id	sid	tid	token	lemma	pos
ejpam-6041	1	1	european	european	PROPN
ejpam-6041	1	2	journal	journal	PROPN
ejpam-6041	1	3	of	of	ADP
ejpam-6041	1	4	pure	pure	ADJ
ejpam-6041	1	5	and	and	CCONJ
ejpam-6041	1	6	applied	applied	ADJ
ejpam-6041	1	7	mathematics	mathematic	NOUN
ejpam-6041	1	8	2025	2025	NUM
ejpam-6041	1	9	,	,	PUNCT
ejpam-6041	1	10	vol	vol	NOUN
ejpam-6041	1	11	.	.	PROPN
ejpam-6041	1	12	18	18	NUM
ejpam-6041	1	13	,	,	PUNCT
ejpam-6041	1	14	issue	issue	NOUN
ejpam-6041	1	15	2	2	NUM
ejpam-6041	1	16	,	,	PUNCT
ejpam-6041	1	17	article	article	NOUN
ejpam-6041	1	18	number	number	NOUN
ejpam-6041	1	19	6041	6041	NUM
ejpam-6041	1	20	issn	issn	PROPN
ejpam-6041	1	21	1307	1307	NUM
ejpam-6041	1	22	-	-	SYM
ejpam-6041	1	23	5543	5543	NUM
ejpam-6041	1	24	–	–	PUNCT
ejpam-6041	1	25	ejpam.com	ejpam.com	X
ejpam-6041	1	26	published	publish	VERB
ejpam-6041	1	27	by	by	ADP
ejpam-6041	1	28	new	new	PROPN
ejpam-6041	1	29	york	york	PROPN
ejpam-6041	1	30	business	business	PROPN
ejpam-6041	1	31	global	global	PROPN
ejpam-6041	1	32	the	the	DET
ejpam-6041	1	33	complete	complete	ADJ
ejpam-6041	1	34	list	list	NOUN
ejpam-6041	1	35	of	of	ADP
ejpam-6041	1	36	solutions	solution	NOUN
ejpam-6041	1	37	to	to	ADP
ejpam-6041	1	38	möbius	möbius	PROPN
ejpam-6041	1	39	’s	’s	PART
ejpam-6041	1	40	exponential	exponential	ADJ
ejpam-6041	1	41	equation	equation	NOUN
ejpam-6041	1	42	rasimate	rasimate	VERB
ejpam-6041	1	43	maungchang1	maungchang1	PROPN
ejpam-6041	1	44	,	,	PUNCT
ejpam-6041	1	45	watchareepan	watchareepan	ADJ
ejpam-6041	1	46	atiponrat2	atiponrat2	NOUN
ejpam-6041	1	47	,	,	PUNCT
ejpam-6041	1	48	tarid	tarid	ADJ
ejpam-6041	1	49	suwansri1	suwansri1	NOUN
ejpam-6041	1	50	,	,	PUNCT
ejpam-6041	1	51	jaturon	jaturon	PROPN
ejpam-6041	2	1	wattanapan2,3	wattanapan2,3	PROPN
ejpam-6041	2	2	,	,	PUNCT
ejpam-6041	2	3	teerapong	teerapong	NOUN
ejpam-6041	2	4	suksumran2,∗	suksumran2,∗	VERB
ejpam-6041	2	5	1	1	NUM
ejpam-6041	2	6	school	school	NOUN
ejpam-6041	2	7	of	of	ADP
ejpam-6041	2	8	science	science	NOUN
ejpam-6041	2	9	,	,	PUNCT
ejpam-6041	2	10	walailak	walailak	ADJ
ejpam-6041	2	11	university	university	NOUN
ejpam-6041	2	12	,	,	PUNCT
ejpam-6041	2	13	nakhon	nakhon	PROPN
ejpam-6041	2	14	si	si	PROPN
ejpam-6041	2	15	thammarat	thammarat	PROPN
ejpam-6041	2	16	80160	80160	NUM
ejpam-6041	2	17	,	,	PUNCT
ejpam-6041	2	18	thailand	thailand	PROPN
ejpam-6041	2	19	2	2	NUM
ejpam-6041	2	20	department	department	NOUN
ejpam-6041	2	21	of	of	ADP
ejpam-6041	2	22	mathematics	mathematic	NOUN
ejpam-6041	2	23	,	,	PUNCT
ejpam-6041	2	24	faculty	faculty	NOUN
ejpam-6041	2	25	of	of	ADP
ejpam-6041	2	26	science	science	NOUN
ejpam-6041	2	27	,	,	PUNCT
ejpam-6041	2	28	chiang	chiang	PROPN
ejpam-6041	2	29	mai	mai	PROPN
ejpam-6041	2	30	university	university	PROPN
ejpam-6041	2	31	,	,	PUNCT
ejpam-6041	2	32	chiang	chiang	PROPN
ejpam-6041	2	33	mai	mai	PROPN
ejpam-6041	2	34	50200	50200	NUM
ejpam-6041	2	35	,	,	PUNCT
ejpam-6041	2	36	thailand	thailand	PROPN
ejpam-6041	2	37	3	3	NUM
ejpam-6041	2	38	office	office	NOUN
ejpam-6041	2	39	of	of	ADP
ejpam-6041	2	40	research	research	PROPN
ejpam-6041	2	41	administration	administration	PROPN
ejpam-6041	2	42	,	,	PUNCT
ejpam-6041	2	43	chiang	chiang	PROPN
ejpam-6041	2	44	mai	mai	PROPN
ejpam-6041	2	45	university	university	PROPN
ejpam-6041	2	46	,	,	PUNCT
ejpam-6041	2	47	chiang	chiang	PROPN
ejpam-6041	2	48	mai	mai	PROPN
ejpam-6041	2	49	50200	50200	NUM
ejpam-6041	2	50	,	,	PUNCT
ejpam-6041	2	51	thailand	thailand	PROPN
ejpam-6041	2	52	abstract	abstract	PROPN
ejpam-6041	2	53	.	.	PUNCT
ejpam-6041	3	1	möbius	möbius	PROPN
ejpam-6041	3	2	addition	addition	NOUN
ejpam-6041	3	3	is	be	AUX
ejpam-6041	3	4	a	a	DET
ejpam-6041	3	5	non	non	ADJ
ejpam-6041	3	6	-	-	ADJ
ejpam-6041	3	7	associative	associative	ADJ
ejpam-6041	3	8	binary	binary	ADJ
ejpam-6041	3	9	operation	operation	NOUN
ejpam-6041	3	10	defined	define	VERB
ejpam-6041	3	11	on	on	ADP
ejpam-6041	3	12	the	the	DET
ejpam-6041	3	13	complex	complex	ADJ
ejpam-6041	3	14	open	open	ADJ
ejpam-6041	3	15	unit	unit	NOUN
ejpam-6041	3	16	disk	disk	NOUN
ejpam-6041	3	17	d	d	NOUN
ejpam-6041	3	18	=	=	PUNCT
ejpam-6041	3	19	{	{	PUNCT
ejpam-6041	3	20	z	z	NOUN
ejpam-6041	3	21	∈	∈	PROPN
ejpam-6041	3	22	c	c	NOUN
ejpam-6041	3	23	:	:	PUNCT
ejpam-6041	3	24	|z|	|z|	NOUN
ejpam-6041	3	25	<	<	X
ejpam-6041	3	26	1	1	NUM
ejpam-6041	3	27	}	}	PUNCT
ejpam-6041	3	28	by	by	ADP
ejpam-6041	3	29	a	a	DET
ejpam-6041	3	30	⊕m	⊕m	NOUN
ejpam-6041	3	31	b	b	PROPN
ejpam-6041	3	32	=	=	PRON
ejpam-6041	3	33	a+	a+	PUNCT
ejpam-6041	3	34	b	b	PROPN
ejpam-6041	3	35	1	1	NUM
ejpam-6041	3	36	+	+	NUM
ejpam-6041	3	37	ab	ab	PROPN
ejpam-6041	3	38	,	,	PUNCT
ejpam-6041	3	39	and	and	CCONJ
ejpam-6041	3	40	möbius	möbius	PROPN
ejpam-6041	3	41	’s	’s	PART
ejpam-6041	3	42	exponential	exponential	ADJ
ejpam-6041	3	43	equation	equation	NOUN
ejpam-6041	3	44	is	be	AUX
ejpam-6041	3	45	a	a	DET
ejpam-6041	3	46	non	non	ADJ
ejpam-6041	3	47	-	-	ADJ
ejpam-6041	3	48	linear	linear	ADJ
ejpam-6041	3	49	functional	functional	ADJ
ejpam-6041	3	50	equation	equation	NOUN
ejpam-6041	3	51	of	of	ADP
ejpam-6041	3	52	the	the	DET
ejpam-6041	3	53	form	form	NOUN
ejpam-6041	3	54	l(a	l(a	PROPN
ejpam-6041	3	55	⊕m	⊕m	PROPN
ejpam-6041	3	56	b	b	X
ejpam-6041	3	57	)	)	PUNCT
ejpam-6041	3	58	=	=	SYM
ejpam-6041	3	59	l(a)l(b	l(a)l(b	NOUN
ejpam-6041	3	60	)	)	PUNCT
ejpam-6041	3	61	,	,	PUNCT
ejpam-6041	3	62	where	where	SCONJ
ejpam-6041	3	63	l	l	NOUN
ejpam-6041	3	64	is	be	AUX
ejpam-6041	3	65	a	a	DET
ejpam-6041	3	66	complex	complex	ADV
ejpam-6041	3	67	-	-	PUNCT
ejpam-6041	3	68	valued	value	VERB
ejpam-6041	3	69	function	function	NOUN
ejpam-6041	3	70	defined	define	VERB
ejpam-6041	3	71	on	on	ADP
ejpam-6041	3	72	d.	d.	PROPN
ejpam-6041	3	73	in	in	ADP
ejpam-6041	3	74	[	[	X
ejpam-6041	3	75	aequat	aequat	PROPN
ejpam-6041	3	76	.	.	PUNCT
ejpam-6041	4	1	math	math	NOUN
ejpam-6041	4	2	.	.	PUNCT
ejpam-6041	5	1	91	91	NUM
ejpam-6041	5	2	(	(	PUNCT
ejpam-6041	5	3	2017	2017	NUM
ejpam-6041	5	4	)	)	PUNCT
ejpam-6041	5	5	,	,	PUNCT
ejpam-6041	6	1	491–503	491–503	NUM
ejpam-6041	6	2	]	]	PUNCT
ejpam-6041	6	3	,	,	PUNCT
ejpam-6041	6	4	the	the	DET
ejpam-6041	6	5	authors	author	NOUN
ejpam-6041	6	6	address	address	VERB
ejpam-6041	6	7	the	the	DET
ejpam-6041	6	8	problem	problem	NOUN
ejpam-6041	6	9	of	of	ADP
ejpam-6041	6	10	determining	determine	VERB
ejpam-6041	6	11	the	the	DET
ejpam-6041	6	12	solutions	solution	NOUN
ejpam-6041	6	13	to	to	ADP
ejpam-6041	6	14	möbius	möbius	PROPN
ejpam-6041	6	15	’s	’s	PART
ejpam-6041	6	16	exponential	exponential	ADJ
ejpam-6041	6	17	equation	equation	NOUN
ejpam-6041	6	18	.	.	PUNCT
ejpam-6041	7	1	in	in	ADP
ejpam-6041	7	2	this	this	DET
ejpam-6041	7	3	paper	paper	NOUN
ejpam-6041	7	4	,	,	PUNCT
ejpam-6041	7	5	we	we	PRON
ejpam-6041	7	6	determine	determine	VERB
ejpam-6041	7	7	the	the	DET
ejpam-6041	7	8	complete	complete	ADJ
ejpam-6041	7	9	list	list	NOUN
ejpam-6041	7	10	of	of	ADP
ejpam-6041	7	11	solutions	solution	NOUN
ejpam-6041	7	12	to	to	ADP
ejpam-6041	7	13	möbius	möbius	PROPN
ejpam-6041	7	14	’s	’s	PART
ejpam-6041	7	15	exponential	exponential	ADJ
ejpam-6041	7	16	equation	equation	NOUN
ejpam-6041	7	17	using	use	VERB
ejpam-6041	7	18	an	an	DET
ejpam-6041	7	19	algebraic	algebraic	ADJ
ejpam-6041	7	20	approach	approach	NOUN
ejpam-6041	7	21	.	.	PUNCT
ejpam-6041	8	1	2020	2020	NUM
ejpam-6041	8	2	mathematics	mathematic	NOUN
ejpam-6041	8	3	subject	subject	NOUN
ejpam-6041	8	4	classifications	classification	NOUN
ejpam-6041	8	5	:	:	PUNCT
ejpam-6041	8	6	39b32	39b32	NUM
ejpam-6041	8	7	,	,	PUNCT
ejpam-6041	8	8	30d05	30d05	NUM
ejpam-6041	8	9	,	,	PUNCT
ejpam-6041	8	10	20n05	20n05	NUM
ejpam-6041	8	11	key	key	ADJ
ejpam-6041	8	12	words	word	NOUN
ejpam-6041	8	13	and	and	CCONJ
ejpam-6041	8	14	phrases	phrase	NOUN
ejpam-6041	8	15	:	:	PUNCT
ejpam-6041	8	16	möbius	möbius	PROPN
ejpam-6041	8	17	addition	addition	NOUN
ejpam-6041	8	18	,	,	PUNCT
ejpam-6041	8	19	möbius	möbius	PROPN
ejpam-6041	8	20	’s	’s	PART
ejpam-6041	8	21	exponential	exponential	ADJ
ejpam-6041	8	22	equation	equation	NOUN
ejpam-6041	8	23	,	,	PUNCT
ejpam-6041	8	24	associator	associator	NOUN
ejpam-6041	8	25	function	function	NOUN
ejpam-6041	8	26	,	,	PUNCT
ejpam-6041	8	27	gyrogroup	gyrogroup	PROPN
ejpam-6041	8	28	,	,	PUNCT
ejpam-6041	8	29	non	non	ADJ
ejpam-6041	8	30	-	-	ADJ
ejpam-6041	8	31	linear	linear	ADJ
ejpam-6041	8	32	equation	equation	NOUN
ejpam-6041	8	33	1	1	NUM
ejpam-6041	8	34	.	.	PUNCT
ejpam-6041	9	1	introduction	introduction	NOUN
ejpam-6041	9	2	möbius	möbius	PROPN
ejpam-6041	9	3	addition	addition	NOUN
ejpam-6041	9	4	arises	arise	VERB
ejpam-6041	9	5	naturally	naturally	ADV
ejpam-6041	9	6	in	in	ADP
ejpam-6041	9	7	the	the	DET
ejpam-6041	9	8	study	study	NOUN
ejpam-6041	9	9	of	of	ADP
ejpam-6041	9	10	hyperbolic	hyperbolic	ADJ
ejpam-6041	9	11	geometry	geometry	NOUN
ejpam-6041	9	12	,	,	PUNCT
ejpam-6041	9	13	special	special	ADJ
ejpam-6041	9	14	relativity	relativity	NOUN
ejpam-6041	9	15	,	,	PUNCT
ejpam-6041	9	16	and	and	CCONJ
ejpam-6041	9	17	non	non	ADJ
ejpam-6041	9	18	-	-	ADJ
ejpam-6041	9	19	euclidean	euclidean	ADJ
ejpam-6041	9	20	structures	structure	NOUN
ejpam-6041	9	21	[	[	X
ejpam-6041	9	22	1	1	NUM
ejpam-6041	9	23	,	,	PUNCT
ejpam-6041	9	24	2	2	NUM
ejpam-6041	9	25	]	]	PUNCT
ejpam-6041	9	26	.	.	PUNCT
ejpam-6041	10	1	in	in	ADP
ejpam-6041	10	2	particular	particular	ADJ
ejpam-6041	10	3	,	,	PUNCT
ejpam-6041	10	4	möbius	möbius	PROPN
ejpam-6041	10	5	addition	addition	NOUN
ejpam-6041	10	6	captures	capture	VERB
ejpam-6041	10	7	key	key	ADJ
ejpam-6041	10	8	aspects	aspect	NOUN
ejpam-6041	10	9	of	of	ADP
ejpam-6041	10	10	gyrogroup	gyrogroup	NOUN
ejpam-6041	10	11	structures	structure	NOUN
ejpam-6041	10	12	,	,	PUNCT
ejpam-6041	10	13	which	which	PRON
ejpam-6041	10	14	have	have	VERB
ejpam-6041	10	15	applications	application	NOUN
ejpam-6041	10	16	in	in	ADP
ejpam-6041	10	17	complex	complex	ADJ
ejpam-6041	10	18	analysis	analysis	NOUN
ejpam-6041	10	19	and	and	CCONJ
ejpam-6041	10	20	differential	differential	NOUN
ejpam-6041	10	21	geometry	geometry	NOUN
ejpam-6041	10	22	,	,	PUNCT
ejpam-6041	10	23	for	for	ADP
ejpam-6041	10	24	instance	instance	NOUN
ejpam-6041	10	25	,	,	PUNCT
ejpam-6041	10	26	and	and	CCONJ
ejpam-6041	10	27	serves	serve	VERB
ejpam-6041	10	28	as	as	ADP
ejpam-6041	10	29	a	a	DET
ejpam-6041	10	30	primary	primary	ADJ
ejpam-6041	10	31	motivation	motivation	NOUN
ejpam-6041	10	32	for	for	ADP
ejpam-6041	10	33	the	the	DET
ejpam-6041	10	34	development	development	NOUN
ejpam-6041	10	35	of	of	ADP
ejpam-6041	10	36	gyrogroup	gyrogroup	PROPN
ejpam-6041	10	37	theory	theory	NOUN
ejpam-6041	10	38	.	.	PUNCT
ejpam-6041	11	1	furthermore	furthermore	ADV
ejpam-6041	11	2	,	,	PUNCT
ejpam-6041	11	3	in	in	ADP
ejpam-6041	11	4	optimization	optimization	NOUN
ejpam-6041	11	5	theory	theory	NOUN
ejpam-6041	11	6	,	,	PUNCT
ejpam-6041	11	7	möbius	möbius	PROPN
ejpam-6041	11	8	transformations	transformation	NOUN
ejpam-6041	11	9	and	and	CCONJ
ejpam-6041	11	10	möbius	möbius	NOUN
ejpam-6041	11	11	addition	addition	NOUN
ejpam-6041	11	12	on	on	ADP
ejpam-6041	11	13	the	the	DET
ejpam-6041	11	14	complex	complex	ADJ
ejpam-6041	11	15	open	open	ADJ
ejpam-6041	11	16	unit	unit	NOUN
ejpam-6041	11	17	disk	disk	NOUN
ejpam-6041	11	18	provide	provide	NOUN
ejpam-6041	11	19	tools	tool	NOUN
ejpam-6041	11	20	to	to	PART
ejpam-6041	11	21	formulate	formulate	VERB
ejpam-6041	11	22	and	and	CCONJ
ejpam-6041	11	23	solve	solve	VERB
ejpam-6041	11	24	problems	problem	NOUN
ejpam-6041	11	25	in	in	ADP
ejpam-6041	11	26	non	non	ADJ
ejpam-6041	11	27	-	-	ADJ
ejpam-6041	11	28	euclidean	euclidean	ADJ
ejpam-6041	11	29	domains	domain	NOUN
ejpam-6041	11	30	,	,	PUNCT
ejpam-6041	11	31	model	model	NOUN
ejpam-6041	11	32	non	non	ADJ
ejpam-6041	11	33	-	-	ADJ
ejpam-6041	11	34	associative	associative	ADJ
ejpam-6041	11	35	dynamics	dynamic	NOUN
ejpam-6041	11	36	,	,	PUNCT
ejpam-6041	11	37	enforce	enforce	VERB
ejpam-6041	11	38	multiplicative	multiplicative	ADJ
ejpam-6041	11	39	structure	structure	NOUN
ejpam-6041	11	40	constraints	constraint	NOUN
ejpam-6041	11	41	,	,	PUNCT
ejpam-6041	11	42	and	and	CCONJ
ejpam-6041	11	43	operate	operate	VERB
ejpam-6041	11	44	within	within	ADP
ejpam-6041	11	45	compact	compact	ADJ
ejpam-6041	11	46	curved	curved	ADJ
ejpam-6041	11	47	spaces	space	NOUN
ejpam-6041	11	48	.	.	PUNCT
ejpam-6041	12	1	these	these	DET
ejpam-6041	12	2	frameworks	framework	NOUN
ejpam-6041	12	3	are	be	AUX
ejpam-6041	12	4	particularly	particularly	ADV
ejpam-6041	12	5	useful	useful	ADJ
ejpam-6041	12	6	for	for	ADP
ejpam-6041	12	7	modern	modern	ADJ
ejpam-6041	12	8	applications	application	NOUN
ejpam-6041	12	9	in	in	ADP
ejpam-6041	12	10	machine	machine	NOUN
ejpam-6041	12	11	learning	learning	NOUN
ejpam-6041	12	12	,	,	PUNCT
ejpam-6041	12	13	signal	signal	ADJ
ejpam-6041	12	14	processing	processing	NOUN
ejpam-6041	12	15	,	,	PUNCT
ejpam-6041	12	16	and	and	CCONJ
ejpam-6041	12	17	geometric	geometric	ADJ
ejpam-6041	12	18	optimization	optimization	NOUN
ejpam-6041	12	19	,	,	PUNCT
ejpam-6041	12	20	where	where	SCONJ
ejpam-6041	12	21	classical	classical	ADJ
ejpam-6041	12	22	euclidean	euclidean	ADJ
ejpam-6041	12	23	assumptions	assumption	NOUN
ejpam-6041	12	24	are	be	AUX
ejpam-6041	12	25	insufficient	insufficient	ADJ
ejpam-6041	12	26	.	.	PUNCT
ejpam-6041	13	1	∗corresponding	∗corresponde	VERB
ejpam-6041	13	2	author	author	NOUN
ejpam-6041	13	3	.	.	PUNCT
ejpam-6041	14	1	doi	doi	NOUN
ejpam-6041	14	2	:	:	PUNCT
ejpam-6041	14	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6041	https://doi.org/10.29020/nybg.ejpam.v18i2.6041	VERB
ejpam-6041	14	4	email	email	NOUN
ejpam-6041	14	5	addresses	address	NOUN
ejpam-6041	14	6	:	:	PUNCT
ejpam-6041	14	7	mate105@gmail.com	mate105@gmail.com	X
ejpam-6041	14	8	(	(	PUNCT
ejpam-6041	14	9	r.	r.	PROPN
ejpam-6041	14	10	maungchang	maungchang	PROPN
ejpam-6041	14	11	)	)	PUNCT
ejpam-6041	14	12	,	,	PUNCT
ejpam-6041	14	13	watchareepan.a@cmu.ac.th	watchareepan.a@cmu.ac.th	PROPN
ejpam-6041	14	14	(	(	PUNCT
ejpam-6041	14	15	w.	w.	PROPN
ejpam-6041	14	16	atiponrat	atiponrat	PROPN
ejpam-6041	14	17	)	)	PUNCT
ejpam-6041	14	18	,	,	PUNCT
ejpam-6041	14	19	tarid.su@mail.wu.ac.th	tarid.su@mail.wu.ac.th	PROPN
ejpam-6041	14	20	(	(	PUNCT
ejpam-6041	14	21	t.	t.	PROPN
ejpam-6041	14	22	suwansri	suwansri	PROPN
ejpam-6041	14	23	)	)	PUNCT
ejpam-6041	14	24	,	,	PUNCT
ejpam-6041	15	1	jaturon.w@cmu.ac.th	jaturon.w@cmu.ac.th	PROPN
ejpam-6041	15	2	(	(	PUNCT
ejpam-6041	15	3	j.	j.	PROPN
ejpam-6041	15	4	wattanapan	wattanapan	PROPN
ejpam-6041	15	5	)	)	PUNCT
ejpam-6041	15	6	,	,	PUNCT
ejpam-6041	15	7	teerapong.suksumran@cmu.ac.th	teerapong.suksumran@cmu.ac.th	PROPN
ejpam-6041	15	8	(	(	PUNCT
ejpam-6041	15	9	t.	t.	PROPN
ejpam-6041	15	10	suksumran	suksumran	PROPN
ejpam-6041	15	11	)	)	PUNCT
ejpam-6041	15	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6041	16	1	1	1	NUM
ejpam-6041	16	2	copyright	copyright	NOUN
ejpam-6041	16	3	:	:	PUNCT
ejpam-6041	16	4	©	©	PROPN
ejpam-6041	16	5	2025	2025	NUM
ejpam-6041	16	6	the	the	DET
ejpam-6041	16	7	author(s	author(s	NOUN
ejpam-6041	16	8	)	)	PUNCT
ejpam-6041	16	9	.	.	PUNCT
ejpam-6041	17	1	(	(	PUNCT
ejpam-6041	17	2	cc	cc	NOUN
ejpam-6041	17	3	by	by	ADP
ejpam-6041	17	4	-	-	PUNCT
ejpam-6041	17	5	nc	nc	PROPN
ejpam-6041	17	6	4.0	4.0	NUM
ejpam-6041	17	7	)	)	PUNCT
ejpam-6041	17	8	r.	r.	PROPN
ejpam-6041	17	9	maungchang	maungchang	PROPN
ejpam-6041	17	10	et	et	PROPN
ejpam-6041	17	11	al	al	PROPN
ejpam-6041	17	12	.	.	PUNCT
ejpam-6041	17	13	/	/	SYM
ejpam-6041	17	14	eur	eur	PROPN
ejpam-6041	17	15	.	.	PUNCT
ejpam-6041	18	1	j.	j.	PROPN
ejpam-6041	18	2	pure	pure	PROPN
ejpam-6041	18	3	appl	appl	PROPN
ejpam-6041	18	4	.	.	PROPN
ejpam-6041	18	5	math	math	PROPN
ejpam-6041	18	6	,	,	PUNCT
ejpam-6041	18	7	18	18	NUM
ejpam-6041	18	8	(	(	PUNCT
ejpam-6041	18	9	2	2	NUM
ejpam-6041	18	10	)	)	PUNCT
ejpam-6041	18	11	(	(	PUNCT
ejpam-6041	18	12	2025	2025	NUM
ejpam-6041	18	13	)	)	PUNCT
ejpam-6041	18	14	,	,	PUNCT
ejpam-6041	18	15	6041	6041	NUM
ejpam-6041	18	16	2	2	NUM
ejpam-6041	18	17	of	of	ADP
ejpam-6041	18	18	8	8	NUM
ejpam-6041	18	19	recall	recall	NOUN
ejpam-6041	18	20	that	that	PRON
ejpam-6041	18	21	möbius	möbius	PROPN
ejpam-6041	18	22	addition	addition	NOUN
ejpam-6041	18	23	is	be	AUX
ejpam-6041	18	24	a	a	DET
ejpam-6041	18	25	non	non	ADJ
ejpam-6041	18	26	-	-	ADJ
ejpam-6041	18	27	associative	associative	ADJ
ejpam-6041	18	28	binary	binary	ADJ
ejpam-6041	18	29	operation	operation	NOUN
ejpam-6041	18	30	defined	define	VERB
ejpam-6041	18	31	on	on	ADP
ejpam-6041	18	32	the	the	DET
ejpam-6041	18	33	complex	complex	ADJ
ejpam-6041	18	34	open	open	ADJ
ejpam-6041	18	35	unit	unit	NOUN
ejpam-6041	18	36	disk	disk	NOUN
ejpam-6041	18	37	d	d	NOUN
ejpam-6041	18	38	=	=	PUNCT
ejpam-6041	18	39	{	{	PUNCT
