id	sid	tid	token	lemma	pos
ap-961	1	1	ap08_2.vp	ap08_2.vp	NOUN
ap-961	1	2	1	1	NUM
ap-961	1	3	what	what	PRON
ap-961	1	4	is	be	AUX
ap-961	1	5	noncommutative	noncommutative	ADJ
ap-961	1	6	geometry	geometry	NOUN
ap-961	1	7	?	?	PUNCT
ap-961	2	1	“	"	PUNCT
ap-961	2	2	...	...	PUNCT
ap-961	2	3	to	to	PART
ap-961	2	4	savour	savour	VERB
ap-961	2	5	the	the	DET
ap-961	2	6	strange	strange	ADJ
ap-961	2	7	warm	warm	ADJ
ap-961	2	8	glow	glow	NOUN
ap-961	2	9	of	of	ADP
ap-961	2	10	being	be	AUX
ap-961	2	11	much	much	ADV
ap-961	2	12	more	more	ADV
ap-961	2	13	ignorant	ignorant	ADJ
ap-961	2	14	than	than	ADP
ap-961	2	15	ordinary	ordinary	ADJ
ap-961	2	16	people	people	NOUN
ap-961	2	17	,	,	PUNCT
ap-961	2	18	who	who	PRON
ap-961	2	19	are	be	AUX
ap-961	2	20	only	only	ADV
ap-961	2	21	ignorant	ignorant	ADJ
ap-961	2	22	of	of	ADP
ap-961	2	23	ordinary	ordinary	ADJ
ap-961	2	24	things	thing	NOUN
ap-961	2	25	.	.	PUNCT
ap-961	3	1	“	"	PUNCT
ap-961	3	2	(	(	PUNCT
ap-961	3	3	terry	terry	PROPN
ap-961	3	4	pratchett	pratchett	PROPN
ap-961	3	5	,	,	PUNCT
ap-961	3	6	“	"	PUNCT
ap-961	3	7	equal	equal	ADJ
ap-961	3	8	rites	rite	NOUN
ap-961	3	9	”	"	PUNCT
ap-961	3	10	)	)	PUNCT
ap-961	3	11	noncommutative	noncommutative	ADJ
ap-961	3	12	geometry	geometry	NOUN
ap-961	3	13	begins	begin	VERB
ap-961	3	14	with	with	ADP
ap-961	3	15	classical	classical	ADJ
ap-961	3	16	geometry	geometry	NOUN
ap-961	3	17	and	and	CCONJ
ap-961	3	18	extends	extend	VERB
ap-961	3	19	into	into	ADP
ap-961	3	20	the	the	DET
ap-961	3	21	realm	realm	NOUN
ap-961	3	22	of	of	ADP
ap-961	3	23	abstract	abstract	ADJ
ap-961	3	24	algebras	algebra	NOUN
ap-961	3	25	and	and	CCONJ
ap-961	3	26	operators	operator	NOUN
ap-961	3	27	.	.	PUNCT
ap-961	4	1	one	one	NUM
ap-961	4	2	may	may	AUX
ap-961	4	3	,	,	PUNCT
ap-961	4	4	of	of	ADP
ap-961	4	5	course	course	NOUN
ap-961	4	6	,	,	PUNCT
ap-961	4	7	say	say	VERB
ap-961	4	8	that	that	SCONJ
ap-961	4	9	noncommutative	noncommutative	ADJ
ap-961	4	10	geometry	geometry	NOUN
ap-961	4	11	studies	study	VERB
ap-961	4	12	the	the	DET
ap-961	4	13	geometry	geometry	NOUN
ap-961	4	14	of	of	ADP
ap-961	4	15	quantum	quantum	ADJ
ap-961	4	16	spaces	space	NOUN
ap-961	4	17	–	–	PUNCT
ap-961	4	18	or	or	CCONJ
ap-961	4	19	,	,	PUNCT
ap-961	4	20	to	to	PART
ap-961	4	21	be	be	AUX
ap-961	4	22	more	more	ADV
ap-961	4	23	explicit	explicit	ADJ
ap-961	4	24	–	–	PUNCT
ap-961	4	25	the	the	DET
ap-961	4	26	geometry	geometry	NOUN
ap-961	4	27	of	of	ADP
ap-961	4	28	noncommutative	noncommutative	ADJ
ap-961	4	29	algebras	algebra	NOUN
ap-961	4	30	.	.	PUNCT
ap-961	5	1	clearly	clearly	ADV
ap-961	5	2	,	,	PUNCT
ap-961	5	3	the	the	DET
ap-961	5	4	word	word	NOUN
ap-961	5	5	quantum	quantum	NOUN
ap-961	5	6	,	,	PUNCT
ap-961	5	7	although	although	SCONJ
ap-961	5	8	at	at	ADP
ap-961	5	9	first	first	ADV
ap-961	5	10	only	only	ADV
ap-961	5	11	superficially	superficially	ADV
ap-961	5	12	related	relate	VERB
ap-961	5	13	to	to	ADP
ap-961	5	14	quantum	quantum	ADJ
ap-961	5	15	mechanics	mechanic	NOUN
ap-961	5	16	or	or	CCONJ
ap-961	5	17	quantum	quantum	NOUN
ap-961	5	18	field	field	NOUN
ap-961	5	19	theory	theory	NOUN
ap-961	5	20	might	might	AUX
ap-961	5	21	be	be	AUX
ap-961	5	22	the	the	DET
ap-961	5	23	right	right	ADJ
ap-961	5	24	one	one	NUM
ap-961	5	25	both	both	DET
ap-961	5	26	physics	physics	NOUN
ap-961	5	27	and	and	CCONJ
ap-961	5	28	mathematics	mathematic	NOUN
ap-961	5	29	are	be	AUX
ap-961	5	30	involved	involve	VERB
ap-961	5	31	in	in	ADP
ap-961	5	32	many	many	ADJ
ap-961	5	33	examples	example	NOUN
ap-961	5	34	and	and	CCONJ
ap-961	5	35	there	there	PRON
ap-961	5	36	is	be	VERB
ap-961	5	37	a	a	DET
ap-961	5	38	huge	huge	ADJ
ap-961	5	39	interplay	interplay	NOUN
ap-961	5	40	between	between	ADP
ap-961	5	41	them	they	PRON
ap-961	5	42	.	.	PUNCT
ap-961	6	1	however	however	ADV
ap-961	6	2	,	,	PUNCT
ap-961	6	3	the	the	DET
ap-961	6	4	notion	notion	NOUN
ap-961	6	5	of	of	ADP
ap-961	6	6	quantum	quantum	ADJ
ap-961	6	7	spaces	space	NOUN
ap-961	6	8	is	be	AUX
ap-961	6	9	a	a	DET
ap-961	6	10	delicate	delicate	ADJ
ap-961	6	11	one	one	NOUN
ap-961	6	12	since	since	SCONJ
ap-961	6	13	the	the	DET
ap-961	6	14	objects	object	NOUN
ap-961	6	15	that	that	SCONJ
ap-961	6	16	noncommutative	noncommutative	ADJ
ap-961	6	17	geometry	geometry	NOUN
ap-961	6	18	attempts	attempt	VERB
ap-961	6	19	to	to	PART
ap-961	6	20	study	study	VERB
ap-961	6	21	are	be	AUX
ap-961	6	22	(	(	PUNCT
ap-961	6	23	usually	usually	ADV
ap-961	6	24	)	)	PUNCT
ap-961	6	25	not	not	PART
ap-961	6	26	spaces	space	VERB
ap-961	6	27	–	–	PUNCT
ap-961	6	28	they	they	PRON
ap-961	6	29	can	can	AUX
ap-961	6	30	not	not	PART
ap-961	6	31	be	be	AUX
ap-961	6	32	visualized	visualize	VERB
ap-961	6	33	.	.	PUNCT
ap-961	7	1	then	then	ADV
ap-961	7	2	why	why	SCONJ
ap-961	7	3	study	study	VERB
ap-961	7	4	noncommutative	noncommutative	ADJ
ap-961	7	5	geometry	geometry	NOUN
ap-961	7	6	?	?	PUNCT
ap-961	8	1	first	first	ADV
ap-961	8	2	of	of	ADP
ap-961	8	3	all	all	PRON
ap-961	8	4	,	,	PUNCT
ap-961	8	5	it	it	PRON
ap-961	8	6	seems	seem	VERB
ap-961	8	7	to	to	PART
ap-961	8	8	be	be	AUX
ap-961	8	9	a	a	DET
ap-961	8	10	natural	natural	ADJ
ap-961	8	11	and	and	CCONJ
ap-961	8	12	rich	rich	ADJ
ap-961	8	13	extension	extension	NOUN
ap-961	8	14	of	of	ADP
ap-961	8	15	the	the	DET
ap-961	8	16	concept	concept	NOUN
ap-961	8	17	of	of	ADP
ap-961	8	18	spaces	space	NOUN
ap-961	8	19	,	,	PUNCT
ap-961	8	20	one	one	NUM
ap-961	8	21	that	that	PRON
ap-961	8	22	can	can	AUX
ap-961	8	23	admit	admit	VERB
ap-961	8	24	the	the	DET
ap-961	8	25	notion	notion	NOUN
ap-961	8	26	of	of	ADP
ap-961	8	27	geometry	geometry	NOUN
ap-961	8	28	in	in	ADP
ap-961	8	29	its	its	PRON
ap-961	8	30	various	various	ADJ
ap-961	8	31	aspects	aspect	NOUN
ap-961	8	32	.	.	PUNCT
ap-961	9	1	moreover	moreover	ADV
ap-961	9	2	,	,	PUNCT
ap-961	9	3	within	within	ADP
ap-961	9	4	noncommutative	noncommutative	ADJ
ap-961	9	5	geometry	geometry	NOUN
ap-961	9	6	one	one	NUM
ap-961	9	7	has	have	VERB
ap-961	9	8	various	various	ADJ
ap-961	9	9	objects	object	NOUN
ap-961	9	10	on	on	ADP
ap-961	9	11	the	the	DET
ap-961	9	12	same	same	ADJ
ap-961	9	13	footing	footing	NOUN
ap-961	9	14	and	and	CCONJ
ap-961	9	15	one	one	PRON
ap-961	9	16	can	can	AUX
ap-961	9	17	do	do	VERB
ap-961	9	18	more	more	ADJ
ap-961	9	19	than	than	ADP
ap-961	9	20	within	within	ADP
ap-961	9	21	classical	classical	ADJ
ap-961	9	22	geometry	geometry	NOUN
ap-961	9	23	.	.	PUNCT
ap-961	10	1	last	last	ADJ
ap-961	10	2	not	not	PART
ap-961	10	3	least	least	ADJ
ap-961	10	4	one	one	NUM
ap-961	10	5	should	should	AUX
ap-961	10	6	mention	mention	VERB
ap-961	10	7	that	that	SCONJ
ap-961	10	8	many	many	ADJ
ap-961	10	9	basic	basic	ADJ
ap-961	10	10	examples	example	NOUN
ap-961	10	11	do	do	AUX
ap-961	10	12	arise	arise	VERB
ap-961	10	13	from	from	ADP
ap-961	10	14	physics	physics	NOUN
ap-961	10	15	:	:	PUNCT
ap-961	10	16	the	the	DET
ap-961	10	17	phase	phase	NOUN
ap-961	10	18	space	space	NOUN
ap-961	10	19	in	in	ADP
ap-961	10	20	quantum	quantum	ADJ
ap-961	10	21	mechanics	mechanic	NOUN
ap-961	10	22	,	,	PUNCT
ap-961	10	23	the	the	DET
ap-961	10	24	brillouin	brillouin	NOUN
ap-961	10	25	zone	zone	NOUN
ap-961	10	26	in	in	ADP
ap-961	10	27	the	the	DET
ap-961	10	28	quantum	quantum	NOUN
ap-961	10	29	hall	hall	NOUN
ap-961	10	30	effect	effect	NOUN
ap-961	10	31	,	,	PUNCT
ap-961	10	32	the	the	DET
ap-961	10	33	geometry	geometry	NOUN
ap-961	10	34	of	of	ADP
ap-961	10	35	finite	finite	ADJ
ap-961	10	36	spaces	space	NOUN
ap-961	10	37	in	in	ADP
ap-961	10	38	the	the	DET
ap-961	10	39	noncommutative	noncommutative	ADJ
ap-961	10	40	description	description	NOUN
ap-961	10	41	of	of	ADP
ap-961	10	42	the	the	DET
ap-961	10	43	standard	standard	ADJ
ap-961	10	44	model	model	NOUN
ap-961	10	45	,	,	PUNCT
ap-961	10	46	quantum	quantum	ADJ
ap-961	10	47	groups	group	NOUN
ap-961	10	48	in	in	ADP
ap-961	10	49	integrable	integrable	ADJ
ap-961	10	50	models	model	NOUN
ap-961	10	51	or	or	CCONJ
ap-961	10	52	the	the	DET
ap-961	10	53	quantized	quantize	VERB
ap-961	10	54	target	target	NOUN
ap-961	10	55	space	space	NOUN
ap-961	10	56	of	of	ADP
ap-961	10	57	string	string	NOUN
ap-961	10	58	theory	theory	NOUN
ap-961	10	59	.	.	PUNCT
ap-961	11	1	before	before	SCONJ
ap-961	11	2	we	we	PRON
ap-961	11	3	begin	begin	VERB
ap-961	11	4	this	this	DET
ap-961	11	5	easy	easy	ADJ
ap-961	11	6	walk	walk	NOUN
ap-961	11	7	through	through	ADP
ap-961	11	8	the	the	DET
ap-961	11	9	noncommutative	noncommutative	ADJ
ap-961	11	10	reality	reality	NOUN
ap-961	11	11	,	,	PUNCT
ap-961	11	12	we	we	PRON
ap-961	11	13	need	need	VERB
ap-961	11	14	to	to	PART
ap-961	11	15	mention	mention	VERB
ap-961	11	16	that	that	PRON
ap-961	11	17	–	–	PUNCT
ap-961	11	18	like	like	ADP
ap-961	11	19	any	any	DET
ap-961	11	20	subject	subject	NOUN
ap-961	11	21	,	,	PUNCT
ap-961	11	22	that	that	PRON
ap-961	11	23	is	be	AUX
ap-961	11	24	in	in	ADP
ap-961	11	25	the	the	DET
ap-961	11	26	early	early	ADJ
ap-961	11	27	stage	stage	NOUN
ap-961	11	28	of	of	ADP
ap-961	11	29	development	development	NOUN
ap-961	11	30	,	,	PUNCT
ap-961	11	31	it	it	PRON
ap-961	11	32	has	have	VERB
ap-961	11	33	many	many	ADJ
ap-961	11	34	branches	branch	NOUN
ap-961	11	35	.	.	PUNCT
ap-961	12	1	the	the	DET
ap-961	12	2	approach	approach	NOUN
ap-961	12	3	presented	present	VERB
ap-961	12	4	here	here	ADV
ap-961	12	5	falls	fall	VERB
ap-961	12	6	close	close	ADV
ap-961	12	7	to	to	ADP
ap-961	12	8	connes	conne	NOUN
ap-961	12	9	’	'	PUNCT
ap-961	12	10	noncommutative	noncommutative	ADJ
ap-961	12	11	geometry	geometry	NOUN
ap-961	12	12	(	(	PUNCT
ap-961	12	13	it	it	PRON
ap-961	12	14	appears	appear	VERB
ap-961	12	15	that	that	SCONJ
ap-961	12	16	the	the	DET
ap-961	12	17	wording	wording	NOUN
ap-961	12	18	connesian	connesian	ADJ
ap-961	12	19	variety	variety	PROPN
ap-961	12	20	has	have	AUX
ap-961	12	21	also	also	ADV
ap-961	12	22	been	be	AUX
ap-961	12	23	used	use	VERB
ap-961	12	24	to	to	PART
ap-961	12	25	describe	describe	VERB
ap-961	12	26	this	this	DET
ap-961	12	27	approach	approach	NOUN
ap-961	12	28	...	...	PUNCT
ap-961	12	29	)	)	PUNCT
ap-961	12	30	–	–	PUNCT
ap-961	12	31	(	(	PUNCT
ap-961	12	32	58b34	58b34	NUM
ap-961	12	33	,	,	PUNCT
ap-961	12	34	to	to	PART
ap-961	12	35	give	give	VERB
ap-961	12	36	mathematical	mathematical	ADJ
ap-961	12	37	subject	subject	ADJ
ap-961	12	38	classification	classification	NOUN
ap-961	12	39	number	number	NOUN
ap-961	12	40	)	)	PUNCT
ap-961	12	41	and	and	CCONJ
ap-961	12	42	noncommutative	noncommutative	ADJ
ap-961	12	43	differential	differential	ADJ
ap-961	12	44	geometry	geometry	NOUN
ap-961	12	45	–	–	PUNCT
ap-961	12	46	(	(	PUNCT
ap-961	12	47	46l87	46l87	NUM
ap-961	12	48	)	)	PUNCT
ap-961	12	49	.	.	PUNCT
ap-961	13	1	however	however	ADV
ap-961	13	2	,	,	PUNCT
ap-961	13	3	the	the	DET
ap-961	13	4	topics	topic	NOUN
ap-961	13	5	that	that	PRON
ap-961	13	6	we	we	PRON
ap-961	13	7	shall	shall	AUX
ap-961	13	8	mention	mention	VERB
ap-961	13	9	range	range	NOUN
ap-961	13	10	fromc*-algebras	fromc*-algebra	NOUN
ap-961	13	11	,	,	PUNCT
ap-961	13	12	differential	differential	NOUN
ap-961	13	13	algebras	algebra	NOUN
ap-961	13	14	to	to	ADP
ap-961	13	15	ideals	ideal	NOUN
ap-961	13	16	and	and	CCONJ
ap-961	13	17	exotic	exotic	ADJ
ap-961	13	18	traces	trace	NOUN
ap-961	13	19	.	.	PUNCT
ap-961	14	1	let	let	VERB
ap-961	14	2	us	we	PRON
ap-961	14	3	also	also	ADV
ap-961	14	4	attempt	attempt	VERB
ap-961	14	5	to	to	PART
ap-961	14	6	place	place	VERB
ap-961	14	7	the	the	DET
ap-961	14	8	subject	subject	ADJ
ap-961	14	9	matter	matter	NOUN
ap-961	14	10	of	of	ADP
ap-961	14	11	noncommutative	noncommutative	ADJ
ap-961	14	12	geometry	geometry	NOUN
ap-961	14	13	in	in	ADP
ap-961	14	14	relation	relation	NOUN
ap-961	14	15	to	to	ADP
ap-961	14	16	other	other	ADJ
ap-961	14	17	subjects	subject	NOUN
ap-961	14	18	in	in	ADP
ap-961	14	19	physics	physics	NOUN
ap-961	14	20	and	and	CCONJ
ap-961	14	21	mathematics	mathematic	NOUN
ap-961	14	22	.	.	PUNCT
ap-961	15	1	clearly	clearly	ADV
ap-961	15	2	,	,	PUNCT
ap-961	15	3	in	in	ADP
ap-961	15	4	mathematics	mathematics	PROPN
ap-961	15	5	noncommutative	noncommutative	PROPN
ap-961	15	6	geometry	geometry	NOUN
ap-961	15	7	lies	lie	VERB
ap-961	15	8	between	between	ADP
ap-961	15	9	algebra	algebra	NOUN
ap-961	15	10	and	and	CCONJ
ap-961	15	11	geometry	geometry	NOUN
ap-961	15	12	(	(	PUNCT
ap-961	15	13	meaning	mean	VERB
ap-961	15	14	rather	rather	ADV
ap-961	15	15	differential	differential	ADJ
ap-961	15	16	geometry	geometry	NOUN
ap-961	15	17	)	)	PUNCT
ap-961	15	18	,	,	PUNCT
ap-961	15	19	based	base	VERB
ap-961	15	20	on	on	ADP
ap-961	15	21	fundamental	fundamental	ADJ
ap-961	15	22	results	result	NOUN
ap-961	15	23	of	of	ADP
ap-961	15	24	operator	operator	NOUN
ap-961	15	25	algebras	algebra	NOUN
ap-961	15	26	.	.	PUNCT
ap-961	16	1	there	there	PRON
ap-961	16	2	are	be	VERB
ap-961	16	3	,	,	PUNCT
ap-961	16	4	however	however	ADV
ap-961	16	5	,	,	PUNCT
ap-961	16	6	many	many	ADJ
ap-961	16	7	other	other	ADJ
ap-961	16	8	,	,	PUNCT
ap-961	16	9	sometimes	sometimes	ADV
ap-961	16	10	not	not	PART
ap-961	16	11	evident	evident	ADJ
ap-961	16	12	connections	connection	NOUN
ap-961	16	13	–	–	PUNCT
ap-961	16	14	with	with	ADP
ap-961	16	15	topology	topology	NOUN
ap-961	16	16	,	,	PUNCT
ap-961	16	17	probability	probability	NOUN
ap-961	16	18	,	,	PUNCT
ap-961	16	19	measure	measure	NOUN
ap-961	16	20	theory	theory	NOUN
ap-961	16	21	,	,	PUNCT
ap-961	16	22	algebraic	algebraic	ADJ
ap-961	16	23	geometry	geometry	NOUN
ap-961	16	24	,	,	PUNCT
ap-961	16	25	ring	ring	NOUN
ap-961	16	26	theory	theory	NOUN
ap-961	16	27	and	and	CCONJ
ap-961	16	28	also	also	ADV
ap-961	16	29	with	with	ADP
ap-961	16	30	number	number	NOUN
ap-961	16	31	theory	theory	NOUN
ap-961	16	32	.	.	PUNCT
ap-961	17	1	in	in	ADP
ap-961	17	2	physics	physics	PROPN
ap-961	17	3	,	,	PUNCT
ap-961	17	4	it	it	PRON
ap-961	17	5	includes	include	VERB
ap-961	17	6	both	both	DET
ap-961	17	7	classical	classical	ADJ
ap-961	17	8	field	field	NOUN
ap-961	17	9	theory	theory	NOUN
ap-961	17	10	,	,	PUNCT
ap-961	17	11	quantum	quantum	ADJ
ap-961	17	12	field	field	NOUN
ap-961	17	13	theory	theory	NOUN
ap-961	17	14	,	,	PUNCT
ap-961	17	15	renormalization	renormalization	NOUN
ap-961	17	16	,	,	PUNCT
ap-961	17	17	quantum	quantum	NOUN
ap-961	17	18	mechanics	mechanic	NOUN
ap-961	17	19	as	as	ADV
ap-961	17	20	well	well	ADV
ap-961	17	21	as	as	ADP
ap-961	17	22	gravity	gravity	NOUN
ap-961	17	23	and	and	CCONJ
ap-961	17	24	string	string	NOUN
ap-961	17	25	theory	theory	NOUN
ap-961	17	26	.	.	PUNCT
ap-961	18	1	of	of	ADP
ap-961	18	2	course	course	NOUN
ap-961	18	3	,	,	PUNCT
ap-961	18	4	one	one	PRON
ap-961	18	5	should	should	AUX
ap-961	18	6	be	be	AUX
ap-961	18	7	aware	aware	ADJ
ap-961	18	8	that	that	SCONJ
ap-961	18	9	we	we	PRON
ap-961	18	10	are	be	AUX
ap-961	18	11	far	far	ADV
ap-961	18	12	from	from	ADP
ap-961	18	13	certain	certain	ADJ
ap-961	18	14	whether	whether	SCONJ
ap-961	18	15	the	the	DET
ap-961	18	16	notion	notion	NOUN
ap-961	18	17	of	of	ADP
ap-961	18	18	space	space	NOUN
ap-961	18	19	-	-	PUNCT
ap-961	18	20	time	time	NOUN
ap-961	18	21	is	be	AUX
ap-961	18	22	indeed	indeed	ADV
ap-961	18	23	best	well	ADV
ap-961	18	24	described	describe	VERB
ap-961	18	25	by	by	ADP
ap-961	18	26	noncommutative	noncommutative	ADJ
ap-961	18	27	geometry	geometry	NOUN
ap-961	18	28	and	and	CCONJ
ap-961	18	29	we	we	PRON
ap-961	18	30	still	still	ADV
ap-961	18	31	need	need	VERB
ap-961	18	32	some	some	DET
ap-961	18	33	crucial	crucial	ADJ
ap-961	18	34	theoretical	theoretical	ADJ
ap-961	18	35	steps	step	NOUN
ap-961	18	36	to	to	PART
ap-961	18	37	pursue	pursue	VERB
ap-961	18	38	this	this	DET
ap-961	18	39	goal	goal	NOUN
ap-961	18	40	.	.	PUNCT
ap-961	19	1	nevertheless	nevertheless	ADV
ap-961	19	2	some	some	DET
ap-961	19	3	qualitative	qualitative	ADJ
ap-961	19	4	considerations	consideration	NOUN
ap-961	19	5	and	and	CCONJ
ap-961	19	6	also	also	ADV
ap-961	19	7	the	the	DET
ap-961	19	8	evidence	evidence	NOUN
ap-961	19	9	that	that	SCONJ
ap-961	19	10	we	we	PRON
ap-961	19	11	already	already	ADV
ap-961	19	12	have	have	VERB
ap-961	19	13	from	from	ADP
ap-961	19	14	high	high	ADJ
ap-961	19	15	energy	energy	NOUN
ap-961	19	16	physics	physics	NOUN
ap-961	19	17	make	make	VERB
ap-961	19	18	this	this	DET
ap-961	19	19	line	line	NOUN
ap-961	19	20	of	of	ADP
ap-961	19	21	research	research	NOUN
ap-961	19	22	quite	quite	ADV
ap-961	19	23	promising	promising	ADJ
ap-961	19	24	.	.	PUNCT
ap-961	20	1	in	in	ADP
ap-961	20	2	this	this	DET
ap-961	20	3	set	set	NOUN
ap-961	20	4	of	of	ADP
ap-961	20	5	lectures	lecture	NOUN
ap-961	20	6	we	we	PRON
ap-961	20	7	shall	shall	AUX
ap-961	20	8	present	present	VERB
ap-961	20	9	an	an	DET
ap-961	20	10	overview	overview	NOUN
ap-961	20	11	of	of	ADP
ap-961	20	12	mathematical	mathematical	ADJ
ap-961	20	13	objects	object	NOUN
ap-961	20	14	and	and	CCONJ
ap-961	20	15	tools	tool	NOUN
ap-961	20	16	leading	lead	VERB
ap-961	20	17	us	we	PRON
ap-961	20	18	towards	towards	ADP
ap-961	20	19	the	the	DET
ap-961	20	20	noncommutative	noncommutative	ADJ
ap-961	20	21	world	world	NOUN
ap-961	20	22	.	.	PUNCT
ap-961	21	1	this	this	DET
ap-961	21	2	set	set	NOUN
ap-961	21	3	of	of	ADP
ap-961	21	4	lectures	lecture	NOUN
ap-961	21	5	is	be	AUX
ap-961	21	6	by	by	ADP
ap-961	21	7	no	no	DET
ap-961	21	8	means	mean	NOUN
ap-961	21	9	self	self	NOUN
ap-961	21	10	-	-	PUNCT
ap-961	21	11	consistent	consistent	ADJ
ap-961	21	12	and	and	CCONJ
ap-961	21	13	many	many	ADJ
ap-961	21	14	statements	statement	NOUN
ap-961	21	15	are	be	AUX
ap-961	21	16	quoted	quote	VERB
ap-961	21	17	without	without	ADP
ap-961	21	18	proofs	proof	NOUN
ap-961	21	19	.	.	PUNCT
ap-961	22	1	some	some	DET
ap-961	22	2	statements	statement	NOUN
ap-961	22	3	are	be	AUX
ap-961	22	4	also	also	ADV
ap-961	22	5	presented	present	VERB
ap-961	22	6	not	not	PART
ap-961	22	7	in	in	ADP
ap-961	22	8	their	their	PRON
ap-961	22	9	most	most	ADV
ap-961	22	10	general	general	ADJ
ap-961	22	11	form	form	NOUN
ap-961	22	12	–	–	PUNCT
ap-961	22	13	we	we	PRON
ap-961	22	14	do	do	VERB
ap-961	22	15	this	this	PRON
ap-961	22	16	on	on	ADP
ap-961	22	17	purpose	purpose	NOUN
ap-961	22	18	,	,	PUNCT
ap-961	22	19	having	have	VERB
ap-961	22	20	as	as	ADP
ap-961	22	21	a	a	DET
ap-961	22	22	basic	basic	ADJ
ap-961	22	23	guide	guide	NOUN
ap-961	22	24	the	the	DET
ap-961	22	25	need	need	NOUN
ap-961	22	26	to	to	PART
ap-961	22	27	explain	explain	VERB
ap-961	22	28	the	the	DET
ap-961	22	29	ideas	idea	NOUN
ap-961	22	30	rather	rather	ADV
ap-961	22	31	than	than	ADP
ap-961	22	32	technical	technical	ADJ
ap-961	22	33	details	detail	NOUN
ap-961	22	34	.	.	PUNCT
ap-961	23	1	we	we	PRON
ap-961	23	2	assume	assume	VERB
ap-961	23	3	some	some	DET
ap-961	23	4	basic	basic	ADJ
ap-961	23	5	knowledge	knowledge	NOUN
ap-961	23	6	of	of	ADP
ap-961	23	7	algebra	algebra	NOUN
ap-961	23	8	,	,	PUNCT
ap-961	23	9	differential	differential	NOUN
ap-961	23	10	geometry	geometry	NOUN
ap-961	23	11	,	,	PUNCT
ap-961	23	12	operator	operator	NOUN
ap-961	23	13	algebras	algebra	NOUN
ap-961	23	14	,	,	PUNCT
ap-961	23	15	hilbert	hilbert	NOUN
ap-961	23	16	spaces	space	VERB
ap-961	23	17	as	as	ADV
ap-961	23	18	well	well	ADV
ap-961	23	19	as	as	ADP
ap-961	23	20	some	some	DET
ap-961	23	21	knowledge	knowledge	NOUN
ap-961	23	22	of	of	ADP
ap-961	23	23	gauge	gauge	ADJ
ap-961	23	24	theory	theory	NOUN
ap-961	23	25	and	and	CCONJ
ap-961	23	26	characteristic	characteristic	ADJ
ap-961	23	27	classes	class	NOUN
ap-961	23	28	.	.	PUNCT
ap-961	24	1	although	although	SCONJ
ap-961	24	2	not	not	PART
ap-961	24	3	essential	essential	ADJ
ap-961	24	4	some	some	DET
ap-961	24	5	knowledge	knowledge	NOUN
ap-961	24	6	in	in	ADP
ap-961	24	7	these	these	DET
ap-961	24	8	topics	topic	NOUN
ap-961	24	9	is	be	AUX
ap-961	24	10	a	a	DET
ap-961	24	11	big	big	ADJ
ap-961	24	12	help	help	NOUN
ap-961	24	13	when	when	SCONJ
ap-961	24	14	learning	learn	VERB
ap-961	24	15	noncommutative	noncommutative	ADJ
ap-961	24	16	geometry	geometry	NOUN
ap-961	24	17	–	–	PUNCT
ap-961	24	18	a	a	DET
ap-961	24	19	good	good	ADJ
ap-961	24	20	example	example	NOUN
ap-961	24	21	of	of	ADP
ap-961	24	22	a	a	DET
ap-961	24	23	textbook	textbook	NOUN
ap-961	24	24	that	that	PRON
ap-961	24	25	can	can	AUX
ap-961	24	26	be	be	AUX
ap-961	24	27	used	use	VERB
ap-961	24	28	as	as	ADP
ap-961	24	29	a	a	DET
ap-961	24	30	starting	starting	NOUN
ap-961	24	31	point	point	NOUN
ap-961	24	32	is	be	AUX
ap-961	24	33	the	the	DET
ap-961	24	34	classical	classical	ADJ
ap-961	24	35	text	text	NOUN
ap-961	24	36	[	[	X
ap-961	24	37	4	4	NUM
ap-961	24	38	]	]	PUNCT
ap-961	24	39	.	.	PUNCT
ap-961	25	1	let	let	VERB
ap-961	25	2	us	we	PRON
ap-961	25	3	finish	finish	VERB
ap-961	25	4	this	this	DET
ap-961	25	5	introduction	introduction	NOUN
ap-961	25	6	with	with	ADP
ap-961	25	7	the	the	DET
ap-961	25	8	quotation	quotation	NOUN
ap-961	25	9	from	from	ADP
ap-961	25	10	alain	alain	PROPN
ap-961	25	11	connes	conne	NOUN
ap-961	25	12	’	'	PUNCT
ap-961	25	13	interview	interview	NOUN
ap-961	25	14	with	with	ADP
ap-961	25	15	george	george	PROPN
ap-961	25	16	skandalis	skandalis	PROPN
ap-961	25	17	(	(	PUNCT
ap-961	25	18	newsletter	newsletter	NOUN
ap-961	25	19	of	of	ADP
ap-961	25	20	the	the	DET
ap-961	25	21	european	european	PROPN
ap-961	25	22	mathematical	mathematical	PROPN
ap-961	25	23	society	society	NOUN
ap-961	25	24	,	,	PUNCT
ap-961	25	25	no	no	DET
ap-961	25	26	3	3	NUM
ap-961	25	27	.	.	PUNCT
ap-961	25	28	(	(	PUNCT
ap-961	25	29	2007	2007	NUM
ap-961	25	30	)	)	PUNCT
ap-961	25	31	)	)	PUNCT
ap-961	25	32	.	.	PUNCT
ap-961	26	1	�	�	PROPN
ap-961	26	2	what	what	PRON
ap-961	26	3	is	be	AUX
ap-961	26	4	noncommutative	noncommutative	ADJ
ap-961	26	5	geometry	geometry	NOUN
ap-961	26	6	?	?	PUNCT
ap-961	27	1	in	in	ADP
ap-961	27	2	your	your	PRON
ap-961	27	3	opinion	opinion	NOUN
ap-961	27	4	,	,	PUNCT
ap-961	27	5	is	be	AUX
ap-961	27	6	“	"	PUNCT
ap-961	27	7	noncommutative	noncommutative	ADJ
ap-961	27	8	geometry	geometry	NOUN
ap-961	27	9	”	"	PUNCT
ap-961	27	10	simply	simply	ADV
ap-961	27	11	a	a	DET
ap-961	27	12	better	well	ADJ
ap-961	27	13	name	name	NOUN
ap-961	27	14	for	for	ADP
ap-961	27	15	operator	operator	NOUN
ap-961	27	16	algebras	algebra	NOUN
ap-961	27	17	or	or	CCONJ
ap-961	27	18	is	be	AUX
ap-961	27	19	it	it	PRON
ap-961	27	20	a	a	DET
ap-961	27	21	close	close	ADJ
ap-961	27	22	but	but	CCONJ
ap-961	27	23	distinct	distinct	ADJ
ap-961	27	24	field	field	NOUN
ap-961	27	25	?	?	PUNCT
ap-961	28	1	�	�	PROPN
ap-961	29	1	yes	yes	INTJ
ap-961	29	2	,	,	PUNCT
ap-961	29	3	it	it	PRON
ap-961	29	4	’s	’	VERB
ap-961	29	5	important	important	ADJ
ap-961	29	6	to	to	PART
ap-961	29	7	be	be	AUX
ap-961	29	8	more	more	ADV
ap-961	29	9	precise	precise	ADJ
ap-961	29	10	.	.	PUNCT
ap-961	30	1	first	first	ADV
ap-961	30	2	,	,	PUNCT
ap-961	30	3	noncommutative	noncommutative	ADJ
ap-961	30	4	geometry	geometry	NOUN
ap-961	30	5	for	for	ADP
ap-961	30	6	me	i	PRON
ap-961	30	7	is	be	AUX
ap-961	30	8	this	this	DET
ap-961	30	9	duality	duality	NOUN
ap-961	30	10	between	between	ADP
ap-961	30	11	geometry	geometry	NOUN
ap-961	30	12	and	and	CCONJ
ap-961	30	13	algebra	algebra	NOUN
ap-961	30	14	,	,	PUNCT
ap-961	30	15	with	with	ADP
ap-961	30	16	a	a	DET
ap-961	30	17	striking	striking	ADJ
ap-961	30	18	coincidence	coincidence	NOUN
ap-961	30	19	between	between	ADP
ap-961	30	20	the	the	DET
ap-961	30	21	algebraic	algebraic	ADJ
ap-961	30	22	rules	rule	NOUN
ap-961	30	23	and	and	CCONJ
ap-961	30	24	the	the	DET
ap-961	30	25	linguistic	linguistic	ADJ
ap-961	30	26	ones	one	NOUN
ap-961	30	27	.	.	PUNCT
ap-961	31	1	ordinary	ordinary	ADJ
ap-961	31	2	language	language	NOUN
ap-961	31	3	never	never	ADV
ap-961	31	4	uses	use	VERB
ap-961	31	5	parentheses	parenthesis	NOUN
ap-961	31	6	inside	inside	ADP
ap-961	31	7	the	the	DET
ap-961	31	8	words	word	NOUN
ap-961	31	9	.	.	PUNCT
ap-961	32	1	this	this	PRON
ap-961	32	2	means	mean	VERB
ap-961	32	3	that	that	SCONJ
ap-961	32	4	associativity	associativity	NOUN
ap-961	32	5	is	be	AUX
ap-961	32	6	taken	take	VERB
ap-961	32	7	into	into	ADP
ap-961	32	8	account	account	NOUN
ap-961	32	9	,	,	PUNCT
ap-961	32	10	but	but	CCONJ
ap-961	32	11	not	not	PART
ap-961	32	12	commutativity	commutativity	NOUN
ap-961	32	13	,	,	PUNCT
ap-961	32	14	which	which	PRON
ap-961	32	15	would	would	AUX
ap-961	32	16	permit	permit	VERB
ap-961	32	17	permuting	permute	VERB
ap-961	32	18	the	the	DET
ap-961	32	19	letters	letter	NOUN
ap-961	32	20	freely	freely	ADV
ap-961	32	21	.	.	PUNCT
ap-961	33	1	36	36	NUM
ap-961	33	2	©	©	PROPN
ap-961	33	3	czech	czech	PROPN
ap-961	33	4	technical	technical	PROPN
ap-961	33	5	university	university	PROPN
ap-961	33	6	publishing	publishing	NOUN
ap-961	33	7	house	house	NOUN
ap-961	33	8	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	33	9	acta	acta	PROPN
ap-961	33	10	polytechnica	polytechnica	PROPN
ap-961	33	11	vol	vol	NOUN
ap-961	33	12	.	.	PUNCT
ap-961	34	1	48	48	NUM
ap-961	34	2	no	no	NOUN
ap-961	34	3	.	.	PUNCT
ap-961	35	1	2/2008	2/2008	PROPN
ap-961	35	2	3	3	NUM
ap-961	35	3	�	�	NOUN
ap-961	35	4	lectures	lecture	NOUN
ap-961	35	5	on	on	ADP
ap-961	35	6	noncommutative	noncommutative	ADJ
ap-961	35	7	geometry	geometry	NOUN
ap-961	35	8	a.	a.	NOUN
ap-961	35	9	sitarz	sitarz	NOUN
ap-961	35	10	we	we	PRON
ap-961	35	11	present	present	VERB
ap-961	35	12	a	a	DET
ap-961	35	13	short	short	ADJ
ap-961	35	14	overview	overview	NOUN
ap-961	35	15	of	of	ADP
ap-961	35	16	noncommutative	noncommutative	ADJ
ap-961	35	17	geometry	geometry	NOUN
ap-961	35	18	.	.	PUNCT
ap-961	36	1	starting	start	VERB
ap-961	36	2	with	with	ADP
ap-961	36	3	c	c	PROPN
ap-961	36	4	*	*	X
ap-961	36	5	algebras	algebra	NOUN
ap-961	36	6	and	and	CCONJ
ap-961	36	7	noncommutative	noncommutative	ADJ
ap-961	36	8	differential	differential	NOUN
ap-961	36	9	forms	form	NOUN
ap-961	36	10	we	we	PRON
ap-961	36	11	pass	pass	VERB
ap-961	36	12	to	to	ADP
ap-961	36	13	k	k	NOUN
ap-961	36	14	-	-	NOUN
ap-961	36	15	theory	theory	NOUN
ap-961	36	16	,	,	PUNCT
ap-961	36	17	k	k	NOUN
ap-961	36	18	-	-	NOUN
ap-961	36	19	homology	homology	NOUN
ap-961	36	20	and	and	CCONJ
ap-961	36	21	cyclic	cyclic	ADJ
ap-961	36	22	(	(	PUNCT
ap-961	36	23	co)homology	co)homology	NOUN
ap-961	36	24	,	,	PUNCT
ap-961	36	25	and	and	CCONJ
ap-961	36	26	we	we	PRON
ap-961	36	27	finish	finish	VERB
ap-961	36	28	with	with	ADP
ap-961	36	29	the	the	DET
ap-961	36	30	notion	notion	NOUN
ap-961	36	31	of	of	ADP
ap-961	36	32	spectral	spectral	ADJ
ap-961	36	33	triples	triple	NOUN
ap-961	36	34	and	and	CCONJ
ap-961	36	35	the	the	DET
ap-961	36	36	spectral	spectral	ADJ
ap-961	36	37	action	action	NOUN
ap-961	36	38	.	.	PUNCT
ap-961	37	1	keywords	keyword	NOUN
ap-961	37	2	:	:	PUNCT
ap-961	37	3	connes	conne	NOUN
ap-961	37	4	’	'	PUNCT
ap-961	37	5	noncommutative	noncommutative	ADJ
ap-961	37	6	geometry	geometry	NOUN
ap-961	37	7	,	,	PUNCT
ap-961	37	8	spectral	spectral	ADJ
ap-961	37	9	triples	triple	NOUN
ap-961	37	10	(	(	PUNCT
ap-961	37	11	mcs	mcs	PROPN
ap-961	37	12	2000	2000	NUM
ap-961	37	13	:	:	PUNCT
ap-961	37	14	58b34	58b34	NUM
ap-961	37	15	,	,	PUNCT
ap-961	37	16	46l85	46l85	NUM
ap-961	37	17	,	,	PUNCT
ap-961	37	18	46l87	46l87	NUM
ap-961	37	19	,	,	PUNCT
ap-961	37	20	81t75	81t75	NUM
ap-961	37	21	)	)	PUNCT
ap-961	37	22	.	.	PUNCT
ap-961	38	1	2	2	NUM
ap-961	38	2	towards	towards	ADP
ap-961	38	3	noncommutative	noncommutative	ADJ
ap-961	38	4	topology	topology	NOUN
ap-961	38	5	mathemata	mathemata	ADJ
ap-961	38	6	mathematicis	mathematicis	NOUN
ap-961	38	7	scribuntur	scribuntur	PROPN
ap-961	38	8	.	.	PUNCT
ap-961	39	1	(	(	PUNCT
ap-961	39	2	mathematics	mathematic	NOUN
ap-961	39	3	is	be	AUX
ap-961	39	4	written	write	VERB
ap-961	39	5	for	for	ADP
ap-961	39	6	mathematicians	mathematician	NOUN
ap-961	39	7	.	.	PUNCT
ap-961	39	8	)	)	PUNCT
ap-961	40	1	nicolaus	nicolaus	PROPN
ap-961	40	2	copernicus	copernicus	PROPN
ap-961	40	3	2.1	2.1	NUM
ap-961	40	4	where	where	SCONJ
ap-961	40	5	it	it	PRON
ap-961	40	6	all	all	PRON
ap-961	40	7	begins	begin	VERB
ap-961	40	8	:	:	PUNCT
ap-961	40	9	gelfand	gelfand	ADJ
ap-961	40	10	-	-	PUNCT
ap-961	40	11	naimark	naimark	PROPN
ap-961	40	12	when	when	SCONJ
ap-961	40	13	we	we	PRON
ap-961	40	14	say	say	VERB
ap-961	40	15	space	space	NOUN
ap-961	40	16	,	,	PUNCT
ap-961	40	17	we	we	PRON
ap-961	40	18	usually	usually	ADV
ap-961	40	19	mean	mean	VERB
ap-961	40	20	a	a	DET
ap-961	40	21	topological	topological	ADJ
ap-961	40	22	space	space	NOUN
ap-961	40	23	.	.	PUNCT
ap-961	41	1	it	it	PRON
ap-961	41	2	may	may	AUX
ap-961	41	3	,	,	PUNCT
ap-961	41	4	of	of	ADP
ap-961	41	5	course	course	NOUN
ap-961	41	6	,	,	PUNCT
ap-961	41	7	have	have	VERB
ap-961	41	8	some	some	DET
ap-961	41	9	additional	additional	ADJ
ap-961	41	10	properties	property	NOUN
ap-961	41	11	and	and	CCONJ
ap-961	41	12	features	feature	NOUN
ap-961	41	13	but	but	CCONJ
ap-961	41	14	the	the	DET
ap-961	41	15	basic	basic	ADJ
ap-961	41	16	ingredient	ingredient	NOUN
ap-961	41	17	,	,	PUNCT
ap-961	41	18	topology	topology	NOUN
ap-961	41	19	,	,	PUNCT
ap-961	41	20	is	be	AUX
ap-961	41	21	there	there	ADV
ap-961	41	22	.	.	PUNCT
ap-961	42	1	we	we	PRON
ap-961	42	2	usually	usually	ADV
ap-961	42	3	assume	assume	VERB
ap-961	42	4	that	that	SCONJ
ap-961	42	5	the	the	DET
ap-961	42	6	topology	topology	NOUN
ap-961	42	7	is	be	AUX
ap-961	42	8	hausdorff	hausdorff	NOUN
ap-961	42	9	,	,	PUNCT
ap-961	42	10	which	which	PRON
ap-961	42	11	means	mean	VERB
ap-961	42	12	that	that	SCONJ
ap-961	42	13	every	every	DET
ap-961	42	14	two	two	NUM
ap-961	42	15	points	point	NOUN
ap-961	42	16	can	can	AUX
ap-961	42	17	be	be	AUX
ap-961	42	18	separated	separate	VERB
ap-961	42	19	by	by	ADP
ap-961	42	20	disjoint	disjoint	ADJ
ap-961	42	21	open	open	ADJ
ap-961	42	22	sets	set	NOUN
ap-961	42	23	.	.	PUNCT
ap-961	43	1	we	we	PRON
ap-961	43	2	know	know	VERB
ap-961	43	3	many	many	ADJ
ap-961	43	4	examples	example	NOUN
ap-961	43	5	of	of	ADP
ap-961	43	6	topological	topological	ADJ
ap-961	43	7	spaces	space	NOUN
ap-961	43	8	and	and	CCONJ
ap-961	43	9	we	we	PRON
ap-961	43	10	have	have	VERB
ap-961	43	11	a	a	DET
ap-961	43	12	good	good	ADJ
ap-961	43	13	notion	notion	NOUN
ap-961	43	14	of	of	ADP
ap-961	43	15	continuous	continuous	ADJ
ap-961	43	16	(	(	PUNCT
ap-961	43	17	complex	complex	ADJ
ap-961	43	18	valued	value	VERB
ap-961	43	19	)	)	PUNCT
ap-961	43	20	functions	function	NOUN
ap-961	43	21	.	.	PUNCT
ap-961	44	1	one	one	NUM
ap-961	44	2	of	of	ADP
ap-961	44	3	the	the	DET
ap-961	44	4	very	very	ADV
ap-961	44	5	basic	basic	ADJ
ap-961	44	6	observations	observation	NOUN
ap-961	44	7	(	(	PUNCT
ap-961	44	8	that	that	SCONJ
ap-961	44	9	we	we	PRON
ap-961	44	10	know	know	VERB
ap-961	44	11	almost	almost	ADV
ap-961	44	12	intuitively	intuitively	ADV
ap-961	44	13	)	)	PUNCT
ap-961	44	14	is	be	AUX
ap-961	44	15	that	that	SCONJ
ap-961	44	16	all	all	DET
ap-961	44	17	continuous	continuous	ADJ
ap-961	44	18	functions	function	NOUN
ap-961	44	19	over	over	ADP
ap-961	44	20	a	a	DET
ap-961	44	21	topological	topological	ADJ
ap-961	44	22	space	space	NOUN
ap-961	44	23	form	form	NOUN
ap-961	44	24	a	a	DET
ap-961	44	25	complex	complex	ADJ
ap-961	44	26	vector	vector	NOUN
ap-961	44	27	space	space	NOUN
ap-961	44	28	and	and	CCONJ
ap-961	44	29	that	that	SCONJ
ap-961	44	30	the	the	DET
ap-961	44	31	product	product	NOUN
ap-961	44	32	of	of	ADP
ap-961	44	33	two	two	NUM
ap-961	44	34	continuous	continuous	ADJ
ap-961	44	35	functions	function	NOUN
ap-961	44	36	is	be	AUX
ap-961	44	37	still	still	ADV
ap-961	44	38	a	a	DET
ap-961	44	39	continuous	continuous	ADJ
ap-961	44	40	function	function	NOUN
ap-961	44	41	.	.	PUNCT
ap-961	45	1	this	this	PRON
ap-961	45	2	says	say	VERB
ap-961	45	3	that	that	SCONJ
ap-961	45	4	we	we	PRON
ap-961	45	5	have	have	VERB
ap-961	45	6	an	an	DET
ap-961	45	7	algebra	algebra	NOUN
ap-961	45	8	of	of	ADP
ap-961	45	9	continuous	continuous	ADJ
ap-961	45	10	functions	function	NOUN
ap-961	45	11	and	and	CCONJ
ap-961	45	12	,	,	PUNCT
ap-961	45	13	since	since	SCONJ
ap-961	45	14	we	we	PRON
ap-961	45	15	can	can	AUX
ap-961	45	16	take	take	VERB
ap-961	45	17	a	a	DET
ap-961	45	18	complex	complex	ADJ
ap-961	45	19	conjugate	conjugate	NOUN
ap-961	45	20	of	of	ADP
ap-961	45	21	a	a	DET
ap-961	45	22	function	function	NOUN
ap-961	45	23	,	,	PUNCT
ap-961	45	24	it	it	PRON
ap-961	45	25	is	be	AUX
ap-961	45	26	a	a	DET
ap-961	45	27	*	*	PUNCT
ap-961	45	28	-algebra	-algebra	NOUN
ap-961	45	29	.	.	PUNCT
ap-961	46	1	let	let	VERB
ap-961	46	2	us	we	PRON
ap-961	46	3	assume	assume	VERB
ap-961	46	4	for	for	ADP
ap-961	46	5	a	a	DET
ap-961	46	6	while	while	NOUN
ap-961	46	7	that	that	SCONJ
ap-961	46	8	our	our	PRON
ap-961	46	9	space	space	NOUN
ap-961	46	10	is	be	AUX
ap-961	46	11	compact	compact	ADJ
ap-961	46	12	,	,	PUNCT
ap-961	46	13	then	then	ADV
ap-961	46	14	clearly	clearly	ADV
ap-961	46	15	to	to	ADP
ap-961	46	16	each	each	DET
ap-961	46	17	function	function	NOUN
ap-961	46	18	we	we	PRON
ap-961	46	19	can	can	AUX
ap-961	46	20	associate	associate	VERB
ap-961	46	21	the	the	DET
ap-961	46	22	supremum	supremum	NOUN
ap-961	46	23	of	of	ADP
ap-961	46	24	its	its	PRON
ap-961	46	25	absolute	absolute	ADJ
ap-961	46	26	value	value	NOUN
ap-961	46	27	.	.	PUNCT
ap-961	47	1	thus	thus	ADV
ap-961	47	2	we	we	PRON
ap-961	47	3	arrive	arrive	VERB
ap-961	47	4	at	at	ADP
ap-961	47	5	the	the	DET
ap-961	47	6	norm	norm	NOUN
ap-961	47	7	of	of	ADP
ap-961	47	8	a	a	DET
ap-961	47	9	function	function	NOUN
ap-961	47	10	.	.	PUNCT
ap-961	48	1	going	go	VERB
ap-961	48	2	a	a	DET
ap-961	48	3	step	step	NOUN
ap-961	48	4	further	far	ADV
ap-961	48	5	we	we	PRON
ap-961	48	6	come	come	VERB
ap-961	48	7	to	to	ADP
ap-961	48	8	the	the	DET
ap-961	48	9	notion	notion	NOUN
ap-961	48	10	of	of	ADP
ap-961	48	11	a	a	DET
ap-961	48	12	c*-algebra	c*-algebra	NOUN
ap-961	48	13	with	with	ADP
ap-961	48	14	the	the	DET
ap-961	48	15	following	follow	VERB
ap-961	48	16	formal	formal	ADJ
ap-961	48	17	definition	definition	NOUN
ap-961	48	18	:	:	PUNCT
ap-961	48	19	definition	definition	NOUN
ap-961	48	20	2.1	2.1	NUM
ap-961	48	21	:	:	PUNCT
ap-961	48	22	an	an	DET
ap-961	48	23	involutive	involutive	ADJ
ap-961	48	24	banach	banach	NOUN
ap-961	48	25	algebra	algebra	NOUN
ap-961	48	26	�	�	PROPN
ap-961	48	27	(	(	PUNCT
ap-961	48	28	that	that	PRON
ap-961	48	29	is	be	AUX
ap-961	48	30	a	a	DET
ap-961	48	31	complex	complex	ADJ
ap-961	48	32	normed	normed	ADJ
ap-961	48	33	algebra	algebra	NOUN
ap-961	48	34	,	,	PUNCT
ap-961	48	35	which	which	PRON
ap-961	48	36	is	be	AUX
ap-961	48	37	complete	complete	ADJ
ap-961	48	38	as	as	ADP
ap-961	48	39	a	a	DET
ap-961	48	40	topological	topological	ADJ
ap-961	48	41	space	space	NOUN
ap-961	48	42	in	in	ADP
ap-961	48	43	the	the	DET
ap-961	48	44	norm	norm	NOUN
ap-961	48	45	)	)	PUNCT
ap-961	48	46	such	such	ADJ
ap-961	48	47	that	that	SCONJ
ap-961	48	48	aa	aa	PROPN
ap-961	48	49	a	a	PROPN
ap-961	48	50	*	*	PUNCT
ap-961	48	51	�	�	PROPN
ap-961	48	52	2	2	NUM
ap-961	48	53	,	,	PUNCT
ap-961	48	54	�	�	PROPN
ap-961	48	55	�	�	PROPN
ap-961	48	56	a	a	DET
ap-961	48	57	�	�	PROPN
ap-961	48	58	is	be	AUX
ap-961	48	59	a	a	DET
ap-961	48	60	c*-algebra	c*-algebra	PROPN
ap-961	48	61	.	.	PUNCT
ap-961	49	1	so	so	ADV
ap-961	49	2	,	,	PUNCT
ap-961	49	3	to	to	PART
ap-961	49	4	cut	cut	VERB
ap-961	49	5	the	the	DET
ap-961	49	6	story	story	NOUN
ap-961	49	7	short	short	ADJ
ap-961	49	8	:	:	PUNCT
ap-961	49	9	the	the	DET
ap-961	49	10	algebra	algebra	NOUN
ap-961	49	11	of	of	ADP
ap-961	49	12	continuous	continuous	ADJ
ap-961	49	13	functions	function	NOUN
ap-961	49	14	on	on	ADP
ap-961	49	15	a	a	DET
ap-961	49	16	compact	compact	ADJ
ap-961	49	17	hausdorff	hausdorff	NOUN
ap-961	49	18	space	space	NOUN
ap-961	49	19	is	be	AUX
ap-961	49	20	a	a	DET
ap-961	49	21	c*-algebra	c*-algebra	NOUN
ap-961	49	22	with	with	ADP
ap-961	49	23	a	a	DET
ap-961	49	24	unit	unit	NOUN
ap-961	49	25	!	!	PUNCT
ap-961	50	1	but	but	CCONJ
ap-961	50	2	does	do	AUX
ap-961	50	3	it	it	PRON
ap-961	50	4	work	work	VERB
ap-961	50	5	the	the	DET
ap-961	50	6	other	other	ADJ
ap-961	50	7	way	way	NOUN
ap-961	50	8	round	round	ADV
ap-961	50	9	?	?	PUNCT
ap-961	51	1	surprisingly	surprisingly	ADV
ap-961	51	2	(	(	PUNCT
ap-961	51	3	or	or	CCONJ
ap-961	51	4	not	not	PART
ap-961	51	5	surprisingly	surprisingly	ADV
ap-961	51	6	)	)	PUNCT
ap-961	52	1	yes	yes	INTJ
ap-961	52	2	,	,	PUNCT
ap-961	52	3	as	as	SCONJ
ap-961	52	4	we	we	PRON
ap-961	52	5	can	can	AUX
ap-961	52	6	state	state	VERB
ap-961	52	7	in	in	ADP
ap-961	52	8	the	the	DET
ap-961	52	9	gelfand	gelfand	ADJ
ap-961	52	10	-	-	PUNCT
ap-961	52	11	naimark	naimark	NOUN
ap-961	52	12	theorem	theorem	NOUN
ap-961	52	13	:	:	PUNCT
ap-961	52	14	theorem	theorem	ADJ
ap-961	52	15	2.2	2.2	NUM
ap-961	52	16	:	:	PUNCT
ap-961	52	17	gelfand	gelfand	NOUN
ap-961	52	18	-	-	PUNCT
ap-961	52	19	naimark	naimark	PROPN
ap-961	52	20	a	a	DET
ap-961	52	21	commutative	commutative	ADJ
ap-961	52	22	unital	unital	ADJ
ap-961	52	23	c*-algebra	c*-algebra	PROPN
ap-961	52	24	is	be	AUX
ap-961	52	25	an	an	DET
ap-961	52	26	algebra	algebra	NOUN
ap-961	52	27	of	of	ADP
ap-961	52	28	continuous	continuous	ADJ
ap-961	52	29	functions	function	NOUN
ap-961	52	30	on	on	ADP
ap-961	52	31	a	a	DET
ap-961	52	32	compact	compact	ADJ
ap-961	52	33	hausdorff	hausdorff	NOUN
ap-961	52	34	space	space	NOUN
ap-961	52	35	.	.	PUNCT
ap-961	53	1	we	we	PRON
ap-961	53	2	shall	shall	AUX
ap-961	53	3	not	not	PART
ap-961	53	4	discuss	discuss	VERB
ap-961	53	5	the	the	DET
ap-961	53	6	details	detail	NOUN
ap-961	53	7	of	of	ADP
ap-961	53	8	the	the	DET
ap-961	53	9	proof	proof	NOUN
ap-961	53	10	as	as	SCONJ
ap-961	53	11	this	this	PRON
ap-961	53	12	can	can	AUX
ap-961	53	13	be	be	AUX
ap-961	53	14	found	find	VERB
ap-961	53	15	in	in	ADP
ap-961	53	16	numerous	numerous	ADJ
ap-961	53	17	textbooks	textbook	NOUN
ap-961	53	18	–	–	PUNCT
ap-961	53	19	and	and	CCONJ
ap-961	53	20	is	be	AUX
ap-961	53	21	(	(	PUNCT
ap-961	53	22	in	in	ADP
ap-961	53	23	principle	principle	NOUN
ap-961	53	24	)	)	PUNCT
ap-961	53	25	quite	quite	ADV
ap-961	53	26	easy	easy	ADJ
ap-961	53	27	.	.	PUNCT
ap-961	54	1	the	the	DET
ap-961	54	2	points	point	NOUN
ap-961	54	3	of	of	ADP
ap-961	54	4	the	the	DET
ap-961	54	5	space	space	NOUN
ap-961	54	6	are	be	AUX
ap-961	54	7	provided	provide	VERB
ap-961	54	8	by	by	ADP
ap-961	54	9	the	the	DET
ap-961	54	10	characters	character	NOUN
ap-961	54	11	of	of	ADP
ap-961	54	12	the	the	DET
ap-961	54	13	algebra	algebra	NOUN
ap-961	54	14	,	,	PUNCT
ap-961	54	15	which	which	PRON
ap-961	54	16	are	be	AUX
ap-961	54	17	continuous	continuous	ADJ
ap-961	54	18	algebra	algebra	NOUN
ap-961	54	19	morphisms	morphism	VERB
ap-961	54	20	from	from	ADP
ap-961	54	21	the	the	DET
ap-961	54	22	algebra	algebra	NOUN
ap-961	54	23	to	to	ADP
ap-961	54	24	the	the	DET
ap-961	54	25	complex	complex	ADJ
ap-961	54	26	numbers	number	NOUN
ap-961	54	27	.	.	PUNCT
ap-961	55	1	this	this	PRON
ap-961	55	2	is	be	AUX
ap-961	55	3	the	the	DET
ap-961	55	4	dawn	dawn	NOUN
ap-961	55	5	of	of	ADP
ap-961	55	6	noncommutative	noncommutative	ADJ
ap-961	55	7	geometry	geometry	NOUN
ap-961	55	8	.	.	PUNCT
ap-961	56	1	why	why	SCONJ
ap-961	56	2	?	?	PUNCT
ap-961	57	1	note	note	VERB
ap-961	57	2	that	that	SCONJ
ap-961	57	3	c	c	X
ap-961	57	4	*	*	PUNCT
ap-961	57	5	algebras	algebra	NOUN
ap-961	57	6	might	might	AUX
ap-961	57	7	not	not	PART
ap-961	57	8	be	be	AUX
ap-961	57	9	commutative	commutative	ADJ
ap-961	57	10	.	.	PUNCT
ap-961	58	1	indeed	indeed	ADV
ap-961	58	2	,	,	PUNCT
ap-961	58	3	take	take	VERB
ap-961	58	4	an	an	DET
ap-961	58	5	example	example	NOUN
ap-961	58	6	:	:	PUNCT
ap-961	58	7	the	the	DET
ap-961	58	8	algebra	algebra	PROPN
ap-961	58	9	mn	mn	PROPN
ap-961	58	10	(	(	PUNCT
ap-961	58	11	)	)	PUNCT
ap-961	58	12	�	�	PROPN
ap-961	58	13	of	of	ADP
ap-961	58	14	matrices	matrix	NOUN
ap-961	58	15	with	with	ADP
ap-961	58	16	complex	complex	ADJ
ap-961	58	17	entries	entry	NOUN
ap-961	58	18	,	,	PUNCT
ap-961	58	19	with	with	ADP
ap-961	58	20	hermitian	hermitian	ADJ
ap-961	58	21	conjugation	conjugation	NOUN
ap-961	58	22	and	and	CCONJ
ap-961	58	23	matrix	matrix	NOUN
ap-961	58	24	multiplication	multiplication	NOUN
ap-961	58	25	is	be	AUX
ap-961	58	26	a	a	DET
ap-961	58	27	very	very	ADV
ap-961	58	28	good	good	ADJ
ap-961	58	29	and	and	CCONJ
ap-961	58	30	simple	simple	ADJ
ap-961	58	31	example	example	NOUN
ap-961	58	32	of	of	ADP
ap-961	58	33	a	a	DET
ap-961	58	34	noncommutative	noncommutative	ADJ
ap-961	58	35	c	c	NOUN
ap-961	58	36	*	*	NOUN
ap-961	58	37	algebra	algebra	PROPN
ap-961	58	38	.	.	PUNCT
ap-961	59	1	then	then	ADV
ap-961	59	2	in	in	ADP
ap-961	59	3	the	the	DET
ap-961	59	4	view	view	NOUN
ap-961	59	5	of	of	ADP
ap-961	59	6	gelfand	gelfand	ADJ
ap-961	59	7	-	-	PUNCT
ap-961	59	8	naimark	naimark	NOUN
ap-961	59	9	’s	’s	PART
ap-961	59	10	theorem	theorem	NOUN
ap-961	59	11	we	we	PRON
ap-961	59	12	can	can	AUX
ap-961	59	13	use	use	VERB
ap-961	59	14	noncommutative	noncommutative	ADJ
ap-961	59	15	c	c	PROPN
ap-961	59	16	*	*	X
ap-961	59	17	algebras	algebras	PROPN
ap-961	59	18	as	as	ADP
ap-961	59	19	the	the	DET
ap-961	59	20	definition	definition	NOUN
ap-961	59	21	of	of	ADP
ap-961	59	22	noncommutative	noncommutative	ADJ
ap-961	59	23	hausdorff	hausdorff	PROPN
ap-961	59	24	compact	compact	ADJ
ap-961	59	25	spaces	space	NOUN
ap-961	59	26	.	.	PUNCT
ap-961	60	1	but	but	CCONJ
ap-961	60	2	these	these	DET
ap-961	60	3	space	space	NOUN
ap-961	60	4	have	have	VERB
ap-961	60	5	no	no	DET
ap-961	60	6	points	point	NOUN
ap-961	60	7	or	or	CCONJ
ap-961	60	8	have	have	VERB
ap-961	60	9	just	just	ADV
ap-961	60	10	a	a	DET
ap-961	60	11	couple	couple	NOUN
ap-961	60	12	of	of	ADP
ap-961	60	13	points	point	NOUN
ap-961	60	14	!	!	PUNCT
ap-961	61	1	coming	come	VERB
ap-961	61	2	back	back	ADV
ap-961	61	3	to	to	ADP
ap-961	61	4	the	the	DET
ap-961	61	5	matrix	matrix	NOUN
ap-961	61	6	algebra	algebra	NOUN
ap-961	61	7	mn	mn	PROPN
ap-961	61	8	(	(	PUNCT
ap-961	61	9	)	)	PUNCT
ap-961	61	10	�	�	PROPN
ap-961	61	11	we	we	PRON
ap-961	61	12	see	see	VERB
ap-961	61	13	that	that	PRON
ap-961	61	14	for	for	ADP
ap-961	61	15	n	n	PRON
ap-961	61	16	�	�	NOUN
ap-961	61	17	1	1	NUM
ap-961	61	18	there	there	PRON
ap-961	61	19	are	be	VERB
ap-961	61	20	no	no	DET
ap-961	61	21	characters	character	NOUN
ap-961	61	22	at	at	ADV
ap-961	61	23	all	all	ADV
ap-961	61	24	.	.	PUNCT
ap-961	62	1	another	another	DET
ap-961	62	2	very	very	ADV
ap-961	62	3	good	good	ADJ
ap-961	62	4	(	(	PUNCT
ap-961	62	5	and	and	CCONJ
ap-961	62	6	generic	generic	ADJ
ap-961	62	7	,	,	PUNCT
ap-961	62	8	as	as	SCONJ
ap-961	62	9	we	we	PRON
ap-961	62	10	shall	shall	AUX
ap-961	62	11	see	see	VERB
ap-961	62	12	)	)	PUNCT
ap-961	62	13	example	example	NOUN
ap-961	62	14	of	of	ADP
ap-961	62	15	a	a	DET
ap-961	62	16	c*-algebra	c*-algebra	PROPN
ap-961	62	17	comes	come	VERB
ap-961	62	18	from	from	ADP
ap-961	62	19	the	the	DET
ap-961	62	20	theory	theory	NOUN
ap-961	62	21	of	of	ADP
ap-961	62	22	hilbert	hilbert	PROPN
ap-961	62	23	spaces	space	NOUN
ap-961	62	24	.	.	PUNCT
ap-961	63	1	the	the	DET
ap-961	63	2	recipe	recipe	NOUN
ap-961	63	3	is	be	AUX
ap-961	63	4	very	very	ADV
ap-961	63	5	easy	easy	ADJ
ap-961	63	6	:	:	PUNCT
ap-961	63	7	take	take	VERB
ap-961	63	8	a	a	DET
ap-961	63	9	(	(	PUNCT
ap-961	63	10	separable	separable	ADJ
ap-961	63	11	)	)	PUNCT
ap-961	63	12	hilbert	hilbert	PROPN
ap-961	63	13	space	space	NOUN
ap-961	63	14	�	�	PROPN
ap-961	63	15	and	and	CCONJ
ap-961	63	16	take	take	VERB
ap-961	63	17	an	an	DET
ap-961	63	18	algebra	algebra	NOUN
ap-961	63	19	�	�	PROPN
ap-961	63	20	(	(	PUNCT
ap-961	63	21	�	�	PROPN
ap-961	63	22	)	)	PUNCT
ap-961	63	23	of	of	ADP
ap-961	63	24	bounded	bounded	ADJ
ap-961	63	25	operators	operator	NOUN
ap-961	63	26	on	on	ADP
ap-961	63	27	�	�	PROPN
ap-961	63	28	with	with	ADP
ap-961	63	29	an	an	DET
ap-961	63	30	operator	operator	NOUN
ap-961	63	31	norm	norm	NOUN
ap-961	63	32	.	.	PUNCT
ap-961	64	1	then	then	ADV
ap-961	64	2	any	any	DET
ap-961	64	3	norm	norm	NOUN
ap-961	64	4	closed	close	VERB
ap-961	64	5	subalgebra	subalgebra	NOUN
ap-961	64	6	of	of	ADP
ap-961	64	7	�	�	PROPN
ap-961	64	8	(	(	PUNCT
ap-961	64	9	�	�	PROPN
ap-961	64	10	)	)	PUNCT
ap-961	64	11	is	be	AUX
ap-961	64	12	a	a	DET
ap-961	64	13	(	(	PUNCT
ap-961	64	14	separable	separable	NOUN
ap-961	64	15	)	)	PUNCT
ap-961	64	16	c	c	NOUN
ap-961	64	17	*	*	PUNCT
ap-961	64	18	algebra	algebra	PROPN
ap-961	64	19	.	.	PUNCT
ap-961	65	1	of	of	ADP
ap-961	65	2	course	course	NOUN
ap-961	65	3	,	,	PUNCT
ap-961	65	4	our	our	PRON
ap-961	65	5	toy	toy	NOUN
ap-961	65	6	-	-	PUNCT
ap-961	65	7	model	model	NOUN
ap-961	65	8	algebra	algebra	PROPN
ap-961	65	9	mn	mn	PROPN
ap-961	65	10	(	(	PUNCT
ap-961	65	11	)	)	PUNCT
ap-961	65	12	�	�	PROPN
ap-961	65	13	is	be	AUX
ap-961	65	14	in	in	ADP
ap-961	65	15	fact	fact	NOUN
ap-961	65	16	of	of	ADP
ap-961	65	17	this	this	DET
ap-961	65	18	type	type	NOUN
ap-961	65	19	:	:	PUNCT
ap-961	65	20	just	just	ADV
ap-961	65	21	take	take	VERB
ap-961	65	22	�	�	PROPN
ap-961	65	23	�	�	PROPN
ap-961	65	24	�	�	PROPN
ap-961	65	25	n	n	PROPN
ap-961	65	26	.	.	PUNCT
ap-961	66	1	the	the	DET
ap-961	66	2	algebra	algebra	NOUN
ap-961	66	3	of	of	ADP
ap-961	66	4	all	all	DET
ap-961	66	5	bounded	bounded	ADJ
ap-961	66	6	operators	operator	NOUN
ap-961	66	7	on	on	ADP
ap-961	66	8	it	it	PRON
ap-961	66	9	is	be	AUX
ap-961	66	10	nothing	nothing	PRON
ap-961	66	11	else	else	ADV
ap-961	66	12	but	but	CCONJ
ap-961	66	13	mn	mn	PROPN
ap-961	66	14	(	(	PUNCT
ap-961	66	15	)	)	PUNCT
ap-961	66	16	�	�	PROPN
ap-961	66	17	.	.	PUNCT
ap-961	67	1	of	of	ADP
ap-961	67	2	course	course	ADV
ap-961	67	3	,	,	PUNCT
ap-961	67	4	we	we	PRON
ap-961	67	5	have	have	AUX
ap-961	67	6	shown	show	VERB
ap-961	67	7	one	one	NUM
ap-961	67	8	of	of	ADP
ap-961	67	9	the	the	DET
ap-961	67	10	most	most	ADV
ap-961	67	11	restrictive	restrictive	ADJ
ap-961	67	12	versions	version	NOUN
ap-961	67	13	of	of	ADP
ap-961	67	14	the	the	DET
ap-961	67	15	theorem	theorem	NOUN
ap-961	67	16	.	.	PUNCT
ap-961	68	1	if	if	SCONJ
ap-961	68	2	we	we	PRON
ap-961	68	3	forget	forget	VERB
ap-961	68	4	about	about	ADP
ap-961	68	5	unital	unital	ADJ
ap-961	68	6	,	,	PUNCT
ap-961	68	7	we	we	PRON
ap-961	68	8	still	still	ADV
ap-961	68	9	get	get	VERB
ap-961	68	10	hausdorff	hausdorff	NOUN
ap-961	68	11	spaces	space	NOUN
ap-961	68	12	but	but	CCONJ
ap-961	68	13	the	the	PRON
ap-961	68	14	are	be	AUX
ap-961	68	15	only	only	ADV
ap-961	68	16	locally	locally	ADV
ap-961	68	17	compact	compact	ADJ
ap-961	68	18	.	.	PUNCT
ap-961	69	1	2.2	2.2	NUM
ap-961	69	2	.	.	PUNCT
ap-961	70	1	into	into	ADP
ap-961	70	2	the	the	DET
ap-961	70	3	c*-world	c*-world	NOUN
ap-961	70	4	.	.	PUNCT
ap-961	71	1	what	what	PRON
ap-961	71	2	are	be	AUX
ap-961	71	3	c	c	NOUN
ap-961	71	4	*	*	NOUN
ap-961	71	5	algebras	algebras	X
ap-961	71	6	?	?	PUNCT
ap-961	72	1	there	there	PRON
ap-961	72	2	are	be	VERB
ap-961	72	3	many	many	ADJ
ap-961	72	4	(	(	PUNCT
ap-961	72	5	isomorphic	isomorphic	ADJ
ap-961	72	6	)	)	PUNCT
ap-961	72	7	definitions	definition	NOUN
ap-961	72	8	out	out	ADP
ap-961	72	9	of	of	ADP
ap-961	72	10	which	which	PRON
ap-961	72	11	we	we	PRON
ap-961	72	12	have	have	AUX
ap-961	72	13	already	already	ADV
ap-961	72	14	presented	present	VERB
ap-961	72	15	one	one	NUM
ap-961	72	16	.	.	PUNCT
ap-961	73	1	we	we	PRON
ap-961	73	2	know	know	VERB
ap-961	73	3	that	that	SCONJ
ap-961	73	4	there	there	PRON
ap-961	73	5	are	be	VERB
ap-961	73	6	many	many	ADJ
ap-961	73	7	c	c	NOUN
ap-961	73	8	*	*	X
ap-961	73	9	algebras	algebra	NOUN
ap-961	73	10	,	,	PUNCT
ap-961	73	11	and	and	CCONJ
ap-961	73	12	what	what	PRON
ap-961	73	13	we	we	PRON
ap-961	73	14	would	would	AUX
ap-961	73	15	like	like	VERB
ap-961	73	16	is	be	AUX
ap-961	73	17	to	to	PART
ap-961	73	18	have	have	VERB
ap-961	73	19	a	a	DET
ap-961	73	20	description	description	NOUN
ap-961	73	21	of	of	ADP
ap-961	73	22	them	they	PRON
ap-961	73	23	that	that	PRON
ap-961	73	24	would	would	AUX
ap-961	73	25	be	be	AUX
ap-961	73	26	on	on	ADP
ap-961	73	27	the	the	DET
ap-961	73	28	same	same	ADJ
ap-961	73	29	footing	footing	NOUN
ap-961	73	30	–	–	PUNCT
ap-961	73	31	independently	independently	ADV
ap-961	73	32	,	,	PUNCT
ap-961	73	33	whether	whether	SCONJ
ap-961	73	34	there	there	PRON
ap-961	73	35	are	be	VERB
ap-961	73	36	commutative	commutative	ADJ
ap-961	73	37	or	or	CCONJ
ap-961	73	38	noncommutative	noncommutative	ADJ
ap-961	73	39	.	.	PUNCT
ap-961	74	1	and	and	CCONJ
ap-961	74	2	we	we	PRON
ap-961	74	3	want	want	VERB
ap-961	74	4	not	not	PART
ap-961	74	5	an	an	DET
ap-961	74	6	abstract	abstract	ADJ
ap-961	74	7	description	description	NOUN
ap-961	74	8	but	but	CCONJ
ap-961	74	9	a	a	DET
ap-961	74	10	concrete	concrete	ADJ
ap-961	74	11	one	one	NUM
ap-961	74	12	.	.	PUNCT
ap-961	75	1	again	again	ADV
ap-961	75	2	,	,	PUNCT
ap-961	75	3	gelfand	gelfand	ADJ
ap-961	75	4	,	,	PUNCT
ap-961	75	5	naimark	naimark	NOUN
ap-961	75	6	and	and	CCONJ
ap-961	75	7	segal	segal	PROPN
ap-961	75	8	come	come	VERB
ap-961	75	9	to	to	ADP
ap-961	75	10	our	our	PRON
ap-961	75	11	aid	aid	NOUN
ap-961	75	12	:	:	PUNCT
ap-961	75	13	theorem	theorem	VERB
ap-961	75	14	2.3	2.3	NUM
ap-961	75	15	(	(	PUNCT
ap-961	75	16	gns	gns	NOUN
ap-961	75	17	):	):	PUNCT
ap-961	75	18	every	every	DET
ap-961	75	19	abstract	abstract	ADJ
ap-961	75	20	c*-algebra	c*-algebra	X
ap-961	75	21	�	�	PROPN
ap-961	75	22	is	be	AUX
ap-961	75	23	isometrically	isometrically	NOUN
ap-961	75	24	*	*	PUNCT
ap-961	75	25	-isomorphic	-isomorphic	ADJ
ap-961	75	26	to	to	ADP
ap-961	75	27	a	a	DET
ap-961	75	28	concrete	concrete	ADJ
ap-961	75	29	c	c	NOUN
ap-961	75	30	*	*	NOUN
ap-961	75	31	algebra	algebra	NOUN
ap-961	75	32	of	of	ADP
ap-961	75	33	operators	operator	NOUN
ap-961	75	34	on	on	ADP
ap-961	75	35	a	a	DET
ap-961	75	36	hilbert	hilbert	NOUN
ap-961	75	37	space	space	NOUN
ap-961	75	38	�	�	PROPN
ap-961	75	39	.	.	PUNCT
ap-961	76	1	if	if	SCONJ
ap-961	76	2	the	the	DET
ap-961	76	3	algebra	algebra	PROPN
ap-961	76	4	�	�	PROPN
ap-961	76	5	is	be	AUX
ap-961	76	6	separable	separable	ADJ
ap-961	76	7	then	then	ADV
ap-961	76	8	we	we	PRON
ap-961	76	9	can	can	AUX
ap-961	76	10	take	take	VERB
ap-961	76	11	�	�	PROPN
ap-961	76	12	to	to	PART
ap-961	76	13	be	be	AUX
ap-961	76	14	separable	separable	ADJ
ap-961	76	15	.	.	PUNCT
ap-961	77	1	now	now	ADV
ap-961	77	2	we	we	PRON
ap-961	77	3	have	have	VERB
ap-961	77	4	a	a	DET
ap-961	77	5	powerful	powerful	ADJ
ap-961	77	6	tool	tool	NOUN
ap-961	77	7	:	:	PUNCT
ap-961	77	8	a	a	DET
ap-961	77	9	description	description	NOUN
ap-961	77	10	of	of	ADP
ap-961	77	11	all	all	DET
ap-961	77	12	c*-algebras	c*-algebra	NOUN
ap-961	77	13	as	as	ADP
ap-961	77	14	operators	operator	NOUN
ap-961	77	15	on	on	ADP
ap-961	77	16	the	the	DET
ap-961	77	17	hilbert	hilbert	NOUN
ap-961	77	18	space	space	NOUN
ap-961	77	19	.	.	PUNCT
ap-961	78	1	this	this	PRON
ap-961	78	2	is	be	AUX
ap-961	78	3	a	a	DET
ap-961	78	4	good	good	ADJ
ap-961	78	5	starting	starting	NOUN
ap-961	78	6	point	point	NOUN
ap-961	78	7	for	for	ADP
ap-961	78	8	noncommutative	noncommutative	ADJ
ap-961	78	9	topology	topology	NOUN
ap-961	78	10	and	and	CCONJ
ap-961	78	11	towards	towards	ADP
ap-961	78	12	many	many	ADJ
ap-961	78	13	other	other	ADJ
ap-961	78	14	notions	notion	NOUN
ap-961	78	15	like	like	ADP
ap-961	78	16	measurable	measurable	ADJ
ap-961	78	17	functions	function	NOUN
ap-961	78	18	,	,	PUNCT
ap-961	78	19	for	for	ADP
ap-961	78	20	instance	instance	NOUN
ap-961	78	21	.	.	PUNCT
ap-961	79	1	to	to	PART
ap-961	79	2	summarize	summarize	VERB
ap-961	79	3	this	this	DET
ap-961	79	4	section	section	NOUN
ap-961	79	5	let	let	VERB
ap-961	79	6	us	we	PRON
ap-961	79	7	quote	quote	VERB
ap-961	79	8	the	the	DET
ap-961	79	9	dictionary	dictionary	PROPN
ap-961	79	10	,	,	PUNCT
ap-961	79	11	which	which	PRON
ap-961	79	12	establishes	establish	VERB
ap-961	79	13	parallel	parallel	ADJ
ap-961	79	14	notions	notion	NOUN
ap-961	79	15	between	between	ADP
ap-961	79	16	standard	standard	ADJ
ap-961	79	17	and	and	CCONJ
ap-961	79	18	noncommutative	noncommutative	ADJ
ap-961	79	19	topology	topology	NOUN
ap-961	79	20	:	:	PUNCT
ap-961	79	21	topology	topology	NOUN
ap-961	79	22	algebra	algebra	NOUN
ap-961	79	23	(	(	PUNCT
ap-961	79	24	locally	locally	ADV
ap-961	79	25	compact	compact	ADJ
ap-961	79	26	)	)	PUNCT
ap-961	79	27	topological	topological	ADJ
ap-961	79	28	space	space	NOUN
ap-961	79	29	commutative	commutative	ADJ
ap-961	79	30	c*-algebra	c*-algebra	PROPN
ap-961	79	31	homeomorphism	homeomorphism	PROPN
ap-961	79	32	automorphism	automorphism	NOUN
ap-961	79	33	continuous	continuous	ADJ
ap-961	79	34	proper	proper	ADJ
ap-961	79	35	map	map	NOUN
ap-961	79	36	morphism	morphism	NOUN
ap-961	79	37	compact	compact	ADJ
ap-961	79	38	space	space	NOUN
ap-961	79	39	unital	unital	ADJ
ap-961	79	40	-algebra	-algebra	PROPN
ap-961	79	41	open	open	ADJ
ap-961	79	42	(	(	PUNCT
ap-961	79	43	dense	dense	ADJ
ap-961	79	44	)	)	PUNCT
ap-961	79	45	subset	subset	NOUN
ap-961	79	46	(	(	PUNCT
ap-961	79	47	essential	essential	ADJ
ap-961	79	48	)	)	PUNCT
ap-961	79	49	ideal	ideal	ADJ
ap-961	79	50	compactification	compactification	NOUN
ap-961	79	51	unitization	unitization	NOUN
ap-961	79	52	stone	stone	NOUN
ap-961	79	53	-	-	PUNCT
ap-961	79	54	čech	čech	NOUN
ap-961	79	55	compactification	compactification	NOUN
ap-961	79	56	multiplier	multipli	ADJ
ap-961	79	57	algebra	algebra	NOUN
ap-961	79	58	cartesian	cartesian	ADJ
ap-961	79	59	product	product	NOUN
ap-961	79	60	tensor	tensor	NOUN
ap-961	79	61	product	product	NOUN
ap-961	79	62	2.3	2.3	NUM
ap-961	79	63	the	the	DET
ap-961	79	64	tricky	tricky	ADJ
ap-961	79	65	bits	bit	NOUN
ap-961	79	66	and	and	CCONJ
ap-961	79	67	examples	example	NOUN
ap-961	79	68	among	among	ADP
ap-961	79	69	the	the	DET
ap-961	79	70	first	first	ADJ
ap-961	79	71	problems	problem	NOUN
ap-961	79	72	that	that	PRON
ap-961	79	73	arise	arise	VERB
ap-961	79	74	in	in	ADP
ap-961	79	75	the	the	DET
ap-961	79	76	noncommutative	noncommutative	ADJ
ap-961	79	77	world	world	NOUN
ap-961	79	78	and	and	CCONJ
ap-961	79	79	have	have	VERB
ap-961	79	80	no	no	DET
ap-961	79	81	classical	classical	ADJ
ap-961	79	82	correspondence	correspondence	NOUN
ap-961	79	83	,	,	PUNCT
ap-961	79	84	are	be	AUX
ap-961	79	85	some	some	DET
ap-961	79	86	ambiguities	ambiguity	NOUN
ap-961	79	87	in	in	ADP
ap-961	79	88	construction	construction	NOUN
ap-961	79	89	.	.	PUNCT
ap-961	80	1	for	for	ADP
ap-961	80	2	this	this	DET
ap-961	80	3	reason	reason	NOUN
ap-961	80	4	,	,	PUNCT
ap-961	80	5	one	one	PRON
ap-961	80	6	should	should	AUX
ap-961	80	7	treat	treat	VERB
ap-961	80	8	some	some	DET
ap-961	80	9	“	"	PUNCT
ap-961	80	10	equivalences	equivalence	NOUN
ap-961	80	11	”	"	PUNCT
ap-961	80	12	from	from	ADP
ap-961	80	13	the	the	DET
ap-961	80	14	above	above	ADJ
ap-961	80	15	dictionary	dictionary	NOUN
ap-961	80	16	with	with	ADP
ap-961	80	17	due	due	ADJ
ap-961	80	18	respect	respect	NOUN
ap-961	80	19	.	.	PUNCT
ap-961	81	1	a	a	DET
ap-961	81	2	good	good	ADJ
ap-961	81	3	example	example	NOUN
ap-961	81	4	is	be	AUX
ap-961	81	5	the	the	DET
ap-961	81	6	case	case	NOUN
ap-961	81	7	of	of	ADP
ap-961	81	8	the	the	DET
ap-961	81	9	tensor	tensor	NOUN
ap-961	81	10	product	product	NOUN
ap-961	81	11	of	of	ADP
ap-961	81	12	noncommutative	noncommutative	PROPN
ap-961	81	13	c	c	PROPN
ap-961	81	14	*	*	X
ap-961	81	15	algebras	algebra	NOUN
ap-961	81	16	,	,	PUNCT
ap-961	81	17	which	which	PRON
ap-961	81	18	might	might	AUX
ap-961	81	19	depend	depend	VERB
ap-961	81	20	on	on	ADP
ap-961	81	21	the	the	DET
ap-961	81	22	completion	completion	NOUN
ap-961	81	23	of	of	ADP
ap-961	81	24	the	the	DET
ap-961	81	25	algebraic	algebraic	ADJ
ap-961	81	26	tensor	tensor	NOUN
ap-961	81	27	product	product	NOUN
ap-961	81	28	of	of	ADP
ap-961	81	29	two	two	NUM
ap-961	81	30	algebras	algebra	NOUN
ap-961	81	31	.	.	PUNCT
ap-961	82	1	let	let	VERB
ap-961	82	2	us	we	PRON
ap-961	82	3	first	first	ADV
ap-961	82	4	have	have	VERB
ap-961	82	5	a	a	DET
ap-961	82	6	look	look	NOUN
ap-961	82	7	at	at	ADP
ap-961	82	8	an	an	DET
ap-961	82	9	example	example	NOUN
ap-961	82	10	:	:	PUNCT
ap-961	82	11	example	example	NOUN
ap-961	82	12	2.4	2.4	NUM
ap-961	82	13	:	:	PUNCT
ap-961	82	14	take	take	VERB
ap-961	82	15	an	an	DET
ap-961	82	16	interval	interval	NOUN
ap-961	82	17	i	i	PRON
ap-961	82	18	�	�	INTJ
ap-961	82	19	(	(	PUNCT
ap-961	82	20	,	,	PUNCT
ap-961	82	21	)	)	PUNCT
ap-961	82	22	0	0	NUM
ap-961	82	23	1	1	NUM
ap-961	82	24	and	and	CCONJ
ap-961	82	25	the	the	DET
ap-961	82	26	algebra	algebra	NOUN
ap-961	82	27	of	of	ADP
ap-961	82	28	continuous	continuous	ADJ
ap-961	82	29	functions	function	NOUN
ap-961	82	30	on	on	ADP
ap-961	82	31	it	it	PRON
ap-961	82	32	,	,	PUNCT
ap-961	82	33	c	c	PROPN
ap-961	82	34	i	i	PRON
ap-961	82	35	(	(	PUNCT
ap-961	82	36	)	)	PUNCT
ap-961	82	37	.	.	PUNCT
ap-961	83	1	of	of	ADP
ap-961	83	2	course	course	ADV
ap-961	83	3	,	,	PUNCT
ap-961	83	4	we	we	PRON
ap-961	83	5	can	can	AUX
ap-961	83	6	interpret	interpret	VERB
ap-961	83	7	©	©	PROPN
ap-961	83	8	czech	czech	PROPN
ap-961	83	9	technical	technical	PROPN
ap-961	83	10	university	university	PROPN
ap-961	83	11	publishing	publishing	NOUN
ap-961	83	12	house	house	NOUN
ap-961	83	13	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	83	14	37	37	NUM
ap-961	83	15	acta	acta	PROPN
ap-961	83	16	polytechnica	polytechnica	PROPN
ap-961	83	17	vol	vol	NOUN
ap-961	83	18	.	.	PUNCT
ap-961	84	1	48	48	NUM
ap-961	84	2	no	no	NOUN
ap-961	84	3	.	.	PUNCT
ap-961	85	1	2/2008	2/2008	NUM
ap-961	85	2	each	each	DET
ap-961	85	3	element	element	NOUN
ap-961	85	4	of	of	ADP
ap-961	85	5	the	the	DET
ap-961	85	6	tensor	tensor	NOUN
ap-961	85	7	product	product	NOUN
ap-961	85	8	c	c	NOUN
ap-961	86	1	i	i	PRON
ap-961	86	2	c	c	VERB
ap-961	86	3	i	i	PRON
ap-961	86	4	(	(	PUNCT
ap-961	86	5	)	)	PUNCT
ap-961	86	6	(	(	PUNCT
ap-961	86	7	)	)	PUNCT
ap-961	86	8	�	�	PROPN
ap-961	86	9	as	as	ADP
ap-961	86	10	a	a	DET
ap-961	86	11	continuous	continuous	ADJ
ap-961	86	12	function	function	NOUN
ap-961	86	13	on	on	ADP
ap-961	86	14	the	the	DET
ap-961	86	15	cartesian	cartesian	ADJ
ap-961	86	16	product	product	NOUN
ap-961	86	17	i	i	PRON
ap-961	86	18	i	i	PROPN
ap-961	86	19	i2	i2	PROPN
ap-961	86	20	�	�	PROPN
ap-961	86	21	�	�	PROPN
ap-961	86	22	,	,	PUNCT
ap-961	86	23	but	but	CCONJ
ap-961	86	24	it	it	PRON
ap-961	86	25	clear	clear	VERB
ap-961	86	26	that	that	SCONJ
ap-961	86	27	not	not	PART
ap-961	86	28	all	all	DET
ap-961	86	29	continuous	continuous	ADJ
ap-961	86	30	functions	function	NOUN
ap-961	86	31	on	on	ADP
ap-961	86	32	i2	i2	PROPN
ap-961	86	33	are	be	AUX
ap-961	86	34	of	of	ADP
ap-961	86	35	this	this	DET
ap-961	86	36	form	form	NOUN
ap-961	86	37	.	.	PUNCT
ap-961	87	1	of	of	ADP
ap-961	87	2	course	course	ADV
ap-961	87	3	,	,	PUNCT
ap-961	87	4	it	it	PRON
ap-961	87	5	is	be	AUX
ap-961	87	6	true	true	ADJ
ap-961	87	7	thatc	thatc	NOUN
ap-961	88	1	i	i	PRON
ap-961	88	2	c	c	VERB
ap-961	88	3	i	i	PRON
ap-961	88	4	(	(	PUNCT
ap-961	88	5	)	)	PUNCT
ap-961	88	6	(	(	PUNCT
ap-961	88	7	)	)	PUNCT
ap-961	88	8	�	�	PROPN
ap-961	88	9	is	be	AUX
ap-961	88	10	dense	dense	ADJ
ap-961	88	11	inc	inc	PROPN
ap-961	88	12	i	i	PROPN
ap-961	88	13	(	(	PUNCT
ap-961	88	14	)	)	PUNCT
ap-961	88	15	2	2	NUM
ap-961	88	16	,	,	PUNCT
ap-961	88	17	so	so	ADV
ap-961	89	1	,	,	PUNCT
ap-961	89	2	in	in	ADP
ap-961	89	3	order	order	NOUN
ap-961	89	4	to	to	PART
ap-961	89	5	work	work	VERB
ap-961	89	6	with	with	ADP
ap-961	89	7	tensor	tensor	NOUN
ap-961	89	8	products	product	NOUN
ap-961	89	9	of	of	ADP
ap-961	89	10	c	c	X
ap-961	89	11	*	*	VERB
ap-961	89	12	we	we	PRON
ap-961	89	13	need	need	VERB
ap-961	89	14	to	to	PART
ap-961	89	15	work	work	VERB
ap-961	89	16	out	out	ADP
ap-961	89	17	a	a	DET
ap-961	89	18	way	way	NOUN
ap-961	89	19	to	to	PART
ap-961	89	20	complete	complete	VERB
ap-961	89	21	the	the	DET
ap-961	89	22	algebraic	algebraic	ADJ
ap-961	89	23	tensor	tensor	NOUN
ap-961	89	24	product	product	NOUN
ap-961	89	25	.	.	PUNCT
ap-961	90	1	this	this	PRON
ap-961	90	2	leads	lead	VERB
ap-961	90	3	to	to	ADP
ap-961	90	4	a	a	DET
ap-961	90	5	rather	rather	ADV
ap-961	90	6	extended	extended	ADJ
ap-961	90	7	and	and	CCONJ
ap-961	90	8	not	not	PART
ap-961	90	9	unique	unique	ADJ
ap-961	90	10	construction	construction	NOUN
ap-961	90	11	unlike	unlike	ADP
ap-961	90	12	the	the	DET
ap-961	90	13	topological	topological	ADJ
ap-961	90	14	cartesian	cartesian	ADJ
ap-961	90	15	product	product	NOUN
ap-961	90	16	of	of	ADP
ap-961	90	17	spaces	space	NOUN
ap-961	90	18	.	.	PUNCT
ap-961	91	1	this	this	PRON
ap-961	91	2	is	be	AUX
ap-961	91	3	rather	rather	ADV
ap-961	91	4	bad	bad	ADJ
ap-961	91	5	news	news	NOUN
ap-961	91	6	but	but	CCONJ
ap-961	91	7	we	we	PRON
ap-961	91	8	might	might	AUX
ap-961	91	9	find	find	VERB
ap-961	91	10	some	some	DET
ap-961	91	11	comfort	comfort	NOUN
ap-961	91	12	in	in	ADP
ap-961	91	13	the	the	DET
ap-961	91	14	fact	fact	NOUN
ap-961	91	15	that	that	SCONJ
ap-961	91	16	many	many	ADJ
ap-961	91	17	algebras	algebra	NOUN
ap-961	91	18	(	(	PUNCT
ap-961	91	19	called	call	VERB
ap-961	91	20	nuclear	nuclear	NOUN
ap-961	91	21	)	)	PUNCT
ap-961	91	22	do	do	AUX
ap-961	91	23	have	have	AUX
ap-961	91	24	unique	unique	ADJ
ap-961	91	25	completions	completion	NOUN
ap-961	91	26	of	of	ADP
ap-961	91	27	tensor	tensor	NOUN
ap-961	91	28	products	product	NOUN
ap-961	91	29	with	with	ADP
ap-961	91	30	them	they	PRON
ap-961	91	31	.	.	PUNCT
ap-961	92	1	the	the	DET
ap-961	92	2	list	list	NOUN
ap-961	92	3	includes	include	VERB
ap-961	92	4	all	all	DET
ap-961	92	5	commutative	commutative	ADJ
ap-961	92	6	algebras	algebra	NOUN
ap-961	92	7	,	,	PUNCT
ap-961	92	8	matrix	matrix	NOUN
ap-961	92	9	algebras	algebra	NOUN
ap-961	92	10	and	and	CCONJ
ap-961	92	11	–	–	PUNCT
ap-961	92	12	last	last	ADJ
ap-961	92	13	but	but	CCONJ
ap-961	92	14	not	not	PART
ap-961	92	15	least	least	ADJ
ap-961	92	16	–	–	PUNCT
ap-961	92	17	the	the	DET
ap-961	92	18	algebra	algebra	NOUN
ap-961	92	19	of	of	ADP
ap-961	92	20	compact	compact	ADJ
ap-961	92	21	operators	operator	NOUN
ap-961	92	22	(	(	PUNCT
ap-961	92	23	which	which	PRON
ap-961	92	24	we	we	PRON
ap-961	92	25	shall	shall	AUX
ap-961	92	26	define	define	VERB
ap-961	92	27	later	later	ADV
ap-961	92	28	)	)	PUNCT
ap-961	92	29	.	.	PUNCT
ap-961	93	1	so	so	ADV
ap-961	93	2	in	in	ADP
ap-961	93	3	the	the	DET
ap-961	93	4	end	end	NOUN
ap-961	93	5	,	,	PUNCT
ap-961	93	6	there	there	PRON
ap-961	93	7	is	be	VERB
ap-961	93	8	no	no	DET
ap-961	93	9	ambiguity	ambiguity	NOUN
ap-961	93	10	in	in	ADP
ap-961	93	11	defining	define	VERB
ap-961	93	12	continuous	continuous	ADJ
ap-961	93	13	functions	function	NOUN
ap-961	93	14	on	on	ADP
ap-961	93	15	the	the	DET
ap-961	93	16	square	square	NOUN
ap-961	93	17	,	,	PUNCT
ap-961	93	18	but	but	CCONJ
ap-961	93	19	it	it	PRON
ap-961	93	20	might	might	AUX
ap-961	93	21	be	be	AUX
ap-961	93	22	a	a	DET
ap-961	93	23	different	different	ADJ
ap-961	93	24	story	story	NOUN
ap-961	93	25	for	for	ADP
ap-961	93	26	a	a	DET
ap-961	93	27	noncommutative	noncommutative	ADJ
ap-961	93	28	square	square	NOUN
ap-961	93	29	!	!	PUNCT
ap-961	94	1	nevertheless	nevertheless	ADV
ap-961	94	2	,	,	PUNCT
ap-961	94	3	we	we	PRON
ap-961	94	4	shall	shall	AUX
ap-961	94	5	always	always	ADV
ap-961	94	6	work	work	VERB
ap-961	94	7	with	with	ADP
ap-961	94	8	the	the	DET
ap-961	94	9	algebraic	algebraic	ADJ
ap-961	94	10	tensor	tensor	NOUN
ap-961	94	11	product	product	NOUN
ap-961	94	12	,	,	PUNCT
ap-961	94	13	keeping	keep	VERB
ap-961	94	14	in	in	ADP
ap-961	94	15	mind	mind	NOUN
ap-961	94	16	that	that	SCONJ
ap-961	94	17	some	some	DET
ap-961	94	18	further	further	ADJ
ap-961	94	19	details	detail	NOUN
ap-961	94	20	and	and	CCONJ
ap-961	94	21	more	more	ADJ
ap-961	94	22	knowledge	knowledge	NOUN
ap-961	94	23	are	be	AUX
ap-961	94	24	needed	need	VERB
ap-961	94	25	when	when	SCONJ
ap-961	94	26	we	we	PRON
ap-961	94	27	try	try	VERB
ap-961	94	28	to	to	PART
ap-961	94	29	think	think	VERB
ap-961	94	30	of	of	ADP
ap-961	94	31	c	c	NOUN
ap-961	94	32	*	*	X
ap-961	94	33	algebras	algebras	PROPN
ap-961	94	34	.	.	PUNCT
ap-961	95	1	we	we	PRON
ap-961	95	2	shall	shall	AUX
ap-961	95	3	very	very	ADV
ap-961	95	4	often	often	ADV
ap-961	95	5	restrict	restrict	VERB
ap-961	95	6	ourselves	ourselves	PRON
ap-961	95	7	to	to	ADP
ap-961	95	8	the	the	DET
ap-961	95	9	c	c	PROPN
ap-961	95	10	*	*	PUNCT
ap-961	95	11	algebras	algebras	PROPN
ap-961	95	12	generated	generate	VERB
ap-961	95	13	by	by	ADP
ap-961	95	14	some	some	DET
ap-961	95	15	concrete	concrete	ADJ
ap-961	95	16	operators	operator	NOUN
ap-961	95	17	,	,	PUNCT
ap-961	95	18	which	which	PRON
ap-961	95	19	are	be	AUX
ap-961	95	20	defined	define	VERB
ap-961	95	21	by	by	ADP
ap-961	95	22	their	their	PRON
ap-961	95	23	actions	action	NOUN
ap-961	95	24	on	on	ADP
ap-961	95	25	the	the	DET
ap-961	95	26	orthonormal	orthonormal	ADJ
ap-961	95	27	basis	basis	NOUN
ap-961	95	28	.	.	PUNCT
ap-961	96	1	so	so	ADV
ap-961	96	2	,	,	PUNCT
ap-961	96	3	if	if	SCONJ
ap-961	96	4	t	t	PROPN
ap-961	96	5	is	be	AUX
ap-961	96	6	a	a	DET
ap-961	96	7	bounded	bounded	ADJ
ap-961	96	8	operator	operator	NOUN
ap-961	96	9	on	on	ADP
ap-961	96	10	a	a	DET
ap-961	96	11	hilbert	hilbert	NOUN
ap-961	96	12	space	space	NOUN
ap-961	96	13	�	�	PROPN
ap-961	96	14	,	,	PUNCT
ap-961	96	15	then	then	ADV
ap-961	96	16	we	we	PRON
ap-961	96	17	shall	shall	AUX
ap-961	96	18	denote	denote	VERB
ap-961	96	19	by	by	ADP
ap-961	96	20	c	c	PROPN
ap-961	96	21	t	t	PROPN
ap-961	96	22	*	*	PUNCT
ap-961	96	23	(	(	PUNCT
ap-961	96	24	)	)	PUNCT
ap-961	96	25	the	the	DET
ap-961	96	26	c	c	PROPN
ap-961	96	27	*	*	PUNCT
ap-961	96	28	algebra	algebra	PROPN
ap-961	96	29	,	,	PUNCT
ap-961	96	30	which	which	PRON
ap-961	96	31	is	be	AUX
ap-961	96	32	the	the	DET
ap-961	96	33	norm	norm	NOUN
ap-961	96	34	closure	closure	NOUN
ap-961	96	35	of	of	ADP
ap-961	96	36	the	the	DET
ap-961	96	37	algebra	algebra	NOUN
ap-961	96	38	of	of	ADP
ap-961	96	39	all	all	DET
ap-961	96	40	polynomials	polynomial	NOUN
ap-961	96	41	in	in	ADP
ap-961	96	42	t	t	PROPN
ap-961	96	43	,	,	PUNCT
ap-961	96	44	t	t	PROPN
ap-961	96	45	*	*	PUNCT
ap-961	96	46	.	.	PUNCT
ap-961	97	1	let	let	VERB
ap-961	97	2	us	we	PRON
ap-961	97	3	consider	consider	VERB
ap-961	97	4	a	a	DET
ap-961	97	5	couple	couple	NOUN
ap-961	97	6	of	of	ADP
ap-961	97	7	examples	example	NOUN
ap-961	97	8	,	,	PUNCT
ap-961	97	9	the	the	DET
ap-961	97	10	first	first	ADJ
ap-961	97	11	of	of	ADP
ap-961	97	12	which	which	PRON
ap-961	97	13	defines	define	VERB
ap-961	97	14	compact	compact	ADJ
ap-961	97	15	operators	operator	NOUN
ap-961	97	16	:	:	PUNCT
ap-961	97	17	example	example	NOUN
ap-961	97	18	2.5	2.5	NUM
ap-961	97	19	:	:	PUNCT
ap-961	97	20	take	take	VERB
ap-961	97	21	pn	pn	NOUN
ap-961	97	22	to	to	PART
ap-961	97	23	be	be	AUX
ap-961	97	24	a	a	DET
ap-961	97	25	one	one	NUM
ap-961	97	26	-	-	PUNCT
ap-961	97	27	dimensional	dimensional	ADJ
ap-961	97	28	projection	projection	NOUN
ap-961	97	29	on	on	ADP
ap-961	97	30	a	a	DET
ap-961	97	31	hilbert	hilbert	NOUN
ap-961	97	32	space	space	NOUN
ap-961	97	33	with	with	ADP
ap-961	97	34	basis	basis	NOUN
ap-961	97	35	{	{	PUNCT
ap-961	97	36	}	}	PUNCT
ap-961	97	37	ek	ek	PROPN
ap-961	97	38	k	k	PROPN
ap-961	97	39	0	0	NUM
ap-961	97	40	:	:	PUNCT
ap-961	97	41	p	p	X
ap-961	97	42	e	e	X
ap-961	97	43	en	en	X
ap-961	97	44	k	k	PROPN
ap-961	97	45	nk	nk	PROPN
ap-961	97	46	k	k	PROPN
ap-961	97	47	�	�	PROPN
ap-961	97	48	�	�	PROPN
ap-961	97	49	,	,	PUNCT
ap-961	97	50	n	n	CCONJ
ap-961	97	51	k	k	NOUN
ap-961	97	52	,	,	PUNCT
ap-961	97	53	0	0	X
ap-961	97	54	.	.	PUNCT
ap-961	98	1	then	then	ADV
ap-961	98	2	the	the	DET
ap-961	98	3	smallest	small	ADJ
ap-961	98	4	c	c	X
ap-961	98	5	*	*	PUNCT
ap-961	98	6	algebra	algebra	NOUN
ap-961	98	7	that	that	PRON
ap-961	98	8	contains	contain	VERB
ap-961	98	9	all	all	DET
ap-961	98	10	projections	projection	NOUN
ap-961	98	11	pn	pn	PROPN
ap-961	98	12	,	,	PUNCT
ap-961	98	13	c	c	PROPN
ap-961	98	14	pn	pn	PROPN
ap-961	98	15	*	*	PUNCT
ap-961	98	16	(	(	PUNCT
ap-961	98	17	{	{	PUNCT
ap-961	98	18	}	}	PUNCT
ap-961	98	19	)	)	PUNCT
ap-961	99	1	is	be	AUX
ap-961	99	2	the	the	DET
ap-961	99	3	algebra	algebra	NOUN
ap-961	99	4	of	of	ADP
ap-961	99	5	compact	compact	ADJ
ap-961	99	6	operators	operator	NOUN
ap-961	99	7	�	�	PROPN
ap-961	99	8	.	.	PUNCT
ap-961	99	9	example	example	NOUN
ap-961	99	10	2.6	2.6	NUM
ap-961	99	11	:	:	PUNCT
ap-961	99	12	let	let	VERB
ap-961	99	13	u	u	PRON
ap-961	99	14	be	be	AUX
ap-961	99	15	a	a	DET
ap-961	99	16	unilateral	unilateral	ADJ
ap-961	99	17	shift	shift	NOUN
ap-961	99	18	on	on	ADP
ap-961	99	19	a	a	DET
ap-961	99	20	hilbert	hilbert	NOUN
ap-961	99	21	space	space	NOUN
ap-961	99	22	with	with	ADP
ap-961	99	23	basis	basis	NOUN
ap-961	99	24	{	{	PUNCT
ap-961	99	25	}	}	PUNCT
ap-961	99	26	ek	ek	PROPN
ap-961	99	27	k	k	PROPN
ap-961	99	28	�	�	PROPN
ap-961	99	29	�	�	PROPN
ap-961	99	30	:	:	PUNCT
ap-961	99	31	u	u	PROPN
ap-961	99	32	e	e	X
ap-961	99	33	en	en	X
ap-961	99	34	n	n	X
ap-961	99	35	�	�	PROPN
ap-961	99	36	1	1	NUM
ap-961	99	37	,	,	PUNCT
ap-961	99	38	n	n	PRON
ap-961	99	39	�	�	PROPN
ap-961	99	40	�	�	PROPN
ap-961	99	41	.	.	PUNCT
ap-961	100	1	then	then	ADV
ap-961	100	2	the	the	DET
ap-961	100	3	algebra	algebra	NOUN
ap-961	100	4	c	c	VERB
ap-961	100	5	u	u	NOUN
ap-961	100	6	*	*	PUNCT
ap-961	100	7	(	(	PUNCT
ap-961	100	8	)	)	PUNCT
ap-961	100	9	is	be	AUX
ap-961	100	10	isomorphic	isomorphic	ADJ
ap-961	100	11	to	to	ADP
ap-961	100	12	the	the	DET
ap-961	100	13	algebra	algebra	NOUN
ap-961	100	14	of	of	ADP
ap-961	100	15	continuous	continuous	ADJ
ap-961	100	16	functions	function	NOUN
ap-961	100	17	on	on	ADP
ap-961	100	18	the	the	DET
ap-961	100	19	circle	circle	NOUN
ap-961	100	20	,	,	PUNCT
ap-961	100	21	c	c	PROPN
ap-961	100	22	s	s	PROPN
ap-961	100	23	(	(	PUNCT
ap-961	100	24	)	)	PUNCT
ap-961	100	25	1	1	NUM
ap-961	100	26	.	.	PUNCT
ap-961	100	27	remark	remark	VERB
ap-961	100	28	2.7	2.7	NUM
ap-961	100	29	:	:	PUNCT
ap-961	100	30	note	note	VERB
ap-961	100	31	that	that	SCONJ
ap-961	100	32	our	our	PRON
ap-961	100	33	intuition	intuition	NOUN
ap-961	100	34	is	be	AUX
ap-961	100	35	that	that	SCONJ
ap-961	100	36	the	the	DET
ap-961	100	37	operator	operator	NOUN
ap-961	100	38	u	u	NOUN
ap-961	100	39	corresponds	correspond	VERB
ap-961	100	40	to	to	ADP
ap-961	100	41	the	the	DET
ap-961	100	42	function	function	NOUN
ap-961	100	43	�	�	PROPN
ap-961	100	44	�	�	PROPN
ap-961	100	45	�	�	PROPN
ap-961	100	46	�	�	PROPN
ap-961	100	47	e	e	PROPN
ap-961	100	48	i2	i2	PROPN
ap-961	100	49	on	on	ADP
ap-961	100	50	the	the	DET
ap-961	100	51	circle	circle	NOUN
ap-961	100	52	,	,	PUNCT
ap-961	100	53	where	where	SCONJ
ap-961	100	54	�	�	PROPN
ap-961	100	55	is	be	AUX
ap-961	100	56	the	the	DET
ap-961	100	57	angle	angle	NOUN
ap-961	100	58	.	.	PUNCT
ap-961	101	1	however	however	ADV
ap-961	101	2	,	,	PUNCT
ap-961	101	3	this	this	PRON
ap-961	101	4	may	may	AUX
ap-961	101	5	not	not	PART
ap-961	101	6	be	be	AUX
ap-961	101	7	the	the	DET
ap-961	101	8	case	case	NOUN
ap-961	101	9	.	.	PUNCT
ap-961	102	1	take	take	VERB
ap-961	102	2	,	,	PUNCT
ap-961	102	3	for	for	ADP
ap-961	102	4	instance	instance	NOUN
ap-961	102	5	,	,	PUNCT
ap-961	102	6	any	any	DET
ap-961	102	7	monotonic	monotonic	ADJ
ap-961	102	8	,	,	PUNCT
ap-961	102	9	continuous	continuous	ADJ
ap-961	102	10	function	function	NOUN
ap-961	102	11	�	�	PROPN
ap-961	102	12	:	:	PUNCT
ap-961	102	13	[	[	PUNCT
ap-961	102	14	,	,	PUNCT
ap-961	102	15	]	]	X
ap-961	102	16	[	[	PUNCT
ap-961	102	17	,	,	PUNCT
ap-961	102	18	]	]	SYM
ap-961	102	19	0	0	NUM
ap-961	102	20	1	1	NUM
ap-961	102	21	0	0	NUM
ap-961	102	22	1	1	NUM
ap-961	102	23	�	�	PROPN
ap-961	102	24	.	.	PUNCT
ap-961	103	1	we	we	PRON
ap-961	103	2	can	can	AUX
ap-961	103	3	now	now	ADV
ap-961	103	4	replace	replace	VERB
ap-961	103	5	one	one	NUM
ap-961	103	6	of	of	ADP
ap-961	103	7	the	the	DET
ap-961	103	8	elements	element	NOUN
ap-961	103	9	of	of	ADP
ap-961	103	10	the	the	DET
ap-961	103	11	basis	basis	NOUN
ap-961	103	12	of	of	ADP
ap-961	103	13	the	the	DET
ap-961	103	14	hilbert	hilbert	NOUN
ap-961	103	15	space	space	NOUN
ap-961	103	16	by	by	ADP
ap-961	103	17	e	e	PROPN
ap-961	103	18	i2	i2	PROPN
ap-961	103	19	�	�	PROPN
ap-961	103	20	�	�	PROPN
ap-961	103	21	�	�	PROPN
ap-961	103	22	(	(	PUNCT
ap-961	103	23	)	)	PUNCT
ap-961	103	24	and	and	CCONJ
ap-961	103	25	using	use	VERB
ap-961	103	26	the	the	DET
ap-961	103	27	standard	standard	ADJ
ap-961	103	28	procedure	procedure	NOUN
ap-961	103	29	we	we	PRON
ap-961	103	30	can	can	AUX
ap-961	103	31	introduce	introduce	VERB
ap-961	103	32	a	a	DET
ap-961	103	33	new	new	ADJ
ap-961	103	34	orthonormal	orthonormal	ADJ
ap-961	103	35	basis	basis	NOUN
ap-961	103	36	of	of	ADP
ap-961	103	37	the	the	DET
ap-961	103	38	hilbert	hilbert	NOUN
ap-961	103	39	space	space	NOUN
ap-961	103	40	,	,	PUNCT
ap-961	103	41	in	in	ADP
ap-961	103	42	which	which	PRON
ap-961	103	43	one	one	NUM
ap-961	103	44	of	of	ADP
ap-961	103	45	the	the	DET
ap-961	103	46	basis	basis	NOUN
ap-961	103	47	vectors	vector	NOUN
ap-961	103	48	is	be	AUX
ap-961	103	49	proportional	proportional	ADJ
ap-961	103	50	to	to	ADP
ap-961	103	51	the	the	DET
ap-961	103	52	chosen	choose	VERB
ap-961	103	53	one	one	NUM
ap-961	103	54	.	.	PUNCT
ap-961	104	1	certainly	certainly	ADV
ap-961	104	2	it	it	PRON
ap-961	104	3	will	will	AUX
ap-961	104	4	not	not	PART
ap-961	104	5	be	be	AUX
ap-961	104	6	the	the	DET
ap-961	104	7	basis	basis	NOUN
ap-961	104	8	we	we	PRON
ap-961	104	9	have	have	VERB
ap-961	104	10	in	in	ADP
ap-961	104	11	mind	mind	NOUN
ap-961	104	12	and	and	CCONJ
ap-961	104	13	the	the	DET
ap-961	104	14	unilateral	unilateral	ADJ
ap-961	104	15	shift	shift	NOUN
ap-961	104	16	operator	operator	NOUN
ap-961	104	17	u	u	NOUN
ap-961	104	18	can	can	AUX
ap-961	104	19	not	not	PART
ap-961	104	20	then	then	ADV
ap-961	104	21	be	be	AUX
ap-961	104	22	identified	identify	VERB
ap-961	104	23	with	with	ADP
ap-961	104	24	e	e	PROPN
ap-961	104	25	i2	i2	PROPN
ap-961	104	26	�	�	PROPN
ap-961	104	27	�	�	PROPN
ap-961	104	28	.	.	PROPN
ap-961	104	29	example	example	NOUN
ap-961	104	30	2.8	2.8	NUM
ap-961	104	31	:	:	PUNCT
ap-961	104	32	take	take	VERB
ap-961	104	33	t	t	PROPN
ap-961	104	34	to	to	PART
ap-961	104	35	be	be	AUX
ap-961	104	36	a	a	DET
ap-961	104	37	unilateral	unilateral	ADJ
ap-961	104	38	shift	shift	NOUN
ap-961	104	39	on	on	ADP
ap-961	104	40	a	a	DET
ap-961	104	41	hilbert	hilbert	NOUN
ap-961	104	42	space	space	NOUN
ap-961	104	43	with	with	ADP
ap-961	104	44	basis	basis	NOUN
ap-961	104	45	{	{	PUNCT
ap-961	104	46	}	}	PUNCT
ap-961	104	47	ek	ek	PROPN
ap-961	104	48	k	k	PROPN
ap-961	104	49	0	0	PROPN
ap-961	104	50	:	:	PUNCT
ap-961	104	51	t	t	PROPN
ap-961	104	52	e	e	PROPN
ap-961	104	53	en	en	X
ap-961	104	54	n	n	X
ap-961	104	55	�	�	PROPN
ap-961	104	56	1	1	NUM
ap-961	104	57	,	,	PUNCT
ap-961	104	58	n	n	PRON
ap-961	104	59	0	0	NUM
ap-961	104	60	.	.	PUNCT
ap-961	105	1	then	then	ADV
ap-961	105	2	the	the	DET
ap-961	105	3	algebra	algebra	NOUN
ap-961	105	4	c	c	PROPN
ap-961	105	5	t	t	PROPN
ap-961	105	6	*	*	PUNCT
ap-961	105	7	(	(	PUNCT
ap-961	105	8	)	)	PUNCT
ap-961	105	9	is	be	AUX
ap-961	105	10	isomorphic	isomorphic	ADJ
ap-961	105	11	to	to	ADP
ap-961	105	12	the	the	DET
ap-961	105	13	algebra	algebra	NOUN
ap-961	105	14	c	c	PROPN
ap-961	105	15	s	s	X
ap-961	105	16	(	(	PUNCT
ap-961	105	17	)	)	SYM
ap-961	105	18	1	1	NUM
ap-961	105	19	�	�	PROPN
ap-961	105	20	,	,	PUNCT
ap-961	105	21	in	in	ADP
ap-961	105	22	the	the	DET
ap-961	105	23	following	follow	VERB
ap-961	105	24	sense	sense	NOUN
ap-961	105	25	:	:	PUNCT
ap-961	105	26	there	there	PRON
ap-961	105	27	exist	exist	VERB
ap-961	105	28	c*-algebra	c*-algebra	PROPN
ap-961	105	29	morphisms	morphism	VERB
ap-961	106	1	i	i	PRON
ap-961	106	2	,	,	PUNCT
ap-961	106	3	�	�	PROPN
ap-961	106	4	such	such	ADJ
ap-961	106	5	that	that	SCONJ
ap-961	106	6	the	the	DET
ap-961	106	7	following	follow	VERB
ap-961	106	8	sequence	sequence	NOUN
ap-961	106	9	is	be	AUX
ap-961	106	10	exact	exact	ADJ
ap-961	106	11	:	:	PUNCT
ap-961	106	12	0	0	NUM
ap-961	106	13	01	01	NUM
ap-961	106	14	�	�	PROPN
ap-961	106	15	�	�	PROPN
ap-961	106	16	�	�	PROPN
ap-961	106	17	�	�	PROPN
ap-961	106	18	�	�	PROPN
ap-961	106	19	�	�	PROPN
ap-961	106	20	�	�	PROPN
ap-961	106	21	�	�	PROPN
ap-961	106	22	k	k	PROPN
ap-961	106	23	c	c	PROPN
ap-961	106	24	t	t	PROPN
ap-961	106	25	c	c	X
ap-961	106	26	si	si	X
ap-961	106	27	*	*	PUNCT
ap-961	106	28	(	(	PUNCT
ap-961	106	29	)	)	PUNCT
ap-961	106	30	(	(	PUNCT
ap-961	106	31	)	)	PUNCT
ap-961	106	32	�	�	PROPN
ap-961	106	33	.	.	PUNCT
ap-961	107	1	example	example	NOUN
ap-961	107	2	2.9	2.9	NUM
ap-961	107	3	:	:	PUNCT
ap-961	107	4	take	take	VERB
ap-961	107	5	u	u	NOUN
ap-961	107	6	,	,	PUNCT
ap-961	107	7	v	v	NOUN
ap-961	107	8	to	to	PART
ap-961	107	9	be	be	AUX
ap-961	107	10	the	the	DET
ap-961	107	11	following	follow	VERB
ap-961	107	12	unitary	unitary	ADJ
ap-961	107	13	operators	operator	NOUN
ap-961	107	14	on	on	ADP
ap-961	107	15	the	the	DET
ap-961	107	16	hilbert	hilbert	NOUN
ap-961	107	17	space	space	NOUN
ap-961	107	18	with	with	ADP
ap-961	107	19	basis	basis	NOUN
ap-961	107	20	{	{	PUNCT
ap-961	107	21	}	}	PUNCT
ap-961	107	22	,	,	PUNCT
ap-961	107	23	,	,	PUNCT
ap-961	107	24	em	em	PRON
ap-961	107	25	n	n	PROPN
ap-961	107	26	m	m	PROPN
ap-961	107	27	n	n	PRON
ap-961	107	28	�	�	PROPN
ap-961	107	29	�	�	PROPN
ap-961	107	30	:	:	PUNCT
ap-961	107	31	u	u	NOUN
ap-961	107	32	e	e	VERB
ap-961	107	33	em	em	PRON
ap-961	107	34	n	n	PROPN
ap-961	107	35	m	m	VERB
ap-961	107	36	n	n	CCONJ
ap-961	107	37	,	,	PUNCT
ap-961	107	38	,	,	PUNCT
ap-961	107	39	�	�	PROPN
ap-961	107	40	1	1	NUM
ap-961	107	41	,	,	PUNCT
ap-961	107	42	v	v	NOUN
ap-961	107	43	e	e	X
ap-961	107	44	e	e	NOUN
ap-961	107	45	em	em	PRON
ap-961	108	1	n	n	ADV
ap-961	109	1	i	i	PRON
ap-961	109	2	m	m	VERB
ap-961	109	3	m	m	VERB
ap-961	109	4	n	n	PRON
ap-961	109	5	,	,	PUNCT
ap-961	109	6	,	,	PUNCT
ap-961	109	7	�	�	PROPN
ap-961	109	8	�	�	PROPN
ap-961	109	9	2	2	NUM
ap-961	109	10	1	1	NUM
ap-961	109	11	�	�	PROPN
ap-961	109	12	�	�	PROPN
ap-961	109	13	,	,	PUNCT
ap-961	109	14	m	m	PROPN
ap-961	109	15	n	n	CCONJ
ap-961	109	16	,	,	PUNCT
ap-961	109	17	�	�	PROPN
ap-961	109	18	�	�	PROPN
ap-961	109	19	,	,	PUNCT
ap-961	109	20	where	where	SCONJ
ap-961	109	21	�	�	PROPN
ap-961	109	22	�	�	PROPN
ap-961	109	23	�	�	PROPN
ap-961	109	24	.	.	PUNCT
ap-961	110	1	if	if	SCONJ
ap-961	110	2	�	�	PROPN
ap-961	110	3	�	�	PROPN
ap-961	110	4	0	0	NUM
ap-961	110	5	we	we	PRON
ap-961	110	6	can	can	AUX
ap-961	110	7	identify	identify	VERB
ap-961	110	8	the	the	DET
ap-961	110	9	algebra	algebra	NOUN
ap-961	110	10	with	with	ADP
ap-961	110	11	the	the	DET
ap-961	110	12	continuous	continuous	ADJ
ap-961	110	13	functions	function	NOUN
ap-961	110	14	on	on	ADP
ap-961	110	15	the	the	DET
ap-961	110	16	torus	torus	NOUN
ap-961	110	17	.	.	PUNCT
ap-961	111	1	if	if	SCONJ
ap-961	111	2	�	�	PROPN
ap-961	111	3	is	be	AUX
ap-961	111	4	irrational	irrational	ADJ
ap-961	111	5	then	then	ADV
ap-961	111	6	the	the	DET
ap-961	111	7	algebrac	algebrac	PROPN
ap-961	111	8	u	u	PROPN
ap-961	111	9	v	v	PROPN
ap-961	111	10	*	*	PUNCT
ap-961	111	11	(	(	PUNCT
ap-961	111	12	,	,	PUNCT
ap-961	111	13	)	)	PUNCT
ap-961	111	14	is	be	AUX
ap-961	111	15	the	the	DET
ap-961	111	16	so	so	ADV
ap-961	111	17	-	-	PUNCT
ap-961	111	18	called	call	VERB
ap-961	111	19	irrational	irrational	ADJ
ap-961	111	20	rotation	rotation	NOUN
ap-961	111	21	algebra	algebra	NOUN
ap-961	111	22	,	,	PUNCT
ap-961	111	23	aka	aka	ADV
ap-961	111	24	functions	function	NOUN
ap-961	111	25	on	on	ADP
ap-961	111	26	the	the	DET
ap-961	111	27	noncommutative	noncommutative	ADJ
ap-961	111	28	torus	torus	NOUN
ap-961	111	29	.	.	PUNCT
ap-961	112	1	it	it	PRON
ap-961	112	2	is	be	AUX
ap-961	112	3	easy	easy	ADJ
ap-961	112	4	to	to	PART
ap-961	112	5	see	see	VERB
ap-961	112	6	that	that	DET
ap-961	112	7	u	u	PROPN
ap-961	112	8	and	and	CCONJ
ap-961	112	9	v	v	NOUN
ap-961	112	10	satisfy	satisfy	NOUN
ap-961	112	11	:	:	PUNCT
ap-961	112	12	u	u	NOUN
ap-961	112	13	v	v	ADP
ap-961	112	14	e	e	PROPN
ap-961	112	15	vui	vui	NOUN
ap-961	112	16	�	�	PROPN
ap-961	112	17	2	2	NUM
ap-961	112	18	�	�	PROPN
ap-961	112	19	�	�	PROPN
ap-961	112	20	clearly	clearly	ADV
ap-961	112	21	,	,	PUNCT
ap-961	112	22	the	the	DET
ap-961	112	23	above	above	ADJ
ap-961	112	24	presentation	presentation	NOUN
ap-961	112	25	of	of	ADP
ap-961	112	26	the	the	DET
ap-961	112	27	noncommutative	noncommutative	ADJ
ap-961	112	28	torus	torus	NOUN
ap-961	112	29	is	be	AUX
ap-961	112	30	not	not	PART
ap-961	112	31	unique	unique	ADJ
ap-961	112	32	.	.	PUNCT
ap-961	113	1	we	we	PRON
ap-961	113	2	can	can	AUX
ap-961	113	3	take	take	VERB
ap-961	113	4	as	as	SCONJ
ap-961	113	5	the	the	DET
ap-961	113	6	hilbert	hilbert	PROPN
ap-961	113	7	space	space	NOUN
ap-961	113	8	l2(s1	l2(s1	PROPN
ap-961	113	9	)	)	PUNCT
ap-961	113	10	and	and	CCONJ
ap-961	113	11	as	as	ADP
ap-961	113	12	operators	operator	NOUN
ap-961	113	13	u	u	NOUN
ap-961	113	14	and	and	CCONJ
ap-961	113	15	v	v	ADP
ap-961	113	16	:	:	PUNCT
ap-961	113	17	u	u	X
ap-961	113	18	f	f	PROPN
ap-961	113	19	z	z	PROPN
ap-961	113	20	z	z	PROPN
ap-961	114	1	f	f	PROPN
ap-961	115	1	z	z	X
ap-961	115	2	(	(	PUNCT
ap-961	115	3	)	)	PUNCT
ap-961	115	4	(	(	PUNCT
ap-961	115	5	)	)	PUNCT
ap-961	115	6	�	�	PROPN
ap-961	115	7	,	,	PUNCT
ap-961	115	8	v	v	NOUN
ap-961	115	9	f	f	PROPN
ap-961	115	10	z	z	NOUN
ap-961	115	11	f	f	PROPN
ap-961	115	12	e	e	X
ap-961	115	13	zi	zi	PROPN
ap-961	115	14	(	(	PUNCT
ap-961	115	15	)	)	PUNCT
ap-961	115	16	(	(	PUNCT
ap-961	115	17	)	)	PUNCT
ap-961	115	18	�	�	PROPN
ap-961	115	19	2	2	NUM
ap-961	115	20	�	�	PROPN
ap-961	115	21	�	�	PROPN
ap-961	115	22	.	.	PUNCT
ap-961	116	1	both	both	DET
ap-961	116	2	operators	operator	NOUN
ap-961	116	3	are	be	AUX
ap-961	116	4	unitary	unitary	ADJ
ap-961	116	5	and	and	CCONJ
ap-961	116	6	both	both	PRON
ap-961	116	7	satisfy	satisfy	VERB
ap-961	116	8	the	the	DET
ap-961	116	9	same	same	ADJ
ap-961	116	10	commutation	commutation	NOUN
ap-961	116	11	relation	relation	NOUN
ap-961	116	12	–	–	PUNCT
ap-961	116	13	is	be	AUX
ap-961	116	14	the	the	DET
ap-961	116	15	c*-algebra	c*-algebra	PROPN
ap-961	116	16	then	then	ADV
ap-961	116	17	the	the	DET
ap-961	116	18	same	same	ADJ
ap-961	116	19	?	?	PUNCT
ap-961	117	1	it	it	PRON
ap-961	117	2	is	be	AUX
ap-961	117	3	reassuring	reassure	VERB
ap-961	117	4	that	that	SCONJ
ap-961	117	5	the	the	DET
ap-961	117	6	answer	answer	NOUN
ap-961	117	7	is	be	AUX
ap-961	117	8	positive	positive	ADJ
ap-961	117	9	(	(	PUNCT
ap-961	117	10	though	though	SCONJ
ap-961	117	11	it	it	PRON
ap-961	117	12	takes	take	VERB
ap-961	117	13	some	some	DET
ap-961	117	14	time	time	NOUN
ap-961	117	15	to	to	PART
ap-961	117	16	prove	prove	VERB
ap-961	117	17	it	it	PRON
ap-961	117	18	)	)	PUNCT
ap-961	117	19	.	.	PUNCT
ap-961	118	1	3	3	NUM
ap-961	118	2	differential	differential	NOUN
ap-961	118	3	geometry	geometry	NOUN
ap-961	118	4	(	(	PUNCT
ap-961	118	5	noncommutative	noncommutative	ADJ
ap-961	118	6	way	way	NOUN
ap-961	118	7	)	)	PUNCT
ap-961	119	1	alice	alice	PROPN
ap-961	119	2	laughed	laugh	VERB
ap-961	119	3	:	:	PUNCT
ap-961	119	4	“	"	PUNCT
ap-961	119	5	there	there	PRON
ap-961	119	6	’s	’	VERB
ap-961	119	7	no	no	DET
ap-961	119	8	use	use	NOUN
ap-961	119	9	trying	try	VERB
ap-961	119	10	,	,	PUNCT
ap-961	119	11	”	"	PUNCT
ap-961	119	12	she	she	PRON
ap-961	119	13	said	say	VERB
ap-961	119	14	;	;	PUNCT
ap-961	119	15	“	"	PUNCT
ap-961	119	16	one	one	NUM
ap-961	119	17	ca	can	AUX
ap-961	119	18	n’t	not	PART
ap-961	119	19	believe	believe	VERB
ap-961	119	20	impossible	impossible	ADJ
ap-961	119	21	things	thing	NOUN
ap-961	119	22	.	.	PUNCT
ap-961	119	23	”	"	PUNCT
ap-961	120	1	–	–	PUNCT
ap-961	120	2	“	"	PUNCT
ap-961	120	3	i	i	PRON
ap-961	120	4	daresay	daresay	VERB
ap-961	120	5	you	you	PRON
ap-961	120	6	have	have	AUX
ap-961	120	7	n’t	not	PART
ap-961	120	8	had	have	VERB
ap-961	120	9	much	much	ADJ
ap-961	120	10	practice	practice	NOUN
ap-961	120	11	,	,	PUNCT
ap-961	120	12	”	"	PUNCT
ap-961	120	13	said	say	VERB
ap-961	120	14	the	the	DET
ap-961	120	15	queen	queen	NOUN
ap-961	120	16	.	.	PUNCT
ap-961	121	1	“	"	PUNCT
ap-961	121	2	when	when	SCONJ
ap-961	121	3	i	i	PRON
ap-961	121	4	was	be	AUX
ap-961	121	5	younger	young	ADJ
ap-961	121	6	,	,	PUNCT
ap-961	121	7	i	i	PRON
ap-961	121	8	always	always	ADV
ap-961	121	9	did	do	VERB
ap-961	121	10	it	it	PRON
ap-961	121	11	for	for	ADP
ap-961	121	12	half	half	DET
ap-961	121	13	an	an	DET
ap-961	121	14	hour	hour	NOUN
ap-961	121	15	a	a	DET
ap-961	121	16	day	day	NOUN
ap-961	121	17	.	.	PUNCT
ap-961	122	1	why	why	SCONJ
ap-961	122	2	,	,	PUNCT
ap-961	122	3	sometimes	sometimes	ADV
ap-961	122	4	i	i	PRON
ap-961	122	5	’ve	’ve	AUX
ap-961	122	6	believed	believe	VERB
ap-961	122	7	as	as	ADV
ap-961	122	8	many	many	ADJ
ap-961	122	9	as	as	ADP
ap-961	122	10	six	six	NUM
ap-961	122	11	impossible	impossible	ADJ
ap-961	122	12	things	thing	NOUN
ap-961	122	13	before	before	ADP
ap-961	122	14	breakfast	breakfast	NOUN
ap-961	122	15	.	.	PUNCT
ap-961	122	16	”	"	PUNCT
ap-961	123	1	(	(	PUNCT
ap-961	123	2	lewis	lewis	PROPN
ap-961	123	3	carroll	carroll	PROPN
ap-961	123	4	,	,	PUNCT
ap-961	123	5	“	"	PUNCT
ap-961	123	6	alice	alice	PROPN
ap-961	123	7	in	in	ADP
ap-961	123	8	wonderland	wonderland	PROPN
ap-961	123	9	”	"	PUNCT
ap-961	123	10	.	.	PUNCT
ap-961	123	11	)	)	PUNCT
ap-961	123	12	having	having	AUX
ap-961	123	13	started	start	VERB
ap-961	123	14	with	with	ADP
ap-961	123	15	topology	topology	NOUN
ap-961	123	16	we	we	PRON
ap-961	123	17	have	have	AUX
ap-961	123	18	established	establish	VERB
ap-961	123	19	a	a	DET
ap-961	123	20	nice	nice	ADJ
ap-961	123	21	setup	setup	NOUN
ap-961	123	22	for	for	ADP
ap-961	123	23	the	the	DET
ap-961	123	24	discussion	discussion	NOUN
ap-961	123	25	of	of	ADP
ap-961	123	26	noncommutative	noncommutative	ADJ
ap-961	123	27	spaces	space	NOUN
ap-961	123	28	.	.	PUNCT
ap-961	124	1	however	however	ADV
ap-961	124	2	,	,	PUNCT
ap-961	124	3	we	we	PRON
ap-961	124	4	are	be	AUX
ap-961	124	5	still	still	ADV
ap-961	124	6	very	very	ADV
ap-961	124	7	far	far	ADV
ap-961	124	8	from	from	ADP
ap-961	124	9	geometry	geometry	NOUN
ap-961	124	10	as	as	SCONJ
ap-961	124	11	topology	topology	NOUN
ap-961	124	12	does	do	AUX
ap-961	124	13	not	not	PART
ap-961	124	14	distinguish	distinguish	VERB
ap-961	124	15	between	between	ADP
ap-961	124	16	a	a	DET
ap-961	124	17	ball	ball	NOUN
ap-961	124	18	and	and	CCONJ
ap-961	124	19	a	a	DET
ap-961	124	20	cube	cube	NOUN
ap-961	124	21	!	!	PUNCT
ap-961	125	1	our	our	PRON
ap-961	125	2	task	task	NOUN
ap-961	125	3	in	in	ADP
ap-961	125	4	this	this	DET
ap-961	125	5	section	section	NOUN
ap-961	125	6	is	be	AUX
ap-961	125	7	to	to	PART
ap-961	125	8	carry	carry	VERB
ap-961	125	9	out	out	ADP
ap-961	125	10	the	the	DET
ap-961	125	11	parallels	parallel	NOUN
ap-961	125	12	built	build	VERB
ap-961	125	13	up	up	ADP
ap-961	125	14	for	for	ADP
ap-961	125	15	c*-algebras	c*-algebra	NOUN
ap-961	125	16	as	as	ADP
ap-961	125	17	noncommutative	noncommutative	ADJ
ap-961	125	18	spaces	space	NOUN
ap-961	125	19	for	for	ADP
ap-961	125	20	some	some	DET
ap-961	125	21	more	more	ADJ
ap-961	125	22	geometric	geometric	ADJ
ap-961	125	23	notions	notion	NOUN
ap-961	125	24	.	.	PUNCT
ap-961	126	1	we	we	PRON
ap-961	126	2	shall	shall	AUX
ap-961	126	3	begin	begin	VERB
ap-961	126	4	with	with	ADP
ap-961	126	5	a	a	DET
ap-961	126	6	pure	pure	ADJ
ap-961	126	7	algebraic	algebraic	ADJ
ap-961	126	8	setup	setup	NOUN
ap-961	126	9	of	of	ADP
ap-961	126	10	differential	differential	ADJ
ap-961	126	11	calculi	calculi	PROPN
