❈á❧❝✉❧♦ ✶
❙✳ ❋r✐❡❞❧✐
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s
❱❡rsã♦ ✶✳✵ ✶✻ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸
❙✉♠ár✐♦
✶ ❋✉♥❞❛♠❡♥t♦s ✸
✶✳✶ ◆ú♠❡r♦s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ❊q✉❛çõ❡s ❞♦ ♣r✐♠❡✐r♦ ❡ s❡❣✉♥❞♦ ❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ❖r❞❡♠ ❡ ✐♥t❡r✈❛❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✸ ❱❛❧♦r ❛❜s♦❧✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✹ ■♥❡q✉❛çõ❡s ❡ s✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❖ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✶ ❘❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✷ ❈ír❝✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸ ❚r✐❣♦♥♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳✶ ▼❡❞✐r â♥❣✉❧♦s ♥♦ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳✷ ❙❡♥♦✱ ❝♦ss❡♥♦ ❡ t❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✸ ■❞❡♥t✐❞❛❞❡s tr✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷ ❋✉♥çõ❡s ✷✶
✷✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✶ ▲✐♠✐t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ●rá✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✳✶ P♦tê♥❝✐❛s ✐♥t❡✐r❛s✿ ①♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✷✳✷✳✷ P❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✸ ❋✉♥çõ❡s ❚r✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷✳✹ ❚r❛♥s❢♦r♠❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸ ▼♦♥t❛r ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹ ❈♦♠♣♦s✐çã♦✱ ❝♦♥tr❛❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✹✳✶ ❇✐❥❡çã♦✱ ❢✉♥çã♦ ✐♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✹✳✷ ■♥✈❡rs♦s ❞❛s ♣♦tê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✹✳✸ ❋✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸ ❊①♣♦♥❡♥❝✐❛❧ ❡ ▲♦❣❛r✐t♠♦ ✹✺
✸✳✶ ❊①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✷ ▲♦❣❛r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✸ ❆ ❜❛s❡ ❡ ❂ ✷❀ ✼✶✽✿✿✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✹ ❋✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❤✐♣❡r❜ó❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
❙❯▼➪❘■❖
✹ ▲✐♠✐t❡s ✺✾
✹✳✶ ▲✐♠✐t❡s ❧✐♠①✦✝✶❢✭①✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✹✳✶✳✶ ❆ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✶✳✷ ▲✐♠✐t❡s ✐♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✹✳✶✳✸ ◗✉♦❝✐❡♥t❡s ❡ ❞✐❢❡r❡♥ç❛s ❞❡ ♥ú♠❡r♦s ❣r❛♥❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✷ ▲✐♠✐t❡s ❧❛t❡r❛✐s✿ ❧✐♠①✦❛✝❢✭①✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
✹✳✸ ▲✐♠✐t❡s ❧✐♠①✦❛❢✭①✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✹✳✹ ■♥❞❡t❡r♠✐♥❛çõ❡s ❞♦ t✐♣♦ ✏✵
✵✑ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
✹✳✹✳✶ ❖ ❧✐♠✐t❡ ❧✐♠❤✦✵✭①✰❤✮❤♥ ①♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
✹✳✹✳✷ ❖ ❧✐♠✐t❡ ❧✐♠①✦✵ s❡♥ ①① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
✹✳✺ ▲✐♠✐t❡s ❧❛t❡r❛✐s ✐♥✜♥✐t♦s✱ ❛ssí♥t♦t❛s ✈❡rt✐❝❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✻ ▼✉❞❛r ❞❡ ✈❛r✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✼ ❖ ❧✐♠✐t❡ ❡ ❂ ❧✐♠①✦✶
✏
✶ ✰ ✶ ①
✑①
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✹✳✽ ❈♦♥t✐♥✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✹✳✽✳✶ ❖ ❚❡♦r❡♠❛ ❞♦ ✈❛❧♦r ✐♥t❡r♠❡❞✐ár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✹✳✾ ▲✐♠✐t❡s ❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✹✳✶✵ ❊①❡r❝í❝✐♦s ❞❡ r❡✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹
✺ ❉❡r✐✈❛❞❛ ✽✼
✺✳✶ ❘❡t❛s ❡ ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✺✳✷ ❘❡t❛ t❛♥❣❡♥t❡ ❡ ❞❡r✐✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✺✳✷✳✶ P♦♥t♦s ❞❡ ♥ã♦✲❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✺✳✷✳✷ ❉❡r✐✈❛❜✐❧✐❞❛❞❡ ❡ ❝♦♥t✐♥✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✺✳✸ ❆ ❞❡r✐✈❛❞❛ ❝♦♠♦ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✺✳✸✳✶ ❉❡r✐✈❛r ❛s ♣♦tê♥❝✐❛s ✐♥t❡✐r❛s✿ ①♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹
✺✳✸✳✷ ❉❡r✐✈❛r ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✺✳✸✳✸ ❉❡r✐✈❛r ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛r✐t♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✺✳✹ ❘❡❣r❛s ❞❡ ❞❡r✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✺✳✹✳✶ ❉❡r✐✈❛r ❛s ♣♦tê♥❝✐❛s ①☛❀ ❡①♣♦♥❡♥❝✐❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾
✺✳✹✳✷ ❉❡r✐✈❛❞❛s ❧♦❣❛rít♠✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✺✳✹✳✸ ❉❡r✐✈❛r ✉♠❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✺✳✺ ❖ ❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ✺✳✻ ❉❡r✐✈❛❞❛ ❡ ❱❛r✐❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸ ✺✳✼ ▲✐♥❡❛r✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻ ✺✳✽ ❉❡r✐✈❛çã♦ ✐♠♣❧í❝✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ✺✳✾ ❚❛①❛ ❞❡ ✈❛r✐❛çã♦✱ ✈❡❧♦❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶ ✺✳✾✳✶ ❚❛①❛s r❡❧❛❝✐♦♥❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✹ ✺✳✶✵ ❈♦♥✈❡①✐❞❛❞❡✱ ❝♦♥❝❛✈✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺ ✺✳✶✶ ❱❛❧♦r❡s ❡①tr❡♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽ ✺✳✶✶✳✶ ❊①tr❡♠♦s ❣❧♦❜❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽ ✺✳✶✶✳✷ ❊①tr❡♠♦s ❧♦❝❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵ ✺✳✶✶✳✸ ❆ ♣r♦❝✉r❛ ❞❡ ❡①tr❡♠♦s ❡♠ ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✷ ✺✳✶✶✳✹ Pr♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸ ✺✳✶✶✳✺ ❆ ▲❡✐ ❞❡ ❙♥❡❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✺✳✶✷ ❆ ❘❡❣r❛ ❞❡ ❇❡r♥♦✉❧❧✐✲❧✬❍ô♣✐t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✽ ✺✳✶✷✳✶ ❙♦❜r❡ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❛s ❢✉♥çõ❡s ♥♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸
❙❯▼➪❘■❖
✺✳✶✸ ❆ssí♥t♦t❛s ♦❜❧íq✉❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✹ ✺✳✶✹ ❊st✉❞♦s ❞❡ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✼
✻ ■♥t❡❣r❛❧ ✶✹✶
✻✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✶ ✻✳✷ ❆ ✐♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✺ ✻✳✸ ❖ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✼ ✻✳✹ ➪r❡❛s ❞❡ r❡❣✐õ❡s ❞♦ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✵ ✻✳✺ Pr✐♠✐t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷ ✻✳✺✳✶ ■♥t❡❣r❛çã♦ ♣♦r ❙✉❜st✐t✉✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✸ ✻✳✺✳✷ ■♥t❡❣r❛çã♦ ♣♦r P❛rt❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻ ✻✳✻ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✾ ✻✳✼ ❙ó❧✐❞♦s ❞❡ r❡✈♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✵ ✻✳✼✳✶ ❆♣r♦①✐♠❛çã♦ ♣♦r ❝✐❧✐♥❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✶ ✻✳✼✳✷ ❆♣r♦①✐♠❛çã♦ ♣♦r ❝❛s❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✹ ✻✳✼✳✸ ❊①❡r❝í❝✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽ ✻✳✽ ➪r❡❛s ❞❡ s✉♣❡r❢í❝✐❡s ❞❡ r❡✈♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽ ✻✳✾ ■♥t❡❣r❛çã♦ ❞❡ ❢✉♥çõ❡s r❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✾ ✻✳✶✵ ■♥t❡❣r❛✐s ■♠♣ró♣r✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✹ ✻✳✶✵✳✶ ■♥t❡❣r❛✐s ✐♠♣ró♣r✐❛s ❡♠ ✐♥t❡r✈❛❧♦s ✐♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✹ ✻✳✶✵✳✷ ❆s ✐♥t❡❣r❛✐s ❘✶
❛ ❞①①♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✻
✻✳✶✵✳✸ ❖ ❝r✐tér✐♦ ❞❡ ❝♦♠♣❛r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✽ ✻✳✶✵✳✹ ■♥t❡❣r❛✐s ✐♠♣ró♣r✐❛s ❡♠ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✾
✻✳✶✵✳✺ ■♥t❡❣r❛✐s ✐♠♣ró♣r✐❛s ❡♠ ✐♥t❡r✈❛❧♦s ✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✵ ✻✳✶✶ ■♥t❡❣r❛r ♣♦tê♥❝✐❛s ❞❡ ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✶ ✻✳✶✶✳✶ Pr✐♠✐t✐✈❛s ❞❛s ❢✉♥çõ❡s s❡♥♠① ❝♦s♥① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✶
✻✳✶✶✳✷ Pr✐♠✐t✐✈❛s ❞❛s ❢✉♥çõ❡s t❛♥♠① s❡❝♥① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✸
✻✳✶✷ ❙✉❜st✐t✉✐çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✹ ✻✳✶✷✳✶ ❆ ♣r✐♠✐t✐✈❛ ❘ ♣✶ ①✷❞① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✹
✻✳✶✷✳✷ ❆ ♣r✐♠✐t✐✈❛ ❘ ♣✶ ✰ ①✷❞① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✼
✻✳✶✷✳✸ ❆ ♣r✐♠✐t✐✈❛ ❘ ♣①✷ ✶ ❞① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✽
❆ ❙♦❧✉çõ❡s ❞♦s ❊①❡r❝í❝✐♦s ✶✽✾
❱❡rsã♦ ✶✳✵ ✭✶✻ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸✮✳ ❙✉❣❡stõ❡s✱ ❝rít✐❝❛s ❡ ❝♦rr❡çõ❡s✿ s❛❝❤❛❅♠❛t✳✉❢♠❣✳❜r
❙❯▼➪❘■❖
Pr❡❢á❝✐♦
❖r✐✉♥❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞♦ ❡st✉❞♦ ❞❛ ♠❡❝â♥✐❝❛ ❡ ❞❛ ❛str♦♥♦♠✐❛✱ ♦ ❈á❧❝✉❧♦✱ ❝❤❛♠❛❞♦ t❛♠❜é♠ ❈á❧❝✉❧♦ ✐♥✜♥✐t❡s✐♠❛❧✱ ♥❛s❝❡✉ ♥♦ ✜♠ ❞♦ sé❝✉❧♦ ❳❱■■✱ ❝♦♠ ♦s tr❛❜❛❧❤♦s ❞❡ ◆❡✇t♦♥ ✶ ❡ ▲❡✐❜♥✐③ ✷✳ ❍♦❥❡ ❡♠ ❞✐❛✱ ❡❧❡ é ✉s❛❞♦ ❡♠ t♦❞❛s ❛s ár❡❛s ❞❛ ❝✐ê♥❝✐❛✱ ❡ ❢✉♥❞❛✲
♠❡♥t❛❧ ♥❛s ár❡❛s ❞❛ ❡♥❣❡♥❤❛r✐❛✳
❆ ♣r❡s❡♥t❡ ❛♣♦st✐❧❛ ❝♦♥té♠ ❛ ❡♠❡♥t❛ ❞❛ ♠❛tér✐❛ ❈á❧❝✉❧♦ ■✱ ❝♦♠♦ ❡♥s✐♥❛❞❛ ♥♦ ❉❡✲ ♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋▼●✳ ❊❧❛ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❢♦r♥❡❝❡r ❛♦ ❛❧✉♥♦ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❜ás✐❝♦ ❞♦s ❝♦♥❝❡✐t♦s ♣r✐♥❝✐♣❛✐s ❞♦ ❈á❧❝✉❧♦ q✉❡ sã♦✿ ❧✐♠✐t❡s✱ ❞❡r✐✈❛❞❛s ❡ ✐♥t❡❣r❛❧✳ ❊❧❛ t❛♠❜é♠ ♣r❡♣❛r❛ ♦ ❛❧✉♥♦ ♣❛r❛ ❛s ♦✉tr❛s ♠❛tér✐❛s q✉❡ ✉s❛♠ ❈á❧❝✉❧♦ ■ ♥♦s ❝✉rs♦s ❞❡ ❝✐ê♥❝✐❛s ❡①❛t❛s ✭❢ís✐❝❛ ❡ ♠❛t❡♠át✐❝❛✮ ❡ ❡♥❣❡♥❤❛r✐❛✱ t❛✐s ❝♦♠♦ ❈á❧❝✉❧♦ ■■ ❡ ■■■✱ ❊❉❆✱ ❊❉❇✱ ❊❉❈✳✳✳
❆ ❛♣♦st✐❧❛ ❝♦♠❡ç❛ ❝♦♠ ✉♠ ❝❛♣ít✉❧♦ s♦❜r❡ ❢✉♥❞❛♠❡♥t♦s✱ ❢❛③❡♥❞♦ ✉♠❛ r❡✈✐sã♦ ❞❡ ✈ár✐♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❡♠ ♣r✐♥❝í♣✐♦ ❥á ❝♦♥❤❡❝✐❞♦s ♣❡❧♦ ❛❧✉♥♦✿ ❡q✉❛çõ❡s✱ ✐♥❡q✉❛çõ❡s✱ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❡ tr✐❣♦♥♦♠❡tr✐❛✳ ❆ ♣❛rt✐r ❞♦ ❈❛♣ít✉❧♦ ✷✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ é ✐♥tr♦❞✉③✐❞♦✳ ❆ ♥♦çã♦ ❝❡♥tr❛❧ ❞❡ ❧✐♠✐t❡ é ❛❜♦r❞❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✹✱ ❡ ❛ ❞❡ ❞❡r✐✈❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✺✳ ❖ r❡st♦ ❞♦ t❡①t♦ é s♦❜r❡ ♦ ♦❜❥❡t♦ ❝❡♥tr❛❧ ❞❡ss❡ ❝✉rs♦✿ ❛ ♥♦çã♦ ❞❡ ✐♥t❡❣r❛❧✱ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦✱ ❡ ❛s s✉❛s ❛♣❧✐❝❛çõ❡s✳
