❯♥✐✈❡%&✐❞❛❞❡ ❋❡❞❡%❛❧ ❞❛ +❛%❛,❜❛
❈❡♥/%♦ ❞❡ ❈✐1♥❝✐❛& ❊①❛/❛& ❡ ❞❛ ◆❛/✉%❡③❛
❉❡♣❛%/❛♠❡♥/♦ ❞❡ ▼❛/❡♠</✐❝❛
▼❡&/%❛❞♦ +%♦✜&&✐♦♥❛❧ ❡♠ ▼❛/❡♠</✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ +❘❖❋▼❆❚
❆❧❣✉♠❛& ❆♣❧✐❝❛*+❡& ❞❡ ❋/&✐❝❛ ❞♦ ❊♥&✐♥♦ ▼4❞✐♦ ❛
♣❛56✐5 ❞♦ ❈8❧❝✉❧♦ ❉✐❢❡5❡♥❝✐❛❧ ❡ ■♥6❡❣5❛❧
†♣♦"
▲✉❝❛& ❈❛✈❛❧❝❛♥6✐ ❈5✉③
#♦❜ ♦"✐❡♥(❛*+♦ ❞♦
?5♦❢✳ ❉5✳ ❈❛5❧♦& ❇♦❝❦❡5 ◆❡6♦
❚!❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉-.♦ ❞❡ ❈✉!-♦ ❛♣!❡-❡♥✲ 1❛❞♦ ❛♦ ❈♦!♣♦ ❉♦❝❡♥1❡ ❞♦ ▼❡-1!❛❞♦ 4!♦✲ ✜--✐♦♥❛❧ ❡♠ ▼❛1❡♠81✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐✲ ♦♥❛❧ 4❘❖❋▼❆❚ ❈❈❊◆✲❯❋4❇✱ ❝♦♠♦ !❡✲ B✉✐-✐1♦ ♣❛!❝✐❛❧ ♣❛!❛ ♦❜1❡♥C.♦ ❞♦ 1D1✉❧♦ ❞❡ ▼❡-1!❡ ❡♠ ▼❛1❡♠81✐❝❛✳
▼❛!C♦✴✷✵✶✸ ❏♦.♦ 4❡--♦❛ ✲ 4❇
†❖ ♣"❡$❡♥&❡ &"❛❜❛❧❤♦ ❢♦✐ "❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆4❊❙✱ ❈♦♦"❞❡♥❛89♦ ❞❡ ❆♣❡"❢❡✐8♦❛♠❡♥&♦ ❞❡
C957a Cruz, Lucas Cavalcanti.
Algumas aplicações de física do ensino médio a partir do cálculo diferencial e integral / Lucas Cavalcanti Cruz.-- João Pessoa, 2013.
85f.
Orientador: Carlos Bocker Neto Dissertação (Mestrado) – UFPB/CCEN
1. Matemática. 2. Física. 3. Cálculo diferencial. 4. Cálculo integral. 5. Limites. 6. Derivadas.
✏❍✐r❛♠ ❢❡③ ❛✐♥❞❛ ♦ ▼❛r✱ t♦❞♦ ❞❡ ♠❡t❛❧ ❢✉♥❞✐❞♦✱ ❝♦♠ ❝✐♥❝♦ ♠❡tr♦s ❞❡ ❞✐â♠❡tr♦✳ ❊r❛ r❡❞♦♥❞♦✱ t✐♥❤❛ ❞♦✐s ♠❡tr♦s ❡ ♠❡✐♦ ❞❡ ❛❧t✉r❛✱ ❡ s✉❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ t✐♥❤❛ q✉✐♥③❡ ♠❡tr♦s✳✑ ✶ ❘s ✼✱✷✸
❆❣r❛❞❡❝✐♠❡♥t♦s
P❛r❛ ♠✐♠✱ s❡♠♣r❡ é ❞✐❢í❝✐❧ ❝♦♠❡ç❛r ❛ ❡s❝r❡✈❡r ♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣♦✐s t❛♥t❛s ❢♦r❛♠ ❛s ♣❡ss♦❛s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ss❡ ♠❡str❛❞♦✱ ❜❡♠ ❝♦♠♦ ♣❛r❛ ❛ ✜♥❛❧✐③❛çã♦ ❞❡ss❛ ❞✐ss❡rt❛çã♦✳ ❙❡r✐❛ ✐♥❥✉st♦ ❞❛ ♠✐♥❤❛ ♣❛rt❡ ❛❣r❛❞❡❝❡r ❛ ❛❧❣✉♠❛s ♣❡ss♦❛s ❡ ❡sq✉❡❝❡r ❞❡ ♦✉tr❛s✱ ♠❛s ❝♦♠♦✱ ❝♦♠ ❝❡rt❡③❛✱ ♥ã♦ ❧❡♠❜r❛r❡✐ t♦❞♦s ♦s ♥♦♠❡s✱ ❛q✉❡❧❡s q✉❡ ♥ã♦ ❢♦r❡♠ ❝✐t❛❞♦s t❡♥❤❛♠ ❝❡rt❡③❛ q✉❡ t❛♠❜é♠ sã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s✳
❈♦♠❡ç♦ ❛❣r❛❞❡❝❡♥❞♦ ❛ t♦❞❛ ♠✐♥❤❛ t✉r♠❛ ❞♦ P❘❖❋▼❆❚ ✲ ❯❋P❇✳ P❡ss♦❛s q✉❡ ❝♦♠ ❛s q✉❛✐s ❡✉ ❝♦♥✈✐✈✐ ❞✉r❛♥t❡ ❝❡r❝❛ ❞❡ ❞♦✐s ❛♥♦s✱ ♣❡ss♦❛s q✉❡ ❜❛t❛❧❤❛r❛♠ ❝♦♠✐❣♦✱ ♠❡ ❛❥✉❞❛♥❞♦✱ q✉❡r s❡❥❛ ❡st✉❞❛♥❞♦ ♣❛r❛ ❛s ♠❛✐s ❞✐✈❡rs❛s ❞✐s❝✐♣❧✐♥❛s ♦✉ ❛♣❡♥❛s ❝♦♥✲ ✈❡rs❛♥❞♦✱ ❞❛♥❞♦ ót✐♠♦s ❝♦♥s❡❧❤♦s✳ ◗✉❛♥❞♦ ❝♦♠❡ç❛♠♦s ér❛♠♦s ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❝✐♥q✉❡♥t❛ ♣❡ss♦❛s✱ ❞❡ ❞✐✈❡rs♦s ❡st❛❞♦s✱ ❞✐❢❡r❡♥t❡s ❝✉❧t✉r❛s ❡ ❝♦st✉♠❡s✱ ❛❧❣✉♥s ♣❡r✲ ❝♦rr✐❛♠ ❞✐stâ♥❝✐❛s ❡♥♦r♠❡s ♣❛r❛ ❝❤❡❣❛r ❛té ❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❡♥❢r❡♥t❛♥❞♦ ✈❡r❞❛❞❡✐r❛s ❜❛t❛❧❤❛s t♦❞♦s ♦s ✜♥❛✐s ❞❡ s❡♠❛♥❛✳ ▼❛s ♠❡s♠♦ ❝♦♠ t✉❞♦ ✐ss♦✱ ❥✉♥t♦s ❜❛t❛❧❤❛♠♦s ❡ s✉♣❡r❛♠♦s t♦❞❛s ❛s ❞✐✜❝✉❧❞❛❞❡s✳ ❏❛♠❛✐s ♠❡ ❡sq✉❡❝❡r❡✐ ❞❡ss❛ t✉r♠❛ ✐♥❝rí✈❡❧✳
❊♠ ♣❛rt✐❝✉❧❛r ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛♦s ▼❡str❡s✿ ❆♠❜rós✐♦ ❊❧✐❛s✱ ❆♥tô♥✐♦ ●❡✲ r❛❧❞♦✱ ❆✉rí❧✐♦ ●✉❡❞❡s✱ ❈í❝❡r♦ ❉❡♠étr✐♦✱ ❊❧❧❡♥ P❛trí❝✐❛✱ ❋❡r♥❛♥❞♦ ❱✐❛♥❛✱ ❍á❧✐ss♦♥ ❇❛rr❡t♦✱ ▼❛r❝❡❧♦ ❘♦❞r✐❣✉❡s ❡ ❙í✈✐♦ ❈r✉③ ♣♦r ❢♦r♠❛r❡♠ ♦ ❣r✉♣♦ ❞♦ q✉❛❧ ❡st✐✈❡ ♠❛✐s ♣ró①✐♠♦ ❡ ❝♦♥✈✐✈✐ ✉♠ ♣♦✉❝♦ ♠❛✐s✳ ❖❜r✐❣❛❞♦ ♣♦r t♦❞❛s ♥♦ss❛s ❝♦♥✈❡rs❛s✱ ❝❛r♦♥❛s✱ ❛❥✉❞❛s ❝♦♠ ❡①❡r❝í❝✐♦s✱ ❡t❝✳
❆❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ ❛♦ ❛♠✐❣♦ ▲✉✐s ❊❞✉❛r❞♦ ❲❛♥❞❡r❧❡② ❇✉❛rq✉❡ ❞❡ ❇❛rr♦s✱ ♣♦✐s s❡♠ ❡❧❡ ♥ã♦ s❡✐ ❝♦♠♦ ❢❛r✐❛ ♣❛r❛ ❝♦♥t✐♥✉❛r✱ ♣♦✐s ❡❧❡ ❢♦✐ ❛ ú♥✐❝❛ ♣❡ss♦❛ q✉❡ s❡ ♣r❡♦❝✉♣♦✉ ❡ ❝♦♥s❡❣✉✐✉ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❛ ♠✐♥❤❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s✳
❊✱ ♣❛r❛ t❡r♠✐♥❛r✱ ❛❣r❛❞❡❝❡r ❛♦ ❣r✉♣♦ ❞❡ ◆❛t❛❧ ✲ ❘◆✱ ❡♠ ♥♦♠❡ ❞♦ ▼❡str❡ ❚❤✐❛❣♦ ❱❛❧❡♥t✐♠✱ q✉❡ ❛rt✐❝✉❧♦✉ ♦ ❝♦♥t❛t♦ ❝♦♠ ♦ ♣r♦❢❡ss♦r ❈❛r❧♦s ●♦♠❡s✱ ❝✉❥❛ ❛❥✉❞❛ ❢♦✐ ✐♥❡st✐♠á✈❡❧ ♥❛ r❡t❛ ✜♥❛❧ ♣❛r❛ ❛ ♣r❡♣❛r❛çã♦ ♣❛r❛ ♦ ❊①❛♠❡ ❞❡ ◗✉❛❧✐✜❝❛çã♦✳
❡①♣❡r✐ê♥❝✐❛ ✐♥❝rí✈❡❧ q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ✈✐✈❡r✳ ❊♠ ❡s♣❡❝✐❛❧ ❛❣r❛❞❡ç♦ ❛♦ ❛♠✐❣♦ ❆r♦❧❞♦ ❆t❤✐❛s ❘♦❞r✐❣✉❡s✱ q✉❡ ♠❡s♠♦ s❡♠ s❛❜❡r✱ ♠❡ ❞✐ss❡ ♣❛❧❛✈r❛s ✐♠♣♦rt❛♥tíss✐♠❛s ♣❛r❛ ❛ ♠✐♥❤❛ ❞❡❝✐sã♦ s♦❜r❡ q✉❡ ❞✐r❡çã♦ t♦♠❛r ♣❛r❛ ❝♦♥❝❧✉✐r ❡ss❛ ❞✐ss❡rt❛çã♦✳
▼✐♥❤❛ t✉r♠❛ ❞♦ ❈✉rs♦ ❞❡ ❊s♣❡❝✐❛❧✐③❛çã♦ ♣❛r❛ Pr♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ t❛♠❜é♠ t❡✈❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❞✉r❛♥t❡ ❡ss❡ ♣❡rí♦❞♦✳ P♦✐s ♥♦ ♣r✐✲ ♠❡✐r♦ ❛♥♦ ❞♦ ❝✉rs♦✱ ❡♠ ✷✵✶✶✱ ❡✉ ❝✉rs❛✈❛ t❛♥t♦ ❛ ❡s♣❡❝✐❛❧✐③❛çã♦ q✉❛♥t♦ ♦ ♠❡str❛❞♦✱ ❛❧é♠ ❞❡ ♠♦r❛r ❧♦♥❣❡ ❞❡ ❝❛s❛✱ ❡♠ ❈❛♠♣✐♥❛ ●r❛♥❞❡ ✲ P❇✳ ❊ t♦❞♦ ♦ ♣❡ss♦❛❧ s❡♠♣r❡ ♠❡ ❞❡✉ ❜❛st❛♥t❡ ❢♦rç❛ ♣❛r❛ ❝♦♥t✐♥✉❛r✳ ❆❣r❛❞❡ç♦ ❛ t♦❞♦s✱ ❞❡ ❝♦r❛çã♦✳
▼✐♥❤❛s t✉r♠❛s ❞❛s ♠❛✐s ❞✐✈❡rs❛s ❣r❛❞✉❛çõ❡s✱ ♣♦✐s s❡♠ ✈♦❝ês ❡✉ ♥ã♦ ❝❤❡❣❛r✐❛ ❛té ❛q✉✐✳ ❚♦❞♦ ♦ ♣❡ss♦❛❧ ❞❛ ♠❛t❡♠át✐❝❛ ❡ ❞❛ ❢ís✐❝❛ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❛ ❝♦♥t✐♥✉❛r ♥♦s ❡st✉❞♦s ❞❛♥❞♦ ❢♦rç❛s ♣❛r❛ ♥ã♦ ❞❡s✐st✐r ❡ ❞❡ss❛ ♠❛♥❡✐r❛ ❡✉ ❝♦♥s❡❣✉✐ ❛❧✐❛r✱ ♥♦ ♠❡✉ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦✱ ❛s ❞✉❛s ❞✐s❝✐♣❧✐♥❛s q✉❡ ♠❛✐s ♠❡ ❞❡❞✐❝♦ ❛ ❡st✉❞❛r✳