ejpam-6041	18	40	z	z	NOUN
ejpam-6041	18	41	∈	∈	PROPN
ejpam-6041	18	42	c	c	NOUN
ejpam-6041	18	43	:	:	PUNCT
ejpam-6041	18	44	|z|	|z|	NOUN
ejpam-6041	18	45	<	<	X
ejpam-6041	18	46	1	1	NUM
ejpam-6041	18	47	}	}	PUNCT
ejpam-6041	18	48	by	by	ADP
ejpam-6041	18	49	the	the	DET
ejpam-6041	18	50	formula	formula	NOUN
ejpam-6041	18	51	a⊕m	a⊕m	PROPN
ejpam-6041	18	52	b	b	PROPN
ejpam-6041	18	53	=	=	SYM
ejpam-6041	18	54	a+	a+	PUNCT
ejpam-6041	18	55	b	b	PROPN
ejpam-6041	18	56	1	1	NUM
ejpam-6041	18	57	+	+	NUM
ejpam-6041	18	58	ab	ab	PROPN
ejpam-6041	18	59	(	(	PUNCT
ejpam-6041	18	60	1	1	NUM
ejpam-6041	18	61	)	)	PUNCT
ejpam-6041	18	62	for	for	ADP
ejpam-6041	18	63	all	all	DET
ejpam-6041	18	64	a	a	DET
ejpam-6041	18	65	,	,	PUNCT
ejpam-6041	18	66	b	b	X
ejpam-6041	18	67	∈	∈	PROPN
ejpam-6041	18	68	d.	d.	NOUN
ejpam-6041	18	69	it	it	PRON
ejpam-6041	18	70	turns	turn	VERB
ejpam-6041	18	71	out	out	ADP
ejpam-6041	18	72	that	that	SCONJ
ejpam-6041	18	73	(	(	PUNCT
ejpam-6041	18	74	d,⊕m	d,⊕m	X
ejpam-6041	18	75	)	)	PUNCT
ejpam-6041	18	76	forms	form	VERB
ejpam-6041	18	77	a	a	DET
ejpam-6041	18	78	non	non	ADJ
ejpam-6041	18	79	-	-	ADJ
ejpam-6041	18	80	associative	associative	ADJ
ejpam-6041	18	81	group	group	NOUN
ejpam-6041	18	82	-	-	PUNCT
ejpam-6041	18	83	like	like	ADJ
ejpam-6041	18	84	structure	structure	NOUN
ejpam-6041	18	85	that	that	PRON
ejpam-6041	18	86	shares	share	VERB
ejpam-6041	18	87	several	several	ADJ
ejpam-6041	18	88	common	common	ADJ
ejpam-6041	18	89	properties	property	NOUN
ejpam-6041	18	90	with	with	ADP
ejpam-6041	18	91	groups	group	NOUN
ejpam-6041	18	92	.	.	PUNCT
ejpam-6041	19	1	in	in	ADP
ejpam-6041	19	2	particular	particular	ADJ
ejpam-6041	19	3	,	,	PUNCT
ejpam-6041	19	4	a⊕m	a⊕m	PROPN
ejpam-6041	19	5	b	b	PROPN
ejpam-6041	19	6	belongs	belong	VERB
ejpam-6041	19	7	to	to	ADP
ejpam-6041	19	8	d	d	PROPN
ejpam-6041	19	9	for	for	ADP
ejpam-6041	19	10	all	all	DET
ejpam-6041	19	11	a	a	PRON
ejpam-6041	19	12	,	,	PUNCT
ejpam-6041	19	13	b	b	X
ejpam-6041	19	14	∈	∈	PROPN
ejpam-6041	19	15	d.	d.	NOUN
ejpam-6041	19	16	in	in	ADP
ejpam-6041	19	17	[	[	X
ejpam-6041	19	18	3	3	NUM
ejpam-6041	19	19	]	]	PUNCT
ejpam-6041	19	20	,	,	PUNCT
ejpam-6041	19	21	the	the	DET
ejpam-6041	19	22	authors	author	NOUN
ejpam-6041	19	23	introduce	introduce	VERB
ejpam-6041	19	24	a	a	DET
ejpam-6041	19	25	functional	functional	ADJ
ejpam-6041	19	26	equation	equation	NOUN
ejpam-6041	19	27	in	in	ADP
ejpam-6041	19	28	connection	connection	NOUN
ejpam-6041	19	29	with	with	ADP
ejpam-6041	19	30	schur	schur	PROPN
ejpam-6041	19	31	’s	’s	PART
ejpam-6041	19	32	lemma	lemma	PROPN
ejpam-6041	19	33	of	of	ADP
ejpam-6041	19	34	the	the	DET
ejpam-6041	19	35	form	form	NOUN
ejpam-6041	19	36	l(a⊕m	l(a⊕m	PROPN
ejpam-6041	19	37	b	b	PROPN
ejpam-6041	19	38	)	)	PUNCT
ejpam-6041	19	39	=	=	SYM
ejpam-6041	19	40	l(a)l(b	l(a)l(b	NOUN
ejpam-6041	19	41	)	)	PUNCT
ejpam-6041	19	42	(	(	PUNCT
ejpam-6041	19	43	me	i	PRON
ejpam-6041	19	44	)	)	PUNCT
ejpam-6041	19	45	called	call	VERB
ejpam-6041	19	46	möbius	möbius	NOUN
ejpam-6041	19	47	’s	’s	PART
ejpam-6041	19	48	exponential	exponential	ADJ
ejpam-6041	19	49	equation	equation	NOUN
ejpam-6041	19	50	,	,	PUNCT
ejpam-6041	19	51	which	which	PRON
ejpam-6041	19	52	is	be	AUX
ejpam-6041	19	53	in	in	ADP
ejpam-6041	19	54	some	some	DET
ejpam-6041	19	55	sense	sense	NOUN
ejpam-6041	19	56	an	an	DET
ejpam-6041	19	57	extension	extension	NOUN
ejpam-6041	19	58	of	of	ADP
ejpam-6041	19	59	cauchy	cauchy	PROPN
ejpam-6041	19	60	’s	’s	PART
ejpam-6041	19	61	exponential	exponential	ADJ
ejpam-6041	19	62	equation	equation	NOUN
ejpam-6041	19	63	,	,	PUNCT
ejpam-6041	19	64	where	where	SCONJ
ejpam-6041	19	65	l	l	NOUN
ejpam-6041	19	66	is	be	AUX
ejpam-6041	19	67	a	a	DET
ejpam-6041	19	68	complex	complex	ADV
ejpam-6041	19	69	-	-	PUNCT
ejpam-6041	19	70	valued	value	VERB
ejpam-6041	19	71	function	function	NOUN
ejpam-6041	19	72	defined	define	VERB
ejpam-6041	19	73	on	on	ADP
ejpam-6041	19	74	d.	d.	PROPN
ejpam-6041	19	75	they	they	PRON
ejpam-6041	19	76	also	also	ADV
ejpam-6041	19	77	give	give	VERB
ejpam-6041	19	78	partial	partial	ADJ
ejpam-6041	19	79	solutions	solution	NOUN
ejpam-6041	19	80	on	on	ADP
ejpam-6041	19	81	the	the	DET
ejpam-6041	19	82	open	open	ADJ
ejpam-6041	19	83	interval	interval	NOUN
ejpam-6041	19	84	(	(	PUNCT
ejpam-6041	19	85	−1	−1	NOUN
ejpam-6041	19	86	,	,	PUNCT
ejpam-6041	19	87	1	1	NUM
ejpam-6041	19	88	)	)	PUNCT
ejpam-6041	19	89	and	and	CCONJ
ejpam-6041	19	90	then	then	ADV
ejpam-6041	19	91	address	address	VERB
ejpam-6041	19	92	the	the	DET
ejpam-6041	19	93	problem	problem	NOUN
ejpam-6041	19	94	of	of	ADP
ejpam-6041	19	95	determining	determine	VERB
ejpam-6041	19	96	the	the	DET
ejpam-6041	19	97	complete	complete	ADJ
ejpam-6041	19	98	solutions	solution	NOUN
ejpam-6041	19	99	to	to	ADP
ejpam-6041	19	100	möbius	möbius	PROPN
ejpam-6041	19	101	’s	’s	PART
ejpam-6041	19	102	exponential	exponential	ADJ
ejpam-6041	19	103	equation	equation	NOUN
ejpam-6041	19	104	on	on	ADP
ejpam-6041	19	105	the	the	DET
ejpam-6041	19	106	whole	whole	ADJ
ejpam-6041	19	107	disk	disk	NOUN
ejpam-6041	19	108	(	(	PUNCT
ejpam-6041	19	109	see	see	VERB
ejpam-6041	19	110	problem	problem	NOUN
ejpam-6041	19	111	1.3	1.3	NUM
ejpam-6041	19	112	of	of	ADP
ejpam-6041	19	113	[	[	X
ejpam-6041	19	114	3	3	NUM
ejpam-6041	19	115	]	]	NUM
ejpam-6041	19	116	)	)	PUNCT
ejpam-6041	19	117	.	.	PUNCT
ejpam-6041	20	1	actually	actually	ADV
ejpam-6041	20	2	,	,	PUNCT
ejpam-6041	20	3	möbius	möbius	PROPN
ejpam-6041	20	4	’s	’s	PART
ejpam-6041	20	5	exponential	exponential	ADJ
ejpam-6041	20	6	equation	equation	NOUN
ejpam-6041	20	7	arises	arise	VERB
ejpam-6041	20	8	when	when	SCONJ
ejpam-6041	20	9	one	one	NUM
ejpam-6041	20	10	attempts	attempt	VERB
ejpam-6041	20	11	to	to	PART
ejpam-6041	20	12	determine	determine	VERB
ejpam-6041	20	13	the	the	DET
ejpam-6041	20	14	irreducible	irreducible	ADJ
ejpam-6041	20	15	linear	linear	ADJ
ejpam-6041	20	16	representations	representation	NOUN
ejpam-6041	20	17	of	of	ADP
ejpam-6041	20	18	(	(	PUNCT
ejpam-6041	20	19	d,⊕m	d,⊕m	X
ejpam-6041	20	20	)	)	PUNCT
ejpam-6041	20	21	,	,	PUNCT
ejpam-6041	20	22	as	as	SCONJ
ejpam-6041	20	23	shown	show	VERB
ejpam-6041	20	24	in	in	ADP
ejpam-6041	20	25	section	section	NOUN
ejpam-6041	20	26	3.2	3.2	NUM
ejpam-6041	20	27	of	of	ADP
ejpam-6041	20	28	[	[	X
ejpam-6041	20	29	3	3	NUM
ejpam-6041	20	30	]	]	PUNCT
ejpam-6041	20	31	.	.	PUNCT
ejpam-6041	21	1	in	in	ADP
ejpam-6041	21	2	this	this	DET
ejpam-6041	21	3	paper	paper	NOUN
ejpam-6041	21	4	,	,	PUNCT
ejpam-6041	21	5	we	we	PRON
ejpam-6041	21	6	give	give	VERB
ejpam-6041	21	7	the	the	DET
ejpam-6041	21	8	complete	complete	ADJ
ejpam-6041	21	9	list	list	NOUN
ejpam-6041	21	10	of	of	ADP
ejpam-6041	21	11	solutions	solution	NOUN
ejpam-6041	21	12	to	to	ADP
ejpam-6041	21	13	möbius	möbius	PROPN
ejpam-6041	21	14	’s	’s	PART
ejpam-6041	21	15	exponential	exponential	ADJ
ejpam-6041	21	16	equation	equation	NOUN
ejpam-6041	21	17	using	use	VERB
ejpam-6041	21	18	the	the	DET
ejpam-6041	21	19	recent	recent	ADJ
ejpam-6041	21	20	notion	notion	NOUN
ejpam-6041	21	21	of	of	ADP
ejpam-6041	21	22	associators	associator	NOUN
ejpam-6041	21	23	formulated	formulate	VERB
ejpam-6041	21	24	in	in	ADP
ejpam-6041	21	25	[	[	X
ejpam-6041	21	26	4	4	NUM
ejpam-6041	21	27	]	]	PUNCT
ejpam-6041	21	28	.	.	PUNCT
ejpam-6041	22	1	2	2	X
ejpam-6041	22	2	.	.	X
ejpam-6041	22	3	the	the	DET
ejpam-6041	22	4	solutions	solution	NOUN
ejpam-6041	22	5	to	to	ADP
ejpam-6041	22	6	möbius	möbius	PROPN
ejpam-6041	22	7	’s	’s	PART
ejpam-6041	22	8	exponential	exponential	ADJ
ejpam-6041	22	9	equation	equation	NOUN
ejpam-6041	22	10	suppose	suppose	VERB
ejpam-6041	22	11	that	that	SCONJ
ejpam-6041	22	12	l	l	NOUN
ejpam-6041	22	13	is	be	AUX
ejpam-6041	22	14	a	a	DET
ejpam-6041	22	15	complex	complex	ADV
ejpam-6041	22	16	-	-	PUNCT
ejpam-6041	22	17	valued	value	VERB
ejpam-6041	22	18	function	function	NOUN
ejpam-6041	22	19	defined	define	VERB
ejpam-6041	22	20	on	on	ADP
ejpam-6041	22	21	the	the	DET
ejpam-6041	22	22	disk	disk	NOUN
ejpam-6041	22	23	d.	d.	NOUN
ejpam-6041	22	24	as	as	SCONJ
ejpam-6041	22	25	noted	note	VERB
ejpam-6041	22	26	on	on	ADP
ejpam-6041	22	27	page	page	NOUN
ejpam-6041	22	28	492	492	NUM
ejpam-6041	22	29	of	of	ADP
ejpam-6041	22	30	[	[	X
ejpam-6041	22	31	3	3	NUM
ejpam-6041	22	32	]	]	PUNCT
ejpam-6041	22	33	,	,	PUNCT
ejpam-6041	22	34	if	if	SCONJ
ejpam-6041	22	35	l	l	NOUN
ejpam-6041	22	36	satisfies	satisfy	VERB
ejpam-6041	22	37	möbius	möbius	PROPN
ejpam-6041	22	38	’s	’s	PART
ejpam-6041	22	39	exponential	exponential	ADJ
ejpam-6041	22	40	equation	equation	NOUN
ejpam-6041	22	41	,	,	PUNCT
ejpam-6041	22	42	then	then	ADV
ejpam-6041	22	43	l	l	NOUN
ejpam-6041	22	44	is	be	AUX
ejpam-6041	22	45	either	either	CCONJ
ejpam-6041	22	46	everywhere	everywhere	ADV
ejpam-6041	22	47	or	or	CCONJ
ejpam-6041	22	48	nowhere	nowhere	ADV
ejpam-6041	22	49	zero	zero	NUM
ejpam-6041	22	50	.	.	PUNCT
ejpam-6041	23	1	furthermore	furthermore	ADV
ejpam-6041	23	2	,	,	PUNCT
ejpam-6041	23	3	if	if	SCONJ
ejpam-6041	23	4	l	l	NOUN
ejpam-6041	23	5	is	be	AUX
ejpam-6041	23	6	assumed	assume	VERB
ejpam-6041	23	7	to	to	PART
ejpam-6041	23	8	be	be	AUX
ejpam-6041	23	9	a	a	DET
ejpam-6041	23	10	borel	borel	NOUN
ejpam-6041	23	11	measurable	measurable	ADJ
ejpam-6041	23	12	function	function	NOUN
ejpam-6041	23	13	,	,	PUNCT
ejpam-6041	23	14	then	then	ADV
ejpam-6041	23	15	either	either	CCONJ
ejpam-6041	23	16	l	l	NOUN
ejpam-6041	23	17	is	be	AUX
ejpam-6041	23	18	the	the	DET
ejpam-6041	23	19	zero	zero	NUM
ejpam-6041	23	20	function	function	NOUN
ejpam-6041	23	21	or	or	CCONJ
ejpam-6041	23	22	there	there	PRON
ejpam-6041	23	23	exists	exist	VERB
ejpam-6041	23	24	a	a	DET
ejpam-6041	23	25	complex	complex	ADJ
ejpam-6041	23	26	constant	constant	ADJ
ejpam-6041	23	27	λ	λ	NOUN
ejpam-6041	23	28	for	for	ADP
ejpam-6041	23	29	which	which	PRON
ejpam-6041	23	30	l(r	l(r	PROPN
ejpam-6041	23	31	)	)	PUNCT
ejpam-6041	23	32	=	=	SYM
ejpam-6041	23	33	eλ	eλ	ADP
ejpam-6041	24	1	tanh−1	tanh−1	NOUN
ejpam-6041	24	2	r	r	NOUN
ejpam-6041	24	3	for	for	ADP
ejpam-6041	24	4	all	all	DET
ejpam-6041	24	5	r	r	NOUN
ejpam-6041	24	6	∈	∈	PROPN
ejpam-6041	24	7	(	(	PUNCT
ejpam-6041	24	8	−1	−1	NOUN
ejpam-6041	24	9	,	,	PUNCT
ejpam-6041	24	10	1	1	NUM
ejpam-6041	24	11	)	)	PUNCT
ejpam-6041	24	12	(	(	PUNCT
ejpam-6041	24	13	see	see	VERB
ejpam-6041	24	14	theorem	theorem	ADJ
ejpam-6041	24	15	1.2	1.2	NUM
ejpam-6041	24	16	of	of	ADP
ejpam-6041	24	17	[	[	X
ejpam-6041	24	18	3	3	NUM
ejpam-6041	24	19	]	]	NUM
ejpam-6041	24	20	)	)	PUNCT
ejpam-6041	24	21	.	.	PUNCT
ejpam-6041	25	1	the	the	DET
ejpam-6041	25	2	authors	author	NOUN
ejpam-6041	25	3	of	of	ADP
ejpam-6041	25	4	[	[	X
ejpam-6041	25	5	3	3	NUM
ejpam-6041	25	6	]	]	PUNCT
ejpam-6041	25	7	establish	establish	VERB
ejpam-6041	25	8	that	that	SCONJ
ejpam-6041	25	9	möbius	möbius	PROPN
ejpam-6041	25	10	’s	’s	PART
ejpam-6041	25	11	exponential	exponential	ADJ
ejpam-6041	25	12	equation	equation	NOUN
ejpam-6041	25	13	does	do	AUX
ejpam-6041	25	14	not	not	PART
ejpam-6041	25	15	have	have	VERB
ejpam-6041	25	16	any	any	DET
ejpam-6041	25	17	non	non	ADJ
ejpam-6041	25	18	-	-	ADJ
ejpam-6041	25	19	trivial	trivial	ADJ
ejpam-6041	25	20	holomorphic	holomorphic	ADJ
ejpam-6041	25	21	solutions	solution	NOUN
ejpam-6041	25	22	using	use	VERB
ejpam-6041	25	23	the	the	DET
ejpam-6041	25	24	open	open	ADJ
ejpam-6041	25	25	mapping	mapping	NOUN
ejpam-6041	25	26	theorem	theorem	VERB
ejpam-6041	25	27	in	in	ADP
ejpam-6041	25	28	complex	complex	ADJ
ejpam-6041	25	29	analysis	analysis	NOUN
ejpam-6041	25	30	.	.	PUNCT
ejpam-6041	26	1	here	here	ADV
ejpam-6041	26	2	,	,	PUNCT
ejpam-6041	26	3	we	we	PRON
ejpam-6041	26	4	will	will	AUX
ejpam-6041	26	5	apply	apply	VERB
ejpam-6041	26	6	an	an	DET
ejpam-6041	26	7	algebraic	algebraic	ADJ
ejpam-6041	26	8	tool	tool	NOUN
ejpam-6041	26	9	,	,	PUNCT
ejpam-6041	26	10	together	together	ADV
ejpam-6041	26	11	with	with	ADP
ejpam-6041	26	12	the	the	DET
ejpam-6041	26	13	well	well	ADV
ejpam-6041	26	14	-	-	PUNCT
ejpam-6041	26	15	known	know	VERB
ejpam-6041	26	16	intermediate	intermediate	ADJ
ejpam-6041	26	17	value	value	NOUN
ejpam-6041	26	18	theorem	theorem	VERB
ejpam-6041	26	19	,	,	PUNCT
ejpam-6041	26	20	to	to	PART
ejpam-6041	26	21	prove	prove	VERB
ejpam-6041	26	22	the	the	DET
ejpam-6041	26	23	following	follow	VERB
ejpam-6041	26	24	theorem	theorem	VERB
ejpam-6041	26	25	.	.	PUNCT
ejpam-6041	26	26	theorem	theorem	NOUN
ejpam-6041	26	27	1	1	NUM
ejpam-6041	26	28	.	.	PUNCT
ejpam-6041	27	1	the	the	DET
ejpam-6041	27	2	solutions	solution	NOUN
ejpam-6041	27	3	to	to	ADP
ejpam-6041	27	4	möbius	möbius	PROPN
ejpam-6041	27	5	’s	’s	PART
ejpam-6041	27	6	exponential	exponential	ADJ
ejpam-6041	27	7	equation	equation	NOUN
ejpam-6041	27	8	are	be	AUX
ejpam-6041	27	9	only	only	ADV
ejpam-6041	27	10	the	the	DET
ejpam-6041	27	11	functions	function	NOUN
ejpam-6041	27	12	l	l	NOUN
ejpam-6041	27	13	:	:	PUNCT
ejpam-6041	27	14	z	z	X
ejpam-6041	27	15	7→	7→	NUM
ejpam-6041	27	16	0	0	NUM
ejpam-6041	27	17	and	and	CCONJ
ejpam-6041	27	18	l	l	NOUN
ejpam-6041	27	19	:	:	PUNCT
ejpam-6041	28	1	z	z	X
ejpam-6041	28	2	7→	7→	NUM
ejpam-6041	28	3	1	1	NUM
ejpam-6041	28	4	,	,	PUNCT
ejpam-6041	28	5	z	z	PROPN
ejpam-6041	28	6	∈	∈	PROPN
ejpam-6041	28	7	d.	d.	PROPN
ejpam-6041	28	8	theorem	theorem	VERB
ejpam-6041	28	9	1	1	NUM
ejpam-6041	28	10	states	state	NOUN
ejpam-6041	28	11	that	that	PRON
ejpam-6041	28	12	möbius	möbius	PROPN
ejpam-6041	28	13	’s	’s	PART
ejpam-6041	28	14	exponential	exponential	ADJ
ejpam-6041	28	15	equation	equation	NOUN
ejpam-6041	28	16	does	do	AUX
ejpam-6041	28	17	not	not	PART
ejpam-6041	28	18	have	have	VERB
ejpam-6041	28	19	any	any	DET
ejpam-6041	28	20	non	non	ADJ
ejpam-6041	28	21	-	-	ADJ
ejpam-6041	28	22	trivial	trivial	ADJ
ejpam-6041	28	23	solutions	solution	NOUN
ejpam-6041	28	24	,	,	PUNCT
ejpam-6041	28	25	which	which	PRON
ejpam-6041	28	26	refines	refine	VERB
ejpam-6041	28	27	the	the	DET
ejpam-6041	28	28	result	result	NOUN
ejpam-6041	28	29	in	in	ADP
ejpam-6041	28	30	[	[	X
ejpam-6041	28	31	3	3	NUM
ejpam-6041	28	32	]	]	PUNCT
ejpam-6041	28	33	as	as	SCONJ
ejpam-6041	28	34	the	the	DET
ejpam-6041	28	35	condition	condition	NOUN
ejpam-6041	28	36	of	of	ADP
ejpam-6041	28	37	being	be	AUX
ejpam-6041	28	38	a	a	DET
ejpam-6041	28	39	holomorphic	holomorphic	ADJ
ejpam-6041	28	40	function	function	NOUN
ejpam-6041	28	41	is	be	AUX
ejpam-6041	28	42	removed	remove	VERB
ejpam-6041	28	43	.	.	PUNCT
ejpam-6041	29	1	in	in	ADP
ejpam-6041	29	2	particular	particular	ADJ
ejpam-6041	29	3	,	,	PUNCT
ejpam-6041	29	4	this	this	PRON
ejpam-6041	29	5	implies	imply	VERB
ejpam-6041	29	6	that	that	SCONJ
ejpam-6041	29	7	the	the	DET
ejpam-6041	29	8	disk	disk	NOUN
ejpam-6041	29	9	d	d	NOUN
ejpam-6041	29	10	,	,	PUNCT
ejpam-6041	29	11	endowed	endow	VERB
ejpam-6041	29	12	with	with	ADP
ejpam-6041	29	13	möbius	möbius	PROPN
ejpam-6041	29	14	addition	addition	NOUN
ejpam-6041	29	15	⊕m	⊕m	NOUN
ejpam-6041	29	16	,	,	PUNCT
ejpam-6041	29	17	has	have	VERB
ejpam-6041	29	18	no	no	DET
ejpam-6041	29	19	non	non	ADJ
ejpam-6041	29	20	-	-	ADJ
ejpam-6041	29	21	trivial	trivial	ADJ
ejpam-6041	29	22	representations	representation	NOUN
ejpam-6041	29	23	(	(	PUNCT
ejpam-6041	29	24	see	see	VERB
ejpam-6041	29	25	the	the	DET
ejpam-6041	29	26	remark	remark	NOUN
ejpam-6041	29	27	at	at	ADP
ejpam-6041	29	28	the	the	DET
ejpam-6041	29	29	end	end	NOUN
ejpam-6041	29	30	of	of	ADP
ejpam-6041	29	31	[	[	X
ejpam-6041	29	32	3	3	X
ejpam-6041	29	33	]	]	PUNCT
ejpam-6041	29	34	for	for	ADP
ejpam-6041	29	35	more	more	ADJ
ejpam-6041	29	36	details	detail	NOUN
ejpam-6041	29	37	)	)	PUNCT
ejpam-6041	29	38	.	.	PUNCT
ejpam-6041	30	1	the	the	DET
ejpam-6041	30	2	proof	proof	NOUN
ejpam-6041	30	3	of	of	ADP
ejpam-6041	30	4	theorem	theorem	ADJ
ejpam-6041	30	5	1	1	NUM
ejpam-6041	30	6	is	be	AUX
ejpam-6041	30	7	presented	present	VERB
ejpam-6041	30	8	in	in	ADP
ejpam-6041	30	9	the	the	DET
ejpam-6041	30	10	next	next	ADJ
ejpam-6041	30	11	section	section	NOUN
ejpam-6041	30	12	for	for	ADP
ejpam-6041	30	13	convenience	convenience	NOUN
ejpam-6041	30	14	.	.	PUNCT
ejpam-6041	31	1	r.	r.	PROPN
ejpam-6041	31	2	maungchang	maungchang	PROPN
ejpam-6041	31	3	et	et	PROPN
ejpam-6041	31	4	al	al	PROPN
ejpam-6041	31	5	.	.	PUNCT
ejpam-6041	31	6	/	/	SYM
ejpam-6041	31	7	eur	eur	PROPN
ejpam-6041	31	8	.	.	PUNCT
ejpam-6041	32	1	j.	j.	PROPN
ejpam-6041	32	2	pure	pure	PROPN
ejpam-6041	32	3	appl	appl	PROPN
ejpam-6041	32	4	.	.	PROPN
ejpam-6041	32	5	math	math	PROPN
ejpam-6041	32	6	,	,	PUNCT
ejpam-6041	32	7	18	18	NUM
ejpam-6041	32	8	(	(	PUNCT
ejpam-6041	32	9	2	2	NUM
ejpam-6041	32	10	)	)	PUNCT
ejpam-6041	32	11	(	(	PUNCT
ejpam-6041	32	12	2025	2025	NUM
ejpam-6041	32	13	)	)	PUNCT
ejpam-6041	32	14	,	,	PUNCT