ap-961	126	12	.	.	PUNCT
ap-961	127	1	3.1	3.1	NUM
ap-961	127	2	differential	differential	ADJ
ap-961	127	3	calculi	calculi	NOUN
ap-961	127	4	in	in	ADP
ap-961	127	5	the	the	DET
ap-961	127	6	course	course	NOUN
ap-961	127	7	of	of	ADP
ap-961	127	8	differential	differential	ADJ
ap-961	127	9	geometry	geometry	NOUN
ap-961	127	10	one	one	PRON
ap-961	127	11	begins	begin	VERB
ap-961	127	12	with	with	ADP
ap-961	127	13	the	the	DET
ap-961	127	14	notion	notion	NOUN
ap-961	127	15	of	of	ADP
ap-961	127	16	a	a	DET
ap-961	127	17	smooth	smooth	ADJ
ap-961	127	18	manifold	manifold	NOUN
ap-961	127	19	,	,	PUNCT
ap-961	127	20	c	c	NOUN
ap-961	127	21	functions	function	NOUN
ap-961	127	22	and	and	CCONJ
ap-961	127	23	vector	vector	NOUN
ap-961	127	24	fields	field	NOUN
ap-961	127	25	.	.	PUNCT
ap-961	128	1	this	this	PRON
ap-961	128	2	is	be	AUX
ap-961	128	3	,	,	PUNCT
ap-961	128	4	however	however	ADV
ap-961	128	5	,	,	PUNCT
ap-961	128	6	reserved	reserve	VERB
ap-961	128	7	for	for	ADP
ap-961	128	8	a	a	DET
ap-961	128	9	purely	purely	ADV
ap-961	128	10	commutative	commutative	ADJ
ap-961	128	11	world	world	NOUN
ap-961	128	12	as	as	SCONJ
ap-961	128	13	can	can	AUX
ap-961	128	14	be	be	AUX
ap-961	128	15	immediately	immediately	ADV
ap-961	128	16	noticed	notice	VERB
ap-961	128	17	when	when	SCONJ
ap-961	128	18	on	on	ADP
ap-961	128	19	takes	take	VERB
ap-961	128	20	the	the	DET
ap-961	128	21	simplest	simple	ADJ
ap-961	128	22	example	example	NOUN
ap-961	128	23	of	of	ADP
ap-961	128	24	a	a	DET
ap-961	128	25	noncommutative	noncommutative	ADJ
ap-961	128	26	space	space	NOUN
ap-961	128	27	,	,	PUNCT
ap-961	128	28	described	describe	VERB
ap-961	128	29	by	by	ADP
ap-961	128	30	the	the	DET
ap-961	128	31	algebra	algebra	NOUN
ap-961	128	32	of	of	ADP
ap-961	128	33	38	38	NUM
ap-961	128	34	©	©	PROPN
ap-961	128	35	czech	czech	PROPN
ap-961	128	36	technical	technical	PROPN
ap-961	128	37	university	university	PROPN
ap-961	128	38	publishing	publishing	NOUN
ap-961	128	39	house	house	NOUN
ap-961	128	40	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	128	41	acta	acta	PROPN
ap-961	128	42	polytechnica	polytechnica	PROPN
ap-961	128	43	vol	vol	NOUN
ap-961	128	44	.	.	PUNCT
ap-961	129	1	48	48	NUM
ap-961	129	2	no	no	NOUN
ap-961	129	3	.	.	PUNCT
ap-961	130	1	2/2008	2/2008	PROPN
ap-961	130	2	matrices	matrix	NOUN
ap-961	130	3	mn	mn	PROPN
ap-961	130	4	(	(	PUNCT
ap-961	130	5	)	)	PUNCT
ap-961	130	6	�	�	PROPN
ap-961	130	7	,	,	PUNCT
ap-961	130	8	n	n	X
ap-961	130	9	�	�	PROPN
ap-961	130	10	0	0	NUM
ap-961	130	11	.	.	PUNCT
ap-961	130	12	suppose	suppose	VERB
ap-961	130	13	we	we	PRON
ap-961	130	14	want	want	VERB
ap-961	130	15	to	to	PART
ap-961	130	16	have	have	VERB
ap-961	130	17	a	a	DET
ap-961	130	18	vector	vector	NOUN
ap-961	130	19	field	field	NOUN
ap-961	130	20	–	–	PUNCT
ap-961	130	21	what	what	PRON
ap-961	130	22	is	be	AUX
ap-961	130	23	a	a	DET
ap-961	130	24	vector	vector	NOUN
ap-961	130	25	field	field	NOUN
ap-961	130	26	then	then	ADV
ap-961	130	27	?	?	PUNCT
ap-961	131	1	a	a	DET
ap-961	131	2	good	good	ADJ
ap-961	131	3	answer	answer	NOUN
ap-961	131	4	is	be	AUX
ap-961	131	5	that	that	SCONJ
ap-961	131	6	a	a	DET
ap-961	131	7	vector	vector	NOUN
ap-961	131	8	field	field	NOUN
ap-961	131	9	is	be	AUX
ap-961	131	10	a	a	DET
ap-961	131	11	derivation	derivation	NOUN
ap-961	131	12	on	on	ADP
ap-961	131	13	the	the	DET
ap-961	131	14	algebra	algebra	NOUN
ap-961	131	15	of	of	ADP
ap-961	131	16	smooth	smooth	ADJ
ap-961	131	17	functions	function	NOUN
ap-961	131	18	:	:	PUNCT
ap-961	131	19	definition	definition	NOUN
ap-961	131	20	3.1	3.1	NUM
ap-961	131	21	:	:	PUNCT
ap-961	131	22	a	a	DET
ap-961	131	23	derivation	derivation	NOUN
ap-961	131	24	�	�	NOUN
ap-961	131	25	on	on	ADP
ap-961	131	26	an	an	DET
ap-961	131	27	algebra	algebra	NOUN
ap-961	131	28	�	�	PROPN
ap-961	131	29	is	be	AUX
ap-961	131	30	a	a	DET
ap-961	131	31	map	map	NOUN
ap-961	131	32	�	�	NOUN
ap-961	131	33	:	:	PUNCT
ap-961	131	34	�	�	PROPN
ap-961	131	35	�	�	PROPN
ap-961	131	36	�	�	PROPN
ap-961	131	37	satisfying	satisfy	VERB
ap-961	131	38	the	the	DET
ap-961	131	39	leibniz	leibniz	PROPN
ap-961	131	40	rule	rule	NOUN
ap-961	131	41	:	:	PUNCT
ap-961	131	42	�	�	PROPN
ap-961	131	43	�	�	PROPN
ap-961	131	44	�	�	PROPN
ap-961	131	45	(	(	PUNCT
ap-961	131	46	)	)	PUNCT
ap-961	131	47	(	(	PUNCT
ap-961	131	48	)	)	PUNCT
ap-961	131	49	(	(	PUNCT
ap-961	131	50	)	)	PUNCT
ap-961	131	51	ab	ab	PROPN
ap-961	131	52	a	a	DET
ap-961	131	53	b	b	PROPN
ap-961	131	54	a	a	DET
ap-961	131	55	b	b	PROPN
ap-961	131	56	�	�	PROPN
ap-961	131	57	,	,	PUNCT
ap-961	131	58	�	�	PROPN
ap-961	131	59	�	�	PROPN
ap-961	131	60	a	a	DET
ap-961	131	61	b	b	PROPN
ap-961	131	62	,	,	PUNCT
ap-961	131	63	�	�	PROPN
ap-961	131	64	.	.	PUNCT
ap-961	132	1	a	a	DET
ap-961	132	2	derivation	derivation	NOUN
ap-961	132	3	is	be	AUX
ap-961	132	4	inner	inner	ADJ
ap-961	132	5	if	if	SCONJ
ap-961	132	6	there	there	PRON
ap-961	132	7	exist	exist	VERB
ap-961	132	8	an	an	DET
ap-961	132	9	elementx	elementx	VERB
ap-961	132	10	�	�	NOUN
ap-961	132	11	�	�	PROPN
ap-961	132	12	such	such	ADJ
ap-961	132	13	that	that	PRON
ap-961	132	14	for	for	ADP
ap-961	132	15	every	every	DET
ap-961	132	16	a	a	DET
ap-961	132	17	�	�	PROPN
ap-961	132	18	�	�	PROPN
ap-961	132	19	,	,	PUNCT
ap-961	132	20	�	�	PROPN
ap-961	132	21	(	(	PUNCT
ap-961	132	22	)	)	PUNCT
ap-961	132	23	[	[	PUNCT
ap-961	132	24	,	,	PUNCT
ap-961	132	25	]	]	X
ap-961	132	26	a	a	X
ap-961	132	27	x	x	X
ap-961	132	28	a	a	DET
ap-961	132	29	�	�	PROPN
ap-961	132	30	.	.	PUNCT
ap-961	133	1	a	a	DET
ap-961	133	2	derivation	derivation	NOUN
ap-961	133	3	which	which	PRON
ap-961	133	4	is	be	AUX
ap-961	133	5	not	not	PART
ap-961	133	6	inner	inner	ADJ
ap-961	133	7	is	be	AUX
ap-961	133	8	called	call	VERB
ap-961	133	9	an	an	DET
ap-961	133	10	outer	outer	ADJ
ap-961	133	11	derivation	derivation	NOUN
ap-961	133	12	.	.	PUNCT
ap-961	134	1	since	since	SCONJ
ap-961	134	2	for	for	ADP
ap-961	134	3	commutative	commutative	ADJ
ap-961	134	4	algebras	algebra	NOUN
ap-961	134	5	there	there	PRON
ap-961	134	6	are	be	VERB
ap-961	134	7	no	no	DET
ap-961	134	8	inner	inner	ADJ
ap-961	134	9	derivations	derivation	NOUN
ap-961	134	10	,	,	PUNCT
ap-961	134	11	vector	vector	NOUN
ap-961	134	12	fields	field	NOUN
ap-961	134	13	are	be	AUX
ap-961	134	14	(	(	PUNCT
ap-961	134	15	in	in	ADP
ap-961	134	16	classical	classical	ADJ
ap-961	134	17	differential	differential	NOUN
ap-961	134	18	geometry	geometry	NOUN
ap-961	134	19	)	)	PUNCT
ap-961	134	20	outer	outer	ADJ
ap-961	134	21	derivations	derivation	NOUN
ap-961	134	22	.	.	PUNCT
ap-961	135	1	but	but	CCONJ
ap-961	135	2	let	let	VERB
ap-961	135	3	us	we	PRON
ap-961	135	4	look	look	VERB
ap-961	135	5	at	at	ADP
ap-961	135	6	mn	mn	PROPN
ap-961	135	7	(	(	PUNCT
ap-961	135	8	)	)	PUNCT
ap-961	135	9	�	�	PROPN
ap-961	135	10	–	–	PUNCT
ap-961	135	11	where	where	SCONJ
ap-961	135	12	every	every	DET
ap-961	135	13	derivation	derivation	NOUN
ap-961	135	14	is	be	AUX
ap-961	135	15	in	in	ADP
ap-961	135	16	fact	fact	NOUN
ap-961	135	17	inner	inner	ADJ
ap-961	135	18	(	(	PUNCT
ap-961	135	19	which	which	PRON
ap-961	135	20	means	mean	VERB
ap-961	135	21	that	that	SCONJ
ap-961	135	22	there	there	PRON
ap-961	135	23	are	be	VERB
ap-961	135	24	no	no	DET
ap-961	135	25	outer	outer	ADJ
ap-961	135	26	derivations	derivation	NOUN
ap-961	135	27	)	)	PUNCT
ap-961	135	28	.	.	PUNCT
ap-961	136	1	so	so	ADV
ap-961	136	2	,	,	PUNCT
ap-961	136	3	can	can	AUX
ap-961	136	4	we	we	PRON
ap-961	136	5	call	call	VERB
ap-961	136	6	them	they	PRON
ap-961	136	7	vector	vector	NOUN
ap-961	136	8	fields	field	NOUN
ap-961	136	9	?	?	PUNCT
ap-961	137	1	we	we	PRON
ap-961	137	2	can	can	AUX
ap-961	137	3	even	even	ADV
ap-961	137	4	take	take	VERB
ap-961	137	5	a	a	DET
ap-961	137	6	much	much	ADV
ap-961	137	7	simpler	simple	ADJ
ap-961	137	8	example	example	NOUN
ap-961	137	9	:	:	PUNCT
ap-961	137	10	a	a	DET
ap-961	137	11	commutative	commutative	ADJ
ap-961	137	12	algebra	algebra	NOUN
ap-961	137	13	of	of	ADP
ap-961	137	14	complex	complex	ADJ
ap-961	137	15	functions	function	NOUN
ap-961	137	16	on	on	ADP
ap-961	137	17	two	two	NUM
ap-961	137	18	points	point	NOUN
ap-961	137	19	.	.	PUNCT
ap-961	138	1	as	as	ADP
ap-961	138	2	a	a	DET
ap-961	138	3	vector	vector	NOUN
ap-961	138	4	space	space	NOUN
ap-961	138	5	it	it	PRON
ap-961	138	6	has	have	VERB
ap-961	138	7	two	two	NUM
ap-961	138	8	basis	basis	NOUN
ap-961	138	9	vectors	vector	NOUN
ap-961	138	10	:	:	PUNCT
ap-961	138	11	a	a	DET
ap-961	138	12	unit	unit	NOUN
ap-961	138	13	(	(	PUNCT
ap-961	138	14	1	1	NUM
ap-961	138	15	)	)	PUNCT
ap-961	138	16	and	and	CCONJ
ap-961	138	17	the	the	DET
ap-961	138	18	function	function	NOUN
ap-961	138	19	which	which	PRON
ap-961	138	20	takes	take	VERB
ap-961	138	21	value	value	NOUN
ap-961	138	22	1	1	NUM
ap-961	138	23	on	on	ADP
ap-961	138	24	the	the	DET
ap-961	138	25	first	first	ADJ
ap-961	138	26	point	point	NOUN
ap-961	138	27	and	and	CCONJ
ap-961	138	28	�	�	NOUN
ap-961	138	29	1	1	NUM
ap-961	138	30	on	on	ADP
ap-961	138	31	the	the	DET
ap-961	138	32	other	other	ADJ
ap-961	138	33	,	,	PUNCT
ap-961	138	34	which	which	PRON
ap-961	138	35	we	we	PRON
ap-961	138	36	shall	shall	AUX
ap-961	138	37	call	call	VERB
ap-961	138	38	e.	e.	PROPN
ap-961	138	39	each	each	DET
ap-961	138	40	function	function	NOUN
ap-961	138	41	is	be	AUX
ap-961	138	42	a	a	DET
ap-961	138	43	linear	linear	ADJ
ap-961	138	44	combination	combination	NOUN
ap-961	138	45	of	of	ADP
ap-961	138	46	these	these	DET
ap-961	138	47	two	two	NUM
ap-961	138	48	,	,	PUNCT
ap-961	138	49	and	and	CCONJ
ap-961	138	50	the	the	DET
ap-961	138	51	algebra	algebra	NOUN
ap-961	138	52	structure	structure	NOUN
ap-961	138	53	is	be	AUX
ap-961	138	54	encoded	encode	VERB
ap-961	138	55	in	in	ADP
ap-961	138	56	one	one	NUM
ap-961	138	57	simple	simple	ADJ
ap-961	138	58	identity	identity	NOUN
ap-961	138	59	e2	e2	NOUN
ap-961	138	60	1	1	NUM
ap-961	138	61	�	�	PROPN
ap-961	138	62	.	.	PUNCT
ap-961	139	1	now	now	ADV
ap-961	139	2	what	what	PRON
ap-961	139	3	is	be	AUX
ap-961	139	4	the	the	DET
ap-961	139	5	space	space	NOUN
ap-961	139	6	of	of	ADP
ap-961	139	7	the	the	DET
ap-961	139	8	derivations	derivation	NOUN
ap-961	139	9	?	?	PUNCT
ap-961	140	1	clearly	clearly	ADV
ap-961	140	2	,	,	PUNCT
ap-961	140	3	there	there	PRON
ap-961	140	4	are	be	VERB
ap-961	140	5	no	no	DET
ap-961	140	6	inner	inner	ADJ
ap-961	140	7	ones	one	NOUN
ap-961	140	8	,	,	PUNCT
ap-961	140	9	as	as	SCONJ
ap-961	140	10	the	the	DET
ap-961	140	11	algebra	algebra	NOUN
ap-961	140	12	is	be	AUX
ap-961	140	13	commutative	commutative	ADJ
ap-961	140	14	.	.	PUNCT
ap-961	141	1	assume	assume	VERB
ap-961	141	2	that	that	SCONJ
ap-961	141	3	�	�	PROPN
ap-961	141	4	is	be	AUX
ap-961	141	5	a	a	DET
ap-961	141	6	derivation	derivation	NOUN
ap-961	141	7	.	.	PUNCT
ap-961	142	1	then	then	ADV
ap-961	142	2	,	,	PUNCT
ap-961	142	3	using	use	VERB
ap-961	142	4	the	the	DET
ap-961	142	5	leibniz	leibniz	NOUN
ap-961	142	6	rule	rule	NOUN
ap-961	142	7	we	we	PRON
ap-961	142	8	have	have	VERB
ap-961	142	9	:	:	PUNCT
ap-961	142	10	0	0	NUM
ap-961	142	11	1	1	NUM
ap-961	142	12	2	2	NUM
ap-961	142	13	�	�	PROPN
ap-961	142	14	�	�	PROPN
ap-961	142	15	�	�	PROPN
ap-961	142	16	�	�	PROPN
ap-961	142	17	(	(	PUNCT
ap-961	142	18	)	)	PUNCT
ap-961	142	19	(	(	PUNCT
ap-961	142	20	)	)	PUNCT
ap-961	142	21	e	e	X
ap-961	142	22	e	e	NOUN
ap-961	142	23	,	,	PUNCT
ap-961	142	24	which	which	PRON
ap-961	142	25	simply	simply	ADV
ap-961	142	26	tells	tell	VERB
ap-961	142	27	us	we	PRON
ap-961	142	28	that	that	SCONJ
ap-961	142	29	apart	apart	ADV
ap-961	142	30	from	from	ADP
ap-961	142	31	the	the	DET
ap-961	142	32	trivial	trivial	ADJ
ap-961	142	33	derivation	derivation	NOUN
ap-961	142	34	,	,	PUNCT
ap-961	142	35	�	�	PROPN
ap-961	142	36	�	�	PROPN
ap-961	142	37	0	0	NUM
ap-961	142	38	,	,	PUNCT
ap-961	142	39	there	there	PRON
ap-961	142	40	are	be	VERB
ap-961	142	41	no	no	DET
ap-961	142	42	derivations	derivation	NOUN
ap-961	142	43	at	at	ADV
ap-961	142	44	all	all	ADV
ap-961	142	45	.	.	PUNCT
ap-961	143	1	of	of	ADP
ap-961	143	2	course	course	NOUN
ap-961	143	3	,	,	PUNCT
ap-961	143	4	a	a	DET
ap-961	143	5	good	good	ADJ
ap-961	143	6	lesson	lesson	NOUN
ap-961	143	7	to	to	PART
ap-961	143	8	learn	learn	VERB
ap-961	143	9	is	be	AUX
ap-961	143	10	that	that	SCONJ
ap-961	143	11	we	we	PRON
ap-961	143	12	have	have	AUX
ap-961	143	13	chosen	choose	VERB
ap-961	143	14	a	a	DET
ap-961	143	15	bad	bad	ADJ
ap-961	143	16	object	object	NOUN
ap-961	143	17	to	to	PART
ap-961	143	18	start	start	VERB
ap-961	143	19	with	with	ADP
ap-961	143	20	.	.	PUNCT
ap-961	144	1	instead	instead	ADV
ap-961	144	2	of	of	ADP
ap-961	144	3	looking	look	VERB
ap-961	144	4	at	at	ADP
ap-961	144	5	the	the	DET
ap-961	144	6	vector	vector	NOUN
ap-961	144	7	fields	field	NOUN
ap-961	144	8	we	we	PRON
ap-961	144	9	need	need	VERB
ap-961	144	10	to	to	PART
ap-961	144	11	look	look	VERB
ap-961	144	12	at	at	ADP
ap-961	144	13	differential	differential	ADJ
ap-961	144	14	algebras	algebra	NOUN
ap-961	144	15	,	,	PUNCT
ap-961	144	16	which	which	PRON
ap-961	144	17	generalize	generalize	VERB
ap-961	144	18	nicely	nicely	ADV
ap-961	144	19	to	to	ADP
ap-961	144	20	the	the	DET
ap-961	144	21	noncommutative	noncommutative	ADJ
ap-961	144	22	world	world	NOUN
ap-961	144	23	.	.	PUNCT
ap-961	145	1	definition	definition	NOUN
ap-961	145	2	3.2	3.2	NUM
ap-961	145	3	:	:	PUNCT
ap-961	145	4	a	a	DET
ap-961	145	5	differential	differential	NOUN
ap-961	145	6	graded	grade	VERB
ap-961	145	7	algebra	algebra	NOUN
ap-961	145	8	(	(	PUNCT
ap-961	145	9	dga	dga	NOUN
ap-961	145	10	)	)	PUNCT
ap-961	145	11	over	over	ADP
ap-961	145	12	an	an	DET
ap-961	145	13	algebra	algebra	NOUN
ap-961	145	14	�	�	PROPN
ap-961	145	15	is	be	AUX
ap-961	145	16	an	an	DET
ap-961	145	17	�	�	NOUN
ap-961	145	18	-graded	-grade	VERB
ap-961	145	19	algebra	algebra	NOUN
ap-961	145	20	,	,	PUNCT
ap-961	145	21	not	not	PART
ap-961	145	22	necessarily	necessarily	ADV
ap-961	145	23	finite	finite	VERB
ap-961	145	24	,	,	PUNCT
ap-961	145	25	such	such	ADJ
ap-961	145	26	that	that	SCONJ
ap-961	145	27	the	the	DET
ap-961	145	28	0	0	NUM
ap-961	145	29	-	-	PUNCT
ap-961	145	30	th	th	VERB
ap-961	145	31	grade	grade	NOUN
ap-961	145	32	is	be	AUX
ap-961	145	33	isomorphic	isomorphic	ADJ
ap-961	145	34	with	with	ADP
ap-961	145	35	�	�	PROPN
ap-961	145	36	and	and	CCONJ
ap-961	145	37	that	that	PRON
ap-961	145	38	is	be	AUX
ap-961	145	39	equipped	equip	VERB
ap-961	145	40	with	with	ADP
ap-961	145	41	a	a	DET
ap-961	145	42	degree	degree	NOUN
ap-961	145	43	linear	linear	NOUN
ap-961	145	44	map	map	NOUN
ap-961	145	45	(	(	PUNCT
ap-961	145	46	grade	grade	NOUN
ap-961	145	47	increasing	increasing	NOUN
ap-961	145	48	)	)	PUNCT
ap-961	145	49	,	,	PUNCT
ap-961	145	50	which	which	PRON
ap-961	145	51	obeys	obey	VERB
ap-961	145	52	the	the	DET
ap-961	145	53	graded	grade	VERB
ap-961	145	54	leibniz	leibniz	NOUN
ap-961	145	55	rule	rule	NOUN
ap-961	145	56	:	:	PUNCT
ap-961	146	1	d	d	X
ap-961	146	2	d	d	X
ap-961	146	3	d	d	PROPN
ap-961	146	4	(	(	PUNCT
ap-961	146	5	)	)	PUNCT
ap-961	146	6	(	(	PUNCT
ap-961	146	7	)	)	PUNCT
ap-961	146	8	�	�	PROPN
ap-961	146	9	�	�	PROPN
ap-961	146	10	�	�	PROPN
ap-961	146	11	�	�	PROPN
ap-961	146	12	�	�	PROPN
ap-961	146	13	�	�	PROPN
ap-961	146	14	�	�	PROPN
ap-961	146	15	�	�	PROPN
ap-961	146	16	�	�	PROPN
ap-961	146	17	1	1	NUM
ap-961	146	18	,	,	PUNCT
ap-961	146	19	for	for	ADP
ap-961	146	20	any	any	DET
ap-961	146	21	elements	element	NOUN
ap-961	146	22	�	�	PROPN
ap-961	146	23	,	,	PUNCT
ap-961	146	24	�	�	PROPN
ap-961	146	25	,	,	PUNCT
ap-961	146	26	where	where	SCONJ
ap-961	146	27	�	�	PROPN
ap-961	146	28	denotes	denote	VERB
ap-961	146	29	the	the	DET
ap-961	146	30	degree	degree	NOUN
ap-961	146	31	of	of	ADP
ap-961	146	32	the	the	DET
ap-961	146	33	form	form	NOUN
ap-961	146	34	�	�	PROPN
ap-961	146	35	.	.	PUNCT
ap-961	147	1	there	there	PRON
ap-961	147	2	is	be	VERB
ap-961	147	3	,	,	PUNCT
ap-961	147	4	however	however	ADV
ap-961	147	5	,	,	PUNCT
ap-961	147	6	no	no	DET
ap-961	147	7	unique	unique	ADJ
ap-961	147	8	way	way	NOUN
ap-961	147	9	to	to	PART
ap-961	147	10	construct	construct	VERB
ap-961	147	11	such	such	DET
ap-961	147	12	an	an	DET
ap-961	147	13	object	object	NOUN
ap-961	147	14	in	in	ADP
ap-961	147	15	the	the	DET
ap-961	147	16	noncommutative	noncommutative	ADJ
ap-961	147	17	situation	situation	NOUN
ap-961	147	18	and	and	CCONJ
ap-961	147	19	one	one	PRON
ap-961	147	20	might	might	AUX
ap-961	147	21	have	have	VERB
ap-961	147	22	many	many	ADJ
ap-961	147	23	different	different	ADJ
ap-961	147	24	dgas	dgas	NOUN
ap-961	147	25	over	over	ADP
ap-961	147	26	a	a	DET
ap-961	147	27	single	single	ADJ
ap-961	147	28	algebra	algebra	NOUN
ap-961	147	29	–	–	PUNCT
ap-961	147	30	even	even	ADV
ap-961	147	31	in	in	ADP
ap-961	147	32	the	the	DET
ap-961	147	33	commutative	commutative	ADJ
ap-961	147	34	case	case	NOUN
ap-961	147	35	.	.	PUNCT
ap-961	148	1	before	before	SCONJ
ap-961	148	2	we	we	PRON
ap-961	148	3	look	look	VERB
ap-961	148	4	at	at	ADP
ap-961	148	5	some	some	DET
ap-961	148	6	interesting	interesting	ADJ
ap-961	148	7	examples	example	NOUN
ap-961	148	8	,	,	PUNCT
ap-961	148	9	let	let	VERB
ap-961	148	10	us	we	PRON
ap-961	148	11	define	define	VERB
ap-961	148	12	an	an	DET
ap-961	148	13	important	important	ADJ
ap-961	148	14	dga	dga	NOUN
ap-961	148	15	,	,	PUNCT
ap-961	148	16	which	which	PRON
ap-961	148	17	can	can	AUX
ap-961	148	18	be	be	AUX
ap-961	148	19	canonically	canonically	ADV
ap-961	148	20	constructed	construct	VERB
ap-961	148	21	for	for	ADP
ap-961	148	22	every	every	DET
ap-961	148	23	algebra	algebra	NOUN
ap-961	148	24	.	.	PUNCT
ap-961	149	1	proposition	proposition	NOUN
ap-961	149	2	3.3	3.3	NUM
ap-961	149	3	:	:	PUNCT
ap-961	149	4	we	we	PRON
ap-961	149	5	assume	assume	VERB
ap-961	149	6	that	that	SCONJ
ap-961	149	7	the	the	DET
ap-961	149	8	algebra	algebra	PROPN
ap-961	149	9	�	�	PROPN
ap-961	149	10	is	be	AUX
ap-961	149	11	unital	unital	ADJ
ap-961	149	12	.	.	PUNCT
ap-961	150	1	let	let	VERB
ap-961	150	2	u	u	PRON
ap-961	150	3	1	1	NUM
ap-961	150	4	(	(	PUNCT
ap-961	150	5	)	)	PUNCT
ap-961	150	6	�	�	PROPN
ap-961	150	7	be	be	AUX
ap-961	150	8	the	the	DET
ap-961	150	9	kernel	kernel	NOUN
ap-961	150	10	of	of	ADP
ap-961	150	11	the	the	DET
ap-961	150	12	multiplication	multiplication	NOUN
ap-961	150	13	map	map	NOUN
ap-961	150	14	in	in	ADP
ap-961	150	15	�	�	PROPN
ap-961	150	16	�	�	PROPN
ap-961	150	17	�	�	PROPN
ap-961	150	18	:	:	PUNCT
ap-961	150	19	u	u	NOUN
ap-961	151	1	i	i	PRON
ap-961	152	1	i	i	PRON
ap-961	153	1	i	i	VERB
ap-961	154	1	a	a	DET
ap-961	154	2	b	b	NOUN
ap-961	154	3	a	a	DET
ap-961	154	4	b	b	NOUN
ap-961	154	5	a	a	DET
ap-961	154	6	b	b	NOUN
ap-961	155	1	i	i	PRON
ap-961	155	2	i	i	PRON
ap-961	156	1	i	i	PRON
ap-961	156	2	ii	ii	VERB
ap-961	156	3	1	1	NUM
ap-961	156	4	0	0	NUM
ap-961	157	1	(	(	PUNCT
ap-961	157	2	)	)	PUNCT
ap-961	157	3	,	,	PUNCT
ap-961	157	4	;	;	PUNCT
ap-961	157	5	�	�	PROPN
ap-961	157	6	�	�	PROPN
ap-961	157	7	�	�	PROPN
ap-961	157	8	�	�	PROPN
ap-961	157	9	�	�	PROPN
ap-961	157	10	�	�	PROPN
ap-961	157	11	�	�	PROPN
ap-961	157	12	�	�	PROPN
ap-961	157	13	�	�	PROPN
ap-961	157	14	�	�	PROPN
ap-961	157	15	�	�	PROPN
ap-961	157	16	�	�	PROPN
ap-961	157	17	�	�	PROPN
ap-961	157	18	�	�	PROPN
ap-961	157	19	�	�	PROPN
ap-961	157	20	�	�	PROPN
ap-961	157	21	�	�	PROPN
ap-961	157	22	�	�	PROPN
ap-961	157	23	let	let	VERB
ap-961	157	24	us	we	PRON
ap-961	157	25	take	take	VERB
ap-961	157	26	as	as	ADP
ap-961	157	27	u	u	NOUN
ap-961	157	28	(	(	PUNCT
ap-961	157	29	)	)	PUNCT
ap-961	157	30	�	�	PROPN
ap-961	157	31	the	the	DET
ap-961	157	32	tensor	tensor	NOUN
ap-961	157	33	algebra	algebra	NOUN
ap-961	157	34	over	over	ADP
ap-961	157	35	�	�	PROPN
ap-961	157	36	of	of	ADP
ap-961	157	37	u	u	NOUN
ap-961	157	38	1	1	NUM
ap-961	157	39	(	(	PUNCT
ap-961	157	40	)	)	PUNCT
ap-961	157	41	�	�	PROPN
ap-961	157	42	.	.	PUNCT
ap-961	158	1	then	then	ADV
ap-961	158	2	the	the	DET
ap-961	158	3	linear	linear	ADJ
ap-961	158	4	map	map	NOUN
ap-961	158	5	defined	define	VERB
ap-961	158	6	on	on	ADP
ap-961	158	7	u	u	NOUN
ap-961	158	8	0	0	NUM
ap-961	158	9	(	(	PUNCT
ap-961	158	10	)	)	PUNCT
ap-961	158	11	�	�	PROPN
ap-961	158	12	�	�	PROPN
ap-961	158	13	�	�	PROPN
ap-961	158	14	as	as	ADP
ap-961	158	15	:	:	PUNCT
ap-961	158	16	d	d	X
ap-961	158	17	a	a	DET
ap-961	158	18	a	a	DET
ap-961	158	19	au	au	PROPN
ap-961	158	20	(	(	PUNCT
ap-961	158	21	)	)	PUNCT
ap-961	158	22	�	�	PROPN
ap-961	158	23	�	�	PROPN
ap-961	158	24	�	�	PROPN
ap-961	158	25	�	�	PROPN
ap-961	158	26	1	1	NUM
ap-961	158	27	1	1	NUM
ap-961	158	28	,	,	PUNCT
ap-961	158	29	extends	extend	VERB
ap-961	158	30	in	in	ADP
ap-961	158	31	a	a	DET
ap-961	158	32	unique	unique	ADJ
ap-961	158	33	way	way	NOUN
ap-961	158	34	to	to	ADP
ap-961	158	35	a	a	DET
ap-961	158	36	degree	degree	NOUN
ap-961	158	37	1	1	NUM
ap-961	158	38	linear	linear	NOUN
ap-961	158	39	operator	operator	NOUN
ap-961	158	40	on	on	ADP
ap-961	158	41	u	u	PROPN
ap-961	158	42	(	(	PUNCT
ap-961	158	43	)	)	PUNCT
ap-961	158	44	�	�	PROPN
ap-961	158	45	,	,	PUNCT
ap-961	158	46	which	which	PRON
ap-961	158	47	satisfies	satisfy	VERB
ap-961	158	48	the	the	DET
ap-961	158	49	graded	grade	VERB
ap-961	158	50	leibniz	leibniz	NOUN
ap-961	158	51	rule	rule	NOUN
ap-961	158	52	and	and	CCONJ
ap-961	158	53	is	be	AUX
ap-961	158	54	nilpotent	nilpotent	ADJ
ap-961	158	55	,	,	PUNCT
ap-961	158	56	du	du	PROPN
ap-961	158	57	2	2	NUM
ap-961	158	58	0	0	NUM
ap-961	158	59	�	�	PROPN
ap-961	158	60	.	.	PUNCT
ap-961	159	1	this	this	DET
ap-961	159	2	dga	dga	NOUN
ap-961	159	3	is	be	AUX
ap-961	159	4	called	call	VERB
ap-961	159	5	a	a	DET
ap-961	159	6	universal	universal	ADJ
ap-961	159	7	dga	dga	NOUN
ap-961	159	8	and	and	CCONJ
ap-961	159	9	du	du	PROPN
ap-961	159	10	is	be	AUX
ap-961	159	11	the	the	DET
ap-961	159	12	universal	universal	ADJ
ap-961	159	13	external	external	ADJ
ap-961	159	14	derivative	derivative	NOUN
ap-961	159	15	.	.	PUNCT
ap-961	160	1	proof	proof	NOUN
ap-961	160	2	:	:	PUNCT
ap-961	160	3	it	it	PRON
ap-961	160	4	is	be	AUX
ap-961	160	5	a	a	DET
ap-961	160	6	good	good	ADJ
ap-961	160	7	exercise	exercise	NOUN
ap-961	160	8	to	to	PART
ap-961	160	9	have	have	VERB
ap-961	160	10	a	a	DET
ap-961	160	11	look	look	NOUN
ap-961	160	12	at	at	ADP
ap-961	160	13	the	the	DET
ap-961	160	14	proof	proof	NOUN
ap-961	160	15	.	.	PUNCT
ap-961	161	1	actually	actually	ADV
ap-961	161	2	,	,	PUNCT
ap-961	161	3	most	most	ADJ
ap-961	161	4	of	of	ADP
ap-961	161	5	the	the	DET
ap-961	161	6	properties	property	NOUN
ap-961	161	7	of	of	ADP
ap-961	161	8	du	du	NOUN
ap-961	161	9	are	be	AUX
ap-961	161	10	used	use	VERB
ap-961	161	11	in	in	ADP
ap-961	161	12	the	the	DET
ap-961	161	13	construction	construction	NOUN
ap-961	161	14	.	.	PUNCT
ap-961	162	1	for	for	ADP
ap-961	162	2	instance	instance	NOUN
ap-961	162	3	,	,	PUNCT
ap-961	162	4	we	we	PRON
ap-961	162	5	extend	extend	VERB
ap-961	162	6	the	the	DET
ap-961	162	7	definition	definition	NOUN
ap-961	162	8	of	of	ADP
ap-961	162	9	du	du	PROPN
ap-961	162	10	in	in	ADP
ap-961	162	11	such	such	DET
ap-961	162	12	a	a	DET
ap-961	162	13	way	way	NOUN
ap-961	162	14	that	that	PRON
ap-961	162	15	the	the	DET
ap-961	162	16	leibniz	leibniz	NOUN
ap-961	162	17	rule	rule	NOUN
ap-961	162	18	is	be	AUX
ap-961	162	19	assured	assure	VERB
ap-961	162	20	.	.	PUNCT
ap-961	163	1	the	the	DET
ap-961	163	2	universal	universal	ADJ
ap-961	163	3	one	one	NUM
ap-961	163	4	-	-	PUNCT
ap-961	163	5	forms	form	NOUN
ap-961	163	6	could	could	AUX
ap-961	163	7	be	be	AUX
ap-961	163	8	always	always	ADV
ap-961	163	9	uniquely	uniquely	ADV
ap-961	163	10	expressed	express	VERB
ap-961	163	11	as	as	ADP
ap-961	163	12	a	a	DET
ap-961	163	13	d	d	PROPN
ap-961	163	14	bi	bi	PROPN
ap-961	163	15	u	u	PROPN
ap-961	163	16	ii	ii	PROPN
ap-961	163	17	�	�	PROPN
ap-961	163	18	,	,	PUNCT
ap-961	163	19	for	for	ADP
ap-961	163	20	a	a	DET
ap-961	163	21	bi	bi	NOUN
ap-961	163	22	i	i	PROPN
ap-961	163	23	,	,	PUNCT
ap-961	163	24	�	�	PROPN
ap-961	163	25	�	�	PROPN
ap-961	163	26	,	,	PUNCT
ap-961	163	27	a	a	DET
ap-961	163	28	bi	bi	PROPN
ap-961	163	29	ii	ii	PROPN
ap-961	163	30	�	�	PROPN
ap-961	163	31	�	�	PROPN
ap-961	163	32	0	0	NUM
ap-961	163	33	.	.	PUNCT
ap-961	164	1	indeed	indeed	ADV
ap-961	164	2	,	,	PUNCT
ap-961	164	3	an	an	DET
ap-961	164	4	arbitrary	arbitrary	ADJ
ap-961	164	5	universal	universal	ADJ
ap-961	164	6	one	one	NUM
ap-961	164	7	-	-	PUNCT
ap-961	164	8	form	form	NOUN
ap-961	164	9	is	be	AUX
ap-961	164	10	of	of	ADP
ap-961	164	11	the	the	DET
ap-961	164	12	type	type	NOUN
ap-961	164	13	a	a	DET
ap-961	164	14	bi	bi	PROPN
ap-961	164	15	ii	ii	PROPN
ap-961	164	16	�	�	PROPN
ap-961	164	17	�	�	PROPN
ap-961	164	18	for	for	ADP
ap-961	164	19	some	some	PRON
ap-961	164	20	a	a	DET
ap-961	164	21	bi	bi	NOUN
ap-961	164	22	i	i	PROPN
ap-961	164	23	,	,	PUNCT
ap-961	164	24	�	�	PROPN
ap-961	164	25	�	�	PROPN
ap-961	164	26	such	such	ADJ
ap-961	164	27	that	that	SCONJ
ap-961	164	28	a	a	DET
ap-961	164	29	bi	bi	NOUN
ap-961	164	30	i	i	PROPN
ap-961	164	31	�	�	PROPN
ap-961	164	32	�	�	PROPN
ap-961	164	33	0	0	NUM
ap-961	164	34	.	.	PUNCT
ap-961	165	1	but	but	CCONJ
ap-961	165	2	:	:	PUNCT
ap-961	165	3	(	(	PUNCT
ap-961	165	4	)	)	PUNCT
ap-961	165	5	(	(	PUNCT
ap-961	165	6	)	)	PUNCT
ap-961	165	7	�	�	PROPN
ap-961	165	8	�	�	PROPN
ap-961	165	9	�	�	PROPN
ap-961	165	10	�	�	PROPN
ap-961	165	11	�	�	PROPN
ap-961	165	12	�	�	PROPN
ap-961	165	13	�	�	PROPN
ap-961	165	14	�	�	PROPN
ap-961	165	15	�	�	PROPN
ap-961	165	16	�	�	PROPN
ap-961	165	17	�	�	PROPN
ap-961	165	18	a	a	PROPN
ap-961	165	19	d	d	PROPN
ap-961	165	20	b	b	PROPN
ap-961	165	21	a	a	DET
ap-961	165	22	b	b	NOUN
ap-961	165	23	a	a	DET
ap-961	165	24	b	b	NOUN
ap-961	165	25	a	a	DET
ap-961	165	26	bi	bi	ADJ
ap-961	165	27	u	u	NOUN
ap-961	166	1	i	i	PRON
ap-961	166	2	i	i	PRON
ap-961	167	1	i	i	PRON
ap-961	167	2	i	i	PRON
ap-961	168	1	i	i	PRON
ap-961	168	2	i	i	PRON
ap-961	169	1	i	i	PRON
ap-961	169	2	i	i	PRON
ap-961	170	1	i	i	PRON
ap-961	170	2	i	i	PRON
ap-961	170	3	i	i	VERB
ap-961	170	4	1	1	NUM
ap-961	170	5	.	.	PUNCT
ap-961	171	1	we	we	PRON
ap-961	171	2	set	set	VERB
ap-961	171	3	:	:	PUNCT
ap-961	172	1	d	d	X
ap-961	172	2	a	a	PRON
ap-961	172	3	d	d	X
ap-961	172	4	b	b	PROPN
ap-961	172	5	d	d	NOUN
ap-961	172	6	a	a	PROPN
ap-961	172	7	d	d	X
ap-961	172	8	bu	bu	INTJ
ap-961	172	9	i	i	PRON
ap-961	172	10	u	u	INTJ
ap-961	173	1	i	i	PRON
ap-961	173	2	i	i	PRON
ap-961	173	3	u	u	VERB
ap-961	173	4	i	i	PRON
ap-961	173	5	u	u	VERB
ap-961	174	1	i	i	PRON
ap-961	174	2	i	i	PRON
ap-961	174	3	�	�	PROPN
ap-961	174	4	�	�	PROPN
ap-961	174	5	�	�	PROPN
ap-961	174	6	�	�	PROPN
ap-961	174	7	�	�	PROPN
ap-961	174	8	�	�	PROPN
ap-961	174	9	�	�	PROPN
ap-961	174	10	�	�	PROPN
ap-961	174	11	�	�	PROPN
ap-961	174	12	�	�	PROPN
ap-961	174	13	�	�	PROPN
ap-961	174	14	�	�	PROPN
ap-961	174	15	�	�	PROPN
ap-961	174	16	.	.	PUNCT
ap-961	175	1	this	this	PRON
ap-961	175	2	assures	assure	VERB
ap-961	175	3	the	the	DET
ap-961	175	4	graded	grade	VERB
ap-961	175	5	leibniz	leibniz	NOUN
ap-961	175	6	rule	rule	NOUN
ap-961	175	7	between	between	ADP
ap-961	175	8	elements	element	NOUN
ap-961	175	9	of	of	ADP
ap-961	175	10	the	the	DET
ap-961	175	11	algebra	algebra	NOUN
ap-961	175	12	and	and	CCONJ
ap-961	175	13	one	one	NUM
ap-961	175	14	-	-	PUNCT
ap-961	175	15	forms	form	NOUN
ap-961	175	16	,	,	PUNCT
ap-961	175	17	and	and	CCONJ
ap-961	175	18	makes	make	VERB
ap-961	175	19	sure	sure	ADJ
ap-961	175	20	that	that	SCONJ
ap-961	175	21	du	du	PROPN
ap-961	175	22	2	2	NUM
ap-961	175	23	0	0	NUM
ap-961	175	24	�	�	PROPN
ap-961	175	25	on	on	ADP
ap-961	175	26	�	�	PROPN
ap-961	175	27	.	.	PUNCT
ap-961	176	1	the	the	DET
ap-961	176	2	rest	rest	NOUN
ap-961	176	3	is	be	AUX
ap-961	176	4	just	just	ADV
ap-961	176	5	the	the	DET
ap-961	176	6	application	application	NOUN
ap-961	176	7	of	of	ADP
ap-961	176	8	the	the	DET
ap-961	176	9	leibniz	leibniz	PROPN
ap-961	176	10	rule	rule	NOUN
ap-961	176	11	.	.	PUNCT
ap-961	177	1	we	we	PRON
ap-961	177	2	extend	extend	VERB
ap-961	177	3	du	du	NOUN
ap-961	177	4	to	to	ADP
ap-961	177	5	products	product	NOUN
ap-961	177	6	of	of	ADP
ap-961	177	7	universal	universal	ADJ
ap-961	177	8	one	one	NUM
ap-961	177	9	-	-	PUNCT
ap-961	177	10	forms	form	NOUN
ap-961	177	11	through	through	ADP
ap-961	177	12	:	:	PUNCT
ap-961	178	1	d	d	X
ap-961	178	2	d	d	PUNCT
ap-961	178	3	u	u	X
ap-961	178	4	k	k	PROPN
ap-961	178	5	j	j	PROPN
ap-961	179	1	j	j	PROPN
ap-961	180	1	k	k	PROPN
ap-961	180	2	j	j	PROPN
ap-961	180	3	k	k	PROPN
ap-961	180	4	(	(	PUNCT
ap-961	180	5	)	)	PUNCT
ap-961	180	6	(	(	PUNCT
ap-961	180	7	)	)	PUNCT
ap-961	180	8	�	�	PROPN
ap-961	180	9	�	�	PROPN
ap-961	180	10	�	�	PROPN
ap-961	180	11	�	�	PROPN
ap-961	180	12	�	�	PROPN
ap-961	180	13	�	�	PROPN
ap-961	180	14	�	�	PROPN
ap-961	180	15	1	1	NUM
ap-961	180	16	2	2	NUM
ap-961	180	17	1	1	NUM
ap-961	180	18	1	1	NUM
ap-961	180	19	2	2	NUM
ap-961	180	20	1	1	NUM
ap-961	180	21	1	1	NUM
ap-961	180	22	�	�	PROPN
ap-961	180	23	�	�	PROPN
ap-961	180	24	�	�	PROPN
ap-961	180	25	�	�	PROPN
ap-961	180	26	�	�	PROPN
ap-961	180	27	�	�	PROPN
ap-961	180	28	�	�	PROPN
ap-961	180	29	�	�	PROPN
ap-961	180	30	�	�	PROPN
ap-961	180	31	�	�	PROPN
ap-961	180	32	�	�	PROPN
ap-961	180	33	�	�	PROPN
ap-961	180	34	�	�	PROPN
ap-961	180	35	�	�	PROPN
ap-961	180	36	�	�	PROPN
ap-961	180	37	�	�	PROPN
ap-961	180	38	�	�	PROPN
ap-961	180	39	�	�	PROPN
ap-961	180	40	�	�	PROPN
ap-961	180	41	for	for	ADP
ap-961	180	42	each	each	DET
ap-961	180	43	�	�	PROPN
ap-961	180	44	i	i	PRON
ap-961	180	45	u	u	NOUN
ap-961	180	46	�	�	PROPN
ap-961	180	47	1	1	NUM
ap-961	180	48	(	(	PUNCT
ap-961	180	49	)	)	PUNCT
ap-961	180	50	�	�	PROPN
ap-961	180	51	,	,	PUNCT
ap-961	180	52	i	i	PRON
ap-961	180	53	k	k	PROPN
ap-961	180	54	�	�	PROPN
ap-961	180	55	1	1	NUM
ap-961	180	56	,	,	PUNCT
ap-961	180	57	,	,	PUNCT
ap-961	180	58	�	�	PROPN
ap-961	180	59	.	.	PUNCT
ap-961	181	1	of	of	ADP
ap-961	181	2	course	course	ADV
ap-961	181	3	,	,	PUNCT
ap-961	181	4	we	we	PRON
ap-961	181	5	need	need	VERB
ap-961	181	6	to	to	PART
ap-961	181	7	check	check	VERB
ap-961	181	8	that	that	SCONJ
ap-961	181	9	the	the	DET
ap-961	181	10	definition	definition	NOUN
ap-961	181	11	is	be	AUX
ap-961	181	12	compatible	compatible	ADJ
ap-961	181	13	with	with	ADP
ap-961	181	14	the	the	DET
ap-961	181	15	tensor	tensor	NOUN
ap-961	181	16	product	product	NOUN
ap-961	181	17	over	over	ADP
ap-961	181	18	�	�	PROPN
ap-961	181	19	,	,	PUNCT
ap-961	181	20	which	which	PRON
ap-961	181	21	mean	mean	VERB
ap-961	181	22	that	that	SCONJ
ap-961	181	23	it	it	PRON
ap-961	181	24	gives	give	VERB
ap-961	181	25	the	the	DET
ap-961	181	26	same	same	ADJ
ap-961	181	27	result	result	NOUN
ap-961	181	28	on	on	ADP
ap-961	181	29	�	�	PROPN
ap-961	181	30	�	�	PROPN
ap-961	181	31	�	�	PROPN
ap-961	181	32	�	�	PROPN
ap-961	181	33	a	a	DET
ap-961	181	34	and	and	CCONJ
ap-961	181	35	�	�	PROPN
ap-961	181	36	�	�	PROPN
ap-961	181	37	a	a	DET
ap-961	181	38	�	�	PROPN
ap-961	181	39	�	�	PROPN
ap-961	181	40	:	:	PUNCT
ap-961	181	41	d	d	ADP
ap-961	181	42	a	a	X
ap-961	181	43	d	d	X
ap-961	181	44	a	a	PROPN
ap-961	181	45	d	d	NOUN
ap-961	181	46	a	a	PROPN
ap-961	181	47	d	d	NOUN
ap-961	181	48	a	a	DET
ap-961	181	49	u	u	NOUN
ap-961	181	50	u	u	X
ap-961	181	51	u	u	NOUN
ap-961	181	52	u	u	NOUN
ap-961	181	53	(	(	PUNCT
ap-961	181	54	(	(	PUNCT
ap-961	181	55	(	(	PUNCT
ap-961	181	56	)	)	PUNCT
ap-961	181	57	(	(	PUNCT
ap-961	181	58	)	)	PUNCT
ap-961	181	59	(	(	PUNCT
ap-961	181	60	(	(	PUNCT
ap-961	181	61	)	)	PUNCT
ap-961	181	62	(	(	PUNCT
ap-961	181	63	�	�	PROPN
ap-961	181	64	�	�	PROPN
ap-961	181	65	�	�	PROPN
ap-961	181	66	�	�	PROPN
ap-961	181	67	�	�	PROPN
ap-961	181	68	�	�	PROPN
ap-961	181	69	�	�	PROPN
ap-961	181	70	�	�	PROPN
ap-961	181	71	�	�	PROPN
ap-961	181	72	�	�	PROPN
ap-961	181	73	�	�	PROPN
ap-961	181	74	�	�	PROPN
ap-961	181	75	�	�	PROPN
ap-961	181	76	�	�	PROPN
ap-961	181	77	�	�	PROPN
ap-961	181	78	�	�	PROPN
ap-961	181	79	�	�	PROPN
ap-961	181	80	�	�	PROPN
ap-961	181	81	�	�	PROPN
ap-961	181	82	�	�	PROPN
ap-961	181	83	�	�	PROPN
ap-961	181	84	1	1	NUM
ap-961	181	85	1	1	NUM
ap-961	181	86	)	)	PUNCT
ap-961	181	87	(	(	PUNCT
ap-961	181	88	)	)	PUNCT
ap-961	181	89	(	(	PUNCT
ap-961	181	90	)	)	PUNCT
ap-961	181	91	(	(	PUNCT
ap-961	181	92	)	)	PUNCT
ap-961	181	93	(	(	PUNCT
ap-961	181	94	(	(	PUNCT
ap-961	181	95	)	)	PUNCT
ap-961	181	96	�	�	PROPN
ap-961	181	97	�	�	PROPN
ap-961	181	98	�	�	PROPN
ap-961	181	99	�	�	PROPN
ap-961	181	100	�	�	PROPN
ap-961	181	101	�	�	PROPN
ap-961	181	102	�	�	PROPN
ap-961	181	103	�	�	PROPN
ap-961	181	104	�	�	PROPN
ap-961	181	105	�	�	PROPN
ap-961	181	106	�	�	PROPN
ap-961	181	107	�	�	PROPN
ap-961	181	108	�	�	PROPN
ap-961	181	109	�	�	PROPN
ap-961	181	110	�	�	PROPN
ap-961	181	111	�	�	PROPN
ap-961	181	112	�	�	PROPN
ap-961	181	113	�	�	PROPN
ap-961	181	114	�	�	PROPN
ap-961	181	115	�	�	PROPN
ap-961	181	116	�	�	PROPN
ap-961	181	117	�	�	PROPN
ap-961	181	118	�	�	PROPN
ap-961	181	119	d	d	PROPN
ap-961	181	120	a	a	DET
ap-961	181	121	a	a	PROPN
ap-961	181	122	d	d	X
ap-961	181	123	d	d	PROPN
ap-961	181	124	a	a	PROPN
ap-961	181	125	d	d	X
ap-961	181	126	a	a	DET
ap-961	181	127	u	u	NOUN
ap-961	181	128	u	u	X
ap-961	181	129	u	u	NOUN
ap-961	181	130	u	u	NOUN
ap-961	181	131	1	1	NUM
ap-961	181	132	1	1	NUM
ap-961	181	133	�	�	PROPN
ap-961	181	134	�	�	PROPN
ap-961	181	135	�	�	PROPN
ap-961	181	136	�	�	PROPN
ap-961	181	137	�	�	PROPN
ap-961	181	138	�	�	PROPN
ap-961	181	139	�	�	PROPN
ap-961	181	140	�	�	PROPN
ap-961	181	141	�	�	PROPN
ap-961	181	142	�	�	PROPN
ap-961	181	143	�	�	PROPN
ap-961	181	144	�	�	PROPN
ap-961	181	145	�	�	PROPN
ap-961	181	146	�	�	PROPN
ap-961	181	147	�	�	PROPN
ap-961	181	148	�	�	PROPN
ap-961	181	149	�	�	PROPN
ap-961	181	150	�	�	PROPN
ap-961	181	151	�	�	PROPN
ap-961	181	152	�	�	PROPN
ap-961	181	153	�	�	PROPN
ap-961	181	154	�	�	PROPN
ap-961	181	155	�	�	PROPN
ap-961	181	156	�	�	PROPN
ap-961	181	157	(	(	PUNCT
ap-961	181	158	)	)	PUNCT
ap-961	181	159	(	(	PUNCT
ap-961	181	160	)	)	PUNCT
ap-961	181	161	(	(	PUNCT
ap-961	181	162	(	(	PUNCT
ap-961	181	163	)	)	PUNCT
ap-961	181	164	(	(	PUNCT
ap-961	181	165	)	)	PUNCT
ap-961	181	166	(	(	PUNCT
ap-961	181	167	1	1	NUM
ap-961	181	168	1	1	NUM
ap-961	181	169	a	a	DET
ap-961	181	170	d	d	X
ap-961	181	171	d	d	PROPN
ap-961	181	172	a	a	DET
ap-961	181	173	a	a	PROPN
ap-961	181	174	d	d	X
ap-961	181	175	d	d	X
ap-961	181	176	u	u	X
ap-961	181	177	u	u	X
ap-961	181	178	u	u	X
ap-961	181	179	u	u	PROPN
ap-961	181	180	�	�	PROPN
ap-961	181	181	�	�	PROPN
ap-961	181	182	a	a	DET
ap-961	181	183	�	�	PROPN
ap-961	181	184	�	�	PROPN
ap-961	181	185	it	it	PRON
ap-961	181	186	is	be	AUX
ap-961	181	187	now	now	ADV
ap-961	181	188	a	a	DET
ap-961	181	189	matter	matter	NOUN
ap-961	181	190	of	of	ADP
ap-961	181	191	easy	easy	ADJ
ap-961	181	192	verification	verification	NOUN
ap-961	181	193	that	that	PRON
ap-961	181	194	du	du	PROPN
ap-961	181	195	2	2	NUM
ap-961	181	196	0	0	NUM
ap-961	181	197	�	�	PROPN
ap-961	181	198	.	.	PUNCT
ap-961	182	1	�	�	PROPN
ap-961	182	2	from	from	ADP
ap-961	182	3	the	the	DET
ap-961	182	4	above	above	ADJ
ap-961	182	5	construction	construction	NOUN
ap-961	182	6	we	we	PRON
ap-961	182	7	obtain	obtain	VERB
ap-961	182	8	a	a	DET
ap-961	182	9	very	very	ADV
ap-961	182	10	convenient	convenient	ADJ
ap-961	182	11	presentation	presentation	NOUN
ap-961	182	12	of	of	ADP
ap-961	182	13	universal	universal	ADJ
ap-961	182	14	differential	differential	ADJ
ap-961	182	15	forms	form	NOUN
ap-961	182	16	:	:	PUNCT
ap-961	182	17	each	each	DET
ap-961	182	18	form	form	NOUN
ap-961	182	19	of	of	ADP
ap-961	182	20	degree	degree	NOUN
ap-961	182	21	k	k	PROPN
ap-961	182	22	can	can	AUX
ap-961	182	23	be	be	AUX
ap-961	182	24	presented	present	VERB
ap-961	182	25	as	as	ADP
ap-961	182	26	a	a	DET
ap-961	182	27	finite	finite	ADJ
ap-961	182	28	sum	sum	NOUN
ap-961	182	29	of	of	ADP
ap-961	182	30	elements	element	NOUN
ap-961	182	31	of	of	ADP
ap-961	182	32	the	the	DET
ap-961	182	33	type	type	NOUN
ap-961	182	34	a	a	DET
ap-961	182	35	da	da	ADJ
ap-961	182	36	dak0	dak0	NOUN
ap-961	182	37	1	1	NUM
ap-961	182	38	�	�	PROPN
ap-961	182	39	.	.	PUNCT
ap-961	183	1	moreover	moreover	ADV
ap-961	183	2	,	,	PUNCT
ap-961	183	3	having	have	VERB
ap-961	183	4	two	two	NUM
ap-961	183	5	forms	form	NOUN
ap-961	183	6	�	�	PRON
ap-961	183	7	a	a	DET
ap-961	183	8	ka	ka	PROPN
ap-961	183	9	da	da	PROPN
ap-961	183	10	da	da	PROPN
ap-961	183	11	�	�	PROPN
ap-961	183	12	0	0	NUM
ap-961	183	13	1	1	NUM
ap-961	183	14	�	�	PROPN
ap-961	183	15	and	and	CCONJ
ap-961	183	16	�	�	PROPN
ap-961	183	17	b	b	PROPN
ap-961	183	18	ka	ka	PROPN
ap-961	183	19	db	db	PROPN
ap-961	183	20	db	db	PROPN
ap-961	183	21	�	�	PROPN
ap-961	183	22	0	0	NUM
ap-961	183	23	1	1	NUM
ap-961	183	24	�	�	PROPN
ap-961	183	25	they	they	PRON
ap-961	183	26	are	be	AUX
ap-961	183	27	different	different	ADJ
ap-961	183	28	un	un	PROPN
ap-961	183	29	©	©	PROPN
ap-961	183	30	czech	czech	PROPN
ap-961	183	31	technical	technical	PROPN
ap-961	183	32	university	university	PROPN
ap-961	183	33	publishing	publishing	NOUN
ap-961	183	34	house	house	NOUN
ap-961	183	35	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	183	36	39	39	NUM
ap-961	183	37	acta	acta	PROPN
ap-961	183	38	polytechnica	polytechnica	PROPN
ap-961	183	39	vol	vol	NOUN
ap-961	183	40	.	.	PUNCT
ap-961	184	1	48	48	NUM
ap-961	184	2	no	no	NOUN
ap-961	184	3	.	.	PUNCT
ap-961	185	1	2/2008	2/2008	NOUN
ap-961	185	2	less	less	ADJ
ap-961	185	3	for	for	ADP
ap-961	185	4	each	each	DET
ap-961	185	5	i	i	PRON
ap-961	185	6	k	k	NOUN
ap-961	185	7	�	�	PROPN
ap-961	185	8	0	0	NUM
ap-961	185	9	1	1	NUM
ap-961	185	10	,	,	PUNCT
ap-961	185	11	,	,	PUNCT
ap-961	185	12	,	,	PUNCT
ap-961	185	13	�	�	PROPN
ap-961	185	14	,	,	PUNCT
ap-961	185	15	ai	ai	VERB
ap-961	185	16	and	and	CCONJ
ap-961	185	17	bi	bi	NOUN
ap-961	185	18	are	be	AUX
ap-961	185	19	linearly	linearly	ADV
ap-961	185	20	dependent	dependent	ADJ
ap-961	185	21	.	.	PUNCT
ap-961	186	1	this	this	PRON
ap-961	186	2	follows	follow	VERB
ap-961	186	3	directly	directly	ADV
ap-961	186	4	from	from	ADP
ap-961	186	5	the	the	DET
ap-961	186	6	definition	definition	NOUN
ap-961	186	7	of	of	ADP
ap-961	186	8	u	u	PROPN
ap-961	186	9	k	k	PROPN
ap-961	186	10	(	(	PUNCT
ap-961	186	11	)	)	PUNCT
ap-961	186	12	�	�	PROPN
ap-961	186	13	as	as	ADP
ap-961	186	14	the	the	DET
ap-961	186	15	products	product	NOUN
ap-961	186	16	of	of	ADP
ap-961	186	17	k	k	PROPN
ap-961	186	18	one	one	NUM
ap-961	186	19	-	-	PUNCT
ap-961	186	20	forms	form	NOUN
ap-961	186	21	and	and	CCONJ
ap-961	186	22	the	the	DET
ap-961	186	23	fact	fact	NOUN
ap-961	186	24	that	that	SCONJ
ap-961	186	25	each	each	DET
ap-961	186	26	one	one	NUM
ap-961	186	27	-	-	PUNCT
ap-961	186	28	form	form	NOUN
ap-961	186	29	is	be	AUX
ap-961	186	30	already	already	ADV
ap-961	186	31	in	in	ADP
ap-961	186	32	that	that	DET
ap-961	186	33	form	form	NOUN
ap-961	186	34	.	.	PUNCT
ap-961	187	1	the	the	DET
ap-961	187	2	rest	rest	NOUN
ap-961	187	3	is	be	AUX
ap-961	187	4	the	the	DET
ap-961	187	5	iterative	iterative	ADJ
ap-961	187	6	application	application	NOUN
ap-961	187	7	of	of	ADP
ap-961	187	8	the	the	DET
ap-961	187	9	leibniz	leibniz	PROPN
ap-961	187	10	rule	rule	NOUN
ap-961	187	11	.	.	PUNCT
ap-961	188	1	note	note	VERB
ap-961	188	2	that	that	SCONJ
ap-961	188	3	this	this	DET
ap-961	188	4	hints	hint	VERB
ap-961	188	5	at	at	ADP
ap-961	188	6	a	a	DET
ap-961	188	7	very	very	ADV
ap-961	188	8	nice	nice	ADJ
ap-961	188	9	feature	feature	NOUN
ap-961	188	10	of	of	ADP
ap-961	188	11	the	the	DET
ap-961	188	12	universal	universal	ADJ
ap-961	188	13	differential	differential	ADJ
ap-961	188	14	algebra	algebra	NOUN
ap-961	188	15	:	:	PUNCT
ap-961	188	16	each	each	DET
ap-961	188	17	space	space	NOUN
ap-961	188	18	of	of	ADP
ap-961	188	19	forms	form	NOUN
ap-961	188	20	of	of	ADP
ap-961	188	21	fixed	fix	VERB
ap-961	188	22	grade	grade	NOUN
ap-961	188	23	is	be	AUX
ap-961	188	24	generated	generate	VERB
ap-961	188	25	as	as	ADP
ap-961	188	26	a	a	DET
ap-961	188	27	left	left	ADJ
ap-961	188	28	-	-	PUNCT
ap-961	188	29	module	module	NOUN
ap-961	188	30	over	over	ADP
ap-961	188	31	�	�	PROPN
ap-961	188	32	by	by	ADP
ap-961	188	33	the	the	DET
ap-961	188	34	image	image	NOUN
ap-961	188	35	of	of	ADP
ap-961	188	36	d.	d.	PROPN
ap-961	188	37	we	we	PRON
ap-961	188	38	did	do	AUX
ap-961	188	39	not	not	PART
ap-961	188	40	assume	assume	VERB
ap-961	188	41	this	this	PRON
ap-961	188	42	in	in	ADP
ap-961	188	43	the	the	DET
ap-961	188	44	definition	definition	NOUN
ap-961	188	45	of	of	ADP
ap-961	188	46	the	the	DET
ap-961	188	47	dga	dga	NOUN
ap-961	188	48	but	but	CCONJ
ap-961	188	49	we	we	PRON
ap-961	188	50	can	can	AUX
ap-961	188	51	use	use	VERB
ap-961	188	52	the	the	DET
ap-961	188	53	(	(	PUNCT
ap-961	188	54	nonstandard	nonstandard	ADJ
ap-961	188	55	)	)	PUNCT
ap-961	188	56	name	name	NOUN
ap-961	188	57	of	of	ADP
ap-961	188	58	a	a	DET
ap-961	188	59	proper	proper	ADJ
ap-961	188	60	dga	dga	NOUN
ap-961	188	61	for	for	ADP
ap-961	188	62	those	those	PRON
ap-961	188	63	that	that	PRON
ap-961	188	64	have	have	VERB
ap-961	188	65	this	this	DET
ap-961	188	66	property	property	NOUN
ap-961	188	67	.	.	PUNCT
ap-961	189	1	finally	finally	ADV
ap-961	189	2	we	we	PRON
ap-961	189	3	come	come	VERB
ap-961	189	4	to	to	ADP
ap-961	189	5	the	the	DET
ap-961	189	6	question	question	NOUN
ap-961	189	7	of	of	ADP
ap-961	189	8	the	the	DET
ap-961	189	9	name	name	NOUN
ap-961	189	10	.	.	PUNCT
ap-961	190	1	universal	universal	ADJ
ap-961	190	2	differential	differential	ADJ
ap-961	190	3	algebra	algebra	PROPN
ap-961	190	4	owes	owe	VERB
ap-961	190	5	its	its	PRON
ap-961	190	6	name	name	NOUN
ap-961	190	7	its	its	PRON
ap-961	190	8	nice	nice	ADJ
ap-961	190	9	property	property	NOUN
ap-961	190	10	:	:	PUNCT
ap-961	190	11	it	it	PRON
ap-961	190	12	is	be	AUX
ap-961	190	13	indeed	indeed	ADV
ap-961	190	14	universal	universal	ADJ
ap-961	190	15	!	!	PUNCT
ap-961	191	1	what	what	PRON
ap-961	191	2	does	do	AUX
ap-961	191	3	this	this	PRON
ap-961	191	4	mean	mean	VERB
ap-961	191	5	?	?	PUNCT
ap-961	192	1	the	the	DET
ap-961	192	2	following	follow	VERB
ap-961	192	3	is	be	AUX
ap-961	192	4	due	due	ADJ
ap-961	192	5	to	to	AUX
ap-961	192	6	karoubi	karoubi	ADV
ap-961	192	7	:	:	PUNCT
ap-961	192	8	theorem	theorem	VERB
ap-961	192	9	3.4	3.4	NUM
ap-961	192	10	:	:	PUNCT
ap-961	192	11	if	if	SCONJ
ap-961	192	12	*	*	ADJ
ap-961	192	13	(	(	PUNCT
ap-961	192	14	)	)	PUNCT
ap-961	192	15	�	�	PROPN
ap-961	192	16	is	be	AUX
ap-961	192	17	a	a	DET
ap-961	192	18	graded	grade	VERB
ap-961	192	19	differential	differential	ADJ
ap-961	192	20	algebra	algebra	NOUN
ap-961	192	21	over	over	ADP
ap-961	192	22	�	�	PROPN
ap-961	192	23	and	and	CCONJ
ap-961	192	24	�	�	PROPN
ap-961	192	25	:	:	PUNCT
ap-961	192	26	�	�	PROPN
ap-961	192	27	�	�	PROPN
ap-961	192	28	�	�	PROPN
ap-961	192	29	an	an	DET
ap-961	192	30	algebra	algebra	NOUN
ap-961	192	31	homomorphism	homomorphism	NOUN
ap-961	192	32	,	,	PUNCT
ap-961	192	33	then	then	ADV
ap-961	192	34	there	there	PRON
ap-961	192	35	exists	exist	VERB
ap-961	192	36	a	a	DET
ap-961	192	37	unique	unique	ADJ
ap-961	192	38	extension	extension	NOUN
ap-961	192	39	of	of	ADP
ap-961	192	40	�	�	PROPN
ap-961	192	41	:	:	PUNCT
ap-961	192	42	~	~	PUNCT
ap-961	192	43	:	:	PUNCT
ap-961	192	44	(	(	PUNCT
ap-961	192	45	)	)	PUNCT
ap-961	192	46	(	(	PUNCT
ap-961	192	47	)	)	PUNCT
ap-961	192	48	*	*	PUNCT
ap-961	192	49	�	�	PROPN
ap-961	192	50	�	�	PROPN
ap-961	192	51	u	u	PROPN
ap-961	192	52	�	�	PROPN
ap-961	192	53	�	�	PROPN
ap-961	192	54	�	�	PROPN
ap-961	192	55	,	,	PUNCT
ap-961	192	56	which	which	PRON
ap-961	192	57	is	be	AUX
ap-961	192	58	a	a	DET
ap-961	192	59	morphism	morphism	NOUN
ap-961	192	60	of	of	ADP
ap-961	192	61	graded	grade	VERB
ap-961	192	62	differential	differential	ADJ
ap-961	192	63	algebras	algebra	NOUN
ap-961	192	64	:	:	PUNCT
ap-961	192	65	d	d	X
ap-961	192	66	d	d	PROPN
ap-961	192	67	�	�	PROPN
ap-961	192	68	�	�	PROPN
ap-961	192	69	�	�	PROPN
ap-961	192	70	�	�	PROPN
ap-961	192	71	�	�	PROPN
ap-961	192	72	.	.	PUNCT
ap-961	193	1	for	for	ADP
ap-961	193	2	our	our	PRON
ap-961	193	3	purposes	purpose	NOUN
ap-961	193	4	,	,	PUNCT
ap-961	193	5	we	we	PRON
ap-961	193	6	shall	shall	AUX
ap-961	193	7	actually	actually	ADV
ap-961	193	8	need	need	VERB
ap-961	193	9	a	a	DET
ap-961	193	10	consequence	consequence	NOUN
ap-961	193	11	of	of	ADP
ap-961	193	12	this	this	DET
ap-961	193	13	statement	statement	NOUN
ap-961	193	14	,	,	PUNCT
ap-961	193	15	corollary	corollary	NOUN
ap-961	193	16	3.5	3.5	NUM
ap-961	193	17	:	:	PUNCT
ap-961	193	18	for	for	ADP
ap-961	193	19	any	any	DET
ap-961	193	20	proper	proper	ADJ
ap-961	193	21	dga	dga	NOUN
ap-961	193	22	,	,	PUNCT
ap-961	193	23	(	(	PUNCT
ap-961	193	24	)	)	PUNCT
ap-961	193	25	�	�	PROPN
ap-961	193	26	,	,	PUNCT
ap-961	193	27	there	there	PRON
ap-961	193	28	exists	exist	VERB
ap-961	193	29	a	a	DET
ap-961	193	30	surjective	surjective	ADJ
ap-961	193	31	morphism	morphism	NOUN
ap-961	193	32	of	of	ADP
ap-961	193	33	differential	differential	ADJ
ap-961	193	34	graded	grade	VERB
ap-961	193	35	algebras	algebra	NOUN
ap-961	193	36	:	:	PUNCT
ap-961	193	37	�	�	PROPN
ap-961	193	38	:	:	PUNCT
ap-961	193	39	(	(	PUNCT
ap-961	193	40	)	)	PUNCT
ap-961	193	41	(	(	PUNCT
ap-961	193	42	)	)	PUNCT
ap-961	193	43	u	u	PROPN
ap-961	193	44	�	�	PROPN
ap-961	193	45	�	�	PROPN
ap-961	193	46	�	�	PROPN
ap-961	193	47	.	.	PUNCT
ap-961	194	1	proof	proof	NOUN
ap-961	194	2	:	:	PUNCT
ap-961	194	3	we	we	PRON
ap-961	194	4	set	set	VERB
ap-961	194	5	:	:	PUNCT
ap-961	194	6	�	�	PROPN
ap-961	194	7	(	(	PUNCT
ap-961	194	8	)	)	PUNCT
ap-961	194	9	(	(	PUNCT
ap-961	194	10	)	)	PUNCT
ap-961	194	11	a	a	PRON
ap-961	195	1	d	d	NOUN
ap-961	195	2	a	a	PROPN
ap-961	195	3	d	d	NOUN
ap-961	195	4	a	a	DET
ap-961	195	5	a	a	DET
ap-961	195	6	da	da	ADJ
ap-961	195	7	dau	dau	NOUN
ap-961	195	8	u	u	NOUN
ap-961	195	9	k	k	PROPN
ap-961	195	10	k0	k0	PROPN
ap-961	195	11	1	1	NUM
ap-961	195	12	0	0	NUM
ap-961	195	13	1	1	NUM
ap-961	195	14	�	�	PROPN
ap-961	195	15	�	�	PROPN
ap-961	195	16	�	�	PROPN
ap-961	195	17	.	.	PUNCT
ap-961	196	1	this	this	PRON
ap-961	196	2	is	be	AUX
ap-961	196	3	a	a	DET
ap-961	196	4	well	well	ADV
ap-961	196	5	-	-	PUNCT
ap-961	196	6	defined	define	VERB
ap-961	196	7	morphism	morphism	NOUN
ap-961	196	8	of	of	ADP
ap-961	196	9	differential	differential	NOUN
ap-961	196	10	algebras	algebra	NOUN
ap-961	196	11	.	.	PUNCT
ap-961	197	1	but	but	CCONJ
ap-961	197	2	since	since	SCONJ
ap-961	197	3	the	the	DET
ap-961	197	4	differential	differential	NOUN
ap-961	197	5	graded	grade	VERB
ap-961	197	6	algebra	algebra	NOUN
ap-961	197	7	(	(	PUNCT
ap-961	197	8	)	)	PUNCT
ap-961	197	9	�	�	PROPN
ap-961	197	10	is	be	AUX
ap-961	197	11	proper	proper	ADJ
ap-961	197	12	all	all	DET
ap-961	197	13	its	its	PRON
ap-961	197	14	elements	element	NOUN
ap-961	197	15	are	be	AUX
ap-961	197	16	of	of	ADP
ap-961	197	17	the	the	DET
ap-961	197	18	form	form	NOUN
ap-961	197	19	a	a	DET
ap-961	197	20	da	da	X
ap-961	197	21	dak0	dak0	NOUN
ap-961	197	22	1	1	NUM
ap-961	197	23	�	�	PROPN
ap-961	197	24	and	and	CCONJ
ap-961	197	25	hence	hence	ADV
ap-961	197	26	in	in	ADP
ap-961	197	27	the	the	DET
ap-961	197	28	image	image	NOUN
ap-961	197	29	of	of	ADP
ap-961	197	30	�	�	PROPN
ap-961	197	31	.	.	PUNCT
ap-961	197	32	�	�	PROPN
ap-961	197	33	briefly	briefly	ADV
ap-961	197	34	,	,	PUNCT
ap-961	197	35	corollary	corollary	ADJ
ap-961	197	36	3.5	3.5	NUM
ap-961	197	37	means	mean	VERB
ap-961	197	38	that	that	SCONJ
ap-961	197	39	every	every	DET
ap-961	197	40	proper	proper	ADJ
ap-961	197	41	differential	differential	NOUN
ap-961	197	42	graded	grade	VERB
ap-961	197	43	algebra	algebra	NOUN
ap-961	197	44	over	over	ADP
ap-961	197	45	the	the	DET
ap-961	197	46	algebra	algebra	NOUN
ap-961	197	47	�	�	PROPN
ap-961	197	48	is	be	AUX
ap-961	197	49	isomorphic	isomorphic	ADJ
ap-961	197	50	to	to	ADP
ap-961	197	51	a	a	DET
ap-961	197	52	quotient	quotient	NOUN
ap-961	197	53	of	of	ADP
ap-961	197	54	u	u	NOUN
ap-961	197	55	(	(	PUNCT
ap-961	197	56	)	)	PUNCT
ap-961	197	57	�	�	PROPN
ap-961	197	58	by	by	ADP
ap-961	197	59	a	a	DET
ap-961	197	60	differential	differential	ADJ
ap-961	197	61	ideal	ideal	NOUN
ap-961	197	62	,	,	PUNCT
ap-961	197	63	which	which	PRON
ap-961	197	64	is	be	AUX
ap-961	197	65	an	an	DET
ap-961	197	66	ideal	ideal	NOUN
ap-961	197	67	of	of	ADP
ap-961	197	68	the	the	DET
ap-961	197	69	universal	universal	ADJ
ap-961	197	70	graded	grade	VERB
ap-961	197	71	algebra	algebra	NOUN
ap-961	197	72	,	,	PUNCT
ap-961	197	73	�	�	PROPN
ap-961	197	74	�	�	PROPN
ap-961	197	75	�	�	PROPN
ap-961	197	76	u	u	PROPN
ap-961	197	77	(	(	PUNCT
ap-961	197	78	)	)	PUNCT
ap-961	197	79	,	,	PUNCT
ap-961	197	80	such	such	ADJ
ap-961	197	81	that	that	SCONJ
ap-961	197	82	du	du	PROPN
ap-961	197	83	(	(	PUNCT
ap-961	197	84	)	)	PUNCT
ap-961	197	85	�	�	PROPN
ap-961	197	86	�	�	PROPN
ap-961	197	87	�	�	PROPN
ap-961	197	88	.	.	PUNCT
ap-961	198	1	to	to	PART
ap-961	198	2	see	see	VERB
ap-961	198	3	how	how	SCONJ
ap-961	198	4	this	this	DET
ap-961	198	5	works	work	NOUN
ap-961	198	6	we	we	PRON
ap-961	198	7	need	need	VERB
ap-961	198	8	to	to	PART
ap-961	198	9	look	look	VERB
ap-961	198	10	at	at	ADP
ap-961	198	11	a	a	DET
ap-961	198	12	couple	couple	NOUN
ap-961	198	13	of	of	ADP
ap-961	198	14	examples	example	NOUN
ap-961	198	15	.	.	PUNCT
ap-961	199	1	example	example	NOUN
ap-961	199	2	3.6	3.6	NUM
ap-961	199	3	:	:	PUNCT
ap-961	199	4	let	let	VERB
ap-961	199	5	�	�	PROPN
ap-961	199	6	be	be	AUX
ap-961	199	7	the	the	DET
ap-961	199	8	algebra	algebra	NOUN
ap-961	199	9	of	of	ADP
ap-961	199	10	functions	function	NOUN
ap-961	199	11	on	on	ADP
ap-961	199	12	two	two	NUM
ap-961	199	13	points	point	NOUN
ap-961	199	14	(	(	PUNCT
ap-961	199	15	described	describe	VERB
ap-961	199	16	earlier	early	ADV
ap-961	199	17	)	)	PUNCT
ap-961	199	18	.	.	PUNCT
ap-961	200	1	the	the	DET
ap-961	200	2	bimodule	bimodule	NOUN
ap-961	200	3	of	of	ADP
ap-961	200	4	universal	universal	ADJ
ap-961	200	5	one	one	NUM
ap-961	200	6	-	-	PUNCT
ap-961	200	7	forms	form	NOUN
ap-961	200	8	is	be	AUX
ap-961	200	9	generated	generate	VERB
ap-961	200	10	by	by	ADP
ap-961	200	11	due	due	ADJ
ap-961	200	12	.	.	PUNCT
ap-961	201	1	applying	apply	VERB
ap-961	201	2	the	the	DET
ap-961	201	3	leibniz	leibniz	NOUN
ap-961	201	4	rule	rule	NOUN
ap-961	201	5	we	we	PRON
ap-961	201	6	see	see	VERB
ap-961	201	7	that	that	PRON
ap-961	201	8	:	:	PUNCT
ap-961	201	9	e	e	X
ap-961	201	10	d	d	X
ap-961	201	11	e	e	X
ap-961	201	12	d	d	X
ap-961	201	13	e	e	X
ap-961	201	14	eu	eu	PROPN
ap-961	201	15	u	u	PROPN
ap-961	201	16	(	(	PUNCT
ap-961	201	17	)	)	PUNCT
ap-961	201	18	(	(	PUNCT
ap-961	201	19	)	)	PUNCT
ap-961	201	20	�	�	PROPN
ap-961	201	21	0	0	NUM
ap-961	201	22	,	,	PUNCT
ap-961	201	23	so	so	SCONJ
ap-961	201	24	there	there	PRON
ap-961	201	25	are	be	VERB
ap-961	201	26	only	only	ADV
ap-961	201	27	two	two	NUM
ap-961	201	28	linearly	linearly	ADV
ap-961	201	29	independent	independent	ADJ
ap-961	201	30	one	one	NUM
ap-961	201	31	forms	form	NOUN
ap-961	201	32	:	:	PUNCT
ap-961	201	33	de	de	X
ap-961	201	34	and	and	CCONJ
ap-961	201	35	ede	ede	PROPN
ap-961	201	36	.	.	PUNCT
ap-961	202	1	similarly	similarly	ADV
ap-961	202	2	one	one	NUM
ap-961	202	3	constructs	construct	VERB
ap-961	202	4	all	all	DET
ap-961	202	5	higher	high	ADJ
ap-961	202	6	-	-	PUNCT
ap-961	202	7	order	order	NOUN
ap-961	202	8	forms	form	NOUN
ap-961	202	9	,	,	PUNCT
ap-961	202	10	which	which	PRON
ap-961	202	11	(	(	PUNCT
ap-961	202	12	as	as	ADP
ap-961	202	13	in	in	ADP
ap-961	202	14	every	every	DET
ap-961	202	15	universal	universal	ADJ
ap-961	202	16	calculus	calculus	NOUN
ap-961	202	17	)	)	PUNCT
ap-961	202	18	can	can	AUX
ap-961	202	19	be	be	AUX
ap-961	202	20	of	of	ADP
ap-961	202	21	arbitrary	arbitrary	ADJ
ap-961	202	22	order	order	NOUN
ap-961	202	23	.	.	PUNCT
ap-961	203	1	in	in	ADP
ap-961	203	2	fact	fact	NOUN
ap-961	203	3	,	,	PUNCT
ap-961	203	4	du	du	PROPN
ap-961	203	5	is	be	AUX
ap-961	203	6	a	a	DET
ap-961	203	7	derivation	derivation	NOUN
ap-961	203	8	on	on	ADP
ap-961	203	9	the	the	DET
ap-961	203	10	algebra	algebra	PROPN
ap-961	203	11	�	�	PROPN
ap-961	203	12	,	,	PUNCT
ap-961	203	13	but	but	CCONJ
ap-961	203	14	a	a	DET
ap-961	203	15	special	special	ADJ
ap-961	203	16	one	one	NUM
ap-961	203	17	.	.	PUNCT
ap-961	204	1	it	it	PRON
ap-961	204	2	takes	take	VERB
ap-961	204	3	values	value	NOUN
ap-961	204	4	not	not	PART
ap-961	204	5	in	in	ADP
ap-961	204	6	�	�	PROPN
ap-961	204	7	itself	itself	PRON
ap-961	204	8	(	(	PUNCT
ap-961	204	9	it	it	PRON
ap-961	204	10	can	can	AUX
ap-961	204	11	not	not	PART
ap-961	204	12	,	,	PUNCT
ap-961	204	13	as	as	SCONJ
ap-961	204	14	we	we	PRON
ap-961	204	15	have	have	AUX
ap-961	204	16	shown	show	VERB
ap-961	204	17	before	before	ADV
ap-961	204	18	)	)	PUNCT
ap-961	204	19	but	but	CCONJ
ap-961	204	20	in	in	ADP
ap-961	204	21	a	a	DET
ap-961	204	22	bimodule	bimodule	NOUN
ap-961	204	23	over	over	ADP
ap-961	204	24	�	�	PROPN
ap-961	204	25	.	.	PUNCT
ap-961	204	26	example	example	NOUN
ap-961	204	27	3.7	3.7	NUM
ap-961	204	28	:	:	PUNCT
ap-961	204	29	let	let	VERB
ap-961	204	30	x	x	PRON
ap-961	204	31	be	be	AUX
ap-961	204	32	a	a	DET
ap-961	204	33	space	space	NOUN
ap-961	204	34	and	and	CCONJ
ap-961	204	35	�	�	PROPN
ap-961	204	36	an	an	DET
ap-961	204	37	algebra	algebra	NOUN
ap-961	204	38	of	of	ADP
ap-961	204	39	functions	function	NOUN
ap-961	204	40	on	on	ADP
ap-961	204	41	x	x	X
ap-961	204	42	(	(	PUNCT
ap-961	204	43	we	we	PRON
ap-961	204	44	shall	shall	AUX
ap-961	204	45	not	not	PART
ap-961	204	46	need	need	VERB
ap-961	204	47	anything	anything	PRON
ap-961	204	48	more	more	ADJ
ap-961	204	49	about	about	ADP
ap-961	204	50	this	this	DET
ap-961	204	51	algebra	algebra	NOUN
ap-961	204	52	,	,	PUNCT
ap-961	204	53	so	so	SCONJ
ap-961	204	54	we	we	PRON
ap-961	204	55	do	do	AUX
ap-961	204	56	not	not	PART
ap-961	204	57	say	say	VERB
ap-961	204	58	whether	whether	SCONJ
ap-961	204	59	the	the	DET
ap-961	204	60	functions	function	NOUN
ap-961	204	61	are	be	AUX
ap-961	204	62	measurable	measurable	ADJ
ap-961	204	63	or	or	CCONJ
ap-961	204	64	smooth	smooth	ADJ
ap-961	204	65	,	,	PUNCT
ap-961	204	66	or	or	CCONJ
ap-961	204	67	just	just	ADV
ap-961	204	68	arbitrary	arbitrary	ADJ
ap-961	204	69	)	)	PUNCT
ap-961	204	70	.	.	PUNCT
ap-961	205	1	now	now	ADV
ap-961	205	2	,	,	PUNCT
ap-961	205	3	consider	consider	VERB
ap-961	205	4	the	the	DET
ap-961	205	5	following	follow	VERB
ap-961	205	6	differential	differential	NOUN
ap-961	205	7	graded	grade	VERB
ap-961	205	8	algebra	algebra	NOUN
ap-961	205	9	:	:	PUNCT
ap-961	205	10	�	�	PROPN
ap-961	205	11	�	�	PROPN
ap-961	205	12	k	k	PROPN
ap-961	206	1	k	k	PROPN
ap-961	206	2	k	k	PROPN
ap-961	207	1	i	i	PRON
ap-961	207	2	jx	jx	PROPN
ap-961	208	1	f	f	PROPN
ap-961	209	1	x	x	X
ap-961	209	2	f	f	NOUN
ap-961	209	3	x	x	PUNCT
ap-961	209	4	x	x	VERB
ap-961	210	1	i	i	PRON
ap-961	210	2	j	j	NOUN
ap-961	210	3	x	x	X
ap-961	210	4	x	x	X
ap-961	210	5	(	(	PUNCT
ap-961	210	6	)	)	PUNCT
ap-961	210	7	:	:	PUNCT
ap-961	210	8	,	,	PUNCT
ap-961	210	9	(	(	PUNCT
ap-961	210	10	,	,	PUNCT
ap-961	210	11	,	,	PUNCT
ap-961	210	12	)	)	PUNCT
ap-961	210	13	,	,	PUNCT
ap-961	210	14	:	:	PUNCT
ap-961	210	15	�	�	PROPN
ap-961	210	16	�	�	PROPN
ap-961	210	17	�	�	PROPN
ap-961	210	18	!	!	PUNCT
ap-961	210	19	�	�	PROPN
ap-961	210	20	�	�	PROPN
ap-961	210	21	1	1	NUM
ap-961	210	22	0	0	NUM
ap-961	210	23	�	�	PROPN
ap-961	210	24	if	if	SCONJ
ap-961	210	25	with	with	ADP
ap-961	210	26	the	the	DET
ap-961	210	27	product	product	NOUN
ap-961	210	28	:	:	PUNCT
ap-961	210	29	(	(	PUNCT
ap-961	210	30	)	)	PUNCT
ap-961	210	31	(	(	PUNCT
ap-961	210	32	,	,	PUNCT
ap-961	210	33	,	,	PUNCT
ap-961	210	34	,	,	PUNCT
ap-961	210	35	,	,	PUNCT
ap-961	210	36	,	,	PUNCT
ap-961	210	37	)	)	PUNCT
ap-961	210	38	(	(	PUNCT
ap-961	210	39	,	,	PUNCT
ap-961	210	40	,	,	PUNCT
ap-961	210	41	)	)	PUNCT
ap-961	210	42	(	(	PUNCT
ap-961	210	43	,	,	PUNCT
ap-961	210	44	,	,	PUNCT
ap-961	210	45	f	f	PROPN
ap-961	210	46	g	g	NOUN
ap-961	210	47	x	x	PUNCT
ap-961	210	48	x	x	PUNCT
ap-961	210	49	x	x	PUNCT
ap-961	210	50	x	x	X
ap-961	210	51	f	f	NOUN
ap-961	210	52	x	x	X
ap-961	210	53	x	x	X
ap-961	210	54	g	g	NOUN
ap-961	210	55	x	x	PROPN
ap-961	210	56	xk	xk	PROPN
ap-961	211	1	k	k	PROPN
ap-961	211	2	k	k	PROPN
ap-961	212	1	p	p	X
ap-961	212	2	k	k	PROPN
ap-961	213	1	k	k	PROPN
ap-961	213	2	k	k	PROPN
ap-961	213	3	p	p	X
ap-961	213	4	"	"	PUNCT
ap-961	213	5	�	�	PROPN
ap-961	213	6	1	1	NUM
ap-961	213	7	1	1	NUM
ap-961	213	8	1	1	NUM
ap-961	213	9	1	1	NUM
ap-961	213	10	�	�	PROPN
ap-961	213	11	�	�	PROPN
ap-961	213	12	�	�	PROPN
ap-961	213	13	�	�	PROPN
ap-961	213	14	)	)	PUNCT
ap-961	213	15	f	f	PROPN
ap-961	214	1	x	x	PUNCT
ap-961	214	2	g	g	PROPN
ap-961	214	3	xk	xk	PROPN
ap-961	214	4	p	p	PROPN
ap-961	214	5	�	�	PROPN
ap-961	214	6	�	�	PROPN
ap-961	214	7	(	(	PUNCT
ap-961	214	8	)	)	PUNCT
ap-961	214	9	,	,	PUNCT
ap-961	214	10	(	(	PUNCT
ap-961	214	11	)	)	PUNCT
ap-961	214	12	.	.	PUNCT
ap-961	215	1	the	the	DET
ap-961	215	2	external	external	ADJ
ap-961	215	3	derivative	derivative	NOUN
ap-961	215	4	d	d	NOUN
ap-961	215	5	is	be	AUX
ap-961	215	6	defined	define	VERB
ap-961	215	7	on	on	ADP
ap-961	215	8	the	the	DET
ap-961	215	9	functions	function	NOUN
ap-961	215	10	f	f	X
ap-961	215	11	x0	x0	PROPN
ap-961	215	12	0	0	NUM
ap-961	215	13	�	�	PROPN
ap-961	215	14	(	(	PUNCT
ap-961	215	15	)	)	PUNCT
ap-961	215	16	as	as	ADP
ap-961	215	17	:	:	PUNCT
ap-961	215	18	d	d	X
ap-961	215	19	f	f	X
ap-961	215	20	x	x	PUNCT
ap-961	215	21	x	x	SYM
ap-961	215	22	f	f	NOUN
ap-961	215	23	x	x	SYM
ap-961	215	24	f	f	PROPN
ap-961	215	25	x	x	X
ap-961	215	26	(	(	PUNCT
ap-961	215	27	)	)	PUNCT
ap-961	215	28	(	(	PUNCT
ap-961	215	29	,	,	PUNCT
ap-961	215	30	)	)	PUNCT
ap-961	215	31	(	(	PUNCT
ap-961	215	32	)	)	PUNCT
ap-961	215	33	(	(	PUNCT
ap-961	215	34	)	)	PUNCT
ap-961	215	35	0	0	NUM
ap-961	215	36	1	1	NUM
ap-961	215	37	2	2	NUM
ap-961	215	38	1	1	NUM
ap-961	215	39	20	20	NUM
ap-961	215	40	0	0	NUM
ap-961	215	41	�	�	PROPN
ap-961	215	42	�	�	PROPN
ap-961	215	43	,	,	PUNCT
ap-961	215	44	and	and	CCONJ
ap-961	215	45	then	then	ADV
ap-961	215	46	extended	extend	VERB
ap-961	215	47	to	to	ADP
ap-961	215	48	forms	form	NOUN
ap-961	215	49	of	of	ADP
ap-961	215	50	arbitrary	arbitrary	ADJ
ap-961	215	51	order	order	NOUN
ap-961	215	52	through	through	ADP
ap-961	215	53	:	:	PUNCT
ap-961	215	54	(	(	PUNCT
ap-961	215	55	)	)	PUNCT
ap-961	215	56	(	(	PUNCT
ap-961	215	57	,	,	PUNCT
ap-961	215	58	,	,	PUNCT
ap-961	215	59	,	,	PUNCT
ap-961	215	60	)	)	PUNCT
ap-961	215	61	(	(	PUNCT
ap-961	215	62	)	)	PUNCT
ap-961	215	63	(	(	PUNCT
ap-961	215	64	,	,	PUNCT
ap-961	215	65	,	,	PUNCT
ap-961	215	66	�	�	PROPN
ap-961	215	67	,	,	PUNCT
ap-961	215	68	,	,	PUNCT
ap-961	215	69	)	)	PUNCT
ap-961	216	1	d	d	X
ap-961	216	2	f	f	NOUN
ap-961	216	3	x	x	PUNCT
ap-961	216	4	x	x	PUNCT
ap-961	216	5	x	x	X
ap-961	216	6	f	f	NOUN
ap-961	216	7	x	x	PUNCT
ap-961	216	8	x	x	SYM
ap-961	216	9	xn	xn	PROPN
ap-961	216	10	n	n	CCONJ
ap-961	216	11	n	n	NOUN
ap-961	217	1	i	i	PRON
ap-961	217	2	i	i	PRON
ap-961	217	3	n	n	VERB
ap-961	217	4	i	i	NOUN
ap-961	217	5	1	1	NUM
ap-961	217	6	1	1	NUM
ap-961	217	7	2	2	NUM
ap-961	217	8	1	1	NUM
ap-961	217	9	21	21	NUM
ap-961	217	10	�	�	PROPN
ap-961	217	11	�	�	PROPN
ap-961	217	12	�	�	PROPN
ap-961	217	13	�	�	PROPN
ap-961	217	14	�	�	PROPN
ap-961	217	15	�	�	PROPN
ap-961	217	16	,	,	PUNCT
ap-961	217	17	where	where	SCONJ
ap-961	217	18	�	�	PROPN
ap-961	217	19	xi	xi	PROPN
ap-961	217	20	denotes	denote	NOUN
ap-961	217	21	that	that	SCONJ
ap-961	217	22	we	we	PRON
ap-961	217	23	just	just	ADV
ap-961	217	24	omit	omit	VERB
ap-961	217	25	the	the	DET
ap-961	217	26	i	i	PROPN
ap-961	217	27	-	-	PUNCT
ap-961	217	28	th	th	VERB
ap-961	217	29	variable	variable	NOUN
ap-961	217	30	.	.	PUNCT
ap-961	218	1	exercise	exercise	VERB
ap-961	218	2	3.8	3.8	NUM
ap-961	218	3	:	:	PUNCT
ap-961	218	4	prove	prove	VERB
ap-961	218	5	that	that	SCONJ
ap-961	218	6	d	d	NOUN
ap-961	218	7	satisfies	satisfy	VERB
ap-961	218	8	the	the	DET
ap-961	218	9	graded	grade	VERB
ap-961	218	10	leibniz	leibniz	PROPN
ap-961	218	11	rule	rule	NOUN
ap-961	218	12	and	and	CCONJ
ap-961	218	13	is	be	AUX
ap-961	218	14	nilpotent	nilpotent	ADJ
ap-961	218	15	,	,	PUNCT
ap-961	218	16	du	du	PROPN
ap-961	218	17	2	2	NUM
ap-961	218	18	0	0	NUM
ap-961	218	19	�	�	PROPN
ap-961	218	20	.	.	PUNCT
ap-961	219	1	if	if	SCONJ
ap-961	219	2	x	x	PRON
ap-961	219	3	is	be	AUX
ap-961	219	4	a	a	DET
ap-961	219	5	space	space	NOUN
ap-961	219	6	consisting	consist	VERB
ap-961	219	7	of	of	ADP
ap-961	219	8	a	a	DET
ap-961	219	9	finite	finite	ADJ
ap-961	219	10	number	number	NOUN
ap-961	219	11	of	of	ADP
ap-961	219	12	points	point	NOUN
ap-961	219	13	,	,	PUNCT
ap-961	219	14	can	can	AUX
ap-961	219	15	we	we	PRON
ap-961	219	16	identify	identify	VERB
ap-961	219	17	the	the	DET
ap-961	219	18	differential	differential	ADJ
ap-961	219	19	algebra	algebra	NOUN
ap-961	219	20	(	(	PUNCT
ap-961	219	21	)	)	PUNCT
ap-961	219	22	x	x	X
ap-961	219	23	?	?	PUNCT
ap-961	220	1	note	note	VERB
ap-961	220	2	that	that	SCONJ
ap-961	220	3	the	the	DET
ap-961	220	4	differential	differential	NOUN
ap-961	220	5	graded	grade	VERB
ap-961	220	6	algebra	algebra	NOUN
ap-961	220	7	(	(	PUNCT
ap-961	220	8	)	)	PUNCT
ap-961	220	9	x	x	X
ap-961	220	10	is	be	AUX
ap-961	220	11	in	in	ADP
ap-961	220	12	general	general	ADJ
ap-961	220	13	not	not	PART
ap-961	220	14	a	a	DET
ap-961	220	15	proper	proper	ADJ
ap-961	220	16	one	one	NOUN
ap-961	220	17	.	.	PUNCT
ap-961	221	1	this	this	PRON
ap-961	221	2	depends	depend	VERB
ap-961	221	3	strongly	strongly	ADV
ap-961	221	4	on	on	ADP
ap-961	221	5	the	the	DET
ap-961	221	6	class	class	NOUN
ap-961	221	7	of	of	ADP
ap-961	221	8	functions	function	NOUN
ap-961	221	9	that	that	PRON
ap-961	221	10	we	we	PRON
ap-961	221	11	consider	consider	VERB
ap-961	221	12	,	,	PUNCT
ap-961	221	13	as	as	SCONJ
ap-961	221	14	we	we	PRON
ap-961	221	15	can	can	AUX
ap-961	221	16	easily	easily	ADV
ap-961	221	17	see	see	VERB
ap-961	221	18	:	:	PUNCT
ap-961	221	19	exercise	exercise	VERB
ap-961	221	20	3.9	3.9	NUM
ap-961	221	21	:	:	PUNCT
ap-961	221	22	assume	assume	VERB
ap-961	221	23	that	that	SCONJ
ap-961	221	24	x	x	PRON
ap-961	221	25	is	be	AUX
ap-961	221	26	the	the	DET
ap-961	221	27	interval	interval	NOUN
ap-961	222	1	i	i	PRON
ap-961	222	2	and	and	CCONJ
ap-961	222	3	k	k	NOUN
ap-961	222	4	x	x	X
ap-961	222	5	(	(	PUNCT
ap-961	222	6	)	)	PUNCT
ap-961	222	7	are	be	AUX
ap-961	222	8	polynomials	polynomial	NOUN
ap-961	222	9	(	(	PUNCT
ap-961	222	10	of	of	ADP
ap-961	222	11	variables	variable	NOUN
ap-961	222	12	)	)	PUNCT
ap-961	222	13	understood	understand	VERB
ap-961	222	14	as	as	SCONJ
ap-961	222	15	functions	function	NOUN
ap-961	222	16	on	on	ADP
ap-961	222	17	i	i	PROPN
ap-961	222	18	k.	k.	PROPN
ap-961	222	19	show	show	VERB
ap-961	222	20	that	that	SCONJ
ap-961	222	21	the	the	DET
ap-961	222	22	differential	differential	NOUN
ap-961	222	23	graded	grade	VERB
ap-961	222	24	algebra	algebra	NOUN
ap-961	222	25	is	be	AUX
ap-961	222	26	a	a	DET
ap-961	222	27	proper	proper	ADJ
ap-961	222	28	one	one	NOUN
ap-961	222	29	in	in	ADP
ap-961	222	30	this	this	DET
ap-961	222	31	case	case	NOUN
ap-961	222	32	(	(	PUNCT
ap-961	222	33	that	that	PRON
ap-961	222	34	is	is	ADV
ap-961	222	35	,	,	PUNCT
ap-961	222	36	for	for	SCONJ
ap-961	222	37	each	each	DET
ap-961	222	38	k	k	PROPN
ap-961	222	39	k	k	PROPN
ap-961	222	40	x	x	X
ap-961	222	41	(	(	PUNCT
ap-961	222	42	)	)	PUNCT
ap-961	222	43	is	be	AUX
ap-961	222	44	generated	generate	VERB
ap-961	222	45	by	by	ADP
ap-961	222	46	the	the	DET
ap-961	222	47	image	image	NOUN
ap-961	222	48	of	of	ADP
ap-961	222	49	d	d	PROPN
ap-961	222	50	)	)	PUNCT
ap-961	222	51	.	.	PUNCT
ap-961	223	1	what	what	PRON
ap-961	223	2	happens	happen	VERB
ap-961	223	3	if	if	SCONJ
ap-961	223	4	we	we	PRON
ap-961	223	5	take	take	VERB
ap-961	223	6	k	k	PROPN
ap-961	223	7	x	x	X
ap-961	223	8	(	(	PUNCT
ap-961	223	9	)	)	PUNCT
ap-961	223	10	to	to	PART
ap-961	223	11	be	be	AUX
ap-961	223	12	all	all	DET
ap-961	223	13	continuous	continuous	ADJ
ap-961	223	14	functions	function	NOUN
ap-961	223	15	on	on	ADP
ap-961	223	16	ik	ik	NOUN
ap-961	223	17	?	?	PUNCT
ap-961	224	1	although	although	SCONJ
ap-961	224	2	(	(	PUNCT
ap-961	224	3	)	)	PUNCT
ap-961	224	4	x	x	X
ap-961	224	5	might	might	AUX
ap-961	224	6	not	not	PART
ap-961	224	7	be	be	AUX
ap-961	224	8	proper	proper	ADJ
ap-961	224	9	,	,	PUNCT
ap-961	224	10	it	it	PRON
ap-961	224	11	appears	appear	VERB
ap-961	224	12	to	to	PART
ap-961	224	13	be	be	AUX
ap-961	224	14	a	a	DET
ap-961	224	15	suitable	suitable	ADJ
ap-961	224	16	generalization	generalization	NOUN
ap-961	224	17	of	of	ADP
ap-961	224	18	the	the	DET
ap-961	224	19	universal	universal	ADJ
ap-961	224	20	differential	differential	NOUN
ap-961	224	21	graded	grade	VERB
ap-961	224	22	algebra	algebra	NOUN
ap-961	224	23	,	,	PUNCT
ap-961	224	24	especially	especially	ADV
ap-961	224	25	in	in	ADP
ap-961	224	26	the	the	DET
ap-961	224	27	case	case	NOUN
ap-961	224	28	,	,	PUNCT
ap-961	224	29	when	when	SCONJ
ap-961	224	30	the	the	DET
ap-961	224	31	algebraic	algebraic	ADJ
ap-961	224	32	tensor	tensor	NOUN
ap-961	224	33	products	product	NOUN
ap-961	224	34	might	might	AUX
ap-961	224	35	not	not	PART
ap-961	224	36	give	give	VERB
ap-961	224	37	us	we	PRON
ap-961	224	38	all	all	DET
ap-961	224	39	elements	element	NOUN
ap-961	224	40	to	to	PART
ap-961	224	41	play	play	VERB
ap-961	224	42	with	with	ADP
ap-961	224	43	(	(	PUNCT
ap-961	224	44	as	as	SCONJ
ap-961	224	45	is	be	AUX
ap-961	224	46	the	the	DET
ap-961	224	47	case	case	NOUN
ap-961	224	48	with	with	ADP
ap-961	224	49	smooth	smooth	ADJ
ap-961	224	50	or	or	CCONJ
ap-961	224	51	continuous	continuous	ADJ
ap-961	224	52	functions	function	NOUN
ap-961	224	53	)	)	PUNCT
ap-961	224	54	.	.	PUNCT
ap-961	225	1	example	example	NOUN
ap-961	226	1	3.10	3.10	NUM
ap-961	226	2	:	:	PUNCT
ap-961	226	3	an	an	DET
ap-961	226	4	interesting	interesting	ADJ
ap-961	226	5	question	question	NOUN
ap-961	226	6	is	be	AUX
ap-961	226	7	whether	whether	SCONJ
ap-961	226	8	we	we	PRON
ap-961	226	9	can	can	AUX
ap-961	226	10	identify	identify	VERB
ap-961	226	11	the	the	DET
ap-961	226	12	standard	standard	ADJ
ap-961	226	13	de	de	PROPN
ap-961	226	14	rham	rham	PROPN
ap-961	226	15	differential	differential	PROPN
ap-961	226	16	graded	grade	VERB
ap-961	226	17	algebra	algebra	NOUN
ap-961	226	18	using	use	VERB
ap-961	226	19	the	the	DET
ap-961	226	20	universal	universal	ADJ
ap-961	226	21	calculus	calculus	NOUN
ap-961	226	22	and	and	CCONJ
ap-961	226	23	choosing	choose	VERB
ap-961	226	24	a	a	DET
ap-961	226	25	suitable	suitable	ADJ
ap-961	226	26	quotient	quotient	NOUN
ap-961	226	27	.	.	PUNCT
ap-961	227	1	the	the	DET
ap-961	227	2	answer	answer	NOUN
ap-961	227	3	is	be	AUX
ap-961	227	4	positive	positive	ADJ
ap-961	227	5	but	but	CCONJ
ap-961	227	6	again	again	ADV
ap-961	227	7	we	we	PRON
ap-961	227	8	need	need	VERB
ap-961	227	9	to	to	PART
ap-961	227	10	be	be	AUX
ap-961	227	11	careful	careful	ADJ
ap-961	228	1	about	about	ADP
ap-961	228	2	40	40	NUM
ap-961	228	3	©	©	PROPN
ap-961	228	4	czech	czech	PROPN
ap-961	228	5	technical	technical	PROPN
ap-961	228	6	university	university	PROPN
ap-961	228	7	publishing	publishing	NOUN
ap-961	228	8	house	house	NOUN
ap-961	228	9	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	228	10	acta	acta	PROPN
ap-961	228	11	polytechnica	polytechnica	PROPN
ap-961	228	12	vol	vol	NOUN
ap-961	228	13	.	.	PUNCT
ap-961	229	1	48	48	NUM
ap-961	229	2	no	no	NOUN
ap-961	229	3	.	.	PUNCT
ap-961	230	1	2/2008	2/2008	PROPN
ap-961	230	2	the	the	DET
ap-961	230	3	tensor	tensor	NOUN
ap-961	230	4	products	product	NOUN
ap-961	230	5	.	.	PUNCT
ap-961	231	1	the	the	DET
ap-961	231	2	construction	construction	NOUN
ap-961	231	3	should	should	AUX
ap-961	231	4	be	be	AUX
ap-961	231	5	similar	similar	ADJ
ap-961	231	6	as	as	ADP
ap-961	231	7	in	in	ADP
ap-961	231	8	example	example	NOUN
ap-961	231	9	3.7	3.7	NUM
ap-961	231	10	with	with	ADP
ap-961	231	11	the	the	DET
ap-961	231	12	algebra	algebra	NOUN
ap-961	231	13	of	of	ADP
ap-961	231	14	smooth	smooth	ADJ
ap-961	231	15	functions	function	NOUN
ap-961	231	16	over	over	ADP
ap-961	231	17	x.	x.	NOUN
ap-961	231	18	the	the	DET
ap-961	231	19	de	de	PROPN
ap-961	231	20	rham	rham	PROPN
ap-961	231	21	differential	differential	PROPN
ap-961	231	22	calculus	calculus	NOUN
ap-961	231	23	is	be	AUX
ap-961	231	24	obtained	obtain	VERB
ap-961	231	25	if	if	SCONJ
ap-961	231	26	we	we	PRON
ap-961	231	27	take	take	VERB
ap-961	231	28	the	the	DET
ap-961	231	29	quotient	quotient	NOUN
ap-961	231	30	of	of	ADP
ap-961	231	31	this	this	DET
ap-961	231	32	differential	differential	NOUN
ap-961	231	33	graded	grade	VERB
ap-961	231	34	algebra	algebra	NOUN
ap-961	231	35	s	s	PART
ap-961	231	36	x	x	NOUN
ap-961	231	37	(	(	PUNCT
ap-961	231	38	)	)	PUNCT
ap-961	231	39	by	by	ADP
ap-961	231	40	the	the	DET
ap-961	231	41	ideal	ideal	NOUN
ap-961	231	42	described	describe	VERB
ap-961	231	43	as	as	ADP
ap-961	231	44	follows	follow	VERB
ap-961	231	45	.	.	PUNCT
ap-961	232	1	let	let	VERB
ap-961	232	2	�	�	PROPN
ap-961	232	3	be	be	AUX
ap-961	232	4	the	the	DET
ap-961	232	5	space	space	NOUN
ap-961	232	6	of	of	ADP
ap-961	232	7	all	all	DET
ap-961	232	8	functions	function	NOUN
ap-961	232	9	on	on	ADP
ap-961	232	10	x2	x2	INTJ
ap-961	232	11	such	such	ADJ
ap-961	232	12	that	that	SCONJ
ap-961	232	13	:	:	PUNCT
ap-961	232	14	lim	lim	PROPN
ap-961	232	15	(	(	PUNCT
ap-961	232	16	,	,	PUNCT
ap-961	232	17	)	)	PUNCT
ap-961	232	18	x	x	PUNCT
ap-961	233	1	y	y	NOUN
ap-961	233	2	f	f	NOUN
ap-961	233	3	x	x	X
ap-961	233	4	y	y	PROPN
ap-961	233	5	x	x	PROPN
ap-961	233	6	y	y	PROPN
ap-961	233	7	�	�	PROPN
ap-961	233	8	�	�	PROPN
ap-961	233	9	�	�	PROPN
ap-961	233	10	0	0	NUM
ap-961	233	11	,	,	PUNCT
ap-961	233	12	x	x	PROPN
ap-961	233	13	y	y	PROPN
ap-961	233	14	x	x	PROPN
ap-961	233	15	,	,	PUNCT
ap-961	233	16	�	�	PROPN
ap-961	233	17	.	.	PUNCT
ap-961	234	1	it	it	PRON
ap-961	234	2	is	be	AUX
ap-961	234	3	not	not	PART
ap-961	234	4	difficult	difficult	ADJ
ap-961	234	5	to	to	PART
ap-961	234	6	observe	observe	VERB
ap-961	234	7	that	that	DET
ap-961	234	8	multiplication	multiplication	NOUN
ap-961	234	9	by	by	ADP
ap-961	234	10	smooth	smooth	ADJ
ap-961	234	11	functions	function	NOUN
ap-961	234	12	from	from	ADP
ap-961	234	13	both	both	DET
ap-961	234	14	sides	side	NOUN
ap-961	234	15	does	do	AUX
ap-961	234	16	not	not	PART
ap-961	234	17	lead	lead	VERB
ap-961	234	18	outside	outside	ADP
ap-961	234	19	�	�	PROPN
ap-961	234	20	.	.	PUNCT
ap-961	235	1	taking	take	VERB
ap-961	235	2	the	the	DET
ap-961	235	3	quotient	quotient	NOUN
ap-961	235	4	s	s	PROPN
ap-961	235	5	i1	i1	PROPN
ap-961	235	6	(	(	PUNCT
ap-961	235	7	)	)	PUNCT
ap-961	235	8	�	�	PROPN
ap-961	235	9	we	we	PRON
ap-961	235	10	first	first	ADV
ap-961	235	11	obtain	obtain	VERB
ap-961	235	12	all	all	PRON
ap-961	235	13	de	de	ADP
ap-961	235	14	rham	rham	PROPN
ap-961	235	15	one	one	NUM
ap-961	235	16	-	-	PUNCT
ap-961	235	17	forms	form	NOUN
ap-961	235	18	on	on	ADP
ap-961	235	19	x.	x.	NOUN
ap-961	235	20	the	the	DET
ap-961	235	21	construction	construction	NOUN
ap-961	235	22	of	of	ADP
ap-961	235	23	higher	high	ADJ
ap-961	235	24	order	order	NOUN
ap-961	235	25	forms	form	NOUN
ap-961	235	26	is	be	AUX
ap-961	235	27	then	then	ADV
ap-961	235	28	a	a	DET
ap-961	235	29	formality	formality	NOUN
ap-961	235	30	.	.	PUNCT
ap-961	236	1	we	we	PRON
ap-961	236	2	already	already	ADV
ap-961	236	3	know	know	VERB
ap-961	236	4	that	that	SCONJ
ap-961	236	5	there	there	PRON
ap-961	236	6	are	be	VERB
ap-961	236	7	many	many	ADJ
ap-961	236	8	noncommutative	noncommutative	ADJ
ap-961	236	9	differential	differential	ADJ
ap-961	236	10	calculi	calculi	NOUN
ap-961	236	11	over	over	ADP
ap-961	236	12	one	one	NUM
ap-961	236	13	algebra	algebra	NOUN
ap-961	236	14	.	.	PUNCT
ap-961	237	1	the	the	DET
ap-961	237	2	universal	universal	ADJ
ap-961	237	3	one	one	NOUN
ap-961	237	4	,	,	PUNCT
ap-961	237	5	apart	apart	ADV
ap-961	237	6	from	from	ADP
ap-961	237	7	its	its	PRON
ap-961	237	8	universality	universality	NOUN
ap-961	237	9	property	property	NOUN
ap-961	237	10	,	,	PUNCT
ap-961	237	11	is	be	AUX
ap-961	237	12	not	not	PART
ap-961	237	13	actually	actually	ADV
ap-961	237	14	very	very	ADV
ap-961	237	15	interesting	interesting	ADJ
ap-961	237	16	.	.	PUNCT
ap-961	238	1	it	it	PRON
ap-961	238	2	carries	carry	VERB
ap-961	238	3	very	very	ADV
ap-961	238	4	little	little	ADJ
ap-961	238	5	information	information	NOUN
ap-961	238	6	about	about	ADP
ap-961	238	7	the	the	DET
ap-961	238	8	“	"	PUNCT
ap-961	238	9	space	space	NOUN
ap-961	238	10	”	"	PUNCT
ap-961	238	11	underneath	underneath	NOUN
ap-961	238	12	.	.	PUNCT
ap-961	239	1	the	the	DET
ap-961	239	2	interesting	interesting	ADJ
ap-961	239	3	examples	example	NOUN
ap-961	239	4	are	be	AUX
ap-961	239	5	“	"	PUNCT
ap-961	239	6	smaller	small	ADJ
ap-961	239	7	”	"	PUNCT
ap-961	239	8	calculi	calculi	NOUN
ap-961	239	9	.	.	PUNCT
ap-961	240	1	actually	actually	ADV
ap-961	240	2	,	,	PUNCT
ap-961	240	3	even	even	ADV
ap-961	240	4	in	in	ADP
ap-961	240	5	the	the	DET
ap-961	240	6	commutative	commutative	ADJ
ap-961	240	7	case	case	NOUN
ap-961	240	8	we	we	PRON
ap-961	240	9	might	might	AUX
ap-961	240	10	have	have	VERB
ap-961	240	11	many	many	ADJ
ap-961	240	12	differential	differential	ADJ
ap-961	240	13	graded	grade	VERB
ap-961	240	14	algebras	algebra	NOUN
ap-961	240	15	,	,	PUNCT
ap-961	240	16	which	which	PRON
ap-961	240	17	are	be	AUX
ap-961	240	18	neither	neither	CCONJ
ap-961	240	19	de	de	PROPN
ap-961	240	20	rham	rham	PROPN
ap-961	240	21	,	,	PUNCT
ap-961	240	22	nor	nor	CCONJ
ap-961	240	23	universal	universal	ADJ
ap-961	240	24	.	.	PUNCT
ap-961	241	1	see	see	VERB
ap-961	241	2	the	the	DET
ap-961	241	3	following	follow	VERB
ap-961	241	4	example	example	NOUN
ap-961	242	1	:	:	PUNCT
ap-961	242	2	example	example	NOUN
ap-961	242	3	3.11	3.11	NUM
ap-961	242	4	:	:	PUNCT
ap-961	242	5	consider	consider	VERB
ap-961	242	6	an	an	DET
ap-961	242	7	algebra	algebra	NOUN
ap-961	242	8	of	of	ADP
ap-961	242	9	smooth	smooth	ADJ
ap-961	242	10	functions	function	NOUN
ap-961	242	11	on	on	ADP
ap-961	242	12	the	the	DET
ap-961	242	13	circle	circle	NOUN
ap-961	242	14	and	and	CCONJ
ap-961	242	15	the	the	DET
ap-961	242	16	following	follow	VERB
ap-961	242	17	differential	differential	NOUN
ap-961	242	18	graded	grade	VERB
ap-961	242	19	algebra	algebra	NOUN
ap-961	242	20	.	.	PUNCT
ap-961	243	1	the	the	DET
ap-961	243	2	zero	zero	NUM
ap-961	243	3	-	-	PUNCT
ap-961	243	4	forms	form	NOUN
ap-961	243	5	are	be	AUX
ap-961	243	6	the	the	DET
ap-961	243	7	smooth	smooth	ADJ
ap-961	243	8	functions	function	NOUN
ap-961	243	9	,	,	PUNCT
ap-961	243	10	k	k	PROPN
ap-961	243	11	s	s	PROPN
ap-961	243	12	c	c	NOUN
ap-961	243	13	s	s	X
ap-961	243	14	(	(	PUNCT
ap-961	243	15	)	)	PUNCT
ap-961	243	16	(	(	PUNCT
ap-961	243	17	)	)	SYM
ap-961	243	18	1	1	NUM
ap-961	243	19	1	1	NUM
ap-961	243	20	�	�	PROPN
ap-961	243	21	.	.	PUNCT
ap-961	244	1	the	the	DET
ap-961	244	2	bimodule	bimodule	NOUN
ap-961	244	3	of	of	ADP
ap-961	244	4	one	one	NUM
ap-961	244	5	-	-	PUNCT
ap-961	244	6	forms	form	NOUN
ap-961	244	7	and	and	CCONJ
ap-961	244	8	the	the	DET
ap-961	244	9	entire	entire	ADJ
ap-961	244	10	differential	differential	NOUN
ap-961	244	11	algebra	algebra	NOUN
ap-961	244	12	is	be	AUX
ap-961	244	13	generated	generate	VERB
ap-961	244	14	by	by	ADP
ap-961	244	15	two	two	NUM
ap-961	244	16	one	one	NUM
ap-961	244	17	-	-	PUNCT
ap-961	244	18	forms	form	NOUN
ap-961	244	19	,	,	PUNCT
ap-961	244	20	�	�	PROPN
ap-961	244	21	and	and	CCONJ
ap-961	244	22	�	�	PROPN
ap-961	244	23	.	.	PUNCT
ap-961	245	1	the	the	DET
ap-961	245	2	following	follow	VERB
ap-961	245	3	gives	give	VERB
ap-961	245	4	the	the	DET
ap-961	245	5	algebra	algebra	NOUN
ap-961	245	6	rules	rule	NOUN
ap-961	245	7	and	and	CCONJ
ap-961	245	8	the	the	DET
ap-961	245	9	action	action	NOUN
ap-961	245	10	of	of	ADP
ap-961	245	11	the	the	DET
ap-961	245	12	external	external	ADJ
ap-961	245	13	derivative	derivative	NOUN
ap-961	245	14	:	:	PUNCT
ap-961	246	1	d	d	X
ap-961	246	2	f	f	X
ap-961	246	3	f	f	X
ap-961	247	1	f	f	PROPN
ap-961	248	1	d	d	X
ap-961	248	2	f	f	X
ap-961	249	1	f	f	X
ap-961	249	2	f	f	PROPN
ap-961	250	1	d	d	X
ap-961	250	2	f	f	PROPN
ap-961	250	3	f	f	PROPN
ap-961	250	4	�	�	PROPN
ap-961	250	5	�	�	PROPN
ap-961	250	6	�	�	PROPN
ap-961	250	7	�	�	PROPN
ap-961	250	8	#	#	NOUN
ap-961	250	9	�	�	PROPN
ap-961	250	10	#	#	NOUN
ap-961	250	11	(	(	PUNCT
ap-961	250	12	)	)	PUNCT
ap-961	250	13	(	(	PUNCT
ap-961	250	14	)	)	PUNCT
ap-961	250	15	,	,	PUNCT
ap-961	250	16	,	,	PUNCT
ap-961	250	17	(	(	PUNCT
ap-961	250	18	)	)	PUNCT
ap-961	250	19	,	,	PUNCT
ap-961	250	20	,	,	PUNCT
ap-961	250	21	,	,	PUNCT
ap-961	250	22	�	�	PROPN
ap-961	250	23	�	�	PROPN
ap-961	250	24	�	�	PROPN
ap-961	250	25	�	�	PROPN
ap-961	250	26	�	�	PROPN
ap-961	250	27	�	�	PROPN
ap-961	250	28	�	�	PROPN
ap-961	250	29	�	�	PROPN
ap-961	250	30	�	�	PROPN
ap-961	250	31	�	�	PROPN
ap-961	250	32	�	�	PROPN
ap-961	250	33	�	�	PROPN
ap-961	250	34	�	�	PROPN
ap-961	250	35	2	2	NUM
ap-961	250	36	0	0	NUM
ap-961	250	37	2	2	NUM
ap-961	250	38	�	�	PROPN
ap-961	250	39	�	�	PROPN
ap-961	250	40	#	#	NOUN
ap-961	250	41	#	#	NOUN
ap-961	250	42	�	�	PROPN
ap-961	250	43	�	�	PROPN
ap-961	250	44	�	�	PROPN
ap-961	250	45	�	�	PROPN
ap-961	250	46	�	�	PROPN
ap-961	250	47	,	,	PUNCT
ap-961	250	48	,	,	PUNCT
ap-961	250	49	0	0	NUM
ap-961	250	50	where	where	SCONJ
ap-961	250	51	denotes	denote	VERB
ap-961	250	52	the	the	DET
ap-961	250	53	standard	standard	ADJ
ap-961	250	54	derivative	derivative	NOUN
ap-961	250	55	of	of	ADP
ap-961	250	56	the	the	DET
ap-961	250	57	function	function	NOUN
ap-961	250	58	on	on	ADP
ap-961	250	59	the	the	DET
ap-961	250	60	circle	circle	NOUN
ap-961	250	61	.	.	PUNCT
ap-961	251	1	note	note	VERB
ap-961	251	2	that	that	SCONJ
ap-961	251	3	a	a	DET
ap-961	251	4	priori	priori	ADJ
ap-961	251	5	the	the	DET
ap-961	251	6	differential	differential	ADJ
ap-961	251	7	algebra	algebra	NOUN
ap-961	251	8	is	be	AUX
ap-961	251	9	infinite	infinite	ADJ
ap-961	251	10	-	-	PUNCT
ap-961	251	11	dimensional	dimensional	ADJ
ap-961	251	12	,	,	PUNCT
ap-961	251	13	as	as	SCONJ
ap-961	251	14	we	we	PRON
ap-961	251	15	can	can	AUX
ap-961	251	16	construct	construct	VERB
ap-961	251	17	product	product	NOUN
ap-961	251	18	of	of	ADP
ap-961	251	19	arbitrary	arbitrary	ADJ
ap-961	251	20	numbers	number	NOUN
ap-961	251	21	of	of	ADP
ap-961	251	22	�	�	PROPN
ap-961	251	23	.	.	PUNCT
ap-961	252	1	however	however	ADV
ap-961	252	2	,	,	PUNCT
ap-961	252	3	since	since	SCONJ
ap-961	252	4	�	�	PROPN
ap-961	252	5	�	�	PROPN
ap-961	252	6	#	#	NOUN
ap-961	252	7	generates	generate	VERB
ap-961	252	8	a	a	DET
ap-961	252	9	differential	differential	ADJ
ap-961	252	10	ideal	ideal	NOUN
ap-961	252	11	,	,	PUNCT
ap-961	252	12	we	we	PRON
ap-961	252	13	might	might	AUX
ap-961	252	14	take	take	VERB
ap-961	252	15	a	a	DET
ap-961	252	16	quotient	quotient	NOUN
ap-961	252	17	and	and	CCONJ
ap-961	252	18	therefore	therefore	ADV
ap-961	252	19	set	set	VERB
ap-961	252	20	�	�	PROPN
ap-961	252	21	�	�	PROPN
ap-961	252	22	#	#	NOUN
ap-961	252	23	�	�	PROPN
ap-961	252	24	0	0	NUM
ap-961	252	25	.	.	PUNCT
ap-961	253	1	a	a	DET
ap-961	253	2	point	point	NOUN
ap-961	253	3	worth	worth	ADJ
ap-961	253	4	mentioning	mention	VERB
ap-961	253	5	is	be	AUX
ap-961	253	6	that	that	SCONJ
ap-961	253	7	although	although	SCONJ
ap-961	253	8	the	the	DET
ap-961	253	9	algebra	algebra	NOUN
ap-961	253	10	is	be	AUX
ap-961	253	11	commutative	commutative	ADJ
ap-961	253	12	the	the	DET
ap-961	253	13	differential	differential	ADJ
ap-961	253	14	forms	form	NOUN
ap-961	253	15	(	(	PUNCT
ap-961	253	16	which	which	PRON
ap-961	253	17	are	be	AUX
ap-961	253	18	not	not	PART
ap-961	253	19	universal	universal	ADJ
ap-961	253	20	)	)	PUNCT
ap-961	253	21	do	do	AUX
ap-961	253	22	not	not	PART
ap-961	253	23	commute	commute	VERB
ap-961	253	24	with	with	ADP
ap-961	253	25	functions	function	NOUN
ap-961	253	26	!	!	PUNCT
ap-961	254	1	3.2	3.2	NUM
ap-961	254	2	involution	involution	NOUN
ap-961	254	3	,	,	PUNCT
ap-961	254	4	tensor	tensor	NOUN
ap-961	254	5	products	product	NOUN
ap-961	254	6	and	and	CCONJ
ap-961	254	7	representations	representation	NOUN
ap-961	254	8	3.2.1	3.2.1	NUM
ap-961	254	9	involution	involution	NOUN
ap-961	254	10	if	if	SCONJ
ap-961	254	11	the	the	DET
ap-961	254	12	algebra	algebra	NOUN
ap-961	254	13	�	�	PROPN
ap-961	254	14	is	be	AUX
ap-961	254	15	equipped	equip	VERB
ap-961	254	16	with	with	ADP
ap-961	254	17	an	an	DET
ap-961	254	18	involution	involution	NOUN
ap-961	254	19	,	,	PUNCT
ap-961	254	20	we	we	PRON
ap-961	254	21	would	would	AUX
ap-961	254	22	like	like	VERB
ap-961	254	23	to	to	PART
ap-961	254	24	have	have	VERB
ap-961	254	25	this	this	DET
ap-961	254	26	operation	operation	NOUN
ap-961	254	27	extended	extend	VERB
ap-961	254	28	onto	onto	ADP
ap-961	254	29	the	the	DET
ap-961	254	30	differential	differential	ADJ
ap-961	254	31	algebra	algebra	NOUN
ap-961	254	32	.	.	PUNCT
ap-961	255	1	this	this	PRON
ap-961	255	2	is	be	AUX
ap-961	255	3	by	by	ADP
ap-961	255	4	no	no	DET
ap-961	255	5	means	means	NOUN
ap-961	255	6	a	a	DET
ap-961	255	7	problem	problem	NOUN
ap-961	255	8	for	for	ADP
ap-961	255	9	the	the	DET
ap-961	255	10	universal	universal	ADJ
ap-961	255	11	calculus	calculus	NOUN
ap-961	255	12	,	,	PUNCT
ap-961	255	13	as	as	ADP
ap-961	255	14	on	on	ADP
ap-961	255	15	the	the	DET
ap-961	255	16	zero	zero	NUM
ap-961	255	17	-	-	PUNCT
ap-961	255	18	forms	form	NOUN
ap-961	255	19	we	we	PRON
ap-961	255	20	take	take	VERB
ap-961	255	21	the	the	DET
ap-961	255	22	assumed	assumed	ADJ
ap-961	255	23	involution	involution	NOUN
ap-961	255	24	on	on	ADP
ap-961	255	25	the	the	DET
ap-961	255	26	algebra	algebra	NOUN
ap-961	255	27	and	and	CCONJ
ap-961	255	28	for	for	ADP
ap-961	255	29	higher	high	ADJ
ap-961	255	30	-	-	PUNCT
ap-961	255	31	order	order	NOUN
ap-961	255	32	universal	universal	ADJ
ap-961	255	33	forms	form	NOUN
ap-961	255	34	we	we	PRON
ap-961	255	35	set	set	VERB
ap-961	255	36	:	:	PUNCT
ap-961	255	37	(	(	PUNCT
ap-961	255	38	)	)	PUNCT
ap-961	255	39	(	(	PUNCT
ap-961	255	40	)	)	PUNCT
ap-961	255	41	*	*	PUNCT
ap-961	256	1	*	*	PUNCT
ap-961	256	2	da	da	PROPN
ap-961	256	3	d	d	X
ap-961	256	4	a	a	DET
ap-961	256	5	�	�	PROPN
ap-961	256	6	�	�	PROPN
ap-961	256	7	.	.	PUNCT
ap-961	257	1	so	so	ADV
ap-961	257	2	the	the	DET
ap-961	257	3	problem	problem	NOUN
ap-961	257	4	,	,	PUNCT
ap-961	257	5	when	when	SCONJ
ap-961	257	6	reduced	reduce	VERB
ap-961	257	7	to	to	ADP
ap-961	257	8	the	the	DET
ap-961	257	9	universal	universal	ADJ
ap-961	257	10	case	case	NOUN
ap-961	257	11	,	,	PUNCT
ap-961	257	12	is	be	AUX
ap-961	257	13	trivial	trivial	ADJ
ap-961	257	14	and	and	CCONJ
ap-961	257	15	the	the	DET
ap-961	257	16	only	only	ADJ
ap-961	257	17	thing	thing	NOUN
ap-961	257	18	we	we	PRON
ap-961	257	19	need	need	VERB
ap-961	257	20	to	to	PART
ap-961	257	21	take	take	VERB
ap-961	257	22	care	care	NOUN
ap-961	257	23	of	of	ADP
ap-961	257	24	is	be	AUX
ap-961	257	25	that	that	SCONJ
ap-961	257	26	the	the	DET
ap-961	257	27	differential	differential	ADJ
ap-961	257	28	ideal	ideal	PROPN
ap-961	257	29	�	�	PROPN
ap-961	257	30	�	�	PROPN
ap-961	257	31	�	�	PROPN
ap-961	257	32	u	u	PROPN
ap-961	257	33	(	(	PUNCT
ap-961	257	34	)	)	PUNCT
ap-961	257	35	is	be	AUX
ap-961	257	36	involution	involution	NOUN
ap-961	257	37	invariant	invariant	PROPN
ap-961	257	38	�	�	PROPN
ap-961	257	39	�	�	PROPN
ap-961	257	40	*	*	PROPN
ap-961	257	41	�	�	PROPN
ap-961	257	42	.	.	PUNCT
ap-961	258	1	example	example	NOUN
ap-961	259	1	3.12	3.12	NUM
ap-961	259	2	:	:	PUNCT
ap-961	259	3	on	on	ADP
ap-961	259	4	the	the	DET
ap-961	259	5	algebras	algebra	NOUN
ap-961	259	6	of	of	ADP
ap-961	259	7	complex	complex	ADJ
ap-961	259	8	functions	function	NOUN
ap-961	259	9	we	we	PRON
ap-961	259	10	have	have	VERB
ap-961	259	11	a	a	DET
ap-961	259	12	natural	natural	ADJ
ap-961	259	13	involution	involution	NOUN
ap-961	259	14	–	–	PUNCT
ap-961	259	15	complex	complex	ADJ
ap-961	259	16	conjugation	conjugation	NOUN
ap-961	259	17	.	.	PUNCT
ap-961	260	1	taking	take	VERB
ap-961	260	2	the	the	DET
ap-961	260	3	trivial	trivial	ADJ
ap-961	260	4	(	(	PUNCT
ap-961	260	5	but	but	CCONJ
ap-961	260	6	still	still	ADV
ap-961	260	7	interesting	interesting	ADJ
ap-961	260	8	)	)	PUNCT
ap-961	260	9	example	example	NOUN
ap-961	260	10	of	of	ADP
ap-961	260	11	two	two	NUM
ap-961	260	12	-	-	PUNCT
ap-961	260	13	point	point	NOUN
ap-961	260	14	geometry	geometry	NOUN
ap-961	260	15	,	,	PUNCT
ap-961	260	16	e	e	X
ap-961	260	17	e	e	PROPN
ap-961	260	18	*	*	PROPN
ap-961	260	19	�	�	PROPN
ap-961	260	20	,	,	PUNCT
ap-961	260	21	we	we	PRON
ap-961	260	22	have	have	VERB
ap-961	260	23	de	de	X
ap-961	260	24	de	de	X
ap-961	260	25	�	�	PROPN
ap-961	260	26	�	�	PROPN
ap-961	260	27	(	(	PUNCT
ap-961	260	28	)	)	PUNCT
ap-961	260	29	*	*	PUNCT
ap-961	260	30	.	.	PUNCT
ap-961	261	1	exercise	exercise	VERB
ap-961	261	2	3.13	3.13	NUM
ap-961	261	3	:	:	PUNCT
ap-961	261	4	verify	verify	VERB
ap-961	261	5	that	that	SCONJ
ap-961	261	6	in	in	ADP
ap-961	261	7	example	example	NOUN
ap-961	261	8	3.11	3.11	NUM
ap-961	261	9	of	of	ADP
ap-961	261	10	the	the	DET
ap-961	261	11	nonstandard	nonstandard	ADJ
ap-961	261	12	differential	differential	ADJ
ap-961	261	13	algebra	algebra	NOUN
ap-961	261	14	over	over	ADP
ap-961	261	15	the	the	DET
ap-961	261	16	circle	circle	NOUN
ap-961	261	17	we	we	PRON
ap-961	261	18	have	have	VERB
ap-961	261	19	:	:	PUNCT
ap-961	261	20	�	�	PROPN
ap-961	261	21	�	�	PROPN
ap-961	261	22	*	*	PROPN
ap-961	261	23	�	�	PROPN
ap-961	261	24	�	�	PROPN
ap-961	261	25	and	and	CCONJ
ap-961	261	26	�	�	PROPN
ap-961	261	27	�	�	PROPN
ap-961	261	28	*	*	PROPN
ap-961	261	29	�	�	PROPN
ap-961	261	30	.	.	PUNCT
ap-961	262	1	3.2.2	3.2.2	NUM
ap-961	262	2	tensor	tensor	NOUN
ap-961	262	3	products	product	NOUN
ap-961	262	4	next	next	ADV
ap-961	262	5	,	,	PUNCT
ap-961	262	6	let	let	VERB
ap-961	262	7	us	we	PRON
ap-961	262	8	discuss	discuss	VERB
ap-961	262	9	the	the	DET
ap-961	262	10	procedure	procedure	NOUN
ap-961	262	11	for	for	ADP
ap-961	262	12	constructing	construct	VERB
ap-961	262	13	the	the	DET
ap-961	262	14	differential	differential	NOUN
ap-961	262	15	graded	grade	VERB
ap-961	262	16	algebra	algebra	NOUN
ap-961	262	17	for	for	ADP
ap-961	262	18	the	the	DET
ap-961	262	19	tensor	tensor	NOUN
ap-961	262	20	product	product	NOUN
ap-961	262	21	of	of	ADP
ap-961	262	22	two	two	NUM
ap-961	262	23	algebras	algebras	PROPN
ap-961	262	24	�	�	PROPN
ap-961	262	25	and	and	CCONJ
ap-961	262	26	�	�	PROPN
ap-961	262	27	.	.	PUNCT
ap-961	263	1	assuming	assume	VERB
ap-961	263	2	that	that	SCONJ
ap-961	263	3	the	the	DET
ap-961	263	4	respective	respective	ADJ
ap-961	263	5	differential	differential	NOUN
ap-961	263	6	algebras	algebra	NOUN
ap-961	263	7	*	*	PUNCT
ap-961	263	8	(	(	PUNCT
ap-961	263	9	)	)	PUNCT
ap-961	263	10	�	�	PROPN
ap-961	263	11	,	,	PUNCT
ap-961	263	12	*	*	PUNCT
ap-961	263	13	(	(	PUNCT
ap-961	263	14	)	)	PUNCT
ap-961	263	15	�	�	PROPN
ap-961	263	16	are	be	AUX
ap-961	263	17	given	give	VERB
ap-961	263	18	,	,	PUNCT
ap-961	263	19	there	there	PRON
ap-961	263	20	exists	exist	VERB
ap-961	263	21	a	a	DET
ap-961	263	22	canonical	canonical	ADJ
ap-961	263	23	procedure	procedure	NOUN
ap-961	263	24	for	for	ADP
ap-961	263	25	creating	create	VERB
ap-961	263	26	a	a	DET
ap-961	263	27	differential	differential	ADJ
