id	sid	tid	token	lemma	pos
ijassa-2042	1	1	adv	adv	PROPN
ijassa-2042	1	2	syst	syst	PROPN
ijassa-2042	1	3	sci	sci	PROPN
ijassa-2042	1	4	appl	appl	PROPN
ijassa-2042	1	5	2025	2025	NUM
ijassa-2042	1	6	;	;	PUNCT
ijassa-2042	1	7	1:44–54	1:44–54	NUM
ijassa-2042	1	8	published	publish	VERB
ijassa-2042	1	9	online	online	ADV
ijassa-2042	1	10	at	at	ADP
ijassa-2042	1	11	https://ijassa.ipu.ru	https://ijassa.ipu.ru	ADV
ijassa-2042	1	12	.	.	PUNCT
ijassa-2042	2	1	legendre	legendre	PROPN
ijassa-2042	2	2	transformation	transformation	PROPN
ijassa-2042	2	3	and	and	CCONJ
ijassa-2042	2	4	its	its	PRON
ijassa-2042	2	5	applications	application	NOUN
ijassa-2042	2	6	natalia	natalia	PROPN
ijassa-2042	2	7	pavlova1,2	pavlova1,2	PROPN
ijassa-2042	2	8	alexey	alexey	PROPN
ijassa-2042	2	9	remizov1	remizov1	PROPN
ijassa-2042	2	10	*	*	PROPN
ijassa-2042	2	11	1moscow	1moscow	PROPN
ijassa-2042	2	12	institute	institute	PROPN
ijassa-2042	2	13	of	of	ADP
ijassa-2042	2	14	physics	physics	PROPN
ijassa-2042	2	15	and	and	CCONJ
ijassa-2042	2	16	technology	technology	NOUN
ijassa-2042	2	17	(	(	PUNCT
ijassa-2042	2	18	state	state	NOUN
ijassa-2042	2	19	university	university	NOUN
ijassa-2042	2	20	)	)	PUNCT
ijassa-2042	2	21	,	,	PUNCT
ijassa-2042	2	22	dolgoprudnyi	dolgoprudnyi	PROPN
ijassa-2042	2	23	,	,	PUNCT
ijassa-2042	2	24	russia	russia	PROPN
ijassa-2042	2	25	2rudn	2rudn	NUM
ijassa-2042	2	26	university	university	NOUN
ijassa-2042	2	27	,	,	PUNCT
ijassa-2042	2	28	moscow	moscow	PROPN
ijassa-2042	2	29	,	,	PUNCT
ijassa-2042	2	30	russia	russia	PROPN
ijassa-2042	2	31	abstract	abstract	NOUN
ijassa-2042	2	32	:	:	PUNCT
ijassa-2042	2	33	we	we	PRON
ijassa-2042	2	34	present	present	VERB
ijassa-2042	2	35	the	the	DET
ijassa-2042	2	36	legendre	legendre	PROPN
ijassa-2042	2	37	transformation	transformation	NOUN
ijassa-2042	2	38	in	in	ADP
ijassa-2042	2	39	a	a	DET
ijassa-2042	2	40	geometric	geometric	ADJ
ijassa-2042	2	41	way	way	NOUN
ijassa-2042	2	42	based	base	VERB
ijassa-2042	2	43	on	on	ADP
ijassa-2042	2	44	the	the	DET
ijassa-2042	2	45	procedure	procedure	NOUN
ijassa-2042	2	46	of	of	ADP
ijassa-2042	2	47	the	the	DET
ijassa-2042	2	48	legendrian	legendrian	ADJ
ijassa-2042	2	49	lift	lift	NOUN
ijassa-2042	2	50	.	.	PUNCT
ijassa-2042	3	1	this	this	DET
ijassa-2042	3	2	approach	approach	NOUN
ijassa-2042	3	3	allows	allow	VERB
ijassa-2042	3	4	us	we	PRON
ijassa-2042	3	5	to	to	PART
ijassa-2042	3	6	understand	understand	VERB
ijassa-2042	3	7	some	some	DET
ijassa-2042	3	8	interesting	interesting	ADJ
ijassa-2042	3	9	properties	property	NOUN
ijassa-2042	3	10	of	of	ADP
ijassa-2042	3	11	it	it	PRON
ijassa-2042	3	12	,	,	PUNCT
ijassa-2042	3	13	in	in	ADP
ijassa-2042	3	14	particular	particular	ADJ
ijassa-2042	3	15	,	,	PUNCT
ijassa-2042	3	16	the	the	DET
ijassa-2042	3	17	reason	reason	NOUN
ijassa-2042	3	18	for	for	ADP
ijassa-2042	3	19	the	the	DET
ijassa-2042	3	20	appearance	appearance	NOUN
ijassa-2042	3	21	of	of	ADP
ijassa-2042	3	22	singularities	singularity	NOUN
ijassa-2042	3	23	of	of	ADP
ijassa-2042	3	24	dual	dual	ADJ
ijassa-2042	3	25	curves	curve	NOUN
ijassa-2042	3	26	.	.	PUNCT
ijassa-2042	4	1	also	also	ADV
ijassa-2042	4	2	we	we	PRON
ijassa-2042	4	3	consider	consider	VERB
ijassa-2042	4	4	application	application	NOUN
ijassa-2042	4	5	of	of	ADP
ijassa-2042	4	6	the	the	DET
ijassa-2042	4	7	legendre	legendre	PROPN
ijassa-2042	4	8	transformation	transformation	NOUN
ijassa-2042	4	9	to	to	ADP
ijassa-2042	4	10	the	the	DET
ijassa-2042	4	11	clairaut	clairaut	PROPN
ijassa-2042	4	12	differential	differential	PROPN
ijassa-2042	4	13	equation	equation	NOUN
ijassa-2042	4	14	.	.	PUNCT
ijassa-2042	5	1	finally	finally	ADV
ijassa-2042	5	2	,	,	PUNCT
ijassa-2042	5	3	we	we	PRON
ijassa-2042	5	4	say	say	VERB
ijassa-2042	5	5	a	a	DET
ijassa-2042	5	6	few	few	ADJ
ijassa-2042	5	7	words	word	NOUN
ijassa-2042	5	8	class	class	NOUN
ijassa-2042	5	9	of	of	ADP
ijassa-2042	5	10	contact	contact	NOUN
ijassa-2042	5	11	transformations	transformation	NOUN
ijassa-2042	5	12	and	and	CCONJ
ijassa-2042	5	13	present	present	VERB
ijassa-2042	5	14	an	an	DET
ijassa-2042	5	15	infinite	infinite	ADJ
ijassa-2042	5	16	group	group	NOUN
ijassa-2042	5	17	of	of	ADP
ijassa-2042	5	18	contact	contact	NOUN
ijassa-2042	5	19	transformations	transformation	NOUN
ijassa-2042	5	20	different	different	ADJ
ijassa-2042	5	21	from	from	ADP
ijassa-2042	5	22	the	the	DET
ijassa-2042	5	23	legendre	legendre	PROPN
ijassa-2042	5	24	transformation	transformation	NOUN
ijassa-2042	5	25	.	.	PUNCT
ijassa-2042	6	1	keywords	keyword	NOUN
ijassa-2042	6	2	:	:	PUNCT
ijassa-2042	6	3	legendre	legendre	PROPN
ijassa-2042	6	4	transformation	transformation	NOUN
ijassa-2042	6	5	,	,	PUNCT
ijassa-2042	6	6	duality	duality	NOUN
ijassa-2042	6	7	,	,	PUNCT
ijassa-2042	6	8	contact	contact	NOUN
ijassa-2042	6	9	transformation	transformation	NOUN
ijassa-2042	6	10	,	,	PUNCT
ijassa-2042	6	11	contact	contact	NOUN
ijassa-2042	6	12	structure	structure	NOUN
ijassa-2042	6	13	,	,	PUNCT
ijassa-2042	6	14	pedal	pedal	ADJ
ijassa-2042	6	15	curve	curve	NOUN
ijassa-2042	6	16	,	,	PUNCT
ijassa-2042	6	17	singular	singular	ADJ
ijassa-2042	6	18	point	point	NOUN
ijassa-2042	6	19	,	,	PUNCT
ijassa-2042	6	20	clairaut	clairaut	PROPN
ijassa-2042	6	21	equation	equation	NOUN
ijassa-2042	6	22	introduction	introduction	NOUN
ijassa-2042	6	23	the	the	DET
ijassa-2042	6	24	paper	paper	NOUN
ijassa-2042	6	25	has	have	VERB
ijassa-2042	6	26	several	several	ADJ
ijassa-2042	6	27	goals	goal	NOUN
ijassa-2042	6	28	.	.	PUNCT
ijassa-2042	7	1	first	first	ADV
ijassa-2042	7	2	,	,	PUNCT
ijassa-2042	7	3	we	we	PRON
ijassa-2042	7	4	present	present	VERB
ijassa-2042	7	5	the	the	DET
ijassa-2042	7	6	legendre	legendre	PROPN
ijassa-2042	7	7	transformation	transformation	NOUN
ijassa-2042	7	8	in	in	ADP
ijassa-2042	7	9	a	a	DET
ijassa-2042	7	10	geometric	geometric	ADJ
ijassa-2042	7	11	way	way	NOUN
ijassa-2042	7	12	based	base	VERB
ijassa-2042	7	13	on	on	ADP
ijassa-2042	7	14	the	the	DET
ijassa-2042	7	15	procedure	procedure	NOUN
ijassa-2042	7	16	of	of	ADP
ijassa-2042	7	17	the	the	DET
ijassa-2042	7	18	legendrian	legendrian	ADJ
ijassa-2042	7	19	lift	lift	NOUN
ijassa-2042	7	20	.	.	PUNCT
ijassa-2042	8	1	therefore	therefore	ADV
ijassa-2042	8	2	,	,	PUNCT
ijassa-2042	8	3	we	we	PRON
ijassa-2042	8	4	start	start	VERB
ijassa-2042	8	5	with	with	ADP
ijassa-2042	8	6	the	the	DET
ijassa-2042	8	7	legendre	legendre	PROPN
ijassa-2042	8	8	transformation	transformation	NOUN
ijassa-2042	8	9	of	of	ADP
ijassa-2042	8	10	planar	planar	ADJ
ijassa-2042	8	11	curves	curve	NOUN
ijassa-2042	8	12	and	and	CCONJ
ijassa-2042	8	13	then	then	ADV
ijassa-2042	8	14	go	go	VERB
ijassa-2042	8	15	to	to	ADP
ijassa-2042	8	16	the	the	DET
ijassa-2042	8	17	legendre	legendre	PROPN
ijassa-2042	8	18	transformation	transformation	NOUN
ijassa-2042	8	19	of	of	ADP
ijassa-2042	8	20	functions	function	NOUN
ijassa-2042	8	21	.	.	PUNCT
ijassa-2042	9	1	as	as	SCONJ
ijassa-2042	9	2	we	we	PRON
ijassa-2042	9	3	shall	shall	AUX
ijassa-2042	9	4	see	see	VERB
ijassa-2042	9	5	below	below	ADV
ijassa-2042	9	6	,	,	PUNCT
ijassa-2042	9	7	this	this	DET
ijassa-2042	9	8	approach	approach	NOUN
ijassa-2042	9	9	has	have	VERB
ijassa-2042	9	10	some	some	DET
ijassa-2042	9	11	important	important	ADJ
ijassa-2042	9	12	advantages	advantage	NOUN
ijassa-2042	9	13	in	in	ADP
ijassa-2042	9	14	compare	compare	NOUN
ijassa-2042	9	15	with	with	ADP
ijassa-2042	9	16	the	the	DET
ijassa-2042	9	17	standard	standard	ADJ
ijassa-2042	9	18	definition	definition	NOUN
ijassa-2042	9	19	of	of	ADP
ijassa-2042	9	20	the	the	DET
ijassa-2042	9	21	legendre	legendre	PROPN
ijassa-2042	9	22	transformation	transformation	NOUN
ijassa-2042	9	23	of	of	ADP
ijassa-2042	9	24	functions	function	NOUN
ijassa-2042	9	25	,	,	PUNCT
ijassa-2042	9	26	which	which	PRON
ijassa-2042	9	27	is	be	AUX
ijassa-2042	9	28	thrown	throw	VERB
ijassa-2042	9	29	in	in	ADP
ijassa-2042	9	30	the	the	DET
ijassa-2042	9	31	reader	reader	NOUN
ijassa-2042	9	32	’s	’s	PART
ijassa-2042	9	33	face	face	NOUN
ijassa-2042	9	34	without	without	ADP
ijassa-2042	9	35	motivations	motivation	NOUN
ijassa-2042	9	36	.	.	PUNCT
ijassa-2042	10	1	moreover	moreover	ADV
ijassa-2042	10	2	,	,	PUNCT
ijassa-2042	10	3	the	the	DET
ijassa-2042	10	4	presented	present	VERB
ijassa-2042	10	5	geometric	geometric	ADJ
ijassa-2042	10	6	approach	approach	NOUN
ijassa-2042	10	7	allows	allow	VERB
ijassa-2042	10	8	us	we	PRON
ijassa-2042	10	9	to	to	PART
ijassa-2042	10	10	understand	understand	VERB
ijassa-2042	10	11	the	the	DET
ijassa-2042	10	12	meaning	meaning	NOUN
ijassa-2042	10	13	of	of	ADP
ijassa-2042	10	14	restrictions	restriction	NOUN
ijassa-2042	10	15	in	in	ADP
ijassa-2042	10	16	the	the	DET
ijassa-2042	10	17	standard	standard	ADJ
ijassa-2042	10	18	definition	definition	NOUN
ijassa-2042	10	19	of	of	ADP
ijassa-2042	10	20	the	the	DET
ijassa-2042	10	21	legendre	legendre	PROPN
ijassa-2042	10	22	transformation	transformation	NOUN
ijassa-2042	10	23	of	of	ADP
ijassa-2042	10	24	functions	function	NOUN
ijassa-2042	10	25	(	(	PUNCT
ijassa-2042	10	26	for	for	ADP
ijassa-2042	10	27	example	example	NOUN
ijassa-2042	10	28	,	,	PUNCT
ijassa-2042	10	29	the	the	DET
ijassa-2042	10	30	convexity	convexity	NOUN
ijassa-2042	10	31	)	)	PUNCT
ijassa-2042	10	32	and	and	CCONJ
ijassa-2042	10	33	the	the	DET
ijassa-2042	10	34	reason	reason	NOUN
ijassa-2042	10	35	for	for	ADP
ijassa-2042	10	36	the	the	DET
ijassa-2042	10	37	appearance	appearance	NOUN
ijassa-2042	10	38	of	of	ADP
ijassa-2042	10	39	singularities	singularity	NOUN
ijassa-2042	10	40	of	of	ADP
ijassa-2042	10	41	dual	dual	ADJ
ijassa-2042	10	42	curves	curve	NOUN
ijassa-2042	10	43	.	.	PUNCT
ijassa-2042	11	1	second	second	ADJ
ijassa-2042	11	2	,	,	PUNCT
ijassa-2042	11	3	we	we	PRON
ijassa-2042	11	4	consider	consider	VERB
ijassa-2042	11	5	application	application	NOUN
ijassa-2042	11	6	of	of	ADP
ijassa-2042	11	7	the	the	DET
ijassa-2042	11	8	legendre	legendre	PROPN
ijassa-2042	11	9	transformation	transformation	NOUN
ijassa-2042	11	10	to	to	ADP
ijassa-2042	11	11	ordinary	ordinary	ADJ
ijassa-2042	11	12	differential	differential	ADJ
ijassa-2042	11	13	equations	equation	NOUN
ijassa-2042	11	14	at	at	ADP
ijassa-2042	11	15	the	the	DET
ijassa-2042	11	16	example	example	NOUN
ijassa-2042	11	17	of	of	ADP
ijassa-2042	11	18	the	the	DET
ijassa-2042	11	19	clairaut	clairaut	PROPN
ijassa-2042	11	20	equation	equation	NOUN
ijassa-2042	11	21	.	.	PUNCT
ijassa-2042	12	1	for	for	ADP
ijassa-2042	12	2	this	this	PRON
ijassa-2042	12	3	,	,	PUNCT
ijassa-2042	12	4	the	the	DET
ijassa-2042	12	5	geometric	geometric	ADJ
ijassa-2042	12	6	interpretation	interpretation	NOUN
ijassa-2042	12	7	of	of	ADP
ijassa-2042	12	8	the	the	DET
ijassa-2042	12	9	legendre	legendre	PROPN
ijassa-2042	12	10	transformation	transformation	NOUN
ijassa-2042	12	11	is	be	AUX
ijassa-2042	12	12	more	more	ADV
ijassa-2042	12	13	suitable	suitable	ADJ
ijassa-2042	12	14	.	.	PUNCT
ijassa-2042	13	1	finally	finally	ADV
ijassa-2042	13	2	,	,	PUNCT
ijassa-2042	13	3	we	we	PRON
ijassa-2042	13	4	embed	embed	VERB
ijassa-2042	13	5	the	the	DET
ijassa-2042	13	6	legendre	legendre	PROPN
ijassa-2042	13	7	transformation	transformation	NOUN
ijassa-2042	13	8	into	into	ADP
ijassa-2042	13	9	a	a	DET
ijassa-2042	13	10	class	class	NOUN
ijassa-2042	13	11	of	of	ADP
ijassa-2042	13	12	contact	contact	NOUN
ijassa-2042	13	13	transformations	transformation	NOUN
ijassa-2042	13	14	and	and	CCONJ
ijassa-2042	13	15	discuss	discuss	VERB
ijassa-2042	13	16	some	some	DET
ijassa-2042	13	17	examples	example	NOUN
ijassa-2042	13	18	of	of	ADP
ijassa-2042	13	19	contact	contact	NOUN
ijassa-2042	13	20	transformations	transformation	NOUN
ijassa-2042	13	21	different	different	ADJ
ijassa-2042	13	22	from	from	ADP
ijassa-2042	13	23	it	it	PRON
ijassa-2042	13	24	.	.	PUNCT
ijassa-2042	14	1	in	in	ADP
ijassa-2042	14	2	particular	particular	ADJ
ijassa-2042	14	3	,	,	PUNCT
ijassa-2042	14	4	we	we	PRON
ijassa-2042	14	5	consider	consider	VERB
ijassa-2042	14	6	the	the	DET
ijassa-2042	14	7	pedal	pedal	ADJ
ijassa-2042	14	8	transformation	transformation	NOUN
ijassa-2042	14	9	(	(	PUNCT
ijassa-2042	14	10	apparently	apparently	ADV
ijassa-2042	14	11	the	the	DET
ijassa-2042	14	12	earliest	early	ADJ
ijassa-2042	14	13	example	example	NOUN
ijassa-2042	14	14	of	of	ADP
ijassa-2042	14	15	contact	contact	NOUN
ijassa-2042	14	16	transformations	transformation	NOUN
ijassa-2042	14	17	)	)	PUNCT
ijassa-2042	14	18	and	and	CCONJ
ijassa-2042	14	19	with	with	ADP
ijassa-2042	14	20	its	its	PRON
ijassa-2042	14	21	aid	aid	NOUN
ijassa-2042	14	22	we	we	PRON
ijassa-2042	14	23	obtain	obtain	VERB
ijassa-2042	14	24	an	an	DET
ijassa-2042	14	25	infinite	infinite	ADJ
ijassa-2042	14	26	group	group	NOUN
ijassa-2042	14	27	of	of	ADP
ijassa-2042	14	28	contact	contact	NOUN
ijassa-2042	14	29	transformations	transformation	NOUN
ijassa-2042	14	30	,	,	PUNCT
ijassa-2042	14	31	which	which	PRON
ijassa-2042	14	32	was	be	AUX
ijassa-2042	14	33	first	first	ADV
ijassa-2042	14	34	constructed	construct	VERB
ijassa-2042	14	35	by	by	ADP
ijassa-2042	14	36	sophus	sophu	NOUN
ijassa-2042	14	37	lie	lie	NOUN
ijassa-2042	14	38	.	.	PUNCT
ijassa-2042	15	1	it	it	PRON
ijassa-2042	15	2	should	should	AUX
ijassa-2042	15	3	be	be	AUX
ijassa-2042	15	4	remarked	remark	VERB
ijassa-2042	15	5	that	that	SCONJ
ijassa-2042	15	6	many	many	ADJ
ijassa-2042	15	7	of	of	ADP
ijassa-2042	15	8	results	result	NOUN
ijassa-2042	15	9	presented	present	VERB
ijassa-2042	15	10	in	in	ADP
ijassa-2042	15	11	the	the	DET
ijassa-2042	15	12	paper	paper	NOUN
ijassa-2042	15	13	were	be	AUX
ijassa-2042	15	14	previously	previously	ADV
ijassa-2042	15	15	known	know	VERB
ijassa-2042	15	16	,	,	PUNCT
ijassa-2042	15	17	but	but	CCONJ
ijassa-2042	15	18	we	we	PRON
ijassa-2042	15	19	presented	present	VERB
ijassa-2042	15	20	these	these	DET
ijassa-2042	15	21	results	result	NOUN
ijassa-2042	15	22	from	from	ADP
ijassa-2042	15	23	a	a	DET
ijassa-2042	15	24	more	more	ADV
ijassa-2042	15	25	general	general	ADJ
ijassa-2042	15	26	point	point	NOUN
ijassa-2042	15	27	of	of	ADP
ijassa-2042	15	28	view	view	NOUN
ijassa-2042	15	29	and	and	CCONJ
ijassa-2042	15	30	explained	explain	VERB
ijassa-2042	15	31	the	the	DET
ijassa-2042	15	32	relationship	relationship	NOUN
ijassa-2042	15	33	between	between	ADP
ijassa-2042	15	34	them	they	PRON
ijassa-2042	15	35	.	.	PUNCT
ijassa-2042	16	1	we	we	PRON
ijassa-2042	16	2	start	start	VERB
ijassa-2042	16	3	with	with	ADP
ijassa-2042	16	4	a	a	DET
ijassa-2042	16	5	small	small	ADJ
ijassa-2042	16	6	section	section	NOUN
ijassa-2042	16	7	presented	present	VERB
ijassa-2042	16	8	some	some	DET
ijassa-2042	16	9	notions	notion	NOUN
ijassa-2042	16	10	and	and	CCONJ
ijassa-2042	16	11	results	result	NOUN
ijassa-2042	16	12	about	about	ADP
ijassa-2042	16	13	planar	planar	ADJ
ijassa-2042	16	14	curves	curve	NOUN
ijassa-2042	16	15	and	and	CCONJ
ijassa-2042	16	16	their	their	PRON
ijassa-2042	16	17	singularities	singularity	NOUN
ijassa-2042	16	18	,	,	PUNCT
ijassa-2042	16	19	which	which	PRON
ijassa-2042	16	20	are	be	AUX
ijassa-2042	16	21	necessary	necessary	ADJ
ijassa-2042	16	22	for	for	ADP
ijassa-2042	16	23	understanding	understand	VERB
ijassa-2042	16	24	the	the	DET
ijassa-2042	16	25	main	main	ADJ
ijassa-2042	16	26	ideas	idea	NOUN
ijassa-2042	16	27	of	of	ADP
ijassa-2042	16	28	the	the	DET
ijassa-2042	16	29	paper	paper	NOUN
ijassa-2042	16	30	.	.	PUNCT
ijassa-2042	17	1	∗corresponding	∗corresponde	VERB
ijassa-2042	17	2	author	author	NOUN
ijassa-2042	17	3	:	:	PUNCT
ijassa-2042	17	4	alexey-remizov@yandex.ru	alexey-remizov@yandex.ru	PROPN
ijassa-2042	17	5	legendre	legendre	PROPN
ijassa-2042	17	6	transformation	transformation	NOUN
ijassa-2042	17	7	and	and	CCONJ
ijassa-2042	17	8	its	its	PRON
ijassa-2042	17	9	applications	application	NOUN
ijassa-2042	17	10	45	45	NUM
ijassa-2042	17	11	1	1	NUM
ijassa-2042	17	12	.	.	PUNCT
ijassa-2042	18	1	curves	curve	NOUN
ijassa-2042	18	2	and	and	CCONJ
ijassa-2042	18	3	singularities	singularity	NOUN
ijassa-2042	18	4	let	let	VERB
ijassa-2042	18	5	γ	γ	NOUN
ijassa-2042	18	6	be	be	AUX
ijassa-2042	18	7	an	an	DET
ijassa-2042	18	8	arbitrary	arbitrary	ADJ
ijassa-2042	18	9	curve	curve	NOUN
ijassa-2042	18	10	on	on	ADP
ijassa-2042	18	11	the	the	DET
ijassa-2042	18	12	plane	plane	NOUN
ijassa-2042	18	13	given	give	VERB
ijassa-2042	18	14	in	in	ADP
ijassa-2042	18	15	the	the	DET
ijassa-2042	18	16	form	form	NOUN
ijassa-2042	18	17	x	x	X
ijassa-2042	18	18	=	=	SYM
ijassa-2042	18	19	ϕ(t	ϕ(t	NUM
ijassa-2042	18	20	)	)	PUNCT
ijassa-2042	18	21	,	,	PUNCT
ijassa-2042	18	22	y	y	PROPN
ijassa-2042	18	23	=	=	SYM
ijassa-2042	18	24	ψ(t	ψ(t	PROPN
ijassa-2042	18	25	)	)	PUNCT
ijassa-2042	18	26	,	,	PUNCT
ijassa-2042	18	27	(	(	PUNCT
ijassa-2042	18	28	1.1	1.1	NUM
ijassa-2042	18	29	)	)	PUNCT
ijassa-2042	18	30	where	where	SCONJ
ijassa-2042	18	31	ϕ	ϕ	NOUN
ijassa-2042	18	32	,	,	PUNCT
ijassa-2042	18	33	ψ	ψ	X
ijassa-2042	18	34	are	be	AUX
ijassa-2042	18	35	smooth	smooth	ADJ
ijassa-2042	18	36	(	(	PUNCT
ijassa-2042	18	37	c∞	c∞	NOUN
ijassa-2042	18	38	)	)	PUNCT
ijassa-2042	18	39	functions	function	NOUN
ijassa-2042	18	40	.	.	PUNCT
ijassa-2042	19	1	further	far	ADV
ijassa-2042	19	2	,	,	PUNCT
ijassa-2042	19	3	we	we	PRON
ijassa-2042	19	4	shall	shall	AUX
ijassa-2042	19	5	assume	assume	VERB
ijassa-2042	19	6	that	that	SCONJ
ijassa-2042	19	7	all	all	DET
ijassa-2042	19	8	considered	consider	VERB
ijassa-2042	19	9	functions	function	NOUN
ijassa-2042	19	10	and	and	CCONJ
ijassa-2042	19	11	mappings	mapping	NOUN
ijassa-2042	19	12	are	be	AUX
ijassa-2042	19	13	smooth	smooth	ADJ
ijassa-2042	19	14	(	(	PUNCT
ijassa-2042	19	15	c∞	c∞	PROPN
ijassa-2042	19	16	)	)	PUNCT
ijassa-2042	19	17	unless	unless	SCONJ
ijassa-2042	19	18	otherwise	otherwise	ADV
ijassa-2042	19	19	stated	state	VERB
ijassa-2042	19	20	.	.	PUNCT
ijassa-2042	20	1	a	a	DET
ijassa-2042	20	2	point	point	NOUN
ijassa-2042	20	3	of	of	ADP
ijassa-2042	20	4	the	the	DET
ijassa-2042	20	5	curve	curve	NOUN
ijassa-2042	20	6	γ	γ	NOUN
ijassa-2042	20	7	corresponding	correspond	VERB
ijassa-2042	20	8	the	the	DET
ijassa-2042	20	9	parameter	parameter	NOUN
ijassa-2042	20	10	t0	t0	PROPN
ijassa-2042	20	11	is	be	AUX
ijassa-2042	20	12	called	call	VERB
ijassa-2042	20	13	or	or	CCONJ
ijassa-2042	20	14	critical	critical	ADJ
ijassa-2042	20	15	if	if	SCONJ
ijassa-2042	20	16	ϕ′(t0	ϕ′(t0	NOUN
ijassa-2042	20	17	)	)	PUNCT
ijassa-2042	20	18	=	=	SYM
ijassa-2042	20	19	ψ′(t0	ψ′(t0	NOUN
ijassa-2042	20	20	)	)	PUNCT
ijassa-2042	20	21	=	=	PUNCT
ijassa-2042	21	1	0	0	X
ijassa-2042	21	2	.	.	PUNCT
ijassa-2042	22	1	(	(	PUNCT
ijassa-2042	22	2	1.2	1.2	NUM
ijassa-2042	22	3	)	)	PUNCT
ijassa-2042	22	4	otherwise	otherwise	ADV
ijassa-2042	22	5	a	a	DET
ijassa-2042	22	6	point	point	NOUN
ijassa-2042	22	7	is	be	AUX
ijassa-2042	22	8	called	call	VERB
ijassa-2042	22	9	regular	regular	ADJ
ijassa-2042	22	10	.	.	PUNCT
ijassa-2042	23	1	the	the	DET
ijassa-2042	23	2	curve	curve	NOUN
ijassa-2042	23	3	γ	γ	PROPN
ijassa-2042	23	4	is	be	AUX
ijassa-2042	23	5	called	call	VERB
ijassa-2042	23	6	regular	regular	ADJ
ijassa-2042	23	7	is	be	AUX
ijassa-2042	23	8	all	all	PRON
ijassa-2042	23	9	its	its	PRON
ijassa-2042	23	10	points	point	NOUN
ijassa-2042	23	11	are	be	AUX
ijassa-2042	23	12	regular	regular	ADJ
ijassa-2042	23	13	.	.	PUNCT
ijassa-2042	24	1	in	in	ADP
ijassa-2042	24	2	a	a	DET
ijassa-2042	24	3	neighborhood	neighborhood	NOUN
ijassa-2042	24	4	of	of	ADP
ijassa-2042	24	5	every	every	DET
ijassa-2042	24	6	regular	regular	ADJ
ijassa-2042	24	7	point	point	NOUN
ijassa-2042	24	8	,	,	PUNCT
ijassa-2042	24	9	the	the	DET
ijassa-2042	24	10	curve	curve	NOUN
ijassa-2042	24	11	is	be	AUX
ijassa-2042	24	12	diffeomorphic	diffeomorphic	ADJ
ijassa-2042	24	13	to	to	ADP
ijassa-2042	24	14	its	its	PRON
ijassa-2042	24	15	tangent	tangent	ADJ
ijassa-2042	24	16	line	line	NOUN
ijassa-2042	24	17	,	,	PUNCT
ijassa-2042	24	18	while	while	SCONJ
ijassa-2042	24	19	for	for	ADP
ijassa-2042	24	20	singular	singular	ADJ
ijassa-2042	24	21	points	point	NOUN
ijassa-2042	24	22	it	it	PRON
ijassa-2042	24	23	is	be	AUX
ijassa-2042	24	24	not	not	PART
ijassa-2042	24	25	true	true	ADJ
ijassa-2042	24	26	.	.	PUNCT
ijassa-2042	25	1	example	example	NOUN
ijassa-2042	25	2	1.1	1.1	NUM
ijassa-2042	25	3	:	:	PUNCT
ijassa-2042	25	4	in	in	ADP
ijassa-2042	25	5	various	various	ADJ
ijassa-2042	25	6	applications	application	NOUN
ijassa-2042	25	7	,	,	PUNCT
ijassa-2042	25	8	the	the	DET
ijassa-2042	25	9	following	follow	VERB
ijassa-2042	25	10	types	type	NOUN
ijassa-2042	25	11	of	of	ADP
ijassa-2042	25	12	singular	singular	ADJ
ijassa-2042	25	13	points	point	NOUN
ijassa-2042	25	14	often	often	ADV
ijassa-2042	25	15	appear	appear	VERB
ijassa-2042	25	16	:	:	PUNCT
ijassa-2042	25	17	x	x	X
ijassa-2042	25	18	=	=	SYM
ijassa-2042	25	19	ϕ(t	ϕ(t	NUM
ijassa-2042	25	20	)	)	PUNCT
ijassa-2042	26	1	=	=	PRON
ijassa-2042	26	2	αtn	αtn	NOUN
ijassa-2042	26	3	+	+	X
ijassa-2042	26	4	o(tn	o(tn	NUM
ijassa-2042	26	5	)	)	PUNCT
ijassa-2042	26	6	,	,	PUNCT
ijassa-2042	26	7	y	y	PROPN
ijassa-2042	26	8	=	=	SYM
ijassa-2042	26	9	ψ(t	ψ(t	PROPN
ijassa-2042	26	10	)	)	PUNCT
ijassa-2042	26	11	=	=	SYM
ijassa-2042	26	12	βtn+1	βtn+1	X
ijassa-2042	26	13	+	+	CCONJ
ijassa-2042	26	14	o(tn+1	o(tn+1	NUM
ijassa-2042	26	15	)	)	PUNCT
ijassa-2042	26	16	,	,	PUNCT
ijassa-2042	27	1	αβ	αβ	CCONJ
ijassa-2042	27	2	̸=	̸=	PROPN
ijassa-2042	27	3	0	0	NUM
ijassa-2042	27	4	.	.	PUNCT
ijassa-2042	28	1	(	(	PUNCT
ijassa-2042	28	2	1.3	1.3	NUM
ijassa-2042	28	3	)	)	PUNCT
ijassa-2042	28	4	fig	fig	NOUN
ijassa-2042	28	5	.	.	PUNCT
ijassa-2042	29	1	1.1	1.1	NUM
ijassa-2042	29	2	presents	present	VERB
ijassa-2042	29	3	the	the	DET
ijassa-2042	29	4	graphs	graph	NOUN
ijassa-2042	29	5	of	of	ADP
ijassa-2042	29	6	such	such	ADJ
ijassa-2042	29	7	curves	curve	NOUN
ijassa-2042	29	8	for	for	ADP
ijassa-2042	29	9	even	even	ADV
ijassa-2042	29	10	and	and	CCONJ
ijassa-2042	29	11	for	for	ADP
ijassa-2042	29	12	off	off	ADP
ijassa-2042	29	13	n.	n.	PROPN
ijassa-2042	29	14	the	the	DET
ijassa-2042	29	15	singular	singular	ADJ
ijassa-2042	29	16	point	point	NOUN
ijassa-2042	29	17	of	of	ADP
ijassa-2042	29	18	the	the	DET
ijassa-2042	29	19	curve	curve	NOUN
ijassa-2042	29	20	(	(	PUNCT
ijassa-2042	29	21	1.3	1.3	NUM
ijassa-2042	29	22	)	)	PUNCT
ijassa-2042	29	23	with	with	ADP
ijassa-2042	29	24	n	n	NOUN
ijassa-2042	29	25	=	=	SYM
ijassa-2042	29	26	2	2	NUM
ijassa-2042	29	27	is	be	AUX
ijassa-2042	29	28	called	call	VERB
ijassa-2042	29	29	semicubic	semicubic	ADJ
ijassa-2042	29	30	casp	casp	NOUN
ijassa-2042	29	31	or	or	CCONJ
ijassa-2042	29	32	simply	simply	ADV
ijassa-2042	29	33	casp	casp	ADJ
ijassa-2042	29	34	.	.	PUNCT
ijassa-2042	30	1	it	it	PRON
ijassa-2042	30	2	is	be	AUX
ijassa-2042	30	3	proved	prove	VERB
ijassa-2042	30	4	that	that	SCONJ
ijassa-2042	30	5	for	for	ADP
ijassa-2042	30	6	n	n	NOUN
ijassa-2042	30	7	=	=	SYM
ijassa-2042	30	8	2	2	NUM
ijassa-2042	30	9	or	or	CCONJ
ijassa-2042	30	10	3	3	NUM
ijassa-2042	30	11	the	the	DET
ijassa-2042	30	12	germ	germ	NOUN
ijassa-2042	30	13	of	of	ADP
ijassa-2042	30	14	any	any	DET
ijassa-2042	30	15	curve	curve	NOUN
ijassa-2042	30	16	(	(	PUNCT
ijassa-2042	30	17	1.3	1.3	NUM
ijassa-2042	30	18	)	)	PUNCT
ijassa-2042	30	19	at	at	ADP
ijassa-2042	30	20	zero	zero	NUM
ijassa-2042	30	21	can	can	AUX
ijassa-2042	30	22	be	be	AUX
ijassa-2042	30	23	brought	bring	VERB
ijassa-2042	30	24	to	to	ADP
ijassa-2042	30	25	the	the	DET
ijassa-2042	30	26	simplest	simple	ADJ
ijassa-2042	30	27	form	form	NOUN
ijassa-2042	30	28	x	x	NOUN
ijassa-2042	30	29	=	=	SYM
ijassa-2042	30	30	tn	tn	PROPN
ijassa-2042	30	31	,	,	PUNCT
ijassa-2042	30	32	y	y	PROPN
ijassa-2042	30	33	=	=	SYM
ijassa-2042	30	34	tn+1	tn+1	PROPN
ijassa-2042	30	35	by	by	ADP
ijassa-2042	30	36	means	mean	NOUN
ijassa-2042	30	37	of	of	ADP
ijassa-2042	30	38	an	an	DET
ijassa-2042	30	39	appropriate	appropriate	ADJ
ijassa-2042	30	40	change	change	NOUN
ijassa-2042	30	41	of	of	ADP
ijassa-2042	30	42	variables	variable	NOUN
ijassa-2042	30	43	(	(	PUNCT
ijassa-2042	30	44	x	x	X
ijassa-2042	30	45	,	,	PUNCT
ijassa-2042	30	46	y	y	PROPN
ijassa-2042	30	47	)	)	PUNCT
ijassa-2042	30	48	and	and	CCONJ
ijassa-2042	30	49	a	a	DET
ijassa-2042	30	50	change	change	NOUN
ijassa-2042	30	51	of	of	ADP
ijassa-2042	30	52	the	the	DET
ijassa-2042	30	53	parameter	parameter	NOUN
ijassa-2042	30	54	t.	t.	PROPN
ijassa-2042	30	55	however	however	ADV
ijassa-2042	30	56	,	,	PUNCT
ijassa-2042	30	57	for	for	ADP
ijassa-2042	30	58	n	n	PRON
ijassa-2042	30	59	≥	≥	NUM
ijassa-2042	30	60	4	4	NUM
