# Repositório UFAL: Teorema de Decomposição de Cheeger-Gromoll.

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❯♥✐✈❡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 ♥✉♥❝❛ é s✉❛✈❡✱ ♠❛s ❡❧❡ ♣♦❞❡ s❡r ❝♦♥tí♥✉♦✳✳✳

❆❣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 ✐

✐❂✶ ❅ ❅

♦♥❞❡ ❢ ✭♣✮❀ ✿ ✿ ✿ ❀ ✭♣✮❣ é ❛ ❜❛s❡ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❡♠ ♣ ✷ ▼ ❛ss♦❝✐❛❞❛

❅① ❅① ✶ ♥

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 ❥

❅① ✐ ✓

❥

✐

✷

✒ ❛

✐

❳

❥

❳

✓ ❂

❥

❅①

✐

❢ ❅①

✷

❅

❢ ❅① ✐ ❅① ❥

❥

✷

❅① ✐ ✿

✓ ❅❢

❜ ❥ ❅❛ ✐ ❅① ❥

❅❜ ❥ ❅① ✐

✒ ❛ ✐

✐

❳

❥

❳❨ ✭❢✮ ❨ ❳✭❢✮ ❂ ❳

✿ ❉❛í✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❙❝❤✇❛rt③✱ ♦❜t❡♠♦s

❢ ❅① ❥ ❅① ✐

❜ ❥ ❛ ✐ ❅

✿ ❆♥❛❧♦❣❛♠❡♥t❡✱ ✈❡♠♦s q✉❡

❥

❳

✐

❳

❅① ✐ ✰

✓ ❅❢

❅❛ ✐ ❅① ❥

✒ ❜ ❥

❥

❳

✐

❨ ❳✭❢✮ ❂ ❳

❅

✰ ❜

♣❡rt❡♥❝❡♥t❡s ❛ ❳ ✭▼✮✳ ❊♥tã♦ ❬❳❀ ❨ ❪ ✿❂ ❳❨ ❨ ❳ ♣❡rt❡♥❝❡ ❛ ❳ ✭▼✮✳

❳ ✒

❥

❳

✦ ❂

❥

❅❢ ❅①

❥

❜

❥

✥ ❳

❉❡ ❢❛t♦✱ ❳❨ ✭❢✮ ❂ ❳

❥ ❅ ❅①

❥

❜

❥

❡ ❨ ❂ P

✐ ❅ ❅①

❛

✐

❊①❡♠♣❧♦ ✶✳✶✳ ❙❡❥❛♠ ❳ ❂ P

❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❡♠ ▼ s❡rá ❞❡♥♦t❛❞♦ ♣♦r ❳ ✭▼✮✳ ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❞❡ ❞♦✐s ❝❛♠♣♦s✳

❅① ✐ ✭♣✮✿

☛ ✐ ✭♣✮ ❅❢

✐❂✶

❳

❜

❅❢ ❅①

❥

✓ ❂

❅❢ ❅①

❅①

✒ ❅❜

✐

❛

✐

❳

❥

❳

❥

❥

❅❢ ❅①

❥

✒ ❜

✐

❅ ❅①

✐

❛

✐

❳

❥

❳

✓ ❂

■ss♦ ♠♦str❛ q✉❡ ❬❳❀ ❨ ❪ é ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✳ ❉❡♥♦♠✐♥❛♠♦s ❬❳❀ ❨ ❪ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❞♦s ❝❛♠♣♦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❛ ❝❛❞❛ ✐❀ ❥ ✷ ❢✶❀ ✿ ✿ ✿ ❀ ♥❣✱ ❉ ❊

❅ ❅

✭q✮❀ ✭q✮ é ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ ❯✳

❅① ❅① ✐ ❥ q

❉ ❊

❅ ❅ ✐❥

❆s ❢✉♥çõ❡s ❣ ✿❂ ❀ sã♦ ❝❤❛♠❛❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♠étr✐❝❛

❅① ❅① ✐ ❥ ♥ ♥

❘✐❡♠❛♥♥✐❛♥❛ ♥♦ 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 ❝♦♥❞✐çõ❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡

✈❡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✱ ❳ ❂ ✱ ♣♦❞❡✲s❡ ❢❛❝✐❧♠❡♥t❡ ✈❡r✱ ♣❛r❛ ❳ ❂

❅①

P P

✐ ✐ ✐ ✐

① ❳ ❡ ❨ ❂ ② ❨ ✱ q✉❡

✐ ✐

✥ ✦ ❳ ❳

❦

r ❳ ❨ ❂ ① ✐ ② ❥ ✰ ❳✭② ❦ ✮ ❳ ❦ ✿

✐❥ ❦ ✐❀❥

P

❦ ❦

❖s ❝♦❡✜❝✐❡♥t❡s ✱ ❞❡✜♥✐❞♦s ♣♦r r ❳ ❳ ❥ ❂ ❳ ❦ ✱ 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 ■ ❂ ❘✳ P♦r

❞t ❞ ✵

❞❡✜♥✐çã♦✱ ☛ ✭ ✮ é ♦ ✈❡t♦r t❛♥❣❡♥t❡ à ❝✉r✈❛ ☛ ❡♠ ☛✭t✮✳ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦✿

t ❞t

✁

❞ ✵

❞☛ ❂ ☛ ✭t✮✳

❞t Pr♦♣♦s✐çã♦ ✶✳✶✳ ❙❡❥❛♠ ▼ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡ ☛ ✿ ■ ✦ ▼ ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ ❝❛♠♣♦ ❱

❉❱

❛♦ ❧♦♥❣♦ ❞❡ ☛✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❝❛♠♣♦ ❞❡♥♦t❛❞♦ ♣♦r t❛❧ q✉❡

❞t ❉ ❉❱ ❉❲

✭❛✮ ✭❱ ✰ ❲ ✮ ❂ ✰ ❀

❞t ❞t ❞t ❞❢ ❉ ❉❱

✭❜✮ ✭❢❱ ✮ ❂ ❱ ✰ ❢ ❀

❞t ❞t ❞t

✭❝✮ ❙❡ ❱ ✭t✮ é ❛ r❡str✐çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ❨ ❞❡✜♥✐❞♦ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡

❉❱

☛✭■✮✱ ❡♥tã♦ ❂ r ☛ ✭t✮ ❨ ✳

❞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♣ ♣ ✮ ✭✈✮ ❂ ✭❡①♣ ♣ ✭t✈✮✮

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 ✱ ♦♥❞❡ ☛ ♣❀q é ✉♠❛ ❝✉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 ✷ ❱ ✱ q