ejpam-6041	32	15	6041	6041	NUM
ejpam-6041	32	16	3	3	NUM
ejpam-6041	32	17	of	of	ADP
ejpam-6041	32	18	8	8	NUM
ejpam-6041	32	19	3	3	NUM
ejpam-6041	32	20	.	.	PUNCT
ejpam-6041	33	1	the	the	DET
ejpam-6041	33	2	complex	complex	ADJ
ejpam-6041	33	3	möbius	möbius	PROPN
ejpam-6041	33	4	gyrogroup	gyrogroup	PROPN
ejpam-6041	33	5	and	and	CCONJ
ejpam-6041	33	6	its	its	PRON
ejpam-6041	33	7	associators	associator	NOUN
ejpam-6041	33	8	it	it	PRON
ejpam-6041	33	9	is	be	AUX
ejpam-6041	33	10	known	know	VERB
ejpam-6041	33	11	that	that	SCONJ
ejpam-6041	33	12	(	(	PUNCT
ejpam-6041	33	13	d,⊕m	d,⊕m	X
ejpam-6041	33	14	)	)	PUNCT
ejpam-6041	33	15	forms	form	VERB
ejpam-6041	33	16	a	a	DET
ejpam-6041	33	17	non	non	ADJ
ejpam-6041	33	18	-	-	ADJ
ejpam-6041	33	19	associative	associative	ADJ
ejpam-6041	33	20	group	group	NOUN
ejpam-6041	33	21	-	-	PUNCT
ejpam-6041	33	22	like	like	ADJ
ejpam-6041	33	23	structure	structure	NOUN
ejpam-6041	33	24	.	.	PUNCT
ejpam-6041	34	1	this	this	DET
ejpam-6041	34	2	system	system	NOUN
ejpam-6041	34	3	is	be	AUX
ejpam-6041	34	4	called	call	VERB
ejpam-6041	34	5	the	the	DET
ejpam-6041	34	6	(	(	PUNCT
ejpam-6041	34	7	complex	complex	ADJ
ejpam-6041	34	8	)	)	PUNCT
ejpam-6041	34	9	möbius	möbius	PROPN
ejpam-6041	34	10	gyrogroup	gyrogroup	NOUN
ejpam-6041	34	11	due	due	ADP
ejpam-6041	34	12	to	to	ADP
ejpam-6041	34	13	the	the	DET
ejpam-6041	34	14	fact	fact	NOUN
ejpam-6041	34	15	that	that	SCONJ
ejpam-6041	34	16	the	the	DET
ejpam-6041	34	17	function	function	NOUN
ejpam-6041	34	18	τa	τa	VERB
ejpam-6041	34	19	,	,	PUNCT
ejpam-6041	34	20	defined	define	VERB
ejpam-6041	34	21	as	as	ADP
ejpam-6041	34	22	τa(b	τa(b	NUM
ejpam-6041	34	23	)	)	PUNCT
ejpam-6041	34	24	=	=	PUNCT
ejpam-6041	35	1	a	a	DET
ejpam-6041	35	2	⊕m	⊕m	NOUN
ejpam-6041	35	3	b	b	NOUN
ejpam-6041	35	4	,	,	PUNCT
ejpam-6041	35	5	b	b	PROPN
ejpam-6041	35	6	∈	∈	PROPN
ejpam-6041	35	7	d	d	NOUN
ejpam-6041	35	8	,	,	PUNCT
ejpam-6041	35	9	is	be	AUX
ejpam-6041	35	10	a	a	DET
ejpam-6041	35	11	familiar	familiar	ADJ
ejpam-6041	35	12	möbius	möbius	ADJ
ejpam-6041	35	13	transformation	transformation	NOUN
ejpam-6041	35	14	on	on	ADP
ejpam-6041	35	15	d	d	PROPN
ejpam-6041	35	16	for	for	ADP
ejpam-6041	35	17	all	all	DET
ejpam-6041	35	18	a	a	DET
ejpam-6041	35	19	∈	∈	PROPN
ejpam-6041	35	20	d.	d.	NOUN
ejpam-6041	35	21	as	as	SCONJ
ejpam-6041	35	22	noted	note	VERB
ejpam-6041	35	23	in	in	ADP
ejpam-6041	35	24	section	section	NOUN
ejpam-6041	35	25	2	2	NUM
ejpam-6041	35	26	,	,	PUNCT
ejpam-6041	35	27	if	if	SCONJ
ejpam-6041	35	28	l	l	NOUN
ejpam-6041	35	29	satisfies	satisfy	VERB
ejpam-6041	35	30	möbius	möbiu	VERB
ejpam-6041	35	31	’s	’s	PART
ejpam-6041	35	32	exponential	exponential	ADJ
ejpam-6041	35	33	equation	equation	NOUN
ejpam-6041	35	34	and	and	CCONJ
ejpam-6041	35	35	l(w	l(w	PROPN
ejpam-6041	35	36	)	)	PUNCT
ejpam-6041	35	37	̸=	̸=	NOUN
ejpam-6041	35	38	0	0	NUM
ejpam-6041	35	39	for	for	ADP
ejpam-6041	35	40	some	some	DET
ejpam-6041	35	41	w	w	PROPN
ejpam-6041	35	42	∈	∈	PROPN
ejpam-6041	35	43	d	d	NOUN
ejpam-6041	35	44	,	,	PUNCT
ejpam-6041	35	45	then	then	ADV
ejpam-6041	35	46	l(z	l(z	NOUN
ejpam-6041	35	47	)	)	PUNCT
ejpam-6041	35	48	̸=	̸=	PROPN
ejpam-6041	35	49	0	0	NUM
ejpam-6041	35	50	for	for	ADP
ejpam-6041	35	51	all	all	DET
ejpam-6041	35	52	z	z	NOUN
ejpam-6041	35	53	∈	∈	PROPN
ejpam-6041	35	54	d.	d.	NOUN
ejpam-6041	35	55	from	from	ADP
ejpam-6041	35	56	this	this	DET
ejpam-6041	35	57	point	point	NOUN
ejpam-6041	35	58	of	of	ADP
ejpam-6041	35	59	view	view	NOUN
ejpam-6041	35	60	,	,	PUNCT
ejpam-6041	35	61	any	any	DET
ejpam-6041	35	62	non	non	ADJ
ejpam-6041	35	63	-	-	ADJ
ejpam-6041	35	64	zero	zero	NUM
ejpam-6041	35	65	solution	solution	NOUN
ejpam-6041	35	66	to	to	ADP
ejpam-6041	35	67	möbius	möbius	PROPN
ejpam-6041	35	68	’s	’s	PART
ejpam-6041	35	69	exponential	exponential	ADJ
ejpam-6041	35	70	equation	equation	NOUN
ejpam-6041	35	71	is	be	AUX
ejpam-6041	35	72	indeed	indeed	ADV
ejpam-6041	35	73	a	a	DET
ejpam-6041	35	74	homomorphism	homomorphism	NOUN
ejpam-6041	35	75	from	from	ADP
ejpam-6041	35	76	the	the	DET
ejpam-6041	35	77	möbius	möbius	PROPN
ejpam-6041	35	78	gyrogroup	gyrogroup	PROPN
ejpam-6041	35	79	to	to	ADP
ejpam-6041	35	80	the	the	DET
ejpam-6041	35	81	multiplicative	multiplicative	ADJ
ejpam-6041	35	82	group	group	NOUN
ejpam-6041	35	83	of	of	ADP
ejpam-6041	35	84	non	non	ADJ
ejpam-6041	35	85	-	-	ADJ
ejpam-6041	35	86	zero	zero	ADJ
ejpam-6041	35	87	complex	complex	ADJ
ejpam-6041	35	88	numbers	number	NOUN
ejpam-6041	35	89	.	.	PUNCT
ejpam-6041	36	1	it	it	PRON
ejpam-6041	36	2	is	be	AUX
ejpam-6041	36	3	clear	clear	ADJ
ejpam-6041	36	4	that	that	SCONJ
ejpam-6041	36	5	the	the	DET
ejpam-6041	36	6	functions	function	NOUN
ejpam-6041	36	7	defined	define	VERB
ejpam-6041	36	8	by	by	ADP
ejpam-6041	36	9	l(z	l(z	NOUN
ejpam-6041	36	10	)	)	PUNCT
ejpam-6041	36	11	=	=	SYM
ejpam-6041	36	12	0	0	NUM
ejpam-6041	36	13	for	for	ADP
ejpam-6041	36	14	all	all	DET
ejpam-6041	36	15	z	z	NOUN
ejpam-6041	36	16	∈	∈	PROPN
ejpam-6041	36	17	d	d	NOUN
ejpam-6041	36	18	and	and	CCONJ
ejpam-6041	36	19	l(z	l(z	PROPN
ejpam-6041	36	20	)	)	PUNCT
ejpam-6041	36	21	=	=	SYM
ejpam-6041	36	22	1	1	NUM
ejpam-6041	36	23	for	for	ADP
ejpam-6041	36	24	all	all	DET
ejpam-6041	36	25	z	z	NOUN
ejpam-6041	36	26	∈	∈	PROPN
ejpam-6041	36	27	d	d	NOUN
ejpam-6041	36	28	are	be	AUX
ejpam-6041	36	29	solutions	solution	NOUN
ejpam-6041	36	30	to	to	ADP
ejpam-6041	36	31	möbius	möbius	PROPN
ejpam-6041	36	32	’s	’s	PART
ejpam-6041	36	33	exponential	exponential	ADJ
ejpam-6041	36	34	equation	equation	NOUN
ejpam-6041	36	35	.	.	PUNCT
ejpam-6041	37	1	in	in	ADP
ejpam-6041	37	2	this	this	DET
ejpam-6041	37	3	section	section	NOUN
ejpam-6041	37	4	,	,	PUNCT
ejpam-6041	37	5	we	we	PRON
ejpam-6041	37	6	prove	prove	VERB
ejpam-6041	37	7	that	that	SCONJ
ejpam-6041	37	8	these	these	DET
ejpam-6041	37	9	functions	function	NOUN
ejpam-6041	37	10	are	be	AUX
ejpam-6041	37	11	the	the	DET
ejpam-6041	37	12	only	only	ADJ
ejpam-6041	37	13	solutions	solution	NOUN
ejpam-6041	37	14	to	to	ADP
ejpam-6041	37	15	möbius	möbius	PROPN
ejpam-6041	37	16	’s	’s	PART
ejpam-6041	37	17	exponential	exponential	ADJ
ejpam-6041	37	18	equation	equation	NOUN
ejpam-6041	37	19	using	use	VERB
ejpam-6041	37	20	an	an	DET
ejpam-6041	37	21	algebraic	algebraic	ADJ
ejpam-6041	37	22	approach	approach	NOUN
ejpam-6041	37	23	.	.	PUNCT
ejpam-6041	38	1	in	in	ADP
ejpam-6041	38	2	fact	fact	NOUN
ejpam-6041	38	3	,	,	PUNCT
ejpam-6041	38	4	we	we	PRON
ejpam-6041	38	5	prove	prove	VERB
ejpam-6041	38	6	a	a	DET
ejpam-6041	38	7	stronger	strong	ADJ
ejpam-6041	38	8	result	result	NOUN
ejpam-6041	38	9	:	:	PUNCT
ejpam-6041	38	10	any	any	DET
ejpam-6041	38	11	homomorphism	homomorphism	NOUN
ejpam-6041	38	12	from	from	ADP
ejpam-6041	38	13	the	the	DET
ejpam-6041	38	14	möbius	möbius	PROPN
ejpam-6041	38	15	gyrogroup	gyrogroup	PROPN
ejpam-6041	38	16	to	to	ADP
ejpam-6041	38	17	a	a	DET
ejpam-6041	38	18	group	group	NOUN
ejpam-6041	38	19	(	(	PUNCT
ejpam-6041	38	20	which	which	PRON
ejpam-6041	38	21	is	be	AUX
ejpam-6041	38	22	viewed	view	VERB
ejpam-6041	38	23	as	as	ADP
ejpam-6041	38	24	a	a	DET
ejpam-6041	38	25	degenerate	degenerate	ADJ
ejpam-6041	38	26	gyrogroup	gyrogroup	NOUN
ejpam-6041	38	27	)	)	PUNCT
ejpam-6041	38	28	is	be	AUX
ejpam-6041	38	29	necessarily	necessarily	ADV
ejpam-6041	38	30	trivial	trivial	ADJ
ejpam-6041	38	31	.	.	PUNCT
ejpam-6041	39	1	for	for	ADP
ejpam-6041	39	2	basic	basic	ADJ
ejpam-6041	39	3	definitions	definition	NOUN
ejpam-6041	39	4	and	and	CCONJ
ejpam-6041	39	5	relevant	relevant	ADJ
ejpam-6041	39	6	notations	notation	NOUN
ejpam-6041	39	7	,	,	PUNCT
ejpam-6041	39	8	we	we	PRON
ejpam-6041	39	9	refer	refer	VERB
ejpam-6041	39	10	the	the	DET
ejpam-6041	39	11	reader	reader	NOUN
ejpam-6041	39	12	to	to	ADP
ejpam-6041	39	13	[	[	X
ejpam-6041	39	14	2	2	NUM
ejpam-6041	39	15	,	,	PUNCT
ejpam-6041	39	16	4	4	NUM
ejpam-6041	39	17	,	,	PUNCT
ejpam-6041	39	18	5	5	NUM
ejpam-6041	39	19	]	]	PUNCT
ejpam-6041	39	20	.	.	PUNCT
ejpam-6041	40	1	as	as	SCONJ
ejpam-6041	40	2	introduced	introduce	VERB
ejpam-6041	40	3	in	in	ADP
ejpam-6041	40	4	[	[	X
ejpam-6041	40	5	4	4	NUM
ejpam-6041	40	6	]	]	PUNCT
ejpam-6041	40	7	,	,	PUNCT
ejpam-6041	40	8	for	for	ADP
ejpam-6041	40	9	all	all	DET
ejpam-6041	40	10	a	a	DET
ejpam-6041	40	11	,	,	PUNCT
ejpam-6041	40	12	b	b	NOUN
ejpam-6041	40	13	,	,	PUNCT
ejpam-6041	40	14	c	c	PROPN
ejpam-6041	40	15	∈	∈	PROPN
ejpam-6041	40	16	d	d	NOUN
ejpam-6041	40	17	,	,	PUNCT
ejpam-6041	40	18	the	the	DET
ejpam-6041	40	19	associator	associator	NOUN
ejpam-6041	40	20	of	of	ADP
ejpam-6041	40	21	the	the	DET
ejpam-6041	40	22	triple	triple	ADJ
ejpam-6041	40	23	(	(	PUNCT
ejpam-6041	40	24	a	a	DET
ejpam-6041	40	25	,	,	PUNCT
ejpam-6041	40	26	b	b	NOUN
ejpam-6041	40	27	,	,	PUNCT
ejpam-6041	40	28	c	c	NOUN
ejpam-6041	40	29	)	)	PUNCT
ejpam-6041	40	30	in	in	ADP
ejpam-6041	40	31	(	(	PUNCT
ejpam-6041	40	32	d,⊕m	d,⊕m	X
ejpam-6041	40	33	)	)	PUNCT
ejpam-6041	40	34	is	be	AUX
ejpam-6041	40	35	denoted	denote	VERB
ejpam-6041	40	36	by	by	ADP
ejpam-6041	40	37	[	[	X
ejpam-6041	40	38	a	a	PRON
ejpam-6041	40	39	,	,	PUNCT
ejpam-6041	40	40	b	b	NOUN
ejpam-6041	40	41	,	,	PUNCT
ejpam-6041	40	42	c	c	NOUN
ejpam-6041	40	43	]	]	PUNCT
ejpam-6041	40	44	and	and	CCONJ
ejpam-6041	40	45	is	be	AUX
ejpam-6041	40	46	defined	define	VERB
ejpam-6041	40	47	by	by	ADP
ejpam-6041	40	48	the	the	DET
ejpam-6041	40	49	formula	formula	NOUN
ejpam-6041	40	50	[	[	X
ejpam-6041	40	51	a	a	PRON
ejpam-6041	40	52	,	,	PUNCT
ejpam-6041	40	53	b	b	NOUN
ejpam-6041	40	54	,	,	PUNCT
ejpam-6041	40	55	c	c	NOUN
ejpam-6041	40	56	]	]	X
ejpam-6041	40	57	=	=	SYM
ejpam-6041	40	58	⊖m	⊖m	X
ejpam-6041	40	59	(	(	PUNCT
ejpam-6041	40	60	a⊕m	a⊕m	NOUN
ejpam-6041	40	61	(	(	PUNCT
ejpam-6041	40	62	b⊕m	b⊕m	NOUN
ejpam-6041	40	63	c))⊕m	c))⊕m	NOUN
ejpam-6041	40	64	(	(	PUNCT
ejpam-6041	40	65	(	(	PUNCT
ejpam-6041	40	66	a⊕m	a⊕m	PROPN
ejpam-6041	40	67	b)⊕m	b)⊕m	SYM
ejpam-6041	40	68	c	c	NOUN
ejpam-6041	40	69	)	)	PUNCT
ejpam-6041	40	70	,	,	PUNCT
ejpam-6041	40	71	(	(	PUNCT
ejpam-6041	40	72	2	2	X
ejpam-6041	40	73	)	)	PUNCT
ejpam-6041	40	74	where	where	SCONJ
ejpam-6041	40	75	⊖m	⊖m	NOUN
ejpam-6041	40	76	is	be	AUX
ejpam-6041	40	77	defined	define	VERB
ejpam-6041	40	78	as	as	ADP
ejpam-6041	40	79	⊖ma	⊖ma	NOUN
ejpam-6041	40	80	=	=	SYM
ejpam-6041	40	81	−a	−a	NOUN
ejpam-6041	40	82	.	.	PUNCT
ejpam-6041	41	1	for	for	ADP
ejpam-6041	41	2	convenience	convenience	NOUN
ejpam-6041	41	3	,	,	PUNCT
ejpam-6041	41	4	we	we	PRON
ejpam-6041	41	5	define	define	VERB
ejpam-6041	41	6	the	the	DET
ejpam-6041	41	7	associator	associator	NOUN
ejpam-6041	41	8	function	function	NOUN
ejpam-6041	41	9	of	of	ADP
ejpam-6041	41	10	the	the	DET
ejpam-6041	41	11	möbius	möbius	PROPN
ejpam-6041	41	12	gyrogroup	gyrogroup	PROPN
ejpam-6041	41	13	,	,	PUNCT
ejpam-6041	41	14	denoted	denote	VERB
ejpam-6041	41	15	by	by	ADP
ejpam-6041	41	16	am	am	NOUN
ejpam-6041	41	17	,	,	PUNCT
ejpam-6041	41	18	by	by	ADP
ejpam-6041	41	19	the	the	DET
ejpam-6041	41	20	formula	formula	NOUN
ejpam-6041	41	21	am	be	AUX
ejpam-6041	41	22	(	(	PUNCT
ejpam-6041	41	23	a	a	DET
ejpam-6041	41	24	,	,	PUNCT
ejpam-6041	41	25	b	b	NOUN
ejpam-6041	41	26	,	,	PUNCT
ejpam-6041	41	27	c	c	NOUN
ejpam-6041	41	28	)	)	PUNCT
ejpam-6041	41	29	=	=	PUNCT
ejpam-6041	42	1	[	[	X
ejpam-6041	42	2	a	a	PRON
ejpam-6041	42	3	,	,	PUNCT
ejpam-6041	42	4	b	b	NOUN
ejpam-6041	42	5	,	,	PUNCT
ejpam-6041	42	6	c	c	NOUN
ejpam-6041	42	7	]	]	X
ejpam-6041	42	8	(	(	PUNCT
ejpam-6041	42	9	3	3	X
ejpam-6041	42	10	)	)	PUNCT
ejpam-6041	42	11	for	for	ADP
ejpam-6041	42	12	all	all	DET
ejpam-6041	42	13	a	a	DET
ejpam-6041	42	14	,	,	PUNCT
ejpam-6041	42	15	b	b	NOUN
ejpam-6041	42	16	,	,	PUNCT
ejpam-6041	42	17	c	c	PROPN
ejpam-6041	42	18	∈	∈	PROPN
ejpam-6041	42	19	d.	d.	PROPN
ejpam-6041	42	20	note	note	VERB
ejpam-6041	42	21	that	that	SCONJ
ejpam-6041	42	22	(	(	PUNCT
ejpam-6041	42	23	a⊕m	a⊕m	PROPN
ejpam-6041	42	24	b)⊕m	b)⊕m	SYM
ejpam-6041	42	25	c	c	NOUN
ejpam-6041	42	26	=	=	SYM
ejpam-6041	42	27	a⊕m	a⊕m	NOUN
ejpam-6041	42	28	(	(	PUNCT
ejpam-6041	42	29	b⊕m	b⊕m	VERB
ejpam-6041	42	30	c	c	NOUN
ejpam-6041	42	31	)	)	PUNCT
ejpam-6041	42	32	if	if	SCONJ
ejpam-6041	42	33	and	and	CCONJ
ejpam-6041	42	34	only	only	ADV
ejpam-6041	42	35	if	if	SCONJ
ejpam-6041	42	36	[	[	X
ejpam-6041	42	37	a	a	X
ejpam-6041	42	38	,	,	PUNCT
ejpam-6041	42	39	b	b	NOUN
ejpam-6041	42	40	,	,	PUNCT
ejpam-6041	42	41	c	c	NOUN
ejpam-6041	42	42	]	]	X
ejpam-6041	42	43	=	=	SYM
ejpam-6041	43	1	0	0	X
ejpam-6041	43	2	.	.	PUNCT
ejpam-6041	44	1	we	we	PRON
ejpam-6041	44	2	present	present	VERB
ejpam-6041	44	3	an	an	DET
ejpam-6041	44	4	alternative	alternative	ADJ
ejpam-6041	44	5	useful	useful	ADJ
ejpam-6041	44	6	form	form	NOUN
ejpam-6041	44	7	of	of	ADP
ejpam-6041	44	8	formula	formula	NOUN
ejpam-6041	44	9	(	(	PUNCT
ejpam-6041	44	10	3	3	NUM
ejpam-6041	44	11	)	)	PUNCT
ejpam-6041	44	12	in	in	ADP
ejpam-6041	44	13	the	the	DET
ejpam-6041	44	14	following	follow	VERB
ejpam-6041	44	15	proposition	proposition	NOUN
ejpam-6041	44	16	.	.	PUNCT
ejpam-6041	45	1	proposition	proposition	NOUN
ejpam-6041	45	2	1	1	NUM
ejpam-6041	45	3	.	.	PUNCT
ejpam-6041	46	1	the	the	DET
ejpam-6041	46	2	associator	associator	NOUN
ejpam-6041	46	3	function	function	NOUN
ejpam-6041	46	4	of	of	ADP
ejpam-6041	46	5	the	the	DET
ejpam-6041	46	6	möbius	möbius	PROPN
ejpam-6041	46	7	gyrogroup	gyrogroup	PROPN
ejpam-6041	46	8	is	be	AUX
ejpam-6041	46	9	given	give	VERB
ejpam-6041	46	10	by	by	ADP
ejpam-6041	46	11	the	the	DET
ejpam-6041	46	12	formula	formula	NOUN
ejpam-6041	46	13	am	be	AUX
ejpam-6041	46	14	(	(	PUNCT
ejpam-6041	46	15	a	a	DET
ejpam-6041	46	16	,	,	PUNCT
ejpam-6041	46	17	b	b	NOUN
ejpam-6041	46	18	,	,	PUNCT
ejpam-6041	46	19	c	c	NOUN
ejpam-6041	46	20	)	)	PUNCT
ejpam-6041	47	1	=	=	PUNCT
ejpam-6041	47	2	c(ab−	c(ab−	ADP
ejpam-6041	47	3	ab)(1	ab)(1	PRON
ejpam-6041	47	4	+	+	CCONJ
ejpam-6041	47	5	ab+	ab+	PROPN
ejpam-6041	47	6	ac+	ac+	PROPN
ejpam-6041	47	7	bc	bc	PROPN
ejpam-6041	47	8	)	)	PUNCT
ejpam-6041	47	9	(	(	PUNCT
ejpam-6041	47	10	1	1	NUM
ejpam-6041	47	11	+	+	NUM
ejpam-6041	47	12	ab−	ab−	NUM
ejpam-6041	47	13	|c|2(1	|c|2(1	NOUN
ejpam-6041	47	14	+	+	CCONJ
ejpam-6041	47	15	ab))(1	ab))(1	NOUN
ejpam-6041	47	16	+	+	CCONJ
ejpam-6041	47	17	ab+	ab+	PROPN
ejpam-6041	47	18	ac+	ac+	PROPN
ejpam-6041	47	19	bc	bc	PROPN
ejpam-6041	47	20	)	)	PUNCT
ejpam-6041	47	21	(	(	PUNCT
ejpam-6041	47	22	4	4	X
ejpam-6041	47	23	)	)	PUNCT
ejpam-6041	47	24	for	for	ADP
ejpam-6041	47	25	all	all	DET
ejpam-6041	47	26	a	a	DET
ejpam-6041	47	27	,	,	PUNCT
ejpam-6041	47	28	b	b	NOUN
ejpam-6041	47	29	,	,	PUNCT
ejpam-6041	47	30	c	c	PROPN
ejpam-6041	47	31	∈	∈	PROPN
ejpam-6041	47	32	d.	d.	PROPN
ejpam-6041	47	33	proof	proof	NOUN
ejpam-6041	47	34	.	.	PUNCT
ejpam-6041	48	1	the	the	DET
ejpam-6041	48	2	proof	proof	NOUN
ejpam-6041	48	3	of	of	ADP
ejpam-6041	48	4	the	the	DET
ejpam-6041	48	5	proposition	proposition	NOUN
ejpam-6041	48	6	can	can	AUX
ejpam-6041	48	7	be	be	AUX
ejpam-6041	48	8	done	do	VERB
ejpam-6041	48	9	by	by	ADP
ejpam-6041	48	10	algebraic	algebraic	ADJ
ejpam-6041	48	11	manipulation	manipulation	NOUN
ejpam-6041	48	12	using	use	VERB
ejpam-6041	48	13	formulas	formula	NOUN
ejpam-6041	48	14	(	(	PUNCT
ejpam-6041	48	15	1	1	NUM
ejpam-6041	48	16	)	)	PUNCT
ejpam-6041	48	17	,	,	PUNCT
ejpam-6041	48	18	(	(	PUNCT
ejpam-6041	48	19	2	2	NUM
ejpam-6041	48	20	)	)	PUNCT
ejpam-6041	48	21	,	,	PUNCT
ejpam-6041	48	22	and	and	CCONJ
ejpam-6041	48	23	(	(	PUNCT
ejpam-6041	48	24	3	3	NUM
ejpam-6041	48	25	)	)	PUNCT
ejpam-6041	48	26	.	.	PUNCT
ejpam-6041	49	1	to	to	PART
ejpam-6041	49	2	see	see	VERB
ejpam-6041	49	3	some	some	DET
ejpam-6041	49	4	aspects	aspect	NOUN
ejpam-6041	49	5	of	of	ADP
ejpam-6041	49	6	the	the	DET
ejpam-6041	49	7	associator	associator	NOUN
ejpam-6041	49	8	function	function	NOUN
ejpam-6041	49	9	of	of	ADP
ejpam-6041	49	10	the	the	DET
ejpam-6041	49	11	möbius	möbius	PROPN
ejpam-6041	49	12	gyrogroup	gyrogroup	PROPN
ejpam-6041	49	13	,	,	PUNCT
ejpam-6041	49	14	we	we	PRON
ejpam-6041	49	15	plot	plot	VERB
ejpam-6041	49	16	a	a	DET
ejpam-6041	49	17	few	few	ADJ
ejpam-6041	49	18	images	image	NOUN
ejpam-6041	49	19	under	under	ADP
ejpam-6041	49	20	am	be	AUX
ejpam-6041	49	21	with	with	ADP
ejpam-6041	49	22	specification	specification	NOUN
ejpam-6041	49	23	of	of	ADP
ejpam-6041	49	24	a	a	PRON
ejpam-6041	49	25	and	and	CCONJ
ejpam-6041	49	26	b	b	NOUN
ejpam-6041	49	27	in	in	ADP
ejpam-6041	49	28	figures	figure	NOUN
ejpam-6041	49	29	1	1	NUM
ejpam-6041	49	30	and	and	CCONJ
ejpam-6041	49	31	2	2	NUM
ejpam-6041	49	32	.	.	X
ejpam-6041	49	33	in	in	ADP
ejpam-6041	49	34	light	light	NOUN
ejpam-6041	49	35	of	of	ADP
ejpam-6041	49	36	proposition	proposition	NOUN
ejpam-6041	49	37	1	1	NUM
ejpam-6041	49	38	,	,	PUNCT
ejpam-6041	49	39	we	we	PRON
