❆◆❚Ô◆■❖ ❋❆❇■❆◆❖ P❆■❱❆
❱❖▲❯▼❊ ❊ ➪❘❊❆ ❉❊ ❙Ó▲■❉❖❙ ●❊❖▼➱❚❘■❈❖❙ ❯❙❆◆❉❖ ❖ P❘■◆❈❮P■❖ ❉❊ ❈❆❱❆▲■❊❘■
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳
❱■➬❖❙❆
11/6/2015 FichaCatalografica :: Fichacatalografica
https://www3.dti.ufv.br/bbt/ficha/Fichacatalografica/aprovada/201505280328029483 2/3
Ficha catalográfica preparada pela Biblioteca Central da Universidade Federal de Viçosa - Campus Viçosa
T
Paiva, Antônio Fabiano, 1971-P149v
2015
Volume e área de sólidos geométricos usando o Princípio de Cavalieri / Antônio Fabiano Paiva. - Viçosa, MG, 2015.
x, 67f. : il. (algumas color.) ; 29 cm. Orientador: Mércio Botelho Faria.
Dissertação (mestrado) - Universidade Federal de Viçosa. Referências bibliográficas: f.66-67.
1. Geometria sólida. 2. Princípio de Cavalieri (Matemática). 3. Cálculo de volume. I. Universidade Federal de Viçosa. Departamento de Matemática. Programa de Pós-graduação em Matemática. II. Título.
❆◆❚Ô◆■❖ ❋❆❇■❆◆❖ P❆■❱❆
❱❖▲❯▼❊ ❊ ➪❘❊❆ ❉❊ ❙Ó▲■❉❖❙ ●❊❖▼➱❚❘■❈❖❙ ❯❙❆◆❉❖ ❖ P❘■◆❈❮P■❖ ❉❊ ❈❆❱❆▲■❊❘■
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳
❆P❘❖❱❆❉❆✿ ✼ ❞❡ ❛❜r✐❧ ❞❡ ✷✵✶✺✳
▼❛r✐♦ ❏♦sé ❞❡ ❙♦✉③❛ ❲❛❧t❡r ❍✉❛r❛❝❛ ❱❛r❣❛s
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣♦r t♦❞♦s ♦s ♠❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ❢♦rç❛ ♣❛r❛ ❝♦♥s❡❣✉✐r ❧✉t❛r✦
➚ ▲❡♥✐✱ ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛✱ ♣❡❧♦ ❛♣♦✐♦✱ ❝♦♠♣r❡❡♥sã♦✱ ❢♦rç❛ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦ ❛♠♦r ❝❡❞✐❞♦ ❛ ♠✐♠ ❞✉r❛♥t❡ t♦❞♦ ❡st❡ t❡♠♣♦✳
❆♦ ♠❡✉ ✜❧❤♦✱ ●❛❜r✐❡❧✱ ♣❡❧♦ ❝❛r✐♥❤♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❞❛❞♦ ❛ ♠✐♠ ❞✉r❛♥t❡ ♦s ♠❡✉s ❡st✉❞♦s✳ ❆♦s ♠❡✉s ♣❛✐s✱ ■✈❛♥ ❡ ❊❞♥❛✱ ♣❡❧❛s ♦r❛çõ❡s✱ ❝♦♥s❡❧❤♦s ❞✉r❛♥t❡ ❛s ♠✐♥❤❛s ✈✐❛❣❡♥s ❡ ❛ ♠✐♥❤❛ ❡st❛❞✐❛ ❡♠ ❱✐ç♦s❛✳
❆ ♠✐♥❤❛ ✐r♠ã✱ ❋❛❜✐❡❧❡✱ q✉❡ ♠❡s♠♦ ❞✐st❛♥t❡ s❡✐ q✉❡ t♦r❝❡✉ ❡ ♠❡ ❛❥✉❞♦✉ ♠❡♥t❛❧♠❡♥t❡ ❡ ❞♦ s❡✉ ❥❡✐t♦✳
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❢♦rç❛ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ✉♠❛ ❞✐✈❡rsã♦ s❡♠ t❛♠❛♥❤♦ ❡♥q✉❛♥t♦ ❡st✉❞á✈❛♠♦s✳
❆♦s ♠❡✉s ♥♦✈♦s ❛♠✐❣♦s ❆♥❞ré✱ ❉❛♥✐❡❧✱ ❊❧✐❛s✱ ●✐❧❜❡rt♦ ❡ ❱✐✈✐❛♥❡✱ ♦s q✉❛✐s ❢♦r♠❛♠♦s ❛♦ ♥♦ss♦ ❥❡✐t♦ ✉♠❛ ✈❡r❞❛❞❡✐r❛ ✐r♠❛♥❞❛❞❡✳
❆s ❡s❝♦❧❛s ❛s q✉❛✐s tr❛❜❛❧❤❡✐ ❞✉r❛♥t❡ ♦s ❞♦✐s ❛♥♦s ❞❡ ❡st✉❞♦ ♣♦r ❝♦♠♣r❡❡♥❞❡r❡♠ ❛s ♠✐♥❤❛s ♥❡❝❡ss✐❞❛❞❡s ❡ ♦r❣❛♥✐③❛çõ❡s ❞❡ ❤♦rár✐♦s✳
❆♦ ♣r♦❢❡ss♦r✱ ❉♦✉t♦r ▼ér❝✐♦ ❇♦t❡❧❤♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛✱ ❡♥s✐♥❛♠❡♥t♦s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✳
➚ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❙✉♠ár✐♦
❘❡s✉♠♦ ✈✐
❆❜str❛❝t ✈✐✐
■♥tr♦❞✉çã♦ ✶
✶ ❈♦♥t❛♥❞♦ ✉♠ ♣♦✉❝♦ ❞❡ ❍✐stór✐❛ ✸
✷ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❡ ♦ ❊◆❊▼ ✾ ✷✳✶ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❊◆❊▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷✳✶ ❊■❳❖❙ ❈❖●◆■❚■❱❖❙ ✭❝♦♠✉♥s ❛ t♦❞❛s ❛s ár❡❛s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦✮ ✳ ✳ ✶✷ ✷✳✷✳✷ ❈♦♠♣❡tê♥❝✐❛ ❞❡ ár❡❛ ✷ ✲ ❯t✐❧✐③❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ♣❛r❛ r❡✲
❛❧✐③❛r ❛ ❧❡✐t✉r❛ ❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ r❡❛❧✐❞❛❞❡ ❡ ❛❣✐r s♦❜r❡ ❡❧❛✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷✳✸ ❈♦♠♣❡tê♥❝✐❛ ❞❡ ár❡❛ ✸ ✲ ❈♦♥str✉✐r ♥♦çõ❡s ❞❡ ❣r❛♥❞❡③❛s ❡ ♠❡❞✐❞❛s ♣❛r❛
❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ r❡❛❧✐❞❛❞❡ ❡ ❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞♦ ❝♦t✐❞✐❛♥♦✳ ✳ ✶✸
✸ ◆♦çõ❡s ■♥✐❝✐❛✐s ❛ r❡s♣❡✐t♦ ❞♦s ♣♦❧✐❡❞r♦s ❞❡ P❧❛tã♦ ❡ ❛❧❣✉♥s só❧✐❞♦s ✶✹ ✸✳✶ P♦❧✐❡❞r♦s ❊s♣❡❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✶✳✶ Pr✐s♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✶✳✷ P✐râ♠✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✷ ❙ó❧✐❞♦s ❢♦r♠❛❞♦s ♣♦r s✉♣❡r❢í❝✐❡s ♥ã♦ ♣❧❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
❙❯▼➪❘■❖ ✐✈
✸✳✷✳✶ ❈✐❧✐♥❞r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷✳✷ ❈♦♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✷✳✸ ❊s❢❡r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✸ ➪r❡❛s ❡ ✈♦❧✉♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✹ ❊①❡♠♣❧♦s ❞❡ P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✹ ❆ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❛ r❡s♣❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠
♣❛r❛❧❡❧❡♣í♣❡❞♦ ✷✵
✺ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✷✹
✺✳✶ Pr♦❝❡❞✐♠❡♥t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✻ ❯s♦ ❞❡ r❡❝✉rs♦s ❡❧❡trô♥✐❝♦s ❡ s♦❢t✇❛r❡s ♥♦ ❡♠♣r❡❣♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✲
✈❛❧✐❡r✐ ✷✽
✼ ❯s♦ ❞❡ ♠❛t❡r✐❛❧ ✏❝♦♥❝r❡t♦✑ ♥♦ ❡st✉❞♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✸✺
✽ ❉❡♠♦♥str❛çõ❡s ❡ ❡①❡♠♣❧✐✜❝❛çõ❡s ✐♠♣♦rt❛♥t❡s ✸✽
✽✳✶ ❉❡♠♦♥str❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s✱ ✉s❛♥❞♦ r❡❝✉rs♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❝♦♠♦ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ ❞♦❝❡♥t❡s✳ ✳ ✸✽ ✽✳✷ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ✜❣✉r❛s ♣❧❛♥❛s ✲ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✽✳✸ ➪r❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ✉♠❛ ❝✉r✈❛y=x2