✷ ✷ ✷

❥① ❫ ②❥ ❂ ❥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✮②❀ ①✐ ❑✭①❀ ②✮ ❂

✷

❥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 ✮✿ ♥ ✶

✐❂✶

P♦rt❛♥t♦✱ s❡ ▼ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧

♥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 ❘

❀ ✭✶✳✷✮ ❛ q✉❛❧✱ ♥❛ ♥♦t❛çã♦ ❞❡ ❊✐♥st❡✐♥✱ s❡ ❡①♣r❡ss❛ ❝♦♠♦

s ✐❦

❅ ❅❡ ❥

s ❥❦

❅❡ ✐

✰ ❅

❧ ✐❦ s ❥❧

❧ ❥❦ s ✐❧

❂

s ✐❥❦

❘

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✱ ❢ ✐❦❧ ❂

❢ ❅❡ ❧ ❅❡ ✉

❅❡ ❧ ✭

✉ ✐❦

✮ ❅❢

❅❡ ✉

✉ ✐❦

❅

✷

✰ ❅

✿ ❆♣❧✐❝❛♥❞♦ ❡ss❛ ú❧t✐♠❛ ❡q✉❛çã♦ ❛ ✭✷✳✶✮✱ ♦❜t❡♠♦s

❅❡ ❦ ✭

✉ ✐❧

✮ ❅❢

❅❡ ✉ ✰

✉ ✐❧

❅

✷

❢ ✐❦❧ ❢ ✐❧❦ ❂ ❅

❢ ❅❡ ❦ ❅❡ ✉

✉ ❧✐

❅

❅❡ ❦ ❂

❅ ❅❡ ❧

✭

✉ ✐❦

✮ ❅❢

❅❡ ✉

✉ ✐❦

✷

✷

❢ ❅❡ ❧ ❅❡ ✉

✰ ❅

❅❡ ❦ ✭

✉ ✐❧

✮ ❅❢

❅❡ ✉ ✰

✉ ✐❧

❅

❢ ❅❡ ❦ ❅❡ ✉

❢ ✉❦ ✰

❅❡ ❧ ❅❢

✓ ✰

✷

❢ ❅❡

✉

❅❡

❦ s ✉❦

❅❢ ❅❡

s

✉ ❦✐

✉ ❧✐

✒ ❅

✷

❢ ❅❡

✉

❅❡

❧ s ✉❧

❅❢ ❅❡

s

✒ ❅

❢ ❅❡ ❦ ❅❡ ✉

✉ ❦✐

❅

❢ ✉❧ ❂

❅ ❅❡ ❧

✭

✉ ✐❦

✮ ❅❢

❅❡ ✉

✉ ✐❦

✷

✷

❢ ❅❡ ❧ ❅❡ ✉

✰ ❅

❅❡ ❦ ✭

✉ ✐❧

✮ ❅❢

❅❡ ✉ ✰

✉ ✐❧

❅

✐❧

✐❦

❅❢

✿ ✭✷✳✶✮ ▼❛s

✉❦

❅❢

✐❧

❅❡ ❦ ✰

✉ ❦✐

❢

✉❧

❅❢

✉ ❧✐

✐❦

❅❡

❧

❂ ❅

❅❡

❧

✒ ❅

✷

❢

❅❡ ❧

✐

❦ ✉ ❦✐

✐❦

❅❡ ❧

✉ ❧✐

❢ ✉❦

✉ ❧❦

❢ ✉✐ ❀ ❡

❢ ✐❧❦ ❂ ❅❢ ✐❧ ❅❡

❢ ✉❧

✐❦

✉ ❦❧

❢ ✉✐ ✿ ❈♦♠♦

✉ ❧❦ ❂ ✉ ❦❧ ✱ s❡❣✉❡✲s❡ q✉❡

❢

✐❦❧

❢

✐❧❦

❂ ❅❢

❢ ❅❡

❡

❅❢

❅ ❅❡ ❦

❅❡

✉

❡ ❅❢ ✐❧

❅❡ ❦ ❂

❅

✸

❢ ❅❡ ❦ ❅❡ ✐ ❅❡ ❧

✭

❢ ❅❡

✉ ✐❧

✮ ❅❢

❅❡ ✉

✉ ✐❧

❅

✷

❢ ❅❡ ❦ ❅❡ ✉

✿ ❆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♦ P

♥ ✷ ✷

♣ ✷ ▼✳ ❆✜r♠❛♠♦s q✉❡ ❥ ❣r❛❞ ❢❥ ❂ ❢ ✐ ✳ ❉❡ ❢❛t♦✱ ❡♠ ♣ ✷ ▼✱

✐❂✶ ✶ ♥

♦ ❝❛♠♣♦ ❣r❛❞ ❢ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❜❛s❡ ❢❊ ❀ ✿✿✿❀ ❊ ❣ ❝♦♠♦ ❣r❛❞ ❢✭♣✮ ❂ P

♥ ✐ ✐ ✐

☛ ✭♣✮❊ ✭♣✮✱ ♦♥❞❡ ❝❛❞❛ ☛ ✿ ❯ ✦ ❘ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠

✐❂✶

❯✳ ❱✐st♦ q✉❡

❢ ✐ ❂ ❊ ✐ ✭❢✮ ❂ ❤❊ ✐ ❀ ❣r❛❞ ❢✐ ❂ ☛ ✐ ❀

P

♥ ✐ ✐

t❡♠♦s q✉❡ ❣r❛❞ ❢ ❂ ❢ ❊ ❡✱ ♣♦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✉❡

❳ ✁

✶

✷ ✷ ✐ ❥❥✐ ✐❥s❥ s

✁ ❥ ❣r❛❞ ❢❥ ❂ ✭❢ ✰ ❢ ✭❢ ✰ ❘ ❢ ✮✮✿

✐❥

✷

✐❀❥

❋✐♥❛❧♠❡♥t❡✱ ♦❜t❡♠♦s q✉❡ ❳ ❳ ❳

✁ ✶

✷ ✷

✁ ❥ ❣r❛❞ ❢❥ ❂ ❢ ✰ ❢ ✐ ❢ ❥❥✐ ✰ ❘ ✐❥s❥ ❢ s ❢ ✐

✐❥

✷

✐❀❥ ✐❀❥ ✐❀❥

❳

✷

❂ ❥ ❍❡ss ❢❥ ✰ ❤❣r❛❞ ❢❀ ❣r❛❞✭✁❢✮✐ ✰ ❘✐❝✭❊ s ❀ ❊ ✐ ✮❢ s ❢ ✐

✐❀s

❳ ❳

✷ s s ✐ ✐

❂ ❥ ❍❡ss ❢❥ ✰ ❤❣r❛❞ ❢❀ ❣r❛❞✭✁❢✮✐ ✰ ❘✐❝✭ ❢ ❊ ❀ ❢ ❊ ✮

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✭ ❀ ✮ ❂ ✵✳ ❆❧é♠ ❞✐ss♦✱ s❡ ✽ ❳ ❄ ✱ ✈❛❧❡

❅r ❅r ❅r ❅ ❅

❍❡ss r✭ ❀ ❳✮ ❂ ✵✱ ♦♥❞❡ é ♦ ✈❡t♦r ✈❡❧♦❝✐❞❛❞❡✳

❅r ❅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 ❅r ❅r ❅

❂ ✭✶✮ ❅r

❂ ✵✿ ❆❣♦r❛✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ●❛✉ss✱ ✈❡r ❬✽❪✱ ♣á❣✐♥❛ ✶✵✷✱ t❡♠♦s q✉❡

❅ ❣r❛❞ r ❂ ✿

❅r

❅

P♦rt❛♥t♦✱ s❡ ❳ ❄ ✱ ♦❜t❡♠♦s

❅r

❅ ❅ ❍❡ss r✭ ❀ ❳✮ ❂ ❍❡ss r✭❳❀ ✮

❅r ❅r ❅ ❅

❂ ❳ r ✭r ❳ ✮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

❢ ✭r✮ ❍❡ss r✭❳❀ ❨ ✮ ❂ ❣✭❳❀ ❨ ✮✿

❢✭r✮ ❆❧é♠ ❞✐ss♦✱

✵

❢ ✭r✮ ✁r✭①✮ ❂ ✭♥ ✶✮ ✿

❢✭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❡❥❛♠ ❂ ♦ ❝❛♠♣♦ ♥♦r♠❛❧ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ r ❂ ❝ ❡ ❀ ✿ ✿ ✿ ❀

