id	sid	tid	token	lemma	pos
ejpam-6909	1	1	european	european	PROPN
ejpam-6909	1	2	journal	journal	PROPN
ejpam-6909	1	3	of	of	ADP
ejpam-6909	1	4	pure	pure	ADJ
ejpam-6909	1	5	and	and	CCONJ
ejpam-6909	1	6	applied	applied	ADJ
ejpam-6909	1	7	mathematics	mathematic	NOUN
ejpam-6909	1	8	2025	2025	NUM
ejpam-6909	1	9	,	,	PUNCT
ejpam-6909	1	10	vol	vol	NOUN
ejpam-6909	1	11	.	.	PROPN
ejpam-6909	1	12	18	18	NUM
ejpam-6909	1	13	,	,	PUNCT
ejpam-6909	1	14	issue	issue	NOUN
ejpam-6909	1	15	4	4	NUM
ejpam-6909	1	16	,	,	PUNCT
ejpam-6909	1	17	article	article	NOUN
ejpam-6909	1	18	number	number	NOUN
ejpam-6909	1	19	6909	6909	NUM
ejpam-6909	1	20	issn	issn	PROPN
ejpam-6909	1	21	1307	1307	NUM
ejpam-6909	1	22	-	-	SYM
ejpam-6909	1	23	5543	5543	NUM
ejpam-6909	1	24	–	–	PUNCT
ejpam-6909	1	25	ejpam.com	ejpam.com	X
ejpam-6909	1	26	published	publish	VERB
ejpam-6909	1	27	by	by	ADP
ejpam-6909	1	28	new	new	PROPN
ejpam-6909	1	29	york	york	PROPN
ejpam-6909	1	30	business	business	PROPN
ejpam-6909	1	31	global	global	ADJ
ejpam-6909	1	32	bipolar	bipolar	ADJ
ejpam-6909	1	33	fuzzy	fuzzy	ADJ
ejpam-6909	1	34	commutative	commutative	ADJ
ejpam-6909	1	35	hyper	hyper	ADJ
ejpam-6909	1	36	bck	bck	NOUN
ejpam-6909	1	37	-	-	PUNCT
ejpam-6909	1	38	ideals	ideal	NOUN
ejpam-6909	1	39	in	in	ADP
ejpam-6909	1	40	hyper	hyper	ADJ
ejpam-6909	1	41	bck	bck	NOUN
ejpam-6909	1	42	-	-	PUNCT
ejpam-6909	1	43	algebras	algebras	PROPN
ejpam-6909	1	44	d.	d.	PROPN
ejpam-6909	1	45	ramesh1	ramesh1	PROPN
ejpam-6909	1	46	,	,	PUNCT
ejpam-6909	1	47	shake	shake	VERB
ejpam-6909	1	48	baji2	baji2	NOUN
ejpam-6909	1	49	,	,	PUNCT
ejpam-6909	1	50	aiyared	aiyare	VERB
ejpam-6909	1	51	iampan3,∗	iampan3,∗	NOUN
ejpam-6909	1	52	,	,	PUNCT
ejpam-6909	1	53	b.	b.	PROPN
ejpam-6909	1	54	satyanarayana4	satyanarayana4	PROPN
ejpam-6909	2	1	1	1	NUM
ejpam-6909	2	2	department	department	NOUN
ejpam-6909	2	3	of	of	ADP
ejpam-6909	2	4	engineering	engineering	NOUN
ejpam-6909	2	5	mathematics	mathematic	NOUN
ejpam-6909	2	6	,	,	PUNCT
ejpam-6909	2	7	college	college	NOUN
ejpam-6909	2	8	of	of	ADP
ejpam-6909	2	9	engineering	engineering	PROPN
ejpam-6909	2	10	,	,	PUNCT
ejpam-6909	2	11	koneru	koneru	PROPN
ejpam-6909	2	12	lakshmaiah	lakshmaiah	PROPN
ejpam-6909	2	13	educational	educational	ADJ
ejpam-6909	2	14	foundation	foundation	PROPN
ejpam-6909	2	15	,	,	PUNCT
ejpam-6909	2	16	vaddeswaram	vaddeswaram	PROPN
ejpam-6909	2	17	,	,	PUNCT
ejpam-6909	2	18	andhra	andhra	PROPN
ejpam-6909	2	19	pradesh-522302	pradesh-522302	NOUN
ejpam-6909	2	20	,	,	PUNCT
ejpam-6909	2	21	india	india	PROPN
ejpam-6909	2	22	2	2	NUM
ejpam-6909	2	23	department	department	NOUN
ejpam-6909	2	24	of	of	ADP
ejpam-6909	2	25	mathematics	mathematic	NOUN
ejpam-6909	2	26	,	,	PUNCT
ejpam-6909	2	27	sir	sir	PROPN
ejpam-6909	2	28	c.r	c.r	PROPN
ejpam-6909	2	29	.	.	PROPN
ejpam-6909	2	30	reddy	reddy	PROPN
ejpam-6909	2	31	college	college	PROPN
ejpam-6909	2	32	of	of	ADP
ejpam-6909	2	33	engineering	engineering	NOUN
ejpam-6909	2	34	,	,	PUNCT
ejpam-6909	2	35	eluru-534007	eluru-534007	NOUN
ejpam-6909	2	36	,	,	PUNCT
ejpam-6909	2	37	andhra	andhra	PROPN
ejpam-6909	2	38	pradesh	pradesh	PROPN
ejpam-6909	2	39	,	,	PUNCT
ejpam-6909	2	40	india	india	PROPN
ejpam-6909	2	41	3	3	PROPN
ejpam-6909	2	42	department	department	PROPN
ejpam-6909	2	43	of	of	ADP
ejpam-6909	2	44	mathematics	mathematic	NOUN
ejpam-6909	2	45	,	,	PUNCT
ejpam-6909	2	46	school	school	NOUN
ejpam-6909	2	47	of	of	ADP
ejpam-6909	2	48	science	science	NOUN
ejpam-6909	2	49	,	,	PUNCT
ejpam-6909	2	50	university	university	NOUN
ejpam-6909	2	51	of	of	ADP
ejpam-6909	2	52	phayao	phayao	NOUN
ejpam-6909	2	53	,	,	PUNCT
ejpam-6909	2	54	mae	mae	PROPN
ejpam-6909	2	55	ka	ka	PROPN
ejpam-6909	2	56	,	,	PUNCT
ejpam-6909	2	57	mueang	mueang	PROPN
ejpam-6909	2	58	,	,	PUNCT
ejpam-6909	2	59	phayao	phayao	NOUN
ejpam-6909	2	60	56000	56000	NUM
ejpam-6909	2	61	,	,	PUNCT
ejpam-6909	2	62	thailand	thailand	PROPN
ejpam-6909	2	63	4	4	NUM
ejpam-6909	2	64	department	department	NOUN
ejpam-6909	2	65	of	of	ADP
ejpam-6909	2	66	mathematics	mathematic	NOUN
ejpam-6909	2	67	,	,	PUNCT
ejpam-6909	2	68	acharya	acharya	PROPN
ejpam-6909	2	69	nagarjuna	nagarjuna	PROPN
ejpam-6909	2	70	university	university	PROPN
ejpam-6909	2	71	,	,	PUNCT
ejpam-6909	2	72	nagarjuna	nagarjuna	PROPN
ejpam-6909	2	73	nagar	nagar	PROPN
ejpam-6909	2	74	,	,	PUNCT
ejpam-6909	2	75	guntur-522	guntur-522	NOUN
ejpam-6909	2	76	510	510	NUM
ejpam-6909	2	77	,	,	PUNCT
ejpam-6909	2	78	andhra	andhra	PROPN
ejpam-6909	2	79	pradesh	pradesh	PROPN
ejpam-6909	2	80	,	,	PUNCT
ejpam-6909	2	81	india	india	PROPN
ejpam-6909	2	82	abstract	abstract	NOUN
ejpam-6909	2	83	.	.	PUNCT
ejpam-6909	3	1	this	this	DET
ejpam-6909	3	2	study	study	NOUN
ejpam-6909	3	3	introduces	introduce	VERB
ejpam-6909	3	4	the	the	DET
ejpam-6909	3	5	concept	concept	NOUN
ejpam-6909	3	6	of	of	ADP
ejpam-6909	3	7	bipolar	bipolar	ADJ
ejpam-6909	3	8	fuzzy	fuzzy	ADJ
ejpam-6909	3	9	commutative	commutative	ADJ
ejpam-6909	3	10	hyper	hyper	ADJ
ejpam-6909	3	11	bck	bck	NOUN
ejpam-6909	3	12	-	-	PUNCT
ejpam-6909	3	13	ideals	ideal	NOUN
ejpam-6909	3	14	(	(	PUNCT
ejpam-6909	3	15	bf	bf	NOUN
ejpam-6909	3	16	-	-	PUNCT
ejpam-6909	3	17	chbckis	chbcki	NOUN
ejpam-6909	3	18	)	)	PUNCT
ejpam-6909	3	19	within	within	ADP
ejpam-6909	3	20	the	the	DET
ejpam-6909	3	21	algebraic	algebraic	ADJ
ejpam-6909	3	22	framework	framework	NOUN
ejpam-6909	3	23	of	of	ADP
ejpam-6909	3	24	hyper	hyper	ADJ
ejpam-6909	3	25	bck	bck	NOUN
ejpam-6909	3	26	-	-	PUNCT
ejpam-6909	3	27	algebras	algebras	X
ejpam-6909	3	28	,	,	PUNCT
ejpam-6909	3	29	offering	offer	VERB
ejpam-6909	3	30	a	a	DET
ejpam-6909	3	31	novel	novel	ADJ
ejpam-6909	3	32	approach	approach	NOUN
ejpam-6909	3	33	to	to	ADP
ejpam-6909	3	34	modeling	model	VERB
ejpam-6909	3	35	dual	dual	ADJ
ejpam-6909	3	36	uncertainty	uncertainty	NOUN
ejpam-6909	3	37	through	through	ADP
ejpam-6909	3	38	bipolar	bipolar	ADJ
ejpam-6909	3	39	fuzzy	fuzzy	ADJ
ejpam-6909	3	40	sets	set	NOUN
ejpam-6909	3	41	.	.	PUNCT
ejpam-6909	4	1	by	by	ADP
ejpam-6909	4	2	defining	define	VERB
ejpam-6909	4	3	and	and	CCONJ
ejpam-6909	4	4	classifying	classify	VERB
ejpam-6909	4	5	bf	bf	NOUN
ejpam-6909	4	6	-	-	PUNCT
ejpam-6909	4	7	chbckis	chbcki	NOUN
ejpam-6909	4	8	across	across	ADP
ejpam-6909	4	9	multiple	multiple	ADJ
ejpam-6909	4	10	types	type	NOUN
ejpam-6909	4	11	and	and	CCONJ
ejpam-6909	4	12	examining	examine	VERB
ejpam-6909	4	13	their	their	PRON
ejpam-6909	4	14	structural	structural	ADJ
ejpam-6909	4	15	relationships	relationship	NOUN
ejpam-6909	4	16	with	with	ADP
ejpam-6909	4	17	reflexive	reflexive	ADJ
ejpam-6909	4	18	,	,	PUNCT
ejpam-6909	4	19	strong	strong	ADJ
ejpam-6909	4	20	,	,	PUNCT
ejpam-6909	4	21	and	and	CCONJ
ejpam-6909	4	22	weak	weak	ADJ
ejpam-6909	4	23	hyper	hyper	ADJ
ejpam-6909	4	24	bck	bck	NOUN
ejpam-6909	4	25	-	-	PUNCT
ejpam-6909	4	26	ideals	ideal	NOUN
ejpam-6909	4	27	,	,	PUNCT
ejpam-6909	4	28	we	we	PRON
ejpam-6909	4	29	establish	establish	VERB
ejpam-6909	4	30	a	a	DET
ejpam-6909	4	31	comprehensive	comprehensive	ADJ
ejpam-6909	4	32	theoretical	theoretical	ADJ
ejpam-6909	4	33	foundation	foundation	NOUN
ejpam-6909	4	34	supported	support	VERB
ejpam-6909	4	35	by	by	ADP
ejpam-6909	4	36	formal	formal	ADJ
ejpam-6909	4	37	theorems	theorem	NOUN
ejpam-6909	4	38	and	and	CCONJ
ejpam-6909	4	39	illustrative	illustrative	ADJ
ejpam-6909	4	40	examples	example	NOUN
ejpam-6909	4	41	.	.	PUNCT
ejpam-6909	5	1	these	these	DET
ejpam-6909	5	2	findings	finding	NOUN
ejpam-6909	5	3	extend	extend	VERB
ejpam-6909	5	4	current	current	ADJ
ejpam-6909	5	5	understandings	understanding	NOUN
ejpam-6909	5	6	in	in	ADP
ejpam-6909	5	7	hyperstructure	hyperstructure	NOUN
ejpam-6909	5	8	theory	theory	NOUN
ejpam-6909	5	9	and	and	CCONJ
ejpam-6909	5	10	fuzzy	fuzzy	ADJ
ejpam-6909	5	11	algebra	algebra	NOUN
ejpam-6909	5	12	,	,	PUNCT
ejpam-6909	5	13	contributing	contribute	VERB
ejpam-6909	5	14	to	to	ADP
ejpam-6909	5	15	the	the	DET
ejpam-6909	5	16	broader	broad	ADJ
ejpam-6909	5	17	landscape	landscape	NOUN
ejpam-6909	5	18	of	of	ADP
ejpam-6909	5	19	abstract	abstract	ADJ
ejpam-6909	5	20	mathematical	mathematical	ADJ
ejpam-6909	5	21	reasoning	reasoning	NOUN
ejpam-6909	5	22	.	.	PUNCT
ejpam-6909	6	1	importantly	importantly	ADV
ejpam-6909	6	2	,	,	PUNCT
ejpam-6909	6	3	this	this	DET
ejpam-6909	6	4	research	research	NOUN
ejpam-6909	6	5	aligns	align	VERB
ejpam-6909	6	6	with	with	ADP
ejpam-6909	6	7	sustainable	sustainable	ADJ
ejpam-6909	6	8	development	development	NOUN
ejpam-6909	6	9	goal	goal	NOUN
ejpam-6909	6	10	4	4	NUM
ejpam-6909	6	11	(	(	PUNCT
ejpam-6909	6	12	sdg-4	sdg-4	X
ejpam-6909	6	13	)	)	PUNCT
ejpam-6909	6	14	by	by	ADP
ejpam-6909	6	15	promoting	promote	VERB
ejpam-6909	6	16	inclusive	inclusive	ADJ
ejpam-6909	6	17	and	and	CCONJ
ejpam-6909	6	18	equitable	equitable	ADJ
ejpam-6909	6	19	quality	quality	NOUN
ejpam-6909	6	20	education	education	NOUN
ejpam-6909	6	21	.	.	PUNCT
ejpam-6909	7	1	the	the	DET
ejpam-6909	7	2	formalization	formalization	NOUN
ejpam-6909	7	3	of	of	ADP
ejpam-6909	7	4	bf	bf	NOUN
ejpam-6909	7	5	-	-	PUNCT
ejpam-6909	7	6	chbckis	chbcki	NOUN
ejpam-6909	7	7	fosters	foster	VERB
ejpam-6909	7	8	advanced	advanced	ADJ
ejpam-6909	7	9	mathematical	mathematical	ADJ
ejpam-6909	7	10	thinking	thinking	NOUN
ejpam-6909	7	11	and	and	CCONJ
ejpam-6909	7	12	provides	provide	VERB
ejpam-6909	7	13	meaningful	meaningful	ADJ
ejpam-6909	7	14	tools	tool	NOUN
ejpam-6909	7	15	for	for	ADP
ejpam-6909	7	16	enhancing	enhance	VERB
ejpam-6909	7	17	learning	learning	NOUN
ejpam-6909	7	18	environments	environment	NOUN
ejpam-6909	7	19	,	,	PUNCT
ejpam-6909	7	20	particularly	particularly	ADV
ejpam-6909	7	21	in	in	ADP
ejpam-6909	7	22	schools	school	NOUN
ejpam-6909	7	23	and	and	CCONJ
ejpam-6909	7	24	institutions	institution	NOUN
ejpam-6909	7	25	that	that	PRON
ejpam-6909	7	26	emphasize	emphasize	VERB
ejpam-6909	7	27	research	research	NOUN
ejpam-6909	7	28	-	-	PUNCT
ejpam-6909	7	29	oriented	orient	VERB
ejpam-6909	7	30	instruction	instruction	NOUN
ejpam-6909	7	31	.	.	PUNCT
ejpam-6909	8	1	by	by	ADP
ejpam-6909	8	2	integrating	integrate	VERB
ejpam-6909	8	3	abstract	abstract	ADJ
ejpam-6909	8	4	algebraic	algebraic	ADJ
ejpam-6909	8	5	structures	structure	NOUN
ejpam-6909	8	6	with	with	ADP
ejpam-6909	8	7	uncertainty	uncertainty	NOUN
ejpam-6909	8	8	modeling	modeling	NOUN
ejpam-6909	8	9	,	,	PUNCT
ejpam-6909	8	10	this	this	DET
ejpam-6909	8	11	work	work	NOUN
ejpam-6909	8	12	supports	support	VERB
ejpam-6909	8	13	the	the	DET
ejpam-6909	8	14	cultivation	cultivation	NOUN
ejpam-6909	8	15	of	of	ADP
ejpam-6909	8	16	analytical	analytical	ADJ
ejpam-6909	8	17	skills	skill	NOUN
ejpam-6909	8	18	,	,	PUNCT
ejpam-6909	8	19	mathematical	mathematical	ADJ
ejpam-6909	8	20	creativity	creativity	NOUN
ejpam-6909	8	21	,	,	PUNCT
ejpam-6909	8	22	and	and	CCONJ
ejpam-6909	8	23	deeper	deep	ADJ
ejpam-6909	8	24	engagement	engagement	NOUN
ejpam-6909	8	25	with	with	ADP
ejpam-6909	8	26	formal	formal	ADJ
ejpam-6909	8	27	logic	logic	NOUN
ejpam-6909	8	28	among	among	ADP
ejpam-6909	8	29	students	student	NOUN
ejpam-6909	8	30	and	and	CCONJ
ejpam-6909	8	31	emerging	emerge	VERB
ejpam-6909	8	32	researchers	researcher	NOUN
ejpam-6909	8	33	.	.	PUNCT
ejpam-6909	9	1	2020	2020	NUM
ejpam-6909	9	2	mathematics	mathematic	NOUN
ejpam-6909	9	3	subject	subject	NOUN
ejpam-6909	9	4	classifications	classification	NOUN
ejpam-6909	9	5	:	:	PUNCT
ejpam-6909	9	6	03e72	03e72	NUM
ejpam-6909	9	7	,	,	PUNCT
ejpam-6909	9	8	06f35	06f35	NUM
ejpam-6909	9	9	,	,	PUNCT
ejpam-6909	9	10	03g25	03g25	NOUN
ejpam-6909	9	11	key	key	ADJ
ejpam-6909	9	12	words	word	NOUN
ejpam-6909	9	13	and	and	CCONJ
ejpam-6909	9	14	phrases	phrase	NOUN
ejpam-6909	9	15	:	:	PUNCT
ejpam-6909	9	16	hyper	hyper	ADJ
ejpam-6909	9	17	bck	bck	NOUN
ejpam-6909	9	18	-	-	PUNCT
ejpam-6909	9	19	algebra	algebra	NOUN
ejpam-6909	9	20	(	(	PUNCT
ejpam-6909	9	21	hbcka	hbcka	NOUN
ejpam-6909	9	22	)	)	PUNCT
ejpam-6909	9	23	,	,	PUNCT
ejpam-6909	9	24	commutative	commutative	ADJ
ejpam-6909	9	25	hyper	hyper	ADJ
ejpam-6909	9	26	bck	bck	NOUN
ejpam-6909	9	27	-	-	PUNCT
ejpam-6909	9	28	ideal	ideal	NOUN
ejpam-6909	9	29	(	(	PUNCT
ejpam-6909	9	30	chbcki	chbcki	ADJ
ejpam-6909	9	31	)	)	PUNCT
ejpam-6909	9	32	,	,	PUNCT
ejpam-6909	9	33	fuzzy	fuzzy	ADJ
ejpam-6909	9	34	commutative	commutative	ADJ
ejpam-6909	9	35	hyper	hyper	ADJ
ejpam-6909	9	36	bck	bck	NOUN
ejpam-6909	9	37	-	-	PUNCT
ejpam-6909	9	38	ideal	ideal	NOUN
ejpam-6909	9	39	(	(	PUNCT
ejpam-6909	9	40	fchbcki	fchbcki	NOUN
ejpam-6909	9	41	)	)	PUNCT
ejpam-6909	9	42	,	,	PUNCT
ejpam-6909	9	43	bipolar	bipolar	ADJ
ejpam-6909	9	44	fuzzy	fuzzy	ADJ
ejpam-6909	9	45	commutative	commutative	ADJ
ejpam-6909	9	46	hyper	hyper	ADJ
ejpam-6909	9	47	bck	bck	NOUN
ejpam-6909	9	48	-	-	PUNCT
ejpam-6909	9	49	ideal	ideal	NOUN
ejpam-6909	9	50	(	(	PUNCT
ejpam-6909	9	51	bf	bf	NOUN
ejpam-6909	9	52	-	-	PUNCT
ejpam-6909	9	53	chbcki	chbcki	NOUN
ejpam-6909	9	54	)	)	PUNCT
ejpam-6909	9	55	∗corresponding	∗corresponde	VERB
ejpam-6909	9	56	author	author	NOUN
ejpam-6909	9	57	.	.	PUNCT
ejpam-6909	10	1	doi	doi	NOUN
ejpam-6909	10	2	:	:	PUNCT
ejpam-6909	10	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6909	https://doi.org/10.29020/nybg.ejpam.v18i4.6909	ADJ
ejpam-6909	10	4	email	email	NOUN
ejpam-6909	10	5	addresses	address	NOUN
ejpam-6909	10	6	:	:	PUNCT
ejpam-6909	10	7	ram.fuzzy@gmail.com	ram.fuzzy@gmail.com	PROPN
ejpam-6909	10	8	(	(	PUNCT
ejpam-6909	10	9	d.	d.	PROPN
ejpam-6909	10	10	ramesh	ramesh	PROPN
ejpam-6909	10	11	)	)	PUNCT
ejpam-6909	10	12	,	,	PUNCT
ejpam-6909	10	13	shakebaji6@gmail.com	shakebaji6@gmail.com	X
ejpam-6909	10	14	(	(	PUNCT
ejpam-6909	10	15	s.	s.	PROPN
ejpam-6909	10	16	baji	baji	PROPN
ejpam-6909	10	17	)	)	PUNCT
ejpam-6909	10	18	,	,	PUNCT
ejpam-6909	10	19	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-6909	10	20	(	(	PUNCT
ejpam-6909	10	21	a.	a.	NOUN
ejpam-6909	10	22	iampan	iampan	PROPN
ejpam-6909	10	23	)	)	PUNCT
ejpam-6909	10	24	,	,	PUNCT
ejpam-6909	10	25	drbsn63@yahoo.co.in	drbsn63@yahoo.co.in	NOUN
ejpam-6909	10	26	(	(	PUNCT
ejpam-6909	10	27	b.	b.	PROPN
ejpam-6909	10	28	satyanarayana	satyanarayana	PROPN
ejpam-6909	10	29	)	)	PUNCT
ejpam-6909	10	30	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6909	11	1	1	1	NUM
ejpam-6909	11	2	copyright	copyright	NOUN
ejpam-6909	11	3	:	:	PUNCT
ejpam-6909	11	4	©	©	PROPN
ejpam-6909	11	5	2025	2025	NUM
ejpam-6909	11	6	the	the	DET
ejpam-6909	11	7	author(s	author(s	NOUN
ejpam-6909	11	8	)	)	PUNCT
ejpam-6909	11	9	.	.	PUNCT
ejpam-6909	12	1	(	(	PUNCT
ejpam-6909	12	2	cc	cc	NOUN
ejpam-6909	12	3	by	by	ADP
ejpam-6909	12	4	-	-	PUNCT
ejpam-6909	12	5	nc	nc	PROPN
ejpam-6909	12	6	4.0	4.0	NUM
ejpam-6909	12	7	)	)	PUNCT
ejpam-6909	12	8	d.	d.	PROPN
ejpam-6909	12	9	ramesh	ramesh	PROPN
ejpam-6909	12	10	et	et	PROPN
ejpam-6909	12	11	al	al	PROPN
ejpam-6909	12	12	.	.	PUNCT
ejpam-6909	12	13	/	/	SYM
ejpam-6909	12	14	eur	eur	PROPN
ejpam-6909	12	15	.	.	PUNCT
ejpam-6909	13	1	j.	j.	PROPN
ejpam-6909	13	2	pure	pure	PROPN
ejpam-6909	13	3	appl	appl	PROPN
ejpam-6909	13	4	.	.	PROPN
ejpam-6909	13	5	math	math	PROPN
ejpam-6909	13	6	,	,	PUNCT
ejpam-6909	13	7	18	18	NUM
ejpam-6909	13	8	(	(	PUNCT
ejpam-6909	13	9	4	4	NUM
ejpam-6909	13	10	)	)	PUNCT
ejpam-6909	13	11	(	(	PUNCT
ejpam-6909	13	12	2025	2025	NUM
ejpam-6909	13	13	)	)	PUNCT
ejpam-6909	13	14	,	,	PUNCT
ejpam-6909	13	15	6909	6909	NUM
ejpam-6909	13	16	2	2	NUM
ejpam-6909	13	17	of	of	ADP
ejpam-6909	13	18	16	16	NUM
ejpam-6909	13	19	1	1	NUM
ejpam-6909	13	20	.	.	PUNCT
ejpam-6909	13	21	introduction	introduction	NOUN
ejpam-6909	13	22	the	the	DET
ejpam-6909	13	23	study	study	NOUN
ejpam-6909	13	24	of	of	ADP
ejpam-6909	13	25	algebraic	algebraic	ADJ
ejpam-6909	13	26	structures	structure	NOUN
ejpam-6909	13	27	provides	provide	VERB
ejpam-6909	13	28	a	a	DET
ejpam-6909	13	29	common	common	ADJ
ejpam-6909	13	30	foundation	foundation	NOUN
ejpam-6909	13	31	for	for	ADP
ejpam-6909	13	32	understanding	understand	VERB
ejpam-6909	13	33	a	a	DET
ejpam-6909	13	34	wide	wide	ADJ
ejpam-6909	13	35	range	range	NOUN
ejpam-6909	13	36	of	of	ADP
ejpam-6909	13	37	mathematical	mathematical	ADJ
ejpam-6909	13	38	concepts	concept	NOUN
ejpam-6909	13	39	.	.	PUNCT
ejpam-6909	14	1	these	these	DET
ejpam-6909	14	2	structures	structure	NOUN
ejpam-6909	14	3	form	form	VERB
ejpam-6909	14	4	the	the	DET
ejpam-6909	14	5	mathematical	mathematical	ADJ
ejpam-6909	14	6	basis	basis	NOUN
ejpam-6909	14	7	for	for	ADP
ejpam-6909	14	8	many	many	ADJ
ejpam-6909	14	9	of	of	ADP
ejpam-6909	14	10	the	the	DET
ejpam-6909	14	11	algorithms	algorithm	NOUN
ejpam-6909	14	12	and	and	CCONJ
ejpam-6909	14	13	protocols	protocol	NOUN
ejpam-6909	14	14	that	that	PRON
ejpam-6909	14	15	support	support	VERB
ejpam-6909	14	16	our	our	PRON
ejpam-6909	14	17	digital	digital	ADJ
ejpam-6909	14	18	world	world	NOUN
ejpam-6909	14	19	.	.	PUNCT
ejpam-6909	15	1	these	these	PRON
ejpam-6909	15	2	provide	provide	VERB
ejpam-6909	15	3	the	the	DET
ejpam-6909	15	4	way	way	NOUN
ejpam-6909	15	5	to	to	PART
ejpam-6909	15	6	study	study	VERB
ejpam-6909	15	7	mathematical	mathematical	ADJ
ejpam-6909	15	8	operations	operation	NOUN
ejpam-6909	15	9	in	in	ADP
ejpam-6909	15	10	their	their	PRON
ejpam-6909	15	11	most	most	ADV
ejpam-6909	15	12	general	general	ADJ
ejpam-6909	15	13	form	form	NOUN
ejpam-6909	15	14	.	.	PUNCT
ejpam-6909	16	1	by	by	ADP
ejpam-6909	16	2	concentrating	concentrate	VERB
ejpam-6909	16	3	on	on	ADP
ejpam-6909	16	4	the	the	DET
ejpam-6909	16	5	essential	essential	ADJ
ejpam-6909	16	6	properties	property	NOUN
ejpam-6909	16	7	of	of	ADP
ejpam-6909	16	8	algebraic	algebraic	ADJ
ejpam-6909	16	9	structures	structure	NOUN
ejpam-6909	16	10	,	,	PUNCT
ejpam-6909	16	11	we	we	PRON
ejpam-6909	16	12	can	can	AUX
ejpam-6909	16	13	gain	gain	VERB
ejpam-6909	16	14	a	a	DET
ejpam-6909	16	15	deeper	deep	ADJ
ejpam-6909	16	16	understanding	understanding	NOUN
ejpam-6909	16	17	of	of	ADP
ejpam-6909	16	18	the	the	DET
ejpam-6909	16	19	underlying	underlie	VERB
ejpam-6909	16	20	ideas	idea	NOUN
ejpam-6909	16	21	of	of	ADP
ejpam-6909	16	22	mathematical	mathematical	ADJ
ejpam-6909	16	23	systems	system	NOUN
ejpam-6909	16	24	.	.	PUNCT
ejpam-6909	17	1	in	in	ADP
ejpam-6909	17	2	1966	1966	NUM
ejpam-6909	17	3	,	,	PUNCT
ejpam-6909	17	4	imai	imai	PROPN
ejpam-6909	17	5	et	et	PROPN
ejpam-6909	17	6	al	al	PROPN
ejpam-6909	17	7	.	.	PUNCT
ejpam-6909	18	1	(	(	PUNCT
ejpam-6909	18	2	see	see	VERB
ejpam-6909	18	3	[	[	X
ejpam-6909	18	4	1–3	1–3	NOUN
ejpam-6909	18	5	]	]	PUNCT
ejpam-6909	18	6	)	)	PUNCT
ejpam-6909	18	7	introduced	introduce	VERB
ejpam-6909	18	8	the	the	DET
ejpam-6909	18	9	algebraic	algebraic	ADJ
ejpam-6909	18	10	structure	structure	NOUN
ejpam-6909	18	11	called	call	VERB
ejpam-6909	18	12	a	a	DET
ejpam-6909	18	13	bck	bck	NOUN
ejpam-6909	18	14	-	-	PUNCT
ejpam-6909	18	15	algebra	algebra	NOUN
ejpam-6909	18	16	as	as	ADP
ejpam-6909	18	17	an	an	DET
ejpam-6909	18	18	extension	extension	NOUN
ejpam-6909	18	19	of	of	ADP
ejpam-6909	18	20	the	the	DET
ejpam-6909	18	21	concepts	concept	NOUN
ejpam-6909	18	22	of	of	ADP
ejpam-6909	18	23	propositional	propositional	ADJ
ejpam-6909	18	24	calculus	calculus	NOUN
ejpam-6909	18	25	and	and	CCONJ
ejpam-6909	18	26	set	set	ADJ
ejpam-6909	18	27	-	-	PUNCT
ejpam-6909	18	28	theoretic	theoretic	NOUN
ejpam-6909	18	29	difference	difference	NOUN
ejpam-6909	18	30	.	.	PUNCT
ejpam-6909	19	1	following	follow	VERB
ejpam-6909	19	2	their	their	PRON
ejpam-6909	19	3	introduction	introduction	NOUN
ejpam-6909	19	4	,	,	PUNCT
ejpam-6909	19	5	many	many	ADJ
ejpam-6909	19	6	researchers	researcher	NOUN
ejpam-6909	19	7	have	have	AUX
ejpam-6909	19	8	extensively	extensively	ADV
ejpam-6909	19	9	studied	study	VERB
ejpam-6909	19	10	bck	bck	NOUN
ejpam-6909	19	11	-	-	PUNCT
ejpam-6909	19	12	algebras	algebra	NOUN
ejpam-6909	19	13	,	,	PUNCT
ejpam-6909	19	14	especially	especially	ADV
ejpam-6909	19	15	on	on	ADP
ejpam-6909	19	16	ideals	ideal	NOUN
ejpam-6909	19	17	.	.	PUNCT
ejpam-6909	20	1	in	in	ADP
ejpam-6909	20	2	1934	1934	NUM
ejpam-6909	20	3	,	,	PUNCT
ejpam-6909	20	4	marty	marty	PROPN
ejpam-6909	20	5	[	[	X
ejpam-6909	20	6	4	4	X
ejpam-6909	20	7	]	]	PUNCT
ejpam-6909	20	8	presented	present	VERB
ejpam-6909	20	9	the	the	DET
ejpam-6909	20	10	idea	idea	NOUN
ejpam-6909	20	11	of	of	ADP
ejpam-6909	20	12	hyperstructure	hyperstructure	PROPN
ejpam-6909	20	13	theory	theory	NOUN
ejpam-6909	20	14	,	,	PUNCT
ejpam-6909	20	15	or	or	CCONJ
ejpam-6909	20	16	multi	multi	NOUN
ejpam-6909	20	17	-	-	NOUN
ejpam-6909	20	18	algebras	algebra	NOUN
ejpam-6909	20	19	,	,	PUNCT
ejpam-6909	20	20	during	during	ADP
ejpam-6909	20	21	the	the	DET
ejpam-6909	20	22	8th	8th	ADJ
ejpam-6909	20	23	congress	congress	NOUN
ejpam-6909	20	24	of	of	ADP
ejpam-6909	20	25	scandinavian	scandinavian	ADJ
ejpam-6909	20	26	mathematicians	mathematician	NOUN
ejpam-6909	20	27	.	.	PUNCT
ejpam-6909	21	1	there	there	PRON
ejpam-6909	21	2	are	be	VERB
ejpam-6909	21	3	numerous	numerous	ADJ
ejpam-6909	21	4	areas	area	NOUN
ejpam-6909	21	5	in	in	ADP
ejpam-6909	21	6	both	both	PRON
ejpam-6909	21	7	applied	apply	VERB
ejpam-6909	21	8	and	and	CCONJ
ejpam-6909	21	9	pure	pure	ADJ
ejpam-6909	21	10	research	research	NOUN
ejpam-6909	21	11	where	where	SCONJ
ejpam-6909	21	12	the	the	DET
ejpam-6909	21	13	idea	idea	NOUN
ejpam-6909	21	14	of	of	ADP
ejpam-6909	21	15	hyperstructures	hyperstructure	NOUN
ejpam-6909	21	16	can	can	AUX
ejpam-6909	21	17	be	be	AUX
ejpam-6909	21	18	useful	useful	ADJ
ejpam-6909	21	19	.	.	PUNCT
ejpam-6909	22	1	by	by	ADP
ejpam-6909	22	2	applying	apply	VERB
ejpam-6909	22	3	hyperstructure	hyperstructure	NOUN
ejpam-6909	22	4	theory	theory	NOUN
ejpam-6909	22	5	to	to	PART
ejpam-6909	22	6	bck	bck	VERB
ejpam-6909	22	7	-	-	PUNCT
ejpam-6909	22	8	algebras	algebras	PROPN
ejpam-6909	22	9	,	,	PUNCT
ejpam-6909	22	10	jun	jun	PROPN
ejpam-6909	22	11	et	et	PROPN
ejpam-6909	22	12	al	al	PROPN
ejpam-6909	22	13	.	.	PUNCT
ejpam-6909	23	1	[	[	X
ejpam-6909	23	2	5	5	NUM
ejpam-6909	23	3	]	]	PUNCT
ejpam-6909	23	4	developed	develop	VERB
ejpam-6909	23	5	a	a	DET
ejpam-6909	23	6	hyper	hyper	ADJ
ejpam-6909	23	7	bck	bck	NOUN
ejpam-6909	23	8	-	-	PUNCT
ejpam-6909	23	9	algebra	algebra	NOUN
ejpam-6909	23	10	,	,	PUNCT
ejpam-6909	23	11	expanded	expand	VERB
ejpam-6909	23	12	the	the	DET
ejpam-6909	23	13	bck	bck	NOUN
ejpam-6909	23	14	-	-	PUNCT
ejpam-6909	23	15	algebra	algebra	NOUN
ejpam-6909	23	16	,	,	PUNCT
ejpam-6909	23	17	and	and	CCONJ
ejpam-6909	23	18	studied	study	VERB
ejpam-6909	23	19	its	its	PRON
ejpam-6909	23	20	related	related	ADJ
ejpam-6909	23	21	properties	property	NOUN
ejpam-6909	23	22	.	.	PUNCT
ejpam-6909	24	1	building	build	VERB
ejpam-6909	24	2	on	on	ADP
ejpam-6909	24	3	the	the	DET
ejpam-6909	24	4	foundational	foundational	ADJ
ejpam-6909	24	5	structure	structure	NOUN
ejpam-6909	24	6	of	of	ADP
ejpam-6909	24	7	hyper	hyper	ADJ
ejpam-6909	24	8	bck	bck	NOUN
ejpam-6909	24	9	-	-	PUNCT
ejpam-6909	24	10	algebras	algebra	NOUN
ejpam-6909	24	11	,	,	PUNCT
ejpam-6909	24	12	borzooei	borzooei	PROPN
ejpam-6909	24	13	and	and	CCONJ
ejpam-6909	24	14	bakhshi	bakhshi	PROPN
ejpam-6909	24	15	[	[	X
ejpam-6909	24	16	6	6	NUM
ejpam-6909	24	17	]	]	PUNCT
ejpam-6909	24	18	systematically	systematically	ADV
ejpam-6909	24	19	introduced	introduce	VERB
ejpam-6909	24	20	four	four	NUM
ejpam-6909	24	21	distinct	distinct	ADJ
ejpam-6909	24	22	types	type	NOUN
ejpam-6909	24	23	of	of	ADP
ejpam-6909	24	24	commutative	commutative	ADJ
ejpam-6909	24	25	hyper	hyper	ADJ
ejpam-6909	24	26	bck	bck	NOUN
ejpam-6909	24	27	-	-	PUNCT
ejpam-6909	24	28	ideals	ideal	NOUN
ejpam-6909	24	29	(	(	PUNCT
ejpam-6909	24	30	chbckis	chbckis	PROPN
ejpam-6909	24	31	)	)	PUNCT
ejpam-6909	24	32	,	,	PUNCT
ejpam-6909	24	33	laying	lay	VERB
ejpam-6909	24	34	the	the	DET
ejpam-6909	24	35	groundwork	groundwork	NOUN
ejpam-6909	24	36	for	for	ADP
ejpam-6909	24	37	a	a	DET
ejpam-6909	24	38	series	series	NOUN
ejpam-6909	24	39	of	of	ADP
ejpam-6909	24	40	significant	significant	ADJ
ejpam-6909	24	41	algebraic	algebraic	ADJ
ejpam-6909	24	42	results	result	NOUN
ejpam-6909	24	43	.	.	PUNCT
ejpam-6909	25	1	expanding	expand	VERB
ejpam-6909	25	2	this	this	DET
ejpam-6909	25	3	framework	framework	NOUN
ejpam-6909	25	4	into	into	ADP
ejpam-6909	25	5	the	the	DET
ejpam-6909	25	6	domain	domain	NOUN
ejpam-6909	25	7	of	of	ADP
ejpam-6909	25	8	uncertainty	uncertainty	NOUN
ejpam-6909	25	9	modeling	modeling	NOUN
ejpam-6909	25	10	,	,	PUNCT
ejpam-6909	25	11	durga	durga	NOUN
ejpam-6909	25	12	prasad	prasad	PROPN
ejpam-6909	25	13	et	et	PROPN
ejpam-6909	25	14	al	al	PROPN
ejpam-6909	25	15	.	.	PUNCT
ejpam-6909	26	1	[	[	X
ejpam-6909	26	2	7	7	X
ejpam-6909	26	3	]	]	PUNCT
ejpam-6909	26	4	proposed	propose	VERB
ejpam-6909	26	5	the	the	DET
ejpam-6909	26	6	concepts	concept	NOUN
ejpam-6909	26	7	of	of	ADP
ejpam-6909	26	8	intuitionistic	intuitionistic	ADJ
ejpam-6909	26	9	fuzzy	fuzzy	ADJ
ejpam-6909	26	10	positive	positive	ADJ
ejpam-6909	26	11	implicative	implicative	ADJ
ejpam-6909	26	12	hyper	hyper	ADJ
ejpam-6909	26	13	bck	bck	NOUN
ejpam-6909	26	14	-	-	PUNCT
ejpam-6909	26	15	ideals	ideal	NOUN
ejpam-6909	26	16	,	,	PUNCT
ejpam-6909	26	17	also	also	ADV
ejpam-6909	26	18	categorized	categorize	VERB
ejpam-6909	26	19	into	into	ADP
ejpam-6909	26	20	types	type	NOUN
ejpam-6909	26	21	1	1	NUM
ejpam-6909	26	22	,	,	PUNCT
ejpam-6909	26	23	2	2	NUM
ejpam-6909	26	24	,	,	PUNCT
ejpam-6909	26	25	3	3	NUM
ejpam-6909	26	26	,	,	PUNCT
ejpam-6909	26	27	and	and	CCONJ
ejpam-6909	26	28	4	4	NUM
ejpam-6909	26	29	,	,	PUNCT
ejpam-6909	26	30	thereby	thereby	ADV
ejpam-6909	26	31	bridging	bridge	VERB
ejpam-6909	26	32	intuitionistic	intuitionistic	ADJ
ejpam-6909	26	33	fuzzy	fuzzy	ADJ
ejpam-6909	26	34	logic	logic	NOUN
ejpam-6909	26	35	with	with	ADP
ejpam-6909	26	36	hyperstructure	hyperstructure	NOUN
ejpam-6909	26	37	theory	theory	NOUN
ejpam-6909	26	38	.	.	PUNCT
ejpam-6909	27	1	further	far	ADV
ejpam-6909	27	2	advancing	advance	VERB
ejpam-6909	27	3	this	this	DET
ejpam-6909	27	4	trajectory	trajectory	NOUN
ejpam-6909	27	5	,	,	PUNCT
ejpam-6909	27	6	satyanarayana	satyanarayana	PROPN
ejpam-6909	27	7	et	et	PROPN
ejpam-6909	27	8	al	al	PROPN
ejpam-6909	27	9	.	.	PUNCT
ejpam-6909	28	1	[	[	X
ejpam-6909	28	2	8	8	NUM
ejpam-6909	28	3	]	]	PUNCT
ejpam-6909	28	4	developed	develop	VERB
ejpam-6909	28	5	the	the	DET
ejpam-6909	28	6	notions	notion	NOUN
ejpam-6909	28	7	of	of	ADP
ejpam-6909	28	8	intuitionistic	intuitionistic	ADJ
ejpam-6909	28	9	fuzzy	fuzzy	ADJ
ejpam-6909	28	10	commutative	commutative	ADJ
ejpam-6909	28	11	hyper	hyper	ADJ
ejpam-6909	28	12	bck	bck	NOUN
ejpam-6909	28	13	-	-	PUNCT
ejpam-6909	28	14	ideals	ideal	NOUN
ejpam-6909	28	15	across	across	ADP
ejpam-6909	28	16	the	the	DET
ejpam-6909	28	17	same	same	ADJ
ejpam-6909	28	18	typological	typological	ADJ
ejpam-6909	28	19	classifications	classification	NOUN
ejpam-6909	28	20	,	,	PUNCT
ejpam-6909	28	21	reinforcing	reinforce	VERB
ejpam-6909	28	22	the	the	DET
ejpam-6909	28	23	depth	depth	NOUN
ejpam-6909	28	24	and	and	CCONJ
ejpam-6909	28	25	versatility	versatility	NOUN
ejpam-6909	28	26	of	of	ADP
ejpam-6909	28	27	fuzzy	fuzzy	ADJ
ejpam-6909	28	28	extensions	extension	NOUN
ejpam-6909	28	29	within	within	ADP
ejpam-6909	28	30	commutative	commutative	ADJ
ejpam-6909	28	31	hyper	hyper	ADJ
ejpam-6909	28	32	bck	bck	VERB
ejpam-6909	28	33	-	-	PUNCT
ejpam-6909	28	34	algebraic	algebraic	ADJ
ejpam-6909	28	35	systems	system	NOUN
ejpam-6909	28	36	.	.	PUNCT
ejpam-6909	29	1	numerous	numerous	ADJ
ejpam-6909	29	2	techniques	technique	NOUN
ejpam-6909	29	3	extend	extend	VERB
ejpam-6909	29	4	the	the	DET
ejpam-6909	29	5	concept	concept	NOUN
ejpam-6909	29	6	of	of	ADP
ejpam-6909	29	7	a	a	DET
ejpam-6909	29	8	set	set	NOUN
ejpam-6909	29	9	,	,	PUNCT
ejpam-6909	29	10	with	with	ADP
ejpam-6909	29	11	zadeh	zadeh	PROPN
ejpam-6909	29	12	’s	’s	PART
ejpam-6909	29	13	fuzzy	fuzzy	ADJ
ejpam-6909	29	14	sets	set	NOUN
ejpam-6909	29	15	[	[	X
ejpam-6909	29	16	9	9	NUM
ejpam-6909	29	17	]	]	PUNCT
ejpam-6909	29	18	being	be	AUX
ejpam-6909	29	19	the	the	DET
ejpam-6909	29	20	most	most	ADV
ejpam-6909	29	21	prominent	prominent	ADJ
ejpam-6909	29	22	example	example	NOUN
ejpam-6909	29	23	.	.	PUNCT
ejpam-6909	30	1	fuzzy	fuzzy	ADJ
ejpam-6909	30	2	sets	set	NOUN
ejpam-6909	30	3	allow	allow	VERB
ejpam-6909	30	4	elements	element	NOUN
ejpam-6909	30	5	to	to	PART
ejpam-6909	30	6	belong	belong	VERB
ejpam-6909	30	7	to	to	ADP
ejpam-6909	30	8	a	a	DET
ejpam-6909	30	9	set	set	NOUN
ejpam-6909	30	10	with	with	ADP
ejpam-6909	30	11	a	a	DET
ejpam-6909	30	12	degree	degree	NOUN
ejpam-6909	30	13	of	of	ADP
ejpam-6909	30	14	membership	membership	NOUN
ejpam-6909	30	15	,	,	PUNCT
ejpam-6909	30	16	known	know	VERB
ejpam-6909	30	17	as	as	ADP
ejpam-6909	30	18	the	the	DET
ejpam-6909	30	19	membership	membership	NOUN
ejpam-6909	30	20	grade	grade	NOUN
ejpam-6909	30	21	(	(	PUNCT
ejpam-6909	30	22	between	between	ADP
ejpam-6909	30	23	0	0	NUM
ejpam-6909	30	24	and	and	CCONJ
ejpam-6909	30	25	1	1	NUM
ejpam-6909	30	26	)	)	PUNCT
ejpam-6909	30	27	.	.	PUNCT
ejpam-6909	31	1	they	they	PRON
ejpam-6909	31	2	provide	provide	VERB
ejpam-6909	31	3	a	a	DET
ejpam-6909	31	4	framework	framework	NOUN
ejpam-6909	31	5	for	for	ADP
ejpam-6909	31	6	handling	handle	VERB
ejpam-6909	31	7	uncertainty	uncertainty	NOUN
ejpam-6909	31	8	.	.	PUNCT
ejpam-6909	32	1	lee	lee	PROPN
ejpam-6909	33	1	[	[	X
ejpam-6909	33	2	10	10	NUM
ejpam-6909	33	3	]	]	PUNCT
ejpam-6909	33	4	initially	initially	ADV
ejpam-6909	33	5	proposed	propose	VERB
ejpam-6909	33	6	the	the	DET
ejpam-6909	33	7	notion	notion	NOUN
ejpam-6909	33	8	of	of	ADP
ejpam-6909	33	9	bipolar	bipolar	ADV
ejpam-6909	33	10	-	-	PUNCT
ejpam-6909	33	11	valued	value	VERB
ejpam-6909	33	12	fuzzy	fuzzy	ADJ
ejpam-6909	33	13	sets	set	NOUN
ejpam-6909	33	14	as	as	ADP
ejpam-6909	33	15	an	an	DET
ejpam-6909	33	16	extension	extension	NOUN
ejpam-6909	33	17	of	of	ADP
ejpam-6909	33	18	fuzzy	fuzzy	ADJ
ejpam-6909	33	19	sets	set	NOUN
ejpam-6909	33	20	.	.	PUNCT
ejpam-6909	34	1	the	the	DET
ejpam-6909	34	2	membership	membership	NOUN
ejpam-6909	34	3	function	function	NOUN
ejpam-6909	34	4	that	that	PRON
ejpam-6909	34	5	characterizes	characterize	VERB
ejpam-6909	34	6	these	these	DET
ejpam-6909	34	7	sets	set	NOUN
ejpam-6909	34	8	assigns	assign	NOUN
ejpam-6909	34	9	each	each	DET
ejpam-6909	34	10	element	element	NOUN
ejpam-6909	34	11	a	a	DET
ejpam-6909	34	12	pair	pair	NOUN
ejpam-6909	34	13	of	of	ADP
ejpam-6909	34	14	values	value	NOUN
ejpam-6909	34	15	,	,	PUNCT
ejpam-6909	34	16	one	one	NUM
ejpam-6909	34	17	from	from	ADP
ejpam-6909	34	18	[	[	X
ejpam-6909	34	19	0	0	NUM
ejpam-6909	34	20	,	,	PUNCT
ejpam-6909	34	21	1	1	NUM
ejpam-6909	34	22	]	]	PUNCT
ejpam-6909	34	23	(	(	PUNCT
ejpam-6909	34	24	indicating	indicate	VERB
ejpam-6909	34	25	positive	positive	ADJ
ejpam-6909	34	26	membership	membership	NOUN
ejpam-6909	34	27	)	)	PUNCT
ejpam-6909	34	28	and	and	CCONJ
ejpam-6909	34	29	one	one	NUM
ejpam-6909	34	30	from	from	ADP
ejpam-6909	34	31	[	[	X
ejpam-6909	34	32	−1	−1	NOUN
ejpam-6909	34	33	,	,	PUNCT
ejpam-6909	34	34	0	0	NUM
ejpam-6909	34	35	]	]	PUNCT
ejpam-6909	34	36	(	(	PUNCT
ejpam-6909	34	37	indicating	indicate	VERB
ejpam-6909	34	38	negative	negative	ADJ
ejpam-6909	34	39	membership	membership	NOUN
ejpam-6909	34	40	)	)	PUNCT
ejpam-6909	34	41	.	.	PUNCT
ejpam-6909	35	1	this	this	DET
ejpam-6909	35	2	representation	representation	NOUN
ejpam-6909	35	3	is	be	AUX
ejpam-6909	35	4	effective	effective	ADJ
ejpam-6909	35	5	when	when	SCONJ
ejpam-6909	35	6	analyzing	analyze	VERB
ejpam-6909	35	7	topics	topic	NOUN
ejpam-6909	35	8	that	that	PRON
ejpam-6909	35	9	require	require	VERB
ejpam-6909	35	10	examining	examine	VERB
ejpam-6909	35	11	both	both	CCONJ
ejpam-6909	35	12	positive	positive	ADJ
ejpam-6909	35	13	and	and	CCONJ
ejpam-6909	35	14	negative	negative	ADJ
ejpam-6909	35	15	components	component	NOUN
ejpam-6909	35	16	.	.	PUNCT
ejpam-6909	36	1	the	the	DET
ejpam-6909	36	2	bipolar	bipolar	ADJ
ejpam-6909	36	3	fuzzy	fuzzy	ADJ
ejpam-6909	36	4	set	set	NOUN
ejpam-6909	36	5	(	(	PUNCT
ejpam-6909	36	6	bfs	bfs	NOUN
ejpam-6909	36	7	)	)	PUNCT
ejpam-6909	36	8	framework	framework	NOUN
ejpam-6909	36	9	has	have	AUX
ejpam-6909	36	10	emerged	emerge	VERB
ejpam-6909	36	11	as	as	ADP
ejpam-6909	36	12	a	a	DET
ejpam-6909	36	13	powerful	powerful	ADJ
ejpam-6909	36	14	extension	extension	NOUN
ejpam-6909	36	15	of	of	ADP
ejpam-6909	36	16	classical	classical	ADJ
ejpam-6909	36	17	fuzzy	fuzzy	ADJ
ejpam-6909	36	18	set	set	NOUN
ejpam-6909	36	19	theory	theory	NOUN
ejpam-6909	36	20	,	,	PUNCT
ejpam-6909	36	21	offering	offer	VERB
ejpam-6909	36	22	a	a	DET
ejpam-6909	36	23	dual	dual	ADV
ejpam-6909	36	24	-	-	PUNCT
ejpam-6909	36	25	valued	value	VERB
ejpam-6909	36	26	approach	approach	NOUN
ejpam-6909	36	27	to	to	ADP
ejpam-6909	36	28	uncertainty	uncertainty	NOUN
ejpam-6909	36	29	by	by	ADP
ejpam-6909	36	30	modeling	model	VERB
ejpam-6909	36	31	both	both	DET
ejpam-6909	36	32	degrees	degree	NOUN
ejpam-6909	36	33	of	of	ADP
ejpam-6909	36	34	satisfaction	satisfaction	NOUN
ejpam-6909	36	35	and	and	CCONJ
ejpam-6909	36	36	dissatisfaction	dissatisfaction	NOUN
ejpam-6909	36	37	.	.	PUNCT
ejpam-6909	37	1	this	this	DET
ejpam-6909	37	2	paradigm	paradigm	NOUN
ejpam-6909	37	3	has	have	AUX
ejpam-6909	37	4	been	be	AUX
ejpam-6909	37	5	fruitfully	fruitfully	ADV
ejpam-6909	37	6	applied	apply	VERB
ejpam-6909	37	7	to	to	ADP
ejpam-6909	37	8	a	a	DET
ejpam-6909	37	9	wide	wide	ADJ
ejpam-6909	37	10	range	range	NOUN
ejpam-6909	37	11	of	of	ADP
ejpam-6909	37	12	algebraic	algebraic	ADJ
ejpam-6909	37	13	structures	structure	NOUN
ejpam-6909	37	14	,	,	PUNCT
ejpam-6909	37	15	demonstrating	demonstrate	VERB
ejpam-6909	37	16	its	its	PRON
ejpam-6909	37	17	versatility	versatility	NOUN
ejpam-6909	37	18	and	and	CCONJ
ejpam-6909	37	19	depth	depth	NOUN
ejpam-6909	37	20	.	.	PUNCT
ejpam-6909	38	1	in	in	ADP
ejpam-6909	38	2	particular	particular	ADJ
ejpam-6909	38	3	,	,	PUNCT
ejpam-6909	38	4	bfs	bfs	NOUN
ejpam-6909	38	5	logic	logic	NOUN
ejpam-6909	38	6	has	have	AUX
ejpam-6909	38	7	been	be	AUX
ejpam-6909	38	8	explored	explore	VERB
ejpam-6909	38	9	in	in	ADP
ejpam-6909	38	10	the	the	DET
ejpam-6909	38	11	context	context	NOUN
ejpam-6909	38	12	of	of	ADP
ejpam-6909	38	13	near	near	ADJ
ejpam-6909	38	14	-	-	PUNCT
ejpam-6909	38	15	rings	ring	NOUN
ejpam-6909	38	16	[	[	X
ejpam-6909	38	17	11	11	NUM
ejpam-6909	38	18	]	]	PUNCT
ejpam-6909	38	19	,	,	PUNCT
ejpam-6909	38	20	where	where	SCONJ
ejpam-6909	38	21	novel	novel	ADJ
ejpam-6909	38	22	classes	class	NOUN
ejpam-6909	38	23	of	of	ADP
ejpam-6909	38	24	bipolar	bipolar	ADJ
ejpam-6909	38	25	fuzzy	fuzzy	ADJ
ejpam-6909	38	26	ideals	ideal	NOUN
ejpam-6909	38	27	have	have	AUX
ejpam-6909	38	28	been	be	AUX
ejpam-6909	38	29	introduced	introduce	VERB
ejpam-6909	38	30	.	.	PUNCT
ejpam-6909	39	1	in	in	ADP
ejpam-6909	39	2	γ	γ	NOUN
ejpam-6909	39	3	-	-	PUNCT
ejpam-6909	39	4	semirings	semiring	NOUN
ejpam-6909	39	5	,	,	PUNCT
ejpam-6909	39	6	bfss	bfss	PRON
ejpam-6909	39	7	provide	provide	VERB
ejpam-6909	39	8	a	a	DET
ejpam-6909	39	9	flexible	flexible	ADJ
ejpam-6909	39	10	tool	tool	NOUN
ejpam-6909	39	11	to	to	PART
ejpam-6909	39	12	define	define	VERB
ejpam-6909	39	13	bipolar	bipolar	ADJ
ejpam-6909	39	14	fuzzy	fuzzy	ADJ
ejpam-6909	39	15	ideals	ideal	NOUN
ejpam-6909	39	16	that	that	PRON
ejpam-6909	39	17	generalize	generalize	VERB
ejpam-6909	39	18	standard	standard	ADJ
ejpam-6909	39	19	multiplicative	multiplicative	ADJ
ejpam-6909	39	20	behaviors	behavior	NOUN
ejpam-6909	39	21	under	under	ADP
ejpam-6909	39	22	uncertainty	uncertainty	NOUN
ejpam-6909	39	23	[	[	X
ejpam-6909	39	24	12	12	NUM
ejpam-6909	39	25	]	]	PUNCT
ejpam-6909	39	26	.	.	PUNCT
ejpam-6909	40	1	the	the	DET
ejpam-6909	40	2	theory	theory	NOUN
ejpam-6909	40	3	has	have	AUX
ejpam-6909	40	4	also	also	ADV
ejpam-6909	40	5	been	be	AUX
ejpam-6909	40	6	extended	extend	VERB
ejpam-6909	40	7	to	to	ADP
ejpam-6909	40	8	semigroups	semigroup	NOUN
ejpam-6909	40	9	,	,	PUNCT
ejpam-6909	40	10	where	where	SCONJ
ejpam-6909	40	11	rough	rough	ADJ
ejpam-6909	40	12	bipolar	bipolar	ADJ
ejpam-6909	40	13	fuzzy	fuzzy	ADJ
ejpam-6909	40	14	ideals	ideal	NOUN
ejpam-6909	40	15	offer	offer	VERB
ejpam-6909	40	16	refined	refined	ADJ
ejpam-6909	40	17	ways	way	NOUN
ejpam-6909	40	18	to	to	PART
ejpam-6909	40	19	handle	handle	VERB
ejpam-6909	40	20	incomplete	incomplete	ADJ
ejpam-6909	40	21	or	or	CCONJ
ejpam-6909	40	22	vague	vague	ADJ
ejpam-6909	40	23	operations	operation	NOUN
ejpam-6909	40	24	[	[	X
ejpam-6909	40	25	13	13	NUM
ejpam-6909	40	26	]	]	PUNCT
ejpam-6909	40	27	.	.	PUNCT
ejpam-6909	41	1	more	more	ADJ
ejpam-6909	41	2	abstract	abstract	ADJ
ejpam-6909	41	3	alged	alged	ADJ
ejpam-6909	41	4	.	.	PUNCT
ejpam-6909	42	1	ramesh	ramesh	PROPN
ejpam-6909	42	2	et	et	PROPN
ejpam-6909	42	3	al	al	PROPN
ejpam-6909	42	4	.	.	PUNCT
ejpam-6909	42	5	/	/	SYM
ejpam-6909	42	6	eur	eur	PROPN
ejpam-6909	42	7	.	.	PUNCT
ejpam-6909	43	1	j.	j.	PROPN
ejpam-6909	43	2	pure	pure	PROPN
ejpam-6909	43	3	appl	appl	PROPN
ejpam-6909	43	4	.	.	PROPN
ejpam-6909	43	5	math	math	PROPN
ejpam-6909	43	6	,	,	PUNCT
ejpam-6909	43	7	18	18	NUM
ejpam-6909	43	8	(	(	PUNCT
ejpam-6909	43	9	4	4	NUM
ejpam-6909	43	10	)	)	PUNCT
ejpam-6909	43	11	(	(	PUNCT
ejpam-6909	43	12	2025	2025	NUM
ejpam-6909	43	13	)	)	PUNCT
ejpam-6909	43	14	,	,	PUNCT
ejpam-6909	43	15	6909	6909	NUM
ejpam-6909	43	16	3	3	NUM
ejpam-6909	43	17	of	of	ADP
ejpam-6909	43	18	16	16	NUM
ejpam-6909	43	19	braic	braic	ADJ
ejpam-6909	43	20	systems	system	NOUN
ejpam-6909	43	21	have	have	AUX
ejpam-6909	43	22	likewise	likewise	ADV
ejpam-6909	43	23	benefited	benefit	VERB
ejpam-6909	43	24	from	from	ADP
ejpam-6909	43	25	bfs	bfs	PROPN
ejpam-6909	43	26	modeling	modeling	NOUN
ejpam-6909	43	27	.	.	PUNCT
ejpam-6909	44	1	for	for	ADP
ejpam-6909	44	2	example	example	NOUN
ejpam-6909	44	3	,	,	PUNCT
ejpam-6909	44	4	tm	tm	NOUN
ejpam-6909	44	5	-	-	PUNCT
ejpam-6909	44	6	algebras	algebras	X
ejpam-6909	45	1	[	[	X
ejpam-6909	45	2	14	14	NUM
ejpam-6909	45	3	]	]	PUNCT
ejpam-6909	45	4	,	,	PUNCT
ejpam-6909	45	5	prime	prime	ADJ
ejpam-6909	45	6	ideals	ideal	NOUN
ejpam-6909	45	7	in	in	ADP
ejpam-6909	45	8	lattices	lattice	NOUN
ejpam-6909	45	9	[	[	X
ejpam-6909	45	10	15	15	NUM
ejpam-6909	45	11	]	]	PUNCT
ejpam-6909	45	12	,	,	PUNCT
ejpam-6909	45	13	and	and	CCONJ
ejpam-6909	45	14	subalgebras	subalgebras	PROPN
ejpam-6909	45	15	and	and	CCONJ
ejpam-6909	45	16	ideals	ideal	NOUN
ejpam-6909	45	17	in	in	ADP
ejpam-6909	45	18	bck	bck	PROPN
ejpam-6909	45	19	/	/	SYM
ejpam-6909	45	20	bci	bci	NOUN
ejpam-6909	45	21	-	-	PUNCT
ejpam-6909	45	22	algebras	algebras	X
ejpam-6909	46	1	[	[	X
ejpam-6909	46	2	16	16	NUM
ejpam-6909	46	3	]	]	PUNCT
ejpam-6909	46	4	have	have	AUX
ejpam-6909	46	5	all	all	PRON
ejpam-6909	46	6	been	be	AUX
ejpam-6909	46	7	studied	study	VERB
ejpam-6909	46	8	using	use	VERB
ejpam-6909	46	9	bipolar	bipolar	ADJ
ejpam-6909	46	10	fuzzy	fuzzy	ADJ
ejpam-6909	46	11	logic	logic	NOUN
ejpam-6909	46	12	.	.	PUNCT
ejpam-6909	47	1	within	within	ADP
ejpam-6909	47	2	the	the	DET
ejpam-6909	47	3	framework	framework	NOUN
ejpam-6909	47	4	of	of	ADP
ejpam-6909	47	5	bck	bck	PROPN
ejpam-6909	47	6	-	-	PUNCT
ejpam-6909	47	7	algebras	algebras	PROPN
ejpam-6909	47	8	,	,	PUNCT
ejpam-6909	47	9	bfss	bfss	NOUN
ejpam-6909	47	10	have	have	AUX
ejpam-6909	47	11	been	be	AUX
