id	sid	tid	token	lemma	pos
ap-1849	1	1	acta	acta	PROPN
ap-1849	1	2	polytechnica	polytechnica	PROPN
ap-1849	1	3	doi:10.14311	doi:10.14311	PROPN
ap-1849	1	4	/	/	SYM
ap-1849	1	5	ap.2013.53.0399	ap.2013.53.0399	PROPN
ap-1849	1	6	acta	acta	PROPN
ap-1849	1	7	polytechnica	polytechnica	PROPN
ap-1849	1	8	53(5):399–404	53(5):399–404	PROPN
ap-1849	1	9	,	,	PUNCT
ap-1849	1	10	2013	2013	NUM
ap-1849	1	11	©	©	PROPN
ap-1849	1	12	czech	czech	PROPN
ap-1849	1	13	technical	technical	PROPN
ap-1849	1	14	university	university	PROPN
ap-1849	1	15	in	in	ADP
ap-1849	1	16	prague	prague	PROPN
ap-1849	1	17	,	,	PUNCT
ap-1849	1	18	2013	2013	NUM
ap-1849	1	19	available	available	ADJ
ap-1849	1	20	online	online	ADV
ap-1849	1	21	at	at	ADP
ap-1849	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-1849	1	23	extremal	extremal	ADJ
ap-1849	1	24	vectors	vector	NOUN
ap-1849	1	25	for	for	ADP
ap-1849	1	26	verma	verma	NOUN
ap-1849	1	27	type	type	NOUN
ap-1849	1	28	representation	representation	NOUN
ap-1849	1	29	of	of	ADP
ap-1849	1	30	b2	b2	NOUN
ap-1849	1	31	čestmír	čestmír	NOUN
ap-1849	1	32	burdíka,∗	burdíka,∗	PROPN
ap-1849	1	33	,	,	PUNCT
ap-1849	1	34	ondřej	ondřej	NOUN
ap-1849	1	35	navrátilb	navrátilb	PROPN
ap-1849	1	36	a	a	DET
ap-1849	1	37	department	department	NOUN
ap-1849	1	38	of	of	ADP
ap-1849	1	39	mathematics	mathematic	NOUN
ap-1849	1	40	,	,	PUNCT
ap-1849	1	41	czech	czech	PROPN
ap-1849	1	42	technical	technical	PROPN
ap-1849	1	43	university	university	PROPN
ap-1849	1	44	in	in	ADP
ap-1849	1	45	prague	prague	PROPN
ap-1849	1	46	,	,	PUNCT
ap-1849	1	47	faculty	faculty	NOUN
ap-1849	1	48	of	of	ADP
ap-1849	1	49	nuclear	nuclear	ADJ
ap-1849	1	50	sciences	science	NOUN
ap-1849	1	51	and	and	CCONJ
ap-1849	1	52	physical	physical	ADJ
ap-1849	1	53	engineering	engineering	NOUN
ap-1849	1	54	,	,	PUNCT
ap-1849	1	55	trojanova	trojanova	X
ap-1849	1	56	13	13	NUM
ap-1849	1	57	,	,	PUNCT
ap-1849	1	58	120	120	NUM
ap-1849	1	59	00	00	NUM
ap-1849	1	60	prague	prague	NOUN
ap-1849	1	61	2	2	NUM
ap-1849	1	62	,	,	PUNCT
ap-1849	1	63	czech	czech	PROPN
ap-1849	1	64	republic	republic	PROPN
ap-1849	1	65	b	b	PROPN
ap-1849	1	66	department	department	PROPN
ap-1849	1	67	of	of	ADP
ap-1849	1	68	mathematics	mathematic	NOUN
ap-1849	1	69	,	,	PUNCT
ap-1849	1	70	czech	czech	PROPN
ap-1849	1	71	technical	technical	PROPN
ap-1849	1	72	university	university	PROPN
ap-1849	1	73	in	in	ADP
ap-1849	1	74	prague	prague	PROPN
ap-1849	1	75	,	,	PUNCT
ap-1849	1	76	faculty	faculty	NOUN
ap-1849	1	77	of	of	ADP
ap-1849	1	78	transportation	transportation	NOUN
ap-1849	1	79	sciences	science	NOUN
ap-1849	1	80	,	,	PUNCT
ap-1849	1	81	na	na	PART
ap-1849	1	82	florenci	florenci	VERB
ap-1849	1	83	25	25	NUM
ap-1849	1	84	,	,	PUNCT
ap-1849	1	85	110	110	NUM
ap-1849	1	86	00	00	NUM
ap-1849	1	87	prague	prague	PROPN
ap-1849	1	88	,	,	PUNCT
ap-1849	1	89	czech	czech	PROPN
ap-1849	1	90	republic	republic	NOUN
ap-1849	1	91	∗	∗	NOUN
ap-1849	1	92	corresponding	correspond	VERB
ap-1849	1	93	author	author	NOUN
ap-1849	1	94	:	:	PUNCT
ap-1849	1	95	burdices@kmlinux.fjfi.cvut.cz	burdices@kmlinux.fjfi.cvut.cz	NOUN
ap-1849	1	96	abstract	abstract	NOUN
ap-1849	1	97	.	.	PUNCT
ap-1849	2	1	starting	start	VERB
ap-1849	2	2	from	from	ADP
ap-1849	2	3	the	the	DET
ap-1849	2	4	verma	verma	PROPN
ap-1849	2	5	modules	module	NOUN
ap-1849	2	6	of	of	ADP
ap-1849	2	7	the	the	DET
ap-1849	2	8	algebra	algebra	NOUN
ap-1849	2	9	b2	b2	NOUN
ap-1849	2	10	we	we	PRON
ap-1849	2	11	explicitly	explicitly	ADV
ap-1849	2	12	construct	construct	VERB
ap-1849	2	13	factor	factor	NOUN
ap-1849	2	14	representations	representation	NOUN
ap-1849	2	15	of	of	ADP
ap-1849	2	16	the	the	DET
ap-1849	2	17	algebra	algebra	NOUN
ap-1849	2	18	b2	b2	NOUN
ap-1849	2	19	which	which	PRON
ap-1849	2	20	are	be	AUX
ap-1849	2	21	connected	connect	VERB
ap-1849	2	22	with	with	ADP
ap-1849	2	23	the	the	DET
ap-1849	2	24	unitary	unitary	ADJ
ap-1849	2	25	representation	representation	NOUN
ap-1849	2	26	of	of	ADP
ap-1849	2	27	the	the	DET
ap-1849	2	28	group	group	NOUN
ap-1849	2	29	so(3	so(3	NOUN
ap-1849	2	30	,	,	PUNCT
ap-1849	2	31	2	2	NUM
ap-1849	2	32	)	)	PUNCT
ap-1849	2	33	.	.	PUNCT
ap-1849	3	1	we	we	PRON
ap-1849	3	2	find	find	VERB
ap-1849	3	3	a	a	DET
ap-1849	3	4	full	full	ADJ
ap-1849	3	5	set	set	NOUN
ap-1849	3	6	of	of	ADP
ap-1849	3	7	extremal	extremal	ADJ
ap-1849	3	8	vectors	vector	NOUN
ap-1849	3	9	for	for	ADP
ap-1849	3	10	representations	representation	NOUN
ap-1849	3	11	of	of	ADP
ap-1849	3	12	this	this	DET
ap-1849	3	13	kind	kind	NOUN
ap-1849	3	14	.	.	PUNCT
ap-1849	4	1	so	so	ADV
ap-1849	4	2	we	we	PRON
ap-1849	4	3	can	can	AUX
ap-1849	4	4	explicitly	explicitly	ADV
ap-1849	4	5	resolve	resolve	VERB
ap-1849	4	6	the	the	DET
ap-1849	4	7	problem	problem	NOUN
ap-1849	4	8	of	of	ADP
ap-1849	4	9	the	the	DET
ap-1849	4	10	irreducibility	irreducibility	NOUN
ap-1849	4	11	of	of	ADP
ap-1849	4	12	these	these	DET
ap-1849	4	13	representations	representation	NOUN
ap-1849	4	14	.	.	PUNCT
ap-1849	5	1	keywords	keyword	NOUN
ap-1849	5	2	:	:	PUNCT
ap-1849	5	3	verma	verma	PROPN
ap-1849	5	4	modules	module	NOUN
ap-1849	5	5	,	,	PUNCT
ap-1849	5	6	height	height	NOUN
ap-1849	5	7	-	-	PUNCT
ap-1849	5	8	weight	weight	NOUN
ap-1849	5	9	representation	representation	NOUN
ap-1849	5	10	,	,	PUNCT
ap-1849	5	11	reducibility	reducibility	NOUN
ap-1849	5	12	,	,	PUNCT
ap-1849	5	13	extremal	extremal	ADJ
ap-1849	5	14	vectors	vector	NOUN
ap-1849	5	15	.	.	PUNCT
ap-1849	6	1	submitted	submit	VERB
ap-1849	6	2	:	:	PUNCT
ap-1849	6	3	16	16	NUM
ap-1849	6	4	july	july	PROPN
ap-1849	6	5	2013	2013	NUM
ap-1849	6	6	.	.	PUNCT
ap-1849	7	1	accepted	accept	VERB
ap-1849	7	2	:	:	PUNCT
ap-1849	7	3	28	28	NUM
ap-1849	7	4	july	july	NOUN
ap-1849	7	5	2013	2013	NUM
ap-1849	7	6	.	.	PUNCT
ap-1849	8	1	1	1	X
ap-1849	8	2	.	.	X
ap-1849	8	3	introduction	introduction	NOUN
ap-1849	8	4	representations	representation	NOUN
ap-1849	8	5	of	of	ADP
ap-1849	8	6	lie	lie	NOUN
ap-1849	8	7	algebras	algebra	NOUN
ap-1849	8	8	are	be	AUX
ap-1849	8	9	important	important	ADJ
ap-1849	8	10	in	in	ADP
ap-1849	8	11	many	many	ADJ
ap-1849	8	12	physical	physical	ADJ
ap-1849	8	13	models	model	NOUN
ap-1849	8	14	.	.	PUNCT
ap-1849	9	1	it	it	PRON
ap-1849	9	2	is	be	AUX
ap-1849	9	3	therefore	therefore	ADV
ap-1849	9	4	useful	useful	ADJ
ap-1849	9	5	to	to	PART
ap-1849	9	6	study	study	VERB
ap-1849	9	7	various	various	ADJ
ap-1849	9	8	methods	method	NOUN
ap-1849	9	9	for	for	ADP
ap-1849	9	10	constructing	construct	VERB
ap-1849	9	11	them	they	PRON
ap-1849	9	12	.	.	PUNCT
ap-1849	10	1	the	the	DET
ap-1849	10	2	general	general	ADJ
ap-1849	10	3	method	method	NOUN
ap-1849	10	4	of	of	ADP
ap-1849	10	5	construction	construction	NOUN
ap-1849	10	6	of	of	ADP
ap-1849	10	7	the	the	DET
ap-1849	10	8	highestweight	highestweight	PROPN
ap-1849	10	9	representation	representation	NOUN
ap-1849	10	10	for	for	ADP
ap-1849	10	11	the	the	DET
ap-1849	10	12	semisimple	semisimple	NOUN
ap-1849	10	13	lie	lie	NOUN
ap-1849	10	14	algebra	algebra	NOUN
ap-1849	10	15	was	be	AUX
ap-1849	10	16	developed	develop	VERB
ap-1849	10	17	in	in	ADP
ap-1849	10	18	[	[	X
ap-1849	10	19	1	1	NUM
ap-1849	10	20	,	,	PUNCT
ap-1849	10	21	2	2	NUM
ap-1849	10	22	]	]	PUNCT
ap-1849	10	23	.	.	PUNCT
ap-1849	11	1	the	the	DET
ap-1849	11	2	irreducibility	irreducibility	NOUN
ap-1849	11	3	of	of	ADP
ap-1849	11	4	such	such	ADJ
ap-1849	11	5	representations	representation	NOUN
ap-1849	11	6	(	(	PUNCT
ap-1849	11	7	now	now	ADV
ap-1849	11	8	called	call	VERB
ap-1849	11	9	verma	verma	PROPN
ap-1849	11	10	modules	module	NOUN
ap-1849	11	11	)	)	PUNCT
ap-1849	11	12	was	be	AUX
ap-1849	11	13	studied	study	VERB
ap-1849	11	14	by	by	ADP
ap-1849	11	15	gelfand	gelfand	PROPN
ap-1849	11	16	in	in	ADP
ap-1849	11	17	[	[	X
ap-1849	11	18	3	3	NUM
ap-1849	11	19	]	]	PUNCT
ap-1849	11	20	.	.	PUNCT
ap-1849	12	1	the	the	DET
ap-1849	12	2	theory	theory	NOUN
ap-1849	12	3	of	of	ADP
ap-1849	12	4	these	these	DET
ap-1849	12	5	representations	representation	NOUN
ap-1849	12	6	is	be	AUX
ap-1849	12	7	included	include	VERB
ap-1849	12	8	in	in	ADP
ap-1849	12	9	dixmier	dixmier	NOUN
ap-1849	12	10	’s	’s	PART
ap-1849	12	11	book	book	NOUN
ap-1849	13	1	[	[	X
ap-1849	13	2	4	4	NUM
ap-1849	13	3	]	]	PUNCT
ap-1849	13	4	.	.	PUNCT
ap-1849	14	1	in	in	ADP
ap-1849	14	2	the	the	DET
ap-1849	14	3	1970	1970	NUM
ap-1849	14	4	’s	’s	PART
ap-1849	14	5	prof	prof	NOUN
ap-1849	14	6	.	.	PUNCT
ap-1849	14	7	havlíček	havlíček	PROPN
ap-1849	14	8	with	with	ADP
ap-1849	14	9	his	his	PRON
ap-1849	14	10	coworkers	coworker	NOUN
ap-1849	14	11	dealt	deal	VERB
ap-1849	14	12	with	with	ADP
ap-1849	14	13	the	the	DET
ap-1849	14	14	construction	construction	NOUN
ap-1849	14	15	of	of	ADP
ap-1849	14	16	realizations	realization	NOUN
ap-1849	14	17	of	of	ADP
ap-1849	14	18	the	the	DET
ap-1849	14	19	classical	classical	ADJ
ap-1849	14	20	lie	lie	NOUN
ap-1849	14	21	algebras	algebras	X
ap-1849	14	22	,	,	PUNCT
ap-1849	14	23	see	see	VERB
ap-1849	14	24	[	[	X
ap-1849	14	25	5	5	NUM
ap-1849	14	26	]	]	PUNCT
ap-1849	14	27	.	.	PUNCT
ap-1849	15	1	our	our	PRON
ap-1849	15	2	aim	aim	NOUN
ap-1849	15	3	in	in	ADP
ap-1849	15	4	this	this	DET
ap-1849	15	5	paper	paper	NOUN
ap-1849	15	6	is	be	AUX
ap-1849	15	7	to	to	PART
ap-1849	15	8	show	show	VERB
ap-1849	15	9	how	how	SCONJ
ap-1849	15	10	one	one	PRON
ap-1849	15	11	can	can	AUX
ap-1849	15	12	use	use	VERB
ap-1849	15	13	realizations	realization	NOUN
ap-1849	15	14	of	of	ADP
ap-1849	15	15	the	the	DET
ap-1849	15	16	lie	lie	NOUN
ap-1849	15	17	algebra	algebra	NOUN
ap-1849	15	18	to	to	PART
ap-1849	15	19	construct	construct	VERB
ap-1849	15	20	so	so	ADV
ap-1849	15	21	called	call	VERB
ap-1849	15	22	extremal	extremal	ADJ
ap-1849	15	23	vectors	vector	NOUN
ap-1849	15	24	of	of	ADP
ap-1849	15	25	the	the	DET
ap-1849	15	26	verma	verma	PROPN
ap-1849	15	27	modules	module	NOUN
ap-1849	15	28	.	.	PUNCT
ap-1849	16	1	to	to	PART
ap-1849	16	2	work	work	VERB
ap-1849	16	3	with	with	ADP
ap-1849	16	4	a	a	DET
ap-1849	16	5	specific	specific	ADJ
ap-1849	16	6	lie	lie	NOUN
ap-1849	16	7	algebra	algebra	NOUN
ap-1849	16	8	,	,	PUNCT
ap-1849	16	9	we	we	PRON
ap-1849	16	10	choose	choose	VERB
ap-1849	16	11	lie	lie	NOUN
ap-1849	16	12	algebra	algebra	NOUN
ap-1849	16	13	so(3	so(3	NOUN
ap-1849	16	14	,	,	PUNCT
ap-1849	16	15	2	2	NUM
ap-1849	16	16	)	)	PUNCT
ap-1849	16	17	,	,	PUNCT
ap-1849	16	18	which	which	PRON
ap-1849	16	19	plays	play	VERB
ap-1849	16	20	an	an	DET
ap-1849	16	21	important	important	ADJ
ap-1849	16	22	role	role	NOUN
ap-1849	16	23	in	in	ADP
ap-1849	16	24	physics	physics	NOUN
ap-1849	16	25	,	,	PUNCT
ap-1849	16	26	e.g.	e.g.	ADV
ap-1849	16	27	in	in	ADP
ap-1849	16	28	ads	ad	NOUN
ap-1849	16	29	/	/	SYM
ap-1849	16	30	cft	cft	PROPN
ap-1849	16	31	theory	theory	NOUN
ap-1849	16	32	,	,	PUNCT
ap-1849	16	33	see	see	VERB
ap-1849	16	34	[	[	X
ap-1849	16	35	6	6	NUM
ap-1849	16	36	,	,	PUNCT
ap-1849	16	37	7	7	NUM
ap-1849	16	38	]	]	PUNCT
ap-1849	16	39	.	.	PUNCT
ap-1849	17	1	in	in	ADP
ap-1849	17	2	the	the	DET
ap-1849	17	3	construction	construction	NOUN
ap-1849	17	4	of	of	ADP
ap-1849	17	5	the	the	DET
ap-1849	17	6	verma	verma	PROPN
ap-1849	17	7	modules	module	NOUN
ap-1849	17	8	for	for	ADP
ap-1849	17	9	b2	b2	NOUN
ap-1849	17	10	,	,	PUNCT
ap-1849	17	11	the	the	DET
ap-1849	17	12	representations	representation	NOUN
ap-1849	17	13	depend	depend	VERB
ap-1849	17	14	on	on	ADP
ap-1849	17	15	parameters	parameter	NOUN
ap-1849	17	16	(	(	PUNCT
ap-1849	17	17	λ1	λ1	ADJ
ap-1849	17	18	,	,	PUNCT
ap-1849	17	19	λ2	λ2	PROPN
ap-1849	17	20	)	)	PUNCT
ap-1849	17	21	.	.	PUNCT
ap-1849	18	1	for	for	ADP
ap-1849	18	2	connection	connection	NOUN
ap-1849	18	3	with	with	ADP
ap-1849	18	4	irreducible	irreducible	ADJ
ap-1849	18	5	unitary	unitary	ADJ
ap-1849	18	6	representations	representation	NOUN
ap-1849	18	7	of	of	ADP
ap-1849	18	8	so(3	so(3	NOUN
ap-1849	18	9	,	,	PUNCT
ap-1849	18	10	2	2	NUM
ap-1849	18	11	)	)	PUNCT
ap-1849	18	12	we	we	PRON
ap-1849	18	13	take	take	VERB
ap-1849	18	14	λ2	λ2	PROPN
ap-1849	18	15	∈	∈	PROPN
ap-1849	18	16	n0	n0	NOUN
ap-1849	18	17	,	,	PUNCT
ap-1849	18	18	and	and	CCONJ
ap-1849	18	19	in	in	ADP
ap-1849	18	20	section	section	NOUN
ap-1849	18	21	3	3	NUM
ap-1849	18	22	we	we	PRON
ap-1849	18	23	explicitly	explicitly	ADV
ap-1849	18	24	construct	construct	VERB
ap-1849	18	25	the	the	DET
ap-1849	18	26	factor	factor	NOUN
ap-1849	18	27	-	-	PUNCT
ap-1849	18	28	verma	verma	NOUN
ap-1849	18	29	representation	representation	NOUN
ap-1849	18	30	.	.	PUNCT
ap-1849	19	1	further	far	ADV
ap-1849	19	2	,	,	PUNCT
ap-1849	19	3	we	we	PRON
ap-1849	19	4	construct	construct	VERB
ap-1849	19	5	a	a	DET
ap-1849	19	6	full	full	ADJ
ap-1849	19	7	set	set	NOUN
ap-1849	19	8	of	of	ADP
ap-1849	19	9	extremal	extremal	ADJ
ap-1849	19	10	vectors	vector	NOUN
ap-1849	19	11	.	.	PUNCT
ap-1849	20	1	these	these	DET
ap-1849	20	2	vectors	vector	NOUN
ap-1849	20	3	are	be	AUX
ap-1849	20	4	called	call	VERB
ap-1849	20	5	subsingular	subsingular	ADJ
ap-1849	20	6	vectors	vector	NOUN
ap-1849	20	7	in	in	ADP
ap-1849	20	8	[	[	X
ap-1849	20	9	8	8	NUM
ap-1849	20	10	]	]	PUNCT
ap-1849	20	11	.	.	PUNCT
ap-1849	21	1	in	in	ADP
ap-1849	21	2	this	this	DET
ap-1849	21	3	paper	paper	NOUN
ap-1849	21	4	,	,	PUNCT
ap-1849	21	5	we	we	PRON
ap-1849	21	6	use	use	VERB
ap-1849	21	7	an	an	DET
ap-1849	21	8	almost	almost	ADV
ap-1849	21	9	elementary	elementary	ADJ
ap-1849	21	10	partial	partial	ADJ
ap-1849	21	11	differential	differential	NOUN
ap-1849	21	12	equation	equation	NOUN
ap-1849	21	13	approach	approach	NOUN
ap-1849	21	14	to	to	PART
ap-1849	21	15	determine	determine	VERB
ap-1849	21	16	the	the	DET
ap-1849	21	17	extremal	extremal	ADJ
ap-1849	21	18	vectors	vector	NOUN
ap-1849	21	19	in	in	ADP
ap-1849	21	20	any	any	DET
ap-1849	21	21	factor	factor	NOUN
ap-1849	21	22	-	-	PUNCT
ap-1849	21	23	verma	verma	NOUN
ap-1849	21	24	module	module	NOUN
ap-1849	21	25	of	of	ADP
ap-1849	21	26	b2	b2	NOUN
ap-1849	21	27	.	.	PUNCT
ap-1849	22	1	it	it	PRON
ap-1849	22	2	should	should	AUX
ap-1849	22	3	be	be	AUX
ap-1849	22	4	noted	note	VERB
ap-1849	22	5	that	that	SCONJ
ap-1849	22	6	our	our	PRON
ap-1849	22	7	approach	approach	NOUN
ap-1849	22	8	differs	differ	VERB
ap-1849	22	9	from	from	ADP
ap-1849	22	10	a	a	DET
ap-1849	22	11	similar	similar	ADJ
ap-1849	22	12	one	one	NOUN
ap-1849	22	13	used	use	VERB
ap-1849	22	14	in	in	ADP
ap-1849	22	15	[	[	X
ap-1849	22	16	9	9	NUM
ap-1849	22	17	]	]	PUNCT
ap-1849	22	18	.	.	PUNCT
ap-1849	23	1	first	first	ADV
ap-1849	23	2	,	,	PUNCT
ap-1849	23	3	we	we	PRON
ap-1849	23	4	identify	identify	VERB
ap-1849	23	5	the	the	DET
ap-1849	23	6	factorverma	factorverma	NOUN
ap-1849	23	7	modules	module	NOUN
ap-1849	23	8	with	with	ADP
ap-1849	23	9	a	a	DET
ap-1849	23	10	space	space	NOUN
ap-1849	23	11	of	of	ADP
ap-1849	23	12	polynomials	polynomial	NOUN
ap-1849	23	13	,	,	PUNCT
ap-1849	23	14	and	and	CCONJ
ap-1849	23	15	the	the	DET
ap-1849	23	16	action	action	NOUN
ap-1849	23	17	of	of	ADP
ap-1849	23	18	b2	b2	NOUN
ap-1849	23	19	on	on	ADP
ap-1849	23	20	the	the	DET
ap-1849	23	21	verma	verma	PROPN
ap-1849	23	22	module	module	NOUN
ap-1849	23	23	is	be	AUX
ap-1849	23	24	identified	identify	VERB
ap-1849	23	25	with	with	ADP
ap-1849	23	26	differential	differential	ADJ
ap-1849	23	27	operators	operator	NOUN
ap-1849	23	28	on	on	ADP
ap-1849	23	29	the	the	DET
ap-1849	23	30	polynomials	polynomial	NOUN
ap-1849	23	31	.	.	PUNCT
ap-1849	24	1	any	any	DET
ap-1849	24	2	extremal	extremal	ADJ
ap-1849	24	3	vector	vector	NOUN
ap-1849	24	4	in	in	ADP
ap-1849	24	5	the	the	DET
ap-1849	24	6	factor	factor	NOUN
ap-1849	24	7	-	-	PUNCT
ap-1849	24	8	verma	verma	NOUN
ap-1849	24	9	module	module	NOUN
ap-1849	24	10	becomes	become	VERB
ap-1849	24	11	a	a	DET
ap-1849	24	12	polynomial	polynomial	ADJ
ap-1849	24	13	solution	solution	NOUN
ap-1849	24	14	of	of	ADP
ap-1849	24	15	a	a	DET
ap-1849	24	16	system	system	NOUN
ap-1849	24	17	of	of	ADP
ap-1849	24	18	variable	variable	ADJ
ap-1849	24	19	-	-	PUNCT
ap-1849	24	20	coefficient	coefficient	NOUN
ap-1849	24	21	second	second	ADJ
ap-1849	24	22	-	-	PUNCT
ap-1849	24	23	order	order	NOUN
ap-1849	24	24	linear	linear	ADJ
ap-1849	24	25	partial	partial	ADJ
ap-1849	24	26	differential	differential	NOUN
ap-1849	24	27	equations	equation	NOUN
ap-1849	24	28	.	.	PUNCT
ap-1849	25	1	2	2	X
ap-1849	25	2	.	.	X
ap-1849	25	3	the	the	DET
ap-1849	25	4	root	root	NOUN
ap-1849	25	5	system	system	NOUN
ap-1849	25	6	for	for	ADP
ap-1849	25	7	lie	lie	NOUN
ap-1849	25	8	algebra	algebra	NOUN
ap-1849	25	9	b2	b2	NOUN
ap-1849	25	10	in	in	ADP
ap-1849	25	11	the	the	DET
ap-1849	25	12	lie	lie	NOUN
ap-1849	25	13	algebra	algebra	NOUN
ap-1849	25	14	g	g	NOUN
ap-1849	25	15	=	=	NOUN
ap-1849	25	16	b2	b2	NOUN
ap-1849	25	17	we	we	PRON
ap-1849	25	18	will	will	AUX
ap-1849	25	19	take	take	VERB
ap-1849	25	20	a	a	DET
ap-1849	25	21	basis	basis	NOUN
ap-1849	25	22	composed	compose	VERB
ap-1849	25	23	by	by	ADP
ap-1849	25	24	elements	element	NOUN
ap-1849	25	25	h1	h1	PROPN
ap-1849	25	26	,	,	PUNCT
ap-1849	25	27	h2	h2	PROPN
ap-1849	25	28	,	,	PUNCT
ap-1849	25	29	ek	ek	PROPN
ap-1849	25	30	and	and	CCONJ
ap-1849	25	31	fk	fk	INTJ
ap-1849	25	32	,	,	PUNCT
ap-1849	25	33	where	where	SCONJ
ap-1849	25	34	k	k	PROPN
ap-1849	25	35	=	=	SYM
ap-1849	25	36	1	1	NUM
ap-1849	25	37	,	,	PUNCT
ap-1849	25	38	.	.	PUNCT
ap-1849	25	39	.	.	PUNCT
ap-1849	25	40	.	.	PUNCT
ap-1849	26	1	,	,	PUNCT
ap-1849	26	2	4	4	NUM
ap-1849	26	3	,	,	PUNCT
ap-1849	26	4	which	which	PRON
ap-1849	26	5	fulfill	fulfill	VERB
ap-1849	26	6	the	the	DET
ap-1849	26	7	commutation	commutation	NOUN
ap-1849	26	8	relations	relation	NOUN
ap-1849	27	1	[	[	X
ap-1849	27	2	h1,e1	h1,e1	X
ap-1849	27	3	]	]	X
ap-1849	27	4	=	=	SYM
ap-1849	27	5	2e1	2e1	NUM
ap-1849	27	6	,	,	PUNCT
ap-1849	27	7	[	[	X
ap-1849	27	8	h1,e2	h1,e2	X
ap-1849	27	9	]	]	X
ap-1849	27	10	=	=	SYM
ap-1849	27	11	−e2	−e2	PROPN
ap-1849	27	12	,	,	PUNCT
ap-1849	27	13	[	[	X
ap-1849	27	14	h1,e3	h1,e3	X
ap-1849	27	15	]	]	X
ap-1849	27	16	=	=	PUNCT
ap-1849	27	17	e3	e3	NOUN
ap-1849	27	18	,	,	PUNCT
ap-1849	27	19	[	[	X