ap-961	263	28	algebra	algebra	NOUN
ap-961	263	29	over	over	ADP
ap-961	263	30	the	the	DET
ap-961	263	31	tensor	tensor	NOUN
ap-961	263	32	product	product	NOUN
ap-961	263	33	,	,	PUNCT
ap-961	263	34	which	which	PRON
ap-961	263	35	uses	use	VERB
ap-961	263	36	the	the	DET
ap-961	263	37	�	�	PROPN
ap-961	263	38	2	2	NUM
ap-961	263	39	-	-	PUNCT
ap-961	263	40	graded	grade	VERB
ap-961	263	41	tensor	tensor	NOUN
ap-961	263	42	product	product	NOUN
ap-961	263	43	:	:	PUNCT
ap-961	264	1	*	*	PUNCT
ap-961	264	2	*	*	PUNCT
ap-961	264	3	*	*	PUNCT
ap-961	264	4	(	(	PUNCT
ap-961	264	5	)	)	PUNCT
ap-961	264	6	(	(	PUNCT
ap-961	264	7	)	)	PUNCT
ap-961	264	8	�	�	PROPN
ap-961	264	9	(	(	PUNCT
ap-961	264	10	)	)	PUNCT
ap-961	264	11	�	�	PROPN
ap-961	264	12	�	�	PROPN
ap-961	264	13	�	�	PROPN
ap-961	264	14	�	�	PROPN
ap-961	264	15	�	�	PROPN
ap-961	264	16	�	�	PROPN
ap-961	264	17	�	�	PROPN
ap-961	264	18	where	where	SCONJ
ap-961	264	19	�	�	PROPN
ap-961	264	20	�	�	PROPN
ap-961	264	21	means	mean	VERB
ap-961	264	22	that	that	SCONJ
ap-961	264	23	we	we	PRON
ap-961	264	24	take	take	VERB
ap-961	264	25	the	the	DET
ap-961	264	26	symmetric	symmetric	ADJ
ap-961	264	27	or	or	CCONJ
ap-961	264	28	antisymmetric	antisymmetric	ADJ
ap-961	264	29	part	part	NOUN
ap-961	264	30	of	of	ADP
ap-961	264	31	the	the	DET
ap-961	264	32	tensor	tensor	NOUN
ap-961	264	33	product	product	NOUN
ap-961	264	34	,	,	PUNCT
ap-961	264	35	with	with	ADP
ap-961	264	36	respect	respect	NOUN
ap-961	264	37	to	to	ADP
ap-961	264	38	the	the	DET
ap-961	264	39	degree	degree	NOUN
ap-961	264	40	of	of	ADP
ap-961	264	41	forms	form	NOUN
ap-961	264	42	.	.	PUNCT
ap-961	265	1	in	in	ADP
ap-961	265	2	other	other	ADJ
ap-961	265	3	words	word	NOUN
ap-961	265	4	the	the	DET
ap-961	265	5	product	product	NOUN
ap-961	265	6	of	of	ADP
ap-961	265	7	two	two	NUM
ap-961	265	8	forms	form	NOUN
ap-961	265	9	of	of	ADP
ap-961	265	10	fixed	fix	VERB
ap-961	265	11	degrees	degree	NOUN
ap-961	265	12	�	�	PROPN
ap-961	265	13	�	�	PROPN
ap-961	265	14	�	�	PROPN
ap-961	265	15	�	�	PROPN
ap-961	265	16	*	*	PUNCT
ap-961	265	17	(	(	PUNCT
ap-961	265	18	)	)	PUNCT
ap-961	265	19	and	and	CCONJ
ap-961	265	20	�	�	PROPN
ap-961	265	21	�	�	PROPN
ap-961	265	22	�	�	PROPN
ap-961	265	23	�	�	PROPN
ap-961	265	24	*	*	PUNCT
ap-961	265	25	(	(	PUNCT
ap-961	265	26	)	)	PUNCT
ap-961	265	27	is	be	AUX
ap-961	265	28	always	always	ADV
ap-961	265	29	commutative	commutative	ADJ
ap-961	265	30	or	or	CCONJ
ap-961	265	31	anticommutative	anticommutative	ADJ
ap-961	265	32	depending	depend	VERB
ap-961	265	33	on	on	ADP
ap-961	265	34	their	their	PRON
ap-961	265	35	degrees	degree	NOUN
ap-961	265	36	:	:	PUNCT
ap-961	265	37	�	�	PROPN
ap-961	265	38	�	�	PROPN
ap-961	265	39	�	�	PROPN
ap-961	265	40	�	�	PROPN
ap-961	265	41	�	�	PROPN
ap-961	265	42	�	�	PROPN
ap-961	265	43	�	�	PROPN
ap-961	265	44	�	�	PROPN
ap-961	265	45	�	�	PROPN
ap-961	265	46	�	�	PROPN
ap-961	265	47	�	�	PROPN
ap-961	265	48	�	�	PROPN
ap-961	265	49	#	#	PROPN
ap-961	265	50	�	�	PROPN
ap-961	265	51	�	�	PROPN
ap-961	265	52	#	#	SYM
ap-961	265	53	(	(	PUNCT
ap-961	265	54	)	)	PUNCT
ap-961	265	55	1	1	NUM
ap-961	265	56	.	.	X
ap-961	265	57	3.3	3.3	NUM
ap-961	265	58	representations	representation	NOUN
ap-961	265	59	of	of	ADP
ap-961	265	60	differential	differential	ADJ
ap-961	265	61	algebras	algebra	NOUN
ap-961	265	62	finally	finally	ADV
ap-961	265	63	,	,	PUNCT
ap-961	265	64	let	let	VERB
ap-961	265	65	us	we	PRON
ap-961	265	66	consider	consider	VERB
ap-961	265	67	a	a	DET
ap-961	265	68	specific	specific	ADJ
ap-961	265	69	way	way	NOUN
ap-961	265	70	of	of	ADP
ap-961	265	71	obtaining	obtain	VERB
ap-961	265	72	differential	differential	NOUN
ap-961	265	73	graded	grade	VERB
ap-961	265	74	algebras	algebra	NOUN
ap-961	265	75	–	–	PUNCT
ap-961	265	76	connected	connect	VERB
ap-961	265	77	with	with	ADP
ap-961	265	78	representations	representation	NOUN
ap-961	265	79	and	and	CCONJ
ap-961	265	80	commutators	commutator	NOUN
ap-961	265	81	.	.	PUNCT
ap-961	266	1	let	let	VERB
ap-961	266	2	�	�	PROPN
ap-961	266	3	be	be	AUX
ap-961	266	4	an	an	DET
ap-961	266	5	algebra	algebra	NOUN
ap-961	266	6	and	and	CCONJ
ap-961	266	7	let	let	VERB
ap-961	266	8	�	�	PROPN
ap-961	266	9	be	be	AUX
ap-961	266	10	its	its	PRON
ap-961	266	11	representation	representation	NOUN
ap-961	266	12	on	on	ADP
ap-961	266	13	a	a	DET
ap-961	266	14	vector	vector	NOUN
ap-961	266	15	space	space	NOUN
ap-961	266	16	(	(	PUNCT
ap-961	266	17	not	not	PART
ap-961	266	18	necessarily	necessarily	ADV
ap-961	266	19	finite	finite	VERB
ap-961	266	20	dimensional	dimensional	ADJ
ap-961	266	21	)	)	PUNCT
ap-961	266	22	.	.	PUNCT
ap-961	267	1	let	let	VERB
ap-961	267	2	f	f	PRON
ap-961	267	3	be	be	AUX
ap-961	267	4	an	an	DET
ap-961	267	5	endomorphism	endomorphism	NOUN
ap-961	267	6	(	(	PUNCT
ap-961	267	7	a	a	DET
ap-961	267	8	linear	linear	ADJ
ap-961	267	9	operator	operator	NOUN
ap-961	267	10	,	,	PUNCT
ap-961	267	11	in	in	ADP
ap-961	267	12	other	other	ADJ
ap-961	267	13	words	word	NOUN
ap-961	267	14	)	)	PUNCT
ap-961	267	15	of	of	ADP
ap-961	267	16	this	this	DET
ap-961	267	17	vector	vector	NOUN
ap-961	267	18	space	space	NOUN
ap-961	267	19	.	.	PUNCT
ap-961	268	1	lemma	lemma	PROPN
ap-961	268	2	3.14	3.14	NUM
ap-961	268	3	:	:	PUNCT
ap-961	268	4	if	if	SCONJ
ap-961	268	5	�	�	PROPN
ap-961	268	6	is	be	AUX
ap-961	268	7	a	a	DET
ap-961	268	8	representation	representation	NOUN
ap-961	268	9	of	of	ADP
ap-961	268	10	the	the	DET
ap-961	268	11	algebra	algebra	PROPN
ap-961	268	12	�	�	PROPN
ap-961	268	13	,	,	PUNCT
ap-961	268	14	then	then	ADV
ap-961	268	15	for	for	ADP
ap-961	268	16	each	each	DET
ap-961	268	17	linear	linear	ADJ
ap-961	268	18	operator	operator	NOUN
ap-961	268	19	f	f	PROPN
ap-961	268	20	the	the	DET
ap-961	268	21	following	follow	VERB
ap-961	268	22	gives	give	VERB
ap-961	268	23	a	a	DET
ap-961	268	24	representation	representation	NOUN
ap-961	268	25	of	of	ADP
ap-961	268	26	the	the	DET
ap-961	268	27	universal	universal	ADJ
ap-961	268	28	differential	differential	PROPN
ap-961	268	29	algebra	algebra	PROPN
ap-961	268	30	u	u	NOUN
ap-961	268	31	(	(	PUNCT
ap-961	268	32	)	)	PUNCT
ap-961	268	33	�	�	PROPN
ap-961	268	34	:	:	PUNCT
ap-961	268	35	�	�	PROPN
ap-961	268	36	�	�	PROPN
ap-961	268	37	�	�	PROPN
ap-961	268	38	�	�	PROPN
ap-961	268	39	�	�	PROPN
ap-961	268	40	f	f	PROPN
ap-961	268	41	n	n	PROPN
ap-961	268	42	n	n	ADV
ap-961	268	43	a	a	DET
ap-961	268	44	da	da	X
ap-961	268	45	da	da	X
ap-961	268	46	da	da	PROPN
ap-961	268	47	a	a	DET
ap-961	268	48	f	f	NOUN
ap-961	268	49	a	a	DET
ap-961	268	50	f	f	X
ap-961	268	51	a	a	DET
ap-961	268	52	f	f	PROPN
ap-961	268	53	a	a	PRON
ap-961	268	54	(	(	PUNCT
ap-961	268	55	)	)	PUNCT
ap-961	268	56	(	(	PUNCT
ap-961	268	57	)	)	PUNCT
ap-961	268	58	[	[	PUNCT
ap-961	268	59	,	,	PUNCT
ap-961	268	60	(	(	PUNCT
ap-961	268	61	)	)	PUNCT
ap-961	268	62	]	]	X
ap-961	268	63	[	[	PUNCT
ap-961	268	64	,	,	PUNCT
ap-961	268	65	(	(	PUNCT
ap-961	268	66	)	)	PUNCT
ap-961	268	67	]	]	PUNCT
ap-961	269	1	[	[	PUNCT
ap-961	269	2	,	,	PUNCT
ap-961	269	3	(	(	PUNCT
ap-961	269	4	)	)	PUNCT
ap-961	269	5	]	]	PUNCT
ap-961	269	6	0	0	NUM
ap-961	269	7	1	1	NUM
ap-961	269	8	2	2	NUM
ap-961	269	9	0	0	NUM
ap-961	269	10	1	1	NUM
ap-961	269	11	2	2	NUM
ap-961	269	12	�	�	PROPN
ap-961	269	13	�	�	PROPN
ap-961	269	14	�	�	PROPN
ap-961	269	15	(	(	PUNCT
ap-961	269	16	1	1	NUM
ap-961	269	17	)	)	PUNCT
ap-961	269	18	note	note	NOUN
ap-961	269	19	that	that	SCONJ
ap-961	269	20	�	�	PROPN
ap-961	269	21	f	f	PROPN
ap-961	269	22	is	be	AUX
ap-961	269	23	nothing	nothing	PRON
ap-961	269	24	else	else	ADV
ap-961	269	25	but	but	CCONJ
ap-961	269	26	a	a	DET
ap-961	269	27	representation	representation	NOUN
ap-961	269	28	of	of	ADP
ap-961	269	29	the	the	DET
ap-961	269	30	algebra	algebra	NOUN
ap-961	269	31	,	,	PUNCT
ap-961	269	32	and	and	CCONJ
ap-961	269	33	neither	neither	CCONJ
ap-961	269	34	the	the	DET
ap-961	269	35	grading	grading	NOUN
ap-961	269	36	nor	nor	CCONJ
ap-961	269	37	the	the	DET
ap-961	269	38	external	external	ADJ
ap-961	269	39	derivative	derivative	NOUN
ap-961	269	40	are	be	AUX
ap-961	269	41	in	in	ADP
ap-961	269	42	any	any	DET
ap-961	269	43	way	way	NOUN
ap-961	269	44	preserved	preserve	VERB
ap-961	269	45	.	.	PUNCT
ap-961	270	1	this	this	PRON
ap-961	270	2	follows	follow	VERB
ap-961	270	3	from	from	ADP
ap-961	270	4	the	the	DET
ap-961	270	5	fact	fact	NOUN
ap-961	270	6	that	that	SCONJ
ap-961	270	7	the	the	DET
ap-961	270	8	kernel	kernel	NOUN
ap-961	270	9	of	of	ADP
ap-961	270	10	�	�	PROPN
ap-961	270	11	f	f	PROPN
ap-961	270	12	might	might	AUX
ap-961	270	13	not	not	PART
ap-961	270	14	be	be	AUX
ap-961	270	15	a	a	DET
ap-961	270	16	differential	differential	ADJ
ap-961	270	17	ideal	ideal	NOUN
ap-961	270	18	.	.	PUNCT
ap-961	271	1	we	we	PRON
ap-961	271	2	also	also	ADV
ap-961	271	3	need	need	VERB
ap-961	271	4	to	to	PART
ap-961	271	5	be	be	AUX
ap-961	271	6	careful	careful	ADJ
ap-961	271	7	while	while	SCONJ
ap-961	271	8	dealing	deal	VERB
ap-961	271	9	with	with	ADP
ap-961	271	10	infinite	infinite	ADJ
ap-961	271	11	dimensional	dimensional	ADJ
ap-961	271	12	representations	representation	NOUN
ap-961	271	13	,	,	PUNCT
ap-961	271	14	such	such	ADJ
ap-961	271	15	as	as	ADP
ap-961	271	16	representations	representation	NOUN
ap-961	271	17	on	on	ADP
ap-961	271	18	the	the	DET
ap-961	271	19	hilbert	hilbert	NOUN
ap-961	271	20	space	space	NOUN
ap-961	271	21	.	.	PUNCT
ap-961	272	1	in	in	ADP
ap-961	272	2	such	such	DET
ap-961	272	3	a	a	DET
ap-961	272	4	©	©	PROPN
ap-961	272	5	czech	czech	PROPN
ap-961	272	6	technical	technical	PROPN
ap-961	272	7	university	university	PROPN
ap-961	272	8	publishing	publishing	NOUN
ap-961	272	9	house	house	NOUN
ap-961	272	10	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	272	11	41	41	NUM
ap-961	272	12	acta	acta	PROPN
ap-961	272	13	polytechnica	polytechnica	PROPN
ap-961	272	14	vol	vol	NOUN
ap-961	272	15	.	.	PUNCT
ap-961	273	1	48	48	NUM
ap-961	273	2	no	no	NOUN
ap-961	273	3	.	.	PUNCT
ap-961	274	1	2/2008	2/2008	NUM
ap-961	274	2	case	case	NOUN
ap-961	274	3	,	,	PUNCT
ap-961	274	4	it	it	PRON
ap-961	274	5	is	be	AUX
ap-961	274	6	natural	natural	ADJ
ap-961	274	7	to	to	PART
ap-961	274	8	assume	assume	VERB
ap-961	274	9	that	that	SCONJ
ap-961	274	10	all	all	DET
ap-961	274	11	operators	operator	NOUN
ap-961	274	12	�	�	PROPN
ap-961	274	13	(	(	PUNCT
ap-961	274	14	a	a	NOUN
ap-961	274	15	)	)	PUNCT
ap-961	274	16	and	and	CCONJ
ap-961	274	17	the	the	DET
ap-961	274	18	commutators	commutator	NOUN
ap-961	274	19	[	[	PUNCT
ap-961	274	20	,	,	PUNCT
ap-961	274	21	(	(	PUNCT
ap-961	274	22	)	)	PUNCT
ap-961	274	23	]	]	X
ap-961	274	24	f	f	X
ap-961	274	25	a	a	DET
ap-961	274	26	�	�	PROPN
ap-961	274	27	are	be	AUX
ap-961	274	28	bounded	bound	VERB
ap-961	274	29	for	for	ADP
ap-961	274	30	all	all	DET
ap-961	274	31	a	a	DET
ap-961	274	32	�	�	PROPN
ap-961	274	33	�	�	PROPN
ap-961	274	34	.	.	PUNCT
ap-961	275	1	this	this	PRON
ap-961	275	2	naturally	naturally	ADV
ap-961	275	3	implies	imply	VERB
ap-961	275	4	that	that	SCONJ
ap-961	275	5	the	the	DET
ap-961	275	6	image	image	NOUN
ap-961	275	7	of	of	ADP
ap-961	275	8	an	an	DET
ap-961	275	9	arbitrary	arbitrary	ADJ
ap-961	275	10	form	form	NOUN
ap-961	275	11	is	be	AUX
ap-961	275	12	a	a	DET
ap-961	275	13	bounded	bounded	ADJ
ap-961	275	14	operator	operator	NOUN
ap-961	275	15	.	.	PUNCT
ap-961	276	1	f	f	PROPN
ap-961	276	2	itself	itself	PRON
ap-961	276	3	might	might	AUX
ap-961	276	4	not	not	PART
ap-961	276	5	be	be	AUX
ap-961	276	6	bounded	bound	VERB
ap-961	276	7	and	and	CCONJ
ap-961	276	8	we	we	PRON
ap-961	276	9	shall	shall	AUX
ap-961	276	10	see	see	VERB
ap-961	276	11	the	the	DET
ap-961	276	12	most	most	ADV
ap-961	276	13	natural	natural	ADJ
ap-961	276	14	examples	example	NOUN
ap-961	276	15	when	when	SCONJ
ap-961	276	16	it	it	PRON
ap-961	276	17	is	be	AUX
ap-961	276	18	not	not	PART
ap-961	276	19	the	the	DET
ap-961	276	20	case	case	NOUN
ap-961	276	21	.	.	PUNCT
ap-961	277	1	there	there	PRON
ap-961	277	2	exists	exist	VERB
ap-961	277	3	a	a	DET
ap-961	277	4	canonical	canonical	ADJ
ap-961	277	5	way	way	NOUN
ap-961	277	6	to	to	PART
ap-961	277	7	obtain	obtain	VERB
ap-961	277	8	a	a	DET
ap-961	277	9	differential	differential	NOUN
ap-961	277	10	graded	grade	VERB
ap-961	277	11	algebra	algebra	NOUN
ap-961	277	12	through	through	ADP
ap-961	277	13	�	�	NOUN
ap-961	277	14	f	f	PROPN
ap-961	277	15	:	:	PUNCT
ap-961	277	16	we	we	PRON
ap-961	277	17	have	have	VERB
ap-961	277	18	to	to	PART
ap-961	277	19	take	take	VERB
ap-961	277	20	�	�	PROPN
ap-961	277	21	�	�	PROPN
ap-961	277	22	ker	ker	PROPN
ap-961	277	23	(	(	PUNCT
ap-961	277	24	ker	ker	PROPN
ap-961	277	25	)	)	PUNCT
ap-961	277	26	�	�	PROPN
ap-961	277	27	�	�	PROPN
ap-961	277	28	f	f	PROPN
ap-961	277	29	fd	fd	PROPN
ap-961	277	30	.	.	PUNCT
ap-961	278	1	this	this	PRON
ap-961	278	2	is	be	AUX
ap-961	278	3	a	a	DET
ap-961	278	4	differential	differential	ADJ
ap-961	278	5	ideal	ideal	NOUN
ap-961	278	6	within	within	ADP
ap-961	278	7	u	u	NOUN
ap-961	278	8	(	(	PUNCT
ap-961	278	9	)	)	PUNCT
ap-961	278	10	�	�	PROPN
ap-961	278	11	and	and	CCONJ
ap-961	278	12	then	then	ADV
ap-961	278	13	u	u	NOUN
ap-961	278	14	(	(	PUNCT
ap-961	278	15	)	)	PUNCT
ap-961	278	16	�	�	PROPN
ap-961	278	17	�	�	PROPN
ap-961	278	18	will	will	AUX
ap-961	278	19	be	be	AUX
ap-961	278	20	a	a	DET
ap-961	278	21	differential	differential	ADJ
ap-961	278	22	algebra	algebra	NOUN
ap-961	278	23	.	.	PUNCT
ap-961	279	1	note	note	VERB
ap-961	279	2	that	that	SCONJ
ap-961	279	3	unless	unless	SCONJ
ap-961	279	4	d	d	PROPN
ap-961	279	5	f	f	PROPN
ap-961	279	6	f(ker	f(ker	NOUN
ap-961	279	7	)	)	PUNCT
ap-961	279	8	ker	ker	PROPN
ap-961	279	9	�	�	PROPN
ap-961	279	10	�	�	PROPN
ap-961	279	11	�	�	PROPN
ap-961	279	12	the	the	DET
ap-961	279	13	obtained	obtain	VERB
ap-961	279	14	differential	differential	NOUN
ap-961	279	15	algebra	algebra	NOUN
ap-961	279	16	will	will	AUX
ap-961	279	17	not	not	PART
ap-961	279	18	have	have	VERB
ap-961	279	19	a	a	DET
ap-961	279	20	representation	representation	NOUN
ap-961	279	21	on	on	ADP
ap-961	279	22	the	the	DET
ap-961	279	23	hilbert	hilbert	NOUN
ap-961	279	24	space	space	NOUN
ap-961	279	25	.	.	PUNCT
ap-961	280	1	one	one	PRON
ap-961	280	2	might	might	AUX
ap-961	280	3	always	always	ADV
ap-961	280	4	choose	choose	VERB
ap-961	280	5	for	for	ADP
ap-961	280	6	a	a	DET
ap-961	280	7	higher	high	ADJ
ap-961	280	8	-	-	PUNCT
ap-961	280	9	order	order	NOUN
ap-961	280	10	form	form	VERB
ap-961	280	11	its	its	PRON
ap-961	280	12	representative	representative	NOUN
ap-961	280	13	in	in	ADP
ap-961	280	14	the	the	DET
ap-961	280	15	image	image	NOUN
ap-961	280	16	�	�	PROPN
ap-961	280	17	f	f	PART
ap-961	280	18	u	u	PROPN
ap-961	280	19	(	(	PUNCT
ap-961	280	20	(	(	PUNCT
ap-961	280	21	)	)	PUNCT
ap-961	280	22	)	)	PUNCT
ap-961	281	1	*	*	PUNCT
ap-961	281	2	�	�	PROPN
ap-961	281	3	,	,	PUNCT
ap-961	281	4	however	however	ADV
ap-961	281	5	,	,	PUNCT
ap-961	281	6	this	this	PRON
ap-961	281	7	can	can	AUX
ap-961	281	8	not	not	PART
ap-961	281	9	be	be	AUX
ap-961	281	10	done	do	VERB
ap-961	281	11	in	in	ADP
ap-961	281	12	a	a	DET
ap-961	281	13	unique	unique	ADJ
ap-961	281	14	way	way	NOUN
ap-961	281	15	.	.	PUNCT
ap-961	282	1	3.4	3.4	NUM
ap-961	282	2	from	from	ADP
ap-961	282	3	representations	representation	NOUN
ap-961	282	4	to	to	PART
ap-961	282	5	differential	differential	VERB
ap-961	282	6	calculi	calculi	NOUN
ap-961	282	7	in	in	ADP
ap-961	282	8	view	view	NOUN
ap-961	282	9	of	of	ADP
ap-961	282	10	the	the	DET
ap-961	282	11	previous	previous	ADJ
ap-961	282	12	section	section	NOUN
ap-961	282	13	we	we	PRON
ap-961	282	14	might	might	AUX
ap-961	282	15	consider	consider	VERB
ap-961	282	16	just	just	ADV
ap-961	282	17	a	a	DET
ap-961	282	18	different	different	ADJ
ap-961	282	19	way	way	NOUN
ap-961	282	20	of	of	ADP
ap-961	282	21	obtaining	obtain	VERB
ap-961	282	22	differential	differential	ADJ
ap-961	282	23	graded	grade	VERB
ap-961	282	24	algebras	algebra	NOUN
ap-961	282	25	.	.	PUNCT
ap-961	283	1	just	just	ADV
ap-961	283	2	start	start	VERB
ap-961	283	3	with	with	ADP
ap-961	283	4	a	a	DET
ap-961	283	5	representation	representation	NOUN
ap-961	283	6	�	�	NOUN
ap-961	283	7	of	of	ADP
ap-961	283	8	the	the	DET
ap-961	283	9	algebra	algebra	NOUN
ap-961	283	10	,	,	PUNCT
ap-961	283	11	choose	choose	VERB
ap-961	283	12	a	a	DET
ap-961	283	13	suitable	suitable	ADJ
ap-961	283	14	operator	operator	NOUN
ap-961	283	15	f	f	NOUN
ap-961	283	16	,	,	PUNCT
ap-961	283	17	and	and	CCONJ
ap-961	283	18	consider	consider	VERB
ap-961	283	19	all	all	DET
ap-961	283	20	commutators	commutator	NOUN
ap-961	283	21	[	[	PUNCT
ap-961	283	22	,	,	PUNCT
ap-961	283	23	(	(	PUNCT
ap-961	283	24	)	)	PUNCT
ap-961	284	1	]	]	X
ap-961	284	2	f	f	X
ap-961	284	3	a	a	DET
ap-961	284	4	�	�	PROPN
ap-961	284	5	as	as	ADP
ap-961	284	6	one	one	NUM
ap-961	284	7	-	-	PUNCT
ap-961	284	8	forms	form	NOUN
ap-961	284	9	.	.	PUNCT
ap-961	285	1	with	with	ADP
ap-961	285	2	a	a	DET
ap-961	285	3	bit	bit	NOUN
ap-961	285	4	of	of	ADP
ap-961	285	5	luck	luck	NOUN
ap-961	285	6	(	(	PUNCT
ap-961	285	7	and	and	CCONJ
ap-961	285	8	some	some	DET
ap-961	285	9	additional	additional	ADJ
ap-961	285	10	assumptions	assumption	NOUN
ap-961	285	11	)	)	PUNCT
ap-961	285	12	we	we	PRON
ap-961	285	13	shall	shall	AUX
ap-961	285	14	the	the	PRON
ap-961	285	15	obtain	obtain	VERB
ap-961	285	16	a	a	DET
ap-961	285	17	good	good	ADJ
ap-961	285	18	example	example	NOUN
ap-961	285	19	of	of	ADP
ap-961	285	20	a	a	DET
ap-961	285	21	differential	differential	ADJ
ap-961	285	22	calculus	calculus	NOUN
ap-961	285	23	.	.	PUNCT
ap-961	286	1	we	we	PRON
ap-961	286	2	shall	shall	AUX
ap-961	286	3	consider	consider	VERB
ap-961	286	4	two	two	NUM
ap-961	286	5	canonical	canonical	ADJ
ap-961	286	6	cases	case	NOUN
ap-961	286	7	.	.	PUNCT
ap-961	287	1	first	first	ADV
ap-961	287	2	,	,	PUNCT
ap-961	287	3	let	let	VERB
ap-961	287	4	f	f	PRON
ap-961	287	5	be	be	AUX
ap-961	287	6	a	a	DET
ap-961	287	7	selfadjoint	selfadjoint	NOUN
ap-961	287	8	f	f	PROPN
ap-961	287	9	f	f	X
ap-961	287	10	�	�	PROPN
ap-961	287	11	†.	†.	PROPN
ap-961	287	12	this	this	PRON
ap-961	287	13	assures	assure	VERB
ap-961	287	14	that	that	SCONJ
ap-961	287	15	for	for	ADP
ap-961	287	16	an	an	DET
ap-961	287	17	involutive	involutive	ADJ
ap-961	287	18	algebra	algebra	NOUN
ap-961	287	19	�	�	PROPN
ap-961	287	20	and	and	CCONJ
ap-961	287	21	a	a	DET
ap-961	287	22	*	*	ADJ
ap-961	287	23	-representation	-representation	NOUN
ap-961	287	24	we	we	PRON
ap-961	287	25	have	have	VERB
ap-961	287	26	d	d	PROPN
ap-961	287	27	a	a	DET
ap-961	287	28	da	da	X
ap-961	287	29	(	(	PUNCT
ap-961	287	30	)	)	PUNCT
ap-961	287	31	(	(	PUNCT
ap-961	287	32	)	)	PUNCT
ap-961	287	33	*	*	PUNCT
ap-961	287	34	*	*	PUNCT
ap-961	287	35	�	�	PROPN
ap-961	287	36	�	�	PROPN
ap-961	287	37	:	:	PUNCT
ap-961	287	38	d	d	ADP
ap-961	287	39	a	a	DET
ap-961	287	40	f	f	X
ap-961	287	41	a	a	PRON
ap-961	287	42	a	a	DET
ap-961	287	43	f	f	NOUN
ap-961	287	44	a	a	DET
ap-961	287	45	f	f	PROPN
ap-961	287	46	d	d	X
ap-961	287	47	a	a	PROPN
ap-961	287	48	(	(	PUNCT
ap-961	287	49	)	)	PUNCT
ap-961	287	50	(	(	PUNCT
ap-961	287	51	[	[	PUNCT
ap-961	287	52	,	,	PUNCT
ap-961	287	53	(	(	PUNCT
ap-961	287	54	)	)	PUNCT
ap-961	287	55	]	]	X
ap-961	287	56	)	)	PUNCT
ap-961	288	1	[	[	PUNCT
ap-961	288	2	(	(	PUNCT
ap-961	288	3	)	)	PUNCT
ap-961	288	4	,	,	PUNCT
ap-961	288	5	]	]	PUNCT
ap-961	288	6	[	[	PUNCT
ap-961	288	7	(	(	PUNCT
ap-961	288	8	)	)	PUNCT
ap-961	288	9	,	,	PUNCT
ap-961	288	10	]	]	PUNCT
ap-961	288	11	(	(	PUNCT
ap-961	288	12	)	)	PUNCT
ap-961	288	13	*	*	PUNCT
ap-961	288	14	†	†	PROPN
ap-961	288	15	†	†	PROPN
ap-961	288	16	†	†	PROPN
ap-961	288	17	*	*	PUNCT
ap-961	288	18	*	*	PUNCT
ap-961	288	19	�	�	PROPN
ap-961	288	20	�	�	PROPN
ap-961	288	21	�	�	PROPN
ap-961	288	22	�	�	PROPN
ap-961	288	23	�	�	PROPN
ap-961	288	24	�	�	PROPN
ap-961	288	25	�	�	PROPN
ap-961	288	26	�	�	PROPN
ap-961	288	27	.	.	PUNCT
ap-961	289	1	moreover	moreover	ADV
ap-961	289	2	,	,	PUNCT
ap-961	289	3	let	let	VERB
ap-961	289	4	us	we	PRON
ap-961	289	5	take	take	VERB
ap-961	289	6	f2	f2	PROPN
ap-961	289	7	1	1	NUM
ap-961	289	8	�	�	PROPN
ap-961	289	9	,	,	PUNCT
ap-961	289	10	which	which	PRON
ap-961	289	11	means	mean	VERB
ap-961	289	12	that	that	SCONJ
ap-961	289	13	(	(	PUNCT
ap-961	289	14	as	as	SCONJ
ap-961	289	15	seen	see	VERB
ap-961	289	16	on	on	ADP
ap-961	289	17	a	a	DET
ap-961	289	18	hilbert	hilbert	NOUN
ap-961	289	19	space	space	NOUN
ap-961	289	20	)	)	PUNCT
ap-961	290	1	f	f	PROPN
ap-961	290	2	is	be	AUX
ap-961	290	3	a	a	DET
ap-961	290	4	sign	sign	NOUN
ap-961	290	5	operator	operator	NOUN
ap-961	290	6	with	with	ADP
ap-961	290	7	eigenvalues	eigenvalue	NOUN
ap-961	290	8	being	be	AUX
ap-961	290	9	1	1	NUM
ap-961	290	10	and	and	CCONJ
ap-961	290	11	�	�	PROPN
ap-961	290	12	1	1	NUM
ap-961	290	13	.	.	PUNCT
ap-961	291	1	we	we	PRON
ap-961	291	2	have	have	VERB
ap-961	291	3	:	:	PUNCT
ap-961	291	4	lemma	lemma	PROPN
ap-961	291	5	3.15	3.15	NUM
ap-961	291	6	.	.	PUNCT
ap-961	292	1	let	let	VERB
ap-961	292	2	f	f	PROPN
ap-961	292	3	f	f	PROPN
ap-961	292	4	�	�	PROPN
ap-961	292	5	†	†	PROPN
ap-961	292	6	and	and	CCONJ
ap-961	292	7	f2	f2	PROPN
ap-961	292	8	1	1	NUM
ap-961	292	9	�	�	NOUN
ap-961	292	10	be	be	AUX
ap-961	292	11	an	an	DET
ap-961	292	12	operator	operator	NOUN
ap-961	292	13	on	on	ADP
ap-961	292	14	the	the	DET
ap-961	292	15	hilbert	hilbert	PROPN
ap-961	292	16	space	space	NOUN
ap-961	292	17	�	�	PROPN
ap-961	292	18	and	and	CCONJ
ap-961	292	19	let	let	VERB
ap-961	292	20	�	�	PROPN
ap-961	292	21	be	be	AUX
ap-961	292	22	the	the	DET
ap-961	292	23	representation	representation	NOUN
ap-961	292	24	of	of	ADP
ap-961	292	25	�	�	PROPN
ap-961	292	26	as	as	ADP
ap-961	292	27	bounded	bound	VERB
ap-961	292	28	operators	operator	NOUN
ap-961	292	29	on	on	ADP
ap-961	292	30	�	�	PROPN
ap-961	292	31	.	.	PUNCT
ap-961	293	1	then	then	ADV
ap-961	293	2	�	�	PROPN
ap-961	293	3	u	u	NOUN
ap-961	293	4	defined	define	VERB
ap-961	293	5	in	in	ADP
ap-961	293	6	1	1	NUM
ap-961	293	7	is	be	AUX
ap-961	293	8	a	a	DET
ap-961	293	9	representation	representation	NOUN
ap-961	293	10	of	of	ADP
ap-961	293	11	the	the	DET
ap-961	293	12	differential	differential	ADJ
ap-961	293	13	algebra	algebra	NOUN
ap-961	293	14	,	,	PUNCT
ap-961	293	15	with	with	ADP
ap-961	293	16	:	:	PUNCT
ap-961	293	17	�	�	PROPN
ap-961	293	18	�	�	PROPN
ap-961	293	19	�	�	PROPN
ap-961	293	20	�	�	PROPN
ap-961	293	21	�	�	PROPN
ap-961	293	22	�	�	PROPN
ap-961	293	23	�	�	PROPN
ap-961	293	24	�	�	PROPN
ap-961	293	25	f	f	PROPN
ap-961	293	26	u	u	NOUN
ap-961	293	27	u	u	PROPN
ap-961	293	28	d	d	X
ap-961	293	29	f	f	PROPN
ap-961	293	30	f	f	PROPN
ap-961	293	31	(	(	PUNCT
ap-961	293	32	)	)	PUNCT
ap-961	293	33	[	[	PUNCT
ap-961	293	34	,	,	PUNCT
ap-961	293	35	(	(	PUNCT
ap-961	293	36	)	)	PUNCT
ap-961	293	37	]	]	PUNCT
ap-961	294	1	[	[	PUNCT
ap-961	294	2	,	,	PUNCT
ap-961	294	3	(	(	PUNCT
ap-961	294	4	)	)	PUNCT
ap-961	294	5	]	]	PUNCT
ap-961	294	6	�	�	PROPN
ap-961	294	7	�	�	PROPN
ap-961	294	8	�	�	PROPN
ap-961	294	9	�	�	PROPN
ap-961	294	10	even	even	ADV
ap-961	294	11	odd	odd	ADJ
ap-961	294	12	for	for	ADP
ap-961	294	13	any	any	DET
ap-961	294	14	universal	universal	ADJ
ap-961	294	15	form	form	NOUN
ap-961	294	16	�	�	PROPN
ap-961	294	17	,	,	PUNCT
ap-961	294	18	[	[	PUNCT
ap-961	294	19	,	,	PUNCT
ap-961	294	20	]	]	X
ap-961	294	21	"	"	PUNCT
ap-961	294	22	"	"	PUNCT
ap-961	294	23	denotes	denote	NOUN
ap-961	294	24	anticommutator	anticommutator	NOUN
ap-961	294	25	.	.	PUNCT
ap-961	295	1	proof	proof	NOUN
ap-961	295	2	:	:	PUNCT
ap-961	295	3	first	first	ADV
ap-961	295	4	observe	observe	VERB
ap-961	295	5	:	:	PUNCT
ap-961	295	6	[	[	PUNCT
ap-961	295	7	,	,	PUNCT
ap-961	295	8	]	]	X
ap-961	295	9	[	[	PUNCT
ap-961	295	10	,	,	PUNCT
ap-961	295	11	]	]	X
ap-961	295	12	f	f	X
ap-961	295	13	x	x	X
ap-961	295	14	f	f	PROPN
ap-961	295	15	f	f	PROPN
ap-961	295	16	f	f	PROPN
ap-961	295	17	x	x	X
ap-961	295	18	�	�	PROPN
ap-961	295	19	�	�	PROPN
ap-961	295	20	,	,	PUNCT
ap-961	295	21	�	�	PROPN
ap-961	295	22	�	�	PROPN
ap-961	295	23	x	x	SYM
ap-961	295	24	b	b	PROPN
ap-961	295	25	(	(	PUNCT
ap-961	295	26	)	)	PUNCT
ap-961	295	27	�	�	PROPN
ap-961	295	28	.	.	PUNCT
ap-961	296	1	then	then	ADV
ap-961	296	2	:	:	PUNCT
ap-961	296	3	�	�	PROPN
ap-961	296	4	�	�	PROPN
ap-961	296	5	�	�	PROPN
ap-961	296	6	�	�	PROPN
ap-961	296	7	�	�	PROPN
ap-961	296	8	f	f	PROPN
ap-961	296	9	n	n	CCONJ
ap-961	296	10	nda	nda	PROPN
ap-961	296	11	da	da	PROPN
ap-961	296	12	da	da	PROPN
ap-961	296	13	da	da	PROPN
ap-961	296	14	f	f	PROPN
ap-961	296	15	a	a	DET
ap-961	296	16	f	f	X
ap-961	296	17	a	a	DET
ap-961	296	18	f	f	X
ap-961	296	19	a	a	DET
ap-961	296	20	f	f	X
ap-961	296	21	(	(	PUNCT
ap-961	296	22	)	)	PUNCT
ap-961	296	23	[	[	PUNCT
ap-961	296	24	,	,	PUNCT
ap-961	296	25	(	(	PUNCT
ap-961	296	26	)	)	PUNCT
ap-961	296	27	]	]	X
ap-961	296	28	[	[	PUNCT
ap-961	296	29	,	,	PUNCT
ap-961	296	30	(	(	PUNCT
ap-961	296	31	)	)	PUNCT
ap-961	296	32	]	]	PUNCT
ap-961	297	1	[	[	PUNCT
ap-961	297	2	,	,	PUNCT
ap-961	297	3	(	(	PUNCT
ap-961	297	4	)	)	PUNCT
ap-961	297	5	]	]	PUNCT
ap-961	297	6	(	(	PUNCT
ap-961	297	7	0	0	NUM
ap-961	297	8	1	1	NUM
ap-961	297	9	2	2	NUM
ap-961	297	10	0	0	NUM
ap-961	297	11	1	1	NUM
ap-961	297	12	�	�	PROPN
ap-961	297	13	�	�	PROPN
ap-961	297	14	�	�	PROPN
ap-961	297	15	�	�	PROPN
ap-961	297	16	a	a	DET
ap-961	297	17	f	f	X
ap-961	297	18	a	a	DET
ap-961	297	19	f	f	X
ap-961	297	20	a	a	DET
ap-961	297	21	a	a	DET
ap-961	297	22	f	f	NOUN
ap-961	297	23	a	a	DET
ap-961	297	24	f	f	X
ap-961	297	25	a	a	DET
ap-961	297	26	f	f	PROPN
ap-961	297	27	n	n	CCONJ
ap-961	297	28	n	n	ADV
ap-961	297	29	0	0	NUM
ap-961	297	30	1	1	NUM
ap-961	297	31	0	0	NUM
ap-961	297	32	1	1	NUM
ap-961	297	33	)	)	PUNCT
ap-961	297	34	[	[	PUNCT
ap-961	297	35	,	,	PUNCT
ap-961	297	36	(	(	PUNCT
ap-961	297	37	)	)	PUNCT
ap-961	297	38	]	]	PUNCT
ap-961	298	1	[	[	PUNCT
ap-961	298	2	,	,	PUNCT
ap-961	298	3	(	(	PUNCT
ap-961	298	4	)	)	PUNCT
ap-961	298	5	]	]	PUNCT
ap-961	298	6	(	(	PUNCT
ap-961	298	7	)	)	PUNCT
ap-961	298	8	[	[	PUNCT
ap-961	298	9	,	,	PUNCT
ap-961	298	10	(	(	PUNCT
ap-961	298	11	)	)	PUNCT
ap-961	298	12	]	]	PUNCT
ap-961	298	13	[	[	PUNCT
ap-961	298	14	,	,	PUNCT
ap-961	298	15	(	(	PUNCT
ap-961	298	16	)	)	PUNCT
ap-961	298	17	]	]	PUNCT
ap-961	298	18	(	(	PUNCT
ap-961	298	19	�	�	PROPN
ap-961	298	20	�	�	PROPN
ap-961	298	21	�	�	PROPN
ap-961	298	22	�	�	PROPN
ap-961	298	23	�	�	PROPN
ap-961	298	24	�	�	PROPN
ap-961	298	25	�	�	PROPN
ap-961	298	26	�	�	PROPN
ap-961	298	27	�	�	PROPN
ap-961	298	28	�	�	PROPN
ap-961	298	29	�	�	PROPN
ap-961	298	30	�	�	PROPN
ap-961	298	31	�	�	PROPN
ap-961	298	32	�	�	PROPN
ap-961	298	33	(	(	PUNCT
ap-961	298	34	)	)	PUNCT
ap-961	298	35	[	[	PUNCT
ap-961	298	36	,	,	PUNCT
ap-961	298	37	(	(	PUNCT
ap-961	298	38	)	)	PUNCT
ap-961	298	39	]	]	PUNCT
ap-961	298	40	[	[	PUNCT
ap-961	298	41	,	,	PUNCT
ap-961	298	42	(	(	PUNCT
ap-961	298	43	)	)	PUNCT
ap-961	298	44	]	]	X
ap-961	298	45	)	)	PUNCT
ap-961	298	46	(	(	PUNCT
ap-961	298	47	)	)	PUNCT
ap-961	298	48	(	(	PUNCT
ap-961	298	49	(	(	PUNCT
ap-961	298	50	)	)	PUNCT
ap-961	298	51	[	[	PUNCT
ap-961	298	52	,	,	PUNCT
ap-961	298	53	(	(	PUNCT
ap-961	298	54	)	)	PUNCT
ap-961	298	55	]	]	PUNCT
ap-961	298	56	[	[	PUNCT
ap-961	298	57	,	,	PUNCT
ap-961	298	58	a	a	DET
ap-961	298	59	f	f	NOUN
ap-961	298	60	a	a	DET
ap-961	298	61	f	f	X
ap-961	298	62	a	a	DET
ap-961	298	63	a	a	DET
ap-961	298	64	f	f	NOUN
ap-961	298	65	a	a	DET
ap-961	298	66	f	f	PROPN
ap-961	298	67	n	n	CCONJ
ap-961	298	68	n	n	ADV
ap-961	298	69	0	0	NUM
ap-961	298	70	1	1	NUM
ap-961	298	71	0	0	NUM
ap-961	298	72	11	11	NUM
ap-961	298	73	�	�	PROPN
ap-961	298	74	�	�	PROPN
ap-961	298	75	�	�	PROPN
ap-961	298	76	�	�	PROPN
ap-961	298	77	�	�	PROPN
ap-961	298	78	�	�	PROPN
ap-961	298	79	�	�	PROPN
ap-961	298	80	�	�	PROPN
ap-961	298	81	(	(	PUNCT
ap-961	298	82	)	)	PUNCT
ap-961	298	83	]	]	X
ap-961	298	84	)	)	PUNCT
ap-961	299	1	[	[	PUNCT
ap-961	299	2	,	,	PUNCT
ap-961	299	3	(	(	PUNCT
ap-961	299	4	)	)	PUNCT
ap-961	299	5	[	[	PUNCT
ap-961	299	6	,	,	PUNCT
ap-961	299	7	(	(	PUNCT
ap-961	299	8	)	)	PUNCT
ap-961	299	9	]	]	PUNCT
ap-961	300	1	[	[	PUNCT
ap-961	300	2	,	,	PUNCT
ap-961	300	3	(	(	PUNCT
ap-961	300	4	)	)	PUNCT
ap-961	300	5	]	]	X
ap-961	300	6	]	]	PUNCT
ap-961	300	7	,	,	PUNCT
ap-961	300	8	a	a	DET
ap-961	300	9	f	f	X
ap-961	300	10	f	f	PROPN
ap-961	300	11	a	a	DET
ap-961	300	12	f	f	X
ap-961	300	13	a	a	DET
ap-961	300	14	f	f	PROPN
ap-961	300	15	a	a	DET
ap-961	300	16	n	n	CCONJ
ap-961	300	17	n	n	CCONJ
ap-961	300	18	�	�	PROPN
ap-961	300	19	$	$	SYM
ap-961	300	20	0	0	NUM
ap-961	300	21	1	1	NUM
ap-961	300	22	�	�	NOUN
ap-961	300	23	where	where	SCONJ
ap-961	300	24	the	the	DET
ap-961	300	25	$	$	SYM
ap-961	300	26	sign	sign	NOUN
ap-961	300	27	at	at	ADP
ap-961	300	28	the	the	DET
ap-961	300	29	last	last	ADJ
ap-961	300	30	bracket	bracket	NOUN
ap-961	300	31	means	mean	VERB
ap-961	300	32	that	that	SCONJ
ap-961	300	33	we	we	PRON
ap-961	300	34	take	take	VERB
ap-961	300	35	the	the	DET
ap-961	300	36	commutator	commutator	NOUN
ap-961	300	37	(	(	PUNCT
ap-961	300	38	or	or	CCONJ
ap-961	300	39	anticommutator	anticommutator	NOUN
ap-961	300	40	)	)	PUNCT
ap-961	300	41	depending	depend	VERB
ap-961	300	42	on	on	ADP
ap-961	300	43	n.	n.	NOUN
ap-961	300	44	clearly	clearly	ADV
ap-961	300	45	d2	d2	PROPN
ap-961	300	46	0	0	NUM
ap-961	300	47	�	�	PROPN
ap-961	300	48	,	,	PUNCT
ap-961	300	49	as	as	ADP
ap-961	300	50	:	:	PUNCT
ap-961	300	51	[	[	PUNCT
ap-961	300	52	,	,	PUNCT
ap-961	300	53	[	[	PUNCT
ap-961	300	54	,	,	PUNCT
ap-961	300	55	]	]	PUNCT
ap-961	300	56	]	]	PUNCT
ap-961	301	1	[	[	PUNCT
ap-961	301	2	,	,	PUNCT
ap-961	301	3	[	[	PUNCT
ap-961	301	4	,	,	PUNCT
ap-961	301	5	]	]	X
ap-961	301	6	]	]	X
ap-961	301	7	f	f	X
ap-961	301	8	f	f	X
ap-961	301	9	x	x	X
ap-961	301	10	f	f	PROPN
ap-961	301	11	f	f	PROPN
ap-961	301	12	x	x	PROPN
ap-961	301	13	�	�	PROPN
ap-961	301	14	�	�	PROPN
ap-961	301	15	0	0	NUM
ap-961	301	16	.	.	PUNCT
ap-961	301	17	�	�	PROPN
ap-961	301	18	we	we	PRON
ap-961	301	19	have	have	VERB
ap-961	301	20	a	a	DET
ap-961	301	21	first	first	ADJ
ap-961	301	22	crude	crude	ADJ
ap-961	301	23	method	method	NOUN
ap-961	301	24	for	for	ADP
ap-961	301	25	obtaining	obtain	VERB
ap-961	301	26	differential	differential	ADJ
ap-961	301	27	algebra	algebra	NOUN
ap-961	301	28	through	through	ADP
ap-961	301	29	the	the	DET
ap-961	301	30	representation	representation	NOUN
ap-961	301	31	of	of	ADP
ap-961	301	32	�	�	PROPN
ap-961	301	33	on	on	ADP
ap-961	301	34	a	a	DET
ap-961	301	35	hilbert	hilbert	NOUN
ap-961	301	36	space	space	NOUN
ap-961	301	37	.	.	PUNCT
ap-961	302	1	of	of	ADP
ap-961	302	2	course	course	NOUN
ap-961	302	3	,	,	PUNCT
ap-961	302	4	the	the	DET
ap-961	302	5	differential	differential	ADJ
ap-961	302	6	calculi	calculi	NOUN
ap-961	302	7	obtained	obtain	VERB
ap-961	302	8	in	in	ADP
ap-961	302	9	that	that	DET
ap-961	302	10	way	way	NOUN
ap-961	302	11	are	be	AUX
ap-961	302	12	not	not	PART
ap-961	302	13	very	very	ADV
ap-961	302	14	nice	nice	ADJ
ap-961	302	15	.	.	PUNCT
ap-961	303	1	even	even	ADV
ap-961	303	2	for	for	ADP
ap-961	303	3	the	the	DET
ap-961	303	4	commutative	commutative	ADJ
ap-961	303	5	algebras	algebra	NOUN
ap-961	303	6	they	they	PRON
ap-961	303	7	are	be	AUX
ap-961	303	8	rather	rather	ADV
ap-961	303	9	awkward	awkward	ADJ
ap-961	303	10	and	and	CCONJ
ap-961	303	11	–	–	PUNCT
ap-961	303	12	in	in	ADP
ap-961	303	13	particular	particular	ADJ
ap-961	303	14	–	–	PUNCT
ap-961	303	15	infinite	infinite	ADJ
ap-961	303	16	dimensional	dimensional	ADJ
ap-961	303	17	.	.	PUNCT
ap-961	304	1	consider	consider	VERB
ap-961	304	2	the	the	DET
ap-961	304	3	example	example	NOUN
ap-961	304	4	of	of	ADP
ap-961	304	5	a	a	DET
ap-961	304	6	circle	circle	NOUN
ap-961	304	7	:	:	PUNCT
ap-961	304	8	example	example	NOUN
ap-961	305	1	3.16	3.16	NUM
ap-961	305	2	:	:	PUNCT
ap-961	305	3	let	let	VERB
ap-961	305	4	�	�	PROPN
ap-961	305	5	be	be	AUX
ap-961	305	6	the	the	DET
ap-961	305	7	algebra	algebra	NOUN
ap-961	305	8	of	of	ADP
ap-961	305	9	functions	function	NOUN
ap-961	305	10	on	on	ADP
ap-961	305	11	the	the	DET
ap-961	305	12	circle	circle	NOUN
ap-961	305	13	(	(	PUNCT
ap-961	305	14	for	for	ADP
ap-961	305	15	our	our	PRON
ap-961	305	16	purposes	purpose	NOUN
ap-961	305	17	we	we	PRON
ap-961	305	18	shall	shall	AUX
ap-961	305	19	take	take	VERB
ap-961	305	20	the	the	DET
ap-961	305	21	only	only	ADJ
ap-961	305	22	polynomials	polynomial	NOUN
ap-961	305	23	in	in	ADP
ap-961	305	24	u	u	NOUN
ap-961	305	25	,	,	PUNCT
ap-961	305	26	with	with	ADP
ap-961	305	27	the	the	DET
ap-961	305	28	representation	representation	NOUN
ap-961	305	29	on	on	ADP
ap-961	305	30	the	the	DET
ap-961	305	31	hilbert	hilbert	NOUN
ap-961	305	32	space	space	NOUN
ap-961	305	33	with	with	ADP
ap-961	305	34	basis	basis	NOUN
ap-961	305	35	{	{	PUNCT
ap-961	305	36	}	}	PUNCT
ap-961	305	37	en	en	X
ap-961	305	38	,	,	PUNCT
ap-961	305	39	n	n	PROPN
ap-961	305	40	�	�	PROPN
ap-961	305	41	�	�	PROPN
ap-961	305	42	given	give	VERB
ap-961	305	43	as	as	ADP
ap-961	305	44	u	u	PROPN
ap-961	305	45	e	e	X
ap-961	305	46	en	en	X
ap-961	305	47	n	n	X
ap-961	305	48	�	�	PROPN
ap-961	305	49	1	1	NUM
ap-961	305	50	.	.	PUNCT
ap-961	306	1	we	we	PRON
ap-961	306	2	take	take	VERB
ap-961	306	3	the	the	DET
ap-961	306	4	operator	operator	NOUN
ap-961	306	5	f	f	PROPN
ap-961	306	6	to	to	PART
ap-961	306	7	be	be	AUX
ap-961	306	8	:	:	PUNCT
ap-961	306	9	f	f	X
ap-961	306	10	e	e	NOUN
ap-961	306	11	sign	sign	VERB
ap-961	306	12	n	n	CCONJ
ap-961	306	13	en	en	X
ap-961	306	14	n	n	CCONJ
ap-961	306	15	�	�	PROPN
ap-961	306	16	(	(	PUNCT
ap-961	306	17	)	)	PUNCT
ap-961	306	18	1	1	NUM
ap-961	306	19	2	2	NUM
ap-961	306	20	.	.	PUNCT
ap-961	307	1	what	what	PRON
ap-961	307	2	are	be	AUX
ap-961	307	3	then	then	ADV
ap-961	307	4	the	the	DET
ap-961	307	5	differential	differential	ADJ
ap-961	307	6	forms	form	NOUN
ap-961	307	7	?	?	PUNCT
ap-961	308	1	we	we	PRON
ap-961	308	2	calculate	calculate	VERB
ap-961	308	3	du	du	PROPN
ap-961	308	4	:	:	PUNCT
ap-961	308	5	du	du	AUX
ap-961	308	6	e	e	PROPN
ap-961	308	7	f	f	PROPN
ap-961	308	8	u	u	PROPN
ap-961	308	9	e	e	PROPN
ap-961	308	10	sign	sign	VERB
ap-961	308	11	n	n	PRON
ap-961	308	12	sign	sign	VERB
ap-961	308	13	n	n	CCONJ
ap-961	308	14	en	en	ADP
ap-961	308	15	n	n	CCONJ
ap-961	308	16	n	n	CCONJ
ap-961	308	17	�	�	PROPN
ap-961	308	18	�	�	PROPN
ap-961	308	19	�	�	PROPN
ap-961	308	20	�	�	PROPN
ap-961	308	21	�	�	PROPN
ap-961	308	22	�	�	PROPN
ap-961	308	23	�	�	PROPN
ap-961	308	24	�	�	PROPN
ap-961	308	25	�	�	PROPN
ap-961	308	26	�	�	PROPN
ap-961	308	27	�	�	PROPN
ap-961	308	28	�	�	PROPN
ap-961	308	29	�	�	PROPN
ap-961	308	30	�	�	PROPN
ap-961	308	31	�	�	PROPN
ap-961	308	32	�	�	PROPN
ap-961	308	33	�	�	PROPN
ap-961	308	34	�	�	PROPN
ap-961	308	35	�	�	PROPN
ap-961	308	36	�	�	PROPN
ap-961	308	37	�	�	PROPN
ap-961	308	38	[	[	PUNCT
ap-961	308	39	,	,	PUNCT
ap-961	308	40	]	]	SYM
ap-961	308	41	3	3	NUM
ap-961	308	42	2	2	NUM
ap-961	308	43	1	1	NUM
ap-961	308	44	2	2	NUM
ap-961	308	45	�	�	PROPN
ap-961	308	46	�	�	PROPN
ap-961	308	47	1	1	NUM
ap-961	308	48	1	1	NUM
ap-961	308	49	12	12	NUM
ap-961	308	50	�	�	PROPN
ap-961	308	51	n	n	SYM
ap-961	308	52	ne	ne	NOUN
ap-961	308	53	,	,	PUNCT
ap-961	308	54	the	the	DET
ap-961	308	55	differential	differential	ADJ
ap-961	308	56	form	form	NOUN
ap-961	308	57	du	du	NOUN
ap-961	308	58	is	be	AUX
ap-961	308	59	an	an	DET
ap-961	308	60	operator	operator	NOUN
ap-961	308	61	of	of	ADP
ap-961	308	62	finite	finite	PROPN
ap-961	308	63	rank	rank	PROPN
ap-961	308	64	and	and	CCONJ
ap-961	308	65	,	,	PUNCT
ap-961	308	66	as	as	SCONJ
ap-961	308	67	we	we	PRON
ap-961	308	68	can	can	AUX
ap-961	308	69	easily	easily	ADV
ap-961	308	70	see	see	VERB
ap-961	308	71	,	,	PUNCT
ap-961	308	72	1	1	NUM
ap-961	308	73	2	2	NUM
ap-961	308	74	u	u	NOUN
ap-961	308	75	du	du	X
ap-961	308	76	*	*	VERB
ap-961	308	77	is	be	AUX
ap-961	308	78	a	a	DET
ap-961	308	79	projection	projection	NOUN
ap-961	308	80	on	on	ADP
ap-961	308	81	the	the	DET
ap-961	308	82	subspace	subspace	NOUN
ap-961	308	83	spanned	span	VERB
ap-961	308	84	by	by	ADP
ap-961	308	85	e	e	PROPN
ap-961	308	86	�	�	PROPN
ap-961	308	87	1	1	NUM
ap-961	308	88	.	.	PUNCT
ap-961	308	89	exercise	exercise	VERB
ap-961	308	90	3.17	3.17	NUM
ap-961	308	91	:	:	PUNCT
ap-961	308	92	show	show	VERB
ap-961	308	93	that	that	SCONJ
ap-961	308	94	every	every	DET
ap-961	308	95	differential	differential	ADJ
ap-961	308	96	form	form	NOUN
ap-961	308	97	over	over	ADP
ap-961	308	98	the	the	DET
ap-961	308	99	algebra	algebra	NOUN
ap-961	308	100	of	of	ADP
ap-961	308	101	polynomials	polynomial	NOUN
ap-961	308	102	in	in	ADP
ap-961	308	103	u	u	NOUN
ap-961	308	104	and	and	CCONJ
ap-961	308	105	u	u	NOUN
ap-961	308	106	*	*	PUNCT
ap-961	308	107	is	be	AUX
ap-961	308	108	in	in	ADP
ap-961	308	109	fact	fact	NOUN
ap-961	308	110	a	a	DET
ap-961	308	111	finite	finite	ADJ
ap-961	308	112	rank	rank	NOUN
ap-961	308	113	operator	operator	NOUN
ap-961	308	114	.	.	PUNCT
ap-961	309	1	can	can	AUX
ap-961	309	2	we	we	PRON
ap-961	309	3	generalize	generalize	VERB
ap-961	309	4	this	this	DET
ap-961	309	5	result	result	NOUN
ap-961	309	6	to	to	ADP
ap-961	309	7	the	the	DET
ap-961	309	8	one	one	NUM
ap-961	309	9	-	-	PUNCT
ap-961	309	10	forms	form	NOUN
ap-961	309	11	constructed	construct	VERB
ap-961	309	12	with	with	ADP
ap-961	309	13	f	f	PROPN
ap-961	309	14	over	over	ADP
ap-961	309	15	the	the	DET
ap-961	309	16	algebra	algebra	NOUN
ap-961	309	17	of	of	ADP
ap-961	309	18	continuous	continuous	ADJ
ap-961	309	19	function	function	NOUN
ap-961	309	20	over	over	ADP
ap-961	309	21	the	the	DET
ap-961	309	22	circle	circle	NOUN
ap-961	309	23	?	?	PUNCT
ap-961	310	1	the	the	DET
ap-961	310	2	obtained	obtain	VERB
ap-961	310	3	calculus	calculus	NOUN
ap-961	310	4	is	be	AUX
ap-961	310	5	somehow	somehow	ADV
ap-961	310	6	strange	strange	ADJ
ap-961	310	7	–	–	PUNCT
ap-961	310	8	but	but	CCONJ
ap-961	310	9	we	we	PRON
ap-961	310	10	shall	shall	AUX
ap-961	310	11	see	see	VERB
ap-961	310	12	that	that	SCONJ
ap-961	310	13	it	it	PRON
ap-961	310	14	plays	play	VERB
ap-961	310	15	an	an	DET
ap-961	310	16	important	important	ADJ
ap-961	310	17	role	role	NOUN
ap-961	310	18	.	.	PUNCT
ap-961	311	1	surely	surely	ADV
ap-961	311	2	enough	enough	ADV
ap-961	311	3	it	it	PRON
ap-961	311	4	is	be	AUX
ap-961	311	5	genuinely	genuinely	ADV
ap-961	311	6	noncommutative	noncommutative	ADJ
ap-961	311	7	even	even	ADV
ap-961	311	8	for	for	ADP
ap-961	311	9	such	such	DET
ap-961	311	10	a	a	DET
ap-961	311	11	commutative	commutative	ADJ
ap-961	311	12	space	space	NOUN
ap-961	311	13	as	as	ADP
ap-961	311	14	a	a	DET
ap-961	311	15	circle	circle	NOUN
ap-961	311	16	.	.	PUNCT
ap-961	312	1	we	we	PRON
ap-961	312	2	may	may	AUX
ap-961	312	3	,	,	PUNCT
ap-961	312	4	however	however	ADV
ap-961	312	5	,	,	PUNCT
ap-961	312	6	look	look	VERB
ap-961	312	7	for	for	ADP
ap-961	312	8	some	some	DET
ap-961	312	9	examples	example	NOUN
ap-961	312	10	of	of	ADP
ap-961	312	11	the	the	DET
ap-961	312	12	differential	differential	ADJ
ap-961	312	13	graded	grade	VERB
ap-961	312	14	algebras	algebra	NOUN
ap-961	312	15	,	,	PUNCT
ap-961	312	16	which	which	PRON
ap-961	312	17	are	be	AUX
ap-961	312	18	more	more	ADV
ap-961	312	19	“	"	PUNCT
ap-961	312	20	reasonable	reasonable	ADJ
ap-961	312	21	”	"	PUNCT
ap-961	312	22	.	.	PUNCT
ap-961	313	1	example	example	NOUN
ap-961	313	2	3.18	3.18	NUM
ap-961	313	3	:	:	PUNCT
ap-961	313	4	let	let	VERB
ap-961	313	5	�	�	PROPN
ap-961	313	6	be	be	AUX
ap-961	313	7	the	the	DET
ap-961	313	8	algebra	algebra	NOUN
ap-961	313	9	of	of	ADP
ap-961	313	10	polynomial	polynomial	ADJ
ap-961	313	11	functions	function	NOUN
ap-961	313	12	on	on	ADP
ap-961	313	13	the	the	DET
ap-961	313	14	circle	circle	NOUN
ap-961	313	15	.	.	PUNCT
ap-961	314	1	we	we	PRON
ap-961	314	2	take	take	VERB
ap-961	314	3	the	the	DET
ap-961	314	4	same	same	ADJ
ap-961	314	5	representation	representation	NOUN
ap-961	314	6	on	on	ADP
ap-961	314	7	the	the	DET
ap-961	314	8	hilbert	hilbert	NOUN
ap-961	314	9	space	space	NOUN
ap-961	314	10	as	as	ADP
ap-961	314	11	in	in	ADP
ap-961	314	12	example	example	NOUN
ap-961	314	13	3.16	3.16	NUM
ap-961	314	14	.	.	PUNCT
ap-961	315	1	but	but	CCONJ
ap-961	315	2	instead	instead	ADV
ap-961	315	3	of	of	ADP
ap-961	315	4	taking	take	VERB
ap-961	315	5	the	the	DET
ap-961	315	6	operator	operator	NOUN
ap-961	315	7	f	f	PROPN
ap-961	315	8	let	let	VERB
ap-961	315	9	us	we	PRON
ap-961	315	10	consider	consider	VERB
ap-961	315	11	an	an	DET
ap-961	315	12	unbounded	unbounded	ADJ
ap-961	315	13	operator	operator	NOUN
ap-961	315	14	d	d	NOUN
ap-961	315	15	,	,	PUNCT
ap-961	315	16	de	de	X
ap-961	315	17	nen	nen	X
ap-961	315	18	n	n	X
ap-961	315	19	�	�	PROPN
ap-961	315	20	.	.	PUNCT
ap-961	316	1	of	of	ADP
ap-961	316	2	course	course	ADV
ap-961	316	3	,	,	PUNCT
ap-961	316	4	d	d	X
ap-961	316	5	,	,	PUNCT
ap-961	316	6	as	as	SCONJ
ap-961	316	7	an	an	DET
ap-961	316	8	unbounded	unbounded	ADJ
ap-961	316	9	operator	operator	NOUN
ap-961	316	10	is	be	AUX
ap-961	316	11	only	only	ADV
ap-961	316	12	densely	densely	ADV
ap-961	316	13	defined	define	VERB
ap-961	316	14	,	,	PUNCT
ap-961	316	15	so	so	SCONJ
ap-961	316	16	we	we	PRON
ap-961	316	17	need	need	VERB
ap-961	316	18	to	to	PART
ap-961	316	19	work	work	VERB
ap-961	316	20	with	with	ADP
ap-961	316	21	a	a	DET
ap-961	316	22	dense	dense	ADJ
ap-961	316	23	subspace	subspace	NOUN
ap-961	316	24	of	of	ADP
ap-961	316	25	�	�	PROPN
ap-961	316	26	.	.	PUNCT
ap-961	317	1	the	the	DET
ap-961	317	2	one	one	NUM
ap-961	317	3	-	-	PUNCT
ap-961	317	4	forms	form	NOUN
ap-961	317	5	are	be	AUX
ap-961	317	6	all	all	PRON
ap-961	317	7	generated	generate	VERB
ap-961	317	8	by	by	ADP
ap-961	317	9	the	the	DET
ap-961	317	10	commutator	commutator	NOUN
ap-961	317	11	[	[	PUNCT
ap-961	317	12	,	,	PUNCT
ap-961	317	13	]	]	X
ap-961	317	14	d	d	X
ap-961	317	15	u	u	NOUN
ap-961	317	16	:	:	PUNCT
ap-961	317	17	[	[	PUNCT
ap-961	317	18	,	,	PUNCT
ap-961	317	19	]	]	X
ap-961	317	20	d	d	X
ap-961	317	21	u	u	X
ap-961	317	22	e	e	X
ap-961	317	23	en	en	X
ap-961	317	24	n	n	X
ap-961	317	25	�	�	PROPN
ap-961	317	26	1	1	NUM
ap-961	317	27	and	and	CCONJ
ap-961	317	28	du	du	PROPN
ap-961	317	29	is	be	AUX
ap-961	317	30	a	a	DET
ap-961	317	31	bounded	bounded	ADJ
ap-961	317	32	operator	operator	NOUN
ap-961	317	33	(	(	PUNCT
ap-961	317	34	and	and	CCONJ
ap-961	317	35	hence	hence	ADV
ap-961	317	36	well	well	ADV
ap-961	317	37	-	-	PUNCT
ap-961	317	38	defined	define	VERB
ap-961	317	39	on	on	ADP
ap-961	317	40	the	the	DET
ap-961	317	41	entire	entire	ADJ
ap-961	317	42	hilbert	hilbert	NOUN
ap-961	317	43	space	space	NOUN
ap-961	317	44	)	)	PUNCT
ap-961	317	45	.	.	PUNCT
ap-961	318	1	since	since	SCONJ
ap-961	318	2	all	all	DET
ap-961	318	3	one	one	NUM
ap-961	318	4	-	-	PUNCT
ap-961	318	5	forms	form	NOUN
ap-961	318	6	and	and	CCONJ
ap-961	318	7	higher	high	ADJ
ap-961	318	8	order	order	NOUN
ap-961	318	9	forms	form	NOUN
ap-961	318	10	arise	arise	VERB
ap-961	318	11	from	from	ADP
ap-961	318	12	products	product	NOUN
ap-961	318	13	of	of	ADP
ap-961	318	14	the	the	DET
ap-961	318	15	elements	element	NOUN
ap-961	318	16	of	of	ADP
ap-961	318	17	the	the	DET
ap-961	318	18	algebra	algebra	NOUN
ap-961	318	19	and	and	CCONJ
ap-961	318	20	du	du	NOUN
ap-961	318	21	,	,	PUNCT
ap-961	318	22	the	the	DET
ap-961	318	23	representation	representation	NOUN
ap-961	318	24	�	�	PROPN
ap-961	318	25	d	d	NOUN
ap-961	318	26	is	be	AUX
ap-961	318	27	into	into	ADP
ap-961	318	28	bounded	bounded	ADJ
ap-961	318	29	operators	operator	NOUN
ap-961	318	30	.	.	PUNCT
ap-961	319	1	observe	observe	VERB
ap-961	319	2	that	that	SCONJ
ap-961	319	3	:	:	PUNCT
ap-961	319	4	u	u	PROPN
ap-961	319	5	du	du	PROPN
ap-961	319	6	du	du	PROPN
ap-961	319	7	u	u	PROPN
ap-961	319	8	�	�	PROPN
ap-961	319	9	,	,	PUNCT
ap-961	319	10	u	u	PROPN
ap-961	319	11	du	du	PROPN
ap-961	319	12	du	du	PROPN
ap-961	319	13	u	u	PROPN
ap-961	319	14	*	*	PROPN
ap-961	319	15	*	*	PUNCT
ap-961	319	16	�	�	PROPN
ap-961	319	17	,	,	PUNCT
ap-961	319	18	42	42	NUM
ap-961	319	19	©	©	PROPN
ap-961	319	20	czech	czech	PROPN
ap-961	319	21	technical	technical	PROPN
ap-961	319	22	university	university	PROPN
ap-961	319	23	publishing	publishing	NOUN
ap-961	319	24	house	house	NOUN
ap-961	319	25	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	319	26	acta	acta	PROPN
ap-961	319	27	polytechnica	polytechnica	PROPN
ap-961	319	28	vol	vol	NOUN
ap-961	319	29	.	.	PUNCT
ap-961	320	1	48	48	NUM
ap-961	320	2	no	no	NOUN
ap-961	320	3	.	.	PUNCT
ap-961	321	1	2/2008	2/2008	NUM
ap-961	322	1	so	so	ADV
ap-961	322	2	,	,	PUNCT
ap-961	322	3	the	the	DET
ap-961	322	4	one	one	NUM
ap-961	322	5	-	-	PUNCT
ap-961	322	6	forms	form	NOUN
ap-961	322	7	really	really	ADV
ap-961	322	8	do	do	AUX
ap-961	322	9	commute	commute	VERB
ap-961	322	10	with	with	ADP
ap-961	322	11	the	the	DET
ap-961	322	12	elements	element	NOUN
ap-961	322	13	of	of	ADP
ap-961	322	14	the	the	DET
ap-961	322	15	algebra	algebra	NOUN
ap-961	322	16	.	.	PUNCT
ap-961	323	1	since	since	SCONJ
ap-961	323	2	d2	d2	PROPN
ap-961	323	3	1	1	NUM
ap-961	323	4	!	!	PUNCT
ap-961	324	1	the	the	DET
ap-961	324	2	image	image	NOUN
ap-961	324	3	of	of	ADP
ap-961	324	4	the	the	DET
ap-961	324	5	universal	universal	ADJ
ap-961	324	6	dga	dga	NOUN
ap-961	324	7	is	be	AUX
ap-961	324	8	not	not	PART
ap-961	324	9	a	a	DET
ap-961	324	10	dga	dga	NOUN
ap-961	324	11	itself	itself	PRON
ap-961	324	12	.	.	PUNCT
ap-961	325	1	for	for	ADP
ap-961	325	2	instance	instance	NOUN
ap-961	325	3	,	,	PUNCT
ap-961	325	4	in	in	ADP
ap-961	325	5	the	the	DET
ap-961	325	6	differential	differential	ADJ
ap-961	325	7	algebra	algebra	NOUN
ap-961	325	8	we	we	PRON
ap-961	325	9	would	would	AUX
ap-961	325	10	have	have	VERB
ap-961	325	11	:	:	PUNCT
ap-961	325	12	0	0	NUM
ap-961	325	13	2	2	NUM
ap-961	325	14	�	�	PROPN
ap-961	325	15	�	�	PROPN
ap-961	325	16	�	�	PROPN
ap-961	325	17	d	d	PROPN
ap-961	325	18	u	u	X
ap-961	325	19	du	du	X
ap-961	325	20	du	du	PROPN
ap-961	325	21	u	u	PROPN
ap-961	325	22	du	du	X
ap-961	325	23	du	du	X
ap-961	325	24	(	(	PUNCT
ap-961	325	25	)	)	PUNCT
ap-961	325	26	,	,	PUNCT
ap-961	325	27	but	but	CCONJ
ap-961	325	28	on	on	ADP
ap-961	325	29	the	the	DET
ap-961	325	30	other	other	ADJ
ap-961	325	31	hand	hand	NOUN
ap-961	325	32	�	�	PROPN
ap-961	325	33	�	�	PROPN
ap-961	325	34	d	d	PROPN
ap-961	325	35	ddu	ddu	PROPN
ap-961	325	36	du	du	PROPN
ap-961	325	37	(	(	PUNCT
ap-961	325	38	)	)	PUNCT
ap-961	325	39	(	(	PUNCT
ap-961	325	40	)	)	PUNCT
ap-961	325	41	is	be	AUX
ap-961	325	42	certainly	certainly	ADV
ap-961	325	43	a	a	DET
ap-961	325	44	non	non	ADJ
ap-961	325	45	-	-	ADJ
ap-961	325	46	zero	zero	NUM
ap-961	325	47	operator	operator	NOUN
ap-961	325	48	:	:	PUNCT
ap-961	325	49	(	(	PUNCT
ap-961	325	50	(	(	PUNCT
ap-961	325	51	)	)	PUNCT
ap-961	325	52	(	(	PUNCT
ap-961	325	53	)	)	PUNCT
ap-961	325	54	)	)	PUNCT
ap-961	325	55	�	�	PROPN
ap-961	325	56	�	�	PROPN
ap-961	325	57	d	d	PROPN
ap-961	325	58	d	d	PROPN
ap-961	325	59	n	n	X
ap-961	325	60	ndu	ndu	NOUN
ap-961	325	61	du	du	PROPN
ap-961	325	62	e	e	PROPN
ap-961	325	63	e	e	PROPN
ap-961	325	64	�	�	PROPN
ap-961	325	65	2	2	NUM
ap-961	325	66	to	to	PART
ap-961	325	67	obtain	obtain	VERB
ap-961	325	68	a	a	DET
ap-961	325	69	true	true	ADJ
ap-961	325	70	differential	differential	NOUN
ap-961	325	71	graded	grade	VERB
ap-961	325	72	algebra	algebra	NOUN
ap-961	325	73	we	we	PRON
ap-961	325	74	need	need	VERB
ap-961	325	75	to	to	PART
ap-961	325	76	quotient	quotient	VERB
ap-961	325	77	the	the	DET
ap-961	325	78	algebra	algebra	NOUN
ap-961	325	79	generated	generate	VERB
ap-961	325	80	by	by	ADP
ap-961	325	81	u	u	NOUN
ap-961	325	82	and	and	CCONJ
ap-961	325	83	du	du	X
ap-961	325	84	by	by	ADP
ap-961	325	85	the	the	DET
ap-961	325	86	differential	differential	ADJ
ap-961	325	87	ideal	ideal	NOUN
ap-961	325	88	.	.	PUNCT
ap-961	326	1	the	the	DET
ap-961	326	2	above	above	ADJ
ap-961	326	3	calculation	calculation	NOUN
ap-961	326	4	was	be	AUX
ap-961	326	5	not	not	PART
ap-961	326	6	a	a	DET
ap-961	326	7	coincidence	coincidence	NOUN
ap-961	326	8	,	,	PUNCT
ap-961	326	9	as	as	SCONJ
ap-961	326	10	it	it	PRON
ap-961	326	11	happens	happen	VERB
ap-961	326	12	that	that	SCONJ
ap-961	326	13	du	du	PROPN
ap-961	326	14	du	du	PROPN
ap-961	326	15	is	be	AUX
ap-961	326	16	exactly	exactly	ADV
ap-961	326	17	the	the	DET
ap-961	326	18	element	element	NOUN
ap-961	326	19	which	which	PRON
ap-961	326	20	should	should	AUX
ap-961	326	21	be	be	AUX
ap-961	326	22	added	add	VERB
ap-961	326	23	to	to	ADP
ap-961	326	24	the	the	DET
ap-961	326	25	ideal	ideal	NOUN
ap-961	326	26	generated	generate	VERB
ap-961	326	27	by	by	ADP
ap-961	326	28	ker	ker	PROPN
ap-961	326	29	�	�	PROPN
ap-961	326	30	d	d	PROPN
ap-961	326	31	in	in	ADP
ap-961	326	32	order	order	NOUN
ap-961	326	33	to	to	PART
ap-961	326	34	make	make	VERB
ap-961	326	35	it	it	PRON
ap-961	326	36	a	a	DET
ap-961	326	37	differential	differential	ADJ
ap-961	326	38	one	one	NOUN
ap-961	326	39	.	.	PUNCT
ap-961	327	1	as	as	ADP
ap-961	327	2	a	a	DET
ap-961	327	3	result	result	NOUN
ap-961	327	4	,	,	PUNCT
ap-961	327	5	we	we	PRON
ap-961	327	6	have	have	VERB
ap-961	327	7	a	a	DET
ap-961	327	8	differential	differential	NOUN
ap-961	327	9	graded	grade	VERB
ap-961	327	10	algebra	algebra	NOUN
ap-961	327	11	with	with	ADP
ap-961	327	12	central	central	ADJ
ap-961	327	13	one	one	NUM
ap-961	327	14	-	-	PUNCT
ap-961	327	15	forms	form	NOUN
ap-961	327	16	(	(	PUNCT
ap-961	327	17	commuting	commute	VERB
ap-961	327	18	with	with	ADP
ap-961	327	19	the	the	DET
ap-961	327	20	elements	element	NOUN
ap-961	327	21	of	of	ADP
ap-961	327	22	the	the	DET
ap-961	327	23	algebra	algebra	NOUN
ap-961	327	24	)	)	PUNCT
ap-961	327	25	and	and	CCONJ
ap-961	327	26	all	all	DET
ap-961	327	27	higher	high	ADJ
ap-961	327	28	-	-	PUNCT
ap-961	327	29	order	order	NOUN
ap-961	327	30	forms	form	NOUN
ap-961	327	31	vanishing	vanish	VERB
ap-961	327	32	.	.	PUNCT
ap-961	328	1	but	but	CCONJ
ap-961	328	2	this	this	PRON
ap-961	328	3	is	be	AUX
ap-961	328	4	nothing	nothing	PRON
ap-961	328	5	else	else	ADV
ap-961	328	6	than	than	ADP
ap-961	328	7	the	the	DET
ap-961	328	8	de	de	X
ap-961	328	9	rham	rham	PROPN
ap-961	328	10	differential	differential	PROPN
ap-961	328	11	algebra	algebra	PROPN
ap-961	328	12	over	over	ADP
ap-961	328	13	a	a	DET
ap-961	328	14	circle	circle	NOUN
ap-961	328	15	,	,	PUNCT
ap-961	328	16	when	when	SCONJ
ap-961	328	17	restricted	restrict	VERB
ap-961	328	18	to	to	ADP
ap-961	328	19	polynomial	polynomial	ADJ
ap-961	328	20	functions	function	NOUN
ap-961	328	21	!	!	PUNCT
ap-961	329	1	finally	finally	ADV
ap-961	329	2	,	,	PUNCT
ap-961	329	3	let	let	VERB
ap-961	329	4	us	we	PRON
ap-961	329	5	come	come	VERB
ap-961	329	6	back	back	ADV
ap-961	329	7	to	to	ADP
ap-961	329	8	the	the	DET
ap-961	329	9	case	case	NOUN
ap-961	329	10	of	of	ADP
ap-961	329	11	functions	function	NOUN
ap-961	329	12	on	on	ADP
ap-961	329	13	two	two	NUM
ap-961	329	14	-	-	PUNCT
ap-961	329	15	points	point	NOUN
ap-961	329	16	:	:	PUNCT
ap-961	329	17	exercise	exercise	VERB
ap-961	329	18	3.19	3.19	NUM
ap-961	329	19	:	:	PUNCT
ap-961	329	20	take	take	VERB
ap-961	329	21	the	the	DET
ap-961	329	22	algebra	algebra	NOUN
ap-961	329	23	of	of	ADP
ap-961	329	24	complex	complex	NOUN
ap-961	329	25	-	-	PUNCT
ap-961	329	26	valued	value	VERB
ap-961	329	27	functions	function	NOUN
ap-961	329	28	on	on	ADP
ap-961	329	29	two	two	NUM
ap-961	329	30	points	point	NOUN
ap-961	329	31	and	and	CCONJ
ap-961	329	32	its	its	PRON
ap-961	329	33	representation	representation	NOUN
ap-961	329	34	on	on	ADP
ap-961	329	35	the	the	DET
ap-961	329	36	hilbert	hilbert	PROPN
ap-961	329	37	space	space	NOUN
ap-961	329	38	�	�	PROPN
ap-961	329	39	2	2	NUM
ap-961	329	40	.	.	PUNCT
ap-961	329	41	show	show	VERB
ap-961	329	42	that	that	SCONJ
ap-961	329	43	the	the	DET
ap-961	329	44	universal	universal	ADJ
ap-961	329	45	differential	differential	ADJ
ap-961	329	46	calculus	calculus	NOUN
ap-961	329	47	is	be	AUX
ap-961	329	48	isomorphic	isomorphic	ADJ
ap-961	329	49	to	to	ADP
ap-961	329	50	the	the	DET
ap-961	329	51	calculus	calculus	NOUN
ap-961	329	52	given	give	VERB
ap-961	329	53	by	by	ADP
ap-961	329	54	the	the	DET
ap-961	329	55	operator	operator	NOUN
ap-961	330	1	f	f	X
ap-961	330	2	:	:	PUNCT
ap-961	330	3	f	f	PROPN
ap-961	330	4	�	�	PROPN
ap-961	330	5	�	�	PROPN
ap-961	330	6	�	�	PROPN
ap-961	330	7	�	�	PROPN
ap-961	330	8	�	�	PROPN
ap-961	330	9	�	�	PROPN
ap-961	330	10	�	�	PROPN
ap-961	330	11	0	0	NUM
ap-961	330	12	1	1	NUM
ap-961	330	13	1	1	NUM
ap-961	330	14	0	0	NUM
ap-961	330	15	.	.	PUNCT
ap-961	330	16	4	4	NUM
ap-961	331	1	the	the	DET
ap-961	331	2	pleasures	pleasure	NOUN
ap-961	331	3	of	of	ADP
ap-961	331	4	geometry	geometry	NOUN
ap-961	331	5	“	"	PUNCT
ap-961	331	6	geometry	geometry	NOUN
ap-961	331	7	is	be	AUX
ap-961	331	8	the	the	DET
ap-961	331	9	only	only	ADJ
ap-961	331	10	science	science	NOUN
ap-961	331	11	that	that	PRON
ap-961	331	12	it	it	PRON
ap-961	331	13	hath	hath	VERB
ap-961	331	14	pleased	please	VERB
ap-961	331	15	god	god	PROPN
ap-961	331	16	hitherto	hitherto	NOUN
ap-961	331	17	to	to	PART
ap-961	331	18	bestow	bestow	VERB
ap-961	331	19	on	on	ADP
ap-961	331	20	mankind	mankind	NOUN
ap-961	331	21	.	.	PUNCT
ap-961	331	22	”	"	PUNCT
ap-961	332	1	(	(	PUNCT
ap-961	332	2	thomas	thomas	PROPN
ap-961	332	3	hobbes	hobbes	PROPN
ap-961	332	4	)	)	PUNCT
ap-961	332	5	so	so	ADV
ap-961	332	6	far	far	ADV
ap-961	332	7	we	we	PRON
ap-961	332	8	have	have	AUX
ap-961	332	9	extended	extend	VERB
ap-961	332	10	one	one	NUM
ap-961	332	11	geometric	geometric	ADJ
ap-961	332	12	notion	notion	NOUN
ap-961	332	13	:	:	PUNCT
ap-961	332	14	that	that	SCONJ
ap-961	332	15	of	of	ADP
ap-961	332	16	differential	differential	ADJ
ap-961	332	17	algebras	algebra	NOUN
ap-961	332	18	and	and	CCONJ
ap-961	332	19	differential	differential	ADJ
ap-961	332	20	forms	form	NOUN
ap-961	332	21	.	.	PUNCT
ap-961	333	1	we	we	PRON
ap-961	333	2	still	still	ADV
ap-961	333	3	have	have	VERB
ap-961	333	4	many	many	ADJ
ap-961	333	5	tasks	task	NOUN
ap-961	333	6	ahead	ahead	ADV
ap-961	333	7	of	of	ADP
ap-961	333	8	us	we	PRON
ap-961	333	9	,	,	PUNCT
ap-961	333	10	at	at	ADP
ap-961	333	11	least	least	ADJ
ap-961	333	12	from	from	ADP
ap-961	333	13	the	the	DET
ap-961	333	14	practical	practical	ADJ
ap-961	333	15	point	point	NOUN
ap-961	333	16	of	of	ADP
ap-961	333	17	view	view	NOUN
ap-961	333	18	of	of	ADP
ap-961	333	19	applications	application	NOUN
ap-961	333	20	to	to	ADP
ap-961	333	21	physics	physics	NOUN
ap-961	333	22	.	.	PUNCT
ap-961	334	1	we	we	PRON
ap-961	334	2	need	need	VERB
ap-961	334	3	to	to	PART
ap-961	334	4	understand	understand	VERB
ap-961	334	5	the	the	DET
ap-961	334	6	noncommutative	noncommutative	ADJ
ap-961	334	7	generalization	generalization	NOUN
ap-961	334	8	of	of	ADP
ap-961	334	9	vector	vector	NOUN
ap-961	334	10	bundles	bundle	NOUN
ap-961	334	11	(	(	PUNCT
ap-961	334	12	so	so	ADV
ap-961	334	13	we	we	PRON
ap-961	334	14	shall	shall	AUX
ap-961	334	15	come	come	VERB
ap-961	334	16	back	back	ADV
ap-961	334	17	to	to	ADP
ap-961	334	18	the	the	DET
ap-961	334	19	notion	notion	NOUN
ap-961	334	20	of	of	ADP
ap-961	334	21	vector	vector	NOUN
ap-961	334	22	fields	field	NOUN
ap-961	334	23	in	in	ADP
ap-961	334	24	the	the	DET
ap-961	334	25	end	end	NOUN
ap-961	334	26	)	)	PUNCT
ap-961	334	27	,	,	PUNCT
ap-961	334	28	connections	connection	NOUN
ap-961	334	29	on	on	ADP
ap-961	334	30	them	they	PRON
ap-961	334	31	,	,	PUNCT
ap-961	334	32	integration	integration	NOUN
ap-961	334	33	and	and	CCONJ
ap-961	334	34	,	,	PUNCT
ap-961	334	35	last	last	ADJ
ap-961	334	36	not	not	PART
ap-961	334	37	least	least	ADJ
ap-961	334	38	,	,	PUNCT
ap-961	334	39	we	we	PRON
ap-961	334	40	need	need	VERB
ap-961	334	41	to	to	PART
ap-961	334	42	recognize	recognize	VERB
ap-961	334	43	whether	whether	SCONJ
ap-961	334	44	our	our	PRON
ap-961	334	45	constructions	construction	NOUN
ap-961	334	46	fall	fall	VERB
ap-961	334	47	into	into	ADP
ap-961	334	48	the	the	DET
ap-961	334	49	same	same	ADJ
ap-961	334	50	classes	class	NOUN
ap-961	334	51	from	from	ADP
ap-961	334	52	the	the	DET
ap-961	334	53	topological	topological	ADJ
ap-961	334	54	point	point	NOUN
ap-961	334	55	of	of	ADP
ap-961	334	56	view	view	NOUN
ap-961	334	57	.	.	PUNCT
ap-961	335	1	4.1	4.1	NUM
ap-961	335	2	.	.	PUNCT
ap-961	335	3	projective	projective	ADJ
ap-961	335	4	modules	module	NOUN
ap-961	335	5	in	in	ADP
ap-961	335	6	the	the	DET
ap-961	335	7	same	same	ADJ
ap-961	335	8	manner	manner	NOUN
ap-961	335	9	as	as	SCONJ
ap-961	335	10	we	we	PRON
ap-961	335	11	have	have	AUX
ap-961	335	12	dealt	deal	VERB
ap-961	335	13	with	with	ADP
ap-961	335	14	spaces	space	NOUN
ap-961	335	15	,	,	PUNCT
ap-961	335	16	which	which	PRON
ap-961	335	17	we	we	PRON
ap-961	335	18	have	have	AUX
ap-961	335	19	replaced	replace	VERB
ap-961	335	20	by	by	ADP
ap-961	335	21	(	(	PUNCT
ap-961	335	22	suitable	suitable	ADJ
ap-961	335	23	)	)	PUNCT
ap-961	335	24	algebras	algebra	NOUN
ap-961	335	25	we	we	PRON
ap-961	335	26	shall	shall	AUX
ap-961	335	27	tackle	tackle	VERB
ap-961	335	28	vector	vector	NOUN
ap-961	335	29	bundles	bundle	NOUN
ap-961	335	30	.	.	PUNCT
ap-961	336	1	so	so	ADV
ap-961	336	2	,	,	PUNCT
ap-961	336	3	instead	instead	ADV
ap-961	336	4	of	of	ADP
ap-961	336	5	taking	take	VERB
ap-961	336	6	the	the	DET
ap-961	336	7	vector	vector	NOUN
ap-961	336	8	bundle	bundle	NOUN
ap-961	336	9	itself	itself	PRON
ap-961	336	10	we	we	PRON
ap-961	336	11	take	take	VERB
ap-961	336	12	the	the	DET
ap-961	336	13	linear	linear	ADJ
ap-961	336	14	space	space	NOUN
ap-961	336	15	of	of	ADP
ap-961	336	16	all	all	DET
ap-961	336	17	its	its	PRON
ap-961	336	18	sections	section	NOUN
ap-961	336	19	.	.	PUNCT
ap-961	337	1	what	what	PRON
ap-961	337	2	is	be	AUX
ap-961	337	3	the	the	DET
ap-961	337	4	structure	structure	NOUN
ap-961	337	5	of	of	ADP
ap-961	337	6	that	that	DET
ap-961	337	7	space	space	NOUN
ap-961	337	8	?	?	PUNCT
ap-961	338	1	clearly	clearly	ADV
ap-961	338	2	,	,	PUNCT
ap-961	338	3	it	it	PRON
ap-961	338	4	is	be	AUX
ap-961	338	5	not	not	PART
ap-961	338	6	an	an	DET
ap-961	338	7	algebra	algebra	NOUN
ap-961	338	8	but	but	CCONJ
ap-961	338	9	since	since	SCONJ
ap-961	338	10	each	each	DET
ap-961	338	11	section	section	NOUN
ap-961	338	12	might	might	AUX
ap-961	338	13	be	be	AUX
ap-961	338	14	multiplied	multiply	VERB
ap-961	338	15	by	by	ADP
ap-961	338	16	the	the	DET
ap-961	338	17	function	function	NOUN
ap-961	338	18	in	in	ADP
ap-961	338	19	the	the	DET
ap-961	338	20	algebra	algebra	NOUN
ap-961	338	21	we	we	PRON
ap-961	338	22	have	have	VERB
ap-961	338	23	the	the	DET
ap-961	338	24	structure	structure	NOUN
ap-961	338	25	of	of	ADP
ap-961	338	26	a	a	DET
ap-961	338	27	module	module	NOUN
ap-961	338	28	.	.	PUNCT
ap-961	339	1	definition	definition	NOUN
ap-961	339	2	4.1	4.1	NUM
ap-961	339	3	:	:	PUNCT
ap-961	339	4	�	�	PROPN
ap-961	339	5	is	be	AUX
ap-961	339	6	a	a	DET
ap-961	339	7	left	left	ADJ
ap-961	339	8	-	-	PUNCT
ap-961	339	9	module	module	NOUN
ap-961	339	10	over	over	ADP
ap-961	339	11	the	the	DET
ap-961	339	12	algebra	algebra	NOUN
ap-961	339	13	�	�	PROPN
ap-961	339	14	if	if	SCONJ
ap-961	339	15	it	it	PRON
ap-961	339	16	is	be	AUX
ap-961	339	17	a	a	DET
ap-961	339	18	linear	linear	ADJ
ap-961	339	19	space	space	NOUN
ap-961	339	20	and	and	CCONJ
ap-961	339	21	there	there	PRON
ap-961	339	22	exists	exist	VERB
ap-961	339	23	an	an	DET
ap-961	339	24	associative	associative	ADJ
ap-961	339	25	action	action	NOUN
ap-961	339	26	:	:	PUNCT
ap-961	339	27	�	�	PROPN
ap-961	339	28	�	�	PROPN
ap-961	339	29	�	�	PROPN
ap-961	339	30	m	m	PROPN
ap-961	339	31	a	a	DET
ap-961	339	32	am	am	NOUN
ap-961	339	33	�	�	PROPN
ap-961	339	34	�	�	PROPN
ap-961	339	35	�	�	PROPN
ap-961	339	36	�	�	PROPN
ap-961	339	37	which	which	PRON
ap-961	339	38	satisfies	satisfy	VERB
ap-961	339	39	:	:	PUNCT
ap-961	339	40	a	a	DET
ap-961	339	41	m	m	VERB
ap-961	339	42	m	m	VERB
ap-961	339	43	am	be	AUX
ap-961	339	44	am	am	ADJ
ap-961	339	45	(	(	PUNCT
ap-961	339	46	)	)	SYM
ap-961	339	47	1	1	NUM
ap-961	339	48	2	2	NUM
ap-961	339	49	1	1	NUM
ap-961	339	50	2	2	NUM
ap-961	339	51	�	�	PROPN
ap-961	339	52	,	,	PUNCT
ap-961	339	53	a	a	DET
ap-961	339	54	bm	bm	PROPN
ap-961	339	55	ab	ab	PROPN
ap-961	339	56	m	m	PROPN
ap-961	339	57	(	(	PUNCT
ap-961	339	58	)	)	PUNCT
ap-961	339	59	(	(	PUNCT
ap-961	339	60	)	)	PUNCT
ap-961	339	61	�	�	PROPN
ap-961	339	62	,	,	PUNCT
ap-961	339	63	for	for	ADP
ap-961	339	64	all	all	DET
ap-961	339	65	a	a	DET
ap-961	339	66	b	b	NOUN
ap-961	339	67	,	,	PUNCT
ap-961	339	68	,	,	PUNCT
ap-961	339	69	�	�	PROPN
ap-961	339	70	�	�	PROPN
ap-961	339	71	and	and	CCONJ
ap-961	339	72	m	m	PROPN
ap-961	339	73	m	m	NOUN
ap-961	339	74	m	m	ADJ
ap-961	339	75	,	,	PUNCT
ap-961	339	76	,	,	PUNCT
ap-961	339	77	1	1	NUM
ap-961	339	78	2	2	NUM
ap-961	339	79	�	�	PROPN
ap-961	339	80	�	�	PROPN
ap-961	339	81	.	.	PUNCT
ap-961	340	1	however	however	ADV
ap-961	340	2	,	,	PUNCT
ap-961	340	3	for	for	ADP
ap-961	340	4	a	a	DET
ap-961	340	5	vector	vector	NOUN
ap-961	340	6	bundle	bundle	NOUN
ap-961	340	7	we	we	PRON
ap-961	340	8	have	have	VERB
ap-961	340	9	a	a	DET
ap-961	340	10	condition	condition	NOUN
ap-961	340	11	of	of	ADP
ap-961	340	12	local	local	ADJ
ap-961	340	13	triviality	triviality	NOUN
ap-961	340	14	,	,	PUNCT
ap-961	340	15	which	which	PRON
ap-961	340	16	is	be	AUX
ap-961	340	17	an	an	DET
ap-961	340	18	essential	essential	ADJ
ap-961	340	19	ingredient	ingredient	NOUN
ap-961	340	20	.	.	PUNCT
ap-961	341	1	cutting	cut	VERB
ap-961	341	2	the	the	DET
ap-961	341	3	story	story	NOUN
ap-961	341	4	short	short	ADJ
ap-961	341	5	,	,	PUNCT
ap-961	341	6	it	it	PRON
ap-961	341	7	could	could	AUX
ap-961	341	8	also	also	ADV
ap-961	341	9	be	be	AUX
ap-961	341	10	nicely	nicely	ADV
ap-961	341	11	translated	translate	VERB
ap-961	341	12	to	to	ADP
ap-961	341	13	the	the	DET
ap-961	341	14	language	language	NOUN
ap-961	341	15	of	of	ADP
ap-961	341	16	modules	module	NOUN
ap-961	341	17	,	,	PUNCT
ap-961	341	18	based	base	VERB
ap-961	341	19	on	on	ADP
ap-961	341	20	the	the	DET
ap-961	341	21	crucial	crucial	ADJ
ap-961	341	22	result	result	NOUN
ap-961	341	23	of	of	ADP
ap-961	341	24	serre	serre	NOUN
ap-961	341	25	-	-	NOUN
ap-961	341	26	swan	swan	NOUN
ap-961	341	27	.	.	PUNCT
ap-961	342	1	first	first	ADV
ap-961	342	2	we	we	PRON
ap-961	342	3	need	need	VERB
ap-961	342	4	a	a	DET
ap-961	342	5	definition	definition	NOUN
ap-961	342	6	:	:	PUNCT
ap-961	342	7	definition	definition	NOUN
ap-961	342	8	4.2	4.2	NUM
ap-961	342	9	:	:	PUNCT
ap-961	342	10	the	the	DET
ap-961	342	11	module	module	NOUN
ap-961	342	12	�	�	PROPN
ap-961	342	13	over	over	ADP
ap-961	342	14	an	an	DET
ap-961	342	15	algebra	algebra	NOUN
ap-961	342	16	�	�	PROPN
ap-961	342	17	is	be	AUX
ap-961	342	18	projective	projective	ADJ
ap-961	342	19	if	if	SCONJ
ap-961	342	20	there	there	PRON
ap-961	342	21	exists	exist	VERB
ap-961	342	22	a	a	DET
ap-961	342	23	module	module	NOUN
ap-961	342	24	�	�	NOUN
ap-961	342	25	%	%	NOUN
ap-961	342	26	such	such	ADJ
ap-961	342	27	that	that	SCONJ
ap-961	342	28	�	�	PROPN
ap-961	342	29	�	�	PROPN
ap-961	342	30	�	�	PROPN
ap-961	342	31	&	&	CCONJ
ap-961	342	32	'	'	PUNCT
ap-961	342	33	%	%	NOUN
ap-961	342	34	n	n	NOUN
ap-961	342	35	for	for	ADP
ap-961	342	36	some	some	DET
ap-961	342	37	n	n	PRON
ap-961	342	38	�	�	PROPN
ap-961	342	39	0	0	NUM
ap-961	342	40	.	.	PUNCT
ap-961	343	1	the	the	DET
ap-961	343	2	module	module	NOUN
ap-961	343	3	is	be	AUX
ap-961	343	4	said	say	VERB
ap-961	343	5	to	to	PART
ap-961	343	6	be	be	AUX
ap-961	343	7	finitely	finitely	ADV
ap-961	343	8	generated	generate	VERB
ap-961	343	9	if	if	SCONJ
ap-961	343	10	there	there	PRON
ap-961	343	11	exist	exist	VERB
ap-961	343	12	a	a	DET
ap-961	343	13	finite	finite	ADJ
ap-961	343	14	number	number	NOUN
ap-961	343	15	of	of	ADP
ap-961	343	16	elements	element	NOUN
ap-961	343	17	m	m	AUX
ap-961	343	18	mk1	mk1	ADJ
ap-961	343	19	,	,	PUNCT
ap-961	343	20	,	,	PUNCT
ap-961	343	21	�	�	PROPN
ap-961	344	1	such	such	ADJ
ap-961	344	2	that	that	SCONJ
ap-961	344	3	�	�	PROPN
ap-961	344	4	�	�	PROPN
ap-961	344	5	�	�	PROPN
ap-961	344	6	�	�	PROPN
ap-961	344	7	�	�	PROPN
ap-961	344	8	�	�	PROPN
ap-961	344	9	�	�	PROPN
ap-961	344	10	�	�	PROPN
ap-961	344	11	�	�	PROPN
ap-961	344	12	�	�	PROPN
ap-961	344	13	�	�	PROPN
ap-961	344	14	�	�	PROPN
ap-961	344	15	�	�	PROPN
ap-961	344	16	�	�	PROPN
ap-961	344	17	�	�	PROPN
ap-961	344	18	�	�	PROPN
ap-961	344	19	a	a	DET
ap-961	344	20	mi	mi	X
ap-961	345	1	i	i	PRON
ap-961	345	2	i	i	PRON
ap-961	346	1	k	k	PROPN
ap-961	347	1	a	a	DET
ap-961	347	2	i	i	NOUN
ap-961	347	3	1	1	NUM
ap-961	347	4	then	then	ADV
ap-961	347	5	we	we	PRON
ap-961	347	6	have	have	VERB
ap-961	347	7	serre	serre	ADJ
ap-961	347	8	-	-	ADJ
ap-961	347	9	swan	swan	ADJ
ap-961	347	10	equivalence	equivalence	NOUN
ap-961	347	11	:	:	PUNCT
ap-961	347	12	theorem	theorem	VERB
ap-961	347	13	4.3	4.3	NUM
ap-961	347	14	:	:	PUNCT
ap-961	347	15	the	the	DET
ap-961	347	16	continuous	continuous	ADJ
ap-961	347	17	sections	section	NOUN
ap-961	347	18	of	of	ADP
ap-961	347	19	a	a	DET
ap-961	347	20	vector	vector	NOUN
ap-961	347	21	bundle	bundle	NOUN
ap-961	347	22	over	over	ADP
ap-961	347	23	a	a	DET
ap-961	347	24	manifold	manifold	ADJ
ap-961	347	25	form	form	NOUN
ap-961	347	26	a	a	DET
ap-961	347	27	finitely	finitely	ADV
ap-961	347	28	generated	generate	VERB
ap-961	347	29	projective	projective	ADJ
ap-961	347	30	module	module	NOUN
ap-961	347	31	over	over	ADP
ap-961	347	32	the	the	DET
ap-961	347	33	algebra	algebra	NOUN
ap-961	347	34	of	of	ADP
ap-961	347	35	continuous	continuous	ADJ
ap-961	347	36	functions	function	NOUN
ap-961	347	37	on	on	ADP
ap-961	347	38	the	the	DET
ap-961	347	39	manifold	manifold	NOUN
ap-961	347	40	.	.	PUNCT
ap-961	348	1	in	in	ADP
ap-961	348	2	turn	turn	NOUN
ap-961	348	3	,	,	PUNCT
ap-961	348	4	every	every	DET
ap-961	348	5	finitely	finitely	ADV
ap-961	348	6	generated	generate	VERB
ap-961	348	7	projective	projective	ADJ
ap-961	348	8	module	module	NOUN
ap-961	348	9	over	over	ADP
ap-961	348	10	a	a	DET
ap-961	348	11	commutative	commutative	ADJ
ap-961	348	12	algebra	algebra	NOUN
ap-961	348	13	of	of	ADP
ap-961	348	14	continuous	continuous	ADJ
ap-961	348	15	functions	function	NOUN
ap-961	348	16	is	be	AUX
ap-961	348	17	of	of	ADP
ap-961	348	18	that	that	DET
ap-961	348	19	form	form	NOUN
ap-961	348	20	.	.	PUNCT
ap-961	349	1	due	due	ADP
ap-961	349	2	to	to	ADP
ap-961	349	3	this	this	DET
ap-961	349	4	theorem	theorem	NOUN
ap-961	349	5	we	we	PRON
ap-961	349	6	have	have	VERB
ap-961	349	7	another	another	DET
ap-961	349	8	straightforward	straightforward	ADJ
ap-961	349	9	generalization	generalization	NOUN
ap-961	349	10	:	:	PUNCT
ap-961	349	11	sections	section	NOUN
ap-961	349	12	of	of	ADP
ap-961	349	13	vector	vector	NOUN
ap-961	349	14	bundles	bundle	NOUN
ap-961	349	15	are	be	AUX
ap-961	349	16	just	just	ADV
ap-961	349	17	elements	element	NOUN
ap-961	349	18	of	of	ADP
ap-961	349	19	projective	projective	ADJ
ap-961	349	20	modules	module	NOUN
ap-961	349	21	!	!	PUNCT
ap-961	350	1	note	note	VERB
ap-961	350	2	that	that	SCONJ
ap-961	350	3	there	there	PRON
ap-961	350	4	are	be	VERB
ap-961	350	5	,	,	PUNCT
ap-961	350	6	of	of	ADP
ap-961	350	7	course	course	NOUN
ap-961	350	8	,	,	PUNCT
ap-961	350	9	several	several	ADJ
ap-961	350	10	equivalent	equivalent	ADJ
ap-961	350	11	but	but	CCONJ
ap-961	350	12	different	different	ADJ
ap-961	350	13	definitions	definition	NOUN
ap-961	350	14	of	of	ADP
ap-961	350	15	projectivity	projectivity	NOUN
ap-961	350	16	in	in	ADP
ap-961	350	17	addition	addition	NOUN
ap-961	350	18	to	to	ADP
ap-961	350	19	4.2	4.2	NUM
ap-961	350	20	.	.	PUNCT
ap-961	351	1	exercise	exercise	VERB
ap-961	351	2	4.4	4.4	NUM
ap-961	351	3	:	:	PUNCT
ap-961	351	4	let	let	VERB
ap-961	351	5	�	�	PROPN
ap-961	351	6	be	be	AUX
ap-961	351	7	a	a	DET
ap-961	351	8	left	left	ADJ
ap-961	351	9	module	module	NOUN
ap-961	351	10	over	over	ADP
ap-961	351	11	�	�	PROPN
ap-961	351	12	.	.	PUNCT
ap-961	352	1	show	show	VERB
ap-961	352	2	that	that	SCONJ
ap-961	352	3	if	if	SCONJ
ap-961	352	4	any	any	DET
ap-961	352	5	surjective	surjective	ADJ
ap-961	352	6	module	module	NOUN
ap-961	352	7	morphism	morphism	NOUN
ap-961	352	8	�	�	PROPN
ap-961	352	9	:	:	PUNCT
ap-961	352	10	�	�	PROPN
ap-961	352	11	�	�	PROPN
ap-961	352	12	splits	split	VERB
ap-961	352	13	for	for	ADP
ap-961	352	14	any	any	DET
ap-961	352	15	�	�	PROPN
ap-961	352	16	left	leave	VERB
ap-961	352	17	module	module	NOUN
ap-961	352	18	,	,	PUNCT
ap-961	352	19	(	(	PUNCT
ap-961	352	20	which	which	PRON
ap-961	352	21	means	mean	VERB
ap-961	352	22	that	that	SCONJ
ap-961	352	23	there	there	PRON
ap-961	352	24	exists	exist	VERB
ap-961	352	25	a	a	DET
ap-961	352	26	morphism	morphism	NOUN
ap-961	352	27	�	�	PROPN
ap-961	352	28	:	:	PUNCT
ap-961	352	29	�	�	PROPN
ap-961	352	30	�	�	PROPN
ap-961	352	31	,	,	PUNCT
ap-961	352	32	such	such	ADJ
ap-961	352	33	that	that	SCONJ
ap-961	352	34	�	�	PROPN
ap-961	352	35	�	�	PROPN
ap-961	352	36	�	�	PROPN
ap-961	352	37	�	�	PROPN
ap-961	352	38	i	i	PROPN
ap-961	352	39	d	d	PROPN
ap-961	352	40	�	�	PROPN
ap-961	352	41	)	)	PUNCT
ap-961	352	42	then	then	ADV
ap-961	352	43	the	the	DET
ap-961	352	44	module	module	NOUN
ap-961	352	45	�	�	PROPN
ap-961	352	46	is	be	AUX
ap-961	352	47	projective	projective	ADJ
ap-961	352	48	.	.	PUNCT
ap-961	353	1	another	another	DET
ap-961	353	2	equivalent	equivalent	ADJ
ap-961	353	3	statement	statement	NOUN
ap-961	353	4	is	be	AUX
ap-961	353	5	that	that	SCONJ
ap-961	353	6	for	for	ADP
ap-961	353	7	any	any	DET
ap-961	353	8	surjective	surjective	ADJ
ap-961	353	9	module	module	NOUN
ap-961	353	10	morphism	morphism	NOUN
ap-961	353	11	�	�	PROPN
ap-961	353	12	:	:	PUNCT
ap-961	353	13	(	(	PUNCT
ap-961	353	14	�	�	PROPN
ap-961	353	15	every	every	DET
ap-961	353	16	homomorphism	homomorphism	PROPN
ap-961	353	17	�	�	PROPN
ap-961	353	18	:	:	PUNCT
ap-961	353	19	�	�	PROPN
ap-961	353	20	�	�	PROPN
ap-961	353	21	can	can	AUX
ap-961	353	22	be	be	AUX
ap-961	353	23	lifted	lift	VERB
ap-961	353	24	to	to	ADP
ap-961	353	25	a	a	DET
ap-961	353	26	homomorphism	homomorphism	NOUN
ap-961	353	27	(	(	PUNCT
ap-961	353	28	�	�	PROPN
ap-961	353	29	(	(	PUNCT
ap-961	353	30	�	�	PROPN
ap-961	353	31	:	:	PUNCT
ap-961	353	32	�	�	PROPN
ap-961	353	33	such	such	ADJ
ap-961	353	34	that	that	SCONJ
ap-961	353	35	�	�	PROPN
ap-961	353	36	�	�	PROPN
ap-961	353	37	�	�	PROPN
ap-961	353	38	�	�	PROPN
ap-961	353	39	(	(	PUNCT
ap-961	353	40	�	�	PROPN
ap-961	353	41	.	.	PUNCT
ap-961	354	1	why	why	SCONJ
ap-961	354	2	don	don	PROPN
ap-961	354	3	we	we	PRON
ap-961	354	4	not	not	PART
ap-961	354	5	play	play	VERB
ap-961	354	6	with	with	ADP
ap-961	354	7	projective	projective	ADJ
ap-961	354	8	modules	module	NOUN
ap-961	354	9	,	,	PUNCT
ap-961	354	10	which	which	PRON
ap-961	354	11	are	be	AUX
ap-961	354	12	infinitely	infinitely	ADV
ap-961	354	13	generated	generate	VERB
ap-961	354	14	?	?	PUNCT
ap-961	355	1	certainly	certainly	ADV
ap-961	355	2	,	,	PUNCT
ap-961	355	3	a	a	DET
ap-961	355	4	good	good	ADJ
ap-961	355	5	reason	reason	NOUN
ap-961	355	6	is	be	AUX
ap-961	355	7	that	that	SCONJ
ap-961	355	8	they	they	PRON
ap-961	355	9	do	do	AUX
ap-961	355	10	not	not	PART
ap-961	355	11	correspond	correspond	VERB
ap-961	355	12	to	to	ADP
ap-961	355	13	locally	locally	ADV
ap-961	355	14	trivial	trivial	ADJ
ap-961	355	15	bundles	bundle	NOUN
ap-961	355	16	.	.	PUNCT
ap-961	356	1	moreover	moreover	ADV
ap-961	356	2	,	,	PUNCT
ap-961	356	3	the	the	DET
ap-961	356	4	world	world	NOUN
ap-961	356	5	of	of	ADP
ap-961	356	6	infinitely	infinitely	ADV
ap-961	356	7	generated	generate	VERB
ap-961	356	8	modules	module	NOUN
ap-961	356	9	is	be	AUX
ap-961	356	10	a	a	DET
ap-961	356	11	strange	strange	ADJ
ap-961	356	12	one	one	NUM
ap-961	356	13	.	.	PUNCT
ap-961	357	1	look	look	VERB
ap-961	357	2	,	,	PUNCT
ap-961	357	3	for	for	ADP
ap-961	357	4	instance	instance	NOUN
ap-961	357	5	,	,	PUNCT
ap-961	357	6	at	at	ADP
ap-961	357	7	the	the	DET
ap-961	357	8	so	so	ADV
ap-961	357	9	-	-	PUNCT
ap-961	357	10	called	call	VERB
ap-961	357	11	eilenberg	eilenberg	PROPN
ap-961	357	12	swindle	swindle	NOUN
ap-961	357	13	.	.	PUNCT
ap-961	358	1	take	take	VERB
ap-961	358	2	a	a	DET
ap-961	358	3	projective	projective	ADJ
ap-961	358	4	module	module	NOUN
ap-961	358	5	�	�	PROPN
ap-961	358	6	and	and	CCONJ
ap-961	358	7	an	an	DET
ap-961	358	8	infinitely	infinitely	ADV
ap-961	358	9	generated	generate	VERB
ap-961	358	10	free	free	ADJ
ap-961	358	11	module	module	NOUN
ap-961	358	12	�	�	PROPN
ap-961	358	13	.	.	PUNCT
ap-961	359	1	then	then	ADV
ap-961	359	2	,	,	PUNCT
ap-961	359	3	if	if	SCONJ
ap-961	359	4	�	�	PROPN
ap-961	359	5	%	%	NOUN
ap-961	359	6	is	be	AUX
ap-961	359	7	the	the	DET
ap-961	359	8	completion	completion	NOUN
ap-961	359	9	of	of	ADP
ap-961	359	10	�	�	PROPN
ap-961	359	11	to	to	ADP
ap-961	359	12	a	a	DET
ap-961	359	13	free	free	ADJ
ap-961	359	14	module	module	NOUN
ap-961	359	15	we	we	PRON
ap-961	359	16	have	have	VERB
ap-961	359	17	:	:	PUNCT
ap-961	359	18	�	�	PROPN
ap-961	359	19	�	�	PROPN
ap-961	359	20	�	�	PROPN
ap-961	359	21	�	�	PROPN
ap-961	359	22	�	�	PROPN
ap-961	359	23	�	�	PROPN
ap-961	359	24	�	�	PROPN
ap-961	359	25	&	&	CCONJ
ap-961	359	26	�	�	PROPN
ap-961	359	27	&	&	CCONJ
ap-961	359	28	&	&	CCONJ
ap-961	359	29	&	&	CCONJ
ap-961	359	30	&	&	CCONJ
ap-961	359	31	&	&	CCONJ
ap-961	359	32	%	%	PROPN
ap-961	359	33	%	%	INTJ
ap-961	359	34	(	(	PUNCT
ap-961	359	35	)	)	PUNCT
ap-961	359	36	(	(	PUNCT
ap-961	359	37	)	)	PUNCT
ap-961	359	38	�	�	PROPN
ap-961	359	39	,	,	PUNCT
ap-961	359	40	so	so	ADV
ap-961	359	41	:	:	PUNCT
ap-961	359	42	�	�	PROPN
ap-961	359	43	�	�	PROPN
ap-961	359	44	�	�	PROPN
ap-961	359	45	�	�	PROPN
ap-961	359	46	�	�	PROPN
ap-961	359	47	�	�	PROPN
ap-961	359	48	&	&	CCONJ
ap-961	359	49	�	�	PROPN
ap-961	359	50	&	&	CCONJ
ap-961	359	51	&	&	CCONJ
ap-961	359	52	&	&	CCONJ
ap-961	359	53	&	&	CCONJ
ap-961	359	54	%	%	PROPN
ap-961	359	55	%	%	INTJ
ap-961	359	56	(	(	PUNCT
ap-961	359	57	)	)	PUNCT
ap-961	359	58	(	(	PUNCT
ap-961	359	59	)	)	PUNCT
ap-961	359	60	�	�	PROPN
ap-961	359	61	and	and	CCONJ
ap-961	359	62	hence	hence	ADV
ap-961	359	63	:	:	PUNCT
ap-961	359	64	©	©	PROPN
ap-961	359	65	czech	czech	PROPN
ap-961	359	66	technical	technical	PROPN
ap-961	359	67	university	university	PROPN
ap-961	359	68	publishing	publishing	NOUN
ap-961	359	69	house	house	NOUN
ap-961	359	70	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	359	71	43	43	NUM
ap-961	359	72	acta	acta	PROPN
ap-961	359	73	polytechnica	polytechnica	PROPN
ap-961	359	74	vol	vol	NOUN
ap-961	359	75	.	.	PUNCT
ap-961	360	1	48	48	NUM
ap-961	360	2	no	no	NOUN
ap-961	360	3	.	.	PUNCT
ap-961	361	1	2/2008	2/2008	PROPN
ap-961	361	2	�	�	PROPN
ap-961	361	3	�	�	PROPN
ap-961	361	4	�	�	PROPN
ap-961	361	5	�	�	PROPN
ap-961	361	6	&	&	CCONJ
ap-961	361	7	�	�	PROPN
ap-961	361	8	.	.	PUNCT
ap-961	362	1	in	in	ADP
ap-961	362	2	the	the	DET
ap-961	362	3	next	next	ADJ
ap-961	362	4	step	step	NOUN
ap-961	362	5	we	we	PRON
ap-961	362	6	need	need	VERB
ap-961	362	7	to	to	PART
ap-961	362	8	learn	learn	VERB
ap-961	362	9	a	a	DET
ap-961	362	10	bit	bit	NOUN
ap-961	362	11	more	more	ADV
ap-961	362	12	:	:	PUNCT
ap-961	362	13	how	how	SCONJ
ap-961	362	14	to	to	PART
ap-961	362	15	construct	construct	VERB
ap-961	362	16	projective	projective	ADJ
ap-961	362	17	modules	module	NOUN
ap-961	362	18	and	and	CCONJ
ap-961	362	19	how	how	SCONJ
ap-961	362	20	to	to	PART
ap-961	362	21	distinguish	distinguish	VERB
ap-961	362	22	between	between	ADP
ap-961	362	23	different	different	ADJ
ap-961	362	24	projective	projective	ADJ
ap-961	362	25	modules	module	NOUN
ap-961	362	26	!	!	PUNCT
ap-961	363	1	4.1.1	4.1.1	NUM
ap-961	363	2	modules	module	NOUN
ap-961	363	3	and	and	CCONJ
ap-961	363	4	projections	projection	NOUN
ap-961	363	5	starting	start	VERB
ap-961	363	6	with	with	ADP
ap-961	363	7	an	an	DET
ap-961	363	8	algebra	algebra	NOUN
ap-961	363	9	�	�	NOUN
ap-961	363	10	we	we	PRON
ap-961	363	11	already	already	ADV
ap-961	363	12	know	know	VERB
ap-961	363	13	how	how	SCONJ
ap-961	363	14	to	to	PART
ap-961	363	15	construct	construct	VERB
ap-961	363	16	a	a	DET
ap-961	363	17	certain	certain	ADJ
ap-961	363	18	class	class	NOUN
ap-961	363	19	of	of	ADP
ap-961	363	20	projective	projective	ADJ
ap-961	363	21	modules	module	NOUN
ap-961	363	22	:	:	PUNCT
ap-961	363	23	�	�	PROPN
ap-961	363	24	n	n	ADP
ap-961	363	25	,	,	PUNCT
ap-961	363	26	called	call	VERB
ap-961	363	27	free	free	ADJ
ap-961	363	28	modules	module	NOUN
ap-961	363	29	.	.	PUNCT
ap-961	364	1	now	now	ADV
ap-961	364	2	,	,	PUNCT
ap-961	364	3	imagine	imagine	VERB
ap-961	364	4	we	we	PRON
ap-961	364	5	have	have	VERB
ap-961	364	6	a	a	DET
ap-961	364	7	projection	projection	NOUN
ap-961	364	8	p	p	PROPN
ap-961	364	9	mn	mn	PROPN
ap-961	364	10	�	�	PROPN
ap-961	364	11	(	(	PUNCT
ap-961	364	12	)	)	PUNCT
ap-961	364	13	�	�	PROPN
ap-961	364	14	,	,	PUNCT
ap-961	364	15	which	which	PRON
ap-961	364	16	is	be	AUX
ap-961	364	17	an	an	DET
ap-961	364	18	n	n	CCONJ
ap-961	364	19	n	n	CCONJ
ap-961	364	20	�	�	PROPN
ap-961	364	21	matrix	matrix	NOUN
ap-961	364	22	with	with	ADP
ap-961	364	23	entries	entry	NOUN
ap-961	364	24	from	from	ADP
ap-961	364	25	the	the	DET
ap-961	364	26	algebra	algebra	PROPN
ap-961	364	27	�	�	PROPN
ap-961	364	28	,	,	PUNCT
ap-961	364	29	such	such	ADJ
ap-961	364	30	that	that	SCONJ
ap-961	364	31	p	p	PROPN
ap-961	364	32	p2	p2	PROPN
ap-961	364	33	�	�	PROPN
ap-961	364	34	.	.	PUNCT
ap-961	365	1	lemma	lemma	PROPN
ap-961	365	2	4.5	4.5	NUM
ap-961	365	3	:	:	PUNCT
ap-961	365	4	let	let	VERB
ap-961	365	5	p	p	PROPN
ap-961	365	6	mn	mn	PROPN
ap-961	365	7	�	�	PROPN
ap-961	365	8	(	(	PUNCT
ap-961	365	9	)	)	PUNCT
ap-961	365	10	�	�	PROPN
ap-961	365	11	be	be	AUX
ap-961	365	12	a	a	DET
ap-961	365	13	projection	projection	NOUN
ap-961	365	14	.	.	PUNCT
ap-961	366	1	let	let	VERB
ap-961	366	2	�	�	PRON
ap-961	366	3	p	p	NOUN
ap-961	366	4	be	be	AUX
ap-961	366	5	defined	define	VERB
ap-961	366	6	as	as	ADP
ap-961	366	7	a	a	DET
ap-961	366	8	subspace	subspace	NOUN
ap-961	366	9	of	of	ADP
ap-961	366	10	all	all	DET
ap-961	366	11	elements	element	NOUN
ap-961	366	12	in	in	ADP
ap-961	366	13	�	�	PROPN
ap-961	366	14	n	n	CCONJ
ap-961	366	15	such	such	ADJ
ap-961	366	16	that	that	SCONJ
ap-961	366	17	mp	mp	PROPN
ap-961	366	18	m	m	PROPN
ap-961	366	19	�	�	PROPN
ap-961	366	20	:	:	PUNCT
ap-961	366	21	m	m	VERB
ap-961	366	22	p	p	X
ap-961	366	23	mi	mi	X
ap-961	367	1	ji	ji	PROPN
ap-961	368	1	i	i	PROPN
ap-961	368	2	n	n	PROPN
ap-961	368	3	j	j	PROPN
ap-961	368	4	�	�	PROPN
ap-961	368	5	�	�	PROPN
ap-961	368	6	�	�	PROPN
ap-961	368	7	1	1	NUM
ap-961	368	8	where	where	SCONJ
ap-961	368	9	we	we	PRON
ap-961	368	10	have	have	AUX
ap-961	368	11	explicitly	explicitly	ADV
ap-961	368	12	taken	take	VERB
ap-961	368	13	m	m	PROPN
ap-961	368	14	n	n	PRON
ap-961	368	15	�	�	PROPN
ap-961	368	16	�	�	PROPN
ap-961	368	17	as	as	ADP
ap-961	368	18	a	a	DET
ap-961	368	19	collection	collection	NOUN
ap-961	368	20	of	of	ADP
ap-961	368	21	elements	element	NOUN
ap-961	368	22	from	from	ADP
ap-961	368	23	�	�	PROPN
ap-961	368	24	,	,	PUNCT
ap-961	368	25	m	m	PROPN
ap-961	368	26	m	m	PROPN
ap-961	368	27	mn	mn	PROPN
ap-961	368	28	�	�	PROPN
ap-961	368	29	{	{	PUNCT
ap-961	368	30	,	,	PUNCT
ap-961	368	31	,	,	PUNCT
ap-961	368	32	}	}	PUNCT
ap-961	368	33	1	1	NUM
ap-961	368	34	�	�	PROPN
ap-961	369	1	and	and	CCONJ
ap-961	369	2	denoted	denote	VERB
ap-961	369	3	the	the	DET
ap-961	369	4	entries	entry	NOUN
ap-961	369	5	pij	pij	NOUN
ap-961	369	6	of	of	ADP
ap-961	369	7	the	the	DET
ap-961	369	8	matrix	matrix	NOUN
ap-961	369	9	p.	p.	NOUN
ap-961	369	10	then	then	ADV
ap-961	369	11	�	�	PROPN
ap-961	370	1	p	p	PROPN
ap-961	370	2	is	be	AUX
ap-961	370	3	a	a	DET
ap-961	370	4	finitely	finitely	ADV
ap-961	370	5	generated	generate	VERB
ap-961	370	6	projective	projective	ADJ
ap-961	370	7	module	module	NOUN
ap-961	370	8	.	.	PUNCT
ap-961	371	1	proof	proof	NOUN
ap-961	371	2	.	.	PUNCT
ap-961	372	1	we	we	PRON
ap-961	372	2	need	need	VERB
ap-961	372	3	to	to	PART
ap-961	372	4	show	show	VERB
ap-961	372	5	that	that	SCONJ
ap-961	372	6	�	�	PROPN
ap-961	372	7	p	p	NOUN
ap-961	372	8	is	be	AUX
ap-961	372	9	indeed	indeed	ADV
ap-961	372	10	a	a	DET
ap-961	372	11	left	left	ADJ
ap-961	372	12	-	-	PUNCT
ap-961	372	13	module	module	NOUN
ap-961	372	14	over	over	ADP
ap-961	372	15	�	�	PROPN
ap-961	372	16	.	.	PUNCT
ap-961	373	1	this	this	PRON
ap-961	373	2	,	,	PUNCT
ap-961	373	3	however	however	ADV
ap-961	373	4	,	,	PUNCT
ap-961	373	5	follows	follow	VERB
ap-961	373	6	immediately	immediately	ADV
ap-961	373	7	from	from	ADP
ap-961	373	8	the	the	DET
ap-961	373	9	definition	definition	NOUN
ap-961	373	10	,	,	PUNCT
ap-961	373	11	if	if	SCONJ
ap-961	373	12	m	m	PROPN
ap-961	373	13	satisfies	satisfy	VERB
ap-961	373	14	mp	mp	PROPN
ap-961	373	15	m	m	PRON
ap-961	373	16	�	�	PROPN
ap-961	373	17	so	so	ADV
ap-961	373	18	does	do	VERB
ap-961	373	19	a	a	DET
ap-961	373	20	m	m	NOUN
ap-961	373	21	(	(	PUNCT
ap-961	373	22	as	as	SCONJ
ap-961	373	23	the	the	DET
ap-961	373	24	multiplication	multiplication	NOUN
ap-961	373	25	in	in	ADP
ap-961	373	26	�	�	PROPN
ap-961	373	27	is	be	AUX
ap-961	373	28	associative	associative	ADJ
ap-961	373	29	)	)	PUNCT
ap-961	373	30	.	.	PUNCT
ap-961	374	1	if	if	SCONJ
ap-961	374	2	ei	ei	PROPN
ap-961	374	3	denotes	denote	VERB
ap-961	374	4	the	the	DET
ap-961	374	5	basis	basis	NOUN
ap-961	374	6	of	of	ADP
ap-961	374	7	�	�	PROPN
ap-961	374	8	n	n	CCONJ
ap-961	374	9	,	,	PUNCT
ap-961	374	10	an	an	DET
ap-961	374	11	element	element	NOUN
ap-961	374	12	with	with	ADP
ap-961	374	13	zeroes	zero	NOUN
ap-961	374	14	at	at	ADP
ap-961	374	15	all	all	DET
ap-961	374	16	entries	entry	NOUN
ap-961	374	17	apart	apart	ADV
ap-961	374	18	from	from	ADP
ap-961	374	19	the	the	DET
ap-961	374	20	i	i	PROPN
ap-961	374	21	-	-	PUNCT
ap-961	374	22	th	th	X
ap-961	374	23	where	where	SCONJ
ap-961	374	24	it	it	PRON
ap-961	374	25	has	have	VERB
ap-961	374	26	1	1	NUM
ap-961	374	27	,	,	PUNCT
ap-961	374	28	then	then	ADV
ap-961	374	29	all	all	DET
ap-961	374	30	elements	element	NOUN
ap-961	374	31	of	of	ADP
ap-961	374	32	�	�	PROPN
ap-961	374	33	p	p	NOUN
ap-961	374	34	are	be	AUX
ap-961	374	35	generated	generate	VERB
ap-961	374	36	by	by	ADP
ap-961	374	37	{	{	PUNCT
ap-961	374	38	}	}	PUNCT
ap-961	374	39	,	,	PUNCT
ap-961	374	40	,	,	PUNCT
ap-961	374	41	e	e	PROPN
ap-961	374	42	pi	pi	NOUN
ap-961	374	43	i	i	PROPN
ap-961	374	44	n	n	X
ap-961	374	45	�	�	PROPN
ap-961	374	46	1	1	NUM
ap-961	374	47	�	�	PROPN
ap-961	374	48	.	.	PUNCT
ap-961	375	1	moreover	moreover	ADV
ap-961	375	2	,	,	PUNCT
ap-961	375	3	because	because	SCONJ
ap-961	375	4	1	1	NUM
ap-961	375	5	�	�	PROPN
ap-961	375	6	p	p	NOUN
ap-961	375	7	is	be	AUX
ap-961	375	8	also	also	ADV
ap-961	375	9	a	a	DET
ap-961	375	10	projection	projection	NOUN
ap-961	375	11	,	,	PUNCT
ap-961	375	12	�	�	PROPN
ap-961	375	13	�	�	PROPN
ap-961	375	14	�	�	PROPN
ap-961	375	15	p	p	PROPN
ap-961	375	16	p	p	PROPN
ap-961	375	17	n	n	CCONJ
ap-961	375	18	&	&	CCONJ
ap-961	375	19	�	�	PROPN
ap-961	375	20	�	�	PROPN
ap-961	375	21	1	1	NUM
ap-961	375	22	,	,	PUNCT
ap-961	375	23	so	so	CCONJ
ap-961	375	24	the	the	DET
ap-961	375	25	module	module	NOUN
ap-961	375	26	�	�	PROPN
ap-961	375	27	p	p	NOUN
ap-961	375	28	is	be	AUX
ap-961	375	29	projective	projective	ADJ
ap-961	375	30	.	.	PUNCT
ap-961	376	1	�	�	PROPN
ap-961	376	2	example	example	NOUN
ap-961	376	3	4.6	4.6	NUM
ap-961	376	4	:	:	PUNCT
ap-961	376	5	take	take	VERB
ap-961	376	6	the	the	DET
ap-961	376	7	algebra	algebra	NOUN
ap-961	376	8	of	of	ADP
ap-961	376	9	functions	function	NOUN
ap-961	376	10	over	over	ADP
ap-961	376	11	sphere	sphere	NOUN
ap-961	376	12	s2	s2	NOUN
ap-961	376	13	and	and	CCONJ
ap-961	376	14	the	the	DET
ap-961	376	15	following	follow	VERB
ap-961	376	16	projection	projection	NOUN
ap-961	376	17	in	in	ADP
ap-961	376	18	m	m	PROPN
ap-961	376	19	c	c	PROPN
ap-961	376	20	s2	s2	NOUN
ap-961	376	21	2	2	NUM
ap-961	376	22	(	(	PUNCT
ap-961	376	23	(	(	PUNCT
ap-961	376	24	)	)	PUNCT
ap-961	376	25	):	):	PUNCT
ap-961	376	26	p	p	X
ap-961	376	27	e	e	X
ap-961	376	28	e	e	X
ap-961	376	29	i	i	PRON
ap-961	376	30	i	i	PROPN
ap-961	376	31	�	�	PROPN
ap-961	376	32	�	�	PROPN
ap-961	376	33	�	�	PROPN
ap-961	376	34	�	�	PROPN
ap-961	376	35	�	�	PROPN
ap-961	376	36	�	�	PROPN
ap-961	376	37	�	�	PROPN
ap-961	376	38	�	�	PROPN
ap-961	376	39	�	�	PROPN
ap-961	376	40	�	�	PROPN
ap-961	376	41	�	�	PROPN
ap-961	376	42	1	1	NUM
ap-961	376	43	2	2	NUM
ap-961	376	44	1	1	NUM
ap-961	376	45	1	1	NUM
ap-961	376	46	cos	cos	X
ap-961	376	47	sin	sin	PROPN
ap-961	376	48	sin	sin	PROPN
ap-961	376	49	cos	cos	PROPN
ap-961	376	50	�	�	PROPN
ap-961	376	51	�	�	PROPN
ap-961	376	52	�	�	PROPN
ap-961	376	53	�	�	PROPN
ap-961	376	54	�	�	PROPN
ap-961	376	55	�	�	PROPN
ap-961	376	56	.	.	PUNCT
ap-961	377	1	the	the	DET
ap-961	377	2	projective	projective	ADJ
ap-961	377	3	module	module	NOUN
ap-961	377	4	defined	define	VERB
ap-961	377	5	by	by	ADP
ap-961	377	6	�	�	PROPN
ap-961	377	7	p	p	NOUN
ap-961	377	8	is	be	AUX
ap-961	377	9	nontrivial	nontrivial	ADJ
ap-961	377	10	(	(	PUNCT
ap-961	377	11	that	that	PRON
ap-961	377	12	is	is	ADV
ap-961	377	13	,	,	PUNCT
ap-961	377	14	it	it	PRON
ap-961	377	15	is	be	AUX
ap-961	377	16	not	not	PART
ap-961	377	17	free	free	ADJ
ap-961	377	18	)	)	PUNCT
ap-961	377	19	–	–	PUNCT
ap-961	377	20	and	and	CCONJ
ap-961	377	21	has	have	VERB
ap-961	377	22	a	a	DET
ap-961	377	23	deep	deep	ADJ
ap-961	377	24	physical	physical	ADJ
ap-961	377	25	meaning	meaning	NOUN
ap-961	377	26	.	.	PUNCT
ap-961	378	1	it	it	PRON
ap-961	378	2	is	be	AUX
ap-961	378	3	a	a	DET
ap-961	378	4	projective	projective	ADJ
ap-961	378	5	module	module	NOUN
ap-961	378	6	associated	associate	VERB
ap-961	378	7	with	with	ADP
ap-961	378	8	the	the	DET
ap-961	378	9	vector	vector	NOUN
ap-961	378	10	bundle	bundle	NOUN
ap-961	378	11	of	of	ADP
ap-961	378	12	the	the	DET
ap-961	378	13	magnetic	magnetic	ADJ
ap-961	378	14	monopole	monopole	NOUN
ap-961	378	15	.	.	PUNCT
ap-961	379	1	the	the	DET
ap-961	379	2	module	module	NOUN
ap-961	379	3	with1	with1	PROPN
ap-961	379	4	�	�	PROPN
ap-961	379	5	p	p	PROPN
ap-961	379	6	is	be	AUX
ap-961	379	7	then	then	ADV
ap-961	379	8	the	the	DET
ap-961	379	9	antimonopole	antimonopole	NOUN
ap-961	379	10	.	.	PUNCT
ap-961	380	1	the	the	DET
ap-961	380	2	next	next	ADJ
ap-961	380	3	example	example	NOUN
ap-961	380	4	shows	show	VERB
ap-961	380	5	that	that	SCONJ
ap-961	380	6	we	we	PRON
ap-961	380	7	need	need	VERB
ap-961	380	8	to	to	PART
ap-961	380	9	be	be	AUX
ap-961	380	10	careful	careful	ADJ
ap-961	380	11	about	about	ADP
ap-961	380	12	the	the	DET
ap-961	380	13	algebra	algebra	NOUN
ap-961	380	14	we	we	PRON
ap-961	380	15	take	take	VERB
ap-961	380	16	.	.	PUNCT
ap-961	381	1	in	in	ADP
ap-961	381	2	the	the	DET
ap-961	381	3	commutative	commutative	ADJ
ap-961	381	4	case	case	NOUN
ap-961	381	5	this	this	PRON
ap-961	381	6	simply	simply	ADV
ap-961	381	7	means	mean	VERB
ap-961	381	8	that	that	SCONJ
ap-961	381	9	it	it	PRON
ap-961	381	10	does	do	AUX
ap-961	381	11	matter	matter	VERB
ap-961	381	12	whether	whether	SCONJ
ap-961	381	13	we	we	PRON
ap-961	381	14	take	take	VERB
ap-961	381	15	polynomials	polynomial	NOUN
ap-961	381	16	,	,	PUNCT
ap-961	381	17	smooth	smooth	ADJ
ap-961	381	18	functions	function	NOUN
ap-961	381	19	or	or	CCONJ
ap-961	381	20	continuous	continuous	ADJ
