id	sid	tid	token	lemma	pos
ejpam-4712	1	1	european	european	PROPN
ejpam-4712	1	2	journal	journal	PROPN
ejpam-4712	1	3	of	of	ADP
ejpam-4712	1	4	pure	pure	ADJ
ejpam-4712	1	5	and	and	CCONJ
ejpam-4712	1	6	applied	apply	VERB
ejpam-4712	1	7	mathematics	mathematic	NOUN
ejpam-4712	1	8	vol	vol	NOUN
ejpam-4712	1	9	.	.	PUNCT
ejpam-4712	2	1	16	16	NUM
ejpam-4712	2	2	,	,	PUNCT
ejpam-4712	2	3	no	no	INTJ
ejpam-4712	2	4	.	.	NOUN
ejpam-4712	2	5	2	2	NUM
ejpam-4712	2	6	,	,	PUNCT
ejpam-4712	2	7	2023	2023	NUM
ejpam-4712	2	8	,	,	PUNCT
ejpam-4712	2	9	784	784	NUM
ejpam-4712	2	10	-	-	SYM
ejpam-4712	2	11	790	790	NUM
ejpam-4712	2	12	issn	issn	PROPN
ejpam-4712	2	13	1307	1307	NUM
ejpam-4712	2	14	-	-	SYM
ejpam-4712	2	15	5543	5543	NUM
ejpam-4712	2	16	–	–	PUNCT
ejpam-4712	2	17	ejpam.com	ejpam.com	X
ejpam-4712	2	18	published	publish	VERB
ejpam-4712	2	19	by	by	ADP
ejpam-4712	2	20	new	new	PROPN
ejpam-4712	2	21	york	york	PROPN
ejpam-4712	2	22	business	business	PROPN
ejpam-4712	2	23	global	global	PROPN
ejpam-4712	3	1	the	the	DET
ejpam-4712	3	2	discrete	discrete	ADJ
ejpam-4712	3	3	lyapunov	lyapunov	ADJ
ejpam-4712	3	4	equation	equation	NOUN
ejpam-4712	3	5	of	of	ADP
ejpam-4712	3	6	the	the	DET
ejpam-4712	3	7	orthogonal	orthogonal	ADJ
ejpam-4712	3	8	matrix	matrix	NOUN
ejpam-4712	3	9	in	in	ADP
ejpam-4712	3	10	semiring	semire	VERB
ejpam-4712	3	11	gregoria	gregoria	PROPN
ejpam-4712	3	12	ariyanti1,∗	ariyanti1,∗	PROPN
ejpam-4712	3	13	,	,	PUNCT
ejpam-4712	3	14	ana	ana	PROPN
ejpam-4712	3	15	easti	easti	PROPN
ejpam-4712	3	16	rahayu	rahayu	PROPN
ejpam-4712	3	17	maya	maya	PROPN
ejpam-4712	3	18	sari1	sari1	PROPN
ejpam-4712	3	19	1	1	NUM
ejpam-4712	3	20	mathematics	mathematics	PROPN
ejpam-4712	3	21	education	education	NOUN
ejpam-4712	3	22	study	study	NOUN
ejpam-4712	3	23	program	program	NOUN
ejpam-4712	3	24	,	,	PUNCT
ejpam-4712	3	25	widya	widya	PROPN
ejpam-4712	3	26	mandala	mandala	PROPN
ejpam-4712	3	27	surabaya	surabaya	PROPN
ejpam-4712	3	28	catholic	catholic	PROPN
ejpam-4712	3	29	university	university	PROPN
ejpam-4712	3	30	,	,	PUNCT
ejpam-4712	3	31	indonesia	indonesia	PROPN
ejpam-4712	3	32	abstract	abstract	NOUN
ejpam-4712	3	33	.	.	PUNCT
ejpam-4712	4	1	semiring	semiring	NOUN
ejpam-4712	4	2	is	be	AUX
ejpam-4712	4	3	an	an	DET
ejpam-4712	4	4	algebraic	algebraic	ADJ
ejpam-4712	4	5	structure	structure	NOUN
ejpam-4712	4	6	of	of	ADP
ejpam-4712	4	7	(	(	PUNCT
ejpam-4712	4	8	s,+,×	s,+,×	PROPN
ejpam-4712	4	9	)	)	PUNCT
ejpam-4712	4	10	.	.	PUNCT
ejpam-4712	5	1	similar	similar	ADJ
ejpam-4712	5	2	to	to	ADP
ejpam-4712	5	3	a	a	DET
ejpam-4712	5	4	ring	ring	NOUN
ejpam-4712	5	5	,	,	PUNCT
ejpam-4712	5	6	but	but	CCONJ
ejpam-4712	5	7	without	without	ADP
ejpam-4712	5	8	the	the	DET
ejpam-4712	5	9	condition	condition	NOUN
ejpam-4712	5	10	that	that	SCONJ
ejpam-4712	5	11	each	each	DET
ejpam-4712	5	12	element	element	NOUN
ejpam-4712	5	13	must	must	AUX
ejpam-4712	5	14	have	have	VERB
ejpam-4712	5	15	an	an	DET
ejpam-4712	5	16	inverse	inverse	NOUN
ejpam-4712	5	17	to	to	ADP
ejpam-4712	5	18	the	the	DET
ejpam-4712	5	19	adding	add	VERB
ejpam-4712	5	20	operation	operation	NOUN
ejpam-4712	5	21	.	.	PUNCT
ejpam-4712	6	1	the	the	DET
ejpam-4712	6	2	forms	form	NOUN
ejpam-4712	6	3	(	(	PUNCT
ejpam-4712	6	4	s,+	s,+	NUM
ejpam-4712	6	5	)	)	PUNCT
ejpam-4712	6	6	and	and	CCONJ
ejpam-4712	6	7	(	(	PUNCT
ejpam-4712	6	8	s,×	s,×	PROPN
ejpam-4712	6	9	)	)	PUNCT
ejpam-4712	6	10	are	be	AUX
ejpam-4712	6	11	semigroups	semigroup	NOUN
ejpam-4712	6	12	that	that	PRON
ejpam-4712	6	13	satisfy	satisfy	VERB
ejpam-4712	6	14	the	the	DET
ejpam-4712	6	15	distributive	distributive	ADJ
ejpam-4712	6	16	law	law	NOUN
ejpam-4712	6	17	of	of	ADP
ejpam-4712	6	18	multiplication	multiplication	NOUN
ejpam-4712	6	19	and	and	CCONJ
ejpam-4712	6	20	addition	addition	NOUN
ejpam-4712	6	21	.	.	PUNCT
ejpam-4712	7	1	in	in	ADP
ejpam-4712	7	2	matrix	matrix	NOUN
ejpam-4712	7	3	theory	theory	NOUN
ejpam-4712	7	4	,	,	PUNCT
ejpam-4712	7	5	there	there	PRON
ejpam-4712	7	6	is	be	VERB
ejpam-4712	7	7	a	a	DET
ejpam-4712	7	8	term	term	NOUN
ejpam-4712	7	9	known	know	VERB
ejpam-4712	7	10	as	as	ADP
ejpam-4712	7	11	the	the	DET
ejpam-4712	7	12	kronecker	kronecker	NOUN
ejpam-4712	7	13	product	product	NOUN
ejpam-4712	7	14	.	.	PUNCT
ejpam-4712	8	1	this	this	DET
ejpam-4712	8	2	operation	operation	NOUN
ejpam-4712	8	3	transforms	transform	VERB
ejpam-4712	8	4	two	two	NUM
ejpam-4712	8	5	matrices	matrix	NOUN
ejpam-4712	8	6	into	into	ADP
ejpam-4712	8	7	a	a	DET
ejpam-4712	8	8	larger	large	ADJ
ejpam-4712	8	9	matrix	matrix	NOUN
ejpam-4712	8	10	containing	contain	VERB
ejpam-4712	8	11	all	all	DET
ejpam-4712	8	12	possible	possible	ADJ
ejpam-4712	8	13	products	product	NOUN
ejpam-4712	8	14	of	of	ADP
ejpam-4712	8	15	the	the	DET
ejpam-4712	8	16	entries	entry	NOUN
ejpam-4712	8	17	in	in	ADP
ejpam-4712	8	18	the	the	DET
ejpam-4712	8	19	two	two	NUM
ejpam-4712	8	20	matrices	matrix	NOUN
ejpam-4712	8	21	.	.	PUNCT
ejpam-4712	9	1	this	this	DET
ejpam-4712	9	2	kronecker	kronecker	NOUN
ejpam-4712	9	3	product	product	NOUN
ejpam-4712	9	4	has	have	VERB
ejpam-4712	9	5	several	several	ADJ
ejpam-4712	9	6	properties	property	NOUN
ejpam-4712	9	7	often	often	ADV
ejpam-4712	9	8	used	use	VERB
ejpam-4712	9	9	to	to	PART
ejpam-4712	9	10	solve	solve	VERB
ejpam-4712	9	11	the	the	DET
ejpam-4712	9	12	complex	complex	ADJ
ejpam-4712	9	13	problems	problem	NOUN
ejpam-4712	9	14	of	of	ADP
ejpam-4712	9	15	linear	linear	PROPN
ejpam-4712	9	16	algebra	algebra	NOUN
ejpam-4712	9	17	and	and	CCONJ
ejpam-4712	9	18	its	its	PRON
ejpam-4712	9	19	applications	application	NOUN
ejpam-4712	9	20	.	.	PUNCT
ejpam-4712	10	1	the	the	DET
ejpam-4712	10	2	kronecker	kronecker	NOUN
ejpam-4712	10	3	product	product	NOUN
ejpam-4712	10	4	is	be	AUX
ejpam-4712	10	5	related	relate	VERB
ejpam-4712	10	6	to	to	ADP
ejpam-4712	10	7	the	the	DET
ejpam-4712	10	8	lyapunov	lyapunov	ADJ
ejpam-4712	10	9	equation	equation	NOUN
ejpam-4712	10	10	of	of	ADP
ejpam-4712	10	11	a	a	DET
ejpam-4712	10	12	linear	linear	ADJ
ejpam-4712	10	13	system	system	NOUN
ejpam-4712	10	14	.	.	PUNCT
ejpam-4712	11	1	based	base	VERB
ejpam-4712	11	2	on	on	ADP
ejpam-4712	11	3	previous	previous	ADJ
ejpam-4712	11	4	research	research	NOUN
ejpam-4712	11	5	in	in	ADP
ejpam-4712	11	6	the	the	DET
ejpam-4712	11	7	lyapunov	lyapunov	ADJ
ejpam-4712	11	8	equation	equation	NOUN
ejpam-4712	11	9	in	in	ADP
ejpam-4712	11	10	conventional	conventional	ADJ
ejpam-4712	11	11	linear	linear	PROPN
ejpam-4712	11	12	algebra	algebra	NOUN
ejpam-4712	11	13	,	,	PUNCT
ejpam-4712	11	14	this	this	DET
ejpam-4712	11	15	paper	paper	NOUN
ejpam-4712	11	16	will	will	AUX
ejpam-4712	11	17	describe	describe	VERB
ejpam-4712	11	18	the	the	DET
ejpam-4712	11	19	characteristics	characteristic	NOUN
ejpam-4712	11	20	of	of	ADP
ejpam-4712	11	21	the	the	DET
ejpam-4712	11	22	lyapunov	lyapunov	ADJ
ejpam-4712	11	23	equation	equation	NOUN
ejpam-4712	11	24	in	in	ADP
ejpam-4712	11	25	a	a	DET
ejpam-4712	11	26	semiring	semire	VERB
ejpam-4712	11	27	linear	linear	NOUN
ejpam-4712	11	28	system	system	NOUN
ejpam-4712	11	29	in	in	ADP
ejpam-4712	11	30	terms	term	NOUN
ejpam-4712	11	31	of	of	ADP
ejpam-4712	11	32	the	the	DET
ejpam-4712	11	33	kronecker	kronecker	NOUN
ejpam-4712	11	34	product	product	NOUN
ejpam-4712	11	35	.	.	PUNCT
ejpam-4712	12	1	2020	2020	NUM
ejpam-4712	12	2	mathematics	mathematic	NOUN
ejpam-4712	12	3	subject	subject	NOUN
ejpam-4712	12	4	classifications	classification	NOUN
ejpam-4712	12	5	:	:	PUNCT
ejpam-4712	12	6	00a64	00a64	NUM
ejpam-4712	12	7	,	,	PUNCT
ejpam-4712	12	8	34d08	34d08	NUM
ejpam-4712	12	9	,	,	PUNCT
ejpam-4712	12	10	37m25	37m25	NUM
ejpam-4712	12	11	,	,	PUNCT
ejpam-4712	12	12	43a46	43a46	NUM
ejpam-4712	12	13	key	key	ADJ
ejpam-4712	12	14	words	word	NOUN
ejpam-4712	12	15	and	and	CCONJ
ejpam-4712	12	16	phrases	phrase	NOUN
ejpam-4712	12	17	:	:	PUNCT
ejpam-4712	12	18	linear	linear	ADJ
ejpam-4712	12	19	system	system	NOUN
ejpam-4712	12	20	,	,	PUNCT
ejpam-4712	12	21	the	the	DET
ejpam-4712	12	22	lyapunov	lyapunov	ADJ
ejpam-4712	12	23	equation	equation	NOUN
ejpam-4712	12	24	,	,	PUNCT
ejpam-4712	12	25	semiring	semiring	NOUN
ejpam-4712	12	26	,	,	PUNCT
ejpam-4712	12	27	the	the	DET
ejpam-4712	12	28	kronecker	kronecker	NOUN
ejpam-4712	12	29	product	product	NOUN
ejpam-4712	12	30	1	1	NUM
ejpam-4712	12	31	.	.	PUNCT
ejpam-4712	12	32	introduction	introduction	NOUN
ejpam-4712	12	33	a	a	DET
ejpam-4712	12	34	non	non	ADJ
ejpam-4712	12	35	-	-	ADJ
ejpam-4712	12	36	empty	empty	ADJ
ejpam-4712	12	37	set	set	VERB
ejpam-4712	12	38	g	g	NOUN
ejpam-4712	12	39	with	with	ADP
ejpam-4712	12	40	a	a	DET
ejpam-4712	12	41	binary	binary	ADJ
ejpam-4712	12	42	operation	operation	NOUN
ejpam-4712	12	43	∗	∗	NOUN
ejpam-4712	12	44	is	be	AUX
ejpam-4712	12	45	known	know	VERB
ejpam-4712	12	46	as	as	ADP
ejpam-4712	12	47	group	group	NOUN
ejpam-4712	12	48	if	if	SCONJ
ejpam-4712	12	49	it	it	PRON
ejpam-4712	12	50	has	have	VERB
ejpam-4712	12	51	the	the	DET
ejpam-4712	12	52	following	follow	VERB
ejpam-4712	12	53	properties	property	NOUN
ejpam-4712	12	54	:	:	PUNCT
ejpam-4712	13	1	associative	associative	ADJ
ejpam-4712	13	2	,	,	PUNCT
ejpam-4712	13	3	a	a	DET
ejpam-4712	13	4	zero	zero	NUM
ejpam-4712	13	5	element	element	NOUN
ejpam-4712	13	6	of	of	ADP
ejpam-4712	13	7	binary	binary	ADJ
ejpam-4712	13	8	operations	operation	NOUN
ejpam-4712	13	9	∗	∗	NOUN
ejpam-4712	13	10	,	,	PUNCT
ejpam-4712	13	11	and	and	CCONJ
ejpam-4712	13	12	every	every	DET
ejpam-4712	13	13	element	element	NOUN
ejpam-4712	13	14	,	,	PUNCT
ejpam-4712	13	15	not	not	PART
ejpam-4712	13	16	a	a	DET
ejpam-4712	13	17	zero	zero	NUM
ejpam-4712	13	18	element	element	NOUN
ejpam-4712	13	19	,	,	PUNCT
ejpam-4712	13	20	has	have	VERB
ejpam-4712	13	21	an	an	DET
ejpam-4712	13	22	inverse	inverse	NOUN
ejpam-4712	13	23	.	.	PUNCT
ejpam-4712	14	1	meanwhile	meanwhile	ADV
ejpam-4712	14	2	,	,	PUNCT
ejpam-4712	14	3	a	a	DET
ejpam-4712	14	4	non	non	ADJ
ejpam-4712	14	5	-	-	ADJ
ejpam-4712	14	6	empty	empty	ADJ
ejpam-4712	14	7	set	set	ADJ
ejpam-4712	14	8	r	r	NOUN
ejpam-4712	14	9	with	with	ADP
ejpam-4712	14	10	two	two	NUM
ejpam-4712	14	11	binary	binary	ADJ
ejpam-4712	14	12	operations	operation	NOUN
ejpam-4712	14	13	,	,	PUNCT
ejpam-4712	14	14	particularly	particularly	ADV
ejpam-4712	14	15	∗	∗	NOUN
ejpam-4712	14	16	and	and	CCONJ
ejpam-4712	14	17	◦	◦	NOUN
ejpam-4712	14	18	is	be	AUX
ejpam-4712	14	19	called	call	VERB
ejpam-4712	14	20	ring	ring	NOUN
ejpam-4712	14	21	.	.	PUNCT
ejpam-4712	15	1	if	if	SCONJ
ejpam-4712	15	2	it	it	PRON
ejpam-4712	15	3	has	have	VERB
ejpam-4712	15	4	the	the	DET
ejpam-4712	15	5	following	follow	VERB
ejpam-4712	15	6	properties	property	NOUN
ejpam-4712	15	7	:	:	PUNCT
ejpam-4712	15	8	(	(	PUNCT
ejpam-4712	15	9	r	r	NOUN
ejpam-4712	15	10	,	,	PUNCT
ejpam-4712	15	11	∗	∗	NOUN
ejpam-4712	15	12	)	)	PUNCT
ejpam-4712	15	13	is	be	AUX
ejpam-4712	15	14	a	a	DET
ejpam-4712	15	15	commutative	commutative	ADJ
ejpam-4712	15	16	group	group	NOUN
ejpam-4712	15	17	,	,	PUNCT
ejpam-4712	15	18	(	(	PUNCT
ejpam-4712	15	19	r	r	NOUN
ejpam-4712	15	20	,	,	PUNCT
ejpam-4712	15	21	◦	◦	NOUN
ejpam-4712	15	22	)	)	PUNCT
ejpam-4712	15	23	is	be	AUX
ejpam-4712	15	24	closed	close	VERB
ejpam-4712	15	25	,	,	PUNCT
ejpam-4712	15	26	(	(	PUNCT
ejpam-4712	15	27	r	r	NOUN
ejpam-4712	15	28	,	,	PUNCT
ejpam-4712	15	29	◦	◦	NOUN
ejpam-4712	15	30	)	)	PUNCT
ejpam-4712	15	31	is	be	AUX
ejpam-4712	15	32	associative	associative	ADJ
ejpam-4712	15	33	,	,	PUNCT
ejpam-4712	15	34	and	and	CCONJ
ejpam-4712	15	35	distributive	distributive	ADJ
ejpam-4712	15	36	.	.	PUNCT
ejpam-4712	16	1	if	if	SCONJ
ejpam-4712	16	2	the	the	DET
ejpam-4712	16	3	ring	ring	NOUN
ejpam-4712	16	4	has	have	VERB
ejpam-4712	16	5	the	the	DET
ejpam-4712	16	6	following	follow	VERB
ejpam-4712	16	7	properties	property	NOUN
ejpam-4712	16	8	:	:	PUNCT
ejpam-4712	16	9	commutative	commutative	ADJ
ejpam-4712	16	10	to	to	ADP
ejpam-4712	16	11	binary	binary	PROPN
ejpam-4712	16	12	operation	operation	NOUN
ejpam-4712	16	13	◦	◦	NOUN
ejpam-4712	16	14	,	,	PUNCT
ejpam-4712	16	15	it	it	PRON
ejpam-4712	16	16	has	have	VERB
ejpam-4712	16	17	a	a	DET
ejpam-4712	16	18	unit	unit	NOUN
ejpam-4712	16	19	element	element	NOUN
ejpam-4712	16	20	of	of	ADP
ejpam-4712	16	21	binary	binary	PROPN
ejpam-4712	16	22	operation	operation	NOUN
ejpam-4712	16	23	◦	◦	NOUN
ejpam-4712	16	24	,	,	PUNCT
ejpam-4712	16	25	and	and	CCONJ
ejpam-4712	16	26	every	every	DET
ejpam-4712	16	27	element	element	NOUN
ejpam-4712	16	28	,	,	PUNCT
ejpam-4712	16	29	not	not	PART
ejpam-4712	16	30	a	a	DET
ejpam-4712	16	31	zero	zero	NUM
ejpam-4712	16	32	element	element	NOUN
ejpam-4712	16	33	,	,	PUNCT
ejpam-4712	16	34	has	have	VERB
ejpam-4712	16	35	an	an	DET
ejpam-4712	16	36	inverse	inverse	NOUN
ejpam-4712	16	37	to	to	ADP
ejpam-4712	16	38	binary	binary	ADJ
ejpam-4712	16	39	operation	operation	NOUN
ejpam-4712	16	40	◦	◦	NOUN
