id	sid	tid	token	lemma	pos
ejpam-3786	1	1	european	european	PROPN
ejpam-3786	1	2	journal	journal	PROPN
ejpam-3786	1	3	of	of	ADP
ejpam-3786	1	4	pure	pure	ADJ
ejpam-3786	1	5	and	and	CCONJ
ejpam-3786	1	6	applied	apply	VERB
ejpam-3786	1	7	mathematics	mathematic	NOUN
ejpam-3786	1	8	vol	vol	NOUN
ejpam-3786	1	9	.	.	PROPN
ejpam-3786	2	1	13	13	NUM
ejpam-3786	2	2	,	,	PUNCT
ejpam-3786	2	3	no	no	INTJ
ejpam-3786	2	4	.	.	NOUN
ejpam-3786	2	5	4	4	NUM
ejpam-3786	2	6	,	,	PUNCT
ejpam-3786	2	7	2020	2020	NUM
ejpam-3786	2	8	,	,	PUNCT
ejpam-3786	2	9	914	914	NUM
ejpam-3786	2	10	-	-	SYM
ejpam-3786	2	11	938	938	NUM
ejpam-3786	2	12	issn	issn	PROPN
ejpam-3786	2	13	1307	1307	NUM
ejpam-3786	2	14	-	-	SYM
ejpam-3786	2	15	5543	5543	NUM
ejpam-3786	2	16	–	–	PUNCT
ejpam-3786	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3786	2	18	published	publish	VERB
ejpam-3786	2	19	by	by	ADP
ejpam-3786	2	20	new	new	PROPN
ejpam-3786	2	21	york	york	PROPN
ejpam-3786	2	22	business	business	PROPN
ejpam-3786	2	23	global	global	PROPN
ejpam-3786	2	24	levin	levin	PROPN
ejpam-3786	2	25	conjecture	conjecture	NOUN
ejpam-3786	2	26	for	for	ADP
ejpam-3786	2	27	group	group	NOUN
ejpam-3786	2	28	equations	equation	NOUN
ejpam-3786	2	29	of	of	ADP
ejpam-3786	2	30	length	length	NOUN
ejpam-3786	2	31	9	9	NUM
ejpam-3786	2	32	muhammad	muhammad	PROPN
ejpam-3786	2	33	saeed	saeed	PROPN
ejpam-3786	2	34	akram1,∗	akram1,∗	PROPN
ejpam-3786	2	35	,	,	PUNCT
ejpam-3786	3	1	maira	maira	PROPN
ejpam-3786	3	2	amjid1	amjid1	PROPN
ejpam-3786	3	3	,	,	PUNCT
ejpam-3786	3	4	sohail	sohail	PROPN
ejpam-3786	3	5	iqbal	iqbal	PROPN
ejpam-3786	3	6	2	2	NUM
ejpam-3786	3	7	1	1	NUM
ejpam-3786	3	8	department	department	NOUN
ejpam-3786	3	9	of	of	ADP
ejpam-3786	3	10	mathematics	mathematics	PROPN
ejpam-3786	3	11	,	,	PUNCT
ejpam-3786	3	12	khwaja	khwaja	PROPN
ejpam-3786	3	13	fareed	fareed	PROPN
ejpam-3786	3	14	university	university	PROPN
ejpam-3786	3	15	of	of	ADP
ejpam-3786	3	16	engineering	engineering	NOUN
ejpam-3786	3	17	and	and	CCONJ
ejpam-3786	3	18	information	information	NOUN
ejpam-3786	3	19	technology	technology	NOUN
ejpam-3786	3	20	,	,	PUNCT
ejpam-3786	3	21	rahim	rahim	PROPN
ejpam-3786	3	22	yar	yar	PROPN
ejpam-3786	3	23	khan	khan	PROPN
ejpam-3786	3	24	,	,	PUNCT
ejpam-3786	3	25	64200	64200	NUM
ejpam-3786	3	26	,	,	PUNCT
ejpam-3786	3	27	pakistan	pakistan	PROPN
ejpam-3786	3	28	2	2	NUM
ejpam-3786	3	29	department	department	NOUN
ejpam-3786	3	30	of	of	ADP
ejpam-3786	3	31	mathematics	mathematics	PROPN
ejpam-3786	3	32	,	,	PUNCT
ejpam-3786	3	33	cui	cui	NOUN
ejpam-3786	3	34	,	,	PUNCT
ejpam-3786	3	35	islamabad	islamabad	PROPN
ejpam-3786	3	36	,	,	PUNCT
ejpam-3786	3	37	pakistan	pakistan	PROPN
ejpam-3786	3	38	abstract	abstract	NOUN
ejpam-3786	3	39	.	.	PUNCT
ejpam-3786	4	1	levin	levin	PROPN
ejpam-3786	4	2	conjecture	conjecture	PROPN
ejpam-3786	4	3	states	state	VERB
ejpam-3786	4	4	that	that	SCONJ
ejpam-3786	4	5	every	every	DET
ejpam-3786	4	6	group	group	NOUN
ejpam-3786	4	7	equation	equation	NOUN
ejpam-3786	4	8	is	be	AUX
ejpam-3786	4	9	solvable	solvable	ADJ
ejpam-3786	4	10	over	over	ADP
ejpam-3786	4	11	any	any	DET
ejpam-3786	4	12	torsion	torsion	NOUN
ejpam-3786	4	13	free	free	ADJ
ejpam-3786	4	14	group	group	NOUN
ejpam-3786	4	15	.	.	PUNCT
ejpam-3786	5	1	the	the	DET
ejpam-3786	5	2	conjecture	conjecture	NOUN
ejpam-3786	5	3	is	be	AUX
ejpam-3786	5	4	shown	show	VERB
ejpam-3786	5	5	to	to	PART
ejpam-3786	5	6	hold	hold	VERB
ejpam-3786	5	7	true	true	ADJ
ejpam-3786	5	8	for	for	ADP
ejpam-3786	5	9	group	group	NOUN
ejpam-3786	5	10	equation	equation	NOUN
ejpam-3786	5	11	of	of	ADP
ejpam-3786	5	12	length	length	NOUN
ejpam-3786	5	13	seven	seven	NUM
ejpam-3786	5	14	using	use	VERB
ejpam-3786	5	15	weight	weight	NOUN
ejpam-3786	5	16	test	test	NOUN
ejpam-3786	5	17	and	and	CCONJ
ejpam-3786	5	18	curvature	curvature	VERB
ejpam-3786	5	19	distribution	distribution	NOUN
ejpam-3786	5	20	method	method	NOUN
ejpam-3786	5	21	.	.	PUNCT
ejpam-3786	6	1	recently	recently	ADV
ejpam-3786	6	2	,	,	PUNCT
ejpam-3786	6	3	these	these	DET
ejpam-3786	6	4	methods	method	NOUN
ejpam-3786	6	5	are	be	AUX
ejpam-3786	6	6	used	use	VERB
ejpam-3786	6	7	to	to	PART
ejpam-3786	6	8	show	show	VERB
ejpam-3786	6	9	that	that	SCONJ
ejpam-3786	6	10	levin	levin	PROPN
ejpam-3786	6	11	conjecture	conjecture	NOUN
ejpam-3786	6	12	is	be	AUX
ejpam-3786	6	13	true	true	ADJ
ejpam-3786	6	14	for	for	SCONJ
ejpam-3786	6	15	some	some	DET
ejpam-3786	6	16	group	group	NOUN
ejpam-3786	6	17	equations	equation	NOUN
ejpam-3786	6	18	of	of	ADP
ejpam-3786	6	19	length	length	NOUN
ejpam-3786	6	20	eight	eight	NUM
ejpam-3786	6	21	and	and	CCONJ
ejpam-3786	6	22	nine	nine	NUM
ejpam-3786	6	23	modulo	modulo	NOUN
ejpam-3786	6	24	some	some	DET
ejpam-3786	6	25	exceptional	exceptional	ADJ
ejpam-3786	6	26	cases	case	NOUN
ejpam-3786	6	27	.	.	PUNCT
ejpam-3786	7	1	in	in	ADP
ejpam-3786	7	2	this	this	DET
ejpam-3786	7	3	paper	paper	NOUN
ejpam-3786	7	4	,	,	PUNCT
ejpam-3786	7	5	we	we	PRON
ejpam-3786	7	6	show	show	VERB
ejpam-3786	7	7	that	that	SCONJ
ejpam-3786	7	8	levin	levin	PROPN
ejpam-3786	7	9	conjecture	conjecture	NOUN
ejpam-3786	7	10	holds	hold	VERB
ejpam-3786	7	11	true	true	ADJ
ejpam-3786	7	12	for	for	ADP
ejpam-3786	7	13	a	a	DET
ejpam-3786	7	14	group	group	NOUN
ejpam-3786	7	15	equation	equation	NOUN
ejpam-3786	7	16	of	of	ADP
ejpam-3786	7	17	length	length	NOUN
ejpam-3786	7	18	nine	nine	NUM
ejpam-3786	7	19	modulo	modulo	NOUN
ejpam-3786	7	20	2	2	NUM
ejpam-3786	7	21	exceptional	exceptional	ADJ
ejpam-3786	7	22	cases	case	NOUN
ejpam-3786	7	23	.	.	PUNCT
ejpam-3786	8	1	in	in	ADP
ejpam-3786	8	2	addition	addition	NOUN
ejpam-3786	8	3	,	,	PUNCT
ejpam-3786	8	4	we	we	PRON
ejpam-3786	8	5	allude	allude	VERB
ejpam-3786	8	6	the	the	DET
ejpam-3786	8	7	list	list	NOUN
ejpam-3786	8	8	of	of	ADP
ejpam-3786	8	9	cases	case	NOUN
ejpam-3786	8	10	that	that	PRON
ejpam-3786	8	11	are	be	AUX
ejpam-3786	8	12	still	still	ADV
ejpam-3786	8	13	open	open	ADJ
ejpam-3786	8	14	for	for	ADP
ejpam-3786	8	15	two	two	NUM
ejpam-3786	8	16	more	more	ADJ
ejpam-3786	8	17	equations	equation	NOUN
ejpam-3786	8	18	of	of	ADP
ejpam-3786	8	19	length	length	NOUN
ejpam-3786	8	20	nine	nine	NUM
ejpam-3786	8	21	.	.	PUNCT
ejpam-3786	9	1	2020	2020	NUM
ejpam-3786	9	2	mathematics	mathematic	NOUN
ejpam-3786	9	3	subject	subject	NOUN
ejpam-3786	9	4	classifications	classification	NOUN
ejpam-3786	9	5	:	:	PUNCT
ejpam-3786	9	6	20f05	20f05	NUM
ejpam-3786	9	7	,	,	PUNCT
ejpam-3786	9	8	20e06	20e06	NUM
ejpam-3786	9	9	,	,	PUNCT
ejpam-3786	9	10	57m05	57m05	NUM
ejpam-3786	9	11	key	key	ADJ
ejpam-3786	9	12	words	word	NOUN
ejpam-3786	9	13	and	and	CCONJ
ejpam-3786	9	14	phrases	phrase	NOUN
ejpam-3786	9	15	:	:	PUNCT
ejpam-3786	9	16	group	group	NOUN
ejpam-3786	9	17	equations	equation	NOUN
ejpam-3786	9	18	,	,	PUNCT
ejpam-3786	9	19	torsion	torsion	NOUN
ejpam-3786	9	20	-	-	PUNCT
ejpam-3786	9	21	free	free	ADJ
ejpam-3786	9	22	groups	group	NOUN
ejpam-3786	9	23	,	,	PUNCT
ejpam-3786	9	24	relative	relative	ADJ
ejpam-3786	9	25	group	group	NOUN
ejpam-3786	9	26	presentations	presentation	NOUN
ejpam-3786	9	27	,	,	PUNCT
ejpam-3786	9	28	asphericity	asphericity	NOUN
ejpam-3786	9	29	,	,	PUNCT
ejpam-3786	9	30	weight	weight	NOUN
ejpam-3786	9	31	test	test	NOUN
ejpam-3786	9	32	,	,	PUNCT
ejpam-3786	9	33	curvature	curvature	VERB
ejpam-3786	9	34	distribution	distribution	NOUN
ejpam-3786	9	35	1	1	NUM
ejpam-3786	9	36	.	.	PUNCT
ejpam-3786	10	1	introduction	introduction	NOUN
ejpam-3786	10	2	:	:	PUNCT
ejpam-3786	10	3	let	let	VERB
ejpam-3786	10	4	g	g	PRON
ejpam-3786	10	5	be	be	AUX
ejpam-3786	10	6	a	a	DET
ejpam-3786	10	7	non	non	ADJ
ejpam-3786	10	8	-	-	ADJ
ejpam-3786	10	9	trivial	trivial	ADJ
ejpam-3786	10	10	group	group	NOUN
ejpam-3786	10	11	and	and	CCONJ
ejpam-3786	10	12	t	t	NOUN
ejpam-3786	10	13	an	an	DET
ejpam-3786	10	14	element	element	NOUN
ejpam-3786	10	15	not	not	PART
ejpam-3786	10	16	in	in	ADP
ejpam-3786	10	17	g.	g.	PROPN
ejpam-3786	10	18	a	a	DET
ejpam-3786	10	19	group	group	NOUN
ejpam-3786	10	20	equation	equation	NOUN
ejpam-3786	10	21	over	over	ADP
ejpam-3786	10	22	g	g	PROPN
ejpam-3786	10	23	is	be	AUX
ejpam-3786	10	24	an	an	DET
ejpam-3786	10	25	equation	equation	NOUN
ejpam-3786	10	26	of	of	ADP
ejpam-3786	10	27	the	the	DET
ejpam-3786	10	28	form	form	NOUN
ejpam-3786	10	29	s(t	s(t	PROPN
ejpam-3786	10	30	)	)	PUNCT
ejpam-3786	11	1	=	=	PUNCT
ejpam-3786	11	2	g1	g1	PROPN
ejpam-3786	11	3	t	t	PROPN
ejpam-3786	12	1	l1g2	l1g2	PROPN
ejpam-3786	12	2	t	t	PROPN
ejpam-3786	12	3	l2	l2	NOUN
ejpam-3786	12	4	...	...	PUNCT
ejpam-3786	13	1	gnt	gnt	PROPN
ejpam-3786	13	2	ln	ln	NOUN
ejpam-3786	14	1	=	=	SYM
ejpam-3786	14	2	1	1	NUM
ejpam-3786	14	3	(	(	PUNCT
ejpam-3786	14	4	gi	gi	VERB
ejpam-3786	14	5	∈	∈	PROPN
ejpam-3786	14	6	g	g	PROPN
ejpam-3786	14	7	,	,	PUNCT
ejpam-3786	14	8	li	li	PROPN
ejpam-3786	14	9	=	=	SYM
ejpam-3786	14	10	±1	±1	PROPN
ejpam-3786	14	11	)	)	PUNCT
ejpam-3786	14	12	such	such	ADJ
ejpam-3786	14	13	that	that	SCONJ
ejpam-3786	14	14	li	li	PROPN
ejpam-3786	15	1	+	+	CCONJ
ejpam-3786	15	2	li+1	li+1	NOUN
ejpam-3786	15	3	=	=	SYM
ejpam-3786	15	4	0	0	NUM
ejpam-3786	15	5	implies	imply	VERB
ejpam-3786	15	6	gi+1	gi+1	NOUN
ejpam-3786	15	7	6=	6=	SYM
ejpam-3786	15	8	1	1	NUM
ejpam-3786	15	9	∈	∈	PROPN
ejpam-3786	15	10	g	g	NOUN
ejpam-3786	15	11	(	(	PUNCT
ejpam-3786	15	12	subscripts	subscript	NOUN
ejpam-3786	15	13	modulo	modulo	NOUN
ejpam-3786	15	14	n	n	CCONJ
ejpam-3786	15	15	)	)	PUNCT
ejpam-3786	15	16	.	.	PUNCT
ejpam-3786	16	1	the	the	DET
ejpam-3786	16	2	non	non	ADJ
ejpam-3786	16	3	-	-	ADJ
ejpam-3786	16	4	negative	negative	ADJ
ejpam-3786	16	5	integer	integer	NOUN
ejpam-3786	16	6	n	n	NUM
ejpam-3786	16	7	is	be	AUX
ejpam-3786	16	8	known	know	VERB
ejpam-3786	16	9	as	as	ADP
ejpam-3786	16	10	the	the	DET
ejpam-3786	16	11	length	length	NOUN
ejpam-3786	16	12	of	of	ADP
ejpam-3786	16	13	equation	equation	NOUN
ejpam-3786	16	14	s(t	s(t	PROPN
ejpam-3786	16	15	)	)	PUNCT
ejpam-3786	16	16	=	=	PUNCT
ejpam-3786	17	1	1	1	X
ejpam-3786	17	2	.	.	PUNCT
ejpam-3786	18	1	the	the	DET
ejpam-3786	18	2	equation	equation	NOUN
ejpam-3786	18	3	s(t	s(t	PROPN
ejpam-3786	18	4	)	)	PUNCT
ejpam-3786	19	1	=	=	PUNCT
ejpam-3786	19	2	1	1	NUM
ejpam-3786	19	3	is	be	AUX
ejpam-3786	19	4	said	say	VERB
ejpam-3786	19	5	to	to	PART
ejpam-3786	19	6	be	be	AUX
ejpam-3786	19	7	solvable	solvable	ADJ
ejpam-3786	19	8	over	over	ADP
ejpam-3786	19	9	g	g	PROPN
ejpam-3786	19	10	if	if	SCONJ
ejpam-3786	19	11	s(h	s(h	PROPN
ejpam-3786	19	12	)	)	PUNCT
ejpam-3786	19	13	=	=	SYM
ejpam-3786	19	14	1	1	NUM
ejpam-3786	19	15	for	for	ADP
ejpam-3786	19	16	some	some	DET
ejpam-3786	19	17	element	element	ADJ
ejpam-3786	19	18	h	h	NOUN
ejpam-3786	19	19	of	of	ADP
ejpam-3786	19	20	a	a	DET
ejpam-3786	19	21	group	group	NOUN
ejpam-3786	19	22	h	h	NOUN
ejpam-3786	19	23	which	which	PRON
ejpam-3786	19	24	contain	contain	VERB
ejpam-3786	19	25	g.	g.	PROPN
ejpam-3786	19	26	equivalently	equivalently	ADV
ejpam-3786	19	27	,	,	PUNCT
ejpam-3786	19	28	s(t	s(t	PROPN
ejpam-3786	19	29	)	)	PUNCT
ejpam-3786	19	30	=	=	PUNCT
ejpam-3786	19	31	1	1	NUM
ejpam-3786	19	32	is	be	AUX
ejpam-3786	19	33	solvable	solvable	ADJ
ejpam-3786	19	34	over	over	ADP
ejpam-3786	19	35	g	g	PROPN
ejpam-3786	19	36	if	if	SCONJ
ejpam-3786	20	1	and	and	CCONJ
ejpam-3786	20	2	only	only	ADV
ejpam-3786	20	3	if	if	SCONJ
ejpam-3786	20	4	the	the	DET
ejpam-3786	20	5	natural	natural	ADJ
ejpam-3786	20	6	homomorphism	homomorphism	NOUN
ejpam-3786	20	7	from	from	ADP
ejpam-3786	20	8	g	g	NOUN
ejpam-3786	20	9	to	to	ADP
ejpam-3786	20	10	g∗(t	g∗(t	NOUN
ejpam-3786	20	11	)	)	PUNCT
ejpam-3786	20	12	n	n	PART
ejpam-3786	20	13	is	be	AUX
ejpam-3786	20	14	injective	injective	ADJ
ejpam-3786	20	15	,	,	PUNCT
ejpam-3786	20	16	in	in	ADP
ejpam-3786	20	17	which	which	PRON
ejpam-3786	20	18	n	n	ADV
ejpam-3786	20	19	is	be	AUX
ejpam-3786	20	20	the	the	DET
ejpam-3786	20	21	normal	normal	ADJ
ejpam-3786	20	22	closure	closure	NOUN
ejpam-3786	20	23	of	of	ADP
ejpam-3786	20	24	s(t	s(t	PROPN
ejpam-3786	20	25	)	)	PUNCT
ejpam-3786	20	26	=	=	PUNCT
ejpam-3786	21	1	1	1	NUM
ejpam-3786	21	2	in	in	ADP
ejpam-3786	21	3	the	the	DET
ejpam-3786	21	4	free	free	ADJ
ejpam-3786	21	5	product	product	NOUN
ejpam-3786	21	6	g∗(t	g∗(t	NOUN
ejpam-3786	21	7	)	)	PUNCT
ejpam-3786	21	8	.	.	PUNCT
ejpam-3786	22	1	the	the	DET
ejpam-3786	22	2	equation	equation	NOUN
ejpam-3786	22	3	s(t	s(t	PROPN
ejpam-3786	22	4	)	)	PUNCT
ejpam-3786	22	5	=	=	PUNCT
ejpam-3786	22	6	1	1	NUM
ejpam-3786	22	7	is	be	AUX
ejpam-3786	22	8	called	call	VERB
ejpam-3786	22	9	singular	singular	ADJ
ejpam-3786	22	10	if	if	SCONJ
ejpam-3786	22	11	∑n	∑n	PROPN
ejpam-3786	22	12	i=1	i=1	PROPN
ejpam-3786	22	13	li	li	PROPN
ejpam-3786	23	1	=	=	SYM
ejpam-3786	23	2	0	0	NUM
ejpam-3786	23	3	and	and	CCONJ
ejpam-3786	23	4	non	non	ADJ
ejpam-3786	23	5	-	-	ADJ
ejpam-3786	23	6	singular	singular	ADJ
ejpam-3786	23	7	otherwise	otherwise	ADV
ejpam-3786	23	8	.	.	PUNCT
ejpam-3786	24	1	the	the	DET
ejpam-3786	24	2	study	study	NOUN
ejpam-3786	24	3	of	of	ADP
ejpam-3786	24	4	group	group	NOUN
ejpam-3786	24	5	equations	equation	NOUN
ejpam-3786	24	6	was	be	AUX
ejpam-3786	24	7	initiated	initiate	VERB
ejpam-3786	24	8	by	by	ADP
ejpam-3786	24	9	neumann	neumann	PROPN
ejpam-3786	24	10	[	[	X
ejpam-3786	24	11	17	17	NUM
ejpam-3786	24	12	]	]	PUNCT
ejpam-3786	24	13	who	who	PRON
ejpam-3786	24	14	solved	solve	VERB
ejpam-3786	24	15	an	an	DET
ejpam-3786	24	16	equation	equation	NOUN
ejpam-3786	24	17	t−1g1tg2	t−1g1tg2	ADJ
ejpam-3786	24	18	=	=	SYM
ejpam-3786	24	19	1	1	NUM
ejpam-3786	24	20	over	over	ADP
ejpam-3786	24	21	any	any	DET
ejpam-3786	24	22	torsion	torsion	NOUN
ejpam-3786	24	23	free	free	ADJ
ejpam-3786	24	24	group	group	NOUN
ejpam-3786	24	25	.	.	PUNCT
ejpam-3786	25	1	motivated	motivate	VERB
ejpam-3786	25	2	by	by	ADP
ejpam-3786	25	3	the	the	DET
ejpam-3786	25	4	solvability	solvability	NOUN
ejpam-3786	25	5	of	of	ADP
ejpam-3786	25	6	the	the	DET
ejpam-3786	25	7	polynomial	polynomial	ADJ
ejpam-3786	25	8	∗corresponding	∗corresponde	VERB
ejpam-3786	25	9	author	author	NOUN
ejpam-3786	25	10	.	.	PUNCT
ejpam-3786	26	1	doi	doi	NOUN
ejpam-3786	26	2	:	:	PUNCT
ejpam-3786	26	3	https://doi.org/10.29020/nybg.ejpam.v13i4.3786	https://doi.org/10.29020/nybg.ejpam.v13i4.3786	PROPN
ejpam-3786	26	4	email	email	NOUN
ejpam-3786	26	5	addresses	address	NOUN
ejpam-3786	26	6	:	:	PUNCT
ejpam-3786	26	7	mrsaeedakram@gmail.com	mrsaeedakram@gmail.com	X
ejpam-3786	26	8	(	(	PUNCT
ejpam-3786	26	9	muhammad	muhammad	PROPN
ejpam-3786	26	10	saeed	saeed	PROPN
ejpam-3786	26	11	akram	akram	PROPN
ejpam-3786	26	12	)	)	PUNCT
ejpam-3786	26	13	,	,	PUNCT
ejpam-3786	26	14	mairaamjad450@gmail.com	mairaamjad450@gmail.com	X
ejpam-3786	26	15	(	(	PUNCT
ejpam-3786	26	16	maira	maira	PROPN
ejpam-3786	26	17	amjid	amjid	PROPN
ejpam-3786	26	18	)	)	PUNCT
ejpam-3786	26	19	,	,	PUNCT
ejpam-3786	26	20	soh.iqbal@gmail.com	soh.iqbal@gmail.com	PROPN
ejpam-3786	26	21	(	(	PUNCT
ejpam-3786	26	22	sohail	sohail	PROPN
ejpam-3786	26	23	iqbal	iqbal	PROPN
ejpam-3786	26	24	)	)	PUNCT
ejpam-3786	26	25	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-3786	26	26	914	914	NUM
ejpam-3786	27	1	c	c	X
ejpam-3786	27	2	©	©	NOUN
ejpam-3786	27	3	2020	2020	NUM
ejpam-3786	27	4	ejpam	ejpam	VERB
ejpam-3786	27	5	all	all	DET
ejpam-3786	27	6	rights	right	NOUN
ejpam-3786	27	7	reserved	reserve	VERB
ejpam-3786	27	8	.	.	PUNCT
ejpam-3786	28	1	muhammad	muhammad	PROPN
ejpam-3786	28	2	saeed	saeed	PROPN
ejpam-3786	28	3	akram	akram	PROPN
ejpam-3786	28	4	,	,	PUNCT
ejpam-3786	28	5	maira	maira	PROPN
ejpam-3786	28	6	amjid	amjid	PROPN
ejpam-3786	28	7	,	,	PUNCT
ejpam-3786	28	8	sohail	sohail	PROPN
ejpam-3786	28	9	iqbal	iqbal	PROPN
ejpam-3786	28	10	/	/	PUNCT
ejpam-3786	28	11	eur	eur	PROPN
ejpam-3786	28	12	.	.	PUNCT
ejpam-3786	29	1	j.	j.	PROPN
ejpam-3786	29	2	pure	pure	PROPN
ejpam-3786	29	3	appl	appl	PROPN
ejpam-3786	29	4	.	.	PROPN
ejpam-3786	29	5	math	math	PROPN
ejpam-3786	29	6	,	,	PUNCT
ejpam-3786	29	7	13	13	NUM
ejpam-3786	29	8	(	(	PUNCT
ejpam-3786	29	9	4	4	NUM
ejpam-3786	29	10	)	)	PUNCT
ejpam-3786	29	11	(	(	PUNCT
ejpam-3786	29	12	2020	2020	NUM
ejpam-3786	29	13	)	)	PUNCT
ejpam-3786	29	14	,	,	PUNCT
ejpam-3786	29	15	914	914	NUM
ejpam-3786	29	16	-	-	SYM
ejpam-3786	29	17	938	938	NUM
ejpam-3786	29	18	915	915	NUM
ejpam-3786	29	19	equations	equation	NOUN
ejpam-3786	29	20	over	over	ADP
ejpam-3786	29	21	fields	field	NOUN
ejpam-3786	29	22	,	,	PUNCT
ejpam-3786	29	23	levin	levin	PROPN
ejpam-3786	30	1	[	[	X
ejpam-3786	30	2	16	16	NUM
ejpam-3786	30	3	]	]	PUNCT
ejpam-3786	30	4	studied	study	VERB
ejpam-3786	30	5	the	the	DET
ejpam-3786	30	6	analogous	analogous	ADJ
ejpam-3786	30	7	problem	problem	NOUN
ejpam-3786	30	8	for	for	ADP
ejpam-3786	30	9	group	group	NOUN
ejpam-3786	30	10	equations	equation	NOUN
ejpam-3786	30	11	and	and	CCONJ
ejpam-3786	30	12	proved	prove	VERB
ejpam-3786	30	13	that	that	SCONJ
ejpam-3786	30	14	the	the	DET
ejpam-3786	30	15	equation	equation	NOUN
ejpam-3786	30	16	s(t	s(t	PROPN
ejpam-3786	30	17	)	)	PUNCT
ejpam-3786	30	18	=	=	SYM
ejpam-3786	30	19	1	1	NUM
ejpam-3786	30	20	(	(	PUNCT
ejpam-3786	30	21	li	li	PROPN
ejpam-3786	30	22	non	non	ADJ
ejpam-3786	30	23	-	-	ADJ
ejpam-3786	30	24	negative	negative	ADJ
ejpam-3786	30	25	and	and	CCONJ
ejpam-3786	30	26	not	not	PART
ejpam-3786	30	27	necessarily	necessarily	ADV
ejpam-3786	30	28	1	1	NUM
ejpam-3786	30	29	)	)	PUNCT
ejpam-3786	30	30	is	be	AUX
ejpam-3786	30	31	solvable	solvable	ADJ
ejpam-3786	30	32	over	over	ADP
ejpam-3786	30	33	any	any	DET
ejpam-3786	30	34	group	group	NOUN
ejpam-3786	30	35	g	g	NOUN
ejpam-3786	30	36	,	,	PUNCT
ejpam-3786	30	37	for	for	ADP
ejpam-3786	30	38	l1	l1	PROPN
ejpam-3786	30	39	+	+	CCONJ
ejpam-3786	30	40	l2	l2	NOUN
ejpam-3786	30	41	+	+	CCONJ
ejpam-3786	30	42	...	...	PUNCT
ejpam-3786	31	1	+	+	PUNCT
ejpam-3786	31	2	ln	ln	ADJ
ejpam-3786	31	3	=	=	NOUN
ejpam-3786	31	4	n	n	NOUN
ejpam-3786	31	5	.	.	PUNCT
ejpam-3786	32	1	these	these	DET
ejpam-3786	32	2	findings	finding	NOUN
ejpam-3786	32	3	of	of	ADP
ejpam-3786	32	4	neumanm	neumanm	PROPN
ejpam-3786	32	5	and	and	CCONJ
ejpam-3786	32	6	levin	levin	PROPN
ejpam-3786	32	7	gave	give	VERB
ejpam-3786	32	8	the	the	DET
ejpam-3786	32	9	hope	hope	NOUN
ejpam-3786	32	10	for	for	ADP
ejpam-3786	32	11	the	the	DET
ejpam-3786	32	12	conjecture	conjecture	NOUN
ejpam-3786	32	13	that	that	SCONJ
ejpam-3786	32	14	every	every	DET
ejpam-3786	32	15	equation	equation	NOUN
ejpam-3786	32	16	is	be	AUX
ejpam-3786	32	17	solvable	solvable	ADJ
ejpam-3786	32	18	over	over	ADP
ejpam-3786	32	19	a	a	DET
ejpam-3786	32	20	torsion	torsion	NOUN
ejpam-3786	32	21	free	free	ADJ
ejpam-3786	32	22	group	group	NOUN
ejpam-3786	32	23	,	,	PUNCT
ejpam-3786	32	24	which	which	PRON
ejpam-3786	32	25	is	be	AUX
ejpam-3786	32	26	known	know	VERB
ejpam-3786	32	27	as	as	ADP
ejpam-3786	32	28	levin	levin	PROPN
ejpam-3786	32	29	conjecture	conjecture	NOUN
ejpam-3786	32	30	.	.	PUNCT
ejpam-3786	33	1	there	there	PRON
ejpam-3786	33	2	has	have	AUX
ejpam-3786	33	3	been	be	AUX
ejpam-3786	33	4	significant	significant	ADJ
ejpam-3786	33	5	work	work	NOUN
ejpam-3786	33	6	to	to	PART
ejpam-3786	33	7	verify	verify	VERB
ejpam-3786	33	8	the	the	DET
ejpam-3786	33	9	levin	levin	PROPN
ejpam-3786	33	10	conjecture	conjecture	NOUN
ejpam-3786	34	1	[	[	X
ejpam-3786	34	2	11	11	NUM
ejpam-3786	34	3	,	,	PUNCT
ejpam-3786	34	4	12	12	NUM
ejpam-3786	34	5	,	,	PUNCT
ejpam-3786	34	6	14	14	NUM
ejpam-3786	34	7	,	,	PUNCT
ejpam-3786	34	8	15	15	NUM
ejpam-3786	34	9	]	]	PUNCT
ejpam-3786	34	10	for	for	ADP
ejpam-3786	34	11	group	group	NOUN
ejpam-3786	34	12	equations	equation	NOUN
ejpam-3786	34	13	of	of	ADP
ejpam-3786	34	14	length	length	NOUN
ejpam-3786	34	15	less	less	ADV
ejpam-3786	34	16	than	than	ADP
ejpam-3786	34	17	or	or	CCONJ
ejpam-3786	34	18	equal	equal	ADJ
ejpam-3786	34	19	to	to	ADP
ejpam-3786	34	20	six	six	NUM
ejpam-3786	34	21	.	.	PUNCT
ejpam-3786	35	1	recently	recently	ADV
ejpam-3786	35	2	,	,	PUNCT
ejpam-3786	35	3	mairaj	mairaj	ADJ
ejpam-3786	35	4	and	and	CCONJ
ejpam-3786	35	5	edjvet	edjvet	VERB
ejpam-3786	36	1	[	[	X
ejpam-3786	36	2	8	8	NUM
ejpam-3786	36	3	]	]	PUNCT
ejpam-3786	36	4	proved	prove	VERB
ejpam-3786	36	5	the	the	DET
ejpam-3786	36	6	levin	levin	PROPN
ejpam-3786	36	7	conjecture	conjecture	NOUN
ejpam-3786	36	8	for	for	ADP
ejpam-3786	36	9	all	all	DET
ejpam-3786	36	10	group	group	NOUN
ejpam-3786	36	11	equations	equation	NOUN
ejpam-3786	36	12	of	of	ADP
ejpam-3786	36	13	length	length	NOUN
ejpam-3786	36	14	seven	seven	NUM
ejpam-3786	36	15	by	by	ADP
ejpam-3786	36	16	using	use	VERB
ejpam-3786	36	17	weight	weight	NOUN
ejpam-3786	36	18	test	test	NOUN
ejpam-3786	36	19	and	and	CCONJ
ejpam-3786	36	20	curvature	curvature	VERB
ejpam-3786	36	21	distribution	distribution	NOUN
ejpam-3786	36	22	method	method	NOUN
ejpam-3786	36	23	.	.	PUNCT
ejpam-3786	37	1	by	by	ADP
ejpam-3786	37	2	employing	employ	VERB
ejpam-3786	37	3	the	the	DET
ejpam-3786	37	4	methods	method	NOUN
ejpam-3786	37	5	used	use	VERB
ejpam-3786	37	6	in	in	ADP
ejpam-3786	37	7	[	[	X
ejpam-3786	37	8	8	8	NUM
ejpam-3786	37	9	]	]	PUNCT
ejpam-3786	37	10	,	,	PUNCT
ejpam-3786	37	11	mairaj	mairaj	PROPN
ejpam-3786	37	12	et	et	PROPN
ejpam-3786	37	13	al	al	PROPN
ejpam-3786	37	14	.	.	PUNCT
ejpam-3786	38	1	[	[	X
ejpam-3786	38	2	7	7	X
ejpam-3786	38	3	]	]	PUNCT
ejpam-3786	38	4	have	have	AUX
ejpam-3786	38	5	proved	prove	VERB
ejpam-3786	38	6	the	the	DET
ejpam-3786	38	7	conjecture	conjecture	NOUN
ejpam-3786	38	8	for	for	ADP
ejpam-3786	38	9	a	a	DET
ejpam-3786	38	10	non	non	ADJ
ejpam-3786	38	11	-	-	ADJ
ejpam-3786	38	12	singular	singular	ADJ
ejpam-3786	38	13	equation	equation	NOUN
ejpam-3786	38	14	of	of	ADP
ejpam-3786	38	15	length	length	NOUN
ejpam-3786	38	16	eight	eight	NUM
ejpam-3786	38	17	modulo	modulo	NOUN
ejpam-3786	38	18	one	one	NUM
ejpam-3786	38	19	exceptional	exceptional	ADJ
ejpam-3786	38	20	case	case	NOUN
ejpam-3786	38	21	.	.	PUNCT
ejpam-3786	39	1	the	the	DET
ejpam-3786	39	2	authors	author	NOUN
ejpam-3786	39	3	have	have	AUX
ejpam-3786	39	4	done	do	VERB
ejpam-3786	39	5	some	some	DET
ejpam-3786	39	6	significant	significant	ADJ
ejpam-3786	39	7	work	work	NOUN
ejpam-3786	39	8	in	in	ADP
ejpam-3786	39	9	[	[	X
ejpam-3786	39	10	4	4	NUM
ejpam-3786	39	11	,	,	PUNCT
ejpam-3786	39	12	5	5	NUM
ejpam-3786	39	13	]	]	PUNCT
ejpam-3786	39	14	using	use	VERB
ejpam-3786	39	15	weight	weight	NOUN
ejpam-3786	39	16	test	test	NOUN
ejpam-3786	39	17	which	which	PRON
ejpam-3786	39	18	establishes	establish	VERB
ejpam-3786	39	19	the	the	DET
ejpam-3786	39	20	conjecture	conjecture	NOUN
ejpam-3786	39	21	up	up	ADP
ejpam-3786	39	22	to	to	ADP
ejpam-3786	39	23	great	great	ADJ
ejpam-3786	39	24	extent	extent	NOUN
ejpam-3786	39	25	for	for	ADP
ejpam-3786	39	26	length	length	NOUN
ejpam-3786	39	27	eight	eight	NUM
ejpam-3786	39	28	.	.	PUNCT
ejpam-3786	40	1	the	the	DET
ejpam-3786	40	2	equations	equation	NOUN
ejpam-3786	40	3	of	of	ADP
ejpam-3786	40	4	length	length	NOUN
ejpam-3786	40	5	nine	nine	NUM
ejpam-3786	40	6	are	be	AUX
ejpam-3786	40	7	considered	consider	VERB
ejpam-3786	40	8	in	in	ADP
ejpam-3786	40	9	[	[	X
ejpam-3786	40	10	6	6	NUM
ejpam-3786	40	11	]	]	PUNCT
ejpam-3786	40	12	,	,	PUNCT
ejpam-3786	40	13	where	where	SCONJ
ejpam-3786	40	14	it	it	PRON
ejpam-3786	40	15	is	be	AUX
ejpam-3786	40	16	proved	prove	VERB
ejpam-3786	40	17	that	that	SCONJ
ejpam-3786	40	18	there	there	PRON
ejpam-3786	40	19	are	be	VERB
ejpam-3786	40	20	only	only	ADV
ejpam-3786	40	21	three	three	NUM
ejpam-3786	40	22	equations	equation	NOUN
ejpam-3786	40	23	of	of	ADP
ejpam-3786	40	24	length	length	NOUN
ejpam-3786	40	25	nine	nine	NUM
ejpam-3786	40	26	which	which	PRON
ejpam-3786	40	27	are	be	AUX
ejpam-3786	40	28	open	open	ADJ
ejpam-3786	40	29	.	.	PUNCT
ejpam-3786	41	1	more	more	ADV
ejpam-3786	41	2	recently	recently	ADV
ejpam-3786	41	3	,	,	PUNCT
ejpam-3786	41	4	fazeel	fazeel	NOUN
ejpam-3786	41	5	et	et	PROPN
ejpam-3786	41	6	al	al	PROPN
ejpam-3786	41	7	.	.	PUNCT
ejpam-3786	42	1	[	[	X
ejpam-3786	42	2	3	3	X
ejpam-3786	42	3	]	]	PUNCT
ejpam-3786	42	4	have	have	AUX
ejpam-3786	42	5	investigated	investigate	VERB
ejpam-3786	42	6	the	the	DET
ejpam-3786	42	7	conjecture	conjecture	NOUN
ejpam-3786	42	8	for	for	ADP
ejpam-3786	42	9	a	a	DET
ejpam-3786	42	10	non	non	ADJ
ejpam-3786	42	11	-	-	ADJ
ejpam-3786	42	12	singular	singular	ADJ
ejpam-3786	42	13	equation	equation	NOUN
ejpam-3786	42	14	of	of	ADP
ejpam-3786	42	15	length	length	NOUN
ejpam-3786	42	16	nine	nine	NUM
ejpam-3786	42	17	(	(	PUNCT
ejpam-3786	42	18	one	one	NUM
ejpam-3786	42	19	of	of	ADP
ejpam-3786	42	20	three	three	NUM
ejpam-3786	42	21	)	)	PUNCT
ejpam-3786	42	22	given	give	VERB
ejpam-3786	42	23	by	by	ADP
ejpam-3786	42	24	s1(t	s1(t	NOUN
ejpam-3786	42	25	)	)	PUNCT
ejpam-3786	42	26	=	=	SYM
ejpam-3786	43	1	atbtctdtetft−1gthtit−1	atbtctdtetft−1gthtit−1	PROPN
ejpam-3786	43	2	=	=	SYM
ejpam-3786	43	3	1	1	NUM
ejpam-3786	43	4	by	by	ADP
ejpam-3786	43	5	applying	apply	VERB
ejpam-3786	43	6	these	these	DET
ejpam-3786	43	7	methods	method	NOUN
ejpam-3786	43	8	.	.	PUNCT
ejpam-3786	44	1	fazeel	fazeel	NOUN
ejpam-3786	44	2	et	et	PROPN
ejpam-3786	44	3	al	al	PROPN
ejpam-3786	44	4	.	.	PUNCT
ejpam-3786	45	1	[	[	X
ejpam-3786	45	2	4	4	X
ejpam-3786	45	3	]	]	PUNCT
ejpam-3786	45	4	solved	solve	VERB
ejpam-3786	45	5	41	41	NUM
ejpam-3786	45	6	cases	case	NOUN
ejpam-3786	45	7	of	of	ADP
ejpam-3786	45	8	this	this	DET
ejpam-3786	45	9	equation	equation	NOUN
ejpam-3786	45	10	.	.	PUNCT
ejpam-3786	46	1	in	in	ADP
ejpam-3786	46	2	this	this	DET
ejpam-3786	46	3	paper	paper	NOUN
ejpam-3786	46	4	we	we	PRON
ejpam-3786	46	5	have	have	AUX
ejpam-3786	46	6	continued	continue	VERB
ejpam-3786	46	7	our	our	PRON
ejpam-3786	46	8	study	study	NOUN
ejpam-3786	46	9	of	of	ADP
ejpam-3786	46	10	exploring	explore	VERB
ejpam-3786	46	11	the	the	DET
ejpam-3786	46	12	validity	validity	NOUN
ejpam-3786	46	13	of	of	ADP
ejpam-3786	46	14	levin	levin	PROPN
