❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ▼❆❘❆◆❍➹❖
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼
▼❆❚❊▼➪❚■❈❆
❆◆❚Ô◆■❖ ❉❊ ❏❊❙❯❙ ❉❊ ❙❖❯❙❆ ❋❊❘❘❊■❘❆
❈➪▲❈❯▲❖ ❉■❋❊❘❊◆❈■❆▲ ❊ ■◆❚❊●❘❆▲✿
❯♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦
❆◆❚Ô◆■❖ ❉❊ ❏❊❙❯❙ ❉❊ ❙❖❯❙❆ ❋❊❘❘❊■❘❆
❈➪▲❈❯▲❖ ❉■❋❊❘❊◆❈■❆▲ ❊ ■◆❚❊●❘❆▲✿
❯♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢♦ ❉r✳ ❏♦ã♦ ❞❡ ❉❡✉s ▼❡♥❞❡s ❞❛ ❙✐❧✈❛✳
❊❧❛❜♦r❛❞❛ ♣❡❧❛ ❇✐❜❧✐♦t❡❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦
❋❡rr❡✐r❛✱ ❆♥tô♥✐♦ ❞❡ ❏❡s✉s ❞❡ ❙♦✉s❛
❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✿ ✉♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ✴ ❆♥tô♥✐♦ ❞❡ ❏❡s✉s ❞❡ ❙♦✉s❛ ❋❡rr❡✐r❛✳ ✕ ✷✵✶✻✳
✶✶✷❢✳
■♠♣r❡ss♦ ♣♦r ❝♦♠♣✉t❛❞♦r ✭❋♦t♦❝ó♣✐❛✮✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢♦ ❉r✳ ❏♦ã♦ ❞❡ ❉❡✉s ▼❡♥❞❡s ❞❛ ❙✐❧✈❛✳
❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦✱ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ✷✵✶✻✳
✶✳ ❈á❧❝✉❧♦ ✕ ❊♥s✐♥♦ ♠é❞✐♦✳ ✷✳ ❊♥s✐♥♦ ❙✉♣❡r✐♦r✳ ✸✳ ▼❡❧❤♦r✐❛s ■✳ ❚ít✉❧♦✳
❆◆❚Ô◆■❖ ❉❊ ❏❊❙❯❙ ❉❊ ❙❖❯❙❆ ❋❊❘❘❊■❘❆
❆ ♣r❡s❡♥t❡ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦ ❡ ❡❧❛❜♦r❛❞❛ ♣♦r ❆♥tô♥✐♦ ❞❡ ❏❡s✉s ❞❡ ❙♦✉s❛ ❋❡rr❡✐r❛ ✱ s♦❜ ♦ tít✉❧♦✿ ✏❈➪▲❈❯▲❖ ❉■❋❊❘❊◆❈■❆▲ ❊ ■◆❚❊●❘❆▲✿ ✉♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♦ ❡♥s✐♥♦
♠é❞✐♦✑✱ ❢♦✐ ❛♣r♦✈❛❞❛ ❡♠ ❞❡ ❞❡
✷✵✶✻ ❡ r❡❝❡❜❡✉ ❝♦♥❝❡✐t♦✿
❈❖▼■❙❙➹❖ ❊❳❆▼■◆❆❉❖❘❆
Pr♦❢✳♦ ❉r✳ ❏♦ã♦ ❞❡ ❉❡✉s ▼❡♥❞❡s ❞❛ ❙✐❧✈❛ ❖r✐❡♥t❛❞♦r
Pr♦❢✳❛ ❉r✳❛ ❱❛❧❡s❦❛ ▼❛rt✐♥s ❞❡ ❙♦✉③❛
Pr♦❢✳❛ ❉r✳❛ ❉❡❛ ◆✉♥❡s ❋❡r♥❛♥❞❡s
❉❊❉■❈❆❚Ó❘■❆
❆ ❉❡✉s✱ q✉❡ ♥♦s ❝r✐♦✉ ❡ ❢♦✐ ❝r✐❛t✐✈♦ ♥❡st❛ t❛r❡❢❛✳ ❙❡✉ ❢ô❧❡❣♦ ❞❡ ✈✐❞❛ ❡♠ ♠✐♠ ♠❡ ❢♦✐ s✉st❡♥t♦ ❡ ♠❡ ❞❡✉ ❝♦r❛❣❡♠ ♣❛r❛ q✉❡st✐♦♥❛r r❡❛❧✐❞❛❞❡s ❡ ♣r♦♣♦r s❡♠♣r❡ ✉♠ ♥♦✈♦ ♠✉♥❞♦ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s✳
❆●❘❆❉❊❈■▼❊◆❚❖❙
❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❉❡✉s✱ r❛③ã♦ ❞❛ ♠✐♥❤❛ ❡①✐stê♥❝✐❛✱ ♣r❡s❡♥t❡ ❝♦♠✐❣♦ ❡♠ t♦❞♦s ♦s ✐♥st❛♥t❡s✳ ▼❡✉s ♣❛✐s✱ ▼❛♥♦❡❧ ❡ ❖❧❞❛r✐♥❛✱ ♣❡❧♦ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠✐♠✳
➚ ♠✐♥❤❛ ❛♠❛❞❛ ❡s♣♦s❛✱ ▼✐r❧❡②✱ ♣❡❧♦ ❛♠♦r✱ ❝❛r✐♥❤♦ ❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❢✉♥❞❛♠❡♥✲ t❛✐s ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s ❞❡st❛ ❝❛♠✐♥❤❛❞❛✳
❆♦s ♠❡✉s ♠❛r❛✈✐❧❤♦s♦s ✜❧❤♦s✱ ❆❧❡①❛♥❞r❡ ▼②❧❡r ❡ ❚❤❛❧❡s ▼②❧❡r✱ s❡✉s s♦rr✐s♦s ❢♦rt❛❧❡❝❡♠✲♠❡ ❛ ❝❛❞❛ ❞✐❛✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢❡ss♦r ❏♦ã♦ ❞❡ ❉❡✉s✱ ♣❡❧♦ ❛✉①í❧✐♦✱ ♣❡❧♦ ❡①❡♠♣❧♦ ❞❡ ❝♦♠♣❛✲ ♥❤❡✐r✐s♠♦ ❡ ♣❡❧❛s t❛♥t❛s s✉❣❡stõ❡s✳
❆♦s ♠❡♠❜r♦s ❞❛ ❇❛♥❝❛✱ Pr♦❢❡ss♦r❛ ❱❛❧❡s❦❛ ▼❛rt✐♥s ❞❡ ❙♦✉③❛ ❡ Pr♦❢❡ss♦r❛ ❉❡❛ ◆✉♥❡s ❋❡r♥❛♥❞❡s✱ ♣❡❧❛s ✈❛❧✐♦s❛s s✉❣❡stõ❡s ❡ ❝♦♥tr✐❜✉✐çõ❡s✱ q✉❡ s❡ t♦r♥❛r❛♠ ♠✉✐t♦ ✐♠♣♦r✲ t❛♥t❡s ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆ ❈♦♦r❞❡♥❛❞♦r❛ ❞♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❞❡ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋✲ ▼❆❚✱ Pr♦❢❡ss♦r❛ ❱❛❧❞✐❛♥❡✱ ♣♦r ❛❝r❡❞✐t❛r q✉❡ ❛ ❢♦rç❛ ❞❡ ✈♦♥t❛❞❡ ♣♦❞❡♠ s✉♣❡r❛r ❛s ❞✐✜✲ ❝✉❧❞❛❞❡s ✐♠♣♦st❛s ♣❡❧❛ ❧♦❣íst✐❝❛✳
❊P❮●❘❆❋❊
✏❖ ♠❛✐♦r ✐♥✐♠✐❣♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥ã♦ é ✐❣♥♦✲ râ♥❝✐❛✱ ♠❛s ❛ ✐❧✉sã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✑
❘❊❙❯▼❖
❊ss❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ t❡♠❛✿ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✿ ❯♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♦ ❊♥✲ s✐♥♦ ▼é❞✐♦✳ ❙✉❛ ❡❧❛❜♦r❛çã♦ t❡✈❡ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♥❛❧✐s❛r ♦ ❝♦♥t❡①t♦ ❢♦r♠❛t✐✈♦ ❞♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✐♥❝❧✉✐♥❞♦ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳ P❛✉t❛❞♦ ♥❡st❡ ♦❜❥❡t✐✈♦✱ ❢♦✐ ❧❡✈❛♥t❛❞♦ ✉♠ q✉❡st✐♦♥❛♠❡♥t♦ ✐♥✈❡st✐❣❛t✐✈♦✱ é ❡❧❡✿ ❆ ❢♦r♠❛çã♦ ❞♦s ❛❧✉♥♦s ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ▼❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛❢❡t❛ t❛♠❜é♠ ♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ♥♦s ❝✉rs♦s ❞❡ ♥í✈❡❧ ❙✉♣❡r✐♦r✱ ❞❡s❡♥❝❛❞❡❛♥❞♦ ❞✐✜❝✉❧❞❛❞❡ ♥♦ ❛♣r❡♥❞✐③❛❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ❡ ❝♦♥t✐♥✉✐❞❛❞❡✱ ♦ ♣r♦❜❧❡♠❛ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✉r✈❛✱ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡❧✐♠✐t❛❞❛s ♣❡❧♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦❄ P❛r❛ r❡s♣♦♥❞❡r ❛ ❡ss❡s q✉❡st✐♦♥❛♠❡♥t♦s ♦♣t❛♠♦s ♣♦r r❡❛❧✐③❛r ✉♠❛ ♣❡sq✉✐s❛ ❜✐❜❧✐♦❣rá✜❝❛✱ ❛ q✉❛❧ ❢✉♥❞❛♠❡♥t♦✉✲s❡ ❡♠ ❛✉t♦r❡s ❝♦♠♦✿ ●❡r❛❧❞♦ ➪✈✐❧❛✱ ❍❛♠✐❧t♦♥ ▲✉✐③ ●✉✐❞♦r✐③③✐ ❡ ❏❛♠❡s ❙t❡✇❛rt✳ ❖ ❞❡❜❛t❡ ❢♦✐ ❛❝❡♥✲ t✉❛❞♦ ♥❛ ♠❡❞✐❞❛ q✉❡ ❛s ❧❡✐t✉r❛s ❡ ♦s ✜❝❤❛♠❡♥t♦s ❢♦r❛♠ s❡♥❞♦ ❛❧✐♥❤❛❞♦s ♥♦ ❝❛♠♣♦ ❞❛ ♣❡sq✉✐s❛ t❡ór✐❝❛✳ ❚❡♥❞♦ tr❛♥s✐t❛❞♦ ♥♦ ❝❛♠♣♦ ❞❛ ✐♥✈❡st✐❣❛çã♦✱ ❝♦♥❝❧✉✐r q✉❡ ❛s ❞✐✜❝✉❧❞❛❞❡s ❞❡ ❛♣r❡♥❞✐③❛✲ ❣❡♠ ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❈á❧❝✉❧♦ ■✱ ♥♦ ❝✐❝❧♦ ❜ás✐❝♦ ❞♦s ❝✉rs♦s ❞❡ ❊♥s✐♥♦ ❙✉♣❡r✐♦r✱ ❡ ❝✐❡♥t❡ ❞♦ ❡❧❡✈❛❞♦ ♥ú♠❡r♦ ❞❡ r❡♣r♦✈❛çõ❡s✱ ♥❡st❛ ❞✐s❝✐♣❧✐♥❛✱ ♦❝♦rr❡✉✲♠❡ ✉♠❛ ♠♦t✐✈❛çã♦ ♥❛t✉r❛❧ ♣❛r❛ tr❛❜❛❧❤❛r ❝♦♠ ♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ♥♦çõ❡s ❞❡ ❈á❧❝✉❧♦✱ ❞❛♥❞♦ ♦♣♦rt✉♥✐❞❛❞❡s ❛ ❡❧❡s ❞❡ ✏❛♠❛❞✉r❡❝✐♠❡♥t♦✑ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦✱ ❝♦♥tr✐❜✉✐♥❞♦ ❛ss✐♠ ♣❛r❛ ❛ ♠❡❧❤♦r✐❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❡ ❝♦♠♦ ❝♦♥✲ s❡q✉ê♥❝✐❛✱ ❛ ♠❡❧❤♦r✐❛ ❞♦ ❊♥s✐♥♦ ❙✉♣❡r✐♦r✳
❆❇❙❚❘❆❈❚
