❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦
❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦
❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧
▼❛r❝✐✉s P❡trú❝✐♦ ❞❡ ❆❧♠❡✐❞❛ ❈❛✈❛❧❝❛♥t❡
▼❛❝❡✐ó✱ ❇r❛s✐❧
✶✹ ❞❡ ❉❡③❡♠❜r♦ ✷✵✵✼
❆♦s ♠❡✉s ♣❛✐s ❋r❛♥❝✐s❝♦ P❡trú❝✐♦ ❈❛✈❛❧❝❛♥t❡ ❡ ▼❛r✐❛ ❞❛s ◆❡✈❡s ❞❡ ❆❧♠❡✐❞❛ ❈❛✈❛❧❝❛♥t❡✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❍✐❧ár✐♦ ❆❧❡♥❝❛r✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣♦r ❡ss❡s ❛♥♦s ❞❡ ♦r✐❡♥t❛çã♦ ❛❝❛❞ê♠✐❝❛ ❡ ♦♣♦rt✉♥✐❞❛❞❡s ❞❛❞❛s✳
❆♦ ♣r♦❢❡ss♦r ❆❞á♥ ❈♦r❝❤♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ❛♠✐③❛❞❡ ❢♦r♥❡❝✐❞❛s ❞❡ s✉❛ ♣❛rt❡ ❞✉r❛♥t❡ ♦s ❛♥♦s ❞❡ ❣r❛❞✉❛çã♦ ❡ ♠❡str❛❞♦✳
❆♦ ♠❡✉ ✐r♠ã♦ ❡ ♣r♦❢❡ss♦r ▼❛r❝♦s P❡trú❝✐♦✱ ♣❡❧♦ ❝✉rs♦ ❞❡ ■♥tr♦❞✉çã♦ à ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ❞❛❞♦ ♥♦ ♣r✐♠❡✐r♦ s❡♠❡str❡ ❞❡ ✷✵✵✼✱ ♣❡❧❛ ♠♦r❛❞✐❛ ♣r♦♣♦r❝✐♦♥❛❞❛ ♥❡st❡s ❞♦✐s ú❧t✐♠♦s ❛♥♦s✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❡t❝✳
❆♦s ♣r♦❢❡ss♦r❡s ❊♥❛❧❞♦ ❱❡r❣❛st❛ ❡ ❋❡r♥❛♥❞♦ ❈♦❞á ▼❛rq✉❡s ♣❡❧❛s ✈❛❧✐♦s❛s ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s ❞❛❞❛s ♣❛r❛ ♠❡❧❤♦r✐❛ ❞❡st❛ ❞✐ss❡rt❛çã♦✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❆▲✱ ❞❡ ❢♦r♠❛ ❡s♣❡❝✐❛❧ ❛ ❆♥❞ré P✐③③❛✐❛✱ ❆r❧②s♦♥ ❆❧✈❡s ❡ ❊✈❡rs♦♥ ❋❡r♥❛♥❞♦✳ ❆♦s ♠❡✉s ❛♠✐❣♦s ❆s❦❡r② ❆❧❡①❛♥❞r❡✱ ❈❛✐♦ ❡ ▼❛♥♦❡❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯❋❆▲ ♣❡❧❛ ❢♦rç❛ ❞❛❞❛✱ ❛ss✐♠ ❝♦♠♦ t❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛ ❉❛r❧✐t♦♥ ❈❡③ár✐♦✱ ●❡❧s✐✈â♥✐♦ ❙♦✉③❛ ❡ ❏♦sé ❊❞✉❛r❞♦✳
❆♦ ♠❡✉ ❛♠✐❣♦ ▼ár❝✐♦ ❍❡♥r✐q✉❡✱ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ t♦❞♦ ❡s♣❡❝✐❛❧ ♣❡❧❛s ✐♠♣♦rt❛♥t❡s ❞ú✈✐❞❛s ❡s❝❧❛r❡❝✐❞❛s✱ ❛s q✉❛✐s ❞❡r❛♠ ✜r♠❡③❛ ❛ ✈ár✐♦s ❛r❣✉♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❞❡st❛ ❞✐ss❡rt❛çã♦✳
❙♦✉ ♠✉✐t♦ ❣r❛t♦ ❛♦s ♣r♦❢❡ss♦r❡s ❆❞r✐❛♥♦ ❆❣✉✐❛r ❡ ❊❞✉❛r❞♦ P❡r❞✐❣ã♦ ♣❡❧♦s s❡♠✐♥ár✐♦s ❞❡ ❊s♣❛ç♦s ▼étr✐❝♦s ❡ ■♥tr♦❞✉çã♦ à ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ♠✐♥✐str❛❞♦s✳
❆♦ ♣r♦❢❡ss♦r ❆❞❡❧❛í❧s♦♥ P❡✐①♦t♦ ❡ s❡✉s ♦r✐❡♥t❛♥❞♦s ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❛♣♦✐♦ ♥❡st❛ ❞✐ss❡rt❛çã♦✱ ✐st♦ ✐♥❝❧✉✐ ❈❧❛r✐ss❛ ❈♦❞á ❡ ❈❧❛✉❞❡♠✐r ❙✐❧✈✐♥♦✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❆▲ q✉❡ ❝♦❧❛❜♦r❛r❛♠ ❝♦♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♠❛t❡♠át✐❝❛✳
❆♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✲ ❈◆Pq ✲ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❆ ❉❡✉s ♣♦r t✉❞♦✳✳✳
❘❡s✉♠♦
❉❡♠♦♥str❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✱ ♦ q✉❛❧ ❣❛r❛♥t❡ q✉❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ♥✲ ❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✱ q✉❡ ♣♦ss✉✐ ✉♠❛ ❧✐♥❤❛✱ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st❛ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ♥✉♠ ♣r♦❞✉t♦ ❘✐❡♠❛♥♥✐❛♥♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✭♥✲✶ ✮✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s r❡❛✐s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ■s♦♠❡tr✐❛✱ ❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✱ ▲❛♣❧❛❝✐❛♥♦✱ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✱ ❋✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥✳
❆❜str❛❝t
❲❡ ❞❡♠♦♥str❛t❡ t❤❡ ❙♣❧✐tt✐♥❣ ❚❤❡♦r❡♠ ❞✉❡ t♦ ❈❤❡❡❣❡r ❛♥❞ ●r♦♠♦❧❧✱ ✇❤✐❝❤ ❡♥s✉r❡s t❤❛t ❛ ❝♦♠♣❧❡t❡ ❘✐❡♠❛♥♥✐❛♥ ♥✲♠❛♥✐❢♦❧❞ ✇❤✐❝❤ ❤❛s ♥♦♥♥❡❣❛t✐✈❡ ❘✐❝❝✐ ❝✉r✈❛t✉r❡ ❛♥❞ ❛ ❧✐♥❡✱ ❝❛♥ ❜❡ s♣❧✐t ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ t❤❡ ❘✐❡♠❛♥♥✐❛♥ ♣r♦❞✉❝t ♦❢ r❡❛❧ ✇✐t❤ ❛ ✭♥✲✶ ✮✲ ♠❛♥✐❢♦❧❞✳
❑❡② ✇♦r❞s✿ ■s♦♠❡tr②✱ ❲❡✐t③❡♥❜ö❝❦✬s ❋♦r♠✉❧❛✱ ▲❛♣❧❛❝✐❛♥✱ ❙♣❧✐tt✐♥❣ ❚❤❡♦r❡♠ ♦❢ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✱ ❇✉s❡♠❛♥♥ ❋✉♥❝t✐♦♥s✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✾
✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✶✷ ✶✳✶ ❱❛r✐❡❞❛❞❡s ❡ ▼étr✐❝❛s ❘✐❡♠❛♥♥✐❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷ ❆ ❈♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸ ❆♣❧✐❝❛çã♦ ❊①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹ ❱❛r✐❡❞❛❞❡s ❈♦♠♣❧❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✺ ❈✉r✈❛t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ✷✹ ✷✳✶ ❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ ❆ ❍❡ss✐❛♥❛ ❞❛ ❋✉♥çã♦ ❉✐stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸ ❖ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧ ✸✾ ✸✳✶ ❋✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷ ❚❡♦r❡♠❛ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✽
❮♥❞✐❝❡ ❘❡♠✐ss✐✈♦ ✺✵
■♥tr♦❞✉çã♦
❖ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧ é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡✱ ❛❧é♠ ❞✐ss♦✱ ♣♦ss✉✐ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❡st❡ t❡♦r❡♠❛ ❢♦✐ ♣r♦✈❛❞♦ ♣♦r ❈♦❤♥✲❱♦ss❡♥✱ ❡♠ ✶✾✸✻✱ ♣❛r❛ s✉♣❡r❢í❝✐❡s ♥♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ ❘✸ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ♣♦r ❚♦♣♦♥♦❣♦✈