ejpam-3786	46	15	conjecture	conjecture	NOUN
ejpam-3786	46	16	for	for	ADP
ejpam-3786	46	17	the	the	DET
ejpam-3786	46	18	group	group	NOUN
ejpam-3786	46	19	equation	equation	NOUN
ejpam-3786	46	20	s1(t	s1(t	PROPN
ejpam-3786	46	21	)	)	PUNCT
ejpam-3786	46	22	=	=	SYM
ejpam-3786	46	23	1	1	NUM
ejpam-3786	46	24	initiated	initiate	VERB
ejpam-3786	46	25	in	in	ADP
ejpam-3786	46	26	[	[	X
ejpam-3786	46	27	4	4	NUM
ejpam-3786	46	28	]	]	PUNCT
ejpam-3786	46	29	and	and	CCONJ
ejpam-3786	46	30	found	find	VERB
ejpam-3786	46	31	that	that	SCONJ
ejpam-3786	46	32	the	the	DET
ejpam-3786	46	33	total	total	ADJ
ejpam-3786	46	34	cases	case	NOUN
ejpam-3786	46	35	of	of	ADP
ejpam-3786	46	36	s1(t	s1(t	ADJ
ejpam-3786	46	37	)	)	PUNCT
ejpam-3786	46	38	=	=	SYM
ejpam-3786	46	39	1	1	NUM
ejpam-3786	46	40	are	be	AUX
ejpam-3786	46	41	245	245	NUM
ejpam-3786	46	42	in	in	ADP
ejpam-3786	46	43	which	which	PRON
ejpam-3786	46	44	183	183	NUM
ejpam-3786	46	45	cases	case	NOUN
ejpam-3786	46	46	are	be	AUX
ejpam-3786	46	47	solved	solve	VERB
ejpam-3786	46	48	by	by	ADP
ejpam-3786	46	49	weight	weight	NOUN
ejpam-3786	46	50	test	test	NOUN
ejpam-3786	46	51	,	,	PUNCT
ejpam-3786	46	52	60	60	NUM
ejpam-3786	46	53	cases	case	NOUN
ejpam-3786	46	54	are	be	AUX
ejpam-3786	46	55	solved	solve	VERB
ejpam-3786	46	56	by	by	ADP
ejpam-3786	46	57	curvature	curvature	NOUN
ejpam-3786	46	58	distribution	distribution	NOUN
ejpam-3786	46	59	method	method	NOUN
ejpam-3786	46	60	and	and	CCONJ
ejpam-3786	46	61	2	2	NUM
ejpam-3786	46	62	cases	case	NOUN
ejpam-3786	46	63	are	be	AUX
ejpam-3786	46	64	still	still	ADV
ejpam-3786	46	65	open	open	ADJ
ejpam-3786	46	66	.	.	PUNCT
ejpam-3786	47	1	these	these	DET
ejpam-3786	47	2	findings	finding	NOUN
ejpam-3786	47	3	are	be	AUX
ejpam-3786	47	4	formulated	formulate	VERB
ejpam-3786	47	5	in	in	ADP
ejpam-3786	47	6	the	the	DET
ejpam-3786	47	7	form	form	NOUN
ejpam-3786	47	8	of	of	ADP
ejpam-3786	47	9	the	the	DET
ejpam-3786	47	10	following	follow	VERB
ejpam-3786	47	11	theorem	theorem	NOUN
ejpam-3786	47	12	which	which	PRON
ejpam-3786	47	13	is	be	AUX
ejpam-3786	47	14	the	the	DET
ejpam-3786	47	15	main	main	ADJ
ejpam-3786	47	16	result	result	NOUN
ejpam-3786	47	17	of	of	ADP
ejpam-3786	47	18	the	the	DET
ejpam-3786	47	19	paper	paper	NOUN
ejpam-3786	47	20	.	.	PUNCT
ejpam-3786	48	1	theorem	theorem	VERB
ejpam-3786	48	2	.	.	PUNCT
ejpam-3786	49	1	the	the	DET
ejpam-3786	49	2	group	group	NOUN
ejpam-3786	49	3	equation	equation	NOUN
ejpam-3786	49	4	s1(t	s1(t	PROPN
ejpam-3786	49	5	)	)	PUNCT
ejpam-3786	49	6	=	=	SYM
ejpam-3786	49	7	1	1	NUM
ejpam-3786	49	8	is	be	AUX
ejpam-3786	49	9	solvable	solvable	ADJ
ejpam-3786	49	10	modulo	modulo	NOUN
ejpam-3786	49	11	two	two	NUM
ejpam-3786	49	12	exceptional	exceptional	ADJ
ejpam-3786	49	13	cases	case	NOUN
ejpam-3786	49	14	:	:	PUNCT
ejpam-3786	49	15	(	(	PUNCT
ejpam-3786	49	16	i	i	NOUN
ejpam-3786	49	17	)	)	PUNCT
ejpam-3786	49	18	a	a	DET
ejpam-3786	49	19	=	=	SYM
ejpam-3786	49	20	g	g	NOUN
ejpam-3786	49	21	,	,	PUNCT
ejpam-3786	49	22	h	h	NOUN
ejpam-3786	49	23	=	=	SYM
ejpam-3786	49	24	e	e	NOUN
ejpam-3786	49	25	,	,	PUNCT
ejpam-3786	49	26	d	d	PROPN
ejpam-3786	49	27	=	=	SYM
ejpam-3786	49	28	b	b	PROPN
ejpam-3786	49	29	,	,	PUNCT
ejpam-3786	49	30	c	c	PROPN
ejpam-3786	49	31	=	=	SYM
ejpam-3786	49	32	b	b	PROPN
ejpam-3786	49	33	,	,	PUNCT
ejpam-3786	49	34	d	d	PROPN
ejpam-3786	49	35	=	=	SYM
ejpam-3786	49	36	c	c	NOUN
ejpam-3786	49	37	;	;	PUNCT
ejpam-3786	49	38	(	(	PUNCT
ejpam-3786	49	39	ii	ii	NOUN
ejpam-3786	49	40	)	)	PUNCT
ejpam-3786	49	41	a	a	DET
ejpam-3786	49	42	=	=	SYM
ejpam-3786	49	43	g	g	NOUN
ejpam-3786	49	44	,	,	PUNCT
ejpam-3786	49	45	e	e	PROPN
ejpam-3786	49	46	=	=	SYM
ejpam-3786	49	47	b	b	PROPN
ejpam-3786	49	48	,	,	PUNCT
ejpam-3786	49	49	e	e	X
ejpam-3786	49	50	=	=	SYM
ejpam-3786	49	51	c	c	X
ejpam-3786	49	52	,	,	PUNCT
ejpam-3786	49	53	c	c	NOUN
ejpam-3786	49	54	=	=	SYM
ejpam-3786	49	55	b	b	PROPN
ejpam-3786	49	56	,	,	PUNCT
ejpam-3786	49	57	e	e	X
ejpam-3786	49	58	=	=	SYM
ejpam-3786	49	59	d	d	PROPN
ejpam-3786	49	60	,	,	PUNCT
ejpam-3786	49	61	d	d	PROPN
ejpam-3786	49	62	=	=	SYM
ejpam-3786	49	63	b	b	PROPN
ejpam-3786	49	64	,	,	PUNCT
ejpam-3786	49	65	d	d	PROPN
ejpam-3786	49	66	=	=	PUNCT
ejpam-3786	49	67	c.	c.	NOUN
ejpam-3786	49	68	the	the	DET
ejpam-3786	49	69	authors	author	NOUN
ejpam-3786	49	70	in	in	ADP
ejpam-3786	49	71	[	[	X
ejpam-3786	49	72	2	2	NUM
ejpam-3786	49	73	]	]	PUNCT
ejpam-3786	49	74	and	and	CCONJ
ejpam-3786	49	75	[	[	X
ejpam-3786	49	76	1	1	X
ejpam-3786	49	77	]	]	PUNCT
ejpam-3786	49	78	have	have	AUX
ejpam-3786	49	79	explored	explore	VERB
ejpam-3786	49	80	the	the	DET
ejpam-3786	49	81	validity	validity	NOUN
ejpam-3786	49	82	of	of	ADP
ejpam-3786	49	83	levin	levin	PROPN
ejpam-3786	49	84	conjecture	conjecture	NOUN
ejpam-3786	49	85	by	by	ADP
ejpam-3786	49	86	applying	apply	VERB
ejpam-3786	49	87	weight	weight	NOUN
ejpam-3786	49	88	test	test	NOUN
ejpam-3786	49	89	and	and	CCONJ
ejpam-3786	49	90	curvature	curvature	VERB
ejpam-3786	49	91	distribution	distribution	NOUN
ejpam-3786	49	92	to	to	ADP
ejpam-3786	49	93	the	the	DET
ejpam-3786	49	94	remaining	remain	VERB
ejpam-3786	49	95	two	two	NUM
ejpam-3786	49	96	equations	equation	NOUN
ejpam-3786	49	97	of	of	ADP
ejpam-3786	49	98	length	length	NOUN
ejpam-3786	49	99	nine	nine	NUM
ejpam-3786	49	100	given	give	VERB
ejpam-3786	49	101	by	by	ADP
ejpam-3786	49	102	s2(t	s2(t	PROPN
ejpam-3786	49	103	)	)	PUNCT
ejpam-3786	49	104	=	=	PUNCT
ejpam-3786	49	105	atbtct−1dtetft−1gthtit−1	atbtct−1dtetft−1gthtit−1	X
ejpam-3786	49	106	=	=	NOUN
ejpam-3786	49	107	1	1	NUM
ejpam-3786	49	108	and	and	CCONJ
ejpam-3786	49	109	s3(t	s3(t	NUM
ejpam-3786	49	110	)	)	PUNCT
ejpam-3786	49	111	=	=	PUNCT
ejpam-3786	50	1	atbtctdtet−1ftgthtit−1	atbtctdtet−1ftgthtit−1	PUNCT
ejpam-3786	50	2	=	=	NOUN
ejpam-3786	50	3	1	1	X
ejpam-3786	50	4	.	.	PUNCT
ejpam-3786	51	1	they	they	PRON
ejpam-3786	51	2	found	find	VERB
ejpam-3786	51	3	that	that	SCONJ
ejpam-3786	51	4	levin	levin	PROPN
ejpam-3786	51	5	conjecture	conjecture	NOUN
ejpam-3786	51	6	holds	hold	VERB
ejpam-3786	51	7	true	true	ADJ
ejpam-3786	51	8	for	for	SCONJ
ejpam-3786	51	9	these	these	DET
ejpam-3786	51	10	equations	equation	NOUN
ejpam-3786	51	11	modulo	modulo	VERB
ejpam-3786	51	12	some	some	DET
ejpam-3786	51	13	exceptional	exceptional	ADJ
ejpam-3786	51	14	cases	case	NOUN
ejpam-3786	51	15	.	.	PUNCT
ejpam-3786	52	1	the	the	DET
ejpam-3786	52	2	authors	author	NOUN
ejpam-3786	52	3	in	in	ADP
ejpam-3786	52	4	[	[	X
ejpam-3786	52	5	2	2	NUM
ejpam-3786	52	6	]	]	PUNCT
ejpam-3786	52	7	have	have	AUX
ejpam-3786	52	8	found	find	VERB
ejpam-3786	52	9	that	that	SCONJ
ejpam-3786	52	10	the	the	DET
ejpam-3786	52	11	total	total	ADJ
ejpam-3786	52	12	cases	case	NOUN
ejpam-3786	52	13	of	of	ADP
ejpam-3786	52	14	s2(t	s2(t	NOUN
ejpam-3786	52	15	)	)	PUNCT
ejpam-3786	52	16	=	=	SYM
ejpam-3786	52	17	1	1	NUM
ejpam-3786	52	18	are	be	AUX
ejpam-3786	52	19	318	318	NUM
ejpam-3786	52	20	in	in	ADP
ejpam-3786	52	21	which	which	PRON
ejpam-3786	52	22	147	147	NUM
ejpam-3786	52	23	cases	case	NOUN
ejpam-3786	52	24	are	be	AUX
ejpam-3786	52	25	solved	solve	VERB
ejpam-3786	52	26	by	by	ADP
ejpam-3786	52	27	weight	weight	NOUN
ejpam-3786	52	28	test	test	NOUN
ejpam-3786	52	29	,	,	PUNCT
ejpam-3786	52	30	117	117	NUM
ejpam-3786	52	31	cases	case	NOUN
ejpam-3786	52	32	are	be	AUX
ejpam-3786	52	33	solved	solve	VERB
ejpam-3786	52	34	by	by	ADP
ejpam-3786	52	35	curvature	curvature	NOUN
ejpam-3786	52	36	distribution	distribution	NOUN
ejpam-3786	52	37	method	method	NOUN
ejpam-3786	52	38	and	and	CCONJ
ejpam-3786	52	39	55	55	NUM
ejpam-3786	52	40	cases	case	NOUN
ejpam-3786	52	41	are	be	AUX
ejpam-3786	52	42	still	still	ADV
ejpam-3786	52	43	open	open	ADJ
ejpam-3786	52	44	.	.	PUNCT
ejpam-3786	53	1	the	the	DET
ejpam-3786	53	2	authors	author	NOUN
ejpam-3786	53	3	in	in	ADP
ejpam-3786	53	4	[	[	X
ejpam-3786	53	5	1	1	NUM
ejpam-3786	53	6	]	]	PUNCT
ejpam-3786	53	7	have	have	AUX
ejpam-3786	53	8	found	find	VERB
ejpam-3786	53	9	that	that	SCONJ
ejpam-3786	53	10	the	the	DET
ejpam-3786	53	11	total	total	ADJ
ejpam-3786	53	12	cases	case	NOUN
ejpam-3786	53	13	of	of	ADP
ejpam-3786	53	14	s3(t	s3(t	NOUN
ejpam-3786	53	15	)	)	PUNCT
ejpam-3786	53	16	=	=	SYM
ejpam-3786	53	17	1	1	NUM
ejpam-3786	53	18	are	be	AUX
ejpam-3786	53	19	245	245	NUM
ejpam-3786	53	20	in	in	ADP
ejpam-3786	53	21	which	which	PRON
ejpam-3786	53	22	136	136	NUM
ejpam-3786	53	23	cases	case	NOUN
ejpam-3786	53	24	are	be	AUX
ejpam-3786	53	25	solved	solve	VERB
ejpam-3786	53	26	by	by	ADP
ejpam-3786	53	27	weight	weight	NOUN
ejpam-3786	53	28	test	test	NOUN
ejpam-3786	53	29	,	,	PUNCT
ejpam-3786	53	30	70	70	NUM
ejpam-3786	53	31	cases	case	NOUN
ejpam-3786	53	32	are	be	AUX
ejpam-3786	53	33	solved	solve	VERB
ejpam-3786	53	34	by	by	ADP
ejpam-3786	53	35	curvature	curvature	NOUN
ejpam-3786	53	36	distribution	distribution	NOUN
ejpam-3786	53	37	method	method	NOUN
ejpam-3786	53	38	and	and	CCONJ
ejpam-3786	53	39	39	39	NUM
ejpam-3786	53	40	cases	case	NOUN
ejpam-3786	53	41	are	be	AUX
ejpam-3786	53	42	still	still	ADV
ejpam-3786	53	43	open	open	ADJ
ejpam-3786	53	44	.	.	PUNCT
ejpam-3786	54	1	muhammad	muhammad	PROPN
ejpam-3786	54	2	saeed	saeed	PROPN
ejpam-3786	54	3	akram	akram	PROPN
ejpam-3786	54	4	,	,	PUNCT
ejpam-3786	54	5	maira	maira	PROPN
ejpam-3786	54	6	amjid	amjid	PROPN
ejpam-3786	54	7	,	,	PUNCT
ejpam-3786	54	8	sohail	sohail	PROPN
ejpam-3786	54	9	iqbal	iqbal	PROPN
ejpam-3786	54	10	/	/	PUNCT
ejpam-3786	54	11	eur	eur	PROPN
ejpam-3786	54	12	.	.	PUNCT
ejpam-3786	55	1	j.	j.	PROPN
ejpam-3786	55	2	pure	pure	PROPN
ejpam-3786	55	3	appl	appl	PROPN
ejpam-3786	55	4	.	.	PROPN
ejpam-3786	55	5	math	math	PROPN
ejpam-3786	55	6	,	,	PUNCT
ejpam-3786	55	7	13	13	NUM
ejpam-3786	55	8	(	(	PUNCT
ejpam-3786	55	9	4	4	NUM
ejpam-3786	55	10	)	)	PUNCT
ejpam-3786	55	11	(	(	PUNCT
ejpam-3786	55	12	2020	2020	NUM
ejpam-3786	55	13	)	)	PUNCT
ejpam-3786	55	14	,	,	PUNCT
ejpam-3786	55	15	914	914	NUM
ejpam-3786	55	16	-	-	SYM
ejpam-3786	55	17	938	938	NUM
ejpam-3786	55	18	916	916	NUM
ejpam-3786	55	19	2	2	NUM
ejpam-3786	55	20	.	.	PUNCT
ejpam-3786	56	1	methodology	methodology	NOUN
ejpam-3786	56	2	:	:	PUNCT
ejpam-3786	56	3	a	a	DET
ejpam-3786	56	4	presentation	presentation	NOUN
ejpam-3786	56	5	is	be	AUX
ejpam-3786	56	6	said	say	VERB
ejpam-3786	56	7	to	to	PART
ejpam-3786	56	8	be	be	AUX
ejpam-3786	56	9	relative	relative	ADJ
ejpam-3786	56	10	(	(	PUNCT
ejpam-3786	56	11	group	group	NOUN
ejpam-3786	56	12	)	)	PUNCT
ejpam-3786	56	13	presentation	presentation	NOUN
ejpam-3786	56	14	p	p	X
ejpam-3786	56	15	=	=	PUNCT
ejpam-3786	56	16	〈	〈	PROPN
ejpam-3786	56	17	g	g	NOUN
ejpam-3786	56	18	,	,	PUNCT
ejpam-3786	56	19	x	x	PROPN
ejpam-3786	57	1	|	|	ADV
ejpam-3786	57	2	r	r	X
ejpam-3786	57	3	〉	〉	NOUN
ejpam-3786	57	4	in	in	ADP
ejpam-3786	57	5	which	which	PRON
ejpam-3786	57	6	r	r	NOUN
ejpam-3786	57	7	is	be	AUX
ejpam-3786	57	8	a	a	DET
ejpam-3786	57	9	set	set	NOUN
ejpam-3786	57	10	of	of	ADP
ejpam-3786	57	11	words	word	NOUN
ejpam-3786	57	12	which	which	PRON
ejpam-3786	57	13	is	be	AUX
ejpam-3786	57	14	cyclically	cyclically	ADV
ejpam-3786	57	15	reduced	reduce	VERB
ejpam-3786	57	16	belongs	belong	VERB
ejpam-3786	57	17	to	to	PART
ejpam-3786	57	18	g.	g.	PROPN
ejpam-3786	57	19	all	all	DET
ejpam-3786	57	20	the	the	DET
ejpam-3786	57	21	definitions	definition	NOUN
ejpam-3786	57	22	concerning	concern	VERB
ejpam-3786	57	23	relative	relative	ADJ
ejpam-3786	57	24	presentations	presentation	NOUN
ejpam-3786	57	25	can	can	AUX
ejpam-3786	57	26	be	be	AUX
ejpam-3786	57	27	seen	see	VERB
ejpam-3786	57	28	in	in	ADP
ejpam-3786	57	29	[	[	X
ejpam-3786	57	30	9	9	NUM
ejpam-3786	57	31	]	]	PUNCT
ejpam-3786	57	32	.	.	PUNCT
ejpam-3786	58	1	in	in	ADP
ejpam-3786	58	2	this	this	DET
ejpam-3786	58	3	paper	paper	NOUN
ejpam-3786	58	4	,	,	PUNCT
ejpam-3786	58	5	we	we	PRON
ejpam-3786	58	6	will	will	AUX
ejpam-3786	58	7	discuss	discuss	VERB
ejpam-3786	58	8	the	the	DET
ejpam-3786	58	9	equation	equation	NOUN
ejpam-3786	58	10	s1(t	s1(t	X
ejpam-3786	58	11	)	)	PUNCT
ejpam-3786	58	12	=	=	SYM
ejpam-3786	59	1	atbtctdtetft−1gthtit−1	atbtctdtetft−1gthtit−1	PROPN
ejpam-3786	59	2	=	=	PUNCT
ejpam-3786	59	3	1	1	NUM
ejpam-3786	59	4	in	in	ADP
ejpam-3786	59	5	detail	detail	NOUN
ejpam-3786	59	6	.	.	PUNCT
ejpam-3786	60	1	it	it	PRON
ejpam-3786	60	2	is	be	AUX
ejpam-3786	60	3	well	well	ADV
ejpam-3786	60	4	known	know	VERB
ejpam-3786	60	5	that	that	SCONJ
ejpam-3786	60	6	the	the	DET
ejpam-3786	60	7	group	group	NOUN
ejpam-3786	60	8	equation	equation	NOUN
ejpam-3786	60	9	s1(t	s1(t	PROPN
ejpam-3786	60	10	)	)	PUNCT
ejpam-3786	60	11	=	=	SYM
ejpam-3786	61	1	atbtctdtetft−1gthtit−1	atbtctdtetft−1gthtit−1	PROPN
ejpam-3786	61	2	is	be	AUX
ejpam-3786	61	3	solvable	solvable	ADJ
ejpam-3786	61	4	if	if	SCONJ
ejpam-3786	61	5	the	the	DET
ejpam-3786	61	6	natural	natural	ADJ
ejpam-3786	61	7	homomorphism	homomorphism	NOUN
ejpam-3786	61	8	τ	τ	X
ejpam-3786	61	9	:	:	PUNCT
ejpam-3786	61	10	g→	g→	PROPN
ejpam-3786	61	11	p(g	p(g	PROPN
ejpam-3786	61	12	)	)	PUNCT
ejpam-3786	61	13	is	be	AUX
ejpam-3786	61	14	injective	injective	ADJ
ejpam-3786	61	15	.	.	PUNCT
ejpam-3786	62	1	the	the	DET
ejpam-3786	62	2	sufficient	sufficient	ADJ
ejpam-3786	62	3	condition	condition	NOUN
ejpam-3786	62	4	for	for	ADP
ejpam-3786	62	5	the	the	DET
ejpam-3786	62	6	injectivity	injectivity	NOUN
ejpam-3786	62	7	of	of	ADP
ejpam-3786	62	8	natural	natural	ADJ
ejpam-3786	62	9	map	map	NOUN
ejpam-3786	62	10	from	from	ADP
ejpam-3786	62	11	g	g	NOUN
ejpam-3786	62	12	to	to	ADP
ejpam-3786	62	13	p	p	NOUN
ejpam-3786	62	14	=	=	PUNCT
ejpam-3786	62	15	〈	〈	PROPN
ejpam-3786	62	16	g	g	NOUN
ejpam-3786	62	17	,	,	PUNCT
ejpam-3786	62	18	x	x	PUNCT
ejpam-3786	62	19	|	|	ADV
ejpam-3786	62	20	r	r	NOUN
ejpam-3786	62	21	〉	〉	NOUN
ejpam-3786	62	22	is	be	AUX
ejpam-3786	62	23	that	that	SCONJ
ejpam-3786	62	24	the	the	DET
ejpam-3786	62	25	relative	relative	ADJ
ejpam-3786	62	26	presentation	presentation	NOUN
ejpam-3786	62	27	is	be	AUX
ejpam-3786	62	28	orientable	orientable	ADJ
ejpam-3786	62	29	and	and	CCONJ
ejpam-3786	62	30	aspherical	aspherical	ADJ
ejpam-3786	62	31	[	[	NOUN
ejpam-3786	62	32	9	9	NUM
ejpam-3786	62	33	]	]	PUNCT
ejpam-3786	62	34	.	.	PUNCT
ejpam-3786	63	1	the	the	DET
ejpam-3786	63	2	notion	notion	NOUN
ejpam-3786	63	3	of	of	ADP
ejpam-3786	63	4	asphericity	asphericity	NOUN
ejpam-3786	63	5	is	be	AUX
ejpam-3786	63	6	discussed	discuss	VERB
ejpam-3786	63	7	in	in	ADP
ejpam-3786	63	8	detail	detail	NOUN
ejpam-3786	63	9	in	in	ADP
ejpam-3786	63	10	[	[	X
ejpam-3786	63	11	9	9	NUM
ejpam-3786	63	12	]	]	PUNCT
ejpam-3786	63	13	.	.	PUNCT
ejpam-3786	64	1	in	in	ADP
ejpam-3786	64	2	our	our	PRON
ejpam-3786	64	3	case	case	NOUN
ejpam-3786	64	4	s1(t	s1(t	NUM
ejpam-3786	64	5	)	)	PUNCT
ejpam-3786	64	6	=	=	SYM
ejpam-3786	64	7	atbtctdtetft−1gthtit−1	atbtctdtetft−1gthtit−1	PROPN
ejpam-3786	64	8	,	,	PUNCT
ejpam-3786	64	9	therefore	therefore	ADV
ejpam-3786	64	10	r	r	NOUN
ejpam-3786	64	11	is	be	AUX
ejpam-3786	64	12	singleton	singleton	NOUN
ejpam-3786	64	13	set	set	NOUN
ejpam-3786	64	14	.	.	PUNCT
ejpam-3786	65	1	as	as	SCONJ
ejpam-3786	65	2	stated	state	VERB
ejpam-3786	65	3	in	in	ADP
ejpam-3786	65	4	[	[	X
ejpam-3786	65	5	9	9	NUM
ejpam-3786	65	6	]	]	PUNCT
ejpam-3786	65	7	,	,	PUNCT
ejpam-3786	65	8	if	if	SCONJ
ejpam-3786	65	9	r	r	NOUN
ejpam-3786	65	10	is	be	AUX
ejpam-3786	65	11	singleton	singleton	PROPN
ejpam-3786	65	12	then	then	ADV
ejpam-3786	65	13	p	p	NOUN
ejpam-3786	65	14	is	be	AUX
ejpam-3786	65	15	always	always	ADV
ejpam-3786	65	16	orientable	orientable	ADJ
ejpam-3786	65	17	,	,	PUNCT
ejpam-3786	65	18	therefore	therefore	ADV
ejpam-3786	65	19	asphericity	asphericity	NOUN
ejpam-3786	65	20	of	of	ADP
ejpam-3786	65	21	p	p	NOUN
ejpam-3786	65	22	establishes	establish	VERB
ejpam-3786	65	23	that	that	SCONJ
ejpam-3786	65	24	s1(t	s1(t	SYM
ejpam-3786	65	25	)	)	PUNCT
ejpam-3786	65	26	=	=	SYM
ejpam-3786	65	27	1	1	NUM
ejpam-3786	65	28	has	have	VERB
ejpam-3786	65	29	solution	solution	NOUN
ejpam-3786	65	30	.	.	PUNCT
ejpam-3786	66	1	in	in	ADP
ejpam-3786	66	2	order	order	NOUN
ejpam-3786	66	3	to	to	PART
ejpam-3786	66	4	establish	establish	VERB
ejpam-3786	66	5	the	the	DET
ejpam-3786	66	6	validity	validity	NOUN
ejpam-3786	66	7	of	of	ADP
ejpam-3786	66	8	levin	levin	PROPN
ejpam-3786	66	9	’s	’s	PART
ejpam-3786	66	10	conjecture	conjecture	NOUN
ejpam-3786	66	11	,	,	PUNCT
ejpam-3786	66	12	it	it	PRON
ejpam-3786	66	13	is	be	AUX
ejpam-3786	66	14	only	only	ADV
ejpam-3786	66	15	left	leave	VERB
ejpam-3786	66	16	to	to	PART
ejpam-3786	66	17	prove	prove	VERB
ejpam-3786	66	18	that	that	SCONJ
ejpam-3786	66	19	the	the	DET
ejpam-3786	66	20	presentation	presentation	NOUN
ejpam-3786	66	21	p	p	NOUN
ejpam-3786	66	22	is	be	AUX
ejpam-3786	66	23	aspherical	aspherical	ADJ
ejpam-3786	66	24	.	.	PUNCT
ejpam-3786	67	1	so	so	ADV
ejpam-3786	67	2	,	,	PUNCT
ejpam-3786	67	3	in	in	ADP
ejpam-3786	67	4	this	this	DET
ejpam-3786	67	5	paper	paper	NOUN
ejpam-3786	67	6	we	we	PRON
ejpam-3786	67	7	apply	apply	VERB
ejpam-3786	67	8	two	two	NUM
ejpam-3786	67	9	tests	test	NOUN
ejpam-3786	67	10	for	for	ADP
ejpam-3786	67	11	showing	show	VERB
ejpam-3786	67	12	asphericity	asphericity	NOUN
ejpam-3786	67	13	of	of	ADP
ejpam-3786	67	14	p	p	X
ejpam-3786	67	15	:	:	PUNCT
ejpam-3786	67	16	weight	weight	NOUN
ejpam-3786	67	17	test	test	NOUN
ejpam-3786	67	18	and	and	CCONJ
ejpam-3786	67	19	curvature	curvature	VERB
ejpam-3786	67	20	distribution	distribution	NOUN
ejpam-3786	67	21	method	method	NOUN
ejpam-3786	67	22	.	.	PUNCT
ejpam-3786	68	1	all	all	DET
ejpam-3786	68	2	the	the	DET
ejpam-3786	68	3	necessary	necessary	ADJ
ejpam-3786	68	4	definitions	definition	NOUN
ejpam-3786	68	5	concerning	concern	VERB
ejpam-3786	68	6	weight	weight	NOUN
ejpam-3786	68	7	test	test	NOUN
ejpam-3786	68	8	can	can	AUX
ejpam-3786	68	9	be	be	AUX
ejpam-3786	68	10	found	find	VERB
ejpam-3786	68	11	in	in	ADP
ejpam-3786	68	12	[	[	X
ejpam-3786	68	13	9	9	NUM
ejpam-3786	68	14	]	]	PUNCT
ejpam-3786	68	15	.	.	PUNCT
ejpam-3786	69	1	the	the	DET
ejpam-3786	69	2	weight	weight	NOUN
ejpam-3786	69	3	test	test	NOUN
ejpam-3786	69	4	states	state	VERB
ejpam-3786	69	5	that	that	SCONJ
ejpam-3786	69	6	if	if	SCONJ
ejpam-3786	69	7	the	the	DET
ejpam-3786	69	8	star	star	NOUN
ejpam-3786	69	9	graph	graph	NOUN
ejpam-3786	69	10	γ	γ	NOUN
ejpam-3786	69	11	of	of	ADP
ejpam-3786	69	12	p	p	PROPN
ejpam-3786	69	13	admits	admit	VERB
ejpam-3786	69	14	an	an	DET
ejpam-3786	69	15	aspherical	aspherical	ADJ
ejpam-3786	69	16	weight	weight	NOUN
ejpam-3786	69	17	function	function	NOUN
ejpam-3786	69	18	θ	θ	NOUN
ejpam-3786	69	19	,	,	PUNCT
ejpam-3786	69	20	then	then	ADV
ejpam-3786	69	21	p	p	NOUN
ejpam-3786	69	22	is	be	AUX
ejpam-3786	69	23	aspherical	aspherical	ADJ
ejpam-3786	69	24	[	[	X
ejpam-3786	69	25	9	9	NUM
ejpam-3786	69	26	]	]	PUNCT
ejpam-3786	69	27	.	.	PUNCT
ejpam-3786	70	1	all	all	DET
ejpam-3786	70	2	the	the	DET
ejpam-3786	70	3	definitions	definition	NOUN
ejpam-3786	70	4	related	relate	VERB
ejpam-3786	70	5	to	to	ADP
ejpam-3786	70	6	pictures	picture	NOUN
ejpam-3786	70	7	can	can	AUX
ejpam-3786	70	8	be	be	AUX
ejpam-3786	70	9	found	find	VERB
ejpam-3786	70	10	in	in	ADP
ejpam-3786	70	11	[	[	X
ejpam-3786	70	12	14	14	NUM
ejpam-3786	70	13	]	]	PUNCT
ejpam-3786	70	14	.	.	PUNCT
ejpam-3786	71	1	the	the	DET
ejpam-3786	71	2	curvature	curvature	NOUN
ejpam-3786	71	3	distribution	distribution	NOUN
ejpam-3786	71	4	asserts	assert	VERB
ejpam-3786	71	5	that	that	SCONJ
ejpam-3786	71	6	if	if	SCONJ
ejpam-3786	71	7	k	k	PROPN
ejpam-3786	71	8	is	be	AUX
ejpam-3786	71	9	a	a	DET
ejpam-3786	71	10	reduced	reduce	VERB
ejpam-3786	71	11	picture	picture	NOUN
ejpam-3786	71	12	over	over	ADP
ejpam-3786	71	13	p	p	NOUN
ejpam-3786	71	14	then	then	ADV
ejpam-3786	71	15	by	by	ADP
ejpam-3786	71	16	euler	euler	PROPN
ejpam-3786	71	17	(	(	PUNCT
ejpam-3786	71	18	or	or	CCONJ
ejpam-3786	71	19	gauss	gauss	ADJ
ejpam-3786	71	20	-	-	PUNCT
ejpam-3786	71	21	bonnet	bonnet	NOUN
ejpam-3786	71	22	)	)	PUNCT
ejpam-3786	71	23	formula	formula	NOUN
ejpam-3786	71	24	,	,	PUNCT
ejpam-3786	71	25	the	the	DET
ejpam-3786	71	26	sum	sum	NOUN
ejpam-3786	71	27	of	of	ADP
ejpam-3786	71	28	the	the	DET
ejpam-3786	71	29	curvature	curvature	NOUN
ejpam-3786	71	30	of	of	ADP
ejpam-3786	71	31	all	all	DET
ejpam-3786	71	32	regions	region	NOUN
ejpam-3786	71	33	of	of	ADP
ejpam-3786	71	34	k	k	PROPN
ejpam-3786	71	35	is	be	AUX
ejpam-3786	71	36	4π	4π	NUM
ejpam-3786	71	37	,	,	PUNCT
ejpam-3786	71	38	that	that	ADV
ejpam-3786	71	39	is	is	ADV
ejpam-3786	71	40	,	,	PUNCT
ejpam-3786	71	41	k	k	PROPN
ejpam-3786	71	42	contains	contain	VERB
ejpam-3786	71	43	regions	region	NOUN
ejpam-3786	71	44	of	of	ADP
ejpam-3786	71	45	positive	positive	ADJ
ejpam-3786	71	46	curvature	curvature	NOUN
ejpam-3786	71	47	[	[	X
ejpam-3786	71	48	14	14	NUM
ejpam-3786	71	49	]	]	PUNCT
ejpam-3786	71	50	.	.	PUNCT
ejpam-3786	72	1	then	then	ADV
ejpam-3786	72	2	,	,	PUNCT
ejpam-3786	72	3	if	if	SCONJ
ejpam-3786	72	4	for	for	ADP
ejpam-3786	72	5	every	every	DET
ejpam-3786	72	6	region	region	NOUN
ejpam-3786	72	7	∆	∆	PROPN
ejpam-3786	72	8	of	of	ADP
ejpam-3786	72	9	k	k	PROPN
ejpam-3786	72	10	of	of	ADP
ejpam-3786	72	11	positive	positive	ADJ
ejpam-3786	72	12	curvature	curvature	NOUN
ejpam-3786	72	13	c(∆	c(∆	PROPN
ejpam-3786	72	14	)	)	PUNCT
ejpam-3786	72	15	,	,	PUNCT
ejpam-3786	72	16	there	there	PRON
ejpam-3786	72	17	is	be	VERB
ejpam-3786	72	18	a	a	DET
ejpam-3786	72	19	neighbouring	neighbouring	ADJ
ejpam-3786	72	20	region	region	NOUN
ejpam-3786	72	21	∆̂	∆̂	NOUN
ejpam-3786	72	22	,	,	PUNCT
ejpam-3786	72	23	uniquely	uniquely	ADV
ejpam-3786	72	24	associated	associate	VERB
ejpam-3786	72	25	with	with	ADP
ejpam-3786	72	26	∆	∆	PROPN
ejpam-3786	72	27	,	,	PUNCT
ejpam-3786	72	28	like	like	ADP
ejpam-3786	72	29	c(∆̂	c(∆̂	PROPN
ejpam-3786	72	30	)	)	PUNCT
ejpam-3786	72	31	+	+	NUM
ejpam-3786	72	32	c(∆	c(∆	NOUN
ejpam-3786	72	33	)	)	PUNCT
ejpam-3786	72	34	≤	≤	NOUN
ejpam-3786	72	35	0	0	NUM
ejpam-3786	72	36	,	,	PUNCT
ejpam-3786	72	37	then	then	ADV
ejpam-3786	72	38	the	the	DET
ejpam-3786	72	39	sum	sum	NOUN
ejpam-3786	72	40	of	of	ADP
ejpam-3786	72	41	the	the	DET
ejpam-3786	72	42	curvature	curvature	NOUN
ejpam-3786	72	43	of	of	ADP
ejpam-3786	72	44	all	all	DET
ejpam-3786	72	45	regions	region	NOUN
ejpam-3786	72	46	of	of	ADP
ejpam-3786	72	47	k	k	PROPN
ejpam-3786	72	48	is	be	AUX
ejpam-3786	72	49	non	non	ADJ
ejpam-3786	72	50	-	-	ADJ
ejpam-3786	72	51	positive	positive	ADJ
ejpam-3786	72	52	,	,	PUNCT
ejpam-3786	72	53	which	which	PRON
ejpam-3786	72	54	implies	imply	VERB
ejpam-3786	72	55	that	that	SCONJ
ejpam-3786	72	56	p	p	PROPN
ejpam-3786	72	57	is	be	AUX
ejpam-3786	72	58	aspherical	aspherical	ADJ
ejpam-3786	72	59	[	[	X
ejpam-3786	72	60	6	6	NUM
ejpam-3786	72	61	,	,	PUNCT
ejpam-3786	72	62	8	8	NUM
ejpam-3786	72	63	]	]	PUNCT
ejpam-3786	72	64	.	.	PUNCT
ejpam-3786	73	1	consider	consider	VERB
ejpam-3786	73	2	a	a	DET
ejpam-3786	73	3	torsion	torsion	NOUN
ejpam-3786	73	4	free	free	ADJ
ejpam-3786	73	5	group	group	NOUN
ejpam-3786	73	6	g.	g.	NOUN
ejpam-3786	73	7	by	by	ADP
ejpam-3786	73	8	applying	apply	VERB
ejpam-3786	73	9	the	the	DET
ejpam-3786	73	10	transformation	transformation	NOUN
ejpam-3786	73	11	u	u	NOUN
ejpam-3786	73	12	=	=	NOUN
ejpam-3786	73	13	tb	tb	X
ejpam-3786	73	14	on	on	ADP
ejpam-3786	73	15	s1(t	s1(t	ADJ
ejpam-3786	73	16	)	)	PUNCT
ejpam-3786	73	17	=	=	SYM
ejpam-3786	74	1	atbtctdtetft−1gthtit−1	atbtctdtetft−1gthtit−1	PROPN
ejpam-3786	74	2	=	=	PUNCT
ejpam-3786	74	3	1	1	NUM
ejpam-3786	74	4	it	it	PRON
ejpam-3786	74	5	can	can	AUX
ejpam-3786	74	6	be	be	AUX
ejpam-3786	74	7	assumed	assume	VERB
ejpam-3786	74	8	that	that	SCONJ
ejpam-3786	74	9	b	b	NOUN
ejpam-3786	74	10	=	=	SYM
ejpam-3786	74	11	1	1	X
ejpam-3786	74	12	.	.	X
ejpam-3786	74	13	recall	recall	VERB
ejpam-3786	74	14	that	that	PRON
ejpam-3786	74	15	p	p	NOUN
ejpam-3786	74	16	=	=	PUNCT
ejpam-3786	74	17	〈	〈	PROPN
ejpam-3786	74	18	g	g	NOUN
ejpam-3786	74	19	,	,	PUNCT
ejpam-3786	74	20	t	t	PROPN
ejpam-3786	74	21	|	|	ADV
ejpam-3786	74	22	s1(t	s1(t	PART
ejpam-3786	74	23	)	)	PUNCT
ejpam-3786	74	24	〉	〉	NOUN
ejpam-3786	74	25	in	in	ADP
ejpam-3786	74	26	which	which	PRON
ejpam-3786	74	27	s1(t	s1(t	SYM
ejpam-3786	74	28	)	)	PUNCT
ejpam-3786	74	29	=	=	SYM
ejpam-3786	74	30	atbtctdtetft−1gthtit−1	atbtctdtetft−1gthtit−1	PROPN
ejpam-3786	74	31	(	(	PUNCT
ejpam-3786	74	32	a	a	PROPN
ejpam-3786	74	33	,	,	PUNCT
ejpam-3786	74	34	f	f	X
ejpam-3786	74	35	,	,	PUNCT
ejpam-3786	74	36	g	g	PROPN
ejpam-3786	74	37	,	,	PUNCT
ejpam-3786	74	38	i	i	PRON
ejpam-3786	74	39	∈	∈	VERB
ejpam-3786	74	40	g	g	PROPN
ejpam-3786	74	41	\	\	PROPN
ejpam-3786	74	42	{	{	PUNCT
ejpam-3786	74	43	1	1	NUM
ejpam-3786	74	44	}	}	PUNCT
ejpam-3786	74	45	,	,	PUNCT
ejpam-3786	74	46	b	b	X
ejpam-3786	74	47	=	=	SYM
ejpam-3786	74	48	1	1	NUM
ejpam-3786	74	49	,	,	PUNCT
ejpam-3786	74	50	c	c	NOUN
ejpam-3786	74	51	,	,	PUNCT
ejpam-3786	74	52	d	d	NOUN
ejpam-3786	74	53	,	,	PUNCT
ejpam-3786	74	54	e	e	NOUN
ejpam-3786	74	55	,	,	PUNCT
ejpam-3786	74	56	h	h	NOUN
ejpam-3786	74	57	∈	∈	PROPN
ejpam-3786	74	58	g	g	PROPN
ejpam-3786	74	59	)	)	PUNCT
ejpam-3786	74	60	.	.	PUNCT
ejpam-3786	75	1	moreover	moreover	ADV
ejpam-3786	75	2	,	,	PUNCT
ejpam-3786	75	3	g	g	PROPN
ejpam-3786	75	4	is	be	AUX
ejpam-3786	75	5	not	not	PART
ejpam-3786	75	6	cyclic	cyclic	ADJ
ejpam-3786	75	7	and	and	CCONJ
ejpam-3786	75	8	g	g	NOUN
ejpam-3786	75	9	=	=	PUNCT
ejpam-3786	75	10	〈	〈	PROPN
ejpam-3786	75	11	a	a	PRON
ejpam-3786	75	12	,	,	PUNCT
ejpam-3786	75	13	b	b	NOUN
ejpam-3786	75	14	,	,	PUNCT
ejpam-3786	75	15	c	c	NOUN
ejpam-3786	75	16	,	,	PUNCT
ejpam-3786	75	17	d	d	NOUN
ejpam-3786	75	18	,	,	PUNCT
ejpam-3786	75	19	e	e	NOUN
ejpam-3786	75	20	,	,	PUNCT
ejpam-3786	75	21	f	f	PROPN
ejpam-3786	75	22	,	,	PUNCT
ejpam-3786	75	23	g	g	PROPN
ejpam-3786	75	24	,	,	PUNCT
ejpam-3786	75	25	h	h	NOUN
ejpam-3786	75	26	,	,	PUNCT
ejpam-3786	75	27	i	i	PROPN
ejpam-3786	75	28	〉	〉	NOUN