❆❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ◆✐❧t♦♥ ❚❡r✉②❛✱ ♣❡❧❛s ❡♥♦r♠❡s ❝♦♥tr✐❜✉✐çõ❡s ❞❛❞❛s ❛♦ tr❛❜❛❧❤♦✱ r❡❝❡❜❡♥❞♦✲♠❡ ❡♠ s✉❛ ❝❛s❛ ❛té t❛r❞❡ ♣❛r❛ ♠❡ ❡①♣❧✐❝❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s q✉❡ ❛✐♥❞❛ ♣❛r❡❝✐❛♠ ❝♦♥❢✉s♦s✳ ❆❣r❛❞❡ç♦ t♦❞❛s ❛s s✉❣❡stõ❡s ❡ ❝rít✐❝❛s q✉❡ t♦r♥❛r❛♠ ♦ tr❛❜❛❧❤♦ ♠❛✐s r✐❝♦✳
■❣✉❛❧♠❡♥t❡ ✐♠♣♦rt❛♥t❡ ❡ q✉❡ ♠❡r❡❝❡ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ é ❛ ♣r♦❢❡ss♦r❛ ❈❛r♦❧✐♥❛ ❞❡ ❆♥❞r❛❞❡ ❆♠♦r✐♠✱ q✉❡ ❛♣❡s❛r ❞♦ ♣♦✉❝♦ t❡♠♣♦ q✉❡ ❝♦♥✈✐✈❡♠♦s ❥✉♥t♦s ♥♦ ❝✉rs♦ ❞❡ ▼❡❞✐❝✐♥❛✱ ❡❧❛ ♣♦❞❡ ❛❜r✐r ♠❡✉s ♦❧❤♦s ♣❛r❛ q✉❡ ❡✉ ❝♦♥s❡❣✉✐ss❡ ❝♦♠♣r❡❡♥❞❡r ❛s♣❡❝t♦s ❞❛ ♠✐♥❤❛ ♣❡rs♦♥❛❧✐❞❛❞❡ q✉❡ ❢♦r❛♠ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛❧❣✉♠❛s ❞❡❝✐sõ❡s ❡ ♠❡ ✜③❡r❛♠ ❝r❡s❝❡r ❝♦♠♦ s❡r ❤✉♠❛♥♦✳
◆❡ss❡ ♠❡s♠♦ ❝♦♥t❡①t♦✱ ❞✉r❛♥t❡ ❛ rá♣✐❞❛ ♣❛ss❛❣❡♠ ♣❡❧♦ ❝✉rs♦ ❞❡ ▼❡❞✐❝✐♥❛ ♥❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ▼é❞✐❝❛s ❞❛ P❛r❛í❜❛✱ ❛❣r❛❞❡ç♦ ❛ t♦❞❛ t✉r♠❛✱ q✉❡ ♠❡ ❛❝♦♠✲ ♣❛♥❤♦✉ ♥♦s ♠♦♠❡♥t♦s ✜♥❛✐s ❞❛ ❞✐ss❡rt❛çã♦ ❡ s❡♠♣r❡ ❝♦♠♣r❡❡♥❞❡✉ ♦ ❡s❢♦rç♦ q✉❡ ❡✉ ✜③ ❡ ❛ss✐♠✱ ♠❡ ❛♣♦✐♦✉ ❜❛st❛♥t❡✳ ❊♠ ❡s♣❡❝✐❛❧ ❛❣r❛❞❡ç♦ ❛ ❙❤❡②❧❛ ❘♦❝❤❛✱ q✉❡ ❥á ❡r❛ ♠✐♥❤❛ ❛♠✐❣❛✱ ♠❛s q✉❡ ❣r❛ç❛s ❛s ✈ár✐❛s ❝❛r♦♥❛s ❞✉r❛♥t❡ ♦ ❝✉rs♦ t✐✈❡♠♦s ❛ ♦♣♦rt✉♥✐✲ ❞❛❞❡ ❞❡ ❝♦♥✈❡rs❛r ✉♠ ♣♦✉❝♦ ♠❛✐s ❡ ❡str❡✐t❛r ♥♦ss❛ ❛♠✐③❛❞❡✳ ❙✉❛s ♣❛❧❛✈r❛s t❛♠❜é♠ ❢♦r❛♠ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❡✉ ❝♦♥❝❧✉✐r ❡ss❛ ❞✐ss❡rt❛çã♦✳
❆❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ ❛ t♦❞♦s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❛ ❊s❝♦❧❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❈✐❞❛❞❡ ❱✐✈❛✱ ❡s❝♦❧❛ ♥❛ q✉❛❧ tr❛❜❛❧❤♦ ❛t✉❛❧♠❡♥t❡✳ ❉❡s❞❡ ✷✵✶✷ ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ✉♠❛ ✐♥❝rí✈❡❧ ❡①♣❡r✐ê♥❝✐❛ ❞❡ ❝♦♥❤❡❝❡r ♥ã♦ ❛♣❡♥❛s ♥♦✈♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ tr❛❜❛❧❤♦✱ ♠❛s ✉♠❛ ♥♦✈❛ ❢❛♠í❧✐❛✱ ❝♦♥❞✉③✐❞❛ ♣♦r ❉❊❯❙✱ ✈❡r❞❛❞❡✐r❛♠❡♥t❡✳
❆ t♦❞♦s ♦s ❛♠✐❣♦s q✉❡ ❡stã♦ ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❡ ♠❡s♠♦ ❛q✉❡❧❡s ❝♦♠ ♦s q✉❛✐s ❥á ♥ã♦ t❡♥❤♦ t❛♥t♦ ❝♦♥t❛t♦✱ q✉❡r s❡❥❛ ❞♦s ♠♦✈✐♠❡♥t♦s ❞❛ ■❣r❡❥❛ ✭❈r✐s♠❛✱ ❊❏❈✱ ❈❛t❡❝✉♠❡♥❛t♦✮✱ q✉❡r s❡❥❛ ❞❛ ❡s❝♦❧❛✱ ❞❛ ❢❛❝✉❧❞❛❞❡ ❞❡ ❉✐r❡✐t♦✱ ❞❛ ❆r❣❡♥t✐♥❛✱ ❞❛ ❋r❛♥ç❛ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣❡ss♦❛s q✉❡ ❛♣❛r❡❝❡r❛♠ ❡♠ ♠✐♥❤❛ ✈✐❞❛✳ ❆♦ ❡st❛r ❝♦♠ q✉❛❧q✉❡r ✉♠ ❞❡ ✈♦❝ês ❡✉ ♣♦ss♦ s❡♥t✐r ❛ ♣r❡s❡♥ç❛ ❞❡ ❈r✐st♦ ❡ ❧❡♠❜r♦ q✉❡ ❥❛♠❛✐s ❞❡✈❡♠♦s ❞❡s✐st✐r✳ ❖❜r✐❣❛❞♦✳
❆♠✐❣♦s ❞❡ ❧♦♥❣❛s ❞❛t❛s ✭❖❧❞ ❙❝❤♦♦❧✮ q✉❡ s❛❜❡♠ ❞❡ s✉❛ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛✳ ◗✉❡ ♠❡ ❛❝♦♠♣❛♥❤❛r❛♠ ♥❡ss❡s ❞♦✐s ú❧t✐♠♦s ❛♥♦s r❡❝❧❛♠❛♥❞♦ ❝♦♠✐❣♦ t♦❞♦ ✜♥❛❧ ❞❡ s❡♠❛♥❛ ♣♦✐s ❡✉ ❥á ♥ã♦ s❛í❛ ♠❛✐s ❝♦♠ ❡❧❡s ♥❛s s❡①t❛s✲❢❡✐r❛s✱ ♣♦rq✉❡ t❡r✐❛ ❛✉❧❛ ♥♦ sá❜❛❞♦ ♣❡❧❛ ♠❛♥❤ã✱ ♥❡♠ s❛í❛ ♠❛✐s ♥♦s sá❜❛❞♦s ♣♦✐s ❡st❛✈❛ ❝❛♥s❛❞♦ ❞❛ s❡♠❛♥❛ ♠✉✐t♦ ❛t❛r❡❢❛❞❛✳ ▼❛s ❡ss❡s ❣r❛♥❞❡s ❛♠✐❣♦s s❡♠♣r❡ ❡st✐✈❡r❛♠ ❛♦ ♠❡✉ ❧❛❞♦✱ ♠❡ ❛♣♦✐❛♥❞♦ ❡ ♠❡ ❛❥✉❞❛♥❞♦ q✉❛♥❞♦ ♣r❡❝✐s❡✐✳ ❆té ♥♦ ❞✐❛ ❞❛ ❞❡❢❡s❛ ❞❛ ❞✐ss❡rt❛çã♦ ❡st❛✈❛♠ ❧á✱ ♠❡s♠♦ s❡♠ ❡♥t❡♥❞❡r ♦ q✉❡ ❡✉ ❡st❛✈❛ ❢❛❧❛♥❞♦✳ ▼❡✉s ❡t❡r♥♦s ❛❣r❛❞❡✲ ❝✐♠❡♥t♦s ❛ t♦❞♦s✿ ❆♥❞ré ❈❧❡♠❡♥t✐♥♦✱ ❇r✉♥♦ ❈❛r✈❛❧❤♦✱ ❉❛♥✐❧♦ ▲❡✐t❡✱ ❉❛✈✐ ❱❡❧♦s♦✱ ❉✐❡❣♦ ❆✉❣✉st♦✱ ●❧ê♥✐♦ ▲❡✐tã♦✱ ▲✐♥❝♦❧♥ P♦♥t❡s✱ P❛✉❧♦ ❍❡♥r✐q✉❡ ❈❛✈❛❧❝❛♥t✐✱ ❘❛❢❛❡❧ ❚❛r❣✐♥♦✱ ❘❛❢❛❡❧ ❯❝❤ô❛ ❡ ❘♦❞r✐❣♦ ●♦❞♦②✳
▼✐♥❤❛ ❢❛♠í❧✐❛ ♠❡r❡❝❡ t♦❞♦s ♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ❞♦ ♠✉♥❞♦✱ ❡❧❛ é ♠❡✉ ♣♦rt♦ s❡❣✉r♦✱ ❢♦r♠❛❞❛ ♣♦r ♣❡ss♦❛s ♥❛s q✉❛✐s s❡✐ q✉❡ s❡♠♣r❡ ♣♦ss♦ ❝♦♥t❛r ❡ q✉❡ ♠❡ ❛♣♦✐❛r❛♠ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✳ ❚✐♦s✱ t✐❛s✱ ♣r✐♠♦s ❡ ♣r✐♠❛s✱ ❞❡ ♣❡rt♦ ♦✉ ❞❡ ❧♦♥❣❡✱ ❞♦ ❇r❛s✐❧ ♦✉ ❞♦ ❡①t❡r✐♦r✳ ❖❜r✐❣❛❞♦ ♣♦r ♠❡ ❛❥✉❞❛r❡♠ ❛ ❝❤❡❣❛r ❛té ❛q✉✐✳
▼✐♥❤❛ ✐r♠ã ◆❛tá❧✐❛ ▲✉❞♠✐❧❛✱ q✉❡ ♠❡s♠♦ ❝♦♠ t♦❞♦ s❡✉ ❥❡✐t♦✱ ❝❛r✐♥❤♦s♦ ❡ ❞❡❧✐✲ ❝❛❞♦✱ ❡✉ s❡✐ q✉❡ ♠❡ ❛♠❛ ♠✉✐t♦ ❡ s❡♠♣r❡ ❞❡s❡❥❛ ♦ ♠❡❧❤♦r ♣❛r❛ ♠✐♠✱ ♠❡s♠♦ q✉❛♥❞♦ ♥ã♦ ❝♦♥❝♦r❞❛♠♦s✳ ❊ss❛ ♠❡♥✐♥❛ ♠❡ ❞á ♠✉✐t♦ tr❛❜❛❧❤♦✱ ♠❛s s❡♠ s✉❛ ♣r❡s❡♥ç❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛✱ s❡♠ s✉❛ ❛❥✉❞❛✱ s❡♠ ♥♦ss❛s ❝♦♥✈❡rs❛s✱ s❡♠ s✉❛s ❜r♦♥❝❛s ❡ ❝♦♥s❡❧❤♦s✱ ❡✉ ♥ã♦ s❡r✐❛ q✉❡♠ s♦✉ ❤♦❥❡✳ ■r♠ã③✐♥❤❛✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳
▼❡✉ ❝✉♥❤❛❞♦ ❘❛♣❤❛❡❧ P❡✐①♦t♦ ❡ ♠✐♥❤❛ ✐r♠ã ♠❛✐s ✈❡❧❤❛ ❇❛rt②r❛ ❈r✉③✳ P♦①❛✱ ♦ ❛♣♦✐♦ q✉❡ r❡❝❡❜♦ ❞❡ ✈♦❝ês ♥ã♦ t❡♠ ♣❛❧❛✈r❛s q✉❡ ♣♦ss❛♠ ❞❡s❝r❡✈ê✲❧♦ ♥❡♠ ✐♥str✉♠❡♥✲ t♦s ❞❡ ♠❡❞✐❞❛s q✉❡ ♣♦ss❛♠ ♠❡♥s✉rá✲❧♦✳ ❱♦❝ês s❡♠♣r❡ ❝♦♥s❡❣✉❡♠ ♠♦str❛r q✉❡ t✉❞♦ ♣♦❞❡ s❡r ✈✐st♦ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡ ❡✱ ❛ss✐♠✱ ❛❜r♦ ♠✐♥❤❛ ♠❡♥t❡ ♣❛r❛ ♥♦✈❛s ♣♦s✲ s✐❜✐❧✐❞❛❞❡s✳ ❊ ❛✐♥❞❛ ❢❛❧t❛ ❢❛❧❛r ❞❡ ♦✉tr♦ ❛♠♦r ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ ♠❡✉ s♦❜r✐♥❤♦ ❙❛♠✉❡❧ ❈r✉③✳ ❖s ♠♦♠❡♥t♦s ❞❡ ❛❧❡❣r✐❛ q✉❡ ❡❧❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛ q✉❛♥❞♦ ❡st❛♠♦s ❜r✐♥❝❛♥❞♦✱ ❥♦❣❛♥❞♦ ✈í❞❡♦✲❣❛♠❡✱ ♥❛ ♣✐s❝✐♥❛✱ ❡t❝✳ sã♦ ❡ss❡♥❝✐❛✐s ♣❛r❛ ♠✐♠✳ ❙✐♥t♦ ♠✉✐t❛ ❢❛❧t❛ ❞❡ ✈♦❝ês ♣♦r ❡st❛r❡♠ tã♦ ❧♦♥❣❡✳