ap-961	381	21	functions	function	NOUN
ap-961	381	22	.	.	PUNCT
ap-961	382	1	example	example	NOUN
ap-961	382	2	4.7	4.7	NUM
ap-961	382	3	:	:	PUNCT
ap-961	382	4	let	let	VERB
ap-961	382	5	c(t2	c(t2	NOUN
ap-961	382	6	)	)	PUNCT
ap-961	382	7	be	be	AUX
ap-961	382	8	the	the	DET
ap-961	382	9	algebra	algebra	NOUN
ap-961	382	10	of	of	ADP
ap-961	382	11	continuous	continuous	ADJ
ap-961	382	12	functions	function	NOUN
ap-961	382	13	over	over	ADP
ap-961	382	14	a	a	DET
ap-961	382	15	torus	torus	NOUN
ap-961	382	16	.	.	PUNCT
ap-961	383	1	consider	consider	VERB
ap-961	383	2	the	the	DET
ap-961	383	3	following	follow	VERB
ap-961	383	4	projection	projection	NOUN
ap-961	383	5	:	:	PUNCT
ap-961	384	1	p	p	X
ap-961	384	2	f	f	X
ap-961	384	3	g	g	NOUN
ap-961	384	4	h	h	NOUN
ap-961	384	5	e	e	NOUN
ap-961	384	6	g	g	NOUN
ap-961	384	7	h	h	NOUN
ap-961	384	8	e	e	NOUN
ap-961	384	9	f	f	PROPN
ap-961	385	1	i	i	PRON
ap-961	385	2	i	i	PROPN
ap-961	385	3	�	�	PROPN
ap-961	385	4	�	�	PROPN
ap-961	385	5	�	�	PROPN
ap-961	385	6	�	�	PROPN
ap-961	385	7	�	�	PROPN
ap-961	385	8	�	�	PROPN
ap-961	385	9	�	�	PROPN
ap-961	385	10	�	�	PROPN
ap-961	385	11	�	�	PROPN
ap-961	385	12	�	�	PROPN
ap-961	385	13	�	�	PROPN
ap-961	385	14	(	(	PUNCT
ap-961	385	15	)	)	PUNCT
ap-961	385	16	(	(	PUNCT
ap-961	385	17	)	)	PUNCT
ap-961	385	18	(	(	PUNCT
ap-961	385	19	)	)	PUNCT
ap-961	385	20	(	(	PUNCT
ap-961	385	21	)	)	PUNCT
ap-961	385	22	(	(	PUNCT
ap-961	385	23	)	)	PUNCT
ap-961	385	24	(	(	PUNCT
ap-961	385	25	)	)	PUNCT
ap-961	385	26	�	�	PROPN
ap-961	385	27	�	�	PROPN
ap-961	385	28	�	�	PROPN
ap-961	385	29	�	�	PROPN
ap-961	385	30	�	�	PROPN
ap-961	385	31	�	�	PROPN
ap-961	385	32	�	�	PROPN
ap-961	385	33	�	�	PROPN
ap-961	385	34	1	1	NUM
ap-961	385	35	,	,	PUNCT
ap-961	386	1	where	where	SCONJ
ap-961	386	2	real	real	ADV
ap-961	386	3	-	-	PUNCT
ap-961	386	4	valued	value	VERB
ap-961	386	5	functions	function	NOUN
ap-961	386	6	f	f	NOUN
ap-961	386	7	,	,	PUNCT
ap-961	386	8	g	g	PROPN
ap-961	386	9	,	,	PUNCT
ap-961	386	10	h	h	NOUN
ap-961	386	11	satisfy	satisfy	NOUN
ap-961	386	12	:	:	PUNCT
ap-961	386	13	gh	gh	PROPN
ap-961	386	14	�	�	PROPN
ap-961	386	15	0	0	PROPN
ap-961	386	16	and	and	CCONJ
ap-961	386	17	g	g	PROPN
ap-961	386	18	h	h	PROPN
ap-961	386	19	f	f	PROPN
ap-961	386	20	f2	f2	PROPN
ap-961	386	21	2	2	NUM
ap-961	386	22	2	2	NUM
ap-961	386	23	�	�	PROPN
ap-961	386	24	�	�	PROPN
ap-961	386	25	,	,	PUNCT
ap-961	386	26	and	and	CCONJ
ap-961	386	27	0	0	NUM
ap-961	386	28	2	2	NUM
ap-961	386	29	)	)	PUNCT
ap-961	386	30	)	)	PUNCT
ap-961	386	31	�	�	PROPN
ap-961	386	32	�	�	PROPN
ap-961	386	33	�	�	PROPN
ap-961	386	34	�	�	PROPN
ap-961	386	35	denotes	denote	VERB
ap-961	386	36	the	the	DET
ap-961	386	37	usual	usual	ADJ
ap-961	386	38	coordinates	coordinate	NOUN
ap-961	386	39	on	on	ADP
ap-961	386	40	the	the	DET
ap-961	386	41	torus	torus	NOUN
ap-961	386	42	.	.	PUNCT
ap-961	387	1	although	although	SCONJ
ap-961	387	2	the	the	DET
ap-961	387	3	verification	verification	NOUN
ap-961	387	4	that	that	PRON
ap-961	387	5	p	p	NOUN
ap-961	387	6	is	be	AUX
ap-961	387	7	a	a	DET
ap-961	387	8	projection	projection	NOUN
ap-961	387	9	is	be	AUX
ap-961	387	10	easy	easy	ADJ
ap-961	387	11	,	,	PUNCT
ap-961	387	12	we	we	PRON
ap-961	387	13	need	need	VERB
ap-961	387	14	to	to	PART
ap-961	387	15	wait	wait	VERB
ap-961	387	16	a	a	DET
ap-961	387	17	while	while	NOUN
ap-961	387	18	to	to	PART
ap-961	387	19	show	show	VERB
ap-961	387	20	that	that	SCONJ
ap-961	387	21	it	it	PRON
ap-961	387	22	might	might	AUX
ap-961	387	23	correspond	correspond	VERB
ap-961	387	24	to	to	ADP
ap-961	387	25	a	a	DET
ap-961	387	26	nontrivial	nontrivial	ADJ
ap-961	387	27	line	line	NOUN
ap-961	387	28	bundle	bundle	NOUN
ap-961	387	29	(	(	PUNCT
ap-961	387	30	a	a	DET
ap-961	387	31	1	1	NUM
ap-961	387	32	-	-	PUNCT
ap-961	387	33	dimensional	dimensional	ADJ
ap-961	387	34	complex	complex	ADJ
ap-961	387	35	vector	vector	NOUN
ap-961	387	36	bundle	bundle	NOUN
ap-961	387	37	)	)	PUNCT
ap-961	387	38	over	over	ADP
ap-961	387	39	the	the	DET
ap-961	387	40	two	two	NUM
ap-961	387	41	-	-	PUNCT
ap-961	387	42	torus	torus	NOUN
ap-961	387	43	!	!	PUNCT
ap-961	388	1	actually	actually	ADV
ap-961	388	2	,	,	PUNCT
ap-961	388	3	we	we	PRON
ap-961	388	4	might	might	AUX
ap-961	388	5	propose	propose	VERB
ap-961	388	6	a	a	DET
ap-961	388	7	different	different	ADJ
ap-961	388	8	projection	projection	NOUN
ap-961	388	9	:	:	PUNCT
ap-961	388	10	example	example	NOUN
ap-961	388	11	4.8	4.8	NUM
ap-961	388	12	:	:	PUNCT
ap-961	388	13	let	let	VERB
ap-961	388	14	c(t2	c(t2	NOUN
ap-961	388	15	)	)	PUNCT
ap-961	388	16	be	be	AUX
ap-961	388	17	again	again	ADV
ap-961	388	18	the	the	DET
ap-961	388	19	algebra	algebra	NOUN
ap-961	388	20	of	of	ADP
ap-961	388	21	continuous	continuous	ADJ
ap-961	388	22	functions	function	NOUN
ap-961	388	23	over	over	ADP
ap-961	388	24	torus	torus	NOUN
ap-961	388	25	.	.	PUNCT
ap-961	389	1	let	let	VERB
ap-961	389	2	us	we	PRON
ap-961	389	3	take	take	VERB
ap-961	389	4	the	the	DET
ap-961	389	5	following	follow	VERB
ap-961	389	6	projection	projection	PROPN
ap-961	389	7	�	�	PROPN
ap-961	389	8	p	p	NOUN
ap-961	389	9	:	:	PUNCT
ap-961	389	10	�	�	PROPN
ap-961	389	11	�	�	PROPN
ap-961	389	12	1	1	NUM
ap-961	389	13	2	2	NUM
ap-961	389	14	2	2	NUM
ap-961	389	15	1	1	NUM
ap-961	389	16	1	1	NUM
ap-961	389	17	1	1	NUM
ap-961	389	18	2	2	NUM
ap-961	389	19	2	2	NUM
ap-961	389	20	2	2	NUM
ap-961	389	21	1	1	NUM
ap-961	389	22	1	1	NUM
ap-961	389	23	2sin	2sin	NUM
ap-961	389	24	(	(	PUNCT
ap-961	389	25	cos	cos	ADJ
ap-961	389	26	)	)	PUNCT
ap-961	389	27	sin	sin	NOUN
ap-961	389	28	(	(	PUNCT
ap-961	389	29	cos	cos	X
ap-961	389	30	(	(	PUNCT
ap-961	389	31	cos	cos	PROPN
ap-961	389	32	)	)	PUNCT
ap-961	389	33	sin	sin	PROPN
ap-961	389	34	�	�	PROPN
ap-961	389	35	�	�	PROPN
ap-961	389	36	�	�	PROPN
ap-961	389	37	�	�	PROPN
ap-961	389	38	�	�	PROPN
ap-961	389	39	�	�	PROPN
ap-961	390	1	i	i	PRON
ap-961	390	2	i	i	VERB
ap-961	390	3	2	2	NUM
ap-961	390	4	2	2	NUM
ap-961	390	5	2	2	NUM
ap-961	390	6	1	1	NUM
ap-961	390	7	1	1	NUM
ap-961	390	8	2	2	NUM
ap-961	390	9	2	2	NUM
ap-961	390	10	12i	12i	NOUN
ap-961	390	11	isin	isin	PROPN
ap-961	390	12	(	(	PUNCT
ap-961	390	13	cos	cos	PROPN
ap-961	390	14	(	(	PUNCT
ap-961	390	15	cos	cos	PROPN
ap-961	390	16	)	)	PUNCT
ap-961	390	17	sin	sin	NOUN
ap-961	390	18	sin	sin	NOUN
ap-961	390	19	(	(	PUNCT
ap-961	390	20	cos	cos	PROPN
ap-961	390	21	)	)	PUNCT
ap-961	390	22	�	�	PROPN
ap-961	390	23	�	�	PROPN
ap-961	390	24	�	�	PROPN
ap-961	390	25	�	�	PROPN
ap-961	390	26	�	�	PROPN
ap-961	390	27	�	�	PROPN
ap-961	390	28	�	�	PROPN
ap-961	390	29	�	�	PROPN
ap-961	390	30	�	�	PROPN
ap-961	390	31	�	�	PROPN
ap-961	390	32	�	�	PROPN
ap-961	390	33	�	�	PROPN
ap-961	390	34	�	�	PROPN
ap-961	390	35	�	�	PROPN
ap-961	390	36	�	�	PROPN
ap-961	390	37	�	�	PROPN
ap-961	390	38	�	�	PROPN
ap-961	390	39	�	�	PROPN
ap-961	390	40	�	�	PROPN
ap-961	390	41	which	which	PRON
ap-961	390	42	(	(	PUNCT
ap-961	390	43	certainly	certainly	ADV
ap-961	390	44	)	)	PUNCT
ap-961	390	45	is	be	AUX
ap-961	390	46	not	not	PART
ap-961	390	47	of	of	ADP
ap-961	390	48	the	the	DET
ap-961	390	49	above	above	ADJ
ap-961	390	50	form	form	NOUN
ap-961	390	51	.	.	PUNCT
ap-961	391	1	again	again	ADV
ap-961	391	2	,	,	PUNCT
ap-961	391	3	to	to	PART
ap-961	391	4	see	see	VERB
ap-961	391	5	what	what	PRON
ap-961	391	6	the	the	DET
ap-961	391	7	vector	vector	NOUN
ap-961	391	8	bundle	bundle	NOUN
ap-961	391	9	is	be	AUX
ap-961	391	10	we	we	PRON
ap-961	391	11	need	need	VERB
ap-961	391	12	to	to	PART
ap-961	391	13	wait	wait	VERB
ap-961	391	14	a	a	DET
ap-961	391	15	bit	bit	NOUN
ap-961	391	16	.	.	PUNCT
ap-961	392	1	an	an	DET
ap-961	392	2	interesting	interesting	ADJ
ap-961	392	3	observation	observation	NOUN
ap-961	392	4	is	be	AUX
ap-961	392	5	that	that	SCONJ
ap-961	392	6	while	while	SCONJ
ap-961	392	7	the	the	DET
ap-961	392	8	projection	projection	NOUN
ap-961	392	9	p	p	PROPN
ap-961	392	10	might	might	AUX
ap-961	392	11	be	be	AUX
ap-961	392	12	smooth	smooth	ADJ
ap-961	392	13	(	(	PUNCT
ap-961	392	14	provided	provide	VERB
ap-961	392	15	that	that	SCONJ
ap-961	392	16	f	f	X
ap-961	392	17	,	,	PUNCT
ap-961	392	18	g	g	PROPN
ap-961	392	19	,	,	PUNCT
ap-961	392	20	h	h	PROPN
ap-961	392	21	are	be	AUX
ap-961	392	22	smooth	smooth	ADJ
ap-961	392	23	)	)	PUNCT
ap-961	392	24	,	,	PUNCT
ap-961	392	25	�	�	PROPN
ap-961	392	26	p	p	NOUN
ap-961	392	27	is	be	AUX
ap-961	392	28	only	only	ADV
ap-961	392	29	continuous	continuous	ADJ
ap-961	392	30	!	!	PUNCT
ap-961	393	1	the	the	DET
ap-961	393	2	advantage	advantage	NOUN
ap-961	393	3	is	be	AUX
ap-961	393	4	,	,	PUNCT
ap-961	393	5	however	however	ADV
ap-961	393	6	,	,	PUNCT
ap-961	393	7	that	that	SCONJ
ap-961	393	8	�	�	PROPN
ap-961	393	9	p	p	NOUN
ap-961	393	10	is	be	AUX
ap-961	393	11	close	close	ADJ
ap-961	393	12	(	(	PUNCT
ap-961	393	13	of	of	ADP
ap-961	393	14	course	course	NOUN
ap-961	393	15	,	,	PUNCT
ap-961	393	16	in	in	ADP
ap-961	393	17	a	a	DET
ap-961	393	18	naive	naive	ADJ
ap-961	393	19	sense	sense	NOUN
ap-961	393	20	)	)	PUNCT
ap-961	393	21	to	to	ADP
ap-961	393	22	the	the	DET
ap-961	393	23	matrix	matrix	NOUN
ap-961	393	24	algebra	algebra	NOUN
ap-961	393	25	over	over	ADP
ap-961	393	26	polynomials	polynomial	NOUN
ap-961	393	27	in	in	ADP
ap-961	393	28	e	e	PROPN
ap-961	393	29	i	i	PRON
ap-961	393	30	�	�	PROPN
ap-961	393	31	and	and	CCONJ
ap-961	393	32	e	e	PROPN
ap-961	393	33	i	i	PROPN
ap-961	393	34	�	�	PROPN
ap-961	393	35	.	.	PUNCT
ap-961	394	1	the	the	DET
ap-961	394	2	notion	notion	NOUN
ap-961	394	3	of	of	ADP
ap-961	394	4	equivalence	equivalence	NOUN
ap-961	394	5	of	of	ADP
ap-961	394	6	projective	projective	ADJ
ap-961	394	7	modules	module	NOUN
ap-961	394	8	is	be	AUX
ap-961	394	9	a	a	DET
ap-961	394	10	natural	natural	ADJ
ap-961	394	11	one	one	NOUN
ap-961	394	12	,	,	PUNCT
ap-961	394	13	given	give	VERB
ap-961	394	14	through	through	ADP
ap-961	394	15	module	module	NOUN
ap-961	394	16	isomorphisms	isomorphism	NOUN
ap-961	394	17	.	.	PUNCT
ap-961	395	1	(	(	PUNCT
ap-961	395	2	to	to	PART
ap-961	395	3	be	be	AUX
ap-961	395	4	more	more	ADV
ap-961	395	5	precise	precise	ADJ
ap-961	395	6	the	the	DET
ap-961	395	7	notion	notion	NOUN
ap-961	395	8	could	could	AUX
ap-961	395	9	be	be	AUX
ap-961	395	10	slightly	slightly	ADV
ap-961	395	11	relaxed	relaxed	ADJ
ap-961	395	12	to	to	ADP
ap-961	395	13	a	a	DET
ap-961	395	14	stable	stable	ADJ
ap-961	395	15	isomorphism	isomorphism	NOUN
ap-961	395	16	:	:	PUNCT
ap-961	395	17	two	two	NUM
ap-961	395	18	modules	module	NOUN
ap-961	395	19	are	be	AUX
ap-961	395	20	stably	stably	ADV
ap-961	395	21	isomorphic	isomorphic	ADJ
ap-961	395	22	,	,	PUNCT
ap-961	395	23	if	if	SCONJ
ap-961	395	24	they	they	PRON
ap-961	395	25	are	be	AUX
ap-961	395	26	isomorphic	isomorphic	ADJ
ap-961	395	27	after	after	ADP
ap-961	395	28	adding	add	VERB
ap-961	395	29	a	a	DET
ap-961	395	30	free	free	ADJ
ap-961	395	31	module	module	NOUN
ap-961	395	32	.	.	PUNCT
ap-961	395	33	)	)	PUNCT
ap-961	396	1	but	but	CCONJ
ap-961	396	2	how	how	SCONJ
ap-961	396	3	can	can	AUX
ap-961	396	4	we	we	PRON
ap-961	396	5	distinguish	distinguish	VERB
ap-961	396	6	that	that	SCONJ
ap-961	396	7	two	two	NUM
ap-961	396	8	modules	module	NOUN
ap-961	396	9	given	give	VERB
ap-961	396	10	by	by	ADP
ap-961	396	11	two	two	NUM
ap-961	396	12	different	different	ADJ
ap-961	396	13	projections	projection	NOUN
ap-961	396	14	are	be	AUX
ap-961	396	15	equivalent	equivalent	ADJ
ap-961	396	16	?	?	PUNCT
ap-961	397	1	for	for	ADP
ap-961	397	2	this	this	DET
ap-961	397	3	purpose	purpose	NOUN
ap-961	397	4	we	we	PRON
ap-961	397	5	need	need	VERB
ap-961	397	6	a	a	DET
ap-961	397	7	notion	notion	NOUN
ap-961	397	8	of	of	ADP
ap-961	397	9	the	the	DET
ap-961	397	10	equivalence	equivalence	NOUN
ap-961	397	11	of	of	ADP
ap-961	397	12	projections	projection	NOUN
ap-961	397	13	,	,	PUNCT
ap-961	397	14	which	which	PRON
ap-961	397	15	is	be	AUX
ap-961	397	16	due	due	ADJ
ap-961	397	17	to	to	ADP
ap-961	397	18	murray	murray	PROPN
ap-961	397	19	-	-	PUNCT
ap-961	397	20	von	von	PROPN
ap-961	397	21	neumann	neumann	PROPN
ap-961	397	22	.	.	PUNCT
ap-961	398	1	definition	definition	NOUN
ap-961	398	2	4.9	4.9	NUM
ap-961	398	3	:	:	PUNCT
ap-961	398	4	we	we	PRON
ap-961	398	5	say	say	VERB
ap-961	398	6	that	that	SCONJ
ap-961	398	7	the	the	DET
ap-961	398	8	projections	projection	NOUN
ap-961	398	9	p	p	NOUN
ap-961	398	10	and	and	CCONJ
ap-961	398	11	q	q	NOUN
ap-961	398	12	are	be	AUX
ap-961	398	13	equivalent	equivalent	ADJ
ap-961	398	14	,	,	PUNCT
ap-961	398	15	if	if	SCONJ
ap-961	398	16	there	there	PRON
ap-961	398	17	exists	exist	VERB
ap-961	398	18	u	u	PROPN
ap-961	398	19	mn	mn	PROPN
ap-961	398	20	�	�	PROPN
ap-961	398	21	(	(	PUNCT
ap-961	398	22	)	)	PUNCT
ap-961	398	23	�	�	PROPN
ap-961	398	24	such	such	ADJ
ap-961	398	25	that	that	SCONJ
ap-961	398	26	u	u	PROPN
ap-961	398	27	u	u	NOUN
ap-961	398	28	p	p	PROPN
ap-961	398	29	*	*	PROPN
ap-961	398	30	�	�	PROPN
ap-961	398	31	and	and	CCONJ
ap-961	398	32	u	u	NOUN
ap-961	398	33	u	u	PROPN
ap-961	398	34	q	q	PROPN
ap-961	398	35	*	*	PROPN
ap-961	398	36	�	�	PROPN
ap-961	398	37	.	.	PUNCT
ap-961	399	1	note	note	VERB
ap-961	399	2	that	that	SCONJ
ap-961	399	3	any	any	DET
ap-961	399	4	projection	projection	NOUN
ap-961	399	5	p	p	PROPN
ap-961	399	6	mn	mn	PROPN
ap-961	399	7	�	�	PROPN
ap-961	399	8	(	(	PUNCT
ap-961	399	9	)	)	PUNCT
ap-961	399	10	�	�	PROPN
ap-961	399	11	can	can	AUX
ap-961	399	12	always	always	ADV
ap-961	399	13	be	be	AUX
ap-961	399	14	embedded	embed	VERB
ap-961	399	15	in	in	ADP
ap-961	399	16	mn	mn	PROPN
ap-961	399	17	(	(	PUNCT
ap-961	399	18	)	)	PUNCT
ap-961	399	19	�	�	PROPN
ap-961	399	20	,	,	PUNCT
ap-961	399	21	n	n	CCONJ
ap-961	399	22	n	n	CCONJ
ap-961	399	23	�	�	PROPN
ap-961	399	24	,	,	PUNCT
ap-961	399	25	putting	put	VERB
ap-961	399	26	p	p	NOUN
ap-961	399	27	in	in	ADP
ap-961	399	28	the	the	DET
ap-961	399	29	upper	upper	ADJ
ap-961	399	30	left	left	ADJ
ap-961	399	31	corner	corner	NOUN
ap-961	399	32	and	and	CCONJ
ap-961	399	33	filling	fill	VERB
ap-961	399	34	the	the	DET
ap-961	399	35	remaining	remain	VERB
ap-961	399	36	entries	entry	NOUN
ap-961	399	37	with	with	ADP
ap-961	399	38	zeroes	zero	NOUN
ap-961	399	39	.	.	PUNCT
ap-961	400	1	this	this	PRON
ap-961	400	2	means	mean	VERB
ap-961	400	3	that	that	SCONJ
ap-961	400	4	the	the	DET
ap-961	400	5	equivalence	equivalence	NOUN
ap-961	400	6	of	of	ADP
ap-961	400	7	projections	projection	NOUN
ap-961	400	8	needs	need	VERB
ap-961	400	9	to	to	PART
ap-961	400	10	be	be	AUX
ap-961	400	11	understood	understand	VERB
ap-961	400	12	in	in	ADP
ap-961	400	13	m	m	PROPN
ap-961	400	14	(	(	PUNCT
ap-961	400	15	)	)	PUNCT
ap-961	400	16	�	�	PROPN
ap-961	400	17	(	(	PUNCT
ap-961	400	18	seen	see	VERB
ap-961	400	19	as	as	ADP
ap-961	400	20	an	an	DET
ap-961	400	21	inductive	inductive	ADJ
ap-961	400	22	limit	limit	NOUN
ap-961	400	23	of	of	ADP
ap-961	400	24	mn	mn	PROPN
ap-961	400	25	(	(	PUNCT
ap-961	400	26	)	)	PUNCT
ap-961	400	27	�	�	PROPN
ap-961	400	28	,	,	PUNCT
ap-961	400	29	n	n	X
ap-961	400	30	�	�	PROPN
ap-961	400	31	)	)	PUNCT
ap-961	400	32	.	.	PUNCT
ap-961	401	1	to	to	PART
ap-961	401	2	finish	finish	VERB
ap-961	401	3	this	this	DET
ap-961	401	4	section	section	NOUN
ap-961	401	5	let	let	VERB
ap-961	401	6	us	we	PRON
ap-961	401	7	mention	mention	VERB
ap-961	401	8	what	what	PRON
ap-961	401	9	we	we	PRON
ap-961	401	10	can	can	AUX
ap-961	401	11	do	do	VERB
ap-961	401	12	with	with	ADP
ap-961	401	13	the	the	DET
ap-961	401	14	newly	newly	ADV
ap-961	401	15	established	establish	VERB
ap-961	401	16	set	set	NOUN
ap-961	401	17	of	of	ADP
ap-961	401	18	all	all	DET
ap-961	401	19	equivalence	equivalence	NOUN
ap-961	401	20	classes	class	NOUN
ap-961	401	21	of	of	ADP
ap-961	401	22	projections	projection	NOUN
ap-961	401	23	.	.	PUNCT
ap-961	402	1	first	first	ADV
ap-961	402	2	,	,	PUNCT
ap-961	402	3	a	a	DET
ap-961	402	4	little	little	ADJ
ap-961	402	5	help	help	NOUN
ap-961	402	6	comes	come	VERB
ap-961	402	7	from	from	ADP
ap-961	402	8	the	the	DET
ap-961	402	9	lemma	lemma	PROPN
ap-961	402	10	:	:	PUNCT
ap-961	402	11	lemma	lemma	PROPN
ap-961	402	12	4.10	4.10	NUM
ap-961	402	13	:	:	PUNCT
ap-961	402	14	the	the	DET
ap-961	402	15	space	space	NOUN
ap-961	402	16	of	of	ADP
ap-961	402	17	equivalence	equivalence	NOUN
ap-961	402	18	classes	class	NOUN
ap-961	402	19	of	of	ADP
ap-961	402	20	projections	projection	NOUN
ap-961	402	21	has	have	VERB
ap-961	402	22	the	the	DET
ap-961	402	23	structure	structure	NOUN
ap-961	402	24	of	of	ADP
ap-961	402	25	a	a	DET
ap-961	402	26	semigroup	semigroup	NOUN
ap-961	402	27	,	,	PUNCT
ap-961	402	28	with	with	ADP
ap-961	402	29	the	the	DET
ap-961	402	30	addition	addition	NOUN
ap-961	402	31	:	:	PUNCT
ap-961	402	32	[	[	PUNCT
ap-961	402	33	]	]	X
ap-961	402	34	[	[	PUNCT
ap-961	402	35	]	]	X
ap-961	402	36	p	p	X
ap-961	402	37	q	q	X
ap-961	402	38	p	p	X
ap-961	402	39	q	q	X
ap-961	402	40	�	�	PROPN
ap-961	402	41	�	�	PROPN
ap-961	402	42	�	�	PROPN
ap-961	402	43	�	�	PROPN
ap-961	402	44	�	�	PROPN
ap-961	402	45	�	�	PROPN
ap-961	402	46	�	�	PROPN
ap-961	402	47	*	*	PUNCT
ap-961	403	1	+	+	CCONJ
ap-961	403	2	,	,	PUNCT
ap-961	403	3	.	.	PUNCT
ap-961	403	4	/	/	SYM
ap-961	404	1	0	0	NUM
ap-961	404	2	0	0	NUM
ap-961	404	3	.	.	PUNCT
ap-961	405	1	and	and	CCONJ
ap-961	405	2	thus	thus	ADV
ap-961	405	3	we	we	PRON
ap-961	405	4	have	have	AUX
ap-961	405	5	met	meet	VERB
ap-961	405	6	the	the	DET
ap-961	405	7	k	k	NOUN
ap-961	405	8	-	-	NOUN
ap-961	405	9	theory	theory	NOUN
ap-961	405	10	.	.	PUNCT
ap-961	406	1	44	44	NUM
ap-961	407	1	©	©	PROPN
ap-961	407	2	czech	czech	PROPN
ap-961	407	3	technical	technical	PROPN
ap-961	407	4	university	university	PROPN
ap-961	407	5	publishing	publishing	NOUN
ap-961	407	6	house	house	NOUN
ap-961	407	7	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	407	8	acta	acta	PROPN
ap-961	407	9	polytechnica	polytechnica	PROPN
ap-961	407	10	vol	vol	NOUN
ap-961	407	11	.	.	PUNCT
ap-961	408	1	48	48	NUM
ap-961	408	2	no	no	NOUN
ap-961	408	3	.	.	PUNCT
ap-961	409	1	2/2008	2/2008	NUM
ap-961	409	2	definition	definition	NOUN
ap-961	409	3	4.11	4.11	NUM
ap-961	409	4	:	:	PUNCT
ap-961	409	5	we	we	PRON
ap-961	409	6	define	define	VERB
ap-961	409	7	the	the	DET
ap-961	409	8	k0	k0	PROPN
ap-961	409	9	group	group	NOUN
ap-961	409	10	of	of	ADP
ap-961	409	11	the	the	DET
ap-961	409	12	algebra	algebra	PROPN
ap-961	409	13	�	�	PROPN
ap-961	409	14	as	as	ADP
ap-961	409	15	the	the	DET
ap-961	409	16	groethendieck	groethendieck	NOUN
ap-961	409	17	group	group	NOUN
ap-961	409	18	associated	associate	VERB
ap-961	409	19	with	with	ADP
ap-961	409	20	the	the	DET
ap-961	409	21	semigroup	semigroup	PROPN
ap-961	409	22	v	v	PROPN
ap-961	409	23	(	(	PUNCT
ap-961	409	24	�	�	PROPN
ap-961	409	25	)	)	PUNCT
ap-961	409	26	.	.	PUNCT
ap-961	410	1	formally	formally	ADV
ap-961	410	2	we	we	PRON
ap-961	410	3	can	can	AUX
ap-961	410	4	speak	speak	VERB
ap-961	410	5	of	of	ADP
ap-961	410	6	abstract	abstract	ADJ
ap-961	410	7	classes	class	NOUN
ap-961	410	8	of	of	ADP
ap-961	410	9	pairs	pair	NOUN
ap-961	410	10	with	with	ADP
ap-961	410	11	the	the	DET
ap-961	410	12	equivalence	equivalence	NOUN
ap-961	410	13	relation	relation	NOUN
ap-961	410	14	:	:	PUNCT
ap-961	410	15	(	(	PUNCT
ap-961	410	16	[	[	PUNCT
ap-961	410	17	]	]	X
ap-961	410	18	,	,	PUNCT
ap-961	410	19	[	[	PUNCT
ap-961	410	20	]	]	X
ap-961	410	21	)	)	PUNCT
ap-961	410	22	~	~	PUNCT
ap-961	410	23	(	(	PUNCT
ap-961	410	24	[	[	PUNCT
ap-961	410	25	]	]	X
ap-961	410	26	,	,	PUNCT
ap-961	410	27	[	[	PUNCT
ap-961	410	28	]	]	X
ap-961	410	29	)	)	PUNCT
ap-961	411	1	[	[	PUNCT
ap-961	411	2	]	]	X
ap-961	411	3	:	:	PUNCT
ap-961	411	4	[	[	PUNCT
ap-961	411	5	]	]	X
ap-961	411	6	[	[	PUNCT
ap-961	411	7	]	]	X
ap-961	411	8	[	[	PUNCT
ap-961	411	9	]	]	X
ap-961	411	10	[	[	PUNCT
ap-961	411	11	]	]	X
ap-961	411	12	[	[	PUNCT
ap-961	411	13	]	]	X
ap-961	411	14	[	[	X
ap-961	411	15	p	p	X
ap-961	411	16	q	q	X
ap-961	411	17	p	p	X
ap-961	411	18	q	q	NOUN
ap-961	411	19	r	r	NOUN
ap-961	411	20	p	p	NOUN
ap-961	411	21	q	q	NOUN
ap-961	411	22	r	r	NOUN
ap-961	411	23	p	p	NOUN
ap-961	411	24	q	q	X
ap-961	411	25	(	(	PUNCT
ap-961	411	26	(	(	PUNCT
ap-961	411	27	0	0	NUM
ap-961	411	28	(	(	PUNCT
ap-961	411	29	�	�	PROPN
ap-961	411	30	(	(	PUNCT
ap-961	411	31	r	r	NOUN
ap-961	411	32	]	]	PUNCT
ap-961	411	33	.	.	PUNCT
ap-961	412	1	the	the	DET
ap-961	412	2	origins	origin	NOUN
ap-961	412	3	of	of	ADP
ap-961	412	4	k	k	NOUN
ap-961	412	5	-	-	NOUN
ap-961	412	6	theory	theory	NOUN
ap-961	412	7	are	be	AUX
ap-961	412	8	based	base	VERB
ap-961	412	9	in	in	ADP
ap-961	412	10	the	the	DET
ap-961	412	11	classification	classification	NOUN
ap-961	412	12	problems	problem	NOUN
ap-961	412	13	of	of	ADP
ap-961	412	14	(	(	PUNCT
ap-961	412	15	real	real	ADJ
ap-961	412	16	)	)	PUNCT
ap-961	412	17	vector	vector	NOUN
ap-961	412	18	bundles	bundle	NOUN
ap-961	412	19	over	over	ADP
ap-961	412	20	manifolds	manifold	NOUN
ap-961	412	21	.	.	PUNCT
ap-961	413	1	the	the	DET
ap-961	413	2	construction	construction	NOUN
ap-961	413	3	of	of	ADP
ap-961	413	4	the	the	DET
ap-961	413	5	k0	k0	PROPN
ap-961	413	6	-	-	PUNCT
ap-961	413	7	group	group	NOUN
ap-961	413	8	in	in	ADP
ap-961	413	9	the	the	DET
ap-961	413	10	seminal	seminal	ADJ
ap-961	413	11	work	work	NOUN
ap-961	413	12	of	of	ADP
ap-961	413	13	atiyah	atiyah	NOUN
ap-961	413	14	was	be	AUX
ap-961	413	15	the	the	DET
ap-961	413	16	breakthrough	breakthrough	NOUN
ap-961	413	17	of	of	ADP
ap-961	413	18	topological	topological	ADJ
ap-961	413	19	k	k	PROPN
ap-961	413	20	-	-	NOUN
ap-961	413	21	theory	theory	NOUN
ap-961	413	22	–	–	PUNCT
ap-961	413	23	this	this	PRON
ap-961	413	24	together	together	ADV
ap-961	413	25	with	with	ADP
ap-961	413	26	the	the	DET
ap-961	413	27	serre	serre	ADJ
ap-961	413	28	-	-	ADJ
ap-961	413	29	swan	swan	NOUN
ap-961	413	30	theorem	theorem	NOUN
ap-961	413	31	,	,	PUNCT
ap-961	413	32	which	which	PRON
ap-961	413	33	formulates	formulate	VERB
ap-961	413	34	the	the	DET
ap-961	413	35	equivalence	equivalence	NOUN
ap-961	413	36	between	between	ADP
ap-961	413	37	vector	vector	NOUN
ap-961	413	38	bundles	bundle	NOUN
ap-961	413	39	and	and	CCONJ
ap-961	413	40	finitely	finitely	ADV
ap-961	413	41	generated	generate	VERB
ap-961	413	42	projective	projective	ADJ
ap-961	413	43	modules	module	NOUN
ap-961	413	44	over	over	ADP
ap-961	413	45	commutative	commutative	ADJ
ap-961	413	46	algebras	algebra	NOUN
ap-961	413	47	enabled	enable	VERB
ap-961	413	48	to	to	PART
ap-961	413	49	push	push	VERB
ap-961	413	50	the	the	DET
ap-961	413	51	theory	theory	NOUN
ap-961	413	52	to	to	ADP
ap-961	413	53	the	the	DET
ap-961	413	54	c*-algebraic	c*-algebraic	PROPN
ap-961	413	55	setup	setup	NOUN
ap-961	413	56	.	.	PUNCT
ap-961	414	1	we	we	PRON
ap-961	414	2	were	be	AUX
ap-961	414	3	very	very	ADV
ap-961	414	4	sloppy	sloppy	ADJ
ap-961	414	5	here	here	ADV
ap-961	414	6	on	on	ADP
ap-961	414	7	the	the	DET
ap-961	414	8	details	detail	NOUN
ap-961	414	9	–	–	PUNCT
ap-961	414	10	for	for	ADP
ap-961	414	11	example	example	NOUN
ap-961	414	12	,	,	PUNCT
ap-961	414	13	whether	whether	SCONJ
ap-961	414	14	we	we	PRON
ap-961	414	15	use	use	VERB
ap-961	414	16	arbitrary	arbitrary	ADJ
ap-961	414	17	algebras	algebra	NOUN
ap-961	414	18	or	or	CCONJ
ap-961	414	19	c*-algebras	c*-algebras	PROPN
ap-961	414	20	.	.	PUNCT
ap-961	415	1	for	for	ADP
ap-961	415	2	the	the	DET
ap-961	415	3	purpose	purpose	NOUN
ap-961	415	4	of	of	ADP
ap-961	415	5	the	the	DET
ap-961	415	6	first	first	ADJ
ap-961	415	7	k	k	ADJ
ap-961	415	8	-	-	ADJ
ap-961	415	9	theory	theory	NOUN
ap-961	415	10	group	group	NOUN
ap-961	415	11	,	,	PUNCT
ap-961	415	12	k0	k0	PROPN
ap-961	415	13	,	,	PUNCT
ap-961	415	14	this	this	PRON
ap-961	415	15	does	do	AUX
ap-961	415	16	not	not	PART
ap-961	415	17	really	really	ADV
ap-961	415	18	matter	matter	VERB
ap-961	415	19	(	(	PUNCT
ap-961	415	20	in	in	ADP
ap-961	415	21	the	the	DET
ap-961	415	22	sense	sense	NOUN
ap-961	415	23	that	that	SCONJ
ap-961	415	24	for	for	ADP
ap-961	415	25	c	c	PROPN
ap-961	415	26	*	*	PUNCT
ap-961	415	27	algebras	algebra	VERB
ap-961	415	28	the	the	DET
ap-961	415	29	algebraic	algebraic	PROPN
ap-961	415	30	k	k	NOUN
ap-961	415	31	-	-	NOUN
ap-961	415	32	theory	theory	NOUN
ap-961	415	33	we	we	PRON
ap-961	415	34	defined	define	VERB
ap-961	415	35	is	be	AUX
ap-961	415	36	the	the	DET
ap-961	415	37	same	same	ADJ
ap-961	415	38	as	as	ADP
ap-961	415	39	the	the	DET
ap-961	415	40	topological	topological	ADJ
ap-961	415	41	k	k	NOUN
ap-961	415	42	-	-	NOUN
ap-961	415	43	theory	theory	NOUN
ap-961	415	44	.	.	PUNCT
ap-961	416	1	the	the	DET
ap-961	416	2	difference	difference	NOUN
ap-961	416	3	arises	arise	VERB
ap-961	416	4	later	later	ADV
ap-961	416	5	,	,	PUNCT
ap-961	416	6	when	when	SCONJ
ap-961	416	7	one	one	PRON
ap-961	416	8	turns	turn	VERB
ap-961	416	9	to	to	ADP
ap-961	416	10	higher	high	ADJ
ap-961	416	11	k	k	NOUN
ap-961	416	12	-	-	NOUN
ap-961	416	13	groups	group	NOUN
ap-961	416	14	–	–	PUNCT
ap-961	416	15	but	but	CCONJ
ap-961	416	16	graciously	graciously	ADV
ap-961	416	17	we	we	PRON
ap-961	416	18	shall	shall	AUX
ap-961	416	19	not	not	PART
ap-961	416	20	touch	touch	VERB
ap-961	416	21	this	this	DET
ap-961	416	22	topic	topic	NOUN
ap-961	416	23	here	here	ADV
ap-961	416	24	.	.	PUNCT
ap-961	417	1	in	in	ADP
ap-961	417	2	principle	principle	NOUN
ap-961	417	3	,	,	PUNCT
ap-961	417	4	k	k	NOUN
ap-961	417	5	-	-	NOUN
ap-961	417	6	theory	theory	NOUN
ap-961	417	7	of	of	ADP
ap-961	417	8	a	a	DET
ap-961	417	9	c*-algebra	c*-algebra	PROPN
ap-961	417	10	is	be	AUX
ap-961	417	11	a	a	DET
ap-961	417	12	functor	functor	NOUN
ap-961	417	13	from	from	ADP
ap-961	417	14	this	this	DET
ap-961	417	15	category	category	NOUN
ap-961	417	16	to	to	ADP
ap-961	417	17	the	the	DET
ap-961	417	18	category	category	NOUN
ap-961	417	19	of	of	ADP
ap-961	417	20	abelian	abelian	ADJ
ap-961	417	21	groups	group	NOUN
ap-961	417	22	:	:	PUNCT
ap-961	417	23	k0	k0	PROPN
ap-961	417	24	is	be	AUX
ap-961	417	25	defined	define	VERB
ap-961	417	26	through	through	ADP
ap-961	417	27	equivalence	equivalence	NOUN
ap-961	417	28	classes	class	NOUN
ap-961	417	29	of	of	ADP
ap-961	417	30	projections	projection	NOUN
ap-961	417	31	,	,	PUNCT
ap-961	417	32	whereas	whereas	SCONJ
ap-961	417	33	k1	k1	PROPN
ap-961	417	34	is	be	AUX
ap-961	417	35	the	the	DET
ap-961	417	36	�	�	PROPN
ap-961	417	37	0	0	NUM
ap-961	417	38	of	of	ADP
ap-961	417	39	the	the	DET
ap-961	417	40	gl	gl	PROPN
ap-961	417	41	group	group	NOUN
ap-961	417	42	of	of	ADP
ap-961	417	43	the	the	DET
ap-961	417	44	algebra	algebra	NOUN
ap-961	417	45	(	(	PUNCT
ap-961	417	46	the	the	DET
ap-961	417	47	inductive	inductive	ADJ
ap-961	417	48	limit	limit	NOUN
ap-961	417	49	of	of	ADP
ap-961	417	50	invertible	invertible	ADJ
ap-961	417	51	matrices	matrix	NOUN
ap-961	417	52	over	over	ADP
ap-961	417	53	the	the	DET
ap-961	417	54	algebra	algebra	NOUN
ap-961	417	55	)	)	PUNCT
ap-961	417	56	.	.	PUNCT
ap-961	418	1	we	we	PRON
ap-961	418	2	skip	skip	VERB
ap-961	418	3	here	here	ADV
ap-961	418	4	the	the	DET
ap-961	418	5	details	detail	NOUN
ap-961	418	6	of	of	ADP
ap-961	418	7	the	the	DET
ap-961	418	8	construction	construction	NOUN
ap-961	418	9	and	and	CCONJ
ap-961	418	10	its	its	PRON
ap-961	418	11	properties	property	NOUN
ap-961	418	12	,	,	PUNCT
ap-961	418	13	which	which	PRON
ap-961	418	14	can	can	AUX
ap-961	418	15	be	be	AUX
ap-961	418	16	found	find	VERB
ap-961	418	17	in	in	ADP
ap-961	418	18	many	many	ADJ
ap-961	418	19	textbooks	textbook	NOUN
ap-961	418	20	.	.	PUNCT
ap-961	419	1	the	the	DET
ap-961	419	2	interested	interested	ADJ
ap-961	419	3	reader	reader	NOUN
ap-961	419	4	is	be	AUX
ap-961	419	5	recommended	recommend	VERB
ap-961	419	6	to	to	PART
ap-961	419	7	consult	consult	VERB
ap-961	419	8	them	they	PRON
ap-961	419	9	(	(	PUNCT
ap-961	419	10	see	see	VERB
ap-961	419	11	the	the	DET
ap-961	419	12	list	list	NOUN
ap-961	419	13	at	at	ADP
ap-961	419	14	the	the	DET
ap-961	419	15	end	end	NOUN
ap-961	419	16	of	of	ADP
ap-961	419	17	the	the	DET
ap-961	419	18	notes	note	NOUN
ap-961	419	19	)	)	PUNCT
ap-961	419	20	.	.	PUNCT
ap-961	420	1	for	for	ADP
ap-961	420	2	us	we	PRON
ap-961	420	3	,	,	PUNCT
ap-961	420	4	there	there	PRON
ap-961	420	5	are	be	VERB
ap-961	420	6	two	two	NUM
ap-961	420	7	important	important	ADJ
ap-961	420	8	things	thing	NOUN
ap-961	420	9	:	:	PUNCT
ap-961	420	10	first	first	ADV
ap-961	420	11	of	of	ADP
ap-961	420	12	all	all	PRON
ap-961	420	13	,	,	PUNCT
ap-961	420	14	k	k	NOUN
ap-961	420	15	-	-	NOUN
ap-961	420	16	theory	theory	NOUN
ap-961	420	17	can	can	AUX
ap-961	420	18	be	be	AUX
ap-961	420	19	calculated	calculate	VERB
ap-961	420	20	(	(	PUNCT
ap-961	420	21	thanks	thank	NOUN
ap-961	420	22	to	to	ADP
ap-961	420	23	advanced	advanced	ADJ
ap-961	420	24	tools	tool	NOUN
ap-961	420	25	such	such	ADJ
ap-961	420	26	as	as	ADP
ap-961	420	27	excision	excision	NOUN
ap-961	420	28	and	and	CCONJ
ap-961	420	29	the	the	DET
ap-961	420	30	connecting	connect	VERB
ap-961	420	31	morphism	morphism	NOUN
ap-961	420	32	between	between	ADP
ap-961	420	33	k0	k0	PROPN
ap-961	420	34	and	and	CCONJ
ap-961	420	35	k1	k1	NOUN
ap-961	420	36	)	)	PUNCT
ap-961	420	37	.	.	PUNCT
ap-961	421	1	furthermore	furthermore	ADV
ap-961	421	2	,	,	PUNCT
ap-961	421	3	it	it	PRON
ap-961	421	4	provides	provide	VERB
ap-961	421	5	important	important	ADJ
ap-961	421	6	information	information	NOUN
ap-961	421	7	about	about	ADP
ap-961	421	8	the	the	DET
ap-961	421	9	algebra	algebra	NOUN
ap-961	421	10	itself	itself	PRON
ap-961	421	11	,	,	PUNCT
ap-961	421	12	like	like	ADP
ap-961	421	13	the	the	DET
ap-961	421	14	existence	existence	NOUN
ap-961	421	15	and	and	CCONJ
ap-961	421	16	classification	classification	NOUN
ap-961	421	17	of	of	ADP
ap-961	421	18	nontrivial	nontrivial	NOUN
ap-961	421	19	(	(	PUNCT
ap-961	421	20	noncommutative	noncommutative	ADJ
ap-961	421	21	)	)	PUNCT
ap-961	421	22	vector	vector	NOUN
ap-961	421	23	bundles	bundle	NOUN
ap-961	421	24	.	.	PUNCT
ap-961	422	1	what	what	PRON
ap-961	422	2	is	be	AUX
ap-961	422	3	also	also	ADV
ap-961	422	4	significant	significant	ADJ
ap-961	422	5	,	,	PUNCT
ap-961	422	6	is	be	AUX
ap-961	422	7	that	that	SCONJ
ap-961	422	8	k	k	NOUN
ap-961	422	9	-	-	NOUN
ap-961	422	10	theory	theory	NOUN
ap-961	422	11	depends	depend	VERB
ap-961	422	12	actually	actually	ADV
ap-961	422	13	on	on	ADP
ap-961	422	14	the	the	DET
ap-961	422	15	dense	dense	ADJ
ap-961	422	16	(	(	PUNCT
ap-961	422	17	and	and	CCONJ
ap-961	422	18	stable	stable	ADJ
ap-961	422	19	under	under	ADP
ap-961	422	20	holomorphic	holomorphic	ADJ
ap-961	422	21	functional	functional	ADJ
ap-961	422	22	calculus	calculus	NOUN
ap-961	422	23	)	)	PUNCT
ap-961	423	1	subalgebra;	subalgebra;	NOUN
ap-961	423	2	that	that	PRON
ap-961	423	3	is	be	AUX
ap-961	423	4	,	,	PUNCT
ap-961	423	5	in	in	ADP
ap-961	423	6	the	the	DET
ap-961	423	7	commutative	commutative	ADJ
ap-961	423	8	case	case	NOUN
ap-961	423	9	one	one	PRON
ap-961	423	10	might	might	AUX
ap-961	423	11	work	work	VERB
ap-961	423	12	with	with	ADP
ap-961	423	13	continuous	continuous	ADJ
ap-961	423	14	as	as	ADV
ap-961	423	15	well	well	ADV
ap-961	423	16	with	with	ADP
ap-961	423	17	smooth	smooth	ADJ
ap-961	423	18	functions	function	NOUN
ap-961	423	19	,	,	PUNCT
ap-961	423	20	and	and	CCONJ
ap-961	423	21	k	k	X
ap-961	423	22	-	-	NOUN
ap-961	423	23	theory	theory	NOUN
ap-961	423	24	still	still	ADV
ap-961	423	25	does	do	AUX
ap-961	423	26	not	not	PART
ap-961	423	27	change	change	VERB
ap-961	423	28	.	.	PUNCT
ap-961	424	1	4.2	4.2	NUM
ap-961	424	2	connections	connection	NOUN
ap-961	424	3	on	on	ADP
ap-961	424	4	projective	projective	ADJ
ap-961	424	5	modules	module	NOUN
ap-961	424	6	finally	finally	ADV
ap-961	424	7	we	we	PRON
ap-961	424	8	can	can	AUX
ap-961	424	9	use	use	VERB
ap-961	424	10	what	what	PRON
ap-961	424	11	we	we	PRON
ap-961	424	12	have	have	AUX
ap-961	424	13	already	already	ADV
ap-961	424	14	learned	learn	VERB
ap-961	424	15	and	and	CCONJ
ap-961	424	16	apply	apply	VERB
ap-961	424	17	to	to	ADP
ap-961	424	18	both	both	PRON
ap-961	424	19	differential	differential	ADJ
ap-961	424	20	algebras	algebra	NOUN
ap-961	424	21	and	and	CCONJ
ap-961	424	22	projective	projective	ADJ
ap-961	424	23	modules	module	NOUN
ap-961	424	24	to	to	PART
ap-961	424	25	construct	construct	VERB
ap-961	424	26	a	a	DET
ap-961	424	27	new	new	ADJ
ap-961	424	28	object	object	NOUN
ap-961	424	29	(	(	PUNCT
ap-961	424	30	known	know	VERB
ap-961	424	31	from	from	ADP
ap-961	424	32	differential	differential	ADJ
ap-961	424	33	geometry	geometry	NOUN
ap-961	424	34	,	,	PUNCT
ap-961	424	35	of	of	ADP
ap-961	424	36	course	course	NOUN
ap-961	424	37	):	):	PUNCT
ap-961	424	38	a	a	DET
ap-961	424	39	connection	connection	NOUN
ap-961	424	40	.	.	PUNCT
ap-961	425	1	in	in	ADP
ap-961	425	2	differential	differential	ADJ
ap-961	425	3	geometry	geometry	NOUN
ap-961	425	4	we	we	PRON
ap-961	425	5	know	know	VERB
ap-961	425	6	that	that	SCONJ
ap-961	425	7	the	the	DET
ap-961	425	8	theory	theory	NOUN
ap-961	425	9	of	of	ADP
ap-961	425	10	connections	connection	NOUN
ap-961	425	11	on	on	ADP
ap-961	425	12	vector	vector	NOUN
ap-961	425	13	bundles	bundle	NOUN
ap-961	425	14	is	be	AUX
ap-961	425	15	equivalent	equivalent	ADJ
ap-961	425	16	to	to	ADP
ap-961	425	17	connections	connection	NOUN
ap-961	425	18	on	on	ADP
ap-961	425	19	principal	principal	ADJ
ap-961	425	20	fibre	fibre	NOUN
ap-961	425	21	bundles	bundle	NOUN
ap-961	425	22	.	.	PUNCT
ap-961	426	1	in	in	ADP
ap-961	426	2	noncommutative	noncommutative	ADJ
ap-961	426	3	geometry	geometry	NOUN
ap-961	426	4	we	we	PRON
ap-961	426	5	have	have	VERB
ap-961	426	6	all	all	DET
ap-961	426	7	the	the	DET
ap-961	426	8	necessary	necessary	ADJ
ap-961	426	9	tools	tool	NOUN
ap-961	426	10	so	so	SCONJ
ap-961	426	11	we	we	PRON
ap-961	426	12	can	can	AUX
ap-961	426	13	start	start	VERB
ap-961	426	14	the	the	DET
ap-961	426	15	theory	theory	NOUN
ap-961	426	16	in	in	ADP
ap-961	426	17	just	just	ADV
ap-961	426	18	the	the	DET
ap-961	426	19	same	same	ADJ
ap-961	426	20	way	way	NOUN
ap-961	426	21	.	.	PUNCT
ap-961	427	1	such	such	ADJ
ap-961	427	2	objects	object	NOUN
ap-961	427	3	are	be	AUX
ap-961	427	4	interesting	interesting	ADJ
ap-961	427	5	from	from	ADP
ap-961	427	6	many	many	ADJ
ap-961	427	7	points	point	NOUN
ap-961	427	8	of	of	ADP
ap-961	427	9	view	view	NOUN
ap-961	427	10	:	:	PUNCT
ap-961	427	11	first	first	ADV
ap-961	427	12	of	of	ADP
ap-961	427	13	all	all	PRON
ap-961	427	14	,	,	PUNCT
ap-961	427	15	they	they	PRON
ap-961	427	16	provide	provide	VERB
ap-961	427	17	an	an	DET
ap-961	427	18	element	element	NOUN
ap-961	427	19	of	of	ADP
ap-961	427	20	the	the	DET
ap-961	427	21	theory	theory	NOUN
ap-961	427	22	that	that	PRON
ap-961	427	23	enables	enable	VERB
ap-961	427	24	some	some	DET
ap-961	427	25	practical	practical	ADJ
ap-961	427	26	calculations	calculation	NOUN
ap-961	427	27	of	of	ADP
ap-961	427	28	noncommutative	noncommutative	ADJ
ap-961	427	29	chern	chern	NOUN
ap-961	427	30	characters	character	NOUN
ap-961	427	31	.	.	PUNCT
ap-961	428	1	this	this	DET
ap-961	428	2	links	link	NOUN
ap-961	428	3	k	k	NOUN
ap-961	428	4	-	-	NOUN
ap-961	428	5	theory	theory	NOUN
ap-961	428	6	with	with	ADP
ap-961	428	7	noncommutative	noncommutative	ADJ
ap-961	428	8	differential	differential	ADJ
ap-961	428	9	forms	form	NOUN
ap-961	428	10	.	.	PUNCT
ap-961	429	1	from	from	ADP
ap-961	429	2	the	the	DET
ap-961	429	3	point	point	NOUN
ap-961	429	4	of	of	ADP
ap-961	429	5	view	view	NOUN
ap-961	429	6	of	of	ADP
ap-961	429	7	physics	physics	NOUN
ap-961	429	8	,	,	PUNCT
ap-961	429	9	connections	connection	NOUN
ap-961	429	10	are	be	AUX
ap-961	429	11	just	just	ADV
ap-961	429	12	a	a	DET
ap-961	429	13	tool	tool	NOUN
ap-961	429	14	for	for	ADP
ap-961	429	15	the	the	DET
ap-961	429	16	gauge	gauge	NOUN
ap-961	429	17	theory	theory	NOUN
ap-961	429	18	and	and	CCONJ
ap-961	429	19	gravity	gravity	NOUN
ap-961	429	20	.	.	PUNCT
ap-961	430	1	therefore	therefore	ADV
ap-961	430	2	we	we	PRON
ap-961	430	3	can	can	AUX
ap-961	430	4	view	view	VERB
ap-961	430	5	this	this	PRON
ap-961	430	6	as	as	ADP
ap-961	430	7	a	a	DET
ap-961	430	8	step	step	NOUN
ap-961	430	9	towards	towards	ADP
ap-961	430	10	the	the	DET
ap-961	430	11	notion	notion	NOUN
ap-961	430	12	of	of	ADP
ap-961	430	13	noncommutative	noncommutative	ADJ
ap-961	430	14	gauge	gauge	NOUN
ap-961	430	15	theory	theory	NOUN
ap-961	430	16	!	!	PUNCT
ap-961	431	1	although	although	SCONJ
ap-961	431	2	the	the	DET
ap-961	431	3	use	use	NOUN
ap-961	431	4	of	of	ADP
ap-961	431	5	connections	connection	NOUN
ap-961	431	6	appears	appear	VERB
ap-961	431	7	in	in	ADP
ap-961	431	8	many	many	ADJ
ap-961	431	9	places	place	NOUN
ap-961	431	10	,	,	PUNCT
ap-961	431	11	including	include	VERB
ap-961	431	12	connes	conne	NOUN
ap-961	431	13	and	and	CCONJ
ap-961	431	14	rieffel	rieffel	VERB
ap-961	431	15	’s	’s	PART
ap-961	431	16	seminal	seminal	ADJ
ap-961	431	17	work	work	NOUN
ap-961	431	18	on	on	ADP
ap-961	431	19	the	the	DET
ap-961	431	20	noncommutative	noncommutative	ADJ
ap-961	431	21	torus	torus	NOUN
ap-961	431	22	[	[	X
ap-961	431	23	24	24	NUM
ap-961	431	24	]	]	PUNCT
ap-961	431	25	,	,	PUNCT
ap-961	431	26	it	it	PRON
ap-961	431	27	is	be	AUX
ap-961	431	28	worth	worth	ADJ
ap-961	431	29	mentioning	mention	VERB
ap-961	431	30	that	that	DET
ap-961	431	31	systematic	systematic	ADJ
ap-961	431	32	analysis	analysis	NOUN
ap-961	431	33	and	and	CCONJ
ap-961	431	34	proof	proof	NOUN
ap-961	431	35	of	of	ADP
ap-961	431	36	the	the	DET
ap-961	431	37	relation	relation	NOUN
ap-961	431	38	between	between	ADP
ap-961	431	39	projectivity	projectivity	NOUN
ap-961	431	40	and	and	CCONJ
ap-961	431	41	the	the	DET
ap-961	431	42	existence	existence	NOUN
ap-961	431	43	of	of	ADP
ap-961	431	44	universal	universal	ADJ
ap-961	431	45	connections	connection	NOUN
ap-961	431	46	is	be	AUX
ap-961	431	47	a	a	DET
ap-961	431	48	great	great	ADJ
ap-961	431	49	result	result	NOUN
ap-961	431	50	of	of	ADP
ap-961	431	51	cuntz	cuntz	PROPN
ap-961	431	52	and	and	CCONJ
ap-961	431	53	quillen	quillen	VERB
ap-961	432	1	[	[	X
ap-961	432	2	33	33	NUM
ap-961	432	3	]	]	PUNCT
ap-961	432	4	.	.	PUNCT
ap-961	433	1	let	let	VERB
ap-961	433	2	us	we	PRON
ap-961	433	3	start	start	VERB
ap-961	433	4	with	with	ADP
ap-961	433	5	a	a	DET
ap-961	433	6	definition	definition	NOUN
ap-961	433	7	and	and	CCONJ
ap-961	433	8	some	some	DET
ap-961	433	9	interesting	interesting	ADJ
ap-961	433	10	properties	property	NOUN
ap-961	433	11	.	.	PUNCT
ap-961	434	1	definition	definition	NOUN
ap-961	434	2	4.12	4.12	NUM
ap-961	434	3	:	:	PUNCT
ap-961	434	4	let	let	VERB
ap-961	434	5	�	�	PROPN
ap-961	434	6	be	be	AUX
ap-961	434	7	a	a	DET
ap-961	434	8	left	left	ADJ
ap-961	434	9	projective	projective	ADJ
ap-961	434	10	module	module	NOUN
ap-961	434	11	over	over	ADP
ap-961	434	12	an	an	DET
ap-961	434	13	algebra	algebra	NOUN
ap-961	434	14	�	�	PROPN
ap-961	434	15	and	and	CCONJ
ap-961	434	16	let	let	VERB
ap-961	434	17	*	*	PUNCT
ap-961	434	18	(	(	PUNCT
ap-961	434	19	)	)	PUNCT
ap-961	434	20	�	�	PROPN
ap-961	434	21	be	be	AUX
ap-961	434	22	a	a	DET
ap-961	434	23	dga	dga	NOUN
ap-961	434	24	over	over	ADP
ap-961	434	25	�	�	PROPN
ap-961	434	26	.	.	PUNCT
ap-961	435	1	a	a	DET
ap-961	435	2	connection1	connection1	NOUN
ap-961	435	3	on	on	ADP
ap-961	435	4	�	�	PROPN
ap-961	435	5	is	be	AUX
ap-961	435	6	a	a	DET
ap-961	435	7	linear	linear	ADJ
ap-961	435	8	map	map	NOUN
ap-961	435	9	:	:	PUNCT
ap-961	435	10	1	1	NUM
ap-961	435	11	�	�	NOUN
ap-961	435	12	:	:	PUNCT
ap-961	435	13	(	(	PUNCT
ap-961	435	14	)	)	PUNCT
ap-961	435	15	�	�	PROPN
ap-961	435	16	�	�	PROPN
ap-961	435	17	�	�	PROPN
ap-961	435	18	�	�	PROPN
ap-961	435	19	�	�	PROPN
ap-961	435	20	1	1	NUM
ap-961	435	21	,	,	PUNCT
ap-961	435	22	such	such	ADJ
ap-961	435	23	that	that	SCONJ
ap-961	435	24	:	:	PUNCT
ap-961	435	25	1	1	NUM
ap-961	435	26	�	�	PROPN
ap-961	435	27	1	1	NUM
ap-961	435	28	�	�	PROPN
ap-961	435	29	(	(	PUNCT
ap-961	435	30	)	)	PUNCT
ap-961	435	31	(	(	PUNCT
ap-961	435	32	)	)	PUNCT
ap-961	435	33	a	a	DET
ap-961	435	34	m	m	VERB
ap-961	435	35	a	a	DET
ap-961	435	36	m	m	NOUN
ap-961	435	37	a	a	DET
ap-961	435	38	da	da	PROPN
ap-961	435	39	m	m	NOUN
ap-961	435	40	�	�	PROPN
ap-961	435	41	where	where	SCONJ
ap-961	435	42	m	m	PROPN
ap-961	435	43	�	�	PROPN
ap-961	435	44	�	�	PROPN
ap-961	435	45	,	,	PUNCT
ap-961	435	46	a	a	DET
ap-961	435	47	�	�	PROPN
ap-961	435	48	�	�	PROPN
ap-961	435	49	.	.	PUNCT
ap-961	436	1	before	before	SCONJ
ap-961	436	2	we	we	PRON
ap-961	436	3	proceed	proceed	VERB
ap-961	436	4	further	far	ADV
ap-961	436	5	,	,	PUNCT
ap-961	436	6	let	let	VERB
ap-961	436	7	us	we	PRON
ap-961	436	8	pose	pose	VERB
ap-961	436	9	the	the	DET
ap-961	436	10	question	question	NOUN
ap-961	436	11	of	of	ADP
ap-961	436	12	existence	existence	NOUN
ap-961	436	13	.	.	PUNCT
ap-961	437	1	evidently	evidently	ADV
ap-961	437	2	,	,	PUNCT
ap-961	437	3	if	if	SCONJ
ap-961	437	4	�	�	PROPN
ap-961	437	5	is	be	AUX
ap-961	437	6	a	a	DET
ap-961	437	7	left	left	ADJ
ap-961	437	8	projective	projective	NOUN
ap-961	437	9	module	module	NOUN
ap-961	437	10	then	then	ADV
ap-961	437	11	we	we	PRON
ap-961	437	12	have	have	VERB
ap-961	437	13	a	a	DET
ap-961	437	14	projection	projection	NOUN
ap-961	437	15	p	p	PROPN
ap-961	437	16	mn	mn	PROPN
ap-961	437	17	�	�	PROPN
ap-961	437	18	(	(	PUNCT
ap-961	437	19	)	)	PUNCT
ap-961	437	20	�	�	PROPN
ap-961	437	21	such	such	ADJ
ap-961	437	22	that	that	SCONJ
ap-961	437	23	�	�	PROPN
ap-961	437	24	�	�	PROPN
ap-961	437	25	�	�	PROPN
ap-961	437	26	n	n	CCONJ
ap-961	437	27	p.	p.	NOUN
ap-961	437	28	then	then	ADV
ap-961	437	29	one	one	PRON
ap-961	437	30	might	might	AUX
ap-961	437	31	always	always	ADV
ap-961	437	32	construct	construct	VERB
ap-961	437	33	a	a	DET
ap-961	437	34	canonical	canonical	ADJ
ap-961	437	35	(	(	PUNCT
ap-961	437	36	so	so	ADV
ap-961	437	37	-	-	PUNCT
ap-961	437	38	called	call	VERB
ap-961	437	39	grassmanian	grassmanian	ADJ
ap-961	437	40	)	)	PUNCT
ap-961	437	41	connection	connection	NOUN
ap-961	437	42	.	.	PUNCT
ap-961	438	1	on	on	ADP
ap-961	438	2	a	a	DET
ap-961	438	3	free	free	ADJ
ap-961	438	4	module	module	NOUN
ap-961	438	5	it	it	PRON
ap-961	438	6	is	be	AUX
ap-961	438	7	a	a	DET
ap-961	438	8	trivial	trivial	ADJ
ap-961	438	9	exercise	exercise	NOUN
ap-961	438	10	to	to	PART
ap-961	438	11	set	set	VERB
ap-961	438	12	:	:	PUNCT
ap-961	438	13	1	1	NUM
ap-961	438	14	�	�	PROPN
ap-961	438	15	(	(	PUNCT
ap-961	438	16	)	)	PUNCT
ap-961	438	17	a	a	DET
ap-961	438	18	da	da	NOUN
ap-961	438	19	,	,	PUNCT
ap-961	438	20	where	where	SCONJ
ap-961	438	21	a	a	DET
ap-961	438	22	n	n	X
ap-961	438	23	�	�	PROPN
ap-961	438	24	�	�	PROPN
ap-961	438	25	and	and	CCONJ
ap-961	438	26	the	the	DET
ap-961	438	27	expression	expression	NOUN
ap-961	438	28	on	on	ADP
ap-961	438	29	the	the	DET
ap-961	438	30	right	right	ADJ
ap-961	438	31	-	-	PUNCT
ap-961	438	32	hand	hand	NOUN
ap-961	438	33	side	side	NOUN
ap-961	438	34	is	be	AUX
ap-961	438	35	understood	understand	VERB
ap-961	438	36	as	as	ADP
ap-961	438	37	an	an	DET
ap-961	438	38	element	element	NOUN
ap-961	438	39	of	of	ADP
ap-961	438	40	�	�	PROPN
ap-961	438	41	n	n	PROPN
ap-961	438	42	.	.	PUNCT
ap-961	439	1	then	then	ADV
ap-961	439	2	using	use	VERB
ap-961	439	3	the	the	DET
ap-961	439	4	inclusion	inclusion	NOUN
ap-961	439	5	of	of	ADP
ap-961	439	6	a	a	DET
ap-961	439	7	projective	projective	ADJ
ap-961	439	8	module	module	NOUN
ap-961	439	9	into	into	ADP
ap-961	439	10	�	�	PROPN
ap-961	439	11	n	n	CCONJ
ap-961	439	12	we	we	PRON
ap-961	439	13	construct	construct	VERB
ap-961	439	14	the	the	DET
ap-961	439	15	grassmanian	grassmanian	ADJ
ap-961	439	16	connection	connection	NOUN
ap-961	439	17	as	as	ADP
ap-961	439	18	a	a	DET
ap-961	439	19	composition	composition	NOUN
ap-961	439	20	of	of	ADP
ap-961	439	21	the	the	DET
ap-961	439	22	connection	connection	NOUN
ap-961	439	23	on	on	ADP
ap-961	439	24	the	the	DET
ap-961	439	25	free	free	ADJ
ap-961	439	26	module	module	NOUN
ap-961	439	27	with	with	ADP
ap-961	439	28	the	the	DET
ap-961	439	29	projection	projection	NOUN
ap-961	439	30	:	:	PUNCT
ap-961	439	31	1	1	NUM
ap-961	439	32	�	�	PROPN
ap-961	439	33	�	�	PROPN
ap-961	439	34	(	(	PUNCT
ap-961	439	35	)	)	PUNCT
ap-961	439	36	ap	ap	PROPN
ap-961	440	1	da	da	PROPN
ap-961	440	2	p	p	X
ap-961	440	3	�	�	PROPN
ap-961	440	4	it	it	PRON
ap-961	440	5	is	be	AUX
ap-961	440	6	a	a	DET
ap-961	440	7	nice	nice	ADJ
ap-961	440	8	feature	feature	NOUN
ap-961	440	9	that	that	SCONJ
ap-961	440	10	the	the	DET
ap-961	440	11	existence	existence	NOUN
ap-961	440	12	of	of	ADP
ap-961	440	13	universal	universal	ADJ
ap-961	440	14	connections	connection	NOUN
ap-961	440	15	,	,	PUNCT
ap-961	440	16	that	that	PRON
ap-961	440	17	is	be	AUX
ap-961	440	18	connections	connection	NOUN
ap-961	440	19	related	relate	VERB
ap-961	440	20	with	with	ADP
ap-961	440	21	the	the	DET
ap-961	440	22	universal	universal	ADJ
ap-961	440	23	differential	differential	ADJ
ap-961	440	24	algebra	algebra	NOUN
ap-961	440	25	,	,	PUNCT
ap-961	440	26	is	be	AUX
ap-961	440	27	actually	actually	ADV
ap-961	440	28	equivalent	equivalent	ADJ
ap-961	440	29	to	to	ADP
ap-961	440	30	the	the	DET
ap-961	440	31	projectivity	projectivity	NOUN
ap-961	440	32	of	of	ADP
ap-961	440	33	the	the	DET
ap-961	440	34	module	module	NOUN
ap-961	440	35	.	.	PUNCT
ap-961	441	1	before	before	SCONJ
ap-961	441	2	we	we	PRON
ap-961	441	3	define	define	VERB
ap-961	441	4	the	the	DET
ap-961	441	5	curvature	curvature	NOUN
ap-961	441	6	of	of	ADP
ap-961	441	7	a	a	DET
ap-961	441	8	connection	connection	NOUN
ap-961	441	9	let	let	VERB
ap-961	441	10	us	we	PRON
ap-961	441	11	observe	observe	VERB
ap-961	441	12	that	that	SCONJ
ap-961	441	13	the	the	DET
ap-961	441	14	space	space	NOUN
ap-961	441	15	of	of	ADP
ap-961	441	16	connections	connection	NOUN
ap-961	441	17	is	be	AUX
ap-961	441	18	an	an	DET
ap-961	441	19	affine	affine	ADJ
ap-961	441	20	space	space	NOUN
ap-961	441	21	over	over	ADP
ap-961	441	22	�	�	PROPN
ap-961	441	23	-linear	-linear	NOUN
ap-961	441	24	module	module	NOUN
ap-961	441	25	endomorphisms	endomorphism	NOUN
ap-961	441	26	of	of	ADP
ap-961	441	27	�	�	PROPN
ap-961	441	28	,	,	PUNCT
ap-961	441	29	that	that	PRON
ap-961	441	30	is	be	AUX
ap-961	441	31	every	every	DET
ap-961	441	32	two	two	NUM
ap-961	441	33	connections	connection	NOUN
ap-961	441	34	differ	differ	VERB
ap-961	441	35	by	by	ADP
ap-961	441	36	an	an	DET
ap-961	441	37	element	element	NOUN
ap-961	441	38	of	of	ADP
ap-961	441	39	end	end	PROPN
ap-961	441	40	�	�	PROPN
ap-961	441	41	�	�	PROPN
ap-961	441	42	�	�	PROPN
ap-961	441	43	�	�	PROPN
ap-961	441	44	�	�	PROPN
ap-961	441	45	(	(	PUNCT
ap-961	441	46	,	,	PUNCT
ap-961	441	47	(	(	PUNCT
ap-961	441	48	)	)	PUNCT
ap-961	441	49	)	)	PUNCT
ap-961	441	50	1	1	NUM
ap-961	441	51	�	�	PROPN
ap-961	441	52	.	.	PUNCT
ap-961	442	1	the	the	DET
ap-961	442	2	connection	connection	NOUN
ap-961	442	3	is	be	AUX
ap-961	442	4	also	also	ADV
ap-961	442	5	easily	easily	ADV
ap-961	442	6	extended	extend	VERB
ap-961	442	7	to	to	ADP
ap-961	442	8	a	a	DET
ap-961	442	9	general	general	ADJ
ap-961	442	10	degree	degree	NOUN
ap-961	442	11	-	-	PUNCT
ap-961	442	12	one	one	NUM
ap-961	442	13	linear	linear	NOUN
ap-961	442	14	map	map	NOUN
ap-961	442	15	satisfying	satisfy	VERB
ap-961	442	16	the	the	DET
ap-961	442	17	graded	grade	VERB
ap-961	442	18	leibniz	leibniz	PROPN
ap-961	442	19	rule	rule	NOUN
ap-961	442	20	on	on	ADP
ap-961	442	21	*	*	PUNCT
ap-961	442	22	(	(	PUNCT
ap-961	442	23	)	)	PUNCT
ap-961	442	24	�	�	PROPN
ap-961	442	25	�	�	PROPN
ap-961	442	26	�	�	PROPN
ap-961	442	27	�	�	PROPN
ap-961	442	28	.	.	PUNCT
ap-961	443	1	the	the	DET
ap-961	443	2	difference	difference	NOUN
ap-961	443	3	of	of	ADP
ap-961	443	4	connections	connection	NOUN
ap-961	443	5	is	be	AUX
ap-961	443	6	always	always	ADV
ap-961	443	7	a	a	DET
ap-961	443	8	*	*	ADJ
ap-961	443	9	(	(	PUNCT
ap-961	443	10	)	)	PUNCT
ap-961	443	11	�	�	PROPN
ap-961	443	12	-module	-module	PROPN
ap-961	443	13	homomorphism	homomorphism	NOUN
ap-961	443	14	.	.	PUNCT
ap-961	444	1	in	in	ADP
ap-961	444	2	particular	particular	ADJ
ap-961	444	3	,	,	PUNCT
ap-961	444	4	for	for	ADP
ap-961	444	5	a	a	DET
ap-961	444	6	free	free	ADJ
ap-961	444	7	module	module	NOUN
ap-961	444	8	the	the	DET
ap-961	444	9	connection	connection	NOUN
ap-961	444	10	is	be	AUX
ap-961	444	11	always	always	ADV
ap-961	444	12	of	of	ADP
ap-961	444	13	the	the	DET
ap-961	444	14	form	form	NOUN
ap-961	444	15	:	:	PUNCT
ap-961	444	16	1	1	NUM
ap-961	444	17	�	�	PROPN
ap-961	444	18	(	(	PUNCT
ap-961	444	19	)	)	PUNCT
ap-961	444	20	(	(	PUNCT
ap-961	444	21	)	)	PUNCT
ap-961	444	22	a	a	DET
ap-961	444	23	da	da	PROPN
ap-961	444	24	a	a	DET
ap-961	444	25	�	�	PROPN
ap-961	444	26	,	,	PUNCT
ap-961	444	27	where	where	SCONJ
ap-961	444	28	�	�	PROPN
ap-961	444	29	�	�	PROPN
ap-961	444	30	�	�	PROPN
ap-961	444	31	end	end	VERB
ap-961	444	32	n	n	CCONJ
ap-961	444	33	n	n	PRON
ap-961	444	34	�	�	PROPN
ap-961	444	35	�	�	PROPN
ap-961	444	36	�	�	PROPN
ap-961	444	37	�	�	PROPN
ap-961	444	38	�	�	PROPN
ap-961	444	39	(	(	PUNCT
ap-961	444	40	,	,	PUNCT
ap-961	444	41	(	(	PUNCT
ap-961	444	42	)	)	PUNCT
ap-961	444	43	)	)	PUNCT
ap-961	444	44	1	1	NUM
ap-961	444	45	can	can	AUX
ap-961	444	46	be	be	AUX
ap-961	444	47	understood	understand	VERB
ap-961	444	48	as	as	ADP
ap-961	444	49	a	a	DET
ap-961	444	50	matrix	matrix	NOUN
ap-961	444	51	of	of	ADP
ap-961	444	52	one	one	NUM
ap-961	444	53	-	-	PUNCT
ap-961	444	54	forms	form	NOUN
ap-961	444	55	over	over	ADP
ap-961	444	56	�	�	PROPN
ap-961	444	57	.	.	PUNCT
ap-961	445	1	©	©	PROPN
ap-961	445	2	czech	czech	PROPN
ap-961	445	3	technical	technical	PROPN
ap-961	445	4	university	university	PROPN
ap-961	445	5	publishing	publishing	NOUN
ap-961	445	6	house	house	NOUN
ap-961	445	7	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	445	8	45	45	NUM
ap-961	445	9	acta	acta	PROPN
ap-961	445	10	polytechnica	polytechnica	PROPN
ap-961	445	11	vol	vol	NOUN
ap-961	445	12	.	.	PUNCT
ap-961	446	1	48	48	NUM
ap-961	446	2	no	no	NOUN
ap-961	446	3	.	.	PUNCT
ap-961	447	1	2/2008	2/2008	PROPN
ap-961	447	2	the	the	DET
ap-961	447	3	curvature	curvature	NOUN
ap-961	447	4	of	of	ADP
ap-961	447	5	a	a	DET
ap-961	447	6	connection	connection	NOUN
ap-961	447	7	is	be	AUX
ap-961	447	8	its	its	PRON
ap-961	447	9	square	square	ADJ
ap-961	447	10	,	,	PUNCT
ap-961	447	11	12	12	NUM
ap-961	447	12	,	,	PUNCT
ap-961	447	13	which	which	PRON
ap-961	447	14	itself	itself	PRON
ap-961	447	15	is	be	AUX
ap-961	447	16	a	a	DET
ap-961	447	17	degree	degree	NOUN
ap-961	447	18	two	two	NUM
ap-961	447	19	�	�	NOUN
ap-961	447	20	-linear	-linear	NOUN
ap-961	447	21	module	module	NOUN
ap-961	447	22	endomorphism	endomorphism	NOUN
ap-961	447	23	of	of	ADP
ap-961	447	24	*	*	PUNCT
ap-961	447	25	(	(	PUNCT
ap-961	447	26	)	)	PUNCT
ap-961	447	27	�	�	PROPN
ap-961	447	28	�	�	PROPN
ap-961	447	29	�	�	PROPN
ap-961	447	30	�	�	PROPN
ap-961	447	31	.	.	PUNCT
ap-961	448	1	for	for	ADP
ap-961	448	2	a	a	DET
ap-961	448	3	free	free	ADJ
ap-961	448	4	module	module	NOUN
ap-961	448	5	we	we	PRON
ap-961	448	6	have	have	VERB
ap-961	448	7	:	:	PUNCT
ap-961	448	8	1	1	NUM
ap-961	448	9	�	�	PROPN
ap-961	448	10	2	2	NUM
ap-961	448	11	2	2	NUM
ap-961	448	12	(	(	PUNCT
ap-961	448	13	)	)	PUNCT
ap-961	448	14	(	(	PUNCT
ap-961	448	15	)	)	PUNCT
ap-961	448	16	(	(	PUNCT
ap-961	448	17	)	)	PUNCT
ap-961	448	18	a	a	PRON
ap-961	448	19	d	d	X
ap-961	448	20	a	a	DET
ap-961	448	21	�	�	PROPN
ap-961	448	22	�	�	PROPN
ap-961	448	23	,	,	PUNCT
ap-961	448	24	which	which	PRON
ap-961	448	25	is	be	AUX
ap-961	448	26	a	a	DET
ap-961	448	27	well	well	ADV
ap-961	448	28	-	-	PUNCT
ap-961	448	29	known	know	VERB
ap-961	448	30	formula	formula	NOUN
ap-961	448	31	from	from	ADP
ap-961	448	32	the	the	DET
ap-961	448	33	gauge	gauge	NOUN
ap-961	448	34	theory	theory	NOUN
ap-961	448	35	but	but	CCONJ
ap-961	448	36	now	now	ADV
ap-961	448	37	applicable	applicable	ADJ
ap-961	448	38	to	to	ADP
ap-961	448	39	a	a	DET
ap-961	448	40	general	general	ADJ
ap-961	448	41	class	class	NOUN
ap-961	448	42	of	of	ADP
ap-961	448	43	noncommutative	noncommutative	ADJ
ap-961	448	44	objects	object	NOUN
ap-961	448	45	!	!	PUNCT
ap-961	449	1	we	we	PRON
ap-961	449	2	shall	shall	AUX
ap-961	449	3	finish	finish	VERB
ap-961	449	4	this	this	DET
ap-961	449	5	short	short	ADJ
ap-961	449	6	review	review	NOUN
ap-961	449	7	of	of	ADP
ap-961	449	8	connections	connection	NOUN
ap-961	449	9	by	by	ADP
ap-961	449	10	mentioning	mention	VERB
ap-961	449	11	that	that	SCONJ
ap-961	449	12	in	in	ADP
ap-961	449	13	case	case	NOUN
ap-961	449	14	the	the	DET
ap-961	449	15	module	module	NOUN
ap-961	449	16	�	�	PROPN
ap-961	449	17	has	have	VERB
ap-961	449	18	an	an	DET
ap-961	449	19	�	�	PROPN
ap-961	449	20	-valued	-value	VERB
ap-961	449	21	hermitian	hermitian	ADJ
ap-961	449	22	inner	inner	ADJ
ap-961	449	23	product	product	NOUN
ap-961	449	24	one	one	PRON
ap-961	449	25	might	might	AUX
ap-961	449	26	require	require	VERB
ap-961	449	27	that	that	SCONJ
ap-961	449	28	the	the	DET
ap-961	449	29	connection	connection	NOUN
ap-961	449	30	is	be	AUX
ap-961	449	31	hermitian	hermitian	ADJ
ap-961	449	32	:	:	PUNCT
ap-961	449	33	1	1	NUM
ap-961	449	34	(	(	PUNCT
ap-961	449	35	1	1	NUM
ap-961	449	36	(	(	PUNCT
ap-961	449	37	�	�	PROPN
ap-961	449	38	(	(	PUNCT
ap-961	449	39	m	m	VERB
ap-961	449	40	m	m	VERB
ap-961	449	41	m	m	VERB
ap-961	449	42	m	m	PROPN
ap-961	449	43	d	d	NOUN
ap-961	449	44	m	m	NOUN
ap-961	449	45	m	m	PRON
ap-961	449	46	,	,	PUNCT
ap-961	449	47	,	,	PUNCT
ap-961	449	48	,	,	PUNCT
ap-961	449	49	,	,	PUNCT
ap-961	449	50	where	where	SCONJ
ap-961	449	51	the	the	DET
ap-961	449	52	left	left	ADJ
ap-961	449	53	-	-	PUNCT
ap-961	449	54	hand	hand	NOUN
ap-961	449	55	side	side	NOUN
ap-961	449	56	is	be	AUX
ap-961	449	57	the	the	DET
ap-961	449	58	natural	natural	ADJ
ap-961	449	59	extension	extension	NOUN
ap-961	449	60	of	of	ADP
ap-961	449	61	the	the	DET
ap-961	449	62	inner	inner	ADJ
ap-961	449	63	product	product	NOUN
ap-961	449	64	on	on	ADP
ap-961	449	65	the	the	DET
ap-961	449	66	tensor	tensor	NOUN
ap-961	449	67	products	product	NOUN
ap-961	449	68	with	with	ADP
ap-961	449	69	one	one	NUM
ap-961	449	70	-	-	PUNCT
ap-961	449	71	forms	form	NOUN
ap-961	449	72	.	.	PUNCT
ap-961	450	1	in	in	ADP
ap-961	450	2	the	the	DET
ap-961	450	3	canonical	canonical	ADJ
ap-961	450	4	example	example	NOUN
ap-961	450	5	of	of	ADP
ap-961	450	6	the	the	DET
ap-961	450	7	free	free	ADJ
ap-961	450	8	module	module	NOUN
ap-961	450	9	(	(	PUNCT
ap-961	450	10	and	and	CCONJ
ap-961	450	11	the	the	DET
ap-961	450	12	canonical	canonical	ADJ
ap-961	450	13	inner	inner	ADJ
ap-961	450	14	product	product	NOUN
ap-961	450	15	)	)	PUNCT
ap-961	450	16	this	this	PRON
ap-961	450	17	translates	translate	VERB
ap-961	450	18	to	to	ADP
ap-961	450	19	the	the	DET
ap-961	450	20	requirement	requirement	NOUN
ap-961	450	21	that	that	SCONJ
ap-961	450	22	�	�	PROPN
ap-961	450	23	�	�	PROPN
ap-961	450	24	*	*	PROPN
ap-961	450	25	�	�	PROPN
ap-961	450	26	�	�	PROPN
ap-961	450	27	(	(	PUNCT
ap-961	450	28	as	as	ADP
ap-961	450	29	a	a	DET
ap-961	450	30	matrix	matrix	NOUN
ap-961	450	31	of	of	ADP
ap-961	450	32	differential	differential	ADJ
ap-961	450	33	forms	form	NOUN
ap-961	450	34	)	)	PUNCT
ap-961	450	35	.	.	PUNCT
ap-961	451	1	to	to	PART
ap-961	451	2	make	make	VERB
ap-961	451	3	the	the	DET
ap-961	451	4	picture	picture	NOUN
ap-961	451	5	a	a	DET
ap-961	451	6	bit	bit	NOUN
ap-961	451	7	more	more	ADV
ap-961	451	8	comprehensible	comprehensible	ADJ
ap-961	451	9	,	,	PUNCT
ap-961	451	10	let	let	VERB
ap-961	451	11	us	we	PRON
ap-961	451	12	study	study	VERB
ap-961	451	13	two	two	NUM
ap-961	451	14	important	important	ADJ
ap-961	451	15	examples	example	NOUN
ap-961	451	16	.	.	PUNCT
ap-961	452	1	example	example	NOUN
ap-961	452	2	4.13	4.13	NUM
ap-961	452	3	:	:	PUNCT
ap-961	452	4	take	take	VERB
ap-961	452	5	the	the	DET
ap-961	452	6	algebra	algebra	NOUN
ap-961	452	7	�	�	PROPN
ap-961	452	8	of	of	ADP
ap-961	452	9	complex	complex	ADJ
ap-961	452	10	valued	value	VERB
ap-961	452	11	functions	function	NOUN
ap-961	452	12	on	on	ADP
ap-961	452	13	the	the	DET
ap-961	452	14	space	space	NOUN
ap-961	452	15	of	of	ADP
ap-961	452	16	two	two	NUM
ap-961	452	17	points	point	NOUN
ap-961	452	18	.	.	PUNCT
ap-961	453	1	it	it	PRON
ap-961	453	2	has	have	VERB
ap-961	453	3	(	(	PUNCT
ap-961	453	4	of	of	ADP
ap-961	453	5	course	course	NOUN
ap-961	453	6	)	)	PUNCT
ap-961	453	7	only	only	ADV
ap-961	453	8	free	free	ADJ
ap-961	453	9	modules	module	NOUN
ap-961	453	10	and	and	CCONJ
ap-961	453	11	we	we	PRON
ap-961	453	12	take	take	VERB
ap-961	453	13	just	just	ADV
ap-961	453	14	the	the	DET
ap-961	453	15	simple	simple	ADJ
ap-961	453	16	one	one	NUM
ap-961	453	17	,	,	PUNCT
ap-961	453	18	�	�	PROPN
ap-961	453	19	itself	itself	PRON
ap-961	453	20	.	.	PUNCT
ap-961	454	1	it	it	PRON
ap-961	454	2	is	be	AUX
ap-961	454	3	equipped	equip	VERB
ap-961	454	4	with	with	ADP
ap-961	454	5	the	the	DET
ap-961	454	6	standard	standard	ADJ
ap-961	454	7	hermitian	hermitian	ADJ
ap-961	454	8	product	product	NOUN
ap-961	454	9	arising	arise	VERB
ap-961	454	10	just	just	ADV
ap-961	454	11	from	from	ADP
ap-961	454	12	the	the	DET
ap-961	454	13	complex	complex	ADJ
ap-961	454	14	conjugation	conjugation	NOUN
ap-961	454	15	and	and	CCONJ
ap-961	454	16	product	product	NOUN
ap-961	454	17	in	in	ADP
ap-961	454	18	�	�	PROPN
ap-961	454	19	.	.	PUNCT
ap-961	455	1	let	let	VERB
ap-961	455	2	us	we	PRON
ap-961	455	3	consider	consider	VERB
ap-961	455	4	all	all	DET
ap-961	455	5	hermitian	hermitian	ADJ
ap-961	455	6	connections	connection	NOUN
ap-961	455	7	with	with	ADP
ap-961	455	8	respect	respect	NOUN
ap-961	455	9	to	to	ADP
ap-961	455	10	universal	universal	ADJ
ap-961	455	11	differential	differential	ADJ
ap-961	455	12	calculi	calculi	NOUN
ap-961	455	13	.	.	PUNCT
ap-961	456	1	we	we	PRON
ap-961	456	2	have	have	VERB
ap-961	456	3	,	,	PUNCT
ap-961	456	4	for	for	ADP
ap-961	456	5	a	a	DET
ap-961	456	6	�	�	PROPN
ap-961	456	7	�	�	PROPN
ap-961	456	8	(	(	PUNCT
ap-961	456	9	but	but	CCONJ
ap-961	456	10	�	�	PROPN
ap-961	456	11	seen	see	VERB
ap-961	456	12	as	as	ADP
ap-961	456	13	a	a	DET
ap-961	456	14	left	left	ADJ
ap-961	456	15	-	-	PUNCT
ap-961	456	16	module	module	NOUN
ap-961	456	17	)	)	PUNCT
ap-961	456	18	1	1	NUM
ap-961	456	19	�	�	PROPN
ap-961	456	20	(	(	PUNCT
ap-961	456	21	)	)	PUNCT
ap-961	456	22	�	�	PROPN
ap-961	456	23	a	a	DET
ap-961	456	24	da	da	PROPN
ap-961	456	25	a	a	DET
ap-961	456	26	�	�	PROPN
ap-961	456	27	,	,	PUNCT
ap-961	456	28	where	where	SCONJ
ap-961	456	29	�	�	PROPN
ap-961	456	30	�	�	PROPN
ap-961	456	31	is	be	AUX
ap-961	456	32	an	an	DET
ap-961	456	33	arbitrary	arbitrary	ADJ
ap-961	456	34	one	one	NUM
ap-961	456	35	-	-	PUNCT
ap-961	456	36	form	form	NOUN
ap-961	456	37	,	,	PUNCT
ap-961	456	38	so	so	ADV
ap-961	456	39	�	�	PROPN
ap-961	456	40	�	�	PROPN
ap-961	456	41	�	�	PROPN
ap-961	456	42	�	�	PROPN
ap-961	456	43	de	de	PROPN
ap-961	456	44	,	,	PUNCT
ap-961	456	45	with	with	ADP
ap-961	456	46	�	�	PROPN
ap-961	456	47	a	a	DET
ap-961	456	48	complex	complex	ADJ
ap-961	456	49	function	function	NOUN
ap-961	456	50	on	on	ADP
ap-961	456	51	two	two	NUM
ap-961	456	52	points	point	NOUN
ap-961	456	53	.	.	PUNCT
ap-961	457	1	since	since	SCONJ
ap-961	457	2	�	�	PROPN
ap-961	457	3	�	�	PROPN
ap-961	457	4	must	must	AUX
ap-961	457	5	be	be	AUX
ap-961	457	6	antiselfadjoint	antiselfadjoint	ADJ
ap-961	457	7	,	,	PUNCT
ap-961	457	8	when	when	SCONJ
ap-961	457	9	we	we	PRON
ap-961	457	10	recall	recall	VERB
ap-961	457	11	all	all	DET
ap-961	457	12	the	the	DET
ap-961	457	13	rules	rule	NOUN
ap-961	457	14	of	of	ADP
ap-961	457	15	differential	differential	ADJ
ap-961	457	16	calculus	calculus	NOUN
ap-961	457	17	(	(	PUNCT
ap-961	457	18	see	see	VERB
ap-961	457	19	examples	example	NOUN
ap-961	457	20	3.6	3.6	NUM
ap-961	457	21	and	and	CCONJ
ap-961	457	22	3.12	3.12	NUM
ap-961	457	23	)	)	PUNCT
ap-961	457	24	,	,	PUNCT
ap-961	457	25	we	we	PRON
ap-961	457	26	have	have	VERB
ap-961	457	27	:	:	PUNCT
ap-961	457	28	(	(	PUNCT
ap-961	457	29	)	)	PUNCT
ap-961	457	30	*	*	PUNCT
ap-961	457	31	*	*	PUNCT
ap-961	457	32	�	�	PROPN
ap-961	457	33	�	�	PROPN
ap-961	457	34	�	�	PROPN
ap-961	457	35	de	de	X
ap-961	457	36	de	de	X
ap-961	457	37	de	de	PROPN
ap-961	457	38	�	�	PROPN
ap-961	457	39	�	�	PROPN
ap-961	457	40	�	�	PROPN
ap-961	457	41	since	since	SCONJ
ap-961	457	42	e	e	PROPN
ap-961	457	43	de	de	X
ap-961	457	44	de	de	X
ap-961	457	45	e	e	PROPN
ap-961	457	46	�	�	PROPN
ap-961	457	47	�	�	PROPN
ap-961	457	48	the	the	DET
ap-961	457	49	function	function	NOUN
ap-961	457	50	�	�	PROPN
ap-961	457	51	is	be	AUX
ap-961	457	52	given	give	VERB
ap-961	457	53	by	by	ADP
ap-961	457	54	a	a	DET
ap-961	457	55	complex	complex	ADJ
ap-961	457	56	number	number	NOUN
ap-961	457	57	�	�	PROPN
ap-961	457	58	:	:	PUNCT
ap-961	457	59	�	�	PROPN
ap-961	457	60	�	�	PROPN
ap-961	457	61	�	�	PROPN
ap-961	457	62	(	(	PUNCT
ap-961	457	63	)	)	PUNCT
ap-961	457	64	(	(	PUNCT
ap-961	457	65	)	)	PUNCT
ap-961	457	66	*	*	PUNCT
ap-961	457	67	*	*	PUNCT
ap-961	457	68	�	�	PROPN
ap-961	457	69	�	�	PROPN
ap-961	457	70	�	�	PROPN
ap-961	457	71	�	�	PROPN
ap-961	457	72	1	1	NUM
ap-961	457	73	e	e	NOUN
ap-961	457	74	.	.	PUNCT
ap-961	458	1	the	the	DET
ap-961	458	2	curvature	curvature	NOUN
ap-961	458	3	of	of	ADP
ap-961	458	4	the	the	DET
ap-961	458	5	connection1,12	connection1,12	PROPN
ap-961	458	6	is	be	AUX
ap-961	458	7	:	:	PUNCT
ap-961	458	8	1	1	NUM
ap-961	458	9	�	�	PROPN
ap-961	458	10	2	2	NUM
ap-961	458	11	2	2	NUM
ap-961	458	12	(	(	PUNCT
ap-961	458	13	)	)	PUNCT
ap-961	458	14	�	�	PROPN
ap-961	458	15	�	�	PROPN
ap-961	458	16	a	a	DET
ap-961	458	17	d	d	PROPN
ap-961	458	18	�	�	PROPN
ap-961	458	19	�	�	PROPN
ap-961	458	20	,	,	PUNCT
ap-961	458	21	which	which	PRON
ap-961	458	22	rewritten	rewrite	VERB
ap-961	458	23	in	in	ADP
ap-961	458	24	the	the	DET
ap-961	458	25	language	language	NOUN
ap-961	458	26	of	of	ADP
ap-961	458	27	forms	form	NOUN
ap-961	458	28	de	de	ADP
ap-961	458	29	gives	give	VERB
ap-961	458	30	:	:	PUNCT
ap-961	458	31	1	1	NUM
ap-961	458	32	�	�	PROPN
ap-961	458	33	�	�	PROPN
ap-961	458	34	2	2	NUM
ap-961	458	35	4	4	NUM
ap-961	458	36	(	(	PUNCT
ap-961	458	37	)	)	PUNCT
ap-961	458	38	(	(	PUNCT
ap-961	458	39	)	)	PUNCT
ap-961	458	40	*	*	PUNCT
ap-961	459	1	*	*	PUNCT
ap-961	459	2	a	a	DET
ap-961	459	3	de	de	X
ap-961	459	4	de	de	X
ap-961	459	5	�	�	PROPN
ap-961	459	6	�	�	PROPN
ap-961	459	7	�	�	PROPN
ap-961	459	8	�	�	PROPN
ap-961	459	9	,	,	PUNCT
ap-961	459	10	next	next	ADV
ap-961	459	11	,	,	PUNCT
ap-961	459	12	by	by	ADP
ap-961	459	13	introducing	introduce	VERB
ap-961	459	14	h	h	PROPN
ap-961	459	15	�	�	PROPN
ap-961	459	16	�	�	PROPN
ap-961	459	17	1	1	NUM
ap-961	459	18	4	4	NUM
ap-961	459	19	�	�	NOUN
ap-961	459	20	we	we	PRON
ap-961	459	21	obtain	obtain	VERB
ap-961	459	22	:	:	PUNCT
ap-961	459	23	1	1	NUM
ap-961	459	24	�	�	PROPN
ap-961	459	25	�	�	PROPN
ap-961	459	26	2	2	NUM
ap-961	459	27	1	1	NUM
ap-961	459	28	4	4	NUM
ap-961	459	29	1	1	NUM
ap-961	459	30	(	(	PUNCT
ap-961	459	31	)	)	PUNCT
ap-961	459	32	(	(	PUNCT
ap-961	459	33	)	)	PUNCT
ap-961	459	34	*	*	PUNCT
ap-961	459	35	a	a	DET
ap-961	459	36	hh	hh	PROPN
ap-961	459	37	de	de	X
ap-961	459	38	de	de	X
ap-961	459	39	.	.	PUNCT
ap-961	459	40	suppose	suppose	VERB
ap-961	459	41	now	now	ADV
ap-961	459	42	that	that	SCONJ
ap-961	459	43	we	we	PRON
ap-961	459	44	think	think	VERB
ap-961	459	45	of	of	ADP
ap-961	459	46	h	h	NOUN
ap-961	459	47	as	as	ADP
ap-961	459	48	a	a	DET
ap-961	459	49	gauge	gauge	NOUN
ap-961	459	50	connection	connection	NOUN
ap-961	459	51	field	field	NOUN
ap-961	459	52	and	and	CCONJ
ap-961	459	53	calculate	calculate	VERB
ap-961	459	54	the	the	DET
ap-961	459	55	square	square	NOUN
ap-961	459	56	of	of	ADP
ap-961	459	57	the	the	DET
ap-961	459	58	curvature	curvature	NOUN
ap-961	459	59	.	.	PUNCT
ap-961	460	1	integrating	integrate	VERB
ap-961	460	2	this	this	DET
ap-961	460	3	square	square	NOUN
ap-961	460	4	over	over	ADP
ap-961	460	5	the	the	DET
ap-961	460	6	space	space	NOUN
ap-961	460	7	(	(	PUNCT
ap-961	460	8	two	two	NUM
ap-961	460	9	points	point	NOUN
ap-961	460	10	)	)	PUNCT
ap-961	460	11	we	we	PRON
ap-961	460	12	get	get	VERB
ap-961	460	13	f	f	PROPN
ap-961	460	14	hh2	hh2	NOUN
ap-961	460	15	1	1	NUM
ap-961	460	16	2	2	NUM
ap-961	460	17	12	12	NUM
ap-961	460	18	�	�	PROPN
ap-961	460	19	�	�	PROPN
ap-961	460	20	(	(	PUNCT
ap-961	460	21	)	)	PUNCT
ap-961	460	22	*	*	PUNCT
ap-961	460	23	.	.	PUNCT
ap-961	461	1	if	if	SCONJ
ap-961	461	2	we	we	PRON
ap-961	461	3	construct	construct	VERB
ap-961	461	4	a	a	DET
ap-961	461	5	weird	weird	ADJ
ap-961	461	6	type	type	NOUN
ap-961	461	7	of	of	ADP
ap-961	461	8	kaluza	kaluza	PROPN
ap-961	461	9	-	-	PUNCT
ap-961	461	10	klein	klein	PROPN
ap-961	461	11	theory	theory	NOUN
ap-961	461	12	with	with	ADP
ap-961	461	13	the	the	DET
ap-961	461	14	extra	extra	ADJ
ap-961	461	15	space	space	NOUN
ap-961	461	16	being	be	AUX
ap-961	461	17	the	the	DET
ap-961	461	18	usual	usual	ADJ
ap-961	461	19	one	one	NOUN
ap-961	461	20	,	,	PUNCT
ap-961	461	21	we	we	PRON
ap-961	461	22	might	might	AUX
ap-961	461	23	interpret	interpret	VERB
ap-961	461	24	this	this	PRON
ap-961	461	25	as	as	ADP
ap-961	461	26	an	an	DET
ap-961	461	27	action	action	NOUN
ap-961	461	28	for	for	ADP
ap-961	461	29	a	a	DET
ap-961	461	30	field	field	NOUN
ap-961	461	31	h(x	h(x	PROPN
ap-961	461	32	)	)	PUNCT
ap-961	461	33	,	,	PUNCT
ap-961	461	34	which	which	PRON
ap-961	461	35	arises	arise	VERB
ap-961	461	36	from	from	ADP
ap-961	461	37	the	the	DET
ap-961	461	38	discrete	discrete	ADJ
ap-961	461	39	geometry	geometry	NOUN
ap-961	461	40	of	of	ADP
ap-961	461	41	two	two	NUM
ap-961	461	42	-	-	PUNCT
ap-961	461	43	points	point	NOUN
ap-961	461	44	.	.	PUNCT
ap-961	462	1	it	it	PRON
ap-961	462	2	is	be	AUX
ap-961	462	3	a	a	DET
ap-961	462	4	surprise	surprise	NOUN
ap-961	462	5	that	that	SCONJ
ap-961	462	6	this	this	DET
ap-961	462	7	type	type	NOUN
ap-961	462	8	of	of	ADP
ap-961	462	9	action	action	NOUN
ap-961	462	10	is	be	AUX
ap-961	462	11	well	well	ADV
ap-961	462	12	known	know	VERB
ap-961	462	13	and	and	CCONJ
ap-961	462	14	has	have	VERB
ap-961	462	15	a	a	DET
ap-961	462	16	very	very	ADV
ap-961	462	17	important	important	ADJ
ap-961	462	18	role	role	NOUN
ap-961	462	19	in	in	ADP
ap-961	462	20	physics	physics	NOUN
ap-961	462	21	as	as	ADP
ap-961	462	22	the	the	DET
ap-961	462	23	action	action	NOUN
ap-961	462	24	for	for	ADP
ap-961	462	25	the	the	DET
ap-961	462	26	higgs	higgs	NOUN
ap-961	462	27	field	field	NOUN
ap-961	462	28	!	!	PUNCT
ap-961	463	1	5	5	NUM
ap-961	463	2	cycles	cycle	NOUN
ap-961	463	3	and	and	CCONJ
ap-961	463	4	cyclic	cyclic	ADJ
ap-961	463	5	cohomology	cohomology	NOUN
ap-961	463	6	“	"	PUNCT
ap-961	463	7	most	most	ADJ
ap-961	463	8	of	of	ADP
ap-961	463	9	the	the	DET
ap-961	463	10	fundamental	fundamental	ADJ
ap-961	463	11	ideas	idea	NOUN
ap-961	463	12	of	of	ADP
ap-961	463	13	science	science	NOUN
ap-961	463	14	are	be	AUX
ap-961	463	15	essentially	essentially	ADV
ap-961	463	16	simple	simple	ADJ
ap-961	463	17	,	,	PUNCT
ap-961	463	18	and	and	CCONJ
ap-961	463	19	may	may	AUX
ap-961	463	20	,	,	PUNCT
ap-961	463	21	as	as	ADP
ap-961	463	22	a	a	DET
ap-961	463	23	rule	rule	NOUN
ap-961	463	24	,	,	PUNCT
ap-961	463	25	be	be	AUX
ap-961	463	26	expressed	express	VERB
ap-961	463	27	in	in	ADP
ap-961	463	28	a	a	DET
ap-961	463	29	language	language	NOUN
ap-961	463	30	comprehensible	comprehensible	ADJ
ap-961	463	31	to	to	ADP
ap-961	463	32	everyone	everyone	PRON
ap-961	463	33	.	.	PUNCT
ap-961	463	34	”	"	PUNCT
ap-961	464	1	(	(	PUNCT
ap-961	464	2	albert	albert	PROPN
ap-961	464	3	einstein	einstein	PROPN
ap-961	464	4	)	)	PUNCT
ap-961	464	5	let	let	VERB
ap-961	464	6	us	we	PRON
ap-961	464	7	recall	recall	VERB
ap-961	464	8	that	that	SCONJ
ap-961	464	9	the	the	DET
ap-961	464	10	classical	classical	ADJ
ap-961	464	11	theory	theory	NOUN
ap-961	464	12	of	of	ADP
ap-961	464	13	connections	connection	NOUN
ap-961	464	14	on	on	ADP
ap-961	464	15	vector	vector	NOUN
ap-961	464	16	bundles	bundle	NOUN
ap-961	464	17	leads	lead	VERB
ap-961	464	18	to	to	ADP
ap-961	464	19	the	the	DET
ap-961	464	20	notion	notion	NOUN
ap-961	464	21	of	of	ADP
ap-961	464	22	characteristic	characteristic	ADJ
ap-961	464	23	classes	class	NOUN
ap-961	464	24	.	.	PUNCT
ap-961	465	1	the	the	DET
ap-961	465	2	square	square	NOUN
ap-961	465	3	of	of	ADP
ap-961	465	4	the	the	DET
ap-961	465	5	connection	connection	NOUN
ap-961	465	6	,	,	PUNCT
ap-961	465	7	curvature	curvature	NOUN
ap-961	465	8	,	,	PUNCT
ap-961	465	9	which	which	PRON
ap-961	465	10	is	be	AUX
ap-961	465	11	a	a	DET
ap-961	465	12	differential	differential	ADJ
ap-961	465	13	two	two	NUM
ap-961	465	14	-	-	PUNCT
ap-961	465	15	form	form	NOUN
ap-961	465	16	valued	value	VERB
ap-961	465	17	in	in	ADP
ap-961	465	18	the	the	DET
ap-961	465	19	algebra	algebra	NOUN
ap-961	465	20	of	of	ADP
ap-961	465	21	endomorphisms	endomorphism	NOUN
ap-961	465	22	of	of	ADP
ap-961	465	23	the	the	DET
ap-961	465	24	vector	vector	NOUN
ap-961	465	25	bundle	bundle	NOUN
ap-961	465	26	is	be	AUX
ap-961	465	27	the	the	DET
ap-961	465	28	basic	basic	ADJ
ap-961	465	29	building	building	NOUN
ap-961	465	30	block	block	NOUN
ap-961	465	31	of	of	ADP
ap-961	465	32	the	the	DET
ap-961	465	33	theory	theory	NOUN
ap-961	465	34	.	.	PUNCT
ap-961	466	1	in	in	ADP
ap-961	466	2	the	the	DET
ap-961	466	3	noncommutative	noncommutative	ADJ
ap-961	466	4	setup	setup	NOUN
ap-961	466	5	we	we	PRON
ap-961	466	6	have	have	VERB
ap-961	466	7	almost	almost	ADV
ap-961	466	8	the	the	DET
ap-961	466	9	same	same	ADJ
ap-961	466	10	possibility	possibility	NOUN
ap-961	466	11	,	,	PUNCT
ap-961	466	12	with	with	ADP
ap-961	466	13	the	the	DET
ap-961	466	14	exception	exception	NOUN
ap-961	466	15	that	that	SCONJ
ap-961	466	16	the	the	DET
ap-961	466	17	differential	differential	ADJ
ap-961	466	18	calculi	calculi	PROPN
ap-961	466	19	are	be	AUX
ap-961	466	20	plentiful	plentiful	ADJ
ap-961	466	21	and	and	CCONJ
ap-961	466	22	therefore	therefore	ADV
ap-961	466	23	some	some	PRON
ap-961	466	24	of	of	ADP
ap-961	466	25	them	they	PRON
ap-961	466	26	(	(	PUNCT
ap-961	466	27	in	in	ADP
ap-961	466	28	particular	particular	ADJ
ap-961	466	29	the	the	DET
ap-961	466	30	universal	universal	ADJ
ap-961	466	31	differential	differential	ADJ
ap-961	466	32	calculus	calculus	NOUN
ap-961	466	33	)	)	PUNCT
ap-961	466	34	may	may	AUX
ap-961	466	35	carry	carry	VERB
ap-961	466	36	no	no	DET
ap-961	466	37	cohomology	cohomology	NOUN
ap-961	466	38	information	information	NOUN
ap-961	466	39	,	,	PUNCT
ap-961	466	40	that	that	PRON
ap-961	466	41	can	can	AUX
ap-961	466	42	be	be	AUX
ap-961	466	43	used	use	VERB
ap-961	466	44	to	to	PART
ap-961	466	45	construct	construct	VERB
ap-961	466	46	characteristic	characteristic	ADJ
ap-961	466	47	classes	class	NOUN
ap-961	466	48	.	.	PUNCT
ap-961	467	1	however	however	ADV
ap-961	467	2	,	,	PUNCT
ap-961	467	3	when	when	SCONJ
ap-961	467	4	we	we	PRON
ap-961	467	5	accept	accept	VERB
ap-961	467	6	this	this	DET
ap-961	467	7	approach	approach	NOUN
ap-961	467	8	we	we	PRON
ap-961	467	9	shall	shall	AUX
ap-961	467	10	have	have	VERB
ap-961	467	11	no	no	DET
ap-961	467	12	general	general	ADJ
ap-961	467	13	principle	principle	NOUN
ap-961	467	14	in	in	ADP
ap-961	467	15	the	the	DET
ap-961	467	16	theory	theory	NOUN
ap-961	467	17	:	:	PUNCT
ap-961	467	18	every	every	DET
ap-961	467	19	single	single	ADJ
ap-961	467	20	case	case	NOUN
ap-961	467	21	needs	need	VERB
ap-961	467	22	to	to	PART
ap-961	467	23	be	be	AUX
ap-961	467	24	studied	study	VERB
ap-961	467	25	separately	separately	ADV
ap-961	467	26	!	!	PUNCT
ap-961	468	1	moreover	moreover	ADV
ap-961	468	2	,	,	PUNCT
ap-961	468	3	it	it	PRON
ap-961	468	4	is	be	AUX
ap-961	468	5	rather	rather	ADV
ap-961	468	6	unrealistic	unrealistic	ADJ
ap-961	468	7	to	to	PART
ap-961	468	8	study	study	VERB
ap-961	468	9	all	all	DET
ap-961	468	10	possible	possible	ADJ
ap-961	468	11	differential	differential	ADJ
ap-961	468	12	calculi	calculi	NOUN
ap-961	468	13	that	that	PRON
ap-961	468	14	can	can	AUX
ap-961	468	15	bring	bring	VERB
ap-961	468	16	some	some	DET
ap-961	468	17	new	new	ADJ
ap-961	468	18	data	datum	NOUN
ap-961	468	19	.	.	PUNCT
ap-961	469	1	again	again	ADV
ap-961	469	2	,	,	PUNCT
ap-961	469	3	the	the	DET
ap-961	469	4	solution	solution	NOUN
ap-961	469	5	to	to	ADP
ap-961	469	6	the	the	DET
ap-961	469	7	problem	problem	NOUN
ap-961	469	8	lies	lie	VERB
ap-961	469	9	in	in	ADP
ap-961	469	10	the	the	DET
ap-961	469	11	approach	approach	NOUN
ap-961	469	12	–	–	PUNCT
ap-961	469	13	we	we	PRON
ap-961	469	14	need	need	VERB
ap-961	469	15	to	to	PART
ap-961	469	16	introduce	introduce	VERB
ap-961	469	17	a	a	DET
ap-961	469	18	more	more	ADV
ap-961	469	19	general	general	ADJ
ap-961	469	20	notion	notion	NOUN
ap-961	469	21	which	which	PRON
ap-961	469	22	replaces	replace	VERB
ap-961	469	23	de	de	ADP
ap-961	469	24	rham	rham	X
ap-961	469	25	cohomology	cohomology	NOUN
ap-961	469	26	in	in	ADP
ap-961	469	27	the	the	DET
ap-961	469	28	noncommutative	noncommutative	ADJ
ap-961	469	29	setup	setup	NOUN
ap-961	469	30	.	.	PUNCT
ap-961	470	1	definition	definition	NOUN
ap-961	470	2	5.1	5.1	NUM
ap-961	470	3	:	:	PUNCT
ap-961	470	4	consider	consider	VERB
ap-961	470	5	cn	cn	PROPN
ap-961	470	6	(	(	PUNCT
ap-961	470	7	)	)	PUNCT
ap-961	470	8	�	�	PROPN
ap-961	470	9	–	–	PUNCT
ap-961	470	10	space	space	NOUN
ap-961	470	11	of	of	ADP
ap-961	470	12	linear	linear	PROPN
ap-961	470	13	maps	map	NOUN
ap-961	470	14	from	from	ADP
ap-961	470	15	�	�	PROPN
ap-961	470	16	�	�	PROPN
ap-961	470	17	(	(	PUNCT
ap-961	470	18	)	)	PUNCT
ap-961	470	19	n	n	X
ap-961	470	20	1	1	NUM
ap-961	470	21	to	to	ADP
ap-961	470	22	�	�	PROPN
ap-961	470	23	,	,	PUNCT
ap-961	470	24	and	and	CCONJ
ap-961	470	25	the	the	DET
ap-961	470	26	linear	linear	PROPN
ap-961	470	27	map	map	NOUN
ap-961	470	28	b	b	PROPN
ap-961	470	29	c	c	X
ap-961	470	30	cn	cn	PROPN
ap-961	470	31	n	n	CCONJ
ap-961	470	32	:	:	PUNCT
ap-961	470	33	(	(	PUNCT
ap-961	470	34	)	)	PUNCT
ap-961	470	35	(	(	PUNCT
ap-961	470	36	)	)	PUNCT
ap-961	470	37	�	�	PROPN
ap-961	470	38	�	�	PROPN
ap-961	470	39	�	�	PROPN
ap-961	470	40	1	1	NUM
ap-961	470	41	:	:	PUNCT
ap-961	470	42	(	(	PUNCT
ap-961	470	43	)	)	PUNCT
ap-961	470	44	(	(	PUNCT
ap-961	470	45	,	,	PUNCT
ap-961	470	46	,	,	PUNCT
ap-961	470	47	,	,	PUNCT
ap-961	470	48	,	,	PUNCT
ap-961	470	49	)	)	PUNCT
ap-961	470	50	:	:	PUNCT
ap-961	470	51	(	(	PUNCT
ap-961	470	52	,	,	PUNCT
ap-961	470	53	,	,	PUNCT
ap-961	470	54	,	,	PUNCT
ap-961	470	55	)	)	PUNCT
ap-961	470	56	(	(	PUNCT
ap-961	470	57	,	,	PUNCT
ap-961	470	58	b	b	X
ap-961	470	59	a	a	DET
ap-961	470	60	a	a	DET
ap-961	470	61	a	a	DET
ap-961	470	62	a	a	DET
ap-961	470	63	a	a	DET
ap-961	470	64	a	a	DET
ap-961	470	65	a	a	DET
ap-961	470	66	a	a	DET
ap-961	470	67	a	a	DET
ap-961	470	68	a	a	DET
ap-961	470	69	a	a	PRON
ap-961	470	70	n	n	CCONJ
ap-961	470	71	n	n	CCONJ
ap-961	470	72	n	n	CCONJ
ap-961	470	73	�	�	PROPN
ap-961	470	74	�	�	PROPN
ap-961	470	75	�	�	PROPN
ap-961	470	76	0	0	NUM
ap-961	470	77	1	1	NUM
ap-961	470	78	1	1	NUM
ap-961	470	79	0	0	NUM
ap-961	470	80	1	1	NUM
ap-961	470	81	2	2	NUM
ap-961	470	82	1	1	NUM
ap-961	470	83	0	0	NUM
ap-961	470	84	1	1	NUM
ap-961	470	85	2	2	NUM
ap-961	470	86	�	�	PROPN
ap-961	470	87	�	�	PROPN
ap-961	470	88	�	�	PROPN
ap-961	470	89	�	�	PROPN
ap-961	470	90	,	,	PUNCT
ap-961	470	91	,	,	PUNCT
ap-961	470	92	)	)	PUNCT
ap-961	470	93	(	(	PUNCT
ap-961	470	94	)	)	PUNCT
ap-961	470	95	(	(	PUNCT
ap-961	470	96	,	,	PUNCT
ap-961	470	97	,	,	PUNCT
ap-961	470	98	,	,	PUNCT
ap-961	470	99	)	)	PUNCT
ap-961	470	100	(	(	PUNCT
ap-961	470	101	)	)	PUNCT
ap-961	470	102	(	(	PUNCT
ap-961	470	103	,	,	PUNCT
ap-961	470	104	�	�	PROPN
ap-961	470	105	�	�	PROPN
ap-961	470	106	�	�	PROPN
ap-961	470	107	a	a	PRON
ap-961	470	108	a	a	PRON
ap-961	470	109	a	a	DET
ap-961	470	110	a	a	DET
ap-961	470	111	a	a	DET
ap-961	470	112	a	a	DET
ap-961	470	113	a	a	DET
ap-961	470	114	n	n	CCONJ
ap-961	470	115	n	n	CCONJ
ap-961	470	116	n	n	CCONJ
ap-961	470	117	n	n	CCONJ
ap-961	470	118	n	n	CCONJ
ap-961	470	119	n	n	CCONJ
ap-961	470	120	�	�	PROPN
ap-961	470	121	�	�	PROPN
ap-961	470	122	1	1	NUM
ap-961	470	123	0	0	NUM
ap-961	470	124	1	1	NUM
ap-961	470	125	1	1	NUM
ap-961	470	126	1	1	NUM
ap-961	470	127	1	1	NUM
ap-961	470	128	0	0	NUM
ap-961	470	129	1	1	NUM
ap-961	470	130	1	1	NUM
ap-961	470	131	�	�	PROPN
ap-961	470	132	�	�	PROPN
ap-961	470	133	a	a	PROPN
ap-961	470	134	an1	an1	PROPN
ap-961	470	135	,	,	PUNCT
ap-961	470	136	,	,	PUNCT
ap-961	470	137	)	)	PUNCT
ap-961	470	138	,	,	PUNCT
ap-961	470	139	�	�	PROPN
ap-961	470	140	where	where	SCONJ
ap-961	470	141	�	�	PROPN
ap-961	470	142	is	be	AUX
ap-961	470	143	an	an	DET
ap-961	470	144	n	n	PRON
ap-961	470	145	1	1	NUM
ap-961	470	146	-	-	PUNCT
ap-961	470	147	linear	linear	NOUN
ap-961	470	148	functional	functional	ADJ
ap-961	470	149	and	and	CCONJ
ap-961	470	150	a	a	DET
ap-961	470	151	a	a	DET
ap-961	470	152	an0	an0	PROPN
ap-961	470	153	1	1	NUM
ap-961	470	154	1	1	NUM
ap-961	470	155	,	,	PUNCT
ap-961	470	156	,	,	PUNCT
ap-961	470	157	,	,	PUNCT
ap-961	470	158	�	�	PROPN
ap-961	470	159	�	�	PROPN
ap-961	470	160	�	�	PROPN
ap-961	470	161	.	.	PUNCT
ap-961	471	1	then	then	ADV
ap-961	471	2	b2	b2	VERB
ap-961	471	3	0	0	NUM
ap-961	471	4	�	�	PROPN
ap-961	471	5	,	,	PUNCT
ap-961	471	6	and	and	CCONJ
ap-961	471	7	we	we	PRON
ap-961	471	8	can	can	AUX
ap-961	471	9	define	define	VERB
ap-961	471	10	the	the	DET
ap-961	471	11	cohomology	cohomology	NOUN
ap-961	471	12	of	of	ADP
ap-961	471	13	the	the	DET
ap-961	471	14	complex	complex	ADJ
ap-961	471	15	{	{	PUNCT
ap-961	471	16	,	,	PUNCT
ap-961	471	17	}	}	PUNCT
ap-961	471	18	c	c	PROPN
ap-961	471	19	bn	bn	ADJ
ap-961	471	20	n	n	CCONJ
ap-961	471	21	�	�	PROPN
ap-961	471	22	�	�	PROPN
ap-961	471	23	,	,	PUNCT
ap-961	471	24	which	which	PRON
ap-961	471	25	is	be	AUX
ap-961	471	26	called	call	VERB
ap-961	471	27	the	the	DET
ap-961	471	28	hochschild	hochschild	ADJ
ap-961	471	29	cohomology	cohomology	NOUN
ap-961	471	30	of	of	ADP
ap-961	471	31	a	a	PRON
ap-961	471	32	and	and	CCONJ
ap-961	471	33	denoted	denote	VERB
ap-961	471	34	hh	hh	PROPN
ap-961	471	35	*	*	ADJ
ap-961	471	36	(	(	PUNCT
ap-961	471	37	)	)	PUNCT
ap-961	471	38	�	�	PROPN
ap-961	471	39	:	:	PUNCT
ap-961	471	40	hh	hh	PROPN
ap-961	471	41	b	b	PROPN
ap-961	471	42	b	b	PROPN
ap-961	471	43	n	n	NUM
ap-961	471	44	c	c	NOUN
ap-961	471	45	c	c	NOUN
ap-961	471	46	n	n	CCONJ
ap-961	471	47	n	n	PROPN
ap-961	472	1	(	(	PUNCT
ap-961	472	2	)	)	PUNCT
ap-961	472	3	ker	ker	NOUN
ap-961	473	1	i	i	PRON
ap-961	473	2	m	m	PROPN
ap-961	473	3	�	�	PROPN
ap-961	473	4	�	�	PROPN
ap-961	473	5	.	.	PUNCT
ap-961	474	1	example	example	NOUN
ap-961	474	2	5.2	5.2	NUM
ap-961	474	3	:	:	PUNCT
ap-961	474	4	let	let	VERB
ap-961	474	5	us	we	PRON
ap-961	474	6	see	see	VERB
ap-961	474	7	what	what	PRON
ap-961	474	8	is	be	AUX
ap-961	474	9	the	the	DET
ap-961	474	10	zeroth	zeroth	PROPN
ap-961	474	11	hochschild	hochschild	ADJ
ap-961	474	12	cohomology	cohomology	NOUN
ap-961	474	13	group	group	NOUN
ap-961	474	14	for	for	ADP
ap-961	474	15	any	any	DET
ap-961	474	16	algebra	algebra	PROPN
ap-961	474	17	�	�	PROPN
ap-961	474	18	.	.	PUNCT
ap-961	475	1	from	from	ADP
ap-961	475	2	the	the	DET
ap-961	475	3	definition	definition	NOUN
ap-961	475	4	,	,	PUNCT
ap-961	475	5	it	it	PRON
ap-961	475	6	consists	consist	VERB
ap-961	475	7	of	of	ADP
ap-961	475	8	all	all	DET
ap-961	475	9	linear	linear	NOUN
ap-961	475	10	functionals	functional	NOUN
ap-961	475	11	�	�	PROPN
ap-961	475	12	,	,	PUNCT
ap-961	475	13	which	which	PRON
ap-961	475	14	obey	obey	VERB
ap-961	475	15	:	:	PUNCT
ap-961	475	16	46	46	NUM
ap-961	475	17	©	©	PROPN
ap-961	475	18	czech	czech	PROPN
ap-961	475	19	technical	technical	PROPN
ap-961	475	20	university	university	PROPN
ap-961	475	21	publishing	publishing	NOUN
ap-961	475	22	house	house	NOUN
ap-961	475	23	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	475	24	acta	acta	PROPN
ap-961	475	25	polytechnica	polytechnica	PROPN
ap-961	475	26	vol	vol	NOUN
ap-961	475	27	.	.	PUNCT
ap-961	476	1	48	48	NUM
ap-961	476	2	no	no	NOUN
ap-961	476	3	.	.	PUNCT
ap-961	477	1	2/2008	2/2008	PROPN
ap-961	477	2	b	b	PROPN
ap-961	477	3	a	a	PRON
ap-961	477	4	a	a	DET
ap-961	477	5	a	a	DET
ap-961	477	6	a	a	DET
ap-961	477	7	a	a	DET
ap-961	477	8	a	a	DET
ap-961	477	9	�	�	PROPN
ap-961	477	10	�	�	PROPN
ap-961	477	11	�	�	PROPN
ap-961	477	12	�	�	PROPN
ap-961	477	13	�	�	PROPN
ap-961	477	14	(	(	PUNCT
ap-961	477	15	,	,	PUNCT
ap-961	477	16	)	)	PUNCT
ap-961	477	17	)	)	PUNCT
ap-961	477	18	)	)	PUNCT
ap-961	478	1	0	0	NUM
ap-961	478	2	1	1	NUM
ap-961	478	3	0	0	NUM
ap-961	478	4	1	1	NUM
ap-961	478	5	1	1	NUM
ap-961	478	6	0	0	NUM
ap-961	478	7	0	0	NUM
ap-961	478	8	�	�	PROPN
ap-961	478	9	�	�	PROPN
ap-961	478	10	�	�	PROPN
ap-961	478	11	.	.	PUNCT
ap-961	479	1	therefore	therefore	ADV
ap-961	479	2	hh	hh	PROPN
ap-961	479	3	0	0	PROPN
ap-961	479	4	(	(	PUNCT
ap-961	479	5	)	)	PUNCT
ap-961	479	6	�	�	PROPN
ap-961	479	7	is	be	AUX
ap-961	479	8	nothing	nothing	PRON
ap-961	479	9	else	else	ADV
ap-961	479	10	but	but	CCONJ
ap-961	479	11	the	the	DET
ap-961	479	12	linear	linear	ADJ
ap-961	479	13	space	space	NOUN
ap-961	479	14	of	of	ADP
ap-961	479	15	all	all	DET
ap-961	479	16	traces	trace	NOUN
ap-961	479	17	on	on	ADP
ap-961	479	18	the	the	DET
ap-961	479	19	algebra	algebra	PROPN
ap-961	479	20	�	�	PROPN
ap-961	479	21	!	!	PUNCT
ap-961	480	1	higher	high	ADJ
ap-961	480	2	hochschild	hochschild	ADJ
ap-961	480	3	cohomology	cohomology	NOUN
ap-961	480	4	groups	group	NOUN
ap-961	480	5	are	be	AUX
ap-961	480	6	more	more	ADV
ap-961	480	7	difficult	difficult	ADJ
ap-961	480	8	to	to	PART
ap-961	480	9	calculate	calculate	VERB
ap-961	480	10	(	(	PUNCT
ap-961	480	11	but	but	CCONJ
ap-961	480	12	still	still	ADV
ap-961	480	13	calculable	calculable	ADJ
ap-961	480	14	in	in	ADP
ap-961	480	15	most	most	ADJ
ap-961	480	16	examples	example	NOUN
ap-961	480	17	)	)	PUNCT
ap-961	480	18	.	.	PUNCT
ap-961	481	1	for	for	ADP
ap-961	481	2	the	the	DET
ap-961	481	3	purpose	purpose	NOUN
ap-961	481	4	of	of	ADP
ap-961	481	5	making	make	VERB
ap-961	481	6	connections	connection	NOUN
ap-961	481	7	with	with	ADP
ap-961	481	8	the	the	DET
ap-961	481	9	commutative	commutative	ADJ
ap-961	481	10	differential	differential	NOUN
ap-961	481	11	geometry	geometry	NOUN
ap-961	481	12	let	let	VERB
ap-961	481	13	us	we	PRON
ap-961	481	14	quote	quote	VERB
ap-961	481	15	:	:	PUNCT
ap-961	481	16	theorem	theorem	VERB
ap-961	481	17	5.3	5.3	NUM
ap-961	481	18	:	:	PUNCT
ap-961	481	19	for	for	ADP
ap-961	481	20	an	an	DET
ap-961	481	21	algebra	algebra	NOUN
ap-961	481	22	of	of	ADP
ap-961	481	23	smooth	smooth	ADJ
ap-961	481	24	functions	function	NOUN
ap-961	481	25	on	on	ADP
ap-961	481	26	a	a	DET
ap-961	481	27	manifold	manifold	ADJ
ap-961	481	28	m	m	NOUN
ap-961	481	29	,	,	PUNCT
ap-961	481	30	the	the	DET
ap-961	481	31	continuous	continuous	ADJ
ap-961	481	32	hochschild	hochschild	ADJ
ap-961	481	33	cohomology	cohomology	NOUN
ap-961	481	34	group	group	NOUN
ap-961	481	35	hh	hh	PROPN
ap-961	481	36	c	c	PROPN
ap-961	481	37	mk	mk	X
ap-961	481	38	(	(	PUNCT
ap-961	481	39	(	(	PUNCT
ap-961	481	40	)	)	PUNCT
ap-961	481	41	)	)	PUNCT
ap-961	481	42	is	be	AUX
ap-961	481	43	canonically	canonically	ADV
ap-961	481	44	isomorphic	isomorphic	ADJ
ap-961	481	45	with	with	ADP
ap-961	481	46	the	the	DET
ap-961	481	47	space	space	NOUN
ap-961	481	48	of	of	ADP
ap-961	481	49	de	de	X
ap-961	481	50	rham	rham	PROPN
ap-961	481	51	currents	current	NOUN
ap-961	481	52	on	on	ADP
ap-961	481	53	the	the	DET
ap-961	481	54	manifold	manifold	ADJ
ap-961	481	55	m	m	NOUN
ap-961	481	56	(	(	PUNCT
ap-961	481	57	which	which	PRON
ap-961	481	58	are	be	AUX
ap-961	481	59	continuous	continuous	ADJ
ap-961	481	60	linear	linear	ADJ
ap-961	481	61	functional	functional	ADJ
ap-961	481	62	on	on	ADP
ap-961	481	63	the	the	DET
ap-961	481	64	space	space	NOUN
ap-961	481	65	of	of	ADP
ap-961	481	66	de	de	X
ap-961	481	67	rham	rham	PROPN
ap-961	481	68	forms	form	NOUN
ap-961	481	69	)	)	PUNCT
ap-961	481	70	.	.	PUNCT
ap-961	482	1	for	for	ADP
ap-961	482	2	proof	proof	NOUN
ap-961	482	3	(	(	PUNCT
ap-961	482	4	and	and	CCONJ
ap-961	482	5	for	for	ADP
ap-961	482	6	more	more	ADJ
ap-961	482	7	details	detail	NOUN
ap-961	482	8	)	)	PUNCT
ap-961	482	9	we	we	PRON
ap-961	482	10	refer	refer	VERB
ap-961	482	11	to	to	ADP
ap-961	482	12	the	the	DET
ap-961	482	13	seminal	seminal	ADJ
ap-961	482	14	work	work	NOUN
ap-961	482	15	of	of	ADP
ap-961	482	16	connes	conne	NOUN
ap-961	482	17	[	[	X
ap-961	482	18	23	23	NUM
ap-961	482	19	]	]	PUNCT
ap-961	482	20	.	.	PUNCT
ap-961	483	1	so	so	ADV
ap-961	483	2	,	,	PUNCT
ap-961	483	3	we	we	PRON
ap-961	483	4	have	have	VERB
ap-961	483	5	a	a	DET
ap-961	483	6	space	space	NOUN
ap-961	483	7	,	,	PUNCT
ap-961	483	8	which	which	PRON
ap-961	483	9	corresponds	correspond	VERB
ap-961	483	10	(	(	PUNCT
ap-961	483	11	roughly	roughly	ADV
ap-961	483	12	)	)	PUNCT
ap-961	483	13	to	to	ADP
ap-961	483	14	the	the	DET
ap-961	483	15	differential	differential	ADJ
ap-961	483	16	forms	form	NOUN
ap-961	483	17	(	(	PUNCT
ap-961	483	18	although	although	SCONJ
ap-961	483	19	it	it	PRON
ap-961	483	20	is	be	AUX
ap-961	483	21	not	not	PART
ap-961	483	22	a	a	DET
ap-961	483	23	differential	differential	ADJ
ap-961	483	24	algebra	algebra	NOUN
ap-961	483	25	)	)	PUNCT
ap-961	483	26	!	!	PUNCT
ap-961	484	1	what	what	PRON
ap-961	484	2	we	we	PRON
ap-961	484	3	need	need	VERB
ap-961	484	4	is	be	AUX
ap-961	484	5	to	to	PART
ap-961	484	6	find	find	VERB
ap-961	484	7	a	a	DET
ap-961	484	8	subcomplex	subcomplex	NOUN
ap-961	484	9	which	which	PRON
ap-961	484	10	would	would	AUX
ap-961	484	11	give	give	VERB
ap-961	484	12	us	we	PRON
ap-961	484	13	in	in	ADP
ap-961	484	14	the	the	DET
ap-961	484	15	commutative	commutative	ADJ
ap-961	484	16	situation	situation	NOUN
ap-961	484	17	some	some	DET
ap-961	484	18	information	information	NOUN
ap-961	484	19	about	about	ADP
ap-961	484	20	the	the	DET
ap-961	484	21	de	de	PROPN
ap-961	484	22	rham	rham	PROPN
ap-961	484	23	cohomology	cohomology	NOUN
ap-961	484	24	.	.	PUNCT
ap-961	485	1	definition	definition	NOUN
ap-961	485	2	5.4	5.4	NUM
ap-961	485	3	:	:	PUNCT
ap-961	485	4	consider	consider	VERB
ap-961	485	5	cn	cn	PROPN
ap-961	485	6	�	�	PROPN
ap-961	485	7	(	(	PUNCT
ap-961	485	8	)	)	PUNCT
ap-961	485	9	�	�	PROPN
ap-961	485	10	space	space	NOUN
ap-961	485	11	of	of	ADP
ap-961	485	12	linear	linear	PROPN
ap-961	485	13	maps	map	NOUN
ap-961	485	14	from	from	ADP
ap-961	485	15	�	�	PROPN
ap-961	485	16	�	�	PROPN
ap-961	485	17	(	(	PUNCT
ap-961	485	18	)	)	PUNCT
ap-961	485	19	n	n	X
ap-961	485	20	1	1	NUM
ap-961	485	21	to	to	ADP
ap-961	485	22	�	�	PROPN
ap-961	485	23	,	,	PUNCT
ap-961	485	24	which	which	PRON
ap-961	485	25	are	be	AUX
ap-961	485	26	cyclic	cyclic	ADJ
ap-961	485	27	:	:	PUNCT
ap-961	485	28	�	�	PROPN
ap-961	485	29	�	�	PROPN
ap-961	485	30	(	(	PUNCT
ap-961	485	31	,	,	PUNCT
ap-961	485	32	,	,	PUNCT
ap-961	485	33	,	,	PUNCT
ap-961	485	34	)	)	PUNCT
ap-961	485	35	(	(	PUNCT
ap-961	485	36	)	)	PUNCT
ap-961	485	37	(	(	PUNCT
ap-961	485	38	,	,	PUNCT
ap-961	485	39	,	,	PUNCT
ap-961	485	40	,	,	PUNCT
ap-961	485	41	,	,	PUNCT
ap-961	485	42	)	)	PUNCT
ap-961	485	43	a	a	DET
ap-961	485	44	a	a	DET
ap-961	485	45	a	a	DET
ap-961	485	46	a	a	DET
ap-961	485	47	a	a	DET
ap-961	485	48	a	a	DET
ap-961	485	49	an	an	DET
