❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
❖s ▲❡♠❛s ❞❡ ❙♣❡r♥❡r ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡ ✉♠❛
♠♦❞❡st❛ ✐♥tr♦❞✉çã♦ à ❚♦♣♦❧♦❣✐❛
❚❤❛❞❡♦ ❆✉❣✉st♦ ❘♦❝❤❛ ❞❡ ❆③❛♠❜✉❥❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❚❤❛ís ❋❡r♥❛♥❞❛ ▼❡♥❞❡s ▼♦♥✐s
✶✶✶ ❳✶✶✶①
❆③❛♠❜✉❥❛✱ ❚❤❛❞❡♦ ❆✉❣✉st♦ ❘♦❝❤❛
❖s ▲❡♠❛s ❞❡ ❙♣❡r♥❡r ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡ ✉♠❛ ♠♦❞❡st❛ ✐♥tr♦❞✉✲ çã♦ à ❚♦♣♦❧♦❣✐❛✴ ❚❤❛❞❡♦ ❆✉❣✉st♦ ❘♦❝❤❛ ❞❡ ❆③❛♠❜✉❥❛✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✹✳
✸✻ ❢✳✿ ✜❣✳✱ t❛❜✳
❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳
❖r✐❡♥t❛❞♦r❛✿ ❚❤❛ís ❋❡r♥❛♥❞❛ ▼❡♥❞❡s ▼♦♥✐s
✶✳ ▲❡♠❛s ❞❡ ❙♣❡r♥❡r✳ ✷✳ ❚♦♣♦❧♦❣✐❛✳ ✸✳ ❍♦♠❡♦♠♦r✜s♠♦✳ ✹✳ P♦♥t♦ ✜①♦ ❞❡ ❇r♦✉✇❡r✳ ■✳ ❚ít✉❧♦
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
❚❤❛❞❡♦ ❆✉❣✉st♦ ❘♦❝❤❛ ❞❡ ❆③❛♠❜✉❥❛
❖s ▲❡♠❛s ❞❡ ❙♣❡r♥❡r ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡ ✉♠❛ ♠♦❞❡st❛
✐♥tr♦❞✉çã♦ à ❚♦♣♦❧♦❣✐❛
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢❛✳ ❉r❛✳ ❚❤❛ís ❋❡r♥❛♥❞❛ ▼❡♥❞❡s ▼♦♥✐s ❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ❯◆❊❙P✲❘✐♦ ❈❧❛r♦
Pr♦❢✳ ❉r✳ ❘♦❞r✐❣♦ ▼❛rt✐♥s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ ❞❡ ▼❛r✐♥❣á
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ♠✐♥❤❛ ❢❛♠í❧✐❛ q✉❡ ❡st❡✈❡ ❥✉♥t♦ ❝♦♠✐❣♦✳
▼✐♥❤❛ ❡s♣♦s❛✱ ▼❛✐♥❛r❛ ▲❡♥③✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣♦r ❡st❛r ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡♥tr♦ ❞♦ ❈✉rs♦ ✭❞❡s❞❡ ❛ ✐♥s❝r✐çã♦ ❛té ♦ ♣r♦t♦❝♦❧♦ ✜♥❛❧ ❞❡st❡ tr❛❜❛❧❤♦✮✳ P♦r✱ ♥❡ss❡s três ❛♥♦s✱ t❡r ♣❛ss❛❞♦ ♣♦r t✉❞♦ ❛♦ ♠❡✉ ❧❛❞♦ ❧✐t❡r❛❧♠❡♥t❡✳
❆s ♠✐♥❤❛s ✜❧❤❛s✿ ❈❡❝í❧✐❛✱ ❆❧✐♥❡ ❡ ❇❡❛tr✐③ q✉❡✱ ❛♦s ♣♦✉❝♦s✱ ❢♦r❛♠ ❡♥t❡♥❞❡♥❞♦ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ tr❛❜❛❧❤♦✳ ❉❛ ♠❛♥❡✐r❛ q✉❡ ✉♠❛ ❝r✐❛♥ç❛ ♣♦❞❡ ❛❝♦♠♣❛♥❤❛r✱ ✜③❡r❛♠ ❞❡ t✉❞♦ ❛♦ ❛❧❝❛♥❝❡✳
▼❡✉s ♣❛✐s✱ ❙✐❧✈✐❛ ❡ ❙✐❧✈✐♦✱ ♠❡✉s ❛✈ós✱ ✈ó ▲✉❝✐❛ ❡ ✈✬Ô❧✐✈✐♦ ❡ ♠❡✉s s♦❣r♦s✱ ❊❧③❛ ❡ P❡❞r♦✱ ♣♦r t♦r❝❡r❡♠ ❡ r❡③❛r❡♠ ♦ t❡♠♣♦ t♦❞♦ ♣♦r ❡st❡ tr❛❜❛❧❤♦✳
❆♦s ❛♠✐❣♦s q✉❡ ❛♣♦✐❛r❛♠ ❡ ❞❡r❛♠ ♠✉✐t♦s ❡♠♣✉rrõ❡s✳ ❊ ❛♦s q✉❡ ❞❡r❛♠ ♠✉✐t❛ ❞✉r❛ t❛♠❜é♠✳ ❆♠✐❣♦s q✉❡ s❡❣✉r❛r❛♠ ❛ ❜❛rr❛ ❞❛ ❢❛♠í❧✐❛ t♦❞❛✿ ❡s♣❡❝✐❛❧♠❡♥t❡ ❋r❛♥❝❡s❝❛✱ ❆❞r✐❛♥❛ ❡ ❋♦♥tã♦✱ ▲✉❝é❧✐❛ ❡ ❍❡❧✐♥❤♦✱ ❆♠❛♥❞❛ ❡ ❚❤✐❛❣♦✳
❆♦s ❛♠✐❣♦s✱ ❝♦❧❡❣❛s ❞❡ ❝❧❛ss❡✱ s❡♠♣r❡ ❡st✉❞❛❞❛♥❞♦✱ ❛♣♦✐❛♥❞♦✱ r✐♥❞♦✱ ❝❤♦r❛♥❞♦✳ ❈❛r❧♦s ❆✉❣✉st♦✱ ✉♠❛ ♣❡ss♦❛ ♣❛r❛ ♠♦♠❡♥t♦s ✐♥❡sq✉❡❝í✈❡✐s✳ ▼❛r❝✐❛ ❡ ❆r②✱ ❝♦❧❡❣❛s ❡s✲ ♣❡❝✐❛✐s✳ ❏✉❧✐❛♥❛✱ ♣♦r ✉♠❛ ❢♦rç❛ ✐♥❝❛❧❝✉❧á✈❡❧✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❈♦❧é❣✐♦ Pr❡✈❡✲❇❛✉r✉ ❡ ❞♦ ❈♦❧é❣✐♦ ❚é❝♥✐❝♦ ■♥❞✉s✲ tr✐❛❧ ✭❈❚■ ❯◆❊❙P✲❇❛✉r✉✮ q✉❡ ♠❡ ✜③❡r❛♠ ❝❤❡❣❛r ❛q✉✐✳ ❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞❛ ❯◆❊❙P✲❇❛✉r✉✳ ❊s♣❡❝✐❛❧♠❡♥t❡ ❛♦s ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❞❡ t♦❞❛ ♠✐♥❤❛ ❢♦r✲ ♠❛çã♦✿ ❘♦s❛♥❛✱ ❆♥❛✱ ❚❡r❡③❛✱ ❏ô✱ ❆❞❡❧✐❛♥❡ ✭✐♥ ♠❡♠♦r✐❛♥✮✱ ▼❛✉rí❝✐♦✱ ❆♥❛ ❈❧á✉❞✐❛✱ ❈❧é❧✐❛✱ ❙✐❧♠❛r❛✱ ●✐❧③❛✱ ▲✉❝✐❛♥❡✳ ❆ ▼❛r❧❡✐ ▼❡♥❡③❡s✱ q✉❡ ♠❡ ❞❡✉ ♦s ♣r✐♠❡✐r♦s ♣❛ss♦s ♥❛ ♣r♦✜ssã♦✳
❆♦s ❝♦❧❡❣❛s ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ✧Pr♦❢❡ss♦r ❏♦ã♦ ◗✉❡✐ró③ ▼❛rq✉❡s✲ ❇♦t✉❝❛t✉ ❡ ❈♦✲ ❧é❣✐♦ ❙❛♥t❛ ▼❛r❝❡❧✐♥❛ ✲ ❇♦t✉❝❛t✉✱ q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✳
❆ t♦❞❛ ❡q✉✐♣❡ P❘❖❋▼❆❚ ❞❛ ❯◆❊❙P✲❘✐♦ ❈❧❛r♦✳ Pr♦❢❡ss♦r❛ ❚❤❛ís ▼♦♥✐s✱ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✳ Pr♦❢❡ss♦r ❚❤✐❛❣♦✱ q✉❡ ♥❡♠ s❛❜❡ q✉❛♥t❛ ❛❥✉❞❛ ❞❡✉✳ Pr♦❢❡ss♦r❛ ❙✉③✐♥❡✐✱ ❝♦♦r❞❡♥❛❞♦r❛ ❞♦ ❈✉rs♦✱ s❡♠♣r❡ ♣r❡s❡♥t❡ ❡ ❝♦♥✜❛♥t❡✳
❆ ▼❛t❡♠át✐❝❛ ❛♣r❡s❡♥t❛ ✐♥✈❡♥çõ❡s tã♦ s✉t✐s q✉❡ ♣♦❞❡rã♦ s❡r✈✐r ♥ã♦ só ♣❛r❛ s❛t✐s❢❛③❡r ♦s ❝✉r✐♦s♦s ❝♦♠♦✱ t❛♠❜é♠ ♣❛r❛ ❛✉①✐❧✐❛r ❛s ❛rt❡s ❡ ♣♦✉♣❛r tr❛❜❛❧❤♦ ❛♦s ❤♦♠❡♥s✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❧❡✈❛r ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛❧❣✉♥s tó♣✐❝♦s ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ ♥ã♦ ❝♦♥st❛♠ ❡♠ ♣❧❛♥♦s ❞❡ ❡♥s✐♥♦ ❡ q✉❡ sã♦ ❞❡ rá♣✐❞❛s ❡①♣❧✐❝❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s✳ ❖❜❥❡t✐✈❛♠❡♥t❡ ❞✐s❝✉t✐r❡♠♦s ♦s ▲❡♠❛s ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ ✐♥t❡r✈❛❧♦ ❡ ♣❛r❛ ♦ tr✐â♥❣✉❧♦✳ ❆❜♦r❞❛r❡♠♦s ❛✐♥❞❛ ❛ r❡❧❛çã♦ ❞❡ t❛✐s ❧❡♠❛s ❝♦♠ ❛ ♥ã♦ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❡♠♣❛t❡s ♣❛r❛ ❞❡t❡r♠✐♥❛❞♦s ❥♦❣♦s✳ P♦r ✜♠✱ ❛♣r❡s❡♥t❛r❡♠♦s ❝♦♠♦ ✉♠❛ ❜❡❧íss✐♠❛ ❛♣❧✐❝❛çã♦ ❞♦s ▲❡♠❛s ❞❡ ❙♣❡r♥❡r ♦ ❢❛♠♦s♦ ❚❡♦r❡♠❛ ❞♦ ♣♦♥t♦ ✜①♦ ❞❡ ❇r♦✉✇❡r✳
❆❜str❛❝t
❚❤✐s st✉❞② ❛✐♠s t♦ t❛❦❡ s♦♠❡ ♠❛t❤❡♠❛t✐❝s t♦♣✐❝s t❤❛t ❛r❡ ♥♦t ✐♥❝❧✉❞❡❞ ❛t ❡❞✉❝❛t✐♦♥ ♣❧❛♥ t♦ ❤✐❣❤ s❝❤♦♦❧✳ ❚❤❡s❡ t♦♣✐❝s ❛r❡ ❢❛st t♦ ❡①♣❧❛✐♥ ❛♥❞ t♦ ❛♣♣❧②✳ ❖❜❥❡❝t✐✈❡❧②✱ ✇❡ ❞✐s❝✉ss ❙♣❡r♥❡r✬s ▲❡♠♠❛ ❢♦r ✐♥t❡r✈❛❧ ❛♥❞ ❢♦r tr✐❛♥❣❧❡✳ ❲❡ ❜r♦❛❝❤ t❤❡ r❡❧❛t✐♦♥ ♦❢ t❤✐s ▲❡♠♠❛ t♦ t❤❡ ♥♦♥✲♣♦ss✐❜✐❧✐t② ♦❢ ❡q✉❛❧✐t② t♦ ❝❡rt❛✐♥ ❣❛♠❡s✳ ❋✐♥❛❧❧②✱ ✇❡ ✐❧❧✉str❛t❡ ❛s ❛ ❜r✐❧❧✐❛♥t ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❙♣❡r♥❡r✬s ▲❡♠♠❛ t❤❡ ❢❛♠♦✉s ❇r♦✉✇❡r✬s ✜①❡❞✲♣♦✐♥t t❤❡♦r❡♠ ✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❯♠ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ▼❛✐s ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸ ❈ô♠♦❞♦s ❞❡ ✉♠❛ ❝❛s❛ ❝♦♠ ✵✱ ✶ ♦✉ ✷ ♣♦rt❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹ P❛ss❡✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✺ ❊①❡♠♣❧♦ ❞❡ tr✐❛♥❣✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✻ ❊①❡♠♣❧♦ ❞❡ ♥ã♦ tr✐❛♥❣✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✼ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✽ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ tr✐â♥❣✉❧♦ ✲ ❝❛♠✐♥❤♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✾ ◗✉❛❞r❛❞♦s ❞❛ ♣❛rt✐çã♦ ❝♦♠ ♣❡❧♦ ♠❡♥♦s três ♠❛r❝❛s ❞✐st✐♥t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶ ❏♦❣♦ ❝♦♠ ✺ ✐♠♣❛❝t♦s ✲ ❥♦❣❛❞♦r í♠♣❛r ❣❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ❊①❡♠♣❧♦ ❞❡ t❛❜✉❧❡✐r♦ ♣❛r❛ ♦ ❏♦❣♦ ✐♠♣❛❝t♦ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸ P♦♥t♦s ❡①tr❡♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹ ❘❡s✉❧t❛❞♦ ❞❡ ✉♠ ❥♦❣♦ s♦❜❡✲❞❡s❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✺ ◆ú♠❡r♦ j ❝♦❧♦❝❛❞♦ ♥♦ t❛❜✉❧❡✐r♦ ♥❛ ❥♦❣❛❞❛ i ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✻ ❊①❡♠♣❧♦ ❞❡ tr✐â♥❣✉❧♦ ❝♦♠ tr✐❛♥❣✉❧❛çã♦ ♣❛r❛ ♦ ❥♦❣♦ ❞❡ ❙♣❡r♥❡r ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✼ ❆❧❣✉♠❛s ❥♦❣❛❞❛s ❡ ♦ ✜♠ ❞♦ ❥♦❣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✶ ❆ ❢✉♥çã♦f é ✐♥✈♦❧✉t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸✳✷ ❆ ❢✉♥çã♦f é ✐♥❥❡t♦r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸✳✸ ❆ ❢✉♥çã♦f é s♦❜r❡❥❡t♦r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
❙✉♠ár✐♦
✶ ▲❡♠❛s ❝♦♠❜✐♥❛t♦r✐❛✐s ✶✶
✶✳✶ Pr✐♠❡✐r♦ ▲❡♠❛ ❈♦♠❜✐♥❛t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❙❡❣✉♥❞♦ ▲❡♠❛ ❈♦♠❜✐♥❛t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸✳✶ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳✷ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ q✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷ ❏♦❣♦s ❞♦ t✐♣♦ ❙♣❡r♥❡r ✶✽
✷✳✶ ❏♦❣♦ ❞♦s ■♠♣❛❝t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✶ ❘❡s✉❧t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳✷ ■♠♣❛❝t♦ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❏♦❣♦ ❞♦ s♦❜❡✲❞❡s❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✶ ❘❡s✉❧t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ❏♦❣♦ ♣❛r❛ ♦ ▲❡♠❛ ❞♦ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸ ❆ t♦♣♦❧♦❣✐❛ ❞❛ r❡t❛ ✷✸
✸✳✶ ❋✉♥çõ❡s ❝♦♥tí♥✉❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ❍♦♠❡♦♠♦r✜s♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✸ ❙❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✹ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❡ ♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ■♥t❡r♠❡❞✐ár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇r♦✉✇❡r ✸✶ ✹✳✶ ❊♠ ✉♠ ✐♥t❡r✈❛❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✷ P❛r❛ ♦ q✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
■♥tr♦❞✉çã♦
◆❡st❡ tr❛❜❛❧❤♦ tr❛t❛r❡♠♦s ♦s ▲❡♠❛s ❈♦♠❜✐♥❛t♦r✐❛✐s ❞❡ ❙♣❡r♥❡r ♣♦r ♠❡✐♦ ❞❡ ❥♦❣♦s ❞❡ r❛❝✐♦❝í♥✐♦✴❡str❛té❣✐❛✳ ❏♦❣♦s s✐♠♣❧❡s ❡ rá♣✐❞♦s ❝♦♠ ❛♣❧✐❝❛çã♦ ♣rát✐❝❛ ❞♦s r❡s✉❧t❛❞♦s✳ ▲❡✈á✲❧♦s ♣❛r❛ ❛ s❛❧❛ ❞❡ ❛✉❧❛ ❞❡ ❊♥s✐♥♦ ❇ás✐❝♦ é ❧❡✈❛r ❛ ❝r✐❛t✐✈✐❞❛❞❡✱ ❛ ❝♦♠♣❡t✐✈✐❞❛❞❡ s❛✉❞á✈❡❧ ❡✱ t❛♠❜é♠✱ ❛ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❙✉♣❡r✐♦r ❝♦♠ s✉❛ ❢♦r♠❛❧✐❞❛❞❡✳ ❆ss✐♠ q✉❡r❡♠♦s ❝♦♠♦ ♣❛rt❡ ❞❡st❡ tr❛❜❛❧❤♦✳
◆ã♦ ♣♦❞❡rí❛♠♦s ❞❡✐①❛r ❞❡ ❝♦❧♦❝❛r ✉♠❛ ♦✉tr❛ ❢✉♥çã♦ ♣❛r❛ ♦s ❧❡♠❛s✿ ♥❡st❡ tr❛❜❛❧❤♦✱ ❛♣♦♥t❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇r♦✉✇❡r ❡ s✉❛ ❞❡♠♦♥str❛çã♦ ❛♣♦✐❛❞❛ ♥♦s r❡s✉❧t❛❞♦s ❞♦s ▲❡♠❛s ❞❡ ❙♣❡r♥❡r✳
❆ ❞❡♠♦♥str❛çã♦ ❛♣♦♥t❛❞❛ ❞♦ ❈❛♣ít✉❧♦ ✹ ♥❡st❡ tr❛❜❛❧❤♦ ♥ã♦ é ❛ ✈❡rsã♦ ✐♥✐❝✐❛❧ ❞♦s tr❛❜❛❧❤♦s ❞❡ ❇r♦✉✇❡r✱ ♣♦✐s ❡st❛ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ à s♦❝✐❡❞❛❞❡ ❛♥t❡s ❞❡ ❙♣❡r♥❡r ❛♣r❡s❡♥t❛r s❡✉s ❧❡♠❛s ❡ r❡s✉❧t❛❞♦s✳ ❆✐♥❞❛ ❛ss✐♠✱ ♣r❡❢❡r✐♠♦s ♦ ❝❛♠✐♥❤♦ ❛♣♦♥t❛❞♦ ♣♦r ✈❡r♠♦s q✉❡ ♥❡❧❡ ❝♦♥st❛♠ ✐❞❡✐❛s s✐♠♣❧❡s ❡ rá♣✐❞❛s q✉❡ ♣♦❞❡♠ s❡r ✉s❛❞❛s ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✳ P❛r❛ t❛❧ ❞❡♠♦♥str❛çã♦ ♥ã♦ sã♦ ♥❡❝❡ssár✐♦s ♠✉✐t♦s ♣ré✲r❡q✉✐s✐t♦s ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ♦✉ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧❀ ♦s ❡❧❡♠❡♥t♦s ♥❡❝❡ssár✐♦s ❡stã♦ ♥♦ ❈❛♣ít✉❧♦ ✸✱ ❜❛s❡❛❞♦ ♥♦ ❡st✉❞♦s ❞❡ ❢✉♥çõ❡s ❡ s❡q✉ê♥❝✐❛s✳
❖ ❣r❛♥❞❡ ♣❛ss♦ ♥♦ ❈❛♣ít✉❧♦ ✸ é ❛ ♣r♦❝✉r❛ ❞❡ ✉♠ ♣♦♥t♦ ✜①♦ ❡♠ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❖ q✉❡ ❛♣r❡s❡♥t❛♠♦s é ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❡♠ ✉♠❛ ♣r♦❝✉r❛ ❞❡ r❛í③❡s ❞❡ ✉♠❛ ❢✉♥çã♦✱ t❡♥❞♦ ❛ss✐♠ ❛❧❣✉♥s t❡r♠♦s q✉❡ ❛❧✉♥♦s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦ ❡♥t❡♥❞❡♠ r❛♣✐❞❛♠❡♥t❡✳ ❊ t♦❞♦ ♦ ❝❛♣ít✉❧♦ tr❛③ t❡♦r❡♠❛s q✉❡ ♣♦❞❡♠ s❡r ❧❡✈❛❞♦s ✐♥t✉✐t✐✈❛♠❡♥t❡ ❛♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳ ❖ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ■♥t❡r♠❡❞✐ár✐♦ é ✉♠ ❞❡❧❡s❀ ❛♣r❡s❡♥tá✲❧♦ r✐❣♦r♦s❛♠❡♥t❡ ❛ ❛❧✉♥♦s q✉❡ ♥ã♦ s❡❣✉✐rã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝❛rr❡✐r❛ ❞❛ ár❡❛ ❞❡ ❡①❛t❛s ♣♦❞❡ s❡r ❝❛♥s❛t✐✈♦✱ ❡♥q✉❛♥t♦ q✉❡ ❛♦ ♠♦str❛r ❧✉❞✐❝❛♠❡♥t❡ ♣♦❞❡ ♠♦t✐✈á✲❧♦s✱ ✐♥❝❧✉✐♥❞♦ ♦s ♣r♦❢❡ss♦r❡s✱ à ♣r♦❝✉r❛ ❞❛ ❧✐♥❣✉❛❣❡♠ ♣❛❞rã♦✳
