❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
P♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ♥♦ ❊♥s✐♥♦
▼é❞✐♦
†♣♦r
❍❡r❝✉❧❡s ❞♦ ◆❛s❝✐♠❡♥t♦ ❙✐❧✈❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞❛
Pr♦❢➟ ❉r➟ ▼✐r✐❛♠ ❞❛ ❙✐❧✈❛ P❡r❡✐r❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥✲ t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ P❘❖❋▼❆❚✲❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❥✉♥t♦ ❛♦ Pr♦✲ ❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ Pr♦✜ss✐♦♥❛❧ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋✲ ▼❆❚✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛✲ r❛í❜❛✳
❆❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❆❣r❛❞❡❝✐♠❡♥t♦s
❖s ❛❣r❛❞❡❝✐♠❡♥t♦s sã♦ ✐♥ú♠❡r♦s✱ ♣♦✐s ❢♦r❛♠ t❛♥t❛s ♣❡ss♦❛s q✉❡ ♠❡ ✜③❡r❛♠ ❝❤❡✲ ❣❛r ❛ ❡ss❡ ♠♦♠❡♥t♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s ❡ ♠❡✐♦ s❡♠♣r❡ ❡st❡✈❡ ♣r❡s❡♥t❡ ❝♦♠♦ ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦ ❡ ♣r♦t❡çã♦ ♥❛s t❛♥t❛s ❡ t❛♥t❛s ✈✐❛❣❡♥s✱ ♣r♦✈❛s✳ ❙❡✐ q✉❡ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❊❧❡ ❡st❛✈❛✱ ❡stá ❡ ❡st❛rá ❣✉✐❛♥❞♦✲♠❡✳
❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ s❡♠♣r❡ ♠❡ ❛❥✉❞❛r❛♠ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s ❞❡ss❛ ❥♦r♥❛❞❛✱ s❡❥❛ ❝♦♠ ✉♠❛ ♣❛❧❛✈r❛ ❞❡ ✐♥❝❡♥t✐✈♦✱ ❝♦♠ ♦r❛çõ❡s ♦✉ ✈✐♥❞♦ ❝♦♠✐❣♦ ❡♠ ❞✐❛s ❞❡ ♣r♦✈❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ t✐❛ ▼❛r✐❛✱ ♠❡✉s ✐r♠ã♦s ❏✉❝❡❧♦ ❡ ■♦♥❛r❛ ❡ ❛♦s ♠❡✉s ♣❛✐s ❏♦sé P❡❞r♦ ❡ ❍♦s❛♥❛ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ♥♦s ❡st✉❞♦s ❡ ♠❡ ❡♥s✐♥❛r❛♠ q✉❡ ♣❛r❛ ❝♦♥q✉✐st❛r ❛❧❣♦ ♥❡ss❛ ✈✐❞❛ é ♣r❡❝✐s♦❀ ♣❛❝✐ê♥❝✐❛✱ ❤♦♥❡st✐❞❛❞❡ ❡ ❞❡❞✐❝❛çã♦✳
❆ ♠✐♥❤❛ ❡s♣♦s❛ ❈r✐st✐♥❛ ❝❡rt❛♠❡♥t❡ ❛ ♣❡ss♦❛ q✉❡ ♠❛✐s ♠❡ ✐♥❝❡♥t✐✈♦✉ ♥❡ss❡ ♣❡✲ rí♦❞♦✱ ♠❡s♠♦ ♥♦s ♠♦♠❡♥t♦s q✉❡ ♣❡♥s❡✐ ❡♠ ❞❡s✐st✐r s✐♠♣❧❡s♠❡♥t❡ ✈♦❝ê ❢❛❧❛✈❛ ❛❧❣♦ ❡ ✐ss♦ ❜❛st❛✈❛ ♣r❛ ♠❡ tr❛③❡r â♥✐♠♦ ❡ ❝♦♥t✐♥✉❛r✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ t✉r♠❛ ❛♦s q✉❡ ❝❤❡❣❛r❛♠ ❛té ❡ss❡ ♠♦♠❡♥t♦ ♦✉ q✉❡ ♣♦r ❛❧❣✉♠ ♠♦t✐✈♦ ❞❡s✐st✐r❛♠ ♥♦ ❝❛♠✐♥❤♦✱ ♥❛s ❜r✐♥❝❛❞❡✐r❛s✱ ♥♦s ❜❛t❡ ♣❛♣♦s✱ ♥♦ ❡st✉❞♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ❛♠✐❣♦s ❙❛❧❛t✐❡❧✱ ❋é❧✐①✱ ❉♦✈❛❧ ❡ ◆❡t♦ s❛✐❜❛♠ q✉❡ ✈♦❝ês ❢♦r❛♠ ❡ss❡♥❝✐❛✐s ♣❛r❛ ❡ss❛ ❝♦♥q✉✐st❛✳
❆♦s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ❡ ❛❧✉♥♦s ❞❛ ❊❘❊▼ ❇❡♥❡❞✐t❛ ❞❡ ▼♦r❛✐s ●✉❡rr❛ q✉❡ ❝♦♠ ♣❛❧❛✈r❛s ❞❡ ✐♥❝❡♥t✐✈♦✱ ♠❡♥s❛❣❡♥s✱ ❧✐❣❛çõ❡s ❡ ♦r❛çõ❡s ❢♦r❛♠ ❛❧✐❝❡r❝❡s ♣❛r❛ ♠❡ ♠❛♥t❡r ✜r♠❡ ❡♠ ♠♦♠❡♥t♦s ❞❡ ❢r❛q✉❡③❛✳
➚ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ♠❡str❛❞♦ q✉❡ ❝♦♠ ❞❡❞✐❝❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛ ♥♦s ♣r♦♣♦r✲ ❝✐♦♥❛r❛♠ ♦ ❝❛♠✐♥❤♦ ❛ ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❛ ♣r♦❢❡ss♦r❛ ▼✐r✐❛♠✱ q✉❡ ❛❧é♠ ❞❡ ♣❛❝✐❡♥t❡ ❡ ❞❡❞✐❝❛❞❛✱ s✉❛s ♦r✐❡♥t❛çõ❡s ❡ ❝♦♥s❡❧❤♦s ❢♦r❛♠ ♠❛✐s q✉❡ ❢✉♥❞❛♠❡♥t❛✐s ❡ ♦ ♣r♦❢❡ss♦r ❋❧❛♥❦ q✉❡ t❛♠❜é♠ ❢♦✐ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ♦r❣❛♥✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
➚ t♦❞♦s q✉❡ ❞✐r❡t❛ ❡ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ss❛ ❡t❛♣❛ ❞❡ ♠✐♥❤❛ ✈✐❞❛✳
❉❡❞✐❝❛tór✐❛
➚ t♦❞♦s q✉❡ ❞❡❞✐❝❛♠✲s❡ ♣❛r❛ ✉♠❛ ❡❞✉❝❛çã♦ ❞❡ q✉❛❧✐❞❛❞❡✳ ✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ ♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✱ ❝♦♠♣❛r❛♥❞♦ ❡ ❞✐s❝✉t✐♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❡ ❛s ❞❡✜♥✐çõ❡s q✉❡ sã♦ ❞❛❞❛s ♥♦ ❡st✉❞♦ ❞♦s ♣♦❧✐❡❞r♦s r❡✲ ❣✉❧❛r❡s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♥❛s ❡s❝♦❧❛s ❜r❛s✐❧❡✐r❛s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦✳ Pr♦✈❛♠♦s ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ❝❛❧❝✉❧❛♠♦s ár❡❛s ❞❡ s✉♣❡r❢í❝✐❡s ❡ ♦s ✈♦❧✉♠❡s ❞♦s ♣♦❧✐✲ ❡❞r♦s r❡❣✉❧❛r❡s✳ P♦r ✜♠✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s s♦❢t✇❛r❡s ♠❛t❡♠át✐❝♦s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ♣❡❧♦s ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ♥❛s ❛✉❧❛s ❞❡ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ ❝♦♠♦ ♠❛t❡r✐❛❧ ❛✉①✐❧✐❛r ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡st❡ t❡♠❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ●❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✱ P♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✱ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ❙♦❢t✇❛r❡s ❞❡ ▼❛t❡♠át✐❝❛✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ ♣r❡s❡♥t ❛ st✉❞② ♦❢ t❤❡ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛✱ ❝♦♠♣❛r✐♥❣ ❛♥❞ ❞✐s❝✉s✲ s✐♥❣ t❤❡ ❝♦♥❝❡♣ts ❛♥❞ ❞❡✜♥✐t✐♦♥s ❣✐✈❡♥ ✐♥ t❤❡ st✉❞② ♦❢ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛ ✐♥ t❡①t❜♦♦❦s ♠♦st ✇✐❞❡❧② ✉s❡❞ ✐♥ ❇r❛③✐❧✐❛♥ ❤✐❣❤ s❝❤♦♦❧s✳ ❲❡ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ♦❢ ❊✉❧❡r✱ ✇❡ ❝❛❧✲ ❝✉❧❛t❡ s✉r❢❛❝❡ ❛r❡❛s ❛♥❞ ✈♦❧✉♠❡s ♦❢ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛✳ ❋✐♥❛❧❧②✱ ✇❡ ♣r❡s❡♥t s♦♠❡ ♠❛t❤❡♠❛t✐❝❛❧ s♦❢t✇❛r❡ t❤❛t ❝❛♥ ❜❡ ✉s❡❞ ❜② st✉❞❡♥ts ❛♥❞ ♠❛t❤❡♠❛t✐❝s t❡❛❝❤❡rs ✐♥ t❤❡ s♣❛t✐❛❧ ❣❡♦♠❡tr② ❝❧❛ss❡s ❛s ❛✉①✐❧✐❛r② ♠❛t❡r✐❛❧ ✐♥ t❤❡ t❡❛❝❤✐♥❣ ❛♥❞ ❧❡❛r♥✐♥❣ ♦❢ t❤✐s s✉❜❥❡❝t ✐♥ t❤❡ ❝❧❛ssr♦♦♠✳