❖ t❡①t♦ ❝♦♥té♠ ❜❛st❛♥t❡ ❡①❡r❝í❝✐♦s✱ ❝✉❥❛ ❝♦♠♣r❡❡♥sã♦ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ❛ss✐♠✐✲ ❧❛çã♦ ❞♦s ❝♦♥❝❡✐t♦s✳ ❆s s♦❧✉çõ❡s✱ às ✈❡③❡s ❞❡t❛❧❤❛❞❛s✱ s❡ ❡♥❝♦♥tr❛♠ ♥✉♠ ❛♣ê♥❞✐❝❡✳
❊ss❛ ❛♣♦st✐❧❛ ❡stá ❡♠ ❢❛s❡ ❞❡ ❡❧❛❜♦r❛çã♦✳ ◗✉❛❧q✉❡r s✉❣❡stã♦✱ ❝rít✐❝❛ ♦✉ ❝♦rr❡çã♦ é ❜❡♠ ✈✐♥❞❛✿ s❛❝❤❛❅♠❛t✳✉❢♠❣✳❜r✳
❆❣r❛❞❡ç♦ às s❡❣✉✐♥t❡ ♣❡ss♦❛s ♣❡❧❛s s✉❛s ❝♦♥tr✐❜✉✐çõ❡s✱ ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s✿ ❊✉❧❧❡r ❚❡r❣✐s ❙❛♥t♦s ❇♦r❣❡s✱ ❋❡❧✐♣❡ ❞❡ ▲✐♠❛ ❍♦rt❛ ❘❛❞✐❝❝❤✐✱ ❋❡r♥❛♥❞❛ ❞❡ ❈❛str♦ ▼❛✐❛✱ ▼❛r✐♥❛ ❲❡r♥❡❝❦ ❘❛❣♦③♦✱ ▼❛r✐❛♥❛ ❈❤❛♠♦♥ ▲❛❞❡✐r❛ ❆♠❛♥❝✐♦✱ P❡❞r♦ ❙✐❧✈❡✐r❛ ●♦♠❡s ❞❡ P❛✐✈❛✱ ❚♦✉✜❝ ▼❛❤♠❡❞ P♦tt✐❡r ▲❛✉❛r✱ Pr♦❢✳ ❈❛r❧♦s ▼❛r✐❛ ❈❛r❜❛❧❧♦✱ Pr♦❢✳ ❋á❜✐♦ ❳❛✈✐❡r P❡♥♥❛ ✭❯◆■❘■❖✮✱ Pr♦❢✳ ❋r❛♥❝✐s❝♦ ❉✉t❡♥❤❡❢♥❡r✱ Pr♦❢✳ ❍❛♠✐❧t♦♥ Pr❛❞♦ ❇✉❡♥♦✱ Pr♦❢✳ ❏♦r❣❡ ❙❛❜❛t✉❝❝✐✱ Pr♦❢❛✳ ❱✐✈✐❛♥❡ ❘✐❜❡✐r♦ ❚♦♠❛③ ❞❛ ❙✐❧✈❛✱ Pr♦❢✳ ❱✐❦t♦r ❇❡❦❦❡rt✳
✶❙✐r ■s❛❛❝ ◆❡✇t♦♥ ✭❲♦♦❧st❤♦r♣❡✲❜②✲❈♦❧st❡r✇♦rt❤✱ ✹ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✶✻✹✸ ✖ ▲♦♥❞r❡s✱ ✸✶ ❞❡ ♠❛rç♦ ❞❡
✶✼✷✼✮✳
✷●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ✈♦♥ ▲❡✐❜♥✐③ ✭▲❡✐♣③✐❣✱ ✶ ❞❡ ❥✉❧❤♦ ❞❡ ✶✻✹✻ ✖ ❍❛♥ô✈❡r✱ ✶✹ ❞❡ ♥♦✈❡♠❜r♦ ❞❡ ✶✼✶✻✮✳
❙❯▼➪❘■❖
❈❛♣ít✉❧♦ ✶
❋✉♥❞❛♠❡♥t♦s
✏❆ ❣♦♦❞ ❝♦✉rs❡ ✐s ❛ ❝♦✉rs❡ ✇✐t❤ ♠❛♥② st✉♣✐❞ q✉❡st✐♦♥s✳✑
❲❡♥❞❡❧✐♥ ❲❡r♥❡r✱ ♠❡❞❛❧❤✐st❛ ❋✐❡❧❞s ✷✵✵✻ ✏◗✉❡♠ ❢❛③ ✉♠❛ ♣❡r❣✉♥t❛ ❜♦❜❛ ✜❝❛ ❝♦♠ ✈❡r❣♦♥❤❛ ✺ s❡❣✉♥❞♦s✳ ◗✉❡♠ ♥ã♦ ♣❡r❣✉♥t❛ ♥❛❞❛ ✜❝❛ ❜♦❜♦ ♣❛r❛ s❡♠♣r❡✳✳✳✑
❯♠ ❢❛①✐♥❡✐r♦ ❞♦ ■❈❊①✱ ✷✵✵✽ ❈á❧❝✉❧♦ ❧✐❞❛ ❝♦♠ ❢✉♥çõ❡s ❞❡ ✉♠❛ ♦✉ ♠❛✐s ✈❛r✐á✈❡✐s r❡❛✐s✳ P♦rt❛♥t♦✱ ❡❧❡ ♥❡❝❡ss✐t❛ ❞❡ ✉♠❛ ❝♦♠♣r❡❡♥sã♦ ❜♦❛ ❞❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❡ s✉❛s ♠❛♥✐♣✉✲ ❧❛çõ❡s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡❧❡♠❡♥t❛r❡s✳
❊ss❡ ❝❛♣ít✉❧♦ ❝♦♥té♠ ❧❡♠❜r❡t❡s s♦❜r❡ ❛ ❛r✐t♠ét✐❝❛ ❡❧❡♠❡♥t❛r ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❛s✲ s✐♠ ❝♦♠♦ ❛ ❞❡s❝r✐çã♦ ❞❡ ❝❡rt♦s ❝♦♥❥✉♥t♦s ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ❝♦♠♦ r❡t❛s ❡ ❝ír❝✉❧♦s✳ ◆ã♦ ♣r❡t❡♥❞❡♠♦s ❞❛r ✉♠❛ ❡①♣♦s✐çã♦ ❝♦♠♣❧❡t❛ s♦❜r❡ ❡ss❡s ❛ss✉♥t♦s✱ ♠❛s ❛♣❡♥❛s ❧❡♠❜r❛r ❛❧❣✉♥s ❢❛t♦s ❡ ❡st❛❜❡❧❡❝❡r ♥♦t❛çõ❡s ❛ r❡s♣❡✐t♦ ❞❡ ❝♦✐s❛s ❡❧❡♠❡♥t❛r❡s ❝♦♥❤❡❝✐❞❛s ♣❡❧♦ ❧❡✐t♦r✳
❆ ♠❛tér✐❛ ❞❡ss❡ ❝❛♣ít✉❧♦ s❡rá ✉s❛❞❛ ❝♦♥st❛♥t❡♠❡♥t❡ ♥♦ r❡st❛♥t❡ ❞❛ ❛♣♦st✐❧❛✿ é ✐♠♣♦r✲ t❛♥t❡ ♦ ❧❡✐t♦r ✈❡r✐✜❝❛r q✉❡ ❡❧❡ ❝♦♥s❡❣✉❡ ❢❛③❡r t♦❞♦s ♦s ❡①❡r❝í❝✐♦s✳
✶✳✶ ◆ú♠❡r♦s r❡❛✐s
❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ R✱ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❛ ❧✐♥❤❛
r❡❛❧✱ q✉❡ s❡rã♦ ❡♠ ❣❡r❛❧ ❞❡♥♦t❛❞♦s ♣♦r ❧❡tr❛s ♠✐♥ús❝✉❧❛s✿ ①❀ ②❀ s❀ t❀ ✉✱ ❡t❝✳ R é ♠✉♥✐❞♦
❞❡ q✉❛tr♦ ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s ❜ás✐❝❛s✿ ❛❞✐çã♦ ✭✰✮✱ s✉❜tr❛çã♦ ✭ ✮✱ ♠✉❧t✐♣❧✐❝❛çã♦ ✭✂ ♦✉ ✁✮ ❡ ❞✐✈✐sã♦ ✭☎✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❂✮✳
▲❡♠❜r❡♠♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❝♦♠ ♣❛♣❡❧ r❡❧❡✈❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ à ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✳ Pr✐♠❡✐r♦✱ ♦ ❡❧❡♠❡♥t♦ ✵ ✭✏③❡r♦✑✮ é t❛❧ q✉❡ ①✰✵ ❂ ✵✰① ❂ ①✱ ①✁✵ ❂ ✵✁① ❂ ✵ ♣❛r❛ t♦❞♦ ①✳ ❯♠ r❡❛❧ ① ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ s❡rá às ✈❡③❡s ❝❤❛♠❛❞♦ ❞❡ ♥ã♦✲♥✉❧♦✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❡❧❡♠❡♥t♦ ✶ ✭✏✉♠✑✮ é t❛❧ q✉❡ ① ✁ ✶ ❂ ✶ ✁ ① ❂ ① ♣❛r❛ t♦❞♦ ① ✷ R✳ ➱
✐♠♣♦rt❛♥t❡ ❧❡♠❜r❛r q✉❡ ❛ ❞✐✈✐sã♦ ♣♦r ③❡r♦ ♥ã♦ é ❞❡✜♥✐❞❛✳ P♦rt❛♥t♦✱ sí♠❜♦❧♦s ❞♦ t✐♣♦
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
①❂✵ ♦✉ ✵❂✵ ♥ã♦ ❢❛③❡♠ s❡♥t✐❞♦✳ ◆♦ ❡♥t❛♥t♦✱ ✵❂① ❂ ✵ ♣❛r❛ t♦❞♦ ① ✻❂ ✵✳
❖s s✉❜❝♦♥❥✉♥t♦s ❞❡Rs❡rã♦ ❡♠ ❣❡r❛❧ ❞❡♥♦t❛❞♦s ✉s❛♥❞♦ ❧❡tr❛s ♠❛✐ús❝✉❧❛s✳ P♦r ❡①❡♠♣❧♦✱
❆ ❂ ❢✵❀ ✶❀ ✷❣ é ♦ ❝♦♥❥✉♥t♦ q✉❡ ❝♦♥té♠ ♦s três ♥ú♠❡r♦s r❡❛✐s ✵❀ ✶ ❡ ✷✱ ❡ ❇ ❂ ✭✵❀ ✷✮ é ♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ q✉❡ ❝♦♥té♠ t♦❞♦s ♦s r❡❛✐s ❡♥tr❡ ✵ ❡ ✷ ✭✈❡r ❛❜❛✐①♦✮✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s é ✶
N✿❂❢✶❀ ✷❀ ✸❀ ✿ ✿ ✿ ❣ ❀
❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s é
Z✿❂❢✿ ✿ ✿ ❀ ✸❀ ✷❀ ✶❀ ✵❀ ✶❀ ✷❀ ✸❀ ✿ ✿ ✿ ❣ ✿
❆s ♦♣❡r❛çõ❡s ❡♥tr❡ ❝♦♥❥✉♥t♦s sã♦✿ ✐♥t❡rs❡çã♦ ✭❭✮✱ ✉♥✐ã♦ ✭❬✮✱ ❞✐❢❡r❡♥ç❛ ✭♥✮✳ ❖ ❝♦♥✲ ❥✉♥t♦ ✈❛③✐♦ s❡rá ❞❡♥♦t❛❞♦ ♣♦r ∅✳
✶✳✶✳✶ ❊q✉❛çõ❡s ❞♦ ♣r✐♠❡✐r♦ ❡ s❡❣✉♥❞♦ ❣r❛✉
❈♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞♦ ♣r✐♠❡✐r♦ ❣r❛✉✿
✶ ✰ ✹① ❂ ✼ ✿ ✭✶✳✶✮
❘❡s♦❧✈❡r ❡ss❛ ❡q✉❛çã♦ s✐❣♥✐✜❝❛ ❛❝❤❛r ♦✭s✮ ✈❛❧♦r✭❡s✮ ❞❛ ✈❛r✐á✈❡❧ ① ♣❛r❛ ♦s q✉❛✐s ❛ ✐❣✉❛❧✲ ❞❛❞❡ ❡♠ ✭✶✳✶✮ é ✈❡r❞❛❞❡✐r❛✳ ❊ss❡ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s s❡rá ❞❡♥♦t❛❞♦ ♣♦r ❙ ❡ ❝❤❛♠❛❞♦ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s✳ ❆ r❡s♦❧✉çã♦ é ❜❡♠ ❝♦♥❤❡❝✐❞❛✿ ✐s♦❧❛♥❞♦ ① ♦❜t❡♠♦s ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ① ❂ ✷✳ P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s s♦❧✉çõ❡s ❞❡ ✭✶✳✶✮ é ❙ ❂ ❢ ✷❣✳