ejpam-6041	49	40	obtain	obtain	VERB
ejpam-6041	49	41	a	a	DET
ejpam-6041	49	42	sufficient	sufficient	ADJ
ejpam-6041	49	43	and	and	CCONJ
ejpam-6041	49	44	necessary	necessary	ADJ
ejpam-6041	49	45	condition	condition	NOUN
ejpam-6041	49	46	for	for	ADP
ejpam-6041	49	47	three	three	NUM
ejpam-6041	49	48	elements	element	NOUN
ejpam-6041	49	49	in	in	ADP
ejpam-6041	49	50	the	the	DET
ejpam-6041	49	51	disk	disk	NOUN
ejpam-6041	49	52	to	to	PART
ejpam-6041	49	53	be	be	AUX
ejpam-6041	49	54	associative	associative	ADJ
ejpam-6041	49	55	with	with	ADP
ejpam-6041	49	56	respect	respect	NOUN
ejpam-6041	49	57	to	to	ADP
ejpam-6041	49	58	möbius	möbius	PROPN
ejpam-6041	49	59	addition	addition	NOUN
ejpam-6041	49	60	,	,	PUNCT
ejpam-6041	49	61	as	as	SCONJ
ejpam-6041	49	62	shown	show	VERB
ejpam-6041	49	63	in	in	ADP
ejpam-6041	49	64	corollary	corollary	ADJ
ejpam-6041	49	65	1	1	NUM
ejpam-6041	49	66	.	.	PUNCT
ejpam-6041	50	1	first	first	ADV
ejpam-6041	50	2	,	,	PUNCT
ejpam-6041	50	3	let	let	VERB
ejpam-6041	50	4	us	we	PRON
ejpam-6041	50	5	prove	prove	VERB
ejpam-6041	50	6	the	the	DET
ejpam-6041	50	7	following	follow	VERB
ejpam-6041	50	8	theorem	theorem	VERB
ejpam-6041	50	9	,	,	PUNCT
ejpam-6041	50	10	which	which	PRON
ejpam-6041	50	11	gives	give	VERB
ejpam-6041	50	12	a	a	DET
ejpam-6041	50	13	sufficient	sufficient	ADJ
ejpam-6041	50	14	and	and	CCONJ
ejpam-6041	50	15	necessary	necessary	ADJ
ejpam-6041	50	16	condition	condition	NOUN
ejpam-6041	50	17	for	for	ADP
ejpam-6041	50	18	an	an	DET
ejpam-6041	50	19	associator	associator	NOUN
ejpam-6041	50	20	to	to	PART
ejpam-6041	50	21	be	be	AUX
ejpam-6041	50	22	zero	zero	NUM
ejpam-6041	50	23	.	.	PUNCT
ejpam-6041	51	1	here	here	ADV
ejpam-6041	51	2	,	,	PUNCT
ejpam-6041	51	3	c	c	PROPN
ejpam-6041	51	4	is	be	AUX
ejpam-6041	51	5	viewed	view	VERB
ejpam-6041	51	6	as	as	ADP
ejpam-6041	51	7	a	a	DET
ejpam-6041	51	8	real	real	ADJ
ejpam-6041	51	9	vector	vector	NOUN
ejpam-6041	51	10	space	space	NOUN
ejpam-6041	51	11	in	in	ADP
ejpam-6041	51	12	the	the	DET
ejpam-6041	51	13	usual	usual	ADJ
ejpam-6041	51	14	way	way	NOUN
ejpam-6041	51	15	.	.	PUNCT
ejpam-6041	52	1	theorem	theorem	NOUN
ejpam-6041	52	2	2	2	NUM
ejpam-6041	52	3	.	.	PUNCT
ejpam-6041	52	4	let	let	VERB
ejpam-6041	52	5	a	a	DET
ejpam-6041	52	6	,	,	PUNCT
ejpam-6041	52	7	b	b	NOUN
ejpam-6041	52	8	,	,	PUNCT
ejpam-6041	52	9	c	c	PROPN
ejpam-6041	52	10	∈	∈	PROPN
ejpam-6041	52	11	d.	d.	NOUN
ejpam-6041	53	1	then	then	ADV
ejpam-6041	53	2	[	[	X
ejpam-6041	53	3	a	a	DET
ejpam-6041	53	4	,	,	PUNCT
ejpam-6041	53	5	b	b	NOUN
ejpam-6041	53	6	,	,	PUNCT
ejpam-6041	53	7	c	c	NOUN
ejpam-6041	53	8	]	]	X
ejpam-6041	53	9	=	=	SYM
ejpam-6041	53	10	0	0	PUNCT
ejpam-6041	54	1	if	if	SCONJ
ejpam-6041	54	2	and	and	CCONJ
ejpam-6041	54	3	only	only	ADV
ejpam-6041	54	4	if	if	SCONJ
ejpam-6041	54	5	c	c	NOUN
ejpam-6041	54	6	=	=	SYM
ejpam-6041	54	7	0	0	NUM
ejpam-6041	54	8	or	or	CCONJ
ejpam-6041	54	9	a	a	PRON
ejpam-6041	54	10	and	and	CCONJ
ejpam-6041	54	11	b	b	NOUN
ejpam-6041	54	12	are	be	AUX
ejpam-6041	54	13	linearly	linearly	ADV
ejpam-6041	54	14	dependent	dependent	ADJ
ejpam-6041	54	15	in	in	ADP
ejpam-6041	54	16	c.	c.	PROPN
ejpam-6041	54	17	r.	r.	PROPN
ejpam-6041	54	18	maungchang	maungchang	PROPN
ejpam-6041	54	19	et	et	PROPN
ejpam-6041	54	20	al	al	PROPN
ejpam-6041	54	21	.	.	PUNCT
ejpam-6041	54	22	/	/	SYM
ejpam-6041	54	23	eur	eur	PROPN
ejpam-6041	54	24	.	.	PUNCT
ejpam-6041	55	1	j.	j.	PROPN
ejpam-6041	55	2	pure	pure	PROPN
ejpam-6041	55	3	appl	appl	PROPN
ejpam-6041	55	4	.	.	PROPN
ejpam-6041	55	5	math	math	PROPN
ejpam-6041	55	6	,	,	PUNCT
ejpam-6041	55	7	18	18	NUM
ejpam-6041	55	8	(	(	PUNCT
ejpam-6041	55	9	2	2	NUM
ejpam-6041	55	10	)	)	PUNCT
ejpam-6041	55	11	(	(	PUNCT
ejpam-6041	55	12	2025	2025	NUM
ejpam-6041	55	13	)	)	PUNCT
ejpam-6041	55	14	,	,	PUNCT
ejpam-6041	55	15	6041	6041	NUM
ejpam-6041	55	16	4	4	NUM
ejpam-6041	55	17	of	of	ADP
ejpam-6041	55	18	8	8	NUM
ejpam-6041	55	19	figure	figure	NOUN
ejpam-6041	55	20	1	1	NUM
ejpam-6041	55	21	:	:	PUNCT
ejpam-6041	55	22	the	the	DET
ejpam-6041	55	23	image	image	NOUN
ejpam-6041	55	24	of	of	ADP
ejpam-6041	55	25	the	the	DET
ejpam-6041	55	26	interval	interval	NOUN
ejpam-6041	55	27	[	[	X
ejpam-6041	55	28	0	0	NUM
ejpam-6041	55	29	,	,	PUNCT
ejpam-6041	55	30	1	1	NUM
ejpam-6041	55	31	)	)	PUNCT
ejpam-6041	55	32	under	under	ADP
ejpam-6041	55	33	am	am	NOUN
ejpam-6041	55	34	,	,	PUNCT
ejpam-6041	55	35	where	where	SCONJ
ejpam-6041	55	36	a	a	PRON
ejpam-6041	55	37	=	=	SYM
ejpam-6041	55	38	0.5	0.5	NUM
ejpam-6041	55	39	and	and	CCONJ
ejpam-6041	55	40	b	b	NOUN
ejpam-6041	55	41	=	=	SYM
ejpam-6041	55	42	0.5i	0.5i	PROPN
ejpam-6041	55	43	.	.	PUNCT
ejpam-6041	56	1	proof	proof	NOUN
ejpam-6041	56	2	.	.	PUNCT
ejpam-6041	57	1	suppose	suppose	VERB
ejpam-6041	57	2	that	that	SCONJ
ejpam-6041	57	3	[	[	X
ejpam-6041	57	4	a	a	DET
ejpam-6041	57	5	,	,	PUNCT
ejpam-6041	57	6	b	b	NOUN
ejpam-6041	57	7	,	,	PUNCT
ejpam-6041	57	8	c	c	NOUN
ejpam-6041	57	9	]	]	X
ejpam-6041	57	10	=	=	SYM
ejpam-6041	57	11	0	0	X
ejpam-6041	57	12	.	.	PUNCT
ejpam-6041	58	1	in	in	ADP
ejpam-6041	58	2	view	view	NOUN
ejpam-6041	58	3	of	of	ADP
ejpam-6041	58	4	(	(	PUNCT
ejpam-6041	58	5	4	4	NUM
ejpam-6041	58	6	)	)	PUNCT
ejpam-6041	58	7	,	,	PUNCT
ejpam-6041	58	8	c(ab	c(ab	PROPN
ejpam-6041	58	9	−	−	NOUN
ejpam-6041	58	10	ab)(1	ab)(1	PRON
ejpam-6041	58	11	+	+	NOUN
ejpam-6041	58	12	ab	ab	PROPN
ejpam-6041	59	1	+	+	CCONJ
ejpam-6041	59	2	ac	ac	PROPN
ejpam-6041	59	3	+	+	CCONJ
ejpam-6041	59	4	bc	bc	PROPN
ejpam-6041	59	5	)	)	PUNCT
ejpam-6041	59	6	=	=	SYM
ejpam-6041	59	7	0	0	X
ejpam-6041	59	8	.	.	PUNCT
ejpam-6041	60	1	we	we	PRON
ejpam-6041	60	2	claim	claim	VERB
ejpam-6041	60	3	that	that	SCONJ
ejpam-6041	60	4	1	1	NUM
ejpam-6041	60	5	+	+	NUM
ejpam-6041	60	6	ab	ab	PROPN
ejpam-6041	60	7	+	+	CCONJ
ejpam-6041	60	8	ac	ac	PROPN
ejpam-6041	60	9	+	+	CCONJ
ejpam-6041	60	10	bc	bc	PROPN
ejpam-6041	60	11	̸=	̸=	PROPN
ejpam-6041	60	12	0	0	NUM
ejpam-6041	60	13	.	.	PUNCT
ejpam-6041	61	1	by	by	ADP
ejpam-6041	61	2	the	the	DET
ejpam-6041	61	3	closure	closure	NOUN
ejpam-6041	61	4	property	property	NOUN
ejpam-6041	61	5	of	of	ADP
ejpam-6041	61	6	⊕m	⊕m	PROPN
ejpam-6041	61	7	,	,	PUNCT
ejpam-6041	61	8	b	b	X
ejpam-6041	61	9	⊕m	⊕m	PROPN
ejpam-6041	61	10	c	c	PROPN
ejpam-6041	61	11	∈	∈	PROPN
ejpam-6041	61	12	d.	d.	PROPN
ejpam-6041	61	13	this	this	PRON
ejpam-6041	61	14	implies	imply	VERB
ejpam-6041	61	15	that	that	SCONJ
ejpam-6041	61	16	1	1	NUM
ejpam-6041	61	17	+	+	NUM
ejpam-6041	61	18	a(b⊕m	a(b⊕m	NOUN
ejpam-6041	61	19	c	c	X
ejpam-6041	61	20	)	)	PUNCT
ejpam-6041	61	21	̸=	̸=	PROPN
ejpam-6041	61	22	0	0	NUM
ejpam-6041	61	23	since	since	SCONJ
ejpam-6041	61	24	otherwise	otherwise	ADV
ejpam-6041	61	25	|a||b⊕m	|a||b⊕m	VERB
ejpam-6041	61	26	c|	c|	PROPN
ejpam-6041	61	27	=	=	SYM
ejpam-6041	61	28	1	1	NUM
ejpam-6041	61	29	,	,	PUNCT
ejpam-6041	61	30	an	an	DET
ejpam-6041	61	31	impossibility	impossibility	NOUN
ejpam-6041	61	32	.	.	PUNCT
ejpam-6041	62	1	a	a	DET
ejpam-6041	62	2	direct	direct	ADJ
ejpam-6041	62	3	computation	computation	NOUN
ejpam-6041	62	4	shows	show	VERB
ejpam-6041	62	5	that	that	SCONJ
ejpam-6041	62	6	1	1	NUM
ejpam-6041	62	7	+	+	NUM
ejpam-6041	62	8	a(b⊕m	a(b⊕m	NOUN
ejpam-6041	62	9	c	c	NOUN
ejpam-6041	62	10	)	)	PUNCT
ejpam-6041	62	11	=	=	SYM
ejpam-6041	63	1	1	1	NUM
ejpam-6041	63	2	+	+	NUM
ejpam-6041	63	3	ab+	ab+	NOUN
ejpam-6041	63	4	ac	ac	PROPN
ejpam-6041	63	5	1	1	NUM
ejpam-6041	63	6	+	+	NUM
ejpam-6041	63	7	bc	bc	X
ejpam-6041	63	8	=	=	SYM
ejpam-6041	63	9	1	1	NUM
ejpam-6041	63	10	+	+	NUM
ejpam-6041	63	11	ab+	ab+	NOUN
ejpam-6041	63	12	ac+	ac+	PROPN
ejpam-6041	63	13	bc	bc	PROPN
ejpam-6041	63	14	1	1	PROPN
ejpam-6041	63	15	+	+	CCONJ
ejpam-6041	63	16	bc	bc	PROPN
ejpam-6041	63	17	,	,	PUNCT
ejpam-6041	63	18	and	and	CCONJ
ejpam-6041	63	19	so	so	ADV
ejpam-6041	63	20	1	1	NUM
ejpam-6041	63	21	+	+	NUM
ejpam-6041	63	22	ab	ab	PROPN
ejpam-6041	63	23	+	+	CCONJ
ejpam-6041	63	24	ac	ac	PROPN
ejpam-6041	63	25	+	+	CCONJ
ejpam-6041	63	26	bc	bc	PROPN
ejpam-6041	63	27	̸=	̸=	PROPN
ejpam-6041	63	28	0	0	NUM
ejpam-6041	63	29	.	.	PUNCT
ejpam-6041	64	1	this	this	PRON
ejpam-6041	64	2	implies	imply	VERB
ejpam-6041	64	3	that	that	SCONJ
ejpam-6041	64	4	1	1	NUM
ejpam-6041	64	5	+	+	NUM
ejpam-6041	64	6	ab	ab	PROPN
ejpam-6041	64	7	+	+	CCONJ
ejpam-6041	64	8	ac	ac	PROPN
ejpam-6041	64	9	+	+	CCONJ
ejpam-6041	64	10	bc	bc	X
ejpam-6041	64	11	=	=	SYM
ejpam-6041	64	12	1	1	NUM
ejpam-6041	64	13	+	+	NUM
ejpam-6041	64	14	ab+	ab+	NOUN
ejpam-6041	64	15	ac+	ac+	PROPN
ejpam-6041	64	16	bc	bc	PROPN
ejpam-6041	64	17	̸=	̸=	PROPN
ejpam-6041	64	18	0	0	NUM
ejpam-6041	64	19	,	,	PUNCT
ejpam-6041	64	20	which	which	PRON
ejpam-6041	64	21	proves	prove	VERB
ejpam-6041	64	22	the	the	DET
ejpam-6041	64	23	claim	claim	NOUN
ejpam-6041	64	24	.	.	PUNCT
ejpam-6041	65	1	hence	hence	ADV
ejpam-6041	65	2	,	,	PUNCT
ejpam-6041	65	3	c	c	NOUN
ejpam-6041	65	4	=	=	SYM
ejpam-6041	65	5	0	0	PROPN
ejpam-6041	65	6	or	or	CCONJ
ejpam-6041	65	7	ab	ab	PROPN
ejpam-6041	65	8	−	−	PROPN
ejpam-6041	65	9	ab	ab	PROPN
ejpam-6041	65	10	=	=	NOUN
ejpam-6041	65	11	0	0	PROPN
ejpam-6041	65	12	.	.	PUNCT
ejpam-6041	66	1	in	in	ADP
ejpam-6041	66	2	the	the	DET
ejpam-6041	66	3	latter	latter	ADJ
ejpam-6041	66	4	case	case	NOUN
ejpam-6041	66	5	,	,	PUNCT
ejpam-6041	66	6	we	we	PRON
ejpam-6041	66	7	show	show	VERB
ejpam-6041	66	8	that	that	SCONJ
ejpam-6041	66	9	a	a	PRON
ejpam-6041	66	10	and	and	CCONJ
ejpam-6041	66	11	b	b	NOUN
ejpam-6041	66	12	are	be	AUX
ejpam-6041	66	13	linearly	linearly	ADV
ejpam-6041	66	14	dependent	dependent	ADJ
ejpam-6041	66	15	.	.	PUNCT
ejpam-6041	67	1	the	the	DET
ejpam-6041	67	2	case	case	NOUN
ejpam-6041	67	3	when	when	SCONJ
ejpam-6041	67	4	a	a	PRON
ejpam-6041	67	5	and	and	CCONJ
ejpam-6041	67	6	b	b	NOUN
ejpam-6041	67	7	are	be	AUX
ejpam-6041	67	8	both	both	PRON
ejpam-6041	67	9	zero	zero	NUM
ejpam-6041	67	10	is	be	AUX
ejpam-6041	67	11	clear	clear	ADJ
ejpam-6041	67	12	.	.	PUNCT
ejpam-6041	68	1	therefore	therefore	ADV
ejpam-6041	68	2	,	,	PUNCT
ejpam-6041	68	3	we	we	PRON
ejpam-6041	68	4	assume	assume	VERB
ejpam-6041	68	5	that	that	SCONJ
ejpam-6041	68	6	a	a	DET
ejpam-6041	68	7	̸=	̸=	PROPN
ejpam-6041	68	8	0	0	NUM
ejpam-6041	68	9	or	or	CCONJ
ejpam-6041	68	10	b	b	X
ejpam-6041	68	11	̸=	̸=	PROPN
ejpam-6041	68	12	0	0	NUM
ejpam-6041	68	13	.	.	PUNCT
ejpam-6041	69	1	now	now	ADV
ejpam-6041	69	2	,	,	PUNCT
ejpam-6041	69	3	suppose	suppose	VERB
ejpam-6041	69	4	that	that	SCONJ
ejpam-6041	69	5	ab	ab	PROPN
ejpam-6041	69	6	−	−	PROPN
ejpam-6041	69	7	ab	ab	PROPN
ejpam-6041	69	8	=	=	NOUN
ejpam-6041	69	9	0	0	X
ejpam-6041	69	10	.	.	PUNCT
ejpam-6041	70	1	set	set	VERB
ejpam-6041	70	2	a	a	DET
ejpam-6041	70	3	=	=	NOUN
ejpam-6041	70	4	a1	a1	NOUN
ejpam-6041	70	5	+	+	CCONJ
ejpam-6041	70	6	a2i	a2i	NOUN
ejpam-6041	70	7	and	and	CCONJ
ejpam-6041	70	8	b	b	X
ejpam-6041	70	9	=	=	SYM
ejpam-6041	70	10	b1	b1	PROPN
ejpam-6041	70	11	+	+	CCONJ
ejpam-6041	70	12	b2i	b2i	PROPN
ejpam-6041	70	13	,	,	PUNCT
ejpam-6041	70	14	where	where	SCONJ
ejpam-6041	70	15	a1	a1	NOUN
ejpam-6041	70	16	,	,	PUNCT
ejpam-6041	70	17	a2	a2	PROPN
ejpam-6041	70	18	,	,	PUNCT
ejpam-6041	70	19	b1	b1	NOUN
ejpam-6041	70	20	,	,	PUNCT
ejpam-6041	70	21	b2	b2	NOUN
ejpam-6041	70	22	∈	∈	PROPN
ejpam-6041	70	23	r.	r.	NOUN
ejpam-6041	70	24	by	by	ADP
ejpam-6041	70	25	assumption	assumption	NOUN
ejpam-6041	70	26	,	,	PUNCT
ejpam-6041	70	27	a1b2	a1b2	PROPN
ejpam-6041	70	28	−	−	PROPN
ejpam-6041	70	29	a2b1	a2b1	SYM
ejpam-6041	70	30	=	=	SYM
ejpam-6041	70	31	0	0	X
ejpam-6041	70	32	.	.	PUNCT
ejpam-6041	71	1	this	this	PRON
ejpam-6041	71	2	implies	imply	VERB
ejpam-6041	71	3	that	that	SCONJ
ejpam-6041	71	4	b2a	b2a	VERB
ejpam-6041	71	5	−	−	PROPN
ejpam-6041	71	6	a2b	a2b	NOUN
ejpam-6041	71	7	=	=	PUNCT
ejpam-6041	71	8	0	0	X
ejpam-6041	71	9	.	.	PUNCT
ejpam-6041	72	1	if	if	SCONJ
ejpam-6041	72	2	b2	b2	NOUN
ejpam-6041	72	3	=	=	SYM
ejpam-6041	72	4	0	0	NUM
ejpam-6041	72	5	and	and	CCONJ
ejpam-6041	72	6	a2	a2	PROPN
ejpam-6041	72	7	=	=	SYM
ejpam-6041	72	8	0	0	PROPN
ejpam-6041	72	9	,	,	PUNCT
ejpam-6041	72	10	then	then	ADV
ejpam-6041	72	11	a	a	DET
ejpam-6041	72	12	=	=	NOUN
ejpam-6041	72	13	a1	a1	NOUN
ejpam-6041	72	14	and	and	CCONJ
ejpam-6041	72	15	b	b	NOUN
ejpam-6041	72	16	=	=	SYM
ejpam-6041	72	17	b1	b1	PROPN
ejpam-6041	72	18	,	,	PUNCT
ejpam-6041	72	19	and	and	CCONJ
ejpam-6041	72	20	furthermore	furthermore	ADV
ejpam-6041	72	21	a1	a1	NOUN
ejpam-6041	72	22	and	and	CCONJ
ejpam-6041	72	23	b1	b1	NOUN
ejpam-6041	72	24	are	be	AUX
ejpam-6041	72	25	not	not	PART
ejpam-6041	72	26	simultaneously	simultaneously	ADV
ejpam-6041	72	27	zero	zero	NUM
ejpam-6041	72	28	.	.	PUNCT
ejpam-6041	73	1	hence	hence	ADV
ejpam-6041	73	2	,	,	PUNCT
ejpam-6041	73	3	−b1a+	−b1a+	VERB
ejpam-6041	73	4	a1b	a1b	NOUN
ejpam-6041	73	5	=	=	SYM
ejpam-6041	73	6	0	0	X
ejpam-6041	73	7	.	.	PUNCT
ejpam-6041	74	1	this	this	PRON
ejpam-6041	74	2	shows	show	VERB
ejpam-6041	74	3	that	that	SCONJ
ejpam-6041	74	4	a	a	PRON
ejpam-6041	74	5	and	and	CCONJ
ejpam-6041	74	6	b	b	NOUN
ejpam-6041	74	7	are	be	AUX
ejpam-6041	74	8	linearly	linearly	ADV
ejpam-6041	74	9	dependent	dependent	ADJ
ejpam-6041	74	10	.	.	PUNCT
ejpam-6041	75	1	suppose	suppose	VERB
ejpam-6041	75	2	conversely	conversely	ADV
ejpam-6041	75	3	that	that	SCONJ
ejpam-6041	75	4	c	c	AUX
ejpam-6041	75	5	=	=	SYM
ejpam-6041	75	6	0	0	NUM
ejpam-6041	75	7	or	or	CCONJ
ejpam-6041	75	8	a	a	PRON
ejpam-6041	75	9	and	and	CCONJ
ejpam-6041	75	10	b	b	NOUN
ejpam-6041	75	11	are	be	AUX
ejpam-6041	75	12	linearly	linearly	ADV
ejpam-6041	75	13	dependent	dependent	ADJ
ejpam-6041	75	14	.	.	PUNCT
ejpam-6041	76	1	the	the	DET
ejpam-6041	76	2	case	case	NOUN
ejpam-6041	76	3	when	when	SCONJ
ejpam-6041	76	4	c	c	PROPN
ejpam-6041	76	5	=	=	SYM
ejpam-6041	76	6	0	0	NUM
ejpam-6041	76	7	is	be	AUX
ejpam-6041	76	8	clear	clear	ADJ
ejpam-6041	76	9	.	.	PUNCT
ejpam-6041	77	1	hence	hence	ADV
ejpam-6041	77	2	,	,	PUNCT
ejpam-6041	77	3	we	we	PRON
ejpam-6041	77	4	assume	assume	VERB
ejpam-6041	77	5	that	that	SCONJ
ejpam-6041	77	6	a	a	PRON
ejpam-6041	77	7	and	and	CCONJ
ejpam-6041	77	8	b	b	NOUN
ejpam-6041	77	9	are	be	AUX
ejpam-6041	77	10	linearly	linearly	ADV
ejpam-6041	77	11	dependent	dependent	ADJ
ejpam-6041	77	12	.	.	PUNCT
ejpam-6041	78	1	thus	thus	ADV
ejpam-6041	78	2	,	,	PUNCT
ejpam-6041	78	3	there	there	PRON
ejpam-6041	78	4	are	be	VERB
ejpam-6041	78	5	real	real	ADJ
ejpam-6041	78	6	numbers	number	NOUN
ejpam-6041	78	7	r	r	NOUN
ejpam-6041	78	8	and	and	CCONJ
ejpam-6041	78	9	s	s	AUX
ejpam-6041	78	10	not	not	PART
ejpam-6041	78	11	simultaneously	simultaneously	ADV
ejpam-6041	78	12	zero	zero	NUM
ejpam-6041	78	13	such	such	ADJ
ejpam-6041	78	14	that	that	DET
ejpam-6041	78	15	ra+	ra+	PROPN
ejpam-6041	78	16	sb	sb	PROPN
ejpam-6041	78	17	=	=	PROPN
ejpam-6041	78	18	0	0	PROPN
ejpam-6041	78	19	.	.	PUNCT
ejpam-6041	79	1	in	in	ADP
ejpam-6041	79	2	the	the	DET
ejpam-6041	79	3	case	case	NOUN
ejpam-6041	79	4	when	when	SCONJ
ejpam-6041	79	5	r	r	NOUN
ejpam-6041	79	6	=	=	SYM
ejpam-6041	79	7	0	0	NUM
ejpam-6041	79	8	,	,	PUNCT
ejpam-6041	79	9	we	we	PRON
ejpam-6041	79	10	obtain	obtain	VERB
ejpam-6041	79	11	that	that	PRON
ejpam-6041	79	12	s	s	VERB
ejpam-6041	79	13	̸=	̸=	PROPN
ejpam-6041	79	14	0	0	NUM
ejpam-6041	79	15	,	,	PUNCT
ejpam-6041	79	16	and	and	CCONJ
ejpam-6041	79	17	so	so	ADV
ejpam-6041	79	18	sb	sb	PROPN
ejpam-6041	79	19	=	=	SYM
ejpam-6041	79	20	0	0	NUM
ejpam-6041	79	21	implies	imply	VERB
ejpam-6041	79	22	b	b	NOUN
ejpam-6041	79	23	=	=	SYM
ejpam-6041	79	24	0	0	PROPN
ejpam-6041	79	25	,	,	PUNCT
ejpam-6041	79	26	which	which	PRON
ejpam-6041	79	27	in	in	ADP
ejpam-6041	79	28	turn	turn	NOUN
ejpam-6041	79	29	implies	imply	VERB
ejpam-6041	79	30	ab−ab	ab−ab	X
ejpam-6041	79	31	=	=	SYM
ejpam-6041	79	32	0	0	X
ejpam-6041	79	33	.	.	PUNCT
ejpam-6041	80	1	now	now	ADV
ejpam-6041	80	2	,	,	PUNCT
ejpam-6041	80	3	suppose	suppose	VERB
ejpam-6041	80	4	that	that	SCONJ
ejpam-6041	80	5	r	r	NOUN
ejpam-6041	80	6	̸=	̸=	PROPN
ejpam-6041	80	7	0	0	NUM
ejpam-6041	80	8	.	.	PUNCT
ejpam-6041	81	1	set	set	VERB
ejpam-6041	81	2	a	a	DET
ejpam-6041	81	3	=	=	PUNCT
ejpam-6041	81	4	a1+a2i	a1+a2i	PROPN
ejpam-6041	81	5	and	and	CCONJ
ejpam-6041	81	6	b	b	X
ejpam-6041	81	7	=	=	SYM
ejpam-6041	81	8	b1+b2i	b1+b2i	PROPN
ejpam-6041	81	9	,	,	PUNCT
ejpam-6041	81	10	where	where	SCONJ
ejpam-6041	81	11	a1	a1	NOUN
ejpam-6041	81	12	,	,	PUNCT
ejpam-6041	81	13	a2	a2	PROPN
ejpam-6041	81	14	,	,	PUNCT
ejpam-6041	81	15	b1	b1	NOUN
ejpam-6041	81	16	,	,	PUNCT
ejpam-6041	81	17	b2	b2	NOUN
ejpam-6041	81	18	∈	∈	PROPN
ejpam-6041	81	19	r.	r.	NOUN
ejpam-6041	81	20	then	then	ADV
ejpam-6041	81	21	solving	solve	VERB
ejpam-6041	81	22	the	the	DET
ejpam-6041	81	23	system	system	NOUN
ejpam-6041	81	24	of	of	ADP
ejpam-6041	81	25	equations	equation	NOUN
ejpam-6041	81	26	induced	induce	VERB