ejpam-4712	16	41	,	,	PUNCT
ejpam-4712	16	42	it	it	PRON
ejpam-4712	16	43	is	be	AUX
ejpam-4712	16	44	called	call	VERB
ejpam-4712	16	45	field	field	NOUN
ejpam-4712	16	46	.	.	PUNCT
ejpam-4712	17	1	different	different	ADJ
ejpam-4712	17	2	algebraic	algebraic	ADJ
ejpam-4712	17	3	systems	system	NOUN
ejpam-4712	17	4	will	will	AUX
ejpam-4712	17	5	appear	appear	VERB
ejpam-4712	17	6	if	if	SCONJ
ejpam-4712	17	7	group	group	NOUN
ejpam-4712	17	8	and	and	CCONJ
ejpam-4712	17	9	ring	ring	NOUN
ejpam-4712	17	10	conditions	condition	NOUN
ejpam-4712	17	11	are	be	AUX
ejpam-4712	17	12	weakened	weaken	VERB
ejpam-4712	17	13	,	,	PUNCT
ejpam-4712	17	14	particularly	particularly	ADV
ejpam-4712	17	15	semigrup	semigrup	NOUN
ejpam-4712	17	16	and	and	CCONJ
ejpam-4712	17	17	semiring	semiring	NOUN
ejpam-4712	17	18	.	.	PUNCT
ejpam-4712	18	1	if	if	SCONJ
ejpam-4712	18	2	a	a	DET
ejpam-4712	18	3	few	few	ADJ
ejpam-4712	18	4	group	group	NOUN
ejpam-4712	18	5	or	or	CCONJ
ejpam-4712	18	6	ring	ring	NOUN
ejpam-4712	18	7	characteristics	characteristic	NOUN
ejpam-4712	18	8	are	be	AUX
ejpam-4712	18	9	removed	remove	VERB
ejpam-4712	18	10	,	,	PUNCT
ejpam-4712	18	11	the	the	DET
ejpam-4712	18	12	algebraic	algebraic	ADJ
ejpam-4712	18	13	structure	structure	NOUN
ejpam-4712	18	14	formed	form	VERB
ejpam-4712	18	15	is	be	AUX
ejpam-4712	18	16	semigrup	semigrup	NOUN
ejpam-4712	18	17	,	,	PUNCT
ejpam-4712	18	18	after	after	ADP
ejpam-4712	18	19	which	which	PRON
ejpam-4712	18	20	semiring	semire	VERB
ejpam-4712	18	21	.	.	PUNCT
ejpam-4712	19	1	one	one	NUM
ejpam-4712	19	2	of	of	ADP
ejpam-4712	19	3	∗corresponding	∗corresponde	VERB
ejpam-4712	19	4	author	author	NOUN
ejpam-4712	19	5	.	.	PUNCT
ejpam-4712	20	1	doi	doi	NOUN
ejpam-4712	20	2	:	:	PUNCT
ejpam-4712	20	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4712	https://doi.org/10.29020/nybg.ejpam.v16i2.4712	VERB
ejpam-4712	20	4	email	email	NOUN
ejpam-4712	20	5	addresses	address	NOUN
ejpam-4712	20	6	:	:	PUNCT
ejpam-4712	20	7	gregoria.ariyanti@ukwms.ac.id	gregoria.ariyanti@ukwms.ac.id	NUM
ejpam-4712	20	8	(	(	PUNCT
ejpam-4712	20	9	g.	g.	PROPN
ejpam-4712	20	10	ariyanti	ariyanti	PROPN
ejpam-4712	20	11	)	)	PUNCT
ejpam-4712	20	12	,	,	PUNCT
ejpam-4712	20	13	ana.easti.rahayu@ukwms.ac.id	ana.easti.rahayu@ukwms.ac.id	PROPN
ejpam-4712	20	14	(	(	PUNCT
ejpam-4712	20	15	a.e.r.m	a.e.r.m	PROPN
ejpam-4712	20	16	.	.	PUNCT
ejpam-4712	20	17	sari	sari	PROPN
ejpam-4712	20	18	)	)	PUNCT
ejpam-4712	20	19	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4712	20	20	784	784	NUM
ejpam-4712	20	21	©	©	PROPN
ejpam-4712	20	22	2023	2023	NUM
ejpam-4712	20	23	ejpam	ejpam	NOUN
ejpam-4712	20	24	all	all	DET
ejpam-4712	20	25	rights	right	NOUN
ejpam-4712	20	26	reserved	reserve	VERB
ejpam-4712	20	27	.	.	PUNCT
ejpam-4712	21	1	g.	g.	PROPN
ejpam-4712	21	2	ariyanti	ariyanti	PROPN
ejpam-4712	21	3	,	,	PUNCT
ejpam-4712	21	4	a.e.r.m	a.e.r.m	PROPN
ejpam-4712	21	5	.	.	PUNCT
ejpam-4712	21	6	sari	sari	PROPN
ejpam-4712	21	7	/	/	SYM
ejpam-4712	21	8	eur	eur	PROPN
ejpam-4712	21	9	.	.	PUNCT
ejpam-4712	22	1	j.	j.	PROPN
ejpam-4712	22	2	pure	pure	PROPN
ejpam-4712	22	3	appl	appl	PROPN
ejpam-4712	22	4	.	.	PROPN
ejpam-4712	22	5	math	math	PROPN
ejpam-4712	22	6	,	,	PUNCT
ejpam-4712	22	7	16	16	NUM
ejpam-4712	22	8	(	(	PUNCT
ejpam-4712	22	9	2	2	NUM
ejpam-4712	22	10	)	)	PUNCT
ejpam-4712	22	11	(	(	PUNCT
ejpam-4712	22	12	2023	2023	NUM
ejpam-4712	22	13	)	)	PUNCT
ejpam-4712	22	14	,	,	PUNCT
ejpam-4712	22	15	784	784	NUM
ejpam-4712	22	16	-	-	SYM
ejpam-4712	22	17	790	790	NUM
ejpam-4712	22	18	785	785	NUM
ejpam-4712	22	19	the	the	DET
ejpam-4712	22	20	problems	problem	NOUN
ejpam-4712	22	21	and	and	CCONJ
ejpam-4712	22	22	applications	application	NOUN
ejpam-4712	22	23	often	often	ADV
ejpam-4712	22	24	encountered	encounter	VERB
ejpam-4712	22	25	in	in	ADP
ejpam-4712	22	26	mathematics	mathematic	NOUN
ejpam-4712	22	27	is	be	AUX
ejpam-4712	22	28	completing	complete	VERB
ejpam-4712	22	29	the	the	DET
ejpam-4712	22	30	linear	linear	ADJ
ejpam-4712	22	31	equations	equation	NOUN
ejpam-4712	22	32	system	system	NOUN
ejpam-4712	22	33	.	.	PUNCT
ejpam-4712	23	1	the	the	DET
ejpam-4712	23	2	kronecker	kronecker	NOUN
ejpam-4712	23	3	product	product	NOUN
ejpam-4712	23	4	is	be	AUX
ejpam-4712	23	5	a	a	DET
ejpam-4712	23	6	binary	binary	ADJ
ejpam-4712	23	7	matrix	matrix	NOUN
ejpam-4712	23	8	operator	operator	NOUN
ejpam-4712	23	9	that	that	PRON
ejpam-4712	23	10	maps	map	VERB
ejpam-4712	23	11	two	two	NUM
ejpam-4712	23	12	arbitrarily	arbitrarily	ADV
ejpam-4712	23	13	dimensioned	dimension	VERB
ejpam-4712	23	14	matrices	matrix	NOUN
ejpam-4712	23	15	into	into	ADP
ejpam-4712	23	16	a	a	DET
ejpam-4712	23	17	larger	large	ADJ
ejpam-4712	23	18	matrix	matrix	NOUN
ejpam-4712	23	19	with	with	ADP
ejpam-4712	23	20	a	a	DET
ejpam-4712	23	21	particular	particular	ADJ
ejpam-4712	23	22	block	block	NOUN
ejpam-4712	23	23	structure	structure	NOUN
ejpam-4712	23	24	(	(	PUNCT
ejpam-4712	23	25	[	[	X
ejpam-4712	23	26	14	14	NUM
ejpam-4712	23	27	]	]	NUM
ejpam-4712	23	28	)	)	PUNCT
ejpam-4712	23	29	.	.	PUNCT
ejpam-4712	24	1	the	the	DET
ejpam-4712	24	2	kronecker	kronecker	NOUN
ejpam-4712	24	3	product	product	NOUN
ejpam-4712	24	4	discussed	discuss	VERB
ejpam-4712	24	5	by	by	ADP
ejpam-4712	24	6	zhou	zhou	PROPN
ejpam-4712	24	7	et	et	PROPN
ejpam-4712	24	8	al	al	PROPN
ejpam-4712	24	9	.	.	PROPN
ejpam-4712	24	10	(	(	PUNCT
ejpam-4712	24	11	1996	1996	NUM
ejpam-4712	24	12	)	)	PUNCT
ejpam-4712	24	13	and	and	CCONJ
ejpam-4712	24	14	whitcomb	whitcomb	NOUN
ejpam-4712	24	15	(	(	PUNCT
ejpam-4712	24	16	2020	2020	NUM
ejpam-4712	24	17	)	)	PUNCT
ejpam-4712	24	18	applies	apply	VERB
ejpam-4712	24	19	to	to	ADP
ejpam-4712	24	20	matrices	matrix	NOUN
ejpam-4712	24	21	whose	whose	DET
ejpam-4712	24	22	elements	element	NOUN
ejpam-4712	24	23	are	be	AUX
ejpam-4712	24	24	real	real	ADJ
ejpam-4712	24	25	numbers	number	NOUN
ejpam-4712	24	26	(	(	PUNCT
ejpam-4712	24	27	[	[	X
ejpam-4712	24	28	4],[11	4],[11	NOUN
ejpam-4712	24	29	]	]	PUNCT
ejpam-4712	24	30	)	)	PUNCT
ejpam-4712	24	31	.	.	PUNCT
ejpam-4712	25	1	the	the	DET
ejpam-4712	25	2	set	set	NOUN
ejpam-4712	25	3	of	of	ADP
ejpam-4712	25	4	real	real	ADJ
ejpam-4712	25	5	numbers	number	NOUN
ejpam-4712	25	6	is	be	AUX
ejpam-4712	25	7	a	a	DET
ejpam-4712	25	8	field	field	NOUN
ejpam-4712	25	9	.	.	PUNCT
ejpam-4712	26	1	the	the	DET
ejpam-4712	26	2	linear	linear	PROPN
ejpam-4712	26	3	equations	equation	NOUN
ejpam-4712	26	4	system	system	NOUN
ejpam-4712	26	5	that	that	SCONJ
ejpam-4712	26	6	researchers	researcher	NOUN
ejpam-4712	26	7	have	have	AUX
ejpam-4712	26	8	developed	develop	VERB
ejpam-4712	26	9	is	be	AUX
ejpam-4712	26	10	a	a	DET
ejpam-4712	26	11	system	system	NOUN
ejpam-4712	26	12	of	of	ADP
ejpam-4712	26	13	linear	linear	PROPN
ejpam-4712	26	14	equations	equation	NOUN
ejpam-4712	26	15	over	over	ADP
ejpam-4712	26	16	field	field	NOUN
ejpam-4712	26	17	,	,	PUNCT
ejpam-4712	26	18	which	which	PRON
ejpam-4712	26	19	include	include	VERB
ejpam-4712	26	20	real	real	ADJ
ejpam-4712	26	21	numbers	number	NOUN
ejpam-4712	26	22	r	r	NOUN
ejpam-4712	26	23	or	or	CCONJ
ejpam-4712	26	24	complex	complex	ADJ
ejpam-4712	26	25	numbers	number	NOUN
ejpam-4712	26	26	c	c	NOUN
ejpam-4712	26	27	(	(	PUNCT
ejpam-4712	26	28	[	[	X
ejpam-4712	26	29	12],[4	12],[4	NUM
ejpam-4712	26	30	]	]	PUNCT
ejpam-4712	26	31	)	)	PUNCT
ejpam-4712	26	32	.	.	PUNCT
ejpam-4712	27	1	in	in	ADP
ejpam-4712	27	2	other	other	ADJ
ejpam-4712	27	3	studies	study	NOUN
ejpam-4712	27	4	,	,	PUNCT
ejpam-4712	27	5	the	the	DET
ejpam-4712	27	6	research	research	NOUN
ejpam-4712	27	7	object	object	NOUN
ejpam-4712	27	8	is	be	AUX
ejpam-4712	27	9	extended	extend	VERB
ejpam-4712	27	10	not	not	PART
ejpam-4712	27	11	to	to	PART
ejpam-4712	27	12	field	field	VERB
ejpam-4712	27	13	anymore	anymore	ADV
ejpam-4712	27	14	but	but	CCONJ
ejpam-4712	27	15	to	to	PART
ejpam-4712	27	16	commutative	commutative	ADJ
ejpam-4712	27	17	ring	ring	NOUN
ejpam-4712	27	18	and	and	CCONJ
ejpam-4712	27	19	linear	linear	ADJ
ejpam-4712	27	20	equations	equation	NOUN
ejpam-4712	27	21	system	system	NOUN
ejpam-4712	27	22	commutative	commutative	ADJ
ejpam-4712	27	23	over	over	ADP
ejpam-4712	27	24	ring	ring	NOUN
ejpam-4712	27	25	have	have	AUX
ejpam-4712	27	26	been	be	AUX
ejpam-4712	27	27	discussed	discuss	VERB
ejpam-4712	27	28	by	by	ADP
ejpam-4712	27	29	brown	brown	NOUN
ejpam-4712	27	30	(	(	PUNCT
ejpam-4712	27	31	[	[	X
ejpam-4712	27	32	4	4	NUM
ejpam-4712	27	33	]	]	NUM
ejpam-4712	27	34	)	)	PUNCT
ejpam-4712	27	35	.	.	PUNCT
ejpam-4712	28	1	likewise	likewise	ADV
ejpam-4712	28	2	,	,	PUNCT
ejpam-4712	28	3	assuming	assume	VERB
ejpam-4712	28	4	an	an	DET
ejpam-4712	28	5	extension	extension	NOUN
ejpam-4712	28	6	from	from	ADP
ejpam-4712	28	7	the	the	DET
ejpam-4712	28	8	ring	ring	NOUN
ejpam-4712	28	9	to	to	PART
ejpam-4712	28	10	ring	ring	NOUN
ejpam-4712	28	11	commutative	commutative	ADJ
ejpam-4712	28	12	does	do	AUX
ejpam-4712	28	13	not	not	PART
ejpam-4712	28	14	generally	generally	ADV
ejpam-4712	28	15	change	change	VERB
ejpam-4712	28	16	the	the	DET
ejpam-4712	28	17	definition	definition	NOUN
ejpam-4712	28	18	.	.	PUNCT
ejpam-4712	29	1	this	this	DET
ejpam-4712	29	2	paper	paper	NOUN
ejpam-4712	29	3	presents	present	VERB
ejpam-4712	29	4	the	the	DET
ejpam-4712	29	5	characteristics	characteristic	NOUN
ejpam-4712	29	6	of	of	ADP
ejpam-4712	29	7	the	the	DET
ejpam-4712	29	8	discrete	discrete	ADJ
ejpam-4712	29	9	lyapunov	lyapunov	ADJ
ejpam-4712	29	10	equation	equation	NOUN
ejpam-4712	29	11	of	of	ADP
ejpam-4712	29	12	a	a	DET
ejpam-4712	29	13	matrix	matrix	NOUN
ejpam-4712	29	14	.	.	PUNCT
ejpam-4712	30	1	the	the	DET
ejpam-4712	30	2	scope	scope	NOUN
ejpam-4712	30	3	of	of	ADP
ejpam-4712	30	4	the	the	DET
ejpam-4712	30	5	topic	topic	NOUN
ejpam-4712	30	6	is	be	AUX
ejpam-4712	30	7	a	a	DET
ejpam-4712	30	8	system	system	NOUN
ejpam-4712	30	9	of	of	ADP
ejpam-4712	30	10	linear	linear	ADJ
ejpam-4712	30	11	equations	equation	NOUN
ejpam-4712	30	12	in	in	ADP
ejpam-4712	30	13	semiring	semire	VERB
ejpam-4712	30	14	in	in	ADP
ejpam-4712	30	15	terms	term	NOUN
ejpam-4712	30	16	of	of	ADP
ejpam-4712	30	17	the	the	DET
ejpam-4712	30	18	kronecker	kronecker	NOUN
ejpam-4712	30	19	product	product	NOUN
ejpam-4712	30	20	.	.	PUNCT
ejpam-4712	31	1	first	first	ADV
ejpam-4712	31	2	,	,	PUNCT
ejpam-4712	31	3	section	section	NOUN
ejpam-4712	31	4	2	2	NUM
ejpam-4712	31	5	will	will	AUX
ejpam-4712	31	6	review	review	VERB
ejpam-4712	31	7	some	some	DET
ejpam-4712	31	8	basic	basic	ADJ
ejpam-4712	31	9	facts	fact	NOUN
ejpam-4712	31	10	about	about	ADP
ejpam-4712	31	11	the	the	DET
ejpam-4712	31	12	semiring	semiring	NOUN
ejpam-4712	31	13	,	,	PUNCT
ejpam-4712	31	14	the	the	DET
ejpam-4712	31	15	kronecker	kronecker	NOUN
ejpam-4712	31	16	product	product	NOUN
ejpam-4712	31	17	of	of	ADP
ejpam-4712	31	18	the	the	DET
ejpam-4712	31	19	matrix	matrix	NOUN
ejpam-4712	31	20	in	in	ADP
ejpam-4712	31	21	semiring	semiring	NOUN
ejpam-4712	31	22	,	,	PUNCT
ejpam-4712	31	23	and	and	CCONJ
ejpam-4712	31	24	the	the	DET
ejpam-4712	31	25	lyapunov	lyapunov	ADJ
ejpam-4712	31	26	equation	equation	NOUN
ejpam-4712	31	27	of	of	ADP
ejpam-4712	31	28	the	the	DET
ejpam-4712	31	29	matrix	matrix	NOUN
ejpam-4712	31	30	in	in	ADP
ejpam-4712	31	31	semiring	semiring	NOUN
ejpam-4712	31	32	.	.	PUNCT
ejpam-4712	32	1	then	then	ADV
ejpam-4712	32	2	,	,	PUNCT
ejpam-4712	32	3	in	in	ADP
ejpam-4712	32	4	section	section	NOUN
ejpam-4712	32	5	3	3	NUM
ejpam-4712	32	6	,	,	PUNCT
ejpam-4712	32	7	we	we	PRON
ejpam-4712	32	8	show	show	VERB
ejpam-4712	32	9	a	a	DET
ejpam-4712	32	10	necessary	necessary	ADJ
ejpam-4712	32	11	or	or	CCONJ
ejpam-4712	32	12	sufficient	sufficient	ADJ
ejpam-4712	32	13	condition	condition	NOUN
ejpam-4712	32	14	of	of	ADP
ejpam-4712	32	15	the	the	DET
ejpam-4712	32	16	lyapunov	lyapunov	ADJ
ejpam-4712	32	17	equation	equation	NOUN
ejpam-4712	32	18	in	in	ADP
ejpam-4712	32	19	a	a	DET
ejpam-4712	32	20	linear	linear	ADJ
ejpam-4712	32	21	system	system	NOUN
ejpam-4712	32	22	over	over	ADP
ejpam-4712	32	23	semiring	semire	VERB
ejpam-4712	32	24	in	in	ADP
ejpam-4712	32	25	terms	term	NOUN
ejpam-4712	32	26	of	of	ADP
ejpam-4712	32	27	the	the	DET
ejpam-4712	32	28	kronecker	kronecker	NOUN
ejpam-4712	32	29	product	product	NOUN
ejpam-4712	32	30	.	.	PUNCT
ejpam-4712	33	1	2	2	X
ejpam-4712	33	2	.	.	X
ejpam-4712	33	3	materials	material	NOUN
ejpam-4712	33	4	and	and	CCONJ
ejpam-4712	33	5	literature	literature	NOUN
ejpam-4712	33	6	review	review	NOUN
ejpam-4712	33	7	2.1	2.1	NUM
ejpam-4712	33	8	.	.	PUNCT
ejpam-4712	34	1	semiring	semiring	NOUN
ejpam-4712	34	2	and	and	CCONJ
ejpam-4712	34	3	matrices	matrix	NOUN
ejpam-4712	34	4	in	in	ADP
ejpam-4712	34	5	semiring	semire	VERB
ejpam-4712	34	6	definition	definition	NOUN
ejpam-4712	34	7	1	1	NUM
ejpam-4712	34	8	.	.	PUNCT
ejpam-4712	35	1	semigroup	semigroup	PROPN