❑❡②✇♦r❞s✿ s♣❛t✐❛❧ ❣❡♦♠❡tr②✱ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛✱ ❊✉❧❡r✬s ❚❤❡♦r❡♠✱ ▼❛t❤❡♠❛t✐❝s ❙♦❢t✇❛r❡✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ①✐✐✐
✶ ❈♦♥❝❡✐t♦ ❍✐stór✐❝♦ ✶
✷ ❖s ♣♦❧✐❡❞r♦s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✺ ✷✳✶ P♦❧✐❡❞r♦s ❝♦♥✈❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❊❧❡♠❡♥t♦s ❞♦s ♣♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸ ❘❡❧❛çõ❡s ❡♥tr❡ ♦s ❊❧❡♠❡♥t♦s ❞♦s P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸✳✶ ❋❛❝❡s ❡ ❆r❡st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸✳✷ ❱ért✐❝❡s ❡ ❆r❡st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸✳✸ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✸ ❆ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r ✶✹ ✹ P♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ✶✼ ✹✳✶ ❊①✐stê♥❝✐❛ ❞❡ ❝✐♥❝♦ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✹✳✷ ➪r❡❛s ❡ ❱♦❧✉♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷✳✶ ➪r❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷✳✷ ❱♦❧✉♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✸ ❖✉tr♦s P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺ Pr♦s♣♦st❛s ❞❡ ❙♦❢t✇❛r❡s ✹✵ ✺✳✶ ❈❛❜r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✷ P♦❧② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✸ ❖✉tr♦s ❙♦❢t✇❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✻
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❋r❛❣♠❡♥t♦ ❞♦ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❙í♠❜♦❧♦ ❞❛ ❡s❝♦❧❛ P✐t❛❣ór✐❝❛✱ ♦ ♣❡♥t❛❣r❛♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ❙ó❧✐❞♦s P❧❛tô♥✐❝♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❡❧❡♠❡♥t♦s ❞❛ ♥❛t✉r❡③❛✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ P✐râ♠✐❞❡s ❞♦ ❊❣✐t♦ ❡ ❛ P✐râ♠✐❞❡ ❞♦ ♠✉s❡✉ ❞♦ ▲♦✉✈r❡✱ ❢♦r♠❛s ♣♦❧✐é✲
❞r✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✶ ❋✐❣✉r❛s P♦❧✐é❞r✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ❋✐❣✉r❛s ♥ã♦ P♦❧✐é❞r✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ P♦❧✐❡❞r♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✹ P♦❧✐❡❞r♦s ♥ã♦ ❝♦♥✈❡①♦ ✭à ❡sq✉❡r❞❛✮ ❡ P♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ✭à ❞✐r❡✐t❛✮✳ ✳ ✳ ✽ ✷✳✺ P♦❧í❣♦♥♦s ♥♦ ♣❧❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✻ P♦❧✐❡❞r♦P ♣❧❛♥✐✜❝❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✷✳✼ ❱✐st❛ s✉♣❡r✐♦r ❞♦ ♣♦❧✐❡❞r♦P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✹✳✶ ❚❡tr❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✶✾ ✹✳✷ ❖❝t❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✸ ■❝♦s❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✹ ❍❡①❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✺ ❉♦❞❡❝❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✷✷ ✹✳✻ ❚r✐â♥❣✉❧♦DEF ❞❡ ❧❛❞♦ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✹✳✼ ◗✉❛❞r❛❞♦ ❞❡ ❧❛❞♦α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✹✳✽ P❡♥tá❣♦♥♦J KLM N ❞❡ ❧❛❞♦ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✹✳✾ ❚❡tr❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✹✳✶✵ ❚r✐â♥❣✉❧♦ ❞❛ ❜❛s❡ ❞♦ t❡tr❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✹✳✶✶ ❖❝t❛❡❞r♦ ❞❡ ❛r❡st❛α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✹✳✶✷ ❉♦❞❡❝❛❡❞r♦ ❞❡❝♦♠♣♦st♦ ❡♠ ♦✉tr♦s ♣♦❧í❣♦♥♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✶✸ ❘❡t❛s s❡❝❝✐♦♥❛♥❞♦ ♦ ♣❡♥tá❣♦♥♦ ❏❑▲▼◆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✹✳✶✹ Pr✐s♠❛ r❡t♦ ♥♦ ♠❡✐♦ ❡ ❛ ❥✉♥çã♦ ❞♦s ♣♦❧✐❡❞r♦s ❞❛s ❧❛t❡r❛✐s ❢♦r♠❛♠ ✉♠❛ ♣✐râ♠✐❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✶✺ ❉✐✈✐❞✐♥❞♦ ♦ ✐❝♦s❛❡❞r♦ ❡♠ ✷✵ ♣✐râ♠✐❞❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✶✻ P♦❧✐❣♦♥❛❧ KP QRM S ❡ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r J KLM N✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✹✳✶✼ ❉✐❛❣♦♥❛❧ d ❞♦ ✐❝♦s❛❡❞r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✹✳✶✽ ❆❧t✉r❛CG❞❡ ✉♠❛ ❞❛s ✷✵ ♣✐râ♠✐❞❡s q✉❡ ❢♦r♠❛♠ ♦ ✐❝♦s❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✹✳✶ P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✷ ➪r❡❛ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✸ ➪r❡❛ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
◆♦t❛çõ❡s
◆♦t❛çõ❡s ●❡r❛✐s
❼ P é ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ q✉❛❧q✉❡r✳
❼ A é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ P✳
❼ F é ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ P✳
❼ V é ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ P✳
❼ Fn é ❛ ♥✲és✐♠❛ ❢❛❝❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦✳
❼ nk é ♦ ❦✲és✐♠♦ ❧❛❞♦ ❞❡ ✉♠❛ ❢❛❝❡✳
❼ Vp é ❛ ♣✲és✐♠❛ ❛r❡st❛ q✉❡ ✐♥❝✐❞❡ ♥✉♠ ✈ért✐❝❡✳
❼ S é ❛ ár❡❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❛s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳
❼ Sb é ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❛ ❜❛s❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳
❼ Sl é ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❛s ❢❛❝❡s ❧❛t❡r❛✐s ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳
❼ V é ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r✳