♠✐♥❤❛ ✈✐❞❛ s❡♠♣r❡ q✉❡ ❡✉ ♣❛r❡ç♦ ❡sq✉❡❝❡r ❞✐ss♦✳ ❉✐✈✐❞❡ ❣r❛♥❞❡s ♠♦♠❡♥t♦s ❝♦♠✐❣♦✱ ♠❡ s✉♣♦rt❛ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ❡♥tr♦ ❡♠ ❞❡s❡s♣❡r♦ ♣♦r ❝❛✉s❛ ❞♦s ♣r♦❜❧❡♠❛s✱ ♠❡ ❛❣✉❡♥t❛ q✉❛♥❞♦ ❡st♦✉ ❝❤❛t♦✱ ♠❡ ❡s❝✉t❛ ❡ ♠❡ ❛❝♦♥s❡❧❤❛✳ ❊ ❛✐♥❞❛ s❡ ❢♦r ❛❣r❛❞❡❝❡r ♣❡❧❛ ❛❥✉❞❛ ❞✐r❡t❛ ♥❡ss❡ tr❛❜❛❧❤♦ ♣r❡❝✐s❛r❡✐ ❡s❝r❡✈❡r ♠✉✐t❛s ♦✉tr❛s ♣á❣✐♥❛s✳ ❙ó ❡❧❛ s❛❜❡ ♦ q✉❛♥t♦ ❢♦✐ ❝♦♠♣❧✐❝❛❞♦ ❝❤❡❣❛r ❛té ❛q✉✐✱ ❡s❝r❡✈❡♥❞♦ ✈ár✐♦s ♦✉tr♦s ❝❛♣ít✉❧♦s q✉❡ ❢♦r❛♠ r❡t✐r❛❞♦s✱ ❛❧é♠ ❞❡ ♠❡ ❛❥✉❞❛r ♥❛s ❝♦rr❡çõ❡s✱ ♥❛s s✉❣❡stõ❡s✱ ♥❛ ❢♦r♠❛t❛çã♦✱ ♥❛s r❡❢❡rê♥❝✐❛s✱ ❡♥✜♠✱ ❡ss❡ tr❛❜❛❧❤♦✱ ❞❡ ❝❡rt❛ ♠❛♥❡✐r❛✱ t❛♠❜é♠ é ❞❡❧❛✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦ ♣♦r t♦❞♦ s❡✉ ❆♠♦r✱ ❝❛r✐♥❤♦✱ ❛♠✐③❛❞❡ ❡ ❞❡❞✐❝❛çã♦✳ ❚❡ ❛♠♦ ♠✉✐t♦✳
P❛r❡❝❡ q✉❡ q✉❛♥❞♦ ❝❤❡❣❛♠♦s ♠❛✐s ♣ró①✐♠♦s ❞♦ ✜♥❛❧ ✜❝❛ ♠❛✐s ❞✐❢í❝✐❧ ❛❣r❛❞❡❝❡r✳ ▼✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ❞❡ ❋át✐♠❛✳ ◆ã♦ t❡♥❤♦ ❝♦♠♦ ❛❣r❛❞❡❝ê✲❧❛✱ ♣♦✐s t❡r✐❛ q✉❡ ❡s❝r❡✈❡r s♦❜r❡ t✉❞♦ ♦ q✉❡ ❡❧❛ ❢❡③ ♥❡ss❡s ú❧t✐♠♦s ✈✐♥t❡ ❡ s❡t❡ ❛♥♦s✳ ❙❡ ❝❤❡❣✉❡✐ ❛té ❛q✉✐ ❢♦✐ ♣♦rq✉❡ s❡♠♣r❡ t✐✈❡ t♦❞♦ ♦ s❡✉ ❛♣♦✐♦✱ t♦❞❛ s✉❛ ❞❡❞✐❝❛çã♦✱ t♦❞♦ s❡✉ ❆♠♦r✳ ▼❛s ❡♠ ❡s♣❡❝✐❛❧ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ ♠❛✐s ❡ss❛ ❡t❛♣❛ ♥❛ ♠✐♥❤❛ ✈✐❞❛✱ ❡ ❡♠ ♣❛rt✐❝✉❧❛r s♦❜r❡ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ q✉❡r✐❛ r❡♣❡t✐r ✉♠❛ ❧✐çã♦ q✉❡ ❛♣r❡♥❞✐ ❝♦♠ ❡❧❛✱ q✉❡ t❛♠❜é♠ ♠❡ ❛❥✉❞♦✉ ❛ t♦♠❛r ❛❧❣✉♠❛s ❞❡❝✐sõ❡s✿ ✏▼❡✉ ✜❧❤♦✱ ♥ã♦ s❡ ♣r❡♦❝✉♣❡✱ t❡r♠✐♥❡ ❡ ❡♥tr❡❣✉❡ ❧♦❣♦ ✐ss♦✱ ❞♦ ❥❡✐t♦ q✉❡ ❡❧❡s q✉❡r❡♠✱ ♣♦rq✉❡ ✉♠ ❞✐❛ ❛ r♦❞❛ ❣r❛♥❞❡ ✈❛✐ r♦❞❛r ❞❡♥tr♦ ❞❛ ♣❡q✉❡♥❛✑✳ ❊❧❛ s❛❜❡ ❞♦ q✉❡ s❡ tr❛t❛ ❡ ❡✉ t❡♥❤♦ ❝❡rt❡③❛ q✉❡ ❡ss❡ ❞✐❛ ❝❤❡❣❛rá✳ ▼ã❡✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦ ♣♦r t✉❞♦✱ t❡ ❛♠♦ ♠✉✐t♦✳
❆❣r❛❞❡❝❡r ❛♦ ♠❡✉ ♣❛✐✱ ❏♦sé ▼❛r❝í❧✐♦ ❈r✉③✱ é ✐❣✉❛❧♠❡♥t❡ ❞✐❢í❝✐❧✱ ♣♦✐s é ❡❧❡ ❝♦♠ q✉❡♠ ❡✉ ❞❡s❛❜❛❢♦✱ ❢♦✐ ❡❧❡ q✉❡♠ ❡s❝✉t♦✉ t♦❞❛s ❛s ♠✐♥❤❛s r❡❝❧❛♠❛çõ❡s✱ ❢♦✐ ❡❧❡ q✉❡♠ s❡♠♣r❡ ❡st❡✈❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ❛♣♦✐❛♥❞♦ ❡ ❝♦♠♣r❛♥❞♦ t♦❞❛s ❛s ❜❛t❛❧❤❛s q✉❡ t✐✈❡ q✉❡ ❞✐s♣✉t❛r✳ ➱ ❡❧❡ q✉❡ ✜❝❛ s❡♠ ❞♦r♠✐r ♣r❡♦❝✉♣❛❞♦ ❡♠ ❝♦♠♦ r❡s♦❧✈❡r ♠❡✉s ♣r♦❜❧❡✲ ♠❛s✱ é ❡❧❡ q✉❡ ❝♦♠ t♦❞❛ ❛ ♣❛❝✐ê♥❝✐❛ ♠♦str❛✲♠❡ q✉❡ ❞❡✈♦ ♠❡ ❛❝❛❧♠❛r ❡ ♥ã♦ ❞❡❝✐❞✐r ♣r❡❝✐♣✐t❛❞❛♠❡♥t❡✱ ❢♦✐ ❡❧❡ q✉❡♠✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♠❡ ♠♦str♦✉ q✉❡ ❛❜r✐r ♠ã♦ ❞❡ ✉♠ ❞✐r❡✐t♦ ♥ã♦ s✐❣♥✐✜❝❛✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♣❡r❞❡r✱ ♠❛s ✐ss♦ ♣♦❞❡ t♦r♥❛r✲s❡ ✉♠❛ ♣♦rt❛ ❛❜❡rt❛ ♣❛r❛ ✉♠❛ ✈✐tór✐❛ ♠❛✐♦r ❛ ❧♦♥❣♦ ♣r❛③♦✳ ❱✐tór✐❛ ❡ ❞❡rr♦t❛ ❞❡♣❡♥❞❡♠ ❞❡ ❝♦♠♦ ❡♥①❡r❣❛♠♦s ❛s s✐t✉❛çõ❡s✳ P❛✐✱ ❛❣♦r❛✱ ♣♦❞❡♠♦s t❡r ♣❡r❞✐❞♦✱ ♠❛s ❝♦♠ s✉❛ ❛❥✉❞❛ ❛❧❝❛♥ç❛r❡♠♦s ✉♠❛ ❣r❛♥❞❡ ✈✐tór✐❛✱ ❡♠ ❜r❡✈❡✳ ❚❡ ❛♠♦ ♠✉✐t♦✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦✳
❈❛❞❛ ✉♠ ❞❡ ✈♦❝ês ❢♦r❛♠ ❡ sã♦ ❡ss❡♥❝✐❛✐s ❡♠ ♠✐♥❤❛ ✈✐❞❛✱ ❡ ❣r❛ç❛s ❛ ✈♦❝ês ❡✉ ♣✉❞❡ ❝♦♥❝❧✉✐r ♠❛✐s ❡ss❛ ❡t❛♣❛ ❡ t♦r♥❛r✲♠❡ ▼❡str❡✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦✳
❉❡❞✐❝❛tór✐❛
❆ ❉❊❯❙ ❚♦❞♦✲❆♠♦r♦s♦✿ ♦ P❛✐✱ ♦ ❋✐❧❤♦ ❡ ♦ ❊s♣ír✐t♦ ❙❛♥t♦✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞♦ ❡♥s✐♥♦ ❞❡ tó♣✐❝♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ♥♦ ❊♥✲ s✐♥♦ ▼é❞✐♦✳ ➱ ❢❡✐t❛ ✉♠❛ ❜r❡✈❡ ❛♥á❧✐s❡ ❤✐stór✐❝❛ ❞♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ♠♦str❛♥❞♦ ❛❧❣✉♠❛s ✐❞❡✐❛s q✉❡ s❡r✈✐r❛♠ ♣❛r❛ s✉❛ ❢♦r♠❛❧✐③❛çã♦ ❝♦♥❢♦r♠❡ t❡♠♦s ❛t✉❛❧♠❡♥t❡✳ ❙ã♦ ❞❡✜♥✐❞♦s ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ❞♦ ❝á❧❝✉❧♦ ♣❛r❛ ❝♦♠♣❛rá✲❧♦s ❝♦♠ ❛❧❣✉♠❛s ♥♦çõ❡s ✐♥t✉✐t✐✲ ✈❛s ♦✉ ✐❞❡✐❛s ❣❡♦♠étr✐❝❛s✳ P❛r❛ ✜♥❛❧✐③❛r✱ s❡rá ❞✐s❝✉t✐❞❛ ❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❛ ✉t✐❧✐❞❛❞❡ ❞❡ss❡s ❛ss✉♥t♦s ♣❛r❛ ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛ ❋ís✐❝❛ ❡ ❝♦♠♦✱ ❛ ♣❛rt✐r ❞❡ ❛❧✲ ❣✉♠❛s ❛♣❧✐❝❛çõ❡s✱ ♦s ❡st✉❞❛♥t❡s ♣♦❞❡r✐❛♠ ❝♦♠♣r❡❡♥❞❡r ✈ár✐♦s ❝♦♥❝❡✐t♦s ❞❡ ♠❛♥❡✐r❛ ♠❛✐s s✐♠♣❧❡s ❡ ♥ã♦ ♥❡❝❡ss✐t❛r✐❛♠ ♠❡♠♦r✐③❛r ✉♠❛ q✉❛♥t✐❞❛❞❡ ❡♥♦r♠❡ ❞❡ ❢ór♠✉❧❛s✳
❆❜str❛❝t
❚❤✐s ♣❛♣❡r ❞❡❛❧s ✇✐t❤ t❤❡ t❡❛❝❤✐♥❣ ♦❢ t♦♣✐❝s ✐♥ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s ✐♥ ❤✐❣❤ s❝❤♦♦❧✳ ❆ ❜r✐❡❢ ❤✐st♦r✐❝❛❧ ❛♥❛❧②s✐s ♦❢ ✐ts ❞❡✈❡❧♦♣♠❡♥t✱ s❤♦✇✐♥❣ s♦♠❡ ✐❞❡❛s t❤❛t s❡r✈❡❞ ❢♦r ✐ts ❝✉rr❡♥t❧② ❢♦r♠❛❧✐③❛t✐♦♥✳ ❙♦♠❡ ❝❛❧❝✉❧✉s✬ ❡❧❡♠❡♥ts ❛r❡ ❛❧s♦ ❞❡✜♥❡❞ t♦ ❝♦♠♣❛r❡ t❤❡♠ ✇✐t❤ s♦♠❡ ✐♥t✉✐t✐✈❡ ♥♦t✐♦♥s ♦r ❣❡♦♠❡tr✐❝ ✐❞❡❛s✳ ❚♦ ✜♥❛❧✐③❡ ✐t ✇✐❧❧ ❞✐s❝✉ss t❤❡ ✐♠♣♦rt❛♥❝❡ ❛♥❞ ✉s❡❢✉❧♥❡ss ♦❢ t❤❡s❡ ✐ss✉❡s t♦ ♦t❤❡r ❞✐s❝✐♣❧✐♥❡s✱ ♣❛rt✐❝✉✲ ❧❛r❧② ♣❤②s✐❝s ❛♥❞ ❤♦✇✱ ❢r♦♠ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s✱ st✉❞❡♥ts ❝♦✉❧❞ ✉♥❞❡rst❛♥❞ ✈❛r✐♦✉s ❝♦♥❝❡♣ts ✐♥ ❛ ♠♦r❡ s✐♠♣❧❡ ✇❛② ❛♥❞ ❞♦❡s ♥♦t r❡q✉✐r❡ t♦ ♠❡♠♦r✐③❡ ❛ ❤✉❣❡ ❛♠♦✉♥t ♦❢ ❢♦r♠✉❧❛s✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ P❛r♠ê♥✐❞❡s ✭❙■❊▼❙✱ ✷✵✶✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷ ❍❡rá❝❧✐t♦ ✭❈❆❇❘❆▲✱ s✳❞✳✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸ ❩❡♥ã♦ ✭❆❘❆Ú❏❖✱ ✷✵✶✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✹ ❆r✐stót❡❧❡s ✭●❆❇❆✱ ✷✵✶✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✺ ❆q✉✐❧❡s ❳ ❚❛rt❛r✉❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✻ P❛r❛❞♦①♦ ❞♦ ▼♦✈✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✶ ❘❡t❛ ❙❡❝❛♥t❡ ♣❡❧♦s ♣♦♥t♦s P ❡ ◗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷ ◗ ❛♣r♦①✐♠❛✲s❡ ❞❡ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸ ◗ ♠❛✐s ♣ró①✐♠♦ ❞❡ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✹ ❘❡t❛ ❚❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ P = (x0, f(x0)) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷✳✺ ❙❡❝❛♥t❡ lP Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✻ ❙♦♠❛s ❞❡ ❘✐❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✸✳✶ ❚✐♣♦s ❞❡ ♠♦✈✐♠❡♥t♦ r❡t✐❧í♥❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✶ ❝✐❝❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✸ ▲❛♥ç❛♠❡♥t♦ ❱❡rt✐❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✹ ❙✐st❡♠❛ ▼❛ss❛✲♠♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ❆❧❣✉♥s ✈❛❧♦r❡s ❞❡ x ❡ ❞❡ f(x)♣❛r❛ x ♣ró①✐♠♦s ❞❡ 3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❆❧❣✉♥s ✈❛❧♦r❡s x ❡ ❞❡ sen(π
x)♣❛r❛ x ♣ró①✐♠♦ ❞❡ 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ❘❡❣r❛s ❞❡ ❞❡r✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✹ ❆❧❣✉♠❛s ■♥t❡❣r❛✐s ■♥❞❡✜♥✐❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✺
✶✳✶ ▼♦t✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷ ❖❜❥❡t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✸ ❊str✉t✉r❛çã♦ ❞♦ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷ ❯♠ ♣♦✉❝♦ ❞❡ ❈á❧❝✉❧♦ ✷✸
✷✳✶ ▲✐♠✐t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶✳✶ ❈♦♥t✐♥✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷ ❉❡r✐✈❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷✳✶ ❚❛♥❣❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✷ ❱❡❧♦❝✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✷✳✸ ❋✉♥çã♦ ❉❡r✐✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✸ ■♥t❡❣r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✸✳✶ ■♥t❡❣r❛✐s ✐♥❞❡✜♥✐❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✸✳✷ ■♥t❡❣r❛✐s ❞❡✜♥✐❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✸✳✸ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✸ ❖ ❈á❧❝✉❧♦ ❡ ❛ ❋ís✐❝❛ ✺✾
✸✳✶ ❖ ▼♦✈✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✶✳✶ ▼♦✈✐♠❡♥t♦ ❘❡t✐❧í♥❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✶✳✷ ▼♦✈✐♠❡♥t♦ ❍❛r♠ô♥✐❝♦ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✷ ❊♥❡r❣✐❛ ❡ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✸✳✷✳✶ ❈❛s♦s P❛rt✐❝✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✽✵
❘❡❢❡rê♥❝✐❛s ✽✹
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
✏❊ ♣✉r s❡ ♠✉♦✈❡✳✑ ●❛❧✐❧❡✉ ●❛❧✐❧❡✐
❍✐r❛♠✱ ❝✐t❛❞♦ ♥❛ ❡♣í❣r❛❢❡ ❞♦ tr❛❜❛❧❤♦✱ ❢♦✐ ♦ ♠❛✐s ❤á❜✐❧ ❛rt❡sã♦ ❢✉♥❞✐❞♦r ❞❡ ♠❡t❛✐s ❞♦ t❡♠♣♦ ❞♦ ❘❡✐ ❙❛❧♦♠ã♦✱ ♦ q✉❛❧ é ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ ❞♦s ♠❛✐♦r❡s r❡✐s ❞❡ t♦❞♦ ♦ ♣♦✈♦ ❥✉❞❡✉ ♣♦r s✉❛ ✐♥✜♥✐t❛ s❛❜❡❞♦r✐❛✳ ❊st❡ ♠❡s♠♦ ✈❡rsí❝✉❧♦ é r❡♣❡t✐❞♦ ❡♠ ✷ ❈r ✹✱ ✷✳ ❊ ♦ q✉❡ ✐ss♦ t❡♠ ❝♦♠ ♦ ❈á❧❝✉❧♦❄ ❚✉❞♦✦ ❊ss❛ ❞❡s❝r✐çã♦ ♥❛rr❛ ♣❛r❛ ♥ós ♦ ✐♥í❝✐♦ ❞♦
❢❛♠♦s♦ ♥ú♠❡r♦π✳ ❈✐♥❝♦ ♠❡tr♦s ❞❡ ❞✐â♠❡tr♦ ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ q✉✐♥③❡ ♠❡tr♦s✳
❆♣❡s❛r ❞❡ t♦❞❛ s✉❛ s❛❜❡❞♦r✐❛✱ ♥♦ t❡♠♣♦ ❞♦ ❘❡✐ ❙❛❧♦♠ã♦ ✉s❛✈❛✲s❡ π = 3✳
❆ ✐♠♣♦rtâ♥❝✐❛ ❡ ❛ r❡❧❛çã♦ ❞♦ π ❝♦♠ ♦ ❈á❧❝✉❧♦ é ♠✉✐t♦ ❣r❛♥❞❡✱ ♣♦✐s ❢♦r❛♠ ♣r♦✲
❝❡ss♦s ❝♦♠♦ ♦s ❞❡ ❞❡t❡r♠✐♥❛r ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ q✉❡ ❞❡r❛♠ ♦r✐❣❡♠ às ✐❞❡✐❛s ✉t✐❧✐③❛❞❛s ♣❛r❛ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✳ ❊ ✉♠❛ ❞❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞❡st❡ ❝á❧❝✉❧♦ é✱ ❡①❛t❛♠❡♥t❡✱ ♦ ♥ú♠❡r♦ π✳
❖ ✜❧ós♦❢♦ ♣ré✲s♦❝rát✐❝♦ P❛r♠ê♥✐❞❡s ✭✺✸✵✲✹✻✵ ❛✳❈✮ ✭❋✐❣✉r❛ ✶✳✶✮ ❛❝r❡❞✐t❛✈❛ q✉❡ só s❡r✐❛ ♣♦ssí✈❡❧ ♦❜t❡r♠♦s ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ✈❡r❞❛❞❡✐r♦ ❞❛q✉✐❧♦ q✉❡ ♥ã♦ ♠✉❞❛✱ q✉❡ é ❡t❡r♥♦ ❡ ✐♥✈❛r✐á✈❡❧✱ ❛♣❡s❛r ❞❛s ❛♣❛rê♥❝✐❛s✳ P♦ré♠ ✉♠ ❛❢♦r✐s♠♦ ❞❡ ❍❡rá❝❧✐t♦ ✭✺✸✺✲✹✼✺ ❛✳❈✮ ✭❋✐❣✉r❛ ✶✳✷✮ ❞✐③ q✉❡ ✏♥ã♦ é ♣♦ssí✈❡❧ ❜❛♥❤❛r✲♥♦s ♥♦ ♠❡s♠♦ r✐♦ ❞✉❛s ✈❡③❡s✱ ♣♦✐s q✉❛♥❞♦ ✈♦❧t❛♠♦s ❛♦ r✐♦✱ ♥❡♠ ♥ós ♥❡♠ ♦ r✐♦ sã♦ ♠❛✐s ♦s ♠❡s♠♦s✑✳ ❊❧❡ ❛❝r❡❞✐t❛✈❛ ♥❛ ✈✐❞❛ ❝♦♠♦ ✉♠❛ ♠✉❞❛♥ç❛ ❝♦♥st❛♥t❡ ❡ ❡t❡r♥❛✱ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ✜❧ós♦❢♦ ❞♦ t✉❞♦ ✢✉✐✳ ❖✉ ❛✐♥❞❛✱ ❝♦♠♦ ❥á ♦✉✈✐ ❝❡rt❛ ✈❡③✿ ✏◆❛ ✈✐❞❛ ❛ ú♥✐❝❛ ❝♦✐s❛ q✉❡ ♥ã♦ ♠✉❞❛✱ é ❛ ❝❡rt❡③❛ ❞❛ ♠✉❞❛♥ç❛✑✳
❊st❛s ✐❞❡✐❛s ❝♦♥tr❛♣õ❡♠✲s❡ ♥♦ ♣❡♥s❛♠❡♥t♦ ❤✉♠❛♥♦ ❤á ❜❛st❛♥t❡ t❡♠♣♦ ❡ ❡ss❡s ✜❧ós♦❢♦s r❡♣r❡s❡♥t❛♠ ♣♦s✐çõ❡s ❛♥t❛❣ô♥✐❝❛s q✉❡ s♦❜r❡✈✐✈❡♠ ❛♦ ❧♦♥❣♦ ❞♦s sé❝✉❧♦s✳ ❆t✉✲ ❛❧♠❡♥t❡ ❡❧❛s sã♦ ♠✉✐t♦ ❜❡♠ r❡♣r❡s❡♥t❛❞❛s ♣❡❧❛ ✏▼❡t❛♠♦r❢♦s❡ ❆♠❜✉❧❛♥t❡✑ r❡❢❡r❡♥t❡
