▼Ô◆■❈❆ ❉❊ P❆❯▲❆ ❚❖▲❊◆❚■◆❖ ❘❊■❙
➪❘❊❆ ❊ ❱❖▲❯▼❊ ❉❊ 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✐❛❡✳
❱■➬❖❙❆
Ficha catalogháfica phepahada pela Seção de Catalogação e Classificação da Biblioteca Centhal da UFV
T
Reis, Mônica de Paula Tolentino,
1972-R375a Área e volume de prisma e pirâmide / Mônica de Paula 2013 Tolentino Reis. – Viçosa, MG, 2013.
xiv, 139 f. : il. (algumas color.) ; 29 cm.
Inclui apêndice.
Orientador: Allan de Oliveira Moura.
Dissertação (mestrado) - Universidade Federal de Viçosa. Referências bibliográficas: f. 136-139.
1. Poliedros. 2. Software educacional. 3. Superfícies
(Matemática). 4. Polígonos. I. Universidade Federal de Viçosa. Departamento de Matemática. Programa de Pós-Graduação do Mestrado Profissional em Matemática em Rede Nacional. II. Título.
▼Ô◆■❈❆ ❉❊ P❆❯▲❆ ❚❖▲❊◆❚■◆❖ ❘❊■❙✳
➪❘❊❆ ❊ ❱❖▲❯▼❊ ❉❊ 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✐❧ ❞❡ ✷✵✶✸✳
❱❛❧ér✐❛ ▼❛tt♦s ❞❛ ❘♦s❛ ▼❛r✐♥ês ●✉❡rr❡✐r♦
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s✱ ❢♦♥t❡ ✐♥❡s❣♦tá✈❡❧ ❞❡ s❛❜❡❞♦r✐❛ q✉❡✱ ❛❝✐♠❛ ❞❡ t✉❞♦ ❡ ❞❡ t♦❞♦s ♠❡ ❞❡✉ ❢♦rç❛ ♣❛r❛ ❡♥❢r❡♥t❛r t♦❞♦s ♦s ♦❜stá❝✉❧♦s✳
❆♦s ♠❡✉s ✜❧❤♦s✿ ❈és❛r ❋✐❧❤♦ ❡ ▲❛r❛ ▲✉ís❛ ❡ ❡s♣♦s♦✿ ❈és❛r q✉❡ s❡♥t✐r❛♠ ❛ ❛✉sê♥❝✐❛ ❡ ❝♦♠♣r❡❡♥❞❡r❛♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛♣♦✐❛r✳
❆♦s ♠❡✉s ♣❛✐s ❩✐❧❞❛ ❡ ●❡r❛❧❞♦ ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣❡❧❛s ♦r❛çõ❡s ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✳
❆♦s ❢❛♠✐❧✐❛r❡s✱ q✉❡ ❝❛❞❛ ✉♠✱ ❛ s❡✉ ♠♦❞♦ ❢♦r❛♠ ❡♠♣❛r❡❧❤❛♥❞♦ ❡♠ s✉♣♦rt❡ ❝♦♥tí♥✉♦ ♣❛r❛ q✉❡ ♦ ❝❛♠✐♥❤❛r ✜❝❛ss❡ ♠❡♥♦s ár❞✉♦✳
❆ ❈❆P❊❙ ❡ ❛♦s ✐❞❡❛❧✐③❛❞♦r❡s ❞♦ P❘❖❋▼❆❚✱ ♣♦r t❡r❡♠ ♠❡ ❞❛❞♦ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ r❡❛❧✐③❛r ✉♠ s♦♥❤♦✳
❆ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛ ❡ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✱ t✉r♠❛ ✷✵✶✶ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❝♦♠ ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ❡♠ ❡s♣❡❝✐❛❧ ♦s q✉❡ t✐✈❡r❛♠ ♠❛✐s ❞✐r❡t❛♠❡♥t❡ ❡♥✈♦❧✈✐❞♦s ❡♠ ♠❡✉ tr❛❜❛❧❤♦ ❡ q✉❡ ❞❡r❛♠ s✉❛ s✐♥❣✉❧❛r ❝♦♥tr✐❜✉✐çã♦✿ ❉r❛✳ ❈❛t❛r✐♥❛ ❞❡ ❏❡s✉s ▼❡♥❞❡s ❡ ❉r✳ ❆❧❛♥ ❞❡ ❖❧✐✈❡✐r❛ ▼♦✉r❛✳
❆ ♠✐♥❤❛ ❝♦❧❡❣❛ ❞❡ ♣r♦✜ssã♦ ▼❛r✐❛ ❞❡ ▲♦✉r❞❡s ♣❡❧❛ ❢♦rç❛✱ ❝♦♥tr✐❜✉✐çã♦ ❡ s✉♣♦rt❡ ♥❛s ❞ú✈✐❞❛s✳
❆♦s ❝♦❧❡❣❛s ❞❛ t✉r♠❛✱ ❝♦♠ ♦s q✉❛✐s ❝♦♥✈✐✈✐ ❡♠ ❡s♣❡❝✐❛❧ às ❏✉❧✐❛♥❛s✭❝♦♠♣❛♥❤❡✐r❛s ❞❡ ❡str❛❞❛✮✱ ❏♦ss❛r❛✱ P❛tr✐❝❦✱ ❱❛♥❡ss❛ ❡ ❑❡②❧❛ q✉❡ t✐✈❡r❛♠ ♣❛rt✐❝✐♣❛çã♦ ❡❢❡t✐✈❛ s❡❥❛ ♥♦ ❡st✉❞♦ ♣❛r❛ ❛s ♣r♦✈❛s✱ ♥♦s tr❛❜❛❧❤♦s ❡ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞❛ ❡❧❛❜♦r❛çã♦ ❞❛s ❛t✐✈✐❞❛❞❡s q✉❡ ♣❡r♠✐t✐r❛♠ ❛ ❡❧❛❜♦r❛çã♦ ❞❛ ♣❡sq✉✐s❛✳
▼❡✉ ❝❛r✐♥❤♦ ❡s♣❡❝✐❛❧ ❛ t♦❞♦s ❛q✉❡❧❡s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡s❡❥❛r❛♠ ❛ ♠✐♥❤❛ ✈✐tór✐❛✳
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✈✐
❘❡s✉♠♦ ①✐✈
❆❜str❛❝t ①✈
■♥tr♦❞✉çã♦ ✶
■♥tr♦❞✉çã♦ ✶
✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ✸
✶✳✶ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ❆❧❣✉♥s P♦st✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✷ P♦s✐çõ❡s r❡❧❛t✐✈❛s ❞❡ ❞✉❛s r❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✸ P♦s✐çõ❡s r❡❧❛t✐✈❛s ❞❡ r❡t❛ ❡ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✹ P♦s✐çõ❡s r❡❧❛t✐✈❛s ❞❡ ❞♦✐s ♣❧❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ P♦❧í❣♦♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹ ◗✉❛❞r✐❧át❡r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✺ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❈ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✻ ➪r❡❛ ❞❡ ❙✉♣❡r❢í❝✐❡s P❧❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
❙❯▼➪❘■❖ ✐✈
✶✳✻✳✶ ➪r❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✻✳✷ ➪r❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✻✳✸ ➪r❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✻✳✹ ➪r❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✻✳✺ ➪r❡❛ ❞♦ ▲♦s❛♥❣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✶✳✻✳✻ ➪r❡❛ ❞♦ ❚r❛♣é③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✳✻✳✼ ➪r❡❛ ❞❡ ✉♠ P♦❧í❣♦♥♦ ❘❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✶✳✼ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✶✳✼✳✶ P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷ ●❡♦♠❡tr✐❛ ♥♦ ❈♦t✐❞✐❛♥♦ ✹✶
✷✳✶ ●❡♦♠❡tr✐❛ ♥❛ ◆❛t✉r❡③❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷ ●❡♦♠❡tr✐❛ ♥❛ ❆rq✉✐t❡t✉r❛ ❡ ❈♦♥str✉çã♦ ❈✐✈✐❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✸ ●❡♦♠❡tr✐❛ ❞♦s P❛r❛❢✉s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✹ ●❡♦♠❡tr✐❛ ❞❛ ❆rt❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✺ ●❡♦♠❡tr✐❛ ❞❛s ❊♠❜❛❧❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✸ ➪r❡❛ ❡ ❱♦❧✉♠❡ ❞❡ Pr✐s♠❛s ❡ P✐râ♠✐❞❡s ✺✽
❙❯▼➪❘■❖ ✈