ijassa-2042	30	61	it	it	PRON
ijassa-2042	30	62	is	be	AUX
ijassa-2042	30	63	not	not	PART
ijassa-2042	30	64	true	true	ADJ
ijassa-2042	30	65	[	[	X
ijassa-2042	30	66	6	6	NUM
ijassa-2042	30	67	]	]	PUNCT
ijassa-2042	30	68	.	.	PUNCT
ijassa-2042	31	1	fig	fig	NOUN
ijassa-2042	31	2	.	.	PUNCT
ijassa-2042	32	1	1.1	1.1	NUM
ijassa-2042	32	2	.	.	PUNCT
ijassa-2042	33	1	the	the	DET
ijassa-2042	33	2	curves	curve	NOUN
ijassa-2042	33	3	(	(	PUNCT
ijassa-2042	33	4	1.3	1.3	NUM
ijassa-2042	33	5	)	)	PUNCT
ijassa-2042	33	6	with	with	ADP
ijassa-2042	33	7	even	even	ADV
ijassa-2042	33	8	n	n	PROPN
ijassa-2042	33	9	(	(	PUNCT
ijassa-2042	33	10	left	left	ADJ
ijassa-2042	33	11	)	)	PUNCT
ijassa-2042	33	12	and	and	CCONJ
ijassa-2042	33	13	odd	odd	ADJ
ijassa-2042	33	14	n	n	CCONJ
ijassa-2042	33	15	(	(	PUNCT
ijassa-2042	33	16	right	right	NOUN
ijassa-2042	33	17	)	)	PUNCT
ijassa-2042	33	18	.	.	PUNCT
ijassa-2042	34	1	the	the	DET
ijassa-2042	34	2	picture	picture	NOUN
ijassa-2042	34	3	is	be	AUX
ijassa-2042	34	4	taken	take	VERB
ijassa-2042	34	5	from	from	ADP
ijassa-2042	34	6	[	[	X
ijassa-2042	34	7	1	1	NUM
ijassa-2042	34	8	]	]	PUNCT
ijassa-2042	34	9	.	.	PUNCT
ijassa-2042	35	1	2	2	X
ijassa-2042	35	2	.	.	X
ijassa-2042	35	3	legendre	legendre	PROPN
ijassa-2042	35	4	transformation	transformation	PROPN
ijassa-2042	35	5	the	the	DET
ijassa-2042	35	6	geometric	geometric	ADJ
ijassa-2042	35	7	definition	definition	NOUN
ijassa-2042	35	8	of	of	ADP
ijassa-2042	35	9	the	the	DET
ijassa-2042	35	10	legendre	legendre	PROPN
ijassa-2042	35	11	transformation	transformation	NOUN
ijassa-2042	35	12	requires	require	VERB
ijassa-2042	35	13	the	the	DET
ijassa-2042	35	14	notion	notion	NOUN
ijassa-2042	35	15	of	of	ADP
ijassa-2042	35	16	the	the	DET
ijassa-2042	35	17	legendrian	legendrian	ADJ
ijassa-2042	35	18	lift	lift	NOUN
ijassa-2042	35	19	.	.	PUNCT
ijassa-2042	36	1	we	we	PRON
ijassa-2042	36	2	start	start	VERB
ijassa-2042	36	3	with	with	ADP
ijassa-2042	36	4	the	the	DET
ijassa-2042	36	5	latter	latter	ADJ
ijassa-2042	36	6	notion	notion	NOUN
ijassa-2042	36	7	,	,	PUNCT
ijassa-2042	36	8	which	which	PRON
ijassa-2042	36	9	is	be	AUX
ijassa-2042	36	10	an	an	DET
ijassa-2042	36	11	important	important	ADJ
ijassa-2042	36	12	mathematical	mathematical	ADJ
ijassa-2042	36	13	procedure	procedure	NOUN
ijassa-2042	36	14	.	.	PUNCT
ijassa-2042	37	1	it	it	PRON
ijassa-2042	37	2	can	can	AUX
ijassa-2042	37	3	be	be	AUX
ijassa-2042	37	4	applied	apply	VERB
ijassa-2042	37	5	to	to	ADP
ijassa-2042	37	6	curves	curve	NOUN
ijassa-2042	37	7	,	,	PUNCT
ijassa-2042	37	8	functions	function	NOUN
ijassa-2042	37	9	and	and	CCONJ
ijassa-2042	37	10	many	many	ADJ
ijassa-2042	37	11	other	other	ADJ
ijassa-2042	37	12	objects	object	NOUN
ijassa-2042	37	13	.	.	PUNCT
ijassa-2042	38	1	the	the	DET
ijassa-2042	38	2	legendrian	legendrian	ADJ
ijassa-2042	38	3	lift	lift	NOUN
ijassa-2042	38	4	appears	appear	VERB
ijassa-2042	38	5	not	not	PART
ijassa-2042	38	6	only	only	ADV
ijassa-2042	38	7	in	in	ADP
ijassa-2042	38	8	mathematics	mathematic	NOUN
ijassa-2042	38	9	and	and	CCONJ
ijassa-2042	38	10	physics	physics	NOUN
ijassa-2042	38	11	,	,	PUNCT
ijassa-2042	38	12	but	but	CCONJ
ijassa-2042	38	13	also	also	ADV
ijassa-2042	38	14	in	in	ADP
ijassa-2042	38	15	various	various	ADJ
ijassa-2042	38	16	technical	technical	ADJ
ijassa-2042	38	17	applications	application	NOUN
ijassa-2042	38	18	and	and	CCONJ
ijassa-2042	38	19	even	even	ADV
ijassa-2042	38	20	in	in	ADP
ijassa-2042	38	21	the	the	DET
ijassa-2042	38	22	living	live	VERB
ijassa-2042	38	23	nature	nature	NOUN
ijassa-2042	38	24	.	.	PUNCT
ijassa-2042	39	1	we	we	PRON
ijassa-2042	39	2	shall	shall	AUX
ijassa-2042	39	3	say	say	VERB
ijassa-2042	39	4	a	a	DET
ijassa-2042	39	5	few	few	ADJ
ijassa-2042	39	6	words	word	NOUN
ijassa-2042	39	7	about	about	ADP
ijassa-2042	39	8	it	it	PRON
ijassa-2042	39	9	below	below	ADV
ijassa-2042	39	10	.	.	PUNCT
ijassa-2042	40	1	2.1	2.1	NUM
ijassa-2042	40	2	.	.	PUNCT
ijassa-2042	41	1	legendrian	legendrian	PROPN
ijassa-2042	41	2	lift	lift	VERB
ijassa-2042	41	3	the	the	DET
ijassa-2042	41	4	lift	lift	NOUN
ijassa-2042	41	5	or	or	CCONJ
ijassa-2042	41	6	1	1	NUM
ijassa-2042	41	7	-	-	PUNCT
ijassa-2042	41	8	graph	graph	NOUN
ijassa-2042	41	9	of	of	ADP
ijassa-2042	41	10	the	the	DET
ijassa-2042	41	11	curve	curve	NOUN
ijassa-2042	41	12	γ	γ	NOUN
ijassa-2042	41	13	given	give	VERB
ijassa-2042	41	14	by	by	ADP
ijassa-2042	41	15	formula	formula	NOUN
ijassa-2042	41	16	(	(	PUNCT
ijassa-2042	41	17	1.1	1.1	NUM
ijassa-2042	41	18	)	)	PUNCT
ijassa-2042	41	19	is	be	AUX
ijassa-2042	41	20	the	the	DET
ijassa-2042	41	21	curve	curve	NOUN
ijassa-2042	41	22	γ	γ	NOUN
ijassa-2042	41	23	:	:	PUNCT
ijassa-2042	41	24	x	x	SYM
ijassa-2042	41	25	=	=	SYM
ijassa-2042	41	26	ϕ(t	ϕ(t	NUM
ijassa-2042	41	27	)	)	PUNCT
ijassa-2042	41	28	,	,	PUNCT
ijassa-2042	41	29	y	y	PROPN
ijassa-2042	41	30	=	=	SYM
ijassa-2042	41	31	ψ(t	ψ(t	PROPN
ijassa-2042	41	32	)	)	PUNCT
ijassa-2042	41	33	,	,	PUNCT
ijassa-2042	41	34	p	p	NOUN
ijassa-2042	41	35	=	=	PUNCT
ijassa-2042	41	36	ψ′(t	ψ′(t	NOUN
ijassa-2042	41	37	)	)	PUNCT
ijassa-2042	41	38	ϕ′(t	ϕ′(t	NOUN
ijassa-2042	41	39	)	)	PUNCT
ijassa-2042	41	40	(	(	PUNCT
ijassa-2042	41	41	2.4	2.4	X
ijassa-2042	41	42	)	)	PUNCT
ijassa-2042	41	43	copyright	copyright	NOUN
ijassa-2042	41	44	©	©	PROPN
ijassa-2042	41	45	2025	2025	NUM
ijassa-2042	41	46	assa	assa	NOUN
ijassa-2042	41	47	.	.	PUNCT
ijassa-2042	42	1	adv	adv	PROPN
ijassa-2042	42	2	syst	syst	PROPN
ijassa-2042	42	3	sci	sci	PROPN
ijassa-2042	42	4	appl	appl	PROPN
ijassa-2042	42	5	(	(	PUNCT
ijassa-2042	42	6	2025	2025	NUM
ijassa-2042	42	7	)	)	PUNCT
ijassa-2042	42	8	46	46	NUM
ijassa-2042	42	9	n.	n.	PROPN
ijassa-2042	42	10	pavlova	pavlova	PROPN
ijassa-2042	42	11	,	,	PUNCT
ijassa-2042	42	12	a.	a.	NOUN
ijassa-2042	42	13	remizov	remizov	NOUN
ijassa-2042	42	14	in	in	ADP
ijassa-2042	42	15	3	3	NUM
ijassa-2042	42	16	-	-	PUNCT
ijassa-2042	42	17	dimensional	dimensional	ADJ
ijassa-2042	42	18	space	space	NOUN
ijassa-2042	42	19	with	with	ADP
ijassa-2042	42	20	the	the	DET
ijassa-2042	42	21	coordinates	coordinate	NOUN
ijassa-2042	42	22	(	(	PUNCT
ijassa-2042	42	23	x	x	X
ijassa-2042	42	24	,	,	PUNCT
ijassa-2042	42	25	y	y	PROPN
ijassa-2042	42	26	,	,	PUNCT
ijassa-2042	42	27	p	p	NOUN
ijassa-2042	42	28	)	)	PUNCT
ijassa-2042	42	29	.	.	PUNCT
ijassa-2042	43	1	this	this	DET
ijassa-2042	43	2	space	space	NOUN
ijassa-2042	43	3	denoted	denote	VERB
ijassa-2042	43	4	by	by	ADP
ijassa-2042	43	5	j1	j1	PROPN
ijassa-2042	43	6	is	be	AUX
ijassa-2042	43	7	called	call	VERB
ijassa-2042	43	8	the	the	DET
ijassa-2042	43	9	space	space	NOUN
ijassa-2042	43	10	of	of	ADP
ijassa-2042	43	11	1	1	NUM
ijassa-2042	43	12	-	-	PUNCT
ijassa-2042	43	13	jets	jet	NOUN
ijassa-2042	43	14	of	of	ADP
ijassa-2042	43	15	functions	function	NOUN
ijassa-2042	43	16	y(x	y(x	NOUN
ijassa-2042	43	17	)	)	PUNCT
ijassa-2042	43	18	.	.	PUNCT
ijassa-2042	44	1	in	in	ADP
ijassa-2042	44	2	the	the	DET
ijassa-2042	44	3	latter	latter	ADJ
ijassa-2042	44	4	equality	equality	NOUN
ijassa-2042	44	5	in	in	ADP
ijassa-2042	44	6	(	(	PUNCT
ijassa-2042	44	7	2.4	2.4	NUM
ijassa-2042	44	8	)	)	PUNCT
ijassa-2042	44	9	the	the	DET
ijassa-2042	44	10	denominator	denominator	NOUN
ijassa-2042	44	11	ϕ′(t	ϕ′(t	PROPN
ijassa-2042	44	12	)	)	PUNCT
ijassa-2042	44	13	may	may	AUX
ijassa-2042	44	14	vanish	vanish	VERB
ijassa-2042	44	15	,	,	PUNCT
ijassa-2042	44	16	and	and	CCONJ
ijassa-2042	44	17	we	we	PRON
ijassa-2042	44	18	have	have	VERB
ijassa-2042	44	19	to	to	PART
ijassa-2042	44	20	discuss	discuss	VERB
ijassa-2042	44	21	the	the	DET
ijassa-2042	44	22	correctness	correctness	NOUN
ijassa-2042	44	23	of	of	ADP
ijassa-2042	44	24	such	such	ADJ
ijassa-2042	44	25	operation	operation	NOUN
ijassa-2042	44	26	.	.	PUNCT
ijassa-2042	45	1	if	if	SCONJ
ijassa-2042	45	2	ϕ′(t	ϕ′(t	VERB
ijassa-2042	45	3	)	)	PUNCT
ijassa-2042	45	4	=	=	SYM
ijassa-2042	45	5	0	0	NUM
ijassa-2042	45	6	,	,	PUNCT
ijassa-2042	45	7	but	but	CCONJ
ijassa-2042	45	8	ψ′(t	ψ′(t	X
ijassa-2042	45	9	)	)	PUNCT
ijassa-2042	45	10	̸=	̸=	PROPN
ijassa-2042	45	11	0	0	NUM
ijassa-2042	45	12	,	,	PUNCT
ijassa-2042	45	13	we	we	PRON
ijassa-2042	45	14	have	have	VERB
ijassa-2042	45	15	p	p	NOUN
ijassa-2042	45	16	=	=	NOUN
ijassa-2042	45	17	∞	∞	NUM
ijassa-2042	45	18	at	at	ADP
ijassa-2042	45	19	the	the	DET
ijassa-2042	45	20	considered	considered	ADJ
ijassa-2042	45	21	point	point	NOUN
ijassa-2042	45	22	,	,	PUNCT
ijassa-2042	45	23	that	that	ADV
ijassa-2042	45	24	is	is	ADV
ijassa-2042	45	25	,	,	PUNCT
ijassa-2042	45	26	the	the	DET
ijassa-2042	45	27	tangent	tangent	ADJ
ijassa-2042	45	28	direction	direction	NOUN
ijassa-2042	45	29	of	of	ADP
ijassa-2042	45	30	the	the	DET
ijassa-2042	45	31	curve	curve	NOUN
ijassa-2042	45	32	γ	γ	NOUN
ijassa-2042	45	33	is	be	AUX
ijassa-2042	45	34	parallel	parallel	ADJ
ijassa-2042	45	35	to	to	ADP
ijassa-2042	45	36	the	the	DET
ijassa-2042	45	37	y	y	NOUN
ijassa-2042	45	38	-	-	PUNCT
ijassa-2042	45	39	axis	axis	NOUN
ijassa-2042	45	40	.	.	PUNCT
ijassa-2042	46	1	the	the	DET
ijassa-2042	46	2	dividing	dividing	NOUN
ijassa-2042	46	3	by	by	ADP
ijassa-2042	46	4	zero	zero	NUM
ijassa-2042	46	5	disappears	disappear	VERB
ijassa-2042	46	6	,	,	PUNCT
ijassa-2042	46	7	if	if	SCONJ
ijassa-2042	46	8	we	we	PRON
ijassa-2042	46	9	interchange	interchange	VERB
ijassa-2042	46	10	the	the	DET
ijassa-2042	46	11	x	x	NOUN
ijassa-2042	46	12	-	-	NOUN
ijassa-2042	46	13	axis	axis	NOUN
ijassa-2042	46	14	with	with	ADP
ijassa-2042	46	15	the	the	DET
ijassa-2042	46	16	y	y	NOUN
ijassa-2042	46	17	-	-	PUNCT
ijassa-2042	46	18	axis	axis	NOUN
ijassa-2042	46	19	.	.	PUNCT
ijassa-2042	47	1	the	the	DET
ijassa-2042	47	2	situation	situation	NOUN
ijassa-2042	47	3	is	be	AUX
ijassa-2042	47	4	more	more	ADV
ijassa-2042	47	5	complicated	complicated	ADJ
ijassa-2042	47	6	if	if	SCONJ
ijassa-2042	47	7	the	the	DET
ijassa-2042	47	8	curve	curve	NOUN
ijassa-2042	47	9	γ	γ	PROPN
ijassa-2042	47	10	has	have	VERB
ijassa-2042	47	11	a	a	DET
ijassa-2042	47	12	singular	singular	ADJ
ijassa-2042	47	13	point	point	NOUN
ijassa-2042	47	14	where	where	SCONJ
ijassa-2042	47	15	ϕ′(t	ϕ′(t	VERB
ijassa-2042	47	16	)	)	PUNCT
ijassa-2042	47	17	=	=	PUNCT
ijassa-2042	48	1	ψ′(t	ψ′(t	X
ijassa-2042	48	2	)	)	PUNCT
ijassa-2042	48	3	=	=	SYM
ijassa-2042	49	1	0	0	X
ijassa-2042	49	2	.	.	PUNCT
ijassa-2042	50	1	however	however	ADV
ijassa-2042	50	2	,	,	PUNCT
ijassa-2042	50	3	in	in	ADP
ijassa-2042	50	4	the	the	DET
ijassa-2042	50	5	most	most	ADJ
ijassa-2042	50	6	of	of	ADP
ijassa-2042	50	7	interesting	interesting	ADJ
ijassa-2042	50	8	cases	case	NOUN
ijassa-2042	50	9	the	the	DET
ijassa-2042	50	10	function	function	NOUN
ijassa-2042	50	11	ψ′(t)/ϕ′(t	ψ′(t)/ϕ′(t	NOUN
ijassa-2042	50	12	)	)	PUNCT
ijassa-2042	50	13	has	have	VERB
ijassa-2042	50	14	the	the	DET
ijassa-2042	50	15	same	same	ADJ
ijassa-2042	50	16	one	one	NUM
ijassa-2042	50	17	-	-	PUNCT
ijassa-2042	50	18	sided	sided	ADJ
ijassa-2042	50	19	limits	limit	NOUN
ijassa-2042	50	20	,	,	PUNCT
ijassa-2042	50	21	and	and	CCONJ
ijassa-2042	50	22	after	after	ADP
ijassa-2042	50	23	an	an	DET
ijassa-2042	50	24	appropriate	appropriate	ADJ
ijassa-2042	50	25	definition	definition	NOUN
ijassa-2042	50	26	at	at	ADP
ijassa-2042	50	27	the	the	DET
ijassa-2042	50	28	point	point	NOUN
ijassa-2042	50	29	itself	itself	PRON
ijassa-2042	50	30	,	,	PUNCT
ijassa-2042	50	31	the	the	DET
ijassa-2042	50	32	function	function	NOUN
ijassa-2042	50	33	ψ′(t)/ϕ′(t	ψ′(t)/ϕ′(t	NOUN
ijassa-2042	50	34	)	)	PUNCT
ijassa-2042	50	35	becomes	become	VERB
ijassa-2042	50	36	continuous	continuous	ADJ
ijassa-2042	50	37	.	.	PUNCT
ijassa-2042	51	1	for	for	ADP
ijassa-2042	51	2	example	example	NOUN
ijassa-2042	51	3	,	,	PUNCT
ijassa-2042	51	4	such	such	DET
ijassa-2042	51	5	a	a	DET
ijassa-2042	51	6	situation	situation	NOUN
ijassa-2042	51	7	takes	take	VERB
ijassa-2042	51	8	place	place	NOUN
ijassa-2042	51	9	for	for	ADP
ijassa-2042	51	10	the	the	DET
ijassa-2042	51	11	semicubic	semicubic	ADJ
ijassa-2042	51	12	parabola	parabola	PROPN
ijassa-2042	51	13	,	,	PUNCT
ijassa-2042	51	14	see	see	VERB
ijassa-2042	51	15	fig	fig	NOUN
ijassa-2042	51	16	.	.	PUNCT
ijassa-2042	52	1	2.2	2.2	NUM
ijassa-2042	52	2	.	.	PUNCT
ijassa-2042	52	3	fig	fig	NOUN
ijassa-2042	52	4	.	.	PUNCT
ijassa-2042	53	1	2.2	2.2	NUM
ijassa-2042	53	2	.	.	PUNCT
ijassa-2042	54	1	legendrian	legendrian	ADJ
ijassa-2042	54	2	lift	lift	NOUN
ijassa-2042	54	3	of	of	ADP
ijassa-2042	54	4	planar	planar	ADJ
ijassa-2042	54	5	curves	curve	NOUN
ijassa-2042	54	6	.	.	PUNCT
ijassa-2042	55	1	remark	remark	NOUN
ijassa-2042	55	2	2.1	2.1	NUM
ijassa-2042	55	3	:	:	PUNCT
ijassa-2042	55	4	the	the	DET
ijassa-2042	55	5	1	1	NUM
ijassa-2042	55	6	-	-	PUNCT
ijassa-2042	55	7	graph	graph	NOUN
ijassa-2042	55	8	of	of	ADP
ijassa-2042	55	9	the	the	DET
ijassa-2042	55	10	curve	curve	NOUN
ijassa-2042	55	11	γ	γ	PROPN
ijassa-2042	55	12	does	do	AUX
ijassa-2042	55	13	not	not	PART
ijassa-2042	55	14	depend	depend	VERB
ijassa-2042	55	15	on	on	ADP
ijassa-2042	55	16	the	the	DET
ijassa-2042	55	17	choice	choice	NOUN
ijassa-2042	55	18	of	of	ADP
ijassa-2042	55	19	the	the	DET
ijassa-2042	55	20	parametrization	parametrization	NOUN
ijassa-2042	55	21	the	the	DET
ijassa-2042	55	22	curve	curve	NOUN
ijassa-2042	55	23	γ	γ	X
ijassa-2042	55	24	.	.	PUNCT
ijassa-2042	56	1	according	accord	VERB
ijassa-2042	56	2	to	to	ADP
ijassa-2042	56	3	modern	modern	ADJ
ijassa-2042	56	4	biological	biological	ADJ
ijassa-2042	56	5	concepts	concept	NOUN
ijassa-2042	56	6	going	go	VERB
ijassa-2042	56	7	back	back	ADV
ijassa-2042	56	8	to	to	ADP
ijassa-2042	56	9	the	the	DET
ijassa-2042	56	10	famous	famous	ADJ
ijassa-2042	56	11	work	work	NOUN
ijassa-2042	56	12	of	of	ADP
ijassa-2042	56	13	hubel	hubel	NOUN
ijassa-2042	56	14	and	and	CCONJ
ijassa-2042	56	15	wiesel	wiesel	PROPN
ijassa-2042	56	16	(	(	PUNCT
ijassa-2042	56	17	1959	1959	NUM
ijassa-2042	56	18	)	)	PUNCT
ijassa-2042	56	19	,	,	PUNCT
ijassa-2042	56	20	the	the	DET
ijassa-2042	56	21	brain	brain	NOUN
ijassa-2042	56	22	of	of	ADP
ijassa-2042	56	23	mammals	mammal	NOUN
ijassa-2042	56	24	performs	perform	VERB
ijassa-2042	56	25	the	the	DET
ijassa-2042	56	26	legendrian	legendrian	ADJ
ijassa-2042	56	27	lift	lift	NOUN
ijassa-2042	56	28	of	of	ADP
ijassa-2042	56	29	apparent	apparent	ADJ
ijassa-2042	56	30	contours	contours	NOUN
ijassa-2042	56	31	of	of	ADP
ijassa-2042	56	32	flat	flat	ADJ
ijassa-2042	56	33	images	image	NOUN
ijassa-2042	56	34	perceived	perceive	VERB
ijassa-2042	56	35	by	by	ADP
ijassa-2042	56	36	the	the	DET
ijassa-2042	56	37	retina	retina	NOUN
ijassa-2042	56	38	.	.	PUNCT
ijassa-2042	57	1	there	there	PRON
ijassa-2042	57	2	exists	exist	VERB
ijassa-2042	57	3	an	an	DET
ijassa-2042	57	4	area	area	NOUN
ijassa-2042	57	5	of	of	ADP
ijassa-2042	57	6	the	the	DET
ijassa-2042	57	7	primary	primary	ADJ
ijassa-2042	57	8	cortex	cortex	NOUN
ijassa-2042	57	9	of	of	ADP
ijassa-2042	57	10	the	the	DET
ijassa-2042	57	11	brain	brain	NOUN
ijassa-2042	57	12	assiciated	assiciate	VERB
ijassa-2042	57	13	with	with	ADP
ijassa-2042	57	14	vision	vision	NOUN
ijassa-2042	57	15	(	(	PUNCT
ijassa-2042	57	16	v1	v1	NOUN
ijassa-2042	57	17	)	)	PUNCT
ijassa-2042	57	18	that	that	PRON
ijassa-2042	57	19	contains	contain	VERB
ijassa-2042	57	20	neurons	neuron	NOUN
ijassa-2042	57	21	united	unite	VERB
ijassa-2042	57	22	in	in	ADP
ijassa-2042	57	23	groups	group	NOUN
ijassa-2042	57	24	with	with	ADP
ijassa-2042	57	25	complex	complex	ADJ
ijassa-2042	57	26	hierarchy	hierarchy	NOUN
ijassa-2042	57	27	(	(	PUNCT
ijassa-2042	57	28	columns	column	NOUN
ijassa-2042	57	29	,	,	PUNCT
ijassa-2042	57	30	hypercolumn	hypercolumn	PROPN
ijassa-2042	57	31	)	)	PUNCT
ijassa-2042	57	32	.	.	PUNCT
ijassa-2042	58	1	each	each	DET
ijassa-2042	58	2	column	column	NOUN
ijassa-2042	58	3	is	be	AUX
ijassa-2042	58	4	sensitive	sensitive	ADJ
ijassa-2042	58	5	to	to	ADP
ijassa-2042	58	6	a	a	DET
ijassa-2042	58	7	certain	certain	ADJ
ijassa-2042	58	8	point	point	NOUN
ijassa-2042	58	9	of	of	ADP
ijassa-2042	58	10	the	the	DET
ijassa-2042	58	11	retina	retina	NOUN
ijassa-2042	58	12	and	and	CCONJ
ijassa-2042	58	13	a	a	DET
ijassa-2042	58	14	certain	certain	ADJ
ijassa-2042	58	15	tangent	tangent	NOUN
ijassa-2042	58	16	direction	direction	NOUN
ijassa-2042	58	17	at	at	ADP
ijassa-2042	58	18	this	this	DET
ijassa-2042	58	19	point	point	NOUN
ijassa-2042	58	20	.	.	PUNCT
ijassa-2042	59	1	flat	flat	ADJ
ijassa-2042	59	2	images	image	NOUN
ijassa-2042	59	3	perceived	perceive	VERB
ijassa-2042	59	4	by	by	ADP
ijassa-2042	59	5	the	the	DET
ijassa-2042	59	6	retina	retina	NOUN
ijassa-2042	59	7	are	be	AUX
ijassa-2042	59	8	lifted	lift	VERB
ijassa-2042	59	9	in	in	ADP
ijassa-2042	59	10	the	the	DET
ijassa-2042	59	11	space	space	NOUN
ijassa-2042	59	12	of	of	ADP
ijassa-2042	59	13	1	1	NUM
ijassa-2042	59	14	-	-	PUNCT
ijassa-2042	59	15	jets	jet	NOUN
ijassa-2042	59	16	,	,	PUNCT
ijassa-2042	59	17	and	and	CCONJ
ijassa-2042	59	18	further	further	ADJ
ijassa-2042	59	19	work	work	NOUN
ijassa-2042	59	20	of	of	ADP
ijassa-2042	59	21	the	the	DET
ijassa-2042	59	22	brain	brain	NOUN
ijassa-2042	59	23	occurs	occur	VERB
ijassa-2042	59	24	with	with	ADP
ijassa-2042	59	25	the	the	DET
ijassa-2042	59	26	lifted	lift	VERB
ijassa-2042	59	27	objects	object	NOUN
ijassa-2042	59	28	.	.	PUNCT
ijassa-2042	60	1	see	see	VERB
ijassa-2042	60	2	[	[	X
ijassa-2042	60	3	13	13	NUM
ijassa-2042	60	4	]	]	PUNCT
ijassa-2042	60	5	and	and	CCONJ
ijassa-2042	60	6	the	the	DET
ijassa-2042	60	7	references	reference	NOUN
ijassa-2042	60	8	therein	therein	ADV
ijassa-2042	60	9	.	.	PUNCT
ijassa-2042	61	1	the	the	DET
ijassa-2042	61	2	only	only	ADJ
ijassa-2042	61	3	difference	difference	NOUN
ijassa-2042	61	4	between	between	ADP
ijassa-2042	61	5	the	the	DET
ijassa-2042	61	6	lift	lift	NOUN
ijassa-2042	61	7	performed	perform	VERB
ijassa-2042	61	8	by	by	ADP
ijassa-2042	61	9	the	the	DET
ijassa-2042	61	10	brain	brain	NOUN
ijassa-2042	61	11	and	and	CCONJ
ijassa-2042	61	12	the	the	DET
ijassa-2042	61	13	lift	lift	NOUN
ijassa-2042	61	14	described	describe	VERB
ijassa-2042	61	15	above	above	ADV
ijassa-2042	61	16	is	be	AUX
ijassa-2042	61	17	that	that	SCONJ
ijassa-2042	61	18	the	the	DET
ijassa-2042	61	19	brain	brain	NOUN
ijassa-2042	61	20	uses	use	VERB
ijassa-2042	61	21	the	the	DET
ijassa-2042	61	22	third	third	ADJ
ijassa-2042	61	23	coordinate	coordinate	NOUN
ijassa-2042	61	24	θ	θ	PROPN
ijassa-2042	61	25	=	=	SYM
ijassa-2042	61	26	arctan	arctan	PROPN
ijassa-2042	61	27	p	p	NOUN
ijassa-2042	61	28	instead	instead	ADV
ijassa-2042	61	29	of	of	ADP
ijassa-2042	62	1	p.	p.	NOUN
ijassa-2042	63	1	it	it	PRON
ijassa-2042	63	2	is	be	AUX
ijassa-2042	63	3	very	very	ADV
ijassa-2042	63	4	natural	natural	ADJ
ijassa-2042	63	5	:	:	PUNCT
ijassa-2042	63	6	θ	θ	NOUN
ijassa-2042	63	7	is	be	AUX
ijassa-2042	63	8	more	more	ADV
ijassa-2042	63	9	appropriate	appropriate	ADJ
ijassa-2042	63	10	for	for	ADP
ijassa-2042	63	11	procedure	procedure	NOUN
ijassa-2042	63	12	performed	perform	VERB
ijassa-2042	63	13	biologically	biologically	ADV
ijassa-2042	63	14	,	,	PUNCT
ijassa-2042	63	15	since	since	SCONJ
ijassa-2042	63	16	it	it	PRON
ijassa-2042	63	17	is	be	AUX
ijassa-2042	63	18	bounded	bound	VERB
ijassa-2042	63	19	,	,	PUNCT
ijassa-2042	63	20	while	while	SCONJ
ijassa-2042	63	21	p	p	NOUN
ijassa-2042	63	22	is	be	AUX
ijassa-2042	63	23	not	not	PART
ijassa-2042	63	24	.	.	PUNCT
ijassa-2042	64	1	it	it	PRON
ijassa-2042	64	2	is	be	AUX
ijassa-2042	64	3	worth	worth	ADJ
ijassa-2042	64	4	observing	observe	VERB
ijassa-2042	64	5	that	that	SCONJ
ijassa-2042	64	6	this	this	DET
ijassa-2042	64	7	theoretical	theoretical	ADJ
ijassa-2042	64	8	concept	concept	NOUN
ijassa-2042	64	9	helped	help	VERB
ijassa-2042	64	10	to	to	PART
ijassa-2042	64	11	construct	construct	VERB
ijassa-2042	64	12	some	some	DET
ijassa-2042	64	13	inpainting	inpainting	ADJ
ijassa-2042	64	14	algorithms	algorithm	NOUN
ijassa-2042	64	15	that	that	PRON
ijassa-2042	64	16	allow	allow	VERB
ijassa-2042	64	17	to	to	PART
ijassa-2042	64	18	recover	recover	VERB
ijassa-2042	64	19	corrupted	corrupted	ADJ
ijassa-2042	64	20	images	image	NOUN
ijassa-2042	64	21	.	.	PUNCT
ijassa-2042	65	1	in	in	ADP
ijassa-2042	65	2	these	these	DET
ijassa-2042	65	3	algorithms	algorithm	NOUN
ijassa-2042	65	4	,	,	PUNCT
ijassa-2042	65	5	the	the	DET
ijassa-2042	65	6	legendrian	legendrian	ADJ
ijassa-2042	65	7	lift	lift	NOUN
ijassa-2042	65	8	also	also	ADV
ijassa-2042	65	9	plays	play	VERB
ijassa-2042	65	10	a	a	DET
ijassa-2042	65	11	crucial	crucial	ADJ
ijassa-2042	65	12	role	role	NOUN
ijassa-2042	65	13	:	:	PUNCT
ijassa-2042	65	14	first	first	ADV
ijassa-2042	65	15	,	,	PUNCT
ijassa-2042	65	16	we	we	PRON
ijassa-2042	65	17	lift	lift	VERB
ijassa-2042	65	18	the	the	DET
ijassa-2042	65	19	image	image	NOUN
ijassa-2042	65	20	,	,	PUNCT
ijassa-2042	65	21	which	which	PRON
ijassa-2042	65	22	is	be	AUX
ijassa-2042	65	23	considered	consider	VERB
ijassa-2042	65	24	as	as	ADP
ijassa-2042	65	25	a	a	DET
ijassa-2042	65	26	family	family	NOUN
ijassa-2042	65	27	of	of	ADP
ijassa-2042	65	28	contours	contours	PROPN
ijassa-2042	65	29	,	,	PUNCT
ijassa-2042	65	30	in	in	ADP
ijassa-2042	65	31	3	3	NUM
ijassa-2042	65	32	-	-	PUNCT
ijassa-2042	65	33	dimensional	dimensional	ADJ
ijassa-2042	65	34	space	space	NOUN
ijassa-2042	65	35	and	and	CCONJ
ijassa-2042	65	36	then	then	ADV
ijassa-2042	65	37	we	we	PRON
ijassa-2042	65	38	act	act	VERB
ijassa-2042	65	39	to	to	ADP
ijassa-2042	65	40	the	the	DET
ijassa-2042	65	41	lifted	lift	VERB
ijassa-2042	65	42	object	object	NOUN
ijassa-2042	65	43	a	a	DET
ijassa-2042	65	44	hypoelliptic	hypoelliptic	ADJ
ijassa-2042	65	45	diffusion	diffusion	NOUN
ijassa-2042	65	46	associated	associate	VERB
ijassa-2042	65	47	with	with	ADP
ijassa-2042	65	48	the	the	DET
ijassa-2042	65	49	contact	contact	NOUN
ijassa-2042	65	50	structure	structure	NOUN
ijassa-2042	65	51	.	.	PUNCT
ijassa-2042	66	1	for	for	ADP
ijassa-2042	66	2	details	detail	NOUN
ijassa-2042	66	3	,	,	PUNCT
ijassa-2042	66	4	see	see	VERB
ijassa-2042	66	5	[	[	X
ijassa-2042	66	6	4	4	NUM
ijassa-2042	66	7	,	,	PUNCT
ijassa-2042	66	8	5	5	NUM
ijassa-2042	66	9	]	]	PUNCT
ijassa-2042	66	10	.	.	PUNCT
ijassa-2042	67	1	2.2	2.2	NUM
ijassa-2042	67	2	.	.	PUNCT
ijassa-2042	68	1	legendre	legendre	PROPN
ijassa-2042	68	2	transformation	transformation	PROPN
ijassa-2042	68	3	and	and	CCONJ
ijassa-2042	68	4	duality	duality	NOUN
ijassa-2042	68	5	legendre	legendre	PROPN
ijassa-2042	68	6	transformation	transformation	NOUN
ijassa-2042	68	7	is	be	AUX
ijassa-2042	68	8	an	an	DET
ijassa-2042	68	9	automorphism	automorphism	NOUN
ijassa-2042	68	10	λ	λ	NOUN
ijassa-2042	68	11	:	:	PUNCT
ijassa-2042	68	12	j1	j1	PROPN
ijassa-2042	68	13	→	→	SYM
ijassa-2042	68	14	j1	j1	PROPN
ijassa-2042	68	15	given	give	VERB
ijassa-2042	68	16	by	by	ADP
ijassa-2042	68	17	the	the	DET
ijassa-2042	68	18	formula	formula	NOUN
ijassa-2042	68	19	(	(	PUNCT
ijassa-2042	68	20	x	x	X
ijassa-2042	68	21	,	,	PUNCT
ijassa-2042	68	22	y	y	PROPN
ijassa-2042	68	23	,	,	PUNCT
ijassa-2042	68	24	p	p	X
ijassa-2042	68	25	)	)	PUNCT
ijassa-2042	68	26	7→	7→	NUM
ijassa-2042	68	27	(	(	PUNCT
ijassa-2042	68	28	x	x	X
ijassa-2042	68	29	,	,	PUNCT
ijassa-2042	68	30	y	y	PROPN
ijassa-2042	68	31	,	,	PUNCT
ijassa-2042	68	32	p	p	NOUN
ijassa-2042	68	33	)	)	PUNCT
ijassa-2042	68	34	,	,	PUNCT
ijassa-2042	68	35	where	where	SCONJ
ijassa-2042	68	36	the	the	DET
ijassa-2042	68	37	coordinates	coordinate	NOUN
ijassa-2042	68	38	are	be	AUX
ijassa-2042	68	39	connected	connect	VERB
ijassa-2042	68	40	by	by	ADP
ijassa-2042	68	41	the	the	DET
ijassa-2042	68	42	equalities	equality	NOUN
ijassa-2042	68	43	:	:	PUNCT
ijassa-2042	68	44	x	x	SYM
ijassa-2042	68	45	=	=	SYM
ijassa-2042	68	46	p	p	X
ijassa-2042	68	47	,	,	PUNCT
ijassa-2042	68	48	p	p	X
ijassa-2042	68	49	=	=	PUNCT
ijassa-2042	68	50	x	x	PROPN
ijassa-2042	68	51	,	,	PUNCT
ijassa-2042	68	52	y	y	PROPN
ijassa-2042	68	53	+	+	NUM
ijassa-2042	68	54	y	y	PROPN
ijassa-2042	68	55	=	=	SYM
ijassa-2042	68	56	xp	xp	PROPN
ijassa-2042	69	1	=	=	SYM
ijassa-2042	69	2	xp	xp	PROPN
ijassa-2042	69	3	.	.	PUNCT
ijassa-2042	70	1	(	(	PUNCT
ijassa-2042	70	2	2.5	2.5	NUM
ijassa-2042	70	3	)	)	PUNCT
ijassa-2042	70	4	the	the	DET
ijassa-2042	70	5	legendre	legendre	PROPN
ijassa-2042	70	6	transformation	transformation	NOUN
ijassa-2042	70	7	can	can	AUX
ijassa-2042	70	8	be	be	AUX
ijassa-2042	70	9	applied	apply	VERB
ijassa-2042	70	10	to	to	ADP
ijassa-2042	70	11	various	various	ADJ
ijassa-2042	70	12	objects	object	NOUN
ijassa-2042	70	13	:	:	PUNCT
ijassa-2042	70	14	curves	curve	NOUN
ijassa-2042	70	15	,	,	PUNCT
