Determinação de comunidades em redes biológicas integradas

▼✐❧❡♥❛ ●✐❣❧✐♦❧✐

❉❡t❡r♠✐♥❛çã♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ❡♠ r❡❞❡s

❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s✳

▼✐❧❡♥❛ ●✐❣❧✐♦❧✐

❉❡t❡r♠✐♥❛çã♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ❡♠ r❡❞❡s

❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s✳

▼♦♥♦❣r❛✜❛ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐✲

t✉t♦ ❞❡ ❇✐♦❝✐ê♥❝✐❛s ❞❛ ❯♥✐✈❡rs✐✲

❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✧❏ú❧✐♦ ❞❡

▼❡sq✉✐t❛ ❋✐❧❤♦✧✱ ❈❛♠♣✉s ❞❡ ❇♦✲

t✉❝❛t✉✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡

❇❛❝❤❛r❡❧ ❡♠ ❋ís✐❝❛ ▼é❞✐❝❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ◆❡② ▲❡♠❦❡

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❡st❛r ❝♦♥❝❧✉✐♥❞♦ ✉♠❛ ❡t❛♣❛ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛ ❡ ♣♦r t❡r ♠❡ ♣r♦♣♦r❝✐♦♥❛❞♦ ✉♠❛ ❢❛♠í❧✐❛ ♠❛r✲ ❛✈✐❧❤♦s❛✱ ❛ q✉❛❧ s❡♠♣r❡ ❡st❡✈❡ ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❡ ❞❡♣♦s✐t♦✉ ❝♦♥✜❛♥ç❛ ❡♠ ♠✐♠ ❛té q✉❛♥❞♦ ❡✉ ♠❡s♠❛ ❥á ♥ã♦ ❛ t✐♥❤❛✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ❛❧✉♥♦s ❞❛ t✉r♠❛ ■■■ ❞❡ ❋ís✐❝❛ ▼é❞✐❝❛ ❞❛ ❯♥❡s♣✲❇♦t✉❝❛t✉✱ ♣♦✐s ❛♣❡s❛r ❞❡ t♦❞♦s ♦s ❛❝♦♥t❡❝✐♠❡♥t♦s✱ s❡❥❛♠ ❡st❡s ❜♦♥s ♦✉ r✉✐♥s✱ ❝❛❞❛ ✉♠ ❝♦♥tr✐❜✉✐✉ ❛ s✉❛ ♠❛♥❡✐r❛ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♠ã❡✲✐r♠ã ❑át❤✐❛ ❚❛❦❡❞❛ ❞❛ ❙✐❧✈❛ ♣❡❧♦ ❛♣♦✐♦✱ ❝❛r✐♥❤♦ ❡ ❝✉✐❞❛❞♦ ❞✉r❛♥t❡ t♦❞♦s ❡ss❡s ❛♥♦s✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s✱ t❛♥t♦ ♦s q✉❡ ❡stã♦ s❡♠♣r❡ ❥✉♥t♦ ❛ ♠✐♠ q✉❛♥t♦ ♦s q✉❡✱ ✐♥❢❡❧✐③♠❡♥t❡✱ ❛ ❞✐stâ♥❝✐❛ ♥ã♦ ♣❡r♠✐t❡ ❛ ❝♦♠♣❛♥❤✐❛✱ ♣♦✐s t♦❞♦s sã♦ ♠✉✐t♦ ❡s♣❡❝✐❛✐s ❡ s❡♠ ♦s q✉❛✐s ❡✉ ♥ã♦ ♣♦❞❡r✐❛ s❡❣✉✐r ❡♠ ❢r❡♥t❡✳

❆❣r❛❞❡ç♦✱ t❛♠❜é♠✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❉r✳ ◆❡② ▲❡♠❦❡✱ q✉❡ ♠❡✉ ❞❡✉ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ r❡❛❧✐③❛r ❡st❡ tr❛❜❛❧❤♦ ❡ s❡♠♣r❡ ❡st❡✈❡ ❞✐s♣♦st♦ ❛ ♠❡ ❛✉①✐❧✐❛r ♥❛s ❞✐✜❝✉❧❞❛❞❡s✱ s❡♠ ❡sq✉❡❝❡r✲s❡ ❞❡ ♠❡ ❡♥s✐♥❛r ❛ s❡r ✐♥❞❡♣❡♥❞❡♥t❡❀ ❛♦ ♣r♦❢❡ss♦r ❉r✳ ❏♦❡❧ ▼❡s❛ ❍♦r♠❛③❛ ❡ ❛ t♦❞♦s ♦s ♠❡✉s ❝♦♠♣❛♥❤❡✐r♦s ❞♦ ▲❇❇❈✱ q✉❡ s❡♠♣r❡ ❡st✐✈❡r❛♠ ❞✐s♣♦st♦s ❛ ♠❡ ❛❥✉❞❛r ❡ ♣r♦♣♦r❝✐♦♥❛r ✉♠ ót✐♠♦ ❛♠❜✐❡♥t❡ ❞❡ tr❛❜❛❧❤♦✱ ❝♦♠ ❞✐❛s ♠✉✐t♦ ❢❡❧✐③❡s ❡ ❡♥❣r❛ç❛❞♦s✳

❆❣r❛❞❡ç♦ ❛♦ ❈◆Pq ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦ q✉❡ ♣♦ss✐❜✐❧✐t♦✉ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ♣❡sq✉✐s❛s✳

❘❡s✉♠♦

❆ ✜♠ ❞❡ ❝♦♠♣r❡❡♥❞❡r ❝♦♠♦ ❢❡♥ô♠❡♥♦s ❜✐♦❧ó❣✐❝♦s ♦❝♦rr❡♠✱ ❛s ✐♥t❡r❛çõ❡s ♥ã♦ ❧✐♥❡❛r❡s ❡♥tr❡ ♦s ❝♦♠♣♦♥❡♥t❡s ❡♥✈♦❧✈✐❞♦s ♥❡st❡s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❡❧❡♠❡♥t♦s✱ t❛✐s ❝♦♠♦ ❣❡♥❡s✱ ♣r♦t❡í♥❛s✱ ❡t❝✳✱ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ♣♦r ✉♠ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦ ❞❡♥♦♠✐♥❛❞♦ ❣r❛❢♦ ♦✉ r❡❞❡✱ ♦♥❞❡ ♦s ❡❧❡♠❡♥t♦s ✐♥t❡r❛❣❡♥t❡s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ♥♦❞♦s q✉❡ sã♦ ❝♦♥❡❝t❛❞♦s ♣♦r ❧✐♥❤❛s✳ ❆ ❛♥á❧✐s❡ ❞❡ ✈ár✐❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛❝✐♦♥❛❞❛s ❛ ❣r❛❢♦s ❞❡ r❡❞❡s ❜✐♦❧ó❣✐❝❛s r❡✈❡❧❛ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦s ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s✱ ❞❡♥tr❡ ❡st❛s s❡ ♣♦❞❡ ❝✐t❛r ❛ ❡str✉t✉r❛ ❞♦s ❣r❛❢♦s✱ q✉❡ s❡ ♦r❣❛♥✐③❛♠ ❡♠ ❣r✉♣♦s ❞❡ ♥♦❞♦s ❞❡♥s❛♠❡♥t❡ ❝♦♥❡❝t❛❞♦s ❡♥tr❡ s✐✱ ♠❛s ❡s♣❛rs❛♠❡♥t❡ ✐♥t❡r❧✐❣❛❞♦s ❛ ♦✉tr♦s ❣r✉♣♦s✱ s❡♥❞♦ ❞❡♥♦♠✐♥❛❞❛ ❝♦♠✉♥✐❞❛❞❡ ♦✉ ♠ó❞✉❧♦✳ ❆ ❞❡t❡❝çã♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ❛tr❛✈és ❞❡ ❞✐✈❡rs♦s ❛❧❣♦r✐t♠♦s✱ ♠❛s ❛ ♠❛✐♦r✐❛ s❡ ❜❛s❡✐❛ ♥❛ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❧✐❣❛çõ❡s ❡♥tr❡ ♦s ♥♦❞♦s ❞❡ ✉♠❛ ♠❡s♠❛ ❝♦♠✉♥✐❞❛❞❡✳ ❆té ♦ ♠♦♠❡♥t♦✱ ❛ ❞❡t❡❝çã♦ ❡ ❛♥á❧✐s❡ ❞❛s ❝♦♠✉♥✐❞❛❞❡s ❡♠ r❡❞❡s ❜✐♦❧ó❣✐❝❛s ❢♦r❛♠ r❡❛❧✐③❛❞❛s ❛♣❡♥❛s ♣❛r❛ r❡❞❡s ❝♦♠♣♦st❛s ❝♦♠ ❛♣❡♥❛s ✉♠ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦ ✭❡①✳✿ ✐♥t❡r❛çõ❡s ♣r♦t❡í♥❛✲♣r♦t❡í♥❛ ♦✉ ✐♥t❡r❛çõ❡s ♠❡t❛❜ó❧✐❝❛s✮✳ ❈♦♠♦ ✉♠❛ r❡❞❡ ❜✐♦❧ó❣✐❝❛ r❡❛❧ é ❝♦♠♣♦st❛✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ♣♦r ✐♥t❡r❛çõ❡s ♣r♦t❡í♥❛✲♣r♦t❡í♥❛✱ ♠❡t❛❜ó❧✐❝❛s ❡ ❞❡ r❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧✱ s❡r✐❛ ✐♥t❡r✲ ❡ss❛♥t❡ ✐♥✈❡st✐❣❛r ❝♦♠♦ ❛s ❝♦♠✉♥✐❞❛❞❡s sã♦ ♦r❣❛♥✐③❛❞❛s ♥❡st❡ t✐♣♦ ❞❡ r❡❞❡✳ P❛r❛ ❡st❡ ♣r♦♣ós✐t♦✱ ♥ós ❞❡t❡❝t❛♠♦s ❛s ❝♦♠✉♥✐❞❛❞❡s ❡♠ r❡❞❡s ❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s ❞❛ ❜❛❝tér✐❛ ❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❡ ❞❛ ❧❡✈❡❞✉r❛ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡ ❛tr❛✈és ❞♦ ❈❧✐q✉❡ P❡r❝♦❧❛t✐♦♥ ▼❡t❤♦❞ ❡ ✈❡r✐✜❝❛♠♦s✱ ❛tr❛✈és ❞♦ ❝á❧❝✉❧♦ ❞❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝❛❞❛ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦ ❡ s✉❛ r❡s♣❡❝t✐✈❛ ❡♥tr♦♣✐❛✱ s❡ ♦s ❝♦♠♣♦♥❡♥t❡s ❞❡ss❛s t❡♥❞❡♠ ❛ s❡r ✐♥t❡r❧✐❣❛❞♦s ❝♦♠ ✉♠ t✐♣♦ ♣r❡❢✲ ❡r❡♥❝✐❛❧ ❞❡ ✐♥t❡r❛çã♦✳ P♦st❡r✐♦r♠❡♥t❡✱ ❝♦♠♣❛r❛♠♦s ❡st❡s r❡s✉❧t❛❞♦s ❝♦♠ ♦❜t✐❞♦s ❞❡ r❡❞❡s ❛❧❡❛tór✐❛s ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ❡ss❛s ❝♦♠✉♥✐❞❛❞❡s tê♠ r❡❧❡✈â♥❝✐❛ ❜✐♦❧ó❣✐❝❛✳ P♦r ✜♠✱ ❢♦✐ ♦❜t✐❞❛ ❛ ❡♥tr♦♣✐❛ ♣❛r❛ ♦s ❝❧✉st❡rs ❛❣r✉♣❛❞♦s ❞❡✈✐❞♦ ❛ s✐♠✐❧❛r✐❞❛❞❡s ♥♦ ♣❡r✜❧ ❞❡ ❡①♣r❡ssã♦ ♥❛ ❡str✉t✉r❛ ❞❛ r❡❞❡ ❞❛ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡ ❣❡r❛❞❛ ❛tr❛✈és ❞❡ ❞❛❞♦s ❞❡ ♠✐❝r♦❛r✲ r❛②✱ ❡ ♦s r❡s✉❧t❛❞♦s ❝♦♠♣❛r❛❞♦s ❝♦♠ ♦s ✈❛❧♦r❡s ♦❜t✐❞♦s ♣❛r❛ ❛s ❝♦♠✉♥✐❞❛❞❡s ❡♥❝♦♥tr❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✳

