❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❡ ❚❡❝♥♦❧♦❣✐❛ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❯♠❛ ❛♥á❧✐s❡ ❞♦ ❡♥s✐♥♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♥♦
❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✿ r❡❛❧✐❞❛❞❡ ❡ ♣❡rs♣❡❝t✐✈❛s
❉❛✈✐❞s♦♥ ▼♦✉r❛ ▲♦♣❡s ❙✐❧✈❛
♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❚♦♠❛s ❊❞s♦♥ ❇❛rr♦s
❯♠❛ ❛♥á❧✐s❡ ❞♦ ❡♥s✐♥♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♥♦
❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✿ r❡❛❧✐❞❛❞❡ ❡ ♣❡rs♣❡❝t✐✈❛s
❉❛✈✐❞s♦♥ ▼♦✉r❛ ▲♦♣❡s ❙✐❧✈❛
♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❚♦♠❛s ❊❞s♦♥ ❇❛rr♦s
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ♣r♦❣r❛♠❛ ❞❡ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧ P❘❖❋▼❆❚ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ♥❛ ár❡❛ ❞❡ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛
❙ã♦ ❈❛r❧♦s ❖✉t✉❜r♦ ❞❡ ✷✵✶✺
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária UFSCar Processamento Técnico
com os dados fornecidos pelo(a) autor(a)
S586a
Silva, Davidson Moura Lopes
Uma análise do ensino de proporcionalidade no ensino fundamental : realidade e perspectivas / Davidson Moura Lopes Silva. -- São Carlos : UFSCar, 2015.
88 p.
Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2015.
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✧❉❡s❝♦❜❡rt❛s ▼❛t❡♠át✐❝❛s✱ ♣❡q✉❡♥❛s ♦✉ ❣r❛♥❞❡s✱ ♥✉♥❝❛ ♥❛s❝❡♠ ♣♦r ❣❡r❛çã♦ ❡s♣♦♥tâ♥❡❛✳ ❊❧❛s ♣r❡ss✉♣õ❡♠ ✉♠ s♦❧♦ ❢❡rt✐❧✐③❛❞♦ ❝♦♠ ❝♦♥❤❡❝✐♠❡♥t♦s ♣r❡❧✐♠✐♥❛r❡s✱ ❡ ❜❡♠ ❧❛✈r❛❞♦ ❝♦♠
tr❛❜❛❧❤♦ ❝♦♥s❝✐❡♥t❡ ❡ ✐♥❝♦♥s❝✐❡♥t❡✧
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ t♦❞♦s ♦s ♠❡✉s ♣r♦❢❡ss♦r❡s✱ q✉❡ ❞❡❞✐❝❛r❛♠ s✉❛s ✈✐❞❛s ❛ ❡♥s✐♥❛r✳ ◗✉❡ ❛❜r❛ç❛r❛♠ ❡ss❛ ♣r♦✜ssã♦ tã♦ ❣r❛♥❞✐♦s❛ ❡ ♠❡ ✐♥s♣✐r❛♠ ❞✐❛r✐❛♠❡♥t❡ ❡♠ ♠✐♥❤❛ ❝❛♠✐♥❤❛❞❛ ❝♦♠ ♠❡✉s ❛❧✉♥♦s✳ ❆❣r❛❞❡ç♦ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♦ ❣r❛♥❞❡ ♣r♦❢❡ss♦r ❡ ❛♠✐❣♦ ❚♦♠❛s ❊❞s♦♥ ❇❛rr♦s ♣♦r t❡r ♠❡ ♦r✐❡♥t❛❞♦✱ ♠❛✐s ✉♠❛ ✈❡③✱ ❝♦♠ ❣r❛♥❞❡ ❞❡❞✐❝❛çã♦✳