❡ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✳ ✹✵ ✽✳✸✳✶ ❈♦♥❝❡✐t♦s ❡ ✐♥❢♦r♠❛çõ❡s ♣r❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✽✳✹ ❱♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ q✉❛❧q✉❡r ✲ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✽✳✺ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r ✲ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✽✳✻ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ q✉❛❧q✉❡r ✭♥ã♦ s❡♥❞♦ tr✐❛♥❣✉❧❛r✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✽✳✼ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✾ Pr♦♣♦st❛ ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❞✐❞át✐❝♦ ❝♦♠ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♥♦ ❊♥s✐♥♦
✈
✾✳✶ ❆t✐✈✐❞❛❞❡s ❛❧t❡r♥❛t✐✈❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✾✳✶✳✶ ❆t✐✈✐❞❛❞❡ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✾✳✶✳✷ ❆t✐✈✐❞❛❞❡ ✲ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✶✵ ❘❡s♦❧✉çã♦ ❞♦s ❡①❡r❝í❝✐♦s t❡ór✐❝♦s ❞♦ ❝❛♣ít✉❧♦ ✾✳ ✺✻
❈♦♥❝❧✉sã♦ ✻✺
❘❡s✉♠♦
P❆■❱❆✱ ❆♥tô♥✐♦ ❋❛❜✐❛♥♦✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❛❜r✐❧ ❞❡ ✷✵✶✺✳ ❱♦❧✉♠❡ ❡ ➪r❡❛ ❞❡ ❙ó❧✐❞♦s ●❡♦♠étr✐❝♦s ❯s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳ ❖r✐❡♥t❛❞♦r✿ ▼ér❝✐♦ ❇♦t❡❧❤♦ ❋❛r✐❛✳
◆❡st❛ ❞✐ss❡rt❛çã♦ ❡st✉❞❛♠♦s ❡ ❡❧❛❜♦r❛♠♦s ❛❧❣✉♠❛s ♣r♦♣♦st❛s ❞❡ ❛t✐✈✐❞❛❞❡s ❡❞✉❝❛❝✐♦♥❛✐s q✉❡♠ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ❜❡♠ ❝♦♠♦ ♦ s❡✉ ✉s♦ ❡ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❢❡✐t♦ ❝♦♠ r❡❝✉rs♦s ❡❧❡trô♥✐❝♦s✱ t❡♥❞♦ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛ ✸❉ ❡ ♠❛t❡r✐❛❧ ❝♦♥❝r❡t♦✳ ❖ ♥♦ss♦ tr❛❜❛❧❤♦ ❝♦♥st❛ ❞❡ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ t❡ór✐❝❛✱ ❜❡♠ ❝♦♠♦ ✉♠❛ r❡✈✐sã♦ ❜✐❜❧✐♦❣rá✜❝❛ ❞❡ ❝♦♥❝❡✐t♦s q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ r❡s♦❧✉çã♦ ❞❡ ❡①❡r❝í❝✐♦s ❡ ♣r♦❜❧❡♠❛s ♣rát✐❝♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ♥♦ ❝♦t✐❞✐❛♥♦✱ ♣♦❞❡♥❞♦ s❡r ❡❧❛❜♦r❛❞♦s ♣❛r❛ q✉❛❧q✉❡r ♥❡❝❡ss✐❞❛❞❡ q✉❡ ♦s ❡❞✉❝❛♥❞♦s ♣♦ss❛♠ t❡r ❡♠ s✉❛ ✈✐❞❛ ❡st✉❞❛♥✲ t✐❧✳ ❆♣r❡s❡♥t❛♠♦s✱ ❛ss✐♠✱ ❛t✐✈✐❞❛❞❡s q✉❡ ♣♦❞❡♠ s❡r✈✐r ❞❡ ❡①❡♠♣❧♦s ❡ ✐♥❝❡♥t✐✈♦ ❛ ❢✉t✉r♦s tr❛❜❛❧❤♦s ❝♦♠ ♦✉tr❛s ♣❛rt❡s ❞❛ ❣❡♦♠❡tr✐❛ ❡ ❡♠ ♦✉tr♦s r❛♠♦s ❞❛ ▼❛t❡♠át✐❝❛✳
❆❜str❛❝t
P❆■❱❆✱ ❆♥tô♥✐♦ ❋❛❜✐❛♥♦✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❆♣r✐❧ ✷✵✶✺✳ ❱♦❧✉♠❡ ❛♥❞ ❆r❡❛ ♦❢ ●❡♦♠❡tr✐❝s ❙♦❧✐❞s ❯s✐♥❣ t❤❡ ❈❛✈❛❧✐❡r✐➫ s Pr✐♥❝✐♣❧❡ ❆❞✈✐s♦r✿ ▼ér❝✐♦ ❇♦t❡❧❤♦ ❋❛r✐❛✳
■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✇❡ st✉❞✐❡❞ ❛♥❞ ❝r❡❛t❡❞ s♦♠❡ ♣r♦♣♦s❛❧s ❢♦r ❡❞✉❝❛t✐♦♥❛❧ ❛❝t✐✈✐t✐❡s ✇❤✐❝❤ ✐♥✈♦❧✈❡ t❤❡ Pr✐♥❝✐♣❧❡ ♦❢ ❈❛✈❛❧✐❡r✐✱ ❛s ✇❡❧❧ ❛s ✐ts ✉s❡ ❛♥❞ ❛♣♣❧✐❝❛❜✐❧✐t② ❞♦♥❡ ✇✐t❤ ❡❧❡❝tr♦♥✐❝ r❡s♦✉r❝❡s✱ ❤❛✈✐♥❣ ❛s t❤❡ ♠❛✐♥ t♦♦❧ ●❡♦❣❡❜r❛ ✸❉ s♦❢t✇❛r❡ ❛♥❞ ❝♦♥❝r❡t❡ ♠❛t❡r✐❛❧s✳ ❖✉r ✇♦r❦ ❝♦♥s✐sts ♦❢ ❛ t❤❡♦r❡t✐❝❛❧ ♣r❡s❡♥t❛t✐♦♥✱ ❛♥❞ ❛ ❧✐t❡r❛t✉r❡ r❡✈✐❡✇ ♦❢ ❝♦♥❝❡♣ts t❤❛t ❛r❡ ✐♠♣♦rt❛♥t ❢♦r ♦❜t❛✐♥✐♥❣ ♥❡✇ ❦♥♦✇❧❡❞❣❡ ❛♥❞ s♦❧✈✐♥❣ ♣r❛❝t✐❝❛❧ ❡①❡r❝✐s❡s ❛♥❞ ♣r♦❜❧❡♠s ✐♥ t❤❡ ❝❧❛ssr♦♦♠ ❛♥❞ ✐♥ ❞❛✐❧② ❧✐❢❡✳ ❚❤❡s❡ ❡①❡r❝✐s❡s ❝❛♥ ❜❡ ♣r❡♣❛r❡❞ ❢♦r ❛♥② ♥❡❡❞ t❤❛t t❤❡ st✉❞❡♥ts ❝❛♥ ❤❛✈❡ ✐♥ t❤❡✐r st✉❞❡♥t ❧✐❢❡✳ ❚❤✐s ✇❛②✱ ✇❡ ♣r❡s❡♥t ❛❝t✐✈✐t✐❡s t❤❛t ❝❛♥ s❡r✈❡ ❛s ❡①❛♠♣❧❡s ❛♥❞ ❡♥❝♦✉✲ r❛❣❡♠❡♥t t♦ ❢✉t✉r❡ ✇♦r❦ ✇✐t❤ ♦t❤❡r ♣❛rts ♦❢ ❣❡♦♠❡tr② ❛♥❞ ♦t❤❡r ❜r❛♥❝❤❡s ♦❢ ♠❛t❤❡♠❛t✐❝s✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❇♦❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐ ✲ ❋♦♥t❡✿ ♠❛t❤❡✉s♠❛t❤✐❝❛✳❜❧♦❣s♣t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❆♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ▲❡♦♥❤❛r❞ P❛✉❧ ❊✉❧❡r ✲ ❋♦♥t❡✿ ✇✇✇✳❡①♣❧✐❝❛t♦r✐✉♠✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✸✳✶ P✐râ♠✐❞❡s ❞♦ ❊❣✐t♦ ✲ ❋♦♥t❡✿ ✇✇✇✳❛s❝♦♥✈❡rs❛✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✷ ❈✐❧✐♥❞r♦ ❈✐r❝✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✸ ❈♦♥❡ ✲ ❋♦♥t❡✿ ❧♦❛❞✐♥❣✐♥❢♦r♠❛t✐♦♥s✳❜❧♦ ❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✹ ❊s❢❡r❛ ❞❡ r❡✈♦❧✉çã♦ ✲ ❋♦♥t❡✿ ✇✇✇✳♠✉♥❞♦❡❞✉❝❛❝❛♦✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✺ P♦❧✐❡❞r♦s ✲ ❋♦♥t❡✿ ❤tt♣✿✴✴✇✇✇✳❡❞✉❝✳❢❝✳✉❧✳♣t✴✐❝♠✴✐❝♠✾✾✴✐❝♠✶✻✴❝✉r✐♦s✐❞❛❞❡s✳❤t♠ ✶✾
✹✳✶ P❛r❛❧❡❧❡♣í♣❡❞♦ ❝♦♠ ❛r❡st❛s ✐❣✉❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✷ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s1✱ 1❡ a ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✸ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛sa✱1 ❡ b ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✹ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛sa✱b ❡c ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✺ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛sa✱b ❡c ✲ ❋♦♥t❡✿ ❛♥♦ss❛❡s❝♦❧❛✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✻ ❘♦♠❜♦❡❞r♦s ✲ ❋♦♥t❡✿ ✇✇✇✳♣♦rt❛❧❡s❝♦❧❛r✳♥❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✼ P❛r❛❧❡❧❡♣í♣❡❞♦ ✲ ❋♦♥t❡✿ ♠❛t❡♠❛t✐❝❛❝♦❡s✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✺✳✶ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛s ✲ ❋♦♥t❡✿ ♠❛✉r♦✇❡✐❣❡❧✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✺✳✷ ❋✐❣✉r❛s P❧❛♥❛s ✲ ❋♦♥t❡✿ ▼❛t❡♠át✐❝❛ ▼✉❧t✐♠í❞✐❛✱ ●✉✐❛ ❞♦ Pr♦❢❡ss♦r✱ ●❡♦♠❡✲
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✐①