ijassa-2042	70	16	functions	function	NOUN
ijassa-2042	70	17	,	,	PUNCT
ijassa-2042	70	18	differential	differential	ADJ
ijassa-2042	70	19	equations	equation	NOUN
ijassa-2042	70	20	,	,	PUNCT
ijassa-2042	70	21	etc	etc	X
ijassa-2042	70	22	.	.	X
ijassa-2042	71	1	the	the	DET
ijassa-2042	71	2	objects	object	NOUN
ijassa-2042	71	3	obtained	obtain	VERB
ijassa-2042	71	4	in	in	ADP
ijassa-2042	71	5	this	this	DET
ijassa-2042	71	6	way	way	NOUN
ijassa-2042	71	7	are	be	AUX
ijassa-2042	71	8	called	call	VERB
ijassa-2042	71	9	dual	dual	ADJ
ijassa-2042	71	10	of	of	ADP
ijassa-2042	71	11	the	the	DET
ijassa-2042	71	12	original	original	NOUN
ijassa-2042	71	13	,	,	PUNCT
ijassa-2042	71	14	they	they	PRON
ijassa-2042	71	15	are	be	AUX
ijassa-2042	71	16	often	often	ADV
ijassa-2042	71	17	denoted	denote	VERB
ijassa-2042	71	18	by	by	ADP
ijassa-2042	71	19	the	the	DET
ijassa-2042	71	20	same	same	ADJ
ijassa-2042	71	21	symbols	symbol	NOUN
ijassa-2042	71	22	with	with	ADP
ijassa-2042	71	23	asterisk	asterisk	NOUN
ijassa-2042	71	24	.	.	PUNCT
ijassa-2042	72	1	dual	dual	ADJ
ijassa-2042	72	2	objects	object	NOUN
ijassa-2042	72	3	reflect	reflect	VERB
ijassa-2042	72	4	many	many	ADJ
ijassa-2042	72	5	important	important	ADJ
ijassa-2042	72	6	properties	property	NOUN
ijassa-2042	72	7	of	of	ADP
ijassa-2042	72	8	their	their	PRON
ijassa-2042	72	9	preimages	preimage	NOUN
ijassa-2042	72	10	.	.	PUNCT
ijassa-2042	73	1	copyright	copyright	NOUN
ijassa-2042	73	2	©	©	PROPN
ijassa-2042	73	3	2025	2025	NUM
ijassa-2042	73	4	assa	assa	NOUN
ijassa-2042	73	5	.	.	PUNCT
ijassa-2042	74	1	adv	adv	PROPN
ijassa-2042	74	2	syst	syst	PROPN
ijassa-2042	74	3	sci	sci	PROPN
ijassa-2042	74	4	appl	appl	PROPN
ijassa-2042	74	5	(	(	PUNCT
ijassa-2042	74	6	2025	2025	NUM
ijassa-2042	74	7	)	)	PUNCT
ijassa-2042	74	8	legendre	legendre	PROPN
ijassa-2042	74	9	transformation	transformation	NOUN
ijassa-2042	74	10	and	and	CCONJ
ijassa-2042	74	11	its	its	PRON
ijassa-2042	74	12	applications	application	NOUN
ijassa-2042	74	13	47	47	NUM
ijassa-2042	74	14	consider	consider	VERB
ijassa-2042	74	15	the	the	DET
ijassa-2042	74	16	application	application	NOUN
ijassa-2042	74	17	of	of	ADP
ijassa-2042	74	18	the	the	DET
ijassa-2042	74	19	legendre	legendre	PROPN
ijassa-2042	74	20	transformation	transformation	NOUN
ijassa-2042	74	21	to	to	ADP
ijassa-2042	74	22	the	the	DET
ijassa-2042	74	23	curve	curve	NOUN
ijassa-2042	74	24	γ	γ	NOUN
ijassa-2042	74	25	given	give	VERB
ijassa-2042	74	26	by	by	ADP
ijassa-2042	74	27	formula	formula	NOUN
ijassa-2042	74	28	(	(	PUNCT
ijassa-2042	74	29	1.1	1.1	NUM
ijassa-2042	74	30	)	)	PUNCT
ijassa-2042	74	31	.	.	PUNCT
ijassa-2042	75	1	it	it	PRON
ijassa-2042	75	2	is	be	AUX
ijassa-2042	75	3	expressed	express	VERB
ijassa-2042	75	4	by	by	ADP
ijassa-2042	75	5	the	the	DET
ijassa-2042	75	6	diagram	diagram	NOUN
ijassa-2042	75	7	presented	present	VERB
ijassa-2042	75	8	in	in	ADP
ijassa-2042	75	9	fig	fig	NOUN
ijassa-2042	75	10	.	.	PUNCT
ijassa-2042	76	1	2.3	2.3	NUM
ijassa-2042	76	2	.	.	PUNCT
ijassa-2042	76	3	fig	fig	NOUN
ijassa-2042	76	4	.	.	PUNCT
ijassa-2042	77	1	2.3	2.3	NUM
ijassa-2042	77	2	.	.	PUNCT
ijassa-2042	78	1	definition	definition	NOUN
ijassa-2042	78	2	of	of	ADP
ijassa-2042	78	3	the	the	DET
ijassa-2042	78	4	legendrian	legendrian	ADJ
ijassa-2042	78	5	lift	lift	NOUN
ijassa-2042	78	6	of	of	ADP
ijassa-2042	78	7	a	a	DET
ijassa-2042	78	8	planar	planar	ADJ
ijassa-2042	78	9	curve	curve	NOUN
ijassa-2042	78	10	.	.	PUNCT
ijassa-2042	79	1	in	in	ADP
ijassa-2042	79	2	fig	fig	NOUN
ijassa-2042	79	3	.	.	PUNCT
ijassa-2042	80	1	2.3	2.3	NUM
ijassa-2042	80	2	,	,	PUNCT
ijassa-2042	80	3	the	the	DET
ijassa-2042	80	4	vertical	vertical	ADJ
ijassa-2042	80	5	upward	upward	ADJ
ijassa-2042	80	6	arrow	arrow	NOUN
ijassa-2042	80	7	denotes	denote	NOUN
ijassa-2042	80	8	the	the	DET
ijassa-2042	80	9	legendrian	legendrian	ADJ
ijassa-2042	80	10	lift	lift	NOUN
ijassa-2042	80	11	of	of	ADP
ijassa-2042	80	12	the	the	DET
ijassa-2042	80	13	curve	curve	NOUN
ijassa-2042	80	14	γ	γ	NOUN
ijassa-2042	80	15	to	to	ADP
ijassa-2042	80	16	the	the	DET
ijassa-2042	80	17	space	space	NOUN
ijassa-2042	80	18	j1	j1	NOUN
ijassa-2042	80	19	with	with	ADP
ijassa-2042	80	20	coordinates	coordinate	NOUN
ijassa-2042	80	21	(	(	PUNCT
ijassa-2042	80	22	x	x	X
ijassa-2042	80	23	,	,	PUNCT
ijassa-2042	80	24	y	y	PROPN
ijassa-2042	80	25	,	,	PUNCT
ijassa-2042	80	26	p	p	NOUN
ijassa-2042	80	27	)	)	PUNCT
ijassa-2042	80	28	.	.	PUNCT
ijassa-2042	81	1	the	the	DET
ijassa-2042	81	2	horizontal	horizontal	ADJ
ijassa-2042	81	3	arrow	arrow	NOUN
ijassa-2042	81	4	denotes	denote	VERB
ijassa-2042	81	5	the	the	DET
ijassa-2042	81	6	legendre	legendre	PROPN
ijassa-2042	81	7	transformation	transformation	PROPN
ijassa-2042	81	8	λ	λ	PROPN
ijassa-2042	81	9	:	:	PUNCT
ijassa-2042	81	10	j1	j1	PROPN
ijassa-2042	81	11	→	→	SYM
ijassa-2042	81	12	j1	j1	PROPN
ijassa-2042	81	13	given	give	VERB
ijassa-2042	81	14	by	by	ADP
ijassa-2042	81	15	formula	formula	NOUN
ijassa-2042	81	16	(	(	PUNCT
ijassa-2042	81	17	2.5	2.5	NUM
ijassa-2042	81	18	)	)	PUNCT
ijassa-2042	81	19	,	,	PUNCT
ijassa-2042	81	20	the	the	DET
ijassa-2042	81	21	vertical	vertical	ADJ
ijassa-2042	81	22	downward	downward	ADJ
ijassa-2042	81	23	arrow	arrow	NOUN
ijassa-2042	81	24	denotes	denote	NOUN
ijassa-2042	81	25	the	the	DET
ijassa-2042	81	26	projection	projection	NOUN
ijassa-2042	81	27	π	π	X
ijassa-2042	81	28	:	:	PUNCT
ijassa-2042	81	29	(	(	PUNCT
ijassa-2042	81	30	x	x	X
ijassa-2042	81	31	,	,	PUNCT
ijassa-2042	81	32	y	y	PROPN
ijassa-2042	81	33	,	,	PUNCT
ijassa-2042	81	34	p	p	NOUN
ijassa-2042	81	35	)	)	PUNCT
ijassa-2042	81	36	7→	7→	NUM
ijassa-2042	81	37	(	(	PUNCT
ijassa-2042	81	38	x	x	X
ijassa-2042	81	39	,	,	PUNCT
ijassa-2042	81	40	y	y	PROPN
ijassa-2042	81	41	)	)	PUNCT
ijassa-2042	81	42	.	.	PUNCT
ijassa-2042	82	1	the	the	DET
ijassa-2042	82	2	composition	composition	NOUN
ijassa-2042	82	3	of	of	ADP
ijassa-2042	82	4	these	these	DET
ijassa-2042	82	5	three	three	NUM
ijassa-2042	82	6	mappings	mapping	NOUN
ijassa-2042	82	7	results	result	NOUN
ijassa-2042	82	8	in	in	ADP
ijassa-2042	82	9	the	the	DET
ijassa-2042	82	10	legendre	legendre	PROPN
ijassa-2042	82	11	transformation	transformation	NOUN
ijassa-2042	82	12	of	of	ADP
ijassa-2042	82	13	the	the	DET
ijassa-2042	82	14	curve	curve	NOUN
ijassa-2042	82	15	γ	γ	NOUN
ijassa-2042	82	16	on	on	ADP
ijassa-2042	82	17	the	the	DET
ijassa-2042	82	18	(	(	PUNCT
ijassa-2042	82	19	x	x	NOUN
ijassa-2042	82	20	,	,	PUNCT
ijassa-2042	82	21	y)-plane	y)-plane	VERB
ijassa-2042	82	22	to	to	ADP
ijassa-2042	82	23	a	a	DET
ijassa-2042	82	24	certain	certain	ADJ
ijassa-2042	82	25	curve	curve	NOUN
ijassa-2042	82	26	γ∗	γ∗	NOUN
ijassa-2042	82	27	on	on	ADP
ijassa-2042	82	28	the	the	DET
ijassa-2042	82	29	(	(	PUNCT
ijassa-2042	82	30	x	x	PROPN
ijassa-2042	82	31	,	,	PUNCT
ijassa-2042	82	32	y	y	NOUN
ijassa-2042	82	33	)	)	PUNCT
ijassa-2042	82	34	-plane	-plane	PROPN
ijassa-2042	82	35	,	,	PUNCT
ijassa-2042	82	36	which	which	PRON
ijassa-2042	82	37	is	be	AUX
ijassa-2042	82	38	called	call	VERB
ijassa-2042	82	39	dual	dual	ADJ
ijassa-2042	82	40	to	to	ADP
ijassa-2042	82	41	γ	γ	PROPN
ijassa-2042	82	42	.	.	PUNCT
ijassa-2042	83	1	the	the	DET
ijassa-2042	83	2	curve	curve	NOUN
ijassa-2042	83	3	γ∗	γ∗	NOUN
ijassa-2042	83	4	dual	dual	ADJ
ijassa-2042	83	5	to	to	ADP
ijassa-2042	83	6	a	a	DET
ijassa-2042	83	7	curve	curve	NOUN
ijassa-2042	83	8	γ	γ	NOUN
ijassa-2042	83	9	can	can	AUX
ijassa-2042	83	10	have	have	VERB
ijassa-2042	83	11	singular	singular	ADJ
ijassa-2042	83	12	points	point	NOUN
ijassa-2042	83	13	even	even	ADV
ijassa-2042	83	14	γ	γ	PROPN
ijassa-2042	83	15	is	be	AUX
ijassa-2042	83	16	regular	regular	ADJ
ijassa-2042	83	17	.	.	PUNCT
ijassa-2042	84	1	obviously	obviously	ADV
ijassa-2042	84	2	,	,	PUNCT
ijassa-2042	84	3	they	they	PRON
ijassa-2042	84	4	appear	appear	VERB
ijassa-2042	84	5	due	due	ADJ
ijassa-2042	84	6	to	to	ADP
ijassa-2042	84	7	the	the	DET
ijassa-2042	84	8	projection	projection	NOUN
ijassa-2042	84	9	π	π	NOUN
ijassa-2042	84	10	:	:	PUNCT
ijassa-2042	84	11	(	(	PUNCT
ijassa-2042	84	12	x	x	X
ijassa-2042	84	13	,	,	PUNCT
ijassa-2042	84	14	y	y	PROPN
ijassa-2042	84	15	,	,	PUNCT
ijassa-2042	84	16	p	p	NOUN
ijassa-2042	84	17	)	)	PUNCT
ijassa-2042	84	18	7→	7→	NUM
ijassa-2042	84	19	(	(	PUNCT
ijassa-2042	84	20	x	x	X
ijassa-2042	84	21	,	,	PUNCT
ijassa-2042	84	22	y	y	PROPN
ijassa-2042	84	23	)	)	PUNCT
ijassa-2042	84	24	.	.	PUNCT
ijassa-2042	85	1	example	example	NOUN
ijassa-2042	85	2	2.1	2.1	NUM
ijassa-2042	85	3	:	:	PUNCT
ijassa-2042	85	4	let	let	VERB
ijassa-2042	85	5	us	we	PRON
ijassa-2042	85	6	find	find	VERB
ijassa-2042	85	7	the	the	DET
ijassa-2042	85	8	dual	dual	ADJ
ijassa-2042	85	9	curves	curve	NOUN
ijassa-2042	85	10	for	for	ADP
ijassa-2042	85	11	the	the	DET
ijassa-2042	85	12	line	line	NOUN
ijassa-2042	85	13	γ1	γ1	NOUN
ijassa-2042	85	14	:	:	PUNCT
ijassa-2042	86	1	y	y	PROPN
ijassa-2042	86	2	=	=	PUNCT
ijassa-2042	86	3	at+	at+	PROPN
ijassa-2042	86	4	b	b	NOUN
ijassa-2042	86	5	,	,	PUNCT
ijassa-2042	86	6	the	the	DET
ijassa-2042	86	7	parabola	parabola	PROPN
ijassa-2042	86	8	γ2	γ2	PROPN
ijassa-2042	86	9	:	:	PUNCT
ijassa-2042	86	10	y	y	PROPN
ijassa-2042	86	11	=	=	SYM
ijassa-2042	86	12	x2	x2	PROPN
ijassa-2042	86	13	,	,	PUNCT
ijassa-2042	86	14	and	and	CCONJ
ijassa-2042	86	15	the	the	DET
ijassa-2042	86	16	cubic	cubic	ADJ
ijassa-2042	86	17	parabola	parabola	PROPN
ijassa-2042	86	18	γ3	γ3	NOUN
ijassa-2042	86	19	:	:	PUNCT
ijassa-2042	86	20	y	y	PROPN
ijassa-2042	86	21	=	=	SYM
ijassa-2042	86	22	x3	x3	PROPN
ijassa-2042	86	23	.	.	PUNCT
ijassa-2042	87	1	the	the	DET
ijassa-2042	87	2	line	line	NOUN
ijassa-2042	87	3	γ1	γ1	NOUN
ijassa-2042	87	4	can	can	AUX
ijassa-2042	87	5	be	be	AUX
ijassa-2042	87	6	given	give	VERB
ijassa-2042	87	7	in	in	ADP
ijassa-2042	87	8	the	the	DET
ijassa-2042	87	9	parametric	parametric	ADJ
ijassa-2042	87	10	form	form	NOUN
ijassa-2042	87	11	x	x	X
ijassa-2042	87	12	=	=	SYM
ijassa-2042	87	13	t	t	PROPN
ijassa-2042	87	14	,	,	PUNCT
ijassa-2042	87	15	y	y	PROPN
ijassa-2042	87	16	=	=	PUNCT
ijassa-2042	87	17	at+	at+	PROPN
ijassa-2042	88	1	b.	b.	PROPN
ijassa-2042	89	1	hence	hence	ADV
ijassa-2042	89	2	p	p	PROPN
ijassa-2042	89	3	=	=	PUNCT
ijassa-2042	89	4	a	a	PROPN
ijassa-2042	89	5	and	and	CCONJ
ijassa-2042	89	6	,	,	PUNCT
ijassa-2042	89	7	consequently	consequently	ADV
ijassa-2042	89	8	,	,	PUNCT
ijassa-2042	89	9	we	we	PRON
ijassa-2042	89	10	get	get	VERB
ijassa-2042	89	11	x	x	X
ijassa-2042	89	12	=	=	SYM
ijassa-2042	89	13	a	a	X
ijassa-2042	89	14	,	,	PUNCT
ijassa-2042	89	15	y	y	NOUN
ijassa-2042	89	16	=	=	PUNCT
ijassa-2042	89	17	−b	−b	VERB
ijassa-2042	89	18	,	,	PUNCT
ijassa-2042	89	19	that	that	ADV
ijassa-2042	89	20	is	is	ADV
ijassa-2042	89	21	,	,	PUNCT
ijassa-2042	89	22	the	the	DET
ijassa-2042	89	23	dual	dual	ADJ
ijassa-2042	89	24	curve	curve	NOUN
ijassa-2042	89	25	γ∗1	γ∗1	NOUN
ijassa-2042	89	26	is	be	AUX
ijassa-2042	89	27	a	a	DET
ijassa-2042	89	28	single	single	ADJ
ijassa-2042	89	29	point	point	NOUN
ijassa-2042	89	30	on	on	ADP
ijassa-2042	89	31	the	the	DET
ijassa-2042	89	32	plane	plane	NOUN
ijassa-2042	89	33	.	.	PUNCT
ijassa-2042	90	1	the	the	DET
ijassa-2042	90	2	parabola	parabola	PROPN
ijassa-2042	90	3	γ2	γ2	PROPN
ijassa-2042	90	4	can	can	AUX
ijassa-2042	90	5	be	be	AUX
ijassa-2042	90	6	presented	present	VERB
ijassa-2042	90	7	in	in	ADP
ijassa-2042	90	8	the	the	DET
ijassa-2042	90	9	parametric	parametric	ADJ
ijassa-2042	90	10	form	form	NOUN
ijassa-2042	90	11	x	x	X
ijassa-2042	90	12	=	=	SYM
ijassa-2042	90	13	t	t	PROPN
ijassa-2042	90	14	,	,	PUNCT
ijassa-2042	90	15	y	y	PROPN
ijassa-2042	90	16	=	=	SYM
ijassa-2042	90	17	t2	t2	PROPN
ijassa-2042	90	18	.	.	PUNCT
ijassa-2042	91	1	hence	hence	ADV
ijassa-2042	91	2	p	p	X
ijassa-2042	91	3	=	=	PROPN
ijassa-2042	91	4	2	2	NUM
ijassa-2042	91	5	t	t	NOUN
ijassa-2042	91	6	and	and	CCONJ
ijassa-2042	91	7	,	,	PUNCT
ijassa-2042	91	8	consequently	consequently	ADV
ijassa-2042	91	9	,	,	PUNCT
ijassa-2042	91	10	we	we	PRON
ijassa-2042	91	11	get	get	VERB
ijassa-2042	91	12	x	x	X
ijassa-2042	91	13	=	=	SYM
ijassa-2042	91	14	2	2	NUM
ijassa-2042	91	15	t	t	NOUN
ijassa-2042	91	16	and	and	CCONJ
ijassa-2042	91	17	y	y	PROPN
ijassa-2042	91	18	=	=	PROPN
ijassa-2042	91	19	t2	t2	PROPN
ijassa-2042	91	20	.	.	PUNCT
ijassa-2042	92	1	thus	thus	ADV
ijassa-2042	92	2	,	,	PUNCT
ijassa-2042	92	3	the	the	DET
ijassa-2042	92	4	dual	dual	ADJ
ijassa-2042	92	5	curve	curve	NOUN
ijassa-2042	92	6	γ∗2	γ∗2	NOUN
ijassa-2042	92	7	is	be	AUX
ijassa-2042	92	8	the	the	DET
ijassa-2042	92	9	parabola	parabola	PROPN
ijassa-2042	92	10	y	y	PROPN
ijassa-2042	92	11	=	=	SYM
ijassa-2042	92	12	x2/4	x2/4	PROPN
ijassa-2042	92	13	.	.	PUNCT
ijassa-2042	93	1	the	the	DET
ijassa-2042	93	2	cubic	cubic	PROPN
ijassa-2042	93	3	parabola	parabola	PROPN
ijassa-2042	93	4	γ3	γ3	PROPN
ijassa-2042	93	5	can	can	AUX
ijassa-2042	93	6	be	be	AUX
ijassa-2042	93	7	presented	present	VERB
ijassa-2042	93	8	in	in	ADP
ijassa-2042	93	9	the	the	DET
ijassa-2042	93	10	parametric	parametric	ADJ
ijassa-2042	93	11	form	form	NOUN
ijassa-2042	93	12	x	x	X
ijassa-2042	93	13	=	=	SYM
ijassa-2042	93	14	t	t	PROPN
ijassa-2042	93	15	,	,	PUNCT
ijassa-2042	93	16	y	y	PROPN
ijassa-2042	93	17	=	=	PROPN
ijassa-2042	93	18	t3	t3	PROPN
ijassa-2042	93	19	.	.	PUNCT
ijassa-2042	94	1	this	this	DET
ijassa-2042	94	2	yields	yield	VERB
ijassa-2042	94	3	p	p	NOUN
ijassa-2042	94	4	=	=	SYM
ijassa-2042	94	5	3t2	3t2	NUM
ijassa-2042	94	6	and	and	CCONJ
ijassa-2042	94	7	x	x	SYM
ijassa-2042	94	8	=	=	SYM
ijassa-2042	94	9	3t2	3t2	NUM
ijassa-2042	94	10	,	,	PUNCT
ijassa-2042	94	11	y	y	PROPN
ijassa-2042	94	12	=	=	SYM
ijassa-2042	94	13	2t3	2t3	PROPN
ijassa-2042	94	14	.	.	PUNCT
ijassa-2042	95	1	this	this	PRON
ijassa-2042	95	2	shows	show	VERB
ijassa-2042	95	3	that	that	SCONJ
ijassa-2042	95	4	the	the	DET
ijassa-2042	95	5	dual	dual	ADJ
ijassa-2042	95	6	curve	curve	NOUN
ijassa-2042	95	7	γ∗3	γ∗3	NOUN
ijassa-2042	95	8	is	be	AUX
ijassa-2042	95	9	the	the	DET
ijassa-2042	95	10	semicubic	semicubic	ADJ
ijassa-2042	95	11	parabola	parabola	NOUN
ijassa-2042	95	12	with	with	ADP
ijassa-2042	95	13	the	the	DET
ijassa-2042	95	14	casp	casp	NOUN
ijassa-2042	95	15	at	at	ADP
ijassa-2042	95	16	the	the	DET
ijassa-2042	95	17	origin	origin	NOUN
ijassa-2042	95	18	.	.	PUNCT
ijassa-2042	96	1	the	the	DET
ijassa-2042	96	2	casp	casp	NOUN
ijassa-2042	96	3	corresponds	correspond	VERB
ijassa-2042	96	4	to	to	ADP
ijassa-2042	96	5	the	the	DET
ijassa-2042	96	6	inflection	inflection	NOUN
ijassa-2042	96	7	point	point	NOUN
ijassa-2042	96	8	of	of	ADP
ijassa-2042	96	9	the	the	DET
ijassa-2042	96	10	initial	initial	ADJ
ijassa-2042	96	11	curve	curve	NOUN
ijassa-2042	96	12	γ3	γ3	NOUN
ijassa-2042	96	13	.	.	PUNCT
ijassa-2042	97	1	lemma	lemma	PROPN
ijassa-2042	97	2	2.1	2.1	NUM
ijassa-2042	97	3	:	:	PUNCT
ijassa-2042	97	4	the	the	DET
ijassa-2042	97	5	legendre	legendre	PROPN
ijassa-2042	97	6	transformation	transformation	NOUN
ijassa-2042	97	7	has	have	VERB
ijassa-2042	97	8	the	the	DET
ijassa-2042	97	9	following	follow	VERB
ijassa-2042	97	10	properties	property	NOUN
ijassa-2042	97	11	.	.	PUNCT
ijassa-2042	98	1	1	1	X
ijassa-2042	98	2	.	.	X
ijassa-2042	98	3	the	the	DET
ijassa-2042	98	4	curve	curve	NOUN
ijassa-2042	98	5	γ∗	γ∗	NOUN
ijassa-2042	98	6	dual	dual	ADJ
ijassa-2042	98	7	to	to	ADP
ijassa-2042	98	8	the	the	DET
ijassa-2042	98	9	curve	curve	NOUN
ijassa-2042	98	10	γ	γ	NOUN
ijassa-2042	98	11	of	of	ADP
ijassa-2042	98	12	the	the	DET
ijassa-2042	98	13	form	form	NOUN
ijassa-2042	98	14	(	(	PUNCT
ijassa-2042	98	15	1.1	1.1	NUM
ijassa-2042	98	16	)	)	PUNCT
ijassa-2042	98	17	is	be	AUX
ijassa-2042	98	18	regular	regular	ADJ
ijassa-2042	98	19	at	at	ADV
ijassa-2042	98	20	all	all	DET
ijassa-2042	98	21	points	point	NOUN
ijassa-2042	98	22	that	that	PRON
ijassa-2042	98	23	correspond	correspond	VERB
ijassa-2042	98	24	to	to	ADP
ijassa-2042	98	25	points	point	NOUN
ijassa-2042	98	26	of	of	ADP
ijassa-2042	98	27	the	the	DET
ijassa-2042	98	28	curve	curve	NOUN
ijassa-2042	98	29	γ	γ	X
ijassa-2042	98	30	with	with	ADP
ijassa-2042	98	31	non	non	ADJ
ijassa-2042	98	32	-	-	ADJ
ijassa-2042	98	33	zero	zero	NUM
ijassa-2042	98	34	curvature	curvature	NOUN
ijassa-2042	98	35	κ(t	κ(t	NOUN
ijassa-2042	98	36	)	)	PUNCT
ijassa-2042	98	37	̸=	̸=	PROPN
ijassa-2042	98	38	0	0	NUM
ijassa-2042	98	39	.	.	PUNCT
ijassa-2042	99	1	the	the	DET
ijassa-2042	99	2	curve	curve	NOUN
ijassa-2042	99	3	γ∗	γ∗	NOUN
ijassa-2042	99	4	has	have	VERB
ijassa-2042	99	5	simicubic	simicubic	ADJ
ijassa-2042	99	6	casp	casp	NOUN
ijassa-2042	99	7	at	at	ADP
ijassa-2042	99	8	points	point	NOUN
ijassa-2042	99	9	that	that	PRON
ijassa-2042	99	10	correspond	correspond	VERB
ijassa-2042	99	11	to	to	ADP
ijassa-2042	99	12	points	point	NOUN
ijassa-2042	99	13	of	of	ADP
ijassa-2042	99	14	γ	γ	PROPN
ijassa-2042	99	15	where	where	SCONJ
ijassa-2042	99	16	κ(t	κ(t	X
ijassa-2042	99	17	)	)	PUNCT
ijassa-2042	99	18	=	=	PUNCT
ijassa-2042	99	19	0	0	NUM
ijassa-2042	99	20	,	,	PUNCT
ijassa-2042	99	21	but	but	CCONJ
ijassa-2042	99	22	κ′(t	κ′(t	ADJ
ijassa-2042	99	23	)	)	PUNCT
ijassa-2042	99	24	̸=	̸=	PROPN
ijassa-2042	99	25	0	0	NUM
ijassa-2042	99	26	.	.	NOUN
ijassa-2042	100	1	2	2	X
ijassa-2042	100	2	.	.	X
ijassa-2042	100	3	assume	assume	VERB
ijassa-2042	100	4	that	that	SCONJ
ijassa-2042	100	5	the	the	DET
ijassa-2042	100	6	curve	curve	NOUN
ijassa-2042	100	7	γ	γ	NOUN
ijassa-2042	100	8	is	be	AUX
ijassa-2042	100	9	the	the	DET
ijassa-2042	100	10	graph	graph	NOUN
ijassa-2042	100	11	of	of	ADP
ijassa-2042	100	12	a	a	DET
ijassa-2042	100	13	functions	function	NOUN
ijassa-2042	100	14	y	y	PROPN
ijassa-2042	100	15	=	=	SYM
ijassa-2042	100	16	f(x	f(x	PROPN
ijassa-2042	100	17	)	)	PUNCT
ijassa-2042	100	18	such	such	ADJ
ijassa-2042	100	19	that	that	SCONJ
ijassa-2042	100	20	df	df	PROPN
ijassa-2042	100	21	dx	dx	PROPN
ijassa-2042	100	22	(	(	PUNCT
ijassa-2042	100	23	0	0	NUM
ijassa-2042	100	24	)	)	PUNCT
ijassa-2042	100	25	=	=	SYM
ijassa-2042	100	26	·	·	PUNCT
ijassa-2042	100	27	·	·	PUNCT
ijassa-2042	100	28	·	·	PUNCT
ijassa-2042	101	1	=	=	SYM
ijassa-2042	101	2	dnf	dnf	PROPN
ijassa-2042	101	3	dxn	dxn	PROPN
ijassa-2042	101	4	(	(	PUNCT
ijassa-2042	101	5	0	0	NUM
ijassa-2042	101	6	)	)	PUNCT
ijassa-2042	101	7	=	=	SYM
ijassa-2042	101	8	0	0	NUM
ijassa-2042	101	9	,	,	PUNCT
ijassa-2042	101	10	dn+1f	dn+1f	ADJ
ijassa-2042	101	11	dxn+1	dxn+1	X
ijassa-2042	101	12	(	(	PUNCT
ijassa-2042	101	13	0	0	X
ijassa-2042	101	14	)	)	PUNCT
ijassa-2042	101	15	̸=	̸=	PROPN
ijassa-2042	101	16	0	0	NUM
ijassa-2042	101	17	.	.	PUNCT
ijassa-2042	102	1	(	(	PUNCT
ijassa-2042	102	2	2.6	2.6	NUM
ijassa-2042	102	3	)	)	PUNCT
ijassa-2042	102	4	then	then	ADV
ijassa-2042	102	5	the	the	DET
ijassa-2042	102	6	curve	curve	NOUN
ijassa-2042	102	7	γ∗	γ∗	NOUN
ijassa-2042	102	8	dual	dual	ADJ
ijassa-2042	102	9	to	to	ADP
ijassa-2042	102	10	γ	γ	PROPN
ijassa-2042	102	11	has	have	VERB
ijassa-2042	102	12	the	the	DET
ijassa-2042	102	13	form	form	NOUN
ijassa-2042	102	14	(	(	PUNCT
ijassa-2042	102	15	1.3	1.3	NUM
ijassa-2042	102	16	)	)	PUNCT
ijassa-2042	102	17	.	.	PUNCT
ijassa-2042	103	1	3	3	X
ijassa-2042	103	2	.	.	X
ijassa-2042	103	3	the	the	DET
ijassa-2042	103	4	legendre	legendre	PROPN
ijassa-2042	103	5	transformation	transformation	PROPN
ijassa-2042	103	6	λ	λ	PROPN
ijassa-2042	103	7	is	be	AUX
ijassa-2042	103	8	an	an	DET
ijassa-2042	103	9	involution	involution	NOUN
ijassa-2042	103	10	,	,	PUNCT
ijassa-2042	103	11	that	that	ADV
ijassa-2042	103	12	is	is	ADV
ijassa-2042	103	13	,	,	PUNCT
ijassa-2042	103	14	λ2	λ2	NOUN
ijassa-2042	103	15	is	be	AUX
ijassa-2042	103	16	the	the	DET
ijassa-2042	103	17	identity	identity	NOUN
ijassa-2042	103	18	transformation	transformation	NOUN
ijassa-2042	103	19	.	.	PUNCT
ijassa-2042	104	1	this	this	PRON
ijassa-2042	104	2	is	be	AUX
ijassa-2042	104	3	equivalent	equivalent	ADJ
ijassa-2042	104	4	to	to	ADP
ijassa-2042	104	5	the	the	DET
ijassa-2042	104	6	relation	relation	NOUN
ijassa-2042	104	7	λ	λ	PROPN
ijassa-2042	104	8	=	=	SYM
ijassa-2042	104	9	λ−1	λ−1	PROPN
ijassa-2042	104	10	,	,	PUNCT
ijassa-2042	104	11	which	which	PRON
ijassa-2042	104	12	yields	yield	VERB
ijassa-2042	104	13	(	(	PUNCT
ijassa-2042	104	14	γ∗)∗	γ∗)∗	NUM
ijassa-2042	104	15	=	=	SYM
ijassa-2042	104	16	γ	γ	PROPN
ijassa-2042	104	17	.	.	PUNCT
ijassa-2042	105	1	the	the	DET
ijassa-2042	105	2	proof	proof	NOUN
ijassa-2042	105	3	is	be	AUX
ijassa-2042	105	4	by	by	ADP
ijassa-2042	105	5	direct	direct	ADJ
ijassa-2042	105	6	calculation	calculation	NOUN
ijassa-2042	105	7	.	.	PUNCT
ijassa-2042	106	1	copyright	copyright	NOUN
ijassa-2042	106	2	©	©	PROPN
ijassa-2042	106	3	2025	2025	NUM
ijassa-2042	106	4	assa	assa	NOUN
ijassa-2042	106	5	.	.	PUNCT
ijassa-2042	107	1	adv	adv	PROPN
ijassa-2042	107	2	syst	syst	PROPN
ijassa-2042	107	3	sci	sci	PROPN
ijassa-2042	107	4	appl	appl	PROPN
ijassa-2042	107	5	(	(	PUNCT
ijassa-2042	107	6	2025	2025	NUM
ijassa-2042	107	7	)	)	PUNCT
ijassa-2042	107	8	48	48	NUM
ijassa-2042	107	9	n.	n.	NOUN
ijassa-2042	107	10	pavlova	pavlova	PROPN
ijassa-2042	107	11	,	,	PUNCT
ijassa-2042	107	12	a.	a.	NOUN
ijassa-2042	107	13	remizov	remizov	PROPN
ijassa-2042	107	14	the	the	DET
ijassa-2042	107	15	lemma	lemma	PROPN
ijassa-2042	107	16	shows	show	VERB
ijassa-2042	107	17	that	that	SCONJ
ijassa-2042	107	18	regular	regular	ADJ
ijassa-2042	107	19	points	point	NOUN
ijassa-2042	107	20	of	of	ADP
ijassa-2042	107	21	γ∗	γ∗	PROPN
ijassa-2042	107	22	correspond	correspond	VERB
ijassa-2042	107	23	to	to	ADP
ijassa-2042	107	24	points	point	NOUN
ijassa-2042	107	25	where	where	SCONJ
ijassa-2042	107	26	f	f	PROPN
ijassa-2042	107	27	′′(x	′′(x	PROPN
ijassa-2042	107	28	)	)	PUNCT
ijassa-2042	107	29	̸=	̸=	PROPN
ijassa-2042	107	30	0	0	NUM
ijassa-2042	107	31	,	,	PUNCT
ijassa-2042	107	32	while	while	SCONJ
ijassa-2042	107	33	semicubic	semicubic	ADJ
ijassa-2042	107	34	casps	casp	NOUN
ijassa-2042	107	35	correspond	correspond	VERB
ijassa-2042	107	36	to	to	ADP
ijassa-2042	107	37	points	point	NOUN
ijassa-2042	107	38	where	where	SCONJ
ijassa-2042	107	39	f	f	PROPN
ijassa-2042	107	40	′′(x	′′(x	PROPN
ijassa-2042	107	41	)	)	PUNCT
ijassa-2042	107	42	=	=	SYM
ijassa-2042	108	1	0	0	NUM
ijassa-2042	108	2	,	,	PUNCT
ijassa-2042	108	3	f	f	PROPN
ijassa-2042	108	4	′′′(x	′′′(x	PROPN
ijassa-2042	108	5	)	)	PUNCT
ijassa-2042	108	6	̸=	̸=	PROPN
ijassa-2042	108	7	0	0	NUM
ijassa-2042	108	8	,	,	PUNCT
ijassa-2042	108	9	i.e.	i.e.	X
ijassa-2042	108	10	,	,	PUNCT
ijassa-2042	108	11	simple	simple	ADJ
ijassa-2042	108	12	(	(	PUNCT
ijassa-2042	108	13	cubic	cubic	ADJ
ijassa-2042	108	14	)	)	PUNCT
ijassa-2042	108	15	inflection	inflection	NOUN
ijassa-2042	108	16	points	point	NOUN
ijassa-2042	108	17	of	of	ADP
ijassa-2042	108	18	the	the	DET
ijassa-2042	108	19	function	function	NOUN
ijassa-2042	108	20	f	f	PROPN
ijassa-2042	108	21	.	.	PUNCT
ijassa-2042	109	1	example	example	NOUN
ijassa-2042	109	2	2.2	2.2	NUM
ijassa-2042	109	3	:	:	PUNCT
ijassa-2042	109	4	in	in	ADP
ijassa-2042	109	5	fig	fig	NOUN
ijassa-2042	109	6	.	.	PUNCT
ijassa-2042	110	1	2.4	2.4	NUM
ijassa-2042	110	2	,	,	PUNCT
ijassa-2042	110	3	the	the	DET
ijassa-2042	110	4	curve	curve	NOUN
ijassa-2042	110	5	dual	dual	ADJ
ijassa-2042	110	6	to	to	ADP
ijassa-2042	110	7	the	the	DET
ijassa-2042	110	8	graph	graph	NOUN
ijassa-2042	110	9	of	of	ADP
ijassa-2042	110	10	sine	sine	NOUN
ijassa-2042	110	11	is	be	AUX
ijassa-2042	110	12	presented	present	VERB
ijassa-2042	110	13	.	.	PUNCT
ijassa-2042	111	1	(	(	PUNCT
ijassa-2042	111	2	for	for	ADP
ijassa-2042	111	3	better	well	ADJ
ijassa-2042	111	4	visibility	visibility	NOUN
ijassa-2042	111	5	,	,	PUNCT
ijassa-2042	111	6	the	the	DET
ijassa-2042	111	7	scale	scale	NOUN
ijassa-2042	111	8	along	along	ADP
ijassa-2042	111	9	the	the	DET
ijassa-2042	111	10	x	x	NOUN
ijassa-2042	111	11	-	-	NOUN
ijassa-2042	111	12	axis	axis	NOUN
ijassa-2042	111	13	is	be	AUX
ijassa-2042	111	14	increased	increase	VERB
ijassa-2042	111	15	approximately	approximately	ADV
ijassa-2042	111	16	20	20	NUM
ijassa-2042	111	17	times	time	NOUN
ijassa-2042	111	18	.	.	PUNCT
ijassa-2042	111	19	)	)	PUNCT
ijassa-2042	112	1	casps	casp	NOUN
ijassa-2042	112	2	of	of	ADP
ijassa-2042	112	3	the	the	DET
ijassa-2042	112	4	dual	dual	ADJ
ijassa-2042	112	5	curve	curve	NOUN
ijassa-2042	112	6	correspond	correspond	NOUN
ijassa-2042	112	7	to	to	ADP
ijassa-2042	112	8	points	point	NOUN
ijassa-2042	112	9	x	x	PUNCT
ijassa-2042	112	10	=	=	SYM
ijassa-2042	112	11	πn	πn	X
ijassa-2042	112	12	with	with	ADP
ijassa-2042	112	13	integer	integer	PROPN
ijassa-2042	112	14	n.	n.	PROPN
