id	sid	tid	token	lemma	pos
ejpam-4983	1	1	european	european	PROPN
ejpam-4983	1	2	journal	journal	PROPN
ejpam-4983	1	3	of	of	ADP
ejpam-4983	1	4	pure	pure	ADJ
ejpam-4983	1	5	and	and	CCONJ
ejpam-4983	1	6	applied	apply	VERB
ejpam-4983	1	7	mathematics	mathematic	NOUN
ejpam-4983	1	8	vol	vol	NOUN
ejpam-4983	1	9	.	.	PROPN
ejpam-4983	2	1	17	17	NUM
ejpam-4983	2	2	,	,	PUNCT
ejpam-4983	2	3	no	no	INTJ
ejpam-4983	2	4	.	.	NOUN
ejpam-4983	2	5	1	1	NUM
ejpam-4983	2	6	,	,	PUNCT
ejpam-4983	2	7	2024	2024	NUM
ejpam-4983	2	8	,	,	PUNCT
ejpam-4983	2	9	11	11	NUM
ejpam-4983	2	10	-	-	SYM
ejpam-4983	2	11	29	29	NUM
ejpam-4983	2	12	issn	issn	PROPN
ejpam-4983	2	13	1307	1307	NUM
ejpam-4983	2	14	-	-	SYM
ejpam-4983	2	15	5543	5543	NUM
ejpam-4983	2	16	–	–	PUNCT
ejpam-4983	3	1	ejpam.com	ejpam.com	X
ejpam-4983	3	2	published	publish	VERB
ejpam-4983	3	3	by	by	ADP
ejpam-4983	3	4	new	new	PROPN
ejpam-4983	3	5	york	york	PROPN
ejpam-4983	3	6	business	business	PROPN
ejpam-4983	3	7	global	global	ADJ
ejpam-4983	3	8	determinants	determinant	NOUN
ejpam-4983	3	9	of	of	ADP
ejpam-4983	3	10	arrowhead	arrowhead	NOUN
ejpam-4983	3	11	matrices	matrix	NOUN
ejpam-4983	3	12	over	over	ADP
ejpam-4983	3	13	finite	finite	PROPN
ejpam-4983	3	14	commutative	commutative	ADJ
ejpam-4983	3	15	chain	chain	NOUN
ejpam-4983	3	16	rings	ring	NOUN
ejpam-4983	3	17	somphong	somphong	PROPN
ejpam-4983	3	18	jitman1,∗	jitman1,∗	PROPN
ejpam-4983	3	19	,	,	PUNCT
ejpam-4983	3	20	pornrudee	pornrudee	NOUN
ejpam-4983	3	21	modjam1	modjam1	PROPN
ejpam-4983	3	22	1	1	NUM
ejpam-4983	3	23	department	department	NOUN
ejpam-4983	3	24	of	of	ADP
ejpam-4983	3	25	mathematics	mathematic	NOUN
ejpam-4983	3	26	,	,	PUNCT
ejpam-4983	3	27	faculty	faculty	NOUN
ejpam-4983	3	28	of	of	ADP
ejpam-4983	3	29	science	science	NOUN
ejpam-4983	3	30	,	,	PUNCT
ejpam-4983	3	31	silpakorn	silpakorn	VERB
ejpam-4983	3	32	university	university	PROPN
ejpam-4983	3	33	,	,	PUNCT
ejpam-4983	3	34	nakhon	nakhon	PROPN
ejpam-4983	3	35	pathom	pathom	PROPN
ejpam-4983	3	36	73000	73000	NUM
ejpam-4983	3	37	,	,	PUNCT
ejpam-4983	3	38	thailand	thailand	PROPN
ejpam-4983	3	39	abstract	abstract	PROPN
ejpam-4983	3	40	.	.	PUNCT
ejpam-4983	4	1	arrowhead	arrowhead	NOUN
ejpam-4983	4	2	matrices	matrix	NOUN
ejpam-4983	4	3	have	have	AUX
ejpam-4983	4	4	attracted	attract	VERB
ejpam-4983	4	5	attention	attention	NOUN
ejpam-4983	4	6	due	due	ADP
ejpam-4983	4	7	to	to	ADP
ejpam-4983	4	8	their	their	PRON
ejpam-4983	4	9	rich	rich	ADJ
ejpam-4983	4	10	algebraic	algebraic	ADJ
ejpam-4983	4	11	structures	structure	NOUN
ejpam-4983	4	12	and	and	CCONJ
ejpam-4983	4	13	numerous	numerous	ADJ
ejpam-4983	4	14	applications	application	NOUN
ejpam-4983	4	15	.	.	PUNCT
ejpam-4983	5	1	in	in	ADP
ejpam-4983	5	2	this	this	DET
ejpam-4983	5	3	paper	paper	NOUN
ejpam-4983	5	4	,	,	PUNCT
ejpam-4983	5	5	we	we	PRON
ejpam-4983	5	6	focus	focus	VERB
ejpam-4983	5	7	on	on	ADP
ejpam-4983	5	8	the	the	DET
ejpam-4983	5	9	enumeration	enumeration	NOUN
ejpam-4983	5	10	of	of	ADP
ejpam-4983	5	11	n	n	NUM
ejpam-4983	5	12	×	×	NOUN
ejpam-4983	5	13	n	n	CCONJ
ejpam-4983	5	14	arrowhead	arrowhead	NOUN
ejpam-4983	5	15	matrices	matrix	NOUN
ejpam-4983	5	16	with	with	ADP
ejpam-4983	5	17	prescribed	prescribe	VERB
ejpam-4983	5	18	determinant	determinant	ADJ
ejpam-4983	5	19	over	over	ADP
ejpam-4983	5	20	a	a	DET
ejpam-4983	5	21	finite	finite	ADJ
ejpam-4983	5	22	field	field	NOUN
ejpam-4983	5	23	fq	fq	PROPN
ejpam-4983	5	24	and	and	CCONJ
ejpam-4983	5	25	over	over	ADP
ejpam-4983	5	26	a	a	DET
ejpam-4983	5	27	finite	finite	ADJ
ejpam-4983	5	28	commutative	commutative	ADJ
ejpam-4983	5	29	chain	chain	NOUN
ejpam-4983	5	30	ring	ring	NOUN
ejpam-4983	5	31	r.	r.	PROPN
ejpam-4983	5	32	the	the	DET
ejpam-4983	5	33	number	number	NOUN
ejpam-4983	5	34	of	of	ADP
ejpam-4983	5	35	n×n	n×n	PROPN
ejpam-4983	5	36	arrowhead	arrowhead	NOUN
ejpam-4983	5	37	matrices	matrix	NOUN
ejpam-4983	5	38	over	over	ADP
ejpam-4983	5	39	fq	fq	PROPN
ejpam-4983	5	40	of	of	ADP
ejpam-4983	5	41	a	a	DET
ejpam-4983	5	42	fixed	fix	VERB
ejpam-4983	5	43	determinant	determinant	NOUN
ejpam-4983	5	44	a	a	PRON
ejpam-4983	5	45	is	be	AUX
ejpam-4983	5	46	determined	determine	VERB
ejpam-4983	5	47	for	for	ADP
ejpam-4983	5	48	all	all	DET
ejpam-4983	5	49	positive	positive	ADJ
ejpam-4983	5	50	integers	integer	NOUN
ejpam-4983	5	51	n	n	PRON
ejpam-4983	5	52	and	and	CCONJ
ejpam-4983	5	53	for	for	ADP
ejpam-4983	5	54	all	all	DET
ejpam-4983	5	55	elements	element	NOUN
ejpam-4983	5	56	a	a	DET
ejpam-4983	5	57	∈	∈	PROPN
ejpam-4983	5	58	fq	fq	NOUN
ejpam-4983	5	59	.	.	PROPN
ejpam-4983	6	1	as	as	ADP
ejpam-4983	6	2	applications	application	NOUN
ejpam-4983	6	3	,	,	PUNCT
ejpam-4983	6	4	this	this	DET
ejpam-4983	6	5	result	result	NOUN
ejpam-4983	6	6	is	be	AUX
ejpam-4983	6	7	used	use	VERB
ejpam-4983	6	8	in	in	ADP
ejpam-4983	6	9	the	the	DET
ejpam-4983	6	10	enumeration	enumeration	NOUN
ejpam-4983	6	11	of	of	ADP
ejpam-4983	6	12	n	n	NUM
ejpam-4983	6	13	×	×	NOUN
ejpam-4983	6	14	n	n	CCONJ
ejpam-4983	6	15	non	non	ADJ
ejpam-4983	6	16	-	-	ADJ
ejpam-4983	6	17	singular	singular	ADJ
ejpam-4983	6	18	arrowhead	arrowhead	NOUN
ejpam-4983	6	19	matrices	matrix	NOUN
ejpam-4983	6	20	with	with	ADP
ejpam-4983	6	21	prescribed	prescribe	VERB
ejpam-4983	6	22	determinant	determinant	ADJ
ejpam-4983	6	23	over	over	ADP
ejpam-4983	6	24	r.	r.	PROPN
ejpam-4983	6	25	subsequently	subsequently	ADV
ejpam-4983	6	26	,	,	PUNCT
ejpam-4983	6	27	some	some	DET
ejpam-4983	6	28	bounds	bound	NOUN
ejpam-4983	6	29	on	on	ADP
ejpam-4983	6	30	the	the	DET
ejpam-4983	6	31	number	number	NOUN
ejpam-4983	6	32	of	of	ADP
ejpam-4983	6	33	n	n	NUM
ejpam-4983	6	34	×	×	NOUN
ejpam-4983	6	35	n	n	CCONJ
ejpam-4983	6	36	singular	singular	ADJ
ejpam-4983	6	37	arrowhead	arrowhead	NOUN
ejpam-4983	6	38	matrices	matrix	NOUN
ejpam-4983	6	39	over	over	ADP
ejpam-4983	6	40	r	r	NOUN
ejpam-4983	6	41	of	of	ADP
ejpam-4983	6	42	a	a	DET
ejpam-4983	6	43	fixed	fix	VERB
ejpam-4983	6	44	determinant	determinant	NOUN
ejpam-4983	6	45	are	be	AUX
ejpam-4983	6	46	given	give	VERB
ejpam-4983	6	47	.	.	PUNCT
ejpam-4983	7	1	finally	finally	ADV
ejpam-4983	7	2	,	,	PUNCT
ejpam-4983	7	3	some	some	DET
ejpam-4983	7	4	open	open	ADJ
ejpam-4983	7	5	problems	problem	NOUN
ejpam-4983	7	6	are	be	AUX
ejpam-4983	7	7	presented	present	VERB
ejpam-4983	7	8	.	.	PUNCT
ejpam-4983	8	1	2020	2020	NUM
ejpam-4983	8	2	mathematics	mathematics	PROPN
ejpam-4983	8	3	subject	subject	NOUN
ejpam-4983	8	4	classifications	classification	NOUN
ejpam-4983	8	5	:	:	PUNCT
ejpam-4983	8	6	11c20	11c20	NUM
ejpam-4983	8	7	,	,	PUNCT
ejpam-4983	8	8	15b33	15b33	NUM
ejpam-4983	8	9	key	key	ADJ
ejpam-4983	8	10	words	word	NOUN
ejpam-4983	8	11	and	and	CCONJ
ejpam-4983	8	12	phrases	phrase	NOUN
ejpam-4983	8	13	:	:	PUNCT
ejpam-4983	8	14	arrowhead	arrowhead	NOUN
ejpam-4983	8	15	matrices	matrix	NOUN
ejpam-4983	8	16	,	,	PUNCT
ejpam-4983	8	17	determinants	determinant	NOUN
ejpam-4983	8	18	,	,	PUNCT
ejpam-4983	8	19	finite	finite	ADJ
ejpam-4983	8	20	fields	field	NOUN
ejpam-4983	8	21	,	,	PUNCT
ejpam-4983	8	22	finite	finite	PROPN
ejpam-4983	8	23	commutative	commutative	ADJ
ejpam-4983	8	24	chain	chain	NOUN
ejpam-4983	8	25	rings	ring	NOUN
ejpam-4983	8	26	,	,	PUNCT
ejpam-4983	8	27	enumeration	enumeration	NOUN
ejpam-4983	8	28	1	1	NUM
ejpam-4983	8	29	.	.	PUNCT
ejpam-4983	9	1	introduction	introduction	NOUN
ejpam-4983	9	2	matrices	matrix	NOUN
ejpam-4983	9	3	and	and	CCONJ
ejpam-4983	9	4	their	their	PRON
ejpam-4983	9	5	determinants	determinant	NOUN
ejpam-4983	9	6	have	have	AUX
ejpam-4983	9	7	been	be	AUX
ejpam-4983	9	8	known	know	VERB
ejpam-4983	9	9	and	and	CCONJ
ejpam-4983	9	10	extensively	extensively	ADV
ejpam-4983	9	11	studied	study	VERB
ejpam-4983	9	12	for	for	ADP
ejpam-4983	9	13	their	their	PRON
ejpam-4983	9	14	nice	nice	ADJ
ejpam-4983	9	15	properties	property	NOUN
ejpam-4983	9	16	and	and	CCONJ
ejpam-4983	9	17	wide	wide	ADJ
ejpam-4983	9	18	applications	application	NOUN
ejpam-4983	9	19	(	(	PUNCT
ejpam-4983	9	20	see	see	VERB
ejpam-4983	9	21	,	,	PUNCT
ejpam-4983	9	22	for	for	ADP
ejpam-4983	9	23	example	example	NOUN
ejpam-4983	9	24	,	,	PUNCT
ejpam-4983	10	1	[	[	X
ejpam-4983	10	2	2	2	NUM
ejpam-4983	10	3	]	]	PUNCT
ejpam-4983	10	4	,	,	PUNCT
ejpam-4983	10	5	[	[	X
ejpam-4983	10	6	9	9	NUM
ejpam-4983	10	7	]	]	PUNCT
ejpam-4983	10	8	,	,	PUNCT
ejpam-4983	10	9	and	and	CCONJ
ejpam-4983	10	10	[	[	X
ejpam-4983	10	11	10	10	NUM
ejpam-4983	10	12	]	]	NUM
ejpam-4983	10	13	)	)	PUNCT
ejpam-4983	10	14	.	.	PUNCT
ejpam-4983	11	1	singularity	singularity	NOUN
ejpam-4983	11	2	of	of	ADP
ejpam-4983	11	3	matrices	matrix	NOUN
ejpam-4983	11	4	is	be	AUX
ejpam-4983	11	5	useful	useful	ADJ
ejpam-4983	11	6	in	in	ADP
ejpam-4983	11	7	applications	application	NOUN
ejpam-4983	11	8	(	(	PUNCT
ejpam-4983	11	9	see	see	VERB
ejpam-4983	11	10	,	,	PUNCT
ejpam-4983	11	11	for	for	ADP
ejpam-4983	11	12	example	example	NOUN
ejpam-4983	11	13	,	,	PUNCT
ejpam-4983	11	14	[	[	X
ejpam-4983	11	15	2	2	NUM
ejpam-4983	11	16	]	]	PUNCT
ejpam-4983	11	17	and	and	CCONJ
ejpam-4983	11	18	[	[	X
ejpam-4983	11	19	11	11	NUM
ejpam-4983	11	20	]	]	NUM
ejpam-4983	11	21	)	)	PUNCT
ejpam-4983	11	22	.	.	PUNCT
ejpam-4983	12	1	the	the	DET
ejpam-4983	12	2	number	number	NOUN
ejpam-4983	12	3	of	of	ADP
ejpam-4983	12	4	n	n	NUM
ejpam-4983	12	5	×	×	NOUN
ejpam-4983	12	6	n	n	CCONJ
ejpam-4983	12	7	singular	singular	NOUN
ejpam-4983	12	8	(	(	PUNCT
ejpam-4983	12	9	resp	resp	NOUN
ejpam-4983	12	10	.	.	PUNCT
ejpam-4983	12	11	,	,	PUNCT
ejpam-4983	12	12	nonsingular	nonsingular	ADJ
ejpam-4983	12	13	)	)	PUNCT
ejpam-4983	12	14	matrices	matrix	NOUN
ejpam-4983	12	15	over	over	ADP
ejpam-4983	12	16	a	a	DET
ejpam-4983	12	17	finite	finite	ADJ
ejpam-4983	12	18	field	field	NOUN
ejpam-4983	12	19	fq	fq	PROPN
ejpam-4983	12	20	has	have	AUX
ejpam-4983	12	21	been	be	AUX
ejpam-4983	12	22	determined	determine	VERB
ejpam-4983	12	23	in	in	ADP
ejpam-4983	12	24	[	[	X
ejpam-4983	12	25	13	13	NUM
ejpam-4983	12	26	]	]	PUNCT
ejpam-4983	12	27	.	.	PUNCT
ejpam-4983	13	1	as	as	ADP
ejpam-4983	13	2	a	a	DET
ejpam-4983	13	3	generalization	generalization	NOUN
ejpam-4983	13	4	of	of	ADP
ejpam-4983	13	5	a	a	DET
ejpam-4983	13	6	prime	prime	ADJ
ejpam-4983	13	7	field	field	NOUN
ejpam-4983	13	8	zp	zp	NOUN
ejpam-4983	13	9	,	,	PUNCT
ejpam-4983	13	10	the	the	DET
ejpam-4983	13	11	number	number	NOUN
ejpam-4983	13	12	of	of	ADP
ejpam-4983	13	13	n	n	NUM
ejpam-4983	13	14	×	×	NOUN
ejpam-4983	13	15	n	n	PRON
ejpam-4983	13	16	matrices	matrix	NOUN
ejpam-4983	13	17	over	over	ADP
ejpam-4983	13	18	zm	zm	PROPN
ejpam-4983	13	19	of	of	ADP
ejpam-4983	13	20	a	a	DET
ejpam-4983	13	21	fixed	fix	VERB
ejpam-4983	13	22	determinant	determinant	NOUN
ejpam-4983	13	23	has	have	AUX
ejpam-4983	13	24	been	be	AUX
ejpam-4983	13	25	first	first	ADV
ejpam-4983	13	26	studied	study	VERB
ejpam-4983	13	27	in	in	ADP
ejpam-4983	13	28	[	[	X
ejpam-4983	13	29	1	1	NUM
ejpam-4983	13	30	]	]	PUNCT
ejpam-4983	13	31	.	.	PUNCT
ejpam-4983	14	1	an	an	DET
ejpam-4983	14	2	alternative	alternative	ADJ
ejpam-4983	14	3	study	study	NOUN
ejpam-4983	14	4	of	of	ADP
ejpam-4983	14	5	the	the	DET
ejpam-4983	14	6	problem	problem	NOUN
ejpam-4983	14	7	in	in	ADP
ejpam-4983	14	8	[	[	X
ejpam-4983	14	9	1	1	X
ejpam-4983	14	10	]	]	PUNCT
ejpam-4983	14	11	has	have	AUX
ejpam-4983	14	12	been	be	AUX
ejpam-4983	14	13	given	give	VERB
ejpam-4983	14	14	in	in	ADP
ejpam-4983	14	15	[	[	X
ejpam-4983	14	16	10	10	NUM
ejpam-4983	14	17	]	]	PUNCT
ejpam-4983	14	18	using	use	VERB
ejpam-4983	14	19	a	a	DET
ejpam-4983	14	20	different	different	ADJ
ejpam-4983	14	21	and	and	CCONJ
ejpam-4983	14	22	simpler	simple	ADJ
ejpam-4983	14	23	approach	approach	NOUN
ejpam-4983	14	24	.	.	PUNCT
ejpam-4983	15	1	a	a	DET
ejpam-4983	15	2	finite	finite	PROPN
ejpam-4983	15	3	commutative	commutative	ADJ
ejpam-4983	15	4	chain	chain	NOUN
ejpam-4983	15	5	ring	ring	NOUN
ejpam-4983	15	6	(	(	PUNCT
ejpam-4983	15	7	fccr	fccr	PROPN
ejpam-4983	15	8	)	)	PUNCT
ejpam-4983	15	9	and	and	CCONJ
ejpam-4983	15	10	a	a	DET
ejpam-4983	15	11	principal	principal	ADJ
ejpam-4983	15	12	ideal	ideal	NOUN
ejpam-4983	15	13	ring	ring	NOUN
ejpam-4983	15	14	are	be	AUX
ejpam-4983	15	15	generalizations	generalization	NOUN
ejpam-4983	15	16	of	of	ADP
ejpam-4983	15	17	the	the	DET
ejpam-4983	15	18	rings	ring	NOUN
ejpam-4983	15	19	zp	zp	PROPN
ejpam-4983	15	20	and	and	CCONJ
ejpam-4983	15	21	zm	zm	PROPN
ejpam-4983	15	22	that	that	PRON
ejpam-4983	15	23	are	be	AUX
ejpam-4983	15	24	useful	useful	ADJ
ejpam-4983	15	25	in	in	ADP
ejpam-4983	15	26	applications	application	NOUN
ejpam-4983	15	27	such	such	ADJ
ejpam-4983	15	28	as	as	ADP
ejpam-4983	15	29	coding	code	VERB
ejpam-4983	15	30	theory	theory	NOUN
ejpam-4983	15	31	and	and	CCONJ
ejpam-4983	15	32	cryptography	cryptography	NOUN
ejpam-4983	15	33	.	.	PUNCT
ejpam-4983	16	1	in	in	ADP
ejpam-4983	16	2	[	[	X
ejpam-4983	16	3	3	3	NUM
ejpam-4983	16	4	]	]	PUNCT
ejpam-4983	16	5	,	,	PUNCT
ejpam-4983	16	6	the	the	DET
ejpam-4983	16	7	techniques	technique	NOUN
ejpam-4983	16	8	in	in	ADP
ejpam-4983	16	9	[	[	X
ejpam-4983	16	10	10	10	NUM
ejpam-4983	16	11	]	]	PUNCT
ejpam-4983	16	12	have	have	AUX
ejpam-4983	16	13	been	be	AUX
ejpam-4983	16	14	extended	extend	VERB
ejpam-4983	16	15	to	to	ADP
ejpam-4983	16	16	matrices	matrix	NOUN
ejpam-4983	16	17	over	over	ADP
ejpam-4983	16	18	fccrs	fccr	NOUN
ejpam-4983	16	19	and	and	CCONJ
ejpam-4983	16	20	principal	principal	ADJ
ejpam-4983	16	21	ideal	ideal	NOUN
ejpam-4983	16	22	rings	ring	NOUN
ejpam-4983	16	23	.	.	PUNCT
ejpam-4983	17	1	precisely	precisely	ADV
ejpam-4983	17	2	,	,	PUNCT
ejpam-4983	17	3	the	the	DET
ejpam-4983	17	4	number	number	NOUN
ejpam-4983	17	5	of	of	ADP
ejpam-4983	17	6	n	n	NUM
ejpam-4983	17	7	×	×	NOUN
ejpam-4983	17	8	n	n	PRON
ejpam-4983	17	9	matrices	matrice	VERB
ejpam-4983	17	10	over	over	ADP
ejpam-4983	17	11	fccrs	fccr	NOUN
ejpam-4983	17	12	and	and	CCONJ
ejpam-4983	17	13	principal	principal	ADJ
ejpam-4983	17	14	ideal	ideal	NOUN
ejpam-4983	17	15	rings	ring	NOUN
ejpam-4983	17	16	of	of	ADP
ejpam-4983	17	17	a	a	DET
ejpam-4983	17	18	fixed	fix	VERB
ejpam-4983	17	19	determinant	determinant	NOUN
ejpam-4983	17	20	has	have	AUX
ejpam-4983	17	21	been	be	AUX
ejpam-4983	17	22	completely	completely	ADV
ejpam-4983	17	23	determined	determine	VERB
ejpam-4983	17	24	.	.	PUNCT
ejpam-4983	18	1	diagonal	diagonal	ADJ
ejpam-4983	18	2	matrices	matrix	NOUN
ejpam-4983	18	3	are	be	AUX
ejpam-4983	18	4	interesting	interesting	ADJ
ejpam-4983	18	5	subfamilies	subfamily	NOUN
ejpam-4983	18	6	of	of	ADP
ejpam-4983	18	7	the	the	DET
ejpam-4983	18	8	ones	one	NOUN
ejpam-4983	18	9	∗corresponding	∗corresponde	VERB
ejpam-4983	18	10	author	author	NOUN
ejpam-4983	18	11	.	.	PUNCT
ejpam-4983	19	1	doi	doi	NOUN
ejpam-4983	19	2	:	:	PUNCT
ejpam-4983	19	3	https://doi.org/10.29020/nybg.ejpam.v17i1.4983	https://doi.org/10.29020/nybg.ejpam.v17i1.4983	PROPN
ejpam-4983	19	4	email	email	NOUN
ejpam-4983	19	5	addresses	address	NOUN
ejpam-4983	19	6	:	:	PUNCT
ejpam-4983	20	1	sjitman@gmail.com	sjitman@gmail.com	PROPN
ejpam-4983	20	2	(	(	PUNCT
ejpam-4983	20	3	s.	s.	PROPN
ejpam-4983	20	4	jitman	jitman	PROPN
ejpam-4983	20	5	)	)	PUNCT
ejpam-4983	20	6	,	,	PUNCT
ejpam-4983	20	7	pornrudee.ole@gmail.com	pornrudee.ole@gmail.com	X
ejpam-4983	20	8	(	(	PUNCT
ejpam-4983	20	9	p.	p.	NOUN
ejpam-4983	20	10	modjam	modjam	PROPN
ejpam-4983	20	11	)	)	PUNCT
ejpam-4983	20	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4983	21	1	11	11	NUM
ejpam-4983	21	2	©	©	ADP
ejpam-4983	21	3	2024	2024	NUM
ejpam-4983	21	4	ejpam	ejpam	NOUN
ejpam-4983	21	5	all	all	DET
ejpam-4983	21	6	rights	right	NOUN
ejpam-4983	21	7	reserved	reserve	VERB
ejpam-4983	21	8	.	.	PUNCT
ejpam-4983	22	1	s.	s.	PROPN
ejpam-4983	22	2	jitman	jitman	PROPN
ejpam-4983	22	3	,	,	PUNCT
ejpam-4983	22	4	p.	p.	PROPN
ejpam-4983	22	5	modjam	modjam	PROPN
ejpam-4983	22	6	/	/	SYM
ejpam-4983	22	7	eur	eur	PROPN
ejpam-4983	22	8	.	.	PUNCT
ejpam-4983	23	1	j.	j.	PROPN
ejpam-4983	23	2	pure	pure	PROPN
ejpam-4983	23	3	appl	appl	PROPN
ejpam-4983	23	4	.	.	PROPN
ejpam-4983	23	5	math	math	PROPN
ejpam-4983	23	6	,	,	PUNCT
ejpam-4983	23	7	17	17	NUM
ejpam-4983	23	8	(	(	PUNCT
ejpam-4983	23	9	1	1	NUM
ejpam-4983	23	10	)	)	PUNCT
ejpam-4983	23	11	(	(	PUNCT
ejpam-4983	23	12	2024	2024	NUM
ejpam-4983	23	13	)	)	PUNCT
ejpam-4983	23	14	,	,	PUNCT
ejpam-4983	23	15	11	11	NUM
ejpam-4983	23	16	-	-	SYM
ejpam-4983	23	17	29	29	NUM
ejpam-4983	23	18	12	12	NUM
ejpam-4983	23	19	in	in	ADP
ejpam-4983	23	20	[	[	X
ejpam-4983	23	21	3	3	NUM
ejpam-4983	23	22	]	]	PUNCT
ejpam-4983	23	23	.	.	PUNCT
ejpam-4983	24	1	the	the	DET
ejpam-4983	24	2	enumeration	enumeration	NOUN
ejpam-4983	24	3	of	of	ADP
ejpam-4983	24	4	diagonal	diagonal	ADJ
ejpam-4983	24	5	matrices	matrix	NOUN
ejpam-4983	24	6	over	over	ADP
ejpam-4983	24	7	fccrs	fccr	NOUN
ejpam-4983	24	8	of	of	ADP
ejpam-4983	24	9	a	a	DET
ejpam-4983	24	10	fixed	fix	VERB
ejpam-4983	24	11	determinant	determinant	NOUN
ejpam-4983	24	12	are	be	AUX
ejpam-4983	24	13	presented	present	VERB
ejpam-4983	24	14	in	in	ADP
ejpam-4983	24	15	[	[	X
ejpam-4983	24	16	8	8	NUM
ejpam-4983	24	17	]	]	PUNCT
ejpam-4983	24	18	and	and	CCONJ
ejpam-4983	24	19	applied	apply	VERB
ejpam-4983	24	20	in	in	ADP
ejpam-4983	24	21	the	the	DET
ejpam-4983	24	22	study	study	NOUN
ejpam-4983	24	23	of	of	ADP
ejpam-4983	24	24	the	the	DET
ejpam-4983	24	25	determinant	determinant	NOUN
ejpam-4983	24	26	of	of	ADP
ejpam-4983	24	27	some	some	DET
ejpam-4983	24	28	circulant	circulant	ADJ
ejpam-4983	24	29	matrices	matrix	NOUN
ejpam-4983	24	30	over	over	ADP
ejpam-4983	24	31	fccrs	fccr	NOUN
ejpam-4983	24	32	.	.	PUNCT
ejpam-4983	25	1	for	for	ADP
ejpam-4983	25	2	a	a	DET
ejpam-4983	25	3	commutative	commutative	ADJ
ejpam-4983	25	4	ring	ring	NOUN
ejpam-4983	25	5	r	r	NOUN
ejpam-4983	25	6	and	and	CCONJ
ejpam-4983	25	7	a	a	DET
ejpam-4983	25	8	positive	positive	ADJ
ejpam-4983	25	9	integer	integer	NOUN
ejpam-4983	25	10	n	n	CCONJ
ejpam-4983	25	11	,	,	PUNCT
ejpam-4983	25	12	an	an	DET
ejpam-4983	25	13	n	n	NUM
ejpam-4983	25	14	×	×	NOUN
ejpam-4983	25	15	n	n	CCONJ
ejpam-4983	25	16	arrowhead	arrowhead	NOUN
ejpam-4983	25	17	matrix	matrix	NOUN
ejpam-4983	25	18	over	over	ADP
ejpam-4983	25	19	r	r	NOUN
ejpam-4983	25	20	is	be	AUX
ejpam-4983	25	21	defined	define	VERB
ejpam-4983	25	22	to	to	PART
ejpam-4983	25	23	be	be	AUX
ejpam-4983	25	24	a	a	DET
ejpam-4983	25	25	square	square	ADJ
ejpam-4983	25	26	matrix	matrix	NOUN
ejpam-4983	25	27	containing	contain	VERB
ejpam-4983	25	28	zeros	zero	NOUN
ejpam-4983	25	29	in	in	ADP
ejpam-4983	25	30	all	all	DET
ejpam-4983	25	31	entries	entry	NOUN
ejpam-4983	25	32	except	except	SCONJ
ejpam-4983	25	33	for	for	ADP
ejpam-4983	25	34	the	the	DET
ejpam-4983	25	35	first	first	ADJ
ejpam-4983	25	36	row	row	NOUN
ejpam-4983	25	37	,	,	PUNCT
ejpam-4983	25	38	first	first	ADJ
ejpam-4983	25	39	column	column	NOUN
ejpam-4983	25	40	,	,	PUNCT
ejpam-4983	25	41	and	and	CCONJ
ejpam-4983	25	42	main	main	ADJ
ejpam-4983	25	43	diagonal	diagonal	NOUN
ejpam-4983	25	44	.	.	PUNCT
ejpam-4983	26	1	precisely	precisely	ADV
ejpam-4983	26	2	,	,	PUNCT
ejpam-4983	26	3	the	the	DET
ejpam-4983	26	4	arrowhead	arrowhead	NOUN
ejpam-4983	26	5	matrix	matrix	NOUN
ejpam-4983	26	6	is	be	AUX
ejpam-4983	26	7	in	in	ADP
ejpam-4983	26	8	the	the	DET
ejpam-4983	26	9	form	form	NOUN
ejpam-4983	26	10	of	of	ADP
ejpam-4983	26	11	a	a	DET
ejpam-4983	26	12	=	=	SYM
ejpam-4983	26	13			NOUN
ejpam-4983	26	14	∗	∗	NOUN
ejpam-4983	26	15	∗	∗	NOUN
ejpam-4983	26	16	∗	∗	NOUN
ejpam-4983	26	17	∗	∗	NOUN
ejpam-4983	26	18	·	·	PUNCT
ejpam-4983	26	19	·	·	PUNCT
ejpam-4983	26	20	·	·	PUNCT
ejpam-4983	27	1	∗	∗	NOUN
ejpam-4983	27	2	∗	∗	NOUN
ejpam-4983	27	3	∗	∗	NOUN
ejpam-4983	27	4	0	0	NUM
ejpam-4983	27	5	0	0	NUM
ejpam-4983	27	6	·	·	PUNCT
ejpam-4983	27	7	·	·	PUNCT
ejpam-4983	27	8	·	·	PUNCT
ejpam-4983	27	9	0	0	NUM
ejpam-4983	28	1	∗	∗	NOUN
ejpam-4983	28	2	0	0	NUM
ejpam-4983	28	3	∗	∗	NOUN
ejpam-4983	28	4	0	0	NUM
ejpam-4983	28	5	·	·	PUNCT
ejpam-4983	28	6	·	·	PUNCT
ejpam-4983	28	7	·	·	PUNCT
ejpam-4983	28	8	0	0	NUM
ejpam-4983	29	1	∗	∗	NOUN
ejpam-4983	29	2	0	0	NUM
ejpam-4983	29	3	0	0	NUM
ejpam-4983	29	4	∗	∗	NOUN
ejpam-4983	29	5	·	·	PUNCT
ejpam-4983	29	6	·	·	PUNCT
ejpam-4983	29	7	·	·	PUNCT
ejpam-4983	29	8	0	0	NUM
ejpam-4983	29	9	...	...	PUNCT
ejpam-4983	29	10	...	...	PUNCT
ejpam-4983	29	11	...	...	PUNCT
ejpam-4983	29	12	...	...	PUNCT
ejpam-4983	29	13	.	.	PUNCT
ejpam-4983	29	14	.	.	PUNCT
ejpam-4983	29	15	.	.	PUNCT
ejpam-4983	30	1	...	...	PUNCT
ejpam-4983	31	1	∗	∗	NOUN
ejpam-4983	31	2	0	0	NUM
ejpam-4983	31	3	0	0	NUM
ejpam-4983	31	4	0	0	NUM
ejpam-4983	31	5	·	·	PUNCT
ejpam-4983	31	6	·	·	PUNCT
ejpam-4983	31	7	·	·	PUNCT
ejpam-4983	32	1	∗	∗	NOUN
ejpam-4983	32	2			NOUN
ejpam-4983	32	3	,	,	PUNCT
ejpam-4983	32	4	where	where	SCONJ
ejpam-4983	32	5	∗	∗	NOUN
ejpam-4983	32	6	’s	’s	PART
ejpam-4983	32	7	are	be	AUX
ejpam-4983	32	8	arbitrary	arbitrary	ADJ
ejpam-4983	32	9	elements	element	NOUN
ejpam-4983	32	10	in	in	ADP
ejpam-4983	32	11	r	r	NOUN
ejpam-4983	33	1	and	and	CCONJ
ejpam-4983	33	2	they	they	PRON
ejpam-4983	33	3	are	be	AUX
ejpam-4983	33	4	not	not	PART
ejpam-4983	33	5	necessarily	necessarily	ADV
ejpam-4983	33	6	the	the	DET
ejpam-4983	33	7	same	same	ADJ
ejpam-4983	33	8	.	.	PUNCT
ejpam-4983	34	1	from	from	ADP
ejpam-4983	34	2	the	the	DET
ejpam-4983	34	3	definition	definition	NOUN
ejpam-4983	34	4	,	,	PUNCT
ejpam-4983	34	5	an	an	DET
ejpam-4983	34	6	arrowhead	arrowhead	NOUN
ejpam-4983	34	7	matrix	matrix	NOUN
ejpam-4983	34	8	is	be	AUX
ejpam-4983	34	9	a	a	DET
ejpam-4983	34	10	generalization	generalization	NOUN
ejpam-4983	34	11	of	of	ADP
ejpam-4983	34	12	a	a	DET
ejpam-4983	34	13	diagonal	diagonal	ADJ
ejpam-4983	34	14	matrix	matrix	NOUN
ejpam-4983	34	15	over	over	ADP
ejpam-4983	34	16	r.	r.	PROPN
ejpam-4983	34	17	it	it	PRON
ejpam-4983	34	18	is	be	AUX
ejpam-4983	34	19	easily	easily	ADV
ejpam-4983	34	20	seen	see	VERB
ejpam-4983	34	21	that	that	SCONJ
ejpam-4983	34	22	the	the	DET
ejpam-4983	34	23	1	1	NUM
ejpam-4983	34	24	×	×	NOUN
ejpam-4983	34	25	1	1	NUM
ejpam-4983	34	26	matrices	matrix	NOUN
ejpam-4983	34	27	,	,	PUNCT
ejpam-4983	34	28	2	2	NUM
ejpam-4983	34	29	×	×	NOUN
ejpam-4983	34	30	2	2	NUM
ejpam-4983	34	31	matrices	matrix	NOUN
ejpam-4983	34	32	,	,	PUNCT
ejpam-4983	34	33	and	and	CCONJ
ejpam-4983	34	34	n	n	CCONJ
ejpam-4983	34	35	×	×	NOUN
ejpam-4983	34	36	n	n	CCONJ
ejpam-4983	34	37	diagonal	diagonal	ADJ
ejpam-4983	34	38	matrices	matrix	NOUN
ejpam-4983	34	39	over	over	ADP
ejpam-4983	34	40	r	r	NOUN
ejpam-4983	34	41	are	be	AUX
ejpam-4983	34	42	arrowhead	arrowhead	NOUN
ejpam-4983	34	43	matrices	matrix	NOUN
ejpam-4983	34	44	for	for	ADP
ejpam-4983	34	45	all	all	DET
ejpam-4983	34	46	positive	positive	ADJ
ejpam-4983	34	47	integers	integer	NOUN
ejpam-4983	34	48	n.	n.	VERB
ejpam-4983	34	49	some	some	DET
ejpam-4983	34	50	properties	property	NOUN
ejpam-4983	34	51	of	of	ADP
ejpam-4983	34	52	arrowhead	arrowhead	NOUN
ejpam-4983	34	53	matrices	matrix	NOUN
ejpam-4983	34	54	such	such	ADJ
ejpam-4983	34	55	as	as	ADP
ejpam-4983	34	56	eigenvalues	eigenvalue	NOUN
ejpam-4983	34	57	,	,	PUNCT
ejpam-4983	34	58	eigenvectors	eigenvector	NOUN
ejpam-4983	34	59	,	,	PUNCT
ejpam-4983	34	60	and	and	CCONJ
ejpam-4983	34	61	inverses	inverse	NOUN
ejpam-4983	34	62	have	have	AUX
ejpam-4983	34	63	been	be	AUX
ejpam-4983	34	64	studied	study	VERB
ejpam-4983	34	65	in	in	ADP
ejpam-4983	34	66	[	[	X
ejpam-4983	34	67	14	14	NUM
ejpam-4983	34	68	]	]	PUNCT
ejpam-4983	34	69	,	,	PUNCT
ejpam-4983	35	1	[	[	X
ejpam-4983	35	2	15	15	NUM
ejpam-4983	35	3	]	]	PUNCT
ejpam-4983	35	4	,	,	PUNCT
ejpam-4983	35	5	and	and	CCONJ
ejpam-4983	35	6	[	[	X
ejpam-4983	35	7	16	16	NUM
ejpam-4983	35	8	]	]	PUNCT
ejpam-4983	35	9	.	.	PUNCT
ejpam-4983	36	1	arrowhead	arrowhead	NOUN
ejpam-4983	36	2	matrices	matrix	NOUN
ejpam-4983	36	3	have	have	VERB
ejpam-4983	36	4	applications	application	NOUN
ejpam-4983	36	5	in	in	ADP
ejpam-4983	36	6	various	various	ADJ
ejpam-4983	36	7	fields	field	NOUN
ejpam-4983	36	8	,	,	PUNCT
ejpam-4983	36	9	e.g.	e.g.	ADV
ejpam-4983	36	10	,	,	PUNCT
ejpam-4983	36	11	wireless	wireless	ADJ
ejpam-4983	36	12	communications	communication	NOUN
ejpam-4983	36	13	in	in	ADP
ejpam-4983	36	14	[	[	X
ejpam-4983	36	15	15	15	NUM
ejpam-4983	36	16	]	]	PUNCT
ejpam-4983	36	17	,	,	PUNCT
ejpam-4983	36	18	eigenvalue	eigenvalue	ADJ
ejpam-4983	36	19	decompositions	decomposition	NOUN
ejpam-4983	36	20	of	of	ADP
ejpam-4983	36	21	some	some	DET
ejpam-4983	36	22	matrices	matrix	NOUN
ejpam-4983	36	23	in	in	ADP
ejpam-4983	36	24	[	[	X
ejpam-4983	36	25	16	16	NUM
ejpam-4983	36	26	]	]	PUNCT
ejpam-4983	36	27	,	,	PUNCT
ejpam-4983	36	28	the	the	DET
ejpam-4983	36	29	study	study	NOUN
ejpam-4983	36	30	of	of	ADP
ejpam-4983	36	31	directed	direct	VERB
ejpam-4983	36	32	multigraphs	multigraph	NOUN
ejpam-4983	36	33	and	and	CCONJ
ejpam-4983	36	34	hub	hub	NOUN
ejpam-4983	36	35	-	-	PUNCT
ejpam-4983	36	36	directed	direct	VERB
ejpam-4983	36	37	multigraphs	multigraph	NOUN
ejpam-4983	36	38	in[12	in[12	NOUN
ejpam-4983	36	39	]	]	PUNCT
ejpam-4983	36	40	,	,	PUNCT
ejpam-4983	36	41	and	and	CCONJ
ejpam-4983	36	42	the	the	DET
ejpam-4983	36	43	study	study	NOUN
ejpam-4983	36	44	of	of	ADP
ejpam-4983	36	45	disordered	disordered	ADJ
ejpam-4983	36	46	quantum	quantum	NOUN
ejpam-4983	36	47	spins	spin	NOUN
ejpam-4983	36	48	in	in	ADP
ejpam-4983	36	49	[	[	X
ejpam-4983	36	50	4	4	NUM
ejpam-4983	36	51	]	]	PUNCT
ejpam-4983	36	52	.	.	PUNCT
ejpam-4983	37	1	as	as	ADP
ejpam-4983	37	2	a	a	DET
ejpam-4983	37	3	generalization	generalization	NOUN
ejpam-4983	37	4	of	of	ADP
ejpam-4983	37	5	[	[	X
ejpam-4983	37	6	8	8	NUM
ejpam-4983	37	7	]	]	PUNCT
ejpam-4983	37	8	,	,	PUNCT
ejpam-4983	37	9	the	the	DET
ejpam-4983	37	10	enumeration	enumeration	NOUN
ejpam-4983	37	11	of	of	ADP
ejpam-4983	37	12	arrowhead	arrowhead	NOUN
ejpam-4983	37	13	matrices	matrix	NOUN
ejpam-4983	37	14	with	with	ADP
ejpam-4983	37	15	prescribed	prescribe	VERB
ejpam-4983	37	16	determinant	determinant	ADJ
ejpam-4983	37	17	over	over	ADP
ejpam-4983	37	18	a	a	DET
ejpam-4983	37	19	fccr	fccr	NOUN
ejpam-4983	37	20	is	be	AUX
ejpam-4983	37	21	investigated	investigate	VERB
ejpam-4983	37	22	in	in	ADP
ejpam-4983	37	23	the	the	DET
ejpam-4983	37	24	following	following	NOUN
ejpam-4983	37	25	set	set	VERB
ejpam-4983	37	26	up	up	ADP
ejpam-4983	37	27	.	.	PUNCT
ejpam-4983	38	1	for	for	ADP
ejpam-4983	38	2	a	a	DET
ejpam-4983	38	3	fccr	fccr	NOUN
ejpam-4983	38	4	r	r	NOUN
ejpam-4983	38	5	,	,	PUNCT
ejpam-4983	38	6	let	let	VERB
ejpam-4983	38	7	u(r	u(r	ADV
ejpam-4983	38	8	)	)	PUNCT
ejpam-4983	38	9	denote	denote	VERB
ejpam-4983	38	10	the	the	DET
ejpam-4983	38	11	set	set	NOUN
ejpam-4983	38	12	of	of	ADP
ejpam-4983	38	13	units	unit	NOUN
ejpam-4983	38	14	in	in	ADP
ejpam-4983	38	15	r	r	NOUN
ejpam-4983	38	16	and	and	CCONJ
ejpam-4983	38	17	let	let	VERB
ejpam-4983	38	18	z(r	z(r	NOUN
ejpam-4983	38	19	)	)	PUNCT
ejpam-4983	38	20	denote	denote	VERB
ejpam-4983	38	21	the	the	DET
ejpam-4983	38	22	set	set	NOUN
ejpam-4983	38	23	of	of	ADP
ejpam-4983	38	24	zero	zero	NUM
ejpam-4983	38	25	-	-	PUNCT
ejpam-4983	38	26	divisors	divisor	NOUN
ejpam-4983	38	27	in	in	ADP
ejpam-4983	38	28	r.	r.	PROPN
ejpam-4983	38	29	let	let	VERB
ejpam-4983	38	30	an(r	an(r	NOUN
ejpam-4983	38	31	)	)	PUNCT
ejpam-4983	38	32	denote	denote	VERB
ejpam-4983	38	33	the	the	DET
ejpam-4983	38	34	set	set	NOUN
ejpam-4983	38	35	of	of	ADP
ejpam-4983	38	36	n×	n×	PRON
ejpam-4983	38	37	n	n	X
ejpam-4983	38	38	arrowhead	arrowhead	NOUN
ejpam-4983	38	39	matrices	matrix	NOUN
ejpam-4983	38	40	over	over	ADP
ejpam-4983	38	41	r.	r.	PROPN
ejpam-4983	38	42	it	it	PRON
ejpam-4983	38	43	is	be	AUX
ejpam-4983	38	44	not	not	PART
ejpam-4983	38	45	difficult	difficult	ADJ
ejpam-4983	38	46	to	to	PART
ejpam-4983	38	47	see	see	VERB
ejpam-4983	38	48	that	that	DET
ejpam-4983	38	49	an(r	an(r	NOUN
ejpam-4983	38	50	)	)	PUNCT
ejpam-4983	38	51	is	be	AUX
ejpam-4983	38	52	a	a	DET
ejpam-4983	38	53	group	group	NOUN
ejpam-4983	38	54	under	under	ADP
ejpam-4983	38	55	addition	addition	NOUN
ejpam-4983	38	56	and	and	CCONJ
ejpam-4983	38	57	|an(r)|	|an(r)|	NOUN
ejpam-4983	38	58	=	=	SYM
ejpam-4983	38	59	|r|3n−2	|r|3n−2	PROPN
ejpam-4983	38	60	.	.	PUNCT
ejpam-4983	39	1	(	(	PUNCT
ejpam-4983	39	2	1	1	X
ejpam-4983	39	3	)	)	PUNCT
ejpam-4983	39	4	an	an	DET
ejpam-4983	39	5	n	n	NUM
ejpam-4983	39	6	×	×	NOUN
ejpam-4983	39	7	n	n	NOUN
ejpam-4983	39	8	matrix	matrix	NOUN
ejpam-4983	39	9	a	a	PRON
ejpam-4983	39	10	over	over	ADP
ejpam-4983	39	11	r	r	NOUN
ejpam-4983	39	12	is	be	AUX
ejpam-4983	39	13	said	say	VERB
ejpam-4983	39	14	to	to	PART
ejpam-4983	39	15	be	be	AUX
ejpam-4983	39	16	non	non	ADJ
ejpam-4983	39	17	-	-	ADJ
ejpam-4983	39	18	singular	singular	ADJ
ejpam-4983	39	19	(	(	PUNCT
ejpam-4983	39	20	or	or	CCONJ
ejpam-4983	39	21	,	,	PUNCT
ejpam-4983	39	22	invertible	invertible	ADJ
ejpam-4983	39	23	)	)	PUNCT
ejpam-4983	39	24	if	if	SCONJ
ejpam-4983	39	25	det(a	det(a	PROPN
ejpam-4983	39	26	)	)	PUNCT
ejpam-4983	39	27	∈	∈	PROPN
ejpam-4983	39	28	u(r	u(r	NOUN
ejpam-4983	39	29	)	)	PUNCT
ejpam-4983	39	30	.	.	PUNCT
ejpam-4983	40	1	otherwise	otherwise	ADV
ejpam-4983	40	2	,	,	PUNCT
ejpam-4983	40	3	a	a	PRON
ejpam-4983	40	4	is	be	AUX
ejpam-4983	40	5	called	call	VERB
ejpam-4983	40	6	a	a	DET
ejpam-4983	40	7	singular	singular	ADJ
ejpam-4983	40	8	matrix	matrix	NOUN
ejpam-4983	40	9	.	.	PUNCT
ejpam-4983	41	1	let	let	VERB
ejpam-4983	41	2	ian(r	ian(r	PRON
ejpam-4983	41	3	)	)	PUNCT
ejpam-4983	41	4	=	=	PRON
ejpam-4983	41	5	{	{	PUNCT
ejpam-4983	41	6	a	a	DET
ejpam-4983	41	7	∈	∈	PROPN
ejpam-4983	41	8	an(r	an(r	NOUN
ejpam-4983	41	9	)	)	PUNCT
ejpam-4983	41	10	|	|	ADV
ejpam-4983	41	11	det(a	det(a	NOUN
ejpam-4983	41	12	)	)	PUNCT
ejpam-4983	41	13	∈	∈	PROPN
ejpam-4983	41	14	u(r	u(r	NOUN
ejpam-4983	41	15	)	)	PUNCT
ejpam-4983	41	16	}	}	PUNCT
ejpam-4983	41	17	be	be	AUX
ejpam-4983	41	18	the	the	DET
ejpam-4983	41	19	set	set	NOUN
ejpam-4983	41	20	of	of	ADP
ejpam-4983	41	21	n×	n×	PROPN
ejpam-4983	41	22	n	n	CCONJ
ejpam-4983	41	23	non	non	ADJ
ejpam-4983	41	24	-	-	ADJ
ejpam-4983	41	25	singular	singular	ADJ
ejpam-4983	41	26	arrowhead	arrowhead	NOUN
ejpam-4983	41	27	matrices	matrix	NOUN
ejpam-4983	41	28	over	over	ADP
ejpam-4983	41	29	r.	r.	PROPN
ejpam-4983	41	30	for	for	ADP
ejpam-4983	41	31	each	each	DET
ejpam-4983	41	32	a	a	DET
ejpam-4983	41	33	∈	∈	PROPN
ejpam-4983	41	34	r	r	NOUN
ejpam-4983	41	35	,	,	PUNCT
ejpam-4983	41	36	let	let	VERB
ejpam-4983	41	37	an(r	an(r	NOUN
ejpam-4983	41	38	,	,	PUNCT
ejpam-4983	41	39	a	a	PRON
ejpam-4983	41	40	)	)	PUNCT
ejpam-4983	42	1	=	=	SYM
ejpam-4983	42	2	{	{	PUNCT
ejpam-4983	42	3	a	a	DET
ejpam-4983	42	4	∈	∈	PROPN
ejpam-4983	42	5	an(r	an(r	NOUN
ejpam-4983	42	6	)	)	PUNCT
ejpam-4983	42	7	|	|	ADV
ejpam-4983	42	8	det(a	det(a	NOUN
ejpam-4983	42	9	)	)	PUNCT
ejpam-4983	42	10	=	=	SYM
ejpam-4983	42	11	a	a	PRON
ejpam-4983	42	12	}	}	PUNCT
ejpam-4983	42	13	.	.	PUNCT
ejpam-4983	43	1	be	be	AUX
ejpam-4983	43	2	the	the	DET
ejpam-4983	43	3	set	set	NOUN
ejpam-4983	43	4	of	of	ADP
ejpam-4983	43	5	all	all	DET
ejpam-4983	43	6	n×	n×	PRON
ejpam-4983	43	7	n	n	CCONJ
ejpam-4983	43	8	arrowhead	arrowhead	NOUN
ejpam-4983	43	9	matrices	matrix	NOUN
ejpam-4983	43	10	over	over	ADP
ejpam-4983	43	11	r	r	NOUN
ejpam-4983	43	12	whose	whose	DET
ejpam-4983	43	13	determinant	determinant	ADJ
ejpam-4983	43	14	is	be	AUX
ejpam-4983	43	15	a.	a.	NOUN
ejpam-4983	43	16	clearly	clearly	ADV
ejpam-4983	43	17	,	,	PUNCT
ejpam-4983	43	18	ian(r	ian(r	NOUN
ejpam-4983	43	19	)	)	PUNCT
ejpam-4983	43	20	=	=	SYM
ejpam-4983	43	21	⋃	⋃	NOUN
ejpam-4983	43	22	a∈u(r	a∈u(r	NOUN
ejpam-4983	43	23	)	)	PUNCT
ejpam-4983	43	24	an(r	an(r	NOUN
ejpam-4983	43	25	,	,	PUNCT
ejpam-4983	43	26	a	a	PRON
ejpam-4983	43	27	)	)	PUNCT
ejpam-4983	43	28	is	be	AUX