◆♦ ❈❛♣ít✉❧♦ ✶ ❡stã♦ ♦s ❧❡♠❛s ❝♦♠❜✐♥❛t♦r✐❛✐s ✭▲❡♠❛s ❞❡ ❙♣❡r♥❡r✮ ♥❛ s✉❛ ✈❡rsã♦ r✐❣♦r♦s❛ ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ ♥♦ ❈❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛♠♦s ❝♦♠♦ ✉♠❛ ❞❡♠♦♥str❛çã♦ ✐♥❢❛♥t♦✲❥✉✈❡♥✐❧✿ ❥♦❣♦s rá♣✐❞♦s✱ s✐♠♣❧❡s ❡ ❞❡s❛✜❛❞♦r❡s✱ ♣♦✐s ♥ã♦ ❤á ❡♠♣❛t❡✱ ❡ ❛ ❞❡rr♦t❛ é ♣♦✉❝❛s ✈❡③❡s ♣♦r ❝♦♥t❛ ❞♦ ❛③❛r✳
✶ ▲❡♠❛s ❝♦♠❜✐♥❛t♦r✐❛✐s
◆❡st❡ ❝❛♣ít✉❧♦✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❧❡♠❛s ♠❛t❡♠át✐❝♦s q✉❡ ❡♥✈♦❧✈❡♠ ✐♥t❡r✈❛❧♦s r❡❛✐s✱ q✉❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ ▲❡♠❛s ❈♦♠❜✐♥❛t♦r✐❛✐s✳ ❖s r❡s✉❧t❛❞♦s ❞❡st❡s ❧❡♠❛s ❡♥✈♦❧✈❡♠ t♦❞❛ ❛ ♣❛rt❡ ❞♦s ❥♦❣♦s ✐♥❞✐❝❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳
✶✳✶ Pr✐♠❡✐r♦ ▲❡♠❛ ❈♦♠❜✐♥❛t♦r✐❛❧
▲❡♠❛ ✶✳✶✳ ❙❡❥❛ ✉♠ ✐♥t❡r✈❛❧♦I ❞✐✈✐❞✐❞♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦s ❡♠ ✐♥t❡r✈❛❧♦s
♠❡♥♦r❡s✳ ❙✉♣♦♥❤❛♠♦s q✉❡ s❡✉ ❡①tr❡♠♦ ✐♥❢❡r✐♦r ❡stá ♠❛r❝❛❞♦ ♣❡❧♦ ♥ú♠❡r♦0✱ ♦ s✉♣❡r✐♦r✱
♣❡❧♦ ♥ú♠❡r♦1❡ ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞✐✈✐sór✐♦s t❡♠ ✉♠❛ ♠❛r❝❛0♦✉1✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱
♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ s✉❜✐♥t❡r✈❛❧♦ ❝✉❥♦s ❡①tr❡♠♦s ❡stã♦ ♠❛r❝❛❞♦s ❝♦♠ ♥ú♠❡r♦s ❞✐st✐♥t♦s✳ ❊ ♠❛✐s✱ ♦ ♥ú♠❡r♦ ❞❡st❡s s✉❜✐♥t❡r✈❛❧♦s é í♠♣❛r✳
❉❡♠♦♥str❛çã♦✳ i. Pr✐♠❡✐r❛♠❡♥t❡✱ ♠♦str❛r❡♠♦s q✉❡ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ s✉❜✐♥t❡r✈❛❧♦
❝♦♠ ♠❛r❝❛s ❞✐st✐♥t❛s✳ ❈❤❛♠❛r❡♠♦s ❞❡ r❛③♦á✈❡❧ ✉♠ t❛❧ s✉❜✐♥t❡r✈❛❧♦✳ ❖s ✐♥t❡r✈❛❧♦s r❛③♦á✈❡✐s ♣♦❞❡♠ s❡r ❞❡ ❞♦✐s t✐♣♦s✿ s❡rá ❝❤❛♠❛❞♦ ❞❡ í♠♣❛r ♦ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧ ❝♦♠ ❡①tr❡♠♦ ✐♥❢❡r✐♦r ♠❛r❝❛❞♦ ♣♦r 0 ❡ s✉♣❡r✐♦r ♠❛r❝❛❞♦ ♣♦r 1 ❡ s❡rá ❝❤❛♠❛❞♦ ❞❡ ♣❛r ♦
✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧ ❝♦♠ ♠❛r❝❛s ✐♥✈❡rt✐❞❛s✳
❉✐✈✐❞✐♥❞♦ ♦ ✐♥t❡r✈❛❧♦ ❡ ♠❛r❝❛♥❞♦ ❛❧❡❛t♦r✐❛♠❡♥t❡ ♦s s✉❜✐♥t❡r✈❛❧♦s ❝♦♠ ❛s ♠❛r❝❛s0
♦✉ 1✿ ♦✉ (a)t♦❞♦s sã♦ ♠❛r❝❛❞♦s ♣♦r 0♦✉ (b) ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s t❡♠ ❛ ♠❛r❝❛ 1✳
◆♦ ❝❛s♦ (a)✱ t❡♠♦s ✉♠ ú♥✐❝♦ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧✿ ♦ s✉❜✐♥t❡r✈❛❧♦ í♠♣❛r ♥♦ ✜♥❛❧ ❞♦
✐♥t❡r✈❛❧♦I✭♠❛r❝❛❞♦ ❝♦♠⋆♥❛ ✜❣✉r❛1.1✮✳ ◆♦ ❝❛s♦(b)✱ ❡s❝♦❧❤❡r❡♠♦s ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s
❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞✐✈✐s♦r❡s ♦ ♣♦♥t♦ ❞✐✈✐sór✐♦ ♠❛r❝❛❞♦ ♣♦r1q✉❡ ❡stá ♠❛✐s ♣❡rt♦ ❞♦
❡①tr❡♠♦ ✐♥❢❡r✐♦r ❞♦ ✐♥t❡r✈❛❧♦ I ❡ ♦ ❞❡♥♦♠✐♥❛r❡♠♦s ♣♦r M✳ ❖ s✉❜✐♥t❡r✈❛❧♦ q✉❡ ♣♦ss✉✐ M t❡♠ ❡①tr❡♠♦ ✐♥❢❡r✐♦r ♠❛r❝❛❞♦ ♣♦r0❡ ♣♦r ✐ss♦ é ✉♠ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧ ❞♦ t✐♣♦ í♠♣❛r✳
❋✐❣✉r❛ ✶✳✶✿ ❯♠ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧
❙❡❣✉♥❞♦ ▲❡♠❛ ❈♦♠❜✐♥❛t♦r✐❛❧ ✶✷
ii.❉❡♠♦♥str❛r❡♠♦s✱ ❛❣♦r❛✱ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ✐♥t❡r✈❛❧♦s r❛③♦á✈❡✐s é í♠♣❛r✳ P♦❞❡♠♦s
♣❡r❝♦rr❡r ♣❡❧♦ ✐♥t❡r✈❛❧♦ ❛ ♣❛rt✐r ❞♦ ❡①tr❡♠♦ ✐♥❢❡r✐♦r ❡ ♥♦♠❡❛♥❞♦ s✉❝❡ss✐✈❛♠❡♥t❡ ♦s ✐♥t❡r✈❛❧♦s r❛③♦á✈❡✐s✿ ♦ ♣r✐♠❡✐r♦ s✉❜✐♥t❡r✈❛❧♦ é ❞♦ t✐♣♦ í♠♣❛r ❡ s❡rá ❝❤❛♠❛❞♦ ❞❡ um✱ ♦
s❡❣✉♥❞♦ q✉❡ é ♣❛r✱ ❞❡dois✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ❛té ♦ ú❧t✐♠♦ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧✳ ❏á
q✉❡ ♦ ♣♦♥t♦ ❡①tr❡♠♦ s✉♣❡r✐♦r ❞❡ ✐♥t❡r✈❛❧♦I ❡stá ♠❛r❝❛❞♦ ❝♦♠1✱ é ✐♠♣♦ssí✈❡❧ ♦ ú❧t✐♠♦
✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧ s❡r ❞♦ t✐♣♦ ♣❛r✱ ♣♦rt❛♥t♦ ♦ ú❧t✐♠♦ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧ t❡♠ ❛ ♠❛r❝❛ 1
à ❞✐r❡✐t❛ ❡ s❡✉ ♥♦♠❡ s❡rá ♦ ❞❡ ✉♠ ♥ú♠❡r♦ í♠♣❛r❀ ❡♥tã♦ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✐♥t❡r✈❛❧♦s r❛③♦á✈❡✐s s❡rá í♠♣❛r✳
❋✐❣✉r❛ ✶✳✷✿ ▼❛✐s ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧
✶✳✷ ❙❡❣✉♥❞♦ ▲❡♠❛ ❈♦♠❜✐♥❛t♦r✐❛❧
❈♦♥s✐❞❡r❡ ✉♠❛ ❝❛s❛ t❛❧ q✉❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s ❝ô♠♦❞♦s ♣♦ss✉❛ ③❡r♦✱ ✉♠❛ ♦✉ ❞✉❛s ♣♦rt❛s✳ ❯♠ q✉❛rt♦ ❝♦♠ ❞✉❛s ♣♦rt❛s ❝❤❛♠❛r❡♠♦s ❞❡ q✉❛rt♦ ❞❡ ♣❛ss❛❣❡♠✳ ❯♠ ❝ô♠♦❞♦ ❝♦♠ s♦♠❡♥t❡ ✉♠❛ ♣♦rt❛ ♣♦❞❡rá s❡r ❞❡ ❞♦✐s t✐♣♦s✿ ♦✉ ❛ ♣♦rt❛ é ✉♠❛ ♣♦rt❛ ❡①t❡r✐♦r ✭✈♦❧t❛❞♦ ♣❛r❛ ❛ r✉❛✮ ♦✉ ✉♥❡ ❛ ❞♦✐s q✉❛rt♦s ✭❝❤❛♠❛❞♦ q✉❛rt♦ s❡♠ s❛í❞❛✮✳ ❙✉♣♦♠♦s t❛♠❜é♠ q✉❡ ✉♠ q✉❛rt♦ só ♣♦❞❡ t❡r ✉♠❛ ♣♦rt❛ ❡①t❡r✐♦r✱ ❡ ❞♦✐s q✉❛rt♦s ✈✐③✐♥❤♦s ♥ã♦ ♣♦❞❡♠ t❡r ♠❛✐s ❞❡ ✉♠❛ ♣♦rt❛ ❝♦♠✉♠✳
❋✐❣✉r❛ ✶✳✸✿ ❈ô♠♦❞♦s ❞❡ ✉♠❛ ❝❛s❛ ❝♦♠ ✵✱ ✶ ♦✉ ✷ ♣♦rt❛s
▲❡♠❛ ✶✳✷✳ ❙❡ ❝❛❞❛ q✉❛rt♦ ❞❡ ✉♠❛ ❝❛s❛ t❡♠ s♦♠❡♥t❡ ✉♠❛✱ ❞✉❛s ♦✉ ♥❡♥❤✉♠❛ ♣♦rt❛✱ ♦ ♥ú♠❡r♦ ❞❡ q✉❛rt♦s s❡♠ s❛í❞❛ ❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦rt❛s ❡①t❡r✐♦r❡s t❡♠ ❛ ♠❡s♠❛ ♣❛r✐❞❛❞❡✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ❛♠❜♦s sã♦ ♣❛r❡s ♦✉ ❛♠❜♦s sã♦ í♠♣❛r❡s✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ ❞❡♠♦♥str❛r ❡st❡ ❧❡♠❛ ✈❛♠♦s ❞❡s❝r❡✈❡r ♦s ♣❛ss❡✐♦s ♣❡❧♦s q✉❛rt♦s ❞❡ ✉♠❛ ❝❛s❛✳ ❊st❡s s❡ r❡❛❧✐③❛rã♦ ❝✉♠♣r✐♥❞♦ ❛s s❡❣✉✐♥t❡s r❡❣r❛s✿
(a)♣♦r ❝❛❞❛ ♣♦rt❛ é ♣♦ssí✈❡❧ ♣❛ss❛r s♦♠❡♥t❡ ✉♠❛ ✈❡③❀
(b)♦s ♣❛ss❡✐♦s ❝♦♠❡ç❛♠ ♣♦r ✉♠❛ ♣♦rt❛ ❡①t❡r✐♦r ♦✉ ❛ ♣❛rt✐r ❞❡ ✉♠ q✉❛rt♦ s❡♠ s❛í❞❛ ❡
❙❡❣✉♥❞♦ ▲❡♠❛ ❈♦♠❜✐♥❛t♦r✐❛❧ ✶✸
(c) q✉❛♥❞♦ s❡ t❡r♠✐♥❛r ✉♠ ♣❛ss❡✐♦✱ ❝♦♠❡ç❛r❡♠♦s ♦✉tr♦ ❛té q✉❡ t❡♥❤❛♠♦s ♣❛ss❛❞♦
♣♦r t♦❞❛s ❛s ♣♦rt❛s ❡①t❡r✐♦r❡s ❡ t♦❞♦s ♦s q✉❛rt♦s s❡♠ s❛í❞❛✳
❊♠ ✈✐rt✉❞❡ ❞❛ s✉♣♦s✐çã♦ ❢❡✐t❛ s♦❜r❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦rt❛s ❞♦s q✉❛rt♦s✱ ❡st❛s r❡❣r❛s ❞❡t❡r♠✐♥❛♠ ✉♠ ú♥✐❝♦ tr❛❥❡t♦ ❛ s❡❣✉✐r ❡♠ ❝❛❞❛ ♣❛ss❡✐♦✱ ❥á q✉❡ ✉♠❛ ✈❡③ q✉❡ s❡ ❡♥tr❛ ❡♠ ✉♠ q✉❛rt♦ ❞❡ ♣❛ss❛❣❡♠ ❤á ✉♠❛ ú♥✐❝❛ ♣♦rt❛ ♣❛r❛ s❛✐r ❞❡❧❡✳ ❈♦♠♦ r❡s✉❧t❛❞♦ s❡ ♦❜té♠ três t✐♣♦s ❞❡ ♣❛ss❡✐♦s✿