ejpam-3786	75	29	given	give	VERB
ejpam-3786	75	30	in	in	ADP
ejpam-3786	75	31	[	[	NOUN
ejpam-3786	75	32	13	13	NUM
ejpam-3786	75	33	]	]	PUNCT
ejpam-3786	75	34	.	.	PUNCT
ejpam-3786	76	1	suppose	suppose	VERB
ejpam-3786	76	2	that	that	SCONJ
ejpam-3786	76	3	k	k	PROPN
ejpam-3786	76	4	is	be	AUX
ejpam-3786	76	5	a	a	DET
ejpam-3786	76	6	reduced	reduce	VERB
ejpam-3786	76	7	spherical	spherical	ADJ
ejpam-3786	76	8	diagram	diagram	NOUN
ejpam-3786	76	9	over	over	ADP
ejpam-3786	76	10	p.	p.	NOUN
ejpam-3786	76	11	up	up	ADP
ejpam-3786	76	12	to	to	ADP
ejpam-3786	76	13	cyclic	cyclic	ADJ
ejpam-3786	76	14	permutation	permutation	NOUN
ejpam-3786	76	15	and	and	CCONJ
ejpam-3786	76	16	inversion	inversion	NOUN
ejpam-3786	76	17	,	,	PUNCT
ejpam-3786	76	18	the	the	DET
ejpam-3786	76	19	regions	region	NOUN
ejpam-3786	76	20	of	of	ADP
ejpam-3786	76	21	k	k	PROPN
ejpam-3786	76	22	are	be	AUX
ejpam-3786	76	23	given	give	VERB
ejpam-3786	76	24	by	by	ADP
ejpam-3786	76	25	∆	∆	PROPN
ejpam-3786	76	26	as	as	SCONJ
ejpam-3786	76	27	shown	show	VERB
ejpam-3786	76	28	in	in	ADP
ejpam-3786	76	29	figure	figure	NOUN
ejpam-3786	76	30	1(i	1(i	NUM
ejpam-3786	76	31	)	)	PUNCT
ejpam-3786	76	32	.	.	PUNCT
ejpam-3786	77	1	the	the	DET
ejpam-3786	77	2	star	star	NOUN
ejpam-3786	77	3	graph	graph	NOUN
ejpam-3786	77	4	γ	γ	NOUN
ejpam-3786	77	5	of	of	ADP
ejpam-3786	77	6	p	p	PROPN
ejpam-3786	77	7	is	be	AUX
ejpam-3786	77	8	shown	show	VERB
ejpam-3786	77	9	in	in	ADP
ejpam-3786	77	10	figure	figure	NOUN
ejpam-3786	77	11	1(ii	1(ii	NUM
ejpam-3786	77	12	)	)	PUNCT
ejpam-3786	77	13	.	.	PUNCT
ejpam-3786	78	1	figure	figure	VERB
ejpam-3786	78	2	1	1	NUM
ejpam-3786	78	3	:	:	PUNCT
ejpam-3786	78	4	region	region	NOUN
ejpam-3786	78	5	∆	∆	PROPN
ejpam-3786	78	6	of	of	ADP
ejpam-3786	78	7	k	k	PROPN
ejpam-3786	78	8	and	and	CCONJ
ejpam-3786	78	9	star	star	NOUN
ejpam-3786	78	10	graph	graph	NOUN
ejpam-3786	78	11	γ	γ	NOUN
ejpam-3786	78	12	of	of	ADP
ejpam-3786	78	13	p	p	PROPN
ejpam-3786	78	14	muhammad	muhammad	PROPN
ejpam-3786	78	15	saeed	saeed	PROPN
ejpam-3786	78	16	akram	akram	PROPN
ejpam-3786	78	17	,	,	PUNCT
ejpam-3786	78	18	maira	maira	PROPN
ejpam-3786	78	19	amjid	amjid	PROPN
ejpam-3786	78	20	,	,	PUNCT
ejpam-3786	78	21	sohail	sohail	PROPN
ejpam-3786	78	22	iqbal	iqbal	PROPN
ejpam-3786	78	23	/	/	PUNCT
ejpam-3786	78	24	eur	eur	PROPN
ejpam-3786	78	25	.	.	PUNCT
ejpam-3786	79	1	j.	j.	PROPN
ejpam-3786	79	2	pure	pure	PROPN
ejpam-3786	79	3	appl	appl	PROPN
ejpam-3786	79	4	.	.	PROPN
ejpam-3786	79	5	math	math	PROPN
ejpam-3786	79	6	,	,	PUNCT
ejpam-3786	79	7	13	13	NUM
ejpam-3786	79	8	(	(	PUNCT
ejpam-3786	79	9	4	4	NUM
ejpam-3786	79	10	)	)	PUNCT
ejpam-3786	79	11	(	(	PUNCT
ejpam-3786	79	12	2020	2020	NUM
ejpam-3786	79	13	)	)	PUNCT
ejpam-3786	79	14	,	,	PUNCT
ejpam-3786	79	15	914	914	NUM
ejpam-3786	79	16	-	-	SYM
ejpam-3786	79	17	938	938	NUM
ejpam-3786	79	18	917	917	NUM
ejpam-3786	79	19	looking	look	VERB
ejpam-3786	79	20	at	at	ADP
ejpam-3786	79	21	closed	closed	ADJ
ejpam-3786	79	22	paths	path	NOUN
ejpam-3786	79	23	in	in	ADP
ejpam-3786	79	24	star	star	NOUN
ejpam-3786	79	25	graph	graph	NOUN
ejpam-3786	79	26	γ	γ	PROPN
ejpam-3786	79	27	,	,	PUNCT
ejpam-3786	79	28	using	use	VERB
ejpam-3786	79	29	the	the	DET
ejpam-3786	79	30	fact	fact	NOUN
ejpam-3786	79	31	that	that	SCONJ
ejpam-3786	79	32	g	g	PROPN
ejpam-3786	79	33	is	be	AUX
ejpam-3786	79	34	torsion	torsion	NOUN
ejpam-3786	79	35	free	free	ADJ
ejpam-3786	79	36	and	and	CCONJ
ejpam-3786	79	37	working	working	NOUN
ejpam-3786	79	38	modulo	modulo	PROPN
ejpam-3786	79	39	cyclic	cyclic	ADJ
ejpam-3786	79	40	permutation	permutation	NOUN
ejpam-3786	79	41	and	and	CCONJ
ejpam-3786	79	42	inversion	inversion	NOUN
ejpam-3786	79	43	,	,	PUNCT
ejpam-3786	79	44	the	the	DET
ejpam-3786	79	45	possible	possible	ADJ
ejpam-3786	79	46	labels	label	NOUN
ejpam-3786	79	47	of	of	ADP
ejpam-3786	79	48	vertices	vertex	NOUN
ejpam-3786	79	49	of	of	ADP
ejpam-3786	79	50	degree	degree	NOUN
ejpam-3786	79	51	2	2	NUM
ejpam-3786	79	52	for	for	ADP
ejpam-3786	79	53	a	a	DET
ejpam-3786	79	54	region	region	NOUN
ejpam-3786	79	55	∆	∆	PROPN
ejpam-3786	79	56	of	of	ADP
ejpam-3786	79	57	k	k	PROPN
ejpam-3786	79	58	are	be	AUX
ejpam-3786	79	59	s	s	PRON
ejpam-3786	79	60	=	=	PUNCT
ejpam-3786	79	61	{	{	PUNCT
ejpam-3786	79	62	ag	ag	PROPN
ejpam-3786	79	63	,	,	PUNCT
ejpam-3786	79	64	ag−1	ag−1	NOUN
ejpam-3786	79	65	,	,	PUNCT
ejpam-3786	79	66	fi	fi	NOUN
ejpam-3786	79	67	,	,	PUNCT
ejpam-3786	79	68	fi−1	fi−1	PROPN
ejpam-3786	79	69	,	,	PUNCT
ejpam-3786	79	70	hb−1	hb−1	NOUN
ejpam-3786	79	71	,	,	PUNCT
ejpam-3786	79	72	hc−1	hc−1	NOUN
ejpam-3786	79	73	,	,	PUNCT
ejpam-3786	79	74	hd−1	hd−1	PROPN
ejpam-3786	79	75	,	,	PUNCT
ejpam-3786	79	76	he−1	he−1	PROPN
ejpam-3786	79	77	,	,	PUNCT
ejpam-3786	79	78	eb−1	eb−1	PROPN
ejpam-3786	79	79	,	,	PUNCT
ejpam-3786	79	80	ec−1	ec−1	NOUN
ejpam-3786	79	81	,	,	PUNCT
ejpam-3786	79	82	ed−1	ed−1	PROPN
ejpam-3786	79	83	,	,	PUNCT
ejpam-3786	79	84	dc−1	dc−1	NOUN
ejpam-3786	79	85	,	,	PUNCT
ejpam-3786	79	86	db−1	db−1	PROPN
ejpam-3786	79	87	,	,	PUNCT
ejpam-3786	79	88	cb−1	cb−1	PROPN
ejpam-3786	79	89	}	}	PUNCT
ejpam-3786	79	90	.	.	PUNCT
ejpam-3786	80	1	we	we	PRON
ejpam-3786	80	2	can	can	AUX
ejpam-3786	80	3	work	work	VERB
ejpam-3786	80	4	modulo	modulo	ADJ
ejpam-3786	80	5	equivalence	equivalence	NOUN
ejpam-3786	80	6	,	,	PUNCT
ejpam-3786	80	7	that	that	ADV
ejpam-3786	80	8	is	is	ADV
ejpam-3786	80	9	,	,	PUNCT
ejpam-3786	80	10	modulo	modulo	PROPN
ejpam-3786	80	11	t↔	t↔	PROPN
ejpam-3786	80	12	t−1	t−1	PROPN
ejpam-3786	80	13	,	,	PUNCT
ejpam-3786	80	14	cyclic	cyclic	ADJ
ejpam-3786	80	15	permutation	permutation	NOUN
ejpam-3786	80	16	,	,	PUNCT
ejpam-3786	80	17	inversion	inversion	NOUN
ejpam-3786	80	18	,	,	PUNCT
ejpam-3786	80	19	and	and	CCONJ
ejpam-3786	80	20	a↔	a↔	PROPN
ejpam-3786	80	21	f−1	f−1	PROPN
ejpam-3786	80	22	,	,	PUNCT
ejpam-3786	80	23	g	g	PROPN
ejpam-3786	80	24	↔	↔	PROPN
ejpam-3786	80	25	i−1	i−1	PROPN
ejpam-3786	80	26	,	,	PUNCT
ejpam-3786	80	27	b↔	b↔	ADV
ejpam-3786	80	28	e−1	e−1	PROPN
ejpam-3786	80	29	,	,	PUNCT
ejpam-3786	80	30	c↔	c↔	PROPN
ejpam-3786	80	31	d−1	d−1	PROPN
ejpam-3786	80	32	,	,	PUNCT
ejpam-3786	80	33	h↔	h↔	X
ejpam-3786	80	34	h−1	h−1	PROPN
ejpam-3786	80	35	.	.	PUNCT
ejpam-3786	81	1	we	we	PRON
ejpam-3786	81	2	will	will	AUX
ejpam-3786	81	3	proceed	proceed	VERB
ejpam-3786	81	4	according	accord	VERB
ejpam-3786	81	5	to	to	ADP
ejpam-3786	81	6	the	the	DET
ejpam-3786	81	7	number	number	NOUN
ejpam-3786	81	8	n	n	ADP
ejpam-3786	81	9	of	of	ADP
ejpam-3786	81	10	labels	label	NOUN
ejpam-3786	81	11	in	in	ADP
ejpam-3786	81	12	s	s	PRON
ejpam-3786	81	13	that	that	PRON
ejpam-3786	81	14	are	be	AUX
ejpam-3786	81	15	admissible	admissible	ADJ
ejpam-3786	81	16	[	[	X
ejpam-3786	81	17	9	9	NUM
ejpam-3786	81	18	]	]	PUNCT
ejpam-3786	81	19	and	and	CCONJ
ejpam-3786	81	20	classify	classify	VERB
ejpam-3786	81	21	the	the	DET
ejpam-3786	81	22	cases	case	NOUN
ejpam-3786	81	23	correspondingly	correspondingly	ADV
ejpam-3786	81	24	[	[	X
ejpam-3786	81	25	8	8	NUM
ejpam-3786	81	26	]	]	PUNCT
ejpam-3786	81	27	.	.	PUNCT
ejpam-3786	82	1	the	the	DET
ejpam-3786	82	2	following	follow	VERB
ejpam-3786	82	3	remark	remark	NOUN
ejpam-3786	82	4	substantially	substantially	ADV
ejpam-3786	82	5	reduce	reduce	VERB
ejpam-3786	82	6	the	the	DET
ejpam-3786	82	7	number	number	NOUN
ejpam-3786	82	8	of	of	ADP
ejpam-3786	82	9	cases	case	NOUN
ejpam-3786	82	10	to	to	PART
ejpam-3786	82	11	be	be	AUX
ejpam-3786	82	12	considered	consider	VERB
ejpam-3786	82	13	.	.	PUNCT
ejpam-3786	83	1	remark	remark	PROPN
ejpam-3786	83	2	1	1	NUM
ejpam-3786	83	3	.	.	PUNCT
ejpam-3786	84	1	the	the	DET
ejpam-3786	84	2	following	follow	VERB
ejpam-3786	84	3	observations	observation	NOUN
ejpam-3786	84	4	holds	hold	VERB
ejpam-3786	84	5	trivially	trivially	ADV
ejpam-3786	84	6	.	.	PUNCT
ejpam-3786	85	1	(	(	PUNCT
ejpam-3786	85	2	i	i	NOUN
ejpam-3786	85	3	)	)	PUNCT
ejpam-3786	85	4	if	if	SCONJ
ejpam-3786	85	5	all	all	DET
ejpam-3786	85	6	the	the	DET
ejpam-3786	85	7	admissible	admissible	ADJ
ejpam-3786	85	8	cycle	cycle	NOUN
ejpam-3786	85	9	has	have	VERB
ejpam-3786	85	10	length	length	NOUN
ejpam-3786	85	11	greater	great	ADJ
ejpam-3786	85	12	than	than	ADP
ejpam-3786	85	13	2	2	NUM
ejpam-3786	85	14	in	in	ADP
ejpam-3786	85	15	region	region	NOUN
ejpam-3786	85	16	∆	∆	PROPN
ejpam-3786	85	17	then	then	ADV
ejpam-3786	85	18	c(∆	c(∆	PROPN
ejpam-3786	85	19	)	)	PUNCT
ejpam-3786	85	20	≤	≤	NOUN
ejpam-3786	85	21	c(3	c(3	ADV
ejpam-3786	85	22	,	,	PUNCT
ejpam-3786	85	23	3	3	NUM
ejpam-3786	85	24	,	,	PUNCT
ejpam-3786	85	25	3	3	NUM
ejpam-3786	85	26	,	,	PUNCT
ejpam-3786	85	27	3	3	NUM
ejpam-3786	85	28	,	,	PUNCT
ejpam-3786	85	29	3	3	NUM
ejpam-3786	85	30	,	,	PUNCT
ejpam-3786	85	31	3	3	NUM
ejpam-3786	85	32	,	,	PUNCT
ejpam-3786	85	33	3	3	NUM
ejpam-3786	85	34	,	,	PUNCT
ejpam-3786	85	35	3	3	NUM
ejpam-3786	85	36	,	,	PUNCT
ejpam-3786	85	37	3	3	NUM
ejpam-3786	85	38	)	)	PUNCT
ejpam-3786	85	39	=	=	SYM
ejpam-3786	85	40	−π	−π	PROPN
ejpam-3786	85	41	.	.	PUNCT
ejpam-3786	86	1	(	(	PUNCT
ejpam-3786	86	2	ii	ii	NOUN
ejpam-3786	86	3	)	)	PUNCT
ejpam-3786	86	4	if	if	SCONJ
ejpam-3786	86	5	ag	ag	PROPN
ejpam-3786	86	6	and	and	CCONJ
ejpam-3786	86	7	ag−1	ag−1	NOUN
ejpam-3786	86	8	are	be	AUX
ejpam-3786	86	9	admissible	admissible	ADJ
ejpam-3786	86	10	then	then	ADV
ejpam-3786	86	11	g2	g2	PROPN
ejpam-3786	86	12	=	=	PUNCT
ejpam-3786	86	13	1	1	NUM
ejpam-3786	86	14	,	,	PUNCT
ejpam-3786	86	15	a	a	DET
ejpam-3786	86	16	contradiction	contradiction	NOUN
ejpam-3786	86	17	.	.	PUNCT
ejpam-3786	87	1	(	(	PUNCT
ejpam-3786	87	2	iii	iii	X
ejpam-3786	87	3	)	)	PUNCT
ejpam-3786	87	4	if	if	SCONJ
ejpam-3786	87	5	fi	fi	NOUN
ejpam-3786	87	6	and	and	CCONJ
ejpam-3786	87	7	fi−1	fi−1	PROPN
ejpam-3786	87	8	are	be	AUX
ejpam-3786	87	9	admissible	admissible	ADJ
ejpam-3786	87	10	then	then	ADV
ejpam-3786	87	11	i2	i2	PROPN
ejpam-3786	87	12	=	=	SYM
ejpam-3786	87	13	1	1	NUM
ejpam-3786	87	14	,	,	PUNCT
ejpam-3786	87	15	a	a	DET
ejpam-3786	87	16	contradiction	contradiction	NOUN
ejpam-3786	87	17	.	.	PUNCT
ejpam-3786	88	1	(	(	PUNCT
ejpam-3786	88	2	iv	iv	X
ejpam-3786	88	3	)	)	PUNCT
ejpam-3786	88	4	at	at	ADP
ejpam-3786	88	5	most	most	ADV
ejpam-3786	88	6	two	two	NUM
ejpam-3786	88	7	of	of	ADP
ejpam-3786	88	8	ag	ag	PROPN
ejpam-3786	88	9	,	,	PUNCT
ejpam-3786	88	10	ag−1	ag−1	NOUN
ejpam-3786	88	11	,	,	PUNCT
ejpam-3786	88	12	fi	fi	NOUN
ejpam-3786	88	13	,	,	PUNCT
ejpam-3786	88	14	fi−1	fi−1	PROPN
ejpam-3786	88	15	are	be	AUX
ejpam-3786	88	16	admissible	admissible	ADJ
ejpam-3786	88	17	.	.	PUNCT
ejpam-3786	89	1	(	(	PUNCT
ejpam-3786	89	2	v	v	NOUN
ejpam-3786	89	3	)	)	PUNCT
ejpam-3786	89	4	if	if	SCONJ
ejpam-3786	89	5	any	any	DET
ejpam-3786	89	6	two	two	NUM
ejpam-3786	89	7	of	of	ADP
ejpam-3786	89	8	hb−1	hb−1	NOUN
ejpam-3786	89	9	,	,	PUNCT
ejpam-3786	89	10	hc−1	hc−1	NOUN
ejpam-3786	89	11	,	,	PUNCT
ejpam-3786	89	12	cb−1	cb−1	NOUN
ejpam-3786	89	13	are	be	AUX
ejpam-3786	89	14	admissible	admissible	ADJ
ejpam-3786	89	15	then	then	ADV
ejpam-3786	89	16	so	so	ADV
ejpam-3786	89	17	is	be	AUX
ejpam-3786	89	18	the	the	DET
ejpam-3786	89	19	third	third	ADJ
ejpam-3786	89	20	.	.	PUNCT
ejpam-3786	90	1	(	(	PUNCT
ejpam-3786	90	2	vi	vi	X
ejpam-3786	90	3	)	)	PUNCT
ejpam-3786	90	4	if	if	SCONJ
ejpam-3786	90	5	any	any	DET
ejpam-3786	90	6	two	two	NUM
ejpam-3786	90	7	of	of	ADP
ejpam-3786	90	8	hb−1	hb−1	NOUN
ejpam-3786	90	9	,	,	PUNCT
ejpam-3786	90	10	hd−1	hd−1	PROPN
ejpam-3786	90	11	,	,	PUNCT
ejpam-3786	90	12	db−1	db−1	PROPN
ejpam-3786	90	13	are	be	AUX
ejpam-3786	90	14	admissible	admissible	ADJ
ejpam-3786	90	15	then	then	ADV
ejpam-3786	90	16	so	so	ADV
ejpam-3786	90	17	is	be	AUX
ejpam-3786	90	18	the	the	DET
ejpam-3786	90	19	third	third	ADJ
ejpam-3786	90	20	.	.	PUNCT
ejpam-3786	91	1	(	(	PUNCT
ejpam-3786	91	2	vii	vii	PROPN
ejpam-3786	91	3	)	)	PUNCT
ejpam-3786	91	4	if	if	SCONJ
ejpam-3786	91	5	any	any	DET
ejpam-3786	91	6	two	two	NUM
ejpam-3786	91	7	of	of	ADP
ejpam-3786	91	8	hb−1	hb−1	NOUN
ejpam-3786	91	9	,	,	PUNCT
ejpam-3786	91	10	he−1	he−1	PROPN
ejpam-3786	91	11	,	,	PUNCT
ejpam-3786	91	12	eb−1	eb−1	PROPN
ejpam-3786	91	13	are	be	AUX
ejpam-3786	91	14	admissible	admissible	ADJ
ejpam-3786	91	15	then	then	ADV
ejpam-3786	91	16	so	so	ADV
ejpam-3786	91	17	is	be	AUX
ejpam-3786	91	18	the	the	DET
ejpam-3786	91	19	third	third	ADJ
ejpam-3786	91	20	.	.	PUNCT
ejpam-3786	92	1	(	(	PUNCT
ejpam-3786	92	2	viii	viii	NOUN
ejpam-3786	92	3	)	)	PUNCT
ejpam-3786	92	4	if	if	SCONJ
ejpam-3786	92	5	any	any	DET
ejpam-3786	92	6	two	two	NUM
ejpam-3786	92	7	of	of	ADP
ejpam-3786	92	8	hc−1	hc−1	PROPN
ejpam-3786	92	9	,	,	PUNCT
ejpam-3786	92	10	hd−1	hd−1	PROPN
ejpam-3786	92	11	,	,	PUNCT
ejpam-3786	92	12	dc−1	dc−1	NOUN
ejpam-3786	92	13	are	be	AUX
ejpam-3786	92	14	admissible	admissible	ADJ
ejpam-3786	92	15	then	then	ADV
ejpam-3786	92	16	so	so	ADV
ejpam-3786	92	17	is	be	AUX
ejpam-3786	92	18	the	the	DET
ejpam-3786	92	19	third	third	ADJ
ejpam-3786	92	20	.	.	PUNCT
ejpam-3786	93	1	(	(	PUNCT
ejpam-3786	93	2	ix	ix	ADP
ejpam-3786	93	3	)	)	PUNCT
ejpam-3786	93	4	if	if	SCONJ
ejpam-3786	93	5	any	any	DET
ejpam-3786	93	6	two	two	NUM
ejpam-3786	93	7	of	of	ADP
ejpam-3786	93	8	hc−1	hc−1	PROPN
ejpam-3786	93	9	,	,	PUNCT
ejpam-3786	93	10	he−1	he−1	PROPN
ejpam-3786	93	11	,	,	PUNCT
ejpam-3786	93	12	ec−1	ec−1	NOUN
ejpam-3786	93	13	are	be	AUX
ejpam-3786	93	14	admissible	admissible	ADJ
ejpam-3786	93	15	then	then	ADV
ejpam-3786	93	16	so	so	ADV
ejpam-3786	93	17	is	be	AUX
ejpam-3786	93	18	the	the	DET
ejpam-3786	93	19	third	third	ADJ
ejpam-3786	93	20	.	.	PUNCT
ejpam-3786	94	1	(	(	PUNCT
ejpam-3786	94	2	x	x	X
ejpam-3786	94	3	)	)	PUNCT
ejpam-3786	94	4	if	if	SCONJ
ejpam-3786	94	5	any	any	DET
ejpam-3786	94	6	two	two	NUM
ejpam-3786	94	7	of	of	ADP
ejpam-3786	94	8	hd−1	hd−1	PROPN
ejpam-3786	94	9	,	,	PUNCT
ejpam-3786	94	10	he−1	he−1	PROPN
ejpam-3786	94	11	,	,	PUNCT
ejpam-3786	94	12	ed−1	ed−1	PROPN
ejpam-3786	94	13	are	be	AUX
ejpam-3786	94	14	admissible	admissible	ADJ
ejpam-3786	94	15	then	then	ADV
ejpam-3786	94	16	so	so	ADV
ejpam-3786	94	17	is	be	AUX
ejpam-3786	94	18	the	the	DET
ejpam-3786	94	19	third	third	ADJ
ejpam-3786	94	20	.	.	PUNCT
ejpam-3786	95	1	(	(	PUNCT
ejpam-3786	95	2	xi	xi	ADP
ejpam-3786	95	3	)	)	PUNCT
ejpam-3786	95	4	if	if	SCONJ
ejpam-3786	95	5	any	any	DET
ejpam-3786	95	6	two	two	NUM
ejpam-3786	95	7	of	of	ADP
ejpam-3786	95	8	eb−1	eb−1	PROPN
ejpam-3786	95	9	,	,	PUNCT
ejpam-3786	95	10	ec−1	ec−1	NOUN
ejpam-3786	95	11	,	,	PUNCT
ejpam-3786	95	12	cb−1	cb−1	NOUN
ejpam-3786	95	13	are	be	AUX
ejpam-3786	95	14	admissible	admissible	ADJ
ejpam-3786	95	15	then	then	ADV
ejpam-3786	95	16	so	so	ADV
ejpam-3786	95	17	is	be	AUX
ejpam-3786	95	18	the	the	DET
ejpam-3786	95	19	third	third	ADJ
ejpam-3786	95	20	.	.	PUNCT
ejpam-3786	96	1	(	(	PUNCT
ejpam-3786	96	2	xii	xii	NOUN
ejpam-3786	96	3	)	)	PUNCT
ejpam-3786	96	4	if	if	SCONJ
ejpam-3786	96	5	any	any	DET
ejpam-3786	96	6	two	two	NUM
ejpam-3786	96	7	of	of	ADP
ejpam-3786	96	8	eb−1	eb−1	PROPN
ejpam-3786	96	9	,	,	PUNCT
ejpam-3786	96	10	ed−1	ed−1	PROPN
ejpam-3786	96	11	,	,	PUNCT
ejpam-3786	96	12	db−1	db−1	PROPN
ejpam-3786	96	13	are	be	AUX
ejpam-3786	96	14	admissible	admissible	ADJ
ejpam-3786	96	15	then	then	ADV
ejpam-3786	96	16	so	so	ADV
ejpam-3786	96	17	is	be	AUX
ejpam-3786	96	18	the	the	DET
ejpam-3786	96	19	third	third	ADJ
ejpam-3786	96	20	.	.	PUNCT
ejpam-3786	97	1	(	(	PUNCT
ejpam-3786	97	2	xiii	xiii	PROPN
ejpam-3786	97	3	)	)	PUNCT
ejpam-3786	97	4	if	if	SCONJ
ejpam-3786	97	5	any	any	DET
ejpam-3786	97	6	two	two	NUM
ejpam-3786	97	7	of	of	ADP
ejpam-3786	97	8	ec−1	ec−1	NOUN
ejpam-3786	97	9	,	,	PUNCT
ejpam-3786	97	10	ed−1	ed−1	PROPN
ejpam-3786	97	11	,	,	PUNCT
ejpam-3786	97	12	dc−1	dc−1	NOUN
ejpam-3786	97	13	are	be	AUX
ejpam-3786	97	14	admissible	admissible	ADJ
ejpam-3786	97	15	then	then	ADV
ejpam-3786	97	16	so	so	ADV
ejpam-3786	97	17	is	be	AUX
ejpam-3786	97	18	the	the	DET
ejpam-3786	97	19	third	third	ADJ
ejpam-3786	97	20	.	.	PUNCT
ejpam-3786	98	1	(	(	PUNCT
ejpam-3786	98	2	xiv	xiv	PROPN
ejpam-3786	98	3	)	)	PUNCT
ejpam-3786	98	4	if	if	SCONJ
ejpam-3786	98	5	any	any	DET
ejpam-3786	98	6	two	two	NUM
ejpam-3786	98	7	of	of	ADP
ejpam-3786	98	8	dc−1	dc−1	NOUN
ejpam-3786	98	9	,	,	PUNCT
ejpam-3786	98	10	db−1	db−1	PROPN
ejpam-3786	98	11	,	,	PUNCT
ejpam-3786	98	12	cb−1	cb−1	NOUN
ejpam-3786	98	13	are	be	AUX
ejpam-3786	98	14	admissible	admissible	ADJ
ejpam-3786	98	15	then	then	ADV
ejpam-3786	98	16	so	so	ADV
ejpam-3786	98	17	is	be	AUX
ejpam-3786	98	18	the	the	DET
ejpam-3786	98	19	third	third	ADJ
ejpam-3786	98	20	.	.	PUNCT
ejpam-3786	99	1	the	the	DET
ejpam-3786	99	2	above	above	ADJ
ejpam-3786	99	3	remark	remark	NOUN
ejpam-3786	99	4	reduces	reduce	VERB
ejpam-3786	99	5	the	the	DET
ejpam-3786	99	6	number	number	NOUN
ejpam-3786	99	7	of	of	ADP
ejpam-3786	99	8	cases	case	NOUN
ejpam-3786	99	9	to	to	ADP
ejpam-3786	99	10	245	245	NUM
ejpam-3786	99	11	for	for	ADP
ejpam-3786	99	12	the	the	DET
ejpam-3786	99	13	group	group	NOUN
ejpam-3786	99	14	equation	equation	NOUN
ejpam-3786	99	15	s1(t	s1(t	PROPN
ejpam-3786	99	16	)	)	PUNCT
ejpam-3786	99	17	=	=	SYM
ejpam-3786	100	1	1	1	X
ejpam-3786	100	2	.	.	X
ejpam-3786	100	3	from	from	ADP
ejpam-3786	100	4	these	these	DET
ejpam-3786	100	5	245	245	NUM
ejpam-3786	100	6	cases	case	NOUN
ejpam-3786	100	7	,	,	PUNCT
ejpam-3786	100	8	41	41	NUM
ejpam-3786	100	9	cases	case	NOUN
ejpam-3786	100	10	are	be	AUX
ejpam-3786	100	11	solved	solve	VERB
ejpam-3786	100	12	in	in	ADP
ejpam-3786	100	13	[	[	X
ejpam-3786	100	14	4	4	NUM
ejpam-3786	100	15	]	]	PUNCT
ejpam-3786	100	16	so	so	SCONJ
ejpam-3786	100	17	there	there	PRON
ejpam-3786	100	18	remains	remain	VERB
ejpam-3786	100	19	204	204	NUM
ejpam-3786	100	20	cases	case	NOUN
ejpam-3786	100	21	that	that	PRON
ejpam-3786	100	22	needs	need	VERB
ejpam-3786	100	23	to	to	PART
ejpam-3786	100	24	be	be	AUX
ejpam-3786	100	25	solved	solve	VERB
ejpam-3786	100	26	.	.	PUNCT
ejpam-3786	101	1	among	among	ADP
ejpam-3786	101	2	these	these	DET
ejpam-3786	101	3	204	204	NUM
ejpam-3786	101	4	remaining	remain	VERB
ejpam-3786	101	5	cases	case	NOUN
ejpam-3786	101	6	,	,	PUNCT
ejpam-3786	101	7	159	159	NUM
ejpam-3786	101	8	cases	case	NOUN
ejpam-3786	101	9	are	be	AUX
ejpam-3786	101	10	solved	solve	VERB
ejpam-3786	101	11	by	by	ADP
ejpam-3786	101	12	weight	weight	NOUN
ejpam-3786	101	13	test	test	NOUN
ejpam-3786	101	14	,	,	PUNCT
ejpam-3786	101	15	43	43	NUM
ejpam-3786	101	16	cases	case	NOUN
ejpam-3786	101	17	are	be	AUX
ejpam-3786	101	18	solved	solve	VERB
ejpam-3786	101	19	by	by	ADP
ejpam-3786	101	20	curvature	curvature	NOUN
ejpam-3786	101	21	distribution	distribution	NOUN
ejpam-3786	101	22	method	method	NOUN
ejpam-3786	101	23	and	and	CCONJ
ejpam-3786	101	24	2	2	NUM
ejpam-3786	101	25	cases	case	NOUN
ejpam-3786	101	26	are	be	AUX
ejpam-3786	101	27	still	still	ADV
ejpam-3786	101	28	open	open	ADJ
ejpam-3786	101	29	.	.	PUNCT
ejpam-3786	102	1	muhammad	muhammad	PROPN
ejpam-3786	102	2	saeed	saeed	PROPN
ejpam-3786	102	3	akram	akram	PROPN
ejpam-3786	102	4	,	,	PUNCT
ejpam-3786	102	5	maira	maira	PROPN
ejpam-3786	102	6	amjid	amjid	PROPN
ejpam-3786	102	7	,	,	PUNCT
ejpam-3786	102	8	sohail	sohail	PROPN
ejpam-3786	102	9	iqbal	iqbal	PROPN
ejpam-3786	102	10	/	/	PUNCT
ejpam-3786	102	11	eur	eur	PROPN
ejpam-3786	102	12	.	.	PUNCT
ejpam-3786	103	1	j.	j.	PROPN
ejpam-3786	103	2	pure	pure	PROPN
ejpam-3786	103	3	appl	appl	PROPN
ejpam-3786	103	4	.	.	PROPN
ejpam-3786	103	5	math	math	PROPN
ejpam-3786	103	6	,	,	PUNCT
ejpam-3786	103	7	13	13	NUM
ejpam-3786	103	8	(	(	PUNCT
ejpam-3786	103	9	4	4	NUM
ejpam-3786	103	10	)	)	PUNCT
ejpam-3786	103	11	(	(	PUNCT
ejpam-3786	103	12	2020	2020	NUM
ejpam-3786	103	13	)	)	PUNCT
ejpam-3786	103	14	,	,	PUNCT
ejpam-3786	103	15	914	914	NUM
ejpam-3786	103	16	-	-	SYM
ejpam-3786	103	17	938	938	NUM
ejpam-3786	103	18	918	918	NUM
ejpam-3786	103	19	3	3	NUM
ejpam-3786	103	20	.	.	PUNCT
ejpam-3786	103	21	main	main	ADJ
ejpam-3786	103	22	results	result	NOUN
ejpam-3786	103	23	:	:	PUNCT
ejpam-3786	103	24	a	a	DET
ejpam-3786	103	25	further	further	ADJ
ejpam-3786	103	26	159	159	NUM
ejpam-3786	103	27	cases	case	NOUN
ejpam-3786	103	28	can	can	AUX
ejpam-3786	103	29	be	be	AUX
ejpam-3786	103	30	straightforwardly	straightforwardly	ADV
ejpam-3786	103	31	solved	solve	VERB
ejpam-3786	103	32	using	use	VERB
ejpam-3786	103	33	the	the	DET
ejpam-3786	103	34	weight	weight	NOUN
ejpam-3786	103	35	test	test	NOUN
ejpam-3786	103	36	.	.	PUNCT
ejpam-3786	104	1	for	for	ADP
ejpam-3786	104	2	example	example	NOUN
ejpam-3786	104	3	,	,	PUNCT
ejpam-3786	104	4	consider	consider	VERB
ejpam-3786	104	5	the	the	DET
ejpam-3786	104	6	case	case	NOUN
ejpam-3786	104	7	,	,	PUNCT
ejpam-3786	104	8	a	a	DET
ejpam-3786	104	9	=	=	X
ejpam-3786	104	10	g−1	g−1	PROPN
ejpam-3786	104	11	,	,	PUNCT
ejpam-3786	104	12	f	f	X
ejpam-3786	104	13	=	=	SYM
ejpam-3786	104	14	i−1	i−1	PROPN
ejpam-3786	104	15	and	and	CCONJ
ejpam-3786	104	16	h	h	NOUN
ejpam-3786	104	17	=	=	PROPN
ejpam-3786	104	18	b.	b.	PROPN
ejpam-3786	104	19	in	in	ADP
ejpam-3786	104	20	this	this	DET
ejpam-3786	104	21	case	case	NOUN
ejpam-3786	104	22	,	,	PUNCT
ejpam-3786	104	23	the	the	DET
ejpam-3786	104	24	relator	relator	NOUN
ejpam-3786	104	25	is	be	AUX
ejpam-3786	104	26	s(t	s(t	PROPN
ejpam-3786	104	27	)	)	PUNCT
ejpam-3786	104	28	=	=	PUNCT
ejpam-3786	105	1	at2ctdtetft−1a−1t2f−1t−1	at2ctdtetft−1a−1t2f−1t−1	NOUN
ejpam-3786	105	2	.	.	PUNCT
ejpam-3786	106	1	we	we	PRON
ejpam-3786	106	2	put	put	VERB
ejpam-3786	106	3	x	x	X
ejpam-3786	106	4	=	=	SYM
ejpam-3786	106	5	t−1a−1	t−1a−1	PROPN
ejpam-3786	106	6	t	t	NOUN
ejpam-3786	106	7	to	to	PART
ejpam-3786	106	8	obtain	obtain	VERB
ejpam-3786	106	9	v1	v1	NOUN
ejpam-3786	106	10	=	=	SYM
ejpam-3786	106	11	x−1tctdtetfxtf−1	x−1tctdtetfxtf−1	PROPN
ejpam-3786	106	12	and	and	CCONJ
ejpam-3786	106	13	v2	v2	PROPN
ejpam-3786	106	14	=	=	SYM
ejpam-3786	106	15	x−1t−1a−1	x−1t−1a−1	NOUN
ejpam-3786	106	16	t.	t.	NOUN
ejpam-3786	106	17	the	the	DET
ejpam-3786	106	18	presentation	presentation	NOUN
ejpam-3786	106	19	p	p	PROPN
ejpam-3786	106	20	has	have	VERB
ejpam-3786	106	21	star	star	NOUN
ejpam-3786	106	22	graph	graph	NOUN
ejpam-3786	106	23	γ	γ	NOUN
ejpam-3786	106	24	which	which	PRON
ejpam-3786	106	25	is	be	AUX
ejpam-3786	106	26	shown	show	VERB
ejpam-3786	106	27	in	in	ADP
ejpam-3786	106	28	figure	figure	NOUN
ejpam-3786	106	29	2	2	NUM
ejpam-3786	106	30	in	in	ADP
ejpam-3786	106	31	which	which	PRON
ejpam-3786	106	32	µ1	µ1	NOUN
ejpam-3786	106	33	=	=	SYM
ejpam-3786	106	34	c	c	NOUN
ejpam-3786	106	35	,	,	PUNCT
ejpam-3786	106	36	µ2	µ2	PROPN
ejpam-3786	106	37	=	=	SYM
ejpam-3786	106	38	d	d	PROPN
ejpam-3786	106	39	,	,	PUNCT
ejpam-3786	106	40	µ3	µ3	NOUN
ejpam-3786	106	41	=	=	SYM
ejpam-3786	106	42	e	e	NOUN
ejpam-3786	106	43	,	,	PUNCT
ejpam-3786	106	44	µ4	µ4	PROPN
ejpam-3786	106	45	=	=	SYM
ejpam-3786	106	46	1	1	NUM
ejpam-3786	106	47	,	,	PUNCT
ejpam-3786	106	48	µ5	µ5	NOUN
ejpam-3786	106	49	=	=	SYM
ejpam-3786	106	50	f	f	PROPN
ejpam-3786	106	51	,	,	PUNCT
ejpam-3786	106	52	µ6	µ6	NOUN
ejpam-3786	106	53	=	=	SYM
ejpam-3786	106	54	f−1	f−1	PROPN
ejpam-3786	106	55	,	,	PUNCT
ejpam-3786	106	56	µ7	µ7	PROPN
ejpam-3786	106	57	=	=	SYM
ejpam-3786	106	58	1	1	NUM
ejpam-3786	106	59	;	;	PUNCT
ejpam-3786	106	60	and	and	CCONJ
ejpam-3786	106	61	ω1	ω1	PROPN
ejpam-3786	106	62	=	=	SYM
ejpam-3786	106	63	a−1	a−1	PROPN
ejpam-3786	106	64	,	,	PUNCT
ejpam-3786	106	65	ω2	ω2	NOUN
ejpam-3786	106	66	=	=	SYM
ejpam-3786	106	67	1	1	NUM
ejpam-3786	106	68	,	,	PUNCT
ejpam-3786	106	69	ω3	ω3	NOUN
ejpam-3786	106	70	=	=	SYM
ejpam-3786	106	71	1	1	X
ejpam-3786	106	72	.	.	X
ejpam-3786	106	73	figure	figure	NOUN
ejpam-3786	106	74	2	2	NUM
ejpam-3786	106	75	:	:	PUNCT
ejpam-3786	106	76	star	star	NOUN
ejpam-3786	106	77	graph	graph	NOUN
ejpam-3786	106	78	γ	γ	PROPN
ejpam-3786	106	79	we	we	PRON
ejpam-3786	106	80	define	define	VERB
ejpam-3786	106	81	a	a	DET
ejpam-3786	106	82	weight	weight	NOUN
ejpam-3786	106	83	function	function	NOUN
ejpam-3786	106	84	θ	θ	NOUN
ejpam-3786	107	1	such	such	ADJ
ejpam-3786	107	2	that	that	PRON
ejpam-3786	107	3	θ(µ4	θ(µ4	NOUN
ejpam-3786	107	4	)	)	PUNCT
ejpam-3786	107	5	=	=	SYM
ejpam-3786	107	6	θ(µ6	θ(µ6	ADJ
ejpam-3786	107	7	)	)	PUNCT
ejpam-3786	107	8	=	=	SYM
ejpam-3786	107	9	θ(ω1	θ(ω1	X
ejpam-3786	107	10	)	)	PUNCT
ejpam-3786	107	11	=	=	SYM
ejpam-3786	107	12	θ(ω2	θ(ω2	X
ejpam-3786	107	13	)	)	PUNCT
ejpam-3786	107	14	=	=	SYM
ejpam-3786	107	15	0	0	NUM
ejpam-3786	107	16	and	and	CCONJ
ejpam-3786	107	17	θ(µ1	θ(µ1	NOUN
ejpam-3786	107	18	)	)	PUNCT
ejpam-3786	107	19	=	=	SYM
ejpam-3786	107	20	θ(µ2	θ(µ2	ADJ
ejpam-3786	107	21	)	)	PUNCT
ejpam-3786	107	22	=	=	SYM
ejpam-3786	107	23	θ(µ3	θ(µ3	X
ejpam-3786	107	24	)	)	PUNCT
ejpam-3786	107	25	=	=	SYM
ejpam-3786	107	26	θ(µ5	θ(µ5	NOUN
ejpam-3786	107	27	)	)	PUNCT
ejpam-3786	107	28	=	=	SYM
ejpam-3786	107	29	θ(µ7	θ(µ7	NOUN
ejpam-3786	107	30	)	)	PUNCT
ejpam-3786	107	31	=	=	SYM
ejpam-3786	107	32	θ(ω3	θ(ω3	PROPN
ejpam-3786	107	33	)	)	PUNCT
ejpam-3786	107	34	=	=	SYM
ejpam-3786	107	35	1	1	X
ejpam-3786	107	36	.	.	PUNCT
ejpam-3786	107	37	then	then	ADV
ejpam-3786	107	38	σ(1−θ(µi	σ(1−θ(µi	NOUN
ejpam-3786	107	39	)	)	PUNCT
ejpam-3786	107	40	)	)	PUNCT
ejpam-3786	107	41	=	=	PUNCT
ejpam-3786	108	1	σ(1−θ(ωj	σ(1−θ(ωj	NUM
ejpam-3786	108	2	)	)	PUNCT
ejpam-3786	108	3	)	)	PUNCT
ejpam-3786	109	1	=	=	SYM
ejpam-3786	109	2	2	2	NUM
ejpam-3786	109	3	indicates	indicate	VERB
ejpam-3786	109	4	that	that	SCONJ
ejpam-3786	109	5	the	the	DET
ejpam-3786	109	6	first	first	ADJ
ejpam-3786	109	7	condition	condition	NOUN
ejpam-3786	109	8	of	of	ADP
ejpam-3786	109	9	weight	weight	NOUN
ejpam-3786	109	10	test	test	NOUN
ejpam-3786	109	11	is	be	AUX
ejpam-3786	109	12	fulfilled	fulfil	VERB
ejpam-3786	109	13	.	.	PUNCT
ejpam-3786	110	1	moreover	moreover	ADV
ejpam-3786	110	2	,	,	PUNCT
ejpam-3786	110	3	every	every	DET
ejpam-3786	110	4	cycle	cycle	NOUN
ejpam-3786	110	5	in	in	ADP
ejpam-3786	110	6	γ	γ	NOUN