❡♠ ✶✾✻✹✱ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❝♦♠♣❧❡t❛s ♥✲❞✐♠❡♥s✐♦♥❛✐s✱ ❝♦♠ ❛ ❤✐♣ót❡s❡ ❞❡ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ♥ã♦✲♥❡❣❛t✐✈❛✳ ❆ ✈❡rsã♦ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛q✉✐✱ ❞❡✈✐❞❛ ❛ ❈❤❡❡❣❡r ❡ ●r♦♠♦❧❧✱ ❣❡♥❡r❛❧✐③❛ ♦ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ❝♦♠♣❧❡t❛s ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✱ ❝✉❥♦ r❡s✉❧t❛❞♦ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✾✼✶ ♥♦ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr② ❝♦♠ ♦ s❡❣✉✐♥t❡ ❡♥✉♥❝✐❛❞♦✿
❚❡♦r❡♠❛ ✵✳✶✳ ❙❡❥❛ ✭▼♥❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠
❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✳ ❊♥tã♦ ▼ é ✐s♦♠étr✐❝❛ ❛ ✉♠ ♣r♦❞✉t♦ ❘✐❡♠❛♥♥✐❛♥♦ ❞❛ ❢♦r♠❛ ❘❦ ✂ ◆✱ ♦♥❞❡ ◆ ♥ã♦ ❝♦♥té♠ ❧✐♥❤❛s ❡ ❘❦ ♣♦ss✉✐
❛ ♠étr✐❝❛ ❝❛♥ô♥✐❝❛✳
◆♦ ❡♥t❛♥t♦✱ ❝♦♠ ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ ✐♥❞✉çã♦ ✜♥✐t❛✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡ s❡ ✭▼♥❀ ❣✮ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛✱ ❛ q✉❛❧ ❝♦♥té♠ ✉♠❛ ❧✐♥❤❛
❡ ❘✐❝✭▼✮ ✕ ✵✱ ❡♥tã♦ ▼ é ✐s♦♠étr✐❝❛ ❛ ✉♠ ♣r♦❞✉t♦ ❘✐❡♠❛♥♥✐❛♥♦ ❞❛ ❢♦r♠❛ ❘ ✂ ◆✳ P♦✐s✱ t❡r❡♠♦s ❘✐❝✭◆✮ ✕ ✵ ❡ ◆ ❝♦♠♣❧❡t❛✳
❊st❛ ❞✐ss❡rt❛çã♦ ❡stá ❞✐✈✐❞✐❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡ ♣r♦❝✉r❛♠♦s ✜①❛r ❛s ♥♦t❛çõ❡s✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s sã♦ ❞❡♠♦♥str❛❞♦s✱ ❛ s❛❜❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✳
Pr♦♣♦s✐çã♦ ✵✳✶✳ ❙❡❥❛ ❢ ✷ ❈✸ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ ▼✳ ❊♥tã♦
✶
✷✁ ❥ ❣r❛❞ ❢❥✷ ✁
❂ ❥ ❍❡ss ❢❥✷ ✰ ❤❣r❛❞ ❢❀ ❣r❛❞✭✁❢✮✐ ✰ ❘✐❝ ✭❣r❛❞ ❢❀ ❣r❛❞ ❢✮ ✿
❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ r❡s✉❧t❛❞♦s s♦❜r❡ ❛ ❍❡ss✐❛♥❛ ❞❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ ❡ ❝♦♥❝❧✉í♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦✿
❚❡♦r❡♠❛ ✵✳✷✳ ❙❡❥❛ ✭▼♥❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛✳ ❙✉♣♦✲
♥❤❛♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ❞❡ ▼ s❛t✐s❢❛③ ❘✐❝✭▼✮ ✕ ✭♥ ✶✮❦❣✳ ❙❡❥❛ r ❛ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ ❛♦ ♣♦♥t♦ ♣ ✷ ▼✳ ❙✉♣♦♥❤❛♠♦s ❛✐♥❞❛ q✉❡ ❛ ❢✉♥çã♦ r é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ ①✳ ❊♥tã♦
✁r✭①✮ ✔ ✁r❦✭⑦①✮❀
♦♥❞❡ r❦✭⑦①✮ ❂ r✭①✮ ❂ r✵ ❡ r✵ ❁ ♣✙❦ s❡ ❦ ❃ ✵✳
❋✐♥❛❧♠❡♥t❡✱ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ❞❡♠♦♥str❛♠♦s ♦ ❚❡♦✲ r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✳ ❆♥t❡s✱ ♣♦ré♠✱ ❛✐♥❞❛ ❞❡♠♦♥s✲ tr❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❛♣r❡s❡♥t❛r♠♦s ♦ t❡♦r❡♠❛ ✜♥❛❧ ❝♦♠ ✉♠❛ ❜♦❛ ♦❜❥❡t✐✈✐❞❛❞❡✳ ❉❡♥tr❡ t❛✐s r❡s✉❧t❛❞♦s✱ ♠♦str❛♠♦s q✉❡ ❛ ❢✉♥çã♦ ❞❡ ❇✉s❡♠❛♥♥ ❜✰ ❛ss♦❝✐❛❞❛ ❛ ✉♠ r❛✐♦ ✌ é s✉♣❡r✲❤❛r♠ô♥✐❝❛ ♥♦ s❡♥t✐❞♦ ❞❛s
❞✐str✐❜✉✐çõ❡s ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳
❘❡ss❛❧t❛♠♦s q✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❢❡✐t❛ ♥❡st❛ ❞✐ss❡rt❛çã♦ é ❜❛s❡❛❞❛ ♥♦ s❡❣✉✐♥t❡ ❛rt✐❣♦✿
❊s❝❤❡♥❜✉r❣✱ ❏✳✲❍✳✱ ❍❡✐♥t③❡✱ ❊✳✱ ❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ ❈❤❡❡❣❡r✲ ●r♦♠♦❧❧ s♣❧✐tt✐♥❣ t❤❡♦r❡♠✱ ❆♥♥✳ ●❧♦❜✳ ❆♥❛❧②s✐s ❛♥❞ ●❡♦♠❡tr②✱ ✈♦❧✳ ✷✱ ✭✶✾✽✹✮✱ ✶✹✶✕✶✺✶✳
❖ q✉❛❧ ❛♣r❡s❡♥t❛ ♥♦✈❛s té❝♥✐❝❛s ❞❡ ❝♦♠♦ ❛t❛❝❛r t❡♦r❡♠❛s ❞❡ ❞❡❝♦♠♣♦s✐çã♦✳
❱❛❧❡ ♦❜s❡r✈❛r q✉❡ ♦ t❡♦r❡♠❛ ♥ã♦ é ✈á❧✐❞♦ ♣❛r❛ ✈❛r✐❡❞❛❞❡s q✉❡ ♣♦ss✉❡♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ♥ã♦ ♥❡❣❛t✐✈❛✳ ❈♦♠♦ ❝♦♥tr❛✲❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s ❝✐t❛r ❛ ✈❛✲ r✐❡❞❛❞❡ ❘✷✂ ❙✷ ❝♦♠ ❛ ♠étr✐❝❛ ❞❡ ❙❝❤✇❛r③s❝❤✐❧❞✳
❈❛♣ít✉❧♦ ✶
❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❞❛ ●❡♦♠❡tr✐❛
❘✐❡♠❛♥♥✐❛♥❛
◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐♠♦s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✱ t❛✐s ❝♦♠♦ ✈❛r✐❡❞❛❞❡s ❡ ♠étr✐❝❛s ❘✐❡♠❛♥♥✐❛♥❛s✱ ❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✱ ❝✉r✈❛t✉r❛s✱ ✈❛r✐❡❞❛❞❡s ❝♦♠♣❧❡t❛s ❡ ♦✉tr♦s ❝♦♥❝❡✐t♦s q✉❡ ♣r❡❝✐✲ s❛r❡♠♦s ♣❛r❛ ♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s ❞❡st❛ ❞✐ss❡rt❛çã♦✳
✶✳✶ ❱❛r✐❡❞❛❞❡s ❡ ▼étr✐❝❛s ❘✐❡♠❛♥♥✐❛♥❛s
❙❛❜❡♠♦s q✉❡ ❞❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼♥✱ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s
❳ ❡♠ ▼ é ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ❛ ❝❛❞❛ ♣♦♥t♦ ♣ ✷ ▼ ❛ss♦❝✐❛ ✉♠ ✈❡t♦r ❳✭♣✮ ♣❡rt❡❝❡♥t❡ ❛♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❚♣▼✳
❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ① ✿ ❯ ✚ ❘♥ ✦ ▼✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✱
♣❛r❛ ❝❛❞❛ ♣ ✷ ①✭❯✮✱
❳✭♣✮ ❂❳♥
✐❂✶
☛✐✭♣✮❅①❅ ✐✭♣✮❀
♦♥❞❡ ❢ ❅
❅①✶✭♣✮❀ ✿ ✿ ✿ ❀
❅
❅①♥✭♣✮❣ é ❛ ❜❛s❡ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❡♠ ♣ ✷ ▼ ❛ss♦❝✐❛❞❛