❆❜str❛❝t

❚♦ ✉♥♦❡rst❛♥♦ ❤♦✇ ❜✐♦❧♦❣✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ❡♠❡r❣❡✱ t❤❡ ♥♦♥❧✐♥❡❛r ✐♥t❡r❛❝t✐♦♥s ❛♠♦♥❣ t❤❡ ❝♦♠♣♦♥❡♥ts ❡♥✈♦❧✈❡♦ ✐♥ t❤❡s❡ ❛♥♦ t❤❡ ❝♦rr❡s♣♦♥♦❡♥t ❝♦♥♥❡❝t❡♦ ❡❧❡♠❡♥ts✱ ❧✐❦❡ ❣❡♥❡s✱ ♣r♦t❡✐♥s✱ ❡t❝✳✱ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡♦ ❜② ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t ❝❛❧❧❡♦ ❣r❛♣❤ ♦r ♥❡t✇♦r❦✱ ✇❤❡r❡ ✐♥t❡r❛❝t✐♥❣ ❡❧❡♠❡♥ts ❛r❡ r❡♣r❡s❡♥t❡♦ ❜② ❡♦❣❡s ❝♦♥♥❡❝t✐♥❣ ♣❛✐rs ♦❢ ♥♦♦❡s✳ ❚❤❡ ❛♥❛❧②s✐s ♦❢ ✈❛r✐♦✉s ❣r❛♣❤✲r❡❧❛t❡♦ ♣r♦♣❡rt✐❡s ♦❢ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s ❤❛s r❡✈❡❛❧❡♦ ♠❛♥② ❝❧✉❡s ❛❜♦✉t ❜✐♦❧♦❣✐❝❛❧ ♣r♦❝❡ss❡s✳ ❆♠♦♥❣ t❤❡s❡ ♣r♦♣❡rt✐❡s✱ t❤❡ ❝♦♠♠✉♥✐t② str✉❝t✉r❡✱ ✐✳❡✳ ❣r♦✉♣s ♦❢ ♥♦♦❡s ♦❡♥s❡❧② ❝♦♥♥❡❝t❡♦ ❛♠♦♥❣ t❤❡♠s❡❧✈❡s✱ ❜✉t s♣❛rs❡❧② ❝♦♥♥❡❝t❡♦ t♦ ♦t❤❡r ❣r♦✉♣s✱ ❛r❡ ✐♠♣♦rt❛♥t ❢♦r ✐♦❡♥t✐❢②✐♥❣ s❡♣❛r❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ♠♦♦✉❧❡s ✇✐t❤✐♥ ❜✐♦❧♦❣✐❝❛❧ s②st❡♠s ❢♦r t❤❡ ❝♦♠♣r❡❤❡♥s✐♦♥ ♦❢ t❤❡ ❤✐❣❤✲❧❡✈❡❧ ♦r❣❛♥✐③❛t✐♦♥ ♦❢ t❤❡ ❝❡❧❧✳ ❈♦♠♠✉♥✐t✐❡s✬ ♦❡t❡❝t✐♦♥ ❝❛♥ ❜❡ ♣❡r❢♦r♠❡♦ ❜② ♠❛♥② ❛❧❣♦r✐t❤♠s✱ ❜✉t ♠♦st ♦❢ t❤❡♠ ❛r❡ ❜❛s❡♦ ♦♥ t❤❡ ♦❡♥s✐t② ♦❢ ✐♥t❡r❛❝✲ t✐♦♥s ❛♠♦♥❣ ♥♦♦❡s ♦❢ t❤❡ s❛♠❡ ❝♦♠♠✉♥✐t②✳ ❙♦ ❢❛r✱ t❤❡ ♦❡t❡❝t✐♦♥ ❛♥♦ ❛♥❛❧②s✐s ♦❢ ♥❡t✇♦r❦ ❝♦♠♠✉♥✐t✐❡s ✐♥ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s ❤❛✈❡ ♦♥❧② ❜❡❡♥ ♣✉rs✉❡♦ ❢♦r ♥❡t✇♦r❦s ❝♦♠♣♦s❡♦ ❜② ♦♥❡ t②♣❡ ♦❢ ✐♥t❡r❛❝t✐♦♥ ✭❡✳❣✳ ♣r♦t❡✐♥✲♣r♦t❡✐♥ ✐♥t❡r❛❝t✐♦♥s ♦r ♠❡t❛❜♦❧✐❝ ✐♥t❡r❛❝t✐♦♥s✮✳ ❙✐♥❝❡ ❛ r❡❛❧ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦ ✐s s✐♠✉❧t❛♥❡♦✉s❧② ❝♦♠♣♦s❡♦ ❜② ♣r♦t❡✐♥✲♣r♦t❡✐♥✱ ♠❡t❛❜♦❧✐❝ ❛♥♦ tr❛♥s❝r✐♣t✐♦♥❛❧ r❡❣✉❧❛t♦r② ✐♥t❡r❛❝t✐♦♥s✱ ✐t ✇♦✉❧♦ ❜❡ ✐♥t❡r❡st✐♥❣ t♦ ✐♥✈❡st✐❣❛t❡ ❤♦✇ ❝♦♠♠✉✲ ♥✐t✐❡s ❛r❡ ♦r❣❛♥✐③❡♦ ✐♥ t❤✐s t②♣❡ ♦❢ ♥❡t✇♦r❦✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ ✇❡ ♦❡t❡❝t❡♦ t❤❡ ❝♦♠♠✉♥✐t✐❡s ✐♥ ❛♥ ✐♥t❡❣r❛t❡♦ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦ ♦❢ t❤❡ ❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛♥♦ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡ ❜② ✉s✐♥❣ t❤❡ ❈❧✐q✉❡ P❡r❝♦❧❛t✐♦♥ ▼❡t❤♦❞ ❛♥♦ ✇❡ ✈❡r✐✜❡♦✱ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❢r❡q✉❡♥❝② ♦❢ ❡❛❝❤ t②♣❡ ♦❢ ✐♥t❡r❛❝t✐♦♥ ❛♥♦ ✐ts r❡❧❛t❡♦ ❡♥tr♦♣②✱ ✐❢ ❝♦♠♣♦♥❡♥ts ♦❢ ❝♦♠♠✉♥✐t✐❡s t❡♥♦ t♦ ❜❡ ✐♥t❡r❝♦♥♥❡❝t❡♦ ✇✐t❤ ❛ ♣r❡❢❡rr❡♦ t②♣❡ ♦❢ ✐♥t❡r❛❝t✐♦♥✳ ▲❛t❡r✱ ✇❡ ❝♦♠♣❛r❡♦ t❤❡s❡ r❡s✉❧ts ✇✐t❤ t❤♦s❡ ♦❜t❛✐♥❡♦ ❢r♦♠ r❛♥♦♦♠✐③❡♦ ♥❡t✇♦r❦s t♦ ♦❡t❡r♠✐♥❡ ✐❢ t❤❡s❡ ❝♦♠♠✉♥✐t✐❡s ❤❛✈❡ ❜✐♦❧♦❣✐❝❛❧ r❡❧❡✈❛♥❝❡✳ ❋✐♥❛❧❧②✱ ❡♥tr♦♣② ✇❛s ♦❜t❛✐♥❡♦ ❢♦r ❝❧✉st❡rs ❣r♦✉♣❡♦ ❢♦r s✐♠✐❧❛r✐t✐❡s ✐♥ t❤❡ ♣r♦✜❧❡ ♦❢ ❡①♣r❡ss✐♦♥ ❢♦r ❙ ❝❡r❡✈✐s✐❛❡✬s ✐♥t❡❣r❛t❡♦ ♥❡t✇♦r❦✱ t❤❛t ✐t ✇❛s ❝r❡❛t❡♦ ❜② ♠✐❝r♦❛rr❛② ♦❛t❛✱ ❛♥♦ r❡s✉❧ts ✇❡r❡ ❝♦♠♣❛r❡♦ ✇✐t❤ ✈❛❧✉❡s ♦❜t❛✐♥❡♦ ❢♦r ♣r❡✈✐♦✉s❧② ✐♦❡♥t✐✜❡♦ ❝♦♠♠✉♥✐t✐❡s✳

❮♥❞✐❝❡

Pá❣✐♥❛

❘❡s✉♠♦ ✸

❆❜str❛❝t ✹

✶ ■♥tr♦❞✉çã♦ ✻

✶✳✶ ❚❡♦r✐❛ ❞❡ ❘❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❇✐♦❧♦❣✐❛ ❞❡ ❙✐st❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷ ✸❜❥❡t✐✈♦s ✶✸

✷✳✶ ❖❜❥❡t✐✈♦s ●❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❖❜❥❡t✐✈♦s ❊s♣❡❝í✜❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✸ ▼❡t♦❞♦❧♦❣✐❛ ✶✹

✸✳✶ ❊t❛♣❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✶✳✶ ❖❜t❡♥çã♦ ❞❡ ❞❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✶✳✷ ❉✐✈✐sã♦ ❡♠ ❝♦♠✉♥✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✶✳✸ ❈á❧❝✉❧♦ ❞❛ ❡♥tr♦♣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✶✳✹ ❘❡♣r❡s❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✷ ❇❛♥❝♦ ❞❡ ❞❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✸ ❆❧❣♦r✐t♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✸✳✶ ❈❧❛✉s❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✸✳✷ ❈❧✐q✉❡ P❡r❝♦❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✸✳✸ ▼✐♥❈✉t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✹ ■♠♣❧❡♠❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✹ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦ ✷✶

✻

✶ ■♥tr♦❞✉çã♦

❯♠❛ q✉❡stã♦ ✐♠♣♦rt❛♥t❡ ♥❛ ❢✉sã♦ ❞❛s ár❡❛s ❝♦♥❤❡❝✐❞❛s ♣♦r ❜✐♦❧♦❣✐❛ s✐stê♠✐❝❛ ❡ t❡♦✲ r✐❛ ❞❡ r❡❞❡s é ❝♦♠♦ ❡ ♣♦rq✉❡ ✐♥t❡❣r❛r ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s✱ ❣❡r❛r r❡❞❡s ❞❡ ❞❛❞♦s ❡✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♥❡st❡ tr❛❜❛❧❤♦✱ ♣♦r q✉❡ ✐❞❡♥t✐✜❝❛r ❝♦♠✉♥✐❞❛❞❡s ♣r❡s❡♥t❡s ♥❡st❛s r❡❞❡s✳

❈♦♥s✐❞❡r❛♥❞♦ ♦ ❡st✉❞♦ ❞❛s ❝é❧✉❧❛s ❞❡ ✉♠ ♦r❣❛♥✐s♠♦✱ ❡st❛s sã♦ ❝♦♠♣♦st❛s ♣♦r ♠✉✐t♦s t✐♣♦s ❞✐st✐♥t♦s ❞❡ ♠♦❧é❝✉❧❛s ❡✱ ❛♣❡s❛r ❞❛s ✐♥❢♦r♠❛çõ❡s ❝❡❧✉❧❛r❡s ❡st❛r❡♠ ❝♦♥t✐❞❛s ♥❛s ♠♦❧é❝✉❧❛s ❞❡ ❉◆❆✱ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❝❡❧✉❧❛r❡s ❞❡♣❡♥❞❡♠ ❞❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ✈ár✐♦s ❝♦♥st✐t✉✐♥t❡s ❡ ❞❡st❡s ❝♦♠ ♦ ♠❡✐♦✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❛♣❧✐❝❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ r❡❞❡s ♥❛ ❜✐♦❧♦❣✐❛ s✐stê♠✐❝❛ ✈✐s❛ ❡st✉❞❛r✱ ❞❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ❝♦♠♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❝❡❧✉❧❛r❡s s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ❛s ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ❡ ❣ê♥✐❝❛s q✉❡ ❛s ❝♦♥st✐t✉❡♠ ❡ q✉❛❧ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❞❡t❡r✲ ♠✐♥❛çã♦ ❞♦s ❣r✉♣♦s ❞❡ ✐♥t❡r❛çõ❡s ❢♦r♠❛❞♦s ❡♠ ❝❛❞❛ r❡❞❡✳

✶✳✶ ❚❡♦r✐❛ ❞❡ ❘❡❞❡s

❆❧❣✉♥s ❢❡♥ô♠❡♥♦s ❜✐♦❧ó❣✐❝♦s sã♦ ♣r♦♣r✐❡❞❛❞❡s ❡♠❡r❣❡♥t❡s ❞❡ ✐♥t❡r❛çõ❡s ♥ã♦ ❧✐♥❡❛r❡s ❡♥tr❡ ♠ú❧t✐♣❧♦s ❝♦♠♣♦♥❡♥t❡s ❞❡ ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s✱ t❛✐s ❝♦♠♦ ❣❡♥❡s✱ ♣r♦t❡í♥❛s✱ ♠❡t❛❜ó✲ ❧✐t♦s ❡ ♦✉tr♦s✳ P❛r❛ ❝♦♠♣r❡❡♥❞❡r ❝♦♠♦ ❡st❛s ♣r♦♣r✐❡❞❛❞❡s s❡ ♦r✐❣✐♥❛♠✱ ❡❧❡♠❡♥t♦s ✐♥t❡r❛❣❡♥✲ t❡s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❝♦♥❡①õ❡s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s ❞❡♥♦♠✐♥❛❞♦s ❣r❛❢♦s ♦✉ r❡❞❡s✱ ♥♦s q✉❛✐s ❛s ✐♥t❡r❛çõ❡s sã♦ r❡♣r❡s❡♥t❛❞❛s ♣♦r ❧✐♥❤❛s ✭❡❞❣❡s✮ q✉❡ ❝♦♥❡❝t❛♠ ♣❛r❡s ❞❡ ♥♦❞♦s ❬❇❛r❛❜ás✐ ❡ ❖❧t✈❛✐ ✷✵✵✹❪✳