❚❤✐s ✇♦r❦ ❤❛s ❛s ✐ts t❤❡♠❡✿ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s✿ ❆ ♣r♦♣♦s❛❧ ❢♦r t❤❡ ❤✐❣❤ s❝❤♦♦❧✳ ❚❤❡✐r ♣r❡♣❛r❛t✐♦♥ ✇❛s t♦ ❛♥❛❧②③❡ t❤❡ ❡❞✉❝❛t✐♦♥❛❧ ❝♦♥t❡①t ♦❢ ❤✐❣❤ s❝❤♦♦❧ st✉❞❡♥ts ✐♥❝❧✉✲ ❞✐♥❣ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s✳ ●✉✐❞❡❞ ❜② t❤✐s ♣✉r♣♦s❡✱ ❛♥ ✐♥✈❡st✐❣❛t✐✈❡ q✉❡st✐♦♥ ✇❛s r❛✐s❡❞✱ ✐s ✐t✿ ❚❤❡ tr❛✐♥✐♥❣ ♦❢ st✉❞❡♥ts ✐♥ ▼❛t❤❡♠❛t✐❝s ✐♥ ❤✐❣❤ s❝❤♦♦❧ ❛❧s♦ ❛✛❡❝ts t❤❡ t❡❛❝❤✐♥❣ ♦❢ ♠❛t❤❡♠❛t✐❝s ✐♥ ❤✐❣❤❡r ❧❡✈❡❧ ❝♦✉rs❡s✱ tr✐❣❣❡r✐♥❣ ❧❡❛r♥✐♥❣ ❞✐s❛❜✐❧✐t✐❡s ❡s♣❡❝✐❛❧❧② t❤❡ ❝♦♥❝❡♣t ♦❢ ❧✐♠✐t ❛♥❞ ❝♦♥t✐♥✉✐t②✱ t❤❡ ♣r♦❜❧❡♠ t❤❡ t❛♥❣❡♥t ❧✐♥❡ t♦ ❛ ❝✉r✈❡✱ ❝❛❧❝✉❧❛t✐♥❣ ❛r❡❛s ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥❄ ❚♦ ❛♥s✇❡r t❤❡s❡ q✉❡st✐♦♥s ✇❡ ❞❡❝✐❞❡❞ t♦ ❝♦♥❞✉❝t ❛ ❧✐t❡r❛t✉r❡ s❡❛r❝❤✱ ✇❤✐❝❤ ✇❛s ❜❛s❡❞ ♦♥ ❛✉t❤♦rs s✉❝❤ ❛s ●❡r❛❧❞♦ ➪✈✐❧❛✱ ❍❛♠✐❧t♦♥ ▲✉✐③ ●✉✐❞♦r✐③③✐ ❛♥❞ ❏❛♠❡s ❙t❡✇❛rt✳ ❚❤❡ ❞❡❜❛t❡ ✇❛s ♠❛r❦❡❞ ❛s t❤❡ r❡❛❞✐♥❣s ❛♥❞ s✉♠♠❛r✐❡s ✇❡r❡ ❜❡✐♥❣ ❧✐♥❡❞ ✉♣ ✐♥ t❤❡ ✜❡❧❞ ♦❢ t❤❡♦r❡t✐❝❛❧ r❡s❡❛r❝❤✳ ❍❛✈✐♥❣ ❝❛rr✐❡❞ ✐♥ t❤❡ ✜❡❧❞ ♦❢ r❡s❡❛r❝❤✱ ❝♦♥❝❧✉❞❡ t❤❛t ❧❡❛r♥✐♥❣ ❞✐✣❝✉❧t✐❡s ✐♥ ❈❛❧❝✉❧✉s ■ ❝♦✉rs❡✱ t❤❡ ❜❛s✐❝ ❝②❝❧❡ ♦❢ ❤✐❣❤❡r ❡❞✉❝❛t✐♦♥ ❝♦✉rs❡s✱ ❛♥❞ ❛✇❛r❡ ♦❢ t❤❡ ❤✐❣❤ ♥✉♠❜❡r ♦❢ ❢❛✐❧✉r❡s ✐♥ t❤✐s ❞✐s❝✐♣❧✐♥❡✱ ✐t ♦❝❝✉rr❡❞ t♦ ♠❡ ❛ ♥❛t✉r❛❧ ♠♦t✐✈❛t✐♦♥ t♦ ✇♦r❦ ✇✐t❤ st✉❞❡♥ts ❤✐❣❤ s❝❤♦♦❧ ❝❛❧❝✉❧✉s ♥♦t✐♦♥s✱ ❣✐✈✐♥❣ ♦♣♣♦rt✉♥✐t✐❡s t♦ t❤❡♠ ✧r✐♣❡♥✐♥❣✧♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❦♥♦✇❧❡❞❣❡✱ t❤✉s ❝♦♥tr✐❜✉t✐♥❣ t♦ t❤❡ ✐♠♣r♦✈❡♠❡♥t ♦❢ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ ❛♥❞ ❛s ❛ r❡s✉❧t✱ t❤❡ ✐♠♣r♦✈❡♠❡♥t ♦❢ ❤✐❣❤❡r ❡❞✉❝❛t✐♦♥✳
❙❯▼➪❘■❖
✶ ◆ú♠❡r♦s ❘❡❛✐s ✶✹
✶✳✶ ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✶✳✶ ❘❡♣r❡s❡♥t❛çã♦ ●❡♦♠étr✐❝❛ ❞♦s ❘❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷ ◆ú♠❡r♦s ■rr❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✷✳✶ ❙❡❣♠❡♥t♦s ❈♦♠❡♥s✉rá✈❡✐s ❡ ■♥❝♦♠❡♥s✉rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸ ❆ ❘❡t❛ ❘❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✸✳✶ ◆ú♠❡r♦s ❘❡❛✐s ♥❛ ❋♦r♠❛ ❉❡❝✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷ ■❞❡✐❛ ■♥t✉✐t✐✈❛ ❞❡ ▲✐♠✐t❡ ✷✽
✷✳✶ ❘❡t❛ t❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✳✶ ❊q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡✿ ♦ ❞❡❝❧✐✈❡ ❝♦♠♦ ❧✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ▲✐♠✐t❡ ❡ ❈♦♥t✐♥✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✶ ▲✐♠✐t❡s ❧❛t❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦s ▲✐♠✐t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸ ❆ ❉❡r✐✈❛❞❛ ✹✵
✸✳✶ ❈♦♥❝❡✐t♦ ❞❡ ❉❡r✐✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✷ ❘❡❣r❛s ❞❡ ❉❡r✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✷✳✶ ❊♠ q✉❡ ❝♦♥s✐st❡♠ ❛s ❘❡❣r❛s ❞❡ ❉❡r✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✸ ❱❛❧♦r❡s ❊①tr❡♠♦s ❞❡ ✉♠❛ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✹ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❛ ❉❡r✐✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✹✳✶ ❋ís✐❝❛ ✲ ❈✐♥❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✹✳✷ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✹✳✸ Pr♦❜❧❡♠❛s ❞❡ ❖t✐♠✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✹✳✹ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✹ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥ ✻✹
✺ ❆ ■♥t❡❣r❛❧ ✻✾
✺✳✶ Pr✐♠✐t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✺✳✷ ❈♦♥❝❡✐t♦ ❞❡ ■♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✺✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❛ ■♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✺✳✹ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✺✳✺ ■♥t❡❣r❛❧ ❞❡✜♥✐❞❛ ❡ ■♥t❡❣r❛❧ ✐♥❞❡✜♥✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✺✳✺✳✶ ❯s❛♥❞♦ ♣r✐♠✐t✐✈❛s ♣❛r❛ ❝❛❧❝✉❧❛r ✐♥t❡❣r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
✻ ❯t✐❧✐③❛çã♦ ❞♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ♥♦ ❊♥s✐♥♦ ❞♦ ❈á❧❝✉❧♦ ✼✾ ✻✳✶ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ❆t✐✈✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✻✳✶✳✶ ❆✉❧❛ ■ ✲ ❆♣r❡s❡♥t❛çã♦ ❞♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✻✳✷ ❆✉❧❛ ■■ ✲ ❈á❧❝✉❧♦ ❞♦ ❧✐♠✐t❡ ❞❡ ❢✉♥çõ❡s ❝♦♠ ♦ ✉s♦ ❞♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✻✳✸ ❆✉❧❛ ■■■ ✲ ❈♦♥str✉✐r ❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ♣❛rt✐r ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛✱ ✉t✐❧✐③❛♥❞♦ ♦
s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✻✳✹ ❆✉❧❛ ■❱ ✲ ❈♦♥❝❡✐t♦ ❞❡ ■♥t❡❣r❛❧ ❉❡✜♥✐❞❛✱ ✉t✐❧✐③❛♥❞♦ ♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✽✻
✼ Pr♦❥❡t♦ ❞❡ ❊①t❡♥sã♦ ✽✼
✽ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✾✸
❘❡❢❡rê♥❝✐❛s ✾✺
❆ ❚❡st❡s ❞❡ ❱❡r✐✜❝❛çã♦ ✾✻
❆✳✶ ❚❡st❡ ❞❡ ❱❡r✐✜❝❛çã♦✿ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ❆✳✷ ❚❡st❡ ❞❡ ❱❡r✐✜❝❛çã♦✿ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ❆✳✸ ❚❡st❡ ❞❡ ❱❡r✐✜❝❛çã♦✿ ❚r✐❣♦♥♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ❇ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❈♦♥❝❧✉sõ❡s ❞❛s ❆t✐✈✐❞❛❞❡s ❞♦ ❈❛♣ít✉❧♦ ✼ ✶✵✸
■♥tr♦❞✉çã♦