à ①✱ ❡ ☛✶❀ ✿ ✿ ✿ ❀ ☛♥ sã♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ ❯✳ ❈♦♠ ❛❜✉s♦ ❞❡ ♥♦t❛çã♦✱
❡s❝r❡✈❡♥❞♦ ❢ ❛♦ ✐♥✈és ❞❡ ❢ ✍ ①✱ ♣♦❞❡♠♦s t❛♠❜é♠ ♣❡♥s❛r ❡♠ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❝♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❳ ✿ ❉ ✦ ❋ ❞♦ ❝♦♥❥✉♥t♦ ❉ ❞❛s ❢✉♥çõ❡s
❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ ▼ ♥♦ ❝♦♥❥✉♥t♦ ❋ ❞❛s ❢✉♥çõ❡s ❡♠ ▼✱ ❞❡✜♥✐❞❛ ♣♦r ❳✭❢✮✭♣✮ ❂
♥
❳
✐❂✶
☛✐✭♣✮❅①❅❢ ✐✭♣✮✿
❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❡♠ ▼ s❡rá ❞❡♥♦t❛❞♦ ♣♦r ❳ ✭▼✮✳ ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❞❡ ❞♦✐s ❝❛♠♣♦s✳
❊①❡♠♣❧♦ ✶✳✶✳ ❙❡❥❛♠ ❳ ❂ P✐❛✐❅①❅✐ ❡ ❨ ❂P❥❜❥❅①❅❥ ♣❡rt❡♥❝❡♥t❡s ❛ ❳ ✭▼✮✳
❊♥tã♦ ❬❳❀ ❨ ❪ ✿❂ ❳❨ ❨ ❳ ♣❡rt❡♥❝❡ ❛ ❳ ✭▼✮✳ ❉❡ ❢❛t♦✱
❳❨ ✭❢✮ ❂ ❳ ✥
❳
❥
❜❥❅①❅❢ ❥ ✦ ❂ ❳ ❥ ❳ ✒
❜❥❅①❅❢ ❥ ✓ ❂ ❳ ❥ ❳ ✐
❛✐❅①❅ ✐
✒ ❜❥❅①❅❢
❥ ✓ ❂ ❳ ❥ ❳ ✐ ❛✐ ✒ ❅❜❥ ❅①✐ ❅❢ ❅①❥ ✰ ❜❥
❅✷❢
❅①✐❅①❥
✓ ❂ ❳ ❥ ❳ ✐ ✒ ❛✐❅①❅❜❥
✐
✓ ❅❢ ❅①❥ ✰
❳ ❥ ❳ ✐ ❛✐❜❥ ❅ ✷❢
❅①✐❅①❥✿
❆♥❛❧♦❣❛♠❡♥t❡✱ ✈❡♠♦s q✉❡ ❨ ❳✭❢✮ ❂ ❳
✐
❳
❥
✒ ❜❥❅①❅❛✐
❥
✓ ❅❢ ❅①✐ ✰
❳ ✐ ❳ ❥ ❜❥❛✐ ❅ ✷❢
❅①❥❅①✐✿
❉❛í✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❙❝❤✇❛rt③✱ ♦❜t❡♠♦s ❳❨ ✭❢✮ ❨ ❳✭❢✮ ❂ ❳
❥
❳
✐
✒ ❛✐❅❜❅①❥
✐ ❜❥
❅❛✐
❅①❥
✓ ❅❢ ❅①✐✿
■ss♦ ♠♦str❛ q✉❡ ❬❳❀ ❨ ❪ é ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✳
P❛ss❡♠♦s ❛❣♦r❛ à s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❯♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼♥ é ✉♠❛
❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ ♣ ✷ ▼ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❤✁❀ ✁✐♣ ❂ ❣♣✭✁❀ ✁✮ ♥♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❚♣▼✱ t❛❧ q✉❡✿ s❡ ① ✿ ❯ ✚ ❘♥ ✦ ▼ é ✉♠ s✐st❡♠❛
❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❡♠ t♦r♥♦ ❞❡ ♣✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ ✐❀ ❥ ✷ ❢✶❀ ✿ ✿ ✿ ❀ ♥❣✱ ❉
❅ ❅①✐✭q✮❀
❅ ❅①❥✭q✮
❊
q é ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ ❯✳
❆s ❢✉♥çõ❡s ❣✐❥ ✿❂
❉
❅ ❅①✐❀
❅ ❅①❥
❊
sã♦ ❝❤❛♠❛❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ♥♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ① ✿ ❯ ✚ ❘♥ ✦ ▼♥✳ ❯♠❛
✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é ❝❤❛♠❛❞❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❉❡♥♦t❛r❡♠♦s ♣♦r ✭▼♥❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡
❞✐♠❡♥sã♦ ♥ ❝♦♠ ✉♠❛ ♠étr✐❝❛ ❣✳
❊①❡♠♣❧♦ ✶✳✷✳ ❆ ❡s❢❡r❛ ✉♥✐tár✐❛ ❙♥ ✶✭✶✮ ❂ ❢① ❂ ✭①
✶❀ ✿ ✿ ✿ ❀ ①♥✮ ✷ ❘♥❀ ❥①❥ ❂ ✶❣
é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❈♦♠ ❡❢❡✐t♦✱ ❜❛st❛ ❞❡✜♥✐r♠♦s ❛ ♠étr✐❝❛ ❣ ♣♦r ❣✭✉❀ ✈✮♣ ❂ ❤✉❀ ✈✐❘♥✱ ♦♥❞❡ ✉❀ ✈ ✷ ❚♣❙♥ ✶✳
❊①❡♠♣❧♦ ✶✳✸✳ ❖ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦ ❍♥✳
❈♦♥s✐❞❡r❡ ❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❯ ❂ ❘♥
✰ ❂ ❢✭①✶❀ ✿ ✿ ✿ ❀ ①♥✮ ✷ ❘♥❀ ①♥ ❃ ✵❣✱
❝♦♠ ❛ ❝❛rt❛ ✐❞ ✿ ❯ ✚ ❘♥ ✦ ❯✳ ❉❛❞♦ ♣ ✷ ❯✱ ♣ ❂ ✭①
✶❀ ✿ ✿ ✿ ❀ ①♥✮✱ s❡❥❛♠
✈❀ ✇ ✷ ❚♣❯ ❂ ❘♥✳
❉❡✜♥✐♠♦s
❤♣✭✈❀ ✇✮ ❂ ①✶✷
♥❤✈❀ ✇✐❘♥✿
❱ê✲s❡ q✉❡ ✭❯❀ ❤✮ ✿❂ ❍♥ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ❞❡♥♦♠✐♥❛❞❛
❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦✳
❊①❡♠♣❧♦ ✶✳✹✳ ❱❛r✐❡❞❛❞❡s ■♠❡rs❛s✳
❙❡❥❛ ❢ ✿ ▼♥ ✦ ◆♥✰❦ ✉♠❛ ✐♠❡rsã♦✱ ✐st♦ é✱ ❢ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧
❡ ❞❢♣ ✿ ❚♣▼ ✦ ❚❢✭♣✮◆ é ✐♥❥❡t✐✈❛ ♣❛r❛ t♦❞♦ ♣ ✷ ▼✳ ❙❡ ◆ t❡♠ ✉♠❛ ♠étr✐❝❛
❘✐❡♠❛♥♥✐❛♥❛✱ ❢ ✐♥❞✉③ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ ▼✱ ❞❛❞❛ ♣♦r ❤✉❀ ✈✐♣ ❂ ❤❞❢♣✭✉✮❀ ❞❢♣✭✈✮✐❢✭♣✮❀ ✉❀ ✈ ✷ ❚♣▼✿
❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ ❞❢♣ é ✐♥❥❡t✐✈❛✱ ❤✁❀ ✁✐ é ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✳ ❆s ❞❡♠❛✐s
✈❡r✐✜❝❛❞❛s✳
❆ ♠étr✐❝❛ ❞❡ ▼ é ❡♥tã♦ ❝❤❛♠❛❞❛ ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ♣♦r ❢ ❡ ❢ é ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛✳
◗✉❛♥❞♦ ❢ ✿ ▼♥ ✦ ◆♥✰❦ é ✉♠❛ ✐♠❡rsã♦ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❢ é ✉♠
❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ ❢✭▼✮ ✚ ◆✱ ♦♥❞❡ ❢✭▼✮ t❡♠ ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣♦r ◆✱ ❞✐③✲s❡ q✉❡ ❢ é ✉♠ ♠❡r❣✉❧❤♦✳ ❙❡ ▼ ✚ ◆ ❡ ❛ ✐♥❝❧✉sã♦ ✐ ✿ ▼ ✱✦ ◆ é ✉♠ ♠❡r❣✉❧❤♦✱ ❞✐③✲s❡ q✉❡ ▼ é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ◆✳
❯♠❛ ❝❧❛ss❡ ✐♠♣♦rt❛♥t❡ ❞❡ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s é ❞❛❞❛ ♥♦ ❡①❡♠♣❧♦ s❡❣✉✐♥t❡✳
❊①❡♠♣❧♦ ✶✳✺✳ ▼étr✐❝❛ Pr♦❞✉t♦✳ ❙❡❥❛♠ ✭▼♥✶
✶ ❀ ❣✶✮ ❡ ✭▼✷♥✷❀ ❣✷✮ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ❡ ❝♦♥s✐❞❡r❡ ♦ ♣r♦❞✉t♦
❝❛rt❡s✐❛♥♦ ▼ ❂ ▼✶ ✂ ▼✷ ❝♦♠ ❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣r♦❞✉t♦✳ ❙❡❥❛♠
✙✶ ✿ ▼✶ ✂ ▼✷ ✦ ▼✶ ❡ ✙✷ ✿ ▼✶ ✂ ▼✷ ✦ ▼✷ ❛s ♣r♦❥❡çõ❡s ♥❛t✉r❛✐s✳
P♦❞❡♠♦s ✐♥tr♦❞✉③✐r ❡♠ ▼✶ ✂ ▼✷ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ♣♦♥❞♦✱ ♣❛r❛
❝❛❞❛ ✭♣❀ q✮ ✷ ▼✶✂ ▼✷ ❡ ✉❀ ✈ ✷ ❚✭♣❀q✮✭▼✶✂ ▼✷✮✱
❤✉❀ ✈✐✭♣❀q✮ ❂ ❤❞✙✶ ✁ ✉❀ ❞✙✶ ✁ ✈✐♣✰ ❤❞✙✷ ✁ ✉❀ ❞✙✷ ✁ ✈✐q✿
❈♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s t❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❙✶ ✂ ❙✶ ❂ ❚✷ ♦✉✱ ♠❛✐s
