174	153	5	)	)	PUNCT
ma-174	153	6	(	(	PUNCT
ma-174	153	7	z	z	X
ma-174	153	8	)	)	PUNCT
ma-174	153	9	=	=	SYM
ma-174	153	10	f	f	PROPN
ma-174	153	11	(	(	PUNCT
ma-174	153	12	e−tz	e−tz	PROPN
ma-174	153	13	)	)	PUNCT
ma-174	153	14	7	7	NUM
ma-174	153	15	it	it	PRON
ma-174	153	16	can	can	AUX
ma-174	153	17	be	be	AUX
ma-174	153	18	easily	easily	ADV
ma-174	153	19	proved	prove	VERB
ma-174	153	20	that	that	SCONJ
ma-174	153	21	(	(	PUNCT
ma-174	153	22	cϕt	cϕt	NOUN
ma-174	153	23	)	)	PUNCT
ma-174	153	24	t∈r	t∈r	PROPN
ma-174	153	25	is	be	AUX
ma-174	153	26	a	a	DET
ma-174	153	27	group	group	NOUN
ma-174	153	28	on	on	ADP
ma-174	153	29	bα0	bα0	PROPN
ma-174	153	30	(	(	PUNCT
ma-174	153	31	u).in	u).in	ADV
ma-174	153	32	what	what	PRON
ma-174	153	33	follows	follow	VERB
ma-174	153	34	,	,	PUNCT
ma-174	153	35	we	we	PRON
ma-174	153	36	prove	prove	VERB
ma-174	153	37	that	that	SCONJ
ma-174	153	38	the	the	DET
ma-174	153	39	composition	composition	NOUN
ma-174	153	40	semigroup	semigroup	NOUN
ma-174	153	41	given	give	VERB
ma-174	153	42	by	by	ADP
ma-174	153	43	equation	equation	NOUN
ma-174	153	44	(	(	PUNCT
ma-174	153	45	4.1	4.1	NUM
ma-174	153	46	)	)	PUNCT
ma-174	153	47	fails	fail	VERB
ma-174	153	48	to	to	PART
ma-174	153	49	be	be	AUX
ma-174	153	50	anisometry	anisometry	NOUN
ma-174	153	51	on	on	ADP
ma-174	153	52	bα0	bα0	PROPN
ma-174	153	53	(	(	PUNCT
ma-174	153	54	u	u	NOUN
ma-174	153	55	)	)	PUNCT
ma-174	153	56	.	.	PUNCT
ma-174	154	1	proposition	proposition	NOUN
ma-174	154	2	4.1	4.1	NUM
ma-174	154	3	.	.	PUNCT
ma-174	155	1	the	the	DET
ma-174	155	2	operator	operator	NOUN
ma-174	155	3	cϕt	cϕt	VERB
ma-174	155	4	fails	fail	VERB
ma-174	155	5	to	to	PART
ma-174	155	6	be	be	AUX
ma-174	155	7	an	an	DET
ma-174	155	8	isometry	isometry	NOUN
ma-174	155	9	on	on	ADP
ma-174	155	10	bα0	bα0	PROPN
ma-174	155	11	(	(	PUNCT
ma-174	155	12	u	u	NOUN
ma-174	155	13	)	)	PUNCT
ma-174	155	14	.	.	PUNCT
ma-174	156	1	proof	proof	NOUN
ma-174	156	2	.	.	PUNCT
ma-174	157	1	by	by	ADP
ma-174	157	2	the	the	DET
ma-174	157	3	definition	definition	NOUN
ma-174	157	4	of	of	ADP
ma-174	157	5	the	the	DET
ma-174	157	6	norm	norm	NOUN
ma-174	157	7	,	,	PUNCT
ma-174	157	8	we	we	PRON
ma-174	157	9	have	have	VERB
ma-174	157	10	for	for	ADP
ma-174	157	11	all	all	DET
ma-174	157	12	f	f	PROPN
ma-174	157	13	∈	∈	PROPN
ma-174	157	14	bα0	bα0	PROPN
ma-174	157	15	(	(	PUNCT
ma-174	157	16	u	u	NOUN
ma-174	157	17	)	)	PUNCT
ma-174	157	18	‖cϕt	‖cϕt	ADP
ma-174	157	19	f	f	PROPN
ma-174	157	20	‖bα(u	‖bα(u	PROPN
ma-174	157	21	)	)	PUNCT
ma-174	158	1	=	=	PUNCT
ma-174	158	2	|cϕt	|cϕt	PROPN
ma-174	158	3	f	f	X
ma-174	158	4	(	(	PUNCT
ma-174	158	5	i)|+	i)|+	ADJ
ma-174	158	6	sup	sup	NOUN
ma-174	158	7	ω∈u	ω∈u	NOUN
ma-174	158	8	=(	=(	NOUN
ma-174	158	9	ω)α|	ω)α|	PROPN
ma-174	158	10	(	(	PUNCT
ma-174	158	11	cϕt	cϕt	NOUN
ma-174	158	12	f	f	PROPN
ma-174	158	13	)	)	PUNCT
ma-174	158	14	′	′	NOUN
ma-174	159	1	(	(	PUNCT
ma-174	159	2	ω)|	ω)|	PROPN
ma-174	159	3	=	=	SYM
ma-174	159	4	|f	|f	PROPN
ma-174	159	5	(	(	PUNCT
ma-174	159	6	e−t	e−t	X
ma-174	159	7	i)|+	i)|+	ADJ
ma-174	159	8	sup	sup	NOUN
ma-174	159	9	ω∈u	ω∈u	NOUN
ma-174	159	10	=(	=(	NOUN
ma-174	159	11	ω)α|e−t	ω)α|e−t	ADP
ma-174	159	12	f	f	PROPN
ma-174	159	13	′(e−tω)|	′(e−tω)|	PROPN
ma-174	159	14	.	.	PUNCT
ma-174	160	1	now	now	ADV
ma-174	160	2	by	by	ADP
ma-174	160	3	change	change	NOUN
ma-174	160	4	of	of	ADP
ma-174	160	5	variables	variable	NOUN
ma-174	160	6	:	:	PUNCT
ma-174	160	7	let	let	VERB
ma-174	160	8	z	z	NOUN
ma-174	160	9	=	=	SYM
ma-174	160	10	e−tω	e−tω	NOUN
ma-174	160	11	,	,	PUNCT
ma-174	160	12	then	then	ADV
ma-174	160	13	ω	ω	X
ma-174	160	14	=	=	PUNCT
ma-174	160	15	etz	etz	PROPN
ma-174	160	16	,	,	PUNCT
ma-174	160	17	and	and	CCONJ
ma-174	160	18	=(	=(	PROPN
ma-174	160	19	ω	ω	NUM
ma-174	160	20	)	)	PUNCT
ma-174	160	21	=	=	SYM
ma-174	160	22	et=(z	et=(z	NUM
ma-174	160	23	)	)	PUNCT
ma-174	160	24	.	.	PUNCT
ma-174	161	1	therefore	therefore	ADV
ma-174	161	2	,	,	PUNCT
ma-174	161	3	‖cϕt	‖cϕt	ADV
ma-174	161	4	f	f	PROPN
ma-174	161	5	‖bα(u	‖bα(u	PROPN
ma-174	161	6	)	)	PUNCT
ma-174	162	1	=	=	PRON
ma-174	162	2	|f	|f	PROPN
ma-174	162	3	(	(	PUNCT
ma-174	162	4	e−t	e−t	X
ma-174	162	5	i)|+	i)|+	ADJ
ma-174	162	6	sup	sup	NOUN
ma-174	162	7	z∈u	z∈u	PROPN
ma-174	162	8	etα=(z)α|e−t	etα=(z)α|e−t	PROPN
ma-174	163	1	f	f	PROPN
ma-174	163	2	′(z)|	′(z)|	PROPN
ma-174	163	3	=	=	PUNCT
ma-174	163	4	|f	|f	PROPN
ma-174	163	5	(	(	PUNCT
ma-174	163	6	e−t	e−t	X
ma-174	163	7	i)|+	i)|+	ADJ
ma-174	163	8	e(α−1)t	e(α−1)t	PROPN
ma-174	163	9	sup	sup	PROPN
ma-174	163	10	z∈u	z∈u	PROPN
ma-174	163	11	=(	=(	PROPN
ma-174	163	12	z)α|f	z)α|f	PROPN
ma-174	163	13	′(z)|	′(z)|	PROPN
ma-174	163	14	6=	6=	PROPN
ma-174	163	15	|f	|f	PROPN
ma-174	163	16	(	(	PUNCT
ma-174	163	17	i)|+	i)|+	ADJ
ma-174	163	18	sup	sup	NOUN
ma-174	163	19	z∈u	z∈u	PROPN
ma-174	163	20	=(	=(	PROPN
ma-174	163	21	z)α|f	z)α|f	PROPN
ma-174	163	22	′(z)|	′(z)|	PROPN
ma-174	163	23	=	=	SYM
ma-174	163	24	‖f	‖f	NOUN
ma-174	163	25	‖bα(u	‖bα(u	NUM
ma-174	163	26	)	)	PUNCT
ma-174	163	27	,	,	PUNCT
ma-174	163	28	which	which	PRON
ma-174	163	29	completes	complete	VERB
ma-174	163	30	the	the	DET
ma-174	163	31	proof	proof	NOUN
ma-174	163	32	.	.	PUNCT
ma-174	164	1	�	�	PROPN
ma-174	164	2	next	next	ADV
ma-174	164	3	,	,	PUNCT
ma-174	164	4	we	we	PRON
ma-174	164	5	prove	prove	VERB
ma-174	164	6	that	that	SCONJ
ma-174	164	7	the	the	DET
ma-174	164	8	operator	operator	NOUN
ma-174	164	9	cϕt	cϕt	VERB
ma-174	164	10	given	give	VERB
ma-174	164	11	by	by	ADP
ma-174	164	12	(	(	PUNCT
ma-174	164	13	4.1	4.1	NUM
ma-174	164	14	)	)	PUNCT
ma-174	164	15	is	be	AUX
ma-174	164	16	strongly	strongly	ADV
ma-174	164	17	continuous	continuous	ADJ
ma-174	164	18	on	on	ADP
ma-174	164	19	bα0	bα0	PROPN
ma-174	164	20	(	(	PUNCT
ma-174	164	21	u	u	NOUN
ma-174	164	22	)	)	PUNCT
ma-174	164	23	.	.	PUNCT
ma-174	165	1	theorem	theorem	VERB
ma-174	165	2	4.2	4.2	NUM
ma-174	165	3	.	.	PUNCT
ma-174	166	1	(	(	PUNCT
ma-174	166	2	cϕt	cϕt	NOUN
ma-174	166	3	)	)	PUNCT
ma-174	166	4	t∈r	t∈r	NOUN
ma-174	166	5	is	be	AUX
ma-174	166	6	strongly	strongly	ADV
ma-174	166	7	continuous	continuous	ADJ
ma-174	166	8	on	on	ADP
ma-174	166	9	bα0	bα0	PROPN
ma-174	166	10	(	(	PUNCT
ma-174	166	11	u	u	NOUN
ma-174	166	12	)	)	PUNCT
ma-174	166	13	.	.	PUNCT
ma-174	167	1	proof	proof	NOUN
ma-174	167	2	.	.	PUNCT
ma-174	168	1	to	to	PART
ma-174	168	2	prove	prove	VERB
ma-174	168	3	strong	strong	ADJ
ma-174	168	4	continuity	continuity	NOUN
ma-174	168	5	of	of	ADP
ma-174	168	6	(	(	PUNCT
ma-174	168	7	cϕt	cϕt	NOUN
ma-174	168	8	)	)	PUNCT
ma-174	168	9	t∈r	t∈r	NOUN
ma-174	168	10	,	,	PUNCT
ma-174	168	11	it	it	PRON
ma-174	168	12	suffices	suffice	VERB
ma-174	168	13	to	to	PART
ma-174	168	14	show	show	VERB
ma-174	168	15	that	that	SCONJ
ma-174	168	16	‖cϕt	‖cϕt	ADV
ma-174	168	17	f	f	PROPN
ma-174	168	18	−f	−f	PROPN
ma-174	168	19	‖bα(u	‖bα(u	PROPN
ma-174	168	20	)	)	PUNCT
ma-174	168	21	→	→	SYM
ma-174	168	22	0	0	NUM
ma-174	168	23	as	as	SCONJ
ma-174	168	24	t	t	PROPN
ma-174	168	25	→	→	SYM
ma-174	168	26	0.that	0.that	X
ma-174	168	27	is	be	AUX
ma-174	168	28	,	,	PUNCT
ma-174	168	29	|	|	ADV
ma-174	168	30	(	(	PUNCT
ma-174	168	31	cϕt	cϕt	NOUN
ma-174	168	32	f	f	PROPN
ma-174	168	33	−	−	PROPN
ma-174	168	34	f	f	PROPN
ma-174	168	35	)	)	PUNCT
ma-174	168	36	(	(	PUNCT
ma-174	168	37	i)|+‖cϕt	i)|+‖cϕt	PROPN
ma-174	168	38	f	f	PROPN
ma-174	168	39	−f	−f	PROPN
ma-174	168	40	‖bα1	‖bα1	PROPN
ma-174	168	41	(	(	PUNCT
ma-174	168	42	u	u	NOUN
ma-174	168	43	)	)	PUNCT
ma-174	168	44	→	→	SYM
ma-174	168	45	0	0	NUM
ma-174	168	46	as	as	ADP
ma-174	168	47	t	t	PROPN
ma-174	168	48	→	→	SYM
ma-174	168	49	0	0	X
ma-174	168	50	.	.	PUNCT
ma-174	169	1	this	this	PRON
ma-174	169	2	is	be	AUX
ma-174	169	3	equivalent	equivalent	ADJ
ma-174	169	4	to	to	ADP
ma-174	169	5	|	|	ADV
ma-174	169	6	(	(	PUNCT
ma-174	169	7	cϕt	cϕt	NOUN
ma-174	169	8	f	f	PROPN
ma-174	169	9	−	−	PROPN
ma-174	169	10	f	f	PROPN
ma-174	169	11	)	)	PUNCT
ma-174	169	12	(	(	PUNCT
ma-174	169	13	i)|	i)|	INTJ
ma-174	169	14	→	→	SYM
ma-174	169	15	0and	0and	NOUN
ma-174	170	1	‖cϕt	‖cϕt	ADP
ma-174	171	1	f	f	PROPN
ma-174	172	1	−	−	PROPN
ma-174	172	2	f	f	PROPN
ma-174	172	3	‖bα1	‖bα1	PROPN
ma-174	172	4	(	(	PUNCT
ma-174	172	5	u	u	NOUN
ma-174	172	6	)	)	PUNCT
ma-174	172	7	→	→	SYM
ma-174	172	8	0	0	NUM
ma-174	172	9	,	,	PUNCT
ma-174	172	10	as	as	ADP
ma-174	172	11	t	t	PROPN
ma-174	172	12	→	→	SYM
ma-174	172	13	0	0	NUM
ma-174	172	14	.	.	PUNCT
ma-174	173	1	for	for	ADP
ma-174	173	2	the	the	DET
ma-174	173	3	former	former	ADJ
ma-174	173	4	,	,	PUNCT
ma-174	173	5	we	we	PRON
ma-174	173	6	have	have	VERB
ma-174	173	7	|	|	ADV
ma-174	173	8	(	(	PUNCT
ma-174	173	9	cϕt	cϕt	NOUN
ma-174	173	10	f	f	PROPN
ma-174	173	11	−	−	PROPN
ma-174	173	12	f	f	PROPN
ma-174	173	13	)	)	PUNCT
ma-174	174	1	(	(	PUNCT
ma-174	174	2	i)|	i)|	INTJ
ma-174	174	3	=	=	PRON
ma-174	174	4	|cϕt	|cϕt	PROPN
ma-174	174	5	f	f	X
ma-174	174	6	(	(	PUNCT
ma-174	174	7	i)−	i)−	PROPN
ma-174	174	8	f	f	PROPN
ma-174	174	9	(	(	PUNCT
ma-174	174	10	i)|	i)|	INTJ
ma-174	174	11	(	(	PUNCT
ma-174	174	12	6	6	NUM
ma-174	174	13	)	)	PUNCT
ma-174	174	14	=	=	PRON
ma-174	174	15	|f	|f	PROPN
ma-174	175	1	(	(	PUNCT
ma-174	175	2	ϕt(i))−	ϕt(i))−	NUM
ma-174	175	3	f	f	PROPN
ma-174	175	4	(	(	PUNCT
ma-174	175	5	i)|	i)|	PROPN
ma-174	175	6	=	=	SYM
ma-174	175	7	|f	|f	PROPN
ma-174	175	8	(	(	PUNCT
ma-174	175	9	e−t	e−t	X
ma-174	175	10	i)−	i)−	PROPN
ma-174	175	11	f	f	PROPN
ma-174	175	12	(	(	PUNCT
ma-174	175	13	i)|	i)|	INTJ
ma-174	175	14	→	→	SYM
ma-174	175	15	0	0	NUM
ma-174	175	16	as	as	ADP
ma-174	175	17	t	t	PROPN
ma-174	175	18	→	→	SYM
ma-174	175	19	0	0	NUM
ma-174	175	20	,	,	PUNCT
ma-174	175	21	as	as	SCONJ
ma-174	175	22	desired	desire	VERB
ma-174	175	23	.	.	PUNCT
ma-174	176	1	we	we	PRON
ma-174	176	2	now	now	ADV
ma-174	176	3	prove	prove	VERB
ma-174	176	4	that	that	SCONJ
ma-174	176	5	‖cϕt	‖cϕt	ADV
ma-174	177	1	f	f	PROPN
ma-174	178	1	−	−	PROPN
ma-174	178	2	f	f	PROPN
ma-174	178	3	‖bα1	‖bα1	PROPN
ma-174	178	4	(	(	PUNCT
ma-174	178	5	u	u	NOUN
ma-174	178	6	)	)	PUNCT
ma-174	178	7	→	→	SYM
ma-174	178	8	0	0	NUM
ma-174	178	9	as	as	ADP
ma-174	178	10	t	t	PROPN
ma-174	178	11	→	→	SYM
ma-174	178	12	0	0	X
ma-174	178	13	.	.	PUNCT
ma-174	178	14	recall	recall	VERB
ma-174	178	15	that	that	PRON
ma-174	178	16	ψ	ψ	X
ma-174	178	17	:	:	PUNCT
ma-174	178	18	d→	d→	NUM
ma-174	178	19	u	u	NOUN
ma-174	178	20	,	,	PUNCT
ma-174	178	21	ϕt	ϕt	ADV
ma-174	178	22	:	:	PUNCT
ma-174	178	23	u→	u→	PUNCT
ma-174	178	24	u	u	NOUN
ma-174	178	25	andψ−1	andψ−1	PROPN
ma-174	178	26	:	:	PUNCT
ma-174	178	27	u	u	PROPN
ma-174	178	28	→	→	PUNCT
ma-174	178	29	d.	d.	PROPN
ma-174	178	30	we	we	PRON
ma-174	178	31	can	can	AUX
ma-174	178	32	therefore	therefore	ADV
ma-174	178	33	have	have	VERB
ma-174	178	34	d	d	NOUN
ma-174	178	35	ψ−→	ψ−→	NOUN
ma-174	178	36	u	u	NOUN
ma-174	178	37	ϕt−→	ϕt−→	CCONJ
ma-174	178	38	u	u	PROPN
ma-174	178	39	ψ−1−−→	ψ−1−−→	PROPN
ma-174	178	40	d.	d.	PROPN
ma-174	178	41	now	now	ADV
ma-174	178	42	,	,	PUNCT
ma-174	178	43	let	let	VERB
ma-174	178	44	xt	xt	X
ma-174	178	45	=	=	SYM
ma-174	179	1	ψ−1	ψ−1	PROPN
ma-174	179	2	◦	◦	NOUN
ma-174	179	3	ϕt	ϕt	ADV
ma-174	179	4	◦	◦	NOUN
ma-174	179	5	ψ	ψ	X
ma-174	179	6	:	:	PUNCT
ma-174	179	7	d→	d→	VERB
ma-174	179	8	d.	d.	PROPN
ma-174	179	9	if	if	SCONJ
ma-174	179	10	(	(	PUNCT
ma-174	179	11	ϕt)t≥0	ϕt)t≥0	NOUN
ma-174	179	12	is	be	AUX
ma-174	179	13	an	an	DET
ma-174	179	14	automorphism	automorphism	NOUN
ma-174	179	15	of	of	ADP
ma-174	179	16	the	the	DET
ma-174	179	17	upper	upper	ADJ
ma-174	179	18	half	half	ADJ
ma-174	179	19	plane	plane	NOUN
ma-174	179	20	u	u	NOUN
ma-174	179	21	,	,	PUNCT
ma-174	179	22	then	then	ADV
ma-174	179	23	(	(	PUNCT
ma-174	179	24	xt)t≥0	xt)t≥0	PROPN
ma-174	179	25	is	be	AUX
ma-174	179	26	an	an	DET
ma-174	179	27	automorphismof	automorphismof	NOUN
ma-174	179	28	the	the	DET
ma-174	179	29	unit	unit	NOUN
ma-174	179	30	disc	disc	VERB
ma-174	179	31	d.	d.	PROPN
ma-174	179	32	since	since	SCONJ
ma-174	179	33	xt	xt	PROPN
ma-174	180	1	=	=	SYM
ma-174	180	2	ψ−1	ψ−1	PROPN
ma-174	180	3	◦	◦	NOUN
ma-174	180	4	ϕt	ϕt	ADV
ma-174	180	5	◦	◦	NOUN
ma-174	180	6	ψ	ψ	NUM
ma-174	180	7	,	,	PUNCT
ma-174	180	8	it	it	PRON
ma-174	180	9	follows	follow	VERB
ma-174	180	10	that	that	SCONJ
ma-174	180	11	‖cϕt	‖cϕt	PROPN
ma-174	181	1	f	f	PROPN
ma-174	182	1	−	−	PROPN
ma-174	182	2	f	f	PROPN
ma-174	182	3	‖bα1	‖bα1	PROPN
ma-174	182	4	(	(	PUNCT
ma-174	182	5	u	u	NOUN
ma-174	182	6	)	)	PUNCT
ma-174	182	7	→	→	SYM
ma-174	182	8	0	0	NUM
ma-174	182	9	as	as	ADP
ma-174	182	10	t	t	PROPN
ma-174	182	11	→	→	SYM
ma-174	182	12	0	0	PUNCT
ma-174	182	13	if	if	SCONJ
ma-174	182	14	andonly	andonly	ADV
ma-174	182	15	if	if	SCONJ
ma-174	182	16	‖cxt	‖cxt	PROPN
ma-174	182	17	f	f	PROPN
ma-174	182	18	∗	∗	VERB
ma-174	182	19	−	−	PROPN
ma-174	182	20	f	f	PROPN
ma-174	182	21	∗‖bα(d	∗‖bα(d	NOUN
ma-174	182	22	)	)	PUNCT
ma-174	182	23	→	→	SYM
ma-174	182	24	0	0	NUM
ma-174	182	25	as	as	SCONJ
ma-174	182	26	t	t	PROPN
ma-174	182	27	→	→	SYM
ma-174	182	28	0	0	NUM
ma-174	182	29	8	8	NUM
ma-174	182	30	cayley	cayley	NOUN
ma-174	182	31	transform	transform	NOUN
ma-174	182	32	is	be	AUX
ma-174	182	33	given	give	VERB
ma-174	182	34	by	by	ADP
ma-174	182	35	ψ(z	ψ(z	PROPN
ma-174	182	36	)	)	PUNCT
ma-174	182	37	=	=	SYM
ma-174	182	38	i(1+z	i(1+z	PROPN
ma-174	182	39	)	)	PUNCT
ma-174	182	40	1−z	1−z	NUM
ma-174	182	41	.	.	PUNCT
ma-174	183	1	we	we	PRON
ma-174	183	2	therefore	therefore	ADV
ma-174	183	3	have	have	VERB
ma-174	183	4	ψ−1	ψ−1	PROPN
ma-174	183	5	◦	◦	NOUN
ma-174	183	6	ϕ−t	ϕ−t	NOUN
ma-174	183	7	◦	◦	NOUN
ma-174	183	8	ψ(z	ψ(z	NOUN
ma-174	183	9	)	)	PUNCT
ma-174	183	10	=	=	SYM
ma-174	183	11	ψ−1	ψ−1	PROPN
ma-174	183	12	(	(	PUNCT
ma-174	183	13	ϕt	ϕt	INTJ
ma-174	183	14	(	(	PUNCT
ma-174	183	15	ψ(z	ψ(z	PROPN
ma-174	183	16	)	)	PUNCT
ma-174	183	17	)	)	PUNCT
ma-174	183	18	)	)	PUNCT
ma-174	183	19	.	.	PUNCT
ma-174	184	1	=	=	PUNCT
ma-174	184	2	ψ−1	ψ−1	PROPN
ma-174	184	3	(	(	PUNCT
ma-174	184	4	ϕt	ϕt	INTJ
ma-174	184	5	(	(	PUNCT
ma-174	184	6	i(1	i(1	PROPN
ma-174	184	7	+	+	PROPN
ma-174	184	8	z	z	NOUN
ma-174	184	9	)	)	PUNCT
ma-174	184	10	1−	1−	NUM
ma-174	184	11	z	z	NOUN
ma-174	184	12	)	)	PUNCT
ma-174	184	13	)	)	PUNCT
ma-174	185	1	=	=	PUNCT
ma-174	185	2	ψ−1	ψ−1	PROPN
ma-174	185	3	(	(	PUNCT
ma-174	185	4	e−t	e−t	X
ma-174	185	5	(	(	PUNCT
ma-174	185	6	i(1	i(1	PROPN
ma-174	185	7	+	+	PROPN
ma-174	185	8	z	z	NOUN
ma-174	185	9	)	)	PUNCT
ma-174	185	10	1−	1−	NUM
ma-174	185	11	z	z	NOUN
ma-174	185	12	)	)	PUNCT
ma-174	185	13	)	)	PUNCT
ma-174	185	14	.	.	PUNCT
ma-174	186	1	substituting	substitute	VERB
ma-174	186	2	ψ−1(z	ψ−1(z	PROPN
ma-174	186	3	)	)	PUNCT