❍✐st♦r✐❝❛♠❡♥t❡ ❛ ❞✐s❝✐♣❧✐♥❛ ▼❛t❡♠át✐❝❛ t❡♠ s✉❛s ♠✐♥ú❝✐❛s ♣❛✉t❛❞❛s ♥❛s ❞✐✜❝✉❧❞❛❞❡s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠✳ ❙♦❜r❡ ❡❧❛ ❡♠❡r❣❡♠ ❝♦♠♦ ♣r✐♥❝í♣✐♦ ❞❡ q✉❡ ♥❡♠ t♦❞♦ s❡r ❤✉♠❛♥♦ ♣♦ss✉✐ ❤❛❜✐❧✐❞❛✲ ❞❡s ✈♦❧t❛❞❛s ❛♦ ❞♦♠í♥✐♦ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s✳ ❊♥tr❡t❛♥t♦✱ ♦ ❡s♣❛ç♦ ❡s❝♦❧❛r✱ ♣♦ss✉✐ ♣r♦♣♦rçõ❡s q✉❡ ❛♠♣❧✐❛♠ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❛✉♠❡♥t❛r ♦ ♥í✈❡❧ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❝✐❡♥tí✜❝♦s ❞♦s s✉❥❡✐t♦s✳ P♦ré♠✱ ♦ ❜❛✐①♦ í♥❞✐❝❡ ❞❡ r❡♥❞✐♠❡♥t♦ ♥❛ ❣r❛❞✉❛çã♦ ❞❡ ▼❛t❡♠át✐❝❛ t❡♠ s❡ t♦r♥❛❞♦ ✉♠❛ r♦t✐♥❛✱ ❣❡r❛♥❞♦ ✉♠❛ ❝✉❧t✉r❛ ♥♦ ♠❡✐♦ ❛❝❛❞ê♠✐❝♦ ❡ ♣r♦✈♦❝❛♥❞♦ ♦ ✐♥s✉❝❡ss♦ ❞❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s ✐♥❣r❡ss❛♥t❡s ♥♦s ❝✉rs♦s ❞❡ ❝✐ê♥❝✐❛s ❡①❛t❛s ❡ ♥❛t✉r❛✐s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧✳ ❊ss❛ ❢r❛❣✐❧✐❞❛❞❡✱ ♣♦r s✉❛ ✈❡③ ❛❝❛❜❛ ❞❡s❡♥❝❛❞❡❛♥❞♦ ♦ ✐♥s✉❝❡ss♦ ❞♦ ❡♥s✐♥♦ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✳
❙❡❣✉♥❞♦ ❛s ❉✐r❡tr✐③❡s ❈✉rr✐❝✉❧❛r❡s ♣❛r❛ ♦ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛❀ ♣r♦♣♦st❛ ♣❡❧❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭✷✵✶✺✮ é ♥❡❝❡ssár✐♦ r❛t✐✜❝❛r q✉❡ é ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ❛ ❝r✐❛çã♦ ❞❡ ✉♠ ❝✉rrí❝✉❧♦ ❞❡ ❜❛s❡ ❝♦♠✉♠✳ ❆ss✐♠✱ ❛ ❞❡♠❛♥❞❛ ❞♦ ❡♥s✐♥❛r ❡ ❞♦ ❛♣r❡♥❞❡r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❣❛♥❤❛rã♦ s✉♣♦rt❡ q✉❛♥t♦ ❛♦ ❛❧✐♥❤❛♠❡♥t♦ ❞♦s ❝♦♥t❡ú❞♦s ❡s❝♦❧❛r❡s✳ ❈♦♥❢♦r♠❡ ❞❛❞♦s ❞♦ ▼❊❈ ✭✷✵✶✺✮ ♥♦ ❞✐❛ ✶✻✴✵✾✴✷✵✶✺ ❢♦r❛♠ ❛❜❡rt♦s ♦s ❞❡❜❛t❡s ❛❝❡r❝❛ ❞❛ ❇❛s❡ ◆❛❝✐♦♥❛❧ ❈♦♠✉♠ ❈✉rr✐❝✉❧❛r✱ tr❛t❛♠ ❞❡ ✉♠❛ ❛❞❡♥tr♦ r❡❧❡✈❛♥t❡ ♣❛r❛ q✉❡ r❡❛❧♠❡♥t❡ ❛ ❞✐s❝✐♣❧✐♥❛ ▼❛t❡♠át✐❝❛ ❣❛♥❤❡ s✉❜stâ♥❝✐❛ ♥♦ ♠❡✐♦ ❛❝❛❞ê♠✐❝♦✱ ♦✉ s❡❥❛✱ q✉❡ ♦s ❝✉rs♦s ❞❡ ❣r❛❞✉❛çã♦ q✉❡ ❢♦r♠❛♠ ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❢♦rt❛❧❡ç❛♠ s✉❛s ❜❛s❡s✱ ❜❡♠ ❝♦♠♦✱ ❛ ♣rát✐❝❛ ❞♦❝❡♥t❡ ♥❛s ❡s❝♦❧❛s ❞❡ ❢♦r♠❛ ❛t❡♥❞❡r ❛s ♥❡❝❡ss✐❞❛❞❡s ❞♦ ❛❧✉♥♦ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❞♦ ♣ró♣r✐♦ ♣r♦❢❡ss♦r✳
◆ã♦ é ♥♦✈✐❞❛❞❡ ♣❛r❛ ♥✐♥❣✉é♠ q✉❡✱ ♥♦ ❡s♣❛ç♦ ❡s❝♦❧❛r✱ ✈❡♠ ✢✉✐♥❞♦ ✉♠❛ ✐♥s❛t✐s❢❛çã♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ♥í✈❡❧ ❞❡ ❛♣r♦✈❡✐t❛♠❡♥t♦ ❞♦s ❛❧✉♥♦s ❝♦♠ r❡❧❛çã♦ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ s❡✉s ❝♦♥t❡ú❞♦s✳ ❊ss❡ ❛s♣❡❝t♦ ✈❡♠ s❡ t♦r♥❛♥❞♦ ✉♠ ❢❛t♦ ♥❛t✉r❛❧✱ ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ ♦s ♣ró♣r✐♦s ❝♦♠♣♦♥❡♥t❡s ❞♦ s✐st❡♠❛ ❞❡ ❡♥s✐♥♦✿ ❝♦♦r❞❡♥❛❞♦r❡s✱ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s✱ ❛❝❛❜❛♠ ♣♦r ♠✐♥✐♠✐③❛r ♦s ❢❛t♦r❡s q✉❡ ❛❝❛rr❡t❛♠ ❡st❡ ♣r♦❜❧❡♠❛✱ ♦ q✉❡ ❣❡r❛ ✉♠❛ ❜❛♥❛❧✐③❛çã♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠✱ ❣❡r❛♥❞♦ ❡✈❛sõ❡s✱ r❡t❡♥çõ❡s ❡ ❝r✐❛♥❞♦ ✉♠❛ ❝✉❧t✉r❛ ❤♦st✐❧ ❡♠ q✉❡ ♦s ú♥✐❝♦s ♣✉♥✐❞♦s sã♦ ♦s ❛❧✉♥♦s✳ ❊ss❛ ❝✉❧t✉r❛ ❡st❛❜❡❧❡❝✐❞❛ ❞❡✐①❛r❛♠ s❡q✉❡❧❛s ❡♠ ❣❡r❛çõ❡s ❞❡ ❥♦✈❡♥s✱ ❤❛❥❛ ✈✐st❛ q✉❡ ♦s ❛❧✉♥♦s sã♦ ✐♠♣❡❞✐❞♦s ❞❡ ❛❣r❡❣❛r ❛♦s s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♦ ✈❛❧♦r ❞❛ ▼❛t❡♠át✐❝❛ ❡♥q✉❛♥t♦ ár❡❛ ❢✉♥❞❛♠❡♥t❛❧ q✉❡ t❡♠ ✐♥✢✉ê♥❝✐❛ ❞✐r❡t❛ ♥♦ s❡✉ s❡r ❡ ❢❛③❡r s♦❝✐❛❧✳ ▼❛✐s ❣r❛✈❡ ❛✐♥❞❛✱ ♦ ✐♥s✉❝❡ss♦ ♥❡ss❛ ❞✐s❝✐♣❧✐♥❛✱ ❛♦ ❧♦♥❣♦ ❞❡ ❝❡♥t❡♥❛s ❞❡ ❛♥♦s✱ ❢❛③ ♣❛r❡❝❡r q✉❡ ❛s r❡♣r♦✈❛çõ❡s sã♦ ♥♦r♠❛✐s✳ ◆♦t❛❞❛♠❡♥t❡✱ ♦s ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛✱ ❞❡✐①❛♠ ❡♠ ❡✈✐❞ê♥❝✐❛ q✉❡ ❛ ❢♦r♠❛çã♦ ❛❞q✉✐r✐❞❛ ♥♦ ♠❡✐♦ ❛❝❛❞ê♠✐❝♦ ♥ã♦ ♦s ♣r❡♣❛r♦✉ ♥♦ ❣r❛✉ ❞❡ ❡①❝❡❧ê♥❝✐❛ ♣❡❞❛❣ó❣✐❝❛✱ ♣♦✐s ♥ã♦ ❛ss❡❣✉r♦✉✲❧❤❡ ♠❡t♦❞♦❧♦❣✐❛s q✉❡ ❡♥✈♦❧✈❛♠ ♦s ❛❧✉♥♦s ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥t❡♥❞❡r✱ ❣♦st❛r ❡ ♣❡sq✉✐s❛r s♦❜r❡ ❛ ✈❛❧✐❞❛❞❡ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s✳
❖s ♣r♦❜❧❡♠❛s ❡♥❢r❡♥t❛❞♦s ♥❛ ❞✐s❝✐♣❧✐♥❛ ♣❡❧♦s ❛❧✉♥♦s tê♠ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❛s ❧❛❝✉♥❛s
❞❡✐①❛❞❛s t❛♠❜é♠ ♥❛ ❢♦r♠❛çã♦ ❞♦ ♣r♦❢❡ss♦r q✉❛♥❞♦ ❡st❡ s❡ ✈ê t❛♠❜é♠ ❝♦♠ ❞✐✜❝✉❧❞❛❞❡s ❡♠ ❛ss✐♠✐❧❛r ❝♦♠ s❡❣✉r❛♥ç❛ ♦s ❝♦♥❤❡❝✐♠❡♥t♦s s♦❜r❡ ❈á❧❝✉❧♦✳ P❛rt✐❝✉❧❛r♠❡♥t❡ ❡ss❛ ❡s♣❡❝✐✜❝✐❞❛❞❡ ❞❛ ▼❛t❡♠át✐❝❛✱ ✈❡♠ ❝r✐❛♥❞♦ ✉♠ ♥ó ♠✉✐t♦ ❣r❛♥❞❡ ♥♦s í♥❞✐❝❡s ❞❡ r❡♣r♦✈❛çã♦ ❡s❝♦❧❛r ❞❡ss❛ r❡❢❡r✐❞❛ ❞✐s❝✐♣❧✐♥❛✱ ♣♦✐s ❛ ❢r❛❣✐❧✐❞❛❞❡ ❞❡ ✉♠ ❛♣r❡♥❞✐③❛❞♦ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♠é❞✐♦ ❛❝❛❜❛ r❡✢❡t✐♥❞♦ ♥♦ ✐♥s✉❝❡ss♦ ❡s❝♦❧❛r q✉❡ ♣♦r s✉❛ ✈❡③✱ ✐♥✢✉❡♥❝✐❛ ♥❛ ❢♦r♠❛çã♦ ❞♦ ♣r♦❢❡ss♦r ❞❛ ❞✐s❝✐♣❧✐♥❛✱ ❥á q✉❡ t❛♠❜é♠ ❡❧❡ ♥ã♦ ♣♦ss✉✐ ✉♠❛ ❢♦r♠❛çã♦ ✐♥✐❝✐❛❧ só❧✐❞❛✳ ❙❡❣✉♥❞♦ ❡st✉❞♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣❡❧♦ ■♥st✐t✉t♦ ◆❛❝✐♦♥❛❧ ❞❡ ❊st✉❞♦s ❡ P❡sq✉✐s❛s ❊❞✉❝❛❝✐♦♥❛✐s ✭■◆❊P✮✱ ♦ ❛❧✉♥♦ ❛♦ ✐♥❣r❡ss❛r ♥♦ ❊♥s✐♥♦ ❙✉♣❡r✐♦r✱ ❞❡❢r♦♥t❛✲s❡ ❝♦♠ ❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ❞✐s❝✐♣❧✐♥❛ ❡st❛ q✉❡ ❢❛③ ♣❛rt❡ ❞♦ ❝✉rrí❝✉❧♦ ❞♦ ❝✉rs♦✳ ❆♦ s❡ ❞❡♣❛r❛r❡♠ ❝♦♠ ❡❧❛ ♣❛ss❛♠ ❛ ✈❡r✐✜❝❛r ♦ ❣r❛✉ ❞❡ ❞✐✜❝✉❧❞❛❞❡ ♣❛r❛ ❡♥t❡♥❞ê✲❧❛✳ ❖ ✐♠♣❛❝t♦ é tã♦ ❣r❛♥❞❡ q✉❡ ♠✉✐t♦s ❞❡s✐st❡♠ ❞♦ ❝✉rs♦✱ r❡✢❡①♦ ❞❡ ✉♠ ❛♣r❡♥❞✐③❛❞♦ ❢r❛❣✐❧✐③❛❞♦ ❞❡ss❡s ❡st✉❞❛♥t❡s ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❡ ▼é❞✐♦✳
◆♦ â♠❜✐t♦ ❞♦s ❞❡❜❛t❡s ❛❧❣✉♥s ♣❡sq✉✐s❛❞♦r❡s ❡♠ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ t❡♠ ❛♠♣❧✐❛❞♦ ♦ t❡♦r ❞❛s ❞✐s❝✉ssõ❡s ❛❝❡r❝❛ ❞♦ ❡♥s✐♥♦ ❞♦ ❈á❧❝✉❧♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❞❡✐①❛♥❞♦ ❡♠ ❡✈✐❞ê♥❝✐❛ q✉❡ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ r❛t✐✜❝❛r ❛ r❡❧❡✈â♥❝✐❛ ❞❛ r❡❢❡r✐❞❛ ❞✐s❝✐♣❧✐♥❛ ❡♥q✉❛♥t♦ ✉♠❛ ❝✐ê♥❝✐❛✳ ❙♦❜r❡ ❡ss❡ ❛s♣❡❝t♦✱ ❘❡③❡♥❞❡ ✭✷✵✵✸✮✱ ❢❛③ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝♦❧♦❝❛çã♦ ❛ ❡ss❡ r❡s♣❡✐t♦✱ ❛✜r♠❛♥❞♦ q✉❡✿
❊♥❣❛♥❛✲s❡ q✉❡♠ ♣❡♥s❛ q✉❡ t❛❧ ♣r♦❜❧❡♠❛ é ❝✉❧t✉r❛❧ ❡ q✉❡ s❡ ❥✉st✐✜❝❛ ♣❡❧❛ ❝♦♥❞✐çã♦ só❝✐♦✲ ❡❝♦♥ô♠✐❝❛ ❞❛ s♦✲ ❝✐❡❞❛❞❡ ❜r❛s✐❧❡✐r❛✳ ❆ s✐t✉❛çã♦ ❞♦ ❡♥s✐♥♦ ❞❡ ❈á❧❝✉❧♦ ♥♦s ♣❛ís❡s ✏❞❡s❡♥✈♦❧✈✐❞♦s✑ ♥ã♦ é ♠✉✐t♦ ❞✐❢❡r❡♥t❡✱ ✈✐st♦ q✉❡ tr❛❜❛❧❤♦s s♦❜r❡ ❡ss❡ t❡♠❛ t❡♠ s✐❞♦ ♣✉❜❧✐❝❛❞♦ ❡ r❡❝❡❜✐❞♦ ♠❡r❡❝✐❞♦ ❞❡st❛q✉❡ ♣♦r ♣❛rt❡ ❞❛ ❧✐t❡r❛t✉r❛ ❡s♣❡❝✐❛❧✐③❛❞❛ ✐♥t❡r♥❛❝✐♦♥❛❧✳