❣❡r❛❧♠❡♥t❡✱ ❙✶ ✂ ✁ ✁ ✁ ✂ ❙✶ ❂ ❚♥✱ ♦s q✉❛✐s tê♠ ✉♠❛ ❡str✉t✉r❛ ❘✐❡♠❛♥♥✐❛♥❛
♦❜t✐❞❛ q✉❛♥❞♦ ❡s❝♦❧❤❡♠♦s ♥♦ ❝ír❝✉❧♦ ❙✶ ✚ ❘✷ ❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛
✐♥❞✉③✐❞❛ ♣♦r ❘✷ ❡✱ ❡♠ s❡❣✉✐❞❛✱ t♦♠❛♠♦s ❛ ♠étr✐❝❛ ♣r♦❞✉t♦ ❡♠ ❚♥✳ ❖ t♦r♦
❚♥ ❝♦♠ ❡st❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é ❝❤❛♠❛❞♦ t♦r♦ ♣❧❛♥♦ ♦✉ t♦r♦ ✢❛t✳
❚❡♦r❡♠❛ ✶✳✶✳ ❚♦❞❛ ✈❛r✐❡❞❛❞❡ ❝♦♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧✱ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❞❡ ❍❛✉s❞♦r✛ ♣♦ss✉✐ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✸❪✱ ♣á❣✐♥❛ ✹✼✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛♠ ✭▼♥❀ ❣✮ ❡ ✭▼♥❀ ❣✮ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s✳ ❯♠
❞✐❢❡♦♠♦r✜s♠♦ ✬ ✿ ▼ ✦ ▼ é ❞✐t♦ ✉♠❛ ✐s♦♠❡tr✐❛ s❡
❣✬✭♣✮✭❞✬♣✭✈✮❀ ❞✬♣✭✇✮✮ ❂ ❣♣✭✈❀ ✇✮❀ ✽ ♣ ✷ ▼❀ ✽ ✈❀ ✇ ✷ ❚♣▼✿
✶✳✷ ❆ ❈♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛
❯♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼ é ✉♠❛ ❛♣❧✐❝❛çã♦ r ✿ ❳ ✭▼✮ ✂ ❳ ✭▼✮ ✦ ❳ ✭▼✮
t❛❧ q✉❡✱ ✽ ❳❀ ❨❀ ❩ ✷ ❳ ✭▼✮ ❡ ✽ ❢❀ ❣ ✷ ❉✭▼✮✱ t❡♠✲s❡ ✭✐✮ r❢❳✰❣❨❩ ❂ ❢r❳❩ ✰ ❣r❳❩
✭✐✐✮ r❳✭❨ ✰ ❩✮ ❂ r❳❨ ✰ r❳❩
✭✐✐✐✮ r❳❢❨ ❂ ❢r❳❨ ✰ ❳✭❢✮❨ ✳
❊♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✱ ❳✐ ❂ ❅①❅✐✱ ♣♦❞❡✲s❡ ❢❛❝✐❧♠❡♥t❡ ✈❡r✱ ♣❛r❛ ❳ ❂
P
✐①✐❳✐ ❡ ❨ ❂
P
✐②✐❨✐✱ q✉❡
r❳❨ ❂
❳
❦
✥ ❳
✐❀❥
①✐②❥ ❦✐❥ ✰ ❳✭②❦✮
✦ ❳❦✿
❖s ❝♦❡✜❝✐❡♥t❡s ❦
✐❥✱ ❞❡✜♥✐❞♦s ♣♦r r❳✐❳❥ ❂
P
❦ ❦✐❥❳❦✱ sã♦ ❝❤❛♠❛❞♦s
sí♠❜♦❧♦s ❞❡ ❈❤r✐st♦✛❡❧ ❞❛ ❝♦♥❡①ã♦✳
❊♠ t❡r♠♦s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✱ ♦s sí♠❜♦❧♦s ❞❡ ❈❤r✐st♦✛❡❧ ♣♦ss✉❡♠ ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿
♠ ✐❥ ❂ ✶✷
❳
❦
✚ ❅ ❅①✐❣❥❦✰
❅ ❅①❥❣❦✐
❅ ❅①❦❣✐❥
✛ ❣❦♠❀
♦♥❞❡ ✭❣❦♠✮ é ❛ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❡ ✭❣
❦♠✮✱ ✈❡r ❬✸❪✱ ♣á❣✐♥❛ ✻✷✳
❉❛❞❛ ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ☛ ✿ ■ ✚ ❘ ✦ ▼✱ ✉♠ ❝❛♠♣♦ ❛♦ ❧♦♥❣♦ ❞❡ ☛ é ✉♠❛ ❢✉♥çã♦ ❱ ✿ ■ ✦ ❚ ▼ t❛❧ q✉❡ ❱ ✭t✮ ✷ ❚☛✭t✮▼✱ ♦♥❞❡
❚ ▼ ❂ ❢✭♣❀ ✈✮❥♣ ✷ ▼❀ ✈ ✷ ❚♣▼❣ é ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ ▼✳
❙❡❥❛ t ✷ ■✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s✱ t❡♠♦s ❛ ❜❛s❡ ❢❞
❞t❣ ❡♠ ❚t■ ❂ ❘✳ P♦r
❞❡✜♥✐çã♦✱ ☛✵
t✭❞t❞✮ é ♦ ✈❡t♦r t❛♥❣❡♥t❡ à ❝✉r✈❛ ☛ ❡♠ ☛✭t✮✳ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦✿
❞☛ ❞ ❞t
✁
❂ ☛✵✭t✮✳
Pr♦♣♦s✐çã♦ ✶✳✶✳ ❙❡❥❛♠ ▼ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡ ☛ ✿ ■ ✦ ▼ ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ ❝❛♠♣♦ ❱ ❛♦ ❧♦♥❣♦ ❞❡ ☛✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❝❛♠♣♦ ❞❡♥♦t❛❞♦ ♣♦r ❉❱
❞t t❛❧ q✉❡
✭❛✮ ❉
❞t✭❱ ✰ ❲ ✮ ❂ ❉❱❞t ✰❉❲❞t ❀
✭❜✮ ❉
❞t✭❢❱ ✮ ❂ ❞❢❞t❱ ✰ ❢❉❱❞t ❀
✭❝✮ ❙❡ ❱ ✭t✮ é ❛ r❡str✐çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ❨ ❞❡✜♥✐❞♦ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ☛✭■✮✱ ❡♥tã♦ ❉❱
❞t ❂ r☛✵✭t✮❨ ✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✸❪✱ ♣á❣✐♥❛ ✺✼✳
❉✐③❡♠♦s q✉❡ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼ é s✐♠étr✐❝❛ s❡
r❳❨ r❨❳ ❂ ❬❳❀ ❨ ❪❀ ✽ ❳❀ ❨ ✷ ❳ ✭▼✮✿
❆ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ t❛❧ ♥♦♠❡♥❝❧❛t✉r❛ ✈❡♠ ❞♦ s❡❣✉✐♥t❡ ❢❛t♦✿ ✽ ✐❀ ❥ ✷ ❢✶❀ ✁ ✁ ✁ ❀ ♥❣✱
r❳✐❳❥ r❳❥❳✐ ❂ ❬❳✐❀ ❳❥❪ ❂ ✵❀
♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❦
✐❥ ❂ ❦❥✐✳
❉✐③❡♠♦s q✉❡ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ▼ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ♠étr✐❝❛ s❡
❳ ❤❨❀ ❩✐ ❂ ❤r❳❨❀ ❩✐ ✰ ❤❨❀ r❳❩✐ ❀ ✽ ❳❀ ❨❀ ❩ ✷ ❳ ✭▼✮✿
❖ ♣ró①✐♠♦ t❡♦r❡♠❛ é ❢✉♥❞❛♠❡♥t❛❧ ❡♠ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✳
❚❡♦r❡♠❛ ✶✳✷ ✭▲❡✈✐✲❈✐✈✐t❛✮✳ ❉❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ▼✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡♠ ▼ q✉❡ é s✐♠étr✐❝❛ ❡ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✸❪✱ ♣á❣✐♥❛ ✻✶✳
❚❛❧ ❝♦♥❡①ã♦ é ❞❡♥♦♠✐♥❛❞❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ✭♦✉ ❘✐❡♠❛♥♥✐❛♥❛✮ ❞❡ ▼✳ ❆ ♣❛rt✐r ❞❛q✉✐ ❡st❛r❡♠♦s s❡♠♣r❡ ❝♦♥s✐❞❡r❛♥❞♦ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ❝♦♠ s✉❛s r❡s♣❡❝t✐✈❛s ❝♦♥❡①õ❡s ❞❡ ▲❡✈✐✲❈✐✈✐t❛✳
✶✳✸ ❆♣❧✐❝❛çã♦ ❊①♣♦♥❡♥❝✐❛❧
❯♠❛ ❝✉r✈❛ ☛ ❡♠ ▼ é ✉♠❛ ❣❡♦❞és✐❝❛ ❡♠ t ✷ ■ s❡ ❉
❞t☛✵✭t✮ ❂ ✵✳
◆❡st❡ ❝❛s♦✱ s❡ ✈✭t✮ ❂♣❤☛✵✭t✮❀ ☛✵✭t✮✐ é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ☛✱ t❡♠♦s q✉❡
❞
❞t✭✈✷✭t✮✮ ❂ ❞
❞t❤☛✵✭t✮❀ ☛✵✭t✮✐ ❂ ✷
✜ ❉
❞t☛✵✭t✮❀ ☛✵✭t✮ ✢ ❂ ✵✿
P♦rt❛♥t♦✱ s❡ ☛ é ❣❡♦❞és✐❝❛✱ ❡♥tã♦ ♦ ✈❡t♦r ✈❡❧♦❝✐❞❛❞❡ ❞❡ ☛ ♣♦ss✉✐ ♥♦r♠❛ ❝♦♥st❛♥t❡✳
❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠ ❝❛♠♣♦ ❱ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❝✉r✈❛ ☛ é ❞✐t♦ ♣❛r❛❧❡❧♦ s❡
❉❱ ❞t ❂ ✵✳
Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡❥❛♠ ☛ ✿ ■ ✦ ▼ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❱✵ ✷ ❚☛✭t✵✮▼✳ ❊♥tã♦