✺✳✸ ❙ó❧✐❞♦s ✲ ❋♦♥t❡✿♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✺✳✹ ▼♦❡❞❛s r❡♣r❡s❡♥t❛♥❞♦ s❡❝çõ❡s✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✷✻ ✺✳✺ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ❈❛✈❛❧✐❡r✐ ■ ✲ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✷✼ ✺✳✻ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ❈❛✈❛❧✐❡r✐ ■■ ✲ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✷✼
✻✳✶ P✐râ♠✐❞❡ ❲✐♥❣❡♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✻✳✷ ❊❞✐çã♦ ❞❛ P✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✻✳✸ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✻✳✹ P♦❧í❣♦♥♦ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✻✳✺ P❛r❛❧❡❧❡♣í♣❡❞♦ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✻✳✻ ❊s❢❡r❛ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✻✳✼ ❉❡❝á❣♦♥♦ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✻✳✽ Pr✐s♠❛ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ✉♠ ❞❡❝á❣♦♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✻✳✾ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✻✳✶✵ P❧❛♥✐✜❝❛çã♦ ❞♦ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✻✳✶✶ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ♥♦ P♦❧② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✻✳✶✷ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ❘❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✼✳✶ P✐râ♠✐❞❡ ❞❡ ❇❛s❡ ◗✉❛❞r❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✼✳✷ P✐râ♠✐❞❡ ❞❡ ❇❛s❡ ❍❡①❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✼✳✸ ❙ó❧✐❞♦s ❞❡ ◆❛t✉r❡③❛ ❉✐❢❡r❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✽✳✶ ❘❡❣✐ã♦ ❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✽✳✷ ❘❡❣✐ã♦ ◗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✽✳✸ ➪r❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✽✳✹ ➪r❡❛ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛ ❝✉r✈❛ y=x2
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①
✽✳✻ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ tr✐❛♥❣✉❧❛r ✲ ❋♦♥t❡✿ ❊❧♦♥ ▲❛❣❡s ▲✐♠❛ ✲ ▼❡❞✐❞❛ ❡ ❋♦r♠❛ ❡♠ ●❡♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✽✳✼ P❡♥tá❣♦♥♦ ❡♠n−2tr✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✽✳✽ P✐râ♠✐❞❡ P❡♥t❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✽✳✾ ❈❧❡♣s✐❞r❛ ❡ ❆♥t✐❝❧❡♣s✐❞r❛✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✹✺ ✽✳✶✵ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✾✳✶ ❊♥❡♠ ✷✵✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✾✳✷ ❯❋❯ ✲ ▼● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✾✳✸ ❯❋❘❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✾✳✹ ❊◆❊▼✱ ✷✵✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✾✳✺ ❈✐❧✐♥❞r♦ ✐♥s❝r✐t♦✳ ❋♦♥t❡✿ ✇✇✇✳♠❛❡str♦♣❧♦♠❡r♦✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✾✳✻ ❙❡♠✐❡s❢❡r❛✳ ❋♦♥t❡✿ ✇✇✇✳♠❛❡str♦♣❧♦♠❡r♦✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✾✳✼ ❊①tr❛t♦ ❞❡ ❚♦♠❛t❡✿ ❘❡❛❧✐❞❛❞❡❀ ❈✐❧✐♥❞r♦✿ ▼♦❞❡❧♦ ▼❛t❡♠át✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
■♥tr♦❞✉çã♦
❆♥❛❧✐s❛♥❞♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭P❈◆✬s✮✱ ✈❡r✐✜❝❛♠♦s q✉❡ ♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞❡✈❡♠ t❡r ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣❧❡♥♦ ❛ r❡s♣❡✐t♦ ❞❛ ●❡♦♠❡tr✐❛ P❛♥❛ ❡ ❊s♣❛❝✐❛❧ ❡ s❡r❡♠ ❝❛♣❛③❡s ❞❡ r❡❧❛❝✐♦♥❛r ❝♦♥❝❡✐t♦s ❞❡st❛s ❞✉❛s ♣❛rt❡s ❛❧é♠ ❞❡ q✉❡st✐♦♥❛r ❛ r❡❛❧✐❞❛❞❡✱ ✉t✐❧✐③❛♥❞♦ ♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦✱ ❛ ❝r✐❛t✐✈✐❞❛❞❡ ❡ t❛♠❜é♠ ❛ t❡❝♥♦❧♦❣✐❛✳ P♦❞❡♠♦s ♣❡r❝❡❜❡r✱ q✉❡ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡♥❡r❛❧✐③❛❞❛✱ ❛ ▼❛t❡♠át✐❝❛✱ ❛ss✐♠ ❝♦♠♦ ✉♠❛ ❞❡ s✉❛s r❛♠✐✜❝❛çõ❡s q✉❡ é ❛ ●❡♦♠❡tr✐❛✱ ❛♣r❡s❡♥t❛ ✉♠❛ ❣r❛♥❞❡ r❡❥❡✐çã♦ ❥✉♥t♦ ❛♦s ❛❧✉♥♦s✳ P♦❞❡ ✲ s❡ ❡①♣❧✐❝❛r ❡ss❡ ❢❛t♦ ❞❡✈✐❞♦ ❛♦ ❡①❝❡ss✐✈♦ ✉s♦ ❞❡ ❢ór♠✉❧❛s ♣r❡❞❡t❡r♠✐♥❛❞❛s ❡ ❛♦ ❡♥s✐♥♦ ♠❡❝â♥✐❝♦ ❝♦♠ r❡s♦❧✉çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s s❡♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ♥♦ ❝♦t✐❞✐❛♥♦ ❡ s❡♠ ✉♠ ❛tr❛t✐✈♦ q✉❡ ♠✉✐t❛s ✈❡③❡s ❡stá ♥♦ ♣ró♣r✐♦ ❞❡s❡♥❤♦ ♦✉ ♥❛ r❡❧❛çã♦ ❝♦♠ ♦ ❞✐❛ ❛ ❞✐❛ ❞♦s ❞✐s❝❡♥t❡s✳ ❯♠ ❞♦s ♣r✐♥❝✐♣❛✐s ❞❡s❛✜♦s ❞♦ ♣r♦❢❡ss♦r é ❜✉s❝❛r ✉♠❛ ❢♦r♠❛ ✐♥♦✈❛❞♦r❛ ♣❛r❛ ✉♠ ❡♥s✐♥♦ ♠❛✐s ❛tr❛t✐✈♦ ❡ q✉❡ ❡st✐♠✉❧❡ ♦s ❛❧✉♥♦s ❛ ❜✉s❝❛r❡♠ r❡s✉❧t❛❞♦s✱ ✈✐s✉❛❧✐③❛♥❞♦ ♦s ♠❡s♠♦s ❡ t❛♠❜é♠ ❝♦♥str✉✐♥❞♦ s✐t✉❛çõ❡s ♥❛s q✉❛✐s ♦ ❛♣r❡♥❞✐③❛❞♦ s❡ t♦r♥❡ ♠❛✐s s✐❣♥✐✜❝❛t✐✈♦✳ ❆ss✐♠✱ ❛ ❡s❝♦❧❤❛ ❞♦s ♠❡❝❛♥✐s♠♦s ❡ r❡❝✉rs♦s ❞✐❞át✐❝♦s ❜❡♠ ❝♦♠♦ ❛ ♠❛♥❡✐r❛ ❞❡ ♠❛♥✐♣✉❧❛r t❛✐s r❡❝✉rs♦s é ❡①tr❡♠❛♠❡♥t❡ ✐♠♣♦rt❛♥t❡✳
P❛r❛ ❡❢❡t✐✈❛♠❡♥t❡ r❡❛❧✐③❛r♠♦s ♦ ❡♥s✐♥♦ ❞❛ ●❡♦♠❡tr✐❛ ❡ ❣❡r❛r♠♦s ✉♠❛ ❛♣r❡♥❞✐③❛❣❡♠ s✐❣♥✐✜❝❛t✐✈❛✱ ❞❡✈❡♠♦s ❡st❛r ❛t❡♥t♦s ❛♦s ❥á ❡①✐st❡♥t❡s ♥♦s ❛❥✉❞❛♠ ❛ ❛♣r❡s❡♥t❛r ❡ ❥✉st✐✜❝❛r ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ♥ã♦ ♠✉✐t♦ ✉t✐❧✐③❛❞❛s ♥♦ ❞✐❛ ❛ ❞✐❛ ❞♦s ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s✳
❉✉r❛♥t❡ ♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✱ ✐r❡♠♦s ❛♣r❡s❡♥t❛r ❝♦♥str✉çõ❡s ❞❡ ❞❡t❡r♠✐♥❛❞♦s só❧✐❞♦s✱ ♦s ❝❤❛♠❛❞♦s só❧✐❞♦s ❞❡ P❧❛tã♦ ❡ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s q✉❡ s❡ ❢❛③❡♠ ✐♠♣♦rt❛♥t❡s✱ ❛❧é♠ ❞❡ ❛❧❣✉♥s ❡①❡r❝í❝✐♦s ♥♦s q✉❛✐s ❛♣❧✐❝❛♠♦s ♦ q✉❡ ❢♦✐ ❛♣r❡s❡♥t❛❞♦✱ t✉❞♦ ✐st♦ ♠♦str❛♥❞♦ t❛♠❜é♠ ♦ ❡♠❜❛s❛♠❡♥t♦ ❞♦s P❈◆s ♣❛r❛ ❥✉st✐✜❝❛r t❛❧ tr❛❜❛❧❤♦ ❡ s✉❛ ❞❡✈✐❞❛ ✐♠♣♦rtâ♥❝✐❛✱ s❡♠ ❡sq✉❡❝❡r ♦ ❊◆❊▼✱ q✉❡ t❡♠ s✐❞♦ ❛❞♦t❛❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♣❛r❛ ♦ ❛❝❡ss♦ ❞❡ ❛❧✉♥♦s ♥❛s ✐♥st✐t✉✐çõ❡s ❞❡ ❡♥s✐♥♦ s✉♣❡r✐♦r✳