ijassa-2042	112	15	fig	fig	PROPN
ijassa-2042	112	16	.	.	PUNCT
ijassa-2042	113	1	2.4	2.4	NUM
ijassa-2042	113	2	.	.	PUNCT
ijassa-2042	114	1	the	the	DET
ijassa-2042	114	2	curve	curve	NOUN
ijassa-2042	114	3	dual	dual	ADJ
ijassa-2042	114	4	to	to	ADP
ijassa-2042	114	5	y	y	PROPN
ijassa-2042	114	6	=	=	SYM
ijassa-2042	114	7	sinx	sinx	PROPN
ijassa-2042	114	8	.	.	PUNCT
ijassa-2042	114	9	from	from	ADP
ijassa-2042	114	10	the	the	DET
ijassa-2042	114	11	left	left	NOUN
ijassa-2042	114	12	to	to	ADP
ijassa-2042	114	13	right	right	NOUN
ijassa-2042	114	14	:	:	PUNCT
ijassa-2042	114	15	x	x	SYM
ijassa-2042	114	16	runs	run	VERB
ijassa-2042	114	17	from	from	ADP
ijassa-2042	114	18	0	0	NUM
ijassa-2042	114	19	to	to	ADP
ijassa-2042	114	20	2π	2π	NOUN
ijassa-2042	114	21	,	,	PUNCT
ijassa-2042	114	22	4π	4π	NUM
ijassa-2042	114	23	,	,	PUNCT
ijassa-2042	114	24	7π	7π	NUM
ijassa-2042	114	25	,	,	PUNCT
ijassa-2042	114	26	respectively	respectively	ADV
ijassa-2042	114	27	.	.	PUNCT
ijassa-2042	115	1	remark	remark	PROPN
ijassa-2042	115	2	2.2	2.2	NUM
ijassa-2042	115	3	:	:	PUNCT
ijassa-2042	115	4	it	it	PRON
ijassa-2042	115	5	is	be	AUX
ijassa-2042	115	6	worth	worth	ADJ
ijassa-2042	115	7	observing	observe	VERB
ijassa-2042	115	8	that	that	SCONJ
ijassa-2042	115	9	there	there	PRON
ijassa-2042	115	10	exist	exist	VERB
ijassa-2042	115	11	other	other	ADJ
ijassa-2042	115	12	ways	way	NOUN
ijassa-2042	115	13	to	to	PART
ijassa-2042	115	14	define	define	VERB
ijassa-2042	115	15	dual	dual	ADJ
ijassa-2042	115	16	curves	curve	NOUN
ijassa-2042	115	17	.	.	PUNCT
ijassa-2042	116	1	for	for	ADP
ijassa-2042	116	2	instance	instance	NOUN
ijassa-2042	116	3	,	,	PUNCT
ijassa-2042	116	4	in	in	ADP
ijassa-2042	116	5	algebraic	algebraic	ADJ
ijassa-2042	116	6	geometry	geometry	NOUN
ijassa-2042	116	7	the	the	DET
ijassa-2042	116	8	definition	definition	NOUN
ijassa-2042	116	9	of	of	ADP
ijassa-2042	116	10	dual	dual	ADJ
ijassa-2042	116	11	curves	curve	NOUN
ijassa-2042	116	12	is	be	AUX
ijassa-2042	116	13	based	base	VERB
ijassa-2042	116	14	on	on	ADP
ijassa-2042	116	15	the	the	DET
ijassa-2042	116	16	same	same	ADJ
ijassa-2042	116	17	construction	construction	NOUN
ijassa-2042	116	18	,	,	PUNCT
ijassa-2042	116	19	where	where	SCONJ
ijassa-2042	116	20	the	the	DET
ijassa-2042	116	21	mapping	mapping	NOUN
ijassa-2042	116	22	λ	λ	NOUN
ijassa-2042	116	23	given	give	VERB
ijassa-2042	116	24	by	by	ADP
ijassa-2042	116	25	formula	formula	NOUN
ijassa-2042	116	26	(	(	PUNCT
ijassa-2042	116	27	2.5	2.5	NUM
ijassa-2042	116	28	)	)	PUNCT
ijassa-2042	116	29	is	be	AUX
ijassa-2042	116	30	replaced	replace	VERB
ijassa-2042	116	31	with	with	ADP
ijassa-2042	116	32	the	the	DET
ijassa-2042	116	33	mapping	mapping	NOUN
ijassa-2042	116	34	(	(	PUNCT
ijassa-2042	116	35	x	x	X
ijassa-2042	116	36	,	,	PUNCT
ijassa-2042	116	37	y	y	PROPN
ijassa-2042	116	38	,	,	PUNCT
ijassa-2042	116	39	p	p	X
ijassa-2042	116	40	)	)	PUNCT
ijassa-2042	116	41	7→	7→	NUM
ijassa-2042	116	42	(	(	PUNCT
ijassa-2042	116	43	x̄	x̄	PROPN
ijassa-2042	116	44	,	,	PUNCT
ijassa-2042	116	45	ȳ	ȳ	PROPN
ijassa-2042	116	46	,	,	PUNCT
ijassa-2042	116	47	p̄	p̄	PROPN
ijassa-2042	116	48	)	)	PUNCT
ijassa-2042	116	49	,	,	PUNCT
ijassa-2042	116	50	where	where	SCONJ
ijassa-2042	116	51	x̄	x̄	PRON
ijassa-2042	116	52	=	=	PUNCT
ijassa-2042	116	53	x	x	X
ijassa-2042	116	54	/	/	SYM
ijassa-2042	116	55	y	y	PROPN
ijassa-2042	116	56	,	,	PUNCT
ijassa-2042	116	57	ȳ	ȳ	NOUN
ijassa-2042	116	58	=	=	NOUN
ijassa-2042	116	59	1	1	NUM
ijassa-2042	116	60	/	/	SYM
ijassa-2042	116	61	y	y	NOUN
ijassa-2042	116	62	,	,	PUNCT
ijassa-2042	116	63	p̄	p̄	NOUN
ijassa-2042	116	64	=	=	PUNCT
ijassa-2042	116	65	dȳ	dȳ	NOUN
ijassa-2042	116	66	/dx̄	/dx̄	NUM
ijassa-2042	116	67	,	,	PUNCT
ijassa-2042	116	68	(	(	PUNCT
ijassa-2042	116	69	2.7	2.7	NUM
ijassa-2042	116	70	)	)	PUNCT
ijassa-2042	116	71	x	x	NOUN
ijassa-2042	116	72	,	,	PUNCT
ijassa-2042	116	73	y	y	PROPN
ijassa-2042	116	74	are	be	AUX
ijassa-2042	116	75	defined	define	VERB
ijassa-2042	116	76	in	in	ADP
ijassa-2042	116	77	(	(	PUNCT
ijassa-2042	116	78	2.5	2.5	NUM
ijassa-2042	116	79	)	)	PUNCT
ijassa-2042	116	80	.	.	PUNCT
ijassa-2042	117	1	example	example	NOUN
ijassa-2042	117	2	2.3	2.3	NUM
ijassa-2042	118	1	:	:	PUNCT
ijassa-2042	118	2	consider	consider	VERB
ijassa-2042	118	3	the	the	DET
ijassa-2042	118	4	curve	curve	NOUN
ijassa-2042	118	5	dual	dual	ADJ
ijassa-2042	118	6	to	to	ADP
ijassa-2042	118	7	the	the	DET
ijassa-2042	118	8	ellipse	ellipse	NOUN
ijassa-2042	118	9	(	(	PUNCT
ijassa-2042	118	10	x	x	NOUN
ijassa-2042	118	11	/	/	SYM
ijassa-2042	118	12	a)2	a)2	NOUN
ijassa-2042	118	13	+	+	CCONJ
ijassa-2042	118	14	(	(	PUNCT
ijassa-2042	118	15	y	y	NOUN
ijassa-2042	118	16	/	/	SYM
ijassa-2042	118	17	b)2	b)2	ADJ
ijassa-2042	118	18	=	=	NOUN
ijassa-2042	118	19	1	1	X
ijassa-2042	118	20	.	.	PUNCT
ijassa-2042	119	1	it	it	PRON
ijassa-2042	119	2	is	be	AUX
ijassa-2042	119	3	easy	easy	ADJ
ijassa-2042	119	4	to	to	PART
ijassa-2042	119	5	see	see	VERB
ijassa-2042	119	6	that	that	SCONJ
ijassa-2042	119	7	this	this	DET
ijassa-2042	119	8	curve	curve	NOUN
ijassa-2042	119	9	is	be	AUX
ijassa-2042	119	10	the	the	DET
ijassa-2042	119	11	hyperbola	hyperbola	PROPN
ijassa-2042	119	12	y	y	PROPN
ijassa-2042	119	13	2	2	NUM
ijassa-2042	119	14	−	−	PROPN
ijassa-2042	119	15	(	(	PUNCT
ijassa-2042	119	16	ax)2	ax)2	NOUN
ijassa-2042	119	17	=	=	SYM
ijassa-2042	119	18	b2	b2	NOUN
ijassa-2042	119	19	in	in	ADP
ijassa-2042	119	20	the	the	DET
ijassa-2042	119	21	coordinates	coordinate	NOUN
ijassa-2042	119	22	(	(	PUNCT
ijassa-2042	119	23	x	x	X
ijassa-2042	119	24	,	,	PUNCT
ijassa-2042	119	25	y	y	PROPN
ijassa-2042	119	26	)	)	PUNCT
ijassa-2042	119	27	and	and	CCONJ
ijassa-2042	119	28	the	the	DET
ijassa-2042	119	29	ellipse	ellipse	NOUN
ijassa-2042	119	30	(	(	PUNCT
ijassa-2042	119	31	ax̄)2	ax̄)2	NUM
ijassa-2042	119	32	+	+	NUM
ijassa-2042	119	33	(	(	PUNCT
ijassa-2042	119	34	bȳ	bȳ	NOUN
ijassa-2042	119	35	)	)	PUNCT
ijassa-2042	119	36	2	2	NUM
ijassa-2042	119	37	=	=	SYM
ijassa-2042	119	38	1	1	NUM
ijassa-2042	119	39	in	in	ADP
ijassa-2042	119	40	the	the	DET
ijassa-2042	119	41	coordinates	coordinate	NOUN
ijassa-2042	119	42	(	(	PUNCT
ijassa-2042	119	43	x̄	x̄	PROPN
ijassa-2042	119	44	,	,	PUNCT
ijassa-2042	119	45	ȳ	ȳ	PROPN
ijassa-2042	119	46	)	)	PUNCT
ijassa-2042	119	47	.	.	PUNCT
ijassa-2042	120	1	it	it	PRON
ijassa-2042	120	2	is	be	AUX
ijassa-2042	120	3	not	not	PART
ijassa-2042	120	4	surprising	surprising	ADJ
ijassa-2042	120	5	,	,	PUNCT
ijassa-2042	120	6	because	because	SCONJ
ijassa-2042	120	7	all	all	DET
ijassa-2042	120	8	conic	conic	ADJ
ijassa-2042	120	9	sections	section	NOUN
ijassa-2042	120	10	(	(	PUNCT
ijassa-2042	120	11	which	which	PRON
ijassa-2042	120	12	include	include	VERB
ijassa-2042	120	13	ellipses	ellipsis	NOUN
ijassa-2042	120	14	and	and	CCONJ
ijassa-2042	120	15	hyperbolas	hyperbola	NOUN
ijassa-2042	120	16	)	)	PUNCT
ijassa-2042	120	17	are	be	AUX
ijassa-2042	120	18	projectively	projectively	ADV
ijassa-2042	120	19	equivalent	equivalent	ADJ
ijassa-2042	120	20	and	and	CCONJ
ijassa-2042	120	21	the	the	DET
ijassa-2042	120	22	coordinates	coordinate	NOUN
ijassa-2042	120	23	(	(	PUNCT
ijassa-2042	120	24	x	x	X
ijassa-2042	120	25	,	,	PUNCT
ijassa-2042	120	26	y	y	PROPN
ijassa-2042	120	27	)	)	PUNCT
ijassa-2042	120	28	and	and	CCONJ
ijassa-2042	120	29	(	(	PUNCT
ijassa-2042	120	30	x̄	x̄	PROPN
ijassa-2042	120	31	,	,	PUNCT
ijassa-2042	120	32	ȳ	ȳ	PROPN
ijassa-2042	120	33	)	)	PUNCT
ijassa-2042	120	34	are	be	AUX
ijassa-2042	120	35	connected	connect	VERB
ijassa-2042	120	36	by	by	ADP
ijassa-2042	120	37	a	a	DET
ijassa-2042	120	38	projective	projective	ADJ
ijassa-2042	120	39	transformation	transformation	NOUN
ijassa-2042	120	40	(	(	PUNCT
ijassa-2042	120	41	2.7	2.7	NUM
ijassa-2042	120	42	)	)	PUNCT
ijassa-2042	120	43	.	.	PUNCT
ijassa-2042	121	1	now	now	ADV
ijassa-2042	121	2	consider	consider	VERB
ijassa-2042	121	3	an	an	DET
ijassa-2042	121	4	important	important	ADJ
ijassa-2042	121	5	case	case	NOUN
ijassa-2042	121	6	that	that	SCONJ
ijassa-2042	121	7	the	the	DET
ijassa-2042	121	8	curve	curve	NOUN
ijassa-2042	121	9	γ	γ	NOUN
ijassa-2042	121	10	is	be	AUX
ijassa-2042	121	11	the	the	DET
ijassa-2042	121	12	graph	graph	NOUN
ijassa-2042	121	13	of	of	ADP
ijassa-2042	121	14	a	a	DET
ijassa-2042	121	15	function	function	NOUN
ijassa-2042	121	16	y	y	PROPN
ijassa-2042	121	17	=	=	SYM
ijassa-2042	121	18	f(x	f(x	PROPN
ijassa-2042	121	19	)	)	PUNCT
ijassa-2042	121	20	.	.	PUNCT
ijassa-2042	122	1	it	it	PRON
ijassa-2042	122	2	is	be	AUX
ijassa-2042	122	3	naturally	naturally	ADV
ijassa-2042	122	4	to	to	PART
ijassa-2042	122	5	ask	ask	VERB
ijassa-2042	122	6	:	:	PUNCT
ijassa-2042	122	7	whether	whether	SCONJ
ijassa-2042	122	8	the	the	DET
ijassa-2042	122	9	dual	dual	ADJ
ijassa-2042	122	10	curve	curve	NOUN
ijassa-2042	122	11	γ∗	γ∗	NOUN
ijassa-2042	122	12	is	be	AUX
ijassa-2042	122	13	the	the	DET
ijassa-2042	122	14	graph	graph	NOUN
ijassa-2042	122	15	of	of	ADP
ijassa-2042	122	16	a	a	DET
ijassa-2042	122	17	certain	certain	ADJ
ijassa-2042	122	18	function	function	NOUN
ijassa-2042	123	1	y	y	PROPN
ijassa-2042	123	2	=	=	PUNCT
ijassa-2042	123	3	f	f	PROPN
ijassa-2042	123	4	∗(x	∗(x	PROPN
ijassa-2042	123	5	)	)	PUNCT
ijassa-2042	123	6	as	as	ADV
ijassa-2042	123	7	well	well	ADV
ijassa-2042	123	8	.	.	PUNCT
ijassa-2042	124	1	if	if	SCONJ
ijassa-2042	124	2	so	so	ADV
ijassa-2042	124	3	,	,	PUNCT
ijassa-2042	124	4	the	the	DET
ijassa-2042	124	5	function	function	NOUN
ijassa-2042	124	6	f	f	PROPN
ijassa-2042	124	7	∗	∗	NOUN
ijassa-2042	124	8	is	be	AUX
ijassa-2042	124	9	called	call	VERB
ijassa-2042	124	10	the	the	DET
ijassa-2042	124	11	dual	dual	ADJ
ijassa-2042	124	12	function	function	NOUN
ijassa-2042	124	13	for	for	ADP
ijassa-2042	124	14	f	f	PROPN
ijassa-2042	124	15	or	or	CCONJ
ijassa-2042	124	16	the	the	DET
ijassa-2042	124	17	legendre	legendre	PROPN
ijassa-2042	124	18	transformation	transformation	NOUN
ijassa-2042	124	19	of	of	ADP
ijassa-2042	124	20	f	f	PROPN
ijassa-2042	124	21	.	.	PUNCT
ijassa-2042	125	1	examples	example	NOUN
ijassa-2042	125	2	considered	consider	VERB
ijassa-2042	125	3	above	above	ADP
ijassa-2042	125	4	show	show	VERB
ijassa-2042	125	5	that	that	SCONJ
ijassa-2042	125	6	the	the	DET
ijassa-2042	125	7	dual	dual	ADJ
ijassa-2042	125	8	curve	curve	NOUN
ijassa-2042	125	9	for	for	ADP
ijassa-2042	125	10	the	the	DET
ijassa-2042	125	11	graph	graph	NOUN
ijassa-2042	125	12	of	of	ADP
ijassa-2042	125	13	a	a	DET
ijassa-2042	125	14	function	function	NOUN
ijassa-2042	125	15	is	be	AUX
ijassa-2042	125	16	not	not	PART
ijassa-2042	125	17	always	always	ADV
ijassa-2042	125	18	the	the	DET
ijassa-2042	125	19	graph	graph	NOUN
ijassa-2042	125	20	of	of	ADP
ijassa-2042	125	21	a	a	DET
ijassa-2042	125	22	function	function	NOUN
ijassa-2042	125	23	.	.	PUNCT
ijassa-2042	126	1	lemma	lemma	PROPN
ijassa-2042	126	2	2.1	2.1	NUM
ijassa-2042	126	3	gives	give	VERB
ijassa-2042	126	4	the	the	DET
ijassa-2042	126	5	following	follow	VERB
ijassa-2042	126	6	sufficient	sufficient	ADJ
ijassa-2042	126	7	condition	condition	NOUN
ijassa-2042	126	8	:	:	PUNCT
ijassa-2042	126	9	if	if	SCONJ
ijassa-2042	126	10	the	the	DET
ijassa-2042	126	11	function	function	NOUN
ijassa-2042	126	12	f	f	PROPN
ijassa-2042	126	13	is	be	AUX
ijassa-2042	126	14	strictly	strictly	ADV
ijassa-2042	126	15	convex	convex	ADJ
ijassa-2042	126	16	(	(	PUNCT
ijassa-2042	126	17	f	f	PROPN
ijassa-2042	126	18	′′	′′	PROPN
ijassa-2042	126	19	>	>	X
ijassa-2042	126	20	0	0	NUM
ijassa-2042	126	21	)	)	PUNCT
ijassa-2042	126	22	or	or	CCONJ
ijassa-2042	126	23	f	f	PROPN
ijassa-2042	126	24	is	be	AUX
ijassa-2042	126	25	strictly	strictly	ADV
ijassa-2042	126	26	concave	concave	VERB
ijassa-2042	126	27	(	(	PUNCT
ijassa-2042	126	28	f	f	X
ijassa-2042	126	29	′′	′′	PROPN
ijassa-2042	126	30	<	<	X
ijassa-2042	126	31	0	0	NUM
ijassa-2042	126	32	)	)	PUNCT
ijassa-2042	126	33	on	on	ADP
ijassa-2042	126	34	its	its	PRON
ijassa-2042	126	35	domain	domain	NOUN
ijassa-2042	126	36	,	,	PUNCT
ijassa-2042	126	37	then	then	ADV
ijassa-2042	126	38	the	the	DET
ijassa-2042	126	39	dual	dual	ADJ
ijassa-2042	126	40	curve	curve	NOUN
ijassa-2042	126	41	is	be	AUX
ijassa-2042	126	42	the	the	DET
ijassa-2042	126	43	graph	graph	NOUN
ijassa-2042	126	44	of	of	ADP
ijassa-2042	126	45	a	a	DET
ijassa-2042	126	46	function	function	NOUN
ijassa-2042	126	47	,	,	PUNCT
ijassa-2042	126	48	whence	whence	SCONJ
ijassa-2042	126	49	the	the	DET
ijassa-2042	126	50	dual	dual	ADJ
ijassa-2042	126	51	function	function	NOUN
ijassa-2042	126	52	f	f	PROPN
ijassa-2042	126	53	∗	∗	NOUN
ijassa-2042	126	54	exists	exist	VERB
ijassa-2042	126	55	.	.	PUNCT
ijassa-2042	127	1	however	however	ADV
ijassa-2042	127	2	,	,	PUNCT
ijassa-2042	127	3	the	the	DET
ijassa-2042	127	4	function	function	NOUN
ijassa-2042	127	5	f	f	X
ijassa-2042	127	6	∗(x	∗(x	PROPN
ijassa-2042	127	7	)	)	PUNCT
ijassa-2042	127	8	can	can	AUX
ijassa-2042	127	9	be	be	AUX
ijassa-2042	127	10	defined	define	VERB
ijassa-2042	127	11	not	not	PART
ijassa-2042	127	12	for	for	ADP
ijassa-2042	127	13	allx	allx	NOUN
ijassa-2042	127	14	.	.	PUNCT
ijassa-2042	128	1	indeed	indeed	ADV
ijassa-2042	128	2	,	,	PUNCT
ijassa-2042	128	3	sincex	sincex	NOUN
ijassa-2042	128	4	=	=	SYM
ijassa-2042	128	5	p	p	NOUN
ijassa-2042	128	6	,	,	PUNCT
ijassa-2042	128	7	the	the	DET
ijassa-2042	128	8	domain	domain	NOUN
ijassa-2042	128	9	of	of	ADP
ijassa-2042	128	10	f	f	PROPN
ijassa-2042	128	11	∗	∗	PROPN
ijassa-2042	128	12	coincides	coincide	NOUN
ijassa-2042	128	13	with	with	ADP
ijassa-2042	128	14	the	the	DET
ijassa-2042	128	15	range	range	NOUN
ijassa-2042	128	16	of	of	ADP
ijassa-2042	128	17	f	f	PROPN
ijassa-2042	128	18	′(x	′(x	PROPN
ijassa-2042	128	19	)	)	PUNCT
ijassa-2042	128	20	,	,	PUNCT
ijassa-2042	128	21	and	and	CCONJ
ijassa-2042	128	22	f	f	PROPN
ijassa-2042	128	23	∗	∗	NOUN
ijassa-2042	128	24	is	be	AUX
ijassa-2042	128	25	defined	define	VERB
ijassa-2042	128	26	for	for	ADP
ijassa-2042	128	27	x	x	SYM
ijassa-2042	128	28	that	that	PRON
ijassa-2042	128	29	belong	belong	VERB
ijassa-2042	128	30	to	to	ADP
ijassa-2042	128	31	the	the	DET
ijassa-2042	128	32	range	range	NOUN
ijassa-2042	128	33	of	of	ADP
ijassa-2042	128	34	f	f	PROPN
ijassa-2042	128	35	′	′	NUM
ijassa-2042	128	36	only	only	ADV
ijassa-2042	128	37	.	.	PUNCT
ijassa-2042	129	1	if	if	SCONJ
ijassa-2042	129	2	the	the	DET
ijassa-2042	129	3	function	function	NOUN
ijassa-2042	129	4	f	f	PROPN
ijassa-2042	129	5	is	be	AUX
ijassa-2042	129	6	strictly	strictly	ADV
ijassa-2042	129	7	convex	convex	ADJ
ijassa-2042	129	8	,	,	PUNCT
ijassa-2042	129	9	its	its	PRON
ijassa-2042	129	10	legendre	legendre	PROPN
ijassa-2042	129	11	transformation	transformation	NOUN
ijassa-2042	129	12	f	f	PROPN
ijassa-2042	129	13	∗	∗	NOUN
ijassa-2042	129	14	can	can	AUX
ijassa-2042	129	15	be	be	AUX
ijassa-2042	129	16	defined	define	VERB
ijassa-2042	129	17	by	by	ADP
ijassa-2042	129	18	the	the	DET
ijassa-2042	129	19	formula	formula	NOUN
ijassa-2042	129	20	f	f	PROPN
ijassa-2042	129	21	∗(p	∗(p	PROPN
ijassa-2042	129	22	)	)	PUNCT
ijassa-2042	130	1	=	=	SYM
ijassa-2042	130	2	sup	sup	NOUN
ijassa-2042	130	3	x	x	SYM
ijassa-2042	130	4	(	(	PUNCT
ijassa-2042	130	5	xp−	xp−	PROPN
ijassa-2042	130	6	f(x	f(x	PROPN
ijassa-2042	130	7	)	)	PUNCT
ijassa-2042	130	8	)	)	PUNCT
ijassa-2042	130	9	,	,	PUNCT
ijassa-2042	130	10	p	p	NOUN
ijassa-2042	130	11	∈	∈	PROPN
ijassa-2042	130	12	r	r	NOUN
ijassa-2042	130	13	,	,	PUNCT
ijassa-2042	130	14	(	(	PUNCT
ijassa-2042	130	15	2.8	2.8	NUM
ijassa-2042	130	16	)	)	PUNCT
ijassa-2042	130	17	copyright	copyright	NOUN
ijassa-2042	130	18	©	©	PROPN
ijassa-2042	130	19	2025	2025	NUM
ijassa-2042	130	20	assa	assa	NOUN
ijassa-2042	130	21	.	.	PUNCT
ijassa-2042	131	1	adv	adv	PROPN
ijassa-2042	131	2	syst	syst	PROPN
ijassa-2042	131	3	sci	sci	PROPN
ijassa-2042	131	4	appl	appl	PROPN
ijassa-2042	131	5	(	(	PUNCT
ijassa-2042	131	6	2025	2025	NUM
ijassa-2042	131	7	)	)	PUNCT
ijassa-2042	131	8	legendre	legendre	PROPN
ijassa-2042	131	9	transformation	transformation	NOUN
ijassa-2042	131	10	and	and	CCONJ
ijassa-2042	131	11	its	its	PRON
ijassa-2042	131	12	applications	application	NOUN
ijassa-2042	131	13	49	49	NUM
ijassa-2042	131	14	and	and	CCONJ
ijassa-2042	131	15	if	if	SCONJ
ijassa-2042	131	16	f	f	PROPN
ijassa-2042	131	17	is	be	AUX
ijassa-2042	131	18	strictly	strictly	ADV
ijassa-2042	131	19	concave	concave	VERB
ijassa-2042	131	20	,	,	PUNCT
ijassa-2042	131	21	its	its	PRON
ijassa-2042	131	22	legendre	legendre	PROPN
ijassa-2042	131	23	transformation	transformation	NOUN
ijassa-2042	131	24	f	f	PROPN
ijassa-2042	131	25	∗	∗	NOUN
ijassa-2042	131	26	can	can	AUX
ijassa-2042	131	27	be	be	AUX
ijassa-2042	131	28	defined	define	VERB
ijassa-2042	131	29	by	by	ADP
ijassa-2042	131	30	the	the	DET
ijassa-2042	131	31	formula	formula	NOUN
ijassa-2042	131	32	f	f	PROPN
ijassa-2042	131	33	∗(p	∗(p	PROPN
ijassa-2042	131	34	)	)	PUNCT
ijassa-2042	132	1	=	=	SYM
ijassa-2042	132	2	sup	sup	NOUN
ijassa-2042	132	3	x	x	SYM
ijassa-2042	132	4	(	(	PUNCT
ijassa-2042	132	5	f(x)−	f(x)−	PROPN
ijassa-2042	132	6	xp	xp	PROPN
ijassa-2042	132	7	)	)	PUNCT
ijassa-2042	132	8	,	,	PUNCT
ijassa-2042	132	9	p	p	NOUN
ijassa-2042	132	10	∈	∈	PROPN
ijassa-2042	132	11	r	r	NOUN
ijassa-2042	132	12	,	,	PUNCT
ijassa-2042	132	13	(	(	PUNCT
ijassa-2042	132	14	2.9	2.9	NUM
ijassa-2042	132	15	)	)	PUNCT
ijassa-2042	132	16	where	where	SCONJ
ijassa-2042	132	17	r	r	NOUN
ijassa-2042	132	18	denoted	denote	VERB
ijassa-2042	132	19	the	the	DET
ijassa-2042	132	20	range	range	NOUN
ijassa-2042	132	21	of	of	ADP
ijassa-2042	132	22	f	f	PROPN
ijassa-2042	132	23	′.	′.	PROPN
ijassa-2042	132	24	remark	remark	VERB
ijassa-2042	132	25	2.3	2.3	NUM
ijassa-2042	132	26	:	:	PUNCT
ijassa-2042	132	27	formulas	formula	NOUN
ijassa-2042	132	28	(	(	PUNCT
ijassa-2042	132	29	2.8	2.8	NUM
ijassa-2042	132	30	)	)	PUNCT
ijassa-2042	132	31	and	and	CCONJ
ijassa-2042	132	32	(	(	PUNCT
ijassa-2042	132	33	2.9	2.9	NUM
ijassa-2042	132	34	)	)	PUNCT
ijassa-2042	132	35	are	be	AUX
ijassa-2042	132	36	often	often	ADV
ijassa-2042	132	37	used	use	VERB
ijassa-2042	132	38	as	as	ADP
ijassa-2042	132	39	the	the	DET
ijassa-2042	132	40	definition	definition	NOUN
ijassa-2042	132	41	of	of	ADP
ijassa-2042	132	42	the	the	DET
ijassa-2042	132	43	legendre	legendre	PROPN
ijassa-2042	132	44	transformation	transformation	NOUN
ijassa-2042	132	45	of	of	ADP
ijassa-2042	132	46	functions	function	NOUN
ijassa-2042	132	47	.	.	PUNCT
ijassa-2042	133	1	in	in	ADP
ijassa-2042	133	2	our	our	PRON
ijassa-2042	133	3	opinion	opinion	NOUN
ijassa-2042	133	4	,	,	PUNCT
ijassa-2042	133	5	such	such	DET
ijassa-2042	133	6	a	a	DET
ijassa-2042	133	7	definition	definition	NOUN
ijassa-2042	133	8	has	have	VERB
ijassa-2042	133	9	a	a	DET
ijassa-2042	133	10	serious	serious	ADJ
ijassa-2042	133	11	drawback	drawback	NOUN
ijassa-2042	133	12	compared	compare	VERB
ijassa-2042	133	13	to	to	ADP
ijassa-2042	133	14	the	the	DET
ijassa-2042	133	15	geometric	geometric	ADJ
ijassa-2042	133	16	approach	approach	NOUN
ijassa-2042	133	17	presented	present	VERB
ijassa-2042	133	18	above	above	ADP
ijassa-2042	133	19	:	:	PUNCT
ijassa-2042	133	20	it	it	PRON
ijassa-2042	133	21	is	be	AUX
ijassa-2042	133	22	not	not	PART
ijassa-2042	133	23	natural	natural	ADJ
ijassa-2042	133	24	and	and	CCONJ
ijassa-2042	133	25	well	well	ADV
ijassa-2042	133	26	motivated	motivated	ADJ
ijassa-2042	133	27	.	.	PUNCT
ijassa-2042	134	1	remark	remark	VERB
ijassa-2042	134	2	2.4	2.4	NUM
ijassa-2042	134	3	:	:	PUNCT
ijassa-2042	134	4	the	the	DET
ijassa-2042	134	5	legendre	legendre	PROPN
ijassa-2042	134	6	transformation	transformation	NOUN
ijassa-2042	134	7	can	can	AUX
ijassa-2042	134	8	be	be	AUX
ijassa-2042	134	9	obviously	obviously	ADV
ijassa-2042	134	10	adapted	adapt	VERB
ijassa-2042	134	11	for	for	ADP
ijassa-2042	134	12	hypersurfaces	hypersurface	NOUN
ijassa-2042	134	13	and	and	CCONJ
ijassa-2042	134	14	,	,	PUNCT
ijassa-2042	134	15	consequently	consequently	ADV
ijassa-2042	134	16	,	,	PUNCT
ijassa-2042	134	17	for	for	ADP
ijassa-2042	134	18	functions	function	NOUN
ijassa-2042	134	19	depending	depend	VERB
ijassa-2042	134	20	on	on	ADP
ijassa-2042	134	21	any	any	DET
ijassa-2042	134	22	number	number	NOUN
ijassa-2042	134	23	of	of	ADP
ijassa-2042	134	24	variables	variable	NOUN
ijassa-2042	134	25	.	.	PUNCT
ijassa-2042	135	1	in	in	ADP
ijassa-2042	135	2	particular	particular	ADJ
ijassa-2042	135	3	,	,	PUNCT
ijassa-2042	135	4	this	this	DET
ijassa-2042	135	5	way	way	NOUN
ijassa-2042	135	6	brings	bring	VERB
ijassa-2042	135	7	us	we	PRON
ijassa-2042	135	8	to	to	ADP
ijassa-2042	135	9	the	the	DET
ijassa-2042	135	10	same	same	ADJ
ijassa-2042	135	11	formulas	formula	NOUN
ijassa-2042	135	12	(	(	PUNCT
ijassa-2042	135	13	2.8	2.8	NUM
ijassa-2042	135	14	)	)	PUNCT
ijassa-2042	135	15	and	and	CCONJ
ijassa-2042	135	16	(	(	PUNCT
ijassa-2042	135	17	2.9	2.9	NUM
ijassa-2042	135	18	)	)	PUNCT
ijassa-2042	135	19	,	,	PUNCT
ijassa-2042	135	20	where	where	SCONJ
ijassa-2042	135	21	x	x	PUNCT
ijassa-2042	135	22	and	and	CCONJ
ijassa-2042	135	23	p	p	NOUN
ijassa-2042	135	24	belong	belong	VERB
ijassa-2042	135	25	to	to	ADP
ijassa-2042	135	26	the	the	DET
ijassa-2042	135	27	spaces	space	NOUN
ijassa-2042	135	28	of	of	ADP
ijassa-2042	135	29	the	the	DET
ijassa-2042	135	30	same	same	ADJ
ijassa-2042	135	31	dimension	dimension	NOUN
ijassa-2042	135	32	and	and	CCONJ
ijassa-2042	135	33	xp	xp	ADV
ijassa-2042	135	34	means	mean	VERB
ijassa-2042	135	35	the	the	DET
ijassa-2042	135	36	dot	dot	NOUN
ijassa-2042	135	37	product	product	NOUN
ijassa-2042	135	38	in	in	ADP
ijassa-2042	135	39	orthonormal	orthonormal	ADJ
ijassa-2042	135	40	coordinates	coordinate	NOUN
ijassa-2042	135	41	.	.	PUNCT
ijassa-2042	136	1	2.3	2.3	NUM
ijassa-2042	136	2	.	.	PUNCT
ijassa-2042	137	1	clairaut	clairaut	PROPN
ijassa-2042	137	2	differential	differential	ADJ
ijassa-2042	137	3	equation	equation	NOUN
ijassa-2042	137	4	the	the	DET
ijassa-2042	137	5	legendre	legendre	PROPN
ijassa-2042	137	6	transformation	transformation	NOUN
ijassa-2042	137	7	can	can	AUX
ijassa-2042	137	8	be	be	AUX
ijassa-2042	137	9	applied	apply	VERB
ijassa-2042	137	10	to	to	ADP
ijassa-2042	137	11	differential	differential	ADJ
ijassa-2042	137	12	equations	equation	NOUN
ijassa-2042	137	13	as	as	ADV
ijassa-2042	137	14	well	well	ADV
ijassa-2042	137	15	.	.	PUNCT
ijassa-2042	138	1	consider	consider	VERB
ijassa-2042	138	2	the	the	DET
ijassa-2042	138	3	first	first	ADJ
ijassa-2042	138	4	-	-	PUNCT
ijassa-2042	138	5	order	order	NOUN
ijassa-2042	138	6	differential	differential	NOUN
ijassa-2042	138	7	equation	equation	NOUN
ijassa-2042	138	8	f	f	X
ijassa-2042	138	9	(	(	PUNCT
ijassa-2042	138	10	x	x	PROPN
ijassa-2042	138	11	,	,	PUNCT
ijassa-2042	138	12	y	y	PROPN
ijassa-2042	138	13	,	,	PUNCT
ijassa-2042	138	14	p	p	NOUN
ijassa-2042	138	15	)	)	PUNCT
ijassa-2042	138	16	=	=	SYM
ijassa-2042	138	17	0	0	NUM
ijassa-2042	138	18	,	,	PUNCT
ijassa-2042	138	19	p	p	NOUN
ijassa-2042	138	20	=	=	PUNCT
ijassa-2042	138	21	dy	dy	X
ijassa-2042	138	22	/	/	SYM
ijassa-2042	138	23	dx	dx	PROPN
ijassa-2042	138	24	.	.	PUNCT
ijassa-2042	139	1	(	(	PUNCT
ijassa-2042	139	2	2.10	2.10	NUM
ijassa-2042	139	3	)	)	PUNCT
ijassa-2042	139	4	passing	pass	VERB
ijassa-2042	139	5	to	to	ADP
ijassa-2042	139	6	the	the	DET
ijassa-2042	139	7	new	new	ADJ
ijassa-2042	139	8	variables	variable	NOUN
ijassa-2042	139	9	x	x	NOUN
ijassa-2042	139	10	,	,	PUNCT
ijassa-2042	139	11	y	y	PROPN
ijassa-2042	139	12	,	,	PUNCT
ijassa-2042	139	13	p	p	NOUN
ijassa-2042	139	14	by	by	ADP
ijassa-2042	139	15	formula	formula	NOUN
ijassa-2042	139	16	(	(	PUNCT
ijassa-2042	139	17	2.5	2.5	NUM
ijassa-2042	139	18	)	)	PUNCT
ijassa-2042	139	19	and	and	CCONJ
ijassa-2042	139	20	substituting	substitute	VERB
ijassa-2042	139	21	the	the	DET
ijassa-2042	139	22	corresponding	corresponding	ADJ
ijassa-2042	139	23	expressions	expression	NOUN
ijassa-2042	139	24	in	in	ADP
ijassa-2042	139	25	(	(	PUNCT
ijassa-2042	139	26	2.10	2.10	NUM
ijassa-2042	139	27	)	)	PUNCT
ijassa-2042	139	28	,	,	PUNCT
ijassa-2042	139	29	we	we	PRON
ijassa-2042	139	30	obtain	obtain	VERB
ijassa-2042	139	31	a	a	DET
ijassa-2042	139	32	new	new	ADJ
ijassa-2042	139	33	first	first	ADJ
ijassa-2042	139	34	-	-	PUNCT
ijassa-2042	139	35	order	order	NOUN
ijassa-2042	139	36	differential	differential	ADJ
ijassa-2042	139	37	equation	equation	NOUN
ijassa-2042	139	38	,	,	PUNCT
ijassa-2042	139	39	which	which	PRON
ijassa-2042	139	40	is	be	AUX
ijassa-2042	139	41	called	call	VERB
ijassa-2042	139	42	dual	dual	ADJ
ijassa-2042	139	43	to	to	ADP