❼ Vpir é ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡✳
❼ d é ❛ ❞✐❛❣♦♥❛❧ ♠❛✐♦r ❞♦ ✐❝♦s❛❡❞r♦✳
❼ h ❛❧t✉r❛ ❞❡ ✉♠❛ ❢❛❝❡ ♦✉ ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳
❼ α é ❛r❡st❛ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r✳
■♥tr♦❞✉çã♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛♥❛❧✐s❛r ❛ ❢♦r♠❛ ❞❡ ❝♦♠♦ ♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s sã♦ ✈✐st♦s ♣♦r ❛❧❣✉♥s ❛✉t♦r❡s ❞❡ ❧✐✈r♦s ❞✐❞át✐❝♦s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ❢♦✐ r❡❛❧✐③❛❞❛ ✉♠❛ ❜r❡✈❡ ♣❛ss❛❣❡♠ ♣♦r ❛❧❣✉♥s ❢❛t♦s q✉❡ ♣♦❞❡♠ t❡r ❞❛❞♦ í♥✐❝✐♦ ♦✉ ♠❛r❝❛r❛♠ ❛ ❤✐stór✐❛ ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ♥❡ss❡ ♠❡s♠♦ ❝❛♣ít✉❧♦ ✈❡r❡♠♦s ♦s ♣r✐♥❝✐♣❛✐s ❣❡ô♠❡tr❛s q✉❡ ❞❡❞✐❝❛r❛♠✲s❡ ❛♦ ❡st✉❞♦ ❞♦s só❧✐❞♦s ❣❡♦♠étr✐❝♦s ❞❡♥tr❡ ❡❧❡s✱ ❞❡st❛❝❛✲s❡ P❧❛tã♦ q✉❡ t❡♠ s❡✉ ♥♦♠❡ ❛ss♦❝✐❛❞♦ ❛♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✳
◆♦ ❝❛♣ít✉❧♦ ❞♦✐s ✐♥✐❝✐❛♠♦s ❝♦♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ♣❛r❛ q✉❛❧q✉❡r ♣♦❧✐❡❞r♦ ❡♠ ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ❞❡♣♦✐s ❞✐✈✐❞✐♠♦s ♦s ♣♦❧✐❡❞r♦s ❡♠ ❞♦✐s ❣r✉♣♦s✿ ♦s ❝♦♥✈❡①♦s ❡ ♦s ♥ã♦ ❝♦♥✈❡①♦s ✭❝ô♥❝❛✈♦s✮✳ ❖ ♣ró①✐♠♦ ♣❛sss♦ ❢♦✐ ✐❞❡♥t✐✜❝❛r ❡ ❝❧❛ss✐✜❝❛r ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ✉♠ ♣♦❧✐❡❞r♦✿ ❢❛❝❡s✱ ✈ért✐❝❡s ❡ ❛r❡st❛s✳ ❈♦♠ ❡ss❡s ❡❧❡♠❡♥t♦s ❞❡✜♥✐❞♦s ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❞✉❛s r❡❧❛çõ❡s ❡♥tr❡ ❡❧❡s✳
❖ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ ✉t✐❧✐③❛♠♦s ❛s r❡❧❛çõ❡s ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞♦s ♣♦❧✐❡❞r♦s ♣❛r❛ ❞❡♠♦♥str❛r♠♦s ♦ ❢❛♠♦s♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡ r❛r❛♠❡♥t❡ é ❞❡♠♦♥str❛❞♦ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ♠❛s ❜❛st❛♥t❡ tr❛❜❛❧❤❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ q✉❛♥❞♦ s❡ ❡st✉❞❛ ♣♦❧✐❡❞r♦s✳
◆♦ ❝❛♣ít✉❧♦ q✉❛tr♦ ❞❡✜♥✐♠♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ✉t✐❧✐③❛♥❞♦✲s❡ ❞❡ ❞✉❛s ❞❡✜♥✐çõ❡s ❡ ❝♦♠ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ❞❡♠♦♥str❛❞♦ ♣r♦✈❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛♣❡♥❛s ❝✐♥❝♦ ♣♦✲ ❧✐❡❞r♦s r❡❣✉❧❛r❡s ♦s só❧✐❞♦s ❞❡ P❧❛tã♦✱ ❛♣r❡s❡♥t❛✲s❡ t❛♠❜é♠ ♦s ✈❛❧♦r❡s ❞❛s ár❡❛s ❞❛s s✉♣❡r❢í❝✐❡s ❡ ✈♦❧✉♠❡s ❞❡ss❡s ❝✐♥❝♦ ♣♦❧✐❡❞r♦s ❡♠ ❢✉♥çã♦ ❞❡ s✉❛ ❛r❡st❛ ❡ á♣♦t❡♠❛✳ ◆♦ ✜♠ ❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❜r❡✈❡ r❡s✉♠♦ ❞❡ ♦✉tr♦s ♣♦❧✐❡❞r♦s ♥ã♦ r❡❣✉❧❛r❡s ♠❛s q✉❡ ❛♣r❡s❡♥t❛♠✲s❡ ❞❡ ❢♦r♠❛ ❞❡st❛❝❛❞❛ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳
❋✐♥❛❧✐③❛♠♦s ♦ tr❛❜❛❧❤♦ ❝♦♠ ✉♠❛ ♣r♦♣♦st❛ ❞❡ ✉s♦ ❞♦s s♦❢t✇❛r❡s ❈❛❜r✐ ❡ P♦❧② ♥❛s ❛♣r❡s❡♥t❛çõ❡s ❞❡ ❝♦♥t❡ú❞♦s ❡♥✈♦❧✈❡♥❞♦ ♣♦❧✐❡❞r♦s ♣❛r❛ ♠❡❧❤♦r ✜①❛çã♦ ❡ ❝♦♠♣r❡❡♥sã♦ ❞♦ ❛ss✉♥t♦✳
❈❛♣ít✉❧♦ ✶
❈♦♥❝❡✐t♦ ❍✐stór✐❝♦
❆ ♠❛t❡♠át✐❝❛ ❞❡s♣❡rt❛ ♦ ✐♥t❡r❡ss❡ ❞❛s ♣❡ss♦❛s ❞❡s❞❡ ❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❡❧❛ é ✉♠❛ ❞❛s ❝✐ê♥❝✐❛s ♠❛✐s ❛♥t✐❣❛s ❡ s✉❛ ♦r✐❣❡♠ ♥ã♦ t❡♠ ❞❛t❛ ✜①❛✱ ❛ ❞❛t❛ ♣r❡❝✐s❛ ❞❡ q✉❛♥❞♦ ♦❝♦rr❡r❛♠ ♦s ♣r✐♠❡✐r♦s ❝á❧❝✉❧♦s q✉❡ ❡♥✈♦❧✈❡r❛♠ ✉♠ r❛❝✐♦❝í♥✐♦ ❣❡♦♠étr✐❝♦ t❛♠❜é♠ ♥ã♦ é ♣r❡❝✐s❛✱ ❛❝r❡❞✐t❛✲s❡ q✉❡ ♥❛ ♣ré✲❤✐stór✐❛ ♦ ❤♦♠❡♠ ❥á ❞❡s❡♥✈♦❧✈✐❛ ✉♠ r❛❝✐♦❝í♥✐♦ ♠❛t❡♠át✐❝♦✱ ❝♦♥t✉❞♦ ♥ã♦ ❤❛✈✐❛ r❡❣✐str♦s ❞❡ t❛✐s ❢❡✐t♦s✳
P♦❞❡♠♦s ❝✐t❛r ❝♦♠♦ ✉♠ ❞♦s ♣r✐♠❡✐r♦s t❡①t♦s ♠❛t❡♠át✐❝♦s ✭♣r♦✈❡♥✐❡♥t❡s ✷✵✵✵ ❛✳❈✳✮ ♦ ♣❛♣✐r♦ ♠❛t❡♠át✐❝♦ ❞❡ ❘❤✐♥❞ ✭♠❛t❡♠át✐❝❛ ❡❣í♣❝✐❛✱ ❝❡r❝❛ ❞❡ ✷✵✵✵ ✲ ✶✽✵✵ ❛✳❈✳✮ ❡ ♦ P❛♣✐r♦ ▼❛t❡♠át✐❝♦ ❞❡ ▼♦s❝♦✈♦ ♦✉ ▼♦s❝♦✉ ✭♠❛t❡♠át✐❝❛ ❡❣í♣❝✐❛✱ ❝❡r❝❛ ❞❡ ✶✽✾✵ ❛✳❈✳✮✳ ❖ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✈♦ q✉❡ ❢♦✐ ❡s❝r✐t♦ ♣♦r ✉♠ ❡s❝r✐❜❛ ❞❡s❝♦♥❤❡❝✐❞♦ ❝♦♥té♠ ✈✐♥t❡ ❡ ❝✐♥❝♦ ❡①❡♠♣❧♦s✱ q✉❛s❡ t♦❞♦s ❞❛ ✈✐❞❛ ♣rát✐❝❛ ❡ ♥ã♦ ❞✐❢❡r❡♥❝✐❛♥❞♦ ♠✉✐t♦ ❞♦ ♣❛♣✐r♦ ❘❤✐♥❞✳ ❆ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✶✹ ❞♦ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✈♦ ❤á ✉♠❛ ✜❣✉r❛ q✉❡ é ♣❛r❡❝✐❞❛ ❝♦♠ ✉♠ tr❛♣é③✐♦✱ ♣♦ré♠ ♦s ❝á❧❝✉❧♦s ❛ss♦❝✐❛❞♦s ❛ ❡❧❛ ♠♦str❛♠ q✉❡ ♦ q✉❡ s❡ r❡♣r❡s❡♥t❛ é ✉♠ tr♦♥❝♦ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛ ✶✹✮✳ ❊ss❡ t❛❧✈❡③ s❡❥❛ ♦ ♣r✐♠❡✐r♦ r❡❣✐str♦ ❞❡ ✉♠ ❝á❧❝✉❧♦ ❡♥✈♦❧✈❡♥❞♦ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✳
❋✐❣✉r❛ ✶✳✶✿ ❋r❛❣♠❡♥t♦ ❞♦ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✈♦✳
❚♦❞❛✈✐❛ ♣♦❞❡rí❛♠♦s t❡r r❡❣✐str♦s ❛♥t❡r✐♦r❡s ❛ ❡ss❡s ❡♥❝♦♥tr❛❞♦s ♥♦ ♣❛♣✐r♦ ▼♦s✲ ❝♦✈♦✱ ❝♦♥t✉❞♦ ❛ ❛✉sê♥❝✐❛ ❞❡ss❡s r❡❣✐str♦s t♦r♥♦✉ ❛ ♦r✐❣❡♠ ❞❛ ●❡♦♠❡tr✐❛ ✐♥❞❡t❡r♠✐✲
♥❛❞❛✱ ♣♦ré♠ ♣♦❞❡♠♦s ❞❡st❛❝❛r ❛❧❣✉♠❛s ✉t✐❧✐③❛çõ❡s ❞❡ ❝♦♥❝❡✐t♦s ❞❡ ❛❧❣✉♥s ♣♦✈♦s ♥❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❡ q✉❡ ❛t✉❛❧♠❡♥t❡ ❛✐♥❞❛ sã♦ út❡✐s ♦✉ s❡r✈❡♠ ❞❡ ✐♥s♣✐r❛çã♦ ♦✉ ❜❛s❡ ❡♠ ♥♦✈❛s ♣r♦♣♦st❛s ❞❡ ✐❞❡✐❛s ❣❡♦♠étr✐❝❛s❀ ❝♦♠♦ ♦s ❡❣í♣❝✐♦s ❡ ♦ ✉s♦ ❞❡ ár❡❛s ❞❡ t❡rr❡♥♦s ♥❛ ❛❣r✐❝✉❧t✉r❛ ❡ s✉❛s ❝♦♥str✉çõ❡s t❡♥❞♦ ❛s ♣✐râ♠✐❞❡s ❝♦♠♦ ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s t❛♠✲ ❜é♠ ❞❡st❛❝❛r ♦✉tr♦s ♣♦✈♦s ❛♥t✐❣♦s ❝♦♠♦ ♦s ❝❤✐♥❡s❡s ❡ s✉❛ ❣r❛♥❞❡ ♠✉r❛❧❤❛✱ ♣♦ré♠ ❢♦✐ ♥❛ ●ré❝✐❛✱ q✉❡ ❛ ●❡♦♠❡tr✐❛ t❡✈❡ s❡✉ ♠❛✐♦r ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✳ ❖s ❣r❡❣♦s ♣❡r❝❡❜❡r❛♠ ❡♠ s❡✉s ❡st✉❞♦s q✉❡ ♦s ❡❣í♣❝✐♦s ❢♦r❛♠ ❝❛♣❛③❡s ❞❡ r❡❛❧✐③❛r ❝á❧❝✉❧♦s ❡ ♠❡❞✐❞❛s ❞❡ ❞✐♠❡♥s✐♦♥❛♠❡♥t♦ ❞❛ t❡rr❛✱ ❡❧❡s ♣r♦❝✉r❛r❛♠ ❞❡♠♦♥str❛r ❧❡✐s ❛❝❡r❝❛ ❞♦ ❡s♣❛ç♦✱ ❞❛í ♦ ♥♦♠❡ ●❊❖▼❊❚❘■❆✱ ❣❡♦ ✲ t❡rr❛ ❡ ♠❡tr✐❛ ✲ ♠❡❞✐❞❛✳