ejpam-6909	47	12	used	use	VERB
ejpam-6909	47	13	to	to	PART
ejpam-6909	47	14	define	define	VERB
ejpam-6909	47	15	commutative	commutative	ADJ
ejpam-6909	47	16	ideals	ideal	NOUN
ejpam-6909	47	17	[	[	X
ejpam-6909	47	18	17	17	NUM
ejpam-6909	47	19	]	]	PUNCT
ejpam-6909	47	20	,	,	PUNCT
ejpam-6909	47	21	as	as	ADV
ejpam-6909	47	22	well	well	ADV
ejpam-6909	47	23	as	as	ADP
ejpam-6909	47	24	bipolar	bipolar	ADJ
ejpam-6909	47	25	intuitionistic	intuitionistic	ADJ
ejpam-6909	47	26	fuzzy	fuzzy	ADJ
ejpam-6909	47	27	implicative	implicative	ADJ
ejpam-6909	48	1	[	[	X
ejpam-6909	48	2	18	18	NUM
ejpam-6909	48	3	]	]	PUNCT
ejpam-6909	48	4	and	and	CCONJ
ejpam-6909	48	5	positive	positive	ADJ
ejpam-6909	48	6	implicative	implicative	ADJ
ejpam-6909	48	7	ideals	ideal	NOUN
ejpam-6909	48	8	[	[	X
ejpam-6909	48	9	19	19	NUM
ejpam-6909	48	10	]	]	PUNCT
ejpam-6909	48	11	,	,	PUNCT
ejpam-6909	48	12	incorporating	incorporate	VERB
ejpam-6909	48	13	a	a	DET
ejpam-6909	48	14	richer	rich	ADJ
ejpam-6909	48	15	uncertainty	uncertainty	NOUN
ejpam-6909	48	16	semantics	semantic	NOUN
ejpam-6909	48	17	that	that	PRON
ejpam-6909	48	18	captures	capture	VERB
ejpam-6909	48	19	both	both	DET
ejpam-6909	48	20	belief	belief	NOUN
ejpam-6909	48	21	and	and	CCONJ
ejpam-6909	48	22	disbelief	disbelief	NOUN
ejpam-6909	48	23	.	.	PUNCT
ejpam-6909	49	1	bfss	bfss	PROPN
ejpam-6909	49	2	have	have	AUX
ejpam-6909	49	3	also	also	ADV
ejpam-6909	49	4	played	play	VERB
ejpam-6909	49	5	a	a	DET
ejpam-6909	49	6	significant	significant	ADJ
ejpam-6909	49	7	role	role	NOUN
ejpam-6909	49	8	in	in	ADP
ejpam-6909	49	9	hyperstructure	hyperstructure	PROPN
ejpam-6909	49	10	theory	theory	NOUN
ejpam-6909	49	11	,	,	PUNCT
ejpam-6909	49	12	particularly	particularly	ADV
ejpam-6909	49	13	in	in	ADP
ejpam-6909	49	14	the	the	DET
ejpam-6909	49	15	study	study	NOUN
ejpam-6909	49	16	of	of	ADP
ejpam-6909	49	17	hyper	hyper	ADJ
ejpam-6909	49	18	bck	bck	NOUN
ejpam-6909	49	19	-	-	PUNCT
ejpam-6909	49	20	ideals	ideal	NOUN
ejpam-6909	49	21	[	[	X
ejpam-6909	49	22	20	20	NUM
ejpam-6909	49	23	]	]	PUNCT
ejpam-6909	49	24	and	and	CCONJ
ejpam-6909	49	25	implicative	implicative	ADJ
ejpam-6909	49	26	hyper	hyper	ADJ
ejpam-6909	49	27	bck	bck	NOUN
ejpam-6909	49	28	-	-	PUNCT
ejpam-6909	49	29	ideals	ideal	NOUN
ejpam-6909	49	30	[	[	X
ejpam-6909	49	31	21	21	NUM
ejpam-6909	49	32	]	]	PUNCT
ejpam-6909	49	33	,	,	PUNCT
ejpam-6909	49	34	where	where	SCONJ
ejpam-6909	49	35	operations	operation	NOUN
ejpam-6909	49	36	yield	yield	VERB
ejpam-6909	49	37	sets	set	NOUN
ejpam-6909	49	38	rather	rather	ADV
ejpam-6909	49	39	than	than	ADP
ejpam-6909	49	40	single	single	ADJ
ejpam-6909	49	41	outcomes	outcome	NOUN
ejpam-6909	49	42	.	.	PUNCT
ejpam-6909	50	1	foundational	foundational	ADJ
ejpam-6909	50	2	work	work	NOUN
ejpam-6909	50	3	by	by	ADP
ejpam-6909	50	4	jun	jun	PROPN
ejpam-6909	50	5	and	and	CCONJ
ejpam-6909	50	6	colleagues	colleague	NOUN
ejpam-6909	51	1	[	[	X
ejpam-6909	51	2	22–24	22–24	NUM
ejpam-6909	51	3	]	]	PUNCT
ejpam-6909	51	4	introduced	introduce	VERB
ejpam-6909	51	5	and	and	CCONJ
ejpam-6909	51	6	classified	classify	VERB
ejpam-6909	51	7	several	several	ADJ
ejpam-6909	51	8	types	type	NOUN
ejpam-6909	51	9	of	of	ADP
ejpam-6909	51	10	bipolar	bipolar	ADJ
ejpam-6909	51	11	fuzzy	fuzzy	ADJ
ejpam-6909	51	12	hyper	hyper	ADJ
ejpam-6909	51	13	bck	bck	NOUN
ejpam-6909	51	14	-	-	PUNCT
ejpam-6909	51	15	ideals	ideal	NOUN
ejpam-6909	51	16	,	,	PUNCT
ejpam-6909	51	17	developing	develop	VERB
ejpam-6909	51	18	rigorous	rigorous	ADJ
ejpam-6909	51	19	definitions	definition	NOUN
ejpam-6909	51	20	and	and	CCONJ
ejpam-6909	51	21	algebraic	algebraic	ADJ
ejpam-6909	51	22	properties	property	NOUN
ejpam-6909	51	23	based	base	VERB
ejpam-6909	51	24	on	on	ADP
ejpam-6909	51	25	cut	cut	VERB
ejpam-6909	51	26	-	-	PUNCT
ejpam-6909	51	27	level	level	NOUN
ejpam-6909	51	28	approaches	approach	NOUN
ejpam-6909	51	29	and	and	CCONJ
ejpam-6909	51	30	structural	structural	ADJ
ejpam-6909	51	31	inclusion	inclusion	NOUN
ejpam-6909	51	32	.	.	PUNCT
ejpam-6909	52	1	these	these	DET
ejpam-6909	52	2	studies	study	NOUN
ejpam-6909	52	3	laid	lay	VERB
ejpam-6909	52	4	the	the	DET
ejpam-6909	52	5	theoretical	theoretical	ADJ
ejpam-6909	52	6	groundwork	groundwork	NOUN
ejpam-6909	52	7	for	for	ADP
ejpam-6909	52	8	further	further	ADJ
ejpam-6909	52	9	generalizations	generalization	NOUN
ejpam-6909	52	10	,	,	PUNCT
ejpam-6909	52	11	including	include	VERB
ejpam-6909	52	12	the	the	DET
ejpam-6909	52	13	integration	integration	NOUN
ejpam-6909	52	14	of	of	ADP
ejpam-6909	52	15	bfs	bfs	ADJ
ejpam-6909	52	16	logic	logic	NOUN
ejpam-6909	52	17	with	with	ADP
ejpam-6909	52	18	soft	soft	ADJ
ejpam-6909	52	19	set	set	NOUN
ejpam-6909	52	20	theory	theory	NOUN
ejpam-6909	52	21	[	[	X
ejpam-6909	52	22	25	25	NUM
ejpam-6909	52	23	]	]	PUNCT
ejpam-6909	52	24	,	,	PUNCT
ejpam-6909	52	25	where	where	SCONJ
ejpam-6909	52	26	muhiuddin	muhiuddin	AUX
ejpam-6909	52	27	et	et	PROPN
ejpam-6909	52	28	al	al	PROPN
ejpam-6909	52	29	.	.	PROPN
ejpam-6909	52	30	introduced	introduce	VERB
ejpam-6909	52	31	the	the	DET
ejpam-6909	52	32	notion	notion	NOUN
ejpam-6909	52	33	of	of	ADP
ejpam-6909	52	34	bipolar	bipolar	ADV
ejpam-6909	52	35	-	-	PUNCT
ejpam-6909	52	36	valued	value	VERB
ejpam-6909	52	37	fuzzy	fuzzy	ADJ
ejpam-6909	52	38	soft	soft	ADJ
ejpam-6909	52	39	hyper	hyper	ADJ
ejpam-6909	52	40	bck	bck	NOUN
ejpam-6909	52	41	-	-	PUNCT
ejpam-6909	52	42	ideals	ideal	NOUN
ejpam-6909	52	43	—	—	PUNCT
ejpam-6909	52	44	a	a	DET
ejpam-6909	52	45	hybrid	hybrid	ADJ
ejpam-6909	52	46	model	model	NOUN
ejpam-6909	52	47	that	that	PRON
ejpam-6909	52	48	enables	enable	VERB
ejpam-6909	52	49	multi	multi	ADJ
ejpam-6909	52	50	-	-	ADJ
ejpam-6909	52	51	criteria	criteria	ADJ
ejpam-6909	52	52	uncertainty	uncertainty	NOUN
ejpam-6909	52	53	reasoning	reason	VERB
ejpam-6909	52	54	in	in	ADP
ejpam-6909	52	55	hyperalgebraic	hyperalgebraic	ADJ
ejpam-6909	52	56	contexts	context	NOUN
ejpam-6909	52	57	.	.	PUNCT
ejpam-6909	53	1	recently	recently	ADV
ejpam-6909	53	2	,	,	PUNCT
ejpam-6909	53	3	this	this	DET
ejpam-6909	53	4	approach	approach	NOUN
ejpam-6909	53	5	has	have	AUX
ejpam-6909	53	6	even	even	ADV
ejpam-6909	53	7	been	be	AUX
ejpam-6909	53	8	applied	apply	VERB
ejpam-6909	53	9	to	to	ADP
ejpam-6909	53	10	fantastic	fantastic	ADJ
ejpam-6909	53	11	ideals	ideal	NOUN
ejpam-6909	53	12	in	in	ADP
ejpam-6909	53	13	bck	bck	PROPN
ejpam-6909	53	14	/	/	SYM
ejpam-6909	53	15	bci	bci	NOUN
ejpam-6909	53	16	-	-	PUNCT
ejpam-6909	53	17	algebras	algebras	X
ejpam-6909	54	1	[	[	X
ejpam-6909	54	2	26	26	NUM
ejpam-6909	54	3	]	]	X
ejpam-6909	54	4	,	,	PUNCT
ejpam-6909	54	5	further	far	ADV
ejpam-6909	54	6	emphasizing	emphasize	VERB
ejpam-6909	54	7	the	the	DET
ejpam-6909	54	8	capacity	capacity	NOUN
ejpam-6909	54	9	of	of	ADP
ejpam-6909	54	10	bfss	bfss	NOUN
ejpam-6909	54	11	to	to	PART
ejpam-6909	54	12	capture	capture	VERB
ejpam-6909	54	13	nuanced	nuanced	ADJ
ejpam-6909	54	14	algebraic	algebraic	ADJ
ejpam-6909	54	15	behavior	behavior	NOUN
ejpam-6909	54	16	under	under	ADP
ejpam-6909	54	17	dual	dual	ADJ
ejpam-6909	54	18	uncertainty	uncertainty	NOUN
ejpam-6909	54	19	.	.	PUNCT
ejpam-6909	55	1	these	these	DET
ejpam-6909	55	2	developments	development	NOUN
ejpam-6909	55	3	collectively	collectively	ADV
ejpam-6909	55	4	highlight	highlight	VERB
ejpam-6909	55	5	the	the	DET
ejpam-6909	55	6	growing	grow	VERB
ejpam-6909	55	7	influence	influence	NOUN
ejpam-6909	55	8	of	of	ADP
ejpam-6909	55	9	bipolar	bipolar	ADJ
ejpam-6909	55	10	fuzzy	fuzzy	ADJ
ejpam-6909	55	11	logic	logic	NOUN
ejpam-6909	55	12	in	in	ADP
ejpam-6909	55	13	algebraic	algebraic	ADJ
ejpam-6909	55	14	reasoning	reasoning	NOUN
ejpam-6909	55	15	,	,	PUNCT
ejpam-6909	55	16	especially	especially	ADV
ejpam-6909	55	17	in	in	ADP
ejpam-6909	55	18	systems	system	NOUN
ejpam-6909	55	19	characterized	characterize	VERB
ejpam-6909	55	20	by	by	ADP
ejpam-6909	55	21	non	non	ADJ
ejpam-6909	55	22	-	-	NOUN
ejpam-6909	55	23	determinism	determinism	ADJ
ejpam-6909	55	24	,	,	PUNCT
ejpam-6909	55	25	duality	duality	NOUN
ejpam-6909	55	26	,	,	PUNCT
ejpam-6909	55	27	and	and	CCONJ
ejpam-6909	55	28	graded	grade	VERB
ejpam-6909	55	29	membership	membership	NOUN
ejpam-6909	55	30	.	.	PUNCT
ejpam-6909	56	1	in	in	ADP
ejpam-6909	56	2	this	this	DET
ejpam-6909	56	3	paper	paper	NOUN
ejpam-6909	56	4	,	,	PUNCT
ejpam-6909	56	5	we	we	PRON
ejpam-6909	56	6	apply	apply	VERB
ejpam-6909	56	7	the	the	DET
ejpam-6909	56	8	concept	concept	NOUN
ejpam-6909	56	9	of	of	ADP
ejpam-6909	56	10	bfss	bfss	NOUN
ejpam-6909	56	11	to	to	ADP
ejpam-6909	56	12	chbckis	chbcki	NOUN
ejpam-6909	56	13	in	in	ADP
ejpam-6909	56	14	hbckas	hbckas	NOUN
ejpam-6909	56	15	and	and	CCONJ
ejpam-6909	56	16	introduce	introduce	VERB
ejpam-6909	56	17	the	the	DET
ejpam-6909	56	18	new	new	ADJ
ejpam-6909	56	19	notion	notion	NOUN
ejpam-6909	56	20	of	of	ADP
ejpam-6909	56	21	bf	bf	NOUN
ejpam-6909	56	22	-	-	PUNCT
ejpam-6909	56	23	chbckis	chbcki	NOUN
ejpam-6909	56	24	.	.	PUNCT
ejpam-6909	57	1	we	we	PRON
ejpam-6909	57	2	then	then	ADV
ejpam-6909	57	3	present	present	VERB
ejpam-6909	57	4	several	several	ADJ
ejpam-6909	57	5	theorems	theorem	NOUN
ejpam-6909	57	6	characterizing	characterize	VERB
ejpam-6909	57	7	these	these	DET
ejpam-6909	57	8	notions	notion	NOUN
ejpam-6909	57	9	in	in	ADP
ejpam-6909	57	10	terms	term	NOUN
ejpam-6909	57	11	of	of	ADP
ejpam-6909	57	12	level	level	NOUN
ejpam-6909	57	13	subsets	subset	NOUN
ejpam-6909	57	14	.	.	PUNCT
ejpam-6909	58	1	furthermore	furthermore	ADV
ejpam-6909	58	2	,	,	PUNCT
ejpam-6909	58	3	we	we	PRON
ejpam-6909	58	4	establish	establish	VERB
ejpam-6909	58	5	the	the	DET
ejpam-6909	58	6	relationship	relationship	NOUN
ejpam-6909	58	7	among	among	ADP
ejpam-6909	58	8	these	these	DET
ejpam-6909	58	9	notions	notion	NOUN
ejpam-6909	58	10	,	,	PUNCT
ejpam-6909	58	11	specifically	specifically	ADV
ejpam-6909	58	12	bf-(strong	bf-(strong	ADP
ejpam-6909	58	13	,	,	PUNCT
ejpam-6909	58	14	weak	weak	ADJ
ejpam-6909	58	15	,	,	PUNCT
ejpam-6909	58	16	reflexive)-hbckis	reflexive)-hbckis	PROPN
ejpam-6909	58	17	and	and	CCONJ
ejpam-6909	58	18	bf	bf	NOUN
ejpam-6909	58	19	-	-	PUNCT
ejpam-6909	58	20	chbckis	chbcki	NOUN
ejpam-6909	58	21	,	,	PUNCT
ejpam-6909	58	22	and	and	CCONJ
ejpam-6909	58	23	investigate	investigate	VERB
ejpam-6909	58	24	some	some	DET
ejpam-6909	58	25	interesting	interesting	ADJ
ejpam-6909	58	26	properties	property	NOUN
ejpam-6909	58	27	.	.	PUNCT
ejpam-6909	59	1	let	let	VERB
ejpam-6909	59	2	h	h	PRON
ejpam-6909	59	3	be	be	AUX
ejpam-6909	59	4	a	a	DET
ejpam-6909	59	5	non	non	ADJ
ejpam-6909	59	6	-	-	ADJ
ejpam-6909	59	7	empty	empty	ADJ
ejpam-6909	59	8	set	set	NOUN
ejpam-6909	59	9	endowed	endow	VERB
ejpam-6909	59	10	with	with	ADP
ejpam-6909	59	11	a	a	DET
ejpam-6909	59	12	hyperoperation	hyperoperation	NOUN
ejpam-6909	59	13	,	,	PUNCT
ejpam-6909	59	14	that	that	ADV
ejpam-6909	59	15	is	is	ADV
ejpam-6909	59	16	,	,	PUNCT
ejpam-6909	59	17	◦	◦	NOUN
ejpam-6909	59	18	is	be	AUX
ejpam-6909	59	19	a	a	DET
ejpam-6909	59	20	function	function	NOUN
ejpam-6909	59	21	from	from	ADP
ejpam-6909	59	22	h×h	h×h	PROPN
ejpam-6909	59	23	to	to	ADP
ejpam-6909	59	24	p∗(h	p∗(h	PROPN
ejpam-6909	59	25	)	)	PUNCT
ejpam-6909	59	26	=	=	PUNCT
ejpam-6909	60	1	p(h)\{∅	p(h)\{∅	PROPN
ejpam-6909	60	2	}	}	PUNCT
ejpam-6909	60	3	.	.	PUNCT
ejpam-6909	61	1	for	for	ADP
ejpam-6909	61	2	any	any	DET
ejpam-6909	61	3	two	two	NUM
ejpam-6909	61	4	subsets	subset	NOUN
ejpam-6909	61	5	t	t	PROPN
ejpam-6909	61	6	and	and	CCONJ
ejpam-6909	61	7	j	j	PROPN
ejpam-6909	61	8	of	of	ADP
ejpam-6909	61	9	h	h	PROPN
ejpam-6909	61	10	,	,	PUNCT
ejpam-6909	61	11	denoted	denote	VERB
ejpam-6909	61	12	by	by	ADP
ejpam-6909	61	13	t	t	PROPN
ejpam-6909	61	14	◦	◦	PROPN
ejpam-6909	61	15	j	j	PROPN
ejpam-6909	61	16	,	,	PUNCT
ejpam-6909	61	17	the	the	DET
ejpam-6909	61	18	set	set	NOUN
ejpam-6909	61	19	∪	∪	ADV
ejpam-6909	61	20	a∈t	a∈t	ADJ
ejpam-6909	61	21	,	,	PUNCT
ejpam-6909	61	22	b∈j	b∈j	NOUN
ejpam-6909	61	23	a	a	DET
ejpam-6909	61	24	◦	◦	NOUN
ejpam-6909	61	25	b.	b.	NOUN
ejpam-6909	61	26	we	we	PRON
ejpam-6909	61	27	will	will	AUX
ejpam-6909	61	28	utilize	utilize	VERB
ejpam-6909	61	29	ℏ1	ℏ1	ADJ
ejpam-6909	61	30	◦	◦	NOUN
ejpam-6909	61	31	ℏ2	ℏ2	NOUN
ejpam-6909	61	32	instead	instead	ADV
ejpam-6909	61	33	of	of	ADP
ejpam-6909	61	34	ℏ1	ℏ1	PROPN
ejpam-6909	61	35	◦	◦	NOUN
ejpam-6909	61	36	{	{	PUNCT
ejpam-6909	61	37	ℏ2	ℏ2	NOUN
ejpam-6909	61	38	}	}	PUNCT
ejpam-6909	61	39	,	,	PUNCT
ejpam-6909	61	40	{	{	PUNCT
ejpam-6909	61	41	ℏ1	ℏ1	ADJ
ejpam-6909	61	42	}	}	PUNCT
ejpam-6909	61	43	◦	◦	NOUN
ejpam-6909	61	44	ℏ2	ℏ2	NOUN
ejpam-6909	61	45	,	,	PUNCT
ejpam-6909	61	46	or	or	CCONJ
ejpam-6909	61	47	{	{	PUNCT
ejpam-6909	61	48	ℏ1	ℏ1	ADJ
ejpam-6909	61	49	}	}	PUNCT
ejpam-6909	61	50	◦	◦	NOUN
ejpam-6909	61	51	{	{	PUNCT
ejpam-6909	61	52	ℏ2	ℏ2	NOUN
ejpam-6909	61	53	}	}	PUNCT
ejpam-6909	61	54	.	.	PUNCT
ejpam-6909	62	1	2	2	X
ejpam-6909	62	2	.	.	X
ejpam-6909	62	3	preliminaries	preliminary	NOUN
ejpam-6909	62	4	in	in	ADP
ejpam-6909	62	5	this	this	DET
ejpam-6909	62	6	section	section	NOUN
ejpam-6909	62	7	,	,	PUNCT
ejpam-6909	62	8	we	we	PRON
ejpam-6909	62	9	recall	recall	VERB
ejpam-6909	62	10	fundamental	fundamental	ADJ
ejpam-6909	62	11	concepts	concept	NOUN
ejpam-6909	62	12	and	and	CCONJ
ejpam-6909	62	13	notation	notation	NOUN
ejpam-6909	62	14	essential	essential	ADJ
ejpam-6909	62	15	to	to	ADP
ejpam-6909	62	16	the	the	DET
ejpam-6909	62	17	development	development	NOUN
ejpam-6909	62	18	of	of	ADP
ejpam-6909	62	19	the	the	DET
ejpam-6909	62	20	main	main	ADJ
ejpam-6909	62	21	results	result	NOUN
ejpam-6909	62	22	in	in	ADP
ejpam-6909	62	23	this	this	DET
ejpam-6909	62	24	paper	paper	NOUN
ejpam-6909	62	25	.	.	PUNCT
ejpam-6909	63	1	these	these	PRON
ejpam-6909	63	2	include	include	VERB
ejpam-6909	63	3	basic	basic	ADJ
ejpam-6909	63	4	definitions	definition	NOUN
ejpam-6909	63	5	related	relate	VERB
ejpam-6909	63	6	to	to	ADP
ejpam-6909	63	7	bckalgebras	bckalgebra	NOUN
ejpam-6909	63	8	,	,	PUNCT
ejpam-6909	63	9	hyper	hyper	ADJ
ejpam-6909	63	10	bck	bck	NOUN
ejpam-6909	63	11	-	-	PUNCT
ejpam-6909	63	12	algebras	algebra	NOUN
ejpam-6909	63	13	,	,	PUNCT
ejpam-6909	63	14	fuzzy	fuzzy	ADJ
ejpam-6909	63	15	sets	set	NOUN
ejpam-6909	63	16	,	,	PUNCT
ejpam-6909	63	17	and	and	CCONJ
ejpam-6909	63	18	bipolar	bipolar	ADJ
ejpam-6909	63	19	fuzzy	fuzzy	ADJ
ejpam-6909	63	20	sets	set	NOUN
ejpam-6909	63	21	,	,	PUNCT
ejpam-6909	63	22	as	as	ADV
ejpam-6909	63	23	well	well	ADV
ejpam-6909	63	24	as	as	ADP
ejpam-6909	63	25	relevant	relevant	ADJ
ejpam-6909	63	26	classes	class	NOUN
ejpam-6909	63	27	of	of	ADP
ejpam-6909	63	28	ideals	ideal	NOUN
ejpam-6909	63	29	.	.	PUNCT
ejpam-6909	64	1	unless	unless	SCONJ
ejpam-6909	64	2	stated	state	VERB
ejpam-6909	64	3	otherwise	otherwise	ADV
ejpam-6909	64	4	,	,	PUNCT
ejpam-6909	64	5	all	all	DET
ejpam-6909	64	6	algebraic	algebraic	ADJ
ejpam-6909	64	7	structures	structure	NOUN
ejpam-6909	64	8	considered	consider	VERB
ejpam-6909	64	9	here	here	ADV
ejpam-6909	64	10	are	be	AUX
ejpam-6909	64	11	assumed	assume	VERB
ejpam-6909	64	12	to	to	PART
ejpam-6909	64	13	be	be	AUX
ejpam-6909	64	14	non	non	ADJ
ejpam-6909	64	15	-	-	ADJ
ejpam-6909	64	16	trivial	trivial	ADJ
ejpam-6909	64	17	.	.	PUNCT
ejpam-6909	65	1	the	the	DET
ejpam-6909	65	2	notion	notion	NOUN
ejpam-6909	65	3	of	of	ADP
ejpam-6909	65	4	bck	bck	PROPN
ejpam-6909	65	5	-	-	PUNCT
ejpam-6909	65	6	algebras	algebras	PROPN
ejpam-6909	65	7	was	be	AUX
ejpam-6909	65	8	first	first	ADV
ejpam-6909	65	9	introduced	introduce	VERB
ejpam-6909	65	10	by	by	ADP
ejpam-6909	65	11	iséki	iséki	NOUN
ejpam-6909	65	12	and	and	CCONJ
ejpam-6909	65	13	tanaka	tanaka	PROPN
ejpam-6909	66	1	[	[	X
ejpam-6909	66	2	1	1	X
ejpam-6909	66	3	]	]	PUNCT
ejpam-6909	66	4	as	as	ADP
ejpam-6909	66	5	an	an	DET
ejpam-6909	66	6	algebraic	algebraic	ADJ
ejpam-6909	66	7	counterpart	counterpart	NOUN
ejpam-6909	66	8	to	to	ADP
ejpam-6909	66	9	certain	certain	ADJ
ejpam-6909	66	10	propositional	propositional	ADJ
ejpam-6909	66	11	calculi	calculi	NOUN
ejpam-6909	66	12	.	.	PUNCT
ejpam-6909	67	1	subsequently	subsequently	ADV
ejpam-6909	67	2	,	,	PUNCT
ejpam-6909	67	3	hyper	hyper	ADJ
ejpam-6909	67	4	bck	bck	NOUN
ejpam-6909	67	5	-	-	PUNCT
ejpam-6909	67	6	algebras	algebras	PROPN
ejpam-6909	67	7	were	be	AUX
ejpam-6909	67	8	developed	develop	VERB
ejpam-6909	67	9	to	to	PART
ejpam-6909	67	10	generalize	generalize	VERB
ejpam-6909	67	11	bck	bck	VERB
ejpam-6909	67	12	-	-	PUNCT
ejpam-6909	67	13	algebras	algebras	PROPN
ejpam-6909	67	14	by	by	ADP
ejpam-6909	67	15	allowing	allow	VERB
ejpam-6909	67	16	hyperoperations	hyperoperation	NOUN
ejpam-6909	67	17	,	,	PUNCT
ejpam-6909	67	18	as	as	SCONJ
ejpam-6909	67	19	introduced	introduce	VERB
ejpam-6909	67	20	in	in	ADP
ejpam-6909	67	21	[	[	X
ejpam-6909	67	22	5	5	NUM
ejpam-6909	67	23	]	]	PUNCT
ejpam-6909	67	24	.	.	PUNCT
ejpam-6909	68	1	meanwhile	meanwhile	ADV
ejpam-6909	68	2	,	,	PUNCT
ejpam-6909	68	3	the	the	DET
ejpam-6909	68	4	concept	concept	NOUN
ejpam-6909	68	5	of	of	ADP
ejpam-6909	68	6	fuzzy	fuzzy	ADJ
ejpam-6909	68	7	sets	set	NOUN
ejpam-6909	68	8	,	,	PUNCT
ejpam-6909	68	9	introduced	introduce	VERB
ejpam-6909	68	10	by	by	ADP
ejpam-6909	68	11	zadeh	zadeh	PROPN
ejpam-6909	69	1	[	[	X
ejpam-6909	69	2	9	9	NUM
ejpam-6909	69	3	]	]	PUNCT
ejpam-6909	69	4	,	,	PUNCT
ejpam-6909	69	5	was	be	AUX
ejpam-6909	69	6	extended	extend	VERB
ejpam-6909	69	7	to	to	ADP
ejpam-6909	69	8	bipolar	bipolar	ADJ
ejpam-6909	69	9	fuzzy	fuzzy	ADJ
ejpam-6909	69	10	sets	set	NOUN
ejpam-6909	69	11	(	(	PUNCT
ejpam-6909	69	12	bfss	bfss	NOUN
ejpam-6909	69	13	)	)	PUNCT
ejpam-6909	69	14	by	by	ADP
ejpam-6909	69	15	zhang	zhang	PROPN
ejpam-6909	70	1	[	[	X
ejpam-6909	70	2	27	27	NUM
ejpam-6909	70	3	]	]	PUNCT
ejpam-6909	70	4	and	and	CCONJ
ejpam-6909	70	5	lee	lee	PROPN
ejpam-6909	70	6	[	[	X
ejpam-6909	70	7	10	10	NUM
ejpam-6909	70	8	]	]	PUNCT
ejpam-6909	70	9	to	to	PART
ejpam-6909	70	10	model	model	VERB
ejpam-6909	70	11	duality	duality	NOUN
ejpam-6909	70	12	in	in	ADP
ejpam-6909	70	13	membership	membership	NOUN
ejpam-6909	70	14	functions	function	NOUN
ejpam-6909	70	15	.	.	PUNCT
ejpam-6909	71	1	for	for	ADP
ejpam-6909	71	2	the	the	DET
ejpam-6909	71	3	reader	reader	NOUN
ejpam-6909	71	4	’s	’s	PART
ejpam-6909	71	5	convenience	convenience	NOUN
ejpam-6909	71	6	,	,	PUNCT
ejpam-6909	71	7	we	we	PRON
ejpam-6909	71	8	summarize	summarize	VERB
ejpam-6909	71	9	here	here	ADV
ejpam-6909	71	10	the	the	DET
ejpam-6909	71	11	essential	essential	ADJ
ejpam-6909	71	12	definitions	definition	NOUN
ejpam-6909	71	13	and	and	CCONJ
ejpam-6909	71	14	properties	property	NOUN
ejpam-6909	71	15	that	that	PRON
ejpam-6909	71	16	will	will	AUX
ejpam-6909	71	17	be	be	AUX
ejpam-6909	71	18	used	use	VERB
ejpam-6909	71	19	throughout	throughout	ADP
ejpam-6909	71	20	the	the	DET
ejpam-6909	71	21	rest	rest	NOUN
ejpam-6909	71	22	of	of	ADP
ejpam-6909	71	23	the	the	DET
ejpam-6909	71	24	paper	paper	NOUN
ejpam-6909	71	25	.	.	PUNCT
ejpam-6909	72	1	d.	d.	PROPN
ejpam-6909	72	2	ramesh	ramesh	PROPN
ejpam-6909	72	3	et	et	PROPN
ejpam-6909	72	4	al	al	PROPN
ejpam-6909	72	5	.	.	PUNCT
ejpam-6909	72	6	/	/	SYM
ejpam-6909	72	7	eur	eur	PROPN
ejpam-6909	72	8	.	.	PUNCT
ejpam-6909	73	1	j.	j.	PROPN
ejpam-6909	73	2	pure	pure	PROPN
ejpam-6909	73	3	appl	appl	PROPN
ejpam-6909	73	4	.	.	PROPN
ejpam-6909	73	5	math	math	PROPN
ejpam-6909	73	6	,	,	PUNCT
ejpam-6909	73	7	18	18	NUM
ejpam-6909	73	8	(	(	PUNCT
ejpam-6909	73	9	4	4	NUM
ejpam-6909	73	10	)	)	PUNCT
ejpam-6909	73	11	(	(	PUNCT
ejpam-6909	73	12	2025	2025	NUM
ejpam-6909	73	13	)	)	PUNCT
ejpam-6909	73	14	,	,	PUNCT
ejpam-6909	73	15	6909	6909	NUM
ejpam-6909	73	16	4	4	NUM
ejpam-6909	73	17	of	of	ADP
ejpam-6909	73	18	16	16	NUM
ejpam-6909	73	19	definition	definition	NOUN
ejpam-6909	73	20	1	1	NUM
ejpam-6909	73	21	.	.	PUNCT
ejpam-6909	74	1	[	[	X
ejpam-6909	74	2	5	5	NUM
ejpam-6909	74	3	]	]	PUNCT
ejpam-6909	74	4	by	by	ADP
ejpam-6909	74	5	a	a	DET
ejpam-6909	74	6	hyper	hyper	ADJ
ejpam-6909	74	7	bck	bck	NOUN
ejpam-6909	74	8	-	-	PUNCT
ejpam-6909	74	9	algebra	algebra	NOUN
ejpam-6909	74	10	(	(	PUNCT
ejpam-6909	74	11	hbcka	hbcka	NOUN
ejpam-6909	74	12	)	)	PUNCT
ejpam-6909	74	13	,	,	PUNCT
ejpam-6909	74	14	we	we	PRON
ejpam-6909	74	15	mean	mean	VERB
ejpam-6909	74	16	a	a	DET
ejpam-6909	74	17	non	non	ADJ
ejpam-6909	74	18	-	-	ADJ
ejpam-6909	74	19	empty	empty	ADJ
ejpam-6909	74	20	collection	collection	NOUN
ejpam-6909	74	21	h	h	NOUN
ejpam-6909	74	22	possessed	possess	VERB
ejpam-6909	74	23	of	of	ADP
ejpam-6909	74	24	a	a	DET
ejpam-6909	74	25	hyperoperation	hyperoperation	NOUN
ejpam-6909	74	26	◦	◦	NOUN
ejpam-6909	74	27	and	and	CCONJ
ejpam-6909	74	28	a	a	DET
ejpam-6909	74	29	constant	constant	ADJ
ejpam-6909	74	30	0	0	NUM
ejpam-6909	74	31	fulfilling	fulfil	VERB
ejpam-6909	74	32	the	the	DET
ejpam-6909	74	33	principles	principle	NOUN
ejpam-6909	74	34	listed	list	VERB
ejpam-6909	74	35	below	below	ADV
ejpam-6909	74	36	:	:	PUNCT
ejpam-6909	74	37	(	(	PUNCT
ejpam-6909	74	38	hbcka-1	hbcka-1	NUM
ejpam-6909	74	39	)	)	PUNCT
ejpam-6909	74	40	(	(	PUNCT
ejpam-6909	74	41	ℏ1	ℏ1	PROPN
ejpam-6909	74	42	◦	◦	PROPN
ejpam-6909	74	43	ℏ3	ℏ3	PROPN
ejpam-6909	74	44	)	)	PUNCT
ejpam-6909	75	1	◦	◦	NOUN
ejpam-6909	75	2	(	(	PUNCT
ejpam-6909	75	3	ℏ2	ℏ2	NOUN
ejpam-6909	75	4	◦	◦	PROPN
ejpam-6909	75	5	ℏ3	ℏ3	PROPN
ejpam-6909	75	6	)	)	PUNCT
ejpam-6909	75	7	≪	≪	PUNCT
ejpam-6909	75	8	ℏ1	ℏ1	ADJ
ejpam-6909	75	9	◦	◦	NOUN
ejpam-6909	75	10	ℏ2	ℏ2	NOUN
ejpam-6909	75	11	,	,	PUNCT
ejpam-6909	75	12	(	(	PUNCT
ejpam-6909	75	13	hbcka-2	hbcka-2	NUM
ejpam-6909	75	14	)	)	PUNCT
ejpam-6909	75	15	(	(	PUNCT
ejpam-6909	75	16	ℏ1	ℏ1	PROPN
ejpam-6909	75	17	◦	◦	NOUN
ejpam-6909	75	18	ℏ2	ℏ2	NOUN
ejpam-6909	75	19	)	)	PUNCT
ejpam-6909	75	20	◦	◦	NOUN
ejpam-6909	75	21	ℏ3	ℏ3	NOUN
ejpam-6909	75	22	=	=	SYM
ejpam-6909	75	23	(	(	PUNCT
ejpam-6909	75	24	ℏ1	ℏ1	PROPN
ejpam-6909	75	25	◦	◦	PROPN
ejpam-6909	75	26	ℏ3	ℏ3	PROPN
ejpam-6909	75	27	)	)	PUNCT
ejpam-6909	75	28	◦	◦	NOUN
ejpam-6909	75	29	ℏ2	ℏ2	NOUN
ejpam-6909	75	30	,	,	PUNCT
ejpam-6909	75	31	(	(	PUNCT
ejpam-6909	75	32	hbcka-3	hbcka-3	NUM
ejpam-6909	75	33	)	)	PUNCT
ejpam-6909	75	34	ℏ1	ℏ1	PROPN
ejpam-6909	75	35	◦	◦	NOUN
ejpam-6909	75	36	h	h	NOUN
ejpam-6909	75	37	≪	≪	VERB
ejpam-6909	75	38	{	{	PUNCT
ejpam-6909	75	39	ℏ1	ℏ1	ADJ
ejpam-6909	75	40	}	}	PUNCT
ejpam-6909	75	41	,	,	PUNCT
ejpam-6909	75	42	(	(	PUNCT
ejpam-6909	75	43	hbcka-4	hbcka-4	NOUN
ejpam-6909	75	44	)	)	PUNCT
ejpam-6909	75	45	ℏ1	ℏ1	NOUN
ejpam-6909	75	46	≪	≪	PUNCT
ejpam-6909	75	47	ℏ2	ℏ2	NOUN
ejpam-6909	75	48	and	and	CCONJ
ejpam-6909	75	49	ℏ2	ℏ2	NOUN
ejpam-6909	75	50	≪	≪	VERB
ejpam-6909	75	51	ℏ1	ℏ1	ADJ
ejpam-6909	75	52	⇒	⇒	NOUN
ejpam-6909	75	53	ℏ1	ℏ1	PROPN
ejpam-6909	75	54	=	=	ADJ
ejpam-6909	75	55	ℏ2	ℏ2	NOUN
ejpam-6909	75	56	,	,	PUNCT
ejpam-6909	75	57	for	for	ADP
ejpam-6909	75	58	all	all	DET
ejpam-6909	75	59	ℏ1	ℏ1	ADJ
ejpam-6909	75	60	,	,	PUNCT
ejpam-6909	75	61	ℏ2	ℏ2	NOUN
ejpam-6909	75	62	,	,	PUNCT
ejpam-6909	75	63	ℏ3	ℏ3	PROPN
ejpam-6909	75	64	∈	∈	PROPN
ejpam-6909	75	65	h.	h.	NOUN
ejpam-6909	75	66	we	we	PRON
ejpam-6909	75	67	denote	denote	VERB
ejpam-6909	75	68	a	a	DET
ejpam-6909	75	69	relationship	relationship	NOUN
ejpam-6909	75	70	≪	≪	VERB
ejpam-6909	75	71	on	on	ADP
ejpam-6909	75	72	h	h	NOUN
ejpam-6909	75	73	by	by	ADP
ejpam-6909	75	74	letting	let	VERB
ejpam-6909	75	75	ℏ1	ℏ1	NOUN
ejpam-6909	75	76	≪	≪	PUNCT
ejpam-6909	75	77	ℏ2	ℏ2	NOUN
ejpam-6909	75	78	⇔	⇔	X
ejpam-6909	75	79	0	0	NUM
ejpam-6909	75	80	∈	∈	PROPN
ejpam-6909	75	81	ℏ1	ℏ1	PROPN
ejpam-6909	75	82	◦	◦	NOUN
ejpam-6909	75	83	ℏ2	ℏ2	NOUN
ejpam-6909	75	84	and	and	CCONJ
ejpam-6909	75	85	every	every	DET
ejpam-6909	75	86	h1,h2	h1,h2	PROPN
ejpam-6909	75	87	⊆	⊆	NUM
ejpam-6909	75	88	h	h	NOUN
ejpam-6909	75	89	,	,	PUNCT
ejpam-6909	75	90	h1	h1	AUX
ejpam-6909	75	91	≪	≪	ADJ
ejpam-6909	75	92	h2	h2	NOUN
ejpam-6909	75	93	is	be	AUX
ejpam-6909	75	94	described	describe	VERB
ejpam-6909	75	95	by	by	ADP
ejpam-6909	75	96	∀r	∀r	PROPN
ejpam-6909	75	97	∈	∈	PROPN
ejpam-6909	75	98	h1	h1	NOUN
ejpam-6909	75	99	,	,	PUNCT
ejpam-6909	75	100	∃j	∃j	PROPN
ejpam-6909	75	101	∈	∈	PROPN
ejpam-6909	75	102	h2	h2	NOUN
ejpam-6909	75	103	such	such	ADJ
ejpam-6909	75	104	that	that	SCONJ
ejpam-6909	75	105	r	r	NOUN
ejpam-6909	75	106	≪	≪	PUNCT
ejpam-6909	75	107	j.	j.	PROPN
ejpam-6909	75	108	in	in	ADP
ejpam-6909	75	109	such	such	DET
ejpam-6909	75	110	a	a	DET
ejpam-6909	75	111	case	case	NOUN
ejpam-6909	75	112	,	,	PUNCT
ejpam-6909	75	113	we	we	PRON
ejpam-6909	75	114	call	call	VERB
ejpam-6909	75	115	≪	≪	VERB
ejpam-6909	75	116	the	the	DET
ejpam-6909	75	117	hyper	hyper	ADJ
ejpam-6909	75	118	order	order	NOUN
ejpam-6909	75	119	in	in	ADP
ejpam-6909	75	120	h.	h.	PROPN
ejpam-6909	75	121	note	note	VERB
ejpam-6909	75	122	that	that	SCONJ
ejpam-6909	75	123	the	the	DET
ejpam-6909	75	124	scenario	scenario	NOUN
ejpam-6909	75	125	(	(	PUNCT
ejpam-6909	75	126	hbcka-3	hbcka-3	NUM
ejpam-6909	75	127	)	)	PUNCT
ejpam-6909	75	128	is	be	AUX
ejpam-6909	75	129	equal	equal	ADJ
ejpam-6909	75	130	to	to	ADP
ejpam-6909	75	131	the	the	DET
ejpam-6909	75	132	condition	condition	NOUN
ejpam-6909	75	133	(	(	PUNCT
ejpam-6909	75	134	p1	p1	NOUN
ejpam-6909	75	135	)	)	PUNCT
ejpam-6909	75	136	.	.	PUNCT
ejpam-6909	76	1	in	in	ADP
ejpam-6909	76	2	any	any	DET
ejpam-6909	76	3	hbcka	hbcka	NOUN
ejpam-6909	76	4	h	h	NOUN
ejpam-6909	76	5	,	,	PUNCT
ejpam-6909	76	6	the	the	DET
ejpam-6909	76	7	following	follow	VERB
ejpam-6909	76	8	is	be	AUX
ejpam-6909	76	9	true	true	ADJ
ejpam-6909	76	10	.	.	PUNCT
ejpam-6909	77	1	(	(	PUNCT
ejpam-6909	77	2	p1	p1	NOUN
ejpam-6909	77	3	)	)	PUNCT
ejpam-6909	77	4	ℏ1	ℏ1	PROPN
ejpam-6909	77	5	◦	◦	NOUN
ejpam-6909	77	6	ℏ2	ℏ2	NOUN
ejpam-6909	77	7	≪	≪	VERB
ejpam-6909	77	8	{	{	PUNCT
ejpam-6909	77	9	ℏ1	ℏ1	ADJ
ejpam-6909	77	10	}	}	PUNCT
ejpam-6909	77	11	,	,	PUNCT
ejpam-6909	77	12	(	(	PUNCT
ejpam-6909	77	13	p2	p2	X
ejpam-6909	77	14	)	)	PUNCT
ejpam-6909	77	15	ℏ1	ℏ1	ADJ
ejpam-6909	77	16	◦	◦	NOUN
ejpam-6909	77	17	0	0	PUNCT
ejpam-6909	77	18	≪	≪	PUNCT
ejpam-6909	77	19	{	{	PUNCT
ejpam-6909	77	20	ℏ1	ℏ1	ADJ
ejpam-6909	77	21	}	}	PUNCT
ejpam-6909	77	22	,	,	PUNCT
ejpam-6909	77	23	0	0	NUM
ejpam-6909	77	24	◦	◦	NOUN
ejpam-6909	77	25	ℏ1	ℏ1	ADJ
ejpam-6909	77	26	≪	≪	VERB
ejpam-6909	77	27	{	{	PUNCT
ejpam-6909	77	28	ℏ1	ℏ1	ADJ
ejpam-6909	77	29	}	}	PUNCT
ejpam-6909	77	30	,	,	PUNCT
ejpam-6909	77	31	and	and	CCONJ
ejpam-6909	77	32	0	0	NUM
ejpam-6909	77	33	◦	◦	NOUN
ejpam-6909	77	34	0	0	NUM
ejpam-6909	77	35	≪	≪	X
ejpam-6909	77	36	{	{	PUNCT
ejpam-6909	77	37	0	0	NUM
ejpam-6909	77	38	}	}	PUNCT
ejpam-6909	77	39	,	,	PUNCT
ejpam-6909	77	40	(	(	PUNCT
ejpam-6909	77	41	p3	p3	PROPN
ejpam-6909	77	42	)	)	PUNCT
ejpam-6909	77	43	(	(	PUNCT
ejpam-6909	77	44	h1	h1	VERB
ejpam-6909	77	45	◦	◦	NOUN
ejpam-6909	77	46	h2	h2	NOUN
ejpam-6909	77	47	)	)	PUNCT
ejpam-6909	77	48	◦	◦	NOUN
ejpam-6909	77	49	h3	h3	NOUN
ejpam-6909	77	50	=	=	SYM
ejpam-6909	77	51	(	(	PUNCT
ejpam-6909	77	52	h1	h1	VERB
ejpam-6909	77	53	◦	◦	NOUN
ejpam-6909	77	54	h3	h3	NOUN
ejpam-6909	77	55	)	)	PUNCT
ejpam-6909	77	56	◦	◦	NOUN
ejpam-6909	77	57	h2	h2	NOUN
ejpam-6909	77	58	,	,	PUNCT
ejpam-6909	77	59	h1	h1	VERB
ejpam-6909	77	60	◦	◦	NOUN
ejpam-6909	77	61	h2	h2	NOUN
ejpam-6909	77	62	≪	≪	VERB
ejpam-6909	77	63	h1	h1	NOUN
ejpam-6909	77	64	,	,	PUNCT
ejpam-6909	77	65	and	and	CCONJ
ejpam-6909	77	66	0	0	NUM
ejpam-6909	77	67	◦	◦	NOUN
ejpam-6909	77	68	h1	h1	NOUN
ejpam-6909	77	69	≪	≪	VERB
ejpam-6909	77	70	{	{	PUNCT
ejpam-6909	77	71	0	0	NUM
ejpam-6909	77	72	}	}	PUNCT
ejpam-6909	77	73	,	,	PUNCT
ejpam-6909	77	74	(	(	PUNCT
ejpam-6909	77	75	p4	p4	ADJ
ejpam-6909	77	76	)	)	PUNCT
ejpam-6909	77	77	0	0	PUNCT
ejpam-6909	78	1	◦	◦	NOUN
ejpam-6909	78	2	0	0	NUM
ejpam-6909	78	3	=	=	SYM
ejpam-6909	78	4	{	{	PUNCT
ejpam-6909	78	5	0	0	NUM
ejpam-6909	78	6	}	}	PUNCT
ejpam-6909	78	7	,	,	PUNCT
ejpam-6909	78	8	(	(	PUNCT
ejpam-6909	78	9	p5	p5	ADJ
ejpam-6909	78	10	)	)	PUNCT
ejpam-6909	78	11	0	0	PUNCT
ejpam-6909	78	12	≪	≪	VERB
ejpam-6909	78	13	ℏ1	ℏ1	PROPN
ejpam-6909	78	14	,	,	PUNCT
ejpam-6909	78	15	(	(	PUNCT
ejpam-6909	78	16	p6	p6	PROPN
ejpam-6909	78	17	)	)	PUNCT
ejpam-6909	78	18	ℏ1	ℏ1	ADJ
ejpam-6909	78	19	≪	≪	VERB
ejpam-6909	78	20	ℏ1	ℏ1	ADJ
ejpam-6909	78	21	,	,	PUNCT
ejpam-6909	78	22	(	(	PUNCT
ejpam-6909	78	23	p7	p7	ADJ
ejpam-6909	78	24	)	)	PUNCT
ejpam-6909	78	25	h1	h1	NOUN
ejpam-6909	78	26	≪	≪	PUNCT
ejpam-6909	78	27	h1	h1	NOUN
ejpam-6909	78	28	,	,	PUNCT
ejpam-6909	78	29	(	(	PUNCT
ejpam-6909	78	30	p8	p8	PROPN
ejpam-6909	78	31	)	)	PUNCT
ejpam-6909	78	32	h1	h1	NOUN
ejpam-6909	78	33	⊆	⊆	NUM
ejpam-6909	78	34	h2	h2	NOUN
ejpam-6909	78	35	⇒	⇒	NOUN
ejpam-6909	78	36	h1	h1	PROPN
ejpam-6909	78	37	≪	≪	PUNCT
ejpam-6909	78	38	h2	h2	NOUN
ejpam-6909	78	39	,	,	PUNCT
ejpam-6909	78	40	(	(	PUNCT
ejpam-6909	78	41	p9	p9	PROPN
ejpam-6909	78	42	)	)	PUNCT
ejpam-6909	78	43	{	{	PUNCT
ejpam-6909	78	44	0	0	NUM
ejpam-6909	78	45	}	}	PUNCT
ejpam-6909	78	46	=	=	SYM
ejpam-6909	78	47	0	0	NUM
ejpam-6909	78	48	◦	◦	NOUN
ejpam-6909	78	49	ℏ1	ℏ1	ADJ
ejpam-6909	78	50	,	,	PUNCT
ejpam-6909	78	51	(	(	PUNCT
ejpam-6909	78	52	p10	p10	NOUN
ejpam-6909	78	53	)	)	PUNCT
ejpam-6909	78	54	ℏ1	ℏ1	PROPN
ejpam-6909	78	55	◦	◦	NOUN
ejpam-6909	78	56	0	0	NUM
ejpam-6909	79	1	=	=	SYM
ejpam-6909	79	2	{	{	PUNCT
ejpam-6909	79	3	ℏ1	ℏ1	PROPN
ejpam-6909	79	4	}	}	PUNCT
ejpam-6909	79	5	,	,	PUNCT
ejpam-6909	79	6	(	(	PUNCT
ejpam-6909	79	7	p11	p11	NOUN
ejpam-6909	79	8	)	)	PUNCT
ejpam-6909	79	9	0	0	PUNCT
ejpam-6909	80	1	◦	◦	NOUN
ejpam-6909	80	2	h1	h1	NOUN
ejpam-6909	80	3	=	=	SYM
ejpam-6909	80	4	{	{	PUNCT
ejpam-6909	80	5	0	0	NUM
ejpam-6909	80	6	}	}	PUNCT
ejpam-6909	80	7	,	,	PUNCT
ejpam-6909	80	8	(	(	PUNCT
ejpam-6909	80	9	p12	p12	NOUN
ejpam-6909	80	10	)	)	PUNCT
ejpam-6909	80	11	ℏ1	ℏ1	NOUN
ejpam-6909	80	12	≪	≪	VERB
ejpam-6909	80	13	{	{	PUNCT
ejpam-6909	80	14	0	0	NUM
ejpam-6909	80	15	}	}	PUNCT
ejpam-6909	80	16	⇒	⇒	VERB
ejpam-6909	80	17	ℏ1	ℏ1	PROPN
ejpam-6909	80	18	=	=	SYM
ejpam-6909	80	19	{	{	PUNCT
ejpam-6909	80	20	0	0	NUM
ejpam-6909	80	21	}	}	PUNCT
ejpam-6909	80	22	,	,	PUNCT
ejpam-6909	80	23	(	(	PUNCT
ejpam-6909	80	24	p13	p13	X
ejpam-6909	80	25	)	)	PUNCT
ejpam-6909	80	26	h1	h1	NOUN
ejpam-6909	80	27	◦	◦	NOUN
ejpam-6909	80	28	h2	h2	NOUN
ejpam-6909	80	29	≪	≪	VERB
ejpam-6909	80	30	h1	h1	NOUN
ejpam-6909	80	31	,	,	PUNCT
ejpam-6909	80	32	(	(	PUNCT
ejpam-6909	80	33	p14	p14	PROPN
ejpam-6909	80	34	)	)	PUNCT
ejpam-6909	80	35	ℏ1	ℏ1	PROPN
ejpam-6909	80	36	∈	∈	PROPN
ejpam-6909	80	37	ℏ1	ℏ1	PROPN
ejpam-6909	80	38	◦	◦	NOUN
ejpam-6909	80	39	0	0	NUM
ejpam-6909	80	40	,	,	PUNCT
ejpam-6909	80	41	(	(	PUNCT
ejpam-6909	80	42	p15	p15	NOUN
ejpam-6909	80	43	)	)	PUNCT
ejpam-6909	80	44	ℏ1	ℏ1	ADJ
ejpam-6909	80	45	◦	◦	NOUN
ejpam-6909	80	46	0	0	PUNCT
ejpam-6909	80	47	≪	≪	PUNCT
ejpam-6909	80	48	{	{	PUNCT
ejpam-6909	80	49	ℏ2	ℏ2	NOUN
ejpam-6909	80	50	}	}	PUNCT
ejpam-6909	80	51	⇒	⇒	VERB
ejpam-6909	80	52	ℏ1	ℏ1	ADJ
ejpam-6909	80	53	≪	≪	ADJ
ejpam-6909	80	54	ℏ2	ℏ2	NOUN
ejpam-6909	80	55	,	,	PUNCT
ejpam-6909	80	56	(	(	PUNCT
ejpam-6909	80	57	p16	p16	NOUN
ejpam-6909	80	58	)	)	PUNCT
ejpam-6909	80	59	ℏ2	ℏ2	NOUN
ejpam-6909	80	60	≪	≪	PUNCT
ejpam-6909	80	61	ℏ3	ℏ3	PROPN
ejpam-6909	80	62	⇒	⇒	VERB
ejpam-6909	80	63	ℏ1	ℏ1	PROPN
ejpam-6909	80	64	◦	◦	PROPN
ejpam-6909	80	65	ℏ3	ℏ3	PROPN
ejpam-6909	80	66	≪	≪	PUNCT
ejpam-6909	80	67	ℏ1	ℏ1	ADJ
ejpam-6909	80	68	◦	◦	NOUN
ejpam-6909	80	69	ℏ2	ℏ2	NOUN
ejpam-6909	80	70	,	,	PUNCT
ejpam-6909	80	71	(	(	PUNCT
ejpam-6909	80	72	p17	p17	NOUN
ejpam-6909	80	73	)	)	PUNCT
ejpam-6909	80	74	ℏ1	ℏ1	PROPN
ejpam-6909	80	75	◦	◦	NOUN
ejpam-6909	80	76	ℏ2	ℏ2	NOUN
ejpam-6909	80	77	=	=	SYM
ejpam-6909	80	78	{	{	PUNCT
ejpam-6909	80	79	0	0	NUM
ejpam-6909	80	80	}	}	PUNCT
ejpam-6909	80	81	⇒	⇒	NOUN
ejpam-6909	80	82	(	(	PUNCT
ejpam-6909	80	83	ℏ1	ℏ1	PROPN
ejpam-6909	80	84	◦	◦	PROPN
ejpam-6909	80	85	ℏ3	ℏ3	PROPN
ejpam-6909	80	86	)	)	PUNCT
ejpam-6909	81	1	◦	◦	NOUN
ejpam-6909	81	2	(	(	PUNCT
ejpam-6909	81	3	ℏ2	ℏ2	NOUN
ejpam-6909	81	4	◦	◦	PROPN
ejpam-6909	81	5	ℏ3	ℏ3	PROPN
ejpam-6909	81	6	)	)	PUNCT
ejpam-6909	82	1	=	=	PUNCT
ejpam-6909	82	2	{	{	PUNCT
ejpam-6909	82	3	0	0	NUM
ejpam-6909	82	4	}	}	PUNCT
ejpam-6909	82	5	and	and	CCONJ
ejpam-6909	82	6	ℏ1	ℏ1	PROPN
ejpam-6909	82	7	◦	◦	PROPN
ejpam-6909	82	8	ℏ3	ℏ3	PROPN
ejpam-6909	82	9	≪	≪	PUNCT
ejpam-6909	82	10	ℏ2	ℏ2	NOUN
ejpam-6909	82	11	◦	◦	PROPN
ejpam-6909	82	12	ℏ3	ℏ3	PROPN
ejpam-6909	82	13	,	,	PUNCT
ejpam-6909	82	14	(	(	PUNCT
ejpam-6909	82	15	p18	p18	NOUN
ejpam-6909	82	16	)	)	PUNCT
ejpam-6909	82	17	h1	h1	PROPN
ejpam-6909	82	18	◦	◦	NOUN
ejpam-6909	82	19	0	0	NUM
ejpam-6909	82	20	=	=	SYM
ejpam-6909	82	21	{	{	PUNCT
ejpam-6909	82	22	0	0	NUM
ejpam-6909	82	23	}	}	PUNCT
ejpam-6909	82	24	⇒	⇒	NOUN
ejpam-6909	82	25	h1	h1	PROPN
ejpam-6909	82	26	=	=	SYM
ejpam-6909	82	27	{	{	PUNCT
ejpam-6909	82	28	0	0	NUM
ejpam-6909	82	29	}	}	PUNCT
ejpam-6909	82	30	,	,	PUNCT
ejpam-6909	82	31	for	for	SCONJ
ejpam-6909	82	32	everyone	everyone	PRON
ejpam-6909	82	33	ℏ1	ℏ1	ADJ
ejpam-6909	82	34	,	,	PUNCT
ejpam-6909	82	35	ℏ2	ℏ2	NOUN
ejpam-6909	82	36	,	,	PUNCT
ejpam-6909	82	37	ℏ3	ℏ3	PROPN
ejpam-6909	82	38	∈	∈	PROPN
ejpam-6909	82	39	h	h	NOUN
ejpam-6909	82	40	in	in	ADP
ejpam-6909	82	41	addition	addition	NOUN
ejpam-6909	82	42	to	to	ADP
ejpam-6909	82	43	every	every	DET
ejpam-6909	82	44	non	non	ADJ
ejpam-6909	82	45	-	-	ADJ
ejpam-6909	82	46	empty	empty	ADJ
ejpam-6909	82	47	subsets	subset	NOUN
ejpam-6909	82	48	h1,h2	h1,h2	PROPN
ejpam-6909	82	49	,	,	PUNCT
ejpam-6909	82	50	and	and	CCONJ
ejpam-6909	82	51	h3	h3	NOUN
ejpam-6909	82	52	of	of	ADP
ejpam-6909	82	53	h.	h.	PROPN
ejpam-6909	82	54	d.	d.	PROPN
ejpam-6909	82	55	ramesh	ramesh	PROPN
ejpam-6909	82	56	et	et	PROPN
ejpam-6909	82	57	al	al	PROPN
ejpam-6909	82	58	.	.	PUNCT
ejpam-6909	82	59	/	/	SYM
ejpam-6909	82	60	eur	eur	PROPN
ejpam-6909	82	61	.	.	PUNCT
ejpam-6909	83	1	j.	j.	PROPN
ejpam-6909	83	2	pure	pure	PROPN
ejpam-6909	83	3	appl	appl	PROPN
ejpam-6909	83	4	.	.	PROPN
ejpam-6909	83	5	math	math	PROPN
ejpam-6909	83	6	,	,	PUNCT
ejpam-6909	83	7	18	18	NUM
ejpam-6909	83	8	(	(	PUNCT
ejpam-6909	83	9	4	4	NUM
ejpam-6909	83	10	)	)	PUNCT
ejpam-6909	83	11	(	(	PUNCT
ejpam-6909	83	12	2025	2025	NUM
ejpam-6909	83	13	)	)	PUNCT
ejpam-6909	83	14	,	,	PUNCT
ejpam-6909	83	15	6909	6909	NUM
ejpam-6909	83	16	5	5	NUM
ejpam-6909	83	17	of	of	ADP
ejpam-6909	83	18	16	16	NUM
ejpam-6909	83	19	definition	definition	NOUN
ejpam-6909	83	20	2	2	NUM
ejpam-6909	83	21	.	.	PUNCT
ejpam-6909	84	1	[	[	X
ejpam-6909	84	2	5	5	X
ejpam-6909	84	3	]	]	PUNCT
ejpam-6909	84	4	let	let	VERB
ejpam-6909	84	5	i	i	PRON
ejpam-6909	84	6	be	be	AUX
ejpam-6909	84	7	a	a	DET
ejpam-6909	84	8	non	non	ADJ
ejpam-6909	84	9	-	-	ADJ
ejpam-6909	84	10	empty	empty	ADJ
ejpam-6909	84	11	subset	subset	NOUN
ejpam-6909	84	12	of	of	ADP
ejpam-6909	84	13	an	an	DET
ejpam-6909	84	14	hbcka	hbcka	NOUN
ejpam-6909	84	15	h	h	NOUN
ejpam-6909	84	16	and	and	CCONJ
ejpam-6909	84	17	0	0	NUM
ejpam-6909	84	18	∈	∈	PROPN
ejpam-6909	84	19	i.	i.	NOUN
ejpam-6909	84	20	then	then	ADV
ejpam-6909	84	21	i	i	PRON
ejpam-6909	84	22	is	be	AUX
ejpam-6909	84	23	known	know	VERB
ejpam-6909	84	24	as	as	ADP
ejpam-6909	84	25	•	•	NUM
ejpam-6909	84	26	an	an	DET
ejpam-6909	84	27	hbcksa	hbcksa	NOUN
ejpam-6909	84	28	of	of	ADP
ejpam-6909	84	29	h	h	NOUN
ejpam-6909	84	30	if	if	SCONJ
ejpam-6909	84	31	ℏ1	ℏ1	ADJ
ejpam-6909	84	32	◦	◦	NOUN
ejpam-6909	84	33	ℏ2	ℏ2	NOUN
ejpam-6909	84	34	⊆	⊆	NUM
ejpam-6909	84	35	i	i	PRON
ejpam-6909	84	36	,	,	PUNCT
ejpam-6909	84	37	for	for	ADP
ejpam-6909	84	38	all	all	DET
ejpam-6909	84	39	ℏ1	ℏ1	NOUN
ejpam-6909	84	40	,	,	PUNCT
ejpam-6909	84	41	ℏ2	ℏ2	NOUN
ejpam-6909	84	42	∈	∈	PROPN
ejpam-6909	84	43	i	i	PRON
ejpam-6909	84	44	,	,	PUNCT
ejpam-6909	84	45	•	•	ADP
ejpam-6909	84	46	a	a	DET
ejpam-6909	84	47	hyper	hyper	ADJ
ejpam-6909	84	48	bck	bck	NOUN
ejpam-6909	84	49	-	-	PUNCT
ejpam-6909	84	50	ideal	ideal	NOUN
ejpam-6909	84	51	of	of	ADP
ejpam-6909	84	52	h	h	NOUN
ejpam-6909	84	53	if	if	SCONJ
ejpam-6909	84	54	for	for	ADP
ejpam-6909	84	55	all	all	DET
ejpam-6909	84	56	ℏ1	ℏ1	NOUN
ejpam-6909	84	57	,	,	PUNCT
ejpam-6909	84	58	ℏ2	ℏ2	NOUN
ejpam-6909	84	59	∈	∈	PROPN
ejpam-6909	84	60	h	h	NOUN
ejpam-6909	84	61	,	,	PUNCT
ejpam-6909	84	62	ℏ1	ℏ1	ADJ
ejpam-6909	84	63	◦	◦	NOUN
ejpam-6909	84	64	ℏ2	ℏ2	NOUN
ejpam-6909	84	65	≪	≪	VERB
ejpam-6909	84	66	i	i	PRON
ejpam-6909	84	67	and	and	CCONJ
ejpam-6909	84	68	ℏ2	ℏ2	NOUN
ejpam-6909	84	69	∈	∈	PROPN
ejpam-6909	84	70	i	i	PRON
ejpam-6909	84	71	⇒	⇒	VERB
ejpam-6909	84	72	ℏ1	ℏ1	PROPN
ejpam-6909	84	73	∈	∈	PROPN
ejpam-6909	85	1	i	i	PRON
ejpam-6909	85	2	,	,	PUNCT
ejpam-6909	85	3	•	•	ADP
ejpam-6909	85	4	a	a	DET
ejpam-6909	85	5	weak	weak	ADJ
ejpam-6909	85	6	hyper	hyper	ADJ
ejpam-6909	85	7	bck	bck	NOUN
ejpam-6909	85	8	-	-	PUNCT
ejpam-6909	85	9	ideal	ideal	NOUN
ejpam-6909	85	10	of	of	ADP
ejpam-6909	85	11	h	h	NOUN
ejpam-6909	85	12	if	if	SCONJ
ejpam-6909	85	13	for	for	ADP
ejpam-6909	85	14	all	all	DET
ejpam-6909	85	15	ℏ1	ℏ1	NOUN
ejpam-6909	85	16	,	,	PUNCT
ejpam-6909	85	17	ℏ2	ℏ2	NOUN
ejpam-6909	85	18	∈	∈	PROPN
ejpam-6909	85	19	h	h	NOUN
ejpam-6909	85	20	,	,	PUNCT
ejpam-6909	85	21	ℏ1	ℏ1	ADJ
ejpam-6909	85	22	◦	◦	NOUN
ejpam-6909	85	23	ℏ2	ℏ2	NOUN
ejpam-6909	85	24	⊆	⊆	NUM
ejpam-6909	85	25	i	i	PRON
ejpam-6909	85	26	and	and	CCONJ
ejpam-6909	85	27	ℏ2	ℏ2	NOUN
ejpam-6909	85	28	∈	∈	PROPN
ejpam-6909	85	29	i	i	PRON
ejpam-6909	85	30	⇒	⇒	VERB
ejpam-6909	85	31	ℏ1	ℏ1	PROPN
ejpam-6909	85	32	∈	∈	PROPN
ejpam-6909	85	33	i	i	PRON
ejpam-6909	85	34	,	,	PUNCT
ejpam-6909	85	35	•	•	ADP
ejpam-6909	85	36	a	a	DET
ejpam-6909	85	37	strong	strong	ADJ
ejpam-6909	85	38	hyper	hyper	ADJ
ejpam-6909	85	39	bck	bck	NOUN
ejpam-6909	85	40	-	-	PUNCT
ejpam-6909	85	41	ideal	ideal	NOUN
ejpam-6909	85	42	of	of	ADP
ejpam-6909	85	43	h	h	NOUN
ejpam-6909	85	44	if	if	SCONJ
ejpam-6909	85	45	for	for	ADP
ejpam-6909	85	46	all	all	DET
ejpam-6909	85	47	ℏ1	ℏ1	NOUN
ejpam-6909	85	48	,	,	PUNCT
ejpam-6909	85	49	ℏ2	ℏ2	NOUN
ejpam-6909	85	50	∈	∈	PROPN
ejpam-6909	85	51	h	h	NOUN
ejpam-6909	85	52	,	,	PUNCT
ejpam-6909	85	53	(	(	PUNCT
ejpam-6909	85	54	ℏ1	ℏ1	PROPN
ejpam-6909	85	55	◦	◦	NOUN
ejpam-6909	85	56	ℏ2	ℏ2	NOUN
ejpam-6909	85	57	)	)	PUNCT
ejpam-6909	85	58	∩	∩	NOUN
ejpam-6909	85	59	i	i	PRON
ejpam-6909	85	60	̸=	̸=	PROPN
ejpam-6909	85	61	∅	∅	NOUN
ejpam-6909	85	62	and	and	CCONJ
ejpam-6909	85	63	ℏ2	ℏ2	NOUN
ejpam-6909	85	64	∈	∈	PROPN
ejpam-6909	85	65	i	i	PRON
ejpam-6909	85	66	⇒	⇒	VERB
ejpam-6909	85	67	ℏ1	ℏ1	PROPN
ejpam-6909	85	68	∈	∈	PROPN
ejpam-6909	85	69	i	i	PRON
ejpam-6909	85	70	,	,	PUNCT
ejpam-6909	85	71	•	•	ADV
ejpam-6909	85	72	reflexive	reflexive	ADJ
ejpam-6909	85	73	if	if	SCONJ
ejpam-6909	85	74	ℏ1	ℏ1	ADJ
ejpam-6909	85	75	◦	◦	NOUN
ejpam-6909	85	76	ℏ1	ℏ1	PROPN
ejpam-6909	85	77	⊆	⊆	NUM
ejpam-6909	85	78	i	i	PROPN
ejpam-6909	85	79	,	,	PUNCT
ejpam-6909	85	80	for	for	ADP
ejpam-6909	85	81	all	all	DET
ejpam-6909	85	82	ℏ1	ℏ1	PROPN
ejpam-6909	85	83	∈	∈	PROPN
ejpam-6909	85	84	h	h	NOUN
ejpam-6909	85	85	,	,	PUNCT
ejpam-6909	85	86	•	•	NOUN
ejpam-6909	85	87	s	s	NOUN
ejpam-6909	85	88	-	-	PUNCT
ejpam-6909	85	89	reflexive	reflexive	ADJ
ejpam-6909	85	90	if	if	SCONJ
ejpam-6909	85	91	for	for	ADP
ejpam-6909	85	92	all	all	DET
ejpam-6909	85	93	ℏ1	ℏ1	NOUN
ejpam-6909	85	94	,	,	PUNCT
ejpam-6909	85	95	ℏ2	ℏ2	NOUN
ejpam-6909	85	96	∈	∈	PROPN
ejpam-6909	85	97	h	h	NOUN
ejpam-6909	85	98	,	,	PUNCT
ejpam-6909	85	99	(	(	PUNCT
ejpam-6909	85	100	ℏ1	ℏ1	PROPN