❅① ❅r ❅① ❅ ♥ ✶

✱ ♦s ❝❛♠♣♦s ❝♦♦r❞❡♥❛❞♦s ❡♠ ❙ ✳

❅① ♥ ✶

❆ss✐♠✱ ♣❛r❛ ✶ ✔ ✐❀ ❥ ✔ ♥ ✶✱ t❡♠♦s q✉❡ ✒ ✓ ✜ ✢

❅ ❅ ❅ ❅ ❍❡ss r ❀ ❂ ❀ r

❅①✐

❅① ✐ ❅① ❥ ❅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♦ ❧❛❞♦✱ ❅❣ ❧✐

✵ ✷ ❦❧ ✷ ✷ ❦❧

❂ ✷❢❢ ✭❞✇ ✮ ❧✐ ❡ ❣ ❂ ❢ ✭❞✇ ✮ ✿ ❅r

❉❛í✱ ❳

✶

❦ ✷ ✷ ❦❧ ✵ ✷

❂ ❢ ✭❞✇ ✮ ✷❢❢ ✭❞✇ ✮ ❧✐

♥✐

✷

❧ ✵

❢ ❂ ✍ ✐❦ ✿

❢ P♦rt❛♥t♦✱

✒ ✓

✵

❅ ❅ ❢

✐❦ ❦❥

❍❡ss r ❀ ❂ ✍ ❣

✐ ❥

❅① ❅① ❢ ✒ ✓

✵

❢ ❅ ❅ ❂ ❣ ❀ ✿

✐ ❥

❢ ❅① ❅①

▲♦❣♦✱ s❡ ❳ ❡ ❨ sã♦ ♦rt♦❣♦♥❛✐s ❛ ✱ ❡♥tã♦

❅r

❢ ❍❡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 ✰ ❢ ❦ ❞✇ ❀

♦♥❞❡ ✽

♣ ✶

❃ ❃

♣ s✐♥✭ ❦r✮❀ s❡ ❦ ❃ ✵ ❃ ❃ ❁

❦ ❢ ❦ ✭r✮ ❂ 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✉❡

✽ ♣ ❃ ❝♦s✭ ❦r✮❀ s❡ ❦ ❃ ✵ ❁

✵

❢ ✭r✮ ❂

❦ ✶❀ s❡ ❦ ❂ ✵

❃ ♣

✿ ❝♦s❤✭ ❦r✮❀ s❡ ❦ ❁ ✵✿

❆ss✐♠✱ ♣❛r❛ ❦ ❃ ✵✱ t❡♠♦s ♣ ♣

❦ ❝♦s✭ ❦r✮ ❍❡ss r ❦ ❂ ♣ ❣ s✐♥✭ ❦r✮

♣ ♣ ❂ ❦ ❝♦t✭ ❦r✮❣✿

◗✉❛♥❞♦ ❦ ❂ ✵✱ ✶

❍❡ss r ❂ ❣✿ r ❊

♣ ♣ ❦ ❝♦s❤✭ ❦r✮

❍❡ss r ❦ ❂ ♣ ❣ s✐♥❤✭ ❦r✮ ♣ ♣

❂ ❦ ❝♦t❤✭ ❦r✮❣❀ s❡ ❦ ❁ ✵✳ P♦rt❛♥t♦ ✽

♣ ♣ ❃

❦ ❝♦t✭ ❦r✮❀ s❡ ❦ ❃ ✵ ❃ ❃ ❁

✶

✶

✁r ❦ ❂ ❀ s❡ ❦ ❂ ✵

r

♥ ✶ ❃ ❃ ❃ ♣ ♣ ✿

❦ ❝♦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 ❀ ❂ ✵ ❡ ❍❡ss r✭ ❀ ❳✮ ❂ ✵❀ ✽ ❳❄ ✳ ❊ss❡ ❢❛t♦ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡

❅r ❅r ❅r ❅r

❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❛♣❧✐❝❛❞❛ às ♠❛tr✐③❡s ❍❡ss✐❛♥❛ ❞❡ r ❡ ■❞❡♥t✐❞❛❞❡ ♥♦s ❞ã♦

✷

✭✁r✮

✷

❥ ❍❡ss 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❡ ❦ ❡ ✽

♣ ✶

❃ ❃

♣ s✐♥✭ ❦r✮❀ s❡ ❦ ❃ ✵ ❃ ❃ ❁ ❦

❦

❢ ✭r✮ ❂ 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✉❡ ✬

✵

♥ ✶ ❡ ❥ ✿

❉❡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

❂

♥ ✶

❳

❦❂✶

☛

❡

❦

✰ ❛ ❅

♥ ✶ ✿

❂ ✥

✭t✮ ✔ ✥

✭✒✭t✮✮✿ ❈♦♠♦ ✥

✵

✭✒✭t✮✮✿ ▼❛s ✬

✵

✭t✮ ❂ ✥

✵

✭✒✭t✮✮✒

✵

✭t✮ ❡✱ ♣♦rt❛♥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❥

❅r ✿

❉❛í✱ ✡ r ❡

❥

✢ ❂

❥

❀ ❡

❅ ❅r

❬❡ ❅r

❅ ❅r r

❅ ❅r r ❅r r❡

✥ ♥ ✶

❡ ❥ r ❅r

❂ ✜ r

❀ ❡ ❥ ✓

❅ ❅r

❅r ❀

❡ ❥ ❀ ❅

❂ ❘ ✒

✜ r ❅r

✓ ❡ ❥ ❀ ❡ ❥

☞☞ ☞

♥ ✶ ❉ r ❅r ❡ ❥ ❀ ❡ ❥

♥ ✶ ✓

✷

✰ ✥

✒ ✥

✶ ♥ ✶

❊ ❂

♥ ✶ ✥

✢ ❂

✵

✥

✢ ❂

✥ ♥ ✶ r ❅r ❡ ❥ ❀ ❡ ❥

♥ ✶ ❡ ❥ ✰

✵

✜ ✥

①

❅r ✓ ☞☞

❅ ❅r

❅ ❅r

❅ ❅r

P❛r❛ ✐ ✻❂ ❥✱ t❡♠♦s ✡ r ❡

✢ ❂ ❛✿

❅ ❅r

❅r ❀

✜ r ❡

✢ ❂

❅r ❀

❂ ❍❡ss r✭❡ ❥ ❀ ❡ ✐ ✮ ❂ ✵✱ ♦✉ s❡❥❛✱ ☛ ✐ ❂ ✵ ♣❛r❛ t♦❞♦ ✐ ✻❂ ❥✳ ❆ss✐♠✱ ♦❜t❡♠♦s q✉❡ r

✜ ❅

❡

✵ ❂ ✶ ✷

☛ ❂ ❛ ❂ ✵✱ ♣♦✐s

❅ ❅r

❀

❀ ❡ ✐ ☛

❡

❡ ❥ ❀ ❅

❂ ❍❡ss r✭❡ ❥ ❀ ❡ ❥ ✮ ❂ ✕✱ t❡♠♦s r

❑ ✒

✿ ❊ ❛ ❛✜r♠❛çã♦ ❡stá ♣r♦✈❛❞❛✳ ❚❡♠♦s ❡♥tã♦

❥

♥ ✶ ❡

❂ ✥

❅ ❅r

❡

❀ ❡ ❥ ☛

❅ ❅r

❅ ❅r

✡ r ❡

✿ ❈♦♠♦ ☛ ❥ ❂

❥

❡

❥

❂ ☛

❂ ❦✿

❉❡✜♥✐çã♦ ✷✳✶✳ ❉✐③❡♠♦s q✉❡ ✁r ✔ ❢ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✱ s❡ ❩ ❩

✶

r✁ ✬ ✔ ✬❢❀ ✽ ✬ ✕ ✵❀ ✬ ✷ ❈ ✭▼✮✿

✵ ▼ ▼

❖ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ t❛♠❜é♠ é ✈á❧✐❞♦ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳

♥

❚❡♦r❡♠❛ ✷✳✷✳ ❙❡❥❛♠ ✭▼ ❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❡ r ❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ❞❡ ❣ s❛✲ t✐s❢❛③ ❘✐❝✭▼✮ ✕ ✭♥ ✶✮❦❣✳ ❊♥tã♦ ✁r✭①✮ ✔ ✁r ❦ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳

P ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ♣ ✷ ▼ ❡ ❝♦♥s✐❞❡r❡♠♦s ♦ ♠❡r❣✉❧❤♦ ❡①♣ ✿ ✦ ▼✳

♣ ♣

❈♦♥s✐❞❡r❡♠♦s t❛♠❜é♠ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❛❜❡rt♦s ❡str❡❧❛❞♦s ❢✡ ✧ ❀ ✧ ❃ ✵❣ t❛❧ P q✉❡✱ q✉❛♥❞♦ ✧ ✦ ✵✱ ✡ ✧ ✦ ✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦✱

♣

❩ ❩ ✁r✬ ✔ ✁r ❦ ✬❀ ✬ ✕ ✵✿ ✭✷✳✺✮

✡ ✡ ✧ ✧

❆❣♦r❛✱ ❛ ♣r✐♠❡✐r❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ●r❡❡♥ ♥♦s ❞á ❩ ❩ ❩

❅r ✁r✬ ❂ ❤❣r❛❞ r❀ ❣r❛❞ ✬✐ ✰ ✬✿

✡ ✡ ❅✡ ❅✑

✧ ✧ ✎ ✎ ✎

❈♦♠♦ ✬ ✕ ✵ ❡ ✡ é ❡str❡❧❛❞♦✱ t❡♠♦s q✉❡ ❛ ✐♥t❡❣r❛❧ s♦❜r❡ ✡ é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ③❡r♦✳ ❉❛í

❩ ❩ ✁r✬ ✕ ❤❣r❛❞ r❀ ❣r❛❞ ✬✐ ✿

✡ ✡ ✧ ✧

❋❛③❡♥❞♦ ✧ ✦ ✵ ❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✺✮✱ ♦❜t❡♠♦s ❩ ❩

❦

❤❣r❛❞ r❀ ❣r❛❞ ✬✐ ✔ ✁r ✬✿

P P

♣ ♣

❈♦♠♦ ❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ é ▲✐♣s❝❤✐t③✐❛♥❛✱ ❝♦♥s❡qü❡♥t❡♠❡♥t❡ é q✿t✿♣✿ ❞✐❢❡✲

✶

r❡♥❝✐á✈❡❧ ❡✱ ♣♦rt❛♥t♦✱ s✉❛ ❞❡r✐✈❛❞❛ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❡♠ ❍ ✱ ✈❡r ❬✻❪✱ ♣á❣✐♥❛s ✽✶ ❡ ✷✸✺✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❡♠ ❍

✶

♥♦s ❞✐③ q✉❡ ❩

▼

❤❣r❛❞ r❀ ❣r❛❞ ✬✐ ❂ ❩

r✁✬❀ ✽ ✬ ✕ ✵❀ ✬ ✷ ❈

✶ ✵

✭▼✮✿ P♦rt❛♥t♦

❩

▼

r✁✬ ✔ ❩

▼

✁r ❦ ✬❀ ✽ ✬ ✕ ✵❀ ✬ ✷ ❈

✶ ✵

✭▼✮✿

❈❛♣ít✉❧♦ ✸ ❖ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧

◆❡st❡ ❝❛♣ít✉❧♦ ❡st❛r❡♠♦s s❡♠♣r❡ ♥♦s ❞✐r❡❝✐♦♥❛♥❞♦ ❛♦ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✱ ♦❜t❡♥❞♦ ❛♥t❡s ✐♠♣♦rt❛♥t❡s r❡s✉❧t❛❞♦s q✉❡ ❢❛r❡♠♦s ✉s♦ ❡♠ s✉❛ ❞❡♠♦♥str❛çã♦✳

✸✳✶ ❋✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥

❆s ❢✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥ ❞❡s❡♠♣❡♥❤❛♠ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ ❞❡♠♦♥s✲ tr❛çã♦ q✉❡ ❞❛r❡♠♦s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧ s♦❜r❡ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✳ ❉❡✜♥✐çã♦ ✸✳✶✳ ❯♠❛ ❣❡♦❞és✐❝❛ ✌ ✿ ✭ ✶❀ ✶✮ ✦ ▼✱ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦

✶ ✷ ✶ ✷

❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱ é ❝❤❛♠❛❞❛ ✉♠❛ ❧✐♥❤❛✱ s❡ ❞✭✌✭t ✮❀ ✌✭t ✮✮ ❂ ❥t t ❥✱

✶ ✷

✽ t ❀ t ✷ ✭ ✶❀ ✶✮✳ ❉❡✜♥✐çã♦ ✸✳✷✳ ❯♠❛ ❣❡♦❞és✐❝❛ ✌ ✿ ❬✵❀ ✶✮ ✦ ▼✱ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱ é ❝❤❛♠❛❞❛ ✉♠ r❛✐♦✱ s❡ ❞✭✌✭t ✶ ✮❀ ✌✭t ✷ ✮✮ ❂ ❥t ✶ t ✷ ❥✱ ✽ t ✶ ❀ t ✷ ✷ ❬✵❀ ✶✮✳

❖❜s❡r✈❛♠♦s q✉❡ s❡ ▼ é ❝♦♠♣❧❡t❛ ❡ ♥ã♦ ❝♦♠♣❛❝t❛✱ ❡♥tã♦✱ ♣❛r❛ t♦❞♦ ♣ ✷ ▼✱ ❡①✐st❡ ✉♠ r❛✐♦ ✌ t❛❧ q✉❡ ✌✭✵✮ ❂ ♣✱ ✈❡r ❬✷❪✱ ♣á❣✐♥❛ ✹✵✺✳ P♦ré♠ ♥❡♠ t♦❞❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❧❡t❛ ❡ ♥ã♦ ❝♦♠♣❛❝t❛ ♣♦ss✉✐ ✉♠❛ ❧✐♥❤❛✱ ❝♦♠♦ é

✸ ✷ ✷

♦ ❝❛s♦ ❞♦ ♣❛r❛❜♦❧ó✐❞❡ ❞❡ r❡✈♦❧✉çã♦ ❢✭①❀ ②❀ ③✮ ✷ ❘ ❥ ③ ❂ ① ✰ ② ❣✳ ❯♠❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ú❧t✐♠♦ ❢❛t♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷❪✱ ♣á❣✐♥❛s ✸✵✾ ❡ ✸✻✸✳ ❉❡✜♥✐çã♦ ✸✳✸✳ ❉❛❞♦ ✉♠ r❛✐♦ ✌✱ ❛ ❢✉♥çã♦ ❞❡ ❇✉s❡♠❛♥♥ ❛ss♦❝✐❛❞❛ ❛ ❡ss❡ r❛✐♦✱ ❜ ✌ ✱ é ❞❡✜♥✐❞❛ ♣♦r

✌

❜ ✭①✮ ❂ ❧✐♠ ✭❞✭①❀ ✌✭t✮✮ t✮✿

t✦✶

❉❛❞❛ ✉♠❛ ❧✐♥❤❛ ✌ ✿ ✭ ✶❀ ✶✮ ✦ ▼✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ❡❧❛ ❞✉❛s ❢✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥✱ ❛ s❛❜❡r✿

✰

❜ ✭①✮ ❂ ❧✐♠ ❞✭①❀ ✌✭t✮✮ t❀

t✦✶

❡ ❜ ✭①✮ ❂ ❧✐♠ ❞✭①❀ ✌✭ t✮✮ t✿

t✦✶

❉❡✜♥✐♥❞♦

✰

❜ ✭①✮ ✿❂ ❞✭①❀ ✌✭t✮✮ t❀

t

t❡♠♦s✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ q✉❡

✰

❜ t ✭①✮ ✔ ❞✭①❀ ✌✭✵✮✮ ✰ ❞✭✌✭✵✮❀ ✌✭t✮✮ t ❂ ❞✭①❀ ✌✭✵✮✮✿

✰

P♦rt❛♥t♦✱ ❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s ❢❜ ❣ t✕✵ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ♥♦s

t

❝♦♠♣❛❝t♦s ❞❡ ▼✳ ❆✜r♠❛♠♦s ❛❣♦r❛ q✉❡ ❡st❛ ❢❛♠í❧✐❛ é ♥ã♦✲❝r❡s❝❡♥t❡✳ ❉❡ ❢❛t♦✱ s❡ t ✶ ❁ t ✷ ✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ t❡♠♦s