❈♦♠ ❧♦❝❛❧✐③❛çã♦ ♣r✐✈✐❧❡❣✐❛❞❛ ❡♥tr❡ ♦s ♠❛r❡s ❊❣❡✉ ❡ ❏ô♥✐♦ ❡ ❝♦❧ô♥✐❛s q✉❡ s❡ ❡♥✲ ❝♦♥tr❛✈❛♠ ♥❛s ♠❛r❣❡♥s ❞♦s ♠❛r❡s ◆❡❣r♦ ❡ ▼❡❞✐t❡rrâ♥❡♦✱ ❛❧é♠ ❞❛ ót✐♠❛ ❧♦❝❛❧✐③❛çã♦ ❣❡♦❣rá✜❝❛ q✉❡ ❣❛r❛♥t✐❛ ♣♦❞❡r❡♠ ✈✐❛❥❛r ❛♦s ❣r❛♥❞❡s ❝❡♥tr♦s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ é♣♦❝❛ ❥✉♥t♦ ❛ ✉♠ ❡s♣ír✐t♦ ♦✉s❛❞♦ ❡ ✐♠❛❣✐♥❛t✐✈♦ ❢❡③ ❝♦♠ q✉❡ ♦s ❛❞q✉✐r✐ss❡♠ ✐♥❢♦r✲ ♠❛çõ❡s ❞❡ ♣r✐♠❡✐r❛ ♠ã♦ s♦❜r❡ ♠❛t❡♠át✐❝❛ ❡ ❛str♦♥♦♠✐❛✳ ❊ ❢♦✐ ♥♦ ❊❣✐t♦ ❞✐③✲s❡ q✉❡ ❛♣r❡♥❞❡r❛♠ ❣❡♦♠❡tr✐❛✳ ❋♦r❛♠ ♠✉✐t♦s ♦s ❣❡ô♠❡tr❛s ❣r❡❣♦s q✉❡ ♣♦❞❡♠♦s ❞❡st❛❝❛r ❡♥tr❡ ❡❧❡s ❚❛❧❡s✱ P✐tá❣♦r❛s✱ ❊✉❝❧✐❞❡s ❡ P❧❛tã♦✳
❚❛❧❡s ❢r❡q✉❡♥t❡♠❡♥t❡ s❛✉❞❛❞♦ ❝♦♠♦ ♦ ♣r✐♠❡✐r♦ ♠❛t❡♠át✐❝♦ ✈❡r❞❛❞❡✐r♦ ✲ ♦r✐❣✐♥❛✲ ❞♦r ❞❛ ♦r❣❛♥✐③❛çã♦ ❞❡❞✉t✐✈❛ ❞❛ ❣❡♦♠❡tr✐❛✳ ◆ã♦ ❤á ❞♦❝✉♠❡♥t♦ q✉❡ ♣r♦✈❡ t❛❧ ❢❡✐t♦✱ ♦ ♠❛✐s ♣ró①✐♠♦ q✉❡ ♣♦❞❡♠♦s ❝❤❡❣❛r ❛ ✐ss♦ ❞✐❣♥♦ ❞❡ ❝♦♥✜❛♥ç❛ é ✉♠❛ ♠❡♥çã♦ ✶✵✵✵ ❛♥♦s ❞❡♣♦✐s ❞♦ t❡♠♣♦ ❞❡ ❚❛❧❡s✳ ❯♠ ❞✐s❝í♣✉❧♦ ❞❡ ❆r✐stót❡❧❡s ❝❤❛♠❛♥❞♦ ❊✉❞❡♠♦ ❡s❝r❡✈❡✉ ✉♠❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡ss❛ s❡ ♣❡r❞❡✉ ♣♦ré♠ ❛❧❣✉é♠ r❡s✉♠✐✉ ♣❛rt❡ ❞❡❧❛✱ ❡ss❡ r❡s✉♠♦ t❛♠❜é♠ ♣❡r❞❡✉✲s❡✱ ♣♦ré♠ Pr♦❝❧✉s ✜❧ós♦❢♦ ♥❡♦✲♣❧❛tô♥✐❝♦ ❡s❝r❡✈❡✉ ♣❛rt❡ ❞❛s ✐♥❢♦r♠❛çõ❡s ❞♦ s✉♠ár✐♦ ❞❡ ❊✉❞❡♠♦s✱ ♥❡❧❡ ❞✐③ ♦ s❡❣✉✐♥t❡ s♦❜r❡ ❚❛❧❡s✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛s ✸✹✲✸✺✮✳
✳✳✳Pr✐♠❡✐r♦ ❢♦✐ ❛♦ ❊❣✐t♦ ❡ ❞❡ ❧á ✐♥tr♦❞✉③✐✉ ❡ss❡ ❡st✉❞♦ ♥❛ ●ré❝✐❛✳ ❉❡s❝♦❜r✐✉ ♠✉✐t❛s ♣r♦♣♦s✐çõ❡s ❡❧❡ ♣ró♣r✐♦ ❡ ✐♥str✉✐✉ s❡✉s s✉❝❡ss♦r❡s ♥♦s ♣rí♥❝✐♣✐♦s q✉❡ r❡❣❡♠ ♠✉✐t❛s ♦✉tr❛s✳✳✳
❏á P✐tá❣♦r❛s✱ ❛❧❣✉♥s ❞✐③❡♠ ❞✐s❝í♣✉❧♦ ❞❡ ❚❛❧❡s✱ t❡✈❡ t❛♠❜é♠ s✉❛s ❝♦♥tr✐❜✉✐çõ❡s ♥❛ ❣❡♦♠❡tr✐❛✳ ❋✉♥❞❛❞♦r ❞❛ ❢❛♠♦s❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛ ❡ ❞♦ ❢❛♠♦s♦ t❡♦r❡♠❛ ✱ ♣r♦✈❛❧✲ ✈❡♠❡♥t❡ ❞❡ ❡st✉❞♦s ♥❛s ♣✐râ♠✐❞❡s ❞♦ ❊❣✐t♦✱ ♥♦ s✉♠ár✐♦ ❞❡ ❊✉❞❡♠♦✲Pr♦❝❧✉s ❧❤❡ é ❛tr✐❜✉í❞❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✧✜❣✉r❛s ❝ós♠✐❝❛s✧ ✭✐st♦ é só❧✐❞♦s r❡❣✉❧❛r❡s✮✱ ♣♦ré♠ ❡①✐s✲ t❡♠ ❛❧❣✉♠❛s ❞ú✈✐❞❛s✳ ❖ ❤❡①❛❡❞r♦✱ ♦ ♦❝t❛❡❞r♦ ❡ ♦ ❞♦❞❡❝❛❡❞r♦ ♣♦❞✐❛♠ t❡r s✐❞♦ ♦❜s❡r✈❛❞♦s ❡♠ ❝r✐st❛✐s✱ ❝♦♠♦ ♦ ❞❛ ♣✐r✐t❛ ✭❞✐ss✉❧✜t♦ ❞❡ ❢❡rr♦✮✱ ♠❛s ❡♠ ❖s ❊❧❡♠❡♥t♦s ❳■■■ ❡stá ❞✐t♦ q✉❡ ♦s ♣✐t❛❣ór✐❝♦s só ❝♦♥❤❡❝✐❛♠ três ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s❀ ♦ t❡tr❛❡❞r♦✱ ♦ ❤❡①❛❡❞r♦ ❡ ♦ ❞♦❞❡❝❛❡❞r♦✳ ❙♦❜r❡ ♦ ú❧t✐♠♦ ♥ã♦ s❡r✐❛ ✐♠♣r♦✈á✈❡❧ ♣♦✐s ❡❧❡s ❝♦♥❤❡❝✐❛♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r✳ ✭❖ ♣❡♥t❛❣r❛♠❛✱ sí♠❜♦❧♦ ❡s♣❡❝✐❛❧ ❞❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛✱ ❢♦r♠❛❞❛ tr❛ç❛♥❞♦ ❛s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❡♥tá❣♦♥♦ q✉❡ é ❢❛❝❡ ❞♦ ❞♦❞❡❝❛❡❞r♦✮✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛ ✸✼✮✳
❋✐❣✉r❛ ✶✳✷✿ ❙í♠❜♦❧♦ ❞❛ ❡s❝♦❧❛ P✐t❛❣ór✐❝❛✱ ♦ ♣❡♥t❛❣r❛♠❛✳
❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❛ss✐♠ ❝♦♥❤❡❝✐❞♦ ♣♦r t❡r ❡♥s✐♥❛❞♦ ♠❛t❡♠át✐❝❛ ♥❛ ❝✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❢♦✐ ♦✉tr♦ ❣r❛♥❞❡ ❣❡ô♠❡tr❛ ❣r❡❣♦✱ ❡s❝r✐t♦r ❞❡ ❞✐✈❡rs❛s ♦❜r❛s ✭❝✐♥❝♦ s♦❜r❡✈✐✈❡♠ ❛té ❤♦❥❡✮ ❞❡♥tr❡ ❡❧❛s ❞❡st❛❝❛✲s❡ ✉♠❛ ♦❜r❛ ❞♦ ✉♥✐✈❡rs♦ ♠❛t❡♠át✐❝♦ q✉❡ ❛té ❤♦❥❡ t❡♠ ❣r❛♥❞❡ ❝♦♥tr✐❜✉✐çã♦ ❡ t❛❧✈❡③ s❡❥❛ ✉♠❛ ❞❛s ♠❛✐♦r❡s ♦❜r❛s ♣✉❜❧✐❝❛❞❛s ✭♣❛r❛ ❛❧❣✉♥s ♠❛t❡♠át✐❝♦s ❛ ♠❛✐♦r✮ é ❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ s❡❣✉♥❞♦ ➪✈✐❧❛ ❬✶❪✿ ✧◆ã♦ s❛❜❡♠♦s s❡ ❊✉❝❧✐❞❡s ❡s❝r❡✈❡✉ ❖s ❊❧❡♠❡♥t♦s ♣❛r❛ ✉s♦ ♥♦ ❡♥s✐♥♦✱ ♦✉ ❛♣❡♥❛s ♣❛r❛ r❡✉♥✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❞❛ é♣♦❝❛✳ ◆❛q✉❡❧❡ t❡♠♣♦ ♥ã♦ ❤❛✈✐❛ ❛ ♣r❡♦❝✉♣❛çã♦ ♣❡❞❛❣ó❣✐❝❛ ❞♦s ❞✐❛s ❞❡ ❤♦❥❡✱ ❞❡ s♦rt❡ q✉❡ ❊✉❝❧✐❞❡s ❛❧❝❛♥ç♦✉ ♦s ❞♦✐s ♦❜❥❡t✐✈♦s❀ ❡ ♦s ❊❧❡♠❡♥t♦s ❢♦r❛♠ ♠✉✐t♦ ✉s❛❞♦s ♥♦ ❛♣r❡♥❞✐③❛❞♦ ❞❛ ▼❛t❡♠át✐❝❛ ♣♦r ♠❛✐s ❞❡ ❞♦✐s ♠✐❧ê♥✐♦s ✧✳ ❖ ú❧t✐♠♦ ❧✐✈r♦ ✭▲✐✈r♦ ❳■■■✮ é t♦t❛❧♠❡♥t❡ ❞❡❞✐❝❛❞♦ ❛♦s ❝✐♥❝♦ só❧✐❞♦s r❡❣✉❧❛r❡s✱ ❢❛t♦ q✉❡ ❧❡✈♦✉ ❛❧❣✉♥s ❤✐st♦r✐❛❞♦r❡s ❛ ❞✐③❡r q✉❡ ❖s ❊❧❡♠❡♥t♦s ❢♦r❛♠ ❝♦♠♣♦st♦s ❝♦♠♦ ✉♠❛ ❣❧♦r✐✜❝❛çã♦ ❞❛s ✜❣✉r❛s ❝ós♠✐❝❛s ♦✉ ♣❧❛tô♥✐❝❛s✳
❖✉tr♦ ❝♦♠ ❡st✉❞♦s ♥♦ ❝❛♠♣♦ ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ❢♦✐ P❧❛tã♦✱ ❡♠❜♦r❛ ♥ã♦ t❡♥❤❛ ❞❛❞♦ ❝♦♥tr✐❜✉✐çã♦ ❞✐❣♥❛ ❞❡ ♥♦t❛ ❛ r❡s✉❧t❛❞♦s ♠❛t❡♠át✐❝♦s té❝♥✐❝♦s✱ ♣♦ré♠ s❡✉ ❡♥t✉s✐❛s♠♦ ♣❡❧♦ ❛ss✉♥t♦ ♦ ❢❡③ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✧❝r✐❛❞♦r ❞❡ ♠❛t❡♠át✐❝♦s✧✳ ❊ss❡ ❢❛s❝í♥✐♦ t♦❞♦ ♣❡❧❛ ♠❛t❡♠át✐❝❛ ❝❡rt❛♠❡♥t❡ ✈❡✐♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ✈✐s✐t❛ ❛ ✉♠ ❛♠✐❣♦ ♥❛ ❙✐❝í❧✐❛✱ ❆rq✉✐t❛s✳ ❚❛❧✈❡③ à ✈❡♥❡r❛çã♦ ❞♦s ♣✐t❛❣ór✐❝♦s ♣❡❧♦ ❞♦❞❡❝❛❡❞r♦ t❡♥❤❛ s✐❞♦ ♦ q✉❡ ❧❡✈♦✉ P❧❛tã♦ ❛ ❝♦♥s✐❞❡rá✲❧♦ ❝♦♠♦ ✉♠ sí♠❜♦❧♦ ❞♦ ✉♥✐✈❡rs♦✳ ❖s ❝✐♥❝♦ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ❢♦r❛♠ ❝♦♥st❛♥t❡♠❡♥t❡ ❝❤❛♠❛❞♦s ❞❡ ✧❝♦r♣♦s ❝ós♠✐❝♦s✧ ♦✉ ✧só❧✐❞♦s ♣❧❛tô✲ ♥✐❝♦s✧ ❞❡✈✐❞♦ ❛ ♠❛♥❡✐r❛ ♣❡❧❛ q✉❛❧ P❧❛tã♦ ♦s ❛♣❧✐❝♦✉ ♣❛r❛ ❡①♣❧✐❝❛çã♦ ❞❡ ❢❡♥ô♠❡♥♦s ❝✐❡♥tí✜❝♦s✱ t❛❧✈❡③ t❛♠❜é♠ s❡❥❛ r❡s♣♦♥sá✈❡❧ ♣♦r ❛❧❣✉♥s ❝á❧❝✉❧♦s ❡♥❝♦♥tr❛❞♦s s♦❜r❡ ♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ♥♦ ❧✐✈r♦ ❖s ❊❧❡♠❡♥t♦s✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛s ✻✷✲✻✸✮✳
❚✐✈❡♠♦s ♦✉tr♦s ♠❛t❡♠át✐❝♦s ❣r❡❣♦s✱ ❡ ❝❧❛r♦ ❞❡ ♦✉tr♦s ♣♦✈♦s✱ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ♣♦ré♠ ♦s ❣r❡❣♦s ❛❜s♦❧✈✐❛♠ ♠✉✐t♦s ❡❧❡♠❡♥t♦s ❞❡ ♦✉tr❛s ❝✉❧t✉r❛s✱ ❞❡ ♦✉tr❛ ❢♦r♠❛ ♥ã♦ t❡r✐❛♠ ❛♣r❡♥❞✐❞♦ tã♦ ❞❡♣r❡ss❛ ❞❡ ♠♦❞♦ ❛ t❡r ♣❛ss❛❞♦ à ❢r❡♥t❡ ❞❡ s❡✉s ♣r❡❞❡❝❡ss♦r❡s✱ ♠❛s ❛ t✉❞♦ q✉❡ t♦❝❛✈❛♠ ❞❛✈❛♠ ♠❛✐s ✈✐❞❛✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛ ✸✹✮✳