✸✳✶✳✽ ❆t✐✈✐❞❛❞❡ ✉s❛♥❞♦ ♦ ❈❛❜r✐ ✸❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✸✳✶✳✾ ❆t✐✈✐❞❛❞❡ ❝♦♠ ❞✐❢❡r❡♥t❡s r❡s♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✸✳✷ P✐râ♠✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✸✳✷✳✶ ➪r❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✸✳✷✳✷ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✸✳✷✳✸ ❱♦❧✉♠❡ ❞❛ P✐râ♠✐❞❡ ❡ ♦ ●❡♦❣❡❜r❛ ✸❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✸✳✷✳✹ ❊①♣❡r✐♠❡♥t♦✿ P✐râ♠✐❞❡ ❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✸✳✷✳✺ ●❡♦❡s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽ ✸✳✷✳✻ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ❝♦♠ ♦ ❲✐♥❣❡♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✸✳✷✳✼ ➪r❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❡ ✈♦❧✉♠❡ ❞♦s t❡tr❛❡❞r♦s r❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✸✳✷✳✽ ❚r♦♥❝♦ ❞❡ ♣✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷
✹ Prát✐❝❛ ❉✐r✐❣✐❞❛ ✶✵✻
✹✳✶ ▼♦str❛ ❞❡ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻ ✹✳✶✳✶ ❖❜❥❡t✐✈♦s ❡ ▼❡t♦❞♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ✹✳✶✳✷ ❘❡s✉❧t❛❞♦s ❊s♣❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ✹✳✶✳✸ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼♦str❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ✹✳✶✳✹ ❆✈❛❧✐❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✹✳✶✳✺ ❘❡s✉❧t❛❞♦s ❆❧❝❛♥ç❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✹✳✷ Prát✐❝❛ ❝♦♠ ♣r♦❢❡ss♦r❡s ✲ ❚r❡✐♥❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶ ✹✳✷✳✶ Pr♦♣♦st❛✱ ❖❜❥❡t✐✈♦ ❡ ▼❡t♦❞♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶ ✹✳✷✳✷ ❘❡s✉❧t❛❞♦s ❊s♣❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷ ✹✳✷✳✸ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼♦str❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷ ✹✳✷✳✹ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✹
❙❯▼➪❘■❖ ✈✐
❆ ◗✉❡st✐♦♥ár✐♦ ✶✸✵
❆✳✶ ❖ Pr♦❢❡ss♦r ❡ ♦ ❊♥s✐♥♦ ❞❡ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ✲ P♦❧✐❡❞r♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✶✸✵ ❆✳✶✳✶ ❉❛❞♦s ♣❡ss♦❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✵ ❆✳✶✳✷ ❈❛r❛❝t❡r✐③❛çã♦ ❞♦s ❈♦♥❤❡❝✐♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✶ ❆✳✶✳✸ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❯t✐❧✐③❛çã♦ ❞❡ ❙♦❢t✇❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✹ ❆✳✶✳✹ ❙♦❜r❡ ♦ ❖❜❥❡t✐✈♦ ●❡r❛❧ ❞♦ ❚r❡✐♥❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✺
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ P♦♥t♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛ ❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❘❡t❛ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ P❧❛♥♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛ ❣r❡❣❛ β ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✻ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✼ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✽ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✾ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✵ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✶ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✷ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✸ ❆s r❡t❛s r ❡ s sã♦ ♣❛r❛❧❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✹ ❆s r❡t❛s r ❡ s sã♦ ♣❛r❛❧❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✺ ❆s r❡t❛s r ❡ s sã♦ r❡✈❡rs❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✻ P❧❛♥♦s α ❡β ♣❛r❛❧❡❧♦s ❞✐st✐♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✼ P❧❛♥♦s α ❡β s❡❝❛♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✽ P❧❛♥♦s α ❡β ❝♦✐♥❝✐❞❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✾ ❋✐❣✉r❛s ❣❡♦♠étr✐❝❛s q✉❡ sã♦ P♦❧í❣♦♥♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✈✐✐✐
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✐①
✶✳✹✺ ❉❡♠♦♥str❛çã♦ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✹✻ ❚r✐â♥❣✉❧♦✱ ❝♦♠ ❧❛❞♦s ♠❡❞✐♥❞♦ ❛✱❜ ❡ ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✹✼ ❚r✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ ❝♦♥❤❡❝❡♥❞♦✲s❡ ❞♦✐s ❧❛❞♦s ❡ ✉♠ â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✹✽ ❚r✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✹✾ ❚r✐â♥❣✉❧♦ ❆❍❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✶✳✺✵ ▲♦s❛♥❣♦ ❞❡ ❞✐❛❣♦♥❛✐s d1 ❡ d2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✶✳✺✶ ▲♦s❛♥❣♦ ❝♦♠ ❛s ♣❛r❛❧❡❧❛s às ❞✐❛❣♦♥❛✐s d1 ❡ d2 ♣❛ss❛♥❞♦ ♣♦r ✉♠ ✈ért✐❝❡ ✳ ✳ ✸✹