■♥tr♦❞✉çã♦ ✷
❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ♠❛♥❡✐r❛s ❛❧t❡r♥❛t✐✈❛s ❡ ♠❛♥❡✐r❛s ✉s✉❛✐s ♣❛r❛ ♦ ❝á❧❝✉❧♦✱ ❛ ✈✐s✉❛❧✐✲ ③❛çã♦ ❞♦ ✈♦❧✉♠❡ ❡ ár❡❛s ❞❡ só❧✐❞♦s✱ ✉s❛♥❞♦ ❝♦♠♦ r❡❝✉rs♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧❧✐❡r✐✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r ❡ s♦❢t✇❛r❡s ❣rá✜❝♦s ❝♦♠♦ ♦ ●❡♦❣❡❜r❛✱ ❛❧é♠ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s✳
❈❛♣ít✉❧♦ ✶
❈♦♥t❛♥❞♦ ✉♠ ♣♦✉❝♦ ❞❡ ❍✐stór✐❛
❆ ❤✐stór✐❛ ♥♦s ❞✐③ q✉❡ ❞❡s❞❡ ♦s ♣r✐♠ór❞✐♦s ❞❛ ♦r❣❛♥✐③❛çã♦ ❤✉♠❛♥❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛s ♦r❣❛♥✐③❛çõ❡s ❛❣rí❝♦❧❛s✱ t❡♠♦s ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝á❧❝✉❧♦✱ ❛♥á❧✐s❡ ❡ ✉s♦ ❞❡ ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✳ ❊✉❝❧✐❞❡s ❥á ❝✐t❛✈❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❡♠ s❡✉s tr❛❜❛❧❤♦s✱ ♥♦ ❧✐✈r♦ ❳■■ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ ♦❜s❡r✈❛♠♦s ❛❧❣✉♥s t❡♦r❡♠❛s q✉❡ r❡❧❛❝✐♦♥❛♠ ❛ ✐❞❡✐❛ ❞❡ ✈♦❧✉♠❡ s❡♠ ♠❡s♠♦ ❞❡✜♥✐✲❧♦✳ ❖ tr❛❜❛❧❤♦ ❝♦♠ ✈♦❧✉♠❡s ❡ ár❡❛s s❡ ❞❡✉ ❞❡✈✐❞♦ ❛ ✉♠❛ ♥❡❝❡ss✐❞❛❞❡ ✜❧♦só✜❝❛ ❡ t❡♦❧ó❣✐❝❛✱ ❝♦♠♦ ♥♦s ❛♣❛r❡❝❡ ❡♠ ✈ár✐❛s s✐t✉❛çõ❡s ♥❛s ❝✐✈✐❧✐③❛çõ❡s ❣r❡❣❛s ❡ ❡♠ ❝✐✲ ✈✐❧✐③❛çõ❡s ♠❛✐s ❛♥t✐❣❛s✱ ♦♥❞❡ ❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ s✉❛s ár❡❛s ❡ s❡✉s ✈♦❧✉♠❡s ❡r❛♠ ✉s❛❞♦s ♣❛r❛ ❥✉st✐✜❝❛r s✐t✉❛çõ❡s ♥❛s q✉❛✐s ♥ã♦ s❡ ❞❡♣❡♥❞✐❛ ❛♣❡♥❛s ❞❛ ❝r❡♥ç❛ t❡♦❧ó❣✐❝❛ ♠❛s s✐♠ ❞❡ s✐t✉❛çõ❡s ❡♠ q✉❡ ♥ú♠❡r♦s ❡ ❢♦r♠❛s r❡♣r❡s❡♥t❛✈❛♠ ❛çõ❡s ❡ ♠❡❞✐❞❛s t♦♠❛❞❛s ♣♦r ❞✐✈✐♥❞❛❞❡s✳ ❖s ❝á❧❝✉❧♦s ❞❡ ár❡❛s ❡ ❞♦s ♣ró♣r✐♦s ✈♦❧✉♠❡s ❝✐t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡ ❡r❛♠ ❢❡✐t♦s✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ❞❡ ❢♦r♠❛ ❛♣r♦①✐♠❛❞❛✱ ♣♦ré♠ ♦s ♠ét♦❞♦s✱ ❢♦r❛♠ s❡♥❞♦ ❛♣❡r❢❡✐ç♦✲ ❛❞♦s ❛té q✉❡ ♥❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ♠❛✐s ❛✈❛♥ç❛❞❛s ❝♦♠♦ ♦s ❡❣í♣❝✐♦s ❡ ♦s ❣r❡❣♦s✱ ♦❜t❡✈❡✲s❡ ❢ór♠✉❧❛s ❡ ❛❜str❛çõ❡s ♠✉✐t♦ s♦✜st✐❝❛❞❛s✳ ❆❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❢ór♠✉❧❛s ❝♦♠♣❧❡①❛s ❡ ❡str✉t✉r❛❞❛s ♣❛r❛ ❛❧❣✉♥s só❧✐❞♦s ❡s♣❡❝í✜❝♦s ❝♦♠♦✿ ♣r✐s♠❛s✱ ♣✐râ♠✐❞❡s ❡ ❝✐❧✐♥❞r♦s✳ P♦❞❡♠♦s ❛✐♥❞❛ ♣❡♥s❛r ♣r✐♠✐t✐✈❛♠❡♥t❡ ❛❝❡r❝❛ ❞❡ ❞❡t❡r♠✐♥❛r ♦ ✈♦❧✉♠❡ ✉s❛♥❞♦ ✉♠❛ ✉♥✐❞❛❞❡ ❝♦♠♦ r❡✲ ❢❡rê♥❝✐❛✱ ❝♦♠♦ ✉♠ ❝✉❜♦ ✉♥✐tár✐♦✱ q✉❡ s❡r✈✐r✐❛ ❞❡ r❡❢❡rê♥❝✐❛ ♣❛r❛ s❛❜❡r♠♦s q✉❛♥t❛s ✈❡③❡s ✉♠ só❧✐❞♦ q✉❛❧q✉❡r ❝♦♥té♠ ❡ss❡ t❛❧ ❝✉❜♦ ✉♥✐tár✐♦✳ ❈♦♠♦ ❞✐t♦✱ ❢♦r♠❛ ❡①tr❡♠❛♠❡♥t❡ ♣r✐♠✐t✐✈❛✳
❖ ✐t❛❧✐❛♥♦✱ ♥❛s❝✐❞♦ ❡♠ ▼✐❧ã♦✱ ❋r❛♥❝❡s❝♦ ❇♦❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐✱ ❢♦✐ ✉♠ ❞♦s ♠❛t❡♠át✐❝♦s ♠❛✐s ✐♥✢✉❡♥t❡s ❞❡ s✉❛ é♣♦❝❛ ❡ ❢♦✐ ♦ ✐♥st✐❣❛❞♦r ❞❡ ✉♠ ♠ét♦❞♦ ♠❛✐s ❛♣✉r❛❞♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s ❡ ár❡❛s✱ ❑❡♣❧❡r ❢♦✐ ❞❡s❛✜❛❞♦ ❛ ❡♥❝♦♥tr❛r ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ ❛❧❣✉♠❛s ✜❣✉r❛s ❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✱ ✐♥❞✐❝❛♥❞♦ ❛ss✐♠ ✉♠ ❡stí♠✉❧♦ ♣❛r❛ q✉❡ ❈❛✈❛❧✐❡r✐ ❡st✉❞❛ss❡ ✉♠❛ ♠❛✲ ♥❡✐r❛ ♣❛r❛ ❡❢❡t✉❛r t❛✐s ❝á❧❝✉❧♦s ❝♦♠♦ ❝✐t❛❞♦ ❡♠ ❬✽❪✳ ❈❛✈❛❧✐❡r✐ ❢♦✐ ✉♠ ❞✐s❝í♣✉❧♦ ❞❡ ●❛❧✐❧❡✉ ●❛❧✐❧❡✐✱ s❡♥❞♦ ❡st❡ q✉❡♠ ♦ ✐♥❞✐❝♦✉ ♣❛r❛ ♦❝✉♣❛r ✉♠❛ ❝át❡❞r❛ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇♦❧♦♥❤❛ ❞❡ ✶✻✷✾ ❛ ✶✻✹✼✱ ❛♥♦ ❞❡ s✉❛ ♠♦rt❡✳ ❊❧❡ ♦❝✉♣❛✈❛ ♣❛r❛❧❡❧❛♠❡♥t❡ ❛♦ ❝❛r❣♦ ❞❡ ♣r♦❢❡ss♦r ✉♠❛ ❛t✐✈✐❞❛❞❡ r❡❧✐❣✐♦s❛ ♥♦ ♠♦♥❛stér✐♦ ❞❡ ❙ã♦ ❏❡rô♥✐♠♦✱ ♣♦✐s ❡r❛ ✉♠ ❥❡s✉❛❞♦ ❡ ♥ã♦ ✉♠ ❥❡s✉ít❛
✹
❝♦♠♦ é ❛✜r♠❛❞♦✱ ❡①❡r❝❡♥❞♦ t❛♠❜é♠ ✉♠❛ ❛t✐✈✐❞❛❞❡ ❝♦♠♦ ❛strô♥♦♠♦✳ ❊st❛❜❡❧❡❝❡✉ ❡♥tã♦ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝♦♥❝❧✉sã♦ ❛ r❡s♣❡✐t♦ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦✱ ❛tr❛✈és ❞❡ ♣❛rt❡s ❞♦ ♠❡s♠♦✱ q✉❡ sã♦ s❡❝çõ❡s ♦❜t✐❞❛s ❛tr❛✈és ❞❡ ❝♦rt❡s ❞♦ só❧✐❞♦ ♦ q✉❡ é ❝❤❛♠❛❞♦ ❤♦❥❡ ❞❡ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ t❡♠ ❝♦♠♦ ❜❛s❡ ♦ ♠ét♦❞♦ ❞♦s ✐♥❞✐✈✐sí✈❡✐s q✉❡ ❝♦♠❡ç♦✉ ❛ s❡ ❞❡s❡♥✈♦❧✈❡r ❛ ♣❛rt✐r ❞❡ ✶✽✷✻ ❡ q✉❡ ❢♦✐ ❞✐✈✉❧❣❛❞♦ ❛tr❛✈és ❞♦ ❧✐✈r♦ ●❡♦♠❡tr✐❛ ■♥❞✐✈✐s✐❜✐❧✐❜✉s ✭●❡♦♠❡tr✐❛ ❞♦s ■♥❞✐✈✐sí✈❡✐s✮✱ ♣✉❜❧✐❝❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ♥♦ ❛♥♦ ❞❡ ✶✻✸✺✳ ❆ ✐❞❡✐❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❈❛✈❛❧✐❡r✐ ♥ã♦ ❡r❛ ❞❡ t♦❞♦ ♥♦✈❛✱ ♣♦✐s ♦ ♣ró♣r✐♦ ●❛❧✐❧❡✉ ❥á ❤❛✈✐❛ t✐❞♦ ❛ ♥❡❝❡ss✐❞❛❞❡ ❡ ♦ ❡stí♠✉❧♦ ♣❛r❛ ♦ ❡♥❝♦♥tr♦ ❞♦s ✈♦❧✉♠❡s ❛❧é♠ ❞♦s ❣r❡❣♦s ❛♥t✐❣♦s q✉❡ t❛♠❜é♠ ✐♥st✐❣❛r❛♠ ❛ ❡♥❝♦♥tr❛r ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ✉s❛♥❞♦ t❛✐s ✐❞❡✐❛s ♣♦ré♠ ❛ ❞❡♠♦♥str❛çã♦ ❞❛s ♠❡s♠❛s ❡r❛♠ ♦❜t✐❞❛s ❞❡ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡ ♣❡❧♦ ❢❛t♦ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s s❡r❡♠ ❜❡♠ ❞✐❢❡r❡♥t❡s✳ ❖ ❜ás✐❝♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❝♦♥s✐st✐❛ ♥♦ s❡❣✉✐♥t❡✿ ❯♠❛ ✜❣✉r❛ ♣❧❛♥❛ s❡r✐❛ ❢♦r♠❛❞❛ ♣♦r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❝♦r❞❛s ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐ ❡ ✉♠❛ ✜❣✉r❛ só❧✐❞❛ ✭✉♠ só❧✐❞♦ ❣❡♦♠étr✐❝♦✮ s❡r✐❛ ❢♦r♠❛❞♦ ♣♦r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ s❡❝çõ❡s ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐✱ ♣♦r ✐ss♦ ❡ss❛s ❝♦r❞❛s ❡ ❡ss❛s s❡❝çõ❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ ✐♥❞✐✈✐sí✈❡✐s✳ ❊ss❛s ✐❞❡✐❛s ❛♣r❡s❡♥t❛❞❛s ♣♦r ❈❛✈❛❧✐❡r✐ s♦❜r❡ ♦s ✐♥❞✐✈✐sí✈❡✐s✱ ❡♥✈♦❧✈✐❛♠ ✉♠❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❡♥t❡♥❞✐♠❡♥t♦✱ ♣♦✐s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❧ó❣✐❝♦ ❝♦♠♦ ♣♦❞❡rí❛♠♦s ✐♠❛❣✐♥❛r ✉♠❛ ✜❣✉r❛ ✜♥✐t❛✱ ❧✐♠✐t❛❞❛✱ ❢♦r♠❛❞❛ ♣♦r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ✐♥❞✐✈✐sí✈❡✐s❄ ❊♠ q✉❛✐s s✐✲ t✉❛çõ❡s ♣♦❞❡♠♦s ✉s❛r ❡st❡ ❝♦♥❝❡✐t♦❄ ❖ ♠ét♦❞♦ ❞♦s ✐♥❞✐✈✐sí✈❡✐s ❡ ♦✉tr♦s ❡q✉✐✈❛❧❡♥t❡s ❛ ❡❧❡✱ ❢♦r❛♠ ❡❢❡t✐✈❛♠❡♥t❡ ✉s❛❞♦s ♣♦r ❚♦rr✐❝❡❧❧✐✱ ❋❡r♠❛t✱ P❛s❝❛❧✱ ❙❛✐♥t✲❱✐♥❝❡♥t✱ ❇❛rr♦✇ ❡ ♦✉tr♦s q✉❡ ♥♦ ❝✉rs♦ ❞❡ s❡✉ tr❛❜❛❧❤♦ ❝❤❡❣❛r❛♠ ❛ r❡s✉❧t❛❞♦s ❡q✉✐✈❛❧❡♥t❡s à ✐♥t❡❣r❛çã♦ ❞❡ ❡①♣r❡ssõ❡s ❝♦♠♦ xn✱senθ✱ sen2