ijassa-2042	139	44	the	the	DET
ijassa-2042	139	45	initial	initial	ADJ
ijassa-2042	139	46	equation	equation	NOUN
ijassa-2042	139	47	(	(	PUNCT
ijassa-2042	139	48	2.10	2.10	NUM
ijassa-2042	139	49	):	):	PUNCT
ijassa-2042	139	50	f	f	PROPN
ijassa-2042	139	51	∗(x	∗(x	PROPN
ijassa-2042	139	52	,	,	PUNCT
ijassa-2042	139	53	y	y	PROPN
ijassa-2042	139	54	,	,	PUNCT
ijassa-2042	139	55	p	p	NOUN
ijassa-2042	139	56	)	)	PUNCT
ijassa-2042	139	57	=	=	SYM
ijassa-2042	139	58	0	0	NUM
ijassa-2042	139	59	,	,	PUNCT
ijassa-2042	139	60	p	p	NOUN
ijassa-2042	139	61	=	=	PUNCT
ijassa-2042	139	62	dy	dy	X
ijassa-2042	139	63	/	/	SYM
ijassa-2042	139	64	dx	dx	PROPN
ijassa-2042	139	65	,	,	PUNCT
ijassa-2042	139	66	(	(	PUNCT
ijassa-2042	139	67	2.11	2.11	NUM
ijassa-2042	139	68	)	)	PUNCT
ijassa-2042	139	69	where	where	SCONJ
ijassa-2042	139	70	f	f	PROPN
ijassa-2042	139	71	∗(x	∗(x	PROPN
ijassa-2042	139	72	,	,	PUNCT
ijassa-2042	139	73	y	y	PROPN
ijassa-2042	139	74	,	,	PUNCT
ijassa-2042	139	75	p	p	NOUN
ijassa-2042	139	76	)	)	PUNCT
ijassa-2042	140	1	=	=	SYM
ijassa-2042	140	2	f	f	X
ijassa-2042	140	3	(	(	PUNCT
ijassa-2042	140	4	p	p	NOUN
ijassa-2042	140	5	,	,	PUNCT
ijassa-2042	140	6	xp	xp	INTJ
ijassa-2042	141	1	−	−	PROPN
ijassa-2042	142	1	y	y	PROPN
ijassa-2042	142	2	,	,	PUNCT
ijassa-2042	142	3	x	x	NOUN
ijassa-2042	142	4	)	)	PUNCT
ijassa-2042	142	5	.	.	PUNCT
ijassa-2042	143	1	in	in	ADP
ijassa-2042	143	2	fact	fact	NOUN
ijassa-2042	143	3	,	,	PUNCT
ijassa-2042	143	4	here	here	ADV
ijassa-2042	143	5	we	we	PRON
ijassa-2042	143	6	have	have	VERB
ijassa-2042	143	7	to	to	PART
ijassa-2042	143	8	verify	verify	VERB
ijassa-2042	143	9	the	the	DET
ijassa-2042	143	10	equality	equality	NOUN
ijassa-2042	143	11	p	p	X
ijassa-2042	143	12	=	=	PUNCT
ijassa-2042	143	13	dy	dy	X
ijassa-2042	143	14	/	/	SYM
ijassa-2042	143	15	dx	dx	PROPN
ijassa-2042	143	16	only	only	ADV
ijassa-2042	143	17	,	,	PUNCT
ijassa-2042	143	18	which	which	PRON
ijassa-2042	143	19	follows	follow	VERB
ijassa-2042	143	20	from	from	ADP
ijassa-2042	143	21	the	the	DET
ijassa-2042	143	22	equality	equality	NOUN
ijassa-2042	143	23	p	p	X
ijassa-2042	143	24	=	=	PUNCT
ijassa-2042	143	25	dy	dy	X
ijassa-2042	143	26	/	/	SYM
ijassa-2042	143	27	dx	dx	PROPN
ijassa-2042	143	28	and	and	CCONJ
ijassa-2042	143	29	formula	formula	NOUN
ijassa-2042	143	30	(	(	PUNCT
ijassa-2042	143	31	2.5	2.5	NUM
ijassa-2042	143	32	)	)	PUNCT
ijassa-2042	143	33	.	.	PUNCT
ijassa-2042	144	1	it	it	PRON
ijassa-2042	144	2	more	more	ADV
ijassa-2042	144	3	convenient	convenient	ADJ
ijassa-2042	144	4	to	to	PART
ijassa-2042	144	5	write	write	VERB
ijassa-2042	144	6	the	the	DET
ijassa-2042	144	7	both	both	DET
ijassa-2042	144	8	equalities	equality	NOUN
ijassa-2042	144	9	via	via	ADP
ijassa-2042	144	10	differential	differential	ADJ
ijassa-2042	144	11	1	1	NUM
ijassa-2042	144	12	-	-	PUNCT
ijassa-2042	144	13	forms	form	NOUN
ijassa-2042	144	14	and	and	CCONJ
ijassa-2042	144	15	check	check	VERB
ijassa-2042	144	16	that	that	DET
ijassa-2042	144	17	1	1	NUM
ijassa-2042	144	18	-	-	PUNCT
ijassa-2042	144	19	forms	form	NOUN
ijassa-2042	144	20	pdx	pdx	NOUN
ijassa-2042	144	21	−	−	PROPN
ijassa-2042	144	22	dy	dy	NOUN
ijassa-2042	144	23	and	and	CCONJ
ijassa-2042	144	24	pdx−	pdx−	PROPN
ijassa-2042	144	25	dy	dy	NOUN
ijassa-2042	144	26	vanish	vanish	VERB
ijassa-2042	144	27	simultaneously	simultaneously	ADV
ijassa-2042	144	28	:	:	PUNCT
ijassa-2042	145	1	pdx	pdx	PROPN
ijassa-2042	145	2	−	−	PROPN
ijassa-2042	145	3	dy	dy	NOUN
ijassa-2042	145	4	=	=	SYM
ijassa-2042	145	5	xdp−	xdp−	PROPN
ijassa-2042	145	6	d(xp−	d(xp−	PROPN
ijassa-2042	145	7	y	y	PROPN
ijassa-2042	145	8	)	)	PUNCT
ijassa-2042	145	9	=	=	SYM
ijassa-2042	145	10	xdp−	xdp−	PROPN
ijassa-2042	145	11	d(xp	d(xp	PROPN
ijassa-2042	145	12	)	)	PUNCT
ijassa-2042	145	13	+	+	NUM
ijassa-2042	145	14	dy	dy	NOUN
ijassa-2042	145	15	=	=	SYM
ijassa-2042	145	16	−(pdx−	−(pdx−	PROPN
ijassa-2042	145	17	dy	dy	NOUN
ijassa-2042	145	18	)	)	PUNCT
ijassa-2042	145	19	.	.	PUNCT
ijassa-2042	146	1	from	from	ADP
ijassa-2042	146	2	geometric	geometric	ADJ
ijassa-2042	146	3	viewpoint	viewpoint	NOUN
ijassa-2042	146	4	,	,	PUNCT
ijassa-2042	146	5	this	this	PRON
ijassa-2042	146	6	means	mean	VERB
ijassa-2042	146	7	that	that	SCONJ
ijassa-2042	146	8	the	the	DET
ijassa-2042	146	9	legendre	legendre	PROPN
ijassa-2042	146	10	transformation	transformation	NOUN
ijassa-2042	146	11	(	(	PUNCT
ijassa-2042	146	12	2.5	2.5	NUM
ijassa-2042	146	13	)	)	PUNCT
ijassa-2042	146	14	sends	send	VERB
ijassa-2042	146	15	contact	contact	NOUN
ijassa-2042	146	16	planes	plane	NOUN
ijassa-2042	146	17	pdx−	pdx−	NOUN
ijassa-2042	146	18	dy	dy	VERB
ijassa-2042	146	19	=	=	NOUN
ijassa-2042	146	20	0	0	NUM
ijassa-2042	146	21	to	to	PART
ijassa-2042	146	22	contact	contact	VERB
ijassa-2042	146	23	planes	plane	NOUN
ijassa-2042	146	24	pdx	pdx	PROPN
ijassa-2042	147	1	−	−	PROPN
ijassa-2042	147	2	dy	dy	X
ijassa-2042	147	3	=	=	SYM
ijassa-2042	147	4	0	0	NUM
ijassa-2042	147	5	,	,	PUNCT
ijassa-2042	147	6	i.e.	i.e.	X
ijassa-2042	147	7	,	,	PUNCT
ijassa-2042	147	8	it	it	PRON
ijassa-2042	147	9	preserves	preserve	VERB
ijassa-2042	147	10	the	the	DET
ijassa-2042	147	11	natural	natural	ADJ
ijassa-2042	147	12	contact	contact	NOUN
ijassa-2042	147	13	structure	structure	NOUN
ijassa-2042	147	14	of	of	ADP
ijassa-2042	147	15	the	the	DET
ijassa-2042	147	16	space	space	NOUN
ijassa-2042	147	17	j1	j1	PROPN
ijassa-2042	147	18	.	.	PUNCT
ijassa-2042	148	1	this	this	PRON
ijassa-2042	148	2	implies	imply	VERB
ijassa-2042	148	3	that	that	SCONJ
ijassa-2042	148	4	the	the	DET
ijassa-2042	148	5	legendre	legendre	PROPN
ijassa-2042	148	6	transformation	transformation	NOUN
ijassa-2042	148	7	sends	send	VERB
ijassa-2042	148	8	integral	integral	ADJ
ijassa-2042	148	9	curves	curve	NOUN
ijassa-2042	148	10	of	of	ADP
ijassa-2042	148	11	equation	equation	NOUN
ijassa-2042	148	12	(	(	PUNCT
ijassa-2042	148	13	2.10	2.10	NUM
ijassa-2042	148	14	)	)	PUNCT
ijassa-2042	148	15	to	to	ADP
ijassa-2042	148	16	integral	integral	ADJ
ijassa-2042	148	17	curves	curve	NOUN
ijassa-2042	148	18	of	of	ADP
ijassa-2042	148	19	its	its	PRON
ijassa-2042	148	20	dual	dual	ADJ
ijassa-2042	148	21	equation	equation	NOUN
ijassa-2042	148	22	(	(	PUNCT
ijassa-2042	148	23	2.11	2.11	NUM
ijassa-2042	148	24	)	)	PUNCT
ijassa-2042	148	25	.	.	PUNCT
ijassa-2042	149	1	in	in	ADP
ijassa-2042	149	2	other	other	ADJ
ijassa-2042	149	3	words	word	NOUN
ijassa-2042	149	4	,	,	PUNCT
ijassa-2042	149	5	integral	integral	ADJ
ijassa-2042	149	6	curves	curve	NOUN
ijassa-2042	149	7	of	of	ADP
ijassa-2042	149	8	equation	equation	NOUN
ijassa-2042	149	9	(	(	PUNCT
ijassa-2042	149	10	2.11	2.11	NUM
ijassa-2042	149	11	)	)	PUNCT
ijassa-2042	149	12	are	be	AUX
ijassa-2042	149	13	dual	dual	ADJ
ijassa-2042	149	14	to	to	ADP
ijassa-2042	149	15	integral	integral	ADJ
ijassa-2042	149	16	curves	curve	NOUN
ijassa-2042	149	17	of	of	ADP
ijassa-2042	149	18	equation	equation	NOUN
ijassa-2042	149	19	(	(	PUNCT
ijassa-2042	149	20	2.10	2.10	NUM
ijassa-2042	149	21	)	)	PUNCT
ijassa-2042	149	22	,	,	PUNCT
ijassa-2042	149	23	and	and	CCONJ
ijassa-2042	149	24	vice	vice	ADV
ijassa-2042	149	25	versa	versa	ADV
ijassa-2042	149	26	(	(	PUNCT
ijassa-2042	149	27	since	since	SCONJ
ijassa-2042	149	28	the	the	DET
ijassa-2042	149	29	legendre	legendre	PROPN
ijassa-2042	149	30	transformation	transformation	NOUN
ijassa-2042	149	31	is	be	AUX
ijassa-2042	149	32	involution	involution	NOUN
ijassa-2042	149	33	)	)	PUNCT
ijassa-2042	149	34	.	.	PUNCT
ijassa-2042	150	1	this	this	DET
ijassa-2042	150	2	fact	fact	NOUN
ijassa-2042	150	3	can	can	AUX
ijassa-2042	150	4	be	be	AUX
ijassa-2042	150	5	used	use	VERB
ijassa-2042	150	6	for	for	ADP
ijassa-2042	150	7	solution	solution	NOUN
ijassa-2042	150	8	of	of	ADP
ijassa-2042	150	9	differential	differential	ADJ
ijassa-2042	150	10	equation	equation	NOUN
ijassa-2042	150	11	:	:	PUNCT
ijassa-2042	150	12	if	if	SCONJ
ijassa-2042	150	13	the	the	DET
ijassa-2042	150	14	dual	dual	ADJ
ijassa-2042	150	15	equation	equation	NOUN
ijassa-2042	150	16	can	can	AUX
ijassa-2042	150	17	be	be	AUX
ijassa-2042	150	18	solved	solve	VERB
ijassa-2042	150	19	easier	easy	ADJ
ijassa-2042	150	20	that	that	SCONJ
ijassa-2042	150	21	the	the	DET
ijassa-2042	150	22	initial	initial	ADJ
ijassa-2042	150	23	one	one	NUM
ijassa-2042	150	24	.	.	PUNCT
ijassa-2042	151	1	the	the	DET
ijassa-2042	151	2	notion	notion	NOUN
ijassa-2042	151	3	of	of	ADP
ijassa-2042	151	4	duality	duality	NOUN
ijassa-2042	151	5	also	also	ADV
ijassa-2042	151	6	helps	help	VERB
ijassa-2042	151	7	to	to	PART
ijassa-2042	151	8	study	study	VERB
ijassa-2042	151	9	singularities	singularity	NOUN
ijassa-2042	151	10	of	of	ADP
ijassa-2042	151	11	implicit	implicit	ADJ
ijassa-2042	151	12	differential	differential	ADJ
ijassa-2042	151	13	equations	equation	NOUN
ijassa-2042	151	14	;	;	PUNCT
ijassa-2042	151	15	see	see	VERB
ijassa-2042	151	16	[	[	X
ijassa-2042	151	17	7	7	NUM
ijassa-2042	151	18	]	]	PUNCT
ijassa-2042	151	19	.	.	PUNCT
ijassa-2042	152	1	a	a	DET
ijassa-2042	152	2	nice	nice	ADJ
ijassa-2042	152	3	example	example	NOUN
ijassa-2042	152	4	of	of	ADP
ijassa-2042	152	5	application	application	NOUN
ijassa-2042	152	6	the	the	DET
ijassa-2042	152	7	legendre	legendre	PROPN
ijassa-2042	152	8	transformation	transformation	NOUN
ijassa-2042	152	9	to	to	PART
ijassa-2042	152	10	differential	differential	VERB
ijassa-2042	152	11	equation	equation	NOUN
ijassa-2042	152	12	is	be	AUX
ijassa-2042	152	13	the	the	DET
ijassa-2042	152	14	clairaut	clairaut	PROPN
ijassa-2042	152	15	equation	equation	NOUN
ijassa-2042	152	16	,	,	PUNCT
ijassa-2042	152	17	which	which	PRON
ijassa-2042	152	18	has	have	VERB
ijassa-2042	152	19	the	the	DET
ijassa-2042	152	20	form	form	NOUN
ijassa-2042	152	21	xp−	xp−	PUNCT
ijassa-2042	152	22	y	y	PROPN
ijassa-2042	152	23	=	=	SYM
ijassa-2042	152	24	f(p	f(p	PROPN
ijassa-2042	152	25	)	)	PUNCT
ijassa-2042	152	26	,	,	PUNCT
ijassa-2042	152	27	p	p	NOUN
ijassa-2042	152	28	=	=	PUNCT
ijassa-2042	152	29	dy	dy	X
ijassa-2042	152	30	/	/	SYM
ijassa-2042	152	31	dx	dx	PROPN
ijassa-2042	152	32	,	,	PUNCT
ijassa-2042	152	33	(	(	PUNCT
ijassa-2042	152	34	2.12	2.12	NUM
ijassa-2042	152	35	)	)	PUNCT
ijassa-2042	152	36	where	where	SCONJ
ijassa-2042	152	37	f	f	PROPN
ijassa-2042	152	38	is	be	AUX
ijassa-2042	152	39	an	an	DET
ijassa-2042	152	40	arbitrary	arbitrary	ADJ
ijassa-2042	152	41	function	function	NOUN
ijassa-2042	152	42	.	.	PUNCT
ijassa-2042	153	1	see	see	VERB
ijassa-2042	153	2	,	,	PUNCT
ijassa-2042	153	3	for	for	ADP
ijassa-2042	153	4	example	example	NOUN
ijassa-2042	153	5	,	,	PUNCT
ijassa-2042	153	6	[	[	X
ijassa-2042	153	7	2	2	NUM
ijassa-2042	153	8	,	,	PUNCT
ijassa-2042	153	9	12	12	NUM
ijassa-2042	153	10	]	]	PUNCT
ijassa-2042	153	11	.	.	PUNCT
ijassa-2042	154	1	the	the	DET
ijassa-2042	154	2	legendre	legendre	PROPN
ijassa-2042	154	3	transformation	transformation	NOUN
ijassa-2042	154	4	sends	send	VERB
ijassa-2042	154	5	differential	differential	ADJ
ijassa-2042	154	6	equation	equation	NOUN
ijassa-2042	154	7	(	(	PUNCT
ijassa-2042	154	8	2.12	2.12	NUM
ijassa-2042	154	9	)	)	PUNCT
ijassa-2042	154	10	to	to	ADP
ijassa-2042	154	11	the	the	DET
ijassa-2042	154	12	equation	equation	NOUN
ijassa-2042	154	13	y	y	PROPN
ijassa-2042	154	14	=	=	SYM
ijassa-2042	154	15	f(x	f(x	PROPN
ijassa-2042	154	16	)	)	PUNCT
ijassa-2042	154	17	,	,	PUNCT
ijassa-2042	154	18	which	which	PRON
ijassa-2042	154	19	does	do	AUX
ijassa-2042	154	20	not	not	PART
ijassa-2042	154	21	contain	contain	VERB
ijassa-2042	154	22	the	the	DET
ijassa-2042	154	23	derivative	derivative	NOUN
ijassa-2042	154	24	.	.	PUNCT
ijassa-2042	155	1	therefore	therefore	ADV
ijassa-2042	155	2	,	,	PUNCT
ijassa-2042	155	3	the	the	DET
ijassa-2042	155	4	legendre	legendre	PROPN
ijassa-2042	155	5	transformation	transformation	NOUN
ijassa-2042	155	6	sends	send	VERB
ijassa-2042	155	7	integral	integral	ADJ
ijassa-2042	155	8	curves	curve	NOUN
ijassa-2042	155	9	of	of	ADP
ijassa-2042	155	10	equation	equation	NOUN
ijassa-2042	155	11	(	(	PUNCT
ijassa-2042	155	12	2.12	2.12	NUM
ijassa-2042	155	13	)	)	PUNCT
ijassa-2042	155	14	to	to	ADP
ijassa-2042	155	15	points	point	NOUN
ijassa-2042	155	16	of	of	ADP
ijassa-2042	155	17	the	the	DET
ijassa-2042	155	18	(	(	PUNCT
ijassa-2042	155	19	x	x	PROPN
ijassa-2042	155	20	,	,	PUNCT
ijassa-2042	155	21	y	y	NOUN
ijassa-2042	155	22	)	)	PUNCT
ijassa-2042	155	23	-plane	-plane	NOUN
ijassa-2042	155	24	filling	fill	VERB
ijassa-2042	155	25	the	the	DET
ijassa-2042	155	26	graph	graph	NOUN
ijassa-2042	155	27	of	of	ADP
ijassa-2042	155	28	the	the	DET
ijassa-2042	155	29	function	function	NOUN
ijassa-2042	155	30	y	y	PROPN
ijassa-2042	155	31	=	=	SYM
ijassa-2042	155	32	f(x	f(x	PROPN
ijassa-2042	155	33	)	)	PUNCT
ijassa-2042	155	34	.	.	PUNCT
ijassa-2042	156	1	taking	take	VERB
ijassa-2042	156	2	into	into	ADP
ijassa-2042	156	3	account	account	NOUN
ijassa-2042	156	4	example	example	NOUN
ijassa-2042	156	5	2.1	2.1	NUM
ijassa-2042	156	6	,	,	PUNCT
ijassa-2042	156	7	one	one	PRON
ijassa-2042	156	8	can	can	AUX
ijassa-2042	156	9	see	see	VERB
ijassa-2042	156	10	that	that	DET
ijassa-2042	156	11	integral	integral	ADJ
ijassa-2042	156	12	curves	curve	NOUN
ijassa-2042	156	13	of	of	ADP
ijassa-2042	156	14	equation	equation	NOUN
ijassa-2042	156	15	(	(	PUNCT
ijassa-2042	156	16	2.12	2.12	NUM
ijassa-2042	156	17	)	)	PUNCT
ijassa-2042	156	18	are	be	AUX
ijassa-2042	156	19	straight	straight	ADJ
ijassa-2042	156	20	lines	line	NOUN
ijassa-2042	156	21	tangent	tangent	ADJ
ijassa-2042	156	22	to	to	ADP
ijassa-2042	156	23	a	a	DET
ijassa-2042	156	24	curve	curve	NOUN
ijassa-2042	156	25	dual	dual	ADJ
ijassa-2042	156	26	to	to	ADP
ijassa-2042	156	27	the	the	DET
ijassa-2042	156	28	graph	graph	NOUN
ijassa-2042	156	29	y	y	PROPN
ijassa-2042	156	30	=	=	SYM
ijassa-2042	156	31	f(x	f(x	PROPN
ijassa-2042	156	32	)	)	PUNCT
ijassa-2042	156	33	.	.	PUNCT
ijassa-2042	157	1	copyright	copyright	NOUN
ijassa-2042	157	2	©	©	PROPN
ijassa-2042	157	3	2025	2025	NUM
ijassa-2042	157	4	assa	assa	NOUN
ijassa-2042	157	5	.	.	PUNCT
ijassa-2042	158	1	adv	adv	PROPN
ijassa-2042	158	2	syst	syst	PROPN
ijassa-2042	158	3	sci	sci	PROPN
ijassa-2042	158	4	appl	appl	PROPN
ijassa-2042	158	5	(	(	PUNCT
ijassa-2042	158	6	2025	2025	NUM
ijassa-2042	158	7	)	)	PUNCT
ijassa-2042	158	8	50	50	NUM
ijassa-2042	158	9	n.	n.	NOUN
ijassa-2042	158	10	pavlova	pavlova	PROPN
ijassa-2042	158	11	,	,	PUNCT
ijassa-2042	158	12	a.	a.	NOUN
ijassa-2042	158	13	remizov	remizov	VERB
ijassa-2042	158	14	a	a	DET
ijassa-2042	158	15	direct	direct	ADJ
ijassa-2042	158	16	calculation	calculation	NOUN
ijassa-2042	158	17	shows	show	VERB
ijassa-2042	158	18	that	that	SCONJ
ijassa-2042	158	19	this	this	DET
ijassa-2042	158	20	curve	curve	NOUN
ijassa-2042	158	21	given	give	VERB
ijassa-2042	158	22	by	by	ADP
ijassa-2042	158	23	formula	formula	NOUN
ijassa-2042	158	24	x	x	X
ijassa-2042	159	1	=	=	PUNCT
ijassa-2042	159	2	f	f	PROPN
ijassa-2042	159	3	′(p	′(p	PROPN
ijassa-2042	159	4	)	)	PUNCT
ijassa-2042	159	5	,	,	PUNCT
ijassa-2042	160	1	y	y	PROPN
ijassa-2042	160	2	=	=	PUNCT
ijassa-2042	160	3	pf	pf	PROPN
ijassa-2042	160	4	′(p)−	′(p)−	PROPN
ijassa-2042	160	5	f(p	f(p	PROPN
ijassa-2042	160	6	)	)	PUNCT
ijassa-2042	160	7	(	(	PUNCT
ijassa-2042	160	8	2.13	2.13	NUM
ijassa-2042	160	9	)	)	PUNCT
ijassa-2042	160	10	is	be	AUX
ijassa-2042	160	11	the	the	DET
ijassa-2042	160	12	discriminant	discriminant	ADJ
ijassa-2042	160	13	curve	curve	NOUN
ijassa-2042	160	14	of	of	ADP
ijassa-2042	160	15	equation	equation	NOUN
ijassa-2042	160	16	(	(	PUNCT
ijassa-2042	160	17	2.12	2.12	NUM
ijassa-2042	160	18	)	)	PUNCT
ijassa-2042	160	19	and	and	CCONJ
ijassa-2042	160	20	the	the	DET
ijassa-2042	160	21	envelope	envelope	NOUN
ijassa-2042	160	22	of	of	ADP
ijassa-2042	160	23	the	the	DET
ijassa-2042	160	24	family	family	NOUN
ijassa-2042	160	25	of	of	ADP
ijassa-2042	160	26	solutions	solution	NOUN
ijassa-2042	160	27	of	of	ADP
ijassa-2042	160	28	equation	equation	NOUN
ijassa-2042	160	29	(	(	PUNCT
ijassa-2042	160	30	2.12	2.12	NUM
ijassa-2042	160	31	)	)	PUNCT
ijassa-2042	160	32	–	–	PUNCT
ijassa-2042	160	33	its	its	PRON
ijassa-2042	160	34	tangent	tangent	NOUN
ijassa-2042	160	35	lines	line	NOUN
ijassa-2042	160	36	.	.	PUNCT
ijassa-2042	161	1	this	this	PRON
ijassa-2042	161	2	gives	give	VERB
ijassa-2042	161	3	a	a	DET
ijassa-2042	161	4	simple	simple	ADJ
ijassa-2042	161	5	way	way	NOUN
ijassa-2042	161	6	to	to	PART
ijassa-2042	161	7	get	get	VERB
ijassa-2042	161	8	all	all	DET
ijassa-2042	161	9	solutions	solution	NOUN
ijassa-2042	161	10	of	of	ADP
ijassa-2042	161	11	equation	equation	NOUN
ijassa-2042	161	12	(	(	PUNCT
ijassa-2042	161	13	2.12	2.12	NUM
ijassa-2042	161	14	):	):	PUNCT
ijassa-2042	161	15	first	first	ADV
ijassa-2042	161	16	,	,	PUNCT
ijassa-2042	161	17	we	we	PRON
ijassa-2042	161	18	find	find	VERB
ijassa-2042	161	19	the	the	DET
ijassa-2042	161	20	discriminant	discriminant	ADJ
ijassa-2042	161	21	curve	curve	NOUN
ijassa-2042	161	22	(	(	PUNCT
ijassa-2042	161	23	2.13	2.13	NUM
ijassa-2042	161	24	)	)	PUNCT
ijassa-2042	161	25	,	,	PUNCT
ijassa-2042	161	26	which	which	PRON
ijassa-2042	161	27	is	be	AUX
ijassa-2042	161	28	called	call	VERB
ijassa-2042	161	29	singular	singular	ADJ
ijassa-2042	161	30	solution	solution	NOUN
ijassa-2042	161	31	of	of	ADP
ijassa-2042	161	32	(	(	PUNCT
ijassa-2042	161	33	2.12	2.12	NUM
ijassa-2042	161	34	)	)	PUNCT
ijassa-2042	161	35	.	.	PUNCT
ijassa-2042	162	1	the	the	DET
ijassa-2042	162	2	remaining	remain	VERB
ijassa-2042	162	3	solutions	solution	NOUN
ijassa-2042	162	4	are	be	AUX
ijassa-2042	162	5	tangent	tangent	ADJ
ijassa-2042	162	6	lines	line	NOUN
ijassa-2042	162	7	to	to	ADP
ijassa-2042	162	8	the	the	DET
ijassa-2042	162	9	discriminant	discriminant	NOUN
ijassa-2042	162	10	curve	curve	NOUN
ijassa-2042	162	11	taken	take	VERB
ijassa-2042	162	12	at	at	ADP
ijassa-2042	162	13	all	all	PRON
ijassa-2042	162	14	its	its	PRON
ijassa-2042	162	15	points	point	NOUN
ijassa-2042	162	16	.	.	PUNCT
ijassa-2042	163	1	moreover	moreover	ADV
ijassa-2042	163	2	,	,	PUNCT
ijassa-2042	163	3	it	it	PRON
ijassa-2042	163	4	is	be	AUX
ijassa-2042	163	5	not	not	PART
ijassa-2042	163	6	hard	hard	ADJ
ijassa-2042	163	7	to	to	PART
ijassa-2042	163	8	see	see	VERB
ijassa-2042	163	9	that	that	SCONJ
ijassa-2042	163	10	these	these	DET
ijassa-2042	163	11	tangent	tangent	NOUN
ijassa-2042	163	12	lines	line	NOUN
ijassa-2042	163	13	are	be	AUX
ijassa-2042	163	14	given	give	VERB
ijassa-2042	163	15	by	by	ADP
ijassa-2042	163	16	the	the	DET
ijassa-2042	163	17	formula	formula	NOUN
ijassa-2042	163	18	y	y	PROPN
ijassa-2042	163	19	=	=	SYM
ijassa-2042	163	20	cx−	cx−	PROPN
ijassa-2042	163	21	f(c	f(c	PROPN
ijassa-2042	163	22	)	)	PUNCT
ijassa-2042	163	23	,	,	PUNCT
ijassa-2042	163	24	c	c	NOUN
ijassa-2042	163	25	=	=	SYM
ijassa-2042	163	26	const	const	PROPN
ijassa-2042	163	27	.	.	PUNCT
ijassa-2042	164	1	(	(	PUNCT
ijassa-2042	164	2	2.14	2.14	NUM
ijassa-2042	164	3	)	)	PUNCT
ijassa-2042	164	4	remark	remark	NOUN
ijassa-2042	164	5	that	that	SCONJ
ijassa-2042	164	6	formula	formula	NOUN
ijassa-2042	164	7	(	(	PUNCT
ijassa-2042	164	8	2.14	2.14	NUM
ijassa-2042	164	9	)	)	PUNCT
ijassa-2042	164	10	can	can	AUX
ijassa-2042	164	11	be	be	AUX
ijassa-2042	164	12	obtained	obtain	VERB
ijassa-2042	164	13	from	from	ADP
ijassa-2042	164	14	equation	equation	NOUN
ijassa-2042	164	15	(	(	PUNCT
ijassa-2042	164	16	2.12	2.12	NUM
ijassa-2042	164	17	)	)	PUNCT
ijassa-2042	164	18	after	after	ADP
ijassa-2042	164	19	replacing	replace	VERB
ijassa-2042	164	20	p	p	NOUN
ijassa-2042	164	21	with	with	ADP
ijassa-2042	164	22	c.	c.	PROPN
ijassa-2042	164	23	example	example	NOUN
ijassa-2042	164	24	2.4	2.4	NUM
ijassa-2042	164	25	:	:	PUNCT
ijassa-2042	164	26	in	in	ADP
ijassa-2042	164	27	fig	fig	NOUN
ijassa-2042	164	28	.	.	PUNCT
ijassa-2042	165	1	2.5	2.5	NUM
ijassa-2042	165	2	,	,	PUNCT
ijassa-2042	165	3	integral	integral	ADJ
ijassa-2042	165	4	curves	curve	NOUN
ijassa-2042	165	5	of	of	ADP
ijassa-2042	165	6	equation	equation	NOUN
ijassa-2042	165	7	(	(	PUNCT
ijassa-2042	165	8	2.12	2.12	NUM
ijassa-2042	165	9	)	)	PUNCT
ijassa-2042	165	10	with	with	ADP
ijassa-2042	165	11	the	the	DET
ijassa-2042	165	12	function	function	NOUN
ijassa-2042	165	13	f(p	f(p	PROPN
ijassa-2042	165	14	)	)	PUNCT
ijassa-2042	166	1	=	=	SYM
ijassa-2042	166	2	p3	p3	PROPN
ijassa-2042	166	3	are	be	AUX
ijassa-2042	166	4	presented	present	VERB
ijassa-2042	166	5	.	.	PUNCT
ijassa-2042	167	1	in	in	ADP
ijassa-2042	167	2	this	this	DET
ijassa-2042	167	3	case	case	NOUN
ijassa-2042	167	4	,	,	PUNCT
ijassa-2042	167	5	the	the	DET
ijassa-2042	167	6	discriminant	discriminant	ADJ
ijassa-2042	167	7	curve	curve	NOUN
ijassa-2042	167	8	(	(	PUNCT
ijassa-2042	167	9	2.13	2.13	NUM
ijassa-2042	167	10	)	)	PUNCT
ijassa-2042	167	11	is	be	AUX
ijassa-2042	167	12	the	the	DET
ijassa-2042	167	13	semicubic	semicubic	ADJ
ijassa-2042	167	14	parabola	parabola	NOUN
ijassa-2042	167	15	x	x	PUNCT
ijassa-2042	167	16	=	=	SYM
ijassa-2042	167	17	3p2	3p2	NUM
ijassa-2042	167	18	,	,	PUNCT
ijassa-2042	167	19	y	y	PROPN
ijassa-2042	167	20	=	=	SYM
ijassa-2042	167	21	2p3	2p3	NUM
ijassa-2042	167	22	(	(	PUNCT
ijassa-2042	167	23	2.15	2.15	NUM
ijassa-2042	167	24	)	)	PUNCT
ijassa-2042	167	25	with	with	ADP
ijassa-2042	167	26	the	the	DET
ijassa-2042	167	27	casp	casp	NOUN
ijassa-2042	167	28	at	at	ADP
ijassa-2042	167	29	the	the	DET
ijassa-2042	167	30	origin	origin	NOUN
ijassa-2042	167	31	.	.	PUNCT
ijassa-2042	168	1	the	the	DET
ijassa-2042	168	2	family	family	NOUN
ijassa-2042	168	3	of	of	ADP
ijassa-2042	168	4	tangent	tangent	ADJ
ijassa-2042	168	5	lines	line	NOUN
ijassa-2042	168	6	to	to	ADP
ijassa-2042	168	7	the	the	DET
ijassa-2042	168	8	curve	curve	NOUN
ijassa-2042	168	9	(	(	PUNCT
ijassa-2042	168	10	2.15	2.15	NUM
ijassa-2042	168	11	)	)	PUNCT
ijassa-2042	168	12	has	have	VERB
ijassa-2042	168	13	concentration	concentration	NOUN
ijassa-2042	168	14	(	(	PUNCT
ijassa-2042	168	15	patch	patch	NOUN
ijassa-2042	168	16	)	)	PUNCT
ijassa-2042	168	17	near	near	ADP
ijassa-2042	168	18	this	this	DET
ijassa-2042	168	19	curve	curve	NOUN
ijassa-2042	168	20	and	and	CCONJ
ijassa-2042	168	21	especially	especially	ADV
ijassa-2042	168	22	near	near	ADP
ijassa-2042	168	23	its	its	PRON
ijassa-2042	168	24	cusp	cusp	NOUN
ijassa-2042	168	25	.	.	PUNCT
ijassa-2042	169	1	this	this	DET
ijassa-2042	169	2	phenomenon	phenomenon	NOUN
ijassa-2042	169	3	is	be	AUX
ijassa-2042	169	4	called	call	VERB
ijassa-2042	169	5	a	a	DET
ijassa-2042	169	6	caustic	caustic	ADJ
ijassa-2042	169	7	[	[	X
ijassa-2042	169	8	3	3	NUM
ijassa-2042	169	9	]	]	PUNCT
ijassa-2042	169	10	.	.	PUNCT
ijassa-2042	170	1	in	in	ADP
ijassa-2042	170	2	physics	physics	PROPN
ijassa-2042	170	3	,	,	PUNCT
ijassa-2042	170	4	a	a	DET
ijassa-2042	170	5	caustic	caustic	NOUN
ijassa-2042	170	6	is	be	AUX
ijassa-2042	170	7	the	the	DET
ijassa-2042	170	8	envelope	envelope	NOUN
ijassa-2042	170	9	of	of	ADP
ijassa-2042	170	10	light	light	ADJ
ijassa-2042	170	11	rays	ray	NOUN
ijassa-2042	170	12	which	which	PRON
ijassa-2042	170	13	have	have	AUX
ijassa-2042	170	14	been	be	AUX
ijassa-2042	170	15	reflected	reflect	VERB
ijassa-2042	170	16	or	or	CCONJ
ijassa-2042	170	17	refracted	refract	VERB
ijassa-2042	170	18	by	by	ADP
ijassa-2042	170	19	a	a	DET
ijassa-2042	170	20	curve	curve	NOUN
ijassa-2042	170	21	(	(	PUNCT
ijassa-2042	170	22	on	on	ADP
ijassa-2042	170	23	the	the	DET
ijassa-2042	170	24	plane	plane	NOUN
ijassa-2042	170	25	)	)	PUNCT
ijassa-2042	170	26	or	or	CCONJ
ijassa-2042	170	27	a	a	DET
ijassa-2042	170	28	surface	surface	NOUN
ijassa-2042	170	29	(	(	PUNCT
ijassa-2042	170	30	in	in	ADP
ijassa-2042	170	31	the	the	DET
ijassa-2042	170	32	space	space	NOUN
ijassa-2042	170	33	)	)	PUNCT
ijassa-2042	170	34	.	.	PUNCT
ijassa-2042	171	1	fig	fig	NOUN
ijassa-2042	171	2	.	.	PUNCT
ijassa-2042	172	1	2.5	2.5	NUM
ijassa-2042	172	2	contains	contain	VERB
ijassa-2042	172	3	a	a	DET
ijassa-2042	172	4	caustic	caustic	ADJ
ijassa-2042	172	5	–	–	PUNCT
ijassa-2042	172	6	the	the	DET
ijassa-2042	172	7	semicubic	semicubic	ADJ
ijassa-2042	172	8	parabola	parabola	NOUN
ijassa-2042	172	9	(	(	PUNCT
ijassa-2042	172	10	2.15	2.15	NUM
ijassa-2042	172	11	)	)	PUNCT
ijassa-2042	172	12	.	.	PUNCT
ijassa-2042	173	1	fig	fig	NOUN
ijassa-2042	173	2	.	.	PUNCT
ijassa-2042	174	1	2.5	2.5	NUM
ijassa-2042	174	2	.	.	PUNCT
ijassa-2042	175	1	integral	integral	ADJ
ijassa-2042	175	2	curves	curve	NOUN
ijassa-2042	175	3	of	of	ADP
ijassa-2042	175	4	the	the	DET
ijassa-2042	175	5	clairaut	clairaut	PROPN
ijassa-2042	175	6	equation	equation	NOUN