✭p1✮ ❞❡ ✉♠❛ ♣♦rt❛ ❡①t❡r✐♦r ❛té ✉♠ q✉❛rt♦ s❡♠ s❛í❞❛ ✭♦✉ ❛ ✐♥✈❡rs❛✱ q✉❡ é ♦ ♠❡s♠♦ ❝❛♠✐♥❤♦✮❀
✭p2✮ ❞❡ ✉♠❛ ♣♦rt❛ ❡①t❡r✐♦r ❛té ♦✉tr❛ ♣♦rt❛ ❡①t❡r✐♦r ❡ ✭p3✮ ❞❡ ✉♠ q✉❛rt♦ s❡♠ s❛í❞❛ ❛té ♦✉tr♦ q✉❛rt♦ s❡♠ s❛í❞❛✳
❋✐❣✉r❛ ✶✳✹✿ P❛ss❡✐♦s
❈❤❛♠❡♠♦s ❞❡ m✱ n ❡ p✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♥ú♠❡r♦s ❞❡ ♣❛ss❡✐♦s ❞♦s t✐♣♦s ✭p1✮✱ ✭p2✮ ❡ ✭p3✮✳ ❈♦♠♦ ❛ ❝❛❞❛ ♣❛ss❡✐♦ ❞♦ t✐♣♦ ✭p1✮ ❝♦rr❡s♣♦♥❞❡ ✉♠❛ ♣♦rt❛ ❡①t❡r✐♦r ❡ ❝❛❞❛ ♣❛ss❡✐♦ ❞♦ t✐♣♦ ✭p2✮ ❝♦rr❡s♣♦♥❞❡ ❛ ❞✉❛s ♣♦rt❛s ❡①t❡r✐♦r❡s t❡♠♦s q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦rt❛s ❡①t❡r✐♦r❡s é ♠✰✷♥✳ ❉♦ ♠❡s♠♦ ❥❡✐t♦✱ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ q✉❛rt♦s s❡♠ s❛í❞❛ s❡rá ♠✰✷♣✳ ❖s ♥ú♠❡r♦s m+ 2n ❡m+ 2pt❡♠ ❛ ♠❡s♠❛ ♣❛r✐❞❛❞❡✱ ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞♦ ✈❛❧♦r ❞❡ m✱ ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳
▲❡♠❛ ❞❡ ❙♣❡r♥❡r ✶✹
✶✳✸ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r
❖s ❧❡♠❛s ❞❡ ❙♣❡r♥❡r sã♦ r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞♦s ❧❡♠❛s ❛♥t❡r✐♦r❡s✳ ▼✉✐t❛s ✈❡③❡s ❛q✉✐ s❡rá ✉t✐❧✐③❛❞♦ ♦ t❡r♠♦ tr✐❛♥❣✉❧❛çã♦✱ q✉❡ ❞❡✜♥✐r❡♠♦s ❛ s❡❣✉✐r✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠ tr✐â♥❣✉❧♦ ❞✐✈✐❞✐❞♦ ❞❡ ❢♦r♠❛ ❛r❜✐trár✐❛ ❡♠ tr✐â♥❣✉❧♦s ♠❡♥♦r❡s✳ ❊st❛ ♣❛rt✐çã♦ ❞❡✈❡ s❛t✐s❢❛③❡r ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ ❞♦✐s tr✐â♥❣✉❧♦ ♠❡♥♦r❡s q✉❛✐sq✉❡r
•♥ã♦ t❡♠ ♣♦♥t♦s ❡♠ ❝♦♠✉♠✱ ♦✉
•tê♠ s♦♠❡♥t❡ ✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✱ ♦✉ •t❡♠ ✉♠ ❧❛❞♦ ❝♦♠✉♠✳
❆ ❡st❛ ♣❛rt✐çã♦ ❞❛r❡♠♦s ♦ ♥♦♠❡ ❞❡ tr✐❛♥❣✉❧❛çã♦✳ ◆❛ ✜❣✉r❛1.5 t❡♠♦s ✉♠ ❡①❡♠♣❧♦
❞❡ ✉♠❛ tr✐❛♥❣✉❧❛çã♦ ♣♦✐s s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✳ ❏á ♥❛ ✜❣✉r❛ 1.6✱ t❡♠♦s ✉♠
❡①❡♠♣❧♦ q✉❡ ♥ã♦ é ❞❡ ✉♠❛ tr✐❛♥❣✉❧❛çã♦ ♣♦✐s ♦s tr✐â♥❣✉❧♦s T1 ❡ T2 tê♠ s♦♠❡♥t❡ ✉♠❛ ♣❛rt❡ ❞❛ ❛r❡st❛ ❡♠ ❝♦♠✉♠✳
❋✐❣✉r❛ ✶✳✺✿ ❊①❡♠♣❧♦ ❞❡ tr✐❛♥❣✉❧❛çã♦
❋✐❣✉r❛ ✶✳✻✿ ❊①❡♠♣❧♦ ❞❡ ♥ã♦ tr✐❛♥❣✉❧❛çã♦
▲❡♠❛ ❞❡ ❙♣❡r♥❡r ✶✺
✶✳✸✳✶ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ tr✐â♥❣✉❧♦
▲❡♠❛ ✶✳✹✳ ▲❊▼❆ ❉❊ ❙P❊❘◆❊❘✿ ❙❡❥❛ ✉♠ tr✐â♥❣✉❧♦ T ❡ ♠❛rq✉❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s
✈ért✐❝❡s ❝♦♠ s♦♠❡♥t❡ ✉♠ ❞♦s ♥ú♠❡r♦s 1✱ 2 ❡ 3✳ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ✉♠❛ tr✐❛♥❣✉❧❛çã♦
q✉❛❧q✉❡r ❞❡ T✳ ❖s ✈ért✐❝❡s ❞❛ tr✐❛♥❣✉❧❛çã♦ s❡rã♦ ♠❛r❝❛❞♦s ♣❡❧♦s ♥ú♠❡r♦s 1✱ 2 ♦✉ 3
❛t❡♥❞❡♥❞♦ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿ s❡ ♦ ✈ért✐❝❡ ❞❛ tr✐❛♥❣✉❧❛çã♦ ❡st✐✈❡r ❡♠ ✉♠ ❧❛❞♦ ❞♦ tr✐â♥❣✉❧♦ T ❡♥tã♦ ❞❡✈❡ s❡r ♠❛r❝❛❞♦ ♣♦r ✉♠ ❞♦s ❞♦✐s ♥ú♠❡r♦s ✐♥❞✐❝❛❞♦s ♥♦s ✈ért✐❝❡s
❞❡ t❛❧ ❧❛❞♦✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❢❛❝❡ ❞❛ tr✐❛♥❣✉❧❛çã♦ t❡rá ✈ért✐❝❡s ❝♦♠ ♠❛r❝❛s ❞✐❢❡r❡♥t❡s 1✱ 2 ❡ 3✳ ❊ ♠❛✐s✱ ♦ ♥ú♠❡r♦ ❞❡ t❛✐s ❢❛❝❡s é ✉♠ ♥ú♠❡r♦ í♠♣❛r✳
❋✐❣✉r❛ ✶✳✼✿ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ tr✐â♥❣✉❧♦
❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ♦s ❡❧❡♠❡♥t♦s ❡ ♦s r❡s✉❧t❛❞♦s ❞♦s ♣r✐♠❡✐r♦ ❡ s❡❣✉♥❞♦ ❧❡♠❛s✱ ❛ss♦❝✐❛r❡♠♦s✿
•♦ tr✐â♥❣✉❧♦ T ❝♦♠ ❛ ❝❛s❛❀
•❛s ❢❛❝❡s ❞❛ tr✐❛♥❣✉❧❛çã♦ ❛♦s q✉❛rt♦s❀ •❛s ❛r❡st❛s ❞♦ tr✐â♥❣✉❧♦ às ♣♦rt❛s✳
❈♦❧♦❝❛r❡♠♦s ❝♦♠♦ ♣♦rt❛s s♦♠❡♥t❡ ❛s ❛r❡st❛s q✉❡ t❡♥❤❛♠ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛s ❞✉❛s ♠❛r❝❛s 1 ❡ 2✱ q✉❡ ✐♥❞✐❝❛r❡♠♦s ❝♦♠♦ t✐♣♦✭✶✱✷✮ ❝♦♠♦ ✜③❡♠♦s ❝♦♠ ♦s ❡❧❡♠❡♥t♦s ❞♦ ♣r✐✲
♠❡✐r♦ ❧❡♠❛ ❝♦♠ ❛ ❞✐❢❡r❡♥ç❛ q✉❡ ❛r❡st❛s t✐♣♦✭✶✱✷✮ ♥ã♦ s❡rã♦ ❞✐❢❡r❡♥❝✐❛❞❛s ❞❛s ❛r❡st❛s ❞♦ t✐♣♦✭✷✱✶✮✳
❖s ❞❡③ ❝❛s♦s ♣♦ssí✈❡✐s ❞❡ ❞✐str✐❜✉✐çã♦ ❞❛s ♠❛r❝❛s 1✱ 2 ❡ 3 ♥♦s ✈ért✐❝❡s ❞❛ ❢❛❝❡s
sã♦✿ t✐♣♦✭✶✱✶✱✶✮✱ t✐♣♦✭✶✱✶✱✷✮✱ t✐♣♦✭✶✱✶✱✸✮✱ t✐♣♦✭✶✱✷✱✷✮✱t✐♣♦✭✶✱✸✱✸✮✱ t✐♣♦✭✶✱✷✱✸✮✱ t✐♣♦✭✷✱✷✱✷✮✱ t✐♣♦✭✷✱✷✱✸✮✱ t✐♣♦✭✷✱✸✱✸✮ ❡ t✐♣♦✭✸✱✸✱✸✮✳
❆s ❢❛❝❡s ❞♦ t✐♣♦✭✶✱✷✱✸✮ tê♠ s♦♠❡♥t❡ ✉♠❛ ❛r❡st❛ ❞♦ t✐♣♦✭✶✱✷✮ ❡✱ ❝♦♠♦ ✐ss♦ só ❛❝♦♥t❡❝❡ ❝♦♠ ❡st❡ t✐♣♦ ❞❡ ❢❛❝❡✱ ❝❤❛♠❛r❡♠♦s ❡st❛s ❢❛❝❡s ❞❡ q✉❛rt♦ s❡♠ s❛í❞❛❀ ❛s ❢❛❝❡s t✐♣♦✭✶✱✶✱✷✮ ❡ t✐♣♦✭✶✱✷✱✷✮ s❡rã♦ ♦s q✉❛rt♦s ❞❡ ♣❛ss❛❣❡♠ ♣♦✐s ❝❛❞❛ ✉♠❛ ❞❡❧❛s tê♠ ❞✉❛s ♣♦rt❛s ✭❛r❡st❛s t✐♣♦✭✶✱✷✮✮❀ t❡♠♦s ❛✐♥❞❛ q✉❡ ♦ t✐♣♦✭✶✱✶✱✶✮✱ ♦ t✐♣♦✭✶✱✶✱✸✮✱ ♦ t✐♣♦✭✶✱✸✱✸✮✱ ♦ t✐♣♦✭✷✱✷✱✷✮✱ ♦ t✐♣♦✭✷✱✷✱✸✮✱ ♦ t✐♣♦✭✷✱✸✱✸✮ ❡ ♦ t✐♣♦✭✸✱✸✱✸✮ sã♦ q✉❛rt♦s s❡♠ ♣♦rt❛s✳
▲❡♠❛ ❞❡ ❙♣❡r♥❡r ✶✻
❋✐❣✉r❛ ✶✳✽✿ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ tr✐â♥❣✉❧♦ ✲ ❝❛♠✐♥❤♦s
P♦❞❡♠♦s ❞❡♠♦♥str❛r q✉❡ ❡st❡ ú❧t✐♠♦ ♥ú♠❡r♦ é í♠♣❛r ✉s❛♥❞♦ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿ ❛s ♣♦rt❛s ❡①t❡r✐♦r❡s ❞♦ t✐♣♦✭✶✱✷✮ s❡ ❡♥❝♦♥tr❛♠ s♦♠❡♥t❡ ❡♠ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦T ❡✱ ❞❡
❛❝♦r❞♦ ❝♦♠ ♦ ♣r✐♠❡✐r♦ ❧❡♠❛ ❝♦♠❜✐♥❛t♦r✐❛❧✱ ♦ ♥ú♠❡r♦ é í♠♣❛r✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❞♦ t✐♣♦✭✶✱✷✱✸✮ é í♠♣❛r✳
✶✳✸✳✷ ▲❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ q✉❛❞r❛❞♦
❯♠ q✉❛❞r❛❞♦ Q ❢♦✐ r❡♣❛rt✐❞♦ ❡♠ ♣❡q✉❡♥♦s q✉❛❞r❛❞♦s ♣♦r r❡t❛s ♣❛r❛❧❡❧❛s ❛ s❡✉s
❧❛❞♦s✳ P❛r❛ ❡st❛ ♣❛rt✐çã♦ t❛♠❜é♠ ✉s❛r❡♠♦s ❛s ❞❡♥♦♠✐♥❛çõ❡s ❢❛❝❡s ✭q✉❡ sã♦ q✉❛❞r❛❞♦s ♣❡q✉❡♥♦s✮✱ ❛r❡st❛s ❡ ✈ért✐❝❡s ❞❛ ♣❛rt✐çã♦✳ ❯s❛r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r ♣❛r❛ ❡❧❛❜♦r❛r ♦ ♣ró①✐♠♦ ❧❡♠❛✳ ❖ ❧❡♠❛ ❞❡ ❙♣❡r♥❡r ♣❛r❛ ♦ tr✐â♥❣✉❧♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ ♠ét♦❞♦ ❞♦s ♣❛ss❡✐♦s ❡ ♦ t❡r♠♦ ♣❛ss❡✐♦ s✐❣♥✐✜❝❛ q✉❡ t❡♠♦s ✉♠❛ tr✐❛♥❣✉❧❛çã♦ ❛t❡♥t❛♥❞♦ ♣❛r❛ ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛r❡st❛s ❡ ♣♦rt❛s ❡ ❡♥tr❡ ❢❛❝❡s ❡ q✉❛rt♦s s❡♠ s❛í❞❛