ap-961	485	50	n	n	ADV
ap-961	485	51	n0	n0	NUM
ap-961	485	52	1	1	NUM
ap-961	485	53	1	1	NUM
ap-961	485	54	2	2	NUM
ap-961	485	55	01	01	NUM
ap-961	485	56	�	�	PROPN
ap-961	485	57	�	�	PROPN
ap-961	485	58	�	�	PROPN
ap-961	485	59	�	�	PROPN
ap-961	485	60	and	and	CCONJ
ap-961	485	61	the	the	DET
ap-961	485	62	linear	linear	PROPN
ap-961	485	63	map	map	NOUN
ap-961	486	1	b	b	PROPN
ap-961	486	2	c	c	X
ap-961	486	3	cn	cn	PROPN
ap-961	486	4	n	n	CCONJ
ap-961	486	5	:	:	PUNCT
ap-961	486	6	(	(	PUNCT
ap-961	486	7	)	)	PUNCT
ap-961	486	8	(	(	PUNCT
ap-961	486	9	)	)	PUNCT
ap-961	486	10	�	�	PROPN
ap-961	486	11	�	�	PROPN
ap-961	486	12	�	�	PROPN
ap-961	486	13	�	�	PROPN
ap-961	486	14	�	�	PROPN
ap-961	486	15	1	1	NUM
ap-961	486	16	,	,	PUNCT
ap-961	486	17	which	which	PRON
ap-961	486	18	is	be	AUX
ap-961	486	19	the	the	DET
ap-961	486	20	restriction	restriction	NOUN
ap-961	486	21	of	of	ADP
ap-961	486	22	the	the	DET
ap-961	486	23	coboundary	coboundary	PROPN
ap-961	486	24	b	b	PROPN
ap-961	486	25	defined	define	VERB
ap-961	486	26	above	above	ADV
ap-961	486	27	.	.	PUNCT
ap-961	487	1	the	the	DET
ap-961	487	2	homology	homology	NOUN
ap-961	487	3	of	of	ADP
ap-961	487	4	the	the	DET
ap-961	487	5	cochain	cochain	NOUN
ap-961	487	6	complex	complex	NOUN
ap-961	487	7	(	(	PUNCT
ap-961	487	8	,	,	PUNCT
ap-961	487	9	)	)	PUNCT
ap-961	487	10	c	c	PROPN
ap-961	487	11	bn	bn	NUM
ap-961	487	12	n	n	CCONJ
ap-961	487	13	�	�	PROPN
ap-961	487	14	�	�	PROPN
ap-961	487	15	�	�	PROPN
ap-961	487	16	is	be	AUX
ap-961	487	17	the	the	DET
ap-961	487	18	cyclic	cyclic	ADJ
ap-961	487	19	cohomology	cohomology	NOUN
ap-961	487	20	of	of	ADP
ap-961	487	21	�	�	PROPN
ap-961	487	22	,	,	PUNCT
ap-961	487	23	denoted	denote	VERB
ap-961	487	24	hc	hc	PROPN
ap-961	487	25	*	*	ADJ
ap-961	487	26	(	(	PUNCT
ap-961	487	27	)	)	PUNCT
ap-961	487	28	�	�	PROPN
ap-961	487	29	.	.	PUNCT
ap-961	488	1	example	example	NOUN
ap-961	488	2	5.5	5.5	NUM
ap-961	488	3	:	:	PUNCT
ap-961	488	4	let	let	VERB
ap-961	488	5	m	m	PRON
ap-961	488	6	be	be	AUX
ap-961	488	7	a	a	DET
ap-961	488	8	manifold	manifold	NOUN
ap-961	488	9	of	of	ADP
ap-961	488	10	dimension	dimension	NOUN
ap-961	488	11	and	and	CCONJ
ap-961	488	12	dr	dr	PROPN
ap-961	488	13	m	m	PROPN
ap-961	488	14	(	(	PUNCT
ap-961	488	15	)	)	PUNCT
ap-961	488	16	the	the	DET
ap-961	488	17	differential	differential	ADJ
ap-961	488	18	algebra	algebra	NOUN
ap-961	488	19	of	of	ADP
ap-961	488	20	de	de	X
ap-961	488	21	rham	rham	PROPN
ap-961	488	22	forms	form	NOUN
ap-961	488	23	over	over	ADP
ap-961	488	24	m.	m.	NOUN
ap-961	488	25	for	for	ADP
ap-961	488	26	smooth	smooth	ADJ
ap-961	488	27	functions	function	NOUN
ap-961	488	28	a	a	DET
ap-961	488	29	a	a	DET
ap-961	488	30	an0	an0	PROPN
ap-961	488	31	1	1	NUM
ap-961	488	32	,	,	PUNCT
ap-961	488	33	,	,	PUNCT
ap-961	488	34	,	,	PUNCT
ap-961	488	35	�	�	PROPN
ap-961	488	36	we	we	PRON
ap-961	488	37	define	define	VERB
ap-961	488	38	:	:	PUNCT
ap-961	488	39	�	�	PROPN
ap-961	488	40	(	(	PUNCT
ap-961	488	41	,	,	PUNCT
ap-961	488	42	,	,	PUNCT
ap-961	488	43	,	,	PUNCT
ap-961	488	44	)	)	PUNCT
ap-961	488	45	:	:	PUNCT
ap-961	488	46	a	a	DET
ap-961	488	47	a	a	DET
ap-961	488	48	a	a	DET
ap-961	488	49	a	a	DET
ap-961	488	50	da	da	PROPN
ap-961	488	51	dan	dan	PROPN
ap-961	488	52	n0	n0	PROPN
ap-961	488	53	1	1	NUM
ap-961	488	54	1	1	NUM
ap-961	488	55	0	0	NUM
ap-961	488	56	1	1	NUM
ap-961	488	57	�	�	PROPN
ap-961	488	58	�	�	PROPN
ap-961	488	59	�	�	PROPN
ap-961	488	60	#	#	PROPN
ap-961	488	61	#	#	SYM
ap-961	488	62	2	2	NUM
ap-961	488	63	,	,	PUNCT
ap-961	488	64	where	where	SCONJ
ap-961	488	65	2	2	NUM
ap-961	488	66	is	be	AUX
ap-961	488	67	the	the	DET
ap-961	488	68	standard	standard	ADJ
ap-961	488	69	integral	integral	ADJ
ap-961	488	70	on	on	ADP
ap-961	488	71	the	the	DET
ap-961	488	72	manifold	manifold	NOUN
ap-961	488	73	.	.	PUNCT
ap-961	489	1	then	then	ADV
ap-961	489	2	�	�	PROPN
ap-961	489	3	is	be	AUX
ap-961	489	4	a	a	DET
ap-961	489	5	cyclic	cyclic	ADJ
ap-961	489	6	cocycle	cocycle	NOUN
ap-961	489	7	of	of	ADP
ap-961	489	8	dimension	dimension	NOUN
ap-961	489	9	n.	n.	PROPN
ap-961	489	10	the	the	DET
ap-961	489	11	cyclicity	cyclicity	NOUN
ap-961	489	12	of	of	ADP
ap-961	489	13	�	�	PROPN
ap-961	489	14	follows	follow	VERB
ap-961	489	15	from	from	ADP
ap-961	489	16	the	the	DET
ap-961	489	17	leibniz	leibniz	NOUN
ap-961	489	18	rule	rule	NOUN
ap-961	489	19	and	and	CCONJ
ap-961	489	20	the	the	DET
ap-961	489	21	fact	fact	NOUN
ap-961	489	22	that	that	SCONJ
ap-961	489	23	the	the	DET
ap-961	489	24	wedge	wedge	NOUN
ap-961	489	25	product	product	NOUN
ap-961	489	26	of	of	ADP
ap-961	489	27	forms	form	NOUN
ap-961	489	28	is	be	AUX
ap-961	489	29	antisymmetric	antisymmetric	ADJ
ap-961	489	30	.	.	PUNCT
ap-961	490	1	we	we	PRON
ap-961	490	2	check	check	VERB
ap-961	490	3	explicitly	explicitly	ADV
ap-961	490	4	that	that	SCONJ
ap-961	490	5	�	�	PROPN
ap-961	490	6	is	be	AUX
ap-961	490	7	a	a	DET
ap-961	490	8	hochschild	hochschild	ADJ
ap-961	490	9	cocycle	cocycle	NOUN
ap-961	490	10	:	:	PUNCT
ap-961	490	11	b	b	ADP
ap-961	490	12	a	a	DET
ap-961	490	13	a	a	DET
ap-961	490	14	a	a	DET
ap-961	490	15	a	a	DET
ap-961	490	16	a	a	DET
ap-961	490	17	a	a	DET
ap-961	490	18	a	a	DET
ap-961	490	19	a	a	DET
ap-961	490	20	a	a	DET
ap-961	490	21	a	a	DET
ap-961	490	22	a	a	PRON
ap-961	490	23	n	n	CCONJ
ap-961	490	24	n	n	CCONJ
ap-961	490	25	�	�	PROPN
ap-961	490	26	�	�	PROPN
ap-961	490	27	�	�	PROPN
ap-961	490	28	(	(	PUNCT
ap-961	490	29	,	,	PUNCT
ap-961	490	30	,	,	PUNCT
ap-961	490	31	,	,	PUNCT
ap-961	490	32	,	,	PUNCT
ap-961	490	33	)	)	PUNCT
ap-961	490	34	(	(	PUNCT
ap-961	490	35	,	,	PUNCT
ap-961	490	36	,	,	PUNCT
ap-961	490	37	,	,	PUNCT
ap-961	490	38	)	)	PUNCT
ap-961	490	39	(	(	PUNCT
ap-961	490	40	,	,	PUNCT
ap-961	490	41	,	,	PUNCT
ap-961	490	42	,	,	PUNCT
ap-961	490	43	0	0	NUM
ap-961	490	44	1	1	NUM
ap-961	490	45	2	2	NUM
ap-961	490	46	1	1	NUM
ap-961	490	47	0	0	NUM
ap-961	490	48	1	1	NUM
ap-961	490	49	2	2	NUM
ap-961	490	50	1	1	NUM
ap-961	490	51	0	0	NUM
ap-961	490	52	1	1	NUM
ap-961	490	53	2	2	NUM
ap-961	490	54	�	�	PROPN
ap-961	490	55	�	�	PROPN
ap-961	490	56	�	�	PROPN
ap-961	490	57	�	�	PROPN
ap-961	490	58	�	�	PROPN
ap-961	490	59	a	a	DET
ap-961	490	60	a	a	DET
ap-961	490	61	a	a	DET
ap-961	490	62	a	a	DET
ap-961	490	63	a	a	PRON
ap-961	490	64	a	a	DET
ap-961	490	65	a	a	DET
ap-961	490	66	n	n	CCONJ
ap-961	490	67	n	n	CCONJ
ap-961	490	68	n	n	CCONJ
ap-961	490	69	n	n	CCONJ
ap-961	490	70	n	n	CCONJ
ap-961	490	71	n	n	CCONJ
ap-961	490	72	n	n	CCONJ
ap-961	490	73	�	�	PROPN
ap-961	490	74	�	�	PROPN
ap-961	490	75	�	�	PROPN
ap-961	490	76	1	1	NUM
ap-961	490	77	0	0	NUM
ap-961	490	78	1	1	NUM
ap-961	490	79	1	1	NUM
ap-961	490	80	1	1	NUM
ap-961	490	81	0	0	NUM
ap-961	490	82	1	1	NUM
ap-961	490	83	1	1	NUM
ap-961	490	84	)	)	PUNCT
ap-961	490	85	(	(	PUNCT
ap-961	490	86	)	)	PUNCT
ap-961	490	87	(	(	PUNCT
ap-961	490	88	,	,	PUNCT
ap-961	490	89	,	,	PUNCT
ap-961	490	90	)	)	PUNCT
ap-961	490	91	(	(	PUNCT
ap-961	490	92	)	)	PUNCT
ap-961	490	93	(	(	PUNCT
ap-961	490	94	,	,	PUNCT
ap-961	490	95	,	,	PUNCT
ap-961	490	96	)	)	PUNCT
ap-961	490	97	�	�	PROPN
ap-961	490	98	�	�	PROPN
ap-961	490	99	�	�	PROPN
ap-961	490	100	�	�	PROPN
ap-961	490	101	�	�	PROPN
ap-961	490	102	(	(	PUNCT
ap-961	490	103	)	)	PUNCT
ap-961	490	104	(	(	PUNCT
ap-961	490	105	)	)	PUNCT
ap-961	490	106	(	(	PUNCT
ap-961	490	107	)	)	PUNCT
ap-961	490	108	a	a	DET
ap-961	490	109	a	a	DET
ap-961	490	110	da	da	NOUN
ap-961	490	111	da	da	PROPN
ap-961	490	112	a	a	PRON
ap-961	490	113	d	d	X
ap-961	490	114	a	a	DET
ap-961	490	115	a	a	DET
ap-961	490	116	da	da	NOUN
ap-961	490	117	a	a	DET
ap-961	490	118	da	da	X
ap-961	490	119	n	n	CCONJ
ap-961	490	120	n	n	CCONJ
ap-961	490	121	n	n	ADV
ap-961	490	122	0	0	NUM
ap-961	490	123	1	1	NUM
ap-961	490	124	2	2	NUM
ap-961	490	125	1	1	NUM
ap-961	490	126	0	0	NUM
ap-961	490	127	1	1	NUM
ap-961	490	128	2	2	NUM
ap-961	490	129	1	1	NUM
ap-961	490	130	0	0	NUM
ap-961	490	131	11	11	NUM
ap-961	490	132	#	#	NOUN
ap-961	490	133	#	#	NOUN
ap-961	490	134	�	�	NOUN
ap-961	490	135	#	#	NOUN
ap-961	490	136	#	#	NOUN
ap-961	490	137	�	�	PROPN
ap-961	490	138	#	#	NOUN
ap-961	490	139	2	2	NUM
ap-961	490	140	�	�	PROPN
ap-961	490	141	�	�	PROPN
ap-961	490	142	�	�	PROPN
ap-961	490	143	�	�	PROPN
ap-961	490	144	�	�	PROPN
ap-961	490	145	�	�	PROPN
ap-961	490	146	#	#	NOUN
ap-961	490	147	�	�	PROPN
ap-961	490	148	#	#	NOUN
ap-961	490	149	#	#	NOUN
ap-961	490	150	�	�	PROPN
ap-961	490	151	�	�	PROPN
ap-961	490	152	#	#	NOUN
ap-961	490	153	#	#	NOUN
ap-961	490	154	d	d	NOUN
ap-961	490	155	a	a	DET
ap-961	490	156	a	a	PRON
ap-961	490	157	a	a	DET
ap-961	490	158	a	a	DET
ap-961	490	159	da	da	NOUN
ap-961	490	160	da	da	PROPN
ap-961	490	161	a	a	DET
ap-961	490	162	da	da	X
ap-961	490	163	da	da	PROPN
ap-961	490	164	n	n	CCONJ
ap-961	490	165	n	n	CCONJ
ap-961	490	166	n	n	CCONJ
ap-961	490	167	n	n	CCONJ
ap-961	490	168	n	n	CCONJ
ap-961	490	169	n	n	PROPN
ap-961	490	170	(	(	PUNCT
ap-961	490	171	)	)	PUNCT
ap-961	490	172	(	(	PUNCT
ap-961	490	173	)	)	PUNCT
ap-961	490	174	(	(	PUNCT
ap-961	490	175	)	)	PUNCT
ap-961	490	176	1	1	NUM
ap-961	490	177	1	1	NUM
ap-961	490	178	1	1	NUM
ap-961	490	179	0	0	NUM
ap-961	490	180	1	1	NUM
ap-961	490	181	0	0	NUM
ap-961	490	182	1	1	NUM
ap-961	490	183	1	1	NUM
ap-961	490	184	1	1	NUM
ap-961	490	185	n	n	CCONJ
ap-961	490	186	n	n	CCONJ
ap-961	490	187	n	n	CCONJ
ap-961	490	188	n	n	PROPN
ap-961	490	189	n	n	ADV
ap-961	490	190	a	a	PRON
ap-961	490	191	a	a	PRON
ap-961	490	192	a	a	DET
ap-961	490	193	da	da	PROPN
ap-961	490	194	da	da	X
ap-961	490	195	�	�	PROPN
ap-961	490	196	#	#	NOUN
ap-961	490	197	#	#	NOUN
ap-961	490	198	�	�	NOUN
ap-961	490	199	1	1	NUM
ap-961	490	200	1	1	NUM
ap-961	490	201	1	1	NUM
ap-961	490	202	0	0	NUM
ap-961	490	203	11	11	NUM
ap-961	490	204	0	0	NUM
ap-961	490	205	(	(	PUNCT
ap-961	490	206	)	)	PUNCT
ap-961	490	207	�	�	PROPN
ap-961	490	208	a	a	DET
ap-961	490	209	cycle	cycle	NOUN
ap-961	490	210	is	be	AUX
ap-961	490	211	a	a	DET
ap-961	490	212	noncommutative	noncommutative	ADJ
ap-961	490	213	generalization	generalization	NOUN
ap-961	490	214	of	of	ADP
ap-961	490	215	what	what	PRON
ap-961	490	216	we	we	PRON
ap-961	490	217	have	have	AUX
ap-961	490	218	in	in	ADP
ap-961	490	219	differential	differential	ADJ
ap-961	490	220	geometry	geometry	NOUN
ap-961	490	221	(	(	PUNCT
ap-961	490	222	as	as	SCONJ
ap-961	490	223	presented	present	VERB
ap-961	490	224	earlier	early	ADV
ap-961	490	225	):	):	PUNCT
ap-961	490	226	functions	function	NOUN
ap-961	490	227	and	and	CCONJ
ap-961	490	228	forms	form	NOUN
ap-961	490	229	together	together	ADV
ap-961	490	230	with	with	ADP
ap-961	490	231	the	the	DET
ap-961	490	232	integral	integral	ADJ
ap-961	490	233	and	and	CCONJ
ap-961	490	234	the	the	DET
ap-961	490	235	(	(	PUNCT
ap-961	490	236	inevitable	inevitable	ADJ
ap-961	490	237	)	)	PUNCT
ap-961	490	238	stokes	stoke	NOUN
ap-961	490	239	theorem	theorem	VERB
ap-961	490	240	.	.	PUNCT
ap-961	491	1	however	however	ADV
ap-961	491	2	,	,	PUNCT
ap-961	491	3	it	it	PRON
ap-961	491	4	is	be	AUX
ap-961	491	5	not	not	PART
ap-961	491	6	at	at	ADV
ap-961	491	7	all	all	ADV
ap-961	491	8	that	that	PRON
ap-961	491	9	easy	easy	ADJ
ap-961	492	1	and	and	CCONJ
ap-961	492	2	if	if	SCONJ
ap-961	492	3	one	one	PRON
ap-961	492	4	goes	go	VERB
ap-961	492	5	noncommutative	noncommutative	ADJ
ap-961	492	6	then	then	ADV
ap-961	492	7	,	,	PUNCT
ap-961	492	8	in	in	ADP
ap-961	492	9	general	general	ADJ
ap-961	492	10	,	,	PUNCT
ap-961	492	11	there	there	PRON
ap-961	492	12	is	be	VERB
ap-961	492	13	no	no	DET
ap-961	492	14	default	default	NOUN
ap-961	492	15	construction	construction	NOUN
ap-961	492	16	of	of	ADP
ap-961	492	17	such	such	DET
ap-961	492	18	a	a	DET
ap-961	492	19	structure	structure	NOUN
ap-961	492	20	.	.	PUNCT
ap-961	493	1	definition	definition	NOUN
ap-961	493	2	5.6	5.6	NUM
ap-961	493	3	:	:	PUNCT
ap-961	493	4	a	a	DET
ap-961	493	5	cycle	cycle	NOUN
ap-961	493	6	of	of	ADP
ap-961	493	7	dimension	dimension	NOUN
ap-961	493	8	n	n	CCONJ
ap-961	493	9	over	over	ADP
ap-961	493	10	�	�	PROPN
ap-961	493	11	is	be	AUX
ap-961	493	12	a	a	DET
ap-961	493	13	graded	grade	VERB
ap-961	493	14	differential	differential	ADJ
ap-961	493	15	algebra	algebra	NOUN
ap-961	493	16	over	over	ADP
ap-961	493	17	�	�	PROPN
ap-961	493	18	together	together	ADV
ap-961	493	19	with	with	ADP
ap-961	493	20	a	a	DET
ap-961	493	21	closed	close	VERB
ap-961	493	22	graded	grade	VERB
ap-961	493	23	trace	trace	NOUN
ap-961	493	24	2	2	NUM
ap-961	493	25	�	�	NOUN
ap-961	493	26	:	:	PUNCT
ap-961	493	27	(	(	PUNCT
ap-961	493	28	)	)	PUNCT
ap-961	493	29	�	�	PROPN
ap-961	493	30	�	�	PROPN
ap-961	493	31	:	:	PUNCT
ap-961	493	32	�	�	PROPN
ap-961	493	33	�	�	PROPN
ap-961	493	34	�	�	PROPN
ap-961	493	35	�	�	PROPN
ap-961	493	36	�	�	PROPN
ap-961	493	37	2	2	NUM
ap-961	493	38	2	2	NUM
ap-961	493	39	�	�	PROPN
ap-961	493	40	�	�	PROPN
ap-961	493	41	(	(	PUNCT
ap-961	493	42	)	)	PUNCT
ap-961	493	43	1	1	NUM
ap-961	493	44	,	,	PUNCT
ap-961	493	45	�	�	PROPN
ap-961	493	46	�	�	PROPN
ap-961	493	47	�	�	PROPN
ap-961	493	48	(	(	PUNCT
ap-961	493	49	)	)	PUNCT
ap-961	493	50	�	�	PROPN
ap-961	493	51	,	,	PUNCT
ap-961	493	52	d	d	X
ap-961	493	53	�	�	PROPN
ap-961	493	54	2	2	NUM
ap-961	493	55	�	�	NOUN
ap-961	493	56	0	0	NUM
ap-961	493	57	,	,	PUNCT
ap-961	493	58	�	�	PROPN
ap-961	493	59	�	�	PROPN
ap-961	493	60	(	(	PUNCT
ap-961	493	61	)	)	PUNCT
ap-961	493	62	�	�	PROPN
ap-961	493	63	.	.	PUNCT
ap-961	494	1	using	use	VERB
ap-961	494	2	cycles	cycle	NOUN
ap-961	494	3	one	one	PRON
ap-961	494	4	might	might	AUX
ap-961	494	5	easily	easily	ADV
ap-961	494	6	construct	construct	VERB
ap-961	494	7	cyclic	cyclic	ADJ
ap-961	494	8	cocycles	cocycle	NOUN
ap-961	494	9	.	.	PUNCT
ap-961	495	1	lemma	lemma	PROPN
ap-961	495	2	5.7	5.7	NUM
ap-961	495	3	:	:	PUNCT
ap-961	495	4	each	each	DET
ap-961	495	5	cycle	cycle	NOUN
ap-961	495	6	of	of	ADP
ap-961	495	7	dimension	dimension	NOUN
ap-961	495	8	n	n	CCONJ
ap-961	495	9	over	over	ADP
ap-961	495	10	algebra	algebra	PROPN
ap-961	495	11	�	�	PROPN
ap-961	495	12	defines	define	VERB
ap-961	495	13	a	a	DET
ap-961	495	14	class	class	NOUN
ap-961	495	15	of	of	ADP
ap-961	495	16	a	a	DET
ap-961	495	17	cyclic	cyclic	ADJ
ap-961	495	18	cocycle	cocycle	NOUN
ap-961	495	19	in	in	ADP
ap-961	495	20	hcn	hcn	PROPN
ap-961	495	21	(	(	PUNCT
ap-961	495	22	)	)	PUNCT
ap-961	495	23	�	�	PROPN
ap-961	495	24	:	:	PUNCT
ap-961	495	25	�	�	PROPN
ap-961	495	26	(	(	PUNCT
ap-961	495	27	,	,	PUNCT
ap-961	495	28	,	,	PUNCT
ap-961	495	29	,	,	PUNCT
ap-961	495	30	)	)	PUNCT
ap-961	495	31	(	(	PUNCT
ap-961	495	32	)	)	PUNCT
ap-961	495	33	(	(	PUNCT
ap-961	495	34	)	)	PUNCT
ap-961	495	35	(	(	PUNCT
ap-961	495	36	)	)	PUNCT
ap-961	495	37	a	a	DET
ap-961	495	38	a	a	PRON
ap-961	495	39	a	a	PRON
ap-961	495	40	i	i	PRON
ap-961	495	41	a	a	DET
ap-961	495	42	di	di	NOUN
ap-961	495	43	a	a	DET
ap-961	495	44	di	di	NOUN
ap-961	495	45	an	an	DET
ap-961	495	46	n0	n0	NUM
ap-961	495	47	1	1	NUM
ap-961	495	48	0	0	NUM
ap-961	495	49	1	1	NUM
ap-961	495	50	�	�	PROPN
ap-961	495	51	�	�	PROPN
ap-961	495	52	�	�	PROPN
ap-961	495	53	2	2	NUM
ap-961	495	54	.	.	PUNCT
ap-961	496	1	for	for	ADP
ap-961	496	2	proof	proof	NOUN
ap-961	496	3	see	see	VERB
ap-961	496	4	[	[	X
ap-961	496	5	5	5	NUM
ap-961	496	6	]	]	PUNCT
ap-961	496	7	,	,	PUNCT
ap-961	496	8	proposition	proposition	NOUN
ap-961	496	9	4	4	NUM
ap-961	496	10	,	,	PUNCT
ap-961	496	11	p.186	p.186	X
ap-961	496	12	.	.	PUNCT
ap-961	497	1	recall	recall	NOUN
ap-961	497	2	that	that	SCONJ
ap-961	497	3	we	we	PRON
ap-961	497	4	have	have	AUX
ap-961	497	5	met	meet	VERB
ap-961	497	6	a	a	DET
ap-961	497	7	very	very	ADV
ap-961	497	8	nice	nice	ADJ
ap-961	497	9	prescription	prescription	NOUN
ap-961	497	10	for	for	ADP
ap-961	497	11	the	the	DET
ap-961	497	12	construction	construction	NOUN
ap-961	497	13	of	of	ADP
ap-961	497	14	differential	differential	NOUN
ap-961	497	15	graded	grade	VERB
ap-961	497	16	algebras	algebra	NOUN
ap-961	497	17	through	through	ADP
ap-961	497	18	the	the	DET
ap-961	497	19	representation	representation	NOUN
ap-961	497	20	and	and	CCONJ
ap-961	497	21	commutators	commutator	NOUN
ap-961	497	22	with	with	ADP
ap-961	497	23	an	an	DET
ap-961	497	24	operator	operator	NOUN
ap-961	497	25	f	f	NOUN
ap-961	497	26	,	,	PUNCT
ap-961	497	27	f2	f2	PROPN
ap-961	497	28	1	1	NUM
ap-961	497	29	�	�	PROPN
ap-961	497	30	.	.	PUNCT
ap-961	498	1	can	can	AUX
ap-961	498	2	we	we	PRON
ap-961	498	3	get	get	VERB
ap-961	498	4	a	a	DET
ap-961	498	5	cycle	cycle	NOUN
ap-961	498	6	in	in	ADP
ap-961	498	7	this	this	DET
ap-961	498	8	way	way	NOUN
ap-961	498	9	?	?	PUNCT
ap-961	499	1	the	the	DET
ap-961	499	2	answer	answer	NOUN
ap-961	499	3	is	be	AUX
ap-961	499	4	positive	positive	ADJ
ap-961	499	5	,	,	PUNCT
ap-961	499	6	provided	provide	VERB
ap-961	499	7	that	that	SCONJ
ap-961	499	8	we	we	PRON
ap-961	499	9	are	be	AUX
ap-961	499	10	ready	ready	ADJ
ap-961	499	11	to	to	PART
ap-961	499	12	add	add	VERB
ap-961	499	13	a	a	DET
ap-961	499	14	couple	couple	NOUN
ap-961	499	15	of	of	ADP
ap-961	499	16	additional	additional	ADJ
ap-961	499	17	features	feature	NOUN
ap-961	499	18	.	.	PUNCT
ap-961	500	1	definition	definition	NOUN
ap-961	500	2	5.8	5.8	NUM
ap-961	500	3	:	:	PUNCT
ap-961	500	4	if	if	SCONJ
ap-961	500	5	�	�	PROPN
ap-961	500	6	is	be	AUX
ap-961	500	7	an	an	DET
ap-961	500	8	algebra	algebra	NOUN
ap-961	500	9	,	,	PUNCT
ap-961	500	10	�	�	PROPN
ap-961	500	11	its	its	PRON
ap-961	500	12	representation	representation	NOUN
ap-961	500	13	as	as	ADP
ap-961	500	14	bounded	bound	VERB
ap-961	500	15	operators	operator	NOUN
ap-961	500	16	on	on	ADP
ap-961	500	17	the	the	DET
ap-961	500	18	hilbert	hilbert	NOUN
ap-961	500	19	space	space	NOUN
ap-961	500	20	,	,	PUNCT
ap-961	500	21	and	and	CCONJ
ap-961	500	22	f	f	PROPN
ap-961	500	23	a	a	DET
ap-961	500	24	selfadjoint	selfadjoint	NOUN
ap-961	500	25	operator	operator	NOUN
ap-961	500	26	such	such	ADJ
ap-961	500	27	that	that	SCONJ
ap-961	500	28	f2	f2	PROPN
ap-961	500	29	1	1	NUM
ap-961	500	30	�	�	PROPN
ap-961	500	31	and	and	CCONJ
ap-961	500	32	for	for	ADP
ap-961	500	33	every	every	DET
ap-961	500	34	a	a	DET
ap-961	500	35	�	�	PROPN
ap-961	500	36	�	�	PROPN
ap-961	500	37	the	the	DET
ap-961	500	38	commutators	commutator	NOUN
ap-961	500	39	[	[	PUNCT
ap-961	500	40	,	,	PUNCT
ap-961	500	41	(	(	PUNCT
ap-961	500	42	)	)	PUNCT
ap-961	500	43	]	]	X
ap-961	500	44	f	f	X
ap-961	500	45	a	a	DET
ap-961	500	46	�	�	PROPN
ap-961	500	47	are	be	AUX
ap-961	500	48	compact	compact	ADJ
ap-961	500	49	then	then	ADV
ap-961	500	50	we	we	PRON
ap-961	500	51	call	call	VERB
ap-961	500	52	(	(	PUNCT
ap-961	500	53	,	,	PUNCT
ap-961	500	54	,	,	PUNCT
ap-961	500	55	)	)	PUNCT
ap-961	500	56	�	�	PROPN
ap-961	500	57	�	�	PROPN
ap-961	500	58	f	f	PROPN
ap-961	501	1	a	a	DET
ap-961	501	2	fredholm	fredholm	NOUN
ap-961	501	3	module	module	NOUN
ap-961	501	4	over	over	ADP
ap-961	501	5	�	�	PROPN
ap-961	501	6	.	.	PUNCT
ap-961	502	1	we	we	PRON
ap-961	502	2	say	say	VERB
ap-961	502	3	that	that	SCONJ
ap-961	502	4	the	the	DET
ap-961	502	5	fredholm	fredholm	NOUN
ap-961	502	6	module	module	NOUN
ap-961	502	7	is	be	AUX
ap-961	502	8	even	even	ADV
ap-961	502	9	if	if	SCONJ
ap-961	502	10	there	there	PRON
ap-961	502	11	exists	exist	VERB
ap-961	502	12	an	an	DET
ap-961	502	13	operator	operator	NOUN
ap-961	502	14	�	�	PROPN
ap-961	502	15	�	�	PROPN
ap-961	502	16	�	�	PROPN
ap-961	502	17	†	†	PROPN
ap-961	502	18	,	,	PUNCT
ap-961	502	19	�	�	PROPN
ap-961	502	20	2	2	NUM
ap-961	502	21	1	1	NUM
ap-961	502	22	�	�	NOUN
ap-961	502	23	which	which	PRON
ap-961	502	24	commutes	commute	VERB
ap-961	502	25	with	with	ADP
ap-961	502	26	the	the	DET
ap-961	502	27	representation	representation	NOUN
ap-961	502	28	�	�	PROPN
ap-961	502	29	and	and	CCONJ
ap-961	502	30	anticommutes	anticommute	NOUN
ap-961	502	31	with	with	ADP
ap-961	502	32	f.	f.	PROPN
ap-961	502	33	actually	actually	ADV
ap-961	502	34	we	we	PRON
ap-961	502	35	have	have	AUX
ap-961	502	36	already	already	ADV
ap-961	502	37	met	meet	VERB
ap-961	502	38	an	an	DET
ap-961	502	39	odd	odd	ADJ
ap-961	502	40	fredholm	fredholm	NOUN
ap-961	502	41	module	module	NOUN
ap-961	502	42	in	in	ADP
ap-961	502	43	the	the	DET
ap-961	502	44	example	example	NOUN
ap-961	502	45	of	of	ADP
ap-961	502	46	the	the	DET
ap-961	502	47	“	"	PUNCT
ap-961	502	48	strange	strange	ADJ
ap-961	502	49	”	"	PUNCT
ap-961	502	50	differential	differential	NOUN
ap-961	502	51	algebra	algebra	NOUN
ap-961	502	52	for	for	ADP
ap-961	502	53	the	the	DET
ap-961	502	54	functions	function	NOUN
ap-961	502	55	on	on	ADP
ap-961	502	56	the	the	DET
ap-961	502	57	circle	circle	NOUN
ap-961	502	58	3.16	3.16	NUM
ap-961	502	59	.	.	PUNCT
ap-961	503	1	and	and	CCONJ
ap-961	503	2	we	we	PRON
ap-961	503	3	have	have	AUX
ap-961	503	4	shown	show	VERB
ap-961	503	5	that	that	SCONJ
ap-961	503	6	indeed	indeed	ADV
ap-961	503	7	all	all	DET
ap-961	503	8	commutators	commutator	NOUN
ap-961	503	9	with	with	ADP
ap-961	503	10	the	the	DET
ap-961	503	11	algebra	algebra	NOUN
ap-961	503	12	elements	element	NOUN
ap-961	503	13	were	be	AUX
ap-961	503	14	compact	compact	ADJ
ap-961	503	15	!	!	PUNCT
ap-961	504	1	to	to	PART
ap-961	504	2	define	define	VERB
ap-961	504	3	the	the	DET
ap-961	504	4	graded	grade	VERB
ap-961	504	5	trace	trace	NOUN
ap-961	504	6	on	on	ADP
ap-961	504	7	the	the	DET
ap-961	504	8	cycle	cycle	NOUN
ap-961	504	9	we	we	PRON
ap-961	504	10	need	need	VERB
ap-961	504	11	to	to	PART
ap-961	504	12	know	know	VERB
ap-961	504	13	something	something	PRON
ap-961	504	14	about	about	ADP
ap-961	504	15	the	the	DET
ap-961	504	16	summability	summability	NOUN
ap-961	504	17	of	of	ADP
ap-961	504	18	the	the	DET
ap-961	504	19	fredholm	fredholm	NOUN
ap-961	504	20	module	module	NOUN
ap-961	504	21	,	,	PUNCT
ap-961	504	22	i.e.	i.e.	X
ap-961	504	23	we	we	PRON
ap-961	504	24	need	need	VERB
ap-961	504	25	to	to	PART
ap-961	504	26	assume	assume	VERB
ap-961	504	27	that	that	SCONJ
ap-961	504	28	the	the	DET
ap-961	504	29	products	product	NOUN
ap-961	504	30	of	of	ADP
ap-961	504	31	commutators	commutator	NOUN
ap-961	504	32	[	[	PUNCT
ap-961	504	33	,	,	PUNCT
ap-961	504	34	(	(	PUNCT
ap-961	504	35	)	)	PUNCT
ap-961	504	36	]	]	X
ap-961	504	37	f	f	X
ap-961	504	38	a	a	DET
ap-961	504	39	�	�	PROPN
ap-961	504	40	fall	fall	VERB
ap-961	504	41	into	into	ADP
ap-961	504	42	the	the	DET
ap-961	504	43	ideal	ideal	NOUN
ap-961	504	44	of	of	ADP
ap-961	504	45	trace	trace	NOUN
ap-961	504	46	class	class	NOUN
ap-961	504	47	operators	operator	NOUN
ap-961	504	48	.	.	PUNCT
ap-961	505	1	trace	trace	NOUN
ap-961	505	2	class	class	NOUN
ap-961	505	3	operators	operator	NOUN
ap-961	505	4	are	be	AUX
ap-961	505	5	operators	operator	NOUN
ap-961	505	6	such	such	ADJ
ap-961	505	7	that	that	SCONJ
ap-961	505	8	the	the	DET
ap-961	505	9	series	series	NOUN
ap-961	505	10	of	of	ADP
ap-961	505	11	partial	partial	ADJ
ap-961	505	12	sums	sum	NOUN
ap-961	505	13	of	of	ADP
ap-961	505	14	their	their	PRON
ap-961	505	15	eigenvalues	eigenvalue	NOUN
ap-961	505	16	is	be	AUX
ap-961	505	17	summable	summable	ADJ
ap-961	505	18	.	.	PUNCT
ap-961	506	1	more	more	ADV
ap-961	506	2	precisely	precisely	ADV
ap-961	506	3	,	,	PUNCT
ap-961	506	4	we	we	PRON
ap-961	506	5	say	say	VERB
ap-961	506	6	that	that	SCONJ
ap-961	506	7	the	the	DET
ap-961	506	8	fredholm	fredholm	NOUN
ap-961	506	9	module	module	NOUN
ap-961	506	10	is	be	AUX
ap-961	506	11	p	p	ADJ
ap-961	506	12	1	1	NUM
ap-961	506	13	-	-	PUNCT
ap-961	506	14	summable	summable	ADJ
ap-961	506	15	,	,	PUNCT
ap-961	506	16	if	if	SCONJ
ap-961	506	17	for	for	ADP
ap-961	506	18	any	any	DET
ap-961	506	19	p	p	ADJ
ap-961	506	20	1	1	NUM
ap-961	506	21	-	-	PUNCT
ap-961	506	22	elements	element	NOUN
ap-961	506	23	the	the	DET
ap-961	506	24	product	product	NOUN
ap-961	506	25	of	of	ADP
ap-961	506	26	compact	compact	ADJ
ap-961	506	27	operators	operator	NOUN
ap-961	506	28	:	:	PUNCT
ap-961	506	29	[	[	PUNCT
ap-961	506	30	,	,	PUNCT
ap-961	506	31	]	]	X
ap-961	506	32	[	[	PUNCT
ap-961	506	33	,	,	PUNCT
ap-961	506	34	]	]	X
ap-961	506	35	[	[	PUNCT
ap-961	506	36	,	,	PUNCT
ap-961	506	37	]	]	X
ap-961	506	38	f	f	X
ap-961	506	39	a	a	DET
ap-961	506	40	f	f	X
ap-961	506	41	a	a	DET
ap-961	506	42	f	f	PROPN
ap-961	506	43	ap1	ap1	PROPN
ap-961	506	44	2	2	NUM
ap-961	506	45	1	1	NUM
ap-961	506	46	�	�	NOUN
ap-961	506	47	is	be	AUX
ap-961	506	48	an	an	DET
ap-961	506	49	operator	operator	NOUN
ap-961	506	50	of	of	ADP
ap-961	506	51	trace	trace	NOUN
ap-961	506	52	class	class	NOUN
ap-961	506	53	.	.	PUNCT
ap-961	507	1	then	then	ADV
ap-961	507	2	we	we	PRON
ap-961	507	3	have	have	VERB
ap-961	507	4	:	:	PUNCT
ap-961	507	5	lemma	lemma	PROPN
ap-961	507	6	5.9	5.9	NUM
ap-961	507	7	:	:	PUNCT
ap-961	507	8	for	for	ADP
ap-961	507	9	a	a	DET
ap-961	507	10	(	(	PUNCT
ap-961	507	11	)	)	PUNCT
ap-961	507	12	n	n	CCONJ
ap-961	507	13	1	1	NUM
ap-961	507	14	summable	summable	ADJ
ap-961	507	15	fredholm	fredholm	NOUN
ap-961	507	16	module	module	NOUN
ap-961	507	17	,	,	PUNCT
ap-961	507	18	the	the	DET
ap-961	507	19	closed	closed	ADJ
ap-961	507	20	graded	grade	VERB
ap-961	507	21	trace	trace	NOUN
ap-961	507	22	of	of	ADP
ap-961	507	23	dimension	dimension	NOUN
ap-961	507	24	is	be	AUX
ap-961	507	25	given	give	VERB
ap-961	507	26	by	by	ADP
ap-961	507	27	:	:	PUNCT
ap-961	507	28	�	�	PROPN
ap-961	507	29	�	�	PROPN
ap-961	507	30	�	�	PROPN
ap-961	507	31	2	2	NUM
ap-961	507	32	�	�	PROPN
ap-961	507	33	1	1	NUM
ap-961	507	34	2	2	NUM
ap-961	507	35	tr	tr	VERB
ap-961	507	36	(	(	PUNCT
ap-961	507	37	)	)	PUNCT
ap-961	507	38	f	f	PROPN
ap-961	507	39	f	f	PROPN
ap-961	507	40	,	,	PUNCT
ap-961	507	41	in	in	ADP
ap-961	507	42	the	the	DET
ap-961	507	43	odd	odd	ADJ
ap-961	507	44	case	case	NOUN
ap-961	507	45	,	,	PUNCT
ap-961	507	46	and	and	CCONJ
ap-961	507	47	for	for	ADP
ap-961	507	48	even	even	ADV
ap-961	507	49	fredholm	fredholm	NOUN
ap-961	507	50	modules	module	NOUN
ap-961	507	51	by	by	ADP
ap-961	507	52	:	:	PUNCT
ap-961	507	53	©	©	PROPN
ap-961	507	54	czech	czech	PROPN
ap-961	507	55	technical	technical	PROPN
ap-961	507	56	university	university	PROPN
ap-961	507	57	publishing	publishing	NOUN
ap-961	507	58	house	house	NOUN
ap-961	507	59	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	507	60	47	47	NUM
ap-961	507	61	acta	acta	PROPN
ap-961	507	62	polytechnica	polytechnica	PROPN
ap-961	507	63	vol	vol	NOUN
ap-961	507	64	.	.	PUNCT
ap-961	508	1	48	48	NUM
ap-961	508	2	no	no	NOUN
ap-961	508	3	.	.	PUNCT
ap-961	509	1	2/2008	2/2008	PROPN
ap-961	509	2	�	�	PROPN
ap-961	509	3	�	�	PROPN
ap-961	509	4	�	�	PROPN
ap-961	509	5	�	�	PROPN
ap-961	509	6	2	2	NUM
ap-961	509	7	�	�	PROPN
ap-961	509	8	1	1	NUM
ap-961	509	9	2	2	NUM
ap-961	509	10	tr	tr	PUNCT
ap-961	509	11	(	(	PUNCT
ap-961	509	12	)	)	PUNCT
ap-961	509	13	f	f	PROPN
ap-961	509	14	f	f	PROPN
ap-961	509	15	.	.	PUNCT
ap-961	510	1	as	as	ADP
ap-961	510	2	a	a	DET
ap-961	510	3	corollary	corollary	NOUN
ap-961	510	4	we	we	PRON
ap-961	510	5	have	have	VERB
ap-961	510	6	:	:	PUNCT
ap-961	510	7	corollary	corollary	ADJ
ap-961	510	8	5.10	5.10	NUM
ap-961	510	9	:	:	PUNCT
ap-961	510	10	each	each	DET
ap-961	510	11	n	n	CCONJ
ap-961	510	12	1	1	NUM
ap-961	510	13	-	-	PUNCT
ap-961	510	14	summable	summable	ADJ
ap-961	510	15	fredholm	fredholm	NOUN
ap-961	510	16	module	module	NOUN
ap-961	510	17	gives	give	VERB
ap-961	510	18	rise	rise	NOUN
ap-961	510	19	to	to	ADP
ap-961	510	20	an	an	DET
ap-961	510	21	n	n	CCONJ
ap-961	510	22	-	-	PUNCT
ap-961	510	23	cyclic	cyclic	ADJ
ap-961	510	24	cocycle	cocycle	NOUN
ap-961	510	25	.	.	PUNCT
ap-961	511	1	to	to	PART
ap-961	511	2	exploit	exploit	VERB
ap-961	511	3	relations	relation	NOUN
ap-961	511	4	with	with	ADP
ap-961	511	5	k	k	NOUN
ap-961	511	6	-	-	NOUN
ap-961	511	7	theory	theory	NOUN
ap-961	511	8	and	and	CCONJ
ap-961	511	9	formulate	formulate	VERB
ap-961	511	10	the	the	DET
ap-961	511	11	chern	chern	NOUN
ap-961	511	12	map	map	NOUN
ap-961	511	13	using	use	VERB
ap-961	511	14	connections	connection	NOUN
ap-961	512	1	one	one	PRON
ap-961	512	2	needs	need	VERB
ap-961	512	3	an	an	DET
ap-961	512	4	additional	additional	ADJ
ap-961	512	5	ingredient	ingredient	NOUN
ap-961	512	6	.	.	PUNCT
ap-961	513	1	this	this	PRON
ap-961	513	2	is	be	AUX
ap-961	513	3	cyclic	cyclic	ADJ
ap-961	513	4	cohomology	cohomology	NOUN
ap-961	513	5	(	(	PUNCT
ap-961	513	6	or	or	CCONJ
ap-961	513	7	rather	rather	ADV
ap-961	513	8	a	a	DET
ap-961	513	9	version	version	NOUN
ap-961	513	10	of	of	ADP
ap-961	513	11	it	it	PRON
ap-961	513	12	,	,	PUNCT
ap-961	513	13	so	so	ADV
ap-961	513	14	-	-	PUNCT
ap-961	513	15	called	call	VERB
ap-961	513	16	periodic	periodic	ADJ
ap-961	513	17	cyclic	cyclic	ADJ
ap-961	513	18	cohomology	cohomology	NOUN
ap-961	513	19	)	)	PUNCT
ap-961	513	20	that	that	PRON
ap-961	513	21	is	be	AUX
ap-961	513	22	the	the	DET
ap-961	513	23	natural	natural	ADJ
ap-961	513	24	receptor	receptor	NOUN
ap-961	513	25	for	for	ADP
ap-961	513	26	the	the	DET
ap-961	513	27	chern	chern	PROPN
ap-961	513	28	map	map	NOUN
ap-961	513	29	.	.	PUNCT
ap-961	514	1	we	we	PRON
ap-961	514	2	will	will	AUX
ap-961	514	3	see	see	VERB
ap-961	514	4	how	how	SCONJ
ap-961	514	5	differential	differential	ADJ
ap-961	514	6	graded	grade	VERB
ap-961	514	7	algebras	algebra	NOUN
ap-961	514	8	can	can	AUX
ap-961	514	9	help	help	VERB
ap-961	514	10	us	we	PRON
ap-961	514	11	to	to	PART
ap-961	514	12	construct	construct	VERB
ap-961	514	13	cyclic	cyclic	ADJ
ap-961	514	14	cohomology	cohomology	NOUN
ap-961	514	15	elements	element	NOUN
ap-961	514	16	.	.	PUNCT
ap-961	515	1	example	example	NOUN
ap-961	515	2	5.11	5.11	NUM
ap-961	515	3	:	:	PUNCT
ap-961	515	4	recall	recall	VERB
ap-961	515	5	again	again	ADV
ap-961	515	6	the	the	DET
ap-961	515	7	construction	construction	NOUN
ap-961	515	8	of	of	ADP
ap-961	515	9	the	the	DET
ap-961	515	10	differential	differential	ADJ
ap-961	515	11	algebra	algebra	NOUN
ap-961	515	12	for	for	ADP
ap-961	515	13	the	the	DET
ap-961	515	14	circle	circle	NOUN
ap-961	515	15	,	,	PUNCT
ap-961	515	16	with	with	ADP
ap-961	515	17	the	the	DET
ap-961	515	18	operator	operator	NOUN
ap-961	515	19	f	f	NOUN
ap-961	515	20	,	,	PUNCT
ap-961	515	21	f2	f2	PROPN
ap-961	515	22	1	1	NUM
ap-961	515	23	�	�	PROPN
ap-961	515	24	in	in	ADP
ap-961	515	25	3.16	3.16	NUM
ap-961	515	26	.	.	PUNCT
ap-961	516	1	the	the	DET
ap-961	516	2	commutators	commutator	NOUN
ap-961	516	3	of	of	ADP
ap-961	516	4	with	with	ADP
ap-961	516	5	the	the	DET
ap-961	516	6	elements	element	NOUN
ap-961	516	7	of	of	ADP
ap-961	516	8	the	the	DET
ap-961	516	9	algebra	algebra	NOUN
ap-961	516	10	are	be	AUX
ap-961	516	11	compact	compact	ADJ
ap-961	516	12	and	and	CCONJ
ap-961	516	13	in	in	ADP
ap-961	516	14	particular	particular	ADJ
ap-961	516	15	,	,	PUNCT
ap-961	516	16	commutators	commutator	NOUN
ap-961	516	17	with	with	ADP
ap-961	516	18	polynomials	polynomial	NOUN
ap-961	516	19	are	be	AUX
ap-961	516	20	finite	finite	ADJ
ap-961	516	21	rank	rank	PROPN
ap-961	516	22	operators	operator	NOUN
ap-961	516	23	.	.	PUNCT
ap-961	517	1	therefore	therefore	ADV
ap-961	517	2	they	they	PRON
ap-961	517	3	are	be	AUX
ap-961	517	4	trace	trace	NOUN
ap-961	517	5	class	class	NOUN
ap-961	517	6	and	and	CCONJ
ap-961	517	7	we	we	PRON
ap-961	517	8	can	can	AUX
ap-961	517	9	easily	easily	ADV
ap-961	517	10	define	define	VERB
ap-961	517	11	a	a	DET
ap-961	517	12	1	1	NUM
ap-961	517	13	-	-	PUNCT
ap-961	517	14	cyclic	cyclic	ADJ
ap-961	517	15	cocycle	cocycle	NOUN
ap-961	517	16	on	on	ADP
ap-961	517	17	the	the	DET
ap-961	517	18	polynomial	polynomial	ADJ
ap-961	517	19	algebra	algebra	NOUN
ap-961	517	20	:	:	PUNCT
ap-961	517	21	�	�	PROPN
ap-961	517	22	(	(	PUNCT
ap-961	517	23	,	,	PUNCT
ap-961	517	24	)	)	PUNCT
ap-961	517	25	(	(	PUNCT
ap-961	517	26	[	[	PUNCT
ap-961	517	27	,	,	PUNCT
ap-961	517	28	]	]	X
ap-961	517	29	[	[	PUNCT
ap-961	517	30	,	,	PUNCT
ap-961	517	31	]	]	PUNCT
ap-961	517	32	)	)	PUNCT
ap-961	517	33	a	a	DET
ap-961	517	34	a	a	DET
ap-961	517	35	a	a	DET
ap-961	517	36	f	f	NOUN
ap-961	517	37	a	a	DET
ap-961	517	38	fa	fa	NOUN
ap-961	517	39	f	f	PROPN
ap-961	517	40	a	a	DET
ap-961	517	41	f0	f0	PROPN
ap-961	517	42	1	1	NUM
ap-961	517	43	0	0	NUM
ap-961	517	44	1	1	NUM
ap-961	517	45	0	0	NUM
ap-961	517	46	1	1	NUM
ap-961	517	47	1	1	NUM
ap-961	517	48	2	2	NUM
ap-961	517	49	�	�	NOUN
ap-961	517	50	tr	tr	VERB
ap-961	517	51	tr	tr	VERB
ap-961	517	52	.	.	PUNCT
ap-961	518	1	we	we	PRON
ap-961	518	2	calculate	calculate	VERB
ap-961	518	3	this	this	PRON
ap-961	518	4	explicitly	explicitly	ADV
ap-961	518	5	for	for	ADP
ap-961	518	6	homogeneous	homogeneous	ADJ
ap-961	518	7	polynomials	polynomial	NOUN
ap-961	518	8	.	.	PUNCT
ap-961	519	1	first	first	ADV
ap-961	519	2	,	,	PUNCT
ap-961	519	3	u	u	PROPN
ap-961	519	4	f	f	PROPN
ap-961	519	5	u	u	NOUN
ap-961	519	6	e	e	PROPN
ap-961	519	7	sign	sign	VERB
ap-961	519	8	p	p	PROPN
ap-961	519	9	k	k	PROPN
ap-961	519	10	sign	sign	PROPN
ap-961	519	11	p	p	PROPN
ap-961	519	12	en	en	ADP
ap-961	519	13	k	k	PROPN
ap-961	519	14	p	p	PROPN
ap-961	519	15	p	p	PROPN
ap-961	519	16	k	k	PROPN
ap-961	519	17	n	n	CCONJ
ap-961	519	18	[	[	X
ap-961	519	19	,	,	PUNCT
ap-961	519	20	]	]	X
ap-961	519	21	(	(	PUNCT
ap-961	519	22	)	)	PUNCT
ap-961	519	23	(	(	PUNCT
ap-961	519	24	)	)	PUNCT
ap-961	519	25	�	�	PROPN
ap-961	519	26	�	�	PROPN
ap-961	519	27	�	�	PROPN
ap-961	519	28	�	�	PROPN
ap-961	519	29	�	�	PROPN
ap-961	519	30	�	�	PROPN
ap-961	519	31	�	�	PROPN
ap-961	519	32	�	�	PROPN
ap-961	519	33	1	1	NUM
ap-961	519	34	2	2	NUM
ap-961	519	35	1	1	NUM
ap-961	519	36	2	2	NUM
ap-961	519	37	.	.	PUNCT
ap-961	520	1	therefore	therefore	ADV
ap-961	520	2	:	:	PUNCT
ap-961	520	3	tru	tru	PROPN
ap-961	520	4	f	f	PROPN
ap-961	520	5	u	u	PROPN
ap-961	520	6	k	k	PROPN
ap-961	520	7	n	n	PROPN
ap-961	520	8	kn	kn	PROPN
ap-961	520	9	k	k	X
ap-961	520	10	[	[	X
ap-961	520	11	,	,	PUNCT
ap-961	520	12	]	]	PUNCT
ap-961	520	13	,	,	PUNCT
ap-961	520	14	�	�	PROPN
ap-961	520	15	�	�	PROPN
ap-961	520	16	�	�	PROPN
ap-961	520	17	�	�	PROPN
ap-961	520	18	�	�	PROPN
ap-961	520	19	2	2	NUM
ap-961	520	20	0	0	NUM
ap-961	520	21	0	0	NUM
ap-961	520	22	if	if	SCONJ
ap-961	520	23	otherwise	otherwise	ADV
ap-961	520	24	.	.	PUNCT
ap-961	521	1	then	then	ADV
ap-961	521	2	:	:	PUNCT
ap-961	521	3	�	�	PROPN
ap-961	521	4	�	�	PROPN
ap-961	521	5	(	(	PUNCT
ap-961	521	6	,	,	PUNCT
ap-961	521	7	)	)	PUNCT
ap-961	521	8	,	,	PUNCT
ap-961	521	9	u	u	NOUN
ap-961	521	10	u	u	NOUN
ap-961	521	11	kn	kn	PROPN
ap-961	521	12	k	k	PROPN
ap-961	521	13	n	n	PROPN
ap-961	521	14	k	k	PROPN
ap-961	521	15	�	�	PROPN
ap-961	521	16	02	02	NUM
ap-961	521	17	.	.	PUNCT
ap-961	522	1	let	let	VERB
ap-961	522	2	us	we	PRON
ap-961	522	3	verify	verify	VERB
ap-961	522	4	that	that	SCONJ
ap-961	522	5	it	it	PRON
ap-961	522	6	is	be	AUX
ap-961	522	7	a	a	DET
ap-961	522	8	cyclic	cyclic	ADJ
ap-961	522	9	cocycle	cocycle	NOUN
ap-961	522	10	.	.	PUNCT
ap-961	523	1	first	first	ADV
ap-961	523	2	of	of	ADP
ap-961	523	3	all	all	PRON
ap-961	523	4	,	,	PUNCT
ap-961	523	5	observe	observe	VERB
ap-961	523	6	that	that	SCONJ
ap-961	523	7	it	it	PRON
ap-961	523	8	is	be	AUX
ap-961	523	9	cyclic	cyclic	ADJ
ap-961	523	10	:	:	PUNCT
ap-961	523	11	�	�	PROPN
ap-961	523	12	�	�	PROPN
ap-961	523	13	�	�	PROPN
ap-961	523	14	�	�	PROPN
ap-961	523	15	(	(	PUNCT
ap-961	523	16	,	,	PUNCT
ap-961	523	17	)	)	PUNCT
ap-961	523	18	(	(	PUNCT
ap-961	523	19	,	,	PUNCT
ap-961	523	20	)	)	PUNCT
ap-961	523	21	,	,	PUNCT
ap-961	523	22	,	,	PUNCT
ap-961	523	23	u	u	NOUN
ap-961	523	24	u	u	X
ap-961	523	25	k	k	PROPN
ap-961	523	26	n	n	CCONJ
ap-961	523	27	u	u	PROPN
ap-961	523	28	un	un	PROPN
ap-961	523	29	k	k	PROPN
ap-961	523	30	n	n	PROPN
ap-961	523	31	k	k	PROPN
ap-961	523	32	n	n	CCONJ
ap-961	523	33	k	k	PROPN
ap-961	523	34	k	k	PROPN
ap-961	523	35	n	n	CCONJ
ap-961	523	36	�	�	PROPN
ap-961	523	37	�	�	PROPN
ap-961	523	38	�	�	PROPN
ap-961	523	39	�	�	PROPN
ap-961	523	40	�	�	PROPN
ap-961	523	41	0	0	NUM
ap-961	523	42	0	0	NUM
ap-961	523	43	.	.	PUNCT
ap-961	524	1	furthermore	furthermore	ADV
ap-961	524	2	:	:	PUNCT
ap-961	524	3	b	b	X
ap-961	524	4	u	u	X
ap-961	524	5	u	u	X
ap-961	524	6	u	u	NOUN
ap-961	524	7	u	u	X
ap-961	524	8	u	u	X
ap-961	524	9	u	u	X
ap-961	524	10	u	u	VERB
ap-961	524	11	u	u	VERB
ap-961	524	12	uk	uk	PROPN
ap-961	524	13	m	m	PROPN
ap-961	524	14	n	n	ADV
ap-961	524	15	k	k	NOUN
ap-961	524	16	m	m	VERB
ap-961	524	17	n	n	ADV
ap-961	524	18	k	k	NOUN
ap-961	524	19	m	m	VERB
ap-961	524	20	n	n	VERB
ap-961	524	21	k	k	PROPN
ap-961	524	22	n	n	ADV
ap-961	524	23	m	m	VERB
ap-961	524	24	k	k	PROPN
ap-961	524	25	�	�	PROPN
ap-961	524	26	�	�	PROPN
ap-961	524	27	�	�	PROPN
ap-961	524	28	�	�	PROPN
ap-961	524	29	�	�	PROPN
ap-961	524	30	(	(	PUNCT
ap-961	524	31	,	,	PUNCT
ap-961	524	32	,	,	PUNCT
ap-961	524	33	)	)	PUNCT
ap-961	524	34	(	(	PUNCT
ap-961	524	35	,	,	PUNCT
ap-961	524	36	)	)	PUNCT
ap-961	524	37	(	(	PUNCT
ap-961	524	38	,	,	PUNCT
ap-961	524	39	)	)	PUNCT
ap-961	524	40	(	(	PUNCT
ap-961	524	41	,	,	PUNCT
ap-961	524	42	)	)	PUNCT
ap-961	524	43	�	�	PROPN
ap-961	524	44	�	�	PROPN
ap-961	524	45	�	�	PROPN
ap-961	524	46	2	2	NUM
ap-961	524	47	m	m	NOUN
ap-961	524	48	n	n	ADP
ap-961	524	49	n	n	NOUN
ap-961	524	50	m	m	VERB
ap-961	524	51	n	n	PRON
ap-961	524	52	m	m	PROPN
ap-961	524	53	�	�	PROPN
ap-961	524	54	�	�	PROPN
ap-961	524	55	(	(	PUNCT
ap-961	524	56	(	(	PUNCT
ap-961	524	57	)	)	PUNCT
ap-961	524	58	)	)	PUNCT
ap-961	524	59	0	0	NUM
ap-961	524	60	note	note	VERB
ap-961	524	61	that	that	SCONJ
ap-961	524	62	up	up	ADP
ap-961	524	63	to	to	ADP
ap-961	524	64	some	some	DET
ap-961	524	65	rescaling	rescaling	NOUN
ap-961	524	66	we	we	PRON
ap-961	524	67	obtain	obtain	VERB
ap-961	524	68	the	the	DET
ap-961	524	69	same	same	ADJ
ap-961	524	70	cocycle	cocycle	NOUN
ap-961	524	71	as	as	ADP
ap-961	524	72	the	the	DET
ap-961	524	73	one	one	NUM
ap-961	524	74	arising	arise	VERB
ap-961	524	75	from	from	ADP
ap-961	524	76	classical	classical	ADJ
ap-961	524	77	de	de	PROPN
ap-961	524	78	rham	rham	PROPN
ap-961	524	79	forms	form	NOUN
ap-961	524	80	and	and	CCONJ
ap-961	524	81	the	the	DET
ap-961	524	82	standard	standard	ADJ
ap-961	524	83	integration	integration	NOUN
ap-961	524	84	:	:	PUNCT
ap-961	524	85	�	�	PROPN
ap-961	524	86	�	�	PROPN
ap-961	524	87	�	�	PROPN
ap-961	524	88	�	�	PROPN
ap-961	524	89	�	�	PROPN
ap-961	524	90	dr	dr	PROPN
ap-961	524	91	n	n	PROPN
ap-961	524	92	k	k	PROPN
ap-961	524	93	in	in	ADP
ap-961	524	94	ik	ik	PROPN
ap-961	524	95	n	n	PROPN
ap-961	524	96	ku	ku	PROPN
ap-961	524	97	u	u	PROPN
ap-961	524	98	d	d	PROPN
ap-961	524	99	e	e	X
ap-961	524	100	de	de	PROPN
ap-961	524	101	i	i	PROPN
ap-961	524	102	k	k	PROPN
ap-961	524	103	(	(	PUNCT
ap-961	524	104	,	,	PUNCT
ap-961	524	105	)	)	PUNCT
ap-961	524	106	(	(	PUNCT
ap-961	524	107	)	)	PUNCT
ap-961	524	108	,	,	PUNCT
ap-961	524	109	�	�	PROPN
ap-961	524	110	�	�	PROPN
ap-961	524	111	2	2	NUM
ap-961	524	112	�	�	PROPN
ap-961	524	113	�	�	PROPN
ap-961	524	114	�	�	PROPN
ap-961	524	115	2	2	NUM
ap-961	524	116	2	2	NUM
ap-961	524	117	0	0	NUM
ap-961	524	118	1	1	NUM
ap-961	524	119	02	02	NUM
ap-961	524	120	.	.	PUNCT
ap-961	525	1	now	now	ADV
ap-961	525	2	we	we	PRON
ap-961	525	3	can	can	AUX
ap-961	525	4	turn	turn	VERB
ap-961	525	5	to	to	ADP
ap-961	525	6	the	the	DET
ap-961	525	7	truly	truly	ADV
ap-961	525	8	noncommutative	noncommutative	ADJ
ap-961	525	9	world	world	NOUN
ap-961	525	10	.	.	PUNCT
ap-961	526	1	example	example	NOUN
ap-961	526	2	5.12	5.12	NUM
ap-961	526	3	:	:	PUNCT
ap-961	526	4	do	do	AUX
ap-961	526	5	you	you	PRON
ap-961	526	6	remember	remember	VERB
ap-961	526	7	the	the	DET
ap-961	526	8	noncommutative	noncommutative	ADJ
ap-961	526	9	torusuv	torusuv	NOUN
ap-961	526	10	e	e	PROPN
ap-961	526	11	vui	vui	ADJ
ap-961	526	12	�	�	PROPN
ap-961	526	13	2	2	NUM
ap-961	526	14	�	�	PROPN
ap-961	526	15	�	�	PROPN
ap-961	526	16	?	?	PUNCT
ap-961	527	1	we	we	PRON
ap-961	527	2	shall	shall	AUX
ap-961	527	3	construct	construct	VERB
ap-961	527	4	a	a	DET
ap-961	527	5	two	two	NUM
ap-961	527	6	-	-	PUNCT
ap-961	527	7	dimensional	dimensional	ADJ
ap-961	527	8	cycle	cycle	NOUN
ap-961	527	9	over	over	ADP
ap-961	527	10	it	it	PRON
ap-961	527	11	.	.	PUNCT
ap-961	528	1	first	first	ADV
ap-961	528	2	,	,	PUNCT
ap-961	528	3	the	the	DET
ap-961	528	4	differential	differential	ADJ
ap-961	528	5	forms	form	NOUN
ap-961	528	6	.	.	PUNCT
ap-961	529	1	we	we	PRON
ap-961	529	2	take	take	VERB
ap-961	529	3	two	two	NUM
ap-961	529	4	generating	generate	VERB
ap-961	529	5	one	one	NUM
ap-961	529	6	-	-	PUNCT
ap-961	529	7	forms	form	NOUN
ap-961	529	8	�	�	NOUN
ap-961	529	9	u	u	NOUN
ap-961	529	10	and	and	CCONJ
ap-961	529	11	�	�	PROPN
ap-961	529	12	v	v	NOUN
ap-961	529	13	,	,	PUNCT
ap-961	529	14	which	which	PRON
ap-961	529	15	are	be	AUX
ap-961	529	16	central	central	ADJ
ap-961	529	17	.	.	PUNCT
ap-961	530	1	the	the	DET
ap-961	530	2	external	external	ADJ
ap-961	530	3	derivative	derivative	NOUN
ap-961	530	4	becomes	become	VERB
ap-961	530	5	:	:	PUNCT
ap-961	530	6	da	da	NOUN
ap-961	530	7	a	a	DET
ap-961	530	8	au	au	X
ap-961	530	9	u	u	PROPN
ap-961	530	10	v	v	PROPN
ap-961	530	11	v	v	PRON
ap-961	530	12	�	�	PROPN
ap-961	530	13	�	�	PROPN
ap-961	530	14	�	�	PROPN
ap-961	530	15	�	�	PROPN
ap-961	530	16	�	�	PROPN
ap-961	530	17	(	(	PUNCT
ap-961	530	18	)	)	PUNCT
ap-961	530	19	(	(	PUNCT
ap-961	530	20	)	)	PUNCT
ap-961	530	21	,	,	PUNCT
ap-961	530	22	where	where	SCONJ
ap-961	530	23	�	�	NOUN
ap-961	530	24	u	u	NOUN
ap-961	530	25	and	and	CCONJ
ap-961	530	26	�	�	PROPN
ap-961	530	27	v	v	NOUN
ap-961	530	28	are	be	AUX
ap-961	530	29	two	two	NUM
ap-961	530	30	outer	outer	ADJ
ap-961	530	31	derivations	derivation	NOUN
ap-961	530	32	on	on	ADP
ap-961	530	33	the	the	DET
ap-961	530	34	algebra	algebra	NOUN
ap-961	530	35	of	of	ADP
ap-961	530	36	the	the	DET
ap-961	530	37	noncommutative	noncommutative	ADJ
ap-961	530	38	torus	torus	NOUN
ap-961	530	39	:	:	PUNCT
ap-961	530	40	�	�	PROPN
ap-961	530	41	n	n	CCONJ
ap-961	530	42	n	n	PRON
ap-961	530	43	m	m	VERB
ap-961	530	44	n	n	PRON
ap-961	530	45	mu	mu	NOUN
ap-961	530	46	v	v	X
ap-961	530	47	nu	nu	INTJ
ap-961	530	48	v	v	PROPN
ap-961	530	49	(	(	PUNCT
ap-961	530	50	,	,	PUNCT
ap-961	530	51	)	)	PUNCT
ap-961	530	52	�	�	PROPN
ap-961	530	53	,	,	PUNCT
ap-961	530	54	�	�	PROPN
ap-961	530	55	m	m	PROPN
ap-961	530	56	n	n	PRON
ap-961	530	57	m	m	PROPN
ap-961	530	58	n	n	PRON
ap-961	530	59	mu	mu	NOUN
ap-961	530	60	v	v	ADP
ap-961	530	61	mu	mu	PROPN
ap-961	530	62	v	v	PROPN
ap-961	530	63	(	(	PUNCT
ap-961	530	64	,	,	PUNCT
ap-961	530	65	)	)	PUNCT
ap-961	530	66	�	�	PROPN
ap-961	530	67	.	.	PUNCT
ap-961	531	1	the	the	DET
ap-961	531	2	two	two	NUM
ap-961	531	3	-	-	PUNCT
ap-961	531	4	forms	form	NOUN
ap-961	531	5	are	be	AUX
ap-961	531	6	generated	generate	VERB
ap-961	531	7	by	by	ADP
ap-961	531	8	the	the	DET
ap-961	531	9	wedge	wedge	NOUN
ap-961	531	10	product	product	NOUN
ap-961	531	11	of	of	ADP
ap-961	531	12	the	the	DET
ap-961	531	13	one	one	NUM
ap-961	531	14	-	-	PUNCT
ap-961	531	15	forms	form	NOUN
ap-961	531	16	:	:	PUNCT
ap-961	531	17	�	�	PROPN
ap-961	531	18	�	�	PROPN
ap-961	531	19	�	�	PROPN
ap-961	531	20	�	�	PROPN
ap-961	531	21	u	u	PROPN
ap-961	531	22	v	v	PROPN
ap-961	531	23	v	v	NUM
ap-961	531	24	u	u	NOUN
ap-961	531	25	#	#	NOUN
ap-961	531	26	�	�	PROPN
ap-961	531	27	�	�	PROPN
ap-961	531	28	#	#	NOUN
ap-961	531	29	.	.	PUNCT
ap-961	532	1	for	for	ADP
ap-961	532	2	the	the	DET
ap-961	532	3	trace	trace	NOUN
ap-961	532	4	we	we	PRON
ap-961	532	5	take	take	VERB
ap-961	532	6	:	:	PUNCT
ap-961	532	7	a	a	DET
ap-961	532	8	au	au	PROPN
ap-961	532	9	v	v	PRON
ap-961	532	10	�	�	PROPN
ap-961	532	11	�	�	PROPN
ap-961	532	12	#	#	NOUN
ap-961	532	13	�	�	PROPN
ap-961	532	14	2	2	NUM
ap-961	532	15	tr	tr	PUNCT
ap-961	532	16	,	,	PUNCT
ap-961	532	17	where	where	SCONJ
ap-961	532	18	tr	tr	PRON
ap-961	532	19	is	be	AUX
ap-961	532	20	the	the	DET
ap-961	532	21	standard	standard	ADJ
ap-961	532	22	trace	trace	NOUN
ap-961	532	23	on	on	ADP
ap-961	532	24	the	the	DET
ap-961	532	25	algebra	algebra	NOUN
ap-961	532	26	,	,	PUNCT
ap-961	532	27	defined	define	VERB
ap-961	532	28	on	on	ADP
ap-961	532	29	the	the	DET
ap-961	532	30	polynomials	polynomial	NOUN
ap-961	532	31	as	as	ADP
ap-961	532	32	:	:	PUNCT
ap-961	532	33	tr	tr	PUNCT
ap-961	532	34	u	u	PROPN
ap-961	532	35	vn	vn	PROPN
ap-961	532	36	m	m	PROPN
ap-961	532	37	n	n	PROPN
ap-961	532	38	m	m	PROPN
ap-961	532	39	�	�	PROPN
ap-961	532	40	�	�	PROPN
ap-961	532	41	�	�	PROPN
ap-961	532	42	,	,	PUNCT
ap-961	532	43	,	,	PUNCT
ap-961	532	44	0	0	NUM
ap-961	532	45	0	0	NUM
ap-961	532	46	.	.	PUNCT
ap-961	533	1	clearly	clearly	ADV
ap-961	533	2	2	2	NUM
ap-961	533	3	is	be	AUX
ap-961	533	4	a	a	DET
ap-961	533	5	graded	grade	VERB
ap-961	533	6	trace	trace	NOUN
ap-961	533	7	(	(	PUNCT
ap-961	533	8	since	since	SCONJ
ap-961	533	9	the	the	DET
ap-961	533	10	basic	basic	ADJ
ap-961	533	11	one	one	NUM
ap-961	533	12	-	-	PUNCT
ap-961	533	13	forms	form	NOUN
ap-961	533	14	anticommute	anticommute	NOUN
ap-961	533	15	and	and	CCONJ
ap-961	533	16	tr	tr	NOUN
ap-961	533	17	is	be	AUX
ap-961	533	18	a	a	DET
ap-961	533	19	trace	trace	NOUN
ap-961	533	20	)	)	PUNCT
ap-961	533	21	,	,	PUNCT
ap-961	533	22	so	so	SCONJ
ap-961	533	23	we	we	PRON
ap-961	533	24	need	need	VERB
ap-961	533	25	only	only	ADV
ap-961	533	26	to	to	PART
ap-961	533	27	show	show	VERB
ap-961	533	28	that	that	SCONJ
ap-961	533	29	it	it	PRON
ap-961	533	30	is	be	AUX
ap-961	533	31	closed	closed	ADJ
ap-961	533	32	:	:	PUNCT
ap-961	533	33	d	d	SYM
ap-961	533	34	u	u	NOUN
ap-961	533	35	v	v	ADP
ap-961	533	36	u	u	NOUN
ap-961	533	37	v	v	ADP
ap-961	533	38	mu	mu	NOUN
ap-961	533	39	v	v	ADP
ap-961	533	40	ku	ku	PROPN
ap-961	533	41	v	v	PROPN
ap-961	533	42	mu	mu	PROPN
ap-961	533	43	v	v	PROPN
ap-961	533	44	k	k	PROPN
ap-961	533	45	n	n	CCONJ
ap-961	533	46	m	m	VERB
ap-961	533	47	u	u	NOUN
ap-961	533	48	k	k	PROPN
ap-961	533	49	l	l	PROPN
ap-961	533	50	v	v	NOUN
ap-961	533	51	n	n	INTJ
ap-961	533	52	m	m	VERB
ap-961	533	53	k	k	NOUN
ap-961	533	54	l	l	PUNCT
ap-961	533	55	u	u	NOUN
ap-961	533	56	v	v	ADP
ap-961	533	57	n	n	INTJ
ap-961	533	58	m	m	VERB
ap-961	533	59	(	(	PUNCT
ap-961	533	60	)	)	PUNCT
ap-961	533	61	(	(	PUNCT
ap-961	533	62	)	)	PUNCT
ap-961	533	63	(	(	PUNCT
ap-961	533	64	�	�	PROPN
ap-961	533	65	�	�	PROPN
ap-961	533	66	�	�	PROPN
ap-961	533	67	�	�	PROPN
ap-961	533	68	�	�	PROPN
ap-961	533	69	�	�	PROPN
ap-961	533	70	#	#	PROPN
ap-961	533	71	�	�	PROPN
ap-961	533	72	�	�	PROPN
ap-961	533	73	2	2	NUM
ap-961	533	74	2	2	NUM
ap-961	533	75	tr	tr	NOUN
ap-961	533	76	u	u	NOUN
ap-961	533	77	v	v	ADP
ap-961	533	78	m	m	NOUN
ap-961	533	79	k	k	PROPN
ap-961	533	80	k	k	PROPN
ap-961	533	81	l	l	PROPN
ap-961	533	82	n	n	CCONJ
ap-961	533	83	m	m	NOUN
ap-961	533	84	k	k	NOUN
ap-961	533	85	l	l	NOUN
ap-961	533	86	)	)	PUNCT
ap-961	533	87	.	.	PUNCT
ap-961	534	1	,	,	PUNCT
ap-961	534	2	,	,	PUNCT
ap-961	534	3	,	,	PUNCT
ap-961	534	4	,	,	PUNCT
ap-961	534	5	�	�	PROPN
ap-961	534	6	�	�	PROPN
ap-961	534	7	�	�	PROPN
ap-961	534	8	�	�	PROPN
ap-961	534	9	�	�	PROPN
ap-961	534	10	�	�	PROPN
ap-961	534	11	�	�	PROPN
ap-961	534	12	0	0	NUM
ap-961	534	13	0	0	NUM
ap-961	534	14	0	0	NUM
ap-961	534	15	0	0	NUM
ap-961	534	16	0	0	NUM
ap-961	535	1	the	the	DET
ap-961	535	2	resulting	result	VERB
ap-961	535	3	two	two	NUM
ap-961	535	4	-	-	PUNCT
ap-961	535	5	cyclic	cyclic	NOUN
ap-961	535	6	cocycle	cocycle	NOUN
ap-961	535	7	is	be	AUX
ap-961	535	8	:	:	PUNCT
ap-961	535	9	�	�	PROPN
ap-961	535	10	�	�	PROPN
ap-961	535	11	�	�	PROPN
ap-961	535	12	�	�	PROPN
ap-961	535	13	�	�	PROPN
ap-961	535	14	�	�	PROPN
ap-961	535	15	(	(	PUNCT
ap-961	535	16	,	,	PUNCT
ap-961	535	17	,	,	PUNCT
ap-961	535	18	)	)	PUNCT
ap-961	535	19	(	(	PUNCT
ap-961	535	20	(	(	PUNCT
ap-961	535	21	)	)	PUNCT
ap-961	535	22	(	(	PUNCT
ap-961	535	23	)	)	PUNCT
ap-961	535	24	)	)	PUNCT
ap-961	536	1	(	(	PUNCT
ap-961	536	2	(	(	PUNCT
ap-961	536	3	a	a	DET
ap-961	536	4	a	a	DET
ap-961	536	5	a	a	DET
ap-961	536	6	a	a	DET
ap-961	536	7	da	da	NOUN
ap-961	536	8	da	da	PROPN
ap-961	536	9	a	a	DET
ap-961	536	10	a	a	DET
ap-961	536	11	a	a	PRON
ap-961	536	12	au	au	X
ap-961	536	13	u	u	NOUN
ap-961	536	14	v	v	NOUN
ap-961	536	15	v	v	NUM
ap-961	536	16	u	u	NOUN
ap-961	536	17	0	0	NUM
ap-961	536	18	1	1	NUM
ap-961	536	19	2	2	NUM
ap-961	536	20	0	0	NUM
ap-961	536	21	1	1	NUM
ap-961	536	22	2	2	NUM
ap-961	536	23	0	0	NUM
ap-961	536	24	1	1	NUM
ap-961	536	25	1	1	NUM
ap-961	536	26	�	�	PROPN
ap-961	536	27	�	�	PROPN
ap-961	536	28	#	#	NOUN
ap-961	536	29	2	2	NUM
ap-961	536	30	2	2	NUM
ap-961	536	31	2	2	NUM
ap-961	536	32	2	2	NUM
ap-961	536	33	0	0	NUM
ap-961	536	34	1	1	NUM
ap-961	536	35	2	2	NUM
ap-961	536	36	2	2	NUM
ap-961	536	37	1	1	NUM
ap-961	536	38	)	)	PUNCT
ap-961	536	39	(	(	PUNCT
ap-961	536	40	)	)	PUNCT
ap-961	536	41	)	)	PUNCT
ap-961	537	1	(	(	PUNCT
ap-961	537	2	(	(	PUNCT
ap-961	537	3	)	)	PUNCT
ap-961	537	4	(	(	PUNCT
ap-961	537	5	)	)	PUNCT
ap-961	537	6	(	(	PUNCT
ap-961	537	7	)	)	PUNCT
ap-961	537	8	(	(	PUNCT
ap-961	537	9	)	)	PUNCT
ap-961	537	10	)	)	PUNCT
ap-961	537	11	�	�	PROPN
ap-961	537	12	�	�	PROPN
ap-961	537	13	�	�	PROPN
ap-961	537	14	�	�	PROPN
ap-961	537	15	�	�	PROPN
ap-961	537	16	�	�	PROPN
ap-961	537	17	�	�	PROPN
ap-961	537	18	�	�	PROPN
ap-961	537	19	�	�	PROPN
ap-961	537	20	u	u	PROPN
ap-961	537	21	v	v	ADP
ap-961	537	22	v	v	ADP
ap-961	537	23	u	u	NOUN
ap-961	537	24	v	v	ADP
ap-961	537	25	u	u	NOUN
ap-961	537	26	v	v	ADP
ap-961	537	27	u	u	NOUN
ap-961	537	28	v	v	ADP
ap-961	537	29	a	a	PRON
ap-961	537	30	a	a	DET
ap-961	537	31	a	a	PRON
ap-961	537	32	a	a	PRON
ap-961	537	33	a	a	DET
ap-961	537	34	a	a	DET
ap-961	537	35	�	�	PROPN
ap-961	537	36	�	�	PROPN
ap-961	537	37	#	#	SYM
ap-961	537	38	2	2	NUM
ap-961	537	39	�	�	PROPN
ap-961	537	40	�	�	PROPN
ap-961	537	41	tr	tr	VERB
ap-961	537	42	(	(	PUNCT
ap-961	537	43	(	(	PUNCT
ap-961	537	44	(	(	PUNCT
ap-961	537	45	)	)	PUNCT
ap-961	537	46	(	(	PUNCT
ap-961	537	47	)	)	PUNCT
ap-961	537	48	(	(	PUNCT
ap-961	537	49	)	)	PUNCT
ap-961	537	50	(	(	PUNCT
ap-961	537	51	)	)	PUNCT
ap-961	537	52	)	)	PUNCT
ap-961	537	53	)	)	PUNCT
ap-961	538	1	.a	.a	NOUN
ap-961	539	1	a	a	DET
ap-961	539	2	a	a	DET
ap-961	539	3	a	a	DET
ap-961	539	4	au	au	NOUN
ap-961	539	5	v	v	ADP
ap-961	539	6	u	u	PROPN
ap-961	539	7	v0	v0	NOUN
ap-961	539	8	1	1	NUM
ap-961	539	9	2	2	NUM
ap-961	539	10	2	2	NUM
ap-961	539	11	1	1	NUM
ap-961	539	12	�	�	PROPN
ap-961	539	13	�	�	PROPN
ap-961	539	14	�	�	PROPN
ap-961	539	15	�	�	PROPN
ap-961	539	16	we	we	PRON
ap-961	539	17	shall	shall	AUX
ap-961	539	18	use	use	VERB
ap-961	539	19	this	this	DET
ap-961	539	20	construction	construction	NOUN
ap-961	539	21	later	later	ADV
ap-961	539	22	–	–	PUNCT
ap-961	539	23	also	also	ADV
ap-961	539	24	in	in	ADP
ap-961	539	25	the	the	DET
ap-961	539	26	case	case	NOUN
ap-961	539	27	�	�	PROPN
ap-961	539	28	�	�	PROPN
ap-961	539	29	0	0	NUM
ap-961	539	30	,	,	PUNCT
ap-961	539	31	which	which	PRON
ap-961	539	32	is	be	AUX
ap-961	539	33	the	the	DET
ap-961	539	34	usual	usual	ADJ
ap-961	539	35	commutative	commutative	ADJ
ap-961	539	36	torus	torus	NOUN
ap-961	539	37	.	.	PUNCT
ap-961	540	1	5.1	5.1	NUM
ap-961	540	2	chern	chern	ADJ
ap-961	540	3	-	-	PUNCT
ap-961	540	4	connes	conne	NOUN
ap-961	540	5	pairing	pair	VERB
ap-961	540	6	in	in	ADP
ap-961	540	7	this	this	DET
ap-961	540	8	section	section	NOUN
ap-961	540	9	we	we	PRON
ap-961	540	10	shall	shall	AUX
ap-961	540	11	use	use	VERB
ap-961	540	12	the	the	DET
ap-961	540	13	cyclic	cyclic	ADJ
ap-961	540	14	cohomology	cohomology	NOUN
ap-961	540	15	to	to	PART
ap-961	540	16	produce	produce	VERB
ap-961	540	17	invariants	invariant	NOUN
ap-961	540	18	of	of	ADP
ap-961	540	19	projective	projective	ADJ
ap-961	540	20	modules	module	NOUN
ap-961	540	21	.	.	PUNCT
ap-961	541	1	we	we	PRON
ap-961	541	2	must	must	AUX
ap-961	541	3	be	be	AUX
ap-961	541	4	aware	aware	ADJ
ap-961	541	5	that	that	SCONJ
ap-961	541	6	in	in	ADP
ap-961	541	7	doing	do	VERB
ap-961	541	8	so	so	ADV
ap-961	541	9	we	we	PRON
ap-961	541	10	cut	cut	VERB
ap-961	541	11	a	a	DET
ap-961	541	12	really	really	ADV
ap-961	541	13	long	long	ADJ
ap-961	541	14	story	story	NOUN
ap-961	541	15	short	short	ADJ
ap-961	541	16	.	.	PUNCT
ap-961	542	1	details	detail	NOUN
ap-961	542	2	and	and	CCONJ
ap-961	542	3	more	more	ADJ
ap-961	542	4	aspects	aspect	NOUN
ap-961	542	5	of	of	ADP
ap-961	542	6	the	the	DET
ap-961	542	7	theory	theory	NOUN
ap-961	542	8	are	be	AUX
ap-961	542	9	left	leave	VERB
ap-961	542	10	for	for	SCONJ
ap-961	542	11	the	the	DET
ap-961	542	12	intrigued	intrigue	VERB
ap-961	542	13	reader	reader	NOUN
ap-961	542	14	to	to	PART
ap-961	542	15	further	far	ADV
ap-961	542	16	self	self	NOUN
ap-961	542	17	-	-	PUNCT
ap-961	542	18	study	study	NOUN
ap-961	542	19	.	.	PUNCT
ap-961	543	1	assume	assume	VERB
ap-961	543	2	now	now	ADV
ap-961	543	3	we	we	PRON
ap-961	543	4	have	have	VERB
ap-961	543	5	an	an	DET
ap-961	543	6	algebra	algebra	PROPN
ap-961	543	7	�	�	NOUN
ap-961	543	8	.	.	PUNCT
ap-961	544	1	let	let	VERB
ap-961	544	2	p	p	PRON
ap-961	544	3	be	be	AUX
ap-961	544	4	a	a	DET
ap-961	544	5	projection	projection	NOUN
ap-961	544	6	in	in	ADP
ap-961	544	7	mn	mn	PROPN
ap-961	544	8	(	(	PUNCT
ap-961	544	9	)	)	PUNCT
ap-961	544	10	�	�	PROPN
ap-961	544	11	which	which	PRON
ap-961	544	12	is	be	AUX
ap-961	544	13	associated	associate	VERB
ap-961	544	14	to	to	ADP
ap-961	544	15	a	a	DET
ap-961	544	16	finitely	finitely	ADV
ap-961	544	17	generated	generate	VERB
ap-961	544	18	projective	projective	ADJ
ap-961	544	19	module	module	NOUN
ap-961	544	20	�	�	PROPN
ap-961	544	21	p.	p.	NOUN
ap-961	544	22	on	on	ADP
ap-961	544	23	the	the	DET
ap-961	544	24	other	other	ADJ
ap-961	544	25	side	side	NOUN
ap-961	544	26	,	,	PUNCT
ap-961	544	27	let	let	VERB
ap-961	544	28	�	�	PROPN
ap-961	544	29	be	be	AUX
ap-961	544	30	an	an	DET
ap-961	544	31	even	even	ADV
ap-961	544	32	cyclic	cyclic	ADJ
ap-961	544	33	cocycle	cocycle	NOUN
ap-961	544	34	over	over	ADP
ap-961	544	35	�	�	PROPN
ap-961	544	36	.	.	PUNCT
ap-961	545	1	even	even	ADV
ap-961	545	2	without	without	ADP
ap-961	545	3	knowing	know	VERB
ap-961	545	4	much	much	ADJ
ap-961	545	5	about	about	ADP
ap-961	545	6	the	the	DET
ap-961	545	7	theory	theory	NOUN
ap-961	545	8	we	we	PRON
ap-961	545	9	can	can	AUX
ap-961	545	10	construct	construct	VERB
ap-961	545	11	the	the	DET
ap-961	545	12	following	follow	VERB
ap-961	545	13	pairing	pair	VERB
ap-961	545	14	:	:	PUNCT
ap-961	545	15	�	�	PROPN
ap-961	545	16	�	�	PROPN
ap-961	545	17	,	,	PUNCT
ap-961	545	18	(	(	PUNCT
ap-961	545	19	,	,	PUNCT
ap-961	545	20	,	,	PUNCT
ap-961	545	21	,	,	PUNCT
ap-961	545	22	)	)	PUNCT
ap-961	545	23	,	,	PUNCT
ap-961	545	24	,	,	PUNCT
ap-961	545	25	p	p	X
ap-961	545	26	p	p	X
ap-961	545	27	p	p	NOUN
ap-961	546	1	pi	pi	NOUN
ap-961	547	1	i	i	PRON
ap-961	547	2	i	i	PRON
ap-961	548	1	i	i	PRON
ap-961	548	2	i	i	PRON
ap-961	549	1	i	i	PRON
ap-961	550	1	i	i	PRON
ap-961	551	1	i	i	VERB
ap-961	551	2	k	k	PROPN
ap-961	551	3	k	k	PROPN
ap-961	551	4	�	�	PROPN
ap-961	551	5	�	�	PROPN
ap-961	551	6	1	1	NUM
ap-961	551	7	2	2	NUM
ap-961	551	8	2	2	NUM
ap-961	551	9	3	3	NUM
ap-961	551	10	1	1	NUM
ap-961	551	11	1	1	NUM
ap-961	551	12	�	�	PROPN
ap-961	551	13	�	�	PROPN
ap-961	551	14	,	,	PUNCT
ap-961	551	15	(	(	PUNCT
ap-961	551	16	2	2	X
ap-961	551	17	)	)	PUNCT
ap-961	551	18	which	which	PRON
ap-961	551	19	(	(	PUNCT
ap-961	551	20	for	for	ADP
ap-961	551	21	brevity	brevity	NOUN
ap-961	551	22	)	)	PUNCT
ap-961	551	23	we	we	PRON
ap-961	551	24	shall	shall	AUX
ap-961	551	25	always	always	ADV
ap-961	551	26	write	write	VERB
ap-961	551	27	as	as	ADP
ap-961	551	28	�	�	PROPN
ap-961	551	29	(	(	PUNCT
ap-961	551	30	,	,	PUNCT
ap-961	551	31	,	,	PUNCT
ap-961	551	32	,	,	PUNCT
ap-961	551	33	)	)	PUNCT
ap-961	552	1	p	p	X
ap-961	552	2	p	p	X
ap-961	552	3	p	p	X
ap-961	552	4	�	�	PROPN
ap-961	552	5	.	.	PUNCT
ap-961	553	1	as	as	ADP
ap-961	553	2	such	such	ADJ
ap-961	553	3	,	,	PUNCT
ap-961	553	4	it	it	PRON
ap-961	553	5	is	be	AUX
ap-961	553	6	only	only	ADV
ap-961	553	7	a	a	DET
ap-961	553	8	number	number	NOUN
ap-961	553	9	.	.	PUNCT
ap-961	554	1	however	however	ADV
ap-961	554	2	,	,	PUNCT
ap-961	554	3	the	the	DET
ap-961	554	4	following	follow	VERB
ap-961	554	5	theorem	theorem	NOUN
ap-961	554	6	makes	make	VERB
ap-961	554	7	it	it	PRON
ap-961	554	8	a	a	DET
ap-961	554	9	really	really	ADV
ap-961	554	10	important	important	ADJ
ap-961	554	11	number	number	NOUN
ap-961	554	12	:	:	PUNCT
ap-961	554	13	theorem	theorem	VERB
ap-961	554	14	5.13	5.13	NUM
ap-961	554	15	:	:	PUNCT
ap-961	554	16	the	the	DET
ap-961	554	17	pairing	pairing	NOUN
ap-961	554	18	(	(	PUNCT
ap-961	554	19	refchern	refchern	PROPN
ap-961	554	20	depends	depend	VERB
ap-961	554	21	only	only	ADV
ap-961	554	22	on	on	ADP
ap-961	554	23	the	the	DET
ap-961	554	24	equivalence	equivalence	NOUN
ap-961	554	25	class	class	NOUN
ap-961	554	26	of	of	ADP
ap-961	554	27	the	the	PRON
ap-961	554	28	of	of	ADP
ap-961	554	29	the	the	DET
ap-961	554	30	projection	projection	NOUN
ap-961	554	31	p	p	NOUN
ap-961	554	32	and	and	CCONJ
ap-961	554	33	also	also	ADV
ap-961	554	34	only	only	ADV
ap-961	554	35	on	on	ADP
ap-961	554	36	48	48	NUM
ap-961	554	37	©	©	PROPN
ap-961	554	38	czech	czech	PROPN
ap-961	554	39	technical	technical	PROPN
ap-961	554	40	university	university	PROPN
ap-961	554	41	publishing	publishing	NOUN
ap-961	554	42	house	house	NOUN
ap-961	554	43	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	554	44	acta	acta	PROPN
ap-961	554	45	polytechnica	polytechnica	PROPN
ap-961	554	46	vol	vol	NOUN
ap-961	554	47	.	.	PUNCT
ap-961	555	1	48	48	NUM
ap-961	555	2	no	no	NOUN
ap-961	555	3	.	.	PUNCT
ap-961	556	1	2/2008	2/2008	PROPN
ap-961	556	2	the	the	DET
ap-961	556	3	class	class	NOUN
ap-961	556	4	of	of	ADP
ap-961	556	5	the	the	DET
ap-961	556	6	cocycle	cocycle	NOUN
ap-961	556	7	�	�	PROPN
ap-961	556	8	within	within	ADP
ap-961	556	9	the	the	DET
ap-961	556	10	cyclic	cyclic	ADJ
ap-961	556	11	cohomology	cohomology	NOUN
ap-961	556	12	group	group	NOUN
ap-961	556	13	hceven	hceven	VERB
ap-961	556	14	(	(	PUNCT
ap-961	556	15	)	)	PUNCT
ap-961	556	16	�	�	PROPN
ap-961	556	17	.	.	PUNCT
ap-961	557	1	for	for	ADP
ap-961	557	2	proof	proof	NOUN
ap-961	557	3	we	we	PRON
ap-961	557	4	refer	refer	VERB
ap-961	557	5	to	to	ADP
ap-961	557	6	connes	conne	NOUN
ap-961	557	7	[	[	X
ap-961	557	8	5	5	NUM
ap-961	557	9	]	]	PUNCT
ap-961	557	10	.	.	PUNCT
ap-961	558	1	what	what	PRON
ap-961	558	2	we	we	PRON
ap-961	558	3	have	have	AUX
ap-961	558	4	gained	gain	VERB
ap-961	558	5	?	?	PUNCT
ap-961	559	1	an	an	DET
ap-961	559	2	extremely	extremely	ADV
ap-961	559	3	useful	useful	ADJ
ap-961	559	4	tool	tool	NOUN
ap-961	559	5	of	of	ADP
ap-961	559	6	noncommutative	noncommutative	ADJ
ap-961	559	7	geometry	geometry	NOUN
ap-961	559	8	applicable	applicable	ADJ
ap-961	559	9	to	to	ADP
ap-961	559	10	noncommutative	noncommutative	ADJ
ap-961	559	11	topology	topology	NOUN
ap-961	559	12	:	:	PUNCT
ap-961	559	13	we	we	PRON
ap-961	559	14	have	have	VERB
ap-961	559	15	a	a	DET
ap-961	559	16	very	very	ADV
ap-961	559	17	precise	precise	ADJ
ap-961	559	18	way	way	NOUN
ap-961	559	19	of	of	ADP
ap-961	559	20	distinguishing	distinguish	VERB
ap-961	559	21	different	different	ADJ
ap-961	559	22	topological	topological	ADJ
ap-961	559	23	classes	class	NOUN
ap-961	559	24	of	of	ADP
ap-961	559	25	projective	projective	ADJ
ap-961	559	26	modules	module	NOUN
ap-961	559	27	.	.	PUNCT
ap-961	560	1	let	let	VERB
ap-961	560	2	us	we	PRON
ap-961	560	3	test	test	VERB
ap-961	560	4	this	this	DET
ap-961	560	5	knowledge	knowledge	NOUN
ap-961	560	6	on	on	ADP
ap-961	560	7	some	some	DET
ap-961	560	8	examples	example	NOUN
ap-961	560	9	.	.	PUNCT
ap-961	561	1	example	example	NOUN
ap-961	561	2	5.14	5.14	NUM
ap-961	561	3	:	:	PUNCT
ap-961	561	4	let	let	VERB
ap-961	561	5	us	we	PRON
ap-961	561	6	take	take	VERB
ap-961	561	7	the	the	DET
ap-961	561	8	two	two	NUM
ap-961	561	9	-	-	PUNCT
ap-961	561	10	dimensional	dimensional	ADJ
ap-961	561	11	sphere	sphere	NOUN
ap-961	561	12	and	and	CCONJ
ap-961	561	13	the	the	DET
ap-961	561	14	projection	projection	NOUN
ap-961	561	15	of	of	ADP
ap-961	561	16	the	the	DET
ap-961	561	17	magnetic	magnetic	ADJ
ap-961	561	18	monopole	monopole	NOUN
ap-961	561	19	from	from	ADP
ap-961	561	20	example	example	NOUN
ap-961	561	21	4.6	4.6	NUM
ap-961	561	22	.	.	PUNCT
ap-961	562	1	taking	take	VERB
ap-961	562	2	the	the	DET
ap-961	562	3	standard	standard	ADJ
ap-961	562	4	cyclic	cyclic	ADJ
ap-961	562	5	cocycle	cocycle	NOUN
ap-961	562	6	that	that	PRON
ap-961	562	7	arises	arise	VERB
ap-961	562	8	from	from	ADP
ap-961	562	9	de	de	PROPN
ap-961	562	10	rham	rham	PROPN
ap-961	562	11	differential	differential	NOUN
ap-961	562	12	forms	form	NOUN
ap-961	562	13	and	and	CCONJ
ap-961	562	14	the	the	DET
ap-961	562	15	standard	standard	ADJ
ap-961	562	16	integration	integration	NOUN
ap-961	562	17	,	,	PUNCT
ap-961	562	18	we	we	PRON
ap-961	562	19	have	have	VERB
ap-961	562	20	:	:	PUNCT
ap-961	562	21	pdpdp	pdpdp	NOUN
ap-961	562	22	e	e	NOUN
ap-961	562	23	e	e	NOUN
ap-961	562	24	d	d	X
ap-961	562	25	i	i	PRON
ap-961	562	26	i	i	PROPN
ap-961	562	27	�	�	PROPN
ap-961	562	28	�	�	PROPN
ap-961	562	29	�	�	PROPN
ap-961	562	30	�	�	PROPN
ap-961	562	31	�	�	PROPN
ap-961	562	32	�	�	PROPN
ap-961	562	33	�	�	PROPN
ap-961	562	34	�	�	PROPN
ap-961	562	35	�	�	PROPN
ap-961	562	36	�	�	PROPN
ap-961	562	37	"	"	PUNCT
ap-961	562	38	�	�	PROPN
ap-961	562	39	�	�	PROPN
ap-961	562	40	1	1	NUM
ap-961	562	41	8	8	NUM
ap-961	562	42	1	1	NUM
ap-961	562	43	1	1	NUM
ap-961	562	44	cos	cos	PROPN
ap-961	562	45	sin	sin	PROPN
ap-961	562	46	sin	sin	NOUN
ap-961	562	47	cos	cos	ADP
ap-961	562	48	sin	sin	PROPN
ap-961	562	49	�	�	PROPN
ap-961	562	50	�	�	PROPN
ap-961	562	51	�	�	PROPN
ap-961	562	52	�	�	PROPN
ap-961	562	53	�	�	PROPN
ap-961	562	54	�	�	PROPN
ap-961	562	55	�	�	PROPN
ap-961	562	56	�	�	PROPN
ap-961	562	57	�	�	PROPN
ap-961	562	58	�	�	PROPN
ap-961	562	59	�	�	PROPN
ap-961	562	60	�	�	PROPN
ap-961	562	61	�	�	PROPN
ap-961	562	62	�	�	PROPN
ap-961	562	63	�	�	PROPN
ap-961	562	64	�	�	PROPN
ap-961	562	65	cos	cos	PROPN
ap-961	562	66	sin	sin	PROPN
ap-961	563	1	cos	cos	PROPN
ap-961	563	2	sin	sin	NOUN
ap-961	563	3	sin	sin	NOUN
ap-961	563	4	e	e	NOUN
ap-961	564	1	d	d	NOUN
ap-961	564	2	i	i	PRON
ap-961	564	3	e	e	NOUN
ap-961	564	4	d	d	X
ap-961	564	5	e	e	X
ap-961	565	1	d	d	NOUN
ap-961	565	2	i	i	NOUN
ap-961	565	3	e	e	PROPN
ap-961	566	1	d	d	NOUN
ap-961	566	2	d	d	VERB
ap-961	567	1	i	i	PRON
ap-961	567	2	i	i	PRON
ap-961	568	1	i	i	PRON
ap-961	569	1	i	i	PRON
ap-961	569	2	�	�	PROPN
ap-961	569	3	�	�	PROPN
ap-961	569	4	�	�	PROPN
ap-961	569	5	�	�	PROPN
ap-961	569	6	�	�	PROPN
ap-961	569	7	�	�	PROPN
ap-961	569	8	�	�	PROPN
ap-961	569	9	�	�	PROPN
ap-961	569	10	�	�	PROPN
ap-961	569	11	�	�	PROPN
ap-961	569	12	�	�	PROPN
ap-961	569	13	�	�	PROPN
ap-961	569	14	�	�	PROPN
ap-961	569	15	�	�	PROPN
ap-961	569	16	�	�	PROPN
ap-961	569	17	"	"	PUNCT
ap-961	569	18	�	�	PROPN
ap-961	569	19	�	�	PROPN
ap-961	569	20	sin	sin	PROPN
ap-961	569	21	cos	cos	PROPN
ap-961	569	22	sin	sin	PROPN
ap-961	569	23	cos	cos	PROPN
ap-961	569	24	sin	sin	PROPN
ap-961	569	25	�	�	PROPN
ap-961	569	26	�	�	PROPN
ap-961	569	27	�	�	PROPN
ap-961	569	28	�	�	PROPN
ap-961	569	29	�	�	PROPN
ap-961	569	30	�	�	PROPN
ap-961	569	31	�	�	PROPN
ap-961	569	32	�	�	PROPN
ap-961	569	33	d	d	PROPN
ap-961	569	34	e	e	PROPN
ap-961	570	1	d	d	NOUN
ap-961	570	2	i	i	PRON
ap-961	570	3	e	e	NOUN
ap-961	570	4	d	d	X
ap-961	570	5	e	e	X
ap-961	571	1	d	d	NOUN
ap-961	571	2	i	i	PRON
ap-961	572	1	e	e	VERB
ap-961	572	2	i	i	PRON
ap-961	573	1	i	i	INTJ
ap-961	574	1	i	i	PRON
ap-961	575	1	i	i	PRON
ap-961	575	2	�	�	PROPN
ap-961	575	3	�	�	PROPN
ap-961	575	4	�	�	PROPN
ap-961	575	5	�	�	PROPN
ap-961	575	6	�	�	PROPN
ap-961	575	7	�	�	PROPN
ap-961	575	8	�	�	PROPN
ap-961	575	9	d	d	PROPN
ap-961	576	1	d	d	NOUN
ap-961	576	2	i	i	PRON
ap-961	577	1	i	i	PRON
ap-961	577	2	e	e	VERB
ap-961	578	1	i	i	PRON
ap-961	578	2	e	e	VERB
ap-961	579	1	i	i	PRON
ap-961	579	2	sin	sin	VERB
ap-961	579	3	sin	sin	NOUN
ap-961	579	4	(	(	PUNCT
ap-961	579	5	cos	cos	ADP
ap-961	579	6	sin	sin	PROPN
ap-961	579	7	sin	sin	PROPN
ap-961	579	8	�	�	PROPN
ap-961	579	9	�	�	PROPN
ap-961	579	10	�	�	PROPN
ap-961	579	11	�	�	PROPN
ap-961	579	12	�	�	PROPN
ap-961	579	13	�	�	PROPN
ap-961	579	14	�	�	PROPN
ap-961	579	15	�	�	PROPN
ap-961	579	16	�	�	PROPN
ap-961	579	17	�	�	PROPN
ap-961	579	18	�	�	PROPN
ap-961	579	19	�	�	PROPN
ap-961	579	20	�	�	PROPN
ap-961	579	21	�	�	PROPN
ap-961	579	22	�	�	PROPN
ap-961	579	23	�	�	PROPN
ap-961	579	24	1	1	NUM
ap-961	579	25	4	4	NUM
ap-961	579	26	1	1	NUM
ap-961	579	27	2	2	NUM
ap-961	579	28	2	2	NUM
ap-961	580	1	i	i	PRON
ap-961	580	2	i	i	NOUN
ap-961	581	1	d	d	PROPN
ap-961	581	2	d	d	X
ap-961	581	3	�	�	PROPN
ap-961	581	4	�	�	PROPN
ap-961	581	5	�	�	PROPN
ap-961	581	6	sin	sin	NOUN
ap-961	581	7	(	(	PUNCT
ap-961	581	8	cos	cos	PROPN
ap-961	581	9	�	�	PROPN
ap-961	581	10	�	�	PROPN
ap-961	581	11	�	�	PROPN
ap-961	581	12	1	1	NUM
ap-961	581	13	�	�	PROPN
ap-961	581	14	�	�	PROPN
ap-961	581	15	�	�	PROPN
ap-961	581	16	�	�	PROPN
ap-961	581	17	�	�	PROPN
ap-961	581	18	�	�	PROPN
ap-961	581	19	�	�	PROPN
ap-961	581	20	�	�	PROPN
ap-961	581	21	�	�	PROPN
ap-961	581	22	#	#	PROPN
ap-961	581	23	the	the	DET
ap-961	581	24	integral	integral	ADJ
ap-961	581	25	over	over	ADP
ap-961	581	26	s2	s2	NOUN
ap-961	581	27	of	of	ADP
ap-961	581	28	the	the	DET
ap-961	581	29	trace	trace	NOUN
ap-961	581	30	of	of	ADP
ap-961	581	31	the	the	DET
ap-961	581	32	above	above	ADJ
ap-961	581	33	expression	expression	NOUN
ap-961	581	34	is	be	AUX
ap-961	581	35	:	:	PUNCT
ap-961	581	36	d	d	X
ap-961	582	1	d	d	X
ap-961	582	2	i	i	NOUN
ap-961	582	3	d	d	PROPN
ap-961	582	4	d	d	X
ap-961	582	5	i	i	PRON
ap-961	582	6	�	�	PROPN
ap-961	582	7	�	�	PROPN
ap-961	582	8	�	�	PROPN
ap-961	582	9	�	�	PROPN
ap-961	582	10	�	�	PROPN
ap-961	582	11	0	0	NUM
ap-961	582	12	0	0	NUM
ap-961	582	13	2	2	NUM
ap-961	582	14	1	1	NUM
ap-961	582	15	2	2	NUM
ap-961	582	16	2	2	NUM
ap-961	582	17	�	�	PROPN
ap-961	582	18	�	�	PROPN
ap-961	582	19	�	�	PROPN
ap-961	582	20	2	2	NUM
ap-961	582	21	2	2	NUM
ap-961	582	22	#	#	NUM
ap-961	582	23	�	�	PROPN
ap-961	582	24	�	�	PROPN
ap-961	582	25	�	�	PROPN
ap-961	582	26	�	�	PROPN
ap-961	582	27	�	�	PROPN
ap-961	582	28	�	�	PROPN
ap-961	582	29	�	�	PROPN
ap-961	582	30	sin	sin	NOUN
ap-961	582	31	.	.	PUNCT
ap-961	583	1	since	since	SCONJ
ap-961	583	2	the	the	DET
ap-961	583	3	trivial	trivial	ADJ
ap-961	583	4	line	line	NOUN
ap-961	583	5	bundle	bundle	NOUN
ap-961	583	6	over	over	ADP
ap-961	583	7	the	the	DET
ap-961	583	8	sphere	sphere	NOUN
ap-961	583	9	s2	s2	NOUN
ap-961	583	10	given	give	VERB
ap-961	583	11	by	by	ADP
ap-961	583	12	the	the	DET
ap-961	583	13	trivial	trivial	ADJ
ap-961	583	14	projection	projection	NOUN
ap-961	583	15	(	(	PUNCT
ap-961	583	16	p	p	NOUN
ap-961	583	17	�	�	PROPN
ap-961	583	18	1	1	NUM
ap-961	583	19	)	)	PUNCT
ap-961	583	20	has	have	VERB
ap-961	583	21	a	a	DET
ap-961	583	22	trivial	trivial	ADJ
ap-961	583	23	pairing	pairing	NOUN
ap-961	583	24	with	with	ADP
ap-961	583	25	the	the	DET
ap-961	583	26	two	two	NUM
ap-961	583	27	-	-	PUNCT
ap-961	583	28	cyclic	cyclic	NOUN
ap-961	583	29	cocycle	cocycle	NOUN
ap-961	583	30	(	(	PUNCT
ap-961	583	31	for	for	ADP
ap-961	583	32	the	the	DET
ap-961	583	33	obvious	obvious	ADJ
ap-961	583	34	reason	reason	NOUN
ap-961	583	35	that	that	SCONJ
ap-961	583	36	p	p	PROPN
ap-961	583	37	�	�	PROPN
ap-961	583	38	1and	1and	NUM
ap-961	583	39	hence	hence	ADV
ap-961	583	40	dp	dp	PROPN
ap-961	583	41	�	�	PROPN
ap-961	583	42	0	0	NUM
ap-961	583	43	)	)	PUNCT
ap-961	583	44	–	–	PUNCT
ap-961	583	45	we	we	PRON
ap-961	583	46	have	have	VERB
ap-961	583	47	the	the	DET
ap-961	583	48	proof	proof	NOUN
ap-961	583	49	(	(	PUNCT
ap-961	583	50	based	base	VERB
ap-961	583	51	,	,	PUNCT
ap-961	583	52	of	of	ADP
ap-961	583	53	course	course	NOUN
ap-961	583	54	,	,	PUNCT
ap-961	583	55	on	on	ADP
ap-961	583	56	theorem	theorem	NOUN
ap-961	583	57	5.13	5.13	NUM
ap-961	583	58	)	)	PUNCT
ap-961	583	59	that	that	SCONJ
ap-961	583	60	the	the	DET
ap-961	583	61	projective	projective	ADJ
ap-961	583	62	module	module	NOUN
ap-961	583	63	of	of	ADP
ap-961	583	64	the	the	DET
ap-961	583	65	magnetic	magnetic	ADJ
ap-961	583	66	monopole	monopole	NOUN
ap-961	583	67	is	be	AUX
ap-961	583	68	not	not	PART
ap-961	583	69	trivial	trivial	ADJ
ap-961	583	70	.	.	PUNCT
ap-961	584	1	example	example	NOUN
ap-961	584	2	5.15	5.15	NUM
ap-961	584	3	:	:	PUNCT
ap-961	584	4	let	let	VERB
ap-961	584	5	us	we	PRON
ap-961	584	6	take	take	VERB
ap-961	584	7	the	the	DET
ap-961	584	8	commutative	commutative	ADJ
ap-961	584	9	two	two	NUM
ap-961	584	10	-	-	PUNCT
ap-961	584	11	torus	torus	NOUN
ap-961	584	12	,	,	PUNCT
ap-961	584	13	with	with	ADP
ap-961	584	14	the	the	DET
ap-961	584	15	cyclic	cyclic	ADJ
ap-961	584	16	cocycle	cocycle	NOUN
ap-961	584	17	given	give	VERB
ap-961	584	18	by	by	ADP
ap-961	584	19	the	the	DET
ap-961	584	20	de	de	PROPN
ap-961	584	21	rham	rham	PROPN
ap-961	584	22	forms	form	NOUN
ap-961	584	23	and	and	CCONJ
ap-961	584	24	integration	integration	NOUN
ap-961	584	25	.	.	PUNCT
ap-961	585	1	first	first	ADV
ap-961	585	2	,	,	PUNCT
ap-961	585	3	we	we	PRON
ap-961	585	4	start	start	VERB
ap-961	585	5	with	with	ADP
ap-961	585	6	the	the	DET
ap-961	585	7	smooth	smooth	ADJ
ap-961	585	8	projection	projection	NOUN
ap-961	585	9	4.7	4.7	NUM
ap-961	585	10	.	.	PUNCT
ap-961	586	1	skipping	skip	VERB
ap-961	586	2	the	the	DET
ap-961	586	3	easy	easy	ADJ
ap-961	586	4	calculation	calculation	NOUN
ap-961	586	5	we	we	PRON
ap-961	586	6	show	show	VERB
ap-961	586	7	the	the	DET
ap-961	586	8	result	result	NOUN
ap-961	586	9	:	:	PUNCT
ap-961	586	10	tr	tr	PUNCT
ap-961	586	11	p	p	NOUN
ap-961	586	12	dp	dp	NOUN
ap-961	587	1	dp	dp	NOUN
ap-961	588	1	i	i	PRON
ap-961	588	2	g	g	PROPN
ap-961	589	1	t	t	PROPN
ap-961	590	1	f	f	PROPN
ap-961	590	2	t	t	PROPN
ap-961	590	3	g	g	PROPN
ap-961	590	4	t	t	PROPN
ap-961	591	1	f	f	PROPN
ap-961	591	2	t	t	PROPN
ap-961	591	3	g	g	PROPN
ap-961	591	4	t	t	PROPN
ap-961	591	5	g	g	PROPN
ap-961	591	6	t	t	PROPN
ap-961	591	7	s	s	PART
ap-961	591	8	�	�	PROPN
ap-961	591	9	(	(	PUNCT
ap-961	591	10	�	�	PROPN
ap-961	591	11	(	(	PUNCT
ap-961	591	12	�	�	PROPN
ap-961	591	13	(	(	PUNCT
ap-961	591	14	2	2	NUM
ap-961	591	15	2	2	NUM
ap-961	591	16	2	2	NUM
ap-961	591	17	(	(	PUNCT
ap-961	591	18	(	(	PUNCT
ap-961	591	19	)	)	PUNCT
ap-961	591	20	(	(	PUNCT
ap-961	591	21	(	(	PUNCT
ap-961	591	22	)	)	PUNCT
ap-961	591	23	(	(	PUNCT
ap-961	591	24	)	)	PUNCT
ap-961	591	25	(	(	PUNCT
ap-961	591	26	)	)	PUNCT
ap-961	591	27	(	(	PUNCT
ap-961	591	28	)	)	PUNCT
ap-961	591	29	(	(	PUNCT
ap-961	591	30	)	)	PUNCT
ap-961	591	31	)	)	PUNCT
ap-961	591	32	cos	cos	PROPN
ap-961	591	33	(	(	PUNCT
ap-961	591	34	)	)	PUNCT
ap-961	591	35	(	(	PUNCT
ap-961	591	36	)	)	PUNCT
ap-961	591	37	(	(	PUNCT
ap-961	591	38	(	(	PUNCT
ap-961	591	39	)	)	PUNCT
ap-961	591	40	(	(	PUNCT
ap-961	591	41	)	)	PUNCT
ap-961	591	42	(	(	PUNCT
ap-961	591	43	)	)	PUNCT
ap-961	591	44	(	(	PUNCT
ap-961	591	45	)	)	PUNCT
ap-961	591	46	(	(	PUNCT
ap-961	591	47	)	)	PUNCT
ap-961	591	48	)	)	PUNCT
ap-961	591	49	)	)	PUNCT
ap-961	591	50	.g	.g	PROPN
ap-961	592	1	t	t	X
ap-961	592	2	f	f	PROPN
ap-961	592	3	t	t	PROPN
ap-961	592	4	h	h	PROPN
ap-961	593	1	t	t	PROPN
ap-961	593	2	h	h	NOUN
ap-961	594	1	t	t	PROPN
ap-961	594	2	f	f	PROPN
ap-961	594	3	t	t	PROPN
ap-961	594	4	h	h	PROPN
ap-961	594	5	t2	t2	PROPN
ap-961	594	6	2	2	NUM
ap-961	594	7	(	(	PUNCT
ap-961	594	8	�	�	PROPN
ap-961	594	9	(	(	PUNCT
ap-961	594	10	�	�	PROPN
ap-961	594	11	(	(	PUNCT
ap-961	594	12	since	since	SCONJ
ap-961	594	13	the	the	DET
ap-961	594	14	integral	integral	NOUN
ap-961	594	15	of	of	ADP
ap-961	594	16	cos	cos	PROPN
ap-961	594	17	(	(	PUNCT
ap-961	594	18	)	)	PUNCT
ap-961	594	19	s	s	AUX
ap-961	594	20	vanishes	vanishe	NOUN
ap-961	594	21	,	,	PUNCT
ap-961	594	22	only	only	ADV
ap-961	594	23	the	the	DET
ap-961	594	24	first	first	ADJ
ap-961	594	25	component	component	NOUN
ap-961	594	26	contributes	contribute	VERB
ap-961	594	27	to	to	ADP
ap-961	594	28	the	the	DET
ap-961	594	29	pairing	pairing	NOUN
ap-961	594	30	.	.	PUNCT
ap-961	595	1	we	we	PRON
ap-961	595	2	rewrite	rewrite	VERB
ap-961	595	3	it	it	PRON
ap-961	595	4	further	far	ADV
ap-961	595	5	as	as	ADP
ap-961	595	6	:	:	PUNCT
ap-961	595	7	(	(	PUNCT
ap-961	595	8	(	(	PUNCT
ap-961	595	9	)	)	PUNCT
ap-961	595	10	(	(	PUNCT
ap-961	595	11	)	)	PUNCT
ap-961	595	12	)	)	PUNCT
ap-961	596	1	(	(	PUNCT
ap-961	596	2	)	)	PUNCT
ap-961	596	3	(	(	PUNCT
ap-961	596	4	)	)	PUNCT
ap-961	596	5	(	(	PUNCT
ap-961	596	6	(	(	PUNCT
ap-961	596	7	)	)	PUNCT
ap-961	596	8	)	)	PUNCT
ap-961	597	1	f	f	PROPN
ap-961	597	2	t	t	PROPN
ap-961	597	3	g	g	PROPN
ap-961	597	4	t	t	PROPN
ap-961	598	1	f	f	PROPN
ap-961	598	2	t	t	PROPN
ap-961	598	3	g	g	PROPN
ap-961	598	4	t	t	PROPN
ap-961	598	5	g	g	PROPN
ap-961	598	6	t2	t2	PROPN
ap-961	598	7	2	2	NUM
ap-961	598	8	23	23	NUM
ap-961	598	9	1	1	NUM
ap-961	598	10	2	2	NUM
ap-961	598	11	(	(	PUNCT
ap-961	598	12	�	�	PROPN
ap-961	598	13	(	(	PUNCT
ap-961	598	14	�	�	PROPN
ap-961	598	15	(	(	PUNCT
ap-961	598	16	.	.	PUNCT
ap-961	599	1	integrating	integrate	VERB
ap-961	599	2	by	by	ADP
ap-961	599	3	parts	part	NOUN
ap-961	599	4	,	,	PUNCT
ap-961	599	5	and	and	CCONJ
ap-961	599	6	using	use	VERB
ap-961	599	7	the	the	DET
ap-961	599	8	fact	fact	NOUN
ap-961	599	9	that	that	SCONJ
ap-961	599	10	all	all	DET
ap-961	599	11	functions	function	NOUN
ap-961	599	12	are	be	AUX
ap-961	599	13	smooth	smooth	ADJ
ap-961	599	14	and	and	CCONJ
ap-961	599	15	periodic	periodic	ADJ
ap-961	599	16	we	we	PRON
ap-961	599	17	get	get	VERB
ap-961	599	18	the	the	DET
ap-961	599	19	result	result	NOUN
ap-961	599	20	for	for	ADP
ap-961	599	21	the	the	DET
ap-961	599	22	pairing	pairing	NOUN
ap-961	599	23	as	as	ADP
ap-961	599	24	:	:	PUNCT
ap-961	599	25	(	(	PUNCT
ap-961	599	26	)	)	PUNCT
ap-961	599	27	(	(	PUNCT
ap-961	599	28	)	)	PUNCT
ap-961	599	29	(	(	PUNCT
ap-961	599	30	)	)	PUNCT
ap-961	599	31	)	)	PUNCT
ap-961	599	32	4	4	NUM
ap-961	599	33	3	3	NUM
ap-961	599	34	2	2	NUM
ap-961	599	35	0	0	NUM
ap-961	599	36	2	2	NUM
ap-961	599	37	�	�	PROPN
ap-961	599	38	�	�	PROPN
ap-961	599	39	i	i	PRON
ap-961	599	40	dt	dt	PROPN
ap-961	599	41	(	(	PUNCT
ap-961	599	42	f	f	PROPN
ap-961	599	43	t	t	PROPN
ap-961	599	44	g	g	PROPN
ap-961	599	45	t(2	t(2	PROPN
ap-961	599	46	.	.	PUNCT
ap-961	600	1	recall	recall	VERB
ap-961	600	2	that	that	SCONJ
ap-961	600	3	the	the	DET
ap-961	600	4	function	function	NOUN
ap-961	600	5	f	f	PROPN
ap-961	600	6	g	g	PROPN
ap-961	600	7	h	h	PROPN
ap-961	600	8	,	,	PUNCT
ap-961	600	9	,	,	PUNCT
ap-961	600	10	must	must	AUX
ap-961	600	11	obey	obey	VERB
ap-961	600	12	:	:	PUNCT
ap-961	600	13	g	g	PROPN
ap-961	600	14	t	t	PROPN
ap-961	600	15	h	h	PROPN
ap-961	600	16	t	t	PROPN
ap-961	600	17	(	(	PUNCT
ap-961	600	18	)	)	PUNCT
ap-961	600	19	(	(	PUNCT
ap-961	600	20	)	)	PUNCT
ap-961	600	21	�	�	PROPN
ap-961	600	22	0	0	NUM
ap-961	600	23	,	,	PUNCT
ap-961	601	1	g	g	PROPN
ap-961	601	2	t	t	PROPN
ap-961	601	3	h	h	NOUN
ap-961	602	1	t	t	PROPN
ap-961	602	2	f	f	PROPN
ap-961	602	3	t	t	PROPN
ap-961	602	4	f	f	PROPN
ap-961	602	5	t	t	PROPN
ap-961	602	6	(	(	PUNCT
ap-961	602	7	)	)	PUNCT
ap-961	602	8	(	(	PUNCT
ap-961	602	9	)	)	PUNCT
ap-961	602	10	(	(	PUNCT
ap-961	602	11	)	)	PUNCT
ap-961	602	12	(	(	PUNCT
ap-961	602	13	)	)	SYM
ap-961	602	14	2	2	NUM
ap-961	602	15	2	2	NUM
ap-961	602	16	2	2	NUM
ap-961	602	17	�	�	PROPN
ap-961	602	18	�	�	PROPN
ap-961	602	19	.	.	PUNCT
ap-961	603	1	then	then	ADV
ap-961	603	2	:	:	PUNCT
ap-961	603	3	(	(	PUNCT
ap-961	603	4	)	)	PUNCT
ap-961	603	5	(	(	PUNCT
ap-961	603	6	)	)	PUNCT
ap-961	603	7	(	(	PUNCT
ap-961	603	8	)	)	PUNCT
ap-961	603	9	)	)	PUNCT
ap-961	603	10	(	(	PUNCT
ap-961	603	11	)	)	PUNCT
ap-961	603	12	(	(	PUNCT
ap-961	603	13	)	)	PUNCT
ap-961	603	14	(	(	PUNCT
ap-961	603	15	(	(	PUNCT
ap-961	603	16	)	)	PUNCT
ap-961	603	17	(	(	PUNCT
ap-961	603	18	)	)	PUNCT
ap-961	603	19	)	)	PUNCT
ap-961	603	20	4	4	NUM
ap-961	603	21	3	3	NUM
ap-961	603	22	4	4	NUM
ap-961	603	23	32	32	NUM
ap-961	603	24	0	0	NUM
ap-961	603	25	2	2	NUM
ap-961	603	26	2	2	NUM
ap-961	603	27	�	�	PROPN
ap-961	603	28	�	�	PROPN
ap-961	603	29	�	�	PROPN
ap-961	603	30	i	i	PRON
ap-961	603	31	dt	dt	PROPN
ap-961	603	32	(	(	PUNCT
ap-961	603	33	f	f	PROPN
ap-961	603	34	t	t	PROPN
ap-961	604	1	g	g	PROPN
ap-961	604	2	t	t	PROPN
ap-961	605	1	i	i	PRON
ap-961	605	2	f	f	PROPN
ap-961	606	1	t	t	PROPN
ap-961	606	2	f	f	PROPN
ap-961	606	3	t	t	PROPN
ap-961	606	4	f	f	PROPN
ap-961	606	5	t	t	PROPN
ap-961	606	6	(	(	PUNCT
ap-961	606	7	�	�	PROPN
ap-961	606	8	(	(	PUNCT
ap-961	606	9	�	�	PROPN
ap-961	606	10	2	2	NUM
ap-961	606	11	supp	supp	NOUN
ap-961	606	12	g	g	PROPN
ap-961	606	13	t	t	PROPN
ap-961	606	14	j	j	PROPN
ap-961	607	1	i	i	PRON
ap-961	607	2	i	i	INTJ
ap-961	607	3	j	j	VERB
ap-961	608	1	n	n	CCONJ
ap-961	608	2	i	i	PRON
ap-961	609	1	f	f	NOUN
ap-961	609	2	x	x	X
ap-961	609	3	f	f	X
ap-961	609	4	x	x	X
ap-961	609	5	(	(	PUNCT
ap-961	609	6	)	)	PUNCT
ap-961	609	7	(	(	PUNCT
ap-961	609	8	)	)	PUNCT
ap-961	609	9	(	(	PUNCT
ap-961	609	10	)	)	PUNCT
ap-961	609	11	(	(	PUNCT
ap-961	609	12	)	)	PUNCT
ap-961	609	13	(	(	PUNCT
ap-961	609	14	)	)	PUNCT
ap-961	609	15	.	.	PUNCT
ap-961	610	1	2	2	NUM
ap-961	610	2	�	�	PROPN
ap-961	610	3	�	�	PROPN
ap-961	610	4	�	�	PROPN
ap-961	610	5	�	�	PROPN
ap-961	610	6	�	�	PROPN
ap-961	610	7	�	�	PROPN
ap-961	610	8	�	�	PROPN
ap-961	610	9	�	�	PROPN
ap-961	610	10	�	�	PROPN
ap-961	610	11	�	�	PROPN
ap-961	610	12	�	�	PROPN
ap-961	610	13	4	4	NUM
ap-961	610	14	1	1	NUM
ap-961	610	15	3	3	NUM
ap-961	610	16	2	2	NUM
ap-961	610	17	1	1	NUM
ap-961	610	18	2	2	NUM
ap-961	610	19	3	3	NUM
ap-961	610	20	1	1	NUM
ap-961	610	21	�	�	NOUN
ap-961	610	22	where	where	SCONJ
ap-961	610	23	the	the	DET
ap-961	610	24	points	point	NOUN
ap-961	610	25	xi	xi	X
ap-961	610	26	are	be	AUX
ap-961	610	27	such	such	ADJ
ap-961	610	28	that	that	SCONJ
ap-961	610	29	:	:	PUNCT
ap-961	610	30	supp	supp	PROPN
ap-961	610	31	g	g	PROPN
ap-961	610	32	t	t	PROPN
ap-961	610	33	x	x	PUNCT
ap-961	611	1	x	x	PUNCT
ap-961	611	2	x	x	PUNCT
ap-961	611	3	x	x	SYM
ap-961	611	4	t	t	NOUN
ap-961	611	5	g	g	PROPN
ap-961	611	6	t	t	PROPN
ap-961	611	7	n	n	CCONJ
ap-961	611	8	(	(	PUNCT
ap-961	611	9	)	)	PUNCT
ap-961	611	10	(	(	PUNCT
ap-961	611	11	,	,	PUNCT
ap-961	611	12	)	)	PUNCT
ap-961	611	13	(	(	PUNCT
ap-961	611	14	,	,	PUNCT
ap-961	611	15	)	)	PUNCT
ap-961	611	16	(	(	PUNCT
ap-961	611	17	,	,	PUNCT
ap-961	611	18	)	)	PUNCT
ap-961	611	19	{	{	PUNCT
ap-961	611	20	:	:	PUNCT
ap-961	611	21	(	(	PUNCT
ap-961	611	22	)	)	PUNCT
ap-961	611	23	}	}	PUNCT
ap-961	611	24	.	.	PUNCT
ap-961	612	1	�	�	PROPN
ap-961	612	2	3	3	NUM
ap-961	612	3	3	3	NUM
ap-961	612	4	3	3	NUM
ap-961	612	5	�	�	PROPN
ap-961	612	6	)	)	PUNCT
ap-961	612	7	)	)	PUNCT
ap-961	612	8	!	!	PUNCT
ap-961	613	1	0	0	NUM
ap-961	613	2	2	2	NUM
ap-961	613	3	0	0	NUM
ap-961	613	4	2	2	NUM
ap-961	613	5	0	0	NUM
ap-961	613	6	1	1	NUM
ap-961	613	7	1	1	NUM
ap-961	613	8	2	2	NUM
ap-961	613	9	�	�	PROPN
ap-961	613	10	�	�	PROPN
ap-961	613	11	�	�	PROPN
ap-961	613	12	since	since	SCONJ
ap-961	613	13	points	point	NOUN
ap-961	613	14	xi	xi	X
ap-961	613	15	are	be	AUX
ap-961	613	16	the	the	DET
ap-961	613	17	boundary	boundary	ADJ
ap-961	613	18	points	point	NOUN
ap-961	613	19	of	of	ADP
ap-961	613	20	a	a	DET
ap-961	613	21	set	set	NOUN
ap-961	613	22	where	where	SCONJ
ap-961	613	23	g	g	PROPN
ap-961	613	24	(	(	PUNCT
ap-961	613	25	and	and	CCONJ
ap-961	613	26	thus	thus	ADV
ap-961	613	27	h	h	X
ap-961	613	28	)	)	PUNCT
ap-961	613	29	do	do	AUX
ap-961	613	30	not	not	PART
ap-961	613	31	vanish	vanish	VERB
ap-961	613	32	,	,	PUNCT
ap-961	613	33	by	by	ADP
ap-961	613	34	continuity	continuity	NOUN
ap-961	613	35	both	both	CCONJ
ap-961	613	36	g	g	PROPN
ap-961	613	37	and	and	CCONJ
ap-961	613	38	h	h	PROPN
ap-961	613	39	must	must	AUX
ap-961	613	40	vanish	vanish	VERB
ap-961	613	41	each	each	DET
ap-961	613	42	xi	xi	PROPN
ap-961	613	43	.	.	PUNCT
ap-961	614	1	so	so	ADV
ap-961	614	2	,	,	PUNCT
ap-961	614	3	f	f	PROPN
ap-961	614	4	satisfies	satisfy	VERB
ap-961	614	5	f	f	PROPN
ap-961	615	1	x	x	SYM
ap-961	615	2	f	f	X
ap-961	615	3	xi	xi	INTJ
ap-961	615	4	i	i	PRON
ap-961	615	5	2	2	NUM
ap-961	615	6	(	(	PUNCT
ap-961	615	7	)	)	PUNCT
ap-961	615	8	(	(	PUNCT
ap-961	615	9	)	)	PUNCT
ap-961	615	10	�	�	PROPN
ap-961	615	11	and	and	CCONJ
ap-961	615	12	therefore	therefore	ADV
ap-961	615	13	it	it	PRON
ap-961	615	14	is	be	AUX
ap-961	615	15	either	either	CCONJ
ap-961	615	16	0	0	NUM
ap-961	615	17	or	or	CCONJ
ap-961	615	18	1	1	NUM
ap-961	615	19	at	at	ADP
ap-961	615	20	each	each	DET
ap-961	615	21	point	point	NOUN
ap-961	615	22	.	.	PUNCT
ap-961	616	1	hence	hence	ADV
ap-961	616	2	,	,	PUNCT
ap-961	616	3	we	we	PRON
ap-961	616	4	introduce	introduce	VERB
ap-961	616	5	�	�	PROPN
ap-961	616	6	j	j	PROPN
ap-961	616	7	i	i	PRON
ap-961	616	8	if	if	SCONJ
ap-961	616	9	x	x	X
ap-961	616	10	f	f	PROPN
ap-961	616	11	x	x	X
ap-961	616	12	�	�	PROPN
ap-961	616	13	�	�	PROPN
ap-961	616	14	�	�	PROPN
ap-961	616	15	�	�	PROPN
ap-961	616	16	�	�	PROPN
ap-961	616	17	�	�	PROPN
ap-961	616	18	�	�	PROPN
ap-961	616	19	�	�	PROPN
ap-961	616	20	�	�	PROPN
ap-961	616	21	�	�	PROPN
ap-961	616	22	�	�	PROPN
ap-961	616	23	�	�	PROPN
ap-961	616	24	2	2	NUM
ap-961	616	25	3	3	NUM
ap-961	616	26	2	2	NUM
ap-961	616	27	0	0	NUM
ap-961	616	28	1	1	NUM
ap-961	616	29	2	2	NUM
ap-961	616	30	3	3	NUM
ap-961	616	31	(	(	PUNCT
ap-961	616	32	)	)	PUNCT
ap-961	616	33	(	(	PUNCT
ap-961	616	34	)	)	PUNCT
ap-961	616	35	since	since	SCONJ
ap-961	616	36	f	f	PROPN
ap-961	616	37	is	be	AUX
ap-961	616	38	periodic	periodic	ADJ
ap-961	616	39	,	,	PUNCT
ap-961	616	40	f	f	PROPN
ap-961	616	41	f	f	X
ap-961	616	42	(	(	PUNCT
ap-961	616	43	)	)	PUNCT
ap-961	616	44	(	(	PUNCT
ap-961	616	45	)	)	PUNCT
ap-961	616	46	0	0	NUM
ap-961	616	47	2	2	NUM
ap-961	616	48	�	�	PROPN
ap-961	616	49	�	�	PROPN
ap-961	616	50	and	and	CCONJ
ap-961	616	51	the	the	DET
ap-961	616	52	points	point	NOUN
ap-961	616	53	0	0	NUM
ap-961	616	54	and	and	CCONJ
ap-961	616	55	2	2	NUM
ap-961	616	56	�	�	NOUN
ap-961	616	57	cancel	cancel	VERB
ap-961	616	58	each	each	DET
ap-961	616	59	other	other	ADJ
ap-961	616	60	in	in	ADP
ap-961	616	61	the	the	DET
ap-961	616	62	sum	sum	NOUN
ap-961	616	63	.	.	PUNCT
ap-961	617	1	in	in	ADP
ap-961	617	2	the	the	DET
ap-961	617	3	end	end	NOUN
ap-961	617	4	we	we	PRON
ap-961	617	5	have	have	VERB
ap-961	617	6	:	:	PUNCT
ap-961	617	7	(	(	PUNCT
ap-961	617	8	)	)	PUNCT
ap-961	617	9	(	(	PUNCT
ap-961	617	10	)	)	SYM
ap-961	617	11	2	2	NUM
ap-961	617	12	1	1	NUM
ap-961	617	13	1	1	NUM
ap-961	617	14	1	1	NUM
ap-961	617	15	�	�	PROPN
ap-961	617	16	�	�	PROPN
ap-961	617	17	i	i	PROPN
ap-961	617	18	j	j	PROPN
ap-961	617	19	j	j	PROPN
ap-961	617	20	j	j	PROPN
ap-961	617	21	n	n	CCONJ
ap-961	617	22	�	�	PROPN
ap-961	617	23	�	�	PROPN
ap-961	617	24	�	�	PROPN
ap-961	617	25	�	�	PROPN
ap-961	617	26	.	.	PUNCT
ap-961	618	1	therefore	therefore	ADV
ap-961	618	2	,	,	PUNCT
ap-961	618	3	independently	independently	ADV
ap-961	618	4	of	of	ADP
ap-961	618	5	the	the	DET
ap-961	618	6	choice	choice	NOUN
ap-961	618	7	of	of	ADP
ap-961	618	8	f	f	PRON
ap-961	618	9	the	the	DET
ap-961	618	10	result	result	NOUN
ap-961	618	11	is	be	AUX
ap-961	618	12	an	an	DET
ap-961	618	13	integral	integral	ADJ
ap-961	618	14	multiple	multiple	NOUN
ap-961	618	15	of	of	ADP
ap-961	618	16	2	2	NUM
ap-961	618	17	�	�	NOUN
ap-961	618	18	i	i	PRON
ap-961	618	19	!	!	PUNCT
ap-961	618	20	example	example	NOUN
ap-961	619	1	5.16	5.16	NUM
ap-961	619	2	:	:	PUNCT
ap-961	619	3	let	let	VERB
ap-961	619	4	us	we	PRON
ap-961	619	5	turn	turn	VERB
ap-961	619	6	to	to	ADP
ap-961	619	7	the	the	DET
ap-961	619	8	second	second	ADJ
ap-961	619	9	presentation	presentation	NOUN
ap-961	619	10	of	of	ADP
ap-961	619	11	the	the	DET
ap-961	619	12	(	(	PUNCT
ap-961	619	13	supposedly	supposedly	ADV
ap-961	619	14	)	)	PUNCT
ap-961	619	15	nontrivial	nontrivial	ADJ
ap-961	619	16	projection	projection	NOUN
ap-961	619	17	on	on	ADP
ap-961	619	18	the	the	DET
ap-961	619	19	torus	torus	NOUN
ap-961	619	20	.	.	PUNCT
ap-961	620	1	we	we	PRON
ap-961	620	2	use	use	VERB
ap-961	620	3	the	the	DET
ap-961	620	4	cyclic	cyclic	ADJ
ap-961	620	5	cocycle	cocycle	NOUN
ap-961	620	6	as	as	SCONJ
ap-961	620	7	derived	derive	VERB
ap-961	620	8	in	in	ADP
ap-961	620	9	in	in	ADP
ap-961	620	10	example	example	NOUN
ap-961	620	11	5.12	5.12	NUM
ap-961	620	12	,	,	PUNCT
ap-961	620	13	keeping	keep	VERB
ap-961	620	14	in	in	ADP
ap-961	620	15	mind	mind	NOUN
ap-961	620	16	that	that	SCONJ
ap-961	620	17	the	the	DET
ap-961	620	18	functions	function	NOUN
ap-961	620	19	that	that	PRON
ap-961	620	20	we	we	PRON
ap-961	620	21	use	use	VERB
ap-961	620	22	in	in	ADP
ap-961	620	23	the	the	DET
ap-961	620	24	projection	projection	NOUN
ap-961	620	25	are	be	AUX
ap-961	620	26	in	in	ADP
ap-961	620	27	fact	fact	NOUN
ap-961	620	28	only	only	ADV
ap-961	620	29	continuous	continuous	ADJ
ap-961	620	30	and	and	CCONJ
ap-961	620	31	are	be	AUX
ap-961	620	32	not	not	PART
ap-961	620	33	differentiable	differentiable	ADJ
ap-961	620	34	at	at	ADP
ap-961	620	35	one	one	NUM
ap-961	620	36	point	point	NOUN
ap-961	620	37	(	(	PUNCT
ap-961	620	38	t	t	PROPN
ap-961	620	39	�	�	PROPN
ap-961	620	40	0	0	NUM
ap-961	620	41	,	,	PUNCT
ap-961	620	42	identified	identify	VERB
ap-961	620	43	with	with	ADP
ap-961	620	44	t	t	PROPN
ap-961	620	45	�	�	PROPN
ap-961	620	46	1	1	NUM
ap-961	620	47	)	)	PUNCT
ap-961	620	48	.	.	PUNCT
ap-961	621	1	we	we	PRON
ap-961	621	2	calculate	calculate	VERB
ap-961	621	3	the	the	DET
ap-961	621	4	pairing	pairing	NOUN
ap-961	621	5	(	(	PUNCT
ap-961	621	6	again	again	ADV
ap-961	621	7	skipping	skip	VERB
ap-961	621	8	the	the	DET
ap-961	621	9	tedious	tedious	ADJ
ap-961	621	10	but	but	CCONJ
ap-961	621	11	uncomplicated	uncomplicated	ADJ
ap-961	621	12	algebraic	algebraic	ADJ
ap-961	621	13	manipulations	manipulation	NOUN
ap-961	621	14	)	)	PUNCT
ap-961	621	15	,	,	PUNCT
ap-961	621	16	obtaining	obtain	VERB
ap-961	621	17	:	:	PUNCT
ap-961	621	18	�	�	PROPN
ap-961	621	19	�	�	PROPN
ap-961	621	20	2	2	NUM
ap-961	621	21	1	1	NUM
ap-961	621	22	4	4	NUM
ap-961	621	23	2	2	NUM
ap-961	621	24	1	1	NUM
ap-961	621	25	2	2	NUM
ap-961	621	26	(	(	PUNCT
ap-961	621	27	,	,	PUNCT
ap-961	621	28	,	,	PUNCT
ap-961	621	29	)	)	PUNCT
ap-961	621	30	sin	sin	NOUN
ap-961	621	31	(	(	PUNCT
ap-961	621	32	cos	cos	PROPN
ap-961	621	33	(	(	PUNCT
ap-961	621	34	)	)	PUNCT
ap-961	621	35	)	)	PUNCT
ap-961	621	36	(	(	PUNCT
ap-961	621	37	)	)	PUNCT
ap-961	621	38	p	p	X
ap-961	621	39	p	p	X
ap-961	622	1	p	p	X
ap-961	623	1	i	i	PRON
ap-961	623	2	i	i	PROPN
ap-961	623	3	�	�	PROPN
ap-961	623	4	�	�	PROPN
ap-961	623	5	�	�	PROPN
ap-961	623	6	�	�	PROPN
ap-961	623	7	�	�	PROPN
ap-961	623	8	�	�	PROPN
ap-961	623	9	�	�	PROPN
ap-961	623	10	�	�	PROPN
ap-961	623	11	�	�	PROPN
ap-961	623	12	�	�	PROPN
ap-961	623	13	tr	tr	VERB
ap-961	623	14	�	�	PROPN
ap-961	623	15	�	�	PROPN
ap-961	623	16	,	,	PUNCT
ap-961	623	17	where	where	SCONJ
ap-961	623	18	we	we	PRON
ap-961	623	19	have	have	AUX
ap-961	623	20	taken	take	VERB
ap-961	623	21	the	the	DET
ap-961	623	22	not	not	PART
ap-961	623	23	-	-	PUNCT
ap-961	623	24	normalized	normalize	VERB
ap-961	623	25	trace	trace	NOUN
ap-961	623	26	coming	come	VERB
ap-961	623	27	from	from	ADP
ap-961	623	28	the	the	DET
ap-961	623	29	integration	integration	NOUN
ap-961	623	30	(	(	PUNCT
ap-961	623	31	tr1	tr1	NOUN
ap-961	623	32	2	2	NUM
ap-961	623	33	2	2	NUM
ap-961	623	34	�	�	PROPN
ap-961	623	35	(	(	PUNCT
ap-961	623	36	)	)	PUNCT
ap-961	623	37	�	�	PROPN
ap-961	623	38	)	)	PUNCT
ap-961	623	39	.	.	PUNCT