ejpam-6041	81	27	by	by	ADP
ejpam-6041	81	28	the	the	DET
ejpam-6041	81	29	equality	equality	NOUN
ejpam-6041	81	30	ra+sb	ra+sb	NOUN
ejpam-6041	81	31	=	=	SYM
ejpam-6041	81	32	0	0	NUM
ejpam-6041	81	33	shows	show	VERB
ejpam-6041	81	34	that	that	SCONJ
ejpam-6041	81	35	r(a1b2−a2b1	r(a1b2−a2b1	NOUN
ejpam-6041	81	36	)	)	PUNCT
ejpam-6041	82	1	=	=	SYM
ejpam-6041	82	2	0	0	NUM
ejpam-6041	82	3	,	,	PUNCT
ejpam-6041	82	4	which	which	PRON
ejpam-6041	82	5	implies	imply	VERB
ejpam-6041	82	6	a1b2	a1b2	PROPN
ejpam-6041	82	7	−	−	X
ejpam-6041	82	8	a2b1	a2b1	SYM
ejpam-6041	82	9	=	=	SYM
ejpam-6041	82	10	0	0	X
ejpam-6041	82	11	.	.	PUNCT
ejpam-6041	83	1	it	it	PRON
ejpam-6041	83	2	follows	follow	VERB
ejpam-6041	83	3	that	that	SCONJ
ejpam-6041	83	4	ab−	ab−	NUM
ejpam-6041	83	5	ab	ab	PROPN
ejpam-6041	83	6	=	=	PUNCT
ejpam-6041	83	7	0	0	PROPN
ejpam-6041	83	8	,	,	PUNCT
ejpam-6041	83	9	and	and	CCONJ
ejpam-6041	84	1	so	so	ADV
ejpam-6041	84	2	[	[	X
ejpam-6041	84	3	a	a	X
ejpam-6041	84	4	,	,	PUNCT
ejpam-6041	84	5	b	b	NOUN
ejpam-6041	84	6	,	,	PUNCT
ejpam-6041	84	7	c	c	NOUN
ejpam-6041	84	8	]	]	X
ejpam-6041	84	9	=	=	SYM
ejpam-6041	84	10	0	0	X
ejpam-6041	84	11	.	.	PUNCT
ejpam-6041	84	12	corollary	corollary	ADJ
ejpam-6041	84	13	1	1	NUM
ejpam-6041	84	14	.	.	PUNCT
ejpam-6041	85	1	let	let	VERB
ejpam-6041	85	2	a	a	DET
ejpam-6041	85	3	,	,	PUNCT
ejpam-6041	85	4	b	b	NOUN
ejpam-6041	85	5	,	,	PUNCT
ejpam-6041	85	6	c	c	PROPN
ejpam-6041	85	7	∈	∈	PROPN
ejpam-6041	85	8	d.	d.	PROPN
ejpam-6041	85	9	then	then	ADV
ejpam-6041	85	10	a	a	DET
ejpam-6041	85	11	⊕m	⊕m	NOUN
ejpam-6041	85	12	(	(	PUNCT
ejpam-6041	85	13	b	b	X
ejpam-6041	85	14	⊕m	⊕m	NOUN
ejpam-6041	85	15	c	c	NOUN
ejpam-6041	85	16	)	)	PUNCT
ejpam-6041	86	1	=	=	SYM
ejpam-6041	86	2	(	(	PUNCT
ejpam-6041	86	3	a	a	DET
ejpam-6041	86	4	⊕m	⊕m	NOUN
ejpam-6041	86	5	b	b	NOUN
ejpam-6041	86	6	)	)	PUNCT
ejpam-6041	86	7	⊕m	⊕m	NOUN
ejpam-6041	87	1	c	c	NOUN
ejpam-6041	87	2	if	if	SCONJ
ejpam-6041	87	3	and	and	CCONJ
ejpam-6041	87	4	only	only	ADV
ejpam-6041	87	5	if	if	SCONJ
ejpam-6041	87	6	c	c	NOUN
ejpam-6041	87	7	=	=	SYM
ejpam-6041	87	8	0	0	NUM
ejpam-6041	87	9	or	or	CCONJ
ejpam-6041	87	10	a	a	PRON
ejpam-6041	87	11	and	and	CCONJ
ejpam-6041	87	12	b	b	NOUN
ejpam-6041	87	13	are	be	AUX
ejpam-6041	87	14	linearly	linearly	ADV
ejpam-6041	87	15	dependent	dependent	ADJ
ejpam-6041	87	16	in	in	ADP
ejpam-6041	87	17	c.	c.	NOUN
ejpam-6041	87	18	proof	proof	NOUN
ejpam-6041	87	19	.	.	PUNCT
ejpam-6041	88	1	the	the	DET
ejpam-6041	88	2	corollary	corollary	NOUN
ejpam-6041	88	3	follows	follow	VERB
ejpam-6041	88	4	from	from	ADP
ejpam-6041	88	5	the	the	DET
ejpam-6041	88	6	fact	fact	NOUN
ejpam-6041	88	7	that	that	SCONJ
ejpam-6041	88	8	(	(	PUNCT
ejpam-6041	88	9	a⊕m	a⊕m	PROPN
ejpam-6041	88	10	b)⊕m	b)⊕m	SYM
ejpam-6041	88	11	c	c	NOUN
ejpam-6041	88	12	=	=	SYM
ejpam-6041	88	13	(	(	PUNCT
ejpam-6041	88	14	a⊕m	a⊕m	NOUN
ejpam-6041	88	15	(	(	PUNCT
ejpam-6041	88	16	b⊕m	b⊕m	VERB
ejpam-6041	88	17	c))⊕m	c))⊕m	NOUN
ejpam-6041	89	1	[	[	X
ejpam-6041	89	2	a	a	DET
ejpam-6041	89	3	,	,	PUNCT
ejpam-6041	89	4	b	b	NOUN
ejpam-6041	89	5	,	,	PUNCT
ejpam-6041	89	6	c	c	X
ejpam-6041	89	7	]	]	X
ejpam-6041	89	8	r.	r.	PROPN
ejpam-6041	89	9	maungchang	maungchang	PROPN
ejpam-6041	89	10	et	et	PROPN
ejpam-6041	89	11	al	al	PROPN
ejpam-6041	89	12	.	.	PUNCT
ejpam-6041	89	13	/	/	SYM
ejpam-6041	89	14	eur	eur	PROPN
ejpam-6041	89	15	.	.	PUNCT
ejpam-6041	90	1	j.	j.	PROPN
ejpam-6041	90	2	pure	pure	PROPN
ejpam-6041	90	3	appl	appl	PROPN
ejpam-6041	90	4	.	.	PROPN
ejpam-6041	90	5	math	math	PROPN
ejpam-6041	90	6	,	,	PUNCT
ejpam-6041	90	7	18	18	NUM
ejpam-6041	90	8	(	(	PUNCT
ejpam-6041	90	9	2	2	NUM
ejpam-6041	90	10	)	)	PUNCT
ejpam-6041	90	11	(	(	PUNCT
ejpam-6041	90	12	2025	2025	NUM
ejpam-6041	90	13	)	)	PUNCT
ejpam-6041	90	14	,	,	PUNCT
ejpam-6041	90	15	6041	6041	NUM
ejpam-6041	90	16	5	5	NUM
ejpam-6041	90	17	of	of	ADP
ejpam-6041	90	18	8	8	NUM
ejpam-6041	90	19	figure	figure	NOUN
ejpam-6041	90	20	2	2	NUM
ejpam-6041	90	21	:	:	PUNCT
ejpam-6041	90	22	the	the	DET
ejpam-6041	90	23	image	image	NOUN
ejpam-6041	90	24	of	of	ADP
ejpam-6041	90	25	the	the	DET
ejpam-6041	90	26	rectangular	rectangular	ADJ
ejpam-6041	90	27	grids	grid	NOUN
ejpam-6041	90	28	under	under	ADP
ejpam-6041	90	29	am	am	NOUN
ejpam-6041	90	30	,	,	PUNCT
ejpam-6041	90	31	where	where	SCONJ
ejpam-6041	90	32	a	a	DET
ejpam-6041	90	33	=	=	SYM
ejpam-6041	90	34	0.5	0.5	NUM
ejpam-6041	90	35	and	and	CCONJ
ejpam-6041	90	36	b	b	NOUN
ejpam-6041	90	37	=	=	SYM
ejpam-6041	90	38	0.5i	0.5i	PROPN
ejpam-6041	90	39	.	.	PUNCT
ejpam-6041	91	1	for	for	ADP
ejpam-6041	91	2	all	all	DET
ejpam-6041	91	3	a	a	DET
ejpam-6041	91	4	,	,	PUNCT
ejpam-6041	91	5	b	b	NOUN
ejpam-6041	91	6	,	,	PUNCT
ejpam-6041	91	7	c	c	PROPN
ejpam-6041	91	8	∈	∈	PROPN
ejpam-6041	91	9	d.	d.	PROPN
ejpam-6041	91	10	next	next	ADV
ejpam-6041	91	11	,	,	PUNCT
ejpam-6041	91	12	we	we	PRON
ejpam-6041	91	13	prove	prove	VERB
ejpam-6041	91	14	that	that	SCONJ
ejpam-6041	91	15	any	any	DET
ejpam-6041	91	16	element	element	NOUN
ejpam-6041	91	17	of	of	ADP
ejpam-6041	91	18	the	the	DET
ejpam-6041	91	19	möbius	möbius	PROPN
ejpam-6041	91	20	gyrogroup	gyrogroup	PROPN
ejpam-6041	91	21	can	can	AUX
ejpam-6041	91	22	be	be	AUX
ejpam-6041	91	23	expressed	express	VERB
ejpam-6041	91	24	as	as	ADP
ejpam-6041	91	25	an	an	DET
ejpam-6041	91	26	associator	associator	NOUN
ejpam-6041	91	27	so	so	SCONJ
ejpam-6041	91	28	that	that	SCONJ
ejpam-6041	91	29	the	the	DET
ejpam-6041	91	30	set	set	NOUN
ejpam-6041	91	31	of	of	ADP
ejpam-6041	91	32	all	all	DET
ejpam-6041	91	33	associators	associator	NOUN
ejpam-6041	91	34	is	be	AUX
ejpam-6041	91	35	d	d	PRON
ejpam-6041	91	36	itself	itself	PRON
ejpam-6041	91	37	.	.	PUNCT
ejpam-6041	92	1	to	to	PART
ejpam-6041	92	2	do	do	VERB
ejpam-6041	92	3	so	so	ADV
ejpam-6041	92	4	,	,	PUNCT
ejpam-6041	92	5	we	we	PRON
ejpam-6041	92	6	need	need	VERB
ejpam-6041	92	7	the	the	DET
ejpam-6041	92	8	following	follow	VERB
ejpam-6041	92	9	lemma	lemma	PROPN
ejpam-6041	92	10	,	,	PUNCT
ejpam-6041	92	11	which	which	PRON
ejpam-6041	92	12	is	be	AUX
ejpam-6041	92	13	important	important	ADJ
ejpam-6041	92	14	in	in	ADP
ejpam-6041	92	15	its	its	PRON
ejpam-6041	92	16	own	own	ADJ
ejpam-6041	92	17	right	right	NOUN
ejpam-6041	92	18	.	.	PUNCT
ejpam-6041	93	1	lemma	lemma	PROPN
ejpam-6041	93	2	1	1	X
ejpam-6041	93	3	.	.	PUNCT
ejpam-6041	94	1	if	if	SCONJ
ejpam-6041	94	2	ω	ω	PROPN
ejpam-6041	94	3	is	be	AUX
ejpam-6041	94	4	a	a	DET
ejpam-6041	94	5	unimodular	unimodular	ADJ
ejpam-6041	94	6	complex	complex	ADJ
ejpam-6041	94	7	number	number	NOUN
ejpam-6041	94	8	(	(	PUNCT
ejpam-6041	94	9	that	that	PRON
ejpam-6041	94	10	is	is	ADV
ejpam-6041	94	11	,	,	PUNCT
ejpam-6041	94	12	|ω|	|ω|	ADP
ejpam-6041	94	13	=	=	NOUN
ejpam-6041	94	14	1	1	NUM
ejpam-6041	94	15	)	)	PUNCT
ejpam-6041	94	16	,	,	PUNCT
ejpam-6041	94	17	then	then	ADV
ejpam-6041	94	18	am	am	VERB
ejpam-6041	94	19	(	(	PUNCT
ejpam-6041	94	20	ωa	ωa	ADJ
ejpam-6041	94	21	,	,	PUNCT
ejpam-6041	94	22	ωb	ωb	NOUN
ejpam-6041	94	23	,	,	PUNCT
ejpam-6041	94	24	ωc	ωc	X
ejpam-6041	94	25	)	)	PUNCT
ejpam-6041	94	26	=	=	SYM
ejpam-6041	94	27	ωam	ωam	X
ejpam-6041	94	28	(	(	PUNCT
ejpam-6041	94	29	a	a	PRON
ejpam-6041	94	30	,	,	PUNCT
ejpam-6041	94	31	b	b	NOUN
ejpam-6041	94	32	,	,	PUNCT
ejpam-6041	94	33	c	c	NOUN
ejpam-6041	94	34	)	)	PUNCT
ejpam-6041	94	35	for	for	ADP
ejpam-6041	94	36	all	all	DET
ejpam-6041	94	37	a	a	DET
ejpam-6041	94	38	,	,	PUNCT
ejpam-6041	94	39	b	b	NOUN
ejpam-6041	94	40	,	,	PUNCT
ejpam-6041	94	41	c	c	PROPN
ejpam-6041	94	42	∈	∈	PROPN
ejpam-6041	94	43	d.	d.	PROPN
ejpam-6041	94	44	proof	proof	PROPN
ejpam-6041	94	45	.	.	PUNCT
ejpam-6041	95	1	suppose	suppose	VERB
ejpam-6041	95	2	that	that	SCONJ
ejpam-6041	95	3	|ω|	|ω|	PROPN
ejpam-6041	95	4	=	=	SYM
ejpam-6041	95	5	1	1	X
ejpam-6041	95	6	.	.	X
ejpam-6041	95	7	note	note	VERB
ejpam-6041	95	8	that	that	SCONJ
ejpam-6041	95	9	if	if	SCONJ
ejpam-6041	95	10	x	x	X
ejpam-6041	95	11	,	,	PUNCT
ejpam-6041	95	12	y	y	PROPN
ejpam-6041	95	13	∈	∈	PROPN
ejpam-6041	95	14	c	c	NOUN
ejpam-6041	95	15	,	,	PUNCT
ejpam-6041	95	16	then	then	ADV
ejpam-6041	95	17	ωxωy	ωxωy	PROPN
ejpam-6041	95	18	=	=	PUNCT
ejpam-6041	95	19	ωxωy	ωxωy	PROPN
ejpam-6041	95	20	=	=	NOUN
ejpam-6041	95	21	|ω|2xy	|ω|2xy	PROPN
ejpam-6041	96	1	=	=	SYM
ejpam-6041	96	2	xy	xy	NOUN
ejpam-6041	96	3	.	.	PUNCT
ejpam-6041	97	1	furthermore	furthermore	ADV
ejpam-6041	97	2	,	,	PUNCT
ejpam-6041	97	3	|ωx|	|ωx|	PROPN
ejpam-6041	97	4	=	=	SYM
ejpam-6041	97	5	|x|	|x|	PROPN
ejpam-6041	97	6	for	for	ADP
ejpam-6041	97	7	all	all	PRON
ejpam-6041	97	8	x	x	SYM
ejpam-6041	97	9	∈	∈	PROPN
ejpam-6041	97	10	c.	c.	NOUN
ejpam-6041	97	11	hence	hence	ADV
ejpam-6041	97	12	,	,	PUNCT
ejpam-6041	97	13	the	the	DET
ejpam-6041	97	14	lemma	lemma	PROPN
ejpam-6041	97	15	follows	follow	VERB
ejpam-6041	97	16	directly	directly	ADV
ejpam-6041	97	17	from	from	ADP
ejpam-6041	97	18	making	make	VERB
ejpam-6041	97	19	substitutions	substitution	NOUN
ejpam-6041	97	20	in	in	ADP
ejpam-6041	97	21	formula	formula	NOUN
ejpam-6041	97	22	(	(	PUNCT
ejpam-6041	97	23	4	4	NUM
ejpam-6041	97	24	)	)	PUNCT
ejpam-6041	97	25	.	.	PUNCT
ejpam-6041	98	1	we	we	PRON
ejpam-6041	98	2	are	be	AUX
ejpam-6041	98	3	now	now	ADV
ejpam-6041	98	4	in	in	ADP
ejpam-6041	98	5	a	a	DET
ejpam-6041	98	6	position	position	NOUN
ejpam-6041	98	7	to	to	PART
ejpam-6041	98	8	prove	prove	VERB
ejpam-6041	98	9	that	that	SCONJ
ejpam-6041	98	10	the	the	DET
ejpam-6041	98	11	associator	associator	NOUN
ejpam-6041	98	12	function	function	NOUN
ejpam-6041	98	13	of	of	ADP
ejpam-6041	98	14	the	the	DET
ejpam-6041	98	15	möbius	möbius	PROPN
ejpam-6041	98	16	gyrogroup	gyrogroup	PROPN
ejpam-6041	98	17	is	be	AUX
ejpam-6041	98	18	surjective	surjective	ADJ
ejpam-6041	98	19	.	.	PUNCT
ejpam-6041	99	1	this	this	PRON
ejpam-6041	99	2	implies	imply	VERB
ejpam-6041	99	3	that	that	SCONJ
ejpam-6041	99	4	every	every	DET
ejpam-6041	99	5	element	element	NOUN
ejpam-6041	99	6	of	of	ADP
ejpam-6041	99	7	the	the	DET
ejpam-6041	99	8	möbius	möbius	PROPN
ejpam-6041	99	9	gyrogroup	gyrogroup	PROPN
ejpam-6041	99	10	is	be	AUX
ejpam-6041	99	11	indeed	indeed	ADV
ejpam-6041	99	12	an	an	DET
ejpam-6041	99	13	associator	associator	NOUN
ejpam-6041	99	14	.	.	PUNCT
ejpam-6041	100	1	theorem	theorem	VERB
ejpam-6041	100	2	3	3	NUM
ejpam-6041	100	3	.	.	PUNCT
ejpam-6041	101	1	the	the	DET
ejpam-6041	101	2	associator	associator	NOUN
ejpam-6041	101	3	function	function	NOUN
ejpam-6041	101	4	of	of	ADP
ejpam-6041	101	5	the	the	DET
ejpam-6041	101	6	möbius	möbius	PROPN
ejpam-6041	101	7	gyrogroup	gyrogroup	PROPN
ejpam-6041	101	8	is	be	AUX
ejpam-6041	101	9	a	a	DET
ejpam-6041	101	10	surjective	surjective	ADJ
ejpam-6041	101	11	function	function	NOUN
ejpam-6041	101	12	from	from	ADP
ejpam-6041	101	13	d3	d3	PROPN
ejpam-6041	101	14	onto	onto	ADP
ejpam-6041	101	15	d.	d.	PROPN
ejpam-6041	101	16	proof	proof	PROPN
ejpam-6041	101	17	.	.	PUNCT
ejpam-6041	102	1	suppose	suppose	VERB
ejpam-6041	102	2	that	that	SCONJ
ejpam-6041	102	3	a	a	DET
ejpam-6041	102	4	,	,	PUNCT
ejpam-6041	102	5	b	b	NOUN
ejpam-6041	102	6	,	,	PUNCT
ejpam-6041	102	7	c	c	PROPN
ejpam-6041	102	8	∈	∈	PROPN
ejpam-6041	102	9	d.	d.	PROPN
ejpam-6041	102	10	first	first	ADV
ejpam-6041	102	11	,	,	PUNCT
ejpam-6041	102	12	we	we	PRON
ejpam-6041	102	13	show	show	VERB
ejpam-6041	102	14	that	that	SCONJ
ejpam-6041	102	15	1	1	NUM
ejpam-6041	102	16	+	+	NUM
ejpam-6041	102	17	ab	ab	PROPN
ejpam-6041	102	18	−	−	PROPN
ejpam-6041	102	19	|c|2(1	|c|2(1	NOUN
ejpam-6041	102	20	+	+	CCONJ
ejpam-6041	102	21	ab	ab	PROPN
ejpam-6041	102	22	)	)	PUNCT
ejpam-6041	102	23	̸=	̸=	PROPN
ejpam-6041	102	24	0	0	NUM
ejpam-6041	102	25	.	.	PUNCT
ejpam-6041	103	1	note	note	VERB
ejpam-6041	103	2	that	that	SCONJ
ejpam-6041	103	3	1+ab	1+ab	NUM
ejpam-6041	103	4	̸=	̸=	PROPN
ejpam-6041	103	5	0	0	PUNCT
ejpam-6041	103	6	since	since	SCONJ
ejpam-6041	103	7	otherwise	otherwise	ADV
ejpam-6041	103	8	−1	−1	NOUN
ejpam-6041	103	9	=	=	SYM
ejpam-6041	103	10	ab	ab	PROPN
ejpam-6041	103	11	implies	imply	VERB
ejpam-6041	103	12	|a||b|	|a||b|	X
ejpam-6041	104	1	=	=	SYM
ejpam-6041	104	2	1	1	NUM
ejpam-6041	104	3	,	,	PUNCT
ejpam-6041	104	4	an	an	DET
ejpam-6041	104	5	impossibility	impossibility	NOUN
ejpam-6041	104	6	.	.	PUNCT
ejpam-6041	105	1	note	note	VERB
ejpam-6041	105	2	also	also	ADV
ejpam-6041	105	3	that	that	SCONJ
ejpam-6041	105	4	1	1	X
ejpam-6041	106	1	+	+	NUM
ejpam-6041	106	2	ab−	ab−	NUM
ejpam-6041	106	3	|c|2(1	|c|2(1	NOUN
ejpam-6041	106	4	+	+	CCONJ
ejpam-6041	106	5	ab	ab	PROPN
ejpam-6041	106	6	)	)	PUNCT
ejpam-6041	106	7	=	=	PUNCT
ejpam-6041	106	8	(	(	PUNCT
ejpam-6041	106	9	1	1	NUM
ejpam-6041	106	10	+	+	NUM
ejpam-6041	106	11	ab	ab	PROPN
ejpam-6041	106	12	)	)	PUNCT
ejpam-6041	106	13	(	(	PUNCT
ejpam-6041	106	14	1	1	NUM
ejpam-6041	106	15	+	+	CCONJ
ejpam-6041	106	16	ab	ab	PROPN
ejpam-6041	106	17	1	1	NUM
ejpam-6041	106	18	+	+	NUM
ejpam-6041	106	19	ab	ab	PROPN
ejpam-6041	106	20	−	−	PROPN
ejpam-6041	106	21	|c|2	|c|2	PROPN
ejpam-6041	106	22	)	)	PUNCT
ejpam-6041	106	23	.	.	PUNCT
ejpam-6041	107	1	r.	r.	PROPN
ejpam-6041	107	2	maungchang	maungchang	PROPN
ejpam-6041	107	3	et	et	PROPN
ejpam-6041	107	4	al	al	PROPN
ejpam-6041	107	5	.	.	PUNCT
ejpam-6041	107	6	/	/	SYM
ejpam-6041	107	7	eur	eur	PROPN
ejpam-6041	107	8	.	.	PUNCT
ejpam-6041	108	1	j.	j.	PROPN
ejpam-6041	108	2	pure	pure	PROPN
ejpam-6041	108	3	appl	appl	PROPN
ejpam-6041	108	4	.	.	PROPN
ejpam-6041	108	5	math	math	PROPN
ejpam-6041	108	6	,	,	PUNCT
ejpam-6041	108	7	18	18	NUM
ejpam-6041	108	8	(	(	PUNCT
ejpam-6041	108	9	2	2	NUM
ejpam-6041	108	10	)	)	PUNCT
ejpam-6041	108	11	(	(	PUNCT
ejpam-6041	108	12	2025	2025	NUM
ejpam-6041	108	13	)	)	PUNCT
ejpam-6041	108	14	,	,	PUNCT
ejpam-6041	108	15	6041	6041	NUM
ejpam-6041	108	16	6	6	NUM
ejpam-6041	108	17	of	of	ADP
ejpam-6041	108	18	8	8	NUM
ejpam-6041	108	19	hence	hence	ADV
ejpam-6041	108	20	,	,	PUNCT
ejpam-6041	108	21	1	1	NUM
ejpam-6041	108	22	+	+	NUM
ejpam-6041	108	23	ab	ab	PROPN
ejpam-6041	108	24	−	−	PROPN
ejpam-6041	108	25	|c|2(1	|c|2(1	NOUN
ejpam-6041	108	26	+	+	CCONJ
ejpam-6041	108	27	ab	ab	PROPN
ejpam-6041	108	28	)	)	PUNCT
ejpam-6041	109	1	=	=	SYM
ejpam-6041	109	2	0	0	NUM
ejpam-6041	109	3	would	would	AUX
ejpam-6041	109	4	imply	imply	VERB
ejpam-6041	109	5	1	1	NUM
ejpam-6041	109	6	+	+	CCONJ
ejpam-6041	109	7	ab	ab	PROPN
ejpam-6041	109	8	1	1	NUM
ejpam-6041	109	9	+	+	CCONJ
ejpam-6041	109	10	ab	ab	PROPN
ejpam-6041	109	11	−	−	PROPN
ejpam-6041	109	12	|c|2	|c|2	PROPN
ejpam-6041	109	13	=	=	SYM
ejpam-6041	109	14	0	0	NUM
ejpam-6041	109	15	,	,	PUNCT
ejpam-6041	109	16	which	which	PRON
ejpam-6041	109	17	would	would	AUX
ejpam-6041	109	18	imply	imply	VERB
ejpam-6041	109	19	|c|2	|c|2	PROPN
ejpam-6041	109	20	=	=	SYM
ejpam-6041	109	21	1	1	NUM
ejpam-6041	109	22	+	+	CCONJ
ejpam-6041	109	23	ab	ab	PROPN
ejpam-6041	109	24	1	1	NUM
ejpam-6041	109	25	+	+	CCONJ
ejpam-6041	109	26	ab	ab	PROPN
ejpam-6041	109	27	,	,	PUNCT
ejpam-6041	109	28	and	and	CCONJ
ejpam-6041	109	29	so	so	ADV
ejpam-6041	109	30	|c|	|c|	PROPN
ejpam-6041	109	31	=	=	SYM
ejpam-6041	109	32	1	1	NUM
ejpam-6041	109	33	,	,	PUNCT
ejpam-6041	109	34	a	a	DET
ejpam-6041	109	35	contradiction	contradiction	NOUN
ejpam-6041	109	36	.	.	PUNCT
ejpam-6041	110	1	next	next	ADJ
ejpam-6041	110	2	,	,	PUNCT
ejpam-6041	110	3	1	1	NUM
ejpam-6041	110	4	+	+	CCONJ
ejpam-6041	110	5	ab+	ab+	NOUN
ejpam-6041	110	6	ac+	ac+	PROPN
ejpam-6041	110	7	bc	bc	PROPN
ejpam-6041	110	8	̸=	̸=	PROPN
ejpam-6041	110	9	0	0	NUM
ejpam-6041	110	10	,	,	PUNCT
ejpam-6041	110	11	as	as	SCONJ
ejpam-6041	110	12	proved	prove	VERB
ejpam-6041	110	13	in	in	ADP
ejpam-6041	110	14	the	the	DET
ejpam-6041	110	15	proof	proof	NOUN
ejpam-6041	110	16	of	of	ADP
ejpam-6041	110	17	theorem	theorem	NOUN
ejpam-6041	110	18	2	2	NUM
ejpam-6041	110	19	.	.	PUNCT
ejpam-6041	110	20	fix	fix	VERB
ejpam-6041	110	21	a	a	PRON
ejpam-6041	110	22	,	,	PUNCT
ejpam-6041	110	23	b	b	X
ejpam-6041	110	24	∈	∈	PROPN
ejpam-6041	110	25	d	d	NOUN
ejpam-6041	110	26	,	,	PUNCT
ejpam-6041	110	27	and	and	CCONJ
ejpam-6041	110	28	suppose	suppose	VERB
ejpam-6041	110	29	that	that	SCONJ
ejpam-6041	110	30	ab−	ab−	NUM
ejpam-6041	110	31	ab	ab	PROPN
ejpam-6041	110	32	̸=	̸=	PROPN
ejpam-6041	110	33	0	0	NUM
ejpam-6041	110	34	.	.	PUNCT
ejpam-6041	111	1	now	now	ADV
ejpam-6041	111	2	,	,	PUNCT
ejpam-6041	111	3	let	let	VERB
ejpam-6041	111	4	us	we	PRON
ejpam-6041	111	5	define	define	VERB
ejpam-6041	111	6	a	a	DET
ejpam-6041	111	7	function	function	NOUN
ejpam-6041	111	8	fa	fa	PROPN
ejpam-6041	111	9	,	,	PUNCT
ejpam-6041	111	10	b	b	PROPN
ejpam-6041	111	11	from	from	ADP
ejpam-6041	111	12	the	the	DET
ejpam-6041	111	13	close	close	ADJ
ejpam-6041	111	14	interval	interval	NOUN
ejpam-6041	111	15	[	[	X
ejpam-6041	111	16	0	0	NUM