ijassa-2042	175	7	xp−	xp−	PUNCT
ijassa-2042	175	8	y	y	PROPN
ijassa-2042	175	9	=	=	SYM
ijassa-2042	175	10	p3	p3	PROPN
ijassa-2042	175	11	are	be	AUX
ijassa-2042	175	12	tangent	tangent	ADJ
ijassa-2042	175	13	lines	line	NOUN
ijassa-2042	175	14	to	to	ADP
ijassa-2042	175	15	the	the	DET
ijassa-2042	175	16	semicubic	semicubic	ADJ
ijassa-2042	175	17	parabola	parabola	NOUN
ijassa-2042	175	18	(	(	PUNCT
ijassa-2042	175	19	2.15	2.15	NUM
ijassa-2042	175	20	)	)	PUNCT
ijassa-2042	175	21	.	.	PUNCT
ijassa-2042	176	1	the	the	DET
ijassa-2042	176	2	picture	picture	NOUN
ijassa-2042	176	3	is	be	AUX
ijassa-2042	176	4	taken	take	VERB
ijassa-2042	176	5	from	from	ADP
ijassa-2042	176	6	[	[	X
ijassa-2042	176	7	10	10	NUM
ijassa-2042	176	8	]	]	PUNCT
ijassa-2042	176	9	finally	finally	ADV
ijassa-2042	176	10	,	,	PUNCT
ijassa-2042	176	11	let	let	VERB
ijassa-2042	176	12	us	we	PRON
ijassa-2042	176	13	consider	consider	VERB
ijassa-2042	176	14	the	the	DET
ijassa-2042	176	15	clairaut	clairaut	NOUN
ijassa-2042	176	16	equation	equation	NOUN
ijassa-2042	176	17	of	of	ADP
ijassa-2042	176	18	more	more	ADJ
ijassa-2042	176	19	general	general	ADJ
ijassa-2042	176	20	type	type	NOUN
ijassa-2042	176	21	:	:	PUNCT
ijassa-2042	176	22	f	f	PROPN
ijassa-2042	176	23	(	(	PUNCT
ijassa-2042	176	24	p	p	X
ijassa-2042	176	25	,	,	PUNCT
ijassa-2042	176	26	xp−	xp−	NUM
ijassa-2042	176	27	y	y	NOUN
ijassa-2042	176	28	)	)	PUNCT
ijassa-2042	176	29	=	=	SYM
ijassa-2042	176	30	0	0	NUM
ijassa-2042	176	31	,	,	PUNCT
ijassa-2042	176	32	p	p	NOUN
ijassa-2042	176	33	=	=	PUNCT
ijassa-2042	176	34	dy	dy	X
ijassa-2042	176	35	/	/	SYM
ijassa-2042	176	36	dx	dx	PROPN
ijassa-2042	176	37	,	,	PUNCT
ijassa-2042	176	38	(	(	PUNCT
ijassa-2042	176	39	2.16	2.16	NUM
ijassa-2042	176	40	)	)	PUNCT
ijassa-2042	176	41	where	where	SCONJ
ijassa-2042	176	42	f	f	PROPN
ijassa-2042	176	43	(	(	PUNCT
ijassa-2042	176	44	u	u	NOUN
ijassa-2042	176	45	,	,	PUNCT
ijassa-2042	176	46	v	v	NOUN
ijassa-2042	176	47	)	)	PUNCT
ijassa-2042	176	48	is	be	AUX
ijassa-2042	176	49	an	an	DET
ijassa-2042	176	50	arbitrary	arbitrary	ADJ
ijassa-2042	176	51	function	function	NOUN
ijassa-2042	176	52	of	of	ADP
ijassa-2042	176	53	two	two	NUM
ijassa-2042	176	54	variables	variable	NOUN
ijassa-2042	176	55	.	.	PUNCT
ijassa-2042	177	1	the	the	DET
ijassa-2042	177	2	legendre	legendre	PROPN
ijassa-2042	177	3	transformation	transformation	NOUN
ijassa-2042	177	4	sends	send	VERB
ijassa-2042	177	5	(	(	PUNCT
ijassa-2042	177	6	2.16	2.16	NUM
ijassa-2042	177	7	)	)	PUNCT
ijassa-2042	177	8	to	to	ADP
ijassa-2042	177	9	the	the	DET
ijassa-2042	177	10	equation	equation	NOUN
ijassa-2042	177	11	f	f	X
ijassa-2042	177	12	(	(	PUNCT
ijassa-2042	177	13	x	x	X
ijassa-2042	177	14	,	,	PUNCT
ijassa-2042	177	15	y	y	PROPN
ijassa-2042	177	16	)	)	PUNCT
ijassa-2042	178	1	=	=	SYM
ijassa-2042	178	2	0	0	NUM
ijassa-2042	178	3	without	without	ADP
ijassa-2042	178	4	derivative	derivative	NOUN
ijassa-2042	178	5	.	.	PUNCT
ijassa-2042	179	1	proceeding	proceed	VERB
ijassa-2042	179	2	in	in	ADP
ijassa-2042	179	3	the	the	DET
ijassa-2042	179	4	same	same	ADJ
ijassa-2042	179	5	way	way	NOUN
ijassa-2042	179	6	as	as	ADP
ijassa-2042	179	7	before	before	ADV
ijassa-2042	179	8	,	,	PUNCT
ijassa-2042	179	9	we	we	PRON
ijassa-2042	179	10	conclude	conclude	VERB
ijassa-2042	179	11	that	that	SCONJ
ijassa-2042	179	12	integral	integral	ADJ
ijassa-2042	179	13	curves	curve	NOUN
ijassa-2042	179	14	of	of	ADP
ijassa-2042	179	15	equation	equation	NOUN
ijassa-2042	179	16	(	(	PUNCT
ijassa-2042	179	17	2.16	2.16	NUM
ijassa-2042	179	18	)	)	PUNCT
ijassa-2042	179	19	are	be	AUX
ijassa-2042	179	20	straight	straight	ADJ
ijassa-2042	179	21	lines	line	NOUN
ijassa-2042	179	22	tangent	tangent	NOUN
ijassa-2042	179	23	to	to	ADP
ijassa-2042	179	24	the	the	DET
ijassa-2042	179	25	curve	curve	NOUN
ijassa-2042	179	26	dual	dual	ADJ
ijassa-2042	179	27	to	to	ADP
ijassa-2042	179	28	f	f	PROPN
ijassa-2042	179	29	(	(	PUNCT
ijassa-2042	179	30	x	x	X
ijassa-2042	179	31	,	,	PUNCT
ijassa-2042	179	32	y	y	PROPN
ijassa-2042	179	33	)	)	PUNCT
ijassa-2042	179	34	=	=	PUNCT
ijassa-2042	180	1	0	0	X
ijassa-2042	180	2	.	.	PUNCT
ijassa-2042	181	1	the	the	DET
ijassa-2042	181	2	equation	equation	NOUN
ijassa-2042	181	3	f	f	X
ijassa-2042	181	4	(	(	PUNCT
ijassa-2042	181	5	x	x	X
ijassa-2042	181	6	,	,	PUNCT
ijassa-2042	181	7	y	y	PROPN
ijassa-2042	181	8	)	)	PUNCT
ijassa-2042	182	1	=	=	SYM
ijassa-2042	182	2	0	0	NUM
ijassa-2042	182	3	defined	define	VERB
ijassa-2042	182	4	the	the	DET
ijassa-2042	182	5	discriminant	discriminant	ADJ
ijassa-2042	182	6	curves	curve	NOUN
ijassa-2042	182	7	of	of	ADP
ijassa-2042	182	8	equation	equation	NOUN
ijassa-2042	182	9	(	(	PUNCT
ijassa-2042	182	10	2.16	2.16	NUM
ijassa-2042	182	11	)	)	PUNCT
ijassa-2042	182	12	,	,	PUNCT
ijassa-2042	182	13	which	which	PRON
ijassa-2042	182	14	is	be	AUX
ijassa-2042	182	15	its	its	PRON
ijassa-2042	182	16	singular	singular	ADJ
ijassa-2042	182	17	solution	solution	NOUN
ijassa-2042	182	18	and	and	CCONJ
ijassa-2042	182	19	the	the	DET
ijassa-2042	182	20	envelope	envelope	NOUN
ijassa-2042	182	21	of	of	ADP
ijassa-2042	182	22	other	other	ADJ
ijassa-2042	182	23	solutions	solution	NOUN
ijassa-2042	182	24	–	–	PUNCT
ijassa-2042	182	25	tangent	tangent	NOUN
ijassa-2042	182	26	lines	line	NOUN
ijassa-2042	182	27	given	give	VERB
ijassa-2042	182	28	by	by	ADP
ijassa-2042	182	29	the	the	DET
ijassa-2042	182	30	formula	formula	NOUN
ijassa-2042	182	31	y	y	NOUN
ijassa-2042	182	32	=	=	PUNCT
ijassa-2042	182	33	ax+	ax+	PROPN
ijassa-2042	182	34	b	b	NUM
ijassa-2042	182	35	,	,	PUNCT
ijassa-2042	182	36	f	f	PROPN
ijassa-2042	182	37	(	(	PUNCT
ijassa-2042	182	38	a,−b	a,−b	NOUN
ijassa-2042	182	39	)	)	PUNCT
ijassa-2042	182	40	=	=	SYM
ijassa-2042	182	41	0	0	NUM
ijassa-2042	182	42	,	,	PUNCT
ijassa-2042	182	43	which	which	PRON
ijassa-2042	182	44	is	be	AUX
ijassa-2042	182	45	a	a	DET
ijassa-2042	182	46	generalization	generalization	NOUN
ijassa-2042	182	47	of	of	ADP
ijassa-2042	182	48	(	(	PUNCT
ijassa-2042	182	49	2.14	2.14	NUM
ijassa-2042	182	50	)	)	PUNCT
ijassa-2042	182	51	.	.	PUNCT
ijassa-2042	183	1	copyright	copyright	NOUN
ijassa-2042	183	2	©	©	PROPN
ijassa-2042	183	3	2025	2025	NUM
ijassa-2042	183	4	assa	assa	NOUN
ijassa-2042	183	5	.	.	PUNCT
ijassa-2042	184	1	adv	adv	PROPN
ijassa-2042	184	2	syst	syst	PROPN
ijassa-2042	184	3	sci	sci	PROPN
ijassa-2042	184	4	appl	appl	PROPN
ijassa-2042	184	5	(	(	PUNCT
ijassa-2042	184	6	2025	2025	NUM
ijassa-2042	184	7	)	)	PUNCT
ijassa-2042	184	8	legendre	legendre	PROPN
ijassa-2042	184	9	transformation	transformation	NOUN
ijassa-2042	184	10	and	and	CCONJ
ijassa-2042	184	11	its	its	PRON
ijassa-2042	184	12	applications	application	NOUN
ijassa-2042	184	13	51	51	NUM
ijassa-2042	184	14	example	example	NOUN
ijassa-2042	184	15	2.5	2.5	NUM
ijassa-2042	184	16	:	:	PUNCT
ijassa-2042	184	17	consider	consider	VERB
ijassa-2042	184	18	the	the	DET
ijassa-2042	184	19	equation	equation	NOUN
ijassa-2042	184	20	p3	p3	NOUN
ijassa-2042	184	21	=	=	SYM
ijassa-2042	184	22	(	(	PUNCT
ijassa-2042	184	23	xp−	xp−	PROPN
ijassa-2042	184	24	y)2	y)2	NOUN
ijassa-2042	184	25	,	,	PUNCT
ijassa-2042	184	26	p	p	NOUN
ijassa-2042	184	27	=	=	PUNCT
ijassa-2042	184	28	dy	dy	X
ijassa-2042	184	29	/	/	SYM
ijassa-2042	184	30	dx	dx	PROPN
ijassa-2042	184	31	,	,	PUNCT
ijassa-2042	184	32	(	(	PUNCT
ijassa-2042	184	33	2.17	2.17	NUM
ijassa-2042	184	34	)	)	PUNCT
ijassa-2042	184	35	which	which	PRON
ijassa-2042	184	36	has	have	VERB
ijassa-2042	184	37	the	the	DET
ijassa-2042	184	38	form	form	NOUN
ijassa-2042	184	39	(	(	PUNCT
ijassa-2042	184	40	2.16	2.16	NUM
ijassa-2042	184	41	)	)	PUNCT
ijassa-2042	184	42	with	with	ADP
ijassa-2042	184	43	f	f	PROPN
ijassa-2042	184	44	(	(	PUNCT
ijassa-2042	184	45	u	u	NOUN
ijassa-2042	184	46	,	,	PUNCT
ijassa-2042	184	47	v	v	NOUN
ijassa-2042	184	48	)	)	PUNCT
ijassa-2042	184	49	=	=	NOUN
ijassa-2042	184	50	u3	u3	NOUN
ijassa-2042	184	51	−	−	PROPN
ijassa-2042	184	52	v2	v2	PROPN
ijassa-2042	184	53	.	.	PUNCT
ijassa-2042	185	1	the	the	DET
ijassa-2042	185	2	curve	curve	NOUN
ijassa-2042	185	3	dual	dual	ADJ
ijassa-2042	185	4	to	to	ADP
ijassa-2042	185	5	the	the	DET
ijassa-2042	185	6	curve	curve	NOUN
ijassa-2042	185	7	u3	u3	NOUN
ijassa-2042	185	8	−	−	PROPN
ijassa-2042	185	9	v2	v2	PROPN
ijassa-2042	185	10	=	=	SYM
ijassa-2042	185	11	0	0	NUM
ijassa-2042	185	12	is	be	AUX
ijassa-2042	185	13	the	the	DET
ijassa-2042	185	14	cubic	cubic	ADJ
ijassa-2042	185	15	parabola	parabola	PROPN
ijassa-2042	185	16	y	y	PROPN
ijassa-2042	185	17	=	=	PROPN
ijassa-2042	185	18	4	4	NUM
ijassa-2042	185	19	27	27	NUM
ijassa-2042	185	20	x3	x3	ADJ
ijassa-2042	185	21	,	,	PUNCT
ijassa-2042	185	22	which	which	PRON
ijassa-2042	185	23	is	be	AUX
ijassa-2042	185	24	the	the	DET
ijassa-2042	185	25	discriminant	discriminant	ADJ
ijassa-2042	185	26	curve	curve	NOUN
ijassa-2042	185	27	of	of	ADP
ijassa-2042	185	28	equation	equation	NOUN
ijassa-2042	185	29	(	(	PUNCT
ijassa-2042	185	30	2.17	2.17	NUM
ijassa-2042	185	31	)	)	PUNCT
ijassa-2042	185	32	and	and	CCONJ
ijassa-2042	185	33	its	its	PRON
ijassa-2042	185	34	singular	singular	ADJ
ijassa-2042	185	35	solutions	solution	NOUN
ijassa-2042	185	36	.	.	PUNCT
ijassa-2042	186	1	the	the	DET
ijassa-2042	186	2	family	family	NOUN
ijassa-2042	186	3	of	of	ADP
ijassa-2042	186	4	non	non	ADJ
ijassa-2042	186	5	-	-	ADJ
ijassa-2042	186	6	singular	singular	ADJ
ijassa-2042	186	7	solutions	solution	NOUN
ijassa-2042	186	8	of	of	ADP
ijassa-2042	186	9	(	(	PUNCT
ijassa-2042	186	10	2.17	2.17	NUM
ijassa-2042	186	11	)	)	PUNCT
ijassa-2042	186	12	consists	consist	VERB
ijassa-2042	186	13	of	of	ADP
ijassa-2042	186	14	all	all	DET
ijassa-2042	186	15	tangent	tangent	ADJ
ijassa-2042	186	16	lines	line	NOUN
ijassa-2042	186	17	to	to	ADP
ijassa-2042	186	18	the	the	DET
ijassa-2042	186	19	singular	singular	ADJ
ijassa-2042	186	20	solution	solution	NOUN
ijassa-2042	186	21	,	,	PUNCT
ijassa-2042	186	22	which	which	PRON
ijassa-2042	186	23	are	be	AUX
ijassa-2042	186	24	given	give	VERB
ijassa-2042	186	25	by	by	ADP
ijassa-2042	186	26	the	the	DET
ijassa-2042	186	27	formula	formula	NOUN
ijassa-2042	186	28	y	y	NOUN
ijassa-2042	186	29	=	=	PUNCT
ijassa-2042	186	30	ax+	ax+	PROPN
ijassa-2042	186	31	b	b	NOUN
ijassa-2042	186	32	,	,	PUNCT
ijassa-2042	186	33	a3	a3	NOUN
ijassa-2042	186	34	=	=	SYM
ijassa-2042	186	35	b2	b2	NOUN
ijassa-2042	186	36	.	.	PUNCT
ijassa-2042	187	1	the	the	DET
ijassa-2042	187	2	latter	latter	ADJ
ijassa-2042	187	3	equation	equation	NOUN
ijassa-2042	187	4	can	can	AUX
ijassa-2042	187	5	be	be	AUX
ijassa-2042	187	6	written	write	VERB
ijassa-2042	187	7	in	in	ADP
ijassa-2042	187	8	the	the	DET
ijassa-2042	187	9	form	form	NOUN
ijassa-2042	187	10	a3	a3	NOUN
ijassa-2042	187	11	=	=	SYM
ijassa-2042	187	12	(	(	PUNCT
ijassa-2042	187	13	ax−	ax−	PUNCT
ijassa-2042	187	14	y)2	y)2	NOUN
ijassa-2042	187	15	,	,	PUNCT
ijassa-2042	187	16	a	a	DET
ijassa-2042	187	17	=	=	SYM
ijassa-2042	187	18	const	const	NOUN
ijassa-2042	187	19	,	,	PUNCT
ijassa-2042	187	20	(	(	PUNCT
ijassa-2042	187	21	2.18	2.18	NUM
ijassa-2042	187	22	)	)	PUNCT
ijassa-2042	187	23	or	or	CCONJ
ijassa-2042	187	24	in	in	ADP
ijassa-2042	187	25	the	the	DET
ijassa-2042	187	26	form	form	NOUN
ijassa-2042	187	27	y	y	PROPN
ijassa-2042	187	28	=	=	SYM
ijassa-2042	187	29	c2x+	c2x+	PROPN
ijassa-2042	187	30	c3	c3	PROPN
ijassa-2042	187	31	,	,	PUNCT
ijassa-2042	187	32	where	where	SCONJ
ijassa-2042	187	33	a	a	DET
ijassa-2042	187	34	=	=	PROPN
ijassa-2042	187	35	c2	c2	PROPN
ijassa-2042	187	36	.	.	PUNCT
ijassa-2042	187	37	formula	formula	NOUN
ijassa-2042	187	38	(	(	PUNCT
ijassa-2042	187	39	2.18	2.18	NUM
ijassa-2042	187	40	)	)	PUNCT
ijassa-2042	187	41	can	can	AUX
ijassa-2042	187	42	be	be	AUX
ijassa-2042	187	43	obtained	obtain	VERB
ijassa-2042	187	44	from	from	ADP
ijassa-2042	187	45	equation	equation	NOUN
ijassa-2042	187	46	(	(	PUNCT
ijassa-2042	187	47	2.17	2.17	NUM
ijassa-2042	187	48	)	)	PUNCT
ijassa-2042	187	49	by	by	ADP
ijassa-2042	187	50	replacing	replace	VERB
ijassa-2042	187	51	the	the	DET
ijassa-2042	187	52	derivative	derivative	ADJ
ijassa-2042	187	53	p	p	NOUN
ijassa-2042	187	54	with	with	ADP
ijassa-2042	187	55	the	the	DET
ijassa-2042	187	56	constant	constant	ADJ
ijassa-2042	187	57	c.	c.	NOUN
ijassa-2042	187	58	3	3	NUM
ijassa-2042	187	59	.	.	PUNCT
ijassa-2042	188	1	contact	contact	NOUN
ijassa-2042	188	2	transformations	transformation	VERB
ijassa-2042	188	3	the	the	DET
ijassa-2042	188	4	legendre	legendre	PROPN
ijassa-2042	188	5	transformation	transformation	NOUN
ijassa-2042	188	6	is	be	AUX
ijassa-2042	188	7	not	not	PART
ijassa-2042	188	8	a	a	DET
ijassa-2042	188	9	unique	unique	ADJ
ijassa-2042	188	10	transformation	transformation	NOUN
ijassa-2042	188	11	j1	j1	PROPN
ijassa-2042	188	12	→	→	SYM
ijassa-2042	188	13	j1	j1	PROPN
ijassa-2042	188	14	preserving	preserve	VERB
ijassa-2042	188	15	the	the	DET
ijassa-2042	188	16	contact	contact	NOUN
ijassa-2042	188	17	structure	structure	NOUN
ijassa-2042	188	18	of	of	ADP
ijassa-2042	188	19	this	this	DET
ijassa-2042	188	20	space	space	NOUN
ijassa-2042	188	21	.	.	PUNCT
ijassa-2042	189	1	there	there	PRON
ijassa-2042	189	2	exist	exist	VERB
ijassa-2042	189	3	a	a	DET
ijassa-2042	189	4	large	large	ADJ
ijassa-2042	189	5	group	group	NOUN
ijassa-2042	189	6	of	of	ADP
ijassa-2042	189	7	diffeomorphisms	diffeomorphism	NOUN
ijassa-2042	189	8	of	of	ADP
ijassa-2042	189	9	the	the	DET
ijassa-2042	189	10	space	space	NOUN
ijassa-2042	189	11	that	that	PRON
ijassa-2042	189	12	also	also	ADV
ijassa-2042	189	13	preserve	preserve	VERB
ijassa-2042	189	14	the	the	DET
ijassa-2042	189	15	contact	contact	NOUN
ijassa-2042	189	16	structure	structure	NOUN
ijassa-2042	189	17	,	,	PUNCT
ijassa-2042	189	18	they	they	PRON
ijassa-2042	189	19	are	be	AUX
ijassa-2042	189	20	called	call	VERB
ijassa-2042	189	21	contact	contact	NOUN
ijassa-2042	189	22	transformations	transformation	NOUN
ijassa-2042	189	23	.	.	PUNCT
ijassa-2042	190	1	equivalent	equivalent	ADJ
ijassa-2042	190	2	definition	definition	NOUN
ijassa-2042	190	3	:	:	PUNCT
ijassa-2042	190	4	contact	contact	NOUN
ijassa-2042	190	5	transformations	transformation	NOUN
ijassa-2042	190	6	are	be	AUX
ijassa-2042	190	7	transformations	transformation	NOUN
ijassa-2042	190	8	of	of	ADP
ijassa-2042	190	9	curves	curve	NOUN
ijassa-2042	190	10	in	in	ADP
ijassa-2042	190	11	the	the	DET
ijassa-2042	190	12	plane	plane	NOUN
ijassa-2042	190	13	in	in	ADP
ijassa-2042	190	14	which	which	PRON
ijassa-2042	190	15	tangent	tangent	NOUN
ijassa-2042	190	16	curves	curve	NOUN
ijassa-2042	190	17	are	be	AUX
ijassa-2042	190	18	transformed	transform	VERB
ijassa-2042	190	19	into	into	ADP
ijassa-2042	190	20	tangent	tangent	ADJ
ijassa-2042	190	21	curves	curve	NOUN
ijassa-2042	190	22	.	.	PUNCT
ijassa-2042	191	1	let	let	VERB
ijassa-2042	191	2	us	we	PRON
ijassa-2042	191	3	try	try	VERB
ijassa-2042	191	4	to	to	PART
ijassa-2042	191	5	determine	determine	VERB
ijassa-2042	191	6	condition	condition	NOUN
ijassa-2042	191	7	under	under	ADP
ijassa-2042	191	8	which	which	PRON
ijassa-2042	191	9	a	a	DET
ijassa-2042	191	10	diffeomorphism	diffeomorphism	NOUN
ijassa-2042	191	11	(	(	PUNCT
ijassa-2042	191	12	x	x	X
ijassa-2042	191	13	,	,	PUNCT
ijassa-2042	191	14	y	y	PROPN
ijassa-2042	191	15	,	,	PUNCT
ijassa-2042	191	16	p	p	NOUN
ijassa-2042	191	17	)	)	PUNCT
ijassa-2042	191	18	→	→	SYM
ijassa-2042	191	19	(	(	PUNCT
ijassa-2042	191	20	x	x	X
ijassa-2042	191	21	,	,	PUNCT
ijassa-2042	191	22	y	y	PROPN
ijassa-2042	191	23	,	,	PUNCT
ijassa-2042	191	24	p	p	NOUN
ijassa-2042	191	25	)	)	PUNCT
ijassa-2042	191	26	(	(	PUNCT
ijassa-2042	191	27	global	global	ADJ
ijassa-2042	191	28	or	or	CCONJ
ijassa-2042	191	29	local	local	ADJ
ijassa-2042	191	30	)	)	PUNCT
ijassa-2042	191	31	preserves	preserve	VERB
ijassa-2042	191	32	the	the	DET
ijassa-2042	191	33	contact	contact	NOUN
ijassa-2042	191	34	structure	structure	NOUN
ijassa-2042	191	35	of	of	ADP
ijassa-2042	191	36	the	the	DET
ijassa-2042	191	37	space	space	NOUN
ijassa-2042	191	38	j1	j1	PROPN
ijassa-2042	191	39	.	.	PUNCT
ijassa-2042	192	1	namely	namely	ADV
ijassa-2042	192	2	,	,	PUNCT
ijassa-2042	192	3	consider	consider	VERB
ijassa-2042	192	4	a	a	DET
ijassa-2042	192	5	diffeomorphism	diffeomorphism	NOUN
ijassa-2042	192	6	(	(	PUNCT
ijassa-2042	192	7	x	x	X
ijassa-2042	192	8	,	,	PUNCT
ijassa-2042	192	9	y	y	PROPN
ijassa-2042	192	10	,	,	PUNCT
ijassa-2042	192	11	p	p	NOUN
ijassa-2042	192	12	)	)	PUNCT
ijassa-2042	192	13	→	→	SYM
ijassa-2042	192	14	(	(	PUNCT
ijassa-2042	192	15	x	x	X
ijassa-2042	192	16	,	,	PUNCT
ijassa-2042	192	17	y	y	PROPN
ijassa-2042	192	18	,	,	PUNCT
ijassa-2042	192	19	p	p	NOUN
ijassa-2042	192	20	)	)	PUNCT
ijassa-2042	192	21	given	give	VERB
ijassa-2042	192	22	by	by	ADP
ijassa-2042	192	23	the	the	DET
ijassa-2042	192	24	formula	formula	NOUN
ijassa-2042	192	25	x	x	PUNCT
ijassa-2042	192	26	=	=	SYM
ijassa-2042	192	27	f	f	X
ijassa-2042	192	28	(	(	PUNCT
ijassa-2042	192	29	x	x	X
ijassa-2042	192	30	,	,	PUNCT
ijassa-2042	192	31	y	y	PROPN
ijassa-2042	192	32	,	,	PUNCT
ijassa-2042	192	33	p	p	NOUN
ijassa-2042	192	34	)	)	PUNCT
ijassa-2042	192	35	,	,	PUNCT
ijassa-2042	192	36	y	y	PROPN
ijassa-2042	192	37	=	=	SYM
ijassa-2042	192	38	g(x	g(x	PROPN
ijassa-2042	192	39	,	,	PUNCT
ijassa-2042	192	40	y	y	PROPN
ijassa-2042	192	41	,	,	PUNCT
ijassa-2042	192	42	p	p	NOUN
ijassa-2042	192	43	)	)	PUNCT
ijassa-2042	192	44	,	,	PUNCT
ijassa-2042	192	45	p	p	NOUN
ijassa-2042	192	46	=	=	SYM
ijassa-2042	192	47	h(x	h(x	PROPN
ijassa-2042	192	48	,	,	PUNCT
ijassa-2042	192	49	y	y	PROPN
ijassa-2042	192	50	,	,	PUNCT
ijassa-2042	192	51	p	p	NOUN
ijassa-2042	192	52	)	)	PUNCT
ijassa-2042	192	53	,	,	PUNCT
ijassa-2042	192	54	(	(	PUNCT
ijassa-2042	192	55	3.19	3.19	NUM
ijassa-2042	192	56	)	)	PUNCT
ijassa-2042	192	57	where	where	SCONJ
ijassa-2042	192	58	the	the	DET
ijassa-2042	192	59	functions	function	NOUN
ijassa-2042	192	60	f	f	X
ijassa-2042	192	61	,	,	PUNCT
ijassa-2042	192	62	g	g	PROPN
ijassa-2042	192	63	satisfy	satisfy	VERB
ijassa-2042	192	64	some	some	DET
ijassa-2042	192	65	relation	relation	NOUN
ijassa-2042	192	66	(	(	PUNCT
ijassa-2042	192	67	that	that	SCONJ
ijassa-2042	192	68	we	we	PRON
ijassa-2042	192	69	shall	shall	AUX
ijassa-2042	192	70	find	find	VERB
ijassa-2042	192	71	)	)	PUNCT
ijassa-2042	192	72	and	and	CCONJ
ijassa-2042	192	73	the	the	DET
ijassa-2042	192	74	function	function	NOUN
ijassa-2042	192	75	h	h	NOUN
ijassa-2042	192	76	is	be	AUX
ijassa-2042	192	77	uniquely	uniquely	ADV
ijassa-2042	192	78	defined	define	VERB
ijassa-2042	192	79	by	by	ADP
ijassa-2042	192	80	f	f	PROPN
ijassa-2042	192	81	,	,	PUNCT
ijassa-2042	192	82	g	g	PROPN
ijassa-2042	192	83	due	due	ADP
ijassa-2042	192	84	to	to	ADP
ijassa-2042	192	85	the	the	DET
ijassa-2042	192	86	condition	condition	NOUN
ijassa-2042	192	87	p	p	NOUN
ijassa-2042	192	88	=	=	PUNCT
ijassa-2042	192	89	dy	dy	X
ijassa-2042	192	90	/	/	SYM
ijassa-2042	192	91	dx	dx	PROPN
ijassa-2042	192	92	.	.	PUNCT
ijassa-2042	193	1	therefore	therefore	ADV
ijassa-2042	193	2	,	,	PUNCT
ijassa-2042	193	3	the	the	DET
ijassa-2042	193	4	condition	condition	NOUN
ijassa-2042	193	5	of	of	ADP
ijassa-2042	193	6	preservation	preservation	NOUN
ijassa-2042	193	7	the	the	DET
ijassa-2042	193	8	contact	contact	NOUN
ijassa-2042	193	9	structure	structure	NOUN
ijassa-2042	193	10	has	have	VERB
ijassa-2042	193	11	the	the	DET
ijassa-2042	193	12	form	form	NOUN
ijassa-2042	193	13	h(x	h(x	PROPN
ijassa-2042	193	14	,	,	PUNCT
ijassa-2042	193	15	y	y	PROPN
ijassa-2042	193	16	,	,	PUNCT
ijassa-2042	193	17	p	p	NOUN
ijassa-2042	193	18	)	)	PUNCT
ijassa-2042	194	1	=	=	PUNCT
ijassa-2042	194	2	dy	dy	NOUN
ijassa-2042	194	3	dx	dx	PROPN
ijassa-2042	194	4	=	=	PROPN
ijassa-2042	194	5	gxdx+gydy	gxdx+gydy	PROPN
ijassa-2042	194	6	+	+	NOUN
ijassa-2042	194	7	gpdp	gpdp	NOUN
ijassa-2042	194	8	fxdx+	fxdx+	X
ijassa-2042	194	9	fydy	fydy	NOUN
ijassa-2042	194	10	+	+	CCONJ
ijassa-2042	194	11	fpdp	fpdp	NOUN
ijassa-2042	194	12	=	=	SYM
ijassa-2042	194	13	gx	gx	PROPN
ijassa-2042	195	1	+	+	NUM
ijassa-2042	195	2	pgy	pgy	NOUN
ijassa-2042	195	3	+	+	CCONJ
ijassa-2042	195	4	p′gp	p′gp	VERB
ijassa-2042	195	5	fx	fx	ADJ
ijassa-2042	195	6	+	+	CCONJ
ijassa-2042	195	7	pfy	pfy	PROPN
ijassa-2042	195	8	+	+	CCONJ
ijassa-2042	195	9	p′fp	p′fp	NOUN
ijassa-2042	195	10	,	,	PUNCT
ijassa-2042	195	11	(	(	PUNCT
ijassa-2042	195	12	3.20	3.20	NUM
ijassa-2042	195	13	)	)	PUNCT
ijassa-2042	196	1	where	where	SCONJ
ijassa-2042	196	2	p′	p′	NOUN
ijassa-2042	196	3	=	=	SYM
ijassa-2042	196	4	dp	dp	PROPN
ijassa-2042	196	5	/	/	SYM
ijassa-2042	196	6	dx	dx	PROPN
ijassa-2042	196	7	.	.	PUNCT
ijassa-2042	197	1	since	since	SCONJ
ijassa-2042	197	2	p	p	PROPN
ijassa-2042	197	3	=	=	PROPN
ijassa-2042	197	4	h(x	h(x	PROPN
ijassa-2042	197	5	,	,	PUNCT
ijassa-2042	197	6	y	y	PROPN
ijassa-2042	197	7	,	,	PUNCT
ijassa-2042	197	8	p	p	NOUN
ijassa-2042	197	9	)	)	PUNCT
ijassa-2042	197	10	does	do	AUX
ijassa-2042	197	11	not	not	PART
ijassa-2042	197	12	depend	depend	VERB
ijassa-2042	197	13	on	on	ADP
ijassa-2042	197	14	p′	p′	PROPN
ijassa-2042	197	15	,	,	PUNCT
ijassa-2042	197	16	the	the	DET
ijassa-2042	197	17	right	right	ADJ
ijassa-2042	197	18	hand	hand	NOUN
ijassa-2042	197	19	side	side	NOUN
ijassa-2042	197	20	of	of	ADP
ijassa-2042	197	21	equality	equality	NOUN
ijassa-2042	197	22	(	(	PUNCT
ijassa-2042	197	23	3.20	3.20	NUM
ijassa-2042	197	24	)	)	PUNCT
ijassa-2042	197	25	does	do	AUX
ijassa-2042	197	26	not	not	PART
ijassa-2042	197	27	depend	depend	VERB
ijassa-2042	197	28	on	on	ADP
ijassa-2042	197	29	p′	p′	NOUN
ijassa-2042	197	30	as	as	ADV
ijassa-2042	197	31	well	well	ADV
ijassa-2042	197	32	.	.	PUNCT
ijassa-2042	198	1	this	this	DET
ijassa-2042	198	2	yields	yield	NOUN
ijassa-2042	198	3	gx	gx	PROPN
ijassa-2042	198	4	+	+	NUM
ijassa-2042	198	5	pgy	pgy	NOUN
ijassa-2042	198	6	fx	fx	PROPN
ijassa-2042	198	7	+	+	CCONJ
ijassa-2042	198	8	pfy	pfy	PROPN
ijassa-2042	198	9	≡	≡	PROPN
ijassa-2042	198	10	gp	gp	NOUN
ijassa-2042	198	11	fp	fp	X
ijassa-2042	198	12	or	or	CCONJ
ijassa-2042	198	13	,	,	PUNCT
ijassa-2042	198	14	equivalently	equivalently	ADV
ijassa-2042	198	15	,	,	PUNCT
ijassa-2042	198	16	fp(gx	fp(gx	VERB
ijassa-2042	198	17	+	+	CCONJ
ijassa-2042	198	18	pgy	pgy	NOUN
ijassa-2042	198	19	)	)	PUNCT
ijassa-2042	198	20	≡	≡	PROPN
ijassa-2042	198	21	gp(fx	gp(fx	PROPN
ijassa-2042	198	22	+	+	CCONJ
ijassa-2042	198	23	pfy	pfy	PROPN
ijassa-2042	198	24	)	)	PUNCT
ijassa-2042	198	25	.	.	PUNCT
ijassa-2042	199	1	(	(	PUNCT
ijassa-2042	199	2	3.21	3.21	NUM
ijassa-2042	199	3	)	)	PUNCT
ijassa-2042	199	4	the	the	DET
ijassa-2042	199	5	identity	identity	NOUN
ijassa-2042	199	6	(	(	PUNCT
ijassa-2042	199	7	3.21	3.21	NUM
ijassa-2042	199	8	)	)	PUNCT
ijassa-2042	199	9	connecting	connect	VERB
ijassa-2042	199	10	the	the	DET
ijassa-2042	199	11	functions	function	NOUN
ijassa-2042	199	12	f	f	PROPN
ijassa-2042	199	13	and	and	CCONJ
ijassa-2042	199	14	g	g	PROPN
ijassa-2042	199	15	is	be	AUX
ijassa-2042	199	16	a	a	DET
ijassa-2042	199	17	necessary	necessary	ADJ
ijassa-2042	199	18	and	and	CCONJ
ijassa-2042	199	19	sufficient	sufficient	ADJ
ijassa-2042	199	20	condition	condition	NOUN
ijassa-2042	199	21	for	for	ADP
ijassa-2042	199	22	the	the	DET
ijassa-2042	199	23	diffeomorphism	diffeomorphism	NOUN
ijassa-2042	199	24	(	(	PUNCT
ijassa-2042	199	25	3.19	3.19	NUM
ijassa-2042	199	26	)	)	PUNCT
ijassa-2042	199	27	be	be	AUX
ijassa-2042	199	28	a	a	DET
ijassa-2042	199	29	contact	contact	NOUN
ijassa-2042	199	30	transformation	transformation	NOUN
ijassa-2042	199	31	.	.	PUNCT
ijassa-2042	200	1	it	it	PRON
ijassa-2042	200	2	is	be	AUX
ijassa-2042	200	3	not	not	PART
ijassa-2042	200	4	hard	hard	ADJ
ijassa-2042	200	5	to	to	PART
ijassa-2042	200	6	see	see	VERB
ijassa-2042	200	7	that	that	SCONJ
ijassa-2042	200	8	the	the	DET
ijassa-2042	200	9	legendre	legendre	PROPN
ijassa-2042	200	10	transformation	transformation	NOUN
ijassa-2042	200	11	satisfies	satisfie	NOUN
ijassa-2042	200	12	(	(	PUNCT
ijassa-2042	200	13	3.21	3.21	NUM
ijassa-2042	200	14	)	)	PUNCT
ijassa-2042	200	15	.	.	PUNCT
ijassa-2042	201	1	the	the	DET
ijassa-2042	201	2	legendre	legendre	PROPN
ijassa-2042	201	3	transform	transform	NOUN
ijassa-2042	201	4	is	be	AUX
ijassa-2042	201	5	the	the	DET
ijassa-2042	201	6	most	most	ADV
ijassa-2042	201	7	commonly	commonly	ADV
ijassa-2042	201	8	used	use	VERB
ijassa-2042	201	9	and	and	CCONJ
ijassa-2042	201	10	the	the	DET
ijassa-2042	201	11	most	most	ADV
ijassa-2042	201	12	useful	useful	ADJ
ijassa-2042	201	13	contact	contact	NOUN