ejpam-3786	110	7	of	of	ADP
ejpam-3786	110	8	weight	weight	NOUN
ejpam-3786	110	9	smaller	small	ADJ
ejpam-3786	110	10	than	than	ADP
ejpam-3786	110	11	2	2	NUM
ejpam-3786	110	12	has	have	VERB
ejpam-3786	110	13	label	label	NOUN
ejpam-3786	110	14	am	am	NOUN
ejpam-3786	110	15	,	,	PUNCT
ejpam-3786	110	16	where	where	SCONJ
ejpam-3786	110	17	m	m	VERB
ejpam-3786	110	18	∈	∈	PROPN
ejpam-3786	110	19	z	z	NOUN
ejpam-3786	110	20	\	\	PUNCT
ejpam-3786	110	21	{	{	PUNCT
ejpam-3786	110	22	0	0	NUM
ejpam-3786	110	23	}	}	PUNCT
ejpam-3786	110	24	and	and	CCONJ
ejpam-3786	110	25	a	a	DET
ejpam-3786	110	26	∈	∈	NOUN
ejpam-3786	110	27	g	g	ADP
ejpam-3786	110	28	\	\	PROPN
ejpam-3786	110	29	{	{	PUNCT
ejpam-3786	110	30	1	1	NUM
ejpam-3786	110	31	}	}	PUNCT
ejpam-3786	110	32	,	,	PUNCT
ejpam-3786	110	33	which	which	PRON
ejpam-3786	110	34	implies	imply	VERB
ejpam-3786	110	35	a	a	DET
ejpam-3786	110	36	is	be	AUX
ejpam-3786	110	37	torsion	torsion	NOUN
ejpam-3786	110	38	element	element	NOUN
ejpam-3786	110	39	in	in	ADP
ejpam-3786	110	40	g	g	PROPN
ejpam-3786	110	41	,	,	PUNCT
ejpam-3786	110	42	a	a	DET
ejpam-3786	110	43	contradiction	contradiction	NOUN
ejpam-3786	110	44	,	,	PUNCT
ejpam-3786	110	45	so	so	CCONJ
ejpam-3786	110	46	the	the	DET
ejpam-3786	110	47	second	second	ADJ
ejpam-3786	110	48	condition	condition	NOUN
ejpam-3786	110	49	of	of	ADP
ejpam-3786	110	50	weight	weight	NOUN
ejpam-3786	110	51	test	test	NOUN
ejpam-3786	110	52	is	be	AUX
ejpam-3786	110	53	fulfilled	fulfil	VERB
ejpam-3786	110	54	.	.	PUNCT
ejpam-3786	111	1	furthermore	furthermore	ADV
ejpam-3786	111	2	,	,	PUNCT
ejpam-3786	111	3	since	since	SCONJ
ejpam-3786	111	4	θ	θ	PROPN
ejpam-3786	111	5	assigns	assign	VERB
ejpam-3786	111	6	non	non	ADJ
ejpam-3786	111	7	-	-	ADJ
ejpam-3786	111	8	negative	negative	ADJ
ejpam-3786	111	9	weights	weight	NOUN
ejpam-3786	111	10	to	to	ADP
ejpam-3786	111	11	each	each	DET
ejpam-3786	111	12	edge	edge	NOUN
ejpam-3786	111	13	,	,	PUNCT
ejpam-3786	111	14	so	so	SCONJ
ejpam-3786	111	15	the	the	DET
ejpam-3786	111	16	third	third	ADJ
ejpam-3786	111	17	condition	condition	NOUN
ejpam-3786	111	18	of	of	ADP
ejpam-3786	111	19	weight	weight	NOUN
ejpam-3786	111	20	test	test	NOUN
ejpam-3786	111	21	is	be	AUX
ejpam-3786	111	22	obviously	obviously	ADV
ejpam-3786	111	23	fulfilled	fulfil	VERB
ejpam-3786	111	24	.	.	PUNCT
ejpam-3786	112	1	a	a	DET
ejpam-3786	112	2	further	further	ADJ
ejpam-3786	112	3	24	24	NUM
ejpam-3786	112	4	cases	case	NOUN
ejpam-3786	112	5	are	be	AUX
ejpam-3786	112	6	solved	solve	VERB
ejpam-3786	112	7	in	in	ADP
ejpam-3786	112	8	lemma	lemma	PROPN
ejpam-3786	112	9	1	1	NUM
ejpam-3786	112	10	by	by	ADP
ejpam-3786	112	11	an	an	DET
ejpam-3786	112	12	immediate	immediate	ADJ
ejpam-3786	112	13	application	application	NOUN
ejpam-3786	112	14	of	of	ADP
ejpam-3786	112	15	curvature	curvature	NOUN
ejpam-3786	112	16	distribution	distribution	NOUN
ejpam-3786	112	17	method	method	NOUN
ejpam-3786	112	18	[	[	X
ejpam-3786	112	19	10	10	NUM
ejpam-3786	112	20	]	]	PUNCT
ejpam-3786	112	21	.	.	PUNCT
ejpam-3786	113	1	in	in	ADP
ejpam-3786	113	2	what	what	PRON
ejpam-3786	113	3	follows	follow	VERB
ejpam-3786	113	4	,	,	PUNCT
ejpam-3786	113	5	the	the	DET
ejpam-3786	113	6	vertex	vertex	NOUN
ejpam-3786	113	7	labels	label	NOUN
ejpam-3786	113	8	correspond	correspond	VERB
ejpam-3786	113	9	to	to	ADP
ejpam-3786	113	10	the	the	DET
ejpam-3786	113	11	closed	closed	ADJ
ejpam-3786	113	12	paths	path	NOUN
ejpam-3786	113	13	in	in	ADP
ejpam-3786	113	14	the	the	DET
ejpam-3786	113	15	star	star	NOUN
ejpam-3786	113	16	graph	graph	NOUN
ejpam-3786	113	17	γ	γ	PROPN
ejpam-3786	113	18	.	.	PROPN
ejpam-3786	113	19	from	from	ADP
ejpam-3786	113	20	now	now	ADV
ejpam-3786	113	21	onward	onward	ADV
ejpam-3786	113	22	,	,	PUNCT
ejpam-3786	113	23	the	the	DET
ejpam-3786	113	24	label	label	NOUN
ejpam-3786	113	25	and	and	CCONJ
ejpam-3786	113	26	the	the	DET
ejpam-3786	113	27	degree	degree	NOUN
ejpam-3786	113	28	of	of	ADP
ejpam-3786	113	29	a	a	DET
ejpam-3786	113	30	vertex	vertex	NOUN
ejpam-3786	113	31	v	v	NOUN
ejpam-3786	113	32	of	of	ADP
ejpam-3786	113	33	region	region	NOUN
ejpam-3786	113	34	∆	∆	PROPN
ejpam-3786	113	35	will	will	AUX
ejpam-3786	113	36	be	be	AUX
ejpam-3786	113	37	denoted	denote	VERB
ejpam-3786	113	38	by	by	ADP
ejpam-3786	113	39	l∆(v	l∆(v	NOUN
ejpam-3786	113	40	)	)	PUNCT
ejpam-3786	113	41	and	and	CCONJ
ejpam-3786	113	42	d∆(v	d∆(v	NOUN
ejpam-3786	113	43	)	)	PUNCT
ejpam-3786	113	44	respectively	respectively	ADV
ejpam-3786	113	45	.	.	PUNCT
ejpam-3786	114	1	furthermore	furthermore	ADV
ejpam-3786	114	2	,	,	PUNCT
ejpam-3786	114	3	l∆	l∆	PROPN
ejpam-3786	114	4	∈	∈	PROPN
ejpam-3786	114	5	{	{	PUNCT
ejpam-3786	114	6	ww1	ww1	PROPN
ejpam-3786	114	7	,	,	PUNCT
ejpam-3786	114	8	.	.	PUNCT
ejpam-3786	114	9	.	.	PUNCT
ejpam-3786	114	10	.	.	PUNCT
ejpam-3786	115	1	,	,	PUNCT
ejpam-3786	115	2	wwk	wwk	PROPN
ejpam-3786	115	3	}	}	PUNCT
ejpam-3786	115	4	will	will	AUX
ejpam-3786	115	5	be	be	AUX
ejpam-3786	115	6	indicated	indicate	VERB
ejpam-3786	115	7	by	by	ADP
ejpam-3786	115	8	l∆(v	l∆(v	NOUN
ejpam-3786	115	9	)	)	PUNCT
ejpam-3786	115	10	=	=	PRON
ejpam-3786	115	11	{	{	PUNCT
ejpam-3786	115	12	ww1	ww1	PROPN
ejpam-3786	115	13	,	,	PUNCT
ejpam-3786	115	14	.	.	PUNCT
ejpam-3786	115	15	.	.	PUNCT
ejpam-3786	116	1	.	.	PUNCT
ejpam-3786	117	1	,	,	PUNCT
ejpam-3786	117	2	wwk	wwk	PROPN
ejpam-3786	117	3	}	}	PUNCT
ejpam-3786	117	4	.	.	PUNCT
ejpam-3786	118	1	lemma	lemma	PROPN
ejpam-3786	118	2	1	1	NUM
ejpam-3786	118	3	.	.	PUNCT
ejpam-3786	119	1	the	the	DET
ejpam-3786	119	2	presentation	presentation	NOUN
ejpam-3786	119	3	p	p	X
ejpam-3786	119	4	=	=	PUNCT
ejpam-3786	119	5	〈	〈	PROPN
ejpam-3786	119	6	g	g	NOUN
ejpam-3786	119	7	,	,	PUNCT
ejpam-3786	119	8	t	t	PROPN
ejpam-3786	119	9	|	|	ADV
ejpam-3786	119	10	s1(t	s1(t	PART
ejpam-3786	119	11	)	)	PUNCT
ejpam-3786	119	12	〉	〉	PROPN
ejpam-3786	119	13	is	be	AUX
ejpam-3786	119	14	aspherical	aspherical	ADJ
ejpam-3786	119	15	if	if	SCONJ
ejpam-3786	119	16	any	any	DET
ejpam-3786	119	17	one	one	NUM
ejpam-3786	119	18	of	of	ADP
ejpam-3786	119	19	the	the	DET
ejpam-3786	119	20	following	follow	VERB
ejpam-3786	119	21	holds	hold	VERB
ejpam-3786	119	22	:	:	PUNCT
ejpam-3786	119	23	(	(	PUNCT
ejpam-3786	119	24	i	i	NOUN
ejpam-3786	119	25	)	)	PUNCT
ejpam-3786	119	26	a	a	DET
ejpam-3786	119	27	=	=	SYM
ejpam-3786	119	28	g	g	NOUN
ejpam-3786	119	29	;	;	PUNCT
ejpam-3786	119	30	(	(	PUNCT
ejpam-3786	119	31	ii	ii	NOUN
ejpam-3786	119	32	)	)	PUNCT
ejpam-3786	119	33	h	h	NOUN
ejpam-3786	120	1	=	=	SYM
ejpam-3786	120	2	b	b	PROPN
ejpam-3786	120	3	;	;	PUNCT
ejpam-3786	120	4	(	(	PUNCT
ejpam-3786	120	5	iii	iii	X
ejpam-3786	120	6	)	)	PUNCT
ejpam-3786	120	7	h	h	NOUN
ejpam-3786	121	1	=	=	SYM
ejpam-3786	121	2	c	c	PROPN
ejpam-3786	121	3	;	;	PUNCT
ejpam-3786	121	4	muhammad	muhammad	PROPN
ejpam-3786	121	5	saeed	saeed	PROPN
ejpam-3786	121	6	akram	akram	PROPN
ejpam-3786	121	7	,	,	PUNCT
ejpam-3786	121	8	maira	maira	PROPN
ejpam-3786	121	9	amjid	amjid	PROPN
ejpam-3786	121	10	,	,	PUNCT
ejpam-3786	121	11	sohail	sohail	PROPN
ejpam-3786	121	12	iqbal	iqbal	PROPN
ejpam-3786	121	13	/	/	PUNCT
ejpam-3786	121	14	eur	eur	PROPN
ejpam-3786	121	15	.	.	PUNCT
ejpam-3786	122	1	j.	j.	PROPN
ejpam-3786	122	2	pure	pure	PROPN
ejpam-3786	122	3	appl	appl	PROPN
ejpam-3786	122	4	.	.	PROPN
ejpam-3786	122	5	math	math	PROPN
ejpam-3786	122	6	,	,	PUNCT
ejpam-3786	122	7	13	13	NUM
ejpam-3786	122	8	(	(	PUNCT
ejpam-3786	122	9	4	4	NUM
ejpam-3786	122	10	)	)	PUNCT
ejpam-3786	122	11	(	(	PUNCT
ejpam-3786	122	12	2020	2020	NUM
ejpam-3786	122	13	)	)	PUNCT
ejpam-3786	122	14	,	,	PUNCT
ejpam-3786	122	15	914	914	NUM
ejpam-3786	122	16	-	-	SYM
ejpam-3786	122	17	938	938	NUM
ejpam-3786	122	18	919	919	NUM
ejpam-3786	122	19	(	(	PUNCT
ejpam-3786	122	20	iv	iv	X
ejpam-3786	122	21	)	)	PUNCT
ejpam-3786	122	22	e	e	NOUN
ejpam-3786	122	23	=	=	SYM
ejpam-3786	122	24	b	b	PROPN
ejpam-3786	122	25	;	;	PUNCT
ejpam-3786	122	26	(	(	PUNCT
ejpam-3786	122	27	v	v	NOUN
ejpam-3786	122	28	)	)	PUNCT
ejpam-3786	122	29	e	e	NOUN
ejpam-3786	122	30	=	=	SYM
ejpam-3786	122	31	c	c	X
ejpam-3786	122	32	;	;	PUNCT
ejpam-3786	122	33	(	(	PUNCT
ejpam-3786	122	34	vi	vi	NOUN
ejpam-3786	122	35	)	)	PUNCT
ejpam-3786	122	36	e	e	NOUN
ejpam-3786	122	37	=	=	SYM
ejpam-3786	122	38	d	d	PROPN
ejpam-3786	122	39	;	;	PUNCT
ejpam-3786	122	40	(	(	PUNCT
ejpam-3786	122	41	vii	vii	PROPN
ejpam-3786	122	42	)	)	PUNCT
ejpam-3786	122	43	d	d	PROPN
ejpam-3786	122	44	=	=	SYM
ejpam-3786	122	45	c	c	NOUN
ejpam-3786	122	46	;	;	PUNCT
ejpam-3786	122	47	(	(	PUNCT
ejpam-3786	122	48	viii	viii	NOUN
ejpam-3786	122	49	)	)	PUNCT
ejpam-3786	122	50	e	e	NOUN
ejpam-3786	122	51	=	=	SYM
ejpam-3786	122	52	b	b	PROPN
ejpam-3786	122	53	,	,	PUNCT
ejpam-3786	122	54	d	d	PROPN
ejpam-3786	122	55	=	=	SYM
ejpam-3786	122	56	c	c	NOUN
ejpam-3786	122	57	;	;	PUNCT
ejpam-3786	122	58	(	(	PUNCT
ejpam-3786	122	59	ix	ix	INTJ
ejpam-3786	122	60	)	)	PUNCT
ejpam-3786	122	61	a	a	DET
ejpam-3786	122	62	=	=	SYM
ejpam-3786	122	63	g	g	NOUN
ejpam-3786	122	64	,	,	PUNCT
ejpam-3786	122	65	h	h	NOUN
ejpam-3786	123	1	=	=	SYM
ejpam-3786	123	2	c	c	X
ejpam-3786	123	3	,	,	PUNCT
ejpam-3786	123	4	d	d	PROPN
ejpam-3786	123	5	=	=	SYM
ejpam-3786	123	6	b	b	PROPN
ejpam-3786	123	7	;	;	PUNCT
ejpam-3786	123	8	(	(	PUNCT
ejpam-3786	123	9	x	x	X
ejpam-3786	123	10	)	)	PUNCT
ejpam-3786	123	11	a	a	DET
ejpam-3786	123	12	=	=	SYM
ejpam-3786	123	13	g	g	NOUN
ejpam-3786	123	14	,	,	PUNCT
ejpam-3786	123	15	h	h	NOUN
ejpam-3786	124	1	=	=	SYM
ejpam-3786	124	2	e	e	NOUN
ejpam-3786	124	3	,	,	PUNCT
ejpam-3786	124	4	d	d	PROPN
ejpam-3786	124	5	=	=	SYM
ejpam-3786	124	6	b	b	PROPN
ejpam-3786	124	7	;	;	PUNCT
ejpam-3786	124	8	(	(	PUNCT
ejpam-3786	124	9	xi	xi	X
ejpam-3786	124	10	)	)	PUNCT
ejpam-3786	124	11	a	a	DET
ejpam-3786	124	12	=	=	SYM
ejpam-3786	124	13	g	g	NOUN
ejpam-3786	124	14	,	,	PUNCT
ejpam-3786	124	15	h	h	NOUN
ejpam-3786	125	1	=	=	SYM
ejpam-3786	125	2	d	d	PROPN
ejpam-3786	125	3	,	,	PUNCT
ejpam-3786	125	4	e	e	PROPN
ejpam-3786	125	5	=	=	SYM
ejpam-3786	125	6	b	b	PROPN
ejpam-3786	125	7	;	;	PUNCT
ejpam-3786	125	8	(	(	PUNCT
ejpam-3786	125	9	xii	xii	NOUN
ejpam-3786	125	10	)	)	PUNCT
ejpam-3786	125	11	a	a	DET
ejpam-3786	125	12	=	=	SYM
ejpam-3786	125	13	g	g	NOUN
ejpam-3786	125	14	,	,	PUNCT
ejpam-3786	125	15	h	h	NOUN
ejpam-3786	126	1	=	=	SYM
ejpam-3786	126	2	d	d	PROPN
ejpam-3786	126	3	,	,	PUNCT
ejpam-3786	126	4	c	c	PROPN
ejpam-3786	126	5	=	=	SYM
ejpam-3786	126	6	b	b	PROPN
ejpam-3786	126	7	;	;	PUNCT
ejpam-3786	126	8	(	(	PUNCT
ejpam-3786	126	9	xiii	xiii	NOUN
ejpam-3786	126	10	)	)	PUNCT
ejpam-3786	126	11	h	h	NOUN
ejpam-3786	127	1	=	=	SYM
ejpam-3786	127	2	b	b	PROPN
ejpam-3786	127	3	,	,	PUNCT
ejpam-3786	127	4	h	h	NOUN
ejpam-3786	128	1	=	=	SYM
ejpam-3786	128	2	c	c	X
ejpam-3786	128	3	,	,	PUNCT
ejpam-3786	128	4	c	c	NOUN
ejpam-3786	128	5	=	=	SYM
ejpam-3786	128	6	b	b	PROPN
ejpam-3786	128	7	;	;	PUNCT
ejpam-3786	128	8	(	(	PUNCT
ejpam-3786	128	9	xiv	xiv	NOUN
ejpam-3786	128	10	)	)	PUNCT
ejpam-3786	128	11	h	h	NOUN
ejpam-3786	129	1	=	=	SYM
ejpam-3786	129	2	b	b	PROPN
ejpam-3786	129	3	,	,	PUNCT
ejpam-3786	129	4	h	h	NOUN
ejpam-3786	130	1	=	=	SYM
ejpam-3786	130	2	d	d	PROPN
ejpam-3786	130	3	,	,	PUNCT
ejpam-3786	130	4	d	d	PROPN
ejpam-3786	130	5	=	=	SYM
ejpam-3786	130	6	b	b	PROPN
ejpam-3786	130	7	;	;	PUNCT
ejpam-3786	130	8	(	(	PUNCT
ejpam-3786	130	9	xv	xv	PROPN
ejpam-3786	130	10	)	)	PUNCT
ejpam-3786	130	11	h	h	NOUN
ejpam-3786	130	12	=	=	SYM
ejpam-3786	130	13	b	b	PROPN
ejpam-3786	130	14	,	,	PUNCT
ejpam-3786	130	15	h	h	NOUN
ejpam-3786	130	16	=	=	SYM
ejpam-3786	130	17	e	e	NOUN
ejpam-3786	130	18	,	,	PUNCT
ejpam-3786	130	19	e	e	PROPN
ejpam-3786	130	20	=	=	SYM
ejpam-3786	130	21	b	b	PROPN
ejpam-3786	130	22	;	;	PUNCT
ejpam-3786	130	23	(	(	PUNCT
ejpam-3786	130	24	xvi	xvi	NOUN
ejpam-3786	130	25	)	)	PUNCT
ejpam-3786	130	26	h	h	NOUN
ejpam-3786	131	1	=	=	SYM
ejpam-3786	131	2	c	c	X
ejpam-3786	131	3	,	,	PUNCT
ejpam-3786	131	4	h	h	NOUN
ejpam-3786	132	1	=	=	SYM
ejpam-3786	132	2	d	d	PROPN
ejpam-3786	132	3	,	,	PUNCT
ejpam-3786	132	4	d	d	PROPN
ejpam-3786	132	5	=	=	SYM
ejpam-3786	132	6	c	c	NOUN
ejpam-3786	132	7	;	;	PUNCT
ejpam-3786	132	8	(	(	PUNCT
ejpam-3786	132	9	xvii	xvii	ADJ
ejpam-3786	132	10	)	)	PUNCT
ejpam-3786	133	1	e	e	NOUN
ejpam-3786	133	2	=	=	SYM
ejpam-3786	133	3	b	b	PROPN
ejpam-3786	133	4	,	,	PUNCT
ejpam-3786	133	5	e	e	X
ejpam-3786	133	6	=	=	SYM
ejpam-3786	133	7	c	c	X
ejpam-3786	133	8	,	,	PUNCT
ejpam-3786	133	9	c	c	NOUN
ejpam-3786	133	10	=	=	SYM
ejpam-3786	133	11	b	b	PROPN
ejpam-3786	133	12	;	;	PUNCT
ejpam-3786	133	13	(	(	PUNCT
ejpam-3786	133	14	xviii	xviii	ADJ
ejpam-3786	133	15	)	)	PUNCT
ejpam-3786	133	16	e	e	NOUN
ejpam-3786	133	17	=	=	SYM
ejpam-3786	133	18	c	c	X
ejpam-3786	133	19	,	,	PUNCT
ejpam-3786	133	20	e	e	X
ejpam-3786	134	1	=	=	SYM
ejpam-3786	134	2	d	d	PROPN
ejpam-3786	134	3	,	,	PUNCT
ejpam-3786	134	4	d	d	PROPN
ejpam-3786	134	5	=	=	SYM
ejpam-3786	134	6	c	c	NOUN
ejpam-3786	134	7	;	;	PUNCT
ejpam-3786	134	8	(	(	PUNCT
ejpam-3786	134	9	xix	xix	NOUN
ejpam-3786	134	10	)	)	PUNCT
ejpam-3786	134	11	a	a	DET
ejpam-3786	134	12	=	=	SYM
ejpam-3786	134	13	g	g	NOUN
ejpam-3786	134	14	,	,	PUNCT
ejpam-3786	134	15	h	h	NOUN
ejpam-3786	134	16	=	=	SYM
ejpam-3786	134	17	c	c	NOUN
ejpam-3786	134	18	,	,	PUNCT
ejpam-3786	134	19	h	h	NOUN
ejpam-3786	135	1	=	=	SYM
ejpam-3786	135	2	d	d	PROPN
ejpam-3786	135	3	,	,	PUNCT
ejpam-3786	135	4	d	d	PROPN
ejpam-3786	135	5	=	=	SYM
ejpam-3786	135	6	c	c	NOUN
ejpam-3786	135	7	;	;	PUNCT
ejpam-3786	135	8	(	(	PUNCT
ejpam-3786	135	9	xx	xx	X
ejpam-3786	135	10	)	)	PUNCT
ejpam-3786	135	11	a	a	DET
ejpam-3786	135	12	=	=	SYM
ejpam-3786	135	13	g	g	NOUN
ejpam-3786	135	14	,	,	PUNCT
ejpam-3786	135	15	h	h	NOUN
ejpam-3786	135	16	=	=	SYM
ejpam-3786	135	17	e	e	NOUN
ejpam-3786	135	18	,	,	PUNCT
ejpam-3786	135	19	h	h	NOUN
ejpam-3786	136	1	=	=	SYM
ejpam-3786	136	2	d	d	PROPN
ejpam-3786	136	3	,	,	PUNCT
ejpam-3786	136	4	e	e	X
ejpam-3786	136	5	=	=	SYM
ejpam-3786	136	6	d	d	PROPN
ejpam-3786	136	7	;	;	PUNCT
ejpam-3786	136	8	(	(	PUNCT
ejpam-3786	136	9	xxi	xxi	PROPN
ejpam-3786	136	10	)	)	PUNCT
ejpam-3786	136	11	a	a	DET
ejpam-3786	136	12	=	=	SYM
ejpam-3786	136	13	g	g	NOUN
ejpam-3786	136	14	,	,	PUNCT
ejpam-3786	136	15	e	e	PROPN
ejpam-3786	136	16	=	=	SYM
ejpam-3786	136	17	b	b	PROPN
ejpam-3786	136	18	,	,	PUNCT
ejpam-3786	136	19	d	d	PROPN
ejpam-3786	136	20	=	=	SYM
ejpam-3786	136	21	b	b	PROPN
ejpam-3786	136	22	,	,	PUNCT
ejpam-3786	137	1	e	e	X
ejpam-3786	137	2	=	=	SYM
ejpam-3786	137	3	d	d	PROPN
ejpam-3786	137	4	;	;	PUNCT
ejpam-3786	137	5	(	(	PUNCT
ejpam-3786	137	6	xxii	xxii	NOUN
ejpam-3786	137	7	)	)	PUNCT
ejpam-3786	137	8	h	h	NOUN
ejpam-3786	138	1	=	=	SYM
ejpam-3786	138	2	b	b	PROPN
ejpam-3786	138	3	,	,	PUNCT
ejpam-3786	138	4	h	h	NOUN
ejpam-3786	139	1	=	=	SYM
ejpam-3786	139	2	c	c	X
ejpam-3786	139	3	,	,	PUNCT
ejpam-3786	139	4	c	c	NOUN
ejpam-3786	139	5	=	=	SYM
ejpam-3786	139	6	b	b	PROPN
ejpam-3786	139	7	,	,	PUNCT
ejpam-3786	139	8	h	h	NOUN
ejpam-3786	140	1	=	=	SYM
ejpam-3786	140	2	d	d	PROPN
ejpam-3786	140	3	,	,	PUNCT
ejpam-3786	140	4	d	d	PROPN
ejpam-3786	140	5	=	=	SYM
ejpam-3786	140	6	b	b	PROPN
ejpam-3786	140	7	,	,	PUNCT
ejpam-3786	140	8	d	d	PROPN
ejpam-3786	140	9	=	=	SYM
ejpam-3786	140	10	c	c	NOUN
ejpam-3786	140	11	;	;	PUNCT
ejpam-3786	140	12	(	(	PUNCT
ejpam-3786	140	13	xxiii	xxiii	ADJ
ejpam-3786	140	14	)	)	PUNCT
ejpam-3786	140	15	h	h	NOUN
ejpam-3786	141	1	=	=	SYM
ejpam-3786	141	2	b	b	PROPN
ejpam-3786	141	3	,	,	PUNCT
ejpam-3786	141	4	h	h	NOUN
ejpam-3786	142	1	=	=	SYM
ejpam-3786	142	2	d	d	PROPN
ejpam-3786	142	3	,	,	PUNCT
ejpam-3786	142	4	d	d	PROPN
ejpam-3786	142	5	=	=	SYM
ejpam-3786	142	6	b	b	PROPN
ejpam-3786	142	7	,	,	PUNCT
ejpam-3786	142	8	h	h	NOUN
ejpam-3786	142	9	=	=	SYM
ejpam-3786	142	10	e	e	NOUN
ejpam-3786	142	11	,	,	PUNCT
ejpam-3786	142	12	e	e	PROPN
ejpam-3786	142	13	=	=	SYM
ejpam-3786	142	14	b	b	PROPN
ejpam-3786	142	15	,	,	PUNCT
ejpam-3786	142	16	e	e	X
ejpam-3786	142	17	=	=	SYM
ejpam-3786	142	18	d	d	PROPN
ejpam-3786	142	19	;	;	PUNCT
ejpam-3786	142	20	(	(	PUNCT
ejpam-3786	142	21	xxiv	xxiv	NUM
ejpam-3786	142	22	)	)	PUNCT
ejpam-3786	143	1	e	e	NOUN
ejpam-3786	143	2	=	=	SYM
ejpam-3786	143	3	b	b	PROPN
ejpam-3786	143	4	,	,	PUNCT
ejpam-3786	143	5	e	e	X
ejpam-3786	143	6	=	=	SYM
ejpam-3786	143	7	c	c	X
ejpam-3786	143	8	,	,	PUNCT
ejpam-3786	143	9	c	c	NOUN
ejpam-3786	143	10	=	=	SYM
ejpam-3786	143	11	b	b	PROPN
ejpam-3786	143	12	,	,	PUNCT
ejpam-3786	143	13	e	e	X
ejpam-3786	143	14	=	=	SYM
ejpam-3786	143	15	d	d	PROPN
ejpam-3786	143	16	,	,	PUNCT
ejpam-3786	143	17	d	d	PROPN
ejpam-3786	143	18	=	=	SYM
ejpam-3786	143	19	b	b	PROPN
ejpam-3786	143	20	,	,	PUNCT
ejpam-3786	143	21	d	d	NOUN
ejpam-3786	143	22	=	=	SYM
ejpam-3786	143	23	c.	c.	NOUN
ejpam-3786	143	24	proof	proof	NOUN
ejpam-3786	143	25	.	.	PUNCT
ejpam-3786	144	1	here	here	ADV
ejpam-3786	144	2	,	,	PUNCT
ejpam-3786	144	3	∆	∆	PROPN
ejpam-3786	144	4	has	have	AUX
ejpam-3786	144	5	at	at	ADP
ejpam-3786	144	6	most	most	ADV
ejpam-3786	144	7	three	three	NUM
ejpam-3786	144	8	vertices	vertex	NOUN
ejpam-3786	144	9	of	of	ADP
ejpam-3786	144	10	degree	degree	NOUN
ejpam-3786	144	11	2	2	NUM
ejpam-3786	144	12	,	,	PUNCT
ejpam-3786	144	13	so	so	ADV
ejpam-3786	144	14	has	have	VERB
ejpam-3786	144	15	non	non	ADJ
ejpam-3786	144	16	-	-	ADJ
ejpam-3786	144	17	positive	positive	ADJ
ejpam-3786	144	18	curvature	curvature	NOUN
ejpam-3786	144	19	for	for	ADP
ejpam-3786	144	20	all	all	PRON
ejpam-3786	144	21	of	of	ADP
ejpam-3786	144	22	these	these	DET
ejpam-3786	144	23	cases	case	NOUN
ejpam-3786	144	24	.	.	PUNCT
ejpam-3786	145	1	consider	consider	VERB
ejpam-3786	145	2	the	the	DET
ejpam-3786	145	3	case	case	NOUN
ejpam-3786	145	4	,	,	PUNCT
ejpam-3786	145	5	•	•	NUM
ejpam-3786	145	6	e	e	NOUN
ejpam-3786	145	7	=	=	SYM
ejpam-3786	145	8	b	b	PROPN
ejpam-3786	145	9	,	,	PUNCT
ejpam-3786	145	10	d	d	PROPN
ejpam-3786	145	11	=	=	SYM
ejpam-3786	145	12	c.	c.	PROPN
ejpam-3786	145	13	muhammad	muhammad	PROPN
ejpam-3786	145	14	saeed	saeed	PROPN
ejpam-3786	145	15	akram	akram	PROPN
ejpam-3786	145	16	,	,	PUNCT
ejpam-3786	145	17	maira	maira	PROPN
ejpam-3786	145	18	amjid	amjid	PROPN
ejpam-3786	145	19	,	,	PUNCT
ejpam-3786	145	20	sohail	sohail	PROPN
ejpam-3786	145	21	iqbal	iqbal	PROPN
ejpam-3786	145	22	/	/	PUNCT
ejpam-3786	145	23	eur	eur	PROPN
ejpam-3786	145	24	.	.	PUNCT
ejpam-3786	146	1	j.	j.	PROPN
ejpam-3786	146	2	pure	pure	PROPN
ejpam-3786	146	3	appl	appl	PROPN
ejpam-3786	146	4	.	.	PROPN
ejpam-3786	146	5	math	math	PROPN
ejpam-3786	146	6	,	,	PUNCT
ejpam-3786	146	7	13	13	NUM
ejpam-3786	146	8	(	(	PUNCT
ejpam-3786	146	9	4	4	NUM
ejpam-3786	146	10	)	)	PUNCT
ejpam-3786	146	11	(	(	PUNCT
ejpam-3786	146	12	2020	2020	NUM
ejpam-3786	146	13	)	)	PUNCT
ejpam-3786	146	14	,	,	PUNCT
ejpam-3786	146	15	914	914	NUM
ejpam-3786	146	16	-	-	SYM
ejpam-3786	146	17	938	938	NUM
ejpam-3786	146	18	920	920	NUM
ejpam-3786	146	19	in	in	ADP
ejpam-3786	146	20	this	this	DET
ejpam-3786	146	21	case	case	NOUN
ejpam-3786	146	22	∆	∆	PROPN
ejpam-3786	146	23	is	be	AUX
ejpam-3786	146	24	given	give	VERB
ejpam-3786	146	25	in	in	ADP
ejpam-3786	146	26	figure	figure	NOUN
ejpam-3786	146	27	3	3	NUM
ejpam-3786	146	28	.	.	PUNCT
ejpam-3786	146	29	figure	figure	VERB
ejpam-3786	146	30	3	3	NUM
ejpam-3786	146	31	:	:	PUNCT
ejpam-3786	146	32	region	region	NOUN
ejpam-3786	146	33	∆	∆	PROPN
ejpam-3786	146	34	since	since	SCONJ
ejpam-3786	146	35	d(vb	d(vb	NOUN
ejpam-3786	146	36	)	)	PUNCT
ejpam-3786	146	37	=	=	SYM
ejpam-3786	146	38	d(vc	d(vc	NOUN
ejpam-3786	146	39	)	)	PUNCT
ejpam-3786	146	40	=	=	SYM
ejpam-3786	146	41	2	2	NUM
ejpam-3786	146	42	or	or	CCONJ
ejpam-3786	146	43	d(vc	d(vc	NOUN
ejpam-3786	146	44	)	)	PUNCT
ejpam-3786	147	1	=	=	SYM
ejpam-3786	147	2	d(vd	d(vd	PROPN
ejpam-3786	147	3	)	)	PUNCT
ejpam-3786	147	4	=	=	SYM
ejpam-3786	147	5	2	2	NUM
ejpam-3786	147	6	or	or	CCONJ
ejpam-3786	147	7	d(vd	d(vd	NUM
ejpam-3786	147	8	)	)	PUNCT
ejpam-3786	147	9	=	=	SYM
ejpam-3786	147	10	d(ve	d(ve	PROPN
ejpam-3786	147	11	)	)	PUNCT
ejpam-3786	147	12	=	=	SYM
ejpam-3786	147	13	2	2	NUM
ejpam-3786	147	14	can	can	AUX
ejpam-3786	147	15	not	not	PART
ejpam-3786	147	16	occur	occur	VERB
ejpam-3786	147	17	together	together	ADV
ejpam-3786	147	18	so	so	SCONJ
ejpam-3786	147	19	c(∆	c(∆	PROPN
ejpam-3786	147	20	)	)	PUNCT
ejpam-3786	147	21	≤	≤	NOUN
ejpam-3786	147	22	0	0	NUM
ejpam-3786	147	23	.	.	PUNCT
ejpam-3786	148	1	a	a	DET
ejpam-3786	148	2	further	further	ADJ
ejpam-3786	148	3	19	19	NUM
ejpam-3786	148	4	cases	case	NOUN
ejpam-3786	148	5	are	be	AUX
ejpam-3786	148	6	solved	solve	VERB
ejpam-3786	148	7	in	in	ADP
ejpam-3786	148	8	lemma	lemma	PROPN
ejpam-3786	148	9	2	2	NUM
ejpam-3786	148	10	by	by	ADP
ejpam-3786	148	11	the	the	DET
ejpam-3786	148	12	application	application	NOUN
ejpam-3786	148	13	of	of	ADP
ejpam-3786	148	14	curvature	curvature	NOUN
ejpam-3786	148	15	distribution	distribution	NOUN
ejpam-3786	148	16	method	method	NOUN
ejpam-3786	148	17	[	[	X
ejpam-3786	148	18	10	10	NUM
ejpam-3786	148	19	]	]	PUNCT
ejpam-3786	148	20	.	.	PUNCT
ejpam-3786	149	1	lemma	lemma	PROPN
ejpam-3786	149	2	2	2	NUM
ejpam-3786	149	3	.	.	PUNCT
ejpam-3786	150	1	the	the	DET
ejpam-3786	150	2	presentation	presentation	NOUN
ejpam-3786	150	3	p	p	X
ejpam-3786	150	4	=	=	PUNCT
ejpam-3786	150	5	〈	〈	PROPN
ejpam-3786	150	6	g	g	NOUN
ejpam-3786	150	7	,	,	PUNCT
ejpam-3786	150	8	t	t	PROPN
ejpam-3786	150	9	|	|	ADV
ejpam-3786	150	10	s1(t	s1(t	PART
ejpam-3786	150	11	)	)	PUNCT
ejpam-3786	150	12	〉	〉	PROPN
ejpam-3786	150	13	is	be	AUX
ejpam-3786	150	14	aspherical	aspherical	ADJ
ejpam-3786	150	15	if	if	SCONJ
ejpam-3786	150	16	any	any	DET
ejpam-3786	150	17	one	one	NUM
ejpam-3786	150	18	of	of	ADP
ejpam-3786	150	19	the	the	DET
ejpam-3786	150	20	following	follow	VERB
ejpam-3786	150	21	holds	hold	VERB
ejpam-3786	150	22	:	:	PUNCT
ejpam-3786	150	23	(	(	PUNCT
ejpam-3786	150	24	i	i	NOUN
ejpam-3786	150	25	)	)	PUNCT
ejpam-3786	150	26	a	a	DET
ejpam-3786	150	27	=	=	SYM
ejpam-3786	150	28	g	g	NOUN
ejpam-3786	150	29	,	,	PUNCT
ejpam-3786	150	30	h	h	NOUN
ejpam-3786	150	31	=	=	SYM
ejpam-3786	150	32	c	c	X
ejpam-3786	150	33	,	,	PUNCT
ejpam-3786	150	34	e	e	X
ejpam-3786	150	35	=	=	SYM
ejpam-3786	150	36	b	b	PROPN
ejpam-3786	150	37	;	;	PUNCT
ejpam-3786	150	38	(	(	PUNCT
ejpam-3786	150	39	ii	ii	NOUN
ejpam-3786	150	40	)	)	PUNCT
ejpam-3786	150	41	a	a	DET
ejpam-3786	150	42	=	=	SYM
ejpam-3786	150	43	g	g	NOUN
ejpam-3786	150	44	,	,	PUNCT
ejpam-3786	150	45	h	h	NOUN
ejpam-3786	151	1	=	=	SYM
ejpam-3786	151	2	c	c	X
ejpam-3786	151	3	,	,	PUNCT
ejpam-3786	151	4	e	e	X
ejpam-3786	152	1	=	=	SYM
ejpam-3786	152	2	d	d	PROPN
ejpam-3786	152	3	;	;	PUNCT
ejpam-3786	152	4	(	(	PUNCT
ejpam-3786	152	5	iii	iii	X
ejpam-3786	152	6	)	)	PUNCT
ejpam-3786	152	7	a	a	DET
ejpam-3786	152	8	=	=	SYM
ejpam-3786	152	9	g	g	NOUN
ejpam-3786	152	10	,	,	PUNCT
ejpam-3786	152	11	h	h	NOUN
ejpam-3786	152	12	=	=	SYM
ejpam-3786	152	13	e	e	NOUN
ejpam-3786	152	14	,	,	PUNCT
ejpam-3786	152	15	c	c	PROPN
ejpam-3786	152	16	=	=	SYM
ejpam-3786	152	17	b	b	PROPN
ejpam-3786	152	18	;	;	PUNCT
ejpam-3786	152	19	(	(	PUNCT
ejpam-3786	152	20	iv	iv	X
ejpam-3786	152	21	)	)	PUNCT
ejpam-3786	152	22	a	a	DET
ejpam-3786	152	23	=	=	SYM
ejpam-3786	152	24	g	g	NOUN
ejpam-3786	152	25	,	,	PUNCT
ejpam-3786	152	26	h	h	NOUN
ejpam-3786	152	27	=	=	SYM
ejpam-3786	152	28	e	e	NOUN
ejpam-3786	152	29	,	,	PUNCT
ejpam-3786	152	30	d	d	PROPN
ejpam-3786	152	31	=	=	SYM
ejpam-3786	152	32	c	c	NOUN
ejpam-3786	152	33	;	;	PUNCT
ejpam-3786	152	34	(	(	PUNCT
ejpam-3786	152	35	v	v	NOUN
ejpam-3786	152	36	)	)	PUNCT
ejpam-3786	152	37	a	a	DET
ejpam-3786	152	38	=	=	SYM
ejpam-3786	152	39	g	g	NOUN
ejpam-3786	152	40	,	,	PUNCT
ejpam-3786	152	41	h	h	NOUN
ejpam-3786	153	1	=	=	SYM
ejpam-3786	153	2	d	d	PROPN
ejpam-3786	153	3	,	,	PUNCT
ejpam-3786	153	4	e	e	X
ejpam-3786	153	5	=	=	SYM
ejpam-3786	153	6	c	c	X
ejpam-3786	153	7	;	;	PUNCT
ejpam-3786	153	8	(	(	PUNCT
ejpam-3786	153	9	vi	vi	NOUN
ejpam-3786	153	10	)	)	PUNCT
ejpam-3786	153	11	a	a	DET
ejpam-3786	153	12	=	=	SYM
ejpam-3786	153	13	g	g	NOUN
ejpam-3786	153	14	,	,	PUNCT
ejpam-3786	153	15	e	e	PROPN
ejpam-3786	153	16	=	=	SYM
ejpam-3786	153	17	b	b	PROPN
ejpam-3786	153	18	,	,	PUNCT
ejpam-3786	153	19	d	d	PROPN
ejpam-3786	153	20	=	=	SYM
ejpam-3786	153	21	c	c	NOUN
ejpam-3786	153	22	;	;	PUNCT
ejpam-3786	153	23	(	(	PUNCT
ejpam-3786	153	24	vii	vii	PROPN
ejpam-3786	153	25	)	)	PUNCT
ejpam-3786	153	26	a	a	DET
ejpam-3786	153	27	=	=	SYM
ejpam-3786	153	28	g	g	NOUN
ejpam-3786	153	29	,	,	PUNCT
ejpam-3786	153	30	e	e	X
ejpam-3786	153	31	=	=	SYM
ejpam-3786	153	32	c	c	X
ejpam-3786	153	33	,	,	PUNCT
ejpam-3786	153	34	d	d	PROPN
ejpam-3786	153	35	=	=	SYM
ejpam-3786	153	36	b	b	PROPN
ejpam-3786	153	37	;	;	PUNCT
ejpam-3786	153	38	(	(	PUNCT
ejpam-3786	153	39	viii	viii	NOUN
ejpam-3786	153	40	)	)	PUNCT
ejpam-3786	153	41	a	a	DET
ejpam-3786	153	42	=	=	SYM
ejpam-3786	153	43	g	g	NOUN
ejpam-3786	153	44	,	,	PUNCT
ejpam-3786	153	45	e	e	X
ejpam-3786	153	46	=	=	SYM
ejpam-3786	153	47	d	d	PROPN
ejpam-3786	153	48	,	,	PUNCT
ejpam-3786	153	49	c	c	PROPN
ejpam-3786	153	50	=	=	SYM
ejpam-3786	153	51	b	b	PROPN
ejpam-3786	153	52	;	;	PUNCT
ejpam-3786	153	53	(	(	PUNCT
ejpam-3786	153	54	ix	ix	INTJ
ejpam-3786	153	55	)	)	PUNCT
ejpam-3786	153	56	a	a	DET
ejpam-3786	153	57	=	=	SYM
ejpam-3786	153	58	g	g	NOUN
ejpam-3786	153	59	,	,	PUNCT
ejpam-3786	153	60	e	e	PROPN
ejpam-3786	153	61	=	=	SYM
ejpam-3786	153	62	b	b	PROPN
ejpam-3786	153	63	,	,	PUNCT
ejpam-3786	153	64	e	e	X
ejpam-3786	153	65	=	=	SYM
ejpam-3786	153	66	c	c	X
ejpam-3786	153	67	,	,	PUNCT
ejpam-3786	153	68	c	c	NOUN
ejpam-3786	153	69	=	=	SYM
ejpam-3786	153	70	b	b	PROPN
ejpam-3786	153	71	;	;	PUNCT
ejpam-3786	153	72	(	(	PUNCT
ejpam-3786	153	73	x	x	X
ejpam-3786	153	74	)	)	PUNCT
ejpam-3786	153	75	a	a	DET
ejpam-3786	153	76	=	=	SYM
ejpam-3786	153	77	g	g	NOUN
ejpam-3786	153	78	,	,	PUNCT
ejpam-3786	153	79	e	e	X
ejpam-3786	153	80	=	=	SYM
ejpam-3786	153	81	c	c	X
ejpam-3786	153	82	,	,	PUNCT