ma-174	186	4	=	=	SYM
ma-174	186	5	z−i	z−i	NUM
ma-174	186	6	z+i	z+i	NUM
ma-174	186	7	,	,	PUNCT
ma-174	186	8	we	we	PRON
ma-174	186	9	obtain	obtain	VERB
ma-174	186	10	ψ−1	ψ−1	PROPN
ma-174	186	11	◦	◦	NOUN
ma-174	186	12	ϕ−t	ϕ−t	NOUN
ma-174	186	13	◦	◦	NOUN
ma-174	186	14	ψ(z	ψ(z	NOUN
ma-174	186	15	)	)	PUNCT
ma-174	186	16	=	=	SYM
ma-174	186	17	e−t	e−t	PROPN
ma-174	186	18	(	(	PUNCT
ma-174	186	19	i(1+z)1−z	i(1+z)1−z	PROPN
ma-174	186	20	)	)	PUNCT
ma-174	186	21	−	−	PROPN
ma-174	187	1	i	i	PRON
ma-174	187	2	e−t	e−t	VERB
ma-174	187	3	(	(	PUNCT
ma-174	187	4	i(1+z)1−z	i(1+z)1−z	PROPN
ma-174	187	5	)	)	PUNCT
ma-174	188	1	+	+	CCONJ
ma-174	188	2	i	i	PRON
ma-174	188	3	.	.	PUNCT
ma-174	189	1	simplifying	simplify	VERB
ma-174	189	2	the	the	DET
ma-174	189	3	fraction	fraction	NOUN
ma-174	189	4	,	,	PUNCT
ma-174	189	5	we	we	PRON
ma-174	189	6	have	have	VERB
ma-174	189	7	ψ−1	ψ−1	PROPN
ma-174	189	8	◦	◦	NOUN
ma-174	189	9	ϕ−t	ϕ−t	NOUN
ma-174	189	10	◦	◦	NOUN
ma-174	189	11	ψ(z	ψ(z	NOUN
ma-174	189	12	)	)	PUNCT
ma-174	189	13	=	=	SYM
ma-174	189	14	z	z	NOUN
ma-174	189	15	+	+	NOUN
ma-174	189	16	e−tz	e−tz	NUM
ma-174	189	17	−	−	NUM
ma-174	189	18	1	1	NUM
ma-174	189	19	+	+	NUM
ma-174	189	20	e−t	e−t	NOUN
ma-174	189	21	−z	−z	NOUN
ma-174	189	22	+	+	CCONJ
ma-174	189	23	e−tz	e−tz	PROPN
ma-174	189	24	+	+	CCONJ
ma-174	189	25	1	1	NUM
ma-174	189	26	+	+	NUM
ma-174	189	27	e−t	e−t	NOUN
ma-174	189	28	.	.	PUNCT
ma-174	190	1	now	now	ADV
ma-174	190	2	,	,	PUNCT
ma-174	190	3	by	by	ADP
ma-174	190	4	factorizing	factorize	VERB
ma-174	190	5	z	z	NOUN
ma-174	190	6	and	and	CCONJ
ma-174	190	7	dividing	divide	VERB
ma-174	190	8	both	both	CCONJ
ma-174	190	9	the	the	DET
ma-174	190	10	numerator	numerator	NOUN
ma-174	190	11	and	and	CCONJ
ma-174	190	12	denominator	denominator	NOUN
ma-174	190	13	by	by	ADP
ma-174	190	14	(	(	PUNCT
ma-174	190	15	1	1	NUM
ma-174	190	16	+	+	NUM
ma-174	190	17	e−t	e−t	NOUN
ma-174	190	18	)	)	PUNCT
ma-174	190	19	,	,	PUNCT
ma-174	190	20	we	we	PRON
ma-174	190	21	obtain	obtain	VERB
ma-174	190	22	ψ−1	ψ−1	PROPN
ma-174	190	23	◦	◦	NOUN
ma-174	190	24	ϕ−t	ϕ−t	NOUN
ma-174	190	25	◦	◦	NOUN
ma-174	190	26	ψ(z	ψ(z	NOUN
ma-174	190	27	)	)	PUNCT
ma-174	190	28	=	=	SYM
ma-174	191	1	z	z	NOUN
ma-174	191	2	−	−	PROPN
ma-174	191	3	(	(	PUNCT
ma-174	191	4	1−e	1−e	NUM
ma-174	191	5	−t	−t	NOUN
ma-174	191	6	)	)	PUNCT
ma-174	191	7	(	(	PUNCT
ma-174	191	8	1+e−t	1+e−t	NUM
ma-174	191	9	)	)	PUNCT
ma-174	191	10	1−	1−	NUM
ma-174	192	1	(	(	PUNCT
ma-174	192	2	1−e	1−e	NUM
ma-174	192	3	−t	−t	NOUN
ma-174	192	4	)	)	PUNCT
ma-174	192	5	1+e−t	1+e−t	PROPN
ma-174	193	1	z	z	NOUN
ma-174	193	2	.	.	PUNCT
ma-174	194	1	let	let	VERB
ma-174	194	2	bt	bt	NOUN
ma-174	194	3	=	=	PUNCT
ma-174	194	4	1−e−t	1−e−t	NOUN
ma-174	194	5	1+e−t	1+e−t	NUM
ma-174	194	6	,	,	PUNCT
ma-174	194	7	and	and	CCONJ
ma-174	194	8	substitute	substitute	VERB
ma-174	194	9	to	to	PART
ma-174	194	10	obtain	obtain	VERB
ma-174	194	11	ψ−1	ψ−1	PROPN
ma-174	194	12	◦	◦	NOUN
ma-174	194	13	ϕ−t	ϕ−t	NOUN
ma-174	194	14	◦	◦	NOUN
ma-174	194	15	ψ(z	ψ(z	NOUN
ma-174	194	16	)	)	PUNCT
ma-174	194	17	=	=	SYM
ma-174	195	1	z	z	NOUN
ma-174	195	2	−	−	NOUN
ma-174	195	3	bt	bt	NOUN
ma-174	195	4	1−	1−	NUM
ma-174	195	5	btz	btz	NOUN
ma-174	195	6	:	:	PUNCT
ma-174	195	7	=	=	SYM
ma-174	195	8	xt(z	xt(z	NUM
ma-174	195	9	)	)	PUNCT
ma-174	195	10	.	.	PUNCT
ma-174	196	1	further	far	ADV
ma-174	196	2	,	,	PUNCT
ma-174	196	3	we	we	PRON
ma-174	196	4	apply	apply	VERB
ma-174	196	5	density	density	NOUN
ma-174	196	6	of	of	ADP
ma-174	196	7	polynomials	polynomial	NOUN
ma-174	196	8	in	in	ADP
ma-174	196	9	bα0	bα0	PROPN
ma-174	196	10	(	(	PUNCT
ma-174	196	11	d	d	NOUN
ma-174	196	12	)	)	PUNCT
ma-174	196	13	to	to	PART
ma-174	196	14	prove	prove	VERB
ma-174	196	15	that	that	SCONJ
ma-174	196	16	for	for	ADP
ma-174	196	17	f	f	PROPN
ma-174	196	18	∗	∗	PROPN
ma-174	196	19	∈	∈	PROPN
ma-174	196	20	bα0	bα0	PROPN
ma-174	196	21	(	(	PUNCT
ma-174	196	22	d	d	PROPN
ma-174	196	23	)	)	PUNCT
ma-174	196	24	,	,	PUNCT
ma-174	196	25	we	we	PRON
ma-174	196	26	have	have	VERB
ma-174	196	27	‖cx	‖cx	PROPN
ma-174	196	28	t	t	PROPN
ma-174	196	29	f	f	PROPN
ma-174	196	30	∗−	∗−	PROPN
ma-174	196	31	f	f	PROPN
ma-174	196	32	∗‖bα1	∗‖bα1	NOUN
ma-174	196	33	(	(	PUNCT
ma-174	196	34	d	d	NOUN
ma-174	196	35	)	)	PUNCT
ma-174	196	36	→	→	SYM
ma-174	196	37	0	0	NUM
ma-174	196	38	as	as	ADP
ma-174	196	39	t	t	PROPN
ma-174	196	40	→	→	SYM
ma-174	196	41	0.by	0.by	NUM
ma-174	196	42	the	the	DET
ma-174	196	43	definition	definition	NOUN
ma-174	196	44	of	of	ADP
ma-174	196	45	the	the	DET
ma-174	196	46	norm	norm	NOUN
ma-174	196	47	,	,	PUNCT
ma-174	196	48	we	we	PRON
ma-174	196	49	have	have	VERB
ma-174	196	50	lim	lim	NOUN
ma-174	196	51	t→0	t→0	PROPN
ma-174	197	1	+	+	NUM
ma-174	197	2	‖cx	‖cx	PROPN
ma-174	197	3	t	t	PROPN
ma-174	197	4	f	f	PROPN
ma-174	197	5	∗	∗	VERB
ma-174	197	6	−	−	PROPN
ma-174	197	7	f	f	PROPN
ma-174	197	8	∗‖bα(d	∗‖bα(d	PROPN
ma-174	197	9	)	)	PUNCT
ma-174	198	1	=	=	SYM
ma-174	198	2	lim	lim	PROPN
ma-174	198	3	t→0	t→0	PROPN
ma-174	199	1	+	+	CCONJ
ma-174	200	1	|(cx	|(cx	NUM
ma-174	200	2	t	t	NOUN
ma-174	200	3	f	f	PROPN
ma-174	200	4	∗	∗	VERB
ma-174	200	5	−	−	PROPN
ma-174	201	1	f	f	PROPN
ma-174	201	2	∗)(0)|+	∗)(0)|+	ADJ
ma-174	201	3	sup	sup	NOUN
ma-174	201	4	z∈d	z∈d	NUM
ma-174	201	5	(	(	PUNCT
ma-174	201	6	1−	1−	NUM
ma-174	201	7	|z	|z	PROPN
ma-174	201	8	|2	|2	NUM
ma-174	201	9	)	)	PUNCT
ma-174	201	10	α	α	PROPN
ma-174	201	11	|(cx	|(cx	PROPN
ma-174	201	12	t	t	PROPN
ma-174	201	13	f	f	PROPN
ma-174	201	14	∗	∗	NOUN
ma-174	201	15	−	−	PROPN
ma-174	201	16	f	f	PROPN
ma-174	201	17	∗)′(z)|	∗)′(z)|	PROPN
ma-174	201	18	.	.	PUNCT
ma-174	202	1	let	let	VERB
ma-174	202	2	f	f	PROPN
ma-174	202	3	∗(z	∗(z	PROPN
ma-174	202	4	)	)	PUNCT
ma-174	202	5	=	=	SYM
ma-174	202	6	zn	zn	PROPN
ma-174	202	7	and	and	CCONJ
ma-174	202	8	z	z	PROPN
ma-174	202	9	∈	∈	PROPN
ma-174	202	10	d.	d.	NOUN
ma-174	202	11	we	we	PRON
ma-174	202	12	need	need	VERB
ma-174	202	13	to	to	PART
ma-174	202	14	show	show	VERB
ma-174	202	15	that	that	SCONJ
ma-174	202	16	‖	‖	PROPN
ma-174	202	17	(	(	PUNCT
ma-174	202	18	cxt	cxt	PROPN
ma-174	202	19	f	f	PROPN
ma-174	202	20	∗	∗	VERB
ma-174	202	21	−	−	PROPN
ma-174	202	22	f	f	PROPN
ma-174	202	23	∗	∗	NOUN
ma-174	202	24	)	)	PUNCT
ma-174	203	1	‖bα1	‖bα1	PROPN
ma-174	203	2	(	(	PUNCT
ma-174	203	3	d	d	NOUN
ma-174	203	4	)	)	PUNCT
ma-174	203	5	→	→	SYM
ma-174	203	6	0	0	NUM
ma-174	203	7	,	,	PUNCT
ma-174	203	8	as	as	SCONJ
ma-174	203	9	t	t	PROPN
ma-174	203	10	→	→	SYM
ma-174	203	11	0.since	0.since	NUM
ma-174	203	12	cxtz	cxtz	NOUN
ma-174	203	13	n	n	CCONJ
ma-174	203	14	−	−	PROPN
ma-174	203	15	zn	zn	NOUN
ma-174	203	16	=	=	SYM
ma-174	203	17	(	(	PUNCT
ma-174	203	18	xt(z))n	xt(z))n	NUM
ma-174	203	19	−	−	PROPN
ma-174	204	1	zn	zn	PROPN
ma-174	204	2	,	,	PUNCT
ma-174	204	3	n	n	PRON
ma-174	204	4	≥	≥	NOUN
ma-174	204	5	1	1	NUM
ma-174	204	6	,	,	PUNCT
ma-174	204	7	differentiating	differentiate	VERB
ma-174	204	8	(	(	PUNCT
ma-174	204	9	xt(z))n	xt(z))n	NUM
ma-174	204	10	−	−	PROPN
ma-174	204	11	zn	zn	INTJ
ma-174	204	12	with	with	ADP
ma-174	204	13	respect	respect	NOUN
ma-174	204	14	to	to	ADP
ma-174	204	15	z	z	NOUN
ma-174	204	16	,	,	PUNCT
ma-174	204	17	we	we	PRON
ma-174	204	18	obtain	obtain	VERB
ma-174	204	19	(	(	PUNCT
ma-174	204	20	cxt	cxt	PROPN
ma-174	204	21	f	f	PROPN
ma-174	204	22	∗	∗	VERB
ma-174	204	23	−	−	PROPN
ma-174	204	24	f	f	PROPN
ma-174	204	25	∗)′(z	∗)′(z	PROPN
ma-174	204	26	)	)	PUNCT
ma-174	204	27	=	=	SYM
ma-174	204	28	n(xt(z))n−1x	n(xt(z))n−1x	PROPN
ma-174	204	29	′t(z)−	′t(z)−	VERB
ma-174	204	30	nzn−1	nzn−1	PROPN
ma-174	204	31	=	=	SYM
ma-174	204	32	n[(xt(z))n−1x	n[(xt(z))n−1x	X
ma-174	204	33	′t(z)−	′t(z)−	ADJ
ma-174	204	34	zn−1	zn−1	PROPN
ma-174	204	35	]	]	PUNCT
ma-174	204	36	.	.	PUNCT
ma-174	205	1	substituting	substitute	VERB
ma-174	205	2	for	for	ADP
ma-174	205	3	xt(z	xt(z	NUM
ma-174	205	4	)	)	PUNCT
ma-174	206	1	=	=	PUNCT
ma-174	206	2	z	z	NOUN
ma-174	207	1	−	−	NOUN
ma-174	207	2	bt	bt	NOUN
ma-174	207	3	1−	1−	NUM
ma-174	207	4	btz9	btz9	NOUN
ma-174	207	5	and	and	CCONJ
ma-174	207	6	x	x	X
ma-174	207	7	′t(z	′t(z	NOUN
ma-174	207	8	)	)	PUNCT
ma-174	207	9	=	=	PUNCT
ma-174	208	1	(	(	PUNCT
ma-174	208	2	1−	1−	NUM
ma-174	208	3	btz)1−	btz)1−	NOUN
ma-174	208	4	(	(	PUNCT
ma-174	208	5	z	z	NOUN
ma-174	208	6	−	−	PROPN
ma-174	208	7	bt)(−bt	bt)(−bt	PROPN
ma-174	208	8	)	)	PUNCT
ma-174	208	9	(	(	PUNCT
ma-174	208	10	1−	1−	NUM
ma-174	208	11	btz)2	btz)2	NOUN
ma-174	208	12	=	=	SYM
ma-174	208	13	(	(	PUNCT
ma-174	208	14	1−	1−	NUM
ma-174	208	15	b2	b2	NOUN
ma-174	208	16	t	t	NOUN
ma-174	208	17	)	)	PUNCT
ma-174	208	18	(	(	PUNCT
ma-174	208	19	1−	1−	NUM
ma-174	208	20	btz)2	btz)2	NOUN
ma-174	208	21	,	,	PUNCT
ma-174	208	22	we	we	PRON
ma-174	208	23	obtain	obtain	VERB
ma-174	208	24	(	(	PUNCT
ma-174	208	25	cxt	cxt	PROPN
ma-174	208	26	f	f	PROPN
ma-174	208	27	∗	∗	VERB
ma-174	208	28	−	−	PROPN
ma-174	208	29	f	f	PROPN
ma-174	208	30	∗)′(z	∗)′(z	PROPN
ma-174	208	31	)	)	PUNCT
ma-174	208	32	=	=	SYM
ma-174	209	1	n	n	PROPN
ma-174	210	1	[	[	X
ma-174	210	2	(	(	PUNCT
ma-174	210	3	z	z	NOUN
ma-174	210	4	−	−	NOUN
ma-174	210	5	bt	bt	NOUN
ma-174	210	6	1−	1−	NUM
ma-174	210	7	btz	btz	NOUN
ma-174	210	8	)	)	PUNCT
ma-174	210	9	n−1	n−1	PROPN
ma-174	210	10	(	(	PUNCT
ma-174	210	11	1−	1−	NUM
ma-174	210	12	b2	b2	NOUN
ma-174	210	13	t	t	NOUN
ma-174	210	14	)	)	PUNCT
ma-174	210	15	(	(	PUNCT
ma-174	210	16	1−	1−	NUM
ma-174	210	17	btz)2	btz)2	PROPN
ma-174	210	18	−	−	PROPN
ma-174	210	19	zn−1	zn−1	PROPN
ma-174	210	20	]	]	PUNCT
ma-174	210	21	=	=	PUNCT
ma-174	211	1	n	n	PART
ma-174	211	2	[	[	PUNCT
ma-174	211	3	(	(	PUNCT
ma-174	211	4	z	z	NOUN
ma-174	211	5	−	−	NOUN
ma-174	211	6	bt)n−1(1−	bt)n−1(1−	NOUN
ma-174	211	7	b2	b2	NOUN
ma-174	211	8	t	t	NOUN
ma-174	211	9	)	)	PUNCT
ma-174	211	10	(	(	PUNCT
ma-174	211	11	1−	1−	NUM
ma-174	211	12	btz)n−1(1−	btz)n−1(1−	VERB
ma-174	211	13	btz)2	btz)2	PROPN
ma-174	211	14	−	−	PROPN
ma-174	211	15	zn−1	zn−1	PROPN
ma-174	211	16	]	]	PUNCT
ma-174	211	17	=	=	PUNCT
ma-174	212	1	n	n	PART
ma-174	212	2	[	[	PUNCT
ma-174	212	3	(	(	PUNCT
ma-174	212	4	z	z	NOUN
ma-174	212	5	−	−	NOUN
ma-174	212	6	bt)n−1(1−	bt)n−1(1−	NOUN
ma-174	212	7	b2	b2	NOUN
ma-174	212	8	t	t	PROPN
ma-174	212	9	)	)	PUNCT
ma-174	212	10	−	−	PROPN
ma-174	213	1	zn−1(1−	zn−1(1−	PROPN
ma-174	213	2	btz)n+1	btz)n+1	NOUN
ma-174	213	3	(	(	PUNCT
ma-174	213	4	1−	1−	NUM
ma-174	213	5	btz)n+1	btz)n+1	NOUN
ma-174	213	6	]	]	PUNCT
ma-174	213	7	.	.	PUNCT
ma-174	214	1	it	it	PRON
ma-174	214	2	therefore	therefore	ADV
ma-174	214	3	follows	follow	VERB
ma-174	214	4	that	that	SCONJ
ma-174	214	5	limt→0	limt→0	PROPN
ma-174	214	6	+	+	CCONJ
ma-174	214	7	‖cx	‖cx	PROPN
ma-174	214	8	t	t	PROPN
ma-174	214	9	f	f	PROPN
ma-174	214	10	∗	∗	VERB
ma-174	214	11	−	−	PROPN
ma-174	214	12	f	f	PROPN
ma-174	214	13	∗‖bα1	∗‖bα1	NOUN
ma-174	214	14	(	(	PUNCT
ma-174	214	15	d	d	X
ma-174	214	16	)	)	PUNCT
ma-174	214	17	is	be	AUX
ma-174	214	18	equivalent	equivalent	ADJ
ma-174	214	19	to	to	ADP
ma-174	214	20	lim	lim	PROPN
ma-174	214	21	t→0	t→0	PROPN
ma-174	215	1	+	+	CCONJ
ma-174	215	2	(	(	PUNCT
ma-174	215	3	(	(	PUNCT
ma-174	215	4	sup	sup	NOUN
ma-174	215	5	z∈d	z∈d	NUM
ma-174	215	6	(	(	PUNCT
ma-174	215	7	1−	1−	NUM
ma-174	215	8	|z	|z	PROPN
ma-174	215	9	|2	|2	NUM
ma-174	215	10	)	)	PUNCT
ma-174	215	11	α	α	PROPN
ma-174	215	12	∣∣∣∣n	∣∣∣∣n	NOUN
ma-174	216	1	[	[	X
ma-174	216	2	(	(	PUNCT
ma-174	216	3	z	z	NOUN
ma-174	216	4	−	−	NOUN
ma-174	216	5	bt)n−1(1−	bt)n−1(1−	NOUN
ma-174	216	6	b2	b2	NOUN
ma-174	216	7	t	t	PROPN
ma-174	216	8	)	)	PUNCT
ma-174	216	9	−	−	PROPN
ma-174	217	1	zn−1(1−	zn−1(1−	PROPN
ma-174	217	2	btz)n+1	btz)n+1	NOUN
ma-174	217	3	(	(	PUNCT
ma-174	217	4	1−	1−	NUM
ma-174	217	5	btz)n+1	btz)n+1	NOUN
ma-174	217	6	]	]	X
ma-174	217	7	∣∣∣∣	∣∣∣∣	NOUN
ma-174	217	8	)	)	PUNCT
ma-174	217	9	.	.	PUNCT
ma-174	218	1	now	now	ADV
ma-174	218	2	,	,	PUNCT
ma-174	218	3	let	let	VERB
ma-174	218	4	bt	bt	PRON
ma-174	218	5	→	→	SYM
ma-174	218	6	0	0	PROPN
ma-174	218	7	as	as	ADP
ma-174	218	8	t	t	PROPN
ma-174	218	9	→	→	SYM
ma-174	218	10	0	0	NUM
ma-174	218	11	,	,	PUNCT
ma-174	218	12	we	we	PRON
ma-174	218	13	obtain	obtain	VERB
ma-174	218	14	lim	lim	NOUN
ma-174	219	1	t→0	t→0	PROPN
ma-174	219	2	+	+	NUM
ma-174	219	3	‖cx	‖cx	PROPN
ma-174	219	4	t	t	PROPN
ma-174	219	5	f	f	PROPN
ma-174	219	6	∗	∗	VERB
ma-174	219	7	−	−	PROPN
ma-174	219	8	f	f	PROPN
ma-174	219	9	∗‖bα1	∗‖bα1	NOUN
ma-174	219	10	(	(	PUNCT
ma-174	219	11	d	d	X
ma-174	219	12	)	)	PUNCT
ma-174	219	13	=	=	SYM
ma-174	219	14	sup	sup	NOUN
ma-174	219	15	z∈d	z∈d	NOUN
ma-174	219	16	(	(	PUNCT
ma-174	219	17	1−	1−	NUM
ma-174	219	18	|z	|z	X
ma-174	219	19	|2)α	|2)α	PUNCT
ma-174	219	20	∣∣n[zn−1	∣∣n[zn−1	PROPN
ma-174	219	21	−	−	PROPN
ma-174	219	22	zn−1	zn−1	PROPN
ma-174	219	23	]	]	PUNCT
ma-174	219	24	∣∣	∣∣	X
ma-174	219	25	=	=	SYM
ma-174	219	26	0	0	X
ma-174	219	27	.	.	PUNCT
ma-174	220	1	since	since	SCONJ
ma-174	220	2	limt→0	limt→0	PROPN
ma-174	220	3	+	+	CCONJ
ma-174	220	4	‖(cxt	‖(cxt	PUNCT
ma-174	220	5	f	f	PROPN
ma-174	220	6	∗	∗	VERB
ma-174	220	7	−	−	PROPN
ma-174	220	8	f	f	PROPN
ma-174	220	9	∗‖bα1	∗‖bα1	NOUN
ma-174	220	10	(	(	PUNCT
ma-174	220	11	d	d	NOUN
ma-174	220	12	)	)	PUNCT
ma-174	220	13	=	=	SYM
ma-174	220	14	0	0	NUM
ma-174	220	15	,	,	PUNCT
ma-174	220	16	it	it	PRON
ma-174	220	17	follows	follow	VERB
ma-174	220	18	that	that	SCONJ
ma-174	220	19	lim	lim	PROPN
ma-174	220	20	t→0	t→0	PROPN
ma-174	221	1	+	+	CCONJ
ma-174	221	2	(	(	PUNCT
ma-174	221	3	‖cϕt	‖cϕt	ADV
ma-174	221	4	f	f	PROPN
ma-174	221	5	−	−	PROPN
ma-174	221	6	f	f	PROPN
ma-174	221	7	‖bα1	‖bα1	PROPN
ma-174	221	8	(	(	PUNCT
ma-174	221	9	u	u	NOUN
ma-174	221	10	)	)	PUNCT
ma-174	221	11	)	)	PUNCT
ma-174	222	1	=	=	PUNCT
ma-174	222	2	0	0	X
ma-174	222	3	.	.	PUNCT
ma-174	223	1	therefore	therefore	ADV
ma-174	223	2	‖cϕt	‖cϕt	ADV
ma-174	223	3	f	f	PROPN
ma-174	224	1	−	−	PROPN
ma-174	224	2	f	f	PROPN
ma-174	224	3	‖bα(u	‖bα(u	PROPN
ma-174	224	4	)	)	PUNCT
ma-174	225	1	=	=	PUNCT
ma-174	225	2	|ϕt	|ϕt	NUM
ma-174	225	3	f	f	X
ma-174	225	4	(	(	PUNCT
ma-174	225	5	i))−	i))−	NOUN
ma-174	225	6	f	f	X
ma-174	225	7	(	(	PUNCT
ma-174	225	8	i)|+	i)|+	ADV
ma-174	225	9	‖cϕt	‖cϕt	VERB
ma-174	225	10	f	f	NOUN
ma-174	225	11	−	−	PROPN
ma-174	225	12	f	f	PROPN
ma-174	225	13	‖bα1	‖bα1	PROPN
ma-174	225	14	(	(	PUNCT
ma-174	225	15	u	u	NOUN
ma-174	225	16	)	)	PUNCT
ma-174	225	17	→	→	SYM
ma-174	225	18	0	0	NUM
ma-174	225	19	as	as	ADP
ma-174	225	20	t→	t→	X
ma-174	225	21	0	0	NUM
ma-174	225	22	,	,	PUNCT
ma-174	225	23	as	as	SCONJ
ma-174	225	24	desired	desire	VERB
ma-174	225	25	.	.	PUNCT
ma-174	226	1	�	�	PROPN
ma-174	226	2	in	in	ADP
