id	sid	tid	token	lemma	pos
ap-3457	1	1	acta	acta	PROPN
ap-3457	1	2	polytechnica	polytechnica	PROPN
ap-3457	1	3	doi:10.14311	doi:10.14311	PROPN
ap-3457	1	4	/	/	SYM
ap-3457	1	5	ap.2016.56.0202	ap.2016.56.0202	PROPN
ap-3457	1	6	acta	acta	PROPN
ap-3457	1	7	polytechnica	polytechnica	PROPN
ap-3457	1	8	56(3):202–213	56(3):202–213	PROPN
ap-3457	1	9	,	,	PUNCT
ap-3457	1	10	2016	2016	NUM
ap-3457	1	11	©	©	PROPN
ap-3457	1	12	czech	czech	PROPN
ap-3457	1	13	technical	technical	PROPN
ap-3457	1	14	university	university	PROPN
ap-3457	1	15	in	in	ADP
ap-3457	1	16	prague	prague	PROPN
ap-3457	1	17	,	,	PUNCT
ap-3457	1	18	2016	2016	NUM
ap-3457	1	19	available	available	ADJ
ap-3457	1	20	online	online	ADV
ap-3457	1	21	at	at	ADP
ap-3457	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-3457	1	23	on	on	ADP
ap-3457	1	24	cubature	cubature	ADJ
ap-3457	1	25	rules	rule	NOUN
ap-3457	1	26	associated	associate	VERB
ap-3457	1	27	to	to	ADP
ap-3457	1	28	weyl	weyl	PROPN
ap-3457	1	29	group	group	PROPN
ap-3457	1	30	orbit	orbit	NOUN
ap-3457	1	31	functions	function	NOUN
ap-3457	1	32	lenka	lenka	PROPN
ap-3457	1	33	hákováa,∗	hákováa,∗	PROPN
ap-3457	1	34	,	,	PUNCT
ap-3457	1	35	jiří	jiří	NOUN
ap-3457	1	36	hrivnákb	hrivnákb	PROPN
ap-3457	1	37	,	,	PUNCT
ap-3457	1	38	lenka	lenka	PROPN
ap-3457	1	39	motlochováb	motlochováb	PROPN
ap-3457	1	40	a	a	DET
ap-3457	1	41	department	department	NOUN
ap-3457	1	42	of	of	ADP
ap-3457	1	43	mathematics	mathematic	NOUN
ap-3457	1	44	,	,	PUNCT
ap-3457	1	45	faculty	faculty	NOUN
ap-3457	1	46	of	of	ADP
ap-3457	1	47	chemical	chemical	ADJ
ap-3457	1	48	engineering	engineering	NOUN
ap-3457	1	49	,	,	PUNCT
ap-3457	1	50	university	university	NOUN
ap-3457	1	51	of	of	ADP
ap-3457	1	52	chemistry	chemistry	NOUN
ap-3457	1	53	and	and	CCONJ
ap-3457	1	54	technology	technology	NOUN
ap-3457	1	55	,	,	PUNCT
ap-3457	1	56	prague	prague	PROPN
ap-3457	1	57	,	,	PUNCT
ap-3457	1	58	technická	technická	NOUN
ap-3457	1	59	5	5	NUM
ap-3457	1	60	,	,	PUNCT
ap-3457	1	61	cz-166	cz-166	NOUN
ap-3457	1	62	28	28	NUM
ap-3457	1	63	prague	prague	PROPN
ap-3457	1	64	,	,	PUNCT
ap-3457	1	65	czech	czech	PROPN
ap-3457	1	66	republic	republic	PROPN
ap-3457	1	67	b	b	PROPN
ap-3457	1	68	department	department	PROPN
ap-3457	1	69	of	of	ADP
ap-3457	1	70	physics	physics	PROPN
ap-3457	1	71	,	,	PUNCT
ap-3457	1	72	faculty	faculty	NOUN
ap-3457	1	73	of	of	ADP
ap-3457	1	74	nuclear	nuclear	ADJ
ap-3457	1	75	sciences	science	NOUN
ap-3457	1	76	and	and	CCONJ
ap-3457	1	77	physical	physical	ADJ
ap-3457	1	78	engineering	engineering	NOUN
ap-3457	1	79	,	,	PUNCT
ap-3457	1	80	czech	czech	PROPN
ap-3457	1	81	technical	technical	PROPN
ap-3457	1	82	university	university	PROPN
ap-3457	1	83	in	in	ADP
ap-3457	1	84	prague	prague	PROPN
ap-3457	1	85	,	,	PUNCT
ap-3457	1	86	břehová	břehová	VERB
ap-3457	1	87	7	7	NUM
ap-3457	1	88	,	,	PUNCT
ap-3457	1	89	cz-115	cz-115	PROPN
ap-3457	1	90	19	19	NUM
ap-3457	1	91	prague	prague	NOUN
ap-3457	1	92	,	,	PUNCT
ap-3457	1	93	czech	czech	PROPN
ap-3457	1	94	republic	republic	NOUN
ap-3457	1	95	∗	∗	NOUN
ap-3457	1	96	corresponding	correspond	VERB
ap-3457	1	97	author	author	NOUN
ap-3457	1	98	:	:	PUNCT
ap-3457	1	99	lenka.hakova@vscht.cz	lenka.hakova@vscht.cz	PROPN
ap-3457	1	100	abstract	abstract	NOUN
ap-3457	1	101	.	.	PUNCT
ap-3457	2	1	the	the	DET
ap-3457	2	2	aim	aim	NOUN
ap-3457	2	3	of	of	ADP
ap-3457	2	4	this	this	DET
ap-3457	2	5	article	article	NOUN
ap-3457	2	6	is	be	AUX
ap-3457	2	7	to	to	PART
ap-3457	2	8	describe	describe	VERB
ap-3457	2	9	several	several	ADJ
ap-3457	2	10	cubature	cubature	ADJ
ap-3457	2	11	formulas	formula	NOUN
ap-3457	2	12	related	relate	VERB
ap-3457	2	13	to	to	ADP
ap-3457	2	14	the	the	DET
ap-3457	2	15	weyl	weyl	PROPN
ap-3457	2	16	group	group	NOUN
ap-3457	2	17	orbit	orbit	NOUN
ap-3457	2	18	functions	function	NOUN
ap-3457	2	19	,	,	PUNCT
ap-3457	2	20	i.e.	i.e.	X
ap-3457	2	21	to	to	ADP
ap-3457	2	22	the	the	DET
ap-3457	2	23	special	special	ADJ
ap-3457	2	24	cases	case	NOUN
ap-3457	2	25	of	of	ADP
ap-3457	2	26	the	the	DET
ap-3457	2	27	jacobi	jacobi	PROPN
ap-3457	2	28	polynomials	polynomial	NOUN
ap-3457	2	29	associated	associate	VERB
ap-3457	2	30	to	to	ADP
ap-3457	2	31	root	root	NOUN
ap-3457	2	32	systems	system	NOUN
ap-3457	2	33	.	.	PUNCT
ap-3457	3	1	the	the	DET
ap-3457	3	2	diagram	diagram	NOUN
ap-3457	3	3	containing	contain	VERB
ap-3457	3	4	the	the	DET
ap-3457	3	5	relations	relation	NOUN
ap-3457	3	6	among	among	ADP
ap-3457	3	7	the	the	DET
ap-3457	3	8	special	special	ADJ
ap-3457	3	9	functions	function	NOUN
ap-3457	3	10	associated	associate	VERB
ap-3457	3	11	to	to	ADP
ap-3457	3	12	the	the	DET
ap-3457	3	13	weyl	weyl	PROPN
ap-3457	3	14	group	group	NOUN
ap-3457	3	15	orbit	orbit	NOUN
ap-3457	3	16	functions	function	NOUN
ap-3457	3	17	is	be	AUX
ap-3457	3	18	presented	present	VERB
ap-3457	3	19	and	and	CCONJ
ap-3457	3	20	the	the	DET
ap-3457	3	21	link	link	NOUN
ap-3457	3	22	between	between	ADP
ap-3457	3	23	the	the	DET
ap-3457	3	24	weyl	weyl	PROPN
ap-3457	3	25	group	group	NOUN
ap-3457	3	26	orbit	orbit	NOUN
ap-3457	3	27	functions	function	NOUN
ap-3457	3	28	and	and	CCONJ
ap-3457	3	29	the	the	DET
ap-3457	3	30	jacobi	jacobi	PROPN
ap-3457	3	31	polynomials	polynomial	NOUN
ap-3457	3	32	is	be	AUX
ap-3457	3	33	explicitly	explicitly	ADV
ap-3457	3	34	derived	derive	VERB
ap-3457	3	35	in	in	ADP
ap-3457	3	36	full	full	ADJ
ap-3457	3	37	generality	generality	NOUN
ap-3457	3	38	.	.	PUNCT
ap-3457	4	1	the	the	DET
ap-3457	4	2	four	four	NUM
ap-3457	4	3	cubature	cubature	ADJ
ap-3457	4	4	rules	rule	NOUN
ap-3457	4	5	corresponding	correspond	VERB
ap-3457	4	6	to	to	ADP
ap-3457	4	7	these	these	DET
ap-3457	4	8	polynomials	polynomial	NOUN
ap-3457	4	9	are	be	AUX
ap-3457	4	10	summarized	summarize	VERB
ap-3457	4	11	for	for	ADP
ap-3457	4	12	all	all	DET
ap-3457	4	13	simple	simple	ADJ
ap-3457	4	14	lie	lie	NOUN
ap-3457	4	15	algebras	algebra	NOUN
ap-3457	4	16	and	and	CCONJ
ap-3457	4	17	their	their	PRON
ap-3457	4	18	properties	property	NOUN
ap-3457	4	19	simultaneously	simultaneously	ADV
ap-3457	4	20	tested	test	VERB
ap-3457	4	21	on	on	ADP
ap-3457	4	22	model	model	NOUN
ap-3457	4	23	functions	function	NOUN
ap-3457	4	24	.	.	PUNCT
ap-3457	5	1	the	the	DET
ap-3457	5	2	clenshaw	clenshaw	ADJ
ap-3457	5	3	-	-	PUNCT
ap-3457	5	4	curtis	curtis	NOUN
ap-3457	5	5	method	method	NOUN
ap-3457	5	6	is	be	AUX
ap-3457	5	7	used	use	VERB
ap-3457	5	8	to	to	PART
ap-3457	5	9	obtain	obtain	VERB
ap-3457	5	10	additional	additional	ADJ
ap-3457	5	11	formulas	formula	NOUN
ap-3457	5	12	connected	connect	VERB
ap-3457	5	13	with	with	ADP
ap-3457	5	14	the	the	DET
ap-3457	5	15	simple	simple	ADJ
ap-3457	5	16	lie	lie	NOUN
ap-3457	5	17	algebra	algebra	PROPN
ap-3457	5	18	c2	c2	PROPN
ap-3457	5	19	.	.	PUNCT
ap-3457	6	1	keywords	keyword	NOUN
ap-3457	6	2	:	:	PUNCT
ap-3457	6	3	weyl	weyl	PROPN
ap-3457	6	4	group	group	PROPN
ap-3457	6	5	orbit	orbit	NOUN
ap-3457	6	6	functions	function	NOUN
ap-3457	6	7	;	;	PUNCT
ap-3457	6	8	jacobi	jacobi	PROPN
ap-3457	6	9	polynomials	polynomial	NOUN
ap-3457	6	10	;	;	PUNCT
ap-3457	6	11	cubature	cubature	NOUN
ap-3457	6	12	formulas	formula	NOUN
ap-3457	6	13	.	.	PUNCT
ap-3457	7	1	1	1	X
ap-3457	7	2	.	.	X
ap-3457	7	3	introduction	introduction	NOUN
ap-3457	7	4	the	the	DET
ap-3457	7	5	purpose	purpose	NOUN
ap-3457	7	6	of	of	ADP
ap-3457	7	7	this	this	DET
ap-3457	7	8	paper	paper	NOUN
ap-3457	7	9	is	be	AUX
ap-3457	7	10	to	to	PART
ap-3457	7	11	explicitly	explicitly	ADV
ap-3457	7	12	overview	overview	VERB
ap-3457	7	13	in	in	ADP
ap-3457	7	14	full	full	ADJ
ap-3457	7	15	generality	generality	NOUN
ap-3457	7	16	the	the	DET
ap-3457	7	17	link	link	NOUN
ap-3457	7	18	between	between	ADP
ap-3457	7	19	the	the	DET
ap-3457	7	20	weyl	weyl	PROPN
ap-3457	7	21	group	group	NOUN
ap-3457	7	22	orbit	orbit	NOUN
ap-3457	7	23	functions	function	NOUN
ap-3457	7	24	and	and	CCONJ
ap-3457	7	25	the	the	DET
ap-3457	7	26	jacobi	jacobi	PROPN
ap-3457	7	27	and	and	CCONJ
ap-3457	7	28	macdonald	macdonald	PROPN
ap-3457	7	29	polynomials	polynomial	NOUN
ap-3457	7	30	and	and	CCONJ
ap-3457	7	31	further	far	ADV
ap-3457	7	32	examine	examine	VERB
ap-3457	7	33	and	and	CCONJ
ap-3457	7	34	compare	compare	VERB
ap-3457	7	35	related	related	ADJ
ap-3457	7	36	methods	method	NOUN
ap-3457	7	37	of	of	ADP
ap-3457	7	38	numerical	numerical	ADJ
ap-3457	7	39	integration	integration	NOUN
ap-3457	7	40	.	.	PUNCT
ap-3457	8	1	these	these	DET
ap-3457	8	2	methods	method	NOUN
ap-3457	8	3	of	of	ADP
ap-3457	8	4	numerical	numerical	ADJ
ap-3457	8	5	integration	integration	NOUN
ap-3457	8	6	,	,	PUNCT
ap-3457	8	7	known	know	VERB
ap-3457	8	8	as	as	ADP
ap-3457	8	9	cubature	cubature	ADJ
ap-3457	8	10	rules	rule	NOUN
ap-3457	8	11	,	,	PUNCT
ap-3457	8	12	emerged	emerge	VERB
ap-3457	8	13	recently	recently	ADV
ap-3457	8	14	for	for	ADP
ap-3457	8	15	all	all	DET
ap-3457	8	16	four	four	NUM
ap-3457	8	17	cases	case	NOUN
ap-3457	8	18	of	of	ADP
ap-3457	8	19	the	the	DET
ap-3457	8	20	weyl	weyl	PROPN
ap-3457	8	21	group	group	NOUN
ap-3457	8	22	orbit	orbit	NOUN
ap-3457	8	23	functions	function	NOUN
ap-3457	8	24	.	.	PUNCT
ap-3457	9	1	the	the	DET
ap-3457	9	2	four	four	NUM
ap-3457	9	3	families	family	NOUN
ap-3457	9	4	of	of	ADP
ap-3457	9	5	the	the	DET
ap-3457	9	6	weyl	weyl	PROPN
ap-3457	9	7	group	group	NOUN
ap-3457	9	8	orbit	orbit	NOUN
ap-3457	9	9	functions	function	NOUN
ap-3457	9	10	[	[	X
ap-3457	9	11	10	10	NUM
ap-3457	9	12	,	,	PUNCT
ap-3457	9	13	18	18	NUM
ap-3457	9	14	,	,	PUNCT
ap-3457	9	15	19	19	NUM
ap-3457	9	16	,	,	PUNCT
ap-3457	9	17	26	26	NUM
ap-3457	9	18	,	,	PUNCT
ap-3457	9	19	30	30	NUM
ap-3457	9	20	]	]	PUNCT
ap-3457	9	21	are	be	AUX
ap-3457	9	22	connected	connect	VERB
ap-3457	9	23	to	to	ADP
ap-3457	9	24	four	four	NUM
ap-3457	9	25	families	family	NOUN
ap-3457	9	26	of	of	ADP
ap-3457	9	27	orthogonal	orthogonal	ADJ
ap-3457	9	28	polynomials	polynomial	NOUN
ap-3457	9	29	via	via	ADP
ap-3457	9	30	similar	similar	ADJ
ap-3457	9	31	relations	relation	NOUN
ap-3457	9	32	between	between	ADP
ap-3457	9	33	chebyshev	chebyshev	NOUN
ap-3457	9	34	polynomials	polynomial	NOUN
ap-3457	9	35	of	of	ADP
ap-3457	9	36	the	the	DET
ap-3457	9	37	first	first	ADJ
ap-3457	9	38	and	and	CCONJ
ap-3457	9	39	second	second	ADJ
ap-3457	9	40	kinds	kind	NOUN
ap-3457	9	41	and	and	CCONJ
ap-3457	9	42	the	the	DET
ap-3457	9	43	ordinary	ordinary	ADJ
ap-3457	9	44	cosine	cosine	NOUN
ap-3457	9	45	and	and	CCONJ
ap-3457	9	46	sine	sine	ADJ
ap-3457	9	47	functions	function	NOUN
ap-3457	9	48	.	.	PUNCT
ap-3457	10	1	a	a	DET
ap-3457	10	2	full	full	ADJ
ap-3457	10	3	set	set	NOUN
ap-3457	10	4	of	of	ADP
ap-3457	10	5	four	four	NUM
ap-3457	10	6	families	family	NOUN
ap-3457	10	7	of	of	ADP
ap-3457	10	8	orbit	orbit	NOUN
ap-3457	10	9	functions	function	NOUN
ap-3457	10	10	,	,	PUNCT
ap-3457	10	11	called	call	VERB
ap-3457	10	12	c-	c-	X
ap-3457	10	13	,	,	PUNCT
ap-3457	10	14	s-	s-	X
ap-3457	10	15	,	,	PUNCT
ap-3457	10	16	ssand	ssand	NOUN
ap-3457	10	17	sl	sl	PROPN
ap-3457	10	18	-	-	PUNCT
ap-3457	10	19	functions	function	NOUN
ap-3457	10	20	,	,	PUNCT
ap-3457	10	21	arises	arise	VERB
ap-3457	10	22	from	from	ADP
ap-3457	10	23	root	root	NOUN
ap-3457	10	24	systems	system	NOUN
ap-3457	10	25	of	of	ADP
ap-3457	10	26	simple	simple	ADJ
ap-3457	10	27	lie	lie	NOUN
ap-3457	10	28	algebras	algebra	NOUN
ap-3457	10	29	with	with	ADP
ap-3457	10	30	two	two	NUM
ap-3457	10	31	different	different	ADJ
ap-3457	10	32	lengths	length	NOUN
ap-3457	10	33	of	of	ADP
ap-3457	10	34	roots	root	NOUN
ap-3457	10	35	.	.	PUNCT
ap-3457	11	1	these	these	DET
ap-3457	11	2	four	four	NUM
ap-3457	11	3	families	family	NOUN
ap-3457	11	4	of	of	ADP
ap-3457	11	5	orthogonal	orthogonal	ADJ
ap-3457	11	6	polynomials	polynomial	NOUN
ap-3457	11	7	are	be	AUX
ap-3457	11	8	in	in	ADP
ap-3457	11	9	fact	fact	NOUN
ap-3457	11	10	special	special	ADJ
ap-3457	11	11	cases	case	NOUN
ap-3457	11	12	of	of	ADP
ap-3457	11	13	the	the	DET
ap-3457	11	14	multivariate	multivariate	NOUN
ap-3457	11	15	jacobi	jacobi	NOUN
ap-3457	11	16	polynomials	polynomial	VERB
ap-3457	11	17	[	[	X
ap-3457	11	18	11	11	NUM
ap-3457	11	19	,	,	PUNCT
ap-3457	11	20	12	12	NUM
ap-3457	11	21	]	]	PUNCT
ap-3457	11	22	.	.	PUNCT
ap-3457	12	1	the	the	DET
ap-3457	12	2	jacobi	jacobi	PROPN
ap-3457	12	3	polynomials	polynomial	NOUN
ap-3457	12	4	associated	associate	VERB
ap-3457	12	5	to	to	PART
ap-3457	12	6	root	root	NOUN
ap-3457	12	7	systems	system	NOUN
ap-3457	12	8	are	be	AUX
ap-3457	12	9	in	in	ADP
ap-3457	12	10	turn	turn	NOUN
ap-3457	12	11	limiting	limit	VERB
ap-3457	12	12	cases	case	NOUN
ap-3457	12	13	of	of	ADP
ap-3457	12	14	the	the	DET
ap-3457	12	15	macdonald	macdonald	PROPN
ap-3457	12	16	polynomials	polynomial	NOUN
ap-3457	12	17	[	[	X
ap-3457	12	18	24	24	NUM
ap-3457	12	19	]	]	PUNCT
ap-3457	12	20	.	.	PUNCT
ap-3457	13	1	this	this	DET
ap-3457	13	2	connection	connection	NOUN
ap-3457	13	3	between	between	ADP
ap-3457	13	4	the	the	DET
ap-3457	13	5	four	four	NUM
ap-3457	13	6	cases	case	NOUN
ap-3457	13	7	of	of	ADP
ap-3457	13	8	the	the	DET
ap-3457	13	9	jacobi	jacobi	PROPN
ap-3457	13	10	polynomials	polynomial	NOUN
ap-3457	13	11	and	and	CCONJ
ap-3457	13	12	the	the	DET
ap-3457	13	13	underlying	underlie	VERB
ap-3457	13	14	orbit	orbit	NOUN
ap-3457	13	15	functions	function	NOUN
ap-3457	13	16	allows	allow	VERB
ap-3457	13	17	to	to	PART
ap-3457	13	18	formulate	formulate	VERB
ap-3457	13	19	the	the	DET
ap-3457	13	20	corresponding	corresponding	ADJ
ap-3457	13	21	methods	method	NOUN
ap-3457	13	22	for	for	ADP
ap-3457	13	23	numerical	numerical	ADJ
ap-3457	13	24	integration	integration	NOUN
ap-3457	13	25	in	in	ADP
ap-3457	13	26	terms	term	NOUN
ap-3457	13	27	of	of	ADP
ap-3457	13	28	the	the	DET
ap-3457	13	29	jacobi	jacobi	PROPN
ap-3457	13	30	polynomials	polynomial	NOUN
ap-3457	13	31	.	.	PUNCT
ap-3457	14	1	among	among	ADP
ap-3457	14	2	methods	method	NOUN
ap-3457	14	3	for	for	ADP
ap-3457	14	4	numerical	numerical	ADJ
ap-3457	14	5	integration	integration	NOUN
ap-3457	14	6	,	,	PUNCT
ap-3457	14	7	the	the	DET
ap-3457	14	8	quadrature	quadrature	NOUN
ap-3457	14	9	and	and	CCONJ
ap-3457	14	10	cubature	cubature	ADJ
ap-3457	14	11	formulas	formula	NOUN
ap-3457	14	12	related	relate	VERB
ap-3457	14	13	to	to	ADP
ap-3457	14	14	polynomials	polynomial	NOUN
ap-3457	14	15	of	of	ADP
ap-3457	14	16	a	a	DET
ap-3457	14	17	bounded	bounded	ADJ
ap-3457	14	18	degree	degree	NOUN
ap-3457	14	19	hold	hold	VERB
ap-3457	14	20	a	a	DET
ap-3457	14	21	prominent	prominent	ADJ
ap-3457	14	22	place	place	NOUN
ap-3457	14	23	[	[	X
ap-3457	14	24	1	1	NUM
ap-3457	14	25	,	,	PUNCT
ap-3457	14	26	6	6	NUM
ap-3457	14	27	–	–	SYM
ap-3457	14	28	8	8	NUM
ap-3457	14	29	,	,	PUNCT
ap-3457	14	30	35	35	NUM
ap-3457	14	31	,	,	PUNCT
ap-3457	14	32	38	38	NUM
ap-3457	14	33	]	]	PUNCT
ap-3457	14	34	.	.	PUNCT
ap-3457	15	1	such	such	ADJ
ap-3457	15	2	formulas	formula	NOUN
ap-3457	15	3	estimate	estimate	VERB
ap-3457	15	4	a	a	DET
ap-3457	15	5	given	give	VERB
ap-3457	15	6	weighted	weight	VERB
ap-3457	15	7	integral	integral	ADJ
ap-3457	15	8	over	over	ADP
ap-3457	15	9	a	a	DET
ap-3457	15	10	fixed	fix	VERB
ap-3457	15	11	domain	domain	NOUN
ap-3457	15	12	in	in	ADP
ap-3457	15	13	euclidean	euclidean	ADJ
ap-3457	15	14	space	space	NOUN
ap-3457	15	15	.	.	PUNCT
ap-3457	16	1	this	this	DET
ap-3457	16	2	estimation	estimation	NOUN
ap-3457	16	3	holds	hold	VERB
ap-3457	16	4	exactly	exactly	ADV
ap-3457	16	5	for	for	ADP
ap-3457	16	6	all	all	DET
ap-3457	16	7	polynomials	polynomial	NOUN
ap-3457	16	8	up	up	ADP
ap-3457	16	9	to	to	ADP
ap-3457	16	10	a	a	DET
ap-3457	16	11	certain	certain	ADJ
ap-3457	16	12	degree	degree	NOUN
ap-3457	16	13	.	.	PUNCT
ap-3457	17	1	a	a	DET
ap-3457	17	2	significant	significant	ADJ
ap-3457	17	3	effort	effort	NOUN
ap-3457	17	4	put	put	VERB
ap-3457	17	5	into	into	ADP
ap-3457	17	6	development	development	NOUN
ap-3457	17	7	of	of	ADP
ap-3457	17	8	various	various	ADJ
ap-3457	17	9	types	type	NOUN
ap-3457	17	10	of	of	ADP
ap-3457	17	11	cubature	cubature	ADJ
ap-3457	17	12	formulas	formula	NOUN
ap-3457	17	13	results	result	NOUN
ap-3457	17	14	in	in	ADP
ap-3457	17	15	multitude	multitude	NOUN
ap-3457	17	16	of	of	ADP
ap-3457	17	17	types	type	NOUN
ap-3457	17	18	of	of	ADP
ap-3457	17	19	integration	integration	NOUN
ap-3457	17	20	domains	domain	NOUN
ap-3457	17	21	with	with	ADP
ap-3457	17	22	varying	vary	VERB
ap-3457	17	23	efficiencies	efficiency	NOUN
ap-3457	17	24	.	.	PUNCT
ap-3457	18	1	the	the	DET
ap-3457	18	2	shapes	shape	NOUN
ap-3457	18	3	of	of	ADP
ap-3457	18	4	the	the	DET
ap-3457	18	5	integration	integration	NOUN
ap-3457	18	6	domains	domain	NOUN
ap-3457	18	7	and	and	CCONJ
ap-3457	18	8	the	the	DET
ap-3457	18	9	nodes	node	NOUN
ap-3457	18	10	for	for	ADP
ap-3457	18	11	cubature	cubature	ADJ
ap-3457	18	12	formulas	formula	NOUN
ap-3457	18	13	corresponding	correspond	VERB
ap-3457	18	14	to	to	ADP
ap-3457	18	15	the	the	DET
ap-3457	18	16	orthogonal	orthogonal	ADJ
ap-3457	18	17	polynomials	polynomial	NOUN
ap-3457	18	18	of	of	ADP
ap-3457	18	19	the	the	DET
ap-3457	18	20	weyl	weyl	PROPN
ap-3457	18	21	group	group	NOUN
ap-3457	18	22	orbit	orbit	NOUN
ap-3457	18	23	functions	function	NOUN
ap-3457	18	24	are	be	AUX
ap-3457	18	25	determined	determine	VERB
ap-3457	18	26	by	by	ADP
ap-3457	18	27	the	the	DET
ap-3457	18	28	symmetries	symmetry	NOUN
ap-3457	18	29	of	of	ADP
ap-3457	18	30	the	the	DET
ap-3457	18	31	affine	affine	NOUN
ap-3457	18	32	weyl	weyl	VERB
ap-3457	18	33	groups	group	NOUN
ap-3457	18	34	and	and	CCONJ
ap-3457	18	35	a	a	DET
ap-3457	18	36	certain	certain	ADJ
ap-3457	18	37	transform	transform	NOUN
ap-3457	18	38	[	[	X
ap-3457	18	39	15	15	NUM
ap-3457	18	40	,	,	PUNCT
ap-3457	18	41	22	22	NUM
ap-3457	18	42	,	,	PUNCT
ap-3457	18	43	23	23	NUM
ap-3457	18	44	,	,	PUNCT
ap-3457	18	45	26	26	NUM
ap-3457	18	46	,	,	PUNCT
ap-3457	18	47	27	27	NUM
ap-3457	18	48	]	]	PUNCT
ap-3457	18	49	.	.	PUNCT
ap-3457	19	1	this	this	DET
ap-3457	19	2	transform	transform	NOUN
ap-3457	19	3	is	be	AUX
ap-3457	19	4	generated	generate	VERB
ap-3457	19	5	by	by	ADP
ap-3457	19	6	the	the	DET
ap-3457	19	7	transform	transform	NOUN
ap-3457	19	8	which	which	PRON
ap-3457	19	9	induces	induce	VERB
ap-3457	19	10	the	the	DET
ap-3457	19	11	given	give	VERB
ap-3457	19	12	set	set	NOUN
ap-3457	19	13	of	of	ADP
ap-3457	19	14	orthogonal	orthogonal	ADJ
ap-3457	19	15	polynomials	polynomial	NOUN
ap-3457	19	16	.	.	PUNCT
ap-3457	20	1	moreover	moreover	ADV
ap-3457	20	2	,	,	PUNCT
ap-3457	20	3	a	a	DET
ap-3457	20	4	specific	specific	ADJ
ap-3457	20	5	notion	notion	NOUN
ap-3457	20	6	of	of	ADP
ap-3457	20	7	the	the	DET
ap-3457	20	8	modified	modify	VERB
ap-3457	20	9	degree	degree	NOUN
ap-3457	20	10	of	of	ADP
ap-3457	20	11	multivariate	multivariate	NOUN
ap-3457	20	12	polynomials	polynomial	NOUN
ap-3457	20	13	is	be	AUX
ap-3457	20	14	essential	essential	ADJ
ap-3457	20	15	for	for	ADP
ap-3457	20	16	establishing	establish	VERB
ap-3457	20	17	the	the	DET
ap-3457	20	18	final	final	ADJ
ap-3457	20	19	cubature	cubature	NOUN
ap-3457	20	20	formulas	formula	NOUN
ap-3457	20	21	.	.	PUNCT
ap-3457	21	1	one	one	NUM
ap-3457	21	2	of	of	ADP
ap-3457	21	3	the	the	DET
ap-3457	21	4	specific	specific	ADJ
ap-3457	21	5	methods	method	NOUN
ap-3457	21	6	of	of	ADP
ap-3457	21	7	deriving	derive	VERB
ap-3457	21	8	quadrature	quadrature	NOUN
ap-3457	21	9	formulas	formula	NOUN
ap-3457	21	10	,	,	PUNCT
ap-3457	21	11	known	know	VERB
ap-3457	21	12	as	as	ADP
ap-3457	21	13	clenshaw	clenshaw	ADJ
ap-3457	21	14	-	-	PUNCT
ap-3457	21	15	curtis	curtis	NOUN
ap-3457	21	16	method	method	NOUN
ap-3457	21	17	[	[	X
ap-3457	21	18	5	5	NUM
ap-3457	21	19	]	]	PUNCT
ap-3457	21	20	,	,	PUNCT
ap-3457	21	21	is	be	AUX
ap-3457	21	22	classically	classically	ADV
ap-3457	21	23	related	relate	VERB
ap-3457	21	24	to	to	ADP
ap-3457	21	25	chebyshev	chebyshev	VERB
ap-3457	21	26	polynomials	polynomial	NOUN
ap-3457	21	27	of	of	ADP
ap-3457	21	28	one	one	NUM
ap-3457	21	29	variable	variable	NOUN
ap-3457	21	30	[	[	X
ap-3457	21	31	9	9	NUM
ap-3457	21	32	]	]	PUNCT
ap-3457	21	33	.	.	PUNCT
ap-3457	22	1	its	its	PRON
ap-3457	22	2	two	two	NUM
ap-3457	22	3	-	-	PUNCT
ap-3457	22	4	dimensional	dimensional	ADJ
ap-3457	22	5	version	version	NOUN
ap-3457	22	6	related	relate	VERB
ap-3457	22	7	to	to	ADP
ap-3457	22	8	twovariable	twovariable	ADJ
ap-3457	22	9	chebyshev	chebyshev	NOUN
ap-3457	22	10	polynomials	polynomial	NOUN
ap-3457	22	11	of	of	ADP
ap-3457	22	12	the	the	DET
ap-3457	22	13	root	root	NOUN
ap-3457	22	14	system	system	NOUN
ap-3457	22	15	a2	a2	PROPN
ap-3457	22	16	is	be	AUX
ap-3457	22	17	also	also	ADV
ap-3457	22	18	developed	develop	VERB
ap-3457	22	19	[	[	X
ap-3457	22	20	31	31	NUM
ap-3457	22	21	]	]	PUNCT
ap-3457	22	22	.	.	PUNCT
ap-3457	23	1	the	the	DET
ap-3457	23	2	importance	importance	NOUN
ap-3457	23	3	of	of	ADP
ap-3457	23	4	this	this	DET
ap-3457	23	5	method	method	NOUN
ap-3457	23	6	lies	lie	VERB
ap-3457	23	7	e.g.	e.g.	ADV
ap-3457	23	8	in	in	ADP
ap-3457	23	9	its	its	PRON
ap-3457	23	10	utilization	utilization	NOUN
ap-3457	23	11	for	for	ADP
ap-3457	23	12	practical	practical	ADJ
ap-3457	23	13	optimization	optimization	NOUN
ap-3457	23	14	of	of	ADP
ap-3457	23	15	the	the	DET
ap-3457	23	16	shapes	shape	NOUN
ap-3457	23	17	of	of	ADP
ap-3457	23	18	integration	integration	NOUN
ap-3457	23	19	domains	domain	NOUN
ap-3457	23	20	.	.	PUNCT
ap-3457	24	1	the	the	DET
ap-3457	24	2	shapes	shape	NOUN
ap-3457	24	3	of	of	ADP
ap-3457	24	4	the	the	DET
ap-3457	24	5	integration	integration	NOUN
ap-3457	24	6	domains	domain	NOUN
ap-3457	24	7	are	be	AUX
ap-3457	24	8	determined	determine	VERB
ap-3457	24	9	by	by	ADP
ap-3457	24	10	the	the	DET
ap-3457	24	11	underlying	underlie	VERB
ap-3457	24	12	lie	lie	NOUN
ap-3457	24	13	algebra	algebra	NOUN
ap-3457	24	14	[	[	X
ap-3457	24	15	15	15	NUM
ap-3457	24	16	]	]	PUNCT
ap-3457	24	17	and	and	CCONJ
ap-3457	24	18	are	be	AUX
ap-3457	24	19	,	,	PUNCT
ap-3457	24	20	however	however	ADV
ap-3457	24	21	,	,	PUNCT
ap-3457	24	22	of	of	ADP
ap-3457	24	23	non	non	ADJ
ap-3457	24	24	-	-	ADJ
ap-3457	24	25	standard	standard	ADJ
ap-3457	24	26	form	form	NOUN
ap-3457	24	27	.	.	PUNCT
ap-3457	25	1	in	in	ADP
ap-3457	25	2	case	case	NOUN
ap-3457	25	3	of	of	ADP
ap-3457	25	4	simple	simple	ADJ
ap-3457	25	5	lie	lie	NOUN
ap-3457	25	6	algebras	algebra	NOUN
ap-3457	25	7	related	relate	VERB
ap-3457	25	8	to	to	ADP
ap-3457	25	9	twovariable	twovariable	ADJ
ap-3457	25	10	functions	function	NOUN
ap-3457	25	11	,	,	PUNCT
ap-3457	25	12	one	one	NUM
ap-3457	25	13	of	of	ADP
ap-3457	25	14	the	the	DET
ap-3457	25	15	possible	possible	ADJ
ap-3457	25	16	optimizations	optimization	NOUN
ap-3457	25	17	of	of	ADP
ap-3457	25	18	these	these	DET
ap-3457	25	19	shapes	shape	NOUN
ap-3457	25	20	is	be	AUX
ap-3457	25	21	,	,	PUNCT
ap-3457	25	22	similarly	similarly	ADV
ap-3457	25	23	to	to	ADP
ap-3457	25	24	[	[	PUNCT
ap-3457	25	25	31	31	NUM
ap-3457	25	26	]	]	PUNCT
ap-3457	25	27	,	,	PUNCT
ap-3457	25	28	inscribing	inscribe	VERB
ap-3457	25	29	a	a	DET
ap-3457	25	30	triangle	triangle	NOUN
ap-3457	25	31	into	into	ADP
ap-3457	25	32	the	the	DET
ap-3457	25	33	original	original	ADJ
ap-3457	25	34	fundamental	fundamental	ADJ
ap-3457	25	35	domain	domain	NOUN
ap-3457	25	36	.	.	PUNCT
ap-3457	26	1	the	the	DET
ap-3457	26	2	focus	focus	NOUN
ap-3457	26	3	of	of	ADP
ap-3457	26	4	the	the	DET
ap-3457	26	5	present	present	ADJ
ap-3457	26	6	article	article	NOUN
ap-3457	26	7	is	be	AUX
ap-3457	26	8	on	on	ADP
ap-3457	26	9	simple	simple	ADJ
ap-3457	26	10	lie	lie	NOUN
ap-3457	26	11	algebra	algebra	NOUN
ap-3457	26	12	c2	c2	PROPN
ap-3457	26	13	and	and	CCONJ
ap-3457	26	14	its	its	PRON
ap-3457	26	15	corresponding	corresponding	ADJ
ap-3457	26	16	cubature	cubature	NOUN
ap-3457	26	17	rules	rule	NOUN
ap-3457	26	18	.	.	PUNCT
ap-3457	27	1	the	the	DET
ap-3457	27	2	integration	integration	NOUN
ap-3457	27	3	domain	domain	NOUN
ap-3457	27	4	in	in	ADP
ap-3457	27	5	the	the	DET
ap-3457	27	6	case	case	NOUN
ap-3457	27	7	of	of	ADP
ap-3457	27	8	c2	c2	PROPN
ap-3457	27	9	is	be	AUX
ap-3457	27	10	a	a	DET
ap-3457	27	11	region	region	NOUN
ap-3457	27	12	bounded	bound	VERB
ap-3457	27	13	by	by	ADP
ap-3457	27	14	two	two	NUM
ap-3457	27	15	lines	line	NOUN
ap-3457	27	16	and	and	CCONJ
ap-3457	27	17	a	a	DET
ap-3457	27	18	parabola	parabola	NOUN
ap-3457	27	19	depicted	depict	VERB
ap-3457	27	20	in	in	ADP
ap-3457	27	21	fig	fig	NOUN
ap-3457	27	22	.	.	PUNCT
ap-3457	28	1	4	4	NUM
ap-3457	28	2	.	.	PUNCT
ap-3457	28	3	except	except	SCONJ
ap-3457	28	4	from	from	ADP
ap-3457	28	5	a	a	DET
ap-3457	28	6	general	general	ADJ
ap-3457	28	7	perspective	perspective	NOUN
ap-3457	28	8	in	in	ADP
ap-3457	28	9	[	[	X
ap-3457	28	10	15	15	NUM
ap-3457	28	11	,	,	PUNCT
ap-3457	28	12	26	26	NUM
ap-3457	28	13	,	,	PUNCT
ap-3457	28	14	27	27	NUM
ap-3457	28	15	]	]	PUNCT
ap-3457	28	16	,	,	PUNCT
ap-3457	28	17	integration	integration	NOUN
ap-3457	28	18	over	over	ADP
ap-3457	28	19	this	this	DET
ap-3457	28	20	region	region	NOUN
ap-3457	28	21	is	be	AUX
ap-3457	28	22	studied	study	VERB
ap-3457	28	23	in	in	ADP
ap-3457	28	24	[	[	X
ap-3457	28	25	32	32	NUM
ap-3457	28	26	]	]	PUNCT
ap-3457	28	27	.	.	PUNCT
ap-3457	29	1	similarly	similarly	ADV
ap-3457	29	2	to	to	ADP
ap-3457	29	3	[	[	X
ap-3457	29	4	31	31	NUM
ap-3457	29	5	]	]	PUNCT
ap-3457	29	6	for	for	ADP
ap-3457	29	7	a2	a2	PROPN
ap-3457	29	8	,	,	PUNCT
ap-3457	29	9	the	the	DET
ap-3457	29	10	non	non	ADJ
ap-3457	29	11	-	-	ADJ
ap-3457	29	12	standard	standard	ADJ
ap-3457	29	13	shape	shape	NOUN
ap-3457	29	14	of	of	ADP
ap-3457	29	15	this	this	DET
ap-3457	29	16	integration	integration	NOUN
ap-3457	29	17	domain	domain	NOUN
ap-3457	29	18	motivates	motivate	VERB
ap-3457	29	19	further	further	ADJ
ap-3457	29	20	exploration	exploration	NOUN
ap-3457	29	21	of	of	ADP
ap-3457	29	22	the	the	DET
ap-3457	29	23	clenshawcurtis	clenshawcurtis	ADJ
ap-3457	29	24	method	method	NOUN
ap-3457	29	25	.	.	PUNCT
ap-3457	30	1	this	this	DET
ap-3457	30	2	method	method	NOUN
ap-3457	30	3	crucially	crucially	ADV
ap-3457	30	4	depends	depend	VERB
ap-3457	30	5	on	on	ADP
ap-3457	30	6	the	the	DET
ap-3457	30	7	choice	choice	NOUN
ap-3457	30	8	of	of	ADP
ap-3457	30	9	the	the	DET
ap-3457	30	10	weight	weight	NOUN
ap-3457	30	11	function	function	NOUN
ap-3457	30	12	and	and	CCONJ
ap-3457	30	13	the	the	DET
ap-3457	30	14	inscribed	inscribe	VERB
ap-3457	30	15	integration	integration	NOUN
ap-3457	30	16	region	region	NOUN
ap-3457	30	17	and	and	CCONJ
ap-3457	30	18	has	have	AUX
ap-3457	30	19	not	not	PART
ap-3457	30	20	yet	yet	ADV
ap-3457	30	21	been	be	AUX
ap-3457	30	22	studied	study	VERB
ap-3457	30	23	in	in	ADP
ap-3457	30	24	detail	detail	NOUN
ap-3457	30	25	for	for	ADP
ap-3457	30	26	202	202	NUM
ap-3457	30	27	http://dx.doi.org/10.14311/ap.2016.56.0202	http://dx.doi.org/10.14311/ap.2016.56.0202	NOUN
ap-3457	30	28	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-3457	30	29	vol	vol	NOUN
ap-3457	30	30	.	.	PUNCT
ap-3457	31	1	56	56	NUM
ap-3457	31	2	no	no	NOUN
ap-3457	31	3	.	.	PUNCT
ap-3457	32	1	3/2016	3/2016	NUM
ap-3457	32	2	on	on	ADP
ap-3457	32	3	cubature	cubature	ADJ
ap-3457	32	4	rules	rule	NOUN
ap-3457	32	5	associated	associate	VERB
ap-3457	32	6	to	to	ADP
ap-3457	32	7	weyl	weyl	PROPN
ap-3457	32	8	group	group	PROPN
ap-3457	32	9	orbit	orbit	NOUN
ap-3457	32	10	functions	function	VERB
ap-3457	32	11	the	the	DET
ap-3457	32	12	case	case	NOUN
ap-3457	32	13	of	of	ADP
ap-3457	32	14	c2	c2	PROPN
ap-3457	32	15	.	.	PUNCT
ap-3457	33	1	in	in	ADP
ap-3457	33	2	this	this	DET
ap-3457	33	3	case	case	NOUN
ap-3457	33	4	,	,	PUNCT
ap-3457	33	5	the	the	DET
ap-3457	33	6	domain	domain	NOUN
ap-3457	33	7	inscribed	inscribe	VERB
ap-3457	33	8	in	in	ADP
ap-3457	33	9	the	the	DET
ap-3457	33	10	original	original	ADJ
ap-3457	33	11	integration	integration	NOUN
ap-3457	33	12	region	region	NOUN
ap-3457	33	13	is	be	AUX
ap-3457	33	14	considered	consider	VERB
ap-3457	33	15	either	either	CCONJ
ap-3457	33	16	the	the	DET
ap-3457	33	17	original	original	ADJ
ap-3457	33	18	domain	domain	NOUN
ap-3457	33	19	itself	itself	PRON
ap-3457	33	20	or	or	CCONJ
ap-3457	33	21	the	the	DET
ap-3457	33	22	triangle	triangle	NOUN
ap-3457	33	23	depicted	depict	VERB
ap-3457	33	24	in	in	ADP
ap-3457	33	25	fig	fig	NOUN
ap-3457	33	26	.	.	PUNCT
ap-3457	34	1	6	6	NUM
ap-3457	34	2	.	.	X
ap-3457	34	3	two	two	NUM
ap-3457	34	4	fundamental	fundamental	ADJ
ap-3457	34	5	choices	choice	NOUN
ap-3457	34	6	of	of	ADP
ap-3457	34	7	the	the	DET
ap-3457	34	8	weight	weight	NOUN
ap-3457	34	9	functions	function	NOUN
ap-3457	34	10	are	be	AUX
ap-3457	34	11	detailed	detail	VERB
ap-3457	34	12	.	.	PUNCT
ap-3457	35	1	prior	prior	ADV
ap-3457	35	2	to	to	ADP
ap-3457	35	3	practical	practical	ADJ
ap-3457	35	4	implementation	implementation	NOUN
ap-3457	35	5	,	,	PUNCT
ap-3457	35	6	exact	exact	ADJ
ap-3457	35	7	values	value	NOUN
ap-3457	35	8	of	of	ADP
ap-3457	35	9	certain	certain	ADJ
ap-3457	35	10	integrals	integral	NOUN
ap-3457	35	11	are	be	AUX
ap-3457	35	12	also	also	ADV
ap-3457	35	13	needed	need	VERB
ap-3457	35	14	.	.	PUNCT
ap-3457	36	1	one	one	NUM
ap-3457	36	2	of	of	ADP
ap-3457	36	3	the	the	DET
ap-3457	36	4	goals	goal	NOUN
ap-3457	36	5	of	of	ADP
ap-3457	36	6	this	this	DET
ap-3457	36	7	article	article	NOUN
ap-3457	36	8	is	be	AUX
ap-3457	36	9	to	to	PART
ap-3457	36	10	provide	provide	VERB
ap-3457	36	11	all	all	DET
ap-3457	36	12	data	datum	NOUN
ap-3457	36	13	necessary	necessary	ADJ
ap-3457	36	14	for	for	ADP
ap-3457	36	15	practical	practical	ADJ
ap-3457	36	16	implementation	implementation	NOUN
ap-3457	36	17	of	of	ADP
ap-3457	36	18	these	these	DET
ap-3457	36	19	cubature	cubature	ADJ
ap-3457	36	20	formulas	formula	NOUN
ap-3457	36	21	.	.	PUNCT
ap-3457	37	1	this	this	PRON
ap-3457	37	2	is	be	AUX
ap-3457	37	3	achieved	achieve	VERB
ap-3457	37	4	by	by	ADP
ap-3457	37	5	tabulating	tabulate	VERB
ap-3457	37	6	and	and	CCONJ
ap-3457	37	7	calculating	calculate	VERB
ap-3457	37	8	all	all	DET
ap-3457	37	9	needed	need	VERB
ap-3457	37	10	stabilizer	stabilizer	NOUN
ap-3457	37	11	coefficients	coefficient	NOUN
ap-3457	37	12	and	and	CCONJ
ap-3457	37	13	exact	exact	ADJ
ap-3457	37	14	integral	integral	ADJ
ap-3457	37	15	values	value	NOUN
ap-3457	37	16	for	for	ADP
ap-3457	37	17	each	each	DET
ap-3457	37	18	choice	choice	NOUN
ap-3457	37	19	it	it	PRON
ap-3457	37	20	the	the	DET
ap-3457	37	21	inscribed	inscribe	VERB
ap-3457	37	22	integration	integration	NOUN
ap-3457	37	23	domain	domain	NOUN
ap-3457	37	24	and	and	CCONJ
ap-3457	37	25	weight	weight	NOUN
ap-3457	37	26	function	function	NOUN
ap-3457	37	27	.	.	PUNCT
ap-3457	38	1	to	to	PART
ap-3457	38	2	demonstrate	demonstrate	VERB
ap-3457	38	3	usefulness	usefulness	NOUN
ap-3457	38	4	and	and	CCONJ
ap-3457	38	5	viability	viability	NOUN
ap-3457	38	6	of	of	ADP
ap-3457	38	7	the	the	DET
ap-3457	38	8	presented	present	VERB
ap-3457	38	9	methods	method	NOUN
ap-3457	38	10	,	,	PUNCT
ap-3457	38	11	the	the	DET
ap-3457	38	12	numerical	numerical	ADJ
ap-3457	38	13	tests	test	NOUN
ap-3457	38	14	on	on	ADP
ap-3457	38	15	model	model	NOUN
ap-3457	38	16	functions	function	NOUN
ap-3457	38	17	,	,	PUNCT
ap-3457	38	18	including	include	VERB
ap-3457	38	19	multidimensional	multidimensional	ADJ
ap-3457	38	20	step	step	NOUN
ap-3457	38	21	-	-	PUNCT
ap-3457	38	22	functions	function	NOUN
ap-3457	38	23	,	,	PUNCT
ap-3457	38	24	are	be	AUX
ap-3457	38	25	also	also	ADV
ap-3457	38	26	performed	perform	VERB
ap-3457	38	27	.	.	PUNCT
ap-3457	39	1	the	the	DET
ap-3457	39	2	development	development	NOUN
ap-3457	39	3	of	of	ADP
ap-3457	39	4	novel	novel	ADJ
ap-3457	39	5	cubature	cubature	NOUN
ap-3457	39	6	formulas	formula	NOUN
ap-3457	39	7	is	be	AUX
ap-3457	39	8	motivated	motivate	VERB
ap-3457	39	9	by	by	ADP
ap-3457	39	10	their	their	PRON
ap-3457	39	11	widespread	widespread	ADJ
ap-3457	39	12	use	use	NOUN
ap-3457	39	13	in	in	ADP
ap-3457	39	14	applied	apply	VERB
ap-3457	39	15	numerical	numerical	ADJ
ap-3457	39	16	simulations	simulation	NOUN
ap-3457	39	17	and	and	CCONJ
ap-3457	39	18	engineering	engineering	NOUN
ap-3457	39	19	problems	problem	NOUN
ap-3457	39	20	.	.	PUNCT
ap-3457	40	1	among	among	ADP
ap-3457	40	2	direct	direct	ADJ
ap-3457	40	3	numerical	numerical	ADJ
ap-3457	40	4	applications	application	NOUN
ap-3457	40	5	of	of	ADP
ap-3457	40	6	the	the	DET
ap-3457	40	7	cubature	cubature	ADJ
ap-3457	40	8	formulas	formula	NOUN
ap-3457	40	9	is	be	AUX
ap-3457	40	10	the	the	DET
ap-3457	40	11	induced	induced	ADJ
ap-3457	40	12	method	method	NOUN
ap-3457	40	13	of	of	ADP
ap-3457	40	14	polynomial	polynomial	ADJ
ap-3457	40	15	approximation	approximation	NOUN
ap-3457	40	16	.	.	PUNCT
ap-3457	41	1	the	the	DET
ap-3457	41	2	cubature	cubature	ADJ
ap-3457	41	3	formulas	formula	NOUN
ap-3457	41	4	are	be	AUX
ap-3457	41	5	ubiquitous	ubiquitous	ADJ
ap-3457	41	6	in	in	ADP
ap-3457	41	7	the	the	DET
ap-3457	41	8	modern	modern	ADJ
ap-3457	41	9	theory	theory	NOUN
ap-3457	41	10	of	of	ADP
ap-3457	41	11	electromagnetism	electromagnetism	NOUN
ap-3457	41	12	,	,	PUNCT
ap-3457	41	13	especially	especially	ADV
ap-3457	41	14	in	in	ADP
ap-3457	41	15	its	its	PRON
ap-3457	41	16	branches	branch	NOUN
ap-3457	41	17	of	of	ADP
ap-3457	41	18	electromagnetic	electromagnetic	ADJ
ap-3457	41	19	wave	wave	NOUN
ap-3457	41	20	propagation	propagation	NOUN
ap-3457	41	21	[	[	X
ap-3457	41	22	33	33	NUM
ap-3457	41	23	]	]	PUNCT
ap-3457	41	24	,	,	PUNCT
ap-3457	41	25	magnetostatic	magnetostatic	ADJ
ap-3457	41	26	modeling	modeling	NOUN
ap-3457	41	27	[	[	X
ap-3457	41	28	39	39	NUM
ap-3457	41	29	]	]	PUNCT
ap-3457	41	30	and	and	CCONJ
ap-3457	41	31	micromagnetic	micromagnetic	ADJ
ap-3457	41	32	simulations	simulation	NOUN
ap-3457	41	33	[	[	X
ap-3457	41	34	4	4	NUM
ap-3457	41	35	]	]	PUNCT
ap-3457	41	36	.	.	PUNCT
ap-3457	42	1	other	other	ADJ
ap-3457	42	2	fields	field	NOUN
ap-3457	42	3	include	include	VERB
ap-3457	42	4	fluid	fluid	ADJ
ap-3457	42	5	flows	flow	NOUN
ap-3457	42	6	simulations	simulation	NOUN
ap-3457	42	7	,	,	PUNCT
ap-3457	42	8	laser	laser	NOUN
ap-3457	42	9	optics	optic	NOUN
ap-3457	42	10	and	and	CCONJ
ap-3457	42	11	stochastic	stochastic	ADJ
ap-3457	42	12	dynamics	dynamic	NOUN
ap-3457	42	13	.	.	PUNCT
ap-3457	43	1	in	in	ADP
ap-3457	43	2	section	section	NOUN
ap-3457	43	3	2	2	NUM
ap-3457	43	4	are	be	AUX
ap-3457	43	5	reviewed	review	VERB
ap-3457	43	6	the	the	DET
ap-3457	43	7	notions	notion	NOUN
ap-3457	43	8	necessary	necessary	ADJ
ap-3457	43	9	for	for	ADP
ap-3457	43	10	definition	definition	NOUN
ap-3457	43	11	of	of	ADP
ap-3457	43	12	the	the	DET
ap-3457	43	13	weyl	weyl	PROPN
ap-3457	43	14	group	group	NOUN
ap-3457	43	15	orbit	orbit	NOUN
ap-3457	43	16	functions	function	NOUN
ap-3457	43	17	.	.	PUNCT
ap-3457	44	1	the	the	DET
ap-3457	44	2	relation	relation	NOUN
ap-3457	44	3	between	between	ADP
ap-3457	44	4	the	the	DET
ap-3457	44	5	orbit	orbit	NOUN
ap-3457	44	6	functions	function	NOUN
ap-3457	44	7	and	and	CCONJ
ap-3457	44	8	the	the	DET
ap-3457	44	9	jacobi	jacobi	PROPN
ap-3457	44	10	polynomials	polynomial	NOUN
ap-3457	44	11	is	be	AUX
ap-3457	44	12	detailed	detail	VERB
ap-3457	44	13	.	.	PUNCT
ap-3457	45	1	in	in	ADP
ap-3457	45	2	section	section	NOUN
ap-3457	45	3	3	3	NUM
ap-3457	45	4	,	,	PUNCT
ap-3457	45	5	the	the	DET
ap-3457	45	6	cubature	cubature	NOUN
ap-3457	45	7	formulas	formula	NOUN
ap-3457	45	8	from	from	ADP
ap-3457	45	9	[	[	X
ap-3457	45	10	15	15	NUM
ap-3457	45	11	,	,	PUNCT
ap-3457	45	12	26	26	NUM
ap-3457	45	13	,	,	PUNCT
ap-3457	45	14	27	27	NUM
ap-3457	45	15	]	]	PUNCT
ap-3457	45	16	are	be	AUX
ap-3457	45	17	summarized	summarize	VERB
ap-3457	45	18	,	,	PUNCT
ap-3457	45	19	clenshawcurtis	clenshawcurtis	ADJ
ap-3457	45	20	method	method	NOUN
ap-3457	45	21	is	be	AUX
ap-3457	45	22	described	describe	VERB
ap-3457	45	23	and	and	CCONJ
ap-3457	45	24	used	use	VERB
ap-3457	45	25	to	to	PART
ap-3457	45	26	derive	derive	VERB
ap-3457	45	27	additional	additional	ADJ
ap-3457	45	28	cubature	cubature	ADJ
ap-3457	45	29	formulas	formula	NOUN
ap-3457	45	30	.	.	PUNCT
ap-3457	46	1	furthermore	furthermore	ADV
ap-3457	46	2	,	,	PUNCT
ap-3457	46	3	numerical	numerical	ADJ
ap-3457	46	4	test	test	NOUN
ap-3457	46	5	results	result	NOUN