θ ❡ θ.senθ✳
❖ tr❛t❛❞♦ ✐♥✐❝✐❛❧ ❞❡ ❈❛✈❛❧✐❡r✐ é ❞❡♠❛s✐❛❞❛♠❡♥t❡ ❝♦♥❢✉s♦✱ s❡♥❞♦ ❞❡ ❞✐❢í❝✐❧ ❡♥t❡♥❞✐♠❡♥t♦✱ ♣♦ré♠ s❡ ❡st✉❞❛❞♦ ❝♦♥s❡❣✉❡✲s❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❝♦❧♦❝❛çã♦ ❞❡♥tr♦ ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡ ❞❡♥tr♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✳ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦ é ♠✉✐t♦ ✉s❛❞♦ ❞❡♥tr♦ ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ❢❛❝✐❧✐t❛♥❞♦ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ❛ ❛❝❡✐t❛çã♦ ❞♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ ❣❡♦♠étr✐❝♦✳
❈♦♥s✐❞❡r❡ ❞♦✐s só❧✐❞♦s X ❡ Y ❝♦♠ ♠❡s♠❛ ❛❧t✉r❛✳ ❖s ✐♥✜♥✐t♦s ♣❧❛♥♦s ♣❛r❛❧❡❧♦s ❡♥tr❡ s✐ ❛♦ ✐♥t❡r❝❡♣t❛r❡♠ ♦s só❧✐❞♦s X ❡ Y ✈ã♦ ❞❡t❡r♠✐♥❛r ✐♥✜♥✐t❛s s❡❝çõ❡s q✉❡ ❞❛rã♦ ❛ ✐❞❡✐❛ ❞♦s ✐♥❞✐✈✐sí✈❡✐s q✉❡ tê♠ ár❡❛s ✐❣✉❛✐s ❡ ♣♦rt❛♥t♦ ♦s só❧✐❞♦s tê♠ ✈♦❧✉♠❡s ✐❣✉❛✐s✳ ❆❧❣✉♥s ♦✉tr♦s t❡♦r❡♠❛s ❡st❛❜❡❧❡❝✐❞♦s ♣♦r ❈❛✈❛❧✐❡r✐ q✉❡ r❡❧❛❝✐♦♥❛♠ ✐♥❞✐✈✐sí✈❡✐s ♣♦❞❡♠ s❡r ✈✐st♦s ❞❡♥tr♦ ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛✱ t❛✐s ❝♦♠♦ ♦s q✉❡ tr❛❜❛❧❤❛♠ ❡ss❡s ✐♥❞✐✈✐sí✈❡✐s ❡♠ ♣❛r❛❧❡❧♦❣r❛♠♦s ❡ tr✐â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣❡❧❛s ❞✐❛❣♦♥❛✐s ❞♦s ♠❡s♠♦s✳
✺
P♦❞❡♠♦s ❞❡❞✉③✐r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✱ ✉s❛♥❞♦ ❛❧❣✉♠❛s ✐❞❡✐❛s ❜ás✐❝❛s ❛ r❡s♣❡✐t♦ ❞♦s ✐♥❞✐✈✐sí✈❡✐s ❡st❛❜❡❧❡❝✐❞♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♣♦r ❈❛✈❛❧✐❡r✐✱ ❡ tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ❞♦✐s só❧✐❞♦s ❞❡ ♥❛t✉r❡③❛ ❞✐❢❡r❡♥t❡✱ ♠❛s s❡❝❝✐♦♥❛♥❞♦ ❛♠❜♦s ♣♦r ✉♠ ♠❡s♠♦ ♣❧❛♥♦ ❡ ♦❜t❡♥❞♦ s❡❝çõ❡s ❜❡♠ ❞✐❢❡r❡♥t❡s✱ ♣♦ré♠ ❝♦♠ ❛ ♠❡s♠❛ ár❡❛ ❡ ❛ss✐♠ ❡♥tã♦ ♦s só❧✐❞♦s ❛♣r❡s❡♥t❛❞♦s ✐♥✐❝✐❛❧♠❡♥t❡ tê♠ ♦ ♠❡s♠♦ ✈♦❧✉♠❡✳ ❉❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡r❛❧✱ ❛❝❡✐t❛✲s❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❝♦♠♦ ✉♠ ❛①✐♦♠❛ ❡ ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛✱ ♥ã♦ ❤❛✈❡♥❞♦ ❛ss✐♠ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❞❡♠♦♥str❛r✳
❋✐❣✉r❛ ✶✳✶✿ ❇♦❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐ ✲ ❋♦♥t❡✿ ♠❛t❤❡✉s♠❛t❤✐❝❛✳❜❧♦❣s♣t✳❝♦♠
❆❧é♠ ❞❛s ✐♥❢♦r♠❛çõ❡s ❛ r❡s♣❡✐t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♣♦❞❡♠♦s t❛♠❜é♠ ✉s❛r ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ♣❛r❛ ❡st❛❜❡❧❡❝❡r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛s ❢❛❝❡s✱ ♦s ❧❛❞♦s ❡ ♦s ✈ért✐❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ♣❛r❛ ♣♦❞❡r♠♦s ❡str✉t✉r❛r ♦ ❝á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞♦ ♠❡s♠♦✱ q✉❛♥❞♦ s❡ tr❛t❛ ❞❡ ✉♠ ♣r✐s♠❛ ✭r❡t♦ ♦✉ ♦❜❧✐q✉♦✮✱ ✈❡r✐✜❝❛♠♦s q✉❡ ❡st❡s sã♦ ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦s✱ ♣♦rt❛♥t♦ ♣♦❧✐❡❞r♦s ❡✉❧❡r✐❛♥♦s✳ ❱❛❧❡ ❧❡♠❜r❛r q✉❡ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r r❡❧❛❝✐♦♥❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢❛❝❡s✱ ✈ért✐❝❡s ❡ ❛r❡st❛s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦✱ ❡ ❡st❛s ✐♥❢♦r♠❛çõ❡s s❡rã♦ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❡♥❝♦♥tr♦ ❞❛s q✉❛♥t✐❞❛❞❡s ❞❡ ❧❛❞♦s ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❞❛s ár❡❛s ❞❡ ♣♦❧í❣♦♥♦s q✉❡ ❢♦r♠❛♠ ❛s s❡❝çõ❡s ❞❡ ✉♠ só❧✐❞♦ ❣❡♦♠étr✐❝♦✳ ❖❜s❡r✈❡ ♦ ❡①❡♠♣❧♦ ❞❛ ❋✐❣✉r❛ ✶✳✷ q✉❡ t❡♠♦s ❛ s❡❣✉✐r ❡♠ q✉❡ ❛♥❛❧✐s❛r❡♠♦s ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ q✉❡ ♣♦❞❡rá s❡r ❛ s❡❝çã♦ ❞❡ ✉♠ só❧✐❞♦ ❝♦♥✈❡①♦ ❞♦ q✉❛❧ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡✳
❋✐❣✉r❛ ✶✳✷✿ ❆♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r
✻
❋✐❣✉r❛ ✶✳✸✿ ▲❡♦♥❤❛r❞ P❛✉❧ ❊✉❧❡r ✲ ❋♦♥t❡✿ ✇✇✇✳❡①♣❧✐❝❛t♦r✐✉♠✳❝♦♠
❉❡✈❡♠♦s ❧❡♠❜r❛r ❞❡ q✉❡✱ ❡♠ s❡ tr❛t❛♥❞♦ ❞❡ ✉♠ ♣♦❧✐❡❞r♦✱ ♣♦❞❡♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❞♦ ♠❡s♠♦ ❡ ❛ss✐♠ s❡♥❞♦ ♦ s❡✉ ✈♦❧✉♠❡ ♣♦❞❡rá s❡r ♦❜t✐❞♦ ❛tr❛✈és ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳
➱ ✐♠♣♦rt❛♥t❡ ❛♣r❡s❡♥t❛r ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡ ❢♦✐ ❞❡s❝♦❜❡rt♦ ❡♠ ✶✼✺✽✱ ❡ q✉❡ ♣♦❞❡ s❡r ❡♥✉♥❝✐❛❞♦ ❛ss✐♠✿ ❙❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ t❡♠ V ✈ért✐❝❡s✱A❛r❡st❛ ❡F ❢❛❝❡s✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿
V +F =A+ 2
❊♠ q✉❡ A é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞♦ ♣♦❧✐❡❞r♦✱ F ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❡ V ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s✳
❆♣❡♥❛s ✉♠❛ ✐♥❢♦r♠❛çã♦ ❤✐stór✐❝❛✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r t❡♠ ✈ár✐❛s ❞❡♠♦♥str❛çõ❡s ❝♦♠♦ ❛ q✉❡ ✐r❡♠♦s ♠♦str❛r ❛❣♦r❛✱ ❡ q✉❡ é ❛♣r❡s❡♥t❛❞❛ ❡♠ ❬✻❪
❉❡✜♥✐çã♦ ✶✳✶ ❙✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❧✐♠✐t❛❞❛ ❝♦♥✈❡①❛ ❙✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❧✐♠✐t❛❞❛ ❝♦♥✈❡①❛ é ❛ r❡✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦❧í❣♦♥♦s ♣❧❛♥♦s ❡ ❝♦♥✈❡①♦s ✭♦✉ r❡❣✐õ❡s ♣♦✲ ❧✐❣♦♥❛✐s ❝♦♥✈❡①❛s ✮✱ t❛✐s q✉❡✿