❋✐❣✉r❛ ✶✳✸✿ ❙ó❧✐❞♦s P❧❛tô♥✐❝♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❡❧❡♠❡♥t♦s ❞❛ ♥❛t✉r❡③❛✳ ❉❡♥tr❡ t♦❞♦s ♦s r❛♠♦s ❞❛ ♠❛t❡♠át✐❝❛ ❛ ●❡♦♠❡tr✐❛ é ❛ q✉❡ ♠❛✐s s♦❢r❡ ♠✉❞❛♥ç❛s ❞❡ ❣♦st♦ ❞❡ ✉♠❛ é♣♦❝❛ ♣❛r❛ ♦✉tr❛✳ ❙❡✉ ❛✉❣❡ ❢♦✐ ♥❛ ●ré❝✐❛ ❝❧áss✐❝❛ ❡ ❝❛✐✉ ❥✉♥t♦ ❝♦♠ ❘♦♠❛✱ t❡✈❡ ✉♠❛ r❡❝✉♣❡r❛çã♦ ♥❛ ➪r❛❜✐❛ ❡ ♥❛ ❊✉r♦♣❛ ❞❛ ❘❡♥❛s❝❡♥ç❛✱ ♥♦ sé❝✉❧♦ ❞❡③❡ss❡t❡ ❡st❡✈❡ ♥♦ ❧✐♠✐❛r ❞❡ ✉♠❛ ♥♦✈❛ ❡r❛✱ ♠❛s ♥♦✈❛♠❡♥t❡ ❢♦✐ ❡sq✉❡❝✐❞❛✱ ❛♦ ♠❡♥♦s ♣❡❧♦s ♣❡sq✉✐s❛❞♦r❡s ♠❛t❡♠át✐❝♦s✱ ♣♦r q✉❛s❡ ♠❛✐s ❞♦✐s sé❝✉❧♦s✱ ❛tr❛✈és ❞❡ ❡s❢♦rç♦s ❞❡ ▼♦♥❣❡ ❡ ❈❛r♥♦t ❤♦✉✈❡ ✉♠ r❡❛✈✐✈❛♠❡♥t♦ ❞❛ ❣❡♦♠❡tr✐❛ ♣✉r❛ ♥♦ ♣❡rí♦❞♦ ❞❛ r❡✈♦❧✉çã♦ ❋r❛♥❝❡s❛✱ ♣♦ré♠ ♥♦ sé❝✉❧♦ ❞❡③❡♥♦✈❡ ❡❧❛ t❡✈❡ ✉♠❛ r❡❞❡s❝♦❜❡rt❛ ❝♦♠♦ r❛♠♦ ✈✐✈♦ ❞❛ ♠❛t❡♠át✐❝❛✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛ ✸✽✼✮✳
❍♦❥❡✱ ❝♦♠♦ ♥♦ ♣❛ss❛❞♦✱ ❛ ●❡♦♠❡tr✐❛ ❝♦♥t✐♥✉❛ ❛ s❡r ✉♠❛ ❢♦♥t❡ ❞❡ ❡♥r✐q✉❡❝✐♠❡♥t♦ ❞❡ r❛❝✐♦❝í♥✐♦ ❡ ❞♦s ❤á❜✐t♦s ❞❡ ♣❡♥s❛r✱ q✉❡ ♣❡r♠✐t❡♠ ❥✉st✐✜❝❛r ❛s ♥♦ss❛s ❛✜r♠❛çõ❡s ✭✈❡❥❛ ❬✶✹❪ ♣á❣✐♥❛ ✸✵✻✮ s✉❛s ❛♣❧✐❝❛çõ❡s sã♦ ✐♥ú♠❡r❛s✳ ❖s só❧✐❞♦s ❣❡♦♠étr✐❝♦s sã♦ ♦❜s❡r✈❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❞✐✈❡rs♦s ❧✉❣❛r❡s s❡❥❛ ❡♠ ❝♦♥str✉çõ❡s ♦✉ ❡♠ ♦❜❥❡t♦s ❞❡ ✉s♦ ❞✐ár✐♦✳
❋✐❣✉r❛ ✶✳✹✿ P✐râ♠✐❞❡s ❞♦ ❊❣✐t♦ ❡ ❛ P✐râ♠✐❞❡ ❞♦ ♠✉s❡✉ ❞♦ ▲♦✉✈r❡✱ ❢♦r♠❛s ♣♦❧✐é❞r✐❝❛s✳
❈❛♣ít✉❧♦ ✷
❖s ♣♦❧✐❡❞r♦s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s
❱ár✐❛s sã♦ ❛s ❞❡✜♥✐çõ❡s ♣❛r❛ ♦s ♣♦❧✐❡❞r♦s ❞✐s♣♦st❛s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳
❙❡❣✉♥❞♦ ❇✐❛♥❝❤✐♥✐ ❡ P❛❝❝♦❧❛ ❬✹❪✱ ♣á❣✐♥❛ ✷✵✺✱ ❞❡✜♥✐♠♦s ♣♦❧✐❡❞r♦ ❝♦♠♦ à r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦ ❧✐♠✐t❛❞❛ ♣♦r ♣♦❧í❣♦♥♦s ♣❧❛♥♦s✱ ❡ t❛✐s q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❛r❡st❛s ❞❡ss❡s ♣♦❧í❣♦♥♦s ♣❡rt❡♥ç❛ ❛ ❞♦✐s ❡ s♦♠❡♥t❡ ❞♦✐s ❞❡❧❡s✳
❊♠ ❇❛rr♦s♦ ❬✸❪✱ ♣á❣✐♥❛ ✶✻✹✱ é ❝❤❛♠❛❞♦ ❞❡ ♣♦❧✐❡❞r♦ ♦ só❧✐❞♦ ❢♦r♠❛❞♦ ♣❡❧❛ r❡✉♥✐ã♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ❝♦♠ t♦❞♦s ♦s ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ❞❡❧✐♠✐t❛❞♦s ♣♦r ❡❧❛✳
❊♠ ❉❛♥t❡ ❬✻❪✱ ♣á❣✐♥❛ ✸✻✵✱ ❞✐③ q✉❡ ♦ ♣♦❧✐❡❞r♦ é ❢♦r♠❛❞♦ ♣❡❧❛ r❡✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ r❡❣✐õ❡s ♣♦❧✐❣♦♥❛✐s ♣❧❛♥❛s ❝❤❛♠❛❞❛s ❢❛❝❡s ❡ ❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦ ❧✐♠✐t❛❞❛ ♣♦r ❡❧❛✱ ❛ss✐♠ ❝♦♠♦ ❘✐❜❡✐r♦ ❬✶✺❪✱ ♣á❣✐♥❛ ✻✽✱ ♣♦❧✐❡❞r♦s sã♦ só❧✐❞♦s ❧✐♠✐t❛❞♦s ♣♦r s✉♣❡r❢í❝✐❡s ♣❧❛♥❛s ♣♦❧✐❣♦♥❛✐s✳
P❛r❛ ■❡③③✐ ❬✾❪✱ ♣á❣✐♥❛ ✹✺✼✱ t❡♠♦s ♣♦❧✐❡❞r♦ ❝♦♠♦ só❧✐❞♦s ❧✐♠✐t❛❞♦s ♣♦r ♣♦rçõ❡s ❞❡ ♣❧❛♥♦s ✲ ♣♦❧í❣♦♥♦s ♣❧❛♥♦s ✲ ❞❡♥♦♠✐♥❛❞♦s ❢❛❝❡s✳
❆s ❞❡✜♥✐çõ❡s ♥❡♠ s❡♠♣r❡ sã♦ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦ t❡♥❤❛♠ ✉♠❛ ✐❞❡✐❛ s♦❜r❡ ♦ q✉❡ é ✉♠ ♣♦❧✐❡❞r♦✱ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✐❧✉str❛çõ❡s ❢❛❝✐❧✐t❛♠ s✉❛ ❝♦♠♣r❡❡♥sã♦✱ ♠❛s ❛✐♥❞❛✱ é ♥❡❝❡ssár✐♦ ✉♠ ❡st✉❞♦ ❞❛s ❢ór♠✉❧❛s q✉❡ ♣r♦❞✉③❡♠ ár❡❛s ❞❡ s✉♣❡r❢í❝✐❡s ❡ ✈♦❧✉♠❡s ❞♦s ♣♦❧✐❡❞r♦s✱ ❜❡♠ ❝♦♠♦ ♦ ❝❧áss✐❝♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡ r❛r❛♠❡♥t❡ é ❞❡♠♦♥str❛❞♦ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❱❛♠♦s ✉t✐❧✐③❛r ❛s ❞❡✜♥✐çõ❡s ❞❡ ▲✐♠❛ ❬✶✷❪ ♣á❣✐♥❛ ✷✽✸✳
❉❡✜♥✐çã♦ ✷✳✶ P♦❧✐❡❞r♦ é ✉♠❛ r❡✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦❧í❣♦♥♦s ♣❧❛♥♦s ❝❤❛♠❛❞♦s ❢❛❝❡s ♦♥❞❡✿
❛✮ ❈❛❞❛ ❧❛❞♦ ❞❡ ✉♠ ❞❡ss❡s ♣♦❧í❣♦♥♦s é t❛♠❜é♠ ❧❛❞♦ ❞❡ ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♦✉tr♦ ♣♦❧í❣♦♥♦✳
❜✮ ❆ ✐♥t❡rs❡çã♦ ❞❡ ❞✉❛s ❢❛❝❡s q✉❛✐sq✉❡r✱ ♦✉ é ✉♠ ❧❛❞♦ ❝♦♠✉♠✱ ♦✉ é ✈ért✐❝❡ ♦✉ é ✈❛③✐❛✳
❝✮ ➱ s❡♠♣r❡ ♣♦ssí✈❡❧ ✐r ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ ❢❛❝❡ ❛ ✉♠ ♣♦♥t♦ ❞❡ q✉❛❧q✉❡r ♦✉tr❛✱ s❡♠ ♣❛ss❛r ♣♦r ♥❡♥❤✉♠ ✈ért✐❝❡✱ ♦✉ s❡❥❛✱ ❝r✉③❛♥❞♦ ❛♣❡♥❛s ❛r❡st❛s✳
❖ q✉❡ ❝♦♥s✐❞❡r❛r❡♠♦s ❝♦♠♦ ♣♦❧✐❡❞r♦ é ❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❢♦r♠❛❞❛ ♣❡❧♦s ♣♦❧í❣♦✲ ♥♦s ♥ã♦ ❛ r❡❣✐ã♦ ✐♥t❡r♥❛ ❧✐♠✐t❛❞❛ ♣♦r ❡❧❡s✳
❋✐❣✉r❛ ✷✳✶✿ ❋✐❣✉r❛s P♦❧✐é❞r✐❝❛s✳
❉❡ ❢❛t♦ é ❢á❝✐❧ ❡♥❝♦♥tr❛r♠♦s ❢♦r♠❛s ♣♦❧✐é❞r✐❝❛s ♥♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛✱ ❛ss✐♠ t♦r♥❛✲ s❡ ❢á❝✐❧ ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❛ss♦❝✐❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ♣♦❧✐❡❞r♦✱ ♣♦ré♠ ❡①✐st❡♠ t❛♠❜é♠ ✜❣✉r❛s q✉❡ ❡stã♦ ♥♦ R3✱ ♠❛s ♥ã♦ s❡❣✉❡♠ ❛❧❣✉♠ ♦✉ ❛❧❣✉♥s ❞♦s ✐t❡♥s ❞❛ ❉❡✜♥✐çã♦
✷✳✶✱ ❝❤❛♠❛r❡♠♦s ❞❡ ✜❣✉r❛s ♥ã♦ ♣♦❧✐é❞r✐❝❛s✳
❋✐❣✉r❛ ✷✳✷✿ ❋✐❣✉r❛s ♥ã♦ P♦❧✐é❞r✐❝❛s✳
❆ss✐♠ t♦❞♦ ♣♦❧✐❡❞r♦ ♥♦ s❡♥t✐❞♦ ❞❛ ❉❡✜♥✐çã♦ ✷✳✶ ❧✐♠✐t❛ ✉♠❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦ ❝❤❛♠❛❞❛ ✐♥t❡r✐♦r ❞❡ss❡ ♣♦❧✐❡❞r♦✱ ❡ss❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡stã♦ ♥♦R3
✳
◆❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♦s ♣♦❧✐❡❞r♦s tê♠ ♠❛✐♦r ê♥❢❛s❡ ♥♦ s❡❣✉♥❞♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣♦ré♠ ❛❧❣✉♥s ❛✉t♦r❡s ❥á ✐♥tr♦❞✉③❡♠ ♦ ❛ss✉♥t♦ ♥♦ ♣r✐♠❡✐r♦ ❛♥♦ ❝♦♠ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❝♦♠♦ ❡❧❡♠❡♥t♦s ❡ ❝❧❛ss✐✜❝❛çõ❡s ❞♦s ♣♦❧✐❡❞r♦s✳ ❙❡❣✉♥❞♦ P❛❧❤❛✲ r❡s ❬✶✹❪ ♣á❣✐♥❛ ✸✵✻✱ é ✉r❣❡♥t❡ ❡ ❢✉♥❞❛♠❡♥t❛❧ q✉❡ ❛ ●❡♦♠❡tr✐❛ r❡✢✐t❛✱ ♣♦r✈❡♥t✉r❛✱ ❤♦❥❡ ♠❛✐s ❞♦ q✉❡ ♥♦ ♣❛ss❛❞♦ r❡❝❡♥t❡✱ ❛s ♣r❡♦❝✉♣❛çõ❡s ❡❞✉❝❛❝✐♦♥❛✐s ❞❡ r❡❧❡✈â♥❝✐❛ ❡ r❡❛❧✐s♠♦✱ ♥♦♠❡❛❞❛♠❡♥t❡ ❛tr❛✈és ❞❡✿
✷✳✶✳ P❖▲■❊❉❘❖❙ ❈❖◆❱❊❳❖❙