ap-1849	27	20	h1,e4	h1,e4	X
ap-1849	27	21	]	]	X
ap-1849	27	22	=	=	SYM
ap-1849	27	23	0	0	NUM
ap-1849	27	24	,	,	PUNCT
ap-1849	27	25	[	[	X
ap-1849	27	26	h2,e1	h2,e1	X
ap-1849	27	27	]	]	X
ap-1849	27	28	=	=	PUNCT
ap-1849	27	29	−2e1	−2e1	X
ap-1849	27	30	,	,	PUNCT
ap-1849	27	31	[	[	X
ap-1849	27	32	h2,e2	h2,e2	X
ap-1849	27	33	]	]	X
ap-1849	27	34	=	=	SYM
ap-1849	27	35	2e2	2e2	PROPN
ap-1849	27	36	,	,	PUNCT
ap-1849	27	37	[	[	X
ap-1849	27	38	h2,e3	h2,e3	X
ap-1849	27	39	]	]	X
ap-1849	27	40	=	=	SYM
ap-1849	27	41	0	0	NUM
ap-1849	27	42	,	,	PUNCT
ap-1849	27	43	[	[	X
ap-1849	27	44	h2,e4	h2,e4	X
ap-1849	27	45	]	]	X
ap-1849	27	46	=	=	SYM
ap-1849	27	47	2e4	2e4	NUM
ap-1849	27	48	,	,	PUNCT
ap-1849	27	49	[	[	X
ap-1849	27	50	h1,f1	h1,f1	X
ap-1849	27	51	]	]	X
ap-1849	27	52	=	=	SYM
ap-1849	27	53	−2f1	−2f1	NOUN
ap-1849	27	54	,	,	PUNCT
ap-1849	27	55	[	[	X
ap-1849	27	56	h1,f2	h1,f2	X
ap-1849	27	57	]	]	X
ap-1849	27	58	=	=	SYM
ap-1849	27	59	f2	f2	PROPN
ap-1849	27	60	,	,	PUNCT
ap-1849	27	61	[	[	X
ap-1849	27	62	h1,f3	h1,f3	X
ap-1849	27	63	]	]	X
ap-1849	27	64	=	=	SYM
ap-1849	27	65	−f3	−f3	PROPN
ap-1849	27	66	,	,	PUNCT
ap-1849	27	67	[	[	X
ap-1849	27	68	h1,f4	h1,f4	X
ap-1849	27	69	]	]	X
ap-1849	27	70	=	=	SYM
ap-1849	27	71	0	0	NUM
ap-1849	27	72	,	,	PUNCT
ap-1849	27	73	[	[	X
ap-1849	27	74	h2,f1	h2,f1	X
ap-1849	27	75	]	]	X
ap-1849	27	76	=	=	SYM
ap-1849	27	77	2f1	2f1	NOUN
ap-1849	27	78	,	,	PUNCT
ap-1849	27	79	[	[	X
ap-1849	27	80	h2,f2	h2,f2	X
ap-1849	27	81	]	]	X
ap-1849	27	82	=	=	SYM
ap-1849	27	83	−2f2	−2f2	PROPN
ap-1849	27	84	,	,	PUNCT
ap-1849	27	85	[	[	X
ap-1849	27	86	h2,f3	h2,f3	X
ap-1849	27	87	]	]	X
ap-1849	27	88	=	=	SYM
ap-1849	27	89	0	0	NUM
ap-1849	27	90	,	,	PUNCT
ap-1849	27	91	[	[	X
ap-1849	27	92	h2,f4	h2,f4	X
ap-1849	27	93	]	]	X
ap-1849	27	94	=	=	PUNCT
ap-1849	27	95	−2f4	−2f4	X
ap-1849	27	96	,	,	PUNCT
ap-1849	27	97	[	[	X
ap-1849	27	98	e1,e2	e1,e2	PROPN
ap-1849	27	99	]	]	X
ap-1849	27	100	=	=	PUNCT
ap-1849	27	101	e3	e3	NOUN
ap-1849	27	102	,	,	PUNCT
ap-1849	27	103	[	[	X
ap-1849	27	104	e1,e3	e1,e3	X
ap-1849	27	105	]	]	X
ap-1849	27	106	=	=	SYM
ap-1849	27	107	0	0	NUM
ap-1849	27	108	,	,	PUNCT
ap-1849	27	109	[	[	X
ap-1849	27	110	e1,e4	e1,e4	X
ap-1849	27	111	]	]	X
ap-1849	27	112	=	=	SYM
ap-1849	27	113	0	0	NUM
ap-1849	27	114	,	,	PUNCT
ap-1849	27	115	[	[	X
ap-1849	27	116	e2,e3	e2,e3	X
ap-1849	27	117	]	]	X
ap-1849	27	118	=	=	SYM
ap-1849	27	119	2e4	2e4	NUM
ap-1849	27	120	,	,	PUNCT
ap-1849	27	121	[	[	X
ap-1849	27	122	e2,e4	e2,e4	X
ap-1849	27	123	]	]	X
ap-1849	27	124	=	=	SYM
ap-1849	27	125	0	0	NUM
ap-1849	27	126	,	,	PUNCT
ap-1849	27	127	[	[	X
ap-1849	27	128	e3,e4	e3,e4	X
ap-1849	27	129	]	]	X
ap-1849	27	130	=	=	PUNCT
ap-1849	27	131	0	0	NUM
ap-1849	27	132	,	,	PUNCT
ap-1849	27	133	[	[	X
ap-1849	27	134	f1,f2	f1,f2	X
ap-1849	27	135	]	]	X
ap-1849	27	136	=	=	SYM
ap-1849	27	137	−f3	−f3	NOUN
ap-1849	27	138	,	,	PUNCT
ap-1849	27	139	[	[	X
ap-1849	27	140	f1,f3	f1,f3	X
ap-1849	27	141	]	]	X
ap-1849	27	142	=	=	SYM
ap-1849	27	143	0	0	NUM
ap-1849	27	144	,	,	PUNCT
ap-1849	27	145	[	[	X
ap-1849	27	146	f1,f4	f1,f4	X
ap-1849	27	147	]	]	X
ap-1849	27	148	=	=	SYM
ap-1849	27	149	0	0	NUM
ap-1849	27	150	,	,	PUNCT
ap-1849	27	151	[	[	X
ap-1849	27	152	f2,f3	f2,f3	X
ap-1849	27	153	]	]	X
ap-1849	27	154	=	=	PUNCT
ap-1849	27	155	−2f4	−2f4	X
ap-1849	27	156	,	,	PUNCT
ap-1849	27	157	[	[	X
ap-1849	27	158	f2,f4	f2,f4	X
ap-1849	27	159	]	]	X
ap-1849	27	160	=	=	PUNCT
ap-1849	27	161	0	0	NUM
ap-1849	27	162	,	,	PUNCT
ap-1849	27	163	[	[	X
ap-1849	27	164	f3,f4	f3,f4	X
ap-1849	27	165	]	]	X
ap-1849	27	166	=	=	PUNCT
ap-1849	27	167	0	0	NUM
ap-1849	27	168	,	,	PUNCT
ap-1849	27	169	[	[	X
ap-1849	27	170	e1,f1	e1,f1	ADP
ap-1849	27	171	]	]	X
ap-1849	27	172	=	=	SYM
ap-1849	27	173	h1	h1	PROPN
ap-1849	27	174	,	,	PUNCT
ap-1849	27	175	[	[	X
ap-1849	27	176	e1,f2	e1,f2	X
ap-1849	27	177	]	]	X
ap-1849	27	178	=	=	SYM
ap-1849	27	179	0	0	NUM
ap-1849	27	180	,	,	PUNCT
ap-1849	27	181	[	[	X
ap-1849	27	182	e1,f3	e1,f3	X
ap-1849	27	183	]	]	X
ap-1849	27	184	=	=	SYM
ap-1849	27	185	−f2	−f2	PROPN
ap-1849	27	186	,	,	PUNCT
ap-1849	27	187	[	[	X
ap-1849	27	188	e1,f4	e1,f4	X
ap-1849	27	189	]	]	X
ap-1849	27	190	=	=	PUNCT
ap-1849	27	191	0	0	NUM
ap-1849	27	192	,	,	PUNCT
ap-1849	27	193	[	[	X
ap-1849	27	194	e2,f1	e2,f1	X
ap-1849	27	195	]	]	X
ap-1849	27	196	=	=	SYM
ap-1849	27	197	0	0	NUM
ap-1849	27	198	,	,	PUNCT
ap-1849	27	199	[	[	X
ap-1849	27	200	e2,f2	e2,f2	X
ap-1849	27	201	]	]	X
ap-1849	27	202	=	=	SYM
ap-1849	27	203	h2	h2	NOUN
ap-1849	27	204	,	,	PUNCT
ap-1849	27	205	[	[	X
ap-1849	27	206	e2,f3	e2,f3	X
ap-1849	27	207	]	]	X
ap-1849	27	208	=	=	SYM
ap-1849	27	209	2f1	2f1	NOUN
ap-1849	27	210	,	,	PUNCT
ap-1849	27	211	[	[	X
ap-1849	27	212	e2,f4	e2,f4	X
ap-1849	27	213	]	]	X
ap-1849	27	214	=	=	SYM
ap-1849	27	215	−f3	−f3	NOUN
ap-1849	27	216	,	,	PUNCT
ap-1849	27	217	[	[	X
ap-1849	27	218	e3,f1	e3,f1	PROPN
ap-1849	27	219	]	]	X
ap-1849	27	220	=	=	SYM
ap-1849	27	221	−e2	−e2	PROPN
ap-1849	27	222	,	,	PUNCT
ap-1849	27	223	[	[	X
ap-1849	27	224	e3,f2	e3,f2	X
ap-1849	27	225	]	]	X
ap-1849	27	226	=	=	SYM
ap-1849	27	227	2e1	2e1	NUM
ap-1849	27	228	,	,	PUNCT
ap-1849	27	229	[	[	X
ap-1849	27	230	e3,f3	e3,f3	X
ap-1849	27	231	]	]	X
ap-1849	27	232	=	=	SYM
ap-1849	27	233	2h1	2h1	NUM
ap-1849	27	234	+	+	CCONJ
ap-1849	27	235	h2	h2	NOUN
ap-1849	27	236	,	,	PUNCT
ap-1849	27	237	[	[	X
ap-1849	27	238	e3,f4	e3,f4	X
ap-1849	27	239	]	]	X
ap-1849	27	240	=	=	SYM
ap-1849	27	241	f2	f2	PROPN
ap-1849	27	242	,	,	PUNCT
ap-1849	27	243	[	[	X
ap-1849	27	244	e4,f1	e4,f1	X
ap-1849	27	245	]	]	X
ap-1849	27	246	=	=	SYM
ap-1849	27	247	0	0	NUM
ap-1849	27	248	,	,	PUNCT
ap-1849	27	249	[	[	X
ap-1849	27	250	e4,f2	e4,f2	X
ap-1849	27	251	]	]	X
ap-1849	27	252	=	=	SYM
ap-1849	27	253	−e3	−e3	PROPN
ap-1849	27	254	,	,	PUNCT
ap-1849	27	255	[	[	X
ap-1849	27	256	e4,f3	e4,f3	X
ap-1849	27	257	]	]	X
ap-1849	27	258	=	=	SYM
ap-1849	27	259	e2	e2	PROPN
ap-1849	27	260	,	,	PUNCT
ap-1849	27	261	[	[	X
ap-1849	27	262	e4,f4	e4,f4	X
ap-1849	27	263	]	]	X
ap-1849	27	264	=	=	X
ap-1849	27	265	h1	h1	PROPN
ap-1849	27	266	+	+	NUM
ap-1849	27	267	h2	h2	NOUN
ap-1849	27	268	.	.	PUNCT
ap-1849	28	1	we	we	PRON
ap-1849	28	2	can	can	AUX
ap-1849	28	3	take	take	VERB
ap-1849	28	4	as	as	ADP
ap-1849	28	5	h	h	NOUN
ap-1849	28	6	the	the	DET
ap-1849	28	7	cartan	cartan	ADJ
ap-1849	28	8	subalgebra	subalgebra	NOUN
ap-1849	28	9	with	with	ADP
ap-1849	28	10	the	the	DET
ap-1849	28	11	bases	basis	NOUN
ap-1849	28	12	h1	h1	VERB
ap-1849	28	13	and	and	CCONJ
ap-1849	28	14	h2	h2	NOUN
ap-1849	28	15	.	.	PUNCT
ap-1849	29	1	we	we	PRON
ap-1849	29	2	will	will	AUX
ap-1849	29	3	denote	denote	VERB
ap-1849	29	4	λ	λ	X
ap-1849	29	5	=	=	SYM
ap-1849	29	6	(	(	PUNCT
ap-1849	29	7	λ1	λ1	ADJ
ap-1849	29	8	,	,	PUNCT
ap-1849	29	9	λ2	λ2	PROPN
ap-1849	29	10	)	)	PUNCT
ap-1849	29	11	∈	∈	PROPN
ap-1849	29	12	h∗	h∗	PROPN
ap-1849	29	13	,	,	PUNCT
ap-1849	29	14	for	for	ADP
ap-1849	29	15	which	which	PRON
ap-1849	29	16	we	we	PRON
ap-1849	29	17	have	have	VERB
ap-1849	29	18	λ(h1	λ(h1	NOUN
ap-1849	29	19	)	)	PUNCT
ap-1849	30	1	=	=	SYM
ap-1849	30	2	λ1	λ1	ADJ
ap-1849	30	3	,	,	PUNCT
ap-1849	30	4	λ(h2	λ(h2	NOUN
ap-1849	30	5	)	)	PUNCT
ap-1849	30	6	=	=	SYM
ap-1849	30	7	λ2	λ2	NOUN
ap-1849	30	8	.	.	PUNCT
ap-1849	31	1	399	399	NUM
ap-1849	31	2	http://dx.doi.org/10.14311/ap.2013.53.0399	http://dx.doi.org/10.14311/ap.2013.53.0399	NOUN
ap-1849	31	3	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-1849	31	4	č	č	NUM
ap-1849	31	5	.	.	PUNCT
ap-1849	31	6	burdík	burdík	PROPN
ap-1849	31	7	,	,	PUNCT
ap-1849	31	8	o.	o.	PROPN
ap-1849	31	9	navrátil	navrátil	PROPN
ap-1849	31	10	acta	acta	PROPN
ap-1849	31	11	polytechnica	polytechnica	PROPN
ap-1849	31	12	the	the	DET
ap-1849	31	13	root	root	NOUN
ap-1849	31	14	systems	system	NOUN
ap-1849	31	15	g	g	NOUN
ap-1849	31	16	=	=	NOUN
ap-1849	31	17	b2	b2	PROPN
ap-1849	31	18	with	with	ADP
ap-1849	31	19	respect	respect	NOUN
ap-1849	31	20	to	to	ADP
ap-1849	31	21	these	these	DET
ap-1849	31	22	bases	basis	NOUN
ap-1849	31	23	h1	h1	ADJ
ap-1849	31	24	and	and	CCONJ
ap-1849	31	25	h2	h2	NOUN
ap-1849	31	26	are	be	AUX
ap-1849	31	27	r	r	NOUN
ap-1849	31	28	=	=	PUNCT
ap-1849	31	29	{	{	PUNCT
ap-1849	31	30	±αk	±αk	NOUN
ap-1849	31	31	;	;	PUNCT
ap-1849	31	32	k	k	PROPN
ap-1849	31	33	=	=	SYM
ap-1849	31	34	1	1	NUM
ap-1849	31	35	,	,	PUNCT
ap-1849	31	36	2	2	NUM
ap-1849	31	37	,	,	PUNCT
ap-1849	31	38	3	3	NUM
ap-1849	31	39	,	,	PUNCT
ap-1849	31	40	4	4	NUM
ap-1849	31	41	}	}	PUNCT
ap-1849	31	42	,	,	PUNCT
ap-1849	31	43	where	where	SCONJ
ap-1849	31	44	α1	α1	PROPN
ap-1849	31	45	=	=	SYM
ap-1849	31	46	(	(	PUNCT
ap-1849	31	47	2,−2	2,−2	NUM
ap-1849	31	48	)	)	PUNCT
ap-1849	31	49	,	,	PUNCT
ap-1849	31	50	α2	α2	PROPN
ap-1849	31	51	=	=	SYM
ap-1849	31	52	(	(	PUNCT
ap-1849	31	53	−1	−1	NOUN
ap-1849	31	54	,	,	PUNCT
ap-1849	31	55	2	2	NUM
ap-1849	31	56	)	)	PUNCT
ap-1849	31	57	,	,	PUNCT
ap-1849	31	58	α3	α3	PROPN
ap-1849	31	59	=	=	SYM
ap-1849	31	60	α1	α1	PROPN
ap-1849	31	61	+	+	CCONJ
ap-1849	31	62	α2	α2	ADJ
ap-1849	31	63	=	=	SYM
ap-1849	31	64	(	(	PUNCT
ap-1849	31	65	1	1	NUM
ap-1849	31	66	,	,	PUNCT
ap-1849	31	67	0	0	NUM
ap-1849	31	68	)	)	PUNCT
ap-1849	31	69	,	,	PUNCT
ap-1849	31	70	α4	α4	NOUN
ap-1849	31	71	=	=	SYM
ap-1849	31	72	α1	α1	PROPN
ap-1849	31	73	+	+	CCONJ
ap-1849	31	74	2α2	2α2	NUM
ap-1849	31	75	=	=	SYM
ap-1849	31	76	(	(	PUNCT
ap-1849	31	77	0	0	NUM
ap-1849	31	78	,	,	PUNCT
ap-1849	31	79	2	2	NUM
ap-1849	31	80	)	)	PUNCT
ap-1849	31	81	.	.	PUNCT
ap-1849	32	1	if	if	SCONJ
ap-1849	32	2	we	we	PRON
ap-1849	32	3	choose	choose	VERB
ap-1849	32	4	positive	positive	ADJ
ap-1849	32	5	roots	root	NOUN
ap-1849	32	6	r+	r+	PUNCT
ap-1849	32	7	=	=	PUNCT
ap-1849	32	8	{	{	PUNCT
ap-1849	32	9	α1,α2,α3,α4	α1,α2,α3,α4	PROPN
ap-1849	32	10	}	}	PUNCT
ap-1849	32	11	,	,	PUNCT
ap-1849	32	12	the	the	DET
ap-1849	32	13	basis	basis	NOUN
ap-1849	32	14	in	in	ADP
ap-1849	32	15	root	root	NOUN
ap-1849	32	16	system	system	NOUN
ap-1849	32	17	r	r	NOUN
ap-1849	32	18	is	be	AUX
ap-1849	32	19	b	b	NOUN
ap-1849	32	20	=	=	PUNCT
ap-1849	32	21	{	{	PUNCT
ap-1849	32	22	α1,α2	α1,α2	PROPN
ap-1849	32	23	}	}	PUNCT
ap-1849	32	24	.	.	PUNCT
ap-1849	33	1	if	if	SCONJ
ap-1849	33	2	we	we	PRON
ap-1849	33	3	define	define	VERB
ap-1849	33	4	h3	h3	NOUN
ap-1849	33	5	=	=	SYM
ap-1849	33	6	2h1	2h1	NUM
ap-1849	33	7	+	+	CCONJ
ap-1849	33	8	h2	h2	NOUN
ap-1849	33	9	and	and	CCONJ
ap-1849	33	10	h4	h4	PROPN
ap-1849	33	11	=	=	SYM
ap-1849	33	12	h1	h1	PROPN
ap-1849	33	13	+	+	CCONJ
ap-1849	33	14	h2	h2	NOUN
ap-1849	33	15	,	,	PUNCT
ap-1849	33	16	the	the	DET
ap-1849	33	17	following	follow	VERB
ap-1849	33	18	relations	relation	NOUN
ap-1849	34	1	[	[	X
ap-1849	34	2	hk	hk	PROPN
ap-1849	34	3	,	,	PUNCT
ap-1849	34	4	ek	ek	X
ap-1849	34	5	]	]	X
ap-1849	34	6	=	=	SYM
ap-1849	34	7	2ek	2ek	NOUN
ap-1849	34	8	,	,	PUNCT
ap-1849	34	9	[	[	X
ap-1849	34	10	hk	hk	PROPN
ap-1849	34	11	,	,	PUNCT
ap-1849	34	12	fk	fk	INTJ
ap-1849	34	13	]	]	X
ap-1849	34	14	=	=	SYM
ap-1849	34	15	−2fk	−2fk	PROPN
ap-1849	34	16	,	,	PUNCT
ap-1849	34	17	[	[	X
ap-1849	34	18	ek	ek	X
ap-1849	34	19	,	,	PUNCT
ap-1849	34	20	fk	fk	INTJ
ap-1849	34	21	]	]	X
ap-1849	34	22	=	=	SYM
ap-1849	34	23	hk	hk	PROPN
ap-1849	34	24	are	be	AUX
ap-1849	34	25	valid	valid	ADJ
ap-1849	34	26	for	for	ADP
ap-1849	34	27	any	any	DET
ap-1849	34	28	k	k	NOUN
ap-1849	34	29	=	=	SYM
ap-1849	34	30	1	1	NUM
ap-1849	34	31	,	,	PUNCT
ap-1849	34	32	.	.	PUNCT
ap-1849	34	33	.	.	PUNCT
ap-1849	35	1	.	.	PUNCT
ap-1849	36	1	,	,	PUNCT
ap-1849	37	1	4	4	X
ap-1849	37	2	.	.	NOUN
ap-1849	37	3	3	3	NUM
ap-1849	37	4	.	.	PUNCT
ap-1849	38	1	the	the	DET
ap-1849	38	2	extremal	extremal	ADJ
ap-1849	38	3	vectors	vector	NOUN
ap-1849	38	4	for	for	ADP
ap-1849	38	5	verma	verma	PROPN
ap-1849	38	6	type	type	NOUN
ap-1849	38	7	representation	representation	NOUN
ap-1849	38	8	we	we	PRON
ap-1849	38	9	denote	denote	VERB
ap-1849	38	10	by	by	ADP
ap-1849	38	11	n+	n+	PROPN
ap-1849	38	12	,	,	PUNCT
ap-1849	38	13	and	and	CCONJ
ap-1849	38	14	n−	n−	NOUN
ap-1849	38	15	the	the	DET
ap-1849	38	16	lie	lie	NOUN
ap-1849	38	17	subalgebras	subalgebras	PROPN
ap-1849	38	18	generated	generate	VERB
ap-1849	38	19	by	by	ADP
ap-1849	38	20	elements	element	NOUN
ap-1849	38	21	ek	ek	PROPN
ap-1849	38	22	,	,	PUNCT
ap-1849	38	23	and	and	CCONJ
ap-1849	38	24	fk	fk	INTJ
ap-1849	38	25	,	,	PUNCT
ap-1849	38	26	respectively	respectively	ADV
ap-1849	38	27	,	,	PUNCT
ap-1849	38	28	where	where	SCONJ
ap-1849	38	29	k	k	PROPN
ap-1849	38	30	=	=	SYM
ap-1849	38	31	1	1	NUM
ap-1849	38	32	,	,	PUNCT
ap-1849	38	33	.	.	PUNCT
ap-1849	38	34	.	.	PUNCT
ap-1849	38	35	.	.	PUNCT
ap-1849	39	1	,	,	PUNCT
ap-1849	39	2	4	4	NUM
ap-1849	39	3	,	,	PUNCT
ap-1849	39	4	and	and	CCONJ
ap-1849	39	5	b+	b+	X
ap-1849	39	6	=	=	PUNCT
ap-1849	39	7	h+n+	h+n+	NOUN
ap-1849	39	8	.	.	PUNCT
ap-1849	40	1	let	let	VERB
ap-1849	40	2	us	we	PRON
ap-1849	40	3	further	far	ADV
ap-1849	40	4	consider	consider	VERB
ap-1849	40	5	λ	λ	X
ap-1849	40	6	=	=	SYM
ap-1849	40	7	(	(	PUNCT
ap-1849	40	8	λ1	λ1	ADJ
ap-1849	40	9	,	,	PUNCT
ap-1849	40	10	λ2	λ2	PROPN
ap-1849	40	11	)	)	PUNCT
ap-1849	40	12	∈	∈	PROPN
ap-1849	40	13	h∗	h∗	VERB
ap-1849	40	14	the	the	DET
ap-1849	40	15	one	one	NUM
ap-1849	40	16	-	-	PUNCT
ap-1849	40	17	dimensional	dimensional	ADJ
ap-1849	40	18	representation	representation	NOUN
ap-1849	40	19	τλ	τλ	ADP
ap-1849	40	20	for	for	ADP
ap-1849	40	21	the	the	DET
ap-1849	40	22	lie	lie	NOUN
ap-1849	40	23	algebra	algebra	NOUN
ap-1849	40	24	b+	b+	ADJ
ap-1849	40	25	such	such	ADJ
ap-1849	40	26	that	that	PRON
ap-1849	40	27	for	for	ADP
ap-1849	40	28	any	any	DET
ap-1849	40	29	h	h	NOUN
ap-1849	40	30	∈	∈	PROPN
ap-1849	40	31	h	h	NOUN
ap-1849	40	32	and	and	CCONJ
ap-1849	40	33	e	e	PROPN
ap-1849	40	34	∈	∈	PROPN
ap-1849	40	35	n+	n+	PUNCT
ap-1849	40	36	τλ(h	τλ(h	X
ap-1849	40	37	+	+	NOUN
ap-1849	40	38	e)|0	e)|0	NOUN
ap-1849	40	39	〉	〉	NUM
ap-1849	40	40	=	=	SYM
ap-1849	40	41	λ(h)|0	λ(h)|0	PROPN
ap-1849	40	42	〉	〉	PROPN
ap-1849	40	43	.	.	PUNCT
ap-1849	41	1	the	the	DET
ap-1849	41	2	element	element	NOUN
ap-1849	41	3	|0	|0	NUM
ap-1849	41	4	〉	〉	PROPN
ap-1849	41	5	will	will	AUX
ap-1849	41	6	be	be	AUX
ap-1849	41	7	called	call	VERB
ap-1849	41	8	the	the	DET
ap-1849	41	9	lowest	low	ADJ
ap-1849	41	10	-	-	PUNCT
ap-1849	41	11	weight	weight	NOUN
ap-1849	41	12	vector	vector	NOUN
ap-1849	41	13	.	.	PUNCT
ap-1849	42	1	let	let	VERB
ap-1849	42	2	further	far	ADV
ap-1849	42	3	be	be	AUX
ap-1849	42	4	w	w	PROPN
ap-1849	42	5	(	(	PUNCT
ap-1849	42	6	λ	λ	NOUN
ap-1849	42	7	)	)	PUNCT
ap-1849	42	8	=	=	SYM
ap-1849	42	9	u(g)⊗u(b+	u(g)⊗u(b+	PROPN
ap-1849	42	10	)	)	PUNCT
ap-1849	43	1	c|0	c|0	NOUN
ap-1849	43	2	〉	〉	PROPN
ap-1849	43	3	,	,	PUNCT
ap-1849	43	4	where	where	SCONJ
ap-1849	43	5	b+-module	b+-module	NOUN
ap-1849	43	6	c|0	c|0	PROPN
ap-1849	43	7	〉	〉	PROPN
ap-1849	43	8	is	be	AUX
ap-1849	43	9	defined	define	VERB
ap-1849	43	10	by	by	ADP
ap-1849	43	11	τλ	τλ	ADP
ap-1849	43	12	.	.	PUNCT
ap-1849	44	1	it	it	PRON
ap-1849	44	2	is	be	AUX
ap-1849	44	3	clear	clear	ADJ
ap-1849	44	4	that	that	SCONJ
ap-1849	44	5	w	w	PROPN
ap-1849	44	6	(	(	PUNCT
ap-1849	44	7	λ	λ	NOUN
ap-1849	44	8	)	)	PUNCT
ap-1849	44	9	∼	∼	NOUN
ap-1849	44	10	u(n−)|0	u(n−)|0	PROPN
ap-1849	44	11	〉	〉	NOUN
ap-1849	44	12	and	and	CCONJ
ap-1849	44	13	it	it	PRON
ap-1849	44	14	is	be	AUX
ap-1849	44	15	the	the	DET
ap-1849	44	16	u(g)module	u(g)module	NOUN
ap-1849	44	17	for	for	ADP
ap-1849	44	18	the	the	DET
ap-1849	44	19	left	left	ADJ
ap-1849	44	20	regular	regular	ADJ
ap-1849	44	21	representation	representation	NOUN
ap-1849	44	22	,	,	PUNCT
ap-1849	44	23	which	which	PRON
ap-1849	44	24	will	will	AUX
ap-1849	44	25	be	be	AUX
ap-1849	44	26	called	call	VERB
ap-1849	44	27	the	the	DET
ap-1849	44	28	verma	verma	PROPN
ap-1849	44	29	module	module	NOUN
ap-1849	44	30	.	.	PUNCT
ap-1849	45	1	1	1	NUM
ap-1849	45	2	it	it	PRON
ap-1849	45	3	is	be	AUX
ap-1849	45	4	a	a	DET
ap-1849	45	5	well	well	ADV
ap-1849	45	6	-	-	PUNCT
ap-1849	45	7	known	know	VERB
ap-1849	45	8	fact	fact	NOUN
ap-1849	45	9	that	that	SCONJ
ap-1849	45	10	every	every	DET
ap-1849	45	11	u(g)-submodule	u(g)-submodule	PROPN
ap-1849	45	12	of	of	ADP
ap-1849	45	13	the	the	DET
ap-1849	45	14	module	module	NOUN
ap-1849	45	15	w	w	PROPN
ap-1849	45	16	(	(	PUNCT
ap-1849	45	17	λ	λ	NOUN
ap-1849	45	18	)	)	PUNCT
ap-1849	45	19	is	be	AUX
ap-1849	45	20	isomorphic	isomorphic	ADJ
ap-1849	45	21	to	to	ADP
ap-1849	45	22	module	module	NOUN
ap-1849	45	23	w	w	PROPN
ap-1849	45	24	(	(	PUNCT
ap-1849	45	25	µ	µ	NOUN
ap-1849	45	26	)	)	PUNCT
ap-1849	45	27	,	,	PUNCT
ap-1849	45	28	where	where	SCONJ
ap-1849	45	29	µ	µ	X
ap-1849	45	30	=	=	SYM
ap-1849	45	31	λ−	λ−	PROPN
ap-1849	45	32	n1α1	n1α1	NOUN
ap-1849	45	33	−	−	PROPN
ap-1849	45	34	n2α2	n2α2	NOUN
ap-1849	45	35	,	,	PUNCT
ap-1849	45	36	for	for	ADP
ap-1849	45	37	n1	n1	NOUN
ap-1849	45	38	,	,	PUNCT
ap-1849	45	39	n2	n2	PROPN
ap-1849	45	40	∈	∈	PROPN
ap-1849	45	41	n0	n0	PROPN
ap-1849	45	42	=	=	SYM
ap-1849	45	43	{	{	PUNCT
ap-1849	45	44	0	0	NUM
ap-1849	45	45	,	,	PUNCT
ap-1849	45	46	1	1	NUM
ap-1849	45	47	,	,	PUNCT
ap-1849	45	48	2	2	NUM
ap-1849	45	49	,	,	PUNCT
ap-1849	45	50	.	.	PUNCT
ap-1849	45	51	.	.	PUNCT
ap-1849	45	52	.	.	PUNCT
ap-1849	46	1	}	}	PUNCT
ap-1849	46	2	.	.	PUNCT
ap-1849	47	1	for	for	ADP
ap-1849	47	2	the	the	DET
ap-1849	47	3	lowest	low	ADJ
ap-1849	47	4	-	-	PUNCT
ap-1849	47	5	weight	weight	NOUN
ap-1849	47	6	vector	vector	NOUN
ap-1849	47	7	of	of	ADP
ap-1849	47	8	the	the	DET
ap-1849	47	9	representation	representation	NOUN
ap-1849	47	10	w	w	PROPN
ap-1849	47	11	(	(	PUNCT
ap-1849	47	12	µ	µ	NOUN
ap-1849	47	13	)	)	PUNCT
ap-1849	47	14	⊂	⊂	PROPN
ap-1849	47	15	w	w	PROPN
ap-1849	47	16	(	(	PUNCT
ap-1849	47	17	λ	λ	NOUN
ap-1849	47	18	)	)	PUNCT
ap-1849	47	19	,	,	PUNCT
ap-1849	47	20	|0〉µ	|0〉µ	ADV
ap-1849	47	21	,	,	PUNCT
ap-1849	47	22	is	be	AUX
ap-1849	47	23	fulfilled	fulfil	VERB
ap-1849	47	24	h|0〉µ	h|0〉µ	PROPN
ap-1849	47	25	=	=	PUNCT
ap-1849	47	26	µ(h)|0〉µ	µ(h)|0〉µ	PROPN
ap-1849	47	27	,	,	PUNCT
ap-1849	47	28	h	h	NOUN
ap-1849	47	29	∈	∈	PROPN