❆ ❛♠♣❧✐t✉❞❡ ❞❡ q✉❡ ♦ ❝✉rrí❝✉❧♦ é ✉♠ ✐♥str✉♠❡♥t♦ ✈❛❧✐♦s♦ ♥♦ â♠❜✐t♦ ❡s❝♦❧❛r✱ ❝♦❧♦❝❛ ❡♠ ♣❛✉t❛ q✉❡ ♦ ❡♥s✐♥♦ ❡ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞♦ ❝á❧❝✉❧♦ ❞❡✈❡♠ s❡r ✉♠ ♣♦♥t♦ ❞❡ ❛♥á❧✐s❡ ❞♦ ❝✉rrí❝✉❧♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡ ❞❡✈❡ s❡r ❡str✉t✉r❛❞♦ ❞❡ ♠♦❞♦ ❛ ❛ss❡❣✉r❛r ❛♦ ❛❧✉♥♦ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❛♠♣❧✐❛r ❡ ❛♣r♦❢✉♥❞❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❛❞q✉✐r✐❞♦s ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳ ◗✉❛♥t♦ ❛ ❡ss❡ ❢♦❝♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭✷✵✵✷✮✱ t❛♠❜é♠ ❡♥❢❛t✐③❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ t♦r♥❛r ❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠❛ ❝✐ê♥❝✐❛ q✉❡ ❢❛③ ♣❛rt❡ ❞❛ ❤✐stór✐❛ ❞❛ ❤✉♠❛♥✐❞❛❞❡ ❡ q✉❡ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♣r♦❞✉③ ♥❛ s♦❝✐❡❞❛❞❡ ♣❡rs♣❡❝t✐✈❛s ❞❡ ❢♦r♠❛çã♦ ❝✐❞❛❞ã✳
❉❛❞♦s ❞❛s ❛✈❛❧✐❛çõ❡s ✐♥st✐t✉❝✐♦♥❛✐s ❝♦♠♦ ♦ ❙❆❊❇ ✭❙✐st❡♠❛ ◆❛❝✐♦♥❛❧ ❞❡ ❆✈❛❧✐❛çã♦ ❊s❝♦❧❛r ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✮ ❡ ♦ ❊◆❊▼ ✭❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✮✱ ♣r♦♠♦✈✐❞♦s ♣❡❧♦ ●♦✈❡r♥♦ ❋❡❞❡r❛❧✱ ❛♣♦♥t❛♠ q✉❡ ❛ ❢r❛❣✐❧✐❞❛❞❡ q✉❛♥t♦ r❡♥❞✐♠❡♥t♦ ❞♦s ❛❧✉♥♦s ❛♦ ❝♦♥❝❧✉ír❡♠ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❛ ❞✐✜❝✉❧❞❛❞❡s ❡♠ ❝♦♥❝❡✐t♦s ❡ ♣r♦❝❡❞✐♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s✱ t❛✐s ❝♦♠♦ ♦♣❡r❛r ❝♦♠ ♥ú♠❡r♦s r❡❛✐s✱ ✐♥t❡r♣r❡t❛r ❣rá✜❝♦s ❡ t❛❜❡❧❛s✱ ❞❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✳
❈♦♠ ✐♥t✉✐t♦ ❞❡ ❛♠♣❧✐❛r ♦ ♥í✈❡❧ ❞❡ ❞❡❜❛t❡ ❛❝❡r❝❛ ❞♦ ❢♦❝♦ ❞❡ ✐♥✈❡st✐❣❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ♦❜t✐✈❡ ❝♦♠♦ ♦❜❥❡t✐✈♦✿ ❛♥❛❧✐s❛r ♦ ❝♦♥t❡①t♦ ❢♦r♠❛t✐✈♦ ❞♦s ♣r♦❢❡ss♦r❡s q✉❡ ❛t✉❛♠ ♥♦ ❝❛♠♣♦ ❞❛ ▼❛✲ t❡♠át✐❝❛ ✐♥❝❧✉✐♥❞♦ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ r❡❧❡✈â♥❝✐❛ ❞❡ ❢✉♥❞❛♠❡♥t❛r ❛s ❞✐s❝✉ssõ❡s✱ ❢♦✐ ❧❡✈❛♥t❛❞❛ ✉♠❛ q✉❡stã♦ ♥♦rt❡❛❞♦r❛✱ é ❡❧❛✿ ❛ ❢♦r♠❛çã♦ ❞♦s ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ♥♦s ❝✉rs♦s ❞❡ ♥í✈❡❧ s✉♣❡r✐♦r ❛❢❡t❛ t❛♠❜é♠ ♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❞❡s❡♥❝❛❞❡❛♥❞♦ ❞✐✜❝✉❧❞❛❞❡ ♥♦ ❛♣r❡♥❞✐③❛❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❧✐♠✐t❡ ❡ ❝♦♥t✐♥✉✐❞❛❞❡✱
♦ ♣r♦❜❧❡♠❛ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✉r✈❛✱ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡❧✐♠✐t❛❞❛s ♣❡❧♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦❄
❈♦♠♦ ❛çã♦ ♠❡t♦❞♦❧ó❣✐❝❛✱ ♦♣t❡✐ ♣♦r r❡❛❧✐③❛r ✉♠❛ ♣❡sq✉✐s❛ ❜✐❜❧✐♦❣rá✜❝❛✱ ❛ q✉❛❧ ♥♦s ♣♦ss✐✲ ❜✐❧✐t♦✉ ✐♥t❡♥s✐✜❝❛r ❛s ❛r❣✉♠❡♥t❛çõ❡s ❛❝❡r❝❛ ❞♦ ✐♥t❡r❡ss❡ ❛❝❛❞ê♠✐❝♦✳ P❛r❛ t❛♥t♦✱ ❛ ❛♥❛❧✐s❡ t❡ór✐❝❛ s❡ ❞❡✉ ❝♦♠ ❧❡✐t✉r❛s ❝♦♥tí♥✉❛s ❡ ❛tr❡❧❛❞❛s ❛s ♦❜s❡r✈❛çõ❡s ❡♠♣ír✐❝❛s ♥♦ ❝❛♠♣♦ ❞❛ ❡①♣❡r✐ê♥❝✐❛ ❞❛ q✉❛❧ ❥á ❡st♦✉ ✐♥s❡r✐❞♦✱ ❝♦♠♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✳ ❙❡❣✉♥❞♦ ▲✉❞❦ ❡ ❆♥❞r❡ ✭✶✾✾✾✮ ♦ ❡st✉❞♦ ❜✐✲ ❜❧✐♦❣rá✜❝♦ ♣♦ss✉✐ ✉♠❛ r✐q✉❡③❛ ♥♦ ❝❛♠♣♦ ❞❛ ♣❡sq✉✐s❛✱ ♣♦✐s ♣♦ss✐❜✐❧✐t❛ ❛♦ ♣❡sq✉✐s❛❞♦r ❛♣r♦❢✉♥❞❛r s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ❛✐♥❞❛ ❛✈❛❧✐❛r s✉❛ ♣♦st✉r❛ ✐♥✈❡st✐❣❛t✐✈❛ ♥❛ ár❡❛ ❛❝❛❞ê♠✐❝❛✳
❖ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❡♠ ✼ ❝❛♣ít✉❧♦s ❡ ❝♦♥❝❧✉sã♦✱ ❛❧é♠ ❞❛s ❝♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ é tr❛t❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ◆ú♠❡r♦s r❡❛✐s✱ ♥♦ s❡❣✉♥❞♦ sã♦ ❡♥❢❛t✐③❛❞❛s ❛ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛ ❞❡ ❧✐♠✐t❡ ❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ é ❢❡✐t♦ ♦ ❡st✉❞♦ ❞❛ ❞❡r✐✈❛❞❛✱ ♦ q✉❛rt♦ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛❞♦ ♦ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ❡st✐♠❛r ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❡ ♥ã♦ é q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✱ ♥♦ q✉✐♥t♦ ❝❛♣ít✉❧♦ é tr❛t❛❞♦ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛ ❝♦♠ ♦ ✉s♦ ❞❛ ✐♥t❡❣r❛❧✱ ♦ s❡①t♦ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ à ✉t✐❧✐③❛çã♦ ❞♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ♥♦ ❊♥s✐♥♦ ❞♦ ❈á❧❝✉❧♦✱ ♥♦ sét✐♠♦ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ❡s❜♦ç♦ ❞❡ ✉♠ ♣r♦❥❡t♦ ❞❡ ❡①t❡♥sã♦ ❞❡❞✐❝❛❞♦ ❛♦ ❡♥s✐♥♦ ❞♦ ❈á❧❝✉❧♦ ♣❛r❛ ❛❧✉♥♦s q✉❡ t❡♥❤❛♠ ♦ ✐♥t❡r❡ss❡ ❡♠ ❢❛③❡r ✉♠ ❝✉rs♦ s✉♣❡r✐♦r ♥❛ ár❡❛ ❞❡ ❡①❛t❛s ❡ ♥❛t✉r❛✐s✳ ❊ ♥❛ ❝♦♥❝❧✉sã♦ ♠❡✉s ❛r❣✉♠❡♥t♦s s✉r❣❡♠ ❝♦♠♦ ❢♦❝♦ ❞❡ ❛♥á❧✐s❡✱ ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ tr❛t♦ ❞❛ ❢r❛❣✐❧✐❞❛❞❡ ❞♦ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ t❡♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❛ ❢♦r♠❛çã♦ ❞♦❝❡♥t❡ ♥❛ ár❡❛✳ ➱ ❡①tr❡♠❛♠❡♥t❡ ✐♠♣♦rt❛♥t❡ ❝♦♥t❡①t✉❛❧✐③❛r ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ❡①♣❡r✐ê♥❝✐❛s ❡s❝♦❧❛r❡s ❡♥✈♦❧✈❡♥❞♦ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ♥♦ ❡①❡r❝í❝✐♦ ❞❛ ❝✐❞❛❞❛♥✐❛ ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ ♣r❡❝✐s❛♠ ✉s❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❝♦♠♦ ♣❛rt❡ ✐♥t❡❣r❛♥t❡ ❞❛ ✈✐❞❛ ❡♠ s♦❝✐❡❞❛❞❡✳ ❉❡ss❡ ♠♦❞♦✱ ♦ r❡❢❡r✐❞♦ tr❛❜❛❧❤♦ ❣❛♥❤❛ s✉❜stâ♥❝✐❛ t❡ór✐❝❛ ❛♦ s❡ t♦r♥❛r ✉♠ ✐♥str✉♠❡♥t♦ ❞❡ ❛♥á❧✐s❡ ❡ r❡✢❡①ã♦ ♥♦ ♠❡✐♦ ❛❝❛❞ê♠✐❝♦✳
❈❛♣ít✉❧♦ ✶
◆ú♠❡r♦s ❘❡❛✐s