ma-174	226	3	the	the	DET
ma-174	226	4	next	next	ADJ
ma-174	226	5	proposition	proposition	NOUN
ma-174	226	6	,	,	PUNCT
ma-174	226	7	we	we	PRON
ma-174	226	8	compute	compute	VERB
ma-174	226	9	the	the	DET
ma-174	226	10	infinitesimal	infinitesimal	ADJ
ma-174	226	11	generator	generator	NOUN
ma-174	226	12	and	and	CCONJ
ma-174	226	13	determine	determine	VERB
ma-174	226	14	the	the	DET
ma-174	226	15	domain	domain	NOUN
ma-174	226	16	of	of	ADP
ma-174	226	17	thecomposition	thecomposition	NOUN
ma-174	226	18	semigroup	semigroup	NOUN
ma-174	226	19	in	in	ADP
ma-174	226	20	equation	equation	NOUN
ma-174	226	21	(	(	PUNCT
ma-174	226	22	4.1	4.1	NUM
ma-174	226	23	)	)	PUNCT
ma-174	226	24	.	.	PUNCT
ma-174	227	1	proposition	proposition	NOUN
ma-174	227	2	4.3	4.3	NUM
ma-174	227	3	.	.	PUNCT
ma-174	228	1	the	the	DET
ma-174	228	2	infinitesimal	infinitesimal	ADJ
ma-174	228	3	generator	generator	NOUN
ma-174	228	4	γ	γ	X
ma-174	228	5	of	of	ADP
ma-174	228	6	(	(	PUNCT
ma-174	228	7	cϕt	cϕt	NOUN
ma-174	228	8	)	)	PUNCT
ma-174	228	9	t≥0	t≥0	NOUN
ma-174	228	10	on	on	ADP
ma-174	228	11	bα0	bα0	PROPN
ma-174	228	12	(	(	PUNCT
ma-174	228	13	u	u	NOUN
ma-174	228	14	)	)	PUNCT
ma-174	228	15	is	be	AUX
ma-174	228	16	given	give	VERB
ma-174	228	17	by	by	ADP
ma-174	228	18	γf	γf	PROPN
ma-174	228	19	(	(	PUNCT
ma-174	228	20	z	z	NOUN
ma-174	228	21	)	)	PUNCT
ma-174	228	22	=	=	SYM
ma-174	228	23	−zf	−zf	NOUN
ma-174	228	24	′(z	′(z	NOUN
ma-174	228	25	)	)	PUNCT
ma-174	228	26	with	with	ADP
ma-174	228	27	the	the	DET
ma-174	228	28	domain	domain	NOUN
ma-174	228	29	dom	dom	NOUN
ma-174	228	30	(	(	PUNCT
ma-174	228	31	γ	γ	NOUN
ma-174	228	32	)	)	PUNCT
ma-174	228	33	=	=	NOUN
ma-174	228	34	{	{	PUNCT
ma-174	228	35	f	f	PROPN
ma-174	228	36	∈	∈	PROPN
ma-174	228	37	bα0	bα0	PROPN
ma-174	228	38	(	(	PUNCT
ma-174	228	39	u	u	NOUN
ma-174	228	40	)	)	PUNCT
ma-174	228	41	:	:	PUNCT
ma-174	228	42	zf	zf	PROPN
ma-174	228	43	′(z	′(z	NOUN
ma-174	228	44	)	)	PUNCT
ma-174	228	45	∈	∈	PROPN
ma-174	228	46	bα0	bα0	PROPN
ma-174	228	47	(	(	PUNCT
ma-174	228	48	u	u	NOUN
ma-174	228	49	)	)	PUNCT
ma-174	228	50	}	}	PUNCT
ma-174	228	51	.	.	PUNCT
ma-174	229	1	proof	proof	NOUN
ma-174	229	2	.	.	PUNCT
ma-174	230	1	using	use	VERB
ma-174	230	2	the	the	DET
ma-174	230	3	definition	definition	NOUN
ma-174	230	4	of	of	ADP
ma-174	230	5	the	the	DET
ma-174	230	6	infinitesimal	infinitesimal	ADJ
ma-174	230	7	generator	generator	NOUN
ma-174	230	8	γ	γ	PROPN
ma-174	230	9	of	of	ADP
ma-174	230	10	(	(	PUNCT
ma-174	230	11	cϕt	cϕt	NOUN
ma-174	230	12	)	)	PUNCT
ma-174	230	13	t≥0	t≥0	NOUN
ma-174	230	14	,	,	PUNCT
ma-174	230	15	for	for	ADP
ma-174	230	16	f	f	PROPN
ma-174	230	17	∈	∈	PROPN
ma-174	230	18	bα0	bα0	PROPN
ma-174	230	19	(	(	PUNCT
ma-174	230	20	u	u	NOUN
ma-174	230	21	)	)	PUNCT
ma-174	230	22	we	we	PRON
ma-174	230	23	have	have	AUX
ma-174	231	1	γf	γf	PROPN
ma-174	231	2	(	(	PUNCT
ma-174	231	3	z	z	NOUN
ma-174	231	4	)	)	PUNCT
ma-174	232	1	=	=	SYM
ma-174	232	2	lim	lim	PROPN
ma-174	232	3	t→0	t→0	PROPN
ma-174	232	4	+	+	PROPN
ma-174	232	5	cϕt	cϕt	PROPN
ma-174	232	6	f	f	X
ma-174	232	7	(	(	PUNCT
ma-174	232	8	z)−	z)−	PROPN
ma-174	232	9	f	f	X
ma-174	232	10	(	(	PUNCT
ma-174	232	11	z	z	NOUN
ma-174	232	12	)	)	PUNCT
ma-174	232	13	t	t	PROPN
ma-174	232	14	=	=	SYM
ma-174	232	15	lim	lim	PROPN
ma-174	232	16	t→0	t→0	PROPN
ma-174	232	17	+	+	PROPN
ma-174	232	18	f	f	X
ma-174	232	19	(	(	PUNCT
ma-174	232	20	e−tz	e−tz	PROPN
ma-174	232	21	)	)	PUNCT
ma-174	233	1	−	−	PROPN
ma-174	233	2	f	f	X
ma-174	233	3	(	(	PUNCT
ma-174	233	4	z	z	NOUN
ma-174	233	5	)	)	PUNCT
ma-174	233	6	t	t	NOUN
ma-174	233	7	=	=	SYM
ma-174	233	8	∂	∂	NOUN
ma-174	233	9	∂t	∂t	PROPN
ma-174	233	10	f	f	PROPN
ma-174	233	11	(	(	PUNCT
ma-174	233	12	e−tz	e−tz	NUM
ma-174	233	13	)	)	PUNCT
ma-174	233	14	∣∣∣∣	∣∣∣∣	NOUN
ma-174	233	15	t=0	t=0	VERB
ma-174	233	16	=	=	SYM
ma-174	234	1	−zf	−zf	NOUN
ma-174	234	2	′(z).10	′(z).10	NOUN
ma-174	234	3	this	this	PRON
ma-174	234	4	implies	imply	VERB
ma-174	234	5	that	that	SCONJ
ma-174	235	1	γf	γf	INTJ
ma-174	235	2	(	(	PUNCT
ma-174	235	3	z	z	NOUN
ma-174	235	4	)	)	PUNCT
ma-174	235	5	=	=	SYM
ma-174	235	6	−zf	−zf	NOUN
ma-174	235	7	′(z	′(z	NOUN
ma-174	235	8	)	)	PUNCT
ma-174	235	9	and	and	CCONJ
ma-174	235	10	therefore	therefore	ADV
ma-174	235	11	dom(γ	dom(γ	PROPN
ma-174	235	12	)	)	PUNCT
ma-174	235	13	⊆	⊆	NUM
ma-174	235	14	{	{	PUNCT
ma-174	235	15	f	f	PROPN
ma-174	235	16	∈	∈	PROPN
ma-174	235	17	bα0	bα0	PROPN
ma-174	235	18	(	(	PUNCT
ma-174	235	19	u	u	NOUN
ma-174	235	20	)	)	PUNCT
ma-174	235	21	:	:	PUNCT
ma-174	235	22	zf	zf	PROPN
ma-174	235	23	′	′	NUM
ma-174	235	24	∈	∈	PROPN
ma-174	235	25	bα0	bα0	PROPN
ma-174	235	26	(	(	PUNCT
ma-174	235	27	u	u	NOUN
ma-174	235	28	)	)	PUNCT
ma-174	235	29	}	}	PUNCT
ma-174	235	30	.	.	PUNCT
ma-174	236	1	to	to	PART
ma-174	236	2	provereverse	provereverse	PROPN
ma-174	236	3	inclusion	inclusion	NOUN
ma-174	236	4	,	,	PUNCT
ma-174	236	5	we	we	PRON
ma-174	236	6	let	let	VERB
ma-174	236	7	f	f	PROPN
ma-174	236	8	∈	∈	PROPN
ma-174	236	9	bα0	bα0	PROPN
ma-174	236	10	(	(	PUNCT
ma-174	236	11	u	u	NOUN
ma-174	236	12	)	)	PUNCT
ma-174	236	13	be	be	VERB
ma-174	236	14	such	such	ADJ
ma-174	236	15	that	that	SCONJ
ma-174	236	16	zf	zf	PROPN
ma-174	236	17	′	′	NUM
ma-174	236	18	∈	∈	PROPN
ma-174	236	19	bα0	bα0	PROPN
ma-174	236	20	(	(	PUNCT
ma-174	236	21	u	u	NOUN
ma-174	236	22	)	)	PUNCT
ma-174	236	23	.	.	PUNCT
ma-174	237	1	then	then	ADV
ma-174	237	2	for	for	ADP
ma-174	237	3	z	z	PROPN
ma-174	237	4	∈	∈	PROPN
ma-174	237	5	u	u	NOUN
ma-174	237	6	,	,	PUNCT
ma-174	237	7	cϕt	cϕt	PROPN
ma-174	237	8	f	f	PROPN
ma-174	237	9	(	(	PUNCT
ma-174	237	10	z)−	z)−	PROPN
ma-174	237	11	f	f	X
ma-174	237	12	(	(	PUNCT
ma-174	237	13	z	z	NOUN
ma-174	237	14	)	)	PUNCT
ma-174	237	15	t	t	NOUN
ma-174	237	16	=	=	SYM
ma-174	237	17	1	1	NUM
ma-174	237	18	t	t	NOUN
ma-174	237	19	∫	∫	PROPN
ma-174	237	20	t	t	PROPN
ma-174	237	21	0	0	NUM
ma-174	237	22	∂	∂	NUM
ma-174	237	23	∂s	∂s	PROPN
ma-174	237	24	(	(	PUNCT
ma-174	237	25	cϕs	cϕs	NOUN
ma-174	237	26	f	f	PROPN
ma-174	237	27	(	(	PUNCT
ma-174	237	28	z))ds	z))ds	NOUN
ma-174	237	29	=	=	SYM
ma-174	238	1	1	1	NUM
ma-174	238	2	t	t	NOUN
ma-174	238	3	∫	∫	PROPN
ma-174	238	4	t	t	PROPN
ma-174	238	5	0	0	NUM
ma-174	239	1	(	(	PUNCT
ma-174	239	2	−e−szf	−e−szf	NOUN
ma-174	239	3	′(e−sz))ds	′(e−sz))ds	X
ma-174	239	4	=	=	SYM
ma-174	239	5	1	1	NUM
ma-174	239	6	t	t	NOUN
ma-174	239	7	∫	∫	PROPN
ma-174	239	8	t	t	PROPN
ma-174	239	9	0	0	NUM
ma-174	239	10	cϕsf	cϕsf	NOUN
ma-174	239	11	(	(	PUNCT
ma-174	239	12	z)ds	z)ds	PROPN
ma-174	239	13	,	,	PUNCT
ma-174	239	14	wheref	wheref	NOUN
ma-174	239	15	(	(	PUNCT
ma-174	239	16	z	z	NOUN
ma-174	239	17	)	)	PUNCT
ma-174	239	18	=	=	SYM
ma-174	239	19	−zf	−zf	NOUN
ma-174	239	20	′(z	′(z	NOUN
ma-174	239	21	)	)	PUNCT
ma-174	239	22	.	.	PUNCT
ma-174	240	1	since	since	SCONJ
ma-174	240	2	f	f	PROPN
ma-174	240	3	(	(	PUNCT
ma-174	240	4	z	z	NOUN
ma-174	240	5	)	)	PUNCT
ma-174	240	6	is	be	AUX
ma-174	240	7	a	a	DET
ma-174	240	8	function	function	NOUN
ma-174	240	9	in	in	ADP
ma-174	240	10	bα0	bα0	PROPN
ma-174	240	11	(	(	PUNCT
ma-174	240	12	u	u	NOUN
ma-174	240	13	)	)	PUNCT
ma-174	240	14	,	,	PUNCT
ma-174	240	15	it	it	PRON
ma-174	240	16	remains	remain	VERB
ma-174	240	17	to	to	PART
ma-174	240	18	show	show	VERB
ma-174	240	19	that	that	SCONJ
ma-174	240	20	the	the	DET
ma-174	240	21	limit	limit	NOUN
ma-174	240	22	of	of	ADP
ma-174	240	23	f	f	PROPN
ma-174	240	24	(	(	PUNCT
ma-174	240	25	z	z	NOUN
ma-174	240	26	)	)	PUNCT
ma-174	240	27	exist	exist	VERB
ma-174	240	28	in	in	ADP
ma-174	240	29	bα0	bα0	PROPN
ma-174	240	30	(	(	PUNCT
ma-174	240	31	u	u	NOUN
ma-174	240	32	)	)	PUNCT
ma-174	240	33	.	.	PUNCT
ma-174	241	1	thus	thus	ADV
ma-174	241	2	lim	lim	PROPN
ma-174	241	3	t→0	t→0	PROPN
ma-174	241	4	+	+	CCONJ
ma-174	241	5	cϕs	cϕs	X
ma-174	241	6	f	f	PROPN
ma-174	241	7	(	(	PUNCT
ma-174	241	8	z)−	z)−	PROPN
ma-174	241	9	f	f	X
ma-174	241	10	(	(	PUNCT
ma-174	241	11	z	z	NOUN
ma-174	241	12	)	)	PUNCT
ma-174	241	13	t	t	PROPN
ma-174	241	14	=	=	SYM
ma-174	241	15	lim	lim	PROPN
ma-174	241	16	t→0	t→0	PROPN
ma-174	241	17	+	+	CCONJ
ma-174	241	18	1	1	NUM
ma-174	241	19	t	t	NOUN
ma-174	241	20	∫	∫	PROPN
ma-174	241	21	t	t	PROPN
ma-174	241	22	0	0	NUM
ma-174	241	23	cϕsf	cϕsf	NOUN
ma-174	241	24	(	(	PUNCT
ma-174	241	25	z)ds	z)ds	PROPN
ma-174	241	26	.	.	PUNCT
ma-174	242	1	by	by	ADP
ma-174	242	2	strong	strong	ADJ
ma-174	242	3	continuity	continuity	NOUN
ma-174	242	4	of	of	ADP
ma-174	242	5	(	(	PUNCT
ma-174	242	6	cϕs	cϕs	NOUN
ma-174	242	7	)	)	PUNCT
ma-174	242	8	s≥0	s≥0	NOUN
ma-174	242	9	we	we	PRON
ma-174	242	10	have	have	VERB
ma-174	242	11	1	1	NUM
ma-174	242	12	t	t	NOUN
ma-174	242	13	∫	∫	PROPN
ma-174	242	14	t	t	PROPN
ma-174	242	15	0	0	NUM
ma-174	243	1	‖cϕsf	‖cϕsf	PROPN
ma-174	243	2	−	−	PROPN
ma-174	244	1	f‖ds	f‖ds	PROPN
ma-174	244	2	→	→	SYM
ma-174	244	3	0	0	PROPN
ma-174	244	4	as	as	ADP
ma-174	244	5	t	t	PROPN
ma-174	244	6	→	→	SYM
ma-174	244	7	0	0	NUM
ma-174	245	1	+	+	NOUN
ma-174	245	2	.	.	PUNCT
ma-174	246	1	hence	hence	ADV
ma-174	246	2	{	{	PUNCT
ma-174	246	3	f	f	PROPN
ma-174	246	4	∈	∈	PROPN
ma-174	246	5	bα0	bα0	PROPN
ma-174	246	6	(	(	PUNCT
ma-174	246	7	u	u	NOUN
ma-174	246	8	)	)	PUNCT
ma-174	246	9	:	:	PUNCT
ma-174	247	1	zf	zf	PROPN
ma-174	247	2	′	′	NUM
ma-174	247	3	∈	∈	PROPN
ma-174	247	4	bα0	bα0	PROPN
ma-174	247	5	(	(	PUNCT
ma-174	247	6	u	u	NOUN
ma-174	247	7	)	)	PUNCT
ma-174	247	8	}	}	PUNCT
ma-174	247	9	⊆	⊆	NUM
ma-174	247	10	dom(γ).this	dom(γ).this	PROPN
ma-174	247	11	completes	complete	VERB
ma-174	247	12	the	the	DET
ma-174	247	13	proof	proof	NOUN
ma-174	247	14	.	.	PUNCT
ma-174	248	1	�	�	PROPN
ma-174	248	2	4.2	4.2	NUM
ma-174	248	3	.	.	PUNCT
ma-174	249	1	translation	translation	NOUN
ma-174	249	2	group	group	NOUN
ma-174	249	3	.	.	PUNCT
ma-174	250	1	in	in	ADP
ma-174	250	2	this	this	DET
ma-174	250	3	case	case	NOUN
ma-174	250	4	the	the	DET
ma-174	250	5	automorphisms	automorphism	NOUN
ma-174	250	6	are	be	AUX
ma-174	250	7	of	of	ADP
ma-174	250	8	the	the	DET
ma-174	250	9	form	form	NOUN
ma-174	250	10	ϕt(z	ϕt(z	NUM
ma-174	250	11	)	)	PUNCT
ma-174	251	1	=	=	SYM
ma-174	251	2	z	z	PROPN
ma-174	252	1	+	+	CCONJ
ma-174	252	2	kt	kt	PROPN
ma-174	252	3	,	,	PUNCT
ma-174	252	4	where	where	SCONJ
ma-174	252	5	z	z	PROPN
ma-174	252	6	∈	∈	PROPN
ma-174	252	7	u	u	NOUN
ma-174	252	8	and	and	CCONJ
ma-174	252	9	k	k	PROPN
ma-174	252	10	,	,	PUNCT
ma-174	252	11	t	t	PROPN
ma-174	252	12	∈	∈	PROPN
ma-174	252	13	r	r	NOUN
ma-174	252	14	with	with	ADP
ma-174	252	15	k	k	PROPN
ma-174	252	16	6=	6=	PROPN
ma-174	252	17	0	0	NUM
ma-174	252	18	.	.	PUNCT
ma-174	253	1	as	as	SCONJ
ma-174	253	2	noted	note	VERB
ma-174	253	3	in	in	ADP
ma-174	253	4	[	[	X
ma-174	253	5	3	3	NUM
ma-174	253	6	]	]	PUNCT
ma-174	253	7	,	,	PUNCT
ma-174	253	8	we	we	PRON
ma-174	253	9	can	can	AUX
ma-174	253	10	consider	consider	VERB
ma-174	253	11	the	the	DET
ma-174	253	12	self	self	NOUN
ma-174	253	13	analytic	analytic	ADJ
ma-174	253	14	maps	map	NOUN
ma-174	253	15	of	of	ADP
ma-174	253	16	u	u	NOUN
ma-174	253	17	of	of	ADP
ma-174	253	18	theform	theform	NOUN
ma-174	253	19	ϕt(z	ϕt(z	NUM
ma-174	253	20	)	)	PUNCT
ma-174	254	1	=	=	SYM
ma-174	254	2	z	z	X
ma-174	255	1	+	+	NUM
ma-174	255	2	t.	t.	X
ma-174	255	3	(	(	PUNCT
ma-174	255	4	7)the	7)the	DET
ma-174	255	5	composition	composition	NOUN
ma-174	255	6	semigroup	semigroup	NOUN
ma-174	255	7	induced	induce	VERB
ma-174	255	8	by	by	ADP
ma-174	255	9	translation	translation	NOUN
ma-174	255	10	group	group	NOUN
ma-174	255	11	on	on	ADP
ma-174	255	12	bα0	bα0	PROPN
ma-174	255	13	(	(	PUNCT
ma-174	255	14	u	u	NOUN
ma-174	255	15	)	)	PUNCT
ma-174	255	16	is	be	AUX
ma-174	255	17	given	give	VERB
ma-174	255	18	by	by	ADP
ma-174	255	19	cϕt	cϕt	PROPN
ma-174	255	20	f	f	PROPN
ma-174	255	21	(	(	PUNCT
ma-174	255	22	z	z	NOUN
ma-174	255	23	)	)	PUNCT
ma-174	255	24	=	=	SYM
ma-174	255	25	f	f	X
ma-174	255	26	(	(	PUNCT
ma-174	255	27	z	z	PROPN
ma-174	255	28	+	+	PROPN
ma-174	255	29	t	t	PROPN
ma-174	255	30	)	)	PUNCT
ma-174	255	31	.	.	PUNCT
ma-174	256	1	(	(	PUNCT
ma-174	256	2	8)	8)	NUM
ma-174	256	3	the	the	DET
ma-174	256	4	proof	proof	NOUN
ma-174	256	5	of	of	ADP
ma-174	256	6	our	our	PRON
ma-174	256	7	results	result	NOUN
ma-174	256	8	given	give	VERB
ma-174	256	9	in	in	ADP
ma-174	256	10	equation	equation	NOUN
ma-174	256	11	(	(	PUNCT
ma-174	256	12	8)	8)	NUM
ma-174	256	13	as	as	ADP
ma-174	256	14	a	a	DET
ma-174	256	15	group	group	NOUN
ma-174	256	16	on	on	ADP
ma-174	256	17	bα0	bα0	PROPN
ma-174	256	18	(	(	PUNCT
ma-174	256	19	u	u	NOUN
ma-174	256	20	)	)	PUNCT
ma-174	256	21	is	be	AUX
ma-174	256	22	basic	basic	ADJ
ma-174	256	23	,	,	PUNCT
ma-174	256	24	we	we	PRON
ma-174	256	25	therefore	therefore	ADV
ma-174	256	26	omit	omit	VERB
ma-174	256	27	thedetails.we	thedetails.we	PRON
ma-174	256	28	shall	shall	AUX
ma-174	256	29	now	now	ADV
ma-174	256	30	prove	prove	VERB
ma-174	256	31	that	that	SCONJ
ma-174	256	32	the	the	DET
ma-174	256	33	composition	composition	NOUN
ma-174	256	34	semigroup	semigroup	NOUN
ma-174	256	35	in	in	ADP
ma-174	256	36	equation	equation	NOUN
ma-174	256	37	(	(	PUNCT
ma-174	256	38	8)	8)	NUM
ma-174	256	39	,	,	PUNCT
ma-174	256	40	fails	fail	VERB
ma-174	256	41	to	to	PART
ma-174	256	42	be	be	AUX
ma-174	256	43	an	an	DET
ma-174	256	44	isometry	isometry	NOUN
ma-174	256	45	on	on	ADP
ma-174	256	46	bα0	bα0	PROPN
ma-174	256	47	(	(	PUNCT
ma-174	256	48	u	u	NOUN
ma-174	256	49	)	)	PUNCT
ma-174	256	50	.	.	PUNCT
ma-174	257	1	proposition	proposition	NOUN
ma-174	257	2	4.4	4.4	NUM
ma-174	257	3	.	.	PUNCT
ma-174	258	1	the	the	DET
ma-174	258	2	operator	operator	NOUN
ma-174	258	3	cϕt	cϕt	VERB
ma-174	258	4	fails	fail	VERB
ma-174	258	5	to	to	PART
ma-174	258	6	be	be	AUX
ma-174	258	7	an	an	DET
ma-174	258	8	isometry	isometry	NOUN
ma-174	258	9	on	on	ADP
ma-174	258	10	bα0	bα0	PROPN
ma-174	258	11	(	(	PUNCT
ma-174	258	12	u	u	NOUN
ma-174	258	13	)	)	PUNCT
ma-174	258	14	.	.	PUNCT
ma-174	259	1	proof	proof	NOUN
ma-174	259	2	.	.	PUNCT
ma-174	260	1	by	by	ADP
ma-174	260	2	norm	norm	NOUN
ma-174	260	3	definition	definition	NOUN
ma-174	260	4	,	,	PUNCT
ma-174	260	5	we	we	PRON
ma-174	260	6	have	have	VERB
ma-174	260	7	‖cϕt	‖cϕt	PROPN
ma-174	260	8	f	f	PROPN
ma-174	260	9	‖bα(u	‖bα(u	PROPN
ma-174	260	10	)	)	PUNCT
ma-174	261	1	=	=	PUNCT
ma-174	262	1	|cϕt	|cϕt	PROPN
ma-174	262	2	f	f	X
ma-174	262	3	(	(	PUNCT
ma-174	262	4	i)|+	i)|+	ADJ
ma-174	262	5	sup	sup	NOUN
ma-174	262	6	z∈u	z∈u	ADJ
ma-174	262	7	=(	=(	NOUN
ma-174	262	8	z)α|	z)α|	PROPN
ma-174	262	9	(	(	PUNCT
ma-174	262	10	cϕt	cϕt	NOUN
ma-174	262	11	f	f	PROPN
ma-174	262	12	)	)	PUNCT
ma-174	262	13	′	′	NOUN
ma-174	263	1	(	(	PUNCT