ejpam-3786	153	83	e	e	X
ejpam-3786	153	84	=	=	SYM
ejpam-3786	153	85	d	d	PROPN
ejpam-3786	153	86	,	,	PUNCT
ejpam-3786	153	87	d	d	PROPN
ejpam-3786	153	88	=	=	SYM
ejpam-3786	153	89	c	c	NOUN
ejpam-3786	153	90	;	;	PUNCT
ejpam-3786	153	91	(	(	PUNCT
ejpam-3786	153	92	xi	xi	X
ejpam-3786	153	93	)	)	PUNCT
ejpam-3786	153	94	a	a	DET
ejpam-3786	153	95	=	=	SYM
ejpam-3786	153	96	g	g	NOUN
ejpam-3786	153	97	,	,	PUNCT
ejpam-3786	153	98	h	h	NOUN
ejpam-3786	153	99	=	=	SYM
ejpam-3786	153	100	e	e	NOUN
ejpam-3786	153	101	,	,	PUNCT
ejpam-3786	153	102	h	h	NOUN
ejpam-3786	153	103	=	=	SYM
ejpam-3786	153	104	c	c	X
ejpam-3786	153	105	,	,	PUNCT
ejpam-3786	153	106	e	e	X
ejpam-3786	153	107	=	=	SYM
ejpam-3786	153	108	c	c	X
ejpam-3786	153	109	;	;	PUNCT
ejpam-3786	153	110	(	(	PUNCT
ejpam-3786	153	111	xii	xii	NOUN
ejpam-3786	153	112	)	)	PUNCT
ejpam-3786	153	113	a	a	DET
ejpam-3786	153	114	=	=	SYM
ejpam-3786	153	115	g	g	NOUN
ejpam-3786	153	116	,	,	PUNCT
ejpam-3786	153	117	d	d	PROPN
ejpam-3786	153	118	=	=	SYM
ejpam-3786	153	119	b	b	PROPN
ejpam-3786	153	120	,	,	PUNCT
ejpam-3786	153	121	c	c	PROPN
ejpam-3786	153	122	=	=	SYM
ejpam-3786	153	123	b	b	PROPN
ejpam-3786	153	124	,	,	PUNCT
ejpam-3786	153	125	d	d	PROPN
ejpam-3786	153	126	=	=	SYM
ejpam-3786	153	127	c	c	NOUN
ejpam-3786	153	128	;	;	PUNCT
ejpam-3786	153	129	(	(	PUNCT
ejpam-3786	153	130	xiii	xiii	NOUN
ejpam-3786	153	131	)	)	PUNCT
ejpam-3786	153	132	a	a	DET
ejpam-3786	153	133	=	=	SYM
ejpam-3786	153	134	g	g	NOUN
ejpam-3786	153	135	,	,	PUNCT
ejpam-3786	153	136	h	h	NOUN
ejpam-3786	153	137	=	=	SYM
ejpam-3786	153	138	c	c	NOUN
ejpam-3786	153	139	,	,	PUNCT
ejpam-3786	153	140	h	h	NOUN
ejpam-3786	154	1	=	=	SYM
ejpam-3786	154	2	d	d	PROPN
ejpam-3786	154	3	,	,	PUNCT
ejpam-3786	154	4	d	d	PROPN
ejpam-3786	154	5	=	=	SYM
ejpam-3786	154	6	c	c	X
ejpam-3786	154	7	,	,	PUNCT
ejpam-3786	154	8	e	e	X
ejpam-3786	154	9	=	=	SYM
ejpam-3786	154	10	b	b	PROPN
ejpam-3786	154	11	;	;	PUNCT
ejpam-3786	154	12	(	(	PUNCT
ejpam-3786	154	13	xiv	xiv	PROPN
ejpam-3786	154	14	)	)	PUNCT
ejpam-3786	155	1	a	a	DET
ejpam-3786	155	2	=	=	SYM
ejpam-3786	155	3	g	g	NOUN
ejpam-3786	155	4	,	,	PUNCT
ejpam-3786	155	5	h	h	NOUN
ejpam-3786	155	6	=	=	SYM
ejpam-3786	155	7	e	e	NOUN
ejpam-3786	155	8	,	,	PUNCT
ejpam-3786	155	9	h	h	NOUN
ejpam-3786	155	10	=	=	SYM
ejpam-3786	155	11	d	d	PROPN
ejpam-3786	155	12	,	,	PUNCT
ejpam-3786	155	13	e	e	X
ejpam-3786	155	14	=	=	SYM
ejpam-3786	155	15	d	d	PROPN
ejpam-3786	155	16	,	,	PUNCT
ejpam-3786	155	17	c	c	PROPN
ejpam-3786	155	18	=	=	SYM
ejpam-3786	155	19	b	b	PROPN
ejpam-3786	155	20	;	;	PUNCT
ejpam-3786	155	21	(	(	PUNCT
ejpam-3786	155	22	xv	xv	PROPN
ejpam-3786	155	23	)	)	PUNCT
ejpam-3786	155	24	a	a	DET
ejpam-3786	155	25	=	=	SYM
ejpam-3786	155	26	g	g	NOUN
ejpam-3786	155	27	,	,	PUNCT
ejpam-3786	155	28	h	h	NOUN
ejpam-3786	155	29	=	=	SYM
ejpam-3786	155	30	e	e	NOUN
ejpam-3786	155	31	,	,	PUNCT
ejpam-3786	155	32	h	h	NOUN
ejpam-3786	155	33	=	=	SYM
ejpam-3786	155	34	c	c	X
ejpam-3786	155	35	,	,	PUNCT
ejpam-3786	155	36	e	e	X
ejpam-3786	155	37	=	=	SYM
ejpam-3786	155	38	c	c	X
ejpam-3786	155	39	,	,	PUNCT
ejpam-3786	155	40	d	d	PROPN
ejpam-3786	155	41	=	=	SYM
ejpam-3786	155	42	b	b	PROPN
ejpam-3786	155	43	;	;	PUNCT
ejpam-3786	155	44	(	(	PUNCT
ejpam-3786	155	45	xvi	xvi	NOUN
ejpam-3786	155	46	)	)	PUNCT
ejpam-3786	155	47	a	a	DET
ejpam-3786	155	48	=	=	SYM
ejpam-3786	155	49	g	g	NOUN
ejpam-3786	155	50	,	,	PUNCT
ejpam-3786	155	51	h	h	NOUN
ejpam-3786	156	1	=	=	SYM
ejpam-3786	156	2	d	d	PROPN
ejpam-3786	156	3	,	,	PUNCT
ejpam-3786	156	4	e	e	PROPN
ejpam-3786	156	5	=	=	SYM
ejpam-3786	156	6	b	b	PROPN
ejpam-3786	156	7	,	,	PUNCT
ejpam-3786	156	8	e	e	X
ejpam-3786	156	9	=	=	SYM
ejpam-3786	156	10	c	c	X
ejpam-3786	156	11	,	,	PUNCT
ejpam-3786	156	12	c	c	NOUN
ejpam-3786	156	13	=	=	SYM
ejpam-3786	156	14	b	b	PROPN
ejpam-3786	156	15	;	;	PUNCT
ejpam-3786	156	16	muhammad	muhammad	PROPN
ejpam-3786	156	17	saeed	saeed	PROPN
ejpam-3786	156	18	akram	akram	PROPN
ejpam-3786	156	19	,	,	PUNCT
ejpam-3786	156	20	maira	maira	PROPN
ejpam-3786	156	21	amjid	amjid	PROPN
ejpam-3786	156	22	,	,	PUNCT
ejpam-3786	156	23	sohail	sohail	PROPN
ejpam-3786	156	24	iqbal	iqbal	PROPN
ejpam-3786	156	25	/	/	PUNCT
ejpam-3786	156	26	eur	eur	PROPN
ejpam-3786	156	27	.	.	PUNCT
ejpam-3786	157	1	j.	j.	PROPN
ejpam-3786	157	2	pure	pure	PROPN
ejpam-3786	157	3	appl	appl	PROPN
ejpam-3786	157	4	.	.	PROPN
ejpam-3786	157	5	math	math	PROPN
ejpam-3786	157	6	,	,	PUNCT
ejpam-3786	157	7	13	13	NUM
ejpam-3786	157	8	(	(	PUNCT
ejpam-3786	157	9	4	4	NUM
ejpam-3786	157	10	)	)	PUNCT
ejpam-3786	157	11	(	(	PUNCT
ejpam-3786	157	12	2020	2020	NUM
ejpam-3786	157	13	)	)	PUNCT
ejpam-3786	157	14	,	,	PUNCT
ejpam-3786	157	15	914	914	NUM
ejpam-3786	157	16	-	-	SYM
ejpam-3786	157	17	938	938	NUM
ejpam-3786	157	18	921	921	NUM
ejpam-3786	157	19	(	(	PUNCT
ejpam-3786	157	20	xvii	xvii	PROPN
ejpam-3786	157	21	)	)	PUNCT
ejpam-3786	157	22	a	a	DET
ejpam-3786	157	23	=	=	SYM
ejpam-3786	157	24	g	g	NOUN
ejpam-3786	157	25	,	,	PUNCT
ejpam-3786	157	26	h	h	NOUN
ejpam-3786	157	27	=	=	SYM
ejpam-3786	157	28	c	c	X
ejpam-3786	157	29	,	,	PUNCT
ejpam-3786	157	30	e	e	X
ejpam-3786	157	31	=	=	SYM
ejpam-3786	157	32	b	b	PROPN
ejpam-3786	157	33	,	,	PUNCT
ejpam-3786	157	34	d	d	PROPN
ejpam-3786	157	35	=	=	SYM
ejpam-3786	157	36	b	b	PROPN
ejpam-3786	157	37	,	,	PUNCT
ejpam-3786	157	38	e	e	X
ejpam-3786	157	39	=	=	SYM
ejpam-3786	157	40	d	d	PROPN
ejpam-3786	157	41	;	;	PUNCT
ejpam-3786	157	42	(	(	PUNCT
ejpam-3786	157	43	xviii	xviii	PROPN
ejpam-3786	157	44	)	)	PUNCT
ejpam-3786	157	45	a	a	DET
ejpam-3786	157	46	=	=	SYM
ejpam-3786	157	47	g	g	NOUN
ejpam-3786	157	48	,	,	PUNCT
ejpam-3786	157	49	h	h	NOUN
ejpam-3786	157	50	=	=	SYM
ejpam-3786	157	51	e	e	NOUN
ejpam-3786	157	52	,	,	PUNCT
ejpam-3786	157	53	h	h	NOUN
ejpam-3786	158	1	=	=	SYM
ejpam-3786	158	2	d	d	PROPN
ejpam-3786	158	3	,	,	PUNCT
ejpam-3786	158	4	e	e	X
ejpam-3786	158	5	=	=	SYM
ejpam-3786	158	6	d	d	PROPN
ejpam-3786	158	7	,	,	PUNCT
ejpam-3786	158	8	h	h	NOUN
ejpam-3786	159	1	=	=	SYM
ejpam-3786	159	2	c	c	X
ejpam-3786	159	3	,	,	PUNCT
ejpam-3786	159	4	e	e	X
ejpam-3786	159	5	=	=	SYM
ejpam-3786	159	6	c	c	X
ejpam-3786	159	7	,	,	PUNCT
ejpam-3786	159	8	d	d	NOUN
ejpam-3786	159	9	=	=	SYM
ejpam-3786	159	10	c	c	NOUN
ejpam-3786	159	11	;	;	PUNCT
ejpam-3786	159	12	(	(	PUNCT
ejpam-3786	159	13	xix	xix	NOUN
ejpam-3786	159	14	)	)	PUNCT
ejpam-3786	159	15	e	e	NOUN
ejpam-3786	159	16	=	=	SYM
ejpam-3786	159	17	b	b	PROPN
ejpam-3786	159	18	,	,	PUNCT
ejpam-3786	159	19	e	e	X
ejpam-3786	159	20	=	=	SYM
ejpam-3786	159	21	c	c	X
ejpam-3786	159	22	,	,	PUNCT
ejpam-3786	159	23	c	c	NOUN
ejpam-3786	159	24	=	=	SYM
ejpam-3786	159	25	b	b	PROPN
ejpam-3786	159	26	,	,	PUNCT
ejpam-3786	159	27	e	e	X
ejpam-3786	159	28	=	=	SYM
ejpam-3786	159	29	d	d	PROPN
ejpam-3786	159	30	,	,	PUNCT
ejpam-3786	159	31	d	d	PROPN
ejpam-3786	159	32	=	=	SYM
ejpam-3786	159	33	b	b	PROPN
ejpam-3786	159	34	,	,	PUNCT
ejpam-3786	159	35	d	d	PROPN
ejpam-3786	159	36	=	=	SYM
ejpam-3786	159	37	c	c	NOUN
ejpam-3786	159	38	,	,	PUNCT
ejpam-3786	159	39	h	h	NOUN
ejpam-3786	159	40	=	=	SYM
ejpam-3786	159	41	b	b	PROPN
ejpam-3786	159	42	,	,	PUNCT
ejpam-3786	159	43	h	h	NOUN
ejpam-3786	159	44	=	=	SYM
ejpam-3786	159	45	e	e	NOUN
ejpam-3786	159	46	,	,	PUNCT
ejpam-3786	159	47	h	h	NOUN
ejpam-3786	159	48	=	=	SYM
ejpam-3786	159	49	c	c	NOUN
ejpam-3786	159	50	,	,	PUNCT
ejpam-3786	159	51	h	h	NOUN
ejpam-3786	159	52	=	=	SYM
ejpam-3786	159	53	d.	d.	PROPN
ejpam-3786	159	54	proof	proof	NOUN
ejpam-3786	159	55	.	.	PUNCT
ejpam-3786	160	1	1	1	X
ejpam-3786	160	2	.	.	X
ejpam-3786	160	3	in	in	ADP
ejpam-3786	160	4	this	this	DET
ejpam-3786	160	5	case	case	NOUN
ejpam-3786	160	6	∆	∆	PROPN
ejpam-3786	160	7	is	be	AUX
ejpam-3786	160	8	given	give	VERB
ejpam-3786	160	9	in	in	ADP
ejpam-3786	160	10	figure	figure	NOUN
ejpam-3786	160	11	4(i	4(i	NUM
ejpam-3786	160	12	)	)	PUNCT
ejpam-3786	160	13	.	.	PUNCT
ejpam-3786	161	1	figure	figure	VERB
ejpam-3786	161	2	4	4	NUM
ejpam-3786	161	3	:	:	PUNCT
ejpam-3786	161	4	region	region	NOUN
ejpam-3786	161	5	∆	∆	PROPN
ejpam-3786	161	6	the	the	DET
ejpam-3786	161	7	subcases	subcase	NOUN
ejpam-3786	161	8	which	which	PRON
ejpam-3786	161	9	are	be	AUX
ejpam-3786	161	10	to	to	PART
ejpam-3786	161	11	be	be	AUX
ejpam-3786	161	12	examined	examine	VERB
ejpam-3786	161	13	are	be	AUX
ejpam-3786	161	14	given	give	VERB
ejpam-3786	161	15	below	below	ADP
ejpam-3786	161	16	:	:	PUNCT
ejpam-3786	161	17	(	(	PUNCT
ejpam-3786	161	18	a	a	X
ejpam-3786	161	19	)	)	PUNCT
ejpam-3786	161	20	d∆(va	d∆(va	NOUN
ejpam-3786	161	21	)	)	PUNCT
ejpam-3786	161	22	=	=	PUNCT
ejpam-3786	162	1	d∆(vc	d∆(vc	NOUN
ejpam-3786	162	2	)	)	PUNCT
ejpam-3786	162	3	=	=	SYM
ejpam-3786	162	4	d∆(ve	d∆(ve	NOUN
ejpam-3786	162	5	)	)	PUNCT
ejpam-3786	162	6	=	=	SYM
ejpam-3786	163	1	d∆(vg	d∆(vg	NOUN
ejpam-3786	163	2	)	)	PUNCT
ejpam-3786	163	3	=	=	SYM
ejpam-3786	164	1	2	2	NUM
ejpam-3786	164	2	;	;	PUNCT
ejpam-3786	164	3	(	(	PUNCT
ejpam-3786	164	4	b	b	X
ejpam-3786	164	5	)	)	PUNCT
ejpam-3786	164	6	d∆(va	d∆(va	NOUN
ejpam-3786	164	7	)	)	PUNCT
ejpam-3786	164	8	=	=	PUNCT
ejpam-3786	164	9	d∆(vc	d∆(vc	NOUN
ejpam-3786	164	10	)	)	PUNCT
ejpam-3786	164	11	=	=	SYM
ejpam-3786	164	12	d∆(ve	d∆(ve	NOUN
ejpam-3786	164	13	)	)	PUNCT
ejpam-3786	164	14	=	=	SYM
ejpam-3786	164	15	d∆(vh	d∆(vh	X
ejpam-3786	164	16	)	)	PUNCT
ejpam-3786	164	17	=	=	SYM
ejpam-3786	165	1	2	2	X
ejpam-3786	165	2	.	.	PUNCT
ejpam-3786	165	3	(	(	PUNCT
ejpam-3786	165	4	a	a	X
ejpam-3786	165	5	)	)	PUNCT
ejpam-3786	165	6	here	here	ADV
ejpam-3786	165	7	d∆(va	d∆(va	NOUN
ejpam-3786	165	8	)	)	PUNCT
ejpam-3786	165	9	=	=	PUNCT
ejpam-3786	165	10	d∆(vc	d∆(vc	NOUN
ejpam-3786	165	11	)	)	PUNCT
ejpam-3786	165	12	=	=	SYM
ejpam-3786	165	13	d∆(ve	d∆(ve	NOUN
ejpam-3786	165	14	)	)	PUNCT
ejpam-3786	165	15	=	=	SYM
ejpam-3786	166	1	d∆(vg	d∆(vg	NOUN
ejpam-3786	166	2	)	)	PUNCT
ejpam-3786	167	1	=	=	SYM
ejpam-3786	167	2	2	2	NUM
ejpam-3786	167	3	which	which	PRON
ejpam-3786	167	4	implies	imply	VERB
ejpam-3786	167	5	l∆(va	l∆(va	NOUN
ejpam-3786	167	6	)	)	PUNCT
ejpam-3786	168	1	=	=	SYM
ejpam-3786	168	2	ag−1	ag−1	NOUN
ejpam-3786	168	3	,	,	PUNCT
ejpam-3786	168	4	l∆(vc	l∆(vc	PROPN
ejpam-3786	168	5	)	)	PUNCT
ejpam-3786	169	1	=	=	PUNCT
ejpam-3786	169	2	ch−1	ch−1	PROPN
ejpam-3786	169	3	,	,	PUNCT
ejpam-3786	169	4	l(ve	l(ve	PROPN
ejpam-3786	169	5	)	)	PUNCT
ejpam-3786	169	6	=	=	SYM
ejpam-3786	169	7	eb−1	eb−1	PROPN
ejpam-3786	169	8	,	,	PUNCT
ejpam-3786	169	9	and	and	CCONJ
ejpam-3786	169	10	l∆(vg	l∆(vg	NOUN
ejpam-3786	169	11	)	)	PUNCT
ejpam-3786	170	1	=	=	VERB
ejpam-3786	170	2	ga−1	ga−1	NOUN
ejpam-3786	170	3	,	,	PUNCT
ejpam-3786	170	4	as	as	SCONJ
ejpam-3786	170	5	given	give	VERB
ejpam-3786	170	6	in	in	ADP
ejpam-3786	170	7	figure	figure	NOUN
ejpam-3786	170	8	4(ii	4(ii	NUM
ejpam-3786	170	9	)	)	PUNCT
ejpam-3786	170	10	.	.	PUNCT
ejpam-3786	171	1	notice	notice	VERB
ejpam-3786	171	2	that	that	SCONJ
ejpam-3786	171	3	l∆(vc	l∆(vc	NOUN
ejpam-3786	171	4	)	)	PUNCT
ejpam-3786	172	1	=	=	PUNCT
ejpam-3786	172	2	ch−1	ch−1	NOUN
ejpam-3786	172	3	and	and	CCONJ
ejpam-3786	172	4	l∆(ve	l∆(ve	NOUN
ejpam-3786	172	5	)	)	PUNCT
ejpam-3786	173	1	=	=	SYM
ejpam-3786	173	2	eb−1	eb−1	PROPN
ejpam-3786	173	3	implies	imply	VERB
ejpam-3786	173	4	that	that	PRON
ejpam-3786	173	5	l∆(vd	l∆(vd	VERB
ejpam-3786	173	6	)	)	PUNCT
ejpam-3786	173	7	=	=	PUNCT
ejpam-3786	174	1	i−1da−1w	i−1da−1w	PROPN
ejpam-3786	174	2	in	in	ADP
ejpam-3786	174	3	which	which	PRON
ejpam-3786	174	4	w	w	PROPN
ejpam-3786	174	5	∈	∈	PROPN
ejpam-3786	174	6	{	{	PUNCT
ejpam-3786	174	7	b−1	b−1	PROPN
ejpam-3786	174	8	,	,	PUNCT
ejpam-3786	174	9	c−1	c−1	PROPN
ejpam-3786	174	10	,	,	PUNCT
ejpam-3786	174	11	e−1	e−1	PROPN
ejpam-3786	174	12	,	,	PUNCT
ejpam-3786	174	13	h−1	h−1	PROPN
ejpam-3786	174	14	}	}	PUNCT
ejpam-3786	174	15	which	which	PRON
ejpam-3786	174	16	implies	imply	VERB
ejpam-3786	174	17	d∆(vd	d∆(vd	NOUN
ejpam-3786	174	18	)	)	PUNCT
ejpam-3786	174	19	>	>	X
ejpam-3786	175	1	3	3	X
ejpam-3786	175	2	.	.	PUNCT
ejpam-3786	175	3	similarly	similarly	ADV
ejpam-3786	175	4	notice	notice	VERB
ejpam-3786	175	5	that	that	SCONJ
ejpam-3786	175	6	l∆(vg	l∆(vg	NOUN
ejpam-3786	175	7	)	)	PUNCT
ejpam-3786	176	1	=	=	PUNCT
ejpam-3786	176	2	ga−1	ga−1	NOUN
ejpam-3786	176	3	and	and	CCONJ
ejpam-3786	176	4	l∆(ve	l∆(ve	PROPN
ejpam-3786	176	5	)	)	PUNCT
ejpam-3786	177	1	=	=	SYM
ejpam-3786	177	2	eb−1	eb−1	PROPN
ejpam-3786	177	3	implies	imply	VERB
ejpam-3786	177	4	that	that	PRON
ejpam-3786	177	5	l∆(vf	l∆(vf	ADJ
ejpam-3786	177	6	)	)	PUNCT
ejpam-3786	177	7	=	=	PUNCT
ejpam-3786	177	8	fi−1c−1w	fi−1c−1w	NOUN
ejpam-3786	177	9	in	in	ADP
ejpam-3786	177	10	which	which	PRON
ejpam-3786	177	11	w	w	PROPN
ejpam-3786	177	12	∈	∈	PROPN
ejpam-3786	177	13	{	{	PUNCT
ejpam-3786	177	14	b	b	NOUN
ejpam-3786	177	15	,	,	PUNCT
ejpam-3786	177	16	d	d	NOUN
ejpam-3786	177	17	,	,	PUNCT
ejpam-3786	177	18	e	e	NOUN
ejpam-3786	177	19	,	,	PUNCT
ejpam-3786	177	20	h	h	NOUN
ejpam-3786	177	21	}	}	PUNCT
ejpam-3786	177	22	which	which	PRON
ejpam-3786	177	23	implies	imply	VERB
ejpam-3786	177	24	d∆(vf	d∆(vf	NOUN
ejpam-3786	177	25	)	)	PUNCT
ejpam-3786	177	26	>	>	X
ejpam-3786	178	1	3	3	X
ejpam-3786	178	2	.	.	PUNCT
ejpam-3786	178	3	since	since	SCONJ
ejpam-3786	178	4	d∆(vd	d∆(vd	ADJ
ejpam-3786	178	5	)	)	PUNCT
ejpam-3786	178	6	>	>	X
ejpam-3786	178	7	3	3	NUM
ejpam-3786	178	8	and	and	CCONJ
ejpam-3786	178	9	d∆(vf	d∆(vf	ADJ
ejpam-3786	178	10	)	)	PUNCT
ejpam-3786	178	11	>	>	X
ejpam-3786	178	12	3	3	NUM
ejpam-3786	178	13	so	so	ADV
ejpam-3786	178	14	c(∆	c(∆	NOUN
ejpam-3786	178	15	)	)	PUNCT
ejpam-3786	178	16	≤	≤	NOUN
ejpam-3786	178	17	0	0	NUM
ejpam-3786	178	18	.	.	PUNCT
ejpam-3786	179	1	(	(	PUNCT
ejpam-3786	179	2	b	b	X
ejpam-3786	179	3	)	)	PUNCT
ejpam-3786	179	4	here	here	ADV
ejpam-3786	179	5	d∆(va	d∆(va	NOUN
ejpam-3786	179	6	)	)	PUNCT
ejpam-3786	179	7	=	=	PUNCT
ejpam-3786	179	8	d∆(vc	d∆(vc	NOUN
ejpam-3786	179	9	)	)	PUNCT
ejpam-3786	179	10	=	=	SYM
ejpam-3786	179	11	d∆(ve	d∆(ve	NOUN
ejpam-3786	179	12	)	)	PUNCT
ejpam-3786	179	13	=	=	SYM
ejpam-3786	179	14	d∆(vh	d∆(vh	X
ejpam-3786	179	15	)	)	PUNCT
ejpam-3786	179	16	=	=	SYM
ejpam-3786	179	17	2	2	NUM
ejpam-3786	179	18	which	which	PRON
ejpam-3786	179	19	implies	imply	VERB
ejpam-3786	179	20	l∆(va	l∆(va	NOUN
ejpam-3786	179	21	)	)	PUNCT
ejpam-3786	180	1	=	=	SYM
ejpam-3786	180	2	ag−1	ag−1	NOUN
ejpam-3786	180	3	,	,	PUNCT
ejpam-3786	180	4	l∆(vc	l∆(vc	PROPN
ejpam-3786	180	5	)	)	PUNCT
ejpam-3786	181	1	=	=	PUNCT
ejpam-3786	181	2	ch−1	ch−1	PROPN
ejpam-3786	181	3	,	,	PUNCT
ejpam-3786	181	4	l(ve	l(ve	PROPN
ejpam-3786	181	5	)	)	PUNCT
ejpam-3786	181	6	=	=	SYM
ejpam-3786	181	7	eb−1	eb−1	PROPN
ejpam-3786	181	8	,	,	PUNCT
ejpam-3786	181	9	and	and	CCONJ
ejpam-3786	181	10	l∆(vh	l∆(vh	PROPN
ejpam-3786	181	11	)	)	PUNCT
ejpam-3786	181	12	=	=	SYM
ejpam-3786	181	13	hc−1	hc−1	NOUN
ejpam-3786	181	14	,	,	PUNCT
ejpam-3786	181	15	as	as	SCONJ
ejpam-3786	181	16	given	give	VERB
ejpam-3786	181	17	in	in	ADP
ejpam-3786	181	18	figure	figure	NOUN
ejpam-3786	181	19	4(iii	4(iii	NUM
ejpam-3786	181	20	)	)	PUNCT
ejpam-3786	181	21	.	.	PUNCT
ejpam-3786	182	1	notice	notice	VERB
ejpam-3786	182	2	that	that	SCONJ
ejpam-3786	182	3	l∆(vc	l∆(vc	NOUN
ejpam-3786	182	4	)	)	PUNCT
ejpam-3786	183	1	=	=	PUNCT
ejpam-3786	183	2	ch−1	ch−1	NOUN
ejpam-3786	183	3	and	and	CCONJ
ejpam-3786	183	4	l∆(ve	l∆(ve	NOUN
ejpam-3786	183	5	)	)	PUNCT
ejpam-3786	184	1	=	=	SYM
ejpam-3786	184	2	eb−1	eb−1	PROPN
ejpam-3786	184	3	implies	imply	VERB
ejpam-3786	184	4	that	that	PRON
ejpam-3786	184	5	l∆(vd	l∆(vd	VERB
ejpam-3786	184	6	)	)	PUNCT
ejpam-3786	184	7	=	=	PUNCT
ejpam-3786	185	1	i−1da−1w	i−1da−1w	PROPN
ejpam-3786	185	2	in	in	ADP
ejpam-3786	185	3	which	which	PRON
ejpam-3786	185	4	w	w	PROPN
ejpam-3786	185	5	∈	∈	PROPN
ejpam-3786	185	6	{	{	PUNCT
ejpam-3786	185	7	b−1	b−1	PROPN
ejpam-3786	185	8	,	,	PUNCT
ejpam-3786	185	9	c−1	c−1	PROPN
ejpam-3786	185	10	,	,	PUNCT
ejpam-3786	185	11	e−1	e−1	PROPN
ejpam-3786	185	12	,	,	PUNCT
ejpam-3786	185	13	h−1	h−1	PROPN
ejpam-3786	185	14	}	}	PUNCT
ejpam-3786	185	15	which	which	PRON
ejpam-3786	185	16	implies	imply	VERB
ejpam-3786	185	17	d∆(vd	d∆(vd	NOUN
ejpam-3786	185	18	)	)	PUNCT
ejpam-3786	185	19	>	>	X
ejpam-3786	186	1	3	3	X
ejpam-3786	186	2	.	.	PUNCT
ejpam-3786	186	3	similarly	similarly	ADV
ejpam-3786	186	4	notice	notice	VERB
ejpam-3786	186	5	that	that	SCONJ
ejpam-3786	186	6	l∆(va	l∆(va	NOUN
ejpam-3786	186	7	)	)	PUNCT
ejpam-3786	187	1	=	=	SYM
ejpam-3786	187	2	ag−1	ag−1	NOUN
ejpam-3786	187	3	and	and	CCONJ
ejpam-3786	187	4	l∆(vh	l∆(vh	PROPN
ejpam-3786	187	5	)	)	PUNCT
ejpam-3786	188	1	=	=	SYM
ejpam-3786	188	2	hc−1	hc−1	NOUN
ejpam-3786	188	3	implies	imply	VERB
ejpam-3786	188	4	that	that	SCONJ
ejpam-3786	188	5	l∆(vi	l∆(vi	X
ejpam-3786	188	6	)	)	PUNCT
ejpam-3786	188	7	=	=	SYM
ejpam-3786	188	8	if−1d−1w	if−1d−1w	NOUN
ejpam-3786	188	9	in	in	ADP
ejpam-3786	188	10	which	which	PRON
ejpam-3786	188	11	w	w	PROPN
ejpam-3786	188	12	∈	∈	PROPN
ejpam-3786	188	13	{	{	PUNCT
ejpam-3786	188	14	b	b	NOUN
ejpam-3786	188	15	,	,	PUNCT
ejpam-3786	188	16	c	c	X
ejpam-3786	188	17	,	,	PUNCT
ejpam-3786	188	18	e	e	NOUN
ejpam-3786	188	19	,	,	PUNCT
ejpam-3786	188	20	h	h	NOUN
ejpam-3786	188	21	}	}	PUNCT
ejpam-3786	188	22	which	which	PRON
ejpam-3786	188	23	implies	imply	VERB
ejpam-3786	188	24	d∆(vi	d∆(vi	NOUN
ejpam-3786	188	25	)	)	PUNCT
ejpam-3786	188	26	>	>	X
ejpam-3786	189	1	3	3	X
ejpam-3786	189	2	.	.	PUNCT
ejpam-3786	189	3	since	since	SCONJ
ejpam-3786	189	4	d∆(vd	d∆(vd	ADJ
ejpam-3786	189	5	)	)	PUNCT
ejpam-3786	189	6	>	>	X
ejpam-3786	189	7	3	3	NUM
ejpam-3786	189	8	and	and	CCONJ
ejpam-3786	189	9	d∆(vi	d∆(vi	NOUN
ejpam-3786	189	10	)	)	PUNCT
ejpam-3786	189	11	>	>	X
ejpam-3786	189	12	3	3	NUM
ejpam-3786	189	13	so	so	ADV
ejpam-3786	189	14	c(∆	c(∆	NOUN
ejpam-3786	189	15	)	)	PUNCT
ejpam-3786	189	16	≤	≤	NOUN
ejpam-3786	189	17	0	0	NUM
ejpam-3786	189	18	.	.	PROPN
ejpam-3786	190	1	2	2	NUM
ejpam-3786	190	2	.	.	X
ejpam-3786	190	3	in	in	ADP
ejpam-3786	190	4	this	this	DET
ejpam-3786	190	5	case	case	NOUN
ejpam-3786	190	6	∆	∆	PROPN
ejpam-3786	190	7	is	be	AUX
ejpam-3786	190	8	given	give	VERB
ejpam-3786	190	9	in	in	ADP
ejpam-3786	190	10	figure	figure	NOUN
ejpam-3786	190	11	5(i	5(i	NUM
ejpam-3786	190	12	)	)	PUNCT
ejpam-3786	190	13	.	.	PUNCT
ejpam-3786	191	1	muhammad	muhammad	PROPN
ejpam-3786	191	2	saeed	saeed	PROPN
ejpam-3786	191	3	akram	akram	PROPN
ejpam-3786	191	4	,	,	PUNCT
ejpam-3786	191	5	maira	maira	PROPN
ejpam-3786	191	6	amjid	amjid	PROPN
ejpam-3786	191	7	,	,	PUNCT
ejpam-3786	191	8	sohail	sohail	PROPN
ejpam-3786	191	9	iqbal	iqbal	PROPN
ejpam-3786	191	10	/	/	PUNCT
ejpam-3786	191	11	eur	eur	PROPN
ejpam-3786	191	12	.	.	PUNCT
ejpam-3786	192	1	j.	j.	PROPN
ejpam-3786	192	2	pure	pure	PROPN
ejpam-3786	192	3	appl	appl	PROPN
ejpam-3786	192	4	.	.	PROPN
ejpam-3786	192	5	math	math	PROPN
ejpam-3786	192	6	,	,	PUNCT
ejpam-3786	192	7	13	13	NUM
ejpam-3786	192	8	(	(	PUNCT
ejpam-3786	192	9	4	4	NUM
ejpam-3786	192	10	)	)	PUNCT
ejpam-3786	192	11	(	(	PUNCT
ejpam-3786	192	12	2020	2020	NUM
ejpam-3786	192	13	)	)	PUNCT
ejpam-3786	192	14	,	,	PUNCT
ejpam-3786	192	15	914	914	NUM
ejpam-3786	192	16	-	-	SYM
ejpam-3786	192	17	938	938	NUM
ejpam-3786	192	18	922	922	NUM
ejpam-3786	192	19	figure	figure	NOUN
ejpam-3786	192	20	5	5	NUM
ejpam-3786	192	21	:	:	PUNCT
ejpam-3786	192	22	regions	region	NOUN
ejpam-3786	192	23	∆	∆	PROPN
ejpam-3786	192	24	and	and	CCONJ
ejpam-3786	192	25	∆̂	∆̂	DET
ejpam-3786	192	26	the	the	DET
ejpam-3786	192	27	subcases	subcase	NOUN
ejpam-3786	192	28	which	which	PRON
ejpam-3786	192	29	are	be	AUX
ejpam-3786	192	30	to	to	PART
ejpam-3786	192	31	be	be	AUX
ejpam-3786	192	32	examined	examine	VERB
ejpam-3786	192	33	are	be	AUX
ejpam-3786	192	34	given	give	VERB
ejpam-3786	192	35	below	below	ADP
ejpam-3786	192	36	:	:	PUNCT
ejpam-3786	192	37	(	(	PUNCT
ejpam-3786	192	38	a	a	X
ejpam-3786	192	39	)	)	PUNCT
ejpam-3786	192	40	d∆(va	d∆(va	NOUN
ejpam-3786	192	41	)	)	PUNCT
ejpam-3786	192	42	=	=	PUNCT
ejpam-3786	192	43	d∆(vc	d∆(vc	NOUN
ejpam-3786	192	44	)	)	PUNCT
ejpam-3786	192	45	=	=	SYM
ejpam-3786	192	46	d∆(ve	d∆(ve	NOUN
ejpam-3786	192	47	)	)	PUNCT
ejpam-3786	192	48	=	=	SYM
ejpam-3786	192	49	d∆(vg	d∆(vg	NOUN
ejpam-3786	192	50	)	)	PUNCT
ejpam-3786	192	51	=	=	SYM
ejpam-3786	192	52	2	2	NUM
ejpam-3786	192	53	;	;	PUNCT
ejpam-3786	192	54	(	(	PUNCT
ejpam-3786	192	55	b	b	X
ejpam-3786	192	56	)	)	PUNCT
ejpam-3786	192	57	d∆(va	d∆(va	NOUN
ejpam-3786	192	58	)	)	PUNCT
ejpam-3786	192	59	=	=	PUNCT
ejpam-3786	192	60	d∆(vc	d∆(vc	NOUN
ejpam-3786	192	61	)	)	PUNCT
ejpam-3786	192	62	=	=	SYM
ejpam-3786	192	63	d∆(ve	d∆(ve	NOUN
ejpam-3786	192	64	)	)	PUNCT
ejpam-3786	192	65	=	=	SYM
ejpam-3786	192	66	d∆(vh	d∆(vh	X
ejpam-3786	192	67	)	)	PUNCT
ejpam-3786	192	68	=	=	SYM
ejpam-3786	193	1	2	2	X
ejpam-3786	193	2	.	.	PUNCT
ejpam-3786	193	3	(	(	PUNCT
ejpam-3786	193	4	a	a	X
ejpam-3786	193	5	)	)	PUNCT
ejpam-3786	193	6	here	here	ADV
ejpam-3786	193	7	d∆(va	d∆(va	NOUN
ejpam-3786	193	8	)	)	PUNCT
ejpam-3786	193	9	=	=	PUNCT
ejpam-3786	193	10	d∆(vc	d∆(vc	NOUN
ejpam-3786	193	11	)	)	PUNCT
ejpam-3786	193	12	=	=	SYM
ejpam-3786	193	13	d∆(ve	d∆(ve	NOUN
ejpam-3786	193	14	)	)	PUNCT
ejpam-3786	193	15	=	=	SYM
ejpam-3786	194	1	d∆(vg	d∆(vg	NOUN
ejpam-3786	194	2	)	)	PUNCT
ejpam-3786	195	1	=	=	SYM
ejpam-3786	195	2	2	2	NUM
ejpam-3786	195	3	which	which	PRON
ejpam-3786	195	4	implies	imply	VERB
ejpam-3786	195	5	l∆(va	l∆(va	NOUN
ejpam-3786	195	6	)	)	PUNCT
ejpam-3786	196	1	=	=	SYM
ejpam-3786	196	2	ag−1	ag−1	NOUN
ejpam-3786	196	3	,	,	PUNCT
ejpam-3786	196	4	l∆(vc	l∆(vc	PROPN
ejpam-3786	196	5	)	)	PUNCT
ejpam-3786	197	1	=	=	PUNCT
ejpam-3786	197	2	ch−1	ch−1	PROPN
ejpam-3786	197	3	,	,	PUNCT
ejpam-3786	197	4	l∆(ve	l∆(ve	NOUN
ejpam-3786	197	5	)	)	PUNCT
ejpam-3786	197	6	=	=	SYM
ejpam-3786	197	7	ed−1	ed−1	PROPN
ejpam-3786	197	8	,	,	PUNCT
ejpam-3786	197	9	l∆(vg	l∆(vg	NOUN
ejpam-3786	197	10	)	)	PUNCT
ejpam-3786	198	1	=	=	VERB
ejpam-3786	198	2	ga−1	ga−1	NOUN
ejpam-3786	198	3	,	,	PUNCT
ejpam-3786	198	4	as	as	SCONJ
ejpam-3786	198	5	shown	show	VERB
ejpam-3786	198	6	in	in	ADP
ejpam-3786	198	7	figure	figure	NOUN
ejpam-3786	198	8	5(ii	5(ii	NUM
ejpam-3786	198	9	)	)	PUNCT
ejpam-3786	198	10	.	.	PUNCT
ejpam-3786	199	1	notice	notice	VERB
ejpam-3786	199	2	that	that	SCONJ
ejpam-3786	199	3	l∆(vg	l∆(vg	NOUN
ejpam-3786	199	4	)	)	PUNCT
ejpam-3786	200	1	=	=	PUNCT
ejpam-3786	200	2	ga−1	ga−1	NOUN
ejpam-3786	200	3	and	and	CCONJ
ejpam-3786	200	4	l∆(ve	l∆(ve	PROPN
ejpam-3786	200	5	)	)	PUNCT
ejpam-3786	201	1	=	=	SYM
ejpam-3786	201	2	ed−1	ed−1	PROPN
ejpam-3786	201	3	implies	imply	VERB
ejpam-3786	201	4	that	that	SCONJ
ejpam-3786	201	5	l∆(vf	l∆(vf	ADJ
ejpam-3786	201	6	)	)	PUNCT
ejpam-3786	202	1	=	=	SYM
ejpam-3786	202	2	fi−1e−1w	fi−1e−1w	NOUN
ejpam-3786	202	3	in	in	ADP
ejpam-3786	202	4	which	which	PRON
ejpam-3786	202	5	w	w	PROPN
ejpam-3786	202	6	∈	∈	PROPN
ejpam-3786	202	7	{	{	PUNCT
ejpam-3786	202	8	b	b	NOUN
ejpam-3786	202	9	,	,	PUNCT
ejpam-3786	202	10	c	c	NOUN
ejpam-3786	202	11	,	,	PUNCT
ejpam-3786	202	12	d	d	NOUN
ejpam-3786	202	13	,	,	PUNCT
ejpam-3786	202	14	h	h	NOUN
ejpam-3786	202	15	}	}	PUNCT
ejpam-3786	202	16	which	which	PRON
ejpam-3786	202	17	implies	imply	VERB
ejpam-3786	202	18	d∆(vf	d∆(vf	NOUN
ejpam-3786	202	19	)	)	PUNCT
ejpam-3786	202	20	>	>	X
ejpam-3786	203	1	3	3	X
ejpam-3786	203	2	.	.	PUNCT
ejpam-3786	203	3	add	add	VERB
ejpam-3786	203	4	c(∆	c(∆	NOUN
ejpam-3786	203	5	)	)	PUNCT
ejpam-3786	203	6	≤	≤	NUM
ejpam-3786	203	7	π	π	PROPN
ejpam-3786	203	8	6	6	NUM
ejpam-3786	203	9	to	to	ADP
ejpam-3786	203	10	c(∆̂	c(∆̂	PROPN
ejpam-3786	203	11	)	)	PUNCT
ejpam-3786	203	12	is	be	AUX
ejpam-3786	203	13	given	give	VERB
ejpam-3786	203	14	by	by	ADP
ejpam-3786	203	15	figure	figure	NOUN
ejpam-3786	203	16	5(iii	5(iii	NUM
ejpam-3786	203	17	)	)	PUNCT
ejpam-3786	203	18	.	.	PUNCT
ejpam-3786	204	1	notice	notice	VERB
ejpam-3786	204	2	that	that	SCONJ
ejpam-3786	204	3	d	d	NOUN
ejpam-3786	204	4	∆̂	∆̂	PUNCT
ejpam-3786	204	5	(	(	PUNCT
ejpam-3786	204	6	va−1	va−1	NOUN
ejpam-3786	204	7	)	)	PUNCT
ejpam-3786	204	8	=	=	SYM
ejpam-3786	205	1	d	d	NOUN
ejpam-3786	205	2	∆̂	∆̂	NOUN
ejpam-3786	205	3	(	(	PUNCT
ejpam-3786	205	4	ve−1	ve−1	PROPN
ejpam-3786	205	5	)	)	PUNCT
ejpam-3786	205	6	=	=	SYM
ejpam-3786	206	1	2	2	X
ejpam-3786	206	2	.	.	X
ejpam-3786	206	3	similarly	similarly	ADV
ejpam-3786	206	4	notice	notice	VERB
ejpam-3786	206	5	that	that	SCONJ
ejpam-3786	206	6	either	either	CCONJ
ejpam-3786	206	7	d	d	ADP
ejpam-3786	206	8	∆̂	∆̂	NOUN
ejpam-3786	206	9	(	(	PUNCT
ejpam-3786	206	10	vg−1	vg−1	PROPN
ejpam-3786	206	11	)	)	PUNCT
ejpam-3786	206	12	=	=	SYM
ejpam-3786	206	13	2	2	NUM
ejpam-3786	206	14	or	or	CCONJ
ejpam-3786	206	15	d	d	NOUN
ejpam-3786	206	16	∆̂	∆̂	NOUN
ejpam-3786	206	17	(	(	PUNCT
ejpam-3786	206	18	vh−1	vh−1	PROPN
ejpam-3786	206	19	)	)	PUNCT
ejpam-3786	206	20	=	=	SYM
ejpam-3786	206	21	2	2	NUM
ejpam-3786	206	22	otherwise	otherwise	ADV
ejpam-3786	206	23	contradiction	contradiction	NOUN
ejpam-3786	206	24	occur	occur	VERB
ejpam-3786	206	25	and	and	CCONJ
ejpam-3786	206	26	all	all	DET
ejpam-3786	206	27	other	other	ADJ
ejpam-3786	206	28	vertices	vertex	NOUN
ejpam-3786	206	29	have	have	VERB
ejpam-3786	206	30	degree	degree	NOUN
ejpam-3786	206	31	atleast	atleast	ADP
ejpam-3786	206	32	3	3	NUM
ejpam-3786	206	33	.	.	X
ejpam-3786	206	34	therefore	therefore	ADV
ejpam-3786	206	35	c(∆̂	c(∆̂	PROPN
ejpam-3786	206	36	)	)	PUNCT
ejpam-3786	206	37	≤	≤	PROPN
ejpam-3786	206	38	c(2	c(2	PROPN
ejpam-3786	206	39	,	,	PUNCT
ejpam-3786	206	40	2	2	NUM
ejpam-3786	206	41	,	,	PUNCT
ejpam-3786	206	42	2	2	NUM
ejpam-3786	206	43	,	,	PUNCT
ejpam-3786	206	44	3	3	NUM
ejpam-3786	206	45	,	,	PUNCT
ejpam-3786	206	46	3	3	NUM
ejpam-3786	206	47	,	,	PUNCT
ejpam-3786	206	48	3	3	NUM
ejpam-3786	206	49	,	,	PUNCT
ejpam-3786	206	50	3	3	NUM
ejpam-3786	206	51	,	,	PUNCT
ejpam-3786	206	52	3	3	NUM
ejpam-3786	206	53	,	,	PUNCT
ejpam-3786	206	54	4	4	NUM