ap-961	624	1	again	again	ADV
ap-961	624	2	,	,	PUNCT
ap-961	624	3	we	we	PRON
ap-961	624	4	obtain	obtain	VERB
ap-961	624	5	a	a	DET
ap-961	624	6	result	result	NOUN
ap-961	624	7	different	different	ADJ
ap-961	624	8	from	from	ADP
ap-961	624	9	zero	zero	NUM
ap-961	624	10	,	,	PUNCT
ap-961	624	11	which	which	PRON
ap-961	624	12	ensures	ensure	VERB
ap-961	624	13	that	that	SCONJ
ap-961	624	14	the	the	DET
ap-961	624	15	projective	projective	ADJ
ap-961	624	16	module	module	NOUN
ap-961	624	17	is	be	AUX
ap-961	624	18	nontrivial	nontrivial	ADJ
ap-961	624	19	.	.	PUNCT
ap-961	625	1	shall	shall	AUX
ap-961	625	2	we	we	PRON
ap-961	625	3	worry	worry	VERB
ap-961	625	4	that	that	SCONJ
ap-961	625	5	the	the	DET
ap-961	625	6	projection	projection	NOUN
ap-961	625	7	was	be	AUX
ap-961	625	8	not	not	PART
ap-961	625	9	actually	actually	ADV
ap-961	625	10	a	a	DET
ap-961	625	11	smooth	smooth	ADJ
ap-961	625	12	one	one	NUM
ap-961	625	13	?	?	PUNCT
ap-961	626	1	yes	yes	INTJ
ap-961	626	2	,	,	PUNCT
ap-961	626	3	but	but	CCONJ
ap-961	626	4	only	only	ADV
ap-961	626	5	a	a	DET
ap-961	626	6	little	little	ADJ
ap-961	626	7	bit	bit	NOUN
ap-961	626	8	.	.	PUNCT
ap-961	627	1	note	note	VERB
ap-961	627	2	that	that	SCONJ
ap-961	627	3	each	each	DET
ap-961	627	4	function	function	NOUN
ap-961	627	5	in	in	ADP
ap-961	627	6	the	the	DET
ap-961	627	7	matrix	matrix	NOUN
ap-961	627	8	elements	element	NOUN
ap-961	627	9	of	of	ADP
ap-961	627	10	the	the	DET
ap-961	627	11	projection	projection	NOUN
ap-961	627	12	has	have	VERB
ap-961	627	13	a	a	DET
ap-961	627	14	well	well	ADV
ap-961	627	15	-	-	PUNCT
ap-961	627	16	defined	define	VERB
ap-961	627	17	left	leave	VERB
ap-961	627	18	derivative	derivative	NOUN
ap-961	627	19	at	at	ADP
ap-961	627	20	each	each	DET
ap-961	627	21	point	point	NOUN
ap-961	627	22	.	.	PUNCT
ap-961	628	1	however	however	ADV
ap-961	628	2	,	,	PUNCT
ap-961	628	3	even	even	ADV
ap-961	628	4	though	though	SCONJ
ap-961	628	5	this	this	DET
ap-961	628	6	derivative	derivative	NOUN
ap-961	628	7	is	be	AUX
ap-961	628	8	not	not	PART
ap-961	628	9	continuous	continuous	ADJ
ap-961	628	10	at	at	ADP
ap-961	628	11	one	one	NUM
ap-961	628	12	point	point	NOUN
ap-961	628	13	,	,	PUNCT
ap-961	628	14	we	we	PRON
ap-961	628	15	can	can	AUX
ap-961	628	16	multiply	multiply	VERB
ap-961	628	17	functions	function	NOUN
ap-961	628	18	with	with	ADP
ap-961	628	19	such	such	ADJ
ap-961	628	20	discontinuities	discontinuity	NOUN
ap-961	628	21	without	without	ADP
ap-961	628	22	problems	problem	NOUN
ap-961	628	23	.	.	PUNCT
ap-961	629	1	since	since	SCONJ
ap-961	629	2	the	the	DET
ap-961	629	3	integration	integration	NOUN
ap-961	629	4	is	be	AUX
ap-961	629	5	also	also	ADV
ap-961	629	6	well	well	ADV
ap-961	629	7	-	-	PUNCT
ap-961	629	8	defined	define	VERB
ap-961	629	9	fur	fur	NOUN
ap-961	629	10	such	such	ADJ
ap-961	629	11	functions	function	NOUN
ap-961	629	12	,	,	PUNCT
ap-961	629	13	we	we	PRON
ap-961	629	14	see	see	VERB
ap-961	629	15	that	that	SCONJ
ap-961	629	16	no	no	DET
ap-961	629	17	significant	significant	ADJ
ap-961	629	18	problems	problem	NOUN
ap-961	629	19	arise	arise	VERB
ap-961	629	20	.	.	PUNCT
ap-961	630	1	finally	finally	ADV
ap-961	630	2	,	,	PUNCT
ap-961	630	3	we	we	PRON
ap-961	630	4	turn	turn	VERB
ap-961	630	5	to	to	ADP
ap-961	630	6	a	a	DET
ap-961	630	7	very	very	ADV
ap-961	630	8	nontrivial	nontrivial	NOUN
ap-961	630	9	(	(	PUNCT
ap-961	630	10	and	and	CCONJ
ap-961	630	11	very	very	ADV
ap-961	630	12	)	)	PUNCT
ap-961	630	13	noncommutative	noncommutative	ADJ
ap-961	630	14	example	example	NOUN
ap-961	630	15	.	.	PUNCT
ap-961	631	1	example	example	NOUN
ap-961	632	1	5.17	5.17	NUM
ap-961	632	2	:	:	PUNCT
ap-961	632	3	consider	consider	VERB
ap-961	632	4	the	the	DET
ap-961	632	5	following	follow	VERB
ap-961	632	6	family	family	NOUN
ap-961	632	7	of	of	ADP
ap-961	632	8	projective	projective	NOUN
ap-961	632	9	(	(	PUNCT
ap-961	632	10	and	and	CCONJ
ap-961	632	11	smooth	smooth	ADJ
ap-961	632	12	)	)	PUNCT
ap-961	632	13	modules	module	VERB
ap-961	632	14	the	the	PRON
ap-961	632	15	over	over	ADP
ap-961	632	16	the	the	DET
ap-961	632	17	noncommutative	noncommutative	ADJ
ap-961	632	18	torus	torus	NOUN
ap-961	632	19	:	:	PUNCT
ap-961	632	20	�	�	PROPN
ap-961	632	21	q	q	PUNCT
ap-961	632	22	qs	qs	PROPN
ap-961	632	23	�	�	PROPN
ap-961	632	24	�	�	PROPN
ap-961	632	25	(	(	PUNCT
ap-961	632	26	)	)	PUNCT
ap-961	632	27	�	�	PROPN
ap-961	632	28	�	�	PROPN
ap-961	632	29	,	,	PUNCT
ap-961	632	30	q	q	NOUN
ap-961	632	31	�	�	PROPN
ap-961	632	32	�	�	PROPN
ap-961	632	33	,	,	PUNCT
ap-961	632	34	©	©	PROPN
ap-961	632	35	czech	czech	PROPN
ap-961	632	36	technical	technical	PROPN
ap-961	632	37	university	university	PROPN
ap-961	632	38	publishing	publishing	NOUN
ap-961	632	39	house	house	NOUN
ap-961	632	40	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	632	41	49	49	NUM
ap-961	632	42	acta	acta	PROPN
ap-961	632	43	polytechnica	polytechnica	PROPN
ap-961	632	44	vol	vol	NOUN
ap-961	632	45	.	.	PUNCT
ap-961	633	1	48	48	NUM
ap-961	633	2	no	no	NOUN
ap-961	633	3	.	.	PUNCT
ap-961	634	1	2/2008	2/2008	NUM
ap-961	634	2	where	where	SCONJ
ap-961	634	3	s	s	X
ap-961	634	4	(	(	PUNCT
ap-961	634	5	)	)	PUNCT
ap-961	634	6	�	�	PROPN
ap-961	634	7	denotes	denote	VERB
ap-961	634	8	the	the	DET
ap-961	634	9	space	space	NOUN
ap-961	634	10	of	of	ADP
ap-961	634	11	schwartz	schwartz	PROPN
ap-961	634	12	functions	function	NOUN
ap-961	634	13	(	(	PUNCT
ap-961	634	14	rapidly	rapidly	ADV
ap-961	634	15	decreasing	decrease	VERB
ap-961	634	16	functions	function	NOUN
ap-961	634	17	on	on	ADP
ap-961	634	18	�	�	NOUN
ap-961	634	19	)	)	PUNCT
ap-961	634	20	with	with	ADP
ap-961	634	21	the	the	DET
ap-961	634	22	module	module	NOUN
ap-961	634	23	structure	structure	NOUN
ap-961	634	24	:	:	PUNCT
ap-961	634	25	u	u	NOUN
ap-961	634	26	s	s	VERB
ap-961	634	27	v	v	ADP
ap-961	634	28	e	e	X
ap-961	634	29	s	s	PROPN
ap-961	634	30	w	w	PROPN
ap-961	634	31	vis	vis	X
ap-961	634	32	u	u	NOUN
ap-961	634	33	(	(	PUNCT
ap-961	634	34	(	(	PUNCT
ap-961	634	35	)	)	PUNCT
ap-961	634	36	)	)	PUNCT
ap-961	635	1	(	(	PUNCT
ap-961	635	2	)	)	PUNCT
ap-961	635	3	(	(	PUNCT
ap-961	635	4	)	)	PUNCT
ap-961	635	5	�	�	PROPN
ap-961	635	6	�	�	PROPN
ap-961	635	7	�	�	PROPN
ap-961	635	8	�	�	PROPN
ap-961	635	9	�	�	PROPN
ap-961	635	10	�	�	PROPN
ap-961	635	11	2	2	NUM
ap-961	635	12	,	,	PUNCT
ap-961	635	13	v	v	PRON
ap-961	635	14	s	s	NOUN
ap-961	635	15	v	v	NOUN
ap-961	635	16	s	s	X
ap-961	635	17	p	p	NOUN
ap-961	635	18	q	q	PROPN
ap-961	635	19	w	w	NOUN
ap-961	635	20	vv	vv	PROPN
ap-961	635	21	(	(	PUNCT
ap-961	635	22	(	(	PUNCT
ap-961	635	23	)	)	PUNCT
ap-961	635	24	)	)	PUNCT
ap-961	635	25	(	(	PUNCT
ap-961	635	26	)	)	PUNCT
ap-961	635	27	�	�	PROPN
ap-961	635	28	�	�	PROPN
ap-961	635	29	�	�	PROPN
ap-961	635	30	�	�	PROPN
ap-961	635	31	�	�	PROPN
ap-961	635	32	�	�	PROPN
ap-961	635	33	�	�	PROPN
ap-961	635	34	�	�	PROPN
ap-961	635	35	�	�	PROPN
ap-961	635	36	�	�	PROPN
ap-961	635	37	�	�	PROPN
ap-961	635	38	�	�	PROPN
ap-961	635	39	�	�	PROPN
ap-961	635	40	�	�	PROPN
ap-961	635	41	�	�	PROPN
ap-961	635	42	,	,	PUNCT
ap-961	635	43	where	where	SCONJ
ap-961	635	44	wu	wu	PROPN
ap-961	635	45	,	,	PUNCT
ap-961	635	46	wv	wv	PROPN
ap-961	635	47	are	be	AUX
ap-961	635	48	matrices	matrix	NOUN
ap-961	635	49	satisfying	satisfy	VERB
ap-961	635	50	w	w	PROPN
ap-961	635	51	w	w	PROPN
ap-961	635	52	e	e	PROPN
ap-961	635	53	w	w	PROPN
ap-961	635	54	wu	wu	PROPN
ap-961	636	1	v	v	INTJ
ap-961	636	2	i	i	PRON
ap-961	636	3	v	v	ADP
ap-961	636	4	u	u	PRON
ap-961	636	5	p	p	PROPN
ap-961	636	6	q	q	PROPN
ap-961	636	7	�	�	PROPN
ap-961	636	8	2	2	NUM
ap-961	636	9	�	�	PROPN
ap-961	636	10	.	.	PUNCT
ap-961	637	1	it	it	PRON
ap-961	637	2	is	be	AUX
ap-961	637	3	not	not	PART
ap-961	637	4	evident	evident	ADJ
ap-961	637	5	that	that	SCONJ
ap-961	637	6	these	these	DET
ap-961	637	7	modules	module	NOUN
ap-961	637	8	are	be	AUX
ap-961	637	9	projective	projective	ADJ
ap-961	637	10	and	and	CCONJ
ap-961	637	11	that	that	SCONJ
ap-961	637	12	every	every	DET
ap-961	637	13	projective	projective	ADJ
ap-961	637	14	module	module	NOUN
ap-961	637	15	over	over	ADP
ap-961	637	16	the	the	DET
ap-961	637	17	nc	nc	PROPN
ap-961	637	18	torus	torus	PROPN
ap-961	637	19	is	be	AUX
ap-961	637	20	either	either	CCONJ
ap-961	637	21	free	free	ADJ
ap-961	637	22	or	or	CCONJ
ap-961	637	23	is	be	AUX
ap-961	637	24	of	of	ADP
ap-961	637	25	this	this	DET
ap-961	637	26	form	form	NOUN
ap-961	637	27	.	.	PUNCT
ap-961	638	1	details	detail	NOUN
ap-961	638	2	are	be	AUX
ap-961	638	3	to	to	PART
ap-961	638	4	be	be	AUX
ap-961	638	5	found	find	VERB
ap-961	638	6	in	in	ADP
ap-961	638	7	papers	paper	NOUN
ap-961	638	8	by	by	ADP
ap-961	638	9	rieffel	rieffel	NOUN
ap-961	638	10	and	and	CCONJ
ap-961	638	11	connes	conne	NOUN
ap-961	638	12	[	[	X
ap-961	638	13	24	24	NUM
ap-961	638	14	]	]	PUNCT
ap-961	638	15	.	.	PUNCT
ap-961	639	1	here	here	ADV
ap-961	639	2	we	we	PRON
ap-961	639	3	shall	shall	AUX
ap-961	639	4	just	just	ADV
ap-961	639	5	use	use	VERB
ap-961	639	6	this	this	DET
ap-961	639	7	knowledge	knowledge	NOUN
ap-961	639	8	to	to	PART
ap-961	639	9	find	find	VERB
ap-961	639	10	the	the	DET
ap-961	639	11	pairing	pairing	NOUN
ap-961	639	12	between	between	ADP
ap-961	639	13	the	the	DET
ap-961	639	14	standard	standard	ADJ
ap-961	639	15	two	two	NUM
ap-961	639	16	-	-	PUNCT
ap-961	639	17	cyclic	cyclic	NOUN
ap-961	639	18	cocycle	cocycle	NOUN
ap-961	639	19	over	over	ADP
ap-961	639	20	the	the	DET
ap-961	639	21	noncommutative	noncommutative	ADJ
ap-961	639	22	torus	torus	NOUN
ap-961	639	23	and	and	CCONJ
ap-961	639	24	the	the	DET
ap-961	639	25	module	module	NOUN
ap-961	639	26	.	.	PUNCT
ap-961	640	1	the	the	DET
ap-961	640	2	drawback	drawback	NOUN
ap-961	640	3	is	be	AUX
ap-961	640	4	that	that	SCONJ
ap-961	640	5	we	we	PRON
ap-961	640	6	do	do	AUX
ap-961	640	7	not	not	PART
ap-961	640	8	know	know	VERB
ap-961	640	9	the	the	DET
ap-961	640	10	projection	projection	NOUN
ap-961	640	11	–	–	PUNCT
ap-961	640	12	and	and	CCONJ
ap-961	640	13	although	although	SCONJ
ap-961	640	14	one	one	PRON
ap-961	640	15	can	can	AUX
ap-961	640	16	construct	construct	VERB
ap-961	640	17	it	it	PRON
ap-961	640	18	(	(	PUNCT
ap-961	640	19	surprisingly	surprisingly	ADV
ap-961	640	20	enough	enough	ADV
ap-961	640	21	it	it	PRON
ap-961	640	22	is	be	AUX
ap-961	640	23	a	a	DET
ap-961	640	24	projection	projection	NOUN
ap-961	640	25	in	in	ADP
ap-961	640	26	the	the	DET
ap-961	640	27	algebra	algebra	NOUN
ap-961	640	28	of	of	ADP
ap-961	640	29	the	the	DET
ap-961	640	30	noncommutative	noncommutative	ADJ
ap-961	640	31	torus	torus	NOUN
ap-961	640	32	itself	itself	PRON
ap-961	640	33	!	!	PUNCT
ap-961	640	34	)	)	PUNCT
ap-961	641	1	–	–	PUNCT
ap-961	641	2	it	it	PRON
ap-961	641	3	is	be	AUX
ap-961	641	4	sufficiently	sufficiently	ADV
ap-961	641	5	complicated	complicated	ADJ
ap-961	641	6	to	to	PART
ap-961	641	7	make	make	VERB
ap-961	641	8	this	this	DET
ap-961	641	9	form	form	NOUN
ap-961	641	10	of	of	ADP
ap-961	641	11	approach	approach	NOUN
ap-961	641	12	rather	rather	ADV
ap-961	641	13	difficult	difficult	ADJ
ap-961	641	14	.	.	PUNCT
ap-961	642	1	in	in	ADP
ap-961	642	2	turn	turn	NOUN
ap-961	642	3	we	we	PRON
ap-961	642	4	shall	shall	AUX
ap-961	642	5	construct	construct	VERB
ap-961	642	6	the	the	DET
ap-961	642	7	connections	connection	NOUN
ap-961	642	8	and	and	CCONJ
ap-961	642	9	curvature	curvature	VERB
ap-961	642	10	over	over	ADP
ap-961	642	11	the	the	DET
ap-961	642	12	module	module	NOUN
ap-961	642	13	.	.	PUNCT
ap-961	643	1	then	then	ADV
ap-961	643	2	using	use	VERB
ap-961	643	3	a	a	DET
ap-961	643	4	function	function	NOUN
ap-961	643	5	of	of	ADP
ap-961	643	6	the	the	DET
ap-961	643	7	curvature	curvature	NOUN
ap-961	643	8	(	(	PUNCT
ap-961	643	9	like	like	INTJ
ap-961	643	10	in	in	ADP
ap-961	643	11	the	the	DET
ap-961	643	12	case	case	NOUN
ap-961	643	13	of	of	ADP
ap-961	643	14	characteristic	characteristic	ADJ
ap-961	643	15	classes	class	NOUN
ap-961	643	16	)	)	PUNCT
ap-961	643	17	we	we	PRON
ap-961	643	18	shall	shall	AUX
ap-961	643	19	recover	recover	VERB
ap-961	643	20	the	the	DET
ap-961	643	21	pairing	pairing	NOUN
ap-961	643	22	.	.	PUNCT
ap-961	644	1	as	as	SCONJ
ap-961	644	2	the	the	DET
ap-961	644	3	connection	connection	NOUN
ap-961	644	4	on	on	ADP
ap-961	644	5	the	the	DET
ap-961	644	6	module	module	NOUN
ap-961	644	7	we	we	PRON
ap-961	644	8	take	take	VERB
ap-961	644	9	:	:	PUNCT
ap-961	644	10	(	(	PUNCT
ap-961	644	11	)	)	PUNCT
ap-961	644	12	(	(	PUNCT
ap-961	644	13	)	)	PUNCT
ap-961	644	14	(	(	PUNCT
ap-961	644	15	)	)	PUNCT
ap-961	644	16	(	(	PUNCT
ap-961	644	17	)	)	PUNCT
ap-961	644	18	1	1	NUM
ap-961	644	19	�	�	PROPN
ap-961	644	20	�	�	PROPN
ap-961	644	21	�	�	PROPN
ap-961	644	22	�	�	PROPN
ap-961	644	23	�	�	PROPN
ap-961	644	24	�	�	PROPN
ap-961	644	25	�	�	PROPN
ap-961	644	26	�	�	PROPN
ap-961	644	27	�	�	PROPN
ap-961	644	28	�	�	PROPN
ap-961	644	29	�	�	PROPN
ap-961	644	30	s	s	PART
ap-961	644	31	i	i	NOUN
ap-961	644	32	sq	sq	PROPN
ap-961	644	33	p	p	X
ap-961	644	34	q	q	PROPN
ap-961	644	35	s	s	PROPN
ap-961	644	36	d	d	NOUN
ap-961	644	37	s	s	VERB
ap-961	644	38	dsu	dsu	PROPN
ap-961	644	39	v2	v2	PROPN
ap-961	644	40	,	,	PUNCT
ap-961	644	41	where	where	SCONJ
ap-961	644	42	we	we	PRON
ap-961	644	43	use	use	VERB
ap-961	644	44	the	the	DET
ap-961	644	45	forms	form	NOUN
ap-961	644	46	�	�	NOUN
ap-961	644	47	u	u	NOUN
ap-961	644	48	,	,	PUNCT
ap-961	644	49	�	�	PROPN
ap-961	644	50	v	v	NOUN
ap-961	644	51	introduced	introduce	VERB
ap-961	644	52	earlier	early	ADV
ap-961	644	53	.	.	PUNCT
ap-961	645	1	the	the	DET
ap-961	645	2	verification	verification	NOUN
ap-961	645	3	that1	that1	NOUN
ap-961	645	4	is	be	AUX
ap-961	645	5	the	the	DET
ap-961	645	6	connection	connection	NOUN
ap-961	645	7	is	be	AUX
ap-961	645	8	explicit	explicit	ADJ
ap-961	645	9	.	.	PUNCT
ap-961	646	1	the	the	DET
ap-961	646	2	curvature	curvature	NOUN
ap-961	646	3	is	be	AUX
ap-961	646	4	f	f	PROPN
ap-961	646	5	iq	iq	NOUN
ap-961	646	6	p	p	NOUN
ap-961	646	7	q	q	PROPN
ap-961	646	8	u	u	PROPN
ap-961	646	9	v	v	PROPN
ap-961	646	10	�	�	PROPN
ap-961	646	11	�	�	PROPN
ap-961	646	12	�	�	PROPN
ap-961	646	13	#	#	PROPN
ap-961	646	14	2	2	NUM
ap-961	646	15	�	�	PROPN
ap-961	646	16	�	�	PROPN
ap-961	646	17	�	�	PROPN
ap-961	646	18	�	�	PROPN
ap-961	646	19	.	.	PUNCT
ap-961	647	1	the	the	DET
ap-961	647	2	pairing	pairing	NOUN
ap-961	647	3	,	,	PUNCT
ap-961	647	4	in	in	ADP
ap-961	647	5	this	this	DET
ap-961	647	6	two	two	NUM
ap-961	647	7	-	-	PUNCT
ap-961	647	8	dimensional	dimensional	ADJ
ap-961	647	9	case	case	NOUN
ap-961	647	10	,	,	PUNCT
ap-961	647	11	is	be	AUX
ap-961	647	12	given	give	VERB
ap-961	647	13	by	by	ADP
ap-961	647	14	applying	apply	VERB
ap-961	647	15	the	the	DET
ap-961	647	16	closed	close	VERB
ap-961	647	17	graded	grade	VERB
ap-961	647	18	trace	trace	NOUN
ap-961	647	19	in	in	ADP
ap-961	647	20	the	the	DET
ap-961	647	21	differential	differential	ADJ
ap-961	647	22	algebra	algebra	NOUN
ap-961	647	23	to	to	ADP
ap-961	647	24	the	the	DET
ap-961	647	25	trace	trace	NOUN
ap-961	647	26	of	of	ADP
ap-961	647	27	the	the	DET
ap-961	647	28	curvature	curvature	NOUN
ap-961	647	29	two	two	NUM
ap-961	647	30	-	-	PUNCT
ap-961	647	31	form	form	NOUN
ap-961	647	32	(	(	PUNCT
ap-961	647	33	remember	remember	VERB
ap-961	647	34	that	that	SCONJ
ap-961	647	35	the	the	DET
ap-961	647	36	curvature	curvature	NOUN
ap-961	647	37	is	be	AUX
ap-961	647	38	a	a	DET
ap-961	647	39	two	two	NUM
ap-961	647	40	-	-	PUNCT
ap-961	647	41	form	form	NOUN
ap-961	647	42	but	but	CCONJ
ap-961	647	43	with	with	ADP
ap-961	647	44	values	value	NOUN
ap-961	647	45	in	in	ADP
ap-961	647	46	the	the	DET
ap-961	647	47	endomorphism	endomorphism	NOUN
ap-961	647	48	of	of	ADP
ap-961	647	49	the	the	DET
ap-961	647	50	projective	projective	ADJ
ap-961	647	51	module	module	NOUN
ap-961	647	52	):	):	PUNCT
ap-961	647	53	1	1	NUM
ap-961	647	54	2	2	NUM
ap-961	647	55	2	2	NUM
ap-961	647	56	�	�	PROPN
ap-961	647	57	�	�	PROPN
ap-961	647	58	�	�	PROPN
ap-961	647	59	i	i	PRON
ap-961	647	60	iq	iq	VERB
ap-961	647	61	p	p	X
ap-961	647	62	q	q	PROPN
ap-961	647	63	i	i	PROPN
ap-961	647	64	d	d	PROPN
ap-961	647	65	�	�	PROPN
ap-961	647	66	tr	tr	VERB
ap-961	647	67	(	(	PUNCT
ap-961	647	68	)	)	PUNCT
ap-961	647	69	�	�	PROPN
ap-961	647	70	,	,	PUNCT
ap-961	647	71	and	and	CCONJ
ap-961	647	72	since	since	SCONJ
ap-961	647	73	:	:	PUNCT
ap-961	647	74	tr	tr	VERB
ap-961	647	75	(	(	PUNCT
ap-961	647	76	)	)	PUNCT
ap-961	647	77	i	i	PROPN
ap-961	647	78	d	d	PROPN
ap-961	647	79	p	p	X
ap-961	647	80	q	q	PROPN
ap-961	647	81	�	�	PROPN
ap-961	647	82	�	�	PROPN
ap-961	647	83	�	�	PROPN
ap-961	647	84	�	�	PROPN
ap-961	647	85	,	,	PUNCT
ap-961	647	86	we	we	PRON
ap-961	647	87	get	get	VERB
ap-961	647	88	the	the	DET
ap-961	647	89	integer	integer	NOUN
ap-961	647	90	q	q	PROPN
ap-961	647	91	as	as	ADP
ap-961	647	92	the	the	DET
ap-961	647	93	value	value	NOUN
ap-961	647	94	of	of	ADP
ap-961	647	95	the	the	DET
ap-961	647	96	pairing	pairing	NOUN
ap-961	647	97	.	.	PUNCT
ap-961	648	1	here	here	ADV
ap-961	648	2	we	we	PRON
ap-961	648	3	smuggled	smuggle	VERB
ap-961	648	4	in	in	ADP
ap-961	648	5	the	the	DET
ap-961	648	6	information	information	NOUN
ap-961	648	7	about	about	ADP
ap-961	648	8	the	the	DET
ap-961	648	9	dimension	dimension	NOUN
ap-961	648	10	of	of	ADP
ap-961	648	11	the	the	DET
ap-961	648	12	projective	projective	ADJ
ap-961	648	13	module	module	NOUN
ap-961	648	14	(	(	PUNCT
ap-961	648	15	or	or	CCONJ
ap-961	648	16	whatever	whatever	PRON
ap-961	648	17	we	we	PRON
ap-961	648	18	might	might	AUX
ap-961	648	19	call	call	VERB
ap-961	648	20	it	it	PRON
ap-961	648	21	)	)	PUNCT
ap-961	648	22	–	–	PUNCT
ap-961	648	23	which	which	PRON
ap-961	648	24	could	could	AUX
ap-961	648	25	be	be	AUX
ap-961	648	26	defined	define	VERB
ap-961	648	27	as	as	ADP
ap-961	648	28	the	the	DET
ap-961	648	29	trace	trace	NOUN
ap-961	648	30	of	of	ADP
ap-961	648	31	the	the	DET
ap-961	648	32	identity	identity	NOUN
ap-961	648	33	endomorphism	endomorphism	NOUN
ap-961	648	34	.	.	PUNCT
ap-961	649	1	in	in	ADP
ap-961	649	2	fact	fact	NOUN
ap-961	649	3	,	,	PUNCT
ap-961	649	4	one	one	PRON
ap-961	649	5	may	may	AUX
ap-961	649	6	use	use	VERB
ap-961	649	7	the	the	DET
ap-961	649	8	pairing	pairing	NOUN
ap-961	649	9	and	and	CCONJ
ap-961	649	10	some	some	DET
ap-961	649	11	facts	fact	NOUN
ap-961	649	12	about	about	ADP
ap-961	649	13	its	its	PRON
ap-961	649	14	integrality	integrality	NOUN
ap-961	649	15	(	(	PUNCT
ap-961	649	16	which	which	PRON
ap-961	649	17	we	we	PRON
ap-961	649	18	do	do	AUX
ap-961	649	19	not	not	PART
ap-961	649	20	mention	mention	VERB
ap-961	649	21	in	in	ADP
ap-961	649	22	these	these	DET
ap-961	649	23	notes	note	NOUN
ap-961	649	24	)	)	PUNCT
ap-961	649	25	to	to	PART
ap-961	649	26	get	get	VERB
ap-961	649	27	this	this	DET
ap-961	649	28	value	value	NOUN
ap-961	649	29	.	.	PUNCT
ap-961	650	1	nevertheless	nevertheless	ADV
ap-961	650	2	,	,	PUNCT
ap-961	650	3	although	although	SCONJ
ap-961	650	4	we	we	PRON
ap-961	650	5	are	be	AUX
ap-961	650	6	dealing	deal	VERB
ap-961	650	7	with	with	ADP
ap-961	650	8	a	a	DET
ap-961	650	9	very	very	ADV
ap-961	650	10	strange	strange	ADJ
ap-961	650	11	family	family	NOUN
ap-961	650	12	of	of	ADP
ap-961	650	13	projective	projective	ADJ
ap-961	650	14	modules	module	NOUN
ap-961	650	15	in	in	ADP
ap-961	650	16	a	a	DET
ap-961	650	17	pure	pure	ADJ
ap-961	650	18	noncommutative	noncommutative	ADJ
ap-961	650	19	setup	setup	NOUN
ap-961	650	20	,	,	PUNCT
ap-961	650	21	we	we	PRON
ap-961	650	22	can	can	AUX
ap-961	650	23	still	still	ADV
ap-961	650	24	say	say	VERB
ap-961	650	25	that	that	SCONJ
ap-961	650	26	we	we	PRON
ap-961	650	27	can	can	AUX
ap-961	650	28	distinguish	distinguish	VERB
ap-961	650	29	between	between	ADP
ap-961	650	30	those	those	PRON
ap-961	650	31	which	which	PRON
ap-961	650	32	are	be	AUX
ap-961	650	33	not	not	PART
ap-961	650	34	equivalent	equivalent	ADJ
ap-961	650	35	to	to	ADP
ap-961	650	36	each	each	DET
ap-961	650	37	other	other	ADJ
ap-961	650	38	with	with	ADP
ap-961	650	39	the	the	DET
ap-961	650	40	help	help	NOUN
ap-961	650	41	of	of	ADP
ap-961	650	42	the	the	DET
ap-961	650	43	chern	chern	ADJ
ap-961	650	44	-	-	PUNCT
ap-961	650	45	connes	conne	NOUN
ap-961	650	46	pairing	pairing	NOUN
ap-961	650	47	.	.	PUNCT
ap-961	651	1	5.2	5.2	NUM
ap-961	651	2	summary	summary	NOUN
ap-961	651	3	we	we	PRON
ap-961	651	4	may	may	AUX
ap-961	651	5	now	now	ADV
ap-961	651	6	summarize	summarize	VERB
ap-961	651	7	all	all	DET
ap-961	651	8	the	the	DET
ap-961	651	9	tools	tool	NOUN
ap-961	651	10	and	and	CCONJ
ap-961	651	11	construction	construction	NOUN
ap-961	651	12	that	that	PRON
ap-961	651	13	we	we	PRON
ap-961	651	14	have	have	AUX
ap-961	651	15	learned	learn	VERB
ap-961	651	16	in	in	ADP
ap-961	651	17	this	this	DET
ap-961	651	18	crash	crash	NOUN
ap-961	651	19	course	course	NOUN
ap-961	651	20	on	on	ADP
ap-961	651	21	noncommutative	noncommutative	ADJ
ap-961	651	22	geometry	geometry	NOUN
ap-961	651	23	.	.	PUNCT
ap-961	652	1	we	we	PRON
ap-961	652	2	just	just	ADV
ap-961	652	3	extend	extend	VERB
ap-961	652	4	the	the	DET
ap-961	652	5	dictionary	dictionary	NOUN
ap-961	652	6	,	,	PUNCT
ap-961	652	7	which	which	PRON
ap-961	652	8	we	we	PRON
ap-961	652	9	constructed	construct	VERB
ap-961	652	10	first	first	ADV
ap-961	652	11	for	for	ADP
ap-961	652	12	noncommutative	noncommutative	ADJ
ap-961	652	13	topology	topology	NOUN
ap-961	652	14	:	:	PUNCT
ap-961	652	15	geometry	geometry	NOUN
ap-961	652	16	algebra	algebra	NOUN
ap-961	652	17	vector	vector	NOUN
ap-961	652	18	bundles	bundle	NOUN
ap-961	652	19	finitely	finitely	ADV
ap-961	652	20	generated	generate	VERB
ap-961	652	21	projective	projective	ADJ
ap-961	652	22	modules	module	NOUN
ap-961	652	23	differential	differential	NOUN
ap-961	652	24	forms	form	NOUN
ap-961	652	25	differential	differential	VERB
ap-961	652	26	graded	grade	VERB
ap-961	652	27	algebra	algebra	NOUN
ap-961	652	28	integration	integration	NOUN
ap-961	652	29	of	of	ADP
ap-961	652	30	differential	differential	ADJ
ap-961	652	31	forms	form	NOUN
ap-961	652	32	closed	close	VERB
ap-961	652	33	graded	grade	VERB
ap-961	652	34	trace	trace	NOUN
ap-961	652	35	on	on	ADP
ap-961	652	36	dga	dga	NOUN
ap-961	652	37	simplicial	simplicial	NOUN
ap-961	652	38	(	(	PUNCT
ap-961	652	39	de	de	X
ap-961	652	40	rham	rham	PROPN
ap-961	652	41	)	)	PUNCT
ap-961	652	42	homology	homology	NOUN
ap-961	652	43	cyclic	cyclic	ADJ
ap-961	652	44	homology	homology	NOUN
ap-961	652	45	infinitesimals	infinitesimal	VERB
ap-961	652	46	compact	compact	ADJ
ap-961	652	47	operators	operator	NOUN
ap-961	652	48	integral	integral	ADJ
ap-961	652	49	trace	trace	NOUN
ap-961	652	50	(	(	PUNCT
ap-961	652	51	exotic	exotic	ADJ
ap-961	652	52	trace	trace	NOUN
ap-961	652	53	)	)	PUNCT
ap-961	652	54	connection	connection	NOUN
ap-961	652	55	on	on	ADP
ap-961	652	56	a	a	DET
ap-961	652	57	vector	vector	NOUN
ap-961	652	58	bundle	bundle	NOUN
ap-961	652	59	connection	connection	NOUN
ap-961	652	60	on	on	ADP
ap-961	652	61	a	a	DET
ap-961	652	62	projective	projective	ADJ
ap-961	652	63	module	module	NOUN
ap-961	652	64	characteristic	characteristic	ADJ
ap-961	652	65	classes	class	NOUN
ap-961	652	66	chern	chern	ADJ
ap-961	652	67	-	-	PUNCT
ap-961	652	68	connes	conne	NOUN
ap-961	652	69	character	character	NOUN
ap-961	652	70	6	6	NUM
ap-961	652	71	spectral	spectral	ADJ
ap-961	652	72	geometry	geometry	NOUN
ap-961	652	73	and	and	CCONJ
ap-961	652	74	its	its	PRON
ap-961	652	75	applications	application	NOUN
ap-961	652	76	ubi	ubi	NOUN
ap-961	652	77	materia	materia	PROPN
ap-961	652	78	,	,	PUNCT
ap-961	652	79	ibi	ibi	PROPN
ap-961	652	80	geometria	geometria	PROPN
ap-961	652	81	.	.	PUNCT
ap-961	653	1	(	(	PUNCT
ap-961	653	2	where	where	SCONJ
ap-961	653	3	there	there	PRON
ap-961	653	4	is	be	VERB
ap-961	653	5	matter	matter	NOUN
ap-961	653	6	,	,	PUNCT
ap-961	653	7	there	there	PRON
ap-961	653	8	is	be	VERB
ap-961	653	9	geometry	geometry	NOUN
ap-961	653	10	.	.	PUNCT
ap-961	653	11	)	)	PUNCT
ap-961	654	1	(	(	PUNCT
ap-961	654	2	johannes	johannes	PROPN
ap-961	654	3	kepler	kepler	PROPN
ap-961	654	4	)	)	PUNCT
ap-961	654	5	in	in	ADP
ap-961	654	6	this	this	DET
ap-961	654	7	last	last	ADJ
ap-961	654	8	part	part	NOUN
ap-961	654	9	of	of	ADP
ap-961	654	10	the	the	DET
ap-961	654	11	lectures	lecture	NOUN
ap-961	654	12	we	we	PRON
ap-961	654	13	shall	shall	AUX
ap-961	654	14	use	use	VERB
ap-961	654	15	(	(	PUNCT
ap-961	654	16	and	and	CCONJ
ap-961	654	17	probably	probably	ADV
ap-961	654	18	overuse	overuse	NOUN
ap-961	654	19	)	)	PUNCT
ap-961	654	20	the	the	DET
ap-961	654	21	word	word	NOUN
ap-961	654	22	spectral	spectral	ADJ
ap-961	654	23	.	.	PUNCT
ap-961	655	1	its	its	PRON
ap-961	655	2	sense	sense	NOUN
ap-961	655	3	will	will	AUX
ap-961	655	4	be	be	AUX
ap-961	655	5	described	describe	VERB
ap-961	655	6	in	in	ADP
ap-961	655	7	the	the	DET
ap-961	655	8	definition	definition	NOUN
ap-961	655	9	of	of	ADP
ap-961	655	10	properties	property	NOUN
ap-961	655	11	of	of	ADP
ap-961	655	12	spectral	spectral	ADJ
ap-961	655	13	triples	triple	NOUN
ap-961	655	14	–	–	PUNCT
ap-961	655	15	a	a	DET
ap-961	655	16	concise	concise	ADJ
ap-961	655	17	proposition	proposition	NOUN
ap-961	655	18	for	for	ADP
ap-961	655	19	noncommutative	noncommutative	ADJ
ap-961	655	20	spin	spin	NOUN
ap-961	655	21	manifolds	manifold	NOUN
ap-961	655	22	.	.	PUNCT
ap-961	656	1	the	the	DET
ap-961	656	2	clue	clue	NOUN
ap-961	656	3	is	be	AUX
ap-961	656	4	that	that	SCONJ
ap-961	656	5	(	(	PUNCT
ap-961	656	6	almost	almost	ADV
ap-961	656	7	)	)	PUNCT
ap-961	656	8	everything	everything	PRON
ap-961	656	9	is	be	AUX
ap-961	656	10	set	set	VERB
ap-961	656	11	by	by	ADP
ap-961	656	12	the	the	DET
ap-961	656	13	dirac	dirac	NOUN
ap-961	656	14	operator	operator	NOUN
ap-961	656	15	and	and	CCONJ
ap-961	656	16	it	it	PRON
ap-961	656	17	,	,	PUNCT
ap-961	656	18	in	in	ADP
ap-961	656	19	turn	turn	NOUN
ap-961	656	20	,	,	PUNCT
ap-961	656	21	is	be	AUX
ap-961	656	22	defined	define	VERB
ap-961	656	23	through	through	ADP
ap-961	656	24	its	its	PRON
ap-961	656	25	set	set	NOUN
ap-961	656	26	of	of	ADP
ap-961	656	27	discrete	discrete	ADJ
ap-961	656	28	eigenvalues	eigenvalue	NOUN
ap-961	656	29	with	with	ADP
ap-961	656	30	multiplicities	multiplicity	NOUN
ap-961	656	31	.	.	PUNCT
ap-961	657	1	we	we	PRON
ap-961	657	2	briefly	briefly	ADV
ap-961	657	3	touch	touch	VERB
ap-961	657	4	the	the	DET
ap-961	657	5	main	main	ADJ
ap-961	657	6	proposition	proposition	NOUN
ap-961	657	7	,	,	PUNCT
ap-961	657	8	which	which	PRON
ap-961	657	9	links	link	VERB
ap-961	657	10	the	the	DET
ap-961	657	11	theory	theory	NOUN
ap-961	657	12	with	with	ADP
ap-961	657	13	physics	physics	PROPN
ap-961	657	14	:	:	PUNCT
ap-961	657	15	the	the	DET
ap-961	657	16	construction	construction	NOUN
ap-961	657	17	of	of	ADP
ap-961	657	18	gauge	gauge	NOUN
ap-961	657	19	theories	theory	NOUN
ap-961	657	20	and	and	CCONJ
ap-961	657	21	the	the	DET
ap-961	657	22	spectral	spectral	ADJ
ap-961	657	23	action	action	NOUN
ap-961	657	24	principle	principle	NOUN
ap-961	657	25	.	.	PUNCT
ap-961	658	1	6.1	6.1	NUM
ap-961	658	2	enter	enter	VERB
ap-961	658	3	:	:	PUNCT
ap-961	658	4	the	the	DET
ap-961	658	5	dirac	dirac	NOUN
ap-961	658	6	operator	operator	NOUN
ap-961	658	7	in	in	ADP
ap-961	658	8	the	the	DET
ap-961	658	9	example	example	NOUN
ap-961	658	10	of	of	ADP
ap-961	658	11	the	the	DET
ap-961	658	12	differential	differential	NOUN
ap-961	658	13	graded	grade	VERB
ap-961	658	14	algebra	algebra	NOUN
ap-961	658	15	on	on	ADP
ap-961	658	16	the	the	DET
ap-961	658	17	circle	circle	NOUN
ap-961	658	18	we	we	PRON
ap-961	658	19	tested	test	VERB
ap-961	658	20	a	a	DET
ap-961	658	21	peculiar	peculiar	ADJ
ap-961	658	22	unbounded	unbounded	ADJ
ap-961	658	23	operator	operator	NOUN
ap-961	659	1	d	d	NOUN
ap-961	659	2	(	(	PUNCT
ap-961	659	3	example	example	NOUN
ap-961	659	4	3.18	3.18	NUM
ap-961	659	5	)	)	PUNCT
ap-961	659	6	.	.	PUNCT
ap-961	660	1	of	of	ADP
ap-961	660	2	course	course	NOUN
ap-961	660	3	,	,	PUNCT
ap-961	660	4	this	this	PRON
ap-961	660	5	was	be	AUX
ap-961	660	6	not	not	PART
ap-961	660	7	a	a	DET
ap-961	660	8	coincidence	coincidence	NOUN
ap-961	660	9	and	and	CCONJ
ap-961	660	10	the	the	DET
ap-961	660	11	story	story	NOUN
ap-961	660	12	could	could	AUX
ap-961	660	13	have	have	AUX
ap-961	660	14	been	be	AUX
ap-961	660	15	repeated	repeat	VERB
ap-961	660	16	for	for	ADP
ap-961	660	17	any	any	DET
ap-961	660	18	compact	compact	ADJ
ap-961	660	19	spin	spin	NOUN
ap-961	660	20	manifold	manifold	ADJ
ap-961	660	21	.	.	PUNCT
ap-961	661	1	taking	take	VERB
ap-961	661	2	as	as	ADP
ap-961	661	3	the	the	DET
ap-961	661	4	hilbert	hilbert	PROPN
ap-961	661	5	space	space	NOUN
ap-961	661	6	the	the	DET
ap-961	661	7	square	square	ADJ
ap-961	661	8	-	-	PUNCT
ap-961	661	9	summable	summable	ADJ
ap-961	661	10	sections	section	NOUN
ap-961	661	11	of	of	ADP
ap-961	661	12	the	the	DET
ap-961	661	13	spinor	spinor	NOUN
ap-961	661	14	bundle	bundle	NOUN
ap-961	661	15	l2(m	l2(m	PROPN
ap-961	661	16	,	,	PUNCT
ap-961	661	17	s	s	PART
ap-961	661	18	)	)	PUNCT
ap-961	661	19	and	and	CCONJ
ap-961	661	20	d	d	NOUN
ap-961	661	21	as	as	ADP
ap-961	661	22	the	the	DET
ap-961	661	23	true	true	ADJ
ap-961	661	24	dirac	dirac	NOUN
ap-961	661	25	operator	operator	NOUN
ap-961	661	26	(	(	PUNCT
ap-961	661	27	for	for	ADP
ap-961	661	28	a	a	DET
ap-961	661	29	given	give	VERB
ap-961	661	30	riemannian	riemannian	ADJ
ap-961	661	31	metric	metric	NOUN
ap-961	661	32	)	)	PUNCT
ap-961	661	33	we	we	PRON
ap-961	661	34	shall	shall	AUX
ap-961	661	35	always	always	ADV
ap-961	661	36	(	(	PUNCT
ap-961	661	37	in	in	ADP
ap-961	661	38	a	a	DET
ap-961	661	39	similar	similar	ADJ
ap-961	661	40	manner	manner	NOUN
ap-961	661	41	)	)	PUNCT
ap-961	661	42	recover	recover	VERB
ap-961	661	43	the	the	DET
ap-961	661	44	de	de	X
ap-961	661	45	rham	rham	PROPN
ap-961	661	46	differential	differential	PROPN
ap-961	661	47	algebra	algebra	PROPN
ap-961	661	48	.	.	PUNCT
ap-961	662	1	the	the	DET
ap-961	662	2	dirac	dirac	NOUN
ap-961	662	3	operator	operator	NOUN
ap-961	662	4	on	on	ADP
ap-961	662	5	a	a	DET
ap-961	662	6	compact	compact	ADJ
ap-961	662	7	spin	spin	NOUN
ap-961	662	8	manifolds	manifold	NOUN
ap-961	662	9	is	be	AUX
ap-961	662	10	indeed	indeed	ADV
ap-961	662	11	a	a	DET
ap-961	662	12	very	very	ADV
ap-961	662	13	elegant	elegant	ADJ
ap-961	662	14	object	object	NOUN
ap-961	662	15	:	:	PUNCT
ap-961	662	16	an	an	DET
ap-961	662	17	unbounded	unbounded	ADJ
ap-961	662	18	,	,	PUNCT
ap-961	662	19	self	self	NOUN
ap-961	662	20	-	-	PUNCT
ap-961	662	21	adjoint	adjoint	NOUN
ap-961	662	22	operator	operator	NOUN
ap-961	662	23	,	,	PUNCT
ap-961	662	24	with	with	ADP
ap-961	662	25	a	a	DET
ap-961	662	26	discrete	discrete	ADJ
ap-961	662	27	spectrum	spectrum	NOUN
ap-961	662	28	and	and	CCONJ
ap-961	662	29	with	with	ADP
ap-961	662	30	the	the	DET
ap-961	662	31	growth	growth	NOUN
ap-961	662	32	of	of	ADP
ap-961	662	33	eigenvalues	eigenvalue	NOUN
ap-961	662	34	governed	govern	VERB
ap-961	662	35	by	by	ADP
ap-961	662	36	the	the	DET
ap-961	662	37	dimension	dimension	NOUN
ap-961	662	38	of	of	ADP
ap-961	662	39	the	the	DET
ap-961	662	40	manifold	manifold	NOUN
ap-961	662	41	.	.	PUNCT
ap-961	663	1	50	50	NUM
ap-961	663	2	©	©	PROPN
ap-961	663	3	czech	czech	PROPN
ap-961	663	4	technical	technical	PROPN
ap-961	663	5	university	university	PROPN
ap-961	663	6	publishing	publishing	NOUN
ap-961	663	7	house	house	NOUN
ap-961	663	8	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	663	9	acta	acta	PROPN
ap-961	663	10	polytechnica	polytechnica	PROPN
ap-961	663	11	vol	vol	NOUN
ap-961	663	12	.	.	PUNCT
ap-961	664	1	48	48	NUM
ap-961	664	2	no	no	NOUN
ap-961	664	3	.	.	PUNCT
ap-961	665	1	2/2008	2/2008	PROPN
ap-961	666	1	the	the	DET
ap-961	666	2	dirac	dirac	NOUN
ap-961	666	3	operator	operator	NOUN
ap-961	666	4	is	be	AUX
ap-961	666	5	also	also	ADV
ap-961	666	6	a	a	DET
ap-961	666	7	very	very	ADV
ap-961	666	8	useful	useful	ADJ
ap-961	666	9	tool	tool	NOUN
ap-961	666	10	:	:	PUNCT
ap-961	666	11	it	it	PRON
ap-961	666	12	encodes	encode	VERB
ap-961	666	13	a	a	DET
ap-961	666	14	lot	lot	NOUN
ap-961	666	15	of	of	ADP
ap-961	666	16	topological	topological	ADJ
ap-961	666	17	and	and	CCONJ
ap-961	666	18	geometrical	geometrical	ADJ
ap-961	666	19	information	information	NOUN
ap-961	666	20	about	about	ADP
ap-961	666	21	the	the	DET
ap-961	666	22	manifold	manifold	NOUN
ap-961	666	23	,	,	PUNCT
ap-961	666	24	in	in	ADP
ap-961	666	25	particular	particular	ADJ
ap-961	666	26	about	about	ADP
ap-961	666	27	the	the	DET
ap-961	666	28	differential	differential	ADJ
ap-961	666	29	algebra	algebra	NOUN
ap-961	666	30	and	and	CCONJ
ap-961	666	31	the	the	DET
ap-961	666	32	metric	metric	NOUN
ap-961	666	33	.	.	PUNCT
ap-961	667	1	we	we	PRON
ap-961	667	2	shall	shall	AUX
ap-961	667	3	mimic	mimic	VERB
ap-961	667	4	this	this	DET
ap-961	667	5	construction	construction	NOUN
ap-961	667	6	in	in	ADP
ap-961	667	7	the	the	DET
ap-961	667	8	noncommutative	noncommutative	ADJ
ap-961	667	9	world	world	NOUN
ap-961	667	10	,	,	PUNCT
ap-961	667	11	assuming	assume	VERB
ap-961	667	12	that	that	SCONJ
ap-961	667	13	it	it	PRON
ap-961	667	14	is	be	AUX
ap-961	667	15	the	the	DET
ap-961	667	16	basic	basic	ADJ
ap-961	667	17	data	datum	NOUN
ap-961	667	18	that	that	PRON
ap-961	667	19	makes	make	VERB
ap-961	667	20	noncommutative	noncommutative	ADJ
ap-961	667	21	geometry	geometry	NOUN
ap-961	667	22	the	the	DET
ap-961	667	23	geometry	geometry	NOUN
ap-961	667	24	.	.	PUNCT
ap-961	668	1	but	but	CCONJ
ap-961	668	2	to	to	PART
ap-961	668	3	do	do	VERB
ap-961	668	4	this	this	PRON
ap-961	668	5	we	we	PRON
ap-961	668	6	shall	shall	AUX
ap-961	668	7	learn	learn	VERB
ap-961	668	8	one	one	NUM
ap-961	668	9	more	more	ADJ
ap-961	668	10	tool	tool	NOUN
ap-961	668	11	:	:	PUNCT
ap-961	668	12	the	the	DET
ap-961	668	13	construction	construction	NOUN
ap-961	668	14	of	of	ADP
ap-961	668	15	exotic	exotic	ADJ
ap-961	668	16	traces	trace	NOUN
ap-961	668	17	.	.	PUNCT
ap-961	669	1	6.2	6.2	NUM
ap-961	669	2	exotic	exotic	ADJ
ap-961	669	3	traces	trace	NOUN
ap-961	669	4	and	and	CCONJ
ap-961	669	5	residues	residue	NOUN
ap-961	669	6	let	let	VERB
ap-961	669	7	us	we	PRON
ap-961	669	8	start	start	VERB
ap-961	669	9	with	with	ADP
ap-961	669	10	a	a	DET
ap-961	669	11	definition	definition	NOUN
ap-961	669	12	:	:	PUNCT
ap-961	669	13	definition	definition	NOUN
ap-961	669	14	6.1	6.1	NUM
ap-961	669	15	.	.	PUNCT
ap-961	670	1	assume	assume	VERB
ap-961	670	2	we	we	PRON
ap-961	670	3	have	have	VERB
ap-961	670	4	a	a	DET
ap-961	670	5	positive	positive	ADJ
ap-961	670	6	,	,	PUNCT
ap-961	670	7	compact	compact	ADJ
ap-961	670	8	operator	operator	NOUN
ap-961	670	9	t	t	PROPN
ap-961	670	10	on	on	ADP
ap-961	670	11	a	a	DET
ap-961	670	12	hilbert	hilbert	NOUN
ap-961	670	13	space	space	NOUN
ap-961	670	14	,	,	PUNCT
ap-961	670	15	with	with	ADP
ap-961	670	16	the	the	DET
ap-961	670	17	eigenvalues	eigenvalue	NOUN
ap-961	670	18	�	�	PROPN
ap-961	670	19	i	i	PRON
ap-961	670	20	t	t	PROPN
ap-961	670	21	(	(	PUNCT
ap-961	670	22	)	)	PUNCT
ap-961	670	23	,	,	PUNCT
ap-961	670	24	i	i	PRON
ap-961	670	25	�	�	VERB
ap-961	670	26	1	1	NUM
ap-961	670	27	2	2	NUM
ap-961	670	28	,	,	PUNCT
ap-961	670	29	,	,	PUNCT
ap-961	670	30	�	�	PROPN
ap-961	670	31	suppose	suppose	VERB
ap-961	670	32	that	that	SCONJ
ap-961	670	33	the	the	DET
ap-961	670	34	eigenvalues	eigenvalue	NOUN
ap-961	670	35	decrease	decrease	VERB
ap-961	670	36	to	to	ADP
ap-961	670	37	0	0	NUM
ap-961	670	38	so	so	ADV
ap-961	670	39	fast	fast	ADV
ap-961	670	40	that	that	SCONJ
ap-961	670	41	the	the	DET
ap-961	670	42	following	follow	VERB
ap-961	670	43	expression	expression	NOUN
ap-961	670	44	makes	make	VERB
ap-961	670	45	sense	sense	NOUN
ap-961	670	46	:	:	PUNCT
ap-961	670	47	tr	tr	VERB
ap-961	670	48	�	�	PROPN
ap-961	670	49	�	�	PROPN
ap-961	670	50	(	(	PUNCT
ap-961	670	51	)	)	PUNCT
ap-961	670	52	lim	lim	PROPN
ap-961	670	53	log	log	PROPN
ap-961	670	54	(	(	PUNCT
ap-961	670	55	)	)	PUNCT
ap-961	670	56	t	t	PROPN
ap-961	670	57	n	n	ADP
ap-961	670	58	t	t	PROPN
ap-961	671	1	n	n	CCONJ
ap-961	672	1	i	i	PRON
ap-961	672	2	i	i	PROPN
ap-961	672	3	n	n	CCONJ
ap-961	672	4	�	�	PROPN
ap-961	672	5	�	�	PROPN
ap-961	672	6	�	�	PROPN
ap-961	672	7	�	�	PROPN
ap-961	672	8	1	1	NUM
ap-961	672	9	1	1	NUM
ap-961	672	10	.	.	PUNCT
ap-961	673	1	if	if	SCONJ
ap-961	673	2	it	it	PRON
ap-961	673	3	exists	exist	VERB
ap-961	673	4	then	then	ADV
ap-961	673	5	we	we	PRON
ap-961	673	6	call	call	VERB
ap-961	673	7	it	it	PRON
ap-961	673	8	a	a	DET
ap-961	673	9	dixmier	dixmier	NOUN
ap-961	673	10	trace	trace	NOUN
ap-961	673	11	.	.	PUNCT
ap-961	674	1	example	example	NOUN
ap-961	674	2	6.2	6.2	NUM
ap-961	674	3	:	:	PUNCT
ap-961	674	4	let	let	VERB
ap-961	674	5	us	we	PRON
ap-961	674	6	take	take	VERB
ap-961	674	7	,	,	PUNCT
ap-961	674	8	for	for	ADP
ap-961	674	9	instance	instance	NOUN
ap-961	674	10	,	,	PUNCT
ap-961	674	11	the	the	DET
ap-961	674	12	operator	operator	NOUN
ap-961	674	13	(	(	PUNCT
ap-961	674	14	)	)	PUNCT
ap-961	674	15	d	d	NOUN
ap-961	674	16	�	�	PROPN
ap-961	674	17	1	1	NUM
ap-961	674	18	1	1	NUM
ap-961	674	19	,	,	PUNCT
ap-961	674	20	where	where	SCONJ
ap-961	674	21	d	d	NOUN
ap-961	674	22	is	be	AUX
ap-961	674	23	the	the	DET
ap-961	674	24	dirac	dirac	NOUN
ap-961	674	25	operator	operator	NOUN
ap-961	674	26	on	on	ADP
ap-961	674	27	the	the	DET
ap-961	674	28	circle	circle	NOUN
ap-961	674	29	,	,	PUNCT
ap-961	674	30	mentioned	mention	VERB
ap-961	674	31	in	in	ADP
ap-961	674	32	the	the	DET
ap-961	674	33	previous	previous	ADJ
ap-961	674	34	example	example	NOUN
ap-961	674	35	.	.	PUNCT
ap-961	675	1	since	since	SCONJ
ap-961	675	2	each	each	DET
ap-961	675	3	positive	positive	ADJ
ap-961	675	4	integer	integer	NOUN
ap-961	675	5	(	(	PUNCT
ap-961	675	6	apart	apart	ADV
ap-961	675	7	from	from	ADP
ap-961	675	8	1	1	NUM
ap-961	675	9	)	)	PUNCT
ap-961	675	10	enters	enter	VERB
ap-961	675	11	into	into	ADP
ap-961	675	12	the	the	DET
ap-961	675	13	spectrum	spectrum	NOUN
ap-961	675	14	of	of	ADP
ap-961	675	15	d	d	PROPN
ap-961	675	16	1exactly	1exactly	ADV
ap-961	675	17	twice	twice	ADV
ap-961	675	18	,	,	PUNCT
ap-961	675	19	we	we	PRON
ap-961	675	20	have	have	AUX
ap-961	675	21	:	:	PUNCT
ap-961	675	22	tr	tr	VERB
ap-961	675	23	�	�	PROPN
ap-961	675	24	(	(	PUNCT
ap-961	675	25	)	)	PUNCT
ap-961	675	26	lim	lim	PROPN
ap-961	675	27	log	log	PROPN
ap-961	675	28	lim	lim	PROPN
ap-961	675	29	log	log	VERB
ap-961	676	1	d	d	PROPN
ap-961	676	2	n	n	PROPN
ap-961	676	3	i	i	PRON
ap-961	676	4	n	n	CCONJ
ap-961	676	5	n	n	CCONJ
ap-961	677	1	i	i	PRON
ap-961	677	2	n	n	CCONJ
ap-961	677	3	n	n	CCONJ
ap-961	677	4	�	�	PROPN
ap-961	677	5	�	�	PROPN
ap-961	677	6	�	�	PROPN
ap-961	677	7	�	�	PROPN
ap-961	677	8	�	�	PROPN
ap-961	677	9	�	�	PROPN
ap-961	677	10	�	�	PROPN
ap-961	677	11	�	�	PROPN
ap-961	677	12	�	�	PROPN
ap-961	677	13	�	�	PROPN
ap-961	677	14	�	�	PROPN
ap-961	677	15	�	�	PROPN
ap-961	677	16	�	�	PROPN
ap-961	677	17	�	�	PROPN
ap-961	677	18	�	�	PROPN
ap-961	677	19	1	1	NUM
ap-961	677	20	1	1	NUM
ap-961	677	21	2	2	NUM
ap-961	677	22	2	2	NUM
ap-961	677	23	1	1	NUM
ap-961	677	24	1	1	NUM
ap-961	677	25	1	1	NUM
ap-961	677	26	(	(	PUNCT
ap-961	677	27	log	log	NOUN
ap-961	677	28	(	(	PUNCT
ap-961	677	29	)	)	PUNCT
ap-961	677	30	(	(	PUNCT
ap-961	677	31	)	)	PUNCT
ap-961	677	32	)	)	PUNCT
ap-961	677	33	,	,	PUNCT
ap-961	677	34	n	n	CCONJ
ap-961	677	35	o	o	PROPN
ap-961	677	36	n	n	X
ap-961	677	37	�	�	PROPN
ap-961	677	38	�	�	PROPN
ap-961	677	39	2	2	NUM
ap-961	677	40	where	where	SCONJ
ap-961	677	41	we	we	PRON
ap-961	677	42	have	have	AUX
ap-961	677	43	used	use	VERB
ap-961	677	44	the	the	DET
ap-961	677	45	formula	formula	NOUN
ap-961	677	46	for	for	ADP
ap-961	677	47	the	the	DET
ap-961	677	48	asymptotic	asymptotic	ADJ
ap-961	677	49	form	form	NOUN
ap-961	677	50	of	of	ADP
ap-961	677	51	the	the	DET
ap-961	677	52	harmonic	harmonic	ADJ
ap-961	677	53	numbers	number	NOUN
ap-961	677	54	,	,	PUNCT
ap-961	677	55	and	and	CCONJ
ap-961	677	56	�	�	PROPN
ap-961	677	57	is	be	AUX
ap-961	677	58	the	the	DET
ap-961	677	59	euler	euler	PROPN
ap-961	677	60	constant	constant	PROPN
ap-961	677	61	.	.	PUNCT
ap-961	678	1	theorem	theorem	VERB
ap-961	678	2	6.3	6.3	NUM
ap-961	678	3	:	:	PUNCT
ap-961	678	4	the	the	DET
ap-961	678	5	space	space	NOUN
ap-961	678	6	of	of	ADP
ap-961	678	7	all	all	DET
ap-961	678	8	operators	operator	NOUN
ap-961	678	9	for	for	ADP
ap-961	678	10	which	which	PRON
ap-961	678	11	the	the	DET
ap-961	678	12	dixmier	dixmi	ADJ
ap-961	678	13	trace	trace	NOUN
ap-961	678	14	can	can	AUX
ap-961	678	15	be	be	AUX
ap-961	678	16	calculated	calculate	VERB
ap-961	678	17	is	be	AUX
ap-961	678	18	a	a	DET
ap-961	678	19	two	two	NUM
ap-961	678	20	-	-	PUNCT
ap-961	678	21	sided	sided	ADJ
ap-961	678	22	ideal	ideal	NOUN
ap-961	678	23	in	in	ADP
ap-961	678	24	the	the	DET
ap-961	678	25	algebra	algebra	NOUN
ap-961	678	26	of	of	ADP
ap-961	678	27	bounded	bounded	ADJ
ap-961	678	28	operators	operator	NOUN
ap-961	678	29	,	,	PUNCT
ap-961	678	30	moreover	moreover	ADV
ap-961	678	31	for	for	SCONJ
ap-961	678	32	r	r	NOUN
ap-961	678	33	bounded	bound	VERB
ap-961	678	34	and	and	CCONJ
ap-961	678	35	t	t	PROPN
ap-961	678	36	in	in	ADP
ap-961	678	37	this	this	DET
ap-961	678	38	ideal	ideal	NOUN
ap-961	678	39	:	:	PUNCT
ap-961	678	40	tr	tr	VERB
ap-961	678	41	tr	tr	VERB
ap-961	678	42	�	�	PROPN
ap-961	678	43	�	�	PROPN
ap-961	678	44	rt	rt	PROPN
ap-961	678	45	tr	tr	VERB
ap-961	678	46	�	�	PROPN
ap-961	678	47	.	.	PUNCT
ap-961	679	1	for	for	ADP
ap-961	679	2	the	the	DET
ap-961	679	3	proof	proof	NOUN
ap-961	679	4	of	of	ADP
ap-961	679	5	the	the	DET
ap-961	679	6	theorem	theorem	NOUN
ap-961	679	7	and	and	CCONJ
ap-961	679	8	also	also	ADV
ap-961	679	9	for	for	ADP
ap-961	679	10	verification	verification	NOUN
ap-961	679	11	that	that	PRON
ap-961	679	12	the	the	DET
ap-961	679	13	dixmier	dixmi	ADJ
ap-961	679	14	trace	trace	NOUN
ap-961	679	15	is	be	AUX
ap-961	679	16	well	well	ADV
ap-961	679	17	defined	define	VERB
ap-961	679	18	and	and	CCONJ
ap-961	679	19	is	be	AUX
ap-961	679	20	indeed	indeed	ADV
ap-961	679	21	a	a	DET
ap-961	679	22	trace	trace	NOUN
ap-961	679	23	,	,	PUNCT
ap-961	679	24	we	we	PRON
ap-961	679	25	refer	refer	VERB
ap-961	679	26	to	to	ADP
ap-961	679	27	connes	conne	NOUN
ap-961	679	28	’	'	PUNCT
ap-961	679	29	book	book	NOUN
ap-961	679	30	[	[	X
ap-961	679	31	5	5	NUM
ap-961	679	32	]	]	PUNCT
ap-961	679	33	.	.	PUNCT
ap-961	680	1	assume	assume	VERB
ap-961	680	2	now	now	ADV
ap-961	680	3	that	that	SCONJ
ap-961	680	4	we	we	PRON
ap-961	680	5	have	have	VERB
ap-961	680	6	an	an	DET
ap-961	680	7	unbounded	unbounded	ADJ
ap-961	680	8	operator	operator	NOUN
ap-961	680	9	d	d	NOUN
ap-961	680	10	such	such	ADJ
ap-961	680	11	that	that	SCONJ
ap-961	680	12	d	d	PROPN
ap-961	680	13	�	�	NOUN
ap-961	680	14	1	1	NUM
ap-961	680	15	is	be	AUX
ap-961	680	16	compact	compact	ADJ
ap-961	680	17	(	(	PUNCT
ap-961	680	18	for	for	ADP
ap-961	680	19	simplicity	simplicity	NOUN
ap-961	680	20	we	we	PRON
ap-961	680	21	eliminate	eliminate	VERB
ap-961	680	22	zeroes	zero	NOUN
ap-961	680	23	from	from	ADP
ap-961	680	24	the	the	DET
ap-961	680	25	spectrum	spectrum	NOUN
ap-961	680	26	of	of	ADP
ap-961	680	27	d	d	PROPN
ap-961	680	28	)	)	PUNCT
ap-961	680	29	.	.	PUNCT
ap-961	681	1	for	for	ADP
ap-961	681	2	sufficiently	sufficiently	ADV
ap-961	681	3	large	large	ADJ
ap-961	681	4	s	s	NOUN
ap-961	681	5	�	�	PROPN
ap-961	681	6	0	0	NUM
ap-961	681	7	the	the	DET
ap-961	681	8	operator	operator	NOUN
ap-961	681	9	d	d	PROPN
ap-961	681	10	s	s	PROPN
ap-961	681	11	�	�	PROPN
ap-961	681	12	will	will	AUX
ap-961	681	13	be	be	AUX
ap-961	681	14	a	a	DET
ap-961	681	15	trace	trace	NOUN
ap-961	681	16	class	class	NOUN
ap-961	681	17	and	and	CCONJ
ap-961	681	18	thus	thus	ADV
ap-961	681	19	the	the	DET
ap-961	681	20	function	function	NOUN
ap-961	681	21	:	:	PUNCT
ap-961	681	22	�	�	PROPN
ap-961	681	23	d	d	NOUN
ap-961	681	24	ss	ss	NOUN
ap-961	681	25	d	d	PROPN
ap-961	681	26	(	(	PUNCT
ap-961	681	27	)	)	PUNCT
ap-961	681	28	�	�	PROPN
ap-961	681	29	�	�	PROPN
ap-961	681	30	tr	tr	VERB
ap-961	681	31	,	,	PUNCT
ap-961	681	32	makes	make	VERB
ap-961	681	33	sense	sense	NOUN
ap-961	681	34	.	.	PUNCT
ap-961	682	1	using	use	VERB
ap-961	682	2	the	the	DET
ap-961	682	3	functional	functional	ADJ
ap-961	682	4	calculus	calculus	NOUN
ap-961	682	5	on	on	ADP
ap-961	682	6	a	a	DET
ap-961	682	7	hilbert	hilbert	NOUN
ap-961	682	8	space	space	NOUN
ap-961	682	9	we	we	PRON
ap-961	682	10	can	can	AUX
ap-961	682	11	define	define	VERB
ap-961	682	12	this	this	DET
ap-961	682	13	function	function	NOUN
ap-961	682	14	at	at	ADP
ap-961	682	15	least	least	ADJ
ap-961	682	16	on	on	ADP
ap-961	682	17	the	the	DET
ap-961	682	18	part	part	NOUN
ap-961	682	19	of	of	ADP
ap-961	682	20	the	the	DET
ap-961	682	21	complex	complex	ADJ
ap-961	682	22	plane	plane	NOUN
ap-961	682	23	forre	forre	NOUN
ap-961	682	24	(	(	PUNCT
ap-961	682	25	)	)	PUNCT
ap-961	682	26	s	s	VERB
ap-961	682	27	big	big	ADJ
ap-961	682	28	enough	enough	ADV
ap-961	682	29	.	.	PUNCT
ap-961	683	1	suppose	suppose	VERB
ap-961	683	2	now	now	ADV
ap-961	683	3	that	that	SCONJ
ap-961	683	4	we	we	PRON
ap-961	683	5	make	make	VERB
ap-961	683	6	an	an	DET
ap-961	683	7	analytic	analytic	ADJ
ap-961	683	8	continuation	continuation	NOUN
ap-961	683	9	of	of	ADP
ap-961	683	10	the	the	DET
ap-961	683	11	function	function	NOUN
ap-961	683	12	�	�	PROPN
ap-961	683	13	d	d	PROPN
ap-961	683	14	and	and	CCONJ
ap-961	683	15	that	that	SCONJ
ap-961	683	16	it	it	PRON
ap-961	683	17	extends	extend	VERB
ap-961	683	18	to	to	ADP
ap-961	683	19	the	the	DET
ap-961	683	20	entire	entire	ADJ
ap-961	683	21	complex	complex	ADJ
ap-961	683	22	plane	plane	NOUN
ap-961	683	23	with	with	ADP
ap-961	683	24	the	the	DET
ap-961	683	25	exception	exception	NOUN
ap-961	683	26	of	of	ADP
ap-961	683	27	several	several	ADJ
ap-961	683	28	isolated	isolated	ADJ
ap-961	683	29	points	point	NOUN
ap-961	683	30	.	.	PUNCT
ap-961	684	1	then	then	ADV
ap-961	684	2	at	at	ADP
ap-961	684	3	each	each	PRON
ap-961	684	4	of	of	ADP
ap-961	684	5	these	these	DET
ap-961	684	6	points	point	NOUN
ap-961	684	7	we	we	PRON
ap-961	684	8	can	can	AUX
ap-961	684	9	calculate	calculate	VERB
ap-961	684	10	the	the	DET
ap-961	684	11	residue	residue	NOUN
ap-961	684	12	of	of	ADP
ap-961	684	13	�	�	PROPN
ap-961	684	14	d	d	SYM
ap-961	684	15	s	s	PROPN
ap-961	684	16	(	(	PUNCT
ap-961	684	17	)	)	PUNCT
ap-961	684	18	–	–	PUNCT
ap-961	684	19	just	just	ADV
ap-961	684	20	as	as	ADP
ap-961	684	21	a	a	DET
ap-961	684	22	functional	functional	ADJ
ap-961	684	23	.	.	PUNCT
ap-961	685	1	we	we	PRON
ap-961	685	2	have	have	AUX
ap-961	685	3	:	:	PUNCT
ap-961	685	4	theorem	theorem	VERB
ap-961	685	5	6.4	6.4	NUM
ap-961	685	6	:	:	PUNCT
ap-961	685	7	if	if	SCONJ
ap-961	685	8	d	d	PROPN
ap-961	685	9	�	�	PROPN
ap-961	685	10	1	1	NUM
ap-961	685	11	is	be	AUX
ap-961	685	12	an	an	DET
ap-961	685	13	operator	operator	NOUN
ap-961	685	14	of	of	ADP
ap-961	685	15	dixmier	dixmi	ADJ
ap-961	685	16	class	class	NOUN
ap-961	685	17	then	then	ADV
ap-961	685	18	:	:	PUNCT
ap-961	686	1	tr	tr	VERB
ap-961	686	2	�	�	PROPN
ap-961	686	3	d	d	NOUN
ap-961	686	4	s	s	X
ap-961	686	5	ds	ds	PROPN
ap-961	686	6	z	z	PROPN
ap-961	686	7	z	z	PROPN
ap-961	686	8	�	�	PROPN
ap-961	686	9	�	�	PROPN
ap-961	686	10	�	�	PROPN
ap-961	686	11	�	�	PROPN
ap-961	686	12	re	re	ADP
ap-961	686	13	1	1	NUM
ap-961	686	14	.	.	PUNCT
ap-961	686	15	example	example	NOUN
ap-961	687	1	6.5	6.5	NUM
ap-961	687	2	:	:	PUNCT
ap-961	687	3	an	an	DET
ap-961	687	4	elliptic	elliptic	ADJ
ap-961	687	5	differential	differential	NOUN
ap-961	687	6	operator	operator	NOUN
ap-961	687	7	on	on	ADP
ap-961	687	8	a	a	DET
ap-961	687	9	manifold	manifold	NOUN
ap-961	687	10	is	be	AUX
ap-961	687	11	just	just	ADV
ap-961	687	12	a	a	DET
ap-961	687	13	special	special	ADJ
ap-961	687	14	case	case	NOUN
ap-961	687	15	of	of	ADP
ap-961	687	16	an	an	DET
ap-961	687	17	unbounded	unbounded	ADJ
ap-961	687	18	operator	operator	NOUN
ap-961	687	19	.	.	PUNCT
ap-961	688	1	for	for	ADP
ap-961	688	2	instance	instance	NOUN
ap-961	688	3	,	,	PUNCT
ap-961	688	4	taking	take	VERB
ap-961	688	5	the	the	DET
ap-961	688	6	laplace	laplace	NOUN
ap-961	688	7	operator	operator	NOUN
ap-961	688	8	�	�	PROPN
ap-961	688	9	on	on	ADP
ap-961	688	10	a	a	DET
ap-961	688	11	compact	compact	ADJ
ap-961	688	12	manifold	manifold	NOUN
ap-961	688	13	of	of	ADP
ap-961	688	14	dimension	dimension	NOUN
ap-961	688	15	n	n	CCONJ
ap-961	688	16	,	,	PUNCT
ap-961	688	17	it	it	PRON
ap-961	688	18	appears	appear	VERB
ap-961	688	19	that	that	SCONJ
ap-961	688	20	�	�	PROPN
ap-961	688	21	�	�	PROPN
ap-961	688	22	n	n	CCONJ
ap-961	688	23	2	2	NUM
ap-961	688	24	is	be	AUX
ap-961	688	25	in	in	ADP
ap-961	688	26	the	the	DET
ap-961	688	27	dixmier	dixmi	ADJ
ap-961	688	28	class	class	NOUN
ap-961	688	29	and	and	CCONJ
ap-961	688	30	its	its	PRON
ap-961	688	31	dixmier	dixmier	NOUN
ap-961	688	32	trace	trace	NOUN
ap-961	688	33	is	be	AUX
ap-961	688	34	related	relate	VERB
ap-961	688	35	to	to	ADP
ap-961	688	36	the	the	DET
ap-961	688	37	wodzicki	wodzicki	PROPN
ap-961	688	38	residue	residue	NOUN
ap-961	688	39	(	(	PUNCT
ap-961	688	40	which	which	PRON
ap-961	688	41	is	be	AUX
ap-961	688	42	a	a	DET
ap-961	688	43	functional	functional	ADJ
ap-961	688	44	on	on	ADP
ap-961	688	45	the	the	DET
ap-961	688	46	space	space	NOUN
ap-961	688	47	of	of	ADP
ap-961	688	48	differential	differential	ADJ
ap-961	688	49	operators	operator	NOUN
ap-961	688	50	given	give	VERB
ap-961	688	51	explicitly	explicitly	ADV
ap-961	688	52	by	by	ADP
ap-961	688	53	the	the	DET
ap-961	688	54	integral	integral	NOUN
ap-961	688	55	of	of	ADP
ap-961	688	56	the	the	DET
ap-961	688	57	principal	principal	ADJ
ap-961	688	58	symbol	symbol	NOUN
ap-961	688	59	)	)	PUNCT
ap-961	688	60	.	.	PUNCT
ap-961	689	1	as	as	ADP
ap-961	689	2	a	a	DET
ap-961	689	3	specific	specific	ADJ
ap-961	689	4	example	example	NOUN
ap-961	689	5	take	take	NOUN
ap-961	689	6	(	(	PUNCT
ap-961	689	7	again	again	ADV
ap-961	689	8	)	)	PUNCT
ap-961	689	9	the	the	DET
ap-961	689	10	2	2	NUM
ap-961	689	11	-	-	PUNCT
ap-961	689	12	dimensional	dimensional	ADJ
ap-961	689	13	torus	torus	NOUN
ap-961	689	14	and	and	CCONJ
ap-961	689	15	its	its	PRON
ap-961	689	16	laplace	laplace	NOUN
ap-961	689	17	operator	operator	NOUN
ap-961	689	18	�	�	PROPN
ap-961	689	19	�	�	PROPN
ap-961	689	20	�	�	PROPN
ap-961	689	21	�	�	PROPN
ap-961	689	22	2	2	NUM
ap-961	689	23	2	2	NUM
ap-961	689	24	,	,	PUNCT
ap-961	689	25	where	where	SCONJ
ap-961	689	26	0	0	NUM
ap-961	689	27	�	�	PROPN
ap-961	689	28	,	,	PUNCT
ap-961	689	29	�	�	PROPN
ap-961	689	30	42	42	NUM
ap-961	689	31	�	�	PROPN
ap-961	689	32	are	be	AUX
ap-961	689	33	again	again	ADV
ap-961	689	34	the	the	DET
ap-961	689	35	standard	standard	ADJ
ap-961	689	36	coordinates	coordinate	NOUN
ap-961	689	37	on	on	ADP
ap-961	689	38	the	the	DET
ap-961	689	39	torus	torus	NOUN
ap-961	689	40	.	.	PUNCT
ap-961	690	1	the	the	DET
ap-961	690	2	principal	principal	ADJ
ap-961	690	3	symbol	symbol	NOUN
ap-961	690	4	of	of	ADP
ap-961	690	5	�	�	PROPN
ap-961	690	6	(	(	PUNCT
ap-961	690	7	and	and	CCONJ
ap-961	690	8	so	so	ADV
ap-961	690	9	�	�	PROPN
ap-961	690	10	�	�	PROPN
ap-961	690	11	1	1	NUM
ap-961	690	12	)	)	PUNCT
ap-961	690	13	is	be	AUX
ap-961	690	14	constant	constant	ADJ
ap-961	690	15	,	,	PUNCT
ap-961	690	16	so	so	SCONJ
ap-961	690	17	the	the	DET
ap-961	690	18	wodzicki	wodzicki	PROPN
ap-961	690	19	residue	residue	NOUN
ap-961	690	20	,	,	PUNCT
ap-961	690	21	which	which	PRON
ap-961	690	22	is	be	AUX
ap-961	690	23	(	(	PUNCT
ap-961	690	24	in	in	ADP
ap-961	690	25	two	two	NUM
ap-961	690	26	dimensions	dimension	NOUN
ap-961	690	27	)	)	PUNCT
ap-961	690	28	its	its	PRON
ap-961	690	29	integral	integral	ADJ
ap-961	690	30	over	over	ADP
ap-961	690	31	the	the	DET
ap-961	690	32	sphere	sphere	NOUN
ap-961	690	33	bundle	bundle	NOUN
ap-961	690	34	and	and	CCONJ
ap-961	690	35	the	the	DET
ap-961	690	36	manifold	manifold	NOUN
ap-961	690	37	,	,	PUNCT
ap-961	690	38	is	be	AUX
ap-961	690	39	:	:	PUNCT
ap-961	690	40	wres	wre	NOUN
ap-961	690	41	(	(	PUNCT
ap-961	690	42	)	)	PUNCT
ap-961	690	43	(	(	PUNCT
ap-961	690	44	)	)	PUNCT
ap-961	690	45	�	�	PROPN
ap-961	690	46	�	�	PROPN
ap-961	690	47	�	�	PROPN
ap-961	690	48	1	1	NUM
ap-961	690	49	24	24	NUM
ap-961	690	50	2	2	NUM
ap-961	690	51	�	�	PROPN
ap-961	690	52	�	�	PROPN
ap-961	690	53	.	.	PUNCT
ap-961	691	1	to	to	PART
ap-961	691	2	calculate	calculate	VERB
ap-961	691	3	the	the	DET
ap-961	691	4	dixmier	dixmi	ADJ
ap-961	691	5	trace	trace	NOUN
ap-961	691	6	of	of	ADP
ap-961	691	7	we	we	PRON
ap-961	691	8	need	need	VERB
ap-961	691	9	to	to	PART
ap-961	691	10	calculate	calculate	VERB
ap-961	691	11	the	the	DET
ap-961	691	12	following	follow	VERB
ap-961	691	13	limit	limit	NOUN
ap-961	691	14	:	:	PUNCT
ap-961	691	15	lim	lim	PROPN
ap-961	691	16	log	log	PROPN
ap-961	691	17	(	(	PUNCT
ap-961	691	18	)	)	PUNCT
ap-961	691	19	n	n	CCONJ
ap-961	691	20	m	m	VERB
ap-961	691	21	n	n	VERB
ap-961	691	22	n	n	CCONJ
ap-961	691	23	n	n	PRON
ap-961	691	24	m	m	PROPN
ap-961	691	25	n	n	CCONJ
ap-961	691	26	�	�	PROPN
ap-961	691	27	)	)	PUNCT
ap-961	691	28	�	�	PROPN
ap-961	691	29	�	�	PROPN
ap-961	691	30	�	�	PROPN
ap-961	691	31	�	�	PROPN
ap-961	691	32	�	�	PROPN
ap-961	691	33	�	�	PROPN
ap-961	691	34	�	�	PROPN
ap-961	691	35	�	�	PROPN
ap-961	691	36	�	�	PROPN
ap-961	691	37	1	1	NUM
ap-961	691	38	1	1	NUM
ap-961	691	39	2	2	NUM
ap-961	691	40	1	1	NUM
ap-961	691	41	2	2	NUM
ap-961	691	42	2	2	NUM
ap-961	691	43	2	2	NUM
ap-961	691	44	2	2	NUM
ap-961	691	45	2	2	NUM
ap-961	691	46	2	2	NUM
ap-961	691	47	�	�	NOUN
ap-961	691	48	.	.	PUNCT
ap-961	692	1	leaving	leave	VERB
ap-961	692	2	the	the	DET
ap-961	692	3	evaluation	evaluation	NOUN
ap-961	692	4	of	of	ADP
ap-961	692	5	the	the	DET
ap-961	692	6	asymptotics	asymptotic	NOUN
ap-961	692	7	of	of	ADP
ap-961	692	8	the	the	DET
ap-961	692	9	sum	sum	NOUN
ap-961	692	10	as	as	ADP
ap-961	692	11	an	an	DET
ap-961	692	12	exercise	exercise	NOUN
ap-961	692	13	(	(	PUNCT
ap-961	692	14	see	see	VERB
ap-961	692	15	[	[	X
ap-961	692	16	2	2	NUM
ap-961	692	17	]	]	PUNCT
ap-961	692	18	,	,	PUNCT
ap-961	692	19	for	for	ADP
ap-961	692	20	hints	hint	NOUN
ap-961	692	21	)	)	PUNCT
ap-961	692	22	let	let	VERB
ap-961	692	23	us	we	PRON
ap-961	692	24	state	state	VERB
ap-961	692	25	the	the	DET
ap-961	692	26	result	result	NOUN
ap-961	692	27	:	:	PUNCT
ap-961	692	28	tr	tr	PUNCT
ap-961	692	29	�	�	PROPN
ap-961	692	30	�	�	PROPN
ap-961	692	31	(	(	PUNCT
ap-961	692	32	)	)	PUNCT
ap-961	692	33	�	�	PROPN
ap-961	692	34	�	�	PROPN
ap-961	692	35	�	�	PROPN
ap-961	692	36	1	1	NUM
ap-961	692	37	1	1	NUM
ap-961	692	38	2	2	NUM
ap-961	692	39	.	.	PUNCT
ap-961	693	1	the	the	DET
ap-961	693	2	ratio	ratio	NOUN
ap-961	693	3	:	:	PUNCT
ap-961	693	4	wres	wre	NOUN
ap-961	693	5	(	(	PUNCT
ap-961	693	6	)	)	PUNCT
ap-961	693	7	(	(	PUNCT
ap-961	693	8	)	)	PUNCT
ap-961	693	9	(	(	PUNCT
ap-961	693	10	)	)	PUNCT
ap-961	693	11	�	�	PROPN
ap-961	693	12	�	�	PROPN
ap-961	693	13	�	�	PROPN
ap-961	693	14	�	�	PROPN
ap-961	693	15	�	�	PROPN
ap-961	693	16	1	1	NUM
ap-961	693	17	1	1	NUM
ap-961	693	18	22	22	NUM
ap-961	693	19	2	2	NUM
ap-961	693	20	tr	tr	VERB
ap-961	693	21	�	�	PROPN
ap-961	693	22	�	�	PROPN
ap-961	693	23	,	,	PUNCT
ap-961	693	24	and	and	CCONJ
ap-961	693	25	is	be	AUX
ap-961	693	26	,	,	PUNCT
ap-961	693	27	in	in	ADP
ap-961	693	28	fact	fact	NOUN
ap-961	693	29	,	,	PUNCT
ap-961	693	30	universal	universal	ADJ
ap-961	693	31	(	(	PUNCT
ap-961	693	32	in	in	ADP
ap-961	693	33	this	this	DET
ap-961	693	34	dimension	dimension	NOUN
ap-961	693	35	2	2	NUM
ap-961	693	36	)	)	PUNCT
ap-961	693	37	.	.	PUNCT
ap-961	694	1	6.3	6.3	NUM
ap-961	694	2	spectral	spectral	ADJ
ap-961	694	3	triples	triple	NOUN
ap-961	694	4	imagine	imagine	VERB
ap-961	694	5	we	we	PRON
ap-961	694	6	want	want	VERB
ap-961	694	7	to	to	PART
ap-961	694	8	encompass	encompass	VERB
ap-961	694	9	everything	everything	PRON
ap-961	694	10	we	we	PRON
ap-961	694	11	have	have	AUX
ap-961	694	12	learned	learn	VERB
ap-961	694	13	in	in	ADP
ap-961	694	14	one	one	NUM
ap-961	694	15	compact	compact	ADJ
ap-961	694	16	definition	definition	NOUN
ap-961	694	17	.	.	PUNCT
ap-961	695	1	so	so	ADV
ap-961	695	2	,	,	PUNCT
ap-961	695	3	we	we	PRON
ap-961	695	4	work	work	VERB
ap-961	695	5	with	with	ADP
ap-961	695	6	a	a	DET
ap-961	695	7	suitable	suitable	ADJ
ap-961	695	8	algebra	algebra	NOUN
ap-961	695	9	,	,	PUNCT
ap-961	695	10	which	which	PRON
ap-961	695	11	is	be	AUX
ap-961	695	12	a	a	DET
ap-961	695	13	subalgebra	subalgebra	NOUN
ap-961	695	14	of	of	ADP
ap-961	695	15	a	a	DET
ap-961	695	16	c	c	NOUN
ap-961	695	17	*	*	NOUN
ap-961	695	18	algebra	algebra	NOUN
ap-961	695	19	.	.	PUNCT
ap-961	696	1	it	it	PRON
ap-961	696	2	is	be	AUX
ap-961	696	3	represented	represent	VERB
ap-961	696	4	(	(	PUNCT
ap-961	696	5	faithfully	faithfully	ADV
ap-961	696	6	)	)	PUNCT
ap-961	696	7	on	on	ADP
ap-961	696	8	a	a	DET
ap-961	696	9	hilbert	hilbert	NOUN
ap-961	696	10	space	space	NOUN
ap-961	696	11	.	.	PUNCT
ap-961	697	1	further	far	ADV
ap-961	697	2	,	,	PUNCT
ap-961	697	3	we	we	PRON
ap-961	697	4	need	need	VERB
ap-961	697	5	to	to	PART
ap-961	697	6	have	have	VERB
ap-961	697	7	a	a	DET
ap-961	697	8	suitable	suitable	ADJ
ap-961	697	9	definition	definition	NOUN
ap-961	697	10	of	of	ADP
ap-961	697	11	a	a	DET
ap-961	697	12	differential	differential	ADJ
ap-961	697	13	algebra	algebra	NOUN
ap-961	697	14	–	–	PUNCT
ap-961	697	15	and	and	CCONJ
ap-961	697	16	here	here	ADV
ap-961	697	17	we	we	PRON
ap-961	697	18	need	need	VERB
ap-961	697	19	just	just	ADV
ap-961	697	20	to	to	PART
ap-961	697	21	choose	choose	VERB
ap-961	697	22	a	a	DET
ap-961	697	23	suitable	suitable	ADJ
ap-961	697	24	unbounded	unbounded	ADJ
ap-961	697	25	operator	operator	NOUN
ap-961	697	26	d	d	NOUN
ap-961	697	27	on	on	ADP
ap-961	697	28	the	the	DET
ap-961	697	29	hilbert	hilbert	NOUN
ap-961	697	30	space	space	NOUN
ap-961	697	31	so	so	SCONJ
ap-961	697	32	that	that	SCONJ
ap-961	697	33	the	the	DET
ap-961	697	34	differential	differential	ADJ
ap-961	697	35	one	one	NUM
ap-961	697	36	-	-	PUNCT
ap-961	697	37	forms	form	NOUN
ap-961	697	38	,	,	PUNCT
ap-961	697	39	the	the	DET
ap-961	697	40	commutators	commutator	NOUN
ap-961	697	41	of	of	ADP
ap-961	697	42	d	d	PROPN
ap-961	697	43	with	with	ADP
ap-961	697	44	the	the	DET
ap-961	697	45	elements	element	NOUN
ap-961	697	46	of	of	ADP
ap-961	697	47	the	the	DET
ap-961	697	48	algebra	algebra	NOUN
ap-961	697	49	remain	remain	VERB
ap-961	697	50	bounded	bound	VERB
ap-961	697	51	.	.	PUNCT
ap-961	698	1	the	the	DET
ap-961	698	2	sign	sign	NOUN
ap-961	698	3	of	of	ADP
ap-961	698	4	d	d	PROPN
ap-961	698	5	will	will	AUX
ap-961	698	6	define	define	VERB
ap-961	698	7	for	for	ADP
ap-961	698	8	us	we	PRON
ap-961	698	9	a	a	DET
ap-961	698	10	fredholm	fredholm	NOUN
ap-961	698	11	module	module	NOUN
ap-961	698	12	,	,	PUNCT
ap-961	698	13	and	and	CCONJ
ap-961	698	14	thus	thus	ADV
ap-961	698	15	a	a	DET
ap-961	698	16	cyclic	cyclic	ADJ
ap-961	698	17	cocycle	cocycle	NOUN
ap-961	698	18	.	.	PUNCT
ap-961	699	1	moreover	moreover	ADV
ap-961	699	2	,	,	PUNCT
ap-961	699	3	suitably	suitably	ADV
ap-961	699	4	chosen	choose	VERB
ap-961	699	5	d	d	NOUN
ap-961	699	6	will	will	AUX
ap-961	699	7	allow	allow	VERB
ap-961	699	8	us	we	PRON
ap-961	699	9	to	to	PART
ap-961	699	10	introduce	introduce	VERB
ap-961	699	11	noncommutative	noncommutative	ADJ
ap-961	699	12	integrations	integration	NOUN
ap-961	699	13	through	through	ADP
ap-961	699	14	the	the	DET
ap-961	699	15	residua	residua	PROPN
ap-961	699	16	of	of	ADP
ap-961	699	17	�	�	PROPN
ap-961	699	18	d.	d.	PROPN
ap-961	699	19	so	so	ADV
ap-961	699	20	,	,	PUNCT
ap-961	699	21	we	we	PRON
ap-961	699	22	are	be	AUX
ap-961	699	23	ready	ready	ADJ
ap-961	699	24	for	for	ADP
ap-961	699	25	the	the	DET
ap-961	699	26	formal	formal	ADJ
ap-961	699	27	definition	definition	NOUN
ap-961	699	28	:	:	PUNCT
ap-961	699	29	©	©	PROPN
ap-961	699	30	czech	czech	PROPN
ap-961	699	31	technical	technical	PROPN
ap-961	699	32	university	university	PROPN
ap-961	699	33	publishing	publishing	NOUN
ap-961	699	34	house	house	NOUN
ap-961	699	35	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	699	36	51	51	NUM
ap-961	699	37	acta	acta	PROPN
ap-961	699	38	polytechnica	polytechnica	PROPN
ap-961	699	39	vol	vol	NOUN
ap-961	699	40	.	.	PUNCT
ap-961	700	1	48	48	NUM
ap-961	700	2	no	no	NOUN
ap-961	700	3	.	.	PUNCT
ap-961	701	1	2/2008	2/2008	NUM
ap-961	701	2	definition	definition	NOUN
ap-961	701	3	6.6	6.6	NUM
ap-961	701	4	:	:	PUNCT
ap-961	701	5	let	let	VERB
ap-961	701	6	us	we	PRON
ap-961	701	7	have	have	VERB
ap-961	701	8	an	an	DET
ap-961	701	9	algebra	algebra	PROPN
ap-961	701	10	�	�	PROPN
ap-961	701	11	,	,	PUNCT
ap-961	701	12	its	its	PRON
ap-961	701	13	faithful	faithful	ADJ
ap-961	701	14	representation	representation	NOUN
ap-961	701	15	�	�	PROPN
ap-961	701	16	on	on	ADP
ap-961	701	17	a	a	DET
ap-961	701	18	hilbert	hilbert	NOUN
ap-961	701	19	space	space	NOUN
ap-961	701	20	�	�	PROPN
ap-961	701	21	,	,	PUNCT
ap-961	701	22	a	a	DET
ap-961	701	23	selfadjoint	selfadjoint	NOUN
ap-961	701	24	unbounded	unbounded	ADJ
ap-961	701	25	operator	operator	NOUN
ap-961	701	26	d	d	NOUN
ap-961	701	27	with	with	ADP
ap-961	701	28	compact	compact	ADJ
ap-961	701	29	resolvent	resolvent	NOUN
ap-961	701	30	,	,	PUNCT
ap-961	701	31	such	such	ADJ
ap-961	701	32	that	that	SCONJ
ap-961	701	33	�	�	PROPN
ap-961	701	34	�	�	PROPN
ap-961	701	35	a	a	DET
ap-961	701	36	�	�	PROPN
ap-961	701	37	,	,	PUNCT
ap-961	701	38	[	[	PUNCT
ap-961	701	39	,	,	PUNCT
ap-961	701	40	(	(	PUNCT
ap-961	701	41	)	)	PUNCT
ap-961	701	42	]	]	PUNCT
ap-961	701	43	(	(	PUNCT
ap-961	701	44	)	)	PUNCT
ap-961	701	45	d	d	X
ap-961	701	46	a	a	DET
ap-961	701	47	b	b	PROPN
ap-961	701	48	�	�	PROPN
ap-961	701	49	�	�	PROPN
ap-961	701	50	�	�	PROPN
ap-961	701	51	,	,	PUNCT
ap-961	701	52	then	then	ADV
ap-961	701	53	we	we	PRON
ap-961	701	54	call	call	VERB
ap-961	701	55	(	(	PUNCT
ap-961	701	56	,	,	PUNCT
ap-961	701	57	,	,	PUNCT
ap-961	701	58	)	)	PUNCT
ap-961	701	59	�	�	PROPN
ap-961	701	60	�	�	PROPN
ap-961	702	1	d	d	PROPN
ap-961	702	2	a	a	DET
ap-961	702	3	spectral	spectral	ADJ
ap-961	702	4	triple	triple	NOUN
ap-961	702	5	.	.	PUNCT
ap-961	703	1	since	since	SCONJ
ap-961	703	2	the	the	DET
ap-961	703	3	definition	definition	NOUN
ap-961	703	4	is	be	AUX
ap-961	703	5	very	very	ADV
ap-961	703	6	basic	basic	ADJ
ap-961	703	7	,	,	PUNCT
ap-961	703	8	we	we	PRON
ap-961	703	9	shall	shall	AUX
ap-961	703	10	need	need	VERB
ap-961	703	11	(	(	PUNCT
ap-961	703	12	in	in	ADP
ap-961	703	13	most	most	ADJ
ap-961	703	14	cases	case	NOUN
ap-961	703	15	)	)	PUNCT
ap-961	703	16	some	some	DET
ap-961	703	17	additional	additional	ADJ
ap-961	703	18	structures	structure	NOUN
ap-961	703	19	.	.	PUNCT
ap-961	704	1	we	we	PRON
ap-961	704	2	say	say	VERB
ap-961	704	3	that	that	SCONJ
ap-961	704	4	the	the	DET
ap-961	704	5	spectral	spectral	ADJ
ap-961	704	6	triple	triple	NOUN
ap-961	704	7	is	be	AUX
ap-961	704	8	even	even	ADV
ap-961	704	9	if	if	SCONJ
ap-961	704	10	there	there	PRON
ap-961	704	11	exists	exist	VERB
ap-961	704	12	an	an	DET
ap-961	704	13	operator	operator	NOUN
ap-961	704	14	�	�	NOUN
ap-961	704	15	such	such	ADJ
ap-961	704	16	that	that	SCONJ
ap-961	704	17	�	�	PROPN
ap-961	704	18	�	�	PROPN
ap-961	704	19	�	�	PROPN
ap-961	704	20	†	†	PROPN
ap-961	704	21	,	,	PUNCT
ap-961	704	22	�	�	PROPN
ap-961	704	23	�	�	PROPN
ap-961	704	24	�	�	PROPN
ap-961	704	25	�	�	PROPN
ap-961	704	26	(	(	PUNCT
ap-961	704	27	)	)	PUNCT
ap-961	704	28	(	(	PUNCT
ap-961	704	29	)	)	PUNCT
ap-961	704	30	a	a	DET
ap-961	704	31	a	a	DET
ap-961	704	32	�	�	PROPN
ap-961	704	33	and	and	CCONJ
ap-961	704	34	�	�	PROPN
ap-961	704	35	�	�	PROPN
ap-961	704	36	d	d	PROPN
ap-961	704	37	d	d	PROPN
ap-961	704	38	�	�	PROPN
ap-961	704	39	0	0	NUM
ap-961	704	40	.	.	PUNCT
ap-961	705	1	we	we	PRON
ap-961	705	2	say	say	VERB
ap-961	705	3	that	that	SCONJ
ap-961	705	4	the	the	DET
ap-961	705	5	spectral	spectral	ADJ
ap-961	705	6	triple	triple	NOUN
ap-961	705	7	is	be	AUX
ap-961	705	8	finitely	finitely	ADV
ap-961	705	9	summable	summable	ADJ
ap-961	705	10	if	if	SCONJ
ap-961	705	11	the	the	DET
ap-961	705	12	operator	operator	NOUN
ap-961	705	13	d	d	SYM
ap-961	705	14	�	�	PROPN
ap-961	705	15	1	1	NUM
ap-961	705	16	has	have	VERB
ap-961	705	17	eigenvalues	eigenvalue	NOUN
ap-961	705	18	of	of	ADP
ap-961	705	19	the	the	DET
ap-961	705	20	order	order	NOUN
ap-961	705	21	o	o	NOUN
ap-961	706	1	n	n	CCONJ
ap-961	706	2	p	p	X
ap-961	706	3	(	(	PUNCT
ap-961	706	4	)	)	PUNCT
ap-961	706	5	�	�	PROPN
ap-961	706	6	for	for	ADP
ap-961	706	7	some	some	DET
ap-961	706	8	p	p	NOUN
ap-961	706	9	0	0	PROPN
ap-961	706	10	.	.	PUNCT
ap-961	707	1	if	if	SCONJ
ap-961	707	2	the	the	DET
ap-961	707	3	growth	growth	NOUN
ap-961	707	4	of	of	ADP
ap-961	707	5	eigenvalues	eigenvalue	NOUN
ap-961	707	6	of	of	ADP
ap-961	707	7	d	d	NOUN
ap-961	707	8	is	be	AUX
ap-961	707	9	exactly	exactly	ADV
ap-961	707	10	of	of	ADP
ap-961	707	11	the	the	DET
ap-961	707	12	order	order	NOUN
ap-961	707	13	np	np	INTJ
ap-961	707	14	,	,	PUNCT
ap-961	707	15	we	we	PRON
ap-961	707	16	say	say	VERB
ap-961	707	17	that	that	SCONJ
ap-961	707	18	the	the	DET
ap-961	707	19	spectral	spectral	ADJ
ap-961	707	20	triple	triple	NOUN
ap-961	707	21	is	be	AUX
ap-961	707	22	of	of	ADP
ap-961	707	23	metric	metric	ADJ
ap-961	707	24	dimension	dimension	NOUN
ap-961	707	25	p.	p.	NOUN
ap-961	707	26	for	for	ADP
ap-961	707	27	such	such	DET
ap-961	707	28	a	a	DET
ap-961	707	29	triple	triple	NOUN
ap-961	707	30	we	we	PRON
ap-961	707	31	might	might	AUX
ap-961	707	32	introduce	introduce	VERB
ap-961	707	33	the	the	DET
ap-961	707	34	noncommutative	noncommutative	ADJ
ap-961	707	35	integral	integral	ADJ
ap-961	707	36	:	:	PUNCT
ap-961	707	37	�	�	PROPN
ap-961	707	38	�	�	PROPN
ap-961	707	39	2	2	NUM
ap-961	707	40	�	�	PROPN
ap-961	707	41	�	�	PROPN
ap-961	707	42	t	t	PROPN
ap-961	707	43	t	t	PROPN
ap-961	707	44	dz	dz	PROPN
ap-961	707	45	zres	zre	NOUN
ap-961	707	46	0tr	0tr	NOUN
ap-961	707	47	(	(	PUNCT
ap-961	707	48	)	)	PUNCT
ap-961	707	49	.	.	PUNCT
ap-961	708	1	this	this	PRON
ap-961	708	2	exists	exist	VERB
ap-961	708	3	for	for	ADP
ap-961	708	4	all	all	DET
ap-961	708	5	operators	operator	NOUN
ap-961	708	6	t	t	PROPN
ap-961	708	7	,	,	PUNCT
ap-961	708	8	which	which	PRON
ap-961	708	9	are	be	AUX
ap-961	708	10	products	product	NOUN
ap-961	708	11	of	of	ADP
ap-961	708	12	�	�	PROPN
ap-961	708	13	(	(	PUNCT
ap-961	708	14	a	a	NOUN
ap-961	708	15	)	)	PUNCT
ap-961	708	16	,	,	PUNCT
ap-961	708	17	powers	power	NOUN
ap-961	708	18	of	of	ADP
ap-961	708	19	d	d	PROPN
ap-961	708	20	and	and	CCONJ
ap-961	708	21	their	their	PRON
ap-961	708	22	commutators	commutator	NOUN
ap-961	708	23	with	with	ADP
ap-961	708	24	d	d	PROPN
ap-961	708	25	and	and	CCONJ
ap-961	708	26	d	d	NOUN
ap-961	708	27	.	.	PUNCT
ap-961	709	1	definition	definition	NOUN
ap-961	709	2	6.7	6.7	NUM
ap-961	709	3	:	:	PUNCT
ap-961	709	4	a	a	DET
ap-961	709	5	real	real	ADJ
ap-961	709	6	spectral	spectral	ADJ
ap-961	709	7	triple	triple	NOUN
ap-961	709	8	of	of	ADP
ap-961	709	9	ko	ko	PROPN
ap-961	709	10	-	-	PUNCT
ap-961	709	11	dimension	dimension	NOUN
ap-961	709	12	n	n	CCONJ
ap-961	709	13	mod	mod	NOUN
ap-961	709	14	8	8	NUM
ap-961	709	15	is	be	AUX
ap-961	709	16	a	a	DET
ap-961	709	17	spectral	spectral	ADJ
ap-961	709	18	triple	triple	NOUN
ap-961	709	19	with	with	ADP
ap-961	709	20	an	an	DET
ap-961	709	21	antilinear	antilinear	ADJ
ap-961	709	22	unitary	unitary	ADJ
ap-961	709	23	operator	operator	NOUN
ap-961	709	24	j	j	PROPN
ap-961	709	25	,	,	PUNCT
ap-961	709	26	j	j	PROPN
ap-961	709	27	j	j	PROPN
ap-961	709	28	*	*	PROPN
ap-961	709	29	�	�	PROPN
ap-961	709	30	1	1	NUM
ap-961	709	31	,	,	PUNCT
ap-961	709	32	such	such	ADJ
ap-961	709	33	that	that	SCONJ
ap-961	709	34	:	:	PUNCT
ap-961	709	35	dj	dj	X
ap-961	709	36	�	�	PROPN
ap-961	709	37	jd	jd	PROPN
ap-961	709	38	,	,	PUNCT
ap-961	709	39	j2	j2	PROPN
ap-961	709	40	�	�	PROPN
ap-961	709	41	'	'	PUNCT
ap-961	709	42	,	,	PUNCT
ap-961	709	43	j	j	PROPN
ap-961	709	44	�	�	PROPN
ap-961	709	45	�	�	PROPN
ap-961	709	46	''	''	PUNCT
ap-961	709	47	�	�	PROPN
ap-961	709	48	j.	j.	PROPN
ap-961	709	49	(	(	PUNCT
ap-961	709	50	3	3	NUM
ap-961	709	51	)	)	PUNCT
ap-961	709	52	and	and	CCONJ
ap-961	709	53	[	[	PUNCT
ap-961	709	54	(	(	PUNCT
ap-961	709	55	)	)	PUNCT
ap-961	709	56	,	,	PUNCT
ap-961	709	57	(	(	PUNCT
ap-961	709	58	)	)	PUNCT
ap-961	709	59	]	]	X
ap-961	709	60	j	j	PROPN
ap-961	709	61	a	a	DET
ap-961	709	62	j	j	PROPN
ap-961	709	63	b	b	PROPN
ap-961	709	64	�	�	PROPN
ap-961	709	65	�	�	PROPN
ap-961	709	66	�	�	PROPN
ap-961	709	67	0	0	NUM
ap-961	709	68	,	,	PUNCT
ap-961	709	69	[	[	X
ap-961	709	70	[	[	PUNCT
ap-961	709	71	,	,	PUNCT
ap-961	709	72	]	]	X
ap-961	709	73	,	,	PUNCT
ap-961	709	74	(	(	PUNCT
ap-961	709	75	)	)	PUNCT
ap-961	709	76	]	]	X
ap-961	709	77	d	d	X
ap-961	709	78	a	a	DET
ap-961	709	79	j	j	PROPN
ap-961	709	80	b	b	PROPN
ap-961	709	81	j	j	PROPN
ap-961	709	82	�	�	PROPN
ap-961	709	83	�	�	PROPN
ap-961	709	84	0	0	NUM
ap-961	709	85	,	,	PUNCT
ap-961	709	86	(	(	PUNCT
ap-961	709	87	4	4	X
ap-961	709	88	)	)	PUNCT
ap-961	709	89	for	for	ADP
ap-961	709	90	all	all	DET
ap-961	709	91	a	a	DET
ap-961	709	92	b	b	PROPN
ap-961	709	93	,	,	PUNCT
ap-961	709	94	�	�	PROPN
ap-961	709	95	�	�	PROPN
ap-961	709	96	,	,	PUNCT
ap-961	709	97	and	and	CCONJ
ap-961	709	98	where	where	SCONJ
ap-961	709	99	the	the	DET
ap-961	709	100	signs	sign	NOUN
ap-961	709	101	,	,	PUNCT
ap-961	709	102	'	'	PUNCT
ap-961	709	103	,	,	PUNCT
ap-961	709	104	''	''	PUNCT
ap-961	709	105	depend	depend	VERB
ap-961	709	106	on	on	ADP
ap-961	709	107	the	the	DET
ap-961	709	108	ko	ko	PROPN
ap-961	709	109	-	-	PUNCT
ap-961	709	110	dimension	dimension	NOUN
ap-961	709	111	modulo	modulo	NOUN
ap-961	709	112	8	8	NUM
ap-961	709	113	according	accord	VERB
ap-961	709	114	to	to	ADP
ap-961	709	115	the	the	DET
ap-961	709	116	following	follow	VERB
ap-961	709	117	rules	rule	NOUN
ap-961	709	118	:	:	PUNCT
ap-961	709	119	n	n	PRON
ap-961	709	120	mod	mod	ADJ
ap-961	709	121	8	8	NUM
ap-961	709	122	0	0	NUM
ap-961	709	123	1	1	NUM
ap-961	709	124	2	2	NUM
ap-961	709	125	3	3	NUM
ap-961	709	126	4	4	NUM
ap-961	709	127	5	5	NUM
ap-961	709	128	6	6	NUM
ap-961	709	129	7	7	NUM
ap-961	709	130	�	�	PROPN
ap-961	709	131	�	�	PROPN
ap-961	709	132	'	'	PART
ap-961	709	133	�	�	PROPN
ap-961	709	134	�	�	PROPN
ap-961	709	135	�	�	PROPN
ap-961	709	136	�	�	PROPN
ap-961	709	137	''	''	PUNCT
ap-961	709	138	�	�	PROPN
ap-961	709	139	�	�	PROPN
ap-961	709	140	the	the	DET
ap-961	709	141	first	first	ADJ
ap-961	709	142	of	of	ADP
ap-961	709	143	conditions	condition	NOUN
ap-961	709	144	4	4	NUM
ap-961	709	145	states	state	NOUN
ap-961	709	146	that	that	SCONJ
ap-961	709	147	conjugation	conjugation	NOUN
ap-961	709	148	by	by	ADP
ap-961	709	149	j	j	PROPN
ap-961	709	150	maps	map	VERB
ap-961	709	151	the	the	DET
ap-961	709	152	algebra	algebra	NOUN
ap-961	709	153	to	to	ADP
ap-961	709	154	its	its	PRON
ap-961	709	155	commutant	commutant	NOUN
ap-961	709	156	,	,	PUNCT
ap-961	709	157	whereas	whereas	SCONJ
ap-961	709	158	the	the	DET
ap-961	709	159	second	second	ADJ
ap-961	709	160	condition	condition	NOUN
ap-961	709	161	,	,	PUNCT
ap-961	709	162	the	the	DET
ap-961	709	163	so	so	ADV
ap-961	709	164	-	-	PUNCT
ap-961	709	165	called	call	VERB
ap-961	709	166	order	order	NOUN
ap-961	709	167	-	-	PUNCT
ap-961	709	168	one	one	NUM
ap-961	709	169	condition	condition	NOUN
ap-961	709	170	,	,	PUNCT
ap-961	709	171	states	state	VERB
ap-961	709	172	that	that	SCONJ
ap-961	709	173	the	the	DET
ap-961	709	174	one	one	NUM
ap-961	709	175	forms	form	NOUN
ap-961	709	176	commute	commute	NOUN
ap-961	709	177	as	as	ADV
ap-961	709	178	well	well	ADV
ap-961	709	179	with	with	ADP
ap-961	709	180	the	the	DET
ap-961	709	181	commutant	commutant	NOUN
ap-961	709	182	of	of	ADP
ap-961	709	183	�	�	PROPN
ap-961	709	184	.	.	PUNCT
ap-961	710	1	the	the	DET
ap-961	710	2	following	follow	VERB
ap-961	710	3	establishes	establish	VERB
ap-961	710	4	(	(	PUNCT
ap-961	710	5	precisely	precisely	ADV
ap-961	710	6	)	)	PUNCT
ap-961	710	7	the	the	DET
ap-961	710	8	relation	relation	NOUN
ap-961	710	9	of	of	ADP
ap-961	710	10	spectral	spectral	ADJ
ap-961	710	11	triples	triple	NOUN
ap-961	710	12	to	to	ADP
ap-961	710	13	classical	classical	ADJ
ap-961	710	14	differential	differential	NOUN
ap-961	710	15	geometry	geometry	NOUN
ap-961	710	16	(	(	PUNCT
ap-961	710	17	for	for	ADP
ap-961	710	18	proof	proof	NOUN
ap-961	710	19	see	see	VERB
ap-961	710	20	[	[	X
ap-961	710	21	2	2	NUM
ap-961	710	22	]	]	NUM
ap-961	710	23	)	)	PUNCT
ap-961	710	24	.	.	PUNCT
ap-961	711	1	theorem	theorem	VERB
ap-961	711	2	6.8	6.8	NUM
ap-961	711	3	:	:	PUNCT
ap-961	711	4	if	if	SCONJ
ap-961	711	5	�	�	PROPN
ap-961	711	6	�	�	PROPN
ap-961	711	7	c	c	PROPN
ap-961	711	8	m	m	PROPN
ap-961	711	9	(	(	PUNCT
ap-961	711	10	)	)	PUNCT
ap-961	711	11	,	,	PUNCT
ap-961	711	12	m	m	PROPN
ap-961	711	13	is	be	AUX
ap-961	711	14	a	a	DET
ap-961	711	15	spin	spin	NOUN
ap-961	711	16	riemannian	riemannian	ADJ
ap-961	711	17	compact	compact	ADJ
ap-961	711	18	manifold	manifold	NOUN
ap-961	711	19	,	,	PUNCT
ap-961	711	20	s	s	VERB
ap-961	711	21	is	be	AUX
ap-961	711	22	a	a	DET
ap-961	711	23	spinor	spinor	NOUN
ap-961	711	24	bundle	bundle	NOUN
ap-961	711	25	over	over	ADP
ap-961	711	26	m	m	PROPN
ap-961	711	27	,	,	PUNCT
ap-961	711	28	�	�	PROPN
ap-961	711	29	�	�	PROPN
ap-961	711	30	l	l	PROPN
ap-961	711	31	s2	s2	PROPN
ap-961	711	32	(	(	PUNCT
ap-961	711	33	)	)	PUNCT
ap-961	711	34	(	(	PUNCT
ap-961	711	35	summable	summable	ADJ
ap-961	711	36	sections	section	NOUN
ap-961	711	37	of	of	ADP
ap-961	711	38	spinor	spinor	NOUN
ap-961	711	39	bundle	bundle	NOUN
ap-961	711	40	)	)	PUNCT
ap-961	711	41	and	and	CCONJ
ap-961	711	42	d	d	PROPN
ap-961	711	43	is	be	AUX
ap-961	711	44	the	the	DET
ap-961	711	45	dirac	dirac	NOUN
ap-961	711	46	operator	operator	NOUN
ap-961	711	47	on	on	ADP
ap-961	711	48	m	m	NOUN
ap-961	711	49	then	then	ADV
ap-961	711	50	to	to	PART
ap-961	711	51	(	(	PUNCT
ap-961	711	52	,	,	PUNCT
ap-961	711	53	,	,	PUNCT
ap-961	711	54	)	)	PUNCT
ap-961	711	55	�	�	PROPN
ap-961	711	56	�	�	PROPN
ap-961	712	1	d	d	PROPN
ap-961	712	2	is	be	AUX
ap-961	712	3	a	a	DET
ap-961	712	4	spectral	spectral	ADJ
ap-961	712	5	triple	triple	NOUN
ap-961	712	6	(	(	PUNCT
ap-961	712	7	with	with	ADP
ap-961	712	8	a	a	DET
ap-961	712	9	real	real	ADJ
ap-961	712	10	structure	structure	NOUN
ap-961	712	11	)	)	PUNCT
ap-961	712	12	,	,	PUNCT
ap-961	712	13	of	of	ADP
ap-961	712	14	ko	ko	PROPN
ap-961	712	15	and	and	CCONJ
ap-961	712	16	metric	metric	ADJ
ap-961	712	17	dimension	dimension	NOUN
ap-961	712	18	dim(m	dim(m	PROPN
ap-961	712	19	)	)	PUNCT
ap-961	712	20	.	.	PUNCT
ap-961	713	1	even	even	ADV
ap-961	713	2	more	more	ADV
ap-961	713	3	interesting	interesting	ADJ
ap-961	713	4	is	be	AUX
ap-961	713	5	the	the	DET
ap-961	713	6	reconstruction	reconstruction	NOUN
ap-961	713	7	theorem	theorem	VERB
ap-961	713	8	,	,	PUNCT
ap-961	713	9	which	which	PRON
ap-961	713	10	states	state	VERB
ap-961	713	11	the	the	DET
ap-961	713	12	inverse	inverse	NOUN
ap-961	713	13	:	:	PUNCT
ap-961	713	14	theorem	theorem	VERB
ap-961	713	15	6.9	6.9	NUM
ap-961	713	16	:	:	PUNCT
ap-961	713	17	if	if	SCONJ
ap-961	713	18	�	�	PROPN
ap-961	713	19	is	be	AUX
ap-961	713	20	a	a	DET
ap-961	713	21	commutative	commutative	ADJ
ap-961	713	22	algebra	algebra	NOUN
ap-961	713	23	and	and	CCONJ
ap-961	713	24	(	(	PUNCT
ap-961	713	25	,	,	PUNCT
ap-961	713	26	,	,	PUNCT
ap-961	713	27	)	)	PUNCT
ap-961	713	28	�	�	PROPN
ap-961	713	29	�	�	PROPN
ap-961	713	30	d	d	PROPN
ap-961	713	31	is	be	AUX
ap-961	713	32	a	a	DET
ap-961	713	33	spectral	spectral	ADJ
ap-961	713	34	triple	triple	NOUN
ap-961	713	35	satisfying	satisfy	VERB
ap-961	713	36	all	all	DET
ap-961	713	37	required	require	VERB
ap-961	713	38	conditions	condition	NOUN
ap-961	713	39	then	then	ADV
ap-961	713	40	m	m	VERB
ap-961	713	41	is	be	AUX
ap-961	713	42	a	a	DET
ap-961	713	43	spin	spin	NOUN
ap-961	713	44	manifold	manifold	ADJ
ap-961	713	45	and	and	CCONJ
ap-961	713	46	�	�	PROPN
ap-961	713	47	�	�	PROPN
ap-961	713	48	c	c	PROPN
ap-961	713	49	m	m	PROPN
ap-961	713	50	(	(	PUNCT
ap-961	713	51	)	)	PUNCT
ap-961	713	52	,	,	PUNCT
ap-961	713	53	�	�	PROPN
ap-961	713	54	�	�	PROPN
ap-961	713	55	l	l	PROPN
ap-961	713	56	s2	s2	PROPN
ap-961	713	57	(	(	PUNCT
ap-961	713	58	)	)	PUNCT
ap-961	713	59	,	,	PUNCT
ap-961	713	60	and	and	CCONJ
ap-961	713	61	d	d	NOUN
ap-961	713	62	is	be	AUX
ap-961	713	63	the	the	DET
ap-961	713	64	dirac	dirac	NOUN
ap-961	713	65	operator	operator	NOUN
ap-961	713	66	on	on	ADP
ap-961	713	67	m.	m.	NOUN
ap-961	713	68	a	a	DET
ap-961	713	69	weaker	weak	ADJ
ap-961	713	70	version	version	NOUN
ap-961	713	71	of	of	ADP
ap-961	713	72	the	the	DET
ap-961	713	73	theorem	theorem	NOUN
ap-961	713	74	is	be	AUX
ap-961	713	75	due	due	ADJ
ap-961	713	76	to	to	ADP
ap-961	713	77	bondia	bondia	NOUN
ap-961	713	78	&	&	CCONJ
ap-961	714	1	varilly	varilly	ADV
ap-961	714	2	[	[	X
ap-961	714	3	2	2	NUM
ap-961	714	4	]	]	PUNCT
ap-961	714	5	,	,	PUNCT
ap-961	714	6	a	a	DET
ap-961	714	7	proof	proof	NOUN
ap-961	714	8	of	of	ADP
ap-961	714	9	the	the	DET
ap-961	714	10	full	full	ADJ
ap-961	714	11	version	version	NOUN
ap-961	714	12	was	be	AUX
ap-961	714	13	proposed	propose	VERB
ap-961	714	14	by	by	ADP
ap-961	714	15	varilly	varilly	PROPN
ap-961	714	16	&	&	CCONJ
ap-961	714	17	rennie	rennie	PROPN
ap-961	714	18	[	[	X
ap-961	714	19	44	44	NUM
ap-961	714	20	]	]	PUNCT
ap-961	714	21	and	and	CCONJ
ap-961	714	22	then	then	ADV
ap-961	714	23	improved	improve	VERB
ap-961	714	24	by	by	ADP
ap-961	714	25	connes	conne	NOUN
ap-961	714	26	.	.	PUNCT
ap-961	715	1	we	we	PRON
ap-961	715	2	did	do	AUX
ap-961	715	3	not	not	PART
ap-961	715	4	present	present	VERB
ap-961	715	5	here	here	ADV
ap-961	715	6	any	any	DET
ap-961	715	7	further	further	ADJ
ap-961	715	8	requirements	requirement	NOUN
ap-961	715	9	for	for	ADP
ap-961	715	10	the	the	DET
ap-961	715	11	spectral	spectral	ADJ
ap-961	715	12	triples	triple	NOUN
ap-961	715	13	.	.	PUNCT
ap-961	716	1	this	this	PRON
ap-961	716	2	includes	include	VERB
ap-961	716	3	further	further	ADJ
ap-961	716	4	algebraic	algebraic	ADJ
ap-961	716	5	conditions	condition	NOUN
ap-961	716	6	,	,	PUNCT
ap-961	716	7	like	like	ADP
ap-961	716	8	the	the	DET
ap-961	716	9	existence	existence	NOUN
ap-961	716	10	of	of	ADP
ap-961	716	11	a	a	DET
ap-961	716	12	certain	certain	ADJ
ap-961	716	13	hochschild	hochschild	ADJ
ap-961	716	14	cycle	cycle	NOUN
ap-961	716	15	,	,	PUNCT
ap-961	716	16	which	which	PRON
ap-961	716	17	is	be	AUX
ap-961	716	18	mapped	map	VERB
ap-961	716	19	to	to	ADP
ap-961	716	20	�	�	PROPN
ap-961	716	21	or	or	CCONJ
ap-961	716	22	1	1	NUM
ap-961	716	23	(	(	PUNCT
ap-961	716	24	depending	depend	VERB
ap-961	716	25	on	on	ADP
ap-961	716	26	the	the	DET
ap-961	716	27	dimension	dimension	NOUN
ap-961	716	28	)	)	PUNCT
ap-961	716	29	,	,	PUNCT
ap-961	716	30	or	or	CCONJ
ap-961	716	31	the	the	DET
ap-961	716	32	poincaré	poincaré	PROPN
ap-961	716	33	duality	duality	NOUN
ap-961	716	34	in	in	ADP
ap-961	716	35	k	k	NOUN
ap-961	716	36	-	-	NOUN
ap-961	716	37	theory	theory	NOUN
ap-961	716	38	.	.	PUNCT
ap-961	717	1	there	there	PRON
ap-961	717	2	are	be	VERB
ap-961	717	3	also	also	ADV
ap-961	717	4	some	some	DET
ap-961	717	5	very	very	ADV
ap-961	717	6	restrictive	restrictive	ADJ
ap-961	717	7	conditions	condition	NOUN
ap-961	717	8	,	,	PUNCT
ap-961	717	9	more	more	ADJ
ap-961	717	10	of	of	ADP
ap-961	717	11	the	the	DET
ap-961	717	12	“	"	PUNCT
ap-961	717	13	analysis	analysis	NOUN
ap-961	717	14	”	"	PUNCT
ap-961	717	15	type	type	NOUN
ap-961	717	16	.	.	PUNCT
ap-961	718	1	they	they	PRON
ap-961	718	2	ensure	ensure	VERB
ap-961	718	3	,	,	PUNCT
ap-961	718	4	for	for	ADP
ap-961	718	5	instance	instance	NOUN
ap-961	718	6	,	,	PUNCT
ap-961	718	7	that	that	SCONJ
ap-961	718	8	the	the	DET
ap-961	718	9	algebra	algebra	NOUN
ap-961	718	10	is	be	AUX
ap-961	718	11	smooth	smooth	ADJ
ap-961	718	12	with	with	ADP
ap-961	718	13	respect	respect	NOUN
ap-961	718	14	to	to	ADP
ap-961	718	15	the	the	DET
ap-961	718	16	derivation	derivation	NOUN
ap-961	718	17	given	give	VERB
ap-961	718	18	by	by	ADP
ap-961	718	19	the	the	DET
ap-961	718	20	commutators	commutator	NOUN
ap-961	718	21	with	with	ADP
ap-961	718	22	d	d	PROPN
ap-961	718	23	and	and	CCONJ
ap-961	718	24	d	d	NOUN
ap-961	718	25	,	,	PUNCT
ap-961	718	26	and	and	CCONJ
ap-961	718	27	the	the	DET
ap-961	718	28	suitable	suitable	ADJ
ap-961	718	29	domain	domain	NOUN
ap-961	718	30	of	of	ADP
ap-961	718	31	definition	definition	NOUN
ap-961	718	32	of	of	ADP
ap-961	718	33	these	these	DET
ap-961	718	34	operators	operator	NOUN
ap-961	718	35	on	on	ADP
ap-961	718	36	the	the	DET
ap-961	718	37	hilbert	hilbert	NOUN
ap-961	718	38	space	space	NOUN
ap-961	718	39	is	be	AUX
ap-961	718	40	a	a	DET
ap-961	718	41	projective	projective	ADJ
ap-961	718	42	module	module	NOUN
ap-961	718	43	over	over	ADP
ap-961	718	44	this	this	DET
ap-961	718	45	algebra	algebra	NOUN
ap-961	718	46	.	.	PUNCT
ap-961	719	1	although	although	SCONJ
ap-961	719	2	all	all	PRON
ap-961	719	3	of	of	ADP
ap-961	719	4	this	this	PRON
ap-961	719	5	plays	play	VERB
ap-961	719	6	a	a	DET
ap-961	719	7	crucial	crucial	ADJ
ap-961	719	8	role	role	NOUN
ap-961	719	9	in	in	ADP
ap-961	719	10	the	the	DET
ap-961	719	11	reconstruction	reconstruction	NOUN
ap-961	719	12	theorem	theorem	NOUN
ap-961	719	13	,	,	PUNCT
ap-961	719	14	it	it	PRON
ap-961	719	15	is	be	AUX
ap-961	719	16	not	not	PART
ap-961	719	17	certain	certain	ADJ
ap-961	719	18	that	that	SCONJ
ap-961	719	19	the	the	DET
ap-961	719	20	formulation	formulation	NOUN
ap-961	719	21	of	of	ADP
ap-961	719	22	these	these	DET
ap-961	719	23	requirements	requirement	NOUN
ap-961	719	24	is	be	AUX
ap-961	719	25	in	in	ADP
ap-961	719	26	its	its	PRON
ap-961	719	27	final	final	ADJ
ap-961	719	28	form	form	NOUN
ap-961	719	29	in	in	ADP
ap-961	719	30	the	the	DET
ap-961	719	31	noncommutative	noncommutative	ADJ
ap-961	719	32	situation	situation	NOUN
ap-961	719	33	.	.	PUNCT
ap-961	720	1	in	in	ADP
ap-961	720	2	fact	fact	NOUN
ap-961	720	3	,	,	PUNCT
ap-961	720	4	some	some	PRON
ap-961	720	5	of	of	ADP
ap-961	720	6	the	the	DET
ap-961	720	7	recent	recent	ADJ
ap-961	720	8	examples	example	NOUN
ap-961	720	9	coming	come	VERB
ap-961	720	10	from	from	ADP
ap-961	720	11	q	q	ADJ
ap-961	720	12	-	-	PUNCT
ap-961	720	13	deformed	deform	VERB
ap-961	720	14	spaces	space	NOUN
ap-961	720	15	can	can	AUX
ap-961	720	16	hardly	hardly	ADV
ap-961	720	17	meet	meet	VERB
ap-961	720	18	these	these	DET
ap-961	720	19	requirement	requirement	NOUN
ap-961	720	20	,	,	PUNCT
ap-961	720	21	yet	yet	CCONJ
ap-961	720	22	they	they	PRON
ap-961	720	23	have	have	VERB
ap-961	720	24	reasonable	reasonable	ADJ
ap-961	720	25	dirac	dirac	NOUN
ap-961	720	26	operators	operator	NOUN
ap-961	720	27	.	.	PUNCT
ap-961	721	1	6.3.1	6.3.1	NUM
ap-961	721	2	the	the	DET
ap-961	721	3	use	use	NOUN
ap-961	721	4	of	of	ADP
ap-961	721	5	spectral	spectral	ADJ
ap-961	721	6	geometries	geometry	NOUN
ap-961	721	7	and	and	CCONJ
ap-961	721	8	examples	example	NOUN
ap-961	721	9	if	if	SCONJ
ap-961	721	10	spectral	spectral	ADJ
ap-961	721	11	geometries	geometry	NOUN
ap-961	721	12	are	be	AUX
ap-961	721	13	applicable	applicable	ADJ
ap-961	721	14	to	to	ADP
ap-961	721	15	noncommutative	noncommutative	ADJ
ap-961	721	16	algebras	algebras	PROPN
ap-961	721	17	then	then	ADV
ap-961	721	18	we	we	PRON
ap-961	721	19	should	should	AUX
ap-961	721	20	learn	learn	VERB
ap-961	721	21	how	how	SCONJ
ap-961	721	22	to	to	PART
ap-961	721	23	extract	extract	VERB
ap-961	721	24	the	the	DET
ap-961	721	25	geometric	geometric	ADJ
ap-961	721	26	data	datum	NOUN
ap-961	721	27	.	.	PUNCT
ap-961	722	1	we	we	PRON
ap-961	722	2	already	already	ADV
ap-961	722	3	know	know	VERB
ap-961	722	4	how	how	SCONJ
ap-961	722	5	to	to	PART
ap-961	722	6	obtain	obtain	VERB
ap-961	722	7	a	a	DET
ap-961	722	8	differential	differential	ADJ
ap-961	722	9	graded	grade	VERB
ap-961	722	10	algebra	algebra	NOUN
ap-961	722	11	.	.	PUNCT
ap-961	723	1	but	but	CCONJ
ap-961	723	2	,	,	PUNCT
ap-961	723	3	how	how	SCONJ
ap-961	723	4	can	can	AUX
ap-961	723	5	we	we	PRON
ap-961	723	6	get	get	VERB
ap-961	723	7	the	the	DET
ap-961	723	8	metric	metric	NOUN
ap-961	723	9	?	?	PUNCT
ap-961	724	1	it	it	PRON
ap-961	724	2	appears	appear	VERB
ap-961	724	3	that	that	SCONJ
ap-961	724	4	a	a	DET
ap-961	724	5	simple	simple	ADJ
ap-961	724	6	formula	formula	NOUN
ap-961	724	7	allows	allow	VERB
ap-961	724	8	us	we	PRON
ap-961	724	9	to	to	PART
ap-961	724	10	recover	recover	VERB
ap-961	724	11	the	the	DET
ap-961	724	12	distances	distance	NOUN
ap-961	724	13	on	on	ADP
ap-961	724	14	the	the	DET
ap-961	724	15	manifold	manifold	NOUN
ap-961	724	16	(	(	PUNCT
ap-961	724	17	and	and	CCONJ
ap-961	724	18	in	in	ADP
ap-961	724	19	general	general	ADJ
ap-961	724	20	,	,	PUNCT
ap-961	724	21	also	also	ADV
ap-961	724	22	introduce	introduce	VERB
ap-961	724	23	a	a	DET
ap-961	724	24	metric	metric	NOUN
ap-961	724	25	on	on	ADP
ap-961	724	26	the	the	DET
ap-961	724	27	space	space	NOUN
ap-961	724	28	of	of	ADP
ap-961	724	29	states	state	NOUN
ap-961	724	30	):	):	PUNCT
ap-961	724	31	d	d	X
ap-961	724	32	x	x	VERB
ap-961	724	33	y	y	NOUN
ap-961	724	34	x	x	PROPN
ap-961	724	35	a	a	DET
ap-961	724	36	y	y	PROPN
ap-961	724	37	a	a	PROPN
ap-961	724	38	d	d	X
ap-961	724	39	a	a	DET
ap-961	724	40	(	(	PUNCT
ap-961	724	41	,	,	PUNCT
ap-961	724	42	)	)	PUNCT
ap-961	724	43	sup	sup	NOUN
ap-961	724	44	(	(	PUNCT
ap-961	724	45	)	)	PUNCT
ap-961	724	46	(	(	PUNCT
ap-961	724	47	)	)	PUNCT
ap-961	724	48	[	[	PUNCT
ap-961	724	49	,	,	PUNCT
ap-961	724	50	]	]	PUNCT
ap-961	724	51	�	�	PROPN
ap-961	724	52	�	�	PROPN
ap-961	724	53	)	)	PUNCT
ap-961	724	54	1	1	NUM
ap-961	724	55	,	,	PUNCT
ap-961	724	56	(	(	PUNCT
ap-961	724	57	5	5	NUM
ap-961	724	58	)	)	PUNCT
ap-961	724	59	where	where	SCONJ
ap-961	724	60	x	x	PUNCT
ap-961	724	61	y	y	PROPN
ap-961	724	62	m	m	PROPN
ap-961	724	63	,	,	PUNCT
ap-961	724	64	�	�	PROPN
ap-961	724	65	and	and	CCONJ
ap-961	724	66	a	a	DET
ap-961	724	67	c	c	PROPN
ap-961	724	68	m	m	VERB
ap-961	724	69	�	�	PROPN
ap-961	724	70	(	(	PUNCT
ap-961	724	71	)	)	PUNCT
ap-961	724	72	.	.	PUNCT
ap-961	725	1	we	we	PRON
ap-961	725	2	already	already	ADV
ap-961	725	3	know	know	VERB
ap-961	725	4	that	that	SCONJ
ap-961	725	5	a	a	DET
ap-961	725	6	a	a	DET
ap-961	725	7	d	d	PROPN
ap-961	725	8	n2	n2	PROPN
ap-961	725	9	�	�	PROPN
ap-961	725	10	�	�	PROPN
ap-961	725	11	tr	tr	VERB
ap-961	725	12	�	�	NOUN
ap-961	725	13	�	�	PROPN
ap-961	725	14	(	(	PUNCT
ap-961	725	15	)	)	PUNCT
ap-961	725	16	is	be	AUX
ap-961	725	17	well	well	ADV
ap-961	725	18	-	-	PUNCT
ap-961	725	19	defined	define	VERB
ap-961	725	20	.	.	PUNCT
ap-961	726	1	once	once	SCONJ
ap-961	726	2	we	we	PRON
ap-961	726	3	are	be	AUX
ap-961	726	4	sure	sure	ADJ
ap-961	726	5	that	that	SCONJ
ap-961	726	6	the	the	DET
ap-961	726	7	dixmier	dixmi	ADJ
ap-961	726	8	trace	trace	NOUN
ap-961	726	9	of	of	ADP
ap-961	726	10	d	d	PROPN
ap-961	726	11	n	n	CCONJ
ap-961	726	12	�	�	PROPN
ap-961	726	13	exists	exist	VERB
ap-961	726	14	we	we	PRON
ap-961	726	15	can	can	AUX
ap-961	726	16	use	use	VERB
ap-961	726	17	this	this	PRON
ap-961	726	18	as	as	ADP
ap-961	726	19	a	a	DET
ap-961	726	20	definition	definition	NOUN
ap-961	726	21	of	of	ADP