ejpam-4712	35	2	s	s	PART
ejpam-4712	35	3	is	be	AUX
ejpam-4712	35	4	an	an	DET
ejpam-4712	35	5	empty	empty	ADJ
ejpam-4712	35	6	set	set	NOUN
ejpam-4712	35	7	equipped	equip	VERB
ejpam-4712	35	8	with	with	ADP
ejpam-4712	35	9	an	an	DET
ejpam-4712	35	10	associative	associative	ADJ
ejpam-4712	35	11	binary	binary	NOUN
ejpam-4712	35	12	∗	∗	NOUN
ejpam-4712	35	13	operation	operation	NOUN
ejpam-4712	35	14	,	,	PUNCT
ejpam-4712	35	15	x	x	X
ejpam-4712	35	16	∗	∗	NOUN
ejpam-4712	35	17	(	(	PUNCT
ejpam-4712	35	18	y	y	PROPN
ejpam-4712	35	19	∗	∗	PROPN
ejpam-4712	35	20	z	z	NOUN
ejpam-4712	35	21	)	)	PUNCT
ejpam-4712	35	22	=	=	SYM
ejpam-4712	36	1	(	(	PUNCT
ejpam-4712	36	2	x	x	X
ejpam-4712	36	3	∗	∗	PROPN
ejpam-4712	36	4	y	y	NOUN
ejpam-4712	36	5	)	)	PUNCT
ejpam-4712	36	6	∗	∗	NOUN
ejpam-4712	36	7	z	z	NOUN
ejpam-4712	36	8	for	for	ADP
ejpam-4712	36	9	every	every	DET
ejpam-4712	36	10	x	x	PROPN
ejpam-4712	36	11	,	,	PUNCT
ejpam-4712	36	12	y	y	PROPN
ejpam-4712	36	13	,	,	PUNCT
ejpam-4712	36	14	z	z	PROPN
ejpam-4712	36	15	∈	∈	PROPN
ejpam-4712	36	16	s.	s.	PROPN
ejpam-4712	36	17	poplin	poplin	PROPN
ejpam-4712	36	18	defines	define	VERB
ejpam-4712	36	19	a	a	DET
ejpam-4712	36	20	semiring	semiring	NOUN
ejpam-4712	36	21	and	and	CCONJ
ejpam-4712	36	22	its	its	PRON
ejpam-4712	36	23	properties	property	NOUN
ejpam-4712	36	24	as	as	ADP
ejpam-4712	36	25	follows([2],[13],[12],[11	follows([2],[13],[12],[11	NOUN
ejpam-4712	36	26	]	]	PUNCT
ejpam-4712	36	27	)	)	PUNCT
ejpam-4712	36	28	.	.	PUNCT
ejpam-4712	37	1	definition	definition	NOUN
ejpam-4712	37	2	2	2	NUM
ejpam-4712	37	3	.	.	PUNCT
ejpam-4712	38	1	semiring	semiring	NOUN
ejpam-4712	38	2	is	be	AUX
ejpam-4712	38	3	a	a	DET
ejpam-4712	38	4	non	non	ADJ
ejpam-4712	38	5	-	-	ADJ
ejpam-4712	38	6	empty	empty	ADJ
ejpam-4712	38	7	set	set	NOUN
ejpam-4712	38	8	s	s	NOUN
ejpam-4712	38	9	with	with	ADP
ejpam-4712	38	10	two	two	NUM
ejpam-4712	38	11	binary	binary	ADJ
ejpam-4712	38	12	operations	operation	NOUN
ejpam-4712	38	13	,	,	PUNCT
ejpam-4712	38	14	addition	addition	NOUN
ejpam-4712	38	15	(	(	PUNCT
ejpam-4712	38	16	+	+	NOUN
ejpam-4712	38	17	)	)	PUNCT
ejpam-4712	38	18	and	and	CCONJ
ejpam-4712	38	19	multiplication	multiplication	NOUN
ejpam-4712	38	20	(	(	PUNCT
ejpam-4712	38	21	×	×	NOUN
ejpam-4712	38	22	)	)	PUNCT
ejpam-4712	38	23	,	,	PUNCT
ejpam-4712	38	24	which	which	PRON
ejpam-4712	38	25	have	have	VERB
ejpam-4712	38	26	the	the	DET
ejpam-4712	38	27	following	follow	VERB
ejpam-4712	38	28	properties	property	NOUN
ejpam-4712	38	29	:	:	PUNCT
ejpam-4712	38	30	commutative	commutative	ADJ
ejpam-4712	38	31	and	and	CCONJ
ejpam-4712	38	32	associative	associative	ADJ
ejpam-4712	38	33	properties	property	NOUN
ejpam-4712	38	34	of	of	ADP
ejpam-4712	38	35	+	+	ADJ
ejpam-4712	38	36	,	,	PUNCT
ejpam-4712	38	37	associative	associative	ADJ
ejpam-4712	38	38	property	property	NOUN
ejpam-4712	38	39	of	of	ADP
ejpam-4712	38	40	×	×	NOUN
ejpam-4712	38	41	,	,	PUNCT
ejpam-4712	38	42	distributive	distributive	ADJ
ejpam-4712	38	43	property	property	NOUN
ejpam-4712	38	44	of	of	ADP
ejpam-4712	38	45	×	×	NOUN
ejpam-4712	38	46	to	to	ADP
ejpam-4712	38	47	+	+	ADP
ejpam-4712	38	48	,	,	PUNCT
ejpam-4712	38	49	the	the	DET
ejpam-4712	38	50	set	set	NOUN
ejpam-4712	38	51	s	s	PART
ejpam-4712	38	52	has	have	AUX
ejpam-4712	39	1	a	a	DET
ejpam-4712	39	2	zero	zero	NUM
ejpam-4712	39	3	element	element	NOUN
ejpam-4712	39	4	0	0	NUM
ejpam-4712	39	5	∈	∈	NOUN
ejpam-4712	39	6	s	s	VERB
ejpam-4712	39	7	so	so	ADV
ejpam-4712	39	8	that	that	SCONJ
ejpam-4712	39	9	0	0	NUM
ejpam-4712	40	1	+	+	CCONJ
ejpam-4712	40	2	a	a	DET
ejpam-4712	40	3	=	=	X
ejpam-4712	40	4	a	a	DET
ejpam-4712	40	5	+	+	NOUN
ejpam-4712	40	6	0	0	NUM
ejpam-4712	40	7	=	=	SYM
ejpam-4712	40	8	a	a	DET
ejpam-4712	40	9	and	and	CCONJ
ejpam-4712	40	10	0	0	NUM
ejpam-4712	40	11	×	×	NOUN
ejpam-4712	40	12	a	a	X
ejpam-4712	40	13	=	=	PUNCT
ejpam-4712	40	14	a	a	DET
ejpam-4712	40	15	×	×	NOUN
ejpam-4712	40	16	0	0	PUNCT
ejpam-4712	40	17	=	=	SYM
ejpam-4712	40	18	0	0	NUM
ejpam-4712	40	19	for	for	ADP
ejpam-4712	40	20	every	every	DET
ejpam-4712	40	21	a	a	DET
ejpam-4712	40	22	∈	∈	PROPN
ejpam-4712	40	23	s.	s.	NOUN
ejpam-4712	40	24	this	this	DET
ejpam-4712	40	25	zero	zero	NUM
ejpam-4712	40	26	element	element	NOUN
ejpam-4712	40	27	is	be	AUX
ejpam-4712	40	28	called	call	VERB
ejpam-4712	40	29	the	the	DET
ejpam-4712	40	30	absorbent	absorbent	ADJ
ejpam-4712	40	31	element	element	NOUN
ejpam-4712	40	32	(	(	PUNCT
ejpam-4712	40	33	absorption	absorption	NOUN
ejpam-4712	40	34	)	)	PUNCT
ejpam-4712	40	35	,	,	PUNCT
ejpam-4712	40	36	and	and	CCONJ
ejpam-4712	40	37	the	the	DET
ejpam-4712	40	38	set	set	NOUN
ejpam-4712	40	39	s	s	PART
ejpam-4712	40	40	has	have	VERB
ejpam-4712	40	41	a	a	DET
ejpam-4712	40	42	unit	unit	NOUN
ejpam-4712	40	43	element	element	NOUN
ejpam-4712	40	44	e	e	PROPN
ejpam-4712	40	45	,	,	PUNCT
ejpam-4712	40	46	e×	e×	PROPN
ejpam-4712	40	47	a	a	DET
ejpam-4712	40	48	=	=	SYM
ejpam-4712	40	49	a×	a×	PUNCT
ejpam-4712	40	50	e	e	NOUN
ejpam-4712	40	51	=	=	PUNCT
ejpam-4712	40	52	a	a	PRON
ejpam-4712	40	53	for	for	ADP
ejpam-4712	40	54	every	every	DET
ejpam-4712	40	55	a	a	DET
ejpam-4712	40	56	∈	∈	PROPN
ejpam-4712	40	57	s.	s.	PROPN
ejpam-4712	40	58	the	the	DET
ejpam-4712	40	59	commutative	commutative	ADJ
ejpam-4712	40	60	and	and	CCONJ
ejpam-4712	40	61	idempotent	idempotent	ADJ
ejpam-4712	40	62	characteristics	characteristic	NOUN
ejpam-4712	40	63	in	in	ADP
ejpam-4712	40	64	group	group	NOUN
ejpam-4712	40	65	and	and	CCONJ
ejpam-4712	40	66	ring	ring	NOUN
ejpam-4712	40	67	also	also	ADV
ejpam-4712	40	68	apply	apply	VERB
ejpam-4712	40	69	in	in	ADP
ejpam-4712	40	70	semiring	semire	VERB
ejpam-4712	40	71	(	(	PUNCT
ejpam-4712	40	72	[	[	X
ejpam-4712	40	73	13	13	NUM
ejpam-4712	40	74	]	]	NUM
ejpam-4712	40	75	)	)	PUNCT
ejpam-4712	40	76	.	.	PUNCT
ejpam-4712	41	1	let	let	VERB
ejpam-4712	41	2	mn×1(s	mn×1(s	PROPN
ejpam-4712	41	3	)	)	PUNCT
ejpam-4712	41	4	be	be	AUX
ejpam-4712	41	5	the	the	DET
ejpam-4712	41	6	set	set	NOUN
ejpam-4712	41	7	of	of	ADP
ejpam-4712	41	8	all	all	DET
ejpam-4712	41	9	vectors	vector	NOUN
ejpam-4712	41	10	n×	n×	ADV
ejpam-4712	41	11	1	1	NUM
ejpam-4712	41	12	with	with	ADP
ejpam-4712	41	13	the	the	DET
ejpam-4712	41	14	elements	element	NOUN
ejpam-4712	41	15	of	of	ADP
ejpam-4712	41	16	semiring	semire	VERB
ejpam-4712	41	17	s.	s.	PROPN
ejpam-4712	41	18	and	and	CCONJ
ejpam-4712	41	19	,	,	PUNCT
ejpam-4712	41	20	let	let	VERB
ejpam-4712	41	21	mn×n(s	mn×n(s	NOUN
ejpam-4712	41	22	)	)	PUNCT
ejpam-4712	41	23	be	be	VERB
ejpam-4712	41	24	the	the	DET
ejpam-4712	41	25	set	set	NOUN
ejpam-4712	41	26	of	of	ADP
ejpam-4712	41	27	all	all	DET
ejpam-4712	41	28	n	n	PRON
ejpam-4712	42	1	×	×	NOUN
ejpam-4712	42	2	n	n	PRON
ejpam-4712	42	3	matrices	matrix	NOUN
ejpam-4712	42	4	with	with	ADP
ejpam-4712	42	5	the	the	DET
ejpam-4712	42	6	elements	element	NOUN
ejpam-4712	42	7	of	of	ADP
ejpam-4712	42	8	semiring	semire	VERB
ejpam-4712	42	9	s	s	X
ejpam-4712	42	10	(	(	PUNCT
ejpam-4712	42	11	[	[	X
ejpam-4712	42	12	4],[1	4],[1	NUM
ejpam-4712	42	13	]	]	PUNCT
ejpam-4712	42	14	)	)	PUNCT
ejpam-4712	42	15	.	.	PUNCT
ejpam-4712	43	1	the	the	DET
ejpam-4712	43	2	+	+	ADJ
ejpam-4712	43	3	and	and	CCONJ
ejpam-4712	43	4	×	×	NOUN
ejpam-4712	43	5	operation	operation	NOUN
ejpam-4712	43	6	for	for	ADP
ejpam-4712	43	7	matrices	matrix	NOUN
ejpam-4712	43	8	over	over	ADP
ejpam-4712	43	9	semiring	semiring	NOUN
ejpam-4712	43	10	is	be	AUX
ejpam-4712	43	11	defined	define	VERB
ejpam-4712	43	12	as	as	ADP
ejpam-4712	43	13	in	in	ADP
ejpam-4712	43	14	definition	definition	NOUN
ejpam-4712	43	15	3	3	NUM
ejpam-4712	43	16	.	.	PUNCT
ejpam-4712	44	1	definition	definition	NOUN
ejpam-4712	44	2	3	3	X
ejpam-4712	44	3	.	.	PUNCT
ejpam-4712	45	1	let	let	VERB
ejpam-4712	45	2	s	s	PRON
ejpam-4712	45	3	semiring	semire	VERB
ejpam-4712	45	4	,	,	PUNCT
ejpam-4712	45	5	a	a	DET
ejpam-4712	45	6	positive	positive	ADJ
ejpam-4712	45	7	integer	integer	NOUN
ejpam-4712	45	8	n	n	CCONJ
ejpam-4712	45	9	,	,	PUNCT
ejpam-4712	45	10	and	and	CCONJ
ejpam-4712	45	11	mn(s	mn(s	ADV
ejpam-4712	45	12	)	)	PUNCT
ejpam-4712	45	13	is	be	AUX
ejpam-4712	45	14	the	the	DET
ejpam-4712	45	15	set	set	NOUN
ejpam-4712	45	16	of	of	ADP
ejpam-4712	45	17	all	all	DET
ejpam-4712	45	18	n	n	PRON
ejpam-4712	45	19	×	×	NOUN
ejpam-4712	45	20	n	n	PRON
ejpam-4712	45	21	matrices	matrice	VERB
ejpam-4712	45	22	over	over	ADP
ejpam-4712	45	23	s.	s.	PROPN
ejpam-4712	45	24	for	for	ADP
ejpam-4712	45	25	every	every	DET
ejpam-4712	45	26	a	a	PROPN
ejpam-4712	45	27	,	,	PUNCT
ejpam-4712	45	28	b	b	PROPN
ejpam-4712	45	29	∈	∈	PROPN
ejpam-4712	45	30	mn(s	mn(	NOUN
ejpam-4712	45	31	)	)	PUNCT
ejpam-4712	45	32	,	,	PUNCT
ejpam-4712	46	1	+	+	PUNCT
ejpam-4712	46	2	and	and	CCONJ
ejpam-4712	46	3	×	×	PROPN
ejpam-4712	46	4	operations	operation	NOUN
ejpam-4712	46	5	over	over	ADP
ejpam-4712	46	6	semiring	semire	VERB
ejpam-4712	46	7	s	s	X
ejpam-4712	46	8	are	be	AUX
ejpam-4712	46	9	defined	define	VERB
ejpam-4712	46	10	c	c	X
ejpam-4712	46	11	=	=	SYM
ejpam-4712	46	12	a+b	a+b	NUM
ejpam-4712	46	13	as	as	ADP
ejpam-4712	46	14	cij	cij	PROPN
ejpam-4712	46	15	=	=	SYM
ejpam-4712	46	16	aij	aij	PROPN
ejpam-4712	46	17	+	+	CCONJ
ejpam-4712	46	18	bij	bij	NOUN
ejpam-4712	46	19	and	and	CCONJ
ejpam-4712	46	20	c	c	NOUN
ejpam-4712	46	21	=	=	SYM
ejpam-4712	47	1	a×b	a×b	PROPN
ejpam-4712	47	2	as	as	ADP
ejpam-4712	47	3	cij	cij	PROPN
ejpam-4712	47	4	=	=	SYM
ejpam-4712	47	5	∑	∑	PUNCT
ejpam-4712	47	6	l	l	PROPN
ejpam-4712	47	7	ail	ail	X
ejpam-4712	47	8	×	×	PROPN
ejpam-4712	47	9	blj	blj	NOUN
ejpam-4712	47	10	.	.	PUNCT
ejpam-4712	48	1	g.	g.	PROPN
ejpam-4712	48	2	ariyanti	ariyanti	PROPN
ejpam-4712	48	3	,	,	PUNCT
ejpam-4712	48	4	a.e.r.m	a.e.r.m	PROPN
ejpam-4712	48	5	.	.	PUNCT
ejpam-4712	49	1	sari	sari	PROPN
ejpam-4712	49	2	/	/	SYM
ejpam-4712	49	3	eur	eur	PROPN
ejpam-4712	49	4	.	.	PUNCT
ejpam-4712	50	1	j.	j.	PROPN
ejpam-4712	50	2	pure	pure	PROPN
ejpam-4712	50	3	appl	appl	PROPN
ejpam-4712	50	4	.	.	PROPN
ejpam-4712	50	5	math	math	PROPN
ejpam-4712	50	6	,	,	PUNCT
ejpam-4712	50	7	16	16	NUM
ejpam-4712	50	8	(	(	PUNCT
ejpam-4712	50	9	2	2	NUM
ejpam-4712	50	10	)	)	PUNCT
ejpam-4712	50	11	(	(	PUNCT
ejpam-4712	50	12	2023	2023	NUM
ejpam-4712	50	13	)	)	PUNCT
ejpam-4712	50	14	,	,	PUNCT
ejpam-4712	50	15	784	784	NUM
ejpam-4712	50	16	-	-	SYM
ejpam-4712	50	17	790	790	NUM
ejpam-4712	50	18	786	786	NUM
ejpam-4712	50	19	the	the	DET
ejpam-4712	50	20	semiring	semiring	NOUN
ejpam-4712	50	21	s	s	X
ejpam-4712	50	22	has	have	VERB
ejpam-4712	50	23	0	0	NUM
ejpam-4712	50	24	as	as	ADP
ejpam-4712	50	25	a	a	DET
ejpam-4712	50	26	zero	zero	NUM
ejpam-4712	50	27	element	element	NOUN
ejpam-4712	50	28	and	and	CCONJ
ejpam-4712	50	29	1	1	NUM
ejpam-4712	50	30	as	as	ADP
ejpam-4712	50	31	an	an	DET
ejpam-4712	50	32	identity	identity	NOUN
ejpam-4712	50	33	element	element	NOUN
ejpam-4712	50	34	,	,	PUNCT
ejpam-4712	50	35	as	as	ADP
ejpam-4712	50	36	in	in	ADP
ejpam-4712	50	37	the	the	DET
ejpam-4712	50	38	matrix	matrix	NOUN
ejpam-4712	50	39	of	of	ADP
ejpam-4712	50	40	conventional	conventional	ADJ
ejpam-4712	50	41	algebra	algebra	NOUN
ejpam-4712	50	42	.	.	PUNCT
ejpam-4712	51	1	we	we	PRON
ejpam-4712	51	2	can	can	AUX
ejpam-4712	51	3	form	form	VERB
ejpam-4712	51	4	a	a	DET
ejpam-4712	51	5	zero	zero	NUM
ejpam-4712	51	6	matrix	matrix	NOUN
ejpam-4712	51	7	and	and	CCONJ
ejpam-4712	51	8	an	an	DET
ejpam-4712	51	9	identity	identity	NOUN
ejpam-4712	51	10	matrix	matrix	NOUN
ejpam-4712	51	11	based	base	VERB
ejpam-4712	51	12	on	on	ADP
ejpam-4712	51	13	the	the	DET
ejpam-4712	51	14	zero	zero	NUM
ejpam-4712	51	15	element	element	NOUN
ejpam-4712	51	16	and	and	CCONJ
ejpam-4712	51	17	the	the	DET
ejpam-4712	51	18	identity	identity	NOUN
ejpam-4712	51	19	element	element	NOUN
ejpam-4712	51	20	(	(	PUNCT
ejpam-4712	51	21	[	[	X
ejpam-4712	51	22	2	2	NUM
ejpam-4712	51	23	]	]	NUM
ejpam-4712	51	24	)	)	PUNCT
ejpam-4712	51	25	.	.	PUNCT
ejpam-4712	52	1	the	the	DET
ejpam-4712	52	2	zero	zero	NUM
ejpam-4712	52	3	matrix	matrix	NOUN
ejpam-4712	52	4	n×	n×	NOUN
ejpam-4712	52	5	n	n	CCONJ
ejpam-4712	52	6	over	over	ADP
ejpam-4712	52	7	semiring	semire	VERB
ejpam-4712	52	8	s	s	PART
ejpam-4712	52	9	is	be	AUX
ejpam-4712	52	10	0n	0n	NUM
ejpam-4712	52	11	and	and	CCONJ
ejpam-4712	52	12	is	be	AUX
ejpam-4712	52	13	defined	define	VERB
ejpam-4712	52	14	as	as	ADP
ejpam-4712	52	15	a	a	DET
ejpam-4712	52	16	matrix	matrix	NOUN
ejpam-4712	52	17	with	with	ADP
ejpam-4712	52	18	all	all	DET
ejpam-4712	52	19	elements	element	NOUN
ejpam-4712	52	20	equal	equal	ADJ
ejpam-4712	52	21	to	to	ADP
ejpam-4712	52	22	the	the	DET
ejpam-4712	52	23	0−element	0−element	NOUN
ejpam-4712	52	24	,	,	PUNCT
ejpam-4712	52	25	that	that	ADV
ejpam-4712	52	26	is	is	ADV
ejpam-4712	52	27	(	(	PUNCT
ejpam-4712	52	28	0n)ij	0n)ij	X
ejpam-4712	52	29	=	=	SYM
ejpam-4712	53	1	0	0	X
ejpam-4712	53	2	.	.	PUNCT
ejpam-4712	54	1	the	the	DET