✶✳✺✷ ❚r❛♣é③✐♦ P◗❘❙ ❡ ❚r❛♣é③✐♦ P✬◗✬❙❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✶✳✺✸ P♦❧í❣♦♥♦ ❘❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✶✳✺✹ P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶✳✺✺ ❙ó❧✐❞♦s q✉❡ ♥ã♦ sã♦ P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶✳✺✻ P♦❧✐❡❞r♦ ❈♦♥✈❡①♦✭♦❝t❛❡❞r♦✮ ❡ ♥ã♦ ❝♦♥✈❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✶✳✺✼ P♦❧✐❡❞r♦s ❘❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✶✳✺✽ P♦❧✐❡❞r♦s ❞❡ P❧❛tã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✷✳✶ ❆s ❛❜❡❧❤❛s ❡ ♦ ♣r✐s♠❛ ❤❡①❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✷ ✏❈❛❧ç❛❞❛ ❞♦s ●✐❣❛♥t❡s✑ ♥❛ ■r❧❛♥❞❛ ❞♦ ◆♦rt❡ ✈✐st❛ ❞❡ ❞♦✐s â♥❣✉❧♦s ❞✐❢❡r❡♥t❡s✳❬✹✷❪ ✹✷ ✷✳✸ ❈r✐st❛✐s ❝♦♠ ❢♦r♠❛s ❣❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✹ ▼♦♦❞② ●❛r❞❡♥s ❆q✉❛r✐✉♠ P②r❛♠✐❞✱ ❚❡①❛s✱ ♥♦s ❊❯❆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✺ ■❣r❡❥❛ ❙ã♦ ❏♦ã♦ ❇❛t✐st❛ ❞❡ ▼❛t✐♣ó ✭✶✾✺✶ ✲ ✶✾✺✼✱ ♣❡rí♦❞♦ ❞❛ ❝♦♥str✉çã♦✮ ✳ ✳ ✹✺ ✷✳✻ P❛r❛❢✉s♦s ❝♦♠ ❛ ❝❛❜❡ç❛ ❡♠ ❢♦r♠❛ ❞❡ ♣r✐s♠❛ ❤❡①❛❣♦♥❛❧ ❡ q✉❛❞r❛♥❣✉❧❛r ✳ ✳ ✹✻ ✷✳✼ ❈❤❛✈❡s ✉t✐❧✐③❛❞❛s ♣❛r❛ ♣❛r❛❢✉s♦s ❝♦♠ ❝❛❜❡ç❛ ❡♠ ❢♦r♠❛ ❞❡ ♣r✐s♠❛ q✉❛❞r❛♥✲
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①
✷✳✶✶ ❱✐st❛ ❞❛ ❝❛❜❡ç❛ ❞♦ ♣❛r❛❢✉s♦ ❤❡①❛❣♦♥❛❧ ❡ ♣❡♥t❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✶✷ ❈♦♠♦ s❡r✐❛ ♦ ❡♥❝❛✐①❡ ❞❛ ❝❤❛✈❡ ♥♦ ♣❛r❛❢✉s♦ ❞❡ ❝❛❜❡ç❛ ♣❡♥t❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✶✸ ❈♦♠♦ s❡r✐❛ ♦ ❡♥❝❛✐①❡ ❞❛ ❝❤❛✈❡ ♥♦ ♣❛r❛❢✉s♦ ❞❡ ❝❛❜❡ç❛ ❤❡①❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✶✹ ❖❜r❛s ❞❡ ❚❛rs✐❧❛ ❞♦ ❆♠❛r❛❧ ♥❛ ❞é❝❛❞❛ ❞❡ ✶✾✷✵✱ r❡tr❛t❛♥❞♦ ❛ ❢♦rt❡ t❡♥❞ê♥❝✐❛
❛ ❣❡♦♠❡tr✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✶✺ ❖❜r❛ ❞❡ ❑❛♥❞✐♥s❦② ♠♦str❛♥❞♦ ❛ ❡①♣❧♦r❛çã♦ ❞❡ ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡ ❡ ❞❛s
❝♦r❡s ❢♦rt❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✶✻ P✐❡t ▼♦♥❞r✐❛♥✳ ❈♦♠♣♦s✐çã♦ ❝♦♠ ✈❡r♠❡❧❤♦✱ ❛♠❛r❡❧♦ ❡ ❛③✉❧✱ ✶✾✷✶✳ Ó❧❡♦
s♦❜r❡ t❡❧❛✱ ✹✵ ① ✸✺❝♠✳ ❈♦❧❡çã♦ ❍❛❛❣s ●❡♠❡❡♥t❡♠✉♥❡✉♠✱ ❍❛✐❛ ❡ ❨✈❡s✳ ▲❛✉r❡♥t✳ ✑▼♦♥❞r✐❛♥✑ ✈❡st✐❞♦ ❞❡ ❞✐❛✱ ❝♦❧❡çã♦ ♦✉t♦♥♦ ❞❡ ✶✾✻✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✶✼ ❙❡r♣❡♥t❡s ✭✶✾✻✾✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✶✽ P♦❧✐❡❞r♦ ❞❡ ❋❧♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✷✳✶✾ ❊✈♦❧✉çã♦ ■■ ✭✶✾✸✾✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✷✵ ❈❛❞❛ ✈❡③ ♠❛✐s P❡q✉❡♥♦ ■ ✭✶✾✺✻✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✷✶ ❉✐❛❣r❛♠❛ ♣❛r❛ ❧✐♠✐t❡s q✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✷✷ ❊s❝❤❡r ❘❡❧❛t✐✈✐t②✱ ✶✾✺✸✳ ▲✐t♦❣r❛✜❛ ✷✽ ① ✷✾✱✶ ❝♠ ❡ ❈♦♥✈❡① ❛♥❞ ❈♦♥❝❛✈❡✱
✶✾✺✺✳ ▲✐t♦❣r❛✜❛✱ ✷✼✺ ① ✸✸✺ ♠♠✳ ❈♦❧❡çã♦ ❈♦r♥❡❧✐✉s ❱❛♥ ❙✳ ❘♦♦s❡✈❡❧t ✳ ✳ ✳ ✺✻ ✷✳✷✸ ❊♠❜❛❧❛❣❡♠ ❞❡ s❛❜ã♦ ❡♠ ♣ó ❛♥t✐❣❛ ❡ ❛t✉❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✸✳✶ ❊❧❡♠❡♥t♦s ❞♦ Pr✐s♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✷ Pr✐s♠❛ ❍❡①❛❣♦♥❛❧ ❘❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✸ Pr✐s♠❛ ❖❜❧íq✉♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✹ Pr✐s♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✺ P❛r❛❧❡❧❡♣í♣❡❞♦ r❡t♦ r❡tâ♥❣✉❧♦✱ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ♦❜❧íq✉♦ ❡ ❝✉❜♦✱ r❡s♣❡❝t✐✈❛✲
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①✐
✸✳✾ ❈❛♣t✉r❛♥❞♦ P♦❧✐❡❞r♦s ✷✳❋♦♥t❡✿ ❙▼❖▲❊✱ ✷✵✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✶✵ ❈❛♣t✉r❛♥❞♦ P♦❧✐❡❞r♦s ✸✳❋♦♥t❡✿ ❙▼❖▲❊✱ ✷✵✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✶✶ P❛r❛❧❡❧❡♣í♣❡❞♦ ❝♦♠ s✉❛ ❞✐❛❣♦♥❛❧ BH ❡ ❛ ❞✐❛❣♦♥❛❧ ❞❛ ❢❛❝❡EH✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✶✷ ❈✉❜♦ ❝♦♠ s✉❛ ❞✐❛❣♦♥❛❧✭d2✮ ❡ ❛ ❞✐❛❣♦♥❛❧ ❞❛ ❢❛❝❡✭d1✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①✐✐
✸✳✸✹ ❘❡s♦❧✉çã♦ ✉t✐❧✐③❛♥❞♦ ♦ ❈❛❜r✐ ✸❉✱ ♣❛ss♦ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✸✳✸✺ ❘❡s♦❧✉çã♦ ✉t✐❧✐③❛♥❞♦ ♦ ❈❛❜r✐ ✸❉✱ ♣❛ss♦ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✸✻ ❘❡s♦❧✉çã♦ ✉t✐❧✐③❛♥❞♦ ♦ ●❡♦❣❡❜r❛ ✸❉✱ ♣❛ss♦ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✸✼ ❘❡s♦❧✉çã♦ ✉t✐❧✐③❛♥❞♦ ♦ ❈❛❜r✐ ✸❉✱♣❛ss♦ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✸✳✸✽ ❘❡s♦❧✉çã♦ ✉t✐❧✐③❛♥❞♦ ♦ ❙❤❛♣❡ ❈❛❧❝✉❧❛t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✸✳✸✾ P✐râ♠✐❞❡s ❞❡ ●✐sé✳ ❋♦♥t❡✿ ■❊❩❩■✱ ✷✵✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✸✳✹✵ P✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✸✳✹✶ ❊❧❡♠❡♥t♦s ❞❛ P✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✸✳✹✷ P✐râ♠✐❞❡ ♦❜❧íq✉❛ ❡ r❡t❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✸✳✹✸ ❚✐♣♦s ❞❡ ♣✐râ♠✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✸✳✹✹ ❚✐♣♦s ❞❡ ♣✐râ♠✐❞❡s ❡ s❡✉s ❡❧❡♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✸✳✹✺ P✐râ♠✐❞❡ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✸✳✹✻ P♦❧í❣♦♥♦s q✉❡ ❢♦r♠❛♠ ❛ ❜❛s❡ ❞❡ ❛❧❣✉♠❛s ♣✐râ♠✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✸✳✹✼ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠ ♣r✐s♠❛ tr✐❛♥❣✉❧❛r ❡♠ três ♣✐râ♠✐❞❡s tr✐â♥❣✉❧❛r❡s ✳ ✳ ✳ ✾✷ ✸✳✹✽ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✸✳✹✾ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✸✳✺✵ ❊①♣❡r✐♠❡♥t♦✱ ♣❛ss♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✸✳✺✶ ❊①♣❡r✐♠❡♥t♦✱ ♣❛ss♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✸✳✺✷ ❊①♣❡r✐♠❡♥t♦✱ ♣❛ss♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✸✳✺✸ ●❡♦❡s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽ ✸✳✺✹ ❆❜r✐♥❞♦ ♦ ♣r♦❣r❛♠❛✱ ♣❛ss♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✸✳✺✺ ❈♦♥str✉çã♦ ❞❡ ♣✐râ♠✐❞❡ ✉s❛♥❞♦ ♦ ❲✐♥❣❡♦♠✱ ♣❛ss♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✸✳✺✻ Pr✐♠❡✐r❛ P❛rt❡ ✲ ❚r❛ç❛♥❞♦ ❛s ❞✐❛❣♦♥❛✐s ❞❛ P✐râ♠✐❞❡✳ ❙❡❣✉♥❞❛ P❛rt❡ ✲