❯♠❛ r❡❞❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❞❡ ❞✐✈❡rs❛s ♠❛♥❡✐r❛s ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❛♥á❧✐s❡ ❛ s❡r r❡❛❧✐③❛❞❛✳ ❆ ❋✐❣✉r❛ ✶ ✐❧✉str❛ ❛❧❣✉♠❛s ❢♦r♠❛s ❞❡ r❡♣r❡s❡♥t❛çã♦ ✉t✐❧✐③❛♥❞♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ r❡❞❡ ❝♦♠ q✉❛tr♦ t✐♣♦s ❞✐❢❡r❡♥t❡s ❞❡ ♥♦❞♦s ✐❞❡♥t✐✜❝❛❞♦s ♣♦r s✉❛s ♣♦s✐çõ❡s ❡str✉t✉r❛✐s✳ ◆❛ ♣❛rt❡ ❛✮ ❞❛ ✜❣✉r❛✱ ♣♦❞❡✲s❡ ✈✐s✉❛❧✐③❛r ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❣r✉♣♦s ❞❡ ♥♦❞♦s✿ ♦s ♥♦❞♦s ❞♦s t✐♣♦s ❆ ❡ ❈ sã♦ ❞❡♥s❛♠❡♥t❡ ❝♦♥❡❝t❛❞♦s ✐♥t❡r♥❛♠❡♥t❡✱ ❥á ♦s ♥♦❞♦s ❞♦ t✐♣♦ ❇ t❡♠ ❝♦♥❡①õ❡s ❝♦♠ ♦s t✐♣♦s ❆ ❡ ❈✱ ♠❛s ♥ã♦ ♣♦ss✉❡♠ ❝♦♥❡①õ❡s ❡♥tr❡ s✐✱ sã♦ r❡s♣♦♥sá✈❡✐s ♣♦r ✐♥t❡r❧✐❣❛r ♦s ❣r✉♣♦s ❆ ❡ ❈❀ ♦s ♥♦❞♦s ❞♦ t✐♣♦ ❉ ♥ã♦ ♣♦ss✉❡♠ ❝♦♥❡①õ❡s ❡♥tr❡ s✐✱ ✐♥t❡r❛❣✐♥❞♦ ❛♣❡♥❛s ❝♦♠ ♥♦❞♦s ❞♦ ❣r✉♣♦ ❈ ❡✱ ❞❡st❡ ♠♦❞♦✱ ❢♦r♠❛♠ ❛ ♣❡r✐❢❡r✐❛ ❞♦s ♥♦❞♦s t✐♣♦ ❈✳ ❆s ♣❛rt❡s ❜✮ ❡ ❝✮ ❞❛ ✜❣✉r❛ ♠♦str❛♠ ♦✉tr❛s ❞✉❛s ♣♦ssí✈❡✐s ♠❛♥❡✐r❛s ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞♦s ❣r❛❢♦s✱ s❡♥❞♦ q✉❡ ❡♠ ❜✮ ❛ r❡❞❡ ❡st❛ ❞✐✈✐❞✐❞❛ ❡♠ ❣r✉♣♦s✱ ♥♦s q✉❛✐s ♦s ♥♦❞♦s ✐♥t❡r❛❣❡♠ ♠❛✐s ✐♥t❡♥s❛♠❡♥t❡✱ ❡♥q✉❛♥t♦ ❡♠ ❝✮ ❛ r❡♣r❡s❡♥t❛çã♦ é ❢❡✐t❛ ❛tr❛✈és ❞♦s t✐♣♦s ❞❡ ♥♦❞♦s✳ ❏á ❛s ♣❛rt❡s ❞✮ ❡ ❡✮ ✐❧✉str❛♠ ✉♠❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡♥♦♠✐♥❛❞❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ❞❛❞♦s ❡♠ ❣r✉♣♦s ♦✉ t✐♣♦s ❞❡ ♥♦❞♦s✱ ❛ss✐♠ ❝♦♠♦ ♥♦s ✐t❡♥s ❜✮ ❡ ❝✮✳ ❆ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ tr❛t❛✲s❡ ❞❡ ✉♠❛ ♠❛tr✐③ ❞❡ ❞✐♠❡♥sã♦𝑛×𝑛✱ ♥❛ q✉❛❧ ❛s ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s r❡♣r❡s❡♥t❛♠ ♦s ♥♦❞♦s

❡ ❛ ✐♥t❡rs❡❝çã♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ss❛ ♠❛tr✐③ r❡♣r❡s❡♥t❛ ❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❡st❡s✳ ❆ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ é ❜✐♥ár✐❛✿ 𝑀(𝑖, 𝑗) = 1✱ ❝❛s♦ ❡①✐st❛ ✐♥t❡r❛çã♦ ❡♥tr❡

✼

❋✐❣✉r❛ ✶✿ ❋♦r♠❛s ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ r❡❞❡s ❬P✐♥❦❡rt✱ ❙❝❤✉❧t③ ❡ ❘❡✐❝❤❛r❞t ✷✵✵✽❪✳

❊♠ ✉♠❛ r❡❞❡ ❜✐♦❧ó❣✐❝❛ ✐♥t❡❣r❛❞❛✱ ✜❣✉r❛ ✷ ✭✜❣✉r❛ ❝❡❞✐❞❛ ♣♦r ◆❡② ▲❡♠❦❡✮✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦s ❣❡♥❡s ✭♥♦❞♦s✮ ❣✶ ❡ ❣✷ ❝♦❞✐✜❝❛♠ ❛s ♣r♦t❡í♥❛s ♣✶ ❡ ♣✷✱ ❡st❡s sã♦ ❝♦♥❡❝t❛❞♦s ♥❛ r❡❞❡✱ s❡ ♦❝♦rr❡✿

✶✳ ✐♥t❡r❛çã♦ ♣r♦t❡í♥❛✲♣r♦t❡í♥❛✿ ♣✶ ❡ ♣✷ ✐♥t❡r❛❣❡♠ ✜s✐❝❛♠❡♥t❡❀

✷✳ r❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧✿ ❣✶ r❡❣✉❧❛ ❛ tr❛♥s❝r✐çã♦ ❞♦ ❣❡♥❡ ❣✷ ♦✉

∙

∙

✾

✐♥t❡r❧✐❣❛❞♦s ❛ ♦✉tr♦s ❣r✉♣♦s✱ s❡♥❞♦ ❡st❛ ❢♦r❣❛çã♦ ❝♦♥❞❡❝✐❞❛ ♣♦r ❣ó❞✉❧♦ ♦✉ ❝♦❣✉♥✐❞❛❞❡✳ P♦❞❡✲s❡ ❝✐t❛r ❞✐✈❡rs♦s ❡✉❡❣♣❧♦s ❞❡ r❡❞❡s ♦r❣❛♥✐③❛❞❛s ❡❣ ❣ó❞✉❧♦s✱ ❛ss✐❣ ❝♦❣♦ ❛ ✇♦r❧❞✇✐❞❡ ✇❡❜ ❢♦r❣❛❞❛ ♣♦r ❣r✉♣♦s ❞❡ ♣á❣✐♥❛s ♥❛ ✐♥t❡r♥❡t ❞✐✈✐❞✐❞♦s ♣❡❧♦ ❛ss✉♥t♦ ❞♦s ❣❡s❣♦s✱ r❡❞❡s s♦❝✐❛✐s q✉❡ sã♦ ♦r❣❛♥✐③❛❞❛s ❡❣ ✉♥✐❞❛❞❡s s♦❝✐❛✐s✱ ❞❡♥tr❡ ♦✉tr♦s ❬◆❡✇❣❛♥ ✷✵✵✻❪✳ ❘❡❞❡s ❜✐♦❧ó❣✐❝❛s ♣♦ss✉❡❣✱ ❡❣ ❣❡r❛❧✱ ❡str✉t✉r❛ ❣♦❞✉❧❛r✳

❆ ✐❣♣♦rtâ♥❝✐❛ ❞❛ ❡str✉t✉r❛ ❣♦❞✉❧❛r ❡❣ ✉❣❛ r❡❞❡ ❜✐♦❧ó❣✐❝❛ é s✉st❡♥t❛❞❛ ♣♦r ❞✐✈❡rs♦s ❢❛t♦r❡s ❬❍❛rt✇❡❧❧ ❡t ❛❧✳ ✶✾✾✾❪✿

∙ ❆♦ s❡ ✐❞❡♥t✐✜❝❛r ✉❣❛ ❝♦❣✉♥✐❞❛❞❡ ❡ ❛tr✐❜✉✐r ❛ ❡st❛ ✉❣❛ ❢✉♥çã♦✱ ♣♦❞❡✲s❡ ❡st✉❞❛r ❛

❢✉♥çã♦ ❡ ❛ r❡❧❛çã♦ ❞❡st❛ ❝♦❣ ❝❛❞❛ ❝♦♥st✐t✉✐♥t❡ ❞❡st❡ ❣ó❞✉❧♦❀

∙ ❆ ❞❛❜✐❧✐❞❛❞❡ ❝❡❧✉❧❛r ❞❡ ✐♥t❡❣r❛r ✐♥❢♦r❣❛çõ❡s ✈✐♥❞❛s ❞❡ ❢♦♥t❡s ❞✐st✐♥t❛s ❞♦ ♦r❣❛♥✐s❣♦

♣♦❞❡ s❡r ❞❡s❝r✐t❛ ♣❡❧♦s ♣❛❞rõ❡s ❞❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❣ó❞✉❧♦s ✐❞❡♥t✐✜❝❛❞♦s❀

∙ ❆ ❡str✉t✉r❛ ❣♦❞✉❧❛r ❢❛❝✐❧✐t❛ ❣✉❞❛♥ç❛s ❡✈♦❧✉t✐✈❛s✱ ❥á q✉❡ ❛ ❢✉♥çã♦ ♣r✐♥❝✐♣❛❧ ❞❡

✉❣ ❣ó❞✉❧♦ é r❡s✐st❡♥t❡ ❛ ❣✉❞❛♥ç❛s✱ ❣❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❢✉♥çõ❡s ❞❡ ✉❣❛ ❝é❧✉❧❛ ✭❢❡♥ót✐♣♦ ❞❡st❛✮ ♣♦❞❡❣ s♦❢r❡r ❣♦❞✐✜❝❛çõ❡s ❡✈♦❧✉t✐✈❛s ❛tr❛✈és ❞❡ ❛❧t❡r❛çõ❡s ♥❛s ✐♥✲ t❡r❛çõ❡s ❡♥tr❡ ♦s ❣ó❞✉❧♦s ❞✐st✐♥t♦s✳ P♦r ❡✉❡❣♣❧♦✱ s❡ ❛ ❢✉♥çã♦ ❞❡ ✉❣❛ ♣r♦t❡í♥❛ ❛❢❡t❛ss❡ ❞✐r❡t❛❣❡♥t❡ t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉❣❛ ❝é❧✉❧❛✱ ♣r♦✈❛✈❡❧❣❡♥t❡ ❡st❛ ♣r♦✲ t❡í♥❛ ♥ã♦ s♦❢r❡r✐❛ ❣✉❞❛♥ç❛s✱ ❣❛s s❡ ❛ ❢✉♥çã♦ ❞❡ ✉❣❛ ♣r♦t❡í♥❛ ❡stá r❡str✐t❛ ❛ ✉❣ ❣r✉♣♦ ❢✉♥❝✐♦♥❛❧✱ ❛♣❡♥❛s✱ ❡ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❣r✉♣♦s ♦❝♦rr❡❣ ❛tr❛✈és ❞❡ ♣r♦t❡í✲ ♥❛s ✐♥❞✐✈✐❞✉❛✐s✱ s❡rá ❣✉✐t♦ ❣❛✐s ❢á❝✐❧ ❣♦❞✐✜❝❛r ❡ s✉♣r✐❣✐r ✐♥t❡r❛çõ❡s ❞❡st❛ ♣r♦t❡í♥❛ ❝♦❣ ♦✉tr♦s ❣ó❞✉❧♦s ❢✉♥❝✐♦♥❛✐s❀

∙ ❖ ❝♦♥❞❡❝✐❣❡♥t♦ ❞❡ ✉❣ ❣❡❣❜r♦ ❞❡ ✉❣❛ ❝♦❣✉♥✐❞❛❞❡ ♣♦❞❡ ❛✉✉✐❧✐❛r ♥♦ ❡st✉❞♦ ❞❡ ✉❣

❣❡❣❜r♦ ❞❡ ♦✉tr❛ ❝♦❣✉♥✐❞❛❞❡ r❡❧❛❝✐♦♥❛❞❛ ❛ ♣r✐❣❡✐r❛ ❛tr❛✈és ❞❛s ✐♥t❡r❛çõ❡s❀

∙ ❉♦ ❣❡s❣♦ ❣♦❞♦ q✉❡ ❛ ❝♦❣♣❛r❛çã♦ ❞❡ s❡q✉❡♥❝✐❛s ♣r♦té✐❝❛s ❞❡ ❡s♣é❝✐❡s r❡❧❛❝✐♦♥❛❞❛s

♣♦❞❡ s✉❣❡r✐r s✉❛s ❢✉♥çõ❡s✱ ❛ ❝♦❣♣❛r❛çã♦ ❡♥tr❡ ❣ó❞✉❧♦s ❞❡ ❢✉♥çõ❡s ❝♦❣✉♥s✱ ❡❣ ♦r❣❛♥✐s❣♦s ❞✐st✐♥t♦s✱ ♣♦❞❡ ❛✉✉✐❧✐❛r ♥❛ ❝♦❣♣r❡❡♥sã♦ ❞❛ ❢✉♥çã♦ ❞❡ss❡s ❣ó❞✉❧♦s✳

✶✵

❣❧♦❜❛❧ ❞❛ r❡❞❡ é ❛♠❡❛ç❛❞❛✱ ♦ q✉❡ ♠♦str❛ s❡r ❢rá❣✐❧ ❛♦ ♠❛✉ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦s ♥♦❞♦s ❬❑✐t❛♥♦ ✷✵✵✼❪✳