◆❡st❡ ❝❛♣ít✉❧♦✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛s♣❡❝t♦s r❡❧❛❝✐♦♥❛❞♦s ❛♦ ❡♥s✐♥♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳ ❚❡♠❛ ✐♠♣♦rt❛♥t❡✱ ♠❛s q✉❡ ♥♦r♠❛❧♠❡♥t❡ é tr❛t❛❞♦ s❡♠ ê♥❢❛s❡ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ❛ ❡✈♦❧✉çã♦ ❤✐stór✐❝❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s s❡ ❞❡✉ ❞❡s❞❡ ❛ ✏❞❡s❝♦❜❡rt❛✑ ❞♦s s❡❣♠❡♥t♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s ❛té ❛ ❝♦♥str✉çã♦ ❛①✐♦♠át✐❝❛ r❡❛❧✐③❛❞❛ ♣♦r ❉❡❞❡❦✐♥❞ ♥♦ sé❝✉❧♦ ❳■❳✱ ❛ ♣❛rt✐❝✐♣❛çã♦ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ✐❞❡✐❛s ❞♦ ❈á❧❝✉❧♦ ❡ ❞❛ ❆♥á❧✐s❡ ❢♦r❛♠ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ q✉❡ ❛ ❝♦♥str✉çã♦ ❞❡✜♥✐t✐✈❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s s❡ ❡❢❡t✐✈❛ss❡ ✭▲■▼❆✱ ✷✵✶✸✮✳ ❉✐❛♥t❡ ❞✐ss♦ s✉r❣❡ ❛ ♣❡r❣✉♥t❛✿ ❝♦♠♦ ❡♥s✐♥❛r ♥ú♠❡r♦s r❡❛✐s ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ ❞✐s♣♦♠♦s ❞❛s ❢❡rr❛♠❡♥t❛s ❞♦ ❈á❧❝✉❧♦ ❡ ❞❛ ❆♥á❧✐s❡❄
❖s P❈◆✬s ✭✷✵✵✷✮ ❥á ❛♣♦♥t❛♠ ❡ss❛s ❞✐✜❝✉❧❞❛❞❡s✳ ◆♦ ❞♦❝✉♠❡♥t♦ ❤á ♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ t❡♠ s❡ ❧✐♠✐t❛❞♦ q✉❛s❡ q✉❡ ❡①❝❧✉s✐✈❛♠❡♥t❡ ❛♦ ❡♥s✐♥♦ ❞♦ ❝á❧❝✉❧♦ ❝♦♠ r❛❞✐❝❛✐s ❡ q✉❡ ♦ tr❛❜❛❧❤♦ ❝♦♠ ❡st❡s ♥ú♠❡r♦s ♣♦✉❝♦ t❡♠ ❝♦♥tr✐❜✉í❞♦ ♣❛r❛ q✉❡ ♦s ❛❧✉♥♦s t❡♥❤❛♠ ✉♠ ❡♥t❡♥❞✐♠❡♥t♦ ❝♦rr❡t♦ ❞❡ s❡✉ ❝♦♥❝❡✐t♦✳ ❖s P❈◆✬❙ s✉❣❡r❡♠ ❛❧❣✉♠❛s ❛çõ❡s r❡❧❛❝✐♦♥❛❞❛s ❛♦ ❡st✉❞♦ ❞♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s ❛ s❡r❡♠ ❞❡s❡♥✈♦❧✈✐❞❛s ♥♦8◦❛♥♦ ❞♦ ❊♥s✐♥♦
❋✉♥❞❛♠❡♥t❛❧ ❡ ♥♦1◦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❙ã♦ ❡❧❛s✿
• ▼♦str❛r q✉❡ ❛ ❤✐stór✐❛ ❞♦s ♥ú♠❡r♦s ❡stá ❧✐❣❛❞❛ à ❞❛s ♥❡❝❡ss✐❞❛❞❡s ❡ ♣r❡♦❝✉♣❛çõ❡s ❞❡ ♣♦✈♦s
q✉❡✱ ❛♦ ❜✉s❝❛r r❡❝❡♥s❡❛r s❡✉s ♠❡♠❜r♦s✱ s❡✉s ❜❡♥s✱ s✉❛s ♣❡r❞❛s✱ ❛♦ ♣r♦❝✉r❛r ❞❛t❛r ❛ ❢✉♥çã♦ ❞❡ s✉❛s ❝✐❞❛❞❡s ❡ s✉❛s ✈✐tór✐❛s✱ ✉s❛♥❞♦ ♦s ♠❡✐♦s ❞✐s♣♦♥í✈❡✐s✱ ❝♦♥str✉ír❛♠ ✐♥t❡r❡ss❛♥t❡s s✐st❡♠❛s ❞❡ ♥✉♠❡r❛çã♦✳
• ❆♦ ❜✉s❝❛r ❛s ♦r✐❡♥t❛çõ❡s ♣❛r❛ tr❛❜❛❧❤❛r ❝♦♠ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❞❡✈❡✲s❡ t❡r ♣r❡s❡♥t❡ q✉❡ ❛s
❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s ♥ã♦ ♣♦❞❡♠ s❡ ❧✐♠✐t❛r às q✉❡ s❡ ❛♣♦✐❛♠ ❛♣❡♥❛s ❡♠ s✐t✉❛çõ❡s ❝♦♥❝r❡t❛s✱ ♣♦✐s ♥❡♠ s❡♠♣r❡ ❡ss❛s ❝♦♥❝r❡t✐③❛çõ❡s ❡①♣❧✐❝❛♠ ♦s s✐❣♥✐✜❝❛❞♦s ❞❛s ♥♦çõ❡s ❡♥✈♦❧✈✐❞❛s✳ ➱ ♣r❡❝✐s♦ ✐r ✉♠ ♣♦✉❝♦ ❛❧é♠ ❡ ♣♦ss✐❜✐❧✐t❛r✱ ♣❡❧❛ ❡①t❡♥sã♦ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❥á ❝♦♥str✉í❞♦s ♣❛r❛ ♦s ♥❛t✉r❛✐s✱ ❝♦♠♣r❡❡♥❞❡r ❡ ❥✉st✐✜❝❛r ❛❧❣✉♠❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳
• P❛r❛ ❛❜♦r❞❛r ♦ ❡st✉❞♦ ❞♦s r❛❝✐♦♥❛✐s✱ ❞❡✈❡♠♦s ❧❡✈❛r ♦ ❛❧✉♥♦ ❛ ♣❡r❝❡❜❡r q✉❡ ♦s ♥ú♠❡r♦s
♥❛t✉r❛✐s ❡ ✐♥t❡✐r♦s sã♦ ✐♥s✉✜❝✐❡♥t❡s ♣❛r❛ r❡s♦❧✈❡r ❞❡t❡r♠✐♥❛❞❛s s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❝♦♠♦ ❛s q✉❡ ❡♥✈♦❧✈❡♠ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❡ ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ❞✐✈✐sã♦✳
• ❖ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ♣♦❞❡ s❡r ✐♥tr♦❞✉③✐❞♦ ♣♦r ♠❡✐♦ ❞❡ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛
♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ ❞❡ ❡♥❝♦♥tr❛r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦✱ t♦♠❛♥❞♦ ♦ ❧❛❞♦ ❝♦♠♦ ✉♥✐❞❛❞❡✱ q✉❡ ❝♦♥❞✉③ ❛♦ ♥ú♠❡r♦ √2✳ ◆❡ss❡ ❝❛s♦✱ ♣♦❞❡✲s❡ ✐♥❢♦r♠❛r ✭♦✉ ✐♥❞✐❝❛r ❛
♣r♦✈❛✮ ❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡ √2✱ ♣♦r ♥ã♦ s❡r ✉♠❛ r❛③ã♦ ❞❡ ✐♥t❡✐r♦s✱ ♦ ♣r♦❜❧❡♠❛ ❞❛s r❛í③❡s
q✉❛❞r❛❞❛s ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s q✉❡ ♥ã♦ sã♦ q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s✱ √3✱ √5✱ ❡t❝✳✱ ♣♦❞❡r✐❛
s❡❣✉✐r✲s❡ ❛♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡√2✳
❊♥tã♦✱ ❝❛❜❡ ❛ ✏♥ós ♣r♦❢❡ss♦r❡s✑ ♠✉❞❛r♠♦s ❡ss❛ r❡❛❧✐❞❛❞❡✱ ✐♥✈❡st✐♥❞♦ ❞❡ ♠♦❞♦ s✐❣♥✐✜❝❛t✐✈♦ ♥♦ ❡♥s✐♥♦ ❞♦ ❝♦♥t❡ú❞♦✿ ♥ú♠❡r♦s r❡❛✐s✳
✶✳✶ ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s
◆❡st❛ s❡çã♦ é ❢❡✐t♦ ✉♠ ❡st✉❞♦ ❜❛s❡❛❞♦ ♥♦s ❧✐✈r♦s ❞❡ ➪✈✐❧❛✱ ●✉✐❞♦r✐③③✐ ❡ ▲✐♠❛✱ s♦❜r❡ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ✐rr❛❝✐♦♥❛✐s ❡ ♥ú♠❡r♦s r❡❛✐s✱ ❞❛♥❞♦ ê♥❢❛s❡ ♥❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❡s ♥ú♠❡r♦s✱ ❛s q✉❛✐s r❡❝❡❜❡♠ ♣♦✉❝❛ ❛t❡♥çã♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❉❡✈❡✲s❡ tr❛❜❛❧❤❛r ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✏✈❛❧♦r ❛❜s♦❧✉t♦✑ ❡ ❛ r❡❧❛çã♦ ❞❡ ✏❞❡s✐❣✉❛❧❞❛❞❡✑✱ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❛ss✉♥t♦s ❡st❡s q✉❡ sã♦ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ ❈á❧❝✉❧♦✳ ❆♥t❡s ❞❡ ❢❛❧❛r♠♦s ❞✐r❡t❛♠❡♥t❡ ♥♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ✈❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ r❡❝♦r❞❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ✐♥t❡✐r♦s✳ ❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❡✈❡✉✲s❡ à ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❝♦♥t❛r ♦❜❥❡t♦s✳ ❖s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s✱ ❡♠ ❣❡r❛❧✱ s✉r❣✐r❛♠ t❛♠❜é♠ ♣♦r ♥❡❝❡ss✐❞❛❞❡✱ ❝♦♠♦ ❡①t❡♥sõ❡s ❞❛q✉❡❧❡s ❛té ❡♥tã♦ ❝♦♥❤❡❝✐❞♦s✳ ❈❤❛♠❛♠♦s ❞❡ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❛♦ ❣r✉♣♦ ❞❡ ♥ú♠❡r♦s✿ ✵✱ ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✽✱ ✾✱ ✶✵✱ ✶✶✱ ✶✷✱· · ·❀
s❡♥❞♦ ❡ss❡ ❝♦♥❥✉♥t♦ r❡♣r❡s❡♥t❛❞♦ ♣♦r N✳ ❖s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ sã♦ ♦s ♥ú♠❡r♦s✿· · ·❀ ✲✹✱ ✲✸✱ ✲✷✱ ✲✶✱ ✵✱ ✶✱ ✷✱ ✸✱ ✹✱· · ·✱ ❡ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦ sí♠❜♦❧♦ Z✳ ❙❡ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❛❝r❡s❝❡♥t❛r♠♦s ❛s ❢r❛çõ❡s✱ ♦❜t❡♠♦s ♦ ❝❤❛♠❛❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳ ❆ss✐♠✱ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ é t♦❞♦ ♥ú♠❡r♦ q✉❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♥❛ ❢♦r♠❛ p
q✱ ❡♠ q✉❡ p ❡ q sã♦ ✐♥t❡✐r♦s ❡ q é
❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ ✭p
0 é ✉♠❛ ✐♥❞❡t❡r♠✐♥❛çã♦✮✳
Q= {p
q |p, q ∈ Z, q 6= 0}.