ejpam-6909	85	101	◦	◦	NOUN
ejpam-6909	85	102	ℏ2	ℏ2	NOUN
ejpam-6909	85	103	)	)	PUNCT
ejpam-6909	85	104	∩	∩	NOUN
ejpam-6909	85	105	i	i	PRON
ejpam-6909	85	106	̸=	̸=	PROPN
ejpam-6909	85	107	∅	∅	VERB
ejpam-6909	85	108	⇒	⇒	NOUN
ejpam-6909	85	109	ℏ1	ℏ1	PROPN
ejpam-6909	85	110	◦	◦	NOUN
ejpam-6909	85	111	ℏ2	ℏ2	NOUN
ejpam-6909	85	112	≪	≪	VERB
ejpam-6909	85	113	i	i	PRON
ejpam-6909	85	114	,	,	PUNCT
ejpam-6909	85	115	•	•	ADV
ejpam-6909	85	116	closed	close	VERB
ejpam-6909	85	117	if	if	SCONJ
ejpam-6909	85	118	for	for	ADP
ejpam-6909	85	119	all	all	DET
ejpam-6909	85	120	ℏ1	ℏ1	NOUN
ejpam-6909	85	121	,	,	PUNCT
ejpam-6909	85	122	ℏ2	ℏ2	NOUN
ejpam-6909	85	123	∈	∈	PROPN
ejpam-6909	85	124	h	h	NOUN
ejpam-6909	85	125	,	,	PUNCT
ejpam-6909	85	126	ℏ1	ℏ1	ADJ
ejpam-6909	85	127	≪	≪	ADJ
ejpam-6909	85	128	ℏ2	ℏ2	NOUN
ejpam-6909	85	129	and	and	CCONJ
ejpam-6909	85	130	ℏ2	ℏ2	NOUN
ejpam-6909	85	131	∈	∈	PROPN
ejpam-6909	85	132	i	i	PRON
ejpam-6909	85	133	⇒	⇒	VERB
ejpam-6909	85	134	ℏ1	ℏ1	PROPN
ejpam-6909	85	135	∈	∈	PROPN
ejpam-6909	85	136	i.	i.	NOUN
ejpam-6909	85	137	it	it	PRON
ejpam-6909	85	138	is	be	AUX
ejpam-6909	85	139	calm	calm	ADJ
ejpam-6909	85	140	to	to	PART
ejpam-6909	85	141	see	see	VERB
ejpam-6909	85	142	that	that	SCONJ
ejpam-6909	85	143	each	each	DET
ejpam-6909	85	144	s	s	NOUN
ejpam-6909	85	145	-	-	ADJ
ejpam-6909	85	146	reflexive	reflexive	ADJ
ejpam-6909	85	147	subset	subset	NOUN
ejpam-6909	85	148	of	of	ADP
ejpam-6909	85	149	h	h	NOUN
ejpam-6909	85	150	is	be	AUX
ejpam-6909	85	151	reflexive	reflexive	ADJ
ejpam-6909	85	152	.	.	PUNCT
ejpam-6909	86	1	definition	definition	NOUN
ejpam-6909	86	2	3	3	NUM
ejpam-6909	86	3	.	.	PUNCT
ejpam-6909	87	1	[	[	X
ejpam-6909	87	2	6	6	NUM
ejpam-6909	87	3	]	]	PUNCT
ejpam-6909	87	4	let	let	VERB
ejpam-6909	87	5	i	i	PRON
ejpam-6909	87	6	be	be	AUX
ejpam-6909	87	7	a	a	DET
ejpam-6909	87	8	non	non	ADJ
ejpam-6909	87	9	-	-	ADJ
ejpam-6909	87	10	empty	empty	ADJ
ejpam-6909	87	11	subset	subset	NOUN
ejpam-6909	87	12	of	of	ADP
ejpam-6909	87	13	an	an	DET
ejpam-6909	87	14	hbcka	hbcka	NOUN
ejpam-6909	87	15	h	h	NOUN
ejpam-6909	87	16	and	and	CCONJ
ejpam-6909	87	17	0	0	NUM
ejpam-6909	87	18	∈	∈	PROPN
ejpam-6909	87	19	i.	i.	NOUN
ejpam-6909	87	20	then	then	ADV
ejpam-6909	87	21	i	i	PRON
ejpam-6909	87	22	is	be	AUX
ejpam-6909	87	23	called	call	VERB
ejpam-6909	87	24	a	a	DET
ejpam-6909	87	25	chbcki	chbcki	NOUN
ejpam-6909	87	26	of	of	ADP
ejpam-6909	87	27	(	(	PUNCT
ejpam-6909	87	28	i	i	NOUN
ejpam-6909	87	29	)	)	PUNCT
ejpam-6909	87	30	type-1	type-1	PROPN
ejpam-6909	87	31	if	if	SCONJ
ejpam-6909	87	32	for	for	ADP
ejpam-6909	87	33	all	all	DET
ejpam-6909	87	34	ℏ1	ℏ1	ADJ
ejpam-6909	87	35	,	,	PUNCT
ejpam-6909	87	36	ℏ2	ℏ2	NOUN
ejpam-6909	87	37	,	,	PUNCT
ejpam-6909	87	38	ℏ3	ℏ3	PROPN
ejpam-6909	87	39	∈	∈	PROPN
ejpam-6909	87	40	h	h	NOUN
ejpam-6909	87	41	,	,	PUNCT
ejpam-6909	87	42	(	(	PUNCT
ejpam-6909	87	43	ℏ1	ℏ1	PROPN
ejpam-6909	87	44	◦	◦	NOUN
ejpam-6909	87	45	ℏ2)	ℏ2)	NOUN
ejpam-6909	87	46	◦	◦	NOUN
ejpam-6909	87	47	ℏ3	ℏ3	PROPN
ejpam-6909	87	48	⊆	⊆	NUM
ejpam-6909	87	49	i	i	PROPN
ejpam-6909	87	50	and	and	CCONJ
ejpam-6909	87	51	ℏ3	ℏ3	PROPN
ejpam-6909	87	52	∈	∈	PROPN
ejpam-6909	87	53	i	i	PRON
ejpam-6909	87	54	⇒	⇒	VERB
ejpam-6909	87	55	ℏ1	ℏ1	PROPN
ejpam-6909	87	56	◦	◦	NOUN
ejpam-6909	87	57	(ℏ2	(ℏ2	NOUN
ejpam-6909	87	58	◦	◦	ADJ
ejpam-6909	87	59	(ℏ2	(ℏ2	NOUN
ejpam-6909	87	60	◦	◦	NOUN
ejpam-6909	87	61	ℏ1	ℏ1	ADJ
ejpam-6909	87	62	)	)	PUNCT
ejpam-6909	87	63	)	)	PUNCT
ejpam-6909	88	1	⊆	⊆	NUM
ejpam-6909	88	2	i	i	PRON
ejpam-6909	88	3	,	,	PUNCT
ejpam-6909	88	4	(	(	PUNCT
ejpam-6909	88	5	ii	ii	NOUN
ejpam-6909	88	6	)	)	PUNCT
ejpam-6909	88	7	type-2	type-2	CCONJ
ejpam-6909	88	8	if	if	SCONJ
ejpam-6909	88	9	for	for	ADP
ejpam-6909	88	10	all	all	DET
ejpam-6909	88	11	ℏ1	ℏ1	ADJ
ejpam-6909	88	12	,	,	PUNCT
ejpam-6909	88	13	ℏ2	ℏ2	NOUN
ejpam-6909	88	14	,	,	PUNCT
ejpam-6909	88	15	ℏ3	ℏ3	PROPN
ejpam-6909	88	16	∈	∈	PROPN
ejpam-6909	88	17	h	h	NOUN
ejpam-6909	88	18	,	,	PUNCT
ejpam-6909	88	19	(	(	PUNCT
ejpam-6909	88	20	ℏ1	ℏ1	PROPN
ejpam-6909	88	21	◦	◦	NOUN
ejpam-6909	88	22	ℏ2)	ℏ2)	NOUN
ejpam-6909	88	23	◦	◦	NOUN
ejpam-6909	88	24	ℏ3	ℏ3	PROPN
ejpam-6909	88	25	⊆	⊆	NUM
ejpam-6909	88	26	i	i	PROPN
ejpam-6909	88	27	and	and	CCONJ
ejpam-6909	88	28	ℏ3	ℏ3	PROPN
ejpam-6909	88	29	∈	∈	PROPN
ejpam-6909	88	30	i	i	PRON
ejpam-6909	88	31	⇒	⇒	VERB
ejpam-6909	88	32	ℏ1	ℏ1	PROPN
ejpam-6909	88	33	◦	◦	NOUN
ejpam-6909	88	34	(ℏ2	(ℏ2	NOUN
ejpam-6909	88	35	◦	◦	ADJ
ejpam-6909	88	36	(ℏ2	(ℏ2	NOUN
ejpam-6909	88	37	◦	◦	NOUN
ejpam-6909	88	38	ℏ1	ℏ1	ADJ
ejpam-6909	88	39	)	)	PUNCT
ejpam-6909	88	40	)	)	PUNCT
ejpam-6909	89	1	≪	≪	PUNCT
ejpam-6909	89	2	i	i	PRON
ejpam-6909	89	3	,	,	PUNCT
ejpam-6909	89	4	(	(	PUNCT
ejpam-6909	89	5	iii	iii	X
ejpam-6909	89	6	)	)	PUNCT
ejpam-6909	89	7	type-3	type-3	NOUN
ejpam-6909	89	8	if	if	SCONJ
ejpam-6909	89	9	for	for	ADP
ejpam-6909	89	10	all	all	DET
ejpam-6909	89	11	ℏ1	ℏ1	ADJ
ejpam-6909	89	12	,	,	PUNCT
ejpam-6909	89	13	ℏ2	ℏ2	NOUN
ejpam-6909	89	14	,	,	PUNCT
ejpam-6909	89	15	ℏ3	ℏ3	PROPN
ejpam-6909	89	16	∈	∈	PROPN
ejpam-6909	89	17	h	h	NOUN
ejpam-6909	89	18	,	,	PUNCT
ejpam-6909	89	19	(	(	PUNCT
ejpam-6909	89	20	ℏ1	ℏ1	PROPN
ejpam-6909	89	21	◦	◦	NOUN
ejpam-6909	89	22	ℏ2)	ℏ2)	NOUN
ejpam-6909	89	23	◦	◦	NOUN
ejpam-6909	89	24	ℏ3	ℏ3	NOUN
ejpam-6909	89	25	≪	≪	VERB
ejpam-6909	89	26	i	i	PRON
ejpam-6909	89	27	and	and	CCONJ
ejpam-6909	89	28	ℏ3	ℏ3	PROPN
ejpam-6909	89	29	∈	∈	PROPN
ejpam-6909	89	30	i	i	PRON
ejpam-6909	89	31	⇒	⇒	VERB
ejpam-6909	89	32	ℏ1	ℏ1	PROPN
ejpam-6909	89	33	◦	◦	NOUN
ejpam-6909	89	34	(ℏ2	(ℏ2	NOUN
ejpam-6909	89	35	◦	◦	ADJ
ejpam-6909	89	36	(ℏ2	(ℏ2	NOUN
ejpam-6909	89	37	◦	◦	NOUN
ejpam-6909	89	38	ℏ1	ℏ1	ADJ
ejpam-6909	89	39	)	)	PUNCT
ejpam-6909	89	40	)	)	PUNCT
ejpam-6909	90	1	⊆	⊆	NUM
ejpam-6909	90	2	i	i	PRON
ejpam-6909	90	3	,	,	PUNCT
ejpam-6909	90	4	(	(	PUNCT
ejpam-6909	90	5	iv	iv	X
ejpam-6909	90	6	)	)	PUNCT
ejpam-6909	90	7	type-4	type-4	NOUN
ejpam-6909	90	8	if	if	SCONJ
ejpam-6909	90	9	for	for	ADP
ejpam-6909	90	10	all	all	DET
ejpam-6909	90	11	ℏ1	ℏ1	ADJ
ejpam-6909	90	12	,	,	PUNCT
ejpam-6909	90	13	ℏ2	ℏ2	NOUN
ejpam-6909	90	14	,	,	PUNCT
ejpam-6909	90	15	ℏ3	ℏ3	PROPN
ejpam-6909	90	16	∈	∈	PROPN
ejpam-6909	90	17	h	h	NOUN
ejpam-6909	90	18	,	,	PUNCT
ejpam-6909	90	19	(	(	PUNCT
ejpam-6909	90	20	ℏ1	ℏ1	PROPN
ejpam-6909	90	21	◦	◦	NOUN
ejpam-6909	90	22	ℏ2)	ℏ2)	NOUN
ejpam-6909	90	23	◦	◦	NOUN
ejpam-6909	90	24	ℏ3	ℏ3	NOUN
ejpam-6909	90	25	≪	≪	VERB
ejpam-6909	90	26	i	i	PRON
ejpam-6909	90	27	and	and	CCONJ
ejpam-6909	90	28	ℏ3	ℏ3	PROPN
ejpam-6909	90	29	∈	∈	PROPN
ejpam-6909	90	30	i	i	PRON
ejpam-6909	90	31	⇒	⇒	VERB
ejpam-6909	90	32	ℏ1	ℏ1	PROPN
ejpam-6909	90	33	◦	◦	NOUN
ejpam-6909	90	34	(ℏ2	(ℏ2	NOUN
ejpam-6909	90	35	◦	◦	ADJ
ejpam-6909	90	36	(ℏ2	(ℏ2	NOUN
ejpam-6909	90	37	◦	◦	NOUN
ejpam-6909	90	38	ℏ1	ℏ1	ADJ
ejpam-6909	90	39	)	)	PUNCT
ejpam-6909	90	40	)	)	PUNCT
ejpam-6909	90	41	≪	≪	PUNCT
ejpam-6909	90	42	i.	i.	NOUN
ejpam-6909	90	43	theorem	theorem	NOUN
ejpam-6909	90	44	1	1	NUM
ejpam-6909	90	45	.	.	PUNCT
ejpam-6909	91	1	[	[	X
ejpam-6909	91	2	28	28	NUM
ejpam-6909	91	3	]	]	PUNCT
ejpam-6909	91	4	let	let	VERB
ejpam-6909	91	5	h1	h1	VERB
ejpam-6909	92	1	and	and	CCONJ
ejpam-6909	92	2	i	i	PRON
ejpam-6909	92	3	be	be	VERB
ejpam-6909	92	4	non	non	ADJ
ejpam-6909	92	5	-	-	ADJ
ejpam-6909	92	6	empty	empty	ADJ
ejpam-6909	92	7	subsets	subset	NOUN
ejpam-6909	92	8	of	of	ADP
ejpam-6909	92	9	an	an	DET
ejpam-6909	92	10	hbcka	hbcka	NOUN
ejpam-6909	92	11	h.	h.	PROPN
ejpam-6909	93	1	then	then	ADV
ejpam-6909	93	2	(	(	PUNCT
ejpam-6909	93	3	i	i	NOUN
ejpam-6909	93	4	)	)	PUNCT
ejpam-6909	93	5	if	if	SCONJ
ejpam-6909	93	6	i	i	PRON
ejpam-6909	93	7	is	be	AUX
ejpam-6909	93	8	an	an	DET
ejpam-6909	93	9	hbcki	hbcki	NOUN
ejpam-6909	93	10	of	of	ADP
ejpam-6909	93	11	h	h	PROPN
ejpam-6909	93	12	and	and	CCONJ
ejpam-6909	93	13	h1	h1	VERB
ejpam-6909	93	14	≪	≪	PUNCT
ejpam-6909	93	15	i	i	PRON
ejpam-6909	93	16	,	,	PUNCT
ejpam-6909	93	17	then	then	ADV
ejpam-6909	93	18	h1	h1	VERB
ejpam-6909	93	19	⊆	⊆	NUM
ejpam-6909	93	20	i.	i.	NOUN
ejpam-6909	93	21	(	(	PUNCT
ejpam-6909	93	22	ii	ii	PROPN
ejpam-6909	93	23	)	)	PUNCT
ejpam-6909	93	24	if	if	SCONJ
ejpam-6909	93	25	i	i	PRON
ejpam-6909	93	26	is	be	AUX
ejpam-6909	93	27	a	a	DET
ejpam-6909	93	28	reflexive	reflexive	ADJ
ejpam-6909	93	29	hbcki	hbcki	NOUN
ejpam-6909	93	30	of	of	ADP
ejpam-6909	93	31	h	h	NOUN
ejpam-6909	93	32	,	,	PUNCT
ejpam-6909	93	33	then	then	ADV
ejpam-6909	93	34	(	(	PUNCT
ejpam-6909	93	35	ℏ1	ℏ1	PROPN
ejpam-6909	93	36	◦	◦	NOUN
ejpam-6909	93	37	ℏ2	ℏ2	NOUN
ejpam-6909	93	38	)	)	PUNCT
ejpam-6909	93	39	∩	∩	NOUN
ejpam-6909	93	40	i	i	PRON
ejpam-6909	93	41	̸=	̸=	PROPN
ejpam-6909	93	42	∅	∅	VERB
ejpam-6909	93	43	⇒	⇒	NOUN
ejpam-6909	93	44	ℏ1	ℏ1	PROPN
ejpam-6909	93	45	◦	◦	NOUN
ejpam-6909	93	46	ℏ2	ℏ2	NOUN
ejpam-6909	93	47	≪	≪	VERB
ejpam-6909	93	48	i	i	PRON
ejpam-6909	93	49	,	,	PUNCT
ejpam-6909	93	50	for	for	ADP
ejpam-6909	93	51	all	all	DET
ejpam-6909	93	52	ℏ1	ℏ1	NOUN
ejpam-6909	93	53	,	,	PUNCT
ejpam-6909	93	54	ℏ2	ℏ2	NOUN
ejpam-6909	93	55	∈	∈	PROPN
ejpam-6909	93	56	h.	h.	NOUN
ejpam-6909	93	57	definition	definition	NOUN
ejpam-6909	93	58	4	4	NUM
ejpam-6909	93	59	.	.	PUNCT
ejpam-6909	94	1	[	[	X
ejpam-6909	94	2	10	10	NUM
ejpam-6909	94	3	]	]	X
ejpam-6909	94	4	a	a	DET
ejpam-6909	94	5	bipolar	bipolar	ADJ
ejpam-6909	94	6	fuzzy	fuzzy	ADJ
ejpam-6909	94	7	set	set	NOUN
ejpam-6909	94	8	(	(	PUNCT
ejpam-6909	94	9	bfs	bfs	NOUN
ejpam-6909	94	10	)	)	PUNCT
ejpam-6909	94	11	a	a	PRON
ejpam-6909	94	12	of	of	ADP
ejpam-6909	94	13	a	a	DET
ejpam-6909	94	14	set	set	NOUN
ejpam-6909	94	15	i	i	PRON
ejpam-6909	94	16	is	be	AUX
ejpam-6909	94	17	defined	define	VERB
ejpam-6909	94	18	as	as	ADP
ejpam-6909	94	19	a	a	DET
ejpam-6909	94	20	=	=	SYM
ejpam-6909	94	21	{	{	PUNCT
ejpam-6909	94	22	(	(	PUNCT
ejpam-6909	94	23	ℏ1	ℏ1	ADJ
ejpam-6909	94	24	,	,	PUNCT
ejpam-6909	94	25	α+	α+	NOUN
ejpam-6909	94	26	a(ℏ1	a(ℏ1	NUM
ejpam-6909	94	27	)	)	PUNCT
ejpam-6909	94	28	,	,	PUNCT
ejpam-6909	94	29	β	β	X
ejpam-6909	94	30	−	−	VERB
ejpam-6909	94	31	a	a	DET
ejpam-6909	94	32	(	(	PUNCT
ejpam-6909	94	33	ℏ1	ℏ1	ADJ
ejpam-6909	94	34	)	)	PUNCT
ejpam-6909	94	35	)	)	PUNCT
ejpam-6909	95	1	|	|	ADV
ejpam-6909	95	2	ℏ1	ℏ1	PROPN
ejpam-6909	95	3	∈	∈	PROPN
ejpam-6909	95	4	i	i	X
ejpam-6909	95	5	}	}	PUNCT
ejpam-6909	95	6	,	,	PUNCT
ejpam-6909	95	7	(	(	PUNCT
ejpam-6909	95	8	1	1	X
ejpam-6909	95	9	)	)	PUNCT
ejpam-6909	95	10	where	where	SCONJ
ejpam-6909	95	11	α+	α+	PRON
ejpam-6909	95	12	a	a	X
ejpam-6909	95	13	:	:	PUNCT
ejpam-6909	95	14	i	i	PRON
ejpam-6909	95	15	→	→	PUNCT
ejpam-6909	95	16	[	[	X
ejpam-6909	95	17	0	0	NUM
ejpam-6909	95	18	,	,	PUNCT
ejpam-6909	95	19	1	1	NUM
ejpam-6909	95	20	]	]	PUNCT
ejpam-6909	95	21	and	and	CCONJ
ejpam-6909	95	22	β−	β−	PRON
ejpam-6909	95	23	a	a	PRON
ejpam-6909	95	24	:	:	PUNCT
ejpam-6909	95	25	i	i	PRON
ejpam-6909	95	26	→	→	PUNCT
ejpam-6909	96	1	[	[	X
ejpam-6909	96	2	−1	−1	NOUN
ejpam-6909	96	3	,	,	PUNCT
ejpam-6909	96	4	0	0	NUM
ejpam-6909	96	5	]	]	PUNCT
ejpam-6909	96	6	are	be	AUX
ejpam-6909	96	7	mappings	mapping	NOUN
ejpam-6909	96	8	.	.	PUNCT
ejpam-6909	97	1	the	the	DET
ejpam-6909	97	2	positive	positive	ADJ
ejpam-6909	97	3	membership	membership	NOUN
ejpam-6909	97	4	degree	degree	NOUN
ejpam-6909	97	5	α+	α+	PUNCT
ejpam-6909	97	6	a	a	DET
ejpam-6909	97	7	denotes	denote	NOUN
ejpam-6909	97	8	the	the	DET
ejpam-6909	97	9	level	level	NOUN
ejpam-6909	97	10	of	of	ADP
ejpam-6909	97	11	satisfaction	satisfaction	NOUN
ejpam-6909	97	12	that	that	SCONJ
ejpam-6909	97	13	the	the	DET
ejpam-6909	97	14	element	element	NOUN
ejpam-6909	97	15	of	of	ADP
ejpam-6909	97	16	i	i	PRON
ejpam-6909	97	17	to	to	ADP
ejpam-6909	97	18	the	the	DET
ejpam-6909	97	19	property	property	NOUN
ejpam-6909	97	20	associated	associate	VERB
ejpam-6909	97	21	with	with	ADP
ejpam-6909	97	22	the	the	DET
ejpam-6909	97	23	bipolar	bipolar	ADJ
ejpam-6909	97	24	fuzzy	fuzzy	NOUN
ejpam-6909	97	25	set	set	VERB
ejpam-6909	97	26	a	a	PRON
ejpam-6909	97	27	,	,	PUNCT
ejpam-6909	97	28	while	while	SCONJ
ejpam-6909	97	29	the	the	DET
ejpam-6909	97	30	negative	negative	ADJ
ejpam-6909	97	31	membership	membership	NOUN
ejpam-6909	97	32	degree	degree	NOUN
ejpam-6909	97	33	β−	β−	PRON
ejpam-6909	97	34	a	a	DET
ejpam-6909	97	35	denotes	denote	NOUN
ejpam-6909	97	36	the	the	DET
ejpam-6909	97	37	level	level	NOUN
ejpam-6909	97	38	of	of	ADP
ejpam-6909	97	39	satisfaction	satisfaction	NOUN
ejpam-6909	97	40	that	that	SCONJ
ejpam-6909	97	41	the	the	DET
ejpam-6909	97	42	element	element	NOUN
ejpam-6909	97	43	of	of	ADP
ejpam-6909	97	44	i	i	PRON
ejpam-6909	97	45	to	to	ADP
ejpam-6909	97	46	some	some	DET
ejpam-6909	97	47	implicit	implicit	ADJ
ejpam-6909	97	48	counter	counter	ADJ
ejpam-6909	97	49	property	property	NOUN
ejpam-6909	97	50	of	of	ADP
ejpam-6909	97	51	a.	a.	NOUN
ejpam-6909	97	52	we	we	PRON
ejpam-6909	97	53	shall	shall	AUX
ejpam-6909	97	54	use	use	VERB
ejpam-6909	97	55	the	the	DET
ejpam-6909	97	56	symbol	symbol	NOUN
ejpam-6909	97	57	(	(	PUNCT
ejpam-6909	97	58	α+	α+	X
ejpam-6909	97	59	a	a	X
ejpam-6909	97	60	,	,	PUNCT
ejpam-6909	97	61	β	β	PROPN
ejpam-6909	97	62	−	−	NOUN
ejpam-6909	97	63	a	a	PRON
ejpam-6909	97	64	)	)	PUNCT
ejpam-6909	97	65	to	to	PART
ejpam-6909	97	66	denote	denote	VERB
ejpam-6909	97	67	a	a	DET
ejpam-6909	97	68	bipolar	bipolar	ADJ
ejpam-6909	97	69	fuzzy	fuzzy	NOUN
ejpam-6909	97	70	set	set	VERB
ejpam-6909	97	71	a	a	PRON
ejpam-6909	97	72	(	(	PUNCT
ejpam-6909	97	73	see	see	VERB
ejpam-6909	97	74	(	(	PUNCT
ejpam-6909	97	75	1	1	NUM
ejpam-6909	97	76	)	)	PUNCT
ejpam-6909	97	77	)	)	PUNCT
ejpam-6909	97	78	.	.	PUNCT
ejpam-6909	98	1	definition	definition	NOUN
ejpam-6909	98	2	5	5	NUM
ejpam-6909	98	3	.	.	PUNCT
ejpam-6909	99	1	[	[	X
ejpam-6909	99	2	20	20	NUM
ejpam-6909	99	3	]	]	SYM
ejpam-6909	99	4	a	a	DET
ejpam-6909	99	5	bfs	bfs	NOUN
ejpam-6909	99	6	(	(	PUNCT
ejpam-6909	99	7	α+	α+	NOUN
ejpam-6909	99	8	a	a	X
ejpam-6909	99	9	,	,	PUNCT
ejpam-6909	99	10	β	β	PROPN
ejpam-6909	99	11	−	−	NOUN
ejpam-6909	99	12	a	a	X
ejpam-6909	99	13	)	)	PUNCT
ejpam-6909	99	14	in	in	ADP
ejpam-6909	99	15	h	h	NOUN
ejpam-6909	99	16	is	be	AUX
ejpam-6909	99	17	called	call	VERB
ejpam-6909	99	18	a	a	DET
ejpam-6909	99	19	bipolar	bipolar	ADJ
ejpam-6909	99	20	fuzzy	fuzzy	ADJ
ejpam-6909	99	21	hyper	hyper	ADJ
ejpam-6909	99	22	bck	bck	NOUN
ejpam-6909	99	23	-	-	PUNCT
ejpam-6909	99	24	ideal	ideal	NOUN
ejpam-6909	99	25	(	(	PUNCT
ejpam-6909	99	26	bfhbcki	bfhbcki	NOUN
ejpam-6909	99	27	)	)	PUNCT
ejpam-6909	99	28	of	of	ADP
ejpam-6909	99	29	h	h	NOUN
ejpam-6909	99	30	if	if	SCONJ
ejpam-6909	99	31	it	it	PRON
ejpam-6909	99	32	satisfies	satisfy	VERB
ejpam-6909	99	33	:	:	PUNCT
ejpam-6909	100	1	d.	d.	PROPN
ejpam-6909	100	2	ramesh	ramesh	PROPN
ejpam-6909	100	3	et	et	PROPN
ejpam-6909	100	4	al	al	PROPN
ejpam-6909	100	5	.	.	PUNCT
ejpam-6909	100	6	/	/	SYM
ejpam-6909	100	7	eur	eur	PROPN
ejpam-6909	100	8	.	.	PUNCT
ejpam-6909	101	1	j.	j.	PROPN
ejpam-6909	101	2	pure	pure	PROPN
ejpam-6909	101	3	appl	appl	PROPN
ejpam-6909	101	4	.	.	PROPN
ejpam-6909	101	5	math	math	PROPN
ejpam-6909	101	6	,	,	PUNCT
ejpam-6909	101	7	18	18	NUM
ejpam-6909	101	8	(	(	PUNCT
ejpam-6909	101	9	4	4	NUM
ejpam-6909	101	10	)	)	PUNCT
ejpam-6909	101	11	(	(	PUNCT
ejpam-6909	101	12	2025	2025	NUM
ejpam-6909	101	13	)	)	PUNCT
ejpam-6909	101	14	,	,	PUNCT
ejpam-6909	101	15	6909	6909	NUM
ejpam-6909	101	16	6	6	NUM
ejpam-6909	101	17	of	of	ADP
ejpam-6909	101	18	16	16	NUM
ejpam-6909	101	19	(	(	PUNCT
ejpam-6909	101	20	i	i	NOUN
ejpam-6909	101	21	)	)	PUNCT
ejpam-6909	101	22	ℏ1	ℏ1	NOUN
ejpam-6909	101	23	≪	≪	PUNCT
ejpam-6909	101	24	ℏ2	ℏ2	NOUN
ejpam-6909	101	25	⇒	⇒	VERB
ejpam-6909	101	26	α+	α+	PRON
ejpam-6909	101	27	a(ℏ1	a(ℏ1	NUM
ejpam-6909	101	28	)	)	PUNCT
ejpam-6909	101	29	≥	≥	NOUN
ejpam-6909	101	30	α+	α+	X
ejpam-6909	101	31	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	101	32	)	)	PUNCT
ejpam-6909	101	33	and	and	CCONJ
ejpam-6909	101	34	β−	β−	PRON
ejpam-6909	101	35	a	a	DET
ejpam-6909	101	36	(	(	PUNCT
ejpam-6909	101	37	ℏ1	ℏ1	PROPN
ejpam-6909	101	38	)	)	PUNCT
ejpam-6909	101	39	≤	≤	NOUN
ejpam-6909	101	40	β−	β−	PUNCT
ejpam-6909	102	1	a	a	DET
ejpam-6909	102	2	(	(	PUNCT
ejpam-6909	102	3	ℏ2	ℏ2	NOUN
ejpam-6909	102	4	)	)	PUNCT
ejpam-6909	102	5	,	,	PUNCT
ejpam-6909	102	6	(	(	PUNCT
ejpam-6909	102	7	ii	ii	NOUN
ejpam-6909	102	8	)	)	PUNCT
ejpam-6909	102	9	α+	α+	X
ejpam-6909	102	10	a(ℏ1	a(ℏ1	NUM
ejpam-6909	102	11	)	)	PUNCT
ejpam-6909	102	12	≥	≥	PROPN
ejpam-6909	102	13	min	min	PROPN
ejpam-6909	102	14	{	{	PUNCT
ejpam-6909	102	15	inf	inf	PROPN
ejpam-6909	102	16	α+	α+	PRON
ejpam-6909	102	17	a(a	a(a	PROPN
ejpam-6909	102	18	)	)	PUNCT
ejpam-6909	102	19	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	102	20	◦	◦	ADJ
ejpam-6909	102	21	ℏ2	ℏ2	NOUN
ejpam-6909	102	22	,	,	PUNCT
ejpam-6909	102	23	α+	α+	PRON
ejpam-6909	102	24	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	102	25	)	)	PUNCT
ejpam-6909	102	26	}	}	PUNCT
ejpam-6909	102	27	,	,	PUNCT
ejpam-6909	102	28	(	(	PUNCT
ejpam-6909	102	29	iii	iii	NOUN
ejpam-6909	102	30	)	)	PUNCT
ejpam-6909	102	31	β−	β−	NOUN
ejpam-6909	103	1	a	a	DET
ejpam-6909	103	2	(	(	PUNCT
ejpam-6909	103	3	ℏ1	ℏ1	PROPN
ejpam-6909	103	4	)	)	PUNCT
ejpam-6909	103	5	≤	≤	NUM
ejpam-6909	103	6	max	max	PROPN
ejpam-6909	103	7	{	{	PUNCT
ejpam-6909	103	8	supβ−	supβ−	PROPN
ejpam-6909	103	9	a	a	DET
ejpam-6909	103	10	(	(	PUNCT
ejpam-6909	103	11	a	a	PRON
ejpam-6909	103	12	)	)	PUNCT
ejpam-6909	103	13	a∈ℏ1	a∈ℏ1	NOUN
ejpam-6909	103	14	◦	◦	NOUN
ejpam-6909	103	15	ℏ2	ℏ2	NOUN
ejpam-6909	103	16	,	,	PUNCT
ejpam-6909	103	17	β−	β−	PRON
ejpam-6909	103	18	a	a	DET
ejpam-6909	103	19	(	(	PUNCT
ejpam-6909	103	20	ℏ2	ℏ2	NOUN
ejpam-6909	103	21	)	)	PUNCT
ejpam-6909	103	22	}	}	PUNCT
ejpam-6909	103	23	,	,	PUNCT
ejpam-6909	103	24	for	for	ADP
ejpam-6909	103	25	all	all	DET
ejpam-6909	103	26	ℏ1	ℏ1	NOUN
ejpam-6909	103	27	,	,	PUNCT
ejpam-6909	103	28	ℏ2	ℏ2	NOUN
ejpam-6909	103	29	∈	∈	PROPN
ejpam-6909	103	30	h.	h.	NOUN
ejpam-6909	103	31	definition	definition	NOUN
ejpam-6909	103	32	6	6	NUM
ejpam-6909	103	33	.	.	PUNCT
ejpam-6909	104	1	[	[	X
ejpam-6909	104	2	20	20	NUM
ejpam-6909	104	3	]	]	SYM
ejpam-6909	104	4	a	a	DET
ejpam-6909	104	5	bfs	bfs	NOUN
ejpam-6909	104	6	(	(	PUNCT
ejpam-6909	104	7	α+	α+	NOUN
ejpam-6909	104	8	a	a	X
ejpam-6909	104	9	,	,	PUNCT
ejpam-6909	104	10	β	β	PROPN
ejpam-6909	104	11	−	−	NOUN
ejpam-6909	104	12	a	a	X
ejpam-6909	104	13	)	)	PUNCT
ejpam-6909	104	14	in	in	ADP
ejpam-6909	104	15	h	h	NOUN
ejpam-6909	104	16	is	be	AUX
ejpam-6909	104	17	known	know	VERB
ejpam-6909	104	18	as	as	ADP
ejpam-6909	104	19	a	a	DET
ejpam-6909	104	20	bipolar	bipolar	ADJ
ejpam-6909	104	21	fuzzy	fuzzy	ADJ
ejpam-6909	104	22	strong	strong	ADJ
ejpam-6909	104	23	hyper	hyper	ADJ
ejpam-6909	104	24	bckideal	bckideal	NOUN
ejpam-6909	104	25	(	(	PUNCT
ejpam-6909	104	26	bfshbcki	bfshbcki	PROPN
ejpam-6909	104	27	)	)	PUNCT
ejpam-6909	104	28	of	of	ADP
ejpam-6909	104	29	h	h	NOUN
ejpam-6909	104	30	if	if	SCONJ
ejpam-6909	104	31	it	it	PRON
ejpam-6909	104	32	satisfies	satisfy	VERB
ejpam-6909	104	33	:	:	PUNCT
ejpam-6909	104	34	(	(	PUNCT
ejpam-6909	104	35	i	i	NOUN
ejpam-6909	104	36	)	)	PUNCT
ejpam-6909	104	37	inf	inf	PROPN
ejpam-6909	104	38	α+	α+	PRON
ejpam-6909	104	39	a(a	a(a	PROPN
ejpam-6909	104	40	)	)	PUNCT
ejpam-6909	104	41	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	104	42	◦	◦	NOUN
ejpam-6909	104	43	ℏ1	ℏ1	PROPN
ejpam-6909	104	44	≥	≥	X
ejpam-6909	104	45	α+	α+	X
ejpam-6909	104	46	a(ℏ1	a(ℏ1	NUM
ejpam-6909	104	47	)	)	PUNCT
ejpam-6909	104	48	≥	≥	PROPN
ejpam-6909	104	49	min	min	PROPN
ejpam-6909	104	50	{	{	PUNCT
ejpam-6909	104	51	inf	inf	NOUN
ejpam-6909	104	52	α+	α+	PUNCT
ejpam-6909	104	53	a(b	a(b	NOUN
ejpam-6909	104	54	)	)	PUNCT
ejpam-6909	104	55	b∈ℏ1	b∈ℏ1	NOUN
ejpam-6909	104	56	◦	◦	NOUN
ejpam-6909	104	57	ℏ2	ℏ2	NOUN
ejpam-6909	104	58	,	,	PUNCT
ejpam-6909	105	1	α+	α+	DET
ejpam-6909	105	2	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	105	3	)	)	PUNCT
ejpam-6909	106	1	}	}	PUNCT
ejpam-6909	106	2	,	,	PUNCT
ejpam-6909	106	3	(	(	PUNCT
ejpam-6909	106	4	ii	ii	NOUN
ejpam-6909	106	5	)	)	PUNCT
ejpam-6909	106	6	supβ−	supβ−	NOUN
ejpam-6909	106	7	a	a	DET
ejpam-6909	106	8	(	(	PUNCT
ejpam-6909	106	9	c	c	NOUN
ejpam-6909	106	10	)	)	PUNCT
ejpam-6909	106	11	c∈ℏ1	c∈ℏ1	PROPN
ejpam-6909	106	12	◦	◦	NOUN
ejpam-6909	106	13	ℏ1	ℏ1	ADJ
ejpam-6909	106	14	≤	≤	X
ejpam-6909	106	15	β−	β−	PUNCT
ejpam-6909	107	1	a	a	DET
ejpam-6909	107	2	(	(	PUNCT
ejpam-6909	107	3	ℏ1	ℏ1	PROPN
ejpam-6909	107	4	)	)	PUNCT
ejpam-6909	107	5	≤	≤	NUM
ejpam-6909	107	6	max	max	PROPN
ejpam-6909	107	7	{	{	PUNCT
ejpam-6909	107	8	supβ−	supβ−	PROPN
ejpam-6909	107	9	a	a	DET
ejpam-6909	107	10	(	(	PUNCT
ejpam-6909	107	11	d	d	NOUN
ejpam-6909	107	12	)	)	PUNCT
ejpam-6909	107	13	d∈ℏ1	d∈ℏ1	NOUN
ejpam-6909	107	14	◦	◦	NOUN
ejpam-6909	107	15	ℏ2	ℏ2	NOUN
ejpam-6909	107	16	,	,	PUNCT
ejpam-6909	107	17	β−	β−	PRON
ejpam-6909	107	18	a	a	DET
ejpam-6909	107	19	(	(	PUNCT
ejpam-6909	107	20	ℏ2	ℏ2	NOUN
ejpam-6909	107	21	)	)	PUNCT
ejpam-6909	107	22	}	}	PUNCT
ejpam-6909	107	23	,	,	PUNCT
ejpam-6909	107	24	for	for	ADP
ejpam-6909	107	25	all	all	DET
ejpam-6909	107	26	ℏ1	ℏ1	NOUN
ejpam-6909	107	27	,	,	PUNCT
ejpam-6909	107	28	ℏ2	ℏ2	NOUN
ejpam-6909	107	29	∈	∈	PROPN
ejpam-6909	107	30	h.	h.	NOUN
ejpam-6909	107	31	definition	definition	NOUN
ejpam-6909	107	32	7	7	NUM
ejpam-6909	107	33	.	.	PUNCT
ejpam-6909	108	1	[	[	X
ejpam-6909	108	2	20	20	NUM
ejpam-6909	108	3	]	]	SYM
ejpam-6909	108	4	a	a	DET
ejpam-6909	108	5	bfs	bfs	NOUN
ejpam-6909	108	6	(	(	PUNCT
ejpam-6909	108	7	α+	α+	NOUN
ejpam-6909	108	8	a	a	X
ejpam-6909	108	9	,	,	PUNCT
ejpam-6909	108	10	β	β	PROPN
ejpam-6909	108	11	−	−	NOUN
ejpam-6909	108	12	a	a	X
ejpam-6909	108	13	)	)	PUNCT
ejpam-6909	108	14	in	in	ADP
ejpam-6909	108	15	h	h	NOUN
ejpam-6909	108	16	is	be	AUX
ejpam-6909	108	17	known	know	VERB
ejpam-6909	108	18	as	as	ADP
ejpam-6909	108	19	a	a	DET
ejpam-6909	108	20	bipolar	bipolar	ADJ
ejpam-6909	108	21	fuzzy	fuzzy	ADJ
ejpam-6909	108	22	s	s	NOUN
ejpam-6909	108	23	-	-	PUNCT
ejpam-6909	108	24	weak	weak	ADJ
ejpam-6909	108	25	hyper	hyper	ADJ
ejpam-6909	108	26	bckideal	bckideal	ADJ
ejpam-6909	108	27	(	(	PUNCT
ejpam-6909	108	28	bfswhbcki	bfswhbcki	NOUN
ejpam-6909	108	29	)	)	PUNCT
ejpam-6909	108	30	of	of	ADP
ejpam-6909	108	31	h	h	NOUN
ejpam-6909	108	32	if	if	SCONJ
ejpam-6909	108	33	it	it	PRON
ejpam-6909	108	34	satisfies	satisfy	VERB
ejpam-6909	108	35	:	:	PUNCT
ejpam-6909	108	36	(	(	PUNCT
ejpam-6909	108	37	i	i	NOUN
ejpam-6909	108	38	)	)	PUNCT
ejpam-6909	108	39	α+	α+	PRON
ejpam-6909	108	40	a(0	a(0	PROPN
ejpam-6909	108	41	)	)	PUNCT
ejpam-6909	108	42	≥	≥	NOUN
ejpam-6909	108	43	α+	α+	X
ejpam-6909	108	44	a(ℏ1	a(ℏ1	NUM
ejpam-6909	108	45	)	)	PUNCT
ejpam-6909	109	1	and	and	CCONJ
ejpam-6909	109	2	β−	β−	PRON
ejpam-6909	109	3	a	a	DET
ejpam-6909	109	4	(	(	PUNCT
ejpam-6909	109	5	0	0	NUM
ejpam-6909	109	6	)	)	PUNCT
ejpam-6909	109	7	≤	≤	NOUN
ejpam-6909	109	8	β−	β−	PUNCT
ejpam-6909	110	1	a	a	DET
ejpam-6909	110	2	(	(	PUNCT
ejpam-6909	110	3	ℏ1	ℏ1	PROPN
ejpam-6909	110	4	)	)	PUNCT
ejpam-6909	110	5	,	,	PUNCT
ejpam-6909	110	6	for	for	ADP
ejpam-6909	110	7	all	all	DET
ejpam-6909	110	8	ℏ1	ℏ1	PROPN
ejpam-6909	110	9	∈	∈	PROPN
ejpam-6909	110	10	h	h	NOUN
ejpam-6909	110	11	,	,	PUNCT
ejpam-6909	110	12	(	(	PUNCT
ejpam-6909	110	13	ii	ii	NOUN
ejpam-6909	110	14	)	)	PUNCT
ejpam-6909	110	15	for	for	ADP
ejpam-6909	110	16	every	every	DET
ejpam-6909	110	17	ℏ1	ℏ1	NOUN
ejpam-6909	110	18	,	,	PUNCT
ejpam-6909	110	19	ℏ2	ℏ2	NOUN
ejpam-6909	110	20	∈	∈	PROPN
ejpam-6909	110	21	h	h	NOUN
ejpam-6909	110	22	,	,	PUNCT
ejpam-6909	110	23	there	there	PRON
ejpam-6909	110	24	exist	exist	VERB
ejpam-6909	110	25	a	a	DET
ejpam-6909	110	26	,	,	PUNCT
ejpam-6909	110	27	b	b	PROPN
ejpam-6909	110	28	∈	∈	PROPN
ejpam-6909	110	29	ℏ1	ℏ1	PROPN
ejpam-6909	110	30	◦	◦	NOUN
ejpam-6909	110	31	ℏ2	ℏ2	NOUN
ejpam-6909	110	32	such	such	ADJ
ejpam-6909	110	33	that	that	SCONJ
ejpam-6909	110	34	α+	α+	PRON
ejpam-6909	110	35	a(ℏ1	a(ℏ1	NUM
ejpam-6909	110	36	)	)	PUNCT
ejpam-6909	110	37	≥	≥	NOUN
ejpam-6909	110	38	min{α+	min{α+	PROPN
ejpam-6909	110	39	a(a	a(a	PROPN
ejpam-6909	110	40	)	)	PUNCT
ejpam-6909	110	41	,	,	PUNCT
ejpam-6909	110	42	α	α	PROPN
ejpam-6909	110	43	+	+	X
ejpam-6909	110	44	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	110	45	)	)	PUNCT
ejpam-6909	110	46	}	}	PUNCT
ejpam-6909	110	47	and	and	CCONJ
ejpam-6909	110	48	β−	β−	PRON
ejpam-6909	110	49	a	a	DET
ejpam-6909	110	50	(	(	PUNCT
ejpam-6909	110	51	ℏ1	ℏ1	PROPN
ejpam-6909	110	52	)	)	PUNCT
ejpam-6909	110	53	≤	≤	NUM
ejpam-6909	111	1	max{β−	max{β−	PROPN
ejpam-6909	111	2	a	a	DET
ejpam-6909	111	3	(	(	PUNCT
ejpam-6909	111	4	b	b	NOUN
ejpam-6909	111	5	)	)	PUNCT
ejpam-6909	111	6	,	,	PUNCT
ejpam-6909	111	7	β	β	PROPN
ejpam-6909	111	8	−	−	PROPN
ejpam-6909	111	9	a	a	DET
ejpam-6909	111	10	(	(	PUNCT
ejpam-6909	111	11	ℏ2	ℏ2	NOUN
ejpam-6909	111	12	)	)	PUNCT
ejpam-6909	111	13	}	}	PUNCT
ejpam-6909	111	14	.	.	PUNCT
ejpam-6909	112	1	definition	definition	NOUN
ejpam-6909	112	2	8	8	NUM
ejpam-6909	112	3	.	.	PUNCT
ejpam-6909	113	1	[	[	X
ejpam-6909	113	2	20	20	NUM
ejpam-6909	113	3	]	]	SYM
ejpam-6909	113	4	a	a	DET
ejpam-6909	113	5	bfs	bfs	NOUN
ejpam-6909	113	6	(	(	PUNCT
ejpam-6909	113	7	α+	α+	NOUN
ejpam-6909	113	8	a	a	X
ejpam-6909	113	9	,	,	PUNCT
ejpam-6909	113	10	β	β	PROPN
ejpam-6909	113	11	−	−	NOUN
ejpam-6909	113	12	a	a	X
ejpam-6909	113	13	)	)	PUNCT
ejpam-6909	113	14	in	in	ADP
ejpam-6909	113	15	h	h	NOUN
ejpam-6909	113	16	is	be	AUX
ejpam-6909	113	17	known	know	VERB
ejpam-6909	113	18	as	as	ADP
ejpam-6909	113	19	a	a	DET
ejpam-6909	113	20	bf	bf	NOUN
ejpam-6909	113	21	-	-	PUNCT
ejpam-6909	113	22	weak	weak	ADJ
ejpam-6909	113	23	hbcki	hbcki	NOUN
ejpam-6909	113	24	of	of	ADP
ejpam-6909	113	25	h	h	NOUN
ejpam-6909	113	26	if	if	SCONJ
ejpam-6909	113	27	it	it	PRON
ejpam-6909	113	28	satisfies	satisfy	VERB
ejpam-6909	113	29	:	:	PUNCT
ejpam-6909	113	30	(	(	PUNCT
ejpam-6909	113	31	i	i	NOUN
ejpam-6909	113	32	)	)	PUNCT
ejpam-6909	114	1	α+	α+	PRON
ejpam-6909	114	2	a(0	a(0	PROPN
ejpam-6909	114	3	)	)	PUNCT
ejpam-6909	114	4	≥	≥	NOUN
ejpam-6909	114	5	α+	α+	X
ejpam-6909	114	6	a(ℏ1	a(ℏ1	NUM
ejpam-6909	114	7	)	)	PUNCT
ejpam-6909	114	8	≥	≥	PROPN
ejpam-6909	114	9	min	min	PROPN
ejpam-6909	114	10	{	{	PUNCT
ejpam-6909	114	11	inf	inf	PROPN
ejpam-6909	114	12	α+	α+	PRON
ejpam-6909	114	13	a(a	a(a	PROPN
ejpam-6909	114	14	)	)	PUNCT
ejpam-6909	114	15	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	114	16	◦	◦	ADJ
ejpam-6909	114	17	ℏ2	ℏ2	NOUN
ejpam-6909	114	18	,	,	PUNCT
ejpam-6909	114	19	α+	α+	PRON
ejpam-6909	114	20	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	114	21	)	)	PUNCT
ejpam-6909	114	22	}	}	PUNCT
ejpam-6909	114	23	,	,	PUNCT
ejpam-6909	114	24	(	(	PUNCT
ejpam-6909	114	25	ii	ii	NOUN
ejpam-6909	114	26	)	)	PUNCT
ejpam-6909	114	27	β−	β−	NOUN
ejpam-6909	115	1	a	a	DET
ejpam-6909	115	2	(	(	PUNCT
ejpam-6909	115	3	0	0	NUM
ejpam-6909	115	4	)	)	PUNCT
ejpam-6909	115	5	≤	≤	NOUN
ejpam-6909	115	6	β−	β−	PUNCT
ejpam-6909	116	1	a	a	DET
ejpam-6909	116	2	(	(	PUNCT
ejpam-6909	116	3	ℏ1	ℏ1	PROPN
ejpam-6909	116	4	)	)	PUNCT
ejpam-6909	116	5	≤	≤	NUM
ejpam-6909	116	6	max	max	PROPN
ejpam-6909	116	7	{	{	PUNCT
ejpam-6909	116	8	supβ−	supβ−	PROPN
ejpam-6909	116	9	a	a	DET
ejpam-6909	116	10	(	(	PUNCT
ejpam-6909	116	11	a	a	PRON
ejpam-6909	116	12	)	)	PUNCT
ejpam-6909	116	13	a∈ℏ1	a∈ℏ1	NOUN
ejpam-6909	116	14	◦	◦	NOUN
ejpam-6909	116	15	ℏ2	ℏ2	NOUN
ejpam-6909	116	16	,	,	PUNCT
ejpam-6909	116	17	β−	β−	PRON
ejpam-6909	116	18	a	a	DET
ejpam-6909	116	19	(	(	PUNCT
ejpam-6909	116	20	ℏ2	ℏ2	NOUN
ejpam-6909	116	21	)	)	PUNCT
ejpam-6909	116	22	}	}	PUNCT
ejpam-6909	116	23	,	,	PUNCT
ejpam-6909	116	24	for	for	ADP
ejpam-6909	116	25	all	all	DET
ejpam-6909	116	26	ℏ1	ℏ1	NOUN
ejpam-6909	116	27	,	,	PUNCT
ejpam-6909	116	28	ℏ2	ℏ2	NOUN
ejpam-6909	116	29	∈	∈	PROPN
ejpam-6909	116	30	h.	h.	NOUN
ejpam-6909	116	31	definition	definition	NOUN
ejpam-6909	116	32	9	9	NUM
ejpam-6909	116	33	.	.	PUNCT
ejpam-6909	117	1	[	[	X
ejpam-6909	117	2	20	20	NUM
ejpam-6909	117	3	]	]	SYM
ejpam-6909	117	4	a	a	DET
ejpam-6909	117	5	bfs	bfs	NOUN
ejpam-6909	117	6	(	(	PUNCT
ejpam-6909	117	7	α+	α+	NOUN
ejpam-6909	117	8	a	a	X
ejpam-6909	117	9	,	,	PUNCT
ejpam-6909	117	10	β	β	PROPN
ejpam-6909	117	11	−	−	NOUN
ejpam-6909	117	12	a	a	X
ejpam-6909	117	13	)	)	PUNCT
ejpam-6909	117	14	in	in	ADP
ejpam-6909	117	15	h	h	NOUN
ejpam-6909	117	16	is	be	AUX
ejpam-6909	117	17	known	know	VERB
ejpam-6909	117	18	as	as	ADP
ejpam-6909	117	19	a	a	DET
ejpam-6909	117	20	bfhbck	bfhbck	NOUN
ejpam-6909	117	21	-	-	PUNCT
ejpam-6909	117	22	subalgebra	subalgebra	NOUN
ejpam-6909	117	23	of	of	ADP
ejpam-6909	117	24	h	h	NOUN
ejpam-6909	117	25	if	if	SCONJ
ejpam-6909	117	26	it	it	PRON
ejpam-6909	117	27	satisfies	satisfy	VERB
ejpam-6909	117	28	:	:	PUNCT
ejpam-6909	117	29	(	(	PUNCT
ejpam-6909	117	30	i	i	NOUN
ejpam-6909	117	31	)	)	PUNCT
ejpam-6909	117	32	inf	inf	PROPN
ejpam-6909	117	33	α+	α+	PRON
ejpam-6909	118	1	a(a	a(a	PROPN
ejpam-6909	118	2	)	)	PUNCT
ejpam-6909	118	3	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	118	4	◦	◦	ADJ
ejpam-6909	118	5	ℏ2	ℏ2	NOUN
ejpam-6909	118	6	≥	≥	VERB
ejpam-6909	118	7	min{α+	min{α+	VERB
ejpam-6909	118	8	a(ℏ1	a(ℏ1	NUM
ejpam-6909	118	9	)	)	PUNCT
ejpam-6909	118	10	,	,	PUNCT
ejpam-6909	118	11	α	α	PROPN
ejpam-6909	118	12	+	+	X
ejpam-6909	118	13	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	118	14	)	)	PUNCT
ejpam-6909	118	15	}	}	PUNCT
ejpam-6909	118	16	(	(	PUNCT
ejpam-6909	118	17	ii	ii	NOUN
ejpam-6909	118	18	)	)	PUNCT
ejpam-6909	118	19	supβ−	supβ−	NOUN
ejpam-6909	118	20	a	a	DET
ejpam-6909	118	21	(	(	PUNCT
ejpam-6909	118	22	a	a	PRON
ejpam-6909	118	23	)	)	PUNCT
ejpam-6909	118	24	a∈ℏ1	a∈ℏ1	NOUN
ejpam-6909	118	25	◦	◦	NOUN
ejpam-6909	118	26	ℏ2	ℏ2	NOUN
ejpam-6909	118	27	≤	≤	X
ejpam-6909	118	28	max{β−	max{β−	PROPN
ejpam-6909	118	29	a	a	DET
ejpam-6909	118	30	(	(	PUNCT
ejpam-6909	118	31	ℏ1	ℏ1	PROPN
ejpam-6909	118	32	)	)	PUNCT
ejpam-6909	118	33	,	,	PUNCT
ejpam-6909	118	34	β	β	PROPN
ejpam-6909	118	35	−	−	PROPN
ejpam-6909	118	36	a	a	DET
ejpam-6909	118	37	(	(	PUNCT
ejpam-6909	118	38	ℏ2	ℏ2	NOUN
ejpam-6909	118	39	)	)	PUNCT
ejpam-6909	118	40	}	}	PUNCT
ejpam-6909	118	41	,	,	PUNCT
ejpam-6909	118	42	for	for	ADP
ejpam-6909	118	43	all	all	DET
ejpam-6909	118	44	ℏ1	ℏ1	NOUN
ejpam-6909	118	45	,	,	PUNCT
ejpam-6909	118	46	ℏ2	ℏ2	NOUN
ejpam-6909	118	47	∈	∈	PROPN
ejpam-6909	118	48	h.	h.	PROPN
ejpam-6909	118	49	definition	definition	NOUN
ejpam-6909	118	50	10	10	NUM
ejpam-6909	118	51	.	.	PUNCT
ejpam-6909	119	1	a	a	DET
ejpam-6909	119	2	bfs	bfs	NOUN
ejpam-6909	119	3	(	(	PUNCT
ejpam-6909	119	4	α+	α+	NOUN
ejpam-6909	119	5	a	a	X
ejpam-6909	119	6	,	,	PUNCT
ejpam-6909	119	7	β	β	PROPN
ejpam-6909	119	8	−	−	NOUN
ejpam-6909	119	9	a	a	X
ejpam-6909	119	10	)	)	PUNCT
ejpam-6909	119	11	in	in	ADP
ejpam-6909	119	12	h	h	NOUN
ejpam-6909	119	13	is	be	AUX
ejpam-6909	119	14	said	say	VERB
ejpam-6909	119	15	to	to	PART
ejpam-6909	119	16	satisfy	satisfy	VERB
ejpam-6909	119	17	sup	sup	NOUN
ejpam-6909	119	18	-	-	PUNCT
ejpam-6909	119	19	inf	inf	NOUN
ejpam-6909	119	20	property	property	NOUN
ejpam-6909	119	21	if	if	SCONJ
ejpam-6909	119	22	for	for	ADP
ejpam-6909	119	23	any	any	DET
ejpam-6909	119	24	subset	subset	NOUN
ejpam-6909	119	25	h1	h1	NOUN
ejpam-6909	119	26	of	of	ADP
ejpam-6909	119	27	h	h	NOUN
ejpam-6909	119	28	,	,	PUNCT
ejpam-6909	119	29	there	there	PRON
ejpam-6909	119	30	exist	exist	VERB
ejpam-6909	119	31	ℏ10	ℏ10	VERB
ejpam-6909	119	32	,	,	PUNCT
ejpam-6909	119	33	ℏ20	ℏ20	PROPN
ejpam-6909	119	34	∈	∈	PROPN
ejpam-6909	119	35	h1	h1	VERB
ejpam-6909	119	36	such	such	ADJ
ejpam-6909	119	37	that	that	SCONJ
ejpam-6909	119	38	α+	α+	NOUN
ejpam-6909	119	39	a(ℏ10	a(ℏ10	ADJ
ejpam-6909	119	40	)	)	PUNCT
ejpam-6909	120	1	=	=	SYM
ejpam-6909	120	2	supα+	supα+	NOUN
ejpam-6909	120	3	a(ℏ1	a(ℏ1	X
ejpam-6909	120	4	)	)	PUNCT
ejpam-6909	121	1	ℏ1∈h1	ℏ1∈h1	PROPN
ejpam-6909	121	2	and	and	CCONJ
ejpam-6909	121	3	β−	β−	PRON
ejpam-6909	121	4	a	a	DET
ejpam-6909	121	5	(	(	PUNCT
ejpam-6909	121	6	ℏ20	ℏ20	PROPN
ejpam-6909	121	7	)	)	PUNCT
ejpam-6909	121	8	=	=	SYM
ejpam-6909	121	9	inf	inf	NOUN
ejpam-6909	121	10	β−	β−	PROPN
ejpam-6909	121	11	a	a	DET
ejpam-6909	121	12	(	(	PUNCT
ejpam-6909	121	13	ℏ2	ℏ2	NOUN
ejpam-6909	121	14	)	)	PUNCT
ejpam-6909	121	15	ℏ2∈h1	ℏ2∈h1	PROPN
ejpam-6909	121	16	.	.	PUNCT
ejpam-6909	122	1	d.	d.	PROPN
ejpam-6909	122	2	ramesh	ramesh	PROPN
ejpam-6909	122	3	et	et	PROPN
ejpam-6909	122	4	al	al	PROPN
ejpam-6909	122	5	.	.	PUNCT
ejpam-6909	122	6	/	/	SYM
ejpam-6909	122	7	eur	eur	PROPN
ejpam-6909	122	8	.	.	PUNCT
ejpam-6909	123	1	j.	j.	PROPN
ejpam-6909	123	2	pure	pure	PROPN
ejpam-6909	123	3	appl	appl	PROPN
ejpam-6909	123	4	.	.	PROPN
ejpam-6909	123	5	math	math	PROPN
ejpam-6909	123	6	,	,	PUNCT
ejpam-6909	123	7	18	18	NUM
ejpam-6909	123	8	(	(	PUNCT
ejpam-6909	123	9	4	4	NUM
ejpam-6909	123	10	)	)	PUNCT
ejpam-6909	123	11	(	(	PUNCT
ejpam-6909	123	12	2025	2025	NUM
ejpam-6909	123	13	)	)	PUNCT
ejpam-6909	123	14	,	,	PUNCT
ejpam-6909	123	15	6909	6909	NUM
ejpam-6909	123	16	7	7	NUM
ejpam-6909	123	17	of	of	ADP
ejpam-6909	123	18	16	16	NUM
ejpam-6909	123	19	3	3	NUM
ejpam-6909	123	20	.	.	PUNCT
ejpam-6909	123	21	bipolar	bipolar	ADJ
ejpam-6909	123	22	fuzzy	fuzzy	ADJ
ejpam-6909	123	23	commutative	commutative	ADJ
ejpam-6909	123	24	hyper	hyper	ADJ
ejpam-6909	123	25	bck	bck	NOUN
ejpam-6909	123	26	-	-	PUNCT
ejpam-6909	123	27	ideals	ideal	NOUN
ejpam-6909	123	28	in	in	ADP
ejpam-6909	123	29	this	this	DET
ejpam-6909	123	30	section	section	NOUN
ejpam-6909	123	31	,	,	PUNCT
ejpam-6909	123	32	we	we	PRON
ejpam-6909	123	33	introduce	introduce	VERB
ejpam-6909	123	34	and	and	CCONJ
ejpam-6909	123	35	investigate	investigate	VERB
ejpam-6909	123	36	a	a	DET
ejpam-6909	123	37	new	new	ADJ
ejpam-6909	123	38	class	class	NOUN
ejpam-6909	123	39	of	of	ADP
ejpam-6909	123	40	ideals	ideal	NOUN
ejpam-6909	123	41	in	in	ADP
ejpam-6909	123	42	the	the	DET
ejpam-6909	123	43	framework	framework	NOUN
ejpam-6909	123	44	of	of	ADP
ejpam-6909	123	45	hyper	hyper	ADJ
ejpam-6909	123	46	bck	bck	NOUN
ejpam-6909	123	47	-	-	PUNCT
ejpam-6909	123	48	algebras	algebra	NOUN
ejpam-6909	123	49	,	,	PUNCT
ejpam-6909	123	50	namely	namely	ADV
ejpam-6909	123	51	bipolar	bipolar	ADJ
ejpam-6909	123	52	fuzzy	fuzzy	ADJ
ejpam-6909	123	53	commutative	commutative	ADJ
ejpam-6909	123	54	hyper	hyper	ADJ
ejpam-6909	123	55	bck	bck	NOUN
ejpam-6909	123	56	-	-	PUNCT
ejpam-6909	123	57	ideals	ideal	NOUN
ejpam-6909	123	58	.	.	PUNCT
ejpam-6909	124	1	this	this	DET
ejpam-6909	124	2	concept	concept	NOUN
ejpam-6909	124	3	is	be	AUX
ejpam-6909	124	4	formulated	formulate	VERB
ejpam-6909	124	5	by	by	ADP
ejpam-6909	124	6	combining	combine	VERB
ejpam-6909	124	7	the	the	DET
ejpam-6909	124	8	structural	structural	ADJ
ejpam-6909	124	9	flexibility	flexibility	NOUN
ejpam-6909	124	10	of	of	ADP
ejpam-6909	124	11	hyperoperations	hyperoperation	NOUN
ejpam-6909	124	12	with	with	ADP
ejpam-6909	124	13	the	the	DET
ejpam-6909	124	14	dualvalued	dualvalue	VERB
ejpam-6909	124	15	semantics	semantic	NOUN
ejpam-6909	124	16	of	of	ADP
ejpam-6909	124	17	bipolar	bipolar	ADJ
ejpam-6909	124	18	fuzzy	fuzzy	ADJ
ejpam-6909	124	19	sets	set	NOUN
ejpam-6909	124	20	.	.	PUNCT
ejpam-6909	125	1	our	our	PRON
ejpam-6909	125	2	motivation	motivation	NOUN
ejpam-6909	125	3	stems	stem	VERB
ejpam-6909	125	4	from	from	ADP
ejpam-6909	125	5	the	the	DET
ejpam-6909	125	6	growing	grow	VERB
ejpam-6909	125	7	demand	demand	NOUN
ejpam-6909	125	8	to	to	PART
ejpam-6909	125	9	model	model	VERB
ejpam-6909	125	10	algebraic	algebraic	ADJ
ejpam-6909	125	11	uncertainty	uncertainty	NOUN
ejpam-6909	125	12	in	in	ADP
ejpam-6909	125	13	a	a	DET
ejpam-6909	125	14	way	way	NOUN
ejpam-6909	125	15	that	that	PRON
ejpam-6909	125	16	simultaneously	simultaneously	ADV
ejpam-6909	125	17	captures	capture	VERB
ejpam-6909	125	18	both	both	CCONJ
ejpam-6909	125	19	supportive	supportive	ADJ
ejpam-6909	125	20	and	and	CCONJ
ejpam-6909	125	21	opposing	opposing	ADJ
ejpam-6909	125	22	evidence	evidence	NOUN
ejpam-6909	125	23	—	—	PUNCT
ejpam-6909	125	24	a	a	DET
ejpam-6909	125	25	task	task	NOUN
ejpam-6909	125	26	well	well	ADV
ejpam-6909	125	27	-	-	PUNCT
ejpam-6909	125	28	suited	suit	VERB
ejpam-6909	125	29	to	to	ADP
ejpam-6909	125	30	the	the	DET
ejpam-6909	125	31	bipolar	bipolar	ADJ
ejpam-6909	125	32	fuzzy	fuzzy	ADJ
ejpam-6909	125	33	paradigm	paradigm	NOUN
ejpam-6909	125	34	.	.	PUNCT
ejpam-6909	126	1	building	build	VERB
ejpam-6909	126	2	upon	upon	SCONJ
ejpam-6909	126	3	the	the	DET
ejpam-6909	126	4	foundational	foundational	ADJ
ejpam-6909	126	5	definitions	definition	NOUN
ejpam-6909	126	6	outlined	outline	VERB
ejpam-6909	126	7	in	in	ADP
ejpam-6909	126	8	section	section	NOUN
ejpam-6909	126	9	2	2	NUM
ejpam-6909	126	10	,	,	PUNCT
ejpam-6909	126	11	we	we	PRON
ejpam-6909	126	12	first	first	ADV
ejpam-6909	126	13	define	define	VERB
ejpam-6909	126	14	bipolar	bipolar	ADJ
ejpam-6909	126	15	fuzzy	fuzzy	ADJ
ejpam-6909	126	16	commutative	commutative	ADJ
ejpam-6909	126	17	hyper	hyper	ADJ
ejpam-6909	126	18	bck	bck	NOUN
ejpam-6909	126	19	-	-	PUNCT
ejpam-6909	126	20	ideals	ideal	NOUN
ejpam-6909	126	21	and	and	CCONJ
ejpam-6909	126	22	examine	examine	VERB
ejpam-6909	126	23	their	their	PRON
ejpam-6909	126	24	basic	basic	ADJ
ejpam-6909	126	25	properties	property	NOUN
ejpam-6909	126	26	.	.	PUNCT
ejpam-6909	127	1	special	special	ADJ
ejpam-6909	127	2	attention	attention	NOUN
ejpam-6909	127	3	is	be	AUX
ejpam-6909	127	4	given	give	VERB
ejpam-6909	127	5	to	to	ADP
ejpam-6909	127	6	the	the	DET
ejpam-6909	127	7	interplay	interplay	NOUN
ejpam-6909	127	8	between	between	ADP
ejpam-6909	127	9	commutativity	commutativity	NOUN
ejpam-6909	127	10	and	and	CCONJ
ejpam-6909	127	11	bipolarity	bipolarity	NOUN
ejpam-6909	127	12	,	,	PUNCT
ejpam-6909	127	13	which	which	PRON
ejpam-6909	127	14	provides	provide	VERB
ejpam-6909	127	15	a	a	DET
ejpam-6909	127	16	nuanced	nuanced	ADJ
ejpam-6909	127	17	perspective	perspective	NOUN
ejpam-6909	127	18	on	on	ADP
ejpam-6909	127	19	membership	membership	NOUN
ejpam-6909	127	20	under	under	ADP
ejpam-6909	127	21	hyperoperations	hyperoperation	NOUN
ejpam-6909	127	22	.	.	PUNCT
ejpam-6909	128	1	several	several	ADJ
ejpam-6909	128	2	illustrative	illustrative	ADJ
ejpam-6909	128	3	examples	example	NOUN
ejpam-6909	128	4	are	be	AUX
ejpam-6909	128	5	provided	provide	VERB
ejpam-6909	128	6	to	to	PART
ejpam-6909	128	7	clarify	clarify	VERB
ejpam-6909	128	8	the	the	DET
ejpam-6909	128	9	behavior	behavior	NOUN
ejpam-6909	128	10	of	of	ADP
ejpam-6909	128	11	such	such	ADJ
ejpam-6909	128	12	ideals	ideal	NOUN
ejpam-6909	128	13	in	in	ADP
ejpam-6909	128	14	various	various	ADJ
ejpam-6909	128	15	algebraic	algebraic	ADJ
ejpam-6909	128	16	settings	setting	NOUN
ejpam-6909	128	17	.	.	PUNCT
ejpam-6909	129	1	we	we	PRON
ejpam-6909	129	2	also	also	ADV
ejpam-6909	129	3	explore	explore	VERB
ejpam-6909	129	4	necessary	necessary	ADJ
ejpam-6909	129	5	and	and	CCONJ
ejpam-6909	129	6	sufficient	sufficient	ADJ
ejpam-6909	129	7	conditions	condition	NOUN
ejpam-6909	129	8	under	under	ADP
ejpam-6909	129	9	which	which	PRON
ejpam-6909	129	10	these	these	DET
ejpam-6909	129	11	ideals	ideal	NOUN
ejpam-6909	129	12	exhibit	exhibit	VERB
ejpam-6909	129	13	structural	structural	ADJ
ejpam-6909	129	14	regularity	regularity	NOUN
ejpam-6909	129	15	,	,	PUNCT