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①✐✐✐
✸✳✺✾ ❚r♦♥❝♦ ❞❡ ♣✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ✸✳✻✵ ❊❧❡♠❡♥t♦s ❞❛ ❢❛❝❡ ❞❡ ✉♠ tr♦♥❝♦ ❞❡ ♣✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸ ✸✳✻✶ P✐râ♠✐❞❡ ❞❡ ❑✉❧❦✉❦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹ ✸✳✻✷ P❧❛♥✐✜❝❛çã♦ ❞♦ tr♦♥❝♦ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ r❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹ ✸✳✻✸ ❚r♦♥❝♦ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ r❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺
✹✳✶ ❆✈❛❧✐❛çã♦ ❞♦ ❚r❡✐♥❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ à s✉❛ r❡❛❧✐❞❛❞❡ ♣r♦✜ss✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✶✷✺ ✹✳✷ ❆✈❛❧✐❛çã♦ ❞♦ ❊q✉✐❧í❜r✐♦ ❚❡♦r✐❛ ❡ Prát✐❝❛ s♦❜r❡ ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞♦s s♦❢t✇❛r❡s
❘❡s✉♠♦
❘❊■❙✱ ▼ô♥✐❝❛ ❞❡ P❛✉❧❛ ❚♦❧❡♥t✐♥♦✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❛❜r✐❧ ❞❡ 2013. ➪r❡❛ ❡ ❱♦❧✉♠❡ ❞❡ Pr✐s♠❛ ❡ P✐râ♠✐❞❡✳ ❖r✐❡♥t❛❞♦r✿ ❆❧❧❛♥ ❞❡ ❖❧✐✈❡✐r❛ ▼♦✉r❛✳ ❈♦✲ ♦r✐❡♥t❛❞♦r❛✿ ❈❛t❛r✐♥❛ ▼❡♥❞❡s ❞❡ ❏❡s✉s✳
❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ❡str❛té❣✐❛s ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ♣♦❧✐❡❞r♦s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❡♥❢♦✲ ❝❛♥❞♦ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡✱ ❝♦♠ ❛♣❧✐❝❛çõ❡s ❞❛s ♥♦✈❛s t❡❝♥♦❧♦❣✐❛s ❝♦♠♦ s♦❢t✇❛r❡s ❣r❛t✉✐t♦s ❡ ❞❡ ❢á❝✐❧ ❛❝❡ss♦ ❡♠ ❛t✐✈✐❞❛❞❡s ❞♦ ❞✐❛ ❛ ❞✐❛✳ ❆q✉✐ t❛♠❜é♠ ❞❡s❝r❡✈❡♠♦s ♦✜❝✐✲ ♥❛s ❝♦♠ ❛t✐✈✐❞❛❞❡s r❡❧❛❝✐♦♥❛❞❛s ❛♦s ♣♦❧✐❡❞r♦s✱ ❝♦♠ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♦♥❞❡ ❝♦♥st❛t♦✉✲s❡ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ♣r♦❢❡ss♦r❡s ♥ã♦ ❢❛③✐❛♠ ✉s♦ ❞❡ s♦❢t✇❛r❡s ❡ ❛t✐✈✐❞❛❞❡s ❝♦♥t✐❞❛s ♥♦ ❧✐✈r♦ ❞✐❞át✐❝♦ ❛❞♦t❛❞♦✳
❆❜str❛❝t
❘❊■❙✱ ▼ô♥✐❝❛ ❞❡ P❛✉❧❛ ❚♦❧❡♥t✐♥♦✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❆♣r✐❧✱ 2013. ❆r❡❛ ❛♥❞ ❱♦❧✉♠❡ ♦❢ Pr✐s♠ ❛♥❞ P②r❛♠✐❞✳ ❆❞✈✐s♦r✿ ❆❧❧❛♥ ❞❡ ❖❧✐✈❡✐r❛ ▼♦✉r❛✳ ❈♦✲ ❛❞✈✐s♦r✿ ❈❛t❛r✐♥❛ ▼❡♥❞❡s ❞❡ ❏❡s✉s✳
❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts str❛t❡❣✐❡s ❢♦r t❡❛❝❤✐♥❣ ♣♦❧②❤❡❞r❛ ✐♥ ❤✐❣❤ s❝❤♦♦❧✱ ❢♦❝✉s✐♥❣ ♦♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❛r❡❛ ❛♥❞ ✈♦❧✉♠❡✱ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ♥❡✇ t❡❝❤♥♦❧♦❣✐❡s ❛s ❢r❡❡ s♦❢t✇❛r❡ ❛♥❞ ❡❛s✐❧② ❛❝❝❡ss✐❜❧❡ ✐♥ t❤❡ ❞❛② t♦ ❞❛② ❛❝t✐✈✐t✐❡s✳ ❍❡r❡ ✇❡ ❞❡s❝r✐❜❡ ❛❧s♦ ✇♦r❦s❤♦♣s ✇✐t❤ ❛❝t✐✈✐t✐❡s r❡❧❛t❡❞ t♦ ♣♦❧②❤❡❞r❛✱ ✇✐t❤ st✉❞❡♥ts ❛♥❞ t❡❛❝❤❡rs ♦❢ t❤❡ s❝❤♦♦❧✱ ✇❤❡r❡ ✐t ✇❛s ❢♦✉♥❞ t❤❛t ♠♦st t❡❛❝❤❡rs ❞✐❞ ♥♦t ✉s❡ s♦❢t✇❛r❡s ❛♥❞ ❛❝t✐✈✐t✐❡s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ t❡①t❜♦♦❦ ❛❞♦♣t❡❞✳
■♥tr♦❞✉çã♦
❖ ❡st✉❞♦ ❞❛ ●❡♦♠❡tr✐❛ ❥á ✈❡♠ s❡♥❞♦ ✐♥❝❡♥t✐✈❛❞♦ ❞❡s❞❡ ✈ár✐♦s sé❝✉❧♦s ❡ s❡♠♣r❡ t❡✈❡ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛s ♠❡❞✐çõ❡s ❡ ♦r✐❡♥t❛çõ❡s✱ s❡❥❛ ♥❛ ♥❛✈❡❣❛çã♦✱ ❝♦♥str✉çã♦ ♦✉ ♣❛r❛ ❡♥t❡♥❞❡r ♦ ✉♥✐✈❡rs♦✳ ❖ ✜❧ós♦❢♦ ❣r❡❣♦ P❧❛tã♦ ✭✹✷✼ ❛✳ ❈ ✲ ✸✹✼ ❛✳ ❈✮✱ ♣♦r ❡①❡♠♣❧♦✱ ❢♦✐ ❣r❛♥❞❡ ✐♥❝❡♥t✐✲ ✈❛❞♦r ❞♦ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ♣❡❧♦ ✐♥t❡r❡ss❡ q✉❡ t✐♥❤❛ ♣❡❧❛ ♠❛t❡♠át✐❝❛✱ ❛❧é♠ ❞❡ ❢✉♥❞❛r ❛ ❆❝❛❞❡♠✐❛✱ ✭s✐t✉❛❞❛ ♥♦s ❥❛r❞✐♥s ❞♦ ❤❡ró✐ ❆❝❛❞❡♠♦✮❬✷❪✳ P❧❛tã♦ t❛♠❜é♠ t❡✈❡ ✈ár✐♦s ❞✐s❝í♣✉❧♦s ❡ ❛♠✐❣♦s q✉❡ ❞❡s❡♥✈♦❧✈❡r❛♠ ✐♠♣♦rt❛♥t❡s tr❛❜❛❧❤♦s ♠❛t❡♠át✐❝♦s ✭♥♦ sé❝✳ ■❱ ❛✳ ❈✳✮✳ ❯♠ ❞❡❧❡s ❢♦✐ ❚❡❡t❡t♦✱ ♦ ❢✉♥❞❛❞♦r ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ❬✶✺❪✳
❆t✉❛❧♠❡♥t❡ s❡ ♣♦❞❡ ♥♦t❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❣❡♦♠❡tr✐❛ ♥❛ ❛rt❡ ❝♦♥t❡♠♣♦râ♥❡❛ ✭❛ tr✐✲ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡✮ ❡ ♥❛ ❡✈♦❧✉çã♦ t❡❝♥♦❧ó❣✐❝❛✱ ❛❧é♠ ❞❡ ❢❛③❡r ♣❛rt❡ ❞♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✳ ❖❜s❡r✈❛r ❡ ❡①♣❧♦r❛r ❡st❛ ❣❡♦♠❡tr✐❛ ❡♠ ❛t✐✈✐❞❛❞❡s ❝♦♠♦ ❢❡✐r❛s ❡ ♦✜❝✐♥❛s✱ ❡♥✈♦❧✈❡♥❞♦ ♣r♦✲ ❢❡ss♦r❡s ❡ ❛❧✉♥♦s✱ ♣♦❞❡ s❡r✈✐r ❝♦♠♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ♥♦✈❛s ❞✐r❡tr✐③❡s ♥❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ♣r♦♣✐❝✐❡ ❛ r❡✢❡①ã♦ ❡ ❛ r❡❡❧❛❜♦r❛çã♦ ❞♦ ♣❡♥s❛♠❡♥t♦ ❛ ♣❛rt✐r ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ r❡❛❧✐❞❛❞❡✱ q✉❡ é ❛ ♠♦t✐✈❛çã♦ ❞♦ t❡♠❛ ❞❡st❡ tr❛❜❛❧❤♦✳