ma-174	263	2	z)|	z)|	NOUN
ma-174	263	3	=	=	PRON
ma-174	263	4	|f	|f	PROPN
ma-174	263	5	(	(	PUNCT
ma-174	263	6	i	i	PRON
ma-174	263	7	+	+	CCONJ
ma-174	263	8	t)|+	t)|+	PRON
ma-174	263	9	sup	sup	NOUN
ma-174	263	10	z∈u	z∈u	PROPN
ma-174	263	11	=(	=(	PROPN
ma-174	263	12	z)α|f	z)α|f	PART
ma-174	263	13	′(z	′(z	NOUN
ma-174	264	1	+	+	X
ma-174	264	2	t)|	t)|	ADV
ma-174	264	3	.	.	PUNCT
ma-174	265	1	now	now	ADV
ma-174	265	2	by	by	ADP
ma-174	265	3	change	change	NOUN
ma-174	265	4	of	of	ADP
ma-174	265	5	variables	variable	NOUN
ma-174	265	6	:	:	PUNCT
ma-174	265	7	let	let	VERB
ma-174	265	8	z	z	NOUN
ma-174	265	9	+	+	NOUN
ma-174	266	1	t	t	X
ma-174	266	2	=	=	SYM
ma-174	266	3	ω	ω	PROPN
ma-174	266	4	then	then	ADV
ma-174	267	1	z	z	PROPN
ma-174	267	2	=	=	SYM
ma-174	267	3	ω	ω	PROPN
ma-174	267	4	−	−	PROPN
ma-174	267	5	t	t	NOUN
ma-174	267	6	,	,	PUNCT
ma-174	267	7	and	and	CCONJ
ma-174	267	8	=(	=(	NOUN
ma-174	267	9	z	z	NOUN
ma-174	267	10	)	)	PUNCT
ma-174	267	11	=	=	SYM
ma-174	267	12	=(	=(	NOUN
ma-174	267	13	ω	ω	PROPN
ma-174	267	14	)	)	PUNCT
ma-174	267	15	.	.	PUNCT
ma-174	268	1	therefore	therefore	ADV
ma-174	268	2	,	,	PUNCT
ma-174	268	3	‖cϕt	‖cϕt	ADV
ma-174	268	4	f	f	PROPN
ma-174	268	5	‖bα(u	‖bα(u	PROPN
ma-174	268	6	)	)	PUNCT
ma-174	269	1	=	=	PRON
ma-174	269	2	|f	|f	PROPN
ma-174	270	1	(	(	PUNCT
ma-174	270	2	i	i	PRON
ma-174	270	3	+	+	CCONJ
ma-174	270	4	t)|+	t)|+	PRON
ma-174	270	5	sup	sup	NOUN
ma-174	270	6	ω∈u	ω∈u	NOUN
ma-174	270	7	=(	=(	PROPN
ma-174	270	8	ω)α|f	ω)α|f	NUM
ma-174	270	9	′(ω)|	′(ω)|	PROPN
ma-174	270	10	.	.	PUNCT
ma-174	270	11	(	(	PUNCT
ma-174	270	12	9	9	NUM
ma-174	270	13	)	)	PUNCT
ma-174	270	14	11	11	NUM
ma-174	270	15	the	the	DET
ma-174	270	16	right	right	ADJ
ma-174	270	17	hand	hand	NOUN
ma-174	270	18	side	side	NOUN
ma-174	270	19	of	of	ADP
ma-174	270	20	equation	equation	NOUN
ma-174	270	21	(	(	PUNCT
ma-174	270	22	9	9	NUM
ma-174	270	23	)	)	PUNCT
ma-174	270	24	is	be	AUX
ma-174	270	25	not	not	PART
ma-174	270	26	equal	equal	ADJ
ma-174	270	27	to	to	ADP
ma-174	270	28	the	the	DET
ma-174	270	29	norm	norm	NOUN
ma-174	270	30	‖f	‖f	ADP
ma-174	270	31	‖bα(u	‖bα(u	NUM
ma-174	270	32	)	)	PUNCT
ma-174	270	33	for	for	ADP
ma-174	270	34	any	any	DET
ma-174	270	35	t	t	NOUN
ma-174	270	36	>	>	X
ma-174	270	37	0	0	X
ma-174	270	38	.	.	PUNCT
ma-174	271	1	this	this	DET
ma-174	271	2	impliesthat	impliesthat	NOUN
ma-174	271	3	(	(	PUNCT
ma-174	271	4	8)	8)	NUM
ma-174	271	5	is	be	AUX
ma-174	271	6	not	not	PART
ma-174	271	7	an	an	DET
ma-174	271	8	isometry	isometry	NOUN
ma-174	271	9	on	on	ADP
ma-174	271	10	bα0	bα0	PROPN
ma-174	271	11	(	(	PUNCT
ma-174	271	12	u	u	NOUN
ma-174	271	13	)	)	PUNCT
ma-174	271	14	.	.	PUNCT
ma-174	272	1	this	this	PRON
ma-174	272	2	completes	complete	VERB
ma-174	272	3	the	the	DET
ma-174	272	4	proof	proof	NOUN
ma-174	272	5	.	.	PUNCT
ma-174	273	1	�	�	PROPN
ma-174	273	2	in	in	ADP
ma-174	273	3	the	the	DET
ma-174	273	4	following	follow	VERB
ma-174	273	5	results	result	NOUN
ma-174	273	6	,	,	PUNCT
ma-174	273	7	we	we	PRON
ma-174	273	8	investigate	investigate	VERB
ma-174	273	9	the	the	DET
ma-174	273	10	strong	strong	ADJ
ma-174	273	11	continuity	continuity	NOUN
ma-174	273	12	of	of	ADP
ma-174	273	13	the	the	DET
ma-174	273	14	composition	composition	NOUN
ma-174	273	15	semigroup	semigroup	NOUN
ma-174	273	16	inequation	inequation	NOUN
ma-174	273	17	(	(	PUNCT
ma-174	273	18	8)	8)	NUM
ma-174	273	19	on	on	ADP
ma-174	273	20	bα0	bα0	PROPN
ma-174	273	21	(	(	PUNCT
ma-174	273	22	u	u	NOUN
ma-174	273	23	)	)	PUNCT
ma-174	273	24	.	.	PUNCT
ma-174	274	1	proposition	proposition	NOUN
ma-174	274	2	4.5	4.5	NUM
ma-174	274	3	.	.	PUNCT
ma-174	275	1	the	the	DET
ma-174	275	2	operator	operator	NOUN
ma-174	275	3	cϕt	cϕt	VERB
ma-174	275	4	is	be	AUX
ma-174	275	5	strongly	strongly	ADV
ma-174	275	6	continuous	continuous	ADJ
ma-174	275	7	on	on	ADP
ma-174	275	8	bα0	bα0	PROPN
ma-174	275	9	(	(	PUNCT
ma-174	275	10	u	u	NOUN
ma-174	275	11	)	)	PUNCT
ma-174	275	12	.	.	PUNCT
ma-174	276	1	proof	proof	NOUN
ma-174	276	2	.	.	PUNCT
ma-174	277	1	we	we	PRON
ma-174	277	2	need	need	VERB
ma-174	277	3	to	to	PART
ma-174	277	4	show	show	VERB
ma-174	277	5	that	that	SCONJ
ma-174	277	6	‖cϕt	‖cϕt	PROPN
ma-174	278	1	f	f	PROPN
ma-174	279	1	−	−	PROPN
ma-174	279	2	f	f	PROPN
ma-174	279	3	‖bα(u	‖bα(u	PROPN
ma-174	279	4	)	)	PUNCT
ma-174	279	5	→	→	SYM
ma-174	279	6	0	0	NUM
ma-174	280	1	as	as	ADP
ma-174	280	2	t	t	PROPN
ma-174	280	3	→	→	SYM
ma-174	280	4	0	0	X
ma-174	280	5	.	.	PUNCT
ma-174	281	1	this	this	DET
ma-174	281	2	approach	approach	NOUN
ma-174	281	3	is	be	AUX
ma-174	281	4	similar	similar	ADJ
ma-174	281	5	to	to	ADP
ma-174	281	6	(	(	PUNCT
ma-174	281	7	7	7	NUM
ma-174	281	8	)	)	PUNCT
ma-174	281	9	.	.	PUNCT
ma-174	282	1	weomit	weomit	VERB
ma-174	282	2	the	the	DET
ma-174	282	3	details	detail	NOUN
ma-174	282	4	.	.	PUNCT
ma-174	283	1	we	we	PRON
ma-174	283	2	compute	compute	VERB
ma-174	283	3	the	the	DET
ma-174	283	4	automorphism	automorphism	NOUN
ma-174	283	5	of	of	ADP
ma-174	283	6	the	the	DET
ma-174	283	7	unit	unit	NOUN
ma-174	283	8	disc	disc	VERB
ma-174	283	9	d	d	PROPN
ma-174	283	10	,	,	PUNCT
ma-174	283	11	denoted	denote	VERB
ma-174	283	12	by	by	ADP
ma-174	283	13	xt	xt	PROPN
ma-174	283	14	as	as	SCONJ
ma-174	283	15	follows	follow	VERB
ma-174	283	16	xt(z	xt(z	PUNCT
ma-174	283	17	)	)	PUNCT
ma-174	284	1	=	=	PUNCT
ma-174	284	2	ψ−1	ψ−1	PROPN
ma-174	284	3	(	(	PUNCT
ma-174	284	4	ϕt	ϕt	INTJ
ma-174	284	5	(	(	PUNCT
ma-174	284	6	ψ(z	ψ(z	PROPN
ma-174	284	7	)	)	PUNCT
ma-174	284	8	)	)	PUNCT
ma-174	284	9	)	)	PUNCT
ma-174	285	1	=	=	PUNCT
ma-174	285	2	ψ−1	ψ−1	PROPN
ma-174	285	3	(	(	PUNCT
ma-174	285	4	ϕt	ϕt	INTJ
ma-174	285	5	(	(	PUNCT
ma-174	285	6	i(1	i(1	PROPN
ma-174	285	7	+	+	PROPN
ma-174	285	8	z	z	NOUN
ma-174	285	9	)	)	PUNCT
ma-174	285	10	1−	1−	NUM
ma-174	285	11	z	z	NOUN
ma-174	285	12	)	)	PUNCT
ma-174	285	13	)	)	PUNCT
ma-174	286	1	=	=	PUNCT
ma-174	286	2	ψ−1	ψ−1	PROPN
ma-174	286	3	(	(	PUNCT
ma-174	286	4	i(1	i(1	PROPN
ma-174	286	5	+	+	PROPN
ma-174	286	6	z	z	X
ma-174	286	7	)	)	PUNCT
ma-174	286	8	1−	1−	NUM
ma-174	286	9	z	z	NOUN
ma-174	286	10	+	+	NUM
ma-174	286	11	t	t	PROPN
ma-174	286	12	)	)	PUNCT
ma-174	286	13	.	.	PUNCT
ma-174	287	1	since	since	SCONJ
ma-174	287	2	the	the	DET
ma-174	287	3	inverse	inverse	NOUN
ma-174	287	4	of	of	ADP
ma-174	287	5	cayley	cayley	ADJ
ma-174	287	6	transform	transform	NOUN
ma-174	287	7	is	be	AUX
ma-174	287	8	given	give	VERB
ma-174	287	9	by	by	ADP
ma-174	287	10	ψ−1	ψ−1	PROPN
ma-174	287	11	=	=	SYM
ma-174	287	12	z−i	z−i	PROPN
ma-174	287	13	z+i	z+i	NUM
ma-174	287	14	,	,	PUNCT
ma-174	287	15	we	we	PRON
ma-174	287	16	substitute	substitute	VERB
ma-174	287	17	to	to	PART
ma-174	287	18	obtain	obtain	VERB
ma-174	287	19	xt	xt	ADP
ma-174	287	20	=	=	SYM
ma-174	287	21	i(1+z	i(1+z	PROPN
ma-174	287	22	)	)	PUNCT
ma-174	287	23	1−z	1−z	NUM
ma-174	288	1	−	−	PROPN
ma-174	288	2	t	t	NOUN
ma-174	288	3	−	−	PROPN
ma-174	289	1	i	i	PRON
ma-174	289	2	i(1+z	i(1+z	PROPN
ma-174	289	3	)	)	PUNCT
ma-174	290	1	1−z	1−z	NUM
ma-174	291	1	−	−	PROPN
ma-174	291	2	t	t	NOUN
ma-174	292	1	+	+	CCONJ
ma-174	292	2	i	i	PROPN
ma-174	292	3	=	=	SYM
ma-174	292	4	i(1+z	i(1+z	PROPN
ma-174	292	5	)	)	PUNCT
ma-174	292	6	1−z	1−z	NUM
ma-174	293	1	−	−	PROPN
ma-174	293	2	(	(	PUNCT
ma-174	293	3	t	t	PROPN
ma-174	293	4	+	+	CCONJ
ma-174	293	5	i	i	PROPN
ma-174	293	6	)	)	PUNCT
ma-174	293	7	i(1+z	i(1+z	PROPN
ma-174	293	8	)	)	PUNCT
ma-174	293	9	1−z	1−z	PROPN
ma-174	294	1	+	+	CCONJ
ma-174	294	2	(	(	PUNCT
ma-174	294	3	i	i	PRON
ma-174	294	4	−	−	PROPN
ma-174	294	5	t	t	PROPN
ma-174	294	6	)	)	PUNCT
ma-174	294	7	.	.	PUNCT
ma-174	295	1	we	we	PRON
ma-174	295	2	simplify	simplify	VERB
ma-174	295	3	further	far	ADV
ma-174	295	4	by	by	ADP
ma-174	295	5	multiplying	multiply	VERB
ma-174	295	6	both	both	DET
ma-174	295	7	the	the	DET
ma-174	295	8	numerator	numerator	NOUN
ma-174	295	9	and	and	CCONJ
ma-174	295	10	denominator	denominator	NOUN
ma-174	295	11	by	by	ADP
ma-174	295	12	(	(	PUNCT
ma-174	295	13	1−	1−	NUM
ma-174	295	14	z	z	NOUN
ma-174	295	15	)	)	PUNCT
ma-174	295	16	to	to	PART
ma-174	295	17	obtain	obtain	VERB
ma-174	295	18	xt(z	xt(z	PUNCT
ma-174	295	19	)	)	PUNCT
ma-174	296	1	=	=	PUNCT
ma-174	296	2	i(1	i(1	PROPN
ma-174	296	3	+	+	PROPN
ma-174	296	4	z	z	X
ma-174	296	5	)	)	PUNCT
ma-174	297	1	+	+	CCONJ
ma-174	297	2	(	(	PUNCT
ma-174	297	3	t	t	PROPN
ma-174	297	4	−	−	PROPN
ma-174	297	5	i)(1−	i)(1−	PROPN
ma-174	297	6	z	z	PROPN
ma-174	297	7	)	)	PUNCT
ma-174	297	8	)	)	PUNCT
ma-174	298	1	i(1	i(1	PROPN
ma-174	298	2	+	+	PROPN
ma-174	298	3	z	z	X
ma-174	298	4	)	)	PUNCT
ma-174	299	1	+	+	CCONJ
ma-174	299	2	(	(	PUNCT
ma-174	299	3	t	t	NOUN
ma-174	299	4	+	+	NUM
ma-174	299	5	i)(1−	i)(1−	PROPN
ma-174	299	6	z	z	PROPN
ma-174	299	7	)	)	PUNCT
ma-174	299	8	=	=	SYM
ma-174	299	9	(	(	PUNCT
ma-174	299	10	2i	2i	NOUN
ma-174	299	11	−	−	NOUN
ma-174	299	12	t)z	t)z	NOUN
ma-174	299	13	−	−	PROPN
ma-174	299	14	t	t	NOUN
ma-174	299	15	(	(	PUNCT
ma-174	299	16	2i	2i	NUM
ma-174	299	17	+	+	CCONJ
ma-174	299	18	t)−	t)−	PROPN
ma-174	299	19	tz	tz	NOUN
ma-174	299	20	.by	.by	PUNCT
ma-174	299	21	dividing	divide	VERB
ma-174	299	22	both	both	CCONJ
ma-174	299	23	the	the	DET
ma-174	299	24	numerator	numerator	NOUN
ma-174	299	25	and	and	CCONJ
ma-174	299	26	denominator	denominator	NOUN
ma-174	299	27	by	by	ADP
ma-174	299	28	2i	2i	NOUN
ma-174	299	29	−	−	PROPN
ma-174	299	30	t	t	NOUN
ma-174	299	31	,	,	PUNCT
ma-174	299	32	we	we	PRON
ma-174	299	33	get	get	VERB
ma-174	299	34	xt	xt	PUNCT
ma-174	300	1	=	=	PUNCT
ma-174	300	2	z	z	PROPN
ma-174	301	1	+	+	NUM
ma-174	301	2	t	t	X
ma-174	301	3	2i−t	2i−t	NUM
ma-174	301	4	2i+t	2i+t	NUM
ma-174	301	5	2i−t	2i−t	NUM
ma-174	301	6	−	−	PROPN
ma-174	301	7	t	t	NOUN
ma-174	301	8	2i−t	2i−t	NOUN
ma-174	301	9	z.letting	z.lette	VERB
ma-174	301	10	kt	kt	PROPN
ma-174	301	11	=	=	SYM
ma-174	301	12	t	t	PROPN
ma-174	301	13	2i−t	2i−t	NUM
ma-174	301	14	and	and	CCONJ
ma-174	301	15	mt	mt	PROPN
ma-174	301	16	=	=	SYM
ma-174	301	17	2i+t	2i+t	NUM
ma-174	301	18	2i−t	2i−t	NUM
ma-174	301	19	.	.	PUNCT
ma-174	302	1	we	we	PRON
ma-174	302	2	have	have	VERB
ma-174	302	3	xt	xt	VERB
ma-174	303	1	=	=	SYM
ma-174	303	2	z	z	PROPN
ma-174	304	1	+	+	CCONJ
ma-174	304	2	kt	kt	PROPN
ma-174	304	3	mt	mt	PROPN
ma-174	304	4	−	−	PROPN
ma-174	304	5	ktz	ktz	PROPN
ma-174	304	6	.	.	PUNCT
ma-174	305	1	next	next	ADV
ma-174	305	2	,	,	PUNCT
ma-174	305	3	we	we	PRON
ma-174	305	4	apply	apply	VERB
ma-174	305	5	density	density	NOUN
ma-174	305	6	of	of	ADP
ma-174	305	7	polynomials	polynomial	NOUN
ma-174	305	8	in	in	ADP
ma-174	305	9	bα0	bα0	PROPN
ma-174	305	10	(	(	PUNCT
ma-174	305	11	d	d	NOUN
ma-174	305	12	)	)	PUNCT
ma-174	305	13	to	to	PART
ma-174	305	14	prove	prove	VERB
ma-174	305	15	that	that	SCONJ
ma-174	305	16	for	for	ADP
ma-174	305	17	f	f	PROPN
ma-174	305	18	∗	∗	PROPN
ma-174	305	19	∈	∈	PROPN
ma-174	305	20	bα0	bα0	PROPN
ma-174	305	21	(	(	PUNCT
ma-174	305	22	d	d	PROPN
ma-174	305	23	)	)	PUNCT
ma-174	305	24	,	,	PUNCT
ma-174	305	25	we	we	PRON
ma-174	305	26	have	have	VERB
ma-174	305	27	‖cx	‖cx	PROPN
ma-174	305	28	t	t	PROPN
ma-174	305	29	f	f	PROPN
ma-174	305	30	∗	∗	NOUN
ma-174	305	31	−	−	PROPN
ma-174	305	32	f	f	PROPN
ma-174	305	33	∗‖bα1	∗‖bα1	NOUN
ma-174	305	34	(	(	PUNCT
ma-174	305	35	d	d	NOUN
ma-174	305	36	)	)	PUNCT
ma-174	305	37	→	→	SYM
ma-174	305	38	0	0	NUM
ma-174	305	39	as	as	ADP
ma-174	305	40	t	t	PROPN
ma-174	305	41	→	→	SYM
ma-174	305	42	0	0	PROPN
ma-174	305	43	.	.	PUNCT
ma-174	306	1	lim	lim	PROPN
ma-174	306	2	t→0	t→0	PROPN
ma-174	307	1	+	+	CCONJ
ma-174	307	2	‖cx	‖cx	PROPN
ma-174	307	3	t	t	PROPN
ma-174	307	4	f	f	PROPN
ma-174	307	5	∗	∗	VERB
ma-174	307	6	−	−	PROPN
ma-174	307	7	f	f	PROPN
ma-174	307	8	∗‖bα1	∗‖bα1	NOUN
ma-174	307	9	(	(	PUNCT
ma-174	307	10	d	d	NOUN
ma-174	307	11	)	)	PUNCT
ma-174	307	12	=	=	SYM
ma-174	307	13	lim	lim	PROPN
ma-174	307	14	t→0	t→0	PROPN
ma-174	307	15	+	+	CCONJ
ma-174	307	16	(	(	PUNCT
ma-174	307	17	sup	sup	NUM
ma-174	307	18	z∈d	z∈d	NUM
ma-174	307	19	(	(	PUNCT
ma-174	307	20	1−	1−	NUM
ma-174	307	21	|z	|z	PROPN
ma-174	307	22	|2	|2	NUM
ma-174	307	23	)	)	PUNCT
ma-174	308	1	α	α	PROPN
ma-174	308	2	|(cx	|(cx	PROPN
ma-174	308	3	t	t	PROPN
ma-174	308	4	f	f	PROPN
ma-174	308	5	∗	∗	NOUN
ma-174	308	6	−	−	PROPN
ma-174	308	7	f	f	PROPN
ma-174	308	8	∗)′(z)|	∗)′(z)|	NOUN
ma-174	308	9	)	)	PUNCT
ma-174	308	10	.	.	PUNCT
ma-174	309	1	using	use	VERB
ma-174	309	2	density	density	NOUN
ma-174	309	3	of	of	ADP
ma-174	309	4	polynomials	polynomial	NOUN
ma-174	309	5	in	in	ADP
ma-174	309	6	bα0	bα0	PROPN
ma-174	309	7	(	(	PUNCT
ma-174	309	8	d	d	PROPN
ma-174	309	9	)	)	PUNCT
ma-174	309	10	,	,	PUNCT
ma-174	309	11	let	let	VERB
ma-174	309	12	f	f	PROPN
ma-174	309	13	∗(z	∗(z	PROPN
ma-174	309	14	)	)	PUNCT
ma-174	309	15	=	=	SYM
ma-174	309	16	zn	zn	PROPN
ma-174	309	17	and	and	CCONJ
ma-174	309	18	z	z	PROPN
ma-174	309	19	∈	∈	PROPN
ma-174	310	1	d	d	AUX
ma-174	310	2	be	be	AUX
ma-174	310	3	such	such	ADJ
ma-174	310	4	that	that	SCONJ
ma-174	310	5	cx	cx	PROPN
ma-174	310	6	tz	tz	PROPN
ma-174	310	7	n	n	CCONJ
ma-174	310	8	−	−	PROPN
ma-174	310	9	zn	zn	NOUN
ma-174	310	10	=	=	SYM
ma-174	310	11	(	(	PUNCT
ma-174	310	12	xt(z))n	xt(z))n	NUM
ma-174	310	13	−	−	PROPN
ma-174	311	1	zn	zn	PROPN
ma-174	311	2	,	,	PUNCT
ma-174	311	3	n	n	PRON
ma-174	311	4	≥	≥	NOUN
ma-174	311	5	1	1	NUM
ma-174	311	6	.	.	PUNCT
ma-174	312	1	(	(	PUNCT
ma-174	312	2	10)12	10)12	NUM
ma-174	312	3	now	now	ADV
ma-174	312	4	,	,	PUNCT
ma-174	312	5	differentiating	differentiate	VERB
ma-174	312	6	(	(	PUNCT
ma-174	312	7	xt(z))n	xt(z))n	NUM
ma-174	312	8	−	−	PROPN
ma-174	313	1	zn	zn	INTJ
ma-174	313	2	with	with	ADP
ma-174	313	3	respect	respect	NOUN
ma-174	313	4	to	to	ADP
ma-174	313	5	z	z	NOUN
ma-174	313	6	,	,	PUNCT
ma-174	313	7	we	we	PRON
ma-174	313	8	get	get	VERB
ma-174	313	9	(	(	PUNCT
ma-174	313	10	cx	cx	PROPN
ma-174	313	11	t	t	PROPN
ma-174	313	12	f	f	PROPN
ma-174	313	13	∗	∗	VERB
ma-174	313	14	−	−	PROPN
ma-174	313	15	f	f	PROPN
ma-174	313	16	∗)′(z	∗)′(z	PROPN
ma-174	313	17	)	)	PUNCT
ma-174	313	18	=	=	SYM
ma-174	313	19	n(xt(z))n−1x	n(xt(z))n−1x	PROPN
ma-174	313	20	′t(z)−	′t(z)−	VERB
ma-174	313	21	nzn−1	nzn−1	PROPN
ma-174	313	22	=	=	SYM
ma-174	313	23	n[(xt(z))n−1x	n[(xt(z))n−1x	X
ma-174	313	24	′t(z)−	′t(z)−	ADJ
ma-174	313	25	zn−1	zn−1	PROPN
ma-174	313	26	]	]	PUNCT
ma-174	313	27	.	.	PUNCT
ma-174	314	1	(	(	PUNCT
ma-174	314	2	11	11	NUM
ma-174	314	3	)	)	PUNCT
ma-174	314	4	we	we	PRON
ma-174	314	5	also	also	ADV
ma-174	314	6	differentiate	differentiate	VERB
ma-174	314	7	xt	xt	NOUN
ma-174	315	1	=	=	SYM
ma-174	315	2	z+kt	z+kt	NOUN
ma-174	315	3	mt−ktz	mt−ktz	NOUN
ma-174	315	4	by	by	ADP