ejpam-6909	129	16	as	as	ADV
ejpam-6909	129	17	well	well	ADV
ejpam-6909	129	18	as	as	ADP
ejpam-6909	129	19	their	their	PRON
ejpam-6909	129	20	relationships	relationship	NOUN
ejpam-6909	129	21	to	to	ADP
ejpam-6909	129	22	existing	exist	VERB
ejpam-6909	129	23	classes	class	NOUN
ejpam-6909	129	24	of	of	ADP
ejpam-6909	129	25	bipolar	bipolar	ADJ
ejpam-6909	129	26	fuzzy	fuzzy	ADJ
ejpam-6909	129	27	and	and	CCONJ
ejpam-6909	129	28	hyper	hyper	ADJ
ejpam-6909	129	29	bck	bck	NOUN
ejpam-6909	129	30	-	-	PUNCT
ejpam-6909	129	31	ideals	ideal	NOUN
ejpam-6909	129	32	.	.	PUNCT
ejpam-6909	130	1	these	these	DET
ejpam-6909	130	2	results	result	NOUN
ejpam-6909	130	3	not	not	PART
ejpam-6909	130	4	only	only	ADV
ejpam-6909	130	5	generalize	generalize	VERB
ejpam-6909	130	6	previous	previous	ADJ
ejpam-6909	130	7	work	work	NOUN
ejpam-6909	130	8	on	on	ADP
ejpam-6909	130	9	commutative	commutative	ADJ
ejpam-6909	130	10	ideals	ideal	NOUN
ejpam-6909	130	11	and	and	CCONJ
ejpam-6909	130	12	bipolar	bipolar	ADJ
ejpam-6909	130	13	fuzzy	fuzzy	ADJ
ejpam-6909	130	14	structures	structure	NOUN
ejpam-6909	130	15	,	,	PUNCT
ejpam-6909	130	16	but	but	CCONJ
ejpam-6909	130	17	also	also	ADV
ejpam-6909	130	18	pave	pave	VERB
ejpam-6909	130	19	the	the	DET
ejpam-6909	130	20	way	way	NOUN
ejpam-6909	130	21	for	for	ADP
ejpam-6909	130	22	further	further	ADJ
ejpam-6909	130	23	developments	development	NOUN
ejpam-6909	130	24	in	in	ADP
ejpam-6909	130	25	multivalued	multivalued	ADJ
ejpam-6909	130	26	algebraic	algebraic	ADJ
ejpam-6909	130	27	logic	logic	NOUN
ejpam-6909	130	28	.	.	PUNCT
ejpam-6909	131	1	definition	definition	NOUN
ejpam-6909	131	2	11	11	NUM
ejpam-6909	131	3	.	.	PUNCT
ejpam-6909	132	1	let	let	VERB
ejpam-6909	132	2	(	(	PUNCT
ejpam-6909	132	3	α+	α+	X
ejpam-6909	132	4	a	a	X
ejpam-6909	132	5	,	,	PUNCT
ejpam-6909	132	6	β	β	PROPN
ejpam-6909	132	7	−	−	NOUN
ejpam-6909	132	8	a	a	DET
ejpam-6909	132	9	)	)	PUNCT
ejpam-6909	132	10	be	be	AUX
ejpam-6909	132	11	a	a	DET
ejpam-6909	132	12	bfs	bfs	NOUN
ejpam-6909	132	13	in	in	ADP
ejpam-6909	132	14	h	h	NOUN
ejpam-6909	132	15	with	with	ADP
ejpam-6909	132	16	α+	α+	DET
ejpam-6909	132	17	a(0	a(0	PROPN
ejpam-6909	132	18	)	)	PUNCT
ejpam-6909	132	19	≥	≥	NOUN
ejpam-6909	132	20	α+	α+	X
ejpam-6909	132	21	a(ℏ1	a(ℏ1	NUM
ejpam-6909	132	22	)	)	PUNCT
ejpam-6909	132	23	and	and	CCONJ
ejpam-6909	132	24	β−	β−	PRON
ejpam-6909	132	25	a	a	DET
ejpam-6909	132	26	(	(	PUNCT
ejpam-6909	132	27	0	0	NUM
ejpam-6909	132	28	)	)	PUNCT
ejpam-6909	132	29	≤	≤	NOUN
ejpam-6909	132	30	β−	β−	PUNCT
ejpam-6909	133	1	a	a	DET
ejpam-6909	133	2	(	(	PUNCT
ejpam-6909	133	3	ℏ1	ℏ1	PROPN
ejpam-6909	133	4	)	)	PUNCT
ejpam-6909	133	5	,	,	PUNCT
ejpam-6909	133	6	for	for	ADP
ejpam-6909	133	7	all	all	DET
ejpam-6909	133	8	ℏ1	ℏ1	PROPN
ejpam-6909	133	9	∈	∈	PROPN
ejpam-6909	133	10	h.	h.	NOUN
ejpam-6909	133	11	then	then	ADV
ejpam-6909	133	12	(	(	PUNCT
ejpam-6909	133	13	α+	α+	X
ejpam-6909	133	14	a	a	X
ejpam-6909	133	15	,	,	PUNCT
ejpam-6909	133	16	β	β	PROPN
ejpam-6909	133	17	−	−	NOUN
ejpam-6909	133	18	a	a	PRON
ejpam-6909	133	19	)	)	PUNCT
ejpam-6909	133	20	is	be	AUX
ejpam-6909	133	21	said	say	VERB
ejpam-6909	133	22	to	to	PART
ejpam-6909	133	23	be	be	AUX
ejpam-6909	133	24	a	a	DET
ejpam-6909	133	25	bipolar	bipolar	ADJ
ejpam-6909	133	26	fuzzy	fuzzy	ADJ
ejpam-6909	133	27	commutative	commutative	ADJ
ejpam-6909	133	28	hyper	hyper	ADJ
ejpam-6909	133	29	bck	bck	NOUN
ejpam-6909	133	30	-	-	PUNCT
ejpam-6909	133	31	ideal	ideal	NOUN
ejpam-6909	133	32	(	(	PUNCT
ejpam-6909	133	33	bf	bf	NOUN
ejpam-6909	133	34	-	-	PUNCT
ejpam-6909	133	35	chbcki	chbcki	NOUN
ejpam-6909	133	36	)	)	PUNCT
ejpam-6909	133	37	of	of	ADP
ejpam-6909	133	38	(	(	PUNCT
ejpam-6909	133	39	i	i	NOUN
ejpam-6909	133	40	)	)	PUNCT
ejpam-6909	133	41	type-1	type-1	PROPN
ejpam-6909	133	42	if	if	SCONJ
ejpam-6909	133	43	for	for	ADP
ejpam-6909	133	44	all	all	DET
ejpam-6909	133	45	t	t	NOUN
ejpam-6909	133	46	∈	∈	PROPN
ejpam-6909	133	47	ℏ1	ℏ1	PROPN
ejpam-6909	133	48	◦	◦	NOUN
ejpam-6909	133	49	(ℏ2	(ℏ2	NOUN
ejpam-6909	133	50	◦	◦	ADJ
ejpam-6909	133	51	(ℏ2	(ℏ2	NOUN
ejpam-6909	133	52	◦	◦	NOUN
ejpam-6909	133	53	ℏ1	ℏ1	ADJ
ejpam-6909	133	54	)	)	PUNCT
ejpam-6909	133	55	)	)	PUNCT
ejpam-6909	133	56	,	,	PUNCT
ejpam-6909	133	57	α+	α+	PRON
ejpam-6909	133	58	a(t	a(t	NOUN
ejpam-6909	133	59	)	)	PUNCT
ejpam-6909	133	60	≥	≥	NOUN
ejpam-6909	133	61	min	min	PROPN
ejpam-6909	133	62	{	{	PUNCT
ejpam-6909	133	63	inf	inf	PROPN
ejpam-6909	133	64	α+	α+	PRON
ejpam-6909	133	65	a(a	a(a	PROPN
ejpam-6909	133	66	)	)	PUNCT
ejpam-6909	134	1	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	134	2	◦	◦	NOUN
ejpam-6909	134	3	ℏ2)	ℏ2)	NOUN
ejpam-6909	134	4	◦	◦	NOUN
ejpam-6909	134	5	ℏ3	ℏ3	PROPN
ejpam-6909	134	6	,	,	PUNCT
ejpam-6909	134	7	α+	α+	DET
ejpam-6909	134	8	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	134	9	)	)	PUNCT
ejpam-6909	134	10	}	}	PUNCT
ejpam-6909	134	11	,	,	PUNCT
ejpam-6909	134	12	β−	β−	PRON
ejpam-6909	134	13	a	a	DET
ejpam-6909	134	14	(	(	PUNCT
ejpam-6909	134	15	t	t	NOUN
ejpam-6909	134	16	)	)	PUNCT
ejpam-6909	134	17	≤	≤	NUM
ejpam-6909	134	18	max	max	PROPN
ejpam-6909	134	19	{	{	PUNCT
ejpam-6909	134	20	supβ−	supβ−	PROPN
ejpam-6909	134	21	a	a	DET
ejpam-6909	134	22	(	(	PUNCT
ejpam-6909	134	23	b	b	NOUN
ejpam-6909	134	24	)	)	PUNCT
ejpam-6909	134	25	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	134	26	◦	◦	NOUN
ejpam-6909	134	27	ℏ2)	ℏ2)	NOUN
ejpam-6909	134	28	◦	◦	NOUN
ejpam-6909	134	29	ℏ3	ℏ3	NOUN
ejpam-6909	134	30	,	,	PUNCT
ejpam-6909	134	31	β−	β−	PRON
ejpam-6909	134	32	a	a	DET
ejpam-6909	134	33	(	(	PUNCT
ejpam-6909	134	34	ℏ3	ℏ3	PROPN
ejpam-6909	134	35	)	)	PUNCT
ejpam-6909	134	36	}	}	PUNCT
ejpam-6909	134	37	(	(	PUNCT
ejpam-6909	134	38	ii	ii	NOUN
ejpam-6909	134	39	)	)	PUNCT
ejpam-6909	135	1	type-2	type-2	CCONJ
ejpam-6909	135	2	if	if	SCONJ
ejpam-6909	135	3	for	for	ADP
ejpam-6909	135	4	all	all	DET
ejpam-6909	135	5	t	t	NOUN
ejpam-6909	135	6	∈	∈	PROPN
ejpam-6909	135	7	ℏ1	ℏ1	PROPN
ejpam-6909	135	8	◦	◦	NOUN
ejpam-6909	135	9	(ℏ2	(ℏ2	NOUN
ejpam-6909	135	10	◦	◦	ADJ
ejpam-6909	135	11	(ℏ2	(ℏ2	NOUN
ejpam-6909	135	12	◦	◦	NOUN
ejpam-6909	135	13	ℏ1	ℏ1	ADJ
ejpam-6909	135	14	)	)	PUNCT
ejpam-6909	135	15	)	)	PUNCT
ejpam-6909	135	16	,	,	PUNCT
ejpam-6909	135	17	α+	α+	PRON
ejpam-6909	135	18	a(t	a(t	NOUN
ejpam-6909	135	19	)	)	PUNCT
ejpam-6909	135	20	≥	≥	NOUN
ejpam-6909	135	21	min	min	PROPN
ejpam-6909	135	22	{	{	PUNCT
ejpam-6909	135	23	inf	inf	PROPN
ejpam-6909	135	24	α+	α+	PRON
ejpam-6909	135	25	a(a	a(a	PROPN
ejpam-6909	135	26	)	)	PUNCT
ejpam-6909	135	27	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	135	28	◦	◦	NOUN
ejpam-6909	135	29	ℏ2)	ℏ2)	NOUN
ejpam-6909	135	30	◦	◦	NOUN
ejpam-6909	135	31	ℏ3	ℏ3	PROPN
ejpam-6909	135	32	,	,	PUNCT
ejpam-6909	135	33	α+	α+	DET
ejpam-6909	135	34	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	135	35	)	)	PUNCT
ejpam-6909	135	36	}	}	PUNCT
ejpam-6909	135	37	,	,	PUNCT
ejpam-6909	135	38	β−	β−	PRON
ejpam-6909	135	39	a	a	DET
ejpam-6909	135	40	(	(	PUNCT
ejpam-6909	135	41	t	t	NOUN
ejpam-6909	135	42	)	)	PUNCT
ejpam-6909	135	43	≤	≤	NUM
ejpam-6909	135	44	max	max	PROPN
ejpam-6909	135	45	{	{	PUNCT
ejpam-6909	135	46	supβ−	supβ−	PROPN
ejpam-6909	135	47	a	a	DET
ejpam-6909	135	48	(	(	PUNCT
ejpam-6909	135	49	b	b	NOUN
ejpam-6909	135	50	)	)	PUNCT
ejpam-6909	135	51	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	135	52	◦	◦	NOUN
ejpam-6909	135	53	ℏ2)	ℏ2)	NOUN
ejpam-6909	135	54	◦	◦	NOUN
ejpam-6909	135	55	ℏ3	ℏ3	NOUN
ejpam-6909	135	56	,	,	PUNCT
ejpam-6909	135	57	β−	β−	PRON
ejpam-6909	135	58	a	a	DET
ejpam-6909	135	59	(	(	PUNCT
ejpam-6909	135	60	ℏ3	ℏ3	PROPN
ejpam-6909	135	61	)	)	PUNCT
ejpam-6909	135	62	}	}	PUNCT
ejpam-6909	135	63	(	(	PUNCT
ejpam-6909	135	64	iii	iii	X
ejpam-6909	135	65	)	)	PUNCT
ejpam-6909	135	66	type-3	type-3	NUM
ejpam-6909	135	67	if	if	SCONJ
ejpam-6909	135	68	for	for	ADP
ejpam-6909	135	69	all	all	DET
ejpam-6909	135	70	t	t	NOUN
ejpam-6909	135	71	∈	∈	PROPN
ejpam-6909	135	72	ℏ1	ℏ1	PROPN
ejpam-6909	135	73	◦	◦	NOUN
ejpam-6909	135	74	(ℏ2	(ℏ2	NOUN
ejpam-6909	135	75	◦	◦	ADJ
ejpam-6909	135	76	(ℏ2	(ℏ2	NOUN
ejpam-6909	135	77	◦	◦	NOUN
ejpam-6909	135	78	ℏ1	ℏ1	ADJ
ejpam-6909	135	79	)	)	PUNCT
ejpam-6909	135	80	)	)	PUNCT
ejpam-6909	135	81	,	,	PUNCT
ejpam-6909	136	1	α+	α+	DET
ejpam-6909	136	2	a(t	a(t	NOUN
ejpam-6909	136	3	)	)	PUNCT
ejpam-6909	136	4	≥	≥	NOUN
ejpam-6909	136	5	min	min	PROPN
ejpam-6909	136	6	{	{	PUNCT
ejpam-6909	136	7	supα+	supα+	X
ejpam-6909	136	8	a(a	a(a	PROPN
ejpam-6909	136	9	)	)	PUNCT
ejpam-6909	136	10	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	136	11	◦	◦	NOUN
ejpam-6909	136	12	ℏ2)	ℏ2)	NOUN
ejpam-6909	136	13	◦	◦	NOUN
ejpam-6909	136	14	ℏ3	ℏ3	PROPN
ejpam-6909	136	15	,	,	PUNCT
ejpam-6909	136	16	α+	α+	DET
ejpam-6909	136	17	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	136	18	)	)	PUNCT
ejpam-6909	136	19	}	}	PUNCT
ejpam-6909	136	20	,	,	PUNCT
ejpam-6909	136	21	β−	β−	PRON
ejpam-6909	136	22	a	a	DET
ejpam-6909	136	23	(	(	PUNCT
ejpam-6909	136	24	t	t	NOUN
ejpam-6909	136	25	)	)	PUNCT
ejpam-6909	136	26	≤	≤	NUM
ejpam-6909	136	27	max	max	PROPN
ejpam-6909	136	28	{	{	PUNCT
ejpam-6909	136	29	inf	inf	NOUN
ejpam-6909	136	30	β−	β−	PROPN
ejpam-6909	136	31	a	a	DET
ejpam-6909	136	32	(	(	PUNCT
ejpam-6909	136	33	b	b	NOUN
ejpam-6909	136	34	)	)	PUNCT
ejpam-6909	136	35	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	136	36	◦	◦	NOUN
ejpam-6909	136	37	ℏ2)	ℏ2)	NOUN
ejpam-6909	136	38	◦	◦	NOUN
ejpam-6909	136	39	ℏ3	ℏ3	NOUN
ejpam-6909	136	40	,	,	PUNCT
ejpam-6909	136	41	β−	β−	PRON
ejpam-6909	136	42	a	a	DET
ejpam-6909	136	43	(	(	PUNCT
ejpam-6909	136	44	ℏ3	ℏ3	PROPN
ejpam-6909	136	45	)	)	PUNCT
ejpam-6909	136	46	}	}	PUNCT
ejpam-6909	136	47	d.	d.	PROPN
ejpam-6909	136	48	ramesh	ramesh	PROPN
ejpam-6909	136	49	et	et	PROPN
ejpam-6909	136	50	al	al	PROPN
ejpam-6909	136	51	.	.	PUNCT
ejpam-6909	136	52	/	/	SYM
ejpam-6909	136	53	eur	eur	PROPN
ejpam-6909	136	54	.	.	PUNCT
ejpam-6909	137	1	j.	j.	PROPN
ejpam-6909	137	2	pure	pure	PROPN
ejpam-6909	137	3	appl	appl	PROPN
ejpam-6909	137	4	.	.	PROPN
ejpam-6909	137	5	math	math	PROPN
ejpam-6909	137	6	,	,	PUNCT
ejpam-6909	137	7	18	18	NUM
ejpam-6909	137	8	(	(	PUNCT
ejpam-6909	137	9	4	4	NUM
ejpam-6909	137	10	)	)	PUNCT
ejpam-6909	137	11	(	(	PUNCT
ejpam-6909	137	12	2025	2025	NUM
ejpam-6909	137	13	)	)	PUNCT
ejpam-6909	137	14	,	,	PUNCT
ejpam-6909	137	15	6909	6909	NUM
ejpam-6909	137	16	8	8	NUM
ejpam-6909	137	17	of	of	ADP
ejpam-6909	137	18	16	16	NUM
ejpam-6909	137	19	(	(	PUNCT
ejpam-6909	137	20	iv	iv	X
ejpam-6909	137	21	)	)	PUNCT
ejpam-6909	137	22	type-4	type-4	NOUN
ejpam-6909	137	23	if	if	SCONJ
ejpam-6909	137	24	for	for	ADP
ejpam-6909	137	25	all	all	DET
ejpam-6909	137	26	t	t	NOUN
ejpam-6909	137	27	∈	∈	PROPN
ejpam-6909	137	28	ℏ1	ℏ1	PROPN
ejpam-6909	137	29	◦	◦	NOUN
ejpam-6909	137	30	(ℏ2	(ℏ2	NOUN
ejpam-6909	137	31	◦	◦	ADJ
ejpam-6909	137	32	(ℏ2	(ℏ2	NOUN
ejpam-6909	137	33	◦	◦	NOUN
ejpam-6909	137	34	ℏ1	ℏ1	ADJ
ejpam-6909	137	35	)	)	PUNCT
ejpam-6909	137	36	)	)	PUNCT
ejpam-6909	137	37	,	,	PUNCT
ejpam-6909	137	38	α+	α+	PRON
ejpam-6909	137	39	a(t	a(t	NOUN
ejpam-6909	137	40	)	)	PUNCT
ejpam-6909	137	41	≥	≥	NOUN
ejpam-6909	137	42	min	min	PROPN
ejpam-6909	137	43	{	{	PUNCT
ejpam-6909	137	44	supα+	supα+	X
ejpam-6909	137	45	a(a	a(a	PROPN
ejpam-6909	137	46	)	)	PUNCT
ejpam-6909	137	47	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	137	48	◦	◦	NOUN
ejpam-6909	137	49	ℏ2)	ℏ2)	NOUN
ejpam-6909	137	50	◦	◦	NOUN
ejpam-6909	137	51	ℏ3	ℏ3	PROPN
ejpam-6909	137	52	,	,	PUNCT
ejpam-6909	137	53	α+	α+	DET
ejpam-6909	137	54	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	137	55	)	)	PUNCT
ejpam-6909	137	56	}	}	PUNCT
ejpam-6909	137	57	,	,	PUNCT
ejpam-6909	137	58	β−	β−	PRON
ejpam-6909	137	59	a	a	DET
ejpam-6909	137	60	(	(	PUNCT
ejpam-6909	137	61	t	t	NOUN
ejpam-6909	137	62	)	)	PUNCT
ejpam-6909	137	63	≤	≤	NUM
ejpam-6909	137	64	max	max	PROPN
ejpam-6909	137	65	{	{	PUNCT
ejpam-6909	137	66	inf	inf	NOUN
ejpam-6909	137	67	β−	β−	PROPN
ejpam-6909	137	68	a	a	DET
ejpam-6909	137	69	(	(	PUNCT
ejpam-6909	137	70	b	b	NOUN
ejpam-6909	137	71	)	)	PUNCT
ejpam-6909	137	72	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	137	73	◦	◦	NOUN
ejpam-6909	137	74	ℏ2)	ℏ2)	NOUN
ejpam-6909	137	75	◦	◦	NOUN
ejpam-6909	137	76	ℏ3	ℏ3	NOUN
ejpam-6909	137	77	,	,	PUNCT
ejpam-6909	137	78	β−	β−	PRON
ejpam-6909	137	79	a	a	DET
ejpam-6909	137	80	(	(	PUNCT
ejpam-6909	137	81	ℏ3	ℏ3	PROPN
ejpam-6909	137	82	)	)	PUNCT
ejpam-6909	137	83	}	}	PUNCT
ejpam-6909	137	84	,	,	PUNCT
ejpam-6909	137	85	for	for	ADP
ejpam-6909	137	86	all	all	DET
ejpam-6909	137	87	ℏ1	ℏ1	ADJ
ejpam-6909	137	88	,	,	PUNCT
ejpam-6909	137	89	ℏ2	ℏ2	NOUN
ejpam-6909	137	90	,	,	PUNCT
ejpam-6909	137	91	ℏ3	ℏ3	PROPN
ejpam-6909	137	92	∈	∈	PROPN
ejpam-6909	137	93	h.	h.	PROPN
ejpam-6909	137	94	example	example	NOUN
ejpam-6909	138	1	1	1	X
ejpam-6909	138	2	.	.	PUNCT
ejpam-6909	138	3	let	let	VERB
ejpam-6909	138	4	h	h	NOUN
ejpam-6909	138	5	=	=	PUNCT
ejpam-6909	138	6	{	{	PUNCT
ejpam-6909	138	7	0	0	NUM
ejpam-6909	138	8	,	,	PUNCT
ejpam-6909	138	9	ℏ1	ℏ1	ADJ
ejpam-6909	138	10	,	,	PUNCT
ejpam-6909	138	11	ℏ2	ℏ2	NOUN
ejpam-6909	138	12	}	}	PUNCT
ejpam-6909	138	13	.	.	PUNCT
ejpam-6909	139	1	consider	consider	VERB
ejpam-6909	139	2	the	the	DET
ejpam-6909	139	3	following	follow	VERB
ejpam-6909	139	4	cayley	cayley	ADJ
ejpam-6909	139	5	table	table	NOUN
ejpam-6909	139	6	:	:	PUNCT
ejpam-6909	139	7	◦	◦	NOUN
ejpam-6909	139	8	0	0	NUM
ejpam-6909	139	9	ℏ1	ℏ1	ADJ
ejpam-6909	139	10	ℏ2	ℏ2	NOUN
ejpam-6909	139	11	0	0	PUNCT
ejpam-6909	139	12	{	{	PUNCT
ejpam-6909	139	13	0	0	NUM
ejpam-6909	139	14	}	}	PUNCT
ejpam-6909	139	15	{	{	PUNCT
ejpam-6909	139	16	0	0	NUM
ejpam-6909	139	17	}	}	PUNCT
ejpam-6909	139	18	{	{	PUNCT
ejpam-6909	139	19	0	0	NUM
ejpam-6909	139	20	}	}	PUNCT
ejpam-6909	139	21	ℏ1	ℏ1	ADJ
ejpam-6909	139	22	{	{	PUNCT
ejpam-6909	139	23	ℏ1	ℏ1	PROPN
ejpam-6909	139	24	}	}	PUNCT
ejpam-6909	139	25	{	{	PUNCT
ejpam-6909	139	26	0	0	NUM
ejpam-6909	139	27	,	,	PUNCT
ejpam-6909	139	28	ℏ1	ℏ1	ADJ
ejpam-6909	139	29	}	}	PUNCT
ejpam-6909	139	30	{	{	PUNCT
ejpam-6909	139	31	0	0	NUM
ejpam-6909	139	32	,	,	PUNCT
ejpam-6909	139	33	ℏ1	ℏ1	ADJ
ejpam-6909	139	34	}	}	PUNCT
ejpam-6909	139	35	ℏ2	ℏ2	NOUN
ejpam-6909	139	36	{	{	PUNCT
ejpam-6909	139	37	ℏ2	ℏ2	NOUN
ejpam-6909	139	38	}	}	PUNCT
ejpam-6909	139	39	{	{	PUNCT
ejpam-6909	139	40	ℏ1	ℏ1	ADJ
ejpam-6909	139	41	,	,	PUNCT
ejpam-6909	139	42	ℏ2	ℏ2	NOUN
ejpam-6909	139	43	}	}	PUNCT
ejpam-6909	139	44	{	{	PUNCT
ejpam-6909	139	45	0	0	NUM
ejpam-6909	139	46	,	,	PUNCT
ejpam-6909	139	47	ℏ1	ℏ1	ADJ
ejpam-6909	139	48	,	,	PUNCT
ejpam-6909	139	49	ℏ2	ℏ2	NOUN
ejpam-6909	139	50	}	}	PUNCT
ejpam-6909	139	51	then	then	ADV
ejpam-6909	139	52	(	(	PUNCT
ejpam-6909	139	53	h	h	NOUN
ejpam-6909	139	54	,	,	PUNCT
ejpam-6909	139	55	◦	◦	NOUN
ejpam-6909	139	56	)	)	PUNCT
ejpam-6909	139	57	is	be	AUX
ejpam-6909	139	58	an	an	DET
ejpam-6909	139	59	hbcka	hbcka	NOUN
ejpam-6909	139	60	.	.	PUNCT
ejpam-6909	140	1	we	we	PRON
ejpam-6909	140	2	define	define	VERB
ejpam-6909	140	3	a	a	DET
ejpam-6909	140	4	bfs	bfs	NOUN
ejpam-6909	140	5	(	(	PUNCT
ejpam-6909	140	6	α+	α+	NOUN
ejpam-6909	140	7	a	a	X
ejpam-6909	140	8	,	,	PUNCT
ejpam-6909	140	9	β	β	PROPN
ejpam-6909	140	10	−	−	NOUN
ejpam-6909	140	11	a	a	X
ejpam-6909	140	12	)	)	PUNCT
ejpam-6909	140	13	in	in	ADP
ejpam-6909	140	14	h	h	NOUN
ejpam-6909	140	15	as	as	SCONJ
ejpam-6909	140	16	follows	follow	VERB
ejpam-6909	140	17	:	:	PUNCT
ejpam-6909	140	18	α+	α+	PUNCT
ejpam-6909	140	19	a(0	a(0	PROPN
ejpam-6909	140	20	)	)	PUNCT
ejpam-6909	140	21	=	=	PROPN
ejpam-6909	140	22	α+	α+	X
ejpam-6909	140	23	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	140	24	)	)	PUNCT
ejpam-6909	140	25	=	=	SYM
ejpam-6909	140	26	1	1	NUM
ejpam-6909	140	27	,	,	PUNCT
ejpam-6909	140	28	α+	α+	X
ejpam-6909	140	29	a(ℏ1	a(ℏ1	NOUN
ejpam-6909	140	30	)	)	PUNCT
ejpam-6909	140	31	=	=	SYM
ejpam-6909	140	32	0.5	0.5	NUM
ejpam-6909	140	33	,	,	PUNCT
ejpam-6909	140	34	β−	β−	PRON
ejpam-6909	140	35	a	a	DET
ejpam-6909	140	36	(	(	PUNCT
ejpam-6909	140	37	0	0	NUM
ejpam-6909	140	38	)	)	PUNCT
ejpam-6909	140	39	=	=	SYM
ejpam-6909	141	1	β−	β−	PUNCT
ejpam-6909	141	2	a	a	DET
ejpam-6909	141	3	(	(	PUNCT
ejpam-6909	141	4	ℏ2	ℏ2	NOUN
ejpam-6909	141	5	)	)	PUNCT
ejpam-6909	141	6	=	=	SYM
ejpam-6909	142	1	−0.6	−0.6	PROPN
ejpam-6909	142	2	,	,	PUNCT
ejpam-6909	142	3	β−	β−	PRON
ejpam-6909	142	4	a	a	PRON
ejpam-6909	142	5	(	(	PUNCT
ejpam-6909	142	6	ℏ1	ℏ1	PROPN
ejpam-6909	142	7	)	)	PUNCT
ejpam-6909	142	8	=	=	SYM
ejpam-6909	142	9	−0.2	−0.2	PROPN
ejpam-6909	142	10	.	.	PUNCT
ejpam-6909	143	1	then	then	ADV
ejpam-6909	143	2	(	(	PUNCT
ejpam-6909	143	3	α+	α+	X
ejpam-6909	143	4	a	a	X
ejpam-6909	143	5	,	,	PUNCT
ejpam-6909	143	6	β	β	PROPN
ejpam-6909	143	7	−	−	NOUN
ejpam-6909	143	8	a	a	PRON
ejpam-6909	143	9	)	)	PUNCT
ejpam-6909	143	10	is	be	AUX
ejpam-6909	143	11	a	a	DET
ejpam-6909	143	12	bf	bf	NOUN
ejpam-6909	143	13	-	-	PUNCT
ejpam-6909	143	14	chbcki	chbcki	NOUN
ejpam-6909	143	15	of	of	ADP
ejpam-6909	143	16	type-1	type-1	PROPN
ejpam-6909	143	17	and	and	CCONJ
ejpam-6909	143	18	,	,	PUNCT
ejpam-6909	143	19	consequently	consequently	ADV
ejpam-6909	143	20	,	,	PUNCT
ejpam-6909	143	21	of	of	ADP
ejpam-6909	143	22	type-2	type-2	PROPN
ejpam-6909	143	23	.	.	PUNCT
ejpam-6909	143	24	example	example	NOUN
ejpam-6909	144	1	2	2	NUM
ejpam-6909	144	2	.	.	PUNCT
ejpam-6909	144	3	let	let	VERB
ejpam-6909	144	4	h	h	NOUN
ejpam-6909	144	5	=	=	PUNCT
ejpam-6909	144	6	{	{	PUNCT
ejpam-6909	144	7	0	0	NUM
ejpam-6909	144	8	,	,	PUNCT
ejpam-6909	144	9	ℏ1	ℏ1	ADJ
ejpam-6909	144	10	,	,	PUNCT
ejpam-6909	144	11	ℏ2	ℏ2	NOUN
ejpam-6909	144	12	}	}	PUNCT
ejpam-6909	144	13	.	.	PUNCT
ejpam-6909	145	1	consider	consider	VERB
ejpam-6909	145	2	the	the	DET
ejpam-6909	145	3	following	follow	VERB
ejpam-6909	145	4	cayley	cayley	ADJ
ejpam-6909	145	5	table	table	NOUN
ejpam-6909	145	6	:	:	PUNCT
ejpam-6909	145	7	◦	◦	NOUN
ejpam-6909	145	8	0	0	NUM
ejpam-6909	145	9	ℏ1	ℏ1	ADJ
ejpam-6909	145	10	ℏ2	ℏ2	NOUN
ejpam-6909	145	11	0	0	PUNCT
ejpam-6909	145	12	{	{	PUNCT
ejpam-6909	145	13	0	0	NUM
ejpam-6909	145	14	}	}	PUNCT
ejpam-6909	145	15	{	{	PUNCT
ejpam-6909	145	16	0	0	NUM
ejpam-6909	145	17	}	}	PUNCT
ejpam-6909	145	18	{	{	PUNCT
ejpam-6909	145	19	0	0	NUM
ejpam-6909	145	20	}	}	PUNCT
ejpam-6909	145	21	ℏ1	ℏ1	ADJ
ejpam-6909	145	22	{	{	PUNCT
ejpam-6909	145	23	ℏ1	ℏ1	PROPN
ejpam-6909	145	24	}	}	PUNCT
ejpam-6909	145	25	{	{	PUNCT
ejpam-6909	145	26	0	0	NUM
ejpam-6909	145	27	}	}	PUNCT
ejpam-6909	145	28	{	{	PUNCT
ejpam-6909	145	29	ℏ1	ℏ1	ADJ
ejpam-6909	145	30	}	}	PUNCT
ejpam-6909	145	31	ℏ2	ℏ2	NOUN
ejpam-6909	145	32	{	{	PUNCT
ejpam-6909	145	33	ℏ2	ℏ2	NOUN
ejpam-6909	145	34	}	}	PUNCT
ejpam-6909	145	35	{	{	PUNCT
ejpam-6909	145	36	ℏ2	ℏ2	NOUN
ejpam-6909	145	37	}	}	PUNCT
ejpam-6909	145	38	{	{	PUNCT
ejpam-6909	145	39	0	0	NUM
ejpam-6909	145	40	,	,	PUNCT
ejpam-6909	145	41	ℏ2	ℏ2	NOUN
ejpam-6909	145	42	}	}	PUNCT
ejpam-6909	145	43	then	then	ADV
ejpam-6909	145	44	(	(	PUNCT
ejpam-6909	145	45	h	h	NOUN
ejpam-6909	145	46	,	,	PUNCT
ejpam-6909	145	47	◦	◦	NOUN
ejpam-6909	145	48	)	)	PUNCT
ejpam-6909	145	49	is	be	AUX
ejpam-6909	145	50	an	an	DET
ejpam-6909	145	51	hbcka	hbcka	NOUN
ejpam-6909	145	52	.	.	PUNCT
ejpam-6909	146	1	we	we	PRON
ejpam-6909	146	2	define	define	VERB
ejpam-6909	146	3	a	a	DET
ejpam-6909	146	4	bfs	bfs	NOUN
ejpam-6909	146	5	(	(	PUNCT
ejpam-6909	146	6	α+	α+	NOUN
ejpam-6909	146	7	a	a	X
ejpam-6909	146	8	,	,	PUNCT
ejpam-6909	146	9	β	β	PROPN
ejpam-6909	146	10	−	−	NOUN
ejpam-6909	146	11	a	a	X
ejpam-6909	146	12	)	)	PUNCT
ejpam-6909	146	13	in	in	ADP
ejpam-6909	146	14	h	h	NOUN
ejpam-6909	146	15	as	as	SCONJ
ejpam-6909	146	16	follows	follow	VERB
ejpam-6909	146	17	:	:	PUNCT
ejpam-6909	146	18	α+	α+	PUNCT
ejpam-6909	146	19	a(0	a(0	PROPN
ejpam-6909	146	20	)	)	PUNCT
ejpam-6909	146	21	=	=	NUM
ejpam-6909	146	22	0.9	0.9	NUM
ejpam-6909	146	23	,	,	PUNCT
ejpam-6909	146	24	α+	α+	X
ejpam-6909	146	25	a(ℏ1	a(ℏ1	NOUN
ejpam-6909	146	26	)	)	PUNCT
ejpam-6909	146	27	=	=	SYM
ejpam-6909	146	28	0.6	0.6	NUM
ejpam-6909	146	29	,	,	PUNCT
ejpam-6909	146	30	α+	α+	PRON
ejpam-6909	146	31	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	146	32	)	)	PUNCT
ejpam-6909	146	33	=	=	SYM
ejpam-6909	146	34	0.3	0.3	NUM
ejpam-6909	146	35	,	,	PUNCT
ejpam-6909	146	36	β−	β−	PROPN
ejpam-6909	146	37	a	a	DET
ejpam-6909	146	38	(	(	PUNCT
ejpam-6909	146	39	0	0	NUM
ejpam-6909	146	40	)	)	PUNCT
ejpam-6909	146	41	=	=	SYM
ejpam-6909	147	1	−0.29	−0.29	NOUN
ejpam-6909	147	2	,	,	PUNCT
ejpam-6909	147	3	β−	β−	PRON
ejpam-6909	147	4	a	a	DET
ejpam-6909	147	5	(	(	PUNCT
ejpam-6909	147	6	ℏ1	ℏ1	PROPN
ejpam-6909	147	7	)	)	PUNCT
ejpam-6909	147	8	=	=	SYM
ejpam-6909	148	1	−0.23	−0.23	PROPN
ejpam-6909	148	2	,	,	PUNCT
ejpam-6909	148	3	β−	β−	PRON
ejpam-6909	148	4	a	a	DET
ejpam-6909	148	5	(	(	PUNCT
ejpam-6909	148	6	ℏ2	ℏ2	NOUN
ejpam-6909	148	7	)	)	PUNCT
ejpam-6909	148	8	=	=	PUNCT
ejpam-6909	149	1	−0.13	−0.13	X
ejpam-6909	149	2	.	.	PUNCT
ejpam-6909	150	1	then	then	ADV
ejpam-6909	150	2	(	(	PUNCT
ejpam-6909	150	3	α+	α+	X
ejpam-6909	150	4	a	a	X
ejpam-6909	150	5	,	,	PUNCT
ejpam-6909	150	6	β	β	PROPN
ejpam-6909	150	7	−	−	NOUN
ejpam-6909	150	8	a	a	PRON
ejpam-6909	150	9	)	)	PUNCT
ejpam-6909	150	10	is	be	AUX
ejpam-6909	150	11	a	a	DET
ejpam-6909	150	12	bf	bf	NOUN
ejpam-6909	150	13	-	-	PUNCT
ejpam-6909	150	14	chbcki	chbcki	NOUN
ejpam-6909	150	15	of	of	ADP
ejpam-6909	150	16	type-3	type-3	NUM
ejpam-6909	150	17	and	and	CCONJ
ejpam-6909	150	18	,	,	PUNCT
ejpam-6909	150	19	consequently	consequently	ADV
ejpam-6909	150	20	,	,	PUNCT
ejpam-6909	150	21	of	of	ADP
ejpam-6909	150	22	type-4	type-4	PROPN
ejpam-6909	150	23	.	.	PUNCT
ejpam-6909	150	24	theorem	theorem	NOUN
ejpam-6909	150	25	2	2	NUM
ejpam-6909	150	26	.	.	X
ejpam-6909	151	1	let	let	VERB
ejpam-6909	151	2	(	(	PUNCT
ejpam-6909	151	3	α+	α+	X
ejpam-6909	151	4	a	a	X
ejpam-6909	151	5	,	,	PUNCT
ejpam-6909	151	6	β	β	PROPN
ejpam-6909	151	7	−	−	NOUN
ejpam-6909	151	8	a	a	DET
ejpam-6909	151	9	)	)	PUNCT
ejpam-6909	151	10	be	be	AUX
ejpam-6909	151	11	a	a	DET
ejpam-6909	151	12	bfs	bfs	NOUN
ejpam-6909	151	13	in	in	ADP
ejpam-6909	151	14	h.	h.	PROPN
ejpam-6909	151	15	then	then	ADV
ejpam-6909	151	16	the	the	DET
ejpam-6909	151	17	following	follow	VERB
ejpam-6909	151	18	statements	statement	NOUN
ejpam-6909	151	19	are	be	AUX
ejpam-6909	151	20	valid	valid	ADJ
ejpam-6909	151	21	.	.	PUNCT
ejpam-6909	152	1	(	(	PUNCT
ejpam-6909	152	2	i	i	NOUN
ejpam-6909	152	3	)	)	PUNCT
ejpam-6909	152	4	if	if	SCONJ
ejpam-6909	152	5	(	(	PUNCT
ejpam-6909	152	6	α+	α+	X
ejpam-6909	152	7	a	a	X
ejpam-6909	152	8	,	,	PUNCT
ejpam-6909	152	9	β	β	PROPN
ejpam-6909	152	10	−	−	NOUN
ejpam-6909	152	11	a	a	PRON
ejpam-6909	152	12	)	)	PUNCT
ejpam-6909	152	13	is	be	AUX
ejpam-6909	152	14	a	a	DET
ejpam-6909	152	15	bf	bf	NOUN
ejpam-6909	152	16	-	-	PUNCT
ejpam-6909	152	17	chbcki	chbcki	NOUN
ejpam-6909	152	18	of	of	ADP
ejpam-6909	152	19	type-3	type-3	NUM
ejpam-6909	152	20	,	,	PUNCT
ejpam-6909	152	21	then	then	ADV
ejpam-6909	152	22	it	it	PRON
ejpam-6909	152	23	is	be	AUX
ejpam-6909	152	24	a	a	DET
ejpam-6909	152	25	bf	bf	NOUN
ejpam-6909	152	26	-	-	PUNCT
ejpam-6909	152	27	chbcki	chbcki	NOUN
ejpam-6909	152	28	of	of	ADP
ejpam-6909	152	29	type-1	type-1	PROPN
ejpam-6909	152	30	and	and	CCONJ
ejpam-6909	152	31	type-4	type-4	PROPN
ejpam-6909	152	32	.	.	PUNCT
ejpam-6909	153	1	(	(	PUNCT
ejpam-6909	153	2	ii	ii	NOUN
ejpam-6909	153	3	)	)	PUNCT
ejpam-6909	153	4	if	if	SCONJ
ejpam-6909	153	5	(	(	PUNCT
ejpam-6909	153	6	α+	α+	X
ejpam-6909	153	7	a	a	X
ejpam-6909	153	8	,	,	PUNCT
ejpam-6909	153	9	β	β	PROPN
ejpam-6909	153	10	−	−	NOUN
ejpam-6909	153	11	a	a	PRON
ejpam-6909	153	12	)	)	PUNCT
ejpam-6909	153	13	is	be	AUX
ejpam-6909	153	14	a	a	DET
ejpam-6909	153	15	bf	bf	NOUN
ejpam-6909	153	16	-	-	PUNCT
ejpam-6909	153	17	chbcki	chbcki	NOUN
ejpam-6909	153	18	of	of	ADP
ejpam-6909	153	19	type-4	type-4	NOUN
ejpam-6909	153	20	(	(	PUNCT
ejpam-6909	153	21	or	or	CCONJ
ejpam-6909	153	22	)	)	PUNCT
ejpam-6909	153	23	1	1	NUM
ejpam-6909	153	24	,	,	PUNCT
ejpam-6909	153	25	then	then	ADV
ejpam-6909	153	26	it	it	PRON
ejpam-6909	153	27	is	be	AUX
ejpam-6909	153	28	a	a	DET
ejpam-6909	153	29	bf	bf	NOUN
ejpam-6909	153	30	-	-	PUNCT
ejpam-6909	153	31	chbcki	chbcki	NOUN
ejpam-6909	153	32	of	of	ADP
ejpam-6909	153	33	type-2	type-2	PROPN
ejpam-6909	153	34	.	.	PUNCT
ejpam-6909	154	1	proof	proof	NOUN
ejpam-6909	154	2	.	.	PUNCT
ejpam-6909	155	1	the	the	DET
ejpam-6909	155	2	proof	proof	NOUN
ejpam-6909	155	3	is	be	AUX
ejpam-6909	155	4	straightforward	straightforward	ADJ
ejpam-6909	155	5	.	.	PUNCT
ejpam-6909	156	1	the	the	DET
ejpam-6909	156	2	converse	converse	NOUN
ejpam-6909	156	3	of	of	ADP
ejpam-6909	156	4	theorem	theorem	ADJ
ejpam-6909	156	5	2	2	NUM
ejpam-6909	156	6	is	be	AUX
ejpam-6909	156	7	not	not	PART
ejpam-6909	156	8	necessarily	necessarily	ADV
ejpam-6909	156	9	true	true	ADJ
ejpam-6909	156	10	in	in	ADP
ejpam-6909	156	11	general	general	ADJ
ejpam-6909	156	12	.	.	PUNCT
ejpam-6909	157	1	this	this	DET
ejpam-6909	157	2	fact	fact	NOUN
ejpam-6909	157	3	is	be	AUX
ejpam-6909	157	4	illustrated	illustrate	VERB
ejpam-6909	157	5	in	in	ADP
ejpam-6909	157	6	the	the	DET
ejpam-6909	157	7	example	example	NOUN
ejpam-6909	157	8	3	3	X
ejpam-6909	157	9	.	.	X
ejpam-6909	157	10	d.	d.	PROPN
ejpam-6909	157	11	ramesh	ramesh	PROPN
ejpam-6909	157	12	et	et	PROPN
ejpam-6909	157	13	al	al	PROPN
ejpam-6909	157	14	.	.	PUNCT
ejpam-6909	157	15	/	/	SYM
ejpam-6909	157	16	eur	eur	PROPN
ejpam-6909	157	17	.	.	PUNCT
ejpam-6909	158	1	j.	j.	PROPN
ejpam-6909	158	2	pure	pure	PROPN
ejpam-6909	158	3	appl	appl	PROPN
ejpam-6909	158	4	.	.	PROPN
ejpam-6909	158	5	math	math	PROPN
ejpam-6909	158	6	,	,	PUNCT
ejpam-6909	158	7	18	18	NUM
ejpam-6909	158	8	(	(	PUNCT
ejpam-6909	158	9	4	4	NUM
ejpam-6909	158	10	)	)	PUNCT
ejpam-6909	158	11	(	(	PUNCT
ejpam-6909	158	12	2025	2025	NUM
ejpam-6909	158	13	)	)	PUNCT
ejpam-6909	158	14	,	,	PUNCT
ejpam-6909	158	15	6909	6909	NUM
ejpam-6909	158	16	9	9	NUM
ejpam-6909	158	17	of	of	ADP
ejpam-6909	158	18	16	16	NUM
ejpam-6909	158	19	example	example	NOUN
ejpam-6909	158	20	3	3	NUM
ejpam-6909	158	21	.	.	PUNCT
ejpam-6909	159	1	let	let	VERB
ejpam-6909	159	2	h	h	NOUN
ejpam-6909	159	3	=	=	PUNCT
ejpam-6909	159	4	{	{	PUNCT
ejpam-6909	159	5	0	0	NUM
ejpam-6909	159	6	,	,	PUNCT
ejpam-6909	159	7	ℏ1	ℏ1	ADJ
ejpam-6909	159	8	,	,	PUNCT
ejpam-6909	159	9	ℏ2	ℏ2	NOUN
ejpam-6909	159	10	,	,	PUNCT
ejpam-6909	159	11	ℏ3	ℏ3	PROPN
ejpam-6909	159	12	}	}	PUNCT
ejpam-6909	159	13	.	.	PUNCT
ejpam-6909	160	1	consider	consider	VERB
ejpam-6909	160	2	the	the	DET
ejpam-6909	160	3	following	follow	VERB
ejpam-6909	160	4	cayley	cayley	ADJ
ejpam-6909	160	5	table	table	NOUN
ejpam-6909	160	6	:	:	PUNCT
ejpam-6909	160	7	◦	◦	NOUN
ejpam-6909	160	8	0	0	NUM
ejpam-6909	160	9	ℏ1	ℏ1	ADJ
ejpam-6909	160	10	ℏ2	ℏ2	NOUN
ejpam-6909	160	11	ℏ3	ℏ3	PROPN
ejpam-6909	160	12	0	0	PUNCT
ejpam-6909	160	13	{	{	PUNCT
ejpam-6909	160	14	0	0	NUM
ejpam-6909	160	15	}	}	PUNCT
ejpam-6909	160	16	{	{	PUNCT
ejpam-6909	160	17	0	0	NUM
ejpam-6909	160	18	}	}	PUNCT
ejpam-6909	160	19	{	{	PUNCT
ejpam-6909	160	20	0	0	NUM
ejpam-6909	160	21	}	}	PUNCT
ejpam-6909	160	22	{	{	PUNCT
ejpam-6909	160	23	0	0	NUM
ejpam-6909	160	24	}	}	PUNCT
ejpam-6909	160	25	ℏ1	ℏ1	ADJ
ejpam-6909	160	26	{	{	PUNCT
ejpam-6909	160	27	ℏ1	ℏ1	PROPN
ejpam-6909	160	28	}	}	PUNCT
ejpam-6909	160	29	{	{	PUNCT
ejpam-6909	160	30	0	0	NUM
ejpam-6909	160	31	}	}	PUNCT
ejpam-6909	160	32	{	{	PUNCT
ejpam-6909	160	33	0	0	NUM
ejpam-6909	160	34	}	}	PUNCT
ejpam-6909	160	35	{	{	PUNCT
ejpam-6909	160	36	0	0	NUM
ejpam-6909	160	37	}	}	PUNCT
ejpam-6909	160	38	ℏ2	ℏ2	NOUN
ejpam-6909	160	39	{	{	PUNCT
ejpam-6909	160	40	ℏ2	ℏ2	NOUN
ejpam-6909	160	41	}	}	PUNCT
ejpam-6909	160	42	{	{	PUNCT
ejpam-6909	160	43	ℏ2	ℏ2	NOUN
ejpam-6909	160	44	}	}	PUNCT
ejpam-6909	160	45	{	{	PUNCT
ejpam-6909	160	46	0	0	NUM
ejpam-6909	160	47	}	}	PUNCT
ejpam-6909	160	48	{	{	PUNCT
ejpam-6909	160	49	0	0	NUM
ejpam-6909	160	50	}	}	PUNCT
ejpam-6909	160	51	ℏ3	ℏ3	PROPN
ejpam-6909	160	52	{	{	PUNCT
ejpam-6909	160	53	ℏ3	ℏ3	PROPN
ejpam-6909	160	54	}	}	PUNCT
ejpam-6909	160	55	{	{	PUNCT
ejpam-6909	160	56	ℏ3	ℏ3	PROPN
ejpam-6909	160	57	}	}	PUNCT
ejpam-6909	160	58	{	{	PUNCT
ejpam-6909	160	59	ℏ2	ℏ2	NOUN
ejpam-6909	160	60	,	,	PUNCT
ejpam-6909	160	61	ℏ3	ℏ3	PROPN
ejpam-6909	160	62	}	}	PUNCT
ejpam-6909	160	63	{	{	PUNCT
ejpam-6909	160	64	0	0	NUM
ejpam-6909	160	65	,	,	PUNCT
ejpam-6909	160	66	ℏ2	ℏ2	NOUN
ejpam-6909	160	67	,	,	PUNCT
ejpam-6909	160	68	ℏ3	ℏ3	PROPN
ejpam-6909	160	69	}	}	PUNCT
ejpam-6909	160	70	then	then	ADV
ejpam-6909	160	71	(	(	PUNCT
ejpam-6909	160	72	h	h	NOUN
ejpam-6909	160	73	,	,	PUNCT
ejpam-6909	160	74	◦	◦	NOUN
ejpam-6909	160	75	)	)	PUNCT
ejpam-6909	160	76	is	be	AUX
ejpam-6909	160	77	an	an	DET
ejpam-6909	160	78	hbcka	hbcka	NOUN
ejpam-6909	160	79	.	.	PUNCT
ejpam-6909	161	1	we	we	PRON
ejpam-6909	161	2	define	define	VERB
ejpam-6909	161	3	a	a	DET
ejpam-6909	161	4	bfs	bfs	NOUN
ejpam-6909	161	5	(	(	PUNCT
ejpam-6909	161	6	α+	α+	NOUN
ejpam-6909	161	7	a	a	X
ejpam-6909	161	8	,	,	PUNCT
ejpam-6909	161	9	β	β	PROPN
ejpam-6909	161	10	−	−	NOUN
ejpam-6909	161	11	a	a	X
ejpam-6909	161	12	)	)	PUNCT
ejpam-6909	161	13	in	in	ADP
ejpam-6909	161	14	h	h	NOUN
ejpam-6909	161	15	as	as	SCONJ
ejpam-6909	161	16	follows	follow	VERB
ejpam-6909	161	17	:	:	PUNCT
ejpam-6909	161	18	α+	α+	PUNCT
ejpam-6909	161	19	a(0	a(0	PROPN
ejpam-6909	161	20	)	)	PUNCT
ejpam-6909	161	21	=	=	SYM
ejpam-6909	161	22	α+	α+	PUNCT
ejpam-6909	161	23	a(ℏ1	a(ℏ1	NOUN
ejpam-6909	161	24	)	)	PUNCT
ejpam-6909	161	25	=	=	SYM
ejpam-6909	161	26	1	1	NUM
ejpam-6909	161	27	,	,	PUNCT
ejpam-6909	161	28	α+	α+	PRON
ejpam-6909	161	29	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	161	30	)	)	PUNCT
ejpam-6909	161	31	=	=	SYM
ejpam-6909	161	32	0.5	0.5	NUM
ejpam-6909	161	33	,	,	PUNCT
ejpam-6909	161	34	α+	α+	PRON
ejpam-6909	161	35	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	161	36	)	)	PUNCT
ejpam-6909	161	37	=	=	NOUN
ejpam-6909	161	38	0.45	0.45	NUM
ejpam-6909	161	39	,	,	PUNCT
ejpam-6909	161	40	β−	β−	PRON
ejpam-6909	161	41	a	a	DET
ejpam-6909	161	42	(	(	PUNCT
ejpam-6909	161	43	0	0	NUM
ejpam-6909	161	44	)	)	PUNCT
ejpam-6909	161	45	=	=	SYM
ejpam-6909	162	1	β−	β−	PUNCT
ejpam-6909	162	2	a	a	DET
ejpam-6909	162	3	(	(	PUNCT
ejpam-6909	162	4	ℏ1	ℏ1	PROPN
ejpam-6909	162	5	)	)	PUNCT
ejpam-6909	162	6	=	=	SYM
ejpam-6909	162	7	−0.7	−0.7	PROPN
ejpam-6909	162	8	,	,	PUNCT
ejpam-6909	162	9	β−	β−	PRON
ejpam-6909	162	10	a	a	DET
ejpam-6909	162	11	(	(	PUNCT
ejpam-6909	162	12	ℏ2	ℏ2	NOUN
ejpam-6909	162	13	)	)	PUNCT
ejpam-6909	162	14	=	=	SYM
ejpam-6909	163	1	−0.3	−0.3	PROPN
ejpam-6909	163	2	,	,	PUNCT
ejpam-6909	163	3	β−	β−	PRON
ejpam-6909	163	4	a	a	DET
ejpam-6909	163	5	(	(	PUNCT
ejpam-6909	163	6	ℏ3	ℏ3	PROPN
ejpam-6909	163	7	)	)	PUNCT
ejpam-6909	163	8	=	=	PUNCT
ejpam-6909	164	1	−0.1	−0.1	PROPN
ejpam-6909	164	2	.	.	PUNCT
ejpam-6909	165	1	then	then	ADV
ejpam-6909	165	2	(	(	PUNCT
ejpam-6909	165	3	α+	α+	X
ejpam-6909	165	4	a	a	X
ejpam-6909	165	5	,	,	PUNCT
ejpam-6909	165	6	β	β	PROPN
ejpam-6909	165	7	−	−	NOUN
ejpam-6909	165	8	a	a	PRON
ejpam-6909	165	9	)	)	PUNCT
ejpam-6909	165	10	is	be	AUX
ejpam-6909	165	11	a	a	DET
ejpam-6909	165	12	bf	bf	NOUN
ejpam-6909	165	13	-	-	PUNCT
ejpam-6909	165	14	chbcki	chbcki	NOUN
ejpam-6909	165	15	of	of	ADP
ejpam-6909	165	16	type-1	type-1	PROPN
ejpam-6909	165	17	.	.	PUNCT
ejpam-6909	166	1	however	however	ADV
ejpam-6909	166	2	,	,	PUNCT
ejpam-6909	166	3	it	it	PRON
ejpam-6909	166	4	does	do	AUX
ejpam-6909	166	5	not	not	PART
ejpam-6909	166	6	follow	follow	VERB
ejpam-6909	166	7	the	the	DET
ejpam-6909	166	8	conditions	condition	NOUN
ejpam-6909	166	9	for	for	ADP
ejpam-6909	166	10	type-3	type-3	NUM
ejpam-6909	166	11	,	,	PUNCT
ejpam-6909	166	12	as	as	SCONJ
ejpam-6909	166	13	ℏ3	ℏ3	PROPN
ejpam-6909	166	14	∈	∈	PROPN
ejpam-6909	166	15	ℏ3	ℏ3	PROPN
ejpam-6909	166	16	◦	◦	PROPN
ejpam-6909	166	17	(	(	PUNCT
ejpam-6909	166	18	ℏ2	ℏ2	NOUN
ejpam-6909	166	19	◦	◦	VERB
ejpam-6909	166	20	(	(	PUNCT
ejpam-6909	166	21	ℏ2	ℏ2	NOUN
ejpam-6909	166	22	◦	◦	VERB
ejpam-6909	166	23	ℏ3	ℏ3	PROPN
ejpam-6909	166	24	)	)	PUNCT
ejpam-6909	166	25	)	)	PUNCT
ejpam-6909	167	1	=	=	PRON
ejpam-6909	167	2	{	{	PUNCT
ejpam-6909	167	3	ℏ2	ℏ2	NOUN
ejpam-6909	167	4	,	,	PUNCT
ejpam-6909	167	5	ℏ3	ℏ3	PROPN
ejpam-6909	167	6	}	}	PUNCT
ejpam-6909	167	7	,	,	PUNCT
ejpam-6909	167	8	α+	α+	DET
ejpam-6909	167	9	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	167	10	)	)	PUNCT
ejpam-6909	167	11	=	=	PUNCT
ejpam-6909	168	1	0.45	0.45	NUM
ejpam-6909	168	2	<	<	X
ejpam-6909	168	3	0.6	0.6	NUM
ejpam-6909	168	4	=	=	SYM
ejpam-6909	168	5	α+	α+	NOUN
ejpam-6909	168	6	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	168	7	)	)	PUNCT
ejpam-6909	168	8	=	=	SYM
ejpam-6909	168	9	min	min	PROPN
ejpam-6909	168	10	{	{	PUNCT
ejpam-6909	168	11	supα+	supα+	X
ejpam-6909	168	12	a(a	a(a	PROPN
ejpam-6909	168	13	)	)	PUNCT
ejpam-6909	168	14	a∈(ℏ3	a∈(ℏ3	PROPN
ejpam-6909	168	15	◦	◦	NOUN
ejpam-6909	168	16	ℏ2)	ℏ2)	NOUN
ejpam-6909	168	17	◦	◦	NOUN
ejpam-6909	168	18	0	0	NUM
ejpam-6909	168	19	,	,	PUNCT
ejpam-6909	168	20	α+	α+	X
ejpam-6909	168	21	a(0	a(0	PROPN
ejpam-6909	168	22	)	)	PUNCT
ejpam-6909	168	23	}	}	PUNCT
ejpam-6909	168	24	,	,	PUNCT
ejpam-6909	168	25	β−	β−	PRON
ejpam-6909	168	26	a	a	DET
ejpam-6909	168	27	(	(	PUNCT
ejpam-6909	168	28	ℏ3	ℏ3	PROPN
ejpam-6909	168	29	)	)	PUNCT
ejpam-6909	168	30	=	=	PUNCT
ejpam-6909	169	1	−0.1	−0.1	PROPN
ejpam-6909	169	2	>	>	X
ejpam-6909	169	3	−0.3	−0.3	PROPN
ejpam-6909	170	1	=	=	PUNCT
ejpam-6909	170	2	β−	β−	PROPN
ejpam-6909	170	3	a	a	DET
ejpam-6909	170	4	(	(	PUNCT
ejpam-6909	170	5	ℏ2	ℏ2	NOUN
ejpam-6909	170	6	)	)	PUNCT
ejpam-6909	170	7	=	=	SYM
ejpam-6909	170	8	max	max	PROPN
ejpam-6909	170	9	{	{	PUNCT
ejpam-6909	170	10	inf	inf	NOUN
ejpam-6909	170	11	β−	β−	PROPN
ejpam-6909	170	12	a	a	DET
ejpam-6909	170	13	(	(	PUNCT
ejpam-6909	170	14	b	b	NOUN
ejpam-6909	170	15	)	)	PUNCT
ejpam-6909	170	16	b∈(ℏ3	b∈(ℏ3	ADP
ejpam-6909	170	17	◦	◦	NOUN
ejpam-6909	170	18	ℏ2)	ℏ2)	NOUN
ejpam-6909	170	19	◦	◦	NOUN
ejpam-6909	170	20	0	0	NUM
ejpam-6909	170	21	,	,	PUNCT
ejpam-6909	170	22	β−	β−	PRON
ejpam-6909	170	23	a	a	DET
ejpam-6909	170	24	(	(	PUNCT
ejpam-6909	170	25	0	0	NUM
ejpam-6909	170	26	)	)	PUNCT
ejpam-6909	170	27	}	}	PUNCT
ejpam-6909	170	28	.	.	PUNCT
ejpam-6909	171	1	theorem	theorem	NOUN
ejpam-6909	171	2	3	3	NUM
ejpam-6909	171	3	.	.	PUNCT
ejpam-6909	172	1	if	if	SCONJ
ejpam-6909	172	2	(	(	PUNCT
ejpam-6909	172	3	α+	α+	X
ejpam-6909	172	4	a	a	X
ejpam-6909	172	5	,	,	PUNCT
ejpam-6909	172	6	β	β	PROPN
ejpam-6909	172	7	−	−	NOUN
ejpam-6909	172	8	a	a	PRON
ejpam-6909	172	9	)	)	PUNCT
ejpam-6909	172	10	is	be	AUX
ejpam-6909	172	11	a	a	DET
ejpam-6909	172	12	bf	bf	NOUN
ejpam-6909	172	13	-	-	PUNCT
ejpam-6909	172	14	chbcki	chbcki	NOUN
ejpam-6909	172	15	of	of	ADP
ejpam-6909	172	16	type-1	type-1	PROPN
ejpam-6909	172	17	of	of	ADP
ejpam-6909	172	18	h	h	NOUN
ejpam-6909	172	19	,	,	PUNCT
ejpam-6909	172	20	then	then	ADV
ejpam-6909	172	21	it	it	PRON
ejpam-6909	172	22	is	be	AUX
ejpam-6909	172	23	a	a	DET
ejpam-6909	172	24	bf	bf	NOUN
ejpam-6909	172	25	-	-	PUNCT
ejpam-6909	172	26	weak	weak	ADJ
ejpam-6909	172	27	hbcki	hbcki	NOUN
ejpam-6909	172	28	.	.	PUNCT
ejpam-6909	173	1	proof	proof	NOUN
ejpam-6909	173	2	.	.	PUNCT
ejpam-6909	174	1	the	the	DET
ejpam-6909	174	2	proof	proof	NOUN
ejpam-6909	174	3	is	be	AUX
ejpam-6909	174	4	straightforward	straightforward	ADJ
ejpam-6909	174	5	.	.	PUNCT
ejpam-6909	175	1	the	the	DET
ejpam-6909	175	2	converse	converse	NOUN
ejpam-6909	175	3	of	of	ADP
ejpam-6909	175	4	theorem	theorem	NOUN
ejpam-6909	175	5	3	3	NUM
ejpam-6909	175	6	is	be	AUX
ejpam-6909	175	7	not	not	PART
ejpam-6909	175	8	necessarily	necessarily	ADV
ejpam-6909	175	9	true	true	ADJ
ejpam-6909	175	10	in	in	ADP
ejpam-6909	175	11	general	general	ADJ
ejpam-6909	175	12	.	.	PUNCT
ejpam-6909	176	1	this	this	DET
ejpam-6909	176	2	fact	fact	NOUN
ejpam-6909	176	3	is	be	AUX
ejpam-6909	176	4	illustrated	illustrate	VERB
ejpam-6909	176	5	in	in	ADP
ejpam-6909	176	6	example	example	NOUN
ejpam-6909	176	7	4	4	NUM
ejpam-6909	176	8	.	.	NOUN
ejpam-6909	176	9	example	example	NOUN
ejpam-6909	176	10	4	4	NUM
ejpam-6909	176	11	.	.	PUNCT
ejpam-6909	176	12	consider	consider	VERB
ejpam-6909	176	13	the	the	DET
ejpam-6909	176	14	cayley	cayley	ADJ
ejpam-6909	176	15	table	table	NOUN
ejpam-6909	176	16	given	give	VERB
ejpam-6909	176	17	in	in	ADP
ejpam-6909	176	18	example	example	NOUN
ejpam-6909	176	19	3.2	3.2	NUM
ejpam-6909	176	20	.	.	PUNCT
ejpam-6909	177	1	then	then	ADV
ejpam-6909	177	2	(	(	PUNCT
ejpam-6909	177	3	h	h	NOUN
ejpam-6909	177	4	,	,	PUNCT
ejpam-6909	177	5	◦	◦	NOUN
ejpam-6909	177	6	)	)	PUNCT
ejpam-6909	177	7	is	be	AUX
ejpam-6909	177	8	an	an	DET
ejpam-6909	177	9	hbcka	hbcka	NOUN
ejpam-6909	177	10	.	.	PUNCT
ejpam-6909	178	1	we	we	PRON
ejpam-6909	178	2	define	define	VERB
ejpam-6909	178	3	a	a	DET
ejpam-6909	178	4	bfs	bfs	NOUN
ejpam-6909	178	5	(	(	PUNCT
ejpam-6909	178	6	α+	α+	NOUN
ejpam-6909	178	7	a	a	X
ejpam-6909	178	8	,	,	PUNCT
ejpam-6909	178	9	β	β	PROPN
ejpam-6909	178	10	−	−	NOUN
ejpam-6909	178	11	a	a	X
ejpam-6909	178	12	)	)	PUNCT
ejpam-6909	178	13	in	in	ADP
ejpam-6909	178	14	h	h	NOUN
ejpam-6909	178	15	as	as	SCONJ
ejpam-6909	178	16	follows	follow	VERB
ejpam-6909	178	17	:	:	PUNCT
ejpam-6909	178	18	α+	α+	PUNCT
ejpam-6909	178	19	a(0	a(0	PROPN
ejpam-6909	178	20	)	)	PUNCT
ejpam-6909	178	21	=	=	NUM
ejpam-6909	178	22	0.8	0.8	NUM
ejpam-6909	178	23	,	,	PUNCT
ejpam-6909	178	24	α+	α+	NOUN
ejpam-6909	178	25	a(ℏ1	a(ℏ1	NOUN
ejpam-6909	178	26	)	)	PUNCT
ejpam-6909	178	27	=	=	SYM
ejpam-6909	178	28	0.4	0.4	NUM
ejpam-6909	178	29	,	,	PUNCT
ejpam-6909	178	30	α+	α+	PRON
ejpam-6909	178	31	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	178	32	)	)	PUNCT
ejpam-6909	178	33	=	=	SYM
ejpam-6909	178	34	0.25	0.25	NUM
ejpam-6909	178	35	,	,	PUNCT
ejpam-6909	178	36	β−	β−	PROPN
ejpam-6909	178	37	a	a	DET
ejpam-6909	178	38	(	(	PUNCT
ejpam-6909	178	39	0	0	NUM
ejpam-6909	178	40	)	)	PUNCT
ejpam-6909	178	41	=	=	SYM
ejpam-6909	178	42	−0.7	−0.7	PROPN
ejpam-6909	178	43	,	,	PUNCT
ejpam-6909	178	44	β−	β−	PRON
ejpam-6909	178	45	a	a	DET
ejpam-6909	178	46	(	(	PUNCT
ejpam-6909	178	47	ℏ1	ℏ1	PROPN
ejpam-6909	178	48	)	)	PUNCT
ejpam-6909	179	1	=	=	SYM
ejpam-6909	179	2	−0.5	−0.5	PROPN
ejpam-6909	179	3	,	,	PUNCT
ejpam-6909	179	4	β−	β−	PRON
ejpam-6909	179	5	a	a	DET
ejpam-6909	179	6	(	(	PUNCT
ejpam-6909	179	7	ℏ2	ℏ2	NOUN
ejpam-6909	179	8	)	)	PUNCT
ejpam-6909	179	9	=	=	SYM
ejpam-6909	180	1	0	0	X
ejpam-6909	180	2	.	.	PUNCT
ejpam-6909	181	1	then	then	ADV
ejpam-6909	181	2	(	(	PUNCT
ejpam-6909	181	3	α+	α+	X
ejpam-6909	181	4	a	a	X
ejpam-6909	181	5	,	,	PUNCT
ejpam-6909	181	6	β	β	PROPN
ejpam-6909	181	7	−	−	NOUN
ejpam-6909	181	8	a	a	PRON
ejpam-6909	181	9	)	)	PUNCT
ejpam-6909	181	10	is	be	AUX
ejpam-6909	181	11	a	a	DET
ejpam-6909	181	12	bf	bf	NOUN
ejpam-6909	181	13	-	-	PUNCT
ejpam-6909	181	14	weak	weak	ADJ
ejpam-6909	181	15	hbcki	hbcki	NOUN
ejpam-6909	181	16	of	of	ADP
ejpam-6909	181	17	h.	h.	PROPN
ejpam-6909	181	18	however	however	ADV
ejpam-6909	181	19	,	,	PUNCT
ejpam-6909	181	20	it	it	PRON
ejpam-6909	181	21	does	do	AUX
ejpam-6909	181	22	not	not	PART
ejpam-6909	181	23	follow	follow	VERB
ejpam-6909	181	24	the	the	DET
ejpam-6909	181	25	conditions	condition	NOUN
ejpam-6909	181	26	for	for	ADP