❛✮ ❉♦✐s ♣♦❧í❣♦♥♦s ♥ã♦ ❡stã♦ ♥♦ ♠❡s♠♦ ♣❧❛♥♦❀
❜✮ ❈❛❞❛ ❧❛❞♦ ❞❡ ♣♦❧í❣♦♥♦ ♥ã♦ ❡stá ❡♠ ♠❛✐s q✉❡ ❞♦✐s ♣♦❧í❣♦♥♦s❀
❝✮ ❍❛✈❡♥❞♦ ❧❛❞♦s ❞❡ ♣♦❧í❣♦♥♦s q✉❡ ❡stã♦ ❡♠ ✉♠ só ♣♦❧í❣♦♥♦✱ ❡❧❡s ❞❡✈❡♠ ❢♦r♠❛r ✉♠❛ ✜❣✉r❛ ♣♦❧✐❣♦♥❛❧ ❢❡❝❤❛❞❛✱ ♣❧❛♥❛ ♦✉ ♥ã♦✱ ❝❤❛♠❛❞❛ ❝♦♥t♦r♥♦❀
❞✮ ❖ ♣❧❛♥♦ ❞❡ ❝❛❞❛ ♣♦❧í❣♦♥♦ ❞❡✐①❛ ♦s ❞❡♠❛✐s ♥✉♠ ♠❡s♠♦ s❡♠✐❡s♣❛ç♦ ✭❝♦♥❞✐çã♦ ❞❡ ❝♦♥✈❡①✐❞❛❞❡✮✳
✼
P❛rt❡ ■
P♦r ✐♥❞✉çã♦ ✜♥✐t❛ r❡❢❡r❡♥t❡ ❛♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s✱ ✈❛♠♦s ♣r♦✈❛r✱ ♣r❡❧✐♠✐♥❛r♠❡♥t❡✱ q✉❡ ♣❛r❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❧✐♠✐t❛❞❛ ❝♦♥✈❡①❛ ❛❜❡rt❛✱ ✈❛❧❡ ❛ r❡❧❛çã♦✿
Va+Fa−Aa= 1
❡♠ q✉❡✿
Va é ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s✱
Fa é ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❡
Aa é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❧✐♠✐t❛❞❛ ❡ ❛❜❡rt❛✳
✐✮ P❛r❛ Fa= 1✳ ◆❡st❡ ❝❛s♦ ❛ s✉♣❡r❢í❝✐❡ s❡ r❡❞✉③ ❛ ✉♠ ♣♦❧í❣♦♥♦ ♣❧❛♥♦ ❝♦♥✈❡①♦ ❞❡ n ❧❛❞♦s
❡✱ ❡♥tã♦✱ Va ❂ n✱ Aa ❂n✳ ❚❡♠♦s ❛ss✐♠✿
❱a−Aa+Fa =n−n+ 1 = 1✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ Va−Aa+Fa= 1
▲♦❣♦✱ ❛ r❡❧❛çã♦ ❡stá ✈❡r✐✜❝❛❞❛ ♣❛r❛✿
Fa= 1
✳
✐✐✮ ❆❞♠✐t✐♥❞♦ q✉❡ ❛ r❡❧❛çã♦ ✈❛❧❡ ♣❛r❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ F′ ❢❛❝❡s✭ q✉❡ ♣♦ss✉✐ V′ ✈ért✐❝❡s ❡ A′ ❛r❡st❛s ✮✱ ✈❛♠♦s ♣r♦✈❛r t❛♠❜é♠ q✉❡ ✈❛❧❡ ♣❛r❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ F′ + 1 ❢❛❝❡s ✭q✉❡ ♣♦ss✉✐ F′+ 1 =F
1 ❢❛❝❡s✱Va ✈ért✐❝❡s ❡ Aa ❛r❡st❛s✮✳
P♦r ❤✐♣ót❡s❡✱ ♣❛r❛ ❛ s✉♣❡r❢í❝✐❡ ❞❡ F′ ❢❛❝❡s✱A′ ❛r❡st❛s ❡ V′ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿ V′ −A′+F′ = 1
❆❝r❡s❝❡♥t❛♥❞♦ ❛ ❡ss❛ s✉♣❡r❢í❝✐❡ ✭q✉❡ é ❛❜❡rt❛✮ ✉♠❛ ❢❛❝❡ ❞❡ p ❛r❡st❛s ✭❧❛❞♦s✮ ❡ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ q ❞❡ss❛s ❛r❡st❛s ✭❧❛❞♦s✮ ❝♦✐♥❝✐❞❡♠ ❝♦♠ ❛s ❛r❡st❛s ❥á ❡①✐st❡♥t❡s✱ ♦❜t❡♠♦s ✉♠❛ ♥♦✈❛ s✉♣❡r❢í❝✐❡ ❝♦♠ Fa ❢❛❝❡s✱ Aa ❛r❡st❛s ❡Va ✈ért✐❝❡s t❛✐s q✉❡✿
Fa=F′+ 1
Aa ❂A′ +p ✲ q✱ ✭q ❛r❡st❛s ❝♦✐♥❝✐❞✐r❛♠✮
✽
❋♦r♠❛♥❞♦ ❛ss✐♠ ❛ ❡①♣r❡ssã♦ Va−Aa+Fa ❡ s✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s ❛♥t❡r✐♦r✲
♠❡♥t❡✱ t❡r❡♠♦s✿
Va−Aa+Fa=V′+p−(q+ 1)−(A′+p−q) + (F′+ 1) ❂
V′ +p−q−1−A′−p+q+F′+ 1 =V′−A′+F′
❊ ❛ss✐♠ ❝♦♠♦ Va−Aa+Fa =V′ −A′+F′ ♣r♦✈❛♠♦s q✉❡ ❡ss❛ ❡①♣r❡ssã♦ ♥ã♦ s❡ ❛❧t❡r❛ s❡
❛❝r❡s❝❡♥t❛♠♦s ✭♦✉ r❡t✐r❛♠♦s✮ ✉♠❛ ❢❛❝❡ ❞❛ s✉♣❡r❢í❝✐❡✳ ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡✱V′ −A′+F′ = 1✱ ✈❡♠ q✉❡✿
Va−Aa+Fa= 1
❖ q✉❡ ♣r♦✈❛ ❛ r❡❧❛çã♦ ♣r❡❧✐♠✐♥❛r✳
P❛rt❡ ■■
❚♦♠❡♠♦s ❛ s✉♣❡r❢í❝✐❡ ❞❡ q✉❛❧q✉❡r ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ♦✉ q✉❛❧q✉❡r s✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❝♦♥✈❡①❛ ❧✐♠✐t❛❞❛ ❡ ❢❡❝❤❛❞❛ ✭❝♦♠V ✈ért✐❝❡s✱A ❛r❡st❛s ❡F ❢❛❝❡s ✮ ❡ ❞❡❧❛ r❡t✐r❛♠♦s ✉♠❛ ❢❛❝❡✳ ❋✐❝❛♠♦s ❡♥tã♦ ❝♦♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❛❜❡rt❛ ✭❝♦♠ Va ✈ért✐❝❡s✱ Aa ❛r❡st❛s ❡ Fa ❢❛❝❡s✮ ♣❡❧❛
q✉❛❧ ✈❛❧❡ ❛ r❡❧❛çã♦✿
Va−Aa+Fa= 1 ❡ ❝♦♠♦ Va =V✱ Aa=A ❡ Fa=F −1✱ ✈❡♠ V −A+F −1 = 1✱ ♦✉
s❡❥❛✿
V −F =A+ 2✳
❈❛♣ít✉❧♦ ✷
❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♦s P❛râ♠❡tr♦s
❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❡ ♦ ❊◆❊▼
✷✳✶ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s
❖s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❝♦♥st✐t✉❡♠ ✉♠ r❡❢❡r❡♥❝✐❛❧ ♣❛r❛ ❛ ❡❞✉❝❛çã♦ ♥♦ ❊♥✲ s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❡ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡♠ t♦❞♦ ♦ ♣❛ís✳ ❙✉❛ ❢✉♥çã♦ é ♦r✐❡♥t❛r ❡ ❣❛r❛♥t✐r ❛ ❝♦❡rê♥❝✐❛ ❞♦s ✐♥✈❡st✐♠❡♥t♦s ♥♦ s✐st❡♠❛ ❡❞✉❝❛❝✐♦♥❛❧✱ s♦❝✐❛❧✐③❛♥❞♦ ❞✐s❝✉ssõ❡s✱ ♣❡sq✉✐s❛s ❡ r❡✲ ❝♦♠❡♥❞❛çõ❡s✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ♣r♦♣♦s✐çõ❡s ❡①♣r❡ss❛s ♥♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s r❡s♣♦♥❞❡ à ♥❡❝❡ss✐❞❛❞❡ ❞❡ r❡❢❡r❡♥❝✐❛✐s ❛ ♣❛rt✐r ❞♦s q✉❛✐s ♦ s✐st❡♠❛ ❡❞✉❝❛❝✐♦♥❛❧ ❞♦ P❛ís s❡ ♦r❣❛♥✐③❡✱ ❛ ✜♠ ❞❡ ❣❛r❛♥t✐r q✉❡ s❡❥❛♠ r❡s♣❡✐t❛❞❛s ❛s ❞✐✈❡rs✐❞❛❞❡s✳
❆ ▲❡✐ ❞❡ ❉✐r❡tr✐③❡s ❡ ❇❛s❡s ❞❛ ❊❞✉❝❛çã♦ ◆❛❝✐♦♥❛❧ ❡①♣❧✐❝✐t❛ q✉❡ ♦ ❊♥s✐♥♦ ▼é❞✐♦ é ❛ ✏❡t❛♣❛ ✜♥❛❧ ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✑ ✭❆rt✳✸✻✮✱ ♦ q✉❡ ❝♦♥❝♦rr❡ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ s✉❛ ✐❞❡♥t✐✲ ❞❛❞❡✳ ❖ ❊♥s✐♥♦ ▼é❞✐♦ ♣❛ss❛ ❛ t❡r ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ t❡r♠✐♥❛❧✐❞❛❞❡✱ ♦ q✉❡ s✐❣♥✐✜❝❛ ❛ss❡❣✉r❛r ❛ t♦❞♦s ♦s ❝✐❞❛❞ã♦s ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❝♦♥s♦❧✐❞❛r ❡ ❛♣r♦❢✉♥❞❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❛❞q✉✐r✐❞♦s ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧❀ ❛♣r✐♠♦r❛r ♦ ❡❞✉❝❛♥❞♦ ❝♦♠♦ ♣❡ss♦❛ ❤✉♠❛♥❛❀ ♣♦ss✐❜✐❧✐t❛r ♦ ♣r♦ss❡✲ ❣✉✐♠❡♥t♦ ❞❡ ❡st✉❞♦s❀ ❣❛r❛♥t✐r ❛ ♣r❡♣❛r❛çã♦ ❜ás✐❝❛ ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❡ ❛ ❝✐❞❛❞❛♥✐❛❀ ❞♦t❛r ♦ ❡❞✉❝❛♥❞♦ ❞♦s ✐♥str✉♠❡♥t♦s q✉❡ ♦ ♣❡r♠✐t❛♠ ✏❝♦♥t✐♥✉❛r ❛♣r❡♥❞❡♥❞♦✑✳ ❖ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣♦r✲ t❛♥t♦✱ é ❛ ❡t❛♣❛ ✜♥❛❧ ❞❡ ✉♠❛ ❡❞✉❝❛çã♦ ❞❡ ❝❛rát❡r ❣❡r❛❧✱ ❛✜♥❛❞❛ ❝♦♠ ❛ ❝♦♥t❡♠♣♦r❛♥❡✐❞❛❞❡✱ ❝♦♠ ❛ ❝♦♥str✉çã♦ ❞❡ ❝♦♠♣❡tê♥❝✐❛s ❜ás✐❝❛s✱ q✉❡ s✐t✉❡♠ ♦ ❛❧✉♥♦ ❝♦♠♦ s✉❥❡✐t♦ ♣r♦❞✉t♦r ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ♣❛rt✐❝✐♣❛♥t❡ ❞♦ ♠✉♥❞♦ ❡♠ q✉❡ ❡❧❡ ❡stá ✐♥s❡r✐❞♦✱ ♣❛rt✐❝✐♣❛♥❞♦ ❞♦ tr❛❜❛❧❤♦ ♥♦ q✉❛❧ ❡❧❡ ❞❡s❡❥❡✱ ❛t✉❛♥❞♦ ❝♦♠♦ ✏s✉❥❡✐t♦ ❡♠ s✐t✉❛çã♦✑ ✲ ❝✐❞❛❞ã♦✳