▲❡♠❛ ✶✳✺✳ ❙❡❥❛ ✉♠ q✉❛❞r❛❞♦ Q ❝♦♠ ✈ért✐❝❡s ♠❛r❝❛❞♦s ♣❡❧♦s ♥ú♠❡r♦s 1✱ 2✱ 3 ❡ 4 ❞❡
t❛❧ ♠♦❞♦ q✉❡ ♥ã♦ t❡♠♦s ❞♦✐s ✈ért✐❝❡s ❝♦♠ ♠❛r❝❛s ✐❞ê♥t✐❝❛s✳ ❉✐✈✐❞✐♥❞♦ ♦ q✉❛❞r❛❞♦ Q
♣♦r r❡t❛s ♣❛r❛❧❡❧❛s ❛♦s s❡✉s ❧❛❞♦s✱ t❡♠♦s ❢❛❝❡s q✉❛❞r❛❞❛s ❞❡ t❛❧ ❢♦r♠❛ q✉❡ s❡✉s ✈ért✐❝❡s s❡rã♦ ♠❛r❝❛❞♦s ♣❡❧♦s ♥ú♠❡r♦s 1✱ 2✱ 3♦✉ 4 ❛t❡♥❞❡♥❞♦ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿ s❡ ♦ ✈ért✐❝❡
❞❛ ♣❛rt✐çã♦ ❡st✐✈❡r ❡♠ ✉♠ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦Q❡♥tã♦ ❞❡✈❡ s❡r ♠❛r❝❛❞♦ ♣♦r ✉♠ ❞♦s ❞♦✐s
♥ú♠❡r♦s r❡❧❛t✐✈♦s ❛♦s ✈ért✐❝❡s ❞❡ t❛❧ ❧❛❞♦✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❢❛❝❡ t❡rá ✈ért✐❝❡s ❝♦♠✱ ♣❡❧♦ ♠❡♥♦s✱ três ♠❛r❝❛s ❞✐❢❡r❡♥t❡s✳ ❊ ♠❛✐s✱ ♦ ♥ú♠❡r♦ ❞❡st❛s ❢❛❝❡s é ✉♠ ♥ú♠❡r♦ í♠♣❛r✳
❉❡♠♦♥str❛çã♦✳ ❈♦♠❡❝❡♠♦s ❞✐✈✐❞✐♥❞♦ ♦s q✉❛❞r❛❞♦s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❝❛❞❛ q✉❛✲ ❞r❛❞♦ s❡rá ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♣❛rt❡s ❢♦r♠❛♥❞♦✱ ♥♦ ✜♥❛❧ ❞❛s ❞✐✈✐sõ❡s✱ ✉♠❛ tr✐❛♥❣✉❧❛çã♦ ❡♠ q✉❡ ❛s ♠❛r❝❛s s❡rã♦ ♦s ♥ú♠❡r♦s 1✱ 2✱ 3❡ 4✳
▲❡♠❛ ❞❡ ❙♣❡r♥❡r ✶✼
q✉❛rt♦s s❡♠ s❛í❞❛ ✐♥❞✐❝❛❞♦s ♣❡❧♦s tr✐â♥❣✉❧♦s ❝♦♠ três ♠❛r❝❛s ❞✐❢❡r❡♥t❡s✳ ❆ss✐♠✱ sã♦ ♣♦ssí✈❡✐s q✉❛rt♦s s❡♠ s❛í❞❛ ❞❡ ❞♦✐s t✐♣♦s✿ t✐♣♦✭✶✱✷✱✸✮ ❡ t✐♣♦✭✶✱✷✱✹✮✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ✐♥❞✐❝❛❞❛s ♥♦ ❧❡♠❛✱ t❡♠♦s q✉❡ t♦❞❛s ❛s ♣♦rt❛s ❡①t❡r✐♦r❡s ❡st❛rã♦ ♥♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦ ♠❛r❝❛❞♦ ❝♦♠ ♦s ♥ú♠❡r♦s 1 ❡ 2✳ ❚❡♠♦s ❛✐♥❞❛✱ ❝♦♠ ❛
♣r♦♣r✐❡❞❛❞❡ ❞♦ s❡❣✉♥❞♦ ❧❡♠❛✱ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦rt❛s ❡①t❡r✐♦r❡s é í♠♣❛r✳
❯s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ♣❛ss❡✐♦✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ♣♦rt❛ ❡①t❡r✐♦r t❡r❡♠♦s ♦s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s✿ (a) s❛✐r ❞♦ q✉❛❞r❛❞♦ ♣♦r ♦✉tr❛ ♣♦rt❛ ❡①t❡r✐♦r ♦✉ (b) ❝❤❡❣❛♥❞♦ ❡♠ ✉♠
q✉❛rt♦ s❡♠ s❛í❞❛✳ ❈♦♠♦ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♣♦rt❛s ❡①t❡r✐♦r❡s é í♠♣❛r✱ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ♣❛ss❡✐♦s ❛❝❛❜❛ ❡♠ ✉♠ tr✐â♥❣✉❧♦ s❡♠ s❛í❞❛✳
P♦rt❛♥t♦✱ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❢❛❝❡ ❞♦ t✐♣♦✭✶✱✷✱✸✮ ♦✉ ❞♦ t✐♣♦✭✶✱✷✱✹✮ ✭♣❡❧♦ ♠❡♥♦s três ♠❛r❝❛s ❞✐❢❡r❡♥t❡s✮✳
✷ ❏♦❣♦s ❞♦ t✐♣♦ ❙♣❡r♥❡r
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❥♦❣♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ♦s ❧❡♠❛s ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✳
✷✳✶ ❏♦❣♦ ❞♦s ■♠♣❛❝t♦s
∗P❡ç❛s✿❯♠ t❛❜✉❧❡✐r♦ r❡tâ♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦rn q✉❛❞r❛❞♦s✳
❋✐❝❤❛s ❞❡ ❞✉❛s ❝♦r❡s✱ ♣♦r ❡①❡♠♣❧♦ ❛s ♣❡ç❛s ❞♦ ❥♦❣♦ ❞❡ ❞❛♠❛s✳
∗■♥str✉çõ❡s✿
(1) ❏♦❣♦ ❝♦♠ ❞♦✐s ❛❞✈❡rsár✐♦s✱ ❡s❝♦❧❤❡♥❞♦ q✉❡♠ s❡rá ♣❛r ♦✉ í♠♣❛r✳
(2) ❆❧t❡r♥❛❞❛♠❡♥t❡✱ ♦s ❥♦❣❛❞♦r❡s ❞❡✈❡♠ ❝♦❧♦❝❛r ✉♠❛ ✜❝❤❛ ❡♠ ✉♠ q✉❛❞r❛❞♦ ✈❛③✐♦✳ ∗❋✐♠ ❞♦ ❥♦❣♦✿
❖ ❥♦❣♦ ❛❝❛❜❛ q✉❛♥❞♦ t♦❞❛s ❛s ❝❛s❛s ❡st✐✈❡r❡♠ ♦❝✉♣❛❞❛s✳
❈♦♥t❛♠✲s❡ q✉❛♥t♦s ✐♠♣❛❝t♦s ♦❝♦rr❡r❛♠✳ ■♠♣❛❝t♦ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ❞✉❛s ❝❛s❛s s❡✲ ❣✉✐❞❛s tê♠ ✜❝❤❛s ❝♦♠ ❝♦r❡s ❞✐❢❡r❡♥t❡s✳
●❛♥❤❛ ♦ ❥♦❣❛❞♦r q✉❡ t✐✈❡r ❡s❝♦❧❤✐❞♦ ❛ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ❞♦ ♥ú♠❡r♦ ❞❡ ✐♠♣❛❝t♦s✳
❯♠❛ ♦✉tr❛ ♦♣çã♦ é ❛♣r❡s❡♥t❛r ♦ ❥♦❣♦ ♥✉♠ ♣❡❞❛ç♦ ❞❡ ♣❛♣❡❧✿
❋❛③✲s❡ ✉♠❛ t✐r❛ ❝♦♠ n q✉❛❞r❛❞♦s ❡ ❞❡✜♥❡♠ ❞♦✐s sí♠❜♦❧♦s✱ ♣♦r ❡①❡♠♣❧♦ ❡ •✳
❈❛❞❛ ✉♠ ❞♦s ❞♦✐s ❥♦❣❛❞♦r❡s ❡s❝♦❧❤❡ s❡ s❡rá ♣❛r ♦✉ í♠♣❛r✳
❆❧t❡r♥❛❞❛♠❡♥t❡✱ ♦s ❥♦❣❛❞♦r❡s ❞❡✈❡♠ ❝♦❧♦❝❛r ✉♠ sí♠❜♦❧♦ ❡♠ ✉♠ q✉❛❞r❛❞♦ ✈❛③✐♦✳ ❖ ❥♦❣♦ ❛❝❛❜❛ q✉❛♥❞♦ t♦❞❛s ❛s ❝❛s❛s ❡st✐✈❡r❡♠ ♦❝✉♣❛❞❛s✳ ❈♦♥t❛♠✲s❡ q✉❛♥t♦s ✐♠✲ ♣❛❝t♦s ♦❝♦rr❡r❛♠✳ ■♠♣❛❝t♦ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ❞♦✐s q✉❛❞r❛❞♦s s❡❣✉✐❞♦s tê♠ sí♠❜♦❧♦s ❞✐❢❡r❡♥t❡s✳ ●❛♥❤❛ ♦ ❥♦❣❛❞♦r q✉❡ t✐✈❡r ❡s❝♦❧❤✐❞♦ ❛ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ❞♦ ♥ú♠❡r♦ ❞❡ ✐♠✲ ♣❛❝t♦s✳
❋✐❣✉r❛ ✷✳✶✿ ❏♦❣♦ ❝♦♠ ✺ ✐♠♣❛❝t♦s ✲ ❥♦❣❛❞♦r í♠♣❛r ❣❛♥❤❛
❏♦❣♦ ❞♦s ■♠♣❛❝t♦s ✶✾
✷✳✶✳✶ ❘❡s✉❧t❛❞♦
❙❡ ❛s ❡①tr❡♠✐❞❛❞❡s ❢♦r❡♠ ♠❛r❝❛❞❛s ❝♦♠ sí♠❜♦❧♦s ❞✐❢❡r❡♥t❡s ✭♦✉ ❝♦r❡s ❞✐❢❡r❡♥t❡s ♥♦ t❛❜✉❧❡✐r♦✮✱ ♥♦ ✜♥❛❧ t❡r❡♠♦s ✐♠♣❛❝t♦s ✐♥❞✐❝❛❞♦s ♣♦r ✉♠ ♥ú♠❡r♦ í♠♣❛r✳ ❖s ✐♠♣❛❝t♦s sã♦ ❝♦♠♦ ♦s ✐♥t❡r✈❛❧♦s r❛③♦á✈❡✐s ❞❡s❝r✐t♦s ♥♦ Pr✐♠❡✐r♦ ▲❡♠❛ ❈♦♠❜✐♥❛t♦r✐❛❧✳
P♦❞❡♠♦s t❛♠❜é♠ ❛✜r♠❛r q✉❡ t❡r❡♠♦s ♣❛r❡s ✐♠♣❛❝t♦s s❡ ❛s ❡①tr❡♠✐❞❛❞❡s ❢♦r❛♠ ♠❛r❝❛❞♦s ♣❡❧♦ ♠❡s♠♦ sí♠❜♦❧♦✳
P❛r❛ ❣❛♥❤❛r✱ ♦ ❥♦❣❛❞♦r q✉❡ ❢♦r í♠♣❛r ❞❡✈❡ ❡s♣❡r❛r ♦ ❛❞✈❡rsár✐♦ ❢❛③❡r ✉♠ sí♠❜♦❧♦ ❡♠ ✉♠❛ ❡①tr❡♠✐❞❛❞❡ ❡ ♥❛ ♣ró①✐♠❛ ❥♦❣❛❞❛ ❝♦♠♣❧❡t❛r ❛ ♦✉tr❛ ❡①tr❡♠✐❞❛❞❡ ❝♦♠ ♦ sí♠✲ ❜♦❧♦ ❞✐❢❡r❡♥t❡✳ ❆♣ós ❡st❛ r♦❞❛❞❛✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s sí♠❜♦❧♦s✱ ♦s ✐♠♣❛❝t♦s s❡rã♦ í♠♣❛r❡s✳ ❉♦ ♠❡s♠♦ ♠♦❞♦ ♣❛r❛ ♦ ❥♦❣❛❞♦r ♣❛r✱ q✉❡ ❞❡✈❡ ♠❛r❝❛r sí♠❜♦❧♦s ✐❣✉❛✐s ♥❛s ❡①tr❡♠✐❞❛❞❡s✳
✷✳✶✳✷ ■♠♣❛❝t♦ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s
P♦❞❡♠♦s ✐♥✈❡♥t❛r ✉♠❛ ♦✉tr❛ ✈❡rsã♦ ♣❛r❛ ♦ ❥♦❣♦ ❞♦s ✐♠♣❛❝t♦s ❞❡♣♦✐s ❞♦ ♠❡s♠♦ t❡r s✐❞♦ ❛ss♦❝✐❛❞♦ ❛♦ r❡s✉❧t❛❞♦ ❞♦s ❧❡♠❛s✳ ❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ❛ ♥♦✈❛ ✈❡rsã♦ é ♦ ❞❡s❛✜♦ ❛♦s ❛❧✉♥♦s ❞❡ ❛ss♦❝✐❛r ♦s ❧❡♠❛s ❝♦♠ ✉♠ t❛❜✉❧❡✐r♦ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ✷✳✷✳