ejpam-6909	181	27	bf	bf	NOUN
ejpam-6909	181	28	-	-	PUNCT
ejpam-6909	181	29	chbcki	chbcki	NOUN
ejpam-6909	181	30	of	of	ADP
ejpam-6909	181	31	type-1	type-1	PROPN
ejpam-6909	181	32	,	,	PUNCT
ejpam-6909	181	33	because	because	SCONJ
ejpam-6909	181	34	ℏ2	ℏ2	NOUN
ejpam-6909	181	35	∈	∈	PROPN
ejpam-6909	181	36	ℏ2	ℏ2	NOUN
ejpam-6909	181	37	◦	◦	NOUN
ejpam-6909	181	38	(	(	PUNCT
ejpam-6909	181	39	ℏ2	ℏ2	NOUN
ejpam-6909	181	40	◦	◦	VERB
ejpam-6909	181	41	(	(	PUNCT
ejpam-6909	181	42	ℏ2	ℏ2	NOUN
ejpam-6909	181	43	◦	◦	VERB
ejpam-6909	181	44	ℏ1	ℏ1	ADJ
ejpam-6909	181	45	)	)	PUNCT
ejpam-6909	181	46	)	)	PUNCT
ejpam-6909	182	1	=	=	PUNCT
ejpam-6909	182	2	{	{	PUNCT
ejpam-6909	182	3	0	0	NUM
ejpam-6909	182	4	,	,	PUNCT
ejpam-6909	182	5	ℏ1	ℏ1	ADJ
ejpam-6909	182	6	}	}	PUNCT
ejpam-6909	182	7	,	,	PUNCT
ejpam-6909	182	8	α+	α+	X
ejpam-6909	182	9	a(ℏ1	a(ℏ1	NOUN
ejpam-6909	182	10	)	)	PUNCT
ejpam-6909	183	1	=	=	PUNCT
ejpam-6909	184	1	0.35	0.35	NUM
ejpam-6909	184	2	<	<	X
ejpam-6909	184	3	0.8	0.8	NUM
ejpam-6909	184	4	=	=	SYM
ejpam-6909	184	5	α+	α+	PUNCT
ejpam-6909	184	6	a(0	a(0	PROPN
ejpam-6909	184	7	)	)	PUNCT
ejpam-6909	184	8	=	=	SYM
ejpam-6909	184	9	min	min	PROPN
ejpam-6909	184	10	{	{	PUNCT
ejpam-6909	184	11	inf	inf	PROPN
ejpam-6909	184	12	α+	α+	PRON
ejpam-6909	184	13	a(a	a(a	PROPN
ejpam-6909	184	14	)	)	PUNCT
ejpam-6909	184	15	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	184	16	◦	◦	NOUN
ejpam-6909	184	17	ℏ2)	ℏ2)	NOUN
ejpam-6909	184	18	◦	◦	NOUN
ejpam-6909	184	19	0	0	NUM
ejpam-6909	184	20	,	,	PUNCT
ejpam-6909	184	21	α+	α+	X
ejpam-6909	184	22	a(0	a(0	PROPN
ejpam-6909	184	23	)	)	PUNCT
ejpam-6909	184	24	}	}	PUNCT
ejpam-6909	184	25	,	,	PUNCT
ejpam-6909	184	26	β−	β−	PRON
ejpam-6909	184	27	a	a	DET
ejpam-6909	184	28	(	(	PUNCT
ejpam-6909	184	29	ℏ1	ℏ1	PROPN
ejpam-6909	184	30	)	)	PUNCT
ejpam-6909	184	31	=	=	PUNCT
ejpam-6909	184	32	−0.5	−0.5	PROPN
ejpam-6909	184	33	>	>	PUNCT
ejpam-6909	185	1	−0.7	−0.7	NOUN
ejpam-6909	185	2	=	=	PUNCT
ejpam-6909	186	1	β−	β−	PUNCT
ejpam-6909	186	2	a	a	DET
ejpam-6909	186	3	(	(	PUNCT
ejpam-6909	186	4	0	0	NUM
ejpam-6909	186	5	)	)	PUNCT
ejpam-6909	186	6	=	=	SYM
ejpam-6909	186	7	max	max	PROPN
ejpam-6909	186	8	{	{	PUNCT
ejpam-6909	186	9	supβ−	supβ−	X
ejpam-6909	186	10	a	a	DET
ejpam-6909	186	11	(	(	PUNCT
ejpam-6909	186	12	b	b	NOUN
ejpam-6909	186	13	)	)	PUNCT
ejpam-6909	186	14	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	186	15	◦	◦	NOUN
ejpam-6909	186	16	ℏ2)	ℏ2)	NOUN
ejpam-6909	186	17	◦	◦	NOUN
ejpam-6909	186	18	0	0	NUM
ejpam-6909	186	19	,	,	PUNCT
ejpam-6909	186	20	β−	β−	PRON
ejpam-6909	186	21	a	a	DET
ejpam-6909	186	22	(	(	PUNCT
ejpam-6909	186	23	0	0	NUM
ejpam-6909	186	24	)	)	PUNCT
ejpam-6909	186	25	}	}	PUNCT
ejpam-6909	186	26	.	.	PUNCT
ejpam-6909	187	1	theorem	theorem	ADJ
ejpam-6909	187	2	4	4	NUM
ejpam-6909	187	3	.	.	PUNCT
ejpam-6909	188	1	if	if	SCONJ
ejpam-6909	188	2	(	(	PUNCT
ejpam-6909	188	3	α+	α+	X
ejpam-6909	188	4	a	a	X
ejpam-6909	188	5	,	,	PUNCT
ejpam-6909	188	6	β	β	PROPN
ejpam-6909	188	7	−	−	NOUN
ejpam-6909	188	8	a	a	PRON
ejpam-6909	188	9	)	)	PUNCT
ejpam-6909	188	10	is	be	AUX
ejpam-6909	188	11	a	a	DET
ejpam-6909	188	12	bf	bf	NOUN
ejpam-6909	188	13	-	-	PUNCT
ejpam-6909	188	14	chbcki	chbcki	NOUN
ejpam-6909	188	15	of	of	ADP
ejpam-6909	188	16	type-3	type-3	NUM
ejpam-6909	188	17	of	of	ADP
ejpam-6909	188	18	h	h	NOUN
ejpam-6909	188	19	,	,	PUNCT
ejpam-6909	188	20	then	then	ADV
ejpam-6909	188	21	it	it	PRON
ejpam-6909	188	22	is	be	AUX
ejpam-6909	188	23	a	a	DET
ejpam-6909	188	24	bfshbcki	bfshbcki	NOUN
ejpam-6909	188	25	.	.	PUNCT
ejpam-6909	189	1	d.	d.	PROPN
ejpam-6909	189	2	ramesh	ramesh	PROPN
ejpam-6909	189	3	et	et	PROPN
ejpam-6909	189	4	al	al	PROPN
ejpam-6909	189	5	.	.	PUNCT
ejpam-6909	189	6	/	/	SYM
ejpam-6909	189	7	eur	eur	PROPN
ejpam-6909	189	8	.	.	PUNCT
ejpam-6909	190	1	j.	j.	PROPN
ejpam-6909	190	2	pure	pure	PROPN
ejpam-6909	190	3	appl	appl	PROPN
ejpam-6909	190	4	.	.	PROPN
ejpam-6909	190	5	math	math	PROPN
ejpam-6909	190	6	,	,	PUNCT
ejpam-6909	190	7	18	18	NUM
ejpam-6909	190	8	(	(	PUNCT
ejpam-6909	190	9	4	4	NUM
ejpam-6909	190	10	)	)	PUNCT
ejpam-6909	190	11	(	(	PUNCT
ejpam-6909	190	12	2025	2025	NUM
ejpam-6909	190	13	)	)	PUNCT
ejpam-6909	190	14	,	,	PUNCT
ejpam-6909	190	15	6909	6909	NUM
ejpam-6909	190	16	10	10	NUM
ejpam-6909	190	17	of	of	ADP
ejpam-6909	190	18	16	16	NUM
ejpam-6909	190	19	proof	proof	NOUN
ejpam-6909	190	20	.	.	PUNCT
ejpam-6909	191	1	suppose	suppose	VERB
ejpam-6909	191	2	(	(	PUNCT
ejpam-6909	191	3	α+	α+	X
ejpam-6909	191	4	a	a	X
ejpam-6909	191	5	,	,	PUNCT
ejpam-6909	191	6	β	β	PROPN
ejpam-6909	191	7	−	−	NOUN
ejpam-6909	191	8	a	a	PRON
ejpam-6909	191	9	)	)	PUNCT
ejpam-6909	191	10	is	be	AUX
ejpam-6909	191	11	a	a	DET
ejpam-6909	191	12	bf	bf	NOUN
ejpam-6909	191	13	-	-	PUNCT
ejpam-6909	191	14	chbcki	chbcki	NOUN
ejpam-6909	191	15	of	of	ADP
ejpam-6909	191	16	type-3	type-3	NUM
ejpam-6909	191	17	of	of	ADP
ejpam-6909	191	18	h.	h.	NOUN
ejpam-6909	191	19	setting	set	VERB
ejpam-6909	191	20	ℏ2	ℏ2	NOUN
ejpam-6909	191	21	=	=	SYM
ejpam-6909	191	22	0	0	NUM
ejpam-6909	192	1	in	in	ADP
ejpam-6909	192	2	definition	definition	NOUN
ejpam-6909	192	3	11	11	NUM
ejpam-6909	192	4	(	(	PUNCT
ejpam-6909	192	5	iii	iii	NOUN
ejpam-6909	192	6	)	)	PUNCT
ejpam-6909	192	7	,	,	PUNCT
ejpam-6909	192	8	we	we	PRON
ejpam-6909	192	9	obtain	obtain	VERB
ejpam-6909	192	10	t	t	PROPN
ejpam-6909	192	11	∈	∈	PROPN
ejpam-6909	192	12	ℏ1	ℏ1	PROPN
ejpam-6909	192	13	◦	◦	NOUN
ejpam-6909	192	14	(	(	PUNCT
ejpam-6909	192	15	0	0	NUM
ejpam-6909	192	16	◦	◦	NOUN
ejpam-6909	192	17	(	(	PUNCT
ejpam-6909	192	18	0	0	NUM
ejpam-6909	192	19	◦	◦	NOUN
ejpam-6909	192	20	ℏ1	ℏ1	ADJ
ejpam-6909	192	21	)	)	PUNCT
ejpam-6909	192	22	)	)	PUNCT
ejpam-6909	193	1	=	=	SYM
ejpam-6909	193	2	ℏ1,	ℏ1,	NOUN
ejpam-6909	193	3	α+	α+	X
ejpam-6909	193	4	a(ℏ1	a(ℏ1	NUM
ejpam-6909	193	5	)	)	PUNCT
ejpam-6909	193	6	≥	≥	PROPN
ejpam-6909	193	7	min	min	PROPN
ejpam-6909	193	8	{	{	PUNCT
ejpam-6909	193	9	supα+	supα+	X
ejpam-6909	193	10	a(a	a(a	PROPN
ejpam-6909	193	11	)	)	PUNCT
ejpam-6909	193	12	a∈(ℏ1	a∈(ℏ1	PROPN
ejpam-6909	193	13	◦	◦	NOUN
ejpam-6909	193	14	0)	0)	NOUN
ejpam-6909	193	15	◦	◦	NOUN
ejpam-6909	193	16	ℏ3	ℏ3	PROPN
ejpam-6909	193	17	,	,	PUNCT
ejpam-6909	193	18	α+	α+	DET
ejpam-6909	193	19	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	193	20	)	)	PUNCT
ejpam-6909	193	21	}	}	PUNCT
ejpam-6909	193	22	=	=	SYM
ejpam-6909	193	23	min	min	NOUN
ejpam-6909	193	24	{	{	PUNCT
ejpam-6909	193	25	supα+	supα+	X
ejpam-6909	193	26	a(a	a(a	PROPN
ejpam-6909	193	27	)	)	PUNCT
ejpam-6909	193	28	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	193	29	◦	◦	NOUN
ejpam-6909	193	30	ℏ3	ℏ3	PROPN
ejpam-6909	193	31	,	,	PUNCT
ejpam-6909	193	32	α+	α+	DET
ejpam-6909	193	33	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	193	34	)	)	PUNCT
ejpam-6909	193	35	}	}	PUNCT
ejpam-6909	193	36	β−	β−	VERB
ejpam-6909	194	1	a	a	DET
ejpam-6909	194	2	(	(	PUNCT
ejpam-6909	194	3	ℏ1	ℏ1	PROPN
ejpam-6909	194	4	)	)	PUNCT
ejpam-6909	194	5	≤	≤	NUM
ejpam-6909	194	6	max	max	PROPN
ejpam-6909	194	7	{	{	PUNCT
ejpam-6909	194	8	inf	inf	NOUN
ejpam-6909	194	9	β−	β−	PROPN
ejpam-6909	194	10	a	a	DET
ejpam-6909	194	11	(	(	PUNCT
ejpam-6909	194	12	b	b	NOUN
ejpam-6909	194	13	)	)	PUNCT
ejpam-6909	194	14	b∈(ℏ1	b∈(ℏ1	X
ejpam-6909	194	15	◦	◦	NOUN
ejpam-6909	194	16	0)	0)	NOUN
ejpam-6909	194	17	◦	◦	NOUN
ejpam-6909	194	18	ℏ3	ℏ3	PROPN
ejpam-6909	194	19	,	,	PUNCT
ejpam-6909	194	20	β−	β−	PRON
ejpam-6909	194	21	a	a	DET
ejpam-6909	194	22	(	(	PUNCT
ejpam-6909	194	23	ℏ3	ℏ3	PROPN
ejpam-6909	194	24	)	)	PUNCT
ejpam-6909	194	25	}	}	PUNCT
ejpam-6909	194	26	=	=	SYM
ejpam-6909	194	27	max	max	PROPN
ejpam-6909	194	28	{	{	PUNCT
ejpam-6909	194	29	inf	inf	NOUN
ejpam-6909	194	30	β−	β−	PROPN
ejpam-6909	194	31	a	a	DET
ejpam-6909	194	32	(	(	PUNCT
ejpam-6909	194	33	b	b	NOUN
ejpam-6909	194	34	)	)	PUNCT
ejpam-6909	194	35	b∈ℏ1	b∈ℏ1	NOUN
ejpam-6909	194	36	◦	◦	NOUN
ejpam-6909	194	37	ℏ3	ℏ3	PROPN
ejpam-6909	194	38	,	,	PUNCT
ejpam-6909	194	39	β−	β−	PRON
ejpam-6909	194	40	a	a	DET
ejpam-6909	194	41	(	(	PUNCT
ejpam-6909	194	42	ℏ3	ℏ3	PROPN
ejpam-6909	194	43	)	)	PUNCT
ejpam-6909	194	44	}	}	PUNCT
ejpam-6909	194	45			NOUN
ejpam-6909	194	46	(	(	PUNCT
ejpam-6909	194	47	2	2	NUM
ejpam-6909	194	48	)	)	PUNCT
ejpam-6909	194	49	for	for	ADP
ejpam-6909	194	50	all	all	DET
ejpam-6909	194	51	ℏ1	ℏ1	ADJ
ejpam-6909	194	52	,	,	PUNCT
ejpam-6909	194	53	ℏ3	ℏ3	PROPN
ejpam-6909	194	54	∈	∈	PROPN
ejpam-6909	194	55	h.	h.	PROPN
ejpam-6909	194	56	first	first	ADV
ejpam-6909	194	57	,	,	PUNCT
ejpam-6909	194	58	we	we	PRON
ejpam-6909	194	59	show	show	VERB
ejpam-6909	194	60	that	that	SCONJ
ejpam-6909	194	61	for	for	ADP
ejpam-6909	194	62	ℏ1	ℏ1	ADJ
ejpam-6909	194	63	,	,	PUNCT
ejpam-6909	194	64	ℏ2	ℏ2	NOUN
ejpam-6909	194	65	∈	∈	NOUN
ejpam-6909	194	66	h	h	NOUN
ejpam-6909	194	67	,	,	PUNCT
ejpam-6909	194	68	if	if	SCONJ
ejpam-6909	194	69	ℏ1	ℏ1	ADJ
ejpam-6909	194	70	≪	≪	ADJ
ejpam-6909	194	71	ℏ2	ℏ2	NOUN
ejpam-6909	194	72	,	,	PUNCT
ejpam-6909	194	73	then	then	ADV
ejpam-6909	194	74	α+	α+	PUNCT
ejpam-6909	194	75	a(ℏ1	a(ℏ1	NUM
ejpam-6909	194	76	)	)	PUNCT
ejpam-6909	194	77	≥	≥	NOUN
ejpam-6909	194	78	α+	α+	X
ejpam-6909	194	79	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	194	80	)	)	PUNCT
ejpam-6909	194	81	and	and	CCONJ
ejpam-6909	194	82	β−	β−	PRON
ejpam-6909	194	83	a	a	DET
ejpam-6909	194	84	(	(	PUNCT
ejpam-6909	194	85	ℏ1	ℏ1	PROPN
ejpam-6909	194	86	)	)	PUNCT
ejpam-6909	194	87	≤	≤	NOUN
ejpam-6909	194	88	β−	β−	PUNCT
ejpam-6909	195	1	a	a	DET
ejpam-6909	195	2	(	(	PUNCT
ejpam-6909	195	3	ℏ2	ℏ2	NOUN
ejpam-6909	195	4	)	)	PUNCT
ejpam-6909	195	5	.	.	PUNCT
ejpam-6909	196	1	to	to	PART
ejpam-6909	196	2	show	show	VERB
ejpam-6909	196	3	this	this	PRON
ejpam-6909	196	4	,	,	PUNCT
ejpam-6909	196	5	let	let	VERB
ejpam-6909	196	6	ℏ1	ℏ1	ADJ
ejpam-6909	196	7	,	,	PUNCT
ejpam-6909	196	8	ℏ2	ℏ2	NOUN
ejpam-6909	196	9	∈	∈	NOUN
ejpam-6909	196	10	h	h	NOUN
ejpam-6909	196	11	be	be	AUX
ejpam-6909	196	12	such	such	ADJ
ejpam-6909	196	13	that	that	SCONJ
ejpam-6909	196	14	ℏ1	ℏ1	ADJ
ejpam-6909	196	15	≪	≪	ADJ
ejpam-6909	196	16	ℏ2	ℏ2	NOUN
ejpam-6909	196	17	.	.	PUNCT
ejpam-6909	197	1	then	then	ADV
ejpam-6909	197	2	0	0	NUM
ejpam-6909	197	3	∈	∈	PROPN
ejpam-6909	197	4	ℏ1	ℏ1	PROPN
ejpam-6909	197	5	◦	◦	NOUN
ejpam-6909	197	6	ℏ2	ℏ2	NOUN
ejpam-6909	197	7	,	,	PUNCT
ejpam-6909	197	8	and	and	CCONJ
ejpam-6909	197	9	by	by	ADP
ejpam-6909	197	10	(	(	PUNCT
ejpam-6909	197	11	2	2	NUM
ejpam-6909	197	12	)	)	PUNCT
ejpam-6909	197	13	,	,	PUNCT
ejpam-6909	197	14	we	we	PRON
ejpam-6909	197	15	have	have	VERB
ejpam-6909	197	16	α+	α+	X
ejpam-6909	197	17	a(ℏ1	a(ℏ1	NUM
ejpam-6909	197	18	)	)	PUNCT
ejpam-6909	197	19	≥	≥	PROPN
ejpam-6909	197	20	min	min	PROPN
ejpam-6909	197	21	{	{	PUNCT
ejpam-6909	197	22	supα+	supα+	X
ejpam-6909	197	23	a(a	a(a	PROPN
ejpam-6909	197	24	)	)	PUNCT
ejpam-6909	197	25	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	197	26	◦	◦	NOUN
ejpam-6909	197	27	ℏ2	ℏ2	NOUN
ejpam-6909	197	28	,	,	PUNCT
ejpam-6909	197	29	α+	α+	DET
ejpam-6909	197	30	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	197	31	)	)	PUNCT
ejpam-6909	197	32	}	}	PUNCT
ejpam-6909	198	1	=	=	PUNCT
ejpam-6909	198	2	min{α+	min{α+	PROPN
ejpam-6909	198	3	a(0	a(0	PROPN
ejpam-6909	198	4	)	)	PUNCT
ejpam-6909	198	5	,	,	PUNCT
ejpam-6909	198	6	α	α	PROPN
ejpam-6909	198	7	+	+	X
ejpam-6909	198	8	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	198	9	)	)	PUNCT
ejpam-6909	198	10	}	}	PUNCT
ejpam-6909	199	1	=	=	SYM
ejpam-6909	199	2	α+	α+	X
ejpam-6909	199	3	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	199	4	)	)	PUNCT
ejpam-6909	199	5	β−	β−	PUNCT
ejpam-6909	200	1	a	a	DET
ejpam-6909	200	2	(	(	PUNCT
ejpam-6909	200	3	ℏ1	ℏ1	PROPN
ejpam-6909	200	4	)	)	PUNCT
ejpam-6909	200	5	≤	≤	NUM
ejpam-6909	200	6	max	max	PROPN
ejpam-6909	200	7	{	{	PUNCT
ejpam-6909	200	8	inf	inf	NOUN
ejpam-6909	200	9	β−	β−	PROPN
ejpam-6909	200	10	a	a	DET
ejpam-6909	200	11	(	(	PUNCT
ejpam-6909	200	12	b	b	NOUN
ejpam-6909	200	13	)	)	PUNCT
ejpam-6909	200	14	b∈ℏ1	b∈ℏ1	NOUN
ejpam-6909	200	15	◦	◦	NOUN
ejpam-6909	200	16	ℏ2	ℏ2	NOUN
ejpam-6909	200	17	,	,	PUNCT
ejpam-6909	200	18	β−	β−	PRON
ejpam-6909	200	19	a	a	DET
ejpam-6909	200	20	(	(	PUNCT
ejpam-6909	200	21	ℏ2	ℏ2	NOUN
ejpam-6909	200	22	)	)	PUNCT
ejpam-6909	200	23	}	}	PUNCT
ejpam-6909	201	1	=	=	PUNCT
ejpam-6909	202	1	max{β−	max{β−	PROPN
ejpam-6909	202	2	a	a	PRON
ejpam-6909	202	3	(	(	PUNCT
ejpam-6909	202	4	0	0	NUM
ejpam-6909	202	5	)	)	PUNCT
ejpam-6909	202	6	,	,	PUNCT
ejpam-6909	202	7	β	β	X
ejpam-6909	202	8	−	−	PROPN
ejpam-6909	202	9	a	a	DET
ejpam-6909	202	10	(	(	PUNCT
ejpam-6909	202	11	ℏ2	ℏ2	NOUN
ejpam-6909	202	12	)	)	PUNCT
ejpam-6909	202	13	}	}	PUNCT
ejpam-6909	203	1	=	=	PUNCT
ejpam-6909	203	2	β−	β−	PUNCT
ejpam-6909	203	3	a	a	DET
ejpam-6909	203	4	(	(	PUNCT
ejpam-6909	203	5	ℏ2	ℏ2	NOUN
ejpam-6909	203	6	)	)	PUNCT
ejpam-6909	203	7			NOUN
ejpam-6909	203	8	(	(	PUNCT
ejpam-6909	203	9	3	3	X
ejpam-6909	203	10	)	)	PUNCT
ejpam-6909	203	11	let	let	VERB
ejpam-6909	203	12	ℏ1	ℏ1	PROPN
ejpam-6909	203	13	∈	∈	PROPN
ejpam-6909	203	14	h	h	NOUN
ejpam-6909	203	15	and	and	CCONJ
ejpam-6909	203	16	a	a	DET
ejpam-6909	203	17	∈	∈	PROPN
ejpam-6909	203	18	ℏ1	ℏ1	PROPN
ejpam-6909	203	19	◦	◦	NOUN
ejpam-6909	203	20	ℏ1	ℏ1	PROPN
ejpam-6909	203	21	.	.	PUNCT
ejpam-6909	204	1	since	since	SCONJ
ejpam-6909	204	2	ℏ1	ℏ1	PROPN
ejpam-6909	204	3	◦	◦	NOUN
ejpam-6909	204	4	ℏ1	ℏ1	ADJ
ejpam-6909	204	5	≪	≪	ADJ
ejpam-6909	204	6	ℏ1	ℏ1	NOUN
ejpam-6909	204	7	,	,	PUNCT
ejpam-6909	204	8	we	we	PRON
ejpam-6909	204	9	have	have	VERB
ejpam-6909	204	10	a	a	DET
ejpam-6909	204	11	≪	≪	ADJ
ejpam-6909	204	12	ℏ1	ℏ1	NOUN
ejpam-6909	204	13	,	,	PUNCT
ejpam-6909	204	14	for	for	ADP
ejpam-6909	204	15	all	all	DET
ejpam-6909	204	16	a	a	DET
ejpam-6909	204	17	∈	∈	PROPN
ejpam-6909	204	18	ℏ1	ℏ1	PROPN
ejpam-6909	204	19	◦	◦	NOUN
ejpam-6909	204	20	ℏ1	ℏ1	PROPN
ejpam-6909	204	21	.	.	PUNCT
ejpam-6909	205	1	hence	hence	ADV
ejpam-6909	205	2	,	,	PUNCT
ejpam-6909	205	3	by	by	ADP
ejpam-6909	205	4	(	(	PUNCT
ejpam-6909	205	5	3	3	NUM
ejpam-6909	205	6	)	)	PUNCT
ejpam-6909	205	7	,	,	PUNCT
ejpam-6909	205	8	we	we	PRON
ejpam-6909	205	9	deduce	deduce	VERB
ejpam-6909	205	10	α+	α+	PRON
ejpam-6909	205	11	a(a	a(a	PROPN
ejpam-6909	205	12	)	)	PUNCT
ejpam-6909	205	13	≥	≥	AUX
ejpam-6909	206	1	α+	α+	X
ejpam-6909	206	2	a(ℏ1	a(ℏ1	NUM
ejpam-6909	206	3	)	)	PUNCT
ejpam-6909	207	1	and	and	CCONJ
ejpam-6909	207	2	β−	β−	PRON
ejpam-6909	207	3	a	a	DET
ejpam-6909	207	4	(	(	PUNCT
ejpam-6909	207	5	a	a	NOUN
ejpam-6909	207	6	)	)	PUNCT
ejpam-6909	207	7	≤	≤	NOUN
ejpam-6909	207	8	β−	β−	PUNCT
ejpam-6909	208	1	a	a	DET
ejpam-6909	208	2	(	(	PUNCT
ejpam-6909	208	3	ℏ1	ℏ1	PROPN
ejpam-6909	208	4	)	)	PUNCT
ejpam-6909	208	5	,	,	PUNCT
ejpam-6909	208	6	for	for	ADP
ejpam-6909	208	7	all	all	DET
ejpam-6909	208	8	a	a	DET
ejpam-6909	208	9	∈	∈	PROPN
ejpam-6909	208	10	ℏ1	ℏ1	PROPN
ejpam-6909	208	11	◦	◦	NOUN
ejpam-6909	208	12	ℏ1	ℏ1	ADJ
ejpam-6909	208	13	.	.	PUNCT
ejpam-6909	209	1	hence	hence	ADV
ejpam-6909	209	2	,	,	PUNCT
ejpam-6909	209	3	inf	inf	PROPN
ejpam-6909	209	4	α+	α+	PRON
ejpam-6909	209	5	a(a	a(a	PROPN
ejpam-6909	209	6	)	)	PUNCT
ejpam-6909	209	7	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	209	8	◦	◦	NOUN
ejpam-6909	209	9	ℏ1	ℏ1	PROPN
ejpam-6909	209	10	≥	≥	X
ejpam-6909	209	11	α+	α+	X
ejpam-6909	209	12	a(ℏ1	a(ℏ1	NUM
ejpam-6909	209	13	)	)	PUNCT
ejpam-6909	209	14	and	and	CCONJ
ejpam-6909	209	15	supβ−	supβ−	X
ejpam-6909	209	16	a	a	DET
ejpam-6909	209	17	(	(	PUNCT
ejpam-6909	209	18	c	c	NOUN
ejpam-6909	209	19	)	)	PUNCT
ejpam-6909	209	20	c∈ℏ1	c∈ℏ1	PROPN
ejpam-6909	209	21	◦	◦	NOUN
ejpam-6909	209	22	ℏ1	ℏ1	ADJ
ejpam-6909	209	23	≤	≤	X
ejpam-6909	209	24	β−	β−	PUNCT
ejpam-6909	210	1	a	a	DET
ejpam-6909	210	2	(	(	PUNCT
ejpam-6909	210	3	ℏ1	ℏ1	PROPN
ejpam-6909	210	4	)	)	PUNCT
ejpam-6909	210	5	.	.	PUNCT
ejpam-6909	211	1	(	(	PUNCT
ejpam-6909	211	2	4	4	NUM
ejpam-6909	211	3	)	)	PUNCT
ejpam-6909	211	4	from	from	ADP
ejpam-6909	211	5	the	the	DET
ejpam-6909	211	6	combination	combination	NOUN
ejpam-6909	211	7	of	of	ADP
ejpam-6909	211	8	(	(	PUNCT
ejpam-6909	211	9	2	2	NUM
ejpam-6909	211	10	)	)	PUNCT
ejpam-6909	211	11	and	and	CCONJ
ejpam-6909	211	12	(	(	PUNCT
ejpam-6909	211	13	4	4	NUM
ejpam-6909	211	14	)	)	PUNCT
ejpam-6909	211	15	,	,	PUNCT
ejpam-6909	211	16	we	we	PRON
ejpam-6909	211	17	obtain	obtain	VERB
ejpam-6909	211	18	inf	inf	PROPN
ejpam-6909	211	19	α+	α+	PRON
ejpam-6909	211	20	a(a	a(a	PROPN
ejpam-6909	211	21	)	)	PUNCT
ejpam-6909	211	22	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	211	23	◦	◦	NOUN
ejpam-6909	211	24	ℏ1	ℏ1	PROPN
ejpam-6909	211	25	≥	≥	X
ejpam-6909	211	26	α+	α+	X
ejpam-6909	211	27	a(ℏ1	a(ℏ1	NUM
ejpam-6909	211	28	)	)	PUNCT
ejpam-6909	211	29	≥	≥	PROPN
ejpam-6909	211	30	min	min	NOUN
ejpam-6909	211	31	{	{	PUNCT
ejpam-6909	211	32	supα+	supα+	NOUN
ejpam-6909	211	33	a(b	a(b	NOUN
ejpam-6909	211	34	)	)	PUNCT
ejpam-6909	211	35	b∈ℏ1	b∈ℏ1	NOUN
ejpam-6909	211	36	◦	◦	NOUN
ejpam-6909	211	37	ℏ2	ℏ2	NOUN
ejpam-6909	211	38	,	,	PUNCT
ejpam-6909	211	39	α+	α+	DET
ejpam-6909	211	40	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	211	41	)	)	PUNCT
ejpam-6909	211	42	}	}	PUNCT
ejpam-6909	211	43	,	,	PUNCT
ejpam-6909	211	44	supβ−	supβ−	PROPN
ejpam-6909	211	45	a	a	DET
ejpam-6909	211	46	(	(	PUNCT
ejpam-6909	211	47	c	c	NOUN
ejpam-6909	211	48	)	)	PUNCT
ejpam-6909	211	49	c∈ℏ1	c∈ℏ1	PROPN
ejpam-6909	211	50	◦	◦	NOUN
ejpam-6909	211	51	ℏ1	ℏ1	ADJ
ejpam-6909	211	52	≤	≤	X
ejpam-6909	211	53	β−	β−	PUNCT
ejpam-6909	212	1	a	a	DET
ejpam-6909	212	2	(	(	PUNCT
ejpam-6909	212	3	ℏ1	ℏ1	PROPN
ejpam-6909	212	4	)	)	PUNCT
ejpam-6909	212	5	≤	≤	NUM
ejpam-6909	212	6	max	max	PROPN
ejpam-6909	212	7	{	{	PUNCT
ejpam-6909	212	8	inf	inf	NOUN
ejpam-6909	212	9	β−	β−	PROPN
ejpam-6909	212	10	a	a	DET
ejpam-6909	212	11	(	(	PUNCT
ejpam-6909	212	12	d	d	NOUN
ejpam-6909	212	13	)	)	PUNCT
ejpam-6909	212	14	d∈ℏ1	d∈ℏ1	NOUN
ejpam-6909	212	15	◦	◦	NOUN
ejpam-6909	212	16	ℏ2	ℏ2	NOUN
ejpam-6909	212	17	,	,	PUNCT
ejpam-6909	212	18	β−	β−	PRON
ejpam-6909	212	19	a	a	DET
ejpam-6909	212	20	(	(	PUNCT
ejpam-6909	212	21	ℏ2	ℏ2	NOUN
ejpam-6909	212	22	)	)	PUNCT
ejpam-6909	212	23	}	}	PUNCT
ejpam-6909	212	24	.	.	PUNCT
ejpam-6909	213	1	thus	thus	ADV
ejpam-6909	213	2	,	,	PUNCT
ejpam-6909	213	3	(	(	PUNCT
ejpam-6909	213	4	α+	α+	X
ejpam-6909	213	5	a	a	X
ejpam-6909	213	6	,	,	PUNCT
ejpam-6909	213	7	β	β	PROPN
ejpam-6909	213	8	−	−	NOUN
ejpam-6909	213	9	a	a	PRON
ejpam-6909	213	10	)	)	PUNCT
ejpam-6909	213	11	is	be	AUX
ejpam-6909	213	12	a	a	DET
ejpam-6909	213	13	bfshbcki	bfshbcki	NOUN
ejpam-6909	213	14	of	of	ADP
ejpam-6909	213	15	h.	h.	PROPN
ejpam-6909	213	16	the	the	DET
ejpam-6909	213	17	converse	converse	NOUN
ejpam-6909	213	18	of	of	ADP
ejpam-6909	213	19	theorem	theorem	NOUN
ejpam-6909	213	20	4	4	NUM
ejpam-6909	213	21	is	be	AUX
ejpam-6909	213	22	not	not	PART
ejpam-6909	213	23	necessarily	necessarily	ADV
ejpam-6909	213	24	true	true	ADJ
ejpam-6909	213	25	in	in	ADP
ejpam-6909	213	26	general	general	ADJ
ejpam-6909	213	27	.	.	PUNCT
ejpam-6909	214	1	this	this	DET
ejpam-6909	214	2	fact	fact	NOUN
ejpam-6909	214	3	is	be	AUX
ejpam-6909	214	4	illustrated	illustrate	VERB
ejpam-6909	214	5	in	in	ADP
ejpam-6909	214	6	example	example	NOUN
ejpam-6909	214	7	5	5	NUM
ejpam-6909	214	8	.	.	PUNCT
ejpam-6909	214	9	example	example	NOUN
ejpam-6909	215	1	5	5	NUM
ejpam-6909	215	2	.	.	PUNCT
ejpam-6909	215	3	consider	consider	VERB
ejpam-6909	215	4	the	the	DET
ejpam-6909	215	5	cayley	cayley	ADJ
ejpam-6909	215	6	table	table	NOUN
ejpam-6909	215	7	given	give	VERB
ejpam-6909	215	8	in	in	ADP
ejpam-6909	215	9	example	example	NOUN
ejpam-6909	215	10	2	2	NUM
ejpam-6909	215	11	.	.	PUNCT
ejpam-6909	216	1	then	then	ADV
ejpam-6909	216	2	(	(	PUNCT
ejpam-6909	216	3	h	h	NOUN
ejpam-6909	216	4	,	,	PUNCT
ejpam-6909	216	5	◦	◦	NOUN
ejpam-6909	216	6	)	)	PUNCT
ejpam-6909	216	7	is	be	AUX
ejpam-6909	216	8	an	an	DET
ejpam-6909	216	9	hbcka	hbcka	NOUN
ejpam-6909	216	10	.	.	PUNCT
ejpam-6909	217	1	we	we	PRON
ejpam-6909	217	2	define	define	VERB
ejpam-6909	217	3	a	a	DET
ejpam-6909	217	4	bfs	bfs	NOUN
ejpam-6909	217	5	(	(	PUNCT
ejpam-6909	217	6	α+	α+	NOUN
ejpam-6909	217	7	a	a	X
ejpam-6909	217	8	,	,	PUNCT
ejpam-6909	217	9	β	β	PROPN
ejpam-6909	217	10	−	−	NOUN
ejpam-6909	217	11	a	a	X
ejpam-6909	217	12	)	)	PUNCT
ejpam-6909	217	13	in	in	ADP
ejpam-6909	217	14	h	h	NOUN
ejpam-6909	217	15	as	as	SCONJ
ejpam-6909	217	16	follows	follow	VERB
ejpam-6909	217	17	:	:	PUNCT
ejpam-6909	217	18	α+	α+	PUNCT
ejpam-6909	217	19	a(0	a(0	PROPN
ejpam-6909	217	20	)	)	PUNCT
ejpam-6909	217	21	=	=	NUM
ejpam-6909	217	22	0.8	0.8	NUM
ejpam-6909	217	23	,	,	PUNCT
ejpam-6909	217	24	α+	α+	NOUN
ejpam-6909	217	25	a(ℏ1	a(ℏ1	NOUN
ejpam-6909	217	26	)	)	PUNCT
ejpam-6909	217	27	=	=	SYM
ejpam-6909	217	28	0.6	0.6	NUM
ejpam-6909	217	29	,	,	PUNCT
ejpam-6909	217	30	α+	α+	PRON
ejpam-6909	217	31	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	217	32	)	)	PUNCT
ejpam-6909	217	33	=	=	SYM
ejpam-6909	217	34	0.3	0.3	NUM
ejpam-6909	217	35	,	,	PUNCT
ejpam-6909	217	36	β−	β−	PROPN
ejpam-6909	217	37	a	a	DET
ejpam-6909	217	38	(	(	PUNCT
ejpam-6909	217	39	0	0	NUM
ejpam-6909	217	40	)	)	PUNCT
ejpam-6909	217	41	=	=	SYM
ejpam-6909	218	1	−0.23	−0.23	PROPN
ejpam-6909	218	2	,	,	PUNCT
ejpam-6909	218	3	β−	β−	PRON
ejpam-6909	218	4	a	a	DET
ejpam-6909	218	5	(	(	PUNCT
ejpam-6909	218	6	ℏ1	ℏ1	PROPN
ejpam-6909	218	7	)	)	PUNCT
ejpam-6909	218	8	=	=	SYM
ejpam-6909	218	9	−0.19	−0.19	PROPN
ejpam-6909	218	10	,	,	PUNCT
ejpam-6909	218	11	β−	β−	PRON
ejpam-6909	218	12	a	a	DET
ejpam-6909	218	13	(	(	PUNCT
ejpam-6909	218	14	ℏ2	ℏ2	NOUN
ejpam-6909	218	15	)	)	PUNCT
ejpam-6909	218	16	=	=	SYM
ejpam-6909	219	1	−0.13	−0.13	X
ejpam-6909	219	2	.	.	PUNCT
ejpam-6909	220	1	d.	d.	PROPN
ejpam-6909	220	2	ramesh	ramesh	PROPN
ejpam-6909	220	3	et	et	PROPN
ejpam-6909	220	4	al	al	PROPN
ejpam-6909	220	5	.	.	PUNCT
ejpam-6909	220	6	/	/	SYM
ejpam-6909	220	7	eur	eur	PROPN
ejpam-6909	220	8	.	.	PUNCT
ejpam-6909	221	1	j.	j.	PROPN
ejpam-6909	221	2	pure	pure	PROPN
ejpam-6909	221	3	appl	appl	PROPN
ejpam-6909	221	4	.	.	PROPN
ejpam-6909	221	5	math	math	PROPN
ejpam-6909	221	6	,	,	PUNCT
ejpam-6909	221	7	18	18	NUM
ejpam-6909	221	8	(	(	PUNCT
ejpam-6909	221	9	4	4	NUM
ejpam-6909	221	10	)	)	PUNCT
ejpam-6909	221	11	(	(	PUNCT
ejpam-6909	221	12	2025	2025	NUM
ejpam-6909	221	13	)	)	PUNCT
ejpam-6909	221	14	,	,	PUNCT
ejpam-6909	221	15	6909	6909	NUM
ejpam-6909	221	16	11	11	NUM
ejpam-6909	221	17	of	of	ADP
ejpam-6909	221	18	16	16	NUM
ejpam-6909	221	19	then	then	ADV
ejpam-6909	221	20	(	(	PUNCT
ejpam-6909	221	21	α+	α+	X
ejpam-6909	221	22	a	a	X
ejpam-6909	221	23	,	,	PUNCT
ejpam-6909	221	24	β	β	PROPN
ejpam-6909	221	25	−	−	NOUN
ejpam-6909	221	26	a	a	PRON
ejpam-6909	221	27	)	)	PUNCT
ejpam-6909	221	28	is	be	AUX
ejpam-6909	221	29	a	a	DET
ejpam-6909	221	30	bfshbcki	bfshbcki	NOUN
ejpam-6909	221	31	of	of	ADP
ejpam-6909	221	32	h.	h.	PROPN
ejpam-6909	221	33	however	however	ADV
ejpam-6909	221	34	,	,	PUNCT
ejpam-6909	221	35	it	it	PRON
ejpam-6909	221	36	does	do	AUX
ejpam-6909	221	37	not	not	PART
ejpam-6909	221	38	follow	follow	VERB
ejpam-6909	221	39	the	the	DET
ejpam-6909	221	40	conditions	condition	NOUN
ejpam-6909	221	41	for	for	ADP
ejpam-6909	221	42	bf	bf	NOUN
ejpam-6909	221	43	-	-	PUNCT
ejpam-6909	221	44	chbcki	chbcki	NOUN
ejpam-6909	221	45	of	of	ADP
ejpam-6909	221	46	type-3	type-3	NUM
ejpam-6909	221	47	,	,	PUNCT
ejpam-6909	221	48	because	because	SCONJ
ejpam-6909	221	49	ℏ2	ℏ2	NOUN
ejpam-6909	221	50	∈	∈	PROPN
ejpam-6909	221	51	ℏ2	ℏ2	NOUN
ejpam-6909	221	52	◦	◦	NOUN
ejpam-6909	221	53	(	(	PUNCT
ejpam-6909	221	54	ℏ2	ℏ2	NOUN
ejpam-6909	221	55	◦	◦	VERB
ejpam-6909	221	56	(	(	PUNCT
ejpam-6909	221	57	ℏ2	ℏ2	NOUN
ejpam-6909	221	58	◦	◦	VERB
ejpam-6909	221	59	ℏ2	ℏ2	NOUN
ejpam-6909	221	60	)	)	PUNCT
ejpam-6909	221	61	)	)	PUNCT
ejpam-6909	222	1	=	=	PRON
ejpam-6909	222	2	{	{	PUNCT
ejpam-6909	222	3	0	0	NUM
ejpam-6909	222	4	,	,	PUNCT
ejpam-6909	222	5	ℏ2	ℏ2	NOUN
ejpam-6909	222	6	}	}	PUNCT
ejpam-6909	222	7	,	,	PUNCT
ejpam-6909	222	8	α+	α+	DET
ejpam-6909	222	9	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	222	10	)	)	PUNCT
ejpam-6909	222	11	=	=	PUNCT
ejpam-6909	222	12	0.3	0.3	NUM
ejpam-6909	222	13	<	<	X
ejpam-6909	222	14	0.8	0.8	NUM
ejpam-6909	222	15	=	=	SYM
ejpam-6909	222	16	α+	α+	PUNCT
ejpam-6909	222	17	a(0	a(0	PROPN
ejpam-6909	222	18	)	)	PUNCT
ejpam-6909	222	19	=	=	SYM
ejpam-6909	222	20	min	min	PROPN
ejpam-6909	222	21	{	{	PUNCT
ejpam-6909	222	22	inf	inf	PROPN
ejpam-6909	222	23	α+	α+	PRON
ejpam-6909	222	24	a(a	a(a	PROPN
ejpam-6909	222	25	)	)	PUNCT
ejpam-6909	222	26	a∈(ℏ2	a∈(ℏ2	PART
ejpam-6909	222	27	◦	◦	NOUN
ejpam-6909	222	28	ℏ2)	ℏ2)	NOUN
ejpam-6909	222	29	◦	◦	NOUN
ejpam-6909	222	30	0	0	NUM
ejpam-6909	222	31	,	,	PUNCT
ejpam-6909	222	32	α+	α+	X
ejpam-6909	222	33	a(0	a(0	PROPN
ejpam-6909	222	34	)	)	PUNCT
ejpam-6909	222	35	}	}	PUNCT
ejpam-6909	222	36	,	,	PUNCT
ejpam-6909	222	37	β−	β−	PRON
ejpam-6909	222	38	a	a	DET
ejpam-6909	222	39	(	(	PUNCT
ejpam-6909	222	40	ℏ2	ℏ2	NOUN
ejpam-6909	222	41	)	)	PUNCT
ejpam-6909	222	42	=	=	PUNCT
ejpam-6909	223	1	−0.13	−0.13	X
ejpam-6909	223	2	>	>	X
ejpam-6909	224	1	−0.23	−0.23	X
ejpam-6909	224	2	=	=	PUNCT
ejpam-6909	225	1	β−	β−	PUNCT
ejpam-6909	225	2	a	a	DET
ejpam-6909	225	3	(	(	PUNCT
ejpam-6909	225	4	0	0	NUM
ejpam-6909	225	5	)	)	PUNCT
ejpam-6909	225	6	=	=	SYM
ejpam-6909	225	7	max	max	PROPN
ejpam-6909	225	8	{	{	PUNCT
ejpam-6909	225	9	supβ−	supβ−	X
ejpam-6909	225	10	a	a	DET
ejpam-6909	225	11	(	(	PUNCT
ejpam-6909	225	12	b	b	NOUN
ejpam-6909	225	13	)	)	PUNCT
ejpam-6909	225	14	b∈(ℏ2	b∈(ℏ2	PROPN
ejpam-6909	225	15	◦	◦	NOUN
ejpam-6909	225	16	ℏ2)	ℏ2)	NOUN
ejpam-6909	225	17	◦	◦	NOUN
ejpam-6909	225	18	0	0	NUM
ejpam-6909	225	19	,	,	PUNCT
ejpam-6909	225	20	β−	β−	PRON
ejpam-6909	225	21	a	a	DET
ejpam-6909	225	22	(	(	PUNCT
ejpam-6909	225	23	0	0	NUM
ejpam-6909	225	24	)	)	PUNCT
ejpam-6909	225	25	}	}	PUNCT
ejpam-6909	225	26	.	.	PUNCT
ejpam-6909	226	1	corollary	corollary	ADJ
ejpam-6909	226	2	1	1	NUM
ejpam-6909	226	3	.	.	PUNCT
ejpam-6909	227	1	let	let	VERB
ejpam-6909	227	2	(	(	PUNCT
ejpam-6909	227	3	α+	α+	X
ejpam-6909	227	4	a	a	X
ejpam-6909	227	5	,	,	PUNCT
ejpam-6909	227	6	β	β	PROPN
ejpam-6909	227	7	−	−	NOUN
ejpam-6909	227	8	a	a	DET
ejpam-6909	227	9	)	)	PUNCT
ejpam-6909	227	10	be	be	AUX
ejpam-6909	227	11	a	a	DET
ejpam-6909	227	12	bf	bf	NOUN
ejpam-6909	227	13	-	-	PUNCT
ejpam-6909	227	14	chbcki	chbcki	NOUN
ejpam-6909	227	15	of	of	ADP
ejpam-6909	227	16	type-3	type-3	NUM
ejpam-6909	227	17	,	,	PUNCT
ejpam-6909	227	18	and	and	CCONJ
ejpam-6909	227	19	let	let	VERB
ejpam-6909	227	20	ℏ1	ℏ1	NOUN
ejpam-6909	227	21	,	,	PUNCT
ejpam-6909	227	22	ℏ2	ℏ2	NOUN
ejpam-6909	227	23	∈	∈	PROPN
ejpam-6909	227	24	h.	h.	PROPN
ejpam-6909	227	25	then	then	ADV
ejpam-6909	227	26	(	(	PUNCT
ejpam-6909	227	27	i	i	NOUN
ejpam-6909	227	28	)	)	PUNCT
ejpam-6909	227	29	ℏ1	ℏ1	NOUN
ejpam-6909	227	30	≪	≪	PUNCT
ejpam-6909	227	31	ℏ2	ℏ2	NOUN
ejpam-6909	227	32	⇒	⇒	VERB
ejpam-6909	227	33	α+	α+	PRON
ejpam-6909	227	34	a(ℏ1	a(ℏ1	NUM
ejpam-6909	227	35	)	)	PUNCT
ejpam-6909	227	36	≥	≥	NOUN
ejpam-6909	227	37	α+	α+	X
ejpam-6909	227	38	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	227	39	)	)	PUNCT
ejpam-6909	227	40	and	and	CCONJ
ejpam-6909	227	41	β−	β−	PRON
ejpam-6909	227	42	a	a	DET
ejpam-6909	227	43	(	(	PUNCT
ejpam-6909	227	44	ℏ1	ℏ1	PROPN
ejpam-6909	227	45	)	)	PUNCT
ejpam-6909	227	46	≤	≤	NOUN
ejpam-6909	227	47	β−	β−	PUNCT
ejpam-6909	228	1	a	a	DET
ejpam-6909	228	2	(	(	PUNCT
ejpam-6909	228	3	ℏ2	ℏ2	NOUN
ejpam-6909	228	4	)	)	PUNCT
ejpam-6909	228	5	,	,	PUNCT
ejpam-6909	228	6	(	(	PUNCT
ejpam-6909	228	7	ii	ii	NOUN
ejpam-6909	228	8	)	)	PUNCT
ejpam-6909	228	9	α+	α+	X
ejpam-6909	228	10	a(ℏ1	a(ℏ1	NUM
ejpam-6909	228	11	)	)	PUNCT
ejpam-6909	228	12	≥	≥	NOUN
ejpam-6909	228	13	min{α+	min{α+	PROPN
ejpam-6909	228	14	a(a	a(a	PROPN
ejpam-6909	228	15	)	)	PUNCT
ejpam-6909	228	16	,	,	PUNCT
ejpam-6909	228	17	α	α	PROPN
ejpam-6909	228	18	+	+	X
ejpam-6909	228	19	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	228	20	)	)	PUNCT
ejpam-6909	228	21	}	}	PUNCT
ejpam-6909	228	22	,	,	PUNCT
ejpam-6909	228	23	(	(	PUNCT
ejpam-6909	228	24	iii	iii	NOUN
ejpam-6909	228	25	)	)	PUNCT
ejpam-6909	228	26	β−	β−	NOUN
ejpam-6909	229	1	a	a	DET
ejpam-6909	229	2	(	(	PUNCT
ejpam-6909	229	3	ℏ1	ℏ1	PROPN
ejpam-6909	229	4	)	)	PUNCT
ejpam-6909	229	5	≤	≤	NUM
ejpam-6909	230	1	max{β−	max{β−	PROPN
ejpam-6909	230	2	a	a	DET
ejpam-6909	230	3	(	(	PUNCT
ejpam-6909	230	4	b	b	NOUN
ejpam-6909	230	5	)	)	PUNCT
ejpam-6909	230	6	,	,	PUNCT
ejpam-6909	230	7	β	β	PROPN
ejpam-6909	230	8	−	−	PROPN
ejpam-6909	230	9	a	a	DET
ejpam-6909	230	10	(	(	PUNCT
ejpam-6909	230	11	ℏ2	ℏ2	NOUN
ejpam-6909	230	12	)	)	PUNCT
ejpam-6909	230	13	}	}	PUNCT
ejpam-6909	230	14	,	,	PUNCT
ejpam-6909	230	15	for	for	ADP
ejpam-6909	230	16	all	all	DET
ejpam-6909	230	17	a	a	DET
ejpam-6909	230	18	,	,	PUNCT
ejpam-6909	230	19	b	b	PROPN
ejpam-6909	230	20	∈	∈	PROPN
ejpam-6909	230	21	ℏ1	ℏ1	PROPN
ejpam-6909	230	22	◦	◦	NOUN
ejpam-6909	230	23	ℏ2	ℏ2	NOUN
ejpam-6909	230	24	.	.	PUNCT
ejpam-6909	231	1	corollary	corollary	ADJ
ejpam-6909	231	2	2	2	NUM
ejpam-6909	231	3	.	.	PUNCT
ejpam-6909	232	1	if	if	SCONJ
ejpam-6909	232	2	(	(	PUNCT
ejpam-6909	232	3	α+	α+	X
ejpam-6909	232	4	a	a	X
ejpam-6909	232	5	,	,	PUNCT
ejpam-6909	232	6	β	β	PROPN
ejpam-6909	232	7	−	−	NOUN
ejpam-6909	232	8	a	a	PRON
ejpam-6909	232	9	)	)	PUNCT
ejpam-6909	232	10	is	be	AUX
ejpam-6909	232	11	a	a	DET
ejpam-6909	232	12	bf	bf	NOUN
ejpam-6909	232	13	-	-	PUNCT
ejpam-6909	232	14	chbcki	chbcki	NOUN
ejpam-6909	232	15	of	of	ADP
ejpam-6909	232	16	type-3	type-3	NUM
ejpam-6909	232	17	,	,	PUNCT
ejpam-6909	232	18	then	then	ADV
ejpam-6909	232	19	(	(	PUNCT
ejpam-6909	232	20	i	i	NOUN
ejpam-6909	232	21	)	)	PUNCT
ejpam-6909	233	1	α+	α+	X
ejpam-6909	233	2	a(ℏ1	a(ℏ1	NUM
ejpam-6909	233	3	)	)	PUNCT
ejpam-6909	233	4	≥	≥	PROPN
ejpam-6909	233	5	min	min	PROPN
ejpam-6909	233	6	{	{	PUNCT
ejpam-6909	233	7	inf	inf	PROPN
ejpam-6909	233	8	α+	α+	PRON
ejpam-6909	233	9	a(a	a(a	PROPN
ejpam-6909	233	10	)	)	PUNCT
ejpam-6909	233	11	a∈ℏ1	a∈ℏ1	PROPN
ejpam-6909	233	12	◦	◦	ADJ
ejpam-6909	233	13	ℏ2	ℏ2	NOUN
ejpam-6909	233	14	,	,	PUNCT
ejpam-6909	233	15	α+	α+	PRON
ejpam-6909	233	16	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	233	17	)	)	PUNCT
ejpam-6909	233	18	}	}	PUNCT
ejpam-6909	233	19	,	,	PUNCT
ejpam-6909	233	20	(	(	PUNCT
ejpam-6909	233	21	ii	ii	NOUN
ejpam-6909	233	22	)	)	PUNCT
ejpam-6909	233	23	β−	β−	PROPN
ejpam-6909	234	1	a	a	DET
ejpam-6909	234	2	(	(	PUNCT
ejpam-6909	234	3	ℏ1	ℏ1	PROPN
ejpam-6909	234	4	)	)	PUNCT
ejpam-6909	234	5	≤	≤	NUM
ejpam-6909	234	6	max	max	PROPN
ejpam-6909	234	7	{	{	PUNCT
ejpam-6909	234	8	supβ−	supβ−	PROPN
ejpam-6909	234	9	a	a	DET
ejpam-6909	234	10	(	(	PUNCT
ejpam-6909	234	11	b	b	NOUN
ejpam-6909	234	12	)	)	PUNCT
ejpam-6909	234	13	b∈ℏ1	b∈ℏ1	NOUN
ejpam-6909	234	14	◦	◦	NOUN
ejpam-6909	234	15	ℏ2	ℏ2	NOUN
ejpam-6909	234	16	,	,	PUNCT
ejpam-6909	234	17	β−	β−	PRON
ejpam-6909	234	18	a	a	DET
ejpam-6909	234	19	(	(	PUNCT
ejpam-6909	234	20	ℏ2	ℏ2	NOUN
ejpam-6909	234	21	)	)	PUNCT
ejpam-6909	234	22	}	}	PUNCT
ejpam-6909	234	23	,	,	PUNCT
ejpam-6909	234	24	for	for	ADP
ejpam-6909	234	25	all	all	DET
ejpam-6909	234	26	ℏ1	ℏ1	NOUN
ejpam-6909	234	27	,	,	PUNCT
ejpam-6909	234	28	ℏ2	ℏ2	NOUN
ejpam-6909	234	29	∈	∈	PROPN
ejpam-6909	234	30	h.	h.	NOUN
ejpam-6909	234	31	proposition	proposition	NOUN
ejpam-6909	234	32	1	1	X
ejpam-6909	234	33	.	.	PUNCT
ejpam-6909	235	1	let	let	VERB
ejpam-6909	235	2	(	(	PUNCT
ejpam-6909	235	3	α+	α+	X
ejpam-6909	235	4	a	a	X
ejpam-6909	235	5	,	,	PUNCT
ejpam-6909	235	6	β	β	PROPN
ejpam-6909	235	7	−	−	NOUN
ejpam-6909	235	8	a	a	DET
ejpam-6909	235	9	)	)	PUNCT
ejpam-6909	235	10	be	be	AUX
ejpam-6909	235	11	a	a	DET
ejpam-6909	235	12	bf	bf	NOUN
ejpam-6909	235	13	-	-	PUNCT
ejpam-6909	235	14	chbcki	chbcki	NOUN
ejpam-6909	235	15	of	of	ADP
ejpam-6909	235	16	type-3	type-3	PROPN
ejpam-6909	235	17	.	.	PUNCT
ejpam-6909	236	1	if	if	SCONJ
ejpam-6909	236	2	(	(	PUNCT
ejpam-6909	236	3	α+	α+	X
ejpam-6909	236	4	a	a	X
ejpam-6909	236	5	,	,	PUNCT
ejpam-6909	236	6	β	β	PROPN
ejpam-6909	236	7	−	−	NOUN
ejpam-6909	236	8	a	a	PRON
ejpam-6909	236	9	)	)	PUNCT
ejpam-6909	236	10	satisfies	satisfy	VERB
ejpam-6909	236	11	the	the	DET
ejpam-6909	236	12	inf	inf	ADJ
ejpam-6909	236	13	-	-	PUNCT
ejpam-6909	236	14	sup	sup	NOUN
ejpam-6909	236	15	property	property	NOUN
ejpam-6909	236	16	,	,	PUNCT
ejpam-6909	236	17	then	then	ADV
ejpam-6909	236	18	it	it	PRON
ejpam-6909	236	19	is	be	AUX
ejpam-6909	236	20	a	a	DET
ejpam-6909	236	21	bfswhbcki	bfswhbcki	NOUN
ejpam-6909	236	22	of	of	ADP
ejpam-6909	236	23	h.	h.	PROPN
ejpam-6909	236	24	proof	proof	NOUN
ejpam-6909	236	25	.	.	PUNCT
ejpam-6909	237	1	the	the	DET
ejpam-6909	237	2	proof	proof	NOUN
ejpam-6909	237	3	is	be	AUX
ejpam-6909	237	4	straightforward	straightforward	ADJ
ejpam-6909	237	5	.	.	PUNCT
ejpam-6909	238	1	definition	definition	NOUN
ejpam-6909	238	2	12	12	NUM
ejpam-6909	238	3	.	.	PUNCT
ejpam-6909	239	1	let	let	VERB
ejpam-6909	239	2	(	(	PUNCT
ejpam-6909	239	3	α+	α+	X
ejpam-6909	239	4	a	a	X
ejpam-6909	239	5	,	,	PUNCT
ejpam-6909	239	6	β	β	PROPN
ejpam-6909	239	7	−	−	NOUN
ejpam-6909	239	8	a	a	DET
ejpam-6909	239	9	)	)	PUNCT
ejpam-6909	239	10	be	be	AUX
ejpam-6909	239	11	a	a	DET
ejpam-6909	239	12	bfs	bfs	NOUN
ejpam-6909	239	13	in	in	ADP
ejpam-6909	239	14	h.	h.	PROPN
ejpam-6909	240	1	then	then	ADV
ejpam-6909	240	2	(	(	PUNCT
ejpam-6909	240	3	α+	α+	X
ejpam-6909	240	4	a	a	X
ejpam-6909	240	5	,	,	PUNCT
ejpam-6909	240	6	β	β	PROPN
ejpam-6909	240	7	−	−	NOUN
ejpam-6909	240	8	a	a	PRON
ejpam-6909	240	9	)	)	PUNCT
ejpam-6909	240	10	is	be	AUX
ejpam-6909	240	11	closed	close	VERB
ejpam-6909	240	12	if	if	SCONJ
ejpam-6909	240	13	for	for	ADP
ejpam-6909	240	14	all	all	DET
ejpam-6909	240	15	ℏ1	ℏ1	NOUN
ejpam-6909	240	16	,	,	PUNCT
ejpam-6909	240	17	ℏ2	ℏ2	NOUN
ejpam-6909	240	18	∈	∈	PROPN
ejpam-6909	240	19	h	h	NOUN
ejpam-6909	240	20	such	such	ADJ
ejpam-6909	240	21	that	that	SCONJ
ejpam-6909	240	22	ℏ1	ℏ1	ADJ
ejpam-6909	240	23	≪	≪	ADJ
ejpam-6909	240	24	ℏ2	ℏ2	NOUN
ejpam-6909	240	25	,	,	PUNCT
ejpam-6909	240	26	we	we	PRON
ejpam-6909	240	27	have	have	VERB
ejpam-6909	240	28	α+	α+	X
ejpam-6909	240	29	a(ℏ1	a(ℏ1	NUM
ejpam-6909	240	30	)	)	PUNCT
ejpam-6909	240	31	≥	≥	NOUN
ejpam-6909	241	1	α+	α+	X
ejpam-6909	241	2	a(ℏ2	a(ℏ2	ADJ
ejpam-6909	241	3	)	)	PUNCT
ejpam-6909	242	1	and	and	CCONJ
ejpam-6909	242	2	β−	β−	PRON
ejpam-6909	242	3	a	a	DET
ejpam-6909	242	4	(	(	PUNCT
ejpam-6909	242	5	ℏ1	ℏ1	PROPN
ejpam-6909	242	6	)	)	PUNCT
ejpam-6909	242	7	≤	≤	NOUN
ejpam-6909	242	8	β−	β−	PUNCT
ejpam-6909	243	1	a	a	DET
ejpam-6909	243	2	(	(	PUNCT
ejpam-6909	243	3	ℏ2	ℏ2	NOUN
ejpam-6909	243	4	)	)	PUNCT
ejpam-6909	243	5	.	.	PUNCT
ejpam-6909	244	1	definition	definition	NOUN
ejpam-6909	244	2	13	13	NUM
ejpam-6909	244	3	.	.	PUNCT
ejpam-6909	245	1	[	[	X
ejpam-6909	245	2	16	16	NUM
ejpam-6909	245	3	]	]	PUNCT
ejpam-6909	245	4	for	for	ADP
ejpam-6909	245	5	a	a	DET
ejpam-6909	245	6	bfs	bfs	NOUN
ejpam-6909	245	7	(	(	PUNCT
ejpam-6909	245	8	α+	α+	NOUN
ejpam-6909	245	9	a	a	X
ejpam-6909	245	10	,	,	PUNCT
ejpam-6909	245	11	β	β	PROPN
ejpam-6909	245	12	−	−	NOUN
ejpam-6909	245	13	a	a	X
ejpam-6909	245	14	)	)	PUNCT
ejpam-6909	245	15	,	,	PUNCT
ejpam-6909	245	16	the	the	DET
ejpam-6909	245	17	positive	positive	ADJ
ejpam-6909	245	18	p	p	NOUN
ejpam-6909	245	19	-	-	PUNCT
ejpam-6909	245	20	cut	cut	NOUN
ejpam-6909	245	21	(	(	PUNCT
ejpam-6909	245	22	where	where	SCONJ
ejpam-6909	245	23	p	p	X
ejpam-6909	245	24	∈	∈	PROPN
ejpam-6909	246	1	[	[	X
ejpam-6909	246	2	0	0	NUM
ejpam-6909	246	3	,	,	PUNCT
ejpam-6909	246	4	1	1	NUM
ejpam-6909	246	5	]	]	PUNCT
ejpam-6909	246	6	)	)	PUNCT
ejpam-6909	246	7	and	and	CCONJ
ejpam-6909	246	8	the	the	DET
ejpam-6909	246	9	negative	negative	ADJ
ejpam-6909	246	10	n	n	CCONJ
ejpam-6909	246	11	-	-	PUNCT
ejpam-6909	246	12	cut	cut	NOUN
ejpam-6909	246	13	(	(	PUNCT
ejpam-6909	246	14	where	where	SCONJ
ejpam-6909	246	15	n	n	PRON
ejpam-6909	246	16	∈	∈	PROPN
ejpam-6909	246	17	[	[	X
ejpam-6909	246	18	−1	−1	NOUN
ejpam-6909	246	19	,	,	PUNCT
ejpam-6909	246	20	0	0	NUM
ejpam-6909	246	21	]	]	PUNCT
ejpam-6909	246	22	)	)	PUNCT
ejpam-6909	246	23	are	be	AUX
ejpam-6909	246	24	defined	define	VERB
ejpam-6909	246	25	as	as	SCONJ
ejpam-6909	246	26	follows	follow	VERB
ejpam-6909	246	27	:	:	PUNCT
ejpam-6909	246	28	u(α+	u(α+	NOUN
ejpam-6909	246	29	a	a	X
ejpam-6909	246	30	;	;	PUNCT
ejpam-6909	246	31	p	p	X
ejpam-6909	246	32	)	)	PUNCT
ejpam-6909	246	33	=	=	SYM
ejpam-6909	246	34	{	{	PUNCT
ejpam-6909	246	35	ℏ1	ℏ1	PROPN
ejpam-6909	246	36	∈	∈	PROPN
ejpam-6909	246	37	h	h	NOUN
ejpam-6909	246	38	|	|	ADV
ejpam-6909	246	39	α+	α+	PUNCT
ejpam-6909	246	40	a(ℏ1	a(ℏ1	NUM
ejpam-6909	246	41	)	)	PUNCT
ejpam-6909	246	42	≥	≥	NOUN
ejpam-6909	247	1	p	p	X
ejpam-6909	247	2	}	}	PUNCT
ejpam-6909	247	3	,	,	PUNCT
ejpam-6909	247	4	l(β−	l(β−	VERB
ejpam-6909	247	5	a	a	PRON
ejpam-6909	247	6	;	;	PUNCT
ejpam-6909	247	7	n	n	CCONJ
ejpam-6909	247	8	)	)	PUNCT
ejpam-6909	247	9	=	=	PRON
ejpam-6909	247	10	{	{	PUNCT
ejpam-6909	247	11	ℏ1	ℏ1	PROPN
ejpam-6909	247	12	∈	∈	PROPN
ejpam-6909	247	13	h	h	NOUN
ejpam-6909	248	1	|	|	ADV
ejpam-6909	248	2	β−	β−	INTJ
ejpam-6909	249	1	a	a	DET
ejpam-6909	249	2	(	(	PUNCT
ejpam-6909	249	3	ℏ1	ℏ1	PROPN
ejpam-6909	249	4	)	)	PUNCT
ejpam-6909	249	5	≤	≤	NOUN
ejpam-6909	249	6	n	n	CCONJ
ejpam-6909	249	7	}	}	PUNCT
ejpam-6909	249	8	.	.	PUNCT
ejpam-6909	250	1	theorem	theorem	NOUN
ejpam-6909	250	2	5	5	NUM
ejpam-6909	250	3	.	.	PUNCT
ejpam-6909	251	1	let	let	VERB
ejpam-6909	251	2	(	(	PUNCT
ejpam-6909	251	3	α+	α+	X
ejpam-6909	251	4	a	a	X
ejpam-6909	251	5	,	,	PUNCT
ejpam-6909	251	6	β	β	PROPN
ejpam-6909	251	7	−	−	NOUN
ejpam-6909	251	8	a	a	DET
ejpam-6909	251	9	)	)	PUNCT
ejpam-6909	251	10	be	be	AUX
ejpam-6909	251	11	a	a	DET
ejpam-6909	251	12	bfs	bfs	NOUN
ejpam-6909	251	13	in	in	ADP
ejpam-6909	251	14	h.	h.	PROPN
ejpam-6909	251	15	then	then	ADV
ejpam-6909	251	16	the	the	DET
ejpam-6909	251	17	following	follow	VERB
ejpam-6909	251	18	statements	statement	NOUN
ejpam-6909	251	19	are	be	AUX
ejpam-6909	251	20	hold	hold	ADJ
ejpam-6909	251	21	:	:	PUNCT
ejpam-6909	251	22	(	(	PUNCT
ejpam-6909	251	23	i	i	NOUN
ejpam-6909	251	24	)	)	PUNCT
ejpam-6909	251	25	a	a	DET
ejpam-6909	251	26	bfs	bfs	NOUN
ejpam-6909	251	27	(	(	PUNCT
ejpam-6909	251	28	α+	α+	NOUN
ejpam-6909	251	29	a	a	X
ejpam-6909	251	30	,	,	PUNCT
ejpam-6909	251	31	β	β	PROPN
ejpam-6909	251	32	−	−	NOUN
ejpam-6909	251	33	a	a	PRON
ejpam-6909	251	34	)	)	PUNCT
ejpam-6909	251	35	is	be	AUX
ejpam-6909	251	36	a	a	DET
ejpam-6909	251	37	bf	bf	NOUN
ejpam-6909	251	38	-	-	PUNCT
ejpam-6909	251	39	chbcki	chbcki	NOUN
ejpam-6909	251	40	of	of	ADP
ejpam-6909	251	41	type-1	type-1	PROPN
ejpam-6909	251	42	of	of	ADP
ejpam-6909	251	43	h	h	NOUN
ejpam-6909	251	44	if	if	SCONJ
ejpam-6909	252	1	and	and	CCONJ
ejpam-6909	252	2	only	only	ADV
ejpam-6909	252	3	if	if	SCONJ
ejpam-6909	252	4	for	for	ADP
ejpam-6909	252	5	all	all	DET
ejpam-6909	252	6	(	(	PUNCT
ejpam-6909	252	7	p	p	X
ejpam-6909	252	8	,	,	PUNCT
ejpam-6909	252	9	n	n	CCONJ
ejpam-6909	252	10	)	)	PUNCT
ejpam-6909	252	11	∈	∈	PROPN
ejpam-6909	253	1	[	[	X
ejpam-6909	253	2	0	0	NUM
ejpam-6909	253	3	,	,	PUNCT
ejpam-6909	253	4	1]×	1]×	NUM
ejpam-6909	253	5	[	[	X
ejpam-6909	253	6	−1	−1	NOUN
ejpam-6909	253	7	,	,	PUNCT