P♦❞❡✲s❡✱ t❛♠❜é♠✱ r❡❛❧✐③❛r ❛♥á❧✐s❡ ❛tr❛✈és ❞❡ ♠❛✐s ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✐❞❡♥t✐✜❝❛r ❛ r❡s✐stê♥❝✐❛ ❞❡ ❝❛❞❛ ❝♦♠✉♥✐❞❛❞❡ ❢♦r♠❛❞❛ ♥❛ r❡❞❡✳ ❊st❡ t✐♣♦ ❞❡ ❡st✉❞♦ é ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r ❣r✉♣♦s q✉❡ s❡r✐❛♠ ♣♦ssí✈❡✐s ❛❧✈♦s ♣❛r❛ ❛♣❧✐❝❛çã♦ ❞❡ ❞r♦❣❛s✱ ❛tr❛✈és ❞❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ❣r✉♣♦s r❡s✐st❡♥t❡s à ❞r♦❣❛ ♦✉ ❣r✉♣♦s q✉❡ ❡st❡❥❛♠ ❝♦♠ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❛♥♦r♠❛❧✱ ♦s q✉❛✐s ♣♦❞❡♠ t❡r s❡✉ ❢✉♥❝✐♦♥❛♠❡♥t♦ ♥♦r♠❛❧ r❡st❛❜❡❧❡❝✐❞♦ s❡ ❢♦r❡♠ ❛❧✈♦s ♣❛r❛ ❞r♦❣❛s ❡♠ ♣❡sq✉✐s❛✳ ❖ s✉❝❡ss♦ ❞❡ ♠✉✐t❛s ❞r♦❣❛s ♥♦ ♣❛ss❛❞♦ ❞❡✉✲ s❡ ❛ ❡st❛s ❛t✐♥❣✐r❡♠ ♦ ♣♦♥t♦ ❞❡ ❢r❛❣✐❧✐❞❛❞❡✱ r❡❣✉❧❛♥❞♦ ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s ❛tr❛✈és ❞❛ ♠♦❞✐✜❝❛çã♦ ❞❡ ❛♣❡♥❛s ✉♠ ♦✉ ♣♦✉❝♦s ❣r✉♣♦s ✐❞❡♥t✐✜❝❛❞♦s ♣r❡✈✐❛♠❡♥t❡ ❬❑✐t❛♥♦ ✷✵✵✼❪✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱ ♣♦❞❡ s❡r ♥❡❝❡ssár✐❛ ❛ ✐♥❛t✐✈❛çã♦ ❞❡ ❞❡t❡r♠✐♥❛❞❛ ❢✉♥çã♦ ❡ ✐st♦ ♣♦❞❡ s❡r ❛❧❝❛♥ç❛❞♦ ❛ ♣❛rt✐r ❞❛ r❡♠♦çã♦ ❞❡ ✉♠ ❣r✉♣♦ ❢✉♥❝✐♦♥❛❧ ✐♥t❡✐r♦ ❬●✉❧❜❛❤❝❡ ❡ ▲❡❤♠❛♥♥ ✷✵✵✽❪✳ ❯♠❛ ♦✉tr❛ ár❡❛ ♥❛ q✉❛❧ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ♠ó❞✉❧♦s é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ é ❛ ❜✐♦❧♦❣✐❛ ♠♦❧❡❝✉❧❛r✱ ♠❛✐s ❡①❛t❛♠❡♥t❡✱ ❛ ár❡❛ r❡❧❛❝✐♦♥❛❞❛ ❛♦ ❡st✉❞♦ ❞❛ ❢✉♥çã♦ ❞♦s ❣❡♥❡s✳ ❯♠ ♠ét♦❞♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡st✉❞❛r ❛ ❢✉♥çã♦ ❣ê♥✐❝❛ é ❛ ❞❡❧❡çã♦ ❞❡ ❣❡♥❡s ❡ ♣♦st❡r✐♦r ♦❜s❡r✈❛çã♦ ❞❛s ♠✉❞❛♥ç❛s ♦❝♦rr✐❞❛s ♥♦ ❢❡♥ót✐♣♦✳ P♦ré♠✱ ❛ ❞❡❧❡çã♦ ❞❡ ✉♠ ú♥✐❝♦ ❣❡♥❡ ♥ã♦ r❡✈❡❧❛ ♠✉✐t❛ ✐♥❢♦r♠❛çã♦✱ ❡♥q✉❛♥t♦ ❛ ❞❡❧❡çã♦ ❞❡ ❣r✉♣♦s ❞❡ ❣❡♥❡s✱ ♣❡❧♦ ❝♦♥trár✐♦✱ ♣♦❞❡ ❣❡r❛r ♠✉✐t❛s ✐♥❢♦r♠❛çõ❡s ✐♠♣♦rt❛♥t❡s✳ ❈♦♠♦ ♦r❣❛♥✐s♠♦s ❡✉❝❛r✐♦♥t❡s ♣♦ss✉❡♠ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❛ ❞❡ ❣❡♥❡s✱ ❛ ❝♦♠❜✐♥❛çã♦ ❞❡st❡s ♣❛r❛ ❞❡❧❡çã♦ ❞❡ ❣r✉♣♦s s❡ t♦r♥❛ ✐♥✈✐á✈❡❧✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ♣r♦♠✐ss♦r❛ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♦s ❣r✉♣♦s✳

❙❡ ❛s ❝♦♠✉♥✐❞❛❞❡s ❡stã♦ ❝❧❛r❛♠❡♥t❡ s❡♣❛r❛❞❛s ♥♦ s✐st❡♠❛✱ ❛ ♠❛✐♦r✐❛ ❞♦s ♠ét♦❞♦s ❞❡ ❛❣❧♦♠❡r❛çã♦ ♣♦❞❡ ✐❞❡♥t✐✜❝á✲❧❛s✳ ❉❡ ❢❛t♦✱ ♠✉✐t♦s ♠ét♦❞♦s tê♠ s✐❞♦ ✐♥tr♦❞✉③✐❞♦s r❡❝❡♥t❡✲ ♠❡♥t❡ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♠ó❞✉❧♦s ❞❡ r❡❞❡s ✈❛r✐❛❞❛s ❬◆❡✇♠❛♥ ✷✵✵✻❪ ❬❆❞❛♠❝s❡❦ ❡t ❛❧✳ ✷✵✵✻❪ ❬❈❧❛✉s❡t✱ ◆❡✇♠❛♥ ❡ ▼♦♦r❡ ✷✵✵✹❪ ❬P❛❧❧❛ ❡t ❛❧✳ ✷✵✵✺❪ ❬❑✉♠♣✉❧❛ ❡t ❛❧✳ ✷✵✵✽❪✱ ✉s❛♥❞♦ ❛✐♥❞❛ ❛ ❞❡s❝r✐çã♦ t♦♣♦❧ó❣✐❝❛ ❞❛s r❡❞❡s ♦✉ ❝♦♠❜✐♥❛♥❞♦ ❛ t♦♣♦❧♦❣✐❛ ❝♦♠ ♦s ❞❛❞♦s ❣❡♥ô♠✐❝♦s ❢✉♥❝✐♦♥❛✐s ✐♥t❡❣r❛❞♦s✳ ❊♥tr❡t❛♥t♦✱ ♠ét♦❞♦s ❞✐❢❡r❡♥t❡s ♣r❡❞✐③❡♠ ❞✐✈✐sõ❡s ❞✐❢❡r❡♥t❡s ❡♥tr❡ ♠ó❞✉❧♦s q✉❡ ♥ã♦ ❡stã♦ ❝❧❛r❛♠❡♥t❡ s❡♣❛r❛❞♦s✳ ❆t✉❛❧♠❡♥t❡✱ ♥ã♦ ❤á ❝r✐tér✐♦ ♠❛t❡♠át✐❝♦ ♦❜❥❡t✐✈♦ ♣❛r❛ ❞❡❝✐❞✐r q✉❡ ✉♠❛ ❞✐✈✐sã♦ é ♠❡❧❤♦r q✉❡ ♦✉tr❛✱ ❛ss✐♠✱ ❛❧❣✉♥s ♣❛râ♠❡tr♦s ✐♥✲ t❡r♥♦s ❝♦♥tr♦❧❛♠ ♦ t❛♠❛♥❤♦ tí♣✐❝♦ ❞❡ ♠ó❞✉❧♦s ♥ã♦ ❝♦♥❤❡❝✐❞♦s ❡ ❛ ♠✉❞❛♥ç❛ ❞❡st❡s ❣❡r❛ ❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ♠ét♦❞♦s ❞✐st✐♥t♦s ❞❡ ❞✐✈✐sã♦ ❡♠ ❝♦♠✉♥✐❞❛❞❡s✳ ❯♠❛ s♦❧✉çã♦✱ ♣♦rt❛♥t♦✱ é ❡✈✐t❛r ♣r♦❝✉r❛r ✉♠❛ s❡♣❛r❛çã♦ ❡♠ ✉♠ ❣r✉♣♦ ❛❜s♦❧✉t♦ ❞❡ ♠ó❞✉❧♦s✱ ♠❛s s✐♠ ✈✐s✉❛❧✐③❛r ✉♠❛ r❡❧❛çã♦ ❤✐❡rárq✉✐❝❛ ❡♥tr❡ ❝♦♠✉♥✐❞❛❞❡s ❞❡ t❛♠❛♥❤♦s ❞✐❢❡r❡♥t❡s✳

✶✶

❬◆❡✇♠❛♥ ✷✵✵✻❪✳ ❆♠❜❛s ❛s ❧✐♥❤❛s ❞❡ ♣❡sq✉✐s❛ ❞❡s❝r✐t❛s sã♦ ❞✐r❡❝✐♦♥❛❞❛s ❛ ♠❡s♠❛ q✉❡stã♦✱ ❡♠❜♦r❛ ♣♦r ♠❡✐♦s ✉♠ ♣♦✉❝♦ ❞✐❢❡r❡♥t❡s✳ ❍á✱ ❡♥tr❡t❛♥t♦✱ ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ♦s ♦❜❥❡t✐✈♦s ❞♦s ❞♦✐s ❝❛♠♣♦s q✉❡ ♦r✐❣✐♥❛♠ ❛ ❞✐❢❡r❡♥ç❛ té❝♥✐❝❛ ❞❛s ❛♣r♦①✐♠❛çõ❡s ❞❡s❡❥❛❞❛s✳

❯♠ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ tí♣✐❝❛ ❞❛ ♣❛rt✐çã♦ ❞❡ ❣r❛❢♦s é ❛ ❞✐✈✐sã♦ ❞❡ t❛r❡❢❛s ❡♥tr❡ ♦s ♣r♦❝❡ss❛❞♦r❡s ♥❛ ❝♦♠♣✉t❛çã♦ ♣❛r❛❧❡❧❛ ♣❛r❛ ♠✐♥✐♠✐③❛r ❛ q✉❛♥t✐❞❛❞❡ ♥❡❝❡ssár✐❛ ❞❡ ❝♦♠✉✲ ♥✐❝❛çã♦ ❡♥tr❡ ♦s ♣r♦❝❡ss❛❞♦r❡s✳ ❊♠ ❝❛❞❛ ❝á❧❝✉❧♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♣r♦❝❡ss❛❞♦r❡s ❡ ❛ q✉❛♥t✐❛ ❞❡ t❛r❡❢❛s q✉❡ ❝❛❞❛ ✉♠ ❞❡st❡s ♣♦❞❡ s✉♣♦rt❛r sã♦✱ ♥♦r♠❛❧♠❡♥t❡✱ ❝♦♥❤❡❝✐❞♦s ❡✱ ❞❡st❡ ♠♦❞♦✱ ♣♦❞❡✲s❡ ✐❞❡♥t✐✜❝❛r ❡♠ q✉❛♥t♦s ❣r✉♣♦s ❛ r❡❞❡ s❡rá❞✐✈✐❞✐❞❛ ❡ ♦ t❛♠❛♥❤♦ ❞❡ ❝❛❞❛ ❣r✉♣♦✳ ❆❧é♠ ❞✐ss♦✱ ♦ ♦❜❥❡t✐✈♦ é✱ ♥♦r♠❛❧♠❡♥t❡✱ ❡♥❝♦♥tr❛r ❛ ♠❡❧❤♦r ❞✐✈✐sã♦ ❞❛ r❡❞❡ ✐♥❞❡♣❡♥❞❡♥t❡ s❡ ✉♠❛ ❜♦❛ ❞✐✈✐sã♦ ❡①✐st❡ ❬◆❡✇♠❛♥ ✷✵✵✻❪✳

❆ ❞❡t❡❝çã♦ ❞❡ ❡str✉t✉r❛ ❞❡ ❝♦♠✉♥✐❞❛❞❡s✱ ♣❡❧♦ ❝♦♥trár✐♦✱ é✱ t❛❧✈❡③✱ ♦ ♠❡❧❤♦r ♠ét♦❞♦ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ❛ ❡str✉t✉r❛ ❞❡ r❡❞❡s ❞❡ ❧❛r❣❛ ❡s❝❛❧❛✱ ❝♦♠♦ r❡❞❡s s♦❝✐❛✐s✱ ❞❛ ✐♥t❡r♥❡t ❡ ❞❛❞♦s ❞❛ ✇❡❜✱ ♦✉ r❡❞❡s ❜✐♦❧ó❣✐❝❛s✳ ❊st❡ ♠ét♦❞♦ ❛ss✉♠❡ q✉❡ ❛ r❡❞❡ ❞❡ ✐♥t❡r❡ss❡ ❡stá❞✐✈✐❞✐❞❛ ♥❛t✉r❛❧♠❡♥t❡ ❡♠ s✉❜❣r✉♣♦s ❡ ♦ tr❛❜❛❧❤♦ ❞♦s ♣❡sq✉✐s❛❞♦r❡s é ✐❞❡♥t✐✜❝á✲❧♦s✳ ❖ ♥ú♠❡r♦ ❡ ♦ t❛♠❛♥❤♦ ❞♦s ❣r✉♣♦s sã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ ♣ró♣r✐❛ r❡❞❡ ❡ ♥ã♦ ♣❡❧♦ ♣❡sq✉✐s❛❞♦r✳ ❆❧é♠ ❞✐ss♦✱ ❡st❡ ♠ét♦❞♦ ♣♦❞❡✱ ❡①♣❧✐❝✐t❛♠❡♥t❡✱ ❛❞♠✐t✐r ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ♥ã♦ ❤❛❥❛ ✉♠❛ ❜♦❛ ❞✐✈✐sã♦ ♣❛r❛ ❛ r❡❞❡ ❬◆❡✇♠❛♥ ✷✵✵✻❪✳