■♥tr♦❞✉çã♦ ❈❛♣ít✉❧♦ ✶
à ♠✉❞❛♥ç❛ ❝♦♥st❛♥t❡ ♦✉ ❛✐♥❞❛ s♦❜r❡ ❛ r❡s✐❣♥❛çã♦ ❞❡ ✏❊✉ ♥❛s❝✐ ❛ss✐♠✱ ❡✉ ❝r❡s❝✐ ❛s✲ s✐♠✱ ❡✉ ✈♦✉ s❡r s❡♠♣r❡ ❛ss✐♠✳✳✳✳✑ ❞❛ ❝❛♥çã♦ ✏▼♦❞✐♥❤❛ P❛r❛ ●❛❜r✐❡❧❛✑ ❞♦s ❢❛♠♦s♦s ❛rt✐st❛s s♦t❡r♦♣♦❧✐t❛♥♦s ❘❛✉❧ ❙❡✐①❛s ❡ ●❛❧ ❈♦st❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❋✐❣✉r❛ ✶✳✶✿ P❛r♠ê♥✐❞❡s ✭❙■❊▼❙✱ ✷✵✶✶✮ ❋✐❣✉r❛ ✶✳✷✿ ❍❡rá❝❧✐t♦ ✭❈❆❇❘❆▲✱ s✳❞✳✮
▼❛s✱ q✉❛❧ ❛ r❡❧❛çã♦ ❞❡ss❡s ♣❡♥s❛♠❡♥t♦s ♦✉ ❝❛♥çõ❡s ❝♦♠ ♦ ❈á❧❝✉❧♦❄
❙❡♠♣r❡ q✉❡ ♣❡♥s❛♠♦s ❡♠ ❣r❛♥❞❡③❛s q✉❡ ✈❛r✐❛♠ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✱ ♦✉ ❣r❛♥❞❡③❛s q✉❡ ✈❛r✐❛♠ ❡♠ ❢✉♥çã♦ ❞❡ ♦✉tr❛s ❣r❛♥❞❡③❛s✱ ❡stã♦ ✐♥trí♥s❡❝❛s ❡♠ ♥♦ss♦s ♣❡♥s❛♠❡♥t♦s ❛s ✐❞❡✐❛s ❞♦ ❈á❧❝✉❧♦✳ ➱ ❡①❛t❛♠❡♥t❡ s♦❜r❡ ❡ss❛ ❞✐❛❧ét✐❝❛ ❡♥tr❡ ✈❛r✐❛çã♦ ❡ ❝♦♥stâ♥❝✐❛ q✉❡ r❡s✐❞❡ ❛ ❡ssê♥❝✐❛ ❞❡ss❡ ❡st✉❞♦✳ ❈♦♥s✐❞❡r❛r ❣r❛♥❞❡③❛s ✈❛r✐á✈❡✐s ❝♦♠♦ ❝♦♥st❛♥t❡s ❡♠ ♣❡q✉❡♥♦s ✐♥t❡r✈❛❧♦s ♦✉ ❛✐♥❞❛ ❝❛r❛❝t❡r✐③❛r ❞❡ q✉❡ ❢♦r♠❛ ❡ss❛s ❣r❛♥❞❡③❛s ✈❛r✐❛♠✳ ❚✉❞♦ ✐ss♦ é ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ q✉❡ s❡rá ❝❤❛♠❛❞♦ ❛♣❡♥❛s ❞❡ ❈á❧❝✉❧♦✳
❖ ❡st✉❞♦ ❞❡ss❡s tó♣✐❝♦s ♦❝♦rr❡✉ ❛♦ ❧♦♥❣♦ ❞❡ ✈ár✐♦s sé❝✉❧♦s ❡ ❝♦♥t♦✉ ❝♦♠ ❛ ❝♦♥✲ tr✐❜✉✐çã♦ ❞❡ ✈ár✐❛s ♣❡ss♦❛s✳ ◆❛❞❛ ❛♣❛r❡❝❡✉ ♣r♦♥t♦✱ ♠❛s ❛ ♣❛rt✐r ❞❡ ♥❡❝❡ss✐❞❛❞❡s ❤✉♠❛♥❛s ❛ ♠❛t❡♠át✐❝❛ ❞❡s❡♥✈♦❧✈❡✲s❡ ❞❡ ♠❛♥❡✐r❛ ❡①tr❛♦r❞✐♥ár✐❛✳ ❍♦❥❡✱ ♣♦ré♠✱ ❡ss❡ ❛ss✉♥t♦ ❢♦✐ r❡t✐r❛❞♦ ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✱ ❡ é ❞❡✐①❛❞♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣❛r❛ ♦s ❝✉rs♦s ❞❡ ♥í✈❡❧ s✉♣❡r✐♦r✳ ❊ss❡ tr❛❜❛❧❤♦ ♠♦str❛rá q✉❡ ❛s ✐❞❡✐❛s ❢✉♥❞❛♠❡♥t❛✐s ❞♦ ❈á❧❝✉❧♦ sã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ t♦❞❛s ❛s ♣❡ss♦❛s ❞❡✈✐❞♦ ❛ s✉❛ ❣r❛♥❞❡ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ ❞✐✲ ✈❡rs❛s ár❡❛s✱ ♣♦❞❡♥❞♦ ❡ ❞❡✈❡♥❞♦ s❡r ✐♥❝❧✉í❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❈♦st✉♠❛♠♦s ♣❡♥s❛r ❛ ♣r✐♥❝í♣✐♦ ❛♣❡♥❛s ❡♠ ár❡❛s t❡❝♥♦❧ó❣✐❝❛s✱ ♠❛s s❛❜❡♠♦s q✉❡ ♦ ❈á❧❝✉❧♦ é ✉♠❛ ❢❡rr❛✲ ♠❡♥t❛ ❡ss❡♥❝✐❛❧ ❡♠ ♦✉tr❛s ár❡❛s✱ ❝♦♠♦ ❛ ár❡❛ ❞❛ s❛ú❞❡ ❡ ❛ ❞❛s ❝✐ê♥❝✐❛s ❤✉♠❛♥❛s✳ ❖s ❡st✉❞❛♥t❡s ❞❡ ♥í✈❡❧ ♠é❞✐♦ t❛♠❜é♠ s❡ ❜❡♥❡✜❝✐❛rã♦ ❝♦♠ ♦ ❡st✉❞♦ ❞❡ss❡s ❝♦♥t❡ú❞♦s ✉♠❛ ✈❡③ q✉❡ ♣♦❞❡rã♦ ✉t✐❧✐③á✲❧♦s ♥ã♦ só ❡♠ q✉❡stõ❡s ♣ró♣r✐❛s ❞❛ ▼❛t❡♠át✐❝❛✱ ♠❛s ❡♠ ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s✱ ❝♦♠♦ ❋ís✐❝❛✱ ◗✉í♠✐❝❛ ❡ ❛té ❇✐♦❧♦❣✐❛✳
❋❛③❡♥❞♦ ✉♠❛ r❡tr♦s♣❡❝t✐✈❛ ❤✐stór✐❝❛ ❞❛ ❡❞✉❝❛çã♦ ♥♦ ❇r❛s✐❧✱ ✈❡♠♦s q✉❡✱ r❡❝❡♥✲ t❡♠❡♥t❡✱ ♣♦r ♣❡❧♦ ♠❡♥♦s ✸ ✈❡③❡s✱ tó♣✐❝♦s ❞❡ ❈á❧❝✉❧♦ ✜③❡r❛♠ ♣❛rt❡ ❞♦ ❝✉rrí❝✉❧♦ ♦✜❝✐❛❧ ❞❛s ❡s❝♦❧❛s s❡❝✉♥❞ár✐❛s✱ ♣r✐♠❡✐r♦ ❝♦♠ ❛ r❡❢♦r♠❛ ❇❡♥❥❛♠✐♠ ❈♦♥st❛♥t ✭✶✽✾✵✮✱ r❛t✐✜❝❛❞❛ ♥❛ r❡❢♦r♠❛ ❋r❛♥❝✐s❝♦ ❈❛♠♣♦s ✭✶✾✸✶✮ ❡ ❞❡♣♦✐s ❝♦♠ ❛ r❡❢♦r♠❛ ●✉st❛✈♦ ❈❛✲
■♥tr♦❞✉çã♦ ❈❛♣ít✉❧♦ ✶
♣❛♥❡♠❛ ✭✶✾✹✷✮✳ ◆♦ ✜♥❛❧ ❞❛ ❞é❝❛❞❛ ❞❡ ✻✵ ❡ ✐♥í❝✐♦ ❞❛ ❞é❝❛❞❛ ❞❡ ✼✵✱ ❝♦♠ ❛ ✐♥✢✉ê♥❝✐❛ ❞♦ ▼♦✈✐♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛ ▼♦❞❡r♥❛ ♦ ❡♥s✐♥♦ ❞❡ss❛ ❞✐s❝✐♣❧✐♥❛ ❢♦✐ ❜❛st❛♥t❡ ❛❧t❡✲ r❛❞♦✱ ♥♦ ❇r❛s✐❧ ❡ ♥♦ ♠✉♥❞♦✳ ❯♠❛ ❞❡ss❛s ❛❧t❡r❛çõ❡s ❢♦✐ ❛ ❡①❝❧✉sã♦ ❞♦s ❝♦♥t❡ú❞♦ ❞❡ ❈á❧❝✉❧♦ ♥♦ ❡♥s✐♥♦ s❡❝✉♥❞ár✐♦✳
✏❖ ❈á❧❝✉❧♦ ✈❡♠ ❞❡s❡♠♣❡♥❤❛♥❞♦ ✉♠ ♣❛♣❡❧ ❞❡ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛ ❡♠ t♦❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❝✐❡♥tí✜❝♦✲t❡❝♥♦❧ó❣✐❝♦✳ P♦rt❛♥t♦✱ ❞❡s❝❛rtá✲❧♦ ♥♦ ❡♥s✐♥♦ é ❣r❛✈❡✱ ♣♦rq✉❡ ❞❡✐①❛ ❞❡ ❧❛❞♦ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ s✐❣♥✐✜❝❛t✐✈❛ ❡ ❝❡r✲ t❛♠❡♥t❡ ❛ ♠❛✐s r❡❧❡✈❛♥t❡ ❞❛ ▼❛t❡♠át✐❝❛ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦ ❛❧✉♥♦ ♥✉♠ ❝♦♥t❡①t♦ ❞❡ ❡♥s✐♥♦ ♠♦❞❡r♥♦ ❡ ❛t✉❛❧✳ ❬✳✳✳❪ ❖ ❈á❧❝✉❧♦ é ♠♦❞❡r♥♦ ♣♦rq✉❡ tr❛③ ✐❞é✐❛s ♥♦✈❛s✱ ❞✐❢❡r❡♥t❡s ❞♦ q✉❡ ♦ ❛❧✉♥♦ ❞❡ ✷♦❣r❛✉ ❡♥❝♦♥tr❛ ♥❛s ♦✉tr❛s ❝♦✐s❛s q✉❡ ❛♣r❡♥❞❡ ❡♠ ❆r✐t♠ét✐❝❛✱ ➪❧❣❡❜r❛✱ ●❡♦♠❡tr✐❛✱ ❚r✐❣♦♥♦♠❡tr✐❛ ❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ◆ã♦ ❛♣❡♥❛s ♥♦✈❛s✱ ♠❛s ✐❞é✐❛s q✉❡ tê♠ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛ ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❛♣❧✐❝❛çõ❡s ❝✐❡♥tí✜❝❛s ♥♦ ♠✉♥❞♦ ♠♦❞❡r♥♦✳ ❖r❛✱ ♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞♦ ❡♥s✐♥♦ ♥ã♦ é ♦✉tr♦ s❡♥ã♦ ♣r❡♣❛r❛r ♦ ❥♦✈❡♠ ♣❛r❛ s❡ ✐♥t❡❣r❛r ♠❛✐s ❛❞❡q✉❛❞❛♠❡♥t❡ à s♦❝✐❡❞❛❞❡✳ ◆ã♦ s❡ ✈✐s❛✱ ❝♦♠
♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ♥♦ ✷♦ ❣r❛✉✱ ❢♦r♠❛r ❡s♣❡❝✐❛❧✐st❛s ♥♦ ❛ss✉♥t♦✳✑
✭➪❱■▲❆✱ ✶✾✾✶✱ ♣✳ ✸✮
❆ss✐♠ ❝♦♠♦ ❛✜r♠❛ ➪✈✐❧❛✱ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭P❈◆✮ ♣r♦♣õ❡♠ q✉❡ ♦ ❝✉rrí❝✉❧♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❣❛r❛♥t❛ ❛♦s ❡st✉❞❛♥t❡s ✉♠❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❛♠✲ ♣❧✐❛r ❡ ❛♣r♦❢✉♥❞❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❛❞q✉✐r✐❞♦s ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ♠❛♥❡✐r❛ ✐♥t❡❣r❛❞❛ ❡♥tr❡ ❛s ❞✐s❝✐♣❧✐♥❛s ♣r❡♣❛r❛♥❞♦✲♦s ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❡ ♣❛r❛ ♦ ❡①❡r❝í❝✐♦ ❞❛ ❝✐❞❛❞❛♥✐❛✱ ❜❡♠ ❝♦♠♦ ❝❛♣❛❝✐t❛♥❞♦✲♦s ♣❛r❛ ❝♦♥t✐♥✉❛r ♦s ❡st✉❞♦s ♥♦ ♥í✈❡❧ s✉♣❡r✐♦r✳