ma-174	315	5	quotient	quotient	NOUN
ma-174	315	6	rule	rule	NOUN
ma-174	315	7	to	to	PART
ma-174	315	8	obtain	obtain	VERB
ma-174	315	9	x	x	SYM
ma-174	315	10	′t(z	′t(z	NOUN
ma-174	315	11	)	)	PUNCT
ma-174	315	12	=	=	PUNCT
ma-174	315	13	(	(	PUNCT
ma-174	315	14	mt	mt	PROPN
ma-174	315	15	−	−	PROPN
ma-174	315	16	ktz)1−	ktz)1−	NOUN
ma-174	315	17	(	(	PUNCT
ma-174	315	18	z	z	NOUN
ma-174	315	19	+	+	NOUN
ma-174	315	20	kt)(−kt	kt)(−kt	NOUN
ma-174	315	21	)	)	PUNCT
ma-174	315	22	(	(	PUNCT
ma-174	315	23	mt	mt	PROPN
ma-174	315	24	−	−	PROPN
ma-174	315	25	ktz)2	ktz)2	PROPN
ma-174	315	26	=	=	PROPN
ma-174	315	27	mt	mt	PROPN
ma-174	315	28	+	+	CCONJ
ma-174	315	29	k2	k2	PROPN
ma-174	315	30	t	t	PROPN
ma-174	315	31	(	(	PUNCT
ma-174	315	32	mt	mt	PROPN
ma-174	315	33	−	−	PROPN
ma-174	315	34	ktz)2	ktz)2	PROPN
ma-174	315	35	.	.	PUNCT
ma-174	316	1	substituting	substitute	VERB
ma-174	316	2	for	for	ADP
ma-174	316	3	xt	xt	NOUN
ma-174	316	4	=	=	SYM
ma-174	316	5	z+kt	z+kt	NOUN
ma-174	316	6	mt+ktz	mt+ktz	NOUN
ma-174	316	7	and	and	CCONJ
ma-174	316	8	x	x	SYM
ma-174	316	9	′t(z	′t(z	NOUN
ma-174	316	10	)	)	PUNCT
ma-174	316	11	=	=	PUNCT
ma-174	317	1	mt−k2	mt−k2	PROPN
ma-174	317	2	t	t	PROPN
ma-174	317	3	(	(	PUNCT
ma-174	317	4	mt−ktz)2	mt−ktz)2	NOUN
ma-174	317	5	in	in	ADP
ma-174	317	6	equation	equation	NOUN
ma-174	317	7	(	(	PUNCT
ma-174	317	8	11	11	NUM
ma-174	317	9	)	)	PUNCT
ma-174	317	10	we	we	PRON
ma-174	317	11	have	have	VERB
ma-174	317	12	(	(	PUNCT
ma-174	317	13	cx	cx	PROPN
ma-174	317	14	t	t	PROPN
ma-174	318	1	f	f	PROPN
ma-174	318	2	∗	∗	VERB
ma-174	318	3	−	−	PROPN
ma-174	318	4	f	f	PROPN
ma-174	318	5	∗)′(z	∗)′(z	PROPN
ma-174	318	6	)	)	PUNCT
ma-174	319	1	=	=	SYM
ma-174	319	2	n[(xt(z))n−1x	n[(xt(z))n−1x	ADP
ma-174	319	3	′t(z)−	′t(z)−	ADJ
ma-174	319	4	zn−1	zn−1	PROPN
ma-174	319	5	]	]	X
ma-174	319	6	=	=	SYM
ma-174	319	7	n	n	PART
ma-174	319	8	[	[	PUNCT
ma-174	319	9	(	(	PUNCT
ma-174	319	10	z	z	NOUN
ma-174	319	11	+	+	CCONJ
ma-174	319	12	kt	kt	ADJ
ma-174	319	13	)	)	PUNCT
ma-174	319	14	n−1(mt	n−1(mt	NOUN
ma-174	319	15	−	−	NOUN
ma-174	319	16	ktz2)−	ktz2)−	ADJ
ma-174	319	17	zn−1(mt	zn−1(mt	NOUN
ma-174	319	18	−	−	PROPN
ma-174	319	19	ktz)n+1	ktz)n+1	PROPN
ma-174	319	20	(	(	PUNCT
ma-174	319	21	mt	mt	PROPN
ma-174	319	22	−	−	PROPN
ma-174	319	23	ktz)n+1	ktz)n+1	NOUN
ma-174	319	24	]	]	PUNCT
ma-174	319	25	.	.	PUNCT
ma-174	320	1	it	it	PRON
ma-174	320	2	therefore	therefore	ADV
ma-174	320	3	follows	follow	VERB
ma-174	320	4	that	that	SCONJ
ma-174	320	5	as	as	ADP
ma-174	320	6	t	t	PROPN
ma-174	320	7	→	→	SYM
ma-174	320	8	0	0	NUM
ma-174	320	9	,	,	PUNCT
ma-174	320	10	we	we	PRON
ma-174	320	11	have	have	VERB
ma-174	320	12	‖cx	‖cx	PROPN
ma-174	320	13	t	t	PROPN
ma-174	320	14	f	f	PROPN
ma-174	320	15	∗	∗	NOUN
ma-174	320	16	−	−	PROPN
ma-174	320	17	f	f	PROPN
ma-174	320	18	∗‖bα(d	∗‖bα(d	PROPN
ma-174	320	19	)	)	PUNCT
ma-174	320	20	=	=	PUNCT
ma-174	320	21	(	(	PUNCT
ma-174	320	22	|(xt(0))n	|(xt(0))n	PRON
ma-174	320	23	−	−	NOUN
ma-174	320	24	0|	0|	NUM
ma-174	320	25	)	)	PUNCT
ma-174	321	1	+	+	CCONJ
ma-174	321	2	(	(	PUNCT
ma-174	321	3	sup	sup	NOUN
ma-174	321	4	z∈d	z∈d	NUM
ma-174	321	5	(	(	PUNCT
ma-174	321	6	1−	1−	NUM
ma-174	321	7	|z	|z	PROPN
ma-174	321	8	|2	|2	NUM
ma-174	321	9	)	)	PUNCT
ma-174	321	10	α	α	PRON
ma-174	321	11	∣∣n[(xt(z))n−1x	∣∣n[(xt(z))n−1x	NOUN
ma-174	321	12	′t(z)−	′t(z)−	PUNCT
ma-174	321	13	zn−1	zn−1	PROPN
ma-174	321	14	]	]	PUNCT
ma-174	321	15	∣∣	∣∣	X
ma-174	321	16	=	=	SYM
ma-174	321	17	0	0	X
ma-174	321	18	.	.	PUNCT
ma-174	322	1	therefore	therefore	ADV
ma-174	322	2	‖cϕt	‖cϕt	ADV
ma-174	322	3	f	f	PROPN
ma-174	323	1	−	−	PROPN
ma-174	323	2	f	f	PROPN
ma-174	323	3	‖bα(u	‖bα(u	PROPN
ma-174	323	4	)	)	PUNCT
ma-174	324	1	=	=	PUNCT
ma-174	324	2	|ϕt	|ϕt	NUM
ma-174	324	3	f	f	X
ma-174	324	4	(	(	PUNCT
ma-174	324	5	i	i	NOUN
ma-174	324	6	)	)	PUNCT
ma-174	324	7	)	)	PUNCT
ma-174	325	1	−	−	PROPN
ma-174	325	2	f	f	X
ma-174	325	3	(	(	PUNCT
ma-174	325	4	i)|	i)|	INTJ
ma-174	326	1	+	+	CCONJ
ma-174	326	2	‖cϕt	‖cϕt	ADV
ma-174	327	1	f	f	NOUN
ma-174	328	1	−	−	PROPN
ma-174	328	2	f	f	PROPN
ma-174	328	3	‖bα1	‖bα1	PROPN
ma-174	328	4	(	(	PUNCT
ma-174	328	5	u	u	NOUN
ma-174	328	6	)	)	PUNCT
ma-174	328	7	→	→	SYM
ma-174	328	8	0	0	NUM
ma-174	328	9	as	as	ADP
ma-174	328	10	t	t	PROPN
ma-174	328	11	→	→	SYM
ma-174	328	12	0	0	NUM
ma-174	328	13	,	,	PUNCT
ma-174	328	14	as	as	SCONJ
ma-174	328	15	desired	desire	VERB
ma-174	328	16	.	.	PUNCT
ma-174	329	1	thiscompletes	thiscomplete	VERB
ma-174	329	2	the	the	DET
ma-174	329	3	proof	proof	NOUN
ma-174	329	4	.	.	PUNCT
ma-174	330	1	�	�	PROPN
ma-174	330	2	in	in	ADP
ma-174	330	3	the	the	DET
ma-174	330	4	next	next	ADJ
ma-174	330	5	theorem	theorem	NOUN
ma-174	330	6	,	,	PUNCT
ma-174	330	7	we	we	PRON
ma-174	330	8	obtain	obtain	VERB
ma-174	330	9	the	the	DET
ma-174	330	10	infinitesimal	infinitesimal	ADJ
ma-174	330	11	generator	generator	NOUN
ma-174	330	12	of	of	ADP
ma-174	330	13	the	the	DET
ma-174	330	14	strongly	strongly	ADV
ma-174	330	15	continuous	continuous	ADJ
ma-174	330	16	compositionsemigroup	compositionsemigroup	NOUN
ma-174	330	17	given	give	VERB
ma-174	330	18	in	in	ADP
ma-174	330	19	equation	equation	NOUN
ma-174	330	20	(	(	PUNCT
ma-174	330	21	8)	8)	NUM
ma-174	330	22	.	.	PUNCT
ma-174	330	23	theorem	theorem	VERB
ma-174	330	24	4.6	4.6	NUM
ma-174	330	25	.	.	PUNCT
ma-174	331	1	the	the	DET
ma-174	331	2	infinitesimal	infinitesimal	ADJ
ma-174	331	3	generator	generator	NOUN
ma-174	331	4	γ	γ	X
ma-174	331	5	of	of	ADP
ma-174	331	6	(	(	PUNCT
ma-174	331	7	cϕt	cϕt	NOUN
ma-174	331	8	)	)	PUNCT
ma-174	331	9	t≥0	t≥0	NOUN
ma-174	331	10	on	on	ADP
ma-174	331	11	bα0	bα0	PROPN
ma-174	331	12	(	(	PUNCT
ma-174	331	13	u	u	NOUN
ma-174	331	14	)	)	PUNCT
ma-174	331	15	is	be	AUX
ma-174	331	16	given	give	VERB
ma-174	331	17	by	by	ADP
ma-174	331	18	γf	γf	PROPN
ma-174	331	19	(	(	PUNCT
ma-174	331	20	z)=f	z)=f	PROPN
ma-174	331	21	′(z	′(z	NOUN
ma-174	331	22	)	)	PUNCT
ma-174	331	23	with	with	ADP
ma-174	331	24	the	the	DET
ma-174	331	25	domain	domain	NOUN
ma-174	331	26	dom(γ	dom(γ	PROPN
ma-174	331	27	)	)	PUNCT
ma-174	331	28	=	=	PRON
ma-174	332	1	{	{	PUNCT
ma-174	332	2	f	f	PROPN
ma-174	332	3	∈	∈	PROPN
ma-174	332	4	bα0	bα0	PROPN
ma-174	332	5	(	(	PUNCT
ma-174	332	6	u	u	NOUN
ma-174	332	7	)	)	PUNCT
ma-174	332	8	:	:	PUNCT
ma-174	332	9	f	f	PROPN
ma-174	332	10	′(z	′(z	NOUN
ma-174	332	11	)	)	PUNCT
ma-174	332	12	∈	∈	PROPN
ma-174	332	13	bα0	bα0	PROPN
ma-174	332	14	(	(	PUNCT
ma-174	332	15	u	u	NOUN
ma-174	332	16	)	)	PUNCT
ma-174	332	17	}	}	PUNCT
ma-174	332	18	.	.	PUNCT
ma-174	333	1	proof	proof	NOUN
ma-174	333	2	.	.	PUNCT
ma-174	334	1	using	use	VERB
ma-174	334	2	the	the	DET
ma-174	334	3	definition	definition	NOUN
ma-174	334	4	of	of	ADP
ma-174	334	5	the	the	DET
ma-174	334	6	infinitesimal	infinitesimal	ADJ
ma-174	334	7	generator	generator	NOUN
ma-174	334	8	γ	γ	PROPN
ma-174	334	9	,	,	PUNCT
ma-174	334	10	for	for	ADP
ma-174	334	11	f	f	PROPN
ma-174	334	12	∈	∈	PROPN
ma-174	334	13	bα0	bα0	PROPN
ma-174	334	14	(	(	PUNCT
ma-174	334	15	u	u	NOUN
ma-174	334	16	)	)	PUNCT
ma-174	334	17	,	,	PUNCT
ma-174	334	18	we	we	PRON
ma-174	334	19	have	have	VERB
ma-174	334	20	;	;	PUNCT
ma-174	334	21	γf	γf	PROPN
ma-174	334	22	(	(	PUNCT
ma-174	334	23	z	z	NOUN
ma-174	334	24	)	)	PUNCT
ma-174	335	1	=	=	SYM
ma-174	335	2	lim	lim	PROPN
ma-174	335	3	t→0	t→0	PROPN
ma-174	335	4	+	+	NUM
ma-174	335	5	f	f	X
ma-174	335	6	(	(	PUNCT
ma-174	335	7	z	z	PROPN
ma-174	335	8	+	+	CCONJ
ma-174	335	9	t)−	t)−	PROPN
ma-174	335	10	f	f	X
ma-174	335	11	(	(	PUNCT
ma-174	335	12	z	z	NOUN
ma-174	335	13	)	)	PUNCT
ma-174	335	14	t	t	NOUN
ma-174	335	15	=	=	SYM
ma-174	335	16	∂	∂	NOUN
ma-174	335	17	∂t	∂t	PROPN
ma-174	335	18	f	f	PROPN
ma-174	335	19	(	(	PUNCT
ma-174	335	20	z	z	PROPN
ma-174	335	21	+	+	NOUN
ma-174	335	22	t	t	PROPN
ma-174	335	23	)	)	PUNCT
ma-174	335	24	∣∣∣∣	∣∣∣∣	NOUN
ma-174	335	25	t=0	t=0	VERB
ma-174	335	26	=	=	SYM
ma-174	335	27	f	f	PROPN
ma-174	335	28	′(z	′(z	NOUN
ma-174	335	29	)	)	PUNCT
ma-174	335	30	.	.	PUNCT
ma-174	336	1	this	this	PRON
ma-174	336	2	means	mean	VERB
ma-174	336	3	that	that	SCONJ
ma-174	336	4	dom(γ	dom(γ	PROPN
ma-174	336	5	)	)	PUNCT
ma-174	336	6	⊂	⊂	PROPN
ma-174	336	7	{	{	PUNCT
ma-174	336	8	f	f	PROPN
ma-174	336	9	∈	∈	PROPN
ma-174	336	10	bα0	bα0	PROPN
ma-174	336	11	(	(	PUNCT
ma-174	336	12	u	u	NOUN
ma-174	336	13	)	)	PUNCT
ma-174	336	14	:	:	PUNCT
ma-174	336	15	f	f	PROPN
ma-174	336	16	′(z	′(z	NOUN
ma-174	336	17	)	)	PUNCT
ma-174	336	18	∈	∈	PROPN
ma-174	336	19	bα0	bα0	NOUN
ma-174	336	20	(	(	PUNCT
ma-174	336	21	u)}.it	u)}.it	NOUN
ma-174	336	22	remains	remain	VERB
ma-174	336	23	to	to	PART
ma-174	336	24	prove	prove	VERB
ma-174	336	25	the	the	DET
ma-174	336	26	reverse	reverse	ADJ
ma-174	336	27	inclusion	inclusion	NOUN
ma-174	336	28	.	.	PUNCT
ma-174	337	1	let	let	VERB
ma-174	337	2	f	f	PROPN
ma-174	337	3	∈	∈	PROPN
ma-174	337	4	bα0	bα0	PROPN
ma-174	337	5	(	(	PUNCT
ma-174	337	6	u	u	NOUN
ma-174	337	7	)	)	PUNCT
ma-174	337	8	be	be	VERB
ma-174	337	9	such	such	ADJ
ma-174	337	10	that	that	SCONJ
ma-174	337	11	f	f	PROPN
ma-174	337	12	′(z	′(z	NOUN
ma-174	337	13	)	)	PUNCT
ma-174	337	14	∈	∈	PROPN
ma-174	337	15	bα0	bα0	NOUN
ma-174	337	16	(	(	PUNCT
ma-174	337	17	u).then	u).then	PROPN
ma-174	337	18	for	for	ADP
ma-174	337	19	z	z	PROPN
ma-174	337	20	∈	∈	PROPN
ma-174	337	21	u	u	NOUN
ma-174	337	22	,	,	PUNCT
ma-174	337	23	we	we	PRON
ma-174	337	24	have	have	VERB
ma-174	337	25	;	;	PUNCT
ma-174	337	26	cϕt	cϕt	PROPN
ma-174	337	27	f	f	X
ma-174	337	28	(	(	PUNCT
ma-174	337	29	z)−	z)−	PROPN
ma-174	337	30	f	f	X
ma-174	337	31	(	(	PUNCT
ma-174	337	32	z	z	NOUN
ma-174	337	33	)	)	PUNCT
ma-174	337	34	=	=	SYM
ma-174	338	1	∫	∫	PROPN
ma-174	338	2	t	t	PROPN
ma-174	338	3	0	0	NUM
ma-174	338	4	∂	∂	NUM
ma-174	339	1	∂s	∂s	PROPN
ma-174	339	2	f	f	PROPN
ma-174	339	3	(	(	PUNCT
ma-174	339	4	z	z	NOUN
ma-174	339	5	+	+	CCONJ
ma-174	340	1	s)ds	s)ds	PROPN
ma-174	340	2	=	=	SYM
ma-174	340	3	∫	∫	PROPN
ma-174	341	1	t	t	PROPN
ma-174	341	2	0	0	NUM
ma-174	341	3	f	f	PROPN
ma-174	341	4	′(z)ds	′(z)ds	PROPN
ma-174	341	5	.	.	NOUN
ma-174	341	6	13	13	NUM
ma-174	341	7	letting	let	VERB
ma-174	341	8	f	f	X
ma-174	341	9	(	(	PUNCT
ma-174	341	10	z	z	NOUN
ma-174	341	11	)	)	PUNCT
ma-174	341	12	=	=	SYM
ma-174	341	13	f	f	PROPN
ma-174	341	14	′(z	′(z	NOUN
ma-174	341	15	)	)	PUNCT
ma-174	341	16	,	,	PUNCT
ma-174	341	17	we	we	PRON
ma-174	341	18	obtain	obtain	VERB
ma-174	341	19	cϕt	cϕt	PROPN
ma-174	341	20	f	f	PROPN
ma-174	341	21	(	(	PUNCT
ma-174	341	22	z)−	z)−	PROPN
ma-174	341	23	f	f	X
ma-174	341	24	(	(	PUNCT
ma-174	341	25	z	z	NOUN
ma-174	341	26	)	)	PUNCT
ma-174	341	27	=	=	SYM
ma-174	342	1	∫	∫	PROPN
ma-174	342	2	t	t	NOUN
ma-174	342	3	0	0	NUM
ma-174	343	1	f	f	PROPN
ma-174	343	2	(	(	PUNCT
ma-174	343	3	z)ds	z)ds	PROPN
ma-174	343	4	.	.	PUNCT
ma-174	344	1	this	this	PRON
ma-174	344	2	implies	imply	VERB
ma-174	344	3	that	that	SCONJ
ma-174	344	4	f	f	PROPN
ma-174	344	5	(	(	PUNCT
ma-174	344	6	z	z	NOUN
ma-174	344	7	)	)	PUNCT
ma-174	344	8	=	=	SYM
ma-174	344	9	f	f	PROPN
ma-174	344	10	′(z	′(z	NOUN
ma-174	344	11	)	)	PUNCT
ma-174	344	12	is	be	AUX
ma-174	344	13	a	a	DET
ma-174	344	14	function	function	NOUN
ma-174	344	15	of	of	ADP
ma-174	344	16	bα0	bα0	PROPN
ma-174	344	17	(	(	PUNCT
ma-174	344	18	u	u	NOUN
ma-174	344	19	)	)	PUNCT
ma-174	344	20	.	.	PUNCT
ma-174	345	1	it	it	PRON
ma-174	345	2	remains	remain	VERB
ma-174	345	3	to	to	PART
ma-174	345	4	show	show	VERB
ma-174	345	5	that	that	SCONJ
ma-174	345	6	the	the	DET
ma-174	345	7	limit	limit	NOUN
ma-174	345	8	of	of	ADP
ma-174	345	9	f	f	PROPN
ma-174	345	10	(	(	PUNCT
ma-174	345	11	z)exists	z)exists	PROPN
ma-174	345	12	in	in	ADP
ma-174	345	13	bα0	bα0	PROPN
ma-174	345	14	(	(	PUNCT
ma-174	345	15	u	u	NOUN
ma-174	345	16	)	)	PUNCT
ma-174	345	17	.	.	PUNCT
ma-174	346	1	since	since	SCONJ
ma-174	346	2	cϕt	cϕt	PROPN
ma-174	346	3	f	f	PROPN
ma-174	346	4	(	(	PUNCT
ma-174	346	5	z)−	z)−	PROPN
ma-174	346	6	f	f	X
ma-174	346	7	(	(	PUNCT
ma-174	346	8	z	z	NOUN
ma-174	346	9	)	)	PUNCT
ma-174	346	10	t	t	NOUN
ma-174	346	11	=	=	SYM
ma-174	346	12	1	1	NUM
ma-174	346	13	t	t	NOUN
ma-174	346	14	∫	∫	PROPN
ma-174	346	15	t	t	PROPN
ma-174	346	16	0	0	NUM
ma-174	346	17	f	f	PROPN
ma-174	346	18	(	(	PUNCT
ma-174	346	19	z)ds	z)ds	PROPN
ma-174	346	20	,	,	PUNCT
ma-174	346	21	we	we	PRON
ma-174	346	22	now	now	ADV
ma-174	346	23	take	take	VERB
ma-174	346	24	limits	limit	NOUN
ma-174	346	25	as	as	ADP
ma-174	346	26	t	t	PROPN
ma-174	346	27	→	→	SYM
ma-174	346	28	0	0	NUM
ma-174	346	29	+	+	CCONJ
ma-174	346	30	and	and	CCONJ
ma-174	346	31	invoke	invoke	VERB
ma-174	346	32	strong	strong	ADJ
ma-174	346	33	continuity	continuity	NOUN
ma-174	346	34	of	of	ADP
ma-174	346	35	(	(	PUNCT
ma-174	346	36	cϕs	cϕs	NOUN
ma-174	346	37	)	)	PUNCT
ma-174	346	38	s≥0	s≥0	VERB
ma-174	346	39	to	to	PART
ma-174	346	40	obtain	obtain	VERB
ma-174	346	41	lim	lim	NOUN
ma-174	346	42	t→0	t→0	PROPN
ma-174	347	1	+	+	CCONJ
ma-174	347	2	1	1	NUM
ma-174	347	3	t	t	NOUN
ma-174	347	4	∫	∫	PROPN
ma-174	347	5	t	t	PROPN
ma-174	347	6	0	0	NUM
ma-174	348	1	‖cϕsfds	‖cϕsfds	NOUN
ma-174	348	2	−	−	NOUN
ma-174	348	3	f‖	f‖	ADP
ma-174	348	4	=	=	SYM
ma-174	348	5	0	0	X
ma-174	348	6	.	.	PUNCT
ma-174	349	1	hence	hence	ADV
ma-174	349	2	dom(γ	dom(γ	PROPN
ma-174	349	3	)	)	PUNCT
ma-174	349	4	⊇	⊇	NOUN
ma-174	349	5	{	{	PUNCT
ma-174	349	6	f	f	PROPN
ma-174	349	7	∈	∈	PROPN
ma-174	349	8	bα0	bα0	PROPN
ma-174	349	9	(	(	PUNCT
ma-174	349	10	u	u	NOUN
ma-174	349	11	)	)	PUNCT
ma-174	349	12	:	:	PUNCT
ma-174	349	13	f	f	PROPN
ma-174	349	14	′(z	′(z	NOUN
ma-174	349	15	)	)	PUNCT
ma-174	349	16	∈	∈	PROPN
ma-174	349	17	bα0	bα0	PROPN
ma-174	349	18	(	(	PUNCT
ma-174	349	19	u	u	NOUN
ma-174	349	20	)	)	PUNCT
ma-174	349	21	}	}	PUNCT
ma-174	349	22	which	which	PRON
ma-174	349	23	completes	complete	VERB
ma-174	349	24	the	the	DET
ma-174	349	25	proof	proof	NOUN
ma-174	349	26	.	.	PUNCT
ma-174	350	1	�	�	PROPN
ma-174	350	2	5	5	NUM
ma-174	350	3	.	.	PUNCT
ma-174	351	1	rotation	rotation	NOUN
ma-174	351	2	group	group	NOUN
ma-174	351	3	the	the	DET
ma-174	351	4	induced	induced	ADJ
ma-174	351	5	composition	composition	NOUN
ma-174	351	6	semigroups	semigroup	NOUN
ma-174	351	7	for	for	ADP
ma-174	351	8	rotation	rotation	NOUN
ma-174	351	9	group	group	NOUN
ma-174	351	10	are	be	AUX
ma-174	351	11	defined	define	VERB
ma-174	351	12	on	on	ADP
ma-174	351	13	the	the	DET
ma-174	351	14	analytic	analytic	ADJ
ma-174	351	15	spaces	space	NOUN
ma-174	351	16	of	of	ADP
ma-174	351	17	theunit	theunit	VERB
ma-174	351	18	disk	disk	NOUN
ma-174	351	19	.	.	PUNCT
ma-174	352	1	we	we	PRON
ma-174	352	2	shall	shall	AUX