ejpam-3786	206	55	)	)	PUNCT
ejpam-3786	206	56	=	=	PUNCT
ejpam-3786	207	1	−π	−π	ADP
ejpam-3786	207	2	6	6	NUM
ejpam-3786	207	3	.	.	PUNCT
ejpam-3786	208	1	(	(	PUNCT
ejpam-3786	208	2	b	b	X
ejpam-3786	208	3	)	)	PUNCT
ejpam-3786	208	4	here	here	ADV
ejpam-3786	208	5	d∆(va	d∆(va	NOUN
ejpam-3786	208	6	)	)	PUNCT
ejpam-3786	208	7	=	=	PUNCT
ejpam-3786	208	8	d∆(vc	d∆(vc	NOUN
ejpam-3786	208	9	)	)	PUNCT
ejpam-3786	208	10	=	=	SYM
ejpam-3786	208	11	d∆(ve	d∆(ve	NOUN
ejpam-3786	208	12	)	)	PUNCT
ejpam-3786	208	13	=	=	SYM
ejpam-3786	208	14	d∆(vh	d∆(vh	X
ejpam-3786	208	15	)	)	PUNCT
ejpam-3786	208	16	=	=	SYM
ejpam-3786	208	17	2	2	NUM
ejpam-3786	208	18	which	which	PRON
ejpam-3786	208	19	implies	imply	VERB
ejpam-3786	208	20	l∆(va	l∆(va	NOUN
ejpam-3786	208	21	)	)	PUNCT
ejpam-3786	209	1	=	=	SYM
ejpam-3786	209	2	ag−1	ag−1	NOUN
ejpam-3786	209	3	,	,	PUNCT
ejpam-3786	209	4	l∆(vc	l∆(vc	PROPN
ejpam-3786	209	5	)	)	PUNCT
ejpam-3786	210	1	=	=	PUNCT
ejpam-3786	210	2	ch−1	ch−1	PROPN
ejpam-3786	210	3	,	,	PUNCT
ejpam-3786	210	4	l∆(ve	l∆(ve	NOUN
ejpam-3786	210	5	)	)	PUNCT
ejpam-3786	210	6	=	=	SYM
ejpam-3786	210	7	ed−1	ed−1	PROPN
ejpam-3786	210	8	,	,	PUNCT
ejpam-3786	210	9	l∆(vh	l∆(vh	NOUN
ejpam-3786	210	10	)	)	PUNCT
ejpam-3786	210	11	=	=	SYM
ejpam-3786	210	12	hc−1	hc−1	NOUN
ejpam-3786	210	13	,	,	PUNCT
ejpam-3786	210	14	as	as	SCONJ
ejpam-3786	210	15	shown	show	VERB
ejpam-3786	210	16	in	in	ADP
ejpam-3786	210	17	figure	figure	NOUN
ejpam-3786	210	18	5(iv	5(iv	NUM
ejpam-3786	210	19	)	)	PUNCT
ejpam-3786	210	20	.	.	PUNCT
ejpam-3786	211	1	notice	notice	VERB
ejpam-3786	211	2	that	that	SCONJ
ejpam-3786	211	3	l∆(va	l∆(va	NOUN
ejpam-3786	211	4	)	)	PUNCT
ejpam-3786	212	1	=	=	SYM
ejpam-3786	212	2	ag−1	ag−1	NOUN
ejpam-3786	212	3	and	and	CCONJ
ejpam-3786	212	4	l∆(vh	l∆(vh	PROPN
ejpam-3786	212	5	)	)	PUNCT
ejpam-3786	213	1	=	=	SYM
ejpam-3786	213	2	hc−1	hc−1	NOUN
ejpam-3786	213	3	implies	imply	VERB
ejpam-3786	213	4	that	that	SCONJ
ejpam-3786	213	5	l∆(vi	l∆(vi	X
ejpam-3786	213	6	)	)	PUNCT
ejpam-3786	213	7	=	=	SYM
ejpam-3786	213	8	if−1d−1w	if−1d−1w	NOUN
ejpam-3786	213	9	in	in	ADP
ejpam-3786	213	10	which	which	PRON
ejpam-3786	213	11	w	w	PROPN
ejpam-3786	213	12	∈	∈	PROPN
ejpam-3786	213	13	{	{	PUNCT
ejpam-3786	213	14	b	b	NOUN
ejpam-3786	213	15	,	,	PUNCT
ejpam-3786	213	16	c	c	X
ejpam-3786	213	17	,	,	PUNCT
ejpam-3786	213	18	e	e	NOUN
ejpam-3786	213	19	,	,	PUNCT
ejpam-3786	213	20	h	h	NOUN
ejpam-3786	213	21	}	}	PUNCT
ejpam-3786	213	22	which	which	PRON
ejpam-3786	213	23	implies	imply	VERB
ejpam-3786	213	24	d∆(vi	d∆(vi	NOUN
ejpam-3786	213	25	)	)	PUNCT
ejpam-3786	213	26	>	>	X
ejpam-3786	214	1	3	3	X
ejpam-3786	214	2	.	.	PUNCT
ejpam-3786	214	3	add	add	VERB
ejpam-3786	214	4	c(∆	c(∆	NOUN
ejpam-3786	214	5	)	)	PUNCT
ejpam-3786	214	6	≤	≤	NUM
ejpam-3786	214	7	π	π	PROPN
ejpam-3786	214	8	6	6	NUM
ejpam-3786	214	9	to	to	ADP
ejpam-3786	214	10	c(∆̂	c(∆̂	PROPN
ejpam-3786	214	11	)	)	PUNCT
ejpam-3786	214	12	is	be	AUX
ejpam-3786	214	13	given	give	VERB
ejpam-3786	214	14	by	by	ADP
ejpam-3786	214	15	figure	figure	NOUN
ejpam-3786	214	16	5(v	5(v	NUM
ejpam-3786	214	17	)	)	PUNCT
ejpam-3786	214	18	.	.	PUNCT
ejpam-3786	215	1	notice	notice	VERB
ejpam-3786	215	2	that	that	SCONJ
ejpam-3786	215	3	d	d	NOUN
ejpam-3786	215	4	∆̂	∆̂	NOUN
ejpam-3786	215	5	(	(	PUNCT
ejpam-3786	215	6	vc−1	vc−1	NOUN
ejpam-3786	215	7	)	)	PUNCT
ejpam-3786	215	8	=	=	SYM
ejpam-3786	216	1	d	d	NOUN
ejpam-3786	216	2	∆̂	∆̂	NOUN
ejpam-3786	216	3	(	(	PUNCT
ejpam-3786	216	4	ve−1	ve−1	PROPN
ejpam-3786	216	5	)	)	PUNCT
ejpam-3786	216	6	=	=	SYM
ejpam-3786	217	1	2	2	X
ejpam-3786	217	2	.	.	X
ejpam-3786	217	3	similarly	similarly	ADV
ejpam-3786	217	4	notice	notice	VERB
ejpam-3786	217	5	that	that	SCONJ
ejpam-3786	217	6	either	either	CCONJ
ejpam-3786	217	7	d	d	ADP
ejpam-3786	217	8	∆̂	∆̂	NOUN
ejpam-3786	217	9	(	(	PUNCT
ejpam-3786	217	10	vg−1	vg−1	PROPN
ejpam-3786	217	11	)	)	PUNCT
ejpam-3786	217	12	=	=	SYM
ejpam-3786	217	13	2	2	NUM
ejpam-3786	217	14	or	or	CCONJ
ejpam-3786	217	15	d	d	NOUN
ejpam-3786	217	16	∆̂	∆̂	NOUN
ejpam-3786	217	17	(	(	PUNCT
ejpam-3786	217	18	vh−1	vh−1	PROPN
ejpam-3786	217	19	)	)	PUNCT
ejpam-3786	217	20	=	=	SYM
ejpam-3786	217	21	2	2	NUM
ejpam-3786	217	22	otherwise	otherwise	ADV
ejpam-3786	217	23	contradiction	contradiction	NOUN
ejpam-3786	217	24	occur	occur	VERB
ejpam-3786	217	25	.	.	PUNCT
ejpam-3786	218	1	observe	observe	VERB
ejpam-3786	218	2	that	that	SCONJ
ejpam-3786	218	3	either	either	CCONJ
ejpam-3786	218	4	d	d	ADP
ejpam-3786	218	5	∆̂	∆̂	NOUN
ejpam-3786	218	6	(	(	PUNCT
ejpam-3786	218	7	vb−1	vb−1	PROPN
ejpam-3786	218	8	)	)	PUNCT
ejpam-3786	218	9	=	=	SYM
ejpam-3786	218	10	3	3	NUM
ejpam-3786	218	11	or	or	CCONJ
ejpam-3786	218	12	d	d	NOUN
ejpam-3786	218	13	∆̂	∆̂	INTJ
ejpam-3786	218	14	(	(	PUNCT
ejpam-3786	218	15	va−1	va−1	NOUN
ejpam-3786	218	16	)	)	PUNCT
ejpam-3786	218	17	=	=	SYM
ejpam-3786	218	18	2	2	NUM
ejpam-3786	218	19	since	since	SCONJ
ejpam-3786	218	20	d	d	PROPN
ejpam-3786	218	21	∆̂	∆̂	NOUN
ejpam-3786	218	22	(	(	PUNCT
ejpam-3786	218	23	vb−1	vb−1	PROPN
ejpam-3786	218	24	)	)	PUNCT
ejpam-3786	218	25	=	=	SYM
ejpam-3786	218	26	3	3	NUM
ejpam-3786	218	27	already	already	ADV
ejpam-3786	218	28	present	present	ADJ
ejpam-3786	218	29	so	so	ADV
ejpam-3786	218	30	d	d	ADP
ejpam-3786	218	31	∆̂	∆̂	NOUN
ejpam-3786	218	32	(	(	PUNCT
ejpam-3786	218	33	va−1	va−1	NOUN
ejpam-3786	218	34	)	)	PUNCT
ejpam-3786	218	35	>	>	X
ejpam-3786	219	1	2	2	NUM
ejpam-3786	219	2	otherwise	otherwise	ADV
ejpam-3786	219	3	contradiction	contradiction	NOUN
ejpam-3786	219	4	occur	occur	VERB
ejpam-3786	219	5	and	and	CCONJ
ejpam-3786	219	6	all	all	DET
ejpam-3786	219	7	other	other	ADJ
ejpam-3786	219	8	vertices	vertex	NOUN
ejpam-3786	219	9	have	have	VERB
ejpam-3786	219	10	degree	degree	NOUN
ejpam-3786	219	11	atleast	atleast	ADP
ejpam-3786	219	12	3	3	NUM
ejpam-3786	219	13	.	.	X
ejpam-3786	219	14	therefore	therefore	ADV
ejpam-3786	219	15	c(∆̂	c(∆̂	PROPN
ejpam-3786	219	16	)	)	PUNCT
ejpam-3786	219	17	≤	≤	PROPN
ejpam-3786	219	18	c(2	c(2	PROPN
ejpam-3786	219	19	,	,	PUNCT
ejpam-3786	219	20	2	2	NUM
ejpam-3786	219	21	,	,	PUNCT
ejpam-3786	219	22	2	2	NUM
ejpam-3786	219	23	,	,	PUNCT
ejpam-3786	219	24	3	3	NUM
ejpam-3786	219	25	,	,	PUNCT
ejpam-3786	219	26	3	3	NUM
ejpam-3786	219	27	,	,	PUNCT
ejpam-3786	219	28	3	3	NUM
ejpam-3786	219	29	,	,	PUNCT
ejpam-3786	219	30	3	3	NUM
ejpam-3786	219	31	,	,	PUNCT
ejpam-3786	219	32	3	3	NUM
ejpam-3786	219	33	,	,	PUNCT
ejpam-3786	219	34	4	4	NUM
ejpam-3786	219	35	)	)	PUNCT
ejpam-3786	219	36	=	=	PUNCT
ejpam-3786	220	1	−π	−π	ADP
ejpam-3786	220	2	3	3	NUM
ejpam-3786	220	3	.	.	PUNCT
ejpam-3786	221	1	3	3	X
ejpam-3786	221	2	.	.	X
ejpam-3786	221	3	in	in	ADP
ejpam-3786	221	4	this	this	DET
ejpam-3786	221	5	case	case	NOUN
ejpam-3786	221	6	∆	∆	PROPN
ejpam-3786	221	7	is	be	AUX
ejpam-3786	221	8	given	give	VERB
ejpam-3786	221	9	in	in	ADP
ejpam-3786	221	10	figure	figure	NOUN
ejpam-3786	221	11	6(i	6(i	NUM
ejpam-3786	221	12	)	)	PUNCT
ejpam-3786	221	13	.	.	PUNCT
ejpam-3786	222	1	muhammad	muhammad	PROPN
ejpam-3786	222	2	saeed	saeed	PROPN
ejpam-3786	222	3	akram	akram	PROPN
ejpam-3786	222	4	,	,	PUNCT
ejpam-3786	222	5	maira	maira	PROPN
ejpam-3786	222	6	amjid	amjid	PROPN
ejpam-3786	222	7	,	,	PUNCT
ejpam-3786	222	8	sohail	sohail	PROPN
ejpam-3786	222	9	iqbal	iqbal	PROPN
ejpam-3786	222	10	/	/	PUNCT
ejpam-3786	222	11	eur	eur	PROPN
ejpam-3786	222	12	.	.	PUNCT
ejpam-3786	223	1	j.	j.	PROPN
ejpam-3786	223	2	pure	pure	PROPN
ejpam-3786	223	3	appl	appl	PROPN
ejpam-3786	223	4	.	.	PROPN
ejpam-3786	223	5	math	math	PROPN
ejpam-3786	223	6	,	,	PUNCT
ejpam-3786	223	7	13	13	NUM
ejpam-3786	223	8	(	(	PUNCT
ejpam-3786	223	9	4	4	NUM
ejpam-3786	223	10	)	)	PUNCT
ejpam-3786	223	11	(	(	PUNCT
ejpam-3786	223	12	2020	2020	NUM
ejpam-3786	223	13	)	)	PUNCT
ejpam-3786	223	14	,	,	PUNCT
ejpam-3786	223	15	914	914	NUM
ejpam-3786	223	16	-	-	SYM
ejpam-3786	223	17	938	938	NUM
ejpam-3786	223	18	923	923	NUM
ejpam-3786	223	19	figure	figure	NOUN
ejpam-3786	223	20	6	6	NUM
ejpam-3786	223	21	:	:	PUNCT
ejpam-3786	223	22	regions	region	NOUN
ejpam-3786	223	23	∆	∆	PROPN
ejpam-3786	223	24	and	and	CCONJ
ejpam-3786	223	25	∆̂	∆̂	X
ejpam-3786	223	26	by	by	ADP
ejpam-3786	223	27	adding	add	VERB
ejpam-3786	223	28	c(∆	c(∆	NOUN
ejpam-3786	223	29	)	)	PUNCT
ejpam-3786	223	30	to	to	ADP
ejpam-3786	223	31	c(∆̂	c(∆̂	PROPN
ejpam-3786	223	32	)	)	PUNCT
ejpam-3786	223	33	we	we	PRON
ejpam-3786	223	34	get	get	VERB
ejpam-3786	223	35	c(∆	c(∆	NOUN
ejpam-3786	223	36	)	)	PUNCT
ejpam-3786	223	37	≤	≤	NOUN
ejpam-3786	223	38	0	0	NUM
ejpam-3786	223	39	.	.	PROPN
ejpam-3786	224	1	4	4	NUM
ejpam-3786	224	2	.	.	X
ejpam-3786	224	3	in	in	ADP
ejpam-3786	224	4	this	this	DET
ejpam-3786	224	5	case	case	NOUN
ejpam-3786	224	6	∆	∆	PROPN
ejpam-3786	224	7	is	be	AUX
ejpam-3786	224	8	given	give	VERB
ejpam-3786	224	9	in	in	ADP
ejpam-3786	224	10	figure	figure	NOUN
ejpam-3786	224	11	7(i	7(i	NUM
ejpam-3786	224	12	)	)	PUNCT
ejpam-3786	224	13	.	.	PUNCT
ejpam-3786	225	1	figure	figure	VERB
ejpam-3786	225	2	7	7	NUM
ejpam-3786	225	3	:	:	PUNCT
ejpam-3786	225	4	regions	region	NOUN
ejpam-3786	225	5	∆	∆	PROPN
ejpam-3786	225	6	and	and	CCONJ
ejpam-3786	225	7	∆̂	∆̂	X
ejpam-3786	225	8	by	by	ADP
ejpam-3786	225	9	adding	add	VERB
ejpam-3786	225	10	c(∆	c(∆	NOUN
ejpam-3786	225	11	)	)	PUNCT
ejpam-3786	225	12	to	to	ADP
ejpam-3786	225	13	c(∆̂	c(∆̂	PROPN
ejpam-3786	225	14	)	)	PUNCT
ejpam-3786	225	15	we	we	PRON
ejpam-3786	225	16	get	get	VERB
ejpam-3786	225	17	c(∆	c(∆	NOUN
ejpam-3786	225	18	)	)	PUNCT
ejpam-3786	225	19	≤	≤	NOUN
ejpam-3786	225	20	0	0	NUM
ejpam-3786	225	21	.	.	PROPN
ejpam-3786	226	1	5	5	NUM
ejpam-3786	226	2	.	.	X
ejpam-3786	226	3	in	in	ADP
ejpam-3786	226	4	this	this	DET
ejpam-3786	226	5	case	case	NOUN
ejpam-3786	226	6	∆	∆	PROPN
ejpam-3786	226	7	is	be	AUX
ejpam-3786	226	8	given	give	VERB
ejpam-3786	226	9	in	in	ADP
ejpam-3786	226	10	figure	figure	NOUN
ejpam-3786	226	11	8(i	8(i	NUM
ejpam-3786	226	12	)	)	PUNCT
ejpam-3786	226	13	.	.	PUNCT
ejpam-3786	227	1	muhammad	muhammad	PROPN
ejpam-3786	227	2	saeed	saeed	PROPN
ejpam-3786	227	3	akram	akram	PROPN
ejpam-3786	227	4	,	,	PUNCT
ejpam-3786	227	5	maira	maira	PROPN
ejpam-3786	227	6	amjid	amjid	PROPN
ejpam-3786	227	7	,	,	PUNCT
ejpam-3786	227	8	sohail	sohail	PROPN
ejpam-3786	227	9	iqbal	iqbal	PROPN
ejpam-3786	227	10	/	/	PUNCT
ejpam-3786	227	11	eur	eur	PROPN
ejpam-3786	227	12	.	.	PUNCT
ejpam-3786	228	1	j.	j.	PROPN
ejpam-3786	228	2	pure	pure	PROPN
ejpam-3786	228	3	appl	appl	PROPN
ejpam-3786	228	4	.	.	PROPN
ejpam-3786	228	5	math	math	PROPN
ejpam-3786	228	6	,	,	PUNCT
ejpam-3786	228	7	13	13	NUM
ejpam-3786	228	8	(	(	PUNCT
ejpam-3786	228	9	4	4	NUM
ejpam-3786	228	10	)	)	PUNCT
ejpam-3786	228	11	(	(	PUNCT
ejpam-3786	228	12	2020	2020	NUM
ejpam-3786	228	13	)	)	PUNCT
ejpam-3786	228	14	,	,	PUNCT
ejpam-3786	228	15	914	914	NUM
ejpam-3786	228	16	-	-	SYM
ejpam-3786	228	17	938	938	NUM
ejpam-3786	228	18	924	924	NUM
ejpam-3786	228	19	figure	figure	NOUN
ejpam-3786	228	20	8	8	NUM
ejpam-3786	228	21	:	:	PUNCT
ejpam-3786	228	22	region	region	NOUN
ejpam-3786	228	23	∆	∆	PROPN
ejpam-3786	228	24	the	the	DET
ejpam-3786	228	25	subcases	subcase	NOUN
ejpam-3786	228	26	which	which	PRON
ejpam-3786	228	27	are	be	AUX
ejpam-3786	228	28	to	to	PART
ejpam-3786	228	29	be	be	AUX
ejpam-3786	228	30	examined	examine	VERB
ejpam-3786	228	31	are	be	AUX
ejpam-3786	228	32	given	give	VERB
ejpam-3786	228	33	below	below	ADP
ejpam-3786	228	34	:	:	PUNCT
ejpam-3786	228	35	(	(	PUNCT
ejpam-3786	228	36	a	a	X
ejpam-3786	228	37	)	)	PUNCT
ejpam-3786	228	38	d∆(va	d∆(va	NOUN
ejpam-3786	228	39	)	)	PUNCT
ejpam-3786	228	40	=	=	PUNCT
ejpam-3786	229	1	d∆(vc	d∆(vc	NOUN
ejpam-3786	229	2	)	)	PUNCT
ejpam-3786	229	3	=	=	SYM
ejpam-3786	229	4	d∆(ve	d∆(ve	NOUN
ejpam-3786	229	5	)	)	PUNCT
ejpam-3786	229	6	=	=	SYM
ejpam-3786	230	1	d∆(vg	d∆(vg	NOUN
ejpam-3786	230	2	)	)	PUNCT
ejpam-3786	230	3	=	=	SYM
ejpam-3786	231	1	2	2	NUM
ejpam-3786	231	2	;	;	PUNCT
ejpam-3786	231	3	(	(	PUNCT
ejpam-3786	231	4	b	b	X
ejpam-3786	231	5	)	)	PUNCT
ejpam-3786	231	6	d∆(va	d∆(va	NOUN
ejpam-3786	231	7	)	)	PUNCT
ejpam-3786	231	8	=	=	PUNCT
ejpam-3786	231	9	d∆(vc	d∆(vc	NOUN
ejpam-3786	231	10	)	)	PUNCT
ejpam-3786	231	11	=	=	SYM
ejpam-3786	231	12	d∆(ve	d∆(ve	NOUN
ejpam-3786	231	13	)	)	PUNCT
ejpam-3786	231	14	=	=	SYM
ejpam-3786	231	15	d∆(vh	d∆(vh	X
ejpam-3786	231	16	)	)	PUNCT
ejpam-3786	231	17	=	=	SYM
ejpam-3786	232	1	2	2	X
ejpam-3786	232	2	.	.	PUNCT
ejpam-3786	232	3	(	(	PUNCT
ejpam-3786	232	4	a	a	X
ejpam-3786	232	5	)	)	PUNCT
ejpam-3786	232	6	here	here	ADV
ejpam-3786	232	7	d∆(va	d∆(va	NOUN
ejpam-3786	232	8	)	)	PUNCT
ejpam-3786	232	9	=	=	PUNCT
ejpam-3786	232	10	d∆(vc	d∆(vc	NOUN
ejpam-3786	232	11	)	)	PUNCT
ejpam-3786	232	12	=	=	SYM
ejpam-3786	232	13	d∆(ve	d∆(ve	NOUN
ejpam-3786	232	14	)	)	PUNCT
ejpam-3786	232	15	=	=	SYM
ejpam-3786	233	1	d∆(vg	d∆(vg	NOUN
ejpam-3786	233	2	)	)	PUNCT
ejpam-3786	234	1	=	=	SYM
ejpam-3786	234	2	2	2	NUM
ejpam-3786	234	3	which	which	PRON
ejpam-3786	234	4	implies	imply	VERB
ejpam-3786	234	5	l∆(va	l∆(va	NOUN
ejpam-3786	234	6	)	)	PUNCT
ejpam-3786	235	1	=	=	SYM
ejpam-3786	235	2	ag−1	ag−1	NOUN
ejpam-3786	235	3	,	,	PUNCT
ejpam-3786	235	4	l∆(vc	l∆(vc	PROPN
ejpam-3786	235	5	)	)	PUNCT
ejpam-3786	236	1	=	=	SYM
ejpam-3786	236	2	ce−1	ce−1	PROPN
ejpam-3786	236	3	,	,	PUNCT
ejpam-3786	236	4	l(ve	l(ve	PROPN
ejpam-3786	236	5	)	)	PUNCT
ejpam-3786	236	6	=	=	SYM
ejpam-3786	236	7	ec−1	ec−1	NOUN
ejpam-3786	236	8	,	,	PUNCT
ejpam-3786	236	9	and	and	CCONJ
ejpam-3786	236	10	l∆(vg	l∆(vg	NOUN
ejpam-3786	236	11	)	)	PUNCT
ejpam-3786	236	12	=	=	VERB
ejpam-3786	236	13	ga−1	ga−1	NOUN
ejpam-3786	236	14	,	,	PUNCT
ejpam-3786	236	15	as	as	SCONJ
ejpam-3786	236	16	shown	show	VERB
ejpam-3786	236	17	in	in	ADP
ejpam-3786	236	18	figure	figure	NOUN
ejpam-3786	236	19	8(ii	8(ii	NUM
ejpam-3786	236	20	)	)	PUNCT
ejpam-3786	236	21	.	.	PUNCT
ejpam-3786	237	1	notice	notice	VERB
ejpam-3786	237	2	that	that	SCONJ
ejpam-3786	237	3	l∆(va	l∆(va	NOUN
ejpam-3786	237	4	)	)	PUNCT
ejpam-3786	238	1	=	=	SYM
ejpam-3786	238	2	ag−1	ag−1	NOUN
ejpam-3786	238	3	and	and	CCONJ
ejpam-3786	238	4	l∆(vc	l∆(vc	NOUN
ejpam-3786	238	5	)	)	PUNCT
ejpam-3786	239	1	=	=	NOUN
ejpam-3786	239	2	ce−1	ce−1	PROPN
ejpam-3786	239	3	implies	imply	VERB
ejpam-3786	239	4	that	that	SCONJ
ejpam-3786	239	5	l∆(vb	l∆(vb	NOUN
ejpam-3786	239	6	)	)	PUNCT
ejpam-3786	239	7	=	=	SYM
ejpam-3786	239	8	bd−1h−1w	bd−1h−1w	NOUN
ejpam-3786	239	9	in	in	ADP
ejpam-3786	239	10	which	which	PRON
ejpam-3786	239	11	w	w	PROPN
ejpam-3786	239	12	∈	∈	PROPN
ejpam-3786	239	13	{	{	PUNCT
ejpam-3786	239	14	b	b	NOUN
ejpam-3786	239	15	,	,	PUNCT
ejpam-3786	239	16	c	c	NOUN
ejpam-3786	239	17	,	,	PUNCT
ejpam-3786	239	18	d	d	NOUN
ejpam-3786	239	19	,	,	PUNCT
ejpam-3786	239	20	e	e	NOUN
ejpam-3786	239	21	}	}	PUNCT
ejpam-3786	239	22	which	which	PRON
ejpam-3786	239	23	implies	imply	VERB
ejpam-3786	239	24	d∆(vb	d∆(vb	NOUN
ejpam-3786	239	25	)	)	PUNCT
ejpam-3786	239	26	>	>	X
ejpam-3786	240	1	3	3	X
ejpam-3786	240	2	.	.	PUNCT
ejpam-3786	240	3	similarly	similarly	ADV
ejpam-3786	240	4	notice	notice	VERB
ejpam-3786	240	5	that	that	SCONJ
ejpam-3786	240	6	l∆(vg	l∆(vg	NOUN
ejpam-3786	240	7	)	)	PUNCT
ejpam-3786	241	1	=	=	PUNCT
ejpam-3786	241	2	ga−1	ga−1	NOUN
ejpam-3786	241	3	and	and	CCONJ
ejpam-3786	241	4	l∆(ve	l∆(ve	NOUN
ejpam-3786	241	5	)	)	PUNCT
ejpam-3786	242	1	=	=	SYM
ejpam-3786	242	2	ec−1	ec−1	NOUN
ejpam-3786	242	3	implies	imply	VERB
ejpam-3786	242	4	that	that	PRON
ejpam-3786	242	5	l∆(vf	l∆(vf	ADJ
ejpam-3786	242	6	)	)	PUNCT
ejpam-3786	242	7	=	=	SYM
ejpam-3786	242	8	fi−1d−1w	fi−1d−1w	NOUN
ejpam-3786	242	9	in	in	ADP
ejpam-3786	242	10	which	which	PRON
ejpam-3786	242	11	w	w	PROPN
ejpam-3786	242	12	∈	∈	PROPN
ejpam-3786	242	13	{	{	PUNCT
ejpam-3786	242	14	b	b	NOUN
ejpam-3786	242	15	,	,	PUNCT
ejpam-3786	242	16	c	c	X
ejpam-3786	242	17	,	,	PUNCT
ejpam-3786	242	18	e	e	NOUN
ejpam-3786	242	19	,	,	PUNCT
ejpam-3786	242	20	h	h	NOUN
ejpam-3786	242	21	}	}	PUNCT
ejpam-3786	242	22	which	which	PRON
ejpam-3786	242	23	implies	imply	VERB
ejpam-3786	242	24	d∆(vf	d∆(vf	NOUN
ejpam-3786	242	25	)	)	PUNCT
ejpam-3786	242	26	>	>	X
ejpam-3786	243	1	3	3	X
ejpam-3786	243	2	.	.	PUNCT
ejpam-3786	243	3	since	since	SCONJ
ejpam-3786	243	4	d∆(vb	d∆(vb	PROPN
ejpam-3786	243	5	)	)	PUNCT
ejpam-3786	243	6	>	>	X
ejpam-3786	243	7	3	3	NUM
ejpam-3786	243	8	and	and	CCONJ
ejpam-3786	243	9	d∆(vf	d∆(vf	ADJ
ejpam-3786	243	10	)	)	PUNCT
ejpam-3786	243	11	>	>	X
ejpam-3786	243	12	3	3	NUM
ejpam-3786	243	13	so	so	ADV
ejpam-3786	243	14	c(∆	c(∆	NOUN
ejpam-3786	243	15	)	)	PUNCT
ejpam-3786	243	16	≤	≤	NOUN
ejpam-3786	243	17	0	0	NUM
ejpam-3786	243	18	.	.	PUNCT
ejpam-3786	244	1	(	(	PUNCT
ejpam-3786	244	2	b	b	X
ejpam-3786	244	3	)	)	PUNCT
ejpam-3786	244	4	here	here	ADV
ejpam-3786	244	5	d∆(va	d∆(va	NOUN
ejpam-3786	244	6	)	)	PUNCT
ejpam-3786	244	7	=	=	PUNCT
ejpam-3786	244	8	d∆(vc	d∆(vc	NOUN
ejpam-3786	244	9	)	)	PUNCT
ejpam-3786	244	10	=	=	SYM
ejpam-3786	244	11	d∆(ve	d∆(ve	NOUN
ejpam-3786	244	12	)	)	PUNCT
ejpam-3786	244	13	=	=	SYM
ejpam-3786	244	14	d∆(vh	d∆(vh	X
ejpam-3786	244	15	)	)	PUNCT
ejpam-3786	244	16	=	=	SYM
ejpam-3786	244	17	2	2	NUM
ejpam-3786	244	18	which	which	PRON
ejpam-3786	244	19	implies	imply	VERB
ejpam-3786	244	20	l∆(va	l∆(va	NOUN
ejpam-3786	244	21	)	)	PUNCT
ejpam-3786	245	1	=	=	SYM
ejpam-3786	245	2	ag−1	ag−1	NOUN
ejpam-3786	245	3	,	,	PUNCT
ejpam-3786	245	4	l∆(vc	l∆(vc	PROPN
ejpam-3786	245	5	)	)	PUNCT
ejpam-3786	246	1	=	=	SYM
ejpam-3786	246	2	ce−1	ce−1	PROPN
ejpam-3786	246	3	,	,	PUNCT
ejpam-3786	246	4	l(ve	l(ve	PROPN
ejpam-3786	246	5	)	)	PUNCT
ejpam-3786	246	6	=	=	SYM
ejpam-3786	246	7	ec−1	ec−1	NOUN
ejpam-3786	246	8	,	,	PUNCT
ejpam-3786	246	9	and	and	CCONJ
ejpam-3786	246	10	l∆(vh	l∆(vh	PROPN
ejpam-3786	246	11	)	)	PUNCT
ejpam-3786	246	12	=	=	SYM
ejpam-3786	246	13	hd−1	hd−1	PROPN
ejpam-3786	246	14	,	,	PUNCT
ejpam-3786	246	15	as	as	SCONJ
ejpam-3786	246	16	shown	show	VERB
ejpam-3786	246	17	in	in	ADP
ejpam-3786	246	18	figure	figure	NOUN
ejpam-3786	246	19	8(iii	8(iii	NUM
ejpam-3786	246	20	)	)	PUNCT
ejpam-3786	246	21	.	.	PUNCT
ejpam-3786	247	1	notice	notice	VERB
ejpam-3786	247	2	that	that	SCONJ
ejpam-3786	247	3	l∆(va	l∆(va	NOUN
ejpam-3786	247	4	)	)	PUNCT
ejpam-3786	248	1	=	=	SYM
ejpam-3786	248	2	ag−1	ag−1	NOUN
ejpam-3786	248	3	and	and	CCONJ
ejpam-3786	248	4	l∆(vc	l∆(vc	NOUN
ejpam-3786	248	5	)	)	PUNCT
ejpam-3786	249	1	=	=	NOUN
ejpam-3786	249	2	ce−1	ce−1	PROPN
ejpam-3786	249	3	implies	imply	VERB
ejpam-3786	249	4	that	that	SCONJ
ejpam-3786	249	5	l∆(vb	l∆(vb	NOUN
ejpam-3786	249	6	)	)	PUNCT
ejpam-3786	249	7	=	=	SYM
ejpam-3786	249	8	bd−1h−1w	bd−1h−1w	NOUN
ejpam-3786	249	9	in	in	ADP
ejpam-3786	249	10	which	which	PRON
ejpam-3786	249	11	w	w	PROPN
ejpam-3786	249	12	∈	∈	PROPN
ejpam-3786	249	13	{	{	PUNCT
ejpam-3786	249	14	b	b	NOUN
ejpam-3786	249	15	,	,	PUNCT
ejpam-3786	249	16	c	c	NOUN
ejpam-3786	249	17	,	,	PUNCT
ejpam-3786	249	18	d	d	NOUN
ejpam-3786	249	19	,	,	PUNCT
ejpam-3786	249	20	e	e	NOUN
ejpam-3786	249	21	}	}	PUNCT
ejpam-3786	249	22	which	which	PRON
ejpam-3786	249	23	implies	imply	VERB
ejpam-3786	249	24	d∆(vb	d∆(vb	NOUN
ejpam-3786	249	25	)	)	PUNCT
ejpam-3786	249	26	>	>	X
ejpam-3786	250	1	3	3	X
ejpam-3786	250	2	.	.	PUNCT
ejpam-3786	250	3	similarly	similarly	ADV
ejpam-3786	250	4	notice	notice	VERB
ejpam-3786	250	5	that	that	SCONJ
ejpam-3786	250	6	l∆(va	l∆(va	NOUN
ejpam-3786	250	7	)	)	PUNCT
ejpam-3786	251	1	=	=	SYM
ejpam-3786	251	2	ag−1	ag−1	NOUN
ejpam-3786	251	3	and	and	CCONJ
ejpam-3786	251	4	l∆(vh	l∆(vh	PROPN
ejpam-3786	251	5	)	)	PUNCT
ejpam-3786	252	1	=	=	SYM
ejpam-3786	252	2	hd−1	hd−1	PROPN
ejpam-3786	252	3	implies	imply	VERB
ejpam-3786	252	4	that	that	SCONJ
ejpam-3786	252	5	l∆(vi	l∆(vi	X
ejpam-3786	252	6	)	)	PUNCT
ejpam-3786	252	7	=	=	PUNCT
ejpam-3786	252	8	if−1e−1w	if−1e−1w	NOUN
ejpam-3786	252	9	in	in	ADP
ejpam-3786	252	10	which	which	PRON
ejpam-3786	252	11	w	w	PROPN
ejpam-3786	252	12	∈	∈	PROPN
ejpam-3786	252	13	{	{	PUNCT
ejpam-3786	252	14	b	b	NOUN
ejpam-3786	252	15	,	,	PUNCT
ejpam-3786	252	16	c	c	NOUN
ejpam-3786	252	17	,	,	PUNCT
ejpam-3786	252	18	d	d	NOUN
ejpam-3786	252	19	,	,	PUNCT
ejpam-3786	252	20	h	h	NOUN
ejpam-3786	252	21	}	}	PUNCT
ejpam-3786	252	22	which	which	PRON
ejpam-3786	252	23	implies	imply	VERB
ejpam-3786	252	24	d∆(vi	d∆(vi	NOUN
ejpam-3786	252	25	)	)	PUNCT
ejpam-3786	252	26	>	>	X
ejpam-3786	253	1	3	3	X
ejpam-3786	253	2	.	.	PUNCT
ejpam-3786	253	3	since	since	SCONJ
ejpam-3786	253	4	d∆(vb	d∆(vb	PROPN
ejpam-3786	253	5	)	)	PUNCT
ejpam-3786	253	6	>	>	X
ejpam-3786	253	7	3	3	NUM
ejpam-3786	253	8	and	and	CCONJ
ejpam-3786	253	9	d∆(vi	d∆(vi	NOUN
ejpam-3786	253	10	)	)	PUNCT
ejpam-3786	253	11	>	>	X
ejpam-3786	253	12	3	3	NUM
ejpam-3786	253	13	so	so	ADV
ejpam-3786	253	14	c(∆	c(∆	NOUN
ejpam-3786	253	15	)	)	PUNCT
ejpam-3786	253	16	≤	≤	NOUN
ejpam-3786	253	17	0	0	NUM
ejpam-3786	253	18	.	.	PROPN
ejpam-3786	254	1	6	6	NUM
ejpam-3786	254	2	.	.	X
ejpam-3786	254	3	in	in	ADP
ejpam-3786	254	4	this	this	DET
ejpam-3786	254	5	case	case	NOUN
ejpam-3786	254	6	∆	∆	PROPN
ejpam-3786	254	7	is	be	AUX
ejpam-3786	254	8	given	give	VERB
ejpam-3786	254	9	in	in	ADP
ejpam-3786	254	10	figure	figure	NOUN
ejpam-3786	254	11	9(i	9(i	NUM
ejpam-3786	254	12	)	)	PUNCT
ejpam-3786	254	13	.	.	PUNCT
ejpam-3786	255	1	figure	figure	VERB
ejpam-3786	255	2	9	9	NUM
ejpam-3786	255	3	:	:	PUNCT
ejpam-3786	255	4	region	region	NOUN
ejpam-3786	255	5	∆	∆	PROPN
ejpam-3786	255	6	here	here	ADV
ejpam-3786	255	7	d∆(va	d∆(va	NOUN
ejpam-3786	255	8	)	)	PUNCT
ejpam-3786	256	1	=	=	PUNCT
ejpam-3786	256	2	d∆(vc	d∆(vc	NOUN
ejpam-3786	256	3	)	)	PUNCT
ejpam-3786	256	4	=	=	SYM
ejpam-3786	256	5	d∆(ve	d∆(ve	NOUN
ejpam-3786	256	6	)	)	PUNCT
ejpam-3786	256	7	=	=	SYM
ejpam-3786	257	1	d∆(vg	d∆(vg	NOUN
ejpam-3786	257	2	)	)	PUNCT
ejpam-3786	258	1	=	=	SYM
ejpam-3786	258	2	2	2	NUM
ejpam-3786	258	3	which	which	PRON
ejpam-3786	258	4	implies	imply	VERB
ejpam-3786	258	5	l∆(va	l∆(va	NOUN
ejpam-3786	258	6	)	)	PUNCT
ejpam-3786	259	1	=	=	SYM
ejpam-3786	259	2	ag−1	ag−1	NOUN
ejpam-3786	259	3	,	,	PUNCT
ejpam-3786	259	4	l∆(vc	l∆(vc	PROPN
ejpam-3786	259	5	)	)	PUNCT
ejpam-3786	260	1	=	=	SYM
ejpam-3786	260	2	cd−1	cd−1	PROPN
ejpam-3786	260	3	,	,	PUNCT
ejpam-3786	260	4	l(ve	l(ve	PROPN
ejpam-3786	260	5	)	)	PUNCT
ejpam-3786	260	6	=	=	SYM
ejpam-3786	260	7	eb−1	eb−1	PROPN
ejpam-3786	260	8	,	,	PUNCT
ejpam-3786	260	9	and	and	CCONJ
ejpam-3786	260	10	l∆(vg	l∆(vg	NOUN
ejpam-3786	260	11	)	)	PUNCT
ejpam-3786	261	1	=	=	VERB
ejpam-3786	261	2	ga−1	ga−1	NOUN
ejpam-3786	261	3	,	,	PUNCT
ejpam-3786	261	4	as	as	SCONJ
ejpam-3786	261	5	shown	show	VERB
ejpam-3786	261	6	in	in	ADP
ejpam-3786	261	7	figure	figure	NOUN
ejpam-3786	261	8	9(ii	9(ii	NUM
ejpam-3786	261	9	)	)	PUNCT
ejpam-3786	261	10	.	.	PUNCT
ejpam-3786	262	1	notice	notice	VERB
ejpam-3786	262	2	that	that	SCONJ
ejpam-3786	262	3	l∆(va	l∆(va	NOUN
ejpam-3786	262	4	)	)	PUNCT
ejpam-3786	263	1	=	=	PROPN
ejpam-3786	263	2	muhammad	muhammad	PROPN
ejpam-3786	263	3	saeed	saeed	PROPN
ejpam-3786	263	4	akram	akram	PROPN
ejpam-3786	263	5	,	,	PUNCT
ejpam-3786	263	6	maira	maira	PROPN
ejpam-3786	263	7	amjid	amjid	PROPN
ejpam-3786	263	8	,	,	PUNCT
ejpam-3786	263	9	sohail	sohail	PROPN
ejpam-3786	263	10	iqbal	iqbal	PROPN
ejpam-3786	263	11	/	/	PUNCT
ejpam-3786	263	12	eur	eur	PROPN
ejpam-3786	263	13	.	.	PUNCT
ejpam-3786	264	1	j.	j.	PROPN
ejpam-3786	264	2	pure	pure	PROPN
ejpam-3786	264	3	appl	appl	PROPN
ejpam-3786	264	4	.	.	PROPN
ejpam-3786	264	5	math	math	PROPN
ejpam-3786	264	6	,	,	PUNCT
ejpam-3786	264	7	13	13	NUM
ejpam-3786	264	8	(	(	PUNCT
ejpam-3786	264	9	4	4	NUM
ejpam-3786	264	10	)	)	PUNCT
ejpam-3786	264	11	(	(	PUNCT
ejpam-3786	264	12	2020	2020	NUM
ejpam-3786	264	13	)	)	PUNCT
ejpam-3786	264	14	,	,	PUNCT
ejpam-3786	264	15	914	914	NUM
ejpam-3786	264	16	-	-	SYM
ejpam-3786	264	17	938	938	NUM
ejpam-3786	264	18	925	925	NUM
ejpam-3786	264	19	ag−1	ag−1	NOUN
ejpam-3786	264	20	and	and	CCONJ
ejpam-3786	264	21	l∆(vc	l∆(vc	NOUN
ejpam-3786	264	22	)	)	PUNCT
ejpam-3786	265	1	=	=	SYM
ejpam-3786	265	2	cd−1	cd−1	PROPN
ejpam-3786	265	3	implies	imply	VERB
ejpam-3786	265	4	that	that	SCONJ
ejpam-3786	265	5	l∆(vb	l∆(vb	NOUN
ejpam-3786	265	6	)	)	PUNCT
ejpam-3786	266	1	=	=	PUNCT
ejpam-3786	266	2	bc−1h−1w	bc−1h−1w	NOUN
ejpam-3786	266	3	in	in	ADP
ejpam-3786	266	4	which	which	PRON
ejpam-3786	266	5	w	w	PROPN
ejpam-3786	266	6	∈	∈	PROPN
ejpam-3786	266	7	{	{	PUNCT
ejpam-3786	266	8	b	b	NOUN
ejpam-3786	266	9	,	,	PUNCT
ejpam-3786	266	10	c	c	NOUN
ejpam-3786	266	11	,	,	PUNCT
ejpam-3786	266	12	d	d	NOUN
ejpam-3786	266	13	,	,	PUNCT
ejpam-3786	266	14	e	e	NOUN
ejpam-3786	266	15	}	}	PUNCT
ejpam-3786	266	16	which	which	PRON
ejpam-3786	266	17	implies	imply	VERB
ejpam-3786	266	18	d∆(vb	d∆(vb	NOUN
ejpam-3786	266	19	)	)	PUNCT
ejpam-3786	266	20	>	>	X
ejpam-3786	267	1	3	3	X
ejpam-3786	267	2	.	.	PUNCT
ejpam-3786	267	3	similarly	similarly	ADV
ejpam-3786	267	4	notice	notice	VERB
ejpam-3786	267	5	that	that	SCONJ
ejpam-3786	267	6	l∆(vg	l∆(vg	NOUN
ejpam-3786	267	7	)	)	PUNCT
ejpam-3786	268	1	=	=	PUNCT
ejpam-3786	268	2	ga−1	ga−1	NOUN
ejpam-3786	268	3	and	and	CCONJ
ejpam-3786	268	4	l∆(ve	l∆(ve	PROPN
ejpam-3786	268	5	)	)	PUNCT
ejpam-3786	269	1	=	=	SYM
ejpam-3786	269	2	eb−1	eb−1	PROPN
ejpam-3786	269	3	implies	imply	VERB
ejpam-3786	269	4	that	that	PRON
ejpam-3786	269	5	l∆(vf	l∆(vf	ADJ
ejpam-3786	269	6	)	)	PUNCT
ejpam-3786	269	7	=	=	PUNCT
ejpam-3786	269	8	fi−1c−1w	fi−1c−1w	NOUN
ejpam-3786	269	9	in	in	ADP
ejpam-3786	269	10	which	which	PRON
ejpam-3786	269	11	w	w	PROPN
ejpam-3786	269	12	∈	∈	PROPN
ejpam-3786	269	13	{	{	PUNCT
ejpam-3786	269	14	b	b	NOUN
ejpam-3786	269	15	,	,	PUNCT
ejpam-3786	269	16	d	d	NOUN
ejpam-3786	269	17	,	,	PUNCT
ejpam-3786	269	18	e	e	NOUN
ejpam-3786	269	19	,	,	PUNCT
ejpam-3786	269	20	h	h	NOUN
ejpam-3786	269	21	}	}	PUNCT
ejpam-3786	269	22	which	which	PRON
ejpam-3786	269	23	implies	imply	VERB
ejpam-3786	269	24	d∆(vf	d∆(vf	NOUN