ap-961	726	22	the	the	DET
ap-961	726	23	integral	integral	ADJ
ap-961	726	24	on	on	ADP
ap-961	726	25	the	the	DET
ap-961	726	26	manifold	manifold	NOUN
ap-961	726	27	.	.	PUNCT
ap-961	727	1	summarizing	summarize	VERB
ap-961	727	2	–	–	PUNCT
ap-961	727	3	all	all	DET
ap-961	727	4	practical	practical	ADJ
ap-961	727	5	information	information	NOUN
ap-961	727	6	of	of	ADP
ap-961	727	7	geometry	geometry	NOUN
ap-961	727	8	is	be	AUX
ap-961	727	9	encoded	encode	VERB
ap-961	727	10	in	in	ADP
ap-961	727	11	the	the	DET
ap-961	727	12	datum	datum	NOUN
ap-961	727	13	of	of	ADP
ap-961	727	14	the	the	DET
ap-961	727	15	spectral	spectral	ADJ
ap-961	727	16	triple	triple	NOUN
ap-961	727	17	.	.	PUNCT
ap-961	728	1	example	example	NOUN
ap-961	729	1	6.10	6.10	NUM
ap-961	729	2	:	:	PUNCT
ap-961	729	3	we	we	PRON
ap-961	729	4	might	might	AUX
ap-961	729	5	come	come	VERB
ap-961	729	6	back	back	ADV
ap-961	729	7	to	to	ADP
ap-961	729	8	the	the	DET
ap-961	729	9	canonical	canonical	ADJ
ap-961	729	10	example	example	NOUN
ap-961	729	11	of	of	ADP
ap-961	729	12	two	two	NUM
ap-961	729	13	points	point	NOUN
ap-961	729	14	.	.	PUNCT
ap-961	730	1	as	as	ADP
ap-961	730	2	the	the	DET
ap-961	730	3	dirac	dirac	NOUN
ap-961	730	4	operator	operator	NOUN
ap-961	730	5	we	we	PRON
ap-961	730	6	take	take	VERB
ap-961	730	7	an	an	DET
ap-961	730	8	arbitrary	arbitrary	ADJ
ap-961	730	9	self	self	NOUN
ap-961	730	10	-	-	PUNCT
ap-961	730	11	adjoint	adjoint	NOUN
ap-961	730	12	off	off	ADP
ap-961	730	13	-	-	PUNCT
ap-961	730	14	diagonal	diagonal	ADJ
ap-961	730	15	operator	operator	NOUN
ap-961	730	16	on	on	ADP
ap-961	730	17	�	�	PROPN
ap-961	730	18	2	2	NUM
ap-961	730	19	.	.	PUNCT
ap-961	730	20	clearly	clearly	ADV
ap-961	730	21	all	all	DET
ap-961	730	22	conditions	condition	NOUN
ap-961	730	23	of	of	ADP
ap-961	730	24	the	the	DET
ap-961	730	25	spectral	spectral	ADJ
ap-961	730	26	triple	triple	NOUN
ap-961	730	27	are	be	AUX
ap-961	730	28	fulfilled	fulfil	VERB
ap-961	730	29	:	:	PUNCT
ap-961	730	30	even	even	ADV
ap-961	730	31	more	more	ADJ
ap-961	730	32	,	,	PUNCT
ap-961	730	33	we	we	PRON
ap-961	730	34	can	can	AUX
ap-961	730	35	construct	construct	VERB
ap-961	730	36	a	a	DET
ap-961	730	37	real	real	ADJ
ap-961	730	38	spectral	spectral	ADJ
ap-961	730	39	triple	triple	NOUN
ap-961	730	40	,	,	PUNCT
ap-961	730	41	of	of	ADP
ap-961	730	42	course	course	NOUN
ap-961	730	43	,	,	PUNCT
ap-961	730	44	of	of	ADP
ap-961	730	45	dimension	dimension	NOUN
ap-961	730	46	0	0	NUM
ap-961	730	47	.	.	PUNCT
ap-961	731	1	for	for	ADP
ap-961	731	2	this	this	PRON
ap-961	731	3	,	,	PUNCT
ap-961	731	4	we	we	PRON
ap-961	731	5	double	double	VERB
ap-961	731	6	the	the	DET
ap-961	731	7	hilbert	hilbert	NOUN
ap-961	731	8	space	space	NOUN
ap-961	731	9	to	to	ADP
ap-961	731	10	�	�	PROPN
ap-961	731	11	�	�	PROPN
ap-961	731	12	2	2	NUM
ap-961	731	13	2	2	NUM
ap-961	731	14	�	�	PROPN
ap-961	731	15	.	.	PUNCT
ap-961	732	1	it	it	PRON
ap-961	732	2	is	be	AUX
ap-961	732	3	now	now	ADV
ap-961	732	4	convenient	convenient	ADJ
ap-961	732	5	to	to	PART
ap-961	732	6	write	write	VERB
ap-961	732	7	every	every	DET
ap-961	732	8	operator	operator	NOUN
ap-961	732	9	in	in	ADP
ap-961	732	10	a	a	DET
ap-961	732	11	block	block	NOUN
ap-961	732	12	diagonal	diagonal	ADJ
ap-961	732	13	form	form	NOUN
ap-961	732	14	:	:	PUNCT
ap-961	732	15	�	�	PROPN
ap-961	732	16	(	(	PUNCT
ap-961	732	17	)	)	PUNCT
ap-961	732	18	a	a	DET
ap-961	732	19	a	a	DET
ap-961	732	20	a	a	DET
ap-961	732	21	�	�	PROPN
ap-961	732	22	�	�	PROPN
ap-961	732	23	�	�	PROPN
ap-961	732	24	�	�	PROPN
ap-961	732	25	�	�	PROPN
ap-961	732	26	�	�	PROPN
ap-961	732	27	�	�	PROPN
ap-961	732	28	0	0	NUM
ap-961	732	29	0	0	NUM
ap-961	732	30	,	,	PUNCT
ap-961	732	31	�	�	PROPN
ap-961	732	32	�	�	PROPN
ap-961	732	33	�	�	PROPN
ap-961	732	34	�	�	PROPN
ap-961	732	35	�	�	PROPN
ap-961	732	36	�	�	PROPN
ap-961	732	37	�	�	PROPN
ap-961	732	38	�	�	PROPN
ap-961	732	39	�	�	PROPN
ap-961	732	40	1	1	NUM
ap-961	732	41	0	0	NUM
ap-961	732	42	0	0	NUM
ap-961	732	43	1	1	NUM
ap-961	732	44	,	,	PUNCT
ap-961	733	1	j	j	PROPN
ap-961	733	2	j	j	PROPN
ap-961	733	3	j	j	PROPN
ap-961	733	4	�	�	PROPN
ap-961	733	5	�	�	PROPN
ap-961	733	6	�	�	PROPN
ap-961	733	7	�	�	PROPN
ap-961	733	8	�	�	PROPN
ap-961	733	9	�	�	PROPN
ap-961	733	10	�	�	PROPN
ap-961	733	11	0	0	NUM
ap-961	733	12	0	0	NUM
ap-961	733	13	0	0	NUM
ap-961	733	14	0	0	NUM
ap-961	733	15	,	,	PUNCT
ap-961	733	16	d	d	PROPN
ap-961	733	17	d	d	X
ap-961	733	18	d	d	X
ap-961	733	19	�	�	PROPN
ap-961	733	20	�	�	PROPN
ap-961	733	21	�	�	PROPN
ap-961	733	22	�	�	PROPN
ap-961	733	23	�	�	PROPN
ap-961	733	24	�	�	PROPN
ap-961	733	25	�	�	PROPN
ap-961	733	26	�	�	PROPN
ap-961	733	27	�	�	PROPN
ap-961	733	28	0	0	NUM
ap-961	733	29	0	0	NUM
ap-961	733	30	0	0	NUM
ap-961	733	31	0	0	NUM
ap-961	733	32	†	†	NOUN
ap-961	733	33	,	,	PUNCT
ap-961	733	34	52	52	NUM
ap-961	733	35	©	©	PROPN
ap-961	733	36	czech	czech	PROPN
ap-961	733	37	technical	technical	PROPN
ap-961	733	38	university	university	PROPN
ap-961	733	39	publishing	publishing	NOUN
ap-961	733	40	house	house	NOUN
ap-961	733	41	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	733	42	acta	acta	PROPN
ap-961	733	43	polytechnica	polytechnica	PROPN
ap-961	733	44	vol	vol	NOUN
ap-961	733	45	.	.	PUNCT
ap-961	734	1	48	48	NUM
ap-961	734	2	no	no	NOUN
ap-961	734	3	.	.	PUNCT
ap-961	735	1	2/2008	2/2008	NUM
ap-961	735	2	where	where	SCONJ
ap-961	735	3	j0	j0	PROPN
ap-961	735	4	is	be	AUX
ap-961	735	5	the	the	DET
ap-961	735	6	standard	standard	ADJ
ap-961	735	7	complex	complex	ADJ
ap-961	735	8	conjugation	conjugation	NOUN
ap-961	735	9	on	on	ADP
ap-961	735	10	�	�	PROPN
ap-961	735	11	2	2	NUM
ap-961	735	12	.	.	PUNCT
ap-961	736	1	the	the	DET
ap-961	736	2	condition	condition	NOUN
ap-961	736	3	d	d	PROPN
ap-961	736	4	j	j	PROPN
ap-961	736	5	j	j	PROPN
ap-961	736	6	d	d	PROPN
ap-961	736	7	�	�	PROPN
ap-961	736	8	gives	give	VERB
ap-961	736	9	:	:	PUNCT
ap-961	736	10	d	d	ADP
ap-961	736	11	dt	dt	X
ap-961	736	12	0	0	NUM
ap-961	736	13	0	0	NUM
ap-961	736	14	�	�	PROPN
ap-961	736	15	,	,	PUNCT
ap-961	736	16	whereas	whereas	SCONJ
ap-961	736	17	the	the	DET
ap-961	736	18	order	order	NOUN
ap-961	736	19	-	-	PUNCT
ap-961	736	20	one	one	NUM
ap-961	736	21	condition	condition	NOUN
ap-961	736	22	is	be	AUX
ap-961	736	23	satisfied	satisfied	ADJ
ap-961	736	24	for	for	ADP
ap-961	736	25	any	any	DET
ap-961	736	26	d.	d.	NOUN
ap-961	736	27	therefore	therefore	ADV
ap-961	736	28	the	the	DET
ap-961	736	29	spectral	spectral	ADJ
ap-961	736	30	triple	triple	NOUN
ap-961	736	31	is	be	AUX
ap-961	736	32	set	set	VERB
ap-961	736	33	by	by	ADP
ap-961	736	34	a	a	DET
ap-961	736	35	symmetric	symmetric	ADJ
ap-961	736	36	two	two	NUM
ap-961	736	37	-	-	PUNCT
ap-961	736	38	by	by	ADP
ap-961	736	39	-	-	PUNCT
ap-961	736	40	two	two	NUM
ap-961	736	41	matrix	matrix	NOUN
ap-961	736	42	d0	d0	NOUN
ap-961	736	43	.	.	PUNCT
ap-961	737	1	of	of	ADP
ap-961	737	2	course	course	NOUN
ap-961	737	3	,	,	PUNCT
ap-961	737	4	since	since	SCONJ
ap-961	737	5	the	the	DET
ap-961	737	6	diagonal	diagonal	ADJ
ap-961	737	7	entries	entry	NOUN
ap-961	737	8	do	do	AUX
ap-961	737	9	not	not	PART
ap-961	737	10	contribute	contribute	VERB
ap-961	737	11	to	to	ADP
ap-961	737	12	the	the	DET
ap-961	737	13	commutators	commutator	NOUN
ap-961	737	14	,	,	PUNCT
ap-961	737	15	it	it	PRON
ap-961	737	16	is	be	AUX
ap-961	737	17	the	the	DET
ap-961	737	18	off	off	ADJ
ap-961	737	19	-	-	PUNCT
ap-961	737	20	diagonal	diagonal	ADJ
ap-961	737	21	term	term	NOUN
ap-961	737	22	that	that	PRON
ap-961	737	23	matters	matter	VERB
ap-961	737	24	and	and	CCONJ
ap-961	737	25	that	that	SCONJ
ap-961	737	26	fixes	fix	NOUN
ap-961	737	27	,	,	PUNCT
ap-961	737	28	for	for	ADP
ap-961	737	29	instance	instance	NOUN
ap-961	737	30	,	,	PUNCT
ap-961	737	31	the	the	DET
ap-961	737	32	metric	metric	NOUN
ap-961	737	33	.	.	PUNCT
ap-961	738	1	indeed	indeed	ADV
ap-961	738	2	,	,	PUNCT
ap-961	738	3	we	we	PRON
ap-961	738	4	might	might	AUX
ap-961	738	5	ask	ask	VERB
ap-961	738	6	what	what	PRON
ap-961	738	7	the	the	DET
ap-961	738	8	distance	distance	NOUN
ap-961	738	9	is	be	AUX
ap-961	738	10	between	between	ADP
ap-961	738	11	these	these	DET
ap-961	738	12	two	two	NUM
ap-961	738	13	points	point	NOUN
ap-961	738	14	?	?	PUNCT
ap-961	739	1	using	use	VERB
ap-961	739	2	the	the	DET
ap-961	739	3	formula	formula	NOUN
ap-961	739	4	5	5	NUM
ap-961	739	5	,	,	PUNCT
ap-961	739	6	we	we	PRON
ap-961	739	7	need	need	VERB
ap-961	739	8	to	to	PART
ap-961	739	9	calculate	calculate	VERB
ap-961	739	10	f	f	PROPN
ap-961	739	11	f	f	PROPN
ap-961	739	12	(	(	PUNCT
ap-961	739	13	)	)	PUNCT
ap-961	739	14	(	(	PUNCT
ap-961	739	15	)	)	PUNCT
ap-961	739	16	�	�	PROPN
ap-961	739	17	�	�	PROPN
ap-961	739	18	,	,	PUNCT
ap-961	739	19	where	where	SCONJ
ap-961	739	20	,	,	PUNCT
ap-961	739	21	�	�	PROPN
ap-961	739	22	denotes	denote	VERB
ap-961	739	23	the	the	DET
ap-961	739	24	two	two	NUM
ap-961	739	25	points	point	NOUN
ap-961	739	26	of	of	ADP
ap-961	739	27	the	the	DET
ap-961	739	28	space	space	NOUN
ap-961	739	29	,	,	PUNCT
ap-961	739	30	for	for	ADP
ap-961	739	31	any	any	DET
ap-961	739	32	function	function	NOUN
ap-961	739	33	such	such	ADJ
ap-961	739	34	that	that	SCONJ
ap-961	739	35	[	[	PUNCT
ap-961	739	36	,	,	PUNCT
ap-961	739	37	]	]	X
ap-961	739	38	d	d	X
ap-961	739	39	f	f	PROPN
ap-961	739	40	)	)	PUNCT
ap-961	739	41	1	1	X
ap-961	739	42	.	.	PUNCT
ap-961	740	1	first	first	ADV
ap-961	740	2	,	,	PUNCT
ap-961	740	3	take	take	VERB
ap-961	740	4	f	f	PROPN
ap-961	740	5	z	z	NOUN
ap-961	740	6	we	we	PRON
ap-961	740	7	�	�	PROPN
ap-961	740	8	,	,	PUNCT
ap-961	740	9	where	where	SCONJ
ap-961	740	10	z	z	PROPN
ap-961	740	11	w	w	PROPN
ap-961	740	12	,	,	PUNCT
ap-961	740	13	�	�	PROPN
ap-961	740	14	�	�	PROPN
ap-961	740	15	and	and	CCONJ
ap-961	740	16	e	e	NOUN
ap-961	740	17	is	be	AUX
ap-961	740	18	the	the	DET
ap-961	740	19	generating	generate	VERB
ap-961	740	20	function	function	NOUN
ap-961	740	21	of	of	ADP
ap-961	740	22	the	the	DET
ap-961	740	23	algebra	algebra	PROPN
ap-961	740	24	e2	e2	PROPN
ap-961	740	25	1	1	NUM
ap-961	740	26	�	�	PROPN
ap-961	740	27	.	.	PUNCT
ap-961	741	1	we	we	PRON
ap-961	741	2	calculate	calculate	VERB
ap-961	741	3	:	:	PUNCT
ap-961	741	4	[	[	PUNCT
ap-961	741	5	,	,	PUNCT
ap-961	741	6	]	]	X
ap-961	741	7	[	[	PUNCT
ap-961	741	8	,	,	PUNCT
ap-961	741	9	]	]	X
ap-961	741	10	[	[	PUNCT
ap-961	741	11	,	,	PUNCT
ap-961	741	12	]	]	PUNCT
ap-961	741	13	†d	†d	X
ap-961	741	14	f	f	PROPN
ap-961	741	15	w	w	PROPN
ap-961	741	16	d	d	PROPN
ap-961	741	17	e	e	PROPN
ap-961	741	18	w	w	PROPN
ap-961	741	19	d	d	PROPN
ap-961	741	20	e	e	X
ap-961	741	21	�	�	PROPN
ap-961	741	22	�	�	PROPN
ap-961	741	23	�	�	PROPN
ap-961	741	24	�	�	PROPN
ap-961	741	25	�	�	PROPN
ap-961	741	26	�	�	PROPN
ap-961	741	27	�	�	PROPN
ap-961	741	28	�	�	PROPN
ap-961	741	29	�	�	PROPN
ap-961	741	30	0	0	NUM
ap-961	741	31	0	0	NUM
ap-961	741	32	0	0	NUM
ap-961	741	33	0	0	NUM
ap-961	741	34	.	.	PUNCT
ap-961	742	1	next	next	ADV
ap-961	742	2	:	:	PUNCT
ap-961	742	3	[	[	PUNCT
ap-961	742	4	,	,	PUNCT
ap-961	742	5	]	]	X
ap-961	742	6	(	(	PUNCT
ap-961	742	7	)	)	PUNCT
ap-961	742	8	(	(	PUNCT
ap-961	742	9	)	)	PUNCT
ap-961	742	10	d	d	X
ap-961	742	11	e	e	X
ap-961	742	12	d	d	X
ap-961	742	13	d0	d0	PROPN
ap-961	742	14	0	0	NUM
ap-961	742	15	12	12	NUM
ap-961	742	16	0	0	NUM
ap-961	742	17	12	12	NUM
ap-961	742	18	2	2	NUM
ap-961	742	19	0	0	NUM
ap-961	742	20	0	0	NUM
ap-961	742	21	�	�	PROPN
ap-961	742	22	�	�	PROPN
ap-961	742	23	�	�	PROPN
ap-961	742	24	�	�	PROPN
ap-961	742	25	�	�	PROPN
ap-961	742	26	�	�	PROPN
ap-961	742	27	�	�	PROPN
ap-961	742	28	�	�	PROPN
ap-961	742	29	.	.	PUNCT
ap-961	742	30	where	where	SCONJ
ap-961	742	31	we	we	PRON
ap-961	742	32	have	have	AUX
ap-961	742	33	used	use	VERB
ap-961	742	34	that	that	PRON
ap-961	742	35	(	(	PUNCT
ap-961	742	36	)	)	PUNCT
ap-961	742	37	(	(	PUNCT
ap-961	742	38	)	)	PUNCT
ap-961	742	39	d	d	SYM
ap-961	742	40	d0	d0	NOUN
ap-961	742	41	12	12	NUM
ap-961	742	42	0	0	NUM
ap-961	742	43	21	21	NUM
ap-961	742	44	�	�	PROPN
ap-961	742	45	(	(	PUNCT
ap-961	742	46	symmetric	symmetric	ADJ
ap-961	742	47	matrix	matrix	NOUN
ap-961	742	48	d0	d0	NOUN
ap-961	742	49	)	)	PUNCT
ap-961	742	50	.	.	PUNCT
ap-961	743	1	the	the	DET
ap-961	743	2	norm	norm	NOUN
ap-961	743	3	of	of	ADP
ap-961	743	4	[	[	PUNCT
ap-961	743	5	,	,	PUNCT
ap-961	743	6	]	]	X
ap-961	743	7	d	d	X
ap-961	743	8	f	f	PROPN
ap-961	743	9	is	be	AUX
ap-961	743	10	:	:	PUNCT
ap-961	743	11	[	[	PUNCT
ap-961	743	12	,	,	PUNCT
ap-961	743	13	]	]	X
ap-961	743	14	(	(	PUNCT
ap-961	743	15	)	)	PUNCT
ap-961	744	1	d	d	NOUN
ap-961	744	2	f	f	PROPN
ap-961	745	1	d	d	PROPN
ap-961	745	2	w	w	PROPN
ap-961	745	3	�	�	PROPN
ap-961	745	4	4	4	NUM
ap-961	745	5	0	0	NUM
ap-961	745	6	12	12	NUM
ap-961	745	7	.	.	PUNCT
ap-961	746	1	for	for	ADP
ap-961	746	2	the	the	DET
ap-961	746	3	function	function	NOUN
ap-961	746	4	f	f	NOUN
ap-961	746	5	given	give	VERB
ap-961	746	6	above	above	ADP
ap-961	746	7	f	f	PROPN
ap-961	746	8	f	f	PROPN
ap-961	746	9	w	w	PROPN
ap-961	746	10	(	(	PUNCT
ap-961	746	11	)	)	PUNCT
ap-961	746	12	(	(	PUNCT
ap-961	746	13	)	)	PUNCT
ap-961	746	14	�	�	PROPN
ap-961	746	15	�	�	PROPN
ap-961	746	16	�	�	PROPN
ap-961	746	17	2	2	NUM
ap-961	746	18	,	,	PUNCT
ap-961	746	19	therefore	therefore	ADV
ap-961	746	20	the	the	DET
ap-961	746	21	distance	distance	NOUN
ap-961	746	22	between	between	ADP
ap-961	746	23	the	the	DET
ap-961	746	24	two	two	NUM
ap-961	746	25	points	point	NOUN
ap-961	746	26	,	,	PUNCT
ap-961	746	27	in	in	ADP
ap-961	746	28	the	the	DET
ap-961	746	29	noncommutative	noncommutative	ADJ
ap-961	746	30	geometry	geometry	NOUN
ap-961	746	31	given	give	VERB
ap-961	746	32	by	by	ADP
ap-961	746	33	the	the	DET
ap-961	746	34	dirac	dirac	NOUN
ap-961	746	35	operator	operator	NOUN
ap-961	746	36	d	d	PROPN
ap-961	746	37	,	,	PUNCT
ap-961	746	38	is	be	AUX
ap-961	746	39	:	:	PUNCT
ap-961	746	40	dist	dist	NOUN
ap-961	746	41	(	(	PUNCT
ap-961	746	42	,	,	PUNCT
ap-961	746	43	)	)	PUNCT
ap-961	746	44	sup	sup	NOUN
ap-961	746	45	(	(	PUNCT
ap-961	746	46	)	)	PUNCT
ap-961	746	47	(	(	PUNCT
ap-961	746	48	)	)	PUNCT
ap-961	746	49	�	�	PROPN
ap-961	746	50	�	�	PROPN
ap-961	746	51	�	�	PROPN
ap-961	746	52	)	)	PUNCT
ap-961	747	1	4	4	NUM
ap-961	747	2	1	1	NUM
ap-961	747	3	0	0	NUM
ap-961	747	4	120	120	NUM
ap-961	747	5	12	12	NUM
ap-961	747	6	2	2	NUM
ap-961	747	7	1	1	NUM
ap-961	747	8	2	2	NUM
ap-961	747	9	1	1	NUM
ap-961	747	10	d	d	PROPN
ap-961	747	11	w	w	PROPN
ap-961	747	12	w	w	PROPN
ap-961	747	13	d	d	PROPN
ap-961	747	14	.	.	PUNCT
ap-961	748	1	it	it	PRON
ap-961	748	2	is	be	AUX
ap-961	748	3	a	a	DET
ap-961	748	4	more	more	ADV
ap-961	748	5	difficult	difficult	ADJ
ap-961	748	6	task	task	NOUN
ap-961	748	7	to	to	PART
ap-961	748	8	calculate	calculate	VERB
ap-961	748	9	the	the	DET
ap-961	748	10	distance	distance	NOUN
ap-961	748	11	for	for	ADP
ap-961	748	12	the	the	DET
ap-961	748	13	circle	circle	NOUN
ap-961	748	14	.	.	PUNCT
ap-961	749	1	there	there	ADV
ap-961	749	2	,	,	PUNCT
ap-961	749	3	we	we	PRON
ap-961	749	4	need	need	VERB
ap-961	749	5	to	to	PART
ap-961	749	6	consider	consider	VERB
ap-961	749	7	all	all	DET
ap-961	749	8	smooth	smooth	ADJ
ap-961	749	9	functions	function	NOUN
ap-961	749	10	on	on	ADP
ap-961	749	11	s1	s1	NOUN
ap-961	749	12	,	,	PUNCT
ap-961	749	13	represent	represent	VERB
ap-961	749	14	them	they	PRON
ap-961	749	15	on	on	ADP
ap-961	749	16	the	the	DET
ap-961	749	17	hilbert	hilbert	NOUN
ap-961	749	18	space	space	NOUN
ap-961	749	19	and	and	CCONJ
ap-961	749	20	calculate	calculate	VERB
ap-961	749	21	their	their	PRON
ap-961	749	22	operator	operator	NOUN
ap-961	749	23	norm	norm	NOUN
ap-961	749	24	!	!	PUNCT
ap-961	750	1	finally	finally	ADV
ap-961	750	2	,	,	PUNCT
ap-961	750	3	let	let	VERB
ap-961	750	4	us	we	PRON
ap-961	750	5	study	study	VERB
ap-961	750	6	the	the	DET
ap-961	750	7	best	well	ADV
ap-961	750	8	known	know	VERB
ap-961	750	9	example	example	NOUN
ap-961	750	10	of	of	ADP
ap-961	750	11	the	the	DET
ap-961	750	12	spectral	spectral	ADJ
ap-961	750	13	triple	triple	NOUN
ap-961	750	14	:	:	PUNCT
ap-961	750	15	that	that	PRON
ap-961	750	16	of	of	ADP
ap-961	750	17	the	the	DET
ap-961	750	18	noncommutative	noncommutative	ADJ
ap-961	750	19	torus	torus	NOUN
ap-961	750	20	.	.	PUNCT
ap-961	751	1	there	there	PRON
ap-961	751	2	are	be	VERB
ap-961	751	3	several	several	ADJ
ap-961	751	4	ways	way	NOUN
ap-961	751	5	to	to	PART
ap-961	751	6	guess	guess	VERB
ap-961	751	7	or	or	CCONJ
ap-961	751	8	derive	derive	VERB
ap-961	751	9	the	the	DET
ap-961	751	10	construction	construction	NOUN
ap-961	751	11	–	–	PUNCT
ap-961	751	12	we	we	PRON
ap-961	751	13	shall	shall	AUX
ap-961	751	14	,	,	PUNCT
ap-961	751	15	however	however	ADV
ap-961	751	16	,	,	PUNCT
ap-961	751	17	be	be	AUX
ap-961	751	18	very	very	ADV
ap-961	751	19	minimalistic	minimalistic	ADJ
ap-961	751	20	and	and	CCONJ
ap-961	751	21	just	just	ADV
ap-961	751	22	provide	provide	VERB
ap-961	751	23	it	it	PRON
ap-961	751	24	.	.	PUNCT
ap-961	752	1	example	example	NOUN
ap-961	752	2	6.11	6.11	NUM
ap-961	752	3	:	:	PUNCT
ap-961	752	4	we	we	PRON
ap-961	752	5	take	take	VERB
ap-961	752	6	the	the	DET
ap-961	752	7	algebra	algebra	NOUN
ap-961	752	8	of	of	ADP
ap-961	752	9	the	the	DET
ap-961	752	10	noncommutative	noncommutative	ADJ
ap-961	752	11	torus	torus	NOUN
ap-961	752	12	as	as	SCONJ
ap-961	752	13	generated	generate	VERB
ap-961	752	14	by	by	ADP
ap-961	752	15	u	u	PROPN
ap-961	752	16	,	,	PUNCT
ap-961	752	17	v	v	NOUN
ap-961	752	18	,	,	PUNCT
ap-961	752	19	with	with	ADP
ap-961	752	20	the	the	DET
ap-961	752	21	relation	relation	NOUN
ap-961	752	22	uv	uv	NOUN
ap-961	752	23	e	e	PROPN
ap-961	752	24	vui	vui	PROPN
ap-961	752	25	�	�	PROPN
ap-961	752	26	2	2	NUM
ap-961	752	27	�	�	PROPN
ap-961	752	28	�	�	PROPN
ap-961	752	29	,	,	PUNCT
ap-961	752	30	together	together	ADV
ap-961	752	31	with	with	ADP
ap-961	752	32	the	the	DET
ap-961	752	33	faithful	faithful	ADJ
ap-961	752	34	representation	representation	NOUN
ap-961	752	35	on	on	ADP
ap-961	752	36	the	the	DET
ap-961	752	37	hilbert	hilbert	PROPN
ap-961	752	38	space	space	PROPN
ap-961	752	39	�	�	PROPN
ap-961	752	40	(	(	PUNCT
ap-961	752	41	presented	present	VERB
ap-961	752	42	in	in	ADP
ap-961	752	43	example	example	NOUN
ap-961	752	44	2.9	2.9	NUM
ap-961	752	45	)	)	PUNCT
ap-961	752	46	.	.	PUNCT
ap-961	753	1	we	we	PRON
ap-961	753	2	double	double	VERB
ap-961	753	3	the	the	DET
ap-961	753	4	hilbert	hilbert	NOUN
ap-961	753	5	space	space	NOUN
ap-961	753	6	,	,	PUNCT
ap-961	753	7	take	take	VERB
ap-961	753	8	the	the	DET
ap-961	753	9	diagonal	diagonal	ADJ
ap-961	753	10	representation	representation	NOUN
ap-961	753	11	of	of	ADP
ap-961	753	12	the	the	DET
ap-961	753	13	algebra	algebra	NOUN
ap-961	753	14	and	and	CCONJ
ap-961	753	15	introduce	introduce	VERB
ap-961	753	16	the	the	DET
ap-961	753	17	operators	operator	NOUN
ap-961	753	18	j	j	PROPN
ap-961	753	19	,	,	PUNCT
ap-961	753	20	d	d	PROPN
ap-961	753	21	,	,	PUNCT
ap-961	753	22	�	�	PROPN
ap-961	753	23	as	as	ADP
ap-961	753	24	block	block	NOUN
ap-961	753	25	-	-	PUNCT
ap-961	753	26	type	type	NOUN
ap-961	753	27	operators	operator	NOUN
ap-961	753	28	.	.	PUNCT
ap-961	754	1	first	first	ADV
ap-961	754	2	we	we	PRON
ap-961	754	3	set	set	VERB
ap-961	754	4	:	:	PUNCT
ap-961	754	5	j	j	PROPN
ap-961	754	6	e	e	PROPN
ap-961	754	7	e	e	VERB
ap-961	754	8	em	em	PRON
ap-961	755	1	n	n	PROPN
ap-961	756	1	i	i	PRON
ap-961	756	2	mn	mn	PROPN
ap-961	756	3	m	m	PROPN
ap-961	756	4	n0	n0	ADJ
ap-961	756	5	2	2	NUM
ap-961	756	6	,	,	PUNCT
ap-961	756	7	,	,	PUNCT
ap-961	756	8	�	�	PROPN
ap-961	756	9	�	�	PROPN
ap-961	756	10	�	�	PROPN
ap-961	756	11	�	�	PROPN
ap-961	756	12	�	�	PROPN
ap-961	756	13	�	�	PROPN
ap-961	756	14	,	,	PUNCT
ap-961	756	15	�	�	PROPN
ap-961	756	16	e	e	NOUN
ap-961	756	17	m	m	PROPN
ap-961	756	18	in	in	ADP
ap-961	756	19	em	em	PRON
ap-961	756	20	n	n	PROPN
ap-961	756	21	m	m	NOUN
ap-961	756	22	n	n	CCONJ
ap-961	756	23	,	,	PUNCT
ap-961	756	24	,	,	PUNCT
ap-961	756	25	(	(	PUNCT
ap-961	756	26	)	)	PUNCT
ap-961	756	27	�	�	PROPN
ap-961	756	28	.	.	PUNCT
ap-961	757	1	then	then	ADV
ap-961	757	2	:	:	PUNCT
ap-961	757	3	�	�	PROPN
ap-961	757	4	�	�	PROPN
ap-961	757	5	�	�	PROPN
ap-961	757	6	�	�	PROPN
ap-961	757	7	�	�	PROPN
ap-961	757	8	�	�	PROPN
ap-961	757	9	�	�	PROPN
ap-961	757	10	�	�	PROPN
ap-961	757	11	�	�	PROPN
ap-961	757	12	1	1	NUM
ap-961	757	13	0	0	NUM
ap-961	757	14	0	0	NUM
ap-961	757	15	1	1	NUM
ap-961	757	16	,	,	PUNCT
ap-961	757	17	j	j	PROPN
ap-961	757	18	j	j	PROPN
ap-961	757	19	j	j	PROPN
ap-961	757	20	�	�	PROPN
ap-961	757	21	�	�	PROPN
ap-961	757	22	�	�	PROPN
ap-961	757	23	�	�	PROPN
ap-961	757	24	�	�	PROPN
ap-961	757	25	�	�	PROPN
ap-961	757	26	�	�	PROPN
ap-961	757	27	�	�	PROPN
ap-961	757	28	0	0	NUM
ap-961	757	29	0	0	NUM
ap-961	757	30	0	0	NUM
ap-961	757	31	0	0	NUM
ap-961	757	32	,	,	PUNCT
ap-961	757	33	d	d	PROPN
ap-961	757	34	�	�	PROPN
ap-961	757	35	�	�	PROPN
ap-961	757	36	�	�	PROPN
ap-961	757	37	�	�	PROPN
ap-961	757	38	�	�	PROPN
ap-961	757	39	�	�	PROPN
ap-961	757	40	�	�	PROPN
ap-961	757	41	�	�	PROPN
ap-961	757	42	�	�	PROPN
ap-961	757	43	0	0	NUM
ap-961	757	44	0	0	NUM
ap-961	757	45	�	�	PROPN
ap-961	757	46	�	�	PROPN
ap-961	757	47	†	†	PROPN
ap-961	757	48	.	.	PUNCT
ap-961	757	49	gives	give	VERB
ap-961	757	50	the	the	DET
ap-961	757	51	data	datum	NOUN
ap-961	757	52	of	of	ADP
ap-961	757	53	the	the	DET
ap-961	757	54	real	real	ADJ
ap-961	757	55	spectral	spectral	ADJ
ap-961	757	56	triple	triple	NOUN
ap-961	757	57	of	of	ADP
ap-961	757	58	ko	ko	PROPN
ap-961	757	59	and	and	CCONJ
ap-961	757	60	metric	metric	ADJ
ap-961	757	61	dimension	dimension	NOUN
ap-961	757	62	2	2	NUM
ap-961	757	63	on	on	ADP
ap-961	757	64	the	the	DET
ap-961	757	65	noncommutative	noncommutative	ADJ
ap-961	757	66	torus	torus	NOUN
ap-961	757	67	.	.	PUNCT
ap-961	758	1	exercise	exercise	VERB
ap-961	758	2	6.12	6.12	NUM
ap-961	758	3	:	:	PUNCT
ap-961	758	4	verify	verify	VERB
ap-961	758	5	that	that	SCONJ
ap-961	758	6	all	all	DET
ap-961	758	7	properties	property	NOUN
ap-961	758	8	of	of	ADP
ap-961	758	9	spectral	spectral	ADJ
ap-961	758	10	triples	triple	NOUN
ap-961	758	11	are	be	AUX
ap-961	758	12	satisfied	satisfied	ADJ
ap-961	758	13	!	!	PUNCT
ap-961	759	1	6.4	6.4	NUM
ap-961	759	2	making	make	VERB
ap-961	759	3	(	(	PUNCT
ap-961	759	4	noncommutative	noncommutative	ADJ
ap-961	759	5	)	)	PUNCT
ap-961	759	6	physics	physics	NOUN
ap-961	759	7	suppose	suppose	VERB
ap-961	759	8	we	we	PRON
ap-961	759	9	accept	accept	VERB
ap-961	759	10	that	that	SCONJ
ap-961	759	11	spectral	spectral	ADJ
ap-961	759	12	triples	triple	NOUN
ap-961	759	13	do	do	AUX
ap-961	759	14	describe	describe	VERB
ap-961	759	15	noncommutative	noncommutative	ADJ
ap-961	759	16	manifolds	manifold	NOUN
ap-961	759	17	.	.	PUNCT
ap-961	760	1	is	be	AUX
ap-961	760	2	there	there	PRON
ap-961	760	3	any	any	DET
ap-961	760	4	physical	physical	ADJ
ap-961	760	5	contents	content	NOUN
ap-961	760	6	in	in	ADP
ap-961	760	7	them	they	PRON
ap-961	760	8	?	?	PUNCT
ap-961	761	1	can	can	AUX
ap-961	761	2	we	we	PRON
ap-961	761	3	use	use	VERB
ap-961	761	4	them	they	PRON
ap-961	761	5	to	to	PART
ap-961	761	6	describe	describe	VERB
ap-961	761	7	some	some	DET
ap-961	761	8	noncommutative	noncommutative	ADJ
ap-961	761	9	physics	physic	NOUN
ap-961	761	10	?	?	PUNCT
ap-961	762	1	the	the	DET
ap-961	762	2	answer	answer	NOUN
ap-961	762	3	is	be	AUX
ap-961	762	4	yes	yes	INTJ
ap-961	762	5	and	and	CCONJ
ap-961	762	6	,	,	PUNCT
ap-961	762	7	indeed	indeed	ADV
ap-961	762	8	,	,	PUNCT
ap-961	762	9	we	we	PRON
ap-961	762	10	shall	shall	AUX
ap-961	762	11	be	be	AUX
ap-961	762	12	able	able	ADJ
ap-961	762	13	to	to	PART
ap-961	762	14	provide	provide	VERB
ap-961	762	15	–	–	PUNCT
ap-961	762	16	at	at	ADP
ap-961	762	17	least	least	ADJ
ap-961	762	18	–	–	PUNCT
ap-961	762	19	some	some	DET
ap-961	762	20	partial	partial	ADJ
ap-961	762	21	answers	answer	NOUN
ap-961	762	22	.	.	PUNCT
ap-961	763	1	6.4.1	6.4.1	NUM
ap-961	763	2	gauge	gauge	NOUN
ap-961	763	3	theory	theory	NOUN
ap-961	763	4	and	and	CCONJ
ap-961	763	5	gravity	gravity	NOUN
ap-961	763	6	if	if	SCONJ
ap-961	763	7	we	we	PRON
ap-961	763	8	have	have	VERB
ap-961	763	9	a	a	DET
ap-961	763	10	spectral	spectral	ADJ
ap-961	763	11	triple	triple	ADJ
ap-961	763	12	(	(	PUNCT
ap-961	763	13	,	,	PUNCT
ap-961	763	14	,	,	PUNCT
ap-961	763	15	)	)	PUNCT
ap-961	763	16	�	�	PROPN
ap-961	763	17	�	�	PROPN
ap-961	764	1	d	d	ADP
ap-961	764	2	we	we	PRON
ap-961	764	3	may	may	AUX
ap-961	764	4	always	always	ADV
ap-961	764	5	wonder	wonder	VERB
ap-961	764	6	whether	whether	SCONJ
ap-961	764	7	the	the	DET
ap-961	764	8	dirac	dirac	NOUN
ap-961	764	9	operator	operator	NOUN
ap-961	764	10	we	we	PRON
ap-961	764	11	have	have	AUX
ap-961	764	12	chosen	choose	VERB
ap-961	764	13	is	be	AUX
ap-961	764	14	a	a	DET
ap-961	764	15	good	good	ADJ
ap-961	764	16	one	one	NOUN
ap-961	764	17	.	.	PUNCT
ap-961	765	1	certainly	certainly	ADV
ap-961	765	2	nothing	nothing	PRON
ap-961	765	3	(	(	PUNCT
ap-961	765	4	apart	apart	ADV
ap-961	765	5	from	from	ADP
ap-961	765	6	some	some	DET
ap-961	765	7	symmetries	symmetry	NOUN
ap-961	765	8	)	)	PUNCT
ap-961	765	9	guarantees	guarantee	VERB
ap-961	765	10	it	it	PRON
ap-961	765	11	and	and	CCONJ
ap-961	765	12	,	,	PUNCT
ap-961	765	13	in	in	ADP
ap-961	765	14	fact	fact	NOUN
ap-961	765	15	,	,	PUNCT
ap-961	765	16	a	a	DET
ap-961	765	17	simple	simple	ADJ
ap-961	765	18	transformation	transformation	NOUN
ap-961	766	1	d	d	PROPN
ap-961	766	2	d	d	PROPN
ap-961	766	3	a	a	DET
ap-961	766	4	�	�	PROPN
ap-961	766	5	,	,	PUNCT
ap-961	766	6	where	where	SCONJ
ap-961	766	7	a	a	PRON
ap-961	766	8	is	be	AUX
ap-961	766	9	a	a	DET
ap-961	766	10	self	self	NOUN
ap-961	766	11	-	-	PUNCT
ap-961	766	12	adjoint	adjoint	NOUN
ap-961	766	13	one	one	NUM
ap-961	766	14	-	-	PUNCT
ap-961	766	15	form	form	NOUN
ap-961	766	16	a	a	DET
ap-961	766	17	a	a	NOUN
ap-961	766	18	d	d	X
ap-961	766	19	bi	bi	PROPN
ap-961	766	20	i	i	PROPN
ap-961	766	21	�	�	PROPN
ap-961	766	22	�	�	PROPN
ap-961	766	23	�	�	PROPN
ap-961	766	24	�	�	PROPN
ap-961	766	25	(	(	PUNCT
ap-961	766	26	)	)	PUNCT
ap-961	766	27	[	[	PUNCT
ap-961	766	28	,	,	PUNCT
ap-961	766	29	(	(	PUNCT
ap-961	766	30	)	)	PUNCT
ap-961	766	31	]	]	PUNCT
ap-961	766	32	allows	allow	VERB
ap-961	766	33	us	we	PRON
ap-961	766	34	to	to	PART
ap-961	766	35	construct	construct	VERB
ap-961	766	36	a	a	DET
ap-961	766	37	spectral	spectral	ADJ
ap-961	766	38	triple	triple	NOUN
ap-961	766	39	with	with	ADP
ap-961	766	40	the	the	DET
ap-961	766	41	same	same	ADJ
ap-961	766	42	algebra	algebra	NOUN
ap-961	766	43	and	and	CCONJ
ap-961	766	44	hilbert	hilbert	NOUN
ap-961	766	45	space	space	NOUN
ap-961	766	46	but	but	CCONJ
ap-961	766	47	a	a	DET
ap-961	766	48	–	–	PUNCT
ap-961	766	49	slightly	slightly	ADV
ap-961	766	50	–	–	PUNCT
ap-961	766	51	modified	modify	VERB
ap-961	766	52	dirac	dirac	NOUN
ap-961	766	53	operator	operator	NOUN
ap-961	766	54	.	.	PUNCT
ap-961	767	1	moreover	moreover	ADV
ap-961	767	2	,	,	PUNCT
ap-961	767	3	when	when	SCONJ
ap-961	767	4	we	we	PRON
ap-961	767	5	look	look	VERB
ap-961	767	6	at	at	ADP
ap-961	767	7	it	it	PRON
ap-961	767	8	,	,	PUNCT
ap-961	767	9	we	we	PRON
ap-961	767	10	see	see	VERB
ap-961	767	11	that	that	SCONJ
ap-961	767	12	in	in	ADP
ap-961	767	13	this	this	DET
ap-961	767	14	way	way	NOUN
ap-961	767	15	we	we	PRON
ap-961	767	16	reconstruct	reconstruct	VERB
ap-961	767	17	the	the	DET
ap-961	767	18	gauge	gauge	NOUN
ap-961	767	19	theory	theory	NOUN
ap-961	767	20	and	and	CCONJ
ap-961	767	21	is	be	AUX
ap-961	767	22	nothing	nothing	PRON
ap-961	767	23	else	else	ADV
ap-961	767	24	but	but	CCONJ
ap-961	767	25	the	the	DET
ap-961	767	26	gauge	gauge	NOUN
ap-961	767	27	potential	potential	NOUN
ap-961	767	28	.	.	PUNCT
ap-961	768	1	we	we	PRON
ap-961	768	2	call	call	VERB
ap-961	768	3	this	this	DET
ap-961	768	4	inner	inner	ADJ
ap-961	768	5	fluctuations	fluctuation	NOUN
ap-961	768	6	of	of	ADP
ap-961	768	7	the	the	DET
ap-961	768	8	dirac	dirac	NOUN
ap-961	768	9	operator	operator	NOUN
ap-961	768	10	.	.	PUNCT
ap-961	769	1	it	it	PRON
ap-961	769	2	is	be	AUX
ap-961	769	3	a	a	DET
ap-961	769	4	small	small	ADJ
ap-961	769	5	step	step	NOUN
ap-961	769	6	to	to	PART
ap-961	769	7	calculate	calculate	NOUN
ap-961	769	8	d2	d2	PROPN
ap-961	769	9	,	,	PUNCT
ap-961	769	10	recover	recover	VERB
ap-961	769	11	the	the	DET
ap-961	769	12	curvature	curvature	NOUN
ap-961	769	13	of	of	ADP
ap-961	769	14	the	the	DET
ap-961	769	15	gauge	gauge	ADJ
ap-961	769	16	potential	potential	NOUN
ap-961	769	17	and	and	CCONJ
ap-961	769	18	construct	construct	VERB
ap-961	769	19	the	the	DET
ap-961	769	20	action	action	NOUN
ap-961	769	21	.	.	PUNCT
ap-961	770	1	in	in	ADP
ap-961	770	2	this	this	DET
ap-961	770	3	approach	approach	NOUN
ap-961	770	4	there	there	PRON
ap-961	770	5	are	be	VERB
ap-961	770	6	,	,	PUNCT
ap-961	770	7	however	however	ADV
ap-961	770	8	,	,	PUNCT
ap-961	770	9	some	some	DET
ap-961	770	10	hidden	hide	VERB
ap-961	770	11	obstacles	obstacle	NOUN
ap-961	770	12	,	,	PUNCT
ap-961	770	13	which	which	PRON
ap-961	770	14	have	have	VERB
ap-961	770	15	origin	origin	NOUN
ap-961	770	16	in	in	ADP
ap-961	770	17	the	the	DET
ap-961	770	18	fact	fact	NOUN
ap-961	770	19	that	that	SCONJ
ap-961	770	20	d	d	PROPN
ap-961	770	21	defines	define	VERB
ap-961	770	22	a	a	DET
ap-961	770	23	differential	differential	ADJ
ap-961	770	24	algebra	algebra	NOUN
ap-961	770	25	only	only	ADV
ap-961	770	26	after	after	SCONJ
ap-961	770	27	we	we	PRON
ap-961	770	28	quotient	quotient	VERB
ap-961	770	29	out	out	ADP
ap-961	770	30	an	an	DET
ap-961	770	31	additional	additional	ADJ
ap-961	770	32	ideal	ideal	NOUN
ap-961	770	33	.	.	PUNCT
ap-961	771	1	then	then	ADV
ap-961	771	2	,	,	PUNCT
ap-961	771	3	there	there	PRON
ap-961	771	4	are	be	VERB
ap-961	771	5	many	many	ADJ
ap-961	771	6	non	non	ADJ
ap-961	771	7	-	-	ADJ
ap-961	771	8	equivalent	equivalent	ADJ
ap-961	771	9	ways	way	NOUN
ap-961	771	10	of	of	ADP
ap-961	771	11	embedding	embed	VERB
ap-961	771	12	the	the	DET
ap-961	771	13	two	two	NUM
ap-961	771	14	-	-	PUNCT
ap-961	771	15	forms	form	NOUN
ap-961	771	16	into	into	ADP
ap-961	771	17	the	the	DET
ap-961	771	18	algebra	algebra	NOUN
ap-961	771	19	of	of	ADP
ap-961	771	20	operators	operator	NOUN
ap-961	771	21	on	on	ADP
ap-961	771	22	the	the	DET
ap-961	771	23	hilbert	hilbert	NOUN
ap-961	771	24	space	space	NOUN
ap-961	771	25	(	(	PUNCT
ap-961	771	26	which	which	PRON
ap-961	771	27	we	we	PRON
ap-961	771	28	need	need	VERB
ap-961	771	29	if	if	SCONJ
ap-961	771	30	we	we	PRON
ap-961	771	31	want	want	VERB
ap-961	771	32	to	to	PART
ap-961	771	33	use	use	VERB
ap-961	771	34	noncommutative	noncommutative	ADJ
ap-961	771	35	integrals	integral	NOUN
ap-961	771	36	to	to	PART
ap-961	771	37	calculate	calculate	VERB
ap-961	771	38	the	the	DET
ap-961	771	39	action	action	NOUN
ap-961	771	40	)	)	PUNCT
ap-961	771	41	.	.	PUNCT
ap-961	772	1	clearly	clearly	ADV
ap-961	772	2	,	,	PUNCT
ap-961	772	3	the	the	DET
ap-961	772	4	information	information	NOUN
ap-961	772	5	that	that	PRON
ap-961	772	6	is	be	AUX
ap-961	772	7	encoded	encode	VERB
ap-961	772	8	in	in	ADP
ap-961	772	9	d	d	NOUN
ap-961	772	10	includes	include	VERB
ap-961	772	11	also	also	ADV
ap-961	772	12	the	the	DET
ap-961	772	13	metric	metric	ADJ
ap-961	772	14	and	and	CCONJ
ap-961	772	15	hence	hence	ADV
ap-961	772	16	the	the	DET
ap-961	772	17	riemannian	riemannian	ADJ
ap-961	772	18	connection	connection	NOUN
ap-961	772	19	.	.	PUNCT
ap-961	773	1	are	be	AUX
ap-961	773	2	we	we	PRON
ap-961	773	3	able	able	ADJ
ap-961	773	4	to	to	PART
ap-961	773	5	construct	construct	VERB
ap-961	773	6	the	the	DET
ap-961	773	7	gravity	gravity	NOUN
ap-961	773	8	action	action	NOUN
ap-961	773	9	as	as	ADV
ap-961	773	10	well	well	ADV
ap-961	773	11	?	?	PUNCT
ap-961	774	1	a	a	DET
ap-961	774	2	partial	partial	ADJ
ap-961	774	3	answer	answer	NOUN
ap-961	774	4	was	be	AUX
ap-961	774	5	provided	provide	VERB
ap-961	774	6	some	some	DET
ap-961	774	7	time	time	NOUN
ap-961	774	8	ago	ago	ADV
ap-961	774	9	by	by	ADP
ap-961	774	10	kastler	kastler	PROPN
ap-961	774	11	,	,	PUNCT
ap-961	774	12	kalau	kalau	PROPN
ap-961	774	13	and	and	CCONJ
ap-961	774	14	walze	walze	PROPN
ap-961	774	15	,	,	PUNCT
ap-961	774	16	who	who	PRON
ap-961	774	17	proved	prove	VERB
ap-961	774	18	that	that	SCONJ
ap-961	774	19	the	the	DET
ap-961	774	20	einstein	einstein	PROPN
ap-961	774	21	-	-	PUNCT
ap-961	774	22	hilbert	hilbert	NOUN
ap-961	774	23	functional	functional	NOUN
ap-961	774	24	(	(	PUNCT
ap-961	774	25	the	the	DET
ap-961	774	26	integral	integral	NOUN
ap-961	774	27	of	of	ADP
ap-961	774	28	the	the	DET
ap-961	774	29	scalar	scalar	NOUN
ap-961	774	30	of	of	ADP
ap-961	774	31	curvature	curvature	NOUN
ap-961	774	32	,	,	PUNCT
ap-961	774	33	in	in	ADP
ap-961	774	34	other	other	ADJ
ap-961	774	35	words	word	NOUN
ap-961	774	36	)	)	PUNCT
ap-961	774	37	on	on	ADP
ap-961	774	38	manifolds	manifold	NOUN
ap-961	774	39	can	can	AUX
ap-961	774	40	be	be	AUX
ap-961	774	41	expressed	express	VERB
ap-961	774	42	as	as	ADP
ap-961	774	43	a	a	DET
ap-961	774	44	wodzicki	wodzicki	ADJ
ap-961	774	45	residue	residue	NOUN
ap-961	774	46	of	of	ADP
ap-961	774	47	a	a	DET
ap-961	774	48	certain	certain	ADJ
ap-961	774	49	power	power	NOUN
ap-961	774	50	of	of	ADP
ap-961	774	51	the	the	DET
ap-961	774	52	operator	operator	NOUN
ap-961	774	53	d	d	PROPN
ap-961	775	1	[	[	X
ap-961	775	2	38	38	NUM
ap-961	775	3	,	,	PUNCT
ap-961	775	4	39	39	NUM
ap-961	775	5	]	]	PUNCT
ap-961	775	6	.	.	PUNCT
ap-961	776	1	now	now	ADV
ap-961	776	2	,	,	PUNCT
ap-961	776	3	we	we	PRON
ap-961	776	4	are	be	AUX
ap-961	776	5	ready	ready	ADJ
ap-961	776	6	to	to	PART
ap-961	776	7	see	see	VERB
ap-961	776	8	the	the	DET
ap-961	776	9	proposition	proposition	NOUN
ap-961	776	10	that	that	PRON
ap-961	776	11	encompasses	encompass	VERB
ap-961	776	12	both	both	DET
ap-961	776	13	contributions	contribution	NOUN
ap-961	776	14	.	.	PUNCT
ap-961	777	1	©	©	PROPN
ap-961	777	2	czech	czech	PROPN
ap-961	777	3	technical	technical	PROPN
ap-961	777	4	university	university	PROPN
ap-961	777	5	publishing	publishing	NOUN
ap-961	777	6	house	house	NOUN
ap-961	777	7	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	777	8	53	53	NUM
ap-961	777	9	acta	acta	PROPN
ap-961	777	10	polytechnica	polytechnica	PROPN
ap-961	777	11	vol	vol	NOUN
ap-961	777	12	.	.	PUNCT
ap-961	778	1	48	48	NUM
ap-961	778	2	no	no	NOUN
ap-961	778	3	.	.	PUNCT
ap-961	779	1	2/2008	2/2008	PROPN
ap-961	779	2	6.4.2	6.4.2	NUM
ap-961	779	3	the	the	DET
ap-961	779	4	spectral	spectral	ADJ
ap-961	779	5	action	action	NOUN
ap-961	779	6	again	again	ADV
ap-961	779	7	,	,	PUNCT
ap-961	779	8	assume	assume	VERB
ap-961	779	9	that	that	SCONJ
ap-961	779	10	we	we	PRON
ap-961	779	11	have	have	VERB
ap-961	779	12	a	a	DET
ap-961	779	13	spectral	spectral	ADJ
ap-961	779	14	triple	triple	NOUN
ap-961	779	15	,	,	PUNCT
ap-961	780	1	that	that	ADV
ap-961	780	2	is	is	ADV
ap-961	780	3	(	(	PUNCT
ap-961	780	4	,	,	PUNCT
ap-961	780	5	,	,	PUNCT
ap-961	780	6	)	)	PUNCT
ap-961	780	7	�	�	PROPN
ap-961	780	8	�	�	PROPN
ap-961	780	9	d	d	PROPN
ap-961	780	10	.	.	PUNCT
ap-961	781	1	the	the	DET
ap-961	781	2	following	follow	VERB
ap-961	781	3	defines	define	VERB
ap-961	781	4	a	a	DET
ap-961	781	5	functional	functional	ADJ
ap-961	781	6	on	on	ADP
ap-961	781	7	the	the	DET
ap-961	781	8	space	space	NOUN
ap-961	781	9	of	of	ADP
ap-961	781	10	all	all	DET
ap-961	781	11	admissible	admissible	ADJ
ap-961	781	12	dirac	dirac	NOUN
ap-961	781	13	operators	operator	NOUN
ap-961	781	14	:	:	PUNCT
ap-961	781	15	s	s	VERB
ap-961	781	16	d	d	X
ap-961	781	17	f	f	X
ap-961	781	18	d	d	PROPN
ap-961	781	19	(	(	PUNCT
ap-961	781	20	)	)	PUNCT
ap-961	781	21	(	(	PUNCT
ap-961	781	22	)	)	PUNCT
ap-961	781	23	�	�	PROPN
ap-961	781	24	tr	tr	VERB
ap-961	781	25	2	2	NUM
ap-961	781	26	,	,	PUNCT
ap-961	781	27	where	where	SCONJ
ap-961	781	28	f	f	PROPN
ap-961	781	29	is	be	AUX
ap-961	781	30	a	a	DET
ap-961	781	31	cut	cut	VERB
ap-961	781	32	-	-	PUNCT
ap-961	781	33	off	off	ADP
ap-961	781	34	function	function	NOUN
ap-961	781	35	,	,	PUNCT
ap-961	781	36	which	which	PRON
ap-961	781	37	,	,	PUNCT
ap-961	781	38	for	for	ADP
ap-961	781	39	instance	instance	NOUN
ap-961	781	40	,	,	PUNCT
ap-961	781	41	vanishes	vanish	VERB
ap-961	781	42	for	for	ADP
ap-961	781	43	arguments	argument	NOUN
ap-961	781	44	bigger	big	ADJ
ap-961	781	45	than	than	ADP
ap-961	781	46	a	a	DET
ap-961	781	47	certain	certain	ADJ
ap-961	781	48	number	number	NOUN
ap-961	781	49	�	�	PROPN
ap-961	781	50	.	.	PUNCT
ap-961	782	1	this	this	DET
ap-961	782	2	idea	idea	NOUN
ap-961	782	3	appeared	appear	VERB
ap-961	782	4	for	for	ADP
ap-961	782	5	the	the	DET
ap-961	782	6	first	first	ADJ
ap-961	782	7	time	time	NOUN
ap-961	782	8	(	(	PUNCT
ap-961	782	9	in	in	ADP
ap-961	782	10	a	a	DET
ap-961	782	11	similar	similar	ADJ
ap-961	782	12	phrasing	phrasing	NOUN
ap-961	782	13	)	)	PUNCT
ap-961	782	14	in	in	ADP
ap-961	782	15	the	the	DET
ap-961	782	16	work	work	NOUN
ap-961	782	17	of	of	ADP
ap-961	782	18	sakharov	sakharov	NOUN
ap-961	782	19	[	[	X
ap-961	782	20	41	41	NUM
ap-961	782	21	]	]	PUNCT
ap-961	782	22	in	in	ADP
ap-961	782	23	1965	1965	NUM
ap-961	782	24	.	.	PUNCT
ap-961	783	1	of	of	ADP
ap-961	783	2	course	course	NOUN
ap-961	783	3	,	,	PUNCT
ap-961	783	4	the	the	DET
ap-961	783	5	action	action	NOUN
ap-961	783	6	functional	functional	ADJ
ap-961	783	7	depends	depend	VERB
ap-961	783	8	on	on	ADP
ap-961	783	9	the	the	DET
ap-961	783	10	choice	choice	NOUN
ap-961	783	11	of	of	ADP
ap-961	783	12	f.	f.	PROPN
ap-961	783	13	however	however	ADV
ap-961	783	14	,	,	PUNCT
ap-961	783	15	it	it	PRON
ap-961	783	16	appears	appear	VERB
ap-961	783	17	that	that	SCONJ
ap-961	783	18	the	the	DET
ap-961	783	19	dependence	dependence	NOUN
ap-961	783	20	is	be	AUX
ap-961	783	21	not	not	PART
ap-961	783	22	as	as	ADV
ap-961	783	23	significant	significant	ADJ
ap-961	783	24	as	as	SCONJ
ap-961	783	25	we	we	PRON
ap-961	783	26	might	might	AUX
ap-961	783	27	suspect	suspect	VERB
ap-961	783	28	,	,	PUNCT
ap-961	783	29	as	as	SCONJ
ap-961	783	30	we	we	PRON
ap-961	783	31	shall	shall	AUX
ap-961	783	32	see	see	VERB
ap-961	783	33	later	later	ADV
ap-961	783	34	.	.	PUNCT
ap-961	784	1	let	let	VERB
ap-961	784	2	us	we	PRON
ap-961	784	3	consider	consider	VERB
ap-961	784	4	now	now	ADV
ap-961	784	5	the	the	DET
ap-961	784	6	easy	easy	ADJ
ap-961	784	7	example	example	NOUN
ap-961	784	8	of	of	ADP
ap-961	784	9	two	two	NUM
ap-961	784	10	-	-	PUNCT
ap-961	784	11	points	point	NOUN
ap-961	784	12	and	and	CCONJ
ap-961	784	13	the	the	DET
ap-961	784	14	spectral	spectral	ADJ
ap-961	784	15	action	action	NOUN
ap-961	784	16	in	in	ADP
ap-961	784	17	this	this	DET
ap-961	784	18	case	case	NOUN
ap-961	784	19	.	.	PUNCT
ap-961	785	1	example	example	NOUN
ap-961	785	2	6.13	6.13	NUM
ap-961	785	3	:	:	PUNCT
ap-961	785	4	recall	recall	VERB
ap-961	785	5	the	the	DET
ap-961	785	6	construction	construction	NOUN
ap-961	785	7	of	of	ADP
ap-961	785	8	the	the	DET
ap-961	785	9	spectral	spectral	ADJ
ap-961	785	10	triple	triple	NOUN
ap-961	785	11	for	for	ADP
ap-961	785	12	the	the	DET
ap-961	785	13	algebra	algebra	NOUN
ap-961	785	14	of	of	ADP
ap-961	785	15	two	two	NUM
ap-961	785	16	-	-	PUNCT
ap-961	785	17	points	point	NOUN
ap-961	785	18	and	and	CCONJ
ap-961	785	19	its	its	PRON
ap-961	785	20	dirac	dirac	NOUN
ap-961	785	21	operator	operator	NOUN
ap-961	785	22	d.	d.	PROPN
ap-961	785	23	the	the	DET
ap-961	785	24	only	only	ADJ
ap-961	785	25	free	free	ADJ
ap-961	785	26	parameter	parameter	NOUN
ap-961	785	27	in	in	ADP
ap-961	785	28	d	d	PROPN
ap-961	785	29	was	be	AUX
ap-961	785	30	a	a	DET
ap-961	785	31	complex	complex	ADJ
ap-961	785	32	number	number	NOUN
ap-961	785	33	(	(	PUNCT
ap-961	785	34	)	)	PUNCT
ap-961	785	35	d0	d0	PROPN
ap-961	785	36	12	12	NUM
ap-961	785	37	.	.	PUNCT
ap-961	786	1	a	a	DET
ap-961	786	2	most	most	ADV
ap-961	786	3	general	general	ADJ
ap-961	786	4	self	self	NOUN
ap-961	786	5	-	-	PUNCT
ap-961	786	6	adjoint	adjoint	NOUN
ap-961	786	7	one	one	NUM
ap-961	786	8	-	-	PUNCT
ap-961	786	9	form	form	NOUN
ap-961	786	10	a	a	NOUN
ap-961	786	11	is	be	AUX
ap-961	786	12	given	give	VERB
ap-961	786	13	as	as	ADP
ap-961	786	14	:	:	PUNCT
ap-961	786	15	a	a	DET
ap-961	786	16	�	�	PROPN
ap-961	786	17	�	�	PROPN
ap-961	786	18	�	�	PROPN
ap-961	786	19	�	�	PROPN
ap-961	786	20	�	�	PROPN
ap-961	786	21	�	�	PROPN
ap-961	786	22	�	�	PROPN
ap-961	786	23	0	0	NUM
ap-961	786	24	0	0	NUM
ap-961	786	25	�	�	PROPN
ap-961	786	26	�	�	PROPN
ap-961	786	27	†	†	PROPN
ap-961	786	28	,	,	PUNCT
ap-961	786	29	where	where	SCONJ
ap-961	786	30	�	�	PROPN
ap-961	786	31	�	�	PROPN
ap-961	786	32	�	�	PROPN
ap-961	786	33	�	�	PROPN
ap-961	786	34	�	�	PROPN
ap-961	786	35	�	�	PROPN
ap-961	786	36	�	�	PROPN
ap-961	786	37	�	�	PROPN
ap-961	786	38	(	(	PUNCT
ap-961	786	39	)	)	PUNCT
ap-961	786	40	d	d	X
ap-961	786	41	w	w	PROPN
ap-961	786	42	z0	z0	PROPN
ap-961	786	43	12	12	NUM
ap-961	786	44	0	0	NUM
ap-961	786	45	0	0	NUM
ap-961	786	46	,	,	PUNCT
ap-961	786	47	for	for	ADP
ap-961	786	48	arbitrary	arbitrary	ADJ
ap-961	786	49	w	w	PROPN
ap-961	786	50	z	z	PROPN
ap-961	786	51	,	,	PUNCT
ap-961	786	52	�	�	PROPN
ap-961	786	53	�	�	PROPN
ap-961	786	54	.	.	PUNCT
ap-961	787	1	since	since	SCONJ
ap-961	787	2	our	our	PRON
ap-961	787	3	triple	triple	NOUN
ap-961	787	4	is	be	AUX
ap-961	787	5	a	a	DET
ap-961	787	6	real	real	ADJ
ap-961	787	7	one	one	NOUN
ap-961	787	8	and	and	CCONJ
ap-961	787	9	jd	jd	PROPN
ap-961	787	10	d	d	PROPN
ap-961	787	11	j	j	PROPN
ap-961	787	12	�	�	PROPN
ap-961	787	13	,	,	PUNCT
ap-961	787	14	we	we	PRON
ap-961	787	15	need	need	VERB
ap-961	787	16	to	to	PART
ap-961	787	17	require	require	VERB
ap-961	787	18	the	the	DET
ap-961	787	19	same	same	ADJ
ap-961	787	20	for	for	ADP
ap-961	787	21	the	the	DET
ap-961	787	22	gauge	gauge	ADJ
ap-961	787	23	potential	potential	NOUN
ap-961	787	24	a	a	PRON
ap-961	787	25	and	and	CCONJ
ap-961	787	26	thus	thus	ADV
ap-961	787	27	we	we	PRON
ap-961	787	28	have	have	VERB
ap-961	787	29	z	z	PROPN
ap-961	787	30	w	w	PROPN
ap-961	787	31	�	�	PROPN
ap-961	787	32	.	.	PUNCT
ap-961	788	1	then	then	ADV
ap-961	788	2	the	the	DET
ap-961	788	3	spectral	spectral	ADJ
ap-961	788	4	action	action	NOUN
ap-961	788	5	is	be	AUX
ap-961	788	6	:	:	PUNCT
ap-961	788	7	tr	tr	VERB
ap-961	788	8	(	(	PUNCT
ap-961	788	9	)	)	PUNCT
ap-961	788	10	(	(	PUNCT
ap-961	788	11	)	)	PUNCT
ap-961	788	12	(	(	PUNCT
ap-961	788	13	)	)	PUNCT
ap-961	788	14	(	(	PUNCT
ap-961	788	15	)	)	PUNCT
ap-961	788	16	d	d	NOUN
ap-961	788	17	a	a	X
ap-961	788	18	d	d	X
ap-961	788	19	z	z	NOUN
ap-961	788	20	z	z	NOUN
ap-961	788	21	�	�	PROPN
ap-961	788	22	2	2	NUM
ap-961	788	23	0	0	NUM
ap-961	788	24	12	12	NUM
ap-961	788	25	24	24	NUM
ap-961	788	26	1	1	NUM
ap-961	788	27	1	1	NUM
ap-961	788	28	.	.	PUNCT
ap-961	789	1	it	it	PRON
ap-961	789	2	is	be	AUX
ap-961	789	3	not	not	PART
ap-961	789	4	an	an	DET
ap-961	789	5	exciting	exciting	ADJ
ap-961	789	6	answer	answer	NOUN
ap-961	789	7	but	but	CCONJ
ap-961	789	8	,	,	PUNCT
ap-961	789	9	in	in	ADP
ap-961	789	10	fact	fact	NOUN
ap-961	789	11	,	,	PUNCT
ap-961	789	12	we	we	PRON
ap-961	789	13	did	do	AUX
ap-961	789	14	not	not	PART
ap-961	789	15	expect	expect	VERB
ap-961	789	16	anything	anything	PRON
ap-961	789	17	exciting	exciting	ADJ
ap-961	789	18	here	here	ADV
ap-961	789	19	.	.	PUNCT
ap-961	790	1	more	more	ADV
ap-961	790	2	interesting	interesting	ADJ
ap-961	790	3	things	thing	NOUN
ap-961	790	4	happen	happen	VERB
ap-961	790	5	when	when	SCONJ
ap-961	790	6	we	we	PRON
ap-961	790	7	consider	consider	VERB
ap-961	790	8	the	the	DET
ap-961	790	9	continuous	continuous	ADJ
ap-961	790	10	geometry	geometry	NOUN
ap-961	790	11	of	of	ADP
ap-961	790	12	the	the	DET
ap-961	790	13	c	c	PROPN
ap-961	790	14	-functions	-function	NOUN
ap-961	790	15	on	on	ADP
ap-961	790	16	a	a	DET
ap-961	790	17	manifold	manifold	NOUN
ap-961	790	18	and	and	CCONJ
ap-961	790	19	the	the	DET
ap-961	790	20	spectral	spectral	ADJ
ap-961	790	21	action	action	NOUN
ap-961	790	22	of	of	ADP
ap-961	790	23	the	the	DET
ap-961	790	24	true	true	ADJ
ap-961	790	25	dirac	dirac	NOUN
ap-961	790	26	operator	operator	NOUN
ap-961	790	27	!	!	PUNCT
ap-961	791	1	lemma	lemma	PROPN
ap-961	791	2	6.14	6.14	NUM
ap-961	791	3	:	:	PUNCT
ap-961	791	4	for	for	ADP
ap-961	791	5	the	the	DET
ap-961	791	6	spectral	spectral	ADJ
ap-961	791	7	triple	triple	ADJ
ap-961	791	8	(	(	PUNCT
ap-961	791	9	(	(	PUNCT
ap-961	791	10	)	)	PUNCT
ap-961	791	11	,	,	PUNCT
ap-961	791	12	(	(	PUNCT
ap-961	791	13	,	,	PUNCT
ap-961	791	14	)	)	PUNCT
ap-961	791	15	,	,	PUNCT
ap-961	791	16	)	)	PUNCT
ap-961	792	1	c	c	NOUN
ap-961	792	2	m	m	VERB
ap-961	792	3	l	l	NOUN
ap-961	792	4	m	m	NOUN
ap-961	792	5	s	s	PROPN
ap-961	792	6	d	d	PROPN
ap-961	792	7	2	2	NUM
ap-961	792	8	over	over	ADP
ap-961	792	9	a	a	DET
ap-961	792	10	4	4	NUM
ap-961	792	11	-	-	PUNCT
ap-961	792	12	dimensional	dimensional	ADJ
ap-961	792	13	manifold	manifold	NOUN
ap-961	792	14	,	,	PUNCT
ap-961	792	15	the	the	DET
ap-961	792	16	spectral	spectral	ADJ
ap-961	792	17	action	action	NOUN
ap-961	792	18	(	(	PUNCT
ap-961	792	19	modulo	modulo	PROPN
ap-961	792	20	topological	topological	ADJ
ap-961	792	21	and	and	CCONJ
ap-961	792	22	boundary	boundary	ADJ
ap-961	792	23	terms	term	NOUN
ap-961	792	24	)	)	PUNCT
ap-961	792	25	has	have	VERB
ap-961	792	26	the	the	DET
ap-961	792	27	following	follow	VERB
ap-961	792	28	asymptotic	asymptotic	ADJ
ap-961	792	29	expansion	expansion	NOUN
ap-961	792	30	:	:	PUNCT
ap-961	792	31	tr	tr	PROPN
ap-961	793	1	f	f	PROPN
ap-961	793	2	d	d	X
ap-961	793	3	f	f	X
ap-961	794	1	f	f	PROPN
ap-961	794	2	f	f	PROPN
ap-961	795	1	g	g	PROPN
ap-961	795	2	d	d	PROPN
ap-961	795	3	x	x	SYM
ap-961	795	4	f	f	NOUN
ap-961	795	5	2	2	NUM
ap-961	795	6	2	2	NUM
ap-961	795	7	2	2	NUM
ap-961	795	8	4	4	NUM
ap-961	795	9	4	4	NUM
ap-961	795	10	2	2	NUM
ap-961	795	11	2	2	NUM
ap-961	795	12	0	0	NUM
ap-961	795	13	4	4	NUM
ap-961	795	14	2	2	NUM
ap-961	795	15	1	1	NUM
ap-961	795	16	16	16	NUM
ap-961	795	17	2	2	NUM
ap-961	795	18	1	1	NUM
ap-961	795	19	96	96	NUM
ap-961	795	20	�	�	PROPN
ap-961	795	21	�	�	PROPN
ap-961	795	22	�	�	PROPN
ap-961	795	23	�	�	PROPN
ap-961	795	24	�	�	PROPN
ap-961	795	25	�	�	PROPN
ap-961	795	26	�	�	PROPN
ap-961	795	27	�	�	PROPN
ap-961	795	28	�	�	PROPN
ap-961	795	29	�	�	PROPN
ap-961	795	30	�	�	PROPN
ap-961	795	31	�	�	PROPN
ap-961	795	32	�	�	PROPN
ap-961	795	33	�	�	PROPN
ap-961	795	34	2	2	NUM
ap-961	795	35	�	�	PROPN
ap-961	795	36	�	�	PROPN
ap-961	795	37	(	(	PUNCT
ap-961	795	38	)	)	PUNCT
ap-961	795	39	(	(	PUNCT
ap-961	795	40	�	�	PROPN
ap-961	795	41	�	�	PROPN
ap-961	795	42	�	�	PROPN
ap-961	795	43	�	�	PROPN
ap-961	795	44	�	�	PROPN
ap-961	795	45	�	�	PROPN
ap-961	795	46	�	�	PROPN
ap-961	795	47	�	�	PROPN
ap-961	795	48	�	�	PROPN
ap-961	795	49	�	�	PROPN
ap-961	795	50	�	�	PROPN
ap-961	795	51	"	"	PUNCT
ap-961	795	52	�	�	PROPN
ap-961	795	53	�	�	PROPN
ap-961	795	54	2f	2f	NUM
ap-961	795	55	r	r	NOUN
ap-961	795	56	g	g	NOUN
ap-961	795	57	d	d	NOUN
ap-961	795	58	x	x	X
ap-961	796	1	f	f	NOUN
ap-961	796	2	r	r	NOUN
ap-961	796	3	r	r	NOUN
ap-961	796	4	r	r	NOUN
ap-961	796	5	r	r	NOUN
ap-961	796	6	r	r	NOUN
ap-961	796	7	0	0	NUM
ap-961	796	8	4	4	NUM
ap-961	796	9	2	2	NUM
ap-961	796	10	0	0	NUM
ap-961	796	11	2	2	NUM
ap-961	796	12	1	1	NUM
ap-961	796	13	4	4	NUM
ap-961	796	14	1	1	NUM
ap-961	796	15	360	360	NUM
ap-961	796	16	5	5	NUM
ap-961	796	17	2	2	NUM
ap-961	796	18	12	12	NUM
ap-961	796	19	2	2	NUM
ap-961	796	20	)	)	PUNCT
ap-961	796	21	(	(	PUNCT
ap-961	796	22	;	;	PUNCT
ap-961	796	23	�	�	PROPN
ap-961	796	24	�	�	PROPN
ap-961	796	25	r	r	PROPN
ap-961	796	26	�	�	PROPN
ap-961	796	27	�	�	PROPN
ap-961	796	28	�	�	PROPN
ap-961	796	29	�	�	PROPN
ap-961	796	30	)	)	PUNCT
ap-961	796	31	,	,	PUNCT
ap-961	796	32	2	2	NUM
ap-961	796	33	where	where	SCONJ
ap-961	796	34	f	f	PROPN
ap-961	796	35	f	f	PROPN
ap-961	796	36	u	u	X
ap-961	796	37	u	u	PROPN
ap-961	796	38	duk	duk	PROPN
ap-961	796	39	k	k	PROPN
ap-961	796	40	�	�	PROPN
ap-961	796	41	�	�	PROPN
ap-961	796	42	2	2	NUM
ap-961	796	43	(	(	PUNCT
ap-961	796	44	)	)	PUNCT
ap-961	796	45	1	1	NUM
ap-961	796	46	0	0	NUM
ap-961	796	47	for	for	ADP
ap-961	796	48	k	k	PROPN
ap-961	796	49	�	�	PROPN
ap-961	796	50	0	0	PROPN
ap-961	796	51	and	and	CCONJ
ap-961	796	52	f	f	PROPN
ap-961	796	53	f0	f0	PROPN
ap-961	796	54	0	0	NUM
ap-961	796	55	�	�	PROPN
ap-961	796	56	(	(	PUNCT
ap-961	796	57	)	)	PUNCT
ap-961	796	58	are	be	AUX
ap-961	796	59	the	the	DET
ap-961	796	60	moments	moment	NOUN
ap-961	796	61	of	of	ADP
ap-961	796	62	f.	f.	PROPN
ap-961	796	63	this	this	PRON
ap-961	796	64	is	be	AUX
ap-961	796	65	a	a	DET
ap-961	796	66	pure	pure	ADJ
ap-961	796	67	gravity	gravity	NOUN
ap-961	796	68	action	action	NOUN
ap-961	796	69	,	,	PUNCT
ap-961	796	70	which	which	PRON
ap-961	796	71	includes	include	VERB
ap-961	796	72	the	the	DET
ap-961	796	73	cosmological	cosmological	ADJ
ap-961	796	74	constant	constant	ADJ
ap-961	796	75	,	,	PUNCT
ap-961	796	76	the	the	DET
ap-961	796	77	einstein	einstein	ADJ
ap-961	796	78	-	-	PUNCT
ap-961	796	79	hilbert	hilbert	NOUN
ap-961	796	80	action	action	NOUN
ap-961	796	81	and	and	CCONJ
ap-961	796	82	some	some	DET
ap-961	796	83	additional	additional	ADJ
ap-961	796	84	term	term	NOUN
ap-961	796	85	,	,	PUNCT
ap-961	796	86	which	which	PRON
ap-961	796	87	depend	depend	VERB
ap-961	796	88	on	on	ADP
ap-961	796	89	the	the	DET
ap-961	796	90	riemannian	riemannian	ADJ
ap-961	796	91	curvature	curvature	NOUN
ap-961	796	92	tensor	tensor	NOUN
ap-961	796	93	,	,	PUNCT
ap-961	796	94	the	the	DET
ap-961	796	95	ricci	ricci	PROPN
ap-961	796	96	tensor	tensor	NOUN
ap-961	796	97	and	and	CCONJ
ap-961	796	98	the	the	DET
ap-961	796	99	scalar	scalar	ADJ
ap-961	796	100	curvature	curvature	NOUN
ap-961	796	101	.	.	PUNCT
ap-961	797	1	if	if	SCONJ
ap-961	797	2	we	we	PRON
ap-961	797	3	introduce	introduce	VERB
ap-961	797	4	just	just	ADV
ap-961	797	5	a	a	DET
ap-961	797	6	bit	bit	NOUN
ap-961	797	7	of	of	ADP
ap-961	797	8	noncommutative	noncommutative	ADJ
ap-961	797	9	geometry	geometry	NOUN
ap-961	797	10	,	,	PUNCT
ap-961	797	11	by	by	ADP
ap-961	797	12	taking	take	VERB
ap-961	797	13	as	as	ADP
ap-961	797	14	the	the	DET
ap-961	797	15	algebra	algebra	NOUN
ap-961	797	16	not	not	PART
ap-961	797	17	c	c	NOUN
ap-961	797	18	m	m	VERB
ap-961	797	19	(	(	PUNCT
ap-961	797	20	)	)	PUNCT
ap-961	797	21	but	but	CCONJ
ap-961	797	22	c	c	X
ap-961	797	23	m	m	PROPN
ap-961	797	24	mn	mn	PROPN
ap-961	797	25	�	�	PROPN
ap-961	797	26	(	(	PUNCT
ap-961	797	27	)	)	PUNCT
ap-961	797	28	(	(	PUNCT
ap-961	797	29	)	)	PUNCT
ap-961	797	30	�	�	PROPN
ap-961	797	31	,	,	PUNCT
ap-961	797	32	the	the	DET
ap-961	797	33	algebra	algebra	NOUN
ap-961	797	34	of	of	ADP
ap-961	797	35	matrix	matrix	NOUN
ap-961	797	36	valued	value	VERB
ap-961	797	37	functions	function	NOUN
ap-961	797	38	on	on	ADP
ap-961	797	39	m	m	ADV
ap-961	797	40	we	we	PRON
ap-961	797	41	obtain	obtain	VERB
ap-961	797	42	the	the	DET
ap-961	797	43	possibility	possibility	NOUN
ap-961	797	44	of	of	ADP
ap-961	797	45	constructing	construct	VERB
ap-961	797	46	an	an	DET
ap-961	797	47	su(n	su(n	NOUN
ap-961	797	48	)	)	PUNCT
ap-961	797	49	gauge	gauge	NOUN
ap-961	797	50	theory	theory	NOUN
ap-961	797	51	.	.	PUNCT
ap-961	798	1	the	the	DET
ap-961	798	2	gauge	gauge	ADJ
ap-961	798	3	connection	connection	NOUN
ap-961	798	4	one	one	NUM
ap-961	798	5	-	-	PUNCT
ap-961	798	6	form	form	NOUN
ap-961	798	7	a	a	DET
ap-961	798	8	a	a	DET
ap-961	798	9	dx	dx	PROPN
ap-961	798	10	�	�	PROPN
ap-961	798	11	�	�	PROPN
ap-961	798	12	�	�	PROPN
ap-961	798	13	(	(	PUNCT
ap-961	798	14	in	in	ADP
ap-961	798	15	local	local	ADJ
ap-961	798	16	coordinates	coordinate	NOUN
ap-961	798	17	)	)	PUNCT
ap-961	798	18	will	will	AUX
ap-961	798	19	enter	enter	VERB
ap-961	798	20	the	the	DET
ap-961	798	21	spectral	spectral	ADJ
ap-961	798	22	action	action	NOUN
ap-961	798	23	as	as	ADV
ap-961	798	24	well	well	ADV
ap-961	798	25	and	and	CCONJ
ap-961	798	26	we	we	PRON
ap-961	798	27	shall	shall	AUX
ap-961	798	28	additionally	additionally	ADV
ap-961	798	29	obtain	obtain	VERB
ap-961	798	30	(	(	PUNCT
ap-961	798	31	apart	apart	ADV
ap-961	798	32	from	from	ADP
ap-961	798	33	some	some	DET
ap-961	798	34	change	change	NOUN
ap-961	798	35	in	in	ADP
ap-961	798	36	the	the	DET
ap-961	798	37	coefficients	coefficient	NOUN
ap-961	798	38	in	in	ADP
ap-961	798	39	the	the	DET
ap-961	798	40	two	two	NUM
ap-961	798	41	first	first	ADJ
ap-961	798	42	terms	term	NOUN
ap-961	798	43	)	)	PUNCT
ap-961	798	44	another	another	DET
ap-961	798	45	term	term	NOUN
ap-961	798	46	in	in	ADP
ap-961	798	47	the	the	DET
ap-961	798	48	�	�	NOUN
ap-961	798	49	-independent	-independent	ADJ
ap-961	798	50	part	part	NOUN
ap-961	798	51	of	of	ADP
ap-961	798	52	the	the	DET
ap-961	798	53	expansion	expansion	NOUN
ap-961	798	54	:	:	PUNCT
ap-961	798	55	the	the	DET
ap-961	798	56	yang	yang	PROPN
ap-961	798	57	-	-	PUNCT
ap-961	798	58	mills	mill	NOUN
ap-961	798	59	action	action	NOUN
ap-961	798	60	:	:	PUNCT
ap-961	798	61	1	1	NUM
ap-961	798	62	4	4	NUM
ap-961	798	63	1	1	NUM
ap-961	798	64	1202	1202	NUM
ap-961	798	65	0	0	NUM
ap-961	798	66	�	�	PROPN
ap-961	798	67	�	�	PROPN
ap-961	798	68	�	�	PROPN
ap-961	798	69	�	�	PROPN
ap-961	798	70	�	�	PROPN
ap-961	798	71	"	"	PUNCT
ap-961	798	72	2f	2f	NUM
ap-961	798	73	f	f	PROPN
ap-961	798	74	ftr	ftr	PROPN
ap-961	798	75	(	(	PUNCT
ap-961	798	76	)	)	PUNCT
ap-961	798	77	.	.	PUNCT
ap-961	799	1	6.5	6.5	NUM
ap-961	799	2	the	the	DET
ap-961	799	3	standard	standard	ADJ
ap-961	799	4	model	model	NOUN
ap-961	799	5	so	so	ADV
ap-961	799	6	far	far	ADV
ap-961	799	7	we	we	PRON
ap-961	799	8	have	have	AUX
ap-961	799	9	recovered	recover	VERB
ap-961	799	10	important	important	ADJ
ap-961	799	11	parts	part	NOUN
ap-961	799	12	of	of	ADP
ap-961	799	13	theoretical	theoretical	ADJ
ap-961	799	14	physics	physics	NOUN
ap-961	799	15	all	all	PRON
ap-961	799	16	encoded	encode	VERB
ap-961	799	17	in	in	ADP
ap-961	799	18	one	one	NUM
ap-961	799	19	simple	simple	ADJ
ap-961	799	20	action	action	NOUN
ap-961	799	21	.	.	PUNCT
ap-961	800	1	there	there	PRON
ap-961	800	2	is	be	VERB
ap-961	800	3	,	,	PUNCT
ap-961	800	4	however	however	ADV
ap-961	800	5	,	,	PUNCT
ap-961	800	6	more	more	ADJ
ap-961	800	7	to	to	ADP
ap-961	800	8	it	it	PRON
ap-961	800	9	as	as	SCONJ
ap-961	800	10	we	we	PRON
ap-961	800	11	shall	shall	AUX
ap-961	800	12	finally	finally	ADV
ap-961	800	13	see	see	VERB
ap-961	800	14	in	in	ADP
ap-961	800	15	this	this	DET
ap-961	800	16	section	section	NOUN
ap-961	800	17	.	.	PUNCT
ap-961	801	1	the	the	DET
ap-961	801	2	crucial	crucial	ADJ
ap-961	801	3	point	point	NOUN
ap-961	801	4	is	be	AUX
ap-961	801	5	to	to	PART
ap-961	801	6	take	take	VERB
ap-961	801	7	the	the	DET
ap-961	801	8	geometries	geometry	NOUN
ap-961	801	9	of	of	ADP
ap-961	801	10	the	the	DET
ap-961	801	11	type	type	NOUN
ap-961	801	12	m	m	PROPN
ap-961	801	13	f	f	X
ap-961	801	14	�	�	PROPN
ap-961	801	15	,	,	PUNCT
ap-961	801	16	where	where	SCONJ
ap-961	801	17	m	m	NOUN
ap-961	801	18	is	be	AUX
ap-961	801	19	a	a	DET
ap-961	801	20	riemannian	riemannian	ADJ
ap-961	801	21	manifold	manifold	NOUN
ap-961	801	22	and	and	CCONJ
ap-961	801	23	f	f	PROPN
ap-961	801	24	is	be	AUX
ap-961	801	25	a	a	DET
ap-961	801	26	discrete	discrete	ADJ
ap-961	801	27	geometry	geometry	NOUN
ap-961	801	28	.	.	PUNCT
ap-961	802	1	it	it	PRON
ap-961	802	2	is	be	AUX
ap-961	802	3	like	like	ADP
ap-961	802	4	a	a	DET
ap-961	802	5	kaluza	kaluza	PROPN
ap-961	802	6	-	-	PUNCT
ap-961	802	7	klein	klein	PROPN
ap-961	802	8	model	model	NOUN
ap-961	802	9	but	but	CCONJ
ap-961	802	10	with	with	ADP
ap-961	802	11	the	the	DET
ap-961	802	12	extra	extra	ADJ
ap-961	802	13	dimensions	dimension	NOUN
ap-961	802	14	being	be	AUX
ap-961	802	15	in	in	ADP
ap-961	802	16	fact	fact	NOUN
ap-961	802	17	of	of	ADP
ap-961	802	18	(	(	PUNCT
ap-961	802	19	classical	classical	ADJ
ap-961	802	20	dimension	dimension	NOUN
ap-961	802	21	)	)	PUNCT
ap-961	802	22	zero	zero	NUM
ap-961	802	23	.	.	PUNCT
ap-961	803	1	we	we	PRON
ap-961	803	2	shall	shall	AUX
ap-961	803	3	study	study	VERB
ap-961	803	4	here	here	ADV
ap-961	803	5	a	a	DET
ap-961	803	6	toy	toy	NOUN
ap-961	803	7	model	model	NOUN
ap-961	803	8	of	of	ADP
ap-961	803	9	the	the	DET
ap-961	803	10	construction	construction	NOUN
ap-961	803	11	,	,	PUNCT
ap-961	803	12	referring	refer	VERB
ap-961	803	13	to	to	ADP
ap-961	803	14	the	the	DET
ap-961	803	15	recent	recent	ADJ
ap-961	803	16	papers	paper	NOUN
ap-961	803	17	by	by	ADP
ap-961	803	18	connes	conne	NOUN
ap-961	803	19	[	[	X
ap-961	803	20	31	31	NUM
ap-961	803	21	,	,	PUNCT
ap-961	803	22	32	32	NUM
ap-961	803	23	]	]	PUNCT
ap-961	803	24	for	for	ADP
ap-961	803	25	the	the	DET
ap-961	803	26	detailed	detail	VERB
ap-961	803	27	advanced	advanced	ADJ
ap-961	803	28	construction	construction	NOUN
ap-961	803	29	,	,	PUNCT
ap-961	803	30	which	which	PRON
ap-961	803	31	makes	make	VERB
ap-961	803	32	contact	contact	NOUN
ap-961	803	33	with	with	ADP
ap-961	803	34	the	the	DET
ap-961	803	35	real	real	ADJ
ap-961	803	36	physical	physical	ADJ
ap-961	803	37	standard	standard	PROPN
ap-961	803	38	model	model	PROPN
ap-961	803	39	.	.	PUNCT
ap-961	803	40	example	example	NOUN
ap-961	804	1	6.15	6.15	NUM
ap-961	804	2	:	:	PUNCT
ap-961	804	3	let	let	VERB
ap-961	804	4	us	we	PRON
ap-961	804	5	begin	begin	VERB
ap-961	804	6	with	with	ADP
ap-961	804	7	the	the	DET
ap-961	804	8	construction	construction	NOUN
ap-961	804	9	of	of	ADP
ap-961	804	10	the	the	DET
ap-961	804	11	(	(	PUNCT
ap-961	804	12	real	real	ADJ
ap-961	804	13	)	)	PUNCT
ap-961	804	14	spectral	spectral	ADJ
ap-961	804	15	triple	triple	NOUN
ap-961	804	16	for	for	ADP
ap-961	804	17	the	the	DET
ap-961	804	18	algebra	algebra	NOUN
ap-961	804	19	of	of	ADP
ap-961	804	20	functions	function	NOUN
ap-961	804	21	on	on	ADP
ap-961	804	22	m	m	PROPN
ap-961	804	23	f	f	X
ap-961	804	24	�	�	PROPN
ap-961	804	25	2	2	NUM
ap-961	804	26	,	,	PUNCT
ap-961	804	27	with	with	ADP
ap-961	804	28	f2	f2	PROPN
ap-961	804	29	the	the	DET
ap-961	804	30	space	space	NOUN
ap-961	804	31	of	of	ADP
ap-961	804	32	two	two	NUM
ap-961	804	33	points	point	NOUN
ap-961	804	34	.	.	PUNCT
ap-961	805	1	the	the	DET
ap-961	805	2	standard	standard	ADJ
ap-961	805	3	procedure	procedure	NOUN
ap-961	805	4	is	be	AUX
ap-961	805	5	to	to	PART
ap-961	805	6	take	take	VERB
ap-961	805	7	the	the	DET
ap-961	805	8	spectral	spectral	ADJ
ap-961	805	9	triples	triple	NOUN
ap-961	805	10	on	on	ADP
ap-961	805	11	both	both	DET
ap-961	805	12	spaces	space	NOUN
ap-961	805	13	and	and	CCONJ
ap-961	805	14	construct	construct	VERB
ap-961	805	15	their	their	PRON
ap-961	805	16	tensor	tensor	NOUN
ap-961	805	17	product	product	NOUN
ap-961	805	18	.	.	PUNCT
ap-961	806	1	however	however	ADV
ap-961	806	2	,	,	PUNCT
ap-961	806	3	we	we	PRON
ap-961	806	4	shall	shall	AUX
ap-961	806	5	simplify	simplify	VERB
ap-961	806	6	the	the	DET
ap-961	806	7	construction	construction	NOUN
ap-961	806	8	:	:	PUNCT
ap-961	806	9	we	we	PRON
ap-961	806	10	shall	shall	AUX
ap-961	806	11	just	just	ADV
ap-961	806	12	postulate	postulate	VERB
ap-961	806	13	the	the	DET
ap-961	806	14	dirac	dirac	NOUN
ap-961	806	15	operator	operator	NOUN
ap-961	806	16	.	.	PUNCT
ap-961	807	1	another	another	DET
ap-961	807	2	simplification	simplification	NOUN
ap-961	807	3	that	that	SCONJ
ap-961	807	4	we	we	PRON
ap-961	807	5	make	make	VERB
ap-961	807	6	is	be	AUX
ap-961	807	7	that	that	SCONJ
ap-961	807	8	we	we	PRON
ap-961	807	9	skip	skip	VERB
ap-961	807	10	the	the	DET
ap-961	807	11	requirement	requirement	NOUN
ap-961	807	12	of	of	ADP
ap-961	807	13	the	the	DET
ap-961	807	14	reality	reality	NOUN
ap-961	807	15	conditions	condition	NOUN
ap-961	807	16	(	(	PUNCT
ap-961	807	17	in	in	ADP
ap-961	807	18	the	the	DET
ap-961	807	19	sense	sense	NOUN
ap-961	807	20	of	of	ADP
ap-961	807	21	j	j	NOUN
ap-961	807	22	-	-	NOUN
ap-961	807	23	reality	reality	NOUN
ap-961	807	24	)	)	PUNCT
ap-961	807	25	and	and	CCONJ
ap-961	807	26	take	take	VERB
ap-961	807	27	the	the	DET
ap-961	807	28	manifold	manifold	NOUN
ap-961	807	29	to	to	PART
ap-961	807	30	be	be	AUX
ap-961	807	31	flat	flat	ADJ
ap-961	807	32	(	(	PUNCT
ap-961	807	33	say	say	VERB
ap-961	807	34	a	a	DET
ap-961	807	35	4	4	NUM
ap-961	807	36	-	-	PUNCT
ap-961	807	37	torus	torus	NOUN
ap-961	807	38	!	!	PUNCT
ap-961	807	39	)	)	PUNCT
ap-961	807	40	.	.	PUNCT
ap-961	808	1	as	as	ADP
ap-961	808	2	a	a	DET
ap-961	808	3	hilbert	hilbert	NOUN
ap-961	808	4	space	space	NOUN
ap-961	808	5	we	we	PRON
ap-961	808	6	take	take	VERB
ap-961	808	7	the	the	DET
ap-961	808	8	two	two	NUM
ap-961	808	9	copies	copy	NOUN
ap-961	808	10	of	of	ADP
ap-961	808	11	the	the	DET
ap-961	808	12	spinorial	spinorial	ADJ
ap-961	808	13	hilbert	hilbert	NOUN
ap-961	808	14	space	space	NOUN
ap-961	808	15	over	over	ADP
ap-961	808	16	manifold	manifold	ADJ
ap-961	808	17	m	m	NOUN
ap-961	808	18	:	:	PUNCT
ap-961	808	19	�	�	PROPN
ap-961	808	20	�	�	PROPN
ap-961	808	21	&	&	CCONJ
ap-961	808	22	l	l	PROPN
ap-961	808	23	m	m	PROPN
ap-961	808	24	s	s	PROPN
ap-961	808	25	l	l	NOUN
ap-961	808	26	m	m	NOUN
ap-961	808	27	s2	s2	NOUN
ap-961	808	28	2	2	NUM
ap-961	808	29	(	(	PUNCT
ap-961	808	30	,	,	PUNCT
ap-961	808	31	)	)	PUNCT
ap-961	808	32	(	(	PUNCT
ap-961	808	33	,	,	PUNCT
ap-961	808	34	)	)	PUNCT
ap-961	808	35	.	.	PUNCT
ap-961	809	1	the	the	DET
ap-961	809	2	functions	function	NOUN
ap-961	809	3	act	act	VERB
ap-961	809	4	on	on	ADP
ap-961	809	5	�	�	PROPN
ap-961	809	6	so	so	SCONJ
ap-961	809	7	that	that	SCONJ
ap-961	809	8	f	f	PROPN
ap-961	809	9	h	h	NOUN
ap-961	809	10	h	h	NOUN
ap-961	810	1	f	f	NOUN
ap-961	810	2	h	h	NOUN
ap-961	811	1	f	f	PROPN
ap-961	811	2	h	h	PROPN
ap-961	811	3	(	(	PUNCT
ap-961	811	4	)	)	PUNCT
ap-961	811	5	(	(	PUNCT
ap-961	811	6	)	)	PUNCT
ap-961	811	7	(	(	PUNCT
ap-961	811	8	)	)	PUNCT
ap-961	811	9	�	�	PROPN
ap-961	811	10	�	�	PROPN
ap-961	811	11	&	&	CCONJ
ap-961	811	12	�	�	PROPN
ap-961	811	13	&	&	CCONJ
ap-961	811	14	�	�	PROPN
ap-961	811	15	.	.	PUNCT
ap-961	812	1	as	as	ADP
ap-961	812	2	the	the	DET
ap-961	812	3	dirac	dirac	NOUN
ap-961	812	4	operator	operator	NOUN
ap-961	812	5	we	we	PRON
ap-961	812	6	take	take	VERB
ap-961	812	7	:	:	PUNCT
ap-961	812	8	d	d	PROPN
ap-961	812	9	�	�	PROPN
ap-961	812	10	�	�	PROPN
ap-961	812	11	�	�	PROPN
ap-961	812	12	�	�	PROPN
ap-961	812	13	�	�	PROPN
ap-961	812	14	�	�	PROPN
ap-961	812	15	�	�	PROPN
ap-961	812	16	�	�	PROPN
ap-961	812	17	�	�	PROPN
ap-961	812	18	�	�	PROPN
ap-961	812	19	�	�	PROPN
ap-961	812	20	�	�	PROPN
ap-961	812	21	�	�	PROPN
ap-961	812	22	�	�	PROPN
ap-961	812	23	�	�	PROPN
ap-961	812	24	�	�	PROPN
ap-961	812	25	�	�	PROPN
ap-961	812	26	5	5	NUM
ap-961	812	27	5	5	NUM
ap-961	812	28	,	,	PUNCT
ap-961	812	29	where	where	SCONJ
ap-961	812	30	by	by	ADP
ap-961	812	31	�	�	PROPN
ap-961	812	32	5	5	NUM
ap-961	812	33	we	we	PRON
ap-961	812	34	denote	denote	VERB
ap-961	812	35	the	the	DET
ap-961	812	36	�	�	PROPN
ap-961	812	37	2	2	NUM
ap-961	812	38	-	-	PUNCT
ap-961	812	39	grading	grading	NOUN
ap-961	812	40	of	of	ADP
ap-961	812	41	the	the	DET
ap-961	812	42	standard	standard	ADJ
ap-961	812	43	spectral	spectral	ADJ
ap-961	812	44	triple	triple	NOUN
ap-961	812	45	over	over	ADP
ap-961	812	46	m	m	PROPN
ap-961	812	47	(	(	PUNCT
ap-961	812	48	which	which	PRON
ap-961	812	49	in	in	ADP
ap-961	812	50	physical	physical	ADJ
ap-961	812	51	notation	notation	NOUN
ap-961	812	52	is	be	AUX
ap-961	812	53	�	�	NOUN
ap-961	812	54	5	5	NUM
ap-961	812	55	)	)	PUNCT
ap-961	812	56	and	and	CCONJ
ap-961	812	57	�	�	PROPN
ap-961	812	58	�	�	PROPN
ap-961	812	59	�	�	PROPN
ap-961	812	60	�	�	PROPN
ap-961	812	61	�	�	PROPN
ap-961	812	62	(	(	PUNCT
ap-961	812	63	)	)	PUNCT
ap-961	812	64	�	�	PROPN
ap-961	812	65	is	be	AUX
ap-961	812	66	the	the	DET
ap-961	812	67	standard	standard	ADJ
ap-961	812	68	dirac	dirac	NOUN
ap-961	812	69	operator	operator	NOUN
ap-961	812	70	expressed	express	VERB
ap-961	812	71	in	in	ADP
ap-961	812	72	local	local	ADJ
ap-961	812	73	coordinates	coordinate	NOUN
ap-961	812	74	with	with	ADP
ap-961	812	75	the	the	DET
ap-961	812	76	spin	spin	NOUN
ap-961	812	77	connection	connection	NOUN
ap-961	812	78	�	�	PROPN
ap-961	812	79	�	�	PROPN
ap-961	812	80	.	.	PUNCT
ap-961	813	1	the	the	DET
ap-961	813	2	most	most	ADV
ap-961	813	3	general	general	ADJ
ap-961	813	4	inner	inner	ADJ
ap-961	813	5	fluctuations	fluctuation	NOUN
ap-961	813	6	of	of	ADP
ap-961	813	7	the	the	DET
ap-961	813	8	dirac	dirac	NOUN
ap-961	813	9	operator	operator	NOUN
ap-961	813	10	are	be	AUX
ap-961	813	11	:	:	PUNCT
ap-961	813	12	a	a	DET
ap-961	813	13	a	a	DET
ap-961	813	14	x	x	SYM
ap-961	813	15	h	h	NOUN
ap-961	813	16	x	x	NOUN
ap-961	813	17	h	h	NOUN
ap-961	813	18	x	x	PUNCT
ap-961	813	19	a	a	DET
ap-961	813	20	x	x	PROPN
ap-961	813	21	�	�	PROPN
ap-961	813	22	�	�	PROPN
ap-961	813	23	�	�	PROPN
ap-961	813	24	�	�	PROPN
ap-961	813	25	�	�	PROPN
ap-961	813	26	�	�	PROPN
ap-961	813	27	�	�	PROPN
ap-961	813	28	�	�	PROPN
ap-961	813	29	�	�	PROPN
ap-961	813	30	�	�	PROPN
ap-961	813	31	�	�	PROPN
ap-961	813	32	�	�	PROPN
ap-961	813	33	�	�	PROPN
ap-961	813	34	�	�	PROPN
ap-961	813	35	�	�	PROPN
ap-961	813	36	�	�	PROPN
ap-961	813	37	�	�	PROPN
ap-961	813	38	�	�	PROPN
ap-961	813	39	(	(	PUNCT
ap-961	813	40	)	)	PUNCT
ap-961	813	41	(	(	PUNCT
ap-961	813	42	)	)	PUNCT
ap-961	813	43	(	(	PUNCT
ap-961	813	44	)	)	PUNCT
ap-961	813	45	(	(	PUNCT
ap-961	813	46	)	)	PUNCT
ap-961	813	47	*	*	SYM
ap-961	813	48	5	5	NUM
ap-961	813	49	5	5	NUM
ap-961	813	50	54	54	NUM
ap-961	813	51	©	©	PROPN
ap-961	813	52	czech	czech	PROPN
ap-961	813	53	technical	technical	PROPN
ap-961	813	54	university	university	PROPN
ap-961	813	55	publishing	publishing	NOUN
ap-961	813	56	house	house	NOUN
ap-961	813	57	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	813	58	acta	acta	PROPN
ap-961	813	59	polytechnica	polytechnica	PROPN
ap-961	813	60	vol	vol	NOUN
ap-961	813	61	.	.	PUNCT
ap-961	814	1	48	48	NUM
ap-961	814	2	no	no	NOUN
ap-961	814	3	.	.	PUNCT
ap-961	815	1	2/2008	2/2008	NOUN
ap-961	815	2	the	the	DET
ap-961	815	3	next	next	ADJ
ap-961	815	4	step	step	NOUN
ap-961	815	5	is	be	AUX
ap-961	815	6	a	a	DET
ap-961	815	7	not	not	PART
ap-961	815	8	a	a	DET
ap-961	815	9	trivial	trivial	ADJ
ap-961	815	10	one	one	NOUN
ap-961	815	11	–	–	PUNCT
ap-961	815	12	but	but	CCONJ
ap-961	815	13	using	use	VERB
ap-961	815	14	the	the	DET
ap-961	815	15	knowledge	knowledge	NOUN
ap-961	815	16	of	of	ADP
ap-961	815	17	the	the	DET
ap-961	815	18	asymptotic	asymptotic	ADJ
ap-961	815	19	expansion	expansion	NOUN
ap-961	815	20	and	and	CCONJ
ap-961	815	21	seeley	seeley	NOUN
ap-961	815	22	-	-	PUNCT
ap-961	815	23	de	de	X
ap-961	815	24	witt	witt	ADJ
ap-961	815	25	coefficients	coefficient	NOUN
ap-961	815	26	(	(	PUNCT
ap-961	815	27	see	see	VERB
ap-961	815	28	[	[	X
ap-961	815	29	22	22	NUM
ap-961	815	30	,	,	PUNCT
ap-961	815	31	25	25	NUM
ap-961	815	32	]	]	PUNCT
ap-961	815	33	for	for	ADP
ap-961	815	34	explanation	explanation	NOUN
ap-961	815	35	and	and	CCONJ
ap-961	815	36	details	detail	NOUN
ap-961	815	37	)	)	PUNCT
ap-961	815	38	we	we	PRON
ap-961	815	39	find	find	VERB
ap-961	815	40	out	out	ADP
ap-961	815	41	what	what	PRON
ap-961	815	42	will	will	AUX
ap-961	815	43	change	change	VERB
ap-961	815	44	in	in	ADP
ap-961	815	45	the	the	DET
ap-961	815	46	expansion	expansion	NOUN
ap-961	815	47	of	of	ADP
ap-961	815	48	the	the	DET
ap-961	815	49	spectral	spectral	ADJ
ap-961	815	50	action	action	NOUN
ap-961	815	51	.	.	PUNCT
ap-961	816	1	we	we	PRON
ap-961	816	2	skip	skip	VERB
ap-961	816	3	the	the	DET
ap-961	816	4	exact	exact	ADJ
ap-961	816	5	calculation	calculation	NOUN
ap-961	816	6	(	(	PUNCT
ap-961	816	7	which	which	PRON
ap-961	816	8	we	we	PRON
ap-961	816	9	recommend	recommend	VERB
ap-961	816	10	,	,	PUNCT
ap-961	816	11	however	however	ADV
ap-961	816	12	,	,	PUNCT
ap-961	816	13	as	as	ADP
ap-961	816	14	a	a	DET
ap-961	816	15	good	good	ADJ
ap-961	816	16	exercise	exercise	NOUN
ap-961	816	17	!	!	PUNCT
ap-961	816	18	)	)	PUNCT
ap-961	817	1	it	it	PRON
ap-961	817	2	is	be	AUX
ap-961	817	3	no	no	DET
ap-961	817	4	surprise	surprise	NOUN
ap-961	817	5	that	that	SCONJ
ap-961	817	6	the	the	DET
ap-961	817	7	additional	additional	ADJ
ap-961	817	8	terms	term	NOUN
ap-961	817	9	involve	involve	VERB
ap-961	817	10	certain	certain	ADJ
ap-961	817	11	covariant	covariant	ADJ
ap-961	817	12	functionals	functional	NOUN
ap-961	817	13	of	of	ADP
ap-961	817	14	h	h	NOUN
ap-961	817	15	:	:	PUNCT
ap-961	817	16	�	�	PROPN
ap-961	817	17	the	the	DET
ap-961	817	18	vacuum	vacuum	NOUN
ap-961	817	19	energy	energy	NOUN
ap-961	817	20	of	of	ADP
ap-961	817	21	field	field	NOUN
ap-961	817	22	h(x	h(x	PROPN
ap-961	817	23	)	)	PUNCT
ap-961	817	24	(	(	PUNCT
ap-961	817	25	both	both	CCONJ
ap-961	817	26	in	in	ADP
ap-961	817	27	�	�	PROPN
ap-961	817	28	2	2	NUM
ap-961	817	29	and	and	CCONJ
ap-961	817	30	�	�	NOUN
ap-961	817	31	0	0	NUM
ap-961	817	32	parts	part	NOUN
ap-961	817	33	)	)	PUNCT
ap-961	817	34	g	g	NOUN
ap-961	817	35	d	d	NOUN
ap-961	817	36	x	x	SYM
ap-961	817	37	h	h	NOUN
ap-961	817	38	x	x	SYM
ap-961	817	39	h	h	NOUN
ap-961	817	40	x4	x4	PROPN
ap-961	817	41	(	(	PUNCT
ap-961	817	42	)	)	PUNCT
ap-961	817	43	(	(	PUNCT
ap-961	817	44	)	)	PUNCT
ap-961	817	45	*	*	SYM
ap-961	817	46	2	2	NUM
ap-961	817	47	,	,	PUNCT
ap-961	817	48	�	�	PROPN
ap-961	817	49	the	the	DET
ap-961	817	50	coupling	coupling	NOUN
ap-961	817	51	of	of	ADP
ap-961	817	52	h	h	NOUN
ap-961	817	53	to	to	ADP
ap-961	817	54	the	the	DET
ap-961	817	55	scalar	scalar	NOUN
ap-961	817	56	of	of	ADP
ap-961	817	57	curvature	curvature	NOUN
ap-961	817	58	(	(	PUNCT
ap-961	817	59	�	�	PROPN
ap-961	817	60	0	0	NUM
ap-961	817	61	part	part	NOUN
ap-961	817	62	)	)	PUNCT
ap-961	817	63	g	g	NOUN
ap-961	818	1	d	d	NOUN
ap-961	818	2	x	x	SYM
ap-961	818	3	h	h	NOUN
ap-961	818	4	x	x	NOUN
ap-961	818	5	h	h	NOUN
ap-961	818	6	x	x	PUNCT
ap-961	818	7	r4	r4	PROPN
ap-961	818	8	(	(	PUNCT
ap-961	818	9	)	)	PUNCT
ap-961	818	10	(	(	PUNCT
ap-961	818	11	)	)	PUNCT
ap-961	818	12	*	*	SYM
ap-961	818	13	2	2	NUM
ap-961	818	14	,	,	PUNCT
ap-961	818	15	�	�	PROPN
ap-961	818	16	the	the	DET
ap-961	818	17	kinetic	kinetic	ADJ
ap-961	818	18	term	term	NOUN
ap-961	818	19	for	for	ADP
ap-961	818	20	the	the	DET
ap-961	818	21	h(x	h(x	PROPN
ap-961	818	22	)	)	PUNCT
ap-961	818	23	field	field	NOUN
ap-961	818	24	:	:	PUNCT
ap-961	818	25	g	g	PROPN
ap-961	818	26	d	d	X
ap-961	818	27	x	x	PROPN
ap-961	818	28	d	d	NOUN
ap-961	818	29	h	h	NOUN
ap-961	818	30	x	x	PUNCT
ap-961	818	31	d	d	NOUN
ap-961	818	32	h	h	NOUN
ap-961	818	33	x4	x4	PROPN
ap-961	818	34	(	(	PUNCT
ap-961	818	35	(	(	PUNCT
ap-961	818	36	)	)	PUNCT
ap-961	818	37	(	(	PUNCT
ap-961	818	38	)	)	PUNCT
ap-961	818	39	)	)	PUNCT
ap-961	819	1	*	*	PUNCT
ap-961	819	2	�	�	PROPN
ap-961	819	3	�	�	PROPN
ap-961	819	4	2	2	NUM
ap-961	819	5	.	.	PUNCT
ap-961	819	6	�	�	PROPN
ap-961	819	7	the	the	DET
ap-961	819	8	potential	potential	NOUN
ap-961	819	9	of	of	ADP
ap-961	819	10	the	the	DET
ap-961	819	11	h(x	h(x	PROPN
ap-961	819	12	)	)	PUNCT
ap-961	819	13	field	field	NOUN
ap-961	819	14	:	:	PUNCT
ap-961	819	15	g	g	PROPN
ap-961	819	16	d	d	X
ap-961	819	17	x	x	SYM
ap-961	819	18	h	h	NOUN
ap-961	819	19	x4	x4	PROPN
ap-961	819	20	4	4	NUM
ap-961	819	21	(	(	PUNCT
ap-961	819	22	)	)	PUNCT
ap-961	819	23	2	2	X
ap-961	819	24	.	.	PUNCT
ap-961	820	1	here	here	ADV
ap-961	820	2	d	d	X
ap-961	820	3	�	�	PROPN
ap-961	820	4	denotes	denote	VERB
ap-961	820	5	there	there	ADV
ap-961	820	6	the	the	DET
ap-961	820	7	u	u	PROPN
ap-961	820	8	u	u	NOUN
ap-961	820	9	(	(	PUNCT
ap-961	820	10	)	)	PUNCT
ap-961	820	11	(	(	PUNCT
ap-961	820	12	)	)	SYM
ap-961	820	13	1	1	NUM
ap-961	820	14	1	1	NUM
ap-961	820	15	�	�	PROPN
ap-961	820	16	-covariant	-covariant	NOUN
ap-961	820	17	derivative	derivative	NOUN
ap-961	820	18	.	.	PUNCT
ap-961	821	1	we	we	PRON
ap-961	821	2	can	can	AUX
ap-961	821	3	interpret	interpret	VERB
ap-961	821	4	these	these	DET
ap-961	821	5	contributions	contribution	NOUN
ap-961	821	6	in	in	ADP
ap-961	821	7	physical	physical	ADJ
ap-961	821	8	terms	term	NOUN
ap-961	821	9	as	as	ADP
ap-961	821	10	the	the	DET
ap-961	821	11	kinetic	kinetic	ADJ
ap-961	821	12	action	action	NOUN
ap-961	821	13	of	of	ADP
ap-961	821	14	the	the	DET
ap-961	821	15	higgs	higgs	PROPN
ap-961	821	16	field	field	PROPN
ap-961	821	17	h(x	h(x	PROPN
ap-961	821	18	)	)	PUNCT
ap-961	821	19	and	and	CCONJ
ap-961	821	20	the	the	DET
ap-961	821	21	higgs	higgs	NOUN
ap-961	821	22	potential	potential	NOUN
ap-961	821	23	,	,	PUNCT
ap-961	821	24	which	which	PRON
ap-961	821	25	after	after	ADP
ap-961	821	26	some	some	DET
ap-961	821	27	rescaling	rescaling	NOUN
ap-961	821	28	can	can	AUX
ap-961	821	29	be	be	AUX
ap-961	821	30	written	write	VERB
ap-961	821	31	as	as	ADP
ap-961	821	32	:	:	PUNCT
ap-961	821	33	g	g	PROPN
ap-961	821	34	d	d	X
ap-961	821	35	x	x	SYM
ap-961	821	36	h	h	NOUN
ap-961	822	1	x	x	NOUN
ap-961	822	2	h	h	NOUN
ap-961	822	3	x4	x4	ADJ
ap-961	822	4	2	2	NUM
ap-961	822	5	2	2	NUM
ap-961	822	6	(	(	PUNCT
ap-961	822	7	(	(	PUNCT
ap-961	822	8	)	)	PUNCT
ap-961	822	9	(	(	PUNCT
ap-961	822	10	)	)	PUNCT
ap-961	822	11	)	)	PUNCT
ap-961	823	1	*	*	PUNCT
ap-961	823	2	�	�	PROPN
ap-961	823	3	$	$	SYM
ap-961	823	4	2	2	NUM
ap-961	823	5	.	.	PUNCT
ap-961	824	1	the	the	DET
ap-961	824	2	crucial	crucial	ADJ
ap-961	824	3	information	information	NOUN
ap-961	824	4	(	(	PUNCT
ap-961	824	5	from	from	ADP
ap-961	824	6	the	the	DET
ap-961	824	7	physical	physical	ADJ
ap-961	824	8	point	point	NOUN
ap-961	824	9	of	of	ADP
ap-961	824	10	view	view	NOUN
ap-961	824	11	)	)	PUNCT
ap-961	824	12	lies	lie	VERB
ap-961	824	13	actually	actually	ADV
ap-961	824	14	not	not	PART
ap-961	824	15	in	in	ADP
ap-961	824	16	the	the	DET
ap-961	824	17	exact	exact	ADJ
ap-961	824	18	values	value	NOUN
ap-961	824	19	of	of	ADP
ap-961	824	20	the	the	DET
ap-961	824	21	coefficients	coefficient	NOUN
ap-961	824	22	but	but	CCONJ
ap-961	824	23	their	their	PRON
ap-961	824	24	relative	relative	ADJ
ap-961	824	25	signs	sign	NOUN
ap-961	824	26	.	.	PUNCT
ap-961	825	1	if	if	SCONJ
ap-961	825	2	h	h	NOUN
ap-961	825	3	2	2	NUM
ap-961	825	4	and	and	CCONJ
ap-961	825	5	h	h	NOUN
ap-961	825	6	4	4	NUM
ap-961	825	7	appear	appear	VERB
ap-961	825	8	with	with	ADP
ap-961	825	9	opposite	opposite	ADJ
ap-961	825	10	signs	sign	NOUN
ap-961	825	11	we	we	PRON
ap-961	825	12	have	have	VERB
ap-961	825	13	the	the	DET
ap-961	825	14	standard	standard	ADJ
ap-961	825	15	higgs	higgs	PROPN
ap-961	825	16	potential	potential	NOUN
ap-961	825	17	leading	lead	VERB
ap-961	825	18	to	to	ADP
ap-961	825	19	the	the	DET
ap-961	825	20	symmetry	symmetry	NOUN
ap-961	825	21	breaking	breaking	NOUN
ap-961	825	22	mechanism	mechanism	NOUN
ap-961	825	23	.	.	PUNCT
ap-961	826	1	since	since	SCONJ
ap-961	826	2	(	(	PUNCT
ap-961	826	3	a	a	DET
ap-961	826	4	priori	priori	X
ap-961	826	5	)	)	PUNCT
ap-961	826	6	�	�	PROPN
ap-961	826	7	and	and	CCONJ
ap-961	826	8	all	all	DET
ap-961	826	9	coefficients	coefficient	NOUN
ap-961	826	10	fk	fk	INTJ
ap-961	826	11	from	from	ADP
ap-961	826	12	the	the	DET
ap-961	826	13	cutoff	cutoff	NOUN
ap-961	826	14	function	function	NOUN
ap-961	826	15	are	be	AUX
ap-961	826	16	free	free	ADJ
ap-961	826	17	parameters	parameter	NOUN
ap-961	826	18	we	we	PRON
ap-961	826	19	can	can	AUX
ap-961	826	20	actually	actually	ADV
ap-961	826	21	fix	fix	VERB
ap-961	826	22	them	they	PRON
ap-961	826	23	so	so	SCONJ
ap-961	826	24	that	that	SCONJ
ap-961	826	25	the	the	DET
ap-961	826	26	signs	sign	NOUN
ap-961	826	27	are	be	AUX
ap-961	826	28	correct	correct	ADJ
ap-961	826	29	.	.	PUNCT
ap-961	827	1	for	for	ADP
ap-961	827	2	the	the	DET
ap-961	827	3	more	more	ADV
ap-961	827	4	realistic	realistic	ADJ
ap-961	827	5	approach	approach	NOUN
ap-961	827	6	to	to	ADP
ap-961	827	7	the	the	DET
ap-961	827	8	standard	standard	ADJ
ap-961	827	9	model	model	NOUN
ap-961	827	10	we	we	PRON
ap-961	827	11	need	need	VERB
ap-961	827	12	to	to	PART
ap-961	827	13	take	take	VERB
ap-961	827	14	a	a	DET
ap-961	827	15	slightly	slightly	ADV
ap-961	827	16	complicated	complicate	VERB
ap-961	827	17	model	model	NOUN
ap-961	827	18	,	,	PUNCT
ap-961	827	19	with	with	ADP
ap-961	827	20	the	the	DET
ap-961	827	21	finite	finite	ADJ
ap-961	827	22	algebra	algebra	NOUN
ap-961	827	23	being	be	AUX
ap-961	827	24	�	�	PROPN
ap-961	827	25	�	�	PROPN
ap-961	827	26	&	&	CCONJ
ap-961	827	27	&	&	CCONJ
ap-961	827	28	�	�	PROPN
ap-961	827	29	m3	m3	PROPN
ap-961	827	30	(	(	PUNCT
ap-961	827	31	)	)	PUNCT
ap-961	827	32	.	.	PUNCT
ap-961	828	1	it	it	PRON
ap-961	828	2	comes	come	VERB
ap-961	828	3	as	as	ADP
ap-961	828	4	no	no	DET
ap-961	828	5	surprise	surprise	NOUN
ap-961	828	6	that	that	SCONJ
ap-961	828	7	this	this	DET
ap-961	828	8	construction	construction	NOUN
ap-961	828	9	leads	lead	VERB
ap-961	828	10	to	to	ADP
ap-961	828	11	the	the	DET
ap-961	828	12	full	full	ADJ
ap-961	828	13	gauge	gauge	NOUN
ap-961	828	14	group	group	NOUN
ap-961	828	15	of	of	ADP
ap-961	828	16	the	the	DET
ap-961	828	17	standard	standard	ADJ
ap-961	828	18	model	model	NOUN
ap-961	828	19	:	:	PUNCT
ap-961	828	20	u	u	PROPN
ap-961	828	21	su	su	PROPN
ap-961	828	22	su	su	PROPN
ap-961	828	23	(	(	PUNCT
ap-961	828	24	)	)	PUNCT
ap-961	828	25	(	(	PUNCT
ap-961	828	26	)	)	PUNCT
ap-961	828	27	(	(	PUNCT
ap-961	828	28	)	)	SYM
ap-961	828	29	1	1	NUM
ap-961	828	30	2	2	NUM
ap-961	828	31	3	3	NUM
ap-961	828	32	�	�	PROPN
ap-961	828	33	�	�	PROPN
ap-961	828	34	.	.	PUNCT
ap-961	829	1	taking	take	VERB
ap-961	829	2	an	an	DET
ap-961	829	3	appropriate	appropriate	ADJ
ap-961	829	4	spectral	spectral	ADJ
ap-961	829	5	triple	triple	NOUN
ap-961	829	6	over	over	ADP
ap-961	829	7	the	the	DET
ap-961	829	8	finite	finite	ADJ
ap-961	829	9	algebra	algebra	PROPN
ap-961	829	10	(	(	PUNCT
ap-961	829	11	which	which	PRON
ap-961	829	12	is	be	AUX
ap-961	829	13	actually	actually	ADV
ap-961	829	14	of	of	ADP
ap-961	829	15	ko	ko	PROPN
ap-961	829	16	-	-	PUNCT
ap-961	829	17	dimension	dimension	NOUN
ap-961	829	18	6	6	NUM
ap-961	829	19	)	)	PUNCT
ap-961	829	20	and	and	CCONJ
ap-961	829	21	tensoring	tensore	VERB
ap-961	829	22	it	it	PRON
ap-961	829	23	with	with	ADP
ap-961	829	24	the	the	DET
ap-961	829	25	usual	usual	ADJ
ap-961	829	26	spectral	spectral	ADJ
ap-961	829	27	triple	triple	NOUN
ap-961	829	28	over	over	ADP
ap-961	829	29	a	a	DET
ap-961	829	30	manifold	manifold	NOUN
ap-961	829	31	,	,	PUNCT
ap-961	829	32	one	one	NUM
ap-961	829	33	obtains	obtain	VERB
ap-961	829	34	as	as	ADP
ap-961	829	35	a	a	DET
ap-961	829	36	spectral	spectral	ADJ
ap-961	829	37	action	action	NOUN
ap-961	829	38	:	:	PUNCT
ap-961	829	39	s	s	VERB
ap-961	830	1	f	f	X
ap-961	830	2	f	f	PROPN
ap-961	830	3	c	c	NOUN
ap-961	830	4	f	f	PROPN
ap-961	831	1	d	d	X
ap-961	831	2	g	g	PROPN
ap-961	831	3	d	d	X
ap-961	831	4	x	x	X
ap-961	831	5	f	f	NOUN
ap-961	831	6	f	f	NOUN
ap-961	831	7	c	c	NOUN
ap-961	831	8	r	r	NOUN
ap-961	831	9	g	g	PROPN
ap-961	831	10	d	d	PROPN
ap-961	831	11	�	�	PROPN
ap-961	831	12	�	�	PROPN
ap-961	831	13	�	�	PROPN
ap-961	831	14	�	�	PROPN
ap-961	831	15	�	�	PROPN
ap-961	831	16	�	�	PROPN
ap-961	831	17	�	�	PROPN
ap-961	831	18	�	�	PROPN
ap-961	831	19	21	21	NUM
ap-961	831	20	48	48	NUM
ap-961	831	21	4	4	NUM
ap-961	831	22	96	96	NUM
ap-961	831	23	24	24	NUM
ap-961	831	24	2	2	NUM
ap-961	831	25	4	4	NUM
ap-961	831	26	4	4	NUM
ap-961	831	27	2	2	NUM
ap-961	831	28	0	0	NUM
ap-961	831	29	4	4	NUM
ap-961	831	30	2	2	NUM
ap-961	831	31	2	2	NUM
ap-961	831	32	0	0	NUM
ap-961	831	33	2	2	NUM
ap-961	831	34	�	�	PROPN
ap-961	831	35	�	�	PROPN
ap-961	831	36	�	�	PROPN
ap-961	831	37	�	�	PROPN
ap-961	831	38	�	�	PROPN
ap-961	831	39	�	�	PROPN
ap-961	831	40	4	4	NUM
ap-961	831	41	0	0	NUM
ap-961	831	42	2	2	NUM
ap-961	831	43	4	4	NUM
ap-961	831	44	2	2	NUM
ap-961	831	45	2	2	NUM
ap-961	831	46	10	10	NUM
ap-961	831	47	11	11	NUM
ap-961	831	48	6	6	NUM
ap-961	831	49	3	3	NUM
ap-961	831	50	2	2	NUM
ap-961	831	51	x	x	SYM
ap-961	831	52	f	f	NOUN
ap-961	831	53	r	r	NOUN
ap-961	831	54	r	r	NOUN
ap-961	831	55	c	c	NOUN
ap-961	831	56	c	c	NOUN
ap-961	831	57	g	g	PROPN
ap-961	831	58	d	d	PROPN
ap-961	831	59	x	x	PROPN
ap-961	831	60	a	a	DET
ap-961	831	61	f	f	PROPN
ap-961	831	62	2	2	NUM
ap-961	831	63	2	2	NUM
ap-961	831	64	�	�	PROPN
ap-961	831	65	�	�	PROPN
ap-961	831	66	�	�	PROPN
ap-961	831	67	�	�	PROPN
ap-961	831	68	�	�	PROPN
ap-961	831	69	�	�	PROPN
ap-961	831	70	�	�	PROPN
ap-961	831	71	�	�	PROPN
ap-961	831	72	�	�	PROPN
ap-961	831	73	�	�	PROPN
ap-961	831	74	�	�	PROPN
ap-961	831	75	�	�	PROPN
ap-961	831	76	�	�	PROPN
ap-961	831	77	�	�	PROPN
ap-961	831	78	�	�	PROPN
ap-961	831	79	�	�	PROPN
ap-961	831	80	�	�	PROPN
ap-961	831	81	*	*	NOUN
ap-961	831	82	*	*	PUNCT
ap-961	831	83	(	(	PUNCT
ap-961	831	84	�	�	PROPN
ap-961	831	85	e	e	NOUN
ap-961	831	86	f	f	PROPN
ap-961	831	87	g	g	PROPN
ap-961	831	88	d	d	PROPN
ap-961	831	89	x	x	PROPN
ap-961	831	90	f	f	PROPN
ap-961	831	91	a	a	X
ap-961	831	92	d	d	X
ap-961	831	93	g	g	PROPN
ap-961	831	94	d	d	X
ap-961	831	95	x	x	X
ap-961	831	96	f	f	PROPN
ap-961	831	97	ar	ar	PROPN
ap-961	831	98	g	g	PROPN
ap-961	831	99	d	d	PROPN
ap-961	831	100	x	x	PROPN
ap-961	831	101	f	f	NOUN
ap-961	831	102	0	0	NUM
ap-961	831	103	2	2	NUM
ap-961	831	104	2	2	NUM
ap-961	831	105	4	4	NUM
ap-961	831	106	0	0	NUM
ap-961	831	107	2	2	NUM
ap-961	831	108	2	2	NUM
ap-961	831	109	4	4	NUM
ap-961	831	110	0	0	NUM
ap-961	831	111	2	2	NUM
ap-961	831	112	2	2	NUM
ap-961	831	113	4	4	NUM
ap-961	831	114	0	0	NUM
ap-961	831	115	2	2	NUM
ap-961	831	116	12	12	NUM
ap-961	831	117	2	2	NUM
ap-961	831	118	)	)	PUNCT
ap-961	831	119	�	�	PROPN
ap-961	831	120	�	�	PROPN
ap-961	831	121	�	�	PROPN
ap-961	831	122	�	�	PROPN
ap-961	831	123	�	�	PROPN
ap-961	831	124	2	2	NUM
ap-961	831	125	2	2	NUM
ap-961	831	126	2	2	NUM
ap-961	831	127	�	�	NOUN
ap-961	831	128	2	2	NUM
ap-961	831	129	3	3	NUM
ap-961	831	130	2	2	NUM
ap-961	831	131	2	2	NUM
ap-961	831	132	2	2	NUM
ap-961	831	133	1	1	NUM
ap-961	831	134	2	2	NUM
ap-961	831	135	4	4	NUM
ap-961	831	136	0	0	NUM
ap-961	831	137	5	5	NUM
ap-961	831	138	3	3	NUM
ap-961	831	139	g	g	NOUN
ap-961	831	140	g	g	PROPN
ap-961	831	141	g	g	PROPN
ap-961	831	142	g	g	PROPN
ap-961	831	143	f	f	PROPN
ap-961	831	144	f	f	PROPN
ap-961	831	145	g	g	PROPN
ap-961	831	146	b	b	PROPN
ap-961	832	1	b	b	PROPN
ap-961	832	2	g	g	PROPN
ap-961	832	3	d	d	PROPN
ap-961	832	4	x	x	X
ap-961	832	5	f	f	X
ap-961	833	1	i	i	PRON
ap-961	833	2	i	i	PROPN
ap-961	833	3	�	�	VERB
ap-961	833	4	�	�	PROPN
ap-961	833	5	�	�	PROPN
ap-961	833	6	�	�	PROPN
ap-961	833	7	�	�	PROPN
ap-961	833	8	�	�	PROPN
ap-961	833	9	�	�	PROPN
ap-961	833	10	�	�	PROPN
ap-961	833	11	�	�	PROPN
ap-961	833	12	�	�	PROPN
ap-961	833	13	�	�	PROPN
ap-961	833	14	�	�	PROPN
ap-961	833	15	�	�	PROPN
ap-961	833	16	�	�	PROPN
ap-961	833	17	�	�	PROPN
ap-961	833	18	�	�	PROPN
ap-961	833	19	�	�	PROPN
ap-961	833	20	�	�	PROPN
ap-961	833	21	�	�	PROPN
ap-961	833	22	�	�	PROPN
ap-961	833	23	2	2	NUM
ap-961	833	24	2	2	NUM
ap-961	833	25	2	2	NUM
ap-961	833	26	4	4	NUM
ap-961	833	27	4	4	NUM
ap-961	833	28	�	�	PROPN
ap-961	833	29	b	b	NOUN
ap-961	833	30	g	g	PROPN
ap-961	833	31	d	d	PROPN
ap-961	833	32	x2	x2	PROPN
ap-961	833	33	,	,	PUNCT
ap-961	833	34	where	where	SCONJ
ap-961	833	35	f	f	X
ap-961	833	36	,	,	PUNCT
ap-961	833	37	b	b	PROPN
ap-961	833	38	,	,	PUNCT
ap-961	833	39	g	g	PROPN
ap-961	833	40	are	be	AUX
ap-961	833	41	curvatures	curvature	NOUN
ap-961	833	42	of	of	ADP
ap-961	833	43	the	the	DET
ap-961	833	44	electro	electro	NOUN
ap-961	833	45	-	-	PUNCT
ap-961	833	46	weak	weak	ADJ
ap-961	833	47	and	and	CCONJ
ap-961	833	48	strong	strong	ADJ
ap-961	833	49	gauge	gauge	NOUN
ap-961	833	50	fields	field	NOUN
ap-961	833	51	and	and	CCONJ
ap-961	833	52	�	�	PROPN
ap-961	833	53	is	be	AUX
ap-961	833	54	the	the	DET
ap-961	833	55	sm	sm	PROPN
ap-961	833	56	higgs	higgs	PROPN
ap-961	833	57	doublet	doublet	PROPN
ap-961	833	58	(	(	PUNCT
ap-961	833	59	seen	see	VERB
ap-961	833	60	as	as	ADP
ap-961	833	61	a	a	DET
ap-961	833	62	quaternionic	quaternionic	ADJ
ap-961	833	63	field	field	NOUN
ap-961	833	64	)	)	PUNCT
ap-961	833	65	.	.	PUNCT
ap-961	834	1	for	for	ADP
ap-961	834	2	more	more	ADJ
ap-961	834	3	details	detail	NOUN
ap-961	834	4	,	,	PUNCT
ap-961	834	5	please	please	INTJ
ap-961	834	6	consult	consult	VERB
ap-961	834	7	[	[	X
ap-961	834	8	6	6	NUM
ap-961	834	9	]	]	PUNCT
ap-961	834	10	.	.	PUNCT
ap-961	835	1	6.6	6.6	NUM
ap-961	835	2	where	where	SCONJ
ap-961	835	3	and	and	CCONJ
ap-961	835	4	why	why	SCONJ
ap-961	835	5	learn	learn	VERB
ap-961	835	6	more	more	ADJ
ap-961	835	7	?	?	PUNCT
ap-961	836	1	in	in	ADP
ap-961	836	2	these	these	DET
ap-961	836	3	three	three	NUM
ap-961	836	4	lectures	lecture	NOUN
ap-961	836	5	we	we	PRON
ap-961	836	6	have	have	AUX
ap-961	836	7	tried	try	VERB
ap-961	836	8	to	to	PART
ap-961	836	9	give	give	VERB
ap-961	836	10	a	a	DET
ap-961	836	11	glimpse	glimpse	NOUN
ap-961	836	12	of	of	ADP
ap-961	836	13	noncommutative	noncommutative	ADJ
ap-961	836	14	geometry	geometry	NOUN
ap-961	836	15	–	–	PUNCT
ap-961	836	16	a	a	DET
ap-961	836	17	theory	theory	NOUN
ap-961	836	18	,	,	PUNCT
ap-961	836	19	which	which	PRON
ap-961	836	20	,	,	PUNCT
ap-961	836	21	motivated	motivate	VERB
ap-961	836	22	by	by	ADP
ap-961	836	23	examples	example	NOUN
ap-961	836	24	,	,	PUNCT
ap-961	836	25	extends	extend	VERB
ap-961	836	26	the	the	DET
ap-961	836	27	notion	notion	NOUN
ap-961	836	28	of	of	ADP
ap-961	836	29	geometry	geometry	NOUN
ap-961	836	30	into	into	ADP
ap-961	836	31	the	the	DET
ap-961	836	32	algebraic	algebraic	ADJ
ap-961	836	33	world	world	NOUN
ap-961	836	34	.	.	PUNCT
ap-961	837	1	what	what	PRON
ap-961	837	2	we	we	PRON
ap-961	837	3	still	still	ADV
ap-961	837	4	need	need	VERB
ap-961	837	5	to	to	PART
ap-961	837	6	supply	supply	VERB
ap-961	837	7	is	be	AUX
ap-961	837	8	a	a	DET
ap-961	837	9	word	word	NOUN
ap-961	837	10	about	about	ADP
ap-961	837	11	prospects	prospect	NOUN
ap-961	837	12	:	:	PUNCT
ap-961	837	13	first	first	ADV
ap-961	837	14	of	of	ADP
ap-961	837	15	learning	learn	VERB
ap-961	837	16	(	(	PUNCT
ap-961	837	17	where	where	SCONJ
ap-961	837	18	to	to	PART
ap-961	837	19	learn	learn	VERB
ap-961	837	20	more	more	ADJ
ap-961	837	21	)	)	PUNCT
ap-961	837	22	but	but	CCONJ
ap-961	837	23	also	also	ADV
ap-961	837	24	the	the	DET
ap-961	837	25	prospects	prospect	NOUN
ap-961	837	26	of	of	ADP
ap-961	837	27	the	the	DET
ap-961	837	28	field	field	NOUN
ap-961	837	29	(	(	PUNCT
ap-961	837	30	why	why	SCONJ
ap-961	837	31	learn	learn	VERB
ap-961	837	32	it	it	PRON
ap-961	837	33	)	)	PUNCT
ap-961	837	34	.	.	PUNCT
ap-961	838	1	6.6.1	6.6.1	NUM
ap-961	838	2	the	the	DET
ap-961	838	3	sources	source	NOUN
ap-961	838	4	for	for	ADP
ap-961	838	5	the	the	DET
ap-961	838	6	more	more	ADV
ap-961	838	7	interested	interested	ADJ
ap-961	838	8	reader	reader	NOUN
ap-961	838	9	we	we	PRON
ap-961	838	10	recommend	recommend	VERB
ap-961	838	11	further	further	ADJ
ap-961	838	12	reading	reading	NOUN
ap-961	838	13	.	.	PUNCT
ap-961	839	1	first	first	ADV
ap-961	839	2	of	of	ADP
ap-961	839	3	all	all	PRON
ap-961	839	4	,	,	PUNCT
ap-961	839	5	there	there	PRON
ap-961	839	6	is	be	VERB
ap-961	839	7	an	an	DET
ap-961	839	8	excellent	excellent	ADJ
ap-961	839	9	introduction	introduction	NOUN
ap-961	839	10	to	to	ADP
ap-961	839	11	almost	almost	ADV
ap-961	839	12	all	all	PRON
ap-961	839	13	of	of	ADP
ap-961	839	14	the	the	DET
ap-961	839	15	topics	topic	NOUN
ap-961	839	16	of	of	ADP
ap-961	839	17	these	these	DET
ap-961	839	18	lectures	lecture	NOUN
ap-961	839	19	:	:	PUNCT
ap-961	839	20	“	"	PUNCT
ap-961	839	21	elements	element	NOUN
ap-961	839	22	of	of	ADP