✶✳✷ ❇✐♦❧♦❣✐❛ ❞❡ ❙✐st❡♠❛s

❆ ❜✐♦❧♦❣✐❛ ❞❡ s✐st❡♠❛s ❛❜r❛♥❣❡ ♦ ❡st✉❞♦ ❞❡ r❡❞❡s ❜✐♦❧ó❣✐❝❛s ❝♦♠♣❧❡①❛s✱ ❝♦♠♦ ♠❡t❛❜ó❧✐✲ ❝❛s✱ ❣ê♥✐❝❛s✱ ♣r♦t❡✐❝❛s✱ ❞❡♥tr❡ ♦✉tr❛s✱ ❡ ✈✐s❛✱ ❛ ♣❛rt✐r ❞❡ ❢❡♥ô♠❡♥♦s ❝♦♠♣❧❡①♦s✱ ♣r♦✈❡r ✉♠❛ ❡str✉t✉r❛ ❛❜str❛t❛ q✉❡ ❛✉♠❡♥t❡ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❞✐♥â♠✐❝❛ ❞❡ r❡❞❡s ❜✐♦❧ó❣✐❝❛s ❬❋❡rr❡❧❧ ✷✵✵✾❪✳ ❊♠ ✉♠ s❡♥s♦ ❧✐♠✐t❛❞♦✱ ❛ ❜✐♦❧♦❣✐❛ ❞❡ s✐st❡♠❛s ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ✉♠❛ ❛♥tít❡s❡ ❛♦ r❡❞✉❝✐♦♥✐s♠♦✱ ♣♦✐s✱ ❛♣❡s❛r ❞❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ✐❞❡♥t✐✜❝❛r ❡ ❝♦♥❤❡❝❡r ❝❛❞❛ ♣❛rt❡ ❞❡ ✉♠ s✐st❡♠❛ ❜✐♦❧ó❣✐❝♦✱ ❡ss❛s ✐♥❢♦r♠❛çõ❡s ♥ã♦ sã♦ s✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❝♦♠✲ ♣❧❡①✐❞❛❞❡ ✐♥trí♥s❡❝❛ ❞❡ ✉♠ s✐st❡♠❛✳ P♦rt❛♥t♦✱ ❞❡✈❡✲s❡ ❝♦♥s✐❞❡r❛r ♥ã♦ só ❛ ❡str✉t✉r❛ ❞❡ ✉♠ s✐st❡♠❛✱ ♠❛s t❛♠❜é♠ ❛ ❞✐♥â♠✐❝❛ ❞❡st❡✱ ♦✉ s❡❥❛✱ ❛s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ ❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡st❡s s♦❜ ❞❡t❡r♠✐♥❛❞❛s ❝♦♥❞✐çõ❡s ❬❋❡rr❡❧❧ ✷✵✵✾❪ ❬❑✐t❛♥♦ ✷✵✵✷❪✳

✶✷

r❡❣✉❧❛çã♦ ❞♦ str❡ss ❝❡❧✉❧❛r✳ ❖s 30❣r✉♣♦s ♦❜t✐❞♦s ♥❡ss❛ ❛♥á❧✐s❡ sã♦ ✉t✐❧✐③❛❞♦s ♥♦ ♣r❡s❡♥t❡

tr❛❜❛❧❤♦✳

❊①✐st❡♠✱ ❛✐♥❞❛✱ ♦✉tr❛s ❢♦r♠❛s ❞❡ ❝♦♥str✉✐r r❡❞❡s ❜✐♦❧ó❣✐❝❛s✱ ♥❡st❡ ❝❛s♦✱ ❛tr❛✈és ❞❡ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❞✐❢❡r❡♥t❡s ❣❡♥❡s q✉❡ ❝♦♠♣♦❡♠ ✉♠ ♦r❣❛♥✐s♠♦✳ ❊st❛s ✐♥t❡r❛çõ❡s ✐♥❝❧✉❡♠ ✐♥t❡r❛çõ❡s r❡❣✉❧❛tór✐❛s✱ ♠❡t❛❜ó❧✐❝❛s ❡ ❢ís✐❝❛s✱ ❡♥tr❡ ♦✉tr❛s ♣♦ss✐✲ ❜✐❧✐❞❛❞❡s✳ ❈♦♠❜✐♥❛♥❞♦ ❡ss❛s ✐♥❢♦r♠❛çõ❡s ❝♦♠ ❛❧❣♦r✐t♠♦s ❞❡ ❞❡❝✐sã♦ q✉❡ t❡♠ ❝♦♠♦ ❜❛s❡ ❛ ❡♥tr♦♣✐❛✱ ❤✐♣ót❡s❡s s♦❜r❡ ❛ ❡str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ r❡❞❡s sã♦ s✉❣❡r✐❞❛s ❝♦♠ ♠❡♥♦r ❛♠❜✐❣✉✐❞❛❞❡ ❬❑✐t❛♥♦ ✷✵✵✷❪ ❬❋❡rr❡❧❧ ✷✵✵✾❪✳ ❊s♣❡r❛✲s❡ q✉❡ ❛s ❡♥tr❡ r❡❞❡s ❝♦♥str✉í❞❛s ✉t✐✲ ❧✐③❛♥❞♦ ❡st❛s ❛❜♦r❞❛❣❡♥s ❞❡✈❛♠ ♣♦ss✉✐r s✐♠✐❧✐❛r✐❞❛❞❡s ♦✉ ❛té✐❞❡❛❧♠❡♥t❡ ❡q✉✐✈❛❧ê♥❝✐❛s✳ ❊st❛ q✉❡stã♦✱ ❝♦♥t✉❞♦✱ ❛✐♥❞❛ ♥ã♦ ❢♦✐ s✉✜❝✐❡♥t❡♠❡♥t❡ ❡①♣❧♦r❛❞❛✳ ◆❡st❡ tr❛❜❛❧❤♦ ✐r❡♠♦s ❝♦♠♣❛r❛r ❛s r❡❞❡s ❝♦♥str✉í❞❛s ♣♦r ❡st❛s ❛❜♦r❞❛❣❡♥s✳

✶✸

✷ ❖❜❥❡t✐✈♦s

✷✳✶ ❖❜❥❡t✐✈♦s ●❡r❛✐s

✶✳ ❉❡t❡r♠✐♥❛r ❛ ❡str✉t✉r❛ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ♣❛r❛ ❛s r❡❞❡s ❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s ❞❛ ❜❛❝✲ tér✐❛ ❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❡ ❞❛ ❧❡✈❡❞✉r❛ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡✳

✷✳✷ ❖❜❥❡t✐✈♦s ❊s♣❡❝í✜❝♦s

✶✳ ❈♦♠♣❛r❛r ❛❧❣♦r✐t♠♦s ❞✐st✐♥t♦s ♣❛r❛ ❛ ❞✐✈✐sã♦ ❞❡ r❡❞❡s ❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s ❡♠ ❝♦♠✉♥✐❞❛❞❡s❀

✶✹

✸ ▼❡t♦❞♦❧♦❣✐❛

✸✳✶ ❊t❛♣❛s

❆s ❡t❛♣❛s r❡❛❧✐③❛❞❛s ♣♦❞❡♠ s❡r ❛❝♦♠♣❛♥❤❛❞❛s ❛tr❛✈és ❞❛ ✜❣✉r❛ ✸✳

❋✐❣✉r❛ ✸✿ ❊sq✉❡♠❛ ❞❛ ♠❡t♦❞♦❧♦❣✐❛ ✉t✐❧✐③❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦✳

✸✳✶✳✶ ❖❜t❡♥çã♦ ❞❡ ❞❛❞♦s

✶✺

❊♠ s❡❣✉✐❞❛✱ ♦s ❞❛❞♦s ❢♦❣❛♠ ❛♥❛❧✐s❛❞♦s✱ ✐♥t❡❣❣❛❞♦s ❡✱ ♦s ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✱ ❝❧❛ss✐✜❝❛❞♦s ❞❡ ❛❝♦❣❞♦ ❝♦♠ s✉❛s ❢✉♥çõ❡s ❡ ❧♦❝❛❧✐③❛çõ❡s✳

✸✳✶✳✷ ❉✐✈✐sã♦ ❡♠ ❝♦♠✉♥✐❞❛❞❡s

❆♠❜❛s ❛s ❣❡❞❡s ❢♦❣❛♠ ❞✐✈✐❞✐❞❛s ❡♠ ❝♦♠✉♥✐❞❛❞❡s ✉t✐❧✐③❛♥❞♦ t❣ês ♠ét♦❞♦s ❞✐st✐♥t♦s✱ s❡♥❞♦ ❞✉❛s ❢✉♥çõ❡s ❝♦♥t✐❞❛s ♥♦ s♦❢t✇❛r❡ ▼❛t❤❡♠❛t✐❝❛✱ ❞❡♥♦♠✐♥❛❞❛s ❈♦♠♠✉♥✐t②❙tr✉❝✲ t✉r❡❆ss✐❣♥♠❡♥t✱ q✉❡ ❛♣❧✐❝❛ ♦ ❛❧❣♦❣✐t♠♦ ♣❣♦♣♦st♦ ♣♦❣ ❈❧❛✉s❡t✱ ❡ ▼✐♥❈✉t✱ ❡ ♦ ✉t✐❧✐③❛❞♦ ♣❡❧♦ s♦❢t✇❛r❡ ❈❋✐♥❞❡r✱ ❞❡♥♦♠✐♥❛❞♦ ❈❧✐q✉❡ P❡r❝♦❧❛t✐♦♥ ▼❡t❤♦❞✳ ❖s ❛❧❣♦❣✐t♠♦s ✉t✐❧✐③❛❞♦s ❡stã♦ ❡s♣❡❝✐✜❝❛❞♦s ♥♦ ✐t❡♠ ✸✳✸ ❡ ♦s ❝ó❞✐❣♦s ♥♦ ✐t❡♠ ✸✳✹✳

✸✳✶✳✸ ❈á❧❝✉❧♦ ❞❛ ❡♥tr♦♣✐❛

◆❡st❡ t❣❛❜❛❧❤♦✱ ❛ ❡♥t❣♦♣✐❛ ❣❡♣❣❡s❡♥t❛ ❛ ♠❡❞✐❞❛ ❞❡ ✐♥❝❡❣t❡③❛ ❞❡ ✉♠❛ ✈❛❣✐á✈❡❧ ❛❧❡❛tó❣✐❛ ❡ é ✉t✐❧✐③❛❞❛ ❝♦♠♦ ❢❡❣❣❛♠❡♥t❛ ❞❡ ❛♥á❧✐s❡ ♣❛❣❛ ♦s ❣❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳ ❙❡❥❛ ♥ ✉♠❛ ✈❛❣✐á✈❡❧ ❛❧❡❛tó❣✐❛ ❞✐s❝❣❡t❛ ❡ ❛ ❢✉♥çã♦ ❞❡ ♣❣♦❜❛❜✐❧✐❞❛❞❡ 𝑝(𝑛)✳ ❙✉♣♦♥❞♦ ① ✈❛❣✐á✈❡✐s {𝑛₁, 𝑛₂, 𝑛𝑥}✱ ❛s

q✉❛✐s ❛♣❛❣❡❝❡♠ ❡♠ ✉♠ s✐st❡♠❛ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ♦✉ s❡❥❛✱ s❡♠ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦s ✈❛❧♦❣❡s ❛♥t❡❣✐♦❣❡s ❡ ❝♦♠ ♣❣♦❜❛❜✐❧✐❞❛❞❡s {𝑝1, 𝑝2, 𝑝𝑛} ❬❈♦✈❡❣ ❡ ❚❤♦♠❛s ✷✵✵✻❪ ❬❙❤❛♥♥♦♥ ✶✾✹✽❪✳

❙❡♥❞♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥❢♦❣♠❛çã♦ ❞❡ ❝❛❞❛ ✈❛❣✐á✈❡❧ ❞♦ s✐st❡♠❛ ❞❛❞❛ ♣♦❣ −log₂(𝑝𝚤)❡♠

✉♠❛ ú♥✐❝❛ ♦❜s❡❣✈❛çã♦✱ ♣❛❣❛ ◆ ♦❜s❡❣✈❛çõ❡s✱ ❝❛❞❛ ✈❛❣✐á✈❡❧ ❛♣❛❣❡❝❡ ♥♦ s✐st❡♠❛ ◆ ✈❡③❡s ❛ s✉❛ ♣❣♦❜❛❜✐❧✐❞❛❞❡✱ ✐st♦ é✱ 𝑁 ×𝑝_𝚤 ❡✱ ❛ss✐♠✱ ♣❛❣❛ ◆ ♦❜s❡❣✈❛çõ❡s ✐♥❞❡♣❡♥❞❡♥t❡s✱ t❡♠✲s❡ ❛