❈♦♥✈é♠ ❞❡st❛❝❛r q✉❡N é s✉❜❝♦♥❥✉♥t♦ ❞❡ Z✱ q✉❡✱ ♣♦r s✉❛ ✈❡③✱ é s✉❜❝♦♥❥✉♥t♦ ❞❡Q❀ ✐st♦ é✱ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ é ♥ú♠❡r♦ ✐♥t❡✐r♦✱ ❡ t♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ é ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳
✵ ❂ ✵✱ ✶ ❂ ✰✶✱ ✷ ❂ ✰✷✱ ✸ ❂ ✰✸✱ ✹ ❂ ✰✹✱ ✺ ❂ ✰✺✱ ✻ ❂ ✰✻✱ ✼ ❂ ✰✼✱ ✽ ❂ ✰✽✱ ✾ ❂ ✰ ✾✱· · ·✳
✵ ❂ 0
1✱ ✶ ❂ 1 1✱ ✷ ❂
2 1✱ ✸ ❂
3 1✱ ✹ ❂
4 1✱ ✺ ❂
5 1✱ ✻ ❂
6 1✱ ✼ ❂
7 1✱ ✽ ❂
8 1✱ ✾ ❂
9 1✱· · ·✳
❙❡❥❛♠ a b ❡
c
d ❞♦✐s r❛❝✐♦♥❛✐s q✉❛✐sq✉❡r✳ ❆ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❡st❡s r❛❝✐♦♥❛✐s sã♦ ❞❡✜♥✐❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
a b +
c d =
❆ ♦♣❡r❛çã♦ q✉❡ ❛ ❝❛❞❛ ♣❛r ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❛ss♦❝✐❛ ❛ s✉❛ s♦♠❛ ❞❡♥♦♠✐♥❛✲s❡ ❛❞✐çã♦✱ ❡ ❛ q✉❡ ❛ss♦❝✐❛ ♦ ♣r♦❞✉t♦ ❞❡♥♦♠✐♥❛✲s❡ ♠✉❧t✐♣❧✐❝❛çã♦✳
❖ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ a
b s❡ ❞✐③ ♣♦s✐t✐✈♦ s❡a·b∈N/{0}❀ a·b /∈N/{0}✱ a
b s❡ ❞✐③ ♥❡❣❛t✐✈♦✳ ❊①❡♠♣❧♦ ✶✳✶
• −2 −3 =
+2 +3 =
2
3 é ♣♦s✐t✐✈♦✳
• −32 = 2
−3 =− 2
3 é ♥❡❣❛t✐✈♦✳
❙❡❥❛♠ r ❡ s ❞♦✐s r❛❝✐♦♥❛✐s❀ ❞✐③❡♠♦s q✉❡ r é ♠❡♥♦r q✉❡ s ✭♦✉ q✉❡ s é ♠❛✐♦r q✉❡ r✮ ❡ ❡s❝r❡✈❡♠♦sr < s ✭r❡s♣❡❝t✐✈❛♠❡♥t❡s > r✮ s❡ ❡①✐st❡ ✉♠ r❛❝✐♦♥❛❧ t ♣♦s✐t✐✈♦ t❛❧ q✉❡ s=r+t✳ ❆ ♥♦t❛çã♦r6s✭❧❡✐❛✿ r ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ s✮ é ✉s❛❞❛ ♣❛r❛ ✐♥❞✐❝❛r ❛ ❛✜r♠❛çã♦ ✏r < s ♦✉r =s✑✳ ❆ ♥♦t❛çã♦r >s✭❧❡✐❛✿ r ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ s✮ é ❡q✉✐✈❛❧❡♥t❡ ❛s6r✳ ❖❜s❡r✈❡ q✉❡ r ♣♦s✐t✐✈♦✱ ❡q✉✐✈❛❧❡ ❛r >0✳ ❙❡r <0✱ ❞✐③❡♠♦s q✉❡ r é ♥❡❣❛t✐✈♦✳
❆♥t❡s ❞❡ ❛✈❛♥ç❛r♠♦s ♥♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ é ♥❡❝❡ssár✐♦ ♥♦s ❢❛♠✐❧✐❛r✐③❛r♠♦s ❝♦♠ ♦s s❡❣✉✐♥t❡s sí♠❜♦❧♦s ✏⇒✑ ❡ ✏⇔✑ ✭✐♠♣❧✐❝❛ ❡ s❡ ❡ s♦♠❡♥t❡ s❡✮ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s t❡r ✉♠
♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♣♦st❡r✐♦r♠❡♥t❡✳ ❖s ❡♥✉♥❝✐❛❞♦s ❡ ❛s ❞❡♠♦♥str❛çõ❡s sã♦ ❢❡✐t♦s ✉s❛♥❞♦✲s❡ ❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✳ P♦r ✐ss♦✱ t♦r♥❛✲s❡ ✐♠♣♦rt❛♥t❡ ❝♦♥❤❡❝❡r ♦s s✐❣♥✐✜❝❛❞♦s ❡ ❝♦♠♦ ✉s❛r ♦s ✈ár✐♦s sí♠❜♦❧♦s ♠❛t❡♠át✐❝♦s✳
■♠♣❧✐❝❛çã♦ ❡ ❊q✉✐✈❛❧ê♥❝✐❛ ❉❡✜♥✐çã♦ ✶✳✶
❈❤❛♠❛✲s❡ ✏♣r♦♣♦s✐çã♦✑ ♦✉ ✏s❡♥t❡♥ç❛✑ t♦❞❛ ♦r❛çã♦ ❞❡❝❧❛r❛t✐✈❛ q✉❡ ♣♦❞❡ s❡r ❝❧❛ss✐✜❝❛❞❛ ❡♠ ✈❡r❞❛❞❡✐r❛ ♦✉ ❡♠ ❢❛❧s❛✳
❖❜s❡r✈❛✲s❡ q✉❡ t♦❞❛ ♣r♦♣♦s✐çã♦ ❛♣r❡s❡♥t❛ três ❝❛r❛❝t❡ríst✐❝❛s ♦❜r✐❣❛tór✐❛s✿ ✶❛✮ s❡♥❞♦ ♦r❛çã♦✱ t❡♠ s✉❥❡✐t♦ ❡ ♣r❡❞✐❝❛❞♦❀
✷❛✮ é ❞❡❝❧❛r❛t✐✈❛ ✭♥ã♦ é ❡①❝❧❛♠❛t✐✈❛ ❡ ♥❡♠ ✐♥t❡rr♦❣❛t✐✈❛✮❀
✸❛✮ t❡♠ ✉♠✱ ❡ s♦♠❡♥t❡ ✉♠✱ ❞♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s✿ ♦✉ é ✈❡r❞❛❞❡✐r❛ ✭❱✮ ♦✉ é ❢❛❧s❛ ✭❋✮✳
❊①❡♠♣❧♦ ✶✳✷ ❛✮ ✶✺ é ❞✐✈✐sí✈❡❧ ♣♦r ✸✳
❜✮ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ q✉❛❧q✉❡r tr✐â♥❣✉❧♦ é s❡♠♣r❡ ✶✽✵◦
✳ ❝✮ s❡t❡ é ♠❛✐♦r q✉❡ três✳
❆s três ♣r♦♣♦s✐çõ❡s ❛❝✐♠❛ sã♦ ✈❡r❞❛❞❡✐r❛s✳
◗✉❛♥❞♦ ❡s❝r❡✈❡♠♦s ✏A⇒ B✑ ✭r❡❧❛çã♦ ❞❡ ✐♠♣❧✐❝❛çã♦✮✱ q✉❡r❡♠♦s ❞✐③❡r q✉❡ ❛ ♣r♦♣♦s✐çã♦ ❆ ✐♠♣❧✐❝❛ ✭♦✉ ❛❝❛rr❡t❛✮ ❛ ♣r♦♣♦s✐çã♦ ❇✳ P♦r ❡①❡♠♣❧♦✱ ❆ ♣♦❞❡ s✐❣♥✐✜❝❛r ✏◆ é ❞✐✈✐sí✈❡❧ ♣♦r ✾ ✑ ❡ ❇ ♣♦❞❡ s✐❣♥✐✜❝❛r ✏◆ é ❞✐✈✐sí✈❡❧ ♣♦r ✸ ✑✳ ◆❡st❡ ❝❛s♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✱ ❝♦♥❝r❡t❛♠❡♥t❡✿
◆ é ❞✐✈✐sí✈❡❧ ♣♦r ✾ ⇒ ◆ é ❞✐✈✐sí✈❡❧ ♣♦r ✸✳
▼❛s ❛ r❡❝í♣r♦❝❛ ♥ã♦ é ✈❡r❞❛❞❡✐r❛❀ ♥ã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ◆ é ❞✐✈✐sí✈❡❧ ♣♦r ✸ ⇒ ◆ é ❞✐✈✐sí✈❡❧ ♣♦r ✾✳
❉❡ ❢❛t♦✱ s❡♠♣r❡ q✉❡ ✉♠ ♥ú♠❡r♦ ❢♦r ❞✐✈✐sí✈❡❧ ♣♦r ✾ ✕ ❝♦♠♦ ✾✱ ✶✽✱ ✷✼✱ ❡t❝✳ ✕✱ ❡❧❡ s❡rá ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐✈✐sí✈❡❧ ♣♦r ✸❀ ♠❛s ✉♠ ♥ú♠❡r♦ ♣♦❞❡ ♠✉✐t♦ ❜❡♠ s❡r ❞✐✈✐sí✈❡❧ ♣♦r ✸ ✕ ❝♦♠♦ ✻✱ ✶✷✱ ✶✺✱ ✷✶✱ ❡t❝✳ ✕✱ s❡♠ s❡r ❞✐✈✐sí✈❡❧ ♣♦r ✾✳
❱❡❥❛✿ ❞✐ss❡♠♦s q✉❡ ◆ s❡rá ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐✈✐sí✈❡❧ ♣♦r ✸ s❡ ❢♦r ❞✐✈✐sí✈❡❧ ♣♦r ✾✳ ❉✐t♦ ❞❡ ♦✉tr❛ ♠❛♥❡✐r❛✱ ✕ s❡r ❞✐✈✐sí✈❡❧ ♣♦r ✸ ✕ é ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❞❡ ✕ s❡r ❞✐✈✐sí✈❡❧ ♣♦r ✾ ✕✳ ▼❛✐s ❢♦r♠❛❧♠❡♥t❡✱ q✉❛♥❞♦ A⇒B✱ ❞✐③❡♠♦s q✉❡ ❇ é ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❞❡ ❆❀ ✈❛❧❡ ❞✐③❡r✱ ❛❝♦♥t❡❝❡♥❞♦ ❆✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛❝♦♥t❡❝❡rá ❇✳ ▼❛s ❇ ♣♦❞❡ ❛❝♦♥t❡❝❡r s❡♠ q✉❡ ❆ ❛❝♦♥t❡ç❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❆ é ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ❞❡ ❇✱ ♣♦✐s ❆ s❡♥❞♦ ✈❡r❞❛❞❡✐r❛✱ ❇ t❛♠❜é♠ ♦ s❡rá✳
◗✉❛♥❞♦ t❡♠♦s✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱
A⇒B ❡ B ⇒A✱
❝♦st✉♠❛✲s❡ ❡s❝r❡✈❡rA⇔B✳ ◆❡st❡ ❝❛s♦✱ q✉❛❧q✉❡r ✉♠❛ ❞❛s ❞✉❛s ♣r♦♣♦s✐çõ❡s ✭❆ ❡ ❇✮ é ❛♦ ♠❡s♠♦ t❡♠♣♦ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ♦✉tr❛✳ ❈♦st✉♠❛✲s❡ t❛♠❜é♠ ❞✐③❡r q✉❡ ❡❧❛s sã♦ ❡q✉✐✲ ✈❛❧❡♥t❡s✳
❊①❡♠♣❧♦ ✶✳✸
x+ 3 = 5 ⇔ x= 2,
♦✉ s❡❥❛✱
(x+ 3 = 5 ⇒ x= 2) ❡ (x= 2 ⇒ x+ 3 = 5)✳
Pr♦♣r✐❡❞❛❞❡s ❞♦s ❘❛❝✐♦♥❛✐s
❆❣♦r❛ ✈❡r❡♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❡♠ t❡r♠♦s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ✭+✮✱ ♠✉❧t✐♣❧✐❝❛çã♦ ✭·✮ ❡ ❞❛ r❡❧❛çã♦ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡ ✭6✮✳