❆❣r❛❞❡ç♦ t❛♠❜é♠ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞❛ ❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✳ ❆❣r❛❞❡ç♦ ❛✐♥❞❛ ❛♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ❝♦♠ q✉❡♠ ❞✐✈✐❞✐ ❞ú✈✐❞❛s✱ ❞✐✜❝✉❧❞❛❞❡s ❡ t❛♠❜é♠ ❝♦♥q✉✐st❛s ❡ s✉♣❡r❛çõ❡s✳ ❆❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦s ❣r❛♥❞❡s ❛♠✐❣♦s ❈és❛r ❈❛♥❝❡✱ ▲ú❝✐❛ ❇❡❧tr❛♠✐ ❡ ❘♦sâ♥❣❡❧❛ ❘♦ss✐ ♣❡❧♦ ❛♣♦✐♦ ❧♦❣íst✐❝♦ ❡ ❤♦r❛s ✐♥❝❛♥sá✈❡✐s ❞❡ ❡st✉❞♦ r❡❝❤❡❛❞❛s ❞❡ ♠✉✐t❛ ❞✐✈❡rsã♦ ♣❛r❛ ❝♦♠♣❡♥s❛r t♦❞♦ ♦ s♦❢r✐♠❡♥t♦ ♣❡r❛♥t❡ ❛s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞❛ ❡s❝♦❧❛ ❆♥♥❛ ❞♦s ❘❡✐s ❙✐❣♥♦r✐♥✐✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❣r❛♥❞❡ ❛♠✐❣♦ ❊✉❞❡ ❉✉q✉❡s ❞♦ ❘❡❣♦ ♣♦r t♦❞❛ ❛ ❞✐s♣♦s✐çã♦ ❡♠ ♠❡ ❛❥✉❞❛r ❞❡♠♦♥str❛❞❛ ♥ã♦ só ❛♦ ❧♦♥❣♦ ❝✉rs♦✱ ♠❛s ♥♦ ❝♦♥✈í✈✐♦ ❞✐ár✐♦✳ ➚s ❞✐r❡t♦r❛s ❑❡❧❧② ▼❛r❝♦♥ ❆r❝❛s ❡ ❙ô♥✐❛ ●❛❧❤❛r❞♦ ❙❝❤♠✐❞t ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ q✉❛♥❞♦✱ ♣♦r ♠♦t✐✈♦ ❞❡ ❡st✉❞♦s✱ ♣r❡❝✐s❡✐ ♠❡ ❛✉s❡♥t❛r✳ ❆t✐t✉❞❡ q✉❡ ❞❡♠♦♥str❛ q✉❡ t♦r❝❡♠ ♣♦r ♠✐♠ ❡ ❝♦♥✜❛♠ ♥♦ ♠❡✉ tr❛❜❛❧❤♦✳
❘❡s✉♠♦
◆❡ss❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ♥♦✈❛ ♣r♦♣♦st❛ ❞❡ ❛❜♦r❞❛❣❡♠ ❞♦ t❡♠❛ ♣r♦♣♦r❝✐♦♥❛✲ ❧✐❞❛❞❡✱ ♣❛r❛ ❛❧✉♥♦s ❞♦ ✼♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳ ❊st❡ tr❛❜❛❧❤♦ é ♦ r❡❣✐str♦ ❞❡ ✉♠❛
❊♥❣❡♥❤❛r✐❛ ❉✐❞át✐❝❛ q✉❡ ❣❡r♦✉ ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♠♣♦st❛ ♣♦r q✉❛tr♦ s❡ssõ❡s ❞✐❞át✐❝❛s✳ ❖ ❢♦❝♦ ❞♦ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ♠❡✐♦s ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❞❡t❡r♠✐♥❡✱ ❜❛s❡❛❞♦ ♥❛s ❞❡✜♥✐çõ❡s ♠❛t❡♠át✐❝❛s✱ s❡ ❞✉❛s ❣r❛♥❞❡③❛s sã♦ ♣r♦♣♦r❝✐♦♥❛✐s ❡✱ ❡♠ ❝❛s♦ ♣♦s✐t✐✈♦✱ s❡ é ✉♠❛ r❡❧❛çã♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞✐r❡t❛ ♦✉ ✐♥✈❡rs❛✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ♣r❡s❡♥ts ❛ ♥❡✇ ♣r♦♣♦s❛❧ ♦❢ ❛♣♣r♦❛❝❤ ❢♦r ♣r♦♣♦r❝✐♦♥❛❧✐t②✱ ✇❤♦s❡ ❢♦❝✉s ✐s t❤❡ ❇r❛③✐❧✐❛♥ ❊❧❡♠❡♥t❛r② ✼t❤ ❣r❛❞❡ st✉❞❡♥t✳ ❚❤✐s ✇♦r❦ ✐s t❤❡ r❡❝♦r❞ ♦❢ ❛ ✇♦r❦ ♦❢ ❞✐❞❛❝t✐❝ ❡♥❣✐♥❡❡❡r✐♥❣ t❤❛t ❣❡♥❡r❛t❡❞ ❛ s❡q✉❡♥❝❡ ♠❛❞❡ ❜② ❢♦✉r ❞✐❞❛❝t✐❝ s❡ss✐♦♥s✳