❈♦♥s✐❞❡r❡ ❡♠ s❡❣✉✐❞❛ ❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✿
①✷ ❂ ✾ ✿ ✭✶✳✷✮
❆q✉✐✱ s❛❜❡♠♦s q✉❡ ❡①✐st❡♠ ❞✉❛s s♦❧✉çõ❡s✱ ① ❂ ✝♣✾ ❂ ✝✸✱ ❧♦❣♦ ❙ ❂ ❢✰✸❀ ✸❣✳
❆❣♦r❛✱ ❥á q✉❡ ✉♠ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦ ♥ã♦ ♣♦ss✉✐ r❛✐③ q✉❛❞r❛❞❛✱ ❛ ❡q✉❛çã♦ ①✷ ❂ ✹
♥ã♦ ♣♦ss✉✐ ♥❡♥❤✉♠❛ s♦❧✉çã♦ r❡❛❧✿ ❙ ❂∅✳ ❋✐♥❛❧♠❡♥t❡✱
①✷ ❂ ✵
♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦✿ ❙ ❂ ❢✵❣✳
❯♠ ♦✉tr♦ ❥❡✐t♦ ❞❡ ❡♥t❡♥❞❡r ✭✶✳✷✮ é ❞❡ ❡s❝r❡✈ê✲❧❛ ①✷ ✾ ❂ ✵ ❡ ❞❡ ❢❛t♦r❛r ♦ ♣♦❧✐♥ô♠✐♦
①✷ ✾✱ ♦❜t❡♥❞♦ ✉♠ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❢❛t♦r❡s✿
✭① ✸✮✭① ✰ ✸✮ ❂ ✵ ✿
✶❆♦ ❧♦♥❣♦ ❞❛ ❛♣♦st✐❧❛✱ ♦ sí♠❜♦❧♦ ✏✿❂✑ s❡rá ✉s❛❞♦ ♣❛r❛ ❞❡✜♥✐r ✉♠ ♦❜❥❡t♦✳ P♦r ❡①❡♠♣❧♦✱ ❆✿❂❢① ✷R✿
①✷❃ ✶❣ s✐❣♥✐✜❝❛ q✉❡ ❆ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❝✉❥♦ q✉❛❞r❛❞♦ é ♠❛✐♦r q✉❡ ✶✳
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
P❛r❛ ♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❢❛t♦r❡s ✭❛q✉✐✱ ① ✸ ❡ ① ✰ ✸✮ s❡r ③❡r♦✱ é ♥❡❝❡ssár✐♦ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s s❡❥❛ ♥✉❧♦✳ ❙❡ ❢♦r ♦ ♣r✐♠❡✐r♦✱ ① ✸ ❂ ✵✱ ❡♥tã♦ ① ❂ ✸✳ ❙❡ ❢♦r ♦ s❡❣✉♥❞♦✱ ① ✰ ✸ ❂ ✵✱ ❧♦❣♦ ① ❂ ✸✳ ❉❡ ♠♦❞♦ ❣❡r❛❧✱ ♣❛r❛ ① s❡r s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❛ ❢♦r♠❛
✭① ☛✮✭① ☞✮ ❂ ✵ ❀ ✭✶✳✸✮
♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❢❛t♦r❡s✱ ✭① ☛✮ ♦✉ ✭① ☞✮✱ ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ ③❡r♦✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ① ❂ ☛ ♦✉ ① ❂ ☞✳ P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s s♦❧✉çõ❡s ❞❡ ✭✶✳✸✮ é ❞❛❞♦ ♣♦r ❙ ❂ ❢☛❀ ☞❣✳
❖❧❤❡♠♦s ❛❣♦r❛ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❞❛ ❢♦r♠❛ ❣❡r❛❧
❛①✷✰ ❜① ✰ ❝ ❂ ✵ ✿ ✭✶✳✹✮
❙❡ ❛ ❂ ✵✱ ❡ss❛ ❡q✉❛çã♦ é ❞♦ ♣r✐♠❡✐r♦ ❣r❛✉✱
❜① ✰ ❝ ❂ ✵ ❀ ❡ ❛ s✉❛ ú♥✐❝❛ s♦❧✉çã♦ é ❞❛❞❛ ♣♦r ① ❂ ❝
❜ ✭s✉♣♦♥❞♦ ❜ ✻❂ ✵✮✳ ■st♦ é✱ ❙ ❂ ❢ ❝❜❣✳ P♦r ♦✉tr♦
❧❛❞♦✱ s❡ ❛ ✻❂ ✵✱ ❡♥tã♦ ❞✐✈✐❞✐♥❞♦ ✭✶✳✹✮ ♣♦r ❛✱ ❡ ❝♦♠♣❧❡t❛♥❞♦ ♦ q✉❛❞r❛❞♦ ♦❜t❡♠♦s✿ ✵ ❂ ①✷✰ ❜
❛① ✰ ❛❝ ❂ ✭① ✰ ✷❛❜ ✮✷ ✭✷❛❜ ✮✷✰ ❝❛✿
P♦rt❛♥t♦✱
✭① ✰ ❜
✷❛✮✷ ❂ ✭✷❛❜ ✮✷ ❛❝ ❂ ❜
✷ ✹❛❝
✹❛✷ ✿
❉❡✜♥❛ ✁✿❂❜✷ ✹❛❝✳ ❙❡ ✁ ❁ ✵✱ ♥ã♦ t❡♠ s♦❧✉çõ❡s✿ ❙ ❂∅✳ ❙❡ ✁ ✕ ✵✱ ♣♦❞❡♠♦s t♦♠❛r ❛
r❛✐③ q✉❛❞r❛❞❛ ❡♠ ❛♠❜♦s ❧❛❞♦s ❞❡ss❛ ú❧t✐♠❛ ❡①♣r❡ssã♦✱ ❡ ♦❜t❡r
① ✰ ❜ ✷❛ ❂ ✝
♣ ✁ ✷❛ ✿
■st♦ é✱
① ❂ ❜✝♣✁
✷❛ ✿ ✭✶✳✺✮
❘❡s✉♠✐♥❞♦✿ q✉❛♥❞♦ ❛ ✻❂ ✵✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s s♦❧✉çõ❡s ❞❡ ✭✶✳✹✮ é ❞❛❞♦ ♣♦r
❙ ❂
✽ ❃ ❃ ❁ ❃ ❃ ✿
∅ s❡ ✁ ❁ ✵ ✭③❡r♦ s♦❧✉çõ❡s✮
❢ ❜
✷❛❣ s❡ ✁ ❂ ✵ ✭✉♠❛ s♦❧✉çã♦✮
❢ ❜✝♣✁
✷❛ ❣ s❡ ✁ ❃ ✵ ✭❞✉❛s s♦❧✉çõ❡s✮ ✿
❊①❡r❝í❝✐♦ ✶✳✶✳ ❘❡s♦❧✈❛ ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✳ ✶✳ ✶ ① ❂ ✶
✷✳ ①✷ ❂ ✶
✸✳ ✶
① ❂ ① ✰ ✶
✹✳ ✭① ✰ ✶✮✭① ✼✮ ❂ ✵ ✺✳ ① ❂ ①
✻✳ ① ❂ ①✷
✼✳ ✶ ❂ ✵
✽✳ ✻①✸ ✶ ❂ ✸①✭✶ ✰ ✷①✷✮
✾✳ ✭① ✰ ✻✮✭① ✰ ✶✮ ❂ ✶
❊①❡r❝í❝✐♦ ✶✳✷✳ ❊①✐st❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❞❡ ár❡❛ ✼ ❡ ❞❡ ♣❡rí♠❡tr♦ ✶✷❄
❱❡rsã♦ ✶✳✵ ✭✶✻ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸✮✳ ❙✉❣❡stõ❡s✱ ❝rít✐❝❛s ❡ ❝♦rr❡çõ❡s✿ s❛❝❤❛❅♠❛t✳✉❢♠❣✳❜r
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
✶✳✶✳✷ ❖r❞❡♠ ❡ ✐♥t❡r✈❛❧♦s
❊①✐st❡ ❡♠ R ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✿ ❞♦✐s r❡❛✐s ①❀ ② ♣♦❞❡♠ s❡r ❝♦♠♣❛r❛❞♦s ✉s❛♥❞♦ ♦s
s❡❣✉✐♥t❡s sí♠❜♦❧♦s✿
✎ ① ❂ ②✿ ✏① é ✐❣✉❛❧ ❛ ②✑✱ ✎ ① ✻❂ ②✿ ✏① é ❞✐❢❡r❡♥t❡ ❞❡ ②✑✱ ✎ ① ✕ ②✿ ✏① é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ②✑✱
✎ ① ❃ ②✿ ✏① é ❡str✐t❛♠❡♥t❡ ♠❛✐♦r q✉❡ ②✑✱ ✎ ① ✔ ②✿ ✏① é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ②✑✱
✎ ① ❁ ②✿ ✏① é ❡str✐t❛♠❡♥t❡ ♠❡♥♦r q✉❡ ②✑✳
❆ ♦r❞❡♠ ♣❡r♠✐t❡ ❞❡✜♥✐r s✉❜❝♦♥❥✉♥t♦s ❡❧❡♠❡♥t❛r❡s ❞❡ R✳ P♦r ❡①❡♠♣❧♦✱ ♦s r❡❛✐s ♥ã♦✲
♥❡❣❛t✐✈♦s R✰ sã♦ ❞❡✜♥✐❞♦s ♣♦r
R✰✿❂❢① ✷R✿ ① ✕ ✵❣ ❀
✭❧❡✐❛✲s❡✿ ✏♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ① ✷R t❛✐s q✉❡ ① s❡❥❛ ✕ ✵✮ ❡ ♦s r❡❛✐s ♣♦s✐t✐✈♦s
♣♦r
R✄
✰✿❂❢① ✷R✿ ① ❃ ✵❣ ✿
P♦❞❡♠ t❛♠❜é♠ s❡r ❞❡✜♥✐❞♦s ❝♦♥❥✉♥t♦s ♣❛rt✐❝✉❧❛r❡s ❝❤❛♠❛❞♦s ✐♥t❡r✈❛❧♦s✳ ❈♦♠❡ç❛r❡✲ ♠♦s ❝♦♠ ♦s ✐♥t❡r✈❛❧♦s ❧✐♠✐t❛❞♦s✳ ❙❡ ❛ ❁ ❜ sã♦ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s✱ ♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ é ❞❡✜♥✐❞♦ ❝♦♠♦
❬❛❀ ❜❪✿❂❢① ✷R✿ ❛ ✔ ① ✔ ❜❣ ✿
▲❡✐❛✲s❡✿ ✏❬❛❀ ❜❪ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ① t❛✐s q✉❡ ① s❡❥❛ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ❛✱ ❡ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ❜✑✳ ❖ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ é ❞❡✜♥✐❞♦ ❝♦♠♦
✭❛❀ ❜✮✿❂❢① ✷R✿ ❛ ❁ ① ❁ ❜❣ ✿
❖❜s❡r✈❡ q✉❡ ✭❛❀ ❜✮ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ ❬❛❀ ❜❪ r❡t✐r❛♥❞♦ ❛s ❡①tr❡♠✐❞❛❞❡s✿ ✭❛❀ ❜✮ ❂ ❬❛❀ ❜❪♥❢❛❀ ❜❣✳ ❉❡✜♥❛♠✲s❡ t❛♠❜é♠ ♦s ✐♥t❡r✈❛❧♦s s❡♠✐✲❛❜❡rt♦s ✭♦✉ s❡♠✐✲❢❡❝❤❛❞♦s✮
❬❛❀ ❜✮✿❂❢① ✷ R✿ ❛ ✔ ① ❁ ❜❣ ❀ ✭❛❀ ❜❪✿❂❢① ✷R✿ ❛ ❁ ① ✔ ❜❣ ✿
●r❛✜❝❛♠❡♥t❡✱ r❡♣r❡s❡♥t❛r❡♠♦s ❡ss❡s ✐♥t❡r✈❛❧♦s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
R
❛p ❜p ❬❛❀ ❜✮
❝p ❞p
❬❝❀ ❞❪
❡p ❢p
✭❡❀ ❢❪
■♥tr♦❞✉③✐r❡♠♦s t❛♠❜é♠ ✐♥t❡r✈❛❧♦s ♥ã♦✲❧✐♠✐t❛❞♦s✿ ♦s s❡♠✐✲✐♥✜♥✐t♦s ❢❡❝❤❛❞♦s ✭ ✶❀ ❛❪✿❂❢① ✷R✿ ① ✔ ❛❣ ❀ ❬❝❀ ✰✶✮✿❂❢① ✷ R✿ ① ✕ ❝❣ ❀