ap-3457	46	6	are	be	AUX
ap-3457	46	7	presented	present	VERB
ap-3457	46	8	.	.	PUNCT
ap-3457	47	1	2	2	X
ap-3457	47	2	.	.	X
ap-3457	47	3	special	special	ADJ
ap-3457	47	4	functions	function	NOUN
ap-3457	47	5	associated	associate	VERB
ap-3457	47	6	to	to	PART
ap-3457	47	7	root	root	VERB
ap-3457	47	8	systems	system	NOUN
ap-3457	47	9	2.1	2.1	NUM
ap-3457	47	10	.	.	PUNCT
ap-3457	48	1	basic	basic	ADJ
ap-3457	48	2	definitions	definition	NOUN
ap-3457	48	3	this	this	DET
ap-3457	48	4	section	section	NOUN
ap-3457	48	5	reviews	review	VERB
ap-3457	48	6	the	the	DET
ap-3457	48	7	basic	basic	ADJ
ap-3457	48	8	concepts	concept	NOUN
ap-3457	48	9	and	and	CCONJ
ap-3457	48	10	notation	notation	NOUN
ap-3457	48	11	from	from	ADP
ap-3457	48	12	the	the	DET
ap-3457	48	13	theory	theory	NOUN
ap-3457	48	14	of	of	ADP
ap-3457	48	15	root	root	NOUN
ap-3457	48	16	systems	system	NOUN
ap-3457	48	17	,	,	PUNCT
ap-3457	48	18	weyl	weyl	VERB
ap-3457	48	19	groups	group	NOUN
ap-3457	48	20	and	and	CCONJ
ap-3457	48	21	weyl	weyl	PROPN
ap-3457	48	22	group	group	NOUN
ap-3457	48	23	orbit	orbit	NOUN
ap-3457	48	24	functions	function	NOUN
ap-3457	48	25	.	.	PUNCT
ap-3457	49	1	it	it	PRON
ap-3457	49	2	is	be	AUX
ap-3457	49	3	consistent	consistent	ADJ
ap-3457	49	4	with	with	ADP
ap-3457	49	5	the	the	DET
ap-3457	49	6	notation	notation	NOUN
ap-3457	49	7	used	use	VERB
ap-3457	49	8	in	in	ADP
ap-3457	49	9	recent	recent	ADJ
ap-3457	49	10	papers	paper	NOUN
ap-3457	49	11	regarding	regard	VERB
ap-3457	49	12	the	the	DET
ap-3457	49	13	topic	topic	NOUN
ap-3457	49	14	,	,	PUNCT
ap-3457	49	15	such	such	ADJ
ap-3457	49	16	as	as	ADP
ap-3457	49	17	[	[	X
ap-3457	49	18	10	10	NUM
ap-3457	49	19	,	,	PUNCT
ap-3457	49	20	13–15	13–15	NUM
ap-3457	49	21	,	,	PUNCT
ap-3457	49	22	26	26	NUM
ap-3457	49	23	]	]	PUNCT
ap-3457	49	24	and	and	CCONJ
ap-3457	49	25	others	other	NOUN
ap-3457	49	26	.	.	PUNCT
ap-3457	50	1	we	we	PRON
ap-3457	50	2	consider	consider	VERB
ap-3457	50	3	simple	simple	ADJ
ap-3457	50	4	lie	lie	NOUN
ap-3457	50	5	algebras	algebra	NOUN
ap-3457	50	6	,	,	PUNCT
ap-3457	50	7	i.e.	i.e.	X
ap-3457	50	8	four	four	NUM
ap-3457	50	9	infinite	infinite	ADJ
ap-3457	50	10	families	family	NOUN
ap-3457	50	11	an(n	an(n	CCONJ
ap-3457	50	12	≥	≥	PROPN
ap-3457	50	13	1	1	NUM
ap-3457	50	14	)	)	PUNCT
ap-3457	50	15	,	,	PUNCT
ap-3457	50	16	bn(n	bn(n	PUNCT
ap-3457	50	17	≥	≥	NOUN
ap-3457	50	18	3	3	NUM
ap-3457	50	19	)	)	PUNCT
ap-3457	50	20	,	,	PUNCT
ap-3457	50	21	cn(n	cn(n	X
ap-3457	50	22	≥	≥	NOUN
ap-3457	50	23	2	2	NUM
ap-3457	50	24	)	)	PUNCT
ap-3457	50	25	and	and	CCONJ
ap-3457	50	26	dn(n	dn(n	NUM
ap-3457	50	27	≥	≥	NOUN
ap-3457	50	28	4	4	NUM
ap-3457	50	29	)	)	PUNCT
ap-3457	50	30	and	and	CCONJ
ap-3457	50	31	five	five	NUM
ap-3457	50	32	exceptional	exceptional	ADJ
ap-3457	50	33	algebras	algebras	PROPN
ap-3457	50	34	e6	e6	PROPN
ap-3457	50	35	,	,	PUNCT
ap-3457	50	36	e7	e7	PROPN
ap-3457	50	37	,	,	PUNCT
ap-3457	50	38	e8	e8	PROPN
ap-3457	50	39	,	,	PUNCT
ap-3457	50	40	f4	f4	PROPN
ap-3457	50	41	and	and	CCONJ
ap-3457	50	42	g2	g2	PROPN
ap-3457	50	43	(	(	PUNCT
ap-3457	50	44	for	for	ADP
ap-3457	50	45	the	the	DET
ap-3457	50	46	classification	classification	NOUN
ap-3457	50	47	see	see	VERB
ap-3457	50	48	[	[	X
ap-3457	50	49	2	2	NUM
ap-3457	50	50	,	,	PUNCT
ap-3457	50	51	16	16	NUM
ap-3457	50	52	]	]	PUNCT
ap-3457	50	53	)	)	PUNCT
ap-3457	50	54	.	.	PUNCT
ap-3457	51	1	in	in	ADP
ap-3457	51	2	particular	particular	ADJ
ap-3457	51	3	,	,	PUNCT
ap-3457	51	4	we	we	PRON
ap-3457	51	5	focus	focus	VERB
ap-3457	51	6	on	on	ADP
ap-3457	51	7	the	the	DET
ap-3457	51	8	algebra	algebra	NOUN
ap-3457	51	9	c2	c2	PROPN
ap-3457	51	10	as	as	ADP
ap-3457	51	11	the	the	DET
ap-3457	51	12	simplest	simple	ADJ
ap-3457	51	13	non	non	ADJ
ap-3457	51	14	-	-	ADJ
ap-3457	51	15	trivial	trivial	ADJ
ap-3457	51	16	example	example	NOUN
ap-3457	51	17	.	.	PUNCT
ap-3457	52	1	each	each	DET
ap-3457	52	2	simple	simple	ADJ
ap-3457	52	3	lie	lie	NOUN
ap-3457	52	4	algebra	algebra	NOUN
ap-3457	52	5	is	be	AUX
ap-3457	52	6	completely	completely	ADV
ap-3457	52	7	described	describe	VERB
ap-3457	52	8	by	by	ADP
ap-3457	52	9	its	its	PRON
ap-3457	52	10	set	set	NOUN
ap-3457	52	11	of	of	ADP
ap-3457	52	12	simple	simple	ADJ
ap-3457	52	13	roots	root	NOUN
ap-3457	52	14	∆	∆	X
ap-3457	52	15	=	=	SYM
ap-3457	52	16	{	{	PUNCT
ap-3457	52	17	α1	α1	PROPN
ap-3457	52	18	,	,	PUNCT
ap-3457	52	19	.	.	PUNCT
ap-3457	52	20	.	.	PUNCT
ap-3457	53	1	.	.	PUNCT
ap-3457	54	1	,	,	PUNCT
ap-3457	54	2	αn	αn	NOUN
ap-3457	54	3	}	}	PUNCT
ap-3457	54	4	which	which	PRON
ap-3457	54	5	forms	form	VERB
ap-3457	54	6	a	a	DET
ap-3457	54	7	nonorthogonal	nonorthogonal	ADJ
ap-3457	54	8	basis	basis	NOUN
ap-3457	54	9	of	of	ADP
ap-3457	54	10	the	the	DET
ap-3457	54	11	euclidean	euclidean	ADJ
ap-3457	54	12	space	space	NOUN
ap-3457	54	13	rn	rn	PROPN
ap-3457	54	14	equipped	equip	VERB
ap-3457	54	15	with	with	ADP
ap-3457	54	16	a	a	DET
ap-3457	54	17	scalar	scalar	ADJ
ap-3457	54	18	product	product	NOUN
ap-3457	54	19	denoted	denote	VERB
ap-3457	54	20	by	by	ADP
ap-3457	54	21	〈	〈	PROPN
ap-3457	54	22	·	·	PROPN
ap-3457	54	23	,	,	PUNCT
ap-3457	54	24	·	·	SYM
ap-3457	54	25	〉	〉	NUM
ap-3457	54	26	.	.	PUNCT
ap-3457	55	1	simple	simple	ADJ
ap-3457	55	2	roots	root	NOUN
ap-3457	55	3	are	be	AUX
ap-3457	55	4	either	either	PRON
ap-3457	55	5	of	of	ADP
ap-3457	55	6	the	the	DET
ap-3457	55	7	same	same	ADJ
ap-3457	55	8	length	length	NOUN
ap-3457	55	9	or	or	CCONJ
ap-3457	55	10	of	of	ADP
ap-3457	55	11	two	two	NUM
ap-3457	55	12	different	different	ADJ
ap-3457	55	13	lengths	length	NOUN
ap-3457	55	14	,	,	PUNCT
ap-3457	55	15	in	in	ADP
ap-3457	55	16	the	the	DET
ap-3457	55	17	latter	latter	ADJ
ap-3457	55	18	case	case	NOUN
ap-3457	55	19	we	we	PRON
ap-3457	55	20	distinguish	distinguish	VERB
ap-3457	55	21	so	so	ADV
ap-3457	55	22	-	-	PUNCT
ap-3457	55	23	called	call	VERB
ap-3457	55	24	short	short	ADJ
ap-3457	55	25	and	and	CCONJ
ap-3457	55	26	long	long	ADJ
ap-3457	55	27	roots	root	NOUN
ap-3457	55	28	and	and	CCONJ
ap-3457	55	29	write	write	VERB
ap-3457	55	30	∆	∆	X
ap-3457	55	31	=	=	SYM
ap-3457	55	32	∆s	∆s	PROPN
ap-3457	55	33	∪∆l	∪∆l	PROPN
ap-3457	55	34	.	.	PUNCT
ap-3457	56	1	the	the	DET
ap-3457	56	2	set	set	NOUN
ap-3457	56	3	of	of	ADP
ap-3457	56	4	dual	dual	ADJ
ap-3457	56	5	roots	root	NOUN
ap-3457	56	6	is	be	AUX
ap-3457	56	7	denoted	denote	VERB
ap-3457	56	8	by	by	ADP
ap-3457	56	9	∆∨	∆∨	NOUN
ap-3457	56	10	=	=	SYM
ap-3457	56	11	{	{	PUNCT
ap-3457	56	12	α∨1	α∨1	NOUN
ap-3457	56	13	,	,	PUNCT
ap-3457	56	14	.	.	PUNCT
ap-3457	56	15	.	.	PUNCT
ap-3457	57	1	.	.	PUNCT
ap-3457	58	1	,	,	PUNCT
ap-3457	58	2	α∨n	α∨n	PROPN
ap-3457	58	3	}	}	PUNCT
ap-3457	58	4	,	,	PUNCT
ap-3457	58	5	where	where	SCONJ
ap-3457	58	6	α∨i	α∨i	NOUN
ap-3457	58	7	=	=	PUNCT
ap-3457	58	8	2αi	2αi	PROPN
ap-3457	59	1	〈	〈	PROPN
ap-3457	59	2	αi	αi	PART
ap-3457	59	3	,	,	PUNCT
ap-3457	59	4	αi	αi	VERB
ap-3457	59	5	〉	〉	NOUN
ap-3457	59	6	.	.	PUNCT
ap-3457	60	1	in	in	ADP
ap-3457	60	2	addition	addition	NOUN
ap-3457	60	3	to	to	ADP
ap-3457	60	4	the	the	DET
ap-3457	60	5	bases	basis	NOUN
ap-3457	60	6	of	of	ADP
ap-3457	60	7	simple	simple	ADJ
ap-3457	60	8	roots	root	NOUN
ap-3457	60	9	and	and	CCONJ
ap-3457	60	10	dual	dual	ADJ
ap-3457	60	11	roots	root	NOUN
ap-3457	60	12	we	we	PRON
ap-3457	60	13	introduce	introduce	VERB
ap-3457	60	14	the	the	DET
ap-3457	60	15	weight	weight	NOUN
ap-3457	60	16	basis	basis	NOUN
ap-3457	60	17	ω1	ω1	PROPN
ap-3457	60	18	,	,	PUNCT
ap-3457	60	19	.	.	PUNCT
ap-3457	60	20	.	.	PUNCT
ap-3457	60	21	.	.	PUNCT
ap-3457	61	1	,	,	PUNCT
ap-3457	61	2	ωn	ωn	NUM
ap-3457	61	3	and	and	CCONJ
ap-3457	61	4	the	the	DET
ap-3457	61	5	dual	dual	ADJ
ap-3457	61	6	weight	weight	NOUN
ap-3457	61	7	basis	basis	NOUN
ap-3457	61	8	ω∨1	ω∨1	NOUN
ap-3457	61	9	,	,	PUNCT
ap-3457	61	10	.	.	PUNCT
ap-3457	61	11	.	.	PUNCT
ap-3457	62	1	.	.	PUNCT
ap-3457	63	1	,	,	PUNCT
ap-3457	63	2	ω∨n	ω∨n	NUM
ap-3457	63	3	where	where	SCONJ
ap-3457	63	4	〈	〈	PROPN
ap-3457	63	5	α∨i	α∨i	NOUN
ap-3457	63	6	,	,	PUNCT
ap-3457	63	7	ωj	ωj	ADP
ap-3457	63	8	〉	〉	NOUN
ap-3457	63	9	=	=	SYM
ap-3457	63	10	〈	〈	PROPN
ap-3457	63	11	αi	αi	NOUN
ap-3457	63	12	,	,	PUNCT
ap-3457	63	13	ω∨j	ω∨j	PROPN
ap-3457	63	14	〉	〉	NOUN
ap-3457	63	15	=	=	SYM
ap-3457	63	16	δij	δij	NOUN
ap-3457	63	17	.	.	PUNCT
ap-3457	64	1	the	the	DET
ap-3457	64	2	cartan	cartan	PROPN
ap-3457	64	3	matrix	matrix	NOUN
ap-3457	64	4	c	c	NOUN
ap-3457	64	5	is	be	AUX
ap-3457	64	6	defined	define	VERB
ap-3457	64	7	as	as	ADP
ap-3457	64	8	cij	cij	PROPN
ap-3457	64	9	=	=	SYM
ap-3457	64	10	2〈αi	2〈αi	PROPN
ap-3457	64	11	,	,	PUNCT
ap-3457	64	12	αj	αj	VERB
ap-3457	64	13	〉	〉	NUM
ap-3457	64	14	〈	〈	PROPN
ap-3457	64	15	αj	αj	PROPN
ap-3457	64	16	,	,	PUNCT
ap-3457	64	17	αj	αj	VERB
ap-3457	64	18	〉	〉	NUM
ap-3457	64	19	and	and	CCONJ
ap-3457	64	20	its	its	PRON
ap-3457	64	21	determinant	determinant	NOUN
ap-3457	64	22	is	be	AUX
ap-3457	64	23	denoted	denote	VERB
ap-3457	64	24	by	by	ADP
ap-3457	64	25	c.	c.	PROPN
ap-3457	64	26	each	each	DET
ap-3457	64	27	simple	simple	ADJ
ap-3457	64	28	root	root	NOUN
ap-3457	64	29	αi	αi	PROPN
ap-3457	64	30	relates	relate	VERB
ap-3457	64	31	to	to	ADP
ap-3457	64	32	a	a	DET
ap-3457	64	33	reflection	reflection	NOUN
ap-3457	64	34	ri	ri	NOUN
ap-3457	64	35	defined	define	VERB
ap-3457	64	36	for	for	ADP
ap-3457	64	37	every	every	DET
ap-3457	64	38	a	a	DET
ap-3457	64	39	∈	∈	PROPN
ap-3457	64	40	rn	rn	NOUN
ap-3457	64	41	as	as	ADP
ap-3457	64	42	ria	ria	PROPN
ap-3457	64	43	=	=	SYM
ap-3457	64	44	a−	a−	PROPN
ap-3457	64	45	2〈a	2〈a	NUM
ap-3457	64	46	,	,	PUNCT
ap-3457	64	47	αi	αi	PROPN
ap-3457	64	48	〉	〉	PROPN
ap-3457	64	49	〈	〈	PROPN
ap-3457	64	50	αi	αi	PART
ap-3457	64	51	,	,	PUNCT
ap-3457	64	52	αi	αi	PROPN
ap-3457	64	53	〉	〉	PROPN
ap-3457	64	54	αi	αi	PROPN
ap-3457	64	55	.	.	PUNCT
ap-3457	65	1	the	the	DET
ap-3457	65	2	set	set	NOUN
ap-3457	65	3	of	of	ADP
ap-3457	65	4	reflections	reflection	NOUN
ap-3457	65	5	{	{	PUNCT
ap-3457	65	6	r1	r1	NOUN
ap-3457	65	7	,	,	PUNCT
ap-3457	65	8	.	.	PUNCT
ap-3457	65	9	.	.	PUNCT
ap-3457	66	1	.	.	PUNCT
ap-3457	67	1	,	,	PUNCT
ap-3457	67	2	rn	rn	PROPN
ap-3457	67	3	}	}	PUNCT
ap-3457	67	4	generates	generate	VERB
ap-3457	67	5	a	a	DET
ap-3457	67	6	finite	finite	ADJ
ap-3457	67	7	groupw	groupw	NOUN
ap-3457	67	8	called	call	VERB
ap-3457	67	9	the	the	DET
ap-3457	67	10	weyl	weyl	PROPN
ap-3457	67	11	group	group	NOUN
ap-3457	67	12	.	.	PUNCT
ap-3457	68	1	by	by	ADP
ap-3457	68	2	the	the	DET
ap-3457	68	3	action	action	NOUN
ap-3457	68	4	ofw	ofw	PROPN
ap-3457	68	5	on	on	ADP
ap-3457	68	6	the	the	DET
ap-3457	68	7	set	set	NOUN
ap-3457	68	8	of	of	ADP
ap-3457	68	9	simple	simple	ADJ
ap-3457	68	10	roots	root	NOUN
ap-3457	68	11	we	we	PRON
ap-3457	68	12	obtain	obtain	VERB
ap-3457	68	13	the	the	DET
ap-3457	68	14	root	root	NOUN
ap-3457	68	15	system	system	NOUN
ap-3457	68	16	π	π	NOUN
ap-3457	68	17	=	=	PUNCT
ap-3457	68	18	w∆.	w∆.	PROPN
ap-3457	68	19	analogously	analogously	ADV
ap-3457	68	20	,	,	PUNCT
ap-3457	68	21	we	we	PRON
ap-3457	68	22	define	define	VERB
ap-3457	68	23	π∨	π∨	PROPN
ap-3457	68	24	=	=	SYM
ap-3457	68	25	w∆∨,πs	w∆∨,πs	PROPN
ap-3457	68	26	=	=	PUNCT
ap-3457	68	27	w∆s	w∆s	PROPN
ap-3457	68	28	and	and	CCONJ
ap-3457	68	29	πl	πl	NOUN
ap-3457	68	30	=	=	NOUN
ap-3457	68	31	w∆l	w∆l	NOUN
ap-3457	68	32	.	.	PUNCT
ap-3457	69	1	every	every	DET
ap-3457	69	2	element	element	NOUN
ap-3457	69	3	of	of	ADP
ap-3457	69	4	π	π	PROPN
ap-3457	69	5	can	can	AUX
ap-3457	69	6	be	be	AUX
ap-3457	69	7	written	write	VERB
ap-3457	69	8	as	as	ADP
ap-3457	69	9	a	a	DET
ap-3457	69	10	combination	combination	NOUN
ap-3457	69	11	of	of	ADP
ap-3457	69	12	simple	simple	ADJ
ap-3457	69	13	roots	root	NOUN
ap-3457	69	14	with	with	ADP
ap-3457	69	15	only	only	ADJ
ap-3457	69	16	non	non	ADJ
ap-3457	69	17	-	-	ADJ
ap-3457	69	18	negative	negative	ADJ
ap-3457	69	19	(	(	PUNCT
ap-3457	69	20	positive	positive	ADJ
ap-3457	69	21	roots	root	NOUN
ap-3457	69	22	)	)	PUNCT
ap-3457	69	23	or	or	CCONJ
ap-3457	69	24	non	non	ADJ
ap-3457	69	25	-	-	ADJ
ap-3457	69	26	positive	positive	ADJ
ap-3457	69	27	integer	integer	NOUN
ap-3457	69	28	coefficients	coefficient	NOUN
ap-3457	69	29	(	(	PUNCT
ap-3457	69	30	negative	negative	ADJ
ap-3457	69	31	roots	root	NOUN
ap-3457	69	32	)	)	PUNCT
ap-3457	69	33	.	.	PUNCT
ap-3457	70	1	the	the	DET
ap-3457	70	2	set	set	NOUN
ap-3457	70	3	of	of	ADP
ap-3457	70	4	positive	positive	ADJ
ap-3457	70	5	roots	root	NOUN
ap-3457	70	6	is	be	AUX
ap-3457	70	7	denoted	denote	VERB
ap-3457	70	8	by	by	ADP
ap-3457	70	9	π+	π+	X
ap-3457	70	10	.	.	PUNCT
ap-3457	71	1	we	we	PRON
ap-3457	71	2	define	define	VERB
ap-3457	71	3	a	a	DET
ap-3457	71	4	partial	partial	ADJ
ap-3457	71	5	ordering	order	VERB
ap-3457	71	6	�	�	PROPN
ap-3457	71	7	of	of	ADP
ap-3457	71	8	roots	root	NOUN
ap-3457	71	9	,	,	PUNCT
ap-3457	71	10	µ	µ	X
ap-3457	71	11	�	�	PROPN
ap-3457	71	12	λ	λ	PROPN
ap-3457	71	13	if	if	SCONJ
ap-3457	71	14	µ	µ	PRON
ap-3457	71	15	−	−	PROPN
ap-3457	71	16	λ	λ	PROPN
ap-3457	71	17	is	be	AUX
ap-3457	71	18	a	a	DET
ap-3457	71	19	sum	sum	NOUN
ap-3457	71	20	of	of	ADP
ap-3457	71	21	simple	simple	ADJ
ap-3457	71	22	roots	root	NOUN
ap-3457	71	23	with	with	ADP
ap-3457	71	24	non	non	ADJ
ap-3457	71	25	-	-	ADJ
ap-3457	71	26	negative	negative	ADJ
ap-3457	71	27	integer	integer	NOUN
ap-3457	71	28	coefficients	coefficient	NOUN
ap-3457	71	29	.	.	PUNCT
ap-3457	72	1	there	there	PRON
ap-3457	72	2	is	be	VERB
ap-3457	72	3	a	a	DET
ap-3457	72	4	unique	unique	ADJ
ap-3457	72	5	highest	high	ADJ
ap-3457	72	6	root	root	NOUN
ap-3457	72	7	ξ	ξ	NOUN
ap-3457	72	8	with	with	ADP
ap-3457	72	9	respect	respect	NOUN
ap-3457	72	10	to	to	ADP
ap-3457	72	11	this	this	DET
ap-3457	72	12	ordering	ordering	NOUN
ap-3457	72	13	,	,	PUNCT
ap-3457	72	14	its	its	PRON
ap-3457	72	15	coordinates	coordinate	NOUN
ap-3457	72	16	in	in	ADP
ap-3457	72	17	the	the	DET
ap-3457	72	18	basis	basis	NOUN
ap-3457	72	19	of	of	ADP
ap-3457	72	20	simple	simple	ADJ
ap-3457	72	21	roots	root	NOUN
ap-3457	72	22	are	be	AUX
ap-3457	72	23	called	call	VERB
ap-3457	72	24	marks	mark	NOUN
ap-3457	72	25	and	and	CCONJ
ap-3457	72	26	denoted	denote	VERB
ap-3457	72	27	by	by	ADP
ap-3457	72	28	m1	m1	PROPN
ap-3457	72	29	,	,	PUNCT
ap-3457	72	30	.	.	PUNCT
ap-3457	72	31	.	.	PUNCT
ap-3457	72	32	.	.	PUNCT
ap-3457	73	1	,	,	PUNCT
ap-3457	73	2	mn	mn	PROPN
ap-3457	73	3	.	.	PROPN
ap-3457	73	4	dual	dual	ADJ
ap-3457	73	5	root	root	NOUN
ap-3457	73	6	system	system	NOUN
ap-3457	73	7	π∨	π∨	PROPN
ap-3457	73	8	contains	contain	VERB
ap-3457	73	9	the	the	DET
ap-3457	73	10	highest	high	ADJ
ap-3457	73	11	root	root	NOUN
ap-3457	73	12	η	η	NOUN
ap-3457	73	13	=	=	PROPN
ap-3457	73	14	m∨1α	m∨1α	PROPN
ap-3457	73	15	∨	∨	NUM
ap-3457	73	16	1	1	NUM
ap-3457	73	17	+	+	CCONJ
ap-3457	73	18	·	·	PUNCT
ap-3457	73	19	·	·	PUNCT
ap-3457	73	20	·	·	PUNCT
ap-3457	74	1	+	+	CCONJ
ap-3457	74	2	m∨nα	m∨nα	PROPN
ap-3457	74	3	∨	∨	NUM
ap-3457	74	4	n	n	NOUN
ap-3457	74	5	with	with	ADP
ap-3457	74	6	the	the	DET
ap-3457	74	7	coefficients	coefficient	NOUN
ap-3457	74	8	m∨i	m∨i	NOUN
ap-3457	74	9	called	call	VERB
ap-3457	74	10	dual	dual	ADJ
ap-3457	74	11	marks	mark	NOUN
ap-3457	74	12	.	.	PUNCT
ap-3457	75	1	an	an	DET
ap-3457	75	2	infinite	infinite	ADJ
ap-3457	75	3	extension	extension	NOUN
ap-3457	75	4	of	of	ADP
ap-3457	75	5	the	the	DET
ap-3457	75	6	weyl	weyl	PROPN
ap-3457	75	7	group	group	NOUN
ap-3457	75	8	w	w	PROPN
ap-3457	75	9	is	be	AUX
ap-3457	75	10	the	the	DET
ap-3457	75	11	affine	affine	NOUN
ap-3457	75	12	weyl	weyl	PROPN
ap-3457	75	13	group	group	PROPN
ap-3457	75	14	w	w	PROPN
ap-3457	75	15	aff	aff	PROPN
ap-3457	75	16	which	which	PRON
ap-3457	75	17	is	be	AUX
ap-3457	75	18	obtained	obtain	VERB
ap-3457	75	19	by	by	ADP
ap-3457	75	20	adding	add	VERB
ap-3457	75	21	to	to	ADP
ap-3457	75	22	the	the	DET
ap-3457	75	23	set	set	NOUN
ap-3457	75	24	of	of	ADP
ap-3457	75	25	generators	generator	NOUN
ap-3457	75	26	of	of	ADP
ap-3457	75	27	w	w	ADP
ap-3457	75	28	the	the	DET
ap-3457	75	29	affine	affine	NOUN
ap-3457	75	30	reflection	reflection	NOUN
ap-3457	75	31	r0	r0	NOUN
ap-3457	75	32	,	,	PUNCT
ap-3457	75	33	r0a	r0a	NOUN
ap-3457	75	34	=	=	PUNCT
ap-3457	76	1	rξa+	rξa+	PROPN
ap-3457	76	2	2ξ	2ξ	NUM
ap-3457	76	3	〈	〈	PROPN
ap-3457	76	4	ξ	ξ	PROPN
ap-3457	76	5	,	,	PUNCT
ap-3457	76	6	ξ	ξ	PROPN
ap-3457	76	7	〉	〉	NUM
ap-3457	76	8	,	,	PUNCT
ap-3457	76	9	rξa	rξa	ADJ
ap-3457	76	10	=	=	SYM
ap-3457	76	11	a−	a−	PROPN
ap-3457	76	12	2〈a	2〈a	NOUN
ap-3457	76	13	,	,	PUNCT
ap-3457	76	14	ξ	ξ	X
ap-3457	76	15	〉	〉	NUM
ap-3457	76	16	〈	〈	PROPN
ap-3457	76	17	ξ	ξ	PROPN
ap-3457	76	18	,	,	PUNCT
ap-3457	76	19	ξ	ξ	PROPN
ap-3457	76	20	〉	〉	PROPN
ap-3457	76	21	ξ	ξ	PROPN
ap-3457	76	22	.	.	PUNCT
ap-3457	77	1	it	it	PRON
ap-3457	77	2	can	can	AUX
ap-3457	77	3	also	also	ADV
ap-3457	77	4	be	be	AUX
ap-3457	77	5	written	write	VERB
ap-3457	77	6	as	as	ADP
ap-3457	77	7	a	a	DET
ap-3457	77	8	semidirect	semidirect	NOUN
ap-3457	77	9	product	product	NOUN
ap-3457	77	10	ofw	ofw	PROPN
ap-3457	77	11	and	and	CCONJ
ap-3457	77	12	a	a	DET
ap-3457	77	13	set	set	NOUN
ap-3457	77	14	of	of	ADP
ap-3457	77	15	shifts	shift	NOUN
ap-3457	77	16	by	by	ADP
ap-3457	77	17	integer	integer	NOUN
ap-3457	77	18	combinations	combination	NOUN
ap-3457	77	19	of	of	ADP
ap-3457	77	20	dual	dual	ADJ
ap-3457	77	21	roots	root	NOUN
ap-3457	77	22	[	[	X
ap-3457	77	23	13	13	NUM
ap-3457	77	24	]	]	PUNCT
ap-3457	77	25	.	.	PUNCT
ap-3457	78	1	we	we	PRON
ap-3457	78	2	denote	denote	VERB
ap-3457	78	3	by	by	ADP
ap-3457	78	4	ψ	ψ	PRON
ap-3457	78	5	the	the	DET
ap-3457	78	6	retraction	retraction	NOUN
ap-3457	78	7	homomorphismw	homomorphismw	NOUN
ap-3457	78	8	aff	aff	PROPN
ap-3457	78	9	→	→	SYM
ap-3457	78	10	w	w	PROPN
ap-3457	79	1	[	[	X
ap-3457	79	2	14	14	NUM
ap-3457	79	3	]	]	PUNCT
ap-3457	79	4	.	.	PUNCT
ap-3457	80	1	the	the	DET
ap-3457	80	2	fundamental	fundamental	ADJ
ap-3457	80	3	domain	domain	NOUN
ap-3457	80	4	f	f	X
ap-3457	80	5	a	a	DET
ap-3457	80	6	set	set	NOUN
ap-3457	80	7	containing	contain	VERB
ap-3457	80	8	exactly	exactly	ADV
ap-3457	80	9	one	one	NUM
ap-3457	80	10	point	point	NOUN
ap-3457	80	11	from	from	ADP
ap-3457	80	12	eachw	eachw	PROPN
ap-3457	80	13	aff	aff	PROPN
ap-3457	80	14	orbit	orbit	NOUN
ap-3457	80	15	can	can	AUX
ap-3457	80	16	be	be	AUX
ap-3457	80	17	chosen	choose	VERB
ap-3457	80	18	as	as	ADP
ap-3457	80	19	f	f	PROPN
ap-3457	80	20	=	=	PROPN
ap-3457	80	21	{	{	PUNCT
ap-3457	80	22	b1ω	b1ω	PROPN
ap-3457	80	23	∨	∨	NUM
ap-3457	80	24	1	1	NUM
ap-3457	80	25	+	+	CCONJ
ap-3457	80	26	·	·	PUNCT
ap-3457	80	27	·	·	PUNCT
ap-3457	80	28	·	·	PUNCT
ap-3457	81	1	+	+	NUM
ap-3457	81	2	bnω	bnω	PROPN
ap-3457	81	3	∨	∨	NUM
ap-3457	81	4	n	n	CCONJ
ap-3457	81	5	|	|	ADV
ap-3457	81	6	bi	bi	PROPN
ap-3457	81	7	∈	∈	PROPN
ap-3457	81	8	r≥0	r≥0	PROPN
ap-3457	81	9	,	,	PUNCT
ap-3457	81	10	b0	b0	NOUN
ap-3457	81	11	+	+	CCONJ
ap-3457	81	12	b1m1	b1m1	NOUN
ap-3457	81	13	+	+	X
ap-3457	81	14	·	·	PUNCT
ap-3457	81	15	·	·	PUNCT
ap-3457	81	16	·	·	PUNCT
ap-3457	82	1	+	+	NUM
ap-3457	82	2	bnmn	bnmn	ADJ
ap-3457	82	3	=	=	SYM
ap-3457	82	4	1	1	NUM
ap-3457	82	5	}	}	PUNCT
ap-3457	82	6	.	.	PUNCT
ap-3457	83	1	(	(	PUNCT
ap-3457	83	2	1	1	X
ap-3457	83	3	)	)	PUNCT
ap-3457	83	4	analogously	analogously	ADV
ap-3457	83	5	we	we	PRON
ap-3457	83	6	define	define	VERB
ap-3457	83	7	dual	dual	ADJ
ap-3457	83	8	affine	affine	NOUN
ap-3457	83	9	weyl	weyl	VERB
ap-3457	83	10	group	group	NOUN
ap-3457	83	11	as	as	ADP
ap-3457	83	12	a	a	DET
ap-3457	83	13	semidirect	semidirect	NOUN
ap-3457	83	14	product	product	NOUN
ap-3457	83	15	of	of	ADP
ap-3457	83	16	w	w	NOUN
ap-3457	83	17	and	and	CCONJ
ap-3457	83	18	shifts	shift	NOUN
ap-3457	83	19	by	by	ADP
ap-3457	83	20	integer	integer	NOUN
ap-3457	83	21	combinations	combination	NOUN
ap-3457	83	22	of	of	ADP
ap-3457	83	23	simple	simple	ADJ
ap-3457	83	24	roots	root	NOUN
ap-3457	83	25	.	.	PUNCT
ap-3457	84	1	we	we	PRON
ap-3457	84	2	introduce	introduce	VERB
ap-3457	84	3	three	three	NUM
ap-3457	84	4	lattices	lattice	NOUN
ap-3457	84	5	p	p	X
ap-3457	84	6	,	,	PUNCT
ap-3457	84	7	p+	p+	NOUN
ap-3457	84	8	and	and	CCONJ
ap-3457	84	9	p∨	p∨	NOUN
ap-3457	84	10	as	as	ADP
ap-3457	84	11	p	p	NOUN
ap-3457	84	12	=	=	PROPN
ap-3457	84	13	zω1	zω1	PROPN
ap-3457	84	14	+	+	CCONJ
ap-3457	84	15	·	·	PUNCT
ap-3457	84	16	·	·	PUNCT
ap-3457	84	17	·	·	PUNCT
ap-3457	85	1	+	+	NUM
ap-3457	85	2	zωn	zωn	NUM
ap-3457	85	3	,	,	PUNCT
ap-3457	85	4	p+	p+	X
ap-3457	85	5	=	=	SYM
ap-3457	85	6	z≥0ω1	z≥0ω1	X
ap-3457	85	7	+	+	X
ap-3457	85	8	·	·	PUNCT
ap-3457	85	9	·	·	PUNCT
ap-3457	85	10	·	·	PUNCT
ap-3457	85	11	+	+	PUNCT
ap-3457	85	12	z≥0ωn	z≥0ωn	NOUN
ap-3457	85	13	,	,	PUNCT
ap-3457	85	14	p∨	p∨	NOUN
ap-3457	85	15	=	=	PUNCT
ap-3457	85	16	zω∨1	zω∨1	PROPN
ap-3457	86	1	+	+	CCONJ
ap-3457	86	2	·	·	PUNCT
ap-3457	86	3	·	·	PUNCT
ap-3457	86	4	·	·	PUNCT
ap-3457	86	5	+	+	CCONJ
ap-3457	86	6	zω∨n	zω∨n	X
ap-3457	86	7	.	.	PUNCT
ap-3457	86	8	note	note	VERB
ap-3457	86	9	that	that	SCONJ
ap-3457	86	10	the	the	DET
ap-3457	86	11	root	root	NOUN
ap-3457	86	12	system	system	NOUN
ap-3457	86	13	π	π	NOUN
ap-3457	86	14	is	be	AUX
ap-3457	86	15	contained	contain	VERB
ap-3457	86	16	in	in	ADP
ap-3457	86	17	p	p	NOUN
ap-3457	86	18	,	,	PUNCT
ap-3457	86	19	therefore	therefore	ADV
ap-3457	86	20	,	,	PUNCT
ap-3457	86	21	the	the	DET
ap-3457	86	22	partial	partial	ADJ
ap-3457	86	23	ordering	order	VERB
ap-3457	86	24	�	�	PROPN
ap-3457	86	25	can	can	AUX
ap-3457	86	26	be	be	AUX
ap-3457	86	27	extended	extend	VERB
ap-3457	86	28	to	to	ADP
ap-3457	86	29	the	the	DET
ap-3457	86	30	lattice	lattice	PROPN
ap-3457	86	31	p	p	PROPN
ap-3457	86	32	.	.	PUNCT
ap-3457	87	1	a	a	DET
ap-3457	87	2	function	function	NOUN
ap-3457	87	3	k	k	NOUN
ap-3457	87	4	:	:	PUNCT
ap-3457	87	5	α	α	PROPN
ap-3457	87	6	∈	∈	PROPN
ap-3457	88	1	π→	π→	INTJ
ap-3457	88	2	kα	kα	PROPN
ap-3457	88	3	∈	∈	PROPN
ap-3457	88	4	r≥0	r≥0	NOUN
ap-3457	88	5	such	such	ADJ
ap-3457	88	6	that	that	SCONJ
ap-3457	88	7	kα	kα	NOUN
ap-3457	88	8	=	=	PUNCT
ap-3457	88	9	kw(α	kw(α	X
ap-3457	88	10	)	)	PUNCT
ap-3457	88	11	for	for	ADP
ap-3457	88	12	all	all	DET
ap-3457	88	13	w	w	NOUN
ap-3457	88	14	∈w	∈w	PROPN
ap-3457	88	15	203	203	NUM
ap-3457	88	16	l.	l.	PROPN
ap-3457	88	17	háková	háková	PROPN
ap-3457	88	18	,	,	PUNCT
ap-3457	88	19	j.	j.	PROPN
ap-3457	88	20	hrivnák	hrivnák	PROPN
ap-3457	88	21	,	,	PUNCT
ap-3457	88	22	l.	l.	PROPN
ap-3457	88	23	motlochová	motlochová	PROPN
ap-3457	88	24	acta	acta	PROPN
ap-3457	88	25	polytechnica	polytechnica	PROPN
ap-3457	88	26	is	be	AUX
ap-3457	88	27	known	know	VERB
ap-3457	88	28	as	as	ADP
ap-3457	88	29	a	a	DET
ap-3457	88	30	multiplicity	multiplicity	NOUN
ap-3457	88	31	function	function	NOUN
ap-3457	88	32	on	on	ADP
ap-3457	88	33	π	π	PROPN
ap-3457	88	34	.	.	PUNCT
ap-3457	89	1	the	the	DET
ap-3457	89	2	trivial	trivial	ADJ
ap-3457	89	3	example	example	NOUN
ap-3457	89	4	is	be	AUX
ap-3457	89	5	to	to	PART
ap-3457	89	6	take	take	VERB
ap-3457	89	7	kα	kα	NOUN
ap-3457	89	8	=	=	PUNCT
ap-3457	89	9	const	const	PROPN
ap-3457	89	10	for	for	ADP
ap-3457	89	11	all	all	PRON
ap-3457	89	12	α	α	DET
ap-3457	89	13	∈	∈	PROPN
ap-3457	89	14	π	π	NOUN
ap-3457	89	15	which	which	PRON
ap-3457	89	16	we	we	PRON
ap-3457	89	17	denote	denote	VERB
ap-3457	89	18	by	by	ADP
ap-3457	89	19	kconst	kconst	PROPN
ap-3457	89	20	.	.	PUNCT
ap-3457	90	1	for	for	ADP
ap-3457	90	2	simple	simple	ADJ
ap-3457	90	3	lie	lie	NOUN
ap-3457	90	4	algebras	algebra	NOUN
ap-3457	90	5	with	with	ADP
ap-3457	90	6	two	two	NUM
ap-3457	90	7	different	different	ADJ
ap-3457	90	8	root	root	NOUN
ap-3457	90	9	lengths	length	NOUN
ap-3457	90	10	,	,	PUNCT
ap-3457	90	11	it	it	PRON
ap-3457	90	12	is	be	AUX
ap-3457	90	13	natural	natural	ADJ
ap-3457	90	14	to	to	PART
ap-3457	90	15	distinguish	distinguish	VERB
ap-3457	90	16	between	between	ADP
ap-3457	90	17	short	short	ADJ
ap-3457	90	18	and	and	CCONJ
ap-3457	90	19	long	long	ADJ
ap-3457	90	20	roots	root	NOUN
ap-3457	90	21	by	by	ADP
ap-3457	90	22	defining	define	VERB
ap-3457	90	23	ksα	ksα	PROPN
ap-3457	90	24	≡	≡	PROPN
ap-3457	90	25	{	{	PUNCT
ap-3457	90	26	1	1	NUM
ap-3457	90	27	if	if	SCONJ
ap-3457	90	28	α	α	PRON
ap-3457	90	29	∈	∈	PROPN
ap-3457	90	30	πs	πs	ADP
ap-3457	90	31	,	,	PUNCT
ap-3457	90	32	0	0	PUNCT
ap-3457	91	1	if	if	SCONJ
ap-3457	91	2	α	α	PROPN
ap-3457	91	3	∈	∈	PROPN
ap-3457	91	4	πl	πl	PROPN
ap-3457	91	5	,	,	PUNCT
ap-3457	91	6	klα	klα	PROPN
ap-3457	91	7	≡	≡	PROPN
ap-3457	91	8	{	{	PUNCT
ap-3457	91	9	0	0	NUM
ap-3457	92	1	if	if	SCONJ
ap-3457	92	2	α	α	PROPN
ap-3457	92	3	∈	∈	PROPN
ap-3457	92	4	πs	πs	ADP
ap-3457	92	5	,	,	PUNCT
ap-3457	92	6	1	1	NUM
ap-3457	92	7	if	if	SCONJ
ap-3457	92	8	α	α	PRON
ap-3457	92	9	∈	∈	PROPN
ap-3457	93	1	πl	πl	PROPN
ap-3457	93	2	.	.	PUNCT
ap-3457	94	1	the	the	DET
ap-3457	94	2	notion	notion	NOUN
ap-3457	94	3	of	of	ADP
ap-3457	94	4	multiplicity	multiplicity	NOUN
ap-3457	94	5	function	function	NOUN
ap-3457	94	6	allows	allow	VERB
ap-3457	94	7	us	we	PRON
ap-3457	94	8	to	to	PART
ap-3457	94	9	define	define	VERB
ap-3457	94	10	sums	sum	NOUN
ap-3457	94	11	of	of	ADP
ap-3457	94	12	positive	positive	ADJ
ap-3457	94	13	roots	root	NOUN
ap-3457	94	14	%	%	NOUN
ap-3457	94	15	(	(	PUNCT
ap-3457	94	16	k	k	NOUN
ap-3457	94	17	)	)	PUNCT
ap-3457	94	18	and	and	CCONJ
ap-3457	94	19	numbers	number	NOUN
ap-3457	94	20	h(k	h(k	PROPN
ap-3457	94	21	)	)	PUNCT
ap-3457	94	22	,	,	PUNCT
ap-3457	94	23	%	%	INTJ
ap-3457	94	24	(	(	PUNCT
ap-3457	94	25	k	k	NOUN
ap-3457	94	26	)	)	PUNCT
ap-3457	94	27	≡	≡	PROPN
ap-3457	94	28	1	1	NUM
ap-3457	94	29	2	2	NUM
ap-3457	94	30	∑	∑	PROPN
ap-3457	94	31	α∈π+	α∈π+	ADJ
ap-3457	94	32	kαα	kαα	PROPN
ap-3457	94	33	,	,	PUNCT
ap-3457	94	34	(	(	PUNCT
ap-3457	94	35	2	2	X
ap-3457	94	36	)	)	PUNCT
ap-3457	94	37	h(k	h(k	PROPN
ap-3457	94	38	)	)	PUNCT
ap-3457	95	1	=	=	PUNCT
ap-3457	95	2	kξ	kξ	INTJ
ap-3457	95	3	+	+	CCONJ
ap-3457	95	4	n∑	n∑	ADJ
ap-3457	95	5	i=1	i=1	PROPN
ap-3457	95	6	mikαi	mikαi	NOUN
ap-3457	95	7	.	.	PUNCT
ap-3457	96	1	in	in	ADP
ap-3457	96	2	particular	particular	ADJ
ap-3457	96	3	,	,	PUNCT
ap-3457	96	4	with	with	ADP
ap-3457	96	5	the	the	DET
ap-3457	96	6	choice	choice	NOUN
ap-3457	96	7	of	of	ADP
ap-3457	96	8	kt	kt	PROPN
ap-3457	96	9	,	,	PUNCT
ap-3457	96	10	where	where	SCONJ
ap-3457	96	11	t	t	PROPN
ap-3457	96	12	is	be	AUX
ap-3457	96	13	one	one	NUM
ap-3457	96	14	of	of	ADP
ap-3457	96	15	the	the	DET
ap-3457	96	16	symbols	symbol	NOUN
ap-3457	96	17	{	{	PUNCT
ap-3457	96	18	0	0	NUM
ap-3457	96	19	,	,	PUNCT
ap-3457	96	20	1	1	NUM
ap-3457	96	21	,	,	PUNCT
ap-3457	96	22	s	s	X
ap-3457	96	23	,	,	PUNCT
ap-3457	96	24	l	l	NOUN
ap-3457	96	25	}	}	PUNCT
ap-3457	96	26	,	,	PUNCT
ap-3457	96	27	we	we	PRON
ap-3457	96	28	have	have	VERB
ap-3457	96	29	%	%	NOUN
ap-3457	96	30	0	0	NUM
ap-3457	97	1	≡	≡	ADJ
ap-3457	97	2	%	%	NOUN
ap-3457	97	3	(	(	PUNCT
ap-3457	97	4	k0	k0	PROPN
ap-3457	97	5	)	)	PUNCT
ap-3457	97	6	=	=	SYM
ap-3457	97	7	0	0	NUM
ap-3457	97	8	,	,	PUNCT
ap-3457	97	9	%	%	INTJ
ap-3457	97	10	1	1	NUM
ap-3457	97	11	≡	≡	ADJ
ap-3457	97	12	%	%	NOUN
ap-3457	97	13	(	(	PUNCT
ap-3457	97	14	k1	k1	NOUN
ap-3457	97	15	)	)	PUNCT
ap-3457	97	16	=	=	SYM
ap-3457	97	17	n∑	n∑	PROPN
ap-3457	97	18	i=1	i=1	PROPN
ap-3457	97	19	ωi	ωi	PROPN
ap-3457	97	20	,	,	PUNCT
ap-3457	97	21	%	%	NOUN
ap-3457	97	22	s	s	X
ap-3457	97	23	≡	≡	PROPN
ap-3457	97	24	%	%	NOUN
ap-3457	97	25	(	(	PUNCT
ap-3457	97	26	ks	ks	NOUN
ap-3457	97	27	)	)	PUNCT
ap-3457	97	28	=	=	PUNCT
ap-3457	97	29	∑	∑	PUNCT
ap-3457	97	30	αi∈∆s	αi∈∆s	NUM
ap-3457	97	31	ωi	ωi	NOUN
ap-3457	97	32	,	,	PUNCT
ap-3457	97	33	%	%	NOUN
ap-3457	97	34	l	l	X
ap-3457	97	35	≡	≡	PROPN
ap-3457	97	36	%	%	INTJ
ap-3457	97	37	(	(	PUNCT
ap-3457	97	38	kl	kl	NOUN
ap-3457	97	39	)	)	PUNCT
ap-3457	97	40	=	=	SYM
ap-3457	97	41	∑	∑	PUNCT
ap-3457	97	42	αi∈∆l	αi∈∆l	NUM
ap-3457	97	43	ωi	ωi	X
ap-3457	97	44	(	(	PUNCT
ap-3457	97	45	3	3	NUM
ap-3457	97	46	)	)	PUNCT
ap-3457	97	47	and	and	CCONJ
ap-3457	97	48	h0	h0	PROPN
ap-3457	97	49	≡	≡	PROPN
ap-3457	97	50	h(k0	h(k0	NOUN
ap-3457	97	51	)	)	PUNCT
ap-3457	97	52	=	=	SYM
ap-3457	97	53	0	0	NUM
ap-3457	97	54	,	,	PUNCT
ap-3457	97	55	h1	h1	PROPN
ap-3457	97	56	≡	≡	PROPN
ap-3457	97	57	h(k1	h(k1	NOUN
ap-3457	97	58	)	)	PUNCT
ap-3457	97	59	=	=	SYM
ap-3457	98	1	1	1	NUM
ap-3457	98	2	+	+	NUM
ap-3457	98	3	n∑	n∑	PROPN
ap-3457	98	4	i=1	i=1	PROPN
ap-3457	98	5	mi	mi	PROPN
ap-3457	98	6	,	,	PUNCT
ap-3457	98	7	hs	hs	PROPN
ap-3457	98	8	≡	≡	PROPN
ap-3457	98	9	h(ks	h(ks	PROPN
ap-3457	98	10	)	)	PUNCT
ap-3457	99	1	=	=	PUNCT
ap-3457	99	2	∑	∑	PUNCT
ap-3457	99	3	αi∈∆s	αi∈∆s	PROPN
ap-3457	99	4	mi	mi	PROPN
ap-3457	99	5	,	,	PUNCT
ap-3457	99	6	hl	hl	PROPN
ap-3457	99	7	≡	≡	PROPN
ap-3457	99	8	h(kl	h(kl	PROPN
ap-3457	99	9	)	)	PUNCT
ap-3457	99	10	=	=	SYM
ap-3457	100	1	1	1	NUM
ap-3457	100	2	+	+	CCONJ
ap-3457	100	3	∑	∑	PART
ap-3457	100	4	αi∈∆l	αi∈∆l	NUM
ap-3457	100	5	mi	mi	PROPN
ap-3457	100	6	.	.	PROPN
ap-3457	100	7	(	(	PUNCT
ap-3457	100	8	4	4	X
ap-3457	100	9	)	)	PUNCT
ap-3457	100	10	the	the	DET
ap-3457	100	11	number	number	NOUN
ap-3457	100	12	h	h	NOUN
ap-3457	100	13	≡	≡	PROPN
ap-3457	100	14	h1	h1	PROPN
ap-3457	100	15	is	be	AUX
ap-3457	100	16	called	call	VERB
ap-3457	100	17	the	the	DET
ap-3457	100	18	coxeter	coxet	ADJ
ap-3457	100	19	number	number	NOUN
ap-3457	100	20	,	,	PUNCT
ap-3457	100	21	analogously	analogously	ADV
ap-3457	100	22	,	,	PUNCT
ap-3457	100	23	we	we	PRON
ap-3457	100	24	call	call	VERB
ap-3457	100	25	hs	hs	PROPN
ap-3457	100	26	and	and	CCONJ
ap-3457	100	27	hl	hl	PROPN
ap-3457	100	28	short	short	ADJ
ap-3457	100	29	and	and	CCONJ
ap-3457	100	30	long	long	ADJ
ap-3457	100	31	coxeter	coxeter	NOUN
ap-3457	100	32	number	number	NOUN
ap-3457	100	33	.	.	PUNCT
ap-3457	101	1	the	the	DET
ap-3457	101	2	set	set	NOUN
ap-3457	101	3	of	of	ADP
ap-3457	101	4	simple	simple	ADJ
ap-3457	101	5	roots	root	NOUN
ap-3457	101	6	∆	∆	X
ap-3457	101	7	=	=	SYM
ap-3457	101	8	{	{	PUNCT
ap-3457	101	9	α1	α1	PROPN
ap-3457	101	10	,	,	PUNCT
ap-3457	101	11	α2	α2	ADJ
ap-3457	101	12	}	}	PUNCT
ap-3457	101	13	of	of	ADP
ap-3457	101	14	the	the	DET
ap-3457	101	15	algebra	algebra	NOUN
ap-3457	101	16	c2	c2	PROPN
ap-3457	101	17	decomposes	decompose	VERB
ap-3457	101	18	into	into	ADP
ap-3457	101	19	the	the	DET
ap-3457	101	20	set	set	NOUN
ap-3457	101	21	of	of	ADP
ap-3457	101	22	the	the	DET
ap-3457	101	23	short	short	ADJ
ap-3457	101	24	simple	simple	ADJ
ap-3457	101	25	roots	root	NOUN
ap-3457	101	26	∆s	∆s	NOUN
ap-3457	101	27	=	=	SYM
ap-3457	101	28	{	{	PUNCT
ap-3457	101	29	α1	α1	PROPN
ap-3457	101	30	}	}	PUNCT
ap-3457	101	31	and	and	CCONJ
ap-3457	101	32	the	the	DET
ap-3457	101	33	set	set	NOUN
ap-3457	101	34	of	of	ADP
ap-3457	101	35	the	the	DET
ap-3457	101	36	long	long	ADJ
ap-3457	101	37	simple	simple	ADJ
ap-3457	101	38	roots	root	NOUN
ap-3457	101	39	∆l	∆l	PROPN
ap-3457	101	40	=	=	PROPN
ap-3457	101	41	{	{	PUNCT
ap-3457	101	42	α2	α2	ADV
ap-3457	101	43	}	}	PUNCT
ap-3457	101	44	.	.	PUNCT
ap-3457	102	1	the	the	DET
ap-3457	102	2	highest	high	ADJ
ap-3457	102	3	root	root	NOUN
ap-3457	102	4	is	be	AUX
ap-3457	102	5	of	of	ADP
ap-3457	102	6	the	the	DET
ap-3457	102	7	form	form	NOUN
ap-3457	102	8	ξ	ξ	X
ap-3457	102	9	=	=	SYM
ap-3457	102	10	2α1	2α1	NUM
ap-3457	102	11	+	+	CCONJ
ap-3457	102	12	α2	α2	ADJ
ap-3457	102	13	and	and	CCONJ
ap-3457	102	14	the	the	DET
ap-3457	102	15	dual	dual	ADJ
ap-3457	102	16	highest	high	ADJ
ap-3457	102	17	root	root	NOUN
ap-3457	102	18	is	be	AUX
ap-3457	102	19	η	η	NOUN
ap-3457	102	20	=	=	ADJ
ap-3457	102	21	α∨1	α∨1	X
ap-3457	103	1	+	+	CCONJ
ap-3457	103	2	2α∨2	2α∨2	NUM
ap-3457	103	3	.	.	PUNCT
ap-3457	104	1	thus	thus	ADV
ap-3457	104	2	,	,	PUNCT
ap-3457	104	3	the	the	DET
ap-3457	104	4	sets	set	NOUN
ap-3457	104	5	of	of	ADP
ap-3457	104	6	mark	mark	NOUN
ap-3457	104	7	and	and	CCONJ
ap-3457	104	8	dual	dual	ADJ
ap-3457	104	9	marks	mark	NOUN
ap-3457	104	10	are	be	AUX
ap-3457	104	11	(	(	PUNCT
ap-3457	104	12	m1,m2	m1,m2	PROPN
ap-3457	104	13	)	)	PUNCT
ap-3457	104	14	=	=	PUNCT
ap-3457	105	1	(	(	PUNCT
ap-3457	105	2	2	2	NUM
ap-3457	105	3	,	,	PUNCT
ap-3457	105	4	1	1	NUM
ap-3457	105	5	)	)	PUNCT
ap-3457	105	6	and	and	CCONJ
ap-3457	105	7	(	(	PUNCT
ap-3457	105	8	m∨1	m∨1	NOUN
ap-3457	105	9	,	,	PUNCT
ap-3457	105	10	m∨2	m∨2	X
ap-3457	105	11	)	)	PUNCT
ap-3457	106	1	=	=	PUNCT
ap-3457	106	2	(	(	PUNCT
ap-3457	106	3	1	1	NUM
ap-3457	106	4	,	,	PUNCT
ap-3457	106	5	2	2	NUM
ap-3457	106	6	)	)	PUNCT
ap-3457	106	7	.	.	PUNCT
ap-3457	107	1	the	the	DET
ap-3457	107	2	vectors	vector	NOUN
ap-3457	107	3	%	%	INTJ
ap-3457	107	4	t	t	PROPN
ap-3457	107	5	are	be	AUX
ap-3457	107	6	of	of	ADP
ap-3457	107	7	the	the	DET
ap-3457	107	8	form	form	NOUN
ap-3457	107	9	(	(	PUNCT
ap-3457	107	10	%	%	NOUN
ap-3457	107	11	1	1	NUM
ap-3457	107	12	,	,	PUNCT
ap-3457	107	13	%	%	NOUN
ap-3457	107	14	s	s	NOUN
ap-3457	107	15	,	,	PUNCT
ap-3457	107	16	%	%	NOUN
ap-3457	107	17	l	l	NOUN
ap-3457	107	18	)	)	PUNCT
ap-3457	108	1	=	=	SYM
ap-3457	108	2	(	(	PUNCT
ap-3457	108	3	ω1	ω1	PROPN
ap-3457	108	4	+	+	CCONJ
ap-3457	108	5	ω2	ω2	ADJ
ap-3457	108	6	,	,	PUNCT
ap-3457	108	7	ω1	ω1	PROPN
ap-3457	108	8	,	,	PUNCT
ap-3457	108	9	ω2	ω2	NUM
ap-3457	108	10	)	)	PUNCT
ap-3457	108	11	and	and	CCONJ
ap-3457	108	12	the	the	DET
ap-3457	108	13	coxeter	coxeter	NOUN
ap-3457	108	14	numbers	number	NOUN
ap-3457	108	15	are	be	AUX
ap-3457	108	16	(	(	PUNCT
ap-3457	108	17	h1	h1	PROPN
ap-3457	108	18	,	,	PUNCT
ap-3457	108	19	hs	hs	PROPN
ap-3457	108	20	,	,	PUNCT
ap-3457	108	21	hl	hl	NOUN
ap-3457	108	22	)	)	PUNCT
ap-3457	108	23	=	=	SYM
ap-3457	108	24	(	(	PUNCT
ap-3457	108	25	4	4	NUM
ap-3457	108	26	,	,	PUNCT
ap-3457	108	27	2	2	NUM
ap-3457	108	28	,	,	PUNCT
ap-3457	108	29	2	2	NUM
ap-3457	108	30	)	)	PUNCT
ap-3457	108	31	.	.	PUNCT
ap-3457	109	1	the	the	DET
ap-3457	109	2	roots	root	NOUN
ap-3457	109	3	and	and	CCONJ
ap-3457	109	4	dual	dual	ADJ
ap-3457	109	5	roots	root	NOUN
ap-3457	109	6	,	,	PUNCT
ap-3457	109	7	the	the	DET
ap-3457	109	8	weights	weight	NOUN
ap-3457	109	9	and	and	CCONJ
ap-3457	109	10	dual	dual	ADJ
ap-3457	109	11	weights	weight	NOUN
ap-3457	109	12	,	,	PUNCT
ap-3457	109	13	together	together	ADV
ap-3457	109	14	with	with	ADP
ap-3457	109	15	the	the	DET
ap-3457	109	16	fundamental	fundamental	ADJ
ap-3457	109	17	domain	domain	NOUN
ap-3457	109	18	f	f	NOUN
ap-3457	109	19	and	and	CCONJ
ap-3457	109	20	the	the	DET
ap-3457	109	21	vectors	vector	NOUN
ap-3457	109	22	%	%	INTJ
ap-3457	109	23	t	t	PROPN
ap-3457	109	24	are	be	AUX
ap-3457	109	25	depicted	depict	VERB
ap-3457	109	26	in	in	ADP
ap-3457	109	27	figure	figure	NOUN
ap-3457	109	28	1	1	NUM
ap-3457	109	29	.	.	X
ap-3457	109	30	2.2	2.2	NUM
ap-3457	109	31	.	.	PUNCT
ap-3457	110	1	weyl	weyl	PROPN
ap-3457	110	2	group	group	PROPN
ap-3457	110	3	orbit	orbit	NOUN
ap-3457	110	4	functions	function	VERB
ap-3457	110	5	the	the	DET
ap-3457	110	6	definition	definition	NOUN
ap-3457	110	7	of	of	ADP
ap-3457	110	8	weyl	weyl	PROPN
ap-3457	110	9	group	group	NOUN
ap-3457	110	10	orbit	orbit	NOUN
ap-3457	110	11	function	function	NOUN
ap-3457	110	12	uses	use	VERB
ap-3457	110	13	the	the	DET
ap-3457	110	14	notion	notion	NOUN
ap-3457	110	15	of	of	ADP
ap-3457	110	16	a	a	DET
ap-3457	110	17	sign	sign	NOUN
ap-3457	110	18	homomorphisms	homomorphism	VERB
ap-3457	110	19	σt	σt	ADP
ap-3457	110	20	:	:	PUNCT
ap-3457	110	21	w	w	PROPN
ap-3457	110	22	7→	7→	NUM
ap-3457	110	23	±1	±1	VERB
ap-3457	110	24	,	,	PUNCT
ap-3457	110	25	where	where	SCONJ
ap-3457	110	26	figure	figure	NOUN
ap-3457	110	27	1	1	NUM
ap-3457	110	28	.	.	PUNCT
ap-3457	110	29	root	root	NOUN
ap-3457	110	30	system	system	NOUN
ap-3457	110	31	of	of	ADP
ap-3457	110	32	c2	c2	PROPN
ap-3457	110	33	.	.	PUNCT
ap-3457	111	1	the	the	DET
ap-3457	111	2	white	white	PROPN
ap-3457	111	3	circles	circle	NOUN
ap-3457	111	4	denote	denote	VERB
ap-3457	111	5	the	the	DET
ap-3457	111	6	roots	root	NOUN
ap-3457	111	7	,	,	PUNCT
ap-3457	111	8	black	black	ADJ
ap-3457	111	9	dots	dot	NOUN
ap-3457	111	10	depict	depict	VERB
ap-3457	111	11	the	the	DET
ap-3457	111	12	dual	dual	ADJ
ap-3457	111	13	roots	root	NOUN
ap-3457	111	14	.	.	PUNCT
ap-3457	112	1	the	the	DET
ap-3457	112	2	triangle	triangle	NOUN
ap-3457	112	3	denotes	denote	VERB
ap-3457	112	4	the	the	DET
ap-3457	112	5	fundamental	fundamental	ADJ
ap-3457	112	6	domain	domain	NOUN
ap-3457	112	7	f	f	X