ejpam-3786	269	25	)	)	PUNCT
ejpam-3786	269	26	>	>	X
ejpam-3786	270	1	3	3	X
ejpam-3786	270	2	.	.	PUNCT
ejpam-3786	270	3	since	since	SCONJ
ejpam-3786	270	4	d∆(vb	d∆(vb	PROPN
ejpam-3786	270	5	)	)	PUNCT
ejpam-3786	270	6	>	>	X
ejpam-3786	270	7	3	3	NUM
ejpam-3786	270	8	and	and	CCONJ
ejpam-3786	270	9	d∆(vf	d∆(vf	ADJ
ejpam-3786	270	10	)	)	PUNCT
ejpam-3786	270	11	>	>	X
ejpam-3786	270	12	3	3	NUM
ejpam-3786	270	13	so	so	ADV
ejpam-3786	270	14	c(∆	c(∆	NOUN
ejpam-3786	270	15	)	)	PUNCT
ejpam-3786	270	16	≤	≤	NOUN
ejpam-3786	270	17	0	0	NUM
ejpam-3786	270	18	.	.	PUNCT
ejpam-3786	271	1	7	7	X
ejpam-3786	271	2	.	.	X
ejpam-3786	271	3	in	in	ADP
ejpam-3786	271	4	this	this	DET
ejpam-3786	271	5	case	case	NOUN
ejpam-3786	271	6	∆	∆	PROPN
ejpam-3786	271	7	is	be	AUX
ejpam-3786	271	8	given	give	VERB
ejpam-3786	271	9	in	in	ADP
ejpam-3786	271	10	figure	figure	NOUN
ejpam-3786	271	11	10(i	10(i	NUM
ejpam-3786	271	12	)	)	PUNCT
ejpam-3786	271	13	.	.	PUNCT
ejpam-3786	272	1	figure	figure	VERB
ejpam-3786	272	2	10	10	NUM
ejpam-3786	272	3	:	:	PUNCT
ejpam-3786	272	4	regions	region	NOUN
ejpam-3786	272	5	∆	∆	PROPN
ejpam-3786	272	6	and	and	CCONJ
ejpam-3786	272	7	∆̂	∆̂	X
ejpam-3786	272	8	by	by	ADP
ejpam-3786	272	9	adding	add	VERB
ejpam-3786	272	10	c(∆	c(∆	NOUN
ejpam-3786	272	11	)	)	PUNCT
ejpam-3786	272	12	to	to	ADP
ejpam-3786	272	13	c(∆̂	c(∆̂	PROPN
ejpam-3786	272	14	)	)	PUNCT
ejpam-3786	272	15	we	we	PRON
ejpam-3786	272	16	get	get	VERB
ejpam-3786	272	17	c(∆	c(∆	NOUN
ejpam-3786	272	18	)	)	PUNCT
ejpam-3786	272	19	≤	≤	NOUN
ejpam-3786	272	20	0	0	NUM
ejpam-3786	272	21	.	.	NOUN
ejpam-3786	272	22	8	8	NUM
ejpam-3786	272	23	.	.	PUNCT
ejpam-3786	273	1	in	in	ADP
ejpam-3786	273	2	this	this	DET
ejpam-3786	273	3	case	case	NOUN
ejpam-3786	273	4	∆	∆	PROPN
ejpam-3786	273	5	is	be	AUX
ejpam-3786	273	6	given	give	VERB
ejpam-3786	273	7	in	in	ADP
ejpam-3786	273	8	figure	figure	NOUN
ejpam-3786	273	9	11(i	11(i	NUM
ejpam-3786	273	10	)	)	PUNCT
ejpam-3786	273	11	.	.	PUNCT
ejpam-3786	274	1	figure	figure	VERB
ejpam-3786	274	2	11	11	NUM
ejpam-3786	274	3	:	:	PUNCT
ejpam-3786	274	4	region	region	NOUN
ejpam-3786	274	5	∆	∆	PROPN
ejpam-3786	274	6	here	here	ADV
ejpam-3786	274	7	d∆(va	d∆(va	NOUN
ejpam-3786	274	8	)	)	PUNCT
ejpam-3786	275	1	=	=	PUNCT
ejpam-3786	275	2	d∆(vc	d∆(vc	NOUN
ejpam-3786	275	3	)	)	PUNCT
ejpam-3786	275	4	=	=	SYM
ejpam-3786	275	5	d∆(ve	d∆(ve	NOUN
ejpam-3786	275	6	)	)	PUNCT
ejpam-3786	275	7	=	=	SYM
ejpam-3786	276	1	d∆(vg	d∆(vg	NOUN
ejpam-3786	276	2	)	)	PUNCT
ejpam-3786	277	1	=	=	SYM
ejpam-3786	277	2	2	2	NUM
ejpam-3786	277	3	which	which	PRON
ejpam-3786	277	4	implies	imply	VERB
ejpam-3786	277	5	l∆(va	l∆(va	NOUN
ejpam-3786	277	6	)	)	PUNCT
ejpam-3786	278	1	=	=	SYM
ejpam-3786	278	2	ag−1	ag−1	NOUN
ejpam-3786	278	3	,	,	PUNCT
ejpam-3786	278	4	l∆(vc	l∆(vc	NOUN
ejpam-3786	278	5	)	)	PUNCT
ejpam-3786	279	1	=	=	SYM
ejpam-3786	279	2	cb−1	cb−1	NOUN
ejpam-3786	279	3	,	,	PUNCT
ejpam-3786	279	4	l(ve	l(ve	PROPN
ejpam-3786	279	5	)	)	PUNCT
ejpam-3786	279	6	=	=	SYM
ejpam-3786	279	7	ed−1	ed−1	PROPN
ejpam-3786	279	8	,	,	PUNCT
ejpam-3786	279	9	and	and	CCONJ
ejpam-3786	279	10	l∆(vg	l∆(vg	NOUN
ejpam-3786	279	11	)	)	PUNCT
ejpam-3786	280	1	=	=	VERB
ejpam-3786	280	2	ga−1	ga−1	NOUN
ejpam-3786	280	3	,	,	PUNCT
ejpam-3786	280	4	as	as	SCONJ
ejpam-3786	280	5	shown	show	VERB
ejpam-3786	280	6	in	in	ADP
ejpam-3786	280	7	figure	figure	NOUN
ejpam-3786	280	8	11(ii	11(ii	NUM
ejpam-3786	280	9	)	)	PUNCT
ejpam-3786	280	10	.	.	PUNCT
ejpam-3786	281	1	notice	notice	VERB
ejpam-3786	281	2	that	that	SCONJ
ejpam-3786	281	3	l∆(ve	l∆(ve	NOUN
ejpam-3786	281	4	)	)	PUNCT
ejpam-3786	281	5	=	=	SYM
ejpam-3786	281	6	ed−1	ed−1	PROPN
ejpam-3786	281	7	and	and	CCONJ
ejpam-3786	281	8	l∆(vc	l∆(vc	NOUN
ejpam-3786	281	9	)	)	PUNCT
ejpam-3786	282	1	=	=	NOUN
ejpam-3786	282	2	cb−1	cb−1	NOUN
ejpam-3786	282	3	implies	imply	VERB
ejpam-3786	282	4	that	that	SCONJ
ejpam-3786	282	5	l∆(vd	l∆(vd	VERB
ejpam-3786	282	6	)	)	PUNCT
ejpam-3786	282	7	=	=	SYM
ejpam-3786	282	8	dc−1c−1w	dc−1c−1w	NOUN
ejpam-3786	282	9	in	in	ADP
ejpam-3786	282	10	which	which	PRON
ejpam-3786	282	11	w	w	PROPN
ejpam-3786	282	12	∈	∈	PROPN
ejpam-3786	282	13	{	{	PUNCT
ejpam-3786	282	14	b	b	NOUN
ejpam-3786	282	15	,	,	PUNCT
ejpam-3786	282	16	d	d	NOUN
ejpam-3786	282	17	,	,	PUNCT
ejpam-3786	282	18	e	e	NOUN
ejpam-3786	282	19	,	,	PUNCT
ejpam-3786	282	20	h	h	NOUN
ejpam-3786	282	21	}	}	PUNCT
ejpam-3786	282	22	which	which	PRON
ejpam-3786	282	23	implies	imply	VERB
ejpam-3786	282	24	d∆(vd	d∆(vd	NOUN
ejpam-3786	282	25	)	)	PUNCT
ejpam-3786	282	26	>	>	X
ejpam-3786	283	1	3	3	X
ejpam-3786	283	2	.	.	PUNCT
ejpam-3786	283	3	similarly	similarly	ADV
ejpam-3786	283	4	notice	notice	VERB
ejpam-3786	283	5	that	that	SCONJ
ejpam-3786	283	6	l∆(vg	l∆(vg	NOUN
ejpam-3786	283	7	)	)	PUNCT
ejpam-3786	284	1	=	=	PUNCT
ejpam-3786	284	2	ga−1	ga−1	NOUN
ejpam-3786	284	3	and	and	CCONJ
ejpam-3786	284	4	l∆(ve	l∆(ve	PROPN
ejpam-3786	284	5	)	)	PUNCT
ejpam-3786	285	1	=	=	SYM
ejpam-3786	285	2	ed−1	ed−1	PROPN
ejpam-3786	285	3	implies	imply	VERB
ejpam-3786	285	4	that	that	SCONJ
ejpam-3786	285	5	l∆(vf	l∆(vf	ADJ
ejpam-3786	285	6	)	)	PUNCT
ejpam-3786	286	1	=	=	SYM
ejpam-3786	286	2	fi−1e−1w	fi−1e−1w	NOUN
ejpam-3786	286	3	in	in	ADP
ejpam-3786	286	4	which	which	PRON
ejpam-3786	286	5	w	w	PROPN
ejpam-3786	286	6	∈	∈	PROPN
ejpam-3786	286	7	{	{	PUNCT
ejpam-3786	286	8	b	b	NOUN
ejpam-3786	286	9	,	,	PUNCT
ejpam-3786	286	10	c	c	NOUN
ejpam-3786	286	11	,	,	PUNCT
ejpam-3786	286	12	d	d	NOUN
ejpam-3786	286	13	,	,	PUNCT
ejpam-3786	286	14	h	h	NOUN
ejpam-3786	286	15	}	}	PUNCT
ejpam-3786	286	16	which	which	PRON
ejpam-3786	286	17	implies	imply	VERB
ejpam-3786	286	18	d∆(vf	d∆(vf	NOUN
ejpam-3786	286	19	)	)	PUNCT
ejpam-3786	286	20	>	>	X
ejpam-3786	287	1	3	3	X
ejpam-3786	287	2	.	.	PUNCT
ejpam-3786	287	3	since	since	SCONJ
ejpam-3786	287	4	d∆(vd	d∆(vd	ADJ
ejpam-3786	287	5	)	)	PUNCT
ejpam-3786	287	6	>	>	X
ejpam-3786	287	7	3	3	NUM
ejpam-3786	287	8	and	and	CCONJ
ejpam-3786	287	9	d∆(vf	d∆(vf	ADJ
ejpam-3786	287	10	)	)	PUNCT
ejpam-3786	287	11	>	>	X
ejpam-3786	287	12	3	3	NUM
ejpam-3786	287	13	so	so	ADV
ejpam-3786	287	14	c(∆	c(∆	NOUN
ejpam-3786	287	15	)	)	PUNCT
ejpam-3786	287	16	≤	≤	NOUN
ejpam-3786	287	17	0	0	NUM
ejpam-3786	287	18	.	.	PROPN
ejpam-3786	288	1	9	9	NUM
ejpam-3786	288	2	.	.	X
ejpam-3786	288	3	in	in	ADP
ejpam-3786	288	4	this	this	DET
ejpam-3786	288	5	case	case	NOUN
ejpam-3786	288	6	∆	∆	PROPN
ejpam-3786	288	7	is	be	AUX
ejpam-3786	288	8	given	give	VERB
ejpam-3786	288	9	in	in	ADP
ejpam-3786	288	10	figure	figure	NOUN
ejpam-3786	288	11	12(i	12(i	NUM
ejpam-3786	288	12	)	)	PUNCT
ejpam-3786	288	13	.	.	PUNCT
ejpam-3786	289	1	muhammad	muhammad	PROPN
ejpam-3786	289	2	saeed	saeed	PROPN
ejpam-3786	289	3	akram	akram	PROPN
ejpam-3786	289	4	,	,	PUNCT
ejpam-3786	289	5	maira	maira	PROPN
ejpam-3786	289	6	amjid	amjid	PROPN
ejpam-3786	289	7	,	,	PUNCT
ejpam-3786	289	8	sohail	sohail	PROPN
ejpam-3786	289	9	iqbal	iqbal	PROPN
ejpam-3786	289	10	/	/	PUNCT
ejpam-3786	289	11	eur	eur	PROPN
ejpam-3786	289	12	.	.	PUNCT
ejpam-3786	290	1	j.	j.	PROPN
ejpam-3786	290	2	pure	pure	PROPN
ejpam-3786	290	3	appl	appl	PROPN
ejpam-3786	290	4	.	.	PROPN
ejpam-3786	290	5	math	math	PROPN
ejpam-3786	290	6	,	,	PUNCT
ejpam-3786	290	7	13	13	NUM
ejpam-3786	290	8	(	(	PUNCT
ejpam-3786	290	9	4	4	NUM
ejpam-3786	290	10	)	)	PUNCT
ejpam-3786	290	11	(	(	PUNCT
ejpam-3786	290	12	2020	2020	NUM
ejpam-3786	290	13	)	)	PUNCT
ejpam-3786	290	14	,	,	PUNCT
ejpam-3786	290	15	914	914	NUM
ejpam-3786	290	16	-	-	SYM
ejpam-3786	290	17	938	938	NUM
ejpam-3786	290	18	926	926	NUM
ejpam-3786	290	19	figure	figure	NOUN
ejpam-3786	290	20	12	12	NUM
ejpam-3786	290	21	:	:	PUNCT
ejpam-3786	290	22	regions	region	NOUN
ejpam-3786	290	23	∆	∆	PROPN
ejpam-3786	290	24	and	and	CCONJ
ejpam-3786	290	25	∆̂	∆̂	X
ejpam-3786	290	26	by	by	ADP
ejpam-3786	290	27	adding	add	VERB
ejpam-3786	290	28	c(∆	c(∆	NOUN
ejpam-3786	290	29	)	)	PUNCT
ejpam-3786	290	30	to	to	ADP
ejpam-3786	290	31	c(∆̂	c(∆̂	PROPN
ejpam-3786	290	32	)	)	PUNCT
ejpam-3786	290	33	we	we	PRON
ejpam-3786	290	34	get	get	VERB
ejpam-3786	290	35	c(∆	c(∆	NOUN
ejpam-3786	290	36	)	)	PUNCT
ejpam-3786	290	37	≤	≤	NOUN
ejpam-3786	290	38	0	0	NUM
ejpam-3786	290	39	.	.	PUNCT
ejpam-3786	291	1	10	10	NUM
ejpam-3786	291	2	.	.	PUNCT
ejpam-3786	292	1	in	in	ADP
ejpam-3786	292	2	this	this	DET
ejpam-3786	292	3	case	case	NOUN
ejpam-3786	292	4	∆	∆	PROPN
ejpam-3786	292	5	is	be	AUX
ejpam-3786	292	6	given	give	VERB
ejpam-3786	292	7	in	in	ADP
ejpam-3786	292	8	figure	figure	NOUN
ejpam-3786	292	9	13(i	13(i	NUM
ejpam-3786	292	10	)	)	PUNCT
ejpam-3786	292	11	.	.	PUNCT
ejpam-3786	293	1	figure	figure	VERB
ejpam-3786	293	2	13	13	NUM
ejpam-3786	293	3	:	:	PUNCT
ejpam-3786	293	4	regions	region	NOUN
ejpam-3786	293	5	∆	∆	PROPN
ejpam-3786	293	6	and	and	CCONJ
ejpam-3786	293	7	∆̂	∆̂	X
ejpam-3786	293	8	by	by	ADP
ejpam-3786	293	9	adding	add	VERB
ejpam-3786	293	10	c(∆	c(∆	NOUN
ejpam-3786	293	11	)	)	PUNCT
ejpam-3786	293	12	to	to	ADP
ejpam-3786	293	13	c(∆̂	c(∆̂	PROPN
ejpam-3786	293	14	)	)	PUNCT
ejpam-3786	293	15	we	we	PRON
ejpam-3786	293	16	get	get	VERB
ejpam-3786	293	17	c(∆	c(∆	NOUN
ejpam-3786	293	18	)	)	PUNCT
ejpam-3786	293	19	≤	≤	NOUN
ejpam-3786	293	20	0	0	NUM
ejpam-3786	293	21	.	.	PUNCT
ejpam-3786	294	1	muhammad	muhammad	PROPN
ejpam-3786	294	2	saeed	saeed	PROPN
ejpam-3786	294	3	akram	akram	PROPN
ejpam-3786	294	4	,	,	PUNCT
ejpam-3786	294	5	maira	maira	PROPN
ejpam-3786	294	6	amjid	amjid	PROPN
ejpam-3786	294	7	,	,	PUNCT
ejpam-3786	294	8	sohail	sohail	PROPN
ejpam-3786	294	9	iqbal	iqbal	PROPN
ejpam-3786	294	10	/	/	PUNCT
ejpam-3786	294	11	eur	eur	PROPN
ejpam-3786	294	12	.	.	PUNCT
ejpam-3786	295	1	j.	j.	PROPN
ejpam-3786	295	2	pure	pure	PROPN
ejpam-3786	295	3	appl	appl	PROPN
ejpam-3786	295	4	.	.	PROPN
ejpam-3786	295	5	math	math	PROPN
ejpam-3786	295	6	,	,	PUNCT
ejpam-3786	295	7	13	13	NUM
ejpam-3786	295	8	(	(	PUNCT
ejpam-3786	295	9	4	4	NUM
ejpam-3786	295	10	)	)	PUNCT
ejpam-3786	295	11	(	(	PUNCT
ejpam-3786	295	12	2020	2020	NUM
ejpam-3786	295	13	)	)	PUNCT
ejpam-3786	295	14	,	,	PUNCT
ejpam-3786	295	15	914	914	NUM
ejpam-3786	295	16	-	-	SYM
ejpam-3786	295	17	938	938	NUM
ejpam-3786	295	18	927	927	NUM
ejpam-3786	295	19	11	11	NUM
ejpam-3786	295	20	.	.	PUNCT
ejpam-3786	296	1	in	in	ADP
ejpam-3786	296	2	this	this	DET
ejpam-3786	296	3	case	case	NOUN
ejpam-3786	296	4	∆	∆	PROPN
ejpam-3786	296	5	is	be	AUX
ejpam-3786	296	6	given	give	VERB
ejpam-3786	296	7	in	in	ADP
ejpam-3786	296	8	figure	figure	NOUN
ejpam-3786	296	9	14(i	14(i	NUM
ejpam-3786	296	10	)	)	PUNCT
ejpam-3786	296	11	.	.	PUNCT
ejpam-3786	297	1	figure	figure	VERB
ejpam-3786	297	2	14	14	NUM
ejpam-3786	297	3	:	:	PUNCT
ejpam-3786	297	4	regions	region	NOUN
ejpam-3786	297	5	∆	∆	PROPN
ejpam-3786	297	6	and	and	CCONJ
ejpam-3786	297	7	∆̂	∆̂	AUX
ejpam-3786	297	8	figure	figure	VERB
ejpam-3786	297	9	15	15	NUM
ejpam-3786	297	10	:	:	PUNCT
ejpam-3786	297	11	regions	region	NOUN
ejpam-3786	297	12	∆	∆	PROPN
ejpam-3786	297	13	and	and	CCONJ
ejpam-3786	297	14	∆̂	∆̂	X
ejpam-3786	297	15	by	by	ADP
ejpam-3786	297	16	adding	add	VERB
ejpam-3786	297	17	c(∆	c(∆	NOUN
ejpam-3786	297	18	)	)	PUNCT
ejpam-3786	297	19	to	to	ADP
ejpam-3786	297	20	c(∆̂	c(∆̂	PROPN
ejpam-3786	297	21	)	)	PUNCT
ejpam-3786	297	22	we	we	PRON
ejpam-3786	297	23	get	get	VERB
ejpam-3786	297	24	c(∆	c(∆	NOUN
ejpam-3786	297	25	)	)	PUNCT
ejpam-3786	297	26	≤	≤	NOUN
ejpam-3786	297	27	0	0	NUM
ejpam-3786	297	28	.	.	PUNCT
ejpam-3786	298	1	12	12	NUM
ejpam-3786	298	2	.	.	PUNCT
ejpam-3786	299	1	in	in	ADP
ejpam-3786	299	2	this	this	DET
ejpam-3786	299	3	case	case	NOUN
ejpam-3786	299	4	∆	∆	PROPN
ejpam-3786	299	5	is	be	AUX
ejpam-3786	299	6	given	give	VERB
ejpam-3786	299	7	in	in	ADP
ejpam-3786	299	8	figure	figure	NOUN
ejpam-3786	299	9	16(i	16(i	NUM
ejpam-3786	299	10	)	)	PUNCT
ejpam-3786	299	11	.	.	PUNCT
ejpam-3786	300	1	muhammad	muhammad	PROPN
ejpam-3786	300	2	saeed	saeed	PROPN
ejpam-3786	300	3	akram	akram	PROPN
ejpam-3786	300	4	,	,	PUNCT
ejpam-3786	300	5	maira	maira	PROPN
ejpam-3786	300	6	amjid	amjid	PROPN
ejpam-3786	300	7	,	,	PUNCT
ejpam-3786	300	8	sohail	sohail	PROPN
ejpam-3786	300	9	iqbal	iqbal	PROPN
ejpam-3786	300	10	/	/	PUNCT
ejpam-3786	300	11	eur	eur	PROPN
ejpam-3786	300	12	.	.	PUNCT
ejpam-3786	301	1	j.	j.	PROPN
ejpam-3786	301	2	pure	pure	PROPN
ejpam-3786	301	3	appl	appl	PROPN
ejpam-3786	301	4	.	.	PROPN
ejpam-3786	301	5	math	math	PROPN
ejpam-3786	301	6	,	,	PUNCT
ejpam-3786	301	7	13	13	NUM
ejpam-3786	301	8	(	(	PUNCT
ejpam-3786	301	9	4	4	NUM
ejpam-3786	301	10	)	)	PUNCT
ejpam-3786	301	11	(	(	PUNCT
ejpam-3786	301	12	2020	2020	NUM
ejpam-3786	301	13	)	)	PUNCT
ejpam-3786	301	14	,	,	PUNCT
ejpam-3786	301	15	914	914	NUM
ejpam-3786	301	16	-	-	SYM
ejpam-3786	301	17	938	938	NUM
ejpam-3786	301	18	928	928	NUM
ejpam-3786	301	19	figure	figure	NOUN
ejpam-3786	301	20	16	16	NUM
ejpam-3786	301	21	:	:	PUNCT
ejpam-3786	301	22	region	region	NOUN
ejpam-3786	301	23	∆	∆	PROPN
ejpam-3786	301	24	by	by	ADP
ejpam-3786	301	25	adding	add	VERB
ejpam-3786	301	26	c(∆	c(∆	PROPN
ejpam-3786	301	27	)	)	PUNCT
ejpam-3786	301	28	to	to	ADP
ejpam-3786	301	29	c(∆̂	c(∆̂	PROPN
ejpam-3786	301	30	)	)	PUNCT
ejpam-3786	301	31	we	we	PRON
ejpam-3786	301	32	get	get	VERB
ejpam-3786	301	33	c(∆	c(∆	NOUN
ejpam-3786	301	34	)	)	PUNCT
ejpam-3786	301	35	≤	≤	NOUN
ejpam-3786	301	36	0	0	NUM
ejpam-3786	301	37	.	.	PROPN
ejpam-3786	301	38	13	13	NUM
ejpam-3786	301	39	.	.	PUNCT
ejpam-3786	302	1	in	in	ADP
ejpam-3786	302	2	this	this	DET
ejpam-3786	302	3	case	case	NOUN
ejpam-3786	302	4	c(∆	c(∆	NOUN
ejpam-3786	302	5	)	)	PUNCT
ejpam-3786	302	6	≤	≤	NOUN
ejpam-3786	302	7	0	0	NUM
ejpam-3786	303	1	for	for	ADP
ejpam-3786	303	2	all	all	PRON
ejpam-3786	303	3	of	of	ADP
ejpam-3786	303	4	its	its	PRON
ejpam-3786	303	5	subcases	subcase	NOUN
ejpam-3786	303	6	as	as	SCONJ
ejpam-3786	303	7	shown	show	VERB
ejpam-3786	303	8	in	in	ADP
ejpam-3786	303	9	figure	figure	NOUN
ejpam-3786	303	10	17	17	NUM
ejpam-3786	303	11	.	.	PUNCT
ejpam-3786	304	1	figure	figure	NOUN
ejpam-3786	304	2	17	17	NUM
ejpam-3786	304	3	:	:	PUNCT
ejpam-3786	304	4	region	region	NOUN
ejpam-3786	304	5	∆	∆	PROPN
ejpam-3786	304	6	14	14	NUM
ejpam-3786	304	7	.	.	PUNCT
ejpam-3786	305	1	in	in	ADP
ejpam-3786	305	2	this	this	DET
ejpam-3786	305	3	case	case	NOUN
ejpam-3786	305	4	∆	∆	VERB
ejpam-3786	305	5	as	as	SCONJ
ejpam-3786	305	6	given	give	VERB
ejpam-3786	305	7	in	in	ADP
ejpam-3786	305	8	figure	figure	NOUN
ejpam-3786	305	9	18(i	18(i	NUM
ejpam-3786	305	10	)	)	PUNCT
ejpam-3786	305	11	.	.	PUNCT
ejpam-3786	306	1	muhammad	muhammad	PROPN
ejpam-3786	306	2	saeed	saeed	PROPN
ejpam-3786	306	3	akram	akram	PROPN
ejpam-3786	306	4	,	,	PUNCT
ejpam-3786	306	5	maira	maira	PROPN
ejpam-3786	306	6	amjid	amjid	PROPN
ejpam-3786	306	7	,	,	PUNCT
ejpam-3786	306	8	sohail	sohail	PROPN
ejpam-3786	306	9	iqbal	iqbal	PROPN
ejpam-3786	306	10	/	/	PUNCT
ejpam-3786	306	11	eur	eur	PROPN
ejpam-3786	306	12	.	.	PUNCT
ejpam-3786	307	1	j.	j.	PROPN
ejpam-3786	307	2	pure	pure	PROPN
ejpam-3786	307	3	appl	appl	PROPN
ejpam-3786	307	4	.	.	PROPN
ejpam-3786	307	5	math	math	PROPN
ejpam-3786	307	6	,	,	PUNCT
ejpam-3786	307	7	13	13	NUM
ejpam-3786	307	8	(	(	PUNCT
ejpam-3786	307	9	4	4	NUM
ejpam-3786	307	10	)	)	PUNCT
ejpam-3786	307	11	(	(	PUNCT
ejpam-3786	307	12	2020	2020	NUM
ejpam-3786	307	13	)	)	PUNCT
ejpam-3786	307	14	,	,	PUNCT
ejpam-3786	307	15	914	914	NUM
ejpam-3786	307	16	-	-	SYM
ejpam-3786	307	17	938	938	NUM
ejpam-3786	307	18	929	929	NUM
ejpam-3786	307	19	figure	figure	NOUN
ejpam-3786	307	20	18	18	NUM
ejpam-3786	307	21	:	:	PUNCT
ejpam-3786	307	22	region	region	NOUN
ejpam-3786	307	23	∆	∆	PROPN
ejpam-3786	307	24	by	by	ADP
ejpam-3786	307	25	adding	add	VERB
ejpam-3786	307	26	c(∆	c(∆	PROPN
ejpam-3786	307	27	)	)	PUNCT
ejpam-3786	307	28	to	to	ADP
ejpam-3786	307	29	c(∆̂	c(∆̂	PROPN
ejpam-3786	307	30	)	)	PUNCT
ejpam-3786	307	31	we	we	PRON
ejpam-3786	307	32	get	get	VERB
ejpam-3786	307	33	c(∆	c(∆	NOUN
ejpam-3786	307	34	)	)	PUNCT
ejpam-3786	307	35	≤	≤	NOUN
ejpam-3786	307	36	0	0	NUM
ejpam-3786	307	37	.	.	PUNCT
ejpam-3786	307	38	15	15	NUM
ejpam-3786	307	39	.	.	PUNCT
ejpam-3786	308	1	in	in	ADP
ejpam-3786	308	2	this	this	DET
ejpam-3786	308	3	case	case	NOUN
ejpam-3786	308	4	∆	∆	PROPN
ejpam-3786	308	5	is	be	AUX
ejpam-3786	308	6	given	give	VERB
ejpam-3786	308	7	in	in	ADP
ejpam-3786	308	8	figure	figure	NOUN
ejpam-3786	308	9	19(i	19(i	NUM
ejpam-3786	308	10	)	)	PUNCT
ejpam-3786	308	11	.	.	PUNCT
ejpam-3786	309	1	figure	figure	VERB
ejpam-3786	309	2	19	19	NUM
ejpam-3786	309	3	:	:	PUNCT
ejpam-3786	309	4	regions	region	NOUN
ejpam-3786	309	5	∆	∆	PROPN
ejpam-3786	309	6	and	and	CCONJ
ejpam-3786	309	7	∆̂	∆̂	PUNCT
ejpam-3786	309	8	muhammad	muhammad	PROPN
ejpam-3786	309	9	saeed	saeed	PROPN
ejpam-3786	309	10	akram	akram	PROPN
ejpam-3786	309	11	,	,	PUNCT
ejpam-3786	309	12	maira	maira	PROPN
ejpam-3786	309	13	amjid	amjid	PROPN
ejpam-3786	309	14	,	,	PUNCT
ejpam-3786	309	15	sohail	sohail	PROPN
ejpam-3786	309	16	iqbal	iqbal	PROPN
ejpam-3786	309	17	/	/	PUNCT
ejpam-3786	309	18	eur	eur	PROPN
ejpam-3786	309	19	.	.	PUNCT
ejpam-3786	310	1	j.	j.	PROPN
ejpam-3786	310	2	pure	pure	PROPN
ejpam-3786	310	3	appl	appl	PROPN
ejpam-3786	310	4	.	.	PROPN
ejpam-3786	310	5	math	math	PROPN
ejpam-3786	310	6	,	,	PUNCT
ejpam-3786	310	7	13	13	NUM
ejpam-3786	310	8	(	(	PUNCT
ejpam-3786	310	9	4	4	NUM
ejpam-3786	310	10	)	)	PUNCT
ejpam-3786	310	11	(	(	PUNCT
ejpam-3786	310	12	2020	2020	NUM
ejpam-3786	310	13	)	)	PUNCT
ejpam-3786	310	14	,	,	PUNCT
ejpam-3786	310	15	914	914	NUM
ejpam-3786	310	16	-	-	SYM
ejpam-3786	310	17	938	938	NUM
ejpam-3786	310	18	930	930	NUM
ejpam-3786	310	19	figure	figure	NOUN
ejpam-3786	310	20	20	20	NUM
ejpam-3786	310	21	:	:	PUNCT
ejpam-3786	310	22	regions	region	NOUN
ejpam-3786	310	23	∆	∆	PROPN
ejpam-3786	310	24	and	and	CCONJ
ejpam-3786	310	25	∆̂	∆̂	PUNCT
ejpam-3786	310	26	figure	figure	VERB
ejpam-3786	310	27	21	21	NUM
ejpam-3786	310	28	:	:	PUNCT
ejpam-3786	310	29	regions	region	NOUN
ejpam-3786	310	30	∆	∆	PROPN
ejpam-3786	310	31	and	and	CCONJ
ejpam-3786	310	32	∆̂	∆̂	X
ejpam-3786	310	33	by	by	ADP
ejpam-3786	310	34	adding	add	VERB
ejpam-3786	310	35	c(∆	c(∆	NOUN
ejpam-3786	310	36	)	)	PUNCT
ejpam-3786	310	37	to	to	ADP
ejpam-3786	310	38	c(∆̂	c(∆̂	PROPN
ejpam-3786	310	39	)	)	PUNCT
ejpam-3786	310	40	we	we	PRON
ejpam-3786	310	41	get	get	VERB
ejpam-3786	310	42	c(∆	c(∆	NOUN
ejpam-3786	310	43	)	)	PUNCT
ejpam-3786	310	44	≤	≤	NOUN
ejpam-3786	310	45	0	0	NUM
ejpam-3786	310	46	.	.	PUNCT
ejpam-3786	311	1	muhammad	muhammad	PROPN
ejpam-3786	311	2	saeed	saeed	PROPN
ejpam-3786	311	3	akram	akram	PROPN
ejpam-3786	311	4	,	,	PUNCT
ejpam-3786	311	5	maira	maira	PROPN
ejpam-3786	311	6	amjid	amjid	PROPN
ejpam-3786	311	7	,	,	PUNCT
ejpam-3786	311	8	sohail	sohail	PROPN
ejpam-3786	311	9	iqbal	iqbal	PROPN
ejpam-3786	311	10	/	/	PUNCT
ejpam-3786	311	11	eur	eur	PROPN
ejpam-3786	311	12	.	.	PUNCT
ejpam-3786	312	1	j.	j.	PROPN
ejpam-3786	312	2	pure	pure	PROPN
ejpam-3786	312	3	appl	appl	PROPN
ejpam-3786	312	4	.	.	PROPN
ejpam-3786	312	5	math	math	PROPN
ejpam-3786	312	6	,	,	PUNCT
ejpam-3786	312	7	13	13	NUM
ejpam-3786	312	8	(	(	PUNCT
ejpam-3786	312	9	4	4	NUM
ejpam-3786	312	10	)	)	PUNCT
ejpam-3786	312	11	(	(	PUNCT
ejpam-3786	312	12	2020	2020	NUM
ejpam-3786	312	13	)	)	PUNCT
ejpam-3786	312	14	,	,	PUNCT
ejpam-3786	312	15	914	914	NUM
ejpam-3786	312	16	-	-	SYM
ejpam-3786	312	17	938	938	NUM
ejpam-3786	312	18	931	931	NUM
ejpam-3786	312	19	16	16	NUM
ejpam-3786	312	20	.	.	PUNCT
ejpam-3786	313	1	in	in	ADP
ejpam-3786	313	2	this	this	DET
ejpam-3786	313	3	case	case	NOUN
ejpam-3786	313	4	c(∆	c(∆	NOUN
ejpam-3786	313	5	)	)	PUNCT
ejpam-3786	313	6	≤	≤	NOUN
ejpam-3786	313	7	0	0	NUM
ejpam-3786	314	1	for	for	ADP
ejpam-3786	314	2	all	all	PRON
ejpam-3786	314	3	of	of	ADP
ejpam-3786	314	4	its	its	PRON
ejpam-3786	314	5	subcases	subcase	NOUN
ejpam-3786	314	6	as	as	SCONJ
ejpam-3786	314	7	shown	show	VERB
ejpam-3786	314	8	in	in	ADP
ejpam-3786	314	9	figure	figure	NOUN
ejpam-3786	314	10	22(i	22(i	NUM
ejpam-3786	314	11	)	)	PUNCT
ejpam-3786	314	12	.	.	PUNCT
ejpam-3786	315	1	figure	figure	VERB
ejpam-3786	315	2	22	22	NUM
ejpam-3786	315	3	:	:	PUNCT
ejpam-3786	315	4	region	region	NOUN
ejpam-3786	315	5	∆	∆	PROPN
ejpam-3786	315	6	17	17	NUM
ejpam-3786	315	7	.	.	PUNCT
ejpam-3786	316	1	in	in	ADP
ejpam-3786	316	2	this	this	DET
ejpam-3786	316	3	case	case	NOUN
ejpam-3786	316	4	∆	∆	PROPN
ejpam-3786	316	5	is	be	AUX
ejpam-3786	316	6	given	give	VERB
ejpam-3786	316	7	in	in	ADP
ejpam-3786	316	8	figure	figure	NOUN
ejpam-3786	316	9	23(i	23(i	NUM
ejpam-3786	316	10	)	)	PUNCT
ejpam-3786	316	11	.	.	PUNCT
ejpam-3786	317	1	figure	figure	VERB
ejpam-3786	317	2	23	23	NUM
ejpam-3786	317	3	:	:	PUNCT
ejpam-3786	317	4	region	region	NOUN
ejpam-3786	317	5	∆	∆	PROPN
ejpam-3786	317	6	by	by	ADP
ejpam-3786	317	7	adding	add	VERB
ejpam-3786	317	8	c(∆	c(∆	PROPN
ejpam-3786	317	9	)	)	PUNCT
ejpam-3786	317	10	to	to	ADP
ejpam-3786	317	11	c(∆̂	c(∆̂	PROPN
ejpam-3786	317	12	)	)	PUNCT
ejpam-3786	317	13	we	we	PRON
ejpam-3786	317	14	get	get	VERB
ejpam-3786	317	15	c(∆	c(∆	NOUN
ejpam-3786	317	16	)	)	PUNCT
ejpam-3786	317	17	≤	≤	NOUN
ejpam-3786	317	18	0	0	NUM
ejpam-3786	317	19	.	.	PROPN
ejpam-3786	317	20	18	18	NUM
ejpam-3786	317	21	.	.	PUNCT
ejpam-3786	318	1	in	in	ADP
ejpam-3786	318	2	this	this	DET
ejpam-3786	318	3	case	case	NOUN
ejpam-3786	318	4	∆	∆	PROPN
ejpam-3786	318	5	is	be	AUX
ejpam-3786	318	6	given	give	VERB
ejpam-3786	318	7	in	in	ADP
ejpam-3786	318	8	figure	figure	NOUN
ejpam-3786	318	9	24(i	24(i	NUM
ejpam-3786	318	10	)	)	PUNCT
ejpam-3786	318	11	.	.	PUNCT
ejpam-3786	319	1	muhammad	muhammad	PROPN
ejpam-3786	319	2	saeed	saeed	PROPN
ejpam-3786	319	3	akram	akram	PROPN
ejpam-3786	319	4	,	,	PUNCT
ejpam-3786	319	5	maira	maira	PROPN
ejpam-3786	319	6	amjid	amjid	PROPN
ejpam-3786	319	7	,	,	PUNCT
ejpam-3786	319	8	sohail	sohail	PROPN
ejpam-3786	319	9	iqbal	iqbal	PROPN
ejpam-3786	319	10	/	/	PUNCT
ejpam-3786	319	11	eur	eur	PROPN
ejpam-3786	319	12	.	.	PUNCT
ejpam-3786	320	1	j.	j.	PROPN
ejpam-3786	320	2	pure	pure	PROPN
ejpam-3786	320	3	appl	appl	PROPN
ejpam-3786	320	4	.	.	PROPN
ejpam-3786	320	5	math	math	PROPN
ejpam-3786	320	6	,	,	PUNCT
ejpam-3786	320	7	13	13	NUM
ejpam-3786	320	8	(	(	PUNCT
ejpam-3786	320	9	4	4	NUM
ejpam-3786	320	10	)	)	PUNCT
ejpam-3786	320	11	(	(	PUNCT
ejpam-3786	320	12	2020	2020	NUM
ejpam-3786	320	13	)	)	PUNCT
ejpam-3786	320	14	,	,	PUNCT
ejpam-3786	320	15	914	914	NUM
ejpam-3786	320	16	-	-	SYM
ejpam-3786	320	17	938	938	NUM
ejpam-3786	320	18	932	932	NUM
ejpam-3786	320	19	figure	figure	NOUN
ejpam-3786	320	20	24	24	NUM
ejpam-3786	320	21	:	:	PUNCT
ejpam-3786	320	22	region	region	NOUN
ejpam-3786	320	23	∆	∆	PROPN
ejpam-3786	320	24	figure	figure	NOUN
ejpam-3786	320	25	25	25	NUM
ejpam-3786	320	26	:	:	PUNCT
ejpam-3786	320	27	region	region	NOUN
ejpam-3786	320	28	∆	∆	PROPN
ejpam-3786	320	29	muhammad	muhammad	PROPN
ejpam-3786	320	30	saeed	saeed	PROPN
ejpam-3786	320	31	akram	akram	PROPN
ejpam-3786	320	32	,	,	PUNCT
ejpam-3786	320	33	maira	maira	PROPN
ejpam-3786	320	34	amjid	amjid	PROPN
ejpam-3786	320	35	,	,	PUNCT
ejpam-3786	320	36	sohail	sohail	PROPN
ejpam-3786	320	37	iqbal	iqbal	PROPN
ejpam-3786	320	38	/	/	PUNCT
ejpam-3786	320	39	eur	eur	PROPN
ejpam-3786	320	40	.	.	PUNCT
ejpam-3786	321	1	j.	j.	PROPN
ejpam-3786	321	2	pure	pure	PROPN
ejpam-3786	321	3	appl	appl	PROPN
ejpam-3786	321	4	.	.	PROPN
ejpam-3786	321	5	math	math	PROPN
ejpam-3786	321	6	,	,	PUNCT
ejpam-3786	321	7	13	13	NUM
ejpam-3786	321	8	(	(	PUNCT
ejpam-3786	321	9	4	4	NUM
ejpam-3786	321	10	)	)	PUNCT
ejpam-3786	321	11	(	(	PUNCT
ejpam-3786	321	12	2020	2020	NUM
ejpam-3786	321	13	)	)	PUNCT
ejpam-3786	321	14	,	,	PUNCT
ejpam-3786	321	15	914	914	NUM
ejpam-3786	321	16	-	-	SYM
ejpam-3786	321	17	938	938	NUM
ejpam-3786	321	18	933	933	NUM
ejpam-3786	321	19	figure	figure	NOUN
ejpam-3786	321	20	26	26	NUM
ejpam-3786	321	21	:	:	PUNCT
ejpam-3786	321	22	region	region	NOUN
ejpam-3786	321	23	∆	∆	PROPN
ejpam-3786	321	24	figure	figure	NOUN
ejpam-3786	321	25	27	27	NUM
ejpam-3786	321	26	:	:	PUNCT
ejpam-3786	321	27	region	region	NOUN
ejpam-3786	321	28	∆	∆	PROPN
ejpam-3786	321	29	by	by	ADP
ejpam-3786	321	30	adding	add	VERB
ejpam-3786	321	31	c(∆	c(∆	PROPN
ejpam-3786	321	32	)	)	PUNCT
ejpam-3786	321	33	to	to	ADP
ejpam-3786	321	34	c(∆̂	c(∆̂	PROPN
ejpam-3786	321	35	)	)	PUNCT
ejpam-3786	321	36	we	we	PRON
ejpam-3786	321	37	get	get	VERB
ejpam-3786	321	38	c(∆	c(∆	NOUN
ejpam-3786	321	39	)	)	PUNCT
ejpam-3786	321	40	≤	≤	NOUN
ejpam-3786	321	41	0	0	NUM
ejpam-3786	321	42	.	.	PUNCT
ejpam-3786	322	1	muhammad	muhammad	PROPN
ejpam-3786	322	2	saeed	saeed	PROPN
ejpam-3786	322	3	akram	akram	PROPN
ejpam-3786	322	4	,	,	PUNCT
ejpam-3786	322	5	maira	maira	PROPN
ejpam-3786	322	6	amjid	amjid	PROPN
ejpam-3786	322	7	,	,	PUNCT
ejpam-3786	322	8	sohail	sohail	PROPN
ejpam-3786	322	9	iqbal	iqbal	PROPN
ejpam-3786	322	10	/	/	PUNCT
ejpam-3786	322	11	eur	eur	PROPN
ejpam-3786	322	12	.	.	PUNCT
ejpam-3786	323	1	j.	j.	PROPN
ejpam-3786	323	2	pure	pure	PROPN
ejpam-3786	323	3	appl	appl	PROPN
ejpam-3786	323	4	.	.	PROPN
ejpam-3786	323	5	math	math	PROPN
ejpam-3786	323	6	,	,	PUNCT
ejpam-3786	323	7	13	13	NUM
ejpam-3786	323	8	(	(	PUNCT
ejpam-3786	323	9	4	4	NUM
ejpam-3786	323	10	)	)	PUNCT
ejpam-3786	323	11	(	(	PUNCT
ejpam-3786	323	12	2020	2020	NUM
ejpam-3786	323	13	)	)	PUNCT
ejpam-3786	323	14	,	,	PUNCT
ejpam-3786	323	15	914	914	NUM
ejpam-3786	323	16	-	-	SYM
ejpam-3786	323	17	938	938	NUM
ejpam-3786	323	18	934	934	NUM
ejpam-3786	323	19	19	19	NUM
ejpam-3786	323	20	.	.	PUNCT
ejpam-3786	324	1	in	in	ADP
ejpam-3786	324	2	this	this	DET
ejpam-3786	324	3	case	case	NOUN
ejpam-3786	324	4	∆	∆	PROPN
ejpam-3786	324	5	is	be	AUX
ejpam-3786	324	6	given	give	VERB
ejpam-3786	324	7	in	in	ADP
ejpam-3786	324	8	figure	figure	NOUN
ejpam-3786	324	9	28(i	28(i	NUM
ejpam-3786	324	10	)	)	PUNCT
ejpam-3786	324	11	.	.	PUNCT