ap-961	839	23	noncommutative	noncommutative	ADJ
ap-961	839	24	geometry	geometry	NOUN
ap-961	839	25	”	"	PUNCT
ap-961	840	1	[	[	X
ap-961	840	2	2	2	NUM
ap-961	840	3	]	]	PUNCT
ap-961	840	4	by	by	ADP
ap-961	840	5	josé	josé	PROPN
ap-961	840	6	gracia	gracia	PROPN
ap-961	840	7	-	-	PUNCT
ap-961	840	8	bondia	bondia	PROPN
ap-961	840	9	,	,	PUNCT
ap-961	840	10	joseph	joseph	PROPN
ap-961	840	11	várilly	várilly	ADV
ap-961	840	12	and	and	CCONJ
ap-961	840	13	hector	hector	PROPN
ap-961	840	14	figueroa	figueroa	PROPN
ap-961	840	15	.	.	PUNCT
ap-961	841	1	it	it	PRON
ap-961	841	2	offers	offer	VERB
ap-961	841	3	a	a	DET
ap-961	841	4	comprehensive	comprehensive	ADJ
ap-961	841	5	and	and	CCONJ
ap-961	841	6	detailed	detailed	ADJ
ap-961	841	7	course	course	NOUN
ap-961	841	8	in	in	ADP
ap-961	841	9	noncommutative	noncommutative	ADJ
ap-961	841	10	geometry	geometry	NOUN
ap-961	841	11	reviewing	reviewing	NOUN
ap-961	841	12	also	also	ADV
ap-961	841	13	the	the	DET
ap-961	841	14	most	most	ADV
ap-961	841	15	recent	recent	ADJ
ap-961	841	16	trends	trend	NOUN
ap-961	841	17	and	and	CCONJ
ap-961	841	18	links	link	NOUN
ap-961	841	19	with	with	ADP
ap-961	841	20	physics	physics	NOUN
ap-961	841	21	.	.	PUNCT
ap-961	842	1	there	there	PRON
ap-961	842	2	are	be	VERB
ap-961	842	3	,	,	PUNCT
ap-961	842	4	of	of	ADP
ap-961	842	5	course	course	NOUN
ap-961	842	6	,	,	PUNCT
ap-961	842	7	the	the	DET
ap-961	842	8	books	book	NOUN
ap-961	842	9	by	by	ADP
ap-961	842	10	alain	alain	PROPN
ap-961	842	11	connes	conne	NOUN
ap-961	842	12	:	:	PUNCT
ap-961	842	13	the	the	DET
ap-961	842	14	seminal	seminal	ADJ
ap-961	842	15	work	work	NOUN
ap-961	842	16	“	"	PUNCT
ap-961	842	17	noncommutative	noncommutative	ADJ
ap-961	842	18	geometry	geometry	NOUN
ap-961	842	19	”	"	PUNCT
ap-961	843	1	[	[	X
ap-961	843	2	5	5	NUM
ap-961	843	3	]	]	PUNCT
ap-961	843	4	and	and	CCONJ
ap-961	843	5	the	the	DET
ap-961	843	6	more	more	ADV
ap-961	843	7	recent	recent	ADJ
ap-961	843	8	book	book	NOUN
ap-961	843	9	with	with	ADP
ap-961	843	10	mathilde	mathilde	PROPN
ap-961	843	11	marcolli	marcolli	PROPN
ap-961	843	12	[	[	X
ap-961	843	13	6	6	NUM
ap-961	843	14	]	]	PUNCT
ap-961	843	15	.	.	PUNCT
ap-961	844	1	other	other	ADJ
ap-961	844	2	textbooks	textbook	NOUN
ap-961	844	3	which	which	PRON
ap-961	844	4	give	give	VERB
ap-961	844	5	a	a	DET
ap-961	844	6	review	review	NOUN
ap-961	844	7	of	of	ADP
ap-961	844	8	selected	select	VERB
ap-961	844	9	topics	topic	NOUN
ap-961	844	10	(	(	PUNCT
ap-961	844	11	and	and	CCONJ
ap-961	844	12	are	be	AUX
ap-961	844	13	written	write	VERB
ap-961	844	14	more	more	ADJ
ap-961	844	15	from	from	ADP
ap-961	844	16	the	the	DET
ap-961	844	17	perspective	perspective	NOUN
ap-961	844	18	of	of	ADP
ap-961	844	19	mathematical	mathematical	ADJ
ap-961	844	20	physicist	physicist	NOUN
ap-961	844	21	)	)	PUNCT
ap-961	844	22	are	be	AUX
ap-961	844	23	“	"	PUNCT
ap-961	844	24	an	an	DET
ap-961	844	25	introduction	introduction	NOUN
ap-961	844	26	to	to	ADP
ap-961	844	27	noncommutative	noncommutative	ADJ
ap-961	844	28	differential	differential	ADJ
ap-961	844	29	geometry	geometry	NOUN
ap-961	844	30	and	and	CCONJ
ap-961	844	31	its	its	PRON
ap-961	844	32	physical	physical	ADJ
ap-961	844	33	applications	application	NOUN
ap-961	844	34	”	"	PUNCT
ap-961	844	35	,	,	PUNCT
ap-961	844	36	[	[	X
ap-961	844	37	17	17	NUM
ap-961	844	38	]	]	PUNCT
ap-961	844	39	,	,	PUNCT
ap-961	844	40	by	by	ADP
ap-961	844	41	john	john	PROPN
ap-961	844	42	madore	madore	PROPN
ap-961	844	43	and	and	CCONJ
ap-961	844	44	“	"	PUNCT
ap-961	844	45	an	an	DET
ap-961	844	46	introduction	introduction	NOUN
ap-961	844	47	to	to	ADP
ap-961	844	48	noncommutative	noncommutative	ADJ
ap-961	844	49	spaces	space	NOUN
ap-961	844	50	and	and	CCONJ
ap-961	844	51	their	their	PRON
ap-961	844	52	geometry	geometry	NOUN
ap-961	844	53	”	"	PUNCT
ap-961	844	54	,	,	PUNCT
ap-961	844	55	[	[	X
ap-961	844	56	14	14	NUM
ap-961	844	57	]	]	PUNCT
ap-961	844	58	,	,	PUNCT
ap-961	844	59	by	by	ADP
ap-961	844	60	giovanni	giovanni	PROPN
ap-961	844	61	landi	landi	PROPN
ap-961	844	62	.	.	PUNCT
ap-961	845	1	©	©	PROPN
ap-961	845	2	czech	czech	PROPN
ap-961	845	3	technical	technical	PROPN
ap-961	845	4	university	university	PROPN
ap-961	845	5	publishing	publishing	NOUN
ap-961	845	6	house	house	NOUN
ap-961	845	7	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	845	8	55	55	NUM
ap-961	845	9	acta	acta	PROPN
ap-961	845	10	polytechnica	polytechnica	PROPN
ap-961	845	11	vol	vol	NOUN
ap-961	845	12	.	.	PUNCT
ap-961	846	1	48	48	NUM
ap-961	846	2	no	no	NOUN
ap-961	846	3	.	.	PUNCT
ap-961	847	1	2/2008	2/2008	NUM
ap-961	847	2	where	where	SCONJ
ap-961	847	3	a	a	DET
ap-961	847	4	x	x	X
ap-961	847	5	�	�	PROPN
ap-961	847	6	(	(	PUNCT
ap-961	847	7	)	)	PUNCT
ap-961	847	8	$	$	SYM
ap-961	847	9	are	be	AUX
ap-961	847	10	two	two	NUM
ap-961	847	11	copies	copy	NOUN
ap-961	847	12	of	of	ADP
ap-961	847	13	the	the	DET
ap-961	847	14	u	u	NOUN
ap-961	847	15	(	(	PUNCT
ap-961	847	16	)	)	PUNCT
ap-961	847	17	1	1	NUM
ap-961	847	18	gauge	gauge	NOUN
ap-961	847	19	potential	potential	NOUN
ap-961	847	20	and	and	CCONJ
ap-961	847	21	h	h	NOUN
ap-961	847	22	x	x	NOUN
ap-961	847	23	(	(	PUNCT
ap-961	847	24	)	)	PUNCT
ap-961	847	25	is	be	AUX
ap-961	847	26	a	a	DET
ap-961	847	27	complex	complex	ADV
ap-961	847	28	-	-	PUNCT
ap-961	847	29	valued	value	VERB
ap-961	847	30	field	field	NOUN
ap-961	847	31	!	!	PUNCT
ap-961	848	1	we	we	PRON
ap-961	848	2	calculate	calculate	VERB
ap-961	848	3	the	the	DET
ap-961	848	4	square	square	NOUN
ap-961	848	5	of	of	ADP
ap-961	848	6	the	the	DET
ap-961	848	7	dirac	dirac	NOUN
ap-961	848	8	operator	operator	NOUN
ap-961	848	9	paying	pay	VERB
ap-961	848	10	particular	particular	ADJ
ap-961	848	11	attention	attention	NOUN
ap-961	848	12	to	to	ADP
ap-961	848	13	the	the	DET
ap-961	848	14	terms	term	NOUN
ap-961	848	15	that	that	PRON
ap-961	848	16	involve	involve	VERB
ap-961	848	17	h.	h.	NOUN
ap-961	848	18	the	the	DET
ap-961	848	19	rest	rest	NOUN
ap-961	848	20	,	,	PUNCT
ap-961	848	21	depending	depend	VERB
ap-961	848	22	only	only	ADV
ap-961	848	23	on	on	ADP
ap-961	848	24	a	a	DET
ap-961	848	25	x	x	X
ap-961	848	26	�	�	PROPN
ap-961	848	27	(	(	PUNCT
ap-961	848	28	,	,	PUNCT
ap-961	848	29	)	)	PUNCT
ap-961	848	30	$	$	NUM
ap-961	848	31	,	,	PUNCT
ap-961	848	32	does	do	AUX
ap-961	848	33	not	not	PART
ap-961	848	34	differ	differ	VERB
ap-961	848	35	from	from	ADP
ap-961	848	36	the	the	DET
ap-961	848	37	usual	usual	ADJ
ap-961	848	38	classical	classical	ADJ
ap-961	848	39	theory	theory	NOUN
ap-961	848	40	.	.	PUNCT
ap-961	849	1	the	the	DET
ap-961	849	2	terms	term	NOUN
ap-961	849	3	that	that	PRON
ap-961	849	4	depend	depend	VERB
ap-961	849	5	on	on	ADP
ap-961	849	6	h	h	NOUN
ap-961	849	7	in	in	ADP
ap-961	849	8	the	the	DET
ap-961	849	9	square	square	NOUN
ap-961	849	10	of	of	ADP
ap-961	849	11	the	the	DET
ap-961	849	12	dirac	dirac	NOUN
ap-961	849	13	operator	operator	NOUN
ap-961	849	14	d	d	PROPN
ap-961	849	15	a	a	PRON
ap-961	849	16	are	be	AUX
ap-961	849	17	:	:	PUNCT
ap-961	849	18	h	h	NOUN
ap-961	849	19	x	x	SYM
ap-961	849	20	h	h	NOUN
ap-961	849	21	x	x	PUNCT
ap-961	849	22	a	a	PRON
ap-961	849	23	x	x	SYM
ap-961	849	24	a	a	DET
ap-961	849	25	x	x	SYM
ap-961	849	26	h	h	NOUN
ap-961	849	27	x	x	NOUN
ap-961	849	28	a	a	X
ap-961	849	29	x	x	X
ap-961	849	30	(	(	PUNCT
ap-961	849	31	)	)	PUNCT
ap-961	849	32	(	(	PUNCT
ap-961	849	33	)	)	PUNCT
ap-961	849	34	(	(	PUNCT
ap-961	849	35	(	(	PUNCT
ap-961	849	36	)	)	PUNCT
ap-961	849	37	(	(	PUNCT
ap-961	849	38	)	)	PUNCT
ap-961	849	39	)	)	PUNCT
ap-961	850	1	(	(	PUNCT
ap-961	850	2	)	)	PUNCT
ap-961	850	3	(	(	PUNCT
ap-961	850	4	(	(	PUNCT
ap-961	850	5	)	)	PUNCT
ap-961	850	6	*	*	PUNCT
ap-961	850	7	�	�	PROPN
ap-961	850	8	�	�	PROPN
ap-961	850	9	�	�	PROPN
ap-961	850	10	�	�	PROPN
ap-961	850	11	�	�	PROPN
ap-961	850	12	�	�	PROPN
ap-961	850	13	�	�	PROPN
ap-961	850	14	�	�	PROPN
ap-961	850	15	�	�	PROPN
ap-961	850	16	�	�	PROPN
ap-961	850	17	�	�	PROPN
ap-961	850	18	5	5	NUM
ap-961	850	19	5	5	NUM
ap-961	850	20	1	1	NUM
ap-961	850	21	�	�	PROPN
ap-961	850	22	�	�	PROPN
ap-961	850	23	�	�	PROPN
ap-961	850	24	�	�	PROPN
ap-961	850	25	�	�	PROPN
ap-961	850	26	�	�	PROPN
ap-961	850	27	�	�	PROPN
ap-961	850	28	�	�	PROPN
ap-961	850	29	�	�	PROPN
ap-961	850	30	�	�	PROPN
ap-961	850	31	�	�	PROPN
ap-961	850	32	�	�	PROPN
ap-961	850	33	�	�	PROPN
ap-961	850	34	a	a	DET
ap-961	850	35	x	x	SYM
ap-961	850	36	h	h	NOUN
ap-961	850	37	x	x	NOUN
ap-961	850	38	h	h	NOUN
ap-961	851	1	x	x	SYM
ap-961	851	2	h	h	NOUN
ap-961	851	3	x	x	X
ap-961	851	4	�	�	PROPN
ap-961	851	5	(	(	PUNCT
ap-961	851	6	)	)	PUNCT
ap-961	851	7	)	)	PUNCT
ap-961	852	1	(	(	PUNCT
ap-961	852	2	)	)	PUNCT
ap-961	852	3	(	(	PUNCT
ap-961	852	4	)	)	PUNCT
ap-961	852	5	(	(	PUNCT
ap-961	852	6	)	)	PUNCT
ap-961	853	1	*	*	PUNCT
ap-961	853	2	*	*	PUNCT
ap-961	853	3	there	there	PRON
ap-961	853	4	are	be	VERB
ap-961	853	5	,	,	PUNCT
ap-961	853	6	of	of	ADP
ap-961	853	7	course	course	NOUN
ap-961	853	8	,	,	PUNCT
ap-961	853	9	numerous	numerous	ADJ
ap-961	853	10	books	book	NOUN
ap-961	853	11	on	on	ADP
ap-961	853	12	some	some	DET
ap-961	853	13	mathematical	mathematical	ADJ
ap-961	853	14	aspects	aspect	NOUN
ap-961	853	15	of	of	ADP
ap-961	853	16	noncommutative	noncommutative	ADJ
ap-961	853	17	geometry	geometry	NOUN
ap-961	853	18	,	,	PUNCT
ap-961	853	19	e.g.	e.g.	ADV
ap-961	853	20	k	k	X
ap-961	853	21	-	-	NOUN
ap-961	853	22	theory	theory	NOUN
ap-961	853	23	,	,	PUNCT
ap-961	853	24	for	for	ADP
ap-961	853	25	instance	instance	NOUN
ap-961	853	26	.	.	PUNCT
ap-961	854	1	we	we	PRON
ap-961	854	2	shall	shall	AUX
ap-961	854	3	not	not	PART
ap-961	854	4	list	list	VERB
ap-961	854	5	here	here	ADV
ap-961	854	6	all	all	DET
ap-961	854	7	possible	possible	ADJ
ap-961	854	8	available	available	ADJ
ap-961	854	9	books	book	NOUN
ap-961	854	10	and	and	CCONJ
ap-961	854	11	monographs	monograph	NOUN
ap-961	854	12	but	but	CCONJ
ap-961	854	13	will	will	AUX
ap-961	854	14	give	give	VERB
ap-961	854	15	only	only	ADV
ap-961	854	16	single	single	ADJ
ap-961	854	17	examples	example	NOUN
ap-961	854	18	.	.	PUNCT
ap-961	855	1	first	first	ADV
ap-961	855	2	of	of	ADP
ap-961	855	3	all	all	PRON
ap-961	855	4	,	,	PUNCT
ap-961	855	5	there	there	PRON
ap-961	855	6	is	be	VERB
ap-961	855	7	an	an	DET
ap-961	855	8	excellent	excellent	ADJ
ap-961	855	9	monograph	monograph	NOUN
ap-961	855	10	of	of	ADP
ap-961	855	11	jean	jean	PROPN
ap-961	855	12	-	-	PUNCT
ap-961	855	13	louis	louis	PROPN
ap-961	855	14	loday	loday	PROPN
ap-961	855	15	[	[	X
ap-961	855	16	16	16	NUM
ap-961	855	17	]	]	PUNCT
ap-961	855	18	on	on	ADP
ap-961	855	19	cyclic	cyclic	ADJ
ap-961	855	20	homology	homology	NOUN
ap-961	855	21	,	,	PUNCT
ap-961	855	22	hochschild	hochschild	ADJ
ap-961	855	23	and	and	CCONJ
ap-961	855	24	related	related	ADJ
ap-961	855	25	subjects	subject	NOUN
ap-961	855	26	.	.	PUNCT
ap-961	856	1	a	a	DET
ap-961	856	2	concise	concise	ADJ
ap-961	856	3	and	and	CCONJ
ap-961	856	4	useful	useful	ADJ
ap-961	856	5	review	review	NOUN
ap-961	856	6	of	of	ADP
ap-961	856	7	the	the	DET
ap-961	856	8	topis	topis	PROPN
ap-961	856	9	is	be	AUX
ap-961	856	10	given	give	VERB
ap-961	856	11	by	by	ADP
ap-961	856	12	husemoller	husemoller	NOUN
ap-961	856	13	[	[	X
ap-961	856	14	12	12	NUM
ap-961	856	15	]	]	PUNCT
ap-961	856	16	.	.	PUNCT
ap-961	857	1	cyclic	cyclic	ADJ
ap-961	857	2	homology	homology	NOUN
ap-961	857	3	within	within	ADP
ap-961	857	4	noncommutative	noncommutative	ADJ
ap-961	857	5	geometry	geometry	NOUN
ap-961	857	6	is	be	AUX
ap-961	857	7	presented	present	VERB
ap-961	857	8	in	in	ADP
ap-961	857	9	[	[	X
ap-961	857	10	8	8	NUM
ap-961	857	11	]	]	PUNCT
ap-961	857	12	.	.	PUNCT
ap-961	858	1	a	a	DET
ap-961	858	2	very	very	ADV
ap-961	858	3	good	good	ADJ
ap-961	858	4	and	and	CCONJ
ap-961	858	5	comprehensible	comprehensible	ADJ
ap-961	858	6	introduction	introduction	NOUN
ap-961	858	7	to	to	ADP
ap-961	858	8	k	k	NOUN
ap-961	858	9	-	-	NOUN
ap-961	858	10	theory	theory	NOUN
ap-961	858	11	can	can	AUX
ap-961	858	12	be	be	AUX
ap-961	858	13	found	find	VERB
ap-961	858	14	in	in	ADP
ap-961	858	15	a	a	DET
ap-961	858	16	friendly	friendly	ADJ
ap-961	858	17	approach	approach	NOUN
ap-961	858	18	to	to	ADP
ap-961	858	19	k	k	NOUN
ap-961	858	20	-	-	NOUN
ap-961	858	21	theory	theory	NOUN
ap-961	858	22	by	by	ADP
ap-961	858	23	wegge	wegge	NOUN
ap-961	858	24	-	-	PUNCT
ap-961	858	25	olsen	olsen	NOUN
ap-961	858	26	[	[	X
ap-961	858	27	19	19	NUM
ap-961	858	28	]	]	PUNCT
ap-961	858	29	and	and	CCONJ
ap-961	858	30	also	also	ADV
ap-961	858	31	in	in	ADP
ap-961	858	32	the	the	DET
ap-961	858	33	book	book	NOUN
ap-961	858	34	by	by	ADP
ap-961	858	35	blackaddar	blackaddar	NOUN
ap-961	859	1	[	[	X
ap-961	859	2	1	1	NUM
ap-961	859	3	]	]	PUNCT
ap-961	859	4	.	.	PUNCT
ap-961	860	1	the	the	DET
ap-961	860	2	overview	overview	NOUN
ap-961	860	3	of	of	ADP
ap-961	860	4	the	the	DET
ap-961	860	5	link	link	NOUN
ap-961	860	6	between	between	ADP
ap-961	860	7	cyclic	cyclic	ADJ
ap-961	860	8	cohomology	cohomology	NOUN
ap-961	860	9	,	,	PUNCT
ap-961	860	10	k	k	NOUN
ap-961	860	11	-	-	NOUN
ap-961	860	12	theory	theory	NOUN
ap-961	860	13	and	and	CCONJ
ap-961	860	14	chern	chern	NOUN
ap-961	860	15	parings	paring	NOUN
ap-961	860	16	is	be	AUX
ap-961	860	17	nicely	nicely	ADV
ap-961	860	18	explained	explain	VERB
ap-961	860	19	in	in	ADP
ap-961	860	20	the	the	DET
ap-961	860	21	book	book	NOUN
ap-961	860	22	of	of	ADP
ap-961	860	23	jacek	jacek	PROPN
ap-961	860	24	brodzki	brodzki	PROPN
ap-961	861	1	[	[	X
ap-961	861	2	3	3	NUM
ap-961	861	3	]	]	PUNCT
ap-961	861	4	.	.	PUNCT
ap-961	862	1	almost	almost	ADV
ap-961	862	2	everything	everything	PRON
ap-961	862	3	on	on	ADP
ap-961	862	4	operator	operator	NOUN
ap-961	862	5	algebras	algebra	NOUN
ap-961	862	6	can	can	AUX
ap-961	862	7	be	be	AUX
ap-961	862	8	found	find	VERB
ap-961	862	9	in	in	ADP
ap-961	862	10	the	the	DET
ap-961	862	11	excellent	excellent	ADJ
ap-961	862	12	monograph	monograph	NOUN
ap-961	862	13	by	by	ADP
ap-961	862	14	richard	richard	PROPN
ap-961	862	15	kadison	kadison	PROPN
ap-961	862	16	,	,	PUNCT
ap-961	862	17	john	john	PROPN
ap-961	862	18	ringrose	ringrose	VERB
ap-961	862	19	[	[	PUNCT
ap-961	862	20	13	13	NUM
ap-961	862	21	]	]	PUNCT
ap-961	862	22	.	.	PUNCT
ap-961	863	1	a	a	DET
ap-961	863	2	review	review	NOUN
ap-961	863	3	of	of	ADP
ap-961	863	4	differential	differential	NOUN
ap-961	863	5	graded	grade	VERB
ap-961	863	6	algebras	algebra	NOUN
ap-961	863	7	can	can	AUX
ap-961	863	8	be	be	AUX
ap-961	863	9	found	find	VERB
ap-961	863	10	in	in	ADP
ap-961	863	11	the	the	DET
ap-961	863	12	lecture	lecture	NOUN
ap-961	863	13	notes	note	NOUN
ap-961	863	14	by	by	ADP
ap-961	863	15	michel	michel	PROPN
ap-961	863	16	dubois	dubois	PROPN
ap-961	863	17	-	-	PUNCT
ap-961	863	18	violette	violette	PROPN
ap-961	863	19	[	[	X
ap-961	863	20	35	35	NUM
ap-961	863	21	,	,	PUNCT
ap-961	863	22	36	36	NUM
ap-961	863	23	]	]	PUNCT
ap-961	863	24	.	.	PUNCT
ap-961	864	1	one	one	NUM
ap-961	864	2	of	of	ADP
ap-961	864	3	the	the	DET
ap-961	864	4	basic	basic	ADJ
ap-961	864	5	classical	classical	ADJ
ap-961	864	6	texts	text	NOUN
ap-961	864	7	on	on	ADP
ap-961	864	8	all	all	DET
ap-961	864	9	aspects	aspect	NOUN
ap-961	864	10	of	of	ADP
ap-961	864	11	topology	topology	NOUN
ap-961	864	12	,	,	PUNCT
ap-961	864	13	differential	differential	NOUN
ap-961	864	14	geometry	geometry	NOUN
ap-961	864	15	,	,	PUNCT
ap-961	864	16	gauge	gauge	NOUN
ap-961	864	17	theories	theory	NOUN
ap-961	864	18	and	and	CCONJ
ap-961	864	19	characteristic	characteristic	ADJ
ap-961	864	20	classes	class	NOUN
ap-961	864	21	is	be	AUX
ap-961	864	22	the	the	DET
ap-961	864	23	textbook	textbook	NOUN
ap-961	864	24	“	"	PUNCT
ap-961	864	25	analysis	analysis	NOUN
ap-961	864	26	,	,	PUNCT
ap-961	864	27	manifolds	manifold	NOUN
ap-961	864	28	and	and	CCONJ
ap-961	864	29	physics	physic	NOUN
ap-961	864	30	”	"	PUNCT
ap-961	864	31	,	,	PUNCT
ap-961	864	32	[	[	X
ap-961	864	33	4	4	NUM
ap-961	864	34	]	]	PUNCT
ap-961	864	35	,	,	PUNCT
ap-961	864	36	by	by	ADP
ap-961	864	37	choquet	choquet	NOUN
ap-961	864	38	-	-	PUNCT
ap-961	864	39	bruhat	bruhat	NOUN
ap-961	864	40	and	and	CCONJ
ap-961	864	41	dewitt	dewitt	PROPN
ap-961	864	42	-	-	PUNCT
ap-961	864	43	morette	morette	NOUN
ap-961	864	44	.	.	PUNCT
ap-961	865	1	everything	everything	PRON
ap-961	865	2	one	one	PRON
ap-961	865	3	wants	want	VERB
ap-961	865	4	to	to	PART
ap-961	865	5	know	know	VERB
ap-961	865	6	about	about	ADP
ap-961	865	7	spin	spin	NOUN
ap-961	865	8	geometry	geometry	NOUN
ap-961	865	9	is	be	AUX
ap-961	865	10	in	in	ADP
ap-961	865	11	the	the	DET
ap-961	865	12	book	book	NOUN
ap-961	865	13	(	(	PUNCT
ap-961	865	14	surprise	surprise	NOUN
ap-961	865	15	,	,	PUNCT
ap-961	865	16	surprise	surprise	NOUN
ap-961	865	17	):	):	PUNCT
ap-961	865	18	“	"	PUNCT
ap-961	865	19	spin	spin	NOUN
ap-961	865	20	geometry	geometry	NOUN
ap-961	865	21	”	"	PUNCT
ap-961	865	22	,	,	PUNCT
ap-961	865	23	[	[	X
ap-961	865	24	15	15	NUM
ap-961	865	25	]	]	X
ap-961	865	26	,	,	PUNCT
ap-961	865	27	by	by	ADP
ap-961	865	28	lawson	lawson	PROPN
ap-961	865	29	and	and	CCONJ
ap-961	865	30	michelsohn	michelsohn	PROPN
ap-961	865	31	.	.	PUNCT
ap-961	866	1	all	all	DET
ap-961	866	2	properties	property	NOUN
ap-961	866	3	of	of	ADP
ap-961	866	4	dirac	dirac	NOUN
ap-961	866	5	operators	operator	NOUN
ap-961	866	6	are	be	AUX
ap-961	866	7	explained	explain	VERB
ap-961	866	8	in	in	ADP
ap-961	866	9	the	the	DET
ap-961	866	10	work	work	NOUN
ap-961	866	11	of	of	ADP
ap-961	866	12	thomas	thomas	PROPN
ap-961	866	13	friedrich	friedrich	PROPN
ap-961	866	14	“	"	PUNCT
ap-961	866	15	dirac	dirac	NOUN
ap-961	866	16	operator	operator	NOUN
ap-961	866	17	”	"	PUNCT
ap-961	867	1	[	[	X
ap-961	867	2	10	10	NUM
ap-961	867	3	]	]	PUNCT
ap-961	867	4	.	.	PUNCT
ap-961	868	1	the	the	DET
ap-961	868	2	most	most	ADV
ap-961	868	3	recent	recent	ADJ
ap-961	868	4	concise	concise	ADJ
ap-961	868	5	introduction	introduction	NOUN
ap-961	868	6	to	to	ADP
ap-961	868	7	spectral	spectral	ADJ
ap-961	868	8	triples	triple	NOUN
ap-961	868	9	(	(	PUNCT
ap-961	868	10	treating	treat	VERB
ap-961	868	11	the	the	DET
ap-961	868	12	classical	classical	ADJ
ap-961	868	13	and	and	CCONJ
ap-961	868	14	noncommutative	noncommutative	ADJ
ap-961	868	15	examples	example	NOUN
ap-961	868	16	)	)	PUNCT
ap-961	868	17	is	be	AUX
ap-961	868	18	contained	contain	VERB
ap-961	868	19	in	in	ADP
ap-961	868	20	the	the	DET
ap-961	868	21	book	book	NOUN
ap-961	868	22	by	by	ADP
ap-961	868	23	joseph	joseph	PROPN
ap-961	868	24	varilly	varilly	ADV
ap-961	868	25	,	,	PUNCT
ap-961	868	26	[	[	X
ap-961	868	27	18	18	NUM
ap-961	868	28	]	]	PUNCT
ap-961	868	29	,	,	PUNCT
ap-961	868	30	“	"	PUNCT
ap-961	868	31	an	an	DET
ap-961	868	32	introduction	introduction	NOUN
ap-961	868	33	to	to	ADP
ap-961	868	34	noncommutative	noncommutative	ADJ
ap-961	868	35	geometry	geometry	NOUN
ap-961	868	36	”	"	PUNCT
ap-961	868	37	.	.	PUNCT
ap-961	869	1	there	there	PRON
ap-961	869	2	are	be	VERB
ap-961	869	3	,	,	PUNCT
ap-961	869	4	of	of	ADP
ap-961	869	5	course	course	NOUN
ap-961	869	6	,	,	PUNCT
ap-961	869	7	numerous	numerous	ADJ
ap-961	869	8	reviews	review	NOUN
ap-961	869	9	and	and	CCONJ
ap-961	869	10	lecture	lecture	NOUN
ap-961	869	11	notes	note	NOUN
ap-961	869	12	from	from	ADP
ap-961	869	13	courses	course	NOUN
ap-961	869	14	at	at	ADP
ap-961	869	15	institutions	institution	NOUN
ap-961	869	16	and	and	CCONJ
ap-961	869	17	schools	school	NOUN
ap-961	869	18	(	(	PUNCT
ap-961	869	19	for	for	ADP
ap-961	869	20	example	example	NOUN
ap-961	869	21	:	:	PUNCT
ap-961	870	1	[	[	X
ap-961	870	2	7	7	NUM
ap-961	870	3	,	,	PUNCT
ap-961	870	4	9	9	NUM
ap-961	870	5	,	,	PUNCT
ap-961	870	6	11	11	NUM
ap-961	870	7	]	]	NUM
ap-961	870	8	)	)	PUNCT
ap-961	870	9	.	.	PUNCT
ap-961	871	1	written	write	VERB
ap-961	871	2	for	for	ADP
ap-961	871	3	different	different	ADJ
ap-961	871	4	purposes	purpose	NOUN
ap-961	871	5	and	and	CCONJ
ap-961	871	6	by	by	ADP
ap-961	871	7	different	different	ADJ
ap-961	871	8	authors	author	NOUN
ap-961	871	9	,	,	PUNCT
ap-961	871	10	they	they	PRON
ap-961	871	11	offer	offer	VERB
ap-961	871	12	views	view	NOUN
ap-961	871	13	of	of	ADP
ap-961	871	14	the	the	DET
ap-961	871	15	topic	topic	NOUN
ap-961	871	16	from	from	ADP
ap-961	871	17	many	many	ADJ
ap-961	871	18	differen	differen	NOUN
ap-961	871	19	angles	angle	NOUN
ap-961	871	20	.	.	PUNCT
ap-961	872	1	a	a	DET
ap-961	872	2	selection	selection	NOUN
ap-961	872	3	can	can	AUX
ap-961	872	4	also	also	ADV
ap-961	872	5	be	be	AUX
ap-961	872	6	found	find	VERB
ap-961	872	7	found	find	VERB
ap-961	872	8	on	on	ADP
ap-961	872	9	the	the	DET
ap-961	872	10	internet	internet	NOUN
ap-961	872	11	,	,	PUNCT
ap-961	872	12	on	on	ADP
ap-961	872	13	the	the	DET
ap-961	872	14	web	web	NOUN
ap-961	872	15	pages	page	NOUN
ap-961	872	16	of	of	ADP
ap-961	872	17	noncommutative	noncommutative	ADJ
ap-961	872	18	geometers	geometer	NOUN
ap-961	872	19	or	or	CCONJ
ap-961	872	20	common	common	ADJ
ap-961	872	21	sites	site	NOUN
ap-961	872	22	like	like	ADP
ap-961	872	23	the	the	DET
ap-961	872	24	„	„	PUNCT
ap-961	872	25	noncommutative	noncommutative	ADJ
ap-961	872	26	blog	blog	NOUN
ap-961	872	27	“	"	PUNCT
ap-961	872	28	.	.	PUNCT
ap-961	873	1	6.6.2	6.6.2	NUM
ap-961	873	2	the	the	DET
ap-961	873	3	outlook	outlook	NOUN
ap-961	873	4	it	it	PRON
ap-961	873	5	is	be	AUX
ap-961	873	6	hard	hard	ADJ
ap-961	873	7	to	to	PART
ap-961	873	8	see	see	VERB
ap-961	873	9	at	at	ADP
ap-961	873	10	the	the	DET
ap-961	873	11	moment	moment	NOUN
ap-961	873	12	whether	whether	SCONJ
ap-961	873	13	noncommutative	noncommutative	ADJ
ap-961	873	14	geometry	geometry	NOUN
ap-961	873	15	will	will	AUX
ap-961	873	16	become	become	VERB
ap-961	873	17	the	the	DET
ap-961	873	18	right	right	ADJ
ap-961	873	19	tool	tool	NOUN
ap-961	873	20	for	for	ADP
ap-961	873	21	describing	describe	VERB
ap-961	873	22	the	the	DET
ap-961	873	23	physics	physics	NOUN
ap-961	873	24	:	:	PUNCT
ap-961	873	25	both	both	DET
ap-961	873	26	known	known	ADJ
ap-961	873	27	physics	physics	NOUN
ap-961	873	28	and	and	CCONJ
ap-961	873	29	physics	physics	NOUN
ap-961	873	30	yet	yet	ADV
ap-961	873	31	to	to	PART
ap-961	873	32	be	be	AUX
ap-961	873	33	discovered	discover	VERB
ap-961	873	34	.	.	PUNCT
ap-961	874	1	we	we	PRON
ap-961	874	2	have	have	AUX
ap-961	874	3	mentioned	mention	VERB
ap-961	874	4	that	that	SCONJ
ap-961	874	5	ncg	ncg	PROPN
ap-961	874	6	finds	find	VERB
ap-961	874	7	applications	application	NOUN
ap-961	874	8	in	in	ADP
ap-961	874	9	a	a	DET
ap-961	874	10	range	range	NOUN
ap-961	874	11	of	of	ADP
ap-961	874	12	topics	topic	NOUN
ap-961	874	13	,	,	PUNCT
ap-961	874	14	from	from	ADP
ap-961	874	15	the	the	DET
ap-961	874	16	quantum	quantum	PROPN
ap-961	874	17	hall	hall	NOUN
ap-961	874	18	effect	effect	NOUN
ap-961	875	1	[	[	X
ap-961	875	2	20	20	NUM
ap-961	875	3	]	]	PUNCT
ap-961	875	4	,	,	PUNCT
ap-961	875	5	standard	standard	ADJ
ap-961	875	6	model	model	NOUN
ap-961	875	7	[	[	X
ap-961	875	8	43	43	NUM
ap-961	875	9	,	,	PUNCT
ap-961	875	10	40	40	NUM
ap-961	875	11	]	]	PUNCT
ap-961	875	12	up	up	ADP
ap-961	875	13	to	to	ADP
ap-961	875	14	string	string	NOUN
ap-961	875	15	theory	theory	NOUN
ap-961	875	16	[	[	X
ap-961	875	17	42	42	NUM
ap-961	875	18	,	,	PUNCT
ap-961	875	19	28	28	NUM
ap-961	875	20	]	]	PUNCT
ap-961	875	21	.	.	PUNCT
ap-961	876	1	there	there	PRON
ap-961	876	2	are	be	VERB
ap-961	876	3	other	other	ADJ
ap-961	876	4	branches	branch	NOUN
ap-961	876	5	of	of	ADP
ap-961	876	6	noncommutative	noncommutative	ADJ
ap-961	876	7	geometry	geometry	NOUN
ap-961	876	8	that	that	PRON
ap-961	876	9	we	we	PRON
ap-961	876	10	have	have	AUX
ap-961	876	11	not	not	PART
ap-961	876	12	even	even	ADV
ap-961	876	13	touched	touch	VERB
ap-961	876	14	:	:	PUNCT
ap-961	876	15	hopf	hopf	ADJ
ap-961	876	16	algebras	algebra	NOUN
ap-961	876	17	,	,	PUNCT
ap-961	876	18	quantum	quantum	ADJ
ap-961	876	19	groups	group	NOUN
ap-961	876	20	and	and	CCONJ
ap-961	876	21	quantum	quantum	NOUN
ap-961	876	22	deformations	deformation	NOUN
ap-961	876	23	,	,	PUNCT
ap-961	876	24	deformation	deformation	NOUN
ap-961	876	25	quantization	quantization	NOUN
ap-961	876	26	,	,	PUNCT
ap-961	876	27	noncommutative	noncommutative	ADJ
ap-961	876	28	field	field	NOUN
ap-961	876	29	theory	theory	NOUN
ap-961	876	30	,	,	PUNCT
ap-961	876	31	hopf	hopf	ADJ
ap-961	876	32	algebras	algebra	NOUN
ap-961	876	33	in	in	ADP
ap-961	876	34	renormalization	renormalization	NOUN
ap-961	876	35	–	–	PUNCT
ap-961	876	36	to	to	PART
ap-961	876	37	list	list	VERB
ap-961	876	38	only	only	ADV
ap-961	876	39	those	those	PRON
ap-961	876	40	,	,	PUNCT
ap-961	876	41	which	which	PRON
ap-961	876	42	(	(	PUNCT
ap-961	876	43	more	more	ADJ
ap-961	876	44	or	or	CCONJ
ap-961	876	45	less	less	ADJ
ap-961	876	46	)	)	PUNCT
ap-961	876	47	have	have	VERB
ap-961	876	48	some	some	DET
ap-961	876	49	links	link	NOUN
ap-961	876	50	to	to	ADP
ap-961	876	51	physics	physics	NOUN
ap-961	876	52	.	.	PUNCT
ap-961	877	1	one	one	PRON
ap-961	877	2	may	may	AUX
ap-961	877	3	say	say	VERB
ap-961	877	4	that	that	SCONJ
ap-961	877	5	we	we	PRON
ap-961	877	6	are	be	AUX
ap-961	877	7	just	just	ADV
ap-961	877	8	at	at	ADP
ap-961	877	9	the	the	DET
ap-961	877	10	dawn	dawn	NOUN
ap-961	877	11	of	of	ADP
ap-961	877	12	noncommutative	noncommutative	ADJ
ap-961	877	13	geometry	geometry	NOUN
ap-961	877	14	:	:	PUNCT
ap-961	877	15	it	it	PRON
ap-961	877	16	is	be	AUX
ap-961	877	17	a	a	DET
ap-961	877	18	world	world	NOUN
ap-961	877	19	still	still	ADV
ap-961	877	20	to	to	PART
ap-961	877	21	be	be	AUX
ap-961	877	22	discovered	discover	VERB
ap-961	877	23	.	.	PUNCT
ap-961	878	1	whether	whether	SCONJ
ap-961	878	2	this	this	DET
ap-961	878	3	geometry	geometry	NOUN
ap-961	878	4	will	will	AUX
ap-961	878	5	be	be	AUX
ap-961	878	6	the	the	DET
ap-961	878	7	geometry	geometry	NOUN
ap-961	878	8	used	use	VERB
ap-961	878	9	to	to	PART
ap-961	878	10	describe	describe	VERB
ap-961	878	11	the	the	DET
ap-961	878	12	world	world	NOUN
ap-961	878	13	is	be	AUX
ap-961	878	14	not	not	PART
ap-961	878	15	known	know	VERB
ap-961	878	16	.	.	PUNCT
ap-961	879	1	but	but	CCONJ
ap-961	879	2	we	we	PRON
ap-961	879	3	might	might	AUX
ap-961	879	4	soon	soon	ADV
ap-961	879	5	find	find	VERB
ap-961	879	6	out	out	ADP
ap-961	879	7	.	.	PUNCT
ap-961	880	1	acknowledgement	acknowledgement	NOUN
ap-961	880	2	partially	partially	ADV
ap-961	880	3	supported	support	VERB
ap-961	880	4	by	by	ADP
ap-961	880	5	polish	polish	ADJ
ap-961	880	6	government	government	NOUN
ap-961	880	7	grants	grant	VERB
ap-961	880	8	115	115	NUM
ap-961	880	9	/	/	SYM
ap-961	880	10	e-343	e-343	NOUN
ap-961	880	11	/	/	SYM
ap-961	880	12	spb/6.prue	spb/6.prue	NOUN
ap-961	880	13	/	/	SYM
ap-961	880	14	die	die	NOUN
ap-961	880	15	50/2005–2008	50/2005–2008	PROPN
ap-961	880	16	and	and	CCONJ
ap-961	880	17	189/6.prue/2007/7	189/6.prue/2007/7	NUM
ap-961	880	18	references	reference	NOUN
ap-961	880	19	[	[	X
ap-961	880	20	1	1	NUM
ap-961	880	21	]	]	X
ap-961	880	22	blackadar	blackadar	NOUN
ap-961	880	23	,	,	PUNCT
ap-961	880	24	b.	b.	PROPN
ap-961	880	25	:	:	PUNCT
ap-961	881	1	k	k	X
ap-961	881	2	-	-	NOUN
ap-961	881	3	theory	theory	NOUN
ap-961	881	4	for	for	ADP
ap-961	881	5	operator	operator	NOUN
ap-961	881	6	algebras	algebra	NOUN
ap-961	881	7	.	.	PUNCT
ap-961	882	1	cambridge	cambridge	PROPN
ap-961	882	2	:	:	PUNCT
ap-961	882	3	mathematical	mathematical	PROPN
ap-961	882	4	sciences	sciences	PROPN
ap-961	882	5	research	research	PROPN
ap-961	882	6	institute	institute	PROPN
ap-961	882	7	publications	publication	NOUN
ap-961	882	8	,	,	PUNCT
ap-961	882	9	5	5	X
ap-961	882	10	.	.	PROPN
ap-961	882	11	cambridge	cambridge	PROPN
ap-961	882	12	university	university	PROPN
ap-961	882	13	press	press	NOUN
ap-961	882	14	,	,	PUNCT
ap-961	882	15	1998	1998	NUM
ap-961	882	16	.	.	PUNCT
ap-961	883	1	[	[	X
ap-961	883	2	2	2	NUM
ap-961	883	3	]	]	PUNCT
ap-961	883	4	bondia	bondia	NOUN
ap-961	883	5	,	,	PUNCT
ap-961	883	6	josé	josé	PROPN
ap-961	883	7	m.	m.	PROPN
ap-961	883	8	,	,	PUNCT
ap-961	883	9	várilly	várilly	ADV
ap-961	883	10	,	,	PUNCT
ap-961	883	11	j.	j.	PROPN
ap-961	883	12	,	,	PUNCT
ap-961	883	13	figueroa	figueroa	PROPN
ap-961	883	14	,	,	PUNCT
ap-961	883	15	h.	h.	NOUN
ap-961	883	16	:	:	PUNCT
ap-961	883	17	elements	element	NOUN
ap-961	883	18	of	of	ADP
ap-961	883	19	noncommutative	noncommutative	ADJ
ap-961	883	20	geometry	geometry	NOUN
ap-961	883	21	.	.	PUNCT
ap-961	884	1	boston	boston	PROPN
ap-961	884	2	(	(	PUNCT
ap-961	884	3	ma	ma	PROPN
ap-961	884	4	):	):	PUNCT
ap-961	884	5	birkhauser	birkhauser	NOUN
ap-961	884	6	advanced	advanced	ADJ
ap-961	884	7	texts	text	NOUN
ap-961	884	8	,	,	PUNCT
ap-961	884	9	birkhauser	birkhauser	PROPN
ap-961	884	10	boston	boston	PROPN
ap-961	884	11	,	,	PUNCT
ap-961	884	12	inc	inc	PROPN
ap-961	884	13	.	.	PROPN
ap-961	884	14	,	,	PUNCT
ap-961	884	15	2001	2001	NUM
ap-961	884	16	.	.	PUNCT
ap-961	885	1	[	[	X
ap-961	885	2	3	3	NUM
ap-961	885	3	]	]	PUNCT
ap-961	885	4	brodzki	brodzki	NOUN
ap-961	885	5	,	,	PUNCT
ap-961	885	6	j.	j.	PROPN
ap-961	885	7	:	:	PUNCT
ap-961	885	8	an	an	DET
ap-961	885	9	introduction	introduction	NOUN
ap-961	885	10	to	to	ADP
ap-961	885	11	k	k	NOUN
ap-961	885	12	-	-	NOUN
ap-961	885	13	theory	theory	NOUN
ap-961	885	14	and	and	CCONJ
ap-961	885	15	cyclic	cyclic	ADJ
ap-961	885	16	cohomology	cohomology	NOUN
ap-961	885	17	.	.	PUNCT
ap-961	886	1	advanced	advanced	ADJ
ap-961	886	2	topics	topic	NOUN
ap-961	886	3	in	in	ADP
ap-961	886	4	mathematics	mathematics	PROPN
ap-961	886	5	.	.	PUNCT
ap-961	887	1	warsaw	warsaw	PROPN
ap-961	887	2	:	:	PUNCT
ap-961	887	3	pwn	pwn	PROPN
ap-961	887	4	–	–	PUNCT
ap-961	887	5	polish	polish	ADJ
ap-961	887	6	scientific	scientific	ADJ
ap-961	887	7	publishers	publisher	NOUN
ap-961	887	8	,	,	PUNCT
ap-961	887	9	1998	1998	NUM
ap-961	887	10	.	.	PUNCT
ap-961	888	1	[	[	X
ap-961	888	2	4	4	NUM
ap-961	888	3	]	]	SYM
ap-961	888	4	choquet	choquet	NOUN
ap-961	888	5	-	-	PUNCT
ap-961	888	6	bruhat	bruhat	NOUN
ap-961	888	7	,	,	PUNCT
ap-961	888	8	y.	y.	PROPN
ap-961	888	9	,	,	PUNCT
ap-961	888	10	c.	c.	PROPN
ap-961	888	11	dewitt	dewitt	PROPN
ap-961	888	12	-	-	PUNCT
ap-961	888	13	morette	morette	PROPN
ap-961	888	14	,	,	PUNCT
ap-961	888	15	c.	c.	NOUN
ap-961	888	16	:	:	PUNCT
ap-961	888	17	analysis	analysis	NOUN
ap-961	888	18	,	,	PUNCT
ap-961	888	19	manifolds	manifold	NOUN
ap-961	888	20	and	and	CCONJ
ap-961	888	21	physics	physic	NOUN
ap-961	888	22	.	.	PUNCT
ap-961	889	1	amsterdam	amsterdam	PROPN
ap-961	889	2	:	:	PUNCT
ap-961	889	3	north	north	PROPN
ap-961	889	4	-	-	PUNCT
ap-961	889	5	holland	holland	PROPN
ap-961	889	6	publishing	publishing	PROPN
ap-961	889	7	co.	co.	PROPN
ap-961	889	8	,	,	PUNCT
ap-961	889	9	1989	1989	NUM
ap-961	889	10	.	.	PUNCT
ap-961	890	1	[	[	X
ap-961	890	2	5	5	NUM
ap-961	890	3	]	]	PUNCT
ap-961	890	4	connes	conne	NOUN
ap-961	890	5	,	,	PUNCT
ap-961	890	6	a.	a.	NOUN
ap-961	890	7	:	:	PUNCT
ap-961	890	8	noncommutative	noncommutative	ADJ
ap-961	890	9	geometry	geometry	NOUN
ap-961	890	10	and	and	CCONJ
ap-961	890	11	physics	physics	NOUN
ap-961	890	12	.	.	PUNCT
ap-961	891	1	san	san	PROPN
ap-961	891	2	diego	diego	PROPN
ap-961	891	3	:	:	PUNCT
ap-961	891	4	academic	academic	ADJ
ap-961	891	5	press	press	PROPN
ap-961	891	6	,	,	PUNCT
ap-961	891	7	inc	inc	PROPN
ap-961	891	8	.	.	PROPN
ap-961	891	9	,	,	PUNCT
ap-961	891	10	1994	1994	NUM
ap-961	891	11	.	.	PUNCT
ap-961	892	1	[	[	X
ap-961	892	2	6	6	NUM
ap-961	892	3	]	]	PUNCT
ap-961	892	4	connes	conne	NOUN
ap-961	892	5	,	,	PUNCT
ap-961	892	6	a.	a.	PROPN
ap-961	892	7	,	,	PUNCT
ap-961	892	8	marcolli	marcolli	PROPN
ap-961	892	9	,	,	PUNCT
ap-961	892	10	m.	m.	NOUN
ap-961	892	11	:	:	PUNCT
ap-961	892	12	noncommutative	noncommutative	ADJ
ap-961	892	13	geometry	geometry	NOUN
ap-961	892	14	,	,	PUNCT
ap-961	892	15	quantum	quantum	NOUN
ap-961	892	16	fields	field	NOUN
ap-961	892	17	and	and	CCONJ
ap-961	892	18	motives	motive	NOUN
ap-961	892	19	.	.	PUNCT
ap-961	893	1	(	(	PUNCT
ap-961	893	2	colloquium	colloquium	NOUN
ap-961	893	3	publications	publication	NOUN
ap-961	893	4	)	)	PUNCT
ap-961	893	5	,	,	PUNCT
ap-961	893	6	providence	providence	NOUN
ap-961	893	7	:	:	PUNCT
ap-961	893	8	american	american	PROPN
ap-961	893	9	mathematical	mathematical	PROPN
ap-961	893	10	society	society	NOUN
ap-961	893	11	,	,	PUNCT
ap-961	893	12	2008	2008	NUM
ap-961	893	13	.	.	PUNCT
ap-961	894	1	[	[	X
ap-961	894	2	7	7	X
ap-961	894	3	]	]	X
ap-961	894	4	cuntz	cuntz	PROPN
ap-961	894	5	,	,	PUNCT
ap-961	894	6	j.	j.	PROPN
ap-961	894	7	,	,	PUNCT
ap-961	894	8	khalkhali	khalkhali	PROPN
ap-961	894	9	,	,	PUNCT
ap-961	894	10	m.	m.	NOUN
ap-961	894	11	(	(	PUNCT
ap-961	894	12	eds	ed	NOUN
ap-961	894	13	.	.	PROPN
ap-961	894	14	):	):	PUNCT
ap-961	894	15	cyclic	cyclic	ADJ
ap-961	894	16	cohomology	cohomology	NOUN
ap-961	894	17	and	and	CCONJ
ap-961	894	18	noncommutative	noncommutative	ADJ
ap-961	894	19	geometry	geometry	NOUN
ap-961	894	20	.	.	PUNCT
ap-961	895	1	(	(	PUNCT
ap-961	895	2	proceedings	proceeding	NOUN
ap-961	895	3	of	of	ADP
ap-961	895	4	a	a	DET
ap-961	895	5	workshop	workshop	NOUN
ap-961	895	6	,	,	PUNCT
ap-961	895	7	fields	fields	PROPN
ap-961	895	8	institute	institute	PROPN
ap-961	895	9	,	,	PUNCT
ap-961	895	10	waterloo	waterloo	PROPN
ap-961	895	11	)	)	PUNCT
ap-961	895	12	.	.	PUNCT
ap-961	896	1	providence	providence	NOUN
ap-961	896	2	:	:	PUNCT
ap-961	896	3	american	american	PROPN
ap-961	896	4	mathematical	mathematical	PROPN
ap-961	896	5	society	society	NOUN
ap-961	896	6	(	(	PUNCT
ap-961	896	7	ams	ams	PROPN
ap-961	896	8	)	)	PUNCT
ap-961	896	9	,	,	PUNCT
ap-961	896	10	1997	1997	NUM
ap-961	896	11	.	.	PUNCT
ap-961	897	1	[	[	X
ap-961	897	2	8	8	NUM
ap-961	897	3	]	]	X
ap-961	897	4	cuntz	cuntz	PROPN
ap-961	897	5	,	,	PUNCT
ap-961	897	6	j.	j.	PROPN
ap-961	897	7	,	,	PUNCT
ap-961	897	8	skandalis	skandalis	PROPN
ap-961	897	9	,	,	PUNCT
ap-961	897	10	g.	g.	PROPN
ap-961	897	11	,	,	PUNCT
ap-961	897	12	tsygan	tsygan	PROPN
ap-961	897	13	,	,	PUNCT
ap-961	897	14	b.	b.	NOUN
ap-961	897	15	:	:	PUNCT
ap-961	897	16	cyclic	cyclic	ADJ
ap-961	897	17	homology	homology	NOUN
ap-961	897	18	in	in	ADP
ap-961	897	19	non	non	ADJ
ap-961	897	20	-	-	ADJ
ap-961	897	21	commutative	commutative	ADJ
ap-961	897	22	geometry	geometry	NOUN
ap-961	897	23	.	.	PUNCT
ap-961	898	1	(	(	PUNCT
ap-961	898	2	encyclopaedia	encyclopaedia	PROPN
ap-961	898	3	of	of	ADP
ap-961	898	4	mathematical	mathematical	ADJ
ap-961	898	5	sciences	sciences	PROPN
ap-961	898	6	121(ii	121(ii	NUM
ap-961	898	7	)	)	PUNCT
ap-961	898	8	)	)	PUNCT
ap-961	898	9	.	.	PUNCT
ap-961	899	1	berlin	berlin	PROPN
ap-961	899	2	:	:	PUNCT
ap-961	899	3	springer	springer	NOUN
ap-961	899	4	,	,	PUNCT
ap-961	899	5	2004	2004	NUM
ap-961	899	6	.	.	PUNCT
ap-961	900	1	[	[	X
ap-961	900	2	9	9	NUM
ap-961	900	3	]	]	SYM
ap-961	900	4	doplicher	doplicher	NOUN
ap-961	900	5	,	,	PUNCT
ap-961	900	6	s.	s.	PROPN
ap-961	900	7	,	,	PUNCT
ap-961	900	8	longo	longo	PROPN
ap-961	900	9	,	,	PUNCT
ap-961	900	10	r.	r.	PROPN
ap-961	900	11	(	(	PUNCT
ap-961	900	12	eds	eds	PROPN
ap-961	900	13	.	.	PROPN
ap-961	900	14	):	):	PUNCT
ap-961	900	15	noncommutative	noncommutative	ADJ
ap-961	900	16	geometry	geometry	NOUN
ap-961	900	17	.	.	PUNCT
ap-961	901	1	(	(	PUNCT
ap-961	901	2	lectures	lecture	NOUN
ap-961	901	3	given	give	VERB
ap-961	901	4	at	at	ADP
ap-961	901	5	the	the	DET
ap-961	901	6	c.	c.	PROPN
ap-961	901	7	i.	i.	PROPN
ap-961	901	8	m.	m.	PROPN
ap-961	901	9	e.	e.	PROPN
ap-961	901	10	summer	summer	PROPN
ap-961	901	11	school	school	PROPN
ap-961	901	12	,	,	PUNCT
ap-961	901	13	martina	martina	PROPN
ap-961	901	14	franca	franca	PROPN
ap-961	901	15	)	)	PUNCT
ap-961	901	16	,	,	PUNCT
ap-961	901	17	berlin	berlin	PROPN
ap-961	901	18	:	:	PUNCT
ap-961	901	19	springer	springer	NOUN
ap-961	901	20	,	,	PUNCT
ap-961	901	21	2004	2004	NUM
ap-961	901	22	.	.	PUNCT
ap-961	902	1	[	[	X
ap-961	902	2	10	10	NUM
ap-961	902	3	]	]	X
ap-961	902	4	friedrich	friedrich	NOUN
ap-961	902	5	,	,	PUNCT
ap-961	902	6	t.	t.	PROPN
ap-961	902	7	:	:	PUNCT
ap-961	902	8	dirac	dirac	NOUN
ap-961	902	9	operators	operator	NOUN
ap-961	902	10	in	in	ADP
ap-961	902	11	riemannian	riemannian	ADJ
ap-961	902	12	geometry	geometry	NOUN
ap-961	902	13	.	.	PUNCT
ap-961	903	1	providence	providence	NOUN
ap-961	903	2	:	:	PUNCT
ap-961	903	3	american	american	PROPN
ap-961	903	4	mathematical	mathematical	PROPN
ap-961	903	5	society	society	NOUN
ap-961	903	6	(	(	PUNCT
ap-961	903	7	ams	ams	PROPN
ap-961	903	8	)	)	PUNCT
ap-961	903	9	,	,	PUNCT
ap-961	903	10	2000	2000	NUM
ap-961	903	11	.	.	PUNCT
ap-961	904	1	[	[	X
ap-961	904	2	11	11	NUM
ap-961	904	3	]	]	PUNCT
ap-961	904	4	higson	higson	NOUN
ap-961	904	5	,	,	PUNCT
ap-961	904	6	n.	n.	NOUN
ap-961	904	7	,	,	PUNCT
ap-961	904	8	nigel	nigel	PROPN
ap-961	904	9	,	,	PUNCT
ap-961	904	10	j.	j.	PROPN
ap-961	904	11	roe	roe	PROPN
ap-961	904	12	(	(	PUNCT
ap-961	904	13	eds	eds	PROPN
ap-961	904	14	.	.	PUNCT
ap-961	904	15	):	):	PUNCT
ap-961	904	16	surveys	survey	NOUN
ap-961	904	17	in	in	ADP
ap-961	904	18	noncommutative	noncommutative	ADJ
ap-961	904	19	geometry	geometry	NOUN
ap-961	904	20	.	.	PUNCT
ap-961	905	1	(	(	PUNCT
ap-961	905	2	proceedings	proceeding	NOUN
ap-961	905	3	from	from	ADP
ap-961	905	4	the	the	DET
ap-961	905	5	clay	clay	NOUN
ap-961	905	6	mathematics	mathematics	PROPN
ap-961	905	7	institute	institute	PROPN
ap-961	905	8	instructional	instructional	ADJ
ap-961	905	9	symposium	symposium	NOUN
ap-961	905	10	)	)	PUNCT
ap-961	905	11	.	.	PUNCT
ap-961	906	1	providence	providence	NOUN
ap-961	906	2	:	:	PUNCT
ap-961	906	3	american	american	PROPN
ap-961	906	4	mathematical	mathematical	PROPN
ap-961	906	5	society	society	NOUN
ap-961	906	6	,	,	PUNCT
ap-961	906	7	2006	2006	NUM
ap-961	906	8	.	.	PUNCT
ap-961	907	1	[	[	X
ap-961	907	2	12	12	NUM
ap-961	907	3	]	]	PUNCT
ap-961	907	4	husemoller	husemoller	NOUN
ap-961	907	5	,	,	PUNCT
ap-961	907	6	d.	d.	PROPN
ap-961	907	7	:	:	PUNCT
ap-961	907	8	lectures	lecture	VERB
ap-961	907	9	on	on	ADP
ap-961	907	10	cyclic	cyclic	ADJ
ap-961	907	11	homology	homology	NOUN
ap-961	907	12	.	.	PUNCT
ap-961	908	1	berlin	berlin	PROPN
ap-961	908	2	:	:	PUNCT
ap-961	908	3	springer	springer	NOUN
ap-961	908	4	-	-	PUNCT
ap-961	908	5	verlag	verlag	PROPN
ap-961	908	6	,	,	PUNCT
ap-961	908	7	1991	1991	NUM
ap-961	908	8	.	.	PUNCT
ap-961	909	1	[	[	X
ap-961	909	2	13	13	NUM
ap-961	909	3	]	]	SYM
ap-961	909	4	kadison	kadison	PROPN
ap-961	909	5	,	,	PUNCT
ap-961	909	6	r.	r.	PROPN
ap-961	909	7	,	,	PUNCT
ap-961	909	8	ringrose	ringrose	PROPN
ap-961	909	9	,	,	PUNCT
ap-961	909	10	j.	j.	PROPN
ap-961	909	11	:	:	PUNCT
ap-961	909	12	fundamentals	fundamental	NOUN
ap-961	909	13	of	of	ADP
ap-961	909	14	the	the	DET
ap-961	909	15	theory	theory	NOUN
ap-961	909	16	of	of	ADP
ap-961	909	17	operator	operator	NOUN
ap-961	909	18	algebras	algebra	NOUN
ap-961	909	19	.	.	PUNCT
ap-961	910	1	advanced	advanced	ADJ
ap-961	910	2	theory	theory	NOUN
ap-961	910	3	,	,	PUNCT
ap-961	910	4	vol	vol	NOUN
ap-961	910	5	.	.	PUNCT
ap-961	911	1	1–2	1–2	NUM
ap-961	911	2	,	,	PUNCT
ap-961	911	3	american	american	PROPN
ap-961	911	4	mathematical	mathematical	ADJ
ap-961	911	5	society	society	NOUN
ap-961	911	6	,	,	PUNCT
ap-961	911	7	1997	1997	NUM
ap-961	911	8	.	.	PUNCT
ap-961	912	1	[	[	X
ap-961	912	2	14	14	NUM
ap-961	912	3	]	]	PUNCT
ap-961	912	4	landi	landi	PROPN
ap-961	912	5	,	,	PUNCT
ap-961	912	6	g.	g.	PROPN
ap-961	912	7	:	:	PUNCT
ap-961	912	8	an	an	DET
ap-961	912	9	introduction	introduction	NOUN
ap-961	912	10	to	to	ADP
ap-961	912	11	noncommutative	noncommutative	ADJ
ap-961	912	12	spaces	space	NOUN
ap-961	912	13	and	and	CCONJ
ap-961	912	14	their	their	PRON
ap-961	912	15	geometry	geometry	NOUN
ap-961	912	16	.	.	PUNCT
ap-961	913	1	lecture	lecture	NOUN
ap-961	913	2	notes	note	NOUN
ap-961	913	3	in	in	ADP
ap-961	913	4	physics	physics	PROPN
ap-961	913	5	.	.	PUNCT
ap-961	914	1	berlin	berlin	PROPN
ap-961	914	2	:	:	PUNCT
ap-961	914	3	springer	springer	NOUN
ap-961	914	4	-	-	PUNCT
ap-961	914	5	verlag	verlag	PROPN
ap-961	914	6	,	,	PUNCT
ap-961	914	7	1997	1997	NUM
ap-961	914	8	.	.	PUNCT
ap-961	915	1	[	[	X
ap-961	915	2	15	15	NUM
ap-961	915	3	]	]	X
ap-961	915	4	lawson	lawson	PROPN
ap-961	915	5	,	,	PUNCT
ap-961	915	6	h.	h.	PROPN
ap-961	915	7	,	,	PUNCT
ap-961	915	8	michelsohn	michelsohn	PROPN
ap-961	915	9	,	,	PUNCT
ap-961	915	10	m	m	PROPN
ap-961	915	11	-	-	PUNCT
ap-961	915	12	l.	l.	PROPN
ap-961	915	13	:	:	PUNCT
ap-961	915	14	spin	spin	NOUN
ap-961	915	15	geometry	geometry	NOUN
ap-961	915	16	.	.	PUNCT
ap-961	916	1	princeton	princeton	PROPN
ap-961	916	2	,	,	PUNCT
ap-961	916	3	nj	nj	PROPN
ap-961	916	4	:	:	PUNCT
ap-961	916	5	princeton	princeton	PROPN
ap-961	916	6	mathematical	mathematical	PROPN
ap-961	916	7	series	series	PROPN
ap-961	916	8	,	,	PUNCT
ap-961	916	9	38	38	NUM
ap-961	916	10	.	.	PUNCT
ap-961	917	1	princeton	princeton	PROPN
ap-961	917	2	university	university	PROPN
ap-961	917	3	press	press	NOUN
ap-961	917	4	,	,	PUNCT
ap-961	917	5	1989	1989	NUM
ap-961	917	6	.	.	PUNCT
ap-961	918	1	[	[	X
ap-961	918	2	16	16	NUM
ap-961	918	3	]	]	PUNCT
ap-961	918	4	loday	loday	PROPN
ap-961	918	5	,	,	PUNCT
ap-961	918	6	j	j	PROPN
ap-961	918	7	-	-	PUNCT
ap-961	918	8	l.	l.	PROPN
ap-961	918	9	:	:	PUNCT
ap-961	918	10	cyclic	cyclic	ADJ
ap-961	918	11	homology	homology	NOUN
ap-961	918	12	.	.	PUNCT
ap-961	919	1	grundlehren	grundlehren	PROPN
ap-961	919	2	der	der	PROPN
ap-961	919	3	mathematischen	mathematischen	PROPN
ap-961	919	4	wissenschaften	wissenschaften	PROPN
ap-961	919	5	,	,	PUNCT
ap-961	919	6	301	301	NUM
ap-961	919	7	.	.	PUNCT
ap-961	920	1	berlin	berlin	PROPN
ap-961	920	2	:	:	PUNCT
ap-961	920	3	springer	springer	NOUN
ap-961	920	4	-	-	PUNCT
ap-961	920	5	verlag	verlag	PROPN
ap-961	920	6	,	,	PUNCT
ap-961	920	7	1998	1998	NUM
ap-961	920	8	.	.	PUNCT
ap-961	921	1	[	[	X
ap-961	921	2	17	17	NUM
ap-961	921	3	]	]	SYM
ap-961	921	4	madore	madore	NOUN
ap-961	921	5	,	,	PUNCT
ap-961	921	6	j.	j.	PROPN
ap-961	921	7	:	:	PUNCT
ap-961	921	8	an	an	DET
ap-961	921	9	introduction	introduction	NOUN
ap-961	921	10	to	to	ADP
ap-961	921	11	noncommutative	noncommutative	ADJ
ap-961	921	12	differential	differential	ADJ
ap-961	921	13	geometry	geometry	NOUN
ap-961	921	14	and	and	CCONJ
ap-961	921	15	its	its	PRON
ap-961	921	16	physical	physical	ADJ
ap-961	921	17	applications	application	NOUN
ap-961	921	18	.	.	PUNCT
ap-961	922	1	cambridge	cambridge	PROPN
ap-961	922	2	:	:	PUNCT
ap-961	922	3	cambridge	cambridge	PROPN
ap-961	922	4	university	university	PROPN
ap-961	922	5	press	press	NOUN
ap-961	922	6	,	,	PUNCT
ap-961	922	7	1995	1995	NUM
ap-961	922	8	,	,	PUNCT
ap-961	922	9	1999	1999	NUM
ap-961	922	10	.	.	PUNCT
ap-961	923	1	[	[	X
ap-961	923	2	18	18	NUM
ap-961	923	3	]	]	PUNCT
ap-961	923	4	várily	várily	ADV
ap-961	923	5	,	,	PUNCT
ap-961	923	6	j.	j.	PROPN
ap-961	923	7	:	:	PUNCT
ap-961	923	8	an	an	DET
ap-961	923	9	introduction	introduction	NOUN
ap-961	923	10	to	to	ADP
ap-961	923	11	noncommutative	noncommutative	ADJ
ap-961	923	12	geometry	geometry	NOUN
ap-961	923	13	.	.	PUNCT
ap-961	924	1	(	(	PUNCT
ap-961	924	2	ems	ems	PROPN
ap-961	924	3	series	series	PROPN
ap-961	924	4	of	of	ADP
ap-961	924	5	lectures	lecture	NOUN
ap-961	924	6	in	in	ADP
ap-961	924	7	mathematics	mathematic	NOUN
ap-961	924	8	)	)	PUNCT
ap-961	924	9	,	,	PUNCT
ap-961	924	10	zürich	zürich	NOUN
ap-961	924	11	:	:	PUNCT
ap-961	924	12	european	european	PROPN
ap-961	924	13	mathematical	mathematical	PROPN
ap-961	924	14	society	society	PROPN
ap-961	924	15	publishing	publishing	PROPN
ap-961	924	16	house	house	NOUN
ap-961	924	17	,	,	PUNCT
ap-961	924	18	2006	2006	NUM
ap-961	924	19	.	.	PUNCT
ap-961	925	1	56	56	NUM
ap-961	925	2	©	©	PROPN
ap-961	925	3	czech	czech	PROPN
ap-961	925	4	technical	technical	PROPN
ap-961	925	5	university	university	PROPN
ap-961	925	6	publishing	publishing	NOUN
ap-961	925	7	house	house	NOUN
ap-961	925	8	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-961	925	9	acta	acta	PROPN
ap-961	925	10	polytechnica	polytechnica	PROPN
ap-961	925	11	vol	vol	NOUN
ap-961	925	12	.	.	PUNCT
ap-961	926	1	48	48	NUM
ap-961	926	2	no	no	NOUN
ap-961	926	3	.	.	PUNCT
ap-961	927	1	2/2008	2/2008	NUM
ap-961	928	1	[	[	X
ap-961	928	2	19	19	NUM
ap-961	928	3	]	]	PUNCT
ap-961	928	4	wegge	wegge	NOUN
ap-961	928	5	-	-	PUNCT
ap-961	928	6	olsen	olsen	NOUN
ap-961	928	7	,	,	PUNCT
ap-961	928	8	n.	n.	NOUN
ap-961	928	9	:	:	PUNCT
ap-961	928	10	k	k	X
ap-961	928	11	-	-	NOUN
ap-961	928	12	theory	theory	NOUN
ap-961	928	13	and	and	CCONJ
ap-961	928	14	c*-algebras	c*-algebras	PROPN
ap-961	928	15	.	.	PUNCT
ap-961	929	1	a	a	DET
ap-961	929	2	friendly	friendly	ADJ
ap-961	929	3	approach	approach	NOUN
ap-961	929	4	.	.	PUNCT
ap-961	930	1	oxford	oxford	PROPN
ap-961	930	2	science	science	PROPN
ap-961	930	3	publications	publication	NOUN
ap-961	930	4	.	.	PUNCT
ap-961	931	1	oxford	oxford	PROPN
ap-961	931	2	,	,	PUNCT
ap-961	931	3	new	new	PROPN
ap-961	931	4	york	york	PROPN
ap-961	931	5	:	:	PUNCT
ap-961	931	6	the	the	DET
ap-961	931	7	clarendon	clarendon	PROPN
ap-961	931	8	press	press	NOUN
ap-961	931	9	,	,	PUNCT
ap-961	931	10	1993	1993	NUM
ap-961	931	11	.	.	PUNCT
ap-961	932	1	[	[	X
ap-961	932	2	20	20	NUM
ap-961	932	3	]	]	X
ap-961	932	4	bellissard	bellissard	NOUN
ap-961	932	5	,	,	PUNCT
ap-961	932	6	j.	j.	PROPN
ap-961	932	7	:	:	PUNCT
ap-961	932	8	noncommutative	noncommutative	ADJ
ap-961	932	9	geometry	geometry	NOUN
ap-961	932	10	and	and	CCONJ
ap-961	932	11	quantum	quantum	NOUN
ap-961	932	12	hall	hall	NOUN
ap-961	932	13	effect	effect	NOUN
ap-961	932	14	.	.	PUNCT
ap-961	933	1	in	in	ADP
ap-961	933	2	proceedings	proceeding	NOUN
ap-961	933	3	of	of	ADP
ap-961	933	4	the	the	DET
ap-961	933	5	international	international	ADJ
ap-961	933	6	congress	congress	PROPN
ap-961	933	7	of	of	ADP
ap-961	933	8	mathematicians	mathematician	NOUN
ap-961	933	9	,	,	PUNCT
ap-961	933	10	vol	vol	NOUN
ap-961	933	11	.	.	PROPN
ap-961	933	12	1	1	NUM
ap-961	933	13	,	,	PUNCT
ap-961	933	14	2	2	NUM
ap-961	933	15	zürich	zürich	NOUN
ap-961	933	16	:	:	PUNCT
ap-961	933	17	1994	1994	NUM
ap-961	933	18	,	,	PUNCT
ap-961	933	19	p.	p.	NOUN
ap-961	933	20	1238–1246	1238–1246	NUM
ap-961	933	21	,	,	PUNCT
ap-961	933	22	basel	basel	PROPN
ap-961	933	23	:	:	PUNCT
ap-961	933	24	birkhauser	birkhauser	PROPN
ap-961	933	25	,	,	PUNCT
ap-961	933	26	1995	1995	NUM
ap-961	933	27	.	.	PUNCT
ap-961	934	1	[	[	X
ap-961	934	2	21	21	NUM
ap-961	934	3	]	]	X
ap-961	934	4	chamseddine	chamseddine	NOUN
ap-961	934	5	,	,	PUNCT
ap-961	934	6	a.	a.	PROPN
ap-961	934	7	h.	h.	PROPN
ap-961	934	8	,	,	PUNCT
ap-961	934	9	connes	conne	NOUN
ap-961	934	10	,	,	PUNCT
ap-961	934	11	a.	a.	NOUN
ap-961	934	12	:	:	PUNCT
ap-961	934	13	an	an	DET
ap-961	934	14	universal	universal	ADJ
ap-961	934	15	action	action	NOUN
ap-961	934	16	formula	formula	NOUN
ap-961	934	17	,	,	PUNCT
ap-961	934	18	phys	phy	NOUN
ap-961	934	19	.	.	PUNCT
ap-961	934	20	rev	rev	PROPN
ap-961	934	21	.	.	PROPN
ap-961	934	22	lett	lett	PROPN
ap-961	934	23	.	.	PROPN
ap-961	934	24	,	,	PUNCT
ap-961	934	25	vol	vol	NOUN
ap-961	934	26	.	.	PROPN
ap-961	935	1	77	77	NUM
ap-961	935	2	(	(	PUNCT
ap-961	935	3	1996	1996	NUM
ap-961	935	4	)	)	PUNCT
ap-961	935	5	,	,	PUNCT
ap-961	936	1	p.	p.	NOUN
ap-961	936	2	4868	4868	NUM
ap-961	936	3	.	.	PUNCT
ap-961	937	1	[	[	X
ap-961	937	2	22	22	NUM
ap-961	937	3	]	]	X
ap-961	937	4	chamseddine	chamseddine	NOUN
ap-961	937	5	,	,	PUNCT
ap-961	937	6	a.	a.	PROPN
ap-961	937	7	h.	h.	PROPN
ap-961	937	8	,	,	PUNCT
ap-961	937	9	connes	conne	NOUN
ap-961	937	10	,	,	PUNCT
ap-961	937	11	a.	a.	NOUN
ap-961	937	12	:	:	PUNCT
ap-961	937	13	the	the	DET
ap-961	937	14	spectral	spectral	ADJ
ap-961	937	15	action	action	NOUN
ap-961	937	16	principle	principle	NOUN
ap-961	937	17	.	.	PUNCT
ap-961	938	1	commun	commun	PROPN
ap-961	938	2	.	.	PUNCT
ap-961	939	1	math	math	NOUN
ap-961	939	2	.	.	PUNCT
ap-961	940	1	phys	phy	NOUN
ap-961	940	2	.	.	PUNCT
ap-961	940	3	,	,	PUNCT
ap-961	940	4	vol	vol	NOUN
ap-961	940	5	.	.	PROPN
ap-961	940	6	186	186	NUM
ap-961	940	7	(	(	PUNCT
ap-961	940	8	1997	1997	NUM
ap-961	940	9	)	)	PUNCT
ap-961	940	10	,	,	PUNCT
ap-961	940	11	p.	p.	NOUN
ap-961	940	12	731	731	NUM
ap-961	940	13	.	.	PUNCT
ap-961	941	1	[	[	X
ap-961	941	2	23	23	NUM
ap-961	941	3	]	]	PUNCT
ap-961	941	4	connes	conne	NOUN
ap-961	941	5	,	,	PUNCT
ap-961	941	6	a.	a.	NOUN
ap-961	941	7	:	:	PUNCT
ap-961	941	8	noncommutative	noncommutative	ADJ
ap-961	941	9	differential	differential	PROPN
ap-961	941	10	geometry	geometry	NOUN
ap-961	941	11	.	.	PUNCT
ap-961	942	1	inst	inst	AUX
ap-961	942	2	.	.	PUNCT
ap-961	942	3	hautes	hautes	PROPN
ap-961	942	4	études	études	PROPN
ap-961	942	5	sci	sci	PROPN
ap-961	942	6	.	.	PROPN
ap-961	942	7	publ	publ	PROPN
ap-961	942	8	.	.	PUNCT
ap-961	943	1	math	math	NOUN
ap-961	943	2	.	.	PUNCT
ap-961	944	1	,	,	PUNCT
ap-961	944	2	(	(	PUNCT
ap-961	944	3	1985	1985	NUM
ap-961	944	4	)	)	PUNCT
ap-961	944	5	,	,	PUNCT
ap-961	945	1	no	no	INTJ
ap-961	945	2	.	.	NOUN
ap-961	945	3	62	62	NUM
ap-961	945	4	,	,	PUNCT
ap-961	945	5	p.	p.	NOUN
ap-961	945	6	257–360	257–360	NUM
ap-961	945	7	.	.	PUNCT
ap-961	946	1	[	[	X
ap-961	946	2	24	24	NUM
ap-961	946	3	]	]	PUNCT
ap-961	946	4	connes	conne	NOUN
ap-961	946	5	,	,	PUNCT
ap-961	946	6	a.	a.	NOUN
ap-961	946	7	,	,	PUNCT
ap-961	946	8	rieffel	rieffel	NOUN
ap-961	946	9	,	,	PUNCT
ap-961	946	10	m.	m.	NOUN
ap-961	946	11	:	:	PUNCT
ap-961	946	12	yang	yang	PROPN
ap-961	946	13	-	-	PUNCT
ap-961	946	14	mills	mill	NOUN
ap-961	946	15	for	for	ADP
ap-961	946	16	noncommutative	noncommutative	ADJ
ap-961	946	17	two	two	NUM
ap-961	946	18	-	-	PUNCT
ap-961	946	19	tori	tori	NOUN
ap-961	946	20	.	.	PUNCT
ap-961	947	1	operator	operator	NOUN
ap-961	947	2	algebras	algebra	NOUN
ap-961	947	3	and	and	CCONJ
ap-961	947	4	mathematical	mathematical	ADJ
ap-961	947	5	physics	physics	NOUN
ap-961	947	6	(	(	PUNCT
ap-961	947	7	1985	1985	NUM
ap-961	947	8	)	)	PUNCT
ap-961	947	9	,	,	PUNCT
ap-961	947	10	p.	p.	NOUN
ap-961	947	11	237–266	237–266	NUM
ap-961	947	12	,	,	PUNCT
ap-961	947	13	contemp	contemp	NOUN
ap-961	947	14	.	.	PUNCT
ap-961	948	1	math	math	NOUN
ap-961	948	2	.	.	PUNCT
ap-961	948	3	,	,	PUNCT
ap-961	948	4	62	62	NUM
ap-961	948	5	.	.	PUNCT
ap-961	949	1	[	[	X
ap-961	949	2	25	25	NUM
ap-961	949	3	]	]	PUNCT
ap-961	949	4	connes	conne	NOUN
ap-961	949	5	,	,	PUNCT
ap-961	949	6	a.	a.	NOUN
ap-961	949	7	:	:	PUNCT
ap-961	949	8	the	the	DET
ap-961	949	9	action	action	NOUN
ap-961	949	10	functional	functional	ADJ
ap-961	949	11	in	in	ADP
ap-961	949	12	noncommutative	noncommutative	ADJ
ap-961	949	13	geometry	geometry	NOUN
ap-961	949	14	.	.	PUNCT
ap-961	950	1	commun	commun	PROPN
ap-961	950	2	.	.	PUNCT
ap-961	951	1	math	math	NOUN
ap-961	951	2	.	.	PUNCT
ap-961	952	1	phys	phy	NOUN
ap-961	952	2	.	.	PUNCT
ap-961	952	3	,	,	PUNCT
ap-961	952	4	vol	vol	NOUN
ap-961	952	5	.	.	PROPN
ap-961	952	6	117	117	NUM
ap-961	952	7	(	(	PUNCT
ap-961	952	8	1988	1988	NUM
ap-961	952	9	)	)	PUNCT
ap-961	952	10	,	,	PUNCT
ap-961	952	11	p.	p.	NOUN
ap-961	952	12	673	673	NUM
ap-961	952	13	.	.	PUNCT
ap-961	953	1	[	[	X
ap-961	953	2	26	26	NUM
ap-961	953	3	]	]	PUNCT
ap-961	953	4	connes	conne	NOUN
ap-961	953	5	,	,	PUNCT
ap-961	953	6	a.	a.	PROPN
ap-961	953	7	,	,	PUNCT
ap-961	953	8	lott	lott	PROPN
ap-961	953	9	,	,	PUNCT
ap-961	953	10	j.	j.	PROPN
ap-961	953	11	:	:	PUNCT
ap-961	953	12	particle	particle	NOUN
ap-961	953	13	models	model	NOUN
ap-961	953	14	and	and	CCONJ
ap-961	953	15	noncommutative	noncommutative	ADJ
ap-961	953	16	geometry	geometry	NOUN
ap-961	953	17	.	.	PUNCT
ap-961	954	1	nucl	nucl	PROPN
ap-961	954	2	.	.	PUNCT
ap-961	955	1	phys	phy	NOUN
ap-961	955	2	.	.	PUNCT
ap-961	956	1	proc	proc	PROPN
ap-961	956	2	.	.	PROPN
ap-961	956	3	,	,	PUNCT
ap-961	956	4	suppl	suppl	PROPN
ap-961	956	5	.	.	PROPN
ap-961	956	6	18b	18b	NUM
ap-961	956	7	(	(	PUNCT
ap-961	956	8	1991	1991	NUM
ap-961	956	9	)	)	PUNCT
ap-961	956	10	,	,	PUNCT
ap-961	957	1	p.	p.	NOUN
ap-961	957	2	29	29	NUM
ap-961	957	3	.	.	PUNCT
ap-961	958	1	[	[	X
ap-961	958	2	27	27	NUM
ap-961	958	3	]	]	PUNCT
ap-961	958	4	connes	conne	NOUN
ap-961	958	5	,	,	PUNCT
ap-961	958	6	a.	a.	NOUN
ap-961	958	7	:	:	PUNCT
ap-961	958	8	noncommutative	noncommutative	ADJ
ap-961	958	9	geometry	geometry	NOUN
ap-961	958	10	and	and	CCONJ
ap-961	958	11	reality	reality	NOUN
ap-961	958	12	.	.	PUNCT
ap-961	959	1	j.	j.	PROPN
ap-961	959	2	math	math	PROPN
ap-961	959	3	.	.	PUNCT
ap-961	960	1	phys	phy	NOUN
ap-961	960	2	.	.	PUNCT
ap-961	960	3	,	,	PUNCT
ap-961	960	4	vol	vol	NOUN
ap-961	960	5	.	.	PROPN
ap-961	960	6	36	36	NUM
ap-961	960	7	(	(	PUNCT
ap-961	960	8	1995	1995	NUM
ap-961	960	9	)	)	PUNCT
ap-961	960	10	,	,	PUNCT
ap-961	960	11	p.	p.	NOUN
ap-961	960	12	6194	6194	NUM
ap-961	960	13	.	.	PUNCT
ap-961	961	1	[	[	X
ap-961	961	2	28	28	NUM
ap-961	961	3	]	]	PUNCT
ap-961	961	4	connes	conne	NOUN
ap-961	961	5	,	,	PUNCT
ap-961	961	6	a.	a.	PROPN
ap-961	961	7	,	,	PUNCT
ap-961	961	8	douglas	douglas	PROPN
ap-961	961	9	,	,	PUNCT
ap-961	961	10	m.	m.	PROPN
ap-961	961	11	r.	r.	PROPN
ap-961	961	12	,	,	PUNCT
ap-961	961	13	schwarz	schwarz	PROPN
ap-961	961	14	,	,	PUNCT
ap-961	961	15	a.	a.	NOUN
ap-961	961	16	:	:	PUNCT
ap-961	961	17	noncommutative	noncommutative	ADJ
ap-961	961	18	geometry	geometry	NOUN
ap-961	961	19	and	and	CCONJ
ap-961	961	20	matrix	matrix	NOUN
ap-961	961	21	theory	theory	NOUN
ap-961	961	22	:	:	PUNCT
ap-961	961	23	compactification	compactification	NOUN
ap-961	961	24	on	on	ADP
ap-961	961	25	tori	tori	NOUN
ap-961	961	26	.	.	PUNCT
ap-961	962	1	jhep	jhep	ADJ
ap-961	962	2	9802	9802	NUM
ap-961	962	3	(	(	PUNCT
ap-961	962	4	1998	1998	NUM
ap-961	962	5	)	)	PUNCT
ap-961	962	6	,	,	PUNCT
ap-961	962	7	003	003	NUM
ap-961	962	8	.	.	PUNCT
ap-961	963	1	[	[	X
ap-961	963	2	29	29	NUM
ap-961	963	3	]	]	PUNCT
ap-961	963	4	connes	conne	NOUN
ap-961	963	5	,	,	PUNCT
ap-961	963	6	a.	a.	NOUN
ap-961	963	7	:	:	PUNCT
ap-961	963	8	a	a	DET
ap-961	963	9	short	short	ADJ
ap-961	963	10	survey	survey	NOUN
ap-961	963	11	of	of	ADP
ap-961	963	12	noncommutative	noncommutative	ADJ
ap-961	963	13	geometry	geometry	NOUN
ap-961	963	14	.	.	PUNCT
ap-961	964	1	j.	j.	PROPN
ap-961	964	2	math	math	PROPN
ap-961	964	3	.	.	PUNCT
ap-961	965	1	phys	phy	NOUN
ap-961	965	2	.	.	PUNCT
ap-961	965	3	,	,	PUNCT
ap-961	965	4	vol	vol	NOUN
ap-961	965	5	.	.	PROPN
ap-961	965	6	41	41	NUM
ap-961	965	7	(	(	PUNCT
ap-961	965	8	2000	2000	NUM
ap-961	965	9	)	)	PUNCT
ap-961	965	10	,	,	PUNCT
ap-961	965	11	p.	p.	NOUN
ap-961	965	12	3832	3832	NUM
ap-961	965	13	.	.	PUNCT
ap-961	966	1	[	[	X
ap-961	966	2	30	30	NUM
ap-961	966	3	]	]	PUNCT
ap-961	966	4	connes	conne	NOUN
ap-961	966	5	,	,	PUNCT
ap-961	966	6	a.	a.	NOUN
ap-961	966	7	:	:	PUNCT
ap-961	966	8	noncommutative	noncommutative	ADJ
ap-961	966	9	geometry	geometry	NOUN
ap-961	966	10	:	:	PUNCT
ap-961	966	11	year	year	NOUN
ap-961	966	12	2000	2000	NUM
ap-961	966	13	.	.	PUNCT
ap-961	967	1	arxiv	arxiv	NOUN
ap-961	967	2	:	:	PUNCT
ap-961	967	3	math.qa/0011193	math.qa/0011193	PROPN
ap-961	967	4	.	.	PUNCT
ap-961	968	1	[	[	X
ap-961	968	2	31	31	NUM
ap-961	968	3	]	]	PUNCT
ap-961	968	4	connes	conne	NOUN
ap-961	968	5	,	,	PUNCT
ap-961	968	6	a.	a.	NOUN
ap-961	968	7	,	,	PUNCT
ap-961	968	8	chamseddine	chamseddine	NOUN
ap-961	968	9	,	,	PUNCT
ap-961	968	10	a.	a.	PROPN
ap-961	968	11	h.	h.	PROPN
ap-961	968	12	,	,	PUNCT
ap-961	968	13	marcolli	marcolli	PROPN
ap-961	968	14	,	,	PUNCT
ap-961	968	15	m.	m.	NOUN
ap-961	968	16	:	:	PUNCT
ap-961	968	17	gravity	gravity	NOUN
ap-961	968	18	and	and	CCONJ
ap-961	968	19	the	the	DET
ap-961	968	20	standard	standard	ADJ
ap-961	968	21	model	model	NOUN
ap-961	968	22	with	with	ADP
ap-961	968	23	neutrino	neutrino	NOUN
ap-961	968	24	mixing	mixing	NOUN
ap-961	968	25	.	.	PUNCT
ap-961	969	1	arxiv	arxiv	NOUN
ap-961	969	2	:	:	PUNCT
ap-961	969	3	hep	hep	NOUN
ap-961	969	4	-	-	PUNCT
ap-961	969	5	th/0610241	th/0610241	X
ap-961	969	6	[	[	X
ap-961	969	7	32	32	NUM
ap-961	969	8	]	]	PUNCT
ap-961	969	9	connes	conne	NOUN
ap-961	969	10	,	,	PUNCT
ap-961	969	11	a.	a.	NOUN
ap-961	969	12	,	,	PUNCT
ap-961	969	13	chamseddine	chamseddine	NOUN
ap-961	969	14	,	,	PUNCT
ap-961	969	15	a.	a.	PROPN
ap-961	969	16	h.	h.	PROPN
ap-961	969	17	:	:	PUNCT
ap-961	969	18	a	a	DET
ap-961	969	19	dress	dress	NOUN
ap-961	969	20	for	for	ADP
ap-961	969	21	sm	sm	PROPN
ap-961	969	22	the	the	DET
ap-961	969	23	beggar	beggar	NOUN
ap-961	969	24	.	.	PUNCT
ap-961	970	1	arxive	arxive	PROPN
ap-961	970	2	:	:	PUNCT
ap-961	970	3	math/07063690	math/07063690	NOUN
ap-961	970	4	.	.	PUNCT
ap-961	971	1	[	[	X
ap-961	971	2	33	33	NUM
ap-961	971	3	]	]	PUNCT
ap-961	971	4	cuntz	cuntz	PROPN
ap-961	971	5	,	,	PUNCT
ap-961	971	6	j.	j.	PROPN
ap-961	971	7	,	,	PUNCT
ap-961	971	8	quillen	quillen	PROPN
ap-961	971	9	,	,	PUNCT
ap-961	971	10	d.	d.	PROPN
ap-961	971	11	:	:	PUNCT
ap-961	971	12	algebra	algebra	NOUN
ap-961	971	13	extensions	extension	NOUN
ap-961	971	14	and	and	CCONJ
ap-961	971	15	nonsingularity	nonsingularity	NOUN
ap-961	971	16	.	.	PUNCT
ap-961	972	1	j.	j.	PROPN
ap-961	972	2	amer	amer	PROPN
ap-961	972	3	.	.	PROPN
ap-961	972	4	math	math	PROPN
ap-961	972	5	.	.	PUNCT
ap-961	973	1	soc	soc	PROPN
ap-961	973	2	.	.	PUNCT
ap-961	974	1	,	,	PUNCT
ap-961	974	2	vol	vol	NOUN
ap-961	974	3	.	.	PROPN
ap-961	974	4	8	8	NUM
ap-961	974	5	(	(	PUNCT
ap-961	974	6	1995	1995	NUM
ap-961	974	7	)	)	PUNCT
ap-961	974	8	,	,	PUNCT
ap-961	974	9	no	no	INTJ
ap-961	974	10	.	.	NOUN
ap-961	974	11	2	2	NUM
ap-961	974	12	,	,	PUNCT
ap-961	975	1	p.	p.	NOUN
ap-961	975	2	251–289	251–289	NUM
ap-961	975	3	.	.	PUNCT
ap-961	976	1	[	[	X
ap-961	976	2	34	34	NUM
ap-961	976	3	]	]	X
ap-961	976	4	doplicher	doplicher	NOUN
ap-961	976	5	,	,	PUNCT
ap-961	976	6	s.	s.	PROPN
ap-961	976	7	,	,	PUNCT
ap-961	976	8	fredenhagen	fredenhagen	PROPN
ap-961	976	9	,	,	PUNCT
ap-961	976	10	k.	k.	PROPN
ap-961	976	11	,	,	PUNCT
ap-961	976	12	roberts	roberts	PROPN
ap-961	976	13	,	,	PUNCT
ap-961	976	14	j.	j.	PROPN
ap-961	976	15	:	:	PUNCT
ap-961	976	16	the	the	DET
ap-961	976	17	quantum	quantum	ADJ
ap-961	976	18	structure	structure	NOUN
ap-961	976	19	of	of	ADP
ap-961	976	20	spacetime	spacetime	NOUN
ap-961	976	21	at	at	ADP
ap-961	976	22	the	the	DET
ap-961	976	23	planck	planck	NOUN
ap-961	976	24	scale	scale	NOUN
ap-961	976	25	and	and	CCONJ
ap-961	976	26	quantum	quantum	NOUN
ap-961	976	27	fields	field	NOUN
ap-961	976	28	.	.	PUNCT
ap-961	977	1	comm	comm	NOUN
ap-961	977	2	.	.	PUNCT
ap-961	977	3	math	math	NOUN
ap-961	977	4	.	.	PUNCT
ap-961	978	1	phys	phy	NOUN
ap-961	978	2	.	.	PUNCT
ap-961	978	3	,	,	PUNCT
ap-961	978	4	vol	vol	NOUN
ap-961	978	5	.	.	PUNCT
ap-961	978	6	172	172	NUM
ap-961	978	7	(	(	PUNCT
ap-961	978	8	1995	1995	NUM
ap-961	978	9	)	)	PUNCT
ap-961	978	10	,	,	PUNCT
ap-961	978	11	no	no	INTJ
ap-961	978	12	.	.	NOUN
ap-961	978	13	1	1	NUM
ap-961	978	14	,	,	PUNCT
ap-961	978	15	p.	p.	NOUN
ap-961	978	16	187–220	187–220	NUM
ap-961	978	17	.	.	PUNCT
ap-961	979	1	[	[	X
ap-961	979	2	35	35	NUM
ap-961	979	3	]	]	X
ap-961	979	4	dubois	dubois	PROPN
ap-961	979	5	-	-	PUNCT
ap-961	979	6	violette	violette	NOUN
ap-961	979	7	,	,	PUNCT
ap-961	979	8	m.	m.	NOUN
ap-961	979	9	:	:	PUNCT
ap-961	979	10	lectures	lecture	NOUN
ap-961	979	11	on	on	ADP
ap-961	979	12	graded	grade	VERB
ap-961	979	13	differential	differential	ADJ
ap-961	979	14	algebras	algebra	NOUN
ap-961	979	15	and	and	CCONJ
ap-961	979	16	noncommutative	noncommutative	ADJ
ap-961	979	17	geometry	geometry	NOUN
ap-961	979	18	.	.	PUNCT
ap-961	980	1	arxiv	arxiv	NOUN
ap-961	980	2	:	:	PUNCT
ap-961	980	3	math.qa/9912017	math.qa/9912017	X
ap-961	980	4	.	.	PUNCT
ap-961	981	1	[	[	X
ap-961	981	2	36	36	NUM
ap-961	981	3	]	]	X
ap-961	981	4	dubois	dubois	PROPN
ap-961	981	5	-	-	PUNCT
ap-961	981	6	violette	violette	NOUN
ap-961	981	7	,	,	PUNCT
ap-961	981	8	m.	m.	NOUN
ap-961	981	9	:	:	PUNCT
ap-961	981	10	lectures	lecture	NOUN
ap-961	981	11	on	on	ADP
ap-961	981	12	differentials	differential	NOUN
ap-961	981	13	,	,	PUNCT
ap-961	981	14	generalized	generalized	ADJ
ap-961	981	15	differentials	differential	NOUN
ap-961	981	16	and	and	CCONJ
ap-961	981	17	on	on	ADP
ap-961	981	18	some	some	DET
ap-961	981	19	examples	example	NOUN
ap-961	981	20	related	relate	VERB
ap-961	981	21	to	to	ADP
ap-961	981	22	theoretical	theoretical	ADJ
ap-961	981	23	physics	physics	NOUN
ap-961	981	24	.	.	PUNCT
ap-961	982	1	arxiv	arxiv	PROPN
ap-961	982	2	:	:	PUNCT
ap-961	982	3	math.qa/0005256	math.qa/0005256	ADJ
ap-961	982	4	.	.	PUNCT
ap-961	983	1	[	[	X
ap-961	983	2	37	37	NUM
ap-961	983	3	]	]	X
ap-961	983	4	gracia	gracia	PROPN
ap-961	983	5	-	-	PUNCT
ap-961	983	6	bondia	bondia	PROPN
ap-961	983	7	,	,	PUNCT
ap-961	983	8	j.	j.	PROPN
ap-961	983	9	m.	m.	PROPN
ap-961	983	10	:	:	PUNCT
ap-961	983	11	noncommutative	noncommutative	ADJ
ap-961	983	12	geometry	geometry	NOUN
ap-961	983	13	and	and	CCONJ
ap-961	983	14	the	the	DET
ap-961	983	15	standard	standard	ADJ
ap-961	983	16	model	model	NOUN
ap-961	983	17	:	:	PUNCT
ap-961	983	18	an	an	DET
ap-961	983	19	overview	overview	NOUN
ap-961	983	20	.	.	PUNCT
ap-961	984	1	arxiv	arxiv	NOUN
ap-961	984	2	:	:	PUNCT
ap-961	984	3	hep	hep	NOUN
ap-961	984	4	-	-	SYM
ap-961	984	5	th/9602134	th/9602134	PROPN
ap-961	984	6	.	.	PUNCT
ap-961	985	1	[	[	X
ap-961	985	2	38	38	NUM
ap-961	985	3	]	]	X
ap-961	985	4	kalau	kalau	PROPN
ap-961	985	5	,	,	PUNCT
ap-961	985	6	w.	w.	PROPN
ap-961	985	7	,	,	PUNCT
ap-961	985	8	walze	walze	PROPN
ap-961	985	9	,	,	PUNCT
ap-961	985	10	m.	m.	NOUN
ap-961	985	11	:	:	PUNCT
ap-961	985	12	gravity	gravity	NOUN
ap-961	985	13	,	,	PUNCT
ap-961	985	14	non	non	ADJ
ap-961	985	15	-	-	ADJ
ap-961	985	16	commutative	commutative	ADJ
ap-961	985	17	geometry	geometry	NOUN
ap-961	985	18	and	and	CCONJ
ap-961	985	19	the	the	DET
ap-961	985	20	wodzicki	wodzicki	PROPN
ap-961	985	21	residue	residue	NOUN
ap-961	985	22	.	.	PUNCT
ap-961	986	1	j.	j.	PROPN
ap-961	986	2	geom	geom	PROPN
ap-961	986	3	.	.	PUNCT
ap-961	987	1	phys	phy	NOUN
ap-961	987	2	.	.	PUNCT
ap-961	987	3	,	,	PUNCT
ap-961	987	4	vol	vol	NOUN
ap-961	987	5	.	.	PUNCT
ap-961	987	6	16	16	NUM
ap-961	987	7	(	(	PUNCT
ap-961	987	8	1995	1995	NUM
ap-961	987	9	)	)	PUNCT
ap-961	987	10	,	,	PUNCT
ap-961	987	11	no	no	INTJ
ap-961	987	12	.	.	NOUN
ap-961	987	13	4	4	NUM
ap-961	987	14	,	,	PUNCT
ap-961	987	15	p.	p.	NOUN
ap-961	987	16	327–344	327–344	NUM
ap-961	987	17	.	.	PUNCT
ap-961	988	1	[	[	X
ap-961	988	2	39	39	NUM
ap-961	988	3	]	]	SYM
ap-961	988	4	kastler	kastler	NOUN
ap-961	988	5	,	,	PUNCT
ap-961	988	6	d.	d.	PROPN
ap-961	988	7	:	:	PUNCT
ap-961	988	8	the	the	DET
ap-961	988	9	dirac	dirac	NOUN
ap-961	988	10	operator	operator	NOUN
ap-961	988	11	and	and	CCONJ
ap-961	988	12	gravitation	gravitation	NOUN
ap-961	988	13	.	.	PUNCT
ap-961	989	1	commun	commun	PROPN
ap-961	989	2	.	.	PUNCT
ap-961	990	1	math	math	NOUN
ap-961	990	2	.	.	PUNCT
ap-961	991	1	phys	phy	NOUN
ap-961	991	2	.	.	PUNCT
ap-961	991	3	,	,	PUNCT
ap-961	991	4	vol	vol	NOUN
ap-961	991	5	.	.	PROPN
ap-961	991	6	166	166	NUM
ap-961	991	7	(	(	PUNCT
ap-961	991	8	1995	1995	NUM
ap-961	991	9	)	)	PUNCT
ap-961	991	10	,	,	PUNCT
ap-961	991	11	p.	p.	NOUN
ap-961	991	12	633	633	NUM
ap-961	991	13	.	.	PUNCT
ap-961	992	1	[	[	X
ap-961	992	2	40	40	NUM
ap-961	992	3	]	]	X
ap-961	992	4	kastler	kastler	PROPN
ap-961	992	5	,	,	PUNCT
ap-961	992	6	d.	d.	PROPN
ap-961	992	7	:	:	PUNCT
ap-961	992	8	noncommutative	noncommutative	ADJ
ap-961	992	9	geometry	geometry	NOUN
ap-961	992	10	and	and	CCONJ
ap-961	992	11	fundamental	fundamental	ADJ
ap-961	992	12	physical	physical	ADJ
ap-961	992	13	interactions	interaction	NOUN
ap-961	992	14	:	:	PUNCT
ap-961	992	15	the	the	DET
ap-961	992	16	lagrangian	lagrangian	ADJ
ap-961	992	17	level	level	NOUN
ap-961	992	18	:	:	PUNCT
ap-961	992	19	historical	historical	ADJ
ap-961	992	20	sketch	sketch	NOUN
ap-961	992	21	and	and	CCONJ
ap-961	992	22	description	description	NOUN
ap-961	992	23	of	of	ADP
ap-961	992	24	the	the	DET
ap-961	992	25	present	present	ADJ
ap-961	992	26	situation	situation	NOUN
ap-961	992	27	.	.	PUNCT
ap-961	993	1	j.	j.	PROPN
ap-961	993	2	math	math	PROPN
ap-961	993	3	.	.	PUNCT
ap-961	994	1	phys	phy	NOUN
ap-961	994	2	.	.	PUNCT
ap-961	994	3	,	,	PUNCT
ap-961	994	4	vol	vol	NOUN
ap-961	994	5	.	.	PROPN
ap-961	994	6	41	41	NUM
ap-961	994	7	(	(	PUNCT
ap-961	994	8	2000	2000	NUM
ap-961	994	9	)	)	PUNCT
ap-961	994	10	,	,	PUNCT
ap-961	994	11	p.	p.	NOUN
ap-961	994	12	3867	3867	NUM
ap-961	994	13	.	.	PUNCT
ap-961	995	1	[	[	X
ap-961	995	2	41	41	NUM
ap-961	995	3	]	]	X
ap-961	995	4	sakharov	sakharov	NOUN
ap-961	995	5	,	,	PUNCT
ap-961	995	6	a.	a.	NOUN
ap-961	995	7	:	:	PUNCT
ap-961	995	8	vacuum	vacuum	NOUN
ap-961	995	9	quantum	quantum	NOUN
ap-961	995	10	fluctuations	fluctuation	NOUN
ap-961	995	11	in	in	ADP
ap-961	995	12	curved	curved	ADJ
ap-961	995	13	space	space	NOUN
ap-961	995	14	and	and	CCONJ
ap-961	995	15	the	the	DET
ap-961	995	16	theory	theory	NOUN
ap-961	995	17	of	of	ADP
ap-961	995	18	gravitation	gravitation	NOUN
ap-961	995	19	.	.	PUNCT
ap-961	996	1	dokl	dokl	NOUN
ap-961	996	2	.	.	PUNCT
ap-961	997	1	akad	akad	PROPN
ap-961	997	2	.	.	PUNCT
ap-961	998	1	nauk	nauk	PROPN
ap-961	998	2	ser	ser	PROPN
ap-961	998	3	.	.	PUNCT
ap-961	999	1	fiz	fiz	PROPN
ap-961	999	2	.	.	PUNCT
ap-961	1000	1	177	177	NUM
ap-961	1000	2	(	(	PUNCT
ap-961	1000	3	1967	1967	NUM
ap-961	1000	4	)	)	PUNCT
ap-961	1000	5	,	,	PUNCT
ap-961	1001	1	p.	p.	NOUN
ap-961	1001	2	70–71	70–71	NOUN
ap-961	1001	3	.	.	PUNCT
ap-961	1002	1	[	[	X
ap-961	1002	2	42	42	NUM
ap-961	1002	3	]	]	X
ap-961	1002	4	seiberg	seiberg	PROPN
ap-961	1002	5	,	,	PUNCT
ap-961	1002	6	n.	n.	PROPN
ap-961	1002	7	,	,	PUNCT
ap-961	1002	8	witten	witten	PROPN
ap-961	1002	9	,	,	PUNCT
ap-961	1002	10	e.	e.	PROPN
ap-961	1002	11	:	:	PUNCT
ap-961	1002	12	string	string	PROPN
ap-961	1002	13	theory	theory	NOUN
ap-961	1002	14	and	and	CCONJ
ap-961	1002	15	noncommutative	noncommutative	ADJ
ap-961	1002	16	geometry	geometry	NOUN
ap-961	1002	17	.	.	PUNCT
ap-961	1003	1	j.	j.	PROPN
ap-961	1003	2	high	high	PROPN
ap-961	1003	3	energy	energy	NOUN
ap-961	1003	4	phys	phy	NOUN
ap-961	1003	5	.	.	PUNCT
ap-961	1003	6	,	,	PUNCT
ap-961	1003	7	(	(	PUNCT
ap-961	1003	8	1999	1999	NUM
ap-961	1003	9	)	)	PUNCT
ap-961	1003	10	,	,	PUNCT
ap-961	1003	11	no	no	INTJ
ap-961	1003	12	.	.	NOUN
ap-961	1003	13	9	9	NUM
ap-961	1003	14	,	,	PUNCT
ap-961	1003	15	paper	paper	NOUN
ap-961	1003	16	32	32	NUM
ap-961	1003	17	.	.	PUNCT
ap-961	1004	1	[	[	X
ap-961	1004	2	43	43	NUM
ap-961	1004	3	]	]	PUNCT
ap-961	1004	4	varilly	varilly	PROPN
ap-961	1004	5	,	,	PUNCT
ap-961	1004	6	j.	j.	PROPN
ap-961	1004	7	,	,	PUNCT
ap-961	1004	8	gracia	gracia	PROPN
ap-961	1004	9	-	-	PUNCT
ap-961	1004	10	bondia	bondia	PROPN
ap-961	1004	11	,	,	PUNCT
ap-961	1004	12	j.	j.	PROPN
ap-961	1004	13	m.	m.	PROPN
ap-961	1004	14	:	:	PUNCT
ap-961	1004	15	connes	conne	NOUN
ap-961	1004	16	’	'	PUNCT
ap-961	1004	17	noncommutative	noncommutative	ADJ
ap-961	1004	18	differential	differential	ADJ
ap-961	1004	19	geometry	geometry	NOUN
ap-961	1004	20	and	and	CCONJ
ap-961	1004	21	the	the	DET
ap-961	1004	22	standard	standard	ADJ
ap-961	1004	23	model	model	NOUN
ap-961	1004	24	.	.	PUNCT
ap-961	1005	1	j.	j.	PROPN
ap-961	1005	2	geom	geom	PROPN
ap-961	1005	3	.	.	PUNCT
ap-961	1006	1	phys	phy	NOUN
ap-961	1006	2	.	.	PUNCT
ap-961	1006	3	,	,	PUNCT
ap-961	1006	4	vol	vol	NOUN
ap-961	1006	5	.	.	PROPN
ap-961	1006	6	12	12	NUM
ap-961	1006	7	(	(	PUNCT
ap-961	1006	8	1993	1993	NUM
ap-961	1006	9	)	)	PUNCT
ap-961	1006	10	,	,	PUNCT
ap-961	1006	11	p.	p.	NOUN
ap-961	1006	12	223	223	NUM
ap-961	1006	13	.	.	PUNCT
ap-961	1007	1	[	[	X
ap-961	1007	2	44	44	NUM
ap-961	1007	3	]	]	PUNCT
ap-961	1007	4	varilly	varilly	PROPN
ap-961	1007	5	,	,	PUNCT
ap-961	1007	6	j.	j.	PROPN
ap-961	1007	7	,	,	PUNCT
ap-961	1007	8	rennie	rennie	PROPN
ap-961	1007	9	,	,	PUNCT
ap-961	1007	10	a.	a.	NOUN
ap-961	1007	11	:	:	PUNCT
ap-961	1007	12	reconstruction	reconstruction	NOUN
ap-961	1007	13	of	of	ADP
ap-961	1007	14	manifolds	manifold	NOUN
ap-961	1007	15	in	in	ADP
ap-961	1007	16	noncommutative	noncommutative	ADJ
ap-961	1007	17	geometry	geometry	NOUN
ap-961	1007	18	.	.	PUNCT
ap-961	1008	1	arxiv	arxiv	PROPN
ap-961	1008	2	:	:	PUNCT
ap-961	1008	3	math/0610418	math/0610418	PROPN
ap-961	1008	4	andrzej	andrzej	PROPN
ap-961	1008	5	sitarz	sitarz	PROPN
ap-961	1008	6	e	e	NOUN
ap-961	1008	7	-	-	NOUN
ap-961	1008	8	mail	mail	NOUN
ap-961	1008	9	:	:	PUNCT
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ap-961	1008	13	physics	physics	PROPN
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ap-961	1008	15	reymonta	reymonta	VERB
ap-961	1008	16	4	4	NUM
ap-961	1008	17	30	30	NUM
ap-961	1008	18	-	-	SYM
ap-961	1008	19	059	059	NUM
ap-961	1008	20	kraków	kraków	NOUN
ap-961	1008	21	,	,	PUNCT
ap-961	1008	22	poland	poland	PROPN
ap-961	1008	23	©	©	PROPN
ap-961	1008	24	czech	czech	PROPN
ap-961	1008	25	technical	technical	PROPN
ap-961	1008	26	university	university	PROPN
ap-961	1008	27	publishing	publishing	NOUN
ap-961	1008	28	house	house	NOUN
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ap-961	1008	30	57	57	NUM
ap-961	1008	31	acta	acta	PROPN
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ap-961	1008	33	vol	vol	NOUN
ap-961	1008	34	.	.	PUNCT
ap-961	1009	1	48	48	NUM
ap-961	1009	2	no	no	NOUN
ap-961	1009	3	.	.	PUNCT
ap-961	1010	1	2/2008	2/2008	NUM
ap-961	1010	2	<	<	X
ap-961	1010	3	<	<	X
ap-961	1010	4	/ascii85encodepages	/ascii85encodepage	NOUN
ap-961	1010	5	false	false	ADJ
ap-961	1010	6	/allowtransparency	/allowtransparency	NOUN
ap-961	1010	7	false	false	ADJ
ap-961	1010	8	/autopositionepsfiles	/autopositionepsfile	NOUN
ap-961	1010	9	true	true	ADJ
ap-961	1010	10	/autorotatepages	/autorotatepage	NOUN
ap-961	1010	11	/none	/none	PUNCT
ap-961	1010	12	/binding	/binding	PUNCT
ap-961	1011	1	/left	/left	ADJ
ap-961	1012	1	/calgrayprofile	/calgrayprofile	VERB
ap-961	1012	2	(	(	PUNCT
ap-961	1012	3	dot	dot	NOUN
ap-961	1012	4	gain	gain	NOUN
ap-961	1012	5	20	20	NUM
ap-961	1012	6	%	%	NOUN
ap-961	1012	7	)	)	PUNCT
ap-961	1012	8	/calrgbprofile	/calrgbprofile	PUNCT
ap-961	1013	1	(	(	PUNCT
ap-961	1013	2	srgb	srgb	PROPN
ap-961	1013	3	iec61966	iec61966	NOUN
ap-961	1013	4	-	-	PUNCT
ap-961	1013	5	2.1	2.1	NUM
ap-961	1013	6	)	)	PUNCT
ap-961	1013	7	/calcmykprofile	/calcmykprofile	PUNCT
ap-961	1014	1	(	(	PUNCT
ap-961	1014	2	u.s	u.s	PROPN
ap-961	1014	3	.	.	PROPN
ap-961	1014	4	web	web	NOUN
ap-961	1014	5	coated	coat	VERB
ap-961	1014	6	\050swop\051	\050swop\051	ADJ
ap-961	1014	7	v2	v2	NOUN
ap-961	1014	8	)	)	PUNCT
ap-961	1014	9	/srgbprofile	/srgbprofile	PUNCT
ap-961	1014	10	(	(	PUNCT
ap-961	1014	11	srgb	srgb	PROPN
ap-961	1014	12	iec61966	iec61966	NOUN
ap-961	1014	13	-	-	PUNCT
ap-961	1014	14	2.1	2.1	NUM
ap-961	1014	15	)	)	PUNCT
ap-961	1014	16	/cannotembedfontpolicy	/cannotembedfontpolicy	PUNCT
ap-961	1015	1	/error	/error	INTJ
ap-961	1015	2	/compatibilitylevel	/compatibilitylevel	NOUN
ap-961	1016	1	1.4	1.4	NUM
ap-961	1016	2	/compressobjects	/compressobject	NOUN
ap-961	1016	3	/tags	/tag	VERB
ap-961	1016	4	/compresspages	/compresspage	NOUN
ap-961	1016	5	true	true	ADJ
ap-961	1016	6	/convertimagestoindexed	/convertimagestoindexed	ADJ
ap-961	1016	7	true	true	ADJ
ap-961	1016	8	/passthroughjpegimages	/passthroughjpegimage	NOUN
ap-961	1016	9	true	true	ADJ
ap-961	1016	10	/createjobticket	/createjobticket	NOUN
ap-961	1016	11	false	false	ADJ
ap-961	1016	12	/defaultrenderingintent	/defaultrenderingintent	NOUN
ap-961	1017	1	/default	/default	PUNCT
ap-961	1017	2	/detectblends	/detectblend	NOUN
ap-961	1018	1	false	false	ADJ
ap-961	1018	2	/detectcurves	/detectcurve	NOUN
ap-961	1018	3	0.0000	0.0000	NUM
ap-961	1018	4	/colorconversionstrategy	/colorconversionstrategy	NOUN
ap-961	1018	5	/cmyk	/cmyk	PUNCT
ap-961	1018	6	/dothumbnails	/dothumbnail	VERB
ap-961	1019	1	false	false	ADJ
ap-961	1019	2	/embedallfonts	/embedallfont	NOUN
ap-961	1019	3	true	true	ADJ
ap-961	1019	4	/embedopentype	/embedopentype	PUNCT
ap-961	1019	5	false	false	ADJ
ap-961	1019	6	/parseiccprofilesincomments	/parseiccprofilesincomment	NOUN
ap-961	1019	7	true	true	ADJ
ap-961	1019	8	/embedjoboptions	/embedjoboption	NOUN
ap-961	1019	9	true	true	ADJ
ap-961	1019	10	/dscreportinglevel	/dscreportinglevel	NOUN
ap-961	1019	11	0	0	NUM
ap-961	1019	12	/emitdscwarnings	/emitdscwarning	NOUN
ap-961	1019	13	false	false	ADJ
ap-961	1019	14	/endpage	/endpage	NOUN
ap-961	1019	15	-1	-1	NOUN
ap-961	1019	16	/imagememory	/imagememory	NUM
ap-961	1019	17	1048576	1048576	NUM
ap-961	1019	18	/lockdistillerparams	/lockdistillerparams	NOUN
ap-961	1019	19	false	false	ADJ
ap-961	1019	20	/maxsubsetpct	/maxsubsetpct	ADJ
ap-961	1019	21	100	100	NUM
ap-961	1019	22	/optimize	/optimize	NOUN
ap-961	1019	23	true	true	ADJ
ap-961	1019	24	/opm	/opm	NOUN
ap-961	1019	25	1	1	NUM
ap-961	1019	26	/parsedsccomments	/parsedsccomment	NOUN
ap-961	1019	27	true	true	ADJ
ap-961	1019	28	/parsedsccommentsfordocinfo	/parsedsccommentsfordocinfo	VERB
ap-961	1019	29	true	true	ADJ
ap-961	1019	30	/preservecopypage	/preservecopypage	NOUN
ap-961	1019	31	true	true	ADJ
ap-961	1019	32	/preservedicmykvalues	/preservedicmykvalue	NOUN
ap-961	1019	33	true	true	ADJ
ap-961	1019	34	/preserveepsinfo	/preserveepsinfo	NOUN
ap-961	1019	35	true	true	ADJ
ap-961	1019	36	/preserveflatness	/preserveflatness	NOUN
ap-961	1019	37	true	true	ADJ
ap-961	1019	38	/preservehalftoneinfo	/preservehalftoneinfo	PUNCT
ap-961	1019	39	false	false	ADJ
ap-961	1019	40	/preserveopicomments	/preserveopicomment	NOUN
ap-961	1019	41	true	true	ADJ
ap-961	1019	42	/preserveoverprintsettings	/preserveoverprintsetting	NOUN
ap-961	1019	43	true	true	ADJ
ap-961	1019	44	/startpage	/startpage	NOUN
ap-961	1019	45	1	1	NUM
ap-961	1019	46	/subsetfonts	/subsetfont	NOUN
ap-961	1019	47	true	true	ADJ
ap-961	1019	48	/transferfunctioninfo	/transferfunctioninfo	PUNCT
ap-961	1019	49	/apply	/apply	PUNCT
ap-961	1019	50	/ucrandbginfo	/ucrandbginfo	PUNCT
ap-961	1020	1	/preserve	/preserve	NOUN
ap-961	1020	2	/useprologue	/useprologue	NOUN
ap-961	1020	3	false	false	ADJ
ap-961	1020	4	/colorsettingsfile	/colorsettingsfile	NOUN
ap-961	1020	5	(	(	PUNCT
ap-961	1020	6	)	)	PUNCT
ap-961	1020	7	/alwaysembed	/alwaysembe	VERB
ap-961	1020	8	[	[	PUNCT
ap-961	1020	9	true	true	ADJ
ap-961	1020	10	]	]	PUNCT
ap-961	1020	11	/neverembed	/neverembed	PUNCT
ap-961	1021	1	[	[	PUNCT
ap-961	1021	2	true	true	ADJ
ap-961	1021	3	]	]	PUNCT
ap-961	1021	4	/antialiascolorimages	/antialiascolorimages	X
ap-961	1021	5	false	false	ADJ
ap-961	1021	6	/cropcolorimages	/cropcolorimages	PUNCT
ap-961	1021	7	true	true	ADJ
ap-961	1021	8	/colorimageminresolution	/colorimageminresolution	PUNCT
ap-961	1021	9	300	300	NUM
ap-961	1021	10	/colorimageminresolutionpolicy	/colorimageminresolutionpolicy	NOUN
ap-961	1021	11	/ok	/ok	ADJ
ap-961	1022	1	/downsamplecolorimages	/downsamplecolorimage	VERB
ap-961	1023	1	true	true	ADJ
ap-961	1023	2	/colorimagedownsampletype	/colorimagedownsampletype	PUNCT
ap-961	1023	3	/bicubic	/bicubic	NOUN
ap-961	1023	4	/colorimageresolution	/colorimageresolution	NOUN
ap-961	1024	1	300	300	NUM
ap-961	1024	2	/colorimagedepth	/colorimagedepth	PRON
ap-961	1024	3	-1	-1	INTJ
ap-961	1024	4	/colorimagemindownsampledepth	/colorimagemindownsampledepth	SYM
ap-961	1024	5	1	1	NUM
ap-961	1024	6	/colorimagedownsamplethreshold	/colorimagedownsamplethreshold	SYM
ap-961	1024	7	1.50000	1.50000	NUM
ap-961	1024	8	/encodecolorimages	/encodecolorimage	NOUN
ap-961	1024	9	true	true	ADJ
ap-961	1024	10	/colorimagefilter	/colorimagefilter	PUNCT
ap-961	1024	11	/dctencode	/dctencode	PUNCT
ap-961	1025	1	/autofiltercolorimages	/autofiltercolorimage	NOUN
ap-961	1025	2	true	true	ADJ
ap-961	1025	3	/colorimageautofilterstrategy	/colorimageautofilterstrategy	PUNCT
ap-961	1025	4	/jpeg	/jpeg	PUNCT
ap-961	1025	5	/coloracsimagedict	/coloracsimagedict	PUNCT
ap-961	1026	1	<	<	X
ap-961	1026	2	<	<	X
ap-961	1026	3	/qfactor	/qfactor	NOUN
ap-961	1026	4	0.15	0.15	NUM
ap-961	1026	5	/hsamples	/hsample	NOUN
ap-961	1027	1	[	[	X
ap-961	1027	2	1	1	NUM
ap-961	1027	3	1	1	NUM
ap-961	1027	4	1	1	NUM
ap-961	1027	5	1	1	NUM
ap-961	1027	6	]	]	PUNCT
ap-961	1027	7	/vsamples	/vsample	NOUN
ap-961	1028	1	[	[	X
ap-961	1028	2	1	1	NUM
ap-961	1028	3	1	1	NUM
ap-961	1028	4	1	1	NUM
ap-961	1028	5	1	1	NUM
ap-961	1028	6	]	]	PUNCT
ap-961	1028	7	>	>	PUNCT
ap-961	1028	8	>	>	X
ap-961	1028	9	/colorimagedict	/colorimagedict	PUNCT
ap-961	1029	1	<	<	X
ap-961	1029	2	<	<	X
ap-961	1029	3	/qfactor	/qfactor	NOUN
ap-961	1029	4	0.15	0.15	NUM
ap-961	1029	5	/hsamples	/hsample	NOUN
ap-961	1030	1	[	[	X
ap-961	1030	2	1	1	NUM
ap-961	1030	3	1	1	NUM
ap-961	1030	4	1	1	NUM
ap-961	1030	5	1	1	NUM
ap-961	1030	6	]	]	PUNCT
ap-961	1030	7	/vsamples	/vsample	NOUN
ap-961	1031	1	[	[	X
ap-961	1031	2	1	1	NUM
ap-961	1031	3	1	1	NUM
ap-961	1031	4	1	1	NUM
ap-961	1031	5	1	1	NUM
ap-961	1031	6	]	]	PUNCT
ap-961	1031	7	>	>	PUNCT
ap-961	1031	8	>	>	X
ap-961	1031	9	/jpeg2000coloracsimagedict	/jpeg2000coloracsimagedict	PUNCT
ap-961	1032	1	<	<	X
ap-961	1032	2	<	<	X
ap-961	1032	3	/tilewidth	/tilewidth	PUNCT
ap-961	1032	4	256	256	NUM
ap-961	1032	5	/tileheight	/tileheight	NUM
ap-961	1032	6	256	256	NUM
ap-961	1032	7	/quality	/quality	NOUN
ap-961	1032	8	30	30	NUM
ap-961	1032	9	>	>	PUNCT
ap-961	1032	10	>	>	X
ap-961	1032	11	/jpeg2000colorimagedict	/jpeg2000colorimagedict	PUNCT
ap-961	1032	12	<	<	X
ap-961	1032	13	<	<	X
ap-961	1032	14	/tilewidth	/tilewidth	PUNCT
ap-961	1032	15	256	256	NUM
ap-961	1032	16	/tileheight	/tileheight	NUM
ap-961	1032	17	256	256	NUM
ap-961	1032	18	/quality	/quality	NOUN
ap-961	1032	19	30	30	NUM
ap-961	1032	20	>	>	PUNCT
ap-961	1032	21	>	>	X
ap-961	1032	22	/antialiasgrayimages	/antialiasgrayimages	X
ap-961	1032	23	false	false	ADJ
ap-961	1032	24	/cropgrayimages	/cropgrayimage	NOUN
ap-961	1032	25	true	true	ADJ
ap-961	1032	26	/grayimageminresolution	/grayimageminresolution	ADV
ap-961	1032	27	300	300	NUM
ap-961	1032	28	/grayimageminresolutionpolicy	/grayimageminresolutionpolicy	NOUN
ap-961	1032	29	/ok	/ok	ADJ
ap-961	1032	30	/downsamplegrayimages	/downsamplegrayimages	PUNCT
ap-961	1033	1	true	true	ADJ
ap-961	1033	2	/grayimagedownsampletype	/grayimagedownsampletype	INTJ
ap-961	1033	3	/bicubic	/bicubic	NOUN
ap-961	1033	4	/grayimageresolution	/grayimageresolution	NOUN
ap-961	1033	5	300	300	NUM
ap-961	1033	6	/grayimagedepth	/grayimagedepth	PUNCT
ap-961	1033	7	-1	-1	PUNCT
ap-961	1033	8	/grayimagemindownsampledepth	/grayimagemindownsampledepth	PUNCT
ap-961	1034	1	2	2	NUM
ap-961	1034	2	/grayimagedownsamplethreshold	/grayimagedownsamplethreshold	NUM
ap-961	1034	3	1.50000	1.50000	NUM
ap-961	1034	4	/encodegrayimages	/encodegrayimage	NOUN
ap-961	1034	5	true	true	ADJ
ap-961	1034	6	/grayimagefilter	/grayimagefilter	NOUN
ap-961	1034	7	/dctencode	/dctencode	PUNCT
ap-961	1035	1	/autofiltergrayimages	/autofiltergrayimages	VERB
ap-961	1035	2	true	true	ADJ
ap-961	1035	3	/grayimageautofilterstrategy	/grayimageautofilterstrategy	INTJ
ap-961	1035	4	/jpeg	/jpeg	PUNCT
ap-961	1035	5	/grayacsimagedict	/grayacsimagedict	PUNCT
ap-961	1036	1	<	<	X
ap-961	1036	2	<	<	X
ap-961	1036	3	/qfactor	/qfactor	NOUN
ap-961	1036	4	0.15	0.15	NUM
ap-961	1036	5	/hsamples	/hsample	NOUN
ap-961	1037	1	[	[	X
ap-961	1037	2	1	1	NUM
ap-961	1037	3	1	1	NUM
ap-961	1037	4	1	1	NUM
ap-961	1037	5	1	1	NUM
ap-961	1037	6	]	]	PUNCT
ap-961	1037	7	/vsamples	/vsample	NOUN
ap-961	1038	1	[	[	X
ap-961	1038	2	1	1	NUM
ap-961	1038	3	1	1	NUM
ap-961	1038	4	1	1	NUM
ap-961	1038	5	1	1	NUM
ap-961	1038	6	]	]	PUNCT
ap-961	1038	7	>	>	PUNCT
ap-961	1038	8	>	>	X
ap-961	1038	9	/grayimagedict	/grayimagedict	PUNCT
ap-961	1039	1	<	<	X
ap-961	1039	2	<	<	X
ap-961	1039	3	/qfactor	/qfactor	NOUN
ap-961	1039	4	0.15	0.15	NUM
ap-961	1039	5	/hsamples	/hsample	NOUN
ap-961	1040	1	[	[	X
ap-961	1040	2	1	1	NUM
ap-961	1040	3	1	1	NUM
ap-961	1040	4	1	1	NUM
ap-961	1040	5	1	1	NUM
ap-961	1040	6	]	]	PUNCT
ap-961	1040	7	/vsamples	/vsample	NOUN
ap-961	1041	1	[	[	X
ap-961	1041	2	1	1	NUM
ap-961	1041	3	1	1	NUM
ap-961	1041	4	1	1	NUM
ap-961	1041	5	1	1	NUM
ap-961	1041	6	]	]	PUNCT
ap-961	1041	7	>	>	X
ap-961	1041	8	>	>	X
ap-961	1041	9	/jpeg2000grayacsimagedict	/jpeg2000grayacsimagedict	PUNCT
ap-961	1042	1	<	<	X
ap-961	1042	2	<	<	X
ap-961	1042	3	/tilewidth	/tilewidth	PUNCT
ap-961	1042	4	256	256	NUM
ap-961	1042	5	/tileheight	/tileheight	NUM
ap-961	1042	6	256	256	NUM
ap-961	1042	7	/quality	/quality	NOUN
ap-961	1042	8	30	30	NUM
ap-961	1042	9	>	>	PUNCT
ap-961	1042	10	>	>	X
ap-961	1042	11	/jpeg2000grayimagedict	/jpeg2000grayimagedict	PUNCT
ap-961	1042	12	<	<	X
ap-961	1042	13	<	<	X
ap-961	1042	14	/tilewidth	/tilewidth	PUNCT
ap-961	1042	15	256	256	NUM
ap-961	1042	16	/tileheight	/tileheight	NUM
ap-961	1042	17	256	256	NUM
ap-961	1042	18	/quality	/quality	NOUN
ap-961	1042	19	30	30	NUM
ap-961	1042	20	>	>	PUNCT
ap-961	1042	21	>	>	X
ap-961	1042	22	/antialiasmonoimages	/antialiasmonoimage	NOUN
ap-961	1042	23	false	false	ADJ
ap-961	1042	24	/cropmonoimages	/cropmonoimage	NOUN
ap-961	1042	25	true	true	ADJ
ap-961	1042	26	/monoimageminresolution	/monoimageminresolution	NUM
ap-961	1042	27	1200	1200	NUM
ap-961	1042	28	/monoimageminresolutionpolicy	/monoimageminresolutionpolicy	PUNCT
ap-961	1043	1	/ok	/ok	PROPN
ap-961	1043	2	/downsamplemonoimages	/downsamplemonoimage	VERB
ap-961	1044	1	true	true	ADJ
ap-961	1044	2	/monoimagedownsampletype	/monoimagedownsampletype	PUNCT
ap-961	1044	3	/bicubic	/bicubic	NOUN
ap-961	1044	4	/monoimageresolution	/monoimageresolution	NOUN
ap-961	1044	5	1200	1200	NUM
ap-961	1044	6	/monoimagedepth	/monoimagedepth	PRON
ap-961	1044	7	-1	-1	PUNCT
ap-961	1045	1	/monoimagedownsamplethreshold	/monoimagedownsamplethreshold	PUNCT
ap-961	1045	2	1.50000	1.50000	NUM
ap-961	1045	3	/encodemonoimages	/encodemonoimage	NOUN
ap-961	1045	4	true	true	ADJ
ap-961	1045	5	/monoimagefilter	/monoimagefilter	ADP
ap-961	1045	6	/ccittfaxencode	/ccittfaxencode	INTJ
ap-961	1045	7	/monoimagedict	/monoimagedict	PUNCT
ap-961	1046	1	<	<	X
ap-961	1046	2	<	<	X
ap-961	1046	3	/k	/k	PUNCT
ap-961	1046	4	-1	-1	INTJ
ap-961	1046	5	>	>	PUNCT
ap-961	1046	6	>	>	X
ap-961	1046	7	/allowpsxobjects	/allowpsxobjects	PROPN
ap-961	1046	8	false	false	ADJ
ap-961	1046	9	/checkcompliance	/checkcompliance	NOUN
ap-961	1046	10	[	[	PUNCT
ap-961	1046	11	/none	/none	X
ap-961	1046	12	]	]	PUNCT
ap-961	1046	13	/pdfx1acheck	/pdfx1acheck	PUNCT
ap-961	1046	14	false	false	ADJ
ap-961	1046	15	/pdfx3check	/pdfx3check	PUNCT
ap-961	1046	16	false	false	ADJ
ap-961	1046	17	/pdfxcompliantpdfonly	/pdfxcompliantpdfonly	ADV
ap-961	1046	18	false	false	ADJ
ap-961	1046	19	/pdfxnotrimboxerror	/pdfxnotrimboxerror	NOUN
ap-961	1046	20	true	true	ADJ
ap-961	1046	21	/pdfxtrimboxtomediaboxoffset	/pdfxtrimboxtomediaboxoffset	NOUN
ap-961	1046	22	[	[	PUNCT
ap-961	1046	23	0.00000	0.00000	NUM
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ap-961	1046	28	/pdfxsetbleedboxtomediabox	/pdfxsetbleedboxtomediabox	PUNCT
ap-961	1047	1	true	true	ADJ
ap-961	1047	2	/pdfxbleedboxtotrimboxoffset	/pdfxbleedboxtotrimboxoffset	PUNCT
ap-961	1047	3	[	[	PUNCT
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ap-961	1047	8	]	]	PUNCT
ap-961	1047	9	/pdfxoutputintentprofile	/pdfxoutputintentprofile	PUNCT
ap-961	1048	1	(	(	PUNCT
ap-961	1048	2	none	none	NOUN
ap-961	1048	3	)	)	PUNCT
ap-961	1048	4	/pdfxoutputconditionidentifier	/pdfxoutputconditionidentifi	ADJ
ap-961	1048	5	(	(	PUNCT
ap-961	1048	6	)	)	PUNCT
ap-961	1048	7	/pdfxoutputcondition	/pdfxoutputcondition	PRON
ap-961	1048	8	(	(	PUNCT
ap-961	1048	9	)	)	PUNCT
ap-961	1048	10	/pdfxregistryname	/pdfxregistryname	PUNCT
ap-961	1048	11	(	(	PUNCT
ap-961	1048	12	)	)	PUNCT
ap-961	1048	13	/pdfxtrapped	/pdfxtrapped	PUNCT
ap-961	1048	14	/false	/false	INTJ
ap-961	1049	1	/createjdffile	/createjdffile	INTJ
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ap-961	1049	3	/description	/description	NOUN
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ap-961	1050	6	>	>	X
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ap-961	1054	11	>	>	X
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ap-961	1055	9	visokokvalitetni	visokokvalitetni	VERB
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ap-961	1081	10	false	false	ADJ
ap-961	1081	11	/addregmarks	/addregmark	NOUN
ap-961	1081	12	false	false	ADJ
ap-961	1081	13	/convertcolors	/convertcolor	NOUN
ap-961	1081	14	/converttocmyk	/converttocmyk	INTJ
ap-961	1081	15	/destinationprofilename	/destinationprofilename	PUNCT
ap-961	1081	16	(	(	PUNCT
ap-961	1081	17	)	)	PUNCT
ap-961	1081	18	/destinationprofileselector	/destinationprofileselector	NOUN
ap-961	1081	19	/documentcmyk	/documentcmyk	PUNCT
ap-961	1081	20	/downsample16bitimages	/downsample16bitimage	NOUN
ap-961	1081	21	true	true	ADJ
ap-961	1081	22	/flattenerpreset	/flattenerpreset	PUNCT
ap-961	1082	1	<	<	X
ap-961	1082	2	<	<	X
ap-961	1082	3	/presetselector	/presetselector	X
ap-961	1082	4	/mediumresolution	/mediumresolution	NOUN
ap-961	1082	5	>	>	PUNCT
ap-961	1082	6	>	>	X
ap-961	1082	7	/formelements	/formelement	NOUN
ap-961	1083	1	false	false	ADJ
ap-961	1083	2	/generatestructure	/generatestructure	PUNCT
ap-961	1083	3	false	false	ADJ
ap-961	1083	4	/includebookmarks	/includebookmark	NOUN
ap-961	1083	5	false	false	ADJ
ap-961	1083	6	/includehyperlinks	/includehyperlinks	INTJ
ap-961	1083	7	false	false	ADJ
ap-961	1083	8	/includeinteractive	/includeinteractive	ADJ
ap-961	1083	9	false	false	ADJ
ap-961	1083	10	/includelayers	/includelayer	NOUN
ap-961	1083	11	false	false	ADJ
ap-961	1083	12	/includeprofiles	/includeprofiles	INTJ
ap-961	1083	13	false	false	ADJ
ap-961	1083	14	/multimediahandling	/multimediahandling	NOUN
ap-961	1084	1	/useobjectsettings	/useobjectsettings	PRON
ap-961	1085	1	/namespace	/namespace	PUNCT
ap-961	1086	1	[	[	PUNCT
ap-961	1086	2	(	(	PUNCT
ap-961	1086	3	adobe	adobe	PROPN
ap-961	1086	4	)	)	PUNCT
ap-961	1086	5	(	(	PUNCT
ap-961	1086	6	creativesuite	creativesuite	NOUN
ap-961	1086	7	)	)	PUNCT
ap-961	1086	8	(	(	PUNCT
ap-961	1086	9	2.0	2.0	NUM
ap-961	1086	10	)	)	PUNCT
ap-961	1086	11	]	]	PUNCT
ap-961	1087	1	/pdfxoutputintentprofileselector	/pdfxoutputintentprofileselector	INTJ
ap-961	1087	2	/documentcmyk	/documentcmyk	PUNCT
ap-961	1088	1	/preserveediting	/preserveedite	VERB
ap-961	1088	2	true	true	ADJ
ap-961	1088	3	/untaggedcmykhandling	/untaggedcmykhandling	NOUN
ap-961	1088	4	/leaveuntagged	/leaveuntagge	VERB
ap-961	1088	5	/untaggedrgbhandling	/untaggedrgbhandling	NOUN
ap-961	1088	6	/usedocumentprofile	/usedocumentprofile	INTJ
ap-961	1088	7	/usedocumentbleed	/usedocumentbleed	NOUN
ap-961	1089	1	false	false	ADJ
ap-961	1089	2	>	>	PUNCT
ap-961	1089	3	>	>	X
ap-961	1089	4	]	]	PUNCT
ap-961	1089	5	>	>	PUNCT
ap-961	1089	6	>	>	X
ap-961	1089	7	setdistillerparams	setdistillerparam	NOUN
ap-961	1090	1	<	<	X
ap-961	1090	2	<	<	X
ap-961	1090	3	/hwresolution	/hwresolution	NOUN
ap-961	1091	1	[	[	X
ap-961	1091	2	2400	2400	NUM
ap-961	1091	3	2400	2400	NUM
ap-961	1091	4	]	]	PUNCT
ap-961	1091	5	/pagesize	/pagesize	X
ap-961	1092	1	[	[	X
ap-961	1092	2	612.000	612.000	NUM
ap-961	1092	3	792.000	792.000	NUM
ap-961	1092	4	]	]	PUNCT
ap-961	1092	5	>	>	PUNCT
ap-961	1092	6	>	>	X
ap-961	1092	7	setpagedevice	setpagedevice	PROPN