❡ ♦s s❡♠✐✲✐♥✜♥✐t♦s ❛❜❡rt♦s
✭ ✶❀ ❛✮✿❂❢① ✷R✿ ① ❁ ❛❣ ❀ ✭❝❀ ✰✶✮✿❂❢① ✷ R✿ ① ❃ ❝❣ ✿
P♦r ❡①❡♠♣❧♦✱
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
R p
❛ ✭ ✶❀ ❛❪
✿ ✿ ✿ ❝p
✭❝❀ ✰✶✮
✿ ✿ ✿
❖❜s❡r✈❡ q✉❡ ✏✰✶✑ ❡ ✏ ✶✑ ♥ã♦ sã♦ ♥ú♠❡r♦s r❡❛✐s ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦s❀ ✰✶ ✭r❡s♣❡❝✲ t✐✈❛♠❡♥t❡ ✶✮ é s♦♠❡♥t❡ ✉♠ sí♠❜♦❧♦ ✉s❛❞♦ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛ ✐❞é✐❛ ✭♠❡✐♦ ❛❜str❛t❛✮ ❞❡ ✉♠ ♥ú♠❡r♦ ♠❛✐♦r ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ♠❡♥♦r✮ ❞♦ q✉❡ q✉❛❧q✉❡r r❡❛❧ ①✳
❊①❡r❝í❝✐♦ ✶✳✸✳ ❙✐♠♣❧✐✜q✉❡ ❛s ❡①♣r❡ssõ❡s✱ ✉s❛♥❞♦ ❛s ♥♦t❛çõ❡s ✐♥tr♦❞✉③✐❞❛s ❛❝✐♠❛✳ ✶✳ ❆ ❂ ❢① ✷R✿ ①✷ ✔ ✹❣
✷✳ ❇ ❂ ❢① ✿ ① ✕ ✵❣ ❭ ❢① ✿ ① ❁ ✶❣ ✸✳ ❈ ❂ ❢① ✿ ① ✔ ✶❣ ❭ ❢① ✿ ① ❁ ✵❣ ✹✳ ❉ ❂ ❢① ✿ ① ✕ ✶❣ ❭ ❢① ✿ ① ✔ ✶❣
✺✳ ❊ ❂ ❢① ✿ ① ✔ ✷❣ ❬ ❬✵❀ ✰✶✮ ✻✳ ❋ ❂ ❬✶❀ ✷❪ ❭ ✭ ✶❀ ✶❪
✼✳ ● ❂ ❬✵❀ ✶❪ ❭ ❬✵❀✶
✷❪ ❭ ❬✵❀✶✸❪ ❭ ❬✵❀✶✹❪ ❭ ✿ ✿ ✿
✽✳ ❍ ❂ ❬✵❀ ✶❪ ❬ ❬✶❀ ✷❪ ❬ ❬✷❀ ✸❪ ❬ ❬✸❀ ✹❪ ❬ ✿ ✿ ✿
✶✳✶✳✸ ❱❛❧♦r ❛❜s♦❧✉t♦
■♥❢♦r♠❛❧♠❡♥t❡✱ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ ①✱ ❞❡♥♦t❛❞♦ ♣♦r ❥①❥✱ r❡♣r❡s❡♥t❛ ♦ s❡✉ ✏✈❛❧♦r ❡q✉✐✈❛❧❡♥t❡ ♣♦s✐t✐✈♦✑✳ P♦r ❡①❡♠♣❧♦✱ ❥✺❥ ❂ ✺✱ ❥ ✸❥ ❂ ✸✱ ❡ ❥✵❥ ❂ ✵✳ ❋♦r♠❛❧♠❡♥t❡✱
❥①❥✿❂
✽ ❁ ✿
① s❡ ① ✕ ✵
① s❡ ① ❁ ✵ ✿ ✭✶✳✻✮
P♦r ❡①❡♠♣❧♦✱ ❝♦♠ ❡ss❛ ❞❡✜♥✐çã♦✱ ❥á q✉❡ ✸ ❁ ✵✱ t❡♠♦s ❥ ✸❥ ❂ ✭ ✸✮ ❂ ✸✳ ❖❜s❡r✈❡ q✉❡ ♣♦r ❞❡✜♥✐çã♦✱
❥①❥ ✔ ❛ ✭✮ ❛ ✔ ① ✔ ❛ ✭✮ ① ✷ ❬ ❛❀ ❛❪ ✿ ✭✶✳✼✮
❊①❡r❝í❝✐♦ ✶✳✹✳ ◗✉❛✐s ❞❛s ❡①♣r❡ssõ❡s ❛❜❛✐①♦ sã♦ ✈❡r❞❛❞❡✐r❛s ✭♣❛r❛ q✉❛❧q✉❡r ①✮❄
❏✉st✐✜q✉❡✳ ♣
①✷ ❂ ① ❀ ♣①✷ ❂ ① ❀ ♣①✷ ❂ ❥①❥ ✿
❯s❛r❡♠♦s ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ♣❛r❛ ❞❡✜♥✐r ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s✿ ❞✭①❀ ②✮✿❂❥① ②❥ ✿
✶✳✶✳✹ ■♥❡q✉❛çõ❡s ❡ s✐♥❛❧
❈♦♥s✐❞❡r❡ ❛ ✐♥❡q✉❛çã♦ ❞♦ ♣r✐♠❡✐r♦ ❣r❛✉✿
✷ ✷① ✕ ✶ ✿ ✭✶✳✽✮
❈♦♠♦ ❛♥t❡s✱ ✏r❡s♦❧✈❡r✑ ❡ss❛ ✐♥❡q✉❛çã♦ s✐❣♥✐✜❝❛ ❛❝❤❛r t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ ① ♣❛r❛ ♦s q✉❛✐s ❛ ❡①♣r❡ssã♦ ❡♠ ✭✶✳✽✮ s❡ t♦r♥❡ ✈❡r❞❛❞❡✐r❛✳ P♦r ❡①❡♠♣❧♦✱ ① ❂ ✵ é s♦❧✉çã♦✱ ♣♦✐s ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ✈❛❧❡ ✷ ✷ ✁ ✵ ❂ ✷✱ q✉❡ é ✕ ✶✳ ▼❛s ❡♠ ❣❡r❛❧ ✉♠❛ ✐♥❡q✉❛çã♦ ♣♦❞❡ ♣♦ss✉✐r ♠❛✐s ❞❡ ✉♠❛ s♦❧✉çã♦✱ às ✈❡③❡s ♣♦ss✉✐ ✐♥✜♥✐t❛s s♦❧✉çõ❡s✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s♦❧✉çõ❡s✱ t❛♠❜é♠ ❞❡♥♦t❛❞♦ ♣♦r ❙✱ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳ Pr✐♠❡✐r♦✱
❱❡rsã♦ ✶✳✵ ✭✶✻ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸✮✳ ❙✉❣❡stõ❡s✱ ❝rít✐❝❛s ❡ ❝♦rr❡çõ❡s✿ s❛❝❤❛❅♠❛t✳✉❢♠❣✳❜r
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
♦ ❝♦♥❥✉♥t♦ ❙ ❞❛s s♦❧✉çõ❡s ♥ã♦ é ♠♦❞✐✜❝❛❞♦ ❛♦ ❛❞✐❝✐♦♥❛r♠♦s ✭♦✉ s✉❜tr❛✐r♠♦s✮ ❡①♣r❡ssõ❡s ✐❣✉❛✐s ❡♠ ❛♠❜♦s ❧❛❞♦s ❞❡ ✉♠❛ ✐♥❡q✉❛çã♦✳ ❆ss✐♠✱ ❛❞✐❝✐♦♥❛♥❞♦ ✷① ❡♠ ❝❛❞❛ ❧❛❞♦ ❞❡ ✭✶✳✽✮✱ ♦❜t❡♠♦s
✷ ✕ ✶ ✰ ✷① ✿ P♦❞❡♠♦s ❡♠ s❡❣✉✐❞❛ s✉❜tr❛✐r ✶ ❡♠ ❛♠❜♦s ❧❛❞♦s✿
✶ ✕ ✷① ✿
❆❣♦r❛✱ ♦ ❝♦♥❥✉♥t♦ ❙ ❞❛s s♦❧✉çõ❡s ♥ã♦ é ♠♦❞✐✜❝❛❞♦ ❛♦ ♠✉❧t✐♣❧✐❝❛r♠♦s ✭♦✉ ❞✐✲ ✈✐❞✐r♠♦s✮ ❛♠❜♦s ❧❛❞♦s ❞❡ ✉♠❛ ✐♥❡q✉❛çã♦ ♣♦r ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦✳ ❆ss✐♠✱ ❞✐✈✐❞✐♥❞♦ ❛♠❜♦s ❧❛❞♦s ❞❛ ✐♥❡q✉❛çã♦ ✶ ✕ ✷① ♣♦r ✷ ♦❜t❡♠♦s ✶
✷ ✕ ①✱ ✐st♦ é ① ✔ ✶✷✳ ❆ss✐♠✱ q✉❛❧q✉❡r
r❡❛❧ ① ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ✶
✷ t♦r♥❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠ ✭✶✳✽✮ ✈❡r❞❛❞❡✐r❛✳ ▲♦❣♦✱ ❙ ❂ ✭ ✶❀✶✷❪✳
❖❜s❡r✈❡ q✉❡ ✭✶✳✽✮ ♣♦❞❡ t❛♠❜é♠ s❡r r❡s♦❧✈✐❞❛ s✉❜tr❛✐♥❞♦ ✷ ❡♠ ❛♠❜♦s ❧❛❞♦s✱
✷① ✕ ✶ ✿ ✭✶✳✾✮
P❛ss❛♥❞♦ ✷① ♣❛r❛ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❡ ✶ ♣❛r❛ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ♦❜t❡♠♦s ✶ ✕ ✷①✱ ♦ q✉❡ ❡q✉✐✈❛❧❡ ❛
✷① ✔ ✶ ✿ ✭✶✳✶✵✮
❱❡♠♦s q✉❡ ✭✶✳✶✵✮ é ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ ✭✶✳✾✮ tr♦❝❛♥❞♦ ♦s s✐♥❛✐s ✭✐✳é✳ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ❧❛❞♦s ♣♦r ✶✮✱ ❡ tr♦❝❛♥❞♦ ♦ s❡♥t✐❞♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡✳
❊①❡♠♣❧♦ ✶✳✶✳ ❘❡s♦❧✈❛♠♦s ❛❣♦r❛ ✉♠❛ ✐♥❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✿
①✷ ✸① ✰ ✷ ❃ ✵ ✿ ✭✶✳✶✶✮
Pr✐♠❡✐r♦✱ ♦ ♣♦❧✐♥ô♠✐♦ ❞♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠ ✭✶✳✶✶✮ ♣♦❞❡ s❡r ❢❛t♦r❛❞♦✿ ①✷ ✸① ✰ ✷ ❂ ✭① ✶✮✭① ✷✮✳ ❆ss✐♠✱ ✭✶✳✶✶✮ é ❡q✉✐✈❛❧❡♥t❡ ❛
✭① ✶✮✭① ✷✮ ❃ ✵ ✿ ✭✶✳✶✷✮
❖❜s❡r✈❡ ❛❣♦r❛ q✉❡ ♣❛r❛ ♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s s❡r ❃ ✵✱ ❡❧❡s tê♠ q✉❡ s❡r ❛♠❜♦s ♥ã♦✲♥✉❧♦s ❡ t❡r ♦ ♠❡s♠♦ s✐♥❛❧✳ P♦rt❛♥t♦✱ ❛ r❡s♦❧✉çã♦ ❞❡ ✭✶✳✶✷✮ ♣❛ss❛ ♣❡❧♦ ❡st✉❞♦ ❞♦ s✐♥❛❧ ❞❡ ① ✶ ❡ ① ✷✳ ■ss♦ ♣♦❞❡ s❡r ❢❡✐t♦ ❝♦♠♦ ❡♠ ✭✶✳✽✮✳ P♦r ✉♠ ❧❛❞♦✱ ① ✶ ❁ ✵ s❡ ① ❁ ✶✱ ① ✶ ❂ ✵ s❡ ① ❂ ✶✱ ❡ ① ✶ ❃ ✵ s❡ ① ❃ ✶✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ① ✷ ❁ ✵ s❡ ① ❁ ✷✱ ① ✷ ❂ ✵ s❡ ① ❂ ✷✱ ❡ ① ✷ ❃ ✵ s❡ ① ❃ ✷✳ ■ss♦ ♣♦❞❡ s❡r r❡s✉♠✐❞♦ ♥❛s ❞✉❛s ♣r✐♠❡✐r❛s ❧✐♥❤❛s ❞❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿
① ✶
① ✷
✭① ✶✮✭① ✷✮
✶ ✷
✵ ✰ ✰
✵ ✰
✰ ✵ ✵ ✰
❆ t❡r❝❡✐r❛ ❧✐♥❤❛ ❢♦✐ ♦❜t✐❞❛ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♦s s✐♥❛✐s ❞❡ ① ✶ ❡ ① ✷✿ ✭① ✶✮✭① ✷✮ ❃ ✵ s❡ ① ❁ ✶✱ ✭① ✶✮✭① ✷✮ ❂ ✵ s❡ ① ❂ ✶✱ ✭① ✶✮✭① ✷✮ ❁ ✵ s❡ ✶ ❁ ① ❁ ✷✱ ✭① ✶✮✭① ✷✮ ❂ ✵ s❡ ① ❂ ✷✱ ❡ ✭① ✶✮✭① ✷✮ ❃ ✵ s❡ ① ❃ ✷✳ ❆ss✐♠✱ ❙ ❂ ✭ ✶❀ ✶✮ ❬ ✭✷❀ ✰✶✮ ❞á t♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ ✭✶✳✶✶✮✳