ejpam-4712	54	2	identity	identity	NOUN
ejpam-4712	54	3	matrix	matrix	NOUN
ejpam-4712	54	4	n	n	CCONJ
ejpam-4712	54	5	×	×	NOUN
ejpam-4712	54	6	n	n	NOUN
ejpam-4712	54	7	over	over	ADP
ejpam-4712	54	8	s	s	NOUN
ejpam-4712	54	9	is	be	AUX
ejpam-4712	54	10	defined	define	VERB
ejpam-4712	54	11	as	as	ADP
ejpam-4712	54	12	the	the	DET
ejpam-4712	54	13	matrix	matrix	NOUN
ejpam-4712	54	14	with	with	ADP
ejpam-4712	54	15	all	all	DET
ejpam-4712	54	16	elements	element	NOUN
ejpam-4712	54	17	equal	equal	ADJ
ejpam-4712	54	18	to	to	ADP
ejpam-4712	54	19	the	the	DET
ejpam-4712	54	20	e−element	e−element	PROPN
ejpam-4712	54	21	,	,	PUNCT
ejpam-4712	54	22	that	that	ADV
ejpam-4712	54	23	is	is	ADV
ejpam-4712	54	24	,	,	PUNCT
ejpam-4712	55	1	[	[	X
ejpam-4712	55	2	in]ij	in]ij	NOUN
ejpam-4712	55	3	=	=	X
ejpam-4712	55	4	{	{	PUNCT
ejpam-4712	55	5	e	e	NOUN
ejpam-4712	55	6	,	,	PUNCT
ejpam-4712	55	7	if	if	SCONJ
ejpam-4712	55	8	i	i	PRON
ejpam-4712	55	9	=	=	SYM
ejpam-4712	55	10	j	j	PROPN
ejpam-4712	55	11	0	0	NUM
ejpam-4712	55	12	,	,	PUNCT
ejpam-4712	55	13	if	if	SCONJ
ejpam-4712	55	14	i	i	PRON
ejpam-4712	55	15	̸=	̸=	PROPN
ejpam-4712	55	16	j	j	PROPN
ejpam-4712	55	17	.	.	PUNCT
ejpam-4712	56	1	in	in	ADP
ejpam-4712	56	2	semiring	semiring	NOUN
ejpam-4712	56	3	,	,	PUNCT
ejpam-4712	56	4	the	the	DET
ejpam-4712	56	5	element	element	NOUN
ejpam-4712	56	6	of	of	ADP
ejpam-4712	56	7	semiring	semiring	NOUN
ejpam-4712	56	8	has	have	VERB
ejpam-4712	56	9	an	an	DET
ejpam-4712	56	10	inverse	inverse	ADJ
ejpam-4712	56	11	operation	operation	NOUN
ejpam-4712	56	12	on	on	ADP
ejpam-4712	56	13	+	+	CCONJ
ejpam-4712	56	14	to	to	PART
ejpam-4712	56	15	determine	determine	VERB
ejpam-4712	56	16	a	a	DET
ejpam-4712	56	17	matrix	matrix	NOUN
ejpam-4712	56	18	determinant	determinant	ADJ
ejpam-4712	56	19	in	in	ADP
ejpam-4712	56	20	semiring	semire	VERB
ejpam-4712	56	21	s.	s.	PROPN
ejpam-4712	56	22	a	a	DET
ejpam-4712	56	23	permutation	permutation	NOUN
ejpam-4712	56	24	characterizes	characterize	VERB
ejpam-4712	56	25	a	a	DET
ejpam-4712	56	26	determinant	determinant	NOUN
ejpam-4712	56	27	of	of	ADP
ejpam-4712	56	28	a	a	DET
ejpam-4712	56	29	matrix	matrix	NOUN
ejpam-4712	56	30	over	over	ADP
ejpam-4712	56	31	a	a	DET
ejpam-4712	56	32	semiring	semire	VERB
ejpam-4712	56	33	s.	s.	PROPN
ejpam-4712	56	34	2.2	2.2	NUM
ejpam-4712	56	35	.	.	PUNCT
ejpam-4712	57	1	kronecker	kronecker	NOUN
ejpam-4712	57	2	product	product	NOUN
ejpam-4712	57	3	kronecker	kronecker	NOUN
ejpam-4712	57	4	’s	’s	PART
ejpam-4712	57	5	product	product	NOUN
ejpam-4712	57	6	is	be	AUX
ejpam-4712	57	7	related	relate	VERB
ejpam-4712	57	8	to	to	ADP
ejpam-4712	57	9	the	the	DET
ejpam-4712	57	10	stack	stack	NOUN
ejpam-4712	57	11	operator	operator	NOUN
ejpam-4712	57	12	.	.	PUNCT
ejpam-4712	58	1	the	the	DET
ejpam-4712	58	2	stack	stack	NOUN
ejpam-4712	58	3	operator	operator	NOUN
ejpam-4712	58	4	maps	map	VERB
ejpam-4712	58	5	an	an	DET
ejpam-4712	58	6	n×m	n×m	PROPN
ejpam-4712	58	7	matrix	matrix	NOUN
ejpam-4712	58	8	to	to	ADP
ejpam-4712	58	9	an	an	DET
ejpam-4712	58	10	nm×	nm×	ADJ
ejpam-4712	58	11	1	1	NUM
ejpam-4712	58	12	vector	vector	NOUN
ejpam-4712	58	13	(	(	PUNCT
ejpam-4712	58	14	[	[	X
ejpam-4712	58	15	14	14	NUM
ejpam-4712	58	16	]	]	NUM
ejpam-4712	58	17	)	)	PUNCT
ejpam-4712	58	18	.	.	PUNCT
ejpam-4712	59	1	the	the	DET
ejpam-4712	59	2	stack	stack	NOUN
ejpam-4712	59	3	of	of	ADP
ejpam-4712	59	4	the	the	DET
ejpam-4712	59	5	n×m	n×m	PROPN
ejpam-4712	59	6	matrix	matrix	NOUN
ejpam-4712	59	7	a	a	PRON
ejpam-4712	59	8	is	be	AUX
ejpam-4712	59	9	represented	represent	VERB
ejpam-4712	59	10	by	by	ADP
ejpam-4712	59	11	vec(a	vec(a	NUM
ejpam-4712	59	12	)	)	PUNCT
ejpam-4712	59	13	,	,	PUNCT
ejpam-4712	59	14	a	a	DET
ejpam-4712	59	15	vector	vector	NOUN
ejpam-4712	59	16	formed	form	VERB
ejpam-4712	59	17	by	by	ADP
ejpam-4712	59	18	stacking	stack	VERB
ejpam-4712	59	19	the	the	DET
ejpam-4712	59	20	columns	column	NOUN
ejpam-4712	59	21	of	of	ADP
ejpam-4712	59	22	a	a	PRON
ejpam-4712	59	23	on	on	ADP
ejpam-4712	59	24	the	the	DET
ejpam-4712	59	25	vector	vector	NOUN
ejpam-4712	59	26	nm×	nm×	ADP
ejpam-4712	59	27	1	1	NUM
ejpam-4712	59	28	.	.	PUNCT
ejpam-4712	59	29	example	example	NOUN
ejpam-4712	60	1	1	1	NUM
ejpam-4712	60	2	.	.	PUNCT
ejpam-4712	60	3	let	let	VERB
ejpam-4712	60	4	a	a	PRON
ejpam-4712	60	5	is	be	AUX
ejpam-4712	60	6	a	a	DET
ejpam-4712	60	7	matrix	matrix	NOUN
ejpam-4712	60	8	with	with	ADP
ejpam-4712	60	9	a	a	PRON
ejpam-4712	60	10	=	=	X
ejpam-4712	60	11	[	[	PUNCT
ejpam-4712	60	12	a	a	PRON
ejpam-4712	60	13	b	b	NOUN
ejpam-4712	60	14	c	c	NOUN
ejpam-4712	60	15	d	d	X
ejpam-4712	60	16	]	]	X
ejpam-4712	60	17	,	,	PUNCT
ejpam-4712	60	18	then	then	ADV
ejpam-4712	60	19	its	its	PRON
ejpam-4712	60	20	stack	stack	NOUN
ejpam-4712	60	21	form	form	NOUN
ejpam-4712	60	22	is	be	AUX
ejpam-4712	60	23	vec(a	vec(a	NOUN
ejpam-4712	60	24	)	)	PUNCT
ejpam-4712	60	25	=	=	SYM
ejpam-4712	60	26			NOUN
ejpam-4712	61	1	a	a	DET
ejpam-4712	61	2	b	b	X
ejpam-4712	61	3	c	c	NOUN
ejpam-4712	61	4	d	d	X
ejpam-4712	61	5			PROPN
ejpam-4712	61	6	.	.	PUNCT
ejpam-4712	62	1	if	if	SCONJ
ejpam-4712	62	2	c	c	PROPN
ejpam-4712	62	3	is	be	AUX
ejpam-4712	62	4	an	an	DET
ejpam-4712	62	5	n×m	n×m	PROPN
ejpam-4712	62	6	matrix	matrix	NOUN
ejpam-4712	62	7	comprising	comprising	NOUN
ejpam-4712	62	8	m	m	PROPN
ejpam-4712	62	9	column	column	NOUN
ejpam-4712	62	10	vectors	vector	NOUN
ejpam-4712	62	11	c1	c1	PROPN
ejpam-4712	62	12	,	,	PUNCT
ejpam-4712	62	13	c2	c2	PROPN
ejpam-4712	62	14	,	,	PUNCT
ejpam-4712	62	15	c3	c3	PROPN
ejpam-4712	62	16	,	,	PUNCT
ejpam-4712	62	17	...	...	PUNCT
ejpam-4712	62	18	,	,	PUNCT
ejpam-4712	62	19	cm	cm	NOUN
ejpam-4712	62	20	,	,	PUNCT
ejpam-4712	62	21	where	where	SCONJ
ejpam-4712	62	22	each	each	DET
ejpam-4712	62	23	ci	ci	NOUN
ejpam-4712	62	24	is	be	AUX
ejpam-4712	62	25	an	an	DET
ejpam-4712	62	26	n×	n×	PROPN
ejpam-4712	62	27	1	1	NUM
ejpam-4712	62	28	vector	vector	NOUN
ejpam-4712	62	29	c	c	NOUN
ejpam-4712	62	30	=	=	PUNCT
ejpam-4712	63	1	[	[	X
ejpam-4712	63	2	c1	c1	PROPN
ejpam-4712	63	3	,	,	PUNCT
ejpam-4712	63	4	c2	c2	PROPN
ejpam-4712	63	5	,	,	PUNCT
ejpam-4712	63	6	c3	c3	PROPN
ejpam-4712	63	7	,	,	PUNCT
ejpam-4712	63	8	...	...	PUNCT
ejpam-4712	63	9	,	,	PUNCT
ejpam-4712	63	10	cm]n×m	cm]n×m	NOUN
ejpam-4712	63	11	,	,	PUNCT
ejpam-4712	63	12	then	then	ADV
ejpam-4712	63	13	vec(c	vec(c	NOUN
ejpam-4712	63	14	)	)	PUNCT
ejpam-4712	63	15	=	=	PUNCT
ejpam-4712	63	16			PROPN
ejpam-4712	63	17	c1	c1	PROPN
ejpam-4712	63	18	c2	c2	PROPN
ejpam-4712	63	19	...	...	PUNCT
ejpam-4712	63	20	cm	cm	PROPN
ejpam-4712	63	21			VERB
ejpam-4712	63	22	nm×1	nm×1	PROPN
ejpam-4712	63	23	.	.	PUNCT
ejpam-4712	64	1	let	let	VERB
ejpam-4712	64	2	x	x	PUNCT
ejpam-4712	64	3	∈	∈	NOUN
ejpam-4712	64	4	mm×n(s	mm×n(s	NOUN
ejpam-4712	64	5	)	)	PUNCT
ejpam-4712	64	6	,	,	PUNCT
ejpam-4712	64	7	the	the	DET
ejpam-4712	64	8	form	form	NOUN
ejpam-4712	64	9	vec(x	vec(x	NOUN
ejpam-4712	64	10	)	)	PUNCT
ejpam-4712	64	11	denote	denote	VERB
ejpam-4712	64	12	the	the	DET
ejpam-4712	64	13	vector	vector	NOUN
ejpam-4712	64	14	formed	form	VERB
ejpam-4712	64	15	by	by	ADP
ejpam-4712	64	16	stacking	stack	VERB
ejpam-4712	64	17	the	the	DET
ejpam-4712	64	18	columns	column	NOUN
ejpam-4712	64	19	of	of	ADP
ejpam-4712	64	20	x	x	PUNCT
ejpam-4712	64	21	into	into	ADP
ejpam-4712	64	22	one	one	NUM
ejpam-4712	64	23	long	long	ADJ
ejpam-4712	64	24	vector	vector	NOUN
ejpam-4712	64	25	:	:	PUNCT
ejpam-4712	64	26	vec(x	vec(x	NOUN
ejpam-4712	64	27	)	)	PUNCT
ejpam-4712	64	28	=	=	PUNCT
ejpam-4712	65	1	[	[	PUNCT
ejpam-4712	65	2	x11	x11	NOUN
ejpam-4712	65	3	x21	x21	PROPN
ejpam-4712	65	4	.	.	PUNCT
ejpam-4712	65	5	.	.	PUNCT
ejpam-4712	65	6	.	.	PUNCT
ejpam-4712	66	1	xm1	xm1	PROPN
ejpam-4712	66	2	x12	x12	NUM
ejpam-4712	66	3	x22	x22	PROPN
ejpam-4712	66	4	.	.	PUNCT
ejpam-4712	66	5	.	.	PUNCT
ejpam-4712	66	6	.	.	PUNCT
ejpam-4712	67	1	x1n	x1n	PUNCT
ejpam-4712	68	1	x2n	x2n	INTJ
ejpam-4712	68	2	.	.	PUNCT
ejpam-4712	68	3	.	.	PUNCT
ejpam-4712	68	4	.	.	PUNCT
ejpam-4712	69	1	xmn	xmn	NOUN
ejpam-4712	69	2	]	]	X
ejpam-4712	69	3	t	t	PROPN
ejpam-4712	69	4	.	.	PUNCT
ejpam-4712	70	1	kronecker	kronecker	NOUN
ejpam-4712	70	2	product	product	NOUN
ejpam-4712	70	3	is	be	AUX
ejpam-4712	70	4	an	an	DET
ejpam-4712	70	5	operation	operation	NOUN
ejpam-4712	70	6	on	on	ADP
ejpam-4712	70	7	two	two	NUM
ejpam-4712	70	8	matrices	matrix	NOUN
ejpam-4712	70	9	that	that	PRON
ejpam-4712	70	10	do	do	AUX
ejpam-4712	70	11	not	not	PART
ejpam-4712	70	12	require	require	VERB
ejpam-4712	70	13	size	size	NOUN
ejpam-4712	70	14	(	(	PUNCT
ejpam-4712	70	15	[	[	X
ejpam-4712	70	16	8	8	NUM
ejpam-4712	70	17	]	]	PUNCT
ejpam-4712	70	18	,	,	PUNCT
ejpam-4712	70	19	[	[	X
ejpam-4712	70	20	9	9	NUM
ejpam-4712	70	21	]	]	NUM
ejpam-4712	70	22	)	)	PUNCT
ejpam-4712	70	23	.	.	PUNCT
ejpam-4712	71	1	the	the	DET
ejpam-4712	71	2	notation	notation	PROPN
ejpam-4712	71	3	⊗	⊗	PROPN
ejpam-4712	71	4	denotes	denotes	PROPN
ejpam-4712	71	5	kronecker	kronecker	NOUN
ejpam-4712	71	6	products	product	NOUN
ejpam-4712	71	7	.	.	PUNCT
ejpam-4712	72	1	with	with	ADP
ejpam-4712	72	2	s	s	AUX
ejpam-4712	72	3	semiring	semire	VERB
ejpam-4712	72	4	,	,	PUNCT
ejpam-4712	72	5	let	let	VERB
ejpam-4712	72	6	a	a	DET
ejpam-4712	72	7	∈	∈	NOUN
ejpam-4712	72	8	mm×n(s	mm×n(s	NOUN
ejpam-4712	72	9	)	)	PUNCT
ejpam-4712	72	10	and	and	CCONJ
ejpam-4712	72	11	b	b	PROPN
ejpam-4712	72	12	∈	∈	PROPN
ejpam-4712	72	13	mp×q(s	mp×q(s	NOUN
ejpam-4712	72	14	)	)	PUNCT
ejpam-4712	72	15	,	,	PUNCT
ejpam-4712	72	16	then	then	ADV
ejpam-4712	72	17	the	the	DET
ejpam-4712	72	18	kronecker	kronecker	NOUN
ejpam-4712	72	19	product	product	NOUN
ejpam-4712	72	20	of	of	ADP
ejpam-4712	72	21	a	a	PRON
ejpam-4712	72	22	and	and	CCONJ
ejpam-4712	72	23	b	b	NOUN
ejpam-4712	72	24	is	be	AUX
ejpam-4712	72	25	defined	define	VERB
ejpam-4712	72	26	as	as	ADP
ejpam-4712	72	27	a⊗b	a⊗b	PROPN
ejpam-4712	72	28	:	:	PUNCT
ejpam-4712	72	29	=	=	SYM
ejpam-4712	72	30			NOUN
ejpam-4712	72	31	a11b	a11b	NOUN
ejpam-4712	72	32	.	.	PUNCT
ejpam-4712	72	33	.	.	PUNCT
ejpam-4712	72	34	.	.	PUNCT
ejpam-4712	73	1	a1nb	a1nb	PUNCT
ejpam-4712	73	2	...	...	PUNCT
ejpam-4712	73	3	.	.	PUNCT
ejpam-4712	73	4	.	.	PUNCT
ejpam-4712	74	1	.	.	PUNCT
ejpam-4712	75	1	...	...	PUNCT
ejpam-4712	76	1	am1b	am1b	ADP
ejpam-4712	76	2	.	.	PUNCT
ejpam-4712	76	3	.	.	PUNCT
ejpam-4712	76	4	.	.	PUNCT
ejpam-4712	77	1	amnb	amnb	VERB
ejpam-4712	77	2			NUM
ejpam-4712	77	3	∈	∈	PROPN
ejpam-4712	77	4	mmp×nq(s	mmp×nq(s	PROPN
ejpam-4712	77	5	)	)	PUNCT
ejpam-4712	77	6	.	.	PUNCT
ejpam-4712	78	1	g.	g.	PROPN
ejpam-4712	78	2	ariyanti	ariyanti	PROPN
ejpam-4712	78	3	,	,	PUNCT
ejpam-4712	78	4	a.e.r.m	a.e.r.m	PROPN
ejpam-4712	78	5	.	.	PUNCT
ejpam-4712	78	6	sari	sari	PROPN
ejpam-4712	78	7	/	/	SYM
ejpam-4712	78	8	eur	eur	PROPN
ejpam-4712	78	9	.	.	PUNCT
ejpam-4712	79	1	j.	j.	PROPN
ejpam-4712	79	2	pure	pure	PROPN
ejpam-4712	79	3	appl	appl	PROPN
ejpam-4712	79	4	.	.	PROPN
ejpam-4712	79	5	math	math	PROPN
ejpam-4712	79	6	,	,	PUNCT
ejpam-4712	79	7	16	16	NUM
ejpam-4712	79	8	(	(	PUNCT
ejpam-4712	79	9	2	2	NUM
ejpam-4712	79	10	)	)	PUNCT
ejpam-4712	79	11	(	(	PUNCT
ejpam-4712	79	12	2023	2023	NUM
ejpam-4712	79	13	)	)	PUNCT
ejpam-4712	79	14	,	,	PUNCT
ejpam-4712	79	15	784	784	NUM
ejpam-4712	79	16	-	-	SYM
ejpam-4712	79	17	790	790	NUM
ejpam-4712	79	18	787	787	NUM
ejpam-4712	79	19	furthermore	furthermore	ADV
ejpam-4712	79	20	,	,	PUNCT
ejpam-4712	79	21	if	if	SCONJ
ejpam-4712	79	22	a	a	PRON
ejpam-4712	79	23	and	and	CCONJ
ejpam-4712	79	24	b	b	NOUN
ejpam-4712	79	25	are	be	AUX
ejpam-4712	79	26	square	square	ADJ
ejpam-4712	79	27	matrices	matrix	NOUN
ejpam-4712	79	28	with	with	ADP
ejpam-4712	79	29	a	a	DET
ejpam-4712	79	30	∈	∈	PROPN
ejpam-4712	79	31	mn×n(s	mn×n(s	NOUN
ejpam-4712	79	32	)	)	PUNCT
ejpam-4712	79	33	and	and	CCONJ
ejpam-4712	79	34	b	b	PROPN
ejpam-4712	79	35	∈	∈	PROPN
ejpam-4712	79	36	mm×m(s	mm×m(s	PROPN
ejpam-4712	79	37	)	)	PUNCT
ejpam-4712	79	38	then	then	ADV
ejpam-4712	79	39	the	the	DET
ejpam-4712	79	40	kronecker	kronecker	NOUN
ejpam-4712	79	41	sum	sum	NOUN
ejpam-4712	79	42	of	of	ADP
ejpam-4712	79	43	a	a	PRON
ejpam-4712	79	44	and	and	CCONJ
ejpam-4712	79	45	b	b	NOUN
ejpam-4712	79	46	is	be	AUX
ejpam-4712	79	47	defined	define	VERB
ejpam-4712	79	48	as	as	ADP
ejpam-4712	79	49	a⊕b	a⊕b	NOUN
ejpam-4712	79	50	:	:	PUNCT
ejpam-4712	79	51	=	=	SYM
ejpam-4712	79	52	(	(	PUNCT
ejpam-4712	79	53	a⊗	a⊗	NOUN
ejpam-4712	79	54	i	i	PRON
ejpam-4712	79	55	m	m	VERB
ejpam-4712	79	56	+	+	X
ejpam-4712	79	57	in	in	ADP
ejpam-4712	79	58	⊗b	⊗b	NOUN
ejpam-4712	79	59	)	)	PUNCT
ejpam-4712	79	60	∈	∈	PROPN
ejpam-4712	79	61	mnm×nm(s	mnm×nm(s	NOUN
ejpam-4712	79	62	)	)	PUNCT
ejpam-4712	79	63	.	.	PUNCT
ejpam-4712	80	1	properties	property	NOUN
ejpam-4712	80	2	of	of	ADP
ejpam-4712	80	3	kronecker	kronecker	NOUN
ejpam-4712	80	4	product	product	NOUN
ejpam-4712	80	5	is	be	AUX