❙❡❥❛♠ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❛✱ ❜ ❡ ❝✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
❈♦♠✉t❛t✐✈❛
✭❆✶✮ a+b=b+a ✭▼✶✮ a·b=b·a
❆ss♦❝✐❛t✐✈❛
✭❆✷✮ (a+b) +c=a+ (b+c) ✭▼✷✮ (a·b)·c=a·(b·c)
❊①✐stê♥❝✐❛ ❞❡ ❊❧❡♠❡♥t♦ ◆❡✉tr♦
✭❆✸✮ a+ 0 =a ✭▼✸✮ a·1 =a
❊①✐stê♥❝✐❛ ❞❡ ❖♣♦st♦ ✭❆✹✮ a+ (−a) = 0
❊①✐stê♥❝✐❛ ❞❡ ■♥✈❡rs♦ ✭▼✹✮ a·1
a = 1 ♦✉ a·a
−1
= 1 (a6= 0)
❉✐str✐❜✉✐t✐✈❛ ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦ ❡♠ r❡❧❛çã♦ à ❆❞✐çã♦ ✭❉✮ a·(b+c) =a·b+a·c
❘❡✢❡①✐✈❛
✭❖✶✮ a=a ♦✉ a6a
❆♥t✐✲s✐♠étr✐❝❛
✭❖✷✮ a6b ❡ b6a ⇒ a=b
❚r❛♥s✐t✐✈❛
✭❖✸✮ a6b ❡ b6c ⇒ a6c
❚r✐❝♦t♦♠✐❛
✭❖✹✮ a=b✱ a < b ♦✉ b < a
❈♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ ❖r❞❡♠ ❝♦♠ ❛ ❆❞✐çã♦ ✭❖❆✮ a6b ⇒ a+c6b+c
❈♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ ❖r❞❡♠ ❝♦♠ ❛ ▼✉❧t✐♣❧✐❝❛çã♦ ✭❖▼✮ a < b ❡ 0< c ⇒ a·c < b·c
❖❜s❡r✈❛çã♦ ✶✳✶✳ ❙❡❥❛K✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r ♥ã♦✲✈❛③✐♦ ❡ s✉♣♦♥❤❛♠♦s q✉❡ ❡♠ K❡st❡✲ ❥❛♠ ❞❡✜♥✐❞❛s ❛s ♦♣❡r❛çõ❡s ✐♥❞✐❝❛❞❛s ♣♦r ✭+✮ ❡ ✭·✮❀ s❡ ❛ t❡r♥❛ (K,+,·) s❛t✐s❢❛③❡r ❛s ♣r♦♣r✐❡❞❛❞❡s
✭❆✶✮ ❛ ✭❆✹✮✱ ✭▼✶✮ ❛ ✭▼✹✮ ❡ ✭❉✮✱ ❞✐r❡♠♦s q✉❡ (K,+,·) é ✉♠ ❝♦r♣♦✳ ❙❡✱ ❛❧é♠ ❞✐ss♦✱ ❡♠K❡st✐✈❡r ❞❡✜♥✐❞❛ ✉♠❛ r❡❧❛çã♦ ✭6✮ ❞❡ ♠♦❞♦ q✉❡ ❛ q✉á❞r✉♣❧❛ (K,+,·,6) s❛t✐s❢❛ç❛ t♦❞❛s ❛s ✶✺ ♣r♦♣r✐❡✲
❞❛❞❡s ❛♥t❡r✐♦r♠❡♥t❡ ❧✐st❛❞❛s✱ ❡♥tã♦ ❞✐r❡♠♦s q✉❡ (K,+,·,6) é ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦✳ ❙❡❣✉❡ q✉❡ (Q,+,·,6) é ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦❀ ❡♥tr❡t❛♥t♦ (Z,+,·,6) ♥ã♦ é ❝♦r♣♦ ♦r❞❡♥❛❞♦✱ ♣♦✐s ✭▼✹✮ ♥ã♦
s❡ ✈❡r✐✜❝❛ ✭♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♥ã♦ ♣♦ss✉❡♠ ✐♥✈❡rs♦✮✳
✶✳✶✳✶ ❘❡♣r❡s❡♥t❛çã♦ ●❡♦♠étr✐❝❛ ❞♦s ❘❛❝✐♦♥❛✐s
❖s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♣♦r ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛✳ P❛r❛ ✐st♦✱ ❡s❝♦❧❤❡♠✲s❡ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ❞❛ ♠❡s♠❛✱ ✉♠ r❡♣r❡s❡♥t❛♥❞♦ ♦ ✵ ❡ ♦ ♦✉tr♦ ♦ ✶✳ ❚♦♠❛♥❞♦✲s❡ ♦ s❡❣♠❡♥t♦ ❞❡ ❡①tr❡♠✐❞❛❞❡s ✵ ❡ ✶ ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✱ ♠❛r❝❛♠✲s❡ ♦s r❡♣r❡s❡♥✲ t❛♥t❡s ❞♦s ❞❡♠❛✐s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳
❋✐❣✉r❛ ✶✳✶✿ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s r❛❝✐♦♥❛✐s
❙❡ ♦ ♣♦♥t♦ P ❢♦r ♦ r❡♣r❡s❡♥t❛♥t❡ ❞♦ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ r✱ ❞✐r❡♠♦s q✉❡ r é ❛ ❛❜s❝✐ss❛ ❞❡ P✳ ◆❛ ✜❣✉r❛ ✶✳✶✱ −12 é ❛ ❛❜s❝✐ss❛ ❞❡ ❆❀ ✹ é ❛ ❛❜s❝✐ss❛ ❞❡ ❇✳ ❚♦❞♦ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ r é ❛ ❛❜s❝✐ss❛
❞❡ ✉♠ ♣♦♥t♦ ❞❛ r❡t❛❀ ❡♥tr❡t❛♥t♦✱ ♥❡♠ t♦❞♦ ♣♦♥t♦ ❞❛ r❡t❛ t❡♠ ❛❜s❝✐ss❛ r❛❝✐♦♥❛❧✳ ❱❡r❡♠♦s ♥❛ ♣ró①✐♠❛ s❡çã♦✱ q✉❡ ♦s ♣♦♥t♦s ❞❛ r❡t❛ q✉❡ ♥ã♦ ♣♦ss✉❡♠ ❛❜s❝✐ss❛ r❛❝✐♦♥❛❧✱ ♣♦ss✉❡♠ ♥❛ ✈❡r❞❛❞❡ ❝♦♠♦ ❛❜s❝✐ss❛ ✉♠ t✐♣♦ ❞❡ ♥ú♠❡r♦ ❝❤❛♠❛❞♦ ✐rr❛❝✐♦♥❛❧✳
✶✳✷ ◆ú♠❡r♦s ■rr❛❝✐♦♥❛✐s
◆❛ s❡çã♦ ❛♥t❡r✐♦r✱ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞♦s ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳ ❱❡r❡♠♦s ❛❣♦r❛ ❞❡ q✉❡ ♠♦❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐çã♦ ❞❛s ❣r❛♥❞❡③❛s ❞✐t❛s ❝♦♥tí♥✉❛s ❝♦♥❞✉③ à ♥♦çã♦ ❞❡ ♥ú♠❡r♦ r❡❛❧✳ ❯s❛r❡♠♦s ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❝♦♠♦ ♠♦t✐✈❛çã♦✳ ❊st❡ ❡①❡♠♣❧♦ é tã♦ s✐❣♥✐✜❝❛t✐✈♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛ r❡t❛ r❡❛❧✳
✶✳✷✳✶ ❙❡❣♠❡♥t♦s ❈♦♠❡♥s✉rá✈❡✐s ❡ ■♥❝♦♠❡♥s✉rá✈❡✐s
❙❡❥❛ ❆❇ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛✳ P❛r❛ ♠❡❞✐✲❧♦✱ é ♥❡❝❡ssár✐♦ ❛❞♦t❛r ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ✉✱ ♦ s❡❣♠❡♥t♦ q✉❡ ♠❡❞❡ ✶✉ é ❝❤❛♠❛❞♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦✳ ❊st✐♣✉❧❛✲s❡ ❛✐♥❞❛ q✉❡ s❡❣♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s t❡♥❤❛♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛ ❡ q✉❡ s❡n−1 ♣♦♥t♦s ✐♥t❡r✐♦r❡s ❞❡❝♦♠♣✉s❡r❡♠ ❆❇ ❡♠ ♥
s❡❣♠❡♥t♦s ❡ s❡ ❡st❡s s❡❣♠❡♥t♦s ♣❛r❝✐❛✐s ❢♦r❡♠ t♦❞♦s ❝♦♥❣r✉❡♥t❡s ❛ ✉✱ ❞✐r❡♠♦s q✉❡ ✉ ❝❛❜❡ ♥ ✈❡③❡s ❡♠ ❆❇ ❡ ❛ ♠❡❞✐❞❛ ❞❡ ❆❇ ✭q✉❡ r❡♣r❡s❡♥t❛✲s❡ ♣♦rAB✮ s❡rá ✐❣✉❛❧ ❛ ♥✳
❋✐❣✉r❛ ✶✳✷✿ s❡❣♠❡♥t♦ ❞❡ r❡t❛
P♦❞❡ ♦❝♦rr❡r q✉❡ ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ ✉ ♥ã♦ ❝❛✐❜❛ ✉♠ ♥ú♠❡r♦ ❡①❛t♦ ❞❡ ✈❡③❡s ❡♠ ❆❇✳ ❊♥tã♦ ❛ ♠❡❞✐❞❛ ❞❡ ❆❇ ♥ã♦ s❡rá ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❊st❛ s✐t✉❛çã♦ ❝♦♥❞✉③ à ✐❞❡✐❛ ❞❡ ❢r❛çã♦✱ ❝♦♥❢♦r♠❡ ♠♦str❛✲s❡ ❛❣♦r❛✳
Pr♦❝✉r❛♠♦s ✉♠ ♣❡q✉❡♥♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ✇✱ q✉❡ ❝❛✐❜❛ ♥ ✈❡③❡s ♥♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ ✉ ❡ ♠ ✈❡③❡s ❡♠ ❆❇✭❋✐❣✉r❛ ✶✳✸✮✳ ❊st❡ s❡❣♠❡♥t♦ ✇ s❡rá ❡♥tã♦ ✉♠❛ ♠❡❞✐❞❛ ❝♦♠✉♠ ❞❡ ✉ ❡ ❆❇✳
❊♥❝♦♥tr❛❞♦ ✇✱ ❞✐r❡♠♦s q✉❡ ❆❇ ❡ ✉ sã♦ ❝♦♠❡♥s✉rá✈❡✐s✳ ❆ ♠❡❞✐❞❛ ❞❡ ✇ s❡rá ❛ ❢r❛çã♦ 1
n ❡ ❛
♠❡❞✐❞❛ ❞❡ ❆❇✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ s❡rá ♠ ✈❡③❡s 1
n✱ ♦✉ s❡❥❛✱ ✐❣✉❛❧ ❛ mn✳
❋✐❣✉r❛ ✶✳✸✿ ✇ ❝❛❜❡ ♥ ✈❡③❡s ❡♠ ✉ ❡ ♠ ✈❡③❡s ❡♠ ❆❇
❖❜s❡r✈❡ q✉❡✿
u=n·w ⇒ w= u
n =
1
n,
AB=m·w ⇒ AB=m·
1
n
= m
n.