❙❡❣✉♥❞♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s✱ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ♥❛ ár❡❛ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛✱ ▼❛t❡♠át✐❝❛ ❡ s✉❛s ❚❡❝♥♦❧♦❣✐❛s ✐♥❞✐❝❛ q✉❡ ♦ ❛❧✉♥♦ ❞❡✈❡ t❡r ❛ ❝♦♠♣r❡❡♥sã♦ ❡
✷✳✶✳ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✶✵
❛ ✉t✐❧✐③❛çã♦ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❝✐❡♥tí✜❝♦s✱ ♣❛r❛ ❡①♣❧✐❝❛r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ♠✉♥❞♦✱ ❜❡♠ ❝♦♠♦ ♣❧❛♥❡❥❛r✱ ❡①❡❝✉t❛r ❡ ❛✈❛❧✐❛r ❛s ❛çõ❡s ❞❡ ✐♥t❡r✈❡♥çã♦ ♥♦ s❡✉ ❝♦t✐❞✐❛♥♦✳
◆♦ tó♣✐❝♦ ✺ ❞❛s ❉✐r❡tr✐③❡s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦✭❉❈◆❊▼ ✮✱ ♦❜✲ s❡r✈❛♠♦s ❛ s❡❣✉✐♥t❡ ♦r✐❡♥t❛çã♦✿
❖ ❛❧✉♥♦ ❞❡✈❡rá s❡r ❝❛♣❛③ ❞❡✿
• ❛♥❛❧✐s❛r q✉❛❧✐t❛t✐✈❛♠❡♥t❡ ❞❛❞♦s q✉❛♥t✐t❛t✐✈♦s✱ r❡♣r❡s❡♥t❛❞♦s ❣rá✜❝❛ ♦✉ ❛❧❣❡❜r✐❝❛♠❡♥t❡✱
r❡❧❛❝✐♦♥❛❞♦s ❛ ❝♦♥t❡①t♦s só❝✐♦ ❡❝♦♥ô♠✐❝♦s✱ ❝✐❡♥tí✜❝♦s ♦✉ ❝♦t✐❞✐❛♥♦s❀
• ✲ ✐❞❡♥t✐✜❝❛r✱ r❡♣r❡s❡♥t❛r ❡ ✉t✐❧✐③❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ♣❛r❛ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❛
❧❡✐t✉r❛✱ ❞❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❞❛ ❛çã♦ s♦❜r❡ ❛ r❡❛❧✐❞❛❞❡✳
❆♥❛❧✐s❛♥❞♦ ♦s P❈◆s✱ ✈❡r✐✜❝❛♠♦s ❛ ✐♠❡♥s❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ tr❛❜❛❧❤❛r ❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡ ❡s♣❛❝✐❛❧ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣♦✐s ♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❛❧✉♥♦s é ❜❛s❡❛❞♦ ❡♠ ❝✐♠❛ ❞❡ ✜❣✉r❛s ❡ ♣r♦❜❧❡♠❛s q✉❡ ❧❤❡s ❞ã♦ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❡st❛r❡♠ ✐♥s❡r✐❞♦s ♥♦ ♣r♦❝❡ss♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ♠❛♥❡✐r❛ ❛ ♠❛♥✐♣✉❧❛r ❢ór♠✉❧❛s ❝♦♠ ❝♦♥❝❡✐t♦s ♣r❡✈✐❛♠❡♥t❡ ❛♣r❡s❡♥t❛❞♦s ❡ ❡st✐♣✉❧❛❞♦s ♣❡❧♦ ♣r♦❢❡ss♦r ❡ ♣❡❧♦s ♠❛t❡r✐❛✐s ❞✐❞át✐❝♦s ❛♣r❡s❡♥t❛❞♦s ❛♦s ♠❡s♠♦s✳ ❚❡♠♦s✱ s❡❣✉♥❞♦ ♦ P❈◆ ❞❡ ▼❛t❡♠át✐❝❛✱ q✉❡ ❛♣r❡s❡♥t❛r s✐t✉❛çõ❡s q✉❡ r❡♣r❡s❡♥t❡♠ ❞❡s❛✜♦s ♣❛r❛ ♦s ❛❧✉♥♦s✱ q✉❡ s✐❣♥✐✜q✉❡♠ ♥ã♦ ❛♣❡♥❛s ❛ ♠❡♠♦r✐③❛çã♦ ❞❡ ❢ór♠✉❧❛s✱ ♠❛s ❛ ❛♣❧✐❝❛çã♦ ❞❛s ♠❡s♠❛s ❥✉♥t♦ ❝♦♠ ✉♠ r❛❝✐♦❝í♥✐♦ ♠❛✐s ❡❧❛❜♦r❛❞♦ ❡ ❡str✉t✉r❛❞♦ ♣❡❧♦ ♣ró♣r✐♦ ❛❧✉♥♦✱ ♥ã♦ t❡♥❞♦ ❡st❡ ❢❡✐t♦ ✉♠❛ s✐♠♣❧❡s r❡♣❡t✐çã♦ ❞❡ ♣r♦❝❡ss♦s ❡ ♥ã♦ ❞❡✐①❛♥❞♦ t♦t❛❧♠❡♥t❡ ❞❡ ❧❛❞♦ ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ♥❛s q✉❛✐s s❡ ❛♣r❡s❡♥t❡ ✐❞❡✐❛s ❞♦ t✐♣♦ ✏❝❛❧❝✉❧❡✑✱ ✏r❡s♦❧✈❛✑✱ ✏❡♥❝♦♥tr❡✑✱ ♣♦✐s ❡st❛s ✐rã♦ s❡r ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❛♣r❡♥❞❛ ♦s ♣r♦❝❡❞✐♠❡♥t♦s té❝♥✐❝♦s ❞❛ s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s✳
➱ ❜♦♠ ♦❜s❡r✈❛r♠♦s t❛♠❜é♠✱ q✉❡ ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦s✱ ❞❡✈❡♠ s❡r ❛❜♦r❞❛❞♦s ❡ ❛♣r❡s❡♥t❛❞♦s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ q✉❡ ♥ã♦ s❡ ♣r❡♥❞❛ s♦♠❡♥t❡ ♥♦ ❝♦t✐❞✐❛♥♦ ❡ ♥✉♠❛ ❛♣❧✐❝❛çã♦ ♣rát✐❝❛ ❡ ❝♦♥❝r❡t❛✱ ♣♦rq✉❡ ❡♠ ❛❧❣✉♠❛s s✐t✉❛çõ❡s ♥ã♦ t❡r❡♠♦s ✉♠❛ ✈✐s✉❛❧✐③❛çã♦ ♥♦ ❡s♣❛ç♦ q✉❡ ♥♦s ❝❡r❝❛ ❞❛q✉❡❧❛s ❞❡✜♥✐çõ❡s ❡ s✐t✉❛çõ❡s ❛♣r❡s❡♥t❛❞❛s✳ ❚❡✲ ♠♦s q✉❡ ✐♥❝❡♥t✐✈❛r ✉♠ ❡♥✈♦❧✈✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❡♠ ♥í✈❡❧ ▼é❞✐♦ ❡ ❋✉♥❞❛♠❡♥t❛❧✱ ❞❡ ❢♦r♠❛ q✉❡ ♦s ❞✐s❝❡♥t❡s s❡ ❢❛♠✐❧✐❛r✐③❡♠ ❝♦♠ ❛ q✉❡stã♦ ❞❡ ❡st❛r ✉s❛♥❞♦ ❝♦♥❝❡✐t♦s ❡ r❛❝✐♦❝í♥✐♦s q✉❡ ♦s ❛❥✉❞❡♠ ❡♠ ♦✉tr❛s s✐t✉❛çõ❡s✱ ❡♠ ♦✉tr♦s ♥í✈❡✐s ♠❛t❡♠át✐❝♦s ❡ t❛♠❜é♠ ❡♠ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♥ã♦ ✜❝❛♥❞♦ ❞❡♣❡♥❞❡♥t❡s ❞❡ t❡r❡♠ s❡♠♣r❡ ✉♠❛ s✐t✉❛çã♦ ♣rát✐❝❛ ♣❛r❛ ♦ ✉s♦ ♦✉ ✉♠❛ ❥✉st✐✜❝❛t✐✈❛ ❞❡ ❡st✉❞♦✱ ♠❛s q✉❡ ❡st✐♠✉❧❡♠ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ♦ ❛♣r❡ç♦ ♣❡❧♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ♠❛t❡♠át✐❝♦✱ ♣❡❧❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ ♦✉tr❛s ár❡❛s ❡ ❞❛ ♣ró♣r✐❛ ♠❛t❡♠át✐❝❛ ❞❛q✉✐❧♦ q✉❡ s❡ ❡stá ❡st✉❞❛♥❞♦✳
✷✳✶✳ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✶✶
❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s sã♦ ❡st✉❞❛❞♦s ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♥❛ s❡❣✉♥❞❛ sér✐❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡ ❛♣r❡s❡♥t❛✲s❡✱ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❜ás✐❝❛ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s s✐♠♣❧❡s ♦✉ ♠❛✐s ❡❧❛❜♦r❛❞♦s ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ❜❡♠ ❝♦♠♦ ♦ ❛♣r✐♠♦r❛♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ❡ ❛♣❧✐❝❛çã♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✳ ❊❧❡ é ❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ❡❧❡♠❡♥t❛r ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ❡①♣r❡ssõ❡s ❞♦ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s✱ é ✉t✐❧✐③❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❞❡❞✉çã♦ ❞❛ ❡①♣r❡ssã♦ ❞♦ ✈♦❧✉♠❡ ❞❡ ❝♦♥❡s ❡ ❞❡ ❡s❢❡r❛s✳ ❆❧é♠ ❞✐ss♦✱ ♣❡r♠✐t❡ ♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ♥ã♦ r❡❣✉❧❛r❡s✳ ❆ss✐♠✱ ♣❡❧❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛✱ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❞❡✈❡ s❡r ❡♥s✐♥❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