✐♥❢♦❣♠❛çã♦ t♦t❛❧ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ✶ ❡ ❛ ✐♥❢♦❣♠❛çã♦ ♠é❞✐❛ ♣♦❣ ✈❛❣✐á✈❡❧ ♣❡❧❛ ❡q✉❛çã♦ ✷ ❬❙❤❛♥♥♦♥ ✶✾✹✽❪✳

𝐼 =−

𝑥 ∑

𝑖=1

𝑁 𝑝𝚤log(𝑝𝚤) ✭✶✮

𝐼 𝑁 =−

𝐼 𝑁

𝑥 ∑

𝑖=1

𝑁 𝑝𝚤log(𝑝𝚤) =− 𝑥 ∑

𝑖=1

𝑝𝚤log(𝑝𝚤) ✭✷✮

❉❡st❡ ♠♦❞♦✱ ❛ ✐♥❢♦❣♠❛çã♦ é ❡st❣✐t❛♠❡♥t❡ ❞❡s❝❣✐t❛ ❡♠ t❡❣♠♦s ❞❡ ♣❣♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡✈❡♥t♦s✳ ❙✉♣♦♥❞♦ ✉♠❛ ❞✐st❣✐❜✉✐çã♦ ❞❡ ♣❣♦❜❛❜✐❧✐❞❛❞❡s 𝑃 ={𝑝₁, 𝑝₂, 𝑝𝑛}✱ ❛ ❡♥t❣♦♣✐❛ ❞❡st❛✱

♣❛❣❛ ✈❛❧♦❣❡s ❞✐s❝❣❡t♦s✱ é ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ✸✳ ◗✉❛♥❞♦ ♦ log é ❞❡ ❜❛s❡ ✷✱ ❛ ❡♥t❣♦♣✐❛ é

❡①♣❣❡ss❛ ❡♠ ❜✐ts✳

𝐻(𝑃) =−

𝑥 ∑

𝑖=1

𝑝_𝚤log₂(𝑝_𝚤) ✭✸✮

✶✻

❞✐str✐❜✉✐çã♦ ❤♦♠♦❣ê♥❡❛ ❞♦s t✐♣♦s ❞❡ ✐♥t❡r❛çõ❡s✱ ♦✉ s❡❥❛✱ ♣r❡s❡♥ç❛ ❞♦s três t✐♣♦s ❞❡ ✐♥t❡✲ r❛çõ❡s ❡①✐st❡♥t❡s✱ s❡♠ ♣r❡❞♦♠✐♥â♥❝✐❛ ❞❡ ♥❡♥❤✉♠ ❞♦s três t✐♣♦s ❬❇✉tt❡ ❡ ❑♦❤❛♥❡ ✷✵✵✵❪✳

❆♣ós ♦ ❝á❧❝✉❧♦ ❞❛ ❡♥tr♦♣✐❛✱ ❛s ❝♦♠✉♥✐❞❛❞❡s ❢♦r❛♠ ❛❧❡❛t♦r✐③❛❞❛s✳ ■st♦ é r❡❛❧✐③❛❞♦ ❛❧t❡r❛♥❞♦ ♦s t✐♣♦s ❞❡ ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❣❡♥❡s ❞❡ ♠❛♥❡✐r❛ ❛❧❡❛tór✐❛✳ ❊♠ s❡❣✉✐❞❛✱ ❛s ❡♥tr♦♣✐❛s ❞❛s ❝♦♠✉♥✐❞❛❞❡s ❞❛s r❡❞❡s ❛❧❡❛tór✐❛s ❢♦r❛♠ ♠❡❞✐❞❛s ❡ ❢♦✐ r❡❛❧✐③❛❞❛ ✉♠❛ ❝♦♠✲ ♣❛r❛çã♦ ❡♥tr❡ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣❛r❛ ❛s r❡❞❡s r❡❛✐s ❡ ❛s ❛❧❡❛t♦r✐③❛❞❛s✳

✸✳✶✳✹ ❘❡♣r❡s❡♥t❛çã♦

❖s ❣❡♥❡s ❞❡ ❛♠❜❛s ❛s r❡❞❡s ❢♦r❛♠ r❡♦r❞❡♥❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐✈✐sã♦ ❡♠ ❝♦♠✉✲ ♥✐❞❛❞❡s ✉t✐❧✐③❛♥❞♦ ❛ ❢✉♥çã♦✱ ❞♦ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❛♣❧✐❝❛ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❈❧❛✉s❡t✱ ❈♦♠✲ ♠✉♥✐t②❙tr✉❝t✉r❡❆ss✐❣♥♠❡♥t✱ ❡ ❛s r❡❞❡s r❡♣r❡s❡♥t❛❞❛s ❛tr❛✈és ❞❛s ♠❛tr✐③❡s ❞❡ ❛❞❥❛❝ê♥❝✐❛s✳ P♦st❡r✐♦r♠❡♥t❡✱ ❛s r❡❞❡s ♦❜t✐❞❛s ❛tr❛✈és ❞❛ ❞✐✈✐sã♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ✉t✐❧✐③❛♥❞♦ ♦s ♦✉tr♦s ❞♦✐s ♠ét♦❞♦s ❢♦r❛♠ r❡♣r❡s❡♥t❛❞❛s ❛tr❛✈és ❞❛ ✉t✐❧✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❈✐r❝✉❧❛r ❊♠✲ ❜❡❞❞✐♥❣✱ ♣r❡s❡♥t❡ ♥♦ ▼❛t❤❡♠❛t✐❝❛✱ ❡ ❢♦r❛♠ ❝❛❧❝✉❧❛❞♦s ♦s t❛♠❛♥❤♦s ❡ ❛s ❡♥tr♦♣✐❛s✱ ❝♦♠♦ ❡s♣❡❝✐✜❝❛❞♦ ♥♦ ✐t❡♠ ✸✳✶✳✸✱ ♣❛r❛ ❡st❛s ❝♦♠✉♥✐❞❛❞❡s✳ ❖s r❡s✉❧t❛❞♦s ❞❡st❡s ❝á❧❝✉❧♦s ❡s✲ tã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦s ❣rá✜❝♦s ❡ ❤✐st♦❣r❛♠❛s ❞❛ s❡çã♦ ✹✳ P❛r❛ ❛ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✲ ✈✐s✐❛❡✱ ❢♦r❛♠ ❝❛❧❝✉❧❛❞♦s ♦s t❛♠❛♥❤♦s ❡ ❛s ❡♥tr♦♣✐❛s ♣❛r❛ ♦s ❝❧✉st❡rs ✐❞❡♥t✐✜❝❛❞♦s ♣♦r ❬●❡ ❡t ❛❧✳ ✷✵✵✶❪✱ ❜❛s❡❛❞♦ ♥❛ ❝♦❡①♣r❡ssã♦ ❞♦s ❣❡♥❡s ❡✱ ❡st❡s r❡s✉❧t❛❞♦s t❛♠❜é♠ ❡stã♦ ♣r❡✲ s❡♥t❡s ♥❛ s❡çã♦ ✹ ❡♠ ❢♦r♠❛ ❞❡ ❣rá✜❝♦ ❡ ❤✐st♦❣r❛♠❛✳

✸✳✷ ❇❛♥❝♦ ❞❡ ❞❛❞♦s

❖s ❞❛❞♦s ✉t✐❧✐③❛❞♦s s♦❜r❡ ❛s ✐♥t❡r❛çõ❡s ❢ís✐❝❛s ❡♥tr❡ ♣r♦t❡í♥❛s✱ ✐♥t❡r❛çõ❡s ♠❡t❛❜ó❧✐❝❛s ❡ ✐♥t❡r❛çõ❡s ❞❡ r❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧✱ r❡❢❡r❡♥t❡s à ❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❡ à ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡✱ ❢♦r❛♠ ♦❜t✐❞♦s ❞❡ ❜❛♥❝♦s ❞❡ ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s ♣ú❜❧✐❝♦s ♥❛ ✇❡❜✳ ❆ ❚❛❜❡❧❛ ✶ ♠♦str❛ ♦s ❜❛♥❝♦s ❞❡ ❞❛❞♦s ✉t✐❧✐③❛❞♦s ♥❛ ❛q✉✐s✐çã♦ ❞❡ ❞❛❞♦s✳

❚❛❜❡❧❛ ✶✿ ❇❛♥❝♦s ❞❡ ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❛q✉✐s✐çã♦ ❞♦s ❞❛❞♦s✳

❇❛♥❝♦ ❞❡ ❞❛❞♦s ❖r❣❛♥✐s♠♦ ❚✐♣♦ ❞❡ ✐♥t❡r❛çã♦ ❘❡❢❡rê♥❝✐❛

❇■●● ❙✳ ❝❡r❡✈✐s✐❛❡ ▼❡t❛❜ó❧✐❝❛ ❤tt♣✿✴✴❜✐❣❣✳✉❝s❞✳❡❞✉✴❤♦♠❡✳♣❧ ❨❊❆❙❚❘❆❈❚ ❙✳ ❝❡r❡✈✐s✐❛❡ ❘❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧ ❤tt♣✿✴✴✇✇✇✳②❡❛str❛❝t✳❝♦♠

❇■❖●❘■❉ ❙✳ ❝❡r❡✈✐s✐❛❡ Pr♦t❡í♥❛✲♣r♦t❡í♥❛ ❤tt♣✿✴✴✇✇✇✳t❤❡❜✐♦❣r✐❞✳♦r❣ ❑❊●● ❊✳ ❝♦❧✐ ▼❡t❛❜ó❧✐❝❛ ❤tt♣✿✴✴✇✇✇✳❣❡♥♦♠❡✳❥♣✴❦❡❣❣ ❘❡❣✉❧♦♥❉❇ ❊✳ ❝♦❧✐ ❘❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧ ❤tt♣✿✴✴✇✇✇✳r❡❣✉❧♦♥❞❜✳❝❝❣✳✉♥❛♠✳♠① ❇❛❝t❡r✐♦♠❡✳♦r❣ ❊✳ ❝♦❧✐ Pr♦t❡í♥❛✲♣r♦t❡í♥❛ ❤tt♣✿✴✴✇✇✇✳❜❛❝t❡r✐♦♠❡✳♦r❣

✶✼

❛rq✉✐✈♦✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ♦ ❡st✉❞♦ ❡ ❛♥á❧✐s❡ ❞❡st❡s ❡♠ ✉♠ ♣r♦❝❡ss♦ ♠❛✐s ❞❡t❛❧❤❛❞♦✳ P♦s✲ t❡r✐♦♠❡♥t❡✱ ❢♦✐ r❡❛❧✐③❛❞❛ ❛ ❧✐♠♣❡③❛ ❞♦s ❞❛❞♦s✱ ✐st♦ é✱ ❝♦rr❡çã♦ ❞❡ ❞❛❞♦s ✐♥❝♦rr❡t♦s ♦✉ ✐♠♣r❡❝✐s♦s ❡ ❡st❛ ♦♣❡r❛çã♦ ❢♦✐ r❡❛❧✐③❛❞❛ ♣♦r ✉♠ ❡s♣❡❝✐❛❧✐st❛ ❞❛ ár❡❛ ❜✐♦❧ó❣✐❝❛✳

✸✳✸ ❆❧❣♦r✐t♠♦s

✸✳✸✳✶ ❈❧❛✉s❡t

❖ ❛❧❣♦r✐t♠♦ ♣✉❜❧✐❝❛❞♦ ♣♦r ❈❧❛✉s❡t ❬❈❧❛✉s❡t✱ ◆❡✇♠❛♥ ❡ ▼♦♦r❡ ✷✵✵✹❪ é ❜❛s❡❛❞♦ ♥❛ ✈❛r✐❛çã♦ ❞♦ ✈❛❧♦r ❞❛ ♠♦❞✉❧❛r✐❞❛❞❡ ❡♥❝♦♥tr❛❞♦ ❡♥tr❡ ♦s ♥♦❞♦s ♥❛ t❡♥t❛t✐✈❛ ❞❡ ♠❛①✐♠✐③❛r ❡ss❛✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❛❞✐❝✐♦♥❛✲s❡ ♦ ♥♦❞♦ ❝❡♥tr❛❧ ✭❤✉❜✮ à ❝♦♠✉♥✐❞❛❞❡ ❈ ❡ s❡✉s ✈✐③✐♥❤♦s à ❯ ❀ ❛ ❝❛❞❛ ♣❛ss♦✱ ♦ ❛❧❣♦r✐t♠♦ ❛❞✐❝✐♦♥❛ à ❈✱ ♦s ♥♦❞♦s q✉❡ ♠❛①✐♠✐③❛♠ ❛ ✈❛r✐❛çã♦ ❞♦ ✈❛❧♦r ❞❡ ♠♦❞✉❧❛r✐❞❛❞❡ ♦✉ q✉❡ ♠❡♥♦s r❡❞✉③❛♠ ❡ss❛✱ ✐st♦ é✱ ♣❛r❛ ❝❛❞❛ ✈✐③✐♥❤♦ ❛❞✐❝✐♦♥❛❞♦ ❛ ❯✱ ❝❛❧❝✉❧❛✲s❡ ♦ ✈❛❧♦r ❞❛ ✈❛r✐❛çã♦ ❞❛ ♠♦❞✉❧❛r✐❞❛❞❡ ❛tr❛✈és ❞❛ ❊q✉❛çã♦ ✹ ❡ ♦s ♥♦❞♦s q✉❡ ♠❛①✐♠✐③❛♠ ❡st❡ sã♦ ❛❞✐❝✐♦♥❛❞♦s à ❈✳ ❊st❡ ♣r♦❝❡ss♦ ❝♦♥t✐♥✉❛ ❛té q✉❡ ❛t✐♥❥❛✲s❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♥ú♠❡r♦ ❞❡ ♥♦❞♦s ❦ ♦✉ ❛té q✉❡ ✉♠❛ ❝♦♠✉♥✐❞❛❞❡ s❡❥❛ t♦t❛❧♠❡♥t❡ ✐❞❡♥t✐✜❝❛❞❛✳