✰

❜ ✭①✮ ❂ ❞✭①❀ ✌✭t ✷ ✮✮ t ✷

t ✶ ✶ ✷ ✷

✔ ❞✭①❀ ✌✭t ✮✮ ✰ ❞✭✌✭t ✮❀ ✌✭t ✮✮ t

✶ ✷ ✶ ✷

❂ ❞✭①❀ ✌✭t ✮✮ ✰ t t t

✰

❂ ❜ t ✭①✮✿

✰ ✰ ✰

❆ ❢❛♠í❧✐❛ t ✼✦ ❜ t é✱ ♣♦rt❛♥t♦✱ ♠♦♥ót♦♥❛ ♥ã♦✲❝r❡s❝❡♥t❡✳ ❆ss✐♠✱ ❜ t ✦ ❜

✰

✉♥✐❢♦r♠❡♠❡♥t❡ ♥♦s ❝♦♠♣❛❝t♦s ❞❡ ▼ ❡ ❜ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❖❜s❡r✈❛♠♦s ❛✐♥❞❛ q✉❡✱ ❝♦♠♦

✰ ✰

❜ ✭①✮ ❜ ✭②✮ ❂ ❞✭①❀ ✌✭t✮✮ ❞✭②❀ ✌✭t✮✮

t t

✔ ❞✭①❀ ②✮ ✰ ❞✭②❀ ✌✭t✮✮ ❞✭②❀ ✌✭t✮✮ ❂ ❞✭①❀ ②✮❀

✰

❛s ❢✉♥çõ❡s ❜ t sã♦ ▲✐♣s❝❤✐t③✐❛♥❛s ♣❛r❛ t ❃ ✵✱ ❝♦♠ ❝♦♥st❛♥t❡ ❞❡ ▲✐♣s❝❤✐t③ ✐❣✉❛❧

✰

❛ ✶✱ ❝❧❛r♦ q✉❡ ❛ ❢✉♥çã♦ ❜ t❛♠❜é♠ é ▲✐♣s❝❤✐t③✐❛♥❛✱ s❡♥❞♦✱ ♣♦rt❛♥t♦✱ q✿t✿♣✿ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ✈❡r ❬✻❪✱ ♣á❣✐♥❛ ✽✶✳

✸✳✷ ❚❡♦r❡♠❛ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧

❆ ✐❞é✐❛ ❞❛ ♣r♦✈❛ ❝♦♥s✐st❡ ❡♠ ❝♦♥str✉✐r ✉♠❛ ❢✉♥çã♦ ❤ s✉❛✈❡ ❡ ❤❛r♠ô♥✐❝❛✱ ❞❡✜♥✐❞❛ ❡♠ ▼✱ t❛❧ q✉❡ ❥ ❣r❛❞ ❤❥ ❂ ✶✳ ❊♥tã♦✱ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✱ ♠♦str❛♠♦s q✉❡ ♦s ❝♦♥❥✉♥t♦s ❞❡ ♥í✈❡❧ ❞❡ ❤ sã♦ s✉❜✈❛r✐❡❞❛❞❡s

✶

t♦t❛❧♠❡♥t❡ ❣❡♦❞és✐❝❛s ❞❡ ▼✳ ❉❡✜♥✐♠♦s ❡♥tã♦ ◆ ❂ ❤ ✭✵✮✳ P❛r❛ t❛❧ ❝♦♥s✲

✰

tr✉çã♦✱ ❝♦♥t❛r❡♠♦s ❝♦♠ ❛s ❢✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥ ❜ ❡ ❜ ❛ss♦❝✐❛❞❛s à ❧✐♥❤❛ ✌✳

♥

Pr♦♣♦s✐çã♦ ✸✳✶ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞♦ ❱❛❧♦r ▼é❞✐♦✮✳ ❙❡❥❛ ✭▼ ❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠ ❘✐❝✭▼✮ ✕ ✵✳ ❙❡❥❛ ❢ ✕ ✵ ✉♠❛ ❢✉♥çã♦ ▲✐♣s❝❤✐t③✐❛♥❛ t❛❧ q✉❡ ✁❢ ✔ ✵ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳ ❊♥tã♦

❩ ✶

❢✭①✮ ✕ ❢✿

✇ ❘ ❇ ✭①✮ ❉❡♠♦♥str❛çã♦✳ ➱ s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ ❛ ❢✉♥çã♦ ❤ ✿ ✭✵❀ ✶✮ ✦ ✭✵❀ ✶✮✱ ❞❡✜♥✐❞❛ ♣♦r

❩ ✶

❤✭❘✮ ❂ ❢❀

♥

✇ ♥ ❘ ❇ ✭①✮ é ♥ã♦ ❝r❡s❝❡♥t❡✳ P♦✐s✱ ❝♦♠♦

❩ ✶

❧✐♠ ❢ ❂ ❢✭①✮❀

♥ ❘✦✵

✇ ♥ ❘

❇ ✭①✮

✉♠❛ ✈❡③ ♠♦str❛❞♦ q✉❡ ❤ é ♥ã♦ ❝r❡s❝❡♥t❡✱ t❡r❡♠♦s ♦ s❡❣✉✐♥t❡✿ s❡ ❘ ❃ ❘ ✶ ❃ ✁ ✁ ✁ ❃ ❘ ♥ ❃ ✁ ✁ ✁ ❃ ✵✱ ❡♥tã♦ ❤✭❘✮ ✔ ❤✭❘ ✶ ✮ ✔ ✁ ✁ ✁ ✔ ❤✭❘ ♥ ✮ ✔ ✁ ✁ ✁ ✔ ❢✭①✮✳

❇

❢✁r

❡①♣ ✭❇ ✭✵✮❭✡

❩

✧✦✵

❂ ❧✐♠

✷

❢✁r

❩

❂ ❧✐♠

✐♥t❡❣r❛❧ ❞❛ ❡sq✉❡r❞❛ ❡ ♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡ ❞❛ ✐♥t❡❣r❛❧ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦✳ ❆❣♦r❛✱

❢✿ ✭✸✳✶✮ ❱❛❧❡ ♦❜s❡r✈❛r q✉❡ ❡st❛♠♦s ♦♠✐t✐♥❞♦ ❡♠ ✭✸✳✶✮ ♦ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ ♥❛

❇

❢ ✔ ♥ ❩

❙

❘ ❩

✮✮✳ Pr❡❝✐s❛♠♦s ❡♥tã♦ ♠♦str❛r q✉❡

✷

✧✦✵

❭ P

❅✑ ✦

❙

❩

☛ ✰ ✷❘

✷

✡ ❣r❛❞ ❢❀ ❣r❛❞ r

❇

✕ ❩

✷

✥ ❩

❢ ❅r

❅ ❡①♣ ✭❇ ✭✵✮❭✡

❩

☛ ✰

✷

✡ ❣r❛❞ ❢❀ ❣r❛❞ r

❡①♣ ✭❇ ✭✵✮❭✡

♣

✵ ❘ ❂ ❡①♣ ♣ ✭✐♥t✭❙ ❘

❖❜s❡r✈❡♠♦s q✉❡ ❤ é ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥❛✱ ♣♦rt❛♥t♦✱ ❤ é q✿t✿♣✿ ❞✐❢❡✲ r❡♥❝✐á✈❡❧✳ ❚❡♠♦s q✉❡