❡①✐st❡ ✉♠ ú♥✐❝♦ ❝❛♠♣♦ ♣❛r❛❧❡❧♦ ❱ ✱ ❛♦ ❧♦♥❣♦ ❞❡ ☛✱ t❛❧ q✉❡ ❱ ✭t✵✮ ❂ ❱✵✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✸❪✱ ♣á❣✐♥❛ ✺✽✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ t❡♦r❡♠❛ ❞❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s✱ ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
Pr♦♣♦s✐çã♦ ✶✳✸✳ ❉❛❞♦s ♣ ✷ ▼ ❡ ✈ ✷ ❚♣▼✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❣❡♦❞és✐❝❛
☛ ✿ ■ ✦ ▼ t❛❧ q✉❡ ☛✭✵✮ ❂ ♣ ❡ ☛✵✭✵✮ ❂ ✈✳
❙❡ ✈ ✷ ❚♣▼✱ ✈❛♠♦s ❞❡♥♦t❛r ♣♦r ✌✈ ❛ ú♥✐❝❛ ❣❡♦❞és✐❝❛ ❞❡ ▼ q✉❡ ♣❛ss❛ ♣♦r
♣ ✷ ▼ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ✈ ✷ ❚♣▼✳
❙❡❥❛ ✝♣ ❂ ❢✈ ✷ ❚♣▼❀ ✌✈ ❡stá ❞❡✜♥✐❞❛ ♥✉♠ ✐♥t❡r✈❛❧♦ ❝♦♥t❡♥❞♦ ❬✵❀ ✶❪❣✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❆ ❛♣❧✐❝❛çã♦ ❡①♣♣ ✿ ✝♣ ✦ ▼✱ ❞❡✜♥✐❞❛ ♣♦r ❡①♣♣✭✈✮ ❂ ✌✈✭✶✮✱
é ❞❡♥♦♠✐♥❛❞❛ ❛♣❧✐❝❛çã♦ ❡①♣♦♥❡♥❝✐❛❧✳
Pr♦♣♦s✐çã♦ ✶✳✹✳ ❆s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s sã♦ s❛t✐s❢❡✐t❛s✿ ✶✳ ❝❛❞❛ ❝♦♥❥✉♥t♦ ✝♣ ✚ ❚♣▼ é ❡str❡❧❛❞♦ ❡♠ r❡❧❛çã♦ ❛ ♣❀
✷✳ ♣❛r❛ ❝❛❞❛ ✈ ✷ ❚♣▼✱ ❛ ❣❡♦❞és✐❝❛ ✌✈ é ❞❛❞❛ ♣♦r ✌✈✭t✮ ❂ ❡①♣♣✭t✈✮✱ ♣❛r❛
t♦❞♦ t ✷ ❘ ♦♥❞❡ ♦s ❞♦✐s ❧❛❞♦s ❡stã♦ ❞❡✜♥✐❞♦s❀ ✸✳ ❛ ❛♣❧✐❝❛çã♦ ❡①♣♣ é ❞✐❢❡r❡♥❝✐á✈❡❧✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✸❪✱ ♣á❣✐♥❛ ✼✶✳
➱ ♣♦ssí✈❡❧ ❛✉♠❡♥t❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ✉♠❛ ❣❡♦❞és✐❝❛ ❞✐♠✐♥✉✐♥❞♦ ♦ s❡✉ ✐♥t❡r✈❛❧♦ ❞❡ ❞❡✜♥✐çã♦ ❡ ✈✐❝❡✲✈❡rs❛✳ ■ss♦ s❡❣✉❡ ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♥❤❡❝✐❞❛✱ ❝❤❛♠❛❞❛ ❤♦♠♦❣❡♥❡✐❞❛❞❡ ❞❛s ❣❡♦❞és✐❝❛s✳ ❊♠ t❡r♠♦s ♠❛t❡♠á✲ t✐❝♦s✱ ❡①♣r❡ss❛♠♦s ✐ss♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
✌✕✈✭t✮ ❂ ✌✈✭✕t✮❀ ✽ ✈ ✷ ❚♣▼ ❡ ✽ ✕❀ t ✷ ❘✿
Pr♦♣♦s✐çã♦ ✶✳✺✳ P❛r❛ t♦❞♦ ♣ ✷ ▼✱ ❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❱ ❞❛ ♦r✐❣❡♠ ❞❡ ❚♣▼ ❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❯ ❞❡ ♣✱ t❛✐s q✉❡ ❡①♣♣ ✿ ❱ ✦ ❯ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✳
❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱
❞✭❡①♣♣✮✵✭✈✮ ❂ ❞t❞✭❡①♣♣✭t✈✮✮☞☞t❂✵
❂ ❞
❞t✭✌✈✭t✮✮☞☞t❂✵
❂ ✌✵ ✈✭✵✮
❂ ✈✿
▲♦❣♦✱ ❞✭❡①♣♣✮✵ é ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❚♣▼✱ s❡❣✉❡✲s❡ ❡♥tã♦ ❞♦ t❡♦r❡♠❛ ❞❛
❢✉♥çã♦ ✐♥✈❡rs❛ q✉❡ ❡①♣♣ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✵ ❡♠ ❚♣▼✳
❖ ❛❜❡rt♦ ❯ ❞❛❞♦ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r é ❝❤❛♠❛❞♦ ✈✐③✐♥❤❛♥ç❛ ♥♦r♠❛❧ ❞❡ ♣✳
✶✳✹ ❱❛r✐❡❞❛❞❡s ❈♦♠♣❧❡t❛s
◗✉❛♥❞♦ s❡ ❡stá ✐♥t❡r❡ss❛❞♦ ❡♠ ❡st✉❞❛r ♣r♦♣r✐❡❞❛❞❡s ❣❧♦❜❛✐s ❞❡ ✉♠❛ ✈❛✲ r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ✜❝❛ ❝♦♥✈❡♥✐❡♥t❡ ❝♦♥s✐❞❡r❛r♠♦s ✈❛r✐❡❞❛❞❡s ❝♦♠♣❧❡t❛s✳
❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❛♠❛♥♥✐❛♥❛ ▼ é ✭❣❡♦❞❡s✐❝❛♠❡♥t❡✮ ❝♦♠♣❧❡t❛ s❡✱ ♣❛r❛ t♦❞♦ ♣ ✷ ▼✱ ❛ ❛♣❧✐❝❛çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡①♣♣ ❡stá ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ ✈ ✷ ❚♣▼✱ ✐st♦ é✱ ❛s ❣❡♦❞és✐❝❛s ✌✭t✮ q✉❡ ♣❛rt❡♠ ❞❡ ♣ ❡stã♦ ❞❡✜♥✐❞❛s
♣❛r❛ t♦❞♦s ♦s ✈❛❧♦r❡s ❞♦ ♣❛râ♠❡tr♦ t ✷ ❘✳
P♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❞✐stâ♥❝✐❛ ❞✭♣❀ q✮ ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ❛ q✉❛❧ ❡st❛r❡♠♦s s❡♠♣r❡ s✉♣♦♥❞♦ ❝♦♥❡①❛✱ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❞✭♣❀ q✮ ❂ í♥✜♠♦ ❞♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡ t♦❞❛s ❛s ❝✉r✈❛s ☛♣❀q✱ ♦♥❞❡ ☛♣❀qé ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧
♣♦r ♣❛rt❡s ❧✐❣❛♥❞♦ ♣ ❛ q✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ ✭▼❀ ❞✮ é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❚❡♦r❡♠❛ ✶✳✸ ✭❍♦♣❢✲❘✐♥♦✇✮✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ é ❝♦♠♣❧❡t❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ é ❝♦♠♣❧❡t❛ ❝♦♠♦ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✽❪✱ ♣á❣✐♥❛ ✶✵✽✳
❖ ❚❡♦r❡♠❛ ✭✶✳✸✮ ♣♦ss✉✐ ✈ár✐❛s ❝♦♥s❡qüê♥❝✐❛s✱ ♣♦r ❡①❡♠♣❧♦✿
✭✐✮ s❡ ▼ é ❝♦♠♣❧❡t❛✱ ❡♥tã♦ ❞♦✐s ♣♦♥t♦s q✉❛✐sq✉❡r ❞❡ ▼ ♣♦❞❡♠ s❡r ❧✐❣❛❞♦s ♣♦r ✉♠ s❡❣♠❡♥t♦ ❞❡ ❣❡♦❞és✐❝❛ ♠✐♥✐♠✐③❛♥t❡❀
✭✐✐✮ s❡ ▼ é ❝♦♠♣❛❝t❛✱ ❡♥tã♦ ▼ é ❝♦♠♣❧❡t❛✳
✶✳✺ ❈✉r✈❛t✉r❛s
◆❡st❛ s❡çã♦✱ ✐♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝✉r✈❛t✉r❛ ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ❝✉❧♠✐♥❛♥❞♦ ❝♦♠ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ❡ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r✳ ▼❛✐s ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ❝✉r✈❛t✉r❛s ♣♦❞❡♠ s❡r ✈✐st❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✽❪✱ ❝❛♣ít✉❧♦ ✼✳
❆ ❝✉r✈❛t✉r❛ ❘ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ é ❛ ❛♣❧✐❝❛çã♦ q✉❡ ❛ ❝❛❞❛ ♣❛r ❳❀ ❨ ✷ ❳ ✭▼✮ ❛ss♦❝✐❛ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛
❘✭❳❀ ❨ ✮ ✿ ❳ ✭▼✮ ✦ ❳ ✭▼✮ ❞❛❞❛ ♣♦r
❘✭❳❀ ❨ ✮❩ ❂ r❳r❨❩ r❨r❳❩ r❬❳❀❨ ❪❩✿
P❛r❛ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❱ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❤❀ ✐✱ ❞❛❞♦s ❞♦✐s ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ①❀ ② ✷ ❱ ✱
❥① ❫ ②❥ ❂q❥①❥✷❥②❥✷ ❤①❀ ②✐✷
é ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❣❡r❛❞♦ ♣♦r ❢①❀ ②❣✳