ejpam-4712	80	6	given	give	VERB
ejpam-4712	80	7	the	the	DET
ejpam-4712	80	8	following	follow	VERB
ejpam-4712	80	9	theorem	theorem	NOUN
ejpam-4712	80	10	(	(	PUNCT
ejpam-4712	80	11	[	[	X
ejpam-4712	80	12	8	8	NUM
ejpam-4712	80	13	]	]	PUNCT
ejpam-4712	80	14	,	,	PUNCT
ejpam-4712	80	15	[	[	X
ejpam-4712	80	16	14	14	NUM
ejpam-4712	80	17	]	]	SYM
ejpam-4712	80	18	)	)	PUNCT
ejpam-4712	80	19	.	.	PUNCT
ejpam-4712	81	1	theorem	theorem	NOUN
ejpam-4712	81	2	1	1	NUM
ejpam-4712	81	3	.	.	X
ejpam-4712	81	4	for	for	ADP
ejpam-4712	81	5	a	a	DET
ejpam-4712	81	6	∈	∈	PROPN
ejpam-4712	81	7	mm×n(s	mm×n(s	NOUN
ejpam-4712	81	8	)	)	PUNCT
ejpam-4712	81	9	and	and	CCONJ
ejpam-4712	81	10	b	b	PROPN
ejpam-4712	81	11	∈	∈	PROPN
ejpam-4712	81	12	mp×q(s	mp×q(s	NOUN
ejpam-4712	81	13	)	)	PUNCT
ejpam-4712	81	14	with	with	ADP
ejpam-4712	81	15	s	s	AUX
ejpam-4712	81	16	semiring	semire	VERB
ejpam-4712	81	17	,	,	PUNCT
ejpam-4712	81	18	we	we	PRON
ejpam-4712	81	19	have	have	VERB
ejpam-4712	81	20	the	the	DET
ejpam-4712	81	21	following	follow	VERB
ejpam-4712	81	22	properties	property	NOUN
ejpam-4712	81	23	:	:	PUNCT
ejpam-4712	81	24	the	the	DET
ejpam-4712	81	25	kronecker	kronecker	NOUN
ejpam-4712	81	26	product	product	NOUN
ejpam-4712	81	27	is	be	AUX
ejpam-4712	81	28	associative	associative	ADJ
ejpam-4712	81	29	,	,	PUNCT
ejpam-4712	81	30	not	not	PART
ejpam-4712	81	31	in	in	ADP
ejpam-4712	81	32	general	general	ADJ
ejpam-4712	81	33	commutative	commutative	ADJ
ejpam-4712	81	34	,	,	PUNCT
ejpam-4712	81	35	that	that	ADV
ejpam-4712	81	36	is	is	ADV
ejpam-4712	81	37	,	,	PUNCT
ejpam-4712	81	38	(	(	PUNCT
ejpam-4712	81	39	a⊗	a⊗	NOUN
ejpam-4712	81	40	b	b	X
ejpam-4712	81	41	)	)	PUNCT
ejpam-4712	81	42	̸=	̸=	PROPN
ejpam-4712	81	43	(	(	PUNCT
ejpam-4712	81	44	b⊗a	b⊗a	PROPN
ejpam-4712	81	45	)	)	PUNCT
ejpam-4712	81	46	,	,	PUNCT
ejpam-4712	81	47	and	and	CCONJ
ejpam-4712	81	48	transpose	transpose	NOUN
ejpam-4712	81	49	distributes	distribute	VERB
ejpam-4712	81	50	over	over	ADP
ejpam-4712	81	51	the	the	DET
ejpam-4712	81	52	kronecker	kronecker	NOUN
ejpam-4712	81	53	product	product	NOUN
ejpam-4712	81	54	(	(	PUNCT
ejpam-4712	81	55	do	do	AUX
ejpam-4712	81	56	not	not	PART
ejpam-4712	81	57	reverse	reverse	VERB
ejpam-4712	81	58	order	order	NOUN
ejpam-4712	81	59	)	)	PUNCT
ejpam-4712	81	60	(	(	PUNCT
ejpam-4712	81	61	a⊗b)t	a⊗b)t	PROPN
ejpam-4712	81	62	=	=	PUNCT
ejpam-4712	81	63	at	at	ADP
ejpam-4712	81	64	⊗bt	⊗bt	NUM
ejpam-4712	81	65	.	.	PUNCT
ejpam-4712	82	1	let	let	VERB
ejpam-4712	82	2	a	a	DET
ejpam-4712	82	3	∈	∈	PROPN
ejpam-4712	82	4	mn×n(s	mn×n(s	NOUN
ejpam-4712	82	5	)	)	PUNCT
ejpam-4712	82	6	and	and	CCONJ
ejpam-4712	82	7	b	b	PROPN
ejpam-4712	82	8	∈	∈	PROPN
ejpam-4712	82	9	mm×m(s	mm×m(s	NOUN
ejpam-4712	82	10	)	)	PUNCT
ejpam-4712	82	11	,	,	PUNCT
ejpam-4712	82	12	and	and	CCONJ
ejpam-4712	82	13	λi	λi	X
ejpam-4712	82	14	with	with	ADP
ejpam-4712	82	15	i	i	NOUN
ejpam-4712	82	16	=	=	SYM
ejpam-4712	82	17	1	1	NUM
ejpam-4712	82	18	,	,	PUNCT
ejpam-4712	82	19	2	2	NUM
ejpam-4712	82	20	,	,	PUNCT
ejpam-4712	82	21	...	...	PUNCT
ejpam-4712	82	22	,	,	PUNCT
ejpam-4712	82	23	n	n	CCONJ
ejpam-4712	82	24	be	be	VERB
ejpam-4712	82	25	the	the	DET
ejpam-4712	82	26	eigenvalues	eigenvalue	NOUN
ejpam-4712	82	27	of	of	ADP
ejpam-4712	82	28	a	a	PRON
ejpam-4712	82	29	and	and	CCONJ
ejpam-4712	82	30	µi	µi	ADV
ejpam-4712	82	31	with	with	ADP
ejpam-4712	82	32	i	i	PROPN
ejpam-4712	82	33	=	=	SYM
ejpam-4712	82	34	1	1	NUM
ejpam-4712	82	35	,	,	PUNCT
ejpam-4712	82	36	2	2	NUM
ejpam-4712	82	37	,	,	PUNCT
ejpam-4712	82	38	...	...	PUNCT
ejpam-4712	82	39	,	,	PUNCT
ejpam-4712	82	40	m	m	VERB
ejpam-4712	82	41	be	be	VERB
ejpam-4712	82	42	the	the	DET
ejpam-4712	82	43	eigenvalues	eigenvalue	NOUN
ejpam-4712	82	44	of	of	ADP
ejpam-4712	82	45	b.	b.	PROPN
ejpam-4712	82	46	then	then	ADV
ejpam-4712	82	47	we	we	PRON
ejpam-4712	82	48	have	have	VERB
ejpam-4712	82	49	the	the	DET
ejpam-4712	82	50	following	follow	VERB
ejpam-4712	82	51	properties	property	NOUN
ejpam-4712	82	52	:	:	PUNCT
ejpam-4712	82	53	the	the	DET
ejpam-4712	82	54	eigenvalues	eigenvalue	NOUN
ejpam-4712	82	55	of	of	ADP
ejpam-4712	82	56	a	a	DET
ejpam-4712	82	57	⊗	⊗	PROPN
ejpam-4712	82	58	b	b	PROPN
ejpam-4712	82	59	are	be	AUX
ejpam-4712	82	60	the	the	DET
ejpam-4712	82	61	mn	mn	PROPN
ejpam-4712	82	62	numbers	number	NOUN
ejpam-4712	82	63	λiµj	λiµj	NOUN
ejpam-4712	82	64	,	,	PUNCT
ejpam-4712	82	65	and	and	CCONJ
ejpam-4712	82	66	the	the	DET
ejpam-4712	82	67	eigenvalues	eigenvalue	NOUN
ejpam-4712	82	68	of	of	ADP
ejpam-4712	82	69	a⊕b	a⊕b	NOUN
ejpam-4712	82	70	=	=	SYM
ejpam-4712	82	71	(	(	PUNCT
ejpam-4712	82	72	a⊗im)+(in⊗b	a⊗im)+(in⊗b	ADV
ejpam-4712	82	73	)	)	PUNCT
ejpam-4712	82	74	are	be	AUX
ejpam-4712	82	75	the	the	DET
ejpam-4712	82	76	mn	mn	PROPN
ejpam-4712	82	77	numbers	number	NOUN
ejpam-4712	82	78	λi+µj	λi+µj	PROPN
ejpam-4712	82	79	,	,	PUNCT
ejpam-4712	82	80	with	with	ADP
ejpam-4712	82	81	i	i	PROPN
ejpam-4712	82	82	=	=	SYM
ejpam-4712	82	83	1	1	NUM
ejpam-4712	82	84	,	,	PUNCT
ejpam-4712	82	85	2	2	NUM
ejpam-4712	82	86	,	,	PUNCT
ejpam-4712	82	87	...	...	PUNCT
ejpam-4712	82	88	,	,	PUNCT
ejpam-4712	82	89	n	n	CCONJ
ejpam-4712	82	90	,	,	PUNCT
ejpam-4712	82	91	j	j	PROPN
ejpam-4712	82	92	=	=	SYM
ejpam-4712	82	93	1	1	NUM
ejpam-4712	82	94	,	,	PUNCT
ejpam-4712	82	95	2	2	NUM
ejpam-4712	82	96	,	,	PUNCT
ejpam-4712	82	97	...	...	PUNCT
ejpam-4712	82	98	,	,	PUNCT
ejpam-4712	82	99	m.	m.	NOUN
ejpam-4712	82	100	2.3	2.3	NUM
ejpam-4712	82	101	.	.	PUNCT
ejpam-4712	83	1	discrete	discrete	ADJ
ejpam-4712	83	2	lyapunov	lyapunov	NOUN
ejpam-4712	83	3	equation	equation	NOUN
ejpam-4712	83	4	the	the	DET
ejpam-4712	83	5	linear	linear	ADJ
ejpam-4712	83	6	system	system	NOUN
ejpam-4712	83	7	is	be	AUX
ejpam-4712	83	8	closely	closely	ADV
ejpam-4712	83	9	related	relate	VERB
ejpam-4712	83	10	to	to	ADP
ejpam-4712	83	11	stability	stability	NOUN
ejpam-4712	83	12	,	,	PUNCT
ejpam-4712	83	13	which	which	PRON
ejpam-4712	83	14	can	can	AUX
ejpam-4712	83	15	be	be	AUX
ejpam-4712	83	16	observed	observe	VERB
ejpam-4712	83	17	using	use	VERB
ejpam-4712	83	18	the	the	DET
ejpam-4712	83	19	eigenvalue	eigenvalue	ADJ
ejpam-4712	83	20	criterion	criterion	NOUN
ejpam-4712	83	21	of	of	ADP
ejpam-4712	83	22	matrix	matrix	NOUN
ejpam-4712	83	23	a.	a.	NOUN
ejpam-4712	83	24	furthermore	furthermore	ADV
ejpam-4712	83	25	,	,	PUNCT
ejpam-4712	83	26	the	the	DET
ejpam-4712	83	27	stability	stability	NOUN
ejpam-4712	83	28	of	of	ADP
ejpam-4712	83	29	the	the	DET
ejpam-4712	83	30	linear	linear	ADJ
ejpam-4712	83	31	system	system	NOUN
ejpam-4712	83	32	is	be	AUX
ejpam-4712	83	33	closely	closely	ADV
ejpam-4712	83	34	related	relate	VERB
ejpam-4712	83	35	to	to	ADP
ejpam-4712	83	36	the	the	DET
ejpam-4712	83	37	existence	existence	NOUN
ejpam-4712	83	38	of	of	ADP
ejpam-4712	83	39	a	a	DET
ejpam-4712	83	40	solution	solution	NOUN
ejpam-4712	83	41	to	to	ADP
ejpam-4712	83	42	the	the	DET
ejpam-4712	83	43	lyapunov	lyapunov	ADJ
ejpam-4712	83	44	equation	equation	NOUN
ejpam-4712	83	45	(	(	PUNCT
ejpam-4712	84	1	[	[	X
ejpam-4712	84	2	6	6	NUM
ejpam-4712	84	3	]	]	NUM
ejpam-4712	84	4	)	)	PUNCT
ejpam-4712	84	5	.	.	PUNCT
ejpam-4712	85	1	therefore	therefore	ADV
ejpam-4712	85	2	,	,	PUNCT
ejpam-4712	85	3	this	this	DET
ejpam-4712	85	4	method	method	NOUN
ejpam-4712	85	5	can	can	AUX
ejpam-4712	85	6	determine	determine	VERB
ejpam-4712	85	7	the	the	DET
ejpam-4712	85	8	system	system	NOUN
ejpam-4712	85	9	’s	’s	PART
ejpam-4712	85	10	stability	stability	NOUN
ejpam-4712	85	11	without	without	ADP
ejpam-4712	85	12	knowing	know	VERB
ejpam-4712	85	13	the	the	DET
ejpam-4712	85	14	system	system	NOUN
ejpam-4712	85	15	’s	’s	PART
ejpam-4712	85	16	solution	solution	NOUN
ejpam-4712	85	17	.	.	PUNCT
ejpam-4712	86	1	lyapunov	lyapunov	PROPN
ejpam-4712	86	2	’s	’s	PART
ejpam-4712	86	3	equation	equation	NOUN
ejpam-4712	86	4	for	for	ADP
ejpam-4712	86	5	a	a	DET
ejpam-4712	86	6	linear	linear	ADJ
ejpam-4712	86	7	system	system	NOUN
ejpam-4712	86	8	over	over	ADP
ejpam-4712	86	9	a	a	DET
ejpam-4712	86	10	field	field	NOUN
ejpam-4712	86	11	was	be	AUX
ejpam-4712	86	12	given	give	VERB
ejpam-4712	86	13	by	by	ADP
ejpam-4712	86	14	zhou	zhou	PROPN
ejpam-4712	86	15	(	(	PUNCT
ejpam-4712	86	16	[	[	X
ejpam-4712	86	17	7	7	NUM
ejpam-4712	86	18	]	]	NUM
ejpam-4712	86	19	)	)	PUNCT
ejpam-4712	86	20	.	.	PUNCT
ejpam-4712	87	1	in	in	ADP
ejpam-4712	87	2	this	this	DET
ejpam-4712	87	3	study	study	NOUN
ejpam-4712	87	4	,	,	PUNCT
ejpam-4712	87	5	the	the	DET
ejpam-4712	87	6	lyapunov	lyapunov	ADJ
ejpam-4712	87	7	equation	equation	NOUN
ejpam-4712	87	8	is	be	AUX
ejpam-4712	87	9	defined	define	VERB
ejpam-4712	87	10	for	for	ADP
ejpam-4712	87	11	a	a	DET
ejpam-4712	87	12	discrete	discrete	ADJ
ejpam-4712	87	13	linear	linear	NOUN
ejpam-4712	87	14	system	system	NOUN
ejpam-4712	87	15	on	on	ADP
ejpam-4712	87	16	a	a	DET
ejpam-4712	87	17	semiring	semiring	NOUN
ejpam-4712	87	18	,	,	PUNCT
ejpam-4712	87	19	adopting	adopt	VERB
ejpam-4712	87	20	the	the	DET
ejpam-4712	87	21	meaning	meaning	NOUN
ejpam-4712	87	22	of	of	ADP
ejpam-4712	87	23	the	the	DET
ejpam-4712	87	24	lyapunov	lyapunov	ADJ
ejpam-4712	87	25	equation	equation	NOUN
ejpam-4712	87	26	for	for	ADP
ejpam-4712	87	27	a	a	DET
ejpam-4712	87	28	linear	linear	ADJ
ejpam-4712	87	29	system	system	NOUN
ejpam-4712	87	30	on	on	ADP
ejpam-4712	87	31	a	a	DET
ejpam-4712	87	32	plane	plane	NOUN
ejpam-4712	87	33	.	.	PUNCT
ejpam-4712	88	1	the	the	DET
ejpam-4712	88	2	discrete	discrete	ADJ
ejpam-4712	88	3	lyapunov	lyapunov	NOUN
ejpam-4712	88	4	equation	equation	NOUN
ejpam-4712	88	5	for	for	ADP
ejpam-4712	88	6	the	the	DET
ejpam-4712	88	7	linear	linear	ADJ
ejpam-4712	88	8	system	system	NOUN
ejpam-4712	88	9	over	over	ADP
ejpam-4712	88	10	semiring	semiring	NOUN
ejpam-4712	88	11	is	be	AUX
ejpam-4712	88	12	defined	define	VERB
ejpam-4712	88	13	as	as	SCONJ
ejpam-4712	88	14	follows	follow	VERB
ejpam-4712	88	15	(	(	PUNCT
ejpam-4712	88	16	[	[	X
ejpam-4712	88	17	10	10	NUM
ejpam-4712	88	18	]	]	PUNCT
ejpam-4712	88	19	,	,	PUNCT
ejpam-4712	89	1	[	[	X
ejpam-4712	89	2	5	5	NUM
ejpam-4712	89	3	]	]	NUM
ejpam-4712	89	4	)	)	PUNCT
ejpam-4712	89	5	.	.	PUNCT
ejpam-4712	90	1	definition	definition	NOUN
ejpam-4712	90	2	4	4	NUM
ejpam-4712	90	3	.	.	PUNCT
ejpam-4712	91	1	given	give	VERB
ejpam-4712	91	2	a	a	DET
ejpam-4712	91	3	matrix	matrix	NOUN
ejpam-4712	91	4	a	a	DET
ejpam-4712	91	5	,	,	PUNCT
ejpam-4712	91	6	x	x	NOUN
ejpam-4712	91	7	,	,	PUNCT
ejpam-4712	91	8	q	q	PROPN
ejpam-4712	91	9	∈	∈	PROPN
ejpam-4712	91	10	mn(s	mn(	NOUN
ejpam-4712	91	11	)	)	PUNCT
ejpam-4712	91	12	.	.	PUNCT
ejpam-4712	92	1	the	the	DET
ejpam-4712	92	2	lyapunov	lyapunov	ADJ
ejpam-4712	92	3	equation	equation	NOUN
ejpam-4712	92	4	for	for	ADP
ejpam-4712	92	5	a	a	DET
ejpam-4712	92	6	linear	linear	ADJ
ejpam-4712	92	7	system	system	NOUN
ejpam-4712	92	8	over	over	ADP
ejpam-4712	92	9	a	a	DET
ejpam-4712	92	10	semiring	semiring	NOUN
ejpam-4712	92	11	is	be	AUX
ejpam-4712	92	12	defined	define	VERB
ejpam-4712	92	13	as	as	ADP
ejpam-4712	92	14	axat	axat	PROPN
ejpam-4712	92	15	−x	−x	NOUN
ejpam-4712	93	1	+	+	NOUN
ejpam-4712	93	2	q	q	NOUN
ejpam-4712	93	3	=	=	ADJ
ejpam-4712	93	4	0	0	NUM
ejpam-4712	93	5	.	.	PUNCT
ejpam-4712	94	1	for	for	ADP
ejpam-4712	94	2	linear	linear	PROPN
ejpam-4712	94	3	systems	system	NOUN
ejpam-4712	94	4	over	over	ADP
ejpam-4712	94	5	the	the	DET
ejpam-4712	94	6	field	field	NOUN
ejpam-4712	94	7	,	,	PUNCT
ejpam-4712	94	8	the	the	DET
ejpam-4712	94	9	existence	existence	NOUN
ejpam-4712	94	10	of	of	ADP
ejpam-4712	94	11	solutions	solution	NOUN
ejpam-4712	94	12	to	to	ADP
ejpam-4712	94	13	the	the	DET
ejpam-4712	94	14	lyapunov	lyapunov	ADJ
ejpam-4712	94	15	equations	equation	NOUN
ejpam-4712	94	16	is	be	AUX
ejpam-4712	94	17	associated	associate	VERB
ejpam-4712	94	18	with	with	ADP
ejpam-4712	94	19	asymptotic	asymptotic	ADJ
ejpam-4712	94	20	stability	stability	NOUN
ejpam-4712	94	21	.	.	PUNCT
ejpam-4712	95	1	the	the	DET
ejpam-4712	95	2	system	system	NOUN
ejpam-4712	95	3	is	be	AUX
ejpam-4712	95	4	asymptotically	asymptotically	ADV
ejpam-4712	95	5	stable	stable	ADJ
ejpam-4712	95	6	if	if	SCONJ
ejpam-4712	95	7	a	a	DET
ejpam-4712	95	8	solution	solution	NOUN
ejpam-4712	95	9	to	to	ADP
ejpam-4712	95	10	the	the	DET
ejpam-4712	95	11	lyapunov	lyapunov	ADJ
ejpam-4712	95	12	equation	equation	NOUN
ejpam-4712	95	13	exists	exist	VERB
ejpam-4712	95	14	.	.	PUNCT
ejpam-4712	96	1	on	on	ADP
ejpam-4712	96	2	the	the	DET
ejpam-4712	96	3	other	other	ADJ
ejpam-4712	96	4	hand	hand	NOUN
ejpam-4712	96	5	,	,	PUNCT
ejpam-4712	96	6	if	if	SCONJ
ejpam-4712	96	7	the	the	DET
ejpam-4712	96	8	system	system	NOUN
ejpam-4712	96	9	is	be	AUX
ejpam-4712	96	10	asymptotically	asymptotically	ADV
ejpam-4712	96	11	stable	stable	ADJ
ejpam-4712	96	12	,	,	PUNCT
ejpam-4712	96	13	a	a	DET
ejpam-4712	96	14	solution	solution	NOUN
ejpam-4712	96	15	to	to	ADP
ejpam-4712	96	16	the	the	DET