❚❤❡ ♠❛✐♥ ❣♦❛❧ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s t♦ ♣r❡s❡♥t ✇❛②s ❢♦r t❤❡ st✉❞❡♥t t♦ ❞❡t❡r♠✐♥❡✱ ❜❛s❡❞ ♦♥ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡✜♥✐t✐♦♥s✱ ✐❢ t✇♦ ♠❛❣♥✐t✉❞❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❛♥❞✱ ✐♥ ❛✣r♠❛t✐✈❡ ❝❛s❡✱ ✐❢ t❤❡ r❡❧❛t✐♦♥ ✐s ❞✐r❡❝t❧② ♦r ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧✳
①✐✐✐
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ▼❡t♦❞♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✷ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❊s❝♦❧❛ ❡ ❞❛s s❛❧❛s ❞❡ ❛✉❧❛ ✺
✷✳✶ ❙ét✐♠♦ ❆♥♦ ❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❙ét✐♠♦ ❆♥♦ ❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❙ét✐♠♦ ❆♥♦ ❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹ ❙ét✐♠♦ ❆♥♦ ● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✸ ❆♥á❧✐s❡ ♣ré✈✐❛✿ ❆ r❡❛❧✐❞❛❞❡ ❞♦ ❊♥s✐♥♦ ❞❡ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♥♦ ❇r❛s✐❧ ✾ ✹ ❆♥á❧✐s❡ ❛ Pr✐♦r✐✿ ❆ ▼❛t❡♠át✐❝❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡
♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✶✸
✹✳✶ ❚r❛t❛♠❡♥t♦ ❋✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✹✳✷ ❆❜♦r❞❛❣❡♠ ♣♦r ♠❡✐♦ ❞❡ ♣r♦♣♦rçõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✹✳✸ ❆ ❙❡q✉ê♥❝✐❛ ❉✐❞át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✹✳✸✳✶ ❙❡ssã♦ ❞✐❞át✐❝❛ ✶✿ ❘❡✈✐s❛♥❞♦ ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✹✳✸✳✷ ❙❡ssã♦ ❞✐❞át✐❝❛ ✷✿ ❊st✉❞♦ ❞❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦
●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✹✳✸✳✸ ❙❡ssã♦ ❞✐❞át✐❝❛ ✸✿ ●rá✜❝♦ ❞❡ ❣r❛♥❞❡③❛s ♣r♦♣♦r❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✹✳✸✳✹ ❙❡ssã♦ ❞✐❞át✐❝❛ ✹✿ ❯s❛♥❞♦ s♦♠❜r❛s ♣❛r❛ ❡st✐♠❛r ❛❧t✉r❛s ✐♥❛t✐♥❣í✈❡✐s ✶✾
✺ ❊①♣❡r✐♠❡♥t❛çã♦✿ ❆♣❧✐❝❛çã♦ ❞❛ ❙❡q✉ê♥❝✐❛ ❉✐❞át✐❝