P♦❞❡✲s❡ ✈❡r✐✜❝❛r q✉❡✱ s❡ ✛ ✚ ❚♣▼ é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ ✷ ❝♦♠
❜❛s❡ ❢①❀ ②❣✱ ❡♥tã♦
❑✭①❀ ②✮ ❂ ❤❘✭①❀ ②✮②❀ ①✐ ❥① ❫ ②❥✷
♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❜❛s❡ ❞❡ ✛✳
❋✐❝❛ ❡♥tã♦ ❜❡♠ ❞❡✜♥✐❞❛ ❛ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❞❡ ✛ ❡♠ ▼ ♥♦ ♣♦♥t♦ ♣ ❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ r❡❛❧ ❞❛❞♦ ♣♦r ❑✭✛✮ ✿❂ ❑✭①❀ ②✮✳
❙❡❥❛ ① ❂ ③♥ ✉♠ ✈❡t♦r ✉♥✐tár✐♦ ❡♠ ❚♣▼✱ t♦♠❡♠♦s ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧
❇ ❂ ❢③✶❀ ✿ ✿ ✿ ❀ ③♥ ✶❣ ❞♦ ❤✐♣❡r♣❧❛♥♦ ❞❡ ❚♣▼ ♦rt♦❣♦♥❛❧ ❛ ① ❡ ❝♦♥s✐❞❡r❡♠♦s ❛s
s❡❣✉✐♥t❡s ♠é❞✐❛s✿
❉❡✜♥✐çã♦ ✶✳✺✳ ❈✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ❘✐❝♣✭①✮ ❂ ♥ ✶✶
❳
✐
❤❘✭①❀ ③✐✮③✐❀ ①✐ ❀ ✐ ❂ ✶❀ ✿ ✿ ✿ ❀ ♥ ✶✿
❉❡✜♥✐çã♦ ✶✳✻✳ ❈✉r✈❛t✉r❛ ❊s❝❛❧❛r ❙✭♣✮ ❂ ✶
♥ ❳
❥
❘✐❝♣✭③❥✮✿
❊♠ ❬✸❪✱ ♣á❣✐♥❛ ✶✵✽✱ é ❞❡♠♦♥str❛❞♦ q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ❡ ❛ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ♥ã♦ ❞❡♣❡♥❞❡♠ ❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❜❛s❡s ♦rt♦♥♦r♠❛✐s✱ ♣♦rt❛♥t♦ ❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s✳
❆✐♥❞❛ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❇✱ ♦❜s❡r✈❛♠♦s q✉❡ ❘✐❝♣✭①✮ ❂ ♥ ✶✶
♥ ✶
❳
✐❂✶
❑✭①❀ ③✐✮✿
♥ã♦✲♥❡❣❛t✐✈❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s✱ ❡♥tã♦ ♦ ♠❡s♠♦ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐✱ ✐st♦ é✱ ❘✐❝✭▼✮ ✕ ✵✳
❖❜s❡r✈❛çã♦ ✶✳✶✳ ❆❧❣✉♠❛s ✈❡③❡s ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦ ❞❡ ❊✐♥st❡✐♥✱ ♦✉ s❡❥❛✱ ♦♠✐t✐r❡♠♦s ♦ s✐♥❛❧ ❞❡ s♦♠❛tór✐♦ ❡♠ s♦♠❛s q✉❡ ❛♣❛r❡❝❡♠ í♥❞✐❝❡s r❡♣❡t✐❞♦s✱ ♣♦r ❡①❡♠♣❧♦
❳
❦ ❦
✐❥❳❦ ❂ ❦✐❥❳❦✿
❙❡❥❛ ❢❡✶❀ ✿ ✿ ✿ ❀ ❡♥❣ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❡♠ t♦r♥♦ ❞❡
✉♠ ♣♦♥t♦ ♣ ✷ ▼✳ ❖s ❝♦❡✜❝✐❡♥t❡s ❘❧
✐❥❦✱ ❞❡✜♥✐❞♦s ♣♦r
❘✭❡✐❀ ❡❥✮❡❦ ❂
❳
❧
❘❧ ✐❥❦❡❧
❂ ❘❧
✐❥❦❡❧❀ ✭✶✳✶✮
sã♦ ❞❡♥♦♠✐♥❛❞♦s ❝♦♠♣♦♥❡♥t❡s ❞♦ t❡♥s♦r ❝✉r✈❛t✉r❛✳ ❚❡♠♦s ❛✐♥❞❛ q✉❡
❤❘✭❡✐❀ ❡❥✮❡❦❀ ❡s✐ ❂
✯ ❳
❧
❘❧ ✐❥❦❡❧❀ ❡s
✰ ❂ ❳
❧
❘❧ ✐❥❦✍❧s
❂ ❘s ✐❥❦
✿❂ ❘✐❥❦s✿
❖s ❝♦❡✜❝✐❡♥t❡s ❘✐❥❦s sã♦ ❞❡♥♦♠✐♥❛❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❘✐❝❝✐✳
❖✉tr❛ ♥♦t❛çã♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s é ❛ s❡❣✉✐♥t❡✿ ❘✐❝✭✈❀ ✇✮ ❂ ❳
✐
❤❘✭✈❀ ❡✐✮❡✐❀ ✇✮✐ ❂ ❤❘✭✈❀ ❡✐✮❡✐❀ ✇✮✐ ✿
❖s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❘✐❝❝✐ tê♠ ✉♠❛ ❡①♣r❡ssã♦ ❡♠ t❡r♠♦s ❞♦s sí♠❜♦❧♦s ❞❡
❈❤r✐st♦✛❡❧ ❞❛❞❛ ♣♦r ❘s
✐❥❦ ❂
❳
❧ ❧ ❥❦ s✐❧
❳
❧ ❧
✐❦ s❥❧✰❅❡❅ ✐
s ❥❦ ❅❡❅
❥ s
✐❦❀ ✭✶✳✷✮
❛ q✉❛❧✱ ♥❛ ♥♦t❛çã♦ ❞❡ ❊✐♥st❡✐♥✱ s❡ ❡①♣r❡ss❛ ❝♦♠♦ ❘s
✐❥❦❂ ❧❥❦ s✐❧ ❧✐❦ s❥❧✰❅❡❅ ✐
s ❥❦ ❅❡❅
❥ s ✐❦✿
❆ ❡①♣r❡ssã♦ ✭✶✳✷✮ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✸❪✱ ♣á❣✐♥❛ ✶✵✸✳
❈❛♣ít✉❧♦ ✷
❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s
❖s r❡s✉❧t❛❞♦s ❞❡st❡ ❝❛♣ít✉❧♦ ❢♦r♠❛rã♦ ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ♣ró①✐♠♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳
✷✳✶ ❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦
❆♣r❡s❡♥t❛r❡♠♦s ❛ s❡❣✉✐r ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛✱ ❛ s❛❜❡r✿ ❛ ❢ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✳ ❆♥t❡s✱ ♣♦ré♠✱ ♠♦str❛r❡♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❘✐❝❝✐✱ ❛ ✜♠ ❞❡ ❢❛③❡r♠♦s ✉s♦ ❛❞✐❛♥t❡✳
▲❡♠❛ ✷✳✶ ✭■❞❡♥t✐❞❛❞❡ ❞❡ ❘✐❝❝✐✮✳ ❉❛❞❛ ❢ ✷ ❈✸✭▼✮✱ ♣❛r❛ q✉❛✐sq✉❡r ✶ ✔
✐❀ ❥❀ ❦ ✔ ♥✱ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡✿
❢✐❦❧❂ ❢✐❧❦✰ ❘❦❧s✐❢s❀
♦♥❞❡ ❘❦❧s✐ sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❘✐❝❝✐✳
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡
❢✐❦ ❂ ❅✭ ❅❢ ❅❡✐✮
❅❡❦
❅❢ ❅❡✉
✉ ❦✐
❂ ❅✷❢ ❅❡✐❅❡❦
❅❢ ❅❡✉
✉ ❦✐✿
❉❛í✱ ❢✐❦❧ ❂ ❅❢❅❡✐❦ ❧ ✉ ❧✐❢✉❦ ✉❧❦❢✉✐❀ ❡ ❢✐❧❦❂ ❅❢❅❡✐❧ ❦ ✉ ❦✐❢✉❧ ✉❦❧❢✉✐✿ ❈♦♠♦ ✉
❧❦ ❂ ✉❦❧✱ s❡❣✉❡✲s❡ q✉❡
❢✐❦❧ ❢✐❧❦ ❂ ❅❢❅❡✐❦ ❧ ✉ ❧✐❢✉❦ ❅❢❅❡✐❧ ❦ ✰ ✉ ❦✐❢✉❧✿ ✭✷✳✶✮ ▼❛s ❅❢✐❦ ❅❡❧ ❂ ❅ ❅❡❧ ✒ ❅✷❢ ❅❡✐❡❦ ✉ ✐❦❅❡❅❢ ✉ ✓ ❂ ❅❡❅✸❢ ❧❅❡✐❅❡❦ ❅ ❅❡❧✭ ✉ ✐❦✮❅❡❅❢ ✉ ✉ ✐❦ ❅ ✷❢ ❅❡❧❅❡✉ ❡ ❅❢✐❧ ❅❡❦ ❂ ❅✸❢ ❅❡❦❅❡✐❅❡❧ ❅ ❅❡❦✭ ✉ ✐❧✮❅❡❅❢ ✉ ✉ ✐❧ ❅ ✷❢ ❅❡❦❅❡✉✿ ❆ss✐♠✱ ❅❢✐❦ ❅❡❧ ❅❢✐❧ ❅❡❦ ❂ ❅ ❅❡❧✭ ✉ ✐❦✮❅❡❅❢ ✉ ✉ ✐❦ ❅ ✷❢ ❅❡❧❅❡✉ ✰ ❅ ❅❡❦✭ ✉ ✐❧✮❅❡❅❢ ✉ ✰ ✉ ✐❧ ❅ ✷❢ ❅❡❦❅❡✉✿
❯s❛♥❞♦ ♠❛✐s ✉♠❛ ✈❡③ ❛ s✐♠❡tr✐❛ ❞♦s sí♠❜♦❧♦s ❞❡ ❈❤r✐st♦✛❡❧✱ ✈❡♠♦s q✉❡ ❢✐❦❧ ❢✐❧❦ ❂
✒
✉
❧✐ s✉❦ ✉❦✐ s✉❧✰ ❅❡❅ ❦✭
s
✐❧✮ ❅❡❅ ❧✭
s ✐❦✮
✓ ❅❢ ❅❡s
❂ ❘❦❧s✐❢s✿
Pr♦♣♦s✐çã♦ ✷✳✶ ✭❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✮✳ ❙❡❥❛ ❢ ✷ ❈✸ ✉♠❛ ❢✉♥çã♦
❞❡✜♥✐❞❛ ❡♠ ▼✳ ❊♥tã♦ ✶
✷✁ ❥ ❣r❛❞ ❢❥✷ ✁
❂ ❥ ❍❡ss ❢❥✷ ✰ ❤❣r❛❞ ❢❀ ❣r❛❞✭✁❢✮✐ ✰ ❘✐❝ ✭❣r❛❞ ❢❀ ❣r❛❞ ❢✮ ✿
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ❢❊✶❀ ✿✿✿❀ ❊♥❣ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦
♣ ✷ ▼✳ ❆✜r♠❛♠♦s q✉❡ ❥ ❣r❛❞ ❢❥✷ ❂ P♥
✐❂✶❢✐✷✳ ❉❡ ❢❛t♦✱ ❡♠ ♣ ✷ ▼✱
♦ ❝❛♠♣♦ ❣r❛❞ ❢ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❜❛s❡ ❢❊✶❀ ✿✿✿❀ ❊♥❣ ❝♦♠♦ ❣r❛❞ ❢✭♣✮ ❂
P♥
✐❂✶☛✐✭♣✮❊✐✭♣✮✱ ♦♥❞❡ ❝❛❞❛ ☛✐ ✿ ❯ ✦ ❘ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠
❯✳
❱✐st♦ q✉❡