ejpam-4983	43	29	a	a	DET
ejpam-4983	43	30	disjoint	disjoint	NOUN
ejpam-4983	43	31	union	union	NOUN
ejpam-4983	43	32	.	.	PUNCT
ejpam-4983	44	1	s.	s.	PROPN
ejpam-4983	44	2	jitman	jitman	PROPN
ejpam-4983	44	3	,	,	PUNCT
ejpam-4983	44	4	p.	p.	PROPN
ejpam-4983	44	5	modjam	modjam	PROPN
ejpam-4983	44	6	/	/	SYM
ejpam-4983	44	7	eur	eur	PROPN
ejpam-4983	44	8	.	.	PUNCT
ejpam-4983	45	1	j.	j.	PROPN
ejpam-4983	45	2	pure	pure	PROPN
ejpam-4983	45	3	appl	appl	PROPN
ejpam-4983	45	4	.	.	PROPN
ejpam-4983	45	5	math	math	PROPN
ejpam-4983	45	6	,	,	PUNCT
ejpam-4983	45	7	17	17	NUM
ejpam-4983	45	8	(	(	PUNCT
ejpam-4983	45	9	1	1	NUM
ejpam-4983	45	10	)	)	PUNCT
ejpam-4983	45	11	(	(	PUNCT
ejpam-4983	45	12	2024	2024	NUM
ejpam-4983	45	13	)	)	PUNCT
ejpam-4983	45	14	,	,	PUNCT
ejpam-4983	45	15	11	11	NUM
ejpam-4983	45	16	-	-	SYM
ejpam-4983	45	17	29	29	NUM
ejpam-4983	45	18	13	13	NUM
ejpam-4983	45	19	the	the	DET
ejpam-4983	45	20	main	main	ADJ
ejpam-4983	45	21	focus	focus	NOUN
ejpam-4983	45	22	of	of	ADP
ejpam-4983	45	23	this	this	DET
ejpam-4983	45	24	paper	paper	NOUN
ejpam-4983	45	25	is	be	AUX
ejpam-4983	45	26	the	the	DET
ejpam-4983	45	27	enumeration	enumeration	NOUN
ejpam-4983	45	28	of	of	ADP
ejpam-4983	45	29	n	n	NUM
ejpam-4983	45	30	×	×	NOUN
ejpam-4983	45	31	n	n	CCONJ
ejpam-4983	45	32	arrowhead	arrowhead	NOUN
ejpam-4983	45	33	matrices	matrix	NOUN
ejpam-4983	45	34	with	with	ADP
ejpam-4983	45	35	prescribed	prescribe	VERB
ejpam-4983	45	36	determinant	determinant	ADJ
ejpam-4983	45	37	over	over	ADP
ejpam-4983	45	38	a	a	DET
ejpam-4983	45	39	finite	finite	ADJ
ejpam-4983	45	40	field	field	NOUN
ejpam-4983	45	41	fq	fq	PROPN
ejpam-4983	45	42	and	and	CCONJ
ejpam-4983	45	43	over	over	ADP
ejpam-4983	45	44	a	a	DET
ejpam-4983	45	45	fccr	fccr	PROPN
ejpam-4983	45	46	r.	r.	PROPN
ejpam-4983	45	47	the	the	DET
ejpam-4983	45	48	paper	paper	NOUN
ejpam-4983	45	49	is	be	AUX
ejpam-4983	45	50	organized	organize	VERB
ejpam-4983	45	51	as	as	SCONJ
ejpam-4983	45	52	follows	follow	VERB
ejpam-4983	45	53	.	.	PUNCT
ejpam-4983	46	1	the	the	DET
ejpam-4983	46	2	number	number	NOUN
ejpam-4983	46	3	|an(fq	|an(fq	NOUN
ejpam-4983	46	4	,	,	PUNCT
ejpam-4983	46	5	a)|	a)|	PROPN
ejpam-4983	46	6	of	of	ADP
ejpam-4983	46	7	n	n	NUM
ejpam-4983	46	8	×	×	NOUN
ejpam-4983	46	9	n	n	CCONJ
ejpam-4983	46	10	arrowhead	arrowhead	NOUN
ejpam-4983	46	11	matrices	matrix	NOUN
ejpam-4983	46	12	over	over	ADP
ejpam-4983	46	13	fq	fq	PROPN
ejpam-4983	46	14	of	of	ADP
ejpam-4983	46	15	determinant	determinant	ADJ
ejpam-4983	46	16	a	a	PRON
ejpam-4983	46	17	is	be	AUX
ejpam-4983	46	18	determined	determine	VERB
ejpam-4983	46	19	for	for	ADP
ejpam-4983	46	20	all	all	DET
ejpam-4983	46	21	positive	positive	ADJ
ejpam-4983	46	22	integers	integer	NOUN
ejpam-4983	46	23	n	n	PRON
ejpam-4983	46	24	and	and	CCONJ
ejpam-4983	46	25	for	for	ADP
ejpam-4983	46	26	all	all	DET
ejpam-4983	46	27	elements	element	NOUN
ejpam-4983	46	28	a	a	DET
ejpam-4983	46	29	∈	∈	PROPN
ejpam-4983	46	30	fq	fq	NOUN
ejpam-4983	46	31	in	in	ADP
ejpam-4983	46	32	section	section	NOUN
ejpam-4983	46	33	2	2	NUM
ejpam-4983	46	34	.	.	PUNCT
ejpam-4983	47	1	as	as	ADP
ejpam-4983	47	2	applications	application	NOUN
ejpam-4983	47	3	,	,	PUNCT
ejpam-4983	47	4	these	these	DET
ejpam-4983	47	5	results	result	NOUN
ejpam-4983	47	6	are	be	AUX
ejpam-4983	47	7	used	use	VERB
ejpam-4983	47	8	in	in	ADP
ejpam-4983	47	9	the	the	DET
ejpam-4983	47	10	enumeration	enumeration	NOUN
ejpam-4983	47	11	of	of	ADP
ejpam-4983	47	12	arrowhead	arrowhead	NOUN
ejpam-4983	47	13	matrices	matrix	NOUN
ejpam-4983	47	14	of	of	ADP
ejpam-4983	47	15	a	a	DET
ejpam-4983	47	16	fixed	fix	VERB
ejpam-4983	47	17	determinant	determinant	ADJ
ejpam-4983	47	18	over	over	ADP
ejpam-4983	47	19	r	r	NOUN
ejpam-4983	47	20	in	in	ADP
ejpam-4983	47	21	section	section	NOUN
ejpam-4983	47	22	3	3	NUM
ejpam-4983	47	23	.	.	PUNCT
ejpam-4983	48	1	the	the	DET
ejpam-4983	48	2	number	number	NOUN
ejpam-4983	48	3	of	of	ADP
ejpam-4983	48	4	n	n	NUM
ejpam-4983	48	5	×	×	NOUN
ejpam-4983	48	6	n	n	CCONJ
ejpam-4983	48	7	non	non	ADJ
ejpam-4983	48	8	-	-	ADJ
ejpam-4983	48	9	singular	singular	ADJ
ejpam-4983	48	10	arrowhead	arrowhead	NOUN
ejpam-4983	48	11	matrices	matrix	NOUN
ejpam-4983	48	12	of	of	ADP
ejpam-4983	48	13	a	a	DET
ejpam-4983	48	14	fixed	fix	VERB
ejpam-4983	48	15	determinant	determinant	ADJ
ejpam-4983	48	16	over	over	ADP
ejpam-4983	48	17	r	r	NOUN
ejpam-4983	48	18	in	in	ADP
ejpam-4983	48	19	subsection	subsection	NOUN
ejpam-4983	48	20	3.1	3.1	NUM
ejpam-4983	48	21	.	.	PUNCT
ejpam-4983	49	1	subsequently	subsequently	ADV
ejpam-4983	49	2	,	,	PUNCT
ejpam-4983	49	3	bounds	bound	VERB
ejpam-4983	49	4	on	on	ADP
ejpam-4983	49	5	the	the	DET
ejpam-4983	49	6	number	number	NOUN
ejpam-4983	49	7	of	of	ADP
ejpam-4983	49	8	n	n	NUM
ejpam-4983	49	9	×	×	NOUN
ejpam-4983	49	10	n	n	CCONJ
ejpam-4983	49	11	singular	singular	ADJ
ejpam-4983	49	12	arrowhead	arrowhead	NOUN
ejpam-4983	49	13	matrices	matrix	NOUN
ejpam-4983	49	14	over	over	ADP
ejpam-4983	49	15	r	r	NOUN
ejpam-4983	49	16	of	of	ADP
ejpam-4983	49	17	some	some	DET
ejpam-4983	49	18	fixed	fix	VERB
ejpam-4983	49	19	determinant	determinant	ADJ
ejpam-4983	49	20	are	be	AUX
ejpam-4983	49	21	discussed	discuss	VERB
ejpam-4983	49	22	in	in	ADP
ejpam-4983	49	23	subsection	subsection	NOUN
ejpam-4983	49	24	3.2	3.2	NUM
ejpam-4983	49	25	.	.	PUNCT
ejpam-4983	50	1	some	some	DET
ejpam-4983	50	2	remarks	remark	NOUN
ejpam-4983	50	3	and	and	CCONJ
ejpam-4983	50	4	open	open	ADJ
ejpam-4983	50	5	problems	problem	NOUN
ejpam-4983	50	6	are	be	AUX
ejpam-4983	50	7	given	give	VERB
ejpam-4983	50	8	in	in	ADP
ejpam-4983	50	9	section	section	NOUN
ejpam-4983	50	10	4	4	NUM
ejpam-4983	50	11	.	.	NOUN
ejpam-4983	50	12	2	2	NUM
ejpam-4983	50	13	.	.	NUM
ejpam-4983	50	14	determinants	determinant	NOUN
ejpam-4983	50	15	of	of	ADP
ejpam-4983	50	16	arrowhead	arrowhead	NOUN
ejpam-4983	50	17	matrices	matrix	NOUN
ejpam-4983	50	18	over	over	ADP
ejpam-4983	50	19	fq	fq	PROPN
ejpam-4983	50	20	in	in	ADP
ejpam-4983	50	21	this	this	DET
ejpam-4983	50	22	section	section	NOUN
ejpam-4983	50	23	,	,	PUNCT
ejpam-4983	50	24	we	we	PRON
ejpam-4983	50	25	focus	focus	VERB
ejpam-4983	50	26	on	on	ADP
ejpam-4983	50	27	the	the	DET
ejpam-4983	50	28	enumeration	enumeration	NOUN
ejpam-4983	50	29	of	of	ADP
ejpam-4983	50	30	arrowhead	arrowhead	NOUN
ejpam-4983	50	31	matrices	matrix	NOUN
ejpam-4983	50	32	of	of	ADP
ejpam-4983	50	33	a	a	DET
ejpam-4983	50	34	fixed	fix	VERB
ejpam-4983	50	35	determinant	determinant	ADJ
ejpam-4983	50	36	over	over	ADP
ejpam-4983	50	37	a	a	DET
ejpam-4983	50	38	finite	finite	ADJ
ejpam-4983	50	39	field	field	NOUN
ejpam-4983	50	40	fq	fq	PROPN
ejpam-4983	50	41	.	.	PROPN
ejpam-4983	50	42	for	for	ADP
ejpam-4983	50	43	an	an	DET
ejpam-4983	50	44	element	element	NOUN
ejpam-4983	50	45	a	a	DET
ejpam-4983	50	46	∈	∈	PROPN
ejpam-4983	50	47	fq	fq	NOUN
ejpam-4983	50	48	,	,	PUNCT
ejpam-4983	50	49	the	the	DET
ejpam-4983	50	50	formula	formula	NOUN
ejpam-4983	50	51	for	for	ADP
ejpam-4983	50	52	the	the	DET
ejpam-4983	50	53	number	number	NOUN
ejpam-4983	50	54	of	of	ADP
ejpam-4983	50	55	n	n	NUM
ejpam-4983	50	56	×	×	NOUN
ejpam-4983	50	57	n	n	CCONJ
ejpam-4983	50	58	arrowhead	arrowhead	NOUN
ejpam-4983	50	59	matrices	matrix	NOUN
ejpam-4983	50	60	over	over	ADP
ejpam-4983	50	61	fq	fq	PROPN
ejpam-4983	50	62	of	of	ADP
ejpam-4983	50	63	determinant	determinant	ADJ
ejpam-4983	50	64	a	a	PRON
ejpam-4983	50	65	is	be	AUX
ejpam-4983	50	66	given	give	VERB
ejpam-4983	50	67	for	for	ADP
ejpam-4983	50	68	all	all	DET
ejpam-4983	50	69	prime	prime	ADJ
ejpam-4983	50	70	powers	power	NOUN
ejpam-4983	50	71	q	q	NOUN
ejpam-4983	50	72	and	and	CCONJ
ejpam-4983	50	73	positive	positive	ADJ
ejpam-4983	50	74	integers	integer	NOUN
ejpam-4983	50	75	n.	n.	VERB
ejpam-4983	50	76	a	a	DET
ejpam-4983	50	77	recursive	recursive	ADJ
ejpam-4983	50	78	formula	formula	NOUN
ejpam-4983	50	79	for	for	ADP
ejpam-4983	50	80	the	the	DET
ejpam-4983	50	81	number	number	NOUN
ejpam-4983	50	82	|ian(fq)|	|ian(fq)|	NUM
ejpam-4983	50	83	of	of	ADP
ejpam-4983	50	84	n×n	n×n	PROPN
ejpam-4983	50	85	non	non	ADJ
ejpam-4983	50	86	-	-	ADJ
ejpam-4983	50	87	singular	singular	ADJ
ejpam-4983	50	88	arrowhead	arrowhead	NOUN
ejpam-4983	50	89	matrices	matrix	NOUN
ejpam-4983	50	90	over	over	ADP
ejpam-4983	50	91	fq	fq	PROPN
ejpam-4983	50	92	is	be	AUX
ejpam-4983	50	93	given	give	VERB
ejpam-4983	50	94	in	in	ADP
ejpam-4983	50	95	proposition	proposition	NOUN
ejpam-4983	50	96	1	1	NUM
ejpam-4983	50	97	.	.	PUNCT
ejpam-4983	51	1	later	later	ADV
ejpam-4983	51	2	,	,	PUNCT
ejpam-4983	51	3	an	an	DET
ejpam-4983	51	4	explicit	explicit	ADJ
ejpam-4983	51	5	formula	formula	NOUN
ejpam-4983	51	6	for	for	ADP
ejpam-4983	51	7	|ian(fq)|	|ian(fq)|	NUM
ejpam-4983	51	8	is	be	AUX
ejpam-4983	51	9	established	establish	VERB
ejpam-4983	51	10	in	in	ADP
ejpam-4983	51	11	theorem	theorem	NOUN
ejpam-4983	51	12	1	1	NUM
ejpam-4983	51	13	based	base	VERB
ejpam-4983	51	14	on	on	ADP
ejpam-4983	51	15	proposition	proposition	NOUN
ejpam-4983	51	16	1	1	NUM
ejpam-4983	51	17	.	.	PUNCT
ejpam-4983	52	1	proposition	proposition	NOUN
ejpam-4983	52	2	1	1	NUM
ejpam-4983	52	3	.	.	PUNCT
ejpam-4983	53	1	let	let	VERB
ejpam-4983	53	2	q	q	PART
ejpam-4983	53	3	be	be	AUX
ejpam-4983	53	4	a	a	DET
ejpam-4983	53	5	prime	prime	ADJ
ejpam-4983	53	6	power	power	NOUN
ejpam-4983	53	7	.	.	PUNCT
ejpam-4983	54	1	then	then	ADV
ejpam-4983	54	2	|ia1(fq)|	|ia1(fq)|	PRON
ejpam-4983	54	3	=	=	SYM
ejpam-4983	54	4	q	q	NOUN
ejpam-4983	55	1	−	−	PROPN
ejpam-4983	55	2	1	1	NUM
ejpam-4983	55	3	and	and	CCONJ
ejpam-4983	55	4	|ian(fq)|	|ian(fq)|	NUM
ejpam-4983	55	5	=	=	SYM
ejpam-4983	55	6	q2n−3(q	q2n−3(q	NUM
ejpam-4983	55	7	−	−	PROPN
ejpam-4983	55	8	1)n	1)n	PUNCT
ejpam-4983	56	1	+	+	CCONJ
ejpam-4983	56	2	q2(q	q2(q	NOUN
ejpam-4983	56	3	−	−	NUM
ejpam-4983	56	4	1)|ian−1(fq)|	1)|ian−1(fq)|	NUM
ejpam-4983	56	5	for	for	ADP
ejpam-4983	56	6	all	all	DET
ejpam-4983	56	7	integers	integer	NOUN
ejpam-4983	56	8	n	n	PRON
ejpam-4983	56	9	≥	≥	NOUN
ejpam-4983	56	10	2	2	NUM
ejpam-4983	56	11	.	.	PUNCT
ejpam-4983	57	1	proof	proof	NOUN
ejpam-4983	57	2	.	.	PUNCT
ejpam-4983	58	1	clearly	clearly	ADV
ejpam-4983	58	2	,	,	PUNCT
ejpam-4983	58	3	|ia1(fq)|	|ia1(fq)|	X
ejpam-4983	58	4	=	=	SYM
ejpam-4983	58	5	|fq	|fq	PRON
ejpam-4983	58	6	\	\	NOUN
ejpam-4983	58	7	{	{	PUNCT
ejpam-4983	58	8	0}|	0}|	X
ejpam-4983	58	9	=	=	SYM
ejpam-4983	58	10	q	q	NOUN
ejpam-4983	59	1	−	−	NOUN
ejpam-4983	59	2	1	1	X
ejpam-4983	59	3	.	.	PUNCT
ejpam-4983	60	1	let	let	VERB
ejpam-4983	60	2	n	n	PRON
ejpam-4983	60	3	≥	≥	X
ejpam-4983	60	4	2	2	NUM
ejpam-4983	60	5	be	be	AUX
ejpam-4983	60	6	an	an	DET
ejpam-4983	60	7	integer	integer	NOUN
ejpam-4983	60	8	and	and	CCONJ
ejpam-4983	60	9	let	let	VERB
ejpam-4983	60	10	a	a	DET
ejpam-4983	60	11	=	=	X
ejpam-4983	60	12			ADJ
ejpam-4983	60	13	a11	a11	PROPN
ejpam-4983	60	14	a12	a12	PROPN
ejpam-4983	60	15	a13	a13	PROPN
ejpam-4983	60	16	·	·	PUNCT
ejpam-4983	60	17	·	·	PUNCT
ejpam-4983	60	18	·	·	PUNCT
ejpam-4983	61	1	a1,n−1	a1,n−1	ADJ
ejpam-4983	61	2	a1n	a1n	ADP
ejpam-4983	61	3	a21	a21	PROPN
ejpam-4983	61	4	a22	a22	PROPN
ejpam-4983	61	5	0	0	NUM
ejpam-4983	61	6	·	·	PUNCT
ejpam-4983	61	7	·	·	PUNCT
ejpam-4983	61	8	·	·	PUNCT
ejpam-4983	61	9	0	0	NUM
ejpam-4983	61	10	0	0	NUM
ejpam-4983	61	11	a31	a31	NOUN
ejpam-4983	61	12	0	0	NUM
ejpam-4983	61	13	a33	a33	PROPN
ejpam-4983	61	14	·	·	PUNCT
ejpam-4983	61	15	·	·	PUNCT
ejpam-4983	61	16	·	·	PUNCT
ejpam-4983	61	17	0	0	NUM
ejpam-4983	61	18	0	0	NUM
ejpam-4983	61	19	...	...	PUNCT
ejpam-4983	61	20	...	...	PUNCT
ejpam-4983	61	21	...	...	PUNCT
ejpam-4983	61	22	.	.	PUNCT
ejpam-4983	61	23	.	.	PUNCT
ejpam-4983	61	24	.	.	PUNCT
ejpam-4983	61	25	...	...	PUNCT
ejpam-4983	61	26	...	...	PUNCT
ejpam-4983	62	1	an−1,1	an−1,1	X
ejpam-4983	62	2	0	0	NUM
ejpam-4983	62	3	0	0	NUM
ejpam-4983	62	4	·	·	PUNCT
ejpam-4983	62	5	·	·	PUNCT
ejpam-4983	62	6	·	·	PUNCT
ejpam-4983	62	7	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	62	8	0	0	PUNCT
ejpam-4983	63	1	an1	an1	NOUN
ejpam-4983	63	2	0	0	NUM
ejpam-4983	63	3	0	0	NUM
ejpam-4983	63	4	·	·	PUNCT
ejpam-4983	63	5	·	·	PUNCT
ejpam-4983	63	6	·	·	PUNCT
ejpam-4983	63	7	0	0	NUM
ejpam-4983	64	1	ann	ann	PROPN
ejpam-4983	64	2			PROPN
ejpam-4983	64	3	∈	∈	PROPN
ejpam-4983	64	4	ian(fq	ian(fq	NOUN
ejpam-4983	64	5	)	)	PUNCT
ejpam-4983	64	6	.	.	PUNCT
ejpam-4983	65	1	for	for	ADP
ejpam-4983	65	2	each	each	DET
ejpam-4983	65	3	i	i	PRON
ejpam-4983	65	4	∈	∈	PROPN
ejpam-4983	65	5	{	{	PUNCT
ejpam-4983	65	6	1	1	NUM
ejpam-4983	65	7	,	,	PUNCT
ejpam-4983	65	8	2	2	NUM
ejpam-4983	65	9	,	,	PUNCT
ejpam-4983	65	10	.	.	PUNCT
ejpam-4983	65	11	.	.	PUNCT
ejpam-4983	65	12	.	.	PUNCT
ejpam-4983	65	13	,	,	PUNCT
ejpam-4983	65	14	n	n	CCONJ
ejpam-4983	65	15	}	}	PUNCT
ejpam-4983	65	16	,	,	PUNCT
ejpam-4983	65	17	let	let	VERB
ejpam-4983	65	18	ri	ri	PROPN
ejpam-4983	65	19	(	(	PUNCT
ejpam-4983	65	20	resp	resp	PROPN
ejpam-4983	65	21	.	.	PROPN
ejpam-4983	65	22	,	,	PUNCT
ejpam-4983	65	23	ci	ci	PROPN
ejpam-4983	65	24	)	)	PUNCT
ejpam-4983	65	25	denote	denote	VERB
ejpam-4983	65	26	the	the	DET
ejpam-4983	65	27	ith	ith	PROPN
ejpam-4983	65	28	row	row	NOUN
ejpam-4983	65	29	(	(	PUNCT
ejpam-4983	65	30	resp	resp	NOUN
ejpam-4983	65	31	,	,	PUNCT
ejpam-4983	65	32	ith	ith	PROPN
ejpam-4983	65	33	column	column	NOUN
ejpam-4983	65	34	)	)	PUNCT
ejpam-4983	65	35	of	of	ADP
ejpam-4983	65	36	a.	a.	NOUN
ejpam-4983	65	37	we	we	PRON
ejpam-4983	65	38	consider	consider	VERB
ejpam-4983	65	39	the	the	DET
ejpam-4983	65	40	two	two	NUM
ejpam-4983	65	41	cases	case	NOUN
ejpam-4983	65	42	.	.	PUNCT
ejpam-4983	66	1	case	case	NOUN
ejpam-4983	66	2	1	1	NUM
ejpam-4983	66	3	:	:	PUNCT
ejpam-4983	66	4	ann	ann	PROPN
ejpam-4983	66	5	̸=	̸=	PROPN
ejpam-4983	66	6	0	0	NUM
ejpam-4983	66	7	.	.	PUNCT
ejpam-4983	67	1	applying	apply	VERB
ejpam-4983	67	2	the	the	DET
ejpam-4983	67	3	elementary	elementary	PROPN
ejpam-4983	67	4	row	row	NOUN
ejpam-4983	67	5	operation	operation	NOUN
ejpam-4983	67	6	r1	r1	PROPN
ejpam-4983	67	7	−	−	PROPN
ejpam-4983	67	8	a1nann	a1nann	PROPN
ejpam-4983	67	9	−1rn	−1rn	PROPN
ejpam-4983	67	10	→	→	SYM
ejpam-4983	67	11	r1	r1	PROPN
ejpam-4983	67	12	and	and	CCONJ
ejpam-4983	67	13	the	the	DET
ejpam-4983	67	14	elementary	elementary	ADJ
ejpam-4983	67	15	column	column	PROPN
ejpam-4983	67	16	operation	operation	PROPN
ejpam-4983	67	17	c1	c1	PROPN
ejpam-4983	67	18	−	−	PROPN
ejpam-4983	68	1	an1ann	an1ann	PROPN
ejpam-4983	68	2	−1cn	−1cn	PROPN
ejpam-4983	68	3	→	→	SYM
ejpam-4983	68	4	c1	c1	PROPN
ejpam-4983	69	1	,	,	PUNCT
ejpam-4983	69	2	it	it	PRON
ejpam-4983	69	3	follows	follow	VERB
ejpam-4983	69	4	that	that	SCONJ
ejpam-4983	69	5	a	a	DET
ejpam-4983	69	6	∼	∼	NOUN
ejpam-4983	69	7			NOUN
ejpam-4983	69	8	0	0	PUNCT
ejpam-4983	69	9	c	c	NOUN
ejpam-4983	69	10	...	...	PUNCT
ejpam-4983	69	11	0	0	NUM
ejpam-4983	69	12	0	0	NUM
ejpam-4983	69	13	·	·	PUNCT
ejpam-4983	69	14	·	·	PUNCT
ejpam-4983	69	15	·	·	PUNCT
ejpam-4983	69	16	0	0	NUM
ejpam-4983	70	1	ann	ann	PROPN
ejpam-4983	70	2			PROPN
ejpam-4983	70	3	,	,	PUNCT
ejpam-4983	70	4	s.	s.	PROPN
ejpam-4983	70	5	jitman	jitman	PROPN
ejpam-4983	70	6	,	,	PUNCT
ejpam-4983	70	7	p.	p.	PROPN
ejpam-4983	70	8	modjam	modjam	PROPN
ejpam-4983	70	9	/	/	SYM
ejpam-4983	70	10	eur	eur	PROPN
ejpam-4983	70	11	.	.	PUNCT
ejpam-4983	71	1	j.	j.	PROPN
ejpam-4983	71	2	pure	pure	PROPN
ejpam-4983	71	3	appl	appl	PROPN
ejpam-4983	71	4	.	.	PROPN
ejpam-4983	71	5	math	math	PROPN
ejpam-4983	71	6	,	,	PUNCT
ejpam-4983	71	7	17	17	NUM
ejpam-4983	71	8	(	(	PUNCT
ejpam-4983	71	9	1	1	NUM
ejpam-4983	71	10	)	)	PUNCT
ejpam-4983	71	11	(	(	PUNCT
ejpam-4983	71	12	2024	2024	NUM
ejpam-4983	71	13	)	)	PUNCT
ejpam-4983	71	14	,	,	PUNCT
ejpam-4983	71	15	11	11	NUM
ejpam-4983	71	16	-	-	SYM
ejpam-4983	71	17	29	29	NUM
ejpam-4983	71	18	14	14	NUM
ejpam-4983	71	19	where	where	SCONJ
ejpam-4983	71	20	c	c	NOUN
ejpam-4983	71	21	=	=	PUNCT
ejpam-4983	71	22			PROPN
ejpam-4983	71	23	a11	a11	PROPN
ejpam-4983	71	24	−	−	PROPN
ejpam-4983	71	25	a1nan1ann	a1nan1ann	PROPN
ejpam-4983	71	26	−1	−1	NOUN
ejpam-4983	71	27	a12	a12	NOUN
ejpam-4983	71	28	a13	a13	NOUN
ejpam-4983	71	29	·	·	PUNCT
ejpam-4983	71	30	·	·	PUNCT
ejpam-4983	71	31	·	·	PUNCT
ejpam-4983	72	1	a1,n−1	a1,n−1	ADJ
ejpam-4983	72	2	a21	a21	PROPN
ejpam-4983	72	3	a22	a22	PROPN
ejpam-4983	72	4	0	0	NUM
ejpam-4983	72	5	·	·	PUNCT
ejpam-4983	72	6	·	·	PUNCT
ejpam-4983	72	7	·	·	PUNCT
ejpam-4983	72	8	0	0	NUM
ejpam-4983	73	1	a31	a31	NOUN
ejpam-4983	73	2	0	0	NUM
ejpam-4983	73	3	a33	a33	PROPN
ejpam-4983	73	4	·	·	PUNCT
ejpam-4983	73	5	·	·	PUNCT
ejpam-4983	73	6	·	·	PUNCT
ejpam-4983	73	7	0	0	NUM
ejpam-4983	73	8	...	...	PUNCT
ejpam-4983	73	9	...	...	PUNCT
ejpam-4983	73	10	...	...	PUNCT
ejpam-4983	73	11	.	.	PUNCT
ejpam-4983	73	12	.	.	PUNCT
ejpam-4983	73	13	.	.	PUNCT
ejpam-4983	74	1	...	...	PUNCT
ejpam-4983	75	1	an−1,1	an−1,1	X
ejpam-4983	75	2	0	0	NUM
ejpam-4983	75	3	0	0	NUM
ejpam-4983	75	4	·	·	PUNCT
ejpam-4983	75	5	·	·	PUNCT
ejpam-4983	75	6	·	·	PUNCT
ejpam-4983	75	7	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	75	8			PROPN
ejpam-4983	75	9	.	.	PUNCT
ejpam-4983	76	1	then	then	ADV
ejpam-4983	76	2	det(a	det(a	PROPN
ejpam-4983	76	3	)	)	PUNCT
ejpam-4983	76	4	=	=	NOUN
ejpam-4983	77	1	(	(	PUNCT
ejpam-4983	77	2	−1)n+nann	−1)n+nann	PROPN
ejpam-4983	77	3	det(c	det(c	PROPN
ejpam-4983	77	4	)	)	PUNCT
ejpam-4983	77	5	=	=	SYM
ejpam-4983	77	6	ann	ann	PROPN
ejpam-4983	77	7	det(c	det(c	PROPN
ejpam-4983	77	8	)	)	PUNCT
ejpam-4983	77	9	.	.	PUNCT
ejpam-4983	78	1	let	let	VERB
ejpam-4983	78	2	s	s	PRON
ejpam-4983	78	3	=	=	PUNCT
ejpam-4983	78	4			X
ejpam-4983	78	5			ADJ
ejpam-4983	78	6	s11	s11	PROPN
ejpam-4983	78	7	s12	s12	PROPN
ejpam-4983	78	8	s13	s13	PROPN
ejpam-4983	78	9	·	·	PUNCT
ejpam-4983	78	10	·	·	PUNCT
ejpam-4983	78	11	·	·	PUNCT
ejpam-4983	78	12	s1,n−1	s1,n−1	ADJ
ejpam-4983	78	13	s21	s21	NOUN
ejpam-4983	78	14	s22	s22	NOUN
ejpam-4983	78	15	0	0	NUM
ejpam-4983	78	16	·	·	PUNCT
ejpam-4983	78	17	·	·	PUNCT
ejpam-4983	78	18	·	·	PUNCT
ejpam-4983	78	19	0	0	NUM
ejpam-4983	78	20	s31	s31	NOUN
ejpam-4983	78	21	0	0	NUM
ejpam-4983	78	22	s33	s33	PROPN
ejpam-4983	78	23	·	·	PUNCT
ejpam-4983	78	24	·	·	PUNCT
ejpam-4983	78	25	·	·	PUNCT
ejpam-4983	78	26	0	0	NUM
ejpam-4983	78	27	...	...	PUNCT
ejpam-4983	78	28	...	...	PUNCT
ejpam-4983	78	29	...	...	PUNCT
ejpam-4983	78	30	.	.	PUNCT
ejpam-4983	78	31	.	.	PUNCT
ejpam-4983	78	32	.	.	PUNCT
ejpam-4983	79	1	...	...	PUNCT
ejpam-4983	80	1	sn−1,1	sn−1,1	NOUN
ejpam-4983	80	2	0	0	NUM
ejpam-4983	80	3	0	0	NUM
ejpam-4983	80	4	·	·	PUNCT
ejpam-4983	80	5	·	·	PUNCT
ejpam-4983	80	6	·	·	PUNCT
ejpam-4983	80	7	sn−1,n−1	sn−1,n−1	ADJ
ejpam-4983	80	8			PROPN
ejpam-4983	80	9	∈	∈	PROPN
ejpam-4983	80	10	an−1(fq	an−1(fq	PROPN
ejpam-4983	80	11	)	)	PUNCT
ejpam-4983	80	12	∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣	PROPN
ejpam-4983	80	13	det	det	PROPN
ejpam-4983	80	14			ADJ
ejpam-4983	80	15			PROPN
ejpam-4983	80	16	s11	s11	PROPN
ejpam-4983	80	17	−	−	PROPN
ejpam-4983	80	18	a1nan1ann	a1nan1ann	PROPN
ejpam-4983	80	19	−1	−1	NOUN
ejpam-4983	80	20	s12	s12	PROPN
ejpam-4983	80	21	s13	s13	PROPN
ejpam-4983	80	22	·	·	PUNCT
ejpam-4983	80	23	·	·	PUNCT
ejpam-4983	80	24	·	·	PUNCT
ejpam-4983	80	25	s1,n−1	s1,n−1	ADJ
ejpam-4983	80	26	s21	s21	NOUN
ejpam-4983	80	27	s22	s22	NOUN
ejpam-4983	80	28	0	0	NUM
ejpam-4983	80	29	·	·	PUNCT
ejpam-4983	80	30	·	·	PUNCT
ejpam-4983	80	31	·	·	PUNCT
ejpam-4983	80	32	0	0	NUM
ejpam-4983	81	1	s31	s31	NOUN
ejpam-4983	81	2	0	0	NUM
ejpam-4983	81	3	s33	s33	PROPN
ejpam-4983	81	4	·	·	PUNCT
ejpam-4983	81	5	·	·	PUNCT
ejpam-4983	81	6	·	·	PUNCT
ejpam-4983	81	7	0	0	NUM
ejpam-4983	81	8	...	...	PUNCT
ejpam-4983	81	9	...	...	PUNCT
ejpam-4983	81	10	...	...	PUNCT
ejpam-4983	81	11	.	.	PUNCT
ejpam-4983	81	12	.	.	PUNCT
ejpam-4983	81	13	.	.	PUNCT
ejpam-4983	82	1	...	...	PUNCT
ejpam-4983	83	1	sn−1,1	sn−1,1	NOUN
ejpam-4983	83	2	0	0	NUM
ejpam-4983	83	3	0	0	NUM
ejpam-4983	83	4	·	·	PUNCT
ejpam-4983	83	5	·	·	PUNCT
ejpam-4983	83	6	·	·	PUNCT
ejpam-4983	83	7	sn−1,n−1	sn−1,n−1	ADJ
ejpam-4983	83	8			PRON
ejpam-4983	83	9			ADJ
ejpam-4983	83	10	̸=	̸=	NOUN
ejpam-4983	83	11	0	0	NUM
ejpam-4983	83	12			NOUN
ejpam-4983	83	13	.	.	PUNCT
ejpam-4983	84	1	it	it	PRON
ejpam-4983	84	2	follows	follow	VERB
ejpam-4983	84	3	that	that	SCONJ
ejpam-4983	84	4			PROPN
ejpam-4983	84	5	s11	s11	PROPN
ejpam-4983	84	6	s12	s12	PROPN
ejpam-4983	84	7	s13	s13	PROPN
ejpam-4983	84	8	·	·	PUNCT
ejpam-4983	84	9	·	·	PUNCT
ejpam-4983	84	10	·	·	PUNCT
ejpam-4983	84	11	s1,n−1	s1,n−1	ADJ
ejpam-4983	84	12	s21	s21	NOUN
ejpam-4983	84	13	s22	s22	NOUN
ejpam-4983	84	14	0	0	NUM
ejpam-4983	84	15	·	·	PUNCT
ejpam-4983	84	16	·	·	PUNCT
ejpam-4983	84	17	·	·	PUNCT
ejpam-4983	84	18	0	0	NUM
ejpam-4983	84	19	s31	s31	NOUN
ejpam-4983	84	20	0	0	NUM
ejpam-4983	84	21	s33	s33	PROPN
ejpam-4983	84	22	·	·	PUNCT
ejpam-4983	84	23	·	·	PUNCT
ejpam-4983	84	24	·	·	PUNCT
ejpam-4983	84	25	0	0	NUM
ejpam-4983	84	26	...	...	PUNCT
ejpam-4983	84	27	...	...	PUNCT
ejpam-4983	84	28	...	...	PUNCT
ejpam-4983	84	29	.	.	PUNCT
ejpam-4983	84	30	.	.	PUNCT
ejpam-4983	84	31	.	.	PUNCT
ejpam-4983	85	1	...	...	PUNCT
ejpam-4983	86	1	sn−1,1	sn−1,1	NOUN
ejpam-4983	86	2	0	0	NUM
ejpam-4983	86	3	0	0	NUM
ejpam-4983	86	4	·	·	PUNCT
ejpam-4983	86	5	·	·	PUNCT
ejpam-4983	86	6	·	·	PUNCT
ejpam-4983	86	7	sn−1,n−1	sn−1,n−1	ADJ
ejpam-4983	86	8			PROPN
ejpam-4983	86	9	∈	∈	PROPN
ejpam-4983	86	10	s	s	VERB
ejpam-4983	86	11	if	if	SCONJ
ejpam-4983	86	12	and	and	CCONJ
ejpam-4983	86	13	only	only	ADV
ejpam-4983	86	14	if	if	SCONJ
ejpam-4983	86	15			ADJ
ejpam-4983	86	16	s11	s11	NOUN
ejpam-4983	86	17	−	−	PROPN
ejpam-4983	86	18	a1nan1ann	a1nan1ann	PROPN
ejpam-4983	86	19	−1	−1	NOUN
ejpam-4983	86	20	s12	s12	PROPN
ejpam-4983	86	21	s13	s13	PROPN
ejpam-4983	86	22	·	·	PUNCT
ejpam-4983	86	23	·	·	PUNCT
ejpam-4983	86	24	·	·	PUNCT
ejpam-4983	86	25	s1,n−1	s1,n−1	ADJ
ejpam-4983	86	26	s21	s21	NOUN
ejpam-4983	86	27	s22	s22	NOUN
ejpam-4983	86	28	0	0	NUM
ejpam-4983	86	29	·	·	PUNCT
ejpam-4983	86	30	·	·	PUNCT
ejpam-4983	86	31	·	·	PUNCT
ejpam-4983	86	32	0	0	NUM
ejpam-4983	87	1	s31	s31	NOUN
ejpam-4983	87	2	0	0	NUM
ejpam-4983	87	3	s33	s33	PROPN
ejpam-4983	87	4	·	·	PUNCT
ejpam-4983	87	5	·	·	PUNCT
ejpam-4983	87	6	·	·	PUNCT
ejpam-4983	87	7	0	0	NUM
ejpam-4983	87	8	...	...	PUNCT
ejpam-4983	87	9	...	...	PUNCT
ejpam-4983	87	10	...	...	PUNCT
ejpam-4983	87	11	.	.	PUNCT
ejpam-4983	87	12	.	.	PUNCT
ejpam-4983	87	13	.	.	PUNCT
ejpam-4983	88	1	...	...	PUNCT
ejpam-4983	89	1	sn−1,1	sn−1,1	NOUN
ejpam-4983	89	2	0	0	NUM
ejpam-4983	89	3	0	0	NUM
ejpam-4983	89	4	·	·	PUNCT
ejpam-4983	89	5	·	·	PUNCT
ejpam-4983	89	6	·	·	PUNCT
ejpam-4983	89	7	sn−1,n−1	sn−1,n−1	ADJ
ejpam-4983	89	8			PROPN
ejpam-4983	89	9	∈	∈	PROPN
ejpam-4983	89	10	ian−1(fq	ian−1(fq	PROPN
ejpam-4983	89	11	)	)	PUNCT
ejpam-4983	89	12	.	.	PUNCT
ejpam-4983	90	1	consequently	consequently	ADV
ejpam-4983	90	2	,	,	PUNCT
ejpam-4983	90	3	we	we	PRON
ejpam-4983	90	4	have	have	VERB
ejpam-4983	90	5	|s|	|s|	NOUN
ejpam-4983	90	6	=	=	SYM
ejpam-4983	90	7	|ian−1(fq)|	|ian−1(fq)|	PROPN
ejpam-4983	90	8	.	.	PUNCT
ejpam-4983	91	1	we	we	PRON
ejpam-4983	91	2	note	note	VERB
ejpam-4983	91	3	that	that	SCONJ
ejpam-4983	91	4	0	0	NUM
ejpam-4983	91	5	̸=	̸=	PROPN
ejpam-4983	91	6	det(a	det(a	PROPN
ejpam-4983	91	7	)	)	PUNCT
ejpam-4983	91	8	=	=	SYM
ejpam-4983	91	9	ann	ann	PROPN
ejpam-4983	91	10	det(c	det(c	PROPN
ejpam-4983	91	11	)	)	PUNCT
ejpam-4983	91	12	if	if	SCONJ
ejpam-4983	91	13	and	and	CCONJ
ejpam-4983	91	14	only	only	ADV
ejpam-4983	91	15	if	if	SCONJ
ejpam-4983	91	16	det(c	det(c	VERB
ejpam-4983	91	17	)	)	PUNCT
ejpam-4983	91	18	̸=	̸=	PROPN
ejpam-4983	91	19	0	0	NUM
ejpam-4983	91	20	,	,	PUNCT
ejpam-4983	91	21	or	or	CCONJ
ejpam-4983	91	22	equivalently,	equivalently,	PROPN
ejpam-4983	91	23	a11	a11	PROPN
ejpam-4983	91	24	a12	a12	PROPN
ejpam-4983	91	25	a13	a13	PROPN
ejpam-4983	91	26	·	·	PUNCT
ejpam-4983	91	27	·	·	PUNCT
ejpam-4983	91	28	·	·	PUNCT
ejpam-4983	91	29	a1,n−1	a1,n−1	ADJ
ejpam-4983	91	30	a21	a21	PROPN
ejpam-4983	91	31	a22	a22	PROPN
ejpam-4983	91	32	0	0	NUM
ejpam-4983	91	33	·	·	PUNCT
ejpam-4983	91	34	·	·	PUNCT
ejpam-4983	91	35	·	·	PUNCT
ejpam-4983	91	36	0	0	NUM
ejpam-4983	92	1	a31	a31	NOUN
ejpam-4983	92	2	0	0	NUM
ejpam-4983	92	3	a33	a33	PROPN
ejpam-4983	92	4	·	·	PUNCT
ejpam-4983	92	5	·	·	PUNCT
ejpam-4983	92	6	·	·	PUNCT
ejpam-4983	92	7	0	0	NUM
ejpam-4983	92	8	...	...	PUNCT
ejpam-4983	92	9	...	...	PUNCT
ejpam-4983	92	10	...	...	PUNCT
ejpam-4983	92	11	.	.	PUNCT
ejpam-4983	92	12	.	.	PUNCT
ejpam-4983	92	13	.	.	PUNCT
ejpam-4983	93	1	...	...	PUNCT
ejpam-4983	94	1	an−1,1	an−1,1	X
ejpam-4983	94	2	0	0	NUM
ejpam-4983	94	3	0	0	NUM
ejpam-4983	94	4	·	·	PUNCT
ejpam-4983	94	5	·	·	PUNCT
ejpam-4983	94	6	·	·	PUNCT
ejpam-4983	94	7	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	94	8			PROPN
ejpam-4983	94	9	∈	∈	PROPN
ejpam-4983	94	10	s.	s.	PROPN
ejpam-4983	94	11	s.	s.	PROPN
ejpam-4983	94	12	jitman	jitman	PROPN
ejpam-4983	94	13	,	,	PUNCT
ejpam-4983	94	14	p.	p.	PROPN
ejpam-4983	94	15	modjam	modjam	PROPN
ejpam-4983	94	16	/	/	SYM
ejpam-4983	94	17	eur	eur	PROPN
ejpam-4983	94	18	.	.	PUNCT
ejpam-4983	95	1	j.	j.	PROPN
ejpam-4983	95	2	pure	pure	PROPN
ejpam-4983	95	3	appl	appl	PROPN
ejpam-4983	95	4	.	.	PROPN
ejpam-4983	95	5	math	math	PROPN
ejpam-4983	95	6	,	,	PUNCT
ejpam-4983	95	7	17	17	NUM
ejpam-4983	95	8	(	(	PUNCT
ejpam-4983	95	9	1	1	NUM
ejpam-4983	95	10	)	)	PUNCT
ejpam-4983	95	11	(	(	PUNCT
ejpam-4983	95	12	2024	2024	NUM
ejpam-4983	95	13	)	)	PUNCT
ejpam-4983	95	14	,	,	PUNCT
ejpam-4983	95	15	11	11	NUM
ejpam-4983	95	16	-	-	SYM
ejpam-4983	95	17	29	29	NUM
ejpam-4983	95	18	15	15	NUM
ejpam-4983	95	19	hence	hence	ADV
ejpam-4983	95	20	,	,	PUNCT
ejpam-4983	95	21	there	there	PRON
ejpam-4983	95	22	are	be	VERB
ejpam-4983	95	23	|s|	|s|	NOUN
ejpam-4983	95	24	=	=	SYM
ejpam-4983	96	1	|ian−1(fq)|	|ian−1(fq)|	PROPN
ejpam-4983	96	2	possibilities	possibility	NOUN
ejpam-4983	96	3	for	for	ADP
ejpam-4983	96	4	c.	c.	NOUN
ejpam-4983	96	5	the	the	DET
ejpam-4983	96	6	number	number	NOUN
ejpam-4983	96	7	of	of	ADP
ejpam-4983	96	8	choices	choice	NOUN
ejpam-4983	96	9	of	of	ADP
ejpam-4983	96	10	a1n	a1n	PUNCT
ejpam-4983	96	11	and	and	CCONJ
ejpam-4983	96	12	an1	an1	PROPN
ejpam-4983	96	13	are	be	AUX
ejpam-4983	96	14	q2	q2	NOUN
ejpam-4983	96	15	and	and	CCONJ
ejpam-4983	96	16	the	the	DET
ejpam-4983	96	17	number	number	NOUN
ejpam-4983	96	18	of	of	ADP
ejpam-4983	96	19	choices	choice	NOUN
ejpam-4983	96	20	for	for	ADP
ejpam-4983	96	21	ann	ann	PROPN
ejpam-4983	96	22	is	be	AUX
ejpam-4983	96	23	q	q	ADJ
ejpam-4983	96	24	−	−	PROPN
ejpam-4983	96	25	1	1	NUM
ejpam-4983	96	26	.	.	PUNCT
ejpam-4983	97	1	hence	hence	ADV
ejpam-4983	97	2	,	,	PUNCT
ejpam-4983	97	3	the	the	DET
ejpam-4983	97	4	number	number	NOUN
ejpam-4983	97	5	of	of	ADP
ejpam-4983	97	6	arrowhead	arrowhead	NOUN
ejpam-4983	97	7	matrices	matrix	NOUN
ejpam-4983	97	8	a	a	PRON
ejpam-4983	97	9	in	in	ADP
ejpam-4983	97	10	ian(fq	ian(fq	NOUN
ejpam-4983	97	11	)	)	PUNCT
ejpam-4983	97	12	is	be	AUX
ejpam-4983	97	13	q2(q	q2(q	VERB
ejpam-4983	97	14	−	−	ADP
ejpam-4983	97	15	1)|ian−1(fq)|	1)|ian−1(fq)|	NUM
ejpam-4983	97	16	.	.	PUNCT
ejpam-4983	97	17	case	case	NOUN
ejpam-4983	97	18	2	2	NUM
ejpam-4983	97	19	:	:	PUNCT
ejpam-4983	97	20	ann	ann	PROPN
ejpam-4983	97	21	=	=	PUNCT
ejpam-4983	97	22	0	0	PROPN
ejpam-4983	97	23	.	.	PUNCT
ejpam-4983	98	1	since	since	SCONJ
ejpam-4983	98	2	det(a	det(a	PROPN
ejpam-4983	98	3	)	)	PUNCT
ejpam-4983	98	4	̸=	̸=	PROPN
ejpam-4983	98	5	0	0	NUM
ejpam-4983	98	6	,	,	PUNCT
ejpam-4983	98	7	we	we	PRON
ejpam-4983	98	8	have	have	VERB
ejpam-4983	98	9	a1n	a1n	ADP
ejpam-4983	98	10	̸=	̸=	PROPN
ejpam-4983	98	11	0	0	NUM
ejpam-4983	98	12	and	and	CCONJ
ejpam-4983	98	13	an1	an1	NOUN
ejpam-4983	98	14	̸=	̸=	PROPN
ejpam-4983	98	15	0	0	NUM
ejpam-4983	98	16	.	.	PUNCT
ejpam-4983	99	1	applying	apply	VERB
ejpam-4983	99	2	the	the	DET
ejpam-4983	99	3	elementary	elementary	PROPN
ejpam-4983	99	4	row	row	NOUN
ejpam-4983	99	5	operation	operation	NOUN
ejpam-4983	99	6	ri	ri	PROPN
ejpam-4983	100	1	−	−	PROPN
ejpam-4983	100	2	ai1an1	ai1an1	NOUN
ejpam-4983	100	3	−1rn	−1rn	X
ejpam-4983	100	4	→	→	SYM
ejpam-4983	100	5	ri	ri	PROPN
ejpam-4983	100	6	for	for	ADP
ejpam-4983	100	7	all	all	PRON
ejpam-4983	100	8	i	i	PRON
ejpam-4983	100	9	∈	∈	PROPN
ejpam-4983	100	10	{	{	PUNCT
ejpam-4983	100	11	1	1	NUM
ejpam-4983	100	12	,	,	PUNCT
ejpam-4983	100	13	2	2	NUM
ejpam-4983	100	14	,	,	PUNCT
ejpam-4983	100	15	.	.	PUNCT
ejpam-4983	100	16	.	.	PUNCT
ejpam-4983	101	1	.	.	PUNCT
ejpam-4983	102	1	,	,	PUNCT
ejpam-4983	103	1	n	n	CCONJ
ejpam-4983	103	2	−	−	PROPN
ejpam-4983	103	3	1	1	NUM
ejpam-4983	103	4	}	}	PUNCT
ejpam-4983	103	5	and	and	CCONJ
ejpam-4983	103	6	the	the	DET
ejpam-4983	103	7	elementary	elementary	ADJ
ejpam-4983	103	8	column	column	PROPN
ejpam-4983	103	9	operation	operation	PROPN
ejpam-4983	103	10	c1	c1	PROPN
ejpam-4983	103	11	↔	↔	PROPN
ejpam-4983	103	12	cn	cn	PROPN
ejpam-4983	103	13	,	,	PUNCT
ejpam-4983	103	14	we	we	PRON
ejpam-4983	103	15	have	have	VERB
ejpam-4983	103	16	a	a	DET
ejpam-4983	103	17	∼	∼	NOUN
ejpam-4983	103	18			NOUN
ejpam-4983	104	1	a1n	a1n	DET
ejpam-4983	104	2	a12	a12	NOUN
ejpam-4983	104	3	a13	a13	NOUN
ejpam-4983	104	4	·	·	PUNCT
ejpam-4983	104	5	·	·	PUNCT
ejpam-4983	104	6	·	·	PUNCT
ejpam-4983	105	1	a1,n−1	a1,n−1	ADJ
ejpam-4983	105	2	0	0	NUM
ejpam-4983	105	3	0	0	NUM
ejpam-4983	105	4	a22	a22	PROPN
ejpam-4983	105	5	0	0	NUM
ejpam-4983	105	6	·	·	PUNCT
ejpam-4983	105	7	·	·	PUNCT
ejpam-4983	105	8	·	·	PUNCT
ejpam-4983	105	9	0	0	NUM
ejpam-4983	105	10	0	0	NUM
ejpam-4983	105	11	0	0	NUM
ejpam-4983	105	12	0	0	NUM
ejpam-4983	105	13	a33	a33	PROPN
ejpam-4983	105	14	·	·	PUNCT
ejpam-4983	105	15	·	·	PUNCT
ejpam-4983	105	16	·	·	PUNCT
ejpam-4983	105	17	0	0	NUM
ejpam-4983	105	18	0	0	NUM
ejpam-4983	105	19	...	...	PUNCT
ejpam-4983	105	20	...	...	PUNCT
ejpam-4983	105	21	...	...	PUNCT
ejpam-4983	105	22	.	.	PUNCT
ejpam-4983	105	23	.	.	PUNCT
ejpam-4983	105	24	.	.	PUNCT
ejpam-4983	105	25	...	...	PUNCT
ejpam-4983	106	1	...	...	PUNCT
ejpam-4983	107	1	0	0	NUM
ejpam-4983	107	2	0	0	NUM
ejpam-4983	107	3	0	0	NUM
ejpam-4983	107	4	·	·	PUNCT
ejpam-4983	107	5	·	·	PUNCT
ejpam-4983	107	6	·	·	PUNCT
ejpam-4983	107	7	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	107	8	0	0	NUM
ejpam-4983	107	9	0	0	NUM
ejpam-4983	107	10	0	0	NUM
ejpam-4983	107	11	0	0	NUM
ejpam-4983	107	12	·	·	PUNCT
ejpam-4983	107	13	·	·	PUNCT
ejpam-4983	107	14	·	·	PUNCT
ejpam-4983	107	15	0	0	NUM
ejpam-4983	108	1	an1	an1	NOUN
ejpam-4983	108	2			PUNCT
ejpam-4983	109	1	=	=	PRON
ejpam-4983	109	2	:	:	PUNCT
ejpam-4983	109	3	a′.	a′.	VERB
ejpam-4983	109	4	since	since	SCONJ
ejpam-4983	109	5	det(a′	det(a′	NUM
ejpam-4983	109	6	)	)	PUNCT
ejpam-4983	110	1	=	=	PUNCT
ejpam-4983	111	1	−det(a	−det(a	NOUN
ejpam-4983	111	2	)	)	PUNCT
ejpam-4983	111	3	̸=	̸=	PROPN
ejpam-4983	111	4	0	0	PUNCT
ejpam-4983	112	1	if	if	SCONJ
ejpam-4983	112	2	and	and	CCONJ
ejpam-4983	112	3	only	only	ADV
ejpam-4983	112	4	if	if	SCONJ
ejpam-4983	112	5	a1n	a1n	NOUN
ejpam-4983	112	6	,	,	PUNCT
ejpam-4983	112	7	a22	a22	PROPN
ejpam-4983	112	8	,	,	PUNCT
ejpam-4983	112	9	.	.	PUNCT
ejpam-4983	112	10	.	.	PUNCT
ejpam-4983	113	1	.	.	PUNCT
ejpam-4983	114	1	,	,	PUNCT
ejpam-4983	114	2	an−1,n−1	an−1,n−1	PROPN
ejpam-4983	114	3	,	,	PUNCT
ejpam-4983	114	4	an1	an1	PROPN
ejpam-4983	114	5	are	be	AUX
ejpam-4983	114	6	non	non	ADJ
ejpam-4983	114	7	-	-	ADJ
ejpam-4983	114	8	zero	zero	NUM
ejpam-4983	114	9	,	,	PUNCT
ejpam-4983	114	10	the	the	DET
ejpam-4983	114	11	number	number	NOUN
ejpam-4983	114	12	of	of	ADP
ejpam-4983	114	13	(	(	PUNCT
ejpam-4983	114	14	a1n	a1n	PROPN
ejpam-4983	114	15	,	,	PUNCT
ejpam-4983	114	16	a22	a22	PROPN
ejpam-4983	114	17	,	,	PUNCT
ejpam-4983	114	18	a33	a33	PROPN