ma-174	352	3	therefore	therefore	ADV
ma-174	352	4	generate	generate	VERB
ma-174	352	5	composition	composition	NOUN
ma-174	352	6	semigroups	semigroup	NOUN
ma-174	352	7	induced	induce	VERB
ma-174	352	8	by	by	ADP
ma-174	352	9	rotation	rotation	NOUN
ma-174	352	10	group	group	NOUN
ma-174	352	11	on	on	ADP
ma-174	352	12	thegeneralized	thegeneralize	VERB
ma-174	352	13	little	little	ADJ
ma-174	352	14	bloch	bloch	PROPN
ma-174	352	15	space	space	NOUN
ma-174	352	16	of	of	ADP
ma-174	352	17	the	the	DET
ma-174	352	18	disc	disc	NOUN
ma-174	352	19	.	.	PUNCT
ma-174	353	1	the	the	DET
ma-174	353	2	results	result	NOUN
ma-174	353	3	obtained	obtain	VERB
ma-174	353	4	can	can	AUX
ma-174	353	5	then	then	ADV
ma-174	353	6	be	be	AUX
ma-174	353	7	mapped	map	VERB
ma-174	353	8	onto	onto	ADP
ma-174	353	9	the	the	DET
ma-174	353	10	upperhalf	upperhalf	ADJ
ma-174	353	11	plane	plane	NOUN
ma-174	353	12	by	by	ADP
ma-174	353	13	use	use	NOUN
ma-174	353	14	of	of	ADP
ma-174	353	15	cayley	cayley	ADJ
ma-174	353	16	transform	transform	NOUN
ma-174	353	17	.	.	PUNCT
ma-174	354	1	in	in	ADP
ma-174	354	2	this	this	DET
ma-174	354	3	case	case	NOUN
ma-174	354	4	,	,	PUNCT
ma-174	354	5	the	the	DET
ma-174	354	6	self	self	NOUN
ma-174	354	7	analytic	analytic	ADJ
ma-174	354	8	maps	map	NOUN
ma-174	354	9	of	of	ADP
ma-174	354	10	d	d	PROPN
ma-174	354	11	are	be	AUX
ma-174	354	12	of	of	ADP
ma-174	354	13	the	the	DET
ma-174	354	14	form	form	NOUN
ma-174	354	15	ϕt(z	ϕt(z	NUM
ma-174	354	16	)	)	PUNCT
ma-174	354	17	=	=	SYM
ma-174	354	18	e	e	X
ma-174	354	19	iktz	iktz	NOUN
ma-174	354	20	.	.	PUNCT
ma-174	355	1	we	we	PRON
ma-174	355	2	consider	consider	VERB
ma-174	355	3	the	the	DET
ma-174	355	4	composition	composition	NOUN
ma-174	355	5	semigroup	semigroup	NOUN
ma-174	355	6	induced	induce	VERB
ma-174	355	7	by	by	ADP
ma-174	355	8	the	the	DET
ma-174	355	9	rotation	rotation	NOUN
ma-174	355	10	group	group	NOUN
ma-174	355	11	on	on	ADP
ma-174	355	12	bα0	bα0	PROPN
ma-174	356	1	(	(	PUNCT
ma-174	356	2	d)given	d)given	VERB
ma-174	356	3	by	by	ADP
ma-174	356	4	cϕt	cϕt	PROPN
ma-174	356	5	f	f	PROPN
ma-174	357	1	(	(	PUNCT
ma-174	357	2	z	z	NOUN
ma-174	357	3	)	)	PUNCT
ma-174	357	4	=	=	PUNCT
ma-174	358	1	(	(	PUNCT
ma-174	358	2	f	f	X
ma-174	358	3	◦	◦	VERB
ma-174	358	4	ϕt	ϕt	PROPN
ma-174	358	5	)	)	PUNCT
ma-174	358	6	(	(	PUNCT
ma-174	358	7	z	z	X
ma-174	358	8	)	)	PUNCT
ma-174	358	9	=	=	SYM
ma-174	358	10	f	f	X
ma-174	358	11	(	(	PUNCT
ma-174	358	12	e	e	NOUN
ma-174	358	13	itz	itz	PROPN
ma-174	358	14	)	)	PUNCT
ma-174	358	15	,	,	PUNCT
ma-174	358	16	(	(	PUNCT
ma-174	358	17	12	12	NUM
ma-174	358	18	)	)	PUNCT
ma-174	358	19	for	for	ADP
ma-174	358	20	all	all	DET
ma-174	358	21	f	f	PROPN
ma-174	358	22	∈	∈	PROPN
ma-174	358	23	bα0	bα0	NOUN
ma-174	358	24	(	(	PUNCT
ma-174	358	25	d).it	d).it	PRON
ma-174	358	26	can	can	AUX
ma-174	358	27	be	be	AUX
ma-174	358	28	easily	easily	ADV
ma-174	358	29	shown	show	VERB
ma-174	358	30	that	that	SCONJ
ma-174	358	31	(	(	PUNCT
ma-174	358	32	cϕt	cϕt	NOUN
ma-174	358	33	)	)	PUNCT
ma-174	358	34	t≥0	t≥0	NOUN
ma-174	358	35	and	and	CCONJ
ma-174	358	36	(	(	PUNCT
ma-174	358	37	cϕ−t)t≥0	cϕ−t)t≥0	NOUN
ma-174	358	38	are	be	AUX
ma-174	358	39	semigroups	semigroup	NOUN
ma-174	358	40	on	on	ADP
ma-174	358	41	bα0	bα0	PROPN
ma-174	358	42	(	(	PUNCT
ma-174	358	43	d	d	NOUN
ma-174	358	44	)	)	PUNCT
ma-174	358	45	thus	thus	ADV
ma-174	358	46	(	(	PUNCT
ma-174	358	47	cϕt	cϕt	NOUN
ma-174	358	48	)	)	PUNCT
ma-174	358	49	t∈rdefines	t∈rdefine	VERB
ma-174	358	50	a	a	DET
ma-174	358	51	group	group	NOUN
ma-174	358	52	on	on	ADP
ma-174	358	53	bα0	bα0	PROPN
ma-174	358	54	(	(	PUNCT
ma-174	358	55	d).moreover	d).moreover	PROPN
ma-174	358	56	,	,	PUNCT
ma-174	358	57	this	this	DET
ma-174	358	58	group	group	NOUN
ma-174	358	59	is	be	AUX
ma-174	358	60	an	an	DET
ma-174	358	61	isometry	isometry	NOUN
ma-174	358	62	,	,	PUNCT
ma-174	358	63	as	as	SCONJ
ma-174	358	64	we	we	PRON
ma-174	358	65	prove	prove	VERB
ma-174	358	66	in	in	ADP
ma-174	358	67	the	the	DET
ma-174	358	68	next	next	ADJ
ma-174	358	69	proposition	proposition	NOUN
ma-174	358	70	.	.	PUNCT
ma-174	359	1	proposition	proposition	NOUN
ma-174	359	2	5.1	5.1	NUM
ma-174	359	3	.	.	PUNCT
ma-174	360	1	the	the	DET
ma-174	360	2	operator	operator	NOUN
ma-174	360	3	cϕt	cϕt	VERB
ma-174	360	4	given	give	VERB
ma-174	360	5	by	by	ADP
ma-174	360	6	(	(	PUNCT
ma-174	360	7	12	12	NUM
ma-174	360	8	)	)	PUNCT
ma-174	360	9	is	be	AUX
ma-174	360	10	an	an	DET
ma-174	360	11	isometry	isometry	NOUN
ma-174	360	12	on	on	ADP
ma-174	360	13	bα0	bα0	PROPN
ma-174	360	14	(	(	PUNCT
ma-174	360	15	d	d	NOUN
ma-174	360	16	)	)	PUNCT
ma-174	360	17	.	.	PUNCT
ma-174	361	1	proof	proof	NOUN
ma-174	361	2	.	.	PUNCT
ma-174	362	1	we	we	PRON
ma-174	362	2	shall	shall	AUX
ma-174	362	3	prove	prove	VERB
ma-174	362	4	that	that	SCONJ
ma-174	362	5	for	for	ADP
ma-174	362	6	each	each	DET
ma-174	362	7	t	t	NOUN
ma-174	362	8	∈	∈	PROPN
ma-174	362	9	r	r	NOUN
ma-174	362	10	,	,	PUNCT
ma-174	362	11	the	the	DET
ma-174	362	12	group	group	NOUN
ma-174	362	13	(	(	PUNCT
ma-174	362	14	cϕt	cϕt	NOUN
ma-174	362	15	)	)	PUNCT
ma-174	362	16	t∈r	t∈r	PROPN
ma-174	362	17	is	be	AUX
ma-174	362	18	an	an	DET
ma-174	362	19	isometry	isometry	NOUN
ma-174	362	20	on	on	ADP
ma-174	362	21	bα0	bα0	PROPN
ma-174	362	22	(	(	PUNCT
ma-174	362	23	d	d	NOUN
ma-174	362	24	)	)	PUNCT
ma-174	362	25	.	.	PUNCT
ma-174	363	1	it	it	PRON
ma-174	363	2	sufficesto	sufficesto	VERB
ma-174	363	3	prove	prove	VERB
ma-174	363	4	that	that	SCONJ
ma-174	363	5	‖cϕt	‖cϕt	PROPN
ma-174	363	6	f	f	PROPN
ma-174	363	7	‖bα(d	‖bα(d	X
ma-174	363	8	)	)	PUNCT
ma-174	364	1	=	=	SYM
ma-174	364	2	‖f	‖f	ADP
ma-174	364	3	‖bα(d).it	‖bα(d).it	NOUN
ma-174	364	4	follows	follow	VERB
ma-174	364	5	from	from	ADP
ma-174	364	6	the	the	DET
ma-174	364	7	definition	definition	NOUN
ma-174	364	8	that	that	SCONJ
ma-174	364	9	‖cϕt	‖cϕt	PROPN
ma-174	364	10	f	f	PROPN
ma-174	364	11	‖bα(d	‖bα(d	X
ma-174	364	12	)	)	PUNCT
ma-174	365	1	=	=	PRON
ma-174	366	1	|cϕt	|cϕt	PROPN
ma-174	366	2	f	f	X
ma-174	366	3	(	(	PUNCT
ma-174	366	4	0)|+	0)|+	NUM
ma-174	366	5	sup	sup	NOUN
ma-174	366	6	z∈d	z∈d	NUM
ma-174	366	7	(	(	PUNCT
ma-174	366	8	1−	1−	NUM
ma-174	366	9	|z	|z	PROPN
ma-174	366	10	|2	|2	NUM
ma-174	366	11	)	)	PUNCT
ma-174	366	12	α	α	PROPN
ma-174	366	13	|(cϕt	|(cϕt	VERB
ma-174	366	14	f	f	NOUN
ma-174	366	15	)	)	PUNCT
ma-174	366	16	′(z)|	′(z)|	NOUN
ma-174	366	17	=	=	SYM
ma-174	366	18	|(e	|(e	ADJ
ma-174	366	19	it)f	it)f	PROPN
ma-174	366	20	(	(	PUNCT
ma-174	366	21	0)|+	0)|+	NUM
ma-174	366	22	sup	sup	NOUN
ma-174	366	23	z∈d	z∈d	NUM
ma-174	366	24	(	(	PUNCT
ma-174	366	25	1−	1−	NUM
ma-174	366	26	|z	|z	PROPN
ma-174	366	27	|2	|2	NUM
ma-174	366	28	)	)	PUNCT
ma-174	366	29	α	α	PROPN
ma-174	366	30	|e	|e	VERB
ma-174	366	31	it	it	PRON
ma-174	366	32	f	f	PROPN
ma-174	366	33	′(e	′(e	NOUN
ma-174	366	34	itz)|	itz)|	ADJ
ma-174	366	35	=	=	X
ma-174	366	36	|f	|f	PROPN
ma-174	366	37	(	(	PUNCT
ma-174	366	38	0)|+	0)|+	NUM
ma-174	366	39	sup	sup	NOUN
ma-174	366	40	z∈d	z∈d	NUM
ma-174	366	41	(	(	PUNCT
ma-174	366	42	1−	1−	NUM
ma-174	366	43	|z	|z	PROPN
ma-174	366	44	|2	|2	NUM
ma-174	366	45	)	)	PUNCT
ma-174	366	46	α	α	PROPN
ma-174	366	47	|f	|f	PROPN
ma-174	366	48	′(e	′(e	PROPN
ma-174	366	49	itz)|	itz)|	PROPN
ma-174	366	50	.	.	PROPN
ma-174	366	51	14	14	NUM
ma-174	366	52	now	now	ADV
ma-174	366	53	,	,	PUNCT
ma-174	366	54	let	let	VERB
ma-174	366	55	ω	ω	NOUN
ma-174	366	56	=	=	PUNCT
ma-174	366	57	e	e	X
ma-174	366	58	itz	itz	PROPN
ma-174	366	59	so	so	SCONJ
ma-174	366	60	that	that	SCONJ
ma-174	366	61	z	z	NOUN
ma-174	366	62	=	=	PUNCT
ma-174	366	63	e−itω	e−itω	NOUN
ma-174	366	64	.	.	PUNCT
ma-174	367	1	then	then	ADV
ma-174	367	2	;	;	PUNCT
ma-174	367	3	‖cϕt	‖cϕt	ADV
ma-174	367	4	f	f	PROPN
ma-174	367	5	‖bα(d	‖bα(d	X
ma-174	367	6	)	)	PUNCT
ma-174	368	1	=	=	PRON
ma-174	368	2	|f	|f	PROPN
ma-174	369	1	(	(	PUNCT
ma-174	369	2	0)|+	0)|+	NOUN
ma-174	369	3	sup	sup	NOUN
ma-174	369	4	ω∈d	ω∈d	NOUN
ma-174	369	5	(	(	PUNCT
ma-174	369	6	1−	1−	NUM
ma-174	369	7	|e−itω|2	|e−itω|2	NOUN
ma-174	369	8	)	)	PUNCT
ma-174	369	9	α	α	PROPN
ma-174	369	10	|f	|f	NOUN
ma-174	369	11	′(ω)|	′(ω)|	NUM
ma-174	369	12	)	)	PUNCT
ma-174	370	1	=	=	SYM
ma-174	370	2	|f	|f	PROPN
ma-174	370	3	(	(	PUNCT
ma-174	370	4	0)|+	0)|+	NOUN
ma-174	370	5	sup	sup	NOUN
ma-174	370	6	ω∈d	ω∈d	NOUN
ma-174	370	7	(	(	PUNCT
ma-174	370	8	1−	1−	NUM
ma-174	370	9	|ω|2)α|f	|ω|2)α|f	NOUN
ma-174	370	10	′(ω)|	′(ω)|	X
ma-174	370	11	=	=	NOUN
ma-174	370	12	‖f	‖f	PRON
ma-174	370	13	‖bα(d	‖bα(d	NUM
ma-174	370	14	)	)	PUNCT
ma-174	370	15	.	.	PUNCT
ma-174	371	1	�	�	PROPN
ma-174	371	2	theorem	theorem	VERB
ma-174	371	3	5.2	5.2	NUM
ma-174	371	4	.	.	PUNCT
ma-174	372	1	the	the	DET
ma-174	372	2	operator	operator	NOUN
ma-174	372	3	cϕt	cϕt	VERB
ma-174	372	4	given	give	VERB
ma-174	372	5	by	by	ADP
ma-174	372	6	(	(	PUNCT
ma-174	372	7	12	12	NUM
ma-174	372	8	)	)	PUNCT
ma-174	372	9	is	be	AUX
ma-174	372	10	strongly	strongly	ADV
ma-174	372	11	continuous	continuous	ADJ
ma-174	372	12	on	on	ADP
ma-174	372	13	bα0	bα0	PROPN
ma-174	372	14	(	(	PUNCT
ma-174	372	15	d	d	NOUN
ma-174	372	16	)	)	PUNCT
ma-174	372	17	.	.	PUNCT
ma-174	373	1	proof	proof	NOUN
ma-174	373	2	.	.	PUNCT
ma-174	374	1	since	since	SCONJ
ma-174	374	2	polynomials	polynomial	NOUN
ma-174	374	3	are	be	AUX
ma-174	374	4	dense	dense	ADJ
ma-174	374	5	in	in	ADP
ma-174	374	6	bα0	bα0	PROPN
ma-174	374	7	(	(	PUNCT
ma-174	374	8	d	d	PROPN
ma-174	374	9	)	)	PUNCT
ma-174	374	10	,	,	PUNCT
ma-174	374	11	it	it	PRON
ma-174	374	12	suffices	suffice	VERB
ma-174	374	13	to	to	PART
ma-174	374	14	show	show	VERB
ma-174	374	15	that	that	SCONJ
ma-174	374	16	(	(	PUNCT
ma-174	374	17	cϕt	cϕt	NOUN
ma-174	374	18	)	)	PUNCT
ma-174	374	19	t∈r	t∈r	NOUN
ma-174	374	20	is	be	AUX
ma-174	374	21	strongly	strongly	ADV
ma-174	374	22	contin	contin	NOUN
ma-174	374	23	-	-	PUNCT
ma-174	374	24	uous	uous	ADJ
ma-174	374	25	on	on	ADP
ma-174	374	26	bα0	bα0	PROPN
ma-174	374	27	(	(	PUNCT
ma-174	374	28	d	d	NOUN
ma-174	374	29	)	)	PUNCT
ma-174	374	30	that	that	PRON
ma-174	374	31	is	be	AUX
ma-174	374	32	,	,	PUNCT
ma-174	374	33	for	for	ADP
ma-174	374	34	a	a	DET
ma-174	374	35	polynomial	polynomial	ADJ
ma-174	374	36	(	(	PUNCT
ma-174	374	37	zn)n≥0	zn)n≥0	NUM
ma-174	374	38	where	where	SCONJ
ma-174	374	39	z	z	PROPN
ma-174	374	40	∈	∈	PROPN
ma-174	375	1	d	d	SCONJ
ma-174	375	2	we	we	PRON
ma-174	375	3	obtain	obtain	VERB
ma-174	375	4	lim	lim	NOUN
ma-174	375	5	t→0	t→0	PROPN
ma-174	375	6	+	+	CCONJ
ma-174	375	7	‖cϕtzn	‖cϕtzn	ADJ
ma-174	375	8	−	−	PROPN
ma-174	375	9	zn‖bα(d	zn‖bα(d	NOUN
ma-174	375	10	)	)	PUNCT
ma-174	376	1	=	=	PUNCT
ma-174	376	2	0	0	X
ma-174	376	3	.	.	PUNCT
ma-174	377	1	clearly	clearly	ADV
ma-174	377	2	,	,	PUNCT
ma-174	377	3	lim	lim	PROPN
ma-174	377	4	t→0	t→0	PROPN
ma-174	377	5	+	+	CCONJ
ma-174	377	6	‖cϕtzn	‖cϕtzn	ADJ
ma-174	377	7	−	−	PROPN
ma-174	377	8	zn‖bα(d	zn‖bα(d	NOUN
ma-174	377	9	)	)	PUNCT
ma-174	378	1	=	=	SYM
ma-174	378	2	lim	lim	PROPN
ma-174	378	3	t→0	t→0	ADP
ma-174	378	4	+	+	CCONJ
ma-174	378	5	|cϕt	|cϕt	PROPN
ma-174	378	6	f	f	X
ma-174	379	1	(	(	PUNCT
ma-174	379	2	0)−	0)−	NUM
ma-174	379	3	f	f	PROPN
ma-174	379	4	(	(	PUNCT
ma-174	379	5	0)|+	0)|+	PUNCT
ma-174	379	6	(	(	PUNCT
ma-174	379	7	sup	sup	NOUN
ma-174	379	8	z∈d	z∈d	NUM
ma-174	379	9	(	(	PUNCT
ma-174	379	10	1−	1−	NUM
ma-174	379	11	|z	|z	PROPN
ma-174	379	12	|2)α|(cϕtzn	|2)α|(cϕtzn	NUM
ma-174	379	13	−	−	PROPN
ma-174	379	14	zn)′|	zn)′|	PROPN
ma-174	379	15	)	)	PUNCT
ma-174	379	16	)	)	PUNCT
ma-174	379	17	.	.	PUNCT
ma-174	380	1	but	but	CCONJ
ma-174	380	2	cϕtz	cϕtz	VERB
ma-174	380	3	n	n	CCONJ
ma-174	380	4	−	−	PROPN
ma-174	380	5	zn	zn	NOUN
ma-174	380	6	=	=	SYM
ma-174	381	1	(	(	PUNCT
ma-174	381	2	e	e	X
ma-174	381	3	int	int	NOUN
ma-174	381	4	−	−	PROPN
ma-174	382	1	1)zn.so	1)zn.so	NOUN
ma-174	382	2	its	its	PRON
ma-174	382	3	derivative	derivative	NOUN
ma-174	382	4	is	be	AUX
ma-174	382	5	given	give	VERB
ma-174	382	6	by	by	ADP
ma-174	382	7	(	(	PUNCT
ma-174	382	8	cϕtz	cϕtz	NOUN
ma-174	382	9	n	n	CCONJ
ma-174	382	10	−	−	PROPN
ma-174	382	11	zn)′	zn)′	NOUN
ma-174	383	1	=	=	SYM
ma-174	384	1	n(e	n(e	NOUN
ma-174	384	2	int	int	VERB
ma-174	384	3	−	−	PROPN
ma-174	384	4	1)zn−1	1)zn−1	PROPN
ma-174	384	5	,	,	PUNCT
ma-174	384	6	implying	imply	VERB
ma-174	384	7	that	that	SCONJ
ma-174	384	8	lim	lim	PROPN
ma-174	384	9	t→0	t→0	AUX
ma-174	384	10	+	+	CCONJ
ma-174	384	11	‖cϕtzn	‖cϕtzn	ADJ
ma-174	384	12	−	−	PROPN
ma-174	384	13	zn‖bα(d	zn‖bα(d	NOUN
ma-174	384	14	)	)	PUNCT
ma-174	385	1	=	=	SYM
ma-174	385	2	lim	lim	PROPN
ma-174	385	3	t→0	t→0	PROPN
ma-174	385	4	+	+	CCONJ
ma-174	385	5	|e	|e	VERB
ma-174	385	6	it	it	PRON
ma-174	385	7	f	f	X
ma-174	386	1	(	(	PUNCT
ma-174	386	2	0)−	0)−	NUM
ma-174	386	3	f	f	PROPN
ma-174	386	4	(	(	PUNCT
ma-174	386	5	0)|+	0)|+	PUNCT
ma-174	386	6	(	(	PUNCT
ma-174	386	7	sup	sup	NOUN
ma-174	386	8	z∈d	z∈d	NUM
ma-174	386	9	(	(	PUNCT
ma-174	386	10	1−	1−	NUM
ma-174	386	11	|z	|z	PROPN
ma-174	386	12	|2)α|nzn−1||(e	|2)α|nzn−1||(e	PROPN
ma-174	386	13	int	int	PROPN
ma-174	386	14	−	−	PROPN
ma-174	386	15	1)|	1)|	NUM
ma-174	386	16	)	)	PUNCT
ma-174	386	17	)	)	PUNCT
ma-174	386	18	.	.	PUNCT
ma-174	387	1	hence	hence	ADV
ma-174	387	2	,	,	PUNCT
ma-174	387	3	lim	lim	PROPN
ma-174	387	4	t→0	t→0	PROPN
ma-174	387	5	+	+	CCONJ
ma-174	387	6	‖cϕtzn	‖cϕtzn	ADJ
ma-174	387	7	−	−	PROPN
ma-174	387	8	zn‖bα(d	zn‖bα(d	NOUN
ma-174	387	9	)	)	PUNCT
ma-174	388	1	=	=	SYM
ma-174	388	2	0	0	PUNCT
ma-174	388	3	as	as	SCONJ
ma-174	388	4	desired	desire	VERB
ma-174	388	5	.	.	PUNCT
ma-174	389	1	�	�	PROPN
ma-174	389	2	proposition	proposition	NOUN
ma-174	389	3	5.3	5.3	NUM
ma-174	389	4	.	.	PUNCT
ma-174	390	1	the	the	DET
ma-174	390	2	infinitesimal	infinitesimal	ADJ
ma-174	390	3	generator	generator	NOUN
ma-174	390	4	γ	γ	PROPN
ma-174	390	5	of	of	ADP
ma-174	390	6	(	(	PUNCT
ma-174	390	7	cϕt	cϕt	PROPN
ma-174	390	8	)	)	PUNCT
ma-174	390	9	is	be	AUX
ma-174	390	10	given	give	VERB
ma-174	390	11	by	by	ADP
ma-174	390	12	γf	γf	PROPN
ma-174	390	13	(	(	PUNCT
ma-174	390	14	z	z	NOUN
ma-174	390	15	)	)	PUNCT
ma-174	390	16	=	=	PUNCT
ma-174	391	1	iz	iz	INTJ
ma-174	391	2	f	f	NOUN
ma-174	391	3	′(z	′(z	NOUN
ma-174	391	4	)	)	PUNCT
ma-174	391	5	with	with	ADP
ma-174	391	6	the	the	DET
ma-174	391	7	domain	domain	NOUN
ma-174	391	8	dom(γ	dom(γ	PROPN
ma-174	391	9	)	)	PUNCT
ma-174	392	1	=	=	PRON
ma-174	392	2	{	{	PUNCT
ma-174	392	3	f	f	PROPN
ma-174	392	4	∈	∈	PROPN
ma-174	392	5	bα0	bα0	NOUN
ma-174	392	6	(	(	PUNCT
ma-174	392	7	d	d	PROPN
ma-174	392	8	)	)	PUNCT
ma-174	392	9	:	:	PUNCT
ma-174	392	10	zf	zf	PROPN
ma-174	392	11	′(z	′(z	NOUN