ap-1849	47	30	h	h	NOUN
ap-1849	47	31	,	,	PUNCT
ap-1849	47	32	e|0〉µ	e|0〉µ	PROPN
ap-1849	47	33	=	=	SYM
ap-1849	47	34	0	0	NUM
ap-1849	47	35	,	,	PUNCT
ap-1849	47	36	e	e	PROPN
ap-1849	47	37	∈	∈	PROPN
ap-1849	47	38	n+	n+	PROPN
ap-1849	47	39	.	.	PUNCT
ap-1849	48	1	such	such	ADJ
ap-1849	48	2	vectors	vector	NOUN
ap-1849	48	3	|0〉µ	|0〉µ	PART
ap-1849	48	4	will	will	AUX
ap-1849	48	5	be	be	AUX
ap-1849	48	6	called	call	VERB
ap-1849	48	7	extremal	extremal	ADJ
ap-1849	48	8	vectors	vector	NOUN
ap-1849	48	9	w	w	PROPN
ap-1849	48	10	(	(	PUNCT
ap-1849	48	11	λ	λ	NOUN
ap-1849	48	12	)	)	PUNCT
ap-1849	48	13	.	.	PUNCT
ap-1849	49	1	from	from	ADP
ap-1849	49	2	the	the	DET
ap-1849	49	3	well	well	ADV
ap-1849	49	4	-	-	PUNCT
ap-1849	49	5	known	know	VERB
ap-1849	49	6	result	result	NOUN
ap-1849	49	7	for	for	ADP
ap-1849	49	8	the	the	DET
ap-1849	49	9	verma	verma	PROPN
ap-1849	49	10	modules	module	NOUN
ap-1849	49	11	we	we	PRON
ap-1849	49	12	know	know	VERB
ap-1849	49	13	that	that	SCONJ
ap-1849	49	14	the	the	DET
ap-1849	49	15	verma	verma	PROPN
ap-1849	49	16	module	module	PROPN
ap-1849	49	17	w	w	PROPN
ap-1849	49	18	(	(	PUNCT
ap-1849	49	19	λ	λ	NOUN
ap-1849	49	20	)	)	PUNCT
ap-1849	49	21	is	be	AUX
ap-1849	49	22	irreducible	irreducible	ADJ
ap-1849	50	1	iff	iff	PROPN
ap-1849	50	2	λ1	λ1	PROPN
ap-1849	50	3	/∈	/∈	PROPN
ap-1849	50	4	n0	n0	PROPN
ap-1849	50	5	,	,	PUNCT
ap-1849	50	6	λ2	λ2	PROPN
ap-1849	50	7	/∈	/∈	PUNCT
ap-1849	50	8	n0	n0	PROPN
ap-1849	50	9	,	,	PUNCT
ap-1849	50	10	λ1	λ1	PROPN
ap-1849	50	11	+	+	NUM
ap-1849	50	12	λ2	λ2	NOUN
ap-1849	50	13	+	+	CCONJ
ap-1849	50	14	1	1	NUM
ap-1849	50	15	/∈	/∈	SYM
ap-1849	50	16	n0	n0	NUM
ap-1849	50	17	,	,	PUNCT
ap-1849	50	18	2λ1	2λ1	NUM
ap-1849	51	1	+	+	NUM
ap-1849	51	2	λ2	λ2	NOUN
ap-1849	51	3	+	+	CCONJ
ap-1849	51	4	2	2	NUM
ap-1849	51	5	/∈	/∈	SYM
ap-1849	51	6	n0	n0	NOUN
ap-1849	51	7	.	.	PUNCT
ap-1849	52	1	if	if	SCONJ
ap-1849	52	2	λ1	λ1	PROPN
ap-1849	52	3	∈	∈	PROPN
ap-1849	52	4	n0	n0	PROPN
ap-1849	52	5	,	,	PUNCT
ap-1849	52	6	resp	resp	NOUN
ap-1849	52	7	.	.	PUNCT
ap-1849	53	1	λ2	λ2	PROPN
ap-1849	53	2	∈	∈	PROPN
ap-1849	53	3	n0	n0	PROPN
ap-1849	53	4	,	,	PUNCT
ap-1849	53	5	then	then	ADV
ap-1849	53	6	the	the	DET
ap-1849	53	7	extremal	extremal	ADJ
ap-1849	53	8	vectors	vector	NOUN
ap-1849	53	9	are	be	AUX
ap-1849	53	10	fλ1	fλ1	ADJ
ap-1849	53	11	+	+	ADJ
ap-1849	53	12	1	1	NUM
ap-1849	53	13	1	1	NUM
ap-1849	53	14	|0	|0	NUM
ap-1849	53	15	〉	〉	NUM
ap-1849	53	16	=	=	SYM
ap-1849	53	17	|0〉µ1	|0〉µ1	NOUN
ap-1849	53	18	,	,	PUNCT
ap-1849	53	19	resp	resp	NOUN
ap-1849	53	20	.	.	PUNCT
ap-1849	54	1	fλ2	fλ2	PROPN
ap-1849	54	2	+	+	PROPN
ap-1849	54	3	1	1	NUM
ap-1849	54	4	2	2	NUM
ap-1849	54	5	|0	|0	NUM
ap-1849	54	6	〉	〉	NOUN
ap-1849	54	7	=	=	SYM
ap-1849	54	8	|0〉µ2	|0〉µ2	NOUN
ap-1849	54	9	,	,	PUNCT
ap-1849	54	10	1	1	NUM
ap-1849	54	11	in	in	ADP
ap-1849	54	12	dixmier	dixmier	NOUN
ap-1849	54	13	’s	’s	PART
ap-1849	54	14	book	book	NOUN
ap-1849	54	15	the	the	DET
ap-1849	54	16	verma	verma	PROPN
ap-1849	54	17	module	module	NOUN
ap-1849	54	18	m(λ	m(λ	PROPN
ap-1849	54	19	)	)	PUNCT
ap-1849	54	20	is	be	AUX
ap-1849	54	21	defined	define	VERB
ap-1849	54	22	with	with	ADP
ap-1849	54	23	respect	respect	NOUN
ap-1849	54	24	to	to	ADP
ap-1849	54	25	τλ−δ	τλ−δ	PROPN
ap-1849	54	26	,	,	PUNCT
ap-1849	54	27	where	where	SCONJ
ap-1849	54	28	δ	δ	PROPN
ap-1849	54	29	=	=	NOUN
ap-1849	54	30	1	1	NUM
ap-1849	54	31	2	2	NUM
ap-1849	55	1	∑4	∑4	NOUN
ap-1849	55	2	k=1αk	k=1αk	PROPN
ap-1849	55	3	=	=	SYM
ap-1849	55	4	(	(	PUNCT
ap-1849	55	5	1	1	NUM
ap-1849	55	6	,	,	PUNCT
ap-1849	55	7	1	1	NUM
ap-1849	55	8	)	)	PUNCT
ap-1849	55	9	.	.	PUNCT
ap-1849	56	1	so	so	ADV
ap-1849	56	2	we	we	PRON
ap-1849	56	3	have	have	VERB
ap-1849	56	4	w	w	PROPN
ap-1849	56	5	(	(	PUNCT
ap-1849	56	6	λ	λ	NOUN
ap-1849	56	7	)	)	PUNCT
ap-1849	56	8	=	=	SYM
ap-1849	57	1	m(λ	m(λ	PROPN
ap-1849	57	2	+	+	NUM
ap-1849	57	3	δ	δ	PROPN
ap-1849	57	4	)	)	PUNCT
ap-1849	57	5	.	.	PUNCT
ap-1849	58	1	where	where	SCONJ
ap-1849	58	2	µ1	µ1	PROPN
ap-1849	58	3	=	=	SYM
ap-1849	58	4	λ−	λ−	PROPN
ap-1849	58	5	(	(	PUNCT
ap-1849	58	6	λ1	λ1	PROPN
ap-1849	58	7	+	+	NUM
ap-1849	58	8	1)α1	1)α1	NUM
ap-1849	58	9	=	=	SYM
ap-1849	58	10	(	(	PUNCT
ap-1849	58	11	−λ1	−λ1	NOUN
ap-1849	58	12	−	−	PROPN
ap-1849	58	13	2	2	NUM
ap-1849	58	14	,	,	PUNCT
ap-1849	58	15	2λ1	2λ1	NUM
ap-1849	58	16	+	+	NUM
ap-1849	58	17	λ2	λ2	NOUN
ap-1849	58	18	+	+	CCONJ
ap-1849	58	19	2	2	NUM
ap-1849	58	20	)	)	PUNCT
ap-1849	58	21	,	,	PUNCT
ap-1849	58	22	µ2	µ2	PROPN
ap-1849	58	23	=	=	PUNCT
ap-1849	58	24	λ−	λ−	PROPN
ap-1849	58	25	(	(	PUNCT
ap-1849	58	26	λ2	λ2	NOUN
ap-1849	58	27	+	+	CCONJ
ap-1849	58	28	1)α2	1)α2	NUM
ap-1849	58	29	=	=	SYM
ap-1849	58	30	(	(	PUNCT
ap-1849	58	31	λ1	λ1	ADJ
ap-1849	58	32	+	+	NUM
ap-1849	58	33	λ2	λ2	NOUN
ap-1849	58	34	+	+	CCONJ
ap-1849	58	35	1,−λ2	1,−λ2	NUM
ap-1849	58	36	−	−	ADP
ap-1849	58	37	2	2	NUM
ap-1849	58	38	)	)	PUNCT
ap-1849	58	39	.	.	PUNCT
ap-1849	59	1	(	(	PUNCT
ap-1849	59	2	1	1	X
ap-1849	59	3	)	)	PUNCT
ap-1849	59	4	if	if	SCONJ
ap-1849	59	5	w	w	PROPN
ap-1849	59	6	(	(	PUNCT
ap-1849	59	7	µ	µ	NOUN
ap-1849	59	8	)	)	PUNCT
ap-1849	59	9	is	be	AUX
ap-1849	59	10	a	a	DET
ap-1849	59	11	submodule	submodule	NOUN
ap-1849	59	12	w	w	PROPN
ap-1849	59	13	(	(	PUNCT
ap-1849	59	14	λ	λ	NOUN
ap-1849	59	15	)	)	PUNCT
ap-1849	59	16	,	,	PUNCT
ap-1849	59	17	we	we	PRON
ap-1849	59	18	will	will	AUX
ap-1849	59	19	define	define	VERB
ap-1849	59	20	the	the	DET
ap-1849	59	21	u(g)-factor	u(g)-factor	PROPN
ap-1849	59	22	-	-	PUNCT
ap-1849	59	23	module	module	NOUN
ap-1849	59	24	w	w	NOUN
ap-1849	59	25	(	(	PUNCT
ap-1849	59	26	λ|µ	λ|µ	ADJ
ap-1849	59	27	)	)	PUNCT
ap-1849	59	28	=	=	SYM
ap-1849	59	29	w	w	X
ap-1849	59	30	(	(	PUNCT
ap-1849	59	31	λ)/w	λ)/w	PROPN
ap-1849	59	32	(	(	PUNCT
ap-1849	59	33	µ	µ	NOUN
ap-1849	59	34	)	)	PUNCT
ap-1849	59	35	.	.	PUNCT
ap-1849	60	1	now	now	ADV
ap-1849	60	2	we	we	PRON
ap-1849	60	3	can	can	AUX
ap-1849	60	4	study	study	VERB
ap-1849	60	5	the	the	DET
ap-1849	60	6	reducibility	reducibility	NOUN
ap-1849	60	7	of	of	ADP
ap-1849	60	8	a	a	DET
ap-1849	60	9	representation	representation	NOUN
ap-1849	60	10	like	like	ADP
ap-1849	60	11	that	that	PRON
ap-1849	60	12	.	.	PUNCT
ap-1849	61	1	again	again	ADV
ap-1849	61	2	,	,	PUNCT
ap-1849	61	3	the	the	DET
ap-1849	61	4	extremal	extremal	ADJ
ap-1849	61	5	vector	vector	NOUN
ap-1849	61	6	is	be	AUX
ap-1849	61	7	called	call	VERB
ap-1849	61	8	any	any	DET
ap-1849	61	9	nonzero	nonzero	PROPN
ap-1849	61	10	vector	vector	NOUN
ap-1849	61	11	v	v	NOUN
ap-1849	61	12	∈w	∈w	NOUN
ap-1849	61	13	(	(	PUNCT
ap-1849	61	14	λ|µ	λ|µ	PROPN
ap-1849	61	15	)	)	PUNCT
ap-1849	61	16	for	for	ADP
ap-1849	61	17	which	which	PRON
ap-1849	61	18	there	there	PRON
ap-1849	61	19	exists	exist	VERB
ap-1849	61	20	ν	ν	PROPN
ap-1849	61	21	∈	∈	PROPN
ap-1849	61	22	h∗	h∗	PROPN
ap-1849	61	23	such	such	ADJ
ap-1849	61	24	that	that	DET
ap-1849	61	25	hkv	hkv	PROPN
ap-1849	61	26	=	=	NOUN
ap-1849	61	27	νkv	νkv	PROPN
ap-1849	61	28	,	,	PUNCT
ap-1849	61	29	ekv	ekv	PROPN
ap-1849	61	30	=	=	SYM
ap-1849	61	31	0	0	PROPN
ap-1849	61	32	,	,	PUNCT
ap-1849	61	33	k	k	NOUN
ap-1849	62	1	=	=	SYM
ap-1849	62	2	1	1	NUM
ap-1849	62	3	,	,	PUNCT
ap-1849	62	4	2	2	NUM
ap-1849	62	5	.	.	PUNCT
ap-1849	63	1	(	(	PUNCT
ap-1849	63	2	2	2	X
ap-1849	63	3	)	)	PUNCT
ap-1849	63	4	it	it	PRON
ap-1849	63	5	is	be	AUX
ap-1849	63	6	clear	clear	ADJ
ap-1849	63	7	that	that	SCONJ
ap-1849	63	8	ekv	ekv	PROPN
ap-1849	63	9	=	=	NOUN
ap-1849	63	10	0	0	NUM
ap-1849	63	11	for	for	ADP
ap-1849	63	12	k	k	PROPN
ap-1849	63	13	=	=	SYM
ap-1849	63	14	1	1	NUM
ap-1849	63	15	,	,	PUNCT
ap-1849	63	16	2	2	NUM
ap-1849	63	17	,	,	PUNCT
ap-1849	63	18	3	3	NUM
ap-1849	63	19	,	,	PUNCT
ap-1849	63	20	4	4	NUM
ap-1849	63	21	.	.	PUNCT
ap-1849	64	1	in	in	ADP
ap-1849	64	2	this	this	DET
ap-1849	64	3	paper	paper	NOUN
ap-1849	64	4	,	,	PUNCT
ap-1849	64	5	we	we	PRON
ap-1849	64	6	find	find	VERB
ap-1849	64	7	all	all	DET
ap-1849	64	8	such	such	ADJ
ap-1849	64	9	extremal	extremal	ADJ
ap-1849	64	10	vectors	vector	NOUN
ap-1849	64	11	in	in	ADP
ap-1849	64	12	the	the	DET
ap-1849	64	13	space	space	NOUN
ap-1849	64	14	w	w	PROPN
ap-1849	64	15	(	(	PUNCT
ap-1849	64	16	λ|µ2	λ|µ2	PROPN
ap-1849	64	17	)	)	PUNCT
ap-1849	64	18	,	,	PUNCT
ap-1849	64	19	where	where	SCONJ
ap-1849	64	20	λ2	λ2	PROPN
ap-1849	64	21	∈	∈	PROPN
ap-1849	64	22	n0	n0	NOUN
ap-1849	64	23	and	and	CCONJ
ap-1849	64	24	µ2	µ2	PROPN
ap-1849	64	25	is	be	AUX
ap-1849	64	26	given	give	VERB
ap-1849	64	27	by	by	ADP
ap-1849	64	28	(	(	PUNCT
ap-1849	64	29	1	1	NUM
ap-1849	64	30	)	)	PUNCT
ap-1849	64	31	.	.	PUNCT
ap-1849	65	1	4	4	X
ap-1849	65	2	.	.	X
ap-1849	65	3	differential	differential	ADJ
ap-1849	65	4	equations	equation	NOUN
ap-1849	65	5	for	for	ADP
ap-1849	65	6	extremal	extremal	ADJ
ap-1849	65	7	vectors	vector	NOUN
ap-1849	65	8	let	let	VERB
ap-1849	65	9	λ2	λ2	PROPN
ap-1849	65	10	∈	∈	PROPN
ap-1849	65	11	n0	n0	NOUN
ap-1849	65	12	and	and	CCONJ
ap-1849	65	13	µ2	µ2	PROPN
ap-1849	65	14	be	be	AUX
ap-1849	65	15	given	give	VERB
ap-1849	65	16	by	by	ADP
ap-1849	65	17	equation	equation	NOUN
ap-1849	65	18	(	(	PUNCT
ap-1849	65	19	1	1	NUM
ap-1849	65	20	)	)	PUNCT
ap-1849	65	21	.	.	PUNCT
ap-1849	66	1	it	it	PRON
ap-1849	66	2	is	be	AUX
ap-1849	66	3	easy	easy	ADJ
ap-1849	66	4	to	to	PART
ap-1849	66	5	see	see	VERB
ap-1849	66	6	that	that	SCONJ
ap-1849	66	7	the	the	DET
ap-1849	66	8	basis	basis	NOUN
ap-1849	66	9	in	in	ADP
ap-1849	66	10	the	the	DET
ap-1849	66	11	space	space	NOUN
ap-1849	66	12	w	w	PROPN
ap-1849	66	13	(	(	PUNCT
ap-1849	66	14	λ|µ2	λ|µ2	PROPN
ap-1849	66	15	)	)	PUNCT
ap-1849	66	16	is	be	AUX
ap-1849	66	17	given	give	VERB
ap-1849	66	18	by	by	ADP
ap-1849	66	19	the	the	DET
ap-1849	66	20	vectors	vector	NOUN
ap-1849	66	21	|n	|n	NOUN
ap-1849	66	22	〉	〉	NOUN
ap-1849	66	23	=	=	PUNCT
ap-1849	66	24	|n1	|n1	PROPN
ap-1849	66	25	,	,	PUNCT
ap-1849	66	26	n3	n3	PROPN
ap-1849	66	27	,	,	PUNCT
ap-1849	66	28	n4	n4	PROPN
ap-1849	66	29	,	,	PUNCT
ap-1849	66	30	n2	n2	ADJ
ap-1849	66	31	〉	〉	NOUN
ap-1849	66	32	=	=	SYM
ap-1849	66	33	(	(	PUNCT
ap-1849	66	34	λ2	λ2	NOUN
ap-1849	66	35	−	−	NOUN
ap-1849	66	36	n2	n2	NOUN
ap-1849	66	37	)	)	PUNCT
ap-1849	66	38	!	!	PUNCT
ap-1849	67	1	fn1	fn1	NOUN
ap-1849	67	2	1	1	NUM
ap-1849	68	1	fn3	fn3	NOUN
ap-1849	68	2	3	3	NUM
ap-1849	68	3	fn4	fn4	NOUN
ap-1849	68	4	4	4	NUM
ap-1849	68	5	fn2	fn2	NOUN
ap-1849	68	6	2	2	NUM
ap-1849	68	7	|0	|0	NUM
ap-1849	68	8	〉	〉	NUM
ap-1849	68	9	,	,	PUNCT
ap-1849	68	10	where	where	SCONJ
ap-1849	68	11	n1	n1	NOUN
ap-1849	68	12	,	,	PUNCT
ap-1849	68	13	n3	n3	PROPN
ap-1849	68	14	,	,	PUNCT
ap-1849	68	15	n4	n4	PROPN
ap-1849	68	16	∈	∈	PROPN
ap-1849	68	17	n0	n0	PROPN
ap-1849	68	18	and	and	CCONJ
ap-1849	68	19	n2	n2	ADJ
ap-1849	68	20	=	=	SYM
ap-1849	68	21	0	0	NUM
ap-1849	68	22	,	,	PUNCT
ap-1849	68	23	1	1	NUM
ap-1849	68	24	,	,	PUNCT
ap-1849	68	25	.	.	PUNCT
ap-1849	68	26	.	.	PUNCT
ap-1849	68	27	.	.	PUNCT
ap-1849	69	1	,	,	PUNCT
ap-1849	69	2	λ2.2	λ2.2	X
ap-1849	69	3	now	now	ADV
ap-1849	69	4	by	by	ADP
ap-1849	69	5	direct	direct	ADJ
ap-1849	69	6	calculation	calculation	NOUN
ap-1849	69	7	we	we	PRON
ap-1849	69	8	obtain	obtain	VERB
ap-1849	69	9	h1|n	h1|n	PROPN
ap-1849	69	10	〉	〉	PROPN
ap-1849	69	11	=	=	SYM
ap-1849	69	12	(	(	PUNCT
ap-1849	69	13	λ1	λ1	PROPN
ap-1849	69	14	−	−	PROPN
ap-1849	69	15	2n1	2n1	PROPN
ap-1849	69	16	+	+	CCONJ
ap-1849	69	17	n2	n2	ADJ
ap-1849	69	18	−	−	PROPN
ap-1849	69	19	n3)|n	n3)|n	PROPN
ap-1849	69	20	〉	〉	PROPN
ap-1849	69	21	,	,	PUNCT
ap-1849	69	22	h2|n	h2|n	PROPN
ap-1849	69	23	〉	〉	NOUN
ap-1849	69	24	=	=	SYM
ap-1849	69	25	(	(	PUNCT
ap-1849	69	26	λ2	λ2	NOUN
ap-1849	69	27	+	+	CCONJ
ap-1849	69	28	2n1	2n1	NUM
ap-1849	69	29	−	−	ADP
ap-1849	69	30	2n2	2n2	NUM
ap-1849	69	31	−	−	PROPN
ap-1849	69	32	2n4)|n	2n4)|n	NUM
ap-1849	69	33	〉	〉	NUM
ap-1849	69	34	,	,	PUNCT
ap-1849	69	35	e1|n	e1|n	VERB
ap-1849	69	36	〉	〉	NOUN
ap-1849	69	37	=	=	PUNCT
ap-1849	69	38	n1(λ1	n1(λ1	ADP
ap-1849	69	39	−	−	PROPN
ap-1849	69	40	n1	n1	PROPN
ap-1849	69	41	+	+	CCONJ
ap-1849	69	42	n2	n2	ADJ
ap-1849	69	43	−	−	PROPN
ap-1849	69	44	n3	n3	NOUN
ap-1849	69	45	+	+	CCONJ
ap-1849	69	46	1)|n1	1)|n1	ADJ
ap-1849	69	47	−	−	PROPN
ap-1849	69	48	1	1	NUM
ap-1849	69	49	,	,	PUNCT
ap-1849	69	50	n3	n3	PROPN
ap-1849	69	51	,	,	PUNCT
ap-1849	69	52	n4	n4	PROPN
ap-1849	69	53	,	,	PUNCT
ap-1849	69	54	n2	n2	ADJ
ap-1849	69	55	〉	〉	NOUN
ap-1849	69	56	−	−	PROPN
ap-1849	69	57	(	(	PUNCT
ap-1849	69	58	λ2	λ2	NOUN
ap-1849	69	59	−	−	NOUN
ap-1849	69	60	n2)n3|n1	n2)n3|n1	NOUN
ap-1849	69	61	,	,	PUNCT
ap-1849	69	62	n3	n3	NOUN
ap-1849	69	63	−	−	PROPN
ap-1849	69	64	1	1	NUM
ap-1849	69	65	,	,	PUNCT
ap-1849	69	66	n4	n4	PROPN
ap-1849	69	67	,	,	PUNCT
ap-1849	69	68	n2	n2	NOUN
ap-1849	69	69	+	+	CCONJ
ap-1849	69	70	1	1	NUM
ap-1849	69	71	〉	〉	NUM
ap-1849	69	72	+	+	NUM
ap-1849	69	73	n3(n3	n3(n3	NUM
ap-1849	69	74	−	−	NOUN
ap-1849	69	75	1)|n1	1)|n1	ADJ
ap-1849	69	76	,	,	PUNCT
ap-1849	69	77	n3	n3	NOUN
ap-1849	69	78	−	−	PROPN
ap-1849	69	79	2	2	NUM
ap-1849	69	80	,	,	PUNCT
ap-1849	69	81	n4	n4	PROPN
ap-1849	69	82	+	+	PROPN
ap-1849	69	83	1	1	NUM
ap-1849	69	84	,	,	PUNCT
ap-1849	69	85	n2	n2	ADJ
ap-1849	69	86	〉	〉	PROPN
ap-1849	69	87	,	,	PUNCT
ap-1849	69	88	e2|n	e2|n	NOUN
ap-1849	69	89	〉	〉	NOUN
ap-1849	69	90	=	=	SYM
ap-1849	69	91	n2|n1	n2|n1	PROPN
ap-1849	69	92	,	,	PUNCT
ap-1849	69	93	n3	n3	PROPN
ap-1849	69	94	,	,	PUNCT
ap-1849	69	95	n4	n4	PROPN
ap-1849	69	96	,	,	PUNCT
ap-1849	69	97	n2	n2	NOUN
ap-1849	69	98	−	−	PROPN
ap-1849	69	99	1	1	NUM
ap-1849	69	100	〉	〉	PROPN
ap-1849	69	101	+	+	NOUN
ap-1849	69	102	2n3|n1	2n3|n1	NOUN
ap-1849	69	103	+	+	CCONJ
ap-1849	69	104	1	1	NUM
ap-1849	69	105	,	,	PUNCT
ap-1849	69	106	n3	n3	NOUN
ap-1849	69	107	−	−	PROPN
ap-1849	69	108	1	1	NUM
ap-1849	69	109	,	,	PUNCT
ap-1849	69	110	n4	n4	PROPN
ap-1849	69	111	,	,	PUNCT
ap-1849	69	112	n2	n2	ADJ
ap-1849	69	113	〉	〉	PROPN
ap-1849	69	114	−	−	PROPN
ap-1849	69	115	n4|n1	n4|n1	ADP
ap-1849	69	116	,	,	PUNCT
ap-1849	69	117	n3	n3	ADJ
ap-1849	69	118	+	+	CCONJ
ap-1849	69	119	1	1	NUM
ap-1849	69	120	,	,	PUNCT
ap-1849	69	121	n4	n4	PROPN
ap-1849	69	122	−	−	PROPN
ap-1849	69	123	1	1	NUM
ap-1849	69	124	,	,	PUNCT
ap-1849	69	125	n2	n2	ADJ
ap-1849	69	126	〉	〉	PROPN
ap-1849	69	127	.	.	PUNCT
ap-1849	70	1	(	(	PUNCT
ap-1849	70	2	3	3	X
ap-1849	70	3	)	)	PUNCT
ap-1849	70	4	it	it	PRON
ap-1849	70	5	is	be	AUX
ap-1849	70	6	possible	possible	ADJ
ap-1849	70	7	to	to	PART
ap-1849	70	8	rewrite	rewrite	VERB
ap-1849	70	9	the	the	DET
ap-1849	70	10	action	action	NOUN
ap-1849	70	11	by	by	ADP
ap-1849	70	12	the	the	DET
ap-1849	70	13	second	second	ADJ
ap-1849	70	14	order	order	NOUN
ap-1849	70	15	differential	differential	NOUN
ap-1849	70	16	operators	operator	NOUN
ap-1849	70	17	(	(	PUNCT
ap-1849	70	18	see	see	VERB
ap-1849	70	19	[	[	X
ap-1849	70	20	10	10	NUM
ap-1849	70	21	,	,	PUNCT
ap-1849	70	22	11	11	NUM
ap-1849	70	23	]	]	PUNCT
ap-1849	70	24	)	)	PUNCT
ap-1849	70	25	on	on	ADP
ap-1849	70	26	the	the	DET
ap-1849	70	27	polynomial	polynomial	ADJ
ap-1849	70	28	functions	function	NOUN
ap-1849	70	29	z1	z1	VERB
ap-1849	70	30	,	,	PUNCT
ap-1849	70	31	z2	z2	PROPN
ap-1849	70	32	,	,	PUNCT
ap-1849	70	33	z3	z3	PROPN
ap-1849	70	34	a	a	DET
ap-1849	70	35	z4	z4	PROPN
ap-1849	70	36	,	,	PUNCT
ap-1849	70	37	which	which	PRON
ap-1849	70	38	are	be	AUX
ap-1849	70	39	in	in	ADP
ap-1849	70	40	variable	variable	ADJ
ap-1849	70	41	z2	z2	PROPN
ap-1849	70	42	up	up	ADP
ap-1849	70	43	to	to	ADP
ap-1849	70	44	the	the	DET
ap-1849	70	45	level	level	NOUN
ap-1849	70	46	λ2	λ2	NOUN
ap-1849	70	47	.	.	PUNCT
ap-1849	71	1	if	if	SCONJ
ap-1849	71	2	we	we	PRON
ap-1849	71	3	put	put	VERB
ap-1849	71	4	|n1	|n1	NOUN
ap-1849	71	5	,	,	PUNCT
ap-1849	71	6	n3	n3	PROPN
ap-1849	71	7	,	,	PUNCT
ap-1849	71	8	n4	n4	PROPN
ap-1849	71	9	,	,	PUNCT
ap-1849	71	10	n2	n2	ADJ
ap-1849	71	11	〉	〉	NOUN
ap-1849	71	12	=	=	SYM
ap-1849	71	13	(	(	PUNCT
ap-1849	71	14	λ2	λ2	NOUN
ap-1849	71	15	−	−	NOUN
ap-1849	71	16	n2	n2	NOUN
ap-1849	71	17	)	)	PUNCT
ap-1849	71	18	!	!	PUNCT
ap-1849	72	1	fn1	fn1	NOUN
ap-1849	72	2	1	1	NUM
ap-1849	73	1	fn3	fn3	NOUN
ap-1849	73	2	3	3	NUM
ap-1849	73	3	fn4	fn4	NOUN
ap-1849	73	4	4	4	NUM
ap-1849	73	5	fn2	fn2	NOUN
ap-1849	73	6	2	2	NUM
ap-1849	73	7	|0	|0	NUM
ap-1849	73	8	〉	〉	PROPN
ap-1849	73	9	↔	↔	PROPN
ap-1849	73	10	zn1	zn1	NOUN
ap-1849	73	11	1	1	NUM
ap-1849	73	12	zn2	zn2	NOUN
ap-1849	73	13	2	2	NUM
ap-1849	73	14	zn3	zn3	NOUN
ap-1849	73	15	3	3	NUM
ap-1849	73	16	zn4	zn4	NOUN
ap-1849	73	17	4	4	NUM
ap-1849	73	18	,	,	PUNCT
ap-1849	73	19	we	we	PRON
ap-1849	73	20	obtain	obtain	VERB
ap-1849	73	21	from	from	ADP
ap-1849	73	22	equations	equation	NOUN
ap-1849	73	23	(	(	PUNCT
ap-1849	73	24	3	3	NUM
ap-1849	73	25	)	)	PUNCT
ap-1849	73	26	for	for	ADP
ap-1849	73	27	the	the	DET
ap-1849	73	28	action	action	NOUN
ap-1849	73	29	on	on	ADP
ap-1849	73	30	polynomials	polynomial	NOUN
ap-1849	73	31	f	f	NOUN
ap-1849	73	32	=	=	SYM
ap-1849	73	33	f(z1	f(z1	ADJ
ap-1849	73	34	,	,	PUNCT
ap-1849	73	35	z2	z2	PROPN
ap-1849	73	36	,	,	PUNCT
ap-1849	73	37	z3	z3	PROPN
ap-1849	73	38	,	,	PUNCT
ap-1849	73	39	z4	z4	PROPN
ap-1849	73	40	)	)	PUNCT
ap-1849	73	41	h1f	h1f	NOUN
ap-1849	73	42	=	=	PUNCT
ap-1849	73	43	λ1f	λ1f	PUNCT
ap-1849	73	44	−	−	NOUN
ap-1849	73	45	2z1f1	2z1f1	NUM
ap-1849	74	1	+	+	CCONJ
ap-1849	74	2	z2f2	z2f2	NUM
ap-1849	74	3	−	−	X
ap-1849	75	1	z3f3	z3f3	NOUN
ap-1849	75	2	,	,	PUNCT
ap-1849	75	3	h2f	h2f	PROPN
ap-1849	75	4	=	=	SYM
ap-1849	75	5	λ2f	λ2f	PROPN
ap-1849	75	6	+	+	NUM
ap-1849	75	7	2z1f1	2z1f1	NUM
ap-1849	75	8	−	−	NOUN
ap-1849	75	9	2z2f2	2z2f2	NUM
ap-1849	75	10	−	−	PROPN
ap-1849	75	11	2z4f4	2z4f4	NUM
ap-1849	75	12	,	,	PUNCT
ap-1849	75	13	e1f	e1f	X
ap-1849	75	14	=	=	PUNCT
ap-1849	76	1	λ1f1	λ1f1	X
ap-1849	76	2	−	−	PROPN
ap-1849	76	3	z1f11	z1f11	PROPN
ap-1849	76	4	+	+	CCONJ
ap-1849	76	5	z2f12	z2f12	PROPN
ap-1849	77	1	−	−	NOUN
ap-1849	77	2	z3f13	z3f13	NUM
ap-1849	77	3	−	−	PROPN
ap-1849	77	4	λ2z2f3	λ2z2f3	PROPN
ap-1849	77	5	+	+	CCONJ
ap-1849	77	6	z2	z2	NUM
ap-1849	77	7	2f23	2f23	NOUN
ap-1849	77	8	+	+	CCONJ
ap-1849	77	9	z4f33	z4f33	NOUN
ap-1849	77	10	,	,	PUNCT
ap-1849	77	11	e2f	e2f	PROPN