ejpam-4983	114	19	,	,	PUNCT
ejpam-4983	114	20	.	.	PUNCT
ejpam-4983	114	21	.	.	PUNCT
ejpam-4983	115	1	.	.	PUNCT
ejpam-4983	116	1	,	,	PUNCT
ejpam-4983	116	2	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	116	3	,	,	PUNCT
ejpam-4983	116	4	an1	an1	X
ejpam-4983	116	5	)	)	PUNCT
ejpam-4983	116	6	is	be	AUX
ejpam-4983	116	7	(	(	PUNCT
ejpam-4983	116	8	q−1)n	q−1)n	X
ejpam-4983	116	9	,	,	PUNCT
ejpam-4983	116	10	the	the	DET
ejpam-4983	116	11	number	number	NOUN
ejpam-4983	116	12	of	of	ADP
ejpam-4983	116	13	(	(	PUNCT
ejpam-4983	116	14	a12	a12	PROPN
ejpam-4983	116	15	,	,	PUNCT
ejpam-4983	116	16	a13	a13	NOUN
ejpam-4983	116	17	,	,	PUNCT
ejpam-4983	116	18	a14	a14	PROPN
ejpam-4983	116	19	,	,	PUNCT
ejpam-4983	116	20	.	.	PUNCT
ejpam-4983	116	21	.	.	PUNCT
ejpam-4983	116	22	.	.	PUNCT
ejpam-4983	117	1	,	,	PUNCT
ejpam-4983	117	2	a1,n−1	a1,n−1	ADJ
ejpam-4983	117	3	)	)	PUNCT
ejpam-4983	117	4	is	be	AUX
ejpam-4983	117	5	qn−1	qn−1	ADJ
ejpam-4983	117	6	,	,	PUNCT
ejpam-4983	117	7	and	and	CCONJ
ejpam-4983	117	8	the	the	DET
ejpam-4983	117	9	number	number	NOUN
ejpam-4983	117	10	of	of	ADP
ejpam-4983	117	11	(	(	PUNCT
ejpam-4983	117	12	a21	a21	PROPN
ejpam-4983	117	13	,	,	PUNCT
ejpam-4983	117	14	a31	a31	PROPN
ejpam-4983	117	15	,	,	PUNCT
ejpam-4983	117	16	a41	a41	NOUN
ejpam-4983	117	17	,	,	PUNCT
ejpam-4983	117	18	.	.	PUNCT
ejpam-4983	117	19	.	.	PUNCT
ejpam-4983	118	1	.	.	PUNCT
ejpam-4983	119	1	,	,	PUNCT
ejpam-4983	119	2	an−2,1	an−2,1	ADV
ejpam-4983	119	3	)	)	PUNCT
ejpam-4983	119	4	is	be	AUX
ejpam-4983	119	5	q	q	PROPN
ejpam-4983	119	6	n−2	n−2	PROPN
ejpam-4983	119	7	in	in	ADP
ejpam-4983	119	8	this	this	DET
ejpam-4983	119	9	case	case	NOUN
ejpam-4983	119	10	,	,	PUNCT
ejpam-4983	119	11	the	the	DET
ejpam-4983	119	12	number	number	NOUN
ejpam-4983	119	13	of	of	ADP
ejpam-4983	119	14	a	a	PRON
ejpam-4983	119	15	in	in	ADP
ejpam-4983	119	16	ian(fq	ian(fq	NOUN
ejpam-4983	119	17	)	)	PUNCT
ejpam-4983	119	18	is	be	AUX
ejpam-4983	119	19	q2n−3(q	q2n−3(q	NUM
ejpam-4983	119	20	−	−	PROPN
ejpam-4983	119	21	1)n	1)n	X
ejpam-4983	119	22	.	.	PUNCT
ejpam-4983	120	1	from	from	ADP
ejpam-4983	120	2	the	the	DET
ejpam-4983	120	3	two	two	NUM
ejpam-4983	120	4	cases	case	NOUN
ejpam-4983	120	5	,	,	PUNCT
ejpam-4983	120	6	it	it	PRON
ejpam-4983	120	7	can	can	AUX
ejpam-4983	120	8	be	be	AUX
ejpam-4983	120	9	deduced	deduce	VERB
ejpam-4983	120	10	that	that	SCONJ
ejpam-4983	120	11	|ian(fq)|	|ian(fq)|	NOUN
ejpam-4983	120	12	=	=	SYM
ejpam-4983	120	13	q2n−3(q	q2n−3(q	NUM
ejpam-4983	120	14	−	−	PROPN
ejpam-4983	120	15	1)n	1)n	PUNCT
ejpam-4983	121	1	+	+	CCONJ
ejpam-4983	121	2	q2(q	q2(q	NOUN
ejpam-4983	121	3	−	−	NUM
ejpam-4983	121	4	1)|ian−1(fq)|	1)|ian−1(fq)|	NUM
ejpam-4983	121	5	as	as	SCONJ
ejpam-4983	121	6	desired	desire	VERB
ejpam-4983	121	7	.	.	PUNCT
ejpam-4983	122	1	■	■	PUNCT
ejpam-4983	122	2	an	an	DET
ejpam-4983	122	3	explicit	explicit	ADJ
ejpam-4983	122	4	expression	expression	NOUN
ejpam-4983	122	5	for	for	ADP
ejpam-4983	122	6	the	the	DET
ejpam-4983	122	7	number	number	NOUN
ejpam-4983	122	8	|ian(fq)|	|ian(fq)|	NUM
ejpam-4983	122	9	can	can	AUX
ejpam-4983	122	10	be	be	AUX
ejpam-4983	122	11	derived	derive	VERB
ejpam-4983	122	12	using	use	VERB
ejpam-4983	122	13	the	the	DET
ejpam-4983	122	14	recursive	recursive	ADJ
ejpam-4983	122	15	formula	formula	NOUN
ejpam-4983	122	16	given	give	VERB
ejpam-4983	122	17	in	in	ADP
ejpam-4983	122	18	proposition	proposition	NOUN
ejpam-4983	122	19	1	1	NUM
ejpam-4983	122	20	and	and	CCONJ
ejpam-4983	122	21	the	the	DET
ejpam-4983	122	22	principle	principle	NOUN
ejpam-4983	122	23	of	of	ADP
ejpam-4983	122	24	mathematical	mathematical	ADJ
ejpam-4983	122	25	induction	induction	NOUN
ejpam-4983	122	26	.	.	PUNCT
ejpam-4983	123	1	theorem	theorem	NOUN
ejpam-4983	123	2	1	1	X
ejpam-4983	123	3	.	.	PUNCT
ejpam-4983	124	1	let	let	VERB
ejpam-4983	124	2	q	q	PART
ejpam-4983	124	3	be	be	AUX
ejpam-4983	124	4	a	a	DET
ejpam-4983	124	5	prime	prime	ADJ
ejpam-4983	124	6	power	power	NOUN
ejpam-4983	124	7	.	.	PUNCT
ejpam-4983	125	1	then	then	ADV
ejpam-4983	125	2	|ian(fq)|	|ian(fq)|	X
ejpam-4983	125	3	=	=	SYM
ejpam-4983	125	4	q2n−3(q	q2n−3(q	NUM
ejpam-4983	125	5	−	−	PROPN
ejpam-4983	125	6	1)n(q	1)n(q	NUM
ejpam-4983	125	7	+	+	CCONJ
ejpam-4983	125	8	(	(	PUNCT
ejpam-4983	125	9	n−	n−	NOUN
ejpam-4983	125	10	1	1	NUM
ejpam-4983	125	11	)	)	PUNCT
ejpam-4983	125	12	)	)	PUNCT
ejpam-4983	125	13	for	for	ADP
ejpam-4983	125	14	all	all	DET
ejpam-4983	125	15	positive	positive	ADJ
ejpam-4983	125	16	integers	integer	NOUN
ejpam-4983	125	17	n.	n.	NOUN
ejpam-4983	125	18	proof	proof	NOUN
ejpam-4983	125	19	.	.	PUNCT
ejpam-4983	126	1	for	for	ADP
ejpam-4983	126	2	n	n	NOUN
ejpam-4983	126	3	=	=	SYM
ejpam-4983	126	4	1	1	NUM
ejpam-4983	126	5	,	,	PUNCT
ejpam-4983	126	6	we	we	PRON
ejpam-4983	126	7	have	have	VERB
ejpam-4983	126	8	|ia1(fq)|	|ia1(fq)|	X
ejpam-4983	126	9	=	=	SYM
ejpam-4983	126	10	q	q	NOUN
ejpam-4983	126	11	−	−	PROPN
ejpam-4983	126	12	1	1	NUM
ejpam-4983	126	13	=	=	NOUN
ejpam-4983	126	14	q2(1)−3(q	q2(1)−3(q	NOUN
ejpam-4983	126	15	−	−	PROPN
ejpam-4983	126	16	1)1(q	1)1(q	PROPN
ejpam-4983	126	17	+	+	CCONJ
ejpam-4983	126	18	(	(	PUNCT
ejpam-4983	126	19	1−	1−	NUM
ejpam-4983	126	20	1	1	NUM
ejpam-4983	126	21	)	)	PUNCT
ejpam-4983	126	22	)	)	PUNCT
ejpam-4983	126	23	.	.	PUNCT
ejpam-4983	127	1	let	let	VERB
ejpam-4983	127	2	k	k	PROPN
ejpam-4983	127	3	≥	≥	NUM
ejpam-4983	127	4	2	2	NUM
ejpam-4983	127	5	be	be	AUX
ejpam-4983	127	6	an	an	DET
ejpam-4983	127	7	integer	integer	NOUN
ejpam-4983	127	8	.	.	PUNCT
ejpam-4983	128	1	assume	assume	VERB
ejpam-4983	128	2	that	that	SCONJ
ejpam-4983	128	3	|iak−1(fq)|	|iak−1(fq)|	PRON
ejpam-4983	128	4	=	=	SYM
ejpam-4983	128	5	q2(k−1)−3(q	q2(k−1)−3(q	PROPN
ejpam-4983	128	6	−	−	NUM
ejpam-4983	128	7	1)k−1(q	1)k−1(q	NUM
ejpam-4983	129	1	+	+	CCONJ
ejpam-4983	129	2	(	(	PUNCT
ejpam-4983	129	3	(	(	PUNCT
ejpam-4983	129	4	k	k	PROPN
ejpam-4983	129	5	−	−	PROPN
ejpam-4983	130	1	1)−	1)−	PROPN
ejpam-4983	130	2	1	1	NUM
ejpam-4983	130	3	)	)	PUNCT
ejpam-4983	130	4	)	)	PUNCT
ejpam-4983	130	5	.	.	PUNCT
ejpam-4983	131	1	using	use	VERB
ejpam-4983	131	2	the	the	DET
ejpam-4983	131	3	recurrent	recurrent	ADJ
ejpam-4983	131	4	relation	relation	NOUN
ejpam-4983	131	5	given	give	VERB
ejpam-4983	131	6	in	in	ADP
ejpam-4983	131	7	proposition	proposition	NOUN
ejpam-4983	131	8	1	1	NUM
ejpam-4983	131	9	,	,	PUNCT
ejpam-4983	131	10	we	we	PRON
ejpam-4983	131	11	have	have	VERB
ejpam-4983	131	12	|iak(fq)|	|iak(fq)|	X
ejpam-4983	131	13	=	=	SYM
ejpam-4983	131	14	q2k−3(q	q2k−3(q	PROPN
ejpam-4983	131	15	−	−	PROPN
ejpam-4983	131	16	1)k	1)k	NUM
ejpam-4983	132	1	+	+	CCONJ
ejpam-4983	132	2	q2(q	q2(q	NOUN
ejpam-4983	132	3	−	−	ADP
ejpam-4983	132	4	1)|iak−1(fq)|	1)|iak−1(fq)|	NUM
ejpam-4983	132	5	=	=	SYM
ejpam-4983	132	6	q2k−3(q	q2k−3(q	PROPN
ejpam-4983	132	7	−	−	PROPN
ejpam-4983	132	8	1)k	1)k	NUM
ejpam-4983	133	1	+	+	CCONJ
ejpam-4983	133	2	q2(q	q2(q	NOUN
ejpam-4983	133	3	−	−	NUM
ejpam-4983	133	4	1)(q2(k−1)−3(q	1)(q2(k−1)−3(q	NUM
ejpam-4983	133	5	−	−	PROPN
ejpam-4983	133	6	1)k−1(q	1)k−1(q	NUM
ejpam-4983	134	1	+	+	CCONJ
ejpam-4983	134	2	(	(	PUNCT
ejpam-4983	134	3	(	(	PUNCT
ejpam-4983	134	4	k	k	PROPN
ejpam-4983	134	5	−	−	PROPN
ejpam-4983	134	6	1)−	1)−	PROPN
ejpam-4983	134	7	1	1	NUM
ejpam-4983	134	8	)	)	PUNCT
ejpam-4983	134	9	)	)	PUNCT
ejpam-4983	134	10	)	)	PUNCT
ejpam-4983	135	1	s.	s.	PROPN
ejpam-4983	135	2	jitman	jitman	PROPN
ejpam-4983	135	3	,	,	PUNCT
ejpam-4983	135	4	p.	p.	PROPN
ejpam-4983	135	5	modjam	modjam	PROPN
ejpam-4983	135	6	/	/	SYM
ejpam-4983	135	7	eur	eur	PROPN
ejpam-4983	135	8	.	.	PUNCT
ejpam-4983	136	1	j.	j.	PROPN
ejpam-4983	136	2	pure	pure	PROPN
ejpam-4983	136	3	appl	appl	PROPN
ejpam-4983	136	4	.	.	PROPN
ejpam-4983	136	5	math	math	PROPN
ejpam-4983	136	6	,	,	PUNCT
ejpam-4983	136	7	17	17	NUM
ejpam-4983	136	8	(	(	PUNCT
ejpam-4983	136	9	1	1	NUM
ejpam-4983	136	10	)	)	PUNCT
ejpam-4983	136	11	(	(	PUNCT
ejpam-4983	136	12	2024	2024	NUM
ejpam-4983	136	13	)	)	PUNCT
ejpam-4983	136	14	,	,	PUNCT
ejpam-4983	136	15	11	11	NUM
ejpam-4983	136	16	-	-	SYM
ejpam-4983	136	17	29	29	NUM
ejpam-4983	136	18	16	16	NUM
ejpam-4983	136	19	=	=	SYM
ejpam-4983	136	20	q2k−3(q	q2k−3(q	PROPN
ejpam-4983	137	1	−	−	NUM
ejpam-4983	137	2	1)k	1)k	NUM
ejpam-4983	138	1	+	+	CCONJ
ejpam-4983	138	2	q2k−3(q	q2k−3(q	PROPN
ejpam-4983	138	3	−	−	PROPN
ejpam-4983	138	4	1)k(q	1)k(q	NUM
ejpam-4983	138	5	+	+	CCONJ
ejpam-4983	138	6	(	(	PUNCT
ejpam-4983	138	7	k	k	NOUN
ejpam-4983	138	8	−	−	PROPN
ejpam-4983	138	9	2	2	NUM
ejpam-4983	138	10	)	)	PUNCT
ejpam-4983	138	11	)	)	PUNCT
ejpam-4983	139	1	=	=	SYM
ejpam-4983	139	2	q2k−3(q	q2k−3(q	PROPN
ejpam-4983	139	3	−	−	NUM
ejpam-4983	139	4	1)k(q	1)k(q	NUM
ejpam-4983	140	1	+	+	CCONJ
ejpam-4983	140	2	(	(	PUNCT
ejpam-4983	140	3	k	k	NOUN
ejpam-4983	140	4	−	−	PROPN
ejpam-4983	140	5	1	1	NUM
ejpam-4983	140	6	)	)	PUNCT
ejpam-4983	140	7	)	)	PUNCT
ejpam-4983	140	8	.	.	PUNCT
ejpam-4983	141	1	therefore	therefore	ADV
ejpam-4983	141	2	,	,	PUNCT
ejpam-4983	141	3	it	it	PRON
ejpam-4983	141	4	follows	follow	VERB
ejpam-4983	141	5	that	that	SCONJ
ejpam-4983	141	6	|ian(fq)|	|ian(fq)|	X
ejpam-4983	141	7	=	=	SYM
ejpam-4983	141	8	q2n−3(q	q2n−3(q	NUM
ejpam-4983	141	9	−	−	PROPN
ejpam-4983	141	10	1)n(q	1)n(q	NUM
ejpam-4983	141	11	+	+	CCONJ
ejpam-4983	141	12	(	(	PUNCT
ejpam-4983	141	13	n−	n−	NOUN
ejpam-4983	141	14	1	1	NUM
ejpam-4983	141	15	)	)	PUNCT
ejpam-4983	141	16	)	)	PUNCT
ejpam-4983	141	17	for	for	SCONJ
ejpam-4983	141	18	all	all	DET
ejpam-4983	141	19	positive	positive	ADJ
ejpam-4983	141	20	integers	integer	NOUN
ejpam-4983	141	21	n.	n.	PROPN
ejpam-4983	141	22	■	■	PROPN
ejpam-4983	141	23	in	in	ADP
ejpam-4983	141	24	the	the	DET
ejpam-4983	141	25	following	follow	VERB
ejpam-4983	141	26	proposition	proposition	NOUN
ejpam-4983	141	27	,	,	PUNCT
ejpam-4983	141	28	a	a	DET
ejpam-4983	141	29	relation	relation	NOUN
ejpam-4983	141	30	between	between	ADP
ejpam-4983	141	31	|an(fq	|an(fq	PROPN
ejpam-4983	141	32	,	,	PUNCT
ejpam-4983	141	33	1)|	1)|	NUM
ejpam-4983	141	34	and	and	CCONJ
ejpam-4983	141	35	|an(fq	|an(fq	PROPN
ejpam-4983	141	36	,	,	PUNCT
ejpam-4983	141	37	a)|	a)|	X
ejpam-4983	141	38	for	for	ADP
ejpam-4983	141	39	all	all	DET
ejpam-4983	141	40	a	a	DET
ejpam-4983	141	41	∈	∈	PROPN
ejpam-4983	141	42	fq	fq	NOUN
ejpam-4983	141	43	\	\	PROPN
ejpam-4983	141	44	{	{	PUNCT
ejpam-4983	141	45	0	0	NUM
ejpam-4983	141	46	}	}	PUNCT
ejpam-4983	141	47	is	be	AUX
ejpam-4983	141	48	key	key	ADJ
ejpam-4983	141	49	to	to	PART
ejpam-4983	141	50	study	study	VERB
ejpam-4983	141	51	the	the	DET
ejpam-4983	141	52	enumeration	enumeration	NOUN
ejpam-4983	141	53	of	of	ADP
ejpam-4983	141	54	|an(fq	|an(fq	PROPN
ejpam-4983	141	55	,	,	PUNCT
ejpam-4983	141	56	a)|	a)|	X
ejpam-4983	141	57	in	in	ADP
ejpam-4983	141	58	corollary	corollary	ADJ
ejpam-4983	141	59	1	1	NUM
ejpam-4983	141	60	.	.	PUNCT
ejpam-4983	141	61	proposition	proposition	NOUN
ejpam-4983	141	62	2	2	NUM
ejpam-4983	141	63	.	.	PUNCT
ejpam-4983	142	1	let	let	VERB
ejpam-4983	142	2	q	q	PROPN
ejpam-4983	142	3	a	a	DET
ejpam-4983	142	4	prime	prime	ADJ
ejpam-4983	142	5	power	power	NOUN
ejpam-4983	142	6	and	and	CCONJ
ejpam-4983	142	7	let	let	VERB
ejpam-4983	142	8	n	n	PRON
ejpam-4983	142	9	be	be	AUX
ejpam-4983	142	10	a	a	DET
ejpam-4983	142	11	positive	positive	ADJ
ejpam-4983	142	12	integer	integer	NOUN
ejpam-4983	142	13	.	.	PUNCT
ejpam-4983	143	1	then	then	ADV
ejpam-4983	143	2	|an(fq	|an(fq	PROPN
ejpam-4983	143	3	,	,	PUNCT
ejpam-4983	143	4	1)|	1)|	NUM
ejpam-4983	143	5	=	=	SYM
ejpam-4983	143	6	|an(fq	|an(fq	PROPN
ejpam-4983	143	7	,	,	PUNCT
ejpam-4983	143	8	a)|	a)|	X
ejpam-4983	143	9	for	for	ADP
ejpam-4983	143	10	all	all	DET
ejpam-4983	143	11	a	a	DET
ejpam-4983	143	12	∈	∈	PROPN
ejpam-4983	143	13	fq	fq	NOUN
ejpam-4983	143	14	\	\	PROPN
ejpam-4983	143	15	{	{	PUNCT
ejpam-4983	143	16	0	0	NUM
ejpam-4983	143	17	}	}	PUNCT
ejpam-4983	143	18	.	.	PUNCT
ejpam-4983	144	1	proof	proof	NOUN
ejpam-4983	144	2	.	.	PUNCT
ejpam-4983	145	1	let	let	VERB
ejpam-4983	145	2	a	a	DET
ejpam-4983	145	3	∈	∈	PROPN
ejpam-4983	145	4	fq	fq	X
ejpam-4983	145	5	\	\	PROPN
ejpam-4983	145	6	{	{	PUNCT
ejpam-4983	145	7	0	0	NUM
ejpam-4983	145	8	}	}	PUNCT
ejpam-4983	145	9	and	and	CCONJ
ejpam-4983	145	10	let	let	VERB
ejpam-4983	145	11	f	f	NOUN
ejpam-4983	145	12	:	:	PUNCT
ejpam-4983	145	13	an(fq	an(fq	PROPN
ejpam-4983	145	14	,	,	PUNCT
ejpam-4983	145	15	1	1	NUM
ejpam-4983	145	16	)	)	PUNCT
ejpam-4983	145	17	→	→	SYM
ejpam-4983	145	18	an(fq	an(fq	PROPN
ejpam-4983	145	19	,	,	PUNCT
ejpam-4983	145	20	a	a	PRON
ejpam-4983	145	21	)	)	PUNCT
ejpam-4983	145	22	be	be	AUX
ejpam-4983	145	23	defined	define	VERB
ejpam-4983	145	24	by	by	ADP
ejpam-4983	145	25	f(a	f(a	PROPN
ejpam-4983	145	26	)	)	PUNCT
ejpam-4983	146	1	=	=	SYM
ejpam-4983	146	2	diag(a	diag(a	PROPN
ejpam-4983	146	3	,	,	PUNCT
ejpam-4983	146	4	1	1	NUM
ejpam-4983	146	5	,	,	PUNCT
ejpam-4983	146	6	1	1	NUM
ejpam-4983	146	7	,	,	PUNCT
ejpam-4983	146	8	.	.	PUNCT
ejpam-4983	146	9	.	.	PUNCT
ejpam-4983	146	10	.	.	PUNCT
ejpam-4983	147	1	,	,	PUNCT
ejpam-4983	147	2	1)a	1)a	NUM
ejpam-4983	147	3	.	.	PUNCT
ejpam-4983	148	1	let	let	VERB
ejpam-4983	148	2	a	a	DET
ejpam-4983	148	3	=	=	X
ejpam-4983	148	4			ADJ
ejpam-4983	148	5	a11	a11	PROPN
ejpam-4983	148	6	a12	a12	PROPN
ejpam-4983	148	7	a13	a13	PROPN
ejpam-4983	148	8	·	·	PUNCT
ejpam-4983	148	9	·	·	PUNCT
ejpam-4983	148	10	·	·	PUNCT
ejpam-4983	149	1	a1n	a1n	ADP
ejpam-4983	149	2	a21	a21	NOUN
ejpam-4983	149	3	a22	a22	PROPN
ejpam-4983	149	4	0	0	NUM
ejpam-4983	149	5	·	·	PUNCT
ejpam-4983	149	6	·	·	PUNCT
ejpam-4983	149	7	·	·	PUNCT
ejpam-4983	149	8	0	0	NUM
ejpam-4983	150	1	a31	a31	NOUN
ejpam-4983	150	2	0	0	NUM
ejpam-4983	150	3	a33	a33	PROPN
ejpam-4983	150	4	·	·	PUNCT
ejpam-4983	150	5	·	·	PUNCT
ejpam-4983	150	6	·	·	PUNCT
ejpam-4983	150	7	0	0	NUM
ejpam-4983	150	8	...	...	PUNCT
ejpam-4983	150	9	...	...	PUNCT
ejpam-4983	150	10	...	...	PUNCT
ejpam-4983	150	11	.	.	PUNCT
ejpam-4983	150	12	.	.	PUNCT
ejpam-4983	150	13	.	.	PUNCT
ejpam-4983	150	14	...	...	PUNCT
ejpam-4983	151	1	an1	an1	PRON
ejpam-4983	151	2	0	0	NUM
ejpam-4983	151	3	0	0	NUM
ejpam-4983	151	4	·	·	PUNCT
ejpam-4983	151	5	·	·	PUNCT
ejpam-4983	151	6	·	·	PUNCT
ejpam-4983	152	1	ann	ann	X
ejpam-4983	152	2			PROPN
ejpam-4983	152	3	∈	∈	PROPN
ejpam-4983	152	4	an(fq	an(fq	PROPN
ejpam-4983	152	5	,	,	PUNCT
ejpam-4983	152	6	1	1	NUM
ejpam-4983	152	7	)	)	PUNCT
ejpam-4983	152	8	.	.	PUNCT
ejpam-4983	153	1	then	then	ADV
ejpam-4983	153	2	det(a	det(a	PROPN
ejpam-4983	153	3	)	)	PUNCT
ejpam-4983	153	4	=	=	SYM
ejpam-4983	153	5	1	1	NUM
ejpam-4983	153	6	,	,	PUNCT
ejpam-4983	153	7	f(a	f(a	NOUN
ejpam-4983	153	8	)	)	PUNCT
ejpam-4983	154	1	=	=	SYM
ejpam-4983	154	2	diag(a	diag(a	PROPN
ejpam-4983	154	3	,	,	PUNCT
ejpam-4983	154	4	1	1	NUM
ejpam-4983	154	5	,	,	PUNCT
ejpam-4983	154	6	1	1	NUM
ejpam-4983	154	7	,	,	PUNCT
ejpam-4983	154	8	.	.	PUNCT
ejpam-4983	154	9	.	.	PUNCT
ejpam-4983	154	10	.	.	PUNCT
ejpam-4983	155	1	,	,	PUNCT
ejpam-4983	155	2	1)a	1)a	NUM
ejpam-4983	155	3	=	=	PUNCT
ejpam-4983	155	4			ADJ
ejpam-4983	155	5	aa11	aa11	PROPN
ejpam-4983	155	6	aa12	aa12	PROPN
ejpam-4983	155	7	aa13	aa13	PROPN
ejpam-4983	155	8	·	·	PUNCT
ejpam-4983	155	9	·	·	PUNCT
ejpam-4983	155	10	·	·	PUNCT
ejpam-4983	155	11	aa1n	aa1n	ADV
ejpam-4983	155	12	a21	a21	PROPN
ejpam-4983	155	13	a22	a22	PROPN
ejpam-4983	155	14	0	0	NUM
ejpam-4983	155	15	·	·	PUNCT
ejpam-4983	155	16	·	·	PUNCT
ejpam-4983	155	17	·	·	PUNCT
ejpam-4983	155	18	0	0	NUM
ejpam-4983	155	19	a31	a31	NOUN
ejpam-4983	155	20	0	0	NUM
ejpam-4983	155	21	a33	a33	PROPN
ejpam-4983	155	22	·	·	PUNCT
ejpam-4983	155	23	·	·	PUNCT
ejpam-4983	155	24	·	·	PUNCT
ejpam-4983	155	25	0	0	NUM
ejpam-4983	155	26	...	...	PUNCT
ejpam-4983	155	27	...	...	PUNCT
ejpam-4983	155	28	...	...	PUNCT
ejpam-4983	155	29	.	.	PUNCT
ejpam-4983	155	30	.	.	PUNCT
ejpam-4983	155	31	.	.	PUNCT
ejpam-4983	155	32	...	...	PUNCT
ejpam-4983	156	1	an1	an1	PRON
ejpam-4983	156	2	0	0	NUM
ejpam-4983	156	3	0	0	NUM
ejpam-4983	156	4	·	·	PUNCT
ejpam-4983	156	5	·	·	PUNCT
ejpam-4983	156	6	·	·	PUNCT
ejpam-4983	157	1	ann	ann	X
ejpam-4983	157	2			PROPN
ejpam-4983	157	3	∈	∈	PROPN
ejpam-4983	157	4	an(fq	an(fq	PROPN
ejpam-4983	157	5	)	)	PUNCT
ejpam-4983	157	6	,	,	PUNCT
ejpam-4983	157	7	(	(	PUNCT
ejpam-4983	157	8	2	2	X
ejpam-4983	157	9	)	)	PUNCT
ejpam-4983	157	10	and	and	CCONJ
ejpam-4983	157	11	det(f(a	det(f(a	NOUN
ejpam-4983	157	12	)	)	PUNCT
ejpam-4983	157	13	)	)	PUNCT
ejpam-4983	158	1	=	=	SYM
ejpam-4983	158	2	det(diag(a	det(diag(a	NOUN
ejpam-4983	158	3	,	,	PUNCT
ejpam-4983	158	4	1	1	NUM
ejpam-4983	158	5	,	,	PUNCT
ejpam-4983	158	6	1	1	NUM
ejpam-4983	158	7	,	,	PUNCT
ejpam-4983	158	8	.	.	PUNCT
ejpam-4983	158	9	.	.	PUNCT
ejpam-4983	158	10	.	.	PUNCT
ejpam-4983	159	1	,	,	PUNCT
ejpam-4983	159	2	1)a	1)a	NUM
ejpam-4983	159	3	)	)	PUNCT
ejpam-4983	159	4	=	=	SYM
ejpam-4983	159	5	det(diag(a	det(diag(a	NOUN
ejpam-4983	159	6	,	,	PUNCT
ejpam-4983	159	7	1	1	NUM
ejpam-4983	159	8	,	,	PUNCT
ejpam-4983	159	9	1	1	NUM
ejpam-4983	159	10	,	,	PUNCT
ejpam-4983	159	11	.	.	PUNCT
ejpam-4983	159	12	.	.	PUNCT
ejpam-4983	159	13	.	.	PUNCT
ejpam-4983	160	1	,	,	PUNCT
ejpam-4983	160	2	1	1	NUM
ejpam-4983	160	3	)	)	PUNCT
ejpam-4983	160	4	)	)	PUNCT
ejpam-4983	160	5	·	·	PUNCT
ejpam-4983	161	1	det(a	det(a	X
ejpam-4983	161	2	)	)	PUNCT
ejpam-4983	161	3	=	=	SYM
ejpam-4983	161	4	a	a	PRON
ejpam-4983	161	5	·	·	SYM
ejpam-4983	161	6	1	1	NUM
ejpam-4983	161	7	=	=	NOUN
ejpam-4983	161	8	a.	a.	NOUN
ejpam-4983	161	9	hence	hence	ADV
ejpam-4983	161	10	,	,	PUNCT
ejpam-4983	161	11	f(a	f(a	PROPN
ejpam-4983	161	12	)	)	PUNCT
ejpam-4983	161	13	∈	∈	PROPN
ejpam-4983	161	14	an(fq	an(fq	PROPN
ejpam-4983	161	15	,	,	PUNCT
ejpam-4983	161	16	a	a	PRON
ejpam-4983	161	17	)	)	PUNCT
ejpam-4983	161	18	.	.	PUNCT
ejpam-4983	162	1	since	since	SCONJ
ejpam-4983	162	2	diag(a	diag(a	PROPN
ejpam-4983	162	3	,	,	PUNCT
ejpam-4983	162	4	1	1	NUM
ejpam-4983	162	5	,	,	PUNCT
ejpam-4983	162	6	1	1	NUM
ejpam-4983	162	7	,	,	PUNCT
ejpam-4983	162	8	.	.	PUNCT
ejpam-4983	162	9	.	.	PUNCT
ejpam-4983	162	10	.	.	PUNCT
ejpam-4983	163	1	,	,	PUNCT
ejpam-4983	163	2	1	1	X
ejpam-4983	163	3	)	)	PUNCT
ejpam-4983	163	4	is	be	AUX
ejpam-4983	163	5	invertible	invertible	ADJ
ejpam-4983	163	6	,	,	PUNCT
ejpam-4983	163	7	we	we	PRON
ejpam-4983	163	8	have	have	VERB
ejpam-4983	163	9	that	that	SCONJ
ejpam-4983	163	10	f	f	PROPN
ejpam-4983	163	11	is	be	AUX
ejpam-4983	163	12	injective	injective	ADJ
ejpam-4983	163	13	.	.	PUNCT
ejpam-4983	164	1	let	let	VERB
ejpam-4983	164	2	x	x	SYM
ejpam-4983	164	3	∈	∈	PROPN
ejpam-4983	164	4	an(fq	an(fq	PROPN
ejpam-4983	164	5	,	,	PUNCT
ejpam-4983	164	6	a	a	PRON
ejpam-4983	164	7	)	)	PUNCT
ejpam-4983	164	8	and	and	CCONJ
ejpam-4983	164	9	let	let	VERB
ejpam-4983	164	10	a	a	DET
ejpam-4983	164	11	=	=	SYM
ejpam-4983	164	12	diag(a−1	diag(a−1	NOUN
ejpam-4983	164	13	,	,	PUNCT
ejpam-4983	164	14	1	1	NUM
ejpam-4983	164	15	,	,	PUNCT
ejpam-4983	164	16	1	1	NUM
ejpam-4983	164	17	,	,	PUNCT
ejpam-4983	164	18	.	.	PUNCT
ejpam-4983	164	19	.	.	PUNCT
ejpam-4983	165	1	.	.	PUNCT
ejpam-4983	166	1	,	,	PUNCT
ejpam-4983	166	2	1)x	1)x	X
ejpam-4983	166	3	.	.	PUNCT
ejpam-4983	167	1	then	then	ADV
ejpam-4983	167	2	we	we	PRON
ejpam-4983	167	3	have	have	VERB
ejpam-4983	167	4	a	a	DET
ejpam-4983	167	5	∈	∈	PROPN
ejpam-4983	167	6	an(fq	an(fq	NOUN
ejpam-4983	167	7	)	)	PUNCT
ejpam-4983	167	8	and	and	CCONJ
ejpam-4983	167	9	det(a	det(a	PROPN
ejpam-4983	167	10	)	)	PUNCT
ejpam-4983	167	11	=	=	SYM
ejpam-4983	167	12	det(diag(a−1	det(diag(a−1	PROPN
ejpam-4983	167	13	,	,	PUNCT
ejpam-4983	167	14	1	1	NUM
ejpam-4983	167	15	,	,	PUNCT
ejpam-4983	167	16	1	1	NUM
ejpam-4983	167	17	,	,	PUNCT
ejpam-4983	167	18	.	.	PUNCT
ejpam-4983	167	19	.	.	PUNCT
ejpam-4983	167	20	.	.	PUNCT
ejpam-4983	168	1	,	,	PUNCT
ejpam-4983	168	2	1)x	1)x	X
ejpam-4983	168	3	)	)	PUNCT
ejpam-4983	168	4	=	=	SYM
ejpam-4983	168	5	a−1	a−1	PROPN
ejpam-4983	168	6	·	·	PUNCT
ejpam-4983	168	7	a	a	PRON
ejpam-4983	168	8	=	=	NOUN
ejpam-4983	168	9	1	1	X
ejpam-4983	168	10	.	.	PUNCT
ejpam-4983	169	1	it	it	PRON
ejpam-4983	169	2	follows	follow	VERB
ejpam-4983	169	3	that	that	SCONJ
ejpam-4983	169	4	a	a	DET
ejpam-4983	169	5	∈	∈	PROPN
ejpam-4983	169	6	an(fq	an(fq	PROPN
ejpam-4983	169	7	,	,	PUNCT
ejpam-4983	169	8	1	1	NUM
ejpam-4983	169	9	)	)	PUNCT
ejpam-4983	169	10	and	and	CCONJ
ejpam-4983	169	11	f(a	f(a	NOUN
ejpam-4983	169	12	)	)	PUNCT
ejpam-4983	169	13	=	=	SYM
ejpam-4983	169	14	f(diag(a−1	f(diag(a−1	PROPN
ejpam-4983	169	15	,	,	PUNCT
ejpam-4983	169	16	1	1	NUM
ejpam-4983	169	17	,	,	PUNCT
ejpam-4983	169	18	1	1	NUM
ejpam-4983	169	19	,	,	PUNCT
ejpam-4983	169	20	.	.	PUNCT
ejpam-4983	169	21	.	.	PUNCT
ejpam-4983	169	22	.	.	PUNCT
ejpam-4983	170	1	,	,	PUNCT
ejpam-4983	170	2	1)x	1)x	NUM
ejpam-4983	170	3	)	)	PUNCT
ejpam-4983	170	4	=	=	SYM
ejpam-4983	170	5	diag(a	diag(a	PROPN
ejpam-4983	170	6	,	,	PUNCT
ejpam-4983	170	7	1	1	NUM
ejpam-4983	170	8	,	,	PUNCT
ejpam-4983	170	9	1	1	NUM
ejpam-4983	170	10	,	,	PUNCT
ejpam-4983	170	11	.	.	PUNCT
ejpam-4983	170	12	.	.	PUNCT
ejpam-4983	170	13	.	.	PUNCT
ejpam-4983	171	1	,	,	PUNCT
ejpam-4983	171	2	1)diag(a−1	1)diag(a−1	NUM
ejpam-4983	171	3	,	,	PUNCT
ejpam-4983	171	4	1	1	NUM
ejpam-4983	171	5	,	,	PUNCT
ejpam-4983	171	6	1	1	NUM
ejpam-4983	171	7	,	,	PUNCT
ejpam-4983	171	8	.	.	PUNCT
ejpam-4983	171	9	.	.	PUNCT
ejpam-4983	171	10	.	.	PUNCT
ejpam-4983	172	1	,	,	PUNCT
ejpam-4983	172	2	1)x	1)x	X
ejpam-4983	172	3	=	=	SYM
ejpam-4983	172	4	x.	x.	NOUN
ejpam-4983	172	5	consequently	consequently	ADV
ejpam-4983	172	6	,	,	PUNCT
ejpam-4983	172	7	f	f	PROPN
ejpam-4983	172	8	is	be	AUX
ejpam-4983	172	9	surjective	surjective	ADJ
ejpam-4983	172	10	.	.	PUNCT
ejpam-4983	173	1	s.	s.	PROPN
ejpam-4983	173	2	jitman	jitman	PROPN
ejpam-4983	173	3	,	,	PUNCT
ejpam-4983	173	4	p.	p.	PROPN
ejpam-4983	173	5	modjam	modjam	PROPN
ejpam-4983	173	6	/	/	SYM
ejpam-4983	173	7	eur	eur	PROPN
ejpam-4983	173	8	.	.	PUNCT
ejpam-4983	174	1	j.	j.	PROPN
ejpam-4983	174	2	pure	pure	PROPN
ejpam-4983	174	3	appl	appl	PROPN
ejpam-4983	174	4	.	.	PROPN
ejpam-4983	174	5	math	math	PROPN
ejpam-4983	174	6	,	,	PUNCT
ejpam-4983	174	7	17	17	NUM
ejpam-4983	174	8	(	(	PUNCT
ejpam-4983	174	9	1	1	NUM
ejpam-4983	174	10	)	)	PUNCT
ejpam-4983	174	11	(	(	PUNCT
ejpam-4983	174	12	2024	2024	NUM
ejpam-4983	174	13	)	)	PUNCT
ejpam-4983	174	14	,	,	PUNCT
ejpam-4983	174	15	11	11	NUM
ejpam-4983	174	16	-	-	SYM
ejpam-4983	174	17	29	29	NUM
ejpam-4983	174	18	17	17	NUM
ejpam-4983	174	19	it	it	PRON
ejpam-4983	174	20	follows	follow	VERB
ejpam-4983	174	21	that	that	SCONJ
ejpam-4983	174	22	f	f	PROPN
ejpam-4983	174	23	is	be	AUX
ejpam-4983	174	24	a	a	DET
ejpam-4983	174	25	bijection	bijection	NOUN
ejpam-4983	174	26	from	from	ADP
ejpam-4983	174	27	an(fq	an(fq	PROPN
ejpam-4983	174	28	,	,	PUNCT
ejpam-4983	174	29	1	1	NUM
ejpam-4983	174	30	)	)	PUNCT
ejpam-4983	174	31	onto	onto	ADP
ejpam-4983	174	32	an(fq	an(fq	PROPN
ejpam-4983	174	33	,	,	PUNCT
ejpam-4983	174	34	a	a	PRON
ejpam-4983	174	35	)	)	PUNCT
ejpam-4983	174	36	,	,	PUNCT
ejpam-4983	174	37	and	and	CCONJ
ejpam-4983	174	38	hence	hence	ADV
ejpam-4983	174	39	,	,	PUNCT
ejpam-4983	174	40	|an(fq	|an(fq	PROPN
ejpam-4983	174	41	,	,	PUNCT
ejpam-4983	174	42	1)|	1)|	NUM
ejpam-4983	174	43	=	=	SYM
ejpam-4983	174	44	|an(fq	|an(fq	PROPN
ejpam-4983	174	45	,	,	PUNCT
ejpam-4983	174	46	a)|	a)|	PROPN
ejpam-4983	174	47	.	.	PUNCT
ejpam-4983	175	1	■	■	PUNCT
ejpam-4983	175	2	from	from	ADP
ejpam-4983	175	3	proposition	proposition	NOUN
ejpam-4983	175	4	2	2	NUM
ejpam-4983	175	5	,	,	PUNCT
ejpam-4983	175	6	we	we	PRON
ejpam-4983	175	7	have	have	VERB
ejpam-4983	175	8	|an(fq	|an(fq	NOUN
ejpam-4983	175	9	,	,	PUNCT
ejpam-4983	175	10	a)|	a)|	X
ejpam-4983	176	1	=	=	SYM
ejpam-4983	176	2	|an(fq	|an(fq	PROPN
ejpam-4983	176	3	,	,	PUNCT
ejpam-4983	176	4	1)|	1)|	NUM
ejpam-4983	176	5	=	=	SYM
ejpam-4983	176	6	|an(fq	|an(fq	PROPN
ejpam-4983	176	7	,	,	PUNCT
ejpam-4983	176	8	b)|	b)|	NOUN
ejpam-4983	176	9	for	for	ADP
ejpam-4983	176	10	all	all	DET
ejpam-4983	176	11	a	a	PRON
ejpam-4983	176	12	,	,	PUNCT
ejpam-4983	176	13	b	b	PROPN
ejpam-4983	176	14	∈	∈	PROPN
ejpam-4983	176	15	fq	fq	PROPN
ejpam-4983	176	16	\	\	PROPN
ejpam-4983	176	17	{	{	PUNCT
ejpam-4983	176	18	0	0	NUM
ejpam-4983	176	19	}	}	PUNCT
ejpam-4983	176	20	.	.	PUNCT
ejpam-4983	177	1	based	base	VERB
ejpam-4983	177	2	on	on	ADP
ejpam-4983	177	3	theorem	theorem	NOUN
ejpam-4983	177	4	1	1	NUM
ejpam-4983	177	5	and	and	CCONJ
ejpam-4983	177	6	proposition	proposition	NOUN
ejpam-4983	177	7	2	2	NUM
ejpam-4983	177	8	,	,	PUNCT
ejpam-4983	177	9	the	the	DET
ejpam-4983	177	10	next	next	ADJ
ejpam-4983	177	11	corollary	corollary	NOUN
ejpam-4983	177	12	can	can	AUX
ejpam-4983	177	13	be	be	AUX
ejpam-4983	177	14	derived	derive	VERB
ejpam-4983	177	15	.	.	PUNCT
ejpam-4983	178	1	corollary	corollary	ADJ
ejpam-4983	178	2	1	1	NUM
ejpam-4983	178	3	.	.	PUNCT
ejpam-4983	179	1	let	let	VERB
ejpam-4983	179	2	q	q	PART
ejpam-4983	179	3	be	be	AUX
ejpam-4983	179	4	a	a	DET
ejpam-4983	179	5	prime	prime	ADJ
ejpam-4983	179	6	power	power	NOUN
ejpam-4983	179	7	and	and	CCONJ
ejpam-4983	179	8	let	let	VERB
ejpam-4983	179	9	n	n	PRON
ejpam-4983	179	10	be	be	AUX
ejpam-4983	179	11	positive	positive	ADJ
ejpam-4983	179	12	integer	integer	NOUN
ejpam-4983	179	13	.	.	PUNCT
ejpam-4983	180	1	then	then	ADV
ejpam-4983	180	2	|an(fq	|an(fq	PROPN
ejpam-4983	180	3	,	,	PUNCT
ejpam-4983	180	4	a)|	a)|	X
ejpam-4983	181	1	=	=	SYM
ejpam-4983	181	2	q2n−3(q	q2n−3(q	NUM
ejpam-4983	181	3	−	−	PROPN
ejpam-4983	182	1	1)n−1(q	1)n−1(q	NUM
ejpam-4983	183	1	+	+	CCONJ
ejpam-4983	183	2	(	(	PUNCT
ejpam-4983	183	3	n−	n−	NOUN
ejpam-4983	183	4	1	1	NUM
ejpam-4983	183	5	)	)	PUNCT
ejpam-4983	183	6	)	)	PUNCT
ejpam-4983	183	7	for	for	ADP
ejpam-4983	183	8	all	all	DET
ejpam-4983	183	9	a	a	DET
ejpam-4983	183	10	∈	∈	PROPN
ejpam-4983	183	11	fq	fq	NOUN
ejpam-4983	183	12	\	\	PROPN
ejpam-4983	183	13	{	{	PUNCT
ejpam-4983	183	14	0	0	NUM
ejpam-4983	183	15	}	}	PUNCT
ejpam-4983	183	16	.	.	PUNCT
ejpam-4983	184	1	proof	proof	NOUN
ejpam-4983	184	2	.	.	PUNCT
ejpam-4983	185	1	from	from	ADP
ejpam-4983	185	2	proposition	proposition	NOUN
ejpam-4983	185	3	2	2	NUM
ejpam-4983	185	4	,	,	PUNCT
ejpam-4983	185	5	it	it	PRON
ejpam-4983	185	6	follows	follow	VERB
ejpam-4983	185	7	that	that	SCONJ
ejpam-4983	185	8	|an(fq	|an(fq	NOUN
ejpam-4983	185	9	,	,	PUNCT
ejpam-4983	185	10	a)|	a)|	X
ejpam-4983	186	1	=	=	SYM
ejpam-4983	186	2	|an(fq	|an(fq	PROPN
ejpam-4983	186	3	,	,	PUNCT
ejpam-4983	186	4	1)|	1)|	NUM
ejpam-4983	186	5	for	for	ADP
ejpam-4983	186	6	all	all	DET
ejpam-4983	186	7	a	a	DET
ejpam-4983	186	8	∈	∈	PROPN
ejpam-4983	186	9	fq	fq	NOUN
ejpam-4983	186	10	\	\	PROPN
ejpam-4983	186	11	{	{	PUNCT
ejpam-4983	186	12	0	0	NUM
ejpam-4983	186	13	}	}	PUNCT
ejpam-4983	186	14	.	.	PUNCT
ejpam-4983	187	1	since	since	SCONJ
ejpam-4983	187	2	ian(fq	ian(fq	NOUN
ejpam-4983	187	3	)	)	PUNCT
ejpam-4983	187	4	=	=	SYM
ejpam-4983	187	5	⋃	⋃	NOUN
ejpam-4983	187	6	a∈fq\{0	a∈fq\{0	PROPN
ejpam-4983	187	7	}	}	PUNCT
ejpam-4983	187	8	an(fq	an(fq	PROPN
ejpam-4983	187	9	,	,	PUNCT
ejpam-4983	187	10	a	a	PRON
ejpam-4983	187	11	)	)	PUNCT
ejpam-4983	187	12	is	be	AUX
ejpam-4983	187	13	a	a	DET
ejpam-4983	187	14	disjoint	disjoint	NOUN
ejpam-4983	187	15	union	union	NOUN
ejpam-4983	187	16	and	and	CCONJ
ejpam-4983	187	17	|fq	|fq	PRON
ejpam-4983	187	18	\	\	NOUN
ejpam-4983	187	19	{	{	PUNCT
ejpam-4983	187	20	0}|	0}|	X
ejpam-4983	187	21	=	=	SYM
ejpam-4983	187	22	q	q	NOUN
ejpam-4983	187	23	−	−	NOUN
ejpam-4983	187	24	1	1	NUM
ejpam-4983	187	25	,	,	PUNCT
ejpam-4983	187	26	it	it	PRON
ejpam-4983	187	27	follows	follow	VERB
ejpam-4983	187	28	that	that	SCONJ
ejpam-4983	187	29	|ian(fq)|	|ian(fq)|	NUM
ejpam-4983	187	30	=	=	SYM
ejpam-4983	187	31	|fq	|fq	PRON
ejpam-4983	187	32	\	\	NOUN
ejpam-4983	187	33	{	{	PUNCT
ejpam-4983	187	34	0}||an(fq	0}||an(fq	PROPN
ejpam-4983	187	35	,	,	PUNCT
ejpam-4983	187	36	1)|	1)|	NUM
ejpam-4983	187	37	=	=	SYM
ejpam-4983	187	38	(	(	PUNCT
ejpam-4983	187	39	q	q	NOUN
ejpam-4983	187	40	−	−	PROPN
ejpam-4983	187	41	1)|an(fq	1)|an(fq	PROPN
ejpam-4983	187	42	,	,	PUNCT
ejpam-4983	187	43	1)|	1)|	NUM
ejpam-4983	187	44	.	.	PUNCT
ejpam-4983	188	1	by	by	ADP
ejpam-4983	188	2	theorem	theorem	NOUN
ejpam-4983	188	3	1	1	NUM
ejpam-4983	188	4	and	and	CCONJ
ejpam-4983	188	5	proposition	proposition	NOUN
ejpam-4983	188	6	2	2	NUM
ejpam-4983	188	7	,	,	PUNCT
ejpam-4983	188	8	we	we	PRON
ejpam-4983	188	9	have	have	VERB
ejpam-4983	188	10	|an(fq	|an(fq	NOUN
ejpam-4983	188	11	,	,	PUNCT
ejpam-4983	188	12	a)|	a)|	X
ejpam-4983	189	1	=	=	SYM
ejpam-4983	189	2	|an(fq	|an(fq	PROPN
ejpam-4983	189	3	,	,	PUNCT
ejpam-4983	189	4	1)|	1)|	NUM
ejpam-4983	189	5	=	=	SYM
ejpam-4983	189	6	|ian(fq)|	|ian(fq)|	X
ejpam-4983	189	7	q	q	NOUN
ejpam-4983	189	8	−	−	PROPN
ejpam-4983	189	9	1	1	NUM
ejpam-4983	189	10	=	=	SYM
ejpam-4983	189	11	q2n−3(q	q2n−3(q	NUM
ejpam-4983	189	12	−	−	PROPN
ejpam-4983	189	13	1)n(q	1)n(q	NUM
ejpam-4983	189	14	+	+	CCONJ
ejpam-4983	189	15	(	(	PUNCT
ejpam-4983	189	16	n−	n−	NOUN
ejpam-4983	189	17	1	1	NUM
ejpam-4983	189	18	)	)	PUNCT
ejpam-4983	189	19	)	)	PUNCT
ejpam-4983	190	1	q	q	NOUN
ejpam-4983	191	1	−	−	NOUN
ejpam-4983	191	2	1	1	NUM
ejpam-4983	191	3	=	=	SYM
ejpam-4983	191	4	q2n−3(q	q2n−3(q	NUM
ejpam-4983	191	5	−	−	PROPN
ejpam-4983	191	6	1)n−1(q	1)n−1(q	NUM
ejpam-4983	192	1	+	+	CCONJ
ejpam-4983	192	2	(	(	PUNCT
ejpam-4983	192	3	n−	n−	NOUN
ejpam-4983	192	4	1	1	NUM
ejpam-4983	192	5	)	)	PUNCT
ejpam-4983	192	6	)	)	PUNCT
ejpam-4983	192	7	.	.	PUNCT
ejpam-4983	193	1	this	this	PRON
ejpam-4983	193	2	completes	complete	VERB
ejpam-4983	193	3	the	the	DET
ejpam-4983	193	4	proof	proof	NOUN
ejpam-4983	193	5	.	.	PUNCT
ejpam-4983	194	1	■	■	PUNCT
ejpam-4983	194	2	we	we	PRON
ejpam-4983	194	3	note	note	VERB
ejpam-4983	194	4	that	that	SCONJ
ejpam-4983	194	5	|an(fq)|	|an(fq)|	NOUN
ejpam-4983	194	6	=	=	PUNCT
ejpam-4983	194	7	q3n−2	q3n−2	PROPN
ejpam-4983	194	8	and	and	CCONJ
ejpam-4983	194	9	|ian(fq)|	|ian(fq)|	NUM
ejpam-4983	194	10	=	=	SYM
ejpam-4983	194	11	q2n−3(q	q2n−3(q	NUM
ejpam-4983	194	12	−	−	PROPN
ejpam-4983	194	13	1)n(q	1)n(q	NUM
ejpam-4983	194	14	+	+	CCONJ
ejpam-4983	194	15	(	(	PUNCT
ejpam-4983	194	16	n−	n−	NOUN
ejpam-4983	194	17	1	1	NUM
ejpam-4983	194	18	)	)	PUNCT
ejpam-4983	194	19	)	)	PUNCT
ejpam-4983	194	20	given	give	VERB
ejpam-4983	194	21	in	in	ADP
ejpam-4983	194	22	(	(	PUNCT
ejpam-4983	194	23	1	1	NUM
ejpam-4983	194	24	)	)	PUNCT
ejpam-4983	194	25	and	and	CCONJ
ejpam-4983	194	26	theorem	theorem	VERB
ejpam-4983	194	27	1	1	NUM
ejpam-4983	194	28	.	.	PUNCT
ejpam-4983	195	1	the	the	DET
ejpam-4983	195	2	number	number	NOUN
ejpam-4983	195	3	|an(fq	|an(fq	NOUN
ejpam-4983	195	4	,	,	PUNCT
ejpam-4983	195	5	0)|	0)|	NOUN
ejpam-4983	195	6	=	=	SYM
ejpam-4983	195	7	|an(fq)|	|an(fq)|	NOUN
ejpam-4983	195	8	−	−	PROPN
ejpam-4983	195	9	|ian(fq)|	|ian(fq)|	NUM
ejpam-4983	195	10	of	of	ADP
ejpam-4983	195	11	n	n	NUM
ejpam-4983	195	12	×	×	NOUN
ejpam-4983	195	13	n	n	CCONJ
ejpam-4983	195	14	singular	singular	ADJ
ejpam-4983	195	15	arrowhead	arrowhead	NOUN
ejpam-4983	195	16	matrices	matrix	NOUN
ejpam-4983	195	17	over	over	ADP
ejpam-4983	195	18	fq	fq	PROPN
ejpam-4983	195	19	follows	follow	VERB
ejpam-4983	195	20	in	in	ADP
ejpam-4983	195	21	the	the	DET
ejpam-4983	195	22	next	next	ADJ
ejpam-4983	195	23	corollary	corollary	NOUN
ejpam-4983	195	24	.	.	PUNCT
ejpam-4983	196	1	corollary	corollary	ADJ
ejpam-4983	196	2	2	2	NUM
ejpam-4983	196	3	.	.	PUNCT
ejpam-4983	197	1	let	let	VERB
ejpam-4983	197	2	q	q	PART
ejpam-4983	197	3	be	be	AUX
ejpam-4983	197	4	a	a	DET
ejpam-4983	197	5	prime	prime	ADJ
ejpam-4983	197	6	power	power	NOUN
ejpam-4983	197	7	.	.	PUNCT
ejpam-4983	198	1	then	then	ADV
ejpam-4983	198	2	|an(fq	|an(fq	PROPN