❢✐ ❂ ❊✐✭❢✮
❂ ❤❊✐❀ ❣r❛❞ ❢✐
❂ ☛✐❀
t❡♠♦s q✉❡ ❣r❛❞ ❢ ❂P♥✐❂✶❢✐❊✐ ❡✱ ♣♦rt❛♥t♦✱
❥ ❣r❛❞ ❢❥✷ ❂ ❤❣r❛❞ ❢❀ ❣r❛❞ ❢✐
❂ ✯ ♥
❳
✐❂✶
❢✐❊✐❀ ♥
❳
✐❂✶
❢✐❊✐
✰
❂
♥
❳
✐❂✶
❢✷ ✐✿
■st♦ ♣r♦✈❛ ❛ ❛✜r♠❛çã♦✳
❆❣♦r❛✱
✭❥ ❣r❛❞ ❢❥✷✮
❥ ❂ ✭
❳
✐
❢✷ ✐✮❥
❂ ❳
✐
✷❢✐❢✐❥❀
♦✉ s❡❥❛✱
✶
✷✭❥ ❣r❛❞ ❢❥✷✮❥ ❂ ❳
✐
❢✐❢✐❥✿
❚❡♠♦s t❛♠❜é♠ q✉❡ ✶
✷✭❥ ❣r❛❞ ❢❥✷✮❥❥ ❂ ❳
✐
✭❢✷
✐❥ ✰ ❢✐❢✐❥❥✮❀
❞♦♥❞❡
✶
✷✁ ❥ ❣r❛❞ ❢❥✷ ✁
❂ ❳
❥
✥ ❳
✐
✭❢✷
✐❥ ✰ ❢✐❢✐❥❥✮
✦ ❂ ❳
✐❀❥
✭❢✷
✐❥ ✰ ❢✐❢✐❥❥✮✿ ✭✷✳✷✮
❆❣♦r❛✱ ❝♦♠♦ ❛ ❍❡ss✐❛♥❛ ❞❡ ✉♠❛ ❢✉♥çã♦ é ✉♠ ✷✲t❡♥s♦r s✐♠étr✐❝♦✱ t❡♠♦s q✉❡ ❢✐❥❥ ❂ ❢❥✐❥✳ ❯s❛♥❞♦ ❡st❡ ❢❛t♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❘✐❝❝✐✱
♦❜t❡♠♦s q✉❡
❢✐❥❥ ❂ ❢❥✐❥ ❂ ❢❥❥✐✰ ❘✐❥s❥❢s✿
❆♣❧✐❝❛♥❞♦ ❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❛ ✭✷✳✷✮✱ ✈❡♠♦s q✉❡ ✶
✷✁ ❥ ❣r❛❞ ❢❥✷ ✁
❂ ❳
✐❀❥
✭❢✷
✐❥ ✰ ❢✐✭❢❥❥✐✰ ❘✐❥s❥❢s✮✮✿
❋✐♥❛❧♠❡♥t❡✱ ♦❜t❡♠♦s q✉❡ ✶
✷✁ ❥ ❣r❛❞ ❢❥✷ ✁
❂ ❳
✐❀❥
❢✷ ✐❥✰
❳
✐❀❥
❢✐❢❥❥✐✰
❳
✐❀❥
❘✐❥s❥❢s❢✐
❂ ❥ ❍❡ss ❢❥✷✰ ❤❣r❛❞ ❢❀ ❣r❛❞✭✁❢✮✐ ✰❳ ✐❀s
❘✐❝✭❊s❀ ❊✐✮❢s❢✐
❂ ❥ ❍❡ss ❢❥✷✰ ❤❣r❛❞ ❢❀ ❣r❛❞✭✁❢✮✐ ✰ ❘✐❝✭❳ s
❢s❊s❀
❳
✐
❢✐❊✐✮
❂ ❥ ❍❡ss ❢❥✷✰ ❤❣r❛❞ ❢❀ ❣r❛❞✭✁❢✮✐ ✰ ❘✐❝✭❣r❛❞ ❢❀ ❣r❛❞ ❢✮✿
✷✳✷ ❆ ❍❡ss✐❛♥❛ ❞❛ ❋✉♥çã♦ ❉✐stâ♥❝✐❛
◆❡st❛ s❡çã♦✱ ♦❜t❡r❡♠♦s ❞♦✐s r❡s✉❧t❛❞♦s s♦❜r❡ ❛ ❍❡ss✐❛♥❛ ❞❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛✳ ❚❛✐s r❡s✉❧t❛❞♦s s❡rã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ♥❛ ♣ró①✐♠❛ s❡çã♦✳
Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡❥❛ r ✿ ▼ ✦ ❘ ❞❡✜♥✐❞❛ ♣♦r r✭①✮ ❂ ❞✭①❀ ♣✮✱ ♦♥❞❡ ❞ é ❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛✳ ❊♥tã♦ ❍❡ss r✭❅
❅r❀❅r❅✮ ❂ ✵✳ ❆❧é♠ ❞✐ss♦✱ s❡ ✽ ❳ ❄ ❅r❅✱ ✈❛❧❡
❍❡ss r✭❅
❅r❀ ❳✮ ❂ ✵✱ ♦♥❞❡ ❅r❅ é ♦ ✈❡t♦r ✈❡❧♦❝✐❞❛❞❡✳
❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❍❡ss✐❛♥❛ ♣❛r❛ ❛ ❢✉♥çã♦ r ❡ ♦ ❢❛t♦ q✉❡ ❛s ❝✉r✈❛s r ✼ ✦ r✇✱ ♣❛r❛ ✇ ✜①❛❞♦✱ sã♦ ❣❡♦❞és✐❝❛s ❝♦♠ ✈❡t♦r ✈❡❧♦❝✐❞❛❞❡ ❅
❅r✱
t❡♠♦s q✉❡
❍❡ss r ✒
❅ ❅r❀
❅ ❅r
✓
❂ ❅r❅ ❅r❅ r r❅ ❅r
❅ ❅rr ❂ ❅r❅ ✭✶✮
❂ ✵✿
❆❣♦r❛✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ●❛✉ss✱ ✈❡r ❬✽❪✱ ♣á❣✐♥❛ ✶✵✷✱ t❡♠♦s q✉❡ ❣r❛❞ r ❂ ❅r❅ ✿
P♦rt❛♥t♦✱ s❡ ❳ ❄ ❅
❅r✱ ♦❜t❡♠♦s
❍❡ss r✭❅r❅ ❀ ❳✮ ❂ ❍❡ss r✭❳❀❅r❅ ✮ ❂ ❳ ❅
❅rr ✭r❳ ❅ ❅r✮r ❂
✜
r❳❅r❅ ❀❅r❅
✢ ❀ ♣♦✐s ❳✭ ❅
❅r✮r ❂ ❳✭✶✮ ❂ ✵✳
❯s❛♥❞♦ ❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ ♠étr✐❝❛✱ ♦❜t❡♠♦s ❍❡ss r✭❅r❅ ❀ ❳✮ ❂ ✶✷❳
✜ ❅ ❅r❀
❅ ❅r
✢ ✿
❋✐♥❛❧♠❡♥t❡✱ ❝♦♠♦ ❛s ❣❡♦❞és✐❝❛s r❛❞✐❛✐s sã♦ ♣❛r❛♠❡tr✐③❛❞❛s ♣♦r ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱
❍❡ss r✭❅r❅ ❀ ❳✮ ❂ ✶✷❳✭✶✮ ❂ ✵✿
Pr♦♣♦s✐çã♦ ✷✳✸✳ ❈♦♥s✐❞❡r❡♠♦s ❘♥ ♠✉♥✐❞♦ ❝♦♠ ❛ ♠étr✐❝❛ ❣✱ ❡s❝r✐t❛ ❡♠
❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s ❝♦♠♦ ❞r✷ ✰ ❢✷✭r✮❞✇✷✱ ♦♥❞❡ ❞✇✷ r❡♣r❡s❡♥t❛ ❛ ♠étr✐❝❛
❝❛♥ô♥✐❝❛ ❡♠ ❙♥ ✶✳ ❙❡❥❛ r✭①✮ ❂ ❞✭①❀ ♣✮✱ ♦♥❞❡ ❞ é ❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛✳ ❊♥tã♦✱
s❡ ① ❂ r✇✱ r ❃ ✵✱ ✇ ✷ ❙♥ ✶✱ ♣❛r❛ q✉❛✐sq✉❡r ❳❀ ❨ ♦rt♦❣♦♥❛✐s ❛ ❅ ❅r✱
❍❡ss r✭❳❀ ❨ ✮ ❂ ❢❢✭r✮✵✭r✮❣✭❳❀ ❨ ✮✿ ❆❧é♠ ❞✐ss♦✱
✁r✭①✮ ❂ ✭♥ ✶✮❢❢✭r✮✵✭r✮✿
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ ❳❀ ❨ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ♥í✈❡❧
r ❂ ❝✱ ♦♥❞❡ ❝ é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛✳ ❚❡♠♦s q✉❡ ❍❡ss r✭❳❀ ❨ ✮ ❂ ❨ ✭❳✭r✮✮ r❨❳✭r✮
❂ ❨ ✜
❳❀❅r❅
✢ ✜
r❨❳❀❅r❅
✢ ❂
✜
r❨❳❀❅r❅
✢ ✰
✜
❳❀ r❨❅r❅
✢ ✜
r❨❳❀❅r❅
✢ ❂
✜
❳❀ r❨ ❅r❅
✢ ❂
✜
❨❀ r❳❅r❅
✢ ✿
◆❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s ❛ s✐♠❡tr✐❛ ❞❛ ❍❡ss✐❛♥❛✳ ❆❣♦r❛✱ s❡❥❛♠ ❅
❅①♥ ❂
❅
❅r ♦ ❝❛♠♣♦ ♥♦r♠❛❧ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ r ❂ ❝ ❡ ❅①❅✶❀ ✿ ✿ ✿ ❀
❅
❅①♥ ✶✱ ♦s ❝❛♠♣♦s ❝♦♦r❞❡♥❛❞♦s ❡♠ ❙
♥ ✶✳
❆ss✐♠✱ ♣❛r❛ ✶ ✔ ✐❀ ❥ ✔ ♥ ✶✱ t❡♠♦s q✉❡ ❍❡ss r
✒ ❅ ❅①✐❀
❅ ❅①❥ ✓ ❂ ✜ ❅
❅①❥❀ r❅①✐❅
❅ ❅①♥ ✢ ❂ ✜ ❦ ♥✐❅①❅
❦❀ ❅ ❅①❥ ✢ ❂ ❦ ♥✐❣❦❥
❯s❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞♦s sí♠❜♦❧♦s ❞❡ ❈❤r✐st♦✛❡❧ ❡♠ t❡r♠♦s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ♠étr✐❝❛ ❡ ✉s❛♥❞♦ ♦ ❢❛t♦ q✉❡ ❣♥✐ ❂
❉
❅ ❅r❀❅①❅✐
❊ ❂ ✵❀ ✽ ✐ ❂ ✶❀ ✿ ✿ ✿ ❀ ♥ ✶✱ ❡♥❝♦♥tr❛♠♦s ❦ ♥✐ ❂ ✶✷ ❳ ❧ ❣❦❧✒❅❣❧♥
❅①✐ ✰
❅❣❧✐ ❅①♥ ❅❣♥✐ ❅①❧ ✓ ❂ ✶✷❳ ❧ ❣❦❧❅❣❧✐ ❅r ✿ ▼❛s✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱
❅❣❧✐
❉❛í✱
❦
♥✐ ❂ ✶✷
❳
❧
❢ ✷✭❞✇✷✮❦❧✷❢❢✵✭❞✇✷✮ ❧✐
❂ ❢✵ ❢ ✍✐❦✿ P♦rt❛♥t♦✱
❍❡ss r ✒
❅ ❅①✐❀
❅ ❅①❥
✓
❂ ❢✵ ❢✍✐❦❣❦❥ ❂ ❢❢✵❣
✒ ❅ ❅①✐❀
❅ ❅①❥
✓ ✿ ▲♦❣♦✱ s❡ ❳ ❡ ❨ sã♦ ♦rt♦❣♦♥❛✐s ❛ ❅
❅r✱ ❡♥tã♦
❍❡ss r✭❳❀ ❨ ✮ ❂ ❢✵
❢ ❣✭❳❀ ❨ ✮✿
❆❣♦r❛✱ s❛❜❡♠♦s q✉❡ ❡①✐st❡ ❆ ✿ ❚ ❙ ✦ ❚ ❙ t❛❧ q✉❡ tr ❍❡ss r ❂ tr❆ ❡ ❍❡ss r✭❳❀ ❨ ✮ ❂ ❤❆❳❀ ❨ ✐✱ ✽ ❳❀ ❨ ✷ ❚ ❙✳ ❖r❛✱ ❆✭❳✐✮ ❂ ❛✐❥❳❥✳ ■st♦ ✐♠♣❧✐❝❛
q✉❡ ❤❆✭❳✐✮❀ ❳❦✐ ❂ ❛✐❥❣❥❦✱ ♦✉ s❡❥❛✱ ❛✐❥ ❂ ❣❥❦❍❡ss r✭❳✐❀ ❳❦✮✳
❆ss✐♠✱
✁r ❂ tr ❍❡ss r ❂ tr❆ ❂ ❳
✐
❣✐❦❢✵
❢ ❣✭❳✐❀ ❳❦✮ ❂ ✭♥ ✶✮❢❢✵✿
❊①❡♠♣❧♦ ✷✳✶✳ ❈❛❧❝✉❧❛r❡♠♦s ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ r❦✱ ❝♦♠ r❡s✲
♣❡✐t♦ ❛ ♠étr✐❝❛
❞r✷✰ ❢✷ ❦❞✇✷❀
♦♥❞❡
❢❦✭r✮ ❂
✽ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ✿ ✶ ♣ ❦s✐♥✭ ♣
❦r✮❀ s❡ ❦ ❃ ✵ r❀ s❡ ❦ ❂ ✵
✶ ♣
❦s✐♥❤✭ ♣
❦r✮❀ s❡ ❦ ❁ ✵✿
❖❜s❡r✈❛♠♦s q✉❡✱ q✉❛♥❞♦ ❦ ❃ ✵✱ ♦❜t❡♠♦s ❛ ♠étr✐❝❛ ❞❛ ❡s❢❡r❛ ❙♥ ❞❡ r❛✐♦ ✶ ❦✷