◆❡st❡ tr❛❜❛❧❤♦✱ ♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❢♦✐ ❛❜♦r❞❛r ❡str❛té❣✐❛s ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ♣♦❧✐❡❞r♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛ ♣❛rt✐r ❞❡ ♦✜❝✐♥❛s ❡ ♠♦str❛ ❞❡ ▼❛t❡♠át✐❝❛ ♣♦r ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞❛ r❡❞❡ ♣ú❜❧✐❝❛ ❡st❛❞✉❛❧ ❞♦ ♠✉♥✐❝í♣✐♦ ❞❡ ▼❛t✐♣ó ✲ ▼●✳ ❖ ✐♥t✉✐t♦ é ❞❡s❡♥✈♦❧✈❡r ♦ s❡♥s♦ ❝rít✐❝♦✱ ♦ ♣❡♥s❛♠❡♥t♦ ❤✐♣♦tét✐❝♦ ❡ ❞❡❞✉t✐✈♦✱ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♦❜s❡r✈❛çã♦ ❡ ❞❡ ♣❡sq✉✐s❛ ❡ ❛s ❡str❛té❣✐❛s ❞❡ ❝♦♠✉♥✐❝❛çã♦✱ s❛✐♥❞♦ ❞❛s ❛t✐✈✐❞❛❞❡s tr❛❞✐❝✐♦♥❛✐s ✉s❛❞❛s ♥♦ ❡♥s✐♥♦ ❞❡ ♣♦❧✐❡❞r♦s ❡ ✐♥❝❡♥t✐✈❛♥❞♦ ❛ ✉t✐❧✐③❛çã♦ ❞♦ ❝♦♠♣✉t❛❞♦r ❝♦♠♦✿ ✭✐✮ ❡❧❡♠❡♥t♦ ❞❡ ❛♥✐♠❛çã♦✱ ❝♦♠ ❝❛♣❛❝✐❞❛❞❡ ♣❛r❛ ♠❡❧❤♦r❛r ♦ ❛♠❜✐❡♥t❡ ❣❡r❛❧ ❞❛ ❛✉❧❛❀ ✭✐✐✮ ❡❧❡♠❡♥t♦ ❢❛❝✐❧✐t❛❞♦r✱ ♣❡r♠✐t✐♥❞♦ r❡❛❧✐③❛r ❞❡t❡r♠✐♥❛❞❛s t❛r❡❢❛s tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ r❡❛❧✐③❛❞❛s à ♠ã♦❀ ❡ ✭✐✐✐✮ ❡❧❡♠❡♥t♦ ❞❡ ♣♦s✲ s✐❜✐❧✐❞❛❞❡✱ ♣❡r♠✐t✐♥❞♦ ❡q✉❛❝✐♦♥❛r ❛ r❡❛❧✐③❛çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s q✉❡ s❡r✐❛♠ ❞✐❢í❝❡✐s ❞❡ ❡❢❡t✉❛r ❞❡ ♦✉tr♦ ♠♦❞♦✳
■♥tr♦❞✉çã♦ ✷
é ❝♦♠♦ ❛ ✈✐rt✉❛❧✐❞❛❞❡✱ ❝❛❞❛ ✏♣✐①❡❧✑ q✉❡ ❢♦r♠❛ ❛ ✐♠❛❣❡♠ é ✉♠ q✉❡❜r❛✲❝❛❜❡ç❛ q✉❡ ♥♦s ❞❡✐①❛♠ ✐♥tr✐❣❛❞♦s ❡ ❝✉r✐♦s♦s ♥❛ ❜✉s❝❛ ❞♦ ❡♥t❡♥❞✐♠❡♥t♦✱ ❞❛s ❞❡s❝♦❜❡rt❛s ❞❡ ✉♠ ♠✉♥❞♦ ✐♥✜♥✐t♦✱ ♠❛s ❝❤❡✐♦ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s ❡ ❡①♣❛♥sã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳
❊st❡ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❖ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛ ♦ ❡st✉❞♦ ❞❡ ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ❞❡ tó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛✱ ♥❡❝❡ssár✐♦s ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ■♥✐❝✐❛✲s❡ ❝♦♠ ♦s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s✱ s❡❣✉✐❞♦ ❞♦ ❡st✉❞♦ ❞❡ ♣♦❧í❣♦♥♦s✱ tr✐â♥❣✉❧♦s✱ q✉❛❞r✐❧át❡r♦s✱ ❝✐r❝✉♥✲ ❢❡rê♥❝✐❛ ❡ ❝ír❝✉❧♦✱ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡ ✜♥❛❧✐③❛♥❞♦✱ ❞❡✜♥✐çõ❡s ❞❡ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✱ ✐♥❝❧✉✐♥❞♦ ♣♦❧✐❡❞r♦s✳
❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ r❡✈❡❧❛ q✉❡✱ ♥♦ ♠✉♥❞♦ ❛t✉❛❧✱ ✐♥ú♠❡r❛s sã♦ ❛s ♦❜r❛s ❞❡ ❡♥❣❡♥❤❛r✐❛✱ ❛rq✉✐t❡t✉r❛✱ ❛rt❡s ♣❧ást✐❝❛s✱ ❡♥tr❡ ♦✉tr❛s✱ ♠♦str❛♥❞♦ ❛ ✐♠❡♥s❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢♦r♠❛s q✉❡ ♦ ❤♦♠❡♠ ❞❡s❡♥✈♦❧✈❡✉ t❡♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛✳ ❊st❡ ❝❛♣ít✉❧♦ ♠♦str❛ ✉♠ ♣♦✉❝♦ ❞❛s ♠❛r❛✈✐❧❤❛s ❞❛ ❣❡♦♠❡tr✐❛ ♥❛ ♥❛t✉r❡③❛✱ ♥❛ ❛rq✉✐t❡t✉r❛ ❡ ❝♦♥str✉çã♦ ❝✐✈✐❧✱ ♥♦ ❝♦♠ér❝✐♦✱ ♥♦s ♣❛r❛❢✉s♦s ❡ ♥❛ ❛rt❡✳
❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❛♦s ❝á❧❝✉❧♦s ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ ♣r✐s♠❛ ❡ ♣✐râ♠✐❞❡s✱ ❛ss✐♠ ❝♦♠♦ ❛♣r❡s❡♥t❛ ❛❧❣✉♠❛s ♠❡t♦❞♦❧♦❣✐❛s ❞✐❢❡r❡♥❝✐❛❞❛s ♣❛r❛ t♦r♥❛r ♠❛✐s ❛tr❛t✐✈❛ ❡ ❞✐♥â♠✐❝❛ ❛s ❛✉❧❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ t❡♠❛✳
❖ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ♠♦str❛ ♦ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈✐❞♦ ❝♦♠ ♦s ❛❧✉♥♦s ❞❛ ✷❛ sér✐❡ ❞♦ ❊♥s✐♥♦
❈❛♣ít✉❧♦ ✶
❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❞❡ ●❡♦♠❡tr✐❛
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛✲s❡ ♦ ❡st✉❞♦ ❞♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ♥❡❝❡ssár✐♦s ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ■♥✐❝✐❛✲s❡ ❝♦♠ ♦s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s✱ s❡❣✉✐❞♦ ❞♦ ❡st✉❞♦ ❞❡ ♣♦❧í❣♦♥♦s✱ tr✐â♥❣✉❧♦s✱ q✉❛❞r✐❧át❡r♦s✱ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❝ír❝✉❧♦✱ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡ ✜♥❛❧✐③❛♥❞♦ ❝♦♠ ❞❡✜♥✐çõ❡s ❞❡ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❞❡ ♣♦❧✐❡❞r♦s✳ ❖s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❞❡ss❡ ❝❛♣ít✉❧♦ ♣♦❞❡rã♦ s❡r ❡♥❝♦♥tr❛❞♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠ ❬✸❪✱ ❬✶✶❪✱ ❬✶✷❪✱ ❬✶✸❪✱ ❬✸✷❪ ❡ ❬✹✵❪✳
✶✳✶ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s
❆ ♣❛rt❡ t❡ór✐❝❛ ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ q✉❡ tr❛t❛ ❞❡ ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✲ ♣♦♥t♦✱ r❡t❛ ❡ ♣❧❛♥♦ ✲ ❡ s✉❛s r❡❧❛çõ❡s é ❝❤❛♠❛❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ♣♦s✐çã♦ q✉❡ ✐♥✐❝✐❛✲s❡ ❝♦♠ ♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❛ ❧✉③ ❞❛ t❡♦r✐❛ ❡✉❝❧✐❞✐❛♥❛✳❬✶✶❪
◆♦ ❡st✉❞♦ ❞❛ ●❡♦♠❡tr✐❛✱ ❡①✐st❡♠ ♥♦çõ❡s ♦✉ ❝♦♥❝❡✐t♦s q✉❡ sã♦ ❛❝❡✐t♦s ❝♦♠♦ ✈❡r❞❛❞❡✐✲ r♦s✳ ❖ ♣♦♥t♦✱ ❛ r❡t❛ ❡ ♦ ♣❧❛♥♦ sã♦ ❡❧❡♠❡♥t♦s ❜ás✐❝♦s ❞♦ ❡s♣❛ç♦✱ t❛♠❜é♠ ❝❤❛♠❛❞♦s ❡♥t❡s ❣❡♦♠étr✐❝♦s ♣r✐♠✐t✐✈♦s✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❊✉❝❧✐❞❡s✱ t❡♠♦s✿
❉❡✜♥✐çã♦ ✶✳✶ ✏P♦♥t♦ é ❛q✉✐❧♦ q✉❡ ❞❡ ♥❛❞❛ é ♣❛rt❡✑✱ ♦✉ s❡❥❛✱ ♦ ♣♦♥t♦ é ✉♠ ❡❧❡♠❡♥t♦ ❞❛ ●❡♦♠❡tr✐❛ q✉❡ ♥ã♦ ❤á ❝♦♠♦ ❞✐♠❡♥s✐♦♥❛r✳ ❖s ♣♦♥t♦s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ✉♠❛ ❧❡tr❛ ♠❛✐ús❝✉❧❛ ❞♦ ♥♦ss♦ ❛❧❢❛❜❡t♦✳
❈♦♠♦ ❡①❡♠♣❧♦ ❝✐t❛✲s❡ ❛❧❣♦ ❧♦❝❛❧✐③❛❞♦ ♥♦ ❡s♣❛ç♦✱ ❝♦♠♦ ✉♠ ❢✉r♦✱ ✉♠❛ ❡str❡❧❛ ♥♦ ❝é✉✱ ♦ ❝❡♥tr♦ ❞♦ ❝❛♠♣♦ ❞❡ ❢✉t❡❜♦❧✱ ❡♥tr❡ ♦✉tr♦s✳
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✹
❋✐❣✉r❛ ✶✳✶✿ P♦♥t♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛ ❆ ❉❡✜♥✐çã♦ ✶✳✸ ✏▲✐♥❤❛ é ❝♦♠♣r✐♠❡♥t♦ s❡♠ ❧❛r❣✉r❛✑✳