ejpam-4712	96	17	lyapunov	lyapunov	ADJ
ejpam-4712	96	18	equation	equation	NOUN
ejpam-4712	96	19	exists	exist	VERB
ejpam-4712	96	20	.	.	PUNCT
ejpam-4712	97	1	(	(	PUNCT
ejpam-4712	97	2	[	[	X
ejpam-4712	97	3	6	6	NUM
ejpam-4712	97	4	]	]	NUM
ejpam-4712	97	5	)	)	PUNCT
ejpam-4712	97	6	.	.	PUNCT
ejpam-4712	98	1	3	3	X
ejpam-4712	98	2	.	.	NOUN
ejpam-4712	98	3	results	result	NOUN
ejpam-4712	98	4	and	and	CCONJ
ejpam-4712	98	5	discussion	discussion	VERB
ejpam-4712	98	6	the	the	DET
ejpam-4712	98	7	problem	problem	NOUN
ejpam-4712	98	8	of	of	ADP
ejpam-4712	98	9	discrete	discrete	ADJ
ejpam-4712	98	10	lyapunov	lyapunov	ADJ
ejpam-4712	98	11	equations	equation	NOUN
ejpam-4712	98	12	over	over	ADP
ejpam-4712	98	13	semirings	semiring	NOUN
ejpam-4712	98	14	is	be	AUX
ejpam-4712	98	15	limited	limit	VERB
ejpam-4712	98	16	to	to	ADP
ejpam-4712	98	17	orthogonal	orthogonal	ADJ
ejpam-4712	98	18	matrices	matrix	NOUN
ejpam-4712	98	19	.	.	PUNCT
ejpam-4712	99	1	these	these	PRON
ejpam-4712	99	2	are	be	AUX
ejpam-4712	99	3	because	because	SCONJ
ejpam-4712	99	4	of	of	ADP
ejpam-4712	99	5	the	the	DET
ejpam-4712	99	6	limited	limited	ADJ
ejpam-4712	99	7	nature	nature	NOUN
ejpam-4712	99	8	of	of	ADP
ejpam-4712	99	9	the	the	DET
ejpam-4712	99	10	semiring	semiring	NOUN
ejpam-4712	99	11	.	.	PUNCT
ejpam-4712	100	1	theorem	theorem	NOUN
ejpam-4712	100	2	2	2	NUM
ejpam-4712	100	3	.	.	PUNCT
ejpam-4712	101	1	let	let	VERB
ejpam-4712	101	2	s	s	PRON
ejpam-4712	101	3	is	be	AUX
ejpam-4712	101	4	semiring	semire	VERB
ejpam-4712	101	5	.	.	PUNCT
ejpam-4712	102	1	then	then	ADV
ejpam-4712	102	2	for	for	ADP
ejpam-4712	102	3	any	any	DET
ejpam-4712	102	4	matrices	matrix	NOUN
ejpam-4712	102	5	a	a	DET
ejpam-4712	102	6	∈	∈	PROPN
ejpam-4712	102	7	mk×m(s	mk×m(s	NOUN
ejpam-4712	102	8	)	)	PUNCT
ejpam-4712	102	9	,	,	PUNCT
ejpam-4712	102	10	b	b	X
ejpam-4712	102	11	∈	∈	PROPN
ejpam-4712	102	12	mn×l(s	mn×l(s	NOUN
ejpam-4712	102	13	)	)	PUNCT
ejpam-4712	102	14	,	,	PUNCT
ejpam-4712	102	15	and	and	CCONJ
ejpam-4712	102	16	x	x	PUNCT
ejpam-4712	102	17	∈	∈	NOUN
ejpam-4712	102	18	mm×n(s	mm×n(s	NOUN
ejpam-4712	102	19	)	)	PUNCT
ejpam-4712	102	20	,	,	PUNCT
ejpam-4712	102	21	we	we	PRON
ejpam-4712	102	22	have	have	VERB
ejpam-4712	102	23	vec(axb	vec(axb	NOUN
ejpam-4712	102	24	)	)	PUNCT
ejpam-4712	103	1	=	=	PRON
ejpam-4712	103	2	(	(	PUNCT
ejpam-4712	103	3	bt	bt	PROPN
ejpam-4712	103	4	⊗a)vec(x	⊗a)vec(x	PROPN
ejpam-4712	103	5	)	)	PUNCT
ejpam-4712	103	6	.	.	PUNCT
ejpam-4712	104	1	g.	g.	PROPN
ejpam-4712	104	2	ariyanti	ariyanti	PROPN
ejpam-4712	104	3	,	,	PUNCT
ejpam-4712	104	4	a.e.r.m	a.e.r.m	PROPN
ejpam-4712	104	5	.	.	PUNCT
ejpam-4712	104	6	sari	sari	PROPN
ejpam-4712	104	7	/	/	SYM
ejpam-4712	104	8	eur	eur	PROPN
ejpam-4712	104	9	.	.	PUNCT
ejpam-4712	105	1	j.	j.	PROPN
ejpam-4712	105	2	pure	pure	PROPN
ejpam-4712	105	3	appl	appl	PROPN
ejpam-4712	105	4	.	.	PROPN
ejpam-4712	105	5	math	math	PROPN
ejpam-4712	105	6	,	,	PUNCT
ejpam-4712	105	7	16	16	NUM
ejpam-4712	105	8	(	(	PUNCT
ejpam-4712	105	9	2	2	NUM
ejpam-4712	105	10	)	)	PUNCT
ejpam-4712	105	11	(	(	PUNCT
ejpam-4712	105	12	2023	2023	NUM
ejpam-4712	105	13	)	)	PUNCT
ejpam-4712	105	14	,	,	PUNCT
ejpam-4712	105	15	784	784	NUM
ejpam-4712	105	16	-	-	SYM
ejpam-4712	105	17	790	790	NUM
ejpam-4712	105	18	788	788	NUM
ejpam-4712	105	19	proof	proof	NOUN
ejpam-4712	105	20	.	.	PUNCT
ejpam-4712	106	1	we	we	PRON
ejpam-4712	106	2	have	have	VERB
ejpam-4712	106	3	(	(	PUNCT
ejpam-4712	106	4	axb).k	axb).k	VERB
ejpam-4712	106	5	=	=	SYM
ejpam-4712	106	6	∑	∑	PUNCT
ejpam-4712	106	7	j	j	PROPN
ejpam-4712	106	8	bjkax.j	bjkax.j	PROPN
ejpam-4712	106	9	=	=	PRON
ejpam-4712	106	10	(	(	PUNCT
ejpam-4712	106	11	b1ka	b1ka	X
ejpam-4712	106	12	b2ka	b2ka	X
ejpam-4712	106	13	.	.	PUNCT
ejpam-4712	106	14	.	.	PUNCT
ejpam-4712	106	15	.	.	PUNCT
ejpam-4712	107	1	bnka	bnka	PROPN
ejpam-4712	107	2	)	)	PUNCT
ejpam-4712	108	1	and	and	PROPN
ejpam-4712	108	2	x.1	x.1	PROPN
ejpam-4712	108	3	x.2	x.2	PUNCT
ejpam-4712	108	4	...	...	PUNCT
ejpam-4712	109	1	x.n	x.n	PRON
ejpam-4712	109	2			ADV
ejpam-4712	109	3	=	=	PUNCT
ejpam-4712	110	1	[	[	PUNCT
ejpam-4712	110	2	bt	bt	PROPN
ejpam-4712	110	3	.k	.k	PROPN
ejpam-4712	110	4	⊗a	⊗a	PROPN
ejpam-4712	110	5	]	]	PUNCT
ejpam-4712	111	1	vec(x	vec(x	X
ejpam-4712	111	2	)	)	PUNCT
ejpam-4712	111	3	=	=	PUNCT
ejpam-4712	112	1	[	[	PUNCT
ejpam-4712	112	2	(	(	PUNCT
ejpam-4712	112	3	bt	bt	INTJ
ejpam-4712	112	4	)	)	PUNCT
ejpam-4712	112	5	tk	tk	PROPN
ejpam-4712	112	6	.	.	PROPN
ejpam-4712	112	7	⊗a	⊗a	PROPN
ejpam-4712	112	8	]	]	PUNCT
ejpam-4712	112	9	vec(x	vec(x	PROPN
ejpam-4712	112	10	)	)	PUNCT
ejpam-4712	112	11	.	.	PUNCT
ejpam-4712	113	1	furthermore	furthermore	ADV
ejpam-4712	113	2	,	,	PUNCT
ejpam-4712	113	3	we	we	PRON
ejpam-4712	113	4	conclude	conclude	VERB
ejpam-4712	113	5	vec(axb	vec(axb	NOUN
ejpam-4712	113	6	)	)	PUNCT
ejpam-4712	113	7	=	=	PUNCT
ejpam-4712	114	1	[	[	X
ejpam-4712	114	2	bt	bt	X
ejpam-4712	114	3	⊗	⊗	PROPN
ejpam-4712	114	4	a]vec(x	a]vec(x	PROPN
ejpam-4712	114	5	)	)	PUNCT
ejpam-4712	114	6	since	since	SCONJ
ejpam-4712	114	7	the	the	DET
ejpam-4712	114	8	transpose	transpose	NOUN
ejpam-4712	114	9	of	of	ADP
ejpam-4712	114	10	the	the	DET
ejpam-4712	114	11	kth	kth	PROPN
ejpam-4712	114	12	column	column	NOUN
ejpam-4712	114	13	of	of	ADP
ejpam-4712	114	14	b	b	PROPN
ejpam-4712	114	15	is	be	AUX
ejpam-4712	114	16	the	the	DET
ejpam-4712	114	17	kth	kth	PROPN
ejpam-4712	114	18	row	row	NOUN
ejpam-4712	114	19	of	of	ADP
ejpam-4712	114	20	bt	bt	NOUN
ejpam-4712	114	21	.	.	PUNCT
ejpam-4712	115	1	in	in	ADP
ejpam-4712	115	2	the	the	DET
ejpam-4712	115	3	following	following	NOUN
ejpam-4712	115	4	,	,	PUNCT
ejpam-4712	115	5	another	another	DET
ejpam-4712	115	6	property	property	NOUN
ejpam-4712	115	7	of	of	ADP
ejpam-4712	115	8	the	the	DET
ejpam-4712	115	9	kronecker	kronecker	NOUN
ejpam-4712	115	10	product	product	NOUN
ejpam-4712	115	11	for	for	ADP
ejpam-4712	115	12	matrix	matrix	NOUN
ejpam-4712	115	13	over	over	ADP
ejpam-4712	115	14	a	a	DET
ejpam-4712	115	15	semiring	semiring	NOUN
ejpam-4712	115	16	is	be	AUX
ejpam-4712	115	17	given	give	VERB
ejpam-4712	115	18	.	.	PUNCT
ejpam-4712	116	1	theorem	theorem	NOUN
ejpam-4712	116	2	3	3	X
ejpam-4712	116	3	.	.	PUNCT
ejpam-4712	117	1	let	let	VERB
ejpam-4712	117	2	a	a	DET
ejpam-4712	117	3	∈	∈	PROPN
ejpam-4712	117	4	mm×m(s	mm×m(s	NOUN
ejpam-4712	117	5	)	)	PUNCT
ejpam-4712	117	6	,	,	PUNCT
ejpam-4712	117	7	b	b	PROPN
ejpam-4712	117	8	∈	∈	PROPN
ejpam-4712	117	9	mn×n(s	mn×n(s	NOUN
ejpam-4712	117	10	)	)	PUNCT
ejpam-4712	117	11	,	,	PUNCT
ejpam-4712	117	12	and	and	CCONJ
ejpam-4712	117	13	x	x	PUNCT
ejpam-4712	117	14	∈	∈	NOUN
ejpam-4712	117	15	mm×n(s	mm×n(s	NOUN
ejpam-4712	117	16	)	)	PUNCT
ejpam-4712	117	17	,	,	PUNCT
ejpam-4712	117	18	we	we	PRON
ejpam-4712	117	19	have	have	VERB
ejpam-4712	117	20	vec(ax	vec(ax	NOUN
ejpam-4712	117	21	+	+	CCONJ
ejpam-4712	117	22	xb	xb	X
ejpam-4712	117	23	)	)	PUNCT
ejpam-4712	118	1	=	=	PRON
ejpam-4712	118	2	(	(	PUNCT
ejpam-4712	118	3	bt	bt	PROPN
ejpam-4712	118	4	⊕a)vec(x	⊕a)vec(x	NOUN
ejpam-4712	118	5	)	)	PUNCT
ejpam-4712	118	6	.	.	PUNCT
ejpam-4712	119	1	proof	proof	NOUN
ejpam-4712	119	2	.	.	PUNCT
ejpam-4712	120	1	from	from	ADP
ejpam-4712	120	2	the	the	DET
ejpam-4712	120	3	definition	definition	NOUN
ejpam-4712	120	4	of	of	ADP
ejpam-4712	120	5	kronecker	kronecker	NOUN
ejpam-4712	120	6	sum	sum	NOUN
ejpam-4712	120	7	,	,	PUNCT
ejpam-4712	120	8	we	we	PRON
ejpam-4712	120	9	have	have	VERB
ejpam-4712	120	10	(	(	PUNCT
ejpam-4712	120	11	bt	bt	NOUN
ejpam-4712	120	12	⊕a)vec(x	⊕a)vec(x	NOUN
ejpam-4712	120	13	)	)	PUNCT
ejpam-4712	121	1	=	=	PRON
ejpam-4712	122	1	(	(	PUNCT
ejpam-4712	122	2	bt	bt	NOUN
ejpam-4712	122	3	⊗	⊗	PROPN
ejpam-4712	123	1	i	i	PRON
ejpam-4712	123	2	m	m	VERB
ejpam-4712	123	3	+	+	X
ejpam-4712	123	4	in	in	ADP
ejpam-4712	123	5	⊗a)vec(x	⊗a)vec(x	NOUN
ejpam-4712	123	6	)	)	PUNCT
ejpam-4712	124	1	=	=	PRON
ejpam-4712	125	1	(	(	PUNCT
ejpam-4712	125	2	bt	bt	PROPN
ejpam-4712	125	3	⊗	⊗	PROPN
ejpam-4712	125	4	im)vec(x	im)vec(x	PROPN
ejpam-4712	125	5	)	)	PUNCT
ejpam-4712	126	1	+	+	CCONJ
ejpam-4712	126	2	(	(	PUNCT
ejpam-4712	126	3	in	in	ADP
ejpam-4712	126	4	⊗a)vec(x	⊗a)vec(x	PROPN
ejpam-4712	126	5	)	)	PUNCT
ejpam-4712	126	6	.	.	PUNCT
ejpam-4712	127	1	according	accord	VERB
ejpam-4712	127	2	to	to	ADP
ejpam-4712	127	3	theorem	theorem	NOUN
ejpam-4712	127	4	2	2	NUM
ejpam-4712	127	5	,	,	PUNCT
ejpam-4712	127	6	we	we	PRON
ejpam-4712	127	7	have	have	VERB
ejpam-4712	127	8	(	(	PUNCT
ejpam-4712	127	9	bt	bt	NOUN
ejpam-4712	127	10	⊕a)vec(x	⊕a)vec(x	NOUN
ejpam-4712	127	11	)	)	PUNCT
ejpam-4712	127	12	=	=	SYM
ejpam-4712	127	13	vec(imxb	vec(imxb	X
ejpam-4712	127	14	)	)	PUNCT
ejpam-4712	127	15	+	+	NUM
ejpam-4712	127	16	vec(axitn	vec(axitn	NOUN
ejpam-4712	127	17	)	)	PUNCT
ejpam-4712	127	18	=	=	SYM
ejpam-4712	127	19	vec(xb	vec(xb	X
ejpam-4712	127	20	)	)	PUNCT
ejpam-4712	127	21	+	+	NUM
ejpam-4712	127	22	vec(ax	vec(ax	NOUN
ejpam-4712	127	23	)	)	PUNCT
ejpam-4712	127	24	.	.	PUNCT
ejpam-4712	128	1	example	example	NOUN
ejpam-4712	129	1	2	2	NUM
ejpam-4712	129	2	.	.	PUNCT
ejpam-4712	129	3	as	as	SCONJ
ejpam-4712	129	4	is	be	AUX
ejpam-4712	129	5	well	well	ADV
ejpam-4712	129	6	known	know	VERB
ejpam-4712	129	7	,	,	PUNCT
ejpam-4712	129	8	the	the	DET
ejpam-4712	129	9	max	max	PROPN
ejpam-4712	129	10	-	-	PUNCT
ejpam-4712	129	11	plus	plus	NOUN
ejpam-4712	129	12	algebra	algebra	NOUN
ejpam-4712	129	13	rϵ	rϵ	VERB
ejpam-4712	129	14	is	be	AUX
ejpam-4712	129	15	semiring	semire	VERB
ejpam-4712	129	16	.	.	PUNCT
ejpam-4712	130	1	let	let	VERB
ejpam-4712	130	2	a	a	DET
ejpam-4712	130	3	,	,	PUNCT
ejpam-4712	130	4	b	b	NOUN
ejpam-4712	130	5	,	,	PUNCT
ejpam-4712	130	6	and	and	CCONJ
ejpam-4712	131	1	x	x	NOUN
ejpam-4712	131	2	matrices	matrix	NOUN
ejpam-4712	131	3	over	over	ADP
ejpam-4712	131	4	max	max	PROPN
ejpam-4712	131	5	-	-	PUNCT
ejpam-4712	131	6	plus	plus	CCONJ
ejpam-4712	131	7	algebra	algebra	NOUN
ejpam-4712	131	8	,	,	PUNCT
ejpam-4712	131	9	with	with	ADP
ejpam-4712	131	10	a	a	PRON
ejpam-4712	131	11	=	=	X
ejpam-4712	131	12	[	[	PUNCT
ejpam-4712	131	13	3	3	NUM
ejpam-4712	131	14	ϵ	ϵ	SYM
ejpam-4712	131	15	1	1	NUM
ejpam-4712	131	16	]	]	PUNCT
ejpam-4712	131	17	,	,	PUNCT
ejpam-4712	131	18	b	b	X
ejpam-4712	131	19	=	=	PUNCT
ejpam-4712	131	20	[	[	PUNCT
ejpam-4712	131	21	5	5	NUM
ejpam-4712	131	22	0	0	NUM
ejpam-4712	131	23	]	]	PUNCT
ejpam-4712	131	24	,	,	PUNCT
ejpam-4712	131	25	and	and	CCONJ
ejpam-4712	131	26	x	x	X
ejpam-4712	131	27	=	=	SYM
ejpam-4712	131	28			NOUN
ejpam-4712	131	29	2	2	NUM
ejpam-4712	131	30	5	5	NUM
ejpam-4712	131	31	1	1	NUM
ejpam-4712	131	32	2	2	NUM
ejpam-4712	131	33	ϵ	ϵ	PRON
ejpam-4712	131	34	−3	−3	PROPN
ejpam-4712	131	35	.	.	NOUN
ejpam-4712	131	36	based	base	VERB
ejpam-4712	131	37	on	on	ADP
ejpam-4712	131	38	binary	binary	ADJ
ejpam-4712	131	39	operations	operation	NOUN
ejpam-4712	131	40	on	on	ADP
ejpam-4712	131	41	max	max	PROPN
ejpam-4712	131	42	-	-	PUNCT
ejpam-4712	131	43	plus	plus	CCONJ
ejpam-4712	131	44	algebra	algebra	NOUN
ejpam-4712	131	45	,	,	PUNCT
ejpam-4712	131	46	as	as	SCONJ
ejpam-4712	131	47	given	give	VERB
ejpam-4712	131	48	in	in	ADP
ejpam-4712	131	49	ariyanti	ariyanti	NOUN
ejpam-4712	131	50	(	(	PUNCT
ejpam-4712	131	51	[	[	X
ejpam-4712	131	52	3	3	NUM
ejpam-4712	131	53	]	]	NUM
ejpam-4712	131	54	)	)	PUNCT
ejpam-4712	131	55	,	,	PUNCT
ejpam-4712	131	56	we	we	PRON
ejpam-4712	131	57	have	have	VERB
ejpam-4712	131	58	axb	axb	PROPN
ejpam-4712	131	59	=	=	PUNCT
ejpam-4712	132	1	[	[	PUNCT
ejpam-4712	132	2	3	3	NUM
ejpam-4712	132	3	ϵ	ϵ	SYM
ejpam-4712	132	4	1	1	NUM
ejpam-4712	132	5	]	]	PUNCT
ejpam-4712	132	6			NOUN
ejpam-4712	132	7	2	2	NUM
ejpam-4712	132	8	5	5	NUM
ejpam-4712	132	9	1	1	NUM
ejpam-4712	132	10	2	2	NUM
ejpam-4712	132	11	ϵ	ϵ	DET
ejpam-4712	132	12	−3	−3	NOUN
ejpam-4712	132	13			NOUN
ejpam-4712	132	14	[	[	PUNCT
ejpam-4712	132	15	5	5	NUM
ejpam-4712	132	16	0	0	NUM
ejpam-4712	132	17	]	]	PUNCT
ejpam-4712	133	1	=	=	PUNCT
ejpam-4712	133	2	[	[	PUNCT
ejpam-4712	133	3	3	3	NUM
ejpam-4712	133	4	ϵ	ϵ	SYM
ejpam-4712	133	5	1	1	NUM
ejpam-4712	133	6	]	]	PUNCT
ejpam-4712	133	7			NOUN
ejpam-4712	133	8	7	7	NUM
ejpam-4712	133	9	6	6	NUM
ejpam-4712	133	10	−3	−3	ADJ
ejpam-4712	133	11			NOUN
ejpam-4712	133	12	=	=	PUNCT
ejpam-4712	133	13	[	[	PUNCT
ejpam-4712	133	14	10	10	NUM
ejpam-4712	133	15	]	]	PUNCT
ejpam-4712	133	16	because	because	SCONJ
ejpam-4712	133	17	axb	axb	PROPN
ejpam-4712	133	18	=	=	PROPN
ejpam-4712	134	1	[	[	PUNCT
ejpam-4712	134	2	10	10	NUM
ejpam-4712	134	3	]	]	PUNCT
ejpam-4712	134	4	,	,	PUNCT
ejpam-4712	134	5	we	we	PRON
ejpam-4712	134	6	get	get	VERB
ejpam-4712	134	7	vec(axb	vec(axb	NOUN
ejpam-4712	134	8	)	)	PUNCT
ejpam-4712	135	1	=	=	PUNCT
ejpam-4712	136	1	[	[	PUNCT
ejpam-4712	136	2	10	10	NUM
ejpam-4712	136	3	]	]	PUNCT
ejpam-4712	136	4	.	.	PUNCT
ejpam-4712	137	1	next	next	ADV
ejpam-4712	137	2	,	,	PUNCT