ejpam-4983	198	3	,	,	PUNCT
ejpam-4983	198	4	0)|	0)|	NOUN
ejpam-4983	198	5	=	=	SYM
ejpam-4983	198	6	q3n−2	q3n−2	PROPN
ejpam-4983	198	7	−	−	PROPN
ejpam-4983	198	8	q2n−3(q	q2n−3(q	NUM
ejpam-4983	198	9	−	−	PROPN
ejpam-4983	198	10	1)n(q	1)n(q	NUM
ejpam-4983	198	11	+	+	CCONJ
ejpam-4983	198	12	(	(	PUNCT
ejpam-4983	198	13	n−	n−	NOUN
ejpam-4983	198	14	1	1	NUM
ejpam-4983	198	15	)	)	PUNCT
ejpam-4983	198	16	)	)	PUNCT
ejpam-4983	198	17	for	for	ADP
ejpam-4983	198	18	all	all	DET
ejpam-4983	198	19	positive	positive	ADJ
ejpam-4983	198	20	integers	integer	NOUN
ejpam-4983	198	21	n.	n.	PROPN
ejpam-4983	198	22	s.	s.	PROPN
ejpam-4983	198	23	jitman	jitman	PROPN
ejpam-4983	198	24	,	,	PUNCT
ejpam-4983	198	25	p.	p.	PROPN
ejpam-4983	198	26	modjam	modjam	PROPN
ejpam-4983	198	27	/	/	SYM
ejpam-4983	198	28	eur	eur	PROPN
ejpam-4983	198	29	.	.	PUNCT
ejpam-4983	199	1	j.	j.	PROPN
ejpam-4983	199	2	pure	pure	PROPN
ejpam-4983	199	3	appl	appl	PROPN
ejpam-4983	199	4	.	.	PROPN
ejpam-4983	199	5	math	math	PROPN
ejpam-4983	199	6	,	,	PUNCT
ejpam-4983	199	7	17	17	NUM
ejpam-4983	199	8	(	(	PUNCT
ejpam-4983	199	9	1	1	NUM
ejpam-4983	199	10	)	)	PUNCT
ejpam-4983	199	11	(	(	PUNCT
ejpam-4983	199	12	2024	2024	NUM
ejpam-4983	199	13	)	)	PUNCT
ejpam-4983	199	14	,	,	PUNCT
ejpam-4983	199	15	11	11	NUM
ejpam-4983	199	16	-	-	SYM
ejpam-4983	199	17	29	29	NUM
ejpam-4983	199	18	18	18	NUM
ejpam-4983	199	19	3	3	NUM
ejpam-4983	199	20	.	.	PUNCT
ejpam-4983	199	21	determinants	determinant	NOUN
ejpam-4983	199	22	of	of	ADP
ejpam-4983	199	23	arrowhead	arrowhead	NOUN
ejpam-4983	199	24	matrices	matrix	NOUN
ejpam-4983	199	25	over	over	ADP
ejpam-4983	199	26	fccrs	fccr	NOUN
ejpam-4983	199	27	in	in	ADP
ejpam-4983	199	28	this	this	DET
ejpam-4983	199	29	section	section	NOUN
ejpam-4983	200	1	,	,	PUNCT
ejpam-4983	200	2	the	the	DET
ejpam-4983	200	3	enumeration	enumeration	NOUN
ejpam-4983	200	4	of	of	ADP
ejpam-4983	200	5	n×n	n×n	PROPN
ejpam-4983	200	6	arrowhead	arrowhead	NOUN
ejpam-4983	200	7	matrices	matrix	NOUN
ejpam-4983	200	8	with	with	ADP
ejpam-4983	200	9	prescribed	prescribe	VERB
ejpam-4983	200	10	determinant	determinant	ADJ
ejpam-4983	200	11	over	over	ADP
ejpam-4983	200	12	r	r	NOUN
ejpam-4983	200	13	is	be	AUX
ejpam-4983	200	14	discussed	discuss	VERB
ejpam-4983	200	15	.	.	PUNCT
ejpam-4983	201	1	the	the	DET
ejpam-4983	201	2	number	number	NOUN
ejpam-4983	201	3	of	of	ADP
ejpam-4983	201	4	n	n	NUM
ejpam-4983	201	5	×	×	NOUN
ejpam-4983	201	6	n	n	CCONJ
ejpam-4983	201	7	non	non	ADJ
ejpam-4983	201	8	-	-	ADJ
ejpam-4983	201	9	singular	singular	ADJ
ejpam-4983	201	10	(	(	PUNCT
ejpam-4983	201	11	resp	resp	NOUN
ejpam-4983	201	12	.	.	PUNCT
ejpam-4983	201	13	,	,	PUNCT
ejpam-4983	201	14	singular	singular	PROPN
ejpam-4983	201	15	)	)	PUNCT
ejpam-4983	201	16	arrowhead	arrowhead	NOUN
ejpam-4983	201	17	matrices	matrix	NOUN
ejpam-4983	201	18	over	over	ADP
ejpam-4983	201	19	r	r	NOUN
ejpam-4983	201	20	is	be	AUX
ejpam-4983	201	21	presented	present	VERB
ejpam-4983	201	22	.	.	PUNCT
ejpam-4983	202	1	for	for	ADP
ejpam-4983	202	2	non	non	ADJ
ejpam-4983	202	3	-	-	ADJ
ejpam-4983	202	4	singular	singular	ADJ
ejpam-4983	202	5	arrowhead	arrowhead	NOUN
ejpam-4983	202	6	matrices	matrix	NOUN
ejpam-4983	202	7	,	,	PUNCT
ejpam-4983	202	8	the	the	DET
ejpam-4983	202	9	number	number	NOUN
ejpam-4983	202	10	of	of	ADP
ejpam-4983	202	11	n×n	n×n	PROPN
ejpam-4983	202	12	arrowhead	arrowhead	NOUN
ejpam-4983	202	13	matrices	matrix	NOUN
ejpam-4983	202	14	over	over	ADP
ejpam-4983	202	15	r	r	NOUN
ejpam-4983	202	16	with	with	ADP
ejpam-4983	202	17	a	a	DET
ejpam-4983	202	18	given	give	VERB
ejpam-4983	202	19	determinant	determinant	NOUN
ejpam-4983	202	20	is	be	AUX
ejpam-4983	202	21	established	establish	VERB
ejpam-4983	202	22	.	.	PUNCT
ejpam-4983	203	1	for	for	ADP
ejpam-4983	203	2	singular	singular	ADJ
ejpam-4983	203	3	arrowhead	arrowhead	NOUN
ejpam-4983	203	4	matrices	matrix	NOUN
ejpam-4983	203	5	,	,	PUNCT
ejpam-4983	203	6	bounds	bound	VERB
ejpam-4983	203	7	on	on	ADP
ejpam-4983	203	8	the	the	DET
ejpam-4983	203	9	number	number	NOUN
ejpam-4983	203	10	of	of	ADP
ejpam-4983	203	11	n	n	NUM
ejpam-4983	203	12	×	×	NOUN
ejpam-4983	203	13	n	n	CCONJ
ejpam-4983	203	14	arrowhead	arrowhead	NOUN
ejpam-4983	203	15	matrices	matrix	NOUN
ejpam-4983	203	16	with	with	ADP
ejpam-4983	203	17	a	a	DET
ejpam-4983	203	18	fixed	fix	VERB
ejpam-4983	203	19	determinant	determinant	ADJ
ejpam-4983	203	20	over	over	ADP
ejpam-4983	203	21	r	r	NOUN
ejpam-4983	203	22	are	be	AUX
ejpam-4983	203	23	presented	present	VERB
ejpam-4983	203	24	in	in	ADP
ejpam-4983	203	25	some	some	DET
ejpam-4983	203	26	cases	case	NOUN
ejpam-4983	203	27	.	.	PUNCT
ejpam-4983	204	1	to	to	PART
ejpam-4983	204	2	be	be	AUX
ejpam-4983	204	3	self	self	NOUN
ejpam-4983	204	4	-	-	PUNCT
ejpam-4983	204	5	contained	contain	VERB
ejpam-4983	204	6	,	,	PUNCT
ejpam-4983	204	7	a	a	DET
ejpam-4983	204	8	brief	brief	ADJ
ejpam-4983	204	9	information	information	NOUN
ejpam-4983	204	10	of	of	ADP
ejpam-4983	204	11	a	a	DET
ejpam-4983	204	12	fccr	fccr	NOUN
ejpam-4983	204	13	is	be	AUX
ejpam-4983	204	14	recalled	recall	VERB
ejpam-4983	204	15	.	.	PUNCT
ejpam-4983	205	1	the	the	DET
ejpam-4983	205	2	reader	reader	NOUN
ejpam-4983	205	3	may	may	AUX
ejpam-4983	205	4	refer	refer	VERB
ejpam-4983	205	5	to	to	ADP
ejpam-4983	205	6	[	[	X
ejpam-4983	205	7	5	5	NUM
ejpam-4983	205	8	]	]	PUNCT
ejpam-4983	205	9	,	,	PUNCT
ejpam-4983	205	10	[	[	X
ejpam-4983	205	11	6	6	NUM
ejpam-4983	205	12	]	]	PUNCT
ejpam-4983	205	13	,	,	PUNCT
ejpam-4983	205	14	and	and	CCONJ
ejpam-4983	205	15	[	[	X
ejpam-4983	205	16	7	7	X
ejpam-4983	205	17	]	]	PUNCT
ejpam-4983	205	18	for	for	ADP
ejpam-4983	205	19	more	more	ADJ
ejpam-4983	205	20	details	detail	NOUN
ejpam-4983	205	21	.	.	PUNCT
ejpam-4983	206	1	a	a	DET
ejpam-4983	206	2	ring	ring	NOUN
ejpam-4983	206	3	r	r	NOUN
ejpam-4983	206	4	with	with	ADP
ejpam-4983	206	5	identity	identity	NOUN
ejpam-4983	206	6	1	1	NUM
ejpam-4983	206	7	̸=	̸=	PROPN
ejpam-4983	206	8	0	0	NUM
ejpam-4983	206	9	is	be	AUX
ejpam-4983	206	10	called	call	VERB
ejpam-4983	206	11	a	a	DET
ejpam-4983	206	12	finite	finite	ADJ
ejpam-4983	206	13	commutative	commutative	ADJ
ejpam-4983	206	14	chain	chain	NOUN
ejpam-4983	206	15	ring	ring	NOUN
ejpam-4983	206	16	(	(	PUNCT
ejpam-4983	206	17	fccr	fccr	PROPN
ejpam-4983	206	18	)	)	PUNCT
ejpam-4983	206	19	if	if	SCONJ
ejpam-4983	206	20	it	it	PRON
ejpam-4983	206	21	is	be	AUX
ejpam-4983	206	22	finite	finite	ADJ
ejpam-4983	206	23	,	,	PUNCT
ejpam-4983	206	24	commutative	commutative	ADJ
ejpam-4983	206	25	,	,	PUNCT
ejpam-4983	206	26	and	and	CCONJ
ejpam-4983	206	27	its	its	PRON
ejpam-4983	206	28	ideals	ideal	NOUN
ejpam-4983	206	29	are	be	AUX
ejpam-4983	206	30	linearly	linearly	ADV
ejpam-4983	206	31	ordered	order	VERB
ejpam-4983	206	32	by	by	ADP
ejpam-4983	206	33	inclusion	inclusion	NOUN
ejpam-4983	206	34	.	.	PUNCT
ejpam-4983	207	1	let	let	VERB
ejpam-4983	207	2	r	r	PRON
ejpam-4983	207	3	be	be	AUX
ejpam-4983	207	4	a	a	DET
ejpam-4983	207	5	fccr	fccr	NOUN
ejpam-4983	207	6	whose	whose	DET
ejpam-4983	207	7	maximal	maximal	ADJ
ejpam-4983	207	8	ideal	ideal	NOUN
ejpam-4983	207	9	is	be	AUX
ejpam-4983	207	10	generated	generate	VERB
ejpam-4983	207	11	by	by	ADP
ejpam-4983	207	12	γ	γ	PROPN
ejpam-4983	207	13	.	.	PROPN
ejpam-4983	207	14	then	then	ADV
ejpam-4983	207	15	the	the	DET
ejpam-4983	207	16	ideals	ideal	NOUN
ejpam-4983	207	17	in	in	ADP
ejpam-4983	207	18	r	r	NOUN
ejpam-4983	207	19	are	be	AUX
ejpam-4983	207	20	of	of	ADP
ejpam-4983	207	21	the	the	DET
ejpam-4983	207	22	form	form	NOUN
ejpam-4983	207	23	r	r	NOUN
ejpam-4983	207	24	⊋	⊋	ADV
ejpam-4983	207	25	γr	γr	X
ejpam-4983	207	26	⊋	⊋	ADV
ejpam-4983	207	27	γ2r	γ2r	PROPN
ejpam-4983	207	28	⊋	⊋	X
ejpam-4983	207	29	·	·	PUNCT
ejpam-4983	207	30	·	·	PUNCT
ejpam-4983	207	31	·	·	PUNCT
ejpam-4983	208	1	⊋	⊋	PROPN
ejpam-4983	208	2	γe−1r	γe−1r	NUM
ejpam-4983	208	3	⊋	⊋	PROPN
ejpam-4983	208	4	γer	γer	NOUN
ejpam-4983	208	5	=	=	PUNCT
ejpam-4983	208	6	{	{	PUNCT
ejpam-4983	208	7	0	0	NUM
ejpam-4983	208	8	}	}	PUNCT
ejpam-4983	208	9	,	,	PUNCT
ejpam-4983	208	10	for	for	ADP
ejpam-4983	208	11	some	some	DET
ejpam-4983	208	12	positive	positive	ADJ
ejpam-4983	208	13	integer	integer	NOUN
ejpam-4983	208	14	e.	e.	PROPN
ejpam-4983	208	15	the	the	DET
ejpam-4983	208	16	smallest	small	ADJ
ejpam-4983	208	17	positive	positive	ADJ
ejpam-4983	208	18	integer	integer	NOUN
ejpam-4983	208	19	e	e	NOUN
ejpam-4983	208	20	such	such	ADJ
ejpam-4983	208	21	that	that	SCONJ
ejpam-4983	208	22	γe	γe	X
ejpam-4983	208	23	=	=	SYM
ejpam-4983	208	24	0	0	PROPN
ejpam-4983	208	25	is	be	AUX
ejpam-4983	208	26	called	call	VERB
ejpam-4983	208	27	the	the	DET
ejpam-4983	208	28	nilpotency	nilpotency	NOUN
ejpam-4983	208	29	index	index	NOUN
ejpam-4983	208	30	of	of	ADP
ejpam-4983	208	31	r.	r.	PROPN
ejpam-4983	208	32	the	the	DET
ejpam-4983	208	33	quotient	quotient	NOUN
ejpam-4983	208	34	ring	ring	NOUN
ejpam-4983	208	35	r	r	PROPN
ejpam-4983	208	36	/	/	SYM
ejpam-4983	208	37	γr	γr	PROPN
ejpam-4983	208	38	is	be	AUX
ejpam-4983	208	39	a	a	DET
ejpam-4983	208	40	finite	finite	ADJ
ejpam-4983	208	41	field	field	NOUN
ejpam-4983	208	42	and	and	CCONJ
ejpam-4983	208	43	it	it	PRON
ejpam-4983	208	44	is	be	AUX
ejpam-4983	208	45	referred	refer	VERB
ejpam-4983	208	46	to	to	ADP
ejpam-4983	208	47	as	as	ADP
ejpam-4983	208	48	the	the	DET
ejpam-4983	208	49	residue	residue	NOUN
ejpam-4983	208	50	field	field	NOUN
ejpam-4983	208	51	of	of	ADP
ejpam-4983	208	52	r.	r.	PROPN
ejpam-4983	208	53	from	from	ADP
ejpam-4983	208	54	[	[	X
ejpam-4983	208	55	6	6	NUM
ejpam-4983	208	56	]	]	PUNCT
ejpam-4983	208	57	and	and	CCONJ
ejpam-4983	208	58	[	[	X
ejpam-4983	208	59	7	7	NUM
ejpam-4983	208	60	]	]	PUNCT
ejpam-4983	208	61	,	,	PUNCT
ejpam-4983	208	62	useful	useful	ADJ
ejpam-4983	208	63	properties	property	NOUN
ejpam-4983	208	64	of	of	ADP
ejpam-4983	208	65	a	a	DET
ejpam-4983	208	66	fccr	fccr	NOUN
ejpam-4983	208	67	(	(	PUNCT
ejpam-4983	208	68	cf	cf	NOUN
ejpam-4983	208	69	.	.	PUNCT
ejpam-4983	209	1	[	[	X
ejpam-4983	209	2	3	3	NUM
ejpam-4983	209	3	]	]	PUNCT
ejpam-4983	209	4	)	)	PUNCT
ejpam-4983	209	5	are	be	AUX
ejpam-4983	209	6	summarized	summarize	VERB
ejpam-4983	209	7	in	in	ADP
ejpam-4983	209	8	the	the	DET
ejpam-4983	209	9	next	next	ADJ
ejpam-4983	209	10	lemma	lemma	PROPN
ejpam-4983	209	11	.	.	PUNCT
ejpam-4983	210	1	lemma	lemma	PROPN
ejpam-4983	210	2	1	1	X
ejpam-4983	210	3	.	.	PUNCT
ejpam-4983	211	1	let	let	VERB
ejpam-4983	211	2	r	r	PRON
ejpam-4983	211	3	be	be	AUX
ejpam-4983	211	4	a	a	DET
ejpam-4983	211	5	fccr	fccr	NOUN
ejpam-4983	211	6	of	of	ADP
ejpam-4983	211	7	nilpotency	nilpotency	NOUN
ejpam-4983	211	8	index	index	NOUN
ejpam-4983	211	9	e	e	NOUN
ejpam-4983	211	10	and	and	CCONJ
ejpam-4983	211	11	let	let	VERB
ejpam-4983	211	12	γ	γ	NOUN
ejpam-4983	211	13	be	be	AUX
ejpam-4983	211	14	a	a	DET
ejpam-4983	211	15	generator	generator	NOUN
ejpam-4983	211	16	of	of	ADP
ejpam-4983	211	17	its	its	PRON
ejpam-4983	211	18	maximal	maximal	ADJ
ejpam-4983	211	19	ideal	ideal	NOUN
ejpam-4983	211	20	.	.	PUNCT
ejpam-4983	212	1	let	let	VERB
ejpam-4983	212	2	v	v	PRON
ejpam-4983	212	3	⊆	⊆	NUM
ejpam-4983	212	4	r	r	NOUN
ejpam-4983	212	5	be	be	AUX
ejpam-4983	212	6	a	a	DET
ejpam-4983	212	7	set	set	NOUN
ejpam-4983	212	8	of	of	ADP
ejpam-4983	212	9	representatives	representative	NOUN
ejpam-4983	212	10	for	for	ADP
ejpam-4983	212	11	the	the	DET
ejpam-4983	212	12	equivalence	equivalence	NOUN
ejpam-4983	212	13	classes	class	NOUN
ejpam-4983	212	14	of	of	ADP
ejpam-4983	212	15	r	r	NOUN
ejpam-4983	212	16	under	under	ADP
ejpam-4983	212	17	congruence	congruence	NOUN
ejpam-4983	212	18	modulo	modulo	PROPN
ejpam-4983	212	19	γ	γ	PROPN
ejpam-4983	212	20	.	.	PROPN
ejpam-4983	212	21	assume	assume	VERB
ejpam-4983	212	22	that	that	SCONJ
ejpam-4983	212	23	the	the	DET
ejpam-4983	212	24	residue	residue	NOUN
ejpam-4983	212	25	field	field	NOUN
ejpam-4983	212	26	r/⟨γ⟩	r/⟨γ⟩	PRON
ejpam-4983	212	27	∼=	∼=	NOUN
ejpam-4983	212	28	fq	fq	NOUN
ejpam-4983	212	29	for	for	ADP
ejpam-4983	212	30	some	some	DET
ejpam-4983	212	31	prime	prime	ADJ
ejpam-4983	212	32	power	power	NOUN
ejpam-4983	212	33	q.	q.	PROPN
ejpam-4983	212	34	then	then	ADV
ejpam-4983	212	35	the	the	DET
ejpam-4983	212	36	following	follow	VERB
ejpam-4983	212	37	statements	statement	NOUN
ejpam-4983	212	38	hold	hold	VERB
ejpam-4983	212	39	.	.	PUNCT
ejpam-4983	213	1	1	1	NUM
ejpam-4983	213	2	)	)	PUNCT
ejpam-4983	213	3	for	for	ADP
ejpam-4983	213	4	each	each	DET
ejpam-4983	213	5	r	r	NOUN
ejpam-4983	213	6	∈	∈	NOUN
ejpam-4983	213	7	r	r	NOUN
ejpam-4983	213	8	,	,	PUNCT
ejpam-4983	213	9	there	there	PRON
ejpam-4983	213	10	exist	exist	VERB
ejpam-4983	213	11	unique	unique	ADJ
ejpam-4983	213	12	a0	a0	NOUN
ejpam-4983	213	13	,	,	PUNCT
ejpam-4983	213	14	a1	a1	NOUN
ejpam-4983	213	15	,	,	PUNCT
ejpam-4983	213	16	.	.	PUNCT
ejpam-4983	213	17	.	.	PUNCT
ejpam-4983	214	1	.	.	PUNCT
ejpam-4983	215	1	ae−1	ae−1	PROPN
ejpam-4983	215	2	∈	∈	PROPN
ejpam-4983	215	3	v	v	ADP
ejpam-4983	215	4	such	such	ADJ
ejpam-4983	215	5	that	that	DET
ejpam-4983	215	6	r	r	NOUN
ejpam-4983	215	7	=	=	SYM
ejpam-4983	215	8	a0	a0	NOUN
ejpam-4983	215	9	+	+	CCONJ
ejpam-4983	215	10	a1γ	a1γ	ADV
ejpam-4983	215	11	+	+	CCONJ
ejpam-4983	215	12	·	·	PUNCT
ejpam-4983	215	13	·	·	PUNCT
ejpam-4983	215	14	·	·	PUNCT
ejpam-4983	215	15	+	+	CCONJ
ejpam-4983	216	1	ae−1γ	ae−1γ	NUM
ejpam-4983	216	2	e−1	e−1	PROPN
ejpam-4983	216	3	.	.	NOUN
ejpam-4983	216	4	2	2	NUM
ejpam-4983	216	5	)	)	PUNCT
ejpam-4983	216	6	|v	|v	NOUN
ejpam-4983	216	7	|	|	ADV
ejpam-4983	216	8	=	=	PUNCT
ejpam-4983	216	9	q.	q.	NOUN
ejpam-4983	216	10	3	3	NUM
ejpam-4983	216	11	)	)	PUNCT
ejpam-4983	216	12	|γjr|	|γjr|	PROPN
ejpam-4983	216	13	=	=	SYM
ejpam-4983	217	1	qe−j	qe−j	ADV
ejpam-4983	217	2	for	for	ADP
ejpam-4983	217	3	all	all	PRON
ejpam-4983	217	4	0	0	NUM
ejpam-4983	217	5	≤	≤	NUM
ejpam-4983	217	6	j	j	PROPN
ejpam-4983	217	7	≤	≤	PROPN
ejpam-4983	217	8	e.	e.	PROPN
ejpam-4983	217	9	4	4	NUM
ejpam-4983	217	10	)	)	PUNCT
ejpam-4983	217	11	u(r	u(r	NOUN
ejpam-4983	217	12	)	)	PUNCT
ejpam-4983	218	1	=	=	SYM
ejpam-4983	218	2	{	{	PUNCT
ejpam-4983	218	3	a+	a+	PUNCT
ejpam-4983	218	4	γb	γb	NOUN
ejpam-4983	218	5	|	|	ADV
ejpam-4983	218	6	a	a	DET
ejpam-4983	218	7	∈	∈	NOUN
ejpam-4983	218	8	v	v	ADP
ejpam-4983	218	9	\	\	NOUN
ejpam-4983	218	10	{	{	PUNCT
ejpam-4983	218	11	0	0	NUM
ejpam-4983	218	12	}	}	PUNCT
ejpam-4983	218	13	and	and	CCONJ
ejpam-4983	219	1	b	b	X
ejpam-4983	219	2	∈	∈	NOUN
ejpam-4983	219	3	r	r	NOUN
ejpam-4983	219	4	}	}	PUNCT
ejpam-4983	219	5	.	.	PUNCT
ejpam-4983	220	1	5	5	X
ejpam-4983	220	2	)	)	PUNCT
ejpam-4983	220	3	|u(r)|	|u(r)|	NOUN
ejpam-4983	220	4	=	=	SYM
ejpam-4983	220	5	(	(	PUNCT
ejpam-4983	220	6	q	q	PROPN
ejpam-4983	220	7	−	−	PROPN
ejpam-4983	220	8	1)qe−1	1)qe−1	PROPN
ejpam-4983	220	9	.	.	PROPN
ejpam-4983	220	10	6	6	NUM
ejpam-4983	220	11	)	)	PUNCT
ejpam-4983	220	12	for	for	ADP
ejpam-4983	220	13	each	each	DET
ejpam-4983	220	14	0	0	NUM
ejpam-4983	220	15	≤	≤	NUM
ejpam-4983	220	16	i	i	PRON
ejpam-4983	220	17	≤	≤	NOUN
ejpam-4983	220	18	e	e	X
ejpam-4983	220	19	,	,	PUNCT
ejpam-4983	220	20	r	r	NOUN
ejpam-4983	220	21	/	/	SYM
ejpam-4983	220	22	γir	γir	CCONJ
ejpam-4983	220	23	is	be	AUX
ejpam-4983	220	24	a	a	DET
ejpam-4983	220	25	fccr	fccr	NOUN
ejpam-4983	220	26	of	of	ADP
ejpam-4983	220	27	nilpotency	nilpotency	NOUN
ejpam-4983	220	28	index	index	NOUN
ejpam-4983	220	29	i	i	PRON
ejpam-4983	220	30	and	and	CCONJ
ejpam-4983	220	31	residue	residue	NOUN
ejpam-4983	220	32	field	field	NOUN
ejpam-4983	220	33	fq	fq	PROPN
ejpam-4983	220	34	.	.	PROPN
ejpam-4983	220	35	3.1	3.1	NUM
ejpam-4983	220	36	.	.	PUNCT
ejpam-4983	221	1	non	non	ADJ
ejpam-4983	221	2	-	-	ADJ
ejpam-4983	221	3	singular	singular	ADJ
ejpam-4983	221	4	arrowhead	arrowhead	NOUN
ejpam-4983	221	5	matrices	matrix	NOUN
ejpam-4983	221	6	over	over	ADP
ejpam-4983	221	7	fccrs	fccr	NOUN
ejpam-4983	221	8	first	first	ADV
ejpam-4983	221	9	,	,	PUNCT
ejpam-4983	221	10	the	the	DET
ejpam-4983	221	11	number	number	NOUN
ejpam-4983	221	12	of	of	ADP
ejpam-4983	221	13	n	n	NUM
ejpam-4983	221	14	×	×	NOUN
ejpam-4983	221	15	n	n	CCONJ
ejpam-4983	221	16	non	non	ADJ
ejpam-4983	221	17	-	-	ADJ
ejpam-4983	221	18	singular	singular	ADJ
ejpam-4983	221	19	arrowhead	arrowhead	NOUN
ejpam-4983	221	20	matrices	matrix	NOUN
ejpam-4983	221	21	over	over	ADP
ejpam-4983	221	22	a	a	DET
ejpam-4983	221	23	fccrs	fccrs	NOUN
ejpam-4983	221	24	r	r	NOUN
ejpam-4983	221	25	is	be	AUX
ejpam-4983	221	26	presented	present	VERB
ejpam-4983	221	27	.	.	PUNCT
ejpam-4983	222	1	then	then	ADV
ejpam-4983	222	2	it	it	PRON
ejpam-4983	222	3	is	be	AUX
ejpam-4983	222	4	followed	follow	VERB
ejpam-4983	222	5	by	by	ADP
ejpam-4983	222	6	the	the	DET
ejpam-4983	222	7	number	number	NOUN
ejpam-4983	222	8	of	of	ADP
ejpam-4983	222	9	n	n	NUM
ejpam-4983	222	10	×	×	NOUN
ejpam-4983	222	11	n	n	CCONJ
ejpam-4983	222	12	arrowhead	arrowhead	NOUN
ejpam-4983	222	13	matrices	matrix	NOUN
ejpam-4983	222	14	over	over	ADP
ejpam-4983	222	15	r	r	NOUN
ejpam-4983	222	16	with	with	ADP
ejpam-4983	222	17	prescribed	prescribe	VERB
ejpam-4983	222	18	determinant	determinant	ADJ
ejpam-4983	222	19	in	in	ADP
ejpam-4983	222	20	u(r	u(r	NOUN
ejpam-4983	222	21	)	)	PUNCT
ejpam-4983	222	22	.	.	PUNCT
ejpam-4983	223	1	an	an	DET
ejpam-4983	223	2	explicit	explicit	ADJ
ejpam-4983	223	3	formula	formula	NOUN
ejpam-4983	223	4	for	for	ADP
ejpam-4983	223	5	the	the	DET
ejpam-4983	223	6	number	number	NOUN
ejpam-4983	223	7	|ian(r)|	|ian(r)|	NUM
ejpam-4983	223	8	of	of	ADP
ejpam-4983	223	9	n	n	NUM
ejpam-4983	223	10	×	×	NOUN
ejpam-4983	223	11	n	n	CCONJ
ejpam-4983	223	12	non	non	ADJ
ejpam-4983	223	13	-	-	ADJ
ejpam-4983	223	14	singular	singular	ADJ
ejpam-4983	223	15	matrices	matrix	NOUN
ejpam-4983	223	16	is	be	AUX
ejpam-4983	223	17	given	give	VERB
ejpam-4983	223	18	in	in	ADP
ejpam-4983	223	19	the	the	DET
ejpam-4983	223	20	following	follow	VERB
ejpam-4983	223	21	theorem	theorem	PROPN
ejpam-4983	223	22	.	.	PUNCT
ejpam-4983	224	1	s.	s.	PROPN
ejpam-4983	224	2	jitman	jitman	PROPN
ejpam-4983	224	3	,	,	PUNCT
ejpam-4983	224	4	p.	p.	PROPN
ejpam-4983	224	5	modjam	modjam	PROPN
ejpam-4983	224	6	/	/	SYM
ejpam-4983	224	7	eur	eur	PROPN
ejpam-4983	224	8	.	.	PUNCT
ejpam-4983	225	1	j.	j.	PROPN
ejpam-4983	225	2	pure	pure	PROPN
ejpam-4983	225	3	appl	appl	PROPN
ejpam-4983	225	4	.	.	PROPN
ejpam-4983	225	5	math	math	PROPN
ejpam-4983	225	6	,	,	PUNCT
ejpam-4983	225	7	17	17	NUM
ejpam-4983	225	8	(	(	PUNCT
ejpam-4983	225	9	1	1	NUM
ejpam-4983	225	10	)	)	PUNCT
ejpam-4983	225	11	(	(	PUNCT
ejpam-4983	225	12	2024	2024	NUM
ejpam-4983	225	13	)	)	PUNCT
ejpam-4983	225	14	,	,	PUNCT
ejpam-4983	225	15	11	11	NUM
ejpam-4983	225	16	-	-	SYM
ejpam-4983	225	17	29	29	NUM
ejpam-4983	225	18	19	19	NUM
ejpam-4983	225	19	theorem	theorem	NOUN
ejpam-4983	225	20	2	2	NUM
ejpam-4983	225	21	.	.	PUNCT
ejpam-4983	226	1	let	let	VERB
ejpam-4983	226	2	r	r	PRON
ejpam-4983	226	3	be	be	AUX
ejpam-4983	226	4	a	a	DET
ejpam-4983	226	5	fccr	fccr	NOUN
ejpam-4983	226	6	with	with	ADP
ejpam-4983	226	7	residue	residue	NOUN
ejpam-4983	226	8	field	field	NOUN
ejpam-4983	226	9	fq	fq	NOUN
ejpam-4983	226	10	and	and	CCONJ
ejpam-4983	226	11	nilpotency	nilpotency	PROPN
ejpam-4983	226	12	index	index	PROPN
ejpam-4983	226	13	e.	e.	PROPN
ejpam-4983	226	14	then	then	ADV
ejpam-4983	227	1	|ian(r)|	|ian(r)|	PROPN
ejpam-4983	227	2	=	=	SYM
ejpam-4983	227	3	qe(3n−2)−(n+1)(q	qe(3n−2)−(n+1)(q	NOUN
ejpam-4983	227	4	−	−	PROPN
ejpam-4983	228	1	1)n(q	1)n(q	NUM
ejpam-4983	229	1	+	+	CCONJ
ejpam-4983	230	1	(	(	PUNCT
ejpam-4983	230	2	n−	n−	NOUN
ejpam-4983	230	3	1	1	NUM
ejpam-4983	230	4	)	)	PUNCT
ejpam-4983	230	5	)	)	PUNCT
ejpam-4983	230	6	for	for	ADP
ejpam-4983	230	7	all	all	DET
ejpam-4983	230	8	positive	positive	ADJ
ejpam-4983	230	9	integers	integer	NOUN
ejpam-4983	230	10	n.	n.	NOUN
ejpam-4983	230	11	proof	proof	NOUN
ejpam-4983	230	12	.	.	PUNCT
ejpam-4983	231	1	let	let	VERB
ejpam-4983	231	2	γ	γ	NOUN
ejpam-4983	231	3	be	be	AUX
ejpam-4983	231	4	a	a	DET
ejpam-4983	231	5	generator	generator	NOUN
ejpam-4983	231	6	of	of	ADP
ejpam-4983	231	7	the	the	DET
ejpam-4983	231	8	maximal	maximal	ADJ
ejpam-4983	231	9	ideal	ideal	NOUN
ejpam-4983	231	10	of	of	ADP
ejpam-4983	231	11	r	r	NOUN
ejpam-4983	231	12	and	and	CCONJ
ejpam-4983	231	13	let	let	VERB
ejpam-4983	231	14	φ	φ	NOUN
ejpam-4983	231	15	:	:	PUNCT
ejpam-4983	231	16	r	r	X
ejpam-4983	231	17	→	→	SYM
ejpam-4983	231	18	fq	fq	PROPN
ejpam-4983	231	19	be	be	AUX
ejpam-4983	231	20	the	the	DET
ejpam-4983	231	21	ring	ring	NOUN
ejpam-4983	231	22	homomorphism	homomorphism	NOUN
ejpam-4983	231	23	defined	define	VERB
ejpam-4983	231	24	by	by	ADP
ejpam-4983	231	25	a	a	DET
ejpam-4983	231	26	7→	7→	PROPN
ejpam-4983	231	27	a	a	DET
ejpam-4983	231	28	+	+	NOUN
ejpam-4983	231	29	⟨γ⟩.	⟨γ⟩.	NOUN
ejpam-4983	231	30	by	by	ADP
ejpam-4983	231	31	considering	consider	VERB
ejpam-4983	231	32	an(r	an(r	NOUN
ejpam-4983	231	33	)	)	PUNCT
ejpam-4983	231	34	and	and	CCONJ
ejpam-4983	231	35	an(fq	an(fq	PROPN
ejpam-4983	231	36	)	)	PUNCT
ejpam-4983	231	37	as	as	ADP
ejpam-4983	231	38	additive	additive	ADJ
ejpam-4983	231	39	groups	group	NOUN
ejpam-4983	231	40	,	,	PUNCT
ejpam-4983	231	41	let	let	VERB
ejpam-4983	231	42	ϕ	ϕ	NOUN
ejpam-4983	231	43	:	:	PUNCT
ejpam-4983	231	44	an(r	an(r	X
ejpam-4983	231	45	)	)	PUNCT
ejpam-4983	231	46	→	→	SYM
ejpam-4983	231	47	an(fq	an(fq	PROPN
ejpam-4983	231	48	)	)	PUNCT
ejpam-4983	231	49	be	be	VERB
ejpam-4983	231	50	the	the	DET
ejpam-4983	231	51	group	group	NOUN
ejpam-4983	231	52	homomorphism	homomorphism	NOUN
ejpam-4983	231	53	defined	define	VERB
ejpam-4983	231	54	by	by	ADP
ejpam-4983	231	55	a	a	DET
ejpam-4983	231	56	=	=	X
ejpam-4983	231	57	[	[	X
ejpam-4983	231	58	aij	aij	X
ejpam-4983	231	59	]	]	PUNCT
ejpam-4983	231	60	7→	7→	PROPN
ejpam-4983	232	1	[	[	X
ejpam-4983	232	2	φ(aij	φ(aij	NOUN
ejpam-4983	232	3	)	)	PUNCT
ejpam-4983	232	4	]	]	PUNCT
ejpam-4983	232	5	.	.	PUNCT
ejpam-4983	233	1	it	it	PRON
ejpam-4983	233	2	is	be	AUX
ejpam-4983	233	3	not	not	PART
ejpam-4983	233	4	difficult	difficult	ADJ
ejpam-4983	233	5	to	to	PART
ejpam-4983	233	6	see	see	VERB
ejpam-4983	233	7	that	that	PRON
ejpam-4983	233	8	ϕ	ϕ	NOUN
ejpam-4983	233	9	is	be	AUX
ejpam-4983	233	10	a	a	DET
ejpam-4983	233	11	surjective	surjective	ADJ
ejpam-4983	233	12	homomorphism	homomorphism	NOUN
ejpam-4983	233	13	.	.	PUNCT
ejpam-4983	234	1	by	by	ADP
ejpam-4983	234	2	the	the	DET
ejpam-4983	234	3	first	first	ADJ
ejpam-4983	234	4	isomorphism	isomorphism	NOUN
ejpam-4983	234	5	theorem	theorem	NOUN
ejpam-4983	234	6	for	for	ADP
ejpam-4983	234	7	groups	group	NOUN
ejpam-4983	234	8	,	,	PUNCT
ejpam-4983	234	9	it	it	PRON
ejpam-4983	234	10	follows	follow	VERB
ejpam-4983	234	11	that	that	SCONJ
ejpam-4983	234	12	an(fq	an(fq	NOUN
ejpam-4983	234	13	)	)	PUNCT
ejpam-4983	235	1	∼=	∼=	PROPN
ejpam-4983	235	2	an(r)/	an(r)/	PROPN
ejpam-4983	235	3	ker(ϕ	ker(ϕ	PROPN
ejpam-4983	235	4	)	)	PUNCT
ejpam-4983	235	5	.	.	PUNCT
ejpam-4983	236	1	hence	hence	ADV
ejpam-4983	236	2	,	,	PUNCT
ejpam-4983	236	3	|	|	ADV
ejpam-4983	236	4	ker(ϕ)|	ker(ϕ)|	NOUN
ejpam-4983	236	5	=	=	SYM
ejpam-4983	236	6	|an(r)|	|an(r)|	NOUN
ejpam-4983	236	7	|an(fq)|	|an(fq)|	NOUN
ejpam-4983	236	8	=	=	PUNCT
ejpam-4983	236	9	qe(3n−2	qe(3n−2	X
ejpam-4983	236	10	)	)	PUNCT
ejpam-4983	236	11	q3n−2	q3n−2	PROPN
ejpam-4983	236	12	=	=	SYM
ejpam-4983	236	13	q(e−1)(3n−2	q(e−1)(3n−2	NOUN
ejpam-4983	236	14	)	)	PUNCT
ejpam-4983	236	15	.	.	PUNCT
ejpam-4983	237	1	for	for	ADP
ejpam-4983	237	2	a	a	DET
ejpam-4983	237	3	∈	∈	PROPN
ejpam-4983	237	4	an(r	an(r	NOUN
ejpam-4983	237	5	)	)	PUNCT
ejpam-4983	237	6	,	,	PUNCT
ejpam-4983	237	7	we	we	PRON
ejpam-4983	237	8	have	have	VERB
ejpam-4983	237	9	det(ϕ(a	det(ϕ(a	NOUN
ejpam-4983	237	10	)	)	PUNCT
ejpam-4983	237	11	)	)	PUNCT
ejpam-4983	238	1	=	=	PUNCT
ejpam-4983	238	2	φ(det(a	φ(det(a	NOUN
ejpam-4983	238	3	)	)	PUNCT
ejpam-4983	238	4	)	)	PUNCT
ejpam-4983	238	5	which	which	PRON
ejpam-4983	238	6	implies	imply	VERB
ejpam-4983	238	7	that	that	SCONJ
ejpam-4983	238	8	det(a	det(a	PROPN
ejpam-4983	238	9	)	)	PUNCT
ejpam-4983	238	10	is	be	AUX
ejpam-4983	238	11	a	a	DET
ejpam-4983	238	12	unit	unit	NOUN
ejpam-4983	238	13	in	in	ADP
ejpam-4983	238	14	r	r	NOUN
ejpam-4983	238	15	if	if	SCONJ
ejpam-4983	239	1	and	and	CCONJ
ejpam-4983	239	2	only	only	ADV
ejpam-4983	239	3	if	if	SCONJ
ejpam-4983	239	4	det(ϕ(a	det(ϕ(a	NOUN
ejpam-4983	239	5	)	)	PUNCT
ejpam-4983	239	6	)	)	PUNCT
ejpam-4983	240	1	̸=	̸=	NOUN
ejpam-4983	240	2	0	0	NUM
ejpam-4983	240	3	in	in	ADP
ejpam-4983	240	4	fq	fq	PROPN
ejpam-4983	240	5	.	.	PUNCT
ejpam-4983	240	6	equivalently	equivalently	PROPN
ejpam-4983	240	7	,	,	PUNCT
ejpam-4983	240	8	a	a	PRON
ejpam-4983	240	9	is	be	AUX
ejpam-4983	240	10	invertible	invertible	ADJ
ejpam-4983	240	11	over	over	ADP
ejpam-4983	240	12	r	r	NOUN
ejpam-4983	240	13	if	if	SCONJ
ejpam-4983	241	1	and	and	CCONJ
ejpam-4983	241	2	only	only	ADV
ejpam-4983	241	3	if	if	SCONJ
ejpam-4983	241	4	ϕ(a	ϕ(a	NOUN
ejpam-4983	241	5	)	)	PUNCT
ejpam-4983	241	6	is	be	AUX
ejpam-4983	241	7	invertible	invertible	ADJ
ejpam-4983	241	8	over	over	ADP
ejpam-4983	241	9	fq	fq	PROPN
ejpam-4983	241	10	.	.	PROPN
ejpam-4983	242	1	then	then	ADV
ejpam-4983	242	2	the	the	DET
ejpam-4983	242	3	restriction	restriction	NOUN
ejpam-4983	242	4	map	map	NOUN
ejpam-4983	242	5	ϕ|ian(r	ϕ|ian(r	NOUN
ejpam-4983	242	6	)	)	PUNCT
ejpam-4983	242	7	:	:	PUNCT
ejpam-4983	242	8	ian(r	ian(r	NOUN
ejpam-4983	242	9	)	)	PUNCT
ejpam-4983	242	10	→	→	SYM
ejpam-4983	242	11	ian(fq	ian(fq	NOUN
ejpam-4983	242	12	)	)	PUNCT
ejpam-4983	242	13	is	be	AUX
ejpam-4983	242	14	surjective	surjective	ADJ
ejpam-4983	242	15	and	and	CCONJ
ejpam-4983	242	16	it	it	PRON
ejpam-4983	242	17	is	be	AUX
ejpam-4983	242	18	|	|	ADV
ejpam-4983	242	19	ker(ϕ)|	ker(ϕ)|	VERB
ejpam-4983	242	20	to	to	ADP
ejpam-4983	242	21	one	one	NUM
ejpam-4983	242	22	map	map	NOUN
ejpam-4983	242	23	.	.	PUNCT
ejpam-4983	243	1	from	from	ADP
ejpam-4983	243	2	theorem	theorem	ADJ
ejpam-4983	243	3	1	1	NUM
ejpam-4983	243	4	,	,	PUNCT
ejpam-4983	243	5	we	we	PRON
ejpam-4983	243	6	have	have	VERB
ejpam-4983	243	7	|ian(fq)|	|ian(fq)|	NUM
ejpam-4983	243	8	=	=	SYM
ejpam-4983	243	9	q2n−3(q	q2n−3(q	NUM
ejpam-4983	243	10	−	−	PROPN
ejpam-4983	243	11	1)n(q	1)n(q	NUM
ejpam-4983	243	12	+	+	CCONJ
ejpam-4983	243	13	(	(	PUNCT
ejpam-4983	243	14	n−	n−	NOUN
ejpam-4983	243	15	1	1	NUM
ejpam-4983	243	16	)	)	PUNCT
ejpam-4983	243	17	)	)	PUNCT
ejpam-4983	243	18	.	.	PUNCT
ejpam-4983	244	1	it	it	PRON
ejpam-4983	244	2	follows	follow	VERB
ejpam-4983	244	3	that	that	SCONJ
ejpam-4983	244	4	|ian(r)|	|ian(r)|	NOUN
ejpam-4983	244	5	=	=	SYM
ejpam-4983	244	6	|	|	ADV
ejpam-4983	244	7	ker(ϕ)||ian(fq)|	ker(ϕ)||ian(fq)|	NOUN
ejpam-4983	244	8	=	=	PUNCT
ejpam-4983	244	9	q(e−1)(3n−2)|ian(fq)|	q(e−1)(3n−2)|ian(fq)|	X
ejpam-4983	244	10	=	=	PUNCT
ejpam-4983	244	11	q(e−1)(3n−2)q2n−3(q	q(e−1)(3n−2)q2n−3(q	PROPN
ejpam-4983	245	1	−	−	NOUN
ejpam-4983	245	2	1)n(q	1)n(q	NUM
ejpam-4983	246	1	+	+	CCONJ
ejpam-4983	246	2	(	(	PUNCT
ejpam-4983	246	3	n−	n−	NOUN
ejpam-4983	246	4	1	1	NUM
ejpam-4983	246	5	)	)	PUNCT
ejpam-4983	246	6	)	)	PUNCT
ejpam-4983	247	1	=	=	PUNCT
ejpam-4983	247	2	qe(3n−2)−(n+1)(q	qe(3n−2)−(n+1)(q	ADP
ejpam-4983	247	3	−	−	PROPN
ejpam-4983	248	1	1)n(q	1)n(q	NUM
ejpam-4983	249	1	+	+	CCONJ
ejpam-4983	250	1	(	(	PUNCT
ejpam-4983	250	2	n−	n−	NOUN
ejpam-4983	250	3	1	1	NUM
ejpam-4983	250	4	)	)	PUNCT
ejpam-4983	250	5	)	)	PUNCT
ejpam-4983	250	6	as	as	SCONJ
ejpam-4983	250	7	desired	desire	VERB
ejpam-4983	250	8	.	.	PUNCT
ejpam-4983	251	1	■	■	PUNCT
ejpam-4983	251	2	for	for	ADP
ejpam-4983	251	3	each	each	DET
ejpam-4983	251	4	a	a	DET
ejpam-4983	251	5	∈	∈	PROPN
ejpam-4983	251	6	u(r	u(r	NOUN
ejpam-4983	251	7	)	)	PUNCT
ejpam-4983	251	8	,	,	PUNCT
ejpam-4983	251	9	the	the	DET
ejpam-4983	251	10	relation	relation	NOUN
ejpam-4983	251	11	between	between	ADP
ejpam-4983	251	12	|an(r	|an(r	PROPN
ejpam-4983	251	13	,	,	PUNCT
ejpam-4983	251	14	1)|	1)|	NUM
ejpam-4983	251	15	and	and	CCONJ
ejpam-4983	251	16	|an(r	|an(r	PROPN
ejpam-4983	251	17	,	,	PUNCT
ejpam-4983	251	18	a)|	a)|	X
ejpam-4983	251	19	in	in	ADP
ejpam-4983	251	20	the	the	DET
ejpam-4983	251	21	following	follow	VERB
ejpam-4983	251	22	proposition	proposition	NOUN
ejpam-4983	251	23	is	be	AUX
ejpam-4983	251	24	key	key	ADJ
ejpam-4983	251	25	to	to	PART
ejpam-4983	251	26	determine	determine	VERB
ejpam-4983	251	27	the	the	DET
ejpam-4983	251	28	number	number	NOUN
ejpam-4983	251	29	|an(r	|an(r	PROPN
ejpam-4983	251	30	,	,	PUNCT
ejpam-4983	251	31	a)|	a)|	X
ejpam-4983	251	32	in	in	ADP
ejpam-4983	251	33	corollary	corollary	ADJ
ejpam-4983	251	34	3	3	NUM
ejpam-4983	251	35	.	.	PUNCT
ejpam-4983	251	36	proposition	proposition	NOUN
ejpam-4983	251	37	3	3	X
ejpam-4983	251	38	.	.	PUNCT
ejpam-4983	252	1	let	let	VERB
ejpam-4983	252	2	r	r	PRON
ejpam-4983	252	3	be	be	AUX
ejpam-4983	252	4	a	a	DET
ejpam-4983	252	5	fccr	fccr	NOUN
ejpam-4983	252	6	and	and	CCONJ
ejpam-4983	252	7	let	let	VERB
ejpam-4983	252	8	n	n	PRON
ejpam-4983	252	9	be	be	AUX
ejpam-4983	252	10	a	a	DET
ejpam-4983	252	11	positive	positive	ADJ
ejpam-4983	252	12	integer	integer	NOUN
ejpam-4983	252	13	.	.	PUNCT
ejpam-4983	253	1	then	then	ADV
ejpam-4983	253	2	|an(r	|an(r	PRON
ejpam-4983	253	3	,	,	PUNCT
ejpam-4983	253	4	a)|	a)|	X
ejpam-4983	253	5	=	=	SYM
ejpam-4983	253	6	|an(r	|an(r	PROPN
ejpam-4983	253	7	,	,	PUNCT
ejpam-4983	253	8	1)|	1)|	NUM
ejpam-4983	253	9	for	for	ADP
ejpam-4983	253	10	all	all	DET
ejpam-4983	253	11	a	a	DET
ejpam-4983	253	12	∈	∈	PROPN
ejpam-4983	253	13	u(r	u(r	NOUN
ejpam-4983	253	14	)	)	PUNCT
ejpam-4983	253	15	.	.	PUNCT
ejpam-4983	254	1	proof	proof	NOUN
ejpam-4983	254	2	.	.	PUNCT
ejpam-4983	255	1	let	let	VERB
ejpam-4983	255	2	a	a	DET
ejpam-4983	255	3	∈	∈	PROPN
ejpam-4983	255	4	u(r	u(r	NOUN
ejpam-4983	255	5	)	)	PUNCT
ejpam-4983	255	6	and	and	CCONJ
ejpam-4983	255	7	let	let	VERB
ejpam-4983	255	8	θ	θ	NOUN
ejpam-4983	255	9	:	:	PUNCT
ejpam-4983	255	10	an(r	an(r	NOUN
ejpam-4983	255	11	,	,	PUNCT
ejpam-4983	255	12	1	1	X
ejpam-4983	255	13	)	)	PUNCT
ejpam-4983	255	14	→	→	NOUN
ejpam-4983	255	15	an(r	an(r	NOUN
ejpam-4983	255	16	,	,	PUNCT
ejpam-4983	255	17	a	a	PRON
ejpam-4983	255	18	)	)	PUNCT
ejpam-4983	255	19	be	be	VERB
ejpam-4983	255	20	the	the	DET
ejpam-4983	255	21	map	map	NOUN
ejpam-4983	255	22	defined	define	VERB
ejpam-4983	255	23	by	by	ADP
ejpam-4983	255	24	θ(a	θ(a	PROPN
ejpam-4983	255	25	)	)	PUNCT
ejpam-4983	256	1	=	=	SYM
ejpam-4983	256	2	diag(a	diag(a	PROPN
ejpam-4983	256	3	,	,	PUNCT
ejpam-4983	256	4	1	1	NUM
ejpam-4983	256	5	,	,	PUNCT
ejpam-4983	256	6	1	1	NUM
ejpam-4983	256	7	,	,	PUNCT
ejpam-4983	256	8	.	.	PUNCT
ejpam-4983	256	9	.	.	PUNCT
ejpam-4983	256	10	.	.	PUNCT
ejpam-4983	257	1	,	,	PUNCT
ejpam-4983	257	2	1)a	1)a	NUM
ejpam-4983	257	3	.	.	PUNCT
ejpam-4983	258	1	s.	s.	PROPN
ejpam-4983	258	2	jitman	jitman	PROPN
ejpam-4983	258	3	,	,	PUNCT
ejpam-4983	258	4	p.	p.	PROPN
ejpam-4983	258	5	modjam	modjam	PROPN
ejpam-4983	258	6	/	/	SYM
ejpam-4983	258	7	eur	eur	PROPN
ejpam-4983	258	8	.	.	PUNCT
ejpam-4983	259	1	j.	j.	PROPN
ejpam-4983	259	2	pure	pure	PROPN
ejpam-4983	259	3	appl	appl	PROPN
ejpam-4983	259	4	.	.	PROPN
ejpam-4983	259	5	math	math	PROPN
ejpam-4983	259	6	,	,	PUNCT
ejpam-4983	259	7	17	17	NUM
ejpam-4983	259	8	(	(	PUNCT
ejpam-4983	259	9	1	1	NUM
ejpam-4983	259	10	)	)	PUNCT
ejpam-4983	259	11	(	(	PUNCT
ejpam-4983	259	12	2024	2024	NUM
ejpam-4983	259	13	)	)	PUNCT
ejpam-4983	259	14	,	,	PUNCT
ejpam-4983	259	15	11	11	NUM
ejpam-4983	259	16	-	-	SYM
ejpam-4983	259	17	29	29	NUM
ejpam-4983	259	18	20	20	NUM
ejpam-4983	259	19	using	use	VERB
ejpam-4983	259	20	arguments	argument	NOUN
ejpam-4983	259	21	similar	similar	ADJ
ejpam-4983	259	22	to	to	ADP