q✉❡ t❡♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ ❦✳ ◗✉❛♥❞♦ ❦ ❁ ✵✱ ♦❜t❡♠♦s ❛ ♠étr✐❝❛ ❞♦ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦ ❍♥✱ t❛♠❜é♠ ❝♦♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛
❦✳ ◗✉❛♥❞♦ ❦ ❂ ✵✱ ♦❜❡t♠♦s ❛ ♠étr✐❝❛ ❝❛♥ô♥✐❝❛ ❞♦ ❘♥ ❡♠ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✳
➱ ❢á❝✐❧ ✈❡r q✉❡
❢✵ ❦✭r✮ ❂
✽ ❃ ❁ ❃ ✿
❝♦s✭♣❦r✮❀ s❡ ❦ ❃ ✵ ✶❀ s❡ ❦ ❂ ✵ ❝♦s❤✭♣ ❦r✮❀ s❡ ❦ ❁ ✵✿ ❆ss✐♠✱ ♣❛r❛ ❦ ❃ ✵✱ t❡♠♦s
❍❡ss r❦ ❂
♣
❦ ❝♦s✭♣❦r✮ s✐♥✭♣❦r✮ ❣ ❂ ♣❦ ❝♦t✭♣❦r✮❣✿ ◗✉❛♥❞♦ ❦ ❂ ✵✱
❍❡ss r ❂ ✶ r❣✿ ❊
❍❡ss r❦ ❂
♣
❦ ❝♦s❤✭♣❦r✮ s✐♥❤✭♣❦r✮ ❣ ❂ ♣❦ ❝♦t❤✭♣❦r✮❣❀ s❡ ❦ ❁ ✵✳
P♦rt❛♥t♦ ✶
♥ ✶✁r❦❂ ✽ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ✿
♣
❦ ❝♦t✭♣❦r✮❀ s❡ ❦ ❃ ✵
✶
r❀ s❡ ❦ ❂ ✵
♣
❦ ❝♦t❤✭♣ ❦r✮❀ s❡ ❦ ❁ ✵✿
✷✳✸ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦
P❛rt✐♠♦s ❛❣♦r❛ ♣❛r❛ ♦ t❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦✱ ♦ q✉❛❧ s❡rá ❞❡♠♦♥str❛❞♦ ❝♦♠ ♦ ✉s♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✳
❚❡♦r❡♠❛ ✷✳✶✳ ❙❡❥❛ ✭▼♥❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛✳ ❙✉♣♦✲
♥❤❛♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ❞❡ ▼ s❛t✐s❢❛③ ❘✐❝✭▼✮ ✕ ✭♥ ✶✮❦❣✳ ❙❡❥❛ r ❛ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ ❛♦ ♣♦♥t♦ ♣ ✷ ▼✳ ❙✉♣♦♥❤❛♠♦s ❛✐♥❞❛ q✉❡ ❛ ❢✉♥çã♦ r é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ ①✳ ❊♥tã♦
✁r✭①✮ ✔ ✁r❦✭⑦①✮❀ ✭✷✳✸✮
♦♥❞❡ r❦✭⑦①✮ ❂ r✭①✮ ❂ r✵ ❡ r✵ ❁ ♣✙❦ s❡ ❦ ❃ ✵✳ ❙❡ ❛ ✐❣✉❛❧❞❛❞❡ é s❛t✐s❢❡✐t❛✱
❡♥tã♦ ❛ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❞❡ q✉❛❧q✉❡r ♣❧❛♥♦ q✉❡ ❝♦♥t❡♥❤❛ ♦ ✈❡t♦r r❛❞✐❛❧✱ ❛♦ ❧♦♥❣♦ ❞❡ ❣❡♦❞és✐❝❛s ❧✐❣❛♥❞♦ ♣ ❡ ①✱ é ❝♦♥st❛♥t❡ ❡ ✐❣✉❛❧ ❛ ❦✳
❉❡♠♦♥str❛çã♦✳ ❆ ❍❡ss✐❛♥❛ ❞❛ ❢✉♥çã♦ r ♣♦ss✉✐ ✉♠ ❛✉t♦✈❛❧♦r ✐❣✉❛❧ ❛ ③❡r♦✱ ♣♦✐s ❍❡ss r ❅
❅r❀❅r❅
✁
❂ ✵ ❡ ❍❡ss r✭❅
❅r❀ ❳✮ ❂ ✵❀ ✽ ❳❄❅r❅✳ ❊ss❡ ❢❛t♦ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡
❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❛♣❧✐❝❛❞❛ às ♠❛tr✐③❡s ❍❡ss✐❛♥❛ ❞❡ r ❡ ■❞❡♥t✐❞❛❞❡ ♥♦s ❞ã♦ ❥ ❍❡ss r❥✷ ✕ ✭✁r✮✷
♥ ✶✿ ✭✷✳✹✮
❯s❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦ ❡ ♦ ❢❛t♦ ❞❡ q✉❡ ❥ ❣r❛❞ r❥ ❂ ✶✱ ♦❜t❡♠♦s ✵ ❂ ❥ ❍❡ss r❥✷✰ ❤❣r❛❞ r❀ ❣r❛❞✭✁r✮✐ ✰ ❘✐❝
✒ ❅ ❅r❀ ❅ ❅r ✓ ❂ ❥ ❍❡ss r❥✷✰
✜ ❅
❅r❀ ❣r❛❞✭✁r✮ ✢ ✰ ❘✐❝ ✒ ❅ ❅r❀ ❅ ❅r ✓ ❂ ❥ ❍❡ss r❥✷✰ ❅
❅r✁r ✰ ❘✐❝ ✒ ❅ ❅r❀ ❅ ❅r ✓ ❂ ❥ ❍❡ss r❥✷✰ ✭✁r✮✵✰ ❘✐❝
✒ ❅ ❅r❀ ❅ ❅r ✓ ✿
❙❡❥❛ ✬ ❂ ✁r✳ ❯s❛♥❞♦ ❛ ❤✐♣ót❡s❡ s♦❜r❡ ❛ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✹✮✱ ✈❡♠♦s q✉❡
✬✵ ✰ ✬✷
♥ ✶ ✰ ✭♥ ✶✮❦ ✔ ✵✿
❊st✉❞❛♥❞♦ ♦ ❝❛s♦ ❞❛ ✐❣✉❛❧❞❛❞❡✱ ♦❜s❡r✈❛♠♦s q✉❡ ✥ ❂ ✁r❦ ❂ ✭♥ ✶✮❢
✵ ❦
❢❦ s❛✲
t✐s❢❛③ ❛ ✐❣✉❛❧❞❛❞❡✱ ♦♥❞❡ r❦é ❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ s♦❜r❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❝✉r✈❛t✉r❛
s❡❝❝✐♦♥❛❧ ❝♦♥s✲t❛♥t❡ ❦ ❡ ❢❦✭r✮ ❂
✽ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ✿ ✶ ♣ ❦s✐♥✭ ♣
❦r✮❀ s❡ ❦ ❃ ✵ r❀ s❡ ❦ ❂ ✵
✶ ♣
❦s✐♥❤✭ ♣
❦r✮❀ s❡ ❦ ❁ ✵✿
❱✐st♦ q✉❡ ✥ é ✐♥❥❡t✐✈❛ ❡♠ s❡✉s r❡s♣❡❝t✐✈♦s ❞♦♠í♥✐♦s✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠❛ ❢✉♥çã♦ ✒✭t✮✱ ❞❡✜♥✐❞❛ ❡♠ ❬✵❀ r✵✮✱ s❛t✐s❢❛③❡♥❞♦
✒✭✵✮ ❂ ✵❀ ✥✭✒✭t✮✮ ❂ ✬✭t✮✿ ❆❣♦r❛✱ ❝♦♠♦
✬✵✭t✮ ✰ ✬✷✭t✮
♥ ✶ ✰ ✭♥ ✶✮❦ ✔ ✵ ❂ ✥✵✭✒✭t✮✮ ✰
✥✷✭✒✭t✮✮
♥ ✶ ✰ ✭♥ ✶✮❦❀
s❡❣✉❡✲s❡ ❞✐r❡t❛♠❡♥t❡ q✉❡
✬✵✭t✮ ✔ ✥✵✭✒✭t✮✮✿
▼❛s ✬✵✭t✮ ❂ ✥✵✭✒✭t✮✮✒✵✭t✮ ❡✱ ♣♦rt❛♥t♦✱
✥✵✭✒✭t✮✮✒✵✭t✮ ✔ ✥✵✭✒✭t✮✮✿
❈♦♠♦ ✥✵✭t✮ ❁ ✵✱ t❡♠♦s ✒✵✭t✮ ✕ ✶✳ ■♥t❡❣r❛♥❞♦ ❡st❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡
s❡❣✉❡✲s❡ q✉❡ ✒✭t✮ ✕ t✱ ❞♦♥❞❡
✬✭t✮ ❂ ✥✭✒✭t✮✮ ✔ ✥✭t✮❀
♣♦✐s ❛ ❢✉♥çã♦ ✥ é ❞❡❝r❡s❝❡♥t❡✳ ■st♦ é✱ ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✸✮✳
◗✉❛♥t♦ à ✐❣✉❛❧❞❛❞❡ ❡♠ ✭✷✳✸✮✱ s❛❜❡♠♦s q✉❡ ❛ ♠❡s♠❛ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❍❡ss✐❛♥❛ é ♠ú❧t✐♣❧❛ ❞❛ ✐❞❡♥t✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ❍❡ss r ❂ ✕■✱ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ ✕✳ ▼❛s ♦ ❢❛t♦ ❞❡ ❍❡ss r ❅
❅r❀❅r❅
✁
❂ ✵ ✐♠♣❧✐❝❛✱ ♥❡st❡ ❝❛s♦✱ q✉❡ ♦ t❡r♠♦ ❛♥♥ ❞❛ ♠❛tr✐③ ❍❡ss✐❛♥❛ ❞❡ r é ♥✉❧♦ ❡✱ ♣♦rt❛♥t♦✱ ❥ ❍❡ss r❥✷ ❂ ✭♥ ✶✮✕✷✳
P♦r ♦✉tr♦ ❧❛❞♦✱
❥ ❍❡ss r❥✷ ❂ ✥✷
♥ ✶✿
❉❡ss❛s ❞✉❛s ú❧t✐♠❛s ✐❣✉❛❧❞❛❞❡s✱ s❡❣✉❡✲s❡ q✉❡ ✕ ❂ ✥
♥ ✶✳ ❆❣♦r❛✱
❝♦♥s✐❞❡r❡♠♦s ❛ ❜❛s❡ ❇ ❂ ✟❡✶❀ ✿ ✿ ✿ ❀ ❡♥ ✶❀❅r❅✠ ❞❡ ✈❡t♦r❡s ♦rt♦♥♦r♠❛✐s q✉❡
❞✐❛❣♦♥❛❧✐③❛ ❛ ❍❡ss✐❛♥❛ ❞❡ r✳ ❆✜r♠❛♠♦s q✉❡
r❡❥
❅ ❅r ❂
✥ ♥ ✶❡❥✿ ❉❡ ❢❛t♦✱ ❡s❝r❡✈❡♥❞♦ r❡❥❅r❅ ♥❛ ❜❛s❡ ❇✱ t❡♠♦s
r❡❥
❅ ❅r ❂
♥ ✶
❳
❦❂✶
☛❦❡❦✰ ❛❅r❅ ✿
❉❛í✱✡r❡❥❅r❅❀❅r❅
☛
❂ ❛ ❂ ✵✱ ♣♦✐s ✵ ❂ ✶✷❡❥ ✜ ❅ ❅r❀ ❅ ❅r ✢ ❂ ✜ r❡❥
❅ ❅r❀ ❅ ❅r ✢ ❂ ❛✿
P❛r❛ ✐ ✻❂ ❥✱ t❡♠♦s ✡r❡❥❅r❅❀ ❡✐
☛
❂ ❍❡ss r✭❡❥❀ ❡✐✮ ❂ ✵✱ ♦✉ s❡❥❛✱ ☛✐ ❂ ✵ ♣❛r❛
t♦❞♦ ✐ ✻❂ ❥✳
❆ss✐♠✱ ♦❜t❡♠♦s q✉❡
r❡❥
❅
❅r ❂ ☛❥❡❥✿ ❈♦♠♦ ☛❥ ❂✡r❡❥❅r❅ ❀ ❡❥
☛
❂ ❍❡ss r✭❡❥❀ ❡❥✮ ❂ ✕✱ t❡♠♦s
r❡❥
❅ ❅r ❂
✥ ♥ ✶❡❥✿ ❊ ❛ ❛✜r♠❛çã♦ ❡stá ♣r♦✈❛❞❛✳
❚❡♠♦s ❡♥tã♦ ❑
✒ ❡❥❀❅r❅
✓ ☞☞ ☞☞ ☞ ① ❂ ❘ ✒
❡❥❀❅r❅ ❀❅r❅ ❀ ❡❥
✓ ❂
✜
r❡❥r❅r❅
❅
❅r r❅r❅ r❡❥
❅
❅r r❬❡❥❀❅r❅❪
❅ ❅r❀ ❡❥
✢ ❂ ✜ r❅ ❅r ✒ ✥ ♥ ✶ ✓ ❡❥❀ ❡❥ ✢ ❂ ✜ ✥✵ ♥ ✶❡❥ ✰ ✥
♥ ✶r❅r❅ ❡❥❀ ❡❥
✢ ❂ ♥ ✶✥✵ ♥ ✶✥ ❉r❅
❅r❡❥❀ ❡❥