ejpam-4712	137	3	we	we	PRON
ejpam-4712	137	4	have	have	VERB
ejpam-4712	137	5	(	(	PUNCT
ejpam-4712	137	6	bt	bt	PROPN
ejpam-4712	137	7	⊗a)vec(x	⊗a)vec(x	PROPN
ejpam-4712	137	8	)	)	PUNCT
ejpam-4712	137	9	=	=	PUNCT
ejpam-4712	138	1	(	(	PUNCT
ejpam-4712	138	2	[	[	PUNCT
ejpam-4712	138	3	5	5	NUM
ejpam-4712	138	4	0	0	NUM
ejpam-4712	138	5	]	]	PUNCT
ejpam-4712	139	1	⊗	⊗	PROPN
ejpam-4712	139	2	[	[	PUNCT
ejpam-4712	139	3	3	3	NUM
ejpam-4712	139	4	ϵ	ϵ	SYM
ejpam-4712	139	5	1	1	NUM
ejpam-4712	139	6	]	]	PUNCT
ejpam-4712	139	7	)	)	PUNCT
ejpam-4712	139	8	[	[	PUNCT
ejpam-4712	139	9	2	2	NUM
ejpam-4712	139	10	5	5	NUM
ejpam-4712	139	11	1	1	NUM
ejpam-4712	139	12	2	2	NUM
ejpam-4712	139	13	ϵ	ϵ	NOUN
ejpam-4712	139	14	−3	−3	NOUN
ejpam-4712	139	15	]	]	X
ejpam-4712	139	16	t	t	NOUN
ejpam-4712	139	17	=	=	PUNCT
ejpam-4712	139	18	[	[	PUNCT
ejpam-4712	139	19	8	8	NUM
ejpam-4712	139	20	ϵ	ϵ	SYM
ejpam-4712	139	21	6	6	NUM
ejpam-4712	139	22	3	3	NUM
ejpam-4712	139	23	ϵ	ϵ	SYM
ejpam-4712	139	24	1	1	NUM
ejpam-4712	139	25	]	]	PUNCT
ejpam-4712	139	26	[	[	PUNCT
ejpam-4712	139	27	2	2	NUM
ejpam-4712	139	28	5	5	NUM
ejpam-4712	139	29	1	1	NUM
ejpam-4712	139	30	2	2	NUM
ejpam-4712	139	31	ϵ	ϵ	NOUN
ejpam-4712	139	32	−3	−3	NOUN
ejpam-4712	139	33	]	]	X
ejpam-4712	139	34	t	t	NOUN
ejpam-4712	139	35	=	=	PUNCT
ejpam-4712	139	36	[	[	PUNCT
ejpam-4712	139	37	10	10	NUM
ejpam-4712	139	38	]	]	PUNCT
ejpam-4712	139	39	references	reference	NOUN
ejpam-4712	139	40	789	789	NUM
ejpam-4712	139	41	due	due	ADP
ejpam-4712	139	42	to	to	ADP
ejpam-4712	139	43	the	the	DET
ejpam-4712	139	44	limitations	limitation	NOUN
ejpam-4712	139	45	of	of	ADP
ejpam-4712	139	46	semirings	semiring	NOUN
ejpam-4712	139	47	,	,	PUNCT
ejpam-4712	139	48	the	the	DET
ejpam-4712	139	49	existence	existence	NOUN
ejpam-4712	139	50	of	of	ADP
ejpam-4712	139	51	discrete	discrete	ADJ
ejpam-4712	139	52	solutions	solution	NOUN
ejpam-4712	139	53	to	to	ADP
ejpam-4712	139	54	the	the	DET
ejpam-4712	139	55	lyapunov	lyapunov	ADJ
ejpam-4712	139	56	equations	equation	NOUN
ejpam-4712	139	57	on	on	ADP
ejpam-4712	139	58	semirings	semiring	NOUN
ejpam-4712	139	59	depends	depend	VERB
ejpam-4712	139	60	on	on	ADP
ejpam-4712	139	61	the	the	DET
ejpam-4712	139	62	orthogonal	orthogonal	ADJ
ejpam-4712	139	63	matrix	matrix	NOUN
ejpam-4712	139	64	.	.	PUNCT
ejpam-4712	140	1	this	this	DET
ejpam-4712	140	2	statement	statement	NOUN
ejpam-4712	140	3	is	be	AUX
ejpam-4712	140	4	given	give	VERB
ejpam-4712	140	5	in	in	ADP
ejpam-4712	140	6	the	the	DET
ejpam-4712	140	7	following	follow	VERB
ejpam-4712	140	8	theorem	theorem	NOUN
ejpam-4712	140	9	.	.	PUNCT
ejpam-4712	140	10	theorem	theorem	NOUN
ejpam-4712	140	11	4	4	NUM
ejpam-4712	140	12	.	.	PUNCT
ejpam-4712	141	1	let	let	VERB
ejpam-4712	141	2	discrete	discrete	ADJ
ejpam-4712	141	3	lyapunov	lyapunov	ADJ
ejpam-4712	141	4	equation	equation	NOUN
ejpam-4712	141	5	over	over	ADP
ejpam-4712	141	6	semiring	semire	VERB
ejpam-4712	141	7	axat	axat	NOUN
ejpam-4712	141	8	−	−	PROPN
ejpam-4712	141	9	x	x	PROPN
ejpam-4712	142	1	+	+	PUNCT
ejpam-4712	142	2	q	q	X
ejpam-4712	142	3	=	=	SYM
ejpam-4712	142	4	0	0	NUM
ejpam-4712	142	5	with	with	ADP
ejpam-4712	142	6	a	a	DET
ejpam-4712	142	7	∈	∈	PROPN
ejpam-4712	142	8	mn×n(s	mn×n(s	NOUN
ejpam-4712	142	9	)	)	PUNCT
ejpam-4712	142	10	,	,	PUNCT
ejpam-4712	142	11	q	q	PROPN
ejpam-4712	142	12	∈	∈	PROPN
ejpam-4712	142	13	mn×n(s	mn×n(s	NOUN
ejpam-4712	142	14	)	)	PUNCT
ejpam-4712	142	15	.	.	PUNCT
ejpam-4712	143	1	for	for	ADP
ejpam-4712	143	2	a	a	DET
ejpam-4712	143	3	is	be	AUX
ejpam-4712	143	4	orthogonal	orthogonal	ADJ
ejpam-4712	143	5	matrice	matrice	NOUN
ejpam-4712	143	6	,	,	PUNCT
ejpam-4712	143	7	there	there	PRON
ejpam-4712	143	8	is	be	VERB
ejpam-4712	143	9	a	a	DET
ejpam-4712	143	10	unique	unique	ADJ
ejpam-4712	143	11	solution	solution	NOUN
ejpam-4712	143	12	x	x	X
ejpam-4712	143	13	∈	∈	NOUN
ejpam-4712	143	14	mn×n(s	mn×n(s	NOUN
ejpam-4712	143	15	)	)	PUNCT
ejpam-4712	143	16	if	if	SCONJ
ejpam-4712	143	17	and	and	CCONJ
ejpam-4712	143	18	only	only	ADV
ejpam-4712	143	19	if	if	SCONJ
ejpam-4712	143	20	λi(a	λi(a	NOUN
ejpam-4712	143	21	)	)	PUNCT
ejpam-4712	144	1	+	+	CCONJ
ejpam-4712	144	2	λj(−a	λj(−a	ADJ
ejpam-4712	144	3	)	)	PUNCT
ejpam-4712	144	4	̸=	̸=	PROPN
ejpam-4712	144	5	0	0	NUM
ejpam-4712	144	6	.	.	PUNCT
ejpam-4712	145	1	proof	proof	NOUN
ejpam-4712	145	2	.	.	PUNCT
ejpam-4712	146	1	a	a	DET
ejpam-4712	146	2	matrice	matrice	NOUN
ejpam-4712	146	3	a	a	PRON
ejpam-4712	146	4	is	be	AUX
ejpam-4712	146	5	orthogonal	orthogonal	ADJ
ejpam-4712	146	6	if	if	SCONJ
ejpam-4712	146	7	at	at	ADP
ejpam-4712	146	8	=	=	PROPN
ejpam-4712	146	9	a−1	a−1	PROPN
ejpam-4712	146	10	;	;	PUNCT
ejpam-4712	146	11	hence	hence	ADV
ejpam-4712	146	12	we	we	PRON
ejpam-4712	146	13	have	have	VERB
ejpam-4712	146	14	axat	axat	NOUN
ejpam-4712	146	15	−	−	PROPN
ejpam-4712	147	1	x	x	PROPN
ejpam-4712	148	1	+	+	PUNCT
ejpam-4712	148	2	q	q	X
ejpam-4712	148	3	=	=	SYM
ejpam-4712	148	4	0	0	PUNCT
ejpam-4712	148	5	and	and	CCONJ
ejpam-4712	148	6	then	then	ADV
ejpam-4712	148	7	(	(	PUNCT
ejpam-4712	148	8	axat	axat	ADJ
ejpam-4712	148	9	)	)	PUNCT
ejpam-4712	148	10	a−xa	a−xa	PROPN
ejpam-4712	149	1	=	=	SYM
ejpam-4712	149	2	−qa	−qa	PROPN
ejpam-4712	149	3	.	.	PUNCT
ejpam-4712	150	1	because	because	SCONJ
ejpam-4712	150	2	a	a	PRON
ejpam-4712	150	3	is	be	AUX
ejpam-4712	150	4	orthogonal	orthogonal	ADJ
ejpam-4712	150	5	,	,	PUNCT
ejpam-4712	150	6	we	we	PRON
ejpam-4712	150	7	have	have	VERB
ejpam-4712	150	8	ax	ax	NOUN
ejpam-4712	150	9	−xa	−xa	PROPN
ejpam-4712	151	1	=	=	SYM
ejpam-4712	152	1	−qa	−qa	NOUN
ejpam-4712	152	2	.	.	PUNCT
ejpam-4712	153	1	therefore	therefore	ADV
ejpam-4712	153	2	,	,	PUNCT
ejpam-4712	153	3	ax+x(−a	ax+x(−a	ADJ
ejpam-4712	153	4	)	)	PUNCT
ejpam-4712	153	5	=	=	SYM
ejpam-4712	153	6	q(−a	q(−a	NOUN
ejpam-4712	153	7	)	)	PUNCT
ejpam-4712	153	8	and	and	CCONJ
ejpam-4712	153	9	vec(ax+x(−a	vec(ax+x(−a	NOUN
ejpam-4712	153	10	)	)	PUNCT
ejpam-4712	153	11	)	)	PUNCT
ejpam-4712	154	1	=	=	SYM
ejpam-4712	154	2	(	(	PUNCT
ejpam-4712	154	3	(	(	PUNCT
ejpam-4712	154	4	−a)t	−a)t	NOUN
ejpam-4712	154	5	⊕a)vec(x	⊕a)vec(x	NOUN
ejpam-4712	154	6	)	)	PUNCT
ejpam-4712	154	7	.	.	PUNCT
ejpam-4712	155	1	finally	finally	ADV
ejpam-4712	155	2	,	,	PUNCT
ejpam-4712	155	3	we	we	PRON
ejpam-4712	155	4	have	have	VERB
ejpam-4712	155	5	(	(	PUNCT
ejpam-4712	155	6	(	(	PUNCT
ejpam-4712	155	7	−a)t	−a)t	NOUN
ejpam-4712	155	8	⊕a)vec(x	⊕a)vec(x	NOUN
ejpam-4712	155	9	)	)	PUNCT
ejpam-4712	155	10	=	=	SYM
ejpam-4712	155	11	vec(−qa	vec(−qa	NOUN
ejpam-4712	155	12	)	)	PUNCT
ejpam-4712	155	13	.	.	PUNCT
ejpam-4712	156	1	based	base	VERB
ejpam-4712	156	2	on	on	ADP
ejpam-4712	156	3	conditions	condition	NOUN
ejpam-4712	156	4	,	,	PUNCT
ejpam-4712	156	5	we	we	PRON
ejpam-4712	156	6	have	have	VERB
ejpam-4712	156	7	a	a	DET
ejpam-4712	156	8	unique	unique	ADJ
ejpam-4712	156	9	solution	solution	NOUN
ejpam-4712	156	10	if	if	SCONJ
ejpam-4712	156	11	and	and	CCONJ
ejpam-4712	156	12	only	only	ADV
ejpam-4712	156	13	if	if	SCONJ
ejpam-4712	156	14	(	(	PUNCT
ejpam-4712	156	15	−a)t	−a)t	NOUN
ejpam-4712	156	16	⊕a	⊕a	PROPN
ejpam-4712	156	17	non	non	PROPN
ejpam-4712	156	18	-	-	ADJ
ejpam-4712	156	19	singular	singular	ADJ
ejpam-4712	156	20	.	.	PUNCT
ejpam-4712	157	1	4	4	X
ejpam-4712	157	2	.	.	X
ejpam-4712	157	3	conclusion	conclusion	NOUN
ejpam-4712	157	4	we	we	PRON
ejpam-4712	157	5	show	show	VERB
ejpam-4712	157	6	that	that	SCONJ
ejpam-4712	157	7	the	the	DET
ejpam-4712	157	8	solutions	solution	NOUN
ejpam-4712	157	9	of	of	ADP
ejpam-4712	157	10	the	the	DET
ejpam-4712	157	11	discrete	discrete	ADJ
ejpam-4712	157	12	lyapunov	lyapunov	ADJ
ejpam-4712	157	13	equations	equation	NOUN
ejpam-4712	157	14	for	for	ADP
ejpam-4712	157	15	matrices	matrix	NOUN
ejpam-4712	157	16	over	over	ADP
ejpam-4712	157	17	semirings	semiring	NOUN
ejpam-4712	157	18	are	be	AUX
ejpam-4712	157	19	also	also	ADV
ejpam-4712	157	20	valid	valid	ADJ
ejpam-4712	157	21	.	.	PUNCT
ejpam-4712	158	1	the	the	DET
ejpam-4712	158	2	linear	linear	ADJ
ejpam-4712	158	3	system	system	NOUN
ejpam-4712	158	4	that	that	PRON
ejpam-4712	158	5	has	have	AUX
ejpam-4712	158	6	been	be	AUX
ejpam-4712	158	7	developed	develop	VERB
ejpam-4712	158	8	is	be	AUX
ejpam-4712	158	9	the	the	DET
ejpam-4712	158	10	semiring	semire	VERB
ejpam-4712	158	11	linear	linear	NOUN
ejpam-4712	158	12	system	system	NOUN
ejpam-4712	158	13	.	.	PUNCT
ejpam-4712	159	1	due	due	ADP
ejpam-4712	159	2	to	to	ADP
ejpam-4712	159	3	its	its	PRON
ejpam-4712	159	4	semiring	semire	VERB
ejpam-4712	159	5	nature	nature	NOUN
ejpam-4712	159	6	,	,	PUNCT
ejpam-4712	159	7	not	not	PART
ejpam-4712	159	8	all	all	DET
ejpam-4712	159	9	elements	element	NOUN
ejpam-4712	159	10	have	have	VERB
ejpam-4712	159	11	inverses	inverse	NOUN
ejpam-4712	159	12	.	.	PUNCT
ejpam-4712	160	1	it	it	PRON
ejpam-4712	160	2	is	be	AUX
ejpam-4712	160	3	necessary	necessary	ADJ
ejpam-4712	160	4	to	to	PART
ejpam-4712	160	5	have	have	VERB
ejpam-4712	160	6	a	a	DET
ejpam-4712	160	7	particular	particular	ADJ
ejpam-4712	160	8	case	case	NOUN
ejpam-4712	160	9	,	,	PUNCT
ejpam-4712	160	10	namely	namely	ADV
ejpam-4712	160	11	by	by	ADP
ejpam-4712	160	12	reviewing	review	VERB
ejpam-4712	160	13	the	the	DET
ejpam-4712	160	14	lyapunov	lyapunov	ADJ
ejpam-4712	160	15	equation	equation	NOUN
ejpam-4712	160	16	of	of	ADP
ejpam-4712	160	17	the	the	DET
ejpam-4712	160	18	system	system	NOUN
ejpam-4712	160	19	.	.	PUNCT
ejpam-4712	161	1	in	in	ADP
ejpam-4712	161	2	addition	addition	NOUN
ejpam-4712	161	3	,	,	PUNCT
ejpam-4712	161	4	it	it	PRON
ejpam-4712	161	5	is	be	AUX
ejpam-4712	161	6	required	require	VERB
ejpam-4712	161	7	to	to	PART
ejpam-4712	161	8	relate	relate	VERB
ejpam-4712	161	9	it	it	PRON
ejpam-4712	161	10	to	to	ADP
ejpam-4712	161	11	the	the	DET
ejpam-4712	161	12	kroner	kroner	NOUN
ejpam-4712	161	13	product	product	NOUN
ejpam-4712	161	14	as	as	SCONJ
ejpam-4712	161	15	given	give	VERB
ejpam-4712	161	16	in	in	ADP
ejpam-4712	161	17	theorem	theorem	ADJ
ejpam-4712	161	18	3	3	NUM
ejpam-4712	161	19	and	and	CCONJ
ejpam-4712	161	20	theorem	theorem	VERB
ejpam-4712	161	21	4	4	NUM
ejpam-4712	161	22	.	.	PUNCT
ejpam-4712	162	1	furthermore	furthermore	ADV
ejpam-4712	162	2	,	,	PUNCT
ejpam-4712	162	3	this	this	DET
ejpam-4712	162	4	condition	condition	NOUN
ejpam-4712	162	5	can	can	AUX
ejpam-4712	162	6	be	be	AUX
ejpam-4712	162	7	used	use	VERB
ejpam-4712	162	8	to	to	PART
ejpam-4712	162	9	find	find	VERB
ejpam-4712	162	10	other	other	ADJ
ejpam-4712	162	11	characteristics	characteristic	NOUN
ejpam-4712	162	12	of	of	ADP
ejpam-4712	162	13	a	a	DET
ejpam-4712	162	14	linear	linear	ADJ
ejpam-4712	162	15	system	system	NOUN
ejpam-4712	162	16	over	over	ADP
ejpam-4712	162	17	a	a	DET
ejpam-4712	162	18	semiring	semiring	NOUN
ejpam-4712	162	19	.	.	PUNCT
ejpam-4712	163	1	acknowledgements	acknowledgement	NOUN
ejpam-4712	163	2	we	we	PRON
ejpam-4712	163	3	gratefully	gratefully	ADV
ejpam-4712	163	4	thank	thank	VERB
ejpam-4712	163	5	the	the	DET
ejpam-4712	163	6	directorate	directorate	NOUN
ejpam-4712	163	7	of	of	ADP
ejpam-4712	163	8	research	research	NOUN
ejpam-4712	163	9	and	and	CCONJ
ejpam-4712	163	10	community	community	NOUN
ejpam-4712	163	11	service	service	NOUN
ejpam-4712	163	12	,	,	PUNCT
ejpam-4712	163	13	deputy	deputy	NOUN
ejpam-4712	163	14	for	for	ADP
ejpam-4712	163	15	research	research	NOUN
ejpam-4712	163	16	and	and	CCONJ
ejpam-4712	163	17	development	development	NOUN
ejpam-4712	163	18	strengthening	strengthening	NOUN
ejpam-4712	163	19	,	,	PUNCT
ejpam-4712	163	20	the	the	DET
ejpam-4712	163	21	ministry	ministry	PROPN
ejpam-4712	163	22	of	of	ADP
ejpam-4712	163	23	education	education	PROPN
ejpam-4712	163	24	,	,	PUNCT
ejpam-4712	163	25	culture	culture	NOUN
ejpam-4712	163	26	,	,	PUNCT
ejpam-4712	163	27	research	research	NOUN
ejpam-4712	163	28	,	,	PUNCT
ejpam-4712	163	29	and	and	CCONJ
ejpam-4712	163	30	technology	technology	NOUN
ejpam-4712	163	31	,	,	PUNCT
ejpam-4712	163	32	indonesia	indonesia	PROPN
ejpam-4712	163	33	.	.	PUNCT
ejpam-4712	164	1	references	reference	NOUN
ejpam-4712	164	2	[	[	X
ejpam-4712	164	3	1	1	NUM
ejpam-4712	164	4	]	]	PUNCT
ejpam-4712	164	5	h	h	NOUN
ejpam-4712	164	6	anton	anton	NOUN
ejpam-4712	164	7	and	and	CCONJ
ejpam-4712	164	8	c	c	NOUN
ejpam-4712	164	9	rorres	rorre	NOUN
ejpam-4712	164	10	.	.	PUNCT
ejpam-4712	165	1	elementary	elementary	ADJ
ejpam-4712	165	2	linear	linear	PROPN
ejpam-4712	165	3	algebra	algebra	PROPN
ejpam-4712	165	4	:	:	PUNCT
ejpam-4712	165	5	applications	application	NOUN
ejpam-4712	165	6	version	version	NOUN
ejpam-4712	165	7	,	,	PUNCT
ejpam-4712	165	8	11th	11th	ADJ
ejpam-4712	165	9	edition	edition	NOUN
ejpam-4712	165	10	.	.	PUNCT
ejpam-4712	166	1	john	john	PROPN
ejpam-4712	166	2	willey	willey	PROPN
ejpam-4712	166	3	and	and	CCONJ
ejpam-4712	166	4	sons	son	NOUN
ejpam-4712	166	5	,	,	PUNCT
ejpam-4712	166	6	the	the	DET