ejpam-6041	111	17	,	,	PUNCT
ejpam-6041	111	18	1	1	NUM
ejpam-6041	111	19	]	]	PUNCT
ejpam-6041	111	20	to	to	ADP
ejpam-6041	111	21	the	the	DET
ejpam-6041	111	22	close	close	ADJ
ejpam-6041	111	23	disk	disk	NOUN
ejpam-6041	111	24	d	d	NOUN
ejpam-6041	111	25	=	=	PUNCT
ejpam-6041	111	26	{	{	PUNCT
ejpam-6041	111	27	z	z	NOUN
ejpam-6041	111	28	∈	∈	PROPN
ejpam-6041	111	29	c	c	NOUN
ejpam-6041	111	30	:	:	PUNCT
ejpam-6041	111	31	|z|	|z|	VERB
ejpam-6041	111	32	≤	≤	NOUN
ejpam-6041	111	33	1	1	NUM
ejpam-6041	111	34	}	}	PUNCT
ejpam-6041	111	35	by	by	ADP
ejpam-6041	111	36	the	the	DET
ejpam-6041	111	37	formula	formula	NOUN
ejpam-6041	111	38	fa	fa	NOUN
ejpam-6041	111	39	,	,	PUNCT
ejpam-6041	111	40	b(λ	b(λ	NOUN
ejpam-6041	111	41	)	)	PUNCT
ejpam-6041	111	42	=	=	PUNCT
ejpam-6041	112	1	λ(ab−	λ(ab−	NOUN
ejpam-6041	112	2	ab)(1	ab)(1	PRON
ejpam-6041	112	3	+	+	NUM
ejpam-6041	112	4	ab+	ab+	NOUN
ejpam-6041	112	5	aλ+	aλ+	PROPN
ejpam-6041	112	6	bλ	bλ	PROPN
ejpam-6041	112	7	)	)	PUNCT
ejpam-6041	112	8	(	(	PUNCT
ejpam-6041	112	9	1	1	NUM
ejpam-6041	112	10	+	+	NUM
ejpam-6041	112	11	ab−	ab−	NUM
ejpam-6041	112	12	|λ|2(1	|λ|2(1	NOUN
ejpam-6041	112	13	+	+	CCONJ
ejpam-6041	112	14	ab))(1	ab))(1	NOUN
ejpam-6041	112	15	+	+	CCONJ
ejpam-6041	112	16	ab+	ab+	NOUN
ejpam-6041	112	17	aλ+	aλ+	PROPN
ejpam-6041	112	18	bλ	bλ	PROPN
ejpam-6041	112	19	)	)	PUNCT
ejpam-6041	112	20	(	(	PUNCT
ejpam-6041	112	21	5	5	NUM
ejpam-6041	112	22	)	)	PUNCT
ejpam-6041	112	23	for	for	ADP
ejpam-6041	112	24	all	all	DET
ejpam-6041	112	25	λ	λ	X
ejpam-6041	112	26	∈	∈	PROPN
ejpam-6041	113	1	[	[	X
ejpam-6041	113	2	0	0	NUM
ejpam-6041	113	3	,	,	PUNCT
ejpam-6041	113	4	1	1	NUM
ejpam-6041	113	5	]	]	PUNCT
ejpam-6041	113	6	.	.	PUNCT
ejpam-6041	114	1	note	note	VERB
ejpam-6041	114	2	that	that	SCONJ
ejpam-6041	114	3	if	if	SCONJ
ejpam-6041	114	4	λ	λ	PROPN
ejpam-6041	114	5	̸=	̸=	PROPN
ejpam-6041	114	6	1	1	NUM
ejpam-6041	114	7	,	,	PUNCT
ejpam-6041	114	8	then	then	ADV
ejpam-6041	114	9	λ	λ	X
ejpam-6041	114	10	∈	∈	PROPN
ejpam-6041	114	11	d	d	NOUN
ejpam-6041	114	12	,	,	PUNCT
ejpam-6041	114	13	and	and	CCONJ
ejpam-6041	114	14	so	so	ADV
ejpam-6041	114	15	(	(	PUNCT
ejpam-6041	114	16	1	1	NUM
ejpam-6041	114	17	+	+	NUM
ejpam-6041	114	18	ab−	ab−	NUM
ejpam-6041	114	19	|λ|2(1	|λ|2(1	NOUN
ejpam-6041	114	20	+	+	CCONJ
ejpam-6041	114	21	ab))(1	ab))(1	NOUN
ejpam-6041	114	22	+	+	CCONJ
ejpam-6041	114	23	ab+	ab+	NOUN
ejpam-6041	114	24	aλ+	aλ+	PROPN
ejpam-6041	114	25	bλ	bλ	PROPN
ejpam-6041	114	26	)	)	PUNCT
ejpam-6041	114	27	̸=	̸=	PROPN
ejpam-6041	114	28	0	0	NUM
ejpam-6041	114	29	,	,	PUNCT
ejpam-6041	114	30	as	as	SCONJ
ejpam-6041	114	31	shown	show	VERB
ejpam-6041	114	32	above	above	ADV
ejpam-6041	114	33	.	.	PUNCT
ejpam-6041	115	1	in	in	ADP
ejpam-6041	115	2	the	the	DET
ejpam-6041	115	3	case	case	NOUN
ejpam-6041	115	4	when	when	SCONJ
ejpam-6041	115	5	λ	λ	X
ejpam-6041	115	6	=	=	SYM
ejpam-6041	115	7	1	1	NUM
ejpam-6041	115	8	,	,	PUNCT
ejpam-6041	115	9	we	we	PRON
ejpam-6041	115	10	obtain	obtain	VERB
ejpam-6041	115	11	that	that	SCONJ
ejpam-6041	115	12	(	(	PUNCT
ejpam-6041	115	13	1	1	NUM
ejpam-6041	115	14	+	+	NUM
ejpam-6041	115	15	ab−	ab−	NUM
ejpam-6041	115	16	|λ|2(1	|λ|2(1	NOUN
ejpam-6041	115	17	+	+	CCONJ
ejpam-6041	115	18	ab))(1	ab))(1	NOUN
ejpam-6041	115	19	+	+	CCONJ
ejpam-6041	115	20	ab+	ab+	NOUN
ejpam-6041	115	21	aλ+	aλ+	PROPN
ejpam-6041	115	22	bλ	bλ	PROPN
ejpam-6041	115	23	)	)	PUNCT
ejpam-6041	115	24	=	=	PUNCT
ejpam-6041	115	25	(	(	PUNCT
ejpam-6041	115	26	ab−	ab−	NUM
ejpam-6041	115	27	ab)(1	ab)(1	X
ejpam-6041	115	28	+	+	CCONJ
ejpam-6041	115	29	ab+	ab+	NOUN
ejpam-6041	115	30	a+	a+	PUNCT
ejpam-6041	115	31	b	b	NOUN
ejpam-6041	115	32	)	)	PUNCT
ejpam-6041	115	33	,	,	PUNCT
ejpam-6041	115	34	which	which	PRON
ejpam-6041	115	35	is	be	AUX
ejpam-6041	115	36	not	not	PART
ejpam-6041	115	37	zero	zero	NUM
ejpam-6041	115	38	since	since	SCONJ
ejpam-6041	115	39	1	1	NUM
ejpam-6041	115	40	+	+	CCONJ
ejpam-6041	115	41	ab+	ab+	NOUN
ejpam-6041	115	42	a+	a+	PRON
ejpam-6041	115	43	b	b	X
ejpam-6041	115	44	=	=	SYM
ejpam-6041	115	45	(	(	PUNCT
ejpam-6041	115	46	1	1	NUM
ejpam-6041	115	47	+	+	NUM
ejpam-6041	115	48	ab	ab	PROPN
ejpam-6041	115	49	)	)	PUNCT
ejpam-6041	115	50	(	(	PUNCT
ejpam-6041	115	51	1	1	NUM
ejpam-6041	115	52	+	+	CCONJ
ejpam-6041	115	53	ab	ab	PROPN
ejpam-6041	115	54	1	1	NUM
ejpam-6041	115	55	+	+	NUM
ejpam-6041	115	56	ab	ab	PROPN
ejpam-6041	116	1	+	+	CCONJ
ejpam-6041	116	2	a+	a+	PRON
ejpam-6041	116	3	b	b	X
ejpam-6041	116	4	1	1	NUM
ejpam-6041	116	5	+	+	NUM
ejpam-6041	116	6	ab	ab	PROPN
ejpam-6041	116	7	)	)	PUNCT
ejpam-6041	116	8	=	=	PUNCT
ejpam-6041	116	9	(	(	PUNCT
ejpam-6041	116	10	1	1	NUM
ejpam-6041	116	11	+	+	NUM
ejpam-6041	116	12	ab	ab	PROPN
ejpam-6041	116	13	)	)	PUNCT
ejpam-6041	116	14	(	(	PUNCT
ejpam-6041	116	15	1	1	NUM
ejpam-6041	116	16	+	+	CCONJ
ejpam-6041	116	17	ab	ab	PROPN
ejpam-6041	116	18	1	1	NUM
ejpam-6041	116	19	+	+	NUM
ejpam-6041	116	20	ab	ab	PROPN
ejpam-6041	116	21	+	+	CCONJ
ejpam-6041	116	22	(	(	PUNCT
ejpam-6041	116	23	a⊕m	a⊕m	PROPN
ejpam-6041	116	24	b	b	PROPN
ejpam-6041	116	25	)	)	PUNCT
ejpam-6041	116	26	)	)	PUNCT
ejpam-6041	116	27	.	.	PUNCT
ejpam-6041	117	1	this	this	PRON
ejpam-6041	117	2	shows	show	VERB
ejpam-6041	117	3	that	that	SCONJ
ejpam-6041	117	4	fa	fa	PROPN
ejpam-6041	117	5	,	,	PUNCT
ejpam-6041	117	6	b	b	PROPN
ejpam-6041	117	7	is	be	AUX
ejpam-6041	117	8	well	well	ADV
ejpam-6041	117	9	defined	define	VERB
ejpam-6041	117	10	.	.	PUNCT
ejpam-6041	118	1	since	since	SCONJ
ejpam-6041	118	2	fa	fa	PROPN
ejpam-6041	118	3	,	,	PUNCT
ejpam-6041	118	4	b	b	PROPN
ejpam-6041	118	5	is	be	AUX
ejpam-6041	118	6	defined	define	VERB
ejpam-6041	118	7	using	use	VERB
ejpam-6041	118	8	only	only	ADJ
ejpam-6041	118	9	addition	addition	NOUN
ejpam-6041	118	10	,	,	PUNCT
ejpam-6041	118	11	multiplication	multiplication	NOUN
ejpam-6041	118	12	,	,	PUNCT
ejpam-6041	118	13	subtraction	subtraction	NOUN
ejpam-6041	118	14	,	,	PUNCT
ejpam-6041	118	15	division	division	NOUN
ejpam-6041	118	16	,	,	PUNCT
ejpam-6041	118	17	and	and	CCONJ
ejpam-6041	118	18	conjugation	conjugation	NOUN
ejpam-6041	118	19	(	(	PUNCT
ejpam-6041	118	20	which	which	PRON
ejpam-6041	118	21	are	be	AUX
ejpam-6041	118	22	all	all	PRON
ejpam-6041	118	23	continuous	continuous	ADJ
ejpam-6041	118	24	functions	function	NOUN
ejpam-6041	118	25	)	)	PUNCT
ejpam-6041	118	26	,	,	PUNCT
ejpam-6041	118	27	it	it	PRON
ejpam-6041	118	28	follows	follow	VERB
ejpam-6041	118	29	that	that	SCONJ
ejpam-6041	118	30	fa	fa	PROPN
ejpam-6041	118	31	,	,	PUNCT
ejpam-6041	118	32	b	b	PROPN
ejpam-6041	118	33	is	be	AUX
ejpam-6041	118	34	continuous	continuous	ADJ
ejpam-6041	118	35	.	.	PUNCT
ejpam-6041	119	1	fix	fix	VERB
ejpam-6041	119	2	a	a	PRON
ejpam-6041	119	3	,	,	PUNCT
ejpam-6041	119	4	b	b	X
ejpam-6041	119	5	∈	∈	PROPN
ejpam-6041	119	6	d	d	NOUN
ejpam-6041	119	7	,	,	PUNCT
ejpam-6041	119	8	and	and	CCONJ
ejpam-6041	119	9	suppose	suppose	VERB
ejpam-6041	119	10	that	that	SCONJ
ejpam-6041	119	11	ab−ab	ab−ab	NOUN
ejpam-6041	119	12	̸=	̸=	PROPN
ejpam-6041	119	13	0	0	NUM
ejpam-6041	119	14	.	.	PUNCT
ejpam-6041	119	15	define	define	VERB
ejpam-6041	119	16	a	a	DET
ejpam-6041	119	17	function	function	NOUN
ejpam-6041	119	18	ga	ga	PROPN
ejpam-6041	119	19	,	,	PUNCT
ejpam-6041	119	20	b	b	PROPN
ejpam-6041	119	21	by	by	ADP
ejpam-6041	119	22	ga	ga	PROPN
ejpam-6041	119	23	,	,	PUNCT
ejpam-6041	119	24	b(λ	b(λ	PROPN
ejpam-6041	119	25	)	)	PUNCT
ejpam-6041	119	26	=	=	SYM
ejpam-6041	119	27	|fa	|fa	NOUN
ejpam-6041	119	28	,	,	PUNCT
ejpam-6041	119	29	b(λ)|	b(λ)|	NOUN
ejpam-6041	119	30	for	for	ADP
ejpam-6041	119	31	all	all	DET
ejpam-6041	119	32	λ	λ	X
ejpam-6041	119	33	∈	∈	PROPN
ejpam-6041	120	1	[	[	X
ejpam-6041	120	2	0	0	NUM
ejpam-6041	120	3	,	,	PUNCT
ejpam-6041	120	4	1	1	NUM
ejpam-6041	120	5	]	]	PUNCT
ejpam-6041	120	6	.	.	PUNCT
ejpam-6041	121	1	since	since	SCONJ
ejpam-6041	121	2	the	the	DET
ejpam-6041	121	3	complex	complex	ADJ
ejpam-6041	121	4	-	-	PUNCT
ejpam-6041	121	5	modulus	modulus	ADJ
ejpam-6041	121	6	function	function	NOUN
ejpam-6041	121	7	is	be	AUX
ejpam-6041	121	8	continuous	continuous	ADJ
ejpam-6041	121	9	,	,	PUNCT
ejpam-6041	121	10	it	it	PRON
ejpam-6041	121	11	follows	follow	VERB
ejpam-6041	121	12	that	that	SCONJ
ejpam-6041	121	13	ga	ga	PROPN
ejpam-6041	121	14	,	,	PUNCT
ejpam-6041	121	15	b	b	PROPN
ejpam-6041	121	16	is	be	AUX
ejpam-6041	121	17	a	a	DET
ejpam-6041	121	18	continuous	continuous	ADJ
ejpam-6041	121	19	function	function	NOUN
ejpam-6041	121	20	from	from	ADP
ejpam-6041	121	21	[	[	X
ejpam-6041	121	22	0	0	NUM
ejpam-6041	121	23	,	,	PUNCT
ejpam-6041	121	24	1	1	NUM
ejpam-6041	121	25	]	]	PUNCT
ejpam-6041	121	26	to	to	ADP
ejpam-6041	121	27	[	[	X
ejpam-6041	121	28	0	0	NUM
ejpam-6041	121	29	,	,	PUNCT
ejpam-6041	121	30	1	1	NUM
ejpam-6041	121	31	]	]	PUNCT
ejpam-6041	121	32	.	.	PUNCT
ejpam-6041	122	1	note	note	VERB
ejpam-6041	122	2	that	that	SCONJ
ejpam-6041	122	3	ga	ga	PROPN
ejpam-6041	122	4	,	,	PUNCT
ejpam-6041	122	5	b(0	b(0	PROPN
ejpam-6041	122	6	)	)	PUNCT
ejpam-6041	122	7	=	=	SYM
ejpam-6041	122	8	0	0	PUNCT
ejpam-6041	122	9	and	and	CCONJ
ejpam-6041	122	10	that	that	DET
ejpam-6041	122	11	ga	ga	PROPN
ejpam-6041	122	12	,	,	PUNCT
ejpam-6041	122	13	b(1	b(1	PROPN
ejpam-6041	122	14	)	)	PUNCT
ejpam-6041	122	15	=	=	NOUN
ejpam-6041	122	16	1	1	NUM
ejpam-6041	122	17	since	since	SCONJ
ejpam-6041	122	18	fa	fa	PROPN
ejpam-6041	122	19	,	,	PUNCT
ejpam-6041	122	20	b(1	b(1	PROPN
ejpam-6041	122	21	)	)	PUNCT
ejpam-6041	122	22	=	=	PUNCT
ejpam-6041	123	1	(	(	PUNCT
ejpam-6041	123	2	ab−	ab−	NUM
ejpam-6041	123	3	ab)(1	ab)(1	X
ejpam-6041	123	4	+	+	CCONJ
ejpam-6041	123	5	ab+	ab+	NOUN
ejpam-6041	123	6	a+	a+	PUNCT
ejpam-6041	123	7	b	b	X
ejpam-6041	123	8	)	)	PUNCT
ejpam-6041	123	9	(	(	PUNCT
ejpam-6041	123	10	ab−	ab−	NOUN
ejpam-6041	123	11	ab)(1	ab)(1	X
ejpam-6041	123	12	+	+	CCONJ
ejpam-6041	123	13	ab+	ab+	NOUN
ejpam-6041	123	14	a+	a+	PUNCT
ejpam-6041	123	15	b	b	X
ejpam-6041	123	16	)	)	PUNCT
ejpam-6041	123	17	=	=	SYM
ejpam-6041	123	18	−1	−1	NOUN
ejpam-6041	123	19	+	+	NUM
ejpam-6041	123	20	ab+	ab+	NOUN
ejpam-6041	123	21	a+	a+	PUNCT
ejpam-6041	123	22	b	b	PROPN
ejpam-6041	123	23	1	1	NUM
ejpam-6041	123	24	+	+	NUM
ejpam-6041	123	25	ab+	ab+	NOUN
ejpam-6041	123	26	a+	a+	PRON
ejpam-6041	123	27	b	b	X
ejpam-6041	124	1	so	so	SCONJ
ejpam-6041	124	2	that	that	SCONJ
ejpam-6041	124	3	|fa	|fa	NOUN
ejpam-6041	124	4	,	,	PUNCT
ejpam-6041	124	5	b(1)|	b(1)|	PROPN
ejpam-6041	124	6	=	=	NOUN
ejpam-6041	124	7	1	1	X
ejpam-6041	124	8	.	.	PUNCT
ejpam-6041	124	9	by	by	ADP
ejpam-6041	124	10	the	the	DET
ejpam-6041	124	11	intermediate	intermediate	ADJ
ejpam-6041	124	12	value	value	NOUN
ejpam-6041	124	13	theorem	theorem	NOUN
ejpam-6041	124	14	(	(	PUNCT
ejpam-6041	124	15	see	see	VERB
ejpam-6041	124	16	,	,	PUNCT
ejpam-6041	124	17	for	for	ADP
ejpam-6041	124	18	instance	instance	NOUN
ejpam-6041	124	19	,	,	PUNCT
ejpam-6041	124	20	page	page	NOUN
ejpam-6041	124	21	26	26	NUM
ejpam-6041	124	22	of	of	ADP
ejpam-6041	124	23	[	[	X
ejpam-6041	124	24	6	6	NUM
ejpam-6041	124	25	]	]	NUM
ejpam-6041	124	26	)	)	PUNCT
ejpam-6041	124	27	,	,	PUNCT
ejpam-6041	124	28	if	if	SCONJ
ejpam-6041	124	29	0	0	NUM
ejpam-6041	124	30	<	<	X
ejpam-6041	124	31	r	r	X
ejpam-6041	124	32	<	<	X
ejpam-6041	124	33	1	1	NUM
ejpam-6041	124	34	,	,	PUNCT
ejpam-6041	124	35	then	then	ADV
ejpam-6041	124	36	ga	ga	PROPN
ejpam-6041	124	37	,	,	PUNCT
ejpam-6041	124	38	b(λ0	b(λ0	NOUN
ejpam-6041	124	39	)	)	PUNCT
ejpam-6041	124	40	=	=	SYM
ejpam-6041	124	41	r	r	NOUN
ejpam-6041	124	42	for	for	ADP
ejpam-6041	124	43	some	some	DET
ejpam-6041	124	44	λ0	λ0	NOUN
ejpam-6041	124	45	∈	∈	NOUN
ejpam-6041	124	46	(	(	PUNCT
ejpam-6041	124	47	0	0	NUM
ejpam-6041	124	48	,	,	PUNCT
ejpam-6041	124	49	1	1	NUM
ejpam-6041	124	50	)	)	PUNCT
ejpam-6041	124	51	.	.	PUNCT
ejpam-6041	125	1	we	we	PRON
ejpam-6041	125	2	are	be	AUX
ejpam-6041	125	3	now	now	ADV
ejpam-6041	125	4	in	in	ADP
ejpam-6041	125	5	a	a	DET
ejpam-6041	125	6	position	position	NOUN
ejpam-6041	125	7	to	to	PART
ejpam-6041	125	8	prove	prove	VERB
ejpam-6041	125	9	that	that	PRON
ejpam-6041	125	10	am	be	AUX
ejpam-6041	125	11	is	be	AUX
ejpam-6041	125	12	surjective	surjective	ADJ
ejpam-6041	125	13	.	.	PUNCT
ejpam-6041	126	1	let	let	VERB
ejpam-6041	126	2	w	w	PROPN
ejpam-6041	126	3	∈	∈	PROPN
ejpam-6041	126	4	d.	d.	PROPN
ejpam-6041	126	5	in	in	ADP
ejpam-6041	126	6	the	the	DET
ejpam-6041	126	7	case	case	NOUN
ejpam-6041	126	8	when	when	SCONJ
ejpam-6041	126	9	w	w	PROPN
ejpam-6041	126	10	=	=	NOUN
ejpam-6041	126	11	0	0	NUM
ejpam-6041	126	12	,	,	PUNCT
ejpam-6041	126	13	we	we	PRON
ejpam-6041	126	14	obtain	obtain	VERB
ejpam-6041	126	15	that	that	PRON
ejpam-6041	126	16	am	be	AUX
ejpam-6041	126	17	(	(	PUNCT
ejpam-6041	126	18	0	0	NUM
ejpam-6041	126	19	,	,	PUNCT
ejpam-6041	126	20	0	0	NUM
ejpam-6041	126	21	,	,	PUNCT
ejpam-6041	126	22	0	0	NUM
ejpam-6041	126	23	)	)	PUNCT
ejpam-6041	126	24	=	=	SYM
ejpam-6041	127	1	0	0	X
ejpam-6041	127	2	.	.	PUNCT
ejpam-6041	128	1	now	now	ADV
ejpam-6041	128	2	,	,	PUNCT
ejpam-6041	128	3	suppose	suppose	VERB
ejpam-6041	128	4	that	that	SCONJ
ejpam-6041	128	5	w	w	PROPN
ejpam-6041	128	6	̸=	̸=	PROPN
ejpam-6041	128	7	0	0	NUM
ejpam-6041	128	8	.	.	PUNCT
ejpam-6041	129	1	using	use	VERB
ejpam-6041	129	2	the	the	DET
ejpam-6041	129	3	polar	polar	ADJ
ejpam-6041	129	4	form	form	NOUN
ejpam-6041	129	5	,	,	PUNCT
ejpam-6041	129	6	we	we	PRON
ejpam-6041	129	7	can	can	AUX
ejpam-6041	129	8	write	write	VERB
ejpam-6041	129	9	w	w	PROPN
ejpam-6041	129	10	=	=	SYM
ejpam-6041	129	11	|w|(cosα+	|w|(cosα+	NOUN
ejpam-6041	129	12	i	i	PRON
ejpam-6041	129	13	sinα	sinα	VERB
ejpam-6041	129	14	)	)	PUNCT
ejpam-6041	129	15	for	for	ADP
ejpam-6041	129	16	some	some	DET
ejpam-6041	129	17	α	α	PROPN
ejpam-6041	129	18	∈	∈	PROPN
ejpam-6041	129	19	r.	r.	PROPN
ejpam-6041	129	20	note	note	NOUN
ejpam-6041	129	21	that	that	PRON
ejpam-6041	129	22	|w|	|w|	VERB
ejpam-6041	129	23	̸=	̸=	PROPN
ejpam-6041	129	24	0	0	NUM
ejpam-6041	129	25	.	.	PUNCT
ejpam-6041	130	1	choose	choose	VERB
ejpam-6041	130	2	a	a	DET
ejpam-6041	130	3	,	,	PUNCT
ejpam-6041	130	4	b	b	X
ejpam-6041	130	5	∈	∈	PROPN
ejpam-6041	130	6	d	d	X
ejpam-6041	130	7	such	such	ADJ
ejpam-6041	130	8	that	that	SCONJ
ejpam-6041	130	9	ab	ab	PROPN
ejpam-6041	130	10	−	−	PROPN
ejpam-6041	130	11	ab	ab	PROPN
ejpam-6041	130	12	̸=	̸=	PROPN
ejpam-6041	130	13	0	0	NUM
ejpam-6041	130	14	(	(	PUNCT
ejpam-6041	130	15	for	for	ADP
ejpam-6041	130	16	example	example	NOUN
ejpam-6041	130	17	,	,	PUNCT
ejpam-6041	130	18	a	a	DET
ejpam-6041	130	19	=	=	SYM
ejpam-6041	130	20	0.5	0.5	NUM
ejpam-6041	130	21	and	and	CCONJ
ejpam-6041	130	22	b	b	NOUN
ejpam-6041	130	23	=	=	SYM
ejpam-6041	130	24	0.5i	0.5i	NUM
ejpam-6041	130	25	)	)	PUNCT
ejpam-6041	130	26	.	.	PUNCT
ejpam-6041	131	1	as	as	ADP
ejpam-6041	131	2	above	above	ADV
ejpam-6041	131	3	,	,	PUNCT
ejpam-6041	131	4	ga	ga	NOUN
ejpam-6041	131	5	,	,	PUNCT
ejpam-6041	131	6	b(λ0	b(λ0	NOUN
ejpam-6041	131	7	)	)	PUNCT
ejpam-6041	131	8	=	=	SYM
ejpam-6041	131	9	|w|	|w|	ADJ
ejpam-6041	131	10	for	for	ADP
ejpam-6041	131	11	some	some	DET
ejpam-6041	131	12	λ0	λ0	NOUN
ejpam-6041	131	13	∈	∈	NOUN
ejpam-6041	131	14	(	(	PUNCT
ejpam-6041	131	15	0	0	NUM
ejpam-6041	131	16	,	,	PUNCT
ejpam-6041	131	17	1	1	NUM
ejpam-6041	131	18	)	)	PUNCT
ejpam-6041	131	19	.	.	PUNCT
ejpam-6041	132	1	note	note	VERB
ejpam-6041	132	2	that	that	SCONJ
ejpam-6041	132	3	|w|	|w|	PROPN
ejpam-6041	132	4	=	=	SYM
ejpam-6041	132	5	ga	ga	PROPN
ejpam-6041	132	6	,	,	PUNCT
ejpam-6041	132	7	b(λ0	b(λ0	NOUN
ejpam-6041	132	8	)	)	PUNCT
ejpam-6041	132	9	=	=	SYM
ejpam-6041	132	10	|fa	|fa	NOUN
ejpam-6041	132	11	,	,	PUNCT
ejpam-6041	132	12	b(λ0)|	b(λ0)|	NOUN
ejpam-6041	132	13	.	.	PUNCT
ejpam-6041	133	1	hence	hence	ADV
ejpam-6041	133	2	,	,	PUNCT
ejpam-6041	133	3	fa	fa	INTJ
ejpam-6041	133	4	,	,	PUNCT
ejpam-6041	133	5	b(λ0	b(λ0	NOUN
ejpam-6041	133	6	)	)	PUNCT
ejpam-6041	133	7	̸=	̸=	PROPN
ejpam-6041	133	8	0	0	NUM
ejpam-6041	133	9	.	.	PUNCT
ejpam-6041	133	10	assume	assume	VERB
ejpam-6041	133	11	that	that	SCONJ
ejpam-6041	133	12	fa	fa	NOUN
ejpam-6041	133	13	,	,	PUNCT
ejpam-6041	133	14	b(λ0	b(λ0	NOUN
ejpam-6041	133	15	)	)	PUNCT
ejpam-6041	133	16	has	have	VERB
ejpam-6041	133	17	a	a	DET
ejpam-6041	133	18	polar	polar	ADJ
ejpam-6041	133	19	form	form	NOUN
ejpam-6041	133	20	as	as	ADP
ejpam-6041	133	21	fa	fa	NOUN
ejpam-6041	133	22	,	,	PUNCT
ejpam-6041	133	23	b(λ0	b(λ0	NOUN
ejpam-6041	133	24	)	)	PUNCT
ejpam-6041	133	25	=	=	PUNCT
ejpam-6041	134	1	r(cosβ	r(cosβ	NOUN
ejpam-6041	135	1	+	+	CCONJ
ejpam-6041	135	2	i	i	PRON
ejpam-6041	135	3	sinβ	sinβ	NOUN
ejpam-6041	135	4	)	)	PUNCT
ejpam-6041	135	5	,	,	PUNCT
ejpam-6041	135	6	where	where	SCONJ
ejpam-6041	135	7	0	0	NUM
ejpam-6041	135	8	<	<	X
ejpam-6041	135	9	r	r	X
ejpam-6041	135	10	<	<	X