❯♠❛ r❡t❛ é ✉♥✐❞✐♠❡♥s✐♦♥❛❧✱ t❡♥❞♦ ❛♣❡♥❛s ❝♦♠♣r✐♠❡♥t♦✱ ❡ ✐❧✐♠✐t❛❞❛✱ ♦✉ s❡❥❛✱ ❛ ✜❣✉r❛ r❡♣r❡s❡♥t❛ ❛♣❡♥❛s ✉♠❛ ♣❛rt❡ ❞❡❧❛✳ P♦❞❡ s❡r ❡①❡♠♣❧✐✜❝❛❞♦ ❝♦♠♦ ✉♠❛ ❝❛♥❡t❛✱ ✉♠❛ ❝♦r❞❛ ❡st✐❝❛❞❛✱ ❧❛❞♦s ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ❢✉t❡❜♦❧✱ ❛s tr❛✈❡s ❞♦ ❣♦❧ ❡ ♦s r❛✐♦s s♦❧❛r❡s✳
❊①❡♠♣❧♦ ✶✳✹ ❯♠ ❡①❡♠♣❧♦ ❞❡ r❡t❛ é ❛ ✜❣✉r❛ ✶✳✷ ❛❜❛✐①♦ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❧❡tr❛ r✳
❋✐❣✉r❛ ✶✳✷✿ ❘❡t❛ r
❈♦♠♦ ❛ r❡t❛ t❡♠ ❝♦♠♣r✐♠❡♥t♦ ✐♥✜♥✐t♦✱ é ❢♦r♠❛❞❛ ♣♦r ✐♥✜♥✐t♦s ♣♦♥t♦s✳ ◗✉❛❧q✉❡r ❞♦s s❡✉s ♣♦♥t♦s ❛ ❞✐✈✐❞❡ ❡♠ ❞✉❛s s❡♠✐rr❡t❛s✳ ❆♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❧♦❝❛❧✐③❛❞♦s ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ✜①♦s ❆ ❡ ❇ ✭❡ ✐♥❝❧✉✐♥❞♦ ❡st❡s✮ ❞á✲s❡ ♦ ♥♦♠❡ ❞❡ s❡❣♠❡♥t♦ ❞❡ r❡t❛ AB✳
❉❡✜♥✐çã♦ ✶✳✺ ✏❙✉♣❡r❢í❝✐❡ é ❛q✉✐❧♦ q✉❡ t❡♠ ❝♦♠♣r✐♠❡♥t♦ ❡ ❧❛r❣✉r❛✳✑
❖ P❧❛♥♦ é ✉♠❛ s✉♣❡r❢í❝✐❡ q✉❡ ❛♣r❡s❡♥t❛ ❞✉❛s ❞✐♠❡♥sõ❡s✱ ❧❛r❣✉r❛ ❡ ❝♦♠♣r✐♠❡♥t♦✱ s❡♥❞♦ ✐❧✐♠✐t❛❞♦ ❡♠ s✉❛s ❞✐♠❡♥sõ❡s✳ ◆♦s ❞á ❛ ✐❞❡✐❛ ❞❡ ♣❛rt❡ ❞❡ ✉♠ ♣❧❛♥♦ ✉♠❛ ♣❛r❡❞❡✱ ♦ ❝❤ã♦✱ ✉♠ q✉❛❞r♦✱ ♦ t❛♠♣♦ ❞❡ ✉♠❛ ♠❡s❛✱ ❡t❝✳
❊①❡♠♣❧♦ ✶✳✻ ❆ ❋✐❣✉r❛ ✶✳✸ ✐❧✉str❛ ❛♣❡♥❛s ✉♠❛ ♣❛rt❡ ❞♦ ♣❧❛♥♦ q✉❡ é r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❧❡tr❛ ❣r❡❣❛ ♠✐♥ús❝✉❧❛ β✳
❋✐❣✉r❛ ✶✳✸✿ P❧❛♥♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛ ❣r❡❣❛ β
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✺
✶✳✶✳✶ ❆❧❣✉♥s P♦st✉❧❛❞♦s
❉❡✜♥✐çã♦ ✶✳✼ P♦st✉❧❛❞♦ ✶
❉❛❞❛ ✉♠❛ r❡t❛ r✱ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛ ❡❧❛ ❡ ❤á ✐♥✜♥✐t♦s ♣♦♥t♦s ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ ❡❧❛✳
❊①❡♠♣❧♦ ✶✳✽ ❆ ❋✐❣✉r❛ ✶✳✹ ❛❜❛✐①♦ ✐❧✉str❛ ✉♠❛ r❡t❛ r ❝♦♠ ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛ r❡t❛ ❡ ♣♦♥t♦s ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ ❡❧❛✳
F ∈r;A /∈r;H ∈r;G∈r;C /∈r;I ∈r;B /∈r;D /∈r
❋✐❣✉r❛ ✶✳✹✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✶
❖s ♣♦♥t♦s F✱ G✱ H✱I ❡ J sã♦ ❝♦❧✐♥❡❛r❡s✱ ♣♦✐s ❡①✐st❡ ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❡❧❡s✳
❉❡✜♥✐çã♦ ✶✳✾ P♦st✉❧❛❞♦ ✷
❉❛❞♦ ✉♠ ♣❧❛♥♦ α✱ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛ ❡❧❡ ❡ ✐♥✜♥✐t♦s ♣♦♥t♦s q✉❡ ♥ã♦ ♣❡rt❡♥❝❡♠✳
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✻
❋✐❣✉r❛ ✶✳✺✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✷ A ∈α;H /∈α;I /∈α;F /∈α;D∈α;G /∈α✳
❉❡✜♥✐çã♦ ✶✳✶✶ P♦st✉❧❛❞♦ ✸
❉❛❞♦s ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s A ❡ T✱ ❡①✐st❡ ✉♠❛✱ ❡ s♦♠❡♥t❡ ✉♠❛✱ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❡ss❡s ❞♦✐s ♣♦♥t♦s✱ ♦✉ s❡❥❛✱ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s s❡♠♣r❡ s❡rã♦ ❝♦❧✐♥❡❛r❡s✳
❊①❡♠♣❧♦ ✶✳✶✷ ❖ ❡①❡♠♣❧♦ ❞❡ss❡ ♣♦st✉❧❛❞♦ ❡stá r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ r❡t❛ ❞❛ ❋✐❣✉r❛ ✶✳✻ ❝♦♥✲ t❡♥❞♦ ❞♦✐s ♣♦♥t♦s A ❡ T✱ ♣❡rt❡♥❝❡♥t❡s ❛ ❡❧❛ ✳
❋✐❣✉r❛ ✶✳✻✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✸
❉❡✜♥✐çã♦ ✶✳✶✸ P♦st✉❧❛❞♦ ✹
❙❡ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s A ❡ B ♣❡rt❡♥❝❡♠ ❛ ✉♠ ♣❧❛♥♦ α✱ ❡♥tã♦ ❛ r❡t❛ r q✉❡ ♣❛ss❛ ♣♦r ❡ss❡s ♣♦♥t♦s ❡stá ❝♦♥t✐❞❛ ♥♦ ♣❧❛♥♦✳
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✼
❋✐❣✉r❛ ✶✳✼✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✹
❉❡✜♥✐çã♦ ✶✳✶✺ P♦st✉❧❛❞♦ ✺
❉❛❞♦s três ♣♦♥t♦s ❞✐st✐♥t♦sA✱B❡C✱ ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ ✉♠❛ ♠❡s♠❛ r❡t❛✭ ♥ã♦ ❝♦❧✐♥❡❛r❡s✮✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣❧❛♥♦ α q✉❡ ♣❛ss❛ ♣♦r ❡ss❡s três ♣♦♥t♦s
❊①❡♠♣❧♦ ✶✳✶✻ ❆q✉✐ ♣♦❞❡✲s❡ ✉t✐❧✐③❛r ♦ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❝❛❞❡✐r❛ tr✐♣é✿ ❛ ❡①tr❡♠✐❞❛❞❡ ❞♦s três ♣és s❡ ❛♣ó✐❛ s❡♠♣r❡ s♦❜r❡ ✉♠ ♣❧❛♥♦✳
❋✐❣✉r❛ ✶✳✽✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✺
❉❡✜♥✐çã♦ ✶✳✶✼ P♦st✉❧❛❞♦ ✻
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✽
❊①❡♠♣❧♦ ✶✳✶✽ ❆ ❋✐❣✉r❛ ✶✳✾ ❛❜❛✐①♦ é ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ ♣♦♥t♦ ❞❛❞♦ P ♣❡rt❡♥❝❡♥t❡ ❛s r❡t❛s r✱ s✱ t ❡ u✳
❋✐❣✉r❛ ✶✳✾✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✻
❉❡✜♥✐çã♦ ✶✳✶✾ P♦st✉❧❛❞♦ ✼
❉❛❞♦s ✉♠❛ r❡t❛ r ❡ ✉♠ ♣♦♥t♦ P✱ ♥ã♦ ♣❡rt❡♥❝❡♥t❡ à r❡t❛ r✱ ❡①✐st❡ ✉♠❛✱ ❡ s♦♠❡♥t❡ ✉♠❛✱ r❡t❛ t ♣❛r❛❧❡❧❛ ❛ r ♣❛ss❛♥❞♦ ♣♦rP✳
❊①❡♠♣❧♦ ✶✳✷✵ ❆ ❋✐❣✉r❛ ✶✳✶✵ ✐❧✉str❛ ❡ss❡ ♣♦st✉❧❛❞♦✳
❋✐❣✉r❛ ✶✳✶✵✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✼
❉❡✜♥✐çã♦ ✶✳✷✶ P♦st✉❧❛❞♦ ✽
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✾
❊①❡♠♣❧♦ ✶✳✷✷ ❆ ❋✐❣✉r❛ ✶✳✶✶ ♠♦str❛ ✉♠❛ r❡t❛ ❝♦♠ ✉♠ ♣♦♥t♦ O ♣❡rt❡♥❝❡♥t❡ ❛ ❡❧❛✱ ❞✐✈✐❞✐♥❞♦✲❛ ❡♠ ❞✉❛s s❡♠✐rr❡t❛s ♦♣♦st❛s ❝✉❥❛ ♦r✐❣❡♠ é ♦ ♣♦♥t♦ O✳ −→OA ❡ −→OC sã♦ ❛s s❡♠✐rr❡t❛s ♦♣♦st❛s✳
❋✐❣✉r❛ ✶✳✶✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✽
❉❡✜♥✐çã♦ ✶✳✷✸ P♦st✉❧❛❞♦ ✾
❯♠❛ r❡t❛ r ❝♦♥t✐❞❛ ❡♠ ✉♠ ♣❧❛♥♦α ❞✐✈✐❞❡✲♦ ❡♠ ❞♦✐s s❡♠✐♣❧❛♥♦s ♦♣♦st♦s ❝✉❥❛ ♦r✐❣❡♠ ér✳
❊①❡♠♣❧♦ ✶✳✷✹ ❈♦♠♦ ❡①❡♠♣❧♦ ♦ ♣❧❛♥♦ α ❞❛ ❋✐❣✉r❛ ✶✳✶✷ ❝♦♥t❡♠ ❛ r❡t❛ r✱ q✉❡ ♦ ❞✐✈✐❞❡ ❡♠ ❞♦✐s s❡♠✐♣❧❛♥♦s α1 ❡ α2✳
❋✐❣✉r❛ ✶✳✶✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ P♦st✉❧❛❞♦ ✾