❆ss✐♠ s❡♥❞♦✱ ❝❛❜❡ ❛♦ ❊♥s✐♥♦ ▼é❞✐♦ ♣r♦✈❡r ❛s ❢❡rr❛♠❡♥t❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ❣❛✲ r❛♥t✐r t✉❞♦ ♦ q✉❡ é ♣r♦♣♦st♦ ♥♦s P❈◆✳ ❈♦♠ ✐ss♦✱ ❞❡✈❡✲s❡ ❞✐s❝✉t✐r ❝♦♠♦ ❡ q✉❛✐s ❝♦♥t❡ú❞♦s ❞❡ ❈á❧❝✉❧♦ ❞❡✈❡♠ s❡r ♠✐♥✐str❛❞♦s✱ ✉♠❛ ✈❡③ q✉❡ ❛ ♣rát✐❝❛ ♠♦str❛ q✉❡ ♦ ❝✉rrí❝✉❧♦ ❞❡ ♠❛t❡♠át✐❝❛ ❥á ❡stá ❜❛st❛♥t❡ ❝♦♠♣r♦♠❡t✐❞♦✱ ❝♦♠ ♠✉✐t♦s ❛ss✉♥t♦s✱ ❞✐✜❝✉❧t❛♥❞♦ ❛ ✐♥s❡rçã♦ ❞❡ ♠❛✐s ❝♦♥t❡ú❞♦✳
❖ ❡st✉❞♦ ❞♦ ❈á❧❝✉❧♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ♣♦❞❡ ❛♣r❡s❡♥t❛r ❞✐✈❡rs❛s ✈❛♥t❛❣❡♥s✱ ♠❛s ♥ã♦ ❞❡✈❡ s❡r ❢❡✐t♦ ❞❡ q✉❛❧q✉❡r ♠❛♥❡✐r❛✳ ❯♠❛ ✐❞❡✐❛ q✉❡ ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞❛ é ♦ ❡♥s✐♥♦ ❛rt✐❝✉❧❛❞♦ ❝♦♠ ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s✱ ♣♦r ❡①❡♠♣❧♦ ❛ ❋ís✐❝❛✳ P♦❞❡✲s❡ ❡♥s✐♥❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ ❛ ♣❛rt✐r ❞❛ ♠❡❝â♥✐❝❛✱ ❝♦♠ ❡①❡♠♣❧♦s ❞❡ ♠♦✈✐♠❡♥t♦ ✉♥✐❢♦r♠❡✱ ♠♦✈✐♠❡♥t♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ✈❛r✐❛❞♦✱ ♣r❡ssã♦✱ ❞❡♥s✐❞❛❞❡ ❞❡ ❝❛r❣❛ ❡❧étr✐❝❛✱ ❞❡♥tr❡ ♦✉✲ tr♦s✳ ❖✉ ❛✐♥❞❛ ♣❛r❛ ✉s♦ ✐♥t❡r♥♦ ♥❛ ♣ró♣r✐❛ ▼❛t❡♠át✐❝❛✱ ❝♦♠ ❛ ✐❞❡✐❛ ❣❡♦♠étr✐❝❛
▼♦t✐✈❛çã♦ ❈❛♣ít✉❧♦ ✶✿ ■♥tr♦❞✉çã♦
❞❡ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ✉♠❛ ❝✉r✈❛ ♥✉♠ ❞❛❞♦ ♣♦♥t♦✱ ❝♦♥t❡①t✉❛❧✐③❛♥❞♦ ♦ ❝♦♥t❡ú❞♦ ♣❛r❛ ❡st✐♠✉❧❛r ♦ ✐♥t❡r❡ss❡ ❞♦ ❡st✉❞❛♥t❡✳
❖ r❡✢❡①♦ ❞❛ ❢❛❧t❛ ❞❡ss❡s ❝♦♥t❡ú❞♦s é s❡♥t✐❞♦ ♥♦s ❝✉rs♦s ✐♥✐❝✐❛✐s ❞❡ ❈á❧❝✉❧♦ ♥♦ ♥í✈❡❧ s✉♣❡r✐♦r✳ ❙❡❣✉♥❞♦ ❡st✉❞♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣❡❧♦ ■♥st✐t✉t♦ ◆❛❝✐♦♥❛❧ ❞❡ ❊st✉❞♦s ❡ P❡sq✉✐s❛s ❊❞✉❝❛❝✐♦♥❛✐s ✭■◆❊P✮✱ ♦s ❝✉rs♦s ❞❡ ❈á❧❝✉❧♦ ❡stã♦ ❝♦♠ í♥❞✐❝❡s ❞❡ r❡♣r♦✈❛çã♦ ❛❧tíss✐♠♦s ❡ ✐ss♦ ♥♦s r❡♠❡t❡ ♥♦✈❛♠❡♥t❡ à q✉❡stã♦ ❞❡ ✐♥tr♦❞✉③✐r ❡ss❡ ❝♦♥t❡ú❞♦ ♥♦ ❝✉rrí❝✉❧♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❉❡✈❡✲s❡ ❝♦♠♣r❡❡♥❞❡r ♦ ♣♦rq✉ê ❞❛ ❡①❝❧✉sã♦ ❞♦ ❡♥s✐♥♦ ❞♦ ❈á❧❝✉❧♦ ♥♦ ♥í✈❡❧ ♠é❞✐♦✱ ❛♣❡s❛r ❞❡ s✉❛ ♥♦tór✐❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ♦ ♣♦rq✉ê ❞❡ t❛♥t❛ ❞✐✜❝✉❧❞❛❞❡ ❡♥❝♦♥tr❛❞❛ ♣❡❧♦s ❡st✉❞❛♥t❡s✳ ❯♠❛ ❤✐♣ót❡s❡✱ é ❛ ❢❛❧t❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ s✉❛ ❤✐stór✐❛✱ ❞❡ s✉❛ ♦r✐❣❡♠ ❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✳ ❊ ♦✉tr❛ ❤✐♣ót❡s❡✱ é ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♥❝♦♥tr❛❞❛ ♣♦rq✉❡ ♦ ❈á❧❝✉❧♦ ❛♣r❡s❡♥t❛ ✉♠❛ ✈❡r❞❛❞❡✐r❛ r❡✈♦❧✉çã♦ ♥❛s ✐❞❡✐❛s às q✉❛✐s ❡stá✈❛♠♦s ♣r❡s♦s✱ ✐♥tr♦❞✉✲ ③✐♥❞♦ ❝♦♥❝❡✐t♦s ❝♦♠♦ ♦s ❞❡ ❧✐♠✐t❡ q✉❡ ❢❛③❡♠ ❛♣❡❧♦ à ✐♥t✉✐çã♦✱ à ✐❞❡✐❛s ❛❜str❛t❛s ❡✱ t❛❧✈❡③✱ ❛ ♠❛✐s ❞✐❢í❝✐❧ ✐❞❡✐❛✱ q✉❡ ❞❡♠♦r♦✉ ♠✉✐t♦ t❡♠♣♦ ♣❛r❛ ❝❤❡❣❛r à ❝♦♠♣r❡❡♥sã♦ ❞♦ s❡r ❤✉♠❛♥♦✿ ❛ ✐❞❡✐❛ ❞❡ ✐♥✜♥✐t♦✳
✶✳✶ ▼♦t✐✈❛çã♦
◆♦ ❊♥s✐♥♦ ❙✉♣❡r✐♦r✱ ♣♦✉❝♦s s❡ ✐♥t❡r❡ss❛♠ ❡♠ ❝♦♠♣r❡❡♥❞❡r t♦❞❛s ❡ss❛s ♥♦✈❛s ✐❞❡✐❛s ❡ ❛♣r❡♥❞❡r ❛ ❢✉♥❞♦ ♦ ❈á❧❝✉❧♦✳ ❆ ♠❛✐♦r✐❛ ❞♦s ❡st✉❞❛♥t❡s ✜❝❛ ♣r❡s❛ ❛♣❡♥❛s às s✉❛s ❛♣❧✐❝❛çõ❡s✱ s❡♠ s❡ ♣r❡♦❝✉♣❛r ❝♦♠ ✉♠ ❡st✉❞♦ ♠❛✐s ❛❜r❛♥❣❡♥t❡ ❡ ❝♦♠♣❧❡t♦✳ ❆ss✐♠✱ ♦ ❡st✉❞❛♥t❡ ❡s❜❛rr❛ ♥❛s ❞✐✜❝✉❧❞❛❞❡s ❞❛s ♦♣❡r❛çõ❡s ❡ ♦ ❡st✉❞♦ ❞❡ss❡ ❝♦♥t❡ú❞♦ t♦r♥❛✲s❡ ❞❡s♣r❛③❡r♦s♦✳
❊ss❡ ♣r♦❜❧❡♠❛ ♣♦❞❡r✐❛ s❡r r❡s♦❧✈✐❞♦ s❡ ❥á ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❢♦ss❡♠ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ❞❡ss❛s ♥♦çõ❡s✱ ❝♦♠♦ ❛s ❞❡ ❧✐♠✐t❡✱ ♠♦str❛♥❞♦ q✉❛✐s t✐♣♦s ❞❡ ♣r♦❜❧❡♠❛s ✐♠✲ ♣✉❧s✐♦♥❛r❛♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❈á❧❝✉❧♦✱ ♣❛r❛ q✉❡ ♦s ❡st✉❞❛♥t❡s s❡ ✐♥t❡r❡ss❛ss❡♠ ♣♦r ❡ss❡s ❛ss✉♥t♦s ❡ s❡♥t✐ss❡♠ ♠❛✐s ✈♦♥t❛❞❡ ♣❛r❛ ❡st✉❞❛r✳ P♦❞❡r✐❛♠ s❡r ❛♣r❡s❡♥t❛✲ ❞♦s ♦s ♣r♦❜❧❡♠❛s ❞❛s q✉❛❞r❛t✉r❛s✱ ❛s ✐❞❡✐❛s ❞♦ ▼ét♦❞♦ ❞❛ ❊①❛✉stã♦ ♦✉ ♠❡s♠♦ ❛s ✐❞❡✐❛s ♣♦r trás ❞❛s ✐♥t❡❣r❛✐s✳
❆ ♣r♦♣♦st❛ ♥ã♦ é ❡♥s✐♥❛r ♦ ❈á❧❝✉❧♦ ❝♦♠ t♦❞♦ s❡✉ r✐❣♦r✱ ✐ss♦ t❛♠❜é♠ s❡r✐❛ ❞❡s✲ ♥❡❝❡ssár✐♦✱ ❛❧é♠ ❞❡ ❜❛st❛♥t❡ ❡♥❢❛❞♦♥❤♦ ♣❛r❛ ♦s ❡st✉❞❛♥t❡s ❞❡ ♥í✈❡❧ ♠é❞✐♦✱ ❡ ❤✐s✲ t♦r✐❝❛♠❡♥t❡✱ ♦ ❈á❧❝✉❧♦ t❛♠❜é♠ ♥ã♦ s✉r❣✐✉ ❛ss✐♠✱ ♠❛s ❝♦♠❡ç♦✉ ❝♦♠ ✐❞❡✐❛s s✐♠♣❧❡s ❡ ❧❡✈♦✉ ❜❛st❛♥t❡ t❡♠♣♦ ♣❛r❛ r❡❝❡❜❡r ♦ ❢♦r♠❛❧✐s♠♦ ❛t✉❛❧✳ ❉❡✈❡✲s❡ ❛♣r❡s❡♥tá✲❧♦ ❛♦s ❡st✉❞❛♥t❡s ❝♦♠♦ ❝r✐❛çã♦ ❤✉♠❛♥❛ q✉❡ ♥♦s ❛❥✉❞❛ ❛ r❡s♦❧✈❡r ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s✳
▼♦t✐✈❛çã♦ ❈❛♣ít✉❧♦ ✶✿ ■♥tr♦❞✉çã♦
❙❡❣✉♥❞♦ ➪✈✐❧❛ ✭✶✾✾✶✮✿
✏❊♥s✐♥❛✲s❡ ▼❛t❡♠át✐❝❛ ♣♦rq✉❡ ❡st❛ é ✉♠❛ ❞✐s❝✐♣❧✐♥❛ q✉❡ ❢❛③ ♣❛rt❡ s✐❣♥✐✜❝❛t✐✈❛ ❞❛ ❡①♣❡r✐ê♥❝✐❛ ❤✉♠❛♥❛ ❛♦ ❧♦♥❣♦ ❞♦s sé❝✉❧♦s✱ ♣♦rq✉❡ ❡❧❛ ❝♦♥t✐♥✉❛ s❡♥❞♦ ❤♦❥❡✱ ❝♦♠ ✐♥t❡♥s✐❞❛❞❡ ❛✐♥❞❛ ♠❛✐♦r ❞♦ q✉❡ ♥♦ ♣❛ss❛❞♦✱ ✉♠ ✐♥str✉♠❡♥t♦ ❡✜❝❛③ ❡ ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ♦s ♦✉tr♦s r❛♠♦s ❞♦ ❝♦♥❤❡❝✐✲ ♠❡♥t♦✳✑
P♦r ✐ss♦ é ✐♠♣♦rt❛♥t❡ ♣r✐✈✐❧❡❣✐❛r ✉♠❛ ▼❛t❡♠át✐❝❛ q✉❡ s❡❥❛ ❝❛♣❛③ ❞❡ ♠♦❞✐✜❝❛r ♥♦ss❛ ✈✐sã♦ ❞❡ ♠✉♥❞♦ ❡ ♥♦s ♣❡r♠✐t✐r ❛♣❧✐❝❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♣❛r❛ ❛s ❞✐✈❡rs❛s ❛t✐✈✐❞❛❞❡s q✉❡ ❢♦r♠♦s r❡❛❧✐③❛r✱ ♥ã♦ ✉♠❛ ▼❛t❡♠át✐❝❛ ❡stát✐❝❛✱ ❞❡ ❞❡❝♦r❛r ❢ór♠✉❧❛s ♦✉ ♣r♦❜❧❡♠❛s ♣r♦♥t♦s✳