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
❊①❡r❝í❝✐♦ ✶✳✺✳ ❘❡s♦❧✈❛ ❛s s❡❣✉✐♥t❡s ✐♥❡q✉❛çõ❡s✳ ✶✳ ① ❃ ✹ ✺
✷✳ ✸① ✔ ① ✰ ✶ ✸✳ ✽① ❁ ✸ ✹① ✹✳ ✶✵ ❃ ✶✵ ① ✺✳ ①✷ ✕ ✶
✻✳ ①✷ ❃ ✶ ✰ ✷①
✼✳ ① ❃ ① ✽✳ ① ✕ ① ✾✳ ① ✔ ①✷
✶✵✳ ✷①✷✰ ✶✵① ✶✷ ❁ ✵
✶✶✳ ①✷✭① ✰ ✼✮ ✔ ✵
✶✷✳ ①✸ ✷①✷ ① ✰ ✷ ❃ ✵
✶✸✳ ①✷ ①✭① ✰ ✸✮ ✔ ✵
✶✹✳ ① ✔ ①✰✸ ① ✶
❊①❡r❝í❝✐♦ ✶✳✻✳ ◗✉❛♥t♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ ❡①✐st❡♠ t❛✐s q✉❡ ✸♥ ✶ ✔ ✺♥ ✷ ❁ ✹❄ ❊①❡r❝í❝✐♦ ✶✳✼✳ ◗✉❛♥t♦s ♥ú♠❡r♦s ♣r✐♠♦s ♣ ❡①✐st❡♠ t❛✐s q✉❡ ✵ ✔ ✷♣ ✸ ✔ ♣ ✰ ✽❄
P♦❞❡♠♦s t❛♠❜é♠ r❡s♦❧✈❡r ✐♥❡q✉❛çõ❡s q✉❡ ❡♥✈♦❧✈❡♠ ✈❛❧♦r❡s ❛❜s♦❧✉t♦s✿ ❊①❡♠♣❧♦ ✶✳✷✳ ❘❡s♦❧✈❛♠♦s
❥① ✷❥ ✕ ✸ ✿ ✭✶✳✶✸✮
❙❛❜❡♠♦s q✉❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦✱
❥① ✷❥ ❂
✽ ❁ ✿
① ✷ s❡ ① ✕ ✷ ❀ ① ✰ ✷ s❡ ① ❁ ✷ ❀
▲♦❣♦✱ ❛ r❡s♦❧✉çã♦ ❞❡ ✭✶✳✶✸✮ ♣❛ss❛ ♣❡❧❛ r❡s♦❧✉çã♦ ❞❡ ❞✉❛s ✐♥❡q✉❛çõ❡s ♠❛✐s s✐♠♣❧❡s✳ ❆ ♣r✐♠❡✐r❛ é
① ✷ ✕ ✸ ❀ ✐st♦ é ① ✕ ✺ ❀
❡ ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞❛ s♦♠❡♥t❡ ♣❛r❛ ♦s ① t❛✐s q✉❡ ① ✕ ✷✳ ■ss♦ ❞á ✉♠ ♣r✐♠❡✐r♦ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s✿ ❙✶ ❂ ❬✺❀ ✰✶✮ ✭♦s r❡❛✐s q✉❡ sã♦ ❛♦ ♠❡s♠♦ t❡♠♣♦ ♠❛✐♦r❡s ♦✉ ✐❣✉❛✐s ❛ ✺ ❡
♠❛✐♦r❡s ♦✉ ✐❣✉❛✐s ❛ ✷✮✳ ❆ s❡❣✉♥❞❛ é
① ✰ ✷ ✕ ✸ ❀ ✐st♦ é ① ✔ ✶ ❀
❡ ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞❛ s♦♠❡♥t❡ ♣❛r❛ ♦s ① t❛✐s q✉❡ ① ✔ ✷✱ ♦ q✉❡ ❞á ✉♠ s❡❣✉♥❞♦ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s ❙✷ ❂ ✭ ✶❀ ✶❪✳ ❆ss✐♠✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ ✭✶✳✶✸✮ é ❞❛❞♦
♣♦r ❙ ❂ ❙✶❬ ❙✷✿ ❙ ❂ ✭ ✶❀ ✶❪ ❬ ❬✺❀ ✰✶✮✳
❯♠ ❥❡✐t♦ ♠❛✐s ❣❡♦♠étr✐❝♦ ✭♠❛s ❡q✉✐✈❛❧❡♥t❡✮ ❞❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ é ❞❡ ❡s❝r❡✈❡r ✭✶✳✶✸✮ ❝♦♠♦✿ ❞✭①❀ ✷✮ ✕ ✸✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ✐♥t❡r♣r❡t❛r ❛s s♦❧✉çõ❡s ❞❡ ✭✶✳✶✸✮ ❝♦♠♦ s❡♥❞♦ ♦s r❡❛✐s ① ❝✉❥❛ ❞✐stâ♥❝✐❛ ❛♦ ♣♦♥t♦ ✷ é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ✸✱ q✉❡ sã♦ t♦❞♦s ♦s r❡❛✐s ❛ ❡sq✉❡r❞❛ ❞❡ ✶ ♦✉ ❛ ❞✐r❡✐t❛ ❞❡ ✺✿ ❙ ❂ ✭ ✶❀ ✶❪ ❬ ❬✺❀ ✰✶✮✳
❊①❡r❝í❝✐♦ ✶✳✽✳ ❘❡s♦❧✈❛ ❛s s❡❣✉✐♥t❡s ✐♥❡q✉❛çõ❡s✳ ✶✳ ❥① ✰ ✷✼❥ ✕ ✵
✷✳ ❥① ✷❥ ❁ ✵ ✸✳ ❥✷① ✰ ✸❥ ❃ ✵
✹✳ ✸ ❁ ❥✸ ①❥ ✺✳ ✷① ✸❥①❥ ✹ ✕ ✵ ✻✳ ❥①✷ ✶❥ ✔ ✶
✼✳ ✶
① ①✰✷✶ ✕ ✵
✽✳ ✶
① ✰ ✷ ①✷ ❁ ✶
✾✳ ①
❥① ✷❥ ❃ ✷✳
❱❡rsã♦ ✶✳✵ ✭✶✻ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸✮✳ ❙✉❣❡stõ❡s✱ ❝rít✐❝❛s ❡ ❝♦rr❡çõ❡s✿ s❛❝❤❛❅♠❛t✳✉❢♠❣✳❜r
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
❊st✉❞❛r ♦ s✐♥❛❧ ❞❡ ✉♠❛ ❡①♣r❡ssã♦ q✉❡ ❞❡♣❡♥❞❡ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ① s✐❣♥✐✜❝❛ ❞❡t❡r✲ ♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❡ ① ♣❛r❛ ♦s q✉❛✐s ❛ ❡①♣r❡ssã♦ é ♣♦s✐t✐✈❛✱ ♥❡❣❛t✐✈❛✱ ♦✉ ♥✉❧❛✳
❊①❡♠♣❧♦ ✶✳✸✳ ❊st✉❞❡♠♦s ♦ s✐♥❛❧ ❞❛ ❡①♣r❡ssã♦ ①✸✰ ✸①✷✳ ❈♦♠♦ ①✸✰ ✸①✷ ❂ ①✷✭① ✰ ✸✮✱
♦ s✐♥❛❧ ❞❛ ❡①♣r❡ssã♦ ✐♥t❡✐r❛ é ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞♦s s✐♥❛✐s ❞❛s ♣❛rt❡s ①✷ ❡ ① ✰ ✸✳
①✷
① ✰ ✸ ①✷✭① ✰ ✸✮
✸ ✵
✰ ✰ ✵ ✰
✵ ✰ ✰
✵ ✰ ✵ ✰
❆ss✐♠ ✈❡♠♦s q✉❡ ①✸✰ ✸①✷ é ❃ ✵ ✭❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛✮ s❡ ① ✷ ✭ ✸❀ ✵✮ ❬ ✭✵❀ ✶✮✱ ❡❧❛ é
❁ ✵ ✭❡str✐t❛♠❡♥t❡ ♥❡❣❛t✐✈❛✮ s❡ ① ❁ ✵✱ ❡ é ❂ ✵ ✭♥✉❧❛✮ s❡ ① ✷ ❢ ✸❀ ✵❣✳
▼❛✐s t❛r❞❡ r❡s♦❧✈❡r❡♠♦s ✐♥❡q✉❛çõ❡s ♦♥❞❡ ❛♣❛r❡❝❡♠✱ ❡ ❡st✉❞❛r❡♠♦s ♦ s✐♥❛❧ ❞❡ ♦✉tr❛s ❡①♣r❡ssõ❡s✱ ❝♦♠♦ ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ r❛í③❡s ♦✉ ❧♦❣❛r✐t♠♦s✳
❊①❡r❝í❝✐♦ ✶✳✾✳ ❊st✉❞❡ ♦ s✐♥❛❧ ❞❛s s❡❣✉✐♥t❡s ❡①♣r❡ssõ❡s ✶✳ ✺ ✰ ①
✷✳ ✺ ✰ ①✷
✸✳ ✭① ✺✮✷
✹✳ ①✷ ✺
✺✳ ①✷✰✷① ✹✽
✷ ①
✻✳ ✭① ✰ ✶✮❥✷① ✶ ①✷❥
✶✳✷ ❖ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦
❖ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ❡♠ ❣❡r❛❧ ❞❡♥♦t❛❞♦ ♣♦r R✷✱ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s P ❂ ✭①❀ ②✮ ❞❡
r❡❛✐s✱ ① ❡ ②✱ ❝❤❛♠❛❞♦s r❡s♣❡❝t✐✈❛♠❡♥t❡ ❞❡ ❛❜s❝✐ss❛ ✭♦✉ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛✮ ❡ ♦r❞❡♥❛❞❛ ✭♦✉ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛✮✳
P ❂ ✭①❀ ②✮
p
① ②
❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❝✉❥❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ é ♥✉❧❛✱ ✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❛ ❢♦r♠❛ P ❂ ✭✵❀ ②✮✱ é ❝❤❛♠❛❞♦ ❞❡ ❡✐①♦ ②✱ ♦✉ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❝✉❥❛ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛ é ♥✉❧❛✱ ✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❛ ❢♦r♠❛ P ❂ ✭①❀ ✵✮✱ é ❝❤❛♠❛❞♦ ❞❡ ❡✐①♦ ①✱ ♦✉ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✳ ❖s ❡✐①♦s ① ❡ ② ❢♦r♠❛♠ ❞✉❛s r❡t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s✱ ❡ ❞✐✈✐❞❡♠ ♦ ♣❧❛♥♦ ❡♠ q✉❛tr♦ q✉❛❞r❛♥t❡s✿
✶♦
✷♦
✸♦ ✹♦
▼❛✐s ❡①♣❧✐❝✐t❛♠❡♥t❡✱ ❡♠ t❡r♠♦s ❞❛s ❝♦♦r❞❡♥❛❞❛s✱
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
✎ ✶♦ ❂ ❢✭①❀ ②✮ ✿ ① ✕ ✵❀ ② ✕ ✵❣✱
✎ ✷♦ ❂ ❢✭①❀ ②✮ ✿ ① ✔ ✵❀ ② ✕ ✵❣✱
✎ ✸♦ ❂ ❢✭①❀ ②✮ ✿ ① ✔ ✵❀ ② ✔ ✵❣✱
✎ ✹♦ ❂ ❢✭①❀ ②✮ ✿ ① ✕ ✵❀ ② ✔ ✵❣✳
❙❡ P ❂ ✭①❀ ②✮ ❡ ◗ ❂ ✭①✵❀ ②✵✮✱ ❛ ❞✐stâ♥❝✐❛ ❈❛rt❡s✐❛♥❛ ❡♥tr❡ P ❡ ◗ é ❝❛❧❝✉❧❛❞❛ ✉s❛♥❞♦
♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✿
❞✭P❀◗✮