ap-1849	77	12	=	=	PUNCT
ap-1849	77	13	f2	f2	PROPN
ap-1849	77	14	+	+	NOUN
ap-1849	77	15	2z1f3	2z1f3	NUM
ap-1849	77	16	−	−	PROPN
ap-1849	77	17	z3f4	z3f4	PROPN
ap-1849	77	18	,	,	PUNCT
ap-1849	77	19	(	(	PUNCT
ap-1849	77	20	4	4	NUM
ap-1849	77	21	)	)	PUNCT
ap-1849	77	22	2if	2if	ADJ
ap-1849	77	23	λ2	λ2	NOUN
ap-1849	77	24	/∈	/∈	PUNCT
ap-1849	78	1	z	z	NOUN
ap-1849	78	2	we	we	PRON
ap-1849	78	3	can	can	AUX
ap-1849	78	4	use	use	VERB
ap-1849	78	5	a	a	DET
ap-1849	78	6	similar	similar	ADJ
ap-1849	78	7	construction	construction	NOUN
ap-1849	78	8	with	with	ADP
ap-1849	78	9	basis	basis	NOUN
ap-1849	78	10	|n	|n	NOUN
ap-1849	78	11	〉	〉	NOUN
ap-1849	78	12	=	=	SYM
ap-1849	78	13	γ(λ2	γ(λ2	NOUN
ap-1849	78	14	−	−	PROPN
ap-1849	78	15	n2	n2	NOUN
ap-1849	78	16	+	+	CCONJ
ap-1849	78	17	1)fn1	1)fn1	NUM
ap-1849	78	18	1	1	NUM
ap-1849	78	19	fn3	fn3	NOUN
ap-1849	78	20	3	3	NUM
ap-1849	78	21	fn4	fn4	NOUN
ap-1849	78	22	4	4	NUM
ap-1849	78	23	fn2	fn2	NOUN
ap-1849	78	24	2	2	NUM
ap-1849	78	25	|0	|0	NUM
ap-1849	78	26	〉	〉	NUM
ap-1849	78	27	,	,	PUNCT
ap-1849	78	28	where	where	SCONJ
ap-1849	78	29	n1	n1	NOUN
ap-1849	78	30	,	,	PUNCT
ap-1849	78	31	n2	n2	NOUN
ap-1849	78	32	,	,	PUNCT
ap-1849	78	33	n3	n3	PROPN
ap-1849	78	34	,	,	PUNCT
ap-1849	78	35	n4	n4	PROPN
ap-1849	78	36	∈	∈	PROPN
ap-1849	78	37	n0	n0	PROPN
ap-1849	78	38	.	.	PROPN
ap-1849	78	39	400	400	NUM
ap-1849	78	40	vol	vol	NOUN
ap-1849	78	41	.	.	PUNCT
ap-1849	79	1	53	53	NUM
ap-1849	79	2	no	no	NOUN
ap-1849	79	3	.	.	PUNCT
ap-1849	80	1	5/2013	5/2013	NUM
ap-1849	80	2	extremal	extremal	ADJ
ap-1849	80	3	vectors	vector	NOUN
ap-1849	80	4	for	for	ADP
ap-1849	80	5	verma	verma	NOUN
ap-1849	80	6	type	type	NOUN
ap-1849	80	7	representation	representation	NOUN
ap-1849	80	8	of	of	ADP
ap-1849	80	9	b2	b2	NOUN
ap-1849	80	10	where	where	SCONJ
ap-1849	80	11	fk	fk	INTJ
ap-1849	80	12	=	=	SYM
ap-1849	80	13	∂f	∂f	PROPN
ap-1849	80	14	∂zk	∂zk	PROPN
ap-1849	80	15	.	.	PUNCT
ap-1849	81	1	the	the	DET
ap-1849	81	2	conditions	condition	NOUN
ap-1849	81	3	for	for	ADP
ap-1849	81	4	extremal	extremal	ADJ
ap-1849	81	5	vectors	vector	NOUN
ap-1849	81	6	(	(	PUNCT
ap-1849	81	7	2	2	X
ap-1849	81	8	)	)	PUNCT
ap-1849	81	9	are	be	AUX
ap-1849	81	10	now	now	ADV
ap-1849	81	11	λ1f	λ1f	PUNCT
ap-1849	81	12	−	−	NOUN
ap-1849	81	13	2z1f1	2z1f1	NUM
ap-1849	82	1	+	+	CCONJ
ap-1849	82	2	z2f2	z2f2	X
ap-1849	82	3	−	−	NOUN
ap-1849	82	4	z3f3	z3f3	NOUN
ap-1849	82	5	=	=	SYM
ap-1849	82	6	ν1f	ν1f	PROPN
ap-1849	82	7	,	,	PUNCT
ap-1849	82	8	λ2f	λ2f	PROPN
ap-1849	82	9	+	+	CCONJ
ap-1849	82	10	2z1f1	2z1f1	NUM
ap-1849	82	11	−	−	NOUN
ap-1849	82	12	2z2f2	2z2f2	NUM
ap-1849	83	1	−	−	PROPN
ap-1849	83	2	2z4f4	2z4f4	NOUN
ap-1849	84	1	=	=	PUNCT
ap-1849	84	2	ν2f	ν2f	PROPN
ap-1849	84	3	,	,	PUNCT
ap-1849	84	4	λ1f1	λ1f1	X
ap-1849	84	5	−	−	PROPN
ap-1849	84	6	z1f11	z1f11	PROPN
ap-1849	84	7	+	+	CCONJ
ap-1849	84	8	z2f12	z2f12	PROPN
ap-1849	84	9	−	−	NOUN
ap-1849	84	10	z3f13	z3f13	NUM
ap-1849	84	11	−	−	PROPN
ap-1849	84	12	λ2z2f3	λ2z2f3	PROPN
ap-1849	84	13	+	+	CCONJ
ap-1849	84	14	z2	z2	NUM
ap-1849	84	15	2f23	2f23	NOUN
ap-1849	84	16	+	+	CCONJ
ap-1849	84	17	z4f33	z4f33	X
ap-1849	84	18	=	=	SYM
ap-1849	84	19	0	0	NUM
ap-1849	84	20	,	,	PUNCT
ap-1849	84	21	f2	f2	PROPN
ap-1849	84	22	+	+	NOUN
ap-1849	84	23	2z1f3	2z1f3	NUM
ap-1849	84	24	−	−	NOUN
ap-1849	84	25	z3f4	z3f4	NOUN
ap-1849	84	26	=	=	SYM
ap-1849	84	27	0	0	PROPN
ap-1849	84	28	,	,	PUNCT
ap-1849	84	29	(	(	PUNCT
ap-1849	84	30	5	5	NUM
ap-1849	84	31	)	)	PUNCT
ap-1849	84	32	where	where	SCONJ
ap-1849	84	33	ν1	ν1	NOUN
ap-1849	84	34	and	and	CCONJ
ap-1849	84	35	ν2	ν2	NOUN
ap-1849	84	36	are	be	AUX
ap-1849	84	37	complex	complex	ADJ
ap-1849	84	38	numbers	number	NOUN
ap-1849	84	39	.	.	PUNCT
ap-1849	85	1	the	the	DET
ap-1849	85	2	condition	condition	NOUN
ap-1849	85	3	on	on	ADP
ap-1849	85	4	the	the	DET
ap-1849	85	5	degree	degree	NOUN
ap-1849	85	6	of	of	ADP
ap-1849	85	7	the	the	DET
ap-1849	85	8	polynomial	polynomial	ADJ
ap-1849	85	9	f(z1	f(z1	NOUN
ap-1849	85	10	,	,	PUNCT
ap-1849	85	11	z2	z2	PROPN
ap-1849	85	12	,	,	PUNCT
ap-1849	85	13	z3	z3	PROPN
ap-1849	85	14	,	,	PUNCT
ap-1849	85	15	z4	z4	PROPN
ap-1849	85	16	)	)	PUNCT
ap-1849	85	17	in	in	ADP
ap-1849	85	18	variable	variable	ADJ
ap-1849	85	19	z2	z2	PROPN
ap-1849	85	20	can	can	AUX
ap-1849	85	21	be	be	AUX
ap-1849	85	22	rewritten	rewrite	VERB
ap-1849	85	23	in	in	ADP
ap-1849	85	24	the	the	DET
ap-1849	85	25	following	following	ADJ
ap-1849	85	26	way	way	NOUN
ap-1849	85	27	∂λ2	∂λ2	PROPN
ap-1849	85	28	+	+	PROPN
ap-1849	85	29	1f	1f	NOUN
ap-1849	85	30	∂zλ2	∂zλ2	NUM
ap-1849	85	31	+	+	NOUN
ap-1849	85	32	1	1	NUM
ap-1849	85	33	2	2	NUM
ap-1849	85	34	=	=	SYM
ap-1849	85	35	0	0	NUM
ap-1849	85	36	.	.	NOUN
ap-1849	86	1	5	5	NUM
ap-1849	86	2	.	.	PUNCT
ap-1849	87	1	the	the	DET
ap-1849	87	2	extremal	extremal	ADJ
ap-1849	87	3	vectors	vector	NOUN
ap-1849	87	4	the	the	DET
ap-1849	87	5	extremal	extremal	ADJ
ap-1849	87	6	vectors	vector	NOUN
ap-1849	87	7	are	be	AUX
ap-1849	87	8	in	in	ADP
ap-1849	87	9	one	one	NUM
ap-1849	87	10	-	-	PUNCT
ap-1849	87	11	to	to	ADP
ap-1849	87	12	-	-	PUNCT
ap-1849	87	13	one	one	NUM
ap-1849	87	14	correspondence	correspondence	NOUN
ap-1849	87	15	to	to	ADP
ap-1849	87	16	polynomial	polynomial	ADJ
ap-1849	87	17	solutions	solution	NOUN
ap-1849	87	18	of	of	ADP
ap-1849	87	19	the	the	DET
ap-1849	87	20	systems	system	NOUN
ap-1849	87	21	of	of	ADP
ap-1849	87	22	equations	equation	NOUN
ap-1849	87	23	(	(	PUNCT
ap-1849	87	24	5	5	NUM
ap-1849	87	25	)	)	PUNCT
ap-1849	87	26	,	,	PUNCT
ap-1849	87	27	which	which	PRON
ap-1849	87	28	are	be	AUX
ap-1849	87	29	in	in	ADP
ap-1849	87	30	variable	variable	ADJ
ap-1849	87	31	z2	z2	NOUN
ap-1849	87	32	of	of	ADP
ap-1849	87	33	maximal	maximal	ADJ
ap-1849	87	34	degree	degree	NOUN
ap-1849	87	35	λ2	λ2	NOUN
ap-1849	87	36	.	.	PUNCT
ap-1849	88	1	you	you	PRON
ap-1849	88	2	can	can	AUX
ap-1849	88	3	find	find	VERB
ap-1849	88	4	all	all	DET
ap-1849	88	5	such	such	ADJ
ap-1849	88	6	solutions	solution	NOUN
ap-1849	88	7	in	in	ADP
ap-1849	88	8	the	the	DET
ap-1849	88	9	appendix	appendix	NOUN
ap-1849	88	10	.	.	PUNCT
ap-1849	89	1	for	for	ADP
ap-1849	89	2	any	any	DET
ap-1849	89	3	λ1	λ1	ADJ
ap-1849	89	4	and	and	CCONJ
ap-1849	89	5	λ2	λ2	NOUN
ap-1849	89	6	there	there	PRON
ap-1849	89	7	exists	exist	VERB
ap-1849	89	8	a	a	DET
ap-1849	89	9	constant	constant	ADJ
ap-1849	89	10	solution	solution	NOUN
ap-1849	89	11	f(z1	f(z1	NOUN
ap-1849	89	12	,	,	PUNCT
ap-1849	89	13	z2	z2	PROPN
ap-1849	89	14	,	,	PUNCT
ap-1849	89	15	z3	z3	PROPN
ap-1849	89	16	,	,	PUNCT
ap-1849	89	17	z4	z4	PROPN
ap-1849	89	18	)	)	PUNCT
ap-1849	89	19	=	=	SYM
ap-1849	90	1	1	1	X
ap-1849	90	2	.	.	PUNCT
ap-1849	91	1	but	but	CCONJ
ap-1849	91	2	such	such	DET
ap-1849	91	3	a	a	DET
ap-1849	91	4	solution	solution	NOUN
ap-1849	91	5	gives	give	VERB
ap-1849	91	6	v	v	NOUN
ap-1849	91	7	=	=	PUNCT
ap-1849	91	8	|0	|0	NUM
ap-1849	91	9	〉	〉	NUM
ap-1849	91	10	,	,	PUNCT
ap-1849	91	11	which	which	PRON
ap-1849	91	12	is	be	AUX
ap-1849	91	13	not	not	PART
ap-1849	91	14	interesting	interesting	ADJ
ap-1849	91	15	.	.	PUNCT
ap-1849	92	1	a	a	DET
ap-1849	92	2	further	further	ADJ
ap-1849	92	3	solution	solution	NOUN
ap-1849	92	4	exists	exist	VERB
ap-1849	92	5	only	only	ADV
ap-1849	92	6	in	in	ADP
ap-1849	92	7	the	the	DET
ap-1849	92	8	cases	case	NOUN
ap-1849	92	9	λ1	λ1	PROPN
ap-1849	92	10	∈	∈	PROPN
ap-1849	92	11	n0	n0	PROPN
ap-1849	92	12	,	,	PUNCT
ap-1849	92	13	λ1	λ1	PROPN
ap-1849	92	14	+	+	NUM
ap-1849	92	15	λ2	λ2	NOUN
ap-1849	92	16	+	+	CCONJ
ap-1849	92	17	1	1	NUM
ap-1849	92	18	∈	∈	PROPN
ap-1849	92	19	n0	n0	NOUN
ap-1849	92	20	or	or	CCONJ
ap-1849	92	21	2λ1	2λ1	NUM
ap-1849	92	22	+	+	CCONJ
ap-1849	92	23	λ2	λ2	NOUN
ap-1849	92	24	+	+	CCONJ
ap-1849	92	25	2	2	NUM
ap-1849	92	26	∈	∈	PROPN
ap-1849	92	27	n0	n0	NOUN
ap-1849	92	28	.	.	PUNCT
ap-1849	93	1	for	for	ADP
ap-1849	93	2	λ1	λ1	PROPN
ap-1849	93	3	∈	∈	PROPN
ap-1849	93	4	n0	n0	NOUN
ap-1849	93	5	there	there	PRON
ap-1849	93	6	is	be	VERB
ap-1849	93	7	a	a	DET
ap-1849	93	8	function	function	NOUN
ap-1849	93	9	f(z1	f(z1	NOUN
ap-1849	93	10	,	,	PUNCT
ap-1849	93	11	z2	z2	PROPN
ap-1849	93	12	,	,	PUNCT
ap-1849	93	13	z3	z3	PROPN
ap-1849	93	14	,	,	PUNCT
ap-1849	93	15	z4	z4	PROPN
ap-1849	93	16	)	)	PUNCT
ap-1849	93	17	=	=	PUNCT
ap-1849	94	1	zλ1	zλ1	NOUN
ap-1849	94	2	+	+	PROPN
ap-1849	94	3	1	1	NUM
ap-1849	94	4	1	1	NUM
ap-1849	94	5	,	,	PUNCT
ap-1849	94	6	and	and	CCONJ
ap-1849	94	7	we	we	PRON
ap-1849	94	8	obtain	obtain	VERB
ap-1849	94	9	the	the	DET
ap-1849	94	10	extremal	extremal	ADJ
ap-1849	94	11	vector	vector	NOUN
ap-1849	94	12	v	v	NOUN
ap-1849	94	13	=	=	PUNCT
ap-1849	94	14	fλ1	fλ1	NOUN
ap-1849	94	15	+	+	ADJ
ap-1849	94	16	1	1	NUM
ap-1849	94	17	1	1	NUM
ap-1849	94	18	|0	|0	NUM
ap-1849	94	19	〉	〉	NUM
ap-1849	94	20	.	.	PUNCT
ap-1849	95	1	for	for	ADP
ap-1849	95	2	λ1	λ1	ADJ
ap-1849	95	3	+	+	CCONJ
ap-1849	95	4	λ2	λ2	NOUN
ap-1849	95	5	+	+	CCONJ
ap-1849	95	6	1	1	NUM
ap-1849	95	7	∈	∈	PROPN
ap-1849	95	8	n0	n0	NOUN
ap-1849	95	9	and	and	CCONJ
ap-1849	95	10	2λ1	2λ1	NUM
ap-1849	95	11	+	+	CCONJ
ap-1849	95	12	λ2	λ2	NOUN
ap-1849	95	13	+	+	CCONJ
ap-1849	95	14	4	4	NUM
ap-1849	95	15	≤	≤	NUM
ap-1849	95	16	0	0	NUM
ap-1849	96	1	we	we	PRON
ap-1849	96	2	find	find	VERB
ap-1849	96	3	the	the	DET
ap-1849	96	4	solution	solution	NOUN
ap-1849	96	5	f(z1	f(z1	NOUN
ap-1849	96	6	,	,	PUNCT
ap-1849	96	7	z2	z2	PROPN
ap-1849	96	8	,	,	PUNCT
ap-1849	96	9	z3	z3	PROPN
ap-1849	96	10	,	,	PUNCT
ap-1849	96	11	z4	z4	PROPN
ap-1849	96	12	)	)	PUNCT
ap-1849	96	13	=	=	PUNCT
ap-1849	96	14	(	(	PUNCT
ap-1849	96	15	z4	z4	PROPN
ap-1849	96	16	+	+	CCONJ
ap-1849	96	17	z2z3	z2z3	NUM
ap-1849	96	18	−	−	PROPN
ap-1849	96	19	z1z	z1z	NOUN
ap-1849	96	20	2	2	NUM
ap-1849	96	21	2)λ1+λ2	2)λ1+λ2	NUM
ap-1849	96	22	+	+	ADJ
ap-1849	96	23	2	2	NUM
ap-1849	96	24	=	=	SYM
ap-1849	96	25	∑	∑	PUNCT
ap-1849	96	26	(	(	PUNCT
ap-1849	96	27	n1,n3)∈dλ	n1,n3)∈dλ	NUM
ap-1849	96	28	(	(	PUNCT
ap-1849	96	29	−1)n1(λ1	−1)n1(λ1	NOUN
ap-1849	96	30	+	+	CCONJ
ap-1849	96	31	λ2	λ2	NOUN
ap-1849	96	32	+	+	CCONJ
ap-1849	96	33	2	2	NUM
ap-1849	96	34	)	)	PUNCT
ap-1849	96	35	!	!	PUNCT
ap-1849	97	1	n1!n3	n1!n3	PROPN
ap-1849	97	2	!	!	PUNCT
ap-1849	98	1	(	(	PUNCT
ap-1849	98	2	λ1	λ1	ADJ
ap-1849	98	3	+	+	NUM
ap-1849	98	4	λ2	λ2	NOUN
ap-1849	98	5	−	−	PROPN
ap-1849	98	6	n1	n1	PROPN
ap-1849	98	7	−	−	PROPN
ap-1849	98	8	n3	n3	NOUN
ap-1849	98	9	+	+	CCONJ
ap-1849	98	10	2	2	NUM
ap-1849	98	11	)	)	PUNCT
ap-1849	98	12	!	!	PUNCT
ap-1849	99	1	×	×	NOUN
ap-1849	99	2	zn1	zn1	NOUN
ap-1849	99	3	1	1	NUM
ap-1849	99	4	z2n1+n3	z2n1+n3	NUM
ap-1849	99	5	2	2	NUM
ap-1849	99	6	zn3	zn3	NOUN
ap-1849	99	7	3	3	NUM
ap-1849	99	8	zλ1+λ2−n1−n3	zλ1+λ2−n1−n3	NOUN
ap-1849	99	9	+	+	PROPN
ap-1849	99	10	2	2	NUM
ap-1849	99	11	4	4	NUM
ap-1849	99	12	,	,	PUNCT
ap-1849	99	13	where	where	SCONJ
ap-1849	99	14	dλ	dλ	NOUN
ap-1849	99	15	=	=	PRON
ap-1849	99	16	{	{	PUNCT
ap-1849	99	17	(	(	PUNCT
ap-1849	99	18	n1	n1	NOUN
ap-1849	99	19	,	,	PUNCT
ap-1849	99	20	n3	n3	ADJ
ap-1849	99	21	)	)	PUNCT
ap-1849	99	22	∈	∈	PROPN
ap-1849	99	23	n2	n2	NOUN
ap-1849	99	24	0	0	NUM
ap-1849	99	25	;	;	PUNCT
ap-1849	99	26	n1	n1	PROPN
ap-1849	99	27	+	+	CCONJ
ap-1849	99	28	n3	n3	ADJ
ap-1849	99	29	≤	≤	PUNCT
ap-1849	99	30	λ1	λ1	PROPN
ap-1849	99	31	+	+	CCONJ
ap-1849	99	32	λ2	λ2	NOUN
ap-1849	99	33	+	+	CCONJ
ap-1849	99	34	2	2	NUM
ap-1849	99	35	}	}	PUNCT
ap-1849	99	36	.	.	PUNCT
ap-1849	100	1	the	the	DET
ap-1849	100	2	extremal	extremal	ADJ
ap-1849	100	3	vector	vector	NOUN
ap-1849	100	4	corresponding	correspond	VERB
ap-1849	100	5	to	to	ADP
ap-1849	100	6	this	this	DET
ap-1849	100	7	solution	solution	NOUN
ap-1849	100	8	is	be	AUX
ap-1849	100	9	v	v	ADJ
ap-1849	100	10	=	=	SYM
ap-1849	100	11	∑	∑	PROPN
ap-1849	100	12	(	(	PUNCT
ap-1849	100	13	n1,n3)∈dλ	n1,n3)∈dλ	NUM
ap-1849	100	14	(	(	PUNCT
ap-1849	100	15	−1)n1(λ2	−1)n1(λ2	PROPN
ap-1849	100	16	−	−	PROPN
ap-1849	100	17	2n1	2n1	NUM
ap-1849	100	18	−	−	NOUN
ap-1849	100	19	n3	n3	NOUN
ap-1849	100	20	)	)	PUNCT
ap-1849	100	21	!	!	PUNCT
ap-1849	101	1	n1!n3	n1!n3	PROPN
ap-1849	101	2	!	!	PUNCT
ap-1849	102	1	(	(	PUNCT
ap-1849	102	2	λ1	λ1	ADJ
ap-1849	102	3	+	+	NUM
ap-1849	102	4	λ2	λ2	NOUN
ap-1849	102	5	−	−	PROPN
ap-1849	102	6	n1	n1	PROPN
ap-1849	102	7	−	−	PROPN
ap-1849	102	8	n3	n3	NOUN
ap-1849	102	9	+	+	CCONJ
ap-1849	102	10	2	2	NUM
ap-1849	102	11	)	)	PUNCT
ap-1849	102	12	!	!	PUNCT
ap-1849	103	1	×	×	NOUN
ap-1849	103	2	fn1	fn1	NOUN
ap-1849	103	3	1	1	NUM
ap-1849	103	4	fn3	fn3	NOUN
ap-1849	103	5	3	3	NUM
ap-1849	103	6	fλ1+λ2−n1−n3	fλ1+λ2−n1−n3	NOUN
ap-1849	103	7	+	+	ADJ
ap-1849	103	8	2	2	NUM
ap-1849	103	9	4	4	NUM
ap-1849	103	10	f2n1+n3	f2n1+n3	NOUN
ap-1849	103	11	2	2	NUM
ap-1849	103	12	|0	|0	NUM
ap-1849	103	13	〉	〉	NUM
ap-1849	103	14	.	.	PUNCT
ap-1849	104	1	if	if	SCONJ
ap-1849	104	2	2λ1	2λ1	NUM
ap-1849	104	3	+	+	NUM
ap-1849	104	4	λ2	λ2	NOUN
ap-1849	104	5	+	+	CCONJ
ap-1849	104	6	2	2	NUM
ap-1849	104	7	∈	∈	PROPN
ap-1849	104	8	n0	n0	NOUN
ap-1849	104	9	,	,	PUNCT
ap-1849	104	10	we	we	PRON
ap-1849	104	11	introduce	introduce	VERB
ap-1849	104	12	n	n	ADV
ap-1849	104	13	=	=	SYM
ap-1849	104	14	2λ1	2λ1	NUM
ap-1849	105	1	+	+	NUM
ap-1849	105	2	λ2	λ2	NOUN
ap-1849	105	3	+	+	CCONJ
ap-1849	105	4	3	3	NUM
ap-1849	105	5	,	,	PUNCT
ap-1849	105	6	`	`	PUNCT
ap-1849	105	7	2	2	X
ap-1849	105	8	=	=	SYM
ap-1849	105	9	[	[	PUNCT
ap-1849	105	10	1	1	NUM
ap-1849	105	11	2λ2	2λ2	NUM
ap-1849	105	12	]	]	PUNCT
ap-1849	105	13	,	,	PUNCT
ap-1849	105	14	m	m	VERB
ap-1849	105	15	=	=	PUNCT
ap-1849	105	16	[	[	PUNCT
ap-1849	105	17	1	1	NUM
ap-1849	105	18	2n	2n	NUM
ap-1849	105	19	]	]	PUNCT
ap-1849	105	20	.	.	PUNCT
ap-1849	106	1	then	then	ADV
ap-1849	106	2	we	we	PRON
ap-1849	106	3	can	can	AUX
ap-1849	106	4	rewrite	rewrite	VERB
ap-1849	106	5	the	the	DET
ap-1849	106	6	solution	solution	NOUN
ap-1849	106	7	from	from	ADP
ap-1849	106	8	the	the	DET
ap-1849	106	9	appendix	appendix	NOUN
ap-1849	106	10	in	in	ADP
ap-1849	106	11	the	the	DET
ap-1849	106	12	following	following	ADJ
ap-1849	106	13	way	way	NOUN
ap-1849	106	14	:	:	PUNCT
ap-1849	106	15	for	for	ADP
ap-1849	106	16	λ1	λ1	PROPN
ap-1849	106	17	being	be	AUX
ap-1849	106	18	a	a	DET
ap-1849	106	19	half	half	NOUN
ap-1849	106	20	integer	integer	NOUN
ap-1849	106	21	,	,	PUNCT
ap-1849	106	22	i.e.	i.e.	X
ap-1849	106	23	λ1	λ1	ADJ
ap-1849	106	24	=	=	SYM
ap-1849	106	25	`	`	PUNCT
ap-1849	106	26	1	1	NUM
ap-1849	106	27	−	−	NUM
ap-1849	106	28	1	1	NUM
ap-1849	106	29	2	2	NUM
ap-1849	106	30	,	,	PUNCT
ap-1849	106	31	where	where	SCONJ
ap-1849	106	32	`	`	PUNCT
ap-1849	106	33	1	1	NUM
ap-1849	106	34	∈	∈	PROPN
ap-1849	106	35	z	z	NOUN
ap-1849	106	36	,	,	PUNCT
ap-1849	106	37	we	we	PRON
ap-1849	106	38	have	have	VERB
ap-1849	106	39	f	f	NOUN
ap-1849	106	40	=	=	PUNCT
ap-1849	106	41	m∑	m∑	PROPN
ap-1849	106	42	n4=0	n4=0	ADJ
ap-1849	106	43	min(λ2,n−2n4)∑	min(λ2,n−2n4)∑	NOUN
ap-1849	106	44	n2=0	n2=0	PROPN
ap-1849	106	45	(	(	PUNCT
ap-1849	106	46	−1)n2	−1)n2	PROPN
ap-1849	106	47	cn2,n4	cn2,n4	PROPN
ap-1849	106	48	n2!n4	n2!n4	PROPN
ap-1849	106	49	!	!	PUNCT
ap-1849	107	1	×	×	PROPN
ap-1849	107	2	zn2+n4	zn2+n4	NUM
ap-1849	107	3	1	1	NUM
ap-1849	107	4	zn2	zn2	PROPN
ap-1849	107	5	2	2	NUM
ap-1849	107	6	zn−n2−2n4	zn−n2−2n4	SYM
ap-1849	107	7	3	3	NUM
ap-1849	107	8	zn4	zn4	NOUN
ap-1849	107	9	4	4	NUM
ap-1849	107	10	,	,	PUNCT
ap-1849	107	11	where	where	SCONJ
ap-1849	107	12	cn2,n4	cn2,n4	PROPN
ap-1849	107	13	=	=	SYM
ap-1849	107	14			NUM
ap-1849	107	15	∑min(`2,m−n4	∑min(`2,m−n4	NOUN
ap-1849	107	16	)	)	PUNCT
ap-1849	107	17	n=	n=	NOUN
ap-1849	107	18	[	[	PUNCT
ap-1849	107	19	1	1	NUM
ap-1849	107	20	2	2	NUM
ap-1849	107	21	(	(	PUNCT
ap-1849	107	22	n2	n2	ADJ
ap-1849	107	23	+	+	NOUN
ap-1849	107	24	1	1	NUM
ap-1849	107	25	)	)	PUNCT
ap-1849	107	26	]	]	PUNCT
ap-1849	108	1	22n+n2	22n+n2	NUM
ap-1849	108	2	+	+	NOUN
ap-1849	108	3	2n4	2n4	NUM
ap-1849	108	4	×	×	NOUN
ap-1849	108	5	`	`	PUNCT
ap-1849	108	6	2!m	2!m	NOUN
ap-1849	108	7	!	!	PUNCT
ap-1849	109	1	(	(	PUNCT
ap-1849	109	2	2n−n2	2n−n2	NUM
ap-1849	109	3	)	)	PUNCT
ap-1849	109	4	!	!	PUNCT
ap-1849	110	1	(	(	PUNCT
ap-1849	110	2	`	`	PUNCT
ap-1849	110	3	2−n	2−n	NUM
ap-1849	110	4	)	)	PUNCT
ap-1849	110	5	!	!	PUNCT
ap-1849	111	1	(	(	PUNCT
ap-1849	111	2	m−n−n4	m−n−n4	PROPN
ap-1849	111	3	)	)	PUNCT
ap-1849	111	4	!	!	PUNCT
ap-1849	112	1	,	,	PUNCT
ap-1849	112	2	λ2	λ2	NOUN
ap-1849	112	3	even,∑min(`2,m−n4	even,∑min(`2,m−n4	PROPN
ap-1849	112	4	)	)	PUNCT
ap-1849	112	5	n=	n=	NOUN
ap-1849	112	6	[	[	PUNCT
ap-1849	112	7	1	1	NUM
ap-1849	112	8	2n2	2n2	NUM
ap-1849	112	9	]	]	PUNCT
ap-1849	112	10	22n+n2	22n+n2	NUM
ap-1849	112	11	+	+	NOUN
ap-1849	112	12	2n4	2n4	NUM
ap-1849	112	13	×	×	NOUN
ap-1849	112	14	`	`	PUNCT
ap-1849	112	15	2!m	2!m	NOUN
ap-1849	112	16	!	!	PUNCT
ap-1849	113	1	(	(	PUNCT
ap-1849	113	2	2n−n2	2n−n2	NUM
ap-1849	113	3	+	+	NOUN
ap-1849	113	4	1	1	NUM
ap-1849	113	5	)	)	PUNCT
ap-1849	113	6	!	!	PUNCT
ap-1849	114	1	(	(	PUNCT
ap-1849	114	2	`	`	PUNCT
ap-1849	114	3	2−n	2−n	NUM
ap-1849	114	4	)	)	PUNCT
ap-1849	114	5	!	!	PUNCT
ap-1849	115	1	(	(	PUNCT
ap-1849	115	2	m−n−n4	m−n−n4	PROPN
ap-1849	115	3	)	)	PUNCT
ap-1849	115	4	!	!	PUNCT
ap-1849	116	1	,	,	PUNCT
ap-1849	116	2	λ2	λ2	NOUN
ap-1849	116	3	odd	odd	ADJ
ap-1849	116	4	.	.	PUNCT
ap-1849	117	1	for	for	ADP
ap-1849	117	2	these	these	DET
ap-1849	117	3	solutions	solution	NOUN
ap-1849	117	4	we	we	PRON
ap-1849	117	5	obtain	obtain	VERB
ap-1849	117	6	the	the	DET
ap-1849	117	7	extremal	extremal	ADJ
ap-1849	117	8	vectors	vector	NOUN
ap-1849	117	9	v	v	ADP
ap-1849	117	10	=	=	SYM
ap-1849	117	11	m∑	m∑	PROPN
ap-1849	117	12	n4=0	n4=0	ADJ
ap-1849	117	13	min(λ2,n−2n4)∑	min(λ2,n−2n4)∑	NOUN
ap-1849	117	14	n2=0	n2=0	PROPN
ap-1849	117	15	(	(	PUNCT
ap-1849	117	16	−1)n2	−1)n2	PROPN
ap-1849	117	17	(	(	PUNCT
ap-1849	117	18	λ2	λ2	PROPN
ap-1849	117	19	−	−	NOUN
ap-1849	117	20	n2	n2	NOUN
ap-1849	117	21	)	)	PUNCT
ap-1849	117	22	!	!	PUNCT
ap-1849	118	1	n2!n4	n2!n4	PROPN
ap-1849	118	2	!	!	PUNCT
ap-1849	119	1	cn2,n4	cn2,n4	PROPN
ap-1849	119	2	×	×	PROPN
ap-1849	120	1	fn2+n4	fn2+n4	PROPN
ap-1849	120	2	1	1	NUM
ap-1849	120	3	fn−n2−2n4	fn−n2−2n4	NOUN
ap-1849	120	4	3	3	NUM
ap-1849	120	5	fn4	fn4	NOUN
ap-1849	120	6	4	4	NUM
ap-1849	120	7	fn2	fn2	NOUN
ap-1849	120	8	2	2	NUM
ap-1849	120	9	|0	|0	NUM
ap-1849	120	10	〉	〉	NUM
ap-1849	120	11	.	.	PUNCT
ap-1849	121	1	if	if	SCONJ
ap-1849	121	2	λ1	λ1	PROPN
ap-1849	121	3	is	be	AUX
ap-1849	121	4	an	an	DET
ap-1849	121	5	integer	integer	NOUN
ap-1849	121	6	we	we	PRON
ap-1849	121	7	have	have	VERB
ap-1849	121	8	λ1	λ1	ADJ
ap-1849	121	9	≤	≤	ADJ
ap-1849	121	10	−2	−2	NOUN
ap-1849	121	11	.	.	PUNCT
ap-1849	122	1	the	the	DET
ap-1849	122	2	solution	solution	NOUN
ap-1849	122	3	of	of	ADP
ap-1849	122	4	the	the	DET
ap-1849	122	5	differential	differential	ADJ
ap-1849	122	6	equations	equation	NOUN
ap-1849	122	7	in	in	ADP
ap-1849	122	8	this	this	DET
ap-1849	122	9	case	case	NOUN
ap-1849	122	10	is	be	AUX
ap-1849	122	11	f	f	PROPN
ap-1849	122	12	=	=	PUNCT
ap-1849	122	13	m∑	m∑	PROPN
ap-1849	122	14	n4=0	n4=0	PROPN
ap-1849	122	15	n−2n4∑	n−2n4∑	PROPN
ap-1849	122	16	n2=0	n2=0	PROPN
ap-1849	122	17	(	(	PUNCT
ap-1849	122	18	−1)n2	−1)n2	PROPN
ap-1849	122	19	dn2,n4	dn2,n4	PROPN
ap-1849	122	20	n2!n4	n2!n4	PROPN
ap-1849	122	21	!	!	PUNCT
ap-1849	123	1	×	×	PROPN
ap-1849	123	2	zn2+n4	zn2+n4	NUM
ap-1849	123	3	1	1	NUM
ap-1849	123	4	zn2	zn2	PROPN
ap-1849	123	5	2	2	NUM
ap-1849	123	6	zn−n2−2n4	zn−n2−2n4	SYM
ap-1849	123	7	3	3	NUM
ap-1849	123	8	zn4	zn4	NOUN
ap-1849	123	9	4	4	NUM
ap-1849	123	10	,	,	PUNCT
ap-1849	123	11	where	where	SCONJ
ap-1849	123	12	dn2,n4	dn2,n4	ADJ
ap-1849	123	13	=	=	SYM
ap-1849	123	14			NUM
ap-1849	123	15	∑m−n4	∑m−n4	PROPN
ap-1849	123	16	n=	n=	NOUN
ap-1849	123	17	[	[	PUNCT
ap-1849	123	18	1	1	NUM
ap-1849	123	19	2	2	NUM
ap-1849	123	20	(	(	PUNCT
ap-1849	123	21	n2	n2	ADJ
ap-1849	123	22	+	+	NOUN
ap-1849	123	23	1	1	NUM
ap-1849	123	24	)	)	PUNCT
ap-1849	123	25	]	]	PUNCT
ap-1849	124	1	2n+n2	2n+n2	NUM
ap-1849	124	2	+	+	NOUN
ap-1849	124	3	2n4	2n4	NUM
ap-1849	124	4	(	(	PUNCT
ap-1849	124	5	2`2−1	2`2−1	NUM
ap-1849	124	6	)	)	PUNCT
ap-1849	124	7	!	!	PUNCT
ap-1849	124	8	!	!	PUNCT
ap-1849	125	1	(	(	PUNCT
ap-1849	125	2	2`2−2n−1	2`2−2n−1	NOUN
ap-1849	125	3	)	)	PUNCT
ap-1849	125	4	!	!	PUNCT
ap-1849	125	5	!	!	PUNCT
ap-1849	126	1	×	×	NOUN
ap-1849	126	2	m	m	INTJ
ap-1849	126	3	!	!	PUNCT
ap-1849	127	1	(	(	PUNCT
ap-1849	127	2	2n−n2	2n−n2	NUM
ap-1849	127	3	)	)	PUNCT
ap-1849	127	4	!	!	PUNCT
ap-1849	128	1	(	(	PUNCT
ap-1849	128	2	m−n−n4	m−n−n4	PROPN
ap-1849	128	3	)	)	PUNCT
ap-1849	128	4	!	!	PUNCT
ap-1849	129	1	,	,	PUNCT
ap-1849	129	2	λ2	λ2	NOUN
ap-1849	129	3	even,∑m−n4	even,∑m−n4	PROPN
ap-1849	129	4	n=	n=	NOUN
ap-1849	129	5	[	[	PUNCT
ap-1849	129	6	1	1	NUM
ap-1849	129	7	2n2	2n2	NUM
ap-1849	129	8	]	]	X
ap-1849	129	9	2n+n2	2n+n2	PRON
ap-1849	129	10	+	+	PROPN
ap-1849	129	11	2n4	2n4	NUM
ap-1849	129	12	(	(	PUNCT
ap-1849	129	13	2`2−1	2`2−1	NUM
ap-1849	129	14	)	)	PUNCT
ap-1849	129	15	!	!	PUNCT
ap-1849	129	16	!	!	PUNCT
ap-1849	130	1	(	(	PUNCT
ap-1849	130	2	2`2−2n−1	2`2−2n−1	NOUN
ap-1849	130	3	)	)	PUNCT
ap-1849	130	4	!	!	PUNCT
ap-1849	130	5	!	!	PUNCT
ap-1849	131	1	×	×	NOUN
ap-1849	131	2	m	m	INTJ
ap-1849	131	3	!	!	PUNCT
ap-1849	132	1	(	(	PUNCT
ap-1849	132	2	2n−n2	2n−n2	NUM
ap-1849	132	3	+	+	NOUN
ap-1849	132	4	1	1	NUM
ap-1849	132	5	)	)	PUNCT
ap-1849	132	6	!	!	PUNCT
ap-1849	133	1	(	(	PUNCT
ap-1849	133	2	m−n−n4	m−n−n4	PROPN
ap-1849	133	3	)	)	PUNCT
ap-1849	133	4	!	!	PUNCT
ap-1849	134	1	,	,	PUNCT
ap-1849	135	1	λ2	λ2	NOUN
ap-1849	135	2	odd	odd	ADJ
ap-1849	135	3	,	,	PUNCT
ap-1849	135	4	and	and	CCONJ
ap-1849	135	5	the	the	DET
ap-1849	135	6	extremal	extremal	ADJ
ap-1849	135	7	vectors	vector	NOUN
ap-1849	135	8	are	be	AUX
ap-1849	135	9	v	v	ADP
ap-1849	135	10	=	=	PUNCT
ap-1849	135	11	m∑	m∑	PROPN
ap-1849	135	12	n4=0	n4=0	PROPN
ap-1849	135	13	n−2n4∑	n−2n4∑	PROPN
ap-1849	135	14	n2=0	n2=0	PROPN
ap-1849	135	15	(	(	PUNCT
ap-1849	135	16	−1)n2	−1)n2	PROPN
ap-1849	135	17	(	(	PUNCT
ap-1849	135	18	λ2	λ2	PROPN
ap-1849	135	19	−	−	NOUN
ap-1849	135	20	n2	n2	NOUN
ap-1849	135	21	)	)	PUNCT
ap-1849	135	22	!	!	PUNCT
ap-1849	136	1	n2!n4	n2!n4	PROPN
ap-1849	136	2	!	!	PUNCT
ap-1849	137	1	dn2,n4	dn2,n4	ADJ
ap-1849	137	2	×	×	NOUN
ap-1849	137	3	fn2+n4	fn2+n4	PROPN
ap-1849	137	4	1	1	NUM
ap-1849	137	5	fn−n2−2n4	fn−n2−2n4	NOUN
ap-1849	137	6	3	3	NUM
ap-1849	137	7	fn4	fn4	NOUN
ap-1849	137	8	4	4	NUM
ap-1849	137	9	fn2	fn2	NOUN
ap-1849	137	10	2	2	NUM
ap-1849	137	11	|0	|0	NUM
ap-1849	137	12	〉	〉	NUM
ap-1849	137	13	.	.	PROPN
ap-1849	138	1	6	6	NUM
ap-1849	138	2	.	.	X
ap-1849	138	3	appendix	appendix	NOUN
ap-1849	138	4	:	:	PUNCT
ap-1849	138	5	polynomial	polynomial	ADJ
ap-1849	138	6	solutions	solution	NOUN
ap-1849	138	7	of	of	ADP
ap-1849	138	8	differential	differential	ADJ
ap-1849	138	9	equations	equation	NOUN
ap-1849	138	10	to	to	PART
ap-1849	138	11	obtain	obtain	VERB
ap-1849	138	12	extremal	extremal	ADJ
ap-1849	138	13	vectors	vector	NOUN
ap-1849	138	14	we	we	PRON
ap-1849	138	15	need	need	VERB
ap-1849	138	16	to	to	PART
ap-1849	138	17	find	find	VERB
ap-1849	138	18	the	the	DET
ap-1849	138	19	polynomial	polynomial	ADJ
ap-1849	138	20	solutions	solution	NOUN
ap-1849	138	21	f(z1	f(z1	ADJ
ap-1849	138	22	,	,	PUNCT
ap-1849	138	23	z2	z2	PROPN
ap-1849	138	24	,	,	PUNCT
ap-1849	138	25	z3	z3	PROPN
ap-1849	138	26	,	,	PUNCT
ap-1849	138	27	z4	z4	X
ap-1849	138	28	)	)	PUNCT
ap-1849	138	29	=	=	PUNCT
ap-1849	138	30	∑	∑	PUNCT
ap-1849	138	31	n1,n2,n3,n4≥0	n1,n2,n3,n4≥0	PROPN
ap-1849	138	32	cn1,n2,n3,n4z	cn1,n2,n3,n4z	ADV
ap-1849	138	33	n1	n1	PROPN
ap-1849	138	34	1	1	NUM
ap-1849	138	35	zn2	zn2	NOUN
ap-1849	138	36	2	2	NUM
ap-1849	138	37	zn3	zn3	NOUN
ap-1849	138	38	3	3	NUM
ap-1849	138	39	zn4	zn4	NOUN
ap-1849	138	40	4	4	NUM
ap-1849	138	41	of	of	ADP
ap-1849	138	42	the	the	DET
ap-1849	138	43	system	system	NOUN
ap-1849	138	44	of	of	ADP
ap-1849	138	45	equations	equation	NOUN
ap-1849	138	46	(	(	PUNCT
ap-1849	138	47	5	5	NUM
ap-1849	138	48	)	)	PUNCT
ap-1849	138	49	,	,	PUNCT
ap-1849	138	50	which	which	PRON
ap-1849	138	51	are	be	AUX
ap-1849	138	52	of	of	ADP
ap-1849	138	53	less	less	ADJ
ap-1849	138	54	degree	degree	NOUN
ap-1849	138	55	than	than	ADP
ap-1849	138	56	(	(	PUNCT
ap-1849	138	57	λ2	λ2	NOUN
ap-1849	138	58	+	+	NOUN
ap-1849	138	59	1	1	NUM
ap-1849	138	60	)	)	PUNCT
ap-1849	138	61	in	in	ADP
ap-1849	138	62	the	the	DET
ap-1849	138	63	variable	variable	ADJ
ap-1849	138	64	z2	z2	PROPN
ap-1849	138	65	.	.	PUNCT
ap-1849	139	1	to	to	PART
ap-1849	139	2	simplify	simplify	VERB
ap-1849	139	3	the	the	DET
ap-1849	139	4	solution	solution	NOUN
ap-1849	139	5	of	of	ADP
ap-1849	139	6	the	the	DET
ap-1849	139	7	first	first	ADJ
ap-1849	139	8	equations	equation	NOUN
ap-1849	139	9	,	,	PUNCT
ap-1849	139	10	we	we	PRON
ap-1849	139	11	put	put	VERB
ap-1849	139	12	f(z1	f(z1	ADJ
ap-1849	139	13	,	,	PUNCT
ap-1849	139	14	z2	z2	PROPN
ap-1849	139	15	,	,	PUNCT
ap-1849	139	16	z3	z3	PROPN
ap-1849	139	17	,	,	PUNCT
ap-1849	139	18	z4	z4	X
ap-1849	139	19	)	)	PUNCT
ap-1849	139	20	=	=	SYM
ap-1849	139	21	z−ρ2	z−ρ2	NOUN
ap-1849	139	22	1	1	NUM
ap-1849	139	23	(	(	PUNCT
ap-1849	139	24	4z1z4	4z1z4	NUM
ap-1849	139	25	+	+	CCONJ
ap-1849	139	26	z2	z2	PROPN
ap-1849	139	27	3)ρ2+ρ1/2g(t	3)ρ2+ρ1/2g(t	NUM
ap-1849	139	28	,	,	PUNCT
ap-1849	139	29	x1	x1	PROPN
ap-1849	139	30	,	,	PUNCT
ap-1849	139	31	x2	x2	PROPN
ap-1849	139	32	,	,	PUNCT
ap-1849	139	33	x3	x3	ADJ
ap-1849	139	34	)	)	PUNCT
ap-1849	139	35	,	,	PUNCT
ap-1849	139	36	where	where	SCONJ
ap-1849	139	37	ρ1	ρ1	NOUN
ap-1849	139	38	=	=	SYM
ap-1849	139	39	λ1−ν1	λ1−ν1	NUM
ap-1849	139	40	,	,	PUNCT
ap-1849	139	41	ρ2	ρ2	NOUN
ap-1849	139	42	=	=	SYM
ap-1849	139	43	1	1	NUM
ap-1849	139	44	2	2	NUM
ap-1849	139	45	(	(	PUNCT
ap-1849	139	46	λ2−ν2	λ2−ν2	NUM
ap-1849	139	47	)	)	PUNCT
ap-1849	139	48	,	,	PUNCT
ap-1849	139	49	x2	x2	PROPN
ap-1849	139	50	=	=	SYM
ap-1849	139	51	z1	z1	PROPN
ap-1849	139	52	,	,	PUNCT
ap-1849	139	53	x3	x3	ADJ
ap-1849	139	54	=	=	SYM
ap-1849	139	55	z2	z2	PROPN
ap-1849	139	56	and	and	CCONJ
ap-1849	139	57	t	t	NOUN
ap-1849	139	58	=	=	SYM
ap-1849	139	59	(	(	PUNCT
ap-1849	139	60	2z1z2	2z1z2	NUM
ap-1849	139	61	−	−	PROPN
ap-1849	139	62	z3)2	z3)2	PROPN
ap-1849	139	63	4z1z4	4z1z4	NOUN
ap-1849	140	1	+	+	CCONJ
ap-1849	140	2	z2	z2	PROPN
ap-1849	140	3	3	3	NUM
ap-1849	140	4	,	,	PUNCT
ap-1849	140	5	x1	x1	PROPN
ap-1849	140	6	=	=	SYM
ap-1849	140	7	2z1z2	2z1z2	NUM
ap-1849	140	8	−	−	PROPN
ap-1849	140	9	z3	z3	PROPN
ap-1849	140	10	z3	z3	PROPN
ap-1849	140	11	,	,	PUNCT
ap-1849	140	12	401	401	NUM
ap-1849	140	13	č	č	NOUN
ap-1849	140	14	.	.	PUNCT
ap-1849	140	15	burdík	burdík	PROPN
ap-1849	140	16	,	,	PUNCT
ap-1849	140	17	o.	o.	PROPN
ap-1849	140	18	navrátil	navrátil	PROPN
ap-1849	140	19	acta	acta	PROPN
ap-1849	140	20	polytechnica	polytechnica	PROPN
ap-1849	140	21	or	or	CCONJ
ap-1849	140	22	z1	z1	ADJ
ap-1849	140	23	=	=	SYM
ap-1849	140	24	x2	x2	PROPN
ap-1849	140	25	,	,	PUNCT
ap-1849	140	26	z2	z2	PROPN
ap-1849	140	27	=	=	SYM
ap-1849	140	28	x3	x3	ADJ
ap-1849	140	29	and	and	CCONJ
ap-1849	140	30	z3	z3	PROPN
ap-1849	140	31	=	=	SYM
ap-1849	140	32	2x2x3	2x2x3	NUM
ap-1849	140	33	1	1	NUM
ap-1849	141	1	+	+	CCONJ
ap-1849	141	2	x1	x1	PROPN
ap-1849	141	3	,	,	PUNCT
ap-1849	141	4	z4	z4	PROPN
ap-1849	141	5	=	=	SYM
ap-1849	141	6	x2x	x2x	PROPN
ap-1849	141	7	2	2	NUM
ap-1849	141	8	3(x2	3(x2	NUM
ap-1849	141	9	1	1	NUM
ap-1849	141	10	−	−	PROPN
ap-1849	141	11	t	t	PROPN
ap-1849	141	12	)	)	PUNCT
ap-1849	141	13	t(1	t(1	NOUN
ap-1849	141	14	+	+	CCONJ
ap-1849	141	15	x1)2	x1)2	PROPN
ap-1849	141	16	.	.	PUNCT
ap-1849	142	1	the	the	DET
ap-1849	142	2	first	first	ADJ
ap-1849	142	3	order	order	NOUN
ap-1849	142	4	equations	equation	NOUN
ap-1849	142	5	are	be	AUX
ap-1849	142	6	equivalent	equivalent	ADJ
ap-1849	142	7	to	to	ADP
ap-1849	142	8	the	the	DET
ap-1849	142	9	conditions	condition	NOUN
ap-1849	143	1	gx1	gx1	PROPN
ap-1849	143	2	=	=	SYM
ap-1849	143	3	gx2	gx2	PROPN
ap-1849	143	4	=	=	PROPN
ap-1849	143	5	gx3	gx3	PROPN
ap-1849	143	6	=	=	SYM
ap-1849	143	7	0	0	NUM
ap-1849	143	8	,	,	PUNCT
ap-1849	143	9	and	and	CCONJ
ap-1849	143	10	so	so	ADV
ap-1849	143	11	g(t	g(t	PROPN
ap-1849	143	12	,	,	PUNCT
ap-1849	143	13	x1	x1	PROPN
ap-1849	143	14	,	,	PUNCT
ap-1849	143	15	x2	x2	PROPN
ap-1849	143	16	,	,	PUNCT
ap-1849	143	17	x3	x3	ADJ
ap-1849	143	18	)	)	PUNCT
ap-1849	143	19	=	=	SYM
ap-1849	143	20	g(t	g(t	PROPN
ap-1849	143	21	)	)	PUNCT
ap-1849	143	22	.	.	PUNCT
ap-1849	144	1	the	the	DET
ap-1849	144	2	equations	equation	NOUN
ap-1849	144	3	of	of	ADP
ap-1849	144	4	the	the	DET
ap-1849	144	5	second	second	ADJ
ap-1849	144	6	order	order	NOUN
ap-1849	144	7	give	give	VERB
ap-1849	144	8	the	the	DET
ap-1849	144	9	system	system	NOUN
ap-1849	144	10	of	of	ADP
ap-1849	144	11	three	three	NUM
ap-1849	144	12	equations	equation	NOUN
ap-1849	144	13	(	(	PUNCT
ap-1849	144	14	2λ1ρ1	2λ1ρ1	NUM
ap-1849	144	15	+	+	CCONJ
ap-1849	144	16	2λ1ρ2	2λ1ρ2	NUM
ap-1849	144	17	+	+	CCONJ
ap-1849	144	18	λ2ρ1	λ2ρ1	PUNCT
ap-1849	145	1	+	+	CCONJ
ap-1849	145	2	2λ2ρ2	2λ2ρ2	NUM
ap-1849	145	3	−	−	PROPN
ap-1849	145	4	ρ2	ρ2	NOUN
ap-1849	145	5	1	1	NUM
ap-1849	145	6	−	−	PROPN
ap-1849	145	7	2ρ1ρ2	2ρ1ρ2	NOUN
ap-1849	145	8	−	−	NOUN
ap-1849	145	9	2ρ2	2ρ2	NUM
ap-1849	145	10	2	2	NUM
ap-1849	145	11	+	+	CCONJ
ap-1849	145	12	3ρ1	3ρ1	NUM
ap-1849	145	13	+	+	SYM
ap-1849	145	14	4ρ2)g	4ρ2)g	NUM
ap-1849	145	15	=	=	SYM
ap-1849	145	16	0	0	NUM
ap-1849	145	17	,	,	PUNCT
ap-1849	145	18	(	(	PUNCT
ap-1849	145	19	2λ1	2λ1	NUM
ap-1849	145	20	+	+	NUM
ap-1849	145	21	λ2	λ2	NOUN
ap-1849	145	22	−	−	PROPN
ap-1849	145	23	ρ1	ρ1	NOUN
ap-1849	145	24	−	−	PROPN
ap-1849	145	25	2ρ2	2ρ2	NUM
ap-1849	145	26	+	+	SYM
ap-1849	145	27	3)(1−	3)(1−	NUM
ap-1849	146	1	t)g′	t)g′	PROPN
ap-1849	146	2	+	+	CCONJ
ap-1849	146	3	ρ2(λ1	ρ2(λ1	PROPN
ap-1849	146	4	−	−	PROPN
ap-1849	146	5	ρ1	ρ1	NOUN
ap-1849	146	6	−	−	PROPN
ap-1849	146	7	ρ2	ρ2	PROPN
ap-1849	146	8	+	+	CCONJ
ap-1849	146	9	1)g	1)g	PROPN
ap-1849	146	10	=	=	SYM
ap-1849	146	11	0	0	NUM
ap-1849	146	12	,	,	PUNCT
ap-1849	146	13	4t(1−	4t(1−	PROPN
ap-1849	146	14	t)g′′	t)g′′	NOUN
ap-1849	146	15	+	+	CCONJ
ap-1849	146	16	2	2	NUM
ap-1849	146	17	(	(	PUNCT
ap-1849	146	18	1	1	NUM
ap-1849	146	19	+	+	CCONJ
ap-1849	146	20	(	(	PUNCT
ap-1849	146	21	2λ1	2λ1	NUM
ap-1849	146	22	+	+	CCONJ
ap-1849	146	23	2λ2	2λ2	NUM
ap-1849	146	24	+	+	CCONJ
ap-1849	146	25	1)t	1)t	NUM
ap-1849	146	26	)	)	PUNCT
ap-1849	146	27	g′	g′	NOUN
ap-1849	147	1	+	+	CCONJ
ap-1849	147	2	(	(	PUNCT
ap-1849	147	3	2λ1ρ1	2λ1ρ1	ADP
ap-1849	147	4	−	−	NOUN
ap-1849	147	5	ρ2	ρ2	NOUN
ap-1849	147	6	1	1	NUM
ap-1849	147	7	+	+	NUM
ap-1849	147	8	3ρ1	3ρ1	NUM
ap-1849	147	9	+	+	SYM
ap-1849	147	10	2ρ2)g	2ρ2)g	NUM
ap-1849	147	11	=	=	SYM
ap-1849	147	12	0	0	NUM
ap-1849	147	13	.	.	PUNCT
ap-1849	148	1	(	(	PUNCT
ap-1849	148	2	6	6	NUM
ap-1849	148	3	)	)	PUNCT
ap-1849	148	4	as	as	SCONJ
ap-1849	148	5	we	we	PRON
ap-1849	148	6	want	want	VERB
ap-1849	148	7	to	to	PART
ap-1849	148	8	obtain	obtain	VERB
ap-1849	148	9	polynomial	polynomial	ADJ
ap-1849	148	10	solutions	solution	NOUN
ap-1849	148	11	f(z1	f(z1	ADJ
ap-1849	148	12	,	,	PUNCT
ap-1849	148	13	z2	z2	PROPN
ap-1849	148	14	,	,	PUNCT
ap-1849	148	15	z3	z3	PROPN
ap-1849	148	16	,	,	PUNCT
ap-1849	148	17	z4	z4	PROPN
ap-1849	148	18	)	)	PUNCT
ap-1849	148	19	,	,	PUNCT
ap-1849	148	20	which	which	PRON
ap-1849	148	21	are	be	AUX
ap-1849	148	22	in	in	ADP
ap-1849	148	23	variable	variable	ADJ
ap-1849	148	24	z2	z2	PROPN
ap-1849	148	25	of	of	ADP
ap-1849	148	26	less	less	ADJ
ap-1849	148	27	or	or	CCONJ
ap-1849	148	28	equal	equal	ADJ
ap-1849	148	29	degree	degree	NOUN
ap-1849	148	30	λ2	λ2	PROPN
ap-1849	148	31	∈	∈	PROPN
ap-1849	148	32	n0	n0	NUM
ap-1849	148	33	,	,	PUNCT
ap-1849	148	34	there	there	PRON
ap-1849	148	35	must	must	AUX
ap-1849	148	36	be	be	AUX
ap-1849	148	37	solution	solution	NOUN
ap-1849	148	38	g(t	g(t	PROPN
ap-1849	148	39	)	)	PUNCT
ap-1849	148	40	of	of	ADP
ap-1849	148	41	the	the	DET
ap-1849	148	42	system	system	NOUN
ap-1849	148	43	(	(	PUNCT
ap-1849	148	44	6	6	NUM
ap-1849	148	45	)	)	PUNCT
ap-1849	148	46	,	,	PUNCT
ap-1849	148	47	which	which	PRON
ap-1849	148	48	is	be	AUX
ap-1849	148	49	the	the	DET
ap-1849	148	50	polynomial	polynomial	ADJ
ap-1849	148	51	in	in	ADP
ap-1849	148	52	√	√	PROPN
ap-1849	148	53	t	t	NOUN
ap-1849	148	54	of	of	ADP
ap-1849	148	55	less	less	ADJ
ap-1849	148	56	or	or	CCONJ
ap-1849	148	57	equal	equal	ADJ
ap-1849	148	58	degree	degree	NOUN
ap-1849	148	59	λ2	λ2	NOUN
ap-1849	148	60	.	.	PUNCT
ap-1849	149	1	if	if	SCONJ
ap-1849	149	2	we	we	PRON
ap-1849	149	3	exclude	exclude	VERB
ap-1849	149	4	derivatives	derivative	NOUN
ap-1849	149	5	of	of	ADP
ap-1849	149	6	g	g	NOUN
ap-1849	149	7	from	from	ADP
ap-1849	149	8	the	the	DET
ap-1849	149	9	second	second	ADJ
ap-1849	149	10	and	and	CCONJ
ap-1849	149	11	the	the	DET
ap-1849	149	12	third	third	ADJ
ap-1849	149	13	equations	equation	NOUN
ap-1849	149	14	,	,	PUNCT
ap-1849	149	15	we	we	PRON
ap-1849	149	16	find	find	VERB
ap-1849	149	17	that	that	SCONJ
ap-1849	149	18	nonzero	nonzero	PROPN
ap-1849	149	19	solutions	solution	NOUN
ap-1849	149	20	can	can	AUX
ap-1849	149	21	exist	exist	VERB
ap-1849	149	22	only	only	ADV
ap-1849	149	23	in	in	ADP
ap-1849	149	24	the	the	DET
ap-1849	149	25	following	follow	VERB
ap-1849	149	26	six	six	NUM
ap-1849	149	27	cases	case	NOUN
ap-1849	149	28	:	:	PUNCT
ap-1849	149	29	(	(	PUNCT
ap-1849	149	30	1	1	NUM
ap-1849	149	31	.	.	PUNCT
ap-1849	149	32	)	)	PUNCT
ap-1849	149	33	ρ1	ρ1	NOUN
ap-1849	149	34	=	=	SYM
ap-1849	149	35	0	0	NUM
ap-1849	149	36	,	,	PUNCT
ap-1849	150	1	ρ2	ρ2	NOUN
ap-1849	150	2	=	=	SYM
ap-1849	150	3	0	0	NUM
ap-1849	150	4	;	;	PUNCT
ap-1849	150	5	(	(	PUNCT
ap-1849	150	6	2	2	NUM
ap-1849	150	7	.	.	PUNCT
ap-1849	150	8	)	)	PUNCT
ap-1849	150	9	ρ1	ρ1	NOUN
ap-1849	150	10	=	=	PUNCT
ap-1849	150	11	2λ1	2λ1	NUM
ap-1849	151	1	+	+	CCONJ
ap-1849	151	2	2	2	NUM
ap-1849	151	3	,	,	PUNCT
ap-1849	151	4	ρ2	ρ2	NOUN
ap-1849	151	5	=	=	SYM
ap-1849	151	6	−λ1	−λ1	PROPN
ap-1849	151	7	−	−	NOUN
ap-1849	151	8	1	1	NUM
ap-1849	152	1	;	;	PUNCT
ap-1849	152	2	(	(	PUNCT
ap-1849	152	3	3	3	NUM
ap-1849	152	4	.	.	PUNCT
ap-1849	152	5	)	)	PUNCT
ap-1849	152	6	ρ1	ρ1	NOUN
ap-1849	152	7	=	=	SYM
ap-1849	152	8	0	0	NUM
ap-1849	152	9	,	,	PUNCT
ap-1849	152	10	ρ2	ρ2	NOUN
ap-1849	152	11	=	=	SYM
ap-1849	152	12	λ1	λ1	PROPN
ap-1849	153	1	+	+	NUM
ap-1849	153	2	λ2	λ2	NOUN
ap-1849	153	3	+	+	CCONJ
ap-1849	153	4	2	2	NUM
ap-1849	153	5	;	;	PUNCT
ap-1849	153	6	(	(	PUNCT
ap-1849	153	7	4	4	NUM
ap-1849	153	8	.	.	PUNCT
ap-1849	153	9	)	)	PUNCT
ap-1849	153	10	ρ1	ρ1	NOUN
ap-1849	153	11	=	=	PUNCT
ap-1849	153	12	2λ1	2λ1	NUM
ap-1849	154	1	+	+	CCONJ
ap-1849	154	2	2	2	NUM
ap-1849	154	3	,	,	PUNCT
ap-1849	154	4	ρ2	ρ2	NOUN
ap-1849	154	5	=	=	SYM
ap-1849	154	6	λ2	λ2	PROPN
ap-1849	154	7	+	+	CCONJ
ap-1849	154	8	1	1	NUM
ap-1849	154	9	;	;	PUNCT
ap-1849	154	10	(	(	PUNCT
ap-1849	154	11	5	5	NUM
ap-1849	154	12	.	.	PUNCT
ap-1849	154	13	)	)	PUNCT
ap-1849	154	14	ρ1	ρ1	NOUN
ap-1849	154	15	=	=	PUNCT
ap-1849	154	16	2λ1	2λ1	NUM
ap-1849	155	1	+	+	NUM
ap-1849	155	2	λ2	λ2	NOUN
ap-1849	155	3	+	+	CCONJ
ap-1849	155	4	3	3	NUM
ap-1849	155	5	,	,	PUNCT
ap-1849	155	6	ρ2	ρ2	NOUN
ap-1849	155	7	=	=	SYM
ap-1849	155	8	0	0	NUM
ap-1849	155	9	;	;	PUNCT
ap-1849	155	10	(	(	PUNCT
ap-1849	155	11	6	6	NUM
ap-1849	155	12	.	.	PUNCT
ap-1849	155	13	)	)	PUNCT
ap-1849	155	14	ρ1	ρ1	NOUN
ap-1849	155	15	=	=	PUNCT
ap-1849	155	16	−λ2	−λ2	NOUN
ap-1849	155	17	−	−	NOUN
ap-1849	155	18	1	1	NUM
ap-1849	155	19	,	,	PUNCT
ap-1849	155	20	ρ2	ρ2	NOUN
ap-1849	155	21	=	=	SYM
ap-1849	155	22	λ1	λ1	PROPN
ap-1849	155	23	+	+	NUM
ap-1849	155	24	λ2	λ2	NOUN
ap-1849	155	25	+	+	CCONJ
ap-1849	155	26	2	2	NUM
ap-1849	155	27	.	.	X
ap-1849	155	28	case	case	NOUN
ap-1849	155	29	1	1	NUM
ap-1849	155	30	(	(	PUNCT
ap-1849	155	31	ρ1	ρ1	NOUN
ap-1849	155	32	=	=	SYM
ap-1849	155	33	ρ2	ρ2	NOUN
ap-1849	155	34	=	=	NOUN
ap-1849	155	35	0	0	NUM
ap-1849	155	36	)	)	PUNCT
ap-1849	155	37	.	.	PUNCT
ap-1849	156	1	a	a	DET
ap-1849	156	2	function	function	NOUN
ap-1849	156	3	that	that	PRON
ap-1849	156	4	corresponds	correspond	VERB
ap-1849	156	5	to	to	ADP
ap-1849	156	6	the	the	DET
ap-1849	156	7	extremal	extremal	ADJ
ap-1849	156	8	vector	vector	NOUN
ap-1849	156	9	is	be	AUX
ap-1849	156	10	f(z1	f(z1	ADJ
ap-1849	156	11	,	,	PUNCT
ap-1849	156	12	z2	z2	PROPN
ap-1849	156	13	,	,	PUNCT
ap-1849	156	14	z3	z3	PROPN
ap-1849	156	15	,	,	PUNCT
ap-1849	156	16	z4	z4	PROPN
ap-1849	156	17	)	)	PUNCT
ap-1849	156	18	=	=	SYM
ap-1849	156	19	g(t	g(t	PROPN
ap-1849	156	20	)	)	PUNCT
ap-1849	156	21	,	,	PUNCT
ap-1849	156	22	where	where	SCONJ
ap-1849	156	23	g(t	g(t	PROPN
ap-1849	156	24	)	)	PUNCT
ap-1849	156	25	is	be	AUX
ap-1849	156	26	the	the	DET
ap-1849	156	27	solution	solution	NOUN
ap-1849	156	28	of	of	ADP
ap-1849	156	29	the	the	DET
ap-1849	156	30	system	system	NOUN
ap-1849	156	31	(	(	PUNCT
ap-1849	156	32	2λ1	2λ1	NUM
ap-1849	156	33	+	+	NUM
ap-1849	156	34	λ2	λ2	NOUN
ap-1849	156	35	+	+	CCONJ
ap-1849	157	1	3)(1−	3)(1−	NUM
ap-1849	157	2	t)g′	t)g′	PROPN
ap-1849	157	3	=	=	SYM
ap-1849	157	4	0	0	NUM
ap-1849	157	5	,	,	PUNCT
ap-1849	157	6	2t(1−	2t(1−	NUM
ap-1849	157	7	t)g′′	t)g′′	NOUN
ap-1849	157	8	+	+	CCONJ
ap-1849	157	9	(	(	PUNCT
ap-1849	157	10	1	1	NUM
ap-1849	157	11	+	+	CCONJ
ap-1849	157	12	(	(	PUNCT
ap-1849	157	13	2λ1	2λ1	NUM
ap-1849	157	14	+	+	CCONJ
ap-1849	157	15	2λ2	2λ2	NUM
ap-1849	157	16	+	+	CCONJ
ap-1849	157	17	1)t	1)t	NUM
ap-1849	157	18	)	)	PUNCT
ap-1849	157	19	g′	g′	NOUN
ap-1849	157	20	=	=	PUNCT
ap-1849	157	21	0	0	PROPN
ap-1849	157	22	.	.	PUNCT
ap-1849	157	23	(	(	PUNCT
ap-1849	157	24	7	7	NUM
ap-1849	157	25	)	)	PUNCT
ap-1849	157	26	for	for	ADP
ap-1849	157	27	each	each	DET
ap-1849	157	28	λ1	λ1	ADJ
ap-1849	157	29	and	and	CCONJ
ap-1849	157	30	λ2	λ2	NOUN
ap-1849	157	31	this	this	DET
ap-1849	157	32	system	system	NOUN
ap-1849	157	33	has	have	VERB
ap-1849	157	34	the	the	DET
ap-1849	157	35	solution	solution	NOUN
ap-1849	157	36	g(t	g(t	PROPN
ap-1849	157	37	)	)	PUNCT
ap-1849	158	1	=	=	SYM
ap-1849	158	2	1	1	NUM
ap-1849	158	3	which	which	PRON
ap-1849	158	4	corresponds	correspond	VERB
ap-1849	158	5	to	to	ADP
ap-1849	158	6	the	the	DET
ap-1849	158	7	extremal	extremal	ADJ
ap-1849	158	8	vector	vector	NOUN
ap-1849	158	9	f(z1	f(z1	NOUN
ap-1849	158	10	,	,	PUNCT
ap-1849	158	11	z2	z2	PROPN
ap-1849	158	12	,	,	PUNCT
ap-1849	158	13	z3	z3	PROPN
ap-1849	158	14	,	,	PUNCT
ap-1849	158	15	z4	z4	PROPN
ap-1849	158	16	)	)	PUNCT
ap-1849	158	17	=	=	SYM
ap-1849	158	18	1	1	X
ap-1849	158	19	.	.	PUNCT
ap-1849	159	1	but	but	CCONJ
ap-1849	159	2	for	for	ADP
ap-1849	159	3	2λ1	2λ1	NUM
ap-1849	159	4	+	+	NUM
ap-1849	159	5	λ2	λ2	NOUN
ap-1849	159	6	+	+	CCONJ
ap-1849	159	7	3	3	NUM
ap-1849	159	8	=	=	SYM
ap-1849	159	9	0	0	NUM
ap-1849	159	10	we	we	PRON
ap-1849	159	11	obtained	obtain	VERB
ap-1849	159	12	for	for	ADP
ap-1849	159	13	g(t	g(t	PROPN
ap-1849	159	14	)	)	PUNCT
ap-1849	159	15	the	the	DET
ap-1849	159	16	equation	equation	NOUN
ap-1849	159	17	2t(1−	2t(1−	NUM
ap-1849	159	18	t)g′′	t)g′′	NOUN
ap-1849	160	1	+	+	CCONJ
ap-1849	160	2	(	(	PUNCT
ap-1849	160	3	1	1	NUM
ap-1849	160	4	+	+	CCONJ
ap-1849	160	5	(	(	PUNCT
ap-1849	160	6	λ2	λ2	PROPN
ap-1849	160	7	−	−	PROPN
ap-1849	160	8	2)t	2)t	NUM
ap-1849	160	9	)	)	PUNCT
ap-1849	160	10	g′	g′	NOUN
ap-1849	160	11	=	=	SYM
ap-1849	160	12	0	0	PROPN
ap-1849	160	13	,	,	PUNCT
ap-1849	160	14	which	which	PRON
ap-1849	160	15	also	also	ADV
ap-1849	160	16	has	have	VERB
ap-1849	160	17	a	a	DET
ap-1849	160	18	non	non	ADJ
ap-1849	160	19	-	-	ADJ
ap-1849	160	20	constant	constant	ADJ
ap-1849	160	21	solution	solution	NOUN
ap-1849	160	22	g(t	g(t	PROPN
ap-1849	160	23	)	)	PUNCT
ap-1849	160	24	=	=	SYM
ap-1849	160	25	g	g	NOUN
ap-1849	160	26	(	(	PUNCT
ap-1849	160	27	√	√	NOUN
ap-1849	160	28	t	t	NUM
ap-1849	160	29	)	)	PUNCT
ap-1849	160	30	,	,	PUNCT
ap-1849	160	31	where	where	SCONJ
ap-1849	160	32	g(x	g(x	NOUN
ap-1849	160	33	)	)	PUNCT
ap-1849	160	34	=	=	SYM
ap-1849	160	35	∫	∫	PROPN
ap-1849	160	36	(	(	PUNCT
ap-1849	160	37	1−x2)(λ2−1)/2dx	1−x2)(λ2−1)/2dx	NUM
ap-1849	160	38	.	.	PUNCT
ap-1849	161	1	however	however	ADV
ap-1849	161	2	this	this	DET
ap-1849	161	3	solution	solution	NOUN
ap-1849	161	4	does	do	AUX
ap-1849	161	5	not	not	PART
ap-1849	161	6	give	give	VERB
ap-1849	161	7	a	a	DET
ap-1849	161	8	polynomial	polynomial	ADJ
ap-1849	161	9	function	function	NOUN
ap-1849	161	10	f(z1	f(z1	NOUN
ap-1849	161	11	,	,	PUNCT
ap-1849	161	12	z2	z2	PROPN
ap-1849	161	13	,	,	PUNCT
ap-1849	161	14	z3	z3	PROPN
ap-1849	161	15	,	,	PUNCT
ap-1849	161	16	z4	z4	PROPN
ap-1849	161	17	)	)	PUNCT
ap-1849	161	18	for	for	ADP
ap-1849	161	19	any	any	DET
ap-1849	161	20	λ2	λ2	NOUN
ap-1849	161	21	.	.	PUNCT
ap-1849	162	1	case	case	NOUN
ap-1849	162	2	2	2	NUM
ap-1849	162	3	(	(	PUNCT
ap-1849	162	4	ρ1	ρ1	NOUN
ap-1849	162	5	=	=	PUNCT
ap-1849	162	6	2λ1	2λ1	NUM
ap-1849	163	1	+	+	CCONJ
ap-1849	163	2	2	2	NUM
ap-1849	163	3	,	,	PUNCT
ap-1849	163	4	ρ2	ρ2	NOUN
ap-1849	163	5	=	=	SYM
ap-1849	163	6	−λ1	−λ1	PROPN
ap-1849	163	7	−	−	NOUN
ap-1849	163	8	1	1	NUM
ap-1849	163	9	)	)	PUNCT
ap-1849	163	10	.	.	PUNCT
ap-1849	164	1	the	the	DET
ap-1849	164	2	function	function	NOUN
ap-1849	164	3	that	that	PRON
ap-1849	164	4	corresponds	correspond	VERB
ap-1849	164	5	to	to	ADP
ap-1849	164	6	the	the	DET
ap-1849	164	7	extremal	extremal	ADJ
ap-1849	164	8	vector	vector	NOUN
ap-1849	164	9	is	be	AUX
ap-1849	164	10	in	in	ADP
ap-1849	164	11	this	this	DET
ap-1849	164	12	case	case	NOUN
ap-1849	164	13	f(z1	f(z1	NOUN
ap-1849	164	14	,	,	PUNCT
ap-1849	164	15	z2	z2	PROPN
ap-1849	164	16	,	,	PUNCT
ap-1849	164	17	z3	z3	PROPN
ap-1849	164	18	,	,	PUNCT
ap-1849	164	19	z4	z4	PROPN
ap-1849	164	20	)	)	PUNCT
ap-1849	164	21	=	=	PUNCT
ap-1849	165	1	zλ1	zλ1	NOUN
ap-1849	165	2	+	+	ADJ
ap-1849	165	3	1	1	NUM
ap-1849	165	4	1	1	NUM
ap-1849	165	5	g(t	g(t	PROPN
ap-1849	165	6	)	)	PUNCT
ap-1849	165	7	,	,	PUNCT
ap-1849	165	8	where	where	SCONJ
ap-1849	165	9	g(t	g(t	PROPN
ap-1849	165	10	)	)	PUNCT
ap-1849	165	11	is	be	AUX
ap-1849	165	12	the	the	DET
ap-1849	165	13	solution	solution	NOUN
ap-1849	165	14	of	of	ADP
ap-1849	165	15	system	system	NOUN
ap-1849	165	16	(	(	PUNCT
ap-1849	165	17	7	7	NUM
ap-1849	165	18	)	)	PUNCT
ap-1849	165	19	.	.	PUNCT
ap-1849	166	1	as	as	ADP
ap-1849	166	2	in	in	ADP
ap-1849	166	3	event	event	NOUN
ap-1849	166	4	1	1	NUM
ap-1849	166	5	we	we	PRON
ap-1849	166	6	find	find	VERB
ap-1849	166	7	that	that	SCONJ
ap-1849	166	8	the	the	DET
ap-1849	166	9	non	non	ADJ
ap-1849	166	10	-	-	ADJ
ap-1849	166	11	constant	constant	ADJ
ap-1849	166	12	polynomial	polynomial	ADJ
ap-1849	166	13	solutions	solution	NOUN
ap-1849	166	14	f(z1	f(z1	NOUN
ap-1849	166	15	,	,	PUNCT
ap-1849	166	16	z2	z2	PROPN
ap-1849	166	17	,	,	PUNCT
ap-1849	166	18	z3	z3	PROPN
ap-1849	166	19	,	,	PUNCT
ap-1849	166	20	z4	z4	PROPN
ap-1849	166	21	)	)	PUNCT
ap-1849	166	22	=	=	PUNCT
ap-1849	167	1	zλ1	zλ1	NOUN
ap-1849	167	2	+	+	PROPN
ap-1849	167	3	1	1	NUM
ap-1849	167	4	1	1	NUM
ap-1849	167	5	get	get	VERB
ap-1849	167	6	only	only	ADV
ap-1849	167	7	λ1	λ1	PROPN
ap-1849	167	8	∈	∈	PROPN
ap-1849	167	9	n0	n0	PROPN
ap-1849	167	10	.	.	PUNCT
ap-1849	167	11	case	case	NOUN
ap-1849	167	12	3	3	NUM
ap-1849	167	13	(	(	PUNCT
ap-1849	167	14	ρ1	ρ1	NOUN
ap-1849	167	15	=	=	SYM
ap-1849	167	16	0	0	NUM
ap-1849	167	17	,	,	PUNCT
ap-1849	168	1	ρ2	ρ2	NOUN
ap-1849	168	2	=	=	SYM
ap-1849	168	3	λ1	λ1	PROPN
ap-1849	168	4	+	+	NUM
ap-1849	168	5	λ2	λ2	NOUN
ap-1849	168	6	+	+	CCONJ
ap-1849	168	7	2	2	NUM
ap-1849	168	8	.	.	NUM
ap-1849	168	9	)	)	PUNCT
ap-1849	168	10	.	.	PUNCT
ap-1849	169	1	the	the	DET
ap-1849	169	2	function	function	NOUN
ap-1849	169	3	for	for	ADP
ap-1849	169	4	the	the	DET
ap-1849	169	5	extremal	extremal	ADJ
ap-1849	169	6	vectors	vector	NOUN
ap-1849	169	7	is	be	AUX
ap-1849	169	8	f(z1	f(z1	ADJ
ap-1849	169	9	,	,	PUNCT
ap-1849	169	10	z2	z2	PROPN
ap-1849	169	11	,	,	PUNCT
ap-1849	169	12	z3	z3	PROPN
ap-1849	169	13	,	,	PUNCT
ap-1849	169	14	z4	z4	PROPN
ap-1849	169	15	)	)	PUNCT
ap-1849	169	16	=	=	PUNCT
ap-1849	170	1	(	(	PUNCT
ap-1849	170	2	4z1z4	4z1z4	NUM
ap-1849	170	3	+	+	CCONJ
ap-1849	170	4	z2	z2	PROPN
ap-1849	170	5	3	3	NUM
ap-1849	170	6	z1	z1	PROPN
ap-1849	170	7	)	)	PUNCT
ap-1849	171	1	λ1+λ2	λ1+λ2	PROPN
ap-1849	171	2	+	+	PROPN
ap-1849	171	3	2	2	NUM
ap-1849	171	4	g(t	g(t	NOUN
ap-1849	171	5	)	)	PUNCT
ap-1849	171	6	,	,	PUNCT
ap-1849	171	7	where	where	SCONJ
ap-1849	171	8	g(t	g(t	PROPN
ap-1849	171	9	)	)	PUNCT
ap-1849	171	10	is	be	AUX
ap-1849	171	11	the	the	DET
ap-1849	171	12	solution	solution	NOUN
ap-1849	171	13	of	of	ADP
ap-1849	171	14	the	the	DET
ap-1849	171	15	system	system	NOUN
ap-1849	171	16	(	(	PUNCT
ap-1849	171	17	λ2	λ2	NOUN
ap-1849	172	1	+	+	NOUN
ap-1849	173	1	1	1	NUM
ap-1849	173	2	)	)	PUNCT
ap-1849	173	3	(	(	PUNCT
ap-1849	173	4	(	(	PUNCT
ap-1849	173	5	1−	1−	NUM
ap-1849	173	6	t)g′	t)g′	PROPN
ap-1849	173	7	+	+	CCONJ
ap-1849	174	1	(	(	PUNCT
ap-1849	174	2	λ1	λ1	ADJ
ap-1849	174	3	+	+	NUM
ap-1849	174	4	λ2	λ2	NOUN
ap-1849	174	5	+	+	CCONJ
ap-1849	174	6	2)g	2)g	NUM
ap-1849	174	7	)	)	PUNCT
ap-1849	175	1	=	=	SYM
ap-1849	175	2	0	0	NUM
ap-1849	175	3	,	,	PUNCT
ap-1849	175	4	2t(1−	2t(1−	NUM
ap-1849	175	5	t)g′′	t)g′′	NOUN
ap-1849	175	6	+	+	CCONJ
ap-1849	175	7	(	(	PUNCT
ap-1849	175	8	1	1	NUM
ap-1849	175	9	+	+	CCONJ
ap-1849	175	10	(	(	PUNCT
ap-1849	175	11	2λ1	2λ1	NUM
ap-1849	175	12	+	+	CCONJ
ap-1849	175	13	2λ2	2λ2	NUM
ap-1849	175	14	+	+	CCONJ
ap-1849	175	15	1)t	1)t	NUM
ap-1849	175	16	)	)	PUNCT
ap-1849	175	17	g′	g′	NOUN
ap-1849	176	1	+	+	PROPN
ap-1849	176	2	(	(	PUNCT
ap-1849	176	3	λ1	λ1	ADJ
ap-1849	176	4	+	+	NUM
ap-1849	176	5	λ2	λ2	NOUN
ap-1849	176	6	+	+	CCONJ
ap-1849	176	7	2)g	2)g	NUM
ap-1849	176	8	=	=	SYM
ap-1849	176	9	0	0	PROPN
ap-1849	176	10	.	.	PUNCT
ap-1849	177	1	(	(	PUNCT
ap-1849	177	2	8)	8)	NUM
ap-1849	177	3	as	as	SCONJ
ap-1849	177	4	we	we	PRON
ap-1849	177	5	assume	assume	VERB
ap-1849	177	6	that	that	SCONJ
ap-1849	177	7	λ2	λ2	PROPN
ap-1849	177	8	∈	∈	PROPN
ap-1849	177	9	n0,for	n0,for	ADP
ap-1849	177	10	each	each	DET
ap-1849	177	11	λ1	λ1	ADJ
ap-1849	177	12	,	,	PUNCT
ap-1849	177	13	λ2	λ2	NOUN
ap-1849	177	14	this	this	DET
ap-1849	177	15	system	system	NOUN
ap-1849	177	16	has	have	VERB
ap-1849	177	17	the	the	DET
ap-1849	177	18	solution	solution	NOUN
ap-1849	177	19	g(t	g(t	PROPN
ap-1849	177	20	)	)	PUNCT
ap-1849	178	1	=	=	PUNCT
ap-1849	178	2	(	(	PUNCT
ap-1849	178	3	1−	1−	NUM
ap-1849	178	4	t)λ1+λ2	t)λ1+λ2	NUM
ap-1849	178	5	+	+	SYM
ap-1849	178	6	2	2	NUM
ap-1849	178	7	.	.	PUNCT
ap-1849	179	1	this	this	DET
ap-1849	179	2	solution	solution	NOUN
ap-1849	179	3	corresponds	correspond	VERB
ap-1849	179	4	to	to	ADP
ap-1849	179	5	the	the	DET
ap-1849	179	6	function	function	NOUN
ap-1849	179	7	f(z1	f(z1	NOUN
ap-1849	179	8	,	,	PUNCT
ap-1849	179	9	z2	z2	PROPN
ap-1849	179	10	,	,	PUNCT
ap-1849	179	11	z3	z3	PROPN
ap-1849	179	12	,	,	PUNCT
ap-1849	179	13	z4	z4	PROPN
ap-1849	179	14	)	)	PUNCT
ap-1849	179	15	=	=	PUNCT
ap-1849	179	16	(	(	PUNCT
ap-1849	179	17	z4	z4	PROPN
ap-1849	179	18	+	+	CCONJ
ap-1849	179	19	z2z3	z2z3	NUM
ap-1849	179	20	−	−	PROPN
ap-1849	179	21	z1z	z1z	NOUN
ap-1849	179	22	2	2	NUM
ap-1849	179	23	2)λ1+λ2	2)λ1+λ2	NUM
ap-1849	179	24	+	+	ADJ
ap-1849	179	25	2	2	NUM
ap-1849	179	26	,	,	PUNCT
ap-1849	179	27	(	(	PUNCT
ap-1849	179	28	9	9	X
ap-1849	179	29	)	)	PUNCT
ap-1849	179	30	which	which	PRON
ap-1849	179	31	is	be	AUX
ap-1849	179	32	a	a	DET
ap-1849	179	33	non	non	ADJ
ap-1849	179	34	-	-	ADJ
ap-1849	179	35	constant	constant	ADJ
ap-1849	179	36	polynomial	polynomial	NOUN
ap-1849	179	37	for	for	ADP
ap-1849	179	38	λ1	λ1	ADJ
ap-1849	180	1	+	+	CCONJ
ap-1849	180	2	λ2	λ2	NOUN
ap-1849	180	3	+	+	CCONJ
ap-1849	180	4	1	1	NUM
ap-1849	180	5	∈	∈	PROPN
ap-1849	180	6	n0	n0	NUM
ap-1849	180	7	.	.	PUNCT
ap-1849	181	1	this	this	DET
ap-1849	181	2	function	function	NOUN
ap-1849	181	3	is	be	AUX
ap-1849	181	4	a	a	DET
ap-1849	181	5	polynomial	polynomial	NOUN
ap-1849	181	6	in	in	ADP
ap-1849	181	7	the	the	DET
ap-1849	181	8	variable	variable	ADJ
ap-1849	181	9	z2	z2	PROPN
ap-1849	181	10	of	of	ADP
ap-1849	181	11	degree	degree	NOUN
ap-1849	181	12	2λ1	2λ1	NUM
ap-1849	182	1	+	+	CCONJ
ap-1849	182	2	2λ2	2λ2	NUM
ap-1849	182	3	+	+	CCONJ
ap-1849	182	4	2	2	NUM
ap-1849	182	5	.	.	X
ap-1849	183	1	it	it	PRON
ap-1849	183	2	gives	give	VERB
ap-1849	183	3	sought	seek	VERB
ap-1849	183	4	solutions	solution	NOUN
ap-1849	183	5	for	for	ADP
ap-1849	183	6	2λ1	2λ1	NUM
ap-1849	183	7	+	+	NUM
ap-1849	183	8	λ2	λ2	NOUN
ap-1849	183	9	+	+	CCONJ
ap-1849	183	10	4	4	NUM
ap-1849	183	11	≤	≤	NUM
ap-1849	183	12	0	0	NUM
ap-1849	183	13	.	.	PUNCT
ap-1849	184	1	thus	thus	ADV
ap-1849	184	2	,	,	PUNCT
ap-1849	184	3	function	function	NOUN
ap-1849	184	4	(	(	PUNCT
ap-1849	184	5	9	9	NUM
ap-1849	184	6	)	)	PUNCT
ap-1849	184	7	provides	provide	VERB
ap-1849	184	8	a	a	DET
ap-1849	184	9	permissible	permissible	ADJ
ap-1849	184	10	solution	solution	NOUN
ap-1849	184	11	for	for	ADP
ap-1849	184	12	the	the	DET
ap-1849	184	13	λ2	λ2	PROPN
ap-1849	184	14	∈	∈	PROPN
ap-1849	184	15	n0	n0	NOUN
ap-1849	184	16	only	only	ADV
ap-1849	184	17	if	if	SCONJ
ap-1849	184	18	λ1	λ1	PROPN
ap-1849	184	19	∈	∈	PROPN
ap-1849	184	20	z	z	PROPN
ap-1849	184	21	,	,	PUNCT
ap-1849	184	22	−λ2	−λ2	NOUN
ap-1849	184	23	−	−	PROPN
ap-1849	184	24	1	1	NUM
ap-1849	184	25	≤	≤	NOUN
ap-1849	184	26	λ1	λ1	ADJ
ap-1849	184	27	≤	≤	NUM
ap-1849	184	28	−	−	PROPN
ap-1849	184	29	1	1	NUM
ap-1849	184	30	2λ2	2λ2	NUM
ap-1849	184	31	−	−	NUM
ap-1849	184	32	2	2	NUM
ap-1849	184	33	,	,	PUNCT
ap-1849	184	34	from	from	ADP
ap-1849	184	35	which	which	PRON
ap-1849	184	36	follows	follow	VERB
ap-1849	184	37	λ2	λ2	PROPN
ap-1849	184	38	≥	≥	NUM
ap-1849	184	39	2	2	NUM
ap-1849	184	40	.	.	PUNCT
ap-1849	184	41	case	case	NOUN
ap-1849	184	42	4	4	NUM
ap-1849	184	43	(	(	PUNCT
ap-1849	184	44	ρ1	ρ1	NOUN
ap-1849	184	45	=	=	SYM
ap-1849	184	46	2λ1	2λ1	NUM
ap-1849	185	1	+	+	CCONJ
ap-1849	185	2	2	2	NUM
ap-1849	185	3	,	,	PUNCT
ap-1849	185	4	ρ2	ρ2	NOUN
ap-1849	185	5	=	=	SYM
ap-1849	185	6	λ2	λ2	PROPN
ap-1849	185	7	+	+	NOUN
ap-1849	185	8	1	1	NUM
ap-1849	185	9	.	.	NUM
ap-1849	185	10	)	)	PUNCT
ap-1849	185	11	.	.	PUNCT
ap-1849	186	1	in	in	ADP
ap-1849	186	2	this	this	DET
ap-1849	186	3	case	case	NOUN
ap-1849	186	4	,	,	PUNCT
ap-1849	186	5	the	the	DET
ap-1849	186	6	function	function	NOUN
ap-1849	186	7	that	that	PRON
ap-1849	186	8	can	can	AUX
ap-1849	186	9	match	match	VERB
ap-1849	186	10	the	the	DET
ap-1849	186	11	extremal	extremal	ADJ
ap-1849	186	12	vector	vector	NOUN
ap-1849	186	13	is	be	AUX
ap-1849	186	14	f(z1	f(z1	ADJ
ap-1849	186	15	,	,	PUNCT
ap-1849	186	16	z2	z2	PROPN
ap-1849	186	17	,	,	PUNCT
ap-1849	186	18	z3	z3	PROPN
ap-1849	186	19	,	,	PUNCT
ap-1849	186	20	z4	z4	X
ap-1849	186	21	)	)	PUNCT
ap-1849	186	22	=	=	SYM
ap-1849	187	1	z−λ2−1	z−λ2−1	NUM
ap-1849	187	2	1	1	NUM
ap-1849	187	3	(	(	PUNCT
ap-1849	187	4	4z1z4	4z1z4	NUM
ap-1849	187	5	+	+	CCONJ
ap-1849	187	6	z2	z2	PROPN
ap-1849	187	7	3)λ1+λ2	3)λ1+λ2	NUM
ap-1849	187	8	+	+	NOUN
ap-1849	187	9	2g(t	2g(t	NUM
ap-1849	187	10	)	)	PUNCT
ap-1849	187	11	,	,	PUNCT
ap-1849	187	12	where	where	SCONJ
ap-1849	187	13	g(t	g(t	PROPN
ap-1849	187	14	)	)	PUNCT
ap-1849	187	15	is	be	AUX
ap-1849	187	16	the	the	DET
ap-1849	187	17	solution	solution	NOUN
ap-1849	187	18	of	of	ADP
ap-1849	187	19	system	system	NOUN
ap-1849	187	20	(	(	PUNCT
ap-1849	187	21	8)	8)	NUM
ap-1849	187	22	.	.	PUNCT
ap-1849	188	1	so	so	ADV
ap-1849	188	2	f(z1	f(z1	ADJ
ap-1849	188	3	,	,	PUNCT
ap-1849	188	4	z2	z2	PROPN
ap-1849	188	5	,	,	PUNCT
ap-1849	188	6	z3	z3	PROPN
ap-1849	188	7	,	,	PUNCT
ap-1849	188	8	z4	z4	PROPN
ap-1849	188	9	)	)	PUNCT
ap-1849	188	10	=	=	PUNCT
ap-1849	189	1	zλ1	zλ1	NOUN
ap-1849	189	2	+	+	PROPN
ap-1849	189	3	1	1	NUM
ap-1849	189	4	1	1	NUM
ap-1849	189	5	(	(	PUNCT
ap-1849	189	6	z4	z4	PROPN
ap-1849	189	7	+	+	CCONJ
ap-1849	189	8	z2z3	z2z3	NUM
ap-1849	189	9	−	−	PROPN
ap-1849	189	10	z1z	z1z	NOUN
ap-1849	189	11	2	2	NUM
ap-1849	189	12	2)λ1+λ2	2)λ1+λ2	NUM
ap-1849	189	13	+	+	NOUN
ap-1849	189	14	2	2	NUM
ap-1849	189	15	.	.	PUNCT
ap-1849	189	16	to	to	PART
ap-1849	189	17	give	give	VERB
ap-1849	189	18	a	a	DET
ap-1849	189	19	polynomial	polynomial	ADJ
ap-1849	189	20	solution	solution	NOUN
ap-1849	189	21	,	,	PUNCT
ap-1849	189	22	which	which	PRON
ap-1849	189	23	we	we	PRON
ap-1849	189	24	have	have	AUX
ap-1849	189	25	found	find	VERB
ap-1849	189	26	,	,	PUNCT
ap-1849	189	27	to	to	ADP
ap-1849	189	28	this	this	DET
ap-1849	189	29	function	function	NOUN
ap-1849	189	30	there	there	PRON
ap-1849	189	31	must	must	AUX
ap-1849	189	32	be	be	AUX
ap-1849	189	33	λ1	λ1	PROPN
ap-1849	189	34	∈	∈	PROPN
ap-1849	189	35	n0	n0	NOUN
ap-1849	189	36	and	and	CCONJ
ap-1849	189	37	λ1	λ1	ADJ
ap-1849	190	1	+	+	PROPN
ap-1849	190	2	λ2	λ2	NOUN
ap-1849	190	3	+	+	CCONJ
ap-1849	190	4	1	1	NUM
ap-1849	190	5	∈	∈	PROPN
ap-1849	190	6	n0	n0	NUM
ap-1849	190	7	.	.	PUNCT
ap-1849	191	1	but	but	CCONJ
ap-1849	191	2	in	in	ADP
ap-1849	191	3	this	this	DET
ap-1849	191	4	case	case	NOUN
ap-1849	191	5	,	,	PUNCT
ap-1849	191	6	the	the	DET
ap-1849	191	7	degree	degree	NOUN
ap-1849	191	8	of	of	ADP
ap-1849	191	9	polynomial	polynomial	ADJ
ap-1849	191	10	f	f	PROPN
ap-1849	191	11	in	in	ADP
ap-1849	191	12	the	the	DET
ap-1849	191	13	variable	variable	ADJ
ap-1849	191	14	z2	z2	PROPN
ap-1849	191	15	is	be	AUX
ap-1849	191	16	greater	great	ADJ
ap-1849	191	17	than	than	ADP
ap-1849	191	18	λ2	λ2	NOUN
ap-1849	191	19	and	and	CCONJ
ap-1849	191	20	,	,	PUNCT
ap-1849	191	21	therefore	therefore	ADV
ap-1849	191	22	,	,	PUNCT
ap-1849	191	23	is	be	AUX
ap-1849	191	24	not	not	PART
ap-1849	191	25	a	a	DET
ap-1849	191	26	permissible	permissible	ADJ
ap-1849	191	27	solution	solution	NOUN
ap-1849	191	28	.	.	PUNCT
ap-1849	192	1	402	402	NUM
ap-1849	192	2	vol	vol	NOUN
ap-1849	192	3	.	.	PUNCT
ap-1849	193	1	53	53	NUM
ap-1849	193	2	no	no	NOUN
ap-1849	193	3	.	.	PUNCT
ap-1849	194	1	5/2013	5/2013	NUM
ap-1849	194	2	extremal	extremal	ADJ
ap-1849	194	3	vectors	vector	NOUN
ap-1849	194	4	for	for	ADP
ap-1849	194	5	verma	verma	NOUN
ap-1849	194	6	type	type	NOUN
ap-1849	194	7	representation	representation	NOUN
ap-1849	194	8	of	of	ADP
ap-1849	194	9	b2	b2	NOUN
ap-1849	194	10	case	case	NOUN
ap-1849	194	11	5	5	NUM
ap-1849	194	12	(	(	PUNCT
ap-1849	194	13	ρ1	ρ1	NOUN
ap-1849	194	14	=	=	SYM
ap-1849	194	15	2λ1	2λ1	NUM
ap-1849	194	16	+	+	NUM
ap-1849	194	17	λ2	λ2	NOUN
ap-1849	194	18	+	+	CCONJ
ap-1849	194	19	3	3	NUM
ap-1849	194	20	,	,	PUNCT
ap-1849	194	21	ρ2	ρ2	NOUN
ap-1849	194	22	=	=	SYM
ap-1849	194	23	0	0	NUM
ap-1849	194	24	.	.	NUM
ap-1849	194	25	)	)	PUNCT
ap-1849	194	26	.	.	PUNCT
ap-1849	195	1	the	the	DET
ap-1849	195	2	function	function	NOUN
ap-1849	195	3	corresponding	correspond	VERB
ap-1849	195	4	to	to	ADP
ap-1849	195	5	the	the	DET
ap-1849	195	6	possible	possible	ADJ
ap-1849	195	7	extremal	extremal	ADJ
ap-1849	195	8	vectors	vector	NOUN
ap-1849	195	9	is	be	AUX
ap-1849	195	10	f(z1	f(z1	ADJ
ap-1849	195	11	,	,	PUNCT
ap-1849	195	12	z2	z2	PROPN
ap-1849	195	13	,	,	PUNCT
ap-1849	195	14	z3	z3	PROPN
ap-1849	195	15	,	,	PUNCT
ap-1849	195	16	z4	z4	PROPN
ap-1849	195	17	)	)	PUNCT
ap-1849	195	18	=	=	SYM
ap-1849	196	1	(	(	PUNCT
ap-1849	196	2	4z1z4	4z1z4	NUM
ap-1849	196	3	+	+	CCONJ
ap-1849	196	4	z2	z2	PROPN
ap-1849	196	5	3)λ1	3)λ1	NUM
ap-1849	196	6	+	+	CCONJ
ap-1849	196	7	1	1	NUM
ap-1849	196	8	2λ2	2λ2	NUM
ap-1849	196	9	+	+	SYM
ap-1849	196	10	3	3	NUM
ap-1849	196	11	2	2	NUM
ap-1849	196	12	g(t	g(t	PROPN
ap-1849	196	13	)	)	PUNCT
ap-1849	196	14	,	,	PUNCT
ap-1849	196	15	where	where	SCONJ
ap-1849	196	16	function	function	NOUN
ap-1849	196	17	g(t	g(t	PROPN
ap-1849	196	18	)	)	PUNCT
ap-1849	196	19	meets	meet	VERB
ap-1849	196	20	the	the	DET
ap-1849	196	21	equation	equation	NOUN
ap-1849	196	22	4t(1−	4t(1−	PROPN
ap-1849	196	23	t)g′′	t)g′′	NOUN
ap-1849	196	24	+	+	CCONJ
ap-1849	196	25	2	2	NUM
ap-1849	196	26	(	(	PUNCT
ap-1849	196	27	1	1	NUM
ap-1849	196	28	+	+	CCONJ
ap-1849	196	29	(	(	PUNCT
ap-1849	196	30	2λ1	2λ1	NUM
ap-1849	196	31	+	+	CCONJ
ap-1849	196	32	2λ2	2λ2	NUM
ap-1849	196	33	+	+	CCONJ
ap-1849	196	34	1)t	1)t	NUM
ap-1849	196	35	)	)	PUNCT
ap-1849	196	36	g′	g′	NOUN
ap-1849	197	1	−	−	PROPN
ap-1849	197	2	λ2(2λ1	λ2(2λ1	NUM
ap-1849	198	1	+	+	NUM
ap-1849	198	2	λ2	λ2	NOUN
ap-1849	198	3	+	+	CCONJ
ap-1849	198	4	3)g	3)g	NUM
ap-1849	198	5	=	=	SYM
ap-1849	198	6	0	0	NUM
ap-1849	198	7	.	.	PUNCT
ap-1849	199	1	(	(	PUNCT
ap-1849	199	2	10	10	NUM
ap-1849	199	3	)	)	PUNCT
ap-1849	199	4	this	this	DET
ap-1849	199	5	equation	equation	NOUN
ap-1849	199	6	has	have	VERB
ap-1849	199	7	two	two	NUM
ap-1849	199	8	linearly	linearly	ADV
ap-1849	199	9	independent	independent	ADJ
ap-1849	199	10	solutions	solution	NOUN
ap-1849	199	11	g1(t	g1(t	NOUN
ap-1849	199	12	)	)	PUNCT
ap-1849	199	13	=	=	SYM
ap-1849	199	14	f	f	X
ap-1849	199	15	(	(	PUNCT
ap-1849	199	16	−	−	PROPN
ap-1849	199	17	1	1	NUM
ap-1849	199	18	2λ2,−λ1	2λ2,−λ1	NUM
ap-1849	199	19	−	−	NUM
ap-1849	199	20	1	1	NUM
ap-1849	199	21	2λ2	2λ2	NUM
ap-1849	199	22	−	−	NUM
ap-1849	199	23	3	3	NUM
ap-1849	199	24	2	2	NUM
ap-1849	199	25	;	;	PUNCT
ap-1849	199	26	1	1	NUM
ap-1849	199	27	2	2	NUM
ap-1849	199	28	;	;	PUNCT
ap-1849	199	29	t	t	PROPN
ap-1849	199	30	)	)	PUNCT
ap-1849	199	31	,	,	PUNCT
ap-1849	199	32	g2(t	g2(t	PROPN
ap-1849	199	33	)	)	PUNCT
ap-1849	199	34	=	=	PUNCT
ap-1849	200	1	√	√	PROPN
ap-1849	200	2	t	t	NOUN
ap-1849	200	3	f	f	PROPN
ap-1849	200	4	(	(	PUNCT
ap-1849	200	5	1	1	NUM
ap-1849	200	6	2	2	NUM
ap-1849	200	7	−	−	PROPN
ap-1849	200	8	1	1	NUM
ap-1849	200	9	2λ2,−λ1	2λ2,−λ1	NUM
ap-1849	200	10	−	−	NUM
ap-1849	200	11	1	1	NUM
ap-1849	200	12	2λ2	2λ2	NUM
ap-1849	200	13	−	−	NUM
ap-1849	200	14	1	1	NUM
ap-1849	200	15	;	;	PUNCT
ap-1849	200	16	3	3	NUM
ap-1849	200	17	2	2	NUM
ap-1849	200	18	;	;	PUNCT
ap-1849	200	19	t	t	PROPN
ap-1849	200	20	)	)	PUNCT
ap-1849	200	21	,	,	PUNCT
ap-1849	200	22	where	where	SCONJ
ap-1849	200	23	f	f	PROPN
ap-1849	200	24	(	(	PUNCT
ap-1849	200	25	α	α	PROPN
ap-1849	200	26	,	,	PUNCT
ap-1849	200	27	β	β	X
ap-1849	200	28	;	;	PUNCT
ap-1849	200	29	γ	γ	X
ap-1849	200	30	;	;	PUNCT
ap-1849	200	31	t	t	PROPN
ap-1849	200	32	)	)	PUNCT
ap-1849	200	33	is	be	AUX
ap-1849	200	34	the	the	DET
ap-1849	200	35	hypergeometric	hypergeometric	ADJ
ap-1849	200	36	function	function	NOUN
ap-1849	200	37	f	f	PROPN
ap-1849	200	38	(	(	PUNCT
ap-1849	200	39	α	α	X
ap-1849	200	40	,	,	PUNCT
ap-1849	200	41	β	β	X
ap-1849	200	42	;	;	PUNCT
ap-1849	200	43	γ	γ	X
ap-1849	200	44	;	;	PUNCT
ap-1849	200	45	t	t	PROPN
ap-1849	200	46	)	)	PUNCT
ap-1849	200	47	=	=	PUNCT
ap-1849	201	1	∞∑	∞∑	NUM
ap-1849	201	2	n=0	n=0	NUM
ap-1849	201	3	(	(	PUNCT
ap-1849	201	4	α)n(β)n	α)n(β)n	PROPN
ap-1849	201	5	n	n	CCONJ
ap-1849	201	6	!	!	PUNCT
ap-1849	202	1	(	(	PUNCT
ap-1849	202	2	γ)n	γ)n	X
ap-1849	202	3	tn	tn	PROPN
ap-1849	202	4	,	,	PUNCT
ap-1849	202	5	where	where	SCONJ
ap-1849	202	6	(	(	PUNCT
ap-1849	202	7	α)n	α)n	ADJ
ap-1849	202	8	=	=	PUNCT
ap-1849	202	9	γ(α+	γ(α+	X
ap-1849	202	10	n	n	CCONJ
ap-1849	202	11	)	)	PUNCT
ap-1849	202	12	γ(α	γ(α	NOUN
ap-1849	202	13	)	)	PUNCT
ap-1849	203	1	=	=	PUNCT
ap-1849	204	1	α(α+	α(α+	NUM
ap-1849	204	2	1	1	NUM
ap-1849	204	3	)	)	PUNCT
ap-1849	204	4	.	.	PUNCT
ap-1849	204	5	.	.	PUNCT
ap-1849	204	6	.	.	PUNCT
ap-1849	205	1	(	(	PUNCT
ap-1849	205	2	α+	α+	X
ap-1849	205	3	n−	n−	NOUN
ap-1849	205	4	1	1	NUM
ap-1849	205	5	)	)	PUNCT
ap-1849	205	6	.	.	PUNCT
ap-1849	206	1	these	these	DET
ap-1849	206	2	solutions	solution	NOUN
ap-1849	206	3	correspond	correspond	VERB
ap-1849	206	4	to	to	ADP
ap-1849	206	5	the	the	DET
ap-1849	206	6	functions	function	NOUN
ap-1849	206	7	f1	f1	NOUN
ap-1849	206	8	=	=	NOUN
ap-1849	206	9	∞∑	∞∑	PROPN
ap-1849	206	10	n=0	n=0	NUM
ap-1849	206	11	(	(	PUNCT
ap-1849	206	12	−	−	PROPN
ap-1849	206	13	1	1	NUM
ap-1849	206	14	2λ2)n(−λ1	2λ2)n(−λ1	NUM
ap-1849	206	15	−	−	PROPN
ap-1849	206	16	1	1	NUM
ap-1849	206	17	2λ2	2λ2	NUM
ap-1849	206	18	−	−	NUM
ap-1849	206	19	3	3	NUM
ap-1849	206	20	2	2	NUM
ap-1849	206	21	)	)	PUNCT
ap-1849	206	22	n	n	PRON
ap-1849	206	23	n	n	CCONJ
ap-1849	206	24	!	!	PUNCT
ap-1849	207	1	(	(	PUNCT
ap-1849	207	2	1	1	NUM
ap-1849	207	3	2	2	NUM
ap-1849	207	4	)	)	PUNCT
ap-1849	207	5	n	n	PRON
ap-1849	207	6	×	×	NOUN
ap-1849	207	7	(	(	PUNCT
ap-1849	207	8	2z1z2	2z1z2	NUM
ap-1849	207	9	−	−	NOUN
ap-1849	207	10	z3)2n(4z1z4	z3)2n(4z1z4	NOUN
ap-1849	207	11	+	+	CCONJ
ap-1849	207	12	z2	z2	PROPN
ap-1849	207	13	3)λ1	3)λ1	NUM
ap-1849	207	14	+	+	PROPN
ap-1849	207	15	1	1	NUM
ap-1849	207	16	2λ2−n+	2λ2−n+	NUM
ap-1849	207	17	3	3	NUM
ap-1849	207	18	2	2	NUM
ap-1849	207	19	,	,	PUNCT
ap-1849	207	20	f2	f2	PROPN
ap-1849	207	21	=	=	NOUN
ap-1849	207	22	∞∑	∞∑	NUM
ap-1849	207	23	n=0	n=0	NUM
ap-1849	207	24	(	(	PUNCT
ap-1849	207	25	1	1	NUM
ap-1849	207	26	2	2	NUM
ap-1849	207	27	−	−	NOUN
ap-1849	207	28	1	1	NUM
ap-1849	207	29	2λ2)n(−λ1	2λ2)n(−λ1	NUM
ap-1849	207	30	−	−	PROPN
ap-1849	207	31	1	1	NUM
ap-1849	207	32	2λ2	2λ2	NUM
ap-1849	207	33	−	−	PROPN
ap-1849	207	34	1)n	1)n	NUM
ap-1849	207	35	n	n	X
ap-1849	207	36	!	!	PUNCT
ap-1849	208	1	(	(	PUNCT
ap-1849	208	2	3	3	NUM
ap-1849	208	3	2	2	NUM
ap-1849	208	4	)	)	PUNCT
ap-1849	208	5	n	n	PRON
ap-1849	208	6	×	×	NOUN
ap-1849	208	7	(	(	PUNCT
ap-1849	208	8	2z1z2	2z1z2	NUM
ap-1849	208	9	−	−	NOUN
ap-1849	208	10	z3)2n(4z1z4	z3)2n(4z1z4	NOUN
ap-1849	208	11	+	+	CCONJ
ap-1849	208	12	z2	z2	PROPN
ap-1849	208	13	3)λ1	3)λ1	NUM
ap-1849	208	14	+	+	CCONJ
ap-1849	208	15	1	1	NUM
ap-1849	208	16	2λ2−n+1	2λ2−n+1	NUM
ap-1849	208	17	.	.	PUNCT
ap-1849	209	1	for	for	ADP
ap-1849	209	2	at	at	ADV
ap-1849	209	3	least	least	ADJ
ap-1849	209	4	one	one	NUM
ap-1849	209	5	of	of	ADP
ap-1849	209	6	these	these	DET
ap-1849	209	7	functions	function	NOUN
ap-1849	209	8	to	to	PART
ap-1849	209	9	be	be	AUX
ap-1849	209	10	a	a	DET
ap-1849	209	11	nonconstant	nonconstant	ADJ
ap-1849	209	12	polynomial	polynomial	NOUN
ap-1849	209	13	,	,	PUNCT
ap-1849	209	14	must	must	AUX
ap-1849	209	15	be	be	AUX
ap-1849	209	16	2λ1+λ2	2λ1+λ2	NUM
ap-1849	209	17	+	+	SYM
ap-1849	209	18	3	3	NUM
ap-1849	209	19	∈	∈	NOUN
ap-1849	209	20	n	n	CCONJ
ap-1849	209	21	,	,	PUNCT
ap-1849	209	22	i.e.	i.e.	X
ap-1849	209	23	2λ1+λ2	2λ1+λ2	NUM
ap-1849	209	24	+	+	SYM
ap-1849	209	25	2	2	NUM
ap-1849	209	26	∈	∈	PROPN
ap-1849	209	27	n0	n0	NUM
ap-1849	209	28	.	.	PUNCT
ap-1849	210	1	if	if	SCONJ
ap-1849	210	2	2λ1	2λ1	NUM
ap-1849	210	3	+	+	NUM
ap-1849	210	4	λ2	λ2	NOUN
ap-1849	210	5	+	+	CCONJ
ap-1849	210	6	3	3	NUM
ap-1849	210	7	is	be	AUX
ap-1849	210	8	even	even	ADV
ap-1849	210	9	,	,	PUNCT
ap-1849	210	10	we	we	PRON
ap-1849	210	11	get	get	VERB
ap-1849	210	12	the	the	DET
ap-1849	210	13	solution	solution	NOUN
ap-1849	210	14	f1	f1	NOUN
ap-1849	210	15	=	=	SYM
ap-1849	210	16	λ1	λ1	PROPN
ap-1849	210	17	+	+	NUM
ap-1849	210	18	1	1	NUM
ap-1849	210	19	2λ2	2λ2	NUM
ap-1849	210	20	+	+	SYM
ap-1849	210	21	3	3	NUM
ap-1849	210	22	2∑	2∑	NUM
ap-1849	210	23	n=0	n=0	NUM
ap-1849	210	24	(	(	PUNCT
ap-1849	210	25	−	−	PROPN
ap-1849	210	26	1	1	NUM
ap-1849	210	27	2λ2)n(−λ1	2λ2)n(−λ1	NUM
ap-1849	210	28	−	−	PROPN
ap-1849	210	29	1	1	NUM
ap-1849	210	30	2λ2	2λ2	NUM
ap-1849	210	31	−	−	NUM
ap-1849	210	32	3	3	NUM
ap-1849	210	33	2	2	NUM
ap-1849	210	34	)	)	PUNCT
ap-1849	210	35	n	n	PRON
ap-1849	210	36	n	n	CCONJ
ap-1849	210	37	!	!	PUNCT
ap-1849	211	1	(	(	PUNCT
ap-1849	211	2	1	1	NUM
ap-1849	211	3	2	2	NUM
ap-1849	211	4	)	)	PUNCT
ap-1849	211	5	n	n	PRON
ap-1849	211	6	×	×	NOUN
ap-1849	211	7	(	(	PUNCT
ap-1849	211	8	2z1z2	2z1z2	NUM
ap-1849	211	9	−	−	NOUN
ap-1849	211	10	z3)2n(4z1z4	z3)2n(4z1z4	NOUN
ap-1849	211	11	+	+	CCONJ
ap-1849	211	12	z2	z2	PROPN
ap-1849	211	13	3)λ1	3)λ1	NUM
ap-1849	211	14	+	+	PROPN
ap-1849	211	15	1	1	NUM
ap-1849	211	16	2λ2−n+	2λ2−n+	NUM
ap-1849	211	17	3	3	NUM
ap-1849	211	18	2	2	NUM
ap-1849	211	19	,	,	PUNCT
ap-1849	211	20	and	and	CCONJ
ap-1849	211	21	for	for	ADP
ap-1849	211	22	2λ1	2λ1	NUM
ap-1849	211	23	+	+	NUM
ap-1849	211	24	λ2	λ2	NOUN
ap-1849	211	25	+	+	CCONJ
ap-1849	211	26	3	3	NUM
ap-1849	211	27	odd	odd	ADJ
ap-1849	211	28	,	,	PUNCT
ap-1849	211	29	we	we	PRON
ap-1849	211	30	have	have	VERB
ap-1849	211	31	the	the	DET
ap-1849	211	32	solution	solution	NOUN
ap-1849	211	33	f2	f2	PROPN
ap-1849	211	34	=	=	SYM
ap-1849	211	35	λ1	λ1	PROPN
ap-1849	211	36	+	+	NUM
ap-1849	211	37	1	1	NUM
ap-1849	211	38	2λ2	2λ2	NUM
ap-1849	211	39	+	+	NOUN
ap-1849	211	40	1∑	1∑	NOUN
ap-1849	211	41	n=0	n=0	SYM
ap-1849	211	42	(	(	PUNCT
ap-1849	211	43	1	1	NUM
ap-1849	211	44	2	2	NUM
ap-1849	211	45	−	−	NOUN
ap-1849	211	46	1	1	NUM
ap-1849	211	47	2λ2)n(−λ1	2λ2)n(−λ1	NUM
ap-1849	211	48	−	−	PROPN
ap-1849	211	49	1	1	NUM
ap-1849	211	50	2λ2	2λ2	NUM
ap-1849	211	51	−	−	PROPN
ap-1849	211	52	1)n	1)n	NUM
ap-1849	211	53	n	n	X
ap-1849	211	54	!	!	PUNCT
ap-1849	212	1	(	(	PUNCT
ap-1849	212	2	3	3	NUM
ap-1849	212	3	2	2	NUM
ap-1849	212	4	)	)	PUNCT
ap-1849	212	5	n	n	PRON
ap-1849	212	6	×	×	NOUN
ap-1849	212	7	(	(	PUNCT
ap-1849	212	8	2z1z2	2z1z2	NUM
ap-1849	212	9	−	−	PROPN
ap-1849	212	10	z3)2n+1(4z1z4	z3)2n+1(4z1z4	NUM
ap-1849	212	11	+	+	CCONJ
ap-1849	212	12	z2	z2	PROPN
ap-1849	212	13	3)λ1	3)λ1	NUM
ap-1849	212	14	+	+	CCONJ
ap-1849	212	15	1	1	NUM
ap-1849	212	16	2λ2−n+1	2λ2−n+1	NUM
ap-1849	212	17	.	.	PUNCT
ap-1849	213	1	if	if	SCONJ
ap-1849	213	2	2λ1	2λ1	NUM
ap-1849	213	3	+	+	NUM
ap-1849	213	4	λ2	λ2	NOUN
ap-1849	213	5	+	+	CCONJ
ap-1849	213	6	3	3	NUM
ap-1849	213	7	is	be	AUX
ap-1849	213	8	even	even	ADV
ap-1849	213	9	and	and	CCONJ
ap-1849	213	10	λ2	λ2	NOUN
ap-1849	213	11	is	be	AUX
ap-1849	213	12	even	even	ADV
ap-1849	213	13	,	,	PUNCT
ap-1849	213	14	then	then	ADV
ap-1849	213	15	λ1	λ1	PROPN
ap-1849	213	16	is	be	AUX
ap-1849	213	17	a	a	DET
ap-1849	213	18	half	half	NOUN
ap-1849	213	19	integer	integer	NOUN
ap-1849	213	20	,	,	PUNCT
ap-1849	213	21	i.e.	i.e.	X
ap-1849	213	22	λ1	λ1	ADJ
ap-1849	213	23	=	=	SYM
ap-1849	213	24	`	`	PUNCT
ap-1849	213	25	1	1	NUM
ap-1849	213	26	−	−	NUM
ap-1849	213	27	1	1	NUM
ap-1849	213	28	2	2	NUM
ap-1849	213	29	,	,	PUNCT
ap-1849	213	30	where	where	SCONJ
ap-1849	213	31	`	`	PUNCT
ap-1849	213	32	1	1	NUM
ap-1849	213	33	∈	∈	PROPN
ap-1849	213	34	z	z	NOUN
ap-1849	213	35	,	,	PUNCT
ap-1849	213	36	counts	count	NOUN
ap-1849	213	37	in	in	ADP
ap-1849	213	38	f1	f1	NOUN
ap-1849	213	39	only	only	ADV
ap-1849	213	40	to	to	ADP
ap-1849	213	41	n	n	NOUN
ap-1849	213	42	≤	≤	NUM
ap-1849	213	43	1	1	NUM
ap-1849	213	44	2λ2	2λ2	NUM
ap-1849	213	45	,	,	PUNCT
ap-1849	213	46	i.e.	i.e.	X
ap-1849	213	47	f	f	X
ap-1849	213	48	=	=	SYM
ap-1849	213	49	min	min	PROPN
ap-1849	213	50	(	(	PUNCT
ap-1849	213	51	1	1	NUM
ap-1849	213	52	2λ2,λ1	2λ2,λ1	NUM
ap-1849	213	53	+	+	SYM
ap-1849	213	54	1	1	NUM
ap-1849	213	55	2λ2	2λ2	NUM
ap-1849	213	56	+	+	NUM
ap-1849	213	57	3	3	NUM
ap-1849	213	58	2	2	NUM
ap-1849	213	59	)	)	PUNCT
ap-1849	213	60	∑	∑	PUNCT
ap-1849	213	61	n=0	n=0	NUM
ap-1849	213	62	(	(	PUNCT
ap-1849	213	63	−	−	PROPN
ap-1849	213	64	1	1	NUM
ap-1849	213	65	2λ2)n(−λ1	2λ2)n(−λ1	NUM
ap-1849	213	66	−	−	PROPN
ap-1849	213	67	1	1	NUM
ap-1849	213	68	2λ2	2λ2	NUM
ap-1849	213	69	−	−	NUM
ap-1849	213	70	3	3	NUM
ap-1849	213	71	2	2	NUM
ap-1849	213	72	)	)	PUNCT
ap-1849	213	73	n	n	PRON
ap-1849	213	74	n	n	CCONJ
ap-1849	213	75	!	!	PUNCT
ap-1849	214	1	(	(	PUNCT
ap-1849	214	2	1	1	NUM
ap-1849	214	3	2	2	NUM
ap-1849	214	4	)	)	PUNCT
ap-1849	214	5	n	n	PRON
ap-1849	214	6	×	×	NOUN
ap-1849	214	7	(	(	PUNCT
ap-1849	214	8	2z1z2	2z1z2	NUM
ap-1849	214	9	−	−	NOUN
ap-1849	214	10	z3)2n(4z1z4	z3)2n(4z1z4	NOUN
ap-1849	214	11	+	+	CCONJ
ap-1849	214	12	z2	z2	PROPN
ap-1849	214	13	3)λ1	3)λ1	NUM
ap-1849	214	14	+	+	PROPN
ap-1849	214	15	1	1	NUM
ap-1849	214	16	2λ2−n+	2λ2−n+	NUM
ap-1849	214	17	3	3	NUM
ap-1849	214	18	2	2	NUM
ap-1849	214	19	,	,	PUNCT
ap-1849	214	20	and	and	CCONJ
ap-1849	214	21	,	,	PUNCT
ap-1849	214	22	therefore	therefore	ADV
ap-1849	214	23	,	,	PUNCT
ap-1849	214	24	f	f	PROPN
ap-1849	214	25	is	be	AUX
ap-1849	214	26	in	in	ADP
ap-1849	214	27	the	the	DET
ap-1849	214	28	variable	variable	ADJ
ap-1849	214	29	z2	z2	PROPN
ap-1849	214	30	of	of	ADP
ap-1849	214	31	a	a	DET
ap-1849	214	32	polynomial	polynomial	NOUN
ap-1849	214	33	of	of	ADP
ap-1849	214	34	degree	degree	NOUN
ap-1849	214	35	not	not	PART
ap-1849	214	36	exceeding	exceed	VERB
ap-1849	214	37	λ2	λ2	NOUN
ap-1849	214	38	.	.	PUNCT
ap-1849	215	1	if	if	SCONJ
ap-1849	215	2	2λ1	2λ1	NUM
ap-1849	215	3	+	+	NUM
ap-1849	215	4	λ2	λ2	NOUN
ap-1849	215	5	+	+	CCONJ
ap-1849	215	6	3	3	NUM
ap-1849	215	7	is	be	AUX
ap-1849	215	8	even	even	ADV
ap-1849	215	9	and	and	CCONJ
ap-1849	215	10	λ2	λ2	NOUN
ap-1849	215	11	is	be	AUX
ap-1849	215	12	odd	odd	ADJ
ap-1849	215	13	,	,	PUNCT
ap-1849	215	14	i.e.	i.e.	X
ap-1849	215	15	λ1	λ1	PROPN
ap-1849	215	16	is	be	AUX
ap-1849	215	17	an	an	DET
ap-1849	215	18	integer	integer	NOUN
ap-1849	215	19	,	,	PUNCT
ap-1849	215	20	the	the	DET
ap-1849	215	21	function	function	NOUN
ap-1849	215	22	f	f	PROPN
ap-1849	215	23	=	=	SYM
ap-1849	215	24	λ1	λ1	PROPN
ap-1849	215	25	+	+	NUM
ap-1849	215	26	1	1	NUM
ap-1849	215	27	2λ2	2λ2	NUM
ap-1849	215	28	+	+	SYM
ap-1849	215	29	3	3	NUM
ap-1849	215	30	2∑	2∑	NUM
ap-1849	215	31	n=0	n=0	NUM
ap-1849	215	32	(	(	PUNCT
ap-1849	215	33	−	−	PROPN
ap-1849	215	34	1	1	NUM
ap-1849	215	35	2λ2)n(−λ1	2λ2)n(−λ1	NUM
ap-1849	215	36	−	−	PROPN
ap-1849	215	37	1	1	NUM
ap-1849	215	38	2λ2	2λ2	NUM
ap-1849	215	39	−	−	NUM
ap-1849	215	40	3	3	NUM
ap-1849	215	41	2	2	NUM
ap-1849	215	42	)	)	PUNCT
ap-1849	215	43	n	n	PRON
ap-1849	215	44	n	n	CCONJ
ap-1849	215	45	!	!	PUNCT
ap-1849	216	1	(	(	PUNCT
ap-1849	216	2	1	1	NUM
ap-1849	216	3	2	2	NUM
ap-1849	216	4	)	)	PUNCT
ap-1849	216	5	n	n	PRON
ap-1849	216	6	×	×	NOUN
ap-1849	216	7	(	(	PUNCT
ap-1849	216	8	2z1z2	2z1z2	NUM
ap-1849	216	9	−	−	NOUN
ap-1849	216	10	z3)2n(4z1z4	z3)2n(4z1z4	NOUN
ap-1849	216	11	+	+	CCONJ
ap-1849	216	12	z2	z2	PROPN
ap-1849	216	13	3)λ1	3)λ1	NUM
ap-1849	216	14	+	+	PROPN
ap-1849	216	15	1	1	NUM
ap-1849	216	16	2λ2−n+	2λ2−n+	NUM
ap-1849	216	17	3	3	NUM
ap-1849	216	18	2	2	NUM
ap-1849	216	19	,	,	PUNCT
ap-1849	216	20	in	in	ADP
ap-1849	216	21	the	the	DET
ap-1849	216	22	variable	variable	ADJ
ap-1849	216	23	z2	z2	PROPN
ap-1849	216	24	is	be	AUX
ap-1849	216	25	a	a	DET
ap-1849	216	26	polynomial	polynomial	NOUN
ap-1849	216	27	of	of	ADP
ap-1849	216	28	degree	degree	NOUN
ap-1849	216	29	2λ1+λ2	2λ1+λ2	NUM
ap-1849	216	30	+	+	NOUN
ap-1849	216	31	3	3	NUM
ap-1849	216	32	.	.	PUNCT
ap-1849	217	1	thus	thus	ADV
ap-1849	217	2	admissible	admissible	ADJ
ap-1849	217	3	solutions	solution	NOUN
ap-1849	217	4	get	get	VERB
ap-1849	217	5	only	only	ADV
ap-1849	217	6	λ1	λ1	ADJ
ap-1849	217	7	≤	≤	ADJ
ap-1849	217	8	−2	−2	NOUN
ap-1849	217	9	.	.	PUNCT
ap-1849	218	1	if	if	SCONJ
ap-1849	218	2	2λ1	2λ1	NUM
ap-1849	218	3	+	+	NUM
ap-1849	218	4	λ2	λ2	NOUN
ap-1849	218	5	+	+	CCONJ
ap-1849	218	6	3	3	NUM
ap-1849	218	7	is	be	AUX
ap-1849	218	8	odd	odd	ADJ
ap-1849	218	9	,	,	PUNCT
ap-1849	218	10	then	then	ADV
ap-1849	218	11	solution	solution	NOUN
ap-1849	218	12	f2	f2	PROPN
ap-1849	218	13	comes	come	VERB
ap-1849	218	14	into	into	ADP
ap-1849	218	15	play	play	NOUN
ap-1849	218	16	.	.	PUNCT
ap-1849	219	1	if	if	SCONJ
ap-1849	219	2	1	1	NUM
ap-1849	219	3	2	2	NUM
ap-1849	219	4	(	(	PUNCT
ap-1849	219	5	λ2−1	λ2−1	NOUN
ap-1849	219	6	)	)	PUNCT
ap-1849	219	7	∈	∈	PROPN
ap-1849	219	8	n0	n0	PROPN
ap-1849	219	9	,	,	PUNCT
ap-1849	219	10	i.e.	i.e.	X
ap-1849	219	11	for	for	ADP
ap-1849	219	12	odd	odd	ADJ
ap-1849	219	13	λ2	λ2	NOUN
ap-1849	219	14	and	and	CCONJ
ap-1849	219	15	half	half	ADJ
ap-1849	219	16	integer	integer	NOUN
ap-1849	219	17	λ1	λ1	ADJ
ap-1849	219	18	sum	sum	NOUN
ap-1849	219	19	in	in	ADP
ap-1849	219	20	f2	f2	PROPN
ap-1849	219	21	only	only	ADV
ap-1849	219	22	n	n	CCONJ
ap-1849	219	23	≤	≤	NUM
ap-1849	219	24	1	1	NUM
ap-1849	219	25	2	2	NUM
ap-1849	219	26	(	(	PUNCT
ap-1849	219	27	λ2	λ2	NOUN
ap-1849	219	28	−	−	PROPN
ap-1849	219	29	1	1	NUM
ap-1849	219	30	)	)	PUNCT
ap-1849	219	31	,	,	PUNCT
ap-1849	219	32	then	then	ADV
ap-1849	219	33	the	the	DET
ap-1849	219	34	solutions	solution	NOUN
ap-1849	219	35	are	be	AUX
ap-1849	219	36	f	f	PROPN
ap-1849	219	37	=	=	SYM
ap-1849	219	38	min	min	PROPN
ap-1849	219	39	(	(	PUNCT
ap-1849	219	40	1	1	NUM
ap-1849	219	41	2λ2−	2λ2−	NUM
ap-1849	219	42	1	1	NUM
ap-1849	219	43	2	2	NUM
ap-1849	219	44	,	,	PUNCT
ap-1849	219	45	λ1	λ1	ADJ
ap-1849	219	46	+	+	CCONJ
ap-1849	219	47	1	1	NUM
ap-1849	219	48	2λ2	2λ2	NUM
ap-1849	219	49	+	+	NOUN
ap-1849	219	50	1)∑	1)∑	NUM
ap-1849	219	51	n=0	n=0	NUM
ap-1849	219	52	(	(	PUNCT
ap-1849	219	53	1	1	NUM
ap-1849	219	54	2	2	NUM
ap-1849	219	55	−	−	NOUN
ap-1849	219	56	1	1	NUM
ap-1849	219	57	2λ2)n(−λ1	2λ2)n(−λ1	NUM
ap-1849	219	58	−	−	PROPN
ap-1849	219	59	1	1	NUM
ap-1849	219	60	2λ2	2λ2	NUM
ap-1849	219	61	−	−	PROPN
ap-1849	219	62	1)n	1)n	NUM
ap-1849	219	63	n	n	X
ap-1849	219	64	!	!	PUNCT
ap-1849	220	1	(	(	PUNCT
ap-1849	220	2	3	3	NUM
ap-1849	220	3	2	2	NUM
ap-1849	220	4	)	)	PUNCT
ap-1849	220	5	n	n	PRON
ap-1849	220	6	×	×	NOUN
ap-1849	220	7	(	(	PUNCT
ap-1849	220	8	2z1z2	2z1z2	NUM
ap-1849	220	9	−	−	PROPN
ap-1849	220	10	z3)2n+1(4z1z4	z3)2n+1(4z1z4	NUM
ap-1849	220	11	+	+	CCONJ
ap-1849	220	12	z2	z2	PROPN
ap-1849	220	13	3)λ1	3)λ1	NUM
ap-1849	220	14	+	+	CCONJ
ap-1849	220	15	1	1	NUM
ap-1849	220	16	2λ2−n+1	2λ2−n+1	NUM
ap-1849	220	17	in	in	ADP
ap-1849	220	18	the	the	DET
ap-1849	220	19	z2	z2	PROPN
ap-1849	220	20	polynomial	polynomial	NOUN
ap-1849	220	21	of	of	ADP
ap-1849	220	22	degree	degree	NOUN
ap-1849	220	23	not	not	PART
ap-1849	220	24	exceeding	exceed	VERB
ap-1849	220	25	λ2	λ2	NOUN
ap-1849	220	26	.	.	PUNCT
ap-1849	221	1	but	but	CCONJ
ap-1849	221	2	for	for	ADP
ap-1849	221	3	2λ1	2λ1	NUM
ap-1849	221	4	+	+	NUM
ap-1849	221	5	λ2	λ2	NOUN
ap-1849	221	6	+	+	CCONJ
ap-1849	221	7	3	3	NUM
ap-1849	221	8	odd	odd	ADJ
ap-1849	221	9	and	and	CCONJ
ap-1849	221	10	λ2	λ2	PRON
ap-1849	221	11	even	even	ADV
ap-1849	221	12	,	,	PUNCT
ap-1849	221	13	i.e.	i.e.	X
ap-1849	221	14	λ1	λ1	PROPN
ap-1849	221	15	∈	∈	PROPN
ap-1849	221	16	z	z	PROPN
ap-1849	221	17	,	,	PUNCT
ap-1849	221	18	the	the	DET
ap-1849	221	19	solution	solution	NOUN
ap-1849	221	20	is	be	AUX
ap-1849	221	21	f	f	PROPN
ap-1849	221	22	=	=	SYM
ap-1849	221	23	λ1	λ1	PROPN
ap-1849	221	24	+	+	NUM
ap-1849	221	25	1	1	NUM
ap-1849	221	26	2λ2	2λ2	NUM
ap-1849	222	1	+	+	NOUN
ap-1849	222	2	1∑	1∑	NOUN
ap-1849	222	3	n=0	n=0	SYM
ap-1849	222	4	(	(	PUNCT
ap-1849	222	5	1	1	NUM
ap-1849	222	6	2	2	NUM
ap-1849	222	7	−	−	NOUN
ap-1849	222	8	1	1	NUM
ap-1849	222	9	2λ2)n(−λ1	2λ2)n(−λ1	NUM
ap-1849	222	10	−	−	PROPN
ap-1849	222	11	1	1	NUM
ap-1849	222	12	2λ2	2λ2	NUM
ap-1849	222	13	−	−	PROPN
ap-1849	222	14	1)n	1)n	NUM
ap-1849	222	15	n	n	X
ap-1849	222	16	!	!	PUNCT
ap-1849	223	1	(	(	PUNCT
ap-1849	223	2	3	3	NUM
ap-1849	223	3	2	2	NUM
ap-1849	223	4	)	)	PUNCT
ap-1849	223	5	n	n	PRON
ap-1849	223	6	×	×	NOUN
ap-1849	223	7	(	(	PUNCT
ap-1849	223	8	2z1z2	2z1z2	NUM
ap-1849	223	9	−	−	PROPN
ap-1849	223	10	z3)2n+1(4z1z4	z3)2n+1(4z1z4	NUM
ap-1849	223	11	+	+	CCONJ
ap-1849	223	12	z2	z2	PROPN
ap-1849	223	13	3)λ1	3)λ1	NUM
ap-1849	223	14	+	+	CCONJ
ap-1849	223	15	1	1	NUM
ap-1849	223	16	2λ2−n+1	2λ2−n+1	NUM
ap-1849	223	17	.	.	PUNCT
ap-1849	224	1	in	in	ADP
ap-1849	224	2	the	the	DET
ap-1849	224	3	variable	variable	ADJ
ap-1849	224	4	z2	z2	NOUN
ap-1849	224	5	it	it	PRON
ap-1849	224	6	is	be	AUX
ap-1849	224	7	a	a	DET
ap-1849	224	8	polynomial	polynomial	NOUN
ap-1849	224	9	of	of	ADP
ap-1849	224	10	degree	degree	NOUN
ap-1849	224	11	2λ1	2λ1	NUM
ap-1849	225	1	+	+	CCONJ
ap-1849	225	2	λ2	λ2	NOUN
ap-1849	225	3	+	+	CCONJ
ap-1849	225	4	3	3	X
ap-1849	225	5	.	.	PUNCT
ap-1849	226	1	therefore	therefore	ADV
ap-1849	226	2	we	we	PRON
ap-1849	226	3	get	get	VERB
ap-1849	226	4	a	a	DET
ap-1849	226	5	permissible	permissible	ADJ
ap-1849	226	6	solution	solution	NOUN
ap-1849	226	7	for	for	ADP
ap-1849	226	8	2λ1	2λ1	NUM
ap-1849	226	9	+	+	NUM
ap-1849	226	10	λ2	λ2	NOUN
ap-1849	226	11	+	+	CCONJ
ap-1849	226	12	3	3	NUM
ap-1849	226	13	≤	≤	NUM
ap-1849	226	14	λ2	λ2	NOUN
ap-1849	226	15	,	,	PUNCT
ap-1849	226	16	i.e.	i.e.	X
ap-1849	226	17	λ1	λ1	ADJ
ap-1849	226	18	≤	≤	ADJ
ap-1849	226	19	−2	−2	NOUN
ap-1849	226	20	.	.	PUNCT
ap-1849	227	1	case	case	NOUN
ap-1849	227	2	6	6	NUM
ap-1849	227	3	(	(	PUNCT
ap-1849	227	4	ρ1	ρ1	NOUN
ap-1849	227	5	=	=	PUNCT
ap-1849	227	6	−λ2	−λ2	NOUN
ap-1849	227	7	−	−	NOUN
ap-1849	227	8	1	1	NUM
ap-1849	227	9	,	,	PUNCT
ap-1849	227	10	ρ2	ρ2	NOUN
ap-1849	227	11	=	=	SYM
ap-1849	227	12	λ1	λ1	PROPN
ap-1849	227	13	+	+	NUM
ap-1849	227	14	λ2	λ2	NOUN
ap-1849	227	15	+	+	CCONJ
ap-1849	227	16	2	2	NUM
ap-1849	227	17	.	.	NUM
ap-1849	227	18	)	)	PUNCT
ap-1849	227	19	.	.	PUNCT
ap-1849	228	1	in	in	ADP
ap-1849	228	2	this	this	DET
ap-1849	228	3	case	case	NOUN
ap-1849	228	4	,	,	PUNCT
ap-1849	228	5	f(z1	f(z1	ADJ
ap-1849	228	6	,	,	PUNCT
ap-1849	228	7	z2	z2	PROPN
ap-1849	228	8	,	,	PUNCT
ap-1849	228	9	z3	z3	PROPN
ap-1849	228	10	,	,	PUNCT
ap-1849	228	11	z4	z4	PROPN
ap-1849	228	12	)	)	PUNCT
ap-1849	228	13	=	=	PUNCT
ap-1849	229	1	z−λ1−λ2−2	z−λ1−λ2−2	NUM
ap-1849	229	2	1	1	NUM
ap-1849	229	3	(	(	PUNCT
ap-1849	229	4	4z1z4	4z1z4	NUM
ap-1849	229	5	+	+	CCONJ
ap-1849	229	6	z2	z2	PROPN
ap-1849	229	7	3)λ1	3)λ1	NUM
ap-1849	229	8	+	+	CCONJ
ap-1849	229	9	1	1	NUM
ap-1849	229	10	2λ2	2λ2	NUM
ap-1849	229	11	+	+	SYM
ap-1849	229	12	3	3	NUM
ap-1849	229	13	2	2	NUM
ap-1849	229	14	g(t	g(t	PROPN
ap-1849	229	15	)	)	PUNCT
ap-1849	229	16	,	,	PUNCT
ap-1849	229	17	where	where	SCONJ
ap-1849	229	18	function	function	NOUN
ap-1849	229	19	g(t	g(t	PROPN
ap-1849	229	20	)	)	PUNCT
ap-1849	229	21	is	be	AUX
ap-1849	229	22	the	the	DET
ap-1849	229	23	solution	solution	NOUN
ap-1849	229	24	of	of	ADP
ap-1849	229	25	equation	equation	NOUN
ap-1849	229	26	(	(	PUNCT
ap-1849	229	27	10	10	NUM
ap-1849	229	28	)	)	PUNCT
ap-1849	229	29	.	.	PUNCT
ap-1849	230	1	for	for	ADP
ap-1849	230	2	this	this	DET
ap-1849	230	3	function	function	NOUN
ap-1849	230	4	f	f	X
ap-1849	230	5	to	to	PART
ap-1849	230	6	be	be	AUX
ap-1849	230	7	polynomial	polynomial	ADJ
ap-1849	230	8	,	,	PUNCT
ap-1849	230	9	must	must	AUX
ap-1849	230	10	be	be	AUX
ap-1849	230	11	2λ1	2λ1	NUM
ap-1849	230	12	+	+	CCONJ
ap-1849	230	13	λ2	λ2	NOUN
ap-1849	230	14	+	+	CCONJ
ap-1849	230	15	3	3	NUM
ap-1849	230	16	∈	∈	PROPN
ap-1849	230	17	n0	n0	NOUN
ap-1849	230	18	and	and	CCONJ
ap-1849	230	19	−λ1	−λ1	PROPN
ap-1849	231	1	−	−	PROPN
ap-1849	231	2	λ2	λ2	NOUN
ap-1849	231	3	−	−	PROPN
ap-1849	231	4	2	2	NUM
ap-1849	231	5	∈	∈	PROPN
ap-1849	231	6	n0	n0	NUM
ap-1849	231	7	.	.	PUNCT
ap-1849	232	1	but	but	CCONJ
ap-1849	232	2	these	these	DET
ap-1849	232	3	conditions	condition	NOUN
ap-1849	232	4	are	be	AUX
ap-1849	232	5	not	not	PART
ap-1849	232	6	fulfilled	fulfil	VERB
ap-1849	232	7	for	for	ADP
ap-1849	232	8	any	any	DET
ap-1849	232	9	λ2	λ2	PROPN
ap-1849	232	10	∈	∈	PROPN
ap-1849	232	11	n0	n0	PROPN
ap-1849	232	12	.	.	PUNCT
ap-1849	233	1	acknowledgements	acknowledgement	VERB
ap-1849	233	2	the	the	DET
ap-1849	233	3	work	work	NOUN
ap-1849	233	4	of	of	ADP
ap-1849	233	5	č.b	č.b	PROPN
ap-1849	233	6	.	.	PROPN
ap-1849	233	7	and	and	CCONJ
ap-1849	233	8	o.n	o.n	PROPN
ap-1849	233	9	.	.	PROPN
ap-1849	233	10	was	be	AUX
ap-1849	233	11	supported	support	VERB
ap-1849	233	12	in	in	ADP
ap-1849	233	13	part	part	NOUN
ap-1849	233	14	by	by	ADP
ap-1849	233	15	the	the	DET
ap-1849	233	16	gacr	gacr	NOUN
ap-1849	233	17	-	-	PUNCT
ap-1849	233	18	p201/10/1509	p201/10/1509	NOUN
ap-1849	233	19	grant	grant	NOUN
ap-1849	233	20	,	,	PUNCT
ap-1849	233	21	and	and	CCONJ
ap-1849	233	22	by	by	ADP
ap-1849	233	23	research	research	NOUN
ap-1849	233	24	plan	plan	NOUN
ap-1849	233	25	msm6840770039	msm6840770039	NOUN
ap-1849	233	26	.	.	PUNCT
ap-1849	234	1	references	reference	NOUN
ap-1849	234	2	[	[	X
ap-1849	234	3	1	1	NUM
ap-1849	234	4	]	]	X
ap-1849	234	5	d.	d.	PROPN
ap-1849	234	6	verma	verma	PROPN
ap-1849	234	7	.	.	PUNCT
ap-1849	235	1	structure	structure	NOUN
ap-1849	235	2	of	of	ADP
ap-1849	235	3	certain	certain	ADJ
ap-1849	235	4	induced	induced	ADJ
ap-1849	235	5	representations	representation	NOUN
ap-1849	235	6	of	of	ADP
ap-1849	235	7	complex	complex	ADJ
ap-1849	235	8	semisimple	semisimple	NOUN
ap-1849	235	9	lie	lie	NOUN
ap-1849	235	10	algebras	algebras	PROPN
ap-1849	235	11	.	.	PUNCT
ap-1849	236	1	yale	yale	PROPN
ap-1849	236	2	university	university	PROPN
ap-1849	236	3	,	,	PUNCT
ap-1849	236	4	1966	1966	NUM
ap-1849	236	5	.	.	PUNCT
ap-1849	237	1	[	[	X
ap-1849	237	2	2	2	NUM
ap-1849	237	3	]	]	X
ap-1849	237	4	d.	d.	PROPN
ap-1849	237	5	verma	verma	PROPN
ap-1849	237	6	.	.	PUNCT
ap-1849	238	1	structure	structure	NOUN
ap-1849	238	2	of	of	ADP
ap-1849	238	3	certain	certain	ADJ
ap-1849	238	4	induced	induced	ADJ
ap-1849	238	5	representations	representation	NOUN
ap-1849	238	6	of	of	ADP
ap-1849	238	7	complex	complex	ADJ
ap-1849	238	8	semisimple	semisimple	NOUN
ap-1849	238	9	lie	lie	NOUN
ap-1849	238	10	algebras	algebra	NOUN
ap-1849	238	11	.	.	PUNCT
ap-1849	239	1	bull	bull	PROPN
ap-1849	239	2	amer	amer	PROPN
ap-1849	239	3	math	math	PROPN
ap-1849	239	4	soc	soc	PROPN
ap-1849	239	5	74:160–166	74:160–166	PROPN
ap-1849	239	6	,	,	PUNCT
ap-1849	239	7	1968	1968	NUM
ap-1849	239	8	.	.	PUNCT
ap-1849	240	1	403	403	NUM
ap-1849	240	2	č	č	NOUN
ap-1849	240	3	.	.	PUNCT
ap-1849	240	4	burdík	burdík	PROPN
ap-1849	240	5	,	,	PUNCT
ap-1849	240	6	o.	o.	PROPN
ap-1849	240	7	navrátil	navrátil	PROPN
ap-1849	240	8	acta	acta	PROPN
ap-1849	240	9	polytechnica	polytechnica	PROPN
ap-1849	241	1	[	[	X
ap-1849	241	2	3	3	NUM
ap-1849	241	3	]	]	X
ap-1849	241	4	i.	i.	PROPN
ap-1849	241	5	bernstein	bernstein	PROPN
ap-1849	241	6	,	,	PUNCT
ap-1849	241	7	i.m	i.m	PROPN
ap-1849	241	8	.	.	PUNCT
ap-1849	242	1	gel’fand	gel’fand	PROPN
ap-1849	242	2	and	and	CCONJ
ap-1849	242	3	s.i	s.i	PROPN
ap-1849	242	4	.	.	PROPN
ap-1849	242	5	gel’fand	gel’fand	PROPN
ap-1849	242	6	.	.	PUNCT
ap-1849	243	1	structure	structure	NOUN
ap-1849	243	2	of	of	ADP
ap-1849	243	3	representations	representation	NOUN
ap-1849	243	4	generated	generate	VERB
ap-1849	243	5	by	by	ADP
ap-1849	243	6	highest	highest	ADJ
ap-1849	243	7	weight	weight	NOUN
ap-1849	243	8	vectors	vector	NOUN
ap-1849	243	9	.	.	PUNCT
ap-1849	244	1	funct	funct	ADJ
ap-1849	244	2	anal	anal	ADJ
ap-1849	244	3	appl	appl	PROPN
ap-1849	244	4	5:1–8	5:1–8	NUM
ap-1849	244	5	,	,	PUNCT
ap-1849	244	6	1971	1971	NUM
ap-1849	244	7	.	.	PUNCT
ap-1849	245	1	[	[	X
ap-1849	245	2	4	4	X
ap-1849	245	3	]	]	PUNCT
ap-1849	245	4	j.	j.	PROPN
ap-1849	245	5	dixmier	dixmier	PROPN
ap-1849	245	6	.	.	PUNCT
ap-1849	246	1	enveloping	envelop	VERB
ap-1849	246	2	algebras	algebras	PROPN
ap-1849	246	3	.	.	PUNCT
ap-1849	247	1	new	new	PROPN
ap-1849	247	2	york	york	PROPN
ap-1849	247	3	:	:	PUNCT
ap-1849	247	4	north	north	PROPN
ap-1849	247	5	holland	holland	PROPN
ap-1849	247	6	,	,	PUNCT
ap-1849	247	7	1977	1977	NUM
ap-1849	247	8	.	.	PUNCT
ap-1849	248	1	[	[	X
ap-1849	248	2	5	5	NUM
ap-1849	248	3	]	]	PUNCT
ap-1849	248	4	p.	p.	NOUN
ap-1849	248	5	exner	exner	NOUN
ap-1849	248	6	,	,	PUNCT
ap-1849	248	7	m.	m.	NOUN
ap-1849	248	8	havlíček	havlíček	PROPN
ap-1849	248	9	and	and	CCONJ
ap-1849	248	10	w.	w.	PROPN
ap-1849	248	11	lassner	lassner	PROPN
ap-1849	248	12	.	.	PUNCT
ap-1849	249	1	canonical	canonical	ADJ
ap-1849	249	2	realizations	realization	NOUN
ap-1849	249	3	of	of	ADP
ap-1849	249	4	clasical	clasical	ADJ
ap-1849	249	5	lie	lie	NOUN
ap-1849	249	6	algebras	algebra	NOUN
ap-1849	249	7	.	.	PUNCT
ap-1849	250	1	czech	czech	PROPN
ap-1849	250	2	.	.	PUNCT
ap-1849	251	1	j.	j.	PROPN
ap-1849	251	2	phys	phys	PROPN
ap-1849	251	3	.	.	PUNCT
ap-1849	252	1	26b:1213–1228	26b:1213–1228	NUM
ap-1849	252	2	,	,	PUNCT
ap-1849	252	3	1976	1976	NUM
ap-1849	252	4	.	.	PUNCT
ap-1849	253	1	[	[	X
ap-1849	253	2	6	6	NUM
ap-1849	253	3	]	]	PUNCT
ap-1849	253	4	e.	e.	PROPN
ap-1849	253	5	witten	witten	PROPN
ap-1849	253	6	.	.	PUNCT
ap-1849	254	1	anti	anti	ADJ
ap-1849	254	2	–	–	PUNCT
ap-1849	254	3	de	de	ADJ
ap-1849	254	4	sitter	sitter	NOUN
ap-1849	254	5	space	space	NOUN
ap-1849	254	6	and	and	CCONJ
ap-1849	254	7	holography	holography	NOUN
ap-1849	254	8	.	.	PUNCT
ap-1849	255	1	adv	adv	PROPN
ap-1849	255	2	.	.	PUNCT
ap-1849	256	1	theor	theor	PROPN
ap-1849	256	2	.	.	PUNCT
ap-1849	256	3	math	math	NOUN
ap-1849	256	4	.	.	PUNCT
ap-1849	257	1	phys	phy	NOUN
ap-1849	257	2	.	.	PUNCT
ap-1849	258	1	2:253–291	2:253–291	NUM
ap-1849	258	2	,	,	PUNCT
ap-1849	258	3	1998	1998	NUM
ap-1849	258	4	.	.	PUNCT
ap-1849	259	1	[	[	X
ap-1849	259	2	7	7	X
ap-1849	259	3	]	]	X
ap-1849	259	4	o.	o.	NOUN
ap-1849	259	5	aharony	aharony	PROPN
ap-1849	259	6	,	,	PUNCT
ap-1849	259	7	s.s	s.s	PROPN
ap-1849	259	8	gubser	gubser	PROPN
ap-1849	259	9	,	,	PUNCT
ap-1849	259	10	j.	j.	PROPN
ap-1849	259	11	maldacena	maldacena	PROPN
ap-1849	259	12	,	,	PUNCT
ap-1849	259	13	h.	h.	PROPN
ap-1849	259	14	ooguri	ooguri	PROPN
ap-1849	259	15	and	and	CCONJ
ap-1849	259	16	y.	y.	PROPN
ap-1849	259	17	oz	oz	PROPN
ap-1849	259	18	.	.	PUNCT
ap-1849	259	19	large	large	ADJ
ap-1849	259	20	n	n	CCONJ
ap-1849	259	21	field	field	NOUN
ap-1849	259	22	theories	theory	NOUN
ap-1849	259	23	,	,	PUNCT
ap-1849	259	24	string	string	NOUN
ap-1849	259	25	theory	theory	NOUN
ap-1849	259	26	and	and	CCONJ
ap-1849	259	27	gravity	gravity	NOUN
ap-1849	259	28	.	.	PUNCT
ap-1849	260	1	phys	phy	NOUN
ap-1849	260	2	.	.	PUNCT
ap-1849	261	1	rept	rept	NOUN
ap-1849	261	2	.	.	PUNCT
ap-1849	262	1	323:183–386	323:183–386	NUM
ap-1849	262	2	,	,	PUNCT
ap-1849	262	3	2000	2000	NUM
ap-1849	262	4	.	.	PUNCT
ap-1849	263	1	[	[	X
ap-1849	263	2	8	8	NUM
ap-1849	263	3	]	]	X
ap-1849	263	4	v.	v.	PROPN
ap-1849	263	5	dobrev	dobrev	PROPN
ap-1849	263	6	.	.	PUNCT
ap-1849	263	7	subsingular	subsingular	ADJ
ap-1849	263	8	vectors	vector	NOUN
ap-1849	263	9	and	and	CCONJ
ap-1849	263	10	conditionally	conditionally	ADV
ap-1849	263	11	invariant	invariant	ADJ
ap-1849	263	12	(	(	PUNCT
ap-1849	263	13	q	q	ADJ
ap-1849	263	14	-	-	PUNCT
ap-1849	263	15	deformed	deformed	ADJ
ap-1849	263	16	)	)	PUNCT
ap-1849	263	17	equations	equation	NOUN
ap-1849	263	18	.	.	PUNCT
ap-1849	264	1	j	j	PROPN
ap-1849	264	2	phys	phy	NOUN
ap-1849	264	3	a	a	PRON
ap-1849	264	4	:	:	PUNCT
ap-1849	264	5	math	math	NOUN
ap-1849	264	6	gen	gen	PROPN
ap-1849	264	7	28:7135–7155	28:7135–7155	PROPN
ap-1849	264	8	,	,	PUNCT
ap-1849	264	9	1995	1995	NUM
ap-1849	264	10	.	.	PUNCT
ap-1849	265	1	[	[	X
ap-1849	265	2	9	9	NUM
ap-1849	265	3	]	]	PUNCT
ap-1849	265	4	x.	x.	NOUN
ap-1849	265	5	xu	xu	PROPN
ap-1849	265	6	.	.	PUNCT
ap-1849	266	1	differential	differential	ADJ
ap-1849	266	2	equations	equation	NOUN
ap-1849	266	3	for	for	ADP
ap-1849	266	4	singular	singular	ADJ
ap-1849	266	5	vectors	vector	NOUN
ap-1849	266	6	of	of	ADP
ap-1849	266	7	sl(n	sl(n	PROPN
ap-1849	266	8	)	)	PUNCT
ap-1849	266	9	.	.	PUNCT
ap-1849	267	1	arxiv	arxiv	NOUN
ap-1849	267	2	:	:	PUNCT
ap-1849	267	3	math/0305180v1	math/0305180v1	NOUN
ap-1849	267	4	.	.	PUNCT
ap-1849	268	1	[	[	X
ap-1849	268	2	10	10	NUM
ap-1849	268	3	]	]	SYM
ap-1849	268	4	č	č	X
ap-1849	268	5	.	.	PUNCT
ap-1849	268	6	burdík	burdík	PROPN
ap-1849	268	7	.	.	PUNCT
ap-1849	269	1	realization	realization	NOUN
ap-1849	269	2	of	of	ADP
ap-1849	269	3	the	the	DET
ap-1849	269	4	real	real	ADJ
ap-1849	269	5	semisimple	semisimple	NOUN
ap-1849	269	6	lie	lie	NOUN
ap-1849	269	7	algebras	algebra	NOUN
ap-1849	269	8	:	:	PUNCT
ap-1849	269	9	method	method	NOUN
ap-1849	269	10	of	of	ADP
ap-1849	269	11	construction	construction	NOUN
ap-1849	269	12	.	.	PUNCT
ap-1849	270	1	j	j	PROPN
ap-1849	270	2	phys	phy	NOUN
ap-1849	270	3	a	a	PRON
ap-1849	270	4	:	:	PUNCT
ap-1849	270	5	math	math	NOUN
ap-1849	270	6	gen	gen	PROPN
ap-1849	270	7	15:3101–3111	15:3101–3111	NUM
ap-1849	270	8	,	,	PUNCT
ap-1849	270	9	1985	1985	NUM
ap-1849	270	10	.	.	PUNCT
ap-1849	271	1	[	[	X
ap-1849	271	2	11	11	NUM
ap-1849	271	3	]	]	SYM
ap-1849	271	4	č	č	X
ap-1849	271	5	.	.	PUNCT
ap-1849	271	6	burdík	burdík	PROPN
ap-1849	271	7	,	,	PUNCT
ap-1849	271	8	p.	p.	PROPN
ap-1849	271	9	grozman	grozman	NOUN
ap-1849	271	10	,	,	PUNCT
ap-1849	271	11	d.	d.	PROPN
ap-1849	271	12	leites	leites	PROPN
ap-1849	271	13	and	and	CCONJ
ap-1849	271	14	a.	a.	PROPN
ap-1849	271	15	sergeev	sergeev	PROPN
ap-1849	271	16	.	.	PUNCT
ap-1849	272	1	realizations	realization	NOUN
ap-1849	272	2	of	of	ADP
ap-1849	272	3	lie	lie	NOUN
ap-1849	272	4	algebras	algebra	NOUN
ap-1849	272	5	and	and	CCONJ
ap-1849	272	6	superalgebras	superalgebra	NOUN
ap-1849	272	7	via	via	ADP
ap-1849	272	8	creation	creation	NOUN
ap-1849	272	9	and	and	CCONJ
ap-1849	272	10	annihilation	annihilation	NOUN
ap-1849	272	11	operators	operator	NOUN
ap-1849	272	12	i.	i.	PROPN
ap-1849	272	13	theoretical	theoretical	PROPN
ap-1849	272	14	and	and	CCONJ
ap-1849	272	15	math	math	PROPN
ap-1849	272	16	physics	physics	NOUN
ap-1849	272	17	124(2	124(2	NUM
ap-1849	272	18	)	)	PUNCT
ap-1849	272	19	,	,	PUNCT
ap-1849	272	20	2000	2000	NUM
ap-1849	272	21	.	.	PUNCT
ap-1849	273	1	404	404	NUM
ap-1849	273	2	http://arxiv.org/abs/math/0305180v1	http://arxiv.org/abs/math/0305180v1	PROPN
ap-1849	273	3	acta	acta	PROPN
ap-1849	273	4	polytechnica	polytechnica	PROPN
ap-1849	273	5	53(5):399–404	53(5):399–404	PROPN
ap-1849	273	6	,	,	PUNCT
ap-1849	273	7	2013	2013	NUM
ap-1849	273	8	1	1	NUM
ap-1849	273	9	introduction	introduction	NOUN
ap-1849	273	10	2	2	NUM
ap-1849	273	11	the	the	DET
ap-1849	273	12	root	root	NOUN
ap-1849	273	13	system	system	NOUN
ap-1849	273	14	for	for	ADP
ap-1849	273	15	lie	lie	NOUN
ap-1849	273	16	algebra	algebra	NOUN
ap-1849	273	17	b2	b2	NOUN
ap-1849	273	18	3	3	NUM
ap-1849	273	19	the	the	DET
ap-1849	273	20	extremal	extremal	ADJ
ap-1849	273	21	vectors	vector	NOUN
ap-1849	273	22	for	for	ADP
ap-1849	273	23	verma	verma	PROPN
ap-1849	273	24	type	type	NOUN
ap-1849	273	25	representation	representation	NOUN
ap-1849	273	26	4	4	NUM
ap-1849	273	27	differential	differential	NOUN
ap-1849	273	28	equations	equation	NOUN
ap-1849	273	29	for	for	ADP
ap-1849	273	30	extremal	extremal	ADJ
ap-1849	273	31	vectors	vector	NOUN
ap-1849	273	32	5	5	NUM
ap-1849	273	33	the	the	DET
ap-1849	273	34	extremal	extremal	ADJ
ap-1849	273	35	vectors	vector	NOUN
ap-1849	273	36	6	6	NUM
ap-1849	273	37	appendix	appendix	NOUN
ap-1849	273	38	:	:	PUNCT
ap-1849	273	39	polynomial	polynomial	ADJ
ap-1849	273	40	solutions	solution	NOUN
ap-1849	273	41	of	of	ADP
ap-1849	273	42	differential	differential	ADJ
ap-1849	273	43	equations	equation	NOUN
ap-1849	273	44	acknowledgements	acknowledgement	NOUN
ap-1849	273	45	references	reference	NOUN