ejpam-3786	325	1	figure	figure	VERB
ejpam-3786	325	2	28	28	NUM
ejpam-3786	325	3	:	:	PUNCT
ejpam-3786	325	4	region	region	NOUN
ejpam-3786	325	5	∆	∆	PROPN
ejpam-3786	325	6	figure	figure	NOUN
ejpam-3786	325	7	29	29	NUM
ejpam-3786	325	8	:	:	PUNCT
ejpam-3786	325	9	region	region	NOUN
ejpam-3786	325	10	∆	∆	PROPN
ejpam-3786	325	11	muhammad	muhammad	PROPN
ejpam-3786	325	12	saeed	saeed	PROPN
ejpam-3786	325	13	akram	akram	PROPN
ejpam-3786	325	14	,	,	PUNCT
ejpam-3786	325	15	maira	maira	PROPN
ejpam-3786	325	16	amjid	amjid	PROPN
ejpam-3786	325	17	,	,	PUNCT
ejpam-3786	325	18	sohail	sohail	PROPN
ejpam-3786	325	19	iqbal	iqbal	PROPN
ejpam-3786	325	20	/	/	PUNCT
ejpam-3786	325	21	eur	eur	PROPN
ejpam-3786	325	22	.	.	PUNCT
ejpam-3786	326	1	j.	j.	PROPN
ejpam-3786	326	2	pure	pure	PROPN
ejpam-3786	326	3	appl	appl	PROPN
ejpam-3786	326	4	.	.	PROPN
ejpam-3786	326	5	math	math	PROPN
ejpam-3786	326	6	,	,	PUNCT
ejpam-3786	326	7	13	13	NUM
ejpam-3786	326	8	(	(	PUNCT
ejpam-3786	326	9	4	4	NUM
ejpam-3786	326	10	)	)	PUNCT
ejpam-3786	326	11	(	(	PUNCT
ejpam-3786	326	12	2020	2020	NUM
ejpam-3786	326	13	)	)	PUNCT
ejpam-3786	326	14	,	,	PUNCT
ejpam-3786	326	15	914	914	NUM
ejpam-3786	326	16	-	-	SYM
ejpam-3786	326	17	938	938	NUM
ejpam-3786	326	18	935	935	NUM
ejpam-3786	326	19	figure	figure	NOUN
ejpam-3786	326	20	30	30	NUM
ejpam-3786	326	21	:	:	PUNCT
ejpam-3786	326	22	region	region	NOUN
ejpam-3786	326	23	∆	∆	PROPN
ejpam-3786	326	24	figure	figure	NOUN
ejpam-3786	326	25	31	31	NUM
ejpam-3786	326	26	:	:	PUNCT
ejpam-3786	326	27	region	region	NOUN
ejpam-3786	326	28	∆	∆	PROPN
ejpam-3786	326	29	muhammad	muhammad	PROPN
ejpam-3786	326	30	saeed	saeed	PROPN
ejpam-3786	326	31	akram	akram	PROPN
ejpam-3786	326	32	,	,	PUNCT
ejpam-3786	326	33	maira	maira	PROPN
ejpam-3786	326	34	amjid	amjid	PROPN
ejpam-3786	326	35	,	,	PUNCT
ejpam-3786	326	36	sohail	sohail	PROPN
ejpam-3786	326	37	iqbal	iqbal	PROPN
ejpam-3786	326	38	/	/	PUNCT
ejpam-3786	326	39	eur	eur	PROPN
ejpam-3786	326	40	.	.	PUNCT
ejpam-3786	327	1	j.	j.	PROPN
ejpam-3786	327	2	pure	pure	PROPN
ejpam-3786	327	3	appl	appl	PROPN
ejpam-3786	327	4	.	.	PROPN
ejpam-3786	327	5	math	math	PROPN
ejpam-3786	327	6	,	,	PUNCT
ejpam-3786	327	7	13	13	NUM
ejpam-3786	327	8	(	(	PUNCT
ejpam-3786	327	9	4	4	NUM
ejpam-3786	327	10	)	)	PUNCT
ejpam-3786	327	11	(	(	PUNCT
ejpam-3786	327	12	2020	2020	NUM
ejpam-3786	327	13	)	)	PUNCT
ejpam-3786	327	14	,	,	PUNCT
ejpam-3786	327	15	914	914	NUM
ejpam-3786	327	16	-	-	SYM
ejpam-3786	327	17	938	938	NUM
ejpam-3786	327	18	936	936	NUM
ejpam-3786	327	19	figure	figure	NOUN
ejpam-3786	327	20	32	32	NUM
ejpam-3786	327	21	:	:	PUNCT
ejpam-3786	327	22	region	region	NOUN
ejpam-3786	327	23	∆	∆	PROPN
ejpam-3786	327	24	figure	figure	NOUN
ejpam-3786	327	25	33	33	NUM
ejpam-3786	327	26	:	:	PUNCT
ejpam-3786	327	27	region	region	NOUN
ejpam-3786	327	28	∆	∆	PROPN
ejpam-3786	327	29	references	reference	NOUN
ejpam-3786	327	30	937	937	NUM
ejpam-3786	327	31	figure	figure	NOUN
ejpam-3786	327	32	34	34	NUM
ejpam-3786	327	33	:	:	PUNCT
ejpam-3786	327	34	region	region	NOUN
ejpam-3786	327	35	∆	∆	PROPN
ejpam-3786	327	36	by	by	ADP
ejpam-3786	327	37	adding	add	VERB
ejpam-3786	327	38	c(∆	c(∆	PROPN
ejpam-3786	327	39	)	)	PUNCT
ejpam-3786	327	40	to	to	ADP
ejpam-3786	327	41	c(∆̂	c(∆̂	PROPN
ejpam-3786	327	42	)	)	PUNCT
ejpam-3786	327	43	we	we	PRON
ejpam-3786	327	44	get	get	VERB
ejpam-3786	327	45	c(∆	c(∆	NOUN
ejpam-3786	327	46	)	)	PUNCT
ejpam-3786	327	47	≤	≤	NOUN
ejpam-3786	327	48	0	0	NUM
ejpam-3786	327	49	.	.	PUNCT
ejpam-3786	328	1	there	there	PRON
ejpam-3786	328	2	remains	remain	VERB
ejpam-3786	328	3	only	only	ADV
ejpam-3786	328	4	2	2	NUM
ejpam-3786	328	5	cases	case	NOUN
ejpam-3786	328	6	given	give	VERB
ejpam-3786	328	7	in	in	ADP
ejpam-3786	328	8	theorem	theorem	NOUN
ejpam-3786	328	9	that	that	PRON
ejpam-3786	328	10	still	still	ADV
ejpam-3786	328	11	needs	need	VERB
ejpam-3786	328	12	to	to	PART
ejpam-3786	328	13	be	be	AUX
ejpam-3786	328	14	solved	solve	VERB
ejpam-3786	328	15	.	.	PUNCT
ejpam-3786	329	1	in	in	ADP
ejpam-3786	329	2	fact	fact	NOUN
ejpam-3786	329	3	,	,	PUNCT
ejpam-3786	329	4	weight	weight	NOUN
ejpam-3786	329	5	test	test	NOUN
ejpam-3786	329	6	and	and	CCONJ
ejpam-3786	329	7	curvature	curvature	NOUN
ejpam-3786	329	8	distribution	distribution	NOUN
ejpam-3786	329	9	method	method	NOUN
ejpam-3786	329	10	can	can	AUX
ejpam-3786	329	11	not	not	PART
ejpam-3786	329	12	be	be	AUX
ejpam-3786	329	13	applied	apply	VERB
ejpam-3786	329	14	to	to	ADP
ejpam-3786	329	15	these	these	DET
ejpam-3786	329	16	cases	case	NOUN
ejpam-3786	329	17	.	.	PUNCT
ejpam-3786	330	1	the	the	DET
ejpam-3786	330	2	weight	weight	NOUN
ejpam-3786	330	3	test	test	NOUN
ejpam-3786	330	4	can	can	AUX
ejpam-3786	330	5	not	not	PART
ejpam-3786	330	6	be	be	AUX
ejpam-3786	330	7	applied	apply	VERB
ejpam-3786	330	8	to	to	ADP
ejpam-3786	330	9	these	these	DET
ejpam-3786	330	10	cases	case	NOUN
ejpam-3786	330	11	as	as	SCONJ
ejpam-3786	330	12	it	it	PRON
ejpam-3786	330	13	is	be	AUX
ejpam-3786	330	14	not	not	PART
ejpam-3786	330	15	possible	possible	ADJ
ejpam-3786	330	16	to	to	PART
ejpam-3786	330	17	find	find	VERB
ejpam-3786	330	18	a	a	DET
ejpam-3786	330	19	weight	weight	NOUN
ejpam-3786	330	20	function	function	NOUN
ejpam-3786	330	21	θ	θ	PROPN
ejpam-3786	330	22	that	that	PRON
ejpam-3786	330	23	satisfies	satisfy	VERB
ejpam-3786	330	24	all	all	DET
ejpam-3786	330	25	the	the	DET
ejpam-3786	330	26	three	three	NUM
ejpam-3786	330	27	conditions	condition	NOUN
ejpam-3786	330	28	of	of	ADP
ejpam-3786	330	29	weight	weight	NOUN
ejpam-3786	330	30	test	test	NOUN
ejpam-3786	330	31	simultaneously	simultaneously	ADV
ejpam-3786	330	32	,	,	PUNCT
ejpam-3786	330	33	whereas	whereas	SCONJ
ejpam-3786	330	34	curvature	curvature	NOUN
ejpam-3786	330	35	test	test	NOUN
ejpam-3786	330	36	can	can	AUX
ejpam-3786	330	37	not	not	PART
ejpam-3786	330	38	be	be	AUX
ejpam-3786	330	39	applied	apply	VERB
ejpam-3786	330	40	to	to	ADP
ejpam-3786	330	41	these	these	DET
ejpam-3786	330	42	cases	case	NOUN
ejpam-3786	330	43	as	as	SCONJ
ejpam-3786	330	44	it	it	PRON
ejpam-3786	330	45	is	be	AUX
ejpam-3786	330	46	impossible	impossible	ADJ
ejpam-3786	330	47	to	to	PART
ejpam-3786	330	48	find	find	VERB
ejpam-3786	330	49	any	any	DET
ejpam-3786	330	50	neighbouring	neighbouring	NOUN
ejpam-3786	330	51	region	region	NOUN
ejpam-3786	330	52	∆̂	∆̂	NOUN
ejpam-3786	330	53	in	in	ADP
ejpam-3786	330	54	the	the	DET
ejpam-3786	330	55	neighbourhood	neighbourhood	NOUN
ejpam-3786	330	56	of	of	ADP
ejpam-3786	330	57	region	region	NOUN
ejpam-3786	330	58	∆	∆	PROPN
ejpam-3786	330	59	that	that	PRON
ejpam-3786	330	60	cancels	cancel	VERB
ejpam-3786	330	61	the	the	DET
ejpam-3786	330	62	curvatures	curvature	NOUN
ejpam-3786	330	63	of	of	ADP
ejpam-3786	330	64	region	region	NOUN
ejpam-3786	330	65	∆.	∆.	NOUN
ejpam-3786	330	66	remark	remark	NOUN
ejpam-3786	331	1	2	2	X
ejpam-3786	331	2	.	.	PUNCT
ejpam-3786	332	1	we	we	PRON
ejpam-3786	332	2	remark	remark	VERB
ejpam-3786	332	3	that	that	SCONJ
ejpam-3786	332	4	the	the	DET
ejpam-3786	332	5	list	list	NOUN
ejpam-3786	332	6	of	of	ADP
ejpam-3786	332	7	exceptional	exceptional	ADJ
ejpam-3786	332	8	cases	case	NOUN
ejpam-3786	332	9	given	give	VERB
ejpam-3786	332	10	in	in	ADP
ejpam-3786	332	11	theorem	theorem	NOUN
ejpam-3786	332	12	in	in	ADP
ejpam-3786	332	13	section	section	NOUN
ejpam-3786	332	14	1	1	NUM
ejpam-3786	332	15	is	be	AUX
ejpam-3786	332	16	open	open	ADJ
ejpam-3786	332	17	for	for	ADP
ejpam-3786	332	18	the	the	DET
ejpam-3786	332	19	equation	equation	NOUN
ejpam-3786	332	20	s1(t	s1(t	NUM
ejpam-3786	332	21	)	)	PUNCT
ejpam-3786	332	22	.	.	PUNCT
ejpam-3786	333	1	since	since	SCONJ
ejpam-3786	333	2	weight	weight	NOUN
ejpam-3786	333	3	test	test	NOUN
ejpam-3786	333	4	and	and	CCONJ
ejpam-3786	333	5	curvature	curvature	NOUN
ejpam-3786	333	6	distribution	distribution	NOUN
ejpam-3786	333	7	can	can	AUX
ejpam-3786	333	8	not	not	PART
ejpam-3786	333	9	be	be	AUX
ejpam-3786	333	10	applied	apply	VERB
ejpam-3786	333	11	to	to	ADP
ejpam-3786	333	12	these	these	DET
ejpam-3786	333	13	cases	case	NOUN
ejpam-3786	333	14	to	to	PART
ejpam-3786	333	15	prove	prove	VERB
ejpam-3786	333	16	levin	levin	PROPN
ejpam-3786	333	17	conjecture	conjecture	NOUN
ejpam-3786	333	18	,	,	PUNCT
ejpam-3786	333	19	therefore	therefore	ADV
ejpam-3786	333	20	,	,	PUNCT
ejpam-3786	333	21	some	some	DET
ejpam-3786	333	22	new	new	ADJ
ejpam-3786	333	23	methods	method	NOUN
ejpam-3786	333	24	needs	need	VERB
ejpam-3786	333	25	to	to	PART
ejpam-3786	333	26	be	be	AUX
ejpam-3786	333	27	developed	develop	VERB
ejpam-3786	333	28	to	to	PART
ejpam-3786	333	29	establish	establish	VERB
ejpam-3786	333	30	the	the	DET
ejpam-3786	333	31	validity	validity	NOUN
ejpam-3786	333	32	of	of	ADP
ejpam-3786	333	33	the	the	DET
ejpam-3786	333	34	remaining	remain	VERB
ejpam-3786	333	35	cases	case	NOUN
ejpam-3786	333	36	of	of	ADP
ejpam-3786	333	37	levin	levin	PROPN
ejpam-3786	333	38	conjecture	conjecture	NOUN
ejpam-3786	333	39	for	for	ADP
ejpam-3786	333	40	these	these	DET
ejpam-3786	333	41	group	group	NOUN
ejpam-3786	333	42	equations	equation	NOUN
ejpam-3786	333	43	of	of	ADP
ejpam-3786	333	44	length	length	NOUN
ejpam-3786	333	45	9	9	NUM
ejpam-3786	333	46	.	.	PUNCT
ejpam-3786	334	1	references	reference	NOUN
ejpam-3786	334	2	[	[	X
ejpam-3786	334	3	1	1	X
ejpam-3786	334	4	]	]	PUNCT
ejpam-3786	334	5	muhammad	muhammad	PROPN
ejpam-3786	334	6	saeed	saeed	PROPN
ejpam-3786	334	7	akram	akram	PROPN
ejpam-3786	334	8	and	and	CCONJ
ejpam-3786	334	9	maira	maira	PROPN
ejpam-3786	334	10	amjid	amjid	PROPN
ejpam-3786	334	11	.	.	PUNCT
ejpam-3786	335	1	solving	solve	VERB
ejpam-3786	335	2	a	a	DET
ejpam-3786	335	3	group	group	NOUN
ejpam-3786	335	4	equation	equation	NOUN
ejpam-3786	335	5	of	of	ADP
ejpam-3786	335	6	length	length	NOUN
ejpam-3786	335	7	nine	nine	NUM
ejpam-3786	335	8	.	.	PUNCT
ejpam-3786	336	1	arxiv	arxiv	PROPN
ejpam-3786	336	2	preprint	preprint	NOUN
ejpam-3786	336	3	arxiv:2008.09508	arxiv:2008.09508	PROPN
ejpam-3786	336	4	,	,	PUNCT
ejpam-3786	336	5	2020	2020	NUM
ejpam-3786	336	6	.	.	PUNCT
ejpam-3786	337	1	[	[	X
ejpam-3786	337	2	2	2	X
ejpam-3786	337	3	]	]	PUNCT
ejpam-3786	337	4	muhammad	muhammad	PROPN
ejpam-3786	337	5	saeed	saeed	PROPN
ejpam-3786	337	6	akram	akram	PROPN
ejpam-3786	337	7	and	and	CCONJ
ejpam-3786	337	8	khawar	khawar	PROPN
ejpam-3786	337	9	hussain	hussain	PROPN
ejpam-3786	337	10	.	.	PUNCT
ejpam-3786	338	1	levin	levin	PROPN
ejpam-3786	338	2	’s	’s	PART
ejpam-3786	338	3	conjecture	conjecture	NOUN
ejpam-3786	338	4	for	for	ADP
ejpam-3786	338	5	an	an	DET
ejpam-3786	338	6	equation	equation	NOUN
ejpam-3786	338	7	of	of	ADP
ejpam-3786	338	8	length	length	NOUN
ejpam-3786	338	9	nine	nine	NUM
ejpam-3786	338	10	.	.	PUNCT
ejpam-3786	339	1	arxiv	arxiv	PROPN
ejpam-3786	339	2	preprint	preprint	NOUN
ejpam-3786	339	3	arxiv:2008.06846	arxiv:2008.06846	PROPN
ejpam-3786	339	4	,	,	PUNCT
ejpam-3786	339	5	2020	2020	NUM
ejpam-3786	339	6	.	.	PUNCT
ejpam-3786	340	1	[	[	X
ejpam-3786	340	2	3	3	NUM
ejpam-3786	340	3	]	]	X
ejpam-3786	340	4	m	m	VERB
ejpam-3786	340	5	fazeel	fazeel	PROPN
ejpam-3786	340	6	anwar	anwar	PROPN
ejpam-3786	340	7	,	,	PUNCT
ejpam-3786	340	8	mairaj	mairaj	ADJ
ejpam-3786	340	9	bibi	bibi	NOUN
ejpam-3786	340	10	,	,	PUNCT
ejpam-3786	340	11	and	and	CCONJ
ejpam-3786	340	12	muhammad	muhammad	PROPN
ejpam-3786	340	13	saeed	saeed	PROPN
ejpam-3786	340	14	akram	akram	PROPN
ejpam-3786	340	15	.	.	PUNCT
ejpam-3786	341	1	on	on	ADP
ejpam-3786	341	2	a	a	DET
ejpam-3786	341	3	nonsingular	nonsingular	ADJ
ejpam-3786	341	4	equation	equation	NOUN
ejpam-3786	341	5	of	of	ADP
ejpam-3786	341	6	length	length	NOUN
ejpam-3786	341	7	9	9	NUM
ejpam-3786	341	8	over	over	ADP
ejpam-3786	341	9	torsion	torsion	NOUN
ejpam-3786	341	10	free	free	ADJ
ejpam-3786	341	11	groups	group	NOUN
ejpam-3786	341	12	.	.	PUNCT
ejpam-3786	342	1	european	european	ADJ
ejpam-3786	342	2	journal	journal	PROPN
ejpam-3786	342	3	of	of	ADP
ejpam-3786	342	4	pure	pure	ADJ
ejpam-3786	342	5	and	and	CCONJ
ejpam-3786	342	6	applied	applied	ADJ
ejpam-3786	342	7	mathematics	mathematic	NOUN
ejpam-3786	342	8	,	,	PUNCT
ejpam-3786	342	9	12(2):590–604	12(2):590–604	NUM
ejpam-3786	342	10	,	,	PUNCT
ejpam-3786	342	11	2019	2019	NUM
ejpam-3786	342	12	.	.	PUNCT
ejpam-3786	343	1	references	reference	NOUN
ejpam-3786	343	2	938	938	NUM
ejpam-3786	344	1	[	[	X
ejpam-3786	344	2	4	4	X
ejpam-3786	344	3	]	]	PUNCT
ejpam-3786	344	4	muhammad	muhammad	PROPN
ejpam-3786	344	5	fazeel	fazeel	PROPN
ejpam-3786	344	6	anwar	anwar	PROPN
ejpam-3786	344	7	,	,	PUNCT
ejpam-3786	344	8	mairaj	mairaj	ADJ
ejpam-3786	344	9	bibi	bibi	NOUN
ejpam-3786	344	10	,	,	PUNCT
ejpam-3786	344	11	and	and	CCONJ
ejpam-3786	344	12	muhammad	muhammad	PROPN
ejpam-3786	344	13	saeed	saeed	PROPN
ejpam-3786	344	14	akram	akram	PROPN
ejpam-3786	344	15	.	.	PUNCT
ejpam-3786	345	1	on	on	ADP
ejpam-3786	345	2	solvability	solvability	NOUN
ejpam-3786	345	3	of	of	ADP
ejpam-3786	345	4	certain	certain	ADJ
ejpam-3786	345	5	equations	equation	NOUN
ejpam-3786	345	6	of	of	ADP
ejpam-3786	345	7	arbitrary	arbitrary	ADJ
ejpam-3786	345	8	length	length	NOUN
ejpam-3786	345	9	over	over	ADP
ejpam-3786	345	10	torsion	torsion	NOUN
ejpam-3786	345	11	-	-	PUNCT
ejpam-3786	345	12	free	free	ADJ
ejpam-3786	345	13	group	group	NOUN
ejpam-3786	345	14	.	.	PUNCT
ejpam-3786	346	1	preprint	preprint	NOUN
ejpam-3786	346	2	,	,	PUNCT
ejpam-3786	346	3	2018	2018	NUM
ejpam-3786	346	4	.	.	PUNCT
ejpam-3786	347	1	[	[	X
ejpam-3786	347	2	5	5	X
ejpam-3786	347	3	]	]	PUNCT
ejpam-3786	347	4	muhammad	muhammad	PROPN
ejpam-3786	347	5	fazeel	fazeel	PROPN
ejpam-3786	347	6	anwar	anwar	PROPN
ejpam-3786	347	7	,	,	PUNCT
ejpam-3786	347	8	mairaj	mairaj	ADJ
ejpam-3786	347	9	bibi	bibi	NOUN
ejpam-3786	347	10	,	,	PUNCT
ejpam-3786	347	11	and	and	CCONJ
ejpam-3786	347	12	sohail	sohail	PROPN
ejpam-3786	347	13	iqbal	iqbal	PROPN
ejpam-3786	347	14	.	.	PUNCT
ejpam-3786	348	1	on	on	ADP
ejpam-3786	348	2	certain	certain	ADJ
ejpam-3786	348	3	equations	equation	NOUN
ejpam-3786	348	4	of	of	ADP
ejpam-3786	348	5	arbitrary	arbitrary	ADJ
ejpam-3786	348	6	length	length	NOUN
ejpam-3786	348	7	over	over	ADP
ejpam-3786	348	8	torsion	torsion	NOUN
ejpam-3786	348	9	-	-	PUNCT
ejpam-3786	348	10	free	free	ADJ
ejpam-3786	348	11	groups	group	NOUN
ejpam-3786	348	12	.	.	PUNCT
ejpam-3786	349	1	preprint	preprint	NOUN
ejpam-3786	349	2	,	,	PUNCT
ejpam-3786	349	3	2019	2019	NUM
ejpam-3786	349	4	.	.	PUNCT
ejpam-3786	350	1	[	[	X
ejpam-3786	350	2	6	6	NUM
ejpam-3786	350	3	]	]	PUNCT
ejpam-3786	350	4	mairaj	mairaj	ADJ
ejpam-3786	350	5	bibi	bibi	NOUN
ejpam-3786	350	6	.	.	PUNCT
ejpam-3786	351	1	equations	equation	NOUN
ejpam-3786	351	2	of	of	ADP
ejpam-3786	351	3	length	length	NOUN
ejpam-3786	351	4	seven	seven	NUM
ejpam-3786	351	5	over	over	ADP
ejpam-3786	351	6	torsion	torsion	NOUN
ejpam-3786	351	7	free	free	ADJ
ejpam-3786	351	8	groups	group	NOUN
ejpam-3786	351	9	.	.	PUNCT
ejpam-3786	352	1	phd	phd	NOUN
ejpam-3786	352	2	thesis	thesis	PROPN
ejpam-3786	352	3	,	,	PUNCT
ejpam-3786	352	4	phd	phd	NOUN
ejpam-3786	352	5	thesis	thesis	NOUN
ejpam-3786	352	6	,	,	PUNCT
ejpam-3786	352	7	university	university	NOUN
ejpam-3786	352	8	of	of	ADP
ejpam-3786	352	9	notingham	notingham	NOUN
ejpam-3786	352	10	,	,	PUNCT
ejpam-3786	352	11	2015	2015	NUM
ejpam-3786	352	12	.	.	PUNCT
ejpam-3786	353	1	[	[	X
ejpam-3786	353	2	7	7	X
ejpam-3786	353	3	]	]	X
ejpam-3786	353	4	mairaj	mairaj	ADJ
ejpam-3786	353	5	bibi	bibi	NOUN
ejpam-3786	353	6	,	,	PUNCT
ejpam-3786	353	7	muhammad	muhammad	PROPN
ejpam-3786	353	8	fazee	fazee	PROPN
ejpam-3786	353	9	anwar	anwar	PROPN
ejpam-3786	353	10	,	,	PUNCT
ejpam-3786	353	11	sohail	sohail	PROPN
ejpam-3786	353	12	iqbal	iqbal	PROPN
ejpam-3786	353	13	,	,	PUNCT
ejpam-3786	353	14	and	and	CCONJ
ejpam-3786	353	15	muhammad	muhammad	PROPN
ejpam-3786	353	16	saeed	saeed	PROPN
ejpam-3786	353	17	akram	akram	PROPN
ejpam-3786	353	18	.	.	PUNCT
ejpam-3786	354	1	solution	solution	NOUN
ejpam-3786	354	2	of	of	ADP
ejpam-3786	354	3	a	a	DET
ejpam-3786	354	4	non	non	ADJ
ejpam-3786	354	5	-	-	ADJ
ejpam-3786	354	6	singular	singular	ADJ
ejpam-3786	354	7	equation	equation	NOUN
ejpam-3786	354	8	of	of	ADP
ejpam-3786	354	9	length	length	NOUN
ejpam-3786	354	10	8	8	NUM
ejpam-3786	354	11	over	over	ADP
ejpam-3786	354	12	torsion	torsion	NOUN
ejpam-3786	354	13	free	free	ADJ
ejpam-3786	354	14	groups	group	NOUN
ejpam-3786	354	15	.	.	PUNCT
ejpam-3786	355	1	preprint	preprint	NOUN
ejpam-3786	355	2	,	,	PUNCT
ejpam-3786	355	3	2019	2019	NUM
ejpam-3786	355	4	.	.	PUNCT
ejpam-3786	356	1	[	[	X
ejpam-3786	356	2	8	8	NUM
ejpam-3786	356	3	]	]	X
ejpam-3786	356	4	mairaj	mairaj	ADJ
ejpam-3786	356	5	bibi	bibi	NOUN
ejpam-3786	356	6	and	and	CCONJ
ejpam-3786	356	7	martin	martin	PROPN
ejpam-3786	356	8	edjvet	edjvet	NOUN
ejpam-3786	356	9	.	.	PUNCT
ejpam-3786	357	1	solving	solve	VERB
ejpam-3786	357	2	equations	equation	NOUN
ejpam-3786	357	3	of	of	ADP
ejpam-3786	357	4	length	length	NOUN
ejpam-3786	357	5	seven	seven	NUM
ejpam-3786	357	6	over	over	ADP
ejpam-3786	357	7	torsion	torsion	NOUN
ejpam-3786	357	8	-	-	PUNCT
ejpam-3786	357	9	free	free	ADJ
ejpam-3786	357	10	groups	group	NOUN
ejpam-3786	357	11	.	.	PUNCT
ejpam-3786	358	1	journal	journal	PROPN
ejpam-3786	358	2	of	of	ADP
ejpam-3786	358	3	group	group	PROPN
ejpam-3786	358	4	theory	theory	NOUN
ejpam-3786	358	5	,	,	PUNCT
ejpam-3786	358	6	21(1):147–164	21(1):147–164	NOUN
ejpam-3786	358	7	,	,	PUNCT
ejpam-3786	358	8	2018	2018	NUM
ejpam-3786	358	9	.	.	PUNCT
ejpam-3786	359	1	[	[	X
ejpam-3786	359	2	9	9	NUM
ejpam-3786	359	3	]	]	X
ejpam-3786	359	4	william	william	PROPN
ejpam-3786	359	5	a	a	DET
ejpam-3786	359	6	bogley	bogley	NOUN
ejpam-3786	359	7	and	and	CCONJ
ejpam-3786	359	8	stephen	stephen	NOUN
ejpam-3786	359	9	j	j	PROPN
ejpam-3786	359	10	pride	pride	PROPN
ejpam-3786	359	11	.	.	PUNCT
ejpam-3786	360	1	aspherical	aspherical	ADJ
ejpam-3786	360	2	relative	relative	ADJ
ejpam-3786	360	3	presentations	presentation	NOUN
ejpam-3786	360	4	.	.	PUNCT
ejpam-3786	361	1	proceedings	proceeding	NOUN
ejpam-3786	361	2	of	of	ADP
ejpam-3786	361	3	the	the	DET
ejpam-3786	361	4	edinburgh	edinburgh	PROPN
ejpam-3786	361	5	mathematical	mathematical	PROPN
ejpam-3786	361	6	society	society	NOUN
ejpam-3786	361	7	,	,	PUNCT
ejpam-3786	361	8	35(1):1–39	35(1):1–39	NUM
ejpam-3786	361	9	,	,	PUNCT
ejpam-3786	361	10	1992	1992	NUM
ejpam-3786	361	11	.	.	PUNCT
ejpam-3786	362	1	[	[	X
ejpam-3786	362	2	10	10	NUM
ejpam-3786	362	3	]	]	X
ejpam-3786	362	4	martin	martin	PROPN
ejpam-3786	362	5	edjvet	edjvet	NOUN
ejpam-3786	362	6	.	.	PUNCT
ejpam-3786	363	1	on	on	ADP
ejpam-3786	363	2	the	the	DET
ejpam-3786	363	3	asphericity	asphericity	NOUN
ejpam-3786	363	4	of	of	ADP
ejpam-3786	363	5	one	one	NUM
ejpam-3786	363	6	-	-	PUNCT
ejpam-3786	363	7	relator	relator	NOUN
ejpam-3786	363	8	relative	relative	ADJ
ejpam-3786	363	9	presentations	presentation	NOUN
ejpam-3786	363	10	.	.	PUNCT
ejpam-3786	364	1	proceedings	proceeding	NOUN
ejpam-3786	364	2	of	of	ADP
ejpam-3786	364	3	the	the	DET
ejpam-3786	364	4	royal	royal	ADJ
ejpam-3786	364	5	society	society	NOUN
ejpam-3786	364	6	of	of	ADP
ejpam-3786	364	7	edinburgh	edinburgh	PROPN
ejpam-3786	364	8	section	section	PROPN
ejpam-3786	364	9	a	a	DET
ejpam-3786	364	10	:	:	PUNCT
ejpam-3786	364	11	mathematics	mathematic	NOUN
ejpam-3786	364	12	,	,	PUNCT
ejpam-3786	364	13	124(4):713–728	124(4):713–728	PROPN
ejpam-3786	364	14	,	,	PUNCT
ejpam-3786	364	15	1994	1994	NUM
ejpam-3786	364	16	.	.	PUNCT
ejpam-3786	365	1	[	[	X
ejpam-3786	365	2	11	11	NUM
ejpam-3786	365	3	]	]	X
ejpam-3786	365	4	martin	martin	PROPN
ejpam-3786	365	5	edjvet	edjvet	PROPN
ejpam-3786	365	6	and	and	CCONJ
ejpam-3786	365	7	james	james	PROPN
ejpam-3786	365	8	howie	howie	PROPN
ejpam-3786	365	9	.	.	PUNCT
ejpam-3786	366	1	the	the	DET
ejpam-3786	366	2	solution	solution	NOUN
ejpam-3786	366	3	of	of	ADP
ejpam-3786	366	4	length	length	NOUN
ejpam-3786	366	5	four	four	NUM
ejpam-3786	366	6	equations	equation	NOUN
ejpam-3786	366	7	over	over	ADP
ejpam-3786	366	8	groups	group	NOUN
ejpam-3786	366	9	.	.	PUNCT
ejpam-3786	367	1	transactions	transaction	NOUN
ejpam-3786	367	2	of	of	ADP
ejpam-3786	367	3	the	the	DET
ejpam-3786	367	4	american	american	PROPN
ejpam-3786	367	5	mathematical	mathematical	PROPN
ejpam-3786	367	6	society	society	NOUN
ejpam-3786	367	7	,	,	PUNCT
ejpam-3786	367	8	326(1):345–369	326(1):345–369	NUM
ejpam-3786	367	9	,	,	PUNCT
ejpam-3786	367	10	1991	1991	NUM
ejpam-3786	367	11	.	.	PUNCT
ejpam-3786	368	1	[	[	X
ejpam-3786	368	2	12	12	NUM
ejpam-3786	368	3	]	]	X
ejpam-3786	368	4	anastasia	anastasia	PROPN
ejpam-3786	368	5	evangelidou	evangelidou	NOUN
ejpam-3786	368	6	.	.	PUNCT
ejpam-3786	369	1	the	the	DET
ejpam-3786	369	2	solution	solution	NOUN
ejpam-3786	369	3	of	of	ADP
ejpam-3786	369	4	length	length	NOUN
ejpam-3786	369	5	five	five	NUM
ejpam-3786	369	6	equations	equation	NOUN
ejpam-3786	369	7	over	over	ADP
ejpam-3786	369	8	groups	group	NOUN
ejpam-3786	369	9	.	.	PUNCT
ejpam-3786	370	1	communications	communication	NOUN
ejpam-3786	370	2	in	in	ADP
ejpam-3786	370	3	algebra	algebra	NOUN
ejpam-3786	370	4	r	r	NOUN
ejpam-3786	370	5	©	©	PROPN
ejpam-3786	370	6	,	,	PUNCT
ejpam-3786	370	7	35(6):1914–1948	35(6):1914–1948	NUM
ejpam-3786	370	8	,	,	PUNCT
ejpam-3786	370	9	2007	2007	NUM
ejpam-3786	370	10	.	.	PUNCT
ejpam-3786	371	1	[	[	X
ejpam-3786	371	2	13	13	NUM
ejpam-3786	371	3	]	]	X
ejpam-3786	371	4	james	james	PROPN
ejpam-3786	371	5	howie	howie	PROPN
ejpam-3786	371	6	.	.	PUNCT
ejpam-3786	372	1	on	on	ADP
ejpam-3786	372	2	pairs	pair	NOUN
ejpam-3786	372	3	of	of	ADP
ejpam-3786	372	4	2	2	NUM
ejpam-3786	372	5	-	-	PUNCT
ejpam-3786	372	6	complexes	complex	NOUN
ejpam-3786	372	7	and	and	CCONJ
ejpam-3786	372	8	systems	system	NOUN
ejpam-3786	372	9	of	of	ADP
ejpam-3786	372	10	equations	equation	NOUN
ejpam-3786	372	11	over	over	ADP
ejpam-3786	372	12	groups	group	NOUN
ejpam-3786	372	13	.	.	PUNCT
ejpam-3786	373	1	journal	journal	PROPN
ejpam-3786	373	2	für	für	AUX
ejpam-3786	373	3	die	die	VERB
ejpam-3786	373	4	reine	reine	PROPN
ejpam-3786	373	5	und	und	PROPN
ejpam-3786	373	6	angewandte	angewandte	PROPN
ejpam-3786	373	7	mathematik	mathematik	PROPN
ejpam-3786	373	8	,	,	PUNCT
ejpam-3786	373	9	1981(324):165–174	1981(324):165–174	ADV
ejpam-3786	373	10	,	,	PUNCT
ejpam-3786	373	11	1981	1981	NUM
ejpam-3786	373	12	.	.	PUNCT
ejpam-3786	374	1	[	[	X
ejpam-3786	374	2	14	14	NUM
ejpam-3786	374	3	]	]	X
ejpam-3786	374	4	james	james	PROPN
ejpam-3786	374	5	howie	howie	PROPN
ejpam-3786	374	6	.	.	PUNCT
ejpam-3786	375	1	the	the	DET
ejpam-3786	375	2	solution	solution	NOUN
ejpam-3786	375	3	of	of	ADP
ejpam-3786	375	4	length	length	NOUN
ejpam-3786	375	5	three	three	NUM
ejpam-3786	375	6	equations	equation	NOUN
ejpam-3786	375	7	over	over	ADP
ejpam-3786	375	8	groups	group	NOUN
ejpam-3786	375	9	.	.	PUNCT
ejpam-3786	376	1	proceedings	proceeding	NOUN
ejpam-3786	376	2	of	of	ADP
ejpam-3786	376	3	the	the	DET
ejpam-3786	376	4	edinburgh	edinburgh	PROPN
ejpam-3786	376	5	mathematical	mathematical	PROPN
ejpam-3786	376	6	society	society	NOUN
ejpam-3786	376	7	,	,	PUNCT
ejpam-3786	376	8	26(2):89–96	26(2):89–96	NUM
ejpam-3786	376	9	,	,	PUNCT
ejpam-3786	376	10	1983	1983	NUM
ejpam-3786	376	11	.	.	PUNCT
ejpam-3786	377	1	[	[	X
ejpam-3786	377	2	15	15	NUM
ejpam-3786	377	3	]	]	X
ejpam-3786	377	4	sergey	sergey	PROPN
ejpam-3786	377	5	v	v	ADP
ejpam-3786	377	6	ivanov	ivanov	PROPN
ejpam-3786	377	7	and	and	CCONJ
ejpam-3786	377	8	anton	anton	NOUN
ejpam-3786	377	9	a	a	DET
ejpam-3786	377	10	klyachko	klyachko	ADJ
ejpam-3786	377	11	.	.	PUNCT
ejpam-3786	378	1	solving	solve	VERB
ejpam-3786	378	2	equations	equation	NOUN
ejpam-3786	378	3	of	of	ADP
ejpam-3786	378	4	length	length	NOUN
ejpam-3786	378	5	at	at	ADP
ejpam-3786	378	6	most	most	ADV
ejpam-3786	378	7	six	six	NUM
ejpam-3786	378	8	over	over	ADP
ejpam-3786	378	9	torsion	torsion	NOUN
ejpam-3786	378	10	-	-	PUNCT
ejpam-3786	378	11	free	free	ADJ
ejpam-3786	378	12	groups	group	NOUN
ejpam-3786	378	13	.	.	PUNCT
ejpam-3786	379	1	journal	journal	PROPN
ejpam-3786	379	2	of	of	ADP
ejpam-3786	379	3	group	group	PROPN
ejpam-3786	379	4	theory	theory	NOUN
ejpam-3786	379	5	,	,	PUNCT
ejpam-3786	379	6	3(3):329–337	3(3):329–337	NOUN
ejpam-3786	379	7	,	,	PUNCT
ejpam-3786	379	8	2000	2000	NUM
ejpam-3786	379	9	.	.	PUNCT
ejpam-3786	380	1	[	[	X
ejpam-3786	380	2	16	16	NUM
ejpam-3786	380	3	]	]	X
ejpam-3786	380	4	f	f	PROPN
ejpam-3786	380	5	levin	levin	PROPN
ejpam-3786	380	6	.	.	PUNCT
ejpam-3786	381	1	solutions	solution	NOUN
ejpam-3786	381	2	of	of	ADP
ejpam-3786	381	3	equations	equation	NOUN
ejpam-3786	381	4	over	over	ADP
ejpam-3786	381	5	groups	group	NOUN
ejpam-3786	381	6	.	.	PUNCT
ejpam-3786	382	1	bulletin	bulletin	NOUN
ejpam-3786	382	2	of	of	ADP
ejpam-3786	382	3	american	american	PROPN
ejpam-3786	382	4	mathematical	mathematical	PROPN
ejpam-3786	382	5	society	society	NOUN
ejpam-3786	382	6	,	,	PUNCT
ejpam-3786	382	7	68:603–604	68:603–604	PROPN
ejpam-3786	382	8	,	,	PUNCT
ejpam-3786	382	9	1962	1962	NUM
ejpam-3786	382	10	.	.	PUNCT
ejpam-3786	383	1	[	[	X
ejpam-3786	383	2	17	17	NUM
ejpam-3786	383	3	]	]	X
ejpam-3786	383	4	bernhard	bernhard	PROPN
ejpam-3786	383	5	hermann	hermann	PROPN
ejpam-3786	383	6	neumann	neumann	PROPN
ejpam-3786	383	7	.	.	PUNCT
ejpam-3786	384	1	adjunction	adjunction	PROPN
ejpam-3786	384	2	of	of	ADP
ejpam-3786	384	3	elements	element	NOUN
ejpam-3786	384	4	to	to	ADP
ejpam-3786	384	5	groups	group	NOUN
ejpam-3786	384	6	.	.	PUNCT
ejpam-3786	385	1	journal	journal	NOUN
ejpam-3786	385	2	of	of	ADP
ejpam-3786	385	3	the	the	DET
ejpam-3786	385	4	london	london	PROPN
ejpam-3786	385	5	mathematical	mathematical	ADJ
ejpam-3786	385	6	society	society	NOUN
ejpam-3786	385	7	,	,	PUNCT
ejpam-3786	385	8	18:4–11	18:4–11	NUM
ejpam-3786	385	9	,	,	PUNCT
ejpam-3786	385	10	1943	1943	NUM
ejpam-3786	385	11	.	.	PUNCT