❥②
②✵
❥
❥① ①✵❥
P
◗ ❞✭P❀ ◗✮✿❂
q
✭① ①✵✮✷✰ ✭② ②✵✮✷✿
❊①❡r❝í❝✐♦ ✶✳✶✵✳ ❉❡s❝r❡✈❛ ♦s s❡❣✉✐♥t❡s s✉❜❝♦♥❥✉♥t♦s ❞♦ ♣❧❛♥♦ ❡♠ t❡r♠♦s ❞❛s s✉❛s ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✳
✶✳ ❙❡♠✐✲♣❧❛♥♦ ❛❝✐♠❛ ❞♦ ❡✐①♦ ①✱ ✷✳ s❡♠✐✲♣❧❛♥♦ ❛ ❡sq✉❡r❞❛ ❞♦ ❡✐①♦ ②✱
✸✳ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✶ ❝❡♥tr❛❞♦ ♥❛ ♦r✐❣❡♠ ✭❝♦♠ ♦s ❧❛❞♦s ♣❛r❛❧❡❧♦s ❛♦s ❡✐①♦s✮✱ ✹✳ r❡t❛ ✈❡rt✐❝❛❧ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ ✭✷❀ ✵✮✱
✺✳ r❡t❛ ❤♦r✐③♦♥t❛❧ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ ✭ ✸❀ ✺✮✱ ✻✳ r❡t❛ ❤♦r✐③♦♥t❛❧ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ ✭✶✸❀ ✺✮✱
✼✳ ❢❛✐①❛ ✈❡rt✐❝❛❧ ❝♦♥t✐❞❛ ❡♥tr❡ ♦ ❡✐①♦ ② ❡ ❛ r❡t❛ ❞♦ ✐t❡♠ ✭✹✮✱ ✽✳ ❝ír❝✉❧♦ ❞❡ r❛✐♦ ✶ ❝❡♥tr❛❞♦ ♥❛ ♦r✐❣❡♠✳
✾✳ ❞✐s❝♦ ✭❝❤❡✐♦✮ ❞❡ r❛✐♦ ✷ ❝❡♥tr❛❞♦ ❡♠ ✭✶❀ ✷✮✳
✶✳✷✳✶ ❘❡t❛s
❏á ✈✐♠♦s✱ ♥♦ ❊①❡r❝í❝✐♦ ✶✳✶✵✱ ❝♦♠♦ ❡①♣r❡ss❛r r❡t❛s ❤♦r✐③♦♥t❛✐s ❡ ✈❡rt✐❝❛✐s✳ ❯♠❛ r❡t❛ ✈❡rt✐❝❛❧ é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s ✭①❀ ②✮ ❝✉❥❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ ① é ✐❣✉❛❧ ❛ ✉♠ ♥ú♠❡r♦ ✜①♦ ❛ ✷R❀ ❛ s✉❛ ❡q✉❛çã♦ s❡ ❡s❝r❡✈❡✿ ① ❂ ❛✳
②
① ✭❛❀ ✵✮
❡q✉❛çã♦ ❞❛ r❡t❛✿ ① ❂ ❛
P♦r ♦✉tr♦ ❧❛❞♦✱ ✉♠❛ r❡t❛ ❤♦r✐③♦♥t❛❧ é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s ✭①❀ ②✮ ❝✉❥❛ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛ ② é ✐❣✉❛❧ ❛ ✉♠ ♥ú♠❡r♦ ✜①♦ ❜ ✷R❀ ❛ s✉❛ ❡q✉❛çã♦ s❡ ❡s❝r❡✈❡✿ ② ❂ ❜✳
❱❡rsã♦ ✶✳✵ ✭✶✻ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸✮✳ ❙✉❣❡stõ❡s✱ ❝rít✐❝❛s ❡ ❝♦rr❡çõ❡s✿ s❛❝❤❛❅♠❛t✳✉❢♠❣✳❜r
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
②
① ✭✵❀ ❜✮
❡q✉❛çã♦ ❞❛ r❡t❛✿ ② ❂ ❜
❆s r❡t❛s ❤♦r✐③♦♥t❛✐s ❡ ✈❡rt✐❝❛✐s sã♦ ❞❡s❝r✐t❛s ♣♦r s♦♠❡♥t❡ ✉♠ ♣❛râ♠❡tr♦ ✭♦ ✏❛✑ ♣❛r❛ ✉♠❛ r❡t❛ ✈❡rt✐❝❛❧✱ ♦✉ ♦ ✏❜✑ ♣❛r❛ ✉♠❛ r❡t❛ ❤♦r✐③♦♥t❛❧✮✳ P❛r❛ ❛s ♦✉tr❛s r❡t❛s ❞♦ ♣❧❛♥♦✱ q✉❡ ♥ã♦ ✜❝❛♠ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❛r❛❧❡❧❛s ❛ ✉♠ ❞♦s ❡✐①♦s✱ é ♣r❡❝✐s♦ ✉s❛r ❞♦✐s ♣❛râ♠❡tr♦s✱ ♠ ❡ ❤✱ ❝❤❛♠❛❞♦s r❡s♣❡❝t✐✈❛♠❡♥t❡ ✐♥❝❧✐♥❛çã♦ ✭♦✉ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r✮ ❡ ♦r❞❡♥❛❞❛ ♥❛ ♦r✐❣❡♠✱ ♣❛r❛ ❡s♣❡❝✐✜❝❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ① ❡ ②✿
② ❂ ♠① ✰ ❤ ✿ ②
① ♦r❞❡♥❛❞❛ ♥❛ ♦r✐❣❡♠✿ ❤
❡q✉❛çã♦ ❞❛ r❡t❛✿ ② ❂ ♠① ✰ ❤ ✐♥❝❧✐♥❛çã♦✿ ♠
❖ s✐❣♥✐✜❝❛❞♦ ❞❛ ✐♥❝❧✐♥❛çã♦ ♠ ❞❡✈❡ s❡r ❡♥t❡♥❞✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ♣❛rt✐♥❞♦ ❞❡ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ r❡t❛✱ ❛♦ ❛♥❞❛r ❤♦r✐③♦♥t❛❧♠❡♥t❡ ✉♠❛ ❞✐stâ♥❝✐❛ ▲ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ✈❡rt✐❝❛❧ ❞❛ r❡t❛ é ❞❡ ♠▲✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠❛ r❡t❛ ❞❡ ✐♥❝❧✐♥❛çã♦ ✶ ✷
✭♦❜s❡r✈❡ q✉❡ t♦❞♦ ♦s tr✐â♥❣✉❧♦s ❞❛ s❡❣✉✐♥t❡ ✜❣✉r❛ sã♦ s❡♠❡❧❤❛♥t❡s✮✱
▲
▲❂✷
✶
✵✿✺
✵✿✻ ✵✿✸
❙❡ ❛ ✐♥❝❧✐♥❛çã♦ é ♥❡❣❛t✐✈❛✱ ❡♥tã♦ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ✈❡rt✐❝❛❧ é ♣❛r❛ ❜❛✐①♦✳
❙❡ P ❂ ✭①✶❀ ②✶✮ ❡ ◗ ❂ ✭①✷❀ ②✷✮ sã♦ ❞♦✐s ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛ ♥ã♦ ✈❡rt✐❝❛❧ ❞❡ ✐♥❝❧✐♥❛çã♦
♠✱ ❡♥tã♦
②✷ ②✶
①✷ ①✶ ❂ ♠ ✿ ✭✶✳✶✹✮
❊ss❛ r❡❧❛çã♦ ♣♦❞❡ s❡r ✉s❛❞❛ t❛♠❜é♠ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ✐♥❝❧✐♥❛çã♦ ❞❡ ✉♠❛ r❡t❛✳
❊①❡♠♣❧♦ ✶✳✹✳ Pr♦❝✉r❡♠♦s ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ r q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s P ❂ ✭ ✶❀ ✸✮ ❡ ◗ ❂ ✭✸❀ ✵✮✿
②
① P
◗
p p p p p
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
❈♦♠♦ r ♥ã♦ é ✈❡rt✐❝❛❧✱ ❛ s✉❛ ❡q✉❛çã♦ é ❞❛ ❢♦r♠❛ ② ❂ ♠① ✰ ❤✳ ❆ ✐♥❝❧✐♥❛çã♦ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ✉s❛♥❞♦ ✭✶✳✶✹✮✿ ♠ ❂ ✸ ✭ ✶✮✵ ✭✸✮ ❂ ✸
✹✳ ✭P♦❞❡ t❛♠❜é♠ ♦❜s❡r✈❛r q✉❡ ♣❛r❛ ❛♥❞❛r ❞❡
P ❛té ◗✱ é ♥❡❝❡ssár✐♦ ❛♥❞❛r ✹ ♣❛ss♦s ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❡ ✸ ♣❛ss♦s ♣❛r❛ ❜❛✐①♦✱ ❧♦❣♦ ♠ ❂ ✸ ✹ ✳✮
P♦rt❛♥t♦✱ ❛ ❡q✉❛çã♦ é ❞❛ ❢♦r♠❛ ② ❂ ✸
✹① ✰ ❤✳ ❋❛❧t❛ ❛❝❤❛r ❤✱ q✉❡ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦
✉s❛♥❞♦ ♦ ❢❛t♦ ❞❡ r ♣❛ss❛r ♣❡❧♦ ♣♦♥t♦ P ✿ ✸ ❂ ✸
✹ ✁ ✭ ✶✮ ✰ ❤ ✭❞❛r✐❛ ♥❛ ♠❡s♠❛ ✉s❛♥❞♦ ♦
♣♦♥t♦ ◗✮✳ ❆ss✐♠✱ ❤ ❂ ✾
✹✱ ❡ r é ❞❡s❝r✐t❛ ♣❡❧❛ ❡q✉❛çã♦✿
② ❂ ✸
✹① ✰ ✾✹✿
❆♦ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛♠❜♦s ❧❛❞♦s ♣♦r ✹ ❡ r❡❛rr❛♥❥❛♥❞♦ ♣♦❞❡♠♦s ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
✸① ✰ ✹② ✾ ❂ ✵ ✿
❊ss❛ é ❛ ❢♦r♠❛ ❣❡♥ér✐❝❛ ❞❛ r❡t❛✳ ❊♠ ❣❡r❛❧✱ q✉❛❧q✉❡r r❡t❛ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ♥❛ ❢♦r♠❛ ❣é♥ér✐❝❛✱
❛① ✰ ❜② ✰ ❝ ❂ ✵ ❀
❡♠ q✉❡ ❛❀ ❜❀ ❝ sã♦ ❝♦♥st❛♥t❡s✳ ❙❡ ❛ ❂ ✵ ❡ ❜ ✻❂ ✵✱ ❛ r❡t❛ é ❤♦r✐③♦♥t❛❧✳ ❙❡ ❛ ✻❂ ✵ ❡ ❜ ❂ ✵✱ ❛ r❡t❛ é ✈❡rt✐❝❛❧✳ ❙❡ ❛ ✻❂ ✵ ❡ ❜ ✻❂ ✵✱ ❛ r❡t❛ é ♦❜❧íq✉❛✳
❊①❡r❝í❝✐♦ ✶✳✶✶✳ ❈♦♥s✐❞❡r❡ ❛ r❡t❛ r ❞♦ ❊①❡♠♣❧♦ ✶✳✹✳ ❊s❝♦❧❤❛ ❛❧❣✉♥s ♣❛r❡s ❞❡ ♣♦♥t♦s P ❡ ◗ ❡♠ r✱ ❡ ✈❡r✐✜q✉❡ ❛ ❢ór♠✉❧❛ ✭✶✳✶✹✮✳ ❆❝❤❡ ♦s ✈❛❧♦r❡s ❞❡ ① ❡ ② ♣❛r❛ q✉❡ ♦s ♣♦♥t♦s ❘ ❂ ✭①❀ ✶✵✵✮ ❡ ❚ ❂ ✭✻❀ ②✮ ♣❡rt❡♥ç❛♠ ❛ r✳
❊①❡r❝í❝✐♦ ✶✳✶✷✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ❞❛❞♦s✳ ✶✳ ✭✵❀ ✵✮✱ ✭✶❀ ✶✮
✷✳ ✭ ✷❀ ✶✮✱ ✭✶✵✵❀ ✶✮ ✸✳ ✭ ✸❀ ✷✶✿✺✼✮✱ ✭ ✸❀ ✸✮
✹✳ ✭✶❀ ✷✮✱ ✭ ✶❀ ✸✮
✺✳ ✭✸✸✸❀ ✷✷✼✮✱ ✭ ✹✵✷❀ ✷✻✸✮