♥ ✇ ♥ ❘

❢ ✓

❇ ✭①✮

✒❩

❢ ✔ ✵ ✱

❇ ✭①✮

❩

♥✰✶

❢ ✓ ✵

✔ ♥

❇ ✭①✮

✒❩

♥

✇ ♥ ❘

✭❘✮ ✔ ✵ ✱ ✶

✵

❤

✵

❘ ❩

❢❞✛ ❣ ❀ ♦♥❞❡ ❙

❢ ♣

❙

❩

❞❡t❣❞✛ ❂

❢ ♣

✐♥t✭❙ ❭ P

✔ ❩

❞❡t❣ ❞① ✧

❇ ❭ P

❇ ✭①✮

❞❡t❣ ❞① ❘

❢ ♣

❇ ✭❘✰✧✮ ❭ P

❘

✧✦✵

❧✐♠ s✉♣

❢ ❆✐♥❞❛✱

❢❀

♦✉ s❡❥❛✱ ❩ ❩ ❩

✡ ☛

✷ ✷

✷❘ ❢ ❣r❛❞ ❢❀ ❣r❛❞ r ✔ ❢✁r ✿

❙ ❇ ❇ ❘ ❘

❖r❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✱

✷

✁r ❂ ✷r✁r ✰ ✷ ❤❣r❛❞ r❀ ❣r❛❞ r✐ ✔ ✷✭♥ ✶✮ ✰ ✷ ❂ ✷♥✿

P♦rt❛♥t♦✱ ❩ ❩ ❩

✡ ☛

✷

✷❘ ❢ ❣r❛❞ ❢❀ ❣r❛❞ r ✔ ✷♥ ❢✿ ✭✸✳✷✮

❙ ❇ ❇

❘ ❘

❚❡♠♦s t❛♠❜é♠ q✉❡ ❩ ❩

✡ ☛ ✡ ☛

✷ ✷ ✷

❣r❛❞ ❢❀ ❣r❛❞ r ❂ ❣r❛❞ ❢❀ ❣r❛❞✭r ❘ ✮

❇ ❇ ❘ ❘

❩

✷ ✷

❂ ✁❢✭r ❘ ✮

❇

✕ ✵❀ ✭✸✳✸✮

✷ ✷

♣♦✐s ✁❢ ✔ ✵ ❡ r ✭①✮ ❘ ✔ ✵❀ ✽ ① ✷ ❇ ❘ ✳ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✸✳✸✮ ❛♣❧✐❝❛❞❛ ❛ ✭✸✳✷✮ ♥♦s ❢♦r♥❡❝❡

❩ ❩ ✷❘ ❢ ✔ ✷♥ ❢✿

❙ ❇

❈♦r♦❧ár✐♦ ✸✳✶✳ ❙❡❥❛♠ ✭▼ ❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠ ❘✐❝✭▼✮ ✕ ✵ ❡ ❢ ✿ ▼ ✦ ❘ ✉♠❛ ❢✉♥çã♦ ♥ã♦✲♥❡❣❛t✐✈❛ ❝♦♠ ✁❢ ✔ ✵✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❡ ① ✵ ✷ ▼ t❛❧ q✉❡ ❢✭① ✵ ✮ ❂ ✵✳ ❊♥tã♦ ❢ ✑ ✵✳ ❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱

❩ ✵ ✕ ❢❀ ✽ ❘✿

❇ ✭① ✮ ❘ ✵

♥

▲❡♠❛ ✸✳✶✳ ❙❡❥❛ ✭▼ ❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠

✰

❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✳ ❆ ❢✉♥çã♦ ❜ ✿ ▼ ✦ ❘✱ é s✉♣❡r✲❤❛r♠ô♥✐❝❛

✰

♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✱ ✐st♦ é✱ ✁❜ ✔ ✵ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳

✶

❉❡♠♦♥str❛çã♦✳ ❖ ❚❡♦r❡♠❛ ✷✿✷ ✐♠♣❧✐❝❛✱ ♣❛r❛ t♦❞♦ t ❃ ✵ ❡ ♣❛r❛ t♦❞❛ ✬ ✷ ❈ ✱

✵

✬ ✕ ✵✱ q✉❡ ❩ ❩

✰ ✰

❜ ✁✬ ❂ ✬✁❜

t t ▼ ▼

❩ ❂ ❞✭①❀ ✌✭t✮✮✁✬

▼

❩ ✶

✔ ✭♥ ✶✮ ✬ ✿

▼ ❞✭①❀ ✌✭t✮✮

❖r❛✱ ❞✭①❀ ✌✭t✮✮ ✕ ❞✭✌✭t✮❀ ✌✭✵✮✮ ❞✭✌✭✵✮❀ ①✮✿

❚❡♠♦s ❡♥tã♦ q✉❡✱ ✜①❛❞❛ ❛ ❢✉♥çã♦ ✬ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ♥ã♦✲♥❡❣❛t✐✈❛ ❡ ❞❡ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ ❩

✶ ❧✐♠ ✬ ❂ ✵✿

t ✦✶

▼ ❞✭①❀ ✌✭t✮✮

▲♦❣♦✱ ♦❜t❡♠♦s ❩

✰

❜ ✁✬ ✔ ✵✿

▼

❱❛❧❡ ♦❜s❡r✈❛r q✉❡✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡ ❜ t❛♠❜é♠ é s✉♣❡r✲❤❛r♠ô♥✐❝❛ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳ ❊st❛♠♦s ❛❣♦r❛ ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❞❡♠♦♥str❛r ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❛

♠♦♥♦❣r❛✜❛✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✳

♥

❚❡♦r❡♠❛ ✸✳✶✳ ❙❡❥❛ ✭▼ ❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✳ ❙❡ ▼ ♣♦ss✉✐ ✉♠❛ ❧✐♥❤❛✳ ❊♥tã♦ ▼ é ✐s♦♠étr✐❝❛ ❛ ◆ ✂ ❘ ♠✉♥✐❞❛ ❞❛ ♠étr✐❝❛ ♣r♦❞✉t♦✱ ♦♥❞❡ ◆ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❞✐♠❡♥sã♦ ✭♥ ✶✮ ❝♦♠ ❘✐❝✭◆✮ ✕ ✵✳ ❉❡♠♦♥str❛çã♦✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ ✈❡♠♦s q✉❡

✰

❛ ❢✉♥çã♦ ❢✭①✮ ❂ ❜ ✭①✮ ✰ ❜ ✭①✮ é ♥ã♦✲♥❡❣❛t✐✈❛✳ ❆❧é♠ ❞✐ss♦✱ ❢✭①✮ ❂ ✵ ♣❛r❛ ① ❂ ✌✭t✮✳ P❡❧♦ ▲❡♠❛ ✸✳✶✱ ❛ ❢✉♥çã♦ ❢ é s✉♣❡r✲❤❛r♠ô♥✐❝❛ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✱ ✉♠❛ ✈❡③ q✉❡ ❢ é s♦♠❛ ❞❡ ❢✉♥çõ❡s s✉♣❡r✲❤❛r♠ô♥✐❝❛s✳ ❙❡❣✉❡ ❡♥tã♦ ❞♦ ❈♦r♦❧ár✐♦ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✶ q✉❡ ❢ ❂ ✵ ❡♠ ▼✳ P♦rt❛♥t♦✱ ✁❢ ❂ ✵✱

✰ ✰

✐st♦ é✱ ✁❜ ✰ ✁❜ ❂ ✵✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡ ✁❜ ❂ ✁❜ ❂ ✵ ♥♦ s❡♥t✐❞♦

✰

❞❛s ❞✐str✐❜✉✐çõ❡s✱ ♣♦✐s ✁❜ ✔ ✵ ❡ ✁❜ ✔ ✵✳ ■st♦ ❣❛r❛♥t❡ ❛ s✉❛✈✐❞❛❞❡ ❞❛s ❢✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ❲❡②❧✳ ❆✜r♠❛♠♦s ❛❣♦r❛ q✉❡ ❛ ❢✉♥çã♦

✰ ✷

❤ ✿❂ ❜ s❛t✐s❢❛③ ❥ ❣r❛❞ ❤❥ ❂ ✶✳ ❉❡ ❢❛t♦✱ ❥ ❣r❛❞ ❤❥ ✔ ✶✱ ♣♦✐s ❥á ✈✐♠♦s q✉❡ ❤ é ▲✐♣s❝❤✐t✐③✐❛♥❛ ❝♦♠ ❝♦♥st❛♥t❡ ❞❡ ▲✐♣s❝❤✐t③ ✐❣✉❛❧ ❛ ✶✳ ▲♦❣♦ ❛ ❢✉♥çã♦

✷

❋ ✿❂ ✶ ❥ ❣r❛❞ ❤❥ é ♥ã♦✲♥❡❣❛t✐✈❛✳ ❯s❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✱ t❡♠♦s