ma-174	392	12	)	)	PUNCT
ma-174	392	13	∈	∈	PROPN
ma-174	392	14	bα0	bα0	NOUN
ma-174	392	15	(	(	PUNCT
ma-174	392	16	d	d	NOUN
ma-174	392	17	)	)	PUNCT
ma-174	392	18	}	}	PUNCT
ma-174	392	19	.	.	PUNCT
ma-174	393	1	proof	proof	NOUN
ma-174	393	2	.	.	PUNCT
ma-174	394	1	we	we	PRON
ma-174	394	2	obtain	obtain	VERB
ma-174	394	3	the	the	DET
ma-174	394	4	infinitesimal	infinitesimal	ADJ
ma-174	394	5	generator	generator	NOUN
ma-174	394	6	as	as	SCONJ
ma-174	394	7	follows	follow	VERB
ma-174	394	8	γf	γf	PROPN
ma-174	394	9	(	(	PUNCT
ma-174	394	10	z	z	NOUN
ma-174	394	11	)	)	PUNCT
ma-174	394	12	=	=	SYM
ma-174	394	13	lim	lim	PROPN
ma-174	394	14	t→0	t→0	PROPN
ma-174	394	15	+	+	CCONJ
ma-174	394	16	cϕt(z)−	cϕt(z)−	PROPN
ma-174	394	17	f	f	PROPN
ma-174	394	18	(	(	PUNCT
ma-174	394	19	z	z	NOUN
ma-174	394	20	)	)	PUNCT
ma-174	394	21	t	t	NOUN
ma-174	394	22	=	=	SYM
ma-174	394	23	∂	∂	NOUN
ma-174	394	24	∂t	∂t	PROPN
ma-174	394	25	f	f	PROPN
ma-174	394	26	(	(	PUNCT
ma-174	394	27	e	e	NOUN
ma-174	394	28	itz	itz	PROPN
ma-174	394	29	)	)	PUNCT
ma-174	394	30	∣∣∣∣	∣∣∣∣	NOUN
ma-174	394	31	t=0	t=0	VERB
ma-174	395	1	=	=	PUNCT
ma-174	396	1	iz	iz	INTJ
ma-174	396	2	f	f	NOUN
ma-174	396	3	′(z	′(z	NOUN
ma-174	396	4	)	)	PUNCT
ma-174	396	5	.	.	PUNCT
ma-174	397	1	15	15	NUM
ma-174	397	2	it	it	PRON
ma-174	397	3	therefore	therefore	ADV
ma-174	397	4	follows	follow	VERB
ma-174	397	5	that	that	SCONJ
ma-174	397	6	dom(γ	dom(γ	PROPN
ma-174	397	7	)	)	PUNCT
ma-174	397	8	⊆	⊆	NUM
ma-174	397	9	{	{	PUNCT
ma-174	397	10	f	f	PROPN
ma-174	397	11	∈	∈	PROPN
ma-174	397	12	bα0	bα0	NOUN
ma-174	397	13	(	(	PUNCT
ma-174	397	14	d	d	NOUN
ma-174	397	15	)	)	PUNCT
ma-174	397	16	}	}	PUNCT
ma-174	397	17	:	:	PUNCT
ma-174	397	18	zf	zf	PROPN
ma-174	397	19	′(z	′(z	NOUN
ma-174	397	20	)	)	PUNCT
ma-174	397	21	∈	∈	PROPN
ma-174	397	22	bα0	bα0	NOUN
ma-174	397	23	(	(	PUNCT
ma-174	397	24	d	d	NOUN
ma-174	397	25	)	)	PUNCT
ma-174	397	26	}	}	PUNCT
ma-174	397	27	.	.	PUNCT
ma-174	398	1	on	on	ADP
ma-174	398	2	the	the	DET
ma-174	398	3	other	other	ADJ
ma-174	398	4	hand	hand	NOUN
ma-174	398	5	,	,	PUNCT
ma-174	398	6	let	let	VERB
ma-174	398	7	f	f	PROPN
ma-174	398	8	∈	∈	PROPN
ma-174	398	9	bα0	bα0	PROPN
ma-174	398	10	(	(	PUNCT
ma-174	398	11	d	d	NOUN
ma-174	398	12	)	)	PUNCT
ma-174	398	13	}	}	PUNCT
ma-174	398	14	be	be	AUX
ma-174	398	15	such	such	ADJ
ma-174	398	16	that	that	SCONJ
ma-174	398	17	zf	zf	PROPN
ma-174	398	18	′(z	′(z	NOUN
ma-174	398	19	)	)	PUNCT
ma-174	398	20	∈	∈	PROPN
ma-174	398	21	bα0	bα0	NOUN
ma-174	398	22	(	(	PUNCT
ma-174	398	23	d	d	NOUN
ma-174	398	24	)	)	PUNCT
ma-174	398	25	}	}	PUNCT
ma-174	398	26	,	,	PUNCT
ma-174	398	27	then	then	ADV
ma-174	398	28	for	for	ADP
ma-174	398	29	z	z	PROPN
ma-174	398	30	∈	∈	PROPN
ma-174	399	1	d	d	X
ma-174	399	2	we	we	PRON
ma-174	399	3	have	have	VERB
ma-174	399	4	by	by	ADP
ma-174	399	5	the	the	DET
ma-174	399	6	fundamental	fundamental	ADJ
ma-174	399	7	theoremof	theoremof	PROPN
ma-174	399	8	calculus	calculus	NOUN
ma-174	399	9	,	,	PUNCT
ma-174	399	10	cϕt	cϕt	NOUN
ma-174	399	11	f	f	X
ma-174	400	1	(	(	PUNCT
ma-174	400	2	z)−	z)−	PROPN
ma-174	400	3	f	f	X
ma-174	400	4	(	(	PUNCT
ma-174	400	5	z	z	NOUN
ma-174	400	6	)	)	PUNCT
ma-174	400	7	=	=	SYM
ma-174	401	1	∫	∫	PROPN
ma-174	401	2	t	t	PROPN
ma-174	401	3	0	0	NUM
ma-174	401	4	∂	∂	NUM
ma-174	401	5	∂s	∂s	PROPN
ma-174	401	6	(	(	PUNCT
ma-174	401	7	cϕs	cϕs	NOUN
ma-174	401	8	f	f	PROPN
ma-174	401	9	(	(	PUNCT
ma-174	401	10	z))ds	z))ds	PROPN
ma-174	401	11	=	=	SYM
ma-174	401	12	∫	∫	PROPN
ma-174	401	13	t	t	PROPN
ma-174	401	14	0	0	PUNCT
ma-174	402	1	ie	ie	X
ma-174	402	2	iszf	iszf	PROPN
ma-174	402	3	′(e	′(e	PROPN
ma-174	402	4	isz)ds	isz)ds	PROPN
ma-174	403	1	=	=	SYM
ma-174	403	2	∫	∫	PROPN
ma-174	403	3	t	t	PROPN
ma-174	403	4	0	0	NUM
ma-174	403	5	cϕsf	cϕsf	NOUN
ma-174	403	6	(	(	PUNCT
ma-174	403	7	z)ds	z)ds	PROPN
ma-174	403	8	,	,	PUNCT
ma-174	403	9	where	where	SCONJ
ma-174	403	10	f	f	PROPN
ma-174	403	11	(	(	PUNCT
ma-174	403	12	z	z	NOUN
ma-174	403	13	)	)	PUNCT
ma-174	403	14	=	=	PUNCT
ma-174	404	1	iz	iz	INTJ
ma-174	404	2	f	f	NOUN
ma-174	404	3	′(z	′(z	NOUN
ma-174	404	4	)	)	PUNCT
ma-174	404	5	is	be	AUX
ma-174	404	6	a	a	DET
ma-174	404	7	function	function	NOUN
ma-174	404	8	in	in	ADP
ma-174	404	9	bα0	bα0	PROPN
ma-174	404	10	(	(	PUNCT
ma-174	404	11	d	d	NOUN
ma-174	404	12	)	)	PUNCT
ma-174	404	13	.	.	PUNCT
ma-174	405	1	thus	thus	ADV
ma-174	405	2	lim	lim	PROPN
ma-174	405	3	t→0	t→0	PROPN
ma-174	405	4	+	+	CCONJ
ma-174	405	5	cϕt	cϕt	PROPN
ma-174	405	6	f	f	PROPN
ma-174	405	7	−	−	PROPN
ma-174	405	8	f	f	PROPN
ma-174	405	9	t	t	PROPN
ma-174	405	10	=	=	SYM
ma-174	405	11	lim	lim	PROPN
ma-174	405	12	t→0	t→0	PROPN
ma-174	405	13	+	+	CCONJ
ma-174	405	14	1	1	NUM
ma-174	405	15	t	t	NOUN
ma-174	405	16	∫	∫	PROPN
ma-174	405	17	t	t	PROPN
ma-174	405	18	0	0	NUM
ma-174	405	19	cϕsfds	cϕsfds	NOUN
ma-174	405	20	and	and	CCONJ
ma-174	405	21	strong	strong	ADJ
ma-174	405	22	continuity	continuity	NOUN
ma-174	405	23	of	of	ADP
ma-174	405	24	(	(	PUNCT
ma-174	405	25	cϕs	cϕs	NOUN
ma-174	405	26	)	)	PUNCT
ma-174	405	27	s≥0	s≥0	NOUN
ma-174	405	28	implies	imply	VERB
ma-174	405	29	that	that	SCONJ
ma-174	405	30	‖1	‖1	PROPN
ma-174	405	31	t	t	PROPN
ma-174	405	32	∫	∫	PROPN
ma-174	405	33	t0	t0	PROPN
ma-174	405	34	cϕsfds	cϕsfd	VERB
ma-174	405	35	−f‖	−f‖	PROPN
ma-174	405	36	≤	≤	NUM
ma-174	405	37	1	1	NUM
ma-174	405	38	t	t	NOUN
ma-174	405	39	∫	∫	PROPN
ma-174	405	40	t	t	PROPN
ma-174	405	41	0	0	NUM
ma-174	406	1	‖cϕsf	‖cϕsf	NOUN
ma-174	406	2	−f‖ds	−f‖ds	PROPN
ma-174	407	1	→	→	SYM
ma-174	408	1	0	0	NUM
ma-174	409	1	+	+	PUNCT
ma-174	409	2	as	as	ADP
ma-174	409	3	t	t	PROPN
ma-174	409	4	→	→	SYM
ma-174	409	5	0	0	NUM
ma-174	409	6	+	+	NOUN
ma-174	409	7	.	.	PUNCT
ma-174	409	8	thus	thus	ADV
ma-174	409	9	dom(γ	dom(γ	PROPN
ma-174	409	10	)	)	PUNCT
ma-174	409	11	⊇	⊇	NOUN
ma-174	409	12	{	{	PUNCT
ma-174	409	13	f	f	PROPN
ma-174	409	14	∈	∈	PROPN
ma-174	409	15	bα0	bα0	NOUN
ma-174	409	16	(	(	PUNCT
ma-174	409	17	d	d	PROPN
ma-174	409	18	)	)	PUNCT
ma-174	409	19	:	:	PUNCT
ma-174	409	20	zf	zf	PROPN
ma-174	409	21	′(z	′(z	NOUN
ma-174	409	22	)	)	PUNCT
ma-174	409	23	∈	∈	PROPN
ma-174	409	24	bα0	bα0	NOUN
ma-174	409	25	(	(	PUNCT
ma-174	409	26	d	d	NOUN
ma-174	409	27	)	)	PUNCT
ma-174	409	28	}	}	PUNCT
ma-174	409	29	,	,	PUNCT
ma-174	409	30	as	as	SCONJ
ma-174	409	31	desired	desire	VERB
ma-174	409	32	.	.	PUNCT
ma-174	410	1	�	�	PROPN
ma-174	410	2	references	reference	NOUN
ma-174	410	3	[	[	X
ma-174	410	4	1	1	NUM
ma-174	410	5	]	]	PUNCT
ma-174	410	6	r.	r.	PROPN
ma-174	410	7	f.	f.	PROPN
ma-174	410	8	allen	allen	PROPN
ma-174	410	9	and	and	CCONJ
ma-174	410	10	f.	f.	PROPN
ma-174	410	11	colonna	colonna	PROPN
ma-174	410	12	,	,	PUNCT
ma-174	410	13	isometries	isometry	NOUN
ma-174	410	14	and	and	CCONJ
ma-174	410	15	spectral	spectral	ADJ
ma-174	410	16	of	of	ADP
ma-174	410	17	multiplication	multiplication	NOUN
ma-174	410	18	operators	operator	NOUN
ma-174	410	19	on	on	ADP
ma-174	410	20	the	the	DET
ma-174	410	21	bloch	bloch	PROPN
ma-174	410	22	space	space	NOUN
ma-174	410	23	,	,	PUNCT
ma-174	410	24	bull	bull	NOUN
ma-174	410	25	.	.	PUNCT
ma-174	411	1	aust	aust	PROPN
ma-174	411	2	.	.	PUNCT
ma-174	412	1	math.soc	math.soc	X
ma-174	412	2	,	,	PUNCT
ma-174	412	3	79	79	NUM
ma-174	412	4	,	,	PUNCT
ma-174	412	5	(	(	PUNCT
ma-174	412	6	2009	2009	NUM
ma-174	412	7	)	)	PUNCT
ma-174	412	8	,	,	PUNCT
ma-174	412	9	147	147	NUM
ma-174	412	10	-	-	SYM
ma-174	412	11	160	160	NUM
ma-174	412	12	.	.	PUNCT
ma-174	412	13	https://doi.org/10.48550/arxiv.0809.3278.[2	https://doi.org/10.48550/arxiv.0809.3278.[2	NOUN
ma-174	412	14	]	]	X
ma-174	412	15	m.	m.	NOUN
ma-174	412	16	bagasa	bagasa	PROPN
ma-174	412	17	,	,	PUNCT
ma-174	412	18	weighted	weight	VERB
ma-174	412	19	composition	composition	NOUN
ma-174	412	20	groups	group	NOUN
ma-174	412	21	on	on	ADP
ma-174	412	22	the	the	DET
ma-174	412	23	little	little	ADJ
ma-174	412	24	bloch	bloch	PROPN
ma-174	412	25	space	space	NOUN
ma-174	412	26	,	,	PUNCT
ma-174	412	27	j.	j.	PROPN
ma-174	412	28	funct	funct	PROPN
ma-174	412	29	.	.	PUNCT
ma-174	413	1	spaces	space	NOUN
ma-174	413	2	,	,	PUNCT
ma-174	413	3	2020	2020	NUM
ma-174	413	4	(	(	PUNCT
ma-174	413	5	2020	2020	NUM
ma-174	413	6	)	)	PUNCT
ma-174	413	7	,	,	PUNCT
ma-174	413	8	5480602	5480602	NUM
ma-174	413	9	.	.	PUNCT
ma-174	414	1	https://doi.org/10.1155/2020/5480602.[3	https://doi.org/10.1155/2020/5480602.[3	NOUN
ma-174	414	2	]	]	X
ma-174	415	1	s.	s.	PROPN
ma-174	415	2	ballamoole	ballamoole	PROPN
ma-174	415	3	,	,	PUNCT
ma-174	415	4	j.o	j.o	PROPN
ma-174	415	5	.	.	PROPN
ma-174	415	6	bonyo	bonyo	PROPN
ma-174	415	7	,	,	PUNCT
ma-174	415	8	t.l	t.l	PROPN
ma-174	415	9	.	.	PROPN
ma-174	415	10	miller	miller	PROPN
ma-174	415	11	,	,	PUNCT
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ma-174	415	13	g.	g.	PROPN
ma-174	415	14	miller	miller	PROPN
ma-174	415	15	,	,	PUNCT
ma-174	415	16	cesaro	cesaro	NOUN
ma-174	415	17	-	-	PUNCT
ma-174	415	18	like	like	ADJ
ma-174	415	19	operators	operator	NOUN
ma-174	415	20	on	on	ADP
ma-174	415	21	the	the	DET
ma-174	415	22	hardy	hardy	ADJ
ma-174	415	23	and	and	CCONJ
ma-174	415	24	bergman	bergman	PROPN
ma-174	415	25	spaces	space	NOUN
ma-174	415	26	of	of	ADP
ma-174	415	27	thehalf	thehalf	NOUN
ma-174	415	28	plane	plane	NOUN
ma-174	415	29	,	,	PUNCT
ma-174	415	30	complex	complex	ADJ
ma-174	415	31	anal	anal	NOUN
ma-174	415	32	.	.	PUNCT
ma-174	416	1	oper	oper	PROPN
ma-174	416	2	.	.	PROPN
ma-174	416	3	theory	theory	NOUN
ma-174	416	4	,	,	PUNCT
ma-174	416	5	10	10	NUM
ma-174	416	6	(	(	PUNCT
ma-174	416	7	2016	2016	NUM
ma-174	416	8	)	)	PUNCT
ma-174	416	9	,	,	PUNCT
ma-174	416	10	187	187	NUM
ma-174	416	11	-	-	SYM
ma-174	416	12	203	203	NUM
ma-174	416	13	.	.	PUNCT
ma-174	417	1	https://link.springer.com/article/10.1007/	https://link.springer.com/article/10.1007/	AUX
ma-174	417	2	s11785	s11785	NOUN
ma-174	417	3	-	-	PUNCT
ma-174	417	4	015	015	NUM
ma-174	417	5	-	-	PUNCT
ma-174	417	6	0481	0481	NUM
ma-174	417	7	-	-	PUNCT
ma-174	417	8	8[4	8[4	NOUN
ma-174	417	9	]	]	X
ma-174	417	10	c.	c.	PROPN
ma-174	417	11	bishop	bishop	PROPN
ma-174	417	12	,	,	PUNCT
ma-174	417	13	bounded	bound	VERB
ma-174	417	14	functions	function	NOUN
ma-174	417	15	in	in	ADP
ma-174	417	16	the	the	DET
ma-174	417	17	little	little	ADJ
ma-174	417	18	bloch	bloch	PROPN
ma-174	417	19	space	space	NOUN
ma-174	417	20	,	,	PUNCT
ma-174	417	21	pac	pac	PROPN
ma-174	417	22	.	.	PUNCT
ma-174	417	23	j.	j.	PROPN
ma-174	417	24	math	math	PROPN
ma-174	417	25	.	.	PUNCT
ma-174	418	1	142	142	NUM
ma-174	418	2	(	(	PUNCT
ma-174	418	3	1990	1990	NUM
ma-174	418	4	)	)	PUNCT
ma-174	418	5	,	,	PUNCT
ma-174	418	6	209	209	NUM
ma-174	418	7	-	-	SYM
ma-174	418	8	225	225	NUM
ma-174	418	9	.	.	PUNCT
ma-174	419	1	https://doi.org/10	https://doi.org/10	PROPN
ma-174	419	2	.	.	PUNCT
ma-174	420	1	2140	2140	NUM
ma-174	420	2	/	/	SYM
ma-174	420	3	pjm.1990.142.209.[5	pjm.1990.142.209.[5	PROPN
ma-174	420	4	]	]	PUNCT
ma-174	420	5	n.	n.	PROPN
ma-174	420	6	dunford	dunford	PROPN
ma-174	420	7	,	,	PUNCT
ma-174	420	8	j.t	j.t	PROPN
ma-174	420	9	.	.	PROPN
ma-174	420	10	schwartz	schwartz	PROPN
ma-174	420	11	,	,	PUNCT
ma-174	420	12	linear	linear	PROPN
ma-174	420	13	operators	operator	NOUN
ma-174	420	14	part	part	VERB
ma-174	420	15	i	i	PRON
ma-174	420	16	,	,	PUNCT
ma-174	420	17	interscience	interscience	NOUN
ma-174	420	18	publishers	publisher	NOUN
ma-174	420	19	,	,	PUNCT
ma-174	420	20	new	new	PROPN
ma-174	420	21	york	york	PROPN
ma-174	420	22	,	,	PUNCT
ma-174	420	23	1958	1958	NUM
ma-174	420	24	.	.	PUNCT
ma-174	421	1	https://doi.org/	https://doi.org/	NOUN
ma-174	421	2	:	:	PUNCT
ma-174	421	3	10.4236	10.4236	NUM
ma-174	421	4	/	/	SYM
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ma-174	421	6	]	]	PUNCT
ma-174	421	7	k.j	k.j	PROPN
ma-174	421	8	.	.	PUNCT
ma-174	421	9	engel	engel	PROPN
ma-174	421	10	,	,	PUNCT
ma-174	421	11	r.	r.	PROPN
ma-174	421	12	nagel	nagel	PROPN
ma-174	421	13	,	,	PUNCT
ma-174	421	14	a	a	DET
ma-174	421	15	short	short	ADJ
ma-174	421	16	course	course	NOUN
ma-174	421	17	on	on	ADP
ma-174	421	18	operator	operator	NOUN
ma-174	421	19	semigroups	semigroup	NOUN
ma-174	421	20	,	,	PUNCT
ma-174	421	21	springer	springer	NOUN
ma-174	421	22	new	new	PROPN
ma-174	421	23	york	york	PROPN
ma-174	421	24	,	,	PUNCT
ma-174	421	25	2006	2006	NUM
ma-174	421	26	.	.	PUNCT
ma-174	422	1	https://doi.org/10	https://doi.org/10	PROPN
ma-174	422	2	.	.	PUNCT
ma-174	423	1	1007/0	1007/0	NUM
ma-174	423	2	-	-	PUNCT
ma-174	423	3	387	387	NUM
ma-174	423	4	-	-	PUNCT
ma-174	423	5	36619	36619	NUM
ma-174	423	6	-	-	PUNCT
ma-174	423	7	9.[7	9.[7	NUM
ma-174	423	8	]	]	X
ma-174	423	9	e.o	e.o	PROPN
ma-174	423	10	.	.	PROPN
ma-174	423	11	gori	gori	PROPN
ma-174	423	12	,	,	PUNCT
ma-174	423	13	j.o	j.o	PROPN
ma-174	423	14	.	.	PROPN
ma-174	423	15	bonyo	bonyo	PROPN
ma-174	423	16	,	,	PUNCT
ma-174	423	17	duality	duality	NOUN
ma-174	423	18	of	of	ADP
ma-174	423	19	the	the	DET
ma-174	423	20	nonreflexive	nonreflexive	ADJ
ma-174	423	21	bergman	bergman	PROPN
ma-174	423	22	space	space	NOUN
ma-174	423	23	of	of	ADP
ma-174	423	24	the	the	DET
ma-174	423	25	upper	upper	ADJ
ma-174	423	26	half	half	NOUN
ma-174	423	27	plane	plane	NOUN
ma-174	423	28	and	and	CCONJ
ma-174	423	29	composition	composition	NOUN
ma-174	423	30	groups	group	NOUN
ma-174	423	31	,	,	PUNCT
ma-174	423	32	arxiv:1901.07780	arxiv:1901.07780	VERB
ma-174	423	33	[	[	X
ma-174	423	34	math.fa	math.fa	X
ma-174	423	35	]	]	PUNCT
ma-174	423	36	,	,	PUNCT
ma-174	423	37	(	(	PUNCT
ma-174	423	38	2019	2019	NUM
ma-174	423	39	)	)	PUNCT
ma-174	423	40	,	,	PUNCT
ma-174	423	41	1	1	NUM
ma-174	423	42	-	-	SYM
ma-174	423	43	16	16	NUM
ma-174	423	44	.	.	PUNCT
ma-174	424	1	https://doi.org/10.48550/arxiv.1901.07780.[8	https://doi.org/10.48550/arxiv.1901.07780.[8	NOUN
ma-174	424	2	]	]	X
ma-174	424	3	b.d	b.d	PROPN
ma-174	424	4	.	.	PROPN
ma-174	424	5	maccluer	maccluer	PROPN
ma-174	424	6	,	,	PUNCT
ma-174	424	7	r.	r.	PROPN
ma-174	424	8	zhao	zhao	PROPN
ma-174	424	9	,	,	PUNCT
ma-174	424	10	essential	essential	ADJ
ma-174	424	11	norms	norm	NOUN
ma-174	424	12	of	of	ADP
ma-174	424	13	composition	composition	NOUN
ma-174	424	14	operators	operator	NOUN
ma-174	424	15	between	between	ADP
ma-174	424	16	bloch	bloch	PROPN
ma-174	424	17	type	type	NOUN
ma-174	424	18	spaces	space	NOUN
ma-174	424	19	,	,	PUNCT
ma-174	424	20	rocky	rocky	ADJ
ma-174	424	21	mountain	mountain	NOUN
ma-174	424	22	j.math	j.math	NOUN
ma-174	424	23	.	.	PUNCT
ma-174	425	1	33	33	NUM
ma-174	425	2	(	(	PUNCT
ma-174	425	3	2003	2003	NUM
ma-174	425	4	)	)	PUNCT
ma-174	425	5	,	,	PUNCT
ma-174	425	6	1437	1437	NUM