ejpam-6909	253	8	0	0	NUM
ejpam-6909	253	9	]	]	PUNCT
ejpam-6909	253	10	,	,	PUNCT
ejpam-6909	253	11	the	the	DET
ejpam-6909	253	12	non	non	ADJ
ejpam-6909	253	13	-	-	ADJ
ejpam-6909	253	14	empty	empty	ADJ
ejpam-6909	253	15	cut	cut	NOUN
ejpam-6909	253	16	sets	set	VERB
ejpam-6909	253	17	u(α+	u(α+	PRON
ejpam-6909	253	18	a	a	X
ejpam-6909	253	19	;	;	PUNCT
ejpam-6909	253	20	p	p	X
ejpam-6909	253	21	)	)	PUNCT
ejpam-6909	253	22	and	and	CCONJ
ejpam-6909	253	23	l(β−	l(β−	VERB
ejpam-6909	253	24	a	a	PRON
ejpam-6909	253	25	;	;	PUNCT
ejpam-6909	253	26	n	n	CCONJ
ejpam-6909	253	27	)	)	PUNCT
ejpam-6909	253	28	are	be	AUX
ejpam-6909	253	29	chbckis	chbcki	NOUN
ejpam-6909	253	30	type-1	type-1	PROPN
ejpam-6909	253	31	of	of	ADP
ejpam-6909	253	32	h.	h.	PROPN
ejpam-6909	253	33	d.	d.	PROPN
ejpam-6909	253	34	ramesh	ramesh	PROPN
ejpam-6909	254	1	et	et	PROPN
ejpam-6909	254	2	al	al	PROPN
ejpam-6909	254	3	.	.	PUNCT
ejpam-6909	254	4	/	/	SYM
ejpam-6909	254	5	eur	eur	PROPN
ejpam-6909	254	6	.	.	PUNCT
ejpam-6909	255	1	j.	j.	PROPN
ejpam-6909	255	2	pure	pure	PROPN
ejpam-6909	255	3	appl	appl	PROPN
ejpam-6909	255	4	.	.	PROPN
ejpam-6909	255	5	math	math	PROPN
ejpam-6909	255	6	,	,	PUNCT
ejpam-6909	255	7	18	18	NUM
ejpam-6909	255	8	(	(	PUNCT
ejpam-6909	255	9	4	4	NUM
ejpam-6909	255	10	)	)	PUNCT
ejpam-6909	255	11	(	(	PUNCT
ejpam-6909	255	12	2025	2025	NUM
ejpam-6909	255	13	)	)	PUNCT
ejpam-6909	255	14	,	,	PUNCT
ejpam-6909	255	15	6909	6909	NUM
ejpam-6909	255	16	12	12	NUM
ejpam-6909	255	17	of	of	ADP
ejpam-6909	255	18	16	16	NUM
ejpam-6909	255	19	(	(	PUNCT
ejpam-6909	255	20	ii	ii	NOUN
ejpam-6909	255	21	)	)	PUNCT
ejpam-6909	255	22	if	if	SCONJ
ejpam-6909	255	23	(	(	PUNCT
ejpam-6909	255	24	α+	α+	X
ejpam-6909	255	25	a	a	X
ejpam-6909	255	26	,	,	PUNCT
ejpam-6909	255	27	β	β	PROPN
ejpam-6909	255	28	−	−	NOUN
ejpam-6909	255	29	a	a	PRON
ejpam-6909	255	30	)	)	PUNCT
ejpam-6909	255	31	is	be	AUX
ejpam-6909	255	32	a	a	DET
ejpam-6909	255	33	bf	bf	NOUN
ejpam-6909	255	34	-	-	PUNCT
ejpam-6909	255	35	chbcki	chbcki	NOUN
ejpam-6909	255	36	of	of	ADP
ejpam-6909	255	37	type-3	type-3	NUM
ejpam-6909	255	38	of	of	ADP
ejpam-6909	255	39	h	h	NOUN
ejpam-6909	255	40	,	,	PUNCT
ejpam-6909	255	41	then	then	ADV
ejpam-6909	255	42	for	for	ADP
ejpam-6909	255	43	all	all	DET
ejpam-6909	255	44	(	(	PUNCT
ejpam-6909	255	45	p	p	X
ejpam-6909	255	46	,	,	PUNCT
ejpam-6909	255	47	n	n	CCONJ
ejpam-6909	255	48	)	)	PUNCT
ejpam-6909	255	49	∈	∈	PROPN
ejpam-6909	256	1	[	[	X
ejpam-6909	256	2	0	0	NUM
ejpam-6909	256	3	,	,	PUNCT
ejpam-6909	256	4	1]×	1]×	NUM
ejpam-6909	256	5	[	[	X
ejpam-6909	256	6	−1	−1	NOUN
ejpam-6909	256	7	,	,	PUNCT
ejpam-6909	256	8	0	0	NUM
ejpam-6909	256	9	]	]	PUNCT
ejpam-6909	256	10	,	,	PUNCT
ejpam-6909	256	11	the	the	DET
ejpam-6909	256	12	non	non	ADJ
ejpam-6909	256	13	-	-	ADJ
ejpam-6909	256	14	empty	empty	ADJ
ejpam-6909	256	15	cut	cut	NOUN
ejpam-6909	256	16	sets	set	VERB
ejpam-6909	256	17	u(α+	u(α+	PRON
ejpam-6909	256	18	a	a	X
ejpam-6909	256	19	;	;	PUNCT
ejpam-6909	256	20	p	p	X
ejpam-6909	256	21	)	)	PUNCT
ejpam-6909	256	22	and	and	CCONJ
ejpam-6909	256	23	l(β−	l(β−	VERB
ejpam-6909	256	24	a	a	PRON
ejpam-6909	256	25	;	;	PUNCT
ejpam-6909	256	26	n	n	CCONJ
ejpam-6909	256	27	)	)	PUNCT
ejpam-6909	256	28	are	be	AUX
ejpam-6909	256	29	chbckis	chbcki	NOUN
ejpam-6909	256	30	type-3	type-3	NUM
ejpam-6909	256	31	of	of	ADP
ejpam-6909	256	32	h.	h.	PROPN
ejpam-6909	256	33	(	(	PUNCT
ejpam-6909	256	34	iii	iii	NOUN
ejpam-6909	256	35	)	)	PUNCT
ejpam-6909	256	36	if	if	SCONJ
ejpam-6909	256	37	(	(	PUNCT
ejpam-6909	256	38	α+	α+	X
ejpam-6909	256	39	a	a	X
ejpam-6909	256	40	,	,	PUNCT
ejpam-6909	256	41	β	β	PROPN
ejpam-6909	256	42	−	−	NOUN
ejpam-6909	256	43	a	a	PRON
ejpam-6909	256	44	)	)	PUNCT
ejpam-6909	256	45	satisfies	satisfy	VERB
ejpam-6909	256	46	the	the	DET
ejpam-6909	256	47	sup	sup	NOUN
ejpam-6909	256	48	-	-	PUNCT
ejpam-6909	256	49	inf	inf	NOUN
ejpam-6909	256	50	property	property	NOUN
ejpam-6909	256	51	and	and	CCONJ
ejpam-6909	256	52	for	for	ADP
ejpam-6909	256	53	all	all	DET
ejpam-6909	256	54	(	(	PUNCT
ejpam-6909	256	55	p	p	X
ejpam-6909	256	56	,	,	PUNCT
ejpam-6909	256	57	n	n	CCONJ
ejpam-6909	256	58	)	)	PUNCT
ejpam-6909	256	59	∈	∈	PROPN
ejpam-6909	257	1	[	[	X
ejpam-6909	257	2	0	0	NUM
ejpam-6909	257	3	,	,	PUNCT
ejpam-6909	257	4	1]×	1]×	NUM
ejpam-6909	257	5	[	[	X
ejpam-6909	257	6	−1	−1	NOUN
ejpam-6909	257	7	,	,	PUNCT
ejpam-6909	257	8	0	0	NUM
ejpam-6909	257	9	]	]	PUNCT
ejpam-6909	257	10	,	,	PUNCT
ejpam-6909	257	11	the	the	DET
ejpam-6909	257	12	nonempty	nonempty	ADJ
ejpam-6909	257	13	cut	cut	NOUN
ejpam-6909	257	14	sets	set	VERB
ejpam-6909	257	15	u(α+	u(α+	PRON
ejpam-6909	257	16	a	a	X
ejpam-6909	257	17	;	;	PUNCT
ejpam-6909	257	18	p	p	X
ejpam-6909	257	19	)	)	PUNCT
ejpam-6909	257	20	and	and	CCONJ
ejpam-6909	257	21	l(β−	l(β−	VERB
ejpam-6909	257	22	a	a	PRON
ejpam-6909	257	23	;	;	PUNCT
ejpam-6909	257	24	n	n	CCONJ
ejpam-6909	257	25	)	)	PUNCT
ejpam-6909	257	26	are	be	AUX
ejpam-6909	257	27	reflexive	reflexive	ADJ
ejpam-6909	257	28	-	-	PUNCT
ejpam-6909	257	29	chbckis	chbcki	NOUN
ejpam-6909	257	30	of	of	ADP
ejpam-6909	257	31	type-3	type-3	NUM
ejpam-6909	257	32	of	of	ADP
ejpam-6909	257	33	h	h	NOUN
ejpam-6909	257	34	,	,	PUNCT
ejpam-6909	257	35	then	then	ADV
ejpam-6909	257	36	(	(	PUNCT
ejpam-6909	257	37	α+	α+	X
ejpam-6909	257	38	a	a	X
ejpam-6909	257	39	,	,	PUNCT
ejpam-6909	257	40	β	β	PROPN
ejpam-6909	257	41	−	−	NOUN
ejpam-6909	257	42	a	a	PRON
ejpam-6909	257	43	)	)	PUNCT
ejpam-6909	257	44	is	be	AUX
ejpam-6909	257	45	a	a	DET
ejpam-6909	257	46	bf	bf	NOUN
ejpam-6909	257	47	-	-	PUNCT
ejpam-6909	257	48	chbcki	chbcki	NOUN
ejpam-6909	257	49	of	of	ADP
ejpam-6909	257	50	type-3	type-3	NUM
ejpam-6909	257	51	of	of	ADP
ejpam-6909	257	52	h.	h.	NOUN
ejpam-6909	257	53	proof	proof	NOUN
ejpam-6909	257	54	.	.	PUNCT
ejpam-6909	258	1	(	(	PUNCT
ejpam-6909	258	2	i	i	NOUN
ejpam-6909	258	3	)	)	PUNCT
ejpam-6909	258	4	suppose	suppose	VERB
ejpam-6909	258	5	(	(	PUNCT
ejpam-6909	258	6	α+	α+	X
ejpam-6909	258	7	a	a	X
ejpam-6909	258	8	,	,	PUNCT
ejpam-6909	258	9	β	β	PROPN
ejpam-6909	258	10	−	−	NOUN
ejpam-6909	258	11	a	a	PRON
ejpam-6909	258	12	)	)	PUNCT
ejpam-6909	258	13	is	be	AUX
ejpam-6909	258	14	a	a	DET
ejpam-6909	258	15	bf	bf	NOUN
ejpam-6909	258	16	-	-	PUNCT
ejpam-6909	258	17	chbcki	chbcki	NOUN
ejpam-6909	258	18	of	of	ADP
ejpam-6909	258	19	type-1	type-1	PROPN
ejpam-6909	258	20	of	of	ADP
ejpam-6909	258	21	h	h	NOUN
ejpam-6909	258	22	and	and	CCONJ
ejpam-6909	258	23	for	for	ADP
ejpam-6909	258	24	all	all	DET
ejpam-6909	258	25	(	(	PUNCT
ejpam-6909	258	26	p	p	X
ejpam-6909	258	27	,	,	PUNCT
ejpam-6909	258	28	n	n	CCONJ
ejpam-6909	258	29	)	)	PUNCT
ejpam-6909	258	30	∈	∈	PROPN
ejpam-6909	259	1	[	[	X
ejpam-6909	259	2	0	0	NUM
ejpam-6909	259	3	,	,	PUNCT
ejpam-6909	259	4	1	1	NUM
ejpam-6909	259	5	]	]	SYM
ejpam-6909	259	6	×	×	NOUN
ejpam-6909	260	1	[	[	X
ejpam-6909	260	2	−1	−1	NOUN
ejpam-6909	260	3	,	,	PUNCT
ejpam-6909	260	4	0	0	NUM
ejpam-6909	260	5	]	]	PUNCT
ejpam-6909	260	6	,	,	PUNCT
ejpam-6909	260	7	u(α+	u(α+	PROPN
ejpam-6909	260	8	a	a	X
ejpam-6909	260	9	;	;	PUNCT
ejpam-6909	260	10	p	p	X
ejpam-6909	260	11	)	)	PUNCT
ejpam-6909	260	12	and	and	CCONJ
ejpam-6909	260	13	l(β−	l(β−	VERB
ejpam-6909	260	14	a	a	PRON
ejpam-6909	260	15	;	;	PUNCT
ejpam-6909	260	16	n	n	CCONJ
ejpam-6909	260	17	)	)	PUNCT
ejpam-6909	260	18	are	be	AUX
ejpam-6909	260	19	non	non	ADJ
ejpam-6909	260	20	-	-	ADJ
ejpam-6909	260	21	empty	empty	ADJ
ejpam-6909	260	22	.	.	PUNCT
ejpam-6909	261	1	by	by	ADP
ejpam-6909	261	2	definition	definition	NOUN
ejpam-6909	261	3	,	,	PUNCT
ejpam-6909	261	4	0	0	NUM
ejpam-6909	261	5	∈	∈	PROPN
ejpam-6909	261	6	u(α+	u(α+	NOUN
ejpam-6909	261	7	a	a	X
ejpam-6909	261	8	;	;	PUNCT
ejpam-6909	261	9	p	p	X
ejpam-6909	261	10	)	)	PUNCT
ejpam-6909	261	11	and	and	CCONJ
ejpam-6909	261	12	0	0	NUM
ejpam-6909	261	13	∈	∈	NOUN
ejpam-6909	261	14	l(β−	l(β−	VERB
ejpam-6909	261	15	a	a	PRON
ejpam-6909	261	16	;	;	PUNCT
ejpam-6909	261	17	n	n	CCONJ
ejpam-6909	261	18	)	)	PUNCT
ejpam-6909	261	19	.	.	PUNCT
ejpam-6909	262	1	therefore	therefore	ADV
ejpam-6909	262	2	,	,	PUNCT
ejpam-6909	262	3	0	0	X
ejpam-6909	262	4	∈	∈	PROPN
ejpam-6909	262	5	u(α+	u(α+	NOUN
ejpam-6909	262	6	a	a	X
ejpam-6909	262	7	;	;	PUNCT
ejpam-6909	262	8	p	p	X
ejpam-6909	262	9	)	)	PUNCT
ejpam-6909	262	10	∩	∩	NOUN
ejpam-6909	262	11	l(β−	l(β−	VERB
ejpam-6909	262	12	a	a	PRON
ejpam-6909	262	13	;	;	PUNCT
ejpam-6909	262	14	n	n	CCONJ
ejpam-6909	262	15	)	)	PUNCT
ejpam-6909	262	16	.	.	PUNCT
ejpam-6909	263	1	let	let	VERB
ejpam-6909	263	2	ℏ1	ℏ1	ADJ
ejpam-6909	263	3	,	,	PUNCT
ejpam-6909	263	4	ℏ2	ℏ2	NOUN
ejpam-6909	263	5	,	,	PUNCT
ejpam-6909	263	6	ℏ3	ℏ3	PROPN
ejpam-6909	263	7	be	be	VERB
ejpam-6909	263	8	elements	element	NOUN
ejpam-6909	263	9	of	of	ADP
ejpam-6909	263	10	h	h	NOUN
ejpam-6909	263	11	such	such	ADJ
ejpam-6909	263	12	that	that	SCONJ
ejpam-6909	263	13	(	(	PUNCT
ejpam-6909	263	14	ℏ1	ℏ1	ADJ
ejpam-6909	263	15	◦	◦	NOUN
ejpam-6909	263	16	ℏ2	ℏ2	NOUN
ejpam-6909	263	17	)	)	PUNCT
ejpam-6909	263	18	◦	◦	NOUN
ejpam-6909	263	19	ℏ3	ℏ3	PROPN
ejpam-6909	263	20	⊆	⊆	NUM
ejpam-6909	263	21	u(α+	u(α+	NOUN
ejpam-6909	263	22	a	a	X
ejpam-6909	263	23	;	;	PUNCT
ejpam-6909	263	24	p	p	X
ejpam-6909	263	25	)	)	PUNCT
ejpam-6909	263	26	and	and	CCONJ
ejpam-6909	263	27	ℏ3	ℏ3	PROPN
ejpam-6909	263	28	∈	∈	PROPN
ejpam-6909	264	1	u(α+	u(α+	NOUN
ejpam-6909	264	2	a	a	X
ejpam-6909	264	3	;	;	PUNCT
ejpam-6909	264	4	p	p	X
ejpam-6909	264	5	)	)	PUNCT
ejpam-6909	264	6	.	.	PUNCT
ejpam-6909	265	1	then	then	ADV
ejpam-6909	265	2	,	,	PUNCT
ejpam-6909	265	3	for	for	ADP
ejpam-6909	265	4	all	all	DET
ejpam-6909	265	5	a	a	DET
ejpam-6909	265	6	∈	∈	NOUN
ejpam-6909	265	7	(	(	PUNCT
ejpam-6909	265	8	ℏ1	ℏ1	PROPN
ejpam-6909	265	9	◦	◦	NOUN
ejpam-6909	265	10	ℏ2	ℏ2	NOUN
ejpam-6909	265	11	)	)	PUNCT
ejpam-6909	265	12	◦	◦	NOUN
ejpam-6909	265	13	ℏ3	ℏ3	PROPN
ejpam-6909	265	14	,	,	PUNCT
ejpam-6909	265	15	we	we	PRON
ejpam-6909	265	16	have	have	VERB
ejpam-6909	265	17	a	a	DET
ejpam-6909	265	18	∈	∈	PROPN
ejpam-6909	265	19	u(α+	u(α+	NOUN
ejpam-6909	265	20	a	a	X
ejpam-6909	265	21	;	;	PUNCT
ejpam-6909	265	22	p	p	X
ejpam-6909	265	23	)	)	PUNCT
ejpam-6909	265	24	and	and	CCONJ
ejpam-6909	265	25	ℏ3	ℏ3	PROPN
ejpam-6909	265	26	∈	∈	PROPN
ejpam-6909	266	1	u(α+	u(α+	NOUN
ejpam-6909	266	2	a	a	X
ejpam-6909	266	3	;	;	PUNCT
ejpam-6909	266	4	p	p	X
ejpam-6909	266	5	)	)	PUNCT
ejpam-6909	266	6	.	.	PUNCT
ejpam-6909	267	1	this	this	PRON
ejpam-6909	267	2	implies	imply	VERB
ejpam-6909	267	3	α+	α+	PRON
ejpam-6909	267	4	a(a	a(a	PROPN
ejpam-6909	267	5	)	)	PUNCT
ejpam-6909	267	6	≥	≥	NOUN
ejpam-6909	268	1	p	p	X
ejpam-6909	268	2	,	,	PUNCT
ejpam-6909	268	3	for	for	ADP
ejpam-6909	268	4	all	all	DET
ejpam-6909	268	5	a	a	DET
ejpam-6909	268	6	∈	∈	NOUN
ejpam-6909	268	7	(	(	PUNCT
ejpam-6909	268	8	ℏ1	ℏ1	PROPN
ejpam-6909	268	9	◦	◦	NOUN
ejpam-6909	268	10	ℏ2	ℏ2	NOUN
ejpam-6909	268	11	)	)	PUNCT
ejpam-6909	268	12	◦	◦	NOUN
ejpam-6909	268	13	ℏ3	ℏ3	PROPN
ejpam-6909	268	14	and	and	CCONJ
ejpam-6909	268	15	α+	α+	PRON
ejpam-6909	268	16	a(ℏ3	a(ℏ3	ADJ
ejpam-6909	268	17	)	)	PUNCT
ejpam-6909	268	18	≥	≥	NOUN
ejpam-6909	268	19	p.	p.	NOUN
ejpam-6909	268	20	consequently	consequently	ADV
ejpam-6909	268	21	,	,	PUNCT
ejpam-6909	268	22	inf	inf	PROPN
ejpam-6909	268	23	α+	α+	PRON
ejpam-6909	268	24	a(a	a(a	PROPN
ejpam-6909	268	25	)	)	PUNCT
ejpam-6909	269	1	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	269	2	◦	◦	NOUN
ejpam-6909	269	3	ℏ2)	ℏ2)	NOUN
ejpam-6909	269	4	◦	◦	NOUN
ejpam-6909	269	5	ℏ3	ℏ3	PROPN
ejpam-6909	269	6	≥	≥	NUM
ejpam-6909	269	7	p	p	NOUN
ejpam-6909	269	8	and	and	CCONJ
ejpam-6909	269	9	α+	α+	PUNCT
ejpam-6909	269	10	a(ℏ3	a(ℏ3	ADJ
ejpam-6909	269	11	)	)	PUNCT
ejpam-6909	269	12	≥	≥	NOUN
ejpam-6909	270	1	p.	p.	NOUN
ejpam-6909	270	2	thus	thus	ADV
ejpam-6909	270	3	,	,	PUNCT
ejpam-6909	270	4	for	for	ADP
ejpam-6909	270	5	all	all	DET
ejpam-6909	270	6	k	k	PROPN
ejpam-6909	270	7	∈	∈	PROPN
ejpam-6909	270	8	ℏ1	ℏ1	PROPN
ejpam-6909	270	9	◦	◦	NOUN
ejpam-6909	270	10	(	(	PUNCT
ejpam-6909	270	11	ℏ2	ℏ2	NOUN
ejpam-6909	270	12	◦	◦	VERB
ejpam-6909	270	13	(	(	PUNCT
ejpam-6909	270	14	ℏ2	ℏ2	NOUN
ejpam-6909	270	15	◦	◦	VERB
ejpam-6909	270	16	ℏ1	ℏ1	ADJ
ejpam-6909	270	17	)	)	PUNCT
ejpam-6909	270	18	)	)	PUNCT
ejpam-6909	270	19	,	,	PUNCT
ejpam-6909	270	20	we	we	PRON
ejpam-6909	270	21	have	have	VERB
ejpam-6909	270	22	α+	α+	PRON
ejpam-6909	270	23	a(k	a(k	PROPN
ejpam-6909	270	24	)	)	PUNCT
ejpam-6909	270	25	≥	≥	PROPN
ejpam-6909	270	26	min	min	PROPN
ejpam-6909	270	27	{	{	PUNCT
ejpam-6909	270	28	inf	inf	PROPN
ejpam-6909	270	29	α+	α+	PRON
ejpam-6909	270	30	a(a	a(a	PROPN
ejpam-6909	270	31	)	)	PUNCT
ejpam-6909	270	32	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	270	33	◦	◦	NOUN
ejpam-6909	270	34	ℏ2)	ℏ2)	NOUN
ejpam-6909	270	35	◦	◦	NOUN
ejpam-6909	270	36	ℏ3	ℏ3	PROPN
ejpam-6909	270	37	,	,	PUNCT
ejpam-6909	270	38	α+	α+	PRON
ejpam-6909	270	39	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	270	40	)	)	PUNCT
ejpam-6909	270	41	}	}	PUNCT
ejpam-6909	270	42	≥	≥	NOUN
ejpam-6909	270	43	min{p	min{p	ADV
ejpam-6909	270	44	,	,	PUNCT
ejpam-6909	270	45	p	p	X
ejpam-6909	270	46	}	}	PUNCT
ejpam-6909	270	47	=	=	PUNCT
ejpam-6909	271	1	p.	p.	NOUN
ejpam-6909	271	2	this	this	PRON
ejpam-6909	271	3	implies	imply	VERB
ejpam-6909	271	4	k	k	PROPN
ejpam-6909	271	5	∈	∈	PROPN
ejpam-6909	271	6	u(α+	u(α+	NOUN
ejpam-6909	271	7	a	a	X
ejpam-6909	271	8	;	;	PUNCT
ejpam-6909	271	9	p	p	X
ejpam-6909	271	10	)	)	PUNCT
ejpam-6909	271	11	,	,	PUNCT
ejpam-6909	271	12	for	for	ADP
ejpam-6909	271	13	all	all	DET
ejpam-6909	271	14	k	k	PROPN
ejpam-6909	271	15	∈	∈	PROPN
ejpam-6909	271	16	ℏ1	ℏ1	PROPN
ejpam-6909	271	17	◦	◦	NOUN
ejpam-6909	271	18	(	(	PUNCT
ejpam-6909	271	19	ℏ2	ℏ2	NOUN
ejpam-6909	271	20	◦	◦	VERB
ejpam-6909	271	21	(	(	PUNCT
ejpam-6909	271	22	ℏ2	ℏ2	NOUN
ejpam-6909	271	23	◦	◦	VERB
ejpam-6909	271	24	ℏ1	ℏ1	ADJ
ejpam-6909	271	25	)	)	PUNCT
ejpam-6909	271	26	)	)	PUNCT
ejpam-6909	271	27	.	.	PUNCT
ejpam-6909	272	1	therefore	therefore	ADV
ejpam-6909	272	2	,	,	PUNCT
ejpam-6909	272	3	ℏ1	ℏ1	PROPN
ejpam-6909	272	4	◦	◦	NOUN
ejpam-6909	272	5	(	(	PUNCT
ejpam-6909	272	6	ℏ2	ℏ2	NOUN
ejpam-6909	272	7	◦	◦	VERB
ejpam-6909	272	8	(	(	PUNCT
ejpam-6909	272	9	ℏ2	ℏ2	NOUN
ejpam-6909	272	10	◦	◦	VERB
ejpam-6909	272	11	ℏ1	ℏ1	ADJ
ejpam-6909	272	12	)	)	PUNCT
ejpam-6909	272	13	)	)	PUNCT
ejpam-6909	273	1	⊆	⊆	NUM
ejpam-6909	273	2	u(α+	u(α+	NOUN
ejpam-6909	273	3	a	a	X
ejpam-6909	273	4	;	;	PUNCT
ejpam-6909	273	5	p	p	X
ejpam-6909	273	6	)	)	PUNCT
ejpam-6909	273	7	.	.	PUNCT
ejpam-6909	274	1	let	let	VERB
ejpam-6909	274	2	(	(	PUNCT
ejpam-6909	274	3	ℏ1	ℏ1	ADJ
ejpam-6909	274	4	◦	◦	NOUN
ejpam-6909	274	5	ℏ2	ℏ2	NOUN
ejpam-6909	274	6	)	)	PUNCT
ejpam-6909	274	7	◦	◦	NOUN
ejpam-6909	274	8	ℏ3	ℏ3	PROPN
ejpam-6909	274	9	⊆	⊆	NUM
ejpam-6909	274	10	l(β−	l(β−	PROPN
ejpam-6909	274	11	a	a	PRON
ejpam-6909	274	12	;	;	PUNCT
ejpam-6909	274	13	n	n	CCONJ
ejpam-6909	274	14	)	)	PUNCT
ejpam-6909	274	15	and	and	CCONJ
ejpam-6909	274	16	ℏ3	ℏ3	PROPN
ejpam-6909	274	17	∈	∈	PROPN
ejpam-6909	274	18	l(β−	l(β−	VERB
ejpam-6909	274	19	a	a	PRON
ejpam-6909	274	20	;	;	PUNCT
ejpam-6909	274	21	n	n	CCONJ
ejpam-6909	274	22	)	)	PUNCT
ejpam-6909	274	23	.	.	PUNCT
ejpam-6909	275	1	then	then	ADV
ejpam-6909	275	2	,	,	PUNCT
ejpam-6909	275	3	for	for	ADP
ejpam-6909	275	4	all	all	DET
ejpam-6909	275	5	b	b	PROPN
ejpam-6909	275	6	∈	∈	PROPN
ejpam-6909	275	7	(	(	PUNCT
ejpam-6909	275	8	ℏ1	ℏ1	PROPN
ejpam-6909	275	9	◦	◦	NOUN
ejpam-6909	275	10	ℏ2	ℏ2	NOUN
ejpam-6909	275	11	)	)	PUNCT
ejpam-6909	275	12	◦	◦	NOUN
ejpam-6909	275	13	ℏ3	ℏ3	PROPN
ejpam-6909	275	14	,	,	PUNCT
ejpam-6909	275	15	we	we	PRON
ejpam-6909	275	16	have	have	VERB
ejpam-6909	275	17	b	b	NOUN
ejpam-6909	275	18	∈	∈	PROPN
ejpam-6909	275	19	l(β−	l(β−	VERB
ejpam-6909	275	20	a	a	PRON
ejpam-6909	275	21	;	;	PUNCT
ejpam-6909	275	22	n	n	CCONJ
ejpam-6909	275	23	)	)	PUNCT
ejpam-6909	275	24	and	and	CCONJ
ejpam-6909	275	25	ℏ3	ℏ3	PROPN
ejpam-6909	275	26	∈	∈	PROPN
ejpam-6909	275	27	l(β−	l(β−	VERB
ejpam-6909	275	28	a	a	PRON
ejpam-6909	275	29	;	;	PUNCT
ejpam-6909	275	30	n	n	CCONJ
ejpam-6909	275	31	)	)	PUNCT
ejpam-6909	275	32	.	.	PUNCT
ejpam-6909	276	1	this	this	PRON
ejpam-6909	276	2	implies	imply	VERB
ejpam-6909	276	3	β−	β−	PRON
ejpam-6909	276	4	a	a	DET
ejpam-6909	276	5	(	(	PUNCT
ejpam-6909	276	6	b	b	NOUN
ejpam-6909	276	7	)	)	PUNCT
ejpam-6909	276	8	≤	≤	NOUN
ejpam-6909	276	9	n	n	CCONJ
ejpam-6909	276	10	,	,	PUNCT
ejpam-6909	276	11	for	for	ADP
ejpam-6909	276	12	all	all	DET
ejpam-6909	276	13	b	b	PROPN
ejpam-6909	276	14	∈	∈	PROPN
ejpam-6909	276	15	(	(	PUNCT
ejpam-6909	276	16	ℏ1	ℏ1	PROPN
ejpam-6909	276	17	◦	◦	NOUN
ejpam-6909	276	18	ℏ2	ℏ2	NOUN
ejpam-6909	276	19	)	)	PUNCT
ejpam-6909	276	20	◦	◦	NOUN
ejpam-6909	276	21	ℏ3	ℏ3	PROPN
ejpam-6909	276	22	and	and	CCONJ
ejpam-6909	276	23	β−	β−	PRON
ejpam-6909	276	24	a	a	DET
ejpam-6909	276	25	(	(	PUNCT
ejpam-6909	276	26	ℏ3	ℏ3	PROPN
ejpam-6909	276	27	)	)	PUNCT
ejpam-6909	276	28	≤	≤	PROPN
ejpam-6909	276	29	n.	n.	NOUN
ejpam-6909	276	30	consequently	consequently	ADV
ejpam-6909	276	31	,	,	PUNCT
ejpam-6909	276	32	supβ−	supβ−	PROPN
ejpam-6909	276	33	a	a	DET
ejpam-6909	276	34	(	(	PUNCT
ejpam-6909	276	35	b	b	NOUN
ejpam-6909	276	36	)	)	PUNCT
ejpam-6909	276	37	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	276	38	◦	◦	NOUN
ejpam-6909	276	39	ℏ2)	ℏ2)	NOUN
ejpam-6909	276	40	◦	◦	NOUN
ejpam-6909	276	41	ℏ3	ℏ3	NOUN
ejpam-6909	276	42	≤	≤	NOUN
ejpam-6909	276	43	n	n	ADV
ejpam-6909	276	44	and	and	CCONJ
ejpam-6909	276	45	β−	β−	PRON
ejpam-6909	276	46	a	a	DET
ejpam-6909	276	47	(	(	PUNCT
ejpam-6909	276	48	ℏ3	ℏ3	PROPN
ejpam-6909	276	49	)	)	PUNCT
ejpam-6909	276	50	≤	≤	PROPN
ejpam-6909	276	51	n.	n.	NOUN
ejpam-6909	276	52	thus	thus	ADV
ejpam-6909	276	53	,	,	PUNCT
ejpam-6909	276	54	for	for	ADP
ejpam-6909	276	55	all	all	DET
ejpam-6909	276	56	k	k	PROPN
ejpam-6909	276	57	∈	∈	PROPN
ejpam-6909	276	58	ℏ1	ℏ1	PROPN
ejpam-6909	276	59	◦	◦	NOUN
ejpam-6909	276	60	(	(	PUNCT
ejpam-6909	276	61	ℏ2	ℏ2	NOUN
ejpam-6909	276	62	◦	◦	VERB
ejpam-6909	276	63	(	(	PUNCT
ejpam-6909	276	64	ℏ2	ℏ2	NOUN
ejpam-6909	276	65	◦	◦	VERB
ejpam-6909	276	66	ℏ1	ℏ1	ADJ
ejpam-6909	276	67	)	)	PUNCT
ejpam-6909	276	68	)	)	PUNCT
ejpam-6909	276	69	,	,	PUNCT
ejpam-6909	276	70	we	we	PRON
ejpam-6909	276	71	have	have	VERB
ejpam-6909	276	72	β−	β−	PRON
ejpam-6909	277	1	a	a	DET
ejpam-6909	277	2	(	(	PUNCT
ejpam-6909	277	3	k	k	NOUN
ejpam-6909	277	4	)	)	PUNCT
ejpam-6909	277	5	≤	≤	NOUN
ejpam-6909	277	6	max	max	PROPN
ejpam-6909	277	7	{	{	PUNCT
ejpam-6909	277	8	supβ−	supβ−	PROPN
ejpam-6909	277	9	a	a	DET
ejpam-6909	277	10	(	(	PUNCT
ejpam-6909	277	11	b	b	NOUN
ejpam-6909	277	12	)	)	PUNCT
ejpam-6909	277	13	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	277	14	◦	◦	NOUN
ejpam-6909	277	15	ℏ2)	ℏ2)	NOUN
ejpam-6909	277	16	◦	◦	NOUN
ejpam-6909	277	17	ℏ3	ℏ3	NOUN
ejpam-6909	277	18	,	,	PUNCT
ejpam-6909	277	19	β−	β−	PRON
ejpam-6909	277	20	a	a	DET
ejpam-6909	277	21	(	(	PUNCT
ejpam-6909	277	22	ℏ3	ℏ3	PROPN
ejpam-6909	277	23	)	)	PUNCT
ejpam-6909	277	24	}	}	PUNCT
ejpam-6909	277	25	=	=	SYM
ejpam-6909	277	26	max{n	max{n	ADJ
ejpam-6909	277	27	,	,	PUNCT
ejpam-6909	277	28	n	n	CCONJ
ejpam-6909	277	29	}	}	PUNCT
ejpam-6909	277	30	=	=	VERB
ejpam-6909	277	31	n.	n.	NOUN
ejpam-6909	277	32	this	this	PRON
ejpam-6909	277	33	implies	imply	VERB
ejpam-6909	277	34	k	k	PROPN
ejpam-6909	277	35	∈	∈	PROPN
ejpam-6909	277	36	l(β−	l(β−	VERB
ejpam-6909	277	37	a	a	PRON
ejpam-6909	277	38	;	;	PUNCT
ejpam-6909	277	39	n	n	CCONJ
ejpam-6909	277	40	)	)	PUNCT
ejpam-6909	277	41	,	,	PUNCT
ejpam-6909	277	42	for	for	ADP
ejpam-6909	277	43	all	all	DET
ejpam-6909	277	44	t	t	NOUN
ejpam-6909	277	45	∈	∈	PROPN
ejpam-6909	277	46	ℏ1	ℏ1	PROPN
ejpam-6909	277	47	◦	◦	NOUN
ejpam-6909	277	48	(	(	PUNCT
ejpam-6909	277	49	ℏ2	ℏ2	NOUN
ejpam-6909	277	50	◦	◦	VERB
ejpam-6909	277	51	(	(	PUNCT
ejpam-6909	277	52	ℏ2	ℏ2	NOUN
ejpam-6909	277	53	◦	◦	VERB
ejpam-6909	277	54	ℏ1	ℏ1	ADJ
ejpam-6909	277	55	)	)	PUNCT
ejpam-6909	277	56	)	)	PUNCT
ejpam-6909	277	57	.	.	PUNCT
ejpam-6909	278	1	therefore	therefore	ADV
ejpam-6909	278	2	,	,	PUNCT
ejpam-6909	278	3	ℏ1	ℏ1	PROPN
ejpam-6909	278	4	◦	◦	NOUN
ejpam-6909	278	5	(ℏ2	(ℏ2	NOUN
ejpam-6909	278	6	◦	◦	ADJ
ejpam-6909	278	7	(ℏ2	(ℏ2	NOUN
ejpam-6909	278	8	◦	◦	NOUN
ejpam-6909	278	9	ℏ1	ℏ1	ADJ
ejpam-6909	278	10	)	)	PUNCT
ejpam-6909	278	11	)	)	PUNCT
ejpam-6909	279	1	⊆	⊆	NUM
ejpam-6909	279	2	l(βn	l(βn	PRON
ejpam-6909	279	3	a	a	PRON
ejpam-6909	279	4	;	;	PUNCT
ejpam-6909	279	5	n	n	CCONJ
ejpam-6909	279	6	)	)	PUNCT
ejpam-6909	279	7	.	.	PUNCT
ejpam-6909	280	1	thus	thus	ADV
ejpam-6909	280	2	,	,	PUNCT
ejpam-6909	280	3	for	for	ADP
ejpam-6909	280	4	all	all	DET
ejpam-6909	280	5	(	(	PUNCT
ejpam-6909	280	6	p	p	X
ejpam-6909	280	7	,	,	PUNCT
ejpam-6909	280	8	n	n	CCONJ
ejpam-6909	280	9	)	)	PUNCT
ejpam-6909	280	10	∈	∈	PROPN
ejpam-6909	281	1	[	[	X
ejpam-6909	281	2	0	0	NUM
ejpam-6909	281	3	,	,	PUNCT
ejpam-6909	281	4	1	1	NUM
ejpam-6909	281	5	]	]	SYM
ejpam-6909	281	6	×	×	NOUN
ejpam-6909	282	1	[	[	X
ejpam-6909	282	2	−1	−1	NOUN
ejpam-6909	282	3	,	,	PUNCT
ejpam-6909	282	4	0	0	NUM
ejpam-6909	282	5	]	]	PUNCT
ejpam-6909	282	6	,	,	PUNCT
ejpam-6909	282	7	the	the	DET
ejpam-6909	282	8	cut	cut	NOUN
ejpam-6909	282	9	sets	set	VERB
ejpam-6909	282	10	u(α−	u(α−	NOUN
ejpam-6909	282	11	a	a	NOUN
ejpam-6909	282	12	;	;	PUNCT
ejpam-6909	282	13	p	p	X
ejpam-6909	282	14	)	)	PUNCT
ejpam-6909	282	15	and	and	CCONJ
ejpam-6909	282	16	(	(	PUNCT
ejpam-6909	282	17	β−	β−	INTJ
ejpam-6909	282	18	a	a	PRON
ejpam-6909	282	19	;	;	PUNCT
ejpam-6909	282	20	n	n	CCONJ
ejpam-6909	282	21	)	)	PUNCT
ejpam-6909	282	22	are	be	AUX
ejpam-6909	282	23	chbckis	chbcki	NOUN
ejpam-6909	282	24	of	of	ADP
ejpam-6909	282	25	type-1	type-1	NUM
ejpam-6909	282	26	of	of	ADP
ejpam-6909	282	27	h.	h.	NOUN
ejpam-6909	282	28	conversely	conversely	ADV
ejpam-6909	282	29	,	,	PUNCT
ejpam-6909	282	30	let	let	VERB
ejpam-6909	282	31	us	we	PRON
ejpam-6909	282	32	assume	assume	VERB
ejpam-6909	282	33	that	that	SCONJ
ejpam-6909	282	34	for	for	ADP
ejpam-6909	282	35	all	all	DET
ejpam-6909	282	36	(	(	PUNCT
ejpam-6909	282	37	p	p	X
ejpam-6909	282	38	,	,	PUNCT
ejpam-6909	282	39	n	n	CCONJ
ejpam-6909	282	40	)	)	PUNCT
ejpam-6909	282	41	∈	∈	PROPN
ejpam-6909	283	1	[	[	X
ejpam-6909	283	2	0	0	NUM
ejpam-6909	283	3	,	,	PUNCT
ejpam-6909	283	4	1	1	NUM
ejpam-6909	283	5	]	]	SYM
ejpam-6909	283	6	×	×	NOUN
ejpam-6909	284	1	[	[	X
ejpam-6909	284	2	−1	−1	NOUN
ejpam-6909	284	3	,	,	PUNCT
ejpam-6909	284	4	0	0	NUM
ejpam-6909	284	5	]	]	PUNCT
ejpam-6909	284	6	,	,	PUNCT
ejpam-6909	284	7	the	the	DET
ejpam-6909	284	8	cut	cut	NOUN
ejpam-6909	284	9	sets	set	VERB
ejpam-6909	284	10	u(α+	u(α+	PRON
ejpam-6909	284	11	a	a	X
ejpam-6909	284	12	;	;	PUNCT
ejpam-6909	284	13	p	p	X
ejpam-6909	284	14	)	)	PUNCT
ejpam-6909	284	15	and	and	CCONJ
ejpam-6909	284	16	l(β−	l(β−	VERB
ejpam-6909	284	17	a	a	PRON
ejpam-6909	284	18	;	;	PUNCT
ejpam-6909	284	19	n	n	CCONJ
ejpam-6909	284	20	)	)	PUNCT
ejpam-6909	284	21	are	be	AUX
ejpam-6909	284	22	chbckis	chbcki	NOUN
ejpam-6909	284	23	of	of	ADP
ejpam-6909	284	24	type-1	type-1	PROPN
ejpam-6909	284	25	of	of	ADP
ejpam-6909	284	26	h.	h.	PROPN
ejpam-6909	284	27	let	let	VERB
ejpam-6909	284	28	ℏ1	ℏ1	ADJ
ejpam-6909	284	29	,	,	PUNCT
ejpam-6909	284	30	ℏ2	ℏ2	NOUN
ejpam-6909	284	31	,	,	PUNCT
ejpam-6909	284	32	ℏ3	ℏ3	PROPN
ejpam-6909	284	33	be	be	VERB
ejpam-6909	284	34	elements	element	NOUN
ejpam-6909	284	35	of	of	ADP
ejpam-6909	284	36	h.	h.	NOUN
ejpam-6909	284	37	we	we	PRON
ejpam-6909	284	38	define	define	VERB
ejpam-6909	284	39	p	p	NOUN
ejpam-6909	284	40	=	=	NOUN
ejpam-6909	284	41	min	min	PROPN
ejpam-6909	284	42	{	{	PUNCT
ejpam-6909	284	43	inf	inf	PROPN
ejpam-6909	284	44	α+	α+	PRON
ejpam-6909	284	45	a(a	a(a	PROPN
ejpam-6909	284	46	)	)	PUNCT
ejpam-6909	285	1	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	285	2	◦	◦	NOUN
ejpam-6909	285	3	ℏ2)	ℏ2)	NOUN
ejpam-6909	285	4	◦	◦	NOUN
ejpam-6909	285	5	ℏ3	ℏ3	PROPN
ejpam-6909	285	6	,	,	PUNCT
ejpam-6909	285	7	α+	α+	DET
ejpam-6909	285	8	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	285	9	)	)	PUNCT
ejpam-6909	285	10	}	}	PUNCT
ejpam-6909	285	11	.	.	PUNCT
ejpam-6909	286	1	then	then	ADV
ejpam-6909	286	2	inf	inf	PROPN
ejpam-6909	286	3	α+	α+	PRON
ejpam-6909	286	4	a(a	a(a	PROPN
ejpam-6909	286	5	)	)	PUNCT
ejpam-6909	287	1	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	287	2	◦	◦	NOUN
ejpam-6909	287	3	ℏ2)	ℏ2)	NOUN
ejpam-6909	287	4	◦	◦	NOUN
ejpam-6909	287	5	ℏ3	ℏ3	PROPN
ejpam-6909	287	6	≥	≥	NUM
ejpam-6909	287	7	p	p	NOUN
ejpam-6909	287	8	and	and	CCONJ
ejpam-6909	287	9	α+	α+	PUNCT
ejpam-6909	287	10	a(ℏ3	a(ℏ3	ADJ
ejpam-6909	287	11	)	)	PUNCT
ejpam-6909	287	12	≥	≥	NOUN
ejpam-6909	288	1	p.	p.	NOUN
ejpam-6909	288	2	⇒	⇒	PROPN
ejpam-6909	288	3	α+	α+	PRON
ejpam-6909	288	4	a(a	a(a	PROPN
ejpam-6909	288	5	)	)	PUNCT
ejpam-6909	288	6	≥	≥	NOUN
ejpam-6909	288	7	p	p	NOUN
ejpam-6909	288	8	for	for	ADP
ejpam-6909	288	9	all	all	DET
ejpam-6909	288	10	a	a	DET
ejpam-6909	288	11	∈	∈	NOUN
ejpam-6909	288	12	(	(	PUNCT
ejpam-6909	288	13	ℏ1	ℏ1	PROPN
ejpam-6909	288	14	◦	◦	NOUN
ejpam-6909	288	15	ℏ2	ℏ2	NOUN
ejpam-6909	288	16	)	)	PUNCT
ejpam-6909	288	17	◦	◦	NOUN
ejpam-6909	288	18	ℏ3	ℏ3	PROPN
ejpam-6909	288	19	and	and	CCONJ
ejpam-6909	288	20	α+	α+	PRON
ejpam-6909	288	21	a(ℏ3	a(ℏ3	ADJ
ejpam-6909	288	22	)	)	PUNCT
ejpam-6909	288	23	≥	≥	NOUN
ejpam-6909	288	24	p	p	NOUN
ejpam-6909	288	25	⇒	⇒	VERB
ejpam-6909	288	26	a	a	DET
ejpam-6909	288	27	∈	∈	PROPN
ejpam-6909	288	28	u(α+	u(α+	NOUN
ejpam-6909	288	29	a	a	X
ejpam-6909	288	30	;	;	PUNCT
ejpam-6909	288	31	p	p	X
ejpam-6909	288	32	)	)	PUNCT
ejpam-6909	288	33	for	for	ADP
ejpam-6909	288	34	all	all	DET
ejpam-6909	288	35	a	a	DET
ejpam-6909	288	36	∈	∈	PROPN
ejpam-6909	288	37	(	(	PUNCT
ejpam-6909	288	38	ℏ1	ℏ1	PROPN
ejpam-6909	288	39	◦	◦	NOUN
ejpam-6909	288	40	ℏ2	ℏ2	NOUN
ejpam-6909	288	41	)	)	PUNCT
ejpam-6909	288	42	◦	◦	NOUN
ejpam-6909	288	43	ℏ3	ℏ3	PROPN
ejpam-6909	288	44	and	and	CCONJ
ejpam-6909	288	45	ℏ3	ℏ3	PROPN
ejpam-6909	288	46	∈	∈	PROPN
ejpam-6909	289	1	u(α+	u(α+	NOUN
ejpam-6909	289	2	a	a	X
ejpam-6909	289	3	;	;	PUNCT
ejpam-6909	289	4	p	p	X
ejpam-6909	289	5	)	)	PUNCT
ejpam-6909	289	6	⇒	⇒	NOUN
ejpam-6909	289	7	(	(	PUNCT
ejpam-6909	289	8	ℏ1	ℏ1	PROPN
ejpam-6909	289	9	◦	◦	NOUN
ejpam-6909	289	10	ℏ2	ℏ2	NOUN
ejpam-6909	289	11	)	)	PUNCT
ejpam-6909	289	12	◦	◦	NOUN
ejpam-6909	289	13	ℏ3	ℏ3	PROPN
ejpam-6909	289	14	⊆	⊆	NUM
ejpam-6909	289	15	u(α+	u(α+	NOUN
ejpam-6909	289	16	a	a	X
ejpam-6909	289	17	;	;	PUNCT
ejpam-6909	289	18	p	p	X
ejpam-6909	289	19	)	)	PUNCT
ejpam-6909	289	20	and	and	CCONJ
ejpam-6909	289	21	ℏ3	ℏ3	PROPN
ejpam-6909	289	22	∈	∈	PROPN
ejpam-6909	290	1	u(α+	u(α+	NOUN
ejpam-6909	290	2	a	a	X
ejpam-6909	290	3	;	;	PUNCT
ejpam-6909	290	4	p	p	X
ejpam-6909	290	5	)	)	PUNCT
ejpam-6909	290	6	.	.	PUNCT
ejpam-6909	291	1	d.	d.	PROPN
ejpam-6909	291	2	ramesh	ramesh	PROPN
ejpam-6909	291	3	et	et	PROPN
ejpam-6909	291	4	al	al	PROPN
ejpam-6909	291	5	.	.	PUNCT
ejpam-6909	291	6	/	/	SYM
ejpam-6909	291	7	eur	eur	PROPN
ejpam-6909	291	8	.	.	PUNCT
ejpam-6909	292	1	j.	j.	PROPN
ejpam-6909	292	2	pure	pure	PROPN
ejpam-6909	292	3	appl	appl	PROPN
ejpam-6909	292	4	.	.	PROPN
ejpam-6909	292	5	math	math	PROPN
ejpam-6909	292	6	,	,	PUNCT
ejpam-6909	292	7	18	18	NUM
ejpam-6909	292	8	(	(	PUNCT
ejpam-6909	292	9	4	4	NUM
ejpam-6909	292	10	)	)	PUNCT
ejpam-6909	292	11	(	(	PUNCT
ejpam-6909	292	12	2025	2025	NUM
ejpam-6909	292	13	)	)	PUNCT
ejpam-6909	292	14	,	,	PUNCT
ejpam-6909	292	15	6909	6909	NUM
ejpam-6909	292	16	13	13	NUM
ejpam-6909	292	17	of	of	ADP
ejpam-6909	292	18	16	16	NUM
ejpam-6909	292	19	by	by	ADP
ejpam-6909	292	20	hypothesis	hypothesis	NOUN
ejpam-6909	292	21	,	,	PUNCT
ejpam-6909	292	22	we	we	PRON
ejpam-6909	292	23	have	have	AUX
ejpam-6909	292	24	ℏ1	ℏ1	ADJ
ejpam-6909	292	25	◦	◦	NOUN
ejpam-6909	292	26	(ℏ2	(ℏ2	NOUN
ejpam-6909	292	27	◦	◦	ADJ
ejpam-6909	292	28	(ℏ2	(ℏ2	NOUN
ejpam-6909	292	29	◦	◦	NOUN
ejpam-6909	292	30	ℏ1	ℏ1	ADJ
ejpam-6909	292	31	)	)	PUNCT
ejpam-6909	292	32	)	)	PUNCT
ejpam-6909	293	1	⊆	⊆	NUM
ejpam-6909	293	2	u(α+	u(α+	NOUN
ejpam-6909	293	3	a	a	X
ejpam-6909	293	4	;	;	PUNCT
ejpam-6909	293	5	p	p	X
ejpam-6909	293	6	)	)	PUNCT
ejpam-6909	293	7	.	.	PUNCT
ejpam-6909	294	1	thus	thus	ADV
ejpam-6909	294	2	,	,	PUNCT
ejpam-6909	294	3	for	for	ADP
ejpam-6909	294	4	all	all	DET
ejpam-6909	294	5	k1	k1	NOUN
ejpam-6909	294	6	∈	∈	PROPN
ejpam-6909	294	7	(	(	PUNCT
ejpam-6909	294	8	ℏ1	ℏ1	PROPN
ejpam-6909	294	9	◦	◦	NOUN
ejpam-6909	294	10	(ℏ2	(ℏ2	NOUN
ejpam-6909	294	11	◦	◦	ADJ
ejpam-6909	294	12	(ℏ2	(ℏ2	NOUN
ejpam-6909	294	13	◦	◦	NOUN
ejpam-6909	294	14	ℏ1	ℏ1	ADJ
ejpam-6909	294	15	)	)	PUNCT
ejpam-6909	294	16	)	)	PUNCT
ejpam-6909	294	17	,	,	PUNCT
ejpam-6909	294	18	α+	α+	PRON
ejpam-6909	294	19	a(k1	a(k1	NOUN
ejpam-6909	294	20	)	)	PUNCT
ejpam-6909	294	21	≥	≥	NOUN
ejpam-6909	294	22	p	p	NOUN
ejpam-6909	294	23	=	=	SYM
ejpam-6909	294	24	min	min	PROPN
ejpam-6909	294	25	{	{	PUNCT
ejpam-6909	294	26	inf	inf	PROPN
ejpam-6909	294	27	α+	α+	PRON
ejpam-6909	294	28	a(a	a(a	PROPN
ejpam-6909	294	29	)	)	PUNCT
ejpam-6909	295	1	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	295	2	◦	◦	NOUN
ejpam-6909	295	3	ℏ2)	ℏ2)	NOUN
ejpam-6909	295	4	◦	◦	NOUN
ejpam-6909	295	5	ℏ3	ℏ3	PROPN
ejpam-6909	295	6	,	,	PUNCT
ejpam-6909	295	7	α+	α+	DET
ejpam-6909	295	8	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	295	9	)	)	PUNCT
ejpam-6909	295	10	}	}	PUNCT
ejpam-6909	295	11	.	.	PUNCT
ejpam-6909	296	1	define	define	VERB
ejpam-6909	296	2	n	n	NOUN
ejpam-6909	296	3	=	=	SYM
ejpam-6909	296	4	max	max	PROPN
ejpam-6909	296	5	{	{	PUNCT
ejpam-6909	296	6	supβ−	supβ−	X
ejpam-6909	296	7	a	a	DET
ejpam-6909	296	8	(	(	PUNCT
ejpam-6909	296	9	b	b	NOUN
ejpam-6909	296	10	)	)	PUNCT
ejpam-6909	296	11	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	296	12	◦	◦	NOUN
ejpam-6909	296	13	ℏ2)	ℏ2)	NOUN
ejpam-6909	296	14	◦	◦	NOUN
ejpam-6909	296	15	ℏ3	ℏ3	NOUN
ejpam-6909	296	16	,	,	PUNCT
ejpam-6909	296	17	β−	β−	PRON
ejpam-6909	296	18	a	a	DET
ejpam-6909	296	19	(	(	PUNCT
ejpam-6909	296	20	ℏ3	ℏ3	PROPN
ejpam-6909	296	21	)	)	PUNCT
ejpam-6909	296	22	}	}	PUNCT
ejpam-6909	296	23	.	.	PUNCT
ejpam-6909	297	1	then	then	ADV
ejpam-6909	297	2	supβ−	supβ−	VERB
ejpam-6909	297	3	a	a	DET
ejpam-6909	297	4	(	(	PUNCT
ejpam-6909	297	5	b	b	NOUN
ejpam-6909	297	6	)	)	PUNCT
ejpam-6909	297	7	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	297	8	◦	◦	NOUN
ejpam-6909	297	9	ℏ2)	ℏ2)	NOUN
ejpam-6909	297	10	◦	◦	NOUN
ejpam-6909	297	11	ℏ3	ℏ3	NOUN
ejpam-6909	297	12	≤	≤	NOUN
ejpam-6909	297	13	n	n	ADV
ejpam-6909	297	14	and	and	CCONJ
ejpam-6909	297	15	β−	β−	PRON
ejpam-6909	297	16	a	a	DET
ejpam-6909	297	17	(	(	PUNCT
ejpam-6909	297	18	ℏ3	ℏ3	PROPN
ejpam-6909	297	19	)	)	PUNCT
ejpam-6909	297	20	≤	≤	PROPN
ejpam-6909	297	21	n.	n.	NOUN
ejpam-6909	297	22	⇒	⇒	PROPN
ejpam-6909	297	23	β−	β−	PUNCT
ejpam-6909	298	1	a	a	DET
ejpam-6909	298	2	(	(	PUNCT
ejpam-6909	298	3	b	b	NOUN
ejpam-6909	298	4	)	)	PUNCT
ejpam-6909	298	5	≤	≤	NOUN
ejpam-6909	298	6	n	n	CCONJ
ejpam-6909	298	7	for	for	ADP
ejpam-6909	298	8	all	all	DET
ejpam-6909	298	9	b	b	PROPN
ejpam-6909	298	10	∈	∈	PROPN
ejpam-6909	298	11	(	(	PUNCT
ejpam-6909	298	12	ℏ1	ℏ1	PROPN
ejpam-6909	298	13	◦	◦	NOUN
ejpam-6909	298	14	ℏ2	ℏ2	NOUN
ejpam-6909	298	15	)	)	PUNCT
ejpam-6909	298	16	◦	◦	NOUN
ejpam-6909	298	17	ℏ3	ℏ3	PROPN
ejpam-6909	298	18	and	and	CCONJ
ejpam-6909	298	19	β−	β−	PRON
ejpam-6909	298	20	a	a	DET
ejpam-6909	298	21	(	(	PUNCT
ejpam-6909	298	22	ℏ3	ℏ3	PROPN
ejpam-6909	298	23	)	)	PUNCT
ejpam-6909	298	24	≤	≤	NOUN
ejpam-6909	298	25	n	n	PRON
ejpam-6909	298	26	⇒	⇒	NOUN
ejpam-6909	298	27	b	b	X
ejpam-6909	298	28	∈	∈	PROPN
ejpam-6909	298	29	l(β−	l(β−	VERB
ejpam-6909	298	30	a	a	PRON
ejpam-6909	298	31	;	;	PUNCT
ejpam-6909	298	32	n	n	CCONJ
ejpam-6909	298	33	)	)	PUNCT
ejpam-6909	298	34	for	for	ADP
ejpam-6909	298	35	all	all	DET
ejpam-6909	298	36	b	b	PROPN
ejpam-6909	298	37	∈	∈	PROPN
ejpam-6909	298	38	(	(	PUNCT
ejpam-6909	298	39	ℏ1	ℏ1	PROPN
ejpam-6909	298	40	◦	◦	NOUN
ejpam-6909	298	41	ℏ2	ℏ2	NOUN
ejpam-6909	298	42	)	)	PUNCT
ejpam-6909	298	43	◦	◦	NOUN
ejpam-6909	298	44	ℏ3	ℏ3	PROPN
ejpam-6909	298	45	and	and	CCONJ
ejpam-6909	298	46	ℏ3	ℏ3	PROPN
ejpam-6909	298	47	∈	∈	PROPN
ejpam-6909	298	48	l(β−	l(β−	VERB
ejpam-6909	298	49	a	a	PRON
ejpam-6909	298	50	;	;	PUNCT
ejpam-6909	298	51	n	n	CCONJ
ejpam-6909	298	52	)	)	PUNCT
ejpam-6909	298	53	⇒	⇒	NOUN
ejpam-6909	298	54	(	(	PUNCT
ejpam-6909	298	55	ℏ1	ℏ1	PROPN
ejpam-6909	298	56	◦	◦	NOUN
ejpam-6909	298	57	ℏ2	ℏ2	NOUN
ejpam-6909	298	58	)	)	PUNCT
ejpam-6909	298	59	◦	◦	NOUN
ejpam-6909	298	60	ℏ3	ℏ3	PROPN
ejpam-6909	298	61	⊆	⊆	NUM
ejpam-6909	298	62	l(β−	l(β−	PROPN
ejpam-6909	298	63	a	a	PRON
ejpam-6909	298	64	;	;	PUNCT
ejpam-6909	298	65	n	n	CCONJ
ejpam-6909	298	66	)	)	PUNCT
ejpam-6909	298	67	and	and	CCONJ
ejpam-6909	298	68	ℏ3	ℏ3	PROPN
ejpam-6909	298	69	∈	∈	PROPN
ejpam-6909	298	70	l(β−	l(β−	VERB
ejpam-6909	298	71	a	a	PRON
ejpam-6909	298	72	;	;	PUNCT
ejpam-6909	298	73	n	n	CCONJ
ejpam-6909	298	74	)	)	PUNCT
ejpam-6909	298	75	.	.	PUNCT
ejpam-6909	299	1	by	by	ADP
ejpam-6909	299	2	hypothesis	hypothesis	NOUN
ejpam-6909	299	3	,	,	PUNCT
ejpam-6909	299	4	we	we	PRON
ejpam-6909	299	5	have	have	VERB
ejpam-6909	299	6	ℏ1	ℏ1	ADJ
ejpam-6909	299	7	◦	◦	NOUN
ejpam-6909	299	8	(ℏ2	(ℏ2	NOUN
ejpam-6909	299	9	◦	◦	ADJ
ejpam-6909	299	10	(ℏ2	(ℏ2	NOUN
ejpam-6909	299	11	◦	◦	NOUN
ejpam-6909	299	12	ℏ1	ℏ1	ADJ
ejpam-6909	299	13	)	)	PUNCT
ejpam-6909	299	14	)	)	PUNCT
ejpam-6909	300	1	⊆	⊆	NUM
ejpam-6909	300	2	l(β−	l(β−	NOUN
ejpam-6909	300	3	a	a	PRON
ejpam-6909	300	4	;	;	PUNCT
ejpam-6909	300	5	n	n	CCONJ
ejpam-6909	300	6	)	)	PUNCT
ejpam-6909	300	7	.	.	PUNCT
ejpam-6909	301	1	thus	thus	ADV
ejpam-6909	301	2	,	,	PUNCT
ejpam-6909	301	3	for	for	ADP
ejpam-6909	301	4	all	all	DET
ejpam-6909	301	5	k2	k2	PROPN
ejpam-6909	301	6	∈	∈	PROPN
ejpam-6909	301	7	(	(	PUNCT
ejpam-6909	301	8	ℏ1	ℏ1	PROPN
ejpam-6909	301	9	◦	◦	NOUN
ejpam-6909	301	10	(ℏ2	(ℏ2	NOUN
ejpam-6909	301	11	◦	◦	ADJ
ejpam-6909	301	12	(ℏ2	(ℏ2	NOUN
ejpam-6909	301	13	◦	◦	NOUN
ejpam-6909	301	14	ℏ1	ℏ1	ADJ
ejpam-6909	301	15	)	)	PUNCT
ejpam-6909	301	16	)	)	PUNCT
ejpam-6909	301	17	,	,	PUNCT
ejpam-6909	301	18	β−	β−	PRON
ejpam-6909	301	19	a	a	DET
ejpam-6909	301	20	(	(	PUNCT
ejpam-6909	301	21	k2	k2	ADJ
ejpam-6909	301	22	)	)	PUNCT
ejpam-6909	301	23	≤	≤	NOUN
ejpam-6909	301	24	n	n	NOUN
ejpam-6909	301	25	=	=	SYM
ejpam-6909	301	26	max	max	PROPN
ejpam-6909	301	27	{	{	PUNCT
ejpam-6909	301	28	supβ−	supβ−	X
ejpam-6909	301	29	a	a	DET
ejpam-6909	301	30	(	(	PUNCT
ejpam-6909	301	31	b	b	NOUN
ejpam-6909	301	32	)	)	PUNCT
ejpam-6909	301	33	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	301	34	◦	◦	NOUN
ejpam-6909	301	35	ℏ2)	ℏ2)	NOUN
ejpam-6909	301	36	◦	◦	NOUN
ejpam-6909	301	37	ℏ3	ℏ3	NOUN
ejpam-6909	301	38	,	,	PUNCT
ejpam-6909	301	39	β−	β−	PRON
ejpam-6909	301	40	a	a	DET
ejpam-6909	301	41	(	(	PUNCT
ejpam-6909	301	42	ℏ3	ℏ3	PROPN
ejpam-6909	301	43	)	)	PUNCT
ejpam-6909	301	44	}	}	PUNCT
ejpam-6909	301	45	.	.	PUNCT
ejpam-6909	302	1	consider	consider	VERB
ejpam-6909	302	2	the	the	DET
ejpam-6909	302	3	case	case	NOUN
ejpam-6909	302	4	where	where	SCONJ
ejpam-6909	302	5	α+	α+	X
ejpam-6909	302	6	a(ℏ1	a(ℏ1	X
ejpam-6909	302	7	)	)	PUNCT
ejpam-6909	303	1	=	=	SYM
ejpam-6909	303	2	p	p	NOUN
ejpam-6909	303	3	and	and	CCONJ
ejpam-6909	303	4	β−	β−	PRON
ejpam-6909	303	5	a	a	DET
ejpam-6909	303	6	(	(	PUNCT
ejpam-6909	303	7	ℏ2	ℏ2	NOUN
ejpam-6909	303	8	)	)	PUNCT
ejpam-6909	303	9	=	=	SYM
ejpam-6909	303	10	n	n	CCONJ
ejpam-6909	303	11	for	for	ADP
ejpam-6909	303	12	some	some	DET
ejpam-6909	303	13	ℏ1	ℏ1	NOUN
ejpam-6909	303	14	,	,	PUNCT
ejpam-6909	303	15	ℏ2	ℏ2	NOUN
ejpam-6909	303	16	∈	∈	PROPN
ejpam-6909	303	17	h.	h.	PROPN
ejpam-6909	303	18	since	since	SCONJ
ejpam-6909	303	19	0	0	NUM
ejpam-6909	303	20	∈	∈	PROPN
ejpam-6909	303	21	u(α+	u(α+	NOUN
ejpam-6909	303	22	a	a	X
ejpam-6909	303	23	;	;	PUNCT
ejpam-6909	303	24	p	p	X
ejpam-6909	303	25	)	)	PUNCT
ejpam-6909	303	26	∩	∩	NOUN
ejpam-6909	303	27	l(β−	l(β−	VERB
ejpam-6909	303	28	a	a	PRON
ejpam-6909	303	29	;	;	PUNCT
ejpam-6909	303	30	n	n	CCONJ
ejpam-6909	303	31	)	)	PUNCT
ejpam-6909	303	32	,	,	PUNCT
ejpam-6909	303	33	we	we	PRON
ejpam-6909	303	34	obtain	obtain	VERB
ejpam-6909	303	35	α+	α+	DET
ejpam-6909	303	36	a(0	a(0	PROPN
ejpam-6909	303	37	)	)	PUNCT
ejpam-6909	303	38	≥	≥	NOUN
ejpam-6909	304	1	p	p	NOUN
ejpam-6909	304	2	=	=	X
ejpam-6909	304	3	α+	α+	PUNCT
ejpam-6909	304	4	a(ℏ1	a(ℏ1	NOUN
ejpam-6909	304	5	)	)	PUNCT
ejpam-6909	304	6	and	and	CCONJ
ejpam-6909	304	7	β−	β−	PRON
ejpam-6909	304	8	a	a	DET
ejpam-6909	304	9	(	(	PUNCT
ejpam-6909	304	10	0	0	NUM
ejpam-6909	304	11	)	)	PUNCT
ejpam-6909	304	12	≤	≤	NOUN
ejpam-6909	305	1	n	n	NOUN
ejpam-6909	305	2	=	=	SYM
ejpam-6909	305	3	β−	β−	PROPN
ejpam-6909	305	4	a	a	DET
ejpam-6909	305	5	(	(	PUNCT
ejpam-6909	305	6	ℏ2	ℏ2	NOUN
ejpam-6909	305	7	)	)	PUNCT
ejpam-6909	305	8	,	,	PUNCT
ejpam-6909	305	9	for	for	ADP
ejpam-6909	305	10	all	all	DET
ejpam-6909	305	11	ℏ1	ℏ1	NOUN
ejpam-6909	305	12	,	,	PUNCT
ejpam-6909	305	13	ℏ2	ℏ2	NOUN
ejpam-6909	305	14	∈	∈	PROPN
ejpam-6909	305	15	h.	h.	PROPN
ejpam-6909	305	16	therefore	therefore	ADV
ejpam-6909	305	17	,	,	PUNCT
ejpam-6909	305	18	(	(	PUNCT
ejpam-6909	305	19	α+	α+	X
ejpam-6909	305	20	a	a	X
ejpam-6909	305	21	,	,	PUNCT
ejpam-6909	305	22	β	β	PROPN
ejpam-6909	305	23	−	−	NOUN
ejpam-6909	305	24	a	a	PRON
ejpam-6909	305	25	)	)	PUNCT
ejpam-6909	305	26	is	be	AUX
ejpam-6909	305	27	a	a	DET
ejpam-6909	305	28	bf	bf	NOUN
ejpam-6909	305	29	-	-	PUNCT
ejpam-6909	305	30	chbcki	chbcki	NOUN
ejpam-6909	305	31	of	of	ADP
ejpam-6909	305	32	type-1	type-1	PROPN
ejpam-6909	305	33	of	of	ADP
ejpam-6909	305	34	h.	h.	PROPN
ejpam-6909	305	35	(	(	PUNCT
ejpam-6909	305	36	ii	ii	NOUN
ejpam-6909	305	37	)	)	PUNCT
ejpam-6909	305	38	assume	assume	VERB
ejpam-6909	305	39	(	(	PUNCT
ejpam-6909	305	40	α+	α+	X
ejpam-6909	305	41	a	a	X
ejpam-6909	305	42	,	,	PUNCT
ejpam-6909	305	43	β	β	PROPN
ejpam-6909	305	44	−	−	NOUN
ejpam-6909	305	45	a	a	PRON
ejpam-6909	305	46	)	)	PUNCT
ejpam-6909	305	47	is	be	AUX
ejpam-6909	305	48	a	a	DET
ejpam-6909	305	49	bf	bf	NOUN
ejpam-6909	305	50	-	-	PUNCT
ejpam-6909	305	51	chbcki	chbcki	NOUN
ejpam-6909	305	52	of	of	ADP
ejpam-6909	305	53	type-3	type-3	NUM
ejpam-6909	305	54	of	of	ADP
ejpam-6909	305	55	h	h	NOUN
ejpam-6909	305	56	,	,	PUNCT
ejpam-6909	305	57	and	and	CCONJ
ejpam-6909	305	58	for	for	ADP
ejpam-6909	305	59	all	all	DET
ejpam-6909	305	60	(	(	PUNCT
ejpam-6909	305	61	p	p	X
ejpam-6909	305	62	,	,	PUNCT
ejpam-6909	305	63	n	n	CCONJ
ejpam-6909	305	64	)	)	PUNCT
ejpam-6909	305	65	∈	∈	PROPN
ejpam-6909	306	1	[	[	X
ejpam-6909	306	2	0	0	NUM
ejpam-6909	306	3	,	,	PUNCT
ejpam-6909	306	4	1	1	NUM
ejpam-6909	306	5	]	]	SYM
ejpam-6909	306	6	×	×	NOUN
ejpam-6909	307	1	[	[	X
ejpam-6909	307	2	−1	−1	NOUN
ejpam-6909	307	3	,	,	PUNCT
ejpam-6909	307	4	0	0	NUM
ejpam-6909	307	5	]	]	PUNCT
ejpam-6909	307	6	,	,	PUNCT
ejpam-6909	307	7	the	the	DET
ejpam-6909	307	8	cut	cut	NOUN
ejpam-6909	307	9	sets	set	VERB
ejpam-6909	307	10	u(α+	u(α+	PRON
ejpam-6909	307	11	a	a	X
ejpam-6909	307	12	;	;	PUNCT
ejpam-6909	307	13	p	p	X
ejpam-6909	307	14	)	)	PUNCT
ejpam-6909	307	15	and	and	CCONJ
ejpam-6909	307	16	l(β−	l(β−	VERB
ejpam-6909	307	17	a	a	PRON
ejpam-6909	307	18	;	;	PUNCT
ejpam-6909	307	19	n	n	CCONJ
ejpam-6909	307	20	)	)	PUNCT
ejpam-6909	307	21	are	be	AUX