✁ ✶

✷ ✷

✁ ❥ ❣r❛❞ ❤❥ ❂ ❥ ❍❡ss ❤❥ ✰ ❤❣r❛❞ ❤❀ ❣r❛❞✭✁❤✮✐ ✰ ❘✐❝ ✭❣r❛❞ ❤❀ ❣r❛❞ ❤✮ ✷

✷

✕ ❥ ❍❡ss ❤❥ ✕ ✵✿

❞

❙❡❣✉❡✲s❡ ❡♥tã♦ q✉❡ ✁❋ ✔ ✵✳ P♦r s✉❛ ✈❡③✱ t❡♠♦s q✉❡ ❤✭✌✭s✮✮ ❂ ✶ ✭✐ss♦

❞s ✰ ✵

s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❜ ✮ ❡✱ ♣♦rt❛♥t♦✱ ❤❣r❛❞ ❤❀ ✌ ✭s✮✐ ❂ ✶✳ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♠♣❧✐❝❛

✵ ✵

✶ ❂ ❥ ❤❣r❛❞ ❤❀ ✌ ✭s✮✐ ❥ ✔ ❥ ❣r❛❞ ❤❥❥✌ ✭s✮❥ ✔ ✶❀

✵

❞♦♥❞❡ ❣r❛❞ ❤ ❂ ✌ ✳ ❉❛í✱ ♦❜t❡♠♦s q✉❡ ❋ ✭✌✭s✮✮ ❂ ✵✳ ❋✐♥❛❧♠❡♥t❡✱ ✉s❛♥❞♦ ♦ ❈♦r♦❧ár✐♦ ✭✸✳✶✮✱ t❡♠♦s ❋ ✑ ✵✳ ■st♦ ♣r♦✈❛ ❛ ❛✜r♠❛çã♦✱ ♦✉ s❡❥❛✱ ❥ ❣r❛❞ ❤❥ ❂ ✶✳ ❈♦♠♦ ❝♦♥s❡qüê♥❝✐❛ ❞✐ss♦✱ ❛✐♥❞❛ ♦❜t❡♠♦s q✉❡ ❥ ❍❡ss ❤❥ ❂ ✵✳ P♦rt❛♥t♦ ❣r❛❞ ❤ é ♣❛r❛❧❡❧♦✳

✶

P♦❞❡♠♦s ❡♥tã♦ ❞❡✜♥✐r ✉♠❛ ✐s♦♠❡tr✐❛ ❡♥tr❡ ❘ ❝❛rt❡s✐❛♥♦ ◆ ✿❂ ❤ ✭✵✮ ❡

▼ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ✬ ✿ ❘ ✂ ◆ ✦ ▼

✭t❀ ①✮ ✦ ❡①♣ ✭t ❣r❛❞ ❤✭①✮✮❀

①

❈♦♠ ❡❢❡✐t♦✱ ♣❛r❛ ✈❡t♦r❡s ✈❡rt✐❝❛✐s ❜❛st❛ ♦❜s❡r✈❛r♠♦s q✉❡ ✜ ✒ ✓ ✒ ✓✢

❅ ❅ ❞✬ ✭t❀①✮ ❀ ❞✬ ✭t❀①✮ ❂ ❤❣r❛❞ ❤❀ ❣r❛❞ ❤✐

❅t ❅t ❂ ✶

✜ ✢ ❅ ❅

❂ ❀ ✿ ❅t ❅t

◗✉❛♥❞♦ t❡♠♦s ✉♠ ✈❡t♦r ✈❡rt✐❝❛❧ ❡ ♦✉tr♦ ❤♦r✐③♦♥t❛❧✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ❞♦ ▲❡♠❛ ❞❡ ●❛✉ss✳

❋✐♥❛❧♠❡♥t❡✱ ♥♦ ❝❛s♦ ❡♠ q✉❡ t❡♠♦s ✉ ❡ ✈✱ ✈❡t♦r❡s ❤♦r✐③♦♥t❛✐s✱ ❛ ❢✉♥çã♦ ✡ ☛

❢✭t✮ ✿❂ ❞✬ ✭t❀①✮ ✭✉✮❀ ❞✬ ✭t❀①✮ ✭✈✮ ❀ é ❝♦♥st❛♥t❡ ❡♠ t✳ ❈♦♠ ❡❢❡✐t♦✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ❙✐♠❡tr✐❛✱ ✈❡r ❬✸❪✱ ♣á❣✐♥❛ ✼✻✱ ❡ ♣♦r ❣r❛❞ ❤ s❡r ✉♠ ❝❛♠♣♦ ♣❛r❛❧❡❧♦✱ ♦❜t❡♠♦s

✡ ☛ ✡ ☛

✵ ❣r❛❞ ❤ ✭t❀①✮ ✭t❀①✮ ✭t❀①✮ ❣r❛❞ ❤ ✭t❀①✮

❢ ✭t✮ ❂ r ❞✬ ✭✉✮❀ ❞✬ ✭✈✮ ✰ ❞✬ ✭✉✮❀ r ❞✬ ✭✈✮ ❉ ❊ ❉ ❊

❞✬ ✭✉✮ ✭t❀①✮ ✭t❀①✮ ❞✬ ✭✈✮

❂ r ❣r❛❞ ❤❀ ❞✬ ✭✈✮ ✰ ❞✬ ✭✉✮❀ r ❣r❛❞ ❤ ✭t❀①✮ ✭t❀①✮ ❂ ✵✿

P♦rt❛♥t♦✱ ✡ ☛

❞✬ ✭t❀①✮ ✭✉✮❀ ❞✬ ✭t❀①✮ ✭✈✮ ❂ ❤❞✬ ✵ ✭✉✮❀ ❞✬ ✵ ✭✈✮✐ ❂ ❤✉❀ ✈✐ ❀

♣❛r❛ q✉❛✐sq✉❡r ✉❀ ✈ ✷ ❚ ✭t❀①✮ ❘ ✂ ◆✳

❈♦r♦❧ár✐♦ ✸✳✷✳ ❙❡❥❛ ▼ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠

❦

❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✳ ❊♥tã♦ ▼ é ✐s♦♠étr✐❝❛ ❛ ✉♠ ♣r♦❞✉t♦ ◆ ✂❘ ✱

❦

♦♥❞❡ ◆ ♥ã♦ ❝♦♥té♠ ❧✐♥❤❛s ❡ ❘ ♣♦ss✉✐ ❛ ♠étr✐❝❛ ❝❛♥ô♥✐❝❛✳

♥

❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ s❡ ▼ ♣♦ss✉✐r ✉♠❛ ❧✐♥❤❛✱ ❡♥tã♦ ♣❡❧♦ t❡♦r❡♠❛

♥ ✶

❛♥t❡r✐♦r✱ ▼ é ✐s♦♠étr✐❝❛ ❛ ✉♠ ♣r♦❞✉t♦ ❘ ✂ ◆ ✳ ❙❡✱ ❛❧é♠ ❞✐ss♦✱ ◆ ♣♦ss✉✐r ✉♠❛ ❧✐♥❤❛✱ ❡♥tã♦ ✉s❛♠♦s ♦ t❡♦r❡♠❛ ♠❛✐s ✉♠❛ ✈❡③✱ ❥á q✉❡ ❘✐❝✭◆✮ ✕ ✵ ❡ ◆ é ❝♦♠♣❧❡t❛✳ ❖ r❡s✉❧t❛❞♦ s❡❣✉❡ ♣♦r ✐♥❞✉çã♦✳

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶❪ ❈❤❡❡❣❡r✱ ❏✳✱ ●r♦♠♦❧❧✱ ❉✳✱ ❚❤❡ ❙♣❧✐tt✐♥❣ ❚❤❡♦r❡♠ ❢♦r ▼❛♥✐❢♦❧❞s ♦❢ ◆♦♥♥❡❣❛t✐✈❡ ❘✐❝❝✐ ❈✉r✈❛t✉r❡✱ ❏✳ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr②✱ ✭✶✾✼✶✮✱ ✶✶✾✕✶✷✽✳

❬✷❪ ❞♦ ❈❛r♠♦✱ ▼✳ P✳✱ ●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧ ❞❡ ❈✉r✈❛s ❡ ❙✉♣❡r❢í❝✐❡s✱ ❚❡①t♦s ❯♥✐✈❡rs✐tár✐♦s✱ ❙❇▼✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ✷