❼ ❱❡r❞❛❞❡✐r♦s ♣r♦❜❧❡♠❛s ❞♦ ❞✐❛✲❛✲❞✐❛ q✉❡ ❡♥✈♦❧✈❛♠ ✐❞❡✐❛s ❣❡♦♠étr✐❝❛s✱ ❡♠ ✈❡③ ❞❡ ❛♣❧✐❝❛çõ❡s ❛rt✐✜❝✐❛s❀
❼ ❊①♣❧♦r❛çã♦ ❞❡ ❢♦r♠❛s ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞♦ ♠❡✐♦ ❛♠❜✐❡♥t❡✱ ♣♦r ♠✉✐t♦ ❝♦♠♣❧✐✲ ❝❛❞♦ q✉❡ ✐ss♦ ♣❛r❡ç❛✳ ❖ ✉s♦ ❞❡ ♣❧❛♥t❛s✱ ❞❡ ♠❛♣❛s ♦✉ ❞❡ ❢♦t♦❣r❛✜❛s ♣❛r❡❝❡✲♥♦s ❛♣r♦♣r✐❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ ♥♦ ♠✉♥❞♦ ♦♥❞❡ ❛ ❝r✐❛♥ç❛ s❡ ♠♦✈✐♠❡♥t❛ ❛ ●❡♦♠❡tr✐❛ é✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❊s♣❛❝✐❛❧✱ ❛♥t❡s ❞❡ s❡r ♣❧❛♥❛❀
❼ ❚r❛❜❛❧❤♦s ❣❡♦♠étr✐❝♦s ❝♦♠ ♦ r❡❝✉rs♦ às ♥♦✈❛s t❡❝♥♦❧♦❣✐❛s✳
✷✳✶ P♦❧✐❡❞r♦s ❝♦♥✈❡①♦s
◆♦ í♥✐❝✐♦ ❞❡ss❡ ❝❛♣ít✉❧♦ r❡✈✐s❛♠♦s ❛s ✈ár✐❛s ❡ ❡q✉✐✈❛❧❡♥t❡s ♠❛♥❡✐r❛s ❝♦♠♦ ♦s ♣♦❧✐✲ ❡❞r♦s sã♦ ❞❡✜♥✐❞♦s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✉s❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❆ s❡❣✉✐r✱ tr❛t❛♠♦s ❞♦s ❝❤❛♠❛❞♦s ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦s✳
❈♦♥❢♦r♠❡ ❏❛♥♦s ❬✶✵❪✱ ♣á❣✐♥❛ ✽✹✱ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ❡♥❝♦♥tr❛✲s❡ ✐♥t❡✐r❛♠❡♥t❡ ❞❡ ✉♠ ❧❛❞♦ ❞♦ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ q✉❛❧q✉❡r ✉♠❛ ❞❡ s✉❛s ❢❛❝❡s✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ♦ ♣♦❧✐❡❞r♦ ♥ã♦ ❝♦♥✈❡①♦ ❛♣r❡s❡♥t❛ ✉♠ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ ✉♠❛ ❞❡ s✉❛s ❢❛❝❡s ❡ ❝♦rt❛ ♦✉tr❛✳
❋✐❣✉r❛ ✷✳✸✿ P♦❧✐❡❞r♦s✳
P❛r❛ ❘✐❜❡✐r♦ ❬✶✺❪✱ ♣á❣✐♥❛ ✼✵✱ ✉♠ ♣♦❧✐❡❞r♦ é ❞✐t♦ ❝♦♥✈❡①♦ q✉❛♥❞♦ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ❧✐❣❛ q✉❛✐sq✉❡r ❞♦✐s ❞❡ s❡✉s ♣♦♥t♦s ❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ♥❡❧❡✳ ❆❧é♠ ❞✐ss♦✱ ✉♠ ♣♦❧✐❡❞r♦ é ❝♦♥✈❡①♦ s❡ t♦❞❛ r❡t❛ ♥ã♦ ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠❛ ❞❛s ❢❛❝❡s ❝♦rt❛ s✉❛s ❢❛❝❡s ❡♠ ❞♦✐s ♣♦♥t♦s ♥♦ ♠á①✐♠♦✳ ❆ ✜❣✉r❛ ❛❜❛✐①♦ ❛♣r❡s❡♥t❛ ❡①❡♠♣❧♦s ❞❡ ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦ ❡ ♥ã♦ ❝♦♥✈❡①♦✳
❙❡❣✉♥❞♦ ▲✐♠❛ ❬✶✶❪✱ ♣á❣✐♥❛ ✷✽✹✱ ✉♠ ❝♦♥❥✉♥t♦ ❈ ❝♦♥t✐❞♦ ♥♦ ♣❧❛♥♦ ♦✉ ♥♦ ❡s♣❛ç♦✱ ❞✐③✲s❡ ❝♦♥✈❡①♦✱ q✉❛♥❞♦ q✉❛❧q✉❡r s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ❧✐❣❛ ❞♦✐s ♣♦♥t♦s ❞❡ ❈ ❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ ❈✳ ❆ss✐♠ ♣♦❞❡♠♦s ✐♠❛❣✐♥❛r ❡ss❡ ❝♦♥❥✉♥t♦ ❈ t❛♥t♦ ♥♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ♦✉ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠♦ ♥♦ss❛ ❢♦♥t❡ ❞❡ ❡st✉❞♦ é ♥♦R3
♣♦❞❡♠♦s ✉s❛r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ t❛♠❜é♠ ♣r♦♣♦st❛ ♣♦r ▲✐♠❛ ❬✶✶❪✱ ♣á❣✐♥❛ ✷✽✹✳
✷✳✷✳ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙
❋✐❣✉r❛ ✷✳✹✿ P♦❧✐❡❞r♦s ♥ã♦ ❝♦♥✈❡①♦ ✭à ❡sq✉❡r❞❛✮ ❡ P♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ✭à ❞✐r❡✐t❛✮✳ ❉❡✜♥✐çã♦ ✷✳✷ ❯♠ ♣♦❧✐❡❞r♦ é ❞✐t♦ ❝♦♥✈❡①♦ s❡ q✉❛❧q✉❡r r❡t❛ ✭♥ã♦ ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠❛ ❞❡ s✉❛s ❢❛❝❡s✮ ♦ ❝♦rt❛ ❡♠✱ ♥♦ ♠á①✐♠♦✱ ❞♦✐s ♣♦♥t♦s✳
✷✳✷ ❊❧❡♠❡♥t♦s ❞♦s ♣♦❧✐❡❞r♦s
◆❡st❛ s❡çã♦ ✈❡r❡♠♦s ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ♦s ♣♦❧✐❡❞r♦s✳ ❼ ❋❛❝❡s
❖s ♣♦❧í❣♦♥♦s ✷✱ ✸✱ ✹ ❡ ✺ ❞❛ ✜❣✉r❛ ❛❜❛✐①♦ sã♦ tr✐â♥❣✉❧♦s✱ ❡♥q✉❛♥t♦ ♦ ♣♦❧í❣♦♥♦ ✶ é ✉♠ q✉❛❞r❛❞♦✱ ♣♦❞❡♠♦s ✉♥í✲❧♦s ❢♦r♠❛♥❞♦ ✉♠ ♣♦❧✐❡❞r♦P✱ ❛♦ ❢♦r♠❛r ♦ ♣♦❧✐❡❞r♦ P ❡ss❡s ♣♦❧í❣♦♥♦s ❧✐♠✐t❛rã♦ ✉♠❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦✱ ❛ss✐♠ ♣♦❞❡♠♦s ❞❡✜♥✐r ❢❛❝❡s
❞❡ ✉♠ ♣♦❧✐❡❞r♦✳
❋✐❣✉r❛ ✷✳✺✿ P♦❧í❣♦♥♦s ♥♦ ♣❧❛♥♦✳
❉❡✜♥✐çã♦ ✷✳✸ ❆s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ sã♦ ♦s ♣♦❧í❣♦♥♦s q✉❡ ❧✐♠✐t❛♠ ♦ ♣♦❧✐❡✲ ❞r♦✳
✷✳✷✳ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙
❈♦♥❢♦r♠❡ ❛ ❉❡✜♥✐çã♦ ✷✳✶ ♦s ♣♦❧✐❡❞r♦s tê♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❢❛❝❡s✳ ❼ ❆r❡st❛s
❖s ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ ✶ A1B1✱ B1C1✱ C1D1 ❡D1A1 s❡rã♦ ❝♦♠✉♠ ❝♦♠ ♦s ❧❛❞♦s
❞♦s ♣♦❧í❣♦♥♦s ✷✱ ✸✱ ✹ ❡ ✺ A3B3✱ B2C2✱ C5D5 ❡ D4A4 r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊ss❡s
❧❛❞♦s ❝♦♠✉♥s ❢♦r♠❛rã♦ ❛s ❛r❡st❛s AB✱ BC✱ CD ❡DA ❞♦ ♣♦❧✐❡❞r♦ P✳
❋✐❣✉r❛ ✷✳✻✿ P♦❧✐❡❞r♦ P ♣❧❛♥✐✜❝❛❞♦✳
❉❡✜♥✐çã♦ ✷✳✹ ❆r❡st❛ é ♦ ♥♦♠❡ ❞❛❞♦ ❛ ❝❛❞❛ ❧❛❞♦ ❞❛ ❢❛❝❡ ❞♦ ♣♦❧✐❡❞r♦✱ ❛ q✉❛❧ é ❝♦♠✉♠ ❛ s♦♠❡♥t❡ ❞✉❛s ❢❛❝❡s✳
❼ ❱ért✐❝❡s
P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ♥❛ ✜❣✉r❛ ❛❝✐♠❛ ♦s ✈ért✐❝❡s ❞♦ ♣♦❧í❣♦♥♦ ✶ s❡rã♦ ❝♦♠✉♥s ❝♦♠ ♦s ✈ért✐❝❡s ❞♦s ♣♦❧í❣♦♥♦s ✷✱ ✸✱ ✹ ❡ ✺❀ ✭♣♦r ❡①❡♠♣❧♦ ♦ ✈ért✐❝❡A1 ❡B1 ❞♦ ❧❛❞♦
A1B1 ❞♦ ♣♦❧í❣♦♥♦ ✶ s❡rã♦ ❝♦♠✉♥s ❝♦♠ ♦s ✈ért✐❝❡s A3 ❡ B3✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱
❞♦ ❧❛❞♦ A3B3 ❞♦ ♣♦❧í❣♦♥♦ ✸✮✳
❋✐❣✉r❛ ✷✳✼✿ ❱✐st❛ s✉♣❡r✐♦r ❞♦ ♣♦❧✐❡❞r♦ P✳
✷✳✸✳ ❘❊▲❆➬Õ❊❙ ❊◆❚❘❊ ❖❙ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙
❉❡✜♥✐çã♦ ✷✳✺ ❱ért✐❝❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ é ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❞❡ ✸ ♦✉ ♠❛✐s ❛r❡st❛s✳
❖ ✈ért✐❝❡ ❞❡ ❝❛❞❛ ❢❛❝❡ t❛♠❜é♠ é ✈ért✐❝❡ ❞♦ ♣♦❧✐❡❞r♦ P✳
❆q✉✐✱ t❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡♥❝♦♥tr❛❞❛s ❡♠ ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ♦ ♣♦❧✐❡❞r♦✳