P♦r ♠✉✐t♦ t❡♠♣♦ ♣❡♥s❛✈❛✲s❡ q✉❡ ❞♦✐s s❡❣♠❡♥t♦s q✉❛✐sq✉❡r ❡r❛♠ s❡♠♣r❡ ❝♦♠❡♥s✉rá✈❡✐s✿ s❡❥❛♠ q✉❛✐s ❢♦ss❡♠ ❆❇ ❡ ❈❉✱ ❛❝❡✐t❛✈❛✲s❡ t❛❝✐t❛♠❡♥t❡ q✉❡ ❤❛✈❡r✐❛ s❡♠♣r❡ ✉♠ s❡❣♠❡♥t♦ ❊❋ q✉❡ ❝❛❜❡r✐❛ ✉♠ ♥ú♠❡r♦ ❡①❛t♦ ♥ ❞❡ ✈❡③❡s ❡♠ ❆❇ ❡ ✉♠ ♥ú♠❡r♦ ❡①❛t♦ ♠ ❞❡ ✈❡③❡s ❡♠ ❈❉✳ ❊st❛ ❝r❡♥ç❛ t❛❧✈❡③ ❛❞✈✐❡ss❡ ❞❛ ❆r✐t♠ét✐❝❛✱ ♦♥❞❡ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s q✉❛✐sq✉❡r tê♠ s❡♠♣r❡ ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ✭♥❛ ♣✐♦r ❤✐♣ót❡s❡✱ ✐❣✉❛❧ ❛ ✶✮✳
❆ ✐❧✉sã♦ ❞❛ ❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡ ❞✉r♦✉ ❛té ♦ q✉❛rt♦ sé❝✉❧♦ ❛♥t❡s ❞❡ ❈r✐st♦✳ ◆❛q✉❡❧❛ é♣♦❝❛✱ ❡♠ ❈r♦t♦♥❛✱ s✉❧ ❞❛ ■tá❧✐❛✱ ❤❛✈✐❛ ✉♠❛ s❡✐t❛ ✜❧♦só✜❝♦✲r❡❧✐❣✐♦s❛✱ ❧✐❞❡r❛❞❛ ♣♦r P✐tá❣♦r❛s✳ ❯♠ ❞♦s ♣♦♥t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ s✉❛ ❞♦✉tr✐♥❛ ❡r❛ ♦ ❧❡♠❛ ✏❖s ♥ú♠❡r♦s ❣♦✈❡r♥❛♠ ♦ ♠✉♥❞♦✑✳ ✭▲❡♠❜r❡♠♦s q✉❡ ♥ú♠❡r♦ ♣❛r❛ ❡❧❡s ❡r❛♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❛❞♠✐t✐♥❞♦✲s❡ t♦♠❛r r❛③õ❡s ❡♥tr❡ ❡ss❡s ♥ú♠❡r♦s✱ ❢♦r♠❛♥❞♦ ❛s ❢r❛çõ❡s✮✳ ❯♠❛ ❡♥♦r♠❡ ❝r✐s❡✱ q✉❡ ❛❜❛❧♦✉ ♦s ❛❧✐❝❡r❝❡s ❞♦ ♣✐t❛❣♦r✐s♠♦ ❡✱ ♣♦r ❛❧❣✉♠ t❡♠♣♦✱ t♦❞❛ ❛ ❡str✉t✉r❛ ❞❛ ▼❛t❡♠át✐❝❛ ❣r❡❣❛✱ s✉r❣✐✉ q✉❛♥❞♦✱ ❡♥tr❡ ♦s ♣ró♣r✐♦s ❞✐s❝í♣✉❧♦s ❞❡ P✐tá❣♦r❛s✱ ❛❧❣✉é♠ ♦❜s❡r✈♦✉ q✉❡ ♦ ❧❛❞♦ ❡ ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ sã♦ s❡❣♠❡♥t♦s ❞❡ r❡t❛ ✐♥❝♦♠❡♥s✉rá✈❡✐s✱ ▲✐♠❛ ✭✷✵✶✸✱ ♣✳ ✺✺✮✳
❙❡ ❤♦✉✈❡ss❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ✉ q✉❡ ❝♦✉❜❡ss❡ ♥ ✈❡③❡s ♥♦ ❧❛❞♦ ❆❇ ❡ ♠ ✈❡③❡s ♥❛ ❞✐❛❣♦♥❛❧ ❆❈ ❞♦ q✉❛❞r❛❞♦ ❆❇❈❉ ❡♥tã♦✱ t♦♠❛♥❞♦ ❆❇ ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ ❛ ♠❡❞✐❞❛ ❞❡ ❆❈ s❡r✐❛ ✐❣✉❛❧ ❛m/n ❡♥q✉❛♥t♦✱ ♥❛t✉r❛❧♠❡♥t❡✱ ❛ ♠❡❞✐❞❛ ❞❡ ❆❇ s❡r✐❛ ✶✳ ❖❜s❡r✈❡✿
AB=n·u ⇒ u= 1
n (AB= 1)
AC =m·u ⇒ AC=m·
1
n
= m
n
P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ t❡rí❛♠♦s (m/n)2 = 12+ 12 ♦♥❞❡ m2/n2 = 2 ❡ m2 = 2n2✳ ▼❛s ❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ é ❛❜s✉r❞❛✱ ♣♦✐s ♥❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡m2 ❡♠ ❢❛t♦r❡s ♣r✐♠♦s ♦ ❡①♣♦❡♥t❡ ❞♦ ❢❛t♦r2 é ♣❛r ❡♥q✉❛♥t♦ ❡♠2n2 é í♠♣❛r✳
❋✐❣✉r❛ ✶✳✹✿ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✉♥✐tár✐♦
❆ ❡①✐stê♥❝✐❛ ❞❡ s❡❣♠❡♥t♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s s✐❣♥✐✜❝❛ q✉❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♠❛✐s ❛s ❢r❛çõ❡s sã♦ ✐♥s✉✜❝✐❡♥t❡s ♣❛r❛ ♠❡❞✐r t♦❞♦s ♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛✳
❆ s♦❧✉çã♦ q✉❡ s❡ ✐♠♣✉♥❤❛✱ ❡ q✉❡ ❢♦✐ ✜♥❛❧♠❡♥t❡ ❛❞♦t❛❞❛✱ ❡r❛ ❛ ❞❡ ❛♠♣❧✐❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦✱ ✐♥tr♦❞✉③✐♥❞♦ ♦s ❝❤❛♠❛❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡✱ ✜①❛❞♦ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❛r❜✐trár✐❛✱ q✉❛❧q✉❡r s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♣♦❞❡ t❡r ✉♠❛ ♠❡❞✐❞❛ ♥✉♠ér✐❝❛✳ ◗✉❛♥❞♦ ♦ s❡❣♠❡♥t♦ ❝♦♥s✐❞❡r❛❞♦ é ❝♦♠❡♥s✉rá✈❡❧ ❝♦♠ ❛ ✉♥✐❞❛❞❡ ❡s❝♦❧❤✐❞❛✱ s✉❛ ♠❡❞✐❞❛ é ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ✭✐♥t❡✐r♦ ♦✉ ❢r❛❝✐♦♥ár✐♦✮✳ ❖s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s r❡♣r❡s❡♥t❛♠ ♠❡❞✐❞❛s ❞❡ s❡❣♠❡♥t♦s q✉❡ sã♦ ✐♥❝♦♠❡♥s✉rá✈❡✐s ❝♦♠ ❛ ✉♥✐❞❛❞❡✳ ◆❛ ❞❡s❝r✐çã♦ ❛❝✐♠❛✱ q✉❛♥❞♦ ♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦ ♠❡❞❡ ✶✱ ❛ ♠❡❞✐❞❛ ❞❛ ❞✐❛❣♦♥❛❧ é ♦ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧√2✳
❋✐❣✉r❛ ✶✳✺✿ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✉♥✐tár✐♦
✶✳✸ ❆ ❘❡t❛ ❘❡❛❧
❉❛ ✉♥✐ã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ♦❜t❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ s✐♠❜♦❧✐❝❛♠❡♥t❡ r❡♣r❡s❡♥t❛❞♦ ♣♦rR✳ ❈♦♠ ❛ ❞❡s❝♦❜❡rt❛ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✱ é ♣♦ssí✈❡❧ s❡ ❞❡t❡r♠✐♥❛r ❛ ❛❜s❝✐ss❛ ❞❡ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛ r❡t❛ ♥✉♠ér✐❝❛✳
❖❜s❡r✈❡ ❛ r❡t❛ ❛❜❛✐①♦ ❝♦♠ ❛ ❧♦❝❛❧✐③❛çã♦ ❞❡ ❛❧❣✉♥s ♥ú♠❡r♦s r❡❛✐s✿
❋✐❣✉r❛ ✶✳✻✿ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s
P♦r ✐ss♦✱ ❞✐③❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ♥ú♠❡r♦s r❡❛✐s ❡ ♦s ♣♦♥t♦s ❞❛ r❡t❛✱ ♦✉ s❡❥❛✱ ❛ ❝❛❞❛ ♣♦♥t♦ ❞❛ r❡t❛ ❝♦rr❡s♣♦♥❞❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ r❡❛❧ ❡✱ r❡❝✐♣r♦❝❛♠❡♥t❡✱ ❛ ❝❛❞❛ ♥ú♠❡r♦ r❡❛❧ ❝♦rr❡s♣♦♥❞❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ❞❛ r❡t❛✳ ❖ ❝♦♥❥✉♥t♦ R ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ♠♦❞❡❧♦ ❛r✐t♠ét✐❝♦ ❞❡ ✉♠❛ r❡t❛✱ ❡♥q✉❛♥t♦ ❡st❛✱ ♣♦r s✉❛ ✈❡③✱ é ♦ ♠♦❞❡❧♦ ❣❡♦♠étr✐❝♦ ❞❡R✳
❊♠R❡stã♦ ❞❡✜♥✐❞❛s ❞✉❛s ♦♣❡r❛çõ❡s✱ ❛❞✐çã♦ ✭+✮ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✭·✮ ❡ ✉♠❛ r❡❧❛çã♦ ✭6✮✳
❆ ❛❞✐çã♦ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣❛r(x, y)❞❡ ♥ú♠❡r♦s r❡❛✐s ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ r❡❛❧ ✐♥❞✐❝❛❞♦ ♣♦rx+y✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ✉♠ ú♥✐❝♦ r❡❛❧ ✐♥❞✐❝❛❞♦ ♣♦rx·y✳ ❆s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛s ❡♠R✱ q✉❛♥❞♦ r❡str✐t❛s ❛ Q✱ ❝♦✐♥❝✐❞❡♠ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ Q❀ ♦ ♠❡s♠♦ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ r❡❧❛çã♦ ✭6✮✳
✶✳✸✳✶ ◆ú♠❡r♦s ❘❡❛✐s ♥❛ ❋♦r♠❛ ❉❡❝✐♠❛❧
❯♠❛ ❡①♣r❡ssã♦ ❞❡❝✐♠❛❧ é ✉♠ sí♠❜♦❧♦ ❞❛ ❢♦r♠❛α=a0, a1a2· · ·an· · ·,
❡♠ q✉❡a0 é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ③❡r♦ ❡ a1, a2, . . . , an, . . . sã♦ ❞í❣✐t♦s✱ ✐st♦
é✱ ♥ú♠❡r♦s ✐♥t❡✐r♦s t❛✐s q✉❡ 0 6 an 6 9✳ P❛r❛ ❝❛❞❛ n ∈ N✱ t❡♠✲s❡ ✉♠ ❞í❣✐t♦ an✱ ❝❤❛♠❛❞♦ ♦
♥✲és✐♠♦ ❞í❣✐t♦ ❞❛ ❡①♣r❡ssã♦ ❞❡❝✐♠❛❧✳ ❖ ♥ú♠❡r♦ ✐♥t❡✐r♦a0 ❝❤❛♠❛✲s❡ ❛ ♣❛rt❡ ✐♥t❡✐r❛ ❞❡α✳ ❱❛♠♦s
❝♦♥s✐❞❡r❛r ♦s ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✱ ♣❛r❛ tr❛t❛r ❞❡ ♥ú♠❡r♦s r❡❛✐s ♥❡❣❛t✐✈♦s✱ s✐♠♣❧❡s♠❡♥t❡ s❡ ❛❝r❡s❝❡♥t❛ ♦ s✐♥❛❧ ♠❡♥♦s✳
❊①❡♠♣❧♦s ✶✳✹
α= 3,12; β = 15,121212. . .; π= 3,14159265. . .