ejpam-4983	259	23	those	those	PRON
ejpam-4983	259	24	in	in	ADP
ejpam-4983	259	25	the	the	DET
ejpam-4983	259	26	proof	proof	NOUN
ejpam-4983	259	27	of	of	ADP
ejpam-4983	259	28	proposition	proposition	NOUN
ejpam-4983	259	29	2	2	NUM
ejpam-4983	259	30	,	,	PUNCT
ejpam-4983	259	31	it	it	PRON
ejpam-4983	259	32	can	can	AUX
ejpam-4983	259	33	be	be	AUX
ejpam-4983	259	34	deduced	deduce	VERB
ejpam-4983	259	35	that	that	SCONJ
ejpam-4983	259	36	θ	θ	PROPN
ejpam-4983	259	37	is	be	AUX
ejpam-4983	259	38	a	a	DET
ejpam-4983	259	39	bijection	bijection	NOUN
ejpam-4983	259	40	from	from	ADP
ejpam-4983	259	41	an(r	an(r	NOUN
ejpam-4983	259	42	,	,	PUNCT
ejpam-4983	259	43	1	1	NUM
ejpam-4983	259	44	)	)	PUNCT
ejpam-4983	259	45	onto	onto	ADP
ejpam-4983	259	46	an(r	an(r	NOUN
ejpam-4983	259	47	,	,	PUNCT
ejpam-4983	259	48	a	a	PRON
ejpam-4983	259	49	)	)	PUNCT
ejpam-4983	259	50	.	.	PUNCT
ejpam-4983	260	1	as	as	SCONJ
ejpam-4983	260	2	desired	desire	VERB
ejpam-4983	260	3	,	,	PUNCT
ejpam-4983	260	4	|an(r	|an(r	PROPN
ejpam-4983	260	5	,	,	PUNCT
ejpam-4983	260	6	a)|	a)|	X
ejpam-4983	260	7	=	=	SYM
ejpam-4983	260	8	|an(r	|an(r	PROPN
ejpam-4983	260	9	,	,	PUNCT
ejpam-4983	260	10	1)|	1)|	NUM
ejpam-4983	260	11	.	.	PUNCT
ejpam-4983	261	1	■	■	PUNCT
ejpam-4983	261	2	from	from	ADP
ejpam-4983	261	3	proposition	proposition	NOUN
ejpam-4983	261	4	3	3	NUM
ejpam-4983	261	5	,	,	PUNCT
ejpam-4983	261	6	it	it	PRON
ejpam-4983	261	7	follows	follow	VERB
ejpam-4983	261	8	that	that	SCONJ
ejpam-4983	261	9	|an(r	|an(r	NOUN
ejpam-4983	261	10	,	,	PUNCT
ejpam-4983	261	11	a)|	a)|	X
ejpam-4983	261	12	=	=	SYM
ejpam-4983	261	13	|an(r	|an(r	PROPN
ejpam-4983	261	14	,	,	PUNCT
ejpam-4983	261	15	1)|	1)|	NUM
ejpam-4983	261	16	=	=	SYM
ejpam-4983	261	17	|an(r	|an(r	PROPN
ejpam-4983	261	18	,	,	PUNCT
ejpam-4983	261	19	b)|	b)|	NOUN
ejpam-4983	261	20	for	for	ADP
ejpam-4983	261	21	all	all	DET
ejpam-4983	261	22	units	unit	NOUN
ejpam-4983	261	23	a	a	PRON
ejpam-4983	261	24	,	,	PUNCT
ejpam-4983	261	25	b	b	PROPN
ejpam-4983	261	26	∈	∈	PROPN
ejpam-4983	261	27	u(r	u(r	NOUN
ejpam-4983	261	28	)	)	PUNCT
ejpam-4983	261	29	.	.	PUNCT
ejpam-4983	262	1	for	for	ADP
ejpam-4983	262	2	a	a	DET
ejpam-4983	262	3	fixed	fix	VERB
ejpam-4983	262	4	unit	unit	NOUN
ejpam-4983	262	5	a	a	DET
ejpam-4983	262	6	∈	∈	PROPN
ejpam-4983	262	7	r	r	NOUN
ejpam-4983	262	8	,	,	PUNCT
ejpam-4983	262	9	the	the	DET
ejpam-4983	262	10	number	number	NOUN
ejpam-4983	262	11	of	of	ADP
ejpam-4983	262	12	n	n	NUM
ejpam-4983	262	13	×	×	NOUN
ejpam-4983	262	14	n	n	CCONJ
ejpam-4983	262	15	arrowhead	arrowhead	NOUN
ejpam-4983	262	16	matrices	matrix	NOUN
ejpam-4983	262	17	over	over	ADP
ejpam-4983	262	18	r	r	NOUN
ejpam-4983	262	19	whose	whose	DET
ejpam-4983	262	20	determinant	determinant	ADJ
ejpam-4983	262	21	is	be	AUX
ejpam-4983	262	22	a	a	PRON
ejpam-4983	262	23	will	will	AUX
ejpam-4983	262	24	be	be	AUX
ejpam-4983	262	25	given	give	VERB
ejpam-4983	262	26	later	later	ADV
ejpam-4983	262	27	in	in	ADP
ejpam-4983	262	28	corollary	corollary	ADJ
ejpam-4983	262	29	3	3	NUM
ejpam-4983	262	30	.	.	PUNCT
ejpam-4983	262	31	corollary	corollary	ADJ
ejpam-4983	262	32	3	3	X
ejpam-4983	262	33	.	.	PUNCT
ejpam-4983	263	1	let	let	VERB
ejpam-4983	263	2	r	r	PRON
ejpam-4983	263	3	be	be	AUX
ejpam-4983	263	4	a	a	DET
ejpam-4983	263	5	fccr	fccr	NOUN
ejpam-4983	263	6	with	with	ADP
ejpam-4983	263	7	residue	residue	NOUN
ejpam-4983	263	8	field	field	NOUN
ejpam-4983	263	9	fq	fq	NOUN
ejpam-4983	263	10	and	and	CCONJ
ejpam-4983	263	11	nilpotency	nilpotency	NOUN
ejpam-4983	263	12	index	index	NOUN
ejpam-4983	263	13	e	e	NOUN
ejpam-4983	263	14	and	and	CCONJ
ejpam-4983	263	15	let	let	VERB
ejpam-4983	263	16	n	n	PRON
ejpam-4983	263	17	be	be	AUX
ejpam-4983	263	18	a	a	DET
ejpam-4983	263	19	positive	positive	ADJ
ejpam-4983	263	20	integer	integer	NOUN
ejpam-4983	263	21	.	.	PUNCT
ejpam-4983	264	1	then	then	ADV
ejpam-4983	264	2	|an(r	|an(r	PRON
ejpam-4983	264	3	,	,	PUNCT
ejpam-4983	264	4	a)|	a)|	X
ejpam-4983	264	5	=	=	NOUN
ejpam-4983	264	6	q3e(n−1)−n(q	q3e(n−1)−n(q	PROPN
ejpam-4983	265	1	−	−	NOUN
ejpam-4983	265	2	1)n−1(q	1)n−1(q	NUM
ejpam-4983	266	1	+	+	CCONJ
ejpam-4983	266	2	(	(	PUNCT
ejpam-4983	266	3	n−	n−	NOUN
ejpam-4983	266	4	1	1	NUM
ejpam-4983	266	5	)	)	PUNCT
ejpam-4983	266	6	)	)	PUNCT
ejpam-4983	266	7	for	for	ADP
ejpam-4983	266	8	all	all	DET
ejpam-4983	266	9	a	a	DET
ejpam-4983	266	10	∈	∈	PROPN
ejpam-4983	266	11	u(r	u(r	NOUN
ejpam-4983	266	12	)	)	PUNCT
ejpam-4983	266	13	.	.	PUNCT
ejpam-4983	267	1	proof	proof	NOUN
ejpam-4983	267	2	.	.	PUNCT
ejpam-4983	268	1	first	first	ADV
ejpam-4983	268	2	,	,	PUNCT
ejpam-4983	268	3	we	we	PRON
ejpam-4983	268	4	note	note	VERB
ejpam-4983	268	5	that	that	SCONJ
ejpam-4983	268	6	ian(r	ian(r	NOUN
ejpam-4983	268	7	)	)	PUNCT
ejpam-4983	268	8	is	be	AUX
ejpam-4983	268	9	disjoint	disjoint	ADJ
ejpam-4983	268	10	union	union	NOUN
ejpam-4983	268	11	of	of	ADP
ejpam-4983	268	12	an(r	an(r	NOUN
ejpam-4983	268	13	,	,	PUNCT
ejpam-4983	268	14	a	a	PRON
ejpam-4983	268	15	)	)	PUNCT
ejpam-4983	268	16	for	for	ADP
ejpam-4983	268	17	all	all	DET
ejpam-4983	268	18	a	a	DET
ejpam-4983	268	19	∈	∈	PROPN
ejpam-4983	268	20	u(r	u(r	NOUN
ejpam-4983	268	21	)	)	PUNCT
ejpam-4983	268	22	.	.	PUNCT
ejpam-4983	269	1	precisely	precisely	ADV
ejpam-4983	269	2	,	,	PUNCT
ejpam-4983	269	3	ian(r	ian(r	NOUN
ejpam-4983	269	4	)	)	PUNCT
ejpam-4983	269	5	=	=	SYM
ejpam-4983	269	6	⋃	⋃	NOUN
ejpam-4983	269	7	a∈u(r	a∈u(r	NOUN
ejpam-4983	269	8	)	)	PUNCT
ejpam-4983	269	9	an(r	an(r	NOUN
ejpam-4983	269	10	,	,	PUNCT
ejpam-4983	269	11	a	a	PRON
ejpam-4983	269	12	)	)	PUNCT
ejpam-4983	269	13	is	be	AUX
ejpam-4983	269	14	a	a	DET
ejpam-4983	269	15	disjoint	disjoint	NOUN
ejpam-4983	269	16	union	union	NOUN
ejpam-4983	269	17	.	.	PUNCT
ejpam-4983	270	1	by	by	ADP
ejpam-4983	270	2	proposition	proposition	NOUN
ejpam-4983	270	3	3	3	NUM
ejpam-4983	270	4	,	,	PUNCT
ejpam-4983	270	5	an(r	an(r	NOUN
ejpam-4983	270	6	,	,	PUNCT
ejpam-4983	270	7	a	a	PRON
ejpam-4983	270	8	)	)	PUNCT
ejpam-4983	270	9	has	have	VERB
ejpam-4983	270	10	the	the	DET
ejpam-4983	270	11	same	same	ADJ
ejpam-4983	270	12	number	number	NOUN
ejpam-4983	270	13	of	of	ADP
ejpam-4983	270	14	elements	element	NOUN
ejpam-4983	270	15	as	as	ADP
ejpam-4983	270	16	an(r	an(r	NOUN
ejpam-4983	270	17	,	,	PUNCT
ejpam-4983	270	18	1	1	NUM
ejpam-4983	270	19	)	)	PUNCT
ejpam-4983	270	20	,	,	PUNCT
ejpam-4983	270	21	and	and	CCONJ
ejpam-4983	270	22	hence	hence	ADV
ejpam-4983	270	23	,	,	PUNCT
ejpam-4983	270	24	|ian(r)|	|ian(r)|	PROPN
ejpam-4983	270	25	=	=	SYM
ejpam-4983	270	26	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ejpam-4983	270	27	⋃	⋃	NOUN
ejpam-4983	270	28	a∈u(r	a∈u(r	NOUN
ejpam-4983	270	29	)	)	PUNCT
ejpam-4983	270	30	an(r	an(r	NOUN
ejpam-4983	270	31	,	,	PUNCT
ejpam-4983	270	32	a	a	PRON
ejpam-4983	270	33	)	)	PUNCT
ejpam-4983	270	34	∣∣∣∣∣∣	∣∣∣∣∣∣	VERB
ejpam-4983	271	1	=	=	SYM
ejpam-4983	271	2	∑	∑	NOUN
ejpam-4983	271	3	a∈u(r	a∈u(r	PROPN
ejpam-4983	271	4	)	)	PUNCT
ejpam-4983	271	5	|an(r	|an(r	PROPN
ejpam-4983	271	6	,	,	PUNCT
ejpam-4983	271	7	a)|	a)|	X
ejpam-4983	271	8	=	=	NOUN
ejpam-4983	271	9	∑	∑	NOUN
ejpam-4983	271	10	a∈u(r	a∈u(r	PROPN
ejpam-4983	271	11	)	)	PUNCT
ejpam-4983	272	1	|an(r	|an(r	PROPN
ejpam-4983	272	2	,	,	PUNCT
ejpam-4983	272	3	1)|	1)|	NUM
ejpam-4983	272	4	=	=	SYM
ejpam-4983	272	5	|u(r)||an(r	|u(r)||an(r	NOUN
ejpam-4983	272	6	,	,	PUNCT
ejpam-4983	272	7	1)|	1)|	NUM
ejpam-4983	272	8	.	.	PUNCT
ejpam-4983	273	1	from	from	ADP
ejpam-4983	273	2	lemma	lemma	PROPN
ejpam-4983	273	3	1	1	NUM
ejpam-4983	273	4	,	,	PUNCT
ejpam-4983	273	5	we	we	PRON
ejpam-4983	273	6	have	have	VERB
ejpam-4983	273	7	|u(r)|	|u(r)|	PROPN
ejpam-4983	273	8	=	=	SYM
ejpam-4983	273	9	(	(	PUNCT
ejpam-4983	273	10	q	q	PROPN
ejpam-4983	273	11	−	−	PROPN
ejpam-4983	273	12	1)qe−1	1)qe−1	PROPN
ejpam-4983	273	13	.	.	PUNCT
ejpam-4983	274	1	by	by	ADP
ejpam-4983	274	2	proposition	proposition	NOUN
ejpam-4983	274	3	3	3	NUM
ejpam-4983	274	4	,	,	PUNCT
ejpam-4983	274	5	it	it	PRON
ejpam-4983	274	6	can	can	AUX
ejpam-4983	274	7	be	be	AUX
ejpam-4983	274	8	deduced	deduce	VERB
ejpam-4983	274	9	that	that	SCONJ
ejpam-4983	274	10	|an(r	|an(r	PROPN
ejpam-4983	274	11	,	,	PUNCT
ejpam-4983	274	12	a)|	a)|	X
ejpam-4983	274	13	=	=	SYM
ejpam-4983	274	14	|an(r	|an(r	PROPN
ejpam-4983	274	15	,	,	PUNCT
ejpam-4983	274	16	1)|	1)|	NUM
ejpam-4983	274	17	=	=	SYM
ejpam-4983	274	18	|ian(r)|	|ian(r)|	PROPN
ejpam-4983	274	19	|u(r)|	|u(r)|	PROPN
ejpam-4983	274	20	=	=	PUNCT
ejpam-4983	274	21	qe(3n−2)−(n+1)(q	qe(3n−2)−(n+1)(q	NOUN
ejpam-4983	274	22	−	−	PROPN
ejpam-4983	274	23	1)n(q	1)n(q	NUM
ejpam-4983	274	24	+	+	CCONJ
ejpam-4983	274	25	(	(	PUNCT
ejpam-4983	274	26	n−	n−	NOUN
ejpam-4983	274	27	1	1	NUM
ejpam-4983	274	28	)	)	PUNCT
ejpam-4983	274	29	)	)	PUNCT
ejpam-4983	275	1	(	(	PUNCT
ejpam-4983	275	2	q	q	NOUN
ejpam-4983	275	3	−	−	PROPN
ejpam-4983	275	4	1)qe−1	1)qe−1	PROPN
ejpam-4983	275	5	=	=	NOUN
ejpam-4983	275	6	q3e(n−1)−n(q	q3e(n−1)−n(q	NOUN
ejpam-4983	276	1	−	−	NOUN
ejpam-4983	276	2	1)n−1(q	1)n−1(q	NUM
ejpam-4983	277	1	+	+	CCONJ
ejpam-4983	277	2	(	(	PUNCT
ejpam-4983	277	3	n−	n−	NOUN
ejpam-4983	277	4	1	1	NUM
ejpam-4983	277	5	)	)	PUNCT
ejpam-4983	277	6	)	)	PUNCT
ejpam-4983	277	7	.	.	PUNCT
ejpam-4983	278	1	the	the	DET
ejpam-4983	278	2	proof	proof	NOUN
ejpam-4983	278	3	is	be	AUX
ejpam-4983	278	4	completed	complete	VERB
ejpam-4983	278	5	.	.	PUNCT
ejpam-4983	279	1	■	■	PUNCT
ejpam-4983	279	2	3.2	3.2	NUM
ejpam-4983	279	3	.	.	PUNCT
ejpam-4983	279	4	singular	singular	PROPN
ejpam-4983	279	5	arrowhead	arrowhead	NOUN
ejpam-4983	279	6	matrices	matrix	NOUN
ejpam-4983	279	7	over	over	ADP
ejpam-4983	279	8	fccrs	fccr	NOUN
ejpam-4983	279	9	in	in	ADP
ejpam-4983	279	10	this	this	DET
ejpam-4983	279	11	subsection	subsection	NOUN
ejpam-4983	279	12	,	,	PUNCT
ejpam-4983	279	13	the	the	DET
ejpam-4983	279	14	enumeration	enumeration	NOUN
ejpam-4983	279	15	of	of	ADP
ejpam-4983	279	16	singular	singular	ADJ
ejpam-4983	279	17	arrowhead	arrowhead	NOUN
ejpam-4983	279	18	matrices	matrix	NOUN
ejpam-4983	279	19	with	with	ADP
ejpam-4983	279	20	prescribed	prescribe	VERB
ejpam-4983	279	21	determinant	determinant	ADJ
ejpam-4983	279	22	over	over	ADP
ejpam-4983	279	23	a	a	DET
ejpam-4983	279	24	fccr	fccr	NOUN
ejpam-4983	279	25	r	r	NOUN
ejpam-4983	279	26	are	be	AUX
ejpam-4983	279	27	studied	study	VERB
ejpam-4983	279	28	.	.	PUNCT
ejpam-4983	280	1	unlike	unlike	ADP
ejpam-4983	280	2	the	the	DET
ejpam-4983	280	3	previous	previous	ADJ
ejpam-4983	280	4	subsection	subsection	NOUN
ejpam-4983	280	5	,	,	PUNCT
ejpam-4983	280	6	only	only	ADV
ejpam-4983	280	7	bounds	bound	NOUN
ejpam-4983	280	8	on	on	ADP
ejpam-4983	280	9	s.	s.	PROPN
ejpam-4983	280	10	jitman	jitman	PROPN
ejpam-4983	280	11	,	,	PUNCT
ejpam-4983	280	12	p.	p.	PROPN
ejpam-4983	280	13	modjam	modjam	PROPN
ejpam-4983	280	14	/	/	SYM
ejpam-4983	280	15	eur	eur	PROPN
ejpam-4983	280	16	.	.	PUNCT
ejpam-4983	281	1	j.	j.	PROPN
ejpam-4983	281	2	pure	pure	PROPN
ejpam-4983	281	3	appl	appl	PROPN
ejpam-4983	281	4	.	.	PROPN
ejpam-4983	281	5	math	math	PROPN
ejpam-4983	281	6	,	,	PUNCT
ejpam-4983	281	7	17	17	NUM
ejpam-4983	281	8	(	(	PUNCT
ejpam-4983	281	9	1	1	NUM
ejpam-4983	281	10	)	)	PUNCT
ejpam-4983	281	11	(	(	PUNCT
ejpam-4983	281	12	2024	2024	NUM
ejpam-4983	281	13	)	)	PUNCT
ejpam-4983	281	14	,	,	PUNCT
ejpam-4983	281	15	11	11	NUM
ejpam-4983	281	16	-	-	SYM
ejpam-4983	281	17	29	29	NUM
ejpam-4983	281	18	21	21	NUM
ejpam-4983	281	19	the	the	DET
ejpam-4983	281	20	number	number	NOUN
ejpam-4983	281	21	of	of	ADP
ejpam-4983	281	22	singular	singular	PROPN
ejpam-4983	281	23	n×	n×	PROPN
ejpam-4983	281	24	n	n	CCONJ
ejpam-4983	281	25	arrowhead	arrowhead	NOUN
ejpam-4983	281	26	matrices	matrix	NOUN
ejpam-4983	281	27	over	over	ADP
ejpam-4983	281	28	r	r	NOUN
ejpam-4983	281	29	with	with	ADP
ejpam-4983	281	30	prescribed	prescribe	VERB
ejpam-4983	281	31	determinant	determinant	ADJ
ejpam-4983	281	32	are	be	AUX
ejpam-4983	281	33	given	give	VERB
ejpam-4983	281	34	.	.	PUNCT
ejpam-4983	282	1	since	since	SCONJ
ejpam-4983	282	2	the	the	DET
ejpam-4983	282	3	number	number	NOUN
ejpam-4983	282	4	of	of	ADP
ejpam-4983	282	5	n	n	NUM
ejpam-4983	282	6	×	×	NOUN
ejpam-4983	282	7	n	n	CCONJ
ejpam-4983	282	8	arrowhead	arrowhead	NOUN
ejpam-4983	282	9	matrices	matrix	NOUN
ejpam-4983	282	10	over	over	ADP
ejpam-4983	282	11	r	r	NOUN
ejpam-4983	282	12	is	be	AUX
ejpam-4983	282	13	qe(3n−2	qe(3n−2	NOUN
ejpam-4983	282	14	)	)	PUNCT
ejpam-4983	282	15	,	,	PUNCT
ejpam-4983	282	16	the	the	DET
ejpam-4983	282	17	next	next	ADJ
ejpam-4983	282	18	corollary	corollary	ADJ
ejpam-4983	282	19	follow	follow	NOUN
ejpam-4983	282	20	immediately	immediately	ADV
ejpam-4983	282	21	from	from	ADP
ejpam-4983	282	22	theorem	theorem	ADJ
ejpam-4983	282	23	2	2	NUM
ejpam-4983	282	24	.	.	PUNCT
ejpam-4983	282	25	corollary	corollary	ADJ
ejpam-4983	282	26	4	4	NUM
ejpam-4983	282	27	.	.	PUNCT
ejpam-4983	283	1	let	let	VERB
ejpam-4983	283	2	r	r	PRON
ejpam-4983	283	3	be	be	AUX
ejpam-4983	283	4	a	a	DET
ejpam-4983	283	5	fccr	fccr	NOUN
ejpam-4983	283	6	with	with	ADP
ejpam-4983	283	7	residue	residue	NOUN
ejpam-4983	283	8	field	field	NOUN
ejpam-4983	283	9	fq	fq	NOUN
ejpam-4983	283	10	and	and	CCONJ
ejpam-4983	283	11	nilpotency	nilpotency	PROPN
ejpam-4983	283	12	index	index	PROPN
ejpam-4983	283	13	e.	e.	PROPN
ejpam-4983	283	14	then	then	ADV
ejpam-4983	283	15	the	the	DET
ejpam-4983	283	16	number	number	NOUN
ejpam-4983	283	17	of	of	ADP
ejpam-4983	283	18	n×	n×	PRON
ejpam-4983	283	19	n	n	CCONJ
ejpam-4983	283	20	singular	singular	ADJ
ejpam-4983	283	21	arrowhead	arrowhead	NOUN
ejpam-4983	283	22	matrices	matrix	NOUN
ejpam-4983	283	23	over	over	ADP
ejpam-4983	283	24	r	r	NOUN
ejpam-4983	283	25	is	be	AUX
ejpam-4983	283	26	qe(3n−2)−(n+1	qe(3n−2)−(n+1	NOUN
ejpam-4983	283	27	)	)	PUNCT
ejpam-4983	283	28	(	(	PUNCT
ejpam-4983	283	29	qn+1	qn+1	NUM
ejpam-4983	283	30	−	−	PROPN
ejpam-4983	283	31	(	(	PUNCT
ejpam-4983	283	32	q	q	NOUN
ejpam-4983	283	33	−	−	PROPN
ejpam-4983	284	1	1)n(q	1)n(q	NUM
ejpam-4983	285	1	+	+	CCONJ
ejpam-4983	286	1	(	(	PUNCT
ejpam-4983	286	2	n−	n−	NOUN
ejpam-4983	286	3	1	1	NUM
ejpam-4983	286	4	)	)	PUNCT
ejpam-4983	286	5	)	)	PUNCT
ejpam-4983	286	6	)	)	PUNCT
ejpam-4983	287	1	for	for	ADP
ejpam-4983	287	2	all	all	DET
ejpam-4983	287	3	positive	positive	ADJ
ejpam-4983	287	4	integers	integer	NOUN
ejpam-4983	287	5	n.	n.	PROPN
ejpam-4983	287	6	3.2.1	3.2.1	NUM
ejpam-4983	287	7	.	.	PUNCT
ejpam-4983	287	8	singular	singular	PROPN
ejpam-4983	287	9	arrowhead	arrowhead	NOUN
ejpam-4983	287	10	matrices	matrix	NOUN
ejpam-4983	287	11	over	over	ADP
ejpam-4983	287	12	fccrs	fccr	NOUN
ejpam-4983	287	13	with	with	ADP
ejpam-4983	287	14	zero	zero	NUM
ejpam-4983	287	15	determinant	determinant	VERB
ejpam-4983	287	16	a	a	DET
ejpam-4983	287	17	general	general	ADJ
ejpam-4983	287	18	recursive	recursive	NOUN
ejpam-4983	287	19	lower	lower	ADV
ejpam-4983	287	20	bound	bind	VERB
ejpam-4983	287	21	on	on	ADP
ejpam-4983	287	22	the	the	DET
ejpam-4983	287	23	number	number	NOUN
ejpam-4983	287	24	of	of	ADP
ejpam-4983	287	25	n×	n×	PRON
ejpam-4983	287	26	n	n	X
ejpam-4983	287	27	arrowhead	arrowhead	NOUN
ejpam-4983	287	28	matrices	matrix	NOUN
ejpam-4983	287	29	over	over	ADP
ejpam-4983	287	30	r	r	NOUN
ejpam-4983	287	31	with	with	ADP
ejpam-4983	287	32	zero	zero	NUM
ejpam-4983	287	33	determinant	determinant	ADJ
ejpam-4983	287	34	is	be	AUX
ejpam-4983	287	35	given	give	VERB
ejpam-4983	287	36	in	in	ADP
ejpam-4983	287	37	the	the	DET
ejpam-4983	287	38	next	next	ADJ
ejpam-4983	287	39	proposition	proposition	NOUN
ejpam-4983	287	40	.	.	PUNCT
ejpam-4983	288	1	for	for	ADP
ejpam-4983	288	2	e	e	NOUN
ejpam-4983	288	3	=	=	SYM
ejpam-4983	288	4	2	2	NUM
ejpam-4983	288	5	,	,	PUNCT
ejpam-4983	288	6	a	a	DET
ejpam-4983	288	7	more	more	ADV
ejpam-4983	288	8	specific	specific	ADJ
ejpam-4983	288	9	bound	bind	VERB
ejpam-4983	288	10	is	be	AUX
ejpam-4983	288	11	derived	derive	VERB
ejpam-4983	288	12	in	in	ADP
ejpam-4983	288	13	corollary	corollary	ADJ
ejpam-4983	288	14	5	5	NUM
ejpam-4983	288	15	.	.	PUNCT
ejpam-4983	289	1	proposition	proposition	NOUN
ejpam-4983	289	2	4	4	NUM
ejpam-4983	289	3	.	.	PUNCT
ejpam-4983	290	1	let	let	VERB
ejpam-4983	290	2	r	r	PRON
ejpam-4983	290	3	be	be	AUX
ejpam-4983	290	4	a	a	DET
ejpam-4983	290	5	fccr	fccr	NOUN
ejpam-4983	290	6	of	of	ADP
ejpam-4983	290	7	nilpotency	nilpotency	NOUN
ejpam-4983	290	8	index	index	NOUN
ejpam-4983	290	9	e	e	NOUN
ejpam-4983	290	10	and	and	CCONJ
ejpam-4983	290	11	residue	residue	NOUN
ejpam-4983	290	12	field	field	NOUN
ejpam-4983	290	13	fq	fq	NOUN
ejpam-4983	290	14	.	.	PROPN
ejpam-4983	291	1	if	if	SCONJ
ejpam-4983	291	2	γ	γ	PROPN
ejpam-4983	291	3	is	be	AUX
ejpam-4983	291	4	a	a	DET
ejpam-4983	291	5	generator	generator	NOUN
ejpam-4983	291	6	of	of	ADP
ejpam-4983	291	7	the	the	DET
ejpam-4983	291	8	maximal	maximal	ADJ
ejpam-4983	291	9	ideal	ideal	NOUN
ejpam-4983	291	10	of	of	ADP
ejpam-4983	291	11	r	r	NOUN
ejpam-4983	291	12	,	,	PUNCT
ejpam-4983	291	13	then	then	ADV
ejpam-4983	291	14	|a1(r	|a1(r	ADJ
ejpam-4983	291	15	,	,	PUNCT
ejpam-4983	291	16	0)|	0)|	NOUN
ejpam-4983	291	17	=	=	SYM
ejpam-4983	291	18	1	1	NUM
ejpam-4983	291	19	and	and	CCONJ
ejpam-4983	291	20	|an(r	|an(r	PROPN
ejpam-4983	291	21	,	,	PUNCT
ejpam-4983	291	22	0)|	0)|	NOUN
ejpam-4983	291	23	≥	≥	NOUN
ejpam-4983	291	24	(	(	PUNCT
ejpam-4983	291	25	q	q	NOUN
ejpam-4983	291	26	−	−	PROPN
ejpam-4983	291	27	1)q2(e−1)(qe+1	1)q2(e−1)(qe+1	NUM
ejpam-4983	291	28	+	+	CCONJ
ejpam-4983	291	29	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	291	30	,	,	PUNCT
ejpam-4983	291	31	0)|+	0)|+	NUM
ejpam-4983	291	32	q3n−4|an(r	q3n−4|an(r	NUM
ejpam-4983	291	33	/	/	SYM
ejpam-4983	291	34	γ	γ	X
ejpam-4983	291	35	e−1r	e−1r	PROPN
ejpam-4983	291	36	,	,	PUNCT
ejpam-4983	291	37	0	0	PUNCT
ejpam-4983	292	1	+	+	CCONJ
ejpam-4983	292	2	γe−1r)|	γe−1r)|	NOUN
ejpam-4983	292	3	for	for	ADP
ejpam-4983	292	4	all	all	DET
ejpam-4983	292	5	integers	integer	NOUN
ejpam-4983	292	6	n	n	PRON
ejpam-4983	292	7	≥	≥	NOUN
ejpam-4983	292	8	2	2	NUM
ejpam-4983	292	9	.	.	PUNCT
ejpam-4983	293	1	proof	proof	NOUN
ejpam-4983	293	2	.	.	PUNCT
ejpam-4983	294	1	clearly	clearly	ADV
ejpam-4983	294	2	,	,	PUNCT
ejpam-4983	294	3	|a1(r	|a1(r	ADJ
ejpam-4983	294	4	,	,	PUNCT
ejpam-4983	294	5	0)|	0)|	NOUN
ejpam-4983	294	6	=	=	SYM
ejpam-4983	294	7	1	1	X
ejpam-4983	294	8	.	.	PUNCT
ejpam-4983	295	1	let	let	VERB
ejpam-4983	295	2	n	n	PRON
ejpam-4983	295	3	≥	≥	X
ejpam-4983	295	4	2	2	NUM
ejpam-4983	295	5	be	be	AUX
ejpam-4983	295	6	an	an	DET
ejpam-4983	295	7	integer	integer	NOUN
ejpam-4983	295	8	and	and	CCONJ
ejpam-4983	295	9	let	let	VERB
ejpam-4983	295	10	a	a	DET
ejpam-4983	295	11	=	=	X
ejpam-4983	295	12			ADJ
ejpam-4983	295	13	a11	a11	PROPN
ejpam-4983	295	14	a12	a12	PROPN
ejpam-4983	295	15	a13	a13	PROPN
ejpam-4983	295	16	·	·	PUNCT
ejpam-4983	295	17	·	·	PUNCT
ejpam-4983	295	18	·	·	PUNCT
ejpam-4983	296	1	a1,n−1	a1,n−1	ADJ
ejpam-4983	296	2	a1n	a1n	ADP
ejpam-4983	296	3	a21	a21	PROPN
ejpam-4983	296	4	a22	a22	PROPN
ejpam-4983	296	5	0	0	NUM
ejpam-4983	296	6	·	·	PUNCT
ejpam-4983	296	7	·	·	PUNCT
ejpam-4983	296	8	·	·	PUNCT
ejpam-4983	296	9	0	0	NUM
ejpam-4983	296	10	0	0	NUM
ejpam-4983	296	11	a31	a31	NOUN
ejpam-4983	296	12	0	0	NUM
ejpam-4983	296	13	a33	a33	PROPN
ejpam-4983	296	14	·	·	PUNCT
ejpam-4983	296	15	·	·	PUNCT
ejpam-4983	296	16	·	·	PUNCT
ejpam-4983	296	17	0	0	NUM
ejpam-4983	296	18	0	0	NUM
ejpam-4983	296	19	...	...	PUNCT
ejpam-4983	296	20	...	...	PUNCT
ejpam-4983	296	21	...	...	PUNCT
ejpam-4983	296	22	.	.	PUNCT
ejpam-4983	296	23	.	.	PUNCT
ejpam-4983	296	24	.	.	PUNCT
ejpam-4983	296	25	...	...	PUNCT
ejpam-4983	296	26	...	...	PUNCT
ejpam-4983	297	1	an−1,1	an−1,1	X
ejpam-4983	297	2	0	0	NUM
ejpam-4983	297	3	0	0	NUM
ejpam-4983	297	4	·	·	PUNCT
ejpam-4983	297	5	·	·	PUNCT
ejpam-4983	297	6	·	·	PUNCT
ejpam-4983	297	7	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	297	8	0	0	PUNCT
ejpam-4983	298	1	an1	an1	NOUN
ejpam-4983	298	2	0	0	NUM
ejpam-4983	298	3	0	0	NUM
ejpam-4983	298	4	·	·	PUNCT
ejpam-4983	298	5	·	·	PUNCT
ejpam-4983	298	6	·	·	PUNCT
ejpam-4983	298	7	0	0	NUM
ejpam-4983	299	1	ann	ann	PROPN
ejpam-4983	299	2			PROPN
ejpam-4983	299	3	∈	∈	PROPN
ejpam-4983	299	4	an(r	an(r	NOUN
ejpam-4983	299	5	,	,	PUNCT
ejpam-4983	299	6	0	0	NUM
ejpam-4983	299	7	)	)	PUNCT
ejpam-4983	299	8	.	.	PUNCT
ejpam-4983	300	1	for	for	ADP
ejpam-4983	300	2	convenience	convenience	NOUN
ejpam-4983	300	3	,	,	PUNCT
ejpam-4983	300	4	for	for	ADP
ejpam-4983	300	5	each	each	DET
ejpam-4983	300	6	i	i	PRON
ejpam-4983	300	7	∈	∈	PROPN
ejpam-4983	300	8	{	{	PUNCT
ejpam-4983	300	9	1	1	NUM
ejpam-4983	300	10	,	,	PUNCT
ejpam-4983	300	11	2	2	NUM
ejpam-4983	300	12	,	,	PUNCT
ejpam-4983	300	13	.	.	PUNCT
ejpam-4983	300	14	.	.	PUNCT
ejpam-4983	300	15	.	.	PUNCT
ejpam-4983	301	1	,	,	PUNCT
ejpam-4983	302	1	n	n	CCONJ
ejpam-4983	302	2	}	}	PUNCT
ejpam-4983	302	3	,	,	PUNCT
ejpam-4983	303	1	denote	denote	VERB
ejpam-4983	303	2	by	by	ADP
ejpam-4983	303	3	ri	ri	PROPN
ejpam-4983	303	4	(	(	PUNCT
ejpam-4983	303	5	resp	resp	PROPN
ejpam-4983	303	6	.	.	PROPN
ejpam-4983	303	7	,	,	PUNCT
ejpam-4983	303	8	ci	ci	PROPN
ejpam-4983	303	9	)	)	PUNCT
ejpam-4983	303	10	the	the	DET
ejpam-4983	303	11	ith	ith	PROPN
ejpam-4983	303	12	row	row	NOUN
ejpam-4983	303	13	(	(	PUNCT
ejpam-4983	303	14	resp	resp	NOUN
ejpam-4983	303	15	,	,	PUNCT
ejpam-4983	303	16	ith	ith	PROPN
ejpam-4983	303	17	column	column	NOUN
ejpam-4983	303	18	)	)	PUNCT
ejpam-4983	303	19	of	of	ADP
ejpam-4983	303	20	a.	a.	NOUN
ejpam-4983	303	21	we	we	PRON
ejpam-4983	303	22	consider	consider	VERB
ejpam-4983	303	23	the	the	DET
ejpam-4983	303	24	following	follow	VERB
ejpam-4983	303	25	two	two	NUM
ejpam-4983	303	26	cases	case	NOUN
ejpam-4983	303	27	.	.	PUNCT
ejpam-4983	304	1	case	case	NOUN
ejpam-4983	304	2	1	1	NUM
ejpam-4983	304	3	:	:	PUNCT
ejpam-4983	304	4	a1n	a1n	NOUN
ejpam-4983	304	5	∈	∈	PROPN
ejpam-4983	304	6	u(r	u(r	PROPN
ejpam-4983	304	7	)	)	PUNCT
ejpam-4983	304	8	or	or	CCONJ
ejpam-4983	304	9	ann	ann	PROPN
ejpam-4983	304	10	∈	∈	PROPN
ejpam-4983	304	11	u(r	u(r	PROPN
ejpam-4983	304	12	)	)	PUNCT
ejpam-4983	304	13	.	.	PUNCT
ejpam-4983	305	1	case	case	NOUN
ejpam-4983	305	2	1.1	1.1	NUM
ejpam-4983	305	3	:	:	PUNCT
ejpam-4983	305	4	ann	ann	PROPN
ejpam-4983	305	5	∈	∈	PROPN
ejpam-4983	305	6	u(r	u(r	PROPN
ejpam-4983	305	7	)	)	PUNCT
ejpam-4983	305	8	.	.	PUNCT
ejpam-4983	306	1	using	use	VERB
ejpam-4983	306	2	the	the	DET
ejpam-4983	306	3	elementary	elementary	PROPN
ejpam-4983	306	4	row	row	NOUN
ejpam-4983	306	5	operation	operation	NOUN
ejpam-4983	306	6	r1	r1	NOUN
ejpam-4983	306	7	−	−	PROPN
ejpam-4983	306	8	a1na	a1na	PUNCT
ejpam-4983	306	9	−1	−1	NOUN
ejpam-4983	306	10	nnrn	nnrn	NOUN
ejpam-4983	306	11	→	→	SYM
ejpam-4983	306	12	r1	r1	PROPN
ejpam-4983	306	13	,	,	PUNCT
ejpam-4983	306	14	we	we	PRON
ejpam-4983	306	15	have	have	VERB
ejpam-4983	306	16	that	that	SCONJ
ejpam-4983	306	17	a	a	DET
ejpam-4983	306	18	∼	∼	NOUN
ejpam-4983	306	19			NOUN
ejpam-4983	306	20	0	0	PUNCT
ejpam-4983	306	21	c	c	NOUN
ejpam-4983	306	22	...	...	PUNCT
ejpam-4983	306	23	0	0	PUNCT
ejpam-4983	307	1	an1	an1	NOUN
ejpam-4983	307	2	0	0	NUM
ejpam-4983	307	3	·	·	PUNCT
ejpam-4983	307	4	·	·	PUNCT
ejpam-4983	307	5	·	·	PUNCT
ejpam-4983	307	6	0	0	NUM
ejpam-4983	308	1	ann	ann	PROPN
ejpam-4983	308	2			PROPN
ejpam-4983	308	3	,	,	PUNCT
ejpam-4983	308	4	s.	s.	PROPN
ejpam-4983	308	5	jitman	jitman	PROPN
ejpam-4983	308	6	,	,	PUNCT
ejpam-4983	308	7	p.	p.	PROPN
ejpam-4983	308	8	modjam	modjam	PROPN
ejpam-4983	308	9	/	/	SYM
ejpam-4983	308	10	eur	eur	PROPN
ejpam-4983	308	11	.	.	PUNCT
ejpam-4983	309	1	j.	j.	PROPN
ejpam-4983	309	2	pure	pure	PROPN
ejpam-4983	309	3	appl	appl	PROPN
ejpam-4983	309	4	.	.	PROPN
ejpam-4983	309	5	math	math	PROPN
ejpam-4983	309	6	,	,	PUNCT
ejpam-4983	309	7	17	17	NUM
ejpam-4983	309	8	(	(	PUNCT
ejpam-4983	309	9	1	1	NUM
ejpam-4983	309	10	)	)	PUNCT
ejpam-4983	309	11	(	(	PUNCT
ejpam-4983	309	12	2024	2024	NUM
ejpam-4983	309	13	)	)	PUNCT
ejpam-4983	309	14	,	,	PUNCT
ejpam-4983	309	15	11	11	NUM
ejpam-4983	309	16	-	-	SYM
ejpam-4983	309	17	29	29	NUM
ejpam-4983	309	18	22	22	NUM
ejpam-4983	309	19	where	where	SCONJ
ejpam-4983	309	20	c	c	NOUN
ejpam-4983	309	21	=	=	PUNCT
ejpam-4983	309	22			PROPN
ejpam-4983	309	23	a11	a11	PROPN
ejpam-4983	309	24	−	−	PROPN
ejpam-4983	309	25	a1nan1ann	a1nan1ann	PROPN
ejpam-4983	309	26	−1	−1	NOUN
ejpam-4983	309	27	a12	a12	NOUN
ejpam-4983	309	28	a13	a13	NOUN
ejpam-4983	309	29	·	·	PUNCT
ejpam-4983	309	30	·	·	PUNCT
ejpam-4983	309	31	·	·	PUNCT
ejpam-4983	310	1	a1,n−1	a1,n−1	ADJ
ejpam-4983	310	2	a21	a21	PROPN
ejpam-4983	310	3	a22	a22	PROPN
ejpam-4983	310	4	0	0	NUM
ejpam-4983	310	5	·	·	PUNCT
ejpam-4983	310	6	·	·	PUNCT
ejpam-4983	310	7	·	·	PUNCT
ejpam-4983	310	8	0	0	NUM
ejpam-4983	311	1	a31	a31	NOUN
ejpam-4983	311	2	0	0	NUM
ejpam-4983	311	3	a33	a33	PROPN
ejpam-4983	311	4	·	·	PUNCT
ejpam-4983	311	5	·	·	PUNCT
ejpam-4983	311	6	·	·	PUNCT
ejpam-4983	311	7	0	0	NUM
ejpam-4983	311	8	...	...	PUNCT
ejpam-4983	311	9	...	...	PUNCT
ejpam-4983	311	10	...	...	PUNCT
ejpam-4983	311	11	.	.	PUNCT
ejpam-4983	311	12	.	.	PUNCT
ejpam-4983	311	13	.	.	PUNCT
ejpam-4983	312	1	...	...	PUNCT
ejpam-4983	313	1	an−1,1	an−1,1	X
ejpam-4983	313	2	0	0	NUM
ejpam-4983	313	3	0	0	NUM
ejpam-4983	313	4	·	·	PUNCT
ejpam-4983	313	5	·	·	PUNCT
ejpam-4983	313	6	·	·	PUNCT
ejpam-4983	313	7	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	313	8			PROPN
ejpam-4983	313	9	.	.	PUNCT
ejpam-4983	314	1	then	then	ADV
ejpam-4983	314	2	det(a	det(a	PROPN
ejpam-4983	314	3	)	)	PUNCT
ejpam-4983	314	4	=	=	NOUN
ejpam-4983	315	1	(	(	PUNCT
ejpam-4983	315	2	−1)n+nann	−1)n+nann	PROPN
ejpam-4983	315	3	det(c	det(c	PROPN
ejpam-4983	315	4	)	)	PUNCT
ejpam-4983	315	5	=	=	SYM
ejpam-4983	315	6	ann	ann	PROPN
ejpam-4983	315	7	det(c	det(c	PROPN
ejpam-4983	315	8	)	)	PUNCT
ejpam-4983	315	9	.	.	PUNCT
ejpam-4983	316	1	(	(	PUNCT
ejpam-4983	316	2	3	3	X
ejpam-4983	316	3	)	)	PUNCT
ejpam-4983	316	4	let	let	VERB
ejpam-4983	316	5	t	t	NOUN
ejpam-4983	316	6	=	=	SYM
ejpam-4983	316	7			X
ejpam-4983	316	8			ADJ
ejpam-4983	316	9	t11	t11	NOUN
ejpam-4983	316	10	t12	t12	PROPN
ejpam-4983	316	11	t13	t13	NOUN
ejpam-4983	316	12	·	·	PUNCT
ejpam-4983	316	13	·	·	PUNCT
ejpam-4983	316	14	·	·	PUNCT
ejpam-4983	316	15	t1,n−1	t1,n−1	ADJ
ejpam-4983	316	16	t21	t21	PROPN
ejpam-4983	316	17	t22	t22	PROPN
ejpam-4983	316	18	0	0	PUNCT
ejpam-4983	316	19	·	·	PUNCT
ejpam-4983	316	20	·	·	PUNCT
ejpam-4983	316	21	·	·	PUNCT
ejpam-4983	316	22	0	0	NUM
ejpam-4983	317	1	t31	t31	NOUN
ejpam-4983	317	2	0	0	NUM
ejpam-4983	317	3	t33	t33	PROPN
ejpam-4983	317	4	·	·	PUNCT
ejpam-4983	317	5	·	·	PUNCT
ejpam-4983	317	6	·	·	PUNCT
ejpam-4983	317	7	0	0	NUM
ejpam-4983	317	8	...	...	PUNCT
ejpam-4983	317	9	...	...	PUNCT
ejpam-4983	317	10	...	...	PUNCT
ejpam-4983	317	11	.	.	PUNCT
ejpam-4983	317	12	.	.	PUNCT
ejpam-4983	317	13	.	.	PUNCT
ejpam-4983	318	1	...	...	PUNCT
ejpam-4983	319	1	tn−1,1	tn−1,1	SYM
ejpam-4983	319	2	0	0	NUM
ejpam-4983	319	3	0	0	NUM
ejpam-4983	319	4	·	·	PUNCT
ejpam-4983	319	5	·	·	PUNCT
ejpam-4983	319	6	·	·	PUNCT
ejpam-4983	319	7	tn−1,n−1	tn−1,n−1	ADP
ejpam-4983	319	8			PROPN
ejpam-4983	319	9	∈	∈	PROPN
ejpam-4983	319	10	an−1(r	an−1(r	NOUN
ejpam-4983	319	11	)	)	PUNCT
ejpam-4983	319	12	∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣	PROPN
ejpam-4983	319	13	det	det	PROPN
ejpam-4983	319	14			ADJ
ejpam-4983	319	15			ADJ
ejpam-4983	319	16	t11	t11	NOUN
ejpam-4983	319	17	−	−	PROPN
ejpam-4983	319	18	a1nan1a	a1nan1a	PROPN
ejpam-4983	319	19	−1	−1	NOUN
ejpam-4983	319	20	nn	nn	PROPN
ejpam-4983	319	21	t12	t12	PROPN
ejpam-4983	319	22	t13	t13	X
ejpam-4983	319	23	·	·	PUNCT
ejpam-4983	319	24	·	·	PUNCT
ejpam-4983	319	25	·	·	PUNCT
ejpam-4983	320	1	t1,n−1	t1,n−1	ADJ
ejpam-4983	320	2	t21	t21	PROPN
ejpam-4983	320	3	t22	t22	PROPN
ejpam-4983	320	4	0	0	PUNCT
ejpam-4983	320	5	·	·	PUNCT
ejpam-4983	320	6	·	·	PUNCT
ejpam-4983	320	7	·	·	PUNCT
ejpam-4983	320	8	0	0	NUM
ejpam-4983	320	9	t31	t31	NOUN
ejpam-4983	320	10	0	0	NUM
ejpam-4983	320	11	t33	t33	PROPN
ejpam-4983	320	12	·	·	PUNCT
ejpam-4983	320	13	·	·	PUNCT
ejpam-4983	320	14	·	·	PUNCT
ejpam-4983	320	15	0	0	NUM
ejpam-4983	320	16	...	...	PUNCT
ejpam-4983	320	17	...	...	PUNCT
ejpam-4983	320	18	...	...	PUNCT
ejpam-4983	320	19	.	.	PUNCT
ejpam-4983	320	20	.	.	PUNCT
ejpam-4983	320	21	.	.	PUNCT
ejpam-4983	321	1	...	...	PUNCT
ejpam-4983	322	1	tn−1,1	tn−1,1	SYM
ejpam-4983	322	2	0	0	NUM
ejpam-4983	322	3	0	0	NUM
ejpam-4983	322	4	·	·	PUNCT
ejpam-4983	322	5	·	·	PUNCT
ejpam-4983	322	6	·	·	PUNCT
ejpam-4983	322	7	tn−1,n−1	tn−1,n−1	ADP
ejpam-4983	322	8			PRON
ejpam-4983	322	9			PUNCT
ejpam-4983	322	10	=	=	SYM
ejpam-4983	322	11	0	0	NUM
ejpam-4983	322	12			NOUN
ejpam-4983	322	13	.	.	PUNCT
ejpam-4983	323	1	since	since	SCONJ
ejpam-4983	323	2			PROPN
ejpam-4983	323	3	t11	t11	PROPN
ejpam-4983	323	4	t12	t12	PROPN
ejpam-4983	323	5	t13	t13	NOUN
ejpam-4983	323	6	·	·	PUNCT
ejpam-4983	323	7	·	·	PUNCT
ejpam-4983	323	8	·	·	PUNCT
ejpam-4983	323	9	t1,n−1	t1,n−1	ADJ
ejpam-4983	323	10	t21	t21	PROPN
ejpam-4983	323	11	t22	t22	PROPN
ejpam-4983	323	12	0	0	PUNCT
ejpam-4983	323	13	·	·	PUNCT
ejpam-4983	323	14	·	·	PUNCT
ejpam-4983	323	15	·	·	PUNCT
ejpam-4983	323	16	0	0	NUM
ejpam-4983	324	1	t31	t31	NOUN
ejpam-4983	324	2	0	0	NUM
ejpam-4983	324	3	t33	t33	PROPN
ejpam-4983	324	4	·	·	PUNCT
ejpam-4983	324	5	·	·	PUNCT
ejpam-4983	324	6	·	·	PUNCT
ejpam-4983	324	7	0	0	NUM
ejpam-4983	324	8	...	...	PUNCT
ejpam-4983	324	9	...	...	PUNCT
ejpam-4983	324	10	...	...	PUNCT
ejpam-4983	324	11	.	.	PUNCT
ejpam-4983	324	12	.	.	PUNCT
ejpam-4983	324	13	.	.	PUNCT
ejpam-4983	325	1	...	...	PUNCT
ejpam-4983	326	1	tn−1,1	tn−1,1	SYM
ejpam-4983	326	2	0	0	NUM
ejpam-4983	326	3	0	0	NUM
ejpam-4983	326	4	·	·	PUNCT
ejpam-4983	326	5	·	·	PUNCT
ejpam-4983	326	6	·	·	PUNCT
ejpam-4983	326	7	tn−1,n−1	tn−1,n−1	ADP
ejpam-4983	326	8			PROPN
ejpam-4983	326	9	∈	∈	PROPN
ejpam-4983	326	10	t	t	NOUN
ejpam-4983	327	1	if	if	SCONJ
ejpam-4983	327	2	and	and	CCONJ
ejpam-4983	327	3	only	only	ADV
ejpam-4983	327	4	if	if	SCONJ
ejpam-4983	327	5			ADJ
ejpam-4983	327	6	t11	t11	NOUN
ejpam-4983	327	7	−	−	PROPN
ejpam-4983	327	8	a1nan1a	a1nan1a	PROPN
ejpam-4983	327	9	−1	−1	NOUN
ejpam-4983	327	10	nn	nn	PROPN
ejpam-4983	327	11	t12	t12	PROPN
ejpam-4983	327	12	t13	t13	X
ejpam-4983	327	13	·	·	PUNCT
ejpam-4983	327	14	·	·	PUNCT
ejpam-4983	327	15	·	·	PUNCT
ejpam-4983	328	1	t1,n−1	t1,n−1	ADJ
ejpam-4983	328	2	t21	t21	PROPN
ejpam-4983	328	3	t22	t22	PROPN
ejpam-4983	328	4	0	0	PUNCT
ejpam-4983	328	5	·	·	PUNCT
ejpam-4983	328	6	·	·	PUNCT
ejpam-4983	328	7	·	·	PUNCT
ejpam-4983	328	8	0	0	NUM
ejpam-4983	328	9	t31	t31	NOUN
ejpam-4983	328	10	0	0	NUM
ejpam-4983	328	11	t33	t33	PROPN
ejpam-4983	328	12	·	·	PUNCT
ejpam-4983	328	13	·	·	PUNCT
ejpam-4983	328	14	·	·	PUNCT
ejpam-4983	328	15	0	0	NUM
ejpam-4983	328	16	...	...	PUNCT
ejpam-4983	328	17	...	...	PUNCT
ejpam-4983	328	18	...	...	PUNCT
ejpam-4983	328	19	.	.	PUNCT
ejpam-4983	328	20	.	.	PUNCT
ejpam-4983	328	21	.	.	PUNCT
ejpam-4983	329	1	...	...	PUNCT
ejpam-4983	330	1	tn−1,1	tn−1,1	SYM
ejpam-4983	330	2	0	0	NUM
ejpam-4983	330	3	0	0	NUM
ejpam-4983	330	4	·	·	PUNCT
ejpam-4983	330	5	·	·	PUNCT
ejpam-4983	330	6	·	·	PUNCT
ejpam-4983	330	7	tn−1,n−1	tn−1,n−1	ADP
ejpam-4983	330	8			PROPN
ejpam-4983	330	9	∈	∈	PROPN
ejpam-4983	330	10	an−1(r	an−1(r	NOUN
ejpam-4983	330	11	,	,	PUNCT
ejpam-4983	330	12	0	0	NUM
ejpam-4983	330	13	)	)	PUNCT
ejpam-4983	330	14	,	,	PUNCT
ejpam-4983	330	15	it	it	PRON
ejpam-4983	330	16	follows	follow	VERB
ejpam-4983	330	17	that	that	PRON
ejpam-4983	330	18	|t	|t	VERB
ejpam-4983	331	1	|	|	ADV
ejpam-4983	331	2	=	=	SYM
ejpam-4983	331	3	|an−1(r	|an−1(r	NOUN
ejpam-4983	331	4	,	,	PUNCT