ma-174	425	7	-	-	SYM
ma-174	425	8	1458	1458	NUM
ma-174	425	9	.	.	PUNCT
ma-174	426	1	https://doi.org/10.1216/rmjm/1181075473.[9	https://doi.org/10.1216/rmjm/1181075473.[9	NOUN
ma-174	426	2	]	]	PUNCT
ma-174	426	3	a.	a.	NOUN
ma-174	426	4	pazy	pazy	NOUN
ma-174	426	5	,	,	PUNCT
ma-174	426	6	semigroups	semigroup	NOUN
ma-174	426	7	of	of	ADP
ma-174	426	8	linear	linear	PROPN
ma-174	426	9	operators	operator	NOUN
ma-174	426	10	and	and	CCONJ
ma-174	426	11	applications	application	NOUN
ma-174	426	12	to	to	ADP
ma-174	426	13	partial	partial	ADJ
ma-174	426	14	differential	differential	NOUN
ma-174	426	15	equations	equation	NOUN
ma-174	426	16	,	,	PUNCT
ma-174	426	17	springer	springer	NOUN
ma-174	426	18	new	new	PROPN
ma-174	426	19	york	york	PROPN
ma-174	426	20	,	,	PUNCT
ma-174	426	21	newyork	newyork	PROPN
ma-174	426	22	,	,	PUNCT
ma-174	426	23	ny	ny	NOUN
ma-174	426	24	,	,	PUNCT
ma-174	426	25	1983	1983	NUM
ma-174	426	26	.	.	PUNCT
ma-174	427	1	https://doi.org/10.1007/978-1-4612-5561-1.[10	https://doi.org/10.1007/978-1-4612-5561-1.[10	PROPN
ma-174	427	2	]	]	PUNCT
ma-174	427	3	s.	s.	PROPN
ma-174	427	4	stevic	stevic	PROPN
ma-174	427	5	,	,	PUNCT
ma-174	427	6	characterizations	characterization	NOUN
ma-174	427	7	of	of	ADP
ma-174	427	8	composition	composition	NOUN
ma-174	427	9	followed	follow	VERB
ma-174	427	10	by	by	ADP
ma-174	427	11	differentiation	differentiation	NOUN
ma-174	427	12	between	between	ADP
ma-174	427	13	bloch	bloch	NOUN
ma-174	427	14	-	-	PUNCT
ma-174	427	15	type	type	NOUN
ma-174	427	16	spaces	space	NOUN
ma-174	427	17	.	.	PUNCT
ma-174	428	1	appl	appl	PROPN
ma-174	428	2	.	.	PUNCT
ma-174	429	1	math.comput	math.comput	PROPN
ma-174	429	2	,	,	PUNCT
ma-174	429	3	218	218	NUM
ma-174	429	4	(	(	PUNCT
ma-174	429	5	2011	2011	NUM
ma-174	429	6	)	)	PUNCT
ma-174	429	7	,	,	PUNCT
ma-174	429	8	4312–4316	4312–4316	NUM
ma-174	429	9	.	.	PUNCT
ma-174	430	1	https://doi.org/10.1016/j.amc.2011.10.004.[11	https://doi.org/10.1016/j.amc.2011.10.004.[11	PROPN
ma-174	430	2	]	]	PUNCT
ma-174	430	3	a.g	a.g	PROPN
ma-174	430	4	.	.	PROPN
ma-174	430	5	siskakis	siskakis	PROPN
ma-174	430	6	,	,	PUNCT
ma-174	430	7	semigroups	semigroup	NOUN
ma-174	430	8	of	of	ADP
ma-174	430	9	composition	composition	NOUN
ma-174	430	10	operators	operator	NOUN
ma-174	430	11	and	and	CCONJ
ma-174	430	12	integral	integral	ADJ
ma-174	430	13	operators	operator	NOUN
ma-174	430	14	in	in	ADP
ma-174	430	15	the	the	DET
ma-174	430	16	spaces	space	NOUN
ma-174	430	17	of	of	ADP
ma-174	430	18	analytic	analytic	ADJ
ma-174	430	19	functions	function	NOUN
ma-174	430	20	,	,	PUNCT
ma-174	430	21	ann.acad	ann.acad	PROPN
ma-174	430	22	.	.	PUNCT
ma-174	431	1	sci	sci	PROPN
ma-174	431	2	.	.	PUNCT
ma-174	431	3	fenn	fenn	PROPN
ma-174	431	4	.	.	PUNCT
ma-174	431	5	math	math	PROPN
ma-174	431	6	.	.	PUNCT
ma-174	432	1	38	38	NUM
ma-174	432	2	(	(	PUNCT
ma-174	432	3	2013	2013	NUM
ma-174	432	4	)	)	PUNCT
ma-174	432	5	,	,	PUNCT
ma-174	432	6	67	67	NUM
ma-174	432	7	-	-	SYM
ma-174	432	8	89	89	NUM
ma-174	432	9	.	.	PUNCT
ma-174	433	1	https://doi.org/10.5186/aasfm.2013.3806.[12	https://doi.org/10.5186/aasfm.2013.3806.[12	PROPN
ma-174	433	2	]	]	X
ma-174	433	3	a.g	a.g	PROPN
ma-174	433	4	.	.	PROPN
ma-174	433	5	siskakis	siskakis	PROPN
ma-174	433	6	,	,	PUNCT
ma-174	433	7	semigroups	semigroup	NOUN
ma-174	433	8	of	of	ADP
ma-174	433	9	composition	composition	NOUN
ma-174	433	10	operators	operator	NOUN
ma-174	433	11	on	on	ADP
ma-174	433	12	the	the	DET
ma-174	433	13	spaces	space	NOUN
ma-174	433	14	of	of	ADP
ma-174	433	15	analytic	analytic	ADJ
ma-174	433	16	functions	function	NOUN
ma-174	433	17	,	,	PUNCT
ma-174	433	18	a	a	DET
ma-174	433	19	review	review	NOUN
ma-174	433	20	,	,	PUNCT
ma-174	433	21	contemp	contemp	NOUN
ma-174	433	22	.	.	PUNCT
ma-174	434	1	math.213	math.213	NOUN
ma-174	434	2	(	(	PUNCT
ma-174	434	3	1998	1998	NUM
ma-174	434	4	)	)	PUNCT
ma-174	434	5	,	,	PUNCT
ma-174	434	6	229	229	NUM
ma-174	434	7	-	-	SYM
ma-174	434	8	252	252	NUM
ma-174	434	9	.	.	PUNCT
ma-174	435	1	https://doi.org/10.1007/bf03322189.16	https://doi.org/10.1007/bf03322189.16	PROPN
ma-174	436	1	https://doi.org/10.48550/arxiv.0809.3278	https://doi.org/10.48550/arxiv.0809.3278	PROPN
ma-174	436	2	https://doi.org/10.1155/2020/5480602	https://doi.org/10.1155/2020/5480602	PROPN
ma-174	436	3	https://link.springer.com/article/10.1007/s11785-015-0481-8	https://link.springer.com/article/10.1007/s11785-015-0481-8	PROPN
ma-174	436	4	https://link.springer.com/article/10.1007/s11785-015-0481-8	https://link.springer.com/article/10.1007/s11785-015-0481-8	PROPN
ma-174	437	1	https://doi.org/10.2140/pjm.1990.142.209	https://doi.org/10.2140/pjm.1990.142.209	PROPN
ma-174	437	2	https://doi.org/10.2140/pjm.1990.142.209	https://doi.org/10.2140/pjm.1990.142.209	PROPN
ma-174	437	3	https://doi.org/:10.4236/jhepgc.2018.43031	https://doi.org/:10.4236/jhepgc.2018.43031	VERB
ma-174	437	4	https://doi.org/:10.4236/jhepgc.2018.43031	https://doi.org/:10.4236/jhepgc.2018.43031	ADP
ma-174	437	5	https://doi.org/10.1007/0-387-36619-9	https://doi.org/10.1007/0-387-36619-9	PROPN
ma-174	437	6	https://doi.org/10.1007/0-387-36619-9	https://doi.org/10.1007/0-387-36619-9	VERB
ma-174	437	7	https://doi.org/10.48550/arxiv.1901.07780	https://doi.org/10.48550/arxiv.1901.07780	PROPN
ma-174	437	8	https://doi.org/10.1216/rmjm/1181075473	https://doi.org/10.1216/rmjm/1181075473	PROPN
ma-174	437	9	https://doi.org/10.1007/978-1-4612-5561-1	https://doi.org/10.1007/978-1-4612-5561-1	PROPN
ma-174	437	10	https://doi.org/10.1016/j.amc.2011.10.004	https://doi.org/10.1016/j.amc.2011.10.004	PROPN
ma-174	438	1	https://doi.org/10.5186/aasfm.2013.3806	https://doi.org/10.5186/aasfm.2013.3806	PRON
ma-174	438	2	https://doi.org/10.1007/bf03322189	https://doi.org/10.1007/bf03322189	X
ma-174	439	1	[	[	X
ma-174	439	2	13	13	NUM
ma-174	439	3	]	]	PUNCT
ma-174	439	4	s.	s.	PROPN
ma-174	439	5	stevic	stevic	PROPN
ma-174	439	6	,	,	PUNCT
ma-174	439	7	a.k	a.k	PROPN
ma-174	439	8	.	.	PROPN
ma-174	439	9	sharma	sharma	PROPN
ma-174	439	10	,	,	PUNCT
ma-174	439	11	composition	composition	NOUN
ma-174	439	12	operators	operator	NOUN
ma-174	439	13	between	between	ADP
ma-174	439	14	hardy	hardy	ADJ
ma-174	439	15	and	and	CCONJ
ma-174	439	16	bloch	bloch	NOUN
ma-174	439	17	-	-	PUNCT
ma-174	439	18	type	type	NOUN
ma-174	439	19	spaces	space	NOUN
ma-174	439	20	of	of	ADP
ma-174	439	21	the	the	DET
ma-174	439	22	upper	upper	ADJ
ma-174	439	23	half	half	ADJ
ma-174	439	24	-	-	PUNCT
ma-174	439	25	plane	plane	NOUN
ma-174	439	26	,	,	PUNCT
ma-174	439	27	bull.korean	bull.korean	ADJ
ma-174	439	28	math	math	NOUN
ma-174	439	29	.	.	PUNCT
ma-174	440	1	soc	soc	PROPN
ma-174	440	2	.	.	PUNCT
ma-174	441	1	44	44	NUM
ma-174	441	2	(	(	PUNCT
ma-174	441	3	2007	2007	NUM
ma-174	441	4	)	)	PUNCT
ma-174	441	5	,	,	PUNCT
ma-174	441	6	475	475	NUM
ma-174	441	7	-	-	SYM
ma-174	441	8	482	482	NUM
ma-174	441	9	.	.	PUNCT
ma-174	442	1	https://doi.org/:10.4134/bkms.2007.44.3.475.[14	https://doi.org/:10.4134/bkms.2007.44.3.475.[14	PROPN
ma-174	442	2	]	]	X
ma-174	442	3	j.	j.	PROPN
ma-174	442	4	zang	zang	PROPN
ma-174	442	5	,	,	PUNCT
ma-174	442	6	x.	x.	PROPN
ma-174	442	7	fu	fu	PROPN
ma-174	442	8	,	,	PUNCT
ma-174	442	9	bloch	bloch	PROPN
ma-174	442	10	type	type	NOUN
ma-174	442	11	space	space	NOUN
ma-174	442	12	on	on	ADP
ma-174	442	13	the	the	DET
ma-174	442	14	upper	upper	ADJ
ma-174	442	15	half	half	NOUN
ma-174	442	16	plane	plane	NOUN
ma-174	442	17	,	,	PUNCT
ma-174	442	18	bull	bull	NOUN
ma-174	442	19	.	.	PUNCT
ma-174	443	1	korean	korean	ADJ
ma-174	443	2	math	math	PROPN
ma-174	443	3	.	.	PUNCT
ma-174	444	1	soc	soc	PROPN
ma-174	444	2	.	.	PUNCT
ma-174	445	1	54	54	NUM
ma-174	445	2	(	(	PUNCT
ma-174	445	3	2017	2017	NUM
ma-174	445	4	)	)	PUNCT
ma-174	445	5	,	,	PUNCT
ma-174	445	6	1337	1337	NUM
ma-174	445	7	-	-	SYM
ma-174	445	8	1346	1346	NUM
ma-174	445	9	.	.	PUNCT
ma-174	446	1	https	https	NOUN
ma-174	446	2	:	:	PUNCT
ma-174	446	3	//doi.org/10.4134	//doi.org/10.4134	X
ma-174	446	4	/	/	SYM
ma-174	446	5	bkms.b160572.[15	bkms.b160572.[15	NOUN
ma-174	446	6	]	]	PUNCT
ma-174	447	1	y.	y.	PROPN
ma-174	447	2	cheng	cheng	PROPN
ma-174	447	3	,	,	PUNCT
ma-174	447	4	t.	t.	PROPN
ma-174	447	5	zhang	zhang	PROPN
ma-174	447	6	,	,	PUNCT
ma-174	447	7	y.	y.	PROPN
ma-174	447	8	jiang	jiang	PROPN
ma-174	447	9	,	,	PUNCT
ma-174	447	10	composition	composition	NOUN
ma-174	447	11	operators	operator	NOUN
ma-174	447	12	and	and	CCONJ
ma-174	447	13	the	the	DET
ma-174	447	14	bloch	bloch	PROPN
ma-174	447	15	spaces	space	VERB
ma-174	447	16	,	,	PUNCT
ma-174	447	17	in	in	ADP
ma-174	447	18	:	:	PUNCT
ma-174	447	19	2010	2010	NUM
ma-174	447	20	third	third	ADJ
ma-174	447	21	international	international	ADJ
ma-174	447	22	conferenceon	conferenceon	NOUN
ma-174	447	23	information	information	NOUN
ma-174	447	24	and	and	CCONJ
ma-174	447	25	computing	computing	NOUN
ma-174	447	26	,	,	PUNCT
ma-174	447	27	ieee	ieee	PROPN
ma-174	447	28	,	,	PUNCT
ma-174	447	29	wuxi	wuxi	PROPN
ma-174	447	30	,	,	PUNCT
ma-174	447	31	tbd	tbd	PROPN
ma-174	447	32	,	,	PUNCT
ma-174	447	33	china	china	PROPN
ma-174	447	34	,	,	PUNCT
ma-174	447	35	2010	2010	NUM
ma-174	447	36	:	:	PUNCT
ma-174	448	1	pp	pp	ADJ
ma-174	448	2	.	.	PUNCT
ma-174	449	1	297	297	NUM
ma-174	449	2	-	-	SYM
ma-174	449	3	300	300	NUM
ma-174	449	4	.	.	PUNCT
ma-174	450	1	https://doi.org/10.1109/icic	https://doi.org/10.1109/icic	PROPN
ma-174	450	2	.	.	PUNCT
ma-174	451	1	2010.170.[16	2010.170.[16	NUM
ma-174	451	2	]	]	PUNCT
ma-174	452	1	k.	k.	PROPN
ma-174	452	2	zhu	zhu	PROPN
ma-174	452	3	,	,	PUNCT
ma-174	452	4	operator	operator	NOUN
ma-174	452	5	theory	theory	NOUN
ma-174	452	6	in	in	ADP
ma-174	452	7	function	function	NOUN
ma-174	452	8	spaces	space	NOUN
ma-174	452	9	,	,	PUNCT
ma-174	452	10	marcel	marcel	PROPN
ma-174	452	11	dekker	dekker	PROPN
ma-174	452	12	,	,	PUNCT
ma-174	452	13	inc	inc	PROPN
ma-174	452	14	.	.	PROPN
ma-174	452	15	,	,	PUNCT
ma-174	452	16	new	new	PROPN
ma-174	452	17	york	york	PROPN
ma-174	452	18	and	and	CCONJ
ma-174	452	19	basel	basel	PROPN
ma-174	452	20	,	,	PUNCT
ma-174	452	21	1990	1990	NUM
ma-174	452	22	.	.	PUNCT
ma-174	453	1	https://doi.org/	https://doi.org/	VERB
ma-174	453	2	10.1090	10.1090	NUM
ma-174	453	3	/	/	SYM
ma-174	453	4	surv/138.[17	surv/138.[17	NUM
ma-174	453	5	]	]	PUNCT
ma-174	453	6	k.	k.	PROPN
ma-174	453	7	zhu	zhu	PROPN
ma-174	453	8	,	,	PUNCT
ma-174	453	9	bloch	bloch	PROPN
ma-174	453	10	type	type	NOUN
ma-174	453	11	spaces	space	NOUN
ma-174	453	12	of	of	ADP
ma-174	453	13	analytic	analytic	ADJ
ma-174	453	14	functions	function	NOUN
ma-174	453	15	,	,	PUNCT
ma-174	453	16	rocky	rocky	ADJ
ma-174	453	17	mountain	mountain	NOUN
ma-174	453	18	j.	j.	PROPN
ma-174	453	19	math	math	PROPN
ma-174	453	20	.	.	PUNCT
ma-174	454	1	23	23	NUM
ma-174	454	2	(	(	PUNCT
ma-174	454	3	1993	1993	NUM
ma-174	454	4	)	)	PUNCT
ma-174	454	5	,	,	PUNCT
ma-174	454	6	1143	1143	NUM
ma-174	454	7	-	-	SYM
ma-174	454	8	1177	1177	NUM
ma-174	454	9	.	.	PUNCT
ma-174	455	1	https://www	https://www	PROPN
ma-174	455	2	.	.	PUNCT
ma-174	456	1	jstor.org/stable/44237763.[18	jstor.org/stable/44237763.[18	PROPN
ma-174	456	2	]	]	X
ma-174	456	3	k.	k.	PROPN
ma-174	456	4	zhu	zhu	PROPN
ma-174	456	5	,	,	PUNCT
ma-174	456	6	spaces	space	NOUN
ma-174	456	7	of	of	ADP
ma-174	456	8	holomorphic	holomorphic	ADJ
ma-174	456	9	functions	function	NOUN
ma-174	456	10	in	in	ADP
ma-174	456	11	the	the	DET
ma-174	456	12	unit	unit	NOUN
ma-174	456	13	ball	ball	NOUN
ma-174	456	14	,	,	PUNCT
ma-174	456	15	springer	springer	NOUN
ma-174	456	16	-	-	PUNCT
ma-174	456	17	verlag	verlag	PROPN
ma-174	456	18	,	,	PUNCT
ma-174	456	19	new	new	PROPN
ma-174	456	20	york	york	PROPN
ma-174	456	21	,	,	PUNCT
ma-174	456	22	2006	2006	NUM
ma-174	456	23	.	.	PUNCT
ma-174	457	1	https://doi.org/10	https://doi.org/10	PROPN
ma-174	457	2	.	.	PUNCT
ma-174	458	1	1007/0	1007/0	NUM
ma-174	458	2	-	-	PUNCT
ma-174	458	3	387	387	NUM
ma-174	458	4	-	-	PUNCT
ma-174	458	5	27539	27539	NUM
ma-174	458	6	-	-	SYM
ma-174	458	7	8	8	NUM
ma-174	458	8	.	.	NUM
ma-174	459	1	17	17	NUM
ma-174	459	2	https://doi.org/:10.4134/bkms.2007.44.3.475	https://doi.org/:10.4134/bkms.2007.44.3.475	NUM
ma-174	459	3	https://doi.org/10.4134/bkms.b160572	https://doi.org/10.4134/bkms.b160572	NOUN
ma-174	459	4	https://doi.org/10.4134/bkms.b160572	https://doi.org/10.4134/bkms.b160572	NOUN
ma-174	460	1	https://doi.org/10.1109/icic.2010.170	https://doi.org/10.1109/icic.2010.170	PROPN
ma-174	460	2	https://doi.org/10.1109/icic.2010.170	https://doi.org/10.1109/icic.2010.170	PROPN
ma-174	460	3	https://doi.org/10.1090/surv/138	https://doi.org/10.1090/surv/138	PROPN
ma-174	460	4	https://doi.org/10.1090/surv/138	https://doi.org/10.1090/surv/138	PROPN
ma-174	460	5	https://www.jstor.org/stable/44237763	https://www.jstor.org/stable/44237763	PROPN
ma-174	460	6	https://www.jstor.org/stable/44237763	https://www.jstor.org/stable/44237763	PROPN
ma-174	460	7	https://doi.org/10.1007/0-387-27539-8	https://doi.org/10.1007/0-387-27539-8	NOUN
ma-174	460	8	https://doi.org/10.1007/0-387-27539-8	https://doi.org/10.1007/0-387-27539-8	NOUN
ma-174	460	9	1	1	NUM
ma-174	460	10	.	.	PUNCT
ma-174	461	1	introduction	introduction	NOUN
ma-174	461	2	2	2	NUM
ma-174	461	3	.	.	PUNCT
ma-174	461	4	preliminaries	preliminary	NOUN
ma-174	461	5	and	and	CCONJ
ma-174	461	6	definitions	definition	NOUN
ma-174	461	7	3	3	NUM
ma-174	461	8	.	.	PUNCT
ma-174	461	9	generalized	generalize	VERB
ma-174	461	10	bloch	bloch	PROPN
ma-174	461	11	spaces	space	NOUN
ma-174	461	12	of	of	ADP
ma-174	461	13	the	the	DET
ma-174	461	14	upper	upper	ADJ
ma-174	461	15	half	half	ADJ
ma-174	461	16	plane	plane	NOUN
ma-174	461	17	4	4	NUM
ma-174	461	18	.	.	PUNCT
ma-174	461	19	composition	composition	NOUN
ma-174	461	20	semigroups	semigroup	NOUN
ma-174	461	21	on	on	ADP
ma-174	461	22	the	the	DET
ma-174	461	23	generalized	generalized	ADJ
ma-174	461	24	little	little	ADJ
ma-174	461	25	bloch	bloch	NOUN
ma-174	461	26	space	space	NOUN
ma-174	461	27	of	of	ADP
ma-174	461	28	the	the	DET
ma-174	461	29	upper	upper	ADJ
ma-174	461	30	half	half	ADJ
ma-174	461	31	plane	plane	NOUN
ma-174	461	32	4.1	4.1	NUM
ma-174	461	33	.	.	PUNCT
ma-174	462	1	scaling	scale	VERB
ma-174	462	2	group	group	NOUN
ma-174	462	3	4.2	4.2	NUM
ma-174	462	4	.	.	PUNCT
ma-174	463	1	translation	translation	NOUN
ma-174	463	2	group	group	PROPN
ma-174	463	3	5	5	NUM
ma-174	463	4	.	.	PUNCT
ma-174	464	1	rotation	rotation	NOUN
ma-174	464	2	group	group	NOUN
ma-174	464	3	references	reference	NOUN