❊①❡r❝í❝✐♦ ✶✳✶✸✳ ❋❛ç❛ ✉♠ ❡s❜♦ç♦✱ ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ❞❛ r❡t❛ ❞❡s❝r✐t❛ ♣❡❧❛ ❡q✉❛çã♦ ❞❛❞❛✳
✶✳ r✶ ✿ ① ❂ ✹
✷✳ r✷ ✿ ② ❂ ✸❂✷
✸✳ r✸ ✿ ① ✰ ✷② ❂ ✵
✹✳ r✹ ✿ ② ❂ ✷① ✸
❖❜s❡r✈❡ q✉❡ r❡t❛s ♣❛r❛❧❡❧❛s tê♠ ❛ ♠❡s♠❛ ✐♥❝❧✐♥❛çã♦✳
❊①❡r❝í❝✐♦ ✶✳✶✹✳ ❉ê ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ r✵✱ ♣❛r❛❧❡❧❛ ❛ r✱ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ P ✳
✶✳ r ✿ ② ❂ ✺① ✰ ✷✱ P ❂ ✭ ✶❀ ✺✮✳ ✷✳ r ✿ ✹① ✸② ✰ ✻ ❂ ✵✱ P ❂ ✭✸❀ ✺✮✳
❊①❡r❝í❝✐♦ ✶✳✶✺✳ ▼♦str❡ q✉❡ s❡ r✶ t❡♠ ✐♥❝❧✐♥❛çã♦ ♠✶ ✻❂ ✵✱ ❡ r✷ t❡♠ ✐♥❝❧✐♥❛çã♦
♠✷ ❂ ♠✶✶✱ ❡♥tã♦ r✶ ❡ r✷ sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳
❱❡rsã♦ ✶✳✵ ✭✶✻ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸✮✳ ❙✉❣❡stõ❡s✱ ❝rít✐❝❛s ❡ ❝♦rr❡çõ❡s✿ s❛❝❤❛❅♠❛t✳✉❢♠❣✳❜r
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
❊①❡r❝í❝✐♦ ✶✳✶✻✳ ❉❡t❡r♠✐♥❡ q✉❛✐s ❞❛s s❡❣✉✐♥t❡s r❡t❛s sã♦ ♣❛r❛❧❡❧❛s ♦✉ ♣❡r♣❡♥❞✐❝✉✲ ❧❛r❡s✳
r✶ ✿ ✷① ✰ ② ✶ ❂ ✵ ❀ r✷ ✿ ① ✰ ✷② ✰ ✶ ❂ ✵ ❀ r✸ ✿ ② ❂ ✷① ✸ ❀ r✹ ✿ ✸① ✰ ✻② ✸ ❂ ✵ ✿
❊♠ s❡❣✉✐❞❛✱ ❡s❜♦❝❡ ❛s r❡t❛s ❡ ✈❡r✐✜q✉❡✳
✶✳✷✳✷ ❈ír❝✉❧♦s
❈♦♥s✐❞❡r❡ ♦ ❝ír❝✉❧♦ ✷ ✌ ❞❡ ❝❡♥tr♦ ❈ ❂ ✭✶❀ ✷✮ ❡ ❞❡ r❛✐♦ ❘ ❂ ✷✿
②
①
p p p p p p
❈ ✌
P♦r ❞❡✜♥✐çã♦ ✭✈❡r ♦ ❊①❡r❝í❝✐♦ ✶✳✶✵✮✱ ✌ é ❞❡✜♥✐❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s P ❝✉❥❛ ❞✐stâ♥❝✐❛ ❡✉❝❧✐❞✐❛♥❛ ❛ ❈ é ✐❣✉❛❧ ❛ ✷✿ ❞✭P❀ ❈✮ ❂ ✷✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ❛s ❝♦♦r❞❡♥❛❞❛s ✭①❀ ②✮ ❞❡ P sã♦ ❧✐❣❛❞❛s ♣❡❧❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿ q✭① ✶✮✷✰ ✭② ✷✮✷ ❂ ✷✳ ❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱
✌ é ❞❡s❝r✐t♦ ♣❡❧❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿
✭① ✶✮✷✰ ✭② ✷✮✷ ❂ ✹ ✿
❖❜s❡r✈❡ q✉❡✱ ❡①♣❛♥❞✐♥❞♦ ♦s ❢❛t♦r❡s ✭① ✶✮✷ ❡ ✭② ✷✮✷✱ ❡ss❛ ú❧t✐♠❛ ❡①♣r❡ssã♦ ♣♦❞❡ s❡r
❡s❝r✐t❛ ♥❛ ❢♦r♠❛ ❣❡♥ér✐❝❛✿
①✷✰ ②✷ ✷① ✹② ✰ ✶ ❂ ✵ ✿
❊♠ ❣❡r❛❧✱ ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ ❘ ❃ ✵ ❝❡♥tr❛❞♦ ❡♠ ❈ ❂ ✭①✵❀ ②✵✮ é ❞❡s❝r✐t♦ ♣❡❧❛ ❡q✉❛çã♦
✭① ①✵✮✷✰ ✭② ②✵✮✷ ❂ ❘✷✿ ✭✶✳✶✺✮
❯♠ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ é ❞❡ ❛❝❤❛r ♦ ❝❡♥tr♦ ❡ ♦ r❛✐♦ ❛ ♣❛rt✐r ❞❛ ❢♦r♠❛ ❣❡♥ér✐❝❛✳ ❊①❡♠♣❧♦ ✶✳✺✳ ❈♦♥s✐❞❡r❡ ♦ ❝ír❝✉❧♦ ✌ ❞❡s❝r✐t♦ ♣❡❧❛ s✉❛ ❡q✉❛çã♦ ❣❡♥ér✐❝❛
①✷✰ ②✷✰ ✻① ✽② ❂ ✵ ✿ ✭✶✳✶✻✮
P❛r❛ ❛❝❤❛r ♦ s❡✉ ❝❡♥tr♦ ❡ ♦ s❡✉ r❛✐♦✱ ❝♦♠♣❧❡t❡♠♦s ♦s q✉❛❞r❛❞♦s✿ ①✷✰ ✻① ❂ ✭① ✰ ✸✮✷ ✾✱
②✷ ✽② ❂ ✭② ✹✮✷ ✶✻✳ ▲♦❣♦✱ ✭✶✳✶✻✮ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ✭①✰✸✮✷ ✾✰✭② ✹✮✷ ✶✻ ❂ ✵✱
✐st♦ é✿
✭① ✰ ✸✮✷✰ ✭② ✹✮✷ ❂ ✷✺ ✑ ✺✷✿
P♦rt❛♥t♦✱ ✌ é ❝❡♥tr❛❞♦ ❡♠ ❈ ❂ ✭ ✸❀ ✹✮✱ ❞❡ r❛✐♦ ❘ ❂ ✺✳
✷➚s ✈❡③❡s✱ ♦ q✉❡ ❝❤❛♠❛♠♦s ❛q✉✐ ❞❡ ❝ír❝✉❧♦ ❝♦rr❡s♣♦♥❞❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡♠ ♦✉tr♦s t❡①t♦s ❞❡
♠❛t❡♠át✐❝❛ ❡❧❡♠❡♥t❛r✳
❈❆P❮❚❯▲❖ ✶✳ ❋❯◆❉❆▼❊◆❚❖❙
❊①❡♠♣❧♦ ✶✳✻✳ ❈♦♥s✐❞❡r❡ ①✷✰✷①✰②✷✰✷ ❂ ✵✳ ❈♦♠♣❧❡t❛♥❞♦ ♦ q✉❛❞r❛❞♦ ❡ r❡❛rr❛♥❥❛♥❞♦✱
♦❜t❡♠♦s ✭① ✰ ✶✮✷✰ ②✷ ❂ ✶✳ ❈♦♠♦ ✏ ✶✑ ♥ã♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠ q✉❛❞r❛❞♦✱ ❡st❛
❡q✉❛çã♦ ♥ã♦ r❡♣r❡s❡♥t❛ ✉♠ ❝ír❝✉❧♦ ✭❡ ♥❛ ✈❡r❞❛❞❡✱ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♣❛r ✭①❀ ②✮ q✉❡ s❡❥❛ s♦❧✉çã♦✮✳
❊①❡r❝í❝✐♦ ✶✳✶✼✳ ❉❡t❡r♠✐♥❡ q✉❛✐s ❞❛s ❡q✉❛çõ❡s ❛ s❡❣✉✐r ❞❡✜♥❡♠ ✉♠ ❝ír❝✉❧♦✳ ◗✉❛♥❞♦ ❢♦r ♦ ❝❛s♦✱ ❝❛❧❝✉❧❡ ♦ ❝❡♥tr♦ ❡ ♦ r❛✐♦✳
✶✳ ①✷ ✰ ✭② ✰ ✶✮✷ ❂ ✾
✷✳ ①✷ ✰ ②✷ ❂ ✶
✸✳ ①✷✰ ②✷ ❂ ✻①
✹✳ ①✷✰ ②✷✰ ① ✰ ② ✰ ✶ ❂ ✵
✺✳ ①✷✰ ②✷✰ ✷① ✰ ✶ ❂ ✵
✻✳ ①✷ ❂ ②✷✰ ✶
✶✳✸ ❚r✐❣♦♥♦♠❡tr✐❛
❆ tr✐❣♦♥♦♠❡tr✐❛ ❡st❛❜❡❧❡❝❡ r❡❧❛çõ❡s ♣r❡❝✐s❛s ❡♥tr❡ ♦s â♥❣✉❧♦s ❡ ♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥✲ ❣✉❧♦✳ ❉❡✜♥✐r❡♠♦s ❛s três ❢✉♥çõ❡s ✭♠❡s♠♦ s❡ ❛ ♣ró♣r✐❛ ♥♦çã♦ ❞❡ ❢✉♥çã♦ s❡rá ❡st✉❞❛❞❛ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✮ tr✐❣♦♥♦♠étr✐❝❛s ❡❧❡♠❡♥t❛r❡s✱ s❡♥ ✭s❡♥♦✮✱ ❝♦s ✭❝♦ss❡♥♦✮ ❡ t❛♥ ✭t❛♥✲ ❣❡♥t❡✮✱ ❡ ❞❛r❡♠♦s ❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s✳ ◆♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s ♦❧❤❛r❡♠♦s ♠❛✐s ❞❡ ♣❡rt♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛♥❛❧ít✐❝❛s ❞❡ss❛s ❢✉♥çõ❡s✳
✶✳✸✳✶ ▼❡❞✐r â♥❣✉❧♦s ♥♦ ♣❧❛♥♦
P❛r❛ ❝♦♠❡ç❛r✱ é ✐♠♣♦rt❛♥t❡ ❡s❝♦❧❤❡r ✉♠❛ ✉♥✐❞❛❞❡ ✭❝♦♠♦ ✏♠❡tr♦s✑ ♣❛r❛ ❝♦♠♣r✐♠❡♥t♦s✱ ♦✉ ✏❧✐tr♦s✑ ♣❛r❛ ✈♦❧✉♠❡s✮ ♣❛r❛ ♠❡❞✐r ✉♠ â♥❣✉❧♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ ❛❜❡rt✉r❛ ❡♥tr❡ ❞✉❛s r❡t❛s✳ ❉❡s❝r❡✈❡r❡♠♦s ❛s ❞✉❛s ✉♥✐❞❛❞❡s ♠❛✐s ✉s❛❞❛s✱ ❣r❛✉s ❡ r❛❞✐❛♥♦s✳
❖s â♥❣✉❧♦s s❡rã♦ ♠❡❞✐❞♦s ❛ ♣❛rt✐r ❞❡ ✉♠❛ r❡t❛ ❤♦r✐③♦♥t❛❧✱ ❡♠ s❡♥t✐❞♦ ❛♥t✐❤♦rár✐♦✳ ❆ ❛❜❡rt✉r❛ ♠í♥✐♠❛✱ ♥❛t✉r❛❧♠❡♥t❡✱ é ❞❡✜♥✐❞❛ ❝♦♠♦ ✈❛❧❡♥❞♦ ③❡r♦✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ ❛ ✉♥✐❞❛❞❡✳ ❖ q✉❡ ♣r❡❝✐s❛ s❡r ❞❡✜♥✐❞♦ é ♦ ✈❛❧♦r ❞♦ â♥❣✉❧♦ t♦t❛❧✳ ❙❡ ♦ â♥❣✉❧♦ ❢♦r ♠❡❞✐❞♦ ❡♠ ❣r❛✉s✱ ❡ss❡ â♥❣✉❧♦ t♦t❛❧ é ❞❡✜♥✐❞♦ ❝♦♠♦ ✈❛❧❡♥❞♦ ✸✻✵ ❣r❛✉s✿
✵♦
✸✻✵♦
❯♠❛ ✈❡③ q✉❡ ♦ â♥❣✉❧♦ t♦t❛❧ ❢♦✐ ✜①❛❞♦✱ ❛ ♠❡❞✐çã♦ ❞♦s ♦✉tr♦s s❡ ❢❛③ ♣r♦♣♦r❝✐♦♥❛❧♠❡♥t❡✿ ❛ ♠❡t❛❞❡ ❞♦ â♥❣✉❧♦ t♦t❛❧ ✈❛❧❡ ✶✽✵ ❣r❛✉s✱ ♦ â♥❣✉❧♦ r❡t♦ ♠❡❞❡ ✾✵ ❣r❛✉s✱ ❡t❝✳ ❆ ✈❛♥t❛❣❡♠ ❞❡ss❛ ✉♥✐❞❛❞❡ é q✉❡ ✈ár✐♦s â♥❣✉❧♦s ❜❛st❛♥t❡ ✉s❛❞♦s ❡♠ ❣❡♦♠❡tr✐❛ t♦♠❛♠ ✈❛❧♦r❡s ✐♥t❡✐r♦s✿ ✸✵✱ ✻✵✱ ✾✵✱ ✶✽✵✱ ✷✼✵✱ ❡t❝✳
❱❡rsã♦ ✶✳✵ ✭✶✻ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸✮✳ ❙✉❣❡stõ❡s✱ ❝rít✐❝❛s ❡ ❝♦rr❡çõ❡s✿ s❛❝❤❛❅♠❛t✳✉❢♠❣✳❜r