ejpam-6909	307	22	non	non	ADJ
ejpam-6909	307	23	-	-	ADJ
ejpam-6909	307	24	empty	empty	ADJ
ejpam-6909	307	25	.	.	PUNCT
ejpam-6909	308	1	by	by	ADP
ejpam-6909	308	2	definition	definition	NOUN
ejpam-6909	308	3	,	,	PUNCT
ejpam-6909	308	4	we	we	PRON
ejpam-6909	308	5	have	have	VERB
ejpam-6909	308	6	0	0	NUM
ejpam-6909	308	7	∈	∈	PROPN
ejpam-6909	308	8	u(α+	u(α+	NOUN
ejpam-6909	308	9	a	a	X
ejpam-6909	308	10	;	;	PUNCT
ejpam-6909	308	11	p	p	X
ejpam-6909	308	12	)	)	PUNCT
ejpam-6909	308	13	∩	∩	NOUN
ejpam-6909	308	14	l(β−	l(β−	VERB
ejpam-6909	308	15	a	a	PRON
ejpam-6909	308	16	;	;	PUNCT
ejpam-6909	308	17	n	n	CCONJ
ejpam-6909	308	18	)	)	PUNCT
ejpam-6909	308	19	.	.	PUNCT
ejpam-6909	309	1	let	let	VERB
ejpam-6909	309	2	ℏ1	ℏ1	ADJ
ejpam-6909	309	3	,	,	PUNCT
ejpam-6909	309	4	ℏ2	ℏ2	NOUN
ejpam-6909	309	5	,	,	PUNCT
ejpam-6909	309	6	ℏ3	ℏ3	PROPN
ejpam-6909	309	7	be	be	VERB
ejpam-6909	309	8	elements	element	NOUN
ejpam-6909	309	9	of	of	ADP
ejpam-6909	309	10	h	h	NOUN
ejpam-6909	309	11	such	such	ADJ
ejpam-6909	309	12	that	that	SCONJ
ejpam-6909	309	13	(	(	PUNCT
ejpam-6909	309	14	ℏ1	ℏ1	ADJ
ejpam-6909	309	15	◦	◦	NOUN
ejpam-6909	309	16	ℏ2	ℏ2	NOUN
ejpam-6909	309	17	)	)	PUNCT
ejpam-6909	309	18	◦	◦	NOUN
ejpam-6909	309	19	ℏ3	ℏ3	PROPN
ejpam-6909	309	20	≪	≪	VERB
ejpam-6909	309	21	u(α+	u(α+	NOUN
ejpam-6909	309	22	a	a	X
ejpam-6909	309	23	;	;	PUNCT
ejpam-6909	309	24	p	p	X
ejpam-6909	309	25	)	)	PUNCT
ejpam-6909	309	26	and	and	CCONJ
ejpam-6909	309	27	ℏ3	ℏ3	PROPN
ejpam-6909	309	28	∈	∈	PROPN
ejpam-6909	310	1	u(α+	u(α+	NOUN
ejpam-6909	310	2	a	a	X
ejpam-6909	310	3	;	;	PUNCT
ejpam-6909	310	4	p	p	X
ejpam-6909	310	5	)	)	PUNCT
ejpam-6909	310	6	.	.	PUNCT
ejpam-6909	311	1	then	then	ADV
ejpam-6909	311	2	,	,	PUNCT
ejpam-6909	311	3	for	for	ADP
ejpam-6909	311	4	every	every	DET
ejpam-6909	311	5	a	a	DET
ejpam-6909	311	6	∈	∈	PROPN
ejpam-6909	311	7	(	(	PUNCT
ejpam-6909	311	8	ℏ1	ℏ1	PROPN
ejpam-6909	311	9	◦	◦	NOUN
ejpam-6909	311	10	ℏ2	ℏ2	NOUN
ejpam-6909	311	11	)	)	PUNCT
ejpam-6909	311	12	◦	◦	NOUN
ejpam-6909	311	13	ℏ3	ℏ3	PROPN
ejpam-6909	311	14	,	,	PUNCT
ejpam-6909	311	15	we	we	PRON
ejpam-6909	311	16	can	can	AUX
ejpam-6909	311	17	find	find	VERB
ejpam-6909	311	18	k	k	X
ejpam-6909	311	19	∈	∈	PROPN
ejpam-6909	311	20	u(α+	u(α+	NOUN
ejpam-6909	311	21	a	a	X
ejpam-6909	311	22	;	;	PUNCT
ejpam-6909	311	23	p	p	X
ejpam-6909	311	24	)	)	PUNCT
ejpam-6909	311	25	such	such	ADJ
ejpam-6909	311	26	that	that	SCONJ
ejpam-6909	311	27	a	a	DET
ejpam-6909	311	28	≪	≪	NOUN
ejpam-6909	311	29	k.	k.	NOUN
ejpam-6909	311	30	by	by	ADP
ejpam-6909	311	31	corollary	corollary	ADJ
ejpam-6909	311	32	1	1	NUM
ejpam-6909	311	33	,	,	PUNCT
ejpam-6909	311	34	we	we	PRON
ejpam-6909	311	35	obtain	obtain	VERB
ejpam-6909	311	36	α+	α+	PRON
ejpam-6909	311	37	a(a	a(a	PROPN
ejpam-6909	311	38	)	)	PUNCT
ejpam-6909	311	39	≥	≥	AUX
ejpam-6909	311	40	α+	α+	X
ejpam-6909	311	41	a(k	a(k	PROPN
ejpam-6909	311	42	)	)	PUNCT
ejpam-6909	311	43	≥	≥	NOUN
ejpam-6909	312	1	p.	p.	NOUN
ejpam-6909	312	2	this	this	PRON
ejpam-6909	312	3	implies	imply	VERB
ejpam-6909	312	4	α+	α+	PRON
ejpam-6909	312	5	a(a	a(a	PROPN
ejpam-6909	312	6	)	)	PUNCT
ejpam-6909	312	7	≥	≥	NOUN
ejpam-6909	313	1	p	p	X
ejpam-6909	313	2	,	,	PUNCT
ejpam-6909	313	3	for	for	ADP
ejpam-6909	313	4	all	all	DET
ejpam-6909	313	5	a	a	DET
ejpam-6909	313	6	∈	∈	NOUN
ejpam-6909	313	7	(	(	PUNCT
ejpam-6909	313	8	ℏ1	ℏ1	PROPN
ejpam-6909	313	9	◦	◦	NOUN
ejpam-6909	313	10	ℏ2	ℏ2	NOUN
ejpam-6909	313	11	)	)	PUNCT
ejpam-6909	313	12	◦	◦	NOUN
ejpam-6909	313	13	ℏ3	ℏ3	PROPN
ejpam-6909	313	14	and	and	CCONJ
ejpam-6909	313	15	ℏ3	ℏ3	PROPN
ejpam-6909	313	16	∈	∈	PROPN
ejpam-6909	313	17	u(α+	u(α+	NOUN
ejpam-6909	313	18	a	a	X
ejpam-6909	313	19	;	;	PUNCT
ejpam-6909	313	20	p	p	X
ejpam-6909	313	21	)	)	PUNCT
ejpam-6909	313	22	,	,	PUNCT
ejpam-6909	313	23	consequently	consequently	ADV
ejpam-6909	313	24	,	,	PUNCT
ejpam-6909	313	25	supα+	supα+	AUX
ejpam-6909	313	26	a(a	a(a	PROPN
ejpam-6909	313	27	)	)	PUNCT
ejpam-6909	313	28	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	313	29	◦	◦	NOUN
ejpam-6909	313	30	ℏ2)	ℏ2)	NOUN
ejpam-6909	313	31	◦	◦	NOUN
ejpam-6909	313	32	ℏ3	ℏ3	PROPN
ejpam-6909	313	33	≥	≥	NUM
ejpam-6909	313	34	p	p	NOUN
ejpam-6909	313	35	and	and	CCONJ
ejpam-6909	313	36	α+	α+	PUNCT
ejpam-6909	313	37	a(ℏ3	a(ℏ3	ADJ
ejpam-6909	313	38	)	)	PUNCT
ejpam-6909	313	39	≥	≥	NOUN
ejpam-6909	314	1	p.	p.	NOUN
ejpam-6909	314	2	thus	thus	ADV
ejpam-6909	314	3	,	,	PUNCT
ejpam-6909	314	4	by	by	ADP
ejpam-6909	314	5	hypothesis	hypothesis	NOUN
ejpam-6909	314	6	,	,	PUNCT
ejpam-6909	314	7	for	for	ADP
ejpam-6909	314	8	all	all	DET
ejpam-6909	314	9	k	k	PROPN
ejpam-6909	314	10	∈	∈	PROPN
ejpam-6909	314	11	ℏ1	ℏ1	PROPN
ejpam-6909	314	12	◦	◦	NOUN
ejpam-6909	314	13	(ℏ2	(ℏ2	NOUN
ejpam-6909	314	14	◦	◦	ADJ
ejpam-6909	314	15	(ℏ2	(ℏ2	NOUN
ejpam-6909	314	16	◦	◦	NOUN
ejpam-6909	314	17	ℏ1	ℏ1	ADJ
ejpam-6909	314	18	)	)	PUNCT
ejpam-6909	314	19	)	)	PUNCT
ejpam-6909	315	1	,	,	PUNCT
ejpam-6909	315	2	α+	α+	X
ejpam-6909	315	3	a(k	a(k	NUM
ejpam-6909	315	4	)	)	PUNCT
ejpam-6909	315	5	≥	≥	PROPN
ejpam-6909	315	6	min	min	PROPN
ejpam-6909	315	7	{	{	PUNCT
ejpam-6909	315	8	supα+	supα+	X
ejpam-6909	315	9	a(a	a(a	PROPN
ejpam-6909	315	10	)	)	PUNCT
ejpam-6909	315	11	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	315	12	◦	◦	NOUN
ejpam-6909	315	13	ℏ2)	ℏ2)	NOUN
ejpam-6909	315	14	◦	◦	NOUN
ejpam-6909	315	15	ℏ3	ℏ3	PROPN
ejpam-6909	315	16	,	,	PUNCT
ejpam-6909	315	17	α+	α+	PRON
ejpam-6909	315	18	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	315	19	)	)	PUNCT
ejpam-6909	315	20	}	}	PUNCT
ejpam-6909	315	21	≥	≥	NOUN
ejpam-6909	315	22	min{p	min{p	ADV
ejpam-6909	315	23	,	,	PUNCT
ejpam-6909	315	24	p	p	X
ejpam-6909	315	25	}	}	PUNCT
ejpam-6909	315	26	=	=	SYM
ejpam-6909	316	1	p.	p.	NOUN
ejpam-6909	316	2	therefore	therefore	ADV
ejpam-6909	316	3	,	,	PUNCT
ejpam-6909	316	4	it	it	PRON
ejpam-6909	316	5	follows	follow	VERB
ejpam-6909	316	6	that	that	SCONJ
ejpam-6909	316	7	ℏ1	ℏ1	ADJ
ejpam-6909	316	8	◦	◦	NOUN
ejpam-6909	316	9	(	(	PUNCT
ejpam-6909	316	10	ℏ2	ℏ2	NOUN
ejpam-6909	316	11	◦	◦	VERB
ejpam-6909	316	12	(	(	PUNCT
ejpam-6909	316	13	ℏ2	ℏ2	NOUN
ejpam-6909	316	14	◦	◦	VERB
ejpam-6909	316	15	ℏ1	ℏ1	ADJ
ejpam-6909	316	16	)	)	PUNCT
ejpam-6909	316	17	)	)	PUNCT
ejpam-6909	317	1	⊆	⊆	NUM
ejpam-6909	317	2	u(α+	u(α+	NOUN
ejpam-6909	317	3	a	a	X
ejpam-6909	317	4	;	;	PUNCT
ejpam-6909	317	5	p	p	X
ejpam-6909	317	6	)	)	PUNCT
ejpam-6909	317	7	.	.	PUNCT
ejpam-6909	318	1	let	let	VERB
ejpam-6909	318	2	ℏ1	ℏ1	ADJ
ejpam-6909	318	3	,	,	PUNCT
ejpam-6909	318	4	ℏ2	ℏ2	NOUN
ejpam-6909	318	5	,	,	PUNCT
ejpam-6909	318	6	ℏ3	ℏ3	PROPN
ejpam-6909	318	7	∈	∈	PROPN
ejpam-6909	318	8	h	h	NOUN
ejpam-6909	318	9	be	be	AUX
ejpam-6909	318	10	such	such	ADJ
ejpam-6909	318	11	that	that	SCONJ
ejpam-6909	318	12	(	(	PUNCT
ejpam-6909	318	13	ℏ1	ℏ1	ADJ
ejpam-6909	318	14	◦	◦	NOUN
ejpam-6909	318	15	ℏ2	ℏ2	NOUN
ejpam-6909	318	16	)	)	PUNCT
ejpam-6909	318	17	◦	◦	NOUN
ejpam-6909	318	18	ℏ3	ℏ3	PROPN
ejpam-6909	318	19	≪	≪	PUNCT
ejpam-6909	318	20	l(β−	l(β−	VERB
ejpam-6909	318	21	a	a	PRON
ejpam-6909	318	22	;	;	PUNCT
ejpam-6909	318	23	n	n	CCONJ
ejpam-6909	318	24	)	)	PUNCT
ejpam-6909	318	25	and	and	CCONJ
ejpam-6909	318	26	ℏ3	ℏ3	PROPN
ejpam-6909	318	27	∈	∈	PROPN
ejpam-6909	318	28	l(β−	l(β−	VERB
ejpam-6909	318	29	a	a	PRON
ejpam-6909	318	30	;	;	PUNCT
ejpam-6909	318	31	n	n	CCONJ
ejpam-6909	318	32	)	)	PUNCT
ejpam-6909	318	33	.	.	PUNCT
ejpam-6909	319	1	then	then	ADV
ejpam-6909	319	2	,	,	PUNCT
ejpam-6909	319	3	for	for	ADP
ejpam-6909	319	4	every	every	DET
ejpam-6909	319	5	b	b	PROPN
ejpam-6909	319	6	∈	∈	PROPN
ejpam-6909	319	7	(	(	PUNCT
ejpam-6909	319	8	ℏ1	ℏ1	PROPN
ejpam-6909	319	9	◦	◦	NOUN
ejpam-6909	319	10	ℏ2	ℏ2	NOUN
ejpam-6909	319	11	)	)	PUNCT
ejpam-6909	319	12	◦	◦	NOUN
ejpam-6909	319	13	ℏ3	ℏ3	PROPN
ejpam-6909	319	14	,	,	PUNCT
ejpam-6909	319	15	we	we	PRON
ejpam-6909	319	16	can	can	AUX
ejpam-6909	319	17	find	find	VERB
ejpam-6909	319	18	l	l	NOUN
ejpam-6909	319	19	∈	∈	PROPN
ejpam-6909	319	20	l(β−	l(β−	VERB
ejpam-6909	319	21	a	a	PRON
ejpam-6909	319	22	;	;	PUNCT
ejpam-6909	319	23	n	n	CCONJ
ejpam-6909	319	24	)	)	PUNCT
ejpam-6909	319	25	such	such	ADJ
ejpam-6909	319	26	that	that	DET
ejpam-6909	319	27	b	b	NOUN
ejpam-6909	319	28	≪	≪	PUNCT
ejpam-6909	319	29	l.	l.	NOUN
ejpam-6909	319	30	by	by	ADP
ejpam-6909	319	31	corollary	corollary	ADJ
ejpam-6909	319	32	1	1	NUM
ejpam-6909	319	33	,	,	PUNCT
ejpam-6909	319	34	we	we	PRON
ejpam-6909	319	35	obtain	obtain	VERB
ejpam-6909	319	36	β−	β−	PUNCT
ejpam-6909	320	1	a	a	DET
ejpam-6909	320	2	(	(	PUNCT
ejpam-6909	320	3	b	b	NOUN
ejpam-6909	320	4	)	)	PUNCT
ejpam-6909	320	5	≤	≤	NOUN
ejpam-6909	320	6	β−	β−	PUNCT
ejpam-6909	321	1	a	a	DET
ejpam-6909	321	2	(	(	PUNCT
ejpam-6909	321	3	l	l	NOUN
ejpam-6909	321	4	)	)	PUNCT
ejpam-6909	321	5	≤	≤	PUNCT
ejpam-6909	321	6	n.	n.	NOUN
ejpam-6909	321	7	this	this	PRON
ejpam-6909	321	8	implies	imply	VERB
ejpam-6909	321	9	β−	β−	PRON
ejpam-6909	321	10	a	a	DET
ejpam-6909	321	11	(	(	PUNCT
ejpam-6909	321	12	b	b	NOUN
ejpam-6909	321	13	)	)	PUNCT
ejpam-6909	321	14	≤	≤	NOUN
ejpam-6909	321	15	n	n	CCONJ
ejpam-6909	321	16	,	,	PUNCT
ejpam-6909	321	17	for	for	SCONJ
ejpam-6909	321	18	all	all	DET
ejpam-6909	321	19	b	b	PROPN
ejpam-6909	321	20	∈	∈	PROPN
ejpam-6909	321	21	(	(	PUNCT
ejpam-6909	321	22	ℏ1	ℏ1	PROPN
ejpam-6909	321	23	◦	◦	NOUN
ejpam-6909	321	24	ℏ2	ℏ2	NOUN
ejpam-6909	321	25	)	)	PUNCT
ejpam-6909	321	26	◦	◦	NOUN
ejpam-6909	321	27	ℏ3	ℏ3	PROPN
ejpam-6909	321	28	and	and	CCONJ
ejpam-6909	321	29	ℏ3	ℏ3	PROPN
ejpam-6909	321	30	∈	∈	PROPN
ejpam-6909	321	31	l(β−	l(β−	VERB
ejpam-6909	321	32	a	a	PRON
ejpam-6909	321	33	;	;	PUNCT
ejpam-6909	321	34	n	n	CCONJ
ejpam-6909	321	35	)	)	PUNCT
ejpam-6909	321	36	.	.	PUNCT
ejpam-6909	322	1	consequently	consequently	ADV
ejpam-6909	322	2	,	,	PUNCT
ejpam-6909	322	3	inf	inf	PROPN
ejpam-6909	322	4	β−	β−	PROPN
ejpam-6909	322	5	a	a	DET
ejpam-6909	322	6	(	(	PUNCT
ejpam-6909	322	7	b	b	NOUN
ejpam-6909	322	8	)	)	PUNCT
ejpam-6909	322	9	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	322	10	◦	◦	NOUN
ejpam-6909	322	11	ℏ2)	ℏ2)	NOUN
ejpam-6909	322	12	◦	◦	NOUN
ejpam-6909	322	13	ℏ3	ℏ3	NOUN
ejpam-6909	322	14	≤	≤	NOUN
ejpam-6909	322	15	n	n	ADV
ejpam-6909	322	16	and	and	CCONJ
ejpam-6909	322	17	β−	β−	PRON
ejpam-6909	322	18	a	a	DET
ejpam-6909	322	19	(	(	PUNCT
ejpam-6909	322	20	ℏ3	ℏ3	PROPN
ejpam-6909	322	21	)	)	PUNCT
ejpam-6909	322	22	≤	≤	PROPN
ejpam-6909	322	23	n.	n.	NOUN
ejpam-6909	322	24	thus	thus	ADV
ejpam-6909	322	25	,	,	PUNCT
ejpam-6909	322	26	by	by	ADP
ejpam-6909	322	27	hypothesis	hypothesis	NOUN
ejpam-6909	322	28	,	,	PUNCT
ejpam-6909	322	29	for	for	ADP
ejpam-6909	322	30	all	all	DET
ejpam-6909	322	31	k	k	PROPN
ejpam-6909	322	32	∈	∈	PROPN
ejpam-6909	322	33	ℏ1	ℏ1	PROPN
ejpam-6909	322	34	◦	◦	NOUN
ejpam-6909	322	35	(ℏ2	(ℏ2	NOUN
ejpam-6909	322	36	◦	◦	ADJ
ejpam-6909	322	37	(ℏ2	(ℏ2	NOUN
ejpam-6909	322	38	◦	◦	NOUN
ejpam-6909	322	39	ℏ1	ℏ1	ADJ
ejpam-6909	322	40	)	)	PUNCT
ejpam-6909	322	41	)	)	PUNCT
ejpam-6909	322	42	,	,	PUNCT
ejpam-6909	322	43	β−	β−	PRON
ejpam-6909	323	1	a	a	DET
ejpam-6909	323	2	(	(	PUNCT
ejpam-6909	323	3	k	k	NOUN
ejpam-6909	323	4	)	)	PUNCT
ejpam-6909	323	5	≤	≤	NOUN
ejpam-6909	323	6	max	max	PROPN
ejpam-6909	323	7	{	{	PUNCT
ejpam-6909	323	8	inf	inf	NOUN
ejpam-6909	323	9	β−	β−	PROPN
ejpam-6909	323	10	a	a	DET
ejpam-6909	323	11	(	(	PUNCT
ejpam-6909	323	12	b	b	NOUN
ejpam-6909	323	13	)	)	PUNCT
ejpam-6909	323	14	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	323	15	◦	◦	NOUN
ejpam-6909	323	16	ℏ2)	ℏ2)	NOUN
ejpam-6909	323	17	◦	◦	NOUN
ejpam-6909	323	18	ℏ3	ℏ3	NOUN
ejpam-6909	323	19	,	,	PUNCT
ejpam-6909	323	20	β−	β−	PRON
ejpam-6909	323	21	a	a	DET
ejpam-6909	323	22	(	(	PUNCT
ejpam-6909	323	23	ℏ3	ℏ3	PROPN
ejpam-6909	323	24	)	)	PUNCT
ejpam-6909	323	25	}	}	PUNCT
ejpam-6909	323	26	≤	≤	NOUN
ejpam-6909	323	27	max{n	max{n	NUM
ejpam-6909	323	28	,	,	PUNCT
ejpam-6909	323	29	n	n	CCONJ
ejpam-6909	323	30	}	}	PUNCT
ejpam-6909	323	31	=	=	VERB
ejpam-6909	323	32	n.	n.	NOUN
ejpam-6909	323	33	therefore	therefore	ADV
ejpam-6909	323	34	,	,	PUNCT
ejpam-6909	323	35	it	it	PRON
ejpam-6909	323	36	follows	follow	VERB
ejpam-6909	323	37	that	that	SCONJ
ejpam-6909	323	38	ℏ1	ℏ1	ADJ
ejpam-6909	323	39	◦	◦	NOUN
ejpam-6909	323	40	(	(	PUNCT
ejpam-6909	323	41	ℏ2	ℏ2	NOUN
ejpam-6909	323	42	◦	◦	VERB
ejpam-6909	323	43	(	(	PUNCT
ejpam-6909	323	44	ℏ2	ℏ2	NOUN
ejpam-6909	323	45	◦	◦	VERB
ejpam-6909	323	46	ℏ1	ℏ1	ADJ
ejpam-6909	323	47	)	)	PUNCT
ejpam-6909	323	48	)	)	PUNCT
ejpam-6909	324	1	⊆	⊆	NUM
ejpam-6909	324	2	l(β−	l(β−	NOUN
ejpam-6909	324	3	a	a	PRON
ejpam-6909	324	4	;	;	PUNCT
ejpam-6909	324	5	n	n	CCONJ
ejpam-6909	324	6	)	)	PUNCT
ejpam-6909	324	7	.	.	PUNCT
ejpam-6909	325	1	therefore	therefore	ADV
ejpam-6909	325	2	,	,	PUNCT
ejpam-6909	325	3	we	we	PRON
ejpam-6909	325	4	conclude	conclude	VERB
ejpam-6909	325	5	that	that	SCONJ
ejpam-6909	325	6	for	for	SCONJ
ejpam-6909	325	7	all	all	DET
ejpam-6909	325	8	(	(	PUNCT
ejpam-6909	325	9	p	p	X
ejpam-6909	325	10	,	,	PUNCT
ejpam-6909	325	11	n	n	CCONJ
ejpam-6909	325	12	)	)	PUNCT
ejpam-6909	325	13	∈	∈	PROPN
ejpam-6909	326	1	[	[	X
ejpam-6909	326	2	0	0	NUM
ejpam-6909	326	3	,	,	PUNCT
ejpam-6909	326	4	1]×	1]×	NUM
ejpam-6909	326	5	[	[	X
ejpam-6909	326	6	−1	−1	NOUN
ejpam-6909	326	7	,	,	PUNCT
ejpam-6909	326	8	0	0	NUM
ejpam-6909	326	9	]	]	PUNCT
ejpam-6909	326	10	,	,	PUNCT
ejpam-6909	326	11	the	the	DET
ejpam-6909	326	12	cut	cut	NOUN
ejpam-6909	326	13	sets	set	VERB
ejpam-6909	326	14	u(α+	u(α+	PRON
ejpam-6909	326	15	a	a	X
ejpam-6909	326	16	;	;	PUNCT
ejpam-6909	326	17	p	p	X
ejpam-6909	326	18	)	)	PUNCT
ejpam-6909	326	19	and	and	CCONJ
ejpam-6909	326	20	l(β−	l(β−	VERB
ejpam-6909	326	21	a	a	PRON
ejpam-6909	326	22	;	;	PUNCT
ejpam-6909	326	23	n	n	CCONJ
ejpam-6909	326	24	)	)	PUNCT
ejpam-6909	326	25	are	be	AUX
ejpam-6909	326	26	chbckis	chbcki	NOUN
ejpam-6909	326	27	of	of	ADP
ejpam-6909	326	28	type-3	type-3	NUM
ejpam-6909	326	29	of	of	ADP
ejpam-6909	326	30	h.	h.	PROPN
ejpam-6909	326	31	d.	d.	PROPN
ejpam-6909	326	32	ramesh	ramesh	PROPN
ejpam-6909	326	33	et	et	PROPN
ejpam-6909	326	34	al	al	PROPN
ejpam-6909	326	35	.	.	PUNCT
ejpam-6909	326	36	/	/	SYM
ejpam-6909	326	37	eur	eur	PROPN
ejpam-6909	326	38	.	.	PUNCT
ejpam-6909	327	1	j.	j.	PROPN
ejpam-6909	327	2	pure	pure	PROPN
ejpam-6909	327	3	appl	appl	PROPN
ejpam-6909	327	4	.	.	PROPN
ejpam-6909	327	5	math	math	PROPN
ejpam-6909	327	6	,	,	PUNCT
ejpam-6909	327	7	18	18	NUM
ejpam-6909	327	8	(	(	PUNCT
ejpam-6909	327	9	4	4	NUM
ejpam-6909	327	10	)	)	PUNCT
ejpam-6909	327	11	(	(	PUNCT
ejpam-6909	327	12	2025	2025	NUM
ejpam-6909	327	13	)	)	PUNCT
ejpam-6909	327	14	,	,	PUNCT
ejpam-6909	327	15	6909	6909	NUM
ejpam-6909	327	16	14	14	NUM
ejpam-6909	327	17	of	of	ADP
ejpam-6909	327	18	16	16	NUM
ejpam-6909	327	19	(	(	PUNCT
ejpam-6909	327	20	iii	iii	NOUN
ejpam-6909	327	21	)	)	PUNCT
ejpam-6909	327	22	assume	assume	VERB
ejpam-6909	327	23	that	that	SCONJ
ejpam-6909	327	24	h	h	NOUN
ejpam-6909	327	25	satisfies	satisfy	VERB
ejpam-6909	327	26	the	the	DET
ejpam-6909	327	27	sup	sup	NOUN
ejpam-6909	327	28	-	-	PUNCT
ejpam-6909	327	29	inf	inf	NOUN
ejpam-6909	327	30	property	property	NOUN
ejpam-6909	327	31	,	,	PUNCT
ejpam-6909	327	32	and	and	CCONJ
ejpam-6909	327	33	for	for	ADP
ejpam-6909	327	34	all	all	DET
ejpam-6909	327	35	(	(	PUNCT
ejpam-6909	327	36	p	p	X
ejpam-6909	327	37	,	,	PUNCT
ejpam-6909	327	38	n	n	CCONJ
ejpam-6909	327	39	)	)	PUNCT
ejpam-6909	327	40	∈	∈	PROPN
ejpam-6909	328	1	[	[	X
ejpam-6909	328	2	0	0	NUM
ejpam-6909	328	3	,	,	PUNCT
ejpam-6909	328	4	1]×	1]×	NUM
ejpam-6909	328	5	[	[	X
ejpam-6909	328	6	−1	−1	NOUN
ejpam-6909	328	7	,	,	PUNCT
ejpam-6909	328	8	0	0	NUM
ejpam-6909	328	9	]	]	PUNCT
ejpam-6909	328	10	,	,	PUNCT
ejpam-6909	328	11	the	the	DET
ejpam-6909	328	12	cut	cut	NOUN
ejpam-6909	328	13	sets	set	VERB
ejpam-6909	328	14	u(α+	u(α+	PRON
ejpam-6909	328	15	a	a	X
ejpam-6909	328	16	;	;	PUNCT
ejpam-6909	328	17	p	p	X
ejpam-6909	328	18	)	)	PUNCT
ejpam-6909	328	19	and	and	CCONJ
ejpam-6909	328	20	l(β−	l(β−	VERB
ejpam-6909	328	21	a	a	PRON
ejpam-6909	328	22	;	;	PUNCT
ejpam-6909	328	23	n	n	CCONJ
ejpam-6909	328	24	)	)	PUNCT
ejpam-6909	328	25	are	be	AUX
ejpam-6909	328	26	reflexive	reflexive	ADJ
ejpam-6909	328	27	chbckis	chbcki	NOUN
ejpam-6909	328	28	of	of	ADP
ejpam-6909	328	29	type-3	type-3	NUM
ejpam-6909	328	30	of	of	ADP
ejpam-6909	328	31	h.	h.	NOUN
ejpam-6909	328	32	let	let	VERB
ejpam-6909	328	33	ℏ1	ℏ1	ADJ
ejpam-6909	328	34	,	,	PUNCT
ejpam-6909	328	35	ℏ2	ℏ2	NOUN
ejpam-6909	328	36	,	,	PUNCT
ejpam-6909	328	37	ℏ3	ℏ3	PROPN
ejpam-6909	328	38	∈	∈	PROPN
ejpam-6909	328	39	h.	h.	PROPN
ejpam-6909	328	40	define	define	VERB
ejpam-6909	328	41	p	p	PROPN
ejpam-6909	328	42	=	=	NOUN
ejpam-6909	328	43	min	min	PROPN
ejpam-6909	328	44	{	{	PUNCT
ejpam-6909	328	45	supα+	supα+	X
ejpam-6909	328	46	a(a	a(a	PROPN
ejpam-6909	328	47	)	)	PUNCT
ejpam-6909	329	1	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	329	2	◦	◦	NOUN
ejpam-6909	329	3	ℏ2)	ℏ2)	NOUN
ejpam-6909	329	4	◦	◦	NOUN
ejpam-6909	329	5	ℏ3	ℏ3	PROPN
ejpam-6909	329	6	,	,	PUNCT
ejpam-6909	329	7	α+	α+	DET
ejpam-6909	329	8	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	329	9	)	)	PUNCT
ejpam-6909	329	10	}	}	PUNCT
ejpam-6909	329	11	.	.	PUNCT
ejpam-6909	330	1	then	then	ADV
ejpam-6909	330	2	supα+	supα+	AUX
ejpam-6909	330	3	a(a	a(a	PROPN
ejpam-6909	330	4	)	)	PUNCT
ejpam-6909	330	5	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	330	6	◦	◦	NOUN
ejpam-6909	330	7	ℏ2)	ℏ2)	NOUN
ejpam-6909	330	8	◦	◦	NOUN
ejpam-6909	330	9	ℏ3	ℏ3	PROPN
ejpam-6909	330	10	≥	≥	NUM
ejpam-6909	330	11	p	p	NOUN
ejpam-6909	330	12	and	and	CCONJ
ejpam-6909	330	13	α+	α+	PUNCT
ejpam-6909	330	14	a(ℏ3	a(ℏ3	ADJ
ejpam-6909	330	15	)	)	PUNCT
ejpam-6909	330	16	≥	≥	NOUN
ejpam-6909	331	1	p.	p.	NOUN
ejpam-6909	331	2	since	since	SCONJ
ejpam-6909	331	3	α+	α+	PRON
ejpam-6909	331	4	a	a	DET
ejpam-6909	331	5	satisfies	satisfie	NOUN
ejpam-6909	331	6	the	the	DET
ejpam-6909	331	7	sup	sup	NOUN
ejpam-6909	331	8	property	property	NOUN
ejpam-6909	331	9	,	,	PUNCT
ejpam-6909	331	10	there	there	PRON
ejpam-6909	331	11	exists	exist	VERB
ejpam-6909	331	12	a0	a0	PROPN
ejpam-6909	331	13	∈	∈	PROPN
ejpam-6909	331	14	(	(	PUNCT
ejpam-6909	331	15	ℏ1	ℏ1	PROPN
ejpam-6909	331	16	◦	◦	NOUN
ejpam-6909	331	17	ℏ2)	ℏ2)	NOUN
ejpam-6909	331	18	◦	◦	NOUN
ejpam-6909	331	19	ℏ3	ℏ3	NOUN
ejpam-6909	332	1	such	such	ADJ
ejpam-6909	332	2	that	that	SCONJ
ejpam-6909	332	3	α+	α+	NOUN
ejpam-6909	332	4	a(a0	a(a0	X
ejpam-6909	332	5	)	)	PUNCT
ejpam-6909	332	6	=	=	SYM
ejpam-6909	332	7	supα+	supα+	X
ejpam-6909	332	8	a(a	a(a	PROPN
ejpam-6909	332	9	)	)	PUNCT
ejpam-6909	332	10	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	332	11	◦	◦	NOUN
ejpam-6909	332	12	ℏ2)	ℏ2)	NOUN
ejpam-6909	332	13	◦	◦	NOUN
ejpam-6909	332	14	ℏ3	ℏ3	NOUN
ejpam-6909	332	15	≥	≥	PROPN
ejpam-6909	333	1	p.	p.	NOUN
ejpam-6909	333	2	this	this	PRON
ejpam-6909	333	3	implies	imply	VERB
ejpam-6909	333	4	α+	α+	X
ejpam-6909	333	5	a(a0	a(a0	X
ejpam-6909	333	6	)	)	PUNCT
ejpam-6909	333	7	≥	≥	PROPN
ejpam-6909	334	1	p.	p.	NOUN
ejpam-6909	334	2	that	that	PRON
ejpam-6909	334	3	is	be	AUX
ejpam-6909	334	4	,	,	PUNCT
ejpam-6909	334	5	the	the	DET
ejpam-6909	334	6	set	set	NOUN
ejpam-6909	334	7	(	(	PUNCT
ejpam-6909	334	8	(	(	PUNCT
ejpam-6909	334	9	ℏ1	ℏ1	PROPN
ejpam-6909	334	10	◦	◦	NOUN
ejpam-6909	334	11	ℏ2)	ℏ2)	NOUN
ejpam-6909	334	12	◦	◦	NOUN
ejpam-6909	334	13	ℏ3)∩u(α+	ℏ3)∩u(α+	PUNCT
ejpam-6909	334	14	a	a	NOUN
ejpam-6909	334	15	;	;	PUNCT
ejpam-6909	334	16	p	p	X
ejpam-6909	334	17	)	)	PUNCT
ejpam-6909	334	18	is	be	AUX
ejpam-6909	334	19	non	non	ADJ
ejpam-6909	334	20	-	-	ADJ
ejpam-6909	334	21	empty	empty	ADJ
ejpam-6909	334	22	.	.	PUNCT
ejpam-6909	335	1	we	we	PRON
ejpam-6909	335	2	know	know	VERB
ejpam-6909	335	3	that	that	SCONJ
ejpam-6909	335	4	every	every	DET
ejpam-6909	335	5	chbcki	chbcki	NOUN
ejpam-6909	335	6	of	of	ADP
ejpam-6909	335	7	type-3	type-3	NUM
ejpam-6909	335	8	is	be	AUX
ejpam-6909	335	9	an	an	DET
ejpam-6909	335	10	hbcki	hbcki	NOUN
ejpam-6909	335	11	of	of	ADP
ejpam-6909	335	12	h	h	PROPN
ejpam-6909	335	13	(	(	PUNCT
ejpam-6909	335	14	see	see	VERB
ejpam-6909	335	15	theorem	theorem	VERB
ejpam-6909	335	16	4.4	4.4	NUM
ejpam-6909	335	17	in	in	ADP
ejpam-6909	335	18	[	[	X
ejpam-6909	335	19	6	6	NUM
ejpam-6909	335	20	]	]	NUM
ejpam-6909	335	21	)	)	PUNCT
ejpam-6909	335	22	.	.	PUNCT
ejpam-6909	336	1	therefore	therefore	ADV
ejpam-6909	336	2	,	,	PUNCT
ejpam-6909	336	3	u(α+	u(α+	PROPN
ejpam-6909	336	4	a	a	X
ejpam-6909	336	5	;	;	PUNCT
ejpam-6909	336	6	p	p	X
ejpam-6909	336	7	)	)	PUNCT
ejpam-6909	336	8	is	be	AUX
ejpam-6909	336	9	a	a	DET
ejpam-6909	336	10	reflexive	reflexive	ADJ
ejpam-6909	336	11	-	-	PUNCT
ejpam-6909	336	12	hbcki	hbcki	NOUN
ejpam-6909	336	13	of	of	ADP
ejpam-6909	336	14	h	h	PROPN
ejpam-6909	336	15	and	and	CCONJ
ejpam-6909	336	16	(	(	PUNCT
ejpam-6909	336	17	(	(	PUNCT
ejpam-6909	336	18	ℏ1	ℏ1	PROPN
ejpam-6909	336	19	◦	◦	NOUN
ejpam-6909	336	20	ℏ2	ℏ2	NOUN
ejpam-6909	336	21	)	)	PUNCT
ejpam-6909	336	22	◦	◦	VERB
ejpam-6909	336	23	ℏ3)∩u(α+	ℏ3)∩u(α+	PUNCT
ejpam-6909	336	24	a	a	NOUN
ejpam-6909	336	25	;	;	PUNCT
ejpam-6909	336	26	p	p	X
ejpam-6909	336	27	)	)	PUNCT
ejpam-6909	336	28	̸=	̸=	PROPN
ejpam-6909	336	29	∅.	∅.	ADV
ejpam-6909	336	30	by	by	ADP
ejpam-6909	336	31	theorem	theorem	ADJ
ejpam-6909	336	32	1	1	NUM
ejpam-6909	336	33	(	(	PUNCT
ejpam-6909	336	34	ii	ii	NOUN
ejpam-6909	336	35	)	)	PUNCT
ejpam-6909	336	36	,	,	PUNCT
ejpam-6909	336	37	it	it	PRON
ejpam-6909	336	38	follows	follow	VERB
ejpam-6909	336	39	that	that	SCONJ
ejpam-6909	336	40	(	(	PUNCT
ejpam-6909	336	41	ℏ1	ℏ1	ADJ
ejpam-6909	336	42	◦	◦	NOUN
ejpam-6909	336	43	ℏ2	ℏ2	NOUN
ejpam-6909	336	44	)	)	PUNCT
ejpam-6909	336	45	◦	◦	NOUN
ejpam-6909	336	46	ℏ3	ℏ3	PROPN
ejpam-6909	336	47	≪	≪	VERB
ejpam-6909	336	48	u(α+	u(α+	NOUN
ejpam-6909	336	49	a	a	X
ejpam-6909	336	50	;	;	PUNCT
ejpam-6909	336	51	p	p	X
ejpam-6909	336	52	)	)	PUNCT
ejpam-6909	336	53	.	.	PUNCT
ejpam-6909	337	1	since	since	SCONJ
ejpam-6909	337	2	(	(	PUNCT
ejpam-6909	337	3	ℏ1	ℏ1	PROPN
ejpam-6909	337	4	◦	◦	NOUN
ejpam-6909	337	5	ℏ2	ℏ2	NOUN
ejpam-6909	337	6	)	)	PUNCT
ejpam-6909	337	7	◦	◦	NOUN
ejpam-6909	337	8	ℏ3	ℏ3	PROPN
ejpam-6909	337	9	≪	≪	VERB
ejpam-6909	337	10	u(α+	u(α+	NOUN
ejpam-6909	337	11	a	a	X
ejpam-6909	337	12	;	;	PUNCT
ejpam-6909	337	13	p	p	X
ejpam-6909	337	14	)	)	PUNCT
ejpam-6909	337	15	,	,	PUNCT
ejpam-6909	337	16	ℏ3	ℏ3	PROPN
ejpam-6909	337	17	∈	∈	PROPN
ejpam-6909	337	18	u(α+	u(α+	NOUN
ejpam-6909	337	19	a	a	X
ejpam-6909	337	20	;	;	PUNCT
ejpam-6909	337	21	p	p	X
ejpam-6909	337	22	)	)	PUNCT
ejpam-6909	337	23	,	,	PUNCT
ejpam-6909	337	24	and	and	CCONJ
ejpam-6909	337	25	u(α+	u(α+	ADJ
ejpam-6909	337	26	a	a	X
ejpam-6909	337	27	;	;	PUNCT
ejpam-6909	337	28	p	p	X
ejpam-6909	337	29	)	)	PUNCT
ejpam-6909	337	30	is	be	AUX
ejpam-6909	337	31	a	a	DET
ejpam-6909	337	32	chbcki	chbcki	NOUN
ejpam-6909	337	33	of	of	ADP
ejpam-6909	337	34	type-3	type-3	NUM
ejpam-6909	337	35	of	of	ADP
ejpam-6909	337	36	h	h	NOUN
ejpam-6909	337	37	,	,	PUNCT
ejpam-6909	337	38	it	it	PRON
ejpam-6909	337	39	follows	follow	VERB
ejpam-6909	337	40	that	that	SCONJ
ejpam-6909	337	41	ℏ1	ℏ1	ADJ
ejpam-6909	337	42	◦	◦	NOUN
ejpam-6909	337	43	(	(	PUNCT
ejpam-6909	337	44	ℏ2	ℏ2	NOUN
ejpam-6909	337	45	◦	◦	VERB
ejpam-6909	337	46	(	(	PUNCT
ejpam-6909	337	47	ℏ2	ℏ2	NOUN
ejpam-6909	337	48	◦	◦	VERB
ejpam-6909	337	49	ℏ1	ℏ1	ADJ
ejpam-6909	337	50	)	)	PUNCT
ejpam-6909	337	51	)	)	PUNCT
ejpam-6909	338	1	⊆	⊆	NUM
ejpam-6909	338	2	u(α+	u(α+	NOUN
ejpam-6909	338	3	a	a	X
ejpam-6909	338	4	;	;	PUNCT
ejpam-6909	338	5	p	p	X
ejpam-6909	338	6	)	)	PUNCT
ejpam-6909	338	7	.	.	PUNCT
ejpam-6909	339	1	this	this	PRON
ejpam-6909	339	2	implies	imply	VERB
ejpam-6909	339	3	that	that	SCONJ
ejpam-6909	339	4	for	for	ADP
ejpam-6909	339	5	all	all	PRON
ejpam-6909	339	6	k	k	PROPN
ejpam-6909	339	7	∈	∈	PROPN
ejpam-6909	339	8	ℏ1	ℏ1	PROPN
ejpam-6909	339	9	◦	◦	NOUN
ejpam-6909	339	10	(	(	PUNCT
ejpam-6909	339	11	ℏ2	ℏ2	NOUN
ejpam-6909	339	12	◦	◦	VERB
ejpam-6909	339	13	(	(	PUNCT
ejpam-6909	339	14	ℏ2	ℏ2	NOUN
ejpam-6909	339	15	◦	◦	VERB
ejpam-6909	339	16	ℏ1	ℏ1	ADJ
ejpam-6909	339	17	)	)	PUNCT
ejpam-6909	339	18	)	)	PUNCT
ejpam-6909	339	19	,	,	PUNCT
ejpam-6909	339	20	we	we	PRON
ejpam-6909	339	21	have	have	VERB
ejpam-6909	339	22	k	k	PROPN
ejpam-6909	339	23	∈	∈	PROPN
ejpam-6909	339	24	u(α+	u(α+	NOUN
ejpam-6909	339	25	a	a	X
ejpam-6909	339	26	;	;	PUNCT
ejpam-6909	339	27	p	p	X
ejpam-6909	339	28	)	)	PUNCT
ejpam-6909	339	29	.	.	PUNCT
ejpam-6909	340	1	thus	thus	ADV
ejpam-6909	340	2	,	,	PUNCT
ejpam-6909	340	3	α+	α+	PRON
ejpam-6909	340	4	a(a	a(a	PROPN
ejpam-6909	340	5	)	)	PUNCT
ejpam-6909	340	6	≥	≥	NOUN
ejpam-6909	340	7	p	p	NOUN
ejpam-6909	340	8	,	,	PUNCT
ejpam-6909	340	9	where	where	SCONJ
ejpam-6909	340	10	p	p	PROPN
ejpam-6909	340	11	=	=	SYM
ejpam-6909	340	12	min	min	PROPN
ejpam-6909	340	13	{	{	PUNCT
ejpam-6909	340	14	supα+	supα+	X
ejpam-6909	340	15	a(a	a(a	PROPN
ejpam-6909	340	16	)	)	PUNCT
ejpam-6909	340	17	a∈(ℏ1	a∈(ℏ1	VERB
ejpam-6909	340	18	◦	◦	NOUN
ejpam-6909	340	19	ℏ2)	ℏ2)	NOUN
ejpam-6909	340	20	◦	◦	NOUN
ejpam-6909	340	21	ℏ3	ℏ3	PROPN
ejpam-6909	340	22	,	,	PUNCT
ejpam-6909	340	23	α+	α+	DET
ejpam-6909	340	24	a(ℏ3	a(ℏ3	NOUN
ejpam-6909	340	25	)	)	PUNCT
ejpam-6909	340	26	}	}	PUNCT
ejpam-6909	340	27	.	.	PUNCT
ejpam-6909	341	1	define	define	VERB
ejpam-6909	341	2	n	n	NOUN
ejpam-6909	341	3	=	=	SYM
ejpam-6909	341	4	max	max	PROPN
ejpam-6909	341	5	{	{	PUNCT
ejpam-6909	341	6	inf	inf	NOUN
ejpam-6909	341	7	β−	β−	PROPN
ejpam-6909	341	8	a	a	DET
ejpam-6909	341	9	(	(	PUNCT
ejpam-6909	341	10	b	b	NOUN
ejpam-6909	341	11	)	)	PUNCT
ejpam-6909	341	12	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	341	13	◦	◦	NOUN
ejpam-6909	341	14	ℏ2)	ℏ2)	NOUN
ejpam-6909	341	15	◦	◦	NOUN
ejpam-6909	341	16	ℏ3	ℏ3	NOUN
ejpam-6909	341	17	,	,	PUNCT
ejpam-6909	341	18	β−	β−	PRON
ejpam-6909	341	19	a	a	DET
ejpam-6909	341	20	(	(	PUNCT
ejpam-6909	341	21	ℏ3	ℏ3	PROPN
ejpam-6909	341	22	)	)	PUNCT
ejpam-6909	341	23	}	}	PUNCT
ejpam-6909	341	24	.	.	PUNCT
ejpam-6909	342	1	then	then	ADV
ejpam-6909	342	2	,	,	PUNCT
ejpam-6909	342	3	inf	inf	PROPN
ejpam-6909	342	4	β−	β−	PROPN
ejpam-6909	342	5	a	a	DET
ejpam-6909	342	6	(	(	PUNCT
ejpam-6909	342	7	b	b	NOUN
ejpam-6909	342	8	)	)	PUNCT
ejpam-6909	342	9	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	342	10	◦	◦	NOUN
ejpam-6909	342	11	ℏ2)	ℏ2)	NOUN
ejpam-6909	342	12	◦	◦	NOUN
ejpam-6909	342	13	ℏ3	ℏ3	NOUN
ejpam-6909	342	14	≤	≤	NOUN
ejpam-6909	342	15	n	n	ADV
ejpam-6909	342	16	and	and	CCONJ
ejpam-6909	342	17	β−	β−	PRON
ejpam-6909	342	18	a	a	DET
ejpam-6909	342	19	(	(	PUNCT
ejpam-6909	342	20	ℏ3	ℏ3	PROPN
ejpam-6909	342	21	)	)	PUNCT
ejpam-6909	342	22	≤	≤	NOUN
ejpam-6909	342	23	n.	n.	NOUN
ejpam-6909	342	24	since	since	SCONJ
ejpam-6909	342	25	β−	β−	PROPN
ejpam-6909	342	26	a	a	DET
ejpam-6909	342	27	satisfies	satisfie	NOUN
ejpam-6909	342	28	the	the	DET
ejpam-6909	342	29	inf	inf	PROPN
ejpam-6909	342	30	property	property	NOUN
ejpam-6909	342	31	,	,	PUNCT
ejpam-6909	342	32	there	there	PRON
ejpam-6909	342	33	exists	exist	VERB
ejpam-6909	342	34	b0	b0	PROPN
ejpam-6909	342	35	∈	∈	PROPN
ejpam-6909	342	36	(	(	PUNCT
ejpam-6909	342	37	ℏ1	ℏ1	PROPN
ejpam-6909	342	38	◦	◦	NOUN
ejpam-6909	342	39	ℏ2)	ℏ2)	NOUN
ejpam-6909	342	40	◦	◦	NOUN
ejpam-6909	342	41	ℏ3	ℏ3	NOUN
ejpam-6909	342	42	such	such	ADJ
ejpam-6909	342	43	that	that	SCONJ
ejpam-6909	342	44	β−	β−	PRON
ejpam-6909	343	1	a	a	DET
ejpam-6909	343	2	(	(	PUNCT
ejpam-6909	343	3	b0	b0	NOUN
ejpam-6909	343	4	)	)	PUNCT
ejpam-6909	343	5	=	=	SYM
ejpam-6909	344	1	inf	inf	NOUN
ejpam-6909	344	2	β−	β−	PROPN
ejpam-6909	344	3	a	a	DET
ejpam-6909	344	4	(	(	PUNCT
ejpam-6909	344	5	b	b	NOUN
ejpam-6909	344	6	)	)	PUNCT
ejpam-6909	344	7	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	344	8	◦	◦	NOUN
ejpam-6909	344	9	ℏ2)	ℏ2)	NOUN
ejpam-6909	344	10	◦	◦	NOUN
ejpam-6909	344	11	ℏ3	ℏ3	NOUN
ejpam-6909	344	12	≤	≤	PROPN
ejpam-6909	344	13	n.	n.	NOUN
ejpam-6909	344	14	this	this	PRON
ejpam-6909	344	15	implies	imply	VERB
ejpam-6909	344	16	β−	β−	PRON
ejpam-6909	344	17	a	a	DET
ejpam-6909	344	18	(	(	PUNCT
ejpam-6909	344	19	b0	b0	NOUN
ejpam-6909	344	20	)	)	PUNCT
ejpam-6909	344	21	≤	≤	NOUN
ejpam-6909	345	1	n.	n.	NOUN
ejpam-6909	345	2	that	that	PRON
ejpam-6909	345	3	is	be	AUX
ejpam-6909	345	4	,	,	PUNCT
ejpam-6909	345	5	the	the	DET
ejpam-6909	345	6	set	set	NOUN
ejpam-6909	345	7	(	(	PUNCT
ejpam-6909	345	8	(	(	PUNCT
ejpam-6909	345	9	ℏ1	ℏ1	PROPN
ejpam-6909	345	10	◦	◦	NOUN
ejpam-6909	345	11	ℏ2	ℏ2	NOUN
ejpam-6909	345	12	)	)	PUNCT
ejpam-6909	345	13	◦	◦	PROPN
ejpam-6909	345	14	ℏ3	ℏ3	PROPN
ejpam-6909	345	15	)	)	PUNCT
ejpam-6909	345	16	∩	∩	NOUN
ejpam-6909	345	17	l(β−	l(β−	VERB
ejpam-6909	345	18	a	a	PRON
ejpam-6909	345	19	;	;	PUNCT
ejpam-6909	345	20	n	n	CCONJ
ejpam-6909	345	21	)	)	PUNCT
ejpam-6909	345	22	is	be	AUX
ejpam-6909	345	23	non	non	ADJ
ejpam-6909	345	24	-	-	ADJ
ejpam-6909	345	25	empty	empty	ADJ
ejpam-6909	345	26	.	.	PUNCT
ejpam-6909	346	1	we	we	PRON
ejpam-6909	346	2	know	know	VERB
ejpam-6909	346	3	that	that	SCONJ
ejpam-6909	346	4	every	every	DET
ejpam-6909	346	5	chbcki	chbcki	NOUN
ejpam-6909	346	6	of	of	ADP
ejpam-6909	346	7	type-3	type-3	NUM
ejpam-6909	346	8	is	be	AUX
ejpam-6909	346	9	an	an	DET
ejpam-6909	346	10	hbcki	hbcki	NOUN
ejpam-6909	346	11	of	of	ADP
ejpam-6909	346	12	h	h	PROPN
ejpam-6909	346	13	(	(	PUNCT
ejpam-6909	346	14	see	see	VERB
ejpam-6909	346	15	theorem	theorem	VERB
ejpam-6909	346	16	4.4	4.4	NUM
ejpam-6909	346	17	in	in	ADP
ejpam-6909	346	18	[	[	X
ejpam-6909	346	19	6	6	NUM
ejpam-6909	346	20	]	]	NUM
ejpam-6909	346	21	)	)	PUNCT
ejpam-6909	346	22	.	.	PUNCT
ejpam-6909	347	1	therefore	therefore	ADV
ejpam-6909	347	2	,	,	PUNCT
ejpam-6909	347	3	l(β−	l(β−	VERB
ejpam-6909	347	4	a	a	PRON
ejpam-6909	347	5	;	;	PUNCT
ejpam-6909	347	6	n	n	CCONJ
ejpam-6909	347	7	)	)	PUNCT
ejpam-6909	347	8	is	be	AUX
ejpam-6909	347	9	a	a	DET
ejpam-6909	347	10	reflexive	reflexive	ADJ
ejpam-6909	347	11	-	-	PUNCT
ejpam-6909	347	12	hbcki	hbcki	NOUN
ejpam-6909	347	13	of	of	ADP
ejpam-6909	347	14	h	h	PROPN
ejpam-6909	347	15	and	and	CCONJ
ejpam-6909	347	16	(	(	PUNCT
ejpam-6909	347	17	(	(	PUNCT
ejpam-6909	347	18	ℏ1	ℏ1	PROPN
ejpam-6909	347	19	◦	◦	NOUN
ejpam-6909	347	20	ℏ2	ℏ2	NOUN
ejpam-6909	347	21	)	)	PUNCT
ejpam-6909	347	22	◦	◦	NOUN
ejpam-6909	347	23	ℏ3)∩l(β−	ℏ3)∩l(β−	PUNCT
ejpam-6909	347	24	a	a	PRON
ejpam-6909	347	25	;	;	PUNCT
ejpam-6909	347	26	n	n	CCONJ
ejpam-6909	347	27	)	)	PUNCT
ejpam-6909	347	28	̸=	̸=	PROPN
ejpam-6909	347	29	∅.	∅.	VERB
ejpam-6909	347	30	by	by	ADP
ejpam-6909	347	31	theorem	theorem	ADJ
ejpam-6909	347	32	1	1	NUM
ejpam-6909	347	33	(	(	PUNCT
ejpam-6909	347	34	ii	ii	NOUN
ejpam-6909	347	35	)	)	PUNCT
ejpam-6909	347	36	,	,	PUNCT
ejpam-6909	347	37	it	it	PRON
ejpam-6909	347	38	follows	follow	VERB
ejpam-6909	347	39	that	that	SCONJ
ejpam-6909	347	40	(	(	PUNCT
ejpam-6909	347	41	ℏ1	ℏ1	PROPN
ejpam-6909	347	42	◦	◦	NOUN
ejpam-6909	347	43	ℏ2)	ℏ2)	NOUN
ejpam-6909	347	44	◦	◦	NOUN
ejpam-6909	347	45	ℏ3	ℏ3	NOUN
ejpam-6909	347	46	≪	≪	PUNCT
ejpam-6909	347	47	l(β−	l(β−	VERB
ejpam-6909	347	48	a	a	PRON
ejpam-6909	347	49	;	;	PUNCT
ejpam-6909	347	50	n	n	CCONJ
ejpam-6909	347	51	)	)	PUNCT
ejpam-6909	347	52	.	.	PUNCT
ejpam-6909	348	1	since	since	SCONJ
ejpam-6909	348	2	(	(	PUNCT
ejpam-6909	348	3	ℏ1	ℏ1	PROPN
ejpam-6909	348	4	◦	◦	NOUN
ejpam-6909	348	5	ℏ2)	ℏ2)	NOUN
ejpam-6909	348	6	◦	◦	NOUN
ejpam-6909	348	7	ℏ3	ℏ3	NOUN
ejpam-6909	348	8	≪	≪	PUNCT
ejpam-6909	348	9	l(β−	l(β−	VERB
ejpam-6909	348	10	a	a	PRON
ejpam-6909	348	11	;	;	PUNCT
ejpam-6909	348	12	n	n	CCONJ
ejpam-6909	348	13	)	)	PUNCT
ejpam-6909	348	14	,	,	PUNCT
ejpam-6909	348	15	ℏ3	ℏ3	PROPN
ejpam-6909	348	16	∈	∈	PROPN
ejpam-6909	348	17	l(β−	l(β−	VERB
ejpam-6909	348	18	a	a	PRON
ejpam-6909	348	19	;	;	PUNCT
ejpam-6909	348	20	n	n	CCONJ
ejpam-6909	348	21	)	)	PUNCT
ejpam-6909	348	22	,	,	PUNCT
ejpam-6909	348	23	and	and	CCONJ
ejpam-6909	348	24	l(β−	l(β−	VERB
ejpam-6909	348	25	a	a	PRON
ejpam-6909	348	26	;	;	PUNCT
ejpam-6909	348	27	n	n	CCONJ
ejpam-6909	348	28	)	)	PUNCT
ejpam-6909	348	29	is	be	AUX
ejpam-6909	348	30	a	a	DET
ejpam-6909	348	31	chbcki	chbcki	NOUN
ejpam-6909	348	32	of	of	ADP
ejpam-6909	348	33	type-3	type-3	NUM
ejpam-6909	348	34	of	of	ADP
ejpam-6909	348	35	h	h	NOUN
ejpam-6909	348	36	,	,	PUNCT
ejpam-6909	348	37	it	it	PRON
ejpam-6909	348	38	follows	follow	VERB
ejpam-6909	348	39	that	that	SCONJ
ejpam-6909	348	40	ℏ1	ℏ1	ADJ
ejpam-6909	348	41	◦	◦	NOUN
ejpam-6909	348	42	(	(	PUNCT
ejpam-6909	348	43	ℏ2	ℏ2	NOUN
ejpam-6909	348	44	◦	◦	VERB
ejpam-6909	348	45	(	(	PUNCT
ejpam-6909	348	46	ℏ2	ℏ2	NOUN
ejpam-6909	348	47	◦	◦	VERB
ejpam-6909	348	48	ℏ1	ℏ1	ADJ
ejpam-6909	348	49	)	)	PUNCT
ejpam-6909	348	50	)	)	PUNCT
ejpam-6909	349	1	⊆	⊆	NUM
ejpam-6909	349	2	l(β−	l(β−	NOUN
ejpam-6909	349	3	a	a	PRON
ejpam-6909	349	4	;	;	PUNCT
ejpam-6909	349	5	n	n	CCONJ
ejpam-6909	349	6	)	)	PUNCT
ejpam-6909	349	7	.	.	PUNCT
ejpam-6909	350	1	this	this	PRON
ejpam-6909	350	2	implies	imply	VERB
ejpam-6909	350	3	that	that	SCONJ
ejpam-6909	350	4	for	for	ADP
ejpam-6909	350	5	all	all	PRON
ejpam-6909	350	6	k	k	PROPN
ejpam-6909	350	7	∈	∈	PROPN
ejpam-6909	350	8	ℏ1	ℏ1	PROPN
ejpam-6909	350	9	◦	◦	NOUN
ejpam-6909	350	10	(	(	PUNCT
ejpam-6909	350	11	ℏ2	ℏ2	NOUN
ejpam-6909	350	12	◦	◦	VERB
ejpam-6909	350	13	(	(	PUNCT
ejpam-6909	350	14	ℏ2	ℏ2	NOUN
ejpam-6909	350	15	◦	◦	VERB
ejpam-6909	350	16	ℏ1	ℏ1	ADJ
ejpam-6909	350	17	)	)	PUNCT
ejpam-6909	350	18	)	)	PUNCT
ejpam-6909	350	19	,	,	PUNCT
ejpam-6909	350	20	we	we	PRON
ejpam-6909	350	21	have	have	VERB
ejpam-6909	350	22	k	k	PROPN
ejpam-6909	350	23	∈	∈	PROPN
ejpam-6909	350	24	l(β−	l(β−	NOUN
ejpam-6909	350	25	a	a	NOUN
ejpam-6909	350	26	;	;	PUNCT
ejpam-6909	350	27	n	n	CCONJ
ejpam-6909	350	28	)	)	PUNCT
ejpam-6909	350	29	.	.	PUNCT
ejpam-6909	351	1	thus	thus	ADV
ejpam-6909	351	2	,	,	PUNCT
ejpam-6909	351	3	β−	β−	PRON
ejpam-6909	351	4	a	a	DET
ejpam-6909	351	5	(	(	PUNCT
ejpam-6909	351	6	t2	t2	NOUN
ejpam-6909	351	7	)	)	PUNCT
ejpam-6909	351	8	≤	≤	NOUN
ejpam-6909	351	9	n	n	CCONJ
ejpam-6909	351	10	,	,	PUNCT
ejpam-6909	351	11	where	where	SCONJ
ejpam-6909	351	12	n	n	NOUN
ejpam-6909	351	13	=	=	SYM
ejpam-6909	351	14	max	max	PROPN
ejpam-6909	351	15	{	{	PUNCT
ejpam-6909	351	16	inf	inf	NOUN
ejpam-6909	351	17	β−	β−	PROPN
ejpam-6909	351	18	a	a	DET
ejpam-6909	351	19	(	(	PUNCT
ejpam-6909	351	20	b	b	NOUN
ejpam-6909	351	21	)	)	PUNCT
ejpam-6909	351	22	b∈(ℏ1	b∈(ℏ1	NOUN
ejpam-6909	351	23	◦	◦	NOUN
ejpam-6909	351	24	ℏ2)	ℏ2)	NOUN
ejpam-6909	351	25	◦	◦	NOUN
ejpam-6909	351	26	ℏ3	ℏ3	NOUN
ejpam-6909	351	27	,	,	PUNCT
ejpam-6909	351	28	β−	β−	PRON
ejpam-6909	351	29	a	a	DET
ejpam-6909	351	30	(	(	PUNCT
ejpam-6909	351	31	ℏ3	ℏ3	PROPN
ejpam-6909	351	32	)	)	PUNCT
ejpam-6909	351	33	}	}	PUNCT
ejpam-6909	351	34	.	.	PUNCT
ejpam-6909	352	1	therefore	therefore	ADV
ejpam-6909	352	2	,	,	PUNCT
ejpam-6909	352	3	we	we	PRON
ejpam-6909	352	4	conclude	conclude	VERB
ejpam-6909	352	5	that	that	SCONJ
ejpam-6909	352	6	(	(	PUNCT
ejpam-6909	352	7	α+	α+	X
ejpam-6909	352	8	a	a	X
ejpam-6909	352	9	,	,	PUNCT
ejpam-6909	352	10	β	β	PROPN
ejpam-6909	352	11	−	−	NOUN
ejpam-6909	352	12	a	a	PRON
ejpam-6909	352	13	)	)	PUNCT
ejpam-6909	352	14	is	be	AUX
ejpam-6909	352	15	a	a	DET
ejpam-6909	352	16	bf	bf	NOUN
ejpam-6909	352	17	-	-	PUNCT
ejpam-6909	352	18	chbcki	chbcki	NOUN
ejpam-6909	352	19	of	of	ADP
ejpam-6909	352	20	type-3	type-3	NUM
ejpam-6909	352	21	of	of	ADP
ejpam-6909	352	22	h.	h.	NOUN
ejpam-6909	352	23	4	4	NUM
ejpam-6909	352	24	.	.	PUNCT
ejpam-6909	352	25	conclusion	conclusion	NOUN
ejpam-6909	352	26	in	in	ADP
ejpam-6909	352	27	this	this	DET
ejpam-6909	352	28	paper	paper	NOUN
ejpam-6909	352	29	,	,	PUNCT
ejpam-6909	352	30	we	we	PRON
ejpam-6909	352	31	propose	propose	VERB
ejpam-6909	352	32	a	a	DET
ejpam-6909	352	33	comprehensive	comprehensive	ADJ
ejpam-6909	352	34	framework	framework	NOUN
ejpam-6909	352	35	for	for	ADP
ejpam-6909	352	36	the	the	DET
ejpam-6909	352	37	bipolar	bipolar	ADJ
ejpam-6909	352	38	fuzzification	fuzzification	NOUN
ejpam-6909	352	39	of	of	ADP
ejpam-6909	352	40	commutative	commutative	ADJ
ejpam-6909	352	41	hyper	hyper	ADJ
ejpam-6909	352	42	bck	bck	NOUN
ejpam-6909	352	43	-	-	PUNCT
ejpam-6909	352	44	ideals	ideal	NOUN
ejpam-6909	352	45	(	(	PUNCT
ejpam-6909	352	46	chbckis	chbcki	NOUN
ejpam-6909	352	47	)	)	PUNCT
ejpam-6909	352	48	within	within	ADP
ejpam-6909	352	49	the	the	DET
ejpam-6909	352	50	context	context	NOUN
ejpam-6909	352	51	of	of	ADP
ejpam-6909	352	52	hyper	hyper	ADJ
ejpam-6909	352	53	bck	bck	NOUN
ejpam-6909	352	54	-	-	PUNCT
ejpam-6909	352	55	algebras	algebras	PROPN
ejpam-6909	352	56	(	(	PUNCT
ejpam-6909	352	57	hbckas	hbckas	PROPN
ejpam-6909	352	58	)	)	PUNCT
ejpam-6909	352	59	.	.	PUNCT
ejpam-6909	353	1	by	by	ADP
ejpam-6909	353	2	formalizing	formalize	VERB
ejpam-6909	353	3	bipolar	bipolar	ADJ
ejpam-6909	353	4	fuzzy	fuzzy	ADJ
ejpam-6909	353	5	commutative	commutative	ADJ
ejpam-6909	353	6	hyper	hyper	ADJ
ejpam-6909	353	7	bck	bck	NOUN
ejpam-6909	353	8	-	-	PUNCT
ejpam-6909	353	9	ideals	ideal	NOUN
ejpam-6909	353	10	(	(	PUNCT
ejpam-6909	353	11	bf	bf	NOUN
ejpam-6909	353	12	-	-	PUNCT
ejpam-6909	353	13	chbckis	chbcki	NOUN
ejpam-6909	353	14	)	)	PUNCT
ejpam-6909	353	15	,	,	PUNCT
ejpam-6909	353	16	we	we	PRON
ejpam-6909	353	17	introduced	introduce	VERB
ejpam-6909	353	18	new	new	ADJ
ejpam-6909	353	19	definitions	definition	NOUN
ejpam-6909	353	20	,	,	PUNCT
ejpam-6909	353	21	classifications	classification	NOUN
ejpam-6909	353	22	,	,	PUNCT
ejpam-6909	353	23	and	and	CCONJ
ejpam-6909	353	24	theorems	theorem	NOUN
ejpam-6909	353	25	that	that	PRON
ejpam-6909	353	26	elucidate	elucidate	VERB
ejpam-6909	353	27	their	their	PRON
ejpam-6909	353	28	structural	structural	ADJ
ejpam-6909	353	29	properties	property	NOUN
ejpam-6909	353	30	and	and	CCONJ
ejpam-6909	353	31	their	their	PRON
ejpam-6909	353	32	interrelations	interrelation	NOUN
ejpam-6909	353	33	with	with	ADP
ejpam-6909	353	34	various	various	ADJ
ejpam-6909	353	35	forms	form	NOUN
ejpam-6909	353	36	of	of	ADP
ejpam-6909	353	37	hyper	hyper	ADJ
ejpam-6909	353	38	bck	bck	NOUN
ejpam-6909	353	39	-	-	PUNCT
ejpam-6909	353	40	ideals	ideal	NOUN
ejpam-6909	353	41	,	,	PUNCT
ejpam-6909	353	42	including	include	VERB
ejpam-6909	353	43	reflexive	reflexive	ADJ
ejpam-6909	353	44	,	,	PUNCT
ejpam-6909	353	45	strong	strong	ADJ
ejpam-6909	353	46	,	,	PUNCT
ejpam-6909	353	47	and	and	CCONJ
ejpam-6909	353	48	weak	weak	ADJ
ejpam-6909	353	49	types	type	NOUN
ejpam-6909	353	50	.	.	PUNCT
ejpam-6909	354	1	the	the	DET
ejpam-6909	354	2	use	use	NOUN
ejpam-6909	354	3	of	of	ADP
ejpam-6909	354	4	bipolar	bipolar	ADJ
ejpam-6909	354	5	fuzzy	fuzzy	ADJ
ejpam-6909	354	6	subsets	subset	NOUN
ejpam-6909	354	7	as	as	ADP
ejpam-6909	354	8	analytical	analytical	ADJ
ejpam-6909	354	9	tools	tool	NOUN
ejpam-6909	354	10	not	not	PART
ejpam-6909	354	11	only	only	ADV
ejpam-6909	354	12	enhances	enhance	VERB
ejpam-6909	354	13	the	the	DET
ejpam-6909	354	14	flexibility	flexibility	NOUN
ejpam-6909	354	15	of	of	ADP
ejpam-6909	354	16	algebraic	algebraic	ADJ
ejpam-6909	354	17	reasoning	reasoning	NOUN
ejpam-6909	354	18	under	under	ADP
ejpam-6909	354	19	uncertainty	uncertainty	NOUN
ejpam-6909	354	20	but	but	CCONJ
ejpam-6909	354	21	also	also	ADV
ejpam-6909	354	22	strengthens	strengthen	VERB
ejpam-6909	354	23	the	the	DET