Δ𝑄𝑖𝑗 =

{ ₁

2𝑚 − 𝑘𝑖𝑘𝑗

(2𝑚)2 s❡ ✐✱ ❥ ❡stã♦ ✐♥t❡r❛❣✐♥❞♦

0 s❡ ✐✱ ❥ ♥ã♦ ❡stã♦ ✐♥t❡r❛❣✐♥❞♦ ✭✹✮

❖♥❞❡✱ 𝑚 é ♦ ♥ú♠❡r♦ ❞❡ ❧✐❣❛çõ❡s ❞❡ ❝❛❞❛ ❝♦♠✉♥✐❞❛❞❡✱ ❞❛❞♦ ♣♦r 𝑚= 1 2

∑

𝑖𝑗𝐴𝑖𝑗✱ s❡♥❞♦

q✉❡𝐴𝑖𝑗 tr❛t❛✲s❡ ❞❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛s ❞❛ r❡❞❡ ❡ 𝑘𝑖 ❡𝑘𝑗 sã♦ ♦s ❣r❛✉s✱ ♦✉ s❡❥❛✱ ♦ ♥ú♠❡r♦

❞❡ ❧✐❣❛çõ❡s✱ ❞♦s ♥♦❞♦s 𝑖 ❡𝑗✳

✸✳✸✳✷ ❈❧✐q✉❡ P❡r❝♦❧❛t✐♦♥

❊st❡ ❛❧❣♦r✐t♠♦ ♣♦ss✉✐ ❞✉❛s ❡t❛♣❛s ❡ ♣❛r❛ ♦t✐♠✐③❛r ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡st❛s ❞❡✈❡✲s❡✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❞❡✜♥✐r ❦✲❝❧✐q✉❡ ❡ ❝♦♠✉♥✐❞❛❞❡ ❦✲❝❧✐q✉❡✳ ❑✲❝❧✐q✉❡ é ✉♠❛ r❡❞❡ ❞❡ ❦ ♥♦✲ ❞♦s✱ ♥❛ q✉❛❧ t♦❞♦s ❡stã♦ ❧✐❣❛❞♦s ❡♥tr❡ s✐❀ ❝♦♠✉♥✐❞❛❞❡ ❦✲❝❧✐q✉❡ ♦✉ ❝♦♠✉♥✐❞❛❞❡✲❦ tr❛t❛✲s❡ ❞❡ ✉♠❛ r❡❞❡ ❞❡ ♥♦❞♦s q✉❡ é ♦❜t✐❞❛ ❛tr❛✈és ❞❛ s♦❜r❡♣♦s✐çã♦ ❞❡ ❦✲❝❧✐q✉❡s ❞❡ ♠♦❞♦ q✉❡ ♦s ❦✲❝❧✐q✉❡s ❝♦♠♣❛rt✐❧❤❡♠ ❦✲✶ ♥♦❞♦s✳ ❖ ❛❧❣♦r✐t♠♦ ❢✉♥❝✐♦♥❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ✐♥✐❝✐❛❧✲ ♠❡♥t❡✱ t♦❞♦s ♦s ❧✐♥❦s ❧ sã♦ r❡♠♦✈✐❞♦s ❞❛ r❡❞❡ ❡ ❡♥tã♦ ✐♥s❡r✐❞♦s ♥♦✈❛♠❡♥t❡ ✉♠ ♣♦r ✉♠ ❡ ❡♠ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ♣❛r❛ r❡❞❡s ❡♠ q✉❡ ❛s ✐♥t❡r❛çõ❡s ♥ã♦ ♣♦ss✉❛♠ ♣❡s♦s ❞✐❢❡r❡♥t❡s✳ ❉❡st❡ ♠♦❞♦✱ ❛ ♣r✐♠❡✐r❛ ❡t❛♣❛ ❞♦ ❛❧❣♦r✐t♠♦ ❝♦♥s✐st❡ ♥❛ ❞❡t❡❝çã♦ ❞♦s ❦✲❝❧✐q✉❡s ❝♦♥❢♦r♠❡ sã♦ ❛❞✐✲ ❝✐♦♥❛❞♦s ♦s ❧✐♥❦s❀ ♥❛ s❡❣✉♥❞❛ ❡t❛♣❛✱ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ❝♦♠✉♥✐❞❛❞❡s é ❞❛❞❛ ❛tr❛✈és ❞❛ r❡❝♦♥str✉çã♦ ❞❛ r❡❞❡ ❝♦♠ ❛ s♦❜r❡♣♦s✐çã♦ ❞♦s ❦✲❝❧✐q✉❡s ✐❞❡♥t✐✜❝❛❞♦s ♥❛ ♣r✐♠❡✐r❛ ❡t❛♣❛✳ ❆ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ❝♦♠✉♥✐❞❛❞❡s é ❢❡✐t❛ ❛tr❛✈és ❞❛ ❞❡t❡❝çã♦ ❧♦❝❛❧ ❞❡ ♣♦ssí✈❡✐s s♦❜r❡♣♦s✐çõ❡s ❝♦♥❢♦r♠❡ ♥♦✈♦s ❦✲❝❧✐q✉❡s sã♦ ❛❞✐❝✐♦♥❛❞♦s à r❡❞❡ q✉❡ ❡stá s❡♥❞♦ r❡❝♦♥str✉í❞❛ ❡ ❛ ❡str✉t✉r❛ ❞❡ ❝♦♠✉♥✐❞❛❞❡s é ❛t✉❛❧✐③❛❞❛ ❝♦♥❢♦r♠❡ sã♦ ✐❞❡♥t✐✜❝❛❞❛s ♥♦✈❛s s♦❜r❡♣♦s✐çõ❡s✳

✶✽

❡①❡♠♣❧♦ ✉♠❛ r❡❞❡ s✐♠♣❧❡s ❝♦♠ 𝑘 ❂ ✸✳

✶✳ r❡❞❡ ♦r✐❣✐♥❛❧ ❝♦♥s✐st❡ ❞❡ três ✸✲❝❧✐q✉❡s r❡♣r❡s❡♥t❛❞♦s ❝♦♠♦ 𝐴✱ 𝐵 ❡𝐶❀

✷✳ r❡♣r❡s❡♥t❛çã♦ ❞❛ r❡❞❡ ❞✐✈✐❞✐❞❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❡str✉t✉r❛s ❞♦s ❝❧✐q✉❡s✳ P♦❞❡✲s❡ ♥♦t❛r q✉❡ ♦s ✸✲❝❧✐q✉❡s𝐵 ❡𝐶✱ q✉❡ ❢♦r♠❛♠ ✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ✸✲❝❧✐q✉❡✱ sã♦ ❝♦♥❡❝t❛❞♦s

♣❡❧♦ ✷✲❝❧✐q✉❡ ❢✳ ❖ ❝❧✐q✉❡ 𝐴 ❣❡r❛ ♦✉tr❛ ❝♦♠✉♥✐❞❛❞❡ ✸✲❝❧✐q✉❡❀

✸✳ ✸✲❝❧✐q✉❡s ✐❞❡♥t✐✜❝❛❞♦s ♥❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ ❛❧❣♦r✐t♠♦ ❝♦♥❢♦r♠❡ ♦s ❧✐♥❦s ❡r❛♠ s❡q✉❡♥✲ ❝✐❛❧♠❡♥t❡ ✐♥s❡r✐❞♦s ♥❛ r❡❞❡✳ ❈❛❞❛ ♥♦✈♦ 𝑘✲❝❧✐q✉❡ é ❞❡♥♦t❛❞♦ ♣♦r ♥♦❞♦s ✈❡r♠❡❧❤♦s

❡♥q✉❛♥t♦ ♦s ♥♦❞♦s ❛ss♦❝✐❛❞♦s ❛ ❡st❡s sã♦ r❡♣r❡s❡♥t❛❞♦s ❝✐♥③❛s❀

✹✳ ❆t✉❛❧✐③❛çõ❡s ❞❛ r❡❞❡ (𝑘− 1)− 𝑐𝑙𝑖𝑞𝑢𝑒 ❝♦♠♦ r❡s✉❧t❛❞♦ ❞❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞♦ ❛❧❣♦✲

r✐t♠♦✳ ❈♦♠✉♥✐❞❛❞❡s 𝑘 −𝑐𝑙𝑖𝑞𝑢𝑒 ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡❝t❛❞♦s ✭ár❡❛s

❝✐r❝✉❧❛❞❛s✮ ❬P❛❧❧❛ ❡t ❛❧✳ ✷✵✵✺❪✳

❋✐❣✉r❛ ✹✿ ■❧✉str❛çã♦ ❞❛s ❡t❛♣❛s ❞♦ ❛❧❣♦r✐t♠♦ ❈❧✐q✉❡ P❡r❝♦❧❛t✐♦♥✱ r❡t✐r❛❞❛ ❞❡ ❬❑✉♠♣✉❧❛ ❡t ❛❧✳ ✷✵✵✽❪✳

✸✳✸✳✸ ▼✐♥❈✉t

❆ ❢✉♥çã♦ ▼✐♥❈✉t✱ ❝♦♥t✐❞❛ ♥♦ ▼❛t❤❡♠❛t✐❝❛✱ é ✉♠ ♠ét♦❞♦ ❞❡ ❞✐✈✐sã♦ ❡♠ ❝♦♠✉♥✐❞❛❞❡s ♣♦r ♣❛rt✐çã♦ ❞❡ ❣r❛❢♦s✱ ♦✉ s❡❥❛✱ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s é ❞❡✜♥✐❞♦ ♣r❡✈✐❛♠❡♥t❡✳ ❆ ❢✉♥çã♦ ❝♦♥s✐❞❡r❛ ❛ r❡❞❡ ❝♦♠♦ s❡♥❞♦ ✐♥❞✐r❡t❛ ❡ ❛ ❞✐✈✐❞❡ ❡♠ ❦ ❣r✉♣♦s ❝♦♠✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♥♦❞♦s✱ ♠✐♥✐♠✐③❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❣r✉♣♦s✳

✶✾

❚❛❜❡❧❛ ✷✿ ❚❛❜❡❧❛ ❝♦♠♣❛r❛t✐✈❛ ♣❛r❛ ♦s ♠ét♦❞♦s ❞❡ ❞✐✈✐sã♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ✉t✐❧✐③❛❞♦s✳

▼ét♦❞♦ ❚✐♣♦ ❞❡ ❞✐✈✐sã♦ ■♥t❡rs❡❝çã♦ ❡♥tr❡ ❝♦♠✉♥✐❞❛❞❡s ❯s♦ ❞❡ ❈P❯ ✭♥ú♠❡r♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s✮

❈❧❛✉s❡t ❉❡t❡❝çã♦ ❞❡ ❡str✉t✉r❛ ❱❛③✐❛ ✷✵ ♠✐♥✉t♦s ❈❧✐q✉❡ P❡r❝♦❧❛t✐♦♥ ❉❡t❡❝çã♦ ❞❡ ❡str✉t✉r❛ ◆ã♦ ✈❛③✐❛ ✶✷ ❤♦r❛s

▼✐♥❈✉t P❛rt✐çã♦ ❞❡ ❣r❛❢♦s ❱❛③✐❛ ✷ ♠✐♥✉t♦s

✸✳✹ ■♠♣❧❡♠❡♥t❛çã♦

P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡s❝r✐t❛✱ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ♦ s♦❢t✇❛r❡ ❈❋✐♥❞❡r ❡ ♦ ▼❛t❤❡♠❛t✐❝❛ ❡ ♣❛❝♦t❡s ❞❡ ❢✉♥çõ❡s ❝♦♠♦ ❈♦♠❜✐♥❛t♦r✐❝❛ ❡ ●r❛♣❤❯t✐❧✐t✐❡s✳ ❖ s ♥♦t❡❜♦♦❦s ✭❝ó❞✐❣♦s ✐♠♣❧❡♠❡♥t❛❞♦s ♥♦ ▼❛t❤❡♠❛t✐❝❛✮ ❡stã♦ ❞✐s♣♦♥í✈❡✐s ♥♦ s✐t❡✶ _{❡ ❞❡s❝r✐t♦s ❛ s❡❣✉✐r✳}

❝♦♠✉♥✐❞❛❞❡s❴❝✐r❝✉❧♦❴❝❢✐♥❞❡r❴❡❝♦❧✐✳♥❜✳♥❜ ∗✷ ❋❛③ ❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❛s ❝♦♠✉✲

♥✐❞❛❞❡s ✐❞❡♥t✐✜❝❛❞❛s ♥❛ r❡❞❡ ❞❛ ❊✳ ❝♦❧✐✱ ♣❡❧♦ ❈❋✐♥❞❡r✱ ✉t✐❧✐③❛♥❞♦ ❈✐r❝✉❧❛r ❊♠❜❡❞✲ ❞✐♥❣✳

❝♦♠✉♥✐❞❛❞❡s❴❡❝♦❧✐❴❝❧❛✉s❡t✳♥❜ ∗ ❉✐✈✐❞❡ ❛ r❡❞❡ ❞❛ ❊✳ ❝♦❧✐ ❡♠ ❝♦♠✉♥✐❞❛❞❡s ✉s❛♥❞♦ ♦

❛❧❣♦r✐t♠♦ ❞❡ ❈❧❛✉s❡t ❬❈❧❛✉s❡t✱ ◆❡✇♠❛♥ ❡ ▼♦♦r❡ ✷✵✵✹❪✱ ❢❛③ ❛ r❡♣r❡s❡♥t❛çã♦ ❛tr❛✈és ❞❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ ❡ ❝❛❧❝✉❧❛ ❛ ❡♥tr♦♣✐❛ ❞❛s ❝♦♠✉♥✐❞❛❞❡s✳

❡♥tr♦♣✐❛❴❡❝♦❧✐❴❝❢✐♥❞❡r✳♥❜∗ ❈❛❧❝✉❧❛ ❛ ❡♥tr♦♣✐❛ ❞❛s ❝♦♠✉♥✐❞❛❞❡s ❞❡t❡r♠✐♥❛❞❛s ♣❡❧♦

❈❋✐♥❞❡r ♣❛r❛ ❛ r❡❞❡ ❞❛ ❊✳ ❝♦❧✐✳

❝♦❧♦r❴♥♦❞♦s❴❢✉♥❝❛♦❴❢✐♥❛❧❴s❝❡✳♥❜∗ ●❡r❛ ♦ ❣r❛❢♦ ❞❛ r❡❞❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ❡ ❝♦❧♦r❡ ♦s

♥♦❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❢✉♥çã♦✳

❝♦❧♦r❴♥♦❞♦s❴❧♦❝❛❧✐③❛❝❛♦❴❢✐♥❛❧❴s❝❡✳♥❜∗ ●❡r❛ ♦ ❣r❛❢♦ ❞❛ r❡❞❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ❡ ❝♦❧♦r❡

♦s ♥♦❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❧♦❝❛❧✐③❛çã♦✳

❝♦♠✉♥✐❞❛❞❡s❴❝✐r❝✉❧♦❴❝❢✐♥❞❡r❴s❝❡✳♥❜∗ ❋❛③ ❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❛s ❝♦♠✉♥✐❞❛❞❡s

✐❞❡♥t✐✜❝❛❞❛s ♥❛ r❡❞❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✱ ♣❡❧♦ ❈❋✐♥❞❡r✱ ✉t✐❧✐③❛♥❞♦ ❈✐r❝✉❧❛r ❊♠❜❡❞❞✐♥❣✳

❝♦♠✉♥✐❞❛❞❡s❴❝✐r❝✉❧♦❴s❝❡✳♥❜ ❉✐✈✐❞❡ ❛ r❡❞❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ❡♠ ❝♦♠✉♥✐❞❛❞❡s ✉s❛♥❞♦ ▼✐♥❈✉t ❡ ❢❛③ ❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❛s ❝♦♠✉♥✐❞❛❞❡s ✉t✐❧✐③❛♥❞♦ ❈✐r❝✉❧❛r ❊♠❜❡❞✲ ❞✐♥❣✳

✷✵

❝♦♠✉♥✐❞❛❞❡s❴❙❝❡❴❝❧❛✉s❡t✳♥❜∗ ❉✐✈✐❞❡ ❛s ❝♦♠✉♥✐❞❛❞❡s ✉s❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❈❧❛✉s❡t

❬❈❧❛✉s❡t✱ ◆❡✇♠❛♥ ❡ ▼♦♦r❡ ✷✵✵✹❪✱ ❢❛③ ❛ r❡♣r❡s❡♥t❛çã♦ ❛tr❛✈és ❞❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥✲ ❝✐❛ ❡ ❝❛❧❝✉❧❛ ❛ ❡♥tr♦♣✐❛ ❞❛s ❝♦♠✉♥✐❞❛❞❡s✳

❡♥tr♦♣✐❛❝❧✉st❡r✳♥❜∗ ❈❛❧❝✉❧❛ ❛ ❡♥tr♦♣✐❛ ❞♦s ❝❧✉st❡rs ❛❣r✉♣❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♣❡r✜❧

❞❡ ❡①♣r❡ssã♦ ❞♦s ❣❡♥❡s ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✳

❡♥tr♦♣✐❛▼✐♥❈✉t✸✵✳♥❜∗ ❈❛❧❝✉❧❛ ❛ ❡♥tr♦♣✐❛ ❞❛s ❝♦♠✉♥✐❞❛❞❡s ❞❡t❡r♠✐♥❛❞❛s ♣❡❧♦ ▼✐♥❈✉t

♣❛r❛ ❛ r❡❞❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡

❡♥tr♦♣✐❛❴s❝❡❴❝❢✐♥❞❡r✳♥❜∗ ❈❛❧❝✉❧❛ ❛ ❡♥tr♦♣✐❛ ❞❛s ❝♦♠✉♥✐❞❛❞❡s ❞❡t❡r♠✐♥❛❞❛s ♣❡❧♦ ❈❋✐♥✲

❞❡r ♣❛r❛ ❛ r❡❞❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✳

✷✶

✹ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦

❚rês ♠ét♦❞♦s ❞✐st✐♥t♦s ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ❝♦♠✉♥✐❞❛❞❡s ❞❛r❛ ❛s r❡❞❡s ❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s ❞❛ ❊✳ ❝♦❧✐ ❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ❡ ❝♦♠♦ ❝❛❞❛ ❛❧❣♦r✐t♠♦ ✉t✐❧✐③❛ ✉♠ ♠ét♦❞♦ ❞❡ ✐❞❡♥t✐✜❝❛çã♦✱ ❞✐❢❡r❡♥ç❛s ❞♦❞❡♠ s❡r ♥♦t❛❞❛s ❡ ❡stã♦ ❛❞r❡s❡♥t❛❞❛s ♥❛s ❚❛❜❡❧❛s ✷ ❡ ✸ ❡ ♥❛ ❋✐❣✉r❛ ✶✻✳

❆s ❞✐❢❡r❡♥ç❛s ♠❛✐s ♥♦tá✈❡✐s ❡♥tr❡ ♦s t✐❞♦s ❞❡ ❞✐✈✐sõ❡s ❡♠ ❝♦♠✉♥✐❞❛❞❡s sã♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ♦❜t✐❞❛s ❞♦r ❝❛❞❛ ♠ét♦❞♦ ❡ ♦ t❛♠❛♥❤♦ ❞❡st❛s✳■st♦ ❞♦❞❡ t❡r r❡❧❛çã♦ ❝♦♠ ♦ ❢❛t♦ ❞❡ ❤❛✈❡r ❞❛❞rõ❡s ❞✐st✐♥t♦s ❞❡ ❞✐✈✐sã♦✿ ❞❛rt✐çã♦ ❞❡ ❣r❛❢♦s ❡ ❞✐✈✐sã♦ ❞♦r ❡str✉t✉r❛ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ✭✈❡r ❚❛❜❡❧❛ ✷✮✳

❚❛❜❡❧❛ ✸✿ ◗✉❛♥t✐❛ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ♦❜t✐❞❛s ❞♦r ❝❛❞❛ ♠ét♦❞♦ ❞❡ ❞✐✈✐sã♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ✉t✐❧✐③❛❞♦✳

◆ú♠❡r♦ ❞❡ ❝♦♠✉♥✐❞❛❞❡s

▼ét♦❞♦ ❊✳❝♦❧✐ ❙✳ ❝❡r❡✈✐s✐❛❡ ❈❧❛✉s❡t ✶✷✼ ✷✽ ❈❧✐q✉❡ P❡r❝♦❧❛t✐♦♥ ✷✷✵ ✶✻✾✶

▼✐♥❈✉t ✸✵ ✸✵

❖ ❛❧❣♦r✐t♠♦ ❞r♦❞♦st♦ ❞♦r ❈❧❛✉s❡t ❬❈❧❛✉s❡t✱ ◆❡✇♠❛♥ ❡ ▼♦♦r❡ ✷✵✵✹❪ ❞✐✈✐❞❡ ❛s r❡❞❡s ❞❡❧♦ ♠ét♦❞♦ ❞❡ ❡str✉t✉r❛ ❞❡ ❝♦♠✉♥✐❞❛❞❡s✱ ✐st♦ é✱ ❛ss✉♠❡ q✉❡ ❛ r❡❞❡ ❞♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ♥❛t✉r❛❧♠❡♥t❡ ❞✐✈✐❞✐❞❛ ❡♠ ♠ó❞✉❧♦s ❡ ♦s ✐❞❡♥t✐✜❝❛✳❊st❡ ♠ét♦❞♦ ❞✐✈✐❞✐✉ ❛ r❡❞❡ ❞❛ ❊✳ ❝♦❧✐✱ ❛ q✉❛❧ ❞♦ss✉✐ ✷✸✹✾ ❣❡♥❡s ❡ ✻✶✶✺ ✐♥t❡r❛çõ❡s✱ ❡♠ ✶✷✼ ❝♦♠✉♥✐❞❛❞❡s ❡ ❛ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✱ q✉❡ ❞♦ss✉✐ ✺✻✻✼ ❣❡♥❡s ❡ ✼✷✽✵✷ ✐♥t❡r❛çõ❡s✱ ❡♠ ✷✽ ❝♦♠✉♥✐❞❛❞❡s ✭✈❡r ❚❛❜❡❧❛ ✸✮✳❆s ❝♦♠✉✲ ♥✐❞❛❞❡s ✐❞❡♥t✐✜❝❛❞❛s ❡stã♦ r❡❞r❡s❡♥t❛❞❛s ♥❛s ❋✐❣✉r❛s ✺ ❡ ✶✵ ❞❛r❛ ❊✳ ❝♦❧✐ ❡ ❙✳ ❝❡r❡✈✐s✐❛❡✱ r❡s❞❡❝t✐✈❛♠❡♥t❡✱ ❛tr❛✈és ❞❛s ♠❛tr✐③❡s ❞❡ ❛❞❥❛❝ê♥❝✐❛✳

◆❛ ❋✐❣✉r❛ ✺✱ r❡❢❡r❡♥t❡ à ❊✳ ❝♦❧✐✱ ❞♦❞❡✲s❡ ♦❜s❡r✈❛r ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ❜❡♠ ❞✐st✐♥t❛ ❝♦♠ ❞r❡❞♦♠✐♥â♥❝✐❛ ❞❡ ✐♥t❡r❛çõ❡s ❞♦ t✐❞♦ ❞r♦t❡í♥❛✲❞r♦t❡í♥❛✱ ✉♠❛ ❧❡✈❡ ❛❣❧♦♠❡r❛çã♦ ❝♦♠ ❞r❡❞♦♠✐♥â♥❝✐❛ ❞❡ ✐♥t❡r❛çõ❡s ♠❡t❛❜ó❧✐❝❛s ❡ ❛s ✐♥t❡r❛çõ❡s r❡❣✉❧❛tór✐❛s ❡stã♦ ❞✐s❞❡rs❛s ❡ sã♦ ❞✐✜❝✐❧♠❡♥t❡ ✈✐s✉❛❧✐③❛❞❛s✳❏á ♥❛ ❋✐❣✉r❛ ✶✵✱ r❡❢❡r❡♥t❡ à ❙✳ ❈❡r❡✈✐s✐❛❡ ❞♦❞❡✲s❡ ✈❡r✐✜❝❛r ❛ ❞r❡s❡♥ç❛ ❞❡ três ❝♦♠✉♥✐❞❛❞❡s ♠❛✐♦r❡s ❡ ❞❡♥tr♦ ❞❡st❛s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝♦♠✉♥✐❞❛❞❡s ♠❡♥♦r❡s✱ s❡♥❞♦ q✉❡ ♥❛ ❞r✐♠❡✐r❛ ❝♦♠✉♥✐❞❛❞❡ ❤á ❞r❡s❡♥ç❛ ❞❡ ✐♥t❡r❛çõ❡s ❞r♦t❡í♥❛✲❞r♦t❡í♥❛ ❡ ♠❡t❛❜ó❧✐❝❛❀ ❥á ♥❛ s❡❣✉♥❞❛✱ ❤á ❞r❡❞♦♠í♥✐♦ ❞❡ ❧✐❣❛çõ❡s ❞r♦t❡í♥❛✲❞r♦t❡í♥❛ ❡ ❛ ú❧t✐♠❛ ❝♦♠✉♥✐❞❛❞❡ ❛❞r❡s❡♥t❛ ✐♥t❡r❛çõ❡s ❞❡ t♦❞♦s ♦s t✐❞♦s ❡ ♠✉✐t❛s ❝♦♠✉♥✐❞❛❞❡s ♠❡♥♦r❡s✳❆s ✐♥t❡r❛çõ❡s ❞♦ t✐❞♦ r❡❣✉❧❛tór✐❛s ❡stã♦ ♠❡♥♦s ❝♦♥❝❡♥tr❛❞❛s ✭♠❛✐s ❡s❞❛❧❤❛❞❛s✮✱ ✐♥❞✐❝❛♥❞♦ q✉❡ ✉♠ ♠❡s♠♦ ❣❡♥❡ ❞♦❞❡ ❞❛rt✐❝✐❞❛r ❞♦ ❞r♦❝❡ss♦ r❡❣✉❧❛tór✐♦ ❞❡ ✈ár✐♦s ♦✉tr♦s✱ ✐♥t❡r❛❣✐♥❞♦ ❝♦♠ ✈ár✐♦s ❣❡♥❡s ❡ ♦r✐❣✐♥❛♥❞♦ ✉♠❛ ❧✐♥❤❛ ❤♦r✐③♦♥t❛❧ ♥❛ ♠❛tr✐③✳