ejpam-4983	331	5	0)|	0)|	NOUN
ejpam-4983	331	6	.	.	PUNCT
ejpam-4983	332	1	from	from	ADP
ejpam-4983	332	2	(	(	PUNCT
ejpam-4983	332	3	3	3	NUM
ejpam-4983	332	4	)	)	PUNCT
ejpam-4983	332	5	,	,	PUNCT
ejpam-4983	332	6	det(a	det(a	PROPN
ejpam-4983	332	7	)	)	PUNCT
ejpam-4983	332	8	=	=	SYM
ejpam-4983	332	9	0	0	PUNCT
ejpam-4983	333	1	if	if	SCONJ
ejpam-4983	333	2	and	and	CCONJ
ejpam-4983	333	3	only	only	ADV
ejpam-4983	333	4	if	if	SCONJ
ejpam-4983	333	5	det(c	det(c	VERB
ejpam-4983	333	6	)	)	PUNCT
ejpam-4983	333	7	=	=	SYM
ejpam-4983	333	8	0	0	X
ejpam-4983	333	9	.	.	PUNCT
ejpam-4983	334	1	the	the	DET
ejpam-4983	334	2	number	number	NOUN
ejpam-4983	334	3	of	of	ADP
ejpam-4983	334	4	matrices	matrix	NOUN
ejpam-4983	334	5	c	c	NOUN
ejpam-4983	334	6	with	with	ADP
ejpam-4983	334	7	determinant	determinant	ADJ
ejpam-4983	334	8	0	0	NUM
ejpam-4983	334	9	is	be	AUX
ejpam-4983	334	10	|t	|t	VERB
ejpam-4983	334	11	|	|	ADV
ejpam-4983	334	12	=	=	SYM
ejpam-4983	334	13	|an−1(r	|an−1(r	NOUN
ejpam-4983	334	14	,	,	PUNCT
ejpam-4983	334	15	0)|	0)|	NOUN
ejpam-4983	334	16	.	.	PUNCT
ejpam-4983	335	1	the	the	DET
ejpam-4983	335	2	number	number	NOUN
ejpam-4983	335	3	of	of	ADP
ejpam-4983	335	4	choices	choice	NOUN
ejpam-4983	335	5	for	for	ADP
ejpam-4983	335	6	an1	an1	PROPN
ejpam-4983	335	7	is	be	AUX
ejpam-4983	335	8	qe	qe	PROPN
ejpam-4983	335	9	,	,	PUNCT
ejpam-4983	335	10	the	the	DET
ejpam-4983	335	11	number	number	NOUN
ejpam-4983	335	12	of	of	ADP
ejpam-4983	335	13	choices	choice	NOUN
ejpam-4983	335	14	for	for	ADP
ejpam-4983	335	15	a1n	a1n	PUNCT
ejpam-4983	335	16	is	be	AUX
ejpam-4983	335	17	qe	qe	PROPN
ejpam-4983	335	18	,	,	PUNCT
ejpam-4983	335	19	and	and	CCONJ
ejpam-4983	335	20	the	the	DET
ejpam-4983	335	21	number	number	NOUN
ejpam-4983	335	22	of	of	ADP
ejpam-4983	335	23	choices	choice	NOUN
ejpam-4983	335	24	for	for	ADP
ejpam-4983	335	25	ann	ann	PROPN
ejpam-4983	335	26	is	be	AUX
ejpam-4983	335	27	(	(	PUNCT
ejpam-4983	335	28	q	q	PROPN
ejpam-4983	335	29	−	−	PROPN
ejpam-4983	335	30	1)qe−1	1)qe−1	PROPN
ejpam-4983	335	31	.	.	PUNCT
ejpam-4983	336	1	in	in	ADP
ejpam-4983	336	2	this	this	DET
ejpam-4983	336	3	case	case	NOUN
ejpam-4983	336	4	,	,	PUNCT
ejpam-4983	336	5	the	the	DET
ejpam-4983	336	6	possible	possible	ADJ
ejpam-4983	336	7	choices	choice	NOUN
ejpam-4983	336	8	for	for	ADP
ejpam-4983	336	9	a	a	DET
ejpam-4983	336	10	is	be	AUX
ejpam-4983	336	11	(	(	PUNCT
ejpam-4983	336	12	q	q	NOUN
ejpam-4983	336	13	−	−	PROPN
ejpam-4983	336	14	1	1	NUM
ejpam-4983	336	15	)	)	PUNCT
ejpam-4983	336	16	q3e−1|an−1(r	q3e−1|an−1(r	ADJ
ejpam-4983	336	17	,	,	PUNCT
ejpam-4983	336	18	0)|	0)|	NOUN
ejpam-4983	336	19	.	.	PUNCT
ejpam-4983	337	1	s.	s.	PROPN
ejpam-4983	337	2	jitman	jitman	PROPN
ejpam-4983	337	3	,	,	PUNCT
ejpam-4983	337	4	p.	p.	PROPN
ejpam-4983	337	5	modjam	modjam	PROPN
ejpam-4983	337	6	/	/	SYM
ejpam-4983	337	7	eur	eur	PROPN
ejpam-4983	337	8	.	.	PUNCT
ejpam-4983	338	1	j.	j.	PROPN
ejpam-4983	338	2	pure	pure	PROPN
ejpam-4983	338	3	appl	appl	PROPN
ejpam-4983	338	4	.	.	PROPN
ejpam-4983	338	5	math	math	PROPN
ejpam-4983	338	6	,	,	PUNCT
ejpam-4983	338	7	17	17	NUM
ejpam-4983	338	8	(	(	PUNCT
ejpam-4983	338	9	1	1	NUM
ejpam-4983	338	10	)	)	PUNCT
ejpam-4983	338	11	(	(	PUNCT
ejpam-4983	338	12	2024	2024	NUM
ejpam-4983	338	13	)	)	PUNCT
ejpam-4983	338	14	,	,	PUNCT
ejpam-4983	338	15	11	11	NUM
ejpam-4983	338	16	-	-	SYM
ejpam-4983	338	17	29	29	NUM
ejpam-4983	338	18	23	23	NUM
ejpam-4983	338	19	case	case	NOUN
ejpam-4983	338	20	1.2	1.2	NUM
ejpam-4983	338	21	:	:	PUNCT
ejpam-4983	338	22	a1n	a1n	NOUN
ejpam-4983	338	23	∈	∈	PROPN
ejpam-4983	338	24	u(r	u(r	PROPN
ejpam-4983	338	25	)	)	PUNCT
ejpam-4983	338	26	and	and	CCONJ
ejpam-4983	338	27	ann	ann	PROPN
ejpam-4983	338	28	/∈	/∈	PUNCT
ejpam-4983	339	1	u(r	u(r	ADV
ejpam-4983	339	2	)	)	PUNCT
ejpam-4983	339	3	.	.	PUNCT
ejpam-4983	340	1	let	let	VERB
ejpam-4983	340	2	d	d	NOUN
ejpam-4983	340	3	=	=	SYM
ejpam-4983	340	4			PROPN
ejpam-4983	340	5	a11	a11	PROPN
ejpam-4983	340	6	a12	a12	PROPN
ejpam-4983	340	7	a13	a13	PROPN
ejpam-4983	340	8	·	·	PUNCT
ejpam-4983	340	9	·	·	PUNCT
ejpam-4983	340	10	·	·	PUNCT
ejpam-4983	340	11	a1,n−1	a1,n−1	ADJ
ejpam-4983	340	12	a21	a21	PROPN
ejpam-4983	340	13	a22	a22	PROPN
ejpam-4983	340	14	0	0	NUM
ejpam-4983	340	15	·	·	PUNCT
ejpam-4983	340	16	·	·	PUNCT
ejpam-4983	340	17	·	·	PUNCT
ejpam-4983	340	18	0	0	NUM
ejpam-4983	341	1	a31	a31	NOUN
ejpam-4983	341	2	0	0	NUM
ejpam-4983	341	3	a33	a33	PROPN
ejpam-4983	341	4	·	·	PUNCT
ejpam-4983	341	5	·	·	PUNCT
ejpam-4983	341	6	·	·	PUNCT
ejpam-4983	341	7	0	0	NUM
ejpam-4983	341	8	...	...	PUNCT
ejpam-4983	341	9	...	...	PUNCT
ejpam-4983	341	10	...	...	PUNCT
ejpam-4983	341	11	.	.	PUNCT
ejpam-4983	341	12	.	.	PUNCT
ejpam-4983	341	13	.	.	PUNCT
ejpam-4983	342	1	...	...	PUNCT
ejpam-4983	343	1	an−1,1	an−1,1	X
ejpam-4983	343	2	0	0	NUM
ejpam-4983	343	3	0	0	NUM
ejpam-4983	343	4	·	·	PUNCT
ejpam-4983	343	5	·	·	PUNCT
ejpam-4983	343	6	·	·	PUNCT
ejpam-4983	343	7	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	343	8			PROPN
ejpam-4983	343	9	.	.	PUNCT
ejpam-4983	344	1	using	use	VERB
ejpam-4983	344	2	the	the	DET
ejpam-4983	344	3	cofactor	cofactor	NOUN
ejpam-4983	344	4	expansion	expansion	NOUN
ejpam-4983	344	5	through	through	ADP
ejpam-4983	344	6	the	the	DET
ejpam-4983	344	7	last	last	ADJ
ejpam-4983	344	8	column	column	NOUN
ejpam-4983	344	9	of	of	ADP
ejpam-4983	344	10	a	a	PRON
ejpam-4983	344	11	,	,	PUNCT
ejpam-4983	344	12	it	it	PRON
ejpam-4983	344	13	follows	follow	VERB
ejpam-4983	344	14	that	that	PRON
ejpam-4983	344	15	det(a	det(a	NOUN
ejpam-4983	344	16	)	)	PUNCT
ejpam-4983	344	17	=	=	NOUN
ejpam-4983	344	18	(	(	PUNCT
ejpam-4983	344	19	−1)n+1(−1)n−1	−1)n+1(−1)n−1	PROPN
ejpam-4983	344	20	+	+	NOUN
ejpam-4983	344	21	1a1nan1diag(a22	1a1nan1diag(a22	NUM
ejpam-4983	344	22	,	,	PUNCT
ejpam-4983	344	23	a33	a33	NOUN
ejpam-4983	344	24	,	,	PUNCT
ejpam-4983	344	25	.	.	PUNCT
ejpam-4983	344	26	.	.	PUNCT
ejpam-4983	345	1	.	.	PUNCT
ejpam-4983	346	1	,	,	PUNCT
ejpam-4983	346	2	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	346	3	)	)	PUNCT
ejpam-4983	346	4	+	+	CCONJ
ejpam-4983	346	5	(	(	PUNCT
ejpam-4983	346	6	−1)n+nann	−1)n+nann	PROPN
ejpam-4983	346	7	det(d	det(d	PROPN
ejpam-4983	346	8	)	)	PUNCT
ejpam-4983	346	9	=	=	SYM
ejpam-4983	347	1	−a1nan1diag(a22	−a1nan1diag(a22	NOUN
ejpam-4983	347	2	,	,	PUNCT
ejpam-4983	347	3	a33	a33	PROPN
ejpam-4983	347	4	,	,	PUNCT
ejpam-4983	347	5	.	.	PUNCT
ejpam-4983	347	6	.	.	PUNCT
ejpam-4983	347	7	.	.	PUNCT
ejpam-4983	348	1	,	,	PUNCT
ejpam-4983	348	2	an−1,n−1	an−1,n−1	ADJ
ejpam-4983	348	3	)	)	PUNCT
ejpam-4983	348	4	+	+	CCONJ
ejpam-4983	348	5	ann	ann	PROPN
ejpam-4983	348	6	det(d	det(d	PROPN
ejpam-4983	348	7	)	)	PUNCT
ejpam-4983	348	8	.	.	PUNCT
ejpam-4983	349	1	(	(	PUNCT
ejpam-4983	349	2	4	4	X
ejpam-4983	349	3	)	)	PUNCT
ejpam-4983	349	4	it	it	PRON
ejpam-4983	349	5	is	be	AUX
ejpam-4983	349	6	easily	easily	ADV
ejpam-4983	349	7	seen	see	VERB
ejpam-4983	349	8	that	that	PRON
ejpam-4983	349	9	det(a	det(a	NOUN
ejpam-4983	349	10	)	)	PUNCT
ejpam-4983	349	11	=	=	SYM
ejpam-4983	350	1	0	0	PUNCT
ejpam-4983	351	1	whenever	whenever	SCONJ
ejpam-4983	351	2	an1	an1	PROPN
ejpam-4983	351	3	=	=	PUNCT
ejpam-4983	351	4	0	0	PUNCT
ejpam-4983	351	5	and	and	CCONJ
ejpam-4983	351	6	d	d	PROPN
ejpam-4983	351	7	∈	∈	PROPN
ejpam-4983	351	8	an−1(r	an−1(r	NOUN
ejpam-4983	351	9	,	,	PUNCT
ejpam-4983	351	10	0	0	NUM
ejpam-4983	351	11	)	)	PUNCT
ejpam-4983	351	12	.	.	PUNCT
ejpam-4983	352	1	the	the	DET
ejpam-4983	352	2	number	number	NOUN
ejpam-4983	352	3	of	of	ADP
ejpam-4983	352	4	choices	choice	NOUN
ejpam-4983	352	5	for	for	ADP
ejpam-4983	352	6	a1n	a1n	PUNCT
ejpam-4983	352	7	is	be	AUX
ejpam-4983	352	8	(	(	PUNCT
ejpam-4983	352	9	q	q	PROPN
ejpam-4983	352	10	−	−	PROPN
ejpam-4983	352	11	1)qe−1	1)qe−1	PROPN
ejpam-4983	352	12	,	,	PUNCT
ejpam-4983	352	13	the	the	DET
ejpam-4983	352	14	number	number	NOUN
ejpam-4983	352	15	of	of	ADP
ejpam-4983	352	16	choices	choice	NOUN
ejpam-4983	352	17	for	for	ADP
ejpam-4983	352	18	ann	ann	PROPN
ejpam-4983	352	19	is	be	AUX
ejpam-4983	352	20	qe−1	qe−1	PROPN
ejpam-4983	352	21	,	,	PUNCT
ejpam-4983	352	22	and	and	CCONJ
ejpam-4983	352	23	the	the	DET
ejpam-4983	352	24	number	number	NOUN
ejpam-4983	352	25	of	of	ADP
ejpam-4983	352	26	choices	choice	NOUN
ejpam-4983	352	27	for	for	ADP
ejpam-4983	352	28	d	d	PROPN
ejpam-4983	352	29	is	be	AUX
ejpam-4983	352	30	|an−1(r	|an−1(r	NOUN
ejpam-4983	352	31	,	,	PUNCT
ejpam-4983	352	32	0)|	0)|	NOUN
ejpam-4983	352	33	.	.	PUNCT
ejpam-4983	353	1	in	in	ADP
ejpam-4983	353	2	this	this	DET
ejpam-4983	353	3	case	case	NOUN
ejpam-4983	353	4	,	,	PUNCT
ejpam-4983	353	5	the	the	DET
ejpam-4983	353	6	possible	possible	ADJ
ejpam-4983	353	7	choices	choice	NOUN
ejpam-4983	353	8	for	for	ADP
ejpam-4983	353	9	a	a	PRON
ejpam-4983	353	10	is	be	AUX
ejpam-4983	353	11	at	at	ADP
ejpam-4983	353	12	least	least	ADJ
ejpam-4983	353	13	(	(	PUNCT
ejpam-4983	353	14	q	q	NOUN
ejpam-4983	353	15	−	−	PROPN
ejpam-4983	353	16	1	1	NUM
ejpam-4983	353	17	)	)	PUNCT
ejpam-4983	353	18	q2(e−1)|an−1(r	q2(e−1)|an−1(r	PROPN
ejpam-4983	353	19	,	,	PUNCT
ejpam-4983	353	20	0)|	0)|	NOUN
ejpam-4983	353	21	.	.	PUNCT
ejpam-4983	354	1	case	case	NOUN
ejpam-4983	354	2	2	2	NUM
ejpam-4983	354	3	:	:	PUNCT
ejpam-4983	354	4	ann	ann	PROPN
ejpam-4983	354	5	/∈	/∈	PUNCT
ejpam-4983	355	1	u(r	u(r	ADV
ejpam-4983	355	2	)	)	PUNCT
ejpam-4983	355	3	and	and	CCONJ
ejpam-4983	355	4	a1n	a1n	PRON
ejpam-4983	355	5	/∈	/∈	PUNCT
ejpam-4983	355	6	u(r	u(r	NOUN
ejpam-4983	355	7	)	)	PUNCT
ejpam-4983	355	8	.	.	PUNCT
ejpam-4983	356	1	then	then	ADV
ejpam-4983	356	2	the	the	DET
ejpam-4983	356	3	elements	element	NOUN
ejpam-4983	356	4	in	in	ADP
ejpam-4983	356	5	the	the	DET
ejpam-4983	356	6	last	last	ADJ
ejpam-4983	356	7	column	column	NOUN
ejpam-4983	356	8	are	be	AUX
ejpam-4983	356	9	in	in	ADP
ejpam-4983	356	10	γr	γr	PROPN
ejpam-4983	356	11	.	.	PUNCT
ejpam-4983	357	1	let	let	VERB
ejpam-4983	357	2	b	b	NOUN
ejpam-4983	358	1	=	=	PUNCT
ejpam-4983	358	2	[	[	X
ejpam-4983	358	3	bij	bij	NOUN
ejpam-4983	358	4	]	]	PUNCT
ejpam-4983	358	5	be	be	AUX
ejpam-4983	358	6	the	the	DET
ejpam-4983	358	7	matrix	matrix	NOUN
ejpam-4983	358	8	in	in	ADP
ejpam-4983	358	9	an(r	an(r	NOUN
ejpam-4983	358	10	)	)	PUNCT
ejpam-4983	358	11	be	be	AUX
ejpam-4983	358	12	defined	define	VERB
ejpam-4983	358	13	by	by	ADP
ejpam-4983	358	14	bij	bij	NOUN
ejpam-4983	358	15	=	=	PUNCT
ejpam-4983	358	16	{	{	PUNCT
ejpam-4983	358	17	wij	wij	X
ejpam-4983	358	18	if	if	SCONJ
ejpam-4983	358	19	(	(	PUNCT
ejpam-4983	358	20	i	i	PROPN
ejpam-4983	358	21	,	,	PUNCT
ejpam-4983	358	22	j	j	PROPN
ejpam-4983	358	23	)	)	PUNCT
ejpam-4983	358	24	∈	∈	PROPN
ejpam-4983	358	25	{	{	PUNCT
ejpam-4983	358	26	(	(	PUNCT
ejpam-4983	358	27	1	1	NUM
ejpam-4983	358	28	,	,	PUNCT
ejpam-4983	358	29	n	n	CCONJ
ejpam-4983	358	30	)	)	PUNCT
ejpam-4983	358	31	,	,	PUNCT
ejpam-4983	358	32	(	(	PUNCT
ejpam-4983	358	33	n	n	X
ejpam-4983	358	34	,	,	PUNCT
ejpam-4983	358	35	n	n	CCONJ
ejpam-4983	358	36	)	)	PUNCT
ejpam-4983	358	37	}	}	PUNCT
ejpam-4983	358	38	aij	aij	PROPN
ejpam-4983	358	39	otherwise	otherwise	ADV
ejpam-4983	358	40	,	,	PUNCT
ejpam-4983	358	41	where	where	SCONJ
ejpam-4983	358	42	a1n	a1n	ADP
ejpam-4983	358	43	=	=	PUNCT
ejpam-4983	358	44	γw1n	γw1n	PROPN
ejpam-4983	358	45	and	and	CCONJ
ejpam-4983	358	46	ann	ann	PROPN
ejpam-4983	358	47	=	=	PROPN
ejpam-4983	358	48	γwnn	γwnn	NOUN
ejpam-4983	358	49	for	for	ADP
ejpam-4983	358	50	some	some	PRON
ejpam-4983	358	51	for	for	ADP
ejpam-4983	358	52	some	some	DET
ejpam-4983	358	53	w1n	w1n	NOUN
ejpam-4983	358	54	,	,	PUNCT
ejpam-4983	358	55	wnn	wnn	PROPN
ejpam-4983	358	56	∈	∈	PROPN
ejpam-4983	358	57	e−2∑	e−2∑	ADV
ejpam-4983	358	58	j=0	j=0	PROPN
ejpam-4983	358	59	γjv	γjv	NOUN
ejpam-4983	358	60	and	and	CCONJ
ejpam-4983	358	61	v	v	NOUN
ejpam-4983	358	62	is	be	AUX
ejpam-4983	358	63	defined	define	VERB
ejpam-4983	358	64	in	in	ADP
ejpam-4983	358	65	lemma	lemma	PROPN
ejpam-4983	358	66	1	1	NUM
ejpam-4983	358	67	.	.	PUNCT
ejpam-4983	359	1	let	let	VERB
ejpam-4983	359	2	c	c	NOUN
ejpam-4983	359	3	=	=	PUNCT
ejpam-4983	360	1	[	[	X
ejpam-4983	360	2	cij	cij	PROPN
ejpam-4983	360	3	]	]	PUNCT
ejpam-4983	360	4	be	be	AUX
ejpam-4983	360	5	the	the	DET
ejpam-4983	360	6	matrix	matrix	NOUN
ejpam-4983	360	7	in	in	ADP
ejpam-4983	360	8	an(r	an(r	NOUN
ejpam-4983	360	9	/	/	SYM
ejpam-4983	360	10	γ	γ	X
ejpam-4983	360	11	e−1r	e−1r	NOUN
ejpam-4983	360	12	)	)	PUNCT
ejpam-4983	360	13	defined	define	VERB
ejpam-4983	360	14	by	by	ADP
ejpam-4983	360	15	cij	cij	PROPN
ejpam-4983	360	16	=	=	SYM
ejpam-4983	360	17	bij	bij	PROPN
ejpam-4983	360	18	+	+	X
ejpam-4983	360	19	γe−1r	γe−1r	NUM
ejpam-4983	360	20	.	.	PUNCT
ejpam-4983	361	1	we	we	PRON
ejpam-4983	361	2	note	note	VERB
ejpam-4983	361	3	that	that	SCONJ
ejpam-4983	361	4	det(a	det(a	NOUN
ejpam-4983	361	5	)	)	PUNCT
ejpam-4983	361	6	=	=	PUNCT
ejpam-4983	361	7	γ	γ	X
ejpam-4983	361	8	det(b	det(b	PROPN
ejpam-4983	361	9	)	)	PUNCT
ejpam-4983	361	10	∈	∈	PROPN
ejpam-4983	361	11	r.	r.	PROPN
ejpam-4983	361	12	then	then	ADV
ejpam-4983	361	13	det(a	det(a	PROPN
ejpam-4983	361	14	)	)	PUNCT
ejpam-4983	361	15	=	=	SYM
ejpam-4983	361	16	0	0	NUM
ejpam-4983	362	1	in	in	ADP
ejpam-4983	362	2	r	r	NOUN
ejpam-4983	362	3	if	if	SCONJ
ejpam-4983	362	4	and	and	CCONJ
ejpam-4983	362	5	only	only	ADV
ejpam-4983	362	6	if	if	SCONJ
ejpam-4983	362	7	det(b	det(b	PROPN
ejpam-4983	362	8	)	)	PUNCT
ejpam-4983	362	9	∈	∈	PROPN
ejpam-4983	362	10	γe−1r	γe−1r	PROPN
ejpam-4983	362	11	which	which	PRON
ejpam-4983	362	12	is	be	AUX
ejpam-4983	362	13	equivalent	equivalent	ADJ
ejpam-4983	362	14	to	to	PART
ejpam-4983	362	15	det(c	det(c	VERB
ejpam-4983	362	16	)	)	PUNCT
ejpam-4983	362	17	=	=	SYM
ejpam-4983	362	18	0	0	PUNCT
ejpam-4983	363	1	+	+	CCONJ
ejpam-4983	363	2	γe−1r	γe−1r	NUM
ejpam-4983	363	3	in	in	ADP
ejpam-4983	363	4	r	r	PROPN
ejpam-4983	363	5	/	/	SYM
ejpam-4983	363	6	γe−1r	γe−1r	PROPN
ejpam-4983	363	7	.	.	PUNCT
ejpam-4983	364	1	for	for	ADP
ejpam-4983	364	2	each	each	DET
ejpam-4983	364	3	matrix	matrix	NOUN
ejpam-4983	364	4	c	c	X
ejpam-4983	364	5	∈	∈	PROPN
ejpam-4983	364	6	an(r	an(r	NOUN
ejpam-4983	364	7	/	/	SYM
ejpam-4983	364	8	γ	γ	X
ejpam-4983	364	9	e−1r	e−1r	PROPN
ejpam-4983	364	10	,	,	PUNCT
ejpam-4983	364	11	0	0	NUM
ejpam-4983	364	12	+	+	CCONJ
ejpam-4983	364	13	γe−1r	γe−1r	NUM
ejpam-4983	364	14	)	)	PUNCT
ejpam-4983	364	15	,	,	PUNCT
ejpam-4983	364	16	there	there	PRON
ejpam-4983	364	17	are	be	VERB
ejpam-4983	364	18	q3n−4	q3n−4	NOUN
ejpam-4983	364	19	corresponding	correspond	VERB
ejpam-4983	364	20	matrices	matrix	NOUN
ejpam-4983	364	21	b	b	PROPN
ejpam-4983	364	22	∈	∈	PROPN
ejpam-4983	364	23	an(r	an(r	NOUN
ejpam-4983	364	24	,	,	PUNCT
ejpam-4983	364	25	0	0	NUM
ejpam-4983	364	26	)	)	PUNCT
ejpam-4983	364	27	.	.	PUNCT
ejpam-4983	365	1	since	since	SCONJ
ejpam-4983	365	2	the	the	DET
ejpam-4983	365	3	number	number	NOUN
ejpam-4983	365	4	of	of	ADP
ejpam-4983	365	5	possible	possible	ADJ
ejpam-4983	365	6	matrices	matrix	NOUN
ejpam-4983	365	7	c	c	NOUN
ejpam-4983	365	8	is	be	AUX
ejpam-4983	365	9	|an(r	|an(r	NOUN
ejpam-4983	365	10	/	/	SYM
ejpam-4983	365	11	γ	γ	X
ejpam-4983	365	12	e−1r	e−1r	PROPN
ejpam-4983	365	13	,	,	PUNCT
ejpam-4983	365	14	0	0	PUNCT
ejpam-4983	366	1	+	+	CCONJ
ejpam-4983	366	2	γe−1r)|	γe−1r)|	NOUN
ejpam-4983	366	3	and	and	CCONJ
ejpam-4983	366	4	the	the	DET
ejpam-4983	366	5	matrix	matrix	NOUN
ejpam-4983	366	6	a	a	PRON
ejpam-4983	366	7	is	be	AUX
ejpam-4983	366	8	uniquely	uniquely	ADV
ejpam-4983	366	9	determined	determine	VERB
ejpam-4983	366	10	by	by	ADP
ejpam-4983	366	11	b	b	NOUN
ejpam-4983	366	12	by	by	ADP
ejpam-4983	366	13	multiplying	multiply	VERB
ejpam-4983	366	14	the	the	DET
ejpam-4983	366	15	last	last	ADJ
ejpam-4983	366	16	column	column	NOUN
ejpam-4983	366	17	by	by	ADP
ejpam-4983	366	18	γ	γ	PROPN
ejpam-4983	366	19	,	,	PUNCT
ejpam-4983	366	20	the	the	DET
ejpam-4983	366	21	number	number	NOUN
ejpam-4983	366	22	of	of	ADP
ejpam-4983	366	23	choices	choice	NOUN
ejpam-4983	366	24	for	for	ADP
ejpam-4983	366	25	a	a	PRON
ejpam-4983	366	26	is	be	AUX
ejpam-4983	366	27	q3n−4|an(r	q3n−4|an(r	PRON
ejpam-4983	366	28	/	/	SYM
ejpam-4983	366	29	γ	γ	X
ejpam-4983	366	30	e−1r	e−1r	PROPN
ejpam-4983	366	31	,	,	PUNCT
ejpam-4983	366	32	0	0	PUNCT
ejpam-4983	367	1	+	+	CCONJ
ejpam-4983	367	2	γe−1r)|	γe−1r)|	NOUN
ejpam-4983	367	3	.	.	PUNCT
ejpam-4983	368	1	in	in	ADP
ejpam-4983	368	2	summary	summary	NOUN
ejpam-4983	368	3	,	,	PUNCT
ejpam-4983	368	4	we	we	PRON
ejpam-4983	368	5	have	have	VERB
ejpam-4983	368	6	|an(r	|an(r	PROPN
ejpam-4983	368	7	,	,	PUNCT
ejpam-4983	368	8	0)|	0)|	NOUN
ejpam-4983	368	9	≥	≥	NOUN
ejpam-4983	368	10	(	(	PUNCT
ejpam-4983	368	11	q	q	NOUN
ejpam-4983	368	12	−	−	PROPN
ejpam-4983	368	13	1	1	NUM
ejpam-4983	368	14	)	)	PUNCT
ejpam-4983	368	15	q2(e−1)(qe+1	q2(e−1)(qe+1	NOUN
ejpam-4983	369	1	+	+	CCONJ
ejpam-4983	369	2	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	369	3	,	,	PUNCT
ejpam-4983	369	4	0)|+	0)|+	NUM
ejpam-4983	369	5	q3n−4|an(r	q3n−4|an(r	NUM
ejpam-4983	369	6	/	/	SYM
ejpam-4983	369	7	γ	γ	X
ejpam-4983	369	8	e−1r	e−1r	PROPN
ejpam-4983	369	9	,	,	PUNCT
ejpam-4983	369	10	0	0	PUNCT
ejpam-4983	370	1	+	+	CCONJ
ejpam-4983	370	2	γe−1r)|	γe−1r)|	NOUN
ejpam-4983	370	3	as	as	SCONJ
ejpam-4983	370	4	desired	desire	VERB
ejpam-4983	370	5	.	.	PUNCT
ejpam-4983	371	1	■	■	PUNCT
ejpam-4983	371	2	for	for	ADP
ejpam-4983	371	3	a	a	DET
ejpam-4983	371	4	fccr	fccr	NOUN
ejpam-4983	371	5	of	of	ADP
ejpam-4983	371	6	nilpotency	nilpotency	NOUN
ejpam-4983	371	7	index	index	NOUN
ejpam-4983	371	8	2	2	NUM
ejpam-4983	371	9	,	,	PUNCT
ejpam-4983	371	10	we	we	PRON
ejpam-4983	371	11	have	have	VERB
ejpam-4983	371	12	the	the	DET
ejpam-4983	371	13	following	follow	VERB
ejpam-4983	371	14	bound	bind	VERB
ejpam-4983	371	15	.	.	PUNCT
ejpam-4983	372	1	s.	s.	PROPN
ejpam-4983	372	2	jitman	jitman	PROPN
ejpam-4983	372	3	,	,	PUNCT
ejpam-4983	372	4	p.	p.	PROPN
ejpam-4983	372	5	modjam	modjam	PROPN
ejpam-4983	372	6	/	/	SYM
ejpam-4983	372	7	eur	eur	PROPN
ejpam-4983	372	8	.	.	PUNCT
ejpam-4983	373	1	j.	j.	PROPN
ejpam-4983	373	2	pure	pure	PROPN
ejpam-4983	373	3	appl	appl	PROPN
ejpam-4983	373	4	.	.	PROPN
ejpam-4983	373	5	math	math	PROPN
ejpam-4983	373	6	,	,	PUNCT
ejpam-4983	373	7	17	17	NUM
ejpam-4983	373	8	(	(	PUNCT
ejpam-4983	373	9	1	1	NUM
ejpam-4983	373	10	)	)	PUNCT
ejpam-4983	373	11	(	(	PUNCT
ejpam-4983	373	12	2024	2024	NUM
ejpam-4983	373	13	)	)	PUNCT
ejpam-4983	373	14	,	,	PUNCT
ejpam-4983	373	15	11	11	NUM
ejpam-4983	373	16	-	-	SYM
ejpam-4983	373	17	29	29	NUM
ejpam-4983	373	18	24	24	NUM
ejpam-4983	373	19	corollary	corollary	NOUN
ejpam-4983	373	20	5	5	NUM
ejpam-4983	373	21	.	.	PUNCT
ejpam-4983	374	1	let	let	VERB
ejpam-4983	374	2	r	r	PRON
ejpam-4983	374	3	be	be	AUX
ejpam-4983	374	4	a	a	DET
ejpam-4983	374	5	fccr	fccr	NOUN
ejpam-4983	374	6	of	of	ADP
ejpam-4983	374	7	nilpotency	nilpotency	NOUN
ejpam-4983	374	8	index	index	NOUN
ejpam-4983	374	9	2	2	NUM
ejpam-4983	374	10	and	and	CCONJ
ejpam-4983	374	11	residue	residue	NOUN
ejpam-4983	374	12	field	field	NOUN
ejpam-4983	374	13	fq	fq	NOUN
ejpam-4983	374	14	.	.	PROPN
ejpam-4983	375	1	if	if	SCONJ
ejpam-4983	375	2	γ	γ	PROPN
ejpam-4983	375	3	is	be	AUX
ejpam-4983	375	4	a	a	DET
ejpam-4983	375	5	generator	generator	NOUN
ejpam-4983	375	6	of	of	ADP
ejpam-4983	375	7	the	the	DET
ejpam-4983	375	8	maximal	maximal	ADJ
ejpam-4983	375	9	ideal	ideal	NOUN
ejpam-4983	375	10	of	of	ADP
ejpam-4983	375	11	r	r	NOUN
ejpam-4983	375	12	,	,	PUNCT
ejpam-4983	375	13	then	then	ADV
ejpam-4983	375	14	|a1(r	|a1(r	ADJ
ejpam-4983	375	15	,	,	PUNCT
ejpam-4983	375	16	0)|	0)|	NOUN
ejpam-4983	375	17	=	=	SYM
ejpam-4983	375	18	1	1	NUM
ejpam-4983	375	19	and	and	CCONJ
ejpam-4983	375	20	|an(r	|an(r	PROPN
ejpam-4983	375	21	,	,	PUNCT
ejpam-4983	375	22	0)|	0)|	NOUN
ejpam-4983	375	23	≥	≥	NOUN
ejpam-4983	375	24	(	(	PUNCT
ejpam-4983	375	25	q	q	NOUN
ejpam-4983	375	26	−	−	PROPN
ejpam-4983	376	1	1)q2(q3	1)q2(q3	NUM
ejpam-4983	377	1	+	+	CCONJ
ejpam-4983	377	2	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	377	3	,	,	PUNCT
ejpam-4983	377	4	0)|+	0)|+	PUNCT
ejpam-4983	378	1	q3n−4	q3n−4	PROPN
ejpam-4983	378	2	(	(	PUNCT
ejpam-4983	378	3	q3n−2	q3n−2	PROPN
ejpam-4983	378	4	−	−	PROPN
ejpam-4983	378	5	q2n−3(q	q2n−3(q	NUM
ejpam-4983	378	6	−	−	PROPN
ejpam-4983	378	7	1)n(q	1)n(q	NUM
ejpam-4983	378	8	+	+	CCONJ
ejpam-4983	378	9	(	(	PUNCT
ejpam-4983	378	10	n−	n−	NOUN
ejpam-4983	378	11	1	1	NUM
ejpam-4983	378	12	)	)	PUNCT
ejpam-4983	378	13	)	)	PUNCT
ejpam-4983	378	14	)	)	PUNCT
ejpam-4983	379	1	for	for	ADP
ejpam-4983	379	2	all	all	DET
ejpam-4983	379	3	integers	integer	NOUN
ejpam-4983	379	4	n	n	PRON
ejpam-4983	379	5	≥	≥	NOUN
ejpam-4983	379	6	2	2	NUM
ejpam-4983	379	7	.	.	PUNCT
ejpam-4983	380	1	proof	proof	NOUN
ejpam-4983	380	2	.	.	PUNCT
ejpam-4983	381	1	clearly	clearly	ADV
ejpam-4983	381	2	,	,	PUNCT
ejpam-4983	381	3	|a1(r	|a1(r	ADJ
ejpam-4983	381	4	,	,	PUNCT
ejpam-4983	381	5	0)|	0)|	NOUN
ejpam-4983	381	6	=	=	SYM
ejpam-4983	381	7	1	1	X
ejpam-4983	381	8	.	.	PUNCT
ejpam-4983	382	1	let	let	VERB
ejpam-4983	382	2	n	n	PRON
ejpam-4983	382	3	≥	≥	X
ejpam-4983	382	4	2	2	NUM
ejpam-4983	382	5	be	be	AUX
ejpam-4983	382	6	an	an	DET
ejpam-4983	382	7	integer	integer	NOUN
ejpam-4983	382	8	.	.	PUNCT
ejpam-4983	383	1	we	we	PRON
ejpam-4983	383	2	note	note	VERB
ejpam-4983	383	3	that	that	SCONJ
ejpam-4983	383	4	r	r	NOUN
ejpam-4983	383	5	/	/	SYM
ejpam-4983	383	6	γe−1r	γe−1r	NUM
ejpam-4983	383	7	∼=	∼=	PROPN
ejpam-4983	383	8	fq	fq	PROPN
ejpam-4983	383	9	.	.	PROPN
ejpam-4983	383	10	from	from	ADP
ejpam-4983	383	11	proposition	proposition	NOUN
ejpam-4983	383	12	4	4	NUM
ejpam-4983	383	13	and	and	CCONJ
ejpam-4983	383	14	corollary	corollary	ADJ
ejpam-4983	383	15	2	2	NUM
ejpam-4983	383	16	,	,	PUNCT
ejpam-4983	383	17	we	we	PRON
ejpam-4983	383	18	have	have	VERB
ejpam-4983	383	19	|an(r	|an(r	PROPN
ejpam-4983	383	20	,	,	PUNCT
ejpam-4983	383	21	0)|	0)|	NOUN
ejpam-4983	383	22	≥	≥	NOUN
ejpam-4983	383	23	(	(	PUNCT
ejpam-4983	383	24	q	q	NOUN
ejpam-4983	383	25	−	−	PROPN
ejpam-4983	383	26	1)q2(q3	1)q2(q3	NUM
ejpam-4983	384	1	+	+	CCONJ
ejpam-4983	384	2	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	384	3	,	,	PUNCT
ejpam-4983	384	4	0)|+	0)|+	PUNCT
ejpam-4983	384	5	q3n−4|an(fq	q3n−4|an(fq	ADJ
ejpam-4983	384	6	,	,	PUNCT
ejpam-4983	384	7	0)|	0)|	NOUN
ejpam-4983	384	8	=	=	SYM
ejpam-4983	384	9	(	(	PUNCT
ejpam-4983	384	10	q	q	NOUN
ejpam-4983	384	11	−	−	PROPN
ejpam-4983	384	12	1)q2(q3	1)q2(q3	NUM
ejpam-4983	385	1	+	+	CCONJ
ejpam-4983	385	2	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	385	3	,	,	PUNCT
ejpam-4983	385	4	0)|+	0)|+	PUNCT
ejpam-4983	386	1	q3n−4	q3n−4	PROPN
ejpam-4983	386	2	(	(	PUNCT
ejpam-4983	386	3	q3n−2	q3n−2	PROPN
ejpam-4983	386	4	−	−	PROPN
ejpam-4983	386	5	q2n−3(q	q2n−3(q	NUM
ejpam-4983	386	6	−	−	PROPN
ejpam-4983	386	7	1)n(q	1)n(q	NUM
ejpam-4983	386	8	+	+	CCONJ
ejpam-4983	386	9	(	(	PUNCT
ejpam-4983	386	10	n−	n−	NOUN
ejpam-4983	386	11	1	1	NUM
ejpam-4983	386	12	)	)	PUNCT
ejpam-4983	386	13	)	)	PUNCT
ejpam-4983	386	14	)	)	PUNCT
ejpam-4983	386	15	as	as	SCONJ
ejpam-4983	386	16	desired	desire	VERB
ejpam-4983	386	17	.	.	PUNCT
ejpam-4983	387	1	■	■	PUNCT
ejpam-4983	387	2	3.2.2	3.2.2	X
ejpam-4983	387	3	.	.	PUNCT
ejpam-4983	387	4	singular	singular	ADJ
ejpam-4983	387	5	arrowhead	arrowhead	NOUN
ejpam-4983	387	6	matrices	matrix	NOUN
ejpam-4983	387	7	over	over	ADP
ejpam-4983	387	8	fccrs	fccr	NOUN
ejpam-4983	387	9	with	with	ADP
ejpam-4983	387	10	non	non	ADJ
ejpam-4983	387	11	-	-	ADJ
ejpam-4983	387	12	zero	zero	NUM
ejpam-4983	387	13	determinant	determinant	ADJ
ejpam-4983	387	14	in	in	ADP
ejpam-4983	387	15	this	this	DET
ejpam-4983	387	16	subsection	subsection	NOUN
ejpam-4983	387	17	,	,	PUNCT
ejpam-4983	387	18	an	an	DET
ejpam-4983	387	19	upper	upper	ADJ
ejpam-4983	387	20	bound	bind	VERB
ejpam-4983	387	21	on	on	ADP
ejpam-4983	387	22	the	the	DET
ejpam-4983	387	23	number	number	NOUN
ejpam-4983	387	24	of	of	ADP
ejpam-4983	387	25	n	n	NUM
ejpam-4983	387	26	×	×	NOUN
ejpam-4983	387	27	n	n	CCONJ
ejpam-4983	387	28	singular	singular	ADJ
ejpam-4983	387	29	arrowhead	arrowhead	NOUN
ejpam-4983	387	30	matrices	matrix	NOUN
ejpam-4983	387	31	over	over	ADP
ejpam-4983	387	32	r	r	NOUN
ejpam-4983	387	33	with	with	ADP
ejpam-4983	387	34	a	a	DET
ejpam-4983	387	35	fixed	fix	VERB
ejpam-4983	387	36	non	non	ADJ
ejpam-4983	387	37	-	-	ADJ
ejpam-4983	387	38	zero	zero	ADJ
ejpam-4983	387	39	determinant	determinant	ADJ
ejpam-4983	387	40	is	be	AUX
ejpam-4983	387	41	presented	present	VERB
ejpam-4983	387	42	.	.	PUNCT
ejpam-4983	388	1	first	first	ADV
ejpam-4983	388	2	,	,	PUNCT
ejpam-4983	388	3	a	a	DET
ejpam-4983	388	4	relation	relation	NOUN
ejpam-4983	388	5	between	between	ADP
ejpam-4983	388	6	|an(r	|an(r	PROPN
ejpam-4983	388	7	,	,	PUNCT
ejpam-4983	388	8	γ	γ	X
ejpam-4983	388	9	i)|	i)|	NOUN
ejpam-4983	388	10	and	and	CCONJ
ejpam-4983	388	11	|an(r	|an(r	PROPN
ejpam-4983	388	12	,	,	PUNCT
ejpam-4983	388	13	b)|	b)|	NOUN
ejpam-4983	388	14	is	be	AUX
ejpam-4983	388	15	derived	derive	VERB
ejpam-4983	388	16	for	for	ADP
ejpam-4983	388	17	all	all	DET
ejpam-4983	388	18	b	b	NOUN
ejpam-4983	388	19	∈	∈	NOUN
ejpam-4983	388	20	γir	γir	ADP
ejpam-4983	388	21	\	\	NOUN
ejpam-4983	388	22	γi+1r	γi+1r	NOUN
ejpam-4983	388	23	.	.	PUNCT
ejpam-4983	389	1	proposition	proposition	NOUN
ejpam-4983	389	2	5	5	NUM
ejpam-4983	389	3	.	.	PUNCT
ejpam-4983	390	1	let	let	VERB
ejpam-4983	390	2	r	r	PRON
ejpam-4983	390	3	be	be	AUX
ejpam-4983	390	4	a	a	DET
ejpam-4983	390	5	fccr	fccr	NOUN
ejpam-4983	390	6	with	with	ADP
ejpam-4983	390	7	maximal	maximal	ADJ
ejpam-4983	390	8	ideal	ideal	NOUN
ejpam-4983	390	9	generated	generate	VERB
ejpam-4983	390	10	by	by	ADP
ejpam-4983	390	11	γ	γ	PROPN
ejpam-4983	390	12	,	,	PUNCT
ejpam-4983	390	13	residue	residue	NOUN
ejpam-4983	390	14	field	field	NOUN
ejpam-4983	390	15	fq	fq	NOUN
ejpam-4983	390	16	,	,	PUNCT
ejpam-4983	390	17	and	and	CCONJ
ejpam-4983	390	18	nilpotency	nilpotency	NOUN
ejpam-4983	390	19	index	index	PROPN
ejpam-4983	390	20	e.	e.	PROPN
ejpam-4983	390	21	then	then	ADV
ejpam-4983	390	22	|an(r	|an(r	PROPN
ejpam-4983	390	23	,	,	PUNCT
ejpam-4983	390	24	γ	γ	NOUN
ejpam-4983	390	25	i)|	i)|	NOUN
ejpam-4983	390	26	=	=	SYM
ejpam-4983	390	27	|an(r	|an(r	PROPN
ejpam-4983	390	28	,	,	PUNCT
ejpam-4983	390	29	b)|	b)|	NOUN
ejpam-4983	390	30	for	for	ADP
ejpam-4983	390	31	all	all	DET
ejpam-4983	390	32	b	b	NOUN
ejpam-4983	390	33	∈	∈	NOUN
ejpam-4983	391	1	γir	γir	ADP
ejpam-4983	391	2	\	\	NOUN
ejpam-4983	391	3	γi+1r	γi+1r	NOUN
ejpam-4983	391	4	and	and	CCONJ
ejpam-4983	391	5	1	1	NUM
ejpam-4983	391	6	≤	≤	NUM
ejpam-4983	391	7	i	i	PRON
ejpam-4983	391	8	<	<	X
ejpam-4983	391	9	e.	e.	PROPN
ejpam-4983	391	10	proof	proof	PROPN
ejpam-4983	391	11	.	.	PUNCT
ejpam-4983	392	1	let	let	VERB
ejpam-4983	392	2	b	b	X
ejpam-4983	392	3	∈	∈	PROPN
ejpam-4983	392	4	γir	γir	CCONJ
ejpam-4983	392	5	\	\	NOUN
ejpam-4983	392	6	γi+1r	γi+1r	NOUN
ejpam-4983	392	7	.	.	PUNCT
ejpam-4983	393	1	then	then	ADV
ejpam-4983	393	2	b	b	X
ejpam-4983	393	3	=	=	PUNCT
ejpam-4983	393	4	aγi	aγi	ADJ
ejpam-4983	393	5	for	for	ADP
ejpam-4983	393	6	some	some	DET
ejpam-4983	393	7	a	a	DET
ejpam-4983	393	8	∈	∈	PROPN
ejpam-4983	393	9	u(r	u(r	NOUN
ejpam-4983	393	10	)	)	PUNCT
ejpam-4983	393	11	.	.	PUNCT
ejpam-4983	394	1	let	let	VERB
ejpam-4983	394	2	ψ	ψ	NOUN
ejpam-4983	394	3	:	:	PUNCT
ejpam-4983	394	4	an(r	an(r	NOUN
ejpam-4983	394	5	,	,	PUNCT
ejpam-4983	394	6	γ	γ	X
ejpam-4983	394	7	i	i	PROPN
ejpam-4983	394	8	)	)	PUNCT
ejpam-4983	394	9	→	→	SYM
ejpam-4983	394	10	an(r	an(r	NOUN
ejpam-4983	394	11	,	,	PUNCT
ejpam-4983	394	12	aγ	aγ	PRON
ejpam-4983	394	13	i	i	PROPN
ejpam-4983	394	14	)	)	PUNCT
ejpam-4983	394	15	be	be	VERB
ejpam-4983	394	16	the	the	DET
ejpam-4983	394	17	function	function	NOUN
ejpam-4983	394	18	defined	define	VERB
ejpam-4983	394	19	by	by	ADP
ejpam-4983	394	20	ψ(a	ψ(a	PROPN
ejpam-4983	394	21	)	)	PUNCT
ejpam-4983	394	22	=	=	SYM
ejpam-4983	394	23	diag(a	diag(a	PROPN
ejpam-4983	394	24	,	,	PUNCT
ejpam-4983	394	25	1	1	NUM
ejpam-4983	394	26	,	,	PUNCT
ejpam-4983	394	27	1	1	NUM
ejpam-4983	394	28	,	,	PUNCT
ejpam-4983	394	29	.	.	PUNCT
ejpam-4983	394	30	.	.	PUNCT
ejpam-4983	395	1	.	.	PUNCT
ejpam-4983	396	1	,	,	PUNCT
ejpam-4983	396	2	1)a	1)a	NUM
ejpam-4983	396	3	.	.	PUNCT
ejpam-4983	397	1	using	use	VERB
ejpam-4983	397	2	the	the	DET
ejpam-4983	397	3	fact	fact	NOUN
ejpam-4983	397	4	that	that	SCONJ
ejpam-4983	397	5	a	a	PRON
ejpam-4983	397	6	is	be	AUX
ejpam-4983	397	7	convertible	convertible	ADJ
ejpam-4983	397	8	and	and	CCONJ
ejpam-4983	397	9	arguments	argument	NOUN
ejpam-4983	397	10	similar	similar	ADJ
ejpam-4983	397	11	to	to	ADP
ejpam-4983	397	12	those	those	PRON
ejpam-4983	397	13	in	in	ADP
ejpam-4983	397	14	the	the	DET
ejpam-4983	397	15	proof	proof	NOUN
ejpam-4983	397	16	of	of	ADP
ejpam-4983	397	17	proposition	proposition	NOUN
ejpam-4983	397	18	2	2	NUM
ejpam-4983	397	19	,	,	PUNCT
ejpam-4983	397	20	it	it	PRON
ejpam-4983	397	21	can	can	AUX
ejpam-4983	397	22	be	be	AUX
ejpam-4983	397	23	deduced	deduce	VERB
ejpam-4983	397	24	that	that	SCONJ
ejpam-4983	397	25	ψ	ψ	NOUN
ejpam-4983	397	26	is	be	AUX
ejpam-4983	397	27	a	a	DET
ejpam-4983	397	28	bijection	bijection	NOUN
ejpam-4983	397	29	from	from	ADP
ejpam-4983	397	30	an(r	an(r	NOUN
ejpam-4983	397	31	,	,	PUNCT
ejpam-4983	397	32	γ	γ	X
ejpam-4983	397	33	i	i	PROPN
ejpam-4983	397	34	)	)	PUNCT
ejpam-4983	397	35	onto	onto	ADP
ejpam-4983	397	36	an(r	an(r	NOUN
ejpam-4983	397	37	,	,	PUNCT
ejpam-4983	397	38	aγ	aγ	PRON
ejpam-4983	397	39	i	i	PROPN
ejpam-4983	397	40	)	)	PUNCT
ejpam-4983	397	41	.	.	PUNCT
ejpam-4983	398	1	as	as	SCONJ
ejpam-4983	398	2	desired	desire	VERB
ejpam-4983	398	3	,	,	PUNCT
ejpam-4983	398	4	|an(r	|an(r	PROPN
ejpam-4983	398	5	,	,	PUNCT
ejpam-4983	398	6	b)|	b)|	NOUN
ejpam-4983	398	7	=	=	SYM
ejpam-4983	398	8	|an(r	|an(r	PROPN
ejpam-4983	398	9	,	,	PUNCT
ejpam-4983	398	10	γ	γ	NOUN
ejpam-4983	398	11	i)|	i)|	NOUN
ejpam-4983	398	12	.	.	PUNCT
ejpam-4983	399	1	■	■	PUNCT
ejpam-4983	399	2	lemma	lemma	PROPN
ejpam-4983	399	3	2	2	X
ejpam-4983	399	4	.	.	PUNCT
ejpam-4983	399	5	let	let	VERB
ejpam-4983	399	6	r	r	PRON
ejpam-4983	399	7	be	be	AUX
ejpam-4983	399	8	a	a	DET
ejpam-4983	399	9	fccr	fccr	NOUN
ejpam-4983	399	10	of	of	ADP
ejpam-4983	399	11	nilpotency	nilpotency	NOUN
ejpam-4983	399	12	index	index	PROPN
ejpam-4983	399	13	e	e	PROPN
ejpam-4983	399	14	≥	≥	NOUN
ejpam-4983	399	15	3	3	NUM
ejpam-4983	399	16	and	and	CCONJ
ejpam-4983	399	17	residue	residue	NOUN
ejpam-4983	399	18	field	field	NOUN
ejpam-4983	399	19	fq	fq	NOUN
ejpam-4983	399	20	and	and	CCONJ
ejpam-4983	399	21	let	let	VERB
ejpam-4983	399	22	n	n	PRON
ejpam-4983	399	23	be	be	AUX
ejpam-4983	399	24	a	a	DET
ejpam-4983	399	25	positive	positive	ADJ
ejpam-4983	399	26	integer	integer	NOUN
ejpam-4983	399	27	.	.	PUNCT
ejpam-4983	400	1	if	if	SCONJ
ejpam-4983	400	2	γ	γ	X
ejpam-4983	400	3	is	be	AUX
ejpam-4983	400	4	a	a	DET
ejpam-4983	400	5	generator	generator	NOUN
ejpam-4983	400	6	of	of	ADP
ejpam-4983	400	7	the	the	DET
ejpam-4983	400	8	maximal	maximal	ADJ
ejpam-4983	400	9	ideal	ideal	NOUN
ejpam-4983	400	10	of	of	ADP
ejpam-4983	400	11	r	r	NOUN
ejpam-4983	400	12	,	,	PUNCT
ejpam-4983	400	13	then	then	ADV
ejpam-4983	400	14	|an(r	|an(r	PROPN
ejpam-4983	400	15	,	,	PUNCT
ejpam-4983	400	16	γ	γ	NOUN
ejpam-4983	400	17	s)|	s)|	NOUN
ejpam-4983	400	18	=	=	SYM
ejpam-4983	400	19	q3(n−1)|an(r	q3(n−1)|an(r	NOUN
ejpam-4983	400	20	/	/	SYM
ejpam-4983	400	21	γ	γ	PROPN
ejpam-4983	400	22	e−1r	e−1r	PROPN
ejpam-4983	400	23	,	,	PUNCT
ejpam-4983	400	24	γs	γ	VERB
ejpam-4983	400	25	+	+	X
ejpam-4983	400	26	γe−1r)|	γe−1r)|	NOUN
ejpam-4983	400	27	for	for	ADP
ejpam-4983	400	28	all	all	DET