ejpam-6041	135	11	1	1	NUM
ejpam-6041	135	12	and	and	CCONJ
ejpam-6041	135	13	β	β	PROPN
ejpam-6041	135	14	∈	∈	PROPN
ejpam-6041	135	15	r.	r.	PROPN
ejpam-6041	135	16	thus	thus	ADV
ejpam-6041	135	17	,	,	PUNCT
ejpam-6041	135	18	fa	fa	INTJ
ejpam-6041	135	19	,	,	PUNCT
ejpam-6041	135	20	b(λ0	b(λ0	NOUN
ejpam-6041	135	21	)	)	PUNCT
ejpam-6041	135	22	=	=	PUNCT
ejpam-6041	136	1	|w|(cosβ	|w|(cosβ	NOUN
ejpam-6041	136	2	+	+	CCONJ
ejpam-6041	136	3	i	i	PRON
ejpam-6041	136	4	sinβ	sinβ	VERB
ejpam-6041	136	5	)	)	PUNCT
ejpam-6041	136	6	.	.	PUNCT
ejpam-6041	137	1	set	set	VERB
ejpam-6041	137	2	ω	ω	PROPN
ejpam-6041	137	3	=	=	PUNCT
ejpam-6041	137	4	cos	cos	PROPN
ejpam-6041	137	5	(	(	PUNCT
ejpam-6041	137	6	α−	α−	ADP
ejpam-6041	137	7	β	β	NOUN
ejpam-6041	137	8	)	)	PUNCT
ejpam-6041	138	1	+	+	CCONJ
ejpam-6041	138	2	i	i	PRON
ejpam-6041	138	3	sin	sin	VERB
ejpam-6041	138	4	(	(	PUNCT
ejpam-6041	138	5	α−	α−	ADP
ejpam-6041	138	6	β	β	NOUN
ejpam-6041	138	7	)	)	PUNCT
ejpam-6041	138	8	.	.	PUNCT
ejpam-6041	139	1	then	then	ADV
ejpam-6041	139	2	|ω|	|ω|	VERB
ejpam-6041	139	3	=	=	SYM
ejpam-6041	139	4	1	1	X
ejpam-6041	139	5	.	.	X
ejpam-6041	139	6	from	from	ADP
ejpam-6041	139	7	lemma	lemma	PROPN
ejpam-6041	139	8	1	1	NUM
ejpam-6041	139	9	,	,	PUNCT
ejpam-6041	139	10	it	it	PRON
ejpam-6041	139	11	follows	follow	VERB
ejpam-6041	139	12	that	that	PRON
ejpam-6041	139	13	am	be	AUX
ejpam-6041	139	14	(	(	PUNCT
ejpam-6041	139	15	ωa	ωa	ADJ
ejpam-6041	139	16	,	,	PUNCT
ejpam-6041	139	17	ωb	ωb	NOUN
ejpam-6041	139	18	,	,	PUNCT
ejpam-6041	139	19	ωλ0	ωλ0	NOUN
ejpam-6041	139	20	)	)	PUNCT
ejpam-6041	139	21	=	=	SYM
ejpam-6041	139	22	ωam	ωam	X
ejpam-6041	139	23	(	(	PUNCT
ejpam-6041	139	24	a	a	PRON
ejpam-6041	139	25	,	,	PUNCT
ejpam-6041	139	26	b	b	NOUN
ejpam-6041	139	27	,	,	PUNCT
ejpam-6041	139	28	λ0	λ0	NOUN
ejpam-6041	139	29	)	)	PUNCT
ejpam-6041	139	30	r.	r.	PROPN
ejpam-6041	139	31	maungchang	maungchang	PROPN
ejpam-6041	139	32	et	et	PROPN
ejpam-6041	139	33	al	al	PROPN
ejpam-6041	139	34	.	.	PUNCT
ejpam-6041	139	35	/	/	SYM
ejpam-6041	139	36	eur	eur	PROPN
ejpam-6041	139	37	.	.	PUNCT
ejpam-6041	140	1	j.	j.	PROPN
ejpam-6041	140	2	pure	pure	PROPN
ejpam-6041	140	3	appl	appl	PROPN
ejpam-6041	140	4	.	.	PROPN
ejpam-6041	140	5	math	math	PROPN
ejpam-6041	140	6	,	,	PUNCT
ejpam-6041	140	7	18	18	NUM
ejpam-6041	140	8	(	(	PUNCT
ejpam-6041	140	9	2	2	NUM
ejpam-6041	140	10	)	)	PUNCT
ejpam-6041	140	11	(	(	PUNCT
ejpam-6041	140	12	2025	2025	NUM
ejpam-6041	140	13	)	)	PUNCT
ejpam-6041	140	14	,	,	PUNCT
ejpam-6041	140	15	6041	6041	NUM
ejpam-6041	140	16	7	7	NUM
ejpam-6041	140	17	of	of	ADP
ejpam-6041	140	18	8	8	NUM
ejpam-6041	140	19	=	=	SYM
ejpam-6041	140	20	ωfa	ωfa	NOUN
ejpam-6041	140	21	,	,	PUNCT
ejpam-6041	140	22	b(λ0	b(λ0	NOUN
ejpam-6041	140	23	)	)	PUNCT
ejpam-6041	140	24	=	=	SYM
ejpam-6041	141	1	(	(	PUNCT
ejpam-6041	141	2	cos	cos	X
ejpam-6041	141	3	(	(	PUNCT
ejpam-6041	141	4	α−	α−	ADP
ejpam-6041	141	5	β	β	NOUN
ejpam-6041	141	6	)	)	PUNCT
ejpam-6041	142	1	+	+	CCONJ
ejpam-6041	142	2	i	i	PRON
ejpam-6041	142	3	sin	sin	VERB
ejpam-6041	142	4	(	(	PUNCT
ejpam-6041	142	5	α−	α−	ADP
ejpam-6041	142	6	β))|w|(cosβ	β))|w|(cosβ	NUM
ejpam-6041	143	1	+	+	NUM
ejpam-6041	143	2	i	i	PRON
ejpam-6041	143	3	sinβ	sinβ	VERB
ejpam-6041	143	4	)	)	PUNCT
ejpam-6041	144	1	=	=	SYM
ejpam-6041	144	2	|w|(cosα+	|w|(cosα+	NOUN
ejpam-6041	145	1	i	i	PRON
ejpam-6041	145	2	sinα	sinα	VERB
ejpam-6041	145	3	)	)	PUNCT
ejpam-6041	146	1	=	=	SYM
ejpam-6041	146	2	w.	w.	NOUN
ejpam-6041	146	3	note	note	VERB
ejpam-6041	146	4	that	that	SCONJ
ejpam-6041	146	5	ωa	ωa	VERB
ejpam-6041	146	6	,	,	PUNCT
ejpam-6041	146	7	ωb	ωb	NOUN
ejpam-6041	146	8	,	,	PUNCT
ejpam-6041	146	9	and	and	CCONJ
ejpam-6041	146	10	ωλ0	ωλ0	NOUN
ejpam-6041	146	11	are	be	AUX
ejpam-6041	146	12	in	in	ADP
ejpam-6041	146	13	d.	d.	PROPN
ejpam-6041	146	14	this	this	PRON
ejpam-6041	146	15	shows	show	VERB
ejpam-6041	146	16	that	that	PRON
ejpam-6041	146	17	am	be	AUX
ejpam-6041	146	18	is	be	AUX
ejpam-6041	146	19	surjective	surjective	ADJ
ejpam-6041	146	20	,	,	PUNCT
ejpam-6041	146	21	as	as	SCONJ
ejpam-6041	146	22	claimed	claim	VERB
ejpam-6041	146	23	.	.	PUNCT
ejpam-6041	147	1	the	the	DET
ejpam-6041	147	2	notion	notion	NOUN
ejpam-6041	147	3	of	of	ADP
ejpam-6041	147	4	associators	associator	NOUN
ejpam-6041	147	5	can	can	AUX
ejpam-6041	147	6	be	be	AUX
ejpam-6041	147	7	used	use	VERB
ejpam-6041	147	8	to	to	PART
ejpam-6041	147	9	measure	measure	VERB
ejpam-6041	147	10	the	the	DET
ejpam-6041	147	11	deviation	deviation	NOUN
ejpam-6041	147	12	from	from	ADP
ejpam-6041	147	13	associativity	associativity	NOUN
ejpam-6041	147	14	of	of	ADP
ejpam-6041	147	15	möbius	möbius	PROPN
ejpam-6041	147	16	addition	addition	NOUN
ejpam-6041	147	17	.	.	PUNCT
ejpam-6041	148	1	following	follow	VERB
ejpam-6041	148	2	[	[	X
ejpam-6041	148	3	4	4	NUM
ejpam-6041	148	4	]	]	PUNCT
ejpam-6041	148	5	,	,	PUNCT
ejpam-6041	148	6	the	the	DET
ejpam-6041	148	7	normal	normal	ADJ
ejpam-6041	148	8	closure	closure	NOUN
ejpam-6041	148	9	of	of	ADP
ejpam-6041	148	10	the	the	DET
ejpam-6041	148	11	set	set	NOUN
ejpam-6041	148	12	of	of	ADP
ejpam-6041	148	13	all	all	DET
ejpam-6041	148	14	associators	associator	NOUN
ejpam-6041	148	15	in	in	ADP
ejpam-6041	148	16	the	the	DET
ejpam-6041	148	17	möbius	möbius	PROPN
ejpam-6041	148	18	gyrogroup	gyrogroup	PROPN
ejpam-6041	148	19	is	be	AUX
ejpam-6041	148	20	called	call	VERB
ejpam-6041	148	21	the	the	DET
ejpam-6041	148	22	associator	associator	NOUN
ejpam-6041	148	23	normal	normal	ADJ
ejpam-6041	148	24	subgyrogroup	subgyrogroup	NOUN
ejpam-6041	148	25	,	,	PUNCT
ejpam-6041	148	26	denoted	denote	VERB
ejpam-6041	148	27	by	by	ADP
ejpam-6041	148	28	da	da	PROPN
ejpam-6041	148	29	,	,	PUNCT
ejpam-6041	148	30	which	which	PRON
ejpam-6041	148	31	is	be	AUX
ejpam-6041	148	32	the	the	DET
ejpam-6041	148	33	smallest	small	ADJ
ejpam-6041	148	34	normal	normal	ADJ
ejpam-6041	148	35	subgyrogroup	subgyrogroup	NOUN
ejpam-6041	148	36	of	of	ADP
ejpam-6041	148	37	d	d	NOUN
ejpam-6041	148	38	containing	contain	VERB
ejpam-6041	148	39	all	all	DET
ejpam-6041	148	40	the	the	DET
ejpam-6041	148	41	associators	associator	NOUN
ejpam-6041	148	42	in	in	ADP
ejpam-6041	148	43	d.	d.	PROPN
ejpam-6041	148	44	according	accord	VERB
ejpam-6041	148	45	to	to	ADP
ejpam-6041	148	46	proposition	proposition	NOUN
ejpam-6041	148	47	3.4	3.4	NUM
ejpam-6041	148	48	of	of	ADP
ejpam-6041	148	49	[	[	X
ejpam-6041	148	50	4	4	NUM
ejpam-6041	148	51	]	]	PUNCT
ejpam-6041	148	52	,	,	PUNCT
ejpam-6041	148	53	da	da	PROPN
ejpam-6041	148	54	is	be	AUX
ejpam-6041	148	55	the	the	DET
ejpam-6041	148	56	unique	unique	ADJ
ejpam-6041	148	57	normal	normal	ADJ
ejpam-6041	148	58	subgyrogroup	subgyrogroup	NOUN
ejpam-6041	148	59	of	of	ADP
ejpam-6041	148	60	d	d	NOUN
ejpam-6041	148	61	such	such	ADJ
ejpam-6041	148	62	that	that	PRON
ejpam-6041	148	63	d	d	NOUN
ejpam-6041	148	64	/	/	SYM
ejpam-6041	148	65	da	da	PROPN
ejpam-6041	148	66	is	be	AUX
ejpam-6041	148	67	a	a	DET
ejpam-6041	148	68	group	group	NOUN
ejpam-6041	148	69	and	and	CCONJ
ejpam-6041	148	70	if	if	SCONJ
ejpam-6041	148	71	φ	φ	PROPN
ejpam-6041	148	72	is	be	AUX
ejpam-6041	148	73	a	a	DET
ejpam-6041	148	74	homomorphism	homomorphism	NOUN
ejpam-6041	148	75	from	from	ADP
ejpam-6041	148	76	d	d	PROPN
ejpam-6041	148	77	to	to	ADP
ejpam-6041	148	78	a	a	DET
ejpam-6041	148	79	group	group	NOUN
ejpam-6041	148	80	,	,	PUNCT
ejpam-6041	148	81	then	then	ADV
ejpam-6041	148	82	da	da	PROPN
ejpam-6041	148	83	lies	lie	VERB
ejpam-6041	148	84	in	in	ADP
ejpam-6041	148	85	the	the	DET
ejpam-6041	148	86	kernel	kernel	NOUN
ejpam-6041	148	87	of	of	ADP
ejpam-6041	148	88	φ	φ	PROPN
ejpam-6041	148	89	.	.	PUNCT
ejpam-6041	149	1	the	the	DET
ejpam-6041	149	2	quotient	quotient	NOUN
ejpam-6041	149	3	d	d	NOUN
ejpam-6041	149	4	/	/	SYM
ejpam-6041	149	5	da	da	PROPN
ejpam-6041	149	6	is	be	AUX
ejpam-6041	149	7	referred	refer	VERB
ejpam-6041	149	8	to	to	ADP
ejpam-6041	149	9	as	as	ADP
ejpam-6041	149	10	the	the	DET
ejpam-6041	149	11	associativization	associativization	NOUN
ejpam-6041	149	12	of	of	ADP
ejpam-6041	149	13	the	the	DET
ejpam-6041	149	14	möbius	möbius	PROPN
ejpam-6041	149	15	gyrogroup	gyrogroup	PROPN
ejpam-6041	149	16	.	.	PUNCT
ejpam-6041	150	1	from	from	ADP
ejpam-6041	150	2	theorem	theorem	ADJ
ejpam-6041	150	3	3	3	NUM
ejpam-6041	150	4	,	,	PUNCT
ejpam-6041	150	5	we	we	PRON
ejpam-6041	150	6	know	know	VERB
ejpam-6041	150	7	that	that	SCONJ
ejpam-6041	150	8	every	every	DET
ejpam-6041	150	9	element	element	NOUN
ejpam-6041	150	10	of	of	ADP
ejpam-6041	150	11	the	the	DET
ejpam-6041	150	12	möbius	möbius	PROPN
ejpam-6041	150	13	gyrogroup	gyrogroup	PROPN
ejpam-6041	150	14	is	be	AUX
ejpam-6041	150	15	an	an	DET
ejpam-6041	150	16	associator	associator	NOUN
ejpam-6041	150	17	.	.	PUNCT
ejpam-6041	151	1	therefore	therefore	ADV
ejpam-6041	151	2	,	,	PUNCT
ejpam-6041	151	3	we	we	PRON
ejpam-6041	151	4	obtain	obtain	VERB
ejpam-6041	151	5	the	the	DET
ejpam-6041	151	6	following	follow	VERB
ejpam-6041	151	7	corollary	corollary	NOUN
ejpam-6041	151	8	immediately	immediately	ADV
ejpam-6041	151	9	.	.	PUNCT
ejpam-6041	152	1	corollary	corollary	ADJ
ejpam-6041	152	2	2	2	NUM
ejpam-6041	152	3	.	.	PUNCT
ejpam-6041	153	1	the	the	DET
ejpam-6041	153	2	associator	associator	NOUN
ejpam-6041	153	3	normal	normal	ADJ
ejpam-6041	153	4	subgyrogroup	subgyrogroup	NOUN
ejpam-6041	153	5	of	of	ADP
ejpam-6041	153	6	the	the	DET
ejpam-6041	153	7	möbius	möbius	PROPN
ejpam-6041	153	8	gyrogroup	gyrogroup	PROPN
ejpam-6041	153	9	is	be	AUX
ejpam-6041	153	10	the	the	DET
ejpam-6041	153	11	möbius	möbius	PROPN
ejpam-6041	153	12	gyrogroup	gyrogroup	PROPN
ejpam-6041	153	13	itself	itself	PRON
ejpam-6041	153	14	.	.	PUNCT
ejpam-6041	154	1	we	we	PRON
ejpam-6041	154	2	gain	gain	VERB
ejpam-6041	154	3	a	a	DET
ejpam-6041	154	4	better	well	ADJ
ejpam-6041	154	5	understanding	understanding	NOUN
ejpam-6041	154	6	of	of	ADP
ejpam-6041	154	7	the	the	DET
ejpam-6041	154	8	algebraic	algebraic	ADJ
ejpam-6041	154	9	structure	structure	NOUN
ejpam-6041	154	10	of	of	ADP
ejpam-6041	154	11	the	the	DET
ejpam-6041	154	12	möbius	möbius	PROPN
ejpam-6041	154	13	gyrogroup	gyrogroup	PROPN
ejpam-6041	154	14	,	,	PUNCT
ejpam-6041	154	15	as	as	SCONJ
ejpam-6041	154	16	stated	state	VERB
ejpam-6041	154	17	in	in	ADP
ejpam-6041	154	18	the	the	DET
ejpam-6041	154	19	next	next	ADJ
ejpam-6041	154	20	theorem	theorem	NOUN
ejpam-6041	154	21	,	,	PUNCT
ejpam-6041	154	22	which	which	PRON
ejpam-6041	154	23	is	be	AUX
ejpam-6041	154	24	an	an	DET
ejpam-6041	154	25	application	application	NOUN
ejpam-6041	154	26	of	of	ADP
ejpam-6041	154	27	the	the	DET
ejpam-6041	154	28	previous	previous	ADJ
ejpam-6041	154	29	corollary	corollary	NOUN
ejpam-6041	154	30	.	.	PUNCT
ejpam-6041	155	1	theorem	theorem	NOUN
ejpam-6041	155	2	4	4	NUM
ejpam-6041	155	3	.	.	PUNCT
ejpam-6041	156	1	there	there	PRON
ejpam-6041	156	2	is	be	VERB
ejpam-6041	156	3	no	no	DET
ejpam-6041	156	4	non	non	ADJ
ejpam-6041	156	5	-	-	ADJ
ejpam-6041	156	6	trivial	trivial	ADJ
ejpam-6041	156	7	homomorphism	homomorphism	NOUN
ejpam-6041	156	8	from	from	ADP
ejpam-6041	156	9	the	the	DET
ejpam-6041	156	10	möbius	möbius	PROPN
ejpam-6041	156	11	gyrogroup	gyrogroup	PROPN
ejpam-6041	156	12	to	to	ADP
ejpam-6041	156	13	a	a	DET
ejpam-6041	156	14	group	group	NOUN
ejpam-6041	156	15	.	.	PUNCT
ejpam-6041	157	1	proof	proof	NOUN
ejpam-6041	157	2	.	.	PUNCT
ejpam-6041	158	1	suppose	suppose	VERB
ejpam-6041	158	2	that	that	SCONJ
ejpam-6041	158	3	γ	γ	PROPN
ejpam-6041	158	4	is	be	AUX
ejpam-6041	158	5	a	a	DET
ejpam-6041	158	6	group	group	NOUN
ejpam-6041	158	7	,	,	PUNCT
ejpam-6041	158	8	and	and	CCONJ
ejpam-6041	158	9	suppose	suppose	VERB
ejpam-6041	158	10	that	that	SCONJ
ejpam-6041	158	11	φ	φ	PROPN
ejpam-6041	158	12	is	be	AUX
ejpam-6041	158	13	a	a	DET
ejpam-6041	158	14	homomorphism	homomorphism	NOUN
ejpam-6041	158	15	from	from	ADP
ejpam-6041	158	16	d	d	PROPN
ejpam-6041	158	17	to	to	ADP
ejpam-6041	158	18	γ	γ	PROPN
ejpam-6041	158	19	.	.	PUNCT
ejpam-6041	158	20	by	by	ADP
ejpam-6041	158	21	proposition	proposition	NOUN
ejpam-6041	158	22	3.4	3.4	NUM
ejpam-6041	158	23	of	of	ADP
ejpam-6041	158	24	[	[	X
ejpam-6041	158	25	4	4	NUM
ejpam-6041	158	26	]	]	PUNCT
ejpam-6041	158	27	,	,	PUNCT
ejpam-6041	158	28	da	da	PROPN
ejpam-6041	158	29	⊆	⊆	NUM
ejpam-6041	158	30	kerφ	kerφ	PROPN
ejpam-6041	158	31	,	,	PUNCT
ejpam-6041	158	32	and	and	CCONJ
ejpam-6041	158	33	so	so	ADV
ejpam-6041	158	34	kerφ	kerφ	PROPN
ejpam-6041	159	1	=	=	PUNCT
ejpam-6041	160	1	d	d	PROPN
ejpam-6041	160	2	because	because	SCONJ
ejpam-6041	160	3	da	da	PROPN
ejpam-6041	160	4	=	=	PUNCT
ejpam-6041	160	5	d.	d.	PROPN
ejpam-6041	160	6	it	it	PRON
ejpam-6041	160	7	follows	follow	VERB
ejpam-6041	160	8	that	that	SCONJ
ejpam-6041	160	9	φ	φ	PROPN
ejpam-6041	160	10	is	be	AUX
ejpam-6041	160	11	trivial	trivial	ADJ
ejpam-6041	160	12	,	,	PUNCT
ejpam-6041	160	13	which	which	PRON
ejpam-6041	160	14	completes	complete	VERB
ejpam-6041	160	15	the	the	DET
ejpam-6041	160	16	proof	proof	NOUN
ejpam-6041	160	17	.	.	PUNCT
ejpam-6041	161	1	as	as	ADP
ejpam-6041	161	2	a	a	DET
ejpam-6041	161	3	consequence	consequence	NOUN
ejpam-6041	161	4	of	of	ADP
ejpam-6041	161	5	theorem	theorem	NOUN
ejpam-6041	161	6	4	4	NUM
ejpam-6041	161	7	,	,	PUNCT
ejpam-6041	161	8	we	we	PRON
ejpam-6041	161	9	conclude	conclude	VERB
ejpam-6041	161	10	that	that	SCONJ
ejpam-6041	161	11	a	a	DET
ejpam-6041	161	12	non	non	ADJ
ejpam-6041	161	13	-	-	ADJ
ejpam-6041	161	14	trivial	trivial	ADJ
ejpam-6041	161	15	homomorphism	homomorphism	NOUN
ejpam-6041	161	16	from	from	ADP
ejpam-6041	161	17	the	the	DET
ejpam-6041	161	18	möbius	möbius	PROPN
ejpam-6041	161	19	gyrogroup	gyrogroup	PROPN
ejpam-6041	161	20	to	to	ADP
ejpam-6041	161	21	the	the	DET
ejpam-6041	161	22	multiplicative	multiplicative	ADJ
ejpam-6041	161	23	group	group	NOUN
ejpam-6041	161	24	of	of	ADP
ejpam-6041	161	25	non	non	ADJ
ejpam-6041	161	26	-	-	ADJ
ejpam-6041	161	27	zero	zero	ADJ
ejpam-6041	161	28	complex	complex	ADJ
ejpam-6041	161	29	numbers	number	NOUN
ejpam-6041	161	30	does	do	AUX
ejpam-6041	161	31	not	not	PART
ejpam-6041	161	32	exist	exist	VERB
ejpam-6041	161	33	.	.	PUNCT
ejpam-6041	162	1	this	this	PRON
ejpam-6041	162	2	proves	prove	VERB
ejpam-6041	162	3	theorem	theorem	ADJ
ejpam-6041	162	4	1	1	NUM
ejpam-6041	162	5	,	,	PUNCT
ejpam-6041	162	6	which	which	PRON
ejpam-6041	162	7	fulfills	fulfill	VERB
ejpam-6041	162	8	the	the	DET
ejpam-6041	162	9	goal	goal	NOUN
ejpam-6041	162	10	of	of	ADP
ejpam-6041	162	11	this	this	DET
ejpam-6041	162	12	paper	paper	NOUN
ejpam-6041	162	13	and	and	CCONJ
ejpam-6041	162	14	completely	completely	ADV
ejpam-6041	162	15	solves	solve	VERB
ejpam-6041	162	16	problem	problem	VERB
ejpam-6041	162	17	1.3	1.3	NUM
ejpam-6041	162	18	of	of	ADP
ejpam-6041	162	19	[	[	X
ejpam-6041	162	20	3	3	NUM
ejpam-6041	162	21	]	]	PUNCT
ejpam-6041	162	22	.	.	PUNCT
ejpam-6041	163	1	moreover	moreover	ADV
ejpam-6041	163	2	,	,	PUNCT
ejpam-6041	163	3	it	it	PRON
ejpam-6041	163	4	follows	follow	VERB
ejpam-6041	163	5	that	that	SCONJ
ejpam-6041	163	6	every	every	DET
ejpam-6041	163	7	representation	representation	NOUN
ejpam-6041	163	8	of	of	ADP
ejpam-6041	163	9	the	the	DET
ejpam-6041	163	10	möbius	möbius	PROPN
ejpam-6041	163	11	gyrogroup	gyrogroup	PROPN
ejpam-6041	163	12	is	be	AUX
ejpam-6041	163	13	trivial	trivial	ADJ
ejpam-6041	163	14	.	.	PUNCT
ejpam-6041	164	1	note	note	VERB
ejpam-6041	164	2	that	that	SCONJ
ejpam-6041	164	3	the	the	DET
ejpam-6041	164	4	associativization	associativization	NOUN
ejpam-6041	164	5	of	of	ADP
ejpam-6041	164	6	the	the	DET
ejpam-6041	164	7	möbius	möbius	PROPN
ejpam-6041	164	8	gyrogroup	gyrogroup	PROPN
ejpam-6041	164	9	is	be	AUX
ejpam-6041	164	10	trivial	trivial	ADJ
ejpam-6041	164	11	,	,	PUNCT
ejpam-6041	164	12	which	which	PRON
ejpam-6041	164	13	indicates	indicate	VERB
ejpam-6041	164	14	that	that	SCONJ
ejpam-6041	164	15	the	the	DET
ejpam-6041	164	16	möbius	möbius	PROPN
ejpam-6041	164	17	gyrogroup	gyrogroup	PROPN
ejpam-6041	164	18	is	be	AUX
ejpam-6041	164	19	,	,	PUNCT
ejpam-6041	164	20	in	in	ADP
ejpam-6041	164	21	some	some	DET
ejpam-6041	164	22	sense	sense	NOUN
ejpam-6041	164	23	,	,	PUNCT
ejpam-6041	164	24	most	most	ADV
ejpam-6041	164	25	far	far	ADV
ejpam-6041	164	26	from	from	ADP
ejpam-6041	164	27	being	be	AUX
ejpam-6041	164	28	a	a	DET
ejpam-6041	164	29	group	group	NOUN
ejpam-6041	164	30	.	.	PUNCT
ejpam-6041	165	1	we	we	PRON
ejpam-6041	165	2	remark	remark	VERB
ejpam-6041	165	3	that	that	SCONJ
ejpam-6041	165	4	extending	extend	VERB
ejpam-6041	165	5	the	the	DET
ejpam-6041	165	6	obtained	obtain	VERB
ejpam-6041	165	7	results	result	NOUN
ejpam-6041	165	8	to	to	ADP
ejpam-6041	165	9	higher	high	ADJ
ejpam-6041	165	10	dimensions	dimension	NOUN