ijassa-2042	201	14	transformation	transformation	NOUN
ijassa-2042	201	15	,	,	PUNCT
ijassa-2042	201	16	but	but	CCONJ
ijassa-2042	201	17	historically	historically	ADV
ijassa-2042	201	18	it	it	PRON
ijassa-2042	201	19	was	be	AUX
ijassa-2042	201	20	not	not	PART
ijassa-2042	201	21	the	the	DET
ijassa-2042	201	22	first	first	ADJ
ijassa-2042	201	23	one	one	NUM
ijassa-2042	201	24	.	.	PUNCT
ijassa-2042	202	1	apparently	apparently	ADV
ijassa-2042	202	2	,	,	PUNCT
ijassa-2042	202	3	the	the	DET
ijassa-2042	202	4	first	first	ADJ
ijassa-2042	202	5	contact	contact	NOUN
ijassa-2042	202	6	transformation	transformation	NOUN
ijassa-2042	202	7	was	be	AUX
ijassa-2042	202	8	the	the	DET
ijassa-2042	202	9	so	so	ADV
ijassa-2042	202	10	-	-	PUNCT
ijassa-2042	202	11	called	call	VERB
ijassa-2042	202	12	pedal	pedal	ADJ
ijassa-2042	202	13	transformation	transformation	NOUN
ijassa-2042	202	14	,	,	PUNCT
ijassa-2042	202	15	which	which	PRON
ijassa-2042	202	16	is	be	AUX
ijassa-2042	202	17	defined	define	VERB
ijassa-2042	202	18	as	as	SCONJ
ijassa-2042	202	19	follows	follow	VERB
ijassa-2042	202	20	.	.	PUNCT
ijassa-2042	203	1	we	we	PRON
ijassa-2042	203	2	shall	shall	AUX
ijassa-2042	203	3	copyright	copyright	NOUN
ijassa-2042	203	4	©	©	PROPN
ijassa-2042	203	5	2025	2025	NUM
ijassa-2042	203	6	assa	assa	NOUN
ijassa-2042	203	7	.	.	PUNCT
ijassa-2042	204	1	adv	adv	PROPN
ijassa-2042	204	2	syst	syst	PROPN
ijassa-2042	204	3	sci	sci	PROPN
ijassa-2042	204	4	appl	appl	PROPN
ijassa-2042	204	5	(	(	PUNCT
ijassa-2042	204	6	2025	2025	NUM
ijassa-2042	204	7	)	)	PUNCT
ijassa-2042	204	8	52	52	NUM
ijassa-2042	204	9	n.	n.	NOUN
ijassa-2042	204	10	pavlova	pavlova	PROPN
ijassa-2042	204	11	,	,	PUNCT
ijassa-2042	204	12	a.	a.	NOUN
ijassa-2042	204	13	remizov	remizov	PROPN
ijassa-2042	204	14	give	give	VERB
ijassa-2042	204	15	another	another	DET
ijassa-2042	204	16	example	example	NOUN
ijassa-2042	204	17	of	of	ADP
ijassa-2042	204	18	contact	contact	NOUN
ijassa-2042	204	19	transformations	transformation	NOUN
ijassa-2042	204	20	different	different	ADJ
ijassa-2042	204	21	from	from	ADP
ijassa-2042	204	22	the	the	DET
ijassa-2042	204	23	legendre	legendre	PROPN
ijassa-2042	204	24	transform	transform	NOUN
ijassa-2042	204	25	–	–	PUNCT
ijassa-2042	204	26	socalled	socalle	VERB
ijassa-2042	204	27	pedal	pedal	ADJ
ijassa-2042	204	28	transformations	transformation	NOUN
ijassa-2042	204	29	,	,	PUNCT
ijassa-2042	204	30	which	which	PRON
ijassa-2042	204	31	were	be	AUX
ijassa-2042	204	32	investigated	investigate	VERB
ijassa-2042	204	33	by	by	ADP
ijassa-2042	204	34	colin	colin	PROPN
ijassa-2042	204	35	maclaurin	maclaurin	PROPN
ijassa-2042	204	36	,	,	PUNCT
ijassa-2042	204	37	arthur	arthur	PROPN
ijassa-2042	204	38	cayley	cayley	PROPN
ijassa-2042	204	39	,	,	PUNCT
ijassa-2042	204	40	and	and	CCONJ
ijassa-2042	204	41	sophus	sophus	PROPN
ijassa-2042	204	42	lie	lie	VERB
ijassa-2042	204	43	.	.	PUNCT
ijassa-2042	205	1	to	to	PART
ijassa-2042	205	2	define	define	VERB
ijassa-2042	205	3	pedal	pedal	ADJ
ijassa-2042	205	4	transformation	transformation	NOUN
ijassa-2042	205	5	,	,	PUNCT
ijassa-2042	205	6	let	let	VERB
ijassa-2042	205	7	us	we	PRON
ijassa-2042	205	8	fixe	fixe	VERB
ijassa-2042	205	9	an	an	DET
ijassa-2042	205	10	arbitrary	arbitrary	ADJ
ijassa-2042	205	11	pointo	pointo	NOUN
ijassa-2042	205	12	on	on	ADP
ijassa-2042	205	13	the	the	DET
ijassa-2042	205	14	plane	plane	NOUN
ijassa-2042	205	15	(	(	PUNCT
ijassa-2042	205	16	without	without	ADP
ijassa-2042	205	17	loss	loss	NOUN
ijassa-2042	205	18	of	of	ADP
ijassa-2042	205	19	generality	generality	NOUN
ijassa-2042	205	20	,	,	PUNCT
ijassa-2042	205	21	o	o	PROPN
ijassa-2042	205	22	can	can	AUX
ijassa-2042	205	23	be	be	AUX
ijassa-2042	205	24	the	the	DET
ijassa-2042	205	25	origin	origin	NOUN
ijassa-2042	205	26	)	)	PUNCT
ijassa-2042	205	27	.	.	PUNCT
ijassa-2042	206	1	the	the	DET
ijassa-2042	206	2	pointo	pointo	NOUN
ijassa-2042	206	3	is	be	AUX
ijassa-2042	206	4	called	call	VERB
ijassa-2042	206	5	the	the	DET
ijassa-2042	206	6	pole	pole	NOUN
ijassa-2042	206	7	or	or	CCONJ
ijassa-2042	206	8	center	center	NOUN
ijassa-2042	206	9	of	of	ADP
ijassa-2042	206	10	the	the	DET
ijassa-2042	206	11	transformation	transformation	NOUN
ijassa-2042	206	12	.	.	PUNCT
ijassa-2042	207	1	let	let	VERB
ijassa-2042	207	2	γ	γ	X
ijassa-2042	207	3	be	be	AUX
ijassa-2042	207	4	a	a	DET
ijassa-2042	207	5	curve	curve	NOUN
ijassa-2042	207	6	on	on	ADP
ijassa-2042	207	7	the	the	DET
ijassa-2042	207	8	plane	plane	NOUN
ijassa-2042	207	9	.	.	PUNCT
ijassa-2042	208	1	then	then	ADV
ijassa-2042	208	2	the	the	DET
ijassa-2042	208	3	pedal	pedal	ADJ
ijassa-2042	208	4	curve	curve	NOUN
ijassa-2042	208	5	of	of	ADP
ijassa-2042	208	6	γ	γ	PROPN
ijassa-2042	208	7	is	be	AUX
ijassa-2042	208	8	the	the	DET
ijassa-2042	208	9	locus	locus	NOUN
ijassa-2042	208	10	of	of	ADP
ijassa-2042	208	11	points	point	NOUN
ijassa-2042	208	12	x	x	VERB
ijassa-2042	208	13	so	so	SCONJ
ijassa-2042	208	14	that	that	SCONJ
ijassa-2042	208	15	the	the	DET
ijassa-2042	208	16	line	line	NOUN
ijassa-2042	208	17	ox	ox	NOUN
ijassa-2042	208	18	is	be	AUX
ijassa-2042	208	19	perpendicular	perpendicular	ADJ
ijassa-2042	208	20	to	to	ADP
ijassa-2042	208	21	the	the	DET
ijassa-2042	208	22	tangent	tangent	ADJ
ijassa-2042	208	23	line	line	NOUN
ijassa-2042	208	24	to	to	ADP
ijassa-2042	208	25	γ	γ	PROPN
ijassa-2042	208	26	passing	pass	VERB
ijassa-2042	208	27	through	through	ADP
ijassa-2042	208	28	the	the	DET
ijassa-2042	208	29	point	point	NOUN
ijassa-2042	208	30	x	x	INTJ
ijassa-2042	208	31	.	.	PUNCT
ijassa-2042	208	32	example	example	NOUN
ijassa-2042	208	33	3.1	3.1	NUM
ijassa-2042	208	34	:	:	PUNCT
ijassa-2042	208	35	the	the	DET
ijassa-2042	208	36	pedal	pedal	ADJ
ijassa-2042	208	37	curve	curve	NOUN
ijassa-2042	208	38	of	of	ADP
ijassa-2042	208	39	the	the	DET
ijassa-2042	208	40	circle	circle	NOUN
ijassa-2042	208	41	is	be	AUX
ijassa-2042	208	42	the	the	DET
ijassa-2042	208	43	limacon	limacon	NOUN
ijassa-2042	208	44	(	(	PUNCT
ijassa-2042	208	45	pascal	pascal	PROPN
ijassa-2042	208	46	’s	’s	PART
ijassa-2042	208	47	snail	snail	NOUN
ijassa-2042	208	48	)	)	PUNCT
ijassa-2042	208	49	,	,	PUNCT
ijassa-2042	208	50	the	the	DET
ijassa-2042	208	51	special	special	ADJ
ijassa-2042	208	52	shape	shape	NOUN
ijassa-2042	208	53	of	of	ADP
ijassa-2042	208	54	which	which	PRON
ijassa-2042	208	55	is	be	AUX
ijassa-2042	208	56	determined	determine	VERB
ijassa-2042	208	57	by	by	ADP
ijassa-2042	208	58	the	the	DET
ijassa-2042	208	59	distance	distance	NOUN
ijassa-2042	208	60	ρ	ρ	NOUN
ijassa-2042	208	61	between	between	ADP
ijassa-2042	208	62	the	the	DET
ijassa-2042	208	63	pole	pole	NOUN
ijassa-2042	208	64	o	o	NOUN
ijassa-2042	208	65	and	and	CCONJ
ijassa-2042	208	66	the	the	DET
ijassa-2042	208	67	circle	circle	NOUN
ijassa-2042	208	68	(	(	PUNCT
ijassa-2042	208	69	fig	fig	NOUN
ijassa-2042	208	70	.	.	PUNCT
ijassa-2042	208	71	3.6	3.6	NUM
ijassa-2042	208	72	)	)	PUNCT
ijassa-2042	208	73	.	.	PUNCT
ijassa-2042	209	1	considering	consider	VERB
ijassa-2042	209	2	ρ	ρ	PROPN
ijassa-2042	209	3	as	as	ADP
ijassa-2042	209	4	a	a	DET
ijassa-2042	209	5	parameter	parameter	NOUN
ijassa-2042	209	6	,	,	PUNCT
ijassa-2042	209	7	we	we	PRON
ijassa-2042	209	8	get	get	VERB
ijassa-2042	209	9	a	a	DET
ijassa-2042	209	10	family	family	NOUN
ijassa-2042	209	11	of	of	ADP
ijassa-2042	209	12	the	the	DET
ijassa-2042	209	13	pedal	pedal	ADJ
ijassa-2042	209	14	curves	curve	NOUN
ijassa-2042	209	15	,	,	PUNCT
ijassa-2042	209	16	some	some	PRON
ijassa-2042	209	17	of	of	ADP
ijassa-2042	209	18	which	which	PRON
ijassa-2042	209	19	have	have	VERB
ijassa-2042	209	20	singular	singular	ADJ
ijassa-2042	209	21	points	point	NOUN
ijassa-2042	209	22	.	.	PUNCT
ijassa-2042	210	1	namely	namely	ADV
ijassa-2042	210	2	,	,	PUNCT
ijassa-2042	210	3	when	when	SCONJ
ijassa-2042	210	4	the	the	DET
ijassa-2042	210	5	pole	pole	NOUN
ijassa-2042	210	6	o	o	NOUN
ijassa-2042	210	7	lies	lie	VERB
ijassa-2042	210	8	on	on	ADP
ijassa-2042	210	9	the	the	DET
ijassa-2042	210	10	circle	circle	NOUN
ijassa-2042	210	11	itself	itself	PRON
ijassa-2042	210	12	(	(	PUNCT
ijassa-2042	210	13	ρ	ρ	PROPN
ijassa-2042	210	14	=	=	SYM
ijassa-2042	210	15	0	0	NUM
ijassa-2042	210	16	)	)	PUNCT
ijassa-2042	210	17	,	,	PUNCT
ijassa-2042	210	18	the	the	DET
ijassa-2042	210	19	pedal	pedal	ADJ
ijassa-2042	210	20	curve	curve	NOUN
ijassa-2042	210	21	has	have	VERB
ijassa-2042	210	22	the	the	DET
ijassa-2042	210	23	casp	casp	NOUN
ijassa-2042	210	24	at	at	ADP
ijassa-2042	210	25	o	o	NOUN
ijassa-2042	210	26	,	,	PUNCT
ijassa-2042	210	27	this	this	DET
ijassa-2042	210	28	curve	curve	NOUN
ijassa-2042	210	29	is	be	AUX
ijassa-2042	210	30	called	call	VERB
ijassa-2042	210	31	cardioid	cardioid	NOUN
ijassa-2042	210	32	(	(	PUNCT
ijassa-2042	210	33	the	the	DET
ijassa-2042	210	34	third	third	ADJ
ijassa-2042	210	35	figure	figure	NOUN
ijassa-2042	210	36	from	from	ADP
ijassa-2042	210	37	the	the	DET
ijassa-2042	210	38	left	left	NOUN
ijassa-2042	210	39	)	)	PUNCT
ijassa-2042	210	40	.	.	PUNCT
ijassa-2042	211	1	fig	fig	NOUN
ijassa-2042	211	2	.	.	PUNCT
ijassa-2042	212	1	3.6	3.6	NUM
ijassa-2042	212	2	.	.	PUNCT
ijassa-2042	213	1	the	the	DET
ijassa-2042	213	2	pedal	pedal	ADJ
ijassa-2042	213	3	curves	curve	NOUN
ijassa-2042	213	4	of	of	ADP
ijassa-2042	213	5	the	the	DET
ijassa-2042	213	6	circle	circle	NOUN
ijassa-2042	213	7	(	(	PUNCT
ijassa-2042	213	8	depicted	depict	VERB
ijassa-2042	213	9	as	as	ADP
ijassa-2042	213	10	the	the	DET
ijassa-2042	213	11	dotted	dotted	ADJ
ijassa-2042	213	12	line	line	NOUN
ijassa-2042	213	13	)	)	PUNCT
ijassa-2042	213	14	are	be	AUX
ijassa-2042	213	15	limacons	limacon	NOUN
ijassa-2042	213	16	,	,	PUNCT
ijassa-2042	213	17	whose	whose	DET
ijassa-2042	213	18	shape	shape	NOUN
ijassa-2042	213	19	depends	depend	VERB
ijassa-2042	213	20	on	on	ADP
ijassa-2042	213	21	the	the	DET
ijassa-2042	213	22	distance	distance	NOUN
ijassa-2042	213	23	between	between	ADP
ijassa-2042	213	24	the	the	DET
ijassa-2042	213	25	pole	pole	NOUN
ijassa-2042	213	26	o	o	NOUN
ijassa-2042	213	27	and	and	CCONJ
ijassa-2042	213	28	the	the	DET
ijassa-2042	213	29	circle	circle	NOUN
ijassa-2042	213	30	lemma	lemma	PROPN
ijassa-2042	213	31	3.1	3.1	NUM
ijassa-2042	213	32	:	:	PUNCT
ijassa-2042	213	33	in	in	ADP
ijassa-2042	213	34	the	the	DET
ijassa-2042	213	35	cartesian	cartesian	ADJ
ijassa-2042	213	36	coordinate	coordinate	NOUN
ijassa-2042	213	37	system	system	NOUN
ijassa-2042	213	38	centered	center	VERB
ijassa-2042	213	39	at	at	ADP
ijassa-2042	213	40	o	o	PROPN
ijassa-2042	213	41	,	,	PUNCT
ijassa-2042	213	42	the	the	DET
ijassa-2042	213	43	pedal	pedal	ADJ
ijassa-2042	213	44	transformation	transformation	NOUN
ijassa-2042	213	45	is	be	AUX
ijassa-2042	213	46	given	give	VERB
ijassa-2042	213	47	by	by	ADP
ijassa-2042	213	48	the	the	DET
ijassa-2042	213	49	formula	formula	NOUN
ijassa-2042	213	50	p	p	X
ijassa-2042	213	51	:	:	PUNCT
ijassa-2042	213	52	(	(	PUNCT
ijassa-2042	213	53	x	x	X
ijassa-2042	213	54	,	,	PUNCT
ijassa-2042	213	55	y	y	PROPN
ijassa-2042	213	56	,	,	PUNCT
ijassa-2042	213	57	p	p	X
ijassa-2042	213	58	)	)	PUNCT
ijassa-2042	213	59	7→	7→	NUM
ijassa-2042	213	60	(	(	PUNCT
ijassa-2042	213	61	x̄	x̄	PROPN
ijassa-2042	213	62	,	,	PUNCT
ijassa-2042	213	63	ȳ	ȳ	PROPN
ijassa-2042	213	64	,	,	PUNCT
ijassa-2042	213	65	p̄	p̄	PROPN
ijassa-2042	213	66	)	)	PUNCT
ijassa-2042	213	67	,	,	PUNCT
ijassa-2042	213	68	where	where	SCONJ
ijassa-2042	213	69	x̄	x̄	NOUN
ijassa-2042	213	70	=	=	PUNCT
ijassa-2042	213	71	xy/(1	xy/(1	PUNCT
ijassa-2042	214	1	+	+	ADJ
ijassa-2042	214	2	x2	x2	NOUN
ijassa-2042	214	3	)	)	PUNCT
ijassa-2042	214	4	,	,	PUNCT
ijassa-2042	214	5	ȳ	ȳ	NOUN
ijassa-2042	214	6	=	=	PUNCT
ijassa-2042	215	1	−y/(1	−y/(1	PROPN
ijassa-2042	216	1	+	+	ADJ
ijassa-2042	216	2	x2	x2	NOUN
ijassa-2042	216	3	)	)	PUNCT
ijassa-2042	216	4	,	,	PUNCT
ijassa-2042	216	5	p̄	p̄	NOUN
ijassa-2042	216	6	=	=	PUNCT
ijassa-2042	216	7	dȳ	dȳ	PROPN
ijassa-2042	216	8	/dx̄	/dx̄	NUM
ijassa-2042	216	9	,	,	PUNCT
ijassa-2042	216	10	(	(	PUNCT
ijassa-2042	216	11	3.22	3.22	NUM
ijassa-2042	216	12	)	)	PUNCT
ijassa-2042	216	13	the	the	DET
ijassa-2042	216	14	relation	relation	NOUN
ijassa-2042	216	15	between	between	ADP
ijassa-2042	216	16	x	x	PROPN
ijassa-2042	216	17	,	,	PUNCT
ijassa-2042	216	18	y	y	PROPN
ijassa-2042	216	19	,	,	PUNCT
ijassa-2042	216	20	p	p	NOUN
ijassa-2042	216	21	and	and	CCONJ
ijassa-2042	216	22	x	x	PROPN
ijassa-2042	216	23	,	,	PUNCT
ijassa-2042	216	24	y	y	PROPN
ijassa-2042	216	25	,	,	PUNCT
ijassa-2042	216	26	p	p	PROPN
ijassa-2042	216	27	is	be	AUX
ijassa-2042	216	28	given	give	VERB
ijassa-2042	216	29	in	in	ADP
ijassa-2042	216	30	(	(	PUNCT
ijassa-2042	216	31	2.5	2.5	NUM
ijassa-2042	216	32	)	)	PUNCT
ijassa-2042	216	33	.	.	PUNCT
ijassa-2042	217	1	the	the	DET
ijassa-2042	217	2	proof	proof	NOUN
ijassa-2042	217	3	is	be	AUX
ijassa-2042	217	4	by	by	ADP
ijassa-2042	217	5	direct	direct	ADJ
ijassa-2042	217	6	calculation	calculation	NOUN
ijassa-2042	217	7	.	.	PUNCT
ijassa-2042	218	1	in	in	ADP
ijassa-2042	218	2	contrast	contrast	NOUN
ijassa-2042	218	3	to	to	ADP
ijassa-2042	218	4	the	the	DET
ijassa-2042	218	5	legendre	legendre	PROPN
ijassa-2042	218	6	transformation	transformation	PROPN
ijassa-2042	218	7	λ	λ	PROPN
ijassa-2042	218	8	,	,	PUNCT
ijassa-2042	218	9	the	the	DET
ijassa-2042	218	10	pedal	pedal	ADJ
ijassa-2042	218	11	transformation	transformation	NOUN
ijassa-2042	218	12	p	p	NOUN
ijassa-2042	218	13	is	be	AUX
ijassa-2042	218	14	not	not	PART
ijassa-2042	218	15	an	an	DET
ijassa-2042	218	16	involution	involution	NOUN
ijassa-2042	218	17	.	.	PUNCT
ijassa-2042	219	1	therefore	therefore	ADV
ijassa-2042	219	2	,	,	PUNCT
ijassa-2042	219	3	one	one	PRON
ijassa-2042	219	4	can	can	AUX
ijassa-2042	219	5	consider	consider	VERB
ijassa-2042	219	6	the	the	DET
ijassa-2042	219	7	degrees	degree	NOUN
ijassa-2042	219	8	(	(	PUNCT
ijassa-2042	219	9	compositions	composition	NOUN
ijassa-2042	219	10	)	)	PUNCT
ijassa-2042	219	11	of	of	ADP
ijassa-2042	219	12	the	the	DET
ijassa-2042	219	13	pedal	pedal	ADJ
ijassa-2042	219	14	transformation	transformation	NOUN
ijassa-2042	219	15	.	.	PUNCT
ijassa-2042	220	1	fixing	fix	VERB
ijassa-2042	220	2	the	the	DET
ijassa-2042	220	3	pole	pole	NOUN
ijassa-2042	220	4	o	o	NOUN
ijassa-2042	220	5	,	,	PUNCT
ijassa-2042	220	6	consider	consider	VERB
ijassa-2042	220	7	the	the	DET
ijassa-2042	220	8	cyclic	cyclic	ADJ
ijassa-2042	220	9	group	group	NOUN
ijassa-2042	220	10	generated	generate	VERB
ijassa-2042	220	11	by	by	ADP
ijassa-2042	220	12	the	the	DET
ijassa-2042	220	13	elements	element	NOUN
ijassa-2042	220	14	pn	pn	VERB
ijassa-2042	220	15	with	with	ADP
ijassa-2042	220	16	all	all	DET
ijassa-2042	220	17	integers	integer	NOUN
ijassa-2042	220	18	n	n	CCONJ
ijassa-2042	220	19	:	:	PUNCT
ijassa-2042	220	20	.	.	PUNCT
ijassa-2042	220	21	.	.	PUNCT
ijassa-2042	220	22	.	.	PUNCT
ijassa-2042	221	1	p−3	p−3	NOUN
ijassa-2042	221	2	,	,	PUNCT
ijassa-2042	221	3	p−2	p−2	PROPN
ijassa-2042	221	4	,	,	PUNCT
ijassa-2042	221	5	p−1	p−1	PROPN
ijassa-2042	221	6	,	,	PUNCT
ijassa-2042	221	7	p0	p0	NOUN
ijassa-2042	221	8	,	,	PUNCT
ijassa-2042	221	9	p1	p1	NOUN
ijassa-2042	221	10	,	,	PUNCT
ijassa-2042	221	11	p2	p2	NOUN
ijassa-2042	221	12	,	,	PUNCT
ijassa-2042	221	13	p3	p3	PROPN
ijassa-2042	221	14	.	.	PUNCT
ijassa-2042	221	15	.	.	PUNCT
ijassa-2042	221	16	.	.	PUNCT
ijassa-2042	222	1	a	a	DET
ijassa-2042	222	2	natural	natural	ADJ
ijassa-2042	222	3	question	question	NOUN
ijassa-2042	222	4	:	:	PUNCT
ijassa-2042	222	5	is	be	AUX
ijassa-2042	222	6	thus	thus	ADV
ijassa-2042	222	7	group	group	NOUN
ijassa-2042	222	8	finite	finite	NOUN
ijassa-2042	222	9	or	or	CCONJ
ijassa-2042	222	10	infinite	infinite	VERB
ijassa-2042	222	11	?	?	PUNCT
ijassa-2042	223	1	we	we	PRON
ijassa-2042	223	2	will	will	AUX
ijassa-2042	223	3	get	get	VERB
ijassa-2042	223	4	an	an	DET
ijassa-2042	223	5	answer	answer	NOUN
ijassa-2042	223	6	below	below	ADV
ijassa-2042	223	7	.	.	PUNCT
ijassa-2042	224	1	sophus	sophus	PROPN
ijassa-2042	224	2	lie	lie	PROPN
ijassa-2042	224	3	suggested	suggest	VERB
ijassa-2042	224	4	a	a	DET
ijassa-2042	224	5	nice	nice	ADJ
ijassa-2042	224	6	way	way	NOUN
ijassa-2042	224	7	how	how	SCONJ
ijassa-2042	224	8	to	to	PART
ijassa-2042	224	9	extend	extend	VERB
ijassa-2042	224	10	this	this	DET
ijassa-2042	224	11	discrete	discrete	ADJ
ijassa-2042	224	12	grout	grout	NOUN
ijassa-2042	224	13	to	to	ADP
ijassa-2042	224	14	a	a	DET
ijassa-2042	224	15	continuous	continuous	ADJ
ijassa-2042	224	16	group	group	NOUN
ijassa-2042	224	17	.	.	PUNCT
ijassa-2042	225	1	for	for	ADP
ijassa-2042	225	2	this	this	PRON
ijassa-2042	225	3	,	,	PUNCT
ijassa-2042	225	4	consider	consider	VERB
ijassa-2042	225	5	the	the	DET
ijassa-2042	225	6	polar	polar	ADJ
ijassa-2042	225	7	coordinates	coordinate	NOUN
ijassa-2042	225	8	(	(	PUNCT
ijassa-2042	225	9	r	r	NOUN
ijassa-2042	225	10	,	,	PUNCT
ijassa-2042	225	11	ϕ	ϕ	NOUN
ijassa-2042	225	12	)	)	PUNCT
ijassa-2042	225	13	on	on	ADP
ijassa-2042	225	14	the	the	DET
ijassa-2042	225	15	(	(	PUNCT
ijassa-2042	225	16	x	x	NOUN
ijassa-2042	225	17	,	,	PUNCT
ijassa-2042	225	18	y)-plane	y)-plane	NOUN
ijassa-2042	225	19	and	and	CCONJ
ijassa-2042	225	20	the	the	DET
ijassa-2042	225	21	polar	polar	ADJ
ijassa-2042	225	22	coordinates	coordinate	NOUN
ijassa-2042	225	23	(	(	PUNCT
ijassa-2042	225	24	r	r	NOUN
ijassa-2042	225	25	,	,	PUNCT
ijassa-2042	225	26	φ	φ	NUM
ijassa-2042	225	27	)	)	PUNCT
ijassa-2042	225	28	on	on	ADP
ijassa-2042	225	29	the	the	DET
ijassa-2042	225	30	(	(	PUNCT
ijassa-2042	225	31	x̄	x̄	PROPN
ijassa-2042	225	32	,	,	PUNCT
ijassa-2042	225	33	ȳ	ȳ	ADJ
ijassa-2042	225	34	)	)	PUNCT
ijassa-2042	225	35	-plane	-plane	NOUN
ijassa-2042	225	36	.	.	PUNCT
ijassa-2042	226	1	as	as	SCONJ
ijassa-2042	226	2	the	the	DET
ijassa-2042	226	3	third	third	ADJ
ijassa-2042	226	4	coordiante	coordiante	NOUN
ijassa-2042	226	5	in	in	ADP
ijassa-2042	226	6	the	the	DET
ijassa-2042	226	7	space	space	NOUN
ijassa-2042	226	8	j1	j1	NOUN
ijassa-2042	226	9	let	let	VERB
ijassa-2042	226	10	us	we	PRON
ijassa-2042	226	11	take	take	VERB
ijassa-2042	226	12	the	the	DET
ijassa-2042	226	13	difference	difference	NOUN
ijassa-2042	226	14	between	between	ADP
ijassa-2042	226	15	the	the	DET
ijassa-2042	226	16	angle	angle	NOUN
ijassa-2042	226	17	of	of	ADP
ijassa-2042	226	18	inclination	inclination	NOUN
ijassa-2042	226	19	of	of	ADP
ijassa-2042	226	20	the	the	DET
ijassa-2042	226	21	radius	radius	NOUN
ijassa-2042	226	22	vector	vector	NOUN
ijassa-2042	226	23	to	to	ADP
ijassa-2042	226	24	a	a	DET
ijassa-2042	226	25	given	give	VERB
ijassa-2042	226	26	point	point	NOUN
ijassa-2042	226	27	on	on	ADP
ijassa-2042	226	28	the	the	DET
ijassa-2042	226	29	plane	plane	NOUN
ijassa-2042	226	30	and	and	CCONJ
ijassa-2042	226	31	the	the	DET
ijassa-2042	226	32	angle	angle	NOUN
ijassa-2042	226	33	of	of	ADP
ijassa-2042	226	34	inclination	inclination	NOUN
ijassa-2042	226	35	of	of	ADP
ijassa-2042	226	36	the	the	DET
ijassa-2042	226	37	tangent	tangent	NOUN
ijassa-2042	226	38	to	to	ADP
ijassa-2042	226	39	the	the	DET
ijassa-2042	226	40	curve	curve	NOUN
ijassa-2042	226	41	at	at	ADP
ijassa-2042	226	42	this	this	DET
ijassa-2042	226	43	point	point	NOUN
ijassa-2042	226	44	such	such	ADJ
ijassa-2042	226	45	that	that	SCONJ
ijassa-2042	226	46	the	the	DET
ijassa-2042	226	47	orders	order	NOUN
ijassa-2042	226	48	of	of	ADP
ijassa-2042	226	49	subtraction	subtraction	NOUN
ijassa-2042	226	50	in	in	ADP
ijassa-2042	226	51	the	the	DET
ijassa-2042	226	52	preimage	preimage	NOUN
ijassa-2042	226	53	and	and	CCONJ
ijassa-2042	226	54	image	image	NOUN
ijassa-2042	226	55	are	be	AUX
ijassa-2042	226	56	opposite	opposite	ADJ
ijassa-2042	226	57	:	:	PUNCT
ijassa-2042	226	58	x	x	SYM
ijassa-2042	226	59	=	=	PUNCT
ijassa-2042	226	60	r	r	NOUN
ijassa-2042	226	61	cosϕ	cosϕ	NOUN
ijassa-2042	226	62	,	,	PUNCT
ijassa-2042	226	63	y	y	NOUN
ijassa-2042	226	64	=	=	SYM
ijassa-2042	226	65	r	r	NOUN
ijassa-2042	226	66	sinϕ	sinϕ	NOUN
ijassa-2042	226	67	,	,	PUNCT
ijassa-2042	226	68	ψ	ψ	X
ijassa-2042	226	69	=	=	SYM
ijassa-2042	226	70	ϕ−	ϕ−	PROPN
ijassa-2042	226	71	arctan	arctan	PROPN
ijassa-2042	226	72	p	p	PROPN
ijassa-2042	226	73	,	,	PUNCT
ijassa-2042	226	74	x̄	x̄	PUNCT
ijassa-2042	227	1	=	=	SYM
ijassa-2042	227	2	r	r	NOUN
ijassa-2042	227	3	cosφ	cosφ	NOUN
ijassa-2042	227	4	,	,	PUNCT
ijassa-2042	227	5	ȳ	ȳ	NOUN
ijassa-2042	227	6	=	=	NOUN
ijassa-2042	227	7	r	r	NOUN
ijassa-2042	227	8	sinφ	sinφ	NOUN
ijassa-2042	227	9	,	,	PUNCT
ijassa-2042	227	10	ψ	ψ	X
ijassa-2042	227	11	=	=	SYM
ijassa-2042	227	12	arctan	arctan	PROPN
ijassa-2042	227	13	p̄	p̄	PROPN
ijassa-2042	228	1	−	−	PROPN
ijassa-2042	228	2	φ	φ	PROPN
ijassa-2042	228	3	,	,	PUNCT
ijassa-2042	228	4	(	(	PUNCT
ijassa-2042	228	5	3.23	3.23	NUM
ijassa-2042	228	6	)	)	PUNCT
ijassa-2042	228	7	copyright	copyright	NOUN
ijassa-2042	228	8	©	©	PROPN
ijassa-2042	228	9	2025	2025	NUM
ijassa-2042	228	10	assa	assa	NOUN
ijassa-2042	228	11	.	.	PUNCT
ijassa-2042	229	1	adv	adv	PROPN
ijassa-2042	229	2	syst	syst	PROPN
ijassa-2042	229	3	sci	sci	PROPN
ijassa-2042	229	4	appl	appl	PROPN
ijassa-2042	229	5	(	(	PUNCT
ijassa-2042	229	6	2025	2025	NUM
ijassa-2042	229	7	)	)	PUNCT
ijassa-2042	229	8	legendre	legendre	PROPN
ijassa-2042	229	9	transformation	transformation	NOUN
ijassa-2042	229	10	and	and	CCONJ
ijassa-2042	229	11	its	its	PRON
ijassa-2042	229	12	applications	application	NOUN
ijassa-2042	229	13	53	53	NUM
ijassa-2042	229	14	where	where	SCONJ
ijassa-2042	229	15	x	x	X
ijassa-2042	229	16	,	,	PUNCT
ijassa-2042	229	17	y	y	PROPN
ijassa-2042	229	18	,	,	PUNCT
ijassa-2042	229	19	p	p	NOUN
ijassa-2042	229	20	and	and	CCONJ
ijassa-2042	229	21	x̄	x̄	PROPN
ijassa-2042	229	22	,	,	PUNCT
ijassa-2042	229	23	ȳ	ȳ	PROPN
ijassa-2042	229	24	,	,	PUNCT
ijassa-2042	229	25	p̄	p̄	PROPN
ijassa-2042	229	26	are	be	AUX
ijassa-2042	229	27	connected	connect	VERB
ijassa-2042	229	28	via	via	ADP
ijassa-2042	229	29	(	(	PUNCT
ijassa-2042	229	30	3.22	3.22	NUM
ijassa-2042	229	31	)	)	PUNCT
ijassa-2042	229	32	.	.	PUNCT
ijassa-2042	230	1	it	it	PRON
ijassa-2042	230	2	is	be	AUX
ijassa-2042	230	3	not	not	PART
ijassa-2042	230	4	hard	hard	ADJ
ijassa-2042	230	5	to	to	PART
ijassa-2042	230	6	check	check	VERB
ijassa-2042	230	7	that	that	SCONJ
ijassa-2042	230	8	in	in	ADP
ijassa-2042	230	9	the	the	DET
ijassa-2042	230	10	coordinates	coordinate	NOUN
ijassa-2042	230	11	(	(	PUNCT
ijassa-2042	230	12	3.23	3.23	NUM
ijassa-2042	230	13	)	)	PUNCT
ijassa-2042	230	14	,	,	PUNCT
ijassa-2042	230	15	the	the	DET
ijassa-2042	230	16	transformation	transformation	NOUN
ijassa-2042	230	17	pn	pn	NOUN
ijassa-2042	230	18	with	with	ADP
ijassa-2042	230	19	every	every	DET
ijassa-2042	230	20	integer	integer	NOUN
ijassa-2042	230	21	n	n	AUX
ijassa-2042	230	22	is	be	AUX
ijassa-2042	230	23	given	give	VERB
ijassa-2042	230	24	by	by	ADP
ijassa-2042	230	25	the	the	DET
ijassa-2042	230	26	formula	formula	NOUN
ijassa-2042	230	27	r	r	NOUN
ijassa-2042	230	28	=	=	SYM
ijassa-2042	230	29	r|	r|	NOUN
ijassa-2042	230	30	sinψ|n	sinψ|n	NOUN
ijassa-2042	230	31	,	,	PUNCT
ijassa-2042	230	32	φ	φ	PROPN
ijassa-2042	230	33	=	=	PUNCT
ijassa-2042	230	34	ϕ−	ϕ−	PROPN
ijassa-2042	230	35	n(ψ	n(ψ	NOUN
ijassa-2042	230	36	+	+	CCONJ
ijassa-2042	230	37	π	π	PROPN
ijassa-2042	230	38	2	2	NUM
ijassa-2042	230	39	)	)	PUNCT
ijassa-2042	230	40	,	,	PUNCT
ijassa-2042	230	41	ψ	ψ	X
ijassa-2042	230	42	=	=	SYM
ijassa-2042	230	43	ψ	ψ	X
ijassa-2042	230	44	.	.	PUNCT
ijassa-2042	231	1	(	(	PUNCT
ijassa-2042	231	2	3.24	3.24	NUM
ijassa-2042	231	3	)	)	PUNCT
ijassa-2042	231	4	the	the	DET
ijassa-2042	231	5	obtained	obtain	VERB
ijassa-2042	231	6	formulas	formula	NOUN
ijassa-2042	231	7	show	show	NOUN
ijassa-2042	231	8	shows	show	VERB
ijassa-2042	231	9	that	that	SCONJ
ijassa-2042	231	10	pn	pn	PROPN
ijassa-2042	231	11	̸=	̸=	PROPN
ijassa-2042	231	12	pm	pm	VERB
ijassa-2042	231	13	for	for	ADP
ijassa-2042	231	14	any	any	DET
ijassa-2042	231	15	n	n	DET
ijassa-2042	231	16	̸=	̸=	PROPN
ijassa-2042	231	17	m.	m.	NOUN
ijassa-2042	231	18	therefore	therefore	ADV
ijassa-2042	231	19	,	,	PUNCT
ijassa-2042	231	20	the	the	DET
ijassa-2042	231	21	cyclic	cyclic	ADJ
ijassa-2042	231	22	group	group	NOUN
ijassa-2042	231	23	{	{	PUNCT
ijassa-2042	231	24	pn	pn	NOUN
ijassa-2042	231	25	}	}	PUNCT
ijassa-2042	231	26	is	be	AUX
ijassa-2042	231	27	infinite	infinite	ADJ
ijassa-2042	231	28	.	.	PUNCT
ijassa-2042	232	1	if	if	SCONJ
ijassa-2042	232	2	we	we	PRON
ijassa-2042	232	3	consider	consider	VERB
ijassa-2042	232	4	the	the	DET
ijassa-2042	232	5	same	same	ADJ
ijassa-2042	232	6	formula	formula	NOUN