❙❡❣✉♥❞♦ ❉❛♥t❡ ❬✻❪✱ ♣á❣✐♥❛ ✸✻✵✱ ❝❛❞❛ ♣♦❧✐❡❞r♦ é ❢♦r♠❛❞♦ ♣❡❧❛ r❡✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ r❡❣✐õ❡s ♣♦❧✐❣♦♥❛✐s ♣❧❛♥❛s ❝❤❛♠❛❞❛s ❢❛❝❡s ❡ ❛ r❡❣✐ã♦ ❞♦ ❧✐♠✐t❛❞❛s ♣♦r ❡❧❛s✳ ❈❛❞❛ ❧❛❞♦ ❞❡ ✉♠❛ ❞❡ss❛s r❡❣✐õ❡s ♣♦❧✐❣♦♥❛❧ é t❛♠❜é♠ ❧❛❞♦ ❞❡ ✉♠❛ ♦✉tr❛ ú♥✐❝❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧✳ ❆ ✐♥t❡rs❡❝çã♦ ❞❡ ❞✉❛s ❢❛❝❡s q✉❛✐sq✉❡r ♦✉ é ✉♠❛ ❧❛❞♦ ❝♦♠✉♠✱ ♦✉ é ✉♠ ✈ért✐❝❡✱ ♦✉ é ✈❛③✐❛✳ ❈❛❞❛ ❧❛❞♦ ❞❡ ✉♠❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧✱ ❝♦♠✉♠ ❛ ❡①❛t❛♠❡♥t❡ ❞✉❛s ❢❛❝❡s é ❝❤❛♠❛❞♦ ❛r❡st❛ ❞♦ ♣♦❧✐❡❞r♦✱ ❡ ❝❛❞❛ ✈ért✐❝❡ ❞❡ ✉♠❛ ❢❛❝❡ é ✉♠ ✈ért✐❝❡ ❞♦ ♣♦❧✐❡❞r♦✳
P❛r❛ ❘✐❜❡✐r♦ ❬✶✺❪✱ ♣á❣✐♥❛ ✻✽✱ ❢❛❝❡s sã♦ ♦s ♣♦❧í❣♦♥♦s q✉❡ ❧✐♠✐t❛♠ ♦s ♣♦❧✐❡❞r♦s✳ ❚♦❞♦ ♣♦❧✐❡❞r♦ t❡♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❢❛❝❡s✳ ❆r❡st❛ é ♦ ♥♦♠❡ q✉❡ s❡ ❞á ❛ ❝❛❞❛ ❧❛❞♦ ❞❡ ✉♠❛ ❢❛❝❡ ❞♦ ♣♦❧✐❡❞r♦✳ ❈❛❞❛ ❛r❡st❛ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ é ❝♦♠✉♠ ❛ s♦♠❡♥t❡ ❞✉❛s ❢❛❝❡s✳ ❱ért✐❝❡ é ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ ✸ ♦✉ ♠❛✐s ❛r❡st❛s✳ ❖ ✈ért✐❝❡ ❞❡ ❝❛❞❛ ❢❛❝❡ t❛♠❜é♠ é ♦ ✈ért✐❝❡ ❞♦ ♣♦❧✐❡❞r♦✳
✷✳✸ ❘❡❧❛çõ❡s ❡♥tr❡ ♦s ❊❧❡♠❡♥t♦s ❞♦s P♦❧✐❡❞r♦s
P♦❞❡♠♦s r❡❧❛❝✐♦♥❛r ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ q✉❛❧q✉❡r✳ ❆s r❡❧❛çõ❡s ❞❡s❝r✐t❛s ❢♦r❛♠ ❜❛s❡❛❞❛s ♥❛q✉❡❧❛s ❛♣r❡s❡♥t❛❞❛s ♣♦r ▲✐♠❛ ❬✶✷❪✱ ♣á❣✐♥❛s ✷✽✹ ✲ ✷✽✺✳
◆❛ s❡çã♦ ❛♥t❡r✐♦r ❞❡✜♥✐♠♦s ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ✉♠ ♣♦❧✐❡❞r♦✱ ✈❛♠♦s ❛q✉✐ r❡♣r❡s❡♥tá✲❧♦s ❝♦♠♦ A✱ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✱ V ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❡ ♣♦r F ♦
♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ q✉❛❧q✉❡r✳
P♦❞❡♠♦s ❛✐♥❞❛ ✈❡r✐✜❝❛r q✉❡ ❛s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ♣♦❞❡♠ s❡r ❝♦♠♣♦st❛s ♣♦r ♣♦❧í❣♦♥♦s ❞✐❢❡r❡♥t❡s✱ r❡♣r❡s❡♥t❛r❡♠♦s ♣♦r Fn ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s q✉❡ ♣♦ss✉❡♠ n
❧❛❞♦s✱ ❝♦♠ n ≥ 3✱ ❛ss✐♠ t❡♠♦s F3 ❢❛❝❡ tr✐❛♥❣✉❧❛r✱ F4 ❢❛❝❡ q✉❛❞r❛♥❣✉❧❛r ❡ ❛ss✐♠
s✉❝❡ss✐✈❛♠❡♥t❡✳ ❙❡❣✉❡ ❞❛ ❉❡✜♥✐çã♦ ✷✳✶ q✉❡ ♦ ♥ú♠❡r♦ F ❞❡ ❢❛❝❡s é ✜♥✐t♦✱ ❡ ❛❧é♠
❞✐ss♦
F =F3+F4+...+Fn.
❉❛ ♠❡s♠❛ ❢♦r♠❛ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ♣❛r❛ ❝❛❞❛ ✈ért✐❝❡ V ❞❡ ✉♠ ♣♦❧✐❡❞r♦
❝♦♥❝♦rr❡♠ p ❛r❡st❛s✱ ❝♦♠♦ ♣❛r❛ ❝❛❞❛ ✈ért✐❝❡ ♦ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥✲
❝♦rr❡♠ ♣❛r❛ ❡❧❡ é três✱ t❡♠♦s q✉❡ p ≥ 3 ❡ Vp ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s q✉❡ ❝♦♥❝♦rr❡♠
p ❛r❡st❛s✱ ♣♦r ❡①❡♠♣❧♦ V3 é ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ♥♦ q✉❛❧ ❝♦♥❝♦rr❡♠ três ❛r❡st❛s✱
V4 ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ♥♦ q✉❛❧ ❝♦♥❝♦rr❡♠ q✉❛tr♦ ❛r❡st❛s ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳
P♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡
✷✳✸✳ ❘❊▲❆➬Õ❊❙ ❊◆❚❘❊ ❖❙ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙
V =V3+V4+...+Vp.
✷✳✸✳✶ ❋❛❝❡s ❡ ❆r❡st❛s
❱❛♠♦s ✐♠❛❣✐♥❛r ✉♠ ♣♦❧✐❡❞r♦ ❝♦♠ t♦❞❛s ❛s s✉❛s ❢❛❝❡s s❡♣❛r❛❞❛s ❡ ❞✐s♣♦st❛s s♦✲ ❜r❡ ✉♠ ♣❧❛♥♦✳ ❙❡ q✉✐s❡r♠♦s s❛❜❡r q✉❛♥t♦s ❧❛❞♦s ♣♦ss✉❡♠ t♦❞♦s ♦s ♣♦❧í❣♦♥♦s q✉❡ ❢♦r♠❛ ✉♠ ♣♦❧✐❡❞r♦ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛s ❢❛❝❡s tr✐❛♥❣✉❧❛r❡s ✭tr✐â♥❣✉❧♦s✮ ♣♦r três✱ ❛s ❢❛❝❡s q✉❛❞r❛♥❣✉❧❛r❡s ✭q✉❛❞r✐❧át❡r♦s✮ ♣♦r q✉❛tr♦✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ❡ ❞❡♣♦✐s s♦♠❛r♠♦s t♦❞♦s ♦s r❡s✉❧t❛❞♦s✱ ♣♦ré♠ ❝❛❞❛ ❛r❡st❛ ❞♦ ♣♦❧✐❡❞r♦ é ❧❛❞♦ ❝♦♠✉♠ ❞❡ ❡①❛✲ t❛♠❡♥t❡ ❞✉❛s ❢❛❝❡s✱ ❛ss✐♠ ❛ s♦♠❛ ❞❡ t♦❞♦s ♦s ❧❛❞♦s ❞❛s ❢❛❝❡s r❡s✉❧t❛ ♦ ❞♦❜r♦ ❞❡ ❛r❡st❛s✳ ▲♦❣♦✱
3F3+ 4F4+ 5F5+· · ·
| {z }
nF
= 2A.
P♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✿ nF = 2A✱ ✈❛❧❡ ♣❛r❛ t♦❞♦ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r✱ ♦♥❞❡ n é ♦
♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞❡ ❝❛❞❛ ❢❛❝❡✳
✷✳✸✳✷ ❱ért✐❝❡s ❡ ❆r❡st❛s
P♦❞❡♠♦s t❛♠❜é♠ ❞❡t❡r♠✐♥❛r ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❝♦♠ ❛ ♦❜s❡r✈❛çã♦ ♥♦s ✈ért✐❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳ ❙❡ ❝♦♥t❛r♠♦s q✉❛♥t❛s ❛r❡st❛s ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡s ✈❛♠♦s ♦❜s❡r✈❛r q✉❡ ❝❛❞❛ ❛r❡st❛ s❡rá ❝♦♥t❛❞❛ ❞✉❛s ✈❡③❡s✱ ❡♠ ✉♠ ❡①tr❡♠♦ ❡ ♥♦ ♦✉tr♦✱ ❛ss✐♠ s❡ s♦♠❛r♠♦s ♦ r❡s✉❧t❛❞♦ ❞❛s ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡s t❡r❡♠♦s ♦ ❞♦❜r♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳ ❆ss✐♠ t❡♠♦s✱
3V3+ 4V4+ 5V5+· · ·
| {z }
pV
= 2A.
P♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✿ pV = 2A✱ ✈❛❧❡ ♣❛r❛ t♦❞♦ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r✱ ♦♥❞❡ p é ♦
♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳
❙❡❣✉❡ ❞❛s r❡❧❛çõ❡s ❛❝✐♠❛ q✉❡ ♣♦❞❡♠♦s ❝♦♠♣❛rá✲❧❛s ❡ ❝♦♥❝❧✉ír♠♦s q✉❡✿
2A=nF =pV.
✷✳✸✳✸ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r
◆❡st❛ s❡çã♦ s❡rá ❞❡♠♦♥str❛❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡ ♣❡❧❛ s✉❛ ❛♣❧✐❝❛çã♦ s✐♠✲ ♣❧❡s✱ ❞❡✐①❛ ♦s ❛❧✉♥♦s ❝✉r✐♦s♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ q✉❛♥❞♦ ❡❧❡s t❡♠ ❡♠ ♠ã♦s ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ q✉❛❧q✉❡r ❡ ❢❛③❡♠ ❛ ❝♦♥t❛❣❡♠ ❞♦s ❡❧❡♠❡♥t♦s✳ P❛r❛ t❛❧ ❞❡♠♦♥str❛çã♦ ❢♦✐ ✉t✐❧✐③❛❞❛ ❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ▲✐♠❛ ❬✶✷❪✱ ♣á❣✐♥❛s ✷✽✼✲✷✾✵✱ ✈❛❧❡ s❛❧✐❡♥t❛r q✉❡ ❡❧❛ s❡❣✉✐✉ q✉❛s❡ ✐♥t❡❣r❛❧♠❡♥t❡ ❛ ♣✉❜❧✐❝❛❞❛ ♣❡❧♦ ♣r♦❢❡ss♦r ❆③❛♠❜✉❥❛ ❋✐❧❤♦ ❬✷❪✳
✷✳✸✳ ❘❊▲❆➬Õ❊❙ ❊◆❚❘❊ ❖❙ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙
❚❡♦r❡♠❛ ✷✳✶ ❉❛❞♦ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ❝♦♠A ❛r❡st❛s✱ V ✈ért✐❝❡s ❡F ❢❛❝❡s✱ t❡♠✲
s❡
V +F −A= 2.