ejpam-6041	165	11	is	be	AUX
ejpam-6041	165	12	quite	quite	ADV
ejpam-6041	165	13	challenging	challenging	ADJ
ejpam-6041	165	14	and	and	CCONJ
ejpam-6041	165	15	not	not	PART
ejpam-6041	165	16	straightforward	straightforward	ADJ
ejpam-6041	165	17	,	,	PUNCT
ejpam-6041	165	18	due	due	ADP
ejpam-6041	165	19	to	to	ADP
ejpam-6041	165	20	the	the	DET
ejpam-6041	165	21	complexity	complexity	NOUN
ejpam-6041	165	22	of	of	ADP
ejpam-6041	165	23	the	the	DET
ejpam-6041	165	24	möbius	möbius	PROPN
ejpam-6041	165	25	addition	addition	NOUN
ejpam-6041	165	26	formula	formula	NOUN
ejpam-6041	165	27	.	.	PUNCT
ejpam-6041	166	1	finally	finally	ADV
ejpam-6041	166	2	,	,	PUNCT
ejpam-6041	166	3	for	for	ADP
ejpam-6041	166	4	the	the	DET
ejpam-6041	166	5	future	future	ADJ
ejpam-6041	166	6	work	work	NOUN
ejpam-6041	166	7	,	,	PUNCT
ejpam-6041	166	8	we	we	PRON
ejpam-6041	166	9	remark	remark	VERB
ejpam-6041	166	10	that	that	SCONJ
ejpam-6041	166	11	möbius	möbius	PROPN
ejpam-6041	166	12	addition	addition	NOUN
ejpam-6041	166	13	on	on	ADP
ejpam-6041	166	14	the	the	DET
ejpam-6041	166	15	complex	complex	ADJ
ejpam-6041	166	16	open	open	ADJ
ejpam-6041	166	17	unit	unit	NOUN
ejpam-6041	166	18	disk	disk	NOUN
ejpam-6041	166	19	enables	enable	VERB
ejpam-6041	166	20	optimization	optimization	NOUN
ejpam-6041	166	21	in	in	ADP
ejpam-6041	166	22	hyperbolic	hyperbolic	ADJ
ejpam-6041	166	23	geometry	geometry	NOUN
ejpam-6041	166	24	,	,	PUNCT
ejpam-6041	166	25	where	where	SCONJ
ejpam-6041	166	26	geodesic	geodesic	NOUN
ejpam-6041	166	27	-	-	PUNCT
ejpam-6041	166	28	based	base	VERB
ejpam-6041	166	29	methods	method	NOUN
ejpam-6041	166	30	replace	replace	VERB
ejpam-6041	166	31	euclidean	euclidean	ADJ
ejpam-6041	166	32	approaches	approach	NOUN
ejpam-6041	166	33	.	.	PUNCT
ejpam-6041	167	1	its	its	PRON
ejpam-6041	167	2	non	non	ADJ
ejpam-6041	167	3	-	-	ADJ
ejpam-6041	167	4	associative	associative	ADJ
ejpam-6041	167	5	nature	nature	NOUN
ejpam-6041	167	6	models	model	VERB
ejpam-6041	167	7	systems	system	NOUN
ejpam-6041	167	8	with	with	ADP
ejpam-6041	167	9	order	order	NOUN
ejpam-6041	167	10	-	-	PUNCT
ejpam-6041	167	11	sensitive	sensitive	ADJ
ejpam-6041	167	12	updates	update	NOUN
ejpam-6041	167	13	,	,	PUNCT
ejpam-6041	167	14	useful	useful	ADJ
ejpam-6041	167	15	in	in	ADP
ejpam-6041	167	16	multi	multi	ADJ
ejpam-6041	167	17	-	-	ADJ
ejpam-6041	167	18	agent	agent	ADJ
ejpam-6041	167	19	and	and	CCONJ
ejpam-6041	167	20	sequential	sequential	ADJ
ejpam-6041	167	21	decision	decision	NOUN
ejpam-6041	167	22	-	-	PUNCT
ejpam-6041	167	23	making	making	NOUN
ejpam-6041	167	24	.	.	PUNCT
ejpam-6041	168	1	möbius	möbiu	VERB
ejpam-6041	168	2	-	-	PUNCT
ejpam-6041	168	3	based	base	VERB
ejpam-6041	168	4	algorithms	algorithm	NOUN
ejpam-6041	168	5	are	be	AUX
ejpam-6041	168	6	suited	suit	VERB
ejpam-6041	168	7	for	for	ADP
ejpam-6041	168	8	learning	learn	VERB
ejpam-6041	168	9	in	in	ADP
ejpam-6041	168	10	non	non	ADJ
ejpam-6041	168	11	-	-	ADJ
ejpam-6041	168	12	euclidean	euclidean	ADJ
ejpam-6041	168	13	spaces	space	NOUN
ejpam-6041	168	14	like	like	ADP
ejpam-6041	168	15	hyperbolic	hyperbolic	ADJ
ejpam-6041	168	16	spaces	space	NOUN
ejpam-6041	168	17	.	.	PUNCT
ejpam-6041	169	1	as	as	SCONJ
ejpam-6041	169	2	möbius	möbius	PROPN
ejpam-6041	169	3	transformations	transformation	NOUN
ejpam-6041	169	4	preserve	preserve	VERB
ejpam-6041	169	5	the	the	DET
ejpam-6041	169	6	disk	disk	NOUN
ejpam-6041	169	7	,	,	PUNCT
ejpam-6041	169	8	they	they	PRON
ejpam-6041	169	9	support	support	VERB
ejpam-6041	169	10	projection	projection	NOUN
ejpam-6041	169	11	-	-	PUNCT
ejpam-6041	169	12	free	free	ADJ
ejpam-6041	169	13	optimization	optimization	NOUN
ejpam-6041	169	14	in	in	ADP
ejpam-6041	169	15	compact	compact	ADJ
ejpam-6041	169	16	domains	domain	NOUN
ejpam-6041	169	17	.	.	PUNCT
ejpam-6041	170	1	applications	application	NOUN
ejpam-6041	170	2	may	may	AUX
ejpam-6041	170	3	span	span	VERB
ejpam-6041	170	4	machine	machine	NOUN
ejpam-6041	170	5	learning	learning	PROPN
ejpam-6041	170	6	,	,	PUNCT
ejpam-6041	170	7	signal	signal	ADJ
ejpam-6041	170	8	processing	processing	NOUN
ejpam-6041	170	9	,	,	PUNCT
ejpam-6041	170	10	and	and	CCONJ
ejpam-6041	170	11	complex	complex	ADJ
ejpam-6041	170	12	domain	domain	NOUN
ejpam-6041	170	13	optimization	optimization	NOUN
ejpam-6041	170	14	,	,	PUNCT
ejpam-6041	170	15	where	where	SCONJ
ejpam-6041	170	16	möbius	möbius	PROPN
ejpam-6041	170	17	transformations	transformation	NOUN
ejpam-6041	170	18	maintain	maintain	VERB
ejpam-6041	170	19	structural	structural	ADJ
ejpam-6041	170	20	properties	property	NOUN
ejpam-6041	170	21	vital	vital	ADJ
ejpam-6041	170	22	for	for	ADP
ejpam-6041	170	23	constrained	constrained	ADJ
ejpam-6041	170	24	and	and	CCONJ
ejpam-6041	170	25	geometric	geometric	ADJ
ejpam-6041	170	26	optimization	optimization	NOUN
ejpam-6041	170	27	tasks	task	NOUN
ejpam-6041	170	28	.	.	PUNCT
ejpam-6041	171	1	r.	r.	PROPN
ejpam-6041	171	2	maungchang	maungchang	PROPN
ejpam-6041	171	3	et	et	PROPN
ejpam-6041	171	4	al	al	PROPN
ejpam-6041	171	5	.	.	PUNCT
ejpam-6041	171	6	/	/	SYM
ejpam-6041	171	7	eur	eur	PROPN
ejpam-6041	171	8	.	.	PUNCT
ejpam-6041	172	1	j.	j.	PROPN
ejpam-6041	172	2	pure	pure	PROPN
ejpam-6041	172	3	appl	appl	PROPN
ejpam-6041	172	4	.	.	PROPN
ejpam-6041	172	5	math	math	PROPN
ejpam-6041	172	6	,	,	PUNCT
ejpam-6041	172	7	18	18	NUM
ejpam-6041	172	8	(	(	PUNCT
ejpam-6041	172	9	2	2	NUM
ejpam-6041	172	10	)	)	PUNCT
ejpam-6041	172	11	(	(	PUNCT
ejpam-6041	172	12	2025	2025	NUM
ejpam-6041	172	13	)	)	PUNCT
ejpam-6041	172	14	,	,	PUNCT
ejpam-6041	172	15	6041	6041	NUM
ejpam-6041	172	16	8	8	NUM
ejpam-6041	172	17	of	of	ADP
ejpam-6041	172	18	8	8	NUM
ejpam-6041	172	19	acknowledgements	acknowledgement	NOUN
ejpam-6041	172	20	this	this	DET
ejpam-6041	172	21	research	research	NOUN
ejpam-6041	172	22	was	be	AUX
ejpam-6041	172	23	supported	support	VERB
ejpam-6041	172	24	by	by	ADP
ejpam-6041	172	25	chiang	chiang	PROPN
ejpam-6041	172	26	mai	mai	PROPN
ejpam-6041	172	27	university	university	PROPN
ejpam-6041	172	28	.	.	PUNCT
ejpam-6041	173	1	conflict	conflict	NOUN
ejpam-6041	173	2	of	of	ADP
ejpam-6041	173	3	interest	interest	NOUN
ejpam-6041	173	4	the	the	DET
ejpam-6041	173	5	authors	author	NOUN
ejpam-6041	173	6	declare	declare	VERB
ejpam-6041	173	7	no	no	DET
ejpam-6041	173	8	conflict	conflict	NOUN
ejpam-6041	173	9	of	of	ADP
ejpam-6041	173	10	interest	interest	NOUN
ejpam-6041	173	11	.	.	PUNCT
ejpam-6041	174	1	data	datum	NOUN
ejpam-6041	174	2	availability	availability	NOUN
ejpam-6041	174	3	data	datum	NOUN
ejpam-6041	174	4	sharing	share	VERB
ejpam-6041	174	5	not	not	PART
ejpam-6041	174	6	applicable	applicable	ADJ
ejpam-6041	174	7	to	to	ADP
ejpam-6041	174	8	this	this	DET
ejpam-6041	174	9	paper	paper	NOUN
ejpam-6041	174	10	as	as	SCONJ
ejpam-6041	174	11	no	no	DET
ejpam-6041	174	12	datasets	dataset	NOUN
ejpam-6041	174	13	were	be	AUX
ejpam-6041	174	14	generated	generate	VERB
ejpam-6041	174	15	or	or	CCONJ
ejpam-6041	174	16	analyzed	analyze	VERB
ejpam-6041	174	17	during	during	ADP
ejpam-6041	174	18	the	the	DET
ejpam-6041	174	19	current	current	ADJ
ejpam-6041	174	20	study	study	NOUN
ejpam-6041	174	21	.	.	PUNCT
ejpam-6041	175	1	author	author	NOUN
ejpam-6041	175	2	contributions	contributions	PROPN
ejpam-6041	175	3	r.	r.	PROPN
ejpam-6041	175	4	maungchang	maungchang	PROPN
ejpam-6041	175	5	:	:	PUNCT
ejpam-6041	175	6	writing	writing	NOUN
ejpam-6041	175	7	—	—	PUNCT
ejpam-6041	175	8	review	review	NOUN
ejpam-6041	175	9	and	and	CCONJ
ejpam-6041	175	10	editing	editing	NOUN
ejpam-6041	175	11	,	,	PUNCT
ejpam-6041	175	12	validation	validation	NOUN
ejpam-6041	175	13	,	,	PUNCT
ejpam-6041	175	14	investigation	investigation	NOUN
ejpam-6041	175	15	,	,	PUNCT
ejpam-6041	175	16	visualization	visualization	NOUN
ejpam-6041	175	17	;	;	PUNCT
ejpam-6041	175	18	w.	w.	PROPN
ejpam-6041	175	19	atiponrat	atiponrat	PROPN
ejpam-6041	175	20	:	:	PUNCT
ejpam-6041	175	21	writing	writing	NOUN
ejpam-6041	175	22	—	—	PUNCT
ejpam-6041	175	23	review	review	NOUN
ejpam-6041	175	24	and	and	CCONJ
ejpam-6041	175	25	editing	editing	NOUN
ejpam-6041	175	26	,	,	PUNCT
ejpam-6041	175	27	validation	validation	NOUN
ejpam-6041	175	28	,	,	PUNCT
ejpam-6041	175	29	investigation	investigation	NOUN
ejpam-6041	175	30	;	;	PUNCT
ejpam-6041	175	31	t.	t.	PROPN
ejpam-6041	175	32	suwansri	suwansri	PROPN
ejpam-6041	175	33	:	:	PUNCT
ejpam-6041	175	34	validation	validation	NOUN
ejpam-6041	175	35	,	,	PUNCT
ejpam-6041	175	36	investigation	investigation	NOUN
ejpam-6041	175	37	,	,	PUNCT
ejpam-6041	175	38	visualization	visualization	NOUN
ejpam-6041	175	39	;	;	PUNCT
ejpam-6041	175	40	j.	j.	PROPN
ejpam-6041	175	41	wattanapan	wattanapan	PROPN
ejpam-6041	175	42	:	:	PUNCT
ejpam-6041	175	43	validation	validation	NOUN
ejpam-6041	175	44	,	,	PUNCT
ejpam-6041	175	45	investigation	investigation	NOUN
ejpam-6041	175	46	;	;	PUNCT
ejpam-6041	175	47	t.	t.	PROPN
ejpam-6041	175	48	suksumran	suksumran	NOUN
ejpam-6041	175	49	:	:	PUNCT
ejpam-6041	175	50	conceptualization	conceptualization	NOUN
ejpam-6041	175	51	,	,	PUNCT
ejpam-6041	175	52	methodology	methodology	NOUN
ejpam-6041	175	53	,	,	PUNCT
ejpam-6041	175	54	validation	validation	NOUN
ejpam-6041	175	55	,	,	PUNCT
ejpam-6041	175	56	investigation	investigation	NOUN
ejpam-6041	175	57	,	,	PUNCT
ejpam-6041	175	58	writing	writing	NOUN
ejpam-6041	175	59	—	—	PUNCT
ejpam-6041	175	60	original	original	ADJ
ejpam-6041	175	61	draft	draft	NOUN
ejpam-6041	175	62	preparation	preparation	NOUN
ejpam-6041	175	63	,	,	PUNCT
ejpam-6041	175	64	supervision	supervision	NOUN
ejpam-6041	175	65	,	,	PUNCT
ejpam-6041	175	66	project	project	NOUN
ejpam-6041	175	67	administration	administration	NOUN
ejpam-6041	175	68	.	.	PUNCT
ejpam-6041	176	1	all	all	DET
ejpam-6041	176	2	authors	author	NOUN
ejpam-6041	176	3	have	have	AUX
ejpam-6041	176	4	read	read	VERB
ejpam-6041	176	5	and	and	CCONJ
ejpam-6041	176	6	agreed	agree	VERB
ejpam-6041	176	7	to	to	ADP
ejpam-6041	176	8	the	the	DET
ejpam-6041	176	9	published	publish	VERB
ejpam-6041	176	10	version	version	NOUN
ejpam-6041	176	11	of	of	ADP
ejpam-6041	176	12	the	the	DET
ejpam-6041	176	13	manuscript	manuscript	NOUN
ejpam-6041	176	14	.	.	PUNCT
ejpam-6041	177	1	references	reference	NOUN
ejpam-6041	177	2	[	[	X
ejpam-6041	177	3	1	1	NUM
ejpam-6041	177	4	]	]	X
ejpam-6041	177	5	y.	y.	PROPN
ejpam-6041	177	6	friedman	friedman	PROPN
ejpam-6041	177	7	and	and	CCONJ
ejpam-6041	177	8	t.	t.	PROPN
ejpam-6041	177	9	scarr	scarr	PROPN
ejpam-6041	177	10	.	.	PUNCT
ejpam-6041	178	1	physical	physical	ADJ
ejpam-6041	178	2	applications	application	NOUN
ejpam-6041	178	3	of	of	ADP
ejpam-6041	178	4	homogeneous	homogeneous	ADJ
ejpam-6041	178	5	balls	ball	NOUN
ejpam-6041	178	6	,	,	PUNCT
ejpam-6041	178	7	volume	volume	NOUN
ejpam-6041	178	8	40	40	NUM
ejpam-6041	178	9	of	of	ADP
ejpam-6041	178	10	progress	progress	NOUN
ejpam-6041	178	11	in	in	ADP
ejpam-6041	178	12	mathematical	mathematical	ADJ
ejpam-6041	178	13	physics	physics	NOUN
ejpam-6041	178	14	.	.	PUNCT
ejpam-6041	179	1	birkhäuser	birkhäuser	NOUN
ejpam-6041	179	2	,	,	PUNCT
ejpam-6041	179	3	boston	boston	PROPN
ejpam-6041	179	4	,	,	PUNCT
ejpam-6041	179	5	2005	2005	NUM
ejpam-6041	179	6	.	.	PUNCT
ejpam-6041	180	1	[	[	X
ejpam-6041	180	2	2	2	NUM
ejpam-6041	180	3	]	]	PUNCT
ejpam-6041	180	4	a.	a.	NOUN
ejpam-6041	180	5	ungar	ungar	NOUN
ejpam-6041	180	6	.	.	PUNCT
ejpam-6041	181	1	analytic	analytic	ADJ
ejpam-6041	181	2	hyperbolic	hyperbolic	ADJ
ejpam-6041	181	3	geometry	geometry	NOUN
ejpam-6041	181	4	and	and	CCONJ
ejpam-6041	181	5	albert	albert	PROPN
ejpam-6041	181	6	einstein	einstein	PROPN
ejpam-6041	181	7	’s	’s	PART
ejpam-6041	181	8	special	special	ADJ
ejpam-6041	181	9	theory	theory	NOUN
ejpam-6041	181	10	of	of	ADP
ejpam-6041	181	11	relativity	relativity	NOUN
ejpam-6041	181	12	.	.	PUNCT
ejpam-6041	182	1	world	world	NOUN
ejpam-6041	182	2	scientific	scientific	PROPN
ejpam-6041	182	3	,	,	PUNCT
ejpam-6041	182	4	hackensack	hackensack	PROPN
ejpam-6041	182	5	,	,	PUNCT
ejpam-6041	182	6	nj	nj	PROPN
ejpam-6041	182	7	,	,	PUNCT
ejpam-6041	182	8	2008	2008	NUM
ejpam-6041	182	9	.	.	PUNCT
ejpam-6041	183	1	[	[	X
ejpam-6041	183	2	3	3	X
ejpam-6041	183	3	]	]	X
ejpam-6041	183	4	t.	t.	NOUN
ejpam-6041	183	5	suksumran	suksumran	PROPN
ejpam-6041	183	6	and	and	CCONJ
ejpam-6041	183	7	k.	k.	PROPN
ejpam-6041	183	8	wiboonton	wiboonton	PROPN
ejpam-6041	183	9	.	.	PUNCT
ejpam-6041	184	1	möbius	möbius	PROPN
ejpam-6041	184	2	’s	’s	PART
ejpam-6041	184	3	functional	functional	ADJ
ejpam-6041	184	4	equation	equation	NOUN
ejpam-6041	184	5	and	and	CCONJ
ejpam-6041	184	6	schur	schur	PROPN
ejpam-6041	184	7	’s	’s	PROPN
ejpam-6041	184	8	lemma	lemma	PROPN
ejpam-6041	184	9	with	with	ADP
ejpam-6041	184	10	applications	application	NOUN
ejpam-6041	184	11	to	to	ADP
ejpam-6041	184	12	the	the	DET
ejpam-6041	184	13	complex	complex	ADJ
ejpam-6041	184	14	unit	unit	NOUN
ejpam-6041	184	15	disk	disk	NOUN
ejpam-6041	184	16	.	.	PUNCT
ejpam-6041	185	1	aequationes	aequatione	NOUN
ejpam-6041	185	2	math	math	PROPN
ejpam-6041	185	3	.	.	PUNCT
ejpam-6041	185	4	,	,	PUNCT
ejpam-6041	185	5	91(3):491–503	91(3):491–503	NUM
ejpam-6041	185	6	,	,	PUNCT
ejpam-6041	185	7	2017	2017	NUM
ejpam-6041	185	8	.	.	PUNCT
ejpam-6041	186	1	[	[	X
ejpam-6041	186	2	4	4	X
ejpam-6041	186	3	]	]	PUNCT
ejpam-6041	186	4	t.	t.	NOUN
ejpam-6041	186	5	suksumran	suksumran	NOUN
ejpam-6041	186	6	.	.	PUNCT
ejpam-6041	187	1	associativization	associativization	NOUN
ejpam-6041	187	2	of	of	ADP
ejpam-6041	187	3	gyrogroups	gyrogroup	NOUN
ejpam-6041	187	4	and	and	CCONJ
ejpam-6041	187	5	the	the	DET
ejpam-6041	187	6	universal	universal	ADJ
ejpam-6041	187	7	property	property	NOUN
ejpam-6041	187	8	.	.	PUNCT
ejpam-6041	188	1	asian	asian	ADJ
ejpam-6041	188	2	-	-	PUNCT
ejpam-6041	188	3	eur	eur	NOUN
ejpam-6041	188	4	.	.	PUNCT
ejpam-6041	189	1	j.	j.	PROPN
ejpam-6041	189	2	math	math	PROPN
ejpam-6041	189	3	.	.	PUNCT
ejpam-6041	189	4	,	,	PUNCT
ejpam-6041	189	5	17(11):article	17(11):article	PROPN
ejpam-6041	189	6	2450066	2450066	NUM
ejpam-6041	189	7	(	(	PUNCT
ejpam-6041	189	8	16	16	NUM
ejpam-6041	189	9	pages	page	NOUN
ejpam-6041	189	10	)	)	PUNCT
ejpam-6041	189	11	,	,	PUNCT
ejpam-6041	189	12	2024	2024	NUM
ejpam-6041	189	13	.	.	PUNCT
ejpam-6041	190	1	[	[	X
ejpam-6041	190	2	5	5	X
ejpam-6041	190	3	]	]	PUNCT
ejpam-6041	190	4	t.	t.	NOUN
ejpam-6041	190	5	suksumran	suksumran	NOUN
ejpam-6041	190	6	.	.	PUNCT
ejpam-6041	191	1	the	the	DET
ejpam-6041	191	2	algebra	algebra	NOUN
ejpam-6041	191	3	of	of	ADP
ejpam-6041	191	4	gyrogroups	gyrogroup	NOUN
ejpam-6041	191	5	:	:	PUNCT
ejpam-6041	191	6	cayley	cayley	PROPN
ejpam-6041	191	7	’s	’s	PART
ejpam-6041	191	8	theorem	theorem	ADJ
ejpam-6041	191	9	,	,	PUNCT
ejpam-6041	191	10	lagrange	lagrange	PROPN
ejpam-6041	191	11	’s	’s	PART
ejpam-6041	191	12	theorem	theorem	NOUN
ejpam-6041	191	13	,	,	PUNCT
ejpam-6041	191	14	and	and	CCONJ
ejpam-6041	191	15	isomorphism	isomorphism	NOUN
ejpam-6041	191	16	theorems	theorem	NOUN
ejpam-6041	191	17	.	.	PUNCT
ejpam-6041	192	1	in	in	ADP
ejpam-6041	192	2	th	th	PROPN
ejpam-6041	192	3	.	.	PUNCT
ejpam-6041	192	4	m.	m.	NOUN
ejpam-6041	192	5	rassias	rassias	PROPN
ejpam-6041	192	6	and	and	CCONJ
ejpam-6041	192	7	p.	p.	NOUN
ejpam-6041	192	8	m.	m.	NOUN
ejpam-6041	192	9	pardalos	pardalo	NOUN
ejpam-6041	192	10	,	,	PUNCT
ejpam-6041	192	11	editors	editor	NOUN
ejpam-6041	192	12	,	,	PUNCT
ejpam-6041	192	13	essays	essay	NOUN
ejpam-6041	192	14	in	in	ADP
ejpam-6041	192	15	mathematics	mathematic	NOUN
ejpam-6041	192	16	and	and	CCONJ
ejpam-6041	192	17	its	its	PRON
ejpam-6041	192	18	applications	application	NOUN
ejpam-6041	192	19	,	,	PUNCT
ejpam-6041	192	20	pages	page	NOUN
ejpam-6041	192	21	369–437	369–437	NUM
ejpam-6041	192	22	.	.	PUNCT
ejpam-6041	192	23	springer	springer	NOUN
ejpam-6041	192	24	,	,	PUNCT
ejpam-6041	192	25	cham	cham	PROPN
ejpam-6041	192	26	,	,	PUNCT
ejpam-6041	192	27	2016	2016	NUM
ejpam-6041	192	28	.	.	PUNCT
ejpam-6041	193	1	[	[	X
ejpam-6041	193	2	6	6	NUM
ejpam-6041	193	3	]	]	PUNCT
ejpam-6041	193	4	h.	h.	NOUN
ejpam-6041	193	5	royden	royden	PROPN
ejpam-6041	193	6	and	and	CCONJ
ejpam-6041	193	7	p.	p.	PROPN
ejpam-6041	193	8	fitzpatrick	fitzpatrick	PROPN
ejpam-6041	193	9	.	.	PUNCT
ejpam-6041	194	1	real	real	ADJ
ejpam-6041	194	2	analysis	analysis	NOUN
ejpam-6041	194	3	.	.	PUNCT
ejpam-6041	195	1	prentice	prentice	PROPN
ejpam-6041	195	2	hall	hall	PROPN
ejpam-6041	195	3	,	,	PUNCT
ejpam-6041	195	4	hoboken	hoboken	PROPN
ejpam-6041	195	5	,	,	PUNCT
ejpam-6041	195	6	nj	nj	PROPN
ejpam-6041	195	7	,	,	PUNCT
ejpam-6041	195	8	4th	4th	ADJ
ejpam-6041	195	9	edition	edition	NOUN
ejpam-6041	195	10	,	,	PUNCT
ejpam-6041	195	11	2010	2010	NUM
ejpam-6041	195	12	.	.	PUNCT