❛

❡❞✐çã♦✱ ✷✵✵✺✳ ❬✸❪ ❞♦ ❈❛r♠♦✱ ▼✳ P✳✱ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✱ Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ❘✐♦ ❞❡

❏❛♥❡✐r♦✱ ✸

❛

❡❞✐çã♦✱ ✷✵✵✺✳ ❬✹❪ ❊s❝❤❡♥❜✉r❣✱ ❏✳✲❍✳✱ ❍❡✐♥t③❡✱ ❊✳✱ ❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ ❈❤❡❡❣❡r✲

●r♦♠♦❧❧ s♣❧✐tt✐♥❣ t❤❡♦r❡♠✱ ❆♥♥✳ ●❧♦❜✳ ❆♥❛❧②s✐s ❛♥❞ ●❡♦♠❡tr②✱ ✈♦❧✳ ✷✱ ✭✶✾✽✹✮✱ ✶✹✶✕✶✺✶✳

❬✺❪ ❊s❝♦❜❛r✱ ❏✳ ❋✳✱ ❚♦♣✐❝s ✐♥ P❉❊✬❙ ❛♥❞ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr②✱ ❳■■ ❊s❝♦❧❛ ❞❡ ●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧✱ ✷✵✵✷✳

❬✻❪ ❊✈❛♥s✱ ▲✳ ❈✳✱ ●❛r✐❡♣②✱ ❘✳ ❋✳✱ ▼❡❛s✉r❡ ❚❤❡♦r② ❛♥❞ ❋✐♥❡ Pr♦♣❡rt✐❡s ♦❢ ❋✉♥❝t✐♦♥s✱ ❈❘❈ Pr❡ss✱ ✶✾✾✷✳

❬✼❪ ❊✈❛♥s✱ ▲✳ ❈✳✱ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ●r❛❞✉❛t❡ ❙t✉❞✐❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✉♠❡ ✶✾✱ ✷✵✵✷✳

❬✽❪ ▲❡❡✱ ❏✳ ▼✳✱ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ✲ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈✉r✈❛t✉r❡✱ ❙♣r✐♥❣❡r✱ ✶✾✾✼✳

❬✾❪ ❙❝❤♦❡♥✱ ❘✳✱ ❨❛✉✱ ❙✳ ❚✳✱ ▲❡❝t✉r❡s ♦♥ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr②✱ ■♥t❡r♥❛t✐♦♥❛❧ Pr❡ss ■♥❝✱ ❇♦st♦♥✱ ✈♦❧✳ ✶✱ ✶✾✾✹✳

❬✶✵❪ ❙♣✐✈❛❦✱ ▼✳✱ ❆ ❈♦♠♣r❡❤❡♥s✐✈❡ ■♥tr♦❞✉❝t✐♦♥ t♦ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr②✱ P✉❜❧✐s❤ ♦r P❡r✐s❤✱ ■◆❈✳ ❍♦✉st♦♥✱ ❚❡①❛s ✭❯✳❙✳❆✳✮✱ ❱♦❧✉♠❡ ■■✱ ✶✾✼✵✳

❮♥❞✐❝❡ ❘❡♠✐ss✐✈♦

❛♣❧✐❝❛çã♦ ❡①♣♦♥❡♥❝✐❛❧✱ ✶✺✱ ✶✼ ❝❛♠♣♦

❞❡ ✈❡t♦r❡s✱ ✾✱ ✶✵✱ ✶✾ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s✱ ✷✻ ♣❛r❛❧❡❧♦✱ ✶✺✱ ✹✷

❝♦❡✜❝✐❡♥t❡s ❞❛ ♠étr✐❝❛✱ ✶✸✱ ✷✼ ❞❡ ❘✐❝❝✐✱ ✶✾✱ ✷✶

❝♦♥❡①ã♦✱ ✾✱ ✶✸ ❛✜♠✱ ✶✸✱ ✶✹ ❝♦♠♣❛tí✈❡❧✱ ✶✹ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✱ ✶✹ s✐♠étr✐❝❛✱ ✶✹

❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s✱ ✶✵

❝✉r✈❛t✉r❛✱ ✶✼ ❞❡ ❘✐❝❝✐✱ ✶✽✱ ✶✾✱ ✸✵✱ ✸✶✱ ✸✹✱ ✸✻✱

✹✶✱ ✹✸ ❡s❝❛❧❛r✱ ✶✽ s❡❝❝✐♦♥❛❧✱ ✶✽✱ ✷✾✱ ✸✶

❞✐❢❡♦♠♦r✜s♠♦✱ ✶✷✱ ✶✻ ❡s❢❡r❛ ✉♥✐tár✐❛✱ ✶✶ ❡s♣❛ç♦

❤✐♣❡r❜ó❧✐❝♦✱ ✶✶✱ ✷✾ ♠étr✐❝♦✱ ✶✼ t❛♥❣❡♥t❡✱ ✾

❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✱ ✷✸ ✜❜r❛❞♦ t❛♥❣❡♥t❡✱ ✶✸ ❢✉♥çã♦ ❞❡ ❇✉s❡♠❛♥♥✱ ✸✺ ❣❡♦❞és✐❝❛✱ ✽✱ ✶✺✕✶✼✱ ✷✺✱ ✷✻✱ ✸✵✱ ✸✹✱ ✸✻ ■❞❡♥t✐❞❛❞❡ ❞❡ ❘✐❝❝✐✱ ✷✶ ✐♠❡rsã♦✱ ✶✶✱ ✶✷

✐s♦♠étr✐❝❛✱ ✶✷ ✐s♦♠❡tr✐❛✱ ✶✷✱ ✹✷ ▲❡♠❛

❞❡ ●❛✉ss✱ ✷✺ ❞❡ ❙✐♠❡tr✐❛✱ ✹✷

❧✐♥❤❛✱ ✸✹✱ ✹✶ ♠étr✐❝❛

❝❛♥ô♥✐❝❛✱ ✷✻✱ ✹✸ ❞❛ ❡s❢❡r❛✱ ✷✾ ❞♦ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦✱ ✷✾ ✐♥❞✉③✐❞❛✱ ✶✷ ♣r♦❞✉t♦✱ ✶✷✱ ✹✶ ❘✐❡♠❛♥♥✐❛♥❛✱ ✶✶✕✶✹

♠❡r❣✉❧❤♦✱ ✶✷ ♥♦t❛çã♦ ❞❡ ❊✐♥st❡✐♥✱ ✶✾ ♣❛r❛♠❡tr✐③❛çã♦✱ ✾

♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ✶✶✱ ✶✽ r❛✐♦✱ ✸✹ sí♠❜♦❧♦s ❞❡ ❈❤r✐st♦✛❡❧✱ ✶✸✱ ✶✾✱ ✷✸✱ ✷✼ s✉❜✈❛r✐❡❞❛❞❡✱ ✶✷✱ ✸✻ ❚❡♦r❡♠❛

❞❛ ❢✉♥çã♦ ✐♥✈❡rs❛✱ ✶✻ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦✱

✸✵✱ ✸✻ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲

●r♦♠♦❧❧✱ ✹✶ ❞❡ ❍♦♣❢✲❘✐♥♦✇✱ ✶✼ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✱ ✶✹ t❡♦r❡♠❛ ❞❡ ❙❝❤✇❛rt③✱ ✶✵ t♦r♦ ♣❧❛♥♦✱ ✶✷

✈❛r✐❡❞❛❞❡ ❞❡ ❍❛✉s❢♦r✛✱ ✶✷ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ✾✱ ✶✶✱ ✶✸ ✐♠❡rs❛✱ ✶✶ ❘✐❡♠❛♥♥✐❛♥❛✱ ✶✶✱ ✶✷✱ ✶✹✱ ✶✼✱ ✶✽✱

✸✹✱ ✹✶ ❝♦♠♣❧❡t❛✱ ✽✱ ✶✼✱ ✸✵✱ ✸✹✱ ✸✻✱ ✸✼✱

✹✵✱ ✹✶✱ ✹✸