❊♥tã♦ ❛ ✐❞❡✐❛ ❞❡ ❛♣r❡s❡♥tá✲❧❛ ♥♦ ♥í✈❡❧ ♠é❞✐♦ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦s ❡st✉❞❛♥t❡s✱ ♣♦✐s ❡❧❡s s❡rã♦ ♦s ♥♦✈♦s ❡♥❣❡♥❤❡✐r♦s✱ ♠é❞✐❝♦s✱ t❡❝♥ó❧♦❣♦s✱ ❢ís✐❝♦s✱ ♠❛t❡✲ ♠át✐❝♦s✱ q✉í♠✐❝♦s✱ ❜✐ó❧♦❣♦s✱ ❡t❝✳ ▼❛s ❝♦♠♦ ❢❛③❡r ✐ss♦ t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ♦ ❝✉rrí❝✉❧♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❥á é ❜❛st❛♥t❡ ❞❡♥s♦✱ ❥á t❡♠ ❛ss✉♥t♦ ❞❡♠❛✐s ♣❛r❛ tr❛❜❛❧❤❛r❄
❙♦❜r❡ ❡ss❡ ♣♦♥t♦ ➪✈✐❧❛ ✭✶✾✾✶✮ ❛✜r♠❛✿
✏❆ ✐❞é✐❛ ❞❡ q✉❡ ♦s ♣r♦❣r❛♠❛s ❞❡ ▼❛t❡♠át✐❝❛ sã♦ ❡①t❡♥s♦s ❡ ♥ã♦ ❝♦♠♣♦rt❛r✐❛♠ ❛ ✐♥❝❧✉sã♦ ❞♦ ❈á❧❝✉❧♦ é ✉♠ ❡q✉í✈♦❝♦✳ ❖s ❛t✉❛✐s ♣r♦❣r❛♠❛s ❡stã♦✱ ✐st♦ s✐♠✱ ♠❛❧ ❡str✉t✉r❛❞♦s✳✑
❖✉ s❡❥❛✱ ♣❡r❞❡✲s❡ ♠✉✐t♦ t❡♠♣♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡♥s✐♥❛♥❞♦ ❢♦r♠❛❧✐s♠♦s✱ r❡❣r❛s ❡ ♥♦♠❡♥❝❧❛t✉r❛s q✉❡ ♥ã♦ sã♦ ❡ss❡♥❝✐❛✐s✳ P♦❞❡r✲s❡✲✐❛ ✉t✐❧✐③❛r ❡ss❡ t❡♠♣♦ ❞❡ ✉♠❛ ♠❛✲ ♥❡✐r❛ ♠❡❧❤♦r✱ ❡st✐♠✉❧❛♥❞♦ ❛ ❝r✐❛t✐✈✐❞❛❞❡ q✉❡ é tã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♣ró♣r✐❛ ▼❛t❡♠át✐❝❛✳ ❊ ❡ss❛s ♥♦✈❛s ✐❞❡✐❛s ❡ ✉♠ ♥♦✈♦ ♣❛r❛❞✐❣♠❛ ❞❛ ✏♠❛t❡♠át✐❝❛ ❞♦ ♠♦✈✐♠❡♥t♦✑ sã♦ ❞❡s❡♥✈♦❧✈✐❞❛s ♣❡❧♦ ❈á❧❝✉❧♦✳
➪✈✐❧❛ r❡ss❛❧t❛ ❛✐♥❞❛ q✉❡ ❣❛st❛✲s❡ ♠✉✐t♦ t❡♠♣♦ ♥♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s ❝♦♠ ♥♦✲ ♠❡♥❝❧❛t✉r❛s ❝♦♠♦✿ ❞♦♠í♥✐♦✱ ❝♦♥tr❛✲❞♦♠í♥✐♦✱ ❢✉♥çã♦ ✐♥❥❡t♦r❛✱ ❢✉♥çã♦ s♦❜r❡❥❡t♦r❛✱ ❢✉♥çã♦ ✐♥✈❡rs❛✳ ❯♠ ❡s❢♦rç♦ q✉❡ ❣❡r❛ ♣♦✉❝♦s r❡s✉❧t❛❞♦s ♣rát✐❝♦s✱ ✉♠❛ ✈❡③ q✉❡ ❡ss❡s ❝♦♥❝❡✐t♦s ♥ã♦ sã♦ s♦❧✐❝✐t❛❞♦s ♣❡❧♦ ❡st✉❞❛♥t❡ ❛♦ ❧♦♥❣♦ ❞♦ ❝✉rs♦✳ ❊ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ✜❝❛✲s❡ ❝♦♠ ♣♦✉❝♦ t❡♠♣♦ ♣❛r❛ tr❛❜❛❧❤❛r ❛s ✐❞❡✐❛s ✐♥❡r❡♥t❡s às ❢✉♥çõ❡s✱ ❝♦♠♦ ❛ r❡✲ ❧❛çã♦ ❡♥tr❡ ❣r❛♥❞❡③❛s q✉❡ ✈❛r✐❛♠✱ q✉❡ ♠✉❞❛♠✳ ❖✉ s❡❥❛✱ ♥❡ss❡ ♠♦♠❡♥t♦ ♣♦❞❡✲s❡ ❝♦♠❡ç❛r ❝♦♠ ✉♠❛ ♣❡q✉❡♥❛ ✐♥tr♦❞✉çã♦ ❞❛s ✐❞❡✐❛s ❞♦ ❈á❧❝✉❧♦✿ ❛ ✐❞❡✐❛ ❞❡ ✈❛r✐❛çã♦✳
❊ss❡ s❡r✐❛ ✉♠ ❜♦♠ ♠♦♠❡♥t♦ ♣❛r❛ ♠♦str❛r ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠ ♣♦♥t♦ ♥♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ♠❛s ♥ã♦ s❡r✐❛ ♣r❡❝✐s♦✱ ♥❡❝❡ss❛r✐❛✲ ♠❡♥t❡ ❛♣r❡s❡♥t❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥tr❛✲❞♦♠í♥✐♦✳ ◆ã♦ é q✉❡ ❡ss❡s ❝♦♥❝❡✐t♦s ♥ã♦ s❡❥❛♠
▼♦t✐✈❛çã♦ ❈❛♣ít✉❧♦ ✶✿ ■♥tr♦❞✉çã♦
✐♠♣♦rt❛♥t❡s✱ ♠❛s ❛ ❢♦r♠❛ ❝♦♠♦ sã♦ ❡♥s✐♥❛❞♦s✱ ❞❡ ♠❛♥❡✐r❛ ❛ ❢❛③❡r ♦s ❡st✉❞❛♥t❡s ❞❡✲ ❝♦r❛r❡♠ ♠✉✐t♦s ♥♦✈♦s ♥♦♠❡s✱ ♥ã♦ é ❛❞❡q✉❛❞❛✱ ❡ ❞❡✐①❛✲s❡ ❞❡ ❧❛❞♦ ♦ ❢✉♥❞❛♠❡♥t❛❧✳
❩❡♥ã♦ ✭✹✾✵✲✹✸✵ ❛✳❈✮ ✭❋✐❣✉r❛ ✶✳✸✮✱ ✜❧ós♦❢♦ ❣r❡❣♦✱ ♣r♦♣ôs q✉❛tr♦ ♣r♦❜❧❡♠❛s q✉❡ ♥♦s ❢♦r❛♠ ❞❡✐①❛❞♦s ♣♦r ❆r✐stót❡❧❡s ✭✸✽✹✲✸✷✷ ❛✳❈✮ ✭❋✐❣✉r❛ ✶✳✹✮ ❡ ✜❝❛r❛♠ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ♦s P❛r❛❞♦①♦s ❞❡ ❩❡♥ã♦✳
❋✐❣✉r❛ ✶✳✸✿ ❩❡♥ã♦ ✭❆❘❆Ú❏❖✱ ✷✵✶✵✮ ❋✐❣✉r❛ ✶✳✹✿ ❆r✐stót❡❧❡s ✭●❆❇❆✱ ✷✵✶✶✮
❙❡❣✉♥❞♦ ❙t❡✇❛rt ✭✷✵✶✷✮ ❡ss❡s ♣❛r❛❞♦①♦s s❡♠♣r❡ ❞❡s❛✜❛✈❛♠ ♦ s❡♥s♦ ❝♦♠✉♠✳ ❖ s❡❣✉♥❞♦ ♣❛r❛❞♦①♦ tr❛t❛ s♦❜r❡ ✉♠❛ ❝♦rr✐❞❛ ❡♥tr❡ ♦ ❤❡ró✐ ❣r❡❣♦ ❆q✉✐❧❡s ❡ ✉♠❛ t❛rt❛✲
r✉❣❛✿ ❆q✉✐❧❡s ✐♥✐❝✐❛❧♠❡♥t❡ ❡♠ ✉♠❛ ♣♦s✐çã♦ a1 ❞❛r✐❛ ✉♠❛ ✈❛♥t❛❣❡♠ ❛ t❛rt❛r✉❣❛ q✉❡
❧❛r❣❛r✐❛ ❞❡ ✉♠❛ ♣♦s✐çã♦ t1✱ à ❢r❡♥t❡✳ ◗✉❛♥❞♦ ❆q✉✐❧❡s ❝❤❡❣❛ss❡ ♥❛ ♣♦s✐çã♦ a2 =t1✱
❛ t❛rt❛r✉❣❛ ❥á ❡st❛r✐❛ ❡♠ ✉♠❛ ♥♦✈❛ ♣♦s✐çã♦ t2✳ ❉❡♣♦✐s ❆q✉✐❧❡s ❛❧❝❛♥ç❛r✐❛ ❛ ♣♦s✐çã♦
a3 = t2✱ ❛ t❛rt❛r✉❣❛✱ ♣♦ré♠✱ ❥á ❡st❛r✐❛ ♥❛ ♣♦s✐çã♦ t3✱ ♠❛✐s ❛ ❢r❡♥t❡ ❡ ❛ss✐♠ ♣♦r
❞✐❛♥t❡✳ ❉❡ss❛ ♠❛♥❡✐r❛ ♦ ❧❡♥❞ár✐♦ ❤❡ró✐ ❆q✉✐❧❡s ❥❛♠❛✐s ✈❡♥❝❡r✐❛ ❛ ❝♦rr✐❞❛ ❝♦♥tr❛ ❛ t❛rt❛r✉❣❛ ✭❋✐❣✉r❛ ✶✳✺✮✳
❋✐❣✉r❛ ✶✳✺✿ ❆q✉✐❧❡s ❳ ❚❛rt❛r✉❣❛
❖✉tr♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛❞♦①♦ ❝♦♥t❛ q✉❡ ♦ ♠♦✈✐♠❡♥t♦ é ✐♠♣♦ssí✈❡❧✦ P♦✐s ♣❛r❛ q✉❡ ❤❛❥❛ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❡♥tr❡ ✉♠ ♣♦♥t♦ ❆ ✭✐♥✐❝✐❛❧✮ ❡ ❇ ✭✜♥❛❧✮✱ ❞❡✈❡✲s❡ ❛♥t❡s ♣❛ss❛r ♣❡❧♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ ❆ ❡ ❇✱ ❞✐❣❛♠♦s ❈✳ ▼❛s ❛♥t❡s ❞❡ ❝❤❡❣❛r ❛♦ ♣♦♥t♦ ❈✱ é ♥❡❝❡ssár✐♦ ♣❛ss❛r ♣❡❧♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ ❆ ❡ ❈✱ q✉❡ s❡❥❛ ❉✱ ♣♦ré♠ ❛♥t❡s ❞❡ ❝❤❡❣❛r ❛♦ ♣♦♥t♦ ❉✱
❊str✉t✉r❛çã♦ ❞♦ ❚r❛❜❛❧❤♦ ❈❛♣ít✉❧♦ ✶✿ ■♥tr♦❞✉çã♦
é ♥❡❝❡ssár✐♦ ❛❧❝❛♥ç❛r ♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ ❆ ❡ ❉✱ ❞✐t♦ ❊✳ ❈♦♠ ❡ss❛ ✐❞❡✐❛ r❡♣❡t✐❞❛s ✈❡③❡s ♠♦str❛✲s❡ q✉❡ ♥♦ ✜♥❛❧ ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧ s❛✐r ❞♦ ♣♦♥t♦ ❆✱ ♥ã♦ ❤❛✈❡♥❞♦✱ ❛ss✐♠✱ ♦ ♠♦✈✐♠❡♥t♦✦
❋✐❣✉r❛ ✶✳✻✿ P❛r❛❞♦①♦ ❞♦ ▼♦✈✐♠❡♥t♦
❖ q✉❡ ❢❛❧t❛✈❛ ♥❡ss❛ é♣♦❝❛ ♣❛r❛ r❡s♣♦♥❞❡r ❛ ❡ss❡s ♣❛r❛❞♦①♦s✱ ❛✜♥❛❧ ❆q✉✐❧❡s ✈❡♥❝❡✉ ❛ t❛rt❛r✉❣❛ ❡ é ♣♦ssí✈❡❧ ♥♦s ♠♦✈❡r♠♦s ❡♥tr❡ ♦s ♣♦♥t♦s ❆ ❡ ❇✱ ❡r❛ ❛ ✐❞❡✐❛ ❞❡ ❧✐♠✐t❡✱ ❡ss❡♥❝✐❛❧ ❛♦ ❈á❧❝✉❧♦✦ ❊ é ♣r❡❝✐s❛♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ q✉❡ ✈❛✐ ❡①tr❛♣♦❧❛r ❛ ❜❛rr❡✐r❛ ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♠✉♠ ♦✉ ❜ás✐❝❛ ❡ ❞❛rá ♥♦✈❛ ✈✐❞❛ à ❝❤❛♠❛❞❛ ▼❛t❡♠át✐❝❛ ❙✉♣❡r✐♦r✳
❊ss❛s ✐❞❡✐❛s✱ ♣♦❞❡♠ s❡r tr❛❜❛❧❤❛❞❛s ❞❡ ♠❛♥❡✐r❛ ♠✉✐t♦ ♣r❛③❡r♦s❛ ❝♦♠ ❡st✉❞❛♥✲ t❡s ❞❡ ♥í✈❡❧ ♠é❞✐♦✳ P♦❞❡✲s❡ ❛té ✐♥✐❝✐❛r ❡ss❡ ❡st✉❞♦ ❛ ♣❛rt✐r ❞❡ss❡s ♣❛r❛❞♦①♦s✳ ❖s
❝♦♥❝❡✐t♦s ♣r❡❝✐s♦s ❡ s✉❛s ❞❡✜♥✐çõ❡s ǫ−δ s❡rã♦ ♠❡❧❤♦r ❝♦♠♣r❡❡♥❞✐❞♦s ♥♦s ❝✉rs♦ ❞❡
♥í✈❡❧ s✉♣❡r✐♦r✱ ♠❛s tr❛❜❛❧❤❛r ✐❞❡✐❛s ❡ ♠♦str❛r ❝♦♠♦ ❡ ♣❛r❛ q✉❡ ♦ ❈á❧❝✉❧♦ ❢♦✐ ❝r✐❛❞♦ é ❡ss❡♥❝✐❛❧ ❛ t♦❞♦s✦