✶✳✶✳✷ P♦s✐çõ❡s r❡❧❛t✐✈❛s ❞❡ ❞✉❛s r❡t❛s
P❛r❛ ❛♥❛❧✐s❛r ❛ ♣♦s✐çã♦ ❞❡ ❞✉❛s r❡t❛s✱ é ♣r❡❝✐s♦ ❝♦♥s✐❞❡r❛r ❞♦✐s ❝❛s♦s✿ ♦ ❞❡ r❡t❛s ❝♦♣❧❛♥❛r❡s ❡ ♦ ❞❡ r❡t❛s r❡✈❡rs❛s✳
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✶✵
❉❡✜♥✐çã♦ ✶✳✷✻ ✳ ❉✉❛s r❡t❛s r ❡ s ❞✐st✐♥t❛s sã♦ ❝♦♥❝♦rr❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣♦ss✉í✲ r❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✳
❊①❡♠♣❧♦ ✶✳✷✼ ❆ ❋✐❣✉r❛ ✶✳✶✸ ❛❜❛✐①♦ ♠♦str❛ ❞✉❛s r❡t❛s r ❡ s ❝♦♣❧❛♥❛r❡s✱ ❝♦♥❝♦rr❡♥t❡s ❝♦♠ ♦ ♣♦♥t♦ P ❡♠ ❝♦♠✉♠✱ ♦✉ s❡❥❛ r∩s = {P}✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ â♥❣✉❧♦ q✉❡ ❢♦r♠❛♠ ❡♥tr❡ s✐✱ ❞✉❛s r❡t❛s ❝♦♥❝♦rr❡♥t❡s ♣♦❞❡♠ s❡r ♣❡r♣❡♥❞✐❝✉❧❛r❡s✱ q✉❛♥❞♦ ❛s r❡t❛s ❢♦r♠❛♠ ✉♠ â♥❣✉❧♦ r❡t♦✱ ♦✉ s❡❥❛✱ ❞❡ 900✱ ♦✉ ♦❜❧íq✉❛s q✉❛♥❞♦ ♥ã♦ sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳
❋✐❣✉r❛ ✶✳✶✸✿ ❆s r❡t❛s r ❡ s sã♦ ♣❛r❛❧❡❧❛s
❉❡✜♥✐çã♦ ✶✳✷✽ ✳ ❉✉❛s r❡t❛s sã♦ ♣❛r❛❧❡❧❛s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❢♦r❡♠ ♦✉ ❝♦✐♥❝✐❞❡♥t❡s ♦✉ ❝♦♣❧❛♥❛r❡s ❡ ♥ã♦ ♣♦ss✉ír❡♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✳ ❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ♣❛r❛❧❡❧❡❧✐s♠♦ ♦❝♦rr❡ s❡ ❛s r❡t❛s ❢♦r❡♠ ❝♦✐♥❝✐❞❡♥t❡s✳
❊①❡♠♣❧♦ ✶✳✷✾ ❆ ❋✐❣✉r❛ ✶✳✶✹ ❛❜❛✐①♦ ✐❧✉str❛ ✉♠ ♣❧❛♥♦ ❡ ❞✉❛s r❡t❛s ♣❛r❛❧❡❧❛sr❡s❞✐st✐♥t❛s ♣❡rt❡♥❝❡♥t❡s ❛ ❡ss❡ ♣❧❛♥♦✳
❋✐❣✉r❛ ✶✳✶✹✿ ❆s r❡t❛s r ❡ s sã♦ ♣❛r❛❧❡❧❛s
❉❡✜♥✐çã♦ ✶✳✸✵ ✳ ❉✉❛s r❡t❛s r ❡ s sã♦ r❡✈❡rs❛s q✉❛♥❞♦ ♥ã♦ sã♦ ❝♦♣❧❛♥❛r❡s ❡ ♥ã♦ ♣♦ss✉❡♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✳
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✶✶
❋✐❣✉r❛ ✶✳✶✺✿ ❆s r❡t❛s r ❡ s sã♦ r❡✈❡rs❛s
❙❡ ❤♦✉✈❡r ✉♠ ♣❧❛♥♦ q✉❡ ❝♦♥t❡♥❤❛ ❛ r❡t❛ r ❡ q✉❡ ❢♦r♠❡ ✉♠ â♥❣✉❧♦ ❞❡ 900 ❝♦♠ ❛ r❡t❛
s✱ ❛s r❡t❛s s❡rã♦ r❡✈❡rs❛s ❡ ♦rt♦❣♦♥❛✐s✳
❚❡♦r❡♠❛ ✶✳✸✷ ❬✹✵❪ ❉❛❞❛s ❞✉❛s r❡t❛s r❡✈❡rs❛s✱ é ♣♦ssí✈❡❧ tr❛ç❛r ✉♠❛ ❡ s♦♠❡♥t❡ ✉♠❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ❛♠❜❛s
✶✳✶✳✸ P♦s✐çõ❡s r❡❧❛t✐✈❛s ❞❡ r❡t❛ ❡ ♣❧❛♥♦
❊①✐st❡♠ três s✐t✉❛çõ❡s ♣♦ssí✈❡✐s ♣❛r❛ ❛ ♣♦s✐çã♦ r❡❧❛t✐✈❛ ❡♥tr❡ ✉♠❛ r❡t❛ ❡ ✉♠ ♣❧❛♥♦✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ❝♦♠✉♥s✿ ♦✉ ❛ r❡t❛ ❡stá ❝♦♥t✐❞❛ ♥♦ ♣❧❛♥♦✱ ♦✉ é ❝♦♥✲ ❝♦rr❡♥t❡ ❛♦ ♣❧❛♥♦✱ ♦✉ é ♣❛r❛❧❡❧❛ ❛ ❡❧❡✳
❉❡✜♥✐çã♦ ✶✳✸✸ ❯♠❛ r❡t❛ ❡stá ❝♦♥t✐❞❛ ❡♠ ✉♠ ♣❧❛♥♦ s❡ t✐✈❡r ❞♦✐s ♣♦♥t♦s ❝♦♠✉♥s ❝♦♠ ♦ ♣❧❛♥♦✳
❉❡✜♥✐çã♦ ✶✳✸✹ ❯♠❛ r❡t❛ é ❝♦♥❝♦rr❡♥t❡ ❛ ✉♠ ♣❧❛♥♦ s❡ t✐✈❡r ✉♠ ú♥✐❝♦ ♣♦♥t♦ ❡♠ ❝♦♠✉♠ ❝♦♠ ♦ ♣❧❛♥♦✳
❉❡✜♥✐çã♦ ✶✳✸✺ ❯♠❛ r❡t❛ ❡ ✉♠ ♣❧❛♥♦ sã♦ ♣❛r❛❧❡❧♦s q✉❛♥❞♦ ♥ã♦ t❡♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✳
❖❜s❡r✈❛çã♦✿ ❙❡ ✉♠❛ r❡t❛ r é ♣❛r❛❧❡❧❛ ❛ ✉♠ ♣❧❛♥♦α✱ ❡♥tã♦ ❡❧❛ s❡rá ♣❛r❛❧❡❧❛ ♦✉ r❡✈❡rs❛ ❛ q✉❛❧q✉❡r r❡t❛ ❞♦ ♣❧❛♥♦✱ ♣♦✐s✱ ♣❛r❛ ✉♠❛ r❡t❛ r✴✴α t❡♠♦sr∩ α=φ✳
✶✳✶✳✹ P♦s✐çõ❡s r❡❧❛t✐✈❛s ❞❡ ❞♦✐s ♣❧❛♥♦s
✶✳✶✳ ❈♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ✶✷
❉❡✜♥✐çã♦ ✶✳✸✻ ❉♦✐s ♣❧❛♥♦s ❞✐st✐♥t♦s sã♦ ♣❛r❛❧❡❧♦s q✉❛♥❞♦ ♥ã♦ ♣♦ss✉ír❡♠ ♣♦♥t♦s ❡♠ ❝♦♠✉♠✳
❖❜s❡r✈❛çã♦✿ ❙❡ ❞♦✐s ♣❧❛♥♦s ❞✐st✐♥t♦s sã♦ ♣❛r❛❧❡❧♦s✱ ❡♥tã♦ q✉❛❧q✉❡r r❡t❛ ❞❡ ✉♠ ❞❡❧❡s é ♣❛r❛❧❡❧❛ ❛♦ ♦✉tr♦✱ ❡ q✉❛❧q✉❡r r❡t❛ ❝♦♥❝♦rr❡♥t❡ ❛ ✉♠ ❞❡❧❡s t❛♠❜é♠ é ❝♦♥❝♦rr❡♥t❡ ❛♦ ♦✉tr♦✳
❊①❡♠♣❧♦ ✶✳✸✼ ❈♦♠♦ ❡①❡♠♣❧♦✱ ❛ ❋✐❣✉r❛ ✶✳✶✻ ♠♦str❛ ❞♦✐s ♣❧❛♥♦s ❞✐st✐♥t♦s α ❡ β✱ ♣❛r❛✲ ❧❡❧♦s✳
❋✐❣✉r❛ ✶✳✶✻✿ P❧❛♥♦s α ❡β ♣❛r❛❧❡❧♦s ❞✐st✐♥t♦s✳
❉❡✜♥✐çã♦ ✶✳✸✽ ❉♦✐s ♣❧❛♥♦s ❞✐st✐♥t♦s sã♦ s❡❝❛♥t❡s ✭♦✉ ❝♦♥❝♦rr❡♥t❡s✮ q✉❛♥❞♦ ♣♦ss✉❡♠ ♣♦♥t♦s ❡♠ ❝♦♠✉♠✳
❖❜s❡r✈❛çã♦✿ ❊st❡s ♣♦♥t♦s ❞❛ ✐♥t❡rs❡❝çã♦ ❞♦s ❞♦✐s ♣❧❛♥♦s ❢♦r♠❛♠ ✉♠❛ r❡t❛✳ ❈❛s♦ ♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣❡❧♦s ❞♦✐s ♣❧❛♥♦s ❢♦r ❞❡ 900 ♦s ♣❧❛♥♦s s❡rã♦ s❡❝❛♥t❡s ❡ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳
❊①❡♠♣❧♦ ✶✳✸✾ ❆ ❋✐❣✉r❛ ✶✳✶✼ ❛❜❛✐①♦ r❡♣r❡s❡♥t❛ ❞♦✐s ♣❧❛♥♦s ❝♦♥❝♦rr❡♥t❡s ♦✉ s❡❝❛♥t❡s α ❡ β ❡ ❛ ✐♥t❡rs❡çã♦ ❞♦s ❞♦✐s ♣❧❛♥♦s é ❛ r❡t❛ r✳
✶✳✷✳ P♦❧í❣♦♥♦s ✶✸
❉❡✜♥✐çã♦ ✶✳✹✵ ❉♦✐s ♣❧❛♥♦s sã♦ ❝♦✐♥❝✐❞❡♥t❡s q✉❛♥❞♦ t✐✈❡r❡♠ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ❡♠ ❝♦♠✉♠✳
❊①❡♠♣❧♦ ✶✳✹✶ ❖s ♣❧❛♥♦sα ❡β ❞❛ ❋✐❣✉r❛ ✶✳✶✽ ❛❜❛✐①♦ sã♦ ❝♦✐♥❝✐❞❡♥t❡s✱ ♦✉ s❡❥❛✱α∩β =
α=β✳
❋✐❣✉r❛ ✶✳✶✽✿ P❧❛♥♦s α ❡β ❝♦✐♥❝✐❞❡♥t❡s✳
✶✳✷ P♦❧í❣♦♥♦s
❉❡✜♥✐çã♦ ✶✳✹✷ ❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ (A1, A2, ..., An)❝♦♠ n≥3✱
t♦❞♦s ❞✐st✐♥t♦s✱ ♦♥❞❡ três ♣♦♥t♦s ❝♦♥s❡❝✉t✐✈♦s ♥ã♦ sã♦ ❝♦❧✐♥❡❛r❡s✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ❝♦♥s❡❝✉✲ t✐✈♦s An−1✱ An e A1✱ ❛ss✐♠ ❝♦♠♦✱ An, A1 e A2✱ ❝❤❛♠❛✲s❡ ♣♦❧í❣♦♥♦ à r❡✉♥✐ã♦ ❞❡ s❡❣♠❡♥t♦s
A1A2, A2A3, ..., An−1An, AnA1✳ ❖s ♣♦♥t♦s A1, A2, A3..., An−1, An sã♦ ♦s ✈ért✐❝❡s ❞♦
♣♦❧í❣♦♥♦✳ ❖s s❡❣♠❡♥t♦s A1A2, A2A3, ..., An−1An, AnA1 sã♦ ♦s ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦✳ ❊ ♦s
â♥❣✉❧♦sAc1 =AnAc1A2,Ac2 =A1Ac2A3, ...Acn =An−1AcnA1✱ sã♦ ♦s â♥❣✉❧♦s ❞♦ ♣♦❧í❣♦♥♦✳ ❬✶✷❪
❊①❡♠♣❧♦ ✶✳✹✸ ❆ ❋✐❣✉r❛ ✶✳✶✾ ♠♦str❛ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ♣♦❧í❣♦♥♦s✳
❋✐❣✉r❛ ✶✳✶✾✿ ❋✐❣✉r❛s ❣❡♦♠étr✐❝❛s q✉❡ sã♦ P♦❧í❣♦♥♦s✳
✶✳✷✳ P♦❧í❣♦♥♦s ✶✹
❊①❡♠♣❧♦ ✶✳✹✺ ❆ ❋✐❣✉r❛ ✶✳✷✵ ♠♦str❛ ❛❧❣✉♠❛s ✜❣✉r❛s ❞❡ ♥ã♦ ♣♦❧í❣♦♥♦s✳
❋✐❣✉r❛ ✶✳✷✵✿ ❋✐❣✉r❛s ❣❡♦♠étr✐❝❛s q✉❡ ♥ã♦ sã♦ P♦❧í❣♦♥♦s✳