ejpam-4983	400	29	1	1	NUM
ejpam-4983	400	30	≤	≤	NUM
ejpam-4983	400	31	s	s	PART
ejpam-4983	400	32	<	<	X
ejpam-4983	400	33	e−	e−	PROPN
ejpam-4983	400	34	1	1	NUM
ejpam-4983	400	35	.	.	PUNCT
ejpam-4983	401	1	proof	proof	NOUN
ejpam-4983	401	2	.	.	PUNCT
ejpam-4983	402	1	let	let	VERB
ejpam-4983	402	2	1	1	NUM
ejpam-4983	402	3	≤	≤	NOUN
ejpam-4983	402	4	s	s	PART
ejpam-4983	402	5	<	<	X
ejpam-4983	402	6	e−	e−	PROPN
ejpam-4983	402	7	1	1	NUM
ejpam-4983	402	8	be	be	AUX
ejpam-4983	402	9	an	an	DET
ejpam-4983	402	10	integer	integer	NOUN
ejpam-4983	402	11	and	and	CCONJ
ejpam-4983	402	12	let	let	VERB
ejpam-4983	402	13	β	β	PRON
ejpam-4983	402	14	:	:	PUNCT
ejpam-4983	402	15	an(r	an(r	X
ejpam-4983	402	16	)	)	PUNCT
ejpam-4983	402	17	→	→	SYM
ejpam-4983	402	18	an(r	an(r	NOUN
ejpam-4983	402	19	/	/	SYM
ejpam-4983	402	20	γ	γ	X
ejpam-4983	402	21	e−1r	e−1r	PROPN
ejpam-4983	402	22	)	)	PUNCT
ejpam-4983	402	23	be	be	VERB
ejpam-4983	402	24	an	an	DET
ejpam-4983	402	25	additive	additive	ADJ
ejpam-4983	402	26	group	group	NOUN
ejpam-4983	402	27	homomorphism	homomorphism	NOUN
ejpam-4983	402	28	defined	define	VERB
ejpam-4983	402	29	by	by	ADP
ejpam-4983	402	30	β(a	β(a	PROPN
ejpam-4983	402	31	)	)	PUNCT
ejpam-4983	402	32	=	=	SYM
ejpam-4983	403	1	a	a	PROPN
ejpam-4983	403	2	,	,	PUNCT
ejpam-4983	403	3	s.	s.	PROPN
ejpam-4983	403	4	jitman	jitman	PROPN
ejpam-4983	403	5	,	,	PUNCT
ejpam-4983	403	6	p.	p.	PROPN
ejpam-4983	403	7	modjam	modjam	PROPN
ejpam-4983	403	8	/	/	SYM
ejpam-4983	403	9	eur	eur	PROPN
ejpam-4983	403	10	.	.	PUNCT
ejpam-4983	404	1	j.	j.	PROPN
ejpam-4983	404	2	pure	pure	PROPN
ejpam-4983	404	3	appl	appl	PROPN
ejpam-4983	404	4	.	.	PROPN
ejpam-4983	404	5	math	math	PROPN
ejpam-4983	404	6	,	,	PUNCT
ejpam-4983	404	7	17	17	NUM
ejpam-4983	404	8	(	(	PUNCT
ejpam-4983	404	9	1	1	NUM
ejpam-4983	404	10	)	)	PUNCT
ejpam-4983	404	11	(	(	PUNCT
ejpam-4983	404	12	2024	2024	NUM
ejpam-4983	404	13	)	)	PUNCT
ejpam-4983	404	14	,	,	PUNCT
ejpam-4983	404	15	11	11	NUM
ejpam-4983	404	16	-	-	SYM
ejpam-4983	404	17	29	29	NUM
ejpam-4983	404	18	25	25	NUM
ejpam-4983	405	1	where	where	SCONJ
ejpam-4983	405	2	[	[	X
ejpam-4983	405	3	aij	aij	X
ejpam-4983	405	4	]	]	X
ejpam-4983	405	5	:	:	PUNCT
ejpam-4983	405	6	=	=	PUNCT
ejpam-4983	406	1	[	[	X
ejpam-4983	406	2	aij	aij	X
ejpam-4983	406	3	+	+	SYM
ejpam-4983	406	4	γe−1r	γe−1r	PROPN
ejpam-4983	406	5	]	]	PUNCT
ejpam-4983	406	6	for	for	ADP
ejpam-4983	406	7	all	all	DET
ejpam-4983	406	8	[	[	PUNCT
ejpam-4983	406	9	aij	aij	X
ejpam-4983	406	10	]	]	PUNCT
ejpam-4983	406	11	∈	∈	PROPN
ejpam-4983	406	12	an(r	an(r	NOUN
ejpam-4983	406	13	)	)	PUNCT
ejpam-4983	406	14	.	.	PUNCT
ejpam-4983	407	1	note	note	VERB
ejpam-4983	407	2	that	that	SCONJ
ejpam-4983	407	3	,	,	PUNCT
ejpam-4983	407	4	for	for	ADP
ejpam-4983	407	5	each	each	PRON
ejpam-4983	407	6	a	a	DET
ejpam-4983	407	7	∈	∈	PROPN
ejpam-4983	407	8	an(r	an(r	NOUN
ejpam-4983	407	9	)	)	PUNCT
ejpam-4983	407	10	,	,	PUNCT
ejpam-4983	407	11	det(β(a	det(β(a	PROPN
ejpam-4983	407	12	)	)	PUNCT
ejpam-4983	407	13	)	)	PUNCT
ejpam-4983	407	14	=	=	PRON
ejpam-4983	407	15	γs	γ	VERB
ejpam-4983	407	16	+	+	X
ejpam-4983	407	17	γe−1r	γe−1r	PROPN
ejpam-4983	408	1	if	if	SCONJ
ejpam-4983	408	2	and	and	CCONJ
ejpam-4983	408	3	only	only	ADV
ejpam-4983	408	4	if	if	SCONJ
ejpam-4983	408	5	det(a	det(a	PROPN
ejpam-4983	408	6	)	)	PUNCT
ejpam-4983	408	7	=	=	PUNCT
ejpam-4983	408	8	γs	γ	VERB
ejpam-4983	408	9	+	+	X
ejpam-4983	408	10	γe−1b	γe−1b	NUM
ejpam-4983	408	11	for	for	ADP
ejpam-4983	408	12	some	some	DET
ejpam-4983	408	13	b	b	PROPN
ejpam-4983	408	14	∈	∈	PROPN
ejpam-4983	408	15	v	v	NOUN
ejpam-4983	408	16	,	,	PUNCT
ejpam-4983	408	17	where	where	SCONJ
ejpam-4983	408	18	v	v	NOUN
ejpam-4983	408	19	is	be	AUX
ejpam-4983	408	20	defined	define	VERB
ejpam-4983	408	21	in	in	ADP
ejpam-4983	408	22	lemma	lemma	PROPN
ejpam-4983	408	23	1	1	NUM
ejpam-4983	408	24	.	.	PUNCT
ejpam-4983	409	1	since	since	SCONJ
ejpam-4983	409	2	1	1	NUM
ejpam-4983	409	3	≤	≤	NUM
ejpam-4983	410	1	e	e	X
ejpam-4983	410	2	−	−	PROPN
ejpam-4983	410	3	s	s	PART
ejpam-4983	410	4	−	−	PROPN
ejpam-4983	410	5	1	1	NUM
ejpam-4983	410	6	<	<	X
ejpam-4983	410	7	e	e	X
ejpam-4983	410	8	−	−	PROPN
ejpam-4983	410	9	1	1	NUM
ejpam-4983	410	10	,	,	PUNCT
ejpam-4983	410	11	it	it	PRON
ejpam-4983	410	12	follows	follow	VERB
ejpam-4983	410	13	that	that	SCONJ
ejpam-4983	410	14	1	1	NUM
ejpam-4983	410	15	+	+	CCONJ
ejpam-4983	410	16	γe−s−1b	γe−s−1b	X
ejpam-4983	410	17	is	be	AUX
ejpam-4983	410	18	a	a	DET
ejpam-4983	410	19	unit	unit	NOUN
ejpam-4983	410	20	in	in	ADP
ejpam-4983	410	21	u(r	u(r	NOUN
ejpam-4983	410	22	)	)	PUNCT
ejpam-4983	410	23	.	.	PUNCT
ejpam-4983	411	1	hence	hence	ADV
ejpam-4983	411	2	,	,	PUNCT
ejpam-4983	411	3	|{a	|{a	PROPN
ejpam-4983	411	4	∈	∈	PROPN
ejpam-4983	411	5	an(r	an(r	NOUN
ejpam-4983	411	6	)	)	PUNCT
ejpam-4983	411	7	|det(a	|det(a	VERB
ejpam-4983	411	8	)	)	PUNCT
ejpam-4983	411	9	=	=	PRON
ejpam-4983	411	10	γs	γ	VERB
ejpam-4983	411	11	+	+	X
ejpam-4983	412	1	γe−1b	γe−1b	NUM
ejpam-4983	413	1	for	for	ADP
ejpam-4983	413	2	some	some	DET
ejpam-4983	413	3	b	b	NOUN
ejpam-4983	413	4	∈	∈	ADJ
ejpam-4983	413	5	v	v	ADP
ejpam-4983	413	6	}	}	PUNCT
ejpam-4983	413	7	|	|	NOUN
ejpam-4983	413	8	=	=	SYM
ejpam-4983	413	9	|{a	|{a	PROPN
ejpam-4983	413	10	∈	∈	PROPN
ejpam-4983	413	11	an(r	an(r	NOUN
ejpam-4983	413	12	)	)	PUNCT
ejpam-4983	413	13	|	|	ADV
ejpam-4983	413	14	det(a	det(a	NOUN
ejpam-4983	413	15	)	)	PUNCT
ejpam-4983	414	1	=	=	SYM
ejpam-4983	415	1	γs(1	γs(1	PROPN
ejpam-4983	415	2	+	+	CCONJ
ejpam-4983	415	3	γe−s−1b	γe−s−1b	NUM
ejpam-4983	415	4	)	)	PUNCT
ejpam-4983	415	5	for	for	ADP
ejpam-4983	415	6	some	some	DET
ejpam-4983	415	7	b	b	PROPN
ejpam-4983	415	8	∈	∈	ADJ
ejpam-4983	415	9	v	v	ADP
ejpam-4983	415	10	}	}	PUNCT
ejpam-4983	415	11	|	|	NOUN
ejpam-4983	415	12	=	=	SYM
ejpam-4983	415	13	|{a	|{a	PROPN
ejpam-4983	415	14	∈	∈	PROPN
ejpam-4983	415	15	an(r	an(r	NOUN
ejpam-4983	415	16	)	)	PUNCT
ejpam-4983	415	17	|	|	ADV
ejpam-4983	415	18	det(a	det(a	NOUN
ejpam-4983	415	19	)	)	PUNCT
ejpam-4983	415	20	=	=	SYM
ejpam-4983	415	21	γs}|	γs}|	X
ejpam-4983	415	22	=	=	SYM
ejpam-4983	415	23	|an(r	|an(r	PROPN
ejpam-4983	415	24	,	,	PUNCT
ejpam-4983	415	25	γ	γ	PROPN
ejpam-4983	415	26	s)|	s)|	PROPN
ejpam-4983	415	27	.	.	PUNCT
ejpam-4983	416	1	equivalently	equivalently	ADV
ejpam-4983	416	2	,	,	PUNCT
ejpam-4983	416	3	|{a	|{a	PROPN
ejpam-4983	416	4	∈	∈	PROPN
ejpam-4983	416	5	an(r	an(r	NOUN
ejpam-4983	416	6	)	)	PUNCT
ejpam-4983	416	7	|	|	ADV
ejpam-4983	416	8	det(β(a	det(β(a	NOUN
ejpam-4983	416	9	)	)	PUNCT
ejpam-4983	416	10	)	)	PUNCT
ejpam-4983	417	1	=	=	PRON
ejpam-4983	417	2	γs	γs	AUX
ejpam-4983	417	3	+	+	CCONJ
ejpam-4983	417	4	γe−1r}|	γe−1r}|	NOUN
ejpam-4983	417	5	=	=	NOUN
ejpam-4983	417	6	|v	|v	PROPN
ejpam-4983	417	7	||an(r	||an(r	PROPN
ejpam-4983	417	8	,	,	PUNCT
ejpam-4983	417	9	γ	γ	NOUN
ejpam-4983	417	10	s)|	s)|	NOUN
ejpam-4983	417	11	=	=	SYM
ejpam-4983	417	12	q|an(r	q|an(r	PROPN
ejpam-4983	417	13	,	,	PUNCT
ejpam-4983	417	14	γ	γ	PROPN
ejpam-4983	417	15	s)|	s)|	PROPN
ejpam-4983	417	16	.	.	PUNCT
ejpam-4983	418	1	(	(	PUNCT
ejpam-4983	418	2	5	5	NUM
ejpam-4983	418	3	)	)	PUNCT
ejpam-4983	418	4	since	since	SCONJ
ejpam-4983	418	5	|	|	ADV
ejpam-4983	418	6	ker(β)|	ker(β)|	PROPN
ejpam-4983	418	7	=	=	SYM
ejpam-4983	418	8	q3n−2	q3n−2	PROPN
ejpam-4983	418	9	,	,	PUNCT
ejpam-4983	418	10	we	we	PRON
ejpam-4983	418	11	have	have	VERB
ejpam-4983	418	12	|{a	|{a	PROPN
ejpam-4983	418	13	∈	∈	PROPN
ejpam-4983	418	14	an(r	an(r	NOUN
ejpam-4983	418	15	)	)	PUNCT
ejpam-4983	418	16	|	|	ADV
ejpam-4983	418	17	det(β(a	det(β(a	NOUN
ejpam-4983	418	18	)	)	PUNCT
ejpam-4983	418	19	)	)	PUNCT
ejpam-4983	419	1	=	=	PRON
ejpam-4983	419	2	γs	γs	AUX
ejpam-4983	419	3	+	+	CCONJ
ejpam-4983	419	4	γe−1r}|	γe−1r}|	NOUN
ejpam-4983	420	1	=	=	NOUN
ejpam-4983	421	1	|	|	ADV
ejpam-4983	422	1	ker(β)||{b	ker(β)||{b	PROPN
ejpam-4983	422	2	∈	∈	PROPN
ejpam-4983	422	3	an(r	an(r	NOUN
ejpam-4983	422	4	/	/	SYM
ejpam-4983	422	5	γ	γ	X
ejpam-4983	422	6	e−1r	e−1r	NOUN
ejpam-4983	422	7	)	)	PUNCT
ejpam-4983	422	8	|	|	ADV
ejpam-4983	422	9	det(b	det(b	NOUN
ejpam-4983	422	10	)	)	PUNCT
ejpam-4983	422	11	=	=	PUNCT
ejpam-4983	422	12	γs	γs	AUX
ejpam-4983	422	13	+	+	CCONJ
ejpam-4983	422	14	γe−1r}|	γe−1r}|	NOUN
ejpam-4983	422	15	=	=	NOUN
ejpam-4983	422	16	q3n−2|an(r	q3n−2|an(r	PROPN
ejpam-4983	422	17	/	/	SYM
ejpam-4983	422	18	γ	γ	X
ejpam-4983	422	19	e−1r	e−1r	PROPN
ejpam-4983	422	20	,	,	PUNCT
ejpam-4983	422	21	γs	γ	VERB
ejpam-4983	422	22	+	+	X
ejpam-4983	422	23	γe−1r)|	γe−1r)|	NUM
ejpam-4983	422	24	.	.	PUNCT
ejpam-4983	423	1	(	(	PUNCT
ejpam-4983	423	2	6	6	X
ejpam-4983	423	3	)	)	PUNCT
ejpam-4983	423	4	combining	combine	VERB
ejpam-4983	423	5	(	(	PUNCT
ejpam-4983	423	6	5	5	NUM
ejpam-4983	423	7	)	)	PUNCT
ejpam-4983	423	8	and	and	CCONJ
ejpam-4983	423	9	(	(	PUNCT
ejpam-4983	423	10	6	6	NUM
ejpam-4983	423	11	)	)	PUNCT
ejpam-4983	423	12	,	,	PUNCT
ejpam-4983	423	13	it	it	PRON
ejpam-4983	423	14	can	can	AUX
ejpam-4983	423	15	be	be	AUX
ejpam-4983	423	16	concluded	conclude	VERB
ejpam-4983	423	17	that	that	SCONJ
ejpam-4983	423	18	q|an(r	q|an(r	NOUN
ejpam-4983	423	19	,	,	PUNCT
ejpam-4983	423	20	γ	γ	X
ejpam-4983	423	21	s)|	s)|	NOUN
ejpam-4983	423	22	=	=	NOUN
ejpam-4983	423	23	q3n−2|an(r	q3n−2|an(r	PROPN
ejpam-4983	423	24	/	/	SYM
ejpam-4983	423	25	γ	γ	X
ejpam-4983	423	26	e−1r	e−1r	PROPN
ejpam-4983	423	27	,	,	PUNCT
ejpam-4983	423	28	γs	γ	VERB
ejpam-4983	423	29	+	+	X
ejpam-4983	423	30	γe−1r)|	γe−1r)|	NOUN
ejpam-4983	423	31	.	.	PUNCT
ejpam-4983	424	1	therefore	therefore	ADV
ejpam-4983	424	2	,	,	PUNCT
ejpam-4983	424	3	|an(r	|an(r	PROPN
ejpam-4983	424	4	,	,	PUNCT
ejpam-4983	424	5	γ	γ	NOUN
ejpam-4983	424	6	s)|	s)|	NOUN
ejpam-4983	424	7	=	=	SYM
ejpam-4983	424	8	q3(n−1)|an(r	q3(n−1)|an(r	NOUN
ejpam-4983	424	9	/	/	SYM
ejpam-4983	424	10	γ	γ	PROPN
ejpam-4983	424	11	e−1r	e−1r	PROPN
ejpam-4983	424	12	,	,	PUNCT
ejpam-4983	424	13	γs	γ	VERB
ejpam-4983	424	14	+	+	X
ejpam-4983	424	15	γe−1r)|	γe−1r)|	NOUN
ejpam-4983	424	16	as	as	SCONJ
ejpam-4983	424	17	desired	desire	VERB
ejpam-4983	424	18	.	.	PUNCT
ejpam-4983	425	1	■	■	PUNCT
ejpam-4983	425	2	applying	apply	VERB
ejpam-4983	425	3	lemma	lemma	PROPN
ejpam-4983	425	4	2	2	NUM
ejpam-4983	425	5	recursively	recursively	NOUN
ejpam-4983	425	6	,	,	PUNCT
ejpam-4983	425	7	the	the	DET
ejpam-4983	425	8	next	next	ADJ
ejpam-4983	425	9	corollary	corollary	NOUN
ejpam-4983	425	10	follows	follow	VERB
ejpam-4983	425	11	.	.	PUNCT
ejpam-4983	426	1	corollary	corollary	ADJ
ejpam-4983	426	2	6	6	NUM
ejpam-4983	426	3	.	.	PUNCT
ejpam-4983	427	1	let	let	VERB
ejpam-4983	427	2	r	r	PRON
ejpam-4983	427	3	be	be	AUX
ejpam-4983	427	4	a	a	DET
ejpam-4983	427	5	fccr	fccr	NOUN
ejpam-4983	427	6	of	of	ADP
ejpam-4983	427	7	nilpotency	nilpotency	NOUN
ejpam-4983	427	8	index	index	NOUN
ejpam-4983	427	9	e+f	e+f	NOUN
ejpam-4983	427	10	and	and	CCONJ
ejpam-4983	427	11	residue	residue	NOUN
ejpam-4983	427	12	field	field	NOUN
ejpam-4983	427	13	fq	fq	NOUN
ejpam-4983	427	14	,	,	PUNCT
ejpam-4983	427	15	where	where	SCONJ
ejpam-4983	427	16	2	2	NUM
ejpam-4983	427	17	≤	≤	NOUN
ejpam-4983	427	18	e	e	NOUN
ejpam-4983	427	19	and	and	CCONJ
ejpam-4983	427	20	1	1	NUM
ejpam-4983	427	21	≤	≤	NOUN
ejpam-4983	427	22	f	f	NOUN
ejpam-4983	427	23	are	be	AUX
ejpam-4983	427	24	integers	integer	NOUN
ejpam-4983	427	25	.	.	PUNCT
ejpam-4983	428	1	if	if	SCONJ
ejpam-4983	428	2	the	the	DET
ejpam-4983	428	3	maximal	maximal	ADJ
ejpam-4983	428	4	ideal	ideal	NOUN
ejpam-4983	428	5	of	of	ADP
ejpam-4983	428	6	r	r	NOUN
ejpam-4983	428	7	is	be	AUX
ejpam-4983	428	8	generated	generate	VERB
ejpam-4983	428	9	by	by	ADP
ejpam-4983	428	10	γ	γ	PROPN
ejpam-4983	428	11	,	,	PUNCT
ejpam-4983	428	12	then	then	ADV
ejpam-4983	428	13	|an(r	|an(r	PROPN
ejpam-4983	428	14	,	,	PUNCT
ejpam-4983	428	15	γ	γ	NOUN
ejpam-4983	428	16	s)|	s)|	NOUN
ejpam-4983	428	17	=	=	PUNCT
ejpam-4983	428	18	q3f(n−1)|an(r	q3f(n−1)|an(r	PROPN
ejpam-4983	428	19	/	/	SYM
ejpam-4983	428	20	γ	γ	X
ejpam-4983	428	21	er	er	INTJ
ejpam-4983	428	22	,	,	PUNCT
ejpam-4983	428	23	γs	γ	VERB
ejpam-4983	428	24	+	+	CCONJ
ejpam-4983	428	25	γer)|	γer)|	ADJ
ejpam-4983	428	26	for	for	ADP
ejpam-4983	428	27	all	all	DET
ejpam-4983	428	28	1	1	NUM
ejpam-4983	428	29	≤	≤	NUM
ejpam-4983	428	30	s	s	AUX
ejpam-4983	428	31	<	<	X
ejpam-4983	428	32	e.	e.	PROPN
ejpam-4983	428	33	a	a	DET
ejpam-4983	428	34	general	general	ADJ
ejpam-4983	428	35	recursive	recursive	ADJ
ejpam-4983	428	36	formula	formula	NOUN
ejpam-4983	428	37	for	for	ADP
ejpam-4983	428	38	the	the	DET
ejpam-4983	428	39	number	number	NOUN
ejpam-4983	428	40	an(r	an(r	NOUN
ejpam-4983	428	41	,	,	PUNCT
ejpam-4983	428	42	γ	γ	X
ejpam-4983	428	43	s	s	PART
ejpam-4983	428	44	)	)	PUNCT
ejpam-4983	428	45	is	be	AUX
ejpam-4983	428	46	presented	present	VERB
ejpam-4983	428	47	for	for	ADP
ejpam-4983	428	48	all	all	DET
ejpam-4983	428	49	s	s	PART
ejpam-4983	428	50	≥	≥	NOUN
ejpam-4983	428	51	1	1	NUM
ejpam-4983	428	52	in	in	ADP
ejpam-4983	428	53	the	the	DET
ejpam-4983	428	54	next	next	ADJ
ejpam-4983	428	55	theorem	theorem	PROPN
ejpam-4983	428	56	.	.	PUNCT
ejpam-4983	429	1	theorem	theorem	NOUN
ejpam-4983	429	2	3	3	X
ejpam-4983	429	3	.	.	PUNCT
ejpam-4983	430	1	let	let	VERB
ejpam-4983	430	2	r	r	PRON
ejpam-4983	430	3	be	be	AUX
ejpam-4983	430	4	a	a	DET
ejpam-4983	430	5	fccr	fccr	NOUN
ejpam-4983	430	6	of	of	ADP
ejpam-4983	430	7	nilpotency	nilpotency	NOUN
ejpam-4983	430	8	index	index	NOUN
ejpam-4983	430	9	e	e	NOUN
ejpam-4983	430	10	and	and	CCONJ
ejpam-4983	430	11	residue	residue	NOUN
ejpam-4983	430	12	field	field	NOUN
ejpam-4983	430	13	fq	fq	NOUN
ejpam-4983	430	14	and	and	CCONJ
ejpam-4983	430	15	let	let	VERB
ejpam-4983	430	16	n	n	PRON
ejpam-4983	430	17	be	be	AUX
ejpam-4983	430	18	a	a	DET
ejpam-4983	430	19	positive	positive	ADJ
ejpam-4983	430	20	integer	integer	NOUN
ejpam-4983	430	21	.	.	PUNCT
ejpam-4983	431	1	if	if	SCONJ
ejpam-4983	431	2	the	the	DET
ejpam-4983	431	3	maximal	maximal	ADJ
ejpam-4983	431	4	ideal	ideal	NOUN
ejpam-4983	431	5	of	of	ADP
ejpam-4983	431	6	r	r	NOUN
ejpam-4983	431	7	is	be	AUX
ejpam-4983	431	8	generated	generate	VERB
ejpam-4983	431	9	by	by	ADP
ejpam-4983	431	10	γ	γ	PROPN
ejpam-4983	431	11	,	,	PUNCT
ejpam-4983	431	12	then	then	ADV
ejpam-4983	431	13	|an(r	|an(r	PROPN
ejpam-4983	431	14	,	,	PUNCT
ejpam-4983	431	15	γ	γ	NOUN
ejpam-4983	431	16	s)|	s)|	NOUN
ejpam-4983	431	17	=	=	SYM
ejpam-4983	431	18	q3(e−s−1)(n−1	q3(e−s−1)(n−1	PROPN
ejpam-4983	431	19	)	)	PUNCT
ejpam-4983	431	20	q	q	NOUN
ejpam-4983	432	1	−	−	NOUN
ejpam-4983	432	2	1	1	NUM
ejpam-4983	432	3	(	(	PUNCT
ejpam-4983	432	4	q3n−2|an(r	q3n−2|an(r	NUM
ejpam-4983	432	5	/	/	SYM
ejpam-4983	432	6	γ	γ	X
ejpam-4983	432	7	sr	sr	PROPN
ejpam-4983	432	8	,	,	PUNCT
ejpam-4983	432	9	0	0	PUNCT
ejpam-4983	433	1	+	+	CCONJ
ejpam-4983	433	2	γsr)|	γsr)|	ADJ
ejpam-4983	433	3	−	−	ADP
ejpam-4983	433	4	|an(r	|an(r	PROPN
ejpam-4983	433	5	/	/	SYM
ejpam-4983	433	6	γ	γ	X
ejpam-4983	433	7	s+1r	s+1r	PROPN
ejpam-4983	433	8	,	,	PUNCT
ejpam-4983	433	9	0	0	NUM
ejpam-4983	434	1	+	+	NUM
ejpam-4983	434	2	γs+1r)|	γs+1r)|	ADJ
ejpam-4983	434	3	)	)	PUNCT
ejpam-4983	434	4	.	.	PUNCT
ejpam-4983	435	1	for	for	ADP
ejpam-4983	435	2	all	all	DET
ejpam-4983	435	3	integers	integer	NOUN
ejpam-4983	435	4	1	1	NUM
ejpam-4983	435	5	≤	≤	NOUN
ejpam-4983	435	6	s	s	PART
ejpam-4983	435	7	<	<	X
ejpam-4983	435	8	e.	e.	PROPN
ejpam-4983	435	9	s.	s.	PROPN
ejpam-4983	435	10	jitman	jitman	PROPN
ejpam-4983	435	11	,	,	PUNCT
ejpam-4983	435	12	p.	p.	PROPN
ejpam-4983	435	13	modjam	modjam	PROPN
ejpam-4983	435	14	/	/	SYM
ejpam-4983	435	15	eur	eur	PROPN
ejpam-4983	435	16	.	.	PUNCT
ejpam-4983	436	1	j.	j.	PROPN
ejpam-4983	436	2	pure	pure	PROPN
ejpam-4983	436	3	appl	appl	PROPN
ejpam-4983	436	4	.	.	PROPN
ejpam-4983	436	5	math	math	PROPN
ejpam-4983	436	6	,	,	PUNCT
ejpam-4983	436	7	17	17	NUM
ejpam-4983	436	8	(	(	PUNCT
ejpam-4983	436	9	1	1	NUM
ejpam-4983	436	10	)	)	PUNCT
ejpam-4983	436	11	(	(	PUNCT
ejpam-4983	436	12	2024	2024	NUM
ejpam-4983	436	13	)	)	PUNCT
ejpam-4983	436	14	,	,	PUNCT
ejpam-4983	436	15	11	11	NUM
ejpam-4983	436	16	-	-	SYM
ejpam-4983	436	17	29	29	NUM
ejpam-4983	436	18	26	26	NUM
ejpam-4983	436	19	proof	proof	NOUN
ejpam-4983	436	20	.	.	PUNCT
ejpam-4983	437	1	let	let	VERB
ejpam-4983	437	2	1	1	NUM
ejpam-4983	437	3	≤	≤	NOUN
ejpam-4983	437	4	s	s	PART
ejpam-4983	437	5	<	<	X
ejpam-4983	437	6	e	e	X
ejpam-4983	437	7	be	be	AUX
ejpam-4983	437	8	an	an	DET
ejpam-4983	437	9	integer	integer	NOUN
ejpam-4983	437	10	and	and	CCONJ
ejpam-4983	437	11	let	let	VERB
ejpam-4983	437	12	µ	µ	NOUN
ejpam-4983	437	13	:	:	PUNCT
ejpam-4983	437	14	an(r	an(r	NOUN
ejpam-4983	437	15	/	/	SYM
ejpam-4983	437	16	γ	γ	X
ejpam-4983	437	17	s+1r	s+1r	PROPN
ejpam-4983	437	18	)	)	PUNCT
ejpam-4983	437	19	→	→	SYM
ejpam-4983	437	20	an(r	an(r	NOUN
ejpam-4983	437	21	/	/	SYM
ejpam-4983	437	22	γ	γ	X
ejpam-4983	437	23	sr	sr	PROPN
ejpam-4983	437	24	)	)	PUNCT
ejpam-4983	437	25	be	be	VERB
ejpam-4983	437	26	an	an	DET
ejpam-4983	437	27	additive	additive	ADJ
ejpam-4983	437	28	group	group	NOUN
ejpam-4983	437	29	homomorphism	homomorphism	NOUN
ejpam-4983	437	30	defined	define	VERB
ejpam-4983	437	31	by	by	ADP
ejpam-4983	437	32	µ(a	µ(a	PROPN
ejpam-4983	437	33	)	)	PUNCT
ejpam-4983	437	34	=	=	SYM
ejpam-4983	438	1	a	a	NOUN
ejpam-4983	438	2	,	,	PUNCT
ejpam-4983	438	3	where	where	SCONJ
ejpam-4983	438	4	[	[	PUNCT
ejpam-4983	438	5	aij	aij	PROPN
ejpam-4983	438	6	+	+	X
ejpam-4983	438	7	γs+1r	γs+1r	ADJ
ejpam-4983	438	8	]	]	X
ejpam-4983	438	9	:	:	PUNCT
ejpam-4983	438	10	=	=	PUNCT
ejpam-4983	439	1	[	[	PUNCT
ejpam-4983	439	2	aij	aij	PROPN
ejpam-4983	439	3	+	+	SYM
ejpam-4983	439	4	γsr	γsr	PROPN
ejpam-4983	439	5	]	]	PUNCT
ejpam-4983	439	6	for	for	ADP
ejpam-4983	439	7	all	all	PRON
ejpam-4983	439	8	[	[	PUNCT
ejpam-4983	439	9	aij	aij	X
ejpam-4983	439	10	+	+	X
ejpam-4983	439	11	γs+1r	γs+1r	ADJ
ejpam-4983	439	12	]	]	X
ejpam-4983	439	13	∈	∈	PROPN
ejpam-4983	439	14	an(r	an(r	NOUN
ejpam-4983	439	15	/	/	SYM
ejpam-4983	439	16	γ	γ	X
ejpam-4983	439	17	s+1r	s+1r	PROPN
ejpam-4983	439	18	)	)	PUNCT
ejpam-4983	439	19	.	.	PUNCT
ejpam-4983	440	1	then	then	ADV
ejpam-4983	440	2	,	,	PUNCT
ejpam-4983	440	3	for	for	ADP
ejpam-4983	440	4	each	each	DET
ejpam-4983	440	5	a	a	DET
ejpam-4983	440	6	∈	∈	PROPN
ejpam-4983	440	7	an(r	an(r	NOUN
ejpam-4983	440	8	/	/	SYM
ejpam-4983	440	9	γ	γ	X
ejpam-4983	440	10	s+1r	s+1r	PROPN
ejpam-4983	440	11	)	)	PUNCT
ejpam-4983	440	12	,	,	PUNCT
ejpam-4983	440	13	det(µ(a	det(µ(a	NOUN
ejpam-4983	440	14	)	)	PUNCT
ejpam-4983	440	15	)	)	PUNCT
ejpam-4983	440	16	=	=	SYM
ejpam-4983	440	17	0	0	PUNCT
ejpam-4983	441	1	+	+	CCONJ
ejpam-4983	441	2	γsr	γsr	PROPN
ejpam-4983	441	3	if	if	SCONJ
ejpam-4983	441	4	and	and	CCONJ
ejpam-4983	441	5	only	only	ADV
ejpam-4983	441	6	if	if	SCONJ
ejpam-4983	441	7	det(a	det(a	PROPN
ejpam-4983	441	8	)	)	PUNCT
ejpam-4983	441	9	=	=	PUNCT
ejpam-4983	442	1	γsb	γsb	NOUN
ejpam-4983	442	2	+	+	CCONJ
ejpam-4983	442	3	γs+1r	γs+1r	NOUN
ejpam-4983	442	4	for	for	ADP
ejpam-4983	442	5	some	some	DET
ejpam-4983	442	6	b	b	PROPN
ejpam-4983	442	7	∈	∈	PROPN
ejpam-4983	442	8	v	v	NOUN
ejpam-4983	442	9	,	,	PUNCT
ejpam-4983	442	10	where	where	SCONJ
ejpam-4983	442	11	v	v	NOUN
ejpam-4983	442	12	is	be	AUX
ejpam-4983	442	13	defined	define	VERB
ejpam-4983	442	14	in	in	ADP
ejpam-4983	442	15	lemma	lemma	PROPN
ejpam-4983	442	16	1	1	NUM
ejpam-4983	442	17	.	.	PUNCT
ejpam-4983	443	1	since	since	SCONJ
ejpam-4983	443	2	|	|	ADV
ejpam-4983	443	3	ker(µ)|	ker(µ)|	PROPN
ejpam-4983	443	4	=	=	SYM
ejpam-4983	443	5	q3n−2	q3n−2	PROPN
ejpam-4983	443	6	,	,	PUNCT
ejpam-4983	443	7	we	we	PRON
ejpam-4983	443	8	have	have	VERB
ejpam-4983	443	9	q3n−2|an(r	q3n−2|an(r	NUM
ejpam-4983	443	10	/	/	SYM
ejpam-4983	443	11	γ	γ	X
ejpam-4983	443	12	sr	sr	PROPN
ejpam-4983	443	13	,	,	PUNCT
ejpam-4983	443	14	0	0	PUNCT
ejpam-4983	444	1	+	+	CCONJ
ejpam-4983	445	1	γsr)|	γsr)|	ADJ
ejpam-4983	445	2	=	=	NOUN
ejpam-4983	445	3	|	|	NOUN
ejpam-4983	445	4	ker(µ)||an(r	ker(µ)||an(r	NOUN
ejpam-4983	445	5	/	/	SYM
ejpam-4983	445	6	γ	γ	X
ejpam-4983	445	7	sr	sr	PROPN
ejpam-4983	445	8	,	,	PUNCT
ejpam-4983	445	9	0	0	PUNCT
ejpam-4983	446	1	+	+	CCONJ
ejpam-4983	446	2	γsr)|	γsr)|	ADJ
ejpam-4983	446	3	=	=	SYM
ejpam-4983	446	4	|an(r	|an(r	NOUN
ejpam-4983	446	5	/	/	SYM
ejpam-4983	446	6	γ	γ	X
ejpam-4983	446	7	s+1r	s+1r	PROPN
ejpam-4983	446	8	,	,	PUNCT
ejpam-4983	446	9	0	0	NUM
ejpam-4983	447	1	+	+	NUM
ejpam-4983	447	2	γs+1r)|	γs+1r)|	ADJ
ejpam-4983	447	3	+	+	CCONJ
ejpam-4983	447	4	∑	∑	PROPN
ejpam-4983	447	5	b∈v	b∈v	NOUN
ejpam-4983	447	6	\{0	\{0	NOUN
ejpam-4983	447	7	}	}	PUNCT
ejpam-4983	447	8	|an(r	|an(r	PROPN
ejpam-4983	447	9	/	/	SYM
ejpam-4983	447	10	γ	γ	X
ejpam-4983	447	11	s+1r	s+1r	PROPN
ejpam-4983	447	12	,	,	PUNCT
ejpam-4983	447	13	γsb+	γsb+	PROPN
ejpam-4983	447	14	γs+1r)|	γs+1r)|	NOUN
ejpam-4983	447	15	=	=	PUNCT
ejpam-4983	447	16	|an(r	|an(r	PROPN
ejpam-4983	447	17	/	/	SYM
ejpam-4983	447	18	γ	γ	X
ejpam-4983	447	19	s+1r	s+1r	PROPN
ejpam-4983	447	20	,	,	PUNCT
ejpam-4983	447	21	0	0	NUM
ejpam-4983	448	1	+	+	NUM
ejpam-4983	448	2	γs+1r)|	γs+1r)|	NOUN
ejpam-4983	448	3	+	+	CCONJ
ejpam-4983	448	4	(	(	PUNCT
ejpam-4983	448	5	q	q	NOUN
ejpam-4983	448	6	−	−	PROPN
ejpam-4983	448	7	1)|an(r	1)|an(r	NUM
ejpam-4983	448	8	/	/	SYM
ejpam-4983	448	9	γ	γ	X
ejpam-4983	448	10	s+1r	s+1r	PROPN
ejpam-4983	448	11	,	,	PUNCT
ejpam-4983	448	12	γs	γ	VERB
ejpam-4983	448	13	+	+	X
ejpam-4983	448	14	γs+1r)|	γs+1r)|	ADJ
ejpam-4983	448	15	by	by	ADP
ejpam-4983	448	16	proposition	proposition	NOUN
ejpam-4983	448	17	5	5	NUM
ejpam-4983	448	18	.	.	PUNCT
ejpam-4983	449	1	hence	hence	ADV
ejpam-4983	449	2	,	,	PUNCT
ejpam-4983	449	3	we	we	PRON
ejpam-4983	449	4	have	have	VERB
ejpam-4983	449	5	|an(r	|an(r	PROPN
ejpam-4983	449	6	/	/	SYM
ejpam-4983	449	7	γ	γ	X
ejpam-4983	449	8	s+1r	s+1r	PROPN
ejpam-4983	449	9	,	,	PUNCT
ejpam-4983	449	10	γs	γ	VERB
ejpam-4983	449	11	+	+	X
ejpam-4983	449	12	γs+1r)|	γs+1r)|	ADJ
ejpam-4983	449	13	=	=	SYM
ejpam-4983	449	14	1	1	NUM
ejpam-4983	449	15	q	q	NOUN
ejpam-4983	449	16	−	−	PROPN
ejpam-4983	449	17	1	1	NUM
ejpam-4983	449	18	(	(	PUNCT
ejpam-4983	449	19	q3n−2|an(r	q3n−2|an(r	NUM
ejpam-4983	449	20	/	/	SYM
ejpam-4983	449	21	γ	γ	X
ejpam-4983	449	22	sr	sr	PROPN
ejpam-4983	449	23	,	,	PUNCT
ejpam-4983	449	24	0	0	PUNCT
ejpam-4983	450	1	+	+	CCONJ
ejpam-4983	450	2	γsr)|	γsr)|	ADJ
ejpam-4983	450	3	−	−	ADP
ejpam-4983	450	4	|an(r	|an(r	PROPN
ejpam-4983	450	5	/	/	SYM
ejpam-4983	450	6	γ	γ	X
ejpam-4983	450	7	s+1r	s+1r	PROPN
ejpam-4983	450	8	,	,	PUNCT
ejpam-4983	450	9	0	0	NUM
ejpam-4983	451	1	+	+	NUM
ejpam-4983	451	2	γs+1r)|	γs+1r)|	ADJ
ejpam-4983	451	3	)	)	PUNCT
ejpam-4983	451	4	.	.	PUNCT
ejpam-4983	452	1	(	(	PUNCT
ejpam-4983	452	2	7	7	X
ejpam-4983	452	3	)	)	PUNCT
ejpam-4983	452	4	by	by	ADP
ejpam-4983	452	5	corollary	corollary	ADJ
ejpam-4983	452	6	6	6	NUM
ejpam-4983	452	7	,	,	PUNCT
ejpam-4983	452	8	we	we	PRON
ejpam-4983	452	9	have	have	VERB
ejpam-4983	452	10	|an(r	|an(r	PROPN
ejpam-4983	452	11	,	,	PUNCT
ejpam-4983	452	12	γ	γ	X
ejpam-4983	452	13	s)|	s)|	NOUN
ejpam-4983	452	14	=	=	SYM
ejpam-4983	452	15	|an(r	|an(r	PROPN
ejpam-4983	452	16	/	/	SYM
ejpam-4983	452	17	γ	γ	X
ejpam-4983	452	18	e+1+(s−e−1)r	e+1+(s−e−1)r	NOUN
ejpam-4983	452	19	,	,	PUNCT
ejpam-4983	452	20	γs	γ	VERB
ejpam-4983	452	21	+	+	NOUN
ejpam-4983	452	22	γe+1+(s−e−1)r)|	γe+1+(s−e−1)r)|	PROPN
ejpam-4983	452	23	=	=	SYM
ejpam-4983	452	24	q3(e−s−1)(n−1)|an(r	q3(e−s−1)(n−1)|an(r	NOUN
ejpam-4983	452	25	/	/	SYM
ejpam-4983	452	26	γ	γ	X
ejpam-4983	452	27	s+1r	s+1r	PROPN
ejpam-4983	452	28	,	,	PUNCT
ejpam-4983	452	29	γs	γ	VERB
ejpam-4983	452	30	+	+	X
ejpam-4983	452	31	γs+1r)|	γs+1r)|	ADJ
ejpam-4983	452	32	.	.	PUNCT
ejpam-4983	453	1	(	(	PUNCT
ejpam-4983	453	2	8)	8)	NUM
ejpam-4983	453	3	combining	combine	VERB
ejpam-4983	453	4	(	(	PUNCT
ejpam-4983	453	5	7	7	NUM
ejpam-4983	453	6	)	)	PUNCT
ejpam-4983	453	7	and	and	CCONJ
ejpam-4983	453	8	(	(	PUNCT
ejpam-4983	453	9	8)	8)	NUM
ejpam-4983	453	10	,	,	PUNCT
ejpam-4983	453	11	we	we	PRON
ejpam-4983	453	12	therefore	therefore	ADV
ejpam-4983	453	13	have	have	VERB
ejpam-4983	453	14	|an(r	|an(r	PROPN
ejpam-4983	453	15	,	,	PUNCT
ejpam-4983	453	16	γ	γ	X
ejpam-4983	453	17	s)|	s)|	NOUN
ejpam-4983	453	18	=	=	SYM
ejpam-4983	453	19	q3(e−s−1)(n−1	q3(e−s−1)(n−1	PROPN
ejpam-4983	453	20	)	)	PUNCT
ejpam-4983	453	21	q	q	NOUN
ejpam-4983	454	1	−	−	NOUN
ejpam-4983	454	2	1	1	NUM
ejpam-4983	454	3	(	(	PUNCT
ejpam-4983	454	4	q3n−2|an(r	q3n−2|an(r	NUM
ejpam-4983	454	5	/	/	SYM
ejpam-4983	454	6	γ	γ	X
ejpam-4983	454	7	sr	sr	PROPN
ejpam-4983	454	8	,	,	PUNCT
ejpam-4983	454	9	0	0	PUNCT
ejpam-4983	455	1	+	+	CCONJ
ejpam-4983	455	2	γsr)|	γsr)|	ADJ
ejpam-4983	455	3	−	−	ADP
ejpam-4983	455	4	|an(r	|an(r	PROPN
ejpam-4983	455	5	/	/	SYM
ejpam-4983	455	6	γ	γ	X
ejpam-4983	455	7	s+1r	s+1r	PROPN
ejpam-4983	455	8	,	,	PUNCT
ejpam-4983	455	9	0	0	NUM
ejpam-4983	456	1	+	+	CCONJ
ejpam-4983	456	2	γs+1r)|	γs+1r)|	ADJ
ejpam-4983	456	3	)	)	PUNCT
ejpam-4983	456	4	as	as	SCONJ
ejpam-4983	456	5	desired	desire	VERB
ejpam-4983	456	6	.	.	PUNCT
ejpam-4983	457	1	■	■	PUNCT
ejpam-4983	457	2	for	for	ADP
ejpam-4983	457	3	a	a	DET
ejpam-4983	457	4	fccr	fccr	NOUN
ejpam-4983	457	5	of	of	ADP
ejpam-4983	457	6	nilpotency	nilpotency	NOUN
ejpam-4983	457	7	index	index	NOUN
ejpam-4983	457	8	2	2	NUM
ejpam-4983	457	9	,	,	PUNCT
ejpam-4983	457	10	the	the	DET
ejpam-4983	457	11	following	follow	VERB
ejpam-4983	457	12	bound	bind	VERB
ejpam-4983	457	13	on	on	ADP
ejpam-4983	457	14	|an(r	|an(r	PROPN
ejpam-4983	457	15	,	,	PUNCT
ejpam-4983	457	16	a)|	a)|	X
ejpam-4983	457	17	is	be	AUX
ejpam-4983	457	18	derived	derive	VERB
ejpam-4983	457	19	for	for	ADP
ejpam-4983	457	20	all	all	DET
ejpam-4983	457	21	a	a	DET
ejpam-4983	457	22	∈	∈	NOUN
ejpam-4983	457	23	r	r	NOUN
ejpam-4983	457	24	\	\	NOUN
ejpam-4983	457	25	fq	fq	NOUN
ejpam-4983	457	26	and	and	CCONJ
ejpam-4983	457	27	positive	positive	ADJ
ejpam-4983	457	28	integers	integer	NOUN
ejpam-4983	457	29	n.	n.	NOUN
ejpam-4983	457	30	corollary	corollary	PROPN
ejpam-4983	457	31	7	7	NUM
ejpam-4983	457	32	.	.	PUNCT
ejpam-4983	458	1	let	let	VERB
ejpam-4983	458	2	r	r	PRON
ejpam-4983	458	3	be	be	AUX
ejpam-4983	458	4	a	a	DET
ejpam-4983	458	5	fccr	fccr	NOUN
ejpam-4983	458	6	of	of	ADP
ejpam-4983	458	7	nilpotency	nilpotency	NOUN
ejpam-4983	458	8	index	index	NOUN
ejpam-4983	458	9	2	2	NUM
ejpam-4983	458	10	and	and	CCONJ
ejpam-4983	458	11	residue	residue	NOUN
ejpam-4983	458	12	field	field	NOUN
ejpam-4983	458	13	fq	fq	NOUN
ejpam-4983	458	14	.	.	PROPN
ejpam-4983	459	1	if	if	SCONJ
ejpam-4983	459	2	the	the	DET
ejpam-4983	459	3	maximal	maximal	ADJ
ejpam-4983	459	4	ideal	ideal	NOUN
ejpam-4983	459	5	of	of	ADP
ejpam-4983	459	6	r	r	NOUN
ejpam-4983	459	7	is	be	AUX
ejpam-4983	459	8	generated	generate	VERB
ejpam-4983	459	9	by	by	ADP
ejpam-4983	459	10	γ	γ	PROPN
ejpam-4983	459	11	,	,	PUNCT
ejpam-4983	459	12	then	then	ADV
ejpam-4983	459	13	|a1(r	|a1(r	NUM
ejpam-4983	459	14	,	,	PUNCT
ejpam-4983	459	15	a)|	a)|	X
ejpam-4983	460	1	=	=	NOUN
ejpam-4983	460	2	1	1	NUM
ejpam-4983	460	3	and	and	CCONJ
ejpam-4983	460	4	|an(r	|an(r	PROPN
ejpam-4983	460	5	,	,	PUNCT
ejpam-4983	460	6	a)|	a)|	X
ejpam-4983	460	7	≤	≤	NOUN
ejpam-4983	460	8	(	(	PUNCT
ejpam-4983	460	9	q	q	NOUN
ejpam-4983	461	1	+	+	CCONJ
ejpam-4983	461	2	1)q5n−7	1)q5n−7	ADJ
ejpam-4983	461	3	(	(	PUNCT
ejpam-4983	461	4	qn+1	qn+1	NUM
ejpam-4983	461	5	−	−	PROPN
ejpam-4983	461	6	(	(	PUNCT
ejpam-4983	461	7	q	q	NOUN
ejpam-4983	461	8	−	−	PROPN
ejpam-4983	461	9	1)n(q	1)n(q	NUM
ejpam-4983	461	10	+	+	CCONJ
ejpam-4983	461	11	(	(	PUNCT
ejpam-4983	461	12	n−	n−	NOUN
ejpam-4983	461	13	1	1	NUM
ejpam-4983	461	14	)	)	PUNCT
ejpam-4983	461	15	)	)	PUNCT
ejpam-4983	461	16	)	)	PUNCT
ejpam-4983	462	1	−	−	PROPN
ejpam-4983	463	1	q2(q3	q2(q3	PRON
ejpam-4983	464	1	+	+	CCONJ
ejpam-4983	464	2	1)|an−1(r	1)|an−1(r	ADJ
ejpam-4983	464	3	,	,	PUNCT
ejpam-4983	464	4	0)|	0)|	NOUN
ejpam-4983	464	5	for	for	ADP
ejpam-4983	464	6	all	all	DET
ejpam-4983	464	7	a	a	DET
ejpam-4983	464	8	∈	∈	NOUN
ejpam-4983	464	9	r	r	NOUN
ejpam-4983	464	10	\	\	NOUN
ejpam-4983	464	11	fq	fq	PROPN
ejpam-4983	464	12	and	and	CCONJ
ejpam-4983	464	13	integers	integer	NOUN
ejpam-4983	464	14	n	n	PRON
ejpam-4983	464	15	≥	≥	NOUN
ejpam-4983	464	16	2	2	NUM
ejpam-4983	464	17	.	.	PUNCT
ejpam-4983	465	1	proof	proof	NOUN
ejpam-4983	465	2	.	.	PUNCT
ejpam-4983	466	1	clearly	clearly	ADV
ejpam-4983	466	2	,	,	PUNCT
ejpam-4983	466	3	|a1(r	|a1(r	NOUN
ejpam-4983	466	4	,	,	PUNCT
ejpam-4983	466	5	a)|	a)|	X
ejpam-4983	467	1	=	=	NOUN
ejpam-4983	467	2	1	1	X
ejpam-4983	467	3	.	.	PUNCT
ejpam-4983	468	1	let	let	VERB
ejpam-4983	468	2	n	n	PRON
ejpam-4983	468	3	≥	≥	X
ejpam-4983	468	4	2	2	NUM
ejpam-4983	468	5	be	be	AUX
ejpam-4983	468	6	an	an	DET
ejpam-4983	468	7	integer	integer	NOUN
ejpam-4983	468	8	.	.	PUNCT
ejpam-4983	469	1	by	by	ADP
ejpam-4983	469	2	setting	set	VERB
ejpam-4983	469	3	s	s	X
ejpam-4983	469	4	=	=	SYM
ejpam-4983	469	5	1	1	NUM
ejpam-4983	469	6	in	in	ADP
ejpam-4983	469	7	(	(	PUNCT
ejpam-4983	469	8	7	7	NUM
ejpam-4983	469	9	)	)	PUNCT
ejpam-4983	469	10	,	,	PUNCT
ejpam-4983	469	11	we	we	PRON
ejpam-4983	469	12	have	have	VERB
ejpam-4983	469	13	|an(r	|an(r	PROPN
ejpam-4983	469	14	,	,	PUNCT
ejpam-4983	469	15	a)|	a)|	X
ejpam-4983	470	1	=	=	SYM
ejpam-4983	470	2	|an(r	|an(r	PROPN
ejpam-4983	470	3	,	,	PUNCT
ejpam-4983	470	4	γ)|	γ)|	NOUN
ejpam-4983	470	5	s.	s.	PROPN
ejpam-4983	470	6	jitman	jitman	PROPN
ejpam-4983	470	7	,	,	PUNCT
ejpam-4983	470	8	p.	p.	PROPN
ejpam-4983	470	9	modjam	modjam	PROPN
ejpam-4983	470	10	/	/	SYM
ejpam-4983	470	11	eur	eur	PROPN
ejpam-4983	470	12	.	.	PUNCT
ejpam-4983	471	1	j.	j.	PROPN
ejpam-4983	471	2	pure	pure	PROPN
ejpam-4983	471	3	appl	appl	PROPN
ejpam-4983	471	4	.	.	PROPN
ejpam-4983	471	5	math	math	PROPN
ejpam-4983	471	6	,	,	PUNCT
ejpam-4983	471	7	17	17	NUM
ejpam-4983	471	8	(	(	PUNCT
ejpam-4983	471	9	1	1	NUM
ejpam-4983	471	10	)	)	PUNCT
ejpam-4983	471	11	(	(	PUNCT
ejpam-4983	471	12	2024	2024	NUM
ejpam-4983	471	13	)	)	PUNCT
ejpam-4983	471	14	,	,	PUNCT
ejpam-4983	471	15	11	11	NUM
ejpam-4983	471	16	-	-	SYM
ejpam-4983	471	17	29	29	NUM
ejpam-4983	471	18	27	27	NUM
ejpam-4983	471	19	=	=	SYM
ejpam-4983	471	20	1	1	NUM
ejpam-4983	471	21	q	q	NOUN
ejpam-4983	471	22	−	−	PROPN
ejpam-4983	471	23	1	1	NUM
ejpam-4983	471	24	(	(	PUNCT
ejpam-4983	471	25	q3n−2|an(r	q3n−2|an(r	X
ejpam-4983	471	26	/	/	SYM
ejpam-4983	471	27	γr	γr	PROPN
ejpam-4983	471	28	,	,	PUNCT
ejpam-4983	471	29	0	0	PUNCT
ejpam-4983	472	1	+	+	CCONJ
ejpam-4983	473	1	γr)|	γr)|	PROPN
ejpam-4983	473	2	−	−	PROPN
ejpam-4983	473	3	|an(r	|an(r	NOUN
ejpam-4983	473	4	,	,	PUNCT
ejpam-4983	473	5	0)|	0)|	NOUN
ejpam-4983	473	6	)	)	PUNCT
ejpam-4983	473	7	=	=	PUNCT
ejpam-4983	474	1	1	1	NUM
ejpam-4983	474	2	q	q	NOUN
ejpam-4983	474	3	−	−	PROPN
ejpam-4983	474	4	1	1	NUM
ejpam-4983	474	5	(	(	PUNCT
ejpam-4983	474	6	q3n−2|an(fq	q3n−2|an(fq	PROPN
ejpam-4983	474	7	,	,	PUNCT
ejpam-4983	474	8	0)|	0)|	NOUN
ejpam-4983	474	9	−	−	NOUN
ejpam-4983	474	10	|an(r	|an(r	NOUN
ejpam-4983	474	11	,	,	PUNCT
ejpam-4983	474	12	0)|	0)|	NOUN
ejpam-4983	474	13	)	)	PUNCT
ejpam-4983	474	14	.	.	PUNCT
ejpam-4983	475	1	form	form	VERB
ejpam-4983	475	2	the	the	DET
ejpam-4983	475	3	proof	proof	NOUN
ejpam-4983	475	4	of	of	ADP
ejpam-4983	475	5	corollary	corollary	ADJ
ejpam-4983	475	6	5	5	NUM
ejpam-4983	475	7	,	,	PUNCT
ejpam-4983	475	8	we	we	PRON
ejpam-4983	475	9	have	have	VERB
ejpam-4983	475	10	|an(r	|an(r	PROPN
ejpam-4983	475	11	,	,	PUNCT
ejpam-4983	475	12	0)|	0)|	NOUN
ejpam-4983	475	13	≥	≥	NOUN
ejpam-4983	475	14	(	(	PUNCT
ejpam-4983	475	15	q	q	NOUN
ejpam-4983	475	16	−	−	PROPN
ejpam-4983	475	17	1)q2(q3	1)q2(q3	NUM
ejpam-4983	476	1	+	+	CCONJ