P❛r❛ ✉♠ ❜♦♠ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ❝♦♠♣r❡ssã♦ ♣❡r❢❡✐t❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ❢❛③✲s❡ ♥❡❝❡s✲ sár✐♦ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❜ás✐❝♦ ❞❡ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ♦ q✉❡ ❞❡✈❡rá s❡r ❛♣r❡s❡♥t❛❞♦ ❛♦s ❛❧✉♥♦s ❝♦♠ ❛♥t❡❝❡❞ê♥❝✐❛ ♣❡❧♦s ♣r♦✜ss✐♦♥❛✐s q✉❡ tr❛❜❛❧❤❛♠ ❡♠ ❛♥♦s ❛♥t❡r✐♦r❡s✱ ♦✉ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠❛✐s s✉❝✐♥t❛✱ ❞✉r❛♥t❡ ♦ ♣ró♣r✐♦ ❛♥♦ ❧❡t✐✈♦ ❡♠ q✉❡ s❡ ❡stá ❛♣r❡s❡♥t❛♥❞♦ ❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✳ ❖s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞♦s q✉❛✐s ♥♦ss♦s ❞✐s❝❡♥t❡s ❥❛♠❛✐s ♣♦❞❡rã♦ s❡r ♣r✐✲ ✈❛❞♦s ❞❡ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦✱ t❛✐s ❝♦♠♦ ❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❡ ♣♦❧í❣♦♥♦s ❡ s✉❛s ár❡❛s✱ ❛s ♥♦çõ❡s ❣❡r❛✐s ❞❡ ♣❧❛♥♦✱ r❡t❛ ❡ ♣♦♥t♦ sã♦ ♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ q✉❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ s❡❥❛ ❛♣r❡s❡♥t❛❞♦✱ ❡♥t❡♥❞✐❞♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❢❛③❡r ❝♦♠ q✉❡ ❡st❡s ❡❞✉❝❛♥❞♦s s❡❥❛♠ ❝❛♣❛③❡s ❞❡ r❡❝♦♥❤❡❝❡r✱ ❞✐❢❡r❡♥❝✐❛r ❡ ♠❛♥✐♣✉❧❛r ❝♦♠ ❝á❧❝✉❧♦s ❞❡ ár❡❛s✱ ♣❡rí♠❡tr♦s ❡ ✈♦❧✉♠❡s ❞❡ ✜❣✉r❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❡ tr✐❞✐♠❡♥s✐♦♥❛✐s✳ ◆ã♦ ♣♦❞❡♠♦s ❞❡✐①❛r ❞❡ ❝✐t❛r t❛♠❜é♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♦s ❛❧✉♥♦s t❡r❡♠ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❛ r❡s♣❡✐t♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ♣♦s✐çã♦ ♣❛r❛ s❛❜❡r ❝♦♠♦ ♦s só❧✐❞♦s ♣♦❞❡♠ ❡st❛r r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ♦ ♣❧❛♥♦ ❛ r❡t❛ ❡ ♦ ♣♦♥t♦✳
❙❛❜❡♠♦s ❞❛ ✐♠❡♥s❛ ❞✐✜❝✉❧❞❛❞❡ q✉❡ t❡♠♦s ♥♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛ ❡ ❞♦s ❞❡s❛✜♦s ❝♦♠ ♦s q✉❛✐s ♥♦s ❞❡♣❛r❛♠♦s ❛♦ tr❛❜❛❧❤❛r♠♦s ❝♦♠ ♥♦ss♦s ❛❧✉♥♦s ❛ ❣❡♦♠❡tr✐❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦ ❞❡s❡♥❤♦✱ ♥❛ s✉❛ ❝♦♥str✉çã♦ ♦✉ ❛té ♥❛ ♣ró♣r✐❛ ❛♥á❧✐s❡ ❞❡ ❢♦r♠❛s q✉❡ ❡stã♦ ♣r♦♥t❛s✳ ◆♦ss♦s ❛❧✉♥♦s✱ ♥❛ s✉❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛✱ tê♠ ✉♠❛ ❞✐✜❝✉❧❞❛❞❡ ♠✉✐t♦ ❣r❛♥❞❡ ❡♠ r❡❛❧✐③❛r ❞❡s❡♥❤♦s ❡ ❛té ❝❡rt♦ ♣♦♥t♦ ❡♠ ❢❛③❡r ✉♠❛ ❛♥á❧✐s❡ ❞♦s ♠❡s♠♦s✳ Pr❡❝✐s❛♠♦s ❡♥tã♦ ❢❛③❡r ❝♦♠ q✉❡ ❡❧❡s ♣❛ss❡♠ ❛ tr❛❜❛❧❤❛r ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠❛✐s ❢❛♠✐❧✐❛r ❝♦♠ ❛s ❢♦r♠❛s✱ ♣❛r❛ ❡♥tã♦ ❡♥t❡♥❞❡r ♠❡❧❤♦r ♦s ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦s✱ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛❧❣✉♠❛s ❞❡❞✉çõ❡s ❡ ❞❡✜♥✐çõ❡s ❛ s❡r❡♠ ❡st✉❞❛❞❛s ❡ ♥ã♦ só ♦s ❡♥❝❛r❡♠ ❝♦♠♦ ✉♠❛ ❞✐✜❝✉❧❞❛❞❡ ❛ ♠❛✐s ♣❛r❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ❡ ♦ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ ❡ ♣❧❛♥❛✳ P♦❞❡♠♦s ❛✐♥❞❛ ✉s❛r✱ ❤♦❥❡ ❡♠ ❞✐❛✱ ♦✉tr♦s r❡❝✉rs♦s q✉❡ ♥ã♦ s❡❥❛♠ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦s ♠❛✐s tr✐✈✐❛✐s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ r❡❝✉rs♦s ❡❧❡trô♥✐❝♦s ❡ s♦❢t✇❛r❡s ♠❛t❡♠át✐❝♦s ♣❛r❛ ♦ ❛✉①í❧✐♦ ♥❛ ❞❡❞✉çã♦ ❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳
✷✳✷✳ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❊◆❊▼ ✶✷
♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ q✉❡ ❛♦ s♦❜r❡♣♦r♠♦s ➇✐♥✜♥✐t❛s➈ ✜❣✉r❛s ❞❡ ♠❡s♠❛ ár❡❛✱ ♠❛s ❞❡ ❢♦r♠❛s ❣❡♦♠étr✐❝❛s ❞✐❢❡r❡♥t❡s✱ ♣♦❞❡r❡♠♦s t❡r ♦ ♠❡s♠♦ ✈♦❧✉♠❡✳ ❈❛❜❡ ❛q✉✐✱ ❡st❛❜❡❧❡❝❡r♠♦s ✉♠❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ✈♦❧✉♠❡ ❡ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✉♠ só❧✐❞♦✳ ❖ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ é ♦ q✉❛♥t♦ ❡❧❡ ♦❝✉♣❛ ♥♦ ❡s♣❛ç♦✱ ❧❡♠❜r❛♥❞♦✲s❡ s❡♠♣r❡ ❞❡ q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦✱ t❡♦r✐❝❛♠❡♥t❡✱ é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ♣r♦❞✉t♦ ❞❛s s✉❛s três ❞✐♠❡♥sõ❡s✱ ❝♦♠♣r✐♠❡♥t♦✱ ❧❛r❣✉r❛ ❡ ❛❧t✉r❛✱ ♣♦❞❡♥❞♦ t❡r ✈❛r✐❛çõ❡s ❞❡♣❡♥❞❡♥❞♦ ❞♦ só❧✐❞♦ ❡st✉❞❛❞♦✳ ❆ ❝❛♣❛❝✐❞❛❞❡ é ♦ q✉❛♥t♦ ♦ só❧✐❞♦ ❛r♠❛③❡♥❛ ❡♠ s❡✉ ✐♥t❡r✐♦r✳
➱ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ r❡ss❛❧t❛r♠♦s q✉❡ ❛♦ ♥♦s ❜❛s❡❛r♠♦s ♥♦s P❈◆s ❞❡ ♠❛✲ t❡♠át✐❝❛✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠♦♠❡♥t♦ ♦ ✉s♦ ❡ ♦ ❛♣♦✐♦ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ❡ ❞❡ r❡❝✉rs♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❡ s❡✉s ❞❡s❞♦❜r❛♠❡♥t♦s ❡ ❛♣❧✐❝❛çõ❡s ♥♦ ❝♦t✐❞✐❛♥♦ ❡ ❝♦♠♦ ❥á ❝✐t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♣❛r❛ ❞❡❞✉çõ❡s ❞❡ r❛❝✐♦❝í♥✐♦s q✉❡ ♣♦ss❛♠ s❡r ✉s❛❞♦s ❡♠ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ❡♠ ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s✱ ❛❧é♠ ❞❡ s❡r❡♠ ❛♣r♦✈❡✐t❛❞♦s ❞❡♥tr♦ ❞❛ ♣ró♣r✐❛ ♠❛t❡♠át✐❝❛✳
✷✳✷ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❊◆❊▼
❊♠ s✉❛ ▼❛tr✐③ ❞❡ ❘❡❢❡rê♥❝✐❛✱ ♦ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣r❡❝♦♥✐③❛ ♦s s❡❣✉✐♥t❡s tó♣✐❝♦s✿
✷✳✷✳✶ ❊■❳❖❙ ❈❖●◆■❚■❱❖❙ ✭❝♦♠✉♥s ❛ t♦❞❛s ❛s ár❡❛s ❞❡ ❝♦♥❤❡❝✐✲
♠❡♥t♦✮
■✳ ❉♦♠✐♥❛r ❧✐♥❣✉❛❣❡♥s ✭❉▲✮✿ ❞♦♠✐♥❛r ❛ ♥♦r♠❛ ❝✉❧t❛ ❞❛ ▲í♥❣✉❛ P♦rt✉❣✉❡s❛ ❡ ❢❛③❡r ✉s♦ ❞❛s ❧✐♥❣✉❛❣❡♥s ♠❛t❡♠át✐❝❛✱ ❛rtíst✐❝❛ ❡ ❝✐❡♥tí✜❝❛ ❡ ❞❛s ❧í♥❣✉❛s ❡s♣❛♥❤♦❧❛ ❡ ✐♥❣❧❡s❛✳ ■■✳ ❈♦♠♣r❡❡♥❞❡r ❢❡♥ô♠❡♥♦s ✭❈❋✮✿ ❝♦♥str✉✐r ❡ ❛♣❧✐❝❛r ❝♦♥❝❡✐t♦s ❞❛s ✈ár✐❛s ár❡❛s ❞♦ ❝♦✲
♥❤❡❝✐♠❡♥t♦ ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ❢❡♥ô♠❡♥♦s ♥❛t✉r❛✐s✱ ❞❡ ♣r♦❝❡ss♦s ❤✐stór✐❝♦✲❣❡♦❣rá✜❝♦s✱ ❞❛ ♣r♦❞✉çã♦ t❡❝♥♦❧ó❣✐❝❛ ❡ ❞❛s ♠❛♥✐❢❡st❛çõ❡s ❛rtíst✐❝❛s✳
■■■✳ ❊♥❢r❡♥t❛r s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ✭❙P✮✿ s❡❧❡❝✐♦♥❛r✱ ♦r❣❛♥✐③❛r✱ r❡❧❛❝✐♦♥❛r✱ ✐♥t❡r♣r❡t❛r ❞❛❞♦s ❡ ✐♥❢♦r♠❛çõ❡s r❡♣r❡s❡♥t❛❞♦s ❞❡ ❞✐❢❡r❡♥t❡s ❢♦r♠❛s✱ ♣❛r❛ t♦♠❛r ❞❡❝✐sõ❡s ❡ ❡♥❢r❡♥t❛r s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛✳
