Cubo mágico: propriedades e resoluções envolvendo Álgebra e Teoria de Grupos

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❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❈✉❜♦ ▼á❣✐❝♦✿ Pr♦♣r✐❡❞❛❞❡s ❡ ❘❡s♦❧✉çõ❡s

❡♥✈♦❧✈❡♥❞♦ ➪❧❣❡❜r❛ ❡ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s

▲✉✐s ●✉st❛✈♦ ❍❛✉✛ ▼❛rt✐♥s ●r✐♠♠

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✕ P❘❖❋▼❆❚ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❈❛r✐♥❛ ❆❧✈❡s

✶✶✶ ❳✶✶✶①

●r✐♠♠✱ ▲✉✐s ●✉st❛✈♦ ❍❛✉✛ ▼❛rt✐♥s

❈✉❜♦ ▼á❣✐❝♦✿ Pr♦♣r✐❡❞❛❞❡s ❡ ❘❡s♦❧✉çõ❡s ❡♥✈♦❧✈❡♥❞♦ ➪❧❣❡❜r❛ ❡ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s✴ ▲✉✐s ●✉st❛✈♦ ❍❛✉✛ ▼❛rt✐♥s ●r✐♠♠✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✻✳

✽✶ ❢✳✿ ✜❣✳✱ t❛❜✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳

❖r✐❡♥t❛❞♦r❛✿ ❈❛r✐♥❛ ❆❧✈❡s

✶✳ ❈✉❜♦ ❞❡ ❘✉❜✐❦✳ ✷✳ ❚❡♦r✐❛ ❞❡ ❣r✉♣♦s✳ ✸✳ ●r✉♣♦s ❞❡ ♣❡r✲ ♠✉t❛çã♦✳ ✹✳ ❈♦♠✉t❛❞♦r❡s ❡ ❝♦♥❥✉❣❛❞♦s✳ ✺✳ Pr♦♣♦st❛ ❞✐❞át✐❝❛✳ ■✳ ❚ít✉❧♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

▲✉✐s ●✉st❛✈♦ ❍❛✉✛ ▼❛rt✐♥s ●r✐♠♠

❈✉❜♦ ▼á❣✐❝♦✿ Pr♦♣r✐❡❞❛❞❡s ❡ ❘❡s♦❧✉çõ❡s ❡♥✈♦❧✈❡♥❞♦

➪❧❣❡❜r❛ ❡ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛✲ ❞♦r❛✿

Pr♦❢❛✳ ❉r❛✳ ❈❛r✐♥❛ ❆❧✈❡s ❖r✐❡♥t❛❞♦r❛

Pr♦❢✳ ❉r✳ ❈r✐st✐❛♥❡ ❆❧❡①❛♥❞r❛ ▲á③❛r♦ ❯◆❊❙P ✲ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ✲ ❇❛✉r✉

Pr♦❢✳ ❉r✳ ❆❣♥❛❧❞♦ ❏♦sé ❋❡rr❛r✐

❯◆❊❙P ✲ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ✲ ❇❛✉r✉

❆❣r❛❞❡❝✐♠❡♥t♦s

➚ ♠✐♥❤❛ ❡s♣♦s❛ ♣❡❧♦ ❛♣♦✐♦✱ ✐♥❝❡♥t✐✈♦✱ ❝♦♠♣❛♥❤✐❛ ♥♦s ❡st✉❞♦s ❡ ♣❡❧♦ ❛✉①í❧✐♦ ❝♦♠ ♦ ▲❆_❚❊❳✳

➚ ♣r♦❢❡ss♦r❛ ❈❛r✐♥❛ ❆❧✈❡s ♣❡❧❛ ❝♦♠♣❡tê♥❝✐❛ ❡ ♣❛❝✐ê♥❝✐❛ ❞✉r❛♥t❡ ❛ ♦r✐❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞❛ ❯◆❊❙P ❞❡ ❘✐♦ ❈❧❛r♦ ♣❡❧❛ ❝♦♥tr✐❜✉✐çã♦ ♥❡ss❡ ♣r♦❝❡ss♦ ❞❡ ❢♦r♠❛çã♦ ❝♦♥t✐♥✉❛❞❛ ❞❡ ♣r♦✜ss✐♦♥❛✐s q✉❡ ❛t✉❛♠ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳

❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞❛ s❡❝r❡t❛r✐❛ ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❞❛ ❯◆❊❙P ❞❡ ❘✐♦ ❈❧❛r♦ ♣❡❧❛ ❛t❡♥✲ çã♦ ❡ ❡♠♣❡♥❤♦ ❡♠ ❛❥✉❞❛r ♣r♦♥t❛♠❡♥t❡✳

❙❡ ✈♦❝ê ❢♦r ❝✉r✐♦s♦✱ ❡♥❝♦♥tr❛rá q✉❡❜r❛✲❝❛❜❡ç❛s ❡♠ t♦r♥♦ ❞❡ ✈♦❝ê✳ ❙❡ ✈♦❝ê ❡st✐✈❡r ❞❡t❡r♠✐♥❛❞♦✱ ✐rá r❡s♦❧✈ê✲❧♦s✳

❘❡s✉♠♦

❖ ❝✉❜♦ ♠á❣✐❝♦ é ✉♠ ❞♦s q✉❡❜r❛✲❝❛❜❡ç❛s ♠❛✐s ❢❛♠♦s♦s ❞♦ ♠✉♥❞♦✱ ❡ ❡♠ ❣❡r❛❧ ❛tr❛✐ ❛ ❛t❡♥çã♦ ❞❡ ♠✉✐t❛ ❣❡♥t❡✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❞♦s ♠❛t❡♠át✐❝♦s✳ ❖ ❞❡s❛✜♦✱ ❛s ❢♦r♠❛s✱ s✐♠❡tr✐❛s ❡ ♠♦✈✐♠❡♥t♦s ✐♥❞✉③❡♠ ❛ ✐❞❡✐❛ ❞❡ ❡st❛r♠♦s ❞✐❛♥t❡ ❞❡ ✉♠ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦✳ ❊ ♣♦❞❡♠♦s ✐r ❛❧é♠✳ ❆s ❛çõ❡s ❡ ♠♦✈✐♠❡♥t♦s ♥♦ ❝✉❜♦ ♠á❣✐❝♦ sã♦ ❡❧❡♠❡♥t♦s q✉❡ ❛t❡♥❞❡♠ ❛ t♦❞❛s ❛s ❝♦♥❞✐çõ❡s ❞❛ ❡str✉t✉r❛ ❞❡ ✉♠ ❣r✉♣♦✱ ❛ss✐♠ ❝♦♠♦ t❛♠❜é♠ s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ✉♠ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çõ❡s✳ ➚ ❧✉③ ❞❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❡ ❞♦s ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✱ ✐r❡♠♦s ❛♥❛❧✐s❛r ❛❧❣✉♠❛s s❡q✉ê♥❝✐❛s ❞❡ ♠♦✈✐♠❡♥t♦s ❝♦♠♦ ♦s ❝♦♠✉t❛❞♦r❡s ❡ ❝♦♥❥✉❣❛❞♦s✳ ❊①✐st❡♠ ✈ár✐♦s ❛❧❣♦r✐t♠♦s q✉❡ r❡s♦❧✈❡♠ ♦ ❝✉❜♦ ♠á❣✐❝♦ ❡ q✉❡ sã♦ ❢á❝❡✐s ❞❡ s❡r❡♠ ♦❜t✐❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ✐♥t❡r♥❡t✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ❛❧é♠ ❞❡ tr❛③❡r ✉♠❛ ♣r♦♣♦st❛ ❞❡ r❡s♦❧✉çã♦✱ é ♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛r ✉♠ ❝❛♠✐♥❤♦ ♣❛r❛ ❛❧é♠ ❞❛ s✐♠♣❧❡s ♠❡♠♦r✐③❛çã♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦✱ ♥♦ s❡♥t✐❞♦ ❞❡ ❝♦♠♣r❡❡♥❞ê✲❧♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ s❡ r❡s♦❧✈❡r ✉♠ ❝✉❜♦ ♠á❣✐❝♦ é ❞❡ ♦r❞❡♠ ♠❛t❡♠át✐❝❛ ❡ ♥ã♦ ❡♠♣ír✐❝❛✳

❆❜str❛❝t

❚❤❡ ❘✉❜✐❦✬s ❈✉❜❡ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ❢❛♠♦✉s ♣✉③③❧❡ ♦❢ t❤❡ ✇♦r❧❞✱ ❛♥❞ ❣❡♥❡r❛❧❧② ❛t✲ tr❛❝ts t❤❡ ❛tt❡♥t✐♦♥ ♦❢ ♠❛♥② ♣❡♦♣❧❡✱ ❡s♣❡❝✐❛❧❧② ♠❛t❤❡♠❛t✐❝✐❛♥s✳ ❚❤❡ ❝❤❛❧❧❡♥❣❡✱ s❤❛♣❡s✱ s②♠♠❡tr✐❡s ❛♥❞ ♠♦✈❡♠❡♥ts ✐♥❞✉❝❡ t❤❡ ✐❞❡❛ ♦❢ ❜❡✐♥❣ ✐♥ ❢r♦♥t ♦❢ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t✳ ❆♥❞ ✇❡ ❝❛♥ ❣♦ ❢✉rt❤❡r✳ ❚❤❡ ❛❝t✐♦♥s ❛♥❞ ♠♦✈❡♠❡♥ts ✐♥ t❤❡ ♠❛❣✐❝ ❝✉❜❡ ❛r❡ ❡❧❡♠❡♥ts t❤❛t ♠❡❡t ❛❧❧ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ str✉❝t✉r❡ ♦❢ ❛ ❣r♦✉♣✱ ❛s ✇❡❧❧ ❛s r❡❧❛t❡ t♦ ❛ ❣r♦✉♣ ♦❢ ♣❡r♠✉t❛t✐♦♥s✳ ■♥ ❧✐❣❤t ♦❢ t❤❡ ●r♦✉♣ ❚❤❡♦r② ❛♥❞ P❡r♠✉t❛t✐♦♥s ❣r♦✉♣s ✇❡ ✇✐❧❧ ❡①❛♠✐♥❡ s♦♠❡ s❡q✉❡♥❝❡s ♦❢ ♠♦✈❡♠❡♥ts s✉❝❤ ❛s ❝♦♠♠✉t❛t♦rs ❛♥❞ ❝♦♥❥✉❣❛t❡s✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❛❧❣♦r✐t❤♠s t❤❛t s♦❧✈❡ t❤❡ ♠❛❣✐❝ ❝✉❜❡ ❛♥❞ ✇❤✐❝❤ ❛r❡ ❡❛s② t♦ ♦❜t❛✐♥✱ ❢♦r ❡①❛♠♣❧❡✱ ❛t t❤❡ ■♥t❡r♥❡t✳ ❚❤❡ ❛✐♠ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ❜❡②♦♥❞ t♦ s❤♦✇ ❛ r❡s♦❧✉t✐♦♥✱ ✐s t♦ ♣r♦✈✐❞❡ ❛ ♣❛t❤ ❜❡②♦♥❞ s✐♠♣❧❡ ♠❡♠♦r✐③❛t✐♦♥ ♦❢ ❛♥ ❛❧❣♦r✐t❤♠ ✐♥ ♦r❞❡r t♦ ✉♥❞❡rst❛♥❞ ✐t✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ s♦❧✈✐♥❣ ❛ ❘✉❜✐❦✬s ❈✉❜❡ ✐s ♠❛t❤ ❛♥❞ ♥♦t ❡♠♣✐r✐❝❛❧✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❱✐sã♦ ❡①♣❧♦❞✐❞❛ ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦3_×3_×3. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷✳✷ ▼❡❝❛♥✐s♠♦ ✐♥t❡r♥♦ ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦ 3_×3_×3. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷✳✸ ❚✐♣♦s ❡ ❢♦r♠❛t♦s ❞❡r✐✈❛❞♦s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹ ❋❛❝❡s ❞♦ ❝✉❜♦3_×3_×3 ❡ s✉❛ ♣❧❛♥✐✜❝❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷✳✺ ◆♦♠❡♥❝❧❛t✉r❛ ❞♦s ❝✉❜✐♥❤♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✻ ❊✐①♦s ❞❡ r♦t❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✼ ❊①❡♠♣❧♦s ❞❡ ♣♦s✐çã♦ ❡ ♦r✐❡♥t❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✽ ❈ó❞✐❣♦s ❞❛s r♦t❛çõ❡s ❞❛s ❢❛❝❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✶ ◗✉❛❞r❛❞♦ r❡♣r❡s❡♥t❛t✐✈♦ ❞♦ ❣r✉♣♦G_Q✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸✳✷ ❈♦♠♣♦s✐çã♦R_r◦R₁✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸✳✸ ❈♦♠♣♦s✐çã♦R_{1 ◦}R_r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸✳✹ ▼❛❝r♦sF R ❡ RF✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✸✳✺ ▼❛❝r♦F4_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵} ✹✳✶ ❈✉❜✐♥❤♦s ❛❢❡t❛❞♦s ♣❡❧❛ ♠❛❝r♦S✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

✹✳✷ ❈✉❜✐♥❤♦s ❛❢❡t❛❞♦s ♣❡❧♦ ♠♦✈✐♠❡♥t♦ F ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✹✳✸ ❊❢❡✐t♦ ❣❡r❛❞♦ ♣❡❧❛s ♠❛❝r♦sS ❡T✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✾

✷ ❖ ❈✉❜♦ ▼á❣✐❝♦ ✷✶

✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦ ❡ s❡✉ ♠❡❝❛♥✐s♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❋♦r♠❛t♦s ❡ ✈❛r✐❛çõ❡s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ▼♦✈✐♠❡♥t♦s ❞❛s ❢❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ▼❛❝r♦ ♦✉ s❡q✉ê♥❝✐❛s ❞❡ ♠♦✈✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸ ➪❧❣❡❜r❛ ♥♦ ❈✉❜♦ ▼á❣✐❝♦ ✷✾ ✸✳✶ ❚❡♦r✐❛ ❇ás✐❝❛ ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ●r✉♣♦s ❛❞✐t✐✈♦s ❡ ♠✉❧t✐♣❧✐❝❛t✐✈♦s ❞❡ ❝❧❛ss❡s ❞❡ r❡st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❈❧❛ss❡s ▲❛t❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✹ ●r✉♣♦ ❞❡ ❘✉❜✐❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✺ ❙✉❜❣r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✺✳✶ ❙✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r ✉♠ s✉❜❝♦♥❥✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✻ ❙✉❜❣r✉♣♦s ◆♦r♠❛✐s ❡ ●r✉♣♦s ◗✉♦❝✐❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✼ ❍♦♠♦♠♦r✜s♠♦s ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✽ P❡r♠✉t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✽✳✶ Pr♦❞✉t♦ ❞❡ ❈✐❝❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✽✳✷ ❘❡♣❡t✐çã♦ ❞❡ ❝✐❝❧♦s ❡ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✾ ❆ss✐♥❛t✉r❛ ❞❡ ✉♠❛ P❡r♠✉t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✾✳✶ P❛r✐❞❛❞❡ ❞❡ ❈✐❝❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹ ❆♣❧✐❝❛çõ❡s ❞❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çã♦ ♥♦ ❈✉❜♦

▼á❣✐❝♦ ✺✺

✺✳✶✳✷ ▼ét♦❞♦ r❡❛❧✐③á✈❡❧ ♣♦r ❤✉♠❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✺✳✷ ❘❡s♦❧✈❡♥❞♦ ♦ ❈✉❜♦ ▼á❣✐❝♦ ♥♦ ♠❡♥♦r t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✺✳✸ ❘❡s♦❧✈❡♥❞♦ ♦ ❈✉❜♦ ▼á❣✐❝♦ ❝♦♠ ❛ ♠❡♥♦r ♠❡♠♦r✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✺✳✸✳✶ ✶❛ _{❊t❛♣❛✿ ❢♦r♠❛r ✉♠❛ ❝r✉③ ♥❛ ❢❛❝❡ ✐♥✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾}

✺✳✸✳✷ ✷❛ _{❊t❛♣❛✿ ❢♦r♠❛r ✉♠❛ ❝r✉③ ♥❛s ❢❛❝❡s ❛❞❥❛❝❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾}

✺✳✸✳✸ ✸❛ _{❊t❛♣❛✿ ❢♦r♠❛r ✉♠❛ ❝r✉③ ♥❛ ❢❛❝❡ ♦♣♦st❛ à ✐♥✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾}

✺✳✸✳✹ ✹❛ _{❊t❛♣❛✿ ♣♦s✐❝✐♦♥❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ❝❛♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵}

✺✳✸✳✺ ✺❛ _{❊t❛♣❛✿ ♦r✐❡♥t❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ❝❛♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵}

✺✳✸✳✻ ❘❡s✉♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

✻ ❆♣❧✐❝❛çã♦ ❊s❝♦❧❛r ✼✸

✻✳✶ ❆♣r❡s❡♥t❛çã♦ ❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ❝✉❜♦ ♠á❣✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✷ ❈ó❞✐❣♦s ❞❡ ❝❛❞❛ ♠♦✈✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✸ ❘❡s♦❧✉çã♦ ❞♦ ❈✉❜♦ ▼á❣✐❝♦ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❝❛♠❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✸✳✶ ❊t❛♣❛ ✶✿ ❋♦r♠❛r ✉♠❛ ❝r✉③ ❜r❛♥❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✸✳✷ ❊t❛♣❛ ✷✿ P♦s✐❝✐♦♥❛r ♦s ❝❛♥t♦s ❜r❛♥❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✸✳✸ ❊t❛♣❛ ✸✿ ❘❡s♦❧✈❡r ❛ ❝❛♠❛❞❛ ❞♦ ♠❡✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✻✳✸✳✹ ❊t❛♣❛ ✹✿ ❋♦r♠❛r ✉♠❛ ❝r✉③ ❛♠❛r❡❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✻✳✸✳✺ ❊t❛♣❛ ✺✿ ❘❡s♦❧✈❡r ❛ ❝❛♠❛❞❛ s✉♣❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✻✳✸✳✻ ❊t❛♣❛ ✻✿ ❖r✐❡♥t❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ❝❛♥t♦ ❞❛ ❝❛♠❛❞❛ s✉♣❡r✐♦r ✳ ✳ ✼✽ ✻✳✸✳✼ ❊t❛♣❛ ✼✿ P♦s✐❝✐♦♥❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ❛r❡st❛ ❞❛ ❝❛♠❛❞❛ s✉♣❡r✐♦r ✳ ✼✽

✶ ■♥tr♦❞✉çã♦

❉✉r❛♥t❡ ❞é❝❛❞❛s ♦ ❞❡s❛✜♦ ❞❛ r❡s♦❧✉çã♦ ❞♦ ❝✉❜♦ ♠á❣✐❝♦✱ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ❝✉❜♦ ❞❡ ❘✉❜✐❦✱ ❡♠ ❤♦♠❡♥❛❣❡♠ ❛♦ s❡✉ ✐♥✈❡♥t♦r✱ ✐♥tr✐❣♦✉ ♠✉✐t❛ ❣❡♥t❡✳ ❊r♥ö ❘✉❜✐❦ ♥❛s❝❡✉ ❡♠ ✶✸ ❞❡ ❥✉❧❤♦ ❞❡ ✶✾✹✹ ❡♠ ❇✉❞❛♣❡st❡✱ ❍✉♥❣r✐❛ ❡ ♥❛ ❞é❝❛❞❛ ❞❡ ✼✵ ❞❛✈❛ ❛✉❧❛s ❞❡ ❛rq✉✐✲ t❡t✉r❛ ♥❛ ❙❝❤♦♦❧ ❢♦r ❈♦♠♠❡r❝✐❛❧ ❆rt✐sts✳ ❊♠ ✶✾✼✹ t❡✈❡ ❛ ✐❞❡✐❛ ❞❡ ❝♦♥str✉✐r ✉♠ ❝✉❜♦ ❝❛♣❛③ ❞❡ r♦t❛❝✐♦♥❛r s✉❛s ❢❛❝❡s ♣❛r❛ q✉❡ ❡♥tã♦ ♣✉❞❡ss❡ ✐❧✉str❛r ♠❡❧❤♦r ♣❛r❛ s❡✉s ❛❧✉♥♦s ♦ ❝♦♥❝❡✐t♦ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ❖ ♣r✐♠❡✐r♦ ❡①❡♠♣❧❛r ❡r❛ ❞❡ ♠❛❞❡✐r❛ ❡ ❝❛❞❛ ✉♠❛ ❞❛s s❡✐s ❢❛❝❡s ❞♦ ❝✉❜♦ ❡r❛ ♣✐♥t❛❞❛ ❞❡ ✉♠❛ ❝♦r ❞✐❢❡r❡♥t❡✳ ❈❛❞❛ ❢❛❝❡ é ❞✐✈✐❞✐❞❛ ❡♠ ✾ ❝✉❜✐♥❤♦s ♠❡♥♦r❡s t♦t❛❧✐③❛♥❞♦ ✷✻ ❝✉❜♦s ✈✐sí✈❡✐s ❡ ✉♠ ✈✐rt✉❛❧ q✉❡ s❡ ❡①✐st✐ss❡ ❡st❛r✐❛ ❧♦❝❛❧✐③❛❞♦ ♥♦ ✐♥t❡r✐♦r ❞♦ ❝✉❜♦✳ ❊st❡ ✷✼♦ _{❝✉❜✐♥❤♦ ✈✐rt✉❛❧ é ❥✉st❛♠❡♥t❡ ♦ ♠❡❝❛♥✐s♠♦ q✉❡ ♣r❡♥❞❡}

❛s ✻ ❢❛❝❡s✳ ◆♦ ♠❡s♠♦ ❛♥♦ ❞❛ ❝r✐❛çã♦✱ ♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦ ❣❛♥❤♦✉ ♦ ♣rê♠✐♦ ❛❧❡♠ã♦ ❞❡ ❥♦❣♦ ❞♦ ❛♥♦✳ ❊r♥ö ❘✉❜✐❦ ♥ã♦ t✐♥❤❛ ❡①❛t❛♠❡♥t❡ ❛ ✐♥t❡♥çã♦ ❞❡ ❝r✐❛r ✉♠ q✉❡❜r❛✕❝❛❜❡ç❛ ❝✉❥❛ s♦❧✉çã♦ ❝♦♥s✐st❡ ❡♠ ❞❡✐①❛r ❝❛❞❛ ✉♠❛ ❞❛s ❢❛❝❡s s♦♠❡♥t❡ ❝♦♠ ✉♠❛ ❝♦r✳ ❖ ♣ró✲ ♣r✐♦ ❝r✐❛❞♦r ❞♦ ❝✉❜♦ ❞❡♠♦r♦✉ ❝❡r❝❛ ❞❡ ✉♠ ♠ês ♣❛r❛ r❡♠♦♥tá✲❧♦ à s✉❛ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧✳ ❊♠ ✶✾✽✵ ♦ ❝✉❜♦ ❝♦♠❡ç♦✉ ❛ s❡r ♣r♦❞✉③✐❞♦ ❞❡ ❢♦r♠❛ ✐♥❞✉str✐❛❧ ❡ ❡st✐♠❛✕s❡ q✉❡ ❞❡s❞❡ ❡♥tã♦ ❥á t❡♥❤❛♠ s✐❞♦ ✈❡♥❞✐❞♦s ♠❛✐s ❞❡ ✸✺✵ ♠✐❧❤õ❡s ❞❡ ✉♥✐❞❛❞❡s ❡♠ t♦❞♦ ♦ ♠✉♥❞♦✳

■♥✐❝✐❛❧♠❡♥t❡✱ ♥♦ ❈❛♣ít✉❧♦ ✷✱ ✐r❡♠♦s tr❛③❡r ♥♦çõ❡s s♦❜r❡ s❡✉ ♠❡❝❛♥✐s♠♦ ❞❡ ❢✉♥❝✐✲ ♦♥❛♠❡♥t♦ ❡ q✉❛✐s ♠♦✈✐♠❡♥t♦s sã♦ ♣❡r♠✐t✐❞♦s ❡❢❡t✉❛r✳ ❊st❛ ♣r✐♠❡✐r❛ ❛♣r❡s❡♥t❛çã♦ é ♥❡❝❡ssár✐❛ ♣❛r❛ ❡♥t❡♥❞❡r♠♦s ♦ q✉❡ é ❡ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ✉♠ ❝✉❜♦ ♠á❣✐❝♦✳ ❊✱ ❛❧é♠ ❞✐ss♦✱ ♦s ♠♦✈✐♠❡♥t♦s ❞❛s ❢❛❝❡s✱ ❜❡♠ ❝♦♠♦ ❞♦s ❝✉❜✐♥❤♦s✱ sã♦ ♦ ♦❜❥❡t✐✈♦ ❛❧✈♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ♣♦✐s sã♦ ❡st❡s ♦s ❡❧❡♠❡♥t♦s q✉❡ ✐r❡♠♦s ❡st✉❞❛r ♠❛✐s ❛❞✐❛♥t❡✳

◆♦ ❈❛♣ít✉❧♦ ✸ ✐r❡♠♦s ❛❜♦r❞❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ ❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✳ ❆ ♣❛rt✐r ❞❡❧❡s ✐❞❡♥t✐✜❝❛r❡♠♦s ♦ ❝✉❜♦ ♠á❣✐❝♦ ❝♦♠♦ ✉♠ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦✱ ❛ss✐♠ ❝♦♠♦ ❛s ❛çõ❡s ❡ ♦s ♠♦✈✐♠❡♥t♦s ♥♦ ❝✉❜♦ s❡rã♦ ✐❞❡♥t✐✜❝❛❞♦s ❝♦♠♦ ♦♣❡r❛çõ❡s q✉❡ ❛t❡♥❞❡♠ às ❞❡✜♥✐çõ❡s q✉❡ s❡rã♦ ✈✐st❛s✳

◆♦ ❈❛♣ít✉❧♦ ✹ ✐r❡♠♦s ❛♣❧✐❝❛r ♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s✱ ♥♦ ✐♥t✉✐t♦ ❞❡ ❡♥t❡♥❞❡r♠♦s ❛❧❣✉♥s ♠♦✈✐♠❡♥t♦s✱ ❡st❛❞♦s ❡ ❛❧❣♦r✐t♠♦s q✉❡ r❡s♦❧✈❡♠ ♦ ❝✉❜♦ ♠á❣✐❝♦✳ ❆❧é♠ ❞✐ss♦✱ ❝❛❧❝✉❧❛r❡♠♦s ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❝♦♥✜❣✉r❛çõ❡s q✉❡ ♦ ❝✉❜♦ ♠á❣✐❝♦ ♣♦❞❡ ❛ss✉♠✐r✱ ❥✉♥✲ t❛♠❡♥t❡ ❝♦♠ ❛ ❥✉st✐✜❝❛t✐✈❛ ❞❡ q✉❡ ❛❧❣✉♠❛s ❝♦♥✜❣✉r❛çõ❡s sã♦ ✐♠♣♦ssí✈❡✐s ❞❡ s❡r❡♠

✷✵ ■♥tr♦❞✉çã♦

❛t✐♥❣✐❞❛s✳

❘❡❢❡rê♥❝✐❛s

❬✶❪ ❚❘❆❱■❙✱ ▼✳ ❚❤❡ ▼❛t❤❡♠❛t✐❝s ♦❢ t❤❡ ❘✉❜✐❦✬s ❈✉❜❡✳ ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦✱ ✷✵✵✼✳ ❬✷❪ ▲❊◗❯❆■◆✱ ❨✳❀ ●❆❘❈■❆✱ ❆✳ ❊❧❡♠❡♥t♦s ❞❡ á❧❣❡❜r❛✳ ✺✳ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✱

✷✵✶✵✳

❬✸❪ ❉❖▼■◆●❯❊❙✱ ❍✳ ❍✳❀ ■❊❩❩■✱ ●✳ ➪❧❣❡❜r❛ ▼♦❞❡r♥❛✳ ✹✳ ❡❞✳ ❙ã♦ P❛✉❧♦✿ ❆t✉❛❧✱ ✷✵✵✸✳ ❬✹❪ ❈❍❊◆✱ ❏✳ ●r♦✉♣ ❚❤❡♦r② ❛♥❞ t❤❡ ❘✉❜✐❦✬s ❈✉❜❡✳ ❍❛r✈❛r❞✱ ✷✵✵✹✳

❬✺❪ ❙❈❍❯▲❚❩❊❘✱ ❲✳ ❆♣r❡♥❞❡♥❞♦ ➪❧❣❡❜r❛ ❝♦♠ ♦ ❈✉❜♦ ▼á❣✐❝♦✳ ❉▼ ✲ ❯❋❙❈❛r✱ ✷✵✵✺✳ ❬✻❪ ❚❘❖◆❚❖✱ ❙✳ ❋❡✇❡st ▼♦✈❡s ❚✉t♦r✐❛❧✳ ✵✶ ✷✵✶✻✳

❬✼❪ ❚❍■❙❚▲❊❚❍❲❆■❚❊✱ ▼✳ ❇✳ ❚❤❡ ✹✺✲✺✷ ♠♦✈❡ str❛t❡❣②✳ P♦❧②t❡❝❤♥✐❝ ♦❢ t❤❡ ❙♦✉t❤ ❇❛♥❦✱ ✶✾✽✶✳