ejpam-4983	476	2	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	476	3	,	,	PUNCT
ejpam-4983	476	4	0)|+	0)|+	PUNCT
ejpam-4983	476	5	q3n−4|an(fq	q3n−4|an(fq	ADJ
ejpam-4983	476	6	,	,	PUNCT
ejpam-4983	476	7	0)|	0)|	NOUN
ejpam-4983	476	8	which	which	PRON
ejpam-4983	476	9	implies	imply	VERB
ejpam-4983	476	10	that	that	SCONJ
ejpam-4983	476	11	|an(r	|an(r	PROPN
ejpam-4983	476	12	,	,	PUNCT
ejpam-4983	476	13	a)|	a)|	X
ejpam-4983	476	14	≤	≤	ADV
ejpam-4983	476	15	1	1	NUM
ejpam-4983	476	16	q	q	NOUN
ejpam-4983	476	17	−	−	PROPN
ejpam-4983	476	18	1	1	NUM
ejpam-4983	476	19	(	(	PUNCT
ejpam-4983	476	20	q3n−2|an(fq	q3n−2|an(fq	PROPN
ejpam-4983	476	21	,	,	PUNCT
ejpam-4983	476	22	0)|	0)|	NOUN
ejpam-4983	476	23	−	−	PROPN
ejpam-4983	476	24	(	(	PUNCT
ejpam-4983	476	25	(	(	PUNCT
ejpam-4983	476	26	q	q	NOUN
ejpam-4983	476	27	−	−	PROPN
ejpam-4983	476	28	1)q2(q3	1)q2(q3	NUM
ejpam-4983	477	1	+	+	CCONJ
ejpam-4983	477	2	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	477	3	,	,	PUNCT
ejpam-4983	477	4	0)|+	0)|+	PUNCT
ejpam-4983	477	5	q3n−4|an(fq	q3n−4|an(fq	ADJ
ejpam-4983	477	6	,	,	PUNCT
ejpam-4983	477	7	0)|	0)|	NOUN
ejpam-4983	477	8	)	)	PUNCT
ejpam-4983	477	9	)	)	PUNCT
ejpam-4983	478	1	=	=	PUNCT
ejpam-4983	479	1	1	1	NUM
ejpam-4983	479	2	q	q	NOUN
ejpam-4983	479	3	−	−	PROPN
ejpam-4983	479	4	1	1	NUM
ejpam-4983	479	5	(	(	PUNCT
ejpam-4983	479	6	(	(	PUNCT
ejpam-4983	479	7	q3n−2	q3n−2	PROPN
ejpam-4983	479	8	−	−	PROPN
ejpam-4983	479	9	q3n−4)|an(fq	q3n−4)|an(fq	PROPN
ejpam-4983	479	10	,	,	PUNCT
ejpam-4983	479	11	0)|	0)|	NOUN
ejpam-4983	479	12	−	−	NOUN
ejpam-4983	480	1	(	(	PUNCT
ejpam-4983	480	2	q	q	NOUN
ejpam-4983	480	3	−	−	NUM
ejpam-4983	480	4	1)q2(q3	1)q2(q3	NUM
ejpam-4983	481	1	+	+	CCONJ
ejpam-4983	481	2	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	481	3	,	,	PUNCT
ejpam-4983	481	4	0)|	0)|	NOUN
ejpam-4983	481	5	)	)	PUNCT
ejpam-4983	482	1	=	=	PUNCT
ejpam-4983	483	1	1	1	NUM
ejpam-4983	483	2	q	q	NOUN
ejpam-4983	483	3	−	−	PROPN
ejpam-4983	483	4	1	1	NUM
ejpam-4983	483	5	(	(	PUNCT
ejpam-4983	483	6	(	(	PUNCT
ejpam-4983	483	7	q2	q2	NOUN
ejpam-4983	483	8	−	−	NOUN
ejpam-4983	483	9	1)q3n−4|an(fq	1)q3n−4|an(fq	NUM
ejpam-4983	483	10	,	,	PUNCT
ejpam-4983	483	11	0)|	0)|	NOUN
ejpam-4983	483	12	−	−	NOUN
ejpam-4983	483	13	(	(	PUNCT
ejpam-4983	483	14	q	q	NOUN
ejpam-4983	483	15	−	−	NUM
ejpam-4983	483	16	1)q2(q3	1)q2(q3	NUM
ejpam-4983	483	17	+	+	CCONJ
ejpam-4983	483	18	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	483	19	,	,	PUNCT
ejpam-4983	483	20	0)|	0)|	NOUN
ejpam-4983	483	21	)	)	PUNCT
ejpam-4983	483	22	=	=	PUNCT
ejpam-4983	484	1	(	(	PUNCT
ejpam-4983	484	2	q	q	PROPN
ejpam-4983	485	1	+	+	NUM
ejpam-4983	485	2	1)q3n−4|an(fq	1)q3n−4|an(fq	NUM
ejpam-4983	485	3	,	,	PUNCT
ejpam-4983	485	4	0)|	0)|	NOUN
ejpam-4983	485	5	−	−	PUNCT
ejpam-4983	485	6	q2(q3	q2(q3	PRON
ejpam-4983	486	1	+	+	CCONJ
ejpam-4983	486	2	1)|an−1(r	1)|an−1(r	NUM
ejpam-4983	486	3	,	,	PUNCT
ejpam-4983	486	4	0)|	0)|	NOUN
ejpam-4983	486	5	.	.	PUNCT
ejpam-4983	487	1	by	by	ADP
ejpam-4983	487	2	corollary	corollary	ADJ
ejpam-4983	487	3	2	2	NUM
ejpam-4983	487	4	,	,	PUNCT
ejpam-4983	487	5	we	we	PRON
ejpam-4983	487	6	have	have	VERB
ejpam-4983	487	7	|an(fq	|an(fq	NOUN
ejpam-4983	487	8	,	,	PUNCT
ejpam-4983	487	9	0)|	0)|	NOUN
ejpam-4983	487	10	=	=	SYM
ejpam-4983	488	1	q3n−2	q3n−2	PROPN
ejpam-4983	488	2	−	−	PROPN
ejpam-4983	488	3	q2n−3(q	q2n−3(q	NUM
ejpam-4983	488	4	−	−	PROPN
ejpam-4983	488	5	1)n(q	1)n(q	NUM
ejpam-4983	488	6	+	+	CCONJ
ejpam-4983	488	7	(	(	PUNCT
ejpam-4983	488	8	n−	n−	NOUN
ejpam-4983	488	9	1	1	NUM
ejpam-4983	488	10	)	)	PUNCT
ejpam-4983	488	11	)	)	PUNCT
ejpam-4983	488	12	,	,	PUNCT
ejpam-4983	488	13	and	and	CCONJ
ejpam-4983	488	14	hence	hence	ADV
ejpam-4983	488	15	,	,	PUNCT
ejpam-4983	488	16	|an(r	|an(r	PROPN
ejpam-4983	488	17	,	,	PUNCT
ejpam-4983	488	18	a)|	a)|	X
ejpam-4983	488	19	≤	≤	NOUN
ejpam-4983	488	20	(	(	PUNCT
ejpam-4983	488	21	q	q	NOUN
ejpam-4983	488	22	+	+	NUM
ejpam-4983	488	23	1)q3n−4	1)q3n−4	NOUN
ejpam-4983	488	24	(	(	PUNCT
ejpam-4983	488	25	q3n−2	q3n−2	PROPN
ejpam-4983	488	26	−	−	PROPN
ejpam-4983	488	27	q2n−3(q	q2n−3(q	NUM
ejpam-4983	488	28	−	−	PROPN
ejpam-4983	488	29	1)n(q	1)n(q	NUM
ejpam-4983	488	30	+	+	CCONJ
ejpam-4983	488	31	(	(	PUNCT
ejpam-4983	488	32	n−	n−	NOUN
ejpam-4983	488	33	1	1	NUM
ejpam-4983	488	34	)	)	PUNCT
ejpam-4983	488	35	)	)	PUNCT
ejpam-4983	488	36	)	)	PUNCT
ejpam-4983	489	1	−	−	PROPN
ejpam-4983	490	1	q2(q3	q2(q3	PRON
ejpam-4983	491	1	+	+	CCONJ
ejpam-4983	491	2	1)|an−1(r	1)|an−1(r	ADJ
ejpam-4983	491	3	,	,	PUNCT
ejpam-4983	491	4	0)|	0)|	NOUN
ejpam-4983	491	5	=	=	SYM
ejpam-4983	491	6	(	(	PUNCT
ejpam-4983	491	7	q	q	PUNCT
ejpam-4983	491	8	+	+	CCONJ
ejpam-4983	491	9	1)q5n−7	1)q5n−7	ADJ
ejpam-4983	491	10	(	(	PUNCT
ejpam-4983	491	11	qn+1	qn+1	NUM
ejpam-4983	491	12	−	−	PROPN
ejpam-4983	491	13	(	(	PUNCT
ejpam-4983	491	14	q	q	NOUN
ejpam-4983	491	15	−	−	PROPN
ejpam-4983	492	1	1)n(q	1)n(q	NUM
ejpam-4983	492	2	+	+	CCONJ
ejpam-4983	492	3	(	(	PUNCT
ejpam-4983	492	4	n−	n−	NOUN
ejpam-4983	492	5	1	1	NUM
ejpam-4983	492	6	)	)	PUNCT
ejpam-4983	492	7	)	)	PUNCT
ejpam-4983	492	8	)	)	PUNCT
ejpam-4983	493	1	−	−	PROPN
ejpam-4983	494	1	q2(q3	q2(q3	PRON
ejpam-4983	495	1	+	+	CCONJ
ejpam-4983	495	2	1)|an−1(r	1)|an−1(r	ADJ
ejpam-4983	495	3	,	,	PUNCT
ejpam-4983	495	4	0)|	0)|	NOUN
ejpam-4983	495	5	as	as	SCONJ
ejpam-4983	495	6	desired	desire	VERB
ejpam-4983	495	7	.	.	PUNCT
ejpam-4983	496	1	■	■	PUNCT
ejpam-4983	496	2	we	we	PRON
ejpam-4983	496	3	note	note	VERB
ejpam-4983	496	4	that	that	SCONJ
ejpam-4983	496	5	,	,	PUNCT
ejpam-4983	496	6	for	for	ADP
ejpam-4983	496	7	a	a	DET
ejpam-4983	496	8	fccr	fccr	NOUN
ejpam-4983	496	9	of	of	ADP
ejpam-4983	496	10	nilpotency	nilpotency	NOUN
ejpam-4983	496	11	index	index	NOUN
ejpam-4983	496	12	e	e	NOUN
ejpam-4983	496	13	=	=	SYM
ejpam-4983	496	14	2	2	NUM
ejpam-4983	496	15	,	,	PUNCT
ejpam-4983	496	16	a	a	DET
ejpam-4983	496	17	bound	bind	VERB
ejpam-4983	496	18	on	on	ADP
ejpam-4983	496	19	|an−1(r	|an−1(r	NOUN
ejpam-4983	496	20	,	,	PUNCT
ejpam-4983	496	21	0)|	0)|	NOUN
ejpam-4983	496	22	is	be	AUX
ejpam-4983	496	23	determined	determine	VERB
ejpam-4983	496	24	recursively	recursively	ADV
ejpam-4983	496	25	in	in	ADP
ejpam-4983	496	26	corollary	corollary	ADJ
ejpam-4983	496	27	5	5	NUM
ejpam-4983	496	28	.	.	NOUN
ejpam-4983	496	29	4	4	NUM
ejpam-4983	496	30	.	.	NOUN
ejpam-4983	496	31	conclusion	conclusion	NOUN
ejpam-4983	496	32	and	and	CCONJ
ejpam-4983	496	33	remarks	remark	VERB
ejpam-4983	496	34	the	the	DET
ejpam-4983	496	35	enumeration	enumeration	NOUN
ejpam-4983	496	36	of	of	ADP
ejpam-4983	496	37	arrowhead	arrowhead	NOUN
ejpam-4983	496	38	matrices	matrix	NOUN
ejpam-4983	496	39	with	with	ADP
ejpam-4983	496	40	prescribed	prescribe	VERB
ejpam-4983	496	41	determinant	determinant	NOUN
ejpam-4983	496	42	has	have	AUX
ejpam-4983	496	43	been	be	AUX
ejpam-4983	496	44	established	establish	VERB
ejpam-4983	496	45	over	over	ADP
ejpam-4983	496	46	a	a	DET
ejpam-4983	496	47	finite	finite	ADJ
ejpam-4983	496	48	field	field	NOUN
ejpam-4983	496	49	fq	fq	PROPN
ejpam-4983	496	50	and	and	CCONJ
ejpam-4983	496	51	a	a	DET
ejpam-4983	496	52	finite	finite	ADJ
ejpam-4983	496	53	commutative	commutative	ADJ
ejpam-4983	496	54	chain	chain	NOUN
ejpam-4983	496	55	ring	ring	NOUN
ejpam-4983	496	56	r.	r.	PROPN
ejpam-4983	496	57	over	over	ADP
ejpam-4983	496	58	fq	fq	PROPN
ejpam-4983	496	59	,	,	PUNCT
ejpam-4983	496	60	the	the	DET
ejpam-4983	496	61	number	number	NOUN
ejpam-4983	496	62	of	of	ADP
ejpam-4983	496	63	n×n	n×n	PROPN
ejpam-4983	496	64	arrowhead	arrowhead	NOUN
ejpam-4983	496	65	matrices	matrix	NOUN
ejpam-4983	496	66	with	with	ADP
ejpam-4983	496	67	prescribed	prescribe	VERB
ejpam-4983	496	68	determinant	determinant	NOUN
ejpam-4983	496	69	has	have	AUX
ejpam-4983	496	70	been	be	AUX
ejpam-4983	496	71	completely	completely	ADV
ejpam-4983	496	72	determined	determine	VERB
ejpam-4983	496	73	for	for	SCONJ
ejpam-4983	496	74	all	all	DET
ejpam-4983	496	75	positive	positive	ADJ
ejpam-4983	496	76	integers	integer	NOUN
ejpam-4983	496	77	n.	n.	VERB
ejpam-4983	496	78	subsequently	subsequently	ADV
ejpam-4983	496	79	,	,	PUNCT
ejpam-4983	496	80	the	the	DET
ejpam-4983	496	81	number	number	NOUN
ejpam-4983	496	82	of	of	ADP
ejpam-4983	496	83	n×n	n×n	PROPN
ejpam-4983	496	84	non	non	ADJ
ejpam-4983	496	85	-	-	ADJ
ejpam-4983	496	86	singular	singular	ADJ
ejpam-4983	496	87	arrowhead	arrowhead	NOUN
ejpam-4983	496	88	matrices	matrix	NOUN
ejpam-4983	496	89	with	with	ADP
ejpam-4983	496	90	prescribed	prescribe	VERB
ejpam-4983	496	91	determinant	determinant	ADJ
ejpam-4983	496	92	over	over	ADP
ejpam-4983	496	93	r	r	NOUN
ejpam-4983	496	94	has	have	AUX
ejpam-4983	496	95	been	be	AUX
ejpam-4983	496	96	given	give	VERB
ejpam-4983	496	97	for	for	ADP
ejpam-4983	496	98	all	all	DET
ejpam-4983	496	99	positive	positive	ADJ
ejpam-4983	496	100	integers	integer	NOUN
ejpam-4983	496	101	n.	n.	VERB
ejpam-4983	496	102	for	for	ADP
ejpam-4983	496	103	singular	singular	ADJ
ejpam-4983	496	104	arrowhead	arrowhead	NOUN
ejpam-4983	496	105	matrices	matrix	NOUN
ejpam-4983	496	106	over	over	ADP
ejpam-4983	496	107	r	r	NOUN
ejpam-4983	496	108	,	,	PUNCT
ejpam-4983	496	109	bounds	bound	VERB
ejpam-4983	496	110	on	on	ADP
ejpam-4983	496	111	the	the	DET
ejpam-4983	496	112	number	number	NOUN
ejpam-4983	496	113	of	of	ADP
ejpam-4983	496	114	n	n	NUM
ejpam-4983	496	115	×	×	NOUN
ejpam-4983	496	116	n	n	CCONJ
ejpam-4983	496	117	singular	singular	ADJ
ejpam-4983	496	118	arrowhead	arrowhead	NOUN
ejpam-4983	496	119	matrices	matrix	NOUN
ejpam-4983	496	120	have	have	AUX
ejpam-4983	496	121	been	be	AUX
ejpam-4983	496	122	presented	present	VERB
ejpam-4983	496	123	.	.	PUNCT
ejpam-4983	497	1	a	a	DET
ejpam-4983	497	2	general	general	NOUN
ejpam-4983	497	3	set	set	VERB
ejpam-4983	497	4	up	up	ADP
ejpam-4983	497	5	for	for	ADP
ejpam-4983	497	6	an	an	DET
ejpam-4983	497	7	upper	upper	ADJ
ejpam-4983	497	8	bound	bind	VERB
ejpam-4983	497	9	for	for	ADP
ejpam-4983	497	10	the	the	DET
ejpam-4983	497	11	number	number	NOUN
ejpam-4983	497	12	of	of	ADP
ejpam-4983	497	13	n	n	NUM
ejpam-4983	497	14	×	×	NOUN
ejpam-4983	497	15	n	n	CCONJ
ejpam-4983	497	16	singular	singular	ADJ
ejpam-4983	497	17	arrowhead	arrowhead	NOUN
ejpam-4983	497	18	matrices	matrix	NOUN
ejpam-4983	497	19	over	over	ADP
ejpam-4983	497	20	r	r	NOUN
ejpam-4983	497	21	with	with	ADP
ejpam-4983	497	22	zero	zero	NUM
ejpam-4983	497	23	determinant	determinant	ADJ
ejpam-4983	497	24	has	have	AUX
ejpam-4983	497	25	been	be	AUX
ejpam-4983	497	26	given	give	VERB
ejpam-4983	497	27	as	as	ADV
ejpam-4983	497	28	well	well	ADV
ejpam-4983	497	29	as	as	ADP
ejpam-4983	497	30	a	a	DET
ejpam-4983	497	31	lower	lower	ADV
ejpam-4983	497	32	bound	bind	VERB
ejpam-4983	497	33	for	for	ADP
ejpam-4983	497	34	the	the	DET
ejpam-4983	497	35	number	number	NOUN
ejpam-4983	497	36	of	of	ADP
ejpam-4983	497	37	n	n	NUM
ejpam-4983	497	38	×	×	NOUN
ejpam-4983	497	39	n	n	CCONJ
ejpam-4983	497	40	singular	singular	ADJ
ejpam-4983	497	41	arrowhead	arrowhead	NOUN
ejpam-4983	497	42	matrices	matrix	NOUN
ejpam-4983	497	43	over	over	ADP
ejpam-4983	497	44	r	r	NOUN
ejpam-4983	497	45	with	with	ADP
ejpam-4983	497	46	a	a	DET
ejpam-4983	497	47	zero	zero	NUM
ejpam-4983	497	48	-	-	PUNCT
ejpam-4983	497	49	divisor	divisor	NOUN
ejpam-4983	497	50	determinant	determinant	ADJ
ejpam-4983	497	51	.	.	PUNCT
ejpam-4983	498	1	for	for	ADP
ejpam-4983	498	2	e	e	NOUN
ejpam-4983	498	3	=	=	SYM
ejpam-4983	498	4	2	2	NUM
ejpam-4983	498	5	,	,	PUNCT
ejpam-4983	498	6	rigorous	rigorous	ADJ
ejpam-4983	498	7	forms	form	NOUN
ejpam-4983	498	8	of	of	ADP
ejpam-4983	498	9	such	such	ADJ
ejpam-4983	498	10	bounds	bound	NOUN
ejpam-4983	498	11	have	have	AUX
ejpam-4983	498	12	been	be	AUX
ejpam-4983	498	13	presented	present	VERB
ejpam-4983	498	14	.	.	PUNCT
ejpam-4983	499	1	references	reference	NOUN
ejpam-4983	499	2	28	28	NUM
ejpam-4983	499	3	it	it	PRON
ejpam-4983	499	4	would	would	AUX
ejpam-4983	499	5	be	be	AUX
ejpam-4983	499	6	interesting	interesting	ADJ
ejpam-4983	499	7	to	to	PART
ejpam-4983	499	8	derive	derive	VERB
ejpam-4983	499	9	an	an	DET
ejpam-4983	499	10	explicit	explicit	ADJ
ejpam-4983	499	11	formula	formula	NOUN
ejpam-4983	499	12	for	for	ADP
ejpam-4983	499	13	the	the	DET
ejpam-4983	499	14	number	number	NOUN
ejpam-4983	499	15	of	of	ADP
ejpam-4983	499	16	n×	n×	PRON
ejpam-4983	499	17	n	n	CCONJ
ejpam-4983	499	18	singular	singular	ADJ
ejpam-4983	499	19	arrowhead	arrowhead	NOUN
ejpam-4983	499	20	matrices	matrix	NOUN
ejpam-4983	499	21	of	of	ADP
ejpam-4983	499	22	a	a	DET
ejpam-4983	499	23	fixed	fix	VERB
ejpam-4983	499	24	determinant	determinant	ADJ
ejpam-4983	499	25	in	in	ADP
ejpam-4983	499	26	a	a	DET
ejpam-4983	499	27	fccr	fccr	PROPN
ejpam-4983	499	28	r.	r.	PROPN
ejpam-4983	499	29	in	in	ADP
ejpam-4983	499	30	general	general	ADJ
ejpam-4983	499	31	,	,	PUNCT
ejpam-4983	499	32	the	the	DET
ejpam-4983	499	33	study	study	NOUN
ejpam-4983	499	34	of	of	ADP
ejpam-4983	499	35	n	n	NUM
ejpam-4983	499	36	×	×	NOUN
ejpam-4983	499	37	n	n	CCONJ
ejpam-4983	499	38	arrowhead	arrowhead	NOUN
ejpam-4983	499	39	matrices	matrix	NOUN
ejpam-4983	499	40	with	with	ADP
ejpam-4983	499	41	prescribed	prescribe	VERB
ejpam-4983	499	42	determinant	determinant	ADJ
ejpam-4983	499	43	over	over	ADP
ejpam-4983	499	44	more	more	ADJ
ejpam-4983	499	45	general	general	ADJ
ejpam-4983	499	46	finite	finite	PROPN
ejpam-4983	499	47	commutative	commutative	ADJ
ejpam-4983	499	48	rings	ring	NOUN
ejpam-4983	499	49	such	such	ADJ
ejpam-4983	499	50	as	as	ADP
ejpam-4983	499	51	principal	principal	ADJ
ejpam-4983	499	52	ideal	ideal	NOUN
ejpam-4983	499	53	rings	ring	NOUN
ejpam-4983	499	54	,	,	PUNCT
ejpam-4983	499	55	local	local	ADJ
ejpam-4983	499	56	rings	ring	NOUN
ejpam-4983	499	57	,	,	PUNCT
ejpam-4983	499	58	and	and	CCONJ
ejpam-4983	499	59	frobenius	frobenius	ADJ
ejpam-4983	499	60	rings	ring	NOUN
ejpam-4983	499	61	is	be	AUX
ejpam-4983	499	62	another	another	DET
ejpam-4983	499	63	interesting	interesting	ADJ
ejpam-4983	499	64	problem	problem	NOUN
ejpam-4983	499	65	.	.	PUNCT
ejpam-4983	500	1	acknowledgements	acknowledgement	VERB
ejpam-4983	500	2	the	the	DET
ejpam-4983	500	3	authors	author	NOUN
ejpam-4983	500	4	wold	wold	VERB
ejpam-4983	500	5	like	like	ADP
ejpam-4983	500	6	to	to	PART
ejpam-4983	500	7	thank	thank	VERB
ejpam-4983	500	8	the	the	DET
ejpam-4983	500	9	anonymous	anonymous	ADJ
ejpam-4983	500	10	referees	referee	NOUN
ejpam-4983	500	11	for	for	ADP
ejpam-4983	500	12	there	there	PRON
ejpam-4983	500	13	helpful	helpful	ADJ
ejpam-4983	500	14	comments	comment	NOUN
ejpam-4983	500	15	.	.	PUNCT
ejpam-4983	501	1	s.	s.	PROPN
ejpam-4983	501	2	jitman	jitman	PROPN
ejpam-4983	501	3	was	be	AUX
ejpam-4983	501	4	funded	fund	VERB
ejpam-4983	501	5	by	by	ADP
ejpam-4983	501	6	national	national	PROPN
ejpam-4983	501	7	research	research	PROPN
ejpam-4983	501	8	council	council	PROPN
ejpam-4983	501	9	of	of	ADP
ejpam-4983	501	10	thailand	thailand	PROPN
ejpam-4983	501	11	and	and	CCONJ
ejpam-4983	501	12	silpakorn	silpakorn	VERB
ejpam-4983	501	13	university	university	NOUN
ejpam-4983	501	14	under	under	ADP
ejpam-4983	501	15	research	research	NOUN
ejpam-4983	501	16	grant	grant	NOUN
ejpam-4983	501	17	n42a650381	n42a650381	PRON
ejpam-4983	501	18	.	.	PUNCT
ejpam-4983	501	19	references	reference	NOUN
ejpam-4983	502	1	[	[	X
ejpam-4983	502	2	1	1	NUM
ejpam-4983	502	3	]	]	X
ejpam-4983	502	4	r	r	NOUN
ejpam-4983	502	5	p	p	X
ejpam-4983	502	6	brent	brent	NOUN
ejpam-4983	502	7	and	and	CCONJ
ejpam-4983	502	8	b	b	PROPN
ejpam-4983	502	9	d	d	PROPN
ejpam-4983	502	10	mckay	mckay	PROPN
ejpam-4983	502	11	.	.	PUNCT
ejpam-4983	503	1	determinants	determinant	NOUN
ejpam-4983	503	2	and	and	CCONJ
ejpam-4983	503	3	ranks	rank	NOUN
ejpam-4983	503	4	of	of	ADP
ejpam-4983	503	5	random	random	ADJ
ejpam-4983	503	6	matrices	matrix	NOUN
ejpam-4983	503	7	over	over	ADP
ejpam-4983	503	8	zm	zm	PROPN
ejpam-4983	503	9	.	.	PUNCT
ejpam-4983	504	1	discrete	discrete	ADJ
ejpam-4983	504	2	mathematics	mathematic	NOUN
ejpam-4983	504	3	,	,	PUNCT
ejpam-4983	504	4	66:35–49	66:35–49	PROPN
ejpam-4983	504	5	,	,	PUNCT
ejpam-4983	504	6	1987	1987	NUM
ejpam-4983	504	7	.	.	PUNCT
ejpam-4983	505	1	[	[	X
ejpam-4983	505	2	2	2	NUM
ejpam-4983	505	3	]	]	PUNCT
ejpam-4983	505	4	w	w	NOUN
ejpam-4983	505	5	cheney	cheney	PROPN
ejpam-4983	505	6	and	and	CCONJ
ejpam-4983	505	7	d	d	NOUN
ejpam-4983	505	8	r	r	PROPN
ejpam-4983	505	9	kincaid	kincaid	PROPN
ejpam-4983	505	10	.	.	PUNCT
ejpam-4983	506	1	linear	linear	PROPN
ejpam-4983	506	2	algebra	algebra	PROPN
ejpam-4983	506	3	theory	theory	NOUN
ejpam-4983	506	4	and	and	CCONJ
ejpam-4983	506	5	applications	application	NOUN
ejpam-4983	506	6	2nd	2nd	PROPN
ejpam-4983	506	7	edition	edition	PROPN
ejpam-4983	506	8	.	.	PUNCT
ejpam-4983	507	1	jones	jones	PROPN
ejpam-4983	507	2	and	and	CCONJ
ejpam-4983	507	3	bartlett	bartlett	PROPN
ejpam-4983	507	4	publishers	publisher	NOUN
ejpam-4983	507	5	,	,	PUNCT
ejpam-4983	507	6	2010	2010	NUM
ejpam-4983	507	7	.	.	PUNCT
ejpam-4983	508	1	[	[	X
ejpam-4983	508	2	3	3	X
ejpam-4983	508	3	]	]	X
ejpam-4983	508	4	p	p	X
ejpam-4983	508	5	choosuwan	choosuwan	PROPN
ejpam-4983	508	6	,	,	PUNCT
ejpam-4983	508	7	s	s	NOUN
ejpam-4983	508	8	jitman	jitman	NOUN
ejpam-4983	508	9	,	,	PUNCT
ejpam-4983	508	10	and	and	CCONJ
ejpam-4983	508	11	p	p	X
ejpam-4983	508	12	udomkavanich	udomkavanich	NOUN
ejpam-4983	508	13	.	.	PUNCT
ejpam-4983	509	1	determinants	determinant	NOUN
ejpam-4983	509	2	of	of	ADP
ejpam-4983	509	3	matrices	matrix	NOUN
ejpam-4983	509	4	over	over	ADP
ejpam-4983	509	5	commutative	commutative	ADJ
ejpam-4983	509	6	finite	finite	ADJ
ejpam-4983	509	7	principal	principal	PROPN
ejpam-4983	509	8	ideal	ideal	PROPN
ejpam-4983	509	9	rings	ring	NOUN
ejpam-4983	509	10	.	.	PUNCT
ejpam-4983	510	1	finite	finite	PROPN
ejpam-4983	510	2	fields	field	NOUN
ejpam-4983	510	3	and	and	CCONJ
ejpam-4983	510	4	their	their	PRON
ejpam-4983	510	5	applications	application	NOUN
ejpam-4983	510	6	,	,	PUNCT
ejpam-4983	510	7	2017:126	2017:126	NOUN
ejpam-4983	510	8	–	–	PUNCT
ejpam-4983	510	9	140	140	NUM
ejpam-4983	510	10	,	,	PUNCT
ejpam-4983	510	11	48	48	NUM
ejpam-4983	510	12	.	.	PUNCT
ejpam-4983	511	1	[	[	X
ejpam-4983	511	2	4	4	X
ejpam-4983	511	3	]	]	PUNCT
ejpam-4983	511	4	j.	j.	PROPN
ejpam-4983	511	5	dubail	dubail	PROPN
ejpam-4983	511	6	,	,	PUNCT
ejpam-4983	511	7	t.	t.	PROPN
ejpam-4983	511	8	botzung	botzung	PROPN
ejpam-4983	511	9	,	,	PUNCT
ejpam-4983	511	10	j.	j.	PROPN
ejpam-4983	511	11	schachenmayer	schachenmayer	PROPN
ejpam-4983	511	12	,	,	PUNCT
ejpam-4983	511	13	g.	g.	PROPN
ejpam-4983	511	14	pupillo	pupillo	PROPN
ejpam-4983	511	15	,	,	PUNCT
ejpam-4983	511	16	and	and	CCONJ
ejpam-4983	511	17	d.	d.	PROPN
ejpam-4983	511	18	hagenmüller	hagenmüller	AUX
ejpam-4983	511	19	.	.	PUNCT
ejpam-4983	512	1	large	large	ADJ
ejpam-4983	512	2	random	random	ADJ
ejpam-4983	512	3	arrowhead	arrowhead	NOUN
ejpam-4983	512	4	matrices	matrix	NOUN
ejpam-4983	512	5	:	:	PUNCT
ejpam-4983	512	6	multifractality	multifractality	NOUN
ejpam-4983	512	7	,	,	PUNCT
ejpam-4983	512	8	semilocalization	semilocalization	NOUN
ejpam-4983	512	9	,	,	PUNCT
ejpam-4983	512	10	and	and	CCONJ
ejpam-4983	512	11	protected	protect	VERB
ejpam-4983	512	12	transport	transport	NOUN
ejpam-4983	512	13	in	in	ADP
ejpam-4983	512	14	disordered	disordered	ADJ
ejpam-4983	512	15	quantum	quantum	NOUN
ejpam-4983	512	16	spins	spin	NOUN
ejpam-4983	512	17	coupled	couple	VERB
ejpam-4983	512	18	to	to	ADP
ejpam-4983	512	19	a	a	DET
ejpam-4983	512	20	cavity	cavity	NOUN
ejpam-4983	512	21	.	.	PUNCT
ejpam-4983	513	1	phys	phy	NOUN
ejpam-4983	513	2	.	.	PUNCT
ejpam-4983	514	1	rev	rev	PROPN
ejpam-4983	514	2	.	.	PROPN
ejpam-4983	515	1	a	a	DET
ejpam-4983	515	2	,	,	PUNCT
ejpam-4983	515	3	105:023714	105:023714	NUM
ejpam-4983	515	4	,	,	PUNCT
ejpam-4983	515	5	feb	feb	NOUN
ejpam-4983	515	6	2022	2022	NUM
ejpam-4983	515	7	.	.	PUNCT
ejpam-4983	516	1	[	[	X
ejpam-4983	516	2	5	5	NUM
ejpam-4983	516	3	]	]	SYM
ejpam-4983	516	4	b	b	NOUN
ejpam-4983	516	5	gilberto	gilberto	PROPN
ejpam-4983	516	6	and	and	CCONJ
ejpam-4983	516	7	f	f	PROPN
ejpam-4983	516	8	flaminio	flaminio	PROPN
ejpam-4983	516	9	.	.	PUNCT
ejpam-4983	517	1	finite	finite	PROPN
ejpam-4983	517	2	commutative	commutative	ADJ
ejpam-4983	517	3	rings	ring	NOUN
ejpam-4983	517	4	and	and	CCONJ
ejpam-4983	517	5	their	their	PRON
ejpam-4983	517	6	applications	application	NOUN
ejpam-4983	517	7	.	.	PUNCT
ejpam-4983	518	1	springer	springer	NOUN
ejpam-4983	518	2	,	,	PUNCT
ejpam-4983	518	3	2002	2002	NUM
ejpam-4983	518	4	.	.	PUNCT
ejpam-4983	519	1	[	[	X
ejpam-4983	519	2	6	6	NUM
ejpam-4983	519	3	]	]	PUNCT
ejpam-4983	519	4	x	x	SYM
ejpam-4983	519	5	hou	hou	PROPN
ejpam-4983	519	6	.	.	PUNCT
ejpam-4983	519	7	finite	finite	PROPN
ejpam-4983	519	8	commutative	commutative	ADJ
ejpam-4983	519	9	chain	chain	NOUN
ejpam-4983	519	10	rings	ring	NOUN
ejpam-4983	519	11	.	.	PUNCT
ejpam-4983	520	1	finite	finite	PROPN
ejpam-4983	520	2	fields	field	NOUN
ejpam-4983	520	3	and	and	CCONJ
ejpam-4983	520	4	their	their	PRON
ejpam-4983	520	5	applications	application	NOUN
ejpam-4983	520	6	,	,	PUNCT
ejpam-4983	520	7	7:382	7:382	NUM
ejpam-4983	520	8	–	–	PUNCT
ejpam-4983	520	9	396	396	NUM
ejpam-4983	520	10	,	,	PUNCT
ejpam-4983	520	11	2001	2001	NUM
ejpam-4983	520	12	.	.	PUNCT
ejpam-4983	521	1	[	[	X
ejpam-4983	521	2	7	7	NUM
ejpam-4983	521	3	]	]	SYM
ejpam-4983	521	4	x	x	SYM
ejpam-4983	521	5	hou	hou	PROPN
ejpam-4983	521	6	,	,	PUNCT
ejpam-4983	521	7	k	k	PROPN
ejpam-4983	521	8	h	h	PROPN
ejpam-4983	521	9	leung	leung	PROPN
ejpam-4983	521	10	,	,	PUNCT
ejpam-4983	521	11	and	and	CCONJ
ejpam-4983	521	12	s	s	PROPN
ejpam-4983	521	13	l	l	NOUN
ejpam-4983	521	14	mab	mab	NOUN
ejpam-4983	521	15	.	.	PUNCT
ejpam-4983	522	1	on	on	ADP
ejpam-4983	522	2	the	the	DET
ejpam-4983	522	3	groups	group	NOUN
ejpam-4983	522	4	of	of	ADP
ejpam-4983	522	5	units	unit	NOUN
ejpam-4983	522	6	of	of	ADP
ejpam-4983	522	7	finite	finite	PROPN
ejpam-4983	522	8	commutative	commutative	ADJ
ejpam-4983	522	9	chain	chain	NOUN
ejpam-4983	522	10	rings	ring	NOUN
ejpam-4983	522	11	.	.	PUNCT
ejpam-4983	523	1	finite	finite	PROPN
ejpam-4983	523	2	fields	field	NOUN
ejpam-4983	523	3	and	and	CCONJ
ejpam-4983	523	4	their	their	PRON
ejpam-4983	523	5	applications	application	NOUN
ejpam-4983	523	6	,	,	PUNCT
ejpam-4983	523	7	9:20–38	9:20–38	NUM
ejpam-4983	523	8	,	,	PUNCT
ejpam-4983	523	9	2003	2003	NUM
ejpam-4983	523	10	.	.	PUNCT
ejpam-4983	524	1	[	[	X
ejpam-4983	524	2	8	8	NUM
ejpam-4983	524	3	]	]	PUNCT
ejpam-4983	524	4	s	s	VERB
ejpam-4983	524	5	jitman	jitman	NOUN
ejpam-4983	524	6	.	.	PUNCT
ejpam-4983	525	1	determinants	determinant	NOUN
ejpam-4983	525	2	of	of	ADP
ejpam-4983	525	3	some	some	DET
ejpam-4983	525	4	special	special	ADJ
ejpam-4983	525	5	matrices	matrix	NOUN
ejpam-4983	525	6	over	over	ADP
ejpam-4983	525	7	commutative	commutative	ADJ
ejpam-4983	525	8	finite	finite	ADJ
ejpam-4983	525	9	chain	chain	NOUN
ejpam-4983	525	10	rings	ring	NOUN
ejpam-4983	525	11	.	.	PUNCT
ejpam-4983	526	1	special	special	ADJ
ejpam-4983	526	2	matrices	matrix	NOUN
ejpam-4983	526	3	,	,	PUNCT
ejpam-4983	526	4	8:242–256	8:242–256	NUM
ejpam-4983	526	5	,	,	PUNCT
ejpam-4983	526	6	2020	2020	NUM
ejpam-4983	526	7	.	.	PUNCT
ejpam-4983	527	1	[	[	X
ejpam-4983	527	2	9	9	NUM
ejpam-4983	527	3	]	]	X
ejpam-4983	527	4	d	d	NOUN
ejpam-4983	527	5	a	a	DET
ejpam-4983	527	6	klain	klain	NOUN
ejpam-4983	527	7	.	.	PUNCT
ejpam-4983	528	1	an	an	DET
ejpam-4983	528	2	intuitive	intuitive	ADJ
ejpam-4983	528	3	derivation	derivation	NOUN
ejpam-4983	528	4	of	of	ADP
ejpam-4983	528	5	heron	heron	NOUN
ejpam-4983	528	6	’s	’s	PART
ejpam-4983	528	7	formula	formula	NOUN
ejpam-4983	528	8	.	.	PUNCT
ejpam-4983	529	1	the	the	DET
ejpam-4983	529	2	american	american	PROPN
ejpam-4983	529	3	mathematical	mathematical	PROPN
ejpam-4983	529	4	monthly	monthly	ADJ
ejpam-4983	529	5	,	,	PUNCT
ejpam-4983	529	6	111:709–712	111:709–712	NUM
ejpam-4983	529	7	,	,	PUNCT
ejpam-4983	529	8	2004	2004	NUM
ejpam-4983	529	9	.	.	PUNCT
ejpam-4983	530	1	[	[	X
ejpam-4983	530	2	10	10	NUM
ejpam-4983	530	3	]	]	X
ejpam-4983	530	4	j	j	PROPN
ejpam-4983	530	5	m	m	VERB
ejpam-4983	530	6	lockhart	lockhart	PROPN
ejpam-4983	530	7	and	and	CCONJ
ejpam-4983	530	8	w	w	PROPN
ejpam-4983	530	9	p	p	NOUN
ejpam-4983	530	10	wardlaw	wardlaw	NOUN
ejpam-4983	530	11	.	.	PUNCT
ejpam-4983	531	1	determinants	determinant	NOUN
ejpam-4983	531	2	of	of	ADP
ejpam-4983	531	3	matrices	matrix	NOUN
ejpam-4983	531	4	over	over	ADP
ejpam-4983	531	5	the	the	DET
ejpam-4983	531	6	integers	integer	NOUN
ejpam-4983	531	7	modulo	modulo	PROPN
ejpam-4983	531	8	m.	m.	NOUN
ejpam-4983	531	9	mathematics	mathematics	PROPN
ejpam-4983	531	10	magazine	magazine	NOUN
ejpam-4983	531	11	,	,	PUNCT
ejpam-4983	531	12	80:207–214	80:207–214	PROPN
ejpam-4983	531	13	,	,	PUNCT
ejpam-4983	531	14	2007	2007	NUM
ejpam-4983	531	15	.	.	PUNCT
ejpam-4983	532	1	references	reference	NOUN
ejpam-4983	532	2	29	29	NUM
ejpam-4983	532	3	[	[	X
ejpam-4983	532	4	11	11	NUM
ejpam-4983	532	5	]	]	PUNCT
ejpam-4983	532	6	l	l	NOUN
ejpam-4983	532	7	lovász	lovász	NOUN
ejpam-4983	532	8	.	.	PUNCT
ejpam-4983	533	1	singular	singular	PROPN
ejpam-4983	533	2	spaces	space	NOUN
ejpam-4983	533	3	of	of	ADP
ejpam-4983	533	4	matrices	matrix	NOUN
ejpam-4983	533	5	and	and	CCONJ
ejpam-4983	533	6	their	their	PRON
ejpam-4983	533	7	application	application	NOUN
ejpam-4983	533	8	in	in	ADP
ejpam-4983	533	9	combinatorics	combinatoric	NOUN
ejpam-4983	533	10	.	.	PUNCT
ejpam-4983	534	1	bulletin	bulletin	PROPN
ejpam-4983	534	2	brazilian	brazilian	PROPN
ejpam-4983	534	3	mathematical	mathematical	ADJ
ejpam-4983	534	4	society	society	NOUN
ejpam-4983	534	5	,	,	PUNCT
ejpam-4983	534	6	20:87–99	20:87–99	NUM
ejpam-4983	534	7	,	,	PUNCT
ejpam-4983	534	8	1989	1989	NUM
ejpam-4983	534	9	.	.	PUNCT
ejpam-4983	535	1	[	[	X
ejpam-4983	535	2	12	12	NUM
ejpam-4983	535	3	]	]	X
ejpam-4983	535	4	j	j	PROPN
ejpam-4983	535	5	a	a	DET
ejpam-4983	535	6	marrero	marrero	PROPN
ejpam-4983	535	7	,	,	PUNCT
ejpam-4983	535	8	j	j	PROPN
ejpam-4983	535	9	n	n	PRON
ejpam-4983	535	10	valdés	valdés	PROPN
ejpam-4983	535	11	,	,	PUNCT
ejpam-4983	535	12	and	and	CCONJ
ejpam-4983	535	13	m	m	PROPN
ejpam-4983	535	14	t	t	PROPN
ejpam-4983	535	15	villar	villar	PROPN
ejpam-4983	535	16	.	.	PUNCT
ejpam-4983	536	1	associating	associate	VERB
ejpam-4983	536	2	hub	hub	NOUN
ejpam-4983	536	3	-	-	PUNCT
ejpam-4983	536	4	directed	direct	VERB
ejpam-4983	536	5	multigraphs	multigraph	NOUN
ejpam-4983	536	6	to	to	ADP
ejpam-4983	536	7	arrowhead	arrowhead	NOUN
ejpam-4983	536	8	matrices	matrix	NOUN
ejpam-4983	536	9	.	.	PUNCT
ejpam-4983	537	1	mathematical	mathematical	ADJ
ejpam-4983	537	2	methods	method	NOUN
ejpam-4983	537	3	in	in	ADP
ejpam-4983	537	4	the	the	DET
ejpam-4983	537	5	applied	apply	VERB
ejpam-4983	537	6	sciences	science	NOUN
ejpam-4983	537	7	,	,	PUNCT
ejpam-4983	537	8	41(6):2360–2369	41(6):2360–2369	NUM
ejpam-4983	537	9	,	,	PUNCT
ejpam-4983	537	10	2018	2018	NUM
ejpam-4983	537	11	.	.	PUNCT
ejpam-4983	538	1	[	[	X
ejpam-4983	538	2	13	13	NUM
ejpam-4983	538	3	]	]	PUNCT
ejpam-4983	538	4	a	a	DET
ejpam-4983	538	5	mukhopadhyay	mukhopadhyay	NOUN
ejpam-4983	538	6	.	.	PUNCT
ejpam-4983	539	1	on	on	ADP
ejpam-4983	539	2	the	the	DET
ejpam-4983	539	3	probability	probability	NOUN
ejpam-4983	539	4	that	that	SCONJ
ejpam-4983	539	5	the	the	DET
ejpam-4983	539	6	determinant	determinant	NOUN
ejpam-4983	539	7	of	of	ADP
ejpam-4983	539	8	an	an	DET
ejpam-4983	539	9	n×	n×	NOUN
ejpam-4983	539	10	n	n	NOUN
ejpam-4983	539	11	matrix	matrix	NOUN
ejpam-4983	539	12	over	over	ADP
ejpam-4983	539	13	a	a	DET
ejpam-4983	539	14	finite	finite	ADJ
ejpam-4983	539	15	field	field	NOUN
ejpam-4983	539	16	vanishes	vanish	VERB
ejpam-4983	539	17	.	.	PUNCT
ejpam-4983	540	1	discrete	discrete	ADJ
ejpam-4983	540	2	mathematics	mathematic	NOUN
ejpam-4983	540	3	,	,	PUNCT
ejpam-4983	540	4	51:311–315	51:311–315	PROPN
ejpam-4983	540	5	,	,	PUNCT
ejpam-4983	540	6	1984	1984	NUM
ejpam-4983	540	7	.	.	PUNCT
ejpam-4983	541	1	[	[	X
ejpam-4983	541	2	14	14	NUM
ejpam-4983	541	3	]	]	X
ejpam-4983	541	4	h	h	PROPN
ejpam-4983	541	5	saberi	saberi	PROPN
ejpam-4983	541	6	najafi	najafi	PROPN
ejpam-4983	541	7	,	,	PUNCT
ejpam-4983	541	8	s	s	VERB
ejpam-4983	541	9	a	a	DET
ejpam-4983	541	10	edalatpanah	edalatpanah	PROPN
ejpam-4983	541	11	,	,	PUNCT
ejpam-4983	541	12	and	and	CCONJ
ejpam-4983	541	13	g	g	ADP
ejpam-4983	541	14	a	a	DET
ejpam-4983	541	15	gravvanis	gravvanis	NOUN
ejpam-4983	541	16	.	.	PUNCT
ejpam-4983	542	1	an	an	DET
ejpam-4983	542	2	efficient	efficient	ADJ
ejpam-4983	542	3	method	method	NOUN
ejpam-4983	542	4	for	for	ADP
ejpam-4983	542	5	computing	compute	VERB
ejpam-4983	542	6	the	the	DET
ejpam-4983	542	7	inverse	inverse	NOUN
ejpam-4983	542	8	of	of	ADP
ejpam-4983	542	9	arrowhead	arrowhead	NOUN
ejpam-4983	542	10	matrices	matrix	NOUN
ejpam-4983	542	11	.	.	PUNCT
ejpam-4983	543	1	applied	apply	VERB
ejpam-4983	543	2	mathematics	mathematics	NOUN
ejpam-4983	543	3	letters	letter	NOUN
ejpam-4983	543	4	,	,	PUNCT
ejpam-4983	543	5	33:1–5	33:1–5	NUM
ejpam-4983	543	6	,	,	PUNCT
ejpam-4983	543	7	2014	2014	NUM
ejpam-4983	543	8	.	.	PUNCT
ejpam-4983	544	1	[	[	X
ejpam-4983	544	2	15	15	NUM
ejpam-4983	544	3	]	]	X
ejpam-4983	544	4	l	l	NOUN
ejpam-4983	544	5	shen	shen	NOUN
ejpam-4983	544	6	and	and	CCONJ
ejpam-4983	544	7	b	b	PROPN
ejpam-4983	544	8	w	w	PROPN
ejpam-4983	544	9	suter	suter	NOUN
ejpam-4983	544	10	.	.	PUNCT
ejpam-4983	545	1	bounds	bound	VERB
ejpam-4983	545	2	for	for	ADP
ejpam-4983	545	3	eigenvalues	eigenvalue	NOUN
ejpam-4983	545	4	of	of	ADP
ejpam-4983	545	5	arrowhead	arrowhead	NOUN
ejpam-4983	545	6	matrices	matrix	NOUN
ejpam-4983	545	7	and	and	CCONJ
ejpam-4983	545	8	their	their	PRON
ejpam-4983	545	9	applications	application	NOUN
ejpam-4983	545	10	to	to	ADP
ejpam-4983	545	11	hub	hub	NOUN
ejpam-4983	545	12	matrices	matrix	NOUN
ejpam-4983	545	13	and	and	CCONJ
ejpam-4983	545	14	wireless	wireless	ADJ
ejpam-4983	545	15	communications	communication	NOUN
ejpam-4983	545	16	.	.	PUNCT
ejpam-4983	546	1	eurasip	eurasip	PROPN
ejpam-4983	546	2	journal	journal	PROPN
ejpam-4983	546	3	on	on	ADP
ejpam-4983	546	4	advances	advance	NOUN
ejpam-4983	546	5	in	in	ADP
ejpam-4983	546	6	signal	signal	ADJ
ejpam-4983	546	7	processing	processing	NOUN
ejpam-4983	546	8	,	,	PUNCT
ejpam-4983	546	9	2009(1):379402	2009(1):379402	NOUN
ejpam-4983	546	10	,	,	PUNCT
ejpam-4983	546	11	2009	2009	NUM
ejpam-4983	546	12	.	.	PUNCT
ejpam-4983	547	1	[	[	X
ejpam-4983	547	2	16	16	NUM
ejpam-4983	547	3	]	]	PUNCT
ejpam-4983	547	4	n	n	PROPN
ejpam-4983	547	5	j	j	PROPN
ejpam-4983	547	6	stor	stor	PROPN
ejpam-4983	547	7	,	,	PUNCT
ejpam-4983	547	8	i	i	PRON
ejpam-4983	547	9	slapničar	slapničar	VERB
ejpam-4983	547	10	,	,	PUNCT
ejpam-4983	547	11	and	and	CCONJ
ejpam-4983	547	12	j	j	PROPN
ejpam-4983	547	13	l	l	PROPN
ejpam-4983	547	14	barlow	barlow	PROPN
ejpam-4983	547	15	.	.	PUNCT
ejpam-4983	548	1	accurate	accurate	ADJ
ejpam-4983	548	2	eigenvalue	eigenvalue	PROPN
ejpam-4983	548	3	decomposition	decomposition	NOUN
ejpam-4983	548	4	of	of	ADP
ejpam-4983	548	5	arrowhead	arrowhead	NOUN
ejpam-4983	548	6	matrices	matrix	NOUN
ejpam-4983	548	7	and	and	CCONJ
ejpam-4983	548	8	applications	application	NOUN
ejpam-4983	548	9	.	.	PUNCT
ejpam-4983	549	1	linear	linear	ADJ
ejpam-4983	549	2	algebra	algebra	NOUN
ejpam-4983	549	3	and	and	CCONJ
ejpam-4983	549	4	its	its	PRON
ejpam-4983	549	5	applications	application	NOUN
ejpam-4983	549	6	,	,	PUNCT
ejpam-4983	549	7	464:62–89	464:62–89	NUM
ejpam-4983	549	8	,	,	PUNCT
ejpam-4983	549	9	2015	2015	NUM
ejpam-4983	549	10	.	.	PUNCT