ap-3457	112	8	.	.	PUNCT
ap-3457	113	1	the	the	DET
ap-3457	113	2	lines	line	NOUN
ap-3457	113	3	denoted	denote	VERB
ap-3457	113	4	r1	r1	NOUN
ap-3457	113	5	,	,	PUNCT
ap-3457	113	6	r2	r2	PROPN
ap-3457	113	7	and	and	CCONJ
ap-3457	113	8	r0	r0	NOUN
ap-3457	113	9	depict	depict	NOUN
ap-3457	113	10	reflecting	reflect	VERB
ap-3457	113	11	mirrors	mirror	NOUN
ap-3457	113	12	which	which	PRON
ap-3457	113	13	realize	realize	VERB
ap-3457	113	14	the	the	DET
ap-3457	113	15	corresponding	corresponding	ADJ
ap-3457	113	16	reflections	reflection	NOUN
ap-3457	113	17	.	.	PUNCT
ap-3457	114	1	t	t	PROPN
ap-3457	114	2	∈	∈	PROPN
ap-3457	114	3	{	{	PUNCT
ap-3457	114	4	0	0	NUM
ap-3457	114	5	,	,	PUNCT
ap-3457	114	6	1	1	NUM
ap-3457	114	7	,	,	PUNCT
ap-3457	114	8	s	s	X
ap-3457	114	9	,	,	PUNCT
ap-3457	114	10	l	l	NOUN
ap-3457	114	11	}	}	PUNCT
ap-3457	114	12	.	.	PUNCT
ap-3457	115	1	these	these	PRON
ap-3457	115	2	can	can	AUX
ap-3457	115	3	be	be	AUX
ap-3457	115	4	defined	define	VERB
ap-3457	115	5	by	by	ADP
ap-3457	115	6	the	the	DET
ap-3457	115	7	values	value	NOUN
ap-3457	115	8	on	on	ADP
ap-3457	115	9	the	the	DET
ap-3457	115	10	reflections	reflection	NOUN
ap-3457	115	11	corresponding	correspond	VERB
ap-3457	115	12	to	to	ADP
ap-3457	115	13	the	the	DET
ap-3457	115	14	simple	simple	ADJ
ap-3457	115	15	roots	root	NOUN
ap-3457	115	16	ri	ri	PROPN
ap-3457	115	17	,	,	PUNCT
ap-3457	115	18	namely	namely	ADV
ap-3457	115	19	σ0(ri	σ0(ri	NUM
ap-3457	115	20	)	)	PUNCT
ap-3457	115	21	=	=	SYM
ap-3457	115	22	1	1	NUM
ap-3457	115	23	,	,	PUNCT
ap-3457	115	24	αi	αi	NOUN
ap-3457	115	25	∈	∈	PROPN
ap-3457	115	26	∆	∆	PROPN
ap-3457	115	27	,	,	PUNCT
ap-3457	115	28	σ1(ri	σ1(ri	PROPN
ap-3457	115	29	)	)	PUNCT
ap-3457	116	1	=	=	SYM
ap-3457	116	2	−1	−1	NOUN
ap-3457	116	3	,	,	PUNCT
ap-3457	116	4	αi	αi	PROPN
ap-3457	116	5	∈	∈	PROPN
ap-3457	116	6	∆	∆	PROPN
ap-3457	116	7	,	,	PUNCT
ap-3457	116	8	σs(ri	σs(ri	PROPN
ap-3457	116	9	)	)	PUNCT
ap-3457	116	10	=	=	PRON
ap-3457	116	11	{	{	PUNCT
ap-3457	116	12	1	1	NUM
ap-3457	116	13	if	if	SCONJ
ap-3457	116	14	αi	αi	PRON
ap-3457	116	15	∈	∈	PROPN
ap-3457	116	16	∆l	∆l	PROPN
ap-3457	116	17	,	,	PUNCT
ap-3457	116	18	−1	−1	VERB
ap-3457	116	19	if	if	SCONJ
ap-3457	116	20	αi	αi	PRON
ap-3457	116	21	∈	∈	PROPN
ap-3457	116	22	∆s	∆s	PROPN
ap-3457	116	23	,	,	PUNCT
ap-3457	116	24	σl(ri	σl(ri	PROPN
ap-3457	116	25	)	)	PUNCT
ap-3457	116	26	=	=	PRON
ap-3457	116	27	{	{	PUNCT
ap-3457	116	28	1	1	NUM
ap-3457	116	29	if	if	SCONJ
ap-3457	116	30	αi	αi	PRON
ap-3457	116	31	∈	∈	PROPN
ap-3457	116	32	∆s	∆s	PROPN
ap-3457	116	33	,	,	PUNCT
ap-3457	116	34	−1	−1	NOUN
ap-3457	116	35	if	if	SCONJ
ap-3457	116	36	αi	αi	PRON
ap-3457	116	37	∈	∈	PROPN
ap-3457	116	38	∆l	∆l	PROPN
ap-3457	116	39	.	.	PUNCT
ap-3457	117	1	several	several	ADJ
ap-3457	117	2	families	family	NOUN
ap-3457	117	3	of	of	ADP
ap-3457	117	4	special	special	ADJ
ap-3457	117	5	functions	function	NOUN
ap-3457	117	6	are	be	AUX
ap-3457	117	7	connected	connect	VERB
ap-3457	117	8	with	with	ADP
ap-3457	117	9	each	each	DET
ap-3457	117	10	weyl	weyl	VERB
ap-3457	117	11	group	group	NOUN
ap-3457	117	12	w	w	PROPN
ap-3457	117	13	.	.	PUNCT
ap-3457	118	1	they	they	PRON
ap-3457	118	2	are	be	AUX
ap-3457	118	3	labelled	label	VERB
ap-3457	118	4	by	by	ADP
ap-3457	118	5	vectors	vector	NOUN
ap-3457	118	6	λ	λ	PROPN
ap-3457	118	7	∈	∈	PROPN
ap-3457	118	8	p+	p+	NOUN
ap-3457	118	9	and	and	CCONJ
ap-3457	118	10	defined	define	VERB
ap-3457	118	11	as	as	SCONJ
ap-3457	118	12	weighed	weigh	VERB
ap-3457	118	13	sums	sum	NOUN
ap-3457	118	14	over	over	ADP
ap-3457	118	15	the	the	DET
ap-3457	118	16	corresponding	corresponding	ADJ
ap-3457	118	17	weyl	weyl	VERB
ap-3457	118	18	group	group	NOUN
ap-3457	118	19	orbit	orbit	NOUN
ap-3457	118	20	,	,	PUNCT
ap-3457	118	21	i.e.	i.e.	X
ap-3457	118	22	the	the	DET
ap-3457	118	23	set	set	VERB
ap-3457	118	24	wλ	wλ	NOUN
ap-3457	118	25	=	=	PUNCT
ap-3457	118	26	{	{	PUNCT
ap-3457	118	27	wλ	wλ	NOUN
ap-3457	118	28	|	|	ADV
ap-3457	118	29	w	w	PROPN
ap-3457	118	30	∈	∈	PROPN
ap-3457	118	31	w	w	ADP
ap-3457	118	32	}	}	PUNCT
ap-3457	118	33	.	.	PUNCT
ap-3457	119	1	for	for	ADP
ap-3457	119	2	every	every	DET
ap-3457	119	3	x	x	PROPN
ap-3457	119	4	∈	∈	PROPN
ap-3457	119	5	rn	rn	PROPN
ap-3457	119	6	and	and	CCONJ
ap-3457	119	7	t	t	PROPN
ap-3457	119	8	∈	∈	PROPN
ap-3457	119	9	{	{	PUNCT
ap-3457	119	10	0	0	NUM
ap-3457	119	11	,	,	PUNCT
ap-3457	119	12	1	1	NUM
ap-3457	119	13	,	,	PUNCT
ap-3457	119	14	s	s	X
ap-3457	119	15	,	,	PUNCT
ap-3457	119	16	l	l	NOUN
ap-3457	119	17	}	}	PUNCT
ap-3457	119	18	we	we	PRON
ap-3457	119	19	define	define	VERB
ap-3457	119	20	stλ+%t(x	stλ+%t(x	NOUN
ap-3457	119	21	)	)	PUNCT
ap-3457	120	1	=	=	PUNCT
ap-3457	120	2	∑	∑	PROPN
ap-3457	120	3	µ∈w	µ∈w	X
ap-3457	120	4	(	(	PUNCT
ap-3457	120	5	λ+%t	λ+%t	NOUN
ap-3457	120	6	)	)	PUNCT
ap-3457	120	7	σt(µ)e2πi〈µ	σt(µ)e2πi〈µ	NOUN
ap-3457	120	8	,	,	PUNCT
ap-3457	120	9	x	x	NOUN
ap-3457	120	10	〉	〉	NUM
ap-3457	120	11	,	,	PUNCT
ap-3457	120	12	(	(	PUNCT
ap-3457	120	13	5	5	NUM
ap-3457	120	14	)	)	PUNCT
ap-3457	120	15	where	where	SCONJ
ap-3457	120	16	%	%	NOUN
ap-3457	120	17	t	t	PROPN
ap-3457	120	18	is	be	AUX
ap-3457	120	19	given	give	VERB
ap-3457	120	20	by	by	ADP
ap-3457	120	21	(	(	PUNCT
ap-3457	120	22	3	3	NUM
ap-3457	120	23	)	)	PUNCT
ap-3457	120	24	and	and	CCONJ
ap-3457	120	25	σt(µ	σt(µ	NUM
ap-3457	120	26	)	)	PUNCT
ap-3457	120	27	≡	≡	PROPN
ap-3457	120	28	σt(w	σt(w	NOUN
ap-3457	120	29	)	)	PUNCT
ap-3457	120	30	for	for	ADP
ap-3457	120	31	w	w	PROPN
ap-3457	120	32	such	such	ADJ
ap-3457	120	33	that	that	PRON
ap-3457	120	34	µ	µ	NOUN
ap-3457	120	35	=	=	X
ap-3457	120	36	w(λ+	w(λ+	ADP
ap-3457	120	37	%	%	NOUN
ap-3457	120	38	t	t	PROPN
ap-3457	120	39	)	)	PUNCT
ap-3457	120	40	.	.	PUNCT
ap-3457	121	1	functions	function	NOUN
ap-3457	121	2	corresponding	correspond	VERB
ap-3457	121	3	to	to	ADP
ap-3457	121	4	the	the	DET
ap-3457	121	5	choice	choice	NOUN
ap-3457	121	6	of	of	ADP
ap-3457	121	7	t	t	NOUN
ap-3457	121	8	=	=	SYM
ap-3457	121	9	0	0	NUM
ap-3457	121	10	and	and	CCONJ
ap-3457	121	11	t	t	NOUN
ap-3457	121	12	=	=	SYM
ap-3457	121	13	1	1	NUM
ap-3457	121	14	are	be	AUX
ap-3457	121	15	usually	usually	ADV
ap-3457	121	16	called	call	VERB
ap-3457	121	17	cand	cand	NOUN
ap-3457	121	18	sfunctions	sfunction	NOUN
ap-3457	121	19	respectively	respectively	ADV
ap-3457	121	20	,	,	PUNCT
ap-3457	121	21	in	in	ADP
ap-3457	121	22	the	the	DET
ap-3457	121	23	formulas	formula	NOUN
ap-3457	121	24	in	in	ADP
ap-3457	121	25	next	next	ADJ
ap-3457	121	26	sections	section	NOUN
ap-3457	121	27	we	we	PRON
ap-3457	121	28	use	use	VERB
ap-3457	121	29	the	the	DET
ap-3457	121	30	notation	notation	NOUN
ap-3457	121	31	s0	s0	NOUN
ap-3457	121	32	and	and	CCONJ
ap-3457	121	33	s1	s1	NOUN
ap-3457	121	34	for	for	ADP
ap-3457	121	35	a	a	DET
ap-3457	121	36	simplicity	simplicity	NOUN
ap-3457	121	37	.	.	PUNCT
ap-3457	122	1	families	family	NOUN
ap-3457	122	2	of	of	ADP
ap-3457	122	3	c-	c-	X
ap-3457	122	4	,	,	PUNCT
ap-3457	122	5	s-	s-	X
ap-3457	122	6	,	,	PUNCT
ap-3457	122	7	ssand	ssand	PROPN
ap-3457	122	8	sl	sl	PROPN
ap-3457	122	9	-	-	PUNCT
ap-3457	122	10	functions	function	NOUN
ap-3457	122	11	are	be	AUX
ap-3457	122	12	complex	complex	ADJ
ap-3457	122	13	multivariate	multivariate	NOUN
ap-3457	122	14	functions	function	NOUN
ap-3457	122	15	with	with	ADP
ap-3457	122	16	remarkable	remarkable	ADJ
ap-3457	122	17	properties	property	NOUN
ap-3457	122	18	such	such	ADJ
ap-3457	122	19	as	as	ADP
ap-3457	122	20	(	(	PUNCT
ap-3457	122	21	anti-)invariance	anti-)invariance	NOUN
ap-3457	122	22	with	with	ADP
ap-3457	122	23	respect	respect	NOUN
ap-3457	122	24	to	to	ADP
ap-3457	122	25	the	the	DET
ap-3457	122	26	action	action	NOUN
ap-3457	122	27	of	of	ADP
ap-3457	122	28	the	the	DET
ap-3457	122	29	affine	affine	NOUN
ap-3457	122	30	weyl	weyl	VERB
ap-3457	122	31	group	group	NOUN
ap-3457	122	32	,	,	PUNCT
ap-3457	122	33	continuous	continuous	ADJ
ap-3457	122	34	and	and	CCONJ
ap-3457	122	35	discrete	discrete	ADJ
ap-3457	122	36	orthogonality	orthogonality	NOUN
ap-3457	122	37	.	.	PUNCT
ap-3457	123	1	they	they	PRON
ap-3457	123	2	were	be	AUX
ap-3457	123	3	studied	study	VERB
ap-3457	123	4	in	in	ADP
ap-3457	123	5	many	many	ADJ
ap-3457	123	6	papers	paper	NOUN
ap-3457	123	7	,	,	PUNCT
ap-3457	123	8	see	see	VERB
ap-3457	123	9	for	for	ADP
ap-3457	123	10	example	example	NOUN
ap-3457	123	11	[	[	X
ap-3457	123	12	18	18	NUM
ap-3457	123	13	,	,	PUNCT
ap-3457	123	14	19	19	NUM
ap-3457	123	15	,	,	PUNCT
ap-3457	123	16	26	26	NUM
ap-3457	123	17	]	]	PUNCT
ap-3457	123	18	for	for	ADP
ap-3457	123	19	the	the	DET
ap-3457	123	20	general	general	ADJ
ap-3457	123	21	properties	property	NOUN
ap-3457	123	22	and	and	CCONJ
ap-3457	123	23	[	[	X
ap-3457	123	24	13	13	NUM
ap-3457	123	25	,	,	PUNCT
ap-3457	123	26	14	14	NUM
ap-3457	123	27	,	,	PUNCT
ap-3457	123	28	28	28	NUM
ap-3457	123	29	]	]	PUNCT
ap-3457	123	30	for	for	ADP
ap-3457	123	31	their	their	PRON
ap-3457	123	32	discretization	discretization	NOUN
ap-3457	123	33	.	.	PUNCT
ap-3457	124	1	204	204	NUM
ap-3457	124	2	vol	vol	NOUN
ap-3457	124	3	.	.	PUNCT
ap-3457	125	1	56	56	NUM
ap-3457	125	2	no	no	NOUN
ap-3457	125	3	.	.	PUNCT
ap-3457	126	1	3/2016	3/2016	NUM
ap-3457	126	2	on	on	ADP
ap-3457	126	3	cubature	cubature	ADJ
ap-3457	126	4	rules	rule	NOUN
ap-3457	126	5	associated	associate	VERB
ap-3457	126	6	to	to	ADP
ap-3457	126	7	weyl	weyl	PROPN
ap-3457	126	8	group	group	PROPN
ap-3457	126	9	orbit	orbit	NOUN
ap-3457	126	10	functions	function	NOUN
ap-3457	126	11	fundamental	fundamental	ADJ
ap-3457	126	12	domains	domain	NOUN
ap-3457	126	13	f	f	PROPN
ap-3457	126	14	t	t	PROPN
ap-3457	126	15	are	be	AUX
ap-3457	126	16	defined	define	VERB
ap-3457	126	17	as	as	ADP
ap-3457	126	18	subsets	subset	NOUN
ap-3457	126	19	of	of	ADP
ap-3457	126	20	f	f	PROPN
ap-3457	126	21	such	such	ADJ
ap-3457	126	22	that	that	SCONJ
ap-3457	126	23	we	we	PRON
ap-3457	126	24	omit	omit	VERB
ap-3457	126	25	the	the	DET
ap-3457	126	26	part	part	NOUN
ap-3457	126	27	of	of	ADP
ap-3457	126	28	the	the	DET
ap-3457	126	29	boundary	boundary	NOUN
ap-3457	126	30	of	of	ADP
ap-3457	126	31	f	f	PROPN
ap-3457	126	32	which	which	PRON
ap-3457	126	33	is	be	AUX
ap-3457	126	34	stabilized	stabilize	VERB
ap-3457	126	35	by	by	ADP
ap-3457	126	36	certain	certain	ADJ
ap-3457	126	37	generating	generating	NOUN
ap-3457	126	38	reflections	reflection	NOUN
ap-3457	126	39	r	r	NOUN
ap-3457	126	40	∈	∈	NOUN
ap-3457	126	41	r	r	NOUN
ap-3457	126	42	=	=	PUNCT
ap-3457	126	43	{	{	PUNCT
ap-3457	126	44	r0	r0	NOUN
ap-3457	126	45	,	,	PUNCT
ap-3457	126	46	r1	r1	PROPN
ap-3457	126	47	.	.	PUNCT
ap-3457	126	48	.	.	PUNCT
ap-3457	127	1	.	.	PUNCT
ap-3457	128	1	,	,	PUNCT
ap-3457	128	2	rn	rn	PROPN
ap-3457	128	3	}	}	PUNCT
ap-3457	128	4	.	.	PUNCT
ap-3457	129	1	more	more	ADV
ap-3457	129	2	precisely	precisely	ADV
ap-3457	129	3	,	,	PUNCT
ap-3457	129	4	with	with	ADP
ap-3457	129	5	the	the	DET
ap-3457	129	6	notation	notation	NOUN
ap-3457	129	7	rt	rt	PROPN
ap-3457	130	1	=	=	PUNCT
ap-3457	130	2	{	{	PUNCT
ap-3457	130	3	r	r	NOUN
ap-3457	130	4	∈	∈	NOUN
ap-3457	130	5	r	r	NOUN
ap-3457	130	6	|	|	ADV
ap-3457	130	7	σt	σt	AUX
ap-3457	130	8	◦	◦	VERB
ap-3457	130	9	ψ(r	ψ(r	NOUN
ap-3457	130	10	)	)	PUNCT
ap-3457	130	11	=	=	SYM
ap-3457	130	12	−1	−1	NOUN
ap-3457	130	13	}	}	PUNCT
ap-3457	130	14	,	,	PUNCT
ap-3457	130	15	ht	ht	PROPN
ap-3457	130	16	=	=	X
ap-3457	130	17	{	{	PUNCT
ap-3457	130	18	a	a	DET
ap-3457	130	19	∈	∈	X
ap-3457	131	1	f	f	NOUN
ap-3457	131	2	|	|	ADV
ap-3457	131	3	∃r	∃r	PROPN
ap-3457	131	4	∈	∈	PROPN
ap-3457	131	5	rt	rt	PROPN
ap-3457	131	6	,	,	PUNCT
ap-3457	131	7	ra	ra	PROPN
ap-3457	131	8	=	=	PUNCT
ap-3457	131	9	a	a	PRON
ap-3457	131	10	}	}	PUNCT
ap-3457	131	11	,	,	PUNCT
ap-3457	131	12	we	we	PRON
ap-3457	131	13	define	define	VERB
ap-3457	131	14	f	f	PROPN
ap-3457	131	15	t	t	PROPN
ap-3457	131	16	=	=	SYM
ap-3457	131	17	f	f	PROPN
ap-3457	131	18	\ht	\ht	PROPN
ap-3457	131	19	.	.	PUNCT
ap-3457	132	1	the	the	DET
ap-3457	132	2	explicit	explicit	ADJ
ap-3457	132	3	forms	form	NOUN
ap-3457	132	4	are	be	AUX
ap-3457	132	5	obtained	obtain	VERB
ap-3457	132	6	from	from	ADP
ap-3457	132	7	(	(	PUNCT
ap-3457	132	8	1	1	NUM
ap-3457	132	9	)	)	PUNCT
ap-3457	132	10	and	and	CCONJ
ap-3457	132	11	can	can	AUX
ap-3457	132	12	be	be	AUX
ap-3457	132	13	found	find	VERB
ap-3457	132	14	in	in	ADP
ap-3457	132	15	[	[	X
ap-3457	132	16	14	14	NUM
ap-3457	132	17	]	]	PUNCT
ap-3457	132	18	.	.	PUNCT
ap-3457	133	1	the	the	DET
ap-3457	133	2	c-	c-	X
ap-3457	133	3	,	,	PUNCT
ap-3457	133	4	s-	s-	X
ap-3457	133	5	,	,	PUNCT
ap-3457	133	6	ssand	ssand	PROPN
ap-3457	133	7	sl	sl	PROPN
ap-3457	133	8	-	-	PUNCT
ap-3457	133	9	functions	function	NOUN
ap-3457	133	10	can	can	AUX
ap-3457	133	11	be	be	AUX
ap-3457	133	12	viewed	view	VERB
ap-3457	133	13	as	as	ADP
ap-3457	133	14	functional	functional	ADJ
ap-3457	133	15	forms	form	NOUN
ap-3457	133	16	of	of	ADP
ap-3457	133	17	elements	element	NOUN
ap-3457	133	18	from	from	ADP
ap-3457	133	19	the	the	DET
ap-3457	133	20	algebra	algebra	NOUN
ap-3457	133	21	c[p	c[p	PROPN
ap-3457	133	22	]	]	PUNCT
ap-3457	133	23	containing	contain	VERB
ap-3457	133	24	all	all	DET
ap-3457	133	25	complex	complex	ADJ
ap-3457	133	26	linear	linear	ADJ
ap-3457	133	27	combinations	combination	NOUN
ap-3457	133	28	of	of	ADP
ap-3457	133	29	formal	formal	ADJ
ap-3457	133	30	exponentials	exponential	NOUN
ap-3457	133	31	ea	ea	ADP
ap-3457	133	32	,	,	PUNCT
ap-3457	133	33	a	a	DET
ap-3457	133	34	∈	∈	PROPN
ap-3457	133	35	p	p	NOUN
ap-3457	133	36	,	,	PUNCT
ap-3457	133	37	with	with	ADP
ap-3457	133	38	multiplication	multiplication	NOUN
ap-3457	133	39	defined	define	VERB
ap-3457	133	40	by	by	ADP
ap-3457	133	41	ea	ea	PROPN
ap-3457	133	42	·	·	PUNCT
ap-3457	133	43	eb	eb	PROPN
ap-3457	133	44	=	=	SYM
ap-3457	133	45	ea+b	ea+b	PROPN
ap-3457	133	46	,	,	PUNCT
ap-3457	133	47	the	the	DET
ap-3457	133	48	inverse	inverse	NOUN
ap-3457	133	49	given	give	VERB
ap-3457	133	50	by	by	ADP
ap-3457	133	51	(	(	PUNCT
ap-3457	133	52	ea)−1	ea)−1	NOUN
ap-3457	133	53	=	=	PUNCT
ap-3457	133	54	e−a	e−a	PROPN
ap-3457	133	55	and	and	CCONJ
ap-3457	133	56	the	the	DET
ap-3457	133	57	identity	identity	NOUN
ap-3457	133	58	e0	e0	NOUN
ap-3457	133	59	=	=	PUNCT
ap-3457	134	1	1	1	X
ap-3457	134	2	.	.	PUNCT
ap-3457	134	3	the	the	DET
ap-3457	134	4	connection	connection	NOUN
ap-3457	134	5	is	be	AUX
ap-3457	134	6	based	base	VERB
ap-3457	134	7	on	on	ADP
ap-3457	134	8	the	the	DET
ap-3457	134	9	exponential	exponential	ADJ
ap-3457	134	10	mapping	mapping	NOUN
ap-3457	134	11	from	from	ADP
ap-3457	134	12	lie	lie	NOUN
ap-3457	134	13	algebra	algebra	NOUN
ap-3457	134	14	to	to	ADP
ap-3457	134	15	the	the	DET
ap-3457	134	16	corresponding	corresponding	ADJ
ap-3457	134	17	lie	lie	NOUN
ap-3457	134	18	group	group	NOUN
ap-3457	134	19	[	[	X
ap-3457	134	20	2	2	NUM
ap-3457	134	21	,	,	PUNCT
ap-3457	134	22	16	16	NUM
ap-3457	134	23	,	,	PUNCT
ap-3457	134	24	27	27	NUM
ap-3457	134	25	]	]	PUNCT
ap-3457	134	26	.	.	PUNCT
ap-3457	135	1	2.3	2.3	NUM
ap-3457	135	2	.	.	PUNCT
ap-3457	136	1	jacobi	jacobi	PROPN
ap-3457	136	2	polynomials	polynomial	VERB
ap-3457	136	3	we	we	PRON
ap-3457	136	4	assume	assume	VERB
ap-3457	136	5	that	that	SCONJ
ap-3457	136	6	the	the	DET
ap-3457	136	7	multiplicity	multiplicity	NOUN
ap-3457	136	8	function	function	NOUN
ap-3457	136	9	k	k	PROPN
ap-3457	136	10	satisfies	satisfy	VERB
ap-3457	136	11	kα	kα	PROPN
ap-3457	136	12	≥	≥	PROPN
ap-3457	136	13	0	0	NUM
ap-3457	136	14	.	.	PUNCT
ap-3457	137	1	the	the	DET
ap-3457	137	2	jacobi	jacobi	PROPN
ap-3457	137	3	polynomial	polynomial	PROPN
ap-3457	137	4	p	p	PROPN
ap-3457	137	5	(	(	PUNCT
ap-3457	137	6	λ	λ	PROPN
ap-3457	137	7	,	,	PUNCT
ap-3457	137	8	k	k	NOUN
ap-3457	137	9	)	)	PUNCT
ap-3457	138	1	[	[	X
ap-3457	138	2	11	11	NUM
ap-3457	138	3	,	,	PUNCT
ap-3457	138	4	12	12	NUM
ap-3457	138	5	]	]	PUNCT
ap-3457	138	6	associated	associate	VERB
ap-3457	138	7	to	to	ADP
ap-3457	138	8	the	the	DET
ap-3457	138	9	root	root	NOUN
ap-3457	138	10	system	system	NOUN
ap-3457	138	11	π	π	PROPN
ap-3457	138	12	with	with	ADP
ap-3457	138	13	highest	high	ADJ
ap-3457	138	14	weight	weight	NOUN
ap-3457	138	15	λ	λ	PROPN
ap-3457	138	16	∈	∈	PROPN
ap-3457	138	17	p+	p+	NOUN
ap-3457	138	18	and	and	CCONJ
ap-3457	138	19	multiplicity	multiplicity	NOUN
ap-3457	138	20	function	function	NOUN
ap-3457	138	21	k	k	PROPN
ap-3457	138	22	as	as	SCONJ
ap-3457	138	23	parameter	parameter	PROPN
ap-3457	138	24	is	be	AUX
ap-3457	138	25	defined	define	VERB
ap-3457	138	26	by	by	ADP
ap-3457	138	27	the	the	DET
ap-3457	138	28	following	follow	VERB
ap-3457	138	29	formulas	formula	NOUN
ap-3457	138	30	.	.	PUNCT
ap-3457	139	1	p	p	X
ap-3457	139	2	(	(	PUNCT
ap-3457	139	3	λ	λ	PROPN
ap-3457	139	4	,	,	PUNCT
ap-3457	139	5	k	k	NOUN
ap-3457	139	6	)	)	PUNCT
ap-3457	139	7	≡	≡	PROPN
ap-3457	139	8	∑	∑	PUNCT
ap-3457	139	9	µ∈p+	µ∈p+	X
ap-3457	139	10	µ	µ	PRON
ap-3457	139	11	�	�	NOUN
ap-3457	139	12	λ	λ	NOUN
ap-3457	139	13	cλµ(k)cµ	cλµ(k)cµ	NOUN
ap-3457	139	14	,	,	PUNCT
ap-3457	139	15	cµ	cµ	ADP
ap-3457	139	16	=	=	SYM
ap-3457	139	17	∑	∑	PUNCT
ap-3457	139	18	µ′∈wµ	µ′∈wµ	PROPN
ap-3457	139	19	eµ	eµ	VERB
ap-3457	139	20	′	′	NOUN
ap-3457	139	21	,	,	PUNCT
ap-3457	139	22	(	(	PUNCT
ap-3457	139	23	6	6	NUM
ap-3457	139	24	)	)	PUNCT
ap-3457	139	25	where	where	SCONJ
ap-3457	139	26	the	the	DET
ap-3457	139	27	coefficients	coefficient	NOUN
ap-3457	139	28	cλµ(k	cλµ(k	NOUN
ap-3457	139	29	)	)	PUNCT
ap-3457	139	30	are	be	AUX
ap-3457	139	31	recursively	recursively	ADV
ap-3457	139	32	given	give	VERB
ap-3457	139	33	by	by	ADP
ap-3457	139	34	(	(	PUNCT
ap-3457	139	35	〈	〈	PROPN
ap-3457	139	36	λ+	λ+	PUNCT
ap-3457	139	37	%	%	INTJ
ap-3457	139	38	(	(	PUNCT
ap-3457	139	39	k	k	NOUN
ap-3457	139	40	)	)	PUNCT
ap-3457	139	41	,	,	PUNCT
ap-3457	139	42	λ+	λ+	PUNCT
ap-3457	139	43	%	%	INTJ
ap-3457	139	44	(	(	PUNCT
ap-3457	139	45	k	k	NOUN
ap-3457	139	46	)	)	PUNCT
ap-3457	139	47	〉	〉	NOUN
ap-3457	139	48	−	−	PROPN
ap-3457	139	49	〈	〈	PROPN
ap-3457	139	50	µ+	µ+	PUNCT
ap-3457	139	51	%	%	NOUN
ap-3457	139	52	(	(	PUNCT
ap-3457	139	53	k	k	NOUN
ap-3457	139	54	)	)	PUNCT
ap-3457	139	55	,	,	PUNCT
ap-3457	139	56	µ+	µ+	PROPN
ap-3457	139	57	%	%	NOUN
ap-3457	139	58	(	(	PUNCT
ap-3457	139	59	k	k	NOUN
ap-3457	139	60	)	)	PUNCT
ap-3457	139	61	〉	〉	NOUN
ap-3457	139	62	)	)	PUNCT
ap-3457	139	63	cλµ(k	cλµ(k	NOUN
ap-3457	139	64	)	)	PUNCT
ap-3457	139	65	=	=	SYM
ap-3457	139	66	2	2	NUM
ap-3457	139	67	∑	∑	ADP
ap-3457	139	68	α∈π+	α∈π+	ADV
ap-3457	139	69	kα	kα	VERB
ap-3457	139	70	∞∑	∞∑	NUM
ap-3457	139	71	j=1	j=1	ADJ
ap-3457	139	72	〈	〈	PROPN
ap-3457	139	73	µ+	µ+	PROPN
ap-3457	139	74	jα	jα	NOUN
ap-3457	139	75	,	,	PUNCT
ap-3457	139	76	α〉cλ,µ+jα(k	α〉cλ,µ+jα(k	ADJ
ap-3457	139	77	)	)	PUNCT
ap-3457	139	78	along	along	ADP
ap-3457	139	79	with	with	ADP
ap-3457	139	80	the	the	DET
ap-3457	139	81	initial	initial	ADJ
ap-3457	139	82	value	value	NOUN
ap-3457	139	83	cλλ	cλλ	NOUN
ap-3457	139	84	=	=	NOUN
ap-3457	139	85	1	1	NUM
ap-3457	139	86	and	and	CCONJ
ap-3457	139	87	the	the	DET
ap-3457	139	88	assumption	assumption	NOUN
ap-3457	139	89	cλµ	cλµ	NOUN
ap-3457	139	90	=	=	SYM
ap-3457	139	91	cλ	cλ	PROPN
ap-3457	139	92	,	,	PUNCT
ap-3457	139	93	w(µ	w(µ	PROPN
ap-3457	139	94	)	)	PUNCT
ap-3457	139	95	for	for	ADP
ap-3457	139	96	all	all	DET
ap-3457	139	97	w	w	NOUN
ap-3457	139	98	∈w	∈w	NOUN
ap-3457	139	99	.	.	PUNCT
ap-3457	140	1	recall	recall	VERB
ap-3457	140	2	that	that	PRON
ap-3457	141	1	%	%	INTJ
ap-3457	141	2	(	(	PUNCT
ap-3457	141	3	k	k	NOUN
ap-3457	141	4	)	)	PUNCT
ap-3457	141	5	is	be	AUX
ap-3457	141	6	defined	define	VERB
ap-3457	141	7	by	by	ADP
ap-3457	141	8	(	(	PUNCT
ap-3457	141	9	2	2	NUM
ap-3457	141	10	)	)	PUNCT
ap-3457	141	11	.	.	PUNCT
ap-3457	142	1	by	by	ADP
ap-3457	142	2	setting	set	VERB
ap-3457	142	3	kα	kα	PROPN
ap-3457	142	4	=	=	PUNCT
ap-3457	142	5	0	0	NUM
ap-3457	142	6	for	for	ADP
ap-3457	142	7	all	all	DET
ap-3457	142	8	α	α	PRON
ap-3457	142	9	∈	∈	PROPN
ap-3457	142	10	π	π	PROPN
ap-3457	142	11	,	,	PUNCT
ap-3457	142	12	the	the	DET
ap-3457	142	13	jacobi	jacobi	PROPN
ap-3457	142	14	polynomials	polynomial	NOUN
ap-3457	142	15	lead	lead	VERB
ap-3457	142	16	trivially	trivially	ADV
ap-3457	142	17	to	to	ADP
ap-3457	142	18	c	c	NOUN
ap-3457	142	19	-	-	PUNCT
ap-3457	142	20	functions	function	NOUN
ap-3457	142	21	.	.	PUNCT
ap-3457	143	1	in	in	ADP
ap-3457	143	2	the	the	DET
ap-3457	143	3	case	case	NOUN
ap-3457	143	4	k	k	PROPN
ap-3457	143	5	=	=	SYM
ap-3457	143	6	k1	k1	PROPN
ap-3457	143	7	,	,	PUNCT
ap-3457	143	8	the	the	DET
ap-3457	143	9	formula	formula	NOUN
ap-3457	143	10	for	for	ADP
ap-3457	143	11	the	the	DET
ap-3457	143	12	calculation	calculation	NOUN
ap-3457	143	13	of	of	ADP
ap-3457	143	14	the	the	DET
ap-3457	143	15	coefficients	coefficient	NOUN
ap-3457	143	16	becomes	become	VERB
ap-3457	143	17	the	the	DET
ap-3457	143	18	freudenthal	freudenthal	NOUN
ap-3457	143	19	’s	’s	PART
ap-3457	143	20	recurrence	recurrence	NOUN
ap-3457	143	21	formula	formula	NOUN
ap-3457	143	22	[	[	X
ap-3457	143	23	16	16	NUM
ap-3457	143	24	]	]	PUNCT
ap-3457	143	25	.	.	PUNCT
ap-3457	144	1	therefore	therefore	ADV
ap-3457	144	2	,	,	PUNCT
ap-3457	144	3	each	each	PRON
ap-3457	144	4	p	p	X
ap-3457	144	5	(	(	PUNCT
ap-3457	144	6	λ	λ	PROPN
ap-3457	144	7	,	,	PUNCT
ap-3457	144	8	k1	k1	NOUN
ap-3457	144	9	)	)	PUNCT
ap-3457	144	10	specializes	specialize	VERB
ap-3457	144	11	to	to	ADP
ap-3457	144	12	a	a	DET
ap-3457	144	13	character	character	NOUN
ap-3457	144	14	χλ	χλ	NOUN
ap-3457	144	15	of	of	ADP
ap-3457	144	16	an	an	DET
ap-3457	144	17	irreducible	irreducible	ADJ
ap-3457	144	18	representation	representation	NOUN
ap-3457	144	19	of	of	ADP
ap-3457	144	20	the	the	DET
ap-3457	144	21	simple	simple	ADJ
ap-3457	144	22	lie	lie	NOUN
ap-3457	144	23	algebra	algebra	NOUN
ap-3457	144	24	of	of	ADP
ap-3457	144	25	the	the	DET
ap-3457	144	26	highest	high	ADJ
ap-3457	144	27	weight	weight	NOUN
ap-3457	144	28	λ	λ	PROPN
ap-3457	144	29	,	,	PUNCT
ap-3457	144	30	i.e.	i.e.	X
ap-3457	144	31	,	,	PUNCT
ap-3457	144	32	p	p	X
ap-3457	144	33	(	(	PUNCT
ap-3457	144	34	λ	λ	PROPN
ap-3457	144	35	,	,	PUNCT
ap-3457	144	36	k1	k1	NOUN
ap-3457	144	37	)	)	PUNCT
ap-3457	144	38	=	=	PUNCT
ap-3457	144	39	χλ	χλ	NOUN
ap-3457	144	40	=	=	SYM
ap-3457	144	41	sλ+%1	sλ+%1	ADJ
ap-3457	144	42	s%1	s%1	NOUN
ap-3457	144	43	.	.	PUNCT
ap-3457	145	1	in	in	ADP
ap-3457	145	2	addition	addition	NOUN
ap-3457	145	3	,	,	PUNCT
ap-3457	145	4	we	we	PRON
ap-3457	145	5	show	show	VERB
ap-3457	145	6	that	that	SCONJ
ap-3457	145	7	the	the	DET
ap-3457	145	8	jacobi	jacobi	PROPN
ap-3457	145	9	polynomials	polynomial	NOUN
ap-3457	145	10	are	be	AUX
ap-3457	145	11	related	relate	VERB
ap-3457	145	12	to	to	ADP
ap-3457	145	13	the	the	DET
ap-3457	145	14	ssand	ssand	NOUN
ap-3457	145	15	sl	sl	NOUN
ap-3457	145	16	-	-	PUNCT
ap-3457	145	17	functions	function	NOUN
ap-3457	145	18	in	in	ADP
ap-3457	145	19	the	the	DET
ap-3457	145	20	following	following	ADJ
ap-3457	145	21	way	way	NOUN
ap-3457	145	22	.	.	PUNCT
ap-3457	146	1	p	p	X
ap-3457	146	2	(	(	PUNCT
ap-3457	146	3	λ	λ	PROPN
ap-3457	146	4	,	,	PUNCT
ap-3457	146	5	ks	ks	NOUN
ap-3457	146	6	)	)	PUNCT
ap-3457	146	7	=	=	NOUN
ap-3457	146	8	ssλ+%s	ssλ+%s	NOUN
ap-3457	146	9	ss%s	ss%s	NOUN
ap-3457	146	10	and	and	CCONJ
ap-3457	146	11	p	p	PROPN
ap-3457	146	12	(	(	PUNCT
ap-3457	146	13	λ	λ	PROPN
ap-3457	146	14	,	,	PUNCT
ap-3457	146	15	kl	kl	NOUN
ap-3457	146	16	)	)	PUNCT
ap-3457	146	17	=	=	SYM
ap-3457	146	18	slλ+%l	slλ+%l	NOUN
ap-3457	146	19	sl	sl	INTJ
ap-3457	146	20	%	%	INTJ
ap-3457	146	21	l	l	NOUN
ap-3457	146	22	.	.	PUNCT
ap-3457	147	1	we	we	PRON
ap-3457	147	2	first	first	ADV
ap-3457	147	3	observe	observe	VERB
ap-3457	147	4	that	that	SCONJ
ap-3457	147	5	ssλ+%s	ssλ+%s	NOUN
ap-3457	147	6	/	/	SYM
ap-3457	147	7	s	s	NOUN
ap-3457	147	8	s	s	NOUN
ap-3457	147	9	%	%	NOUN
ap-3457	147	10	s	s	NOUN
ap-3457	147	11	are	be	AUX
ap-3457	147	12	weyl	weyl	VERB
ap-3457	147	13	group	group	NOUN
ap-3457	147	14	invariant	invariant	ADJ
ap-3457	147	15	elements	element	NOUN
ap-3457	147	16	of	of	ADP
ap-3457	147	17	c[p	c[p	PROPN
ap-3457	147	18	]	]	PUNCT
ap-3457	147	19	(	(	PUNCT
ap-3457	147	20	see	see	VERB
ap-3457	147	21	proposition	proposition	NOUN
ap-3457	147	22	4.2	4.2	NUM
ap-3457	147	23	of	of	ADP
ap-3457	147	24	[	[	X
ap-3457	147	25	26	26	NUM
ap-3457	147	26	]	]	PUNCT
ap-3457	147	27	)	)	PUNCT
ap-3457	147	28	with	with	ADP
ap-3457	147	29	well	well	ADV
ap-3457	147	30	-	-	PUNCT
ap-3457	147	31	known	know	VERB
ap-3457	147	32	basis	basis	NOUN
ap-3457	147	33	formed	form	VERB
ap-3457	147	34	by	by	ADP
ap-3457	147	35	c	c	NOUN
ap-3457	147	36	-	-	PUNCT
ap-3457	147	37	functions	function	NOUN
ap-3457	147	38	[	[	X
ap-3457	147	39	2	2	NUM
ap-3457	147	40	]	]	PUNCT
ap-3457	147	41	.	.	PUNCT
ap-3457	148	1	therefore	therefore	ADV
ap-3457	148	2	,	,	PUNCT
ap-3457	148	3	each	each	DET
ap-3457	148	4	ssλ+%s	ssλ+%s	NOUN
ap-3457	148	5	/	/	SYM
ap-3457	148	6	s	s	NOUN
ap-3457	148	7	s	s	NOUN
ap-3457	148	8	%	%	NOUN
ap-3457	148	9	s	s	PART
ap-3457	148	10	can	can	AUX
ap-3457	148	11	be	be	AUX
ap-3457	148	12	expressed	express	VERB
ap-3457	148	13	as	as	ADP
ap-3457	148	14	a	a	DET
ap-3457	148	15	linear	linear	ADJ
ap-3457	148	16	combination	combination	NOUN
ap-3457	148	17	of	of	ADP
ap-3457	148	18	c	c	NOUN
ap-3457	148	19	-	-	PUNCT
ap-3457	148	20	functions	function	NOUN
ap-3457	148	21	.	.	PUNCT
ap-3457	149	1	by	by	ADP
ap-3457	149	2	the	the	DET
ap-3457	149	3	definition	definition	NOUN
ap-3457	149	4	of	of	ADP
ap-3457	149	5	the	the	DET
ap-3457	149	6	weyl	weyl	VERB
ap-3457	149	7	group	group	NOUN
ap-3457	149	8	,	,	PUNCT
ap-3457	149	9	the	the	DET
ap-3457	149	10	weights	weight	NOUN
ap-3457	149	11	of	of	ADP
ap-3457	149	12	exponentials	exponential	NOUN
ap-3457	149	13	in	in	ADP
ap-3457	149	14	ssλ+%s	ssλ+%s	NOUN
ap-3457	149	15	are	be	AUX
ap-3457	149	16	of	of	ADP
ap-3457	149	17	the	the	DET
ap-3457	149	18	form	form	NOUN
ap-3457	149	19	λ+%s−g(α1	λ+%s−g(α1	PROPN
ap-3457	149	20	,	,	PUNCT
ap-3457	149	21	.	.	PUNCT
ap-3457	149	22	.	.	PUNCT
ap-3457	150	1	.	.	PUNCT
ap-3457	151	1	,	,	PUNCT
ap-3457	151	2	αn	αn	NOUN
ap-3457	151	3	)	)	PUNCT
ap-3457	151	4	,	,	PUNCT
ap-3457	151	5	where	where	SCONJ
ap-3457	151	6	g(α1	g(α1	NOUN
ap-3457	151	7	,	,	PUNCT
ap-3457	151	8	.	.	PUNCT
ap-3457	151	9	.	.	PUNCT
ap-3457	152	1	.	.	PUNCT
ap-3457	153	1	,	,	PUNCT
ap-3457	153	2	αn	αn	X
ap-3457	153	3	)	)	PUNCT
ap-3457	153	4	denotes	denote	VERB
ap-3457	153	5	a	a	DET
ap-3457	153	6	sum	sum	NOUN
ap-3457	153	7	of	of	ADP
ap-3457	153	8	simple	simple	ADJ
ap-3457	153	9	roots	root	NOUN
ap-3457	153	10	with	with	ADP
ap-3457	153	11	non	non	ADJ
ap-3457	153	12	-	-	ADJ
ap-3457	153	13	negative	negative	ADJ
ap-3457	153	14	integer	integer	NOUN
ap-3457	153	15	coefficients	coefficient	NOUN
ap-3457	153	16	,	,	PUNCT
ap-3457	153	17	and	and	CCONJ
ap-3457	153	18	the	the	DET
ap-3457	153	19	unique	unique	ADJ
ap-3457	153	20	maximal	maximal	ADJ
ap-3457	153	21	weight	weight	NOUN
ap-3457	153	22	is	be	AUX
ap-3457	153	23	λ+	λ+	NUM
ap-3457	153	24	%	%	ADV
ap-3457	153	25	s.	s.	PROPN
ap-3457	153	26	similarly	similarly	ADV
ap-3457	153	27	,	,	PUNCT
ap-3457	153	28	the	the	DET
ap-3457	153	29	unique	unique	ADJ
ap-3457	153	30	maximal	maximal	ADJ
ap-3457	153	31	weight	weight	NOUN
ap-3457	153	32	of	of	ADP
ap-3457	153	33	ss%s	ss%s	NOUN
ap-3457	153	34	is	be	AUX
ap-3457	153	35	%	%	PROPN
ap-3457	153	36	s.	s.	PROPN
ap-3457	153	37	therefore	therefore	ADV
ap-3457	153	38	,	,	PUNCT
ap-3457	153	39	we	we	PRON
ap-3457	153	40	have	have	VERB
ap-3457	153	41	ssλ+%s	ssλ+%s	NOUN
ap-3457	153	42	ss%s	ss%s	NOUN
ap-3457	153	43	=	=	PUNCT
ap-3457	153	44	∑	∑	ADP
ap-3457	153	45	µ∈p+	µ∈p+	X
ap-3457	153	46	µ	µ	PRON
ap-3457	153	47	�	�	NOUN
ap-3457	153	48	λ	λ	NOUN
ap-3457	153	49	bµcµ	bµcµ	NOUN
ap-3457	153	50	,	,	PUNCT
ap-3457	153	51	bλ	bλ	NOUN
ap-3457	153	52	=	=	SYM
ap-3457	153	53	1	1	X
ap-3457	153	54	.	.	PUNCT
ap-3457	153	55	to	to	PART
ap-3457	153	56	prove	prove	VERB
ap-3457	153	57	bµ	bµ	PROPN
ap-3457	153	58	=	=	SYM
ap-3457	153	59	cλµ(ks	cλµ(ks	PROPN
ap-3457	153	60	)	)	PUNCT
ap-3457	153	61	,	,	PUNCT
ap-3457	153	62	we	we	PRON
ap-3457	153	63	proceed	proceed	VERB
ap-3457	153	64	by	by	ADP
ap-3457	153	65	using	use	VERB
ap-3457	153	66	an	an	DET
ap-3457	153	67	equivalent	equivalent	ADJ
ap-3457	153	68	definition	definition	NOUN
ap-3457	153	69	of	of	ADP
ap-3457	153	70	jacobi	jacobi	PROPN
ap-3457	153	71	polynomials	polynomial	VERB
ap-3457	153	72	with	with	ADP
ap-3457	153	73	the	the	DET
ap-3457	153	74	multiplicity	multiplicity	NOUN
ap-3457	153	75	function	function	NOUN
ap-3457	153	76	satisfying	satisfy	VERB
ap-3457	153	77	kα	kα	PROPN
ap-3457	153	78	∈	∈	PROPN
ap-3457	153	79	z≥0	z≥0	PROPN
ap-3457	154	1	[	[	X
ap-3457	154	2	12	12	NUM
ap-3457	154	3	]	]	PUNCT
ap-3457	154	4	.	.	PUNCT
ap-3457	155	1	for	for	ADP
ap-3457	155	2	any	any	DET
ap-3457	155	3	f	f	NOUN
ap-3457	155	4	=	=	SYM
ap-3457	155	5	∑	∑	PUNCT
ap-3457	155	6	λ	λ	X
ap-3457	155	7	aλe	aλe	X
ap-3457	155	8	λ	λ	PROPN
ap-3457	155	9	,	,	PUNCT
ap-3457	155	10	we	we	PRON
ap-3457	155	11	define	define	VERB
ap-3457	155	12	f	f	PROPN
ap-3457	155	13	≡	≡	PROPN
ap-3457	155	14	∑	∑	PROPN
ap-3457	155	15	λ	λ	PROPN
ap-3457	155	16	aλe	aλe	VERB
ap-3457	155	17	−λ	−λ	PROPN
ap-3457	155	18	and	and	CCONJ
ap-3457	155	19	ct	ct	PROPN
ap-3457	155	20	(	(	PUNCT
ap-3457	155	21	f	f	X
ap-3457	155	22	)	)	PUNCT
ap-3457	155	23	≡	≡	PROPN
ap-3457	155	24	a0	a0	PROPN
ap-3457	155	25	.	.	PUNCT
ap-3457	156	1	if	if	SCONJ
ap-3457	156	2	we	we	PRON
ap-3457	156	3	introduce	introduce	VERB
ap-3457	156	4	the	the	DET
ap-3457	156	5	scalar	scalar	ADJ
ap-3457	156	6	product	product	NOUN
ap-3457	156	7	(	(	PUNCT
ap-3457	156	8	·	·	PUNCT
ap-3457	156	9	,	,	PUNCT
ap-3457	156	10	·	·	PUNCT
ap-3457	156	11	)	)	PUNCT
ap-3457	156	12	on	on	ADP
ap-3457	156	13	c[p	c[p	PROPN
ap-3457	156	14	]	]	PUNCT
ap-3457	156	15	by	by	ADP
ap-3457	156	16	(	(	PUNCT
ap-3457	156	17	f	f	X
ap-3457	156	18	,	,	PUNCT
ap-3457	156	19	g	g	NOUN
ap-3457	156	20	)	)	PUNCT
ap-3457	156	21	≡	≡	PROPN
ap-3457	156	22	ct	ct	PROPN
ap-3457	156	23	(	(	PUNCT
ap-3457	156	24	fgδ(k	fgδ(k	PROPN
ap-3457	156	25	)	)	PUNCT
ap-3457	156	26	1	1	NUM
ap-3457	156	27	2	2	NUM
ap-3457	156	28	δ(k	δ(k	NOUN
ap-3457	156	29	)	)	PUNCT
ap-3457	156	30	1	1	NUM
ap-3457	156	31	2	2	NUM
ap-3457	156	32	)	)	PUNCT
ap-3457	156	33	,	,	PUNCT
ap-3457	156	34	f	f	X
ap-3457	156	35	,	,	PUNCT
ap-3457	156	36	g	g	PROPN
ap-3457	156	37	∈	∈	PROPN
ap-3457	156	38	c[p	c[p	PROPN
ap-3457	156	39	]	]	X
ap-3457	156	40	,	,	PUNCT
ap-3457	156	41	δ(k	δ(k	NOUN
ap-3457	156	42	)	)	PUNCT
ap-3457	156	43	1	1	NUM
ap-3457	156	44	2	2	NUM
ap-3457	156	45	≡	≡	PROPN
ap-3457	156	46	∏	∏	PROPN
ap-3457	156	47	α∈π+	α∈π+	ADV
ap-3457	156	48	(	(	PUNCT
ap-3457	156	49	e	e	NOUN
ap-3457	156	50	1	1	NUM
ap-3457	156	51	2α	2α	NOUN
ap-3457	156	52	−	−	NOUN
ap-3457	156	53	e−	e−	PROPN
ap-3457	156	54	1	1	NUM
ap-3457	156	55	2α	2α	NOUN
ap-3457	156	56	)	)	PUNCT
ap-3457	157	1	kα	kα	PROPN
ap-3457	157	2	,	,	PUNCT
ap-3457	157	3	then	then	ADV
ap-3457	157	4	the	the	DET
ap-3457	157	5	jacobi	jacobi	PROPN
ap-3457	157	6	polynomials	polynomial	VERB
ap-3457	157	7	p	p	X
ap-3457	157	8	(	(	PUNCT
ap-3457	157	9	λ	λ	PROPN
ap-3457	157	10	,	,	PUNCT
ap-3457	157	11	k	k	NOUN
ap-3457	157	12	)	)	PUNCT
ap-3457	157	13	are	be	AUX
ap-3457	157	14	the	the	DET
ap-3457	157	15	unique	unique	ADJ
ap-3457	157	16	polynomials	polynomial	NOUN
ap-3457	157	17	of	of	ADP
ap-3457	157	18	the	the	DET
ap-3457	157	19	form	form	NOUN
ap-3457	157	20	(	(	PUNCT
ap-3457	157	21	6	6	NUM
ap-3457	157	22	)	)	PUNCT
ap-3457	157	23	satisfying	satisfy	VERB
ap-3457	157	24	the	the	DET
ap-3457	157	25	requirement	requirement	NOUN
ap-3457	157	26	(	(	PUNCT
ap-3457	157	27	p	p	X
ap-3457	157	28	(	(	PUNCT
ap-3457	157	29	λ	λ	PROPN
ap-3457	157	30	,	,	PUNCT
ap-3457	157	31	k	k	NOUN
ap-3457	157	32	)	)	PUNCT
ap-3457	157	33	,	,	PUNCT
ap-3457	157	34	p	p	X
ap-3457	157	35	(	(	PUNCT
ap-3457	157	36	µ	µ	X
ap-3457	157	37	,	,	PUNCT
ap-3457	157	38	k	k	NOUN
ap-3457	157	39	)	)	PUNCT
ap-3457	157	40	)	)	PUNCT
ap-3457	158	1	=	=	SYM
ap-3457	158	2	0	0	NUM
ap-3457	158	3	for	for	SCONJ
ap-3457	158	4	all	all	DET
ap-3457	158	5	µ	µ	PROPN
ap-3457	158	6	∈	∈	NOUN
ap-3457	158	7	p+	p+	NOUN
ap-3457	158	8	such	such	ADJ
ap-3457	158	9	that	that	SCONJ
ap-3457	158	10	µ	µ	PRON
ap-3457	158	11	�	�	PROPN
ap-3457	158	12	λ	λ	PROPN
ap-3457	158	13	and	and	CCONJ
ap-3457	158	14	λ	λ	PROPN
ap-3457	158	15	6=	6=	ADP
ap-3457	158	16	µ	µ	X
ap-3457	158	17	assuming	assume	VERB
ap-3457	158	18	cλλ	cλλ	PROPN
ap-3457	158	19	=	=	NOUN
ap-3457	158	20	1	1	X
ap-3457	158	21	.	.	X
ap-3457	158	22	using	use	VERB
ap-3457	158	23	proposition	proposition	NOUN
ap-3457	158	24	4.1	4.1	NUM
ap-3457	158	25	of	of	ADP
ap-3457	158	26	[	[	X
ap-3457	158	27	26	26	NUM
ap-3457	158	28	]	]	PUNCT
ap-3457	158	29	,	,	PUNCT
ap-3457	158	30	i.e.	i.e.	X
ap-3457	158	31	δ(ks	δ(ks	NOUN
ap-3457	158	32	)	)	PUNCT
ap-3457	158	33	1	1	NUM
ap-3457	158	34	2	2	NUM
ap-3457	158	35	=	=	NOUN
ap-3457	158	36	ss%s	ss%s	NOUN
ap-3457	158	37	,	,	PUNCT
ap-3457	158	38	we	we	PRON
ap-3457	158	39	obtain	obtain	VERB
ap-3457	158	40	(	(	PUNCT
ap-3457	158	41	ssλ+%s	ssλ+%s	NOUN
ap-3457	158	42	ss%s	ss%s	NOUN
ap-3457	158	43	,	,	PUNCT
ap-3457	158	44	ssµ+%s	ssµ+%s	NOUN
ap-3457	158	45	ss%s	ss%s	NOUN
ap-3457	158	46	)	)	PUNCT
ap-3457	159	1	=	=	PUNCT
ap-3457	159	2	ct	ct	PROPN
ap-3457	159	3	(	(	PUNCT
ap-3457	159	4	ssλ+%ss	ssλ+%ss	NOUN
ap-3457	159	5	s	s	PART
ap-3457	159	6	µ+%s	µ+%s	PRON
ap-3457	159	7	)	)	PUNCT
ap-3457	159	8	=	=	SYM
ap-3457	160	1	ct	ct	PROPN
ap-3457	160	2	(	(	PUNCT
ap-3457	160	3	∑	∑	PUNCT
ap-3457	160	4	λ′∈w	λ′∈w	X
ap-3457	160	5	(	(	PUNCT
ap-3457	160	6	λ+%s	λ+%s	NOUN
ap-3457	160	7	)	)	PUNCT
ap-3457	160	8	∑	∑	ADV
ap-3457	160	9	µ′∈w	µ′∈w	PROPN
ap-3457	160	10	(	(	PUNCT
ap-3457	160	11	µ+%s	µ+%s	NUM
ap-3457	160	12	)	)	PUNCT
ap-3457	160	13	σs(λ′)σs(µ′)eλ	σs(λ′)σs(µ′)eλ	NOUN
ap-3457	160	14	′−µ′	′−µ′	NOUN
ap-3457	160	15	)	)	PUNCT
ap-3457	160	16	.	.	PUNCT
ap-3457	161	1	clearly	clearly	ADV
ap-3457	161	2	λ′	λ′	X
ap-3457	161	3	=	=	SYM
ap-3457	161	4	µ′	µ′	NOUN
ap-3457	161	5	if	if	SCONJ
ap-3457	161	6	and	and	CCONJ
ap-3457	161	7	only	only	ADV
ap-3457	161	8	if	if	SCONJ
ap-3457	161	9	there	there	PRON
ap-3457	161	10	exists	exist	VERB
ap-3457	161	11	w	w	NOUN
ap-3457	161	12	∈w	∈w	NOUN
ap-3457	161	13	such	such	ADJ
ap-3457	161	14	that	that	DET
ap-3457	161	15	λ	λ	PROPN
ap-3457	162	1	+	+	NUM
ap-3457	163	1	%	%	NOUN
ap-3457	163	2	s	s	X
ap-3457	163	3	=	=	PUNCT
ap-3457	163	4	w(µ	w(µ	PROPN
ap-3457	163	5	+	+	NUM
ap-3457	163	6	%	%	NOUN
ap-3457	163	7	s	s	NOUN
ap-3457	163	8	)	)	PUNCT
ap-3457	163	9	.	.	PUNCT
ap-3457	164	1	for	for	SCONJ
ap-3457	164	2	we	we	PRON
ap-3457	164	3	consider	consider	VERB
ap-3457	164	4	λ	λ	X
ap-3457	164	5	∈	∈	PROPN
ap-3457	164	6	p+	p+	VERB
ap-3457	164	7	different	different	ADJ
ap-3457	164	8	from	from	ADP
ap-3457	164	9	µ	µ	PRON
ap-3457	164	10	∈	∈	PROPN
ap-3457	164	11	p+	p+	NOUN
ap-3457	164	12	,	,	PUNCT
ap-3457	164	13	it	it	PRON
ap-3457	164	14	is	be	AUX
ap-3457	164	15	not	not	PART
ap-3457	164	16	possible	possible	ADJ
ap-3457	164	17	to	to	PART
ap-3457	164	18	have	have	VERB
ap-3457	164	19	λ′	λ′	X
ap-3457	164	20	=	=	SYM
ap-3457	164	21	µ′.	µ′.	NOUN
ap-3457	164	22	this	this	PRON
ap-3457	164	23	implies	imply	VERB
ap-3457	164	24	that	that	SCONJ
ap-3457	164	25	(	(	PUNCT
ap-3457	164	26	ssλ+%s	ssλ+%s	NOUN
ap-3457	164	27	ss%s	ss%s	NOUN
ap-3457	164	28	,	,	PUNCT
ap-3457	164	29	ssµ+%s	ssµ+%s	NOUN
ap-3457	164	30	ss%s	ss%s	NOUN
ap-3457	164	31	)	)	PUNCT
ap-3457	165	1	=	=	SYM
ap-3457	165	2	0	0	NUM
ap-3457	166	1	and	and	CCONJ
ap-3457	166	2	p	p	X
ap-3457	166	3	(	(	PUNCT
ap-3457	166	4	λ	λ	PROPN
ap-3457	166	5	,	,	PUNCT
ap-3457	166	6	ks	ks	NOUN
ap-3457	166	7	)	)	PUNCT
ap-3457	166	8	=	=	NOUN
ap-3457	166	9	ssλ+%s	ssλ+%s	NOUN
ap-3457	166	10	ss%s	ss%s	NOUN
ap-3457	166	11	.	.	PUNCT
ap-3457	167	1	the	the	DET
ap-3457	167	2	proof	proof	NOUN
ap-3457	167	3	for	for	ADP
ap-3457	167	4	the	the	DET
ap-3457	167	5	long	long	ADJ
ap-3457	167	6	root	root	NOUN
ap-3457	167	7	case	case	NOUN
ap-3457	167	8	is	be	AUX
ap-3457	167	9	similar	similar	ADJ
ap-3457	167	10	.	.	PUNCT
ap-3457	168	1	finally	finally	ADV