❏♦❣♦ ❞♦ s♦❜❡✲❞❡s❝❡ ✷✵
P♦❞❡♠♦s ♣r♦♣♦r ♦ ❥♦❣♦ t❛♠❜é♠ ❡♠ ✉♠ t❛❜✉❧❡✐r♦ ❝♦♠♦ ♦ t❛❜✉❧❡✐r♦ ❞❡ ❞❛♠❛s✳ ❆ ❡str❛té❣✐❛ ❝♦♥t✐♥✉❛ ❛ ♠❡s♠❛ ❝♦♠ ❛ ❞✐✜❝✉❧❞❛❞❡ ♣❛r❛ ♦s ❥♦❣❛❞♦r❡s ❡st❛r❡♠ ❛t❡♥t♦s ❛♦s ❡①tr❡♠♦s ❡♠ t♦❞❛s ❛s ❧✐♥❤❛ ❡ t♦❞❛s ❛s ❝♦❧✉♥❛s✳
✷✳✷ ❏♦❣♦ ❞♦ s♦❜❡✲❞❡s❝❡
◆st❡ ❥♦❣♦ ♣r❡❝✐s❛♠♦s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ♣♦♥t♦s ❡①tr❡♠♦s✱ ❛ s❡❣✉✐r✿
❋✐❣✉r❛ ✷✳✸✿ P♦♥t♦s ❡①tr❡♠♦s
❉❡✜♥✐çã♦ ✷✳✶✳ ◆♦ ❣rá✜❝♦ ❞❛ ✜❣✉r❛ ✷✳✸✱ ♦s ♣♦♥t♦s x1✱ x2✱ x3 ❡ x4 sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦♥t♦s ❡①tr❡♠♦s q✉❡ ♣♦❞❡♠ s❡r ❞❡ ❞♦✐s t✐♣♦s✿ ♦s ♣♦♥t♦s x1 ❡ x3 sã♦ ♣♦♥t♦s ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ ❡♥q✉❛♥t♦ ♦s ♣♦♥t♦s x2 ❡x4 sã♦ ❝❤❛♠❛❞♦s ♣♦♥t♦s ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧✱ ♣♦✐s✿
• ❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❝♦♥tí♥✉❛ t❡♠ ♠á①✐♠♦ ❧♦❝❛❧ ♥♦ ♣♦♥t♦ c✱ s❡ ❡①✐st✐r ✉♠
✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I✱ ❝♦♥t❡♥❞♦ c✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♦ ✈❛❧♦r ❛ss♦❝✐❛❞♦ ❛ c é ♠❛✐♦r q✉❡
q✉❛✐sq✉❡r ❞♦s ✈❛❧♦r❡s ❛ss♦❝✐❛❞♦s ❛♦s ❞❡♠❛✐s ♥ú♠❡r♦s ❞♦ ✐♥t❡r✈❛❧♦ I✳
• ❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❝♦♥tí♥✉❛ t❡♠ ♠í♥✐♠♦ ❧♦❝❛❧ ♥♦ ♣♦♥t♦ c✱ s❡ ❡①✐st✐r ✉♠
✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I✱ ❝♦♥t❡♥❞♦ c✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♦ ✈❛❧♦r ❛ss♦❝✐❛❞♦ ❛ c é ♠❡♥♦r q✉❡
q✉❛✐sq✉❡r ❞♦s ✈❛❧♦r❡s ❛ss♦❝✐❛❞♦s ❛♦s ❞❡♠❛✐s ♥ú♠❡r♦s ❞♦ ✐♥t❡r✈❛❧♦ I✳
P❛r❛ ♦ ❥♦❣♦✱ ♣♦❞❡♠♦s ✉s❛r ❛q✉✐ ♦ t❛❜✉❧❡✐r♦ ❞♦ ❥♦❣♦ ❞♦ ✐♠♣❛❝t♦ ❡ ♦s ❥♦❣❛❞♦r❡s ❞❡✈❡♠ ❞❡❝✐❞✐r q✉❡♠ s❡rá ♣❛r ❡ q✉❡♠ s❡rá í♠♣❛r✳
❆♦ ✐♥✈és ❞❡ ✜❝❤❛s ♦✉ sí♠❜♦❧♦s✱ ♦s ❥♦❣❛❞♦r❡s ❞❡✈❡♠ ♠❛r❝❛r ♥ú♠❡r♦s✳ ❆❧t❡r♥❛❞❛✲ ♠❡♥t❡✱ ♦s ❥♦❣❛❞♦r❡s ♠❛r❝❛♠ ♦s ♥ú♠❡r♦s ♥♦s q✉❛❞r❛❞♦s ✈❛③✐♦s✱ ♣♦❞❡♥❞♦ ♠❛r❝❛r q✉❛❧✲ q✉❡r ♥ú♠❡r♦ q✉❡ ♥ã♦ t❡♥❤❛ s✐❞♦ ✉s❛❞♦✳ ❖s ♥ú♠❡r♦s ❞❡✈❡♠ s❡r ♣♦s✐t✐✈♦s ❞❡ ✶ ❛ n t❛❧
q✉❡ n é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ q✉❛❞r❛❞♦s ♥♦ t❛❜✉❧❡✐r♦✳
❖ ❥♦❣♦ ❛❝❛❜❛ q✉❛♥❞♦ t♦❞♦s ♦snq✉❛❞r❛❞♦s ❡st✐✈❡r❡♠ ♣r❡❡♥❝❤✐❞♦s ❝♦♠ ♦sn♥ú♠❡r♦s
❡s♣❡r❛❞♦s ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ❞❛ ✜❣✉r❛ ✷✳✹✳
❋✐❣✉r❛ ✷✳✹✿ ❘❡s✉❧t❛❞♦ ❞❡ ✉♠ ❥♦❣♦ s♦❜❡✲❞❡s❝❡
❙ã♦ ❝♦♥t❛❞♦s ♦s ♣♦♥t♦s ❡①tr❡♠♦s ❞♦ ❥♦❣♦✱ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ✷✳✶✳ ❯s❛♥❞♦ ♦ ❡①❡♠♣❧♦✱ ♦❜t❡♠♦s ♦ ❣rá✜❝♦ ♥❛ ✜❣✉r❛ ✷✳✺✳
❏♦❣♦ ♣❛r❛ ♦ ▲❡♠❛ ❞♦ tr✐â♥❣✉❧♦ ✷✶
❋✐❣✉r❛ ✷✳✺✿ ◆ú♠❡r♦ j ❝♦❧♦❝❛❞♦ ♥♦ t❛❜✉❧❡✐r♦ ♥❛ ❥♦❣❛❞❛i
✷✳✷✳✶ ❘❡s✉❧t❛❞♦
❖ r❡s✉❧t❛❞♦ ❡s♣❡r❛❞♦ é✿ s❡ ✉♠❛ ❧✐♥❤❛ ❝♦♥tí♥✉❛ ❢♦r♠❛❞❛ ♣♦r s❡❣♠❡♥t♦s ❞❡ r❡t❛ t❡♠ ♦ ♣r✐♠❡✐r♦ s❡❣♠❡♥t♦ ❝r❡s❝❡♥t❡ ❡ ♦ ú❧t✐♠♦ s❡❣♠❡♥t♦ ❞❡s❝r❡s❝❡♥t❡✱ ❡♥tã♦ ❡st❛ ❧✐♥❤❛ t❡♠ ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ❡①tr❡♠♦s✳ ❉❡ ❢❛t♦✱ ♦s s❡❣♠❡♥t♦s ❝r❡s❝❡♥t❡s sã♦ ❝♦♠♦ ♦s ✐♥t❡r✈❛❧♦s r❛③♦á✈❡✐s ❞♦ t✐♣♦ ✭✵✱✶✮ ❞♦ Pr✐♠❡✐r♦ ▲❡♠❛ ❈♦♠❜✐♥❛t♦r✐❛❧✱ ❡ ❛s r❡t❛s ❞❡❝r❡s❝❡♥t❡s sã♦ ❝♦♠♦ ♦s ✐♥t❡r✈❛❧♦s r❛③♦á✈❡✐s ❞♦ t✐♣♦ ✭✶✱✵✮✳ ❆ss✐♠✱ t❡r❡♠♦s í♠♣❛r❡s ✐♥t❡r✈❛❧♦s r❛③♦á✈❡✐s s❡ t✐✈❡r♠♦s s❡❣♠❡♥t♦s ❝♦♠ ❞✐r❡çõ❡s ♦♣♦st❛s ♥❛s ❡①tr❡♠✐❞❛❞❡s ❞❛ ❧✐♥❤❛✱ ❝♦♠♦ ❞❡✜♥✐❞♦s ♠❛r❝❛s ❞✐❢❡r❡♥t❡s ♥♦s ❡①tr❡♠♦s ❞♦ ✐♥t❡r✈❛❧♦ ❞♦ ❧❡♠❛ ✶✳✶✳
✷✳✸ ❏♦❣♦ ♣❛r❛ ♦ ▲❡♠❛ ❞♦ tr✐â♥❣✉❧♦
❙ã♦ ❞♦✐s ❥♦❣❛❞♦r❡s✳ P♦❞❡♠ ✉s❛r ♦ t❛❜✉❧❡✐r♦ ♦✉ ❢❛③ê✲❧♦ ♥♦ ♣❛♣❡❧✳
❖ t❛❜✉❧❡✐r♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✉♠ tr✐â♥❣✉❧♦ ❝✉❥♦s ✈é❡rt✐❝❡s ❞❡✈❡♠ t❡r ♠❛r❝❛s ❞✐s✲ t✐♥t❛s ❡ ❞❡✈❡ ❝♦♥t❡r ✉♠❛ tr✐❛♥❣✉❧❛çã♦ q✉❛❧q✉❡r ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡✿ ❞♦✐s s✉❜tr✐â♥❣✉❧♦s q✉❛✐sq✉❡r (a)♥ã♦ t❡♠ ♣♦♥t♦s ❡♠ ❝♦♠✉♠ ♦✉ (b) tê♠ s♦♠❡♥t❡ ✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠ ♦✉ (c)t❡♠ ✉♠ ❧❛❞♦ ❝♦♠✉♠✳
❆❧t❡r♥❛❞❛♠❡♥t❡✱ ✉♠ ❥♦❣❛❞♦r ❞❡✈❡ ❢❛③❡r ✉♠❛ ♠❛r❝❛çã♦ ❡♠ q✉❛❧q✉❡r ✈ért✐❝❡ ❞❛ tr✐✲ ❛♥❣✉❧❛çã♦ s❡❣✉✐♥❞♦ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿ s❡ ✉♠ ✈ért✐❝❡ ❢♦r ♠❛r❝❛❞♦ ❡♠ ✉♠ ❧❛❞♦ ❞♦ tr✐â♥❣✉❧♦ ❞❡✈❡ s❡r ♠❛r❝❛❞♦ ♣♦r ✉♠❛ ❞❛s ❞✉❛s ♠❛r❝❛s ❞❛ ❡①tr❡♠✐❞❛❞❡ ❞❡st❡✳ ●❛♥❤❛ ♦ ❥♦❣❛❞♦r q✉❡ ❝♦♥s❡❣✉✐r ♣r✐♠❡✐r♦ ❞❡✐①❛r ✉♠ s✉❜tr✐â♥❣✉❧♦ ❝♦♠ três ♠❛r❝❛s ❞✐st✐♥t❛s✳
❏♦❣♦ ♣❛r❛ ♦ ▲❡♠❛ ❞♦ tr✐â♥❣✉❧♦ ✷✷
❈♦♠♦ ❢♦✐ ♣r♦✈❛❞♦✱ ❡①✐st✐rá ✉♠ s✉❜tr✐â♥❣✉❧♦ ❝♦♠ ♦s três ✈ért✐❝❡s ❝♦♠ ♠❛r❝❛r ❞✐✲ ❢❡r❡♥t❡s❀ ❞❡✈❡✲s❡ ❡♥tã♦ t♦♠❛r ❝✉✐❞❛❞♦ ♣❛r❛ ♥ã♦ ❞❡✐①❛r ❛ ❥♦❣❛❞❛ ❞♦ ❛❞✈❡rsár✐♦ ❝♦♠ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❢❡❝❤❛r ♦ tr✐â♥❣✉❧♦✳
✸ ❆ t♦♣♦❧♦❣✐❛ ❞❛ r❡t❛
❈♦❧♦❝❛r❡♠♦s✱ ♥❡st❡ ❝❛♣ít✉❧♦✱ ♦s ❡❧❡♠❡♥t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡✲ ♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇r♦✉✇❡r✳
✸✳✶ ❋✉♥çõ❡s ❝♦♥tí♥✉❛s
❈♦♥s✐❞❡r❡ ✉♠ ✐♥t❡r✈❛❧♦I ❞❛ r❡t❛ ♥✉♠ér✐❝❛ R ❡ ✉♠❛ r❡❧❛çã♦ f q✉❡ ❛ss♦❝✐❛ ✈❛❧♦r❡s ❞❡ I ❡♠ R✳ ❙❡ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ x ❞❡st❡ ✐♥t❡r✈❛❧♦ I✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❝❡rt❛ r❡❣r❛✱ s❡ ❝♦rr❡s♣♦♥❞❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ r❡❛❧ f(x)✱ ❞✐③❡♠♦s q✉❡ ❡st❛ r❡❧❛çã♦ é ✉♠❛ ❢✉♥çã♦✳ ❖
✐♥t❡r✈❛❧♦ I é ❝❤❛♠❛❞♦ ❞❡ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡R♥❛ ❢♦r♠❛
f(x)é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠✳
■♥t✉✐t✐✈❛♠❡♥t❡✱ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❢✉♥çã♦ é ❝❧❛r❛✿ ♦ ❣rá✜❝♦ ❞❡st❛ ❢✉♥çã♦ é ✉♠❛ ❧✐♥❤❛ q✉❡ s❡ ♣♦❞❡ tr❛ç❛r s❡♠ ❧❡✈❛♥t❛r ♦ ❧á♣✐s ❞♦ ♣❛♣❡❧✳ ❉✐③❡♠♦s t❛♠❜é♠ q✉❡ ✉♠❛ ❢✉♥çã♦ f : R → R é ❝♦♥tí♥✉❛ s❡ ❡❧❛ ❛ss♦❝✐❛ ♣♦♥t♦s ♣ró①✐♠♦s ❞❡ x ❛ ♣♦♥t♦s ♣ró①✐♠♦s ❞❡ y=f(x)✳ ❆ ♣r♦①✐♠✐❞❛❞❡ ❞♦s ♣♦♥t♦s é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ❞✐stâ♥❝✐❛ ❝❛rt❡s✐❛♥❛ ❡♥tr❡
❡❧❡s✳
❉❡✜♥✐çã♦ ✸✳✶✳ ❚♦❞♦s ♦s ♣♦♥t♦s ♣ró①✐♠♦s ❛♦ ♣♦♥t♦ x❢♦r♠❛♠ s✉❛δ✲✈✐③✐♥❤❛♥ç❛✳ ❙❡X
é ✉♠ ❝♦♥❥✉♥t♦ q✉❡ s❡ ❡♥❝♦♥tr❛ ♥♦ ♣❧❛♥♦ ❡ x é ✉♠ ❞❡ s❡✉s ♣♦♥t♦s✱ ❛ δ✲✈✐③✐♥❤❛♥ç❛ ❞❡ x é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞❡ X ❝✉❥❛ ❞✐stâ♥❝✐❛ ❛ x é ♠❡♥♦r q✉❡ δ✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛
❣❡♦♠étr✐❝♦✱ t❡♠♦s q✉❡ ❛ δ✲✈✐③✐♥❤❛♥ç❛ ❞❡ x✱ ♥♦ ♣❧❛♥♦✱ é ✉♠ ❝ír❝✉❧♦ ✭s❡♠ ♦ ❝♦♥t♦r♥♦✮
❝✉❥♦ r❛✐♦ é δ ❡ ♦ ❝❡♥tr♦ é ♦ ♣♦♥t♦ x❀ ♥♦ ❡s♣❛ç♦✱ ❡st❛ ✈✐③✐♥❤❛♥ç❛ é ✉♠❛ ❡s❢❡r❛ s❡♠ ❛
❧✐♥❤❛ ❧✐♠✐t❛❞♦r❛✳
❙❡ ✉♠❛ ❢✉♥çã♦ f :R→Ré ❝♦♥tí♥✉❛ ❡♠ ✉♠ ♣♦♥t♦ xs❡ ♣❛r❛ q✉❛❧q✉❡r ε >0 ❡①✐st❡ ✉♠ ♥ú♠❡r♦ δ > 0 t❛❧ q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❛ δ✲✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ x s❡
❡♥❝♦♥tr❛ ♥❛ ε✲✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ y=f(x)✳ ❆ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❢✉♥çã♦ ♥♦ ♣♦♥t♦
xs✐❣♥✐✜❝❛ q✉❡ s❡ t♦♠❛r♠♦s ✉♠❛ ✈✐③✐♥❤❛♥ç❛ q✉❛❧q✉❡r ❞♦ ♣♦♥t♦y✱ ♣♦❞❡r❡♠♦s ❡♥❝♦♥tr❛r
✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ❡❧❡ ❡ ❞❡❧❡ ❞❡♣❡♥❞❡♥t❡✳ ❙❡ ✉♠❛ ❢✉♥çã♦ f : R→ R é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞♦ ❞♦♠í♥✐♦ ❞✐③❡♠♦s q✉❡ f é ❝♦♥tí♥✉❛✳
Pr♦♣♦s✐çã♦ ✸✳✷✳ ❆ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s t❛♠❜é♠ é ❝♦♥tí♥✉❛✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ g :X →Y ❡ f :Y →Z ❢✉♥çõ❡s r❡❛✐s ❝♦♥tí♥✉❛s✳
❍♦♠❡♦♠♦r✜s♠♦ ✷✹
❈♦♠♦ f é ❝♦♥tí♥✉❛ ❡♠ g(c)✱ ♣❛r❛ ❝❛❞❛ ε > 0 ❡①✐st❡ δ1 > 0 t❛❧ q✉❡ ❛ ✐♠❛❣❡♠ ❞❛
δ1✲✈✐③✐♥❤❛♥ç❛ ❞❡ g(c) s❡ ❡♥❝♦♥tr❛ ♥❛ε✲✈✐③✐♥❤❛♥ç❛ ❞❡f(g((c))✳
❉❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ c✱ ❛ ❛ ❢✉♥çã♦ g t❡♠♦s q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ❛❧❣✉♠❛ δ2✲✈✐③✐♥❤❛♥ç❛ ❞❡c❡stá ❝♦♥t✐❞❛ ♥❛δ1✲✈✐③✐♥❤❛♥ç❛ ❞❡g(c)✳ ❆ss✐♠✱ ❛ ✐♠❛❣❡♠ ❞❛δ2✲✈✐③✐♥❤❛♥ç❛ ❞❡c♣❡❧❛ ❢✉♥çã♦ f ◦g ❡stá ❝♦♥t✐❞❛ ♥❛ε✲✈✐③✐♥❤❛♥ç❛ ❞❡ f(g(c))
✸✳✷ ❍♦♠❡♦♠♦r✜s♠♦
❉❡✜♥✐çã♦ ✸✳✸✳ ❯♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ f :X →X é ❝❤❛♠❛❞❛ ✉♠❛ ✐♥✈♦❧✉çã♦ s❡ f2
(x)
❂ f◦f(x)❂ x ♣❛r❛ t♦❞♦x ∈ X✳
❋✐❣✉r❛ ✸✳✶✿ ❆ ❢✉♥çã♦f é ✐♥✈♦❧✉t✐✈❛
❉❡✜♥✐çã♦ ✸✳✹✳ ❯♠❛ ❢✉♥çã♦ f : X →Y s❡ ❞✐③ ✐♥❥❡t♦r❛ s❡ ❛ss♦❝✐❛ ♣♦♥t♦s ❞✐st✐♥t♦s ❞♦
❝♦♥❥✉♥t♦ X ❛ ♣♦♥t♦s ❞✐st✐♥t♦s ❞♦ ❝♦♥❥✉♥t♦ Y✳
❋✐❣✉r❛ ✸✳✷✿ ❆ ❢✉♥çã♦ f é ✐♥❥❡t♦r❛
❉❡✜♥✐çã♦ ✸✳✺✳ ❯♠❛ ❢✉♥çã♦ f : X → Y s❡ ❝❤❛♠❛ s♦❜r❡❥❡t♦r❛ s❡ ❝❛❞❛ ♣♦♥t♦ ❞♦
❝♦♥❥✉♥t♦ Y é ❛ ✐♠❛❣❡♠ ❞❡ ✉♠ ♣♦♥t♦ ✭♦✉ ❞❡ ✈ár✐♦s ♣♦♥t♦s✮ ❞♦ ❝♦♥❥✉♥t♦ X✳
❋✐❣✉r❛ ✸✳✸✿ ❆ ❢✉♥çã♦ f é s♦❜r❡❥❡t♦r❛
❙❡q✉ê♥❝✐❛s ✷✺
s❡ ❛ss♦❝✐❛ ❝♦♠ y ♣❡❧❛ ❢✉♥çã♦ f✳ ❆ ❢✉♥çã♦ s♦❜r❡❥❡t♦r❛ ❡ ✐♥❥❡t♦r❛ f :X → Y s❡ ❝❤❛♠❛
❤♦♠❡♦♠♦r✜s♠♦ s❡ ❡❧❛ é ❝♦♥tí♥✉❛ ❡ s❡ ❛ ❢✉♥çã♦ f−1 : Y → X t❛♠❜é♠ é✳ ◆❡st❡ ❝❛s♦✱
❞✐③❡♠♦s q✉❡ ♦s ❝♦♥❥✉♥t♦s X ❡Y sã♦ ❤♦♠❡♦♠♦r❢♦s✳
✸✳✸ ❙❡q✉ê♥❝✐❛s
❉❡✜♥✐çã♦ ✸✳✻✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ ❢✉♥çã♦ s:N→R q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ♥ú♠❡r♦ ♥❛t✉r❛❧ n❛ ✉♠ ♥ú♠❡r♦ r❡❛❧ sn t❛❧ q✉❡n✐♥❞✐❝❛ ❛ ♣♦s✐çã♦ ❞♦ ❡❧❡♠❡♥t♦ sn✳ P❛r❛ r❡♣r❡s❡♥t❛r ✉♠❛ s❡q✉ê♥❝✐❛ ♣♦❞❡♠♦s ✉s❛r (s1, s2, ..., sn, ...) ♦✉✱ ❛❜r❡✈✐❛❞❛♠❡♥t❡✱
(sn)✳
❈♦♥✈❡r❣ê♥❝✐❛
❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✭xn✮ é ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ✉♠ ♥ú♠❡r♦x0s❡ ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ ε ❃ ✵ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ m t❛❧ q✉❡ ♣❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s n > m ❛ ❞✐stâ♥❝✐❛
❡♥tr❡ ♦s ♣♦♥t♦s x0 ❡xn é ♠❡♥♦r q✉❡ ε✳
❉❡ ♦✉tr❛ ❢♦r♠❛✱ xn → x0 s✐❣♥✐✜❝❛ q✉❡ q✉❛❧q✉❡r ε✲✈✐③✐♥❤❛ç❛ ❞♦ ♣♦♥t♦ x0 ♣♦ss✉✐ t♦❞♦s ♦s ♣♦♥t♦s ❞❛ s❡q✉ê♥❝✐❛ ❛ ♣❛rt✐r ❞❡ ❝❡rt❛ ♣♦s✐çã♦✳
❊①❡♠♣❧♦ ✶✿ ❛ s❡q✉ê♥❝✐❛ r❡❛❧ ❞❡✜♥✐❞❛ ♣♦rxn =
1 + (−1)n
n é ❝♦♥✈❡r❣❡♥t❡✳ P♦✐s✿
• ❚♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ ♣♦s✐çã♦ í♠♣❛r sã♦ ✐❣✉❛✐s ❛ ③❡r♦✳
• P❛r❛ ♦s ❡❧❡♠❡♥t♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦s✐çã♦ ♣❛r ✭n = 2k✮✱ t❡♠♦s xn =
1 + (−1)2k
2k =
1 + 1
2k =
1
k✳ ❈♦♠♦
1
k < ε =⇒ k >
1
ε ❡✱ n = 2k =⇒ k = n
2✱
❡♥tã♦ n
2 > 1
ε✳ P♦rt❛♥t♦✱n >
2
ε✳ ❙❡ ♣❡❣❛r♠♦s m✐♥t❡✐r♦ t❛❧ q✉❡
2
ε < m≤
2
ε+ 1✱
♦ ♣♦♥t♦xn ♣❡rt❡♥❝❡ ❛ ε✲✈✐③✐♥❤❛♥ç❛ ❞❡ 0✱ ♣❛r❛ t♦❞♦ n > m
Pr✐♥❝í♣✐♦ ❞♦s ✐♥t❡r✈❛❧♦s ❡♥❝❛✐①❛♥t❡s
❙❡❥❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s{I0, I1, ..., In} t❛❧ q✉❡ ❝❛❞❛ ✉♠ ❞❡❧❡s ❡stá ❝♦♥t✐❞♦ ♥♦ ❛♥t❡r✐♦r ✭In ⊇In−1✱ ∀n✮✳ ❙❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ✐♥t❡r✈❛❧♦s t❡♥❞❡♠ ❛ ③❡r♦ q✉❛♥❞♦ n t❡♥❞❡ ❛ ✐♥✜♥✐t♦✱ t♦❞♦s ♦s ✐♥t❡r✈❛❧♦s t❡♠ ✉♠✱ ❡ s♦♠❡♥t❡ ✉♠✱ ♣♦♥t♦ ❝♦♠✉♠✳
❈♦♠♣❛❝✐❞❛❞❡
❙❡ t♦❞❛ s❡q✉ê♥❝✐❛ ✭s1, s2, ..., sn, ...✮ ❢♦r♠❛❞❛ ♣♦r ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ S ♣♦ss✉✐r ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ✭sn1, sn2, ..., snk, ...✮ ❝♦♥✈❡r❣❡♥t❡ ❛ ✉♠ ❡❧❡♠❡♥t♦s0 ❞❡S✱ ❞✐③❡♠♦s q✉❡
S é ❝♦♠♣❛❝t♦✳
Pr♦♣♦s✐çã♦ ✸✳✼✳ ◗✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ✜♥✐t♦ X ❞❡R é ❝♦♠♣❛❝t♦✳
❙❡q✉ê♥❝✐❛s ✷✻
Pr♦♣♦s✐çã♦ ✸✳✽✳ ❖ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I = (0,1) ♥ã♦ é ❝♦♠♣❛❝t♦✳