❖s ♣♦❧í❣♦♥♦s ♣♦❞❡♠ s❡r ❝♦♥✈❡①♦s ♦✉ ♥ã♦ ❝♦♥✈❡①♦s✳
❉❡✜♥✐çã♦ ✶✳✹✻ ❯♠ ♣♦❧í❣♦♥♦ é ❝♦♥✈❡①♦ q✉❛♥❞♦ t♦❞♦ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛✱ ❝✉❥❛s ❡①tr❡♠✐✲ ❞❛❞❡s ♣❡rt❡♥❝❡♠ ❛ ❡ss❡ ♣♦❧í❣♦♥♦✱ t❡♠ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ♥♦ ✐♥t❡r✐♦r ❞♦ ♣♦❧í❣♦♥♦✳
❊①❡♠♣❧♦ ✶✳✹✼ ❯♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ é ♦ ❤❡♣tá❣♦♥♦ ❛❜❛✐①♦✭❋✐❣✉r❛ ✶✳✷✶✮✱ ♠♦str❛♥❞♦ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❝♦♠ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ♥♦ ✐♥t❡r✐♦r ❞♦ ♣♦❧í❣♦♥♦✳
❋✐❣✉r❛ ✶✳✷✶✿ ❖❝tó❣♦♥♦ ✲ P♦❧í❣♦♥♦ ❈♦♥✈❡①♦✳
❉❡✜♥✐çã♦ ✶✳✹✽ ❯♠ ♣♦❧í❣♦♥♦ é ♥ã♦ ❝♦♥✈❡①♦ q✉❛♥❞♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛✱ ❝✉❥❛s ❡①tr❡♠✐❞❛❞❡s ♣❡rt❡♥❝❡♠ ❛ ❡ss❡ ♣♦❧í❣♦♥♦✱ ♣♦ré♠ ♥ã♦ t❡♠ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ♥♦ ✐♥t❡r✐♦r ❞♦ ♣♦❧í❣♦♥♦✳
✶✳✷✳ P♦❧í❣♦♥♦s ✶✺
❋✐❣✉r❛ ✶✳✷✷✿ P♦❧í❣♦♥♦ ♥ã♦ ❝♦♥✈❡①♦✳
❯♠ ♣♦❧í❣♦♥♦ é ❢♦r♠❛❞♦ ♣♦r ✈ár✐♦s ❡❧❡♠❡♥t♦s✳ ❆ ❋✐❣✉r❛✶✳✷✸ ✐❧✉str❛ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♠ s❡✉s r❡s♣❡❝t✐✈♦s ❡❧❡♠❡♥t♦s q✉❡ ❧✐st❛♠♦s ❛ s❡❣✉✐r✳
❋✐❣✉r❛ ✶✳✷✸✿ P♦❧í❣♦♥♦ ❡ s❡✉s ❡❧❡♠❡♥t♦s✳
✶✳ ▲❛❞♦s✿
❈❛❞❛ ✉♠ ❞♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛ q✉❡ ✉♥❡ ✈ért✐❝❡s ❝♦♥s❡❝✉t✐✈♦s✳ ◆❛ ❋✐❣✉r❛ ✶✳✷✸ sã♦ ❡❧❡s✿ AB✱ BC✱ CD✱ DE ❡EA✳
✷✳ ❱ért✐❝❡s✿
P♦♥t♦ ❢♦r♠❛❞♦s ♣♦r ❞♦✐s ❧❛❞♦s✳ ◆❛ ❋✐❣✉r❛ ✶✳✷✸ sã♦ ✈ért✐❝❡s ❞♦ ♣♦❧í❣♦♥♦ ♦s ♣♦♥t♦s✿ A✱ B✱ C✱D ❡ E✳
✸✳ ❉✐❛❣♦♥❛✐s✿
✶✳✷✳ P♦❧í❣♦♥♦s ✶✻
✹✳ ➶♥❣✉❧♦s ✐♥t❡r♥♦s✿
➶♥❣✉❧♦s ❢♦r♠❛❞♦s ♣♦r ❞♦✐s ❧❛❞♦s ❝♦♥s❡❝✉t✐✈♦s✳ ◆❛ ❋✐❣✉r❛ ✶✳✷✸ sã♦ ❡❧❡s✿ ba,bb, bc, d eb be✳ ✺✳ ➶♥❣✉❧♦s ❡①t❡r♥♦s✿
➶♥❣✉❧♦s ❢♦r♠❛❞♦s ♣♦r ✉♠ ❧❛❞♦ ❡ ♣❡❧♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞♦ ❧❛❞♦ ❛ ❡❧❡ ❝♦♥s❡❝✉t✐✈♦✳ ◆❛ ❋✐❣✉r❛ ✶✳✷✸ sã♦ â♥❣✉❧♦s ❡①t❡r♥♦s✿ ab1, bb1,cb1, db1eeb1;
❖s ♣♦❧í❣♦♥♦s r❡❝❡❜❡♠ ♥♦♠❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s q✉❡ ♣♦ss✉❡♠✳
❋✐❣✉r❛ ✶✳✷✹✿ P♦❧í❣♦♥♦s ❡ ♥♦♠❡♥❝❧❛t✉r❛
❖❜s❡r✈❛çã♦ ✶✳✺✵ ❖s q✉❡ ♥ã♦ ❛♣❛r❡❝❡♠ ❧✐st❛❞♦s ❛❝✐♠❛ sã♦ ❞❡♥♦♠✐♥❛❞♦s t❛♠❜é♠ ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s q✉❡ ♣♦ss✉❡♠✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ♣♦❧í❣♦♥♦ ❞❡ tr❡③❡ ❧❛❞♦s✱ ♣♦❧í❣♦♥♦ ❞❡ tr✐♥t❛ ❧❛❞♦s✱ ❡t❝✳
❉❡✜♥✐çã♦ ✶✳✺✶ ❯♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ é r❡❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t❡♠ t♦❞♦s ♦s ❧❛❞♦s ❝♦♥❣r✉❡♥t❡s ❡ t♦❞♦s ♦s â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s✳
❚❡♦r❡♠❛ ✶✳✺✷ ❬✹✵❪ ❙❡❥❛ P ✉♠ ♣♦❧í❣♦♥♦ ❝♦♠ ♥ ❧❛❞♦s✳ ❊♥tã♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❞✐❛❣♦♥❛✐s ❞❡ss❡ ♣♦❧í❣♦♥♦ é✿
d= n(n−3)
✶✳✷✳ P♦❧í❣♦♥♦s ✶✼
❚❡♦r❡♠❛ ✶✳✺✸ ❬✶✷❪ ❆ s♦♠❛ Si ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❞❡ ♥ ❧❛❞♦s (n≥0) é ❞❛❞❛ ♣♦r✿
Si = (n−2)·1800.
❉❡❞✉çã♦✿
❙❡❥❛ A1A2A3...An ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❞❡ ♥ ❧❛❞♦s✳
❉❡ ✉♠ ✈ért✐❝❡ q✉❛❧q✉❡r ❝♦♥❞✉③✐♠♦s t♦❞❛s ❛s ❞✐❛❣♦♥❛✐s q✉❡ t❡♠ ❡st❡ ✈ért✐❝❡ ❝♦♠♦ ❡①tr❡♠♦✳
❖ ♣♦❧í❣♦♥♦ ✜❝❛ ❡♥tã♦ ❞✐✈✐❞✐❞♦ ❡♠(n−2) tr✐â♥❣✉❧♦s ❡ ❛ s♦♠❛Si ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s
❞♦ ♣♦❧í❣♦♥♦ Si = i1 +i2 +i3 +...+in é ✐❣✉❛❧ à s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦s (n−2)
tr✐â♥❣✉❧♦s✳ ▲♦❣♦✿
Si = (n−2)·1800.
❙❡ ♦ ♣♦❧í❣♦♥♦ ❢♦r r❡❣✉❧❛r✱ ❡♥tã♦✿
ai =
(n−2)·1800
n .
❚❡♦r❡♠❛ ✶✳✺✹ ❬✶✷❪ ❆ s♦♠❛ Se ❞♦s â♥❣✉❧♦s ❡①t❡r♥♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ é✿
Se = 3600.
❉❡❞✉çã♦✿
❙❡❥❛ A1A2A3...An ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❞❡n ❧❛❞♦s✳
❈♦♥s✐❞❡r❛♥❞♦ ♦s e1, e2, e3, ..., en s✉♣❧❡♠❡♥t❛r❡s ❛❞❥❛❝❡♥t❡s ❛♦s r❡s♣❡❝t✐✈♦s â♥❣✉❧♦s ✐♥✲
t❡r♥♦s
i1, i2, i3, ..., in
❚❡♠♦s✿
✶✳✸✳ ❚r✐â♥❣✉❧♦s ✶✽
e2+i2 = 1800
e3+i3 = 1800
.
.
.
en+in=n·1800
❙♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s n ✐❣✉❛❧❞❛❞❡s✿
Se+Si =n·1800
✳
❙✉❜st✐t✉✐♥❞♦ Si ♣♦r(n−2)·1800✱ ✈❡♠✿
Se+ (n−2)·1800 =n·1800
Se+n·1800−3600 =n·1800
Se= 3600
❙❡ ♦ ♣♦❧í❣♦♥♦ ❢♦r r❡❣✉❧❛r✱ ❡♥tã♦✿
ae= 3600
n .
✶✳✸ ❚r✐â♥❣✉❧♦s
❉❡✜♥✐çã♦ ✶✳✺✺ ❯♠ tr✐â♥❣✉❧♦ é ✉♠ ♣♦❧í❣♦♥♦ ❞❡ três ❧❛❞♦s✳
❊①❡♠♣❧♦ ✶✳✺✻ ❆ ✜❣✉r❛ ❛❜❛✐①♦ r❡♣r❡s❡♥t❛ ✉♠ tr✐â♥❣✉❧♦ ❆❇❈ ❝♦♠ s❡✉s ❡❧❡♠❡♥t♦s✿ ❱ért✐❝❡s✿ sã♦ ♦s ♣♦♥t♦s A✱ B ❡ C✳
✶✳✸✳ ❚r✐â♥❣✉❧♦s ✶✾
❋✐❣✉r❛ ✶✳✷✺✿ ❚r✐â♥❣✉❧♦ ❡ s❡✉s ❡❧❡♠❡♥t♦s
❖s tr✐â♥❣✉❧♦s ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦s ❧❛❞♦s ❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦s â♥❣✉❧♦s✳ ❊♠ r❡❧❛çã♦ ❛♦s ❧❛❞♦s✱ ✉♠ tr✐â♥❣✉❧♦ ♣♦❞❡ s❡r✿ ❡q✉✐❧át❡r♦✱ ✐sós❝❡❧❡s ❡ ❡s❝❛❧❡♥♦✳
❉❡✜♥✐çã♦ ✶✳✺✼ ❯♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ é ❛q✉❡❧❡ q✉❡ t❡♠ ♦s três ❧❛❞♦s ❝♦♥❣r✉❡♥t❡s✳
❊①❡♠♣❧♦ ✶✳✺✽ ❖ tr✐â♥❣✉❧♦ ❆❇❈ ❞❛ ❋✐❣✉r❛✶✳✷✻ ❛❜❛✐①♦ t❡♠ t♦❞♦s ♦s ❧❛❞♦s ❝♦♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛✳
❋✐❣✉r❛ ✶✳✷✻✿ ❚r✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦
❉❡✜♥✐çã♦ ✶✳✺✾ ❯♠ tr✐â♥❣✉❧♦ q✉❡ ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❧❛❞♦s ❝♦♥❣r✉❡♥t❡s é ❞❡♥♦♠✐✲ ♥❛❞♦ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✳