ap-3457	168	2	,	,	PUNCT
ap-3457	168	3	note	note	VERB
ap-3457	168	4	that	that	SCONJ
ap-3457	168	5	the	the	DET
ap-3457	168	6	jacobi	jacobi	PROPN
ap-3457	168	7	polynomials	polynomial	NOUN
ap-3457	168	8	can	can	AUX
ap-3457	168	9	be	be	AUX
ap-3457	168	10	viewed	view	VERB
ap-3457	168	11	as	as	ADP
ap-3457	168	12	the	the	DET
ap-3457	168	13	limiting	limit	VERB
ap-3457	168	14	case	case	NOUN
ap-3457	168	15	of	of	ADP
ap-3457	168	16	the	the	DET
ap-3457	168	17	macdonald	macdonald	PROPN
ap-3457	168	18	polynomials	polynomial	NOUN
ap-3457	168	19	pλ(q	pλ(q	NUM
ap-3457	168	20	,	,	PUNCT
ap-3457	168	21	tα	tα	PROPN
ap-3457	168	22	)	)	PUNCT
ap-3457	168	23	when	when	SCONJ
ap-3457	168	24	tα	tα	PROPN
ap-3457	168	25	=	=	SYM
ap-3457	168	26	qkα	qkα	PROPN
ap-3457	168	27	with	with	ADP
ap-3457	168	28	kα	kα	PRON
ap-3457	168	29	fixed	fix	VERB
ap-3457	168	30	and	and	CCONJ
ap-3457	168	31	q	q	NOUN
ap-3457	168	32	→	→	SYM
ap-3457	168	33	1	1	X
ap-3457	168	34	.	.	X
ap-3457	168	35	see	see	VERB
ap-3457	168	36	[	[	X
ap-3457	168	37	24	24	NUM
ap-3457	168	38	]	]	PUNCT
ap-3457	168	39	for	for	ADP
ap-3457	168	40	more	more	ADJ
ap-3457	168	41	details	detail	NOUN
ap-3457	168	42	.	.	PUNCT
ap-3457	169	1	the	the	DET
ap-3457	169	2	relations	relation	NOUN
ap-3457	169	3	among	among	ADP
ap-3457	169	4	several	several	ADJ
ap-3457	169	5	special	special	ADJ
ap-3457	169	6	functions	function	NOUN
ap-3457	169	7	associated	associate	VERB
ap-3457	169	8	with	with	ADP
ap-3457	169	9	the	the	DET
ap-3457	169	10	weyl	weyl	VERB
ap-3457	169	11	groups	group	NOUN
ap-3457	169	12	,	,	PUNCT
ap-3457	169	13	which	which	PRON
ap-3457	169	14	are	be	AUX
ap-3457	169	15	summarized	summarize	VERB
ap-3457	169	16	in	in	ADP
ap-3457	169	17	[	[	X
ap-3457	169	18	29	29	NUM
ap-3457	169	19	]	]	PUNCT
ap-3457	169	20	,	,	PUNCT
ap-3457	169	21	are	be	AUX
ap-3457	169	22	depicted	depict	VERB
ap-3457	169	23	in	in	ADP
ap-3457	169	24	figure	figure	NOUN
ap-3457	169	25	2	2	NUM
ap-3457	169	26	.	.	SYM
ap-3457	169	27	205	205	NUM
ap-3457	169	28	l.	l.	PROPN
ap-3457	169	29	háková	háková	PROPN
ap-3457	169	30	,	,	PUNCT
ap-3457	169	31	j.	j.	PROPN
ap-3457	169	32	hrivnák	hrivnák	PROPN
ap-3457	169	33	,	,	PUNCT
ap-3457	169	34	l.	l.	PROPN
ap-3457	169	35	motlochová	motlochová	PROPN
ap-3457	169	36	acta	acta	PROPN
ap-3457	169	37	polytechnica	polytechnica	PROPN
ap-3457	169	38	macdonald	macdonald	PROPN
ap-3457	169	39	polynomials	polynomials	PROPN
ap-3457	169	40	[	[	X
ap-3457	169	41	24	24	NUM
ap-3457	169	42	]	]	PUNCT
ap-3457	169	43	pλ(q	pλ(q	NUM
ap-3457	169	44	,	,	PUNCT
ap-3457	169	45	tα	tα	PROPN
ap-3457	169	46	)	)	PUNCT
ap-3457	169	47	,	,	PUNCT
ap-3457	169	48	tα	tα	PROPN
ap-3457	169	49	=	=	PUNCT
ap-3457	169	50	qkα	qkα	PROPN
ap-3457	169	51	jacobi	jacobi	PROPN
ap-3457	169	52	polynomials	polynomial	NOUN
ap-3457	169	53	associated	associate	VERB
ap-3457	169	54	to	to	PART
ap-3457	169	55	root	root	VERB
ap-3457	169	56	systems	system	NOUN
ap-3457	170	1	[	[	X
ap-3457	170	2	11	11	NUM
ap-3457	170	3	]	]	X
ap-3457	170	4	p	p	X
ap-3457	170	5	(	(	PUNCT
ap-3457	170	6	λ	λ	PROPN
ap-3457	170	7	,	,	PUNCT
ap-3457	170	8	k	k	NOUN
ap-3457	170	9	)	)	PUNCT
ap-3457	170	10	(	(	PUNCT
ap-3457	170	11	section	section	NOUN
ap-3457	170	12	2.3	2.3	NUM
ap-3457	170	13	)	)	PUNCT
ap-3457	170	14	c	c	NOUN
ap-3457	170	15	-	-	PUNCT
ap-3457	170	16	functions	function	NOUN
ap-3457	170	17	and	and	CCONJ
ap-3457	170	18	s	s	NOUN
ap-3457	170	19	-	-	PUNCT
ap-3457	170	20	functions	function	NOUN
ap-3457	170	21	[	[	X
ap-3457	170	22	18	18	NUM
ap-3457	170	23	,	,	PUNCT
ap-3457	170	24	19	19	NUM
ap-3457	170	25	]	]	PUNCT
ap-3457	170	26	cλ	cλ	PROPN
ap-3457	170	27	=	=	SYM
ap-3457	170	28	p	p	X
ap-3457	170	29	(	(	PUNCT
ap-3457	170	30	λ	λ	PROPN
ap-3457	170	31	,	,	PUNCT
ap-3457	170	32	k0	k0	PROPN
ap-3457	170	33	)	)	PUNCT
ap-3457	170	34	sλ+%1	sλ+%1	PROPN
ap-3457	171	1	=	=	PUNCT
ap-3457	171	2	p	p	X
ap-3457	171	3	(	(	PUNCT
ap-3457	171	4	λ	λ	PROPN
ap-3457	171	5	,	,	PUNCT
ap-3457	171	6	k1)s%1	k1)s%1	NOUN
ap-3457	171	7	ss	ss	NOUN
ap-3457	171	8	-	-	PUNCT
ap-3457	171	9	functions	function	NOUN
ap-3457	171	10	and	and	CCONJ
ap-3457	171	11	sl	sl	NOUN
ap-3457	171	12	-	-	PUNCT
ap-3457	171	13	functions	function	NOUN
ap-3457	171	14	[	[	X
ap-3457	171	15	26	26	NUM
ap-3457	171	16	]	]	PUNCT
ap-3457	171	17	ssλ+%s	ssλ+%s	NOUN
ap-3457	171	18	=	=	PUNCT
ap-3457	171	19	p	p	X
ap-3457	171	20	(	(	PUNCT
ap-3457	171	21	λ	λ	PROPN
ap-3457	171	22	,	,	PUNCT
ap-3457	171	23	ks)ss%s	ks)ss%s	NOUN
ap-3457	171	24	slλ+%l	slλ+%l	NOUN
ap-3457	171	25	=	=	SYM
ap-3457	171	26	p	p	X
ap-3457	171	27	(	(	PUNCT
ap-3457	171	28	λ	λ	PROPN
ap-3457	171	29	,	,	PUNCT
ap-3457	171	30	kl)sl%l	kl)sl%l	PROPN
ap-3457	171	31	cck	cck	PROPN
ap-3457	171	32	and	and	CCONJ
ap-3457	171	33	ssk	ssk	NOUN
ap-3457	172	1	[	[	X
ap-3457	172	2	21	21	NUM
ap-3457	172	3	]	]	PUNCT
ap-3457	172	4	?	?	PUNCT
ap-3457	173	1	cos+	cos+	PROPN
ap-3457	173	2	and	and	CCONJ
ap-3457	173	3	sin−	sin−	NOUN
ap-3457	173	4	[	[	X
ap-3457	173	5	17	17	NUM
ap-3457	173	6	]	]	X
ap-3457	173	7	sck	sck	PROPN
ap-3457	173	8	and	and	CCONJ
ap-3457	173	9	csk	csk	X
ap-3457	174	1	[	[	X
ap-3457	174	2	21	21	NUM
ap-3457	174	3	]	]	X
ap-3457	174	4	cos−	cos−	PROPN
ap-3457	174	5	and	and	CCONJ
ap-3457	174	6	sin+	sin+	PROPN
ap-3457	175	1	[	[	X
ap-3457	175	2	17	17	NUM
ap-3457	175	3	]	]	X
ap-3457	175	4	chebyshev	chebyshev	NOUN
ap-3457	175	5	polynomials	polynomial	NOUN
ap-3457	175	6	[	[	X
ap-3457	175	7	9	9	NUM
ap-3457	175	8	]	]	SYM
ap-3457	175	9	tm	tm	NOUN
ap-3457	175	10	,	,	PUNCT
ap-3457	175	11	um	um	INTJ
ap-3457	175	12	,	,	PUNCT
ap-3457	175	13	vm	vm	PROPN
ap-3457	175	14	and	and	CCONJ
ap-3457	175	15	wm	wm	PROPN
ap-3457	175	16	2	2	NUM
ap-3457	175	17	-	-	PUNCT
ap-3457	175	18	d	d	NOUN
ap-3457	175	19	jacobi	jacobi	PROPN
ap-3457	175	20	polynomials	polynomial	NOUN
ap-3457	175	21	pα	pα	VERB
ap-3457	175	22	,	,	PUNCT
ap-3457	175	23	β	β	NOUN
ap-3457	175	24	,	,	PUNCT
ap-3457	175	25	γk1,k2	γk1,k2	NOUN
ap-3457	175	26	with	with	ADP
ap-3457	175	27	α	α	PROPN
ap-3457	175	28	,	,	PUNCT
ap-3457	175	29	β	β	X
ap-3457	175	30	,	,	PUNCT
ap-3457	175	31	γ	γ	PROPN
ap-3457	175	32	∈	∈	PROPN
ap-3457	175	33	{	{	PUNCT
ap-3457	175	34	±	±	NOUN
ap-3457	175	35	1	1	NUM
ap-3457	175	36	2	2	NUM
ap-3457	175	37	}	}	PUNCT
ap-3457	175	38	[	[	X
ap-3457	175	39	20	20	NUM
ap-3457	175	40	]	]	SYM
ap-3457	175	41	2	2	NUM
ap-3457	175	42	-	-	SYM
ap-3457	175	43	d	d	NOUN
ap-3457	175	44	jacobi	jacobi	PROPN
ap-3457	175	45	polynomials	polynomial	NOUN
ap-3457	175	46	on	on	ADP
ap-3457	175	47	steiner	steiner	PROPN
ap-3457	175	48	’s	’s	PART
ap-3457	175	49	hypocycloid	hypocycloid	NOUN
ap-3457	176	1	[	[	X
ap-3457	176	2	20	20	NUM
ap-3457	176	3	]	]	X
ap-3457	176	4	jacobi	jacobi	PROPN
ap-3457	176	5	polynomials	polynomial	VERB
ap-3457	176	6	[	[	X
ap-3457	176	7	36	36	NUM
ap-3457	176	8	]	]	X
ap-3457	176	9	p	p	X
ap-3457	176	10	(	(	PUNCT
ap-3457	176	11	α	α	X
ap-3457	176	12	,	,	PUNCT
ap-3457	176	13	β	β	NOUN
ap-3457	176	14	)	)	PUNCT
ap-3457	176	15	m	m	VERB
ap-3457	176	16	q	q	NOUN
ap-3457	176	17	→	→	SYM
ap-3457	176	18	1	1	NUM
ap-3457	176	19	,	,	PUNCT
ap-3457	176	20	kα	kα	PROPN
ap-3457	176	21	fixed	fix	VERB
ap-3457	176	22	k	k	PROPN
ap-3457	176	23	∈	∈	PROPN
ap-3457	176	24	{	{	PUNCT
ap-3457	176	25	ks	ks	NOUN
ap-3457	176	26	,	,	PUNCT
ap-3457	176	27	kl}k	kl}k	PROPN
ap-3457	176	28	∈	∈	PROPN
ap-3457	176	29	{	{	PUNCT
ap-3457	176	30	k0	k0	PROPN
ap-3457	176	31	,	,	PUNCT
ap-3457	176	32	k1	k1	NOUN
ap-3457	176	33	}	}	PUNCT
ap-3457	176	34	g2an	g2an	NOUN
ap-3457	176	35	bn	bn	NOUN
ap-3457	176	36	and	and	CCONJ
ap-3457	176	37	cn	cn	PROPN
ap-3457	176	38	g2	g2	PROPN
ap-3457	176	39	bn	bn	PROPN
ap-3457	176	40	and	and	CCONJ
ap-3457	176	41	cn	cn	PROPN
ap-3457	176	42	a1	a1	PROPN
ap-3457	176	43	a2	a2	PROPN
ap-3457	176	44	c2	c2	PROPN
ap-3457	176	45	c2	c2	PROPN
ap-3457	176	46	α	α	PROPN
ap-3457	176	47	,	,	PUNCT
ap-3457	176	48	β	β	X
ap-3457	176	49	∈	∈	PROPN
ap-3457	176	50	{	{	PUNCT
ap-3457	176	51	±	±	NOUN
ap-3457	176	52	1	1	NUM
ap-3457	176	53	2	2	NUM
ap-3457	176	54	}	}	PUNCT
ap-3457	176	55	figure	figure	NOUN
ap-3457	176	56	2	2	NUM
ap-3457	176	57	.	.	PUNCT
ap-3457	177	1	the	the	DET
ap-3457	177	2	diagram	diagram	NOUN
ap-3457	177	3	of	of	ADP
ap-3457	177	4	relations	relation	NOUN
ap-3457	177	5	among	among	ADP
ap-3457	177	6	several	several	ADJ
ap-3457	177	7	special	special	ADJ
ap-3457	177	8	functions	function	NOUN
ap-3457	177	9	associated	associate	VERB
ap-3457	177	10	with	with	ADP
ap-3457	177	11	weyl	weyl	VERB
ap-3457	177	12	groups	group	NOUN
ap-3457	177	13	.	.	PUNCT
ap-3457	178	1	3	3	X
ap-3457	178	2	.	.	X
ap-3457	178	3	cubature	cubature	NOUN
ap-3457	178	4	formulas	formula	NOUN
ap-3457	178	5	3.1	3.1	NUM
ap-3457	178	6	.	.	PUNCT
ap-3457	178	7	general	general	ADJ
ap-3457	178	8	form	form	NOUN
ap-3457	178	9	of	of	ADP
ap-3457	178	10	cubature	cubature	ADJ
ap-3457	178	11	formulas	formula	NOUN
ap-3457	178	12	analogously	analogously	ADV
ap-3457	178	13	to	to	ADP
ap-3457	178	14	chebyshev	chebyshev	NOUN
ap-3457	178	15	polynomials	polynomial	NOUN
ap-3457	178	16	,	,	PUNCT
ap-3457	178	17	we	we	PRON
ap-3457	178	18	identify	identify	VERB
ap-3457	178	19	polynomial	polynomial	ADJ
ap-3457	178	20	variables	variable	NOUN
ap-3457	178	21	y1	y1	PROPN
ap-3457	178	22	,	,	PUNCT
ap-3457	178	23	.	.	PUNCT
ap-3457	178	24	.	.	PUNCT
ap-3457	179	1	.	.	PUNCT
ap-3457	180	1	,	,	PUNCT
ap-3457	180	2	yn	yn	PROPN
ap-3457	180	3	with	with	ADP
ap-3457	180	4	real	real	ADV
ap-3457	180	5	-	-	PUNCT
ap-3457	180	6	valued	value	VERB
ap-3457	180	7	functions	function	NOUN
ap-3457	180	8	in	in	ADP
ap-3457	180	9	the	the	DET
ap-3457	180	10	following	following	ADJ
ap-3457	180	11	way	way	NOUN
ap-3457	180	12	.	.	PUNCT
ap-3457	181	1	let	let	VERB
ap-3457	181	2	zj	zj	PROPN
ap-3457	181	3	≡	≡	PROPN
ap-3457	181	4	cωj	cωj	NOUN
ap-3457	181	5	,	,	PUNCT
ap-3457	181	6	then	then	ADV
ap-3457	181	7	a2k	a2k	X
ap-3457	181	8	:	:	PUNCT
ap-3457	182	1	yj	yj	PROPN
ap-3457	182	2	=	=	SYM
ap-3457	182	3	<	<	X
ap-3457	182	4	(	(	PUNCT
ap-3457	182	5	zj	zj	PROPN
ap-3457	182	6	)	)	PUNCT
ap-3457	182	7	,	,	PUNCT
ap-3457	182	8	y2k−j+1	y2k−j+1	PROPN
ap-3457	182	9	=	=	SYM
ap-3457	182	10	=(	=(	PROPN
ap-3457	182	11	zj	zj	PROPN
ap-3457	182	12	)	)	PUNCT
ap-3457	182	13	,	,	PUNCT
ap-3457	182	14	j	j	PROPN
ap-3457	182	15	=	=	SYM
ap-3457	182	16	1	1	NUM
ap-3457	182	17	,	,	PUNCT
ap-3457	182	18	.	.	PUNCT
ap-3457	182	19	.	.	PUNCT
ap-3457	182	20	.	.	PUNCT
ap-3457	183	1	,	,	PUNCT
ap-3457	183	2	k	k	X
ap-3457	183	3	,	,	PUNCT
ap-3457	183	4	a2k+1	a2k+1	VERB
ap-3457	183	5	:	:	PUNCT
ap-3457	184	1	yj	yj	PROPN
ap-3457	184	2	=	=	SYM
ap-3457	184	3	<	<	X
ap-3457	184	4	(	(	PUNCT
ap-3457	184	5	zj	zj	PROPN
ap-3457	184	6	)	)	PUNCT
ap-3457	184	7	,	,	PUNCT
ap-3457	184	8	yk+1	yk+1	NOUN
ap-3457	184	9	=	=	SYM
ap-3457	184	10	zk+1	zk+1	PROPN
ap-3457	184	11	,	,	PUNCT
ap-3457	184	12	y2k−j+2	y2k−j+2	PROPN
ap-3457	184	13	=	=	SYM
ap-3457	184	14	=(	=(	PROPN
ap-3457	184	15	zj	zj	PROPN
ap-3457	184	16	)	)	PUNCT
ap-3457	184	17	,	,	PUNCT
ap-3457	184	18	j	j	PROPN
ap-3457	184	19	=	=	SYM
ap-3457	184	20	1	1	NUM
ap-3457	184	21	,	,	PUNCT
ap-3457	184	22	.	.	PUNCT
ap-3457	184	23	.	.	PUNCT
ap-3457	184	24	.	.	PUNCT
ap-3457	185	1	,	,	PUNCT
ap-3457	185	2	k	k	X
ap-3457	185	3	,	,	PUNCT
ap-3457	185	4	d2k+1	d2k+1	PUNCT
ap-3457	185	5	:	:	PUNCT
ap-3457	185	6	yj	yj	PROPN
ap-3457	185	7	=	=	PROPN
ap-3457	185	8	zj	zj	PROPN
ap-3457	185	9	,	,	PUNCT
ap-3457	185	10	y2k	y2k	PROPN
ap-3457	185	11	=	=	SYM
ap-3457	185	12	<	<	X
ap-3457	185	13	(	(	PUNCT
ap-3457	185	14	z2k	z2k	NOUN
ap-3457	185	15	)	)	PUNCT
ap-3457	185	16	,	,	PUNCT
ap-3457	185	17	y2k+1	y2k+1	VERB
ap-3457	185	18	=	=	SYM
ap-3457	185	19	=(	=(	NOUN
ap-3457	185	20	z2k	z2k	NUM
ap-3457	185	21	)	)	PUNCT
ap-3457	185	22	,	,	PUNCT
ap-3457	185	23	j	j	PROPN
ap-3457	185	24	=	=	SYM
ap-3457	185	25	1	1	NUM
ap-3457	185	26	,	,	PUNCT
ap-3457	185	27	.	.	PUNCT
ap-3457	185	28	.	.	PUNCT
ap-3457	185	29	.	.	PUNCT
ap-3457	186	1	,	,	PUNCT
ap-3457	186	2	2k	2k	NOUN
ap-3457	186	3	−	−	NOUN
ap-3457	186	4	1	1	NUM
ap-3457	186	5	,	,	PUNCT
ap-3457	186	6	e6	e6	NOUN
ap-3457	186	7	:	:	PUNCT
ap-3457	186	8	y1	y1	INTJ
ap-3457	186	9	=	=	PUNCT
ap-3457	186	10	<	<	X
ap-3457	186	11	(	(	PUNCT
ap-3457	186	12	z1	z1	NOUN
ap-3457	186	13	)	)	PUNCT
ap-3457	186	14	,	,	PUNCT
ap-3457	187	1	y2	y2	NOUN
ap-3457	187	2	=	=	PUNCT
ap-3457	188	1	<	<	X
ap-3457	188	2	(	(	PUNCT
ap-3457	188	3	z2	z2	NOUN
ap-3457	188	4	)	)	PUNCT
ap-3457	188	5	,	,	PUNCT
ap-3457	188	6	y3	y3	NOUN
ap-3457	188	7	=	=	SYM
ap-3457	188	8	z3	z3	PROPN
ap-3457	188	9	,	,	PUNCT
ap-3457	188	10	y4	y4	PROPN
ap-3457	188	11	=	=	SYM
ap-3457	188	12	=(	=(	PROPN
ap-3457	188	13	z2	z2	PROPN
ap-3457	188	14	)	)	PUNCT
ap-3457	188	15	,	,	PUNCT
ap-3457	188	16	y5	y5	NOUN
ap-3457	188	17	=	=	SYM
ap-3457	188	18	=(	=(	NOUN
ap-3457	188	19	z1	z1	PROPN
ap-3457	188	20	)	)	PUNCT
ap-3457	188	21	,	,	PUNCT
ap-3457	188	22	y6	y6	NOUN
ap-3457	188	23	=	=	SYM
ap-3457	188	24	z6	z6	PROPN
ap-3457	188	25	,	,	PUNCT
ap-3457	188	26	otherwise	otherwise	ADV
ap-3457	188	27	we	we	PRON
ap-3457	188	28	put	put	VERB
ap-3457	188	29	yj	yj	PROPN
ap-3457	188	30	=	=	PROPN
ap-3457	188	31	zj	zj	PROPN
ap-3457	188	32	.	.	PUNCT
ap-3457	189	1	we	we	PRON
ap-3457	189	2	say	say	VERB
ap-3457	189	3	that	that	SCONJ
ap-3457	189	4	a	a	DET
ap-3457	189	5	monomial	monomial	ADJ
ap-3457	189	6	yλ1	yλ1	NOUN
ap-3457	189	7	1	1	NUM
ap-3457	189	8	.	.	PUNCT
ap-3457	189	9	.	.	PUNCT
ap-3457	189	10	.	.	PUNCT
ap-3457	190	1	yλnn	yλnn	PROPN
ap-3457	190	2	has	have	VERB
ap-3457	190	3	an	an	DET
ap-3457	190	4	m	m	NOUN
ap-3457	190	5	-	-	PUNCT
ap-3457	190	6	degree	degree	NOUN
ap-3457	190	7	degm	degm	ADJ
ap-3457	190	8	yλ1	yλ1	NOUN
ap-3457	190	9	1	1	NUM
ap-3457	190	10	.	.	PUNCT
ap-3457	190	11	.	.	PUNCT
ap-3457	190	12	.	.	PUNCT
ap-3457	191	1	yλnn	yλnn	X
ap-3457	191	2	=	=	PUNCT
ap-3457	192	1	m∨1	m∨1	X
ap-3457	193	1	λ1	λ1	ADJ
ap-3457	193	2	+	+	CCONJ
ap-3457	193	3	·	·	PUNCT
ap-3457	193	4	·	·	PUNCT
ap-3457	193	5	·	·	PUNCT
ap-3457	194	1	+	+	NOUN
ap-3457	194	2	m∨nλn	m∨nλn	NOUN
ap-3457	194	3	and	and	CCONJ
ap-3457	194	4	any	any	DET
ap-3457	194	5	polynomial	polynomial	ADJ
ap-3457	194	6	p	p	NOUN
ap-3457	194	7	in	in	ADP
ap-3457	194	8	c[y1	c[y1	NOUN
ap-3457	194	9	,	,	PUNCT
ap-3457	194	10	.	.	PUNCT
ap-3457	194	11	.	.	PUNCT
ap-3457	195	1	.	.	PUNCT
ap-3457	196	1	,	,	PUNCT
ap-3457	196	2	yn	yn	PRON
ap-3457	196	3	]	]	PUNCT
ap-3457	196	4	has	have	AUX
ap-3457	196	5	anm	anm	NOUN
ap-3457	196	6	-	-	PUNCT
ap-3457	196	7	degree	degree	NOUN
ap-3457	196	8	equal	equal	ADJ
ap-3457	196	9	to	to	ADP
ap-3457	196	10	the	the	DET
ap-3457	196	11	largest	large	ADJ
ap-3457	196	12	m	m	NOUN
ap-3457	196	13	-	-	PUNCT
ap-3457	196	14	degree	degree	NOUN
ap-3457	196	15	of	of	ADP
ap-3457	196	16	the	the	DET
ap-3457	196	17	monomials	monomial	NOUN
ap-3457	196	18	occurring	occur	VERB
ap-3457	196	19	in	in	ADP
ap-3457	196	20	p.	p.	NOUN
ap-3457	196	21	we	we	PRON
ap-3457	196	22	denote	denote	VERB
ap-3457	196	23	the	the	DET
ap-3457	196	24	subspace	subspace	NOUN
ap-3457	196	25	containing	contain	VERB
ap-3457	196	26	all	all	DET
ap-3457	196	27	polynomials	polynomial	NOUN
ap-3457	196	28	of	of	ADP
ap-3457	196	29	m	m	NOUN
ap-3457	196	30	-	-	PUNCT
ap-3457	196	31	degree	degree	NOUN
ap-3457	196	32	at	at	ADP
ap-3457	196	33	most	most	ADJ
ap-3457	196	34	m	m	VERB
ap-3457	196	35	by	by	ADP
ap-3457	196	36	πm	πm	INTJ
ap-3457	196	37	.	.	PUNCT
ap-3457	197	1	for	for	SCONJ
ap-3457	197	2	λ	λ	X
ap-3457	197	3	=	=	PUNCT
ap-3457	197	4	λ1ω1	λ1ω1	PUNCT
ap-3457	197	5	+	+	CCONJ
ap-3457	197	6	·	·	PUNCT
ap-3457	197	7	·	·	PUNCT
ap-3457	197	8	·	·	PUNCT
ap-3457	197	9	+	+	CCONJ
ap-3457	197	10	λnωn	λnωn	ADJ
ap-3457	197	11	,	,	PUNCT
ap-3457	197	12	the	the	DET
ap-3457	197	13	c	c	NOUN
ap-3457	197	14	-	-	PUNCT
ap-3457	197	15	functions	function	NOUN
ap-3457	197	16	cλ	cλ	NOUN
ap-3457	197	17	and	and	CCONJ
ap-3457	197	18	thus	thus	ADV
ap-3457	197	19	all	all	DET
ap-3457	197	20	jacobi	jacobi	NOUN
ap-3457	197	21	polynomials	polynomial	VERB
ap-3457	197	22	p	p	X
ap-3457	197	23	(	(	PUNCT
ap-3457	197	24	λ	λ	PROPN
ap-3457	197	25	,	,	PUNCT
ap-3457	197	26	k	k	NOUN
ap-3457	197	27	)	)	PUNCT
ap-3457	197	28	can	can	AUX
ap-3457	197	29	be	be	AUX
ap-3457	197	30	rewritten	rewrite	VERB
ap-3457	197	31	as	as	ADP
ap-3457	197	32	orthogonal	orthogonal	ADJ
ap-3457	197	33	polynomials	polynomial	NOUN
ap-3457	197	34	in	in	ADP
ap-3457	197	35	variables	variable	NOUN
ap-3457	197	36	y1	y1	PROPN
ap-3457	197	37	,	,	PUNCT
ap-3457	197	38	.	.	PUNCT
ap-3457	197	39	.	.	PUNCT
ap-3457	198	1	.	.	PUNCT
ap-3457	199	1	,	,	PUNCT
ap-3457	199	2	yn	yn	PROPN
ap-3457	199	3	of	of	ADP
ap-3457	199	4	mdegree	mdegree	PROPN
ap-3457	199	5	equal	equal	ADJ
ap-3457	199	6	to	to	ADP
ap-3457	199	7	m∨1	m∨1	VERB
ap-3457	200	1	λ1	λ1	ADJ
ap-3457	200	2	+	+	CCONJ
ap-3457	200	3	·	·	PUNCT
ap-3457	200	4	·	·	PUNCT
ap-3457	200	5	·	·	PUNCT
ap-3457	201	1	+	+	NOUN
ap-3457	201	2	m∨nλn	m∨nλn	NOUN
ap-3457	201	3	[	[	X
ap-3457	201	4	15	15	NUM
ap-3457	201	5	]	]	PUNCT
ap-3457	201	6	.	.	PUNCT
ap-3457	202	1	the	the	DET
ap-3457	202	2	variables	variable	NOUN
ap-3457	202	3	y1	y1	PROPN
ap-3457	202	4	,	,	PUNCT
ap-3457	202	5	.	.	PUNCT
ap-3457	202	6	.	.	PUNCT
ap-3457	202	7	.	.	PUNCT
ap-3457	203	1	,	,	PUNCT
ap-3457	203	2	yn	yn	PRON
ap-3457	203	3	viewed	view	VERB
ap-3457	203	4	as	as	ADP
ap-3457	203	5	functions	function	NOUN
ap-3457	203	6	induce	induce	VERB
ap-3457	203	7	a	a	DET
ap-3457	203	8	map	map	NOUN
ap-3457	203	9	ξ	ξ	NOUN
ap-3457	203	10	:	:	PUNCT
ap-3457	203	11	rn	rn	PROPN
ap-3457	203	12	→	→	SYM
ap-3457	203	13	rn	rn	PROPN
ap-3457	203	14	,	,	PUNCT
ap-3457	203	15	ξ(x	ξ(x	NOUN
ap-3457	203	16	)	)	PUNCT
ap-3457	203	17	=	=	PRON
ap-3457	203	18	(	(	PUNCT
ap-3457	203	19	y1(x	y1(x	NOUN
ap-3457	203	20	)	)	PUNCT
ap-3457	203	21	,	,	PUNCT
ap-3457	203	22	.	.	PUNCT
ap-3457	203	23	.	.	PUNCT
ap-3457	204	1	.	.	PUNCT
ap-3457	205	1	,	,	PUNCT
ap-3457	205	2	yn(x	yn(x	PROPN
ap-3457	205	3	)	)	PUNCT
ap-3457	205	4	)	)	PUNCT
ap-3457	205	5	.	.	PUNCT
ap-3457	206	1	the	the	DET
ap-3457	206	2	map	map	NOUN
ap-3457	206	3	ξ	ξ	NOUN
ap-3457	206	4	is	be	AUX
ap-3457	206	5	used	use	VERB
ap-3457	206	6	to	to	PART
ap-3457	206	7	define	define	VERB
ap-3457	206	8	the	the	DET
ap-3457	206	9	integration	integration	NOUN
ap-3457	206	10	region	region	NOUN
ap-3457	206	11	ω	ω	PROPN
ap-3457	206	12	and	and	CCONJ
ap-3457	206	13	the	the	DET
ap-3457	206	14	sets	set	NOUN
ap-3457	206	15	of	of	ADP
ap-3457	206	16	nodes	node	NOUN
ap-3457	206	17	ωtm	ωtm	NOUN
ap-3457	206	18	,	,	PUNCT
ap-3457	206	19	t	t	PROPN
ap-3457	206	20	∈	∈	PROPN
ap-3457	206	21	{	{	PUNCT
ap-3457	206	22	0	0	NUM
ap-3457	206	23	,	,	PUNCT
ap-3457	206	24	1	1	NUM
ap-3457	206	25	,	,	PUNCT
ap-3457	206	26	s	s	X
ap-3457	206	27	,	,	PUNCT
ap-3457	206	28	l	l	NOUN
ap-3457	206	29	}	}	PUNCT
ap-3457	206	30	by	by	ADP
ap-3457	206	31	ω	ω	PROPN
ap-3457	206	32	≡	≡	PROPN
ap-3457	206	33	ξ(f	ξ(f	PROPN
ap-3457	206	34	1	1	NUM
ap-3457	206	35	)	)	PUNCT
ap-3457	206	36	,	,	PUNCT
ap-3457	206	37	ωtm	ωtm	PROPN
ap-3457	206	38	≡	≡	PROPN
ap-3457	206	39	ξ	ξ	PROPN
ap-3457	206	40	(	(	PUNCT
ap-3457	206	41	1	1	NUM
ap-3457	206	42	m	m	NOUN
ap-3457	206	43	+	+	ADJ
ap-3457	206	44	ht	ht	PROPN
ap-3457	206	45	p∨	p∨	PROPN
ap-3457	206	46	∩	∩	NOUN
ap-3457	206	47	f	f	PROPN
ap-3457	206	48	t	t	PROPN
ap-3457	206	49	)	)	PUNCT
ap-3457	206	50	.	.	PUNCT
ap-3457	207	1	let	let	VERB
ap-3457	207	2	stabw	stabw	ADV
ap-3457	207	3	aff	aff	PROPN
ap-3457	207	4	(	(	PUNCT
ap-3457	207	5	x	x	NOUN
ap-3457	207	6	)	)	PUNCT
ap-3457	207	7	≡	≡	PROPN
ap-3457	207	8	{	{	PUNCT
ap-3457	207	9	w	w	PROPN
ap-3457	207	10	∈w	∈w	PROPN
ap-3457	207	11	aff	aff	X
ap-3457	207	12	∣∣	∣∣	X
ap-3457	207	13	wx	wx	X
ap-3457	207	14	=	=	PUNCT
ap-3457	207	15	x	x	SYM
ap-3457	207	16	}	}	PUNCT
ap-3457	207	17	,	,	PUNCT
ap-3457	207	18	ε(x	ε(x	NOUN
ap-3457	207	19	)	)	PUNCT
ap-3457	207	20	≡	≡	PROPN
ap-3457	207	21	|w	|w	NOUN
ap-3457	207	22	|	|	ADV
ap-3457	207	23	|stabw	|stabw	VERB
ap-3457	207	24	aff	aff	PROPN
ap-3457	207	25	(	(	PUNCT
ap-3457	207	26	x)|	x)|	PROPN
ap-3457	207	27	,	,	PUNCT
ap-3457	207	28	and	and	CCONJ
ap-3457	207	29	define	define	VERB
ap-3457	207	30	a	a	DET
ap-3457	207	31	map	map	NOUN
ap-3457	207	32	ε̃	ε̃	PROPN
ap-3457	207	33	:	:	PUNCT
ap-3457	207	34	ω0	ω0	PROPN
ap-3457	207	35	m	m	VERB
ap-3457	207	36	→	→	SYM
ap-3457	207	37	n	n	X
ap-3457	207	38	by	by	ADP
ap-3457	207	39	ε̃(y	ε̃(y	ADJ
ap-3457	207	40	)	)	PUNCT
ap-3457	207	41	≡	≡	PROPN
ap-3457	207	42	ε(ξ−1	ε(ξ−1	PROPN
ap-3457	207	43	m	m	VERB
ap-3457	207	44	y	y	NOUN
ap-3457	207	45	)	)	PUNCT
ap-3457	207	46	,	,	PUNCT
ap-3457	207	47	ξm	ξm	PROPN
ap-3457	207	48	=	=	SYM
ap-3457	207	49	ξ	ξ	PROPN
ap-3457	207	50	�	�	PROPN
ap-3457	207	51	1	1	NUM
ap-3457	207	52	m+ht	m+ht	NOUN
ap-3457	207	53	p	p	NOUN
ap-3457	207	54	∨∩f	∨∩f	NOUN
ap-3457	207	55	0	0	PUNCT
ap-3457	207	56	.	.	PUNCT
ap-3457	208	1	denoting	denote	VERB
ap-3457	208	2	k(y1	k(y1	NOUN
ap-3457	208	3	,	,	PUNCT
ap-3457	208	4	.	.	PUNCT
ap-3457	208	5	.	.	PUNCT
ap-3457	209	1	.	.	PUNCT
ap-3457	210	1	,	,	PUNCT
ap-3457	210	2	yn	yn	PROPN
ap-3457	210	3	)	)	PUNCT
ap-3457	210	4	≡	≡	PROPN
ap-3457	210	5	√	√	PROPN
ap-3457	210	6	s%1s%1	s%1s%1	PROPN
ap-3457	210	7	,	,	PUNCT
ap-3457	210	8	the	the	DET
ap-3457	210	9	weight	weight	NOUN
ap-3457	210	10	functions	function	NOUN
ap-3457	210	11	are	be	AUX
ap-3457	210	12	given	give	VERB
ap-3457	210	13	by	by	ADP
ap-3457	210	14	st(y1	st(y1	ADJ
ap-3457	210	15	,	,	PUNCT
ap-3457	210	16	.	.	PUNCT
ap-3457	210	17	.	.	PUNCT
ap-3457	211	1	.	.	PUNCT
ap-3457	212	1	,	,	PUNCT
ap-3457	212	2	yn	yn	PROPN
ap-3457	212	3	)	)	PUNCT
ap-3457	212	4	≡	≡	PROPN
ap-3457	212	5	st%tst%t	st%tst%t	PROPN
ap-3457	212	6	,	,	PUNCT
ap-3457	212	7	wt(y1	wt(y1	PROPN
ap-3457	212	8	,	,	PUNCT
ap-3457	212	9	.	.	PUNCT
ap-3457	212	10	.	.	PUNCT
ap-3457	213	1	.	.	PUNCT
ap-3457	213	2	,	,	PUNCT
ap-3457	213	3	yn	yn	PROPN
ap-3457	213	4	)	)	PUNCT
ap-3457	213	5	≡	≡	PROPN
ap-3457	213	6	st(y1	st(y1	NOUN
ap-3457	213	7	,	,	PUNCT
ap-3457	213	8	.	.	PUNCT
ap-3457	213	9	.	.	PUNCT
ap-3457	214	1	.	.	PUNCT
ap-3457	215	1	,	,	PUNCT
ap-3457	215	2	yn	yn	PROPN
ap-3457	215	3	)	)	PUNCT
ap-3457	215	4	k(y1	k(y1	NOUN
ap-3457	215	5	,	,	PUNCT
ap-3457	215	6	.	.	PUNCT
ap-3457	215	7	.	.	PUNCT
ap-3457	216	1	.	.	PUNCT
ap-3457	217	1	,	,	PUNCT
ap-3457	217	2	yn	yn	PROPN
ap-3457	217	3	)	)	PUNCT
ap-3457	217	4	,	,	PUNCT
ap-3457	217	5	t	t	PROPN
ap-3457	217	6	∈	∈	PROPN
ap-3457	217	7	{	{	PUNCT
ap-3457	217	8	0	0	NUM
ap-3457	217	9	,	,	PUNCT
ap-3457	217	10	1	1	NUM
ap-3457	217	11	,	,	PUNCT
ap-3457	217	12	s	s	X
ap-3457	217	13	,	,	PUNCT
ap-3457	217	14	l	l	NOUN
ap-3457	217	15	}	}	PUNCT
ap-3457	217	16	.	.	PUNCT
ap-3457	218	1	206	206	NUM
ap-3457	218	2	vol	vol	NOUN
ap-3457	218	3	.	.	PUNCT
ap-3457	219	1	56	56	NUM
ap-3457	219	2	no	no	NOUN
ap-3457	219	3	.	.	PUNCT
ap-3457	220	1	3/2016	3/2016	NUM
ap-3457	220	2	on	on	ADP
ap-3457	220	3	cubature	cubature	ADJ
ap-3457	220	4	rules	rule	NOUN
ap-3457	220	5	associated	associate	VERB
ap-3457	220	6	to	to	ADP
ap-3457	220	7	weyl	weyl	PROPN
ap-3457	220	8	group	group	PROPN
ap-3457	220	9	orbit	orbit	NOUN
ap-3457	220	10	functions	function	NOUN
ap-3457	220	11	figure	figure	VERB
ap-3457	220	12	3	3	NUM
ap-3457	220	13	.	.	PUNCT
ap-3457	221	1	the	the	DET
ap-3457	221	2	fundamental	fundamental	ADJ
ap-3457	221	3	domain	domain	NOUN
ap-3457	221	4	f	f	X
ap-3457	221	5	=	=	SYM
ap-3457	221	6	f	f	PROPN
ap-3457	221	7	0	0	NUM
ap-3457	221	8	of	of	ADP
ap-3457	221	9	c2	c2	PROPN
ap-3457	221	10	is	be	AUX
ap-3457	221	11	depicted	depict	VERB
ap-3457	221	12	as	as	ADP
ap-3457	221	13	the	the	DET
ap-3457	221	14	triangle	triangle	NOUN
ap-3457	221	15	with	with	ADP
ap-3457	221	16	dashed	dash	VERB
ap-3457	221	17	boundary	boundary	PROPN
ap-3457	221	18	hs	hs	PROPN
ap-3457	221	19	and	and	CCONJ
ap-3457	221	20	dot	dot	NOUN
ap-3457	221	21	-	-	PUNCT
ap-3457	221	22	and	and	CCONJ
ap-3457	221	23	-	-	PUNCT
ap-3457	221	24	dashed	dash	VERB
ap-3457	221	25	boundary	boundary	ADJ
ap-3457	221	26	hl	hl	NOUN
ap-3457	221	27	.	.	PUNCT
ap-3457	222	1	the	the	DET
ap-3457	222	2	black	black	ADJ
ap-3457	222	3	dots	dot	NOUN
ap-3457	222	4	correspond	correspond	VERB
ap-3457	222	5	to	to	ADP
ap-3457	222	6	the	the	DET
ap-3457	222	7	points	point	NOUN
ap-3457	222	8	from	from	ADP
ap-3457	222	9	1	1	NUM
ap-3457	222	10	10p	10p	NUM
ap-3457	222	11	∨∩f	∨∩f	NOUN
ap-3457	222	12	.	.	PUNCT
ap-3457	223	1	the	the	DET
ap-3457	223	2	numbers	number	NOUN
ap-3457	223	3	1	1	NUM
ap-3457	223	4	,	,	PUNCT
ap-3457	223	5	2	2	NUM
ap-3457	223	6	,	,	PUNCT
ap-3457	223	7	4	4	NUM
ap-3457	223	8	of	of	ADP
ap-3457	223	9	the	the	DET
ap-3457	223	10	dots	dot	NOUN
ap-3457	223	11	are	be	AUX
ap-3457	223	12	the	the	DET
ap-3457	223	13	values	value	NOUN
ap-3457	223	14	of	of	ADP
ap-3457	223	15	ε(x	ε(x	NOUN
ap-3457	223	16	)	)	PUNCT
ap-3457	223	17	,	,	PUNCT
ap-3457	223	18	the	the	DET
ap-3457	223	19	inner	inner	ADJ
ap-3457	223	20	dots	dot	NOUN
ap-3457	223	21	have	have	VERB
ap-3457	223	22	ε(x	ε(x	NOUN
ap-3457	223	23	)	)	PUNCT
ap-3457	223	24	=	=	SYM
ap-3457	223	25	8	8	X
ap-3457	223	26	.	.	PUNCT
ap-3457	224	1	since	since	SCONJ
ap-3457	224	2	the	the	DET
ap-3457	224	3	products	product	NOUN
ap-3457	224	4	st%ts	st%ts	PROPN
ap-3457	224	5	t	t	PROPN
ap-3457	224	6	%	%	NOUN
ap-3457	224	7	t	t	PROPN
ap-3457	224	8	,	,	PUNCT
ap-3457	224	9	t	t	PROPN
ap-3457	224	10	∈	∈	PROPN
ap-3457	224	11	{	{	PUNCT
ap-3457	224	12	0	0	NUM
ap-3457	224	13	,	,	PUNCT
ap-3457	224	14	1	1	NUM
ap-3457	224	15	,	,	PUNCT
ap-3457	224	16	s	s	X
ap-3457	224	17	,	,	PUNCT
ap-3457	224	18	l	l	NOUN
ap-3457	224	19	}	}	PUNCT
ap-3457	224	20	are	be	AUX
ap-3457	224	21	w	w	ADP
ap-3457	224	22	invariant	invariant	ADJ
ap-3457	224	23	sums	sum	NOUN
ap-3457	224	24	of	of	ADP
ap-3457	224	25	exponentials	exponential	NOUN
ap-3457	224	26	from	from	ADP
ap-3457	224	27	c[p	c[p	PROPN
ap-3457	224	28	]	]	PUNCT
ap-3457	224	29	,	,	PUNCT
ap-3457	224	30	they	they	PRON
ap-3457	224	31	are	be	AUX
ap-3457	224	32	expressible	expressible	ADJ
ap-3457	224	33	as	as	ADP
ap-3457	224	34	functions	function	NOUN
ap-3457	224	35	in	in	ADP
ap-3457	224	36	polynomial	polynomial	ADJ
ap-3457	224	37	variables	variable	NOUN
ap-3457	224	38	y1	y1	PROPN
ap-3457	224	39	,	,	PUNCT
ap-3457	224	40	.	.	PUNCT
ap-3457	224	41	.	.	PUNCT
ap-3457	225	1	.	.	PUNCT
ap-3457	226	1	,	,	PUNCT
ap-3457	226	2	yn	yn	INTJ
ap-3457	226	3	.	.	PUNCT
ap-3457	227	1	in	in	ADP
ap-3457	227	2	[	[	X
ap-3457	227	3	15	15	NUM
ap-3457	227	4	,	,	PUNCT
ap-3457	227	5	26	26	NUM
ap-3457	227	6	,	,	PUNCT
ap-3457	227	7	27	27	NUM
ap-3457	227	8	]	]	PUNCT
ap-3457	227	9	is	be	AUX
ap-3457	227	10	shown	show	VERB
ap-3457	227	11	that	that	SCONJ
ap-3457	227	12	the	the	DET
ap-3457	227	13	following	follow	VERB
ap-3457	227	14	cubature	cubature	ADJ
ap-3457	227	15	formulas	formula	NOUN
ap-3457	227	16	are	be	AUX
ap-3457	227	17	exact	exact	ADJ
ap-3457	227	18	equalities	equality	NOUN
ap-3457	227	19	for	for	ADP
ap-3457	227	20	any	any	DET
ap-3457	227	21	m	m	NOUN
ap-3457	227	22	∈	∈	NOUN
ap-3457	227	23	n	n	NOUN
ap-3457	227	24	and	and	CCONJ
ap-3457	227	25	any	any	DET
ap-3457	227	26	polynomial	polynomial	ADJ
ap-3457	227	27	p	p	NOUN
ap-3457	227	28	which	which	PRON
ap-3457	227	29	satisfies	satisfy	VERB
ap-3457	227	30	the	the	DET
ap-3457	227	31	following	follow	VERB
ap-3457	227	32	constraints	constraint	NOUN
ap-3457	227	33	,	,	PUNCT
ap-3457	227	34	•	•	NUM
ap-3457	227	35	degm	degm	NOUN
ap-3457	227	36	p	p	NOUN
ap-3457	227	37	≤	≤	NUM
ap-3457	227	38	2	2	NUM
ap-3457	227	39	m	m	NOUN
ap-3457	227	40	−	−	NOUN
ap-3457	227	41	1	1	NUM
ap-3457	227	42	for	for	ADP
ap-3457	227	43	t	t	PROPN
ap-3457	227	44	∈	∈	PROPN
ap-3457	227	45	{	{	PUNCT
ap-3457	227	46	0	0	NUM
ap-3457	227	47	,	,	PUNCT
ap-3457	227	48	l	l	NOUN
ap-3457	227	49	}	}	PUNCT
ap-3457	227	50	,	,	PUNCT
ap-3457	227	51	•	•	NUM
ap-3457	227	52	degm	degm	NOUN
ap-3457	227	53	p	p	NOUN
ap-3457	227	54	≤	≤	NUM
ap-3457	227	55	2	2	NUM
ap-3457	227	56	m	m	NOUN
ap-3457	227	57	+	+	NOUN
ap-3457	227	58	1	1	NUM
ap-3457	227	59	for	for	ADP
ap-3457	227	60	t	t	PROPN
ap-3457	227	61	∈	∈	PROPN
ap-3457	227	62	{	{	PUNCT
ap-3457	227	63	1	1	NUM
ap-3457	227	64	,	,	PUNCT
ap-3457	227	65	s	s	PART
ap-3457	227	66	}	}	PUNCT
ap-3457	227	67	.	.	PUNCT
ap-3457	228	1	thus	thus	ADV
ap-3457	228	2	,	,	PUNCT
ap-3457	228	3	it	it	PRON
ap-3457	228	4	holds	hold	VERB
ap-3457	228	5	that∫	that∫	PROPN
ap-3457	228	6	ω	ω	PROPN
ap-3457	228	7	p(y)wt(y	p(y)wt(y	NOUN
ap-3457	228	8	)	)	PUNCT
ap-3457	228	9	dy	dy	NOUN
ap-3457	228	10	=	=	SYM
ap-3457	228	11	κ	κ	X
ap-3457	228	12	c|w	c|w	NOUN
ap-3457	229	1	|	|	ADV
ap-3457	229	2	(	(	PUNCT
ap-3457	229	3	2π	2π	NOUN
ap-3457	229	4	m	m	VERB
ap-3457	229	5	+	+	ADJ
ap-3457	229	6	ht	ht	X
ap-3457	229	7	)	)	PUNCT
ap-3457	229	8	n	n	CCONJ
ap-3457	229	9	∑	∑	ADP
ap-3457	229	10	y∈ωt	y∈ωt	PROPN
ap-3457	229	11	m	m	PROPN
ap-3457	229	12	ε̃(y)st(y)p(y	ε̃(y)st(y)p(y	NOUN
ap-3457	229	13	)	)	PUNCT
ap-3457	229	14	,	,	PUNCT
ap-3457	229	15	(	(	PUNCT
ap-3457	229	16	7	7	X
ap-3457	229	17	)	)	PUNCT
ap-3457	229	18	where	where	SCONJ
ap-3457	229	19	κ	κ	NOUN
ap-3457	229	20	=	=	SYM
ap-3457	229	21			PROPN
ap-3457	229	22	2−bn2	2−bn2	NUM
ap-3457	229	23	c	c	PROPN
ap-3457	229	24	for	for	ADP
ap-3457	229	25	an	an	DET
ap-3457	229	26	,	,	PUNCT
ap-3457	229	27	1	1	NUM
ap-3457	229	28	2	2	NUM
ap-3457	229	29	for	for	ADP
ap-3457	229	30	d2k+1	d2k+1	NOUN
ap-3457	229	31	,	,	PUNCT
ap-3457	229	32	1	1	NUM
ap-3457	229	33	4	4	NUM
ap-3457	229	34	for	for	ADP
ap-3457	229	35	e6	e6	NOUN
ap-3457	229	36	,	,	PUNCT
ap-3457	229	37	1	1	NUM
ap-3457	229	38	otherwise	otherwise	ADV
ap-3457	229	39	.	.	PUNCT
ap-3457	230	1	to	to	PART
ap-3457	230	2	numerically	numerically	ADV
ap-3457	230	3	compare	compare	VERB
ap-3457	230	4	the	the	DET
ap-3457	230	5	efficiency	efficiency	NOUN
ap-3457	230	6	of	of	ADP
ap-3457	230	7	these	these	DET
ap-3457	230	8	cubature	cubature	ADJ
ap-3457	230	9	formulas	formula	NOUN
ap-3457	230	10	,	,	PUNCT
ap-3457	230	11	we	we	PRON
ap-3457	230	12	may	may	AUX
ap-3457	230	13	consider	consider	VERB
ap-3457	230	14	an	an	DET
ap-3457	230	15	integrable	integrable	ADJ
ap-3457	230	16	function	function	NOUN
ap-3457	230	17	f	f	PROPN
ap-3457	230	18	,	,	PUNCT
ap-3457	230	19	such	such	ADJ
ap-3457	230	20	that	that	SCONJ
ap-3457	230	21	f	f	NOUN
ap-3457	230	22	/	/	SYM
ap-3457	230	23	wt	wt	PROPN
ap-3457	230	24	is	be	AUX
ap-3457	230	25	well	well	ADV
ap-3457	230	26	defined	define	VERB
ap-3457	230	27	on	on	ADP
ap-3457	230	28	ωtm	ωtm	NOUN
ap-3457	230	29	,	,	PUNCT
ap-3457	230	30	and	and	CCONJ
ap-3457	230	31	rewrite	rewrite	VERB
ap-3457	230	32	the	the	DET
ap-3457	230	33	cubatures	cubature	NOUN
ap-3457	230	34	(	(	PUNCT
ap-3457	230	35	7	7	NUM
ap-3457	230	36	)	)	PUNCT
ap-3457	230	37	in	in	ADP
ap-3457	230	38	the	the	DET
ap-3457	230	39	following	follow	VERB
ap-3457	230	40	form	form	NOUN
ap-3457	230	41	:	:	PUNCT
ap-3457	230	42	i(f	i(f	NOUN
ap-3457	230	43	)	)	PUNCT
ap-3457	231	1	≡	≡	PROPN
ap-3457	231	2	∫	∫	PROPN
ap-3457	231	3	ω	ω	PROPN
ap-3457	231	4	f(y	f(y	PROPN
ap-3457	231	5	)	)	PUNCT
ap-3457	232	1	dy	dy	NOUN
ap-3457	232	2	≈	≈	PROPN
ap-3457	232	3	itm	itm	PROPN
ap-3457	232	4	(	(	PUNCT
ap-3457	232	5	f	f	X
ap-3457	232	6	)	)	PUNCT
ap-3457	232	7	,	,	PUNCT
ap-3457	232	8	(	(	PUNCT
ap-3457	232	9	8)	8)	NUM
ap-3457	232	10	itm	itm	NOUN
ap-3457	232	11	(	(	PUNCT
ap-3457	232	12	f	f	X
ap-3457	232	13	)	)	PUNCT
ap-3457	232	14	≡	≡	PROPN
ap-3457	232	15	κ	κ	PROPN
ap-3457	232	16	c|w	c|w	PROPN
ap-3457	233	1	|	|	ADV
ap-3457	233	2	(	(	PUNCT
ap-3457	233	3	2π	2π	NOUN
ap-3457	233	4	m	m	VERB
ap-3457	233	5	+	+	ADJ
ap-3457	233	6	ht	ht	X
ap-3457	233	7	)	)	PUNCT
ap-3457	233	8	n	n	CCONJ
ap-3457	233	9	∑	∑	PROPN
ap-3457	233	10	y∈ωt	y∈ωt	PROPN
ap-3457	233	11	m	m	PROPN
ap-3457	233	12	ε̃(y)k(y)f(y	ε̃(y)k(y)f(y	NUM
ap-3457	233	13	)	)	PUNCT
ap-3457	233	14	.	.	PUNCT
ap-3457	234	1	3.2	3.2	NUM
ap-3457	234	2	.	.	PUNCT
ap-3457	235	1	cubature	cubature	ADJ
ap-3457	235	2	formulas	formula	NOUN
ap-3457	235	3	of	of	ADP
ap-3457	235	4	c2	c2	PROPN
ap-3457	235	5	in	in	ADP
ap-3457	235	6	this	this	DET
ap-3457	235	7	section	section	NOUN
ap-3457	235	8	,	,	PUNCT
ap-3457	235	9	the	the	DET
ap-3457	235	10	general	general	ADJ
ap-3457	235	11	cubature	cubature	NOUN
ap-3457	235	12	formulas	formula	NOUN
ap-3457	235	13	(	(	PUNCT
ap-3457	235	14	8)	8)	NUM
ap-3457	235	15	are	be	AUX
ap-3457	235	16	specialized	specialize	VERB
ap-3457	235	17	and	and	CCONJ
ap-3457	235	18	tested	test	VERB
ap-3457	235	19	on	on	ADP
ap-3457	235	20	model	model	NOUN
ap-3457	235	21	examples	example	NOUN
ap-3457	235	22	for	for	ADP
ap-3457	235	23	the	the	DET
ap-3457	235	24	case	case	NOUN
ap-3457	235	25	of	of	ADP
ap-3457	235	26	algebra	algebra	PROPN
ap-3457	235	27	c2	c2	PROPN
ap-3457	235	28	.	.	PUNCT
ap-3457	236	1	the	the	DET
ap-3457	236	2	region	region	NOUN
ap-3457	236	3	f	f	PROPN
ap-3457	236	4	t	t	PROPN
ap-3457	236	5	of	of	ADP
ap-3457	236	6	c2	c2	PROPN
ap-3457	236	7	with	with	ADP
ap-3457	236	8	the	the	DET
ap-3457	236	9	points	point	NOUN
ap-3457	236	10	from	from	ADP
ap-3457	236	11	1	1	NUM
ap-3457	236	12	10p	10p	NOUN
ap-3457	236	13	∨	∨	NUM
ap-3457	236	14	∩	∩	NOUN
ap-3457	236	15	f	f	PROPN
ap-3457	236	16	t	t	PROPN
ap-3457	236	17	is	be	AUX
ap-3457	236	18	depicted	depict	VERB
ap-3457	236	19	in	in	ADP
ap-3457	236	20	fig	fig	NOUN
ap-3457	236	21	.	.	PUNCT
ap-3457	237	1	3	3	NUM
ap-3457	237	2	,	,	PUNCT
ap-3457	237	3	whereas	whereas	SCONJ
ap-3457	237	4	the	the	DET
ap-3457	237	5	corresponding	corresponding	ADJ
ap-3457	237	6	integration	integration	NOUN
ap-3457	237	7	region	region	NOUN
ap-3457	237	8	ω	ω	PROPN
ap-3457	237	9	with	with	ADP
ap-3457	237	10	the	the	DET
ap-3457	237	11	transformed	transform	VERB
ap-3457	237	12	grid	grid	NOUN
ap-3457	237	13	points	point	NOUN
ap-3457	237	14	is	be	AUX
ap-3457	237	15	depicted	depict	VERB
ap-3457	237	16	in	in	ADP
ap-3457	237	17	fig	fig	NOUN
ap-3457	237	18	.	.	PUNCT
ap-3457	238	1	4	4	X
ap-3457	238	2	.	.	X
ap-3457	238	3	note	note	VERB
ap-3457	238	4	that	that	DET
ap-3457	238	5	figure	figure	NOUN
ap-3457	238	6	4	4	NUM
ap-3457	238	7	.	.	PUNCT
ap-3457	239	1	the	the	DET
ap-3457	239	2	integration	integration	NOUN
ap-3457	239	3	region	region	NOUN
ap-3457	239	4	ω	ω	PROPN
ap-3457	239	5	of	of	ADP
ap-3457	239	6	c2	c2	PROPN
ap-3457	239	7	contains	contain	VERB
ap-3457	239	8	the	the	DET
ap-3457	239	9	points	point	NOUN
ap-3457	239	10	of	of	ADP
ap-3457	239	11	the	the	DET
ap-3457	239	12	grid	grid	NOUN
ap-3457	239	13	ω0	ω0	ADP
ap-3457	239	14	10	10	NUM
ap-3457	239	15	.	.	PUNCT
ap-3457	240	1	the	the	DET
ap-3457	240	2	inner	inner	ADJ
ap-3457	240	3	points	point	NOUN
ap-3457	240	4	of	of	ADP
ap-3457	240	5	ω	ω	PROPN
ap-3457	240	6	corresponds	correspond	NOUN
ap-3457	240	7	to	to	ADP
ap-3457	240	8	the	the	DET
ap-3457	240	9	grid	grid	NOUN
ap-3457	240	10	ω1	ω1	PROPN
ap-3457	240	11	6	6	NUM
ap-3457	240	12	,	,	PUNCT
ap-3457	240	13	the	the	DET
ap-3457	240	14	points	point	NOUN
ap-3457	240	15	not	not	PART
ap-3457	240	16	lying	lie	VERB
ap-3457	240	17	on	on	ADP
ap-3457	240	18	the	the	DET
ap-3457	240	19	dashed	dash	VERB
ap-3457	240	20	boundary	boundary	ADJ
ap-3457	240	21	corresponds	correspond	NOUN
ap-3457	240	22	to	to	ADP
ap-3457	240	23	the	the	DET
ap-3457	240	24	grid	grid	NOUN
ap-3457	240	25	ωs8	ωs8	PROPN
ap-3457	240	26	and	and	CCONJ
ap-3457	240	27	finally	finally	ADV
ap-3457	240	28	,	,	PUNCT
ap-3457	240	29	the	the	DET
ap-3457	240	30	points	point	NOUN
ap-3457	240	31	not	not	PART
ap-3457	240	32	lying	lie	VERB
ap-3457	240	33	on	on	ADP
ap-3457	240	34	the	the	DET
ap-3457	240	35	dot	dot	NOUN
ap-3457	240	36	-	-	PUNCT
ap-3457	240	37	and	and	CCONJ
ap-3457	240	38	-	-	PUNCT
ap-3457	240	39	dashed	dash	VERB
ap-3457	240	40	boundary	boundary	ADJ
ap-3457	240	41	corresponds	correspond	NOUN
ap-3457	240	42	to	to	ADP
ap-3457	240	43	the	the	DET
ap-3457	240	44	grid	grid	NOUN
ap-3457	240	45	ωl	ωl	ADP
ap-3457	240	46	8	8	NUM
ap-3457	240	47	.	.	PUNCT
ap-3457	241	1	the	the	DET
ap-3457	241	2	numbers	number	NOUN
ap-3457	241	3	1	1	NUM
ap-3457	241	4	,	,	PUNCT
ap-3457	241	5	2	2	NUM
ap-3457	241	6	,	,	PUNCT
ap-3457	241	7	4	4	NUM
ap-3457	241	8	are	be	AUX
ap-3457	241	9	the	the	DET
ap-3457	241	10	values	value	NOUN
ap-3457	241	11	of	of	ADP
ap-3457	241	12	ε̃(y	ε̃(y	PROPN
ap-3457	241	13	)	)	PUNCT
ap-3457	241	14	,	,	PUNCT
ap-3457	241	15	the	the	DET
ap-3457	241	16	inner	inner	ADJ
ap-3457	241	17	dots	dot	NOUN
ap-3457	241	18	have	have	VERB
ap-3457	241	19	ε̃(y	ε̃(y	ADJ
ap-3457	241	20	)	)	PUNCT
ap-3457	241	21	=	=	SYM
ap-3457	241	22	8	8	X
ap-3457	241	23	.	.	PUNCT
ap-3457	242	1	the	the	DET
ap-3457	242	2	numbers	number	NOUN
ap-3457	242	3	of	of	ADP
ap-3457	242	4	points	point	NOUN
ap-3457	242	5	in	in	ADP
ap-3457	242	6	grids	grid	NOUN
ap-3457	242	7	ωtm	ωtm	NOUN
ap-3457	242	8	,	,	PUNCT
ap-3457	242	9	t	t	PROPN
ap-3457	242	10	∈	∈	PROPN
ap-3457	242	11	{	{	PUNCT
ap-3457	242	12	0	0	NUM
ap-3457	242	13	,	,	PUNCT
ap-3457	242	14	1	1	NUM
ap-3457	242	15	,	,	PUNCT
ap-3457	242	16	s	s	X
ap-3457	242	17	,	,	PUNCT
ap-3457	242	18	l	l	NOUN
ap-3457	242	19	}	}	PUNCT
ap-3457	242	20	are	be	AUX
ap-3457	242	21	the	the	DET
ap-3457	242	22	same	same	ADJ
ap-3457	242	23	.	.	PUNCT
ap-3457	243	1	fixing	fix	VERB
ap-3457	243	2	the	the	DET
ap-3457	243	3	basis	basis	NOUN
ap-3457	243	4	x	x	NOUN
ap-3457	243	5	=	=	SYM
ap-3457	243	6	b1ω	b1ω	PUNCT
ap-3457	243	7	∨	∨	NUM
ap-3457	243	8	1	1	NUM
ap-3457	243	9	+	+	NUM
ap-3457	243	10	b2ω	b2ω	NOUN
ap-3457	243	11	∨	∨	NUM
ap-3457	243	12	2	2	NUM
ap-3457	243	13	results	result	NOUN
ap-3457	243	14	in	in	ADP
ap-3457	243	15	the	the	DET
ap-3457	243	16	polynomial	polynomial	ADJ
ap-3457	243	17	variables	variable	NOUN
ap-3457	243	18	expressed	express	VERB
ap-3457	243	19	as	as	ADP
ap-3457	243	20	y1	y1	NOUN
ap-3457	243	21	=	=	SYM
ap-3457	243	22	2	2	NUM
ap-3457	243	23	(	(	PUNCT
ap-3457	243	24	cosπ(2b1	cosπ(2b1	ADP
ap-3457	243	25	+	+	NUM
ap-3457	243	26	b2	b2	NOUN
ap-3457	243	27	)	)	PUNCT
ap-3457	243	28	+	+	NUM
ap-3457	243	29	cosπb2	cosπb2	ADJ
ap-3457	243	30	)	)	PUNCT
ap-3457	243	31	,	,	PUNCT
ap-3457	243	32	y2	y2	NOUN
ap-3457	243	33	=	=	SYM
ap-3457	243	34	2	2	NUM
ap-3457	243	35	(	(	PUNCT
ap-3457	243	36	cos	cos	PROPN
ap-3457	243	37	2π(b1	2π(b1	NUM
ap-3457	243	38	+	+	CCONJ
ap-3457	243	39	b2	b2	NOUN
ap-3457	243	40	)	)	PUNCT
ap-3457	244	1	+	+	CCONJ
ap-3457	244	2	cos	cos	PROPN
ap-3457	244	3	2πb1	2πb1	NUM
ap-3457	244	4	)	)	PUNCT
ap-3457	244	5	.	.	PUNCT
ap-3457	245	1	formula	formula	NOUN
ap-3457	245	2	(	(	PUNCT
ap-3457	245	3	8)	8)	NUM
ap-3457	245	4	specializes	specialize	VERB
ap-3457	245	5	into	into	ADP
ap-3457	245	6	itm	itm	PROPN
ap-3457	245	7	(	(	PUNCT
ap-3457	245	8	f	f	X
ap-3457	245	9	)	)	PUNCT
ap-3457	246	1	=	=	PUNCT
ap-3457	246	2	π2	π2	NOUN
ap-3457	246	3	4(m	4(m	NUM
ap-3457	246	4	+	+	CCONJ
ap-3457	246	5	ht)2	ht)2	PROPN
ap-3457	246	6	∑	∑	PROPN
ap-3457	246	7	y∈ωt	y∈ωt	PROPN
ap-3457	246	8	m	m	PROPN
ap-3457	246	9	ε̃(y)k(y)f(y	ε̃(y)k(y)f(y	NUM
ap-3457	246	10	)	)	PUNCT
ap-3457	246	11	,	,	PUNCT
ap-3457	246	12	(	(	PUNCT
ap-3457	246	13	9	9	X
ap-3457	246	14	)	)	PUNCT
ap-3457	247	1	where	where	SCONJ
ap-3457	247	2	:	:	PUNCT
ap-3457	247	3	•	•	NUM
ap-3457	247	4	m	m	VERB
ap-3457	247	5	∈	∈	NOUN
ap-3457	247	6	n	n	NOUN
ap-3457	247	7	is	be	AUX
ap-3457	247	8	arbitrary	arbitrary	ADJ
ap-3457	247	9	;	;	PUNCT
ap-3457	247	10	•	•	NUM
ap-3457	247	11	h0	h0	NOUN
ap-3457	247	12	=	=	SYM
ap-3457	247	13	1	1	NUM
ap-3457	247	14	,	,	PUNCT
ap-3457	247	15	h1	h1	NOUN
ap-3457	247	16	=	=	SYM
ap-3457	247	17	4	4	NUM
ap-3457	247	18	and	and	CCONJ
ap-3457	247	19	hs	hs	NOUN
ap-3457	247	20	=	=	NOUN
ap-3457	247	21	hl	hl	NOUN
ap-3457	247	22	=	=	NOUN
ap-3457	247	23	2	2	NUM
ap-3457	247	24	;	;	PUNCT
ap-3457	247	25	•	•	NUM
ap-3457	247	26	the	the	DET
ap-3457	247	27	integration	integration	NOUN
ap-3457	247	28	region	region	NOUN
ap-3457	247	29	ω	ω	PROPN
ap-3457	247	30	,	,	PUNCT
ap-3457	247	31	depicted	depict	VERB
ap-3457	247	32	in	in	ADP
ap-3457	247	33	fig	fig	NOUN
ap-3457	247	34	.	.	PUNCT
ap-3457	248	1	4	4	NUM
ap-3457	248	2	,	,	PUNCT
ap-3457	248	3	is	be	AUX
ap-3457	248	4	bounded	bound	VERB
ap-3457	248	5	by	by	ADP
ap-3457	248	6	two	two	NUM
ap-3457	248	7	lines	line	NOUN
ap-3457	248	8	y2	y2	NOUN
ap-3457	248	9	=	=	SYM
ap-3457	248	10	±y1−4	±y1−4	PROPN
ap-3457	248	11	and	and	CCONJ
ap-3457	249	1	the	the	DET
ap-3457	249	2	parabola	parabola	PROPN
ap-3457	249	3	y2	y2	PROPN
ap-3457	249	4	=	=	PUNCT
ap-3457	249	5	y2	y2	NOUN
ap-3457	250	1	1	1	NUM
ap-3457	250	2	4	4	NUM
ap-3457	250	3	;	;	PUNCT
ap-3457	250	4	•	•	NUM
ap-3457	250	5	the	the	DET
ap-3457	250	6	finite	finite	PROPN
ap-3457	250	7	grid	grid	NOUN
ap-3457	250	8	ωt	ωt	ADP
ap-3457	250	9	m	m	PROPN
ap-3457	250	10	,	,	PUNCT
ap-3457	250	11	depicted	depict	VERB
ap-3457	250	12	for	for	ADP
ap-3457	250	13	m	m	NOUN
ap-3457	250	14	=	=	NOUN
ap-3457	250	15	10	10	NUM
ap-3457	250	16	in	in	ADP
ap-3457	250	17	fig	fig	NOUN
ap-3457	250	18	.	.	PUNCT
ap-3457	251	1	4	4	NUM
ap-3457	251	2	)	)	PUNCT
ap-3457	252	1	,	,	PUNCT
ap-3457	252	2	consists	consist	VERB
ap-3457	252	3	of	of	ADP
ap-3457	252	4	points	point	NOUN
ap-3457	252	5	(	(	PUNCT
ap-3457	252	6	y1(x	y1(x	NOUN
ap-3457	252	7	)	)	PUNCT
ap-3457	252	8	,	,	PUNCT
ap-3457	252	9	y2(x	y2(x	PROPN
ap-3457	252	10	)	)	PUNCT
ap-3457	252	11	)	)	PUNCT
ap-3457	252	12	where	where	SCONJ
ap-3457	252	13	x	x	X
ap-3457	252	14	=	=	SYM
ap-3457	252	15	st1	st1	PROPN
ap-3457	252	16	m+htω	m+htω	PROPN
ap-3457	252	17	∨	∨	NUM
ap-3457	252	18	1	1	NUM
ap-3457	252	19	+	+	CCONJ
ap-3457	252	20	st2	st2	NOUN
ap-3457	252	21	m+htω	m+htω	PROPN
ap-3457	252	22	∨	∨	NUM
ap-3457	252	23	2	2	NUM
ap-3457	252	24	with	with	ADP
ap-3457	252	25	sti	sti	PROPN
ap-3457	252	26	satisfying	satisfy	VERB
ap-3457	252	27	s0	s0	PROPN
ap-3457	252	28	i	i	PROPN
ap-3457	252	29	∈	∈	PROPN
ap-3457	252	30	z≥0	z≥0	PROPN
ap-3457	252	31	,	,	PUNCT
ap-3457	252	32	2s0	2s0	NUM
ap-3457	252	33	1	1	NUM
ap-3457	252	34	+	+	NUM
ap-3457	252	35	s0	s0	PROPN
ap-3457	252	36	2	2	NUM
ap-3457	252	37	≤m	≤m	NOUN
ap-3457	252	38	,	,	PUNCT
ap-3457	252	39	s1	s1	PROPN
ap-3457	252	40	i	i	PROPN
ap-3457	252	41	∈	∈	PROPN
ap-3457	252	42	z>0	z>0	NOUN
ap-3457	252	43	,	,	PUNCT
ap-3457	252	44	2s1	2s1	NUM
ap-3457	252	45	1	1	NUM
ap-3457	252	46	+	+	NUM
ap-3457	252	47	s1	s1	NOUN
ap-3457	252	48	2	2	NUM
ap-3457	252	49	<	<	X
ap-3457	252	50	m	m	PROPN
ap-3457	252	51	+	+	NOUN
ap-3457	252	52	4	4	NUM
ap-3457	252	53	,	,	PUNCT
ap-3457	252	54	ss1	ss1	PROPN
ap-3457	252	55	∈	∈	PROPN
ap-3457	252	56	z≥0	z≥0	PROPN
ap-3457	252	57	,	,	PUNCT
ap-3457	252	58	ss2	ss2	PROPN
ap-3457	252	59	∈	∈	PROPN
ap-3457	252	60	z>0	z>0	NOUN
ap-3457	252	61	,	,	PUNCT
ap-3457	252	62	2ss1	2ss1	NUM
ap-3457	252	63	+	+	CCONJ
ap-3457	252	64	ss2	ss2	PROPN
ap-3457	252	65	≤m	≤m	PROPN
ap-3457	252	66	+	+	CCONJ
ap-3457	252	67	2	2	NUM
ap-3457	252	68	,	,	PUNCT
ap-3457	252	69	sl1	sl1	PROPN
ap-3457	252	70	∈	∈	PROPN
ap-3457	252	71	z>0	z>0	PROPN
ap-3457	252	72	,	,	PUNCT
ap-3457	252	73	sl2	sl2	PROPN
ap-3457	252	74	∈	∈	PROPN
ap-3457	252	75	z≥0	z≥0	PROPN
ap-3457	252	76	,	,	PUNCT
ap-3457	252	77	2sl1	2sl1	NUM
ap-3457	252	78	+	+	CCONJ
ap-3457	252	79	sl2	sl2	PROPN
ap-3457	252	80	<	<	X
ap-3457	252	81	m	m	PROPN
ap-3457	252	82	+	+	ADJ
ap-3457	252	83	2	2	NUM
ap-3457	252	84	;	;	PUNCT
ap-3457	252	85	•	•	NUM
ap-3457	252	86	the	the	DET
ap-3457	252	87	weight	weight	NOUN
ap-3457	252	88	function	function	NOUN
ap-3457	252	89	k	k	PROPN
ap-3457	252	90	becomes	become	VERB
ap-3457	252	91	k(y1	k(y1	NOUN
ap-3457	252	92	,	,	PUNCT
ap-3457	252	93	y2	y2	NOUN
ap-3457	252	94	)	)	PUNCT
ap-3457	253	1	=	=	SYM
ap-3457	254	1	√	√	PROPN
ap-3457	255	1	(	(	PUNCT
ap-3457	255	2	y2	y2	PROPN
ap-3457	255	3	1	1	NUM
ap-3457	255	4	−	−	PROPN
ap-3457	255	5	4y2)((y2	4y2)((y2	PROPN
ap-3457	255	6	+	+	SYM
ap-3457	255	7	4)2	4)2	PROPN
ap-3457	255	8	−	−	NOUN
ap-3457	255	9	4y2	4y2	NUM
ap-3457	255	10	1	1	NUM
ap-3457	255	11	)	)	PUNCT
ap-3457	255	12	;	;	PUNCT
ap-3457	255	13	•	•	ADP
ap-3457	255	14	the	the	DET
ap-3457	255	15	weight	weight	NOUN
ap-3457	255	16	function	function	NOUN
ap-3457	255	17	ε̃	ε̃	PROPN
ap-3457	255	18	is	be	AUX
ap-3457	255	19	equal	equal	ADJ
ap-3457	255	20	to	to	ADP
ap-3457	255	21	ε̃(y	ε̃(y	ADJ
ap-3457	255	22	)	)	PUNCT
ap-3457	255	23	=	=	PUNCT
ap-3457	256	1			PROPN
ap-3457	256	2	1	1	NUM
ap-3457	257	1	if	if	SCONJ
ap-3457	257	2	(	(	PUNCT
ap-3457	257	3	y1	y1	INTJ
ap-3457	257	4	,	,	PUNCT
ap-3457	257	5	y2	y2	NOUN
ap-3457	257	6	)	)	PUNCT
ap-3457	257	7	=	=	SYM
ap-3457	257	8	(	(	PUNCT
ap-3457	257	9	±4	±4	NOUN
ap-3457	257	10	,	,	PUNCT
ap-3457	257	11	4	4	NUM
ap-3457	257	12	)	)	PUNCT
ap-3457	257	13	,	,	PUNCT
ap-3457	257	14	2	2	NUM
ap-3457	257	15	if	if	SCONJ
ap-3457	257	16	(	(	PUNCT
ap-3457	257	17	y1	y1	INTJ
ap-3457	257	18	,	,	PUNCT
ap-3457	257	19	y2	y2	NOUN
ap-3457	257	20	)	)	PUNCT
ap-3457	257	21	=	=	SYM
ap-3457	257	22	(	(	PUNCT
ap-3457	257	23	0,−4	0,−4	NUM
ap-3457	257	24	)	)	PUNCT
ap-3457	257	25	,	,	PUNCT
ap-3457	257	26	8	8	NUM
ap-3457	257	27	if	if	SCONJ
ap-3457	257	28	(	(	PUNCT
ap-3457	257	29	y1	y1	INTJ
ap-3457	257	30	,	,	PUNCT
ap-3457	257	31	y2	y2	PROPN
ap-3457	257	32	)	)	PUNCT
ap-3457	257	33	is	be	AUX
ap-3457	257	34	an	an	DET
ap-3457	257	35	inner	inner	ADJ
ap-3457	257	36	point	point	NOUN
ap-3457	257	37	of	of	ADP
ap-3457	257	38	ω	ω	NUM
ap-3457	257	39	,	,	PUNCT
ap-3457	257	40	4	4	NUM
ap-3457	257	41	otherwise	otherwise	ADV
ap-3457	257	42	.	.	PUNCT
ap-3457	258	1	207	207	NUM
ap-3457	258	2	l.	l.	PROPN
ap-3457	258	3	háková	háková	PROPN
ap-3457	258	4	,	,	PUNCT
ap-3457	258	5	j.	j.	PROPN
ap-3457	258	6	hrivnák	hrivnák	PROPN
ap-3457	258	7	,	,	PUNCT
ap-3457	258	8	l.	l.	PROPN
ap-3457	258	9	motlochová	motlochová	PROPN
ap-3457	258	10	acta	acta	PROPN
ap-3457	258	11	polytechnica	polytechnica	PROPN
ap-3457	258	12	figure	figure	NOUN
ap-3457	258	13	5	5	NUM
ap-3457	258	14	.	.	PUNCT
ap-3457	259	1	the	the	DET
ap-3457	259	2	graphs	graph	NOUN
ap-3457	259	3	of	of	ADP
ap-3457	259	4	error	error	NOUN
ap-3457	259	5	values	value	NOUN
ap-3457	259	6	|1	|1	PUNCT
ap-3457	259	7	−	−	X
ap-3457	259	8	itm	itm	NOUN
ap-3457	259	9	(	(	PUNCT
ap-3457	259	10	fi)|	fi)|	NOUN
ap-3457	259	11	of	of	ADP
ap-3457	259	12	the	the	DET
ap-3457	259	13	integral	integral	ADJ
ap-3457	259	14	i(fi	i(fi	NOUN
ap-3457	259	15	)	)	PUNCT
ap-3457	259	16	=	=	SYM
ap-3457	259	17	1	1	NUM
ap-3457	259	18	and	and	CCONJ
ap-3457	259	19	its	its	PRON
ap-3457	259	20	estimations	estimation	NOUN
ap-3457	259	21	itm	itm	NOUN
ap-3457	259	22	(	(	PUNCT
ap-3457	259	23	fi	fi	NOUN
ap-3457	259	24	)	)	PUNCT
ap-3457	259	25	,	,	PUNCT
ap-3457	259	26	m	m	VERB
ap-3457	259	27	=	=	NOUN
ap-3457	259	28	10	10	NUM
ap-3457	259	29	,	,	PUNCT
ap-3457	259	30	15	15	NUM
ap-3457	259	31	,	,	PUNCT
ap-3457	259	32	20	20	NUM
ap-3457	259	33	,	,	PUNCT
ap-3457	259	34	.	.	PUNCT
ap-3457	259	35	.	.	PUNCT
ap-3457	260	1	.	.	PUNCT
ap-3457	261	1	,	,	PUNCT
ap-3457	261	2	195	195	NUM
ap-3457	261	3	given	give	VERB
ap-3457	261	4	by	by	ADP
ap-3457	261	5	(	(	PUNCT
ap-3457	261	6	8)	8)	NUM
ap-3457	261	7	.	.	PUNCT
ap-3457	262	1	the	the	DET
ap-3457	262	2	values	value	NOUN
ap-3457	262	3	for	for	ADP
ap-3457	262	4	t	t	NOUN
ap-3457	262	5	=	=	SYM
ap-3457	262	6	0	0	NUM
ap-3457	262	7	,	,	PUNCT
ap-3457	262	8	1	1	NUM
ap-3457	262	9	,	,	PUNCT
ap-3457	262	10	s	s	VERB
ap-3457	262	11	are	be	AUX
ap-3457	262	12	depicted	depict	VERB
ap-3457	262	13	as	as	ADP
ap-3457	262	14	circles	circle	NOUN
ap-3457	262	15	,	,	PUNCT
ap-3457	262	16	“	"	PUNCT
ap-3457	262	17	+	+	ADJ
ap-3457	262	18	”	"	PUNCT
ap-3457	262	19	signs	sign	NOUN
ap-3457	262	20	and	and	CCONJ
ap-3457	262	21	diamonds	diamond	NOUN
ap-3457	262	22	,	,	PUNCT
ap-3457	262	23	respectively	respectively	ADV
ap-3457	262	24	.	.	PUNCT
ap-3457	263	1	for	for	ADP
ap-3457	263	2	the	the	DET
ap-3457	263	3	purpose	purpose	NOUN
ap-3457	263	4	of	of	ADP
ap-3457	263	5	numerical	numerical	ADJ
ap-3457	263	6	tests	test	NOUN
ap-3457	263	7	and	and	CCONJ
ap-3457	263	8	comparison	comparison	NOUN
ap-3457	263	9	,	,	PUNCT
ap-3457	263	10	we	we	PRON
ap-3457	263	11	choose	choose	VERB
ap-3457	263	12	f1(y1	f1(y1	NOUN
ap-3457	263	13	,	,	PUNCT
ap-3457	263	14	y2	y2	NOUN
ap-3457	263	15	)	)	PUNCT
ap-3457	264	1	=	=	PRON
ap-3457	264	2	q1(y20	q1(y20	VERB
ap-3457	264	3	1	1	NUM
ap-3457	264	4	−	−	PROPN
ap-3457	264	5	y1y2	y1y2	PROPN
ap-3457	265	1	+	+	NUM
ap-3457	265	2	y20	y20	NOUN
ap-3457	265	3	2	2	NUM
ap-3457	265	4	)	)	PUNCT
ap-3457	265	5	,	,	PUNCT
ap-3457	265	6	f2(y1	f2(y1	X
ap-3457	265	7	,	,	PUNCT
ap-3457	265	8	y2	y2	NOUN
ap-3457	265	9	)	)	PUNCT
ap-3457	265	10	=	=	SYM
ap-3457	265	11	q2	q2	NOUN
ap-3457	265	12	(	(	PUNCT
ap-3457	265	13	e−(y2	e−(y2	NUM
ap-3457	265	14	1+(y2	1+(y2	PROPN
ap-3457	265	15	+	+	PROPN
ap-3457	265	16	1.8)2)/2×0.352	1.8)2)/2×0.352	PROPN
ap-3457	265	17	)	)	PUNCT
ap-3457	265	18	,	,	PUNCT
ap-3457	265	19	f3(y1	f3(y1	PROPN
ap-3457	265	20	,	,	PUNCT
ap-3457	265	21	y2	y2	NOUN
ap-3457	265	22	)	)	PUNCT
ap-3457	265	23	=	=	SYM
ap-3457	265	24	q3	q3	NOUN
ap-3457	265	25	1	1	NUM
ap-3457	265	26	+	+	CCONJ
ap-3457	265	27	y2	y2	PROPN
ap-3457	265	28	1	1	NUM
ap-3457	265	29	+	+	CCONJ
ap-3457	265	30	y2	y2	SYM
ap-3457	265	31	2	2	NUM
ap-3457	265	32	,	,	PUNCT
ap-3457	265	33	f4(y1	f4(y1	PROPN
ap-3457	265	34	,	,	PUNCT
ap-3457	265	35	y2	y2	NOUN
ap-3457	265	36	)	)	PUNCT
ap-3457	266	1	=	=	PUNCT
ap-3457	266	2	q4e	q4e	PROPN
ap-3457	266	3	y1+y2	y1+y2	PROPN
ap-3457	266	4	,	,	PUNCT
ap-3457	266	5	f5(y1	f5(y1	NOUN
ap-3457	266	6	,	,	PUNCT
ap-3457	266	7	y2	y2	NOUN
ap-3457	266	8	)	)	PUNCT
ap-3457	266	9	=	=	PRON
ap-3457	266	10	{	{	PUNCT
ap-3457	266	11	q5	q5	PROPN
ap-3457	266	12	if	if	SCONJ
ap-3457	266	13	y2	y2	PROPN
ap-3457	266	14	1	1	NUM
ap-3457	266	15	+	+	CCONJ
ap-3457	266	16	(	(	PUNCT
ap-3457	266	17	y2	y2	INTJ
ap-3457	266	18	+	+	CCONJ
ap-3457	266	19	1.5)2	1.5)2	NUM
ap-3457	266	20	≤	≤	NUM
ap-3457	266	21	1	1	NUM
ap-3457	266	22	,	,	PUNCT
ap-3457	266	23	0	0	NUM
ap-3457	266	24	otherwise	otherwise	ADV
ap-3457	266	25	.	.	PUNCT
ap-3457	267	1	as	as	ADP
ap-3457	267	2	model	model	NOUN
ap-3457	267	3	functions	function	NOUN
ap-3457	267	4	.	.	PUNCT
ap-3457	268	1	each	each	DET
ap-3457	268	2	value	value	NOUN
ap-3457	268	3	of	of	ADP
ap-3457	268	4	qi	qi	PROPN
ap-3457	268	5	∈	∈	PROPN
ap-3457	268	6	r	r	NOUN
ap-3457	268	7	is	be	AUX
ap-3457	268	8	set	set	VERB
ap-3457	268	9	to	to	PART
ap-3457	268	10	satisfy	satisfy	VERB
ap-3457	268	11	the	the	DET
ap-3457	268	12	condition	condition	NOUN
ap-3457	268	13	i(fi	i(fi	ADJ
ap-3457	268	14	)	)	PUNCT
ap-3457	268	15	=	=	SYM
ap-3457	268	16	1	1	X
ap-3457	268	17	.	.	X
ap-3457	268	18	fig	fig	NOUN
ap-3457	268	19	.	.	PUNCT
ap-3457	269	1	5	5	NUM
ap-3457	269	2	shows	show	NOUN
ap-3457	269	3	for	for	ADP
ap-3457	269	4	m	m	PROPN
ap-3457	269	5	=	=	NOUN
ap-3457	269	6	10	10	NUM
ap-3457	269	7	,	,	PUNCT
ap-3457	269	8	15	15	NUM
ap-3457	269	9	,	,	PUNCT
ap-3457	269	10	20	20	NUM
ap-3457	269	11	,	,	PUNCT
ap-3457	269	12	.	.	PUNCT
ap-3457	269	13	.	.	PUNCT
ap-3457	269	14	.	.	PUNCT
ap-3457	270	1	,	,	PUNCT
ap-3457	270	2	195	195	NUM
ap-3457	270	3	and	and	CCONJ
ap-3457	270	4	t	t	NOUN
ap-3457	270	5	∈	∈	PROPN
ap-3457	270	6	{	{	PUNCT
ap-3457	270	7	0	0	NUM
ap-3457	270	8	,	,	PUNCT
ap-3457	270	9	1	1	NUM
ap-3457	270	10	,	,	PUNCT
ap-3457	270	11	s	s	X
ap-3457	270	12	}	}	PUNCT
ap-3457	270	13	the	the	DET
ap-3457	270	14	graphs	graph	NOUN
ap-3457	270	15	of	of	ADP
ap-3457	270	16	the	the	DET
ap-3457	270	17	absolute	absolute	ADJ
ap-3457	270	18	value	value	NOUN
ap-3457	270	19	of	of	ADP
ap-3457	270	20	the	the	DET
ap-3457	270	21	difference	difference	NOUN
ap-3457	271	1	|1	|1	PRON
ap-3457	271	2	−	−	NOUN
ap-3457	271	3	itm	itm	NOUN
ap-3457	271	4	(	(	PUNCT
ap-3457	271	5	fi)|	fi)|	PROPN
ap-3457	271	6	.	.	PUNCT
ap-3457	271	7	note	note	VERB
ap-3457	271	8	that	that	SCONJ
ap-3457	271	9	the	the	DET
ap-3457	271	10	cases	case	NOUN
ap-3457	271	11	t	t	X
ap-3457	271	12	=	=	SYM
ap-3457	271	13	s	s	PROPN
ap-3457	271	14	and	and	CCONJ
ap-3457	271	15	t	t	NOUN
ap-3457	271	16	=	=	SYM
ap-3457	271	17	l	l	NOUN
ap-3457	271	18	give	give	VERB
ap-3457	271	19	the	the	DET
ap-3457	271	20	same	same	ADJ
ap-3457	271	21	results	result	NOUN
ap-3457	271	22	since	since	SCONJ
ap-3457	271	23	hs	hs	PROPN
ap-3457	271	24	=	=	NOUN
ap-3457	271	25	hl	hl	NOUN
ap-3457	271	26	=	=	SYM
ap-3457	271	27	2	2	NUM
ap-3457	271	28	and	and	CCONJ
ap-3457	271	29	k(y)fi(y	k(y)fi(y	NOUN
ap-3457	271	30	)	)	PUNCT
ap-3457	271	31	vanish	vanish	VERB
ap-3457	271	32	on	on	ADP
ap-3457	271	33	the	the	DET
ap-3457	271	34	boundary	boundary	NOUN
ap-3457	271	35	of	of	ADP
ap-3457	271	36	ω	ω	PROPN
ap-3457	271	37	.	.	PROPN
ap-3457	271	38	4	4	NUM
ap-3457	271	39	.	.	NOUN
ap-3457	271	40	clenshaw	clenshaw	ADJ
ap-3457	271	41	-	-	PUNCT
ap-3457	271	42	curtis	curtis	NOUN
ap-3457	271	43	cubature	cubature	NOUN
ap-3457	271	44	formulas	formula	VERB
ap-3457	271	45	4.1	4.1	NUM
ap-3457	271	46	.	.	PUNCT
ap-3457	272	1	clenshaw	clenshaw	ADJ
ap-3457	272	2	-	-	PUNCT
ap-3457	272	3	curtis	curtis	NOUN
ap-3457	272	4	method	method	NOUN
ap-3457	272	5	assuming	assume	VERB
ap-3457	272	6	that	that	SCONJ
ap-3457	272	7	we	we	PRON
ap-3457	272	8	have	have	VERB
ap-3457	272	9	an	an	DET
ap-3457	272	10	interpolation	interpolation	NOUN
ap-3457	272	11	of	of	ADP
ap-3457	272	12	a	a	DET
ap-3457	272	13	function	function	NOUN
ap-3457	272	14	f	f	X
ap-3457	272	15	in	in	ADP
ap-3457	272	16	terms	term	NOUN
ap-3457	272	17	of	of	ADP
ap-3457	272	18	p	p	NOUN
ap-3457	272	19	(	(	PUNCT
ap-3457	272	20	λ	λ	PROPN
ap-3457	272	21	,	,	PUNCT
ap-3457	272	22	kt	kt	PROPN
ap-3457	272	23	)	)	PUNCT
ap-3457	272	24	∈	∈	NOUN
ap-3457	272	25	πm	πm	ADP
ap-3457	272	26	in	in	ADP
ap-3457	272	27	points	point	NOUN
ap-3457	272	28	ωtm	ωtm	NOUN
ap-3457	272	29	,	,	PUNCT
ap-3457	272	30	i.e.	i.e.	X
ap-3457	272	31	f	f	X
ap-3457	272	32	≈	≈	PROPN
ap-3457	272	33	∑	∑	PUNCT
ap-3457	272	34	λ∈p+	λ∈p+	PUNCT
ap-3457	272	35	〈	〈	PROPN
ap-3457	272	36	λ	λ	PROPN
ap-3457	272	37	,	,	PUNCT
ap-3457	272	38	η〉≤m	η〉≤m	NOUN
ap-3457	272	39	btλp	btλp	NOUN
ap-3457	272	40	(	(	PUNCT
ap-3457	272	41	λ	λ	PROPN
ap-3457	272	42	,	,	PUNCT
ap-3457	272	43	kt	kt	PROPN
ap-3457	272	44	)	)	PUNCT
ap-3457	272	45	,	,	PUNCT
ap-3457	272	46	f(y	f(y	NOUN
ap-3457	272	47	)	)	PUNCT
ap-3457	272	48	=	=	PUNCT
ap-3457	272	49	∑	∑	PUNCT
ap-3457	272	50	λ∈p+	λ∈p+	PUNCT
ap-3457	272	51	〈	〈	PROPN
ap-3457	272	52	λ	λ	PROPN
ap-3457	272	53	,	,	PUNCT
ap-3457	272	54	η〉≤m	η〉≤m	NOUN
ap-3457	272	55	btλp	btλp	NOUN
ap-3457	272	56	(	(	PUNCT
ap-3457	272	57	λ	λ	PROPN
ap-3457	272	58	,	,	PUNCT
ap-3457	272	59	kt	kt	PROPN
ap-3457	272	60	;	;	PUNCT
ap-3457	272	61	y	y	PROPN
ap-3457	272	62	)	)	PUNCT
ap-3457	272	63	,	,	PUNCT
ap-3457	272	64	y	y	PROPN
ap-3457	272	65	∈	∈	PROPN
ap-3457	272	66	ωtm	ωtm	NOUN
ap-3457	272	67	,	,	PUNCT
ap-3457	272	68	we	we	PRON
ap-3457	272	69	estimate	estimate	VERB
ap-3457	272	70	a	a	DET
ap-3457	272	71	weighted	weight	VERB
ap-3457	272	72	integral	integral	ADJ
ap-3457	272	73	of	of	ADP
ap-3457	272	74	f	f	PROPN
ap-3457	272	75	with	with	ADP
ap-3457	272	76	a	a	DET
ap-3457	272	77	weight	weight	NOUN
ap-3457	272	78	function	function	NOUN
ap-3457	272	79	w	w	NOUN
ap-3457	272	80	over	over	ADP
ap-3457	272	81	a	a	DET
ap-3457	272	82	domain	domain	NOUN
ap-3457	273	1	d	d	X
ap-3457	273	2	⊂	⊂	PROPN
ap-3457	273	3	ω	ω	PROPN
ap-3457	273	4	by∑	by∑	PROPN
ap-3457	273	5	λ∈p+	λ∈p+	NUM
ap-3457	273	6	〈	〈	PROPN
ap-3457	273	7	λ	λ	PROPN
ap-3457	273	8	,	,	PUNCT
ap-3457	273	9	η〉≤m	η〉≤m	NOUN
ap-3457	273	10	btλ	btλ	NOUN
ap-3457	273	11	∫	∫	PROPN
ap-3457	274	1	d	d	X
ap-3457	274	2	p	p	X
ap-3457	274	3	(	(	PUNCT
ap-3457	274	4	λ	λ	PROPN
ap-3457	274	5	,	,	PUNCT
ap-3457	274	6	kt	kt	PROPN
ap-3457	274	7	;	;	PUNCT
ap-3457	274	8	y)w(y	y)w(y	NOUN
ap-3457	274	9	)	)	PUNCT
ap-3457	274	10	dy	dy	NOUN
ap-3457	274	11	.	.	PUNCT
ap-3457	275	1	such	such	ADJ
ap-3457	275	2	construction	construction	NOUN
ap-3457	275	3	of	of	ADP
ap-3457	275	4	the	the	DET
ap-3457	275	5	clenshaw	clenshaw	ADJ
ap-3457	275	6	-	-	PUNCT
ap-3457	275	7	curtis	curtis	NOUN
ap-3457	275	8	cubature	cubature	ADJ
ap-3457	275	9	rule	rule	NOUN
ap-3457	275	10	implies	imply	VERB
ap-3457	275	11	the	the	DET
ap-3457	275	12	exact	exact	ADJ
ap-3457	275	13	equality	equality	NOUN
ap-3457	275	14	for	for	ADP
ap-3457	275	15	any	any	DET
ap-3457	275	16	polynomial	polynomial	ADJ
ap-3457	275	17	f	f	PROPN
ap-3457	275	18	of	of	ADP
ap-3457	275	19	m	m	NOUN
ap-3457	275	20	-	-	PUNCT
ap-3457	275	21	degree	degree	NOUN
ap-3457	275	22	at	at	ADP
ap-3457	275	23	most	most	ADJ
ap-3457	275	24	m	m	VERB
ap-3457	275	25	.	.	PUNCT
ap-3457	276	1	denoting	denote	VERB
ap-3457	276	2	atλ(w	atλ(w	PROPN
ap-3457	276	3	)	)	PUNCT
ap-3457	276	4	≡	≡	PROPN
ap-3457	276	5	∫	∫	PROPN
ap-3457	277	1	d	d	X
ap-3457	277	2	p	p	X
ap-3457	277	3	(	(	PUNCT
ap-3457	277	4	λ	λ	PROPN
ap-3457	277	5	,	,	PUNCT
ap-3457	277	6	kt	kt	PROPN
ap-3457	277	7	;	;	PUNCT
ap-3457	277	8	y)w(y	y)w(y	NOUN
ap-3457	277	9	)	)	PUNCT
ap-3457	277	10	dy	dy	NOUN
ap-3457	277	11	,	,	PUNCT
ap-3457	277	12	the	the	DET
ap-3457	277	13	clenshaw	clenshaw	ADJ
ap-3457	277	14	-	-	PUNCT
ap-3457	277	15	curtis	curtis	NOUN
ap-3457	277	16	cubature	cubature	NOUN
ap-3457	277	17	is	be	AUX
ap-3457	277	18	thus	thus	ADV
ap-3457	277	19	given	give	VERB
ap-3457	277	20	by∫	by∫	PROPN
ap-3457	277	21	d	d	X
ap-3457	277	22	f(y)wt(y	f(y)wt(y	NOUN
ap-3457	277	23	)	)	PUNCT
ap-3457	277	24	dy	dy	NOUN
ap-3457	277	25	≈	≈	PROPN
ap-3457	277	26	∑	∑	PUNCT
ap-3457	277	27	λ∈p+	λ∈p+	PUNCT
ap-3457	277	28	〈	〈	PROPN
ap-3457	277	29	λ	λ	PROPN
ap-3457	277	30	,	,	PUNCT
ap-3457	277	31	η〉≤m	η〉≤m	NOUN
ap-3457	277	32	btλa	btλa	PROPN
ap-3457	277	33	t	t	PROPN
ap-3457	277	34	λ(w	λ(w	PROPN
ap-3457	277	35	)	)	PUNCT
ap-3457	277	36	,	,	PUNCT
ap-3457	277	37	where	where	SCONJ
ap-3457	277	38	the	the	DET
ap-3457	277	39	coefficients	coefficient	NOUN
ap-3457	277	40	btλ	btλ	PROPN
ap-3457	277	41	and	and	CCONJ
ap-3457	277	42	atλ(w	atλ(w	PROPN
ap-3457	277	43	)	)	PUNCT
ap-3457	277	44	need	need	VERB
ap-3457	277	45	to	to	PART
ap-3457	277	46	be	be	AUX
ap-3457	277	47	determined	determine	VERB
ap-3457	277	48	.	.	PUNCT
ap-3457	278	1	the	the	DET
ap-3457	278	2	coefficients	coefficient	NOUN
ap-3457	278	3	btλ	btλ	NOUN
ap-3457	278	4	are	be	AUX
ap-3457	278	5	readily	readily	ADV
ap-3457	278	6	obtained	obtain	VERB
ap-3457	278	7	using	use	VERB
ap-3457	278	8	the	the	DET
ap-3457	278	9	discrete	discrete	ADJ
ap-3457	278	10	orthogonality	orthogonality	NOUN
ap-3457	278	11	relations	relation	NOUN
ap-3457	278	12	of	of	ADP
ap-3457	278	13	the	the	DET
ap-3457	278	14	orbit	orbit	NOUN
ap-3457	278	15	functions	function	NOUN
ap-3457	278	16	from	from	ADP
ap-3457	278	17	[	[	X
ap-3457	278	18	13	13	NUM
ap-3457	278	19	,	,	PUNCT
ap-3457	278	20	14	14	NUM
ap-3457	278	21	]	]	PUNCT
ap-3457	278	22	.	.	PUNCT
ap-3457	279	1	denoting	denote	VERB
ap-3457	279	2	the	the	DET
ap-3457	279	3	order	order	NOUN
ap-3457	279	4	of	of	ADP
ap-3457	279	5	the	the	DET
ap-3457	279	6	stabilizer	stabilizer	NOUN
ap-3457	279	7	of	of	ADP
ap-3457	279	8	λ+%t	λ+%t	NOUN
ap-3457	279	9	m+ht	m+ht	NOUN
ap-3457	279	10	with	with	ADP
ap-3457	279	11	respect	respect	NOUN
ap-3457	279	12	to	to	ADP
ap-3457	279	13	the	the	DET
ap-3457	279	14	dual	dual	ADJ
ap-3457	279	15	affine	affine	NOUN
ap-3457	279	16	weyl	weyl	VERB
ap-3457	279	17	group	group	NOUN
ap-3457	279	18	by	by	ADP
ap-3457	279	19	h∨λ+%t	h∨λ+%t	PROPN
ap-3457	279	20	,	,	PUNCT
ap-3457	279	21	it	it	PRON
ap-3457	279	22	holds	hold	VERB
ap-3457	279	23	that	that	DET
ap-3457	279	24	btλ	btλ	NOUN
ap-3457	279	25	=	=	SYM
ap-3457	279	26	|stabw	|stabw	NOUN
ap-3457	279	27	(	(	PUNCT
ap-3457	279	28	λ+	λ+	NUM
ap-3457	279	29	%	%	NOUN
ap-3457	279	30	t)|2	t)|2	NOUN
ap-3457	279	31	c|w	c|w	PUNCT
ap-3457	280	1	|(m	|(m	PROPN
ap-3457	280	2	+	+	CCONJ
ap-3457	280	3	ht)nh∨λ+%t	ht)nh∨λ+%t	NOUN
ap-3457	280	4	×	×	NOUN
ap-3457	280	5	∑	∑	PUNCT
ap-3457	280	6	y∈ωtm	y∈ωtm	PROPN
ap-3457	280	7	ε̃(y)st(y)f(y)p	ε̃(y)st(y)f(y)p	PROPN
ap-3457	280	8	(	(	PUNCT
ap-3457	280	9	λ	λ	PROPN
ap-3457	280	10	,	,	PUNCT
ap-3457	280	11	kt	kt	PROPN
ap-3457	280	12	;	;	PUNCT
ap-3457	280	13	y	y	PROPN
ap-3457	280	14	)	)	PUNCT
ap-3457	280	15	.	.	PUNCT
ap-3457	281	1	(	(	PUNCT
ap-3457	281	2	10	10	NUM
ap-3457	281	3	)	)	SYM
ap-3457	281	4	208	208	NUM
ap-3457	281	5	vol	vol	NOUN
ap-3457	281	6	.	.	PUNCT
ap-3457	282	1	56	56	NUM
ap-3457	282	2	no	no	NOUN
ap-3457	282	3	.	.	PUNCT
ap-3457	283	1	3/2016	3/2016	NUM
ap-3457	283	2	on	on	ADP
ap-3457	283	3	cubature	cubature	ADJ
ap-3457	283	4	rules	rule	NOUN
ap-3457	283	5	associated	associate	VERB
ap-3457	283	6	to	to	ADP
ap-3457	283	7	weyl	weyl	PROPN
ap-3457	283	8	group	group	PROPN
ap-3457	283	9	orbit	orbit	NOUN
ap-3457	283	10	functions	function	NOUN
ap-3457	283	11	a0	a0	PROPN
ap-3457	283	12	(	(	PUNCT
ap-3457	283	13	λ1,λ2	λ1,λ2	PROPN
ap-3457	283	14	)	)	PUNCT
ap-3457	283	15	(	(	PUNCT
ap-3457	283	16	λ1	λ1	ADJ
ap-3457	283	17	,	,	PUNCT
ap-3457	283	18	λ2	λ2	NOUN
ap-3457	283	19	)	)	PUNCT
ap-3457	283	20	=	=	SYM
ap-3457	283	21	(	(	PUNCT
ap-3457	283	22	2i	2i	NUM
ap-3457	283	23	,	,	PUNCT
ap-3457	283	24	2j	2j	NUM
ap-3457	283	25	)	)	PUNCT
ap-3457	283	26	∑	∑	PUNCT
ap-3457	283	27	(	(	PUNCT
ap-3457	283	28	µ1,µ2)∈m(2i,2j	µ1,µ2)∈m(2i,2j	NOUN
ap-3457	283	29	)	)	PUNCT
ap-3457	284	1	64(4µ2	64(4µ2	NUM
ap-3457	284	2	2	2	NUM
ap-3457	284	3	+	+	NOUN
ap-3457	284	4	4µ1µ2−3	4µ1µ2−3	NOUN
ap-3457	284	5	)	)	PUNCT
ap-3457	284	6	|	|	ADV
ap-3457	284	7	stabw	stabw	INTJ
ap-3457	284	8	(	(	PUNCT
ap-3457	284	9	2i,2j)|(µ2	2i,2j)|(µ2	NUM
ap-3457	284	10	1−1)(4µ2	1−1)(4µ2	NUM
ap-3457	284	11	2−1)(4µ2	2−1)(4µ2	NUM
ap-3457	284	12	2−9	2−9	NUM
ap-3457	284	13	)	)	PUNCT
ap-3457	284	14	(	(	PUNCT
ap-3457	284	15	λ1	λ1	ADJ
ap-3457	284	16	,	,	PUNCT
ap-3457	284	17	λ2	λ2	NOUN
ap-3457	284	18	)	)	PUNCT
ap-3457	284	19	=	=	SYM
ap-3457	284	20	(	(	PUNCT
ap-3457	284	21	2i	2i	NUM
ap-3457	284	22	,	,	PUNCT
ap-3457	284	23	2j	2j	X
ap-3457	284	24	+	+	CCONJ
ap-3457	284	25	1	1	X
ap-3457	284	26	)	)	PUNCT
ap-3457	284	27	∑	∑	PUNCT
ap-3457	284	28	(	(	PUNCT
ap-3457	284	29	µ1,µ2)∈m(2i,2j+1	µ1,µ2)∈m(2i,2j+1	PROPN
ap-3457	284	30	)	)	PUNCT
ap-3457	284	31	−64(4µ2	−64(4µ2	NOUN
ap-3457	284	32	2	2	NUM
ap-3457	284	33	+	+	NOUN
ap-3457	284	34	4µ1µ2	4µ1µ2	ADJ
ap-3457	284	35	+	+	NOUN
ap-3457	284	36	3	3	NUM
ap-3457	284	37	)	)	PUNCT
ap-3457	284	38	|	|	ADV
ap-3457	284	39	stabw	stabw	INTJ
ap-3457	284	40	(	(	PUNCT
ap-3457	284	41	2i,2j+1)|(µ2	2i,2j+1)|(µ2	NUM
ap-3457	284	42	1−4)(4µ2	1−4)(4µ2	NUM
ap-3457	284	43	2−1)(4µ2	2−1)(4µ2	NUM
ap-3457	284	44	2−9	2−9	NUM
ap-3457	284	45	)	)	PUNCT
ap-3457	284	46	otherwise	otherwise	ADV
ap-3457	284	47	0	0	NUM
ap-3457	284	48	a1	a1	NOUN
ap-3457	284	49	(	(	PUNCT
ap-3457	284	50	λ1,λ2	λ1,λ2	PROPN
ap-3457	284	51	)	)	PUNCT
ap-3457	284	52	(	(	PUNCT
ap-3457	284	53	λ1	λ1	ADJ
ap-3457	284	54	,	,	PUNCT
ap-3457	284	55	λ2	λ2	NOUN
ap-3457	284	56	)	)	PUNCT
ap-3457	284	57	=	=	SYM
ap-3457	284	58	(	(	PUNCT
ap-3457	284	59	2i	2i	NUM
ap-3457	284	60	,	,	PUNCT
ap-3457	284	61	2j	2j	NUM
ap-3457	284	62	)	)	PUNCT
ap-3457	284	63	32(i+j+1	32(i+j+1	X
ap-3457	284	64	)	)	PUNCT
ap-3457	284	65	(	(	PUNCT
ap-3457	284	66	2i+4j+3)(2j+1)(2i+1	2i+4j+3)(2j+1)(2i+1	NOUN
ap-3457	284	67	)	)	PUNCT
ap-3457	284	68	(	(	PUNCT
ap-3457	284	69	λ1	λ1	ADJ
ap-3457	284	70	,	,	PUNCT
ap-3457	284	71	λ2	λ2	NOUN
ap-3457	284	72	)	)	PUNCT
ap-3457	284	73	=	=	SYM
ap-3457	284	74	(	(	PUNCT
ap-3457	284	75	2i	2i	NUM
ap-3457	284	76	,	,	PUNCT
ap-3457	284	77	2j	2j	X
ap-3457	284	78	+	+	CCONJ
ap-3457	284	79	1	1	X
ap-3457	284	80	)	)	PUNCT
ap-3457	284	81	32(j+1	32(j+1	NUM
ap-3457	284	82	)	)	PUNCT
ap-3457	284	83	(	(	PUNCT
ap-3457	284	84	2i+4j+5)(2i+2j+3)(2i+1	2i+4j+5)(2i+2j+3)(2i+1	NOUN
ap-3457	284	85	)	)	PUNCT
ap-3457	284	86	otherwise	otherwise	ADV
ap-3457	284	87	0	0	NUM
ap-3457	284	88	as(λ1,λ2	as(λ1,λ2	NOUN
ap-3457	284	89	)	)	PUNCT
ap-3457	284	90	(	(	PUNCT
ap-3457	284	91	λ1	λ1	ADJ
ap-3457	284	92	,	,	PUNCT
ap-3457	284	93	λ2	λ2	NOUN
ap-3457	284	94	)	)	PUNCT
ap-3457	284	95	=	=	SYM
ap-3457	284	96	(	(	PUNCT
ap-3457	284	97	2i	2i	NUM
ap-3457	284	98	,	,	PUNCT
ap-3457	284	99	2j	2j	NUM
ap-3457	284	100	)	)	PUNCT
ap-3457	284	101	∑	∑	PUNCT
ap-3457	284	102	(	(	PUNCT
ap-3457	284	103	µ1,µ2)∈ms	µ1,µ2)∈ms	NUM
ap-3457	284	104	1(2i+1,2j	1(2i+1,2j	NUM
ap-3457	284	105	)	)	PUNCT
ap-3457	284	106	8(µ1+µ2	8(µ1+µ2	NUM
ap-3457	284	107	)	)	PUNCT
ap-3457	284	108	|	|	ADV
ap-3457	284	109	stabw	stabw	INTJ
ap-3457	284	110	(	(	PUNCT
ap-3457	284	111	2i+1,2j)|µ2(µ2	2i+1,2j)|µ2(µ2	NUM
ap-3457	284	112	1−1)(µ2	1−1)(µ2	NUM
ap-3457	284	113	2−1	2−1	NUM
ap-3457	284	114	)	)	PUNCT
ap-3457	284	115	(	(	PUNCT
ap-3457	284	116	λ1	λ1	ADJ
ap-3457	284	117	,	,	PUNCT
ap-3457	284	118	λ2	λ2	NOUN
ap-3457	284	119	)	)	PUNCT
ap-3457	284	120	=	=	SYM
ap-3457	284	121	(	(	PUNCT
ap-3457	284	122	2i	2i	NUM
ap-3457	284	123	,	,	PUNCT
ap-3457	284	124	2j	2j	X
ap-3457	284	125	+	+	CCONJ
ap-3457	284	126	1	1	NUM
ap-3457	284	127	)	)	PUNCT
ap-3457	284	128	∑	∑	PUNCT
ap-3457	284	129	(	(	PUNCT
ap-3457	284	130	µ1,µ2)∈ms	µ1,µ2)∈ms	NUM
ap-3457	284	131	2(2i+1,2j+1	2(2i+1,2j+1	NUM
ap-3457	284	132	)	)	PUNCT
ap-3457	284	133	−8(µ1+µ2	−8(µ1+µ2	PROPN
ap-3457	284	134	)	)	PUNCT
ap-3457	284	135	µ2(µ2	µ2(µ2	ADJ
ap-3457	284	136	1−1)(µ2	1−1)(µ2	NUM
ap-3457	284	137	2−1	2−1	NUM
ap-3457	284	138	)	)	PUNCT
ap-3457	284	139	otherwise	otherwise	ADV
ap-3457	284	140	0	0	X
ap-3457	284	141	al(λ1,λ2	al(λ1,λ2	PROPN
ap-3457	284	142	)	)	PUNCT
ap-3457	284	143	(	(	PUNCT
ap-3457	284	144	λ1	λ1	ADJ
ap-3457	284	145	,	,	PUNCT
ap-3457	284	146	λ2	λ2	NOUN
ap-3457	284	147	)	)	PUNCT
ap-3457	284	148	=	=	SYM
ap-3457	284	149	(	(	PUNCT
ap-3457	284	150	2i	2i	NUM
ap-3457	284	151	,	,	PUNCT
ap-3457	284	152	2j	2j	NUM
ap-3457	284	153	)	)	PUNCT
ap-3457	284	154	(	(	PUNCT
ap-3457	284	155	∑	∑	PUNCT
ap-3457	284	156	(	(	PUNCT
ap-3457	284	157	µ1,µ2)∈ml	µ1,µ2)∈ml	NOUN
ap-3457	284	158	1(2i,2j+1)−	1(2i,2j+1)−	PROPN
ap-3457	284	159	∑	∑	PUNCT
ap-3457	284	160	(	(	PUNCT
ap-3457	284	161	µ1,µ2)∈ml	µ1,µ2)∈ml	NOUN
ap-3457	284	162	2(2i,2j+1	2(2i,2j+1	NOUN
ap-3457	284	163	)	)	PUNCT
ap-3457	284	164	)	)	PUNCT
ap-3457	285	1	32µ2	32µ2	NUM
ap-3457	285	2	|	|	ADV
ap-3457	285	3	stabw	stabw	INTJ
ap-3457	285	4	(	(	PUNCT
ap-3457	285	5	2i,2j+1)|µ1(4µ2	2i,2j+1)|µ1(4µ2	NUM
ap-3457	285	6	2−1	2−1	NUM
ap-3457	285	7	)	)	PUNCT
ap-3457	285	8	(	(	PUNCT
ap-3457	285	9	λ1	λ1	ADJ
ap-3457	285	10	,	,	PUNCT
ap-3457	285	11	λ2	λ2	NOUN
ap-3457	285	12	)	)	PUNCT
ap-3457	285	13	=	=	SYM
ap-3457	285	14	(	(	PUNCT
ap-3457	285	15	2i	2i	NUM
ap-3457	285	16	,	,	PUNCT
ap-3457	285	17	2j	2j	X
ap-3457	285	18	+	+	CCONJ
ap-3457	285	19	1	1	X
ap-3457	285	20	)	)	PUNCT
ap-3457	285	21	(	(	PUNCT
ap-3457	285	22	∑	∑	PUNCT
ap-3457	285	23	(	(	PUNCT
ap-3457	285	24	µ1,µ2)∈ml	µ1,µ2)∈ml	VERB
ap-3457	285	25	2(2i,2j+2)−	2(2i,2j+2)−	NUM
ap-3457	285	26	∑	∑	PUNCT
ap-3457	285	27	(	(	PUNCT
ap-3457	285	28	µ1,µ2)∈ml	µ1,µ2)∈ml	NOUN
ap-3457	285	29	1(2i,2j+2	1(2i,2j+2	NUM
ap-3457	285	30	)	)	PUNCT
ap-3457	285	31	)	)	PUNCT
ap-3457	286	1	16(2µ1µ2	16(2µ1µ2	NUM
ap-3457	287	1	+	+	NOUN
ap-3457	287	2	1	1	NUM
ap-3457	287	3	)	)	PUNCT
ap-3457	287	4	|	|	ADV
ap-3457	287	5	stabw	stabw	INTJ
ap-3457	287	6	(	(	PUNCT
ap-3457	287	7	2i,2j+2)|(µ2	2i,2j+2)|(µ2	PROPN
ap-3457	287	8	1−1)(4µ2	1−1)(4µ2	NUM
ap-3457	287	9	2−1	2−1	NUM
ap-3457	287	10	)	)	PUNCT
ap-3457	287	11	otherwise	otherwise	ADV
ap-3457	287	12	0	0	NUM
ap-3457	287	13	m(λ1	m(λ1	NOUN
ap-3457	287	14	,	,	PUNCT
ap-3457	287	15	λ2	λ2	PROPN
ap-3457	287	16	)	)	PUNCT
ap-3457	287	17	{	{	PUNCT
ap-3457	287	18	(	(	PUNCT
ap-3457	287	19	λ1	λ1	ADJ
ap-3457	287	20	+	+	SYM
ap-3457	287	21	λ2	λ2	NOUN
ap-3457	287	22	,	,	PUNCT
ap-3457	287	23	λ1	λ1	PROPN
ap-3457	287	24	2	2	NUM
ap-3457	287	25	+	+	SYM
ap-3457	287	26	λ2	λ2	NOUN
ap-3457	287	27	)	)	PUNCT
ap-3457	287	28	,	,	PUNCT
ap-3457	287	29	(	(	PUNCT
ap-3457	287	30	λ2	λ2	PROPN
ap-3457	287	31	,	,	PUNCT
ap-3457	287	32	λ1	λ1	PROPN
ap-3457	287	33	2	2	NUM
ap-3457	287	34	+	+	SYM
ap-3457	287	35	λ2	λ2	NOUN
ap-3457	287	36	)	)	PUNCT
ap-3457	287	37	,	,	PUNCT
ap-3457	287	38	(	(	PUNCT
ap-3457	287	39	λ1	λ1	ADJ
ap-3457	287	40	+	+	SYM
ap-3457	287	41	λ2	λ2	NOUN
ap-3457	287	42	,	,	PUNCT
ap-3457	287	43	λ1	λ1	PROPN
ap-3457	287	44	2	2	NUM
ap-3457	287	45	)	)	PUNCT
ap-3457	287	46	,	,	PUNCT
ap-3457	287	47	(	(	PUNCT
ap-3457	287	48	λ2,−λ1	λ2,−λ1	NOUN
ap-3457	287	49	2	2	NUM
ap-3457	287	50	)	)	PUNCT
ap-3457	287	51	}	}	PUNCT
ap-3457	287	52	ms	ms	NOUN
ap-3457	287	53	1(λ1	1(λ1	NUM
ap-3457	287	54	,	,	PUNCT
ap-3457	287	55	λ2	λ2	PROPN
ap-3457	287	56	)	)	PUNCT
ap-3457	287	57	{	{	PUNCT
ap-3457	287	58	(	(	PUNCT
ap-3457	287	59	λ2	λ2	PROPN
ap-3457	287	60	,	,	PUNCT
ap-3457	287	61	λ1	λ1	PROPN
ap-3457	287	62	2	2	NUM
ap-3457	287	63	+	+	SYM
ap-3457	287	64	λ2	λ2	NOUN
ap-3457	287	65	)	)	PUNCT
ap-3457	287	66	,	,	PUNCT
ap-3457	287	67	(	(	PUNCT
ap-3457	287	68	λ2,−λ1	λ2,−λ1	NOUN
ap-3457	287	69	2	2	NUM
ap-3457	287	70	)	)	PUNCT
ap-3457	287	71	}	}	PUNCT
ap-3457	287	72	ms	ms	PROPN
ap-3457	287	73	2(λ1	2(λ1	PROPN
ap-3457	287	74	,	,	PUNCT
ap-3457	287	75	λ2	λ2	PROPN
ap-3457	287	76	)	)	PUNCT
ap-3457	287	77	{	{	PUNCT
ap-3457	287	78	(	(	PUNCT
ap-3457	287	79	λ1	λ1	ADJ
ap-3457	287	80	+	+	SYM
ap-3457	287	81	λ2	λ2	NOUN
ap-3457	287	82	,	,	PUNCT
ap-3457	287	83	λ1	λ1	PROPN
ap-3457	287	84	2	2	NUM
ap-3457	287	85	+	+	SYM
ap-3457	287	86	λ2	λ2	NOUN
ap-3457	287	87	)	)	PUNCT
ap-3457	287	88	,	,	PUNCT
ap-3457	287	89	(	(	PUNCT
ap-3457	287	90	λ1	λ1	ADJ
ap-3457	287	91	+	+	SYM
ap-3457	287	92	λ2	λ2	NOUN
ap-3457	287	93	,	,	PUNCT
ap-3457	287	94	λ1	λ1	PROPN
ap-3457	287	95	2	2	NUM
ap-3457	287	96	)	)	PUNCT
ap-3457	287	97	}	}	PUNCT
ap-3457	287	98	ml	ml	ADP
ap-3457	287	99	1(λ1	1(λ1	NUM
ap-3457	287	100	,	,	PUNCT
ap-3457	287	101	λ2	λ2	PROPN
ap-3457	287	102	)	)	PUNCT
ap-3457	287	103	{	{	PUNCT
ap-3457	287	104	(	(	PUNCT
ap-3457	287	105	λ1	λ1	ADJ
ap-3457	287	106	+	+	SYM
ap-3457	287	107	λ2	λ2	NOUN
ap-3457	287	108	,	,	PUNCT
ap-3457	287	109	λ1	λ1	PROPN
ap-3457	287	110	2	2	NUM
ap-3457	287	111	+	+	SYM
ap-3457	287	112	λ2	λ2	NOUN
ap-3457	287	113	)	)	PUNCT
ap-3457	287	114	,	,	PUNCT
ap-3457	287	115	(	(	PUNCT
ap-3457	287	116	λ2	λ2	PROPN
ap-3457	287	117	,	,	PUNCT
ap-3457	287	118	λ1	λ1	PROPN
ap-3457	287	119	2	2	NUM
ap-3457	287	120	+	+	SYM
ap-3457	287	121	λ2	λ2	NOUN
ap-3457	287	122	)	)	PUNCT
ap-3457	287	123	}	}	PUNCT
ap-3457	287	124	ml	ml	ADP
ap-3457	287	125	2(λ1	2(λ1	NUM
ap-3457	287	126	,	,	PUNCT
ap-3457	287	127	λ2	λ2	PROPN
ap-3457	287	128	)	)	PUNCT
ap-3457	287	129	{	{	PUNCT
ap-3457	287	130	(	(	PUNCT
ap-3457	287	131	λ1	λ1	ADJ
ap-3457	287	132	+	+	SYM
ap-3457	287	133	λ2	λ2	NOUN
ap-3457	287	134	,	,	PUNCT
ap-3457	287	135	λ1	λ1	PROPN
ap-3457	287	136	2	2	NUM
ap-3457	287	137	)	)	PUNCT
ap-3457	287	138	,	,	PUNCT
ap-3457	287	139	(	(	PUNCT
ap-3457	287	140	λ2,−λ1	λ2,−λ1	NOUN
ap-3457	287	141	2	2	NUM
ap-3457	287	142	)	)	PUNCT
ap-3457	287	143	}	}	PUNCT
ap-3457	287	144	table	table	NOUN
ap-3457	287	145	2	2	NUM
ap-3457	287	146	.	.	PUNCT
ap-3457	287	147	values	value	NOUN
ap-3457	287	148	of	of	ADP
ap-3457	287	149	atλ	atλ	NOUN
ap-3457	287	150	(	(	PUNCT
ap-3457	287	151	11	11	NUM
ap-3457	287	152	)	)	PUNCT
ap-3457	287	153	for	for	ADP
ap-3457	287	154	t	t	PROPN
ap-3457	287	155	∈	∈	PROPN
ap-3457	287	156	{	{	PUNCT
ap-3457	287	157	0	0	NUM
ap-3457	287	158	,	,	PUNCT
ap-3457	287	159	1	1	NUM
ap-3457	287	160	,	,	PUNCT
ap-3457	287	161	s	s	X
ap-3457	287	162	,	,	PUNCT
ap-3457	287	163	l	l	NOUN
ap-3457	287	164	}	}	PUNCT
ap-3457	287	165	,	,	PUNCT
ap-3457	287	166	λ	λ	X
ap-3457	287	167	=	=	PUNCT
ap-3457	288	1	λ1ω1	λ1ω1	PUNCT
ap-3457	288	2	+	+	NUM
ap-3457	288	3	λ2ω2	λ2ω2	PUNCT
ap-3457	288	4	and	and	CCONJ
ap-3457	288	5	i	i	PRON
ap-3457	288	6	,	,	PUNCT
ap-3457	288	7	j	j	PROPN
ap-3457	288	8	are	be	AUX
ap-3457	288	9	non	non	ADJ
ap-3457	288	10	-	-	ADJ
ap-3457	288	11	negative	negative	ADJ
ap-3457	288	12	integers	integer	NOUN
ap-3457	288	13	.	.	PUNCT
ap-3457	289	1	[	[	X
ap-3457	289	2	λ0	λ0	NOUN
ap-3457	289	3	,	,	PUNCT
ap-3457	289	4	λ1	λ1	ADJ
ap-3457	289	5	,	,	PUNCT
ap-3457	289	6	λ2	λ2	PROPN
ap-3457	289	7	]	]	PUNCT
ap-3457	289	8	|stabw	|stabw	NOUN
ap-3457	289	9	(	(	PUNCT
ap-3457	289	10	λ+	λ+	X
ap-3457	289	11	%	%	INTJ
ap-3457	289	12	t)|	t)|	INTJ
ap-3457	289	13	h∨λ+%t	h∨λ+%t	NOUN
ap-3457	289	14	(	(	PUNCT
ap-3457	289	15	?	?	PUNCT
ap-3457	289	16	,	,	PUNCT
ap-3457	289	17	?	?	PUNCT
ap-3457	289	18	,	,	PUNCT
ap-3457	289	19	?	?	PUNCT
ap-3457	289	20	)	)	PUNCT
ap-3457	289	21	1	1	NUM
ap-3457	289	22	1	1	NUM
ap-3457	289	23	(	(	PUNCT
ap-3457	289	24	0	0	NUM
ap-3457	289	25	,	,	PUNCT
ap-3457	289	26	?	?	PUNCT
ap-3457	289	27	,	,	PUNCT
ap-3457	289	28	?	?	PUNCT
ap-3457	289	29	)	)	PUNCT
ap-3457	289	30	1	1	NUM
ap-3457	289	31	2	2	NUM
ap-3457	289	32	(	(	PUNCT
ap-3457	289	33	?	?	NUM
ap-3457	289	34	,	,	PUNCT
ap-3457	289	35	0	0	NUM
ap-3457	289	36	,	,	PUNCT
ap-3457	289	37	?	?	PUNCT
ap-3457	289	38	)	)	PUNCT
ap-3457	289	39	2	2	NUM
ap-3457	289	40	2	2	NUM
ap-3457	289	41	(	(	PUNCT
ap-3457	289	42	?	?	PUNCT
ap-3457	289	43	,	,	PUNCT
ap-3457	289	44	?	?	PUNCT
ap-3457	289	45	,	,	PUNCT
ap-3457	289	46	0	0	X
ap-3457	289	47	)	)	PUNCT
ap-3457	289	48	2	2	NUM
ap-3457	289	49	2	2	NUM
ap-3457	289	50	(	(	PUNCT
ap-3457	289	51	0	0	NUM
ap-3457	289	52	,	,	PUNCT
ap-3457	289	53	0	0	NUM
ap-3457	289	54	,	,	PUNCT
ap-3457	289	55	?	?	PUNCT
ap-3457	289	56	)	)	PUNCT
ap-3457	289	57	2	2	NUM
ap-3457	289	58	4	4	NUM
ap-3457	289	59	(	(	PUNCT
ap-3457	289	60	0	0	NUM
ap-3457	289	61	,	,	PUNCT
ap-3457	289	62	?	?	PUNCT
ap-3457	289	63	,	,	PUNCT
ap-3457	289	64	0	0	X
ap-3457	289	65	)	)	PUNCT
ap-3457	289	66	2	2	NUM
ap-3457	289	67	8	8	NUM
ap-3457	289	68	(	(	PUNCT
ap-3457	289	69	?	?	NUM
ap-3457	289	70	,	,	PUNCT
ap-3457	289	71	0	0	NUM
ap-3457	289	72	,	,	PUNCT
ap-3457	289	73	0	0	NUM
ap-3457	289	74	)	)	PUNCT
ap-3457	289	75	8	8	NUM
ap-3457	289	76	8	8	NUM
ap-3457	289	77	table	table	NOUN
ap-3457	289	78	1	1	NUM
ap-3457	289	79	.	.	PUNCT
ap-3457	290	1	the	the	DET
ap-3457	290	2	values	value	NOUN
ap-3457	290	3	of	of	ADP
ap-3457	290	4	|stabw	|stabw	PROPN
ap-3457	290	5	(	(	PUNCT
ap-3457	290	6	λ+	λ+	X
ap-3457	290	7	%	%	INTJ
ap-3457	290	8	t)|	t)|	NOUN
ap-3457	290	9	and	and	CCONJ
ap-3457	290	10	h∨	h∨	PROPN
ap-3457	290	11	λ+%t	λ+%t	PROPN
ap-3457	290	12	of	of	ADP
ap-3457	290	13	c2	c2	PROPN
ap-3457	290	14	,	,	PUNCT
ap-3457	290	15	where	where	SCONJ
ap-3457	290	16	λ+%t	λ+%t	NOUN
ap-3457	290	17	=	=	SYM
ap-3457	290	18	λ1ω1+λ2ω2	λ1ω1+λ2ω2	ADJ
ap-3457	290	19	and	and	CCONJ
ap-3457	290	20	λ0	λ0	NOUN
ap-3457	290	21	≡m+ht−λ1−	≡m+ht−λ1−	X
ap-3457	290	22	2λ2	2λ2	NUM
ap-3457	290	23	.	.	PUNCT
ap-3457	291	1	asterisks	asterisk	NOUN
ap-3457	291	2	denote	denote	VERB
ap-3457	291	3	non	non	ADJ
ap-3457	291	4	-	-	ADJ
ap-3457	291	5	zero	zero	ADJ
ap-3457	291	6	positive	positive	ADJ
ap-3457	291	7	integers	integer	NOUN
ap-3457	291	8	.	.	PUNCT
ap-3457	292	1	it	it	PRON
ap-3457	292	2	remains	remain	VERB
ap-3457	292	3	to	to	PART
ap-3457	292	4	evaluate	evaluate	VERB
ap-3457	292	5	the	the	DET
ap-3457	292	6	integrals	integral	NOUN
ap-3457	292	7	atλ(w	atλ(w	PRON
ap-3457	292	8	)	)	PUNCT
ap-3457	292	9	which	which	PRON
ap-3457	292	10	depend	depend	VERB
ap-3457	292	11	on	on	ADP
ap-3457	292	12	the	the	DET
ap-3457	292	13	chosen	choose	VERB
ap-3457	292	14	weight	weight	NOUN
ap-3457	292	15	and	and	CCONJ
ap-3457	293	1	the	the	DET
ap-3457	293	2	integration	integration	NOUN
ap-3457	293	3	domain	domain	NOUN
ap-3457	293	4	d	d	PROPN
ap-3457	293	5	⊂	⊂	PROPN
ap-3457	293	6	ω	ω	PROPN
ap-3457	293	7	.	.	PUNCT
ap-3457	294	1	since	since	SCONJ
ap-3457	294	2	the	the	DET
ap-3457	294	3	jacobi	jacobi	PROPN
ap-3457	294	4	polynomials	polynomial	NOUN
ap-3457	294	5	have	have	VERB
ap-3457	294	6	several	several	ADJ
ap-3457	294	7	properties	property	NOUN
ap-3457	294	8	connected	connect	VERB
ap-3457	294	9	to	to	ADP
ap-3457	294	10	the	the	DET
ap-3457	294	11	domain	domain	NOUN
ap-3457	294	12	ω	ω	NOUN
ap-3457	294	13	(	(	PUNCT
ap-3457	294	14	e.g.	e.g.	ADV
ap-3457	294	15	continuous	continuous	ADJ
ap-3457	294	16	and	and	CCONJ
ap-3457	294	17	discrete	discrete	ADJ
ap-3457	294	18	orthogonality	orthogonality	NOUN
ap-3457	294	19	)	)	PUNCT
ap-3457	294	20	,	,	PUNCT
ap-3457	294	21	we	we	PRON
ap-3457	294	22	firstly	firstly	ADV
ap-3457	294	23	take	take	VERB
ap-3457	294	24	d	d	NOUN
ap-3457	294	25	=	=	SYM
ap-3457	294	26	ω	ω	PROPN
ap-3457	294	27	.	.	PUNCT
ap-3457	295	1	the	the	DET
ap-3457	295	2	cubature	cubature	ADJ
ap-3457	295	3	rules	rule	VERB
ap-3457	295	4	with	with	ADP
ap-3457	295	5	the	the	DET
ap-3457	295	6	choice	choice	NOUN
ap-3457	295	7	w	w	NOUN
ap-3457	295	8	=	=	NOUN
ap-3457	295	9	wt	wt	NOUN
ap-3457	295	10	coincide	coincide	NOUN
ap-3457	295	11	for	for	ADP
ap-3457	295	12	any	any	DET
ap-3457	295	13	simple	simple	ADJ
ap-3457	295	14	lie	lie	NOUN
ap-3457	295	15	algebra	algebra	NOUN
ap-3457	295	16	with	with	ADP
ap-3457	295	17	the	the	DET
ap-3457	295	18	formulas	formula	NOUN
ap-3457	295	19	(	(	PUNCT
ap-3457	295	20	7	7	NUM
ap-3457	295	21	)	)	PUNCT
ap-3457	295	22	.	.	PUNCT
ap-3457	296	1	the	the	DET
ap-3457	296	2	difference	difference	NOUN
ap-3457	296	3	lies	lie	VERB
ap-3457	296	4	in	in	ADP
ap-3457	296	5	the	the	DET
ap-3457	296	6	fact	fact	NOUN
ap-3457	296	7	that	that	SCONJ
ap-3457	296	8	clenshaw	clenshaw	ADJ
ap-3457	296	9	-	-	PUNCT
ap-3457	296	10	curtis	curtis	NOUN
ap-3457	296	11	method	method	NOUN
ap-3457	296	12	guarantees	guarantee	VERB
ap-3457	296	13	the	the	DET
ap-3457	296	14	exact	exact	ADJ
ap-3457	296	15	equality	equality	NOUN
ap-3457	296	16	only	only	ADV
ap-3457	296	17	for	for	ADP
ap-3457	296	18	polynomials	polynomial	NOUN
ap-3457	296	19	up	up	ADP
ap-3457	296	20	to	to	ADP
ap-3457	296	21	m	m	NOUN
ap-3457	296	22	-	-	PUNCT
ap-3457	296	23	degree	degree	NOUN
ap-3457	296	24	m	m	NOUN
ap-3457	296	25	.	.	PUNCT
ap-3457	297	1	4.2	4.2	NUM
ap-3457	297	2	.	.	PUNCT
ap-3457	298	1	integration	integration	NOUN
ap-3457	298	2	domain	domain	NOUN
ap-3457	298	3	ω	ω	PROPN
ap-3457	298	4	of	of	ADP
ap-3457	298	5	c2	c2	PROPN
ap-3457	298	6	in	in	ADP
ap-3457	298	7	this	this	DET
ap-3457	298	8	section	section	NOUN
ap-3457	298	9	,	,	PUNCT
ap-3457	298	10	the	the	DET
ap-3457	298	11	clenshaw	clenshaw	ADJ
ap-3457	298	12	-	-	PUNCT
ap-3457	298	13	curtis	curtis	NOUN
ap-3457	298	14	integration	integration	NOUN
ap-3457	298	15	method	method	NOUN
ap-3457	298	16	is	be	AUX
ap-3457	298	17	applied	apply	VERB
ap-3457	298	18	to	to	ADP
ap-3457	298	19	the	the	DET
ap-3457	298	20	algebra	algebra	PROPN
ap-3457	298	21	c2	c2	PROPN
ap-3457	298	22	.	.	PUNCT
ap-3457	299	1	the	the	DET
ap-3457	299	2	values	value	NOUN
ap-3457	299	3	of	of	ADP
ap-3457	299	4	|stabw	|stabw	PROPN
ap-3457	299	5	(	(	PUNCT
ap-3457	299	6	λ	λ	PROPN
ap-3457	299	7	+	+	NOUN
ap-3457	299	8	%	%	INTJ
ap-3457	299	9	t)|	t)|	NOUN
ap-3457	299	10	and	and	CCONJ
ap-3457	299	11	h∨λ+%t	h∨λ+%t	PROPN
ap-3457	299	12	,	,	PUNCT
ap-3457	299	13	needed	need	VERB
ap-3457	299	14	in	in	ADP
ap-3457	299	15	(	(	PUNCT
ap-3457	299	16	10	10	NUM
ap-3457	299	17	)	)	PUNCT
ap-3457	299	18	,	,	PUNCT
ap-3457	299	19	are	be	AUX
ap-3457	299	20	tabulated	tabulate	VERB
ap-3457	299	21	in	in	ADP
ap-3457	299	22	tab	tab	NOUN
ap-3457	299	23	.	.	PUNCT
ap-3457	300	1	1	1	X
ap-3457	300	2	.	.	PUNCT
ap-3457	301	1	since	since	SCONJ
ap-3457	301	2	the	the	DET
ap-3457	301	3	choice	choice	NOUN
ap-3457	301	4	of	of	ADP
ap-3457	301	5	w	w	NOUN
ap-3457	301	6	=	=	NOUN
ap-3457	301	7	wt	wt	PROPN
ap-3457	301	8	gives	give	VERB
ap-3457	301	9	standard	standard	ADJ
ap-3457	301	10	cubature	cubature	ADJ
ap-3457	301	11	formulas	formula	NOUN
ap-3457	301	12	,	,	PUNCT
ap-3457	301	13	the	the	DET
ap-3457	301	14	next	next	ADJ
ap-3457	301	15	natural	natural	ADJ
ap-3457	301	16	choice	choice	NOUN
ap-3457	301	17	of	of	ADP
ap-3457	301	18	the	the	DET
ap-3457	301	19	weight	weight	NOUN
ap-3457	301	20	function	function	NOUN
ap-3457	301	21	is	be	AUX
ap-3457	301	22	to	to	PART
ap-3457	301	23	set	set	VERB
ap-3457	301	24	w	w	NOUN
ap-3457	301	25	=	=	NOUN
ap-3457	301	26	1	1	X
ap-3457	301	27	.	.	PUNCT
ap-3457	302	1	in	in	ADP
ap-3457	302	2	this	this	DET
ap-3457	302	3	case	case	NOUN
ap-3457	302	4	are	be	AUX
ap-3457	302	5	the	the	DET
ap-3457	302	6	coefficients	coefficient	NOUN
ap-3457	302	7	atλ(1	atλ(1	PROPN
ap-3457	302	8	)	)	PUNCT
ap-3457	302	9	,	,	PUNCT
ap-3457	302	10	denoted	denote	VERB
ap-3457	302	11	by	by	ADP
ap-3457	302	12	atλ	atλ	NOUN
ap-3457	302	13	,	,	PUNCT
ap-3457	302	14	expressed	express	VERB
ap-3457	302	15	as	as	ADP
ap-3457	302	16	the	the	DET
ap-3457	302	17	following	follow	VERB
ap-3457	302	18	integrals	integral	NOUN
ap-3457	302	19	:	:	PUNCT
ap-3457	302	20	atλ	atλ	VERB
ap-3457	302	21	=	=	PUNCT
ap-3457	302	22	2π2	2π2	NUM
ap-3457	303	1			NUM
ap-3457	303	2	∫	∫	PROPN
ap-3457	303	3	f	f	PROPN
ap-3457	303	4	cλ(x)s%1(x	cλ(x)s%1(x	PROPN
ap-3457	303	5	)	)	PUNCT
ap-3457	303	6	dx	dx	PROPN
ap-3457	303	7	if	if	SCONJ
ap-3457	303	8	t	t	PROPN
ap-3457	303	9	=	=	PUNCT
ap-3457	303	10	0,∫	0,∫	PROPN
ap-3457	303	11	f	f	PROPN
ap-3457	303	12	sλ+%1(x	sλ+%1(x	PROPN
ap-3457	303	13	)	)	PUNCT
ap-3457	303	14	dx	dx	PROPN
ap-3457	303	15	if	if	SCONJ
ap-3457	303	16	t	t	PROPN
ap-3457	303	17	=	=	SYM
ap-3457	303	18	1,∫	1,∫	NUM
ap-3457	303	19	f	f	PROPN
ap-3457	303	20	ssλ+%s(x)sl%l(x	ssλ+%s(x)sl%l(x	PROPN
ap-3457	303	21	)	)	PUNCT
ap-3457	303	22	dx	dx	PROPN
ap-3457	304	1	if	if	SCONJ
ap-3457	304	2	t	t	PROPN
ap-3457	304	3	=	=	PUNCT
ap-3457	304	4	s,∫	s,∫	PROPN
ap-3457	304	5	f	f	PROPN
ap-3457	304	6	slλ+%l(x)ss%s(x	slλ+%l(x)ss%s(x	PROPN
ap-3457	304	7	)	)	PUNCT
ap-3457	304	8	dx	dx	PROPN
ap-3457	304	9	if	if	SCONJ
ap-3457	304	10	t	t	PROPN
ap-3457	304	11	=	=	SYM
ap-3457	304	12	l.	l.	PROPN
ap-3457	304	13	(	(	PUNCT
ap-3457	304	14	11	11	NUM
ap-3457	304	15	)	)	PUNCT
ap-3457	304	16	the	the	DET
ap-3457	304	17	exact	exact	ADJ
ap-3457	304	18	values	value	NOUN
ap-3457	304	19	of	of	ADP
ap-3457	304	20	atλ	atλ	NOUN
ap-3457	304	21	are	be	AUX
ap-3457	304	22	explicitly	explicitly	ADV
ap-3457	304	23	calculated	calculate	VERB
ap-3457	304	24	in	in	ADP
ap-3457	304	25	tab	tab	NOUN
ap-3457	304	26	.	.	PUNCT
ap-3457	305	1	2	2	NUM
ap-3457	305	2	.	.	X
ap-3457	305	3	4.3	4.3	NUM
ap-3457	305	4	.	.	PUNCT
ap-3457	305	5	triangular	triangular	NOUN
ap-3457	305	6	domain	domain	NOUN
ap-3457	305	7	of	of	ADP
ap-3457	305	8	c2	c2	PROPN
ap-3457	305	9	the	the	DET
ap-3457	305	10	next	next	ADJ
ap-3457	305	11	choice	choice	NOUN
ap-3457	305	12	of	of	ADP
ap-3457	305	13	the	the	DET
ap-3457	305	14	domain	domain	NOUN
ap-3457	305	15	d	d	NOUN
ap-3457	305	16	,	,	PUNCT
ap-3457	305	17	for	for	ADP
ap-3457	305	18	which	which	PRON
ap-3457	305	19	we	we	PRON
ap-3457	305	20	derive	derive	VERB
ap-3457	305	21	the	the	DET
ap-3457	305	22	clenshaw	clenshaw	ADJ
ap-3457	305	23	-	-	PUNCT
ap-3457	305	24	curtis	curtis	NOUN
ap-3457	305	25	cubature	cubature	NOUN
ap-3457	305	26	rules	rule	NOUN
ap-3457	305	27	,	,	PUNCT
ap-3457	305	28	is	be	AUX
ap-3457	305	29	the	the	DET
ap-3457	305	30	triangle	triangle	NOUN
ap-3457	305	31	t	t	PROPN
ap-3457	305	32	⊂	⊂	PROPN
ap-3457	305	33	ω	ω	PROPN
ap-3457	305	34	depicted	depict	VERB
ap-3457	305	35	on	on	ADP
ap-3457	305	36	fig	fig	NOUN
ap-3457	305	37	.	.	PUNCT
ap-3457	306	1	6	6	NUM
ap-3457	307	1	and	and	CCONJ
ap-3457	307	2	given	give	VERB
ap-3457	307	3	explicitly	explicitly	ADV
ap-3457	307	4	by	by	ADP
ap-3457	307	5	t	t	PROPN
ap-3457	307	6	≡	≡	PROPN
ap-3457	307	7	{	{	PUNCT
ap-3457	307	8	(	(	PUNCT
ap-3457	307	9	y1	y1	INTJ
ap-3457	307	10	,	,	PUNCT
ap-3457	307	11	y2	y2	NOUN
ap-3457	307	12	)	)	PUNCT
ap-3457	307	13	∣∣∣	∣∣∣	ADP
ap-3457	307	14	y2	y2	PROPN
ap-3457	307	15	≤	≤	ADJ
ap-3457	307	16	0	0	NUM
ap-3457	307	17	,	,	PUNCT
ap-3457	307	18	−y2	−y2	PROPN
ap-3457	307	19	2	2	NUM
ap-3457	307	20	−	−	PROPN
ap-3457	307	21	2	2	NUM
ap-3457	307	22	≤	≤	NUM
ap-3457	307	23	y1	y1	NOUN
ap-3457	307	24	≤	≤	NUM
ap-3457	308	1	y2	y2	NOUN
ap-3457	308	2	2	2	NUM
ap-3457	308	3	+	+	CCONJ
ap-3457	308	4	2	2	NUM
ap-3457	308	5	}	}	PUNCT
ap-3457	308	6	.	.	PUNCT
ap-3457	309	1	209	209	NUM
ap-3457	309	2	l.	l.	PROPN
ap-3457	309	3	háková	háková	PROPN
ap-3457	309	4	,	,	PUNCT
ap-3457	309	5	j.	j.	PROPN
ap-3457	309	6	hrivnák	hrivnák	PROPN
ap-3457	309	7	,	,	PUNCT
ap-3457	309	8	l.	l.	PROPN
ap-3457	309	9	motlochová	motlochová	PROPN
ap-3457	309	10	acta	acta	PROPN
ap-3457	309	11	polytechnica	polytechnica	PROPN
ap-3457	309	12	a0	a0	PROPN
ap-3457	309	13	(	(	PUNCT
ap-3457	309	14	λ1,λ2	λ1,λ2	PROPN
ap-3457	309	15	)	)	PUNCT
ap-3457	309	16	(	(	PUNCT
ap-3457	309	17	λ1	λ1	ADJ
ap-3457	309	18	,	,	PUNCT
ap-3457	309	19	λ2	λ2	NOUN
ap-3457	309	20	)	)	PUNCT
ap-3457	309	21	=	=	SYM
ap-3457	309	22	(	(	PUNCT
ap-3457	309	23	0	0	NUM
ap-3457	309	24	,	,	PUNCT
ap-3457	309	25	0	0	NUM
ap-3457	309	26	)	)	PUNCT
ap-3457	309	27	π2	π2	ADV
ap-3457	309	28	4	4	NUM
ap-3457	309	29	(	(	PUNCT
ap-3457	309	30	λ1	λ1	ADJ
ap-3457	309	31	,	,	PUNCT
ap-3457	309	32	λ2	λ2	NOUN
ap-3457	309	33	)	)	PUNCT
ap-3457	309	34	=	=	SYM
ap-3457	309	35	(	(	PUNCT
ap-3457	309	36	2i	2i	NUM
ap-3457	309	37	,	,	PUNCT
ap-3457	309	38	2j	2j	X
ap-3457	309	39	+	+	CCONJ
ap-3457	309	40	1	1	NUM
ap-3457	309	41	)	)	PUNCT
ap-3457	309	42	(	(	PUNCT
ap-3457	309	43	−1)i+1	−1)i+1	NOUN
ap-3457	309	44	8	8	NUM
ap-3457	309	45	|	|	ADV
ap-3457	309	46	stabw	stabw	INTJ
ap-3457	309	47	(	(	PUNCT
ap-3457	309	48	2i,2j+1)|(2i+2j+1)(2j+1	2i,2j+1)|(2i+2j+1)(2j+1	X
ap-3457	309	49	)	)	PUNCT
ap-3457	309	50	otherwise	otherwise	ADV
ap-3457	309	51	0	0	NUM
ap-3457	309	52	a1	a1	NOUN
ap-3457	309	53	(	(	PUNCT
ap-3457	309	54	λ1,λ2	λ1,λ2	PROPN
ap-3457	309	55	)	)	PUNCT
ap-3457	309	56	(	(	PUNCT
ap-3457	309	57	λ1	λ1	ADJ
ap-3457	309	58	,	,	PUNCT
ap-3457	309	59	λ2	λ2	NOUN
ap-3457	309	60	)	)	PUNCT
ap-3457	309	61	=	=	SYM
ap-3457	309	62	(	(	PUNCT
ap-3457	309	63	0	0	NUM
ap-3457	309	64	,	,	PUNCT
ap-3457	309	65	0	0	NUM
ap-3457	309	66	)	)	PUNCT
ap-3457	309	67	2π2	2π2	NUM
ap-3457	310	1	+	+	CCONJ
ap-3457	310	2	128	128	NUM
ap-3457	310	3	9	9	NUM
ap-3457	310	4	(	(	PUNCT
ap-3457	310	5	λ1	λ1	ADJ
ap-3457	310	6	,	,	PUNCT
ap-3457	310	7	λ2	λ2	NOUN
ap-3457	310	8	)	)	PUNCT
ap-3457	310	9	=	=	SYM
ap-3457	310	10	(	(	PUNCT
ap-3457	310	11	2i	2i	NUM
ap-3457	310	12	,	,	PUNCT
ap-3457	310	13	2j	2j	NUM
ap-3457	310	14	)	)	PUNCT
ap-3457	310	15	,	,	PUNCT
ap-3457	311	1	i+	i+	NUM
ap-3457	311	2	j	j	PROPN
ap-3457	311	3	6=	6=	PROPN
ap-3457	311	4	0	0	NUM
ap-3457	311	5	(	(	PUNCT
ap-3457	311	6	−1)i+1	−1)i+1	NOUN
ap-3457	311	7	128(i+j+1	128(i+j+1	NUM
ap-3457	311	8	)	)	PUNCT
ap-3457	311	9	(	(	PUNCT
ap-3457	311	10	2i+2j+1)(2j−1)(2i+2j+3)(2j+3	2i+2j+1)(2j−1)(2i+2j+3)(2j+3	NUM
ap-3457	311	11	)	)	PUNCT
ap-3457	311	12	(	(	PUNCT
ap-3457	311	13	λ1	λ1	ADJ
ap-3457	311	14	,	,	PUNCT
ap-3457	311	15	λ2	λ2	NOUN
ap-3457	311	16	)	)	PUNCT
ap-3457	311	17	=	=	SYM
ap-3457	311	18	(	(	PUNCT
ap-3457	311	19	2i	2i	NUM
ap-3457	311	20	,	,	PUNCT
ap-3457	311	21	2j	2j	X
ap-3457	311	22	+	+	CCONJ
ap-3457	311	23	1	1	NUM
ap-3457	311	24	)	)	PUNCT
ap-3457	311	25	(	(	PUNCT
ap-3457	311	26	−1)i+1	−1)i+1	NOUN
ap-3457	311	27	128(j+1	128(j+1	NUM
ap-3457	311	28	)	)	PUNCT
ap-3457	311	29	(	(	PUNCT
ap-3457	311	30	2j+1)(2i+2j+1)(2j+3)(2i+2j+5	2j+1)(2i+2j+1)(2j+3)(2i+2j+5	NUM
ap-3457	311	31	)	)	PUNCT
ap-3457	311	32	otherwise	otherwise	ADV
ap-3457	311	33	0	0	NUM
ap-3457	311	34	as(λ1,λ2	as(λ1,λ2	NOUN
ap-3457	311	35	)	)	PUNCT
ap-3457	311	36	(	(	PUNCT
ap-3457	311	37	λ1	λ1	ADJ
ap-3457	311	38	,	,	PUNCT
ap-3457	311	39	λ2	λ2	NOUN
ap-3457	311	40	)	)	PUNCT
ap-3457	311	41	=	=	SYM
ap-3457	311	42	(	(	PUNCT
ap-3457	311	43	0	0	NUM
ap-3457	311	44	,	,	PUNCT
ap-3457	311	45	0	0	NUM
ap-3457	311	46	)	)	PUNCT
ap-3457	311	47	π2	π2	NOUN
ap-3457	311	48	+	+	CCONJ
ap-3457	311	49	8	8	NUM
ap-3457	311	50	(	(	PUNCT
ap-3457	311	51	λ1	λ1	ADJ
ap-3457	311	52	,	,	PUNCT
ap-3457	311	53	λ2	λ2	NOUN
ap-3457	311	54	)	)	PUNCT
ap-3457	311	55	=	=	SYM
ap-3457	311	56	(	(	PUNCT
ap-3457	311	57	2i	2i	NUM
ap-3457	311	58	,	,	PUNCT
ap-3457	311	59	2j	2j	NUM
ap-3457	311	60	)	)	PUNCT
ap-3457	311	61	,	,	PUNCT
ap-3457	312	1	i+	i+	NUM
ap-3457	312	2	j	j	PROPN
ap-3457	312	3	6=	6=	PROPN
ap-3457	312	4	0	0	NUM
ap-3457	312	5	(	(	PUNCT
ap-3457	312	6	−1)i+1	−1)i+1	NOUN
ap-3457	312	7	16	16	NUM
ap-3457	312	8	|	|	ADV
ap-3457	312	9	stabw	stabw	INTJ
ap-3457	312	10	(	(	PUNCT
ap-3457	312	11	2i+1,2j)|(2j+1)(2i+2j+1)(2j−1	2i+1,2j)|(2j+1)(2i+2j+1)(2j−1	NUM
ap-3457	312	12	)	)	PUNCT
ap-3457	312	13	(	(	PUNCT
ap-3457	312	14	λ1	λ1	ADJ
ap-3457	312	15	,	,	PUNCT
ap-3457	312	16	λ2	λ2	NOUN
ap-3457	312	17	)	)	PUNCT
ap-3457	312	18	=	=	SYM
ap-3457	312	19	(	(	PUNCT
ap-3457	312	20	2i	2i	NUM
ap-3457	312	21	,	,	PUNCT
ap-3457	312	22	2j	2j	X
ap-3457	312	23	+	+	CCONJ
ap-3457	312	24	1	1	NUM
ap-3457	312	25	)	)	PUNCT
ap-3457	312	26	(	(	PUNCT
ap-3457	312	27	−1)i+1	−1)i+1	NOUN
ap-3457	312	28	16	16	NUM
ap-3457	312	29	(	(	PUNCT
ap-3457	312	30	2j+1)(2i+2j+3)(2i+2j+1	2j+1)(2i+2j+3)(2i+2j+1	NUM
ap-3457	312	31	)	)	PUNCT
ap-3457	312	32	otherwise	otherwise	ADV
ap-3457	312	33	0	0	X
ap-3457	312	34	al(λ1,λ2	al(λ1,λ2	PROPN
ap-3457	312	35	)	)	PUNCT
ap-3457	312	36	(	(	PUNCT
ap-3457	312	37	λ1	λ1	ADJ
ap-3457	312	38	,	,	PUNCT
ap-3457	312	39	λ2	λ2	NOUN
ap-3457	312	40	)	)	PUNCT
ap-3457	312	41	=	=	SYM
ap-3457	312	42	(	(	PUNCT
ap-3457	312	43	0	0	NUM
ap-3457	312	44	,	,	PUNCT
ap-3457	312	45	0	0	NUM
ap-3457	312	46	)	)	PUNCT
ap-3457	312	47	π2	π2	ADV
ap-3457	312	48	(	(	PUNCT
ap-3457	312	49	λ1	λ1	ADJ
ap-3457	312	50	,	,	PUNCT
ap-3457	312	51	λ2	λ2	NOUN
ap-3457	312	52	)	)	PUNCT
ap-3457	312	53	=	=	SYM
ap-3457	312	54	(	(	PUNCT
ap-3457	312	55	2i	2i	NUM
ap-3457	312	56	,	,	PUNCT
ap-3457	312	57	2j	2j	X
ap-3457	312	58	+	+	CCONJ
ap-3457	312	59	1	1	NUM
ap-3457	312	60	)	)	PUNCT
ap-3457	312	61	(	(	PUNCT
ap-3457	312	62	−1)i+1	−1)i+1	NOUN
ap-3457	312	63	128(i+j+1)(j+1	128(i+j+1)(j+1	NUM
ap-3457	312	64	)	)	PUNCT
ap-3457	312	65	|	|	ADV
ap-3457	312	66	stabw	stabw	INTJ
ap-3457	312	67	(	(	PUNCT
ap-3457	312	68	2i,2j+2)|(2i+2j+3)(2j+3)(2i+2j+1)(2j+1	2i,2j+2)|(2i+2j+3)(2j+3)(2i+2j+1)(2j+1	NUM
ap-3457	312	69	)	)	PUNCT
ap-3457	312	70	otherwise	otherwise	ADV
ap-3457	312	71	0	0	NUM
ap-3457	312	72	table	table	NOUN
ap-3457	312	73	3	3	NUM
ap-3457	312	74	.	.	PUNCT
ap-3457	313	1	values	value	NOUN
ap-3457	313	2	of	of	ADP
ap-3457	313	3	atλ	atλ	NOUN
ap-3457	313	4	,	,	PUNCT
ap-3457	313	5	given	give	VERB
ap-3457	313	6	by	by	ADP
ap-3457	313	7	(	(	PUNCT
ap-3457	313	8	12	12	NUM
ap-3457	313	9	)	)	PUNCT
ap-3457	313	10	,	,	PUNCT
ap-3457	313	11	for	for	ADP
ap-3457	313	12	t	t	PROPN
ap-3457	313	13	∈	∈	PROPN
ap-3457	313	14	{	{	PUNCT
ap-3457	313	15	0	0	NUM
ap-3457	313	16	,	,	PUNCT
ap-3457	313	17	1	1	NUM
ap-3457	313	18	,	,	PUNCT
ap-3457	313	19	s	s	X
ap-3457	313	20	,	,	PUNCT
ap-3457	313	21	l	l	NOUN
ap-3457	313	22	}	}	PUNCT
ap-3457	313	23	and	and	CCONJ
ap-3457	313	24	λ	λ	X
ap-3457	313	25	=	=	PUNCT
ap-3457	314	1	λ1ω1	λ1ω1	X
ap-3457	314	2	+	+	NUM
ap-3457	314	3	λ2ω2	λ2ω2	NOUN
ap-3457	314	4	.	.	PUNCT
ap-3457	315	1	the	the	DET
ap-3457	315	2	indices	index	NOUN
ap-3457	315	3	i	i	PRON
ap-3457	315	4	,	,	PUNCT
ap-3457	315	5	j	j	PROPN
ap-3457	315	6	are	be	AUX
ap-3457	315	7	non	non	ADJ
ap-3457	315	8	-	-	ADJ
ap-3457	315	9	negative	negative	ADJ
ap-3457	315	10	integers	integer	NOUN
ap-3457	315	11	.	.	PUNCT
ap-3457	316	1	figure	figure	VERB
ap-3457	316	2	6	6	NUM
ap-3457	316	3	.	.	PUNCT
ap-3457	317	1	the	the	DET
ap-3457	317	2	domain	domain	NOUN
ap-3457	317	3	bounded	bound	VERB
ap-3457	317	4	by	by	ADP
ap-3457	317	5	the	the	DET
ap-3457	317	6	two	two	NUM
ap-3457	317	7	lines	line	NOUN
ap-3457	317	8	and	and	CCONJ
ap-3457	317	9	the	the	DET
ap-3457	317	10	parabola	parabola	NOUN
ap-3457	317	11	with	with	ADP
ap-3457	317	12	the	the	DET
ap-3457	317	13	inscribed	inscribe	VERB
ap-3457	317	14	triangle	triangle	NOUN
ap-3457	317	15	t	t	PROPN
ap-3457	317	16	corresponds	correspond	VERB
ap-3457	317	17	to	to	ADP
ap-3457	317	18	the	the	DET
ap-3457	317	19	integration	integration	NOUN
ap-3457	317	20	region	region	NOUN
ap-3457	317	21	ω	ω	PROPN
ap-3457	317	22	of	of	ADP
ap-3457	317	23	c2	c2	PROPN
ap-3457	317	24	.	.	PUNCT
ap-3457	318	1	choosing	choose	VERB
ap-3457	318	2	the	the	DET
ap-3457	318	3	weight	weight	NOUN
ap-3457	318	4	function	function	NOUN
ap-3457	318	5	w	w	PROPN
ap-3457	318	6	=	=	PUNCT
ap-3457	318	7	wt	wt	PROPN
ap-3457	318	8	,	,	PUNCT
ap-3457	318	9	the	the	DET
ap-3457	318	10	integrals	integral	NOUN
ap-3457	318	11	atλ(wt	atλ(wt	NOUN
ap-3457	318	12	)	)	PUNCT
ap-3457	318	13	are	be	AUX
ap-3457	318	14	calculated	calculate	VERB
ap-3457	318	15	by	by	ADP
ap-3457	318	16	a	a	DET
ap-3457	318	17	change	change	NOUN
ap-3457	318	18	of	of	ADP
ap-3457	318	19	variables	variable	NOUN
ap-3457	318	20	induced	induce	VERB
ap-3457	318	21	by	by	ADP
ap-3457	318	22	the	the	DET
ap-3457	318	23	map	map	NOUN
ap-3457	318	24	ξ	ξ	X
ap-3457	318	25	.	.	PUNCT
ap-3457	319	1	denoting	denote	VERB
ap-3457	319	2	atλ	atλ	PROPN
ap-3457	319	3	≡	≡	PROPN
ap-3457	319	4	atλ(wt	atλ(wt	PROPN
ap-3457	319	5	)	)	PUNCT
ap-3457	319	6	,	,	PUNCT
ap-3457	319	7	it	it	PRON
ap-3457	319	8	holds	hold	VERB
ap-3457	319	9	that	that	DET
ap-3457	319	10	atλ	atλ	NOUN
ap-3457	319	11	=	=	PUNCT
ap-3457	319	12	2π2	2π2	NUM
ap-3457	319	13	∫	∫	PROPN
ap-3457	319	14	p	p	PROPN
ap-3457	319	15	stλ+%t(x)st%t(x	stλ+%t(x)st%t(x	PROPN
ap-3457	319	16	)	)	PUNCT
ap-3457	319	17	dx	dx	PROPN
ap-3457	319	18	,	,	PUNCT
ap-3457	319	19	(	(	PUNCT
ap-3457	319	20	12	12	NUM
ap-3457	319	21	)	)	PUNCT
ap-3457	319	22	where	where	SCONJ
ap-3457	319	23	p	p	NOUN
ap-3457	319	24	is	be	AUX
ap-3457	319	25	the	the	DET
ap-3457	319	26	pre	pre	NOUN
ap-3457	319	27	-	-	NOUN
ap-3457	319	28	image	image	NOUN
ap-3457	319	29	of	of	ADP
ap-3457	319	30	the	the	DET
ap-3457	319	31	triangle	triangle	NOUN
ap-3457	319	32	t	t	NOUN
ap-3457	319	33	under	under	ADP
ap-3457	319	34	the	the	DET
ap-3457	319	35	map	map	NOUN
ap-3457	320	1	ξ	ξ	X
ap-3457	320	2	.	.	PUNCT
ap-3457	321	1	this	this	DET
ap-3457	321	2	pre	pre	ADJ
ap-3457	321	3	-	-	NOUN
ap-3457	321	4	image	image	ADJ
ap-3457	321	5	p	p	NOUN
ap-3457	321	6	,	,	PUNCT
ap-3457	321	7	depicted	depict	VERB
ap-3457	321	8	as	as	ADP
ap-3457	321	9	a	a	DET
ap-3457	321	10	square	square	NOUN
ap-3457	321	11	in	in	ADP
ap-3457	321	12	fig	fig	NOUN
ap-3457	321	13	.	.	PUNCT
ap-3457	322	1	7	7	NUM
ap-3457	322	2	,	,	PUNCT
ap-3457	322	3	contains	contain	VERB
ap-3457	322	4	the	the	DET
ap-3457	322	5	points	point	NOUN
ap-3457	322	6	b1ω∨1	b1ω∨1	NOUN
ap-3457	323	1	+	+	NUM
ap-3457	323	2	b2ω	b2ω	NOUN
ap-3457	323	3	∨	∨	NOUN
ap-3457	323	4	2	2	NUM
ap-3457	323	5	satisfying	satisfy	VERB
ap-3457	323	6	2b1	2b1	NUM
ap-3457	323	7	+	+	CCONJ
ap-3457	323	8	b2	b2	PROPN
ap-3457	323	9	≥	≥	NUM
ap-3457	323	10	1/2	1/2	NUM
ap-3457	323	11	,	,	PUNCT
ap-3457	323	12	2b1	2b1	NUM
ap-3457	323	13	+	+	CCONJ
ap-3457	323	14	b2	b2	NOUN
ap-3457	323	15	≤	≤	NUM
ap-3457	323	16	1	1	NUM
ap-3457	323	17	,	,	PUNCT
ap-3457	323	18	b2	b2	NOUN
ap-3457	323	19	≥	≥	NOUN
ap-3457	323	20	0	0	NUM
ap-3457	323	21	and	and	CCONJ
ap-3457	323	22	b2	b2	VERB
ap-3457	323	23	≤	≤	NUM
ap-3457	323	24	1/2	1/2	NUM
ap-3457	323	25	.	.	PUNCT
ap-3457	324	1	the	the	DET
ap-3457	324	2	exact	exact	ADJ
ap-3457	324	3	values	value	NOUN
ap-3457	324	4	of	of	ADP
ap-3457	324	5	atλ	atλ	NOUN
ap-3457	324	6	are	be	AUX
ap-3457	324	7	tabulated	tabulate	VERB
ap-3457	324	8	in	in	ADP
ap-3457	324	9	tab	tab	NOUN
ap-3457	324	10	.	.	PUNCT
ap-3457	325	1	3	3	X
ap-3457	325	2	.	.	X
ap-3457	325	3	finally	finally	ADV
ap-3457	325	4	,	,	PUNCT
ap-3457	325	5	choosing	choose	VERB
ap-3457	325	6	w	w	PROPN
ap-3457	325	7	=	=	NOUN
ap-3457	325	8	1	1	NUM
ap-3457	325	9	,	,	PUNCT
ap-3457	325	10	we	we	PRON
ap-3457	325	11	calculate	calculate	VERB
ap-3457	325	12	the	the	DET
ap-3457	325	13	coefficients	coefficient	NOUN
ap-3457	325	14	figure	figure	NOUN
ap-3457	325	15	7	7	NUM
ap-3457	325	16	.	.	PUNCT
ap-3457	326	1	the	the	DET
ap-3457	326	2	fundamental	fundamental	ADJ
ap-3457	326	3	domain	domain	NOUN
ap-3457	326	4	f	f	PROPN
ap-3457	326	5	corresponding	correspond	VERB
ap-3457	326	6	to	to	ADP
ap-3457	326	7	c2	c2	PROPN
ap-3457	326	8	is	be	AUX
ap-3457	326	9	depicted	depict	VERB
ap-3457	326	10	as	as	ADP
ap-3457	326	11	the	the	DET
ap-3457	326	12	triangle	triangle	NOUN
ap-3457	326	13	containing	contain	VERB
ap-3457	326	14	the	the	DET
ap-3457	326	15	square	square	NOUN
ap-3457	326	16	p	p	NOUN
ap-3457	326	17	with	with	ADP
ap-3457	326	18	the	the	DET
ap-3457	326	19	boundaries	boundary	NOUN
ap-3457	326	20	α	α	NOUN
ap-3457	326	21	,	,	PUNCT
ap-3457	326	22	β	β	X
ap-3457	326	23	,	,	PUNCT
ap-3457	326	24	γ	γ	PROPN
ap-3457	326	25	and	and	CCONJ
ap-3457	326	26	δ	δ	PROPN
ap-3457	326	27	.	.	PUNCT
ap-3457	327	1	btλ	btλ	PROPN
ap-3457	327	2	≡	≡	PROPN
ap-3457	327	3	atλ(1	atλ(1	PROPN
ap-3457	327	4	)	)	PUNCT
ap-3457	327	5	on	on	ADP
ap-3457	327	6	t	t	PROPN
ap-3457	327	7	as	as	ADP
ap-3457	327	8	the	the	DET
ap-3457	327	9	following	follow	VERB
ap-3457	327	10	integrals	integral	NOUN
ap-3457	327	11	btλ	btλ	NOUN
ap-3457	327	12	=	=	PUNCT
ap-3457	327	13	2π2	2π2	NUM
ap-3457	328	1			NUM
ap-3457	328	2	∫	∫	PROPN
ap-3457	328	3	p	p	PROPN
ap-3457	328	4	cλ(x)s%1(x	cλ(x)s%1(x	NOUN
ap-3457	328	5	)	)	PUNCT
ap-3457	328	6	dx	dx	PROPN
ap-3457	328	7	if	if	SCONJ
ap-3457	328	8	t	t	PROPN
ap-3457	328	9	=	=	PUNCT
ap-3457	328	10	0,∫	0,∫	NOUN
ap-3457	328	11	p	p	NOUN
ap-3457	328	12	sλ+%1(x	sλ+%1(x	PROPN
ap-3457	328	13	)	)	PUNCT
ap-3457	328	14	dx	dx	PROPN
ap-3457	329	1	if	if	SCONJ
ap-3457	329	2	t	t	PROPN
ap-3457	329	3	=	=	SYM
ap-3457	329	4	1,∫	1,∫	NUM
ap-3457	329	5	p	p	PROPN
ap-3457	329	6	ssλ+%s(x)sl%l(x	ssλ+%s(x)sl%l(x	PROPN
ap-3457	329	7	)	)	PUNCT
ap-3457	329	8	dx	dx	PROPN
ap-3457	329	9	if	if	SCONJ
ap-3457	329	10	t	t	PROPN
ap-3457	329	11	=	=	PUNCT
ap-3457	329	12	s,∫	s,∫	PROPN
ap-3457	329	13	p	p	ADJ
ap-3457	329	14	slλ+%l(x)ss%s(x	slλ+%l(x)ss%s(x	PROPN
ap-3457	329	15	)	)	PUNCT
ap-3457	329	16	dx	dx	PROPN
ap-3457	329	17	if	if	SCONJ
ap-3457	329	18	t	t	PROPN
ap-3457	329	19	=	=	SYM
ap-3457	329	20	l.	l.	PROPN
ap-3457	329	21	(	(	PUNCT
ap-3457	329	22	13	13	NUM
ap-3457	329	23	)	)	PUNCT
ap-3457	329	24	the	the	DET
ap-3457	329	25	exact	exact	ADJ
ap-3457	329	26	values	value	NOUN
ap-3457	329	27	of	of	ADP
ap-3457	329	28	btλ	btλ	NOUN
ap-3457	329	29	are	be	AUX
ap-3457	329	30	tabulated	tabulate	VERB
ap-3457	329	31	in	in	ADP
ap-3457	329	32	tab	tab	NOUN
ap-3457	329	33	.	.	PUNCT
ap-3457	330	1	4	4	X
ap-3457	330	2	.	.	X
ap-3457	330	3	we	we	PRON
ap-3457	330	4	choose	choose	VERB
ap-3457	330	5	the	the	DET
ap-3457	330	6	following	follow	VERB
ap-3457	330	7	functions	function	NOUN
ap-3457	330	8	as	as	ADP
ap-3457	330	9	model	model	NOUN
ap-3457	330	10	functions	function	NOUN
ap-3457	330	11	for	for	ADP
ap-3457	330	12	numerical	numerical	ADJ
ap-3457	330	13	tests	test	NOUN
ap-3457	330	14	,	,	PUNCT
ap-3457	330	15	g1(y1	g1(y1	NOUN
ap-3457	330	16	,	,	PUNCT
ap-3457	330	17	y2	y2	NOUN
ap-3457	330	18	)	)	PUNCT
ap-3457	331	1	=	=	PRON
ap-3457	331	2	r1(y20	r1(y20	ADJ
ap-3457	331	3	1	1	NUM
ap-3457	331	4	−	−	PROPN
ap-3457	331	5	y1y2	y1y2	PROPN
ap-3457	332	1	+	+	NUM
ap-3457	332	2	y20	y20	NOUN
ap-3457	332	3	2	2	NUM
ap-3457	332	4	)	)	PUNCT
ap-3457	332	5	,	,	PUNCT
ap-3457	332	6	g2(y1	g2(y1	NOUN
ap-3457	332	7	,	,	PUNCT
ap-3457	332	8	y2	y2	NOUN
ap-3457	332	9	)	)	PUNCT
ap-3457	332	10	=	=	SYM
ap-3457	332	11	r2	r2	PROPN
ap-3457	332	12	(	(	PUNCT
ap-3457	332	13	e−(y2	e−(y2	NUM
ap-3457	332	14	1+(y2	1+(y2	PROPN
ap-3457	332	15	+	+	PROPN
ap-3457	332	16	1.8)2)/2×0.352	1.8)2)/2×0.352	PROPN
ap-3457	332	17	)	)	PUNCT
ap-3457	332	18	,	,	PUNCT
ap-3457	332	19	g3(y1	g3(y1	NOUN
ap-3457	332	20	,	,	PUNCT
ap-3457	332	21	y2	y2	NOUN
ap-3457	332	22	)	)	PUNCT
ap-3457	332	23	=	=	SYM
ap-3457	332	24	r3	r3	PROPN
ap-3457	332	25	1	1	NUM
ap-3457	332	26	+	+	CCONJ
ap-3457	332	27	y2	y2	PROPN
ap-3457	332	28	1	1	NUM
ap-3457	332	29	+	+	CCONJ
ap-3457	332	30	y2	y2	SYM
ap-3457	332	31	2	2	NUM
ap-3457	332	32	,	,	PUNCT
ap-3457	332	33	g4(y1	g4(y1	NOUN
ap-3457	332	34	,	,	PUNCT
ap-3457	332	35	y2	y2	NOUN
ap-3457	332	36	)	)	PUNCT
ap-3457	332	37	=	=	SYM
ap-3457	332	38	r4e	r4e	NOUN
ap-3457	332	39	x+y	x+y	NUM
ap-3457	332	40	,	,	PUNCT
ap-3457	332	41	210	210	NUM
ap-3457	332	42	vol	vol	NOUN
ap-3457	332	43	.	.	PUNCT
ap-3457	333	1	56	56	NUM
ap-3457	333	2	no	no	NOUN
ap-3457	333	3	.	.	PUNCT
ap-3457	334	1	3/2016	3/2016	NUM
ap-3457	334	2	on	on	ADP
ap-3457	334	3	cubature	cubature	ADJ
ap-3457	334	4	rules	rule	NOUN
ap-3457	334	5	associated	associate	VERB
ap-3457	334	6	to	to	ADP
ap-3457	334	7	weyl	weyl	PROPN
ap-3457	334	8	group	group	PROPN
ap-3457	334	9	orbit	orbit	NOUN
ap-3457	334	10	functions	function	NOUN
ap-3457	334	11	b0	b0	NOUN
ap-3457	334	12	(	(	PUNCT
ap-3457	334	13	λ1,λ2	λ1,λ2	PROPN
ap-3457	334	14	)	)	PUNCT
ap-3457	334	15	(	(	PUNCT
ap-3457	334	16	λ1	λ1	ADJ
ap-3457	334	17	,	,	PUNCT
ap-3457	334	18	λ2	λ2	NOUN
ap-3457	334	19	)	)	PUNCT
ap-3457	334	20	=	=	SYM
ap-3457	334	21	(	(	PUNCT
ap-3457	334	22	0	0	NUM
ap-3457	334	23	,	,	PUNCT
ap-3457	334	24	1	1	NUM
ap-3457	334	25	)	)	PUNCT
ap-3457	334	26	−32	−32	X
ap-3457	334	27	3	3	X
ap-3457	334	28	(	(	PUNCT
ap-3457	334	29	λ1	λ1	ADJ
ap-3457	334	30	,	,	PUNCT
ap-3457	334	31	λ2	λ2	NOUN
ap-3457	334	32	)	)	PUNCT
ap-3457	334	33	=	=	PUNCT
ap-3457	334	34	(	(	PUNCT
ap-3457	334	35	2	2	NUM
ap-3457	334	36	,	,	PUNCT
ap-3457	334	37	0	0	NUM
ap-3457	334	38	)	)	PUNCT
ap-3457	334	39	−16	−16	PRON
ap-3457	334	40	3	3	NUM
ap-3457	334	41	(	(	PUNCT
ap-3457	334	42	λ1	λ1	ADJ
ap-3457	334	43	,	,	PUNCT
ap-3457	334	44	λ2	λ2	NOUN
ap-3457	334	45	)	)	PUNCT
ap-3457	334	46	=	=	SYM
ap-3457	334	47	(	(	PUNCT
ap-3457	334	48	2i	2i	NUM
ap-3457	334	49	,	,	PUNCT
ap-3457	334	50	2	2	NUM
ap-3457	334	51	)	)	PUNCT
ap-3457	334	52	,	,	PUNCT
ap-3457	334	53	i	i	PROPN
ap-3457	334	54	6=	6=	ADP
ap-3457	334	55	0	0	NUM
ap-3457	334	56	16(1+(−1)i+1	16(1+(−1)i+1	NUM
ap-3457	334	57	)	)	PUNCT
ap-3457	334	58	3i(i+1	3i(i+1	NOUN
ap-3457	334	59	)	)	PUNCT
ap-3457	334	60	(	(	PUNCT
ap-3457	334	61	λ1	λ1	ADJ
ap-3457	334	62	,	,	PUNCT
ap-3457	334	63	λ2	λ2	NOUN
ap-3457	334	64	)	)	PUNCT
ap-3457	334	65	=	=	SYM
ap-3457	334	66	(	(	PUNCT
ap-3457	334	67	2i	2i	NUM
ap-3457	334	68	,	,	PUNCT
ap-3457	334	69	1	1	NUM
ap-3457	334	70	)	)	PUNCT
ap-3457	334	71	,	,	PUNCT
ap-3457	334	72	i	i	PROPN
ap-3457	334	73	6=	6=	NUM
ap-3457	334	74	0	0	NUM
ap-3457	334	75	16	16	NUM
ap-3457	334	76	[	[	PUNCT
ap-3457	334	77	2	2	NUM
ap-3457	334	78	(	(	PUNCT
ap-3457	334	79	2i+3)(2i−1	2i+3)(2i−1	NUM
ap-3457	334	80	)	)	PUNCT
ap-3457	334	81	+	+	CCONJ
ap-3457	334	82	1+(2i+1)(−1)i+1	1+(2i+1)(−1)i+1	DET
ap-3457	334	83	3i(i+1	3i(i+1	NOUN
ap-3457	334	84	)	)	PUNCT
ap-3457	334	85	]	]	PUNCT
ap-3457	335	1	(	(	PUNCT
ap-3457	335	2	λ1	λ1	ADJ
ap-3457	335	3	,	,	PUNCT
ap-3457	335	4	λ2	λ2	NOUN
ap-3457	335	5	)	)	PUNCT
ap-3457	335	6	=	=	SYM
ap-3457	335	7	(	(	PUNCT
ap-3457	335	8	2i	2i	NUM
ap-3457	335	9	,	,	PUNCT
ap-3457	335	10	2j	2j	X
ap-3457	335	11	+	+	CCONJ
ap-3457	335	12	1	1	NUM
ap-3457	335	13	)	)	PUNCT
ap-3457	335	14	,	,	PUNCT
ap-3457	335	15	j	j	PROPN
ap-3457	335	16	6=	6=	ADP
ap-3457	335	17	0	0	NUM
ap-3457	335	18	64	64	NUM
ap-3457	336	1	|	|	ADV
ap-3457	336	2	stabw	stabw	INTJ
ap-3457	336	3	(	(	PUNCT
ap-3457	336	4	2i,2j+1)|	2i,2j+1)|	NUM
ap-3457	336	5	[	[	PUNCT
ap-3457	336	6	−1+(2j+1)(−1)j	−1+(2j+1)(−1)j	NOUN
ap-3457	336	7	(	(	PUNCT
ap-3457	336	8	(	(	PUNCT
ap-3457	336	9	2i+2j+1)2−4)((2j+1)2−1	2i+2j+1)2−4)((2j+1)2−1	NOUN
ap-3457	336	10	)	)	PUNCT
ap-3457	337	1	+	+	NUM
ap-3457	337	2	−1+(2i+2j+1)(−1)i+j	−1+(2i+2j+1)(−1)i+j	X
ap-3457	337	3	(	(	PUNCT
ap-3457	337	4	(	(	PUNCT
ap-3457	337	5	2i+2j+1)2−1)((2j+1)2−4	2i+2j+1)2−1)((2j+1)2−4	NOUN
ap-3457	337	6	)	)	PUNCT
ap-3457	337	7	]	]	PUNCT
ap-3457	337	8	(	(	PUNCT
ap-3457	337	9	λ1	λ1	ADJ
ap-3457	337	10	,	,	PUNCT
ap-3457	337	11	λ2	λ2	NOUN
ap-3457	337	12	)	)	PUNCT
ap-3457	337	13	=	=	SYM
ap-3457	337	14	(	(	PUNCT
ap-3457	337	15	2i	2i	NUM
ap-3457	337	16	,	,	PUNCT
ap-3457	337	17	2j	2j	NUM
ap-3457	337	18	)	)	PUNCT
ap-3457	337	19	,	,	PUNCT
ap-3457	337	20	j	j	PROPN
ap-3457	337	21	6=	6=	PROPN
ap-3457	337	22	1	1	NUM
ap-3457	337	23	,	,	PUNCT
ap-3457	337	24	i+	i+	NUM
ap-3457	337	25	j	j	PROPN
ap-3457	337	26	6=	6=	SYM
ap-3457	337	27	1	1	NUM
ap-3457	337	28	16	16	NUM
ap-3457	338	1	|	|	ADV
ap-3457	338	2	stabw	stabw	INTJ
ap-3457	338	3	(	(	PUNCT
ap-3457	338	4	2i,2j)|	2i,2j)|	NUM
ap-3457	338	5	[	[	PUNCT
ap-3457	338	6	1+(−1)i+j	1+(−1)i+j	NUM
ap-3457	338	7	(	(	PUNCT
ap-3457	338	8	(	(	PUNCT
ap-3457	338	9	i+j)2−1)(4j2−1	i+j)2−1)(4j2−1	NOUN
ap-3457	338	10	)	)	PUNCT
ap-3457	338	11	+	+	NUM
ap-3457	338	12	1+(−1)j	1+(−1)j	NUM
ap-3457	338	13	(	(	PUNCT
ap-3457	338	14	4(i+j)2−1)(j2−1	4(i+j)2−1)(j2−1	NUM
ap-3457	338	15	)	)	PUNCT
ap-3457	338	16	]	]	PUNCT
ap-3457	339	1	otherwise	otherwise	ADV
ap-3457	339	2	0	0	NUM
ap-3457	339	3	b1	b1	NOUN
ap-3457	339	4	(	(	PUNCT
ap-3457	339	5	λ1,λ2	λ1,λ2	PROPN
ap-3457	339	6	)	)	PUNCT
ap-3457	339	7	(	(	PUNCT
ap-3457	339	8	λ1	λ1	ADJ
ap-3457	339	9	,	,	PUNCT
ap-3457	339	10	λ2	λ2	NOUN
ap-3457	339	11	)	)	PUNCT
ap-3457	339	12	=	=	SYM
ap-3457	339	13	(	(	PUNCT
ap-3457	339	14	2i	2i	NUM
ap-3457	339	15	,	,	PUNCT
ap-3457	339	16	2j	2j	NUM
ap-3457	339	17	)	)	PUNCT
ap-3457	339	18	4(1+(−1)i+j	4(1+(−1)i+j	NUM
ap-3457	339	19	)	)	PUNCT
ap-3457	339	20	(	(	PUNCT
ap-3457	339	21	i+j+1)(2j+1	i+j+1)(2j+1	PROPN
ap-3457	339	22	)	)	PUNCT
ap-3457	339	23	(	(	PUNCT
ap-3457	339	24	λ1	λ1	ADJ
ap-3457	339	25	,	,	PUNCT
ap-3457	339	26	λ2	λ2	NOUN
ap-3457	339	27	)	)	PUNCT
ap-3457	339	28	=	=	SYM
ap-3457	339	29	(	(	PUNCT
ap-3457	339	30	2i	2i	NUM
ap-3457	339	31	,	,	PUNCT
ap-3457	339	32	2j	2j	X
ap-3457	339	33	+	+	CCONJ
ap-3457	339	34	1	1	X
ap-3457	339	35	)	)	PUNCT
ap-3457	339	36	−4(1+(−1)j	−4(1+(−1)j	NOUN
ap-3457	339	37	)	)	PUNCT
ap-3457	339	38	(	(	PUNCT
ap-3457	339	39	2i+2j+3)(j+1	2i+2j+3)(j+1	NUM
ap-3457	339	40	)	)	PUNCT
ap-3457	339	41	otherwise	otherwise	ADV
ap-3457	339	42	0	0	NUM
ap-3457	339	43	bs(λ1,λ2	bs(λ1,λ2	PROPN
ap-3457	339	44	)	)	PUNCT
ap-3457	339	45	(	(	PUNCT
ap-3457	339	46	λ1	λ1	ADJ
ap-3457	339	47	,	,	PUNCT
ap-3457	339	48	λ2	λ2	NOUN
ap-3457	339	49	)	)	PUNCT
ap-3457	339	50	=	=	SYM
ap-3457	339	51	(	(	PUNCT
ap-3457	339	52	0	0	NUM
ap-3457	339	53	,	,	PUNCT
ap-3457	339	54	0	0	NUM
ap-3457	339	55	)	)	PUNCT
ap-3457	339	56	8	8	NUM
ap-3457	339	57	(	(	PUNCT
ap-3457	339	58	λ1	λ1	ADJ
ap-3457	339	59	,	,	PUNCT
ap-3457	339	60	λ2	λ2	NOUN
ap-3457	339	61	)	)	PUNCT
ap-3457	339	62	=	=	SYM
ap-3457	339	63	(	(	PUNCT
ap-3457	339	64	2i	2i	NUM
ap-3457	339	65	,	,	PUNCT
ap-3457	339	66	1	1	NUM
ap-3457	339	67	)	)	PUNCT
ap-3457	339	68	16	16	NUM
ap-3457	339	69	(	(	PUNCT
ap-3457	339	70	2i+3)(2i+1	2i+3)(2i+1	NUM
ap-3457	339	71	)	)	PUNCT
ap-3457	339	72	(	(	PUNCT
ap-3457	339	73	λ1	λ1	ADJ
ap-3457	339	74	,	,	PUNCT
ap-3457	339	75	λ2	λ2	NOUN
ap-3457	339	76	)	)	PUNCT
ap-3457	339	77	=	=	SYM
ap-3457	339	78	(	(	PUNCT
ap-3457	339	79	2i	2i	NUM
ap-3457	339	80	,	,	PUNCT
ap-3457	339	81	2j	2j	NUM
ap-3457	339	82	)	)	PUNCT
ap-3457	339	83	,	,	PUNCT
ap-3457	339	84	i+	i+	NUM
ap-3457	339	85	j	j	PROPN
ap-3457	339	86	6=	6=	ADP
ap-3457	339	87	0	0	NUM
ap-3457	339	88	8(1+(2i+2j+1)(−1)i+j+1	8(1+(2i+2j+1)(−1)i+j+1	NUM
ap-3457	339	89	)	)	PUNCT
ap-3457	340	1	|	|	ADV
ap-3457	340	2	stabw	stabw	INTJ
ap-3457	340	3	(	(	PUNCT
ap-3457	340	4	2i+1,2j)|(i+j+1)(i+j)(4j2−1	2i+1,2j)|(i+j+1)(i+j)(4j2−1	NUM
ap-3457	340	5	)	)	PUNCT
ap-3457	340	6	(	(	PUNCT
ap-3457	340	7	λ1	λ1	ADJ
ap-3457	340	8	,	,	PUNCT
ap-3457	340	9	λ2	λ2	NOUN
ap-3457	340	10	)	)	PUNCT
ap-3457	340	11	=	=	SYM
ap-3457	340	12	(	(	PUNCT
ap-3457	340	13	2i	2i	NUM
ap-3457	340	14	,	,	PUNCT
ap-3457	340	15	2j	2j	X
ap-3457	340	16	+	+	CCONJ
ap-3457	340	17	1	1	NUM
ap-3457	340	18	)	)	PUNCT
ap-3457	340	19	,	,	PUNCT
ap-3457	340	20	j	j	PROPN
ap-3457	340	21	6=	6=	ADP
ap-3457	340	22	0	0	NUM
ap-3457	340	23	8(−1+(2j+1)(−1)j	8(−1+(2j+1)(−1)j	NUM
ap-3457	340	24	)	)	PUNCT
ap-3457	340	25	(	(	PUNCT
ap-3457	340	26	2i+2j+3)(2i+2j+1)j(j+1	2i+2j+3)(2i+2j+1)j(j+1	NUM
ap-3457	340	27	)	)	PUNCT
ap-3457	340	28	otherwise	otherwise	ADV
ap-3457	340	29	0	0	NUM
ap-3457	340	30	bl(λ1,λ2	bl(λ1,λ2	PROPN
ap-3457	340	31	)	)	PUNCT
ap-3457	340	32	(	(	PUNCT
ap-3457	340	33	λ1	λ1	ADJ
ap-3457	340	34	,	,	PUNCT
ap-3457	340	35	λ2	λ2	NOUN
ap-3457	340	36	)	)	PUNCT
ap-3457	340	37	=	=	SYM
ap-3457	340	38	(	(	PUNCT
ap-3457	340	39	0	0	NUM
ap-3457	340	40	,	,	PUNCT
ap-3457	340	41	0	0	NUM
ap-3457	340	42	)	)	PUNCT
ap-3457	340	43	8	8	NUM
ap-3457	340	44	(	(	PUNCT
ap-3457	340	45	λ1	λ1	ADJ
ap-3457	340	46	,	,	PUNCT
ap-3457	340	47	λ2	λ2	NOUN
ap-3457	340	48	)	)	PUNCT
ap-3457	340	49	=	=	SYM
ap-3457	340	50	(	(	PUNCT
ap-3457	340	51	2i	2i	NUM
ap-3457	340	52	,	,	PUNCT
ap-3457	340	53	0	0	NUM
ap-3457	340	54	)	)	PUNCT
ap-3457	340	55	,	,	PUNCT
ap-3457	340	56	i	i	PROPN
ap-3457	340	57	6=	6=	ADP
ap-3457	340	58	0	0	NUM
ap-3457	340	59	8	8	NUM
ap-3457	340	60	[	[	PUNCT
ap-3457	340	61	2i+1+(−1)i+1	2i+1+(−1)i+1	NUM
ap-3457	340	62	2i(i+1	2i(i+1	NUM
ap-3457	340	63	)	)	PUNCT
ap-3457	341	1	+	+	CCONJ
ap-3457	341	2	1	1	NUM
ap-3457	341	3	2i+1	2i+1	NOUN
ap-3457	341	4	]	]	PUNCT
ap-3457	341	5	(	(	PUNCT
ap-3457	341	6	λ1	λ1	ADJ
ap-3457	341	7	,	,	PUNCT
ap-3457	341	8	λ2	λ2	NOUN
ap-3457	341	9	)	)	PUNCT
ap-3457	341	10	=	=	SYM
ap-3457	341	11	(	(	PUNCT
ap-3457	341	12	2i	2i	NUM
ap-3457	341	13	,	,	PUNCT
ap-3457	341	14	2j	2j	NUM
ap-3457	341	15	)	)	PUNCT
ap-3457	341	16	,	,	PUNCT
ap-3457	341	17	j	j	PROPN
ap-3457	341	18	6=	6=	ADP
ap-3457	341	19	0	0	NUM
ap-3457	341	20	4	4	NUM
ap-3457	342	1	|	|	ADV
ap-3457	342	2	stabw	stabw	INTJ
ap-3457	342	3	(	(	PUNCT
ap-3457	342	4	2i,2j+1)|	2i,2j+1)|	NUM
ap-3457	342	5	[	[	PUNCT
ap-3457	342	6	2i+2j+1+(−1)i+j+1	2i+2j+1+(−1)i+j+1	NUM
ap-3457	342	7	(	(	PUNCT
ap-3457	342	8	i+j+1)(2j+1)(i+j	i+j+1)(2j+1)(i+j	NOUN
ap-3457	342	9	)	)	PUNCT
ap-3457	343	1	+	+	CCONJ
ap-3457	344	1	2j+1+(−1)j+1	2j+1+(−1)j+1	NUM
ap-3457	344	2	(	(	PUNCT
ap-3457	344	3	2i+2j+1)(j+1)j	2i+2j+1)(j+1)j	NOUN
ap-3457	344	4	]	]	PUNCT
ap-3457	344	5	(	(	PUNCT
ap-3457	344	6	λ1	λ1	ADJ
ap-3457	344	7	,	,	PUNCT
ap-3457	344	8	λ2	λ2	NOUN
ap-3457	344	9	)	)	PUNCT
ap-3457	344	10	=	=	SYM
ap-3457	344	11	(	(	PUNCT
ap-3457	344	12	2i	2i	NUM
ap-3457	344	13	,	,	PUNCT
ap-3457	344	14	2j	2j	X
ap-3457	344	15	+	+	CCONJ
ap-3457	344	16	1	1	NUM
ap-3457	344	17	)	)	PUNCT
ap-3457	344	18	,	,	PUNCT
ap-3457	344	19	16	16	NUM
ap-3457	345	1	|	|	ADV
ap-3457	345	2	stabw	stabw	INTJ
ap-3457	345	3	(	(	PUNCT
ap-3457	345	4	2i,2j+2)|	2i,2j+2)|	PROPN
ap-3457	345	5	[	[	PUNCT
ap-3457	345	6	(	(	PUNCT
ap-3457	345	7	−1+(−1)j+1)(i+j+1	−1+(−1)j+1)(i+j+1	PROPN
ap-3457	345	8	)	)	PUNCT
ap-3457	345	9	(	(	PUNCT
ap-3457	345	10	2i+2j+3)(j+1)(2i+2j+1	2i+2j+3)(j+1)(2i+2j+1	NUM
ap-3457	345	11	)	)	PUNCT
ap-3457	345	12	+	+	CCONJ
ap-3457	345	13	(	(	PUNCT
ap-3457	345	14	−1+(−1)i+j+1)(j+1	−1+(−1)i+j+1)(j+1	PROPN
ap-3457	345	15	)	)	PUNCT
ap-3457	345	16	(	(	PUNCT
ap-3457	345	17	i+j+1)(2j+3)(2j+1	i+j+1)(2j+3)(2j+1	NUM
ap-3457	345	18	)	)	PUNCT
ap-3457	345	19	]	]	PUNCT
ap-3457	345	20	otherwise	otherwise	ADV
ap-3457	345	21	0	0	NUM
ap-3457	345	22	table	table	NOUN
ap-3457	345	23	4	4	NUM
ap-3457	345	24	.	.	PUNCT
ap-3457	346	1	values	value	NOUN
ap-3457	346	2	of	of	ADP
ap-3457	346	3	btλ	btλ	NOUN
ap-3457	346	4	,	,	PUNCT
ap-3457	346	5	given	give	VERB
ap-3457	346	6	by	by	ADP
ap-3457	346	7	(	(	PUNCT
ap-3457	346	8	13	13	NUM
ap-3457	346	9	)	)	PUNCT
ap-3457	346	10	,	,	PUNCT
ap-3457	346	11	for	for	ADP
ap-3457	346	12	t	t	PROPN
ap-3457	346	13	∈	∈	PROPN
ap-3457	346	14	{	{	PUNCT
ap-3457	346	15	0	0	NUM
ap-3457	346	16	,	,	PUNCT
ap-3457	346	17	1	1	NUM
ap-3457	346	18	,	,	PUNCT
ap-3457	346	19	s	s	X
ap-3457	346	20	,	,	PUNCT
ap-3457	346	21	l	l	NOUN
ap-3457	346	22	}	}	PUNCT
ap-3457	346	23	and	and	CCONJ
ap-3457	346	24	λ	λ	X
ap-3457	346	25	=	=	PUNCT
ap-3457	347	1	λ1ω1	λ1ω1	X
ap-3457	347	2	+	+	NUM
ap-3457	347	3	λ2ω2	λ2ω2	NOUN
ap-3457	347	4	.	.	PUNCT
ap-3457	348	1	the	the	DET
ap-3457	348	2	indices	index	NOUN
ap-3457	348	3	i	i	PRON
ap-3457	348	4	,	,	PUNCT
ap-3457	348	5	j	j	PROPN
ap-3457	348	6	are	be	AUX
ap-3457	348	7	non	non	ADJ
ap-3457	348	8	-	-	ADJ
ap-3457	348	9	negative	negative	ADJ
ap-3457	348	10	integers	integer	NOUN
ap-3457	348	11	.	.	PUNCT
ap-3457	349	1	g5(y1	g5(y1	VERB
ap-3457	349	2	,	,	PUNCT
ap-3457	349	3	y2	y2	PROPN
ap-3457	349	4	)	)	PUNCT
ap-3457	350	1	=	=	PRON
ap-3457	350	2	{	{	PUNCT
ap-3457	350	3	r5	r5	PROPN
ap-3457	350	4	if	if	SCONJ
ap-3457	350	5	y2	y2	PROPN
ap-3457	350	6	1	1	NUM
ap-3457	350	7	+	+	CCONJ
ap-3457	350	8	(	(	PUNCT
ap-3457	350	9	y2	y2	INTJ
ap-3457	350	10	+	+	CCONJ
ap-3457	350	11	1.5)2	1.5)2	NUM
ap-3457	350	12	≤	≤	NUM
ap-3457	350	13	1	1	NUM
ap-3457	350	14	,	,	PUNCT
ap-3457	350	15	0	0	NUM
ap-3457	350	16	otherwise	otherwise	ADV
ap-3457	350	17	.	.	PUNCT
ap-3457	351	1	each	each	DET
ap-3457	351	2	value	value	NOUN
ap-3457	351	3	of	of	ADP
ap-3457	351	4	ri	ri	PROPN
ap-3457	351	5	∈	∈	PROPN
ap-3457	351	6	r	r	NOUN
ap-3457	351	7	is	be	AUX
ap-3457	351	8	set	set	VERB
ap-3457	351	9	to	to	PART
ap-3457	351	10	satisfy	satisfy	VERB
ap-3457	351	11	the	the	DET
ap-3457	351	12	normalization	normalization	NOUN
ap-3457	351	13	condition	condition	NOUN
ap-3457	351	14	∫	∫	PROPN
ap-3457	351	15	t	t	PROPN
ap-3457	351	16	gi(y	gi(y	NOUN
ap-3457	351	17	)	)	PUNCT
ap-3457	351	18	dy	dy	NOUN
ap-3457	351	19	=	=	SYM
ap-3457	352	1	1	1	X
ap-3457	352	2	.	.	X
ap-3457	353	1	we	we	PRON
ap-3457	353	2	compute	compute	VERB
ap-3457	353	3	the	the	DET
ap-3457	353	4	approximations	approximation	NOUN
ap-3457	353	5	itm	itm	NOUN
ap-3457	353	6	(	(	PUNCT
ap-3457	353	7	gi	gi	INTJ
ap-3457	353	8	)	)	PUNCT
ap-3457	353	9	=	=	PUNCT
ap-3457	353	10	∑	∑	PUNCT
ap-3457	353	11	λ∈p+	λ∈p+	PUNCT
ap-3457	353	12	〈	〈	PROPN
ap-3457	353	13	λ	λ	PROPN
ap-3457	353	14	,	,	PUNCT
ap-3457	353	15	η〉≤m	η〉≤m	NOUN
ap-3457	353	16	btλbtλ	btλbtλ	NOUN
ap-3457	353	17	(	(	PUNCT
ap-3457	353	18	14	14	NUM
ap-3457	353	19	)	)	PUNCT
ap-3457	353	20	of	of	ADP
ap-3457	353	21	∫	∫	PROPN
ap-3457	353	22	t	t	PROPN
ap-3457	353	23	gi(y	gi(y	NOUN
ap-3457	353	24	)	)	PUNCT
ap-3457	353	25	dy	dy	NOUN
ap-3457	353	26	=	=	NOUN
ap-3457	353	27	1	1	NUM
ap-3457	353	28	with	with	ADP
ap-3457	353	29	the	the	DET
ap-3457	353	30	formula	formula	NOUN
ap-3457	353	31	for	for	ADP
ap-3457	353	32	btλ	btλ	NOUN
ap-3457	353	33	given	give	VERB
ap-3457	353	34	by	by	ADP
ap-3457	353	35	(	(	PUNCT
ap-3457	353	36	13	13	NUM
ap-3457	353	37	)	)	PUNCT
ap-3457	353	38	.	.	PUNCT
ap-3457	354	1	figs	fig	NOUN
ap-3457	354	2	.	.	PUNCT
ap-3457	354	3	8	8	NUM
ap-3457	354	4	and	and	CCONJ
ap-3457	354	5	9	9	NUM
ap-3457	354	6	show	show	NOUN
ap-3457	354	7	for	for	ADP
ap-3457	354	8	t	t	PROPN
ap-3457	354	9	∈	∈	PROPN
ap-3457	354	10	{	{	PUNCT
ap-3457	354	11	0	0	NUM
ap-3457	354	12	,	,	PUNCT
ap-3457	354	13	1	1	NUM
ap-3457	354	14	,	,	PUNCT
ap-3457	354	15	s	s	X
ap-3457	354	16	,	,	PUNCT
ap-3457	354	17	l	l	NOUN
ap-3457	354	18	}	}	PUNCT
ap-3457	354	19	the	the	DET
ap-3457	354	20	graphs	graph	NOUN
ap-3457	354	21	of	of	ADP
ap-3457	354	22	of	of	ADP
ap-3457	354	23	the	the	DET
ap-3457	354	24	absolute	absolute	ADJ
ap-3457	354	25	value	value	NOUN
ap-3457	354	26	of	of	ADP
ap-3457	354	27	the	the	DET
ap-3457	354	28	difference	difference	NOUN
ap-3457	354	29	|1−	|1−	INTJ
ap-3457	354	30	itm	itm	PROPN
ap-3457	354	31	(	(	PUNCT
ap-3457	354	32	gi)|	gi)|	NOUN
ap-3457	354	33	.	.	NOUN
ap-3457	355	1	5	5	NUM
ap-3457	355	2	.	.	X
ap-3457	355	3	concluding	conclude	VERB
ap-3457	355	4	remarks	remark	NOUN
ap-3457	355	5	(	(	PUNCT
ap-3457	355	6	1	1	NUM
ap-3457	355	7	.	.	PUNCT
ap-3457	355	8	)	)	PUNCT
ap-3457	355	9	establishing	establish	VERB
ap-3457	355	10	the	the	DET
ap-3457	355	11	explicit	explicit	ADJ
ap-3457	355	12	connection	connection	NOUN
ap-3457	355	13	between	between	ADP
ap-3457	355	14	the	the	DET
ap-3457	355	15	jacobi	jacobi	PROPN
ap-3457	355	16	and	and	CCONJ
ap-3457	355	17	macdonald	macdonald	PROPN
ap-3457	355	18	polynomials	polynomial	NOUN
ap-3457	355	19	and	and	CCONJ
ap-3457	355	20	the	the	DET
ap-3457	355	21	weyl	weyl	PROPN
ap-3457	355	22	group	group	NOUN
ap-3457	355	23	orbit	orbit	NOUN
ap-3457	355	24	functions	function	NOUN
ap-3457	355	25	in	in	ADP
ap-3457	355	26	section	section	NOUN
ap-3457	355	27	2.3	2.3	NUM
ap-3457	355	28	forms	form	NOUN
ap-3457	355	29	a	a	DET
ap-3457	355	30	crucial	crucial	ADJ
ap-3457	355	31	step	step	NOUN
ap-3457	355	32	for	for	ADP
ap-3457	355	33	generalizing	generalize	VERB
ap-3457	355	34	known	know	VERB
ap-3457	355	35	cubature	cubature	NOUN
ap-3457	355	36	formulas	formula	NOUN
ap-3457	355	37	to	to	ADP
ap-3457	355	38	the	the	DET
ap-3457	355	39	entire	entire	ADJ
ap-3457	355	40	class	class	NOUN
ap-3457	355	41	of	of	ADP
ap-3457	355	42	the	the	DET
ap-3457	355	43	jacobi	jacobi	PROPN
ap-3457	355	44	polynomials	polynomial	NOUN
ap-3457	355	45	.	.	PUNCT
ap-3457	356	1	(	(	PUNCT
ap-3457	356	2	2	2	NUM
ap-3457	356	3	.	.	PUNCT
ap-3457	356	4	)	)	PUNCT
ap-3457	356	5	numerical	numerical	ADJ
ap-3457	356	6	tests	test	NOUN
ap-3457	356	7	results	result	VERB
ap-3457	356	8	in	in	ADP
ap-3457	356	9	figs	fig	NOUN
ap-3457	356	10	.	.	PUNCT
ap-3457	357	1	5	5	NUM
ap-3457	357	2	and	and	CCONJ
ap-3457	357	3	8	8	NUM
ap-3457	357	4	indicate	indicate	VERB
ap-3457	357	5	in	in	ADP
ap-3457	357	6	general	general	ADJ
ap-3457	357	7	excellent	excellent	ADJ
ap-3457	357	8	convergence	convergence	NOUN
ap-3457	357	9	rates	rate	NOUN
ap-3457	357	10	of	of	ADP
ap-3457	357	11	the	the	DET
ap-3457	357	12	developed	develop	VERB
ap-3457	357	13	cubature	cubature	ADJ
ap-3457	357	14	rules	rule	NOUN
ap-3457	357	15	including	include	VERB
ap-3457	357	16	their	their	PRON
ap-3457	357	17	clenshaw	clenshaw	ADJ
ap-3457	357	18	-	-	PUNCT
ap-3457	357	19	curtis	curtis	NOUN
ap-3457	357	20	211	211	NUM
ap-3457	357	21	l.	l.	PROPN
ap-3457	357	22	háková	háková	PROPN
ap-3457	357	23	,	,	PUNCT
ap-3457	357	24	j.	j.	PROPN
ap-3457	357	25	hrivnák	hrivnák	PROPN
ap-3457	357	26	,	,	PUNCT
ap-3457	357	27	l.	l.	PROPN
ap-3457	357	28	motlochová	motlochová	PROPN
ap-3457	357	29	acta	acta	PROPN
ap-3457	357	30	polytechnica	polytechnica	PROPN
ap-3457	357	31	figure	figure	NOUN
ap-3457	357	32	8	8	NUM
ap-3457	357	33	.	.	PUNCT
ap-3457	358	1	the	the	DET
ap-3457	358	2	graphs	graph	NOUN
ap-3457	358	3	of	of	ADP
ap-3457	358	4	error	error	NOUN
ap-3457	358	5	values	value	NOUN
ap-3457	358	6	|1−itm	|1−itm	PUNCT
ap-3457	358	7	(	(	PUNCT
ap-3457	358	8	gi)|	gi)|	NOUN
ap-3457	358	9	of	of	ADP
ap-3457	358	10	the	the	DET
ap-3457	358	11	integral	integral	ADJ
ap-3457	358	12	∫	∫	PROPN
ap-3457	358	13	t	t	PROPN
ap-3457	358	14	gi(y	gi(y	NOUN
ap-3457	358	15	)	)	PUNCT
ap-3457	358	16	dy	dy	NOUN
ap-3457	358	17	=	=	SYM
ap-3457	358	18	1	1	NUM
ap-3457	358	19	,	,	PUNCT
ap-3457	358	20	i	i	PRON
ap-3457	358	21	=	=	NOUN
ap-3457	358	22	1	1	NUM
ap-3457	358	23	,	,	PUNCT
ap-3457	358	24	.	.	PUNCT
ap-3457	358	25	.	.	PUNCT
ap-3457	358	26	.	.	PUNCT
ap-3457	359	1	,	,	PUNCT
ap-3457	359	2	4	4	NUM
ap-3457	359	3	and	and	CCONJ
ap-3457	359	4	its	its	PRON
ap-3457	359	5	approximations	approximation	NOUN
ap-3457	359	6	itm	itm	NOUN
ap-3457	359	7	(	(	PUNCT
ap-3457	359	8	gi	gi	NOUN
ap-3457	359	9	)	)	PUNCT
ap-3457	359	10	,	,	PUNCT
ap-3457	359	11	m	m	VERB
ap-3457	359	12	=	=	NOUN
ap-3457	359	13	10	10	NUM
ap-3457	359	14	,	,	PUNCT
ap-3457	359	15	11	11	NUM
ap-3457	359	16	,	,	PUNCT
ap-3457	359	17	.	.	PUNCT
ap-3457	359	18	.	.	PUNCT
ap-3457	360	1	.	.	PUNCT
ap-3457	361	1	,	,	PUNCT
ap-3457	361	2	50	50	NUM
ap-3457	361	3	given	give	VERB
ap-3457	361	4	by	by	ADP
ap-3457	361	5	(	(	PUNCT
ap-3457	361	6	14	14	NUM
ap-3457	361	7	)	)	PUNCT
ap-3457	361	8	.	.	PUNCT
ap-3457	362	1	the	the	DET
ap-3457	362	2	values	value	NOUN
ap-3457	362	3	for	for	ADP
ap-3457	362	4	t	t	NOUN
ap-3457	362	5	=	=	SYM
ap-3457	362	6	0	0	NUM
ap-3457	362	7	,	,	PUNCT
ap-3457	362	8	1	1	NUM
ap-3457	362	9	,	,	PUNCT
ap-3457	362	10	s	s	AUX
ap-3457	362	11	,	,	PUNCT
ap-3457	362	12	l	l	NOUN
ap-3457	362	13	are	be	AUX
ap-3457	362	14	depicted	depict	VERB
ap-3457	362	15	as	as	ADP
ap-3457	362	16	circles	circle	NOUN
ap-3457	362	17	,	,	PUNCT
ap-3457	362	18	“	"	PUNCT
ap-3457	362	19	+	+	ADJ
ap-3457	362	20	”	"	PUNCT
ap-3457	362	21	,	,	PUNCT
ap-3457	362	22	diamonds	diamond	NOUN
ap-3457	362	23	and	and	CCONJ
ap-3457	362	24	“	"	PUNCT
ap-3457	362	25	×	×	NOUN
ap-3457	362	26	”	"	PUNCT
ap-3457	362	27	,	,	PUNCT
ap-3457	362	28	respectively	respectively	ADV
ap-3457	362	29	.	.	PUNCT
ap-3457	363	1	figure	figure	NOUN
ap-3457	363	2	9	9	NUM
ap-3457	363	3	.	.	PUNCT
ap-3457	364	1	the	the	DET
ap-3457	364	2	graphs	graph	NOUN
ap-3457	364	3	of	of	ADP
ap-3457	364	4	error	error	NOUN
ap-3457	364	5	values	value	NOUN
ap-3457	364	6	|1	|1	PUNCT
ap-3457	364	7	−	−	X
ap-3457	364	8	itm	itm	NOUN
ap-3457	364	9	(	(	PUNCT
ap-3457	364	10	g5)|	g5)|	NOUN
ap-3457	364	11	of	of	ADP
ap-3457	364	12	the	the	DET
ap-3457	364	13	integral	integral	ADJ
ap-3457	364	14	∫	∫	PROPN
ap-3457	364	15	t	t	PROPN
ap-3457	364	16	g5(y	g5(y	NOUN
ap-3457	364	17	)	)	PUNCT
ap-3457	364	18	dy	dy	NOUN
ap-3457	364	19	=	=	SYM
ap-3457	364	20	1	1	NUM
ap-3457	364	21	and	and	CCONJ
ap-3457	364	22	its	its	PRON
ap-3457	364	23	approximations	approximation	NOUN
ap-3457	364	24	itm	itm	NOUN
ap-3457	364	25	(	(	PUNCT
ap-3457	364	26	g5	g5	PROPN
ap-3457	364	27	)	)	PUNCT
ap-3457	364	28	,	,	PUNCT
ap-3457	364	29	m	m	VERB
ap-3457	364	30	=	=	NOUN
ap-3457	364	31	10	10	NUM
ap-3457	364	32	,	,	PUNCT
ap-3457	364	33	15	15	NUM
ap-3457	364	34	,	,	PUNCT
ap-3457	364	35	20	20	NUM
ap-3457	364	36	,	,	PUNCT
ap-3457	364	37	.	.	PUNCT
ap-3457	364	38	.	.	PUNCT
ap-3457	364	39	.	.	PUNCT
ap-3457	365	1	,	,	PUNCT
ap-3457	365	2	170	170	NUM
ap-3457	365	3	given	give	VERB
ap-3457	365	4	by	by	ADP
ap-3457	365	5	(	(	PUNCT
ap-3457	365	6	14	14	NUM
ap-3457	365	7	)	)	PUNCT
ap-3457	365	8	.	.	PUNCT
ap-3457	366	1	the	the	DET
ap-3457	366	2	values	value	NOUN
ap-3457	366	3	for	for	ADP
ap-3457	366	4	t	t	NOUN
ap-3457	366	5	=	=	SYM
ap-3457	366	6	0	0	NUM
ap-3457	366	7	,	,	PUNCT
ap-3457	366	8	1	1	NUM
ap-3457	366	9	,	,	PUNCT
ap-3457	366	10	s	s	AUX
ap-3457	366	11	,	,	PUNCT
ap-3457	366	12	l	l	NOUN
ap-3457	366	13	are	be	AUX
ap-3457	366	14	depicted	depict	VERB
ap-3457	366	15	as	as	ADP
ap-3457	366	16	circles	circle	NOUN
ap-3457	366	17	,	,	PUNCT
ap-3457	366	18	“	"	PUNCT
ap-3457	366	19	+	+	ADJ
ap-3457	366	20	”	"	PUNCT
ap-3457	366	21	,	,	PUNCT
ap-3457	366	22	diamonds	diamond	NOUN
ap-3457	366	23	and	and	CCONJ
ap-3457	366	24	“	"	PUNCT
ap-3457	366	25	×	×	NOUN
ap-3457	366	26	”	"	PUNCT
ap-3457	366	27	,	,	PUNCT
ap-3457	366	28	respectively	respectively	ADV
ap-3457	366	29	.	.	PUNCT
ap-3457	367	1	versions	version	NOUN
ap-3457	367	2	for	for	ADP
ap-3457	367	3	the	the	DET
ap-3457	367	4	case	case	NOUN
ap-3457	367	5	c2	c2	PROPN
ap-3457	367	6	.	.	PUNCT
ap-3457	368	1	the	the	DET
ap-3457	368	2	convergence	convergence	NOUN
ap-3457	368	3	rate	rate	NOUN
ap-3457	368	4	of	of	ADP
ap-3457	368	5	the	the	DET
ap-3457	368	6	multidimensional	multidimensional	ADJ
ap-3457	368	7	step	step	NOUN
ap-3457	368	8	-	-	PUNCT
ap-3457	368	9	functions	function	NOUN
ap-3457	368	10	,	,	PUNCT
ap-3457	368	11	even	even	ADV
ap-3457	368	12	though	though	SCONJ
ap-3457	368	13	less	less	ADJ
ap-3457	368	14	uniform	uniform	ADJ
ap-3457	368	15	,	,	PUNCT
ap-3457	368	16	still	still	ADV
ap-3457	368	17	appears	appear	VERB
ap-3457	368	18	to	to	PART
ap-3457	368	19	be	be	AUX
ap-3457	368	20	very	very	ADV
ap-3457	368	21	good	good	ADJ
ap-3457	368	22	.	.	PUNCT
ap-3457	369	1	developing	develop	VERB
ap-3457	369	2	similar	similar	ADJ
ap-3457	369	3	methods	method	NOUN
ap-3457	369	4	for	for	ADP
ap-3457	369	5	the	the	DET
ap-3457	369	6	two	two	NUM
ap-3457	369	7	-	-	PUNCT
ap-3457	369	8	variable	variable	NOUN
ap-3457	369	9	case	case	NOUN
ap-3457	369	10	g2	g2	PROPN
ap-3457	369	11	and	and	CCONJ
ap-3457	369	12	extending	extend	VERB
ap-3457	369	13	the	the	DET
ap-3457	369	14	rules	rule	NOUN
ap-3457	369	15	to	to	ADP
ap-3457	369	16	higher	high	ADJ
ap-3457	369	17	dimensions	dimension	NOUN
ap-3457	369	18	poses	pose	VERB
ap-3457	369	19	an	an	DET
ap-3457	369	20	open	open	ADJ
ap-3457	369	21	problem	problem	NOUN
ap-3457	369	22	.	.	PUNCT
ap-3457	370	1	(	(	PUNCT
ap-3457	370	2	3	3	NUM
ap-3457	370	3	.	.	PUNCT
ap-3457	370	4	)	)	PUNCT
ap-3457	371	1	the	the	DET
ap-3457	371	2	hyperinterpolation	hyperinterpolation	NOUN
ap-3457	371	3	methods	method	NOUN
ap-3457	371	4	[	[	X
ap-3457	371	5	3	3	NUM
ap-3457	371	6	,	,	PUNCT
ap-3457	371	7	25	25	NUM
ap-3457	371	8	,	,	PUNCT
ap-3457	371	9	34	34	NUM
ap-3457	371	10	,	,	PUNCT
ap-3457	371	11	35	35	NUM
ap-3457	371	12	]	]	PUNCT
ap-3457	371	13	are	be	AUX
ap-3457	371	14	among	among	ADP
ap-3457	371	15	the	the	DET
ap-3457	371	16	tools	tool	NOUN
ap-3457	371	17	which	which	PRON
ap-3457	371	18	directly	directly	ADV
ap-3457	371	19	use	use	VERB
ap-3457	371	20	cubature	cubature	ADJ
ap-3457	371	21	rules	rule	NOUN
ap-3457	371	22	.	.	PUNCT
ap-3457	372	1	for	for	ADP
ap-3457	372	2	the	the	DET
ap-3457	372	3	standard	standard	ADJ
ap-3457	372	4	cubature	cubature	ADJ
ap-3457	372	5	rules	rule	NOUN
ap-3457	372	6	of	of	ADP
ap-3457	372	7	the	the	DET
ap-3457	372	8	weyl	weyl	PROPN
ap-3457	372	9	group	group	NOUN
ap-3457	372	10	orbit	orbit	NOUN
ap-3457	372	11	functions	function	NOUN
ap-3457	372	12	,	,	PUNCT
ap-3457	372	13	several	several	ADJ
ap-3457	372	14	tests	test	NOUN
ap-3457	372	15	with	with	ADP
ap-3457	372	16	very	very	ADV
ap-3457	372	17	good	good	ADJ
ap-3457	372	18	results	result	NOUN
ap-3457	372	19	are	be	AUX
ap-3457	372	20	also	also	ADV
ap-3457	372	21	performed	perform	VERB
ap-3457	372	22	in	in	ADP
ap-3457	372	23	[	[	X
ap-3457	372	24	15	15	NUM
ap-3457	372	25	]	]	PUNCT
ap-3457	372	26	.	.	PUNCT
ap-3457	373	1	developing	develop	VERB
ap-3457	373	2	and	and	CCONJ
ap-3457	373	3	testing	testing	NOUN
ap-3457	373	4	hyperinterpolation	hyperinterpolation	NOUN
ap-3457	373	5	methods	method	NOUN
ap-3457	373	6	for	for	ADP
ap-3457	373	7	the	the	DET
ap-3457	373	8	presented	present	VERB
ap-3457	373	9	cubature	cubature	NOUN
ap-3457	373	10	rules	rule	NOUN
ap-3457	373	11	merits	merit	VERB
ap-3457	373	12	further	further	ADJ
ap-3457	373	13	study	study	NOUN
ap-3457	373	14	.	.	PUNCT
ap-3457	374	1	(	(	PUNCT
ap-3457	374	2	4	4	NUM
ap-3457	374	3	.	.	PUNCT
ap-3457	374	4	)	)	PUNCT
ap-3457	375	1	the	the	DET
ap-3457	375	2	present	present	ADJ
ap-3457	375	3	work	work	NOUN
ap-3457	375	4	demonstrates	demonstrate	VERB
ap-3457	375	5	wide	wide	ADJ
ap-3457	375	6	variety	variety	NOUN
ap-3457	375	7	of	of	ADP
ap-3457	375	8	possibilities	possibility	NOUN
ap-3457	375	9	of	of	ADP
ap-3457	375	10	constructing	construct	VERB
ap-3457	375	11	the	the	DET
ap-3457	375	12	cubature	cubature	ADJ
ap-3457	375	13	rules	rule	NOUN
ap-3457	375	14	in	in	ADP
ap-3457	375	15	the	the	DET
ap-3457	375	16	orbit	orbit	NOUN
ap-3457	375	17	functions	function	NOUN
ap-3457	375	18	setting	set	VERB
ap-3457	375	19	.	.	PUNCT
ap-3457	376	1	comparison	comparison	NOUN
ap-3457	376	2	of	of	ADP
ap-3457	376	3	the	the	DET
ap-3457	376	4	developed	develop	VERB
ap-3457	376	5	methods	method	NOUN
ap-3457	376	6	is	be	AUX
ap-3457	376	7	necessary	necessary	ADJ
ap-3457	376	8	for	for	ADP
ap-3457	376	9	establishing	establish	VERB
ap-3457	376	10	range	range	NOUN
ap-3457	376	11	of	of	ADP
ap-3457	376	12	their	their	PRON
ap-3457	376	13	viable	viable	ADJ
ap-3457	376	14	applications	application	NOUN
ap-3457	376	15	.	.	PUNCT
ap-3457	377	1	especially	especially	ADV
ap-3457	377	2	,	,	PUNCT
ap-3457	377	3	comparison	comparison	NOUN
ap-3457	377	4	of	of	ADP
ap-3457	377	5	the	the	DET
ap-3457	377	6	gauss	gauss	ADJ
ap-3457	377	7	and	and	CCONJ
ap-3457	377	8	clenshaw	clenshaw	ADJ
ap-3457	377	9	-	-	PUNCT
ap-3457	377	10	curtis	curtis	NOUN
ap-3457	377	11	cubature	cubature	ADJ
ap-3457	377	12	methods	method	NOUN
ap-3457	377	13	,	,	PUNCT
ap-3457	377	14	similar	similar	ADJ
ap-3457	377	15	to	to	ADP
ap-3457	377	16	[	[	X
ap-3457	377	17	37	37	NUM
ap-3457	377	18	]	]	PUNCT
ap-3457	377	19	,	,	PUNCT
ap-3457	377	20	regarding	regard	VERB
ap-3457	377	21	their	their	PRON
ap-3457	377	22	efficiency	efficiency	NOUN
ap-3457	377	23	,	,	PUNCT
ap-3457	377	24	speed	speed	NOUN
ap-3457	377	25	,	,	PUNCT
ap-3457	377	26	model	model	NOUN
ap-3457	377	27	function	function	NOUN
ap-3457	377	28	and	and	CCONJ
ap-3457	377	29	integration	integration	NOUN
ap-3457	377	30	domain	domain	NOUN
ap-3457	377	31	dependence	dependence	NOUN
ap-3457	377	32	merits	merit	VERB
ap-3457	377	33	further	further	ADJ
ap-3457	377	34	research	research	NOUN
ap-3457	377	35	.	.	PUNCT
ap-3457	378	1	6	6	X
ap-3457	378	2	.	.	X
ap-3457	378	3	acknowledgments	acknowledgment	NOUN
ap-3457	378	4	lm	lm	INTJ
ap-3457	378	5	and	and	CCONJ
ap-3457	378	6	jh	jh	PROPN
ap-3457	378	7	gratefully	gratefully	ADV
ap-3457	378	8	acknowledge	acknowledge	VERB
ap-3457	378	9	the	the	DET
ap-3457	378	10	support	support	NOUN
ap-3457	378	11	of	of	ADP
ap-3457	378	12	this	this	DET
ap-3457	378	13	work	work	NOUN
ap-3457	378	14	by	by	ADP
ap-3457	378	15	rvo68407700	rvo68407700	NOUN
ap-3457	378	16	.	.	PUNCT
ap-3457	379	1	references	reference	NOUN
ap-3457	379	2	[	[	X
ap-3457	379	3	1	1	NUM
ap-3457	379	4	]	]	PUNCT
ap-3457	379	5	h.	h.	PROPN
ap-3457	379	6	berens	berens	PROPN
ap-3457	379	7	,	,	PUNCT
ap-3457	379	8	h.	h.	PROPN
ap-3457	379	9	j.	j.	PROPN
ap-3457	379	10	schmid	schmid	PROPN
ap-3457	379	11	,	,	PUNCT
ap-3457	379	12	y.	y.	PROPN
ap-3457	379	13	xu	xu	PROPN
ap-3457	379	14	,	,	PUNCT
ap-3457	379	15	multivariate	multivariate	NOUN
ap-3457	379	16	gaussian	gaussian	ADJ
ap-3457	379	17	cubature	cubature	NOUN
ap-3457	379	18	formulae	formulae	NOUN
ap-3457	379	19	,	,	PUNCT
ap-3457	379	20	arch	arch	NOUN
ap-3457	379	21	.	.	PUNCT
ap-3457	380	1	math	math	NOUN
ap-3457	380	2	.	.	PUNCT
ap-3457	381	1	(	(	PUNCT
ap-3457	381	2	basel	basel	PROPN
ap-3457	381	3	)	)	PUNCT
ap-3457	381	4	64	64	NUM
ap-3457	381	5	(	(	PUNCT
ap-3457	381	6	1995	1995	NUM
ap-3457	381	7	)	)	PUNCT
ap-3457	381	8	,	,	PUNCT
ap-3457	381	9	no	no	INTJ
ap-3457	381	10	.	.	NOUN
ap-3457	381	11	1	1	NUM
ap-3457	381	12	,	,	PUNCT
ap-3457	381	13	26–32	26–32	NUM
ap-3457	381	14	,	,	PUNCT
ap-3457	381	15	doi:10.1007	doi:10.1007	ADJ
ap-3457	381	16	/	/	SYM
ap-3457	381	17	bf01193547	bf01193547	NOUN
ap-3457	381	18	.	.	PUNCT
ap-3457	382	1	[	[	X
ap-3457	382	2	2	2	NUM
ap-3457	382	3	]	]	X
ap-3457	382	4	n.	n.	NOUN
ap-3457	382	5	bourbaki	bourbaki	PROPN
ap-3457	382	6	,	,	PUNCT
ap-3457	382	7	groupes	groupe	NOUN
ap-3457	382	8	et	et	NOUN
ap-3457	382	9	algèbres	algèbre	NOUN
ap-3457	382	10	de	de	ADP
ap-3457	382	11	lie	lie	NOUN
ap-3457	382	12	,	,	PUNCT
ap-3457	382	13	chapitres	chapitre	VERB
ap-3457	382	14	iv	iv	NUM
ap-3457	382	15	,	,	PUNCT
ap-3457	382	16	v	v	NOUN
ap-3457	382	17	,	,	PUNCT
ap-3457	382	18	vi	vi	PROPN
ap-3457	382	19	,	,	PUNCT
ap-3457	382	20	hermann	hermann	PROPN
ap-3457	382	21	,	,	PUNCT
ap-3457	382	22	paris	paris	PROPN
ap-3457	382	23	,	,	PUNCT
ap-3457	382	24	1968	1968	NUM
ap-3457	382	25	.	.	PUNCT
ap-3457	383	1	[	[	X
ap-3457	383	2	3	3	X
ap-3457	383	3	]	]	X
ap-3457	383	4	m.	m.	NOUN
ap-3457	383	5	caliari	caliari	PROPN
ap-3457	383	6	,	,	PUNCT
ap-3457	383	7	s.	s.	PROPN
ap-3457	383	8	de	de	PROPN
ap-3457	383	9	marchi	marchi	PROPN
ap-3457	383	10	,	,	PUNCT
ap-3457	383	11	m.	m.	NOUN
ap-3457	383	12	vianello	vianello	PROPN
ap-3457	383	13	,	,	PUNCT
ap-3457	383	14	hyperinterpolation	hyperinterpolation	NOUN
ap-3457	383	15	in	in	ADP
ap-3457	383	16	the	the	DET
ap-3457	383	17	cube	cube	NOUN
ap-3457	383	18	,	,	PUNCT
ap-3457	383	19	comput	comput	NOUN
ap-3457	383	20	.	.	PUNCT
ap-3457	383	21	&	&	CCONJ
ap-3457	383	22	math	math	PROPN
ap-3457	383	23	.	.	PUNCT
ap-3457	384	1	with	with	ADP
ap-3457	384	2	appl	appl	NOUN
ap-3457	384	3	.	.	PUNCT
ap-3457	385	1	55	55	NUM
ap-3457	385	2	(	(	PUNCT
ap-3457	385	3	2008	2008	NUM
ap-3457	385	4	)	)	PUNCT
ap-3457	385	5	,	,	PUNCT
ap-3457	385	6	2490–2497	2490–2497	NUM
ap-3457	385	7	,	,	PUNCT
ap-3457	385	8	doi:10.1016	doi:10.1016	PROPN
ap-3457	385	9	/	/	SYM
ap-3457	385	10	j.camwa.2007.10.003	j.camwa.2007.10.003	PROPN
ap-3457	385	11	.	.	PUNCT
ap-3457	386	1	[	[	X
ap-3457	386	2	4	4	X
ap-3457	386	3	]	]	X
ap-3457	386	4	d.	d.	PROPN
ap-3457	386	5	chernyshenko	chernyshenko	PROPN
ap-3457	386	6	,	,	PUNCT
ap-3457	386	7	h.	h.	PROPN
ap-3457	386	8	fangohr	fangohr	PROPN
ap-3457	386	9	,	,	PUNCT
ap-3457	386	10	computing	compute	VERB
ap-3457	386	11	the	the	DET
ap-3457	386	12	demagnetizing	demagnetizing	NOUN
ap-3457	386	13	tensor	tensor	NOUN
ap-3457	386	14	for	for	ADP
ap-3457	386	15	finite	finite	ADJ
ap-3457	386	16	difference	difference	NOUN
ap-3457	386	17	micromagnetic	micromagnetic	ADJ
ap-3457	386	18	simulations	simulation	NOUN
ap-3457	386	19	via	via	ADP
ap-3457	386	20	numerical	numerical	ADJ
ap-3457	386	21	integration	integration	NOUN
ap-3457	386	22	,	,	PUNCT
ap-3457	386	23	j.	j.	PROPN
ap-3457	386	24	magn	magn	PROPN
ap-3457	386	25	.	.	PUNCT
ap-3457	387	1	magn	magn	PROPN
ap-3457	387	2	.	.	PUNCT
ap-3457	388	1	mat	mat	NOUN
ap-3457	388	2	.	.	NOUN
ap-3457	388	3	381	381	NUM
ap-3457	388	4	(	(	PUNCT
ap-3457	388	5	2015	2015	NUM
ap-3457	388	6	)	)	PUNCT
ap-3457	388	7	440–445	440–445	NUM
ap-3457	389	1	,	,	PUNCT
ap-3457	389	2	doi:10.1016	doi:10.1016	PROPN
ap-3457	389	3	/	/	SYM
ap-3457	389	4	j.jmmm.2015.01.013	j.jmmm.2015.01.013	PROPN
ap-3457	389	5	.	.	PUNCT
ap-3457	390	1	[	[	X
ap-3457	390	2	5	5	X
ap-3457	390	3	]	]	X
ap-3457	390	4	c.w	c.w	PROPN
ap-3457	390	5	.	.	PROPN
ap-3457	390	6	clenshaw	clenshaw	PROPN
ap-3457	390	7	,	,	PUNCT
ap-3457	390	8	a.r	a.r	PROPN
ap-3457	390	9	.	.	PROPN
ap-3457	390	10	curtis	curtis	PROPN
ap-3457	390	11	,	,	PUNCT
ap-3457	390	12	a	a	DET
ap-3457	390	13	method	method	NOUN
ap-3457	390	14	for	for	ADP
ap-3457	390	15	numerical	numerical	ADJ
ap-3457	390	16	integration	integration	NOUN
ap-3457	390	17	on	on	ADP
ap-3457	390	18	an	an	DET
ap-3457	390	19	automatic	automatic	ADJ
ap-3457	390	20	computer	computer	NOUN
ap-3457	390	21	,	,	PUNCT
ap-3457	390	22	numer	numer	PROPN
ap-3457	390	23	.	.	PUNCT
ap-3457	390	24	math	math	NOUN
ap-3457	390	25	.	.	PUNCT
ap-3457	391	1	2	2	NUM
ap-3457	391	2	(	(	PUNCT
ap-3457	391	3	1960	1960	NUM
ap-3457	391	4	)	)	PUNCT
ap-3457	391	5	,	,	PUNCT
ap-3457	391	6	197–205	197–205	NUM
ap-3457	391	7	,	,	PUNCT
ap-3457	391	8	doi:10.1007	doi:10.1007	NOUN
ap-3457	391	9	/	/	SYM
ap-3457	391	10	bf01386223	bf01386223	NOUN
ap-3457	391	11	.	.	PUNCT
ap-3457	392	1	[	[	X
ap-3457	392	2	6	6	NUM
ap-3457	392	3	]	]	PUNCT
ap-3457	392	4	r.	r.	PROPN
ap-3457	392	5	cools	cools	PROPN
ap-3457	392	6	,	,	PUNCT
ap-3457	392	7	an	an	DET
ap-3457	392	8	encyclopaedia	encyclopaedia	NOUN
ap-3457	392	9	of	of	ADP
ap-3457	392	10	cubature	cubature	ADJ
ap-3457	392	11	formulas	formula	NOUN
ap-3457	392	12	,	,	PUNCT
ap-3457	392	13	journal	journal	NOUN
ap-3457	392	14	of	of	ADP
ap-3457	392	15	complexity	complexity	NOUN
ap-3457	392	16	19	19	NUM
ap-3457	392	17	(	(	PUNCT
ap-3457	392	18	2003	2003	NUM
ap-3457	392	19	)	)	PUNCT
ap-3457	392	20	,	,	PUNCT
ap-3457	392	21	445–453	445–453	NUM
ap-3457	392	22	,	,	PUNCT
ap-3457	392	23	doi:10.1016	doi:10.1016	PROPN
ap-3457	392	24	/	/	SYM
ap-3457	392	25	s0885	s0885	NOUN
ap-3457	392	26	-	-	PUNCT
ap-3457	392	27	064x(03)00011	064x(03)00011	NOUN
ap-3457	392	28	-	-	PUNCT
ap-3457	392	29	6	6	NUM
ap-3457	392	30	.	.	PUNCT
ap-3457	393	1	[	[	X
ap-3457	393	2	7	7	X
ap-3457	393	3	]	]	X
ap-3457	393	4	r.	r.	PROPN
ap-3457	393	5	cools	cools	PROPN
ap-3457	393	6	,	,	PUNCT
ap-3457	393	7	i.	i.	PROPN
ap-3457	393	8	p.	p.	PROPN
ap-3457	393	9	mysovskikh	mysovskikh	PROPN
ap-3457	393	10	,	,	PUNCT
ap-3457	393	11	h.	h.	PROPN
ap-3457	393	12	j.	j.	PROPN
ap-3457	393	13	schmid	schmid	PROPN
ap-3457	393	14	,	,	PUNCT
ap-3457	393	15	cubature	cubature	ADJ
ap-3457	393	16	formulae	formulae	ADJ
ap-3457	393	17	and	and	CCONJ
ap-3457	393	18	orthogonal	orthogonal	ADJ
ap-3457	393	19	polynomials	polynomial	NOUN
ap-3457	393	20	,	,	PUNCT
ap-3457	393	21	j.	j.	PROPN
ap-3457	393	22	comput	comput	PROPN
ap-3457	393	23	.	.	PUNCT
ap-3457	394	1	appl	appl	PROPN
ap-3457	394	2	.	.	PROPN
ap-3457	394	3	math	math	NOUN
ap-3457	394	4	.	.	PUNCT
ap-3457	395	1	127	127	NUM
ap-3457	395	2	(	(	PUNCT
ap-3457	395	3	2001	2001	NUM
ap-3457	395	4	)	)	PUNCT
ap-3457	395	5	,	,	PUNCT
ap-3457	395	6	no	no	INTJ
ap-3457	395	7	.	.	NOUN
ap-3457	395	8	1	1	NUM
ap-3457	395	9	-	-	SYM
ap-3457	395	10	2	2	NUM
ap-3457	395	11	,	,	PUNCT
ap-3457	395	12	121–152	121–152	NUM
ap-3457	395	13	,	,	PUNCT
ap-3457	395	14	doi:10.1016	doi:10.1016	PROPN
ap-3457	395	15	/	/	SYM
ap-3457	395	16	s0377	s0377	NOUN
ap-3457	395	17	-	-	PUNCT
ap-3457	395	18	0427(00)00495	0427(00)00495	NUM
ap-3457	395	19	-	-	PUNCT
ap-3457	395	20	7	7	NUM
ap-3457	395	21	.	.	NOUN
ap-3457	395	22	212	212	NUM
ap-3457	395	23	http://dx.doi.org/10.1007/bf01193547	http://dx.doi.org/10.1007/bf01193547	NOUN
ap-3457	395	24	http://dx.doi.org/10.1016/j.camwa.2007.10.003	http://dx.doi.org/10.1016/j.camwa.2007.10.003	PROPN
ap-3457	395	25	http://dx.doi.org/10.1016/j.jmmm.2015.01.013	http://dx.doi.org/10.1016/j.jmmm.2015.01.013	PROPN
ap-3457	395	26	http://dx.doi.org/10.1007/bf01386223	http://dx.doi.org/10.1007/bf01386223	NOUN
ap-3457	395	27	http://dx.doi.org/10.1016/s0885-064x(03)00011-6	http://dx.doi.org/10.1016/s0885-064x(03)00011-6	NOUN
ap-3457	395	28	http://dx.doi.org/10.1016/s0377-0427(00)00495-7	http://dx.doi.org/10.1016/s0377-0427(00)00495-7	X
ap-3457	395	29	vol	vol	NOUN
ap-3457	395	30	.	.	PUNCT
ap-3457	396	1	56	56	NUM
ap-3457	396	2	no	no	NOUN
ap-3457	396	3	.	.	PUNCT
ap-3457	397	1	3/2016	3/2016	NUM
ap-3457	397	2	on	on	ADP
ap-3457	397	3	cubature	cubature	ADJ
ap-3457	397	4	rules	rule	NOUN
ap-3457	397	5	associated	associate	VERB
ap-3457	397	6	to	to	ADP
ap-3457	397	7	weyl	weyl	PROPN
ap-3457	397	8	group	group	PROPN
ap-3457	397	9	orbit	orbit	NOUN
ap-3457	397	10	functions	function	NOUN
ap-3457	397	11	[	[	X
ap-3457	397	12	8	8	NUM
ap-3457	397	13	]	]	PUNCT
ap-3457	397	14	p.	p.	NOUN
ap-3457	397	15	de	de	X
ap-3457	397	16	la	la	PROPN
ap-3457	397	17	harpe	harpe	PROPN
ap-3457	397	18	,	,	PUNCT
ap-3457	397	19	c.	c.	PROPN
ap-3457	397	20	pache	pache	PROPN
ap-3457	397	21	,	,	PUNCT
ap-3457	397	22	b.	b.	PROPN
ap-3457	397	23	venkov	venkov	PROPN
ap-3457	397	24	,	,	PUNCT
ap-3457	397	25	construction	construction	NOUN
ap-3457	397	26	of	of	ADP
ap-3457	397	27	spherical	spherical	ADJ
ap-3457	397	28	cubature	cubature	NOUN
ap-3457	397	29	formulas	formula	NOUN
ap-3457	397	30	using	use	VERB
ap-3457	397	31	lattices	lattice	NOUN
ap-3457	397	32	,	,	PUNCT
ap-3457	397	33	algebra	algebra	VERB
ap-3457	397	34	i	i	PRON
ap-3457	397	35	analiz	analiz	VERB
ap-3457	397	36	18	18	NUM
ap-3457	397	37	(	(	PUNCT
ap-3457	397	38	2006	2006	NUM
ap-3457	397	39	)	)	PUNCT
ap-3457	397	40	,	,	PUNCT
ap-3457	397	41	no	no	INTJ
ap-3457	397	42	.	.	NOUN
ap-3457	397	43	1	1	NUM
ap-3457	397	44	,	,	PUNCT
ap-3457	397	45	162–186	162–186	NUM
ap-3457	397	46	;	;	PUNCT
ap-3457	397	47	reprinted	reprint	VERB
ap-3457	397	48	in	in	ADP
ap-3457	397	49	st	st	PROPN
ap-3457	397	50	.	.	PROPN
ap-3457	397	51	petersburg	petersburg	PROPN
ap-3457	397	52	math	math	PROPN
ap-3457	397	53	.	.	PUNCT
ap-3457	398	1	j.	j.	PROPN
ap-3457	398	2	18	18	NUM
ap-3457	398	3	(	(	PUNCT
ap-3457	398	4	2007	2007	NUM
ap-3457	398	5	)	)	PUNCT
ap-3457	398	6	,	,	PUNCT
ap-3457	398	7	no	no	INTJ
ap-3457	398	8	.	.	NOUN
ap-3457	398	9	1	1	NUM
ap-3457	398	10	,	,	PUNCT
ap-3457	398	11	119–139	119–139	NUM
ap-3457	398	12	,	,	PUNCT
ap-3457	398	13	doi:10.1090	doi:10.1090	NOUN
ap-3457	398	14	/	/	SYM
ap-3457	398	15	s1061	s1061	NOUN
ap-3457	398	16	-	-	PUNCT
ap-3457	398	17	0022	0022	NUM
ap-3457	398	18	-	-	PUNCT
ap-3457	398	19	07	07	NUM
ap-3457	398	20	-	-	PUNCT
ap-3457	398	21	00946	00946	NUM
ap-3457	398	22	-	-	PUNCT
ap-3457	398	23	6	6	NUM
ap-3457	398	24	.	.	PUNCT
ap-3457	399	1	[	[	X
ap-3457	399	2	9	9	NUM
ap-3457	399	3	]	]	X
ap-3457	399	4	d.	d.	PROPN
ap-3457	399	5	c.	c.	PROPN
ap-3457	399	6	handscomb	handscomb	PROPN
ap-3457	399	7	,	,	PUNCT
ap-3457	399	8	j.	j.	PROPN
ap-3457	399	9	c.	c.	PROPN
ap-3457	399	10	mason	mason	PROPN
ap-3457	399	11	,	,	PUNCT
ap-3457	399	12	chebyshev	chebyshev	PROPN
ap-3457	399	13	polynomials	polynomial	NOUN
ap-3457	399	14	,	,	PUNCT
ap-3457	399	15	chapman&hall	chapman&hall	PROPN
ap-3457	399	16	/	/	SYM
ap-3457	399	17	crc	crc	PROPN
ap-3457	399	18	,	,	PUNCT
ap-3457	399	19	usa	usa	PROPN
ap-3457	399	20	,	,	PUNCT
ap-3457	399	21	2003	2003	NUM
ap-3457	399	22	,	,	PUNCT
ap-3457	399	23	doi:10.1201/9781420036114	doi:10.1201/9781420036114	NOUN
ap-3457	399	24	.	.	PUNCT
ap-3457	400	1	[	[	X
ap-3457	400	2	10	10	NUM
ap-3457	400	3	]	]	X
ap-3457	400	4	l.	l.	PROPN
ap-3457	400	5	háková	háková	PROPN
ap-3457	400	6	,	,	PUNCT
ap-3457	400	7	j.	j.	PROPN
ap-3457	400	8	hrivnák	hrivnák	PROPN
ap-3457	400	9	,	,	PUNCT
ap-3457	400	10	j.	j.	PROPN
ap-3457	400	11	patera	patera	PROPN
ap-3457	400	12	,	,	PUNCT
ap-3457	400	13	four	four	NUM
ap-3457	400	14	families	family	NOUN
ap-3457	400	15	of	of	ADP
ap-3457	400	16	weyl	weyl	PROPN
ap-3457	400	17	group	group	NOUN
ap-3457	400	18	orbit	orbit	NOUN
ap-3457	400	19	functions	function	NOUN
ap-3457	400	20	of	of	ADP
ap-3457	400	21	b3	b3	PROPN
ap-3457	400	22	and	and	CCONJ
ap-3457	400	23	c3	c3	PROPN
ap-3457	400	24	,	,	PUNCT
ap-3457	400	25	j.	j.	PROPN
ap-3457	400	26	math	math	PROPN
ap-3457	400	27	.	.	PUNCT
ap-3457	401	1	phys	phy	NOUN
ap-3457	401	2	.	.	PUNCT
ap-3457	402	1	54	54	NUM
ap-3457	402	2	(	(	PUNCT
ap-3457	402	3	2013	2013	NUM
ap-3457	402	4	)	)	PUNCT
ap-3457	402	5	,	,	PUNCT
ap-3457	402	6	083501	083501	NUM
ap-3457	402	7	,	,	PUNCT
ap-3457	402	8	19	19	NUM
ap-3457	402	9	,	,	PUNCT
ap-3457	402	10	doi:10.1063/1.4817340	doi:10.1063/1.4817340	NOUN
ap-3457	402	11	.	.	PUNCT
ap-3457	403	1	[	[	X
ap-3457	403	2	11	11	NUM
ap-3457	403	3	]	]	X
ap-3457	403	4	g.	g.	PROPN
ap-3457	403	5	heckman	heckman	PROPN
ap-3457	403	6	,	,	PUNCT
ap-3457	403	7	h.	h.	PROPN
ap-3457	403	8	schlichtkrull	schlichtkrull	PROPN
ap-3457	403	9	,	,	PUNCT
ap-3457	403	10	harmonic	harmonic	VERB
ap-3457	403	11	analysis	analysis	NOUN
ap-3457	403	12	and	and	CCONJ
ap-3457	403	13	special	special	ADJ
ap-3457	403	14	functions	function	NOUN
ap-3457	403	15	on	on	ADP
ap-3457	403	16	symmetric	symmetric	ADJ
ap-3457	403	17	spaces	space	NOUN
ap-3457	403	18	,	,	PUNCT
ap-3457	403	19	academic	academic	PROPN
ap-3457	403	20	press	press	PROPN
ap-3457	403	21	inc	inc	PROPN
ap-3457	403	22	.	.	PROPN
ap-3457	403	23	,	,	PUNCT
ap-3457	403	24	san	san	PROPN
ap-3457	403	25	diego	diego	PROPN
ap-3457	403	26	,	,	PUNCT
ap-3457	403	27	1994	1994	NUM
ap-3457	403	28	.	.	PUNCT
ap-3457	404	1	[	[	X
ap-3457	404	2	12	12	NUM
ap-3457	404	3	]	]	X
ap-3457	404	4	g.	g.	PROPN
ap-3457	404	5	heckman	heckman	PROPN
ap-3457	404	6	,	,	PUNCT
ap-3457	404	7	e.	e.	PROPN
ap-3457	404	8	m.	m.	PROPN
ap-3457	404	9	opdam	opdam	PROPN
ap-3457	404	10	,	,	PUNCT
ap-3457	404	11	root	root	NOUN
ap-3457	404	12	systems	system	NOUN
ap-3457	404	13	and	and	CCONJ
ap-3457	404	14	hypergeometric	hypergeometric	ADJ
ap-3457	404	15	functions	function	NOUN
ap-3457	404	16	.	.	PUNCT
ap-3457	405	1	i	i	PRON
ap-3457	405	2	,	,	PUNCT
ap-3457	405	3	ii	ii	PROPN
ap-3457	405	4	,	,	PUNCT
ap-3457	405	5	composition	composition	NOUN
ap-3457	405	6	math	math	NOUN
ap-3457	405	7	.	.	PUNCT
ap-3457	406	1	64	64	NUM
ap-3457	406	2	(	(	PUNCT
ap-3457	406	3	1987	1987	NUM
ap-3457	406	4	)	)	PUNCT
ap-3457	406	5	,	,	PUNCT
ap-3457	406	6	329	329	NUM
ap-3457	406	7	-	-	SYM
ap-3457	406	8	373	373	NUM
ap-3457	406	9	.	.	PUNCT
ap-3457	407	1	[	[	X
ap-3457	407	2	13	13	NUM
ap-3457	407	3	]	]	PUNCT
ap-3457	407	4	j.	j.	PROPN
ap-3457	407	5	hrivnák	hrivnák	PROPN
ap-3457	407	6	,	,	PUNCT
ap-3457	407	7	j.	j.	PROPN
ap-3457	407	8	patera	patera	PROPN
ap-3457	407	9	,	,	PUNCT
ap-3457	407	10	on	on	ADP
ap-3457	407	11	discretization	discretization	NOUN
ap-3457	407	12	of	of	ADP
ap-3457	407	13	tori	tori	NOUN
ap-3457	407	14	of	of	ADP
ap-3457	407	15	compact	compact	ADJ
ap-3457	407	16	simple	simple	ADJ
ap-3457	407	17	lie	lie	NOUN
ap-3457	407	18	groups	group	NOUN
ap-3457	407	19	,	,	PUNCT
ap-3457	407	20	j.	j.	PROPN
ap-3457	407	21	phys	phys	PROPN
ap-3457	407	22	.	.	PUNCT
ap-3457	408	1	a	a	DET
ap-3457	408	2	:	:	PUNCT
ap-3457	408	3	math	math	NOUN
ap-3457	408	4	.	.	PUNCT
ap-3457	409	1	theor	theor	PROPN
ap-3457	409	2	.	.	PUNCT
ap-3457	410	1	42	42	NUM
ap-3457	410	2	(	(	PUNCT
ap-3457	410	3	2009	2009	NUM
ap-3457	410	4	)	)	PUNCT
ap-3457	410	5	385208	385208	NUM
ap-3457	410	6	,	,	PUNCT
ap-3457	410	7	doi:10.1088/1751	doi:10.1088/1751	ADV
ap-3457	410	8	-	-	PUNCT
ap-3457	410	9	8113/42/38/385208	8113/42/38/385208	NOUN
ap-3457	410	10	.	.	PUNCT
ap-3457	411	1	[	[	X
ap-3457	411	2	14	14	NUM
ap-3457	411	3	]	]	X
ap-3457	411	4	j.	j.	PROPN
ap-3457	411	5	hrivnák	hrivnák	PROPN
ap-3457	411	6	,	,	PUNCT
ap-3457	411	7	l.	l.	PROPN
ap-3457	411	8	motlochová	motlochová	PROPN
ap-3457	411	9	,	,	PUNCT
ap-3457	411	10	j.	j.	PROPN
ap-3457	411	11	patera	patera	PROPN
ap-3457	411	12	,	,	PUNCT
ap-3457	411	13	on	on	ADP
ap-3457	411	14	discretization	discretization	NOUN
ap-3457	411	15	of	of	ADP
ap-3457	411	16	tori	tori	NOUN
ap-3457	411	17	of	of	ADP
ap-3457	411	18	compact	compact	ADJ
ap-3457	411	19	simple	simple	ADJ
ap-3457	411	20	lie	lie	NOUN
ap-3457	411	21	groups	groups	PROPN
ap-3457	411	22	ii	ii	PROPN
ap-3457	411	23	.	.	PROPN
ap-3457	411	24	,	,	PUNCT
ap-3457	411	25	j.	j.	PROPN
ap-3457	411	26	phys	phys	PROPN
ap-3457	411	27	.	.	PUNCT
ap-3457	412	1	a	a	DET
ap-3457	412	2	45	45	NUM
ap-3457	412	3	(	(	PUNCT
ap-3457	412	4	2012	2012	NUM
ap-3457	412	5	)	)	PUNCT
ap-3457	412	6	,	,	PUNCT
ap-3457	412	7	255201	255201	NUM
ap-3457	412	8	,	,	PUNCT
ap-3457	412	9	18	18	NUM
ap-3457	412	10	,	,	PUNCT
ap-3457	412	11	doi:10.1088/1751	doi:10.1088/1751	ADV
ap-3457	412	12	-	-	PUNCT
ap-3457	412	13	8113/45/25/255201	8113/45/25/255201	NUM
ap-3457	412	14	.	.	PUNCT
ap-3457	413	1	[	[	X
ap-3457	413	2	15	15	NUM
ap-3457	413	3	]	]	X
ap-3457	413	4	j.	j.	PROPN
ap-3457	413	5	hrivnák	hrivnák	PROPN
ap-3457	413	6	,	,	PUNCT
ap-3457	413	7	l.	l.	PROPN
ap-3457	413	8	motlochová	motlochová	PROPN
ap-3457	413	9	,	,	PUNCT
ap-3457	413	10	j.	j.	PROPN
ap-3457	413	11	patera	patera	PROPN
ap-3457	413	12	,	,	PUNCT
ap-3457	413	13	cubature	cubature	ADJ
ap-3457	413	14	formulas	formula	NOUN
ap-3457	413	15	of	of	ADP
ap-3457	413	16	multivariate	multivariate	NOUN
ap-3457	413	17	polynomials	polynomial	NOUN
ap-3457	413	18	arising	arise	VERB
ap-3457	413	19	from	from	ADP
ap-3457	413	20	symmetric	symmetric	ADJ
ap-3457	413	21	orbit	orbit	NOUN
ap-3457	413	22	functions	function	NOUN
ap-3457	413	23	,	,	PUNCT
ap-3457	413	24	arxiv:1512.01710	arxiv:1512.01710	ADJ
ap-3457	413	25	.	.	PUNCT
ap-3457	414	1	[	[	X
ap-3457	414	2	16	16	NUM
ap-3457	414	3	]	]	PUNCT
ap-3457	414	4	j.	j.	PROPN
ap-3457	414	5	e.	e.	PROPN
ap-3457	414	6	humphreys	humphreys	PROPN
ap-3457	414	7	,	,	PUNCT
ap-3457	414	8	introduction	introduction	NOUN
ap-3457	414	9	to	to	PART
ap-3457	414	10	lie	lie	VERB
ap-3457	414	11	algebras	algebra	NOUN
ap-3457	414	12	and	and	CCONJ
ap-3457	414	13	representation	representation	NOUN
ap-3457	414	14	theory	theory	NOUN
ap-3457	414	15	,	,	PUNCT
ap-3457	414	16	spinger	spinger	NOUN
ap-3457	414	17	-	-	PUNCT
ap-3457	414	18	verlag	verlag	NOUN
ap-3457	414	19	,	,	PUNCT
ap-3457	414	20	new	new	PROPN
ap-3457	414	21	york	york	PROPN
ap-3457	414	22	,	,	PUNCT
ap-3457	414	23	1978	1978	NUM
ap-3457	414	24	,	,	PUNCT
ap-3457	414	25	doi:10.1007/978	doi:10.1007/978	PROPN
ap-3457	414	26	-	-	PUNCT
ap-3457	414	27	1	1	NUM
ap-3457	414	28	-	-	PUNCT
ap-3457	414	29	4612	4612	NUM
ap-3457	414	30	-	-	PUNCT
ap-3457	414	31	6398	6398	NUM
ap-3457	414	32	-	-	PUNCT
ap-3457	414	33	2	2	NUM
ap-3457	414	34	.	.	PUNCT
ap-3457	415	1	[	[	X
ap-3457	415	2	17	17	NUM
ap-3457	415	3	]	]	PUNCT
ap-3457	415	4	a.	a.	NOUN
ap-3457	415	5	klimyk	klimyk	PROPN
ap-3457	415	6	,	,	PUNCT
ap-3457	415	7	j.	j.	PROPN
ap-3457	415	8	patera	patera	PROPN
ap-3457	415	9	,	,	PUNCT
ap-3457	415	10	(	(	PUNCT
ap-3457	415	11	anti)symmetric	anti)symmetric	ADJ
ap-3457	415	12	multivariate	multivariate	VERB
ap-3457	415	13	trigonometric	trigonometric	NOUN
ap-3457	415	14	functions	function	NOUN
ap-3457	415	15	and	and	CCONJ
ap-3457	415	16	corresponding	corresponding	ADJ
ap-3457	415	17	fourier	fourier	NOUN
ap-3457	415	18	transforms	transform	VERB
ap-3457	415	19	,	,	PUNCT
ap-3457	415	20	j.	j.	PROPN
ap-3457	415	21	math	math	PROPN
ap-3457	415	22	.	.	PUNCT
ap-3457	416	1	phys	phy	NOUN
ap-3457	416	2	.	.	PUNCT
ap-3457	417	1	48	48	NUM
ap-3457	417	2	(	(	PUNCT
ap-3457	417	3	2007	2007	NUM
ap-3457	417	4	)	)	PUNCT
ap-3457	417	5	,	,	PUNCT
ap-3457	417	6	093504	093504	NUM
ap-3457	417	7	,	,	PUNCT
ap-3457	417	8	24	24	NUM
ap-3457	417	9	,	,	PUNCT
ap-3457	417	10	doi:10.1063/1.2779768	doi:10.1063/1.2779768	NOUN
ap-3457	417	11	.	.	PUNCT
ap-3457	418	1	[	[	X
ap-3457	418	2	18	18	NUM
ap-3457	418	3	]	]	PUNCT
ap-3457	418	4	a.	a.	NOUN
ap-3457	418	5	u.	u.	PROPN
ap-3457	418	6	klimyk	klimyk	PROPN
ap-3457	418	7	,	,	PUNCT
ap-3457	418	8	j.	j.	PROPN
ap-3457	418	9	patera	patera	PROPN
ap-3457	418	10	,	,	PUNCT
ap-3457	418	11	orbit	orbit	NOUN
ap-3457	418	12	functions	function	NOUN
ap-3457	418	13	,	,	PUNCT
ap-3457	418	14	sigma	sigma	NOUN
ap-3457	418	15	2	2	NUM
ap-3457	418	16	(	(	PUNCT
ap-3457	418	17	2006	2006	NUM
ap-3457	418	18	)	)	PUNCT
ap-3457	418	19	,	,	PUNCT
ap-3457	418	20	006	006	NUM
ap-3457	418	21	,	,	PUNCT
ap-3457	418	22	60	60	NUM
ap-3457	418	23	pages	page	NOUN
ap-3457	418	24	,	,	PUNCT
ap-3457	418	25	doi:10.3842	doi:10.3842	NOUN
ap-3457	418	26	/	/	SYM
ap-3457	418	27	sigma.2006.006	sigma.2006.006	NOUN
ap-3457	418	28	.	.	PUNCT
ap-3457	419	1	[	[	X
ap-3457	419	2	19	19	NUM
ap-3457	419	3	]	]	PUNCT
ap-3457	419	4	a.	a.	NOUN
ap-3457	419	5	u.	u.	PROPN
ap-3457	419	6	klimyk	klimyk	PROPN
ap-3457	419	7	,	,	PUNCT
ap-3457	419	8	j.	j.	PROPN
ap-3457	419	9	patera	patera	PROPN
ap-3457	419	10	,	,	PUNCT
ap-3457	419	11	antisymmetric	antisymmetric	PROPN
ap-3457	419	12	orbit	orbit	NOUN
ap-3457	419	13	functions	function	NOUN
ap-3457	419	14	,	,	PUNCT
ap-3457	419	15	sigma	sigma	X
ap-3457	419	16	3	3	NUM
ap-3457	419	17	(	(	PUNCT
ap-3457	419	18	2007	2007	NUM
ap-3457	419	19	)	)	PUNCT
ap-3457	419	20	,	,	PUNCT
ap-3457	419	21	paper	paper	NOUN
ap-3457	419	22	023	023	NUM
ap-3457	419	23	,	,	PUNCT
ap-3457	419	24	83	83	NUM
ap-3457	419	25	pages	page	NOUN
ap-3457	419	26	,	,	PUNCT
ap-3457	419	27	doi:10.3842	doi:10.3842	NOUN
ap-3457	419	28	/	/	SYM
ap-3457	419	29	sigma.2007.023	sigma.2007.023	PROPN
ap-3457	419	30	.	.	PUNCT
ap-3457	420	1	[	[	X
ap-3457	420	2	20	20	NUM
ap-3457	420	3	]	]	PUNCT
ap-3457	420	4	t.	t.	PROPN
ap-3457	420	5	h.	h.	PROPN
ap-3457	420	6	koornwinder	koornwinder	PROPN
ap-3457	420	7	,	,	PUNCT
ap-3457	420	8	two	two	NUM
ap-3457	420	9	-	-	PUNCT
ap-3457	420	10	variable	variable	ADJ
ap-3457	420	11	analogues	analogue	NOUN
ap-3457	420	12	of	of	ADP
ap-3457	420	13	the	the	DET
ap-3457	420	14	classical	classical	ADJ
ap-3457	420	15	orthogonal	orthogonal	ADJ
ap-3457	420	16	polynomials	polynomial	NOUN
ap-3457	420	17	,	,	PUNCT
ap-3457	420	18	theory	theory	NOUN
ap-3457	420	19	and	and	CCONJ
ap-3457	420	20	application	application	NOUN
ap-3457	420	21	of	of	ADP
ap-3457	420	22	special	special	ADJ
ap-3457	420	23	functions	function	NOUN
ap-3457	420	24	,	,	PUNCT
ap-3457	420	25	edited	edit	VERB
ap-3457	420	26	by	by	ADP
ap-3457	420	27	r.	r.	PROPN
ap-3457	420	28	a.	a.	PROPN
ap-3457	420	29	askey	askey	PROPN
ap-3457	420	30	,	,	PUNCT
ap-3457	420	31	academic	academic	ADJ
ap-3457	420	32	press	press	NOUN
ap-3457	420	33	,	,	PUNCT
ap-3457	420	34	new	new	PROPN
ap-3457	420	35	york	york	PROPN
ap-3457	420	36	(	(	PUNCT
ap-3457	420	37	1975	1975	NUM
ap-3457	420	38	)	)	PUNCT
ap-3457	420	39	435–495	435–495	NUM
ap-3457	420	40	,	,	PUNCT
ap-3457	420	41	doi:10.1016	doi:10.1016	PROPN
ap-3457	420	42	/	/	SYM
ap-3457	420	43	b978	b978	PROPN
ap-3457	420	44	-	-	PUNCT
ap-3457	420	45	0	0	NUM
ap-3457	420	46	-	-	PUNCT
ap-3457	420	47	12	12	NUM
ap-3457	420	48	-	-	PUNCT
ap-3457	420	49	064850	064850	NUM
ap-3457	420	50	-	-	PUNCT
ap-3457	420	51	4.50015	4.50015	NUM
ap-3457	420	52	-	-	PUNCT
ap-3457	420	53	x.	x.	NOUN
ap-3457	421	1	[	[	X
ap-3457	421	2	21	21	NUM
ap-3457	421	3	]	]	X
ap-3457	421	4	h.	h.	PROPN
ap-3457	421	5	li	li	PROPN
ap-3457	421	6	,	,	PUNCT
ap-3457	421	7	j.	j.	PROPN
ap-3457	421	8	sun	sun	PROPN
ap-3457	421	9	,	,	PUNCT
ap-3457	421	10	y.	y.	PROPN
ap-3457	421	11	xu	xu	PROPN
ap-3457	421	12	,	,	PUNCT
ap-3457	421	13	discrete	discrete	ADJ
ap-3457	421	14	fourier	fourier	NOUN
ap-3457	421	15	analysis	analysis	NOUN
ap-3457	421	16	and	and	CCONJ
ap-3457	421	17	chebyshev	chebyshev	NOUN
ap-3457	421	18	polynomials	polynomial	NOUN
ap-3457	421	19	with	with	ADP
ap-3457	421	20	g2	g2	PROPN
ap-3457	421	21	group	group	NOUN
ap-3457	421	22	,	,	PUNCT
ap-3457	421	23	sigma	sigma	PROPN
ap-3457	421	24	8	8	NUM
ap-3457	421	25	,	,	PUNCT
ap-3457	421	26	paper	paper	NOUN
ap-3457	421	27	067	067	NUM
ap-3457	421	28	,	,	PUNCT
ap-3457	421	29	29	29	NUM
ap-3457	421	30	,	,	PUNCT
ap-3457	421	31	2012	2012	NUM
ap-3457	421	32	,	,	PUNCT
ap-3457	421	33	doi:10.3842	doi:10.3842	NOUN
ap-3457	421	34	/	/	SYM
ap-3457	421	35	sigma.2012.067	sigma.2012.067	PROPN
ap-3457	421	36	.	.	PUNCT
ap-3457	422	1	[	[	X
ap-3457	422	2	22	22	NUM
ap-3457	422	3	]	]	X
ap-3457	422	4	h.	h.	PROPN
ap-3457	422	5	li	li	PROPN
ap-3457	422	6	,	,	PUNCT
ap-3457	422	7	j.	j.	PROPN
ap-3457	422	8	sun	sun	PROPN
ap-3457	422	9	,	,	PUNCT
ap-3457	422	10	y.	y.	PROPN
ap-3457	422	11	xu	xu	PROPN
ap-3457	422	12	,	,	PUNCT
ap-3457	422	13	discrete	discrete	ADJ
ap-3457	422	14	fourier	fourier	NOUN
ap-3457	422	15	analysis	analysis	NOUN
ap-3457	422	16	,	,	PUNCT
ap-3457	422	17	cubature	cubature	NOUN
ap-3457	422	18	and	and	CCONJ
ap-3457	422	19	interpolation	interpolation	NOUN
ap-3457	422	20	on	on	ADP
ap-3457	422	21	a	a	DET
ap-3457	422	22	hexagon	hexagon	NOUN
ap-3457	422	23	and	and	CCONJ
ap-3457	422	24	a	a	DET
ap-3457	422	25	triangle	triangle	NOUN
ap-3457	422	26	,	,	PUNCT
ap-3457	422	27	siam	siam	PROPN
ap-3457	422	28	j.	j.	PROPN
ap-3457	422	29	numer	numer	PROPN
ap-3457	422	30	.	.	PUNCT
ap-3457	423	1	anal	anal	PROPN
ap-3457	423	2	.	.	PUNCT
ap-3457	424	1	46	46	NUM
ap-3457	424	2	(	(	PUNCT
ap-3457	424	3	2008	2008	NUM
ap-3457	424	4	)	)	PUNCT
ap-3457	424	5	,	,	PUNCT
ap-3457	424	6	1653–1681	1653–1681	NUM
ap-3457	424	7	,	,	PUNCT
ap-3457	424	8	doi:10.1137/060671851	doi:10.1137/060671851	NOUN
ap-3457	424	9	.	.	PUNCT
ap-3457	425	1	[	[	X
ap-3457	425	2	23	23	NUM
ap-3457	425	3	]	]	X
ap-3457	425	4	h.	h.	PROPN
ap-3457	425	5	li	li	PROPN
ap-3457	425	6	,	,	PUNCT
ap-3457	425	7	y.	y.	PROPN
ap-3457	425	8	xu	xu	PROPN
ap-3457	425	9	,	,	PUNCT
ap-3457	425	10	discrete	discrete	ADJ
ap-3457	425	11	fourier	fourier	NOUN
ap-3457	425	12	analysis	analysis	NOUN
ap-3457	425	13	on	on	ADP
ap-3457	425	14	fundamental	fundamental	ADJ
ap-3457	425	15	domain	domain	NOUN
ap-3457	425	16	and	and	CCONJ
ap-3457	425	17	simplex	simplex	NOUN
ap-3457	425	18	of	of	ADP
ap-3457	425	19	ad	ad	NOUN
ap-3457	425	20	lattice	lattice	NOUN
ap-3457	425	21	in	in	ADP
ap-3457	425	22	d	d	NOUN
ap-3457	425	23	-	-	NOUN
ap-3457	425	24	variables	variable	NOUN
ap-3457	425	25	,	,	PUNCT
ap-3457	425	26	j.	j.	PROPN
ap-3457	425	27	fourier	fourier	PROPN
ap-3457	425	28	anal	anal	PROPN
ap-3457	425	29	.	.	PUNCT
ap-3457	426	1	appl	appl	PROPN
ap-3457	426	2	.	.	PROPN
ap-3457	427	1	16	16	NUM
ap-3457	427	2	,	,	PUNCT
ap-3457	427	3	383–433	383–433	NUM
ap-3457	427	4	,	,	PUNCT
ap-3457	427	5	(	(	PUNCT
ap-3457	427	6	2010	2010	NUM
ap-3457	427	7	)	)	PUNCT
ap-3457	427	8	,	,	PUNCT
ap-3457	427	9	doi:10.1007	doi:10.1007	VERB
ap-3457	427	10	/	/	SYM
ap-3457	427	11	s00041	s00041	NOUN
ap-3457	427	12	-	-	PUNCT
ap-3457	427	13	009	009	NUM
ap-3457	427	14	-	-	PUNCT
ap-3457	427	15	9106	9106	NUM
ap-3457	427	16	-	-	SYM
ap-3457	427	17	9	9	NUM
ap-3457	427	18	.	.	PUNCT
ap-3457	428	1	[	[	X
ap-3457	428	2	24	24	NUM
ap-3457	428	3	]	]	PUNCT
ap-3457	428	4	i.	i.	PROPN
ap-3457	428	5	g.	g.	PROPN
ap-3457	428	6	macdonald	macdonald	PROPN
ap-3457	428	7	,	,	PUNCT
ap-3457	428	8	orthogonal	orthogonal	ADJ
ap-3457	428	9	polynomials	polynomial	NOUN
ap-3457	428	10	associated	associate	VERB
ap-3457	428	11	with	with	ADP
ap-3457	428	12	root	root	NOUN
ap-3457	428	13	systems	system	NOUN
ap-3457	428	14	,	,	PUNCT
ap-3457	428	15	sém	sém	NOUN
ap-3457	428	16	.	.	PUNCT
ap-3457	429	1	lothar	lothar	PROPN
ap-3457	429	2	.	.	PUNCT
ap-3457	430	1	combin	combin	NOUN
ap-3457	430	2	.	.	PUNCT
ap-3457	431	1	45	45	NUM
ap-3457	431	2	(	(	PUNCT
ap-3457	431	3	2000/01	2000/01	NUM
ap-3457	431	4	)	)	PUNCT
ap-3457	431	5	,	,	PUNCT
ap-3457	431	6	art	art	NOUN
ap-3457	431	7	.	.	PUNCT
ap-3457	432	1	b45a	b45a	PROPN
ap-3457	432	2	,	,	PUNCT
ap-3457	432	3	40	40	NUM
ap-3457	432	4	,	,	PUNCT
ap-3457	432	5	doi:10.1007/978	doi:10.1007/978	PROPN
ap-3457	432	6	-	-	PUNCT
ap-3457	432	7	94	94	NUM
ap-3457	432	8	-	-	PUNCT
ap-3457	432	9	009	009	NUM
ap-3457	432	10	-	-	PUNCT
ap-3457	432	11	0501	0501	NUM
ap-3457	432	12	-	-	PUNCT
ap-3457	432	13	6_14	6_14	NUM
ap-3457	432	14	.	.	PUNCT
ap-3457	433	1	[	[	X
ap-3457	433	2	25	25	NUM
ap-3457	433	3	]	]	PUNCT
ap-3457	433	4	s.	s.	PROPN
ap-3457	433	5	de	de	PROPN
ap-3457	433	6	marchi	marchi	PROPN
ap-3457	433	7	,	,	PUNCT
ap-3457	433	8	m.	m.	NOUN
ap-3457	433	9	vianello	vianello	PROPN
ap-3457	433	10	,	,	PUNCT
ap-3457	433	11	y.	y.	PROPN
ap-3457	433	12	xu	xu	PROPN
ap-3457	433	13	,	,	PUNCT
ap-3457	433	14	new	new	ADJ
ap-3457	433	15	cubature	cubature	NOUN
ap-3457	433	16	formulae	formulae	NOUN
ap-3457	433	17	and	and	CCONJ
ap-3457	433	18	hyperinterpolation	hyperinterpolation	NOUN
ap-3457	433	19	in	in	ADP
ap-3457	433	20	three	three	NUM
ap-3457	433	21	variables	variable	NOUN
ap-3457	433	22	,	,	PUNCT
ap-3457	433	23	bit	bit	NOUN
ap-3457	433	24	.	.	PUNCT
ap-3457	434	1	numerical	numerical	ADJ
ap-3457	434	2	mathematics	mathematics	PROPN
ap-3457	434	3	49	49	NUM
ap-3457	434	4	(	(	PUNCT
ap-3457	434	5	2009	2009	NUM
ap-3457	434	6	)	)	PUNCT
ap-3457	434	7	,	,	PUNCT
ap-3457	434	8	number	number	NOUN
ap-3457	434	9	1	1	NUM
ap-3457	434	10	,	,	PUNCT
ap-3457	434	11	55–73	55–73	NUM
ap-3457	434	12	,	,	PUNCT
ap-3457	434	13	doi:10.1007	doi:10.1007	ADJ
ap-3457	434	14	/	/	SYM
ap-3457	434	15	s10543	s10543	NOUN
ap-3457	434	16	-	-	PUNCT
ap-3457	434	17	009	009	NUM
ap-3457	434	18	-	-	PUNCT
ap-3457	434	19	0210	0210	NUM
ap-3457	434	20	-	-	PUNCT
ap-3457	434	21	7	7	NUM
ap-3457	434	22	.	.	PUNCT
ap-3457	435	1	[	[	X
ap-3457	435	2	26	26	NUM
ap-3457	435	3	]	]	X
ap-3457	435	4	r.	r.	PROPN
ap-3457	435	5	v.	v.	PROPN
ap-3457	435	6	moody	moody	PROPN
ap-3457	435	7	,	,	PUNCT
ap-3457	435	8	l.	l.	PROPN
ap-3457	435	9	motlochová	motlochová	PROPN
ap-3457	435	10	,	,	PUNCT
ap-3457	435	11	j.	j.	PROPN
ap-3457	435	12	patera	patera	PROPN
ap-3457	435	13	,	,	PUNCT
ap-3457	435	14	gaussian	gaussian	ADJ
ap-3457	435	15	cubature	cubature	NOUN
ap-3457	435	16	arising	arise	VERB
ap-3457	435	17	from	from	ADP
ap-3457	435	18	hybrid	hybrid	ADJ
ap-3457	435	19	characters	character	NOUN
ap-3457	435	20	of	of	ADP
ap-3457	435	21	simple	simple	ADJ
ap-3457	435	22	lie	lie	NOUN
ap-3457	435	23	groups	group	NOUN
ap-3457	435	24	,	,	PUNCT
ap-3457	435	25	j.	j.	PROPN
ap-3457	435	26	fourier	fourier	PROPN
ap-3457	435	27	anal	anal	PROPN
ap-3457	435	28	.	.	PUNCT
ap-3457	436	1	appl	appl	PROPN
ap-3457	436	2	.	.	PROPN
ap-3457	437	1	20	20	NUM
ap-3457	437	2	(	(	PUNCT
ap-3457	437	3	2014	2014	NUM
ap-3457	437	4	)	)	PUNCT
ap-3457	438	1	,	,	PUNCT
ap-3457	438	2	issue	issue	NOUN
ap-3457	438	3	6	6	NUM
ap-3457	438	4	,	,	PUNCT
ap-3457	438	5	1257–1290	1257–1290	NUM
ap-3457	438	6	,	,	PUNCT
ap-3457	438	7	doi:10.1007	doi:10.1007	NOUN
ap-3457	438	8	/	/	SYM
ap-3457	438	9	s00041	s00041	NOUN
ap-3457	438	10	-	-	PUNCT
ap-3457	438	11	014	014	NUM
ap-3457	438	12	-	-	PUNCT
ap-3457	438	13	9355	9355	NUM
ap-3457	438	14	-	-	SYM
ap-3457	438	15	0	0	NUM
ap-3457	438	16	.	.	PUNCT
ap-3457	439	1	[	[	X
ap-3457	439	2	27	27	NUM
ap-3457	439	3	]	]	X
ap-3457	439	4	r.	r.	PROPN
ap-3457	439	5	v.	v.	PROPN
ap-3457	439	6	moody	moody	PROPN
ap-3457	439	7	,	,	PUNCT
ap-3457	439	8	j.	j.	PROPN
ap-3457	439	9	patera	patera	PROPN
ap-3457	439	10	,	,	PUNCT
ap-3457	439	11	cubature	cubature	NOUN
ap-3457	439	12	formulae	formulae	NOUN
ap-3457	439	13	for	for	ADP
ap-3457	439	14	orthogonal	orthogonal	ADJ
ap-3457	439	15	polynomials	polynomial	NOUN
ap-3457	439	16	in	in	ADP
ap-3457	439	17	terms	term	NOUN
ap-3457	439	18	of	of	ADP
ap-3457	439	19	elements	element	NOUN
ap-3457	439	20	of	of	ADP
ap-3457	439	21	finite	finite	ADJ
ap-3457	439	22	order	order	NOUN
ap-3457	439	23	of	of	ADP
ap-3457	439	24	compact	compact	ADJ
ap-3457	439	25	simple	simple	ADJ
ap-3457	439	26	lie	lie	NOUN
ap-3457	439	27	groups	group	NOUN
ap-3457	439	28	,	,	PUNCT
ap-3457	439	29	adv	adv	PROPN
ap-3457	439	30	.	.	PUNCT
ap-3457	440	1	in	in	ADP
ap-3457	440	2	appl	appl	PROPN
ap-3457	440	3	.	.	PUNCT
ap-3457	440	4	math	math	NOUN
ap-3457	440	5	.	.	PUNCT
ap-3457	441	1	47	47	NUM
ap-3457	441	2	(	(	PUNCT
ap-3457	441	3	2011	2011	NUM
ap-3457	441	4	)	)	PUNCT
ap-3457	441	5	509–535	509–535	NUM
ap-3457	441	6	,	,	PUNCT
ap-3457	441	7	doi:10.1016	doi:10.1016	PROPN
ap-3457	441	8	/	/	SYM
ap-3457	441	9	j.aam.2010.11.005	j.aam.2010.11.005	PROPN
ap-3457	441	10	.	.	PUNCT
ap-3457	442	1	[	[	X
ap-3457	442	2	28	28	NUM
ap-3457	442	3	]	]	X
ap-3457	442	4	r.	r.	PROPN
ap-3457	442	5	v.	v.	CCONJ
ap-3457	442	6	moody	moody	PROPN
ap-3457	442	7	and	and	CCONJ
ap-3457	442	8	j.	j.	PROPN
ap-3457	442	9	patera	patera	PROPN
ap-3457	442	10	,	,	PUNCT
ap-3457	442	11	orthogonality	orthogonality	NOUN
ap-3457	442	12	within	within	ADP
ap-3457	442	13	the	the	DET
ap-3457	442	14	families	family	NOUN
ap-3457	442	15	of	of	ADP
ap-3457	442	16	c-	c-	X
ap-3457	442	17	,	,	PUNCT
ap-3457	442	18	s-	s-	X
ap-3457	442	19	,	,	PUNCT
ap-3457	442	20	and	and	CCONJ
ap-3457	442	21	e	e	NOUN
ap-3457	442	22	-	-	NOUN
ap-3457	442	23	functions	function	NOUN
ap-3457	442	24	of	of	ADP
ap-3457	442	25	any	any	DET
ap-3457	442	26	compact	compact	ADJ
ap-3457	442	27	semisimple	semisimple	NOUN
ap-3457	442	28	lie	lie	NOUN
ap-3457	442	29	group	group	NOUN
ap-3457	442	30	,	,	PUNCT
ap-3457	442	31	sigma	sigma	NOUN
ap-3457	442	32	2	2	NUM
ap-3457	442	33	(	(	PUNCT
ap-3457	442	34	2006	2006	NUM
ap-3457	442	35	)	)	PUNCT
ap-3457	442	36	076	076	NUM
ap-3457	442	37	,	,	PUNCT
ap-3457	442	38	14	14	NUM
ap-3457	442	39	pages	page	NOUN
ap-3457	442	40	,	,	PUNCT
ap-3457	442	41	doi:10.3842	doi:10.3842	NOUN
ap-3457	442	42	/	/	SYM
ap-3457	442	43	sigma.2006.076	sigma.2006.076	NOUN
ap-3457	442	44	.	.	PUNCT
ap-3457	443	1	[	[	X
ap-3457	443	2	29	29	NUM
ap-3457	443	3	]	]	X
ap-3457	443	4	l.	l.	PROPN
ap-3457	443	5	motlochová	motlochová	PROPN
ap-3457	443	6	,	,	PUNCT
ap-3457	443	7	special	special	ADJ
ap-3457	443	8	functions	function	NOUN
ap-3457	443	9	of	of	ADP
ap-3457	443	10	weyl	weyl	VERB
ap-3457	443	11	groups	group	NOUN
ap-3457	443	12	and	and	CCONJ
ap-3457	443	13	their	their	PRON
ap-3457	443	14	continuous	continuous	ADJ
ap-3457	443	15	and	and	CCONJ
ap-3457	443	16	discrete	discrete	ADJ
ap-3457	443	17	orthogonality	orthogonality	NOUN
ap-3457	443	18	,	,	PUNCT
ap-3457	443	19	ph.d	ph.d	PROPN
ap-3457	443	20	.	.	PUNCT
ap-3457	444	1	thesis	thesis	PROPN
ap-3457	444	2	,	,	PUNCT
ap-3457	444	3	université	université	PROPN
ap-3457	444	4	de	de	PROPN
ap-3457	444	5	montréal	montréal	PROPN
ap-3457	444	6	(	(	PUNCT
ap-3457	444	7	2014	2014	NUM
ap-3457	444	8	)	)	PUNCT
ap-3457	444	9	,	,	PUNCT
ap-3457	444	10	http://hdl.handle.net/1866/11153	http://hdl.handle.net/1866/11153	VERB
ap-3457	444	11	[	[	X
ap-3457	444	12	30	30	NUM
ap-3457	444	13	]	]	PUNCT
ap-3457	444	14	h.	h.	PROPN
ap-3457	444	15	z.	z.	PROPN
ap-3457	444	16	munthe	munthe	PROPN
ap-3457	444	17	-	-	PUNCT
ap-3457	444	18	kaas	kaas	NOUN
ap-3457	444	19	,	,	PUNCT
ap-3457	444	20	m.	m.	NOUN
ap-3457	444	21	nome	nome	PROPN
ap-3457	444	22	,	,	PUNCT
ap-3457	444	23	b.	b.	PROPN
ap-3457	444	24	n.	n.	PROPN
ap-3457	444	25	ryland	ryland	PROPN
ap-3457	444	26	,	,	PUNCT
ap-3457	444	27	through	through	ADP
ap-3457	444	28	the	the	DET
ap-3457	444	29	kaleidoscope	kaleidoscope	NOUN
ap-3457	444	30	:	:	PUNCT
ap-3457	445	1	symmetries	symmetry	NOUN
ap-3457	445	2	,	,	PUNCT
ap-3457	445	3	groups	group	NOUN
ap-3457	445	4	and	and	CCONJ
ap-3457	445	5	chebyshevapproximations	chebyshevapproximation	NOUN
ap-3457	445	6	from	from	ADP
ap-3457	445	7	a	a	DET
ap-3457	445	8	computational	computational	ADJ
ap-3457	445	9	point	point	NOUN
ap-3457	445	10	of	of	ADP
ap-3457	445	11	view	view	NOUN
ap-3457	445	12	,	,	PUNCT
ap-3457	445	13	foundations	foundation	NOUN
ap-3457	445	14	of	of	ADP
ap-3457	445	15	computational	computational	ADJ
ap-3457	445	16	mathematics	mathematic	NOUN
ap-3457	445	17	,	,	PUNCT
ap-3457	445	18	budapest	budapest	NOUN
ap-3457	445	19	2011	2011	NUM
ap-3457	445	20	,	,	PUNCT
ap-3457	445	21	188–229	188–229	NUM
ap-3457	445	22	,	,	PUNCT
ap-3457	445	23	london	london	PROPN
ap-3457	445	24	math	math	NOUN
ap-3457	445	25	.	.	PUNCT
ap-3457	446	1	soc	soc	PROPN
ap-3457	446	2	.	.	PUNCT
ap-3457	447	1	lecture	lecture	NOUN
ap-3457	447	2	note	note	NOUN
ap-3457	447	3	ser	ser	PROPN
ap-3457	447	4	.	.	PROPN
ap-3457	447	5	,	,	PUNCT
ap-3457	447	6	403	403	NUM
ap-3457	447	7	,	,	PUNCT
ap-3457	447	8	cambridge	cambridge	PROPN
ap-3457	447	9	univ	univ	PROPN
ap-3457	447	10	.	.	PUNCT
ap-3457	448	1	press	press	PROPN
ap-3457	448	2	,	,	PUNCT
ap-3457	448	3	cambridge	cambridge	PROPN
ap-3457	448	4	,	,	PUNCT
ap-3457	448	5	2013	2013	NUM
ap-3457	448	6	.	.	PUNCT
ap-3457	449	1	[	[	X
ap-3457	449	2	31	31	NUM
ap-3457	449	3	]	]	PUNCT
ap-3457	449	4	b.	b.	PROPN
ap-3457	449	5	n.	n.	PROPN
ap-3457	449	6	ryland	ryland	PROPN
ap-3457	449	7	,	,	PUNCT
ap-3457	449	8	h.	h.	PROPN
ap-3457	449	9	z.	z.	PROPN
ap-3457	449	10	munthe	munthe	PROPN
ap-3457	449	11	-	-	PUNCT
ap-3457	449	12	kaas	kaas	NOUN
ap-3457	449	13	,	,	PUNCT
ap-3457	449	14	on	on	ADP
ap-3457	449	15	multivariate	multivariate	NOUN
ap-3457	449	16	chebyshev	chebyshev	NOUN
ap-3457	449	17	polynomials	polynomial	NOUN
ap-3457	449	18	and	and	CCONJ
ap-3457	449	19	spectral	spectral	ADJ
ap-3457	449	20	approximation	approximation	NOUN
ap-3457	449	21	on	on	ADP
ap-3457	449	22	triangles	triangle	NOUN
ap-3457	449	23	,	,	PUNCT
ap-3457	449	24	spectral	spectral	ADJ
ap-3457	449	25	and	and	CCONJ
ap-3457	449	26	high	high	ADJ
ap-3457	449	27	order	order	NOUN
ap-3457	449	28	methods	method	NOUN
ap-3457	449	29	for	for	ADP
ap-3457	449	30	partial	partial	ADJ
ap-3457	449	31	differential	differential	ADJ
ap-3457	449	32	equations	equation	NOUN
ap-3457	449	33	,	,	PUNCT
ap-3457	449	34	lecture	lecture	NOUN
ap-3457	449	35	notes	note	NOUN
ap-3457	449	36	in	in	ADP
ap-3457	449	37	computational	computational	ADJ
ap-3457	449	38	science	science	NOUN
ap-3457	449	39	and	and	CCONJ
ap-3457	449	40	engineering	engineering	NOUN
ap-3457	449	41	,	,	PUNCT
ap-3457	449	42	springer	springer	NOUN
ap-3457	449	43	,	,	PUNCT
ap-3457	449	44	2011	2011	NUM
ap-3457	449	45	,	,	PUNCT
ap-3457	449	46	doi:10.1007/978	doi:10.1007/978	PROPN
ap-3457	449	47	-	-	PUNCT
ap-3457	449	48	3	3	NUM
ap-3457	449	49	-	-	PUNCT
ap-3457	449	50	642	642	NUM
ap-3457	449	51	-	-	PUNCT
ap-3457	449	52	15337	15337	NUM
ap-3457	449	53	-	-	PUNCT
ap-3457	449	54	2_2	2_2	NUM
ap-3457	449	55	.	.	PUNCT
ap-3457	450	1	[	[	X
ap-3457	450	2	32	32	NUM
ap-3457	450	3	]	]	PUNCT
ap-3457	450	4	h.	h.	PROPN
ap-3457	450	5	j.	j.	PROPN
ap-3457	450	6	schmid	schmid	PROPN
ap-3457	450	7	,	,	PUNCT
ap-3457	450	8	y.	y.	PROPN
ap-3457	450	9	xu	xu	PROPN
ap-3457	450	10	,	,	PUNCT
ap-3457	450	11	on	on	ADP
ap-3457	450	12	bivariate	bivariate	ADJ
ap-3457	450	13	gaussian	gaussian	ADJ
ap-3457	450	14	cubature	cubature	NOUN
ap-3457	450	15	formulae	formulae	NOUN
ap-3457	450	16	,	,	PUNCT
ap-3457	450	17	proc	proc	NOUN
ap-3457	450	18	.	.	PUNCT
ap-3457	451	1	amer	amer	PROPN
ap-3457	451	2	.	.	PUNCT
ap-3457	451	3	math	math	PROPN
ap-3457	451	4	.	.	PUNCT
ap-3457	452	1	soc	soc	PROPN
ap-3457	452	2	.	.	PUNCT
ap-3457	452	3	,	,	PUNCT
ap-3457	452	4	122	122	NUM
ap-3457	452	5	(	(	PUNCT
ap-3457	452	6	1994	1994	NUM
ap-3457	452	7	)	)	PUNCT
ap-3457	452	8	,	,	PUNCT
ap-3457	452	9	833–841	833–841	NUM
ap-3457	452	10	,	,	PUNCT
ap-3457	452	11	doi:10.2307/2160762	doi:10.2307/2160762	NOUN
ap-3457	452	12	.	.	PUNCT
ap-3457	453	1	[	[	X
ap-3457	453	2	33	33	NUM
ap-3457	453	3	]	]	X
ap-3457	453	4	i.	i.	NOUN
ap-3457	453	5	sfevanovic	sfevanovic	PROPN
ap-3457	453	6	,	,	PUNCT
ap-3457	453	7	f.	f.	PROPN
ap-3457	453	8	merli	merli	PROPN
ap-3457	453	9	,	,	PUNCT
ap-3457	453	10	p.	p.	PROPN
ap-3457	453	11	crespo	crespo	PROPN
ap-3457	453	12	-	-	PUNCT
ap-3457	453	13	valero	valero	PROPN
ap-3457	453	14	,	,	PUNCT
ap-3457	453	15	w.	w.	PROPN
ap-3457	453	16	simon	simon	PROPN
ap-3457	453	17	,	,	PUNCT
ap-3457	453	18	s.	s.	PROPN
ap-3457	453	19	holzwarth	holzwarth	PROPN
ap-3457	453	20	,	,	PUNCT
ap-3457	453	21	m.	m.	NOUN
ap-3457	453	22	mattes	mattes	PROPN
ap-3457	453	23	,	,	PUNCT
ap-3457	453	24	j.	j.	PROPN
ap-3457	453	25	r.	r.	PROPN
ap-3457	453	26	mosig	mosig	PROPN
ap-3457	453	27	,	,	PUNCT
ap-3457	453	28	integral	integral	ADJ
ap-3457	453	29	equation	equation	NOUN
ap-3457	453	30	modeling	modeling	NOUN
ap-3457	453	31	of	of	ADP
ap-3457	453	32	waveguide	waveguide	NOUN
ap-3457	453	33	-	-	PUNCT
ap-3457	453	34	fed	feed	VERB
ap-3457	453	35	planar	planar	PROPN
ap-3457	453	36	antennas	antennas	PROPN
ap-3457	453	37	,	,	PUNCT
ap-3457	453	38	ieee	ieee	PROPN
ap-3457	453	39	antenn	antenn	PROPN
ap-3457	453	40	.	.	PUNCT
ap-3457	454	1	propag	propag	PROPN
ap-3457	454	2	.	.	PUNCT
ap-3457	455	1	m.	m.	NOUN
ap-3457	455	2	51	51	NUM
ap-3457	455	3	(	(	PUNCT
ap-3457	455	4	2009	2009	NUM
ap-3457	455	5	)	)	PUNCT
ap-3457	455	6	,	,	PUNCT
ap-3457	455	7	82–92	82–92	NUM
ap-3457	455	8	,	,	PUNCT
ap-3457	455	9	doi:10.1109	doi:10.1109	VERB
ap-3457	455	10	/	/	SYM
ap-3457	455	11	map.2009.5433099	map.2009.5433099	PROPN
ap-3457	455	12	.	.	PUNCT
ap-3457	456	1	[	[	X
ap-3457	456	2	34	34	NUM
ap-3457	456	3	]	]	X
ap-3457	456	4	i.	i.	PROPN
ap-3457	456	5	h.	h.	PROPN
ap-3457	456	6	sloan	sloan	PROPN
ap-3457	456	7	,	,	PUNCT
ap-3457	456	8	polynomial	polynomial	ADJ
ap-3457	456	9	interpolation	interpolation	NOUN
ap-3457	456	10	and	and	CCONJ
ap-3457	456	11	hyperinterpolation	hyperinterpolation	NOUN
ap-3457	456	12	over	over	ADP
ap-3457	456	13	general	general	ADJ
ap-3457	456	14	regions	region	NOUN
ap-3457	456	15	,	,	PUNCT
ap-3457	456	16	j.	j.	PROPN
ap-3457	456	17	approx	approx	PROPN
ap-3457	456	18	.	.	PUNCT
ap-3457	457	1	theory	theory	NOUN
ap-3457	457	2	83	83	NUM
ap-3457	457	3	(	(	PUNCT
ap-3457	457	4	1995	1995	NUM
ap-3457	457	5	)	)	PUNCT
ap-3457	457	6	,	,	PUNCT
ap-3457	457	7	no	no	INTJ
ap-3457	457	8	.	.	NOUN
ap-3457	457	9	2	2	NUM
ap-3457	457	10	,	,	PUNCT
ap-3457	457	11	238–254	238–254	NUM
ap-3457	457	12	,	,	PUNCT
ap-3457	457	13	doi:10.1006	doi:10.1006	PROPN
ap-3457	457	14	/	/	SYM
ap-3457	457	15	jath.1995.1119	jath.1995.1119	PROPN
ap-3457	457	16	.	.	PUNCT
ap-3457	458	1	[	[	X
ap-3457	458	2	35	35	NUM
ap-3457	458	3	]	]	PUNCT
ap-3457	458	4	a.	a.	NOUN
ap-3457	458	5	sommariva	sommariva	PROPN
ap-3457	458	6	,	,	PUNCT
ap-3457	458	7	m.	m.	NOUN
ap-3457	458	8	vianello	vianello	PROPN
ap-3457	458	9	,	,	PUNCT
ap-3457	458	10	r.	r.	PROPN
ap-3457	458	11	zanovello	zanovello	PROPN
ap-3457	458	12	,	,	PUNCT
ap-3457	458	13	nontensorial	nontensorial	ADJ
ap-3457	458	14	clenshaw	clenshaw	ADJ
ap-3457	458	15	-	-	PUNCT
ap-3457	458	16	curtis	curtis	NOUN
ap-3457	458	17	cubature	cubature	NOUN
ap-3457	458	18	,	,	PUNCT
ap-3457	458	19	numer	numer	PROPN
ap-3457	458	20	.	.	PUNCT
ap-3457	459	1	algorithms	algorithms	PROPN
ap-3457	459	2	49	49	NUM
ap-3457	459	3	(	(	PUNCT
ap-3457	459	4	2008	2008	NUM
ap-3457	459	5	)	)	PUNCT
ap-3457	459	6	,	,	PUNCT
ap-3457	459	7	number	number	NOUN
ap-3457	459	8	1	1	NUM
ap-3457	459	9	-	-	SYM
ap-3457	459	10	4	4	NUM
ap-3457	459	11	,	,	PUNCT
ap-3457	459	12	409–427	409–427	NUM
ap-3457	459	13	,	,	PUNCT
ap-3457	459	14	doi:10.1007	doi:10.1007	ADJ
ap-3457	459	15	/	/	SYM
ap-3457	459	16	s11075	s11075	NOUN
ap-3457	459	17	-	-	PUNCT
ap-3457	459	18	008	008	NUM
ap-3457	459	19	-	-	PUNCT
ap-3457	459	20	9203	9203	NUM
ap-3457	459	21	-	-	PUNCT
ap-3457	459	22	x.	x.	NOUN
ap-3457	460	1	[	[	X
ap-3457	460	2	36	36	NUM
ap-3457	460	3	]	]	X
ap-3457	460	4	g.	g.	PROPN
ap-3457	460	5	szegő	szegő	PROPN
ap-3457	460	6	,	,	PUNCT
ap-3457	460	7	orthogonal	orthogonal	ADJ
ap-3457	460	8	polynomials	polynomial	NOUN
ap-3457	460	9	,	,	PUNCT
ap-3457	460	10	american	american	PROPN
ap-3457	460	11	mathematical	mathematical	ADJ
ap-3457	460	12	society	society	NOUN
ap-3457	460	13	,	,	PUNCT
ap-3457	460	14	providence	providence	NOUN
ap-3457	460	15	,	,	PUNCT
ap-3457	460	16	r.i	r.i	PROPN
ap-3457	460	17	.	.	PROPN
ap-3457	460	18	,	,	PUNCT
ap-3457	460	19	1975	1975	NUM
ap-3457	460	20	.	.	PUNCT
ap-3457	461	1	[	[	X
ap-3457	461	2	37	37	NUM
ap-3457	461	3	]	]	PUNCT
ap-3457	461	4	l.	l.	PROPN
ap-3457	461	5	n.	n.	PROPN
ap-3457	461	6	trefethen	trefethen	PROPN
ap-3457	461	7	,	,	PUNCT
ap-3457	461	8	is	be	AUX
ap-3457	461	9	gauss	gauss	ADJ
ap-3457	461	10	quadrature	quadrature	NOUN
ap-3457	461	11	better	well	ADV
ap-3457	461	12	than	than	ADP
ap-3457	461	13	clenshaw	clenshaw	ADJ
ap-3457	461	14	-	-	PUNCT
ap-3457	461	15	curtis	curtis	NOUN
ap-3457	461	16	?	?	PUNCT
ap-3457	462	1	siam	siam	PROPN
ap-3457	462	2	rev	rev	PROPN
ap-3457	462	3	.	.	PROPN
ap-3457	462	4	50	50	NUM
ap-3457	462	5	(	(	PUNCT
ap-3457	462	6	2008	2008	NUM
ap-3457	462	7	)	)	PUNCT
ap-3457	462	8	67–87	67–87	NUM
ap-3457	462	9	,	,	PUNCT
ap-3457	462	10	doi:10.1137/060659831	doi:10.1137/060659831	NOUN
ap-3457	462	11	.	.	PUNCT
ap-3457	463	1	[	[	X
ap-3457	463	2	38	38	NUM
ap-3457	463	3	]	]	PUNCT
ap-3457	463	4	j.	j.	PROPN
ap-3457	463	5	waldvogel	waldvogel	PROPN
ap-3457	463	6	,	,	PUNCT
ap-3457	463	7	fast	fast	ADJ
ap-3457	463	8	construction	construction	NOUN
ap-3457	463	9	of	of	ADP
ap-3457	463	10	the	the	DET
ap-3457	463	11	fejér	fejér	NOUN
ap-3457	463	12	and	and	CCONJ
ap-3457	463	13	clenshaw	clenshaw	ADJ
ap-3457	463	14	-	-	PUNCT
ap-3457	463	15	curtis	curtis	NOUN
ap-3457	463	16	quadrature	quadrature	NOUN
ap-3457	463	17	rules	rule	NOUN
ap-3457	463	18	,	,	PUNCT
ap-3457	463	19	bit	bit	NOUN
ap-3457	463	20	46	46	NUM
ap-3457	463	21	(	(	PUNCT
ap-3457	463	22	2006	2006	NUM
ap-3457	463	23	)	)	PUNCT
ap-3457	463	24	,	,	PUNCT
ap-3457	463	25	no	no	INTJ
ap-3457	463	26	.	.	NOUN
ap-3457	463	27	1	1	NUM
ap-3457	463	28	,	,	PUNCT
ap-3457	463	29	195–202	195–202	NUM
ap-3457	463	30	,	,	PUNCT
ap-3457	463	31	doi:10.1007	doi:10.1007	ADJ
ap-3457	463	32	/	/	SYM
ap-3457	463	33	s10543	s10543	NOUN
ap-3457	463	34	-	-	PUNCT
ap-3457	463	35	006	006	NUM
ap-3457	463	36	-	-	PUNCT
ap-3457	463	37	0045	0045	NUM
ap-3457	463	38	-	-	PUNCT
ap-3457	463	39	4	4	NUM
ap-3457	463	40	.	.	PUNCT
ap-3457	464	1	[	[	X
ap-3457	464	2	39	39	NUM
ap-3457	464	3	]	]	PUNCT
ap-3457	464	4	j.	j.	PROPN
ap-3457	464	5	c.	c.	PROPN
ap-3457	464	6	young	young	PROPN
ap-3457	464	7	,	,	PUNCT
ap-3457	464	8	s.	s.	PROPN
ap-3457	464	9	d.	d.	PROPN
ap-3457	464	10	gedney	gedney	PROPN
ap-3457	464	11	,	,	PUNCT
ap-3457	464	12	r.	r.	PROPN
ap-3457	464	13	j.	j.	PROPN
ap-3457	464	14	adams	adams	PROPN
ap-3457	464	15	,	,	PUNCT
ap-3457	464	16	quasimixed	quasimixe	VERB
ap-3457	464	17	-	-	PUNCT
ap-3457	464	18	order	order	NOUN
ap-3457	464	19	prism	prism	NOUN
ap-3457	464	20	basis	basis	NOUN
ap-3457	464	21	functions	function	NOUN
ap-3457	464	22	for	for	ADP
ap-3457	464	23	nyström	nyström	ADV
ap-3457	464	24	-	-	PUNCT
ap-3457	464	25	based	base	VERB
ap-3457	464	26	volume	volume	NOUN
ap-3457	464	27	integral	integral	ADJ
ap-3457	464	28	equations	equation	NOUN
ap-3457	464	29	,	,	PUNCT
ap-3457	464	30	ieee	ieee	PROPN
ap-3457	464	31	trans	trans	PROPN
ap-3457	464	32	.	.	PUNCT
ap-3457	465	1	magn	magn	PROPN
ap-3457	465	2	.	.	PUNCT
ap-3457	466	1	48	48	NUM
ap-3457	466	2	(	(	PUNCT
ap-3457	466	3	2012	2012	NUM
ap-3457	466	4	)	)	PUNCT
ap-3457	466	5	,	,	PUNCT
ap-3457	466	6	2560–2566	2560–2566	NUM
ap-3457	466	7	,	,	PUNCT
ap-3457	466	8	doi:10.1109	doi:10.1109	ADJ
ap-3457	466	9	/	/	SYM
ap-3457	466	10	tmag.2012.2197634	tmag.2012.2197634	PROPN
ap-3457	466	11	.	.	PUNCT
ap-3457	467	1	213	213	NUM
ap-3457	467	2	http://dx.doi.org/10.1090/s1061-0022-07-00946-6	http://dx.doi.org/10.1090/s1061-0022-07-00946-6	PROPN
ap-3457	467	3	http://dx.doi.org/10.1201/9781420036114	http://dx.doi.org/10.1201/9781420036114	VERB
ap-3457	467	4	http://dx.doi.org/10.1063/1.4817340	http://dx.doi.org/10.1063/1.4817340	VERB
ap-3457	467	5	http://dx.doi.org/10.1088/1751-8113/42/38/385208	http://dx.doi.org/10.1088/1751-8113/42/38/385208	NUM
ap-3457	467	6	http://dx.doi.org/10.1088/1751-8113/45/25/255201	http://dx.doi.org/10.1088/1751-8113/45/25/255201	NOUN
ap-3457	467	7	http://dx.doi.org/10.1007/978-1-4612-6398-2	http://dx.doi.org/10.1007/978-1-4612-6398-2	NOUN
ap-3457	467	8	http://dx.doi.org/10.1063/1.2779768	http://dx.doi.org/10.1063/1.2779768	PROPN
ap-3457	467	9	http://dx.doi.org/10.3842/sigma.2006.006	http://dx.doi.org/10.3842/sigma.2006.006	NOUN
ap-3457	467	10	http://dx.doi.org/10.3842/sigma.2007.023	http://dx.doi.org/10.3842/sigma.2007.023	X
ap-3457	467	11	http://dx.doi.org/10.1016/b978-0-12-064850-4.50015-x	http://dx.doi.org/10.1016/b978-0-12-064850-4.50015-x	PROPN
ap-3457	467	12	http://dx.doi.org/10.3842/sigma.2012.067	http://dx.doi.org/10.3842/sigma.2012.067	NOUN
ap-3457	467	13	http://dx.doi.org/10.1137/060671851	http://dx.doi.org/10.1137/060671851	NOUN
ap-3457	467	14	http://dx.doi.org/10.1007/s00041-009-9106-9	http://dx.doi.org/10.1007/s00041-009-9106-9	NOUN
ap-3457	467	15	http://dx.doi.org/10.1007/978-94-009-0501-6_14	http://dx.doi.org/10.1007/978-94-009-0501-6_14	NOUN
ap-3457	467	16	http://dx.doi.org/10.1007/s10543-009-0210-7	http://dx.doi.org/10.1007/s10543-009-0210-7	NUM
ap-3457	467	17	http://dx.doi.org/10.1007/s00041-014-9355-0	http://dx.doi.org/10.1007/s00041-014-9355-0	X
ap-3457	468	1	http://dx.doi.org/10.1016/j.aam.2010.11.005	http://dx.doi.org/10.1016/j.aam.2010.11.005	PROPN
ap-3457	468	2	http://dx.doi.org/10.3842/sigma.2006.076	http://dx.doi.org/10.3842/sigma.2006.076	PROPN
ap-3457	468	3	http://hdl.handle.net/1866/11153	http://hdl.handle.net/1866/11153	VERB
ap-3457	468	4	http://dx.doi.org/10.1007/978-3-642-15337-2_2	http://dx.doi.org/10.1007/978-3-642-15337-2_2	PRON
ap-3457	468	5	http://dx.doi.org/10.2307/2160762	http://dx.doi.org/10.2307/2160762	NOUN
ap-3457	469	1	http://dx.doi.org/10.1109/map.2009.5433099	http://dx.doi.org/10.1109/map.2009.5433099	ADP
ap-3457	469	2	http://dx.doi.org/10.1006/jath.1995.1119	http://dx.doi.org/10.1006/jath.1995.1119	PROPN
ap-3457	469	3	http://dx.doi.org/10.1007/s11075-008-9203-x	http://dx.doi.org/10.1007/s11075-008-9203-x	PROPN
ap-3457	469	4	http://dx.doi.org/10.1137/060659831	http://dx.doi.org/10.1137/060659831	X
ap-3457	469	5	http://dx.doi.org/10.1007/s10543-006-0045-4	http://dx.doi.org/10.1007/s10543-006-0045-4	NUM
ap-3457	469	6	http://dx.doi.org/10.1109/tmag.2012.2197634	http://dx.doi.org/10.1109/tmag.2012.2197634	PROPN
ap-3457	469	7	acta	acta	PROPN
ap-3457	469	8	polytechnica	polytechnica	PROPN
ap-3457	469	9	56(3):202–213	56(3):202–213	PROPN
ap-3457	469	10	,	,	PUNCT
ap-3457	469	11	2016	2016	NUM
ap-3457	469	12	1	1	NUM
ap-3457	469	13	introduction	introduction	NOUN
ap-3457	469	14	2	2	NUM
ap-3457	469	15	special	special	ADJ
ap-3457	469	16	functions	function	NOUN
ap-3457	469	17	associated	associate	VERB
ap-3457	469	18	to	to	PART
ap-3457	469	19	root	root	VERB
ap-3457	469	20	systems	system	NOUN
ap-3457	469	21	2.1	2.1	NUM
ap-3457	469	22	basic	basic	ADJ
ap-3457	469	23	definitions	definition	NOUN
ap-3457	469	24	2.2	2.2	NUM
ap-3457	469	25	weyl	weyl	VERB
ap-3457	469	26	group	group	NOUN
ap-3457	469	27	orbit	orbit	NOUN
ap-3457	469	28	functions	function	NOUN
ap-3457	469	29	2.3	2.3	NUM
ap-3457	469	30	jacobi	jacobi	NOUN
ap-3457	469	31	polynomials	polynomial	VERB
ap-3457	469	32	3	3	NUM
ap-3457	469	33	cubature	cubature	NOUN
ap-3457	469	34	formulas	formula	VERB
ap-3457	469	35	3.1	3.1	NUM
ap-3457	469	36	general	general	ADJ
ap-3457	469	37	form	form	NOUN
ap-3457	469	38	of	of	ADP
ap-3457	469	39	cubature	cubature	ADJ
ap-3457	469	40	formulas	formula	NOUN
ap-3457	469	41	3.2	3.2	NUM
ap-3457	469	42	cubature	cubature	ADJ
ap-3457	469	43	formulas	formula	NOUN
ap-3457	469	44	of	of	ADP
ap-3457	469	45	c2	c2	PROPN
ap-3457	469	46	4	4	NUM
ap-3457	469	47	clenshaw	clenshaw	ADJ
ap-3457	469	48	-	-	PUNCT
ap-3457	469	49	curtis	curtis	NOUN
ap-3457	469	50	cubature	cubature	NOUN
ap-3457	469	51	formulas	formula	VERB
ap-3457	469	52	4.1	4.1	NUM
ap-3457	469	53	clenshaw	clenshaw	ADJ
ap-3457	469	54	-	-	PUNCT
ap-3457	469	55	curtis	curtis	NOUN
ap-3457	469	56	method	method	NOUN
ap-3457	469	57	4.2	4.2	NUM
ap-3457	469	58	integration	integration	NOUN
ap-3457	469	59	domain	domain	NOUN
ap-3457	469	60	omega	omega	NOUN
ap-3457	469	61	of	of	ADP
ap-3457	469	62	c2	c2	PROPN
ap-3457	469	63	4.3	4.3	NUM
ap-3457	469	64	triangular	triangular	NOUN
ap-3457	469	65	domain	domain	NOUN
ap-3457	469	66	of	of	ADP
ap-3457	469	67	c2	c2	PROPN
ap-3457	469	68	5	5	NUM
ap-3457	469	69	concluding	concluding	NOUN
ap-3457	469	70	remarks	remark	VERB
ap-3457	469	71	6	6	NUM
ap-3457	469	72	acknowledgments	acknowledgment	NOUN
ap-3457	469	73	references	reference	NOUN