✶✳✷ ❖❜❥❡t✐✈♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r✱ ❞❡ ♠❛♥❡✐r❛ ❜r❡✈❡✱ ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ s♦❜r❡ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❛s ♣r✐♥❝✐✲ ♣❛✐s ✐❞❡✐❛s q✉❡ ♠♦t✐✈❛r❛♠ s❡✉ ❡st✉❞♦✳ ❉❡s❡♥✈♦❧✈❡r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞♦ ❈á❧❝✉❧♦ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ✐♥t✉✐t✐✈❛ ♣❛r❛ ♠♦str❛r q✉❡ é ♣♦ssí✈❡❧ ❛♣r❡s❡♥t❛r ❡ss❛ ❢❡rr❛♠❡♥t❛ ♠❛t❡♠át✐❝❛ ❞❡ ❢♦r♠❛ ❝♦♠♣r❡❡♥sí✈❡❧ ♣❛r❛ ❡st✉❞❛♥t❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛❧✐❛♥❞♦ ❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♠ ❛♣❧✐❝❛çõ❡s ❞❛ ❋ís✐❝❛✳
❊str✉t✉r❛çã♦ ❞♦ ❚r❛❜❛❧❤♦ ❈❛♣ít✉❧♦ ✶✿ ■♥tr♦❞✉çã♦
✶✳✸ ❊str✉t✉r❛çã♦ ❞♦ ❚r❛❜❛❧❤♦
❊st❡ tr❛❜❛❧❤♦ ❡stá ❡str✉t✉r❛❞♦ ❡♠ ✹ ❝❛♣ít✉❧♦s✱ ❛ s❛❜❡r✿
• ❈❛♣ít✉❧♦ ✶ ✲ ■♥tr♦❞✉çã♦✿
◆❡st❡ ❝❛♣ít✉❧♦ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ✐❞❡✐❛s q✉❡ ♠♦t✐✈❛r❛♠ ♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞♦ ❈á❧❝✉❧♦ ❡ ❜r❡✈❡s ❝♦♠❡♥tár✐♦s ❞❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ♦ ❡♥s✐♥♦ ❜r❛s✐❧❡✐r♦✳ ❙ã♦ ❛♣r❡s❡♥t❛❞❛s✱ ❛✐♥❞❛✱ ❛❧❣✉♠❛s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r✐❞❛❞❡✱ ❡♠ ♣❛rt✐❝✉❧❛r ❝♦♠ ❛ ❋ís✐❝❛✳
• ❈❛♣ít✉❧♦ ✷ ✲ ❯♠ ♣♦✉❝♦ ❞❡ ❈á❧❝✉❧♦✿
❖ ❈❛♣ít✉❧♦ ✷ ❝♦♥t❡♠♣❧❛rá tó♣✐❝♦s ❞❡ ❈á❧❝✉❧♦ ♣❛r❛ ❢✉♥çõ❡s ❞❡ ✉♠❛ ú♥✐❝❛ ✈❛✲ r✐á✈❡❧✿ ❧✐♠✐t❡s✱ ❞❡r✐✈❛❞❛s ❡ ✐♥t❡❣r❛✐s✳ ❉❡st❛ ♠❛♥❡✐r❛ s❡rã♦ ♠♦str❛❞♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ ❞❡✜♥✐çõ❡s ❡ ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ❧✐❣❛çã♦ ❝♦♠ ✐❞❡✐❛s ✐♥t✉✐t✐✈❛s ❡ ❝♦♥s✲ tr✉çõ❡s ❣❡♦♠étr✐❝❛s ❛ ✜♠ ❞❡ ♠♦str❛r q✉❡ é ♣♦ssí✈❡❧ tr❛❜❛❧❤❛r ❛❧❣✉♥s ❞❡ss❡s tó♣✐❝♦s ❞❡ ♠❛♥❡✐r❛ s✐♠♣❧❡s s❡♠ ♠✉✐t♦ r✐❣♦r ♠❛t❡♠át✐❝♦✳
• ❈❛♣ít✉❧♦ ✸ ✲ ❖ ❈á❧❝✉❧♦ ❡ ❛ ❋ís✐❝❛✿
❙❡rã♦ ❞✐s❝✉t✐❞❛s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❋ís✐❝❛ q✉❡ ✉t✐❧✐③❛♠ ♦s ❝♦♥❝❡✐t♦s ❞♦ ❈á❧✲ ❝✉❧♦✱ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✷✳ ➱ ❢❡✐t♦ ❛✐♥❞❛ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦ ❝♦♥t❡ú❞♦ ❝♦♠♦ tr❛❜❛❧❤❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ s❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞♦ ❈á❧❝✉❧♦✱ ❝♦♠♦ ♠✉✐t❛s ❢ór♠✉❧❛s q✉❡ só ♣♦❞❡♠ s❡r ♠❡♠♦r✐③❛❞❛s✱ ❞✐✜❝✉❧t❛♥❞♦ ❛ ❝♦♠♣r❡✲ ❡♥sã♦✳ ❉❡ss❛ ♠❛♥❡✐r❛ ♠♦str❛✲s❡ q✉❡ ❛ ✉t✐❧✐③❛çã♦ ❞♦ ❈á❧❝✉❧♦ é ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ❛❥✉❞❛r ♦s ❡st✉❞❛♥t❡s✱ ❝♦♠ ê♥❢❛s❡ ♥❛ ❋ís✐❝❛✳
• ❈❛♣ít✉❧♦ ✹ ✲ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s✿
◆❡st❡ ❈❛♣ít✉❧♦ s❡rã♦ ❢❡✐t♦s ❛❧❣✉♥s ❝♦♠❡♥tár✐♦s ❣❡r❛✐s ❛ r❡s♣❡✐t♦ ❞♦ q✉❡ ❢♦✐ ❞✐s❝✉t✐❞♦ ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✱ r❡ss❛❧t❛♥❞♦ ♦s ♣♦♥t♦s ✐♠♣♦rt❛♥t❡s ❞❛ ❧✐❣❛çã♦ ❞♦ ❈á❧❝✉❧♦ ❝♦♠ ♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♠♦str❛♥❞♦ q✉❡ é ♣♦ssí✈❡❧ ♦ ❡st✉❞❛♥t❡ ❞❡ss❡ ♥í✈❡❧ ❝♦♠♣r❡❡♥❞❡r s✉❛s ✐❞❡✐❛s ❡ ✉t✐❧✐③á✲❧❛s ❞❡ ♠❛♥❡✐r❛ ❛ ❛✉①✐❧✐á✲❧♦ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ út✐❧ ❡♠ ❞✐✈❡rs❛s ❞✐s❝✐♣❧✐♥❛s✳
❈❛♣ít✉❧♦ ✷
❯♠ ♣♦✉❝♦ ❞❡ ❈á❧❝✉❧♦
✏■♥t❡❣r❡ ❉❊❯❙✱ ♣♦✐s ❞❡❧❡ t✉❞♦ ❞❡r✐✈❛✳✑ ❆✉t♦r ❞❡s❝♦♥❤❡❝✐❞♦
◆❡st❡ ❝❛♣ít✉❧♦ s❡rá ❢❡✐t♦ ✉♠ ❡st✉❞♦✱ s✐♠♣❧❡s✱ ❞❡ ❝♦♥❝❡✐t♦s ❞♦ ❈á❧❝✉❧♦✿ ❧✐♠✐t❡s✱ ❞❡r✐✈❛❞❛s ❡ ✐♥t❡❣r❛✐s✳ ❙❡rã♦ ❞❡♠♦♥str❛❞♦s ❛❧❣✉♥s t❡♦r❡♠❛s ✐♠♣♦rt❛♥t❡s✱ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❡①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ss❡ ❝♦♥t❡ú❞♦✳ ❊st❡ tr❛❜❛❧❤♦ ♥ã♦ ♣r❡t❡♥❞❡ r❡♣r♦❞✉③✐r ❡ss❡s ❛ss✉♥t♦s ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦s ❡♠ ❧✐✈r♦s ❞✐❞át✐❝♦s✱ q✉❡ sã♦ ♠✉✐t♦ ♠❛✐s ❛♠♣❧♦s✱ ♠❛s ❛♣❡♥❛s t❡❝❡r ❛❧❣✉♥s ❝♦♠❡♥tár✐♦s ❡ ❛♣r❡s❡♥t❛r ❡❧❡♠❡♥t♦s ❡ss❡♥❝✐❛✐s q✉❡ ♣♦❞❡♠ s❡r ❡♥s✐♥❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❙❡rã♦ ✉s❛❞♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛s ♣r✐♥❝✐♣❛✐s ❙t❡✇❛rt ✭✷✵✶✷✮✱ ❙✇♦❦♦✇s❦✐ ✭✶✾✾✹✮ ❡ ▲■▼❆ ✭✷✵✵✹✮✳
✷✳✶ ▲✐♠✐t❡s
❖ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦ f é ♦ q✉❡ s❡♣❛r❛ ❛ ♠❛t❡♠át✐❝❛ ❡❧❡♠❡♥t❛r ❞❛
♠❛t❡♠át✐❝❛ s✉♣❡r✐♦r✱ s✉❛s ✐❞❡✐❛s sã♦ ❜❛st❛♥t❡ ❞✐❢❡r❡♥t❡s ❞❛ á❧❣❡❜r❛ ❡ ❞❛ ❣❡♦♠❡tr✐❛✳ ◆♦ ✐♥í❝✐♦ ❞❡ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ❞✉r❛♥t❡ ♦ sé❝✉❧♦ ❳❱■■■✱ ❡ss❛s ✐❞❡✐❛s ❢♦r❛♠ ✉t✐❧✐✲ ③❛❞❛s ❞❡ ❢♦r♠❛ ❜❛st❛♥t❡ ✐♥t✉✐t✐✈❛s✱ s❡♠ ❣r❛♥❞❡s ❢♦r♠❛❧✐s♠♦s ♠❛t❡♠át✐❝♦s ❡♠❜♦r❛ t❛✐s ✐❞❡✐❛s s❡❥❛♠ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❈á❧❝✉❧♦✱ s❡♥❞♦ ❡①❛t❛♠❡♥t❡ ♣♦r ✐ss♦ q✉❡ ❡❧❛s ❞❡✈❡♠ s❡r ❞✐s❝✉t✐❞❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❆ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛ ❞❡ ❧✐♠✐t❡ é ❛ s❡❣✉✐♥t❡✿ ❉❛❞♦s f ✉♠❛ ❢✉♥çã♦ ❞❡ ✈❛❧♦r❡s r❡❛✐s
❞❡✜♥✐❞❛ ❡♠ ✉♠ s✉❜❝♦♥❥✉♥t♦ I ❞❡ R ❡ x0 ✉♠ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❡ I ✭♦✉ s❡❥❛✱
x0 é t❛❧ q✉❡✱ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦ ❞❛ ❢♦r♠❛ (x0 −r, x0 +r)✱ ❝♦♠ r > 0✱ ✐♥t❡rs❡❝t❛
I \ {x0}✮✱ ❞✐③❡♠♦s q✉❡ ♦ ❧✐♠✐t❡ ❞❡ f(x) q✉❛♥❞♦ x t❡♥❞❡ ❛ x0 é ✐❣✉❛❧ ❛ ✉♠ ❝❡rt♦
♥ú♠❡r♦L∈Rs❡ q✉❛♥t♦ ♠❛✐s ♣ró①✐♠♦ ❞❡ss❡ x0 ❡st✐✈❡rx✱ ♠❛✐s ♣ró①✐♠♦ ❞❡ L❡st❛rá