ijassa-2042	232	7	(	(	PUNCT
ijassa-2042	232	8	3.24	3.24	NUM
ijassa-2042	232	9	)	)	PUNCT
ijassa-2042	232	10	with	with	ADP
ijassa-2042	232	11	real	real	ADJ
ijassa-2042	232	12	n	n	NOUN
ijassa-2042	232	13	and	and	CCONJ
ijassa-2042	232	14	the	the	DET
ijassa-2042	232	15	operation	operation	NOUN
ijassa-2042	232	16	pn	pn	PROPN
ijassa-2042	232	17	·	·	PUNCT
ijassa-2042	232	18	pm	pm	PROPN
ijassa-2042	232	19	=	=	SYM
ijassa-2042	232	20	pn+m	pn+m	PROPN
ijassa-2042	232	21	,	,	PUNCT
ijassa-2042	232	22	we	we	PRON
ijassa-2042	232	23	obtain	obtain	VERB
ijassa-2042	232	24	a	a	DET
ijassa-2042	232	25	continuous	continuous	ADJ
ijassa-2042	232	26	group	group	NOUN
ijassa-2042	232	27	of	of	ADP
ijassa-2042	232	28	contact	contact	NOUN
ijassa-2042	232	29	transformations	transformation	NOUN
ijassa-2042	232	30	.	.	PUNCT
ijassa-2042	233	1	this	this	DET
ijassa-2042	233	2	group	group	NOUN
ijassa-2042	233	3	was	be	AUX
ijassa-2042	233	4	constructed	construct	VERB
ijassa-2042	233	5	by	by	ADP
ijassa-2042	233	6	sophus	sophus	NOUN
ijassa-2042	233	7	lie	lie	NOUN
ijassa-2042	233	8	.	.	PUNCT
ijassa-2042	234	1	we	we	PRON
ijassa-2042	234	2	will	will	AUX
ijassa-2042	234	3	not	not	PART
ijassa-2042	234	4	go	go	VERB
ijassa-2042	234	5	deep	deep	ADV
ijassa-2042	234	6	into	into	ADP
ijassa-2042	234	7	the	the	DET
ijassa-2042	234	8	theory	theory	NOUN
ijassa-2042	234	9	of	of	ADP
ijassa-2042	234	10	contact	contact	NOUN
ijassa-2042	234	11	transformations	transformation	NOUN
ijassa-2042	234	12	,	,	PUNCT
ijassa-2042	234	13	referring	refer	VERB
ijassa-2042	234	14	the	the	DET
ijassa-2042	234	15	interested	interested	ADJ
ijassa-2042	234	16	reader	reader	NOUN
ijassa-2042	234	17	to	to	ADP
ijassa-2042	234	18	the	the	DET
ijassa-2042	234	19	specialized	specialized	ADJ
ijassa-2042	234	20	literature	literature	NOUN
ijassa-2042	234	21	,	,	PUNCT
ijassa-2042	234	22	which	which	PRON
ijassa-2042	234	23	is	be	AUX
ijassa-2042	234	24	quite	quite	ADV
ijassa-2042	234	25	extensive	extensive	ADJ
ijassa-2042	234	26	.	.	PUNCT
ijassa-2042	235	1	for	for	ADP
ijassa-2042	235	2	example	example	NOUN
ijassa-2042	235	3	,	,	PUNCT
ijassa-2042	235	4	see	see	VERB
ijassa-2042	235	5	[	[	X
ijassa-2042	235	6	8	8	NUM
ijassa-2042	235	7	,	,	PUNCT
ijassa-2042	235	8	9	9	NUM
ijassa-2042	235	9	,	,	PUNCT
ijassa-2042	235	10	11	11	NUM
ijassa-2042	235	11	]	]	PUNCT
ijassa-2042	235	12	.	.	PUNCT
ijassa-2042	236	1	4	4	X
ijassa-2042	236	2	.	.	X
ijassa-2042	236	3	conclusion	conclusion	NOUN
ijassa-2042	236	4	we	we	PRON
ijassa-2042	236	5	presented	present	VERB
ijassa-2042	236	6	the	the	DET
ijassa-2042	236	7	legendre	legendre	PROPN
ijassa-2042	236	8	transformation	transformation	NOUN
ijassa-2042	236	9	in	in	ADP
ijassa-2042	236	10	a	a	DET
ijassa-2042	236	11	geometric	geometric	ADJ
ijassa-2042	236	12	way	way	NOUN
ijassa-2042	236	13	,	,	PUNCT
ijassa-2042	236	14	which	which	PRON
ijassa-2042	236	15	is	be	AUX
ijassa-2042	236	16	based	base	VERB
ijassa-2042	236	17	on	on	ADP
ijassa-2042	236	18	the	the	DET
ijassa-2042	236	19	notion	notion	NOUN
ijassa-2042	236	20	of	of	ADP
ijassa-2042	236	21	the	the	DET
ijassa-2042	236	22	legendrian	legendrian	ADJ
ijassa-2042	236	23	lift	lift	NOUN
ijassa-2042	236	24	.	.	PUNCT
ijassa-2042	237	1	this	this	DET
ijassa-2042	237	2	approach	approach	NOUN
ijassa-2042	237	3	is	be	AUX
ijassa-2042	237	4	more	more	ADV
ijassa-2042	237	5	natural	natural	ADJ
ijassa-2042	237	6	than	than	ADP
ijassa-2042	237	7	the	the	DET
ijassa-2042	237	8	standard	standard	ADJ
ijassa-2042	237	9	definition	definition	NOUN
ijassa-2042	237	10	for	for	ADP
ijassa-2042	237	11	the	the	DET
ijassa-2042	237	12	legendre	legendre	PROPN
ijassa-2042	237	13	transformation	transformation	NOUN
ijassa-2042	237	14	of	of	ADP
ijassa-2042	237	15	functions	function	NOUN
ijassa-2042	237	16	.	.	PUNCT
ijassa-2042	238	1	indeed	indeed	ADV
ijassa-2042	238	2	,	,	PUNCT
ijassa-2042	238	3	it	it	PRON
ijassa-2042	238	4	is	be	AUX
ijassa-2042	238	5	applicable	applicable	ADJ
ijassa-2042	238	6	to	to	ADP
ijassa-2042	238	7	various	various	ADJ
ijassa-2042	238	8	objects	object	NOUN
ijassa-2042	238	9	(	(	PUNCT
ijassa-2042	238	10	including	include	VERB
ijassa-2042	238	11	functions	function	NOUN
ijassa-2042	238	12	,	,	PUNCT
ijassa-2042	238	13	curves	curve	NOUN
ijassa-2042	238	14	,	,	PUNCT
ijassa-2042	238	15	differential	differential	ADJ
ijassa-2042	238	16	equations	equation	NOUN
ijassa-2042	238	17	)	)	PUNCT
ijassa-2042	238	18	and	and	CCONJ
ijassa-2042	238	19	it	it	PRON
ijassa-2042	238	20	shows	show	VERB
ijassa-2042	238	21	that	that	SCONJ
ijassa-2042	238	22	the	the	DET
ijassa-2042	238	23	legendre	legendre	PROPN
ijassa-2042	238	24	transformation	transformation	NOUN
ijassa-2042	238	25	of	of	ADP
ijassa-2042	238	26	functions	function	NOUN
ijassa-2042	238	27	is	be	AUX
ijassa-2042	238	28	a	a	DET
ijassa-2042	238	29	partial	partial	ADJ
ijassa-2042	238	30	case	case	NOUN
ijassa-2042	238	31	of	of	ADP
ijassa-2042	238	32	the	the	DET
ijassa-2042	238	33	legendre	legendre	PROPN
ijassa-2042	238	34	transformation	transformation	NOUN
ijassa-2042	238	35	of	of	ADP
ijassa-2042	238	36	curves	curve	NOUN
ijassa-2042	238	37	.	.	PUNCT
ijassa-2042	239	1	we	we	PRON
ijassa-2042	239	2	also	also	ADV
ijassa-2042	239	3	considered	consider	VERB
ijassa-2042	239	4	the	the	DET
ijassa-2042	239	5	legendre	legendre	PROPN
ijassa-2042	239	6	transformation	transformation	NOUN
ijassa-2042	239	7	to	to	ADP
ijassa-2042	239	8	ordinary	ordinary	ADJ
ijassa-2042	239	9	differential	differential	ADJ
ijassa-2042	239	10	equations	equation	NOUN
ijassa-2042	239	11	at	at	ADP
ijassa-2042	239	12	the	the	DET
ijassa-2042	239	13	example	example	NOUN
ijassa-2042	239	14	of	of	ADP
ijassa-2042	239	15	the	the	DET
ijassa-2042	239	16	clairaut	clairaut	PROPN
ijassa-2042	239	17	equation	equation	NOUN
ijassa-2042	239	18	and	and	CCONJ
ijassa-2042	239	19	discussed	discuss	VERB
ijassa-2042	239	20	its	its	PRON
ijassa-2042	239	21	wonderful	wonderful	ADJ
ijassa-2042	239	22	properties	property	NOUN
ijassa-2042	239	23	.	.	PUNCT
ijassa-2042	240	1	it	it	PRON
ijassa-2042	240	2	was	be	AUX
ijassa-2042	240	3	shown	show	VERB
ijassa-2042	240	4	that	that	SCONJ
ijassa-2042	240	5	the	the	DET
ijassa-2042	240	6	legendre	legendre	PROPN
ijassa-2042	240	7	transformation	transformation	NOUN
ijassa-2042	240	8	is	be	AUX
ijassa-2042	240	9	a	a	DET
ijassa-2042	240	10	member	member	NOUN
ijassa-2042	240	11	of	of	ADP
ijassa-2042	240	12	a	a	DET
ijassa-2042	240	13	class	class	NOUN
ijassa-2042	240	14	(	(	PUNCT
ijassa-2042	240	15	group	group	NOUN
ijassa-2042	240	16	)	)	PUNCT
ijassa-2042	240	17	of	of	ADP
ijassa-2042	240	18	contact	contact	NOUN
ijassa-2042	240	19	transformations	transformation	NOUN
ijassa-2042	240	20	.	.	PUNCT
ijassa-2042	241	1	we	we	PRON
ijassa-2042	241	2	presented	present	VERB
ijassa-2042	241	3	several	several	ADJ
ijassa-2042	241	4	examples	example	NOUN
ijassa-2042	241	5	of	of	ADP
ijassa-2042	241	6	different	different	ADJ
ijassa-2042	241	7	contact	contact	NOUN
ijassa-2042	241	8	transformations	transformation	NOUN
ijassa-2042	241	9	:	:	PUNCT
ijassa-2042	241	10	the	the	DET
ijassa-2042	241	11	pedal	pedal	ADJ
ijassa-2042	241	12	transformation	transformation	NOUN
ijassa-2042	241	13	and	and	CCONJ
ijassa-2042	241	14	an	an	DET
ijassa-2042	241	15	infinite	infinite	ADJ
ijassa-2042	241	16	group	group	NOUN
ijassa-2042	241	17	of	of	ADP
ijassa-2042	241	18	contact	contact	NOUN
ijassa-2042	241	19	transformations	transformation	NOUN
ijassa-2042	241	20	.	.	PUNCT
ijassa-2042	242	1	references	reference	NOUN
ijassa-2042	242	2	1	1	NUM
ijassa-2042	242	3	.	.	X
ijassa-2042	242	4	agakhanova	agakhanova	PROPN
ijassa-2042	242	5	,	,	PUNCT
ijassa-2042	242	6	ya	ya	PROPN
ijassa-2042	242	7	.	.	PUNCT
ijassa-2042	242	8	s.	s.	PROPN
ijassa-2042	242	9	,	,	PUNCT
ijassa-2042	242	10	nikachev	nikachev	PROPN
ijassa-2042	242	11	,	,	PUNCT
ijassa-2042	242	12	a.	a.	NOUN
ijassa-2042	242	13	s.	s.	PROPN
ijassa-2042	242	14	,	,	PUNCT
ijassa-2042	242	15	&	&	CCONJ
ijassa-2042	242	16	oblasova	oblasova	PROPN
ijassa-2042	242	17	,	,	PUNCT
ijassa-2042	242	18	i.	i.	PROPN
ijassa-2042	242	19	n.	n.	PROPN
ijassa-2042	242	20	(	(	PUNCT
ijassa-2042	242	21	2023	2023	NUM
ijassa-2042	242	22	)	)	PUNCT
ijassa-2042	242	23	on	on	ADP
ijassa-2042	242	24	codimension	codimension	NOUN
ijassa-2042	242	25	3	3	NUM
ijassa-2042	242	26	singular	singular	ADJ
ijassa-2042	242	27	points	point	NOUN
ijassa-2042	242	28	of	of	ADP
ijassa-2042	242	29	first	first	ADJ
ijassa-2042	242	30	-	-	PUNCT
ijassa-2042	242	31	order	order	NOUN
ijassa-2042	242	32	implicit	implicit	ADJ
ijassa-2042	242	33	differential	differential	ADJ
ijassa-2042	242	34	equations	equation	NOUN
ijassa-2042	242	35	,	,	PUNCT
ijassa-2042	242	36	adv	adv	PROPN
ijassa-2042	242	37	.	.	PUNCT
ijassa-2042	242	38	syst	syst	PROPN
ijassa-2042	242	39	.	.	PUNCT
ijassa-2042	243	1	sci	sci	PROPN
ijassa-2042	243	2	.	.	PUNCT
ijassa-2042	243	3	appl	appl	PROPN
ijassa-2042	243	4	.	.	PROPN
ijassa-2042	243	5	,	,	PUNCT
ijassa-2042	243	6	23(1	23(1	X
ijassa-2042	243	7	)	)	PUNCT
ijassa-2042	243	8	,	,	PUNCT
ijassa-2042	243	9	8–21	8–21	NOUN
ijassa-2042	243	10	.	.	PUNCT
ijassa-2042	244	1	2	2	X
ijassa-2042	244	2	.	.	X
ijassa-2042	244	3	arnol’d	arnol’d	NOUN
ijassa-2042	244	4	,	,	PUNCT
ijassa-2042	244	5	v.	v.	PROPN
ijassa-2042	244	6	i.	i.	PROPN
ijassa-2042	244	7	(	(	PUNCT
ijassa-2042	244	8	1988	1988	NUM
ijassa-2042	244	9	)	)	PUNCT
ijassa-2042	244	10	geometrical	geometrical	ADJ
ijassa-2042	244	11	methods	method	NOUN
ijassa-2042	244	12	in	in	ADP
ijassa-2042	244	13	the	the	DET
ijassa-2042	244	14	theory	theory	NOUN
ijassa-2042	244	15	of	of	ADP
ijassa-2042	244	16	ordinary	ordinary	ADJ
ijassa-2042	244	17	differential	differential	ADJ
ijassa-2042	244	18	equations	equation	NOUN
ijassa-2042	244	19	,	,	PUNCT
ijassa-2042	244	20	berlin	berlin	PROPN
ijassa-2042	244	21	:	:	PUNCT
ijassa-2042	244	22	springer	springer	NOUN
ijassa-2042	244	23	.	.	PUNCT
ijassa-2042	245	1	3	3	X
ijassa-2042	245	2	.	.	X
ijassa-2042	245	3	arnol’d	arnol’d	NOUN
ijassa-2042	245	4	,	,	PUNCT
ijassa-2042	245	5	v.	v.	PROPN
ijassa-2042	245	6	i.	i.	PROPN
ijassa-2042	245	7	(	(	PUNCT
ijassa-2042	245	8	1990	1990	NUM
ijassa-2042	245	9	)	)	PUNCT
ijassa-2042	245	10	singularities	singularity	NOUN
ijassa-2042	245	11	of	of	ADP
ijassa-2042	245	12	caustics	caustic	NOUN
ijassa-2042	245	13	and	and	CCONJ
ijassa-2042	245	14	wave	wave	NOUN
ijassa-2042	245	15	fronts	front	NOUN
ijassa-2042	245	16	,	,	PUNCT
ijassa-2042	245	17	math	math	NOUN
ijassa-2042	245	18	appl	appl	NOUN
ijassa-2042	245	19	.	.	PUNCT
ijassa-2042	246	1	(	(	PUNCT
ijassa-2042	246	2	soviet	soviet	ADJ
ijassa-2042	246	3	series	series	NOUN
ijassa-2042	246	4	)	)	PUNCT
ijassa-2042	246	5	,	,	PUNCT
ijassa-2042	246	6	62	62	NUM
ijassa-2042	246	7	.	.	PUNCT
ijassa-2042	247	1	kluwer	kluwer	NOUN
ijassa-2042	247	2	academic	academic	ADJ
ijassa-2042	247	3	publishers	publisher	NOUN
ijassa-2042	247	4	.	.	PUNCT
ijassa-2042	248	1	4	4	X
ijassa-2042	248	2	.	.	X
ijassa-2042	248	3	boscain	boscain	VERB
ijassa-2042	248	4	,	,	PUNCT
ijassa-2042	248	5	u.	u.	PROPN
ijassa-2042	248	6	,	,	PUNCT
ijassa-2042	248	7	chertovskih	chertovskih	PROPN
ijassa-2042	248	8	,	,	PUNCT
ijassa-2042	248	9	r.	r.	PROPN
ijassa-2042	248	10	a.	a.	PROPN
ijassa-2042	248	11	,	,	PUNCT
ijassa-2042	248	12	gauthier	gauthier	PROPN
ijassa-2042	248	13	,	,	PUNCT
ijassa-2042	248	14	j.	j.	PROPN
ijassa-2042	248	15	p.	p.	PROPN
ijassa-2042	248	16	,	,	PUNCT
ijassa-2042	248	17	&	&	CCONJ
ijassa-2042	248	18	remizov	remizov	PROPN
ijassa-2042	248	19	,	,	PUNCT
ijassa-2042	248	20	a.	a.	NOUN
ijassa-2042	248	21	o.	o.	PROPN
ijassa-2042	248	22	(	(	PUNCT
ijassa-2042	248	23	2014	2014	NUM
ijassa-2042	248	24	)	)	PUNCT
ijassa-2042	248	25	hypoelliptic	hypoelliptic	ADJ
ijassa-2042	248	26	diffusion	diffusion	NOUN
ijassa-2042	248	27	and	and	CCONJ
ijassa-2042	248	28	human	human	ADJ
ijassa-2042	248	29	vision	vision	NOUN
ijassa-2042	248	30	:	:	PUNCT
ijassa-2042	248	31	a	a	DET
ijassa-2042	248	32	semidiscrete	semidiscrete	ADJ
ijassa-2042	248	33	new	new	ADJ
ijassa-2042	248	34	twist	twist	NOUN
ijassa-2042	248	35	,	,	PUNCT
ijassa-2042	248	36	siam	siam	PROPN
ijassa-2042	248	37	j.	j.	PROPN
ijassa-2042	248	38	imaging	imaging	PROPN
ijassa-2042	248	39	sci	sci	PROPN
ijassa-2042	248	40	.	.	PROPN
ijassa-2042	248	41	,	,	PUNCT
ijassa-2042	248	42	7(2	7(2	NUM
ijassa-2042	248	43	)	)	PUNCT
ijassa-2042	248	44	,	,	PUNCT
ijassa-2042	248	45	669	669	NUM
ijassa-2042	248	46	–	–	PUNCT
ijassa-2042	248	47	695	695	NUM
ijassa-2042	248	48	.	.	X
ijassa-2042	249	1	5	5	NUM
ijassa-2042	249	2	.	.	X
ijassa-2042	249	3	boscain	boscain	NOUN
ijassa-2042	249	4	,	,	PUNCT
ijassa-2042	249	5	u.	u.	PROPN
ijassa-2042	249	6	,	,	PUNCT
ijassa-2042	249	7	chertovskih	chertovskih	PROPN
ijassa-2042	249	8	,	,	PUNCT
ijassa-2042	249	9	r.	r.	PROPN
ijassa-2042	249	10	a.	a.	PROPN
ijassa-2042	249	11	,	,	PUNCT
ijassa-2042	249	12	gauthier	gauthier	PROPN
ijassa-2042	249	13	,	,	PUNCT
ijassa-2042	249	14	j.	j.	PROPN
ijassa-2042	249	15	p.	p.	PROPN
ijassa-2042	249	16	,	,	PUNCT
ijassa-2042	249	17	prandi	prandi	PROPN
ijassa-2042	249	18	,	,	PUNCT
ijassa-2042	249	19	d.	d.	PROPN
ijassa-2042	249	20	,	,	PUNCT
ijassa-2042	249	21	&	&	CCONJ
ijassa-2042	249	22	remizov	remizov	PROPN
ijassa-2042	249	23	,	,	PUNCT
ijassa-2042	249	24	a.	a.	NOUN
ijassa-2042	249	25	o.	o.	PROPN
ijassa-2042	250	1	(	(	PUNCT
ijassa-2042	250	2	2018	2018	NUM
ijassa-2042	250	3	)	)	PUNCT
ijassa-2042	250	4	cortical	cortical	ADJ
ijassa-2042	250	5	-	-	PUNCT
ijassa-2042	250	6	inspired	inspire	VERB
ijassa-2042	250	7	image	image	NOUN
ijassa-2042	250	8	reconstruction	reconstruction	NOUN
ijassa-2042	250	9	via	via	ADP
ijassa-2042	250	10	sub	sub	ADJ
ijassa-2042	250	11	-	-	ADJ
ijassa-2042	250	12	riemannian	riemannian	ADJ
ijassa-2042	250	13	geometry	geometry	NOUN
ijassa-2042	250	14	and	and	CCONJ
ijassa-2042	250	15	hypoelliptic	hypoelliptic	ADJ
ijassa-2042	250	16	diffusion	diffusion	NOUN
ijassa-2042	250	17	,	,	PUNCT
ijassa-2042	250	18	esaim	esaim	NOUN
ijassa-2042	250	19	,	,	PUNCT
ijassa-2042	250	20	proc	proc	NOUN
ijassa-2042	250	21	.	.	PUNCT
ijassa-2042	251	1	surv	surv	PROPN
ijassa-2042	251	2	.	.	PUNCT
ijassa-2042	251	3	,	,	PUNCT
ijassa-2042	251	4	64	64	NUM
ijassa-2042	251	5	,	,	PUNCT
ijassa-2042	251	6	37–53	37–53	NUM
ijassa-2042	251	7	.	.	PUNCT
ijassa-2042	252	1	6	6	NUM
ijassa-2042	252	2	.	.	X
ijassa-2042	252	3	bruce	bruce	PROPN
ijassa-2042	252	4	,	,	PUNCT
ijassa-2042	252	5	j.	j.	PROPN
ijassa-2042	252	6	w.	w.	PROPN
ijassa-2042	252	7	,	,	PUNCT
ijassa-2042	252	8	gaffney	gaffney	PROPN
ijassa-2042	252	9	t.	t.	PROPN
ijassa-2042	252	10	j.	j.	PROPN
ijassa-2042	252	11	(	(	PUNCT
ijassa-2042	252	12	1982	1982	NUM
ijassa-2042	252	13	)	)	PUNCT
ijassa-2042	252	14	simple	simple	ADJ
ijassa-2042	252	15	singularities	singularity	NOUN
ijassa-2042	252	16	of	of	ADP
ijassa-2042	252	17	mappings	mapping	NOUN
ijassa-2042	252	18	c	c	NOUN
ijassa-2042	252	19	,	,	PUNCT
ijassa-2042	252	20	0	0	NUM
ijassa-2042	252	21	→	→	SYM
ijassa-2042	252	22	c2	c2	PROPN
ijassa-2042	252	23	,	,	PUNCT
ijassa-2042	252	24	0	0	NUM
ijassa-2042	252	25	,	,	PUNCT
ijassa-2042	252	26	j.	j.	PROPN
ijassa-2042	252	27	lond	lond	PROPN
ijassa-2042	252	28	.	.	PUNCT
ijassa-2042	253	1	math	math	PROPN
ijassa-2042	253	2	.	.	PUNCT
ijassa-2042	254	1	soc	soc	PROPN
ijassa-2042	254	2	.	.	PROPN
ijassa-2042	254	3	,	,	PUNCT
ijassa-2042	254	4	ii	ii	PROPN
ijassa-2042	254	5	.	.	PUNCT
ijassa-2042	254	6	ser	ser	PROPN
ijassa-2042	254	7	.	.	PROPN
ijassa-2042	254	8	,	,	PUNCT
ijassa-2042	254	9	26	26	NUM
ijassa-2042	254	10	,	,	PUNCT
ijassa-2042	254	11	465–474	465–474	NUM
ijassa-2042	254	12	.	.	PUNCT
ijassa-2042	255	1	7	7	NUM
ijassa-2042	255	2	.	.	X
ijassa-2042	255	3	bruce	bruce	PROPN
ijassa-2042	255	4	,	,	PUNCT
ijassa-2042	255	5	j.	j.	PROPN
ijassa-2042	255	6	w.	w.	PROPN
ijassa-2042	255	7	,	,	PUNCT
ijassa-2042	255	8	tari	tari	PROPN
ijassa-2042	255	9	,	,	PUNCT
ijassa-2042	255	10	f.	f.	PROPN
ijassa-2042	255	11	(	(	PUNCT
ijassa-2042	255	12	2000	2000	NUM
ijassa-2042	255	13	)	)	PUNCT
ijassa-2042	255	14	duality	duality	NOUN
ijassa-2042	255	15	and	and	CCONJ
ijassa-2042	255	16	implicit	implicit	ADJ
ijassa-2042	255	17	differential	differential	ADJ
ijassa-2042	255	18	equations	equation	NOUN
ijassa-2042	255	19	,	,	PUNCT
ijassa-2042	255	20	nonlinearity	nonlinearity	NOUN
ijassa-2042	255	21	,	,	PUNCT
ijassa-2042	255	22	13:3	13:3	NUM
ijassa-2042	255	23	,	,	PUNCT
ijassa-2042	255	24	791–811	791–811	NUM
ijassa-2042	255	25	.	.	NOUN
ijassa-2042	255	26	8	8	NUM
ijassa-2042	255	27	.	.	X
ijassa-2042	256	1	feld	feld	PROPN
ijassa-2042	256	2	,	,	PUNCT
ijassa-2042	256	3	j.	j.	PROPN
ijassa-2042	256	4	m.	m.	PROPN
ijassa-2042	256	5	(	(	PUNCT
ijassa-2042	256	6	1940	1940	NUM
ijassa-2042	256	7	)	)	PUNCT
ijassa-2042	256	8	a	a	DET
ijassa-2042	256	9	continuous	continuous	ADJ
ijassa-2042	256	10	group	group	NOUN
ijassa-2042	256	11	of	of	ADP
ijassa-2042	256	12	contact	contact	NOUN
ijassa-2042	256	13	transformations	transformation	NOUN
ijassa-2042	256	14	containing	contain	VERB
ijassa-2042	256	15	the	the	DET
ijassa-2042	256	16	generalized	generalize	VERB
ijassa-2042	256	17	pedal	pedal	NOUN
ijassa-2042	256	18	transformation	transformation	NOUN
ijassa-2042	256	19	,	,	PUNCT
ijassa-2042	256	20	tohoku	tohoku	PROPN
ijassa-2042	256	21	math	math	PROPN
ijassa-2042	256	22	.	.	PUNCT
ijassa-2042	257	1	j.	j.	PROPN
ijassa-2042	257	2	,	,	PUNCT
ijassa-2042	257	3	46	46	NUM
ijassa-2042	257	4	,	,	PUNCT
ijassa-2042	257	5	252–260	252–260	NUM
ijassa-2042	257	6	.	.	PUNCT
ijassa-2042	257	7	9	9	NUM
ijassa-2042	257	8	.	.	X
ijassa-2042	258	1	gizatullin	gizatullin	PROPN
ijassa-2042	258	2	,	,	PUNCT
ijassa-2042	258	3	m.	m.	PROPN
ijassa-2042	258	4	kh.(2008	kh.(2008	PROPN
ijassa-2042	258	5	)	)	PUNCT
ijassa-2042	258	6	klein	klein	PROPN
ijassa-2042	258	7	’s	’s	PART
ijassa-2042	258	8	conjecture	conjecture	NOUN
ijassa-2042	258	9	for	for	ADP
ijassa-2042	258	10	contact	contact	NOUN
ijassa-2042	258	11	automorphisms	automorphism	NOUN
ijassa-2042	258	12	of	of	ADP
ijassa-2042	258	13	the	the	DET
ijassa-2042	258	14	threedimensional	threedimensional	ADJ
ijassa-2042	258	15	affine	affine	NOUN
ijassa-2042	258	16	space	space	NOUN
ijassa-2042	258	17	,	,	PUNCT
ijassa-2042	258	18	mich	mich	PROPN
ijassa-2042	258	19	.	.	PUNCT
ijassa-2042	258	20	math	math	PROPN
ijassa-2042	258	21	.	.	PUNCT
ijassa-2042	259	1	j.	j.	PROPN
ijassa-2042	259	2	,	,	PUNCT
ijassa-2042	259	3	56(1	56(1	NUM
ijassa-2042	259	4	)	)	PUNCT
ijassa-2042	259	5	,	,	PUNCT
ijassa-2042	259	6	89–98	89–98	NUM
ijassa-2042	259	7	.	.	PUNCT
ijassa-2042	260	1	10	10	NUM
ijassa-2042	260	2	.	.	X
ijassa-2042	261	1	izumiya	izumiya	PROPN
ijassa-2042	261	2	,	,	PUNCT
ijassa-2042	261	3	s.	s.	PROPN
ijassa-2042	261	4	,	,	PUNCT
ijassa-2042	261	5	kurokawa	kurokawa	PROPN
ijassa-2042	261	6	,	,	PUNCT
ijassa-2042	261	7	y.	y.	PROPN
ijassa-2042	261	8	(	(	PUNCT
ijassa-2042	261	9	1994	1994	NUM
ijassa-2042	261	10	)	)	PUNCT
ijassa-2042	261	11	on	on	ADP
ijassa-2042	261	12	systems	system	NOUN
ijassa-2042	261	13	of	of	ADP
ijassa-2042	261	14	clairaut	clairaut	PROPN
ijassa-2042	261	15	type	type	PROPN
ijassa-2042	261	16	,	,	PUNCT
ijassa-2042	261	17	kodai	kodai	PROPN
ijassa-2042	261	18	math	math	PROPN
ijassa-2042	261	19	.	.	PUNCT
ijassa-2042	262	1	j.	j.	PROPN
ijassa-2042	262	2	,	,	PUNCT
ijassa-2042	262	3	17(3	17(3	NUM
ijassa-2042	262	4	)	)	PUNCT
ijassa-2042	262	5	,	,	PUNCT
ijassa-2042	262	6	636–643	636–643	NUM
ijassa-2042	262	7	.	.	PUNCT
ijassa-2042	263	1	copyright	copyright	NOUN
ijassa-2042	263	2	©	©	PROPN
ijassa-2042	263	3	2025	2025	NUM
ijassa-2042	263	4	assa	assa	NOUN
ijassa-2042	263	5	.	.	PUNCT
ijassa-2042	264	1	adv	adv	PROPN
ijassa-2042	264	2	syst	syst	PROPN
ijassa-2042	264	3	sci	sci	PROPN
ijassa-2042	264	4	appl	appl	PROPN
ijassa-2042	264	5	(	(	PUNCT
ijassa-2042	264	6	2025	2025	NUM
ijassa-2042	264	7	)	)	PUNCT
ijassa-2042	264	8	54	54	NUM
ijassa-2042	264	9	n.	n.	NOUN
ijassa-2042	264	10	pavlova	pavlova	PROPN
ijassa-2042	264	11	,	,	PUNCT
ijassa-2042	264	12	a.	a.	NOUN
ijassa-2042	264	13	remizov	remizov	PROPN
ijassa-2042	264	14	11	11	NUM
ijassa-2042	264	15	.	.	PUNCT
ijassa-2042	265	1	lie	lie	NOUN
ijassa-2042	265	2	,	,	PUNCT
ijassa-2042	265	3	s.	s.	PROPN
ijassa-2042	265	4	(	(	PUNCT
ijassa-2042	265	5	1875	1875	NUM
ijassa-2042	265	6	)	)	PUNCT
ijassa-2042	265	7	foundations	foundation	NOUN
ijassa-2042	265	8	of	of	ADP
ijassa-2042	265	9	an	an	DET
ijassa-2042	265	10	invariant	invariant	ADJ
ijassa-2042	265	11	theory	theory	NOUN
ijassa-2042	265	12	of	of	ADP
ijassa-2042	265	13	contact	contact	NOUN
ijassa-2042	265	14	transformations	transformation	NOUN
ijassa-2042	265	15	,	,	PUNCT
ijassa-2042	265	16	math	math	NOUN
ijassa-2042	265	17	.	.	PUNCT
ijassa-2042	266	1	ann	ann	PROPN
ijassa-2042	266	2	.	.	PROPN
ijassa-2042	266	3	,	,	PUNCT
ijassa-2042	266	4	8	8	NUM
ijassa-2042	266	5	,	,	PUNCT
ijassa-2042	266	6	215–303	215–303	NUM
ijassa-2042	266	7	.	.	PUNCT
ijassa-2042	267	1	12	12	NUM
ijassa-2042	267	2	.	.	PUNCT
ijassa-2042	268	1	mitrinović	mitrinović	ADJ
ijassa-2042	268	2	,	,	PUNCT
ijassa-2042	268	3	d.	d.	PROPN
ijassa-2042	268	4	s.	s.	PROPN
ijassa-2042	268	5	,	,	PUNCT
ijassa-2042	268	6	kečkić	kečkić	PROPN
ijassa-2042	268	7	,	,	PUNCT
ijassa-2042	268	8	j.	j.	PROPN
ijassa-2042	268	9	d.	d.	PROPN
ijassa-2042	268	10	(	(	PUNCT
ijassa-2042	268	11	1981	1981	NUM
ijassa-2042	268	12	)	)	PUNCT
ijassa-2042	268	13	variations	variation	NOUN
ijassa-2042	268	14	and	and	CCONJ
ijassa-2042	268	15	generalisations	generalisation	NOUN
ijassa-2042	268	16	of	of	ADP
ijassa-2042	268	17	clairaut	clairaut	PROPN
ijassa-2042	268	18	’s	’s	PART
ijassa-2042	268	19	equations	equation	NOUN
ijassa-2042	268	20	,	,	PUNCT
ijassa-2042	268	21	univ	univ	PROPN
ijassa-2042	268	22	.	.	PUNCT
ijassa-2042	269	1	beograd	beograd	PROPN
ijassa-2042	269	2	.	.	PUNCT
ijassa-2042	270	1	publ	publ	PROPN
ijassa-2042	270	2	.	.	PUNCT
ijassa-2042	271	1	elektrotehn	elektrotehn	NOUN
ijassa-2042	271	2	.	.	PUNCT
ijassa-2042	272	1	fak	fak	PROPN
ijassa-2042	272	2	.	.	PUNCT
ijassa-2042	272	3	ser	ser	PROPN
ijassa-2042	272	4	.	.	PROPN
ijassa-2042	273	1	mat	mat	PROPN
ijassa-2042	273	2	.	.	PUNCT
ijassa-2042	274	1	fiz	fiz	PROPN
ijassa-2042	274	2	.	.	PROPN
ijassa-2042	274	3	,	,	PUNCT
ijassa-2042	274	4	716/734	716/734	NUM
ijassa-2042	274	5	,	,	PUNCT
ijassa-2042	274	6	11–21	11–21	NUM
ijassa-2042	274	7	.	.	NOUN
ijassa-2042	275	1	13	13	NUM
ijassa-2042	275	2	.	.	X
ijassa-2042	275	3	petitot	petitot	PROPN
ijassa-2042	275	4	,	,	PUNCT
ijassa-2042	275	5	j.	j.	PROPN
ijassa-2042	275	6	(	(	PUNCT
ijassa-2042	275	7	2003	2003	NUM
ijassa-2042	275	8	)	)	PUNCT
ijassa-2042	275	9	the	the	DET
ijassa-2042	275	10	neurogeometry	neurogeometry	NOUN
ijassa-2042	275	11	of	of	ADP
ijassa-2042	275	12	pinwheels	pinwheel	NOUN
ijassa-2042	275	13	as	as	ADP
ijassa-2042	275	14	a	a	DET
ijassa-2042	275	15	sub	sub	ADJ
ijassa-2042	275	16	-	-	ADJ
ijassa-2042	275	17	riemannian	riemannian	ADJ
ijassa-2042	275	18	contact	contact	NOUN
ijassa-2042	275	19	structure	structure	NOUN
ijassa-2042	275	20	,	,	PUNCT
ijassa-2042	275	21	j.	j.	PROPN
ijassa-2042	275	22	physiol	physiol	PROPN
ijassa-2042	275	23	.	.	PUNCT
ijassa-2042	276	1	paris	paris	PROPN
ijassa-2042	276	2	,	,	PUNCT
ijassa-2042	276	3	97	97	NUM
ijassa-2042	276	4	,	,	PUNCT
ijassa-2042	276	5	265–309	265–309	NUM
ijassa-2042	276	6	.	.	PUNCT
ijassa-2042	277	1	copyright	copyright	NOUN
ijassa-2042	277	2	©	©	PROPN
ijassa-2042	277	3	2025	2025	NUM
ijassa-2042	277	4	assa	assa	NOUN
ijassa-2042	277	5	.	.	PUNCT
ijassa-2042	278	1	adv	adv	PROPN
ijassa-2042	278	2	syst	syst	PROPN
ijassa-2042	278	3	sci	sci	PROPN
ijassa-2042	278	4	appl	appl	PROPN
ijassa-2042	278	5	(	(	PUNCT
ijassa-2042	278	6	2025	2025	NUM
ijassa-2042	278	7	)	)	PUNCT
ijassa-2042	278	8	curves	curve	NOUN
ijassa-2042	278	9	and	and	CCONJ
ijassa-2042	278	10	singularities	singularity	NOUN
ijassa-2042	278	11	legendre	legendre	PROPN
ijassa-2042	278	12	transformation	transformation	PROPN
ijassa-2042	278	13	legendrian	legendrian	PROPN
ijassa-2042	278	14	lift	lift	PROPN
ijassa-2042	278	15	legendre	legendre	PROPN
ijassa-2042	278	16	transformation	transformation	NOUN
ijassa-2042	278	17	and	and	CCONJ
ijassa-2042	278	18	duality	duality	NOUN
ijassa-2042	278	19	clairaut	clairaut	PROPN
ijassa-2042	278	20	differential	differential	ADJ
ijassa-2042	278	21	equation	equation	NOUN
ijassa-2042	278	22	contact	contact	NOUN
ijassa-2042	278	23	transformations	transformation	NOUN
ijassa-2042	278	24	conclusion	conclusion	NOUN