Pr♦✈❛✿ ■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ t♦❞❛s ❛s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦P✱ ❡ss❛s ❢❛❝❡s s❡rã♦ ♥✉♠❡r❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛F1✱F2✱F3✱
✳✳✳✱Fk ❡ ✉♠ ♥ú♠❡r♦nq✉❡ r❡♣r❡s❡♥t❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛s ❢❛❝❡s n1✱ n2✱ n3✱ ✳✳✳✱ nk✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ t❡♠♦s
Si = 180
◦
·(n−2)❢ór♠✉❧❛ ✈✐st❛ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ ♦✉ s❡❥❛✱ Si =π(n−2)✳ ❆
s♦♠❛ ❞❡ t♦❞♦s ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣♦❧✐❡❞r♦ P é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ Si = π(n1−2) +π(n2−2) +...+π(nk−2)
= π(n1+n2+...+nk)−π(2 + 2 +...+ 2).
❈♦♠♦ ❝❛❞❛ ❛r❡st❛ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ é ❝♦♠✉♠ ❛ ❛♣❡♥❛s ❞✉❛s ❢❛❝❡s✱ ❧♦❣♦ ❛ s♦♠❛ ❞♦s
ni✬s ❧❛❞♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ é ✐❣✉❛❧ ❛♦ ❞♦❜r♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✱ ♦✉ s❡❥❛(n1+n2+
...+nk) = 2A✳ ❖ ♦✉tr♦ ♣❛rê♥t❡s❡ ❝♦♠♣♦st♦s ♣♦r s♦♠❛ ❞❡ ✷ ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❛ss✐♠
✿ 2(1 + 1 +...1)♦♥❞❡ ❛ s♦♠❛ ❞❡ ♣❛r❝❡❧❛s ✉♥s é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢❛❝❡s q✉❡ ❝♦♠♣õ❡♠
♦ ♣♦❧✐❡❞r♦✱ ❞❛í ❝♦♥❝❧✉í♠♦s q✉❡ (2 + 2 +...+ 2) = 2F✳ ❊♥tã♦ t❡♠♦s Si = π2A−π2F
= 2π(A−F). ✭✷✳✶✮
P❛r❛ ♦ ♣ró①✐♠♦ ♣❛ss♦ ❞❛ ❞❡♠♦♥str❛çã♦ ✈❛♠♦s t♦♠❛r ✉♠ ♣❧❛♥♦H ❤♦r✐③♦♥t❛❧ s♦❜
♦ ♣♦❧✐❡❞r♦P ❡ ✉♠❛ r❡t❛r♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦Ht❛❧ q✉❡ ❛ r❡t❛r♥ã♦ s❡❥❛ ♣❛r❛❧❡❧❛
❛ ♥❡♥❤✉♠❛ ❞❛s ❢❛❝❡s ❞♦ ♣♦❧✐❡❞r♦ P✳
❈♦❧♦❝❛♠♦s s♦❜r❡ ♦ ♣♦❧✐❡❞r♦ P ✉♠❛ ❢♦♥t❡ ❧✉♠✐♥♦s❛ ✭♣❛r❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥ssã♦
❞♦ ❡①❡♠♣❧♦ ✐♠❛❣✐♥❡ ♦ s♦❧ ❛ ♣✐♥♦✱ ♦✉ s❡❥❛✱ ♦ ♣♦♥t♦ ♠❛✐s ❡❧❡✈❛❞♦ ❞♦ s♦❧✮ ❡ss❛ ❢♦♥t❡ ♣r♦❞✉③✐rá s♦❜r❡ ♦ ♣❧❛♥♦ H ✉♠❛ s♦♠❜r❛ P′ ❞♦ ♣♦❧✐❡❞r♦ P✳ ❊ss❛ s♦♠❜r❛ P′ t❡♠
❝♦♥t♦r♥♦ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ K′ s♦♠❜r❛ ❞❡ ✉♠ ♣♦❧✐❣♦♥❛❧ ❢❡❝❤❛❞❛ K✱ ❛ q✉❛❧
❝❤❛♠❛r❡♠♦s ❞❡ ❝♦♥t♦r♥♦ ❛♣❛r❡♥t❡ ❞♦ ♣♦❧✐❡❞r♦ P✱ ❢♦r♠❛❞❛ ♣♦r ❛r❡st❛s ❞❡P✳
❈❛❞❛ ♣♦♥t♦ ❞♦ ♣♦❧í❣♦♥♦ K′ é ✉♠ ú♥✐❝♦ ♣♦♥t♦ ❞♦ ♣♦❧✐❡❞r♦ P✱ ❥á ❛ s♦♠❜r❛ P′ é
s♦♠❜r❛ ❞❡ ❡①❛t❛♠❡♥t❡ ❞♦✐s ♣♦♥t♦s ❞❡P✳ ❯♠ ❞❡ss❡s ♣♦♥t♦s ✜❝❛ ♥❛ r❡❣✐ã♦ ✐❧✉♠✐♥❛❞❛
❞❡P✱ r❡❣✐ã♦ q✉❡ ✜❝❛ ♠❛✐s ♣ró①✐♠❛ ❞❛ ❢♦♥t❡ ❧✉♠✐♥♦s❛ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡P1 t♦❞♦
♣♦♥t♦ ❞❡ss❛ r❡❣✐ã♦ é ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ ✐❧✉♠✐♥❛❞♦ ❡ ♦ ♦✉tr♦ ♣♦♥t♦ ✜❝❛rá ♥❛ r❡❣✐ã♦ ♠❡♥♦s ✐❧✉♠✐♥❛❞❛ ❞❡P✱ r❡❣✐ã♦ q✉❡ ✜❝❛ ♠❛✐s ♣ró①✐♠❛ ❞♦ ♣❧❛♥♦H✱ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s
❞❡ P2 t♦❞♦ ♣♦♥t♦ ❞❡ss❛ r❡❣✐ã♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ s♦♠❜r✐♦✳
❉❡♣♦✐s ❞❡ss❛s ❝♦♥s✐❞❡r❛çõ❡s ✈❛♠♦s ❝❛❧❝✉❧❛r ♥♦✈❛♠❡♥t❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥✲ t❡r♥♦s ❞❡ t♦❞❛s ❛s ❢❛❝❡s ❞❡ P✳ ❉❡✈❡♠♦s ❧❡♠❜r❛r q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s
❞❡ ✉♠❛ ❢❛❝❡ ❞❡ P é ✐❣✉❛❧ s♦♠❛ â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ s♦♠❜r❛ ❞❡ss❛ ❢❛❝❡✳
✷✳✸✳ ❘❊▲❆➬Õ❊❙ ❊◆❚❘❊ ❖❙ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙
❆ss✐♠ t❡♠♦s✱V =V0+V1+V2 ♦♥❞❡V0✱V1✱V2 sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛ q✉❛♥t✐❞❛❞❡
❞❡ ✈ért✐❝❡s ❞♦ ❝♦♥t♦r♥♦ ❛♣❛r❡♥t❡✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈ért✐❝❡s ❞❡ P1 ❡ ❛ q✉❛♥t✐❞❛❞❡
❞❡ ✈ért✐❝❡s ❞❡ P2✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣♦❧✐❡❞r♦ P t❡♠♦s
Si = S1+S2✱ ♦♥❞❡ S1 é ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ r❡❣✐ã♦ ✐❧✉♠✐♥❛❞❛ ❡ S2 é ❛
s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ r❡❣✐ã♦ s♦♠❜r✐❛✳ ❉❛❞❛s ♣♦r
S1 = 2πV1+π(V0−2).
❆♥❛❧♦❣❛♠❡♥t❡ t❡♠♦s
S2 = 2πV2+π(V0−2).
❊♥tã♦
Si = S1 +S2
= [2πV1+π(V0−2)] + [2πV2+π(V0−2)]
= 2πV1+ 2πV2+π(V0−2) +π(V0−2)
= 2πV1+ 2πV2+ 2π(V0−2)
= 2π(V1+V2+V0−2).
❖r❛✱ V1+V2+V0 =V✳ ❉❛í t❡♠♦s
Si = 2π(V −2). ✭✷✳✷✮
❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✸✳✶✮ ❡ ✭✸✳✷✮✿
Si = Si
2π(A−F) = 2π(V −2)
A−F = V −2
V −A+F = 2,
❞❡ss❛ ❢♦r♠❛ ❝♦♥❝❧✉í♠♦s ❛ ♣r♦✈❛✳
❈❛♣ít✉❧♦ ✸
❆ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r
◆❡st❡ ❝❛♣ít✉❧♦ s❡rá ❞❡♠♦♥str❛❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ♣❡❧❛ s✉❛ ❛♣❧✐❝❛çã♦ s✐♠✲ ♣❧❡s ❞❡✐①❛ ♦s ❛❧✉♥♦s ❝✉r✐♦s♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ q✉❛♥❞♦ ❡❧❡s t❡♠ ❡♠ ♠ã♦s ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ q✉❛❧q✉❡r❡ ❢❛③❡♠ ❛ ❝♦♥t❛❣❡♠ ❞♦s ❡❧❡♠❡♥t♦s✳ P❛r❛ t❛❧ ❞❡♠♦♥str❛çã♦ ❢♦✐ ✉t✐✲ ❧✐③❛❞❛ ❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ▲✐♠❛ ❬✶✷❪ ♣á❣✐♥❛s ✷✽✼✲✷✾✵✱ ✈❛❧❡ s❛❧✐❡♥t❛r q✉❡ ❡❧❛ s❡❣✉✐✉ q✉❛s❡ ✐♥t❡❣r❛❧♠❡♥t❡ ❛ ♣✉❜❧✐❝❛❞❛ ♣❡❧♦ ♣r♦❢❡ss♦r ❆③❛♠❜✉❥❛ ❋✐❧❤♦ ❬✷❪✳
❚❡♦r❡♠❛ ✸✳✶ ❉❛❞♦ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ❝♦♠A ❛r❡st❛s✱ V ✈ért✐❝❡s ❡F ❢❛❝❡s✱ t❡♠✲
s❡
V +F −A= 2.
Pr♦✈❛✿ ■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ t♦❞❛s ❛s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ P✱ ❡ss❛s ❢❛❝❡s s❡rã♦ ♥✉♠❡r❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ F1✱ F2✱
F3✱ ✳✳✳✱ Fk ❡ ✉♠ ♥ú♠❡r♦ n q✉❡ r❡♣r❡s❡♥t❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛s
❢❛❝❡s n1✱ n2✱ n3✱ ✳✳✳✱ nk✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦
t❡♠♦s Si = 180
◦
·(n−2) ✈✐st❛ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♦✉ s❡❥❛✱ Si = π(n−2)✳ ❆
s♦♠❛ ❞❡ t♦❞♦s ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣♦❧✐❡❞r♦ P é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ Si = π(n1−2) +π(n2−2) +...+π(nk−2)
= π(n1+n2+...+nk)−π(2 + 2 +...+ 2).
❈♦♠♦ ❝❛❞❛ ❛r❡st❛ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ é ❝♦♠✉♠ ❛ ❛♣❡♥❛s ❞✉❛s ❢❛❝❡s✱ ❧♦❣♦ ❛ s♦♠❛ ❞♦s n ✭❡♥❡s✮ ❧❛❞♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ é ✐❣✉❛❧ ❛♦ ❞♦❜r♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✱ ♦✉ s❡❥❛
(n1 +n2 +...+nk) = 2A✳ ❖ ♦✉tr♦ ♣❛rê♥t❡s❡ ❝♦♠♣♦st♦s ♣♦r ✷ ♣♦❞❡ s❡r r❡❡s❝r✐t♦
❛ss✐♠ ✿ 2(1 + 1 +...1) ♦♥❞❡ ❛ s♦♠❛ ❞❡ ♣❛r❝❡❧❛s ✉♥s é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢❛❝❡s q✉❡
❝♦♠♣õ❡♠ ♦ ♣♦❧✐❡❞r♦✱ ❞❛í ❝♦♥❝❧✉í♠♦s q✉❡ (2 + 2 +...+ 2) = 2F✳ ❊♥tã♦ t❡♠♦s Si = π2A−π2F
= 2π(A−F). ✭✸✳✶✮