❉❡♠♦♥str❛çã♦✳ ❊①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ I t❛❧ q✉❡ ♥❡♥❤✉♠❛ s✉❜✲
s❡q✉ê♥❝✐❛ ❞❡❧❛ é ❝♦♥✈❡r❣❡♥t❡ ❛ ✉♠ ♣♦♥t♦ ❞❡ I✿ ❛ s❡q✉ê♥❝✐❛ sn = 1
n t❡♠ s✉❜s❡q✉ê♥❝✐❛ ✭1 2, 1 3, 1 4, 1
n, ...✮ q✉❡ é ❝♦♥✈❡r❣❡♥t❡ ❛ ✵✳
Pr♦♣♦s✐çã♦ ✸✳✾✳ ❖ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ I = [a, b] é ❝♦♠♣❛❝t♦✳
❉❡♠♦♥str❛çã♦✳ i. ❙❡❥❛ ✭sn✮ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s ❞❡ I = [a, b]✳ ❉✐✈✐❞✐♥❞♦ ♦ ✐♥t❡r✲ ✈❛❧♦ [a, b] ♥❛ ♠❡t❛❞❡✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s ❞✉❛s ♠❡t❛❞❡s✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ b−a
2 ✱ ♣♦ss✉✐ ✐♥✜♥✐t♦s ♣♦♥t♦s ❞❡ ✭sn✮✳ ❈❤❛♠❡♠♦s ❛ t❛❧ ♠❡t❛❞❡ ♣♦rI1 ❡ t♦♠❡ ✉♠ ♣♦♥t♦ ❞❛ s❡q✉ê♥❝✐❛
sn1 ∈ I1✳ ❆❣♦r❛✱ ❞✐✈✐❞✐♥❞♦ ♦ ✐♥t❡r✈❛❧♦ I1 ♥❛ ♠❡t❛❞❡✱ t❡♠♦s ✉♠❛ ❞❡ss❛s ♠❡t❛❞❡s ❝♦♥✲ t❡♥❞♦ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ✭sn✮✳ ❉❡♥♦t❡♠♦s t❛❧ ♠❡t❛❞❡✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦
b−a 22 ✱
♣♦r I2 ❡ t♦♠❡♠♦s sn2 ❞❛ s❡q✉ê♥❝✐❛ sn q✉❡ ♣❡rt❡♥❝❡ ❛ I2✱ ❝♦♠ n2 > n1✳ P♦r ✐♥❞✉çã♦✱ ♦❜t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♥t❡r✈❛❧♦s I0 = I = [a, b] ⊃ I1 ⊃ I2 ⊃ ... ⊃ Ik...✱ ♦♥❞❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ Ik é
b−a
2k ✳ ❈♦♠♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦s ✐♥t❡r✈❛❧♦s t❡♥❞❡ ❛ ③❡r♦ ✭ b−a
2k → 0 q✉❛♥❞♦ k → ∞✮✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞♦s ✐♥t❡r✈❛❧♦s ❡♥❝❛✐①❛♥t❡s✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ s0 ♣❡rt❡❝❡♥t❡ ❛ t♦❞♦s ♦s ✐♥t❡r✈❛❧♦s Ik✱ k ={0,1,2, ...}✳
ii. P❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ ε > 0✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ m t❛❧ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦
❞♦ ✐♥t❡r✈❛❧♦ Im✱ b−a
2m✱ é ♠❡♥♦r q✉❡ ε✳ ❈♦♠♦ s0 ∈ Im✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ Im sã♦ ❡❧❡♠❡♥t♦s ❞❛ε✲✈✐③✐♥❤❛♥ç❛ ❞❡st❡ ♣♦♥t♦ ❡ t❛♠❜é♠ t❡♠♦ss0 ∈Ik✱ ♣❛r❛ q✉❛❧q✉❡rk ≥m✳ ❆ ε✲✈✐③✐♥❤❛♥ç❛ ❞❡ s0 ♣♦ss✉✐ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❛ s✉❜s❡q✉ê♥❝✐❛ ✭snk✮ ❛ ♣❛rt✐r ❞♦ ♥ú♠❡r♦ nm✳
❈♦♠♦ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ I = [a, b] ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ✉♠
♣♦♥t♦ s0 ❞❡ I✱ t❡♠♦s q✉❡ I é ❝♦♠♣❛❝t♦✳
Pr♦♣♦s✐çã♦ ✸✳✶✵✳ ❖ q✉❛❞r❛❞♦ ♥♦ ♣❧❛♥♦ R2 é ❝♦♠♣❛❝t♦✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ Q ✉♠ q✉❛❞r❛❞♦ ❝✉❥♦s ❧❛❞♦s ♠❡❞❡♠ ✶ ❡ sã♦ ♣❛r❛❧❡❧♦s ❛♦s ❡✐①♦s Ox ❡ Oy ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳ ❙❡❥❛✱ t❛♠❜é♠✱ ✭pn✮ ✉♠❛ s❡q✉ê♥❝✐❛ q✉❛❧q✉❡r ❞❡ ♣♦♥t♦s ❞❡ Q✳ ❉✐✈✐❞✐♥❞♦ Q ♣♦r r❡t❛s ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s ❡ ♣❡❧♦s s❡✉s ♣♦♥t♦s ♠é❞✐♦s✱ t❡♠♦s
q✉❛tr♦ q✉❛❞r❛❞♦s ✭❞❡ ❧❛❞♦ ♠❡❞✐♥❞♦ 1
2✮ s❡♥❞♦ ✉♠ ❞❡❧❡s ❝♦♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❛ s❡q✉ê♥❝✐❛ pn✳ ❈❤❛♠❡♠♦s ❞❡ Q1 = I1xJ1 ❡st❡ q✉❛❞r❛❞♦ ❡ s❡❥❛ pn1 ✉♠ ♣♦♥t♦ ❞❛ s❡q✉ê♥❝✐❛ ❝♦♥t✐❞❛ ❡♠ Q1✳ ❉✐✈✐❞✐♥❞♦ Q1 ♣♦r ♠❡✐♦ ❞❡ r❡t❛s ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s ♣❡❧♦s s❡✉s ♣♦♥t♦s ♠é❞✐♦s✱ t❡r❡♠♦s ♦✉tr♦s q✉❛❞r❛❞♦s ✭❝♦♠ ❧❛❞♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ 1
22✮✳ ❙✉❝❡s✲
s✐✈❛♠❡♥t❡✱ ❡♥❝♦♥tr❛r❡♠♦s q✉❛❞r❛❞♦s ❡♥❝❛✐①❛❞♦s Qk = IkxJk ❝♦♠ ❝♦♠♣r✐♠❡♥t♦s ❞♦s ❧❛❞♦s t❡♥❞❡♥❞♦ ❛ ③❡r♦ q✉❛♥❞♦ k t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✳ ❈♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞♦s ✐♥t❡r✈❛❧♦s ❡♥✲
❝❛✐①❛❞♦s✱ t❡r❡♠♦s ✉♠ ♣♦♥t♦ x0 ❞❡ Q ♣❡rt❡♥❝❡♥t❡ ❛ t♦❞♦s ♦s q✉❛❞r❛❞♦s Qk✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ 3.4✳
❙❡q✉ê♥❝✐❛s ✷✼
❋✐❣✉r❛ ✸✳✹✿ ❙❡q✉ê♥❝✐❛ ❞❡ q✉❛❞r❛❞♦s Qk ❡♥❝❛✐①❛❞♦s
Pr♦♣♦s✐çã♦ ✸✳✶✶✳ ❯♠❛ ❢✉♥çã♦f ❝♦♥tí♥✉❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦[a, b]❡stá ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r
❡ ✐♥❢❡r✐♦r♠❡♥t❡✱ ♦✉ s❡❥❛✱ ❡①✐st❡♠ ♦s ✈❛❧♦r❡s r❡❛✐s m ❡ M t❛❧ q✉❡ m ≤f(x)≤ M ♣❛r❛
q✉❛❧q✉❡r x∈[a, b]✳
❉❡♠♦♥str❛çã♦✳ P♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✱ ✈❛♠♦s s✉♣♦r q✉❡ ♥ã♦ ❡①✐st❛ M✱ ♦✉ s❡❥❛✱ q✉❡
♥ã♦ ❡①✐st❛ ♥♦ ✐♥t❡r✈❛❧♦[a, b]✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r ♣❛r❛ ❛ ❢✉♥çã♦f✳ P♦❞❡♠♦s s✉♣♦r✱ ❡♥tã♦✱
q✉❡ ❡①✐st❡♠ ♦s s❡❣✉✐♥t❡s ♣♦♥t♦s ❞❡[a, b]✿ x1 t❛❧ q✉❡f(x1)>1✱x2 t❛❧ q✉❡f(x2)>2✱x3 t❛❧ q✉❡f(x3)>3✱ ✳✳✳✱xnt❛❧ q✉❡f(xn)> n✳ ❈♦♠♦ ♦ ✐♥t❡r✈❛❧♦ é ❝♦♠♣❛❝t♦✱ ❞❛ s❡q✉ê♥❝✐❛
(x1, x2, ..., xn, ...) é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ✉♠ ♣♦♥t♦ x0 ❞❡
[a, b]✭♣❛r❛ s✐♠♣❧✐✜❝❛r ❝♦♥s✐❞❡r❛r❡♠♦s ❛ ♣ró♣r✐❛ s❡q✉ê♥❝✐❛(x1, x2, ..., xn, ...)❝♦♥✈❡r❣❡♥t❡ ♣❛r❛x0✮✳ ❈♦♠♦f é ❝♦♥tí♥✉❛ ❡♠x0✱ ♣❛r❛ q✉❛❧q✉❡rε >0❡①✐st❡δ >0t❛❧ q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❛ δ✲✈✐③✐♥❤❛♥ç❛ ❞❡ x0 s❡ ❡♥❝♦♥tr❛ ♥❛ ε✲✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ f(x0)✱ ♦✉ s❡❥❛✱ ♣❛r❛ t♦❞♦ x ❞❡st❛ ✈✐③✐♥❤❛♥ç❛
f(x0)−ε < f(x)< f(x0) +ε
❈♦♠♦ ❛ s❡q✉ê♥❝✐❛xn❝♦♥✈❡r❣❡ ♣❛r❛x0✱ t♦❞♦s ♦s ♣♦♥t♦s ❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ ♥ú♠❡r♦
n0 s❡ ❡♥❝♦♥tr❛♠ ♥❛δ✲✈✐③✐♥❤❛♥ç❛ ❞❡x0 ❡ ♣♦rt❛♥t♦ ♣❛r❛ t♦❞♦n ≥n0✱f(xn)< f(x0) +ε ❝♦♥tr❛r✐❛♥❞♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s f(xn)> n✳
P♦rt❛♥t♦✱ f ♣♦ss✉✐ ✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r✳