ejpam-6909	354	24	foundation	foundation	NOUN
ejpam-6909	354	25	for	for	ADP
ejpam-6909	354	26	advanced	advanced	ADJ
ejpam-6909	354	27	research	research	NOUN
ejpam-6909	354	28	in	in	ADP
ejpam-6909	354	29	fuzzy	fuzzy	ADJ
ejpam-6909	354	30	hyperstructure	hyperstructure	NOUN
ejpam-6909	354	31	theory	theory	NOUN
ejpam-6909	354	32	.	.	PUNCT
ejpam-6909	355	1	moreover	moreover	ADV
ejpam-6909	355	2	,	,	PUNCT
ejpam-6909	355	3	the	the	DET
ejpam-6909	355	4	findings	finding	NOUN
ejpam-6909	355	5	of	of	ADP
ejpam-6909	355	6	this	this	DET
ejpam-6909	355	7	study	study	NOUN
ejpam-6909	355	8	contribute	contribute	VERB
ejpam-6909	355	9	to	to	ADP
ejpam-6909	355	10	the	the	DET
ejpam-6909	355	11	development	development	NOUN
ejpam-6909	355	12	of	of	ADP
ejpam-6909	355	13	research	research	NOUN
ejpam-6909	355	14	-	-	PUNCT
ejpam-6909	355	15	oriented	orient	VERB
ejpam-6909	355	16	mathematical	mathematical	ADJ
ejpam-6909	355	17	pedagogy	pedagogy	NOUN
ejpam-6909	355	18	,	,	PUNCT
ejpam-6909	355	19	offering	offer	VERB
ejpam-6909	355	20	students	student	NOUN
ejpam-6909	355	21	in	in	ADP
ejpam-6909	355	22	inquiry	inquiry	NOUN
ejpam-6909	355	23	-	-	PUNCT
ejpam-6909	355	24	driven	drive	VERB
ejpam-6909	355	25	learning	learning	NOUN
ejpam-6909	355	26	environments	environment	NOUN
ejpam-6909	355	27	access	access	NOUN
ejpam-6909	355	28	to	to	ADP
ejpam-6909	355	29	complex	complex	ADJ
ejpam-6909	355	30	yet	yet	ADV
ejpam-6909	355	31	structured	structured	ADJ
ejpam-6909	355	32	models	model	NOUN
ejpam-6909	355	33	of	of	ADP
ejpam-6909	355	34	logic	logic	NOUN
ejpam-6909	355	35	and	and	CCONJ
ejpam-6909	355	36	abstraction	abstraction	NOUN
ejpam-6909	355	37	.	.	PUNCT
ejpam-6909	356	1	in	in	ADP
ejpam-6909	356	2	alignment	alignment	NOUN
ejpam-6909	356	3	with	with	ADP
ejpam-6909	356	4	sustainable	sustainable	ADJ
ejpam-6909	356	5	development	development	NOUN
ejpam-6909	356	6	d.	d.	PROPN
ejpam-6909	356	7	ramesh	ramesh	PROPN
ejpam-6909	356	8	et	et	PROPN
ejpam-6909	356	9	al	al	PROPN
ejpam-6909	356	10	.	.	PUNCT
ejpam-6909	356	11	/	/	SYM
ejpam-6909	356	12	eur	eur	PROPN
ejpam-6909	356	13	.	.	PUNCT
ejpam-6909	357	1	j.	j.	PROPN
ejpam-6909	357	2	pure	pure	PROPN
ejpam-6909	357	3	appl	appl	PROPN
ejpam-6909	357	4	.	.	PROPN
ejpam-6909	357	5	math	math	PROPN
ejpam-6909	357	6	,	,	PUNCT
ejpam-6909	357	7	18	18	NUM
ejpam-6909	357	8	(	(	PUNCT
ejpam-6909	357	9	4	4	NUM
ejpam-6909	357	10	)	)	PUNCT
ejpam-6909	357	11	(	(	PUNCT
ejpam-6909	357	12	2025	2025	NUM
ejpam-6909	357	13	)	)	PUNCT
ejpam-6909	357	14	,	,	PUNCT
ejpam-6909	357	15	6909	6909	NUM
ejpam-6909	357	16	15	15	NUM
ejpam-6909	357	17	of	of	ADP
ejpam-6909	357	18	16	16	NUM
ejpam-6909	357	19	goal	goal	NOUN
ejpam-6909	357	20	4	4	NUM
ejpam-6909	357	21	(	(	PUNCT
ejpam-6909	357	22	sdg-4	sdg-4	X
ejpam-6909	357	23	)	)	PUNCT
ejpam-6909	357	24	,	,	PUNCT
ejpam-6909	357	25	this	this	DET
ejpam-6909	357	26	work	work	NOUN
ejpam-6909	357	27	supports	support	VERB
ejpam-6909	357	28	the	the	DET
ejpam-6909	357	29	creation	creation	NOUN
ejpam-6909	357	30	of	of	ADP
ejpam-6909	357	31	equitable	equitable	ADJ
ejpam-6909	357	32	and	and	CCONJ
ejpam-6909	357	33	inclusive	inclusive	ADJ
ejpam-6909	357	34	educational	educational	ADJ
ejpam-6909	357	35	pathways	pathway	NOUN
ejpam-6909	357	36	by	by	ADP
ejpam-6909	357	37	enabling	enable	VERB
ejpam-6909	357	38	learners	learner	NOUN
ejpam-6909	357	39	to	to	PART
ejpam-6909	357	40	engage	engage	VERB
ejpam-6909	357	41	with	with	ADP
ejpam-6909	357	42	high	high	ADJ
ejpam-6909	357	43	-	-	PUNCT
ejpam-6909	357	44	level	level	NOUN
ejpam-6909	357	45	mathematical	mathematical	ADJ
ejpam-6909	357	46	constructs	construct	NOUN
ejpam-6909	357	47	early	early	ADV
ejpam-6909	357	48	in	in	ADP
ejpam-6909	357	49	their	their	PRON
ejpam-6909	357	50	academic	academic	ADJ
ejpam-6909	357	51	journey	journey	NOUN
ejpam-6909	357	52	.	.	PUNCT
ejpam-6909	358	1	future	future	ADJ
ejpam-6909	358	2	research	research	NOUN
ejpam-6909	358	3	may	may	AUX
ejpam-6909	358	4	expand	expand	VERB
ejpam-6909	358	5	on	on	ADP
ejpam-6909	358	6	these	these	DET
ejpam-6909	358	7	ideas	idea	NOUN
ejpam-6909	358	8	by	by	ADP
ejpam-6909	358	9	exploring	explore	VERB
ejpam-6909	358	10	analogous	analogous	ADJ
ejpam-6909	358	11	ideals	ideal	NOUN
ejpam-6909	358	12	in	in	ADP
ejpam-6909	358	13	other	other	ADJ
ejpam-6909	358	14	algebraic	algebraic	ADJ
ejpam-6909	358	15	systems	system	NOUN
ejpam-6909	358	16	,	,	PUNCT
ejpam-6909	358	17	further	far	ADV
ejpam-6909	358	18	enriching	enrich	VERB
ejpam-6909	358	19	both	both	DET
ejpam-6909	358	20	theoretical	theoretical	ADJ
ejpam-6909	358	21	knowledge	knowledge	NOUN
ejpam-6909	358	22	and	and	CCONJ
ejpam-6909	358	23	educational	educational	ADJ
ejpam-6909	358	24	practice	practice	NOUN
ejpam-6909	358	25	.	.	PUNCT
ejpam-6909	359	1	acknowledgements	acknowledgement	NOUN
ejpam-6909	359	2	this	this	DET
ejpam-6909	359	3	research	research	NOUN
ejpam-6909	359	4	was	be	AUX
ejpam-6909	359	5	supported	support	VERB
ejpam-6909	359	6	by	by	ADP
ejpam-6909	359	7	university	university	NOUN
ejpam-6909	359	8	of	of	ADP
ejpam-6909	359	9	phayao	phayao	NOUN
ejpam-6909	359	10	and	and	CCONJ
ejpam-6909	359	11	thailand	thailand	PROPN
ejpam-6909	359	12	science	science	PROPN
ejpam-6909	359	13	research	research	PROPN
ejpam-6909	359	14	and	and	CCONJ
ejpam-6909	359	15	innovation	innovation	NOUN
ejpam-6909	359	16	fund	fund	NOUN
ejpam-6909	359	17	(	(	PUNCT
ejpam-6909	359	18	fundamental	fundamental	ADJ
ejpam-6909	359	19	fund	fund	NOUN
ejpam-6909	359	20	2026	2026	NUM
ejpam-6909	359	21	,	,	PUNCT
ejpam-6909	359	22	grant	grant	VERB
ejpam-6909	359	23	no	no	NOUN
ejpam-6909	359	24	.	.	PUNCT
ejpam-6909	359	25	2252/2568	2252/2568	NUM
ejpam-6909	359	26	)	)	PUNCT
ejpam-6909	359	27	.	.	PUNCT
ejpam-6909	360	1	references	reference	NOUN
ejpam-6909	360	2	[	[	X
ejpam-6909	360	3	1	1	NUM
ejpam-6909	360	4	]	]	PUNCT
ejpam-6909	360	5	k.	k.	PROPN
ejpam-6909	360	6	iséki	iséki	PROPN
ejpam-6909	360	7	and	and	CCONJ
ejpam-6909	360	8	s.	s.	PROPN
ejpam-6909	360	9	tanaka	tanaka	PROPN
ejpam-6909	360	10	.	.	PUNCT
ejpam-6909	361	1	an	an	DET
ejpam-6909	361	2	introduction	introduction	NOUN
ejpam-6909	361	3	to	to	ADP
ejpam-6909	361	4	the	the	DET
ejpam-6909	361	5	theory	theory	NOUN
ejpam-6909	361	6	of	of	ADP
ejpam-6909	361	7	bck	bck	PROPN
ejpam-6909	361	8	-	-	PUNCT
ejpam-6909	361	9	algebras	algebras	PROPN
ejpam-6909	361	10	.	.	PUNCT
ejpam-6909	362	1	math	math	PROPN
ejpam-6909	362	2	.	.	PUNCT
ejpam-6909	363	1	japon	japon	PROPN
ejpam-6909	363	2	.	.	PROPN
ejpam-6909	363	3	,	,	PUNCT
ejpam-6909	363	4	23:1–26	23:1–26	NUM
ejpam-6909	363	5	,	,	PUNCT
ejpam-6909	363	6	1978	1978	NUM
ejpam-6909	363	7	.	.	PUNCT
ejpam-6909	364	1	[	[	X
ejpam-6909	364	2	2	2	NUM
ejpam-6909	364	3	]	]	X
ejpam-6909	364	4	y.	y.	PROPN
ejpam-6909	364	5	imai	imai	PROPN
ejpam-6909	364	6	and	and	CCONJ
ejpam-6909	364	7	k.	k.	PROPN
ejpam-6909	364	8	iséki	iséki	PROPN
ejpam-6909	364	9	.	.	PROPN
ejpam-6909	365	1	on	on	ADP
ejpam-6909	365	2	axiom	axiom	NOUN
ejpam-6909	365	3	systems	system	NOUN
ejpam-6909	365	4	of	of	ADP
ejpam-6909	365	5	propositional	propositional	ADJ
ejpam-6909	365	6	calculi	calculi	PROPN
ejpam-6909	365	7	xiv	xiv	PROPN
ejpam-6909	365	8	.	.	PUNCT
ejpam-6909	365	9	proc	proc	PROPN
ejpam-6909	365	10	.	.	PUNCT
ejpam-6909	366	1	japan	japan	PROPN
ejpam-6909	366	2	acad	acad	PROPN
ejpam-6909	366	3	.	.	PROPN
ejpam-6909	366	4	,	,	PUNCT
ejpam-6909	366	5	42(1):19–22	42(1):19–22	NUM
ejpam-6909	366	6	,	,	PUNCT
ejpam-6909	366	7	1966	1966	NUM
ejpam-6909	366	8	.	.	PUNCT
ejpam-6909	367	1	[	[	X
ejpam-6909	367	2	3	3	X
ejpam-6909	367	3	]	]	PUNCT
ejpam-6909	367	4	k.	k.	PROPN
ejpam-6909	367	5	iséki	iséki	PROPN
ejpam-6909	367	6	.	.	PUNCT
ejpam-6909	368	1	an	an	DET
ejpam-6909	368	2	algebra	algebra	NOUN
ejpam-6909	368	3	related	relate	VERB
ejpam-6909	368	4	with	with	ADP
ejpam-6909	368	5	a	a	DET
ejpam-6909	368	6	propositional	propositional	ADJ
ejpam-6909	368	7	calculus	calculus	NOUN
ejpam-6909	368	8	.	.	PUNCT
ejpam-6909	369	1	proc	proc	PROPN
ejpam-6909	369	2	.	.	PUNCT
ejpam-6909	370	1	japan	japan	PROPN
ejpam-6909	370	2	acad	acad	PROPN
ejpam-6909	370	3	.	.	PROPN
ejpam-6909	370	4	,	,	PUNCT
ejpam-6909	370	5	42(1):26–29	42(1):26–29	NUM
ejpam-6909	370	6	,	,	PUNCT
ejpam-6909	370	7	1966	1966	NUM
ejpam-6909	370	8	.	.	PUNCT
ejpam-6909	371	1	[	[	X
ejpam-6909	371	2	4	4	X
ejpam-6909	371	3	]	]	X
ejpam-6909	371	4	f.	f.	PROPN
ejpam-6909	371	5	marty	marty	PROPN
ejpam-6909	371	6	.	.	PUNCT
ejpam-6909	372	1	sur	sur	PROPN
ejpam-6909	372	2	une	une	PROPN
ejpam-6909	372	3	generalization	generalization	PROPN
ejpam-6909	372	4	de	de	X
ejpam-6909	372	5	la	la	PROPN
ejpam-6909	372	6	notion	notion	PROPN
ejpam-6909	372	7	de	de	X
ejpam-6909	372	8	groupe	groupe	PROPN
ejpam-6909	372	9	.	.	PUNCT
ejpam-6909	373	1	in	in	ADP
ejpam-6909	373	2	8th	8th	ADJ
ejpam-6909	373	3	congress	congress	PROPN
ejpam-6909	373	4	math	math	NOUN
ejpam-6909	373	5	.	.	PUNCT
ejpam-6909	374	1	scandinaves	scandinave	NOUN
ejpam-6909	374	2	,	,	PUNCT
ejpam-6909	374	3	pages	page	NOUN
ejpam-6909	374	4	45–49	45–49	PROPN
ejpam-6909	374	5	,	,	PUNCT
ejpam-6909	374	6	stockholm	stockholm	PROPN
ejpam-6909	374	7	,	,	PUNCT
ejpam-6909	374	8	sweden	sweden	PROPN
ejpam-6909	374	9	,	,	PUNCT
ejpam-6909	374	10	1934	1934	NUM
ejpam-6909	374	11	.	.	PUNCT
ejpam-6909	375	1	[	[	X
ejpam-6909	375	2	5	5	X
ejpam-6909	375	3	]	]	X
ejpam-6909	375	4	y.	y.	PROPN
ejpam-6909	375	5	b.	b.	PROPN
ejpam-6909	375	6	jun	jun	PROPN
ejpam-6909	375	7	,	,	PUNCT
ejpam-6909	375	8	r.	r.	PROPN
ejpam-6909	375	9	a.	a.	PROPN
ejpam-6909	375	10	borzooei	borzooei	PROPN
ejpam-6909	375	11	,	,	PUNCT
ejpam-6909	375	12	m.	m.	NOUN
ejpam-6909	375	13	m.	m.	PROPN
ejpam-6909	375	14	zahedi	zahedi	PROPN
ejpam-6909	375	15	,	,	PUNCT
ejpam-6909	375	16	and	and	CCONJ
ejpam-6909	375	17	x.	x.	PROPN
ejpam-6909	375	18	l.	l.	PROPN
ejpam-6909	375	19	xin	xin	PROPN
ejpam-6909	375	20	.	.	PUNCT
ejpam-6909	376	1	on	on	ADP
ejpam-6909	376	2	hyper	hyper	ADJ
ejpam-6909	376	3	bck	bck	NOUN
ejpam-6909	376	4	-	-	PUNCT
ejpam-6909	376	5	algebras	algebras	PROPN
ejpam-6909	376	6	.	.	PUNCT
ejpam-6909	377	1	ital	ital	PROPN
ejpam-6909	377	2	.	.	PUNCT
ejpam-6909	378	1	j.	j.	PROPN
ejpam-6909	378	2	pure	pure	PROPN
ejpam-6909	378	3	appl	appl	PROPN
ejpam-6909	378	4	.	.	PUNCT
ejpam-6909	378	5	math	math	PROPN
ejpam-6909	378	6	.	.	PUNCT
ejpam-6909	378	7	,	,	PUNCT
ejpam-6909	378	8	10:127–136	10:127–136	NUM
ejpam-6909	378	9	,	,	PUNCT
ejpam-6909	378	10	2000	2000	NUM
ejpam-6909	378	11	.	.	PUNCT
ejpam-6909	379	1	[	[	X
ejpam-6909	379	2	6	6	NUM
ejpam-6909	379	3	]	]	PUNCT
ejpam-6909	379	4	r.	r.	PROPN
ejpam-6909	379	5	a.	a.	PROPN
ejpam-6909	379	6	borzooei	borzooei	PROPN
ejpam-6909	379	7	and	and	CCONJ
ejpam-6909	379	8	m.	m.	PROPN
ejpam-6909	379	9	bakhshi	bakhshi	PROPN
ejpam-6909	379	10	.	.	PUNCT
ejpam-6909	380	1	some	some	DET
ejpam-6909	380	2	results	result	NOUN
ejpam-6909	380	3	on	on	ADP
ejpam-6909	380	4	hyper	hyper	ADJ
ejpam-6909	380	5	bck	bck	NOUN
ejpam-6909	380	6	-	-	PUNCT
ejpam-6909	380	7	algebras	algebras	PROPN
ejpam-6909	380	8	.	.	PUNCT
ejpam-6909	381	1	quasigroups	quasigroups	PROPN
ejpam-6909	381	2	relat	relat	PROPN
ejpam-6909	381	3	.	.	PUNCT
ejpam-6909	382	1	syst	syst	PROPN
ejpam-6909	382	2	.	.	PUNCT
ejpam-6909	382	3	,	,	PUNCT
ejpam-6909	382	4	11:9–24	11:9–24	NUM
ejpam-6909	382	5	,	,	PUNCT
ejpam-6909	382	6	2004	2004	NUM
ejpam-6909	382	7	.	.	PUNCT
ejpam-6909	383	1	[	[	X
ejpam-6909	383	2	7	7	X
ejpam-6909	383	3	]	]	X
ejpam-6909	383	4	r.	r.	PROPN
ejpam-6909	383	5	durga	durga	PROPN
ejpam-6909	383	6	prasad	prasad	PROPN
ejpam-6909	383	7	,	,	PUNCT
ejpam-6909	383	8	b.	b.	PROPN
ejpam-6909	383	9	satyanarayana	satyanarayana	PROPN
ejpam-6909	383	10	,	,	PUNCT
ejpam-6909	383	11	d.	d.	PROPN
ejpam-6909	383	12	ramesh	ramesh	PROPN
ejpam-6909	383	13	,	,	PUNCT
ejpam-6909	383	14	and	and	CCONJ
ejpam-6909	383	15	m.	m.	NOUN
ejpam-6909	383	16	gyaneswara	gyaneswara	PROPN
ejpam-6909	383	17	reddy	reddy	PROPN
ejpam-6909	383	18	.	.	PUNCT
ejpam-6909	384	1	on	on	ADP
ejpam-6909	384	2	intuitionistic	intuitionistic	ADJ
ejpam-6909	384	3	fuzzy	fuzzy	ADJ
ejpam-6909	384	4	positive	positive	ADJ
ejpam-6909	384	5	implicative	implicative	ADJ
ejpam-6909	384	6	hyper	hyper	ADJ
ejpam-6909	384	7	bck	bck	NOUN
ejpam-6909	384	8	-	-	PUNCT
ejpam-6909	384	9	ideals	ideal	NOUN
ejpam-6909	384	10	of	of	ADP
ejpam-6909	384	11	hyper	hyper	ADJ
ejpam-6909	384	12	bck	bck	NOUN
ejpam-6909	384	13	-	-	PUNCT
ejpam-6909	384	14	algebras	algebras	PROPN
ejpam-6909	384	15	.	.	PUNCT
ejpam-6909	385	1	int	int	NOUN
ejpam-6909	385	2	.	.	PUNCT
ejpam-6909	386	1	j.	j.	PROPN
ejpam-6909	386	2	math	math	PROPN
ejpam-6909	386	3	.	.	PUNCT
ejpam-6909	387	1	sci	sci	PROPN
ejpam-6909	387	2	.	.	PROPN
ejpam-6909	387	3	engg	engg	PROPN
ejpam-6909	387	4	.	.	PUNCT
ejpam-6909	388	1	appls	appls	PROPN
ejpam-6909	388	2	.	.	PUNCT
ejpam-6909	388	3	,	,	PUNCT
ejpam-6909	388	4	6(1):175–196	6(1):175–196	NOUN
ejpam-6909	388	5	,	,	PUNCT
ejpam-6909	388	6	2012	2012	NUM
ejpam-6909	388	7	.	.	PUNCT
ejpam-6909	389	1	[	[	X
ejpam-6909	389	2	8	8	NUM
ejpam-6909	389	3	]	]	X
ejpam-6909	389	4	b.	b.	PROPN
ejpam-6909	389	5	satyanarayana	satyanarayana	PROPN
ejpam-6909	389	6	,	,	PUNCT
ejpam-6909	389	7	r.	r.	PROPN
ejpam-6909	389	8	durga	durga	PROPN
ejpam-6909	389	9	prasad	prasad	PROPN
ejpam-6909	389	10	,	,	PUNCT
ejpam-6909	389	11	and	and	CCONJ
ejpam-6909	389	12	d.	d.	PROPN
ejpam-6909	389	13	ramesh	ramesh	PROPN
ejpam-6909	389	14	.	.	PUNCT
ejpam-6909	390	1	on	on	ADP
ejpam-6909	390	2	intuitionistic	intuitionistic	ADJ
ejpam-6909	390	3	fuzzy	fuzzy	ADJ
ejpam-6909	390	4	commutative	commutative	ADJ
ejpam-6909	390	5	hyper	hyper	ADJ
ejpam-6909	390	6	bck	bck	NOUN
ejpam-6909	390	7	-	-	PUNCT
ejpam-6909	390	8	ideals	ideal	NOUN
ejpam-6909	390	9	of	of	ADP
ejpam-6909	390	10	hyper	hyper	ADJ
ejpam-6909	390	11	bck	bck	NOUN
ejpam-6909	390	12	-	-	PUNCT
ejpam-6909	390	13	algebras	algebras	PROPN
ejpam-6909	390	14	.	.	PUNCT
ejpam-6909	391	1	int	int	NOUN
ejpam-6909	391	2	.	.	PUNCT
ejpam-6909	392	1	j.	j.	PROPN
ejpam-6909	392	2	algebra	algebra	PROPN
ejpam-6909	392	3	stat	stat	PROPN
ejpam-6909	392	4	.	.	PUNCT
ejpam-6909	392	5	,	,	PUNCT
ejpam-6909	392	6	1(1):110–119	1(1):110–119	NUM
ejpam-6909	392	7	,	,	PUNCT
ejpam-6909	392	8	2012	2012	NUM
ejpam-6909	392	9	.	.	PUNCT
ejpam-6909	393	1	[	[	X
ejpam-6909	393	2	9	9	NUM
ejpam-6909	393	3	]	]	PUNCT
ejpam-6909	393	4	l.	l.	PROPN
ejpam-6909	393	5	a.	a.	PROPN
ejpam-6909	393	6	zadeh	zadeh	PROPN
ejpam-6909	393	7	.	.	PUNCT
ejpam-6909	393	8	fuzzy	fuzzy	ADJ
ejpam-6909	393	9	sets	set	NOUN
ejpam-6909	393	10	.	.	PUNCT
ejpam-6909	394	1	inf	inf	PROPN
ejpam-6909	394	2	.	.	PUNCT
ejpam-6909	394	3	control	control	PROPN
ejpam-6909	394	4	,	,	PUNCT
ejpam-6909	394	5	8(3):338–353	8(3):338–353	NUM
ejpam-6909	394	6	,	,	PUNCT
ejpam-6909	394	7	1965	1965	NUM
ejpam-6909	394	8	.	.	PUNCT
ejpam-6909	395	1	[	[	X
ejpam-6909	395	2	10	10	NUM
ejpam-6909	395	3	]	]	PUNCT
ejpam-6909	395	4	k.	k.	PROPN
ejpam-6909	395	5	m.	m.	PROPN
ejpam-6909	395	6	lee	lee	PROPN
ejpam-6909	395	7	.	.	PUNCT
ejpam-6909	396	1	bipolar	bipolar	ADJ
ejpam-6909	396	2	-	-	PUNCT
ejpam-6909	396	3	valued	value	VERB
ejpam-6909	396	4	fuzzy	fuzzy	ADJ
ejpam-6909	396	5	sets	set	NOUN
ejpam-6909	396	6	and	and	CCONJ
ejpam-6909	396	7	their	their	PRON
ejpam-6909	396	8	operations	operation	NOUN
ejpam-6909	396	9	.	.	PUNCT
ejpam-6909	397	1	in	in	ADP
ejpam-6909	397	2	proc	proc	PROPN
ejpam-6909	397	3	.	.	PUNCT
ejpam-6909	398	1	int	int	NOUN
ejpam-6909	398	2	.	.	PUNCT
ejpam-6909	398	3	conf	conf	PROPN
ejpam-6909	398	4	.	.	PUNCT
ejpam-6909	399	1	intell	intell	PROPN
ejpam-6909	399	2	.	.	PUNCT
ejpam-6909	400	1	technol	technol	PROPN
ejpam-6909	400	2	.	.	PROPN
ejpam-6909	400	3	,	,	PUNCT
ejpam-6909	400	4	pages	page	NOUN
ejpam-6909	400	5	307–312	307–312	NUM
ejpam-6909	400	6	,	,	PUNCT
ejpam-6909	400	7	bangkok	bangkok	PROPN
ejpam-6909	400	8	,	,	PUNCT
ejpam-6909	400	9	thailand	thailand	PROPN
ejpam-6909	400	10	,	,	PUNCT
ejpam-6909	400	11	2000	2000	NUM
ejpam-6909	400	12	.	.	PUNCT
ejpam-6909	401	1	[	[	X
ejpam-6909	401	2	11	11	NUM
ejpam-6909	401	3	]	]	X
ejpam-6909	401	4	h.	h.	PROPN
ejpam-6909	401	5	g.	g.	PROPN
ejpam-6909	401	6	baik	baik	PROPN
ejpam-6909	401	7	.	.	PUNCT
ejpam-6909	402	1	bipolar	bipolar	ADJ
ejpam-6909	402	2	fuzzy	fuzzy	ADJ
ejpam-6909	402	3	ideals	ideal	NOUN
ejpam-6909	402	4	of	of	ADP
ejpam-6909	402	5	near	near	ADJ
ejpam-6909	402	6	rings	ring	NOUN
ejpam-6909	402	7	.	.	PUNCT
ejpam-6909	403	1	j.	j.	PROPN
ejpam-6909	403	2	korean	korean	PROPN
ejpam-6909	403	3	inst	inst	PROPN
ejpam-6909	403	4	.	.	PUNCT
ejpam-6909	404	1	intell	intell	PROPN
ejpam-6909	404	2	.	.	PUNCT
ejpam-6909	405	1	syst	syst	PROPN
ejpam-6909	405	2	.	.	PROPN
ejpam-6909	405	3	,	,	PUNCT
ejpam-6909	405	4	22(3):394	22(3):394	NUM
ejpam-6909	405	5	–	–	PUNCT
ejpam-6909	405	6	398	398	NUM
ejpam-6909	405	7	,	,	PUNCT
ejpam-6909	405	8	2012	2012	NUM
ejpam-6909	405	9	.	.	PUNCT
ejpam-6909	406	1	[	[	X
ejpam-6909	406	2	12	12	NUM
ejpam-6909	406	3	]	]	PUNCT
ejpam-6909	406	4	p.	p.	NOUN
ejpam-6909	406	5	madhu	madhu	PROPN
ejpam-6909	407	1	latha	latha	PROPN
ejpam-6909	407	2	,	,	PUNCT
ejpam-6909	407	3	y.	y.	PROPN
ejpam-6909	407	4	bhargavi	bhargavi	PROPN
ejpam-6909	407	5	,	,	PUNCT
ejpam-6909	407	6	and	and	CCONJ
ejpam-6909	407	7	a.	a.	NOUN
ejpam-6909	407	8	iampan	iampan	PROPN
ejpam-6909	407	9	.	.	PUNCT
ejpam-6909	408	1	bipolar	bipolar	ADJ
ejpam-6909	408	2	fuzzy	fuzzy	ADJ
ejpam-6909	408	3	ideals	ideal	NOUN
ejpam-6909	408	4	of	of	ADP
ejpam-6909	408	5	γ	γ	NOUN
ejpam-6909	408	6	-	-	NOUN
ejpam-6909	408	7	semirings	semiring	NOUN
ejpam-6909	408	8	.	.	PUNCT
ejpam-6909	409	1	asia	asia	PROPN
ejpam-6909	409	2	pac	pac	PROPN
ejpam-6909	409	3	.	.	PUNCT
ejpam-6909	410	1	j.	j.	PROPN
ejpam-6909	410	2	math	math	PROPN
ejpam-6909	410	3	.	.	PUNCT
ejpam-6909	410	4	,	,	PUNCT
ejpam-6909	410	5	10:38	10:38	NUM
ejpam-6909	410	6	,	,	PUNCT
ejpam-6909	410	7	2023	2023	NUM
ejpam-6909	410	8	.	.	PUNCT
ejpam-6909	411	1	[	[	X
ejpam-6909	411	2	13	13	NUM
ejpam-6909	411	3	]	]	X
ejpam-6909	411	4	n.	n.	PROPN
ejpam-6909	411	5	malik	malik	PROPN
ejpam-6909	411	6	,	,	PUNCT
ejpam-6909	411	7	m.	m.	NOUN
ejpam-6909	411	8	shabir	shabir	PROPN
ejpam-6909	411	9	,	,	PUNCT
ejpam-6909	411	10	t.	t.	PROPN
ejpam-6909	411	11	m.	m.	PROPN
ejpam-6909	411	12	al	al	PROPN
ejpam-6909	411	13	-	-	PUNCT
ejpam-6909	411	14	shami	shami	PROPN
ejpam-6909	411	15	,	,	PUNCT
ejpam-6909	411	16	g.	g.	PROPN
ejpam-6909	411	17	rizwan	rizwan	PROPN
ejpam-6909	411	18	,	,	PUNCT
ejpam-6909	411	19	m.	m.	NOUN
ejpam-6909	411	20	arar	arar	PROPN
ejpam-6909	411	21	,	,	PUNCT
ejpam-6909	411	22	and	and	CCONJ
ejpam-6909	411	23	m.	m.	PROPN
ejpam-6909	411	24	hosny	hosny	PROPN
ejpam-6909	411	25	.	.	PUNCT
ejpam-6909	412	1	rough	rough	ADJ
ejpam-6909	412	2	bipolar	bipolar	ADJ
ejpam-6909	412	3	fuzzy	fuzzy	ADJ
ejpam-6909	412	4	ideals	ideal	NOUN
ejpam-6909	412	5	in	in	ADP
ejpam-6909	412	6	semigroups	semigroup	NOUN
ejpam-6909	412	7	.	.	PUNCT
ejpam-6909	413	1	complex	complex	ADJ
ejpam-6909	413	2	intell	intell	PROPN
ejpam-6909	413	3	.	.	PUNCT
ejpam-6909	414	1	syst	syst	PROPN
ejpam-6909	414	2	.	.	PROPN
ejpam-6909	414	3	,	,	PUNCT
ejpam-6909	414	4	9:7197–7212	9:7197–7212	NUM
ejpam-6909	414	5	,	,	PUNCT
ejpam-6909	414	6	2023	2023	NUM
ejpam-6909	414	7	.	.	PUNCT
ejpam-6909	415	1	[	[	X
ejpam-6909	415	2	14	14	NUM
ejpam-6909	415	3	]	]	PUNCT
ejpam-6909	415	4	m.	m.	NOUN
ejpam-6909	415	5	g.	g.	PROPN
ejpam-6909	415	6	fatima	fatima	PROPN
ejpam-6909	415	7	and	and	CCONJ
ejpam-6909	415	8	f.	f.	PROPN
ejpam-6909	415	9	k.	k.	PROPN
ejpam-6909	415	10	fatema	fatema	PROPN
ejpam-6909	415	11	.	.	PUNCT
ejpam-6909	416	1	bipolar	bipolar	ADJ
ejpam-6909	416	2	fuzzy	fuzzy	ADJ
ejpam-6909	416	3	ideals	ideal	NOUN
ejpam-6909	416	4	of	of	ADP
ejpam-6909	416	5	tm	tm	NOUN
ejpam-6909	416	6	-	-	PUNCT
ejpam-6909	416	7	algebras	algebras	PROPN
ejpam-6909	416	8	.	.	PUNCT
ejpam-6909	417	1	aip	aip	PROPN
ejpam-6909	417	2	conf	conf	PROPN
ejpam-6909	417	3	.	.	PUNCT
ejpam-6909	418	1	proc	proc	PROPN
ejpam-6909	418	2	.	.	PROPN
ejpam-6909	418	3	,	,	PUNCT
ejpam-6909	418	4	2834(1):080102	2834(1):080102	NUM
ejpam-6909	418	5	,	,	PUNCT
ejpam-6909	418	6	2023	2023	NUM
ejpam-6909	418	7	.	.	PUNCT
ejpam-6909	419	1	d.	d.	PROPN
ejpam-6909	419	2	ramesh	ramesh	PROPN
ejpam-6909	419	3	et	et	PROPN
ejpam-6909	419	4	al	al	PROPN
ejpam-6909	419	5	.	.	PUNCT
ejpam-6909	419	6	/	/	SYM
ejpam-6909	419	7	eur	eur	PROPN
ejpam-6909	419	8	.	.	PUNCT
ejpam-6909	420	1	j.	j.	PROPN
ejpam-6909	420	2	pure	pure	PROPN
ejpam-6909	420	3	appl	appl	PROPN
ejpam-6909	420	4	.	.	PROPN
ejpam-6909	420	5	math	math	PROPN
ejpam-6909	420	6	,	,	PUNCT
ejpam-6909	420	7	18	18	NUM
ejpam-6909	420	8	(	(	PUNCT
ejpam-6909	420	9	4	4	NUM
ejpam-6909	420	10	)	)	PUNCT
ejpam-6909	420	11	(	(	PUNCT
ejpam-6909	420	12	2025	2025	NUM
ejpam-6909	420	13	)	)	PUNCT
ejpam-6909	420	14	,	,	PUNCT
ejpam-6909	420	15	6909	6909	NUM
ejpam-6909	420	16	16	16	NUM
ejpam-6909	420	17	of	of	ADP
ejpam-6909	420	18	16	16	NUM
ejpam-6909	421	1	[	[	X
ejpam-6909	421	2	15	15	NUM
ejpam-6909	421	3	]	]	X
ejpam-6909	421	4	u.	u.	PROPN
ejpam-6909	421	5	venkata	venkata	PROPN
ejpam-6909	421	6	kalyani	kalyani	PROPN
ejpam-6909	421	7	,	,	PUNCT
ejpam-6909	421	8	b.	b.	PROPN
ejpam-6909	421	9	v.	v.	PROPN
ejpam-6909	421	10	s.	s.	PROPN
ejpam-6909	421	11	n.	n.	PROPN
ejpam-6909	421	12	hari	hari	PROPN
ejpam-6909	421	13	prasad	prasad	PROPN
ejpam-6909	421	14	,	,	PUNCT
ejpam-6909	421	15	t.	t.	PROPN
ejpam-6909	421	16	eswarlal	eswarlal	PROPN
ejpam-6909	421	17	,	,	PUNCT
ejpam-6909	421	18	and	and	CCONJ
ejpam-6909	421	19	a.	a.	NOUN
ejpam-6909	421	20	iampan	iampan	PROPN
ejpam-6909	421	21	.	.	PUNCT
ejpam-6909	422	1	a	a	DET
ejpam-6909	422	2	study	study	NOUN
ejpam-6909	422	3	of	of	ADP
ejpam-6909	422	4	bipolar	bipolar	ADJ
ejpam-6909	422	5	fuzzy	fuzzy	ADJ
ejpam-6909	422	6	prime	prime	ADJ
ejpam-6909	422	7	ideals	ideal	NOUN
ejpam-6909	422	8	of	of	ADP
ejpam-6909	422	9	a	a	DET
ejpam-6909	422	10	lattice	lattice	NOUN
ejpam-6909	422	11	.	.	PUNCT
ejpam-6909	423	1	commun	commun	PROPN
ejpam-6909	423	2	.	.	PUNCT
ejpam-6909	424	1	appl	appl	PROPN
ejpam-6909	424	2	.	.	PUNCT
ejpam-6909	425	1	nonlinear	nonlinear	PROPN
ejpam-6909	425	2	anal	anal	PROPN
ejpam-6909	425	3	.	.	PUNCT
ejpam-6909	425	4	,	,	PUNCT
ejpam-6909	425	5	32(6s):241	32(6s):241	NUM
ejpam-6909	425	6	–	–	PUNCT
ejpam-6909	425	7	249	249	NUM
ejpam-6909	425	8	,	,	PUNCT
ejpam-6909	425	9	2025	2025	NUM
ejpam-6909	425	10	.	.	PUNCT
ejpam-6909	426	1	[	[	X
ejpam-6909	426	2	16	16	NUM
ejpam-6909	426	3	]	]	PUNCT
ejpam-6909	426	4	k.	k.	PROPN
ejpam-6909	426	5	j.	j.	PROPN
ejpam-6909	426	6	lee	lee	PROPN
ejpam-6909	426	7	.	.	PUNCT
ejpam-6909	427	1	bipolar	bipolar	ADJ
ejpam-6909	427	2	fuzzy	fuzzy	ADJ
ejpam-6909	427	3	subalgebras	subalgebra	NOUN
ejpam-6909	427	4	and	and	CCONJ
ejpam-6909	427	5	bipolar	bipolar	ADJ
ejpam-6909	427	6	fuzzy	fuzzy	ADJ
ejpam-6909	427	7	ideals	ideal	NOUN
ejpam-6909	427	8	of	of	ADP
ejpam-6909	427	9	bck	bck	PROPN
ejpam-6909	427	10	/	/	SYM
ejpam-6909	427	11	bci	bci	NOUN
ejpam-6909	427	12	-	-	PUNCT
ejpam-6909	427	13	algebras	algebra	NOUN
ejpam-6909	427	14	.	.	PUNCT
ejpam-6909	428	1	bull	bull	NOUN
ejpam-6909	428	2	.	.	PUNCT
ejpam-6909	429	1	malays	malays	PROPN
ejpam-6909	429	2	.	.	PUNCT
ejpam-6909	430	1	math	math	NOUN
ejpam-6909	430	2	.	.	PUNCT
ejpam-6909	431	1	sci	sci	PROPN
ejpam-6909	431	2	.	.	PROPN
ejpam-6909	431	3	soc	soc	PROPN
ejpam-6909	431	4	.	.	PUNCT
ejpam-6909	432	1	(	(	PUNCT
ejpam-6909	432	2	2	2	NUM
ejpam-6909	432	3	)	)	PUNCT
ejpam-6909	432	4	,	,	PUNCT
ejpam-6909	432	5	32(3):361–373	32(3):361–373	PROPN
ejpam-6909	432	6	,	,	PUNCT
ejpam-6909	432	7	2009	2009	NUM
ejpam-6909	432	8	.	.	PUNCT
ejpam-6909	433	1	[	[	X
ejpam-6909	433	2	17	17	NUM
ejpam-6909	433	3	]	]	PUNCT
ejpam-6909	433	4	a.	a.	NOUN
ejpam-6909	433	5	almuhaimeed	almuhaimeed	NOUN
ejpam-6909	433	6	and	and	CCONJ
ejpam-6909	433	7	a.	a.	NOUN
ejpam-6909	433	8	halimah	halimah	PROPN
ejpam-6909	433	9	.	.	PUNCT
ejpam-6909	434	1	bipolar	bipolar	ADJ
ejpam-6909	434	2	fuzzy	fuzzy	ADJ
ejpam-6909	434	3	commutative	commutative	ADJ
ejpam-6909	434	4	ideals	ideal	NOUN
ejpam-6909	434	5	in	in	ADP
ejpam-6909	434	6	bck	bck	NOUN
ejpam-6909	434	7	-	-	PUNCT
ejpam-6909	434	8	algebras	algebras	PROPN
ejpam-6909	434	9	.	.	PUNCT
ejpam-6909	435	1	eur	eur	PROPN
ejpam-6909	435	2	.	.	PUNCT
ejpam-6909	436	1	j.	j.	PROPN
ejpam-6909	436	2	pure	pure	PROPN
ejpam-6909	436	3	appl	appl	PROPN
ejpam-6909	436	4	.	.	PUNCT
ejpam-6909	436	5	math	math	PROPN
ejpam-6909	436	6	.	.	PUNCT
ejpam-6909	436	7	,	,	PUNCT
ejpam-6909	436	8	17(3):1831–1841	17(3):1831–1841	NUM
ejpam-6909	436	9	,	,	PUNCT
ejpam-6909	436	10	2024	2024	NUM
ejpam-6909	436	11	.	.	PUNCT
ejpam-6909	437	1	[	[	X
ejpam-6909	437	2	18	18	NUM
ejpam-6909	437	3	]	]	X
ejpam-6909	437	4	b.	b.	PROPN
ejpam-6909	437	5	satyanarayana	satyanarayana	PROPN
ejpam-6909	437	6	,	,	PUNCT
ejpam-6909	437	7	s.	s.	PROPN
ejpam-6909	437	8	baji	baji	PROPN
ejpam-6909	437	9	,	,	PUNCT
ejpam-6909	437	10	and	and	CCONJ
ejpam-6909	437	11	d.	d.	PROPN
ejpam-6909	437	12	ramesh	ramesh	PROPN
ejpam-6909	437	13	.	.	PUNCT
ejpam-6909	438	1	bipolar	bipolar	ADJ
ejpam-6909	438	2	intuitionistic	intuitionistic	ADJ
ejpam-6909	438	3	fuzzy	fuzzy	ADJ
ejpam-6909	438	4	implicative	implicative	ADJ
ejpam-6909	438	5	ideals	ideal	NOUN
ejpam-6909	438	6	of	of	ADP
ejpam-6909	438	7	bck	bck	NOUN
ejpam-6909	438	8	-	-	PUNCT
ejpam-6909	438	9	algebra	algebra	NOUN
ejpam-6909	438	10	.	.	PUNCT
ejpam-6909	439	1	asia	asia	PROPN
ejpam-6909	439	2	pac	pac	PROPN
ejpam-6909	439	3	.	.	PUNCT
ejpam-6909	440	1	j.	j.	PROPN
ejpam-6909	440	2	math	math	PROPN
ejpam-6909	440	3	.	.	PUNCT
ejpam-6909	440	4	,	,	PUNCT
ejpam-6909	440	5	10:47	10:47	NUM
ejpam-6909	440	6	,	,	PUNCT
ejpam-6909	440	7	2023	2023	NUM
ejpam-6909	440	8	.	.	PUNCT
ejpam-6909	441	1	[	[	X
ejpam-6909	441	2	19	19	NUM
ejpam-6909	441	3	]	]	X
ejpam-6909	441	4	d.	d.	PROPN
ejpam-6909	441	5	ramesh	ramesh	PROPN
ejpam-6909	441	6	,	,	PUNCT
ejpam-6909	441	7	s.	s.	PROPN
ejpam-6909	441	8	baji	baji	PROPN
ejpam-6909	441	9	,	,	PUNCT
ejpam-6909	441	10	a.	a.	PROPN
ejpam-6909	441	11	iampan	iampan	PROPN
ejpam-6909	441	12	,	,	PUNCT
ejpam-6909	441	13	r.	r.	PROPN
ejpam-6909	441	14	durga	durga	PROPN
ejpam-6909	441	15	prasad	prasad	PROPN
ejpam-6909	441	16	,	,	PUNCT
ejpam-6909	441	17	and	and	CCONJ
ejpam-6909	441	18	b.	b.	PROPN
ejpam-6909	441	19	satyanarayana	satyanarayana	PROPN
ejpam-6909	441	20	.	.	PROPN
ejpam-6909	441	21	bipolarvalued	bipolarvalue	VERB
ejpam-6909	441	22	intuitionistic	intuitionistic	ADJ
ejpam-6909	441	23	fuzzy	fuzzy	ADJ
ejpam-6909	441	24	positive	positive	ADJ
ejpam-6909	441	25	implicative	implicative	ADJ
ejpam-6909	441	26	ideals	ideal	NOUN
ejpam-6909	441	27	in	in	ADP
ejpam-6909	441	28	bck	bck	NOUN
ejpam-6909	441	29	-	-	PUNCT
ejpam-6909	441	30	algebras	algebras	PROPN
ejpam-6909	441	31	.	.	PUNCT
ejpam-6909	442	1	eur	eur	PROPN
ejpam-6909	442	2	.	.	PUNCT
ejpam-6909	443	1	j.	j.	PROPN
ejpam-6909	443	2	pure	pure	PROPN
ejpam-6909	443	3	appl	appl	PROPN
ejpam-6909	443	4	.	.	PUNCT
ejpam-6909	443	5	math	math	PROPN
ejpam-6909	443	6	.	.	PUNCT
ejpam-6909	443	7	,	,	PUNCT
ejpam-6909	444	1	18(1):5699	18(1):5699	NUM
ejpam-6909	444	2	,	,	PUNCT
ejpam-6909	444	3	2025	2025	NUM
ejpam-6909	444	4	.	.	PUNCT
ejpam-6909	445	1	[	[	X
ejpam-6909	445	2	20	20	NUM
ejpam-6909	445	3	]	]	X
ejpam-6909	445	4	y.	y.	PROPN
ejpam-6909	445	5	b.	b.	PROPN
ejpam-6909	445	6	jun	jun	PROPN
ejpam-6909	445	7	,	,	PUNCT
ejpam-6909	445	8	m.	m.	PROPN
ejpam-6909	445	9	s.	s.	PROPN
ejpam-6909	445	10	kang	kang	PROPN
ejpam-6909	445	11	,	,	PUNCT
ejpam-6909	445	12	and	and	CCONJ
ejpam-6909	445	13	h.	h.	PROPN
ejpam-6909	445	14	s.	s.	PROPN
ejpam-6909	445	15	kim	kim	PROPN
ejpam-6909	445	16	.	.	PUNCT
ejpam-6909	446	1	bipolar	bipolar	ADJ
ejpam-6909	446	2	fuzzy	fuzzy	ADJ
ejpam-6909	446	3	hyper	hyper	ADJ
ejpam-6909	446	4	bck	bck	NOUN
ejpam-6909	446	5	-	-	PUNCT
ejpam-6909	446	6	ideals	ideal	NOUN
ejpam-6909	446	7	in	in	ADP
ejpam-6909	446	8	hyper	hyper	ADJ
ejpam-6909	446	9	bck	bck	NOUN
ejpam-6909	446	10	-	-	PUNCT
ejpam-6909	446	11	algebras	algebras	PROPN
ejpam-6909	446	12	.	.	PUNCT
ejpam-6909	447	1	iran	iran	PROPN
ejpam-6909	447	2	.	.	PUNCT
ejpam-6909	448	1	j.	j.	PROPN
ejpam-6909	448	2	fuzzy	fuzzy	PROPN
ejpam-6909	448	3	syst	syst	PROPN
ejpam-6909	448	4	.	.	PROPN
ejpam-6909	448	5	,	,	PUNCT
ejpam-6909	448	6	8(2):105–120	8(2):105–120	NUM
ejpam-6909	448	7	,	,	PUNCT
ejpam-6909	448	8	2011	2011	NUM
ejpam-6909	448	9	.	.	PUNCT
ejpam-6909	449	1	[	[	X
ejpam-6909	449	2	21	21	NUM
ejpam-6909	449	3	]	]	X
ejpam-6909	449	4	y.	y.	PROPN
ejpam-6909	449	5	b.	b.	PROPN
ejpam-6909	449	6	jun	jun	PROPN
ejpam-6909	449	7	,	,	PUNCT
ejpam-6909	449	8	m.	m.	PROPN
ejpam-6909	449	9	s.	s.	PROPN
ejpam-6909	449	10	kang	kang	PROPN
ejpam-6909	449	11	,	,	PUNCT
ejpam-6909	449	12	and	and	CCONJ
ejpam-6909	449	13	h.	h.	PROPN
ejpam-6909	449	14	s.	s.	PROPN
ejpam-6909	449	15	kim	kim	PROPN
ejpam-6909	449	16	.	.	PUNCT
ejpam-6909	450	1	bipolar	bipolar	ADJ
ejpam-6909	450	2	fuzzy	fuzzy	ADJ
ejpam-6909	450	3	implicative	implicative	ADJ
ejpam-6909	450	4	hyper	hyper	ADJ
ejpam-6909	450	5	bck	bck	NOUN
ejpam-6909	450	6	-	-	PUNCT
ejpam-6909	450	7	ideals	ideal	NOUN
ejpam-6909	450	8	in	in	ADP
ejpam-6909	450	9	hyper	hyper	ADJ
ejpam-6909	450	10	bck	bck	NOUN
ejpam-6909	450	11	-	-	PUNCT
ejpam-6909	450	12	algebras	algebra	NOUN
ejpam-6909	450	13	.	.	PUNCT
ejpam-6909	451	1	sci	sci	PROPN
ejpam-6909	451	2	.	.	PROPN
ejpam-6909	451	3	math	math	PROPN
ejpam-6909	451	4	.	.	PUNCT
ejpam-6909	452	1	jpn	jpn	PROPN
ejpam-6909	452	2	.	.	PROPN
ejpam-6909	452	3	,	,	PUNCT
ejpam-6909	453	1	69(2):175–186	69(2):175–186	PROPN
ejpam-6909	453	2	,	,	PUNCT
ejpam-6909	453	3	2009	2009	NUM
ejpam-6909	453	4	.	.	PUNCT
ejpam-6909	454	1	[	[	X
ejpam-6909	454	2	22	22	NUM
ejpam-6909	454	3	]	]	X
ejpam-6909	454	4	y.	y.	PROPN
ejpam-6909	454	5	b.	b.	PROPN
ejpam-6909	454	6	jun	jun	PROPN
ejpam-6909	454	7	,	,	PUNCT
ejpam-6909	454	8	m.	m.	PROPN
ejpam-6909	454	9	s.	s.	PROPN
ejpam-6909	454	10	kang	kang	PROPN
ejpam-6909	454	11	,	,	PUNCT
ejpam-6909	454	12	and	and	CCONJ
ejpam-6909	454	13	h.	h.	PROPN
ejpam-6909	454	14	s.	s.	PROPN
ejpam-6909	454	15	kim	kim	PROPN
ejpam-6909	454	16	.	.	PUNCT
ejpam-6909	455	1	bipolar	bipolar	ADJ
ejpam-6909	455	2	fuzzy	fuzzy	ADJ
ejpam-6909	455	3	implicative	implicative	ADJ
ejpam-6909	455	4	hyper	hyper	ADJ
ejpam-6909	455	5	bck	bck	NOUN
ejpam-6909	455	6	-	-	PUNCT
ejpam-6909	455	7	ideals	ideal	NOUN
ejpam-6909	455	8	in	in	ADP
ejpam-6909	455	9	hyper	hyper	ADJ
ejpam-6909	455	10	bck	bck	NOUN
ejpam-6909	455	11	-	-	PUNCT
ejpam-6909	455	12	algebras	algebra	NOUN
ejpam-6909	455	13	.	.	PUNCT
ejpam-6909	456	1	sci	sci	PROPN
ejpam-6909	456	2	.	.	PROPN
ejpam-6909	456	3	math	math	PROPN
ejpam-6909	456	4	.	.	PUNCT
ejpam-6909	457	1	jpn	jpn	PROPN
ejpam-6909	457	2	.	.	PROPN
ejpam-6909	457	3	,	,	PUNCT
ejpam-6909	458	1	69(2):175–186	69(2):175–186	PROPN
ejpam-6909	458	2	,	,	PUNCT
ejpam-6909	458	3	2009	2009	NUM
ejpam-6909	458	4	.	.	PUNCT
ejpam-6909	459	1	[	[	X
ejpam-6909	459	2	23	23	NUM
ejpam-6909	459	3	]	]	X
ejpam-6909	459	4	y.	y.	PROPN
ejpam-6909	459	5	b.	b.	PROPN
ejpam-6909	459	6	jun	jun	PROPN
ejpam-6909	459	7	,	,	PUNCT
ejpam-6909	459	8	m.	m.	PROPN
ejpam-6909	459	9	s.	s.	PROPN
ejpam-6909	459	10	kang	kang	PROPN
ejpam-6909	459	11	,	,	PUNCT
ejpam-6909	459	12	and	and	CCONJ
ejpam-6909	459	13	h.	h.	PROPN
ejpam-6909	459	14	s.	s.	PROPN
ejpam-6909	459	15	kim	kim	PROPN
ejpam-6909	459	16	.	.	PUNCT
ejpam-6909	460	1	bipolar	bipolar	ADJ
ejpam-6909	460	2	fuzzy	fuzzy	ADJ
ejpam-6909	460	3	structures	structure	NOUN
ejpam-6909	460	4	of	of	ADP
ejpam-6909	460	5	some	some	DET
ejpam-6909	460	6	types	type	NOUN
ejpam-6909	460	7	of	of	ADP
ejpam-6909	460	8	ideals	ideal	NOUN
ejpam-6909	460	9	in	in	ADP
ejpam-6909	460	10	hyper	hyper	ADJ
ejpam-6909	460	11	bck	bck	NOUN
ejpam-6909	460	12	-	-	PUNCT
ejpam-6909	460	13	algebras	algebra	NOUN
ejpam-6909	460	14	.	.	PUNCT
ejpam-6909	461	1	sci	sci	PROPN
ejpam-6909	461	2	.	.	PROPN
ejpam-6909	461	3	math	math	PROPN
ejpam-6909	461	4	.	.	PUNCT
ejpam-6909	462	1	jpn	jpn	PROPN
ejpam-6909	462	2	.	.	PROPN
ejpam-6909	462	3	,	,	PUNCT
ejpam-6909	463	1	70(1):109–121	70(1):109–121	NUM
ejpam-6909	463	2	,	,	PUNCT
ejpam-6909	463	3	2009	2009	NUM
ejpam-6909	463	4	.	.	PUNCT
ejpam-6909	464	1	[	[	X
ejpam-6909	464	2	24	24	NUM
ejpam-6909	464	3	]	]	X
ejpam-6909	464	4	y.	y.	PROPN
ejpam-6909	464	5	b.	b.	PROPN
ejpam-6909	464	6	jun	jun	PROPN
ejpam-6909	464	7	,	,	PUNCT
ejpam-6909	464	8	m.	m.	PROPN
ejpam-6909	464	9	s.	s.	PROPN
ejpam-6909	464	10	kang	kang	PROPN
ejpam-6909	464	11	,	,	PUNCT
ejpam-6909	464	12	and	and	CCONJ
ejpam-6909	464	13	s.	s.	PROPN
ejpam-6909	464	14	z.	z.	PROPN
ejpam-6909	464	15	song	song	PROPN
ejpam-6909	464	16	.	.	PUNCT
ejpam-6909	465	1	several	several	ADJ
ejpam-6909	465	2	types	type	NOUN
ejpam-6909	465	3	of	of	ADP
ejpam-6909	465	4	bipolar	bipolar	ADJ
ejpam-6909	465	5	fuzzy	fuzzy	ADJ
ejpam-6909	465	6	hyper	hyper	ADJ
ejpam-6909	465	7	bckideals	bckideal	NOUN
ejpam-6909	465	8	in	in	ADP
ejpam-6909	465	9	hyper	hyper	ADJ
ejpam-6909	465	10	bck	bck	NOUN
ejpam-6909	465	11	-	-	PUNCT
ejpam-6909	465	12	algebras	algebras	PROPN
ejpam-6909	465	13	.	.	PUNCT
ejpam-6909	466	1	honam	honam	PROPN
ejpam-6909	466	2	math	math	PROPN
ejpam-6909	466	3	.	.	PUNCT
ejpam-6909	467	1	j.	j.	PROPN
ejpam-6909	467	2	,	,	PUNCT
ejpam-6909	467	3	34(2):145–159	34(2):145–159	PROPN
ejpam-6909	467	4	,	,	PUNCT
ejpam-6909	467	5	2012	2012	NUM
ejpam-6909	467	6	.	.	PUNCT
ejpam-6909	468	1	[	[	X
ejpam-6909	468	2	25	25	NUM
ejpam-6909	468	3	]	]	X
ejpam-6909	468	4	g.	g.	PROPN
ejpam-6909	468	5	muhiuddin	muhiuddin	PROPN
ejpam-6909	468	6	,	,	PUNCT
ejpam-6909	468	7	h.	h.	PROPN
ejpam-6909	468	8	harizavi	harizavi	PROPN
ejpam-6909	468	9	,	,	PUNCT
ejpam-6909	468	10	and	and	CCONJ
ejpam-6909	468	11	y.	y.	PROPN
ejpam-6909	468	12	b.	b.	PROPN
ejpam-6909	468	13	jun	jun	PROPN
ejpam-6909	468	14	.	.	PROPN
ejpam-6909	469	1	bipolar	bipolar	ADJ
ejpam-6909	469	2	-	-	PUNCT
ejpam-6909	469	3	valued	value	VERB
ejpam-6909	469	4	fuzzy	fuzzy	ADJ
ejpam-6909	469	5	soft	soft	ADJ
ejpam-6909	469	6	hyper	hyper	ADJ
ejpam-6909	469	7	bck	bck	NOUN
ejpam-6909	469	8	ideals	ideal	NOUN
ejpam-6909	469	9	in	in	ADP
ejpam-6909	469	10	hyper	hyper	ADJ
ejpam-6909	469	11	bck	bck	PROPN
ejpam-6909	469	12	algebras	algebra	NOUN
ejpam-6909	469	13	.	.	PUNCT
ejpam-6909	470	1	discrete	discrete	ADJ
ejpam-6909	470	2	math	math	NOUN
ejpam-6909	470	3	.	.	PUNCT
ejpam-6909	471	1	algorithms	algorithms	PROPN
ejpam-6909	471	2	appl	appl	PROPN
ejpam-6909	471	3	.	.	PROPN
ejpam-6909	471	4	,	,	PUNCT
ejpam-6909	471	5	12(2):2050018	12(2):2050018	NUM
ejpam-6909	471	6	,	,	PUNCT
ejpam-6909	471	7	2020	2020	NUM
ejpam-6909	471	8	.	.	PUNCT
ejpam-6909	472	1	[	[	X
ejpam-6909	472	2	26	26	NUM
ejpam-6909	472	3	]	]	PUNCT
ejpam-6909	472	4	m.	m.	NOUN
ejpam-6909	472	5	balamurugan	balamurugan	NOUN
ejpam-6909	472	6	,	,	PUNCT
ejpam-6909	472	7	k.	k.	PROPN
ejpam-6909	472	8	h.	h.	PROPN
ejpam-6909	472	9	hakami	hakami	PROPN
ejpam-6909	472	10	,	,	PUNCT
ejpam-6909	472	11	m.	m.	NOUN
ejpam-6909	472	12	a.	a.	NOUN
ejpam-6909	472	13	ansari	ansari	PROPN
ejpam-6909	472	14	,	,	PUNCT
ejpam-6909	472	15	and	and	CCONJ
ejpam-6909	472	16	k.	k.	PROPN
ejpam-6909	472	17	loganathan	loganathan	PROPN
ejpam-6909	472	18	.	.	PUNCT
ejpam-6909	473	1	an	an	DET
ejpam-6909	473	2	innovative	innovative	ADJ
ejpam-6909	473	3	perspective	perspective	NOUN
ejpam-6909	473	4	on	on	ADP
ejpam-6909	473	5	bipolar	bipolar	ADJ
ejpam-6909	473	6	fuzzy	fuzzy	ADJ
ejpam-6909	473	7	fantastic	fantastic	ADJ
ejpam-6909	473	8	ideals	ideal	NOUN
ejpam-6909	473	9	in	in	ADP
ejpam-6909	473	10	bck	bck	PROPN
ejpam-6909	473	11	/	/	SYM
ejpam-6909	473	12	bci	bci	NOUN
ejpam-6909	473	13	-	-	PUNCT
ejpam-6909	473	14	algebras	algebra	NOUN
ejpam-6909	473	15	.	.	PUNCT
ejpam-6909	474	1	eur	eur	PROPN
ejpam-6909	474	2	.	.	PUNCT
ejpam-6909	475	1	j.	j.	PROPN
ejpam-6909	475	2	pure	pure	PROPN
ejpam-6909	475	3	appl	appl	PROPN
ejpam-6909	475	4	.	.	PUNCT
ejpam-6909	475	5	math	math	PROPN
ejpam-6909	475	6	.	.	PUNCT
ejpam-6909	475	7	,	,	PUNCT
ejpam-6909	475	8	17(4):3973–3983	17(4):3973–3983	NUM
ejpam-6909	475	9	,	,	PUNCT
ejpam-6909	475	10	2024	2024	NUM
ejpam-6909	475	11	.	.	PUNCT
ejpam-6909	476	1	[	[	X
ejpam-6909	476	2	27	27	NUM
ejpam-6909	476	3	]	]	PUNCT
ejpam-6909	476	4	w.-r	w.-r	PROPN
ejpam-6909	476	5	.	.	PUNCT
ejpam-6909	477	1	zhang	zhang	PROPN
ejpam-6909	477	2	.	.	PUNCT
ejpam-6909	478	1	(	(	PUNCT
ejpam-6909	478	2	yin	yin	PROPN
ejpam-6909	478	3	)	)	PUNCT
ejpam-6909	478	4	(	(	PUNCT
ejpam-6909	478	5	yang	yang	NOUN
ejpam-6909	478	6	)	)	PUNCT
ejpam-6909	478	7	bipolar	bipolar	ADJ
ejpam-6909	478	8	fuzzy	fuzzy	ADJ
ejpam-6909	478	9	sets	set	NOUN
ejpam-6909	478	10	.	.	PUNCT
ejpam-6909	479	1	in	in	ADP
ejpam-6909	479	2	1998	1998	NUM
ejpam-6909	479	3	ieee	ieee	NOUN
ejpam-6909	479	4	international	international	ADJ
ejpam-6909	479	5	conference	conference	NOUN
ejpam-6909	479	6	on	on	ADP
ejpam-6909	479	7	fuzzy	fuzzy	ADJ
ejpam-6909	479	8	systems	system	NOUN
ejpam-6909	479	9	proceedings	proceeding	NOUN
ejpam-6909	479	10	.	.	PUNCT
ejpam-6909	480	1	ieee	ieee	PROPN
ejpam-6909	480	2	world	world	PROPN
ejpam-6909	480	3	congress	congress	PROPN
ejpam-6909	480	4	on	on	ADP
ejpam-6909	480	5	computational	computational	ADJ
ejpam-6909	480	6	intelligence	intelligence	NOUN
ejpam-6909	480	7	(	(	PUNCT
ejpam-6909	480	8	cat	cat	NOUN
ejpam-6909	480	9	.	.	PUNCT
ejpam-6909	481	1	no.98ch36228	no.98ch36228	PROPN
ejpam-6909	481	2	)	)	PUNCT
ejpam-6909	482	1	,	,	PUNCT
ejpam-6909	482	2	volume	volume	NOUN
ejpam-6909	482	3	1	1	NUM
ejpam-6909	482	4	,	,	PUNCT
ejpam-6909	482	5	pages	page	NOUN
ejpam-6909	482	6	835–840	835–840	NUM
ejpam-6909	482	7	,	,	PUNCT
ejpam-6909	482	8	anchorage	anchorage	PROPN
ejpam-6909	482	9	,	,	PUNCT
ejpam-6909	482	10	ak	ak	PROPN
ejpam-6909	482	11	,	,	PUNCT
ejpam-6909	482	12	usa	usa	PROPN
ejpam-6909	482	13	,	,	PUNCT
ejpam-6909	482	14	1998	1998	NUM
ejpam-6909	482	15	.	.	PUNCT
ejpam-6909	483	1	[	[	X
ejpam-6909	483	2	28	28	NUM
ejpam-6909	483	3	]	]	X
ejpam-6909	483	4	y.	y.	PROPN
ejpam-6909	483	5	b.	b.	PROPN
ejpam-6909	483	6	jun	jun	PROPN
ejpam-6909	483	7	,	,	PUNCT
ejpam-6909	483	8	x.	x.	PROPN
ejpam-6909	483	9	l.	l.	PROPN
ejpam-6909	483	10	xin	xin	PROPN
ejpam-6909	483	11	,	,	PUNCT
ejpam-6909	483	12	m.	m.	PROPN
ejpam-6909	483	13	m.	m.	PROPN
ejpam-6909	483	14	zahedi	zahedi	PROPN
ejpam-6909	483	15	,	,	PUNCT
ejpam-6909	483	16	and	and	CCONJ
ejpam-6909	483	17	e.	e.	PROPN
ejpam-6909	483	18	h.	h.	PROPN
ejpam-6909	483	19	roh	roh	PROPN
ejpam-6909	483	20	.	.	PUNCT
ejpam-6909	484	1	strong	strong	ADJ
ejpam-6909	484	2	hyper	hyper	ADJ
ejpam-6909	484	3	bck	bck	NOUN
ejpam-6909	484	4	-	-	PUNCT
ejpam-6909	484	5	ideals	ideal	NOUN
ejpam-6909	484	6	of	of	ADP
ejpam-6909	484	7	hyper	hyper	ADJ
ejpam-6909	484	8	bck	bck	NOUN
ejpam-6909	484	9	-	-	PUNCT
ejpam-6909	484	10	algebras	algebras	PROPN
ejpam-6909	484	11	.	.	PUNCT
ejpam-6909	485	1	math	math	PROPN
ejpam-6909	485	2	.	.	PUNCT
ejpam-6909	486	1	japon	japon	PROPN
ejpam-6909	486	2	.	.	PROPN
ejpam-6909	486	3	,	,	PUNCT
ejpam-6909	487	1	51(3):493–498	51(3):493–498	PROPN
ejpam-6909	487	2	,	,	PUNCT
ejpam-6909	487	3	2000	2000	NUM
ejpam-6909	487	4	.	.	PUNCT