ejpam-4712	166	7	united	united	PROPN
ejpam-4712	166	8	states	states	PROPN
ejpam-4712	166	9	of	of	ADP
ejpam-4712	166	10	america	america	PROPN
ejpam-4712	166	11	,	,	PUNCT
ejpam-4712	166	12	2013	2013	NUM
ejpam-4712	166	13	.	.	PUNCT
ejpam-4712	167	1	[	[	X
ejpam-4712	167	2	2	2	NUM
ejpam-4712	167	3	]	]	X
ejpam-4712	167	4	g	g	PROPN
ejpam-4712	167	5	ariyanti	ariyanti	NOUN
ejpam-4712	167	6	.	.	PUNCT
ejpam-4712	168	1	a	a	DET
ejpam-4712	168	2	note	note	NOUN
ejpam-4712	168	3	of	of	ADP
ejpam-4712	168	4	the	the	DET
ejpam-4712	168	5	linear	linear	ADJ
ejpam-4712	168	6	equation	equation	NOUN
ejpam-4712	168	7	ax	ax	NOUN
ejpam-4712	168	8	=	=	NOUN
ejpam-4712	168	9	b	b	NOUN
ejpam-4712	168	10	with	with	ADP
ejpam-4712	168	11	multiplicatively	multiplicatively	ADV
ejpam-4712	168	12	-	-	PUNCT
ejpam-4712	168	13	reguler	reguler	NOUN
ejpam-4712	168	14	matrix	matrix	NOUN
ejpam-4712	168	15	a	a	PRON
ejpam-4712	168	16	in	in	ADP
ejpam-4712	168	17	semiring	semiring	NOUN
ejpam-4712	168	18	.	.	PUNCT
ejpam-4712	169	1	international	international	ADJ
ejpam-4712	169	2	conference	conference	NOUN
ejpam-4712	169	3	on	on	ADP
ejpam-4712	169	4	applied	apply	VERB
ejpam-4712	169	5	&	&	CCONJ
ejpam-4712	169	6	industrial	industrial	ADJ
ejpam-4712	169	7	mathematics	mathematic	NOUN
ejpam-4712	169	8	and	and	CCONJ
ejpam-4712	169	9	statistics	statistic	NOUN
ejpam-4712	169	10	(	(	PUNCT
ejpam-4712	169	11	icoaims	icoaim	NOUN
ejpam-4712	169	12	)	)	PUNCT
ejpam-4712	169	13	,	,	PUNCT
ejpam-4712	169	14	journal	journal	NOUN
ejpam-4712	169	15	of	of	ADP
ejpam-4712	169	16	physics	physics	PROPN
ejpam-4712	169	17	:	:	PUNCT
ejpam-4712	169	18	conference	conference	NOUN
ejpam-4712	169	19	series	series	NOUN
ejpam-4712	169	20	,	,	PUNCT
ejpam-4712	169	21	2019	2019	NUM
ejpam-4712	169	22	.	.	PUNCT
ejpam-4712	170	1	[	[	X
ejpam-4712	170	2	3	3	X
ejpam-4712	170	3	]	]	X
ejpam-4712	170	4	g	g	PROPN
ejpam-4712	170	5	ariyanti	ariyanti	NOUN
ejpam-4712	170	6	.	.	PUNCT
ejpam-4712	171	1	a	a	DET
ejpam-4712	171	2	note	note	NOUN
ejpam-4712	171	3	on	on	ADP
ejpam-4712	171	4	the	the	DET
ejpam-4712	171	5	solution	solution	NOUN
ejpam-4712	171	6	of	of	ADP
ejpam-4712	171	7	the	the	DET
ejpam-4712	171	8	characteristic	characteristic	ADJ
ejpam-4712	171	9	equation	equation	NOUN
ejpam-4712	171	10	over	over	ADP
ejpam-4712	171	11	the	the	DET
ejpam-4712	171	12	symmetrized	symmetrize	VERB
ejpam-4712	171	13	max	max	PROPN
ejpam-4712	171	14	-	-	PUNCT
ejpam-4712	171	15	plus	plus	CCONJ
ejpam-4712	171	16	algebra	algebra	NOUN
ejpam-4712	171	17	.	.	PUNCT
ejpam-4712	172	1	barekeng	barekeng	NOUN
ejpam-4712	172	2	:	:	PUNCT
ejpam-4712	172	3	journal	journal	PROPN
ejpam-4712	172	4	of	of	ADP
ejpam-4712	172	5	mathematics	mathematic	NOUN
ejpam-4712	172	6	and	and	CCONJ
ejpam-4712	172	7	its	its	PRON
ejpam-4712	172	8	application	application	NOUN
ejpam-4712	172	9	.	.	PUNCT
ejpam-4712	172	10	,	,	PUNCT
ejpam-4712	172	11	16:1347–1354	16:1347–1354	NUM
ejpam-4712	172	12	,	,	PUNCT
ejpam-4712	172	13	2022	2022	NUM
ejpam-4712	172	14	.	.	PUNCT
ejpam-4712	173	1	[	[	X
ejpam-4712	173	2	4	4	NUM
ejpam-4712	173	3	]	]	PUNCT
ejpam-4712	173	4	wc	wc	PROPN
ejpam-4712	173	5	brown	brown	PROPN
ejpam-4712	173	6	.	.	PUNCT
ejpam-4712	174	1	matrices	matrix	NOUN
ejpam-4712	174	2	over	over	ADP
ejpam-4712	174	3	commutative	commutative	ADJ
ejpam-4712	174	4	rings	ring	NOUN
ejpam-4712	174	5	.	.	PUNCT
ejpam-4712	175	1	new	new	PROPN
ejpam-4712	175	2	york	york	PROPN
ejpam-4712	175	3	:	:	PUNCT
ejpam-4712	175	4	marcel	marcel	PROPN
ejpam-4712	175	5	dekker	dekker	PROPN
ejpam-4712	175	6	,	,	PUNCT
ejpam-4712	175	7	inc	inc	PROPN
ejpam-4712	175	8	.	.	PROPN
ejpam-4712	175	9	,	,	PUNCT
ejpam-4712	175	10	1993	1993	NUM
ejpam-4712	175	11	.	.	PUNCT
ejpam-4712	176	1	references	reference	NOUN
ejpam-4712	176	2	790	790	NUM
ejpam-4712	177	1	[	[	X
ejpam-4712	177	2	5	5	NUM
ejpam-4712	177	3	]	]	PUNCT
ejpam-4712	177	4	as	as	ADP
ejpam-4712	177	5	hodel	hodel	NOUN
ejpam-4712	177	6	and	and	CCONJ
ejpam-4712	177	7	b	b	NOUN
ejpam-4712	177	8	tenison	tenison	NOUN
ejpam-4712	177	9	.	.	PUNCT
ejpam-4712	178	1	linear	linear	PROPN
ejpam-4712	178	2	algebra	algebra	PROPN
ejpam-4712	178	3	and	and	CCONJ
ejpam-4712	178	4	its	its	PRON
ejpam-4712	178	5	applications	application	NOUN
ejpam-4712	178	6	.	.	PUNCT
ejpam-4712	179	1	elsevier	elsevier	PROPN
ejpam-4712	179	2	science	science	PROPN
ejpam-4712	179	3	inc	inc	PROPN
ejpam-4712	179	4	.	.	PROPN
ejpam-4712	179	5	,	,	PUNCT
ejpam-4712	179	6	1996	1996	NUM
ejpam-4712	179	7	.	.	PUNCT
ejpam-4712	180	1	[	[	X
ejpam-4712	180	2	6	6	NUM
ejpam-4712	180	3	]	]	PUNCT
ejpam-4712	180	4	a	a	DET
ejpam-4712	180	5	ibrahim	ibrahim	NOUN
ejpam-4712	180	6	,	,	PUNCT
ejpam-4712	180	7	si	si	PROPN
ejpam-4712	180	8	bala	bala	PROPN
ejpam-4712	180	9	,	,	PUNCT
ejpam-4712	180	10	i	i	PRON
ejpam-4712	180	11	ahmed	ahme	VERB
ejpam-4712	180	12	,	,	PUNCT
ejpam-4712	180	13	mj	mj	PROPN
ejpam-4712	180	14	ibrahim	ibrahim	PROPN
ejpam-4712	180	15	,	,	PUNCT
ejpam-4712	180	16	and	and	CCONJ
ejpam-4712	180	17	f	f	PROPN
ejpam-4712	180	18	jarad	jarad	PROPN
ejpam-4712	180	19	.	.	PUNCT
ejpam-4712	181	1	numerical	numerical	ADJ
ejpam-4712	181	2	construction	construction	NOUN
ejpam-4712	181	3	of	of	ADP
ejpam-4712	181	4	lyapunov	lyapunov	ADJ
ejpam-4712	181	5	functions	function	NOUN
ejpam-4712	181	6	using	use	VERB
ejpam-4712	181	7	homotopy	homotopy	NOUN
ejpam-4712	181	8	continuation	continuation	NOUN
ejpam-4712	181	9	method	method	NOUN
ejpam-4712	181	10	.	.	PUNCT
ejpam-4712	182	1	advances	advance	NOUN
ejpam-4712	182	2	in	in	ADP
ejpam-4712	182	3	the	the	DET
ejpam-4712	182	4	theory	theory	NOUN
ejpam-4712	182	5	of	of	ADP
ejpam-4712	182	6	nonlinear	nonlinear	ADJ
ejpam-4712	182	7	analysis	analysis	NOUN
ejpam-4712	182	8	and	and	CCONJ
ejpam-4712	182	9	its	its	PRON
ejpam-4712	182	10	applications	application	NOUN
ejpam-4712	182	11	,	,	PUNCT
ejpam-4712	182	12	6(3):354–363	6(3):354–363	NUM
ejpam-4712	182	13	,	,	PUNCT
ejpam-4712	182	14	2022	2022	NUM
ejpam-4712	182	15	.	.	PUNCT
ejpam-4712	183	1	[	[	X
ejpam-4712	183	2	7	7	X
ejpam-4712	183	3	]	]	X
ejpam-4712	183	4	k	k	PROPN
ejpam-4712	183	5	glover	glover	PROPN
ejpam-4712	183	6	k	k	PROPN
ejpam-4712	183	7	zhou	zhou	PROPN
ejpam-4712	183	8	,	,	PUNCT
ejpam-4712	183	9	jc	jc	PROPN
ejpam-4712	183	10	doyle	doyle	PROPN
ejpam-4712	183	11	.	.	PUNCT
ejpam-4712	184	1	robust	robust	ADJ
ejpam-4712	184	2	and	and	CCONJ
ejpam-4712	184	3	optimal	optimal	ADJ
ejpam-4712	184	4	control	control	NOUN
ejpam-4712	184	5	.	.	PUNCT
ejpam-4712	185	1	prentice	prentice	PROPN
ejpam-4712	185	2	hall	hall	PROPN
ejpam-4712	185	3	,	,	PUNCT
ejpam-4712	185	4	new	new	PROPN
ejpam-4712	185	5	york	york	PROPN
ejpam-4712	185	6	,	,	PUNCT
ejpam-4712	185	7	1996	1996	NUM
ejpam-4712	185	8	.	.	PUNCT
ejpam-4712	186	1	[	[	X
ejpam-4712	186	2	8	8	NUM
ejpam-4712	186	3	]	]	PUNCT
ejpam-4712	186	4	s	s	VERB
ejpam-4712	186	5	kathrin	kathrin	PROPN
ejpam-4712	186	6	.	.	PUNCT
ejpam-4712	187	1	on	on	ADP
ejpam-4712	187	2	the	the	DET
ejpam-4712	187	3	kronecker	kronecker	NOUN
ejpam-4712	187	4	product	product	NOUN
ejpam-4712	187	5	.	.	PUNCT
ejpam-4712	188	1	phd	phd	NOUN
ejpam-4712	188	2	thesis	thesis	PROPN
ejpam-4712	188	3	,	,	PUNCT
ejpam-4712	188	4	university	university	NOUN
ejpam-4712	188	5	of	of	ADP
ejpam-4712	188	6	london	london	PROPN
ejpam-4712	188	7	,	,	PUNCT
ejpam-4712	188	8	2013	2013	NUM
ejpam-4712	188	9	.	.	PUNCT
ejpam-4712	189	1	[	[	X
ejpam-4712	189	2	9	9	NUM
ejpam-4712	189	3	]	]	PUNCT
ejpam-4712	189	4	cfv	cfv	PROPN
ejpam-4712	189	5	loan	loan	PROPN
ejpam-4712	189	6	.	.	PUNCT
ejpam-4712	190	1	the	the	DET
ejpam-4712	190	2	ubiquitous	ubiquitous	ADJ
ejpam-4712	190	3	kronecker	kronecker	NOUN
ejpam-4712	190	4	product	product	NOUN
ejpam-4712	190	5	.	.	PUNCT
ejpam-4712	191	1	journal	journal	NOUN
ejpam-4712	191	2	of	of	ADP
ejpam-4712	191	3	computational	computational	ADJ
ejpam-4712	191	4	and	and	CCONJ
ejpam-4712	191	5	applied	applied	ADJ
ejpam-4712	191	6	mathematics	mathematic	NOUN
ejpam-4712	191	7	,	,	PUNCT
ejpam-4712	191	8	123:85–100	123:85–100	NUM
ejpam-4712	191	9	,	,	PUNCT
ejpam-4712	191	10	2000	2000	NUM
ejpam-4712	191	11	.	.	PUNCT
ejpam-4712	192	1	[	[	X
ejpam-4712	192	2	10	10	NUM
ejpam-4712	192	3	]	]	X
ejpam-4712	192	4	s	s	VERB
ejpam-4712	192	5	massei	massei	NOUN
ejpam-4712	192	6	,	,	PUNCT
ejpam-4712	192	7	d	d	NOUN
ejpam-4712	192	8	palitta	palitta	NOUN
ejpam-4712	192	9	,	,	PUNCT
ejpam-4712	192	10	and	and	CCONJ
ejpam-4712	192	11	robol	robol	NOUN
ejpam-4712	192	12	.	.	PUNCT
ejpam-4712	193	1	solving	solve	VERB
ejpam-4712	193	2	rank	rank	NOUN
ejpam-4712	193	3	structured	structure	VERB
ejpam-4712	193	4	sylvester	sylvester	NOUN
ejpam-4712	193	5	and	and	CCONJ
ejpam-4712	193	6	lyapunov	lyapunov	PROPN
ejpam-4712	193	7	equations	equation	NOUN
ejpam-4712	193	8	.	.	PUNCT
ejpam-4712	194	1	siam	siam	PROPN
ejpam-4712	194	2	journal	journal	PROPN
ejpam-4712	194	3	on	on	ADP
ejpam-4712	194	4	matrix	matrix	NOUN
ejpam-4712	194	5	analysis	analysis	NOUN
ejpam-4712	194	6	and	and	CCONJ
ejpam-4712	194	7	applications	application	NOUN
ejpam-4712	194	8	,	,	PUNCT
ejpam-4712	194	9	39(4	39(4	NOUN
ejpam-4712	194	10	)	)	PUNCT
ejpam-4712	194	11	,	,	PUNCT
ejpam-4712	194	12	2018	2018	NUM
ejpam-4712	194	13	.	.	PUNCT
ejpam-4712	195	1	[	[	X
ejpam-4712	195	2	11	11	NUM
ejpam-4712	195	3	]	]	X
ejpam-4712	195	4	w	w	PROPN
ejpam-4712	195	5	mora	mora	PROPN
ejpam-4712	195	6	,	,	PUNCT
ejpam-4712	195	7	a	a	DET
ejpam-4712	195	8	wasanawichit	wasanawichit	NOUN
ejpam-4712	195	9	,	,	PUNCT
ejpam-4712	195	10	and	and	CCONJ
ejpam-4712	195	11	y	y	PROPN
ejpam-4712	195	12	kemprasit	kemprasit	PROPN
ejpam-4712	195	13	.	.	PUNCT
ejpam-4712	196	1	invertible	invertible	ADJ
ejpam-4712	196	2	matrices	matrix	NOUN
ejpam-4712	196	3	over	over	ADP
ejpam-4712	196	4	idempotent	idempotent	ADJ
ejpam-4712	196	5	semirings	semiring	NOUN
ejpam-4712	196	6	.	.	PUNCT
ejpam-4712	197	1	chamcuri	chamcuri	PROPN
ejpam-4712	197	2	journal	journal	PROPN
ejpam-4712	197	3	of	of	ADP
ejpam-4712	197	4	mathematics	mathematic	NOUN
ejpam-4712	197	5	,	,	PUNCT
ejpam-4712	197	6	1(2):55–61	1(2):55–61	NUM
ejpam-4712	197	7	,	,	PUNCT
ejpam-4712	197	8	2009	2009	NUM
ejpam-4712	197	9	.	.	PUNCT
ejpam-4712	198	1	[	[	X
ejpam-4712	198	2	12	12	NUM
ejpam-4712	198	3	]	]	X
ejpam-4712	198	4	p	p	X
ejpam-4712	198	5	narendran	narendran	NOUN
ejpam-4712	198	6	.	.	PUNCT
ejpam-4712	199	1	solving	solve	VERB
ejpam-4712	199	2	linear	linear	ADJ
ejpam-4712	199	3	equations	equation	NOUN
ejpam-4712	199	4	over	over	ADP
ejpam-4712	199	5	polynomial	polynomial	ADJ
ejpam-4712	199	6	semirings	semiring	NOUN
ejpam-4712	199	7	.	.	PUNCT
ejpam-4712	200	1	”	"	PUNCT
ejpam-4712	200	2	11	11	NUM
ejpam-4712	200	3	th	th	X
ejpam-4712	200	4	anual	anual	PROPN
ejpam-4712	200	5	ieee	ieee	NOUN
ejpam-4712	200	6	symposium	symposium	NOUN
ejpam-4712	200	7	on	on	ADP
ejpam-4712	200	8	logic	logic	NOUN
ejpam-4712	200	9	in	in	ADP
ejpam-4712	200	10	computer	computer	NOUN
ejpam-4712	200	11	science	science	NOUN
ejpam-4712	200	12	”	"	PUNCT
ejpam-4712	200	13	,	,	PUNCT
ejpam-4712	200	14	ieee	ieee	NOUN
ejpam-4712	200	15	computer	computer	NOUN
ejpam-4712	200	16	society	society	PROPN
ejpam-4712	200	17	press	press	PROPN
ejpam-4712	200	18	,	,	PUNCT
ejpam-4712	200	19	los	los	PROPN
ejpam-4712	200	20	alamitos	alamitos	PROPN
ejpam-4712	200	21	,	,	PUNCT
ejpam-4712	200	22	ca	ca	NOUN
ejpam-4712	200	23	,	,	PUNCT
ejpam-4712	200	24	pages	page	NOUN
ejpam-4712	200	25	466–472	466–472	NUM
ejpam-4712	200	26	,	,	PUNCT
ejpam-4712	200	27	1996	1996	NUM
ejpam-4712	200	28	.	.	PUNCT
ejpam-4712	201	1	[	[	X
ejpam-4712	201	2	13	13	NUM
ejpam-4712	201	3	]	]	SYM
ejpam-4712	201	4	pl	pl	X
ejpam-4712	201	5	poplin	poplin	NOUN
ejpam-4712	201	6	.	.	PUNCT
ejpam-4712	202	1	the	the	DET
ejpam-4712	202	2	semiring	semiring	NOUN
ejpam-4712	202	3	of	of	ADP
ejpam-4712	202	4	multisets	multiset	NOUN
ejpam-4712	202	5	.	.	PUNCT
ejpam-4712	203	1	phd	phd	NOUN
ejpam-4712	203	2	thesis	thesis	NOUN
ejpam-4712	203	3	,	,	PUNCT
ejpam-4712	203	4	faculty	faculty	NOUN
ejpam-4712	203	5	of	of	ADP
ejpam-4712	203	6	north	north	PROPN
ejpam-4712	203	7	caroline	caroline	PROPN
ejpam-4712	203	8	state	state	PROPN
ejpam-4712	203	9	university	university	PROPN
ejpam-4712	203	10	,	,	PUNCT
ejpam-4712	203	11	2000	2000	NUM
ejpam-4712	203	12	.	.	PUNCT
ejpam-4712	204	1	[	[	X
ejpam-4712	204	2	14	14	NUM
ejpam-4712	204	3	]	]	X
ejpam-4712	204	4	ll	ll	PROPN
ejpam-4712	204	5	whitcomb	whitcomb	PROPN
ejpam-4712	204	6	.	.	PUNCT
ejpam-4712	205	1	notes	note	NOUN
ejpam-4712	205	2	on	on	ADP
ejpam-4712	205	3	kronecker	kronecker	NOUN
ejpam-4712	205	4	products	product	NOUN
ejpam-4712	205	5	.	.	PUNCT
ejpam-4712	206	1	johns	johns	PROPN
ejpam-4712	206	2	hopkins	hopkins	PROPN
ejpam-4712	206	3	university	university	PROPN
ejpam-4712	206	4	,	,	PUNCT
ejpam-4712	206	5	2020	2020	NUM
ejpam-4712	206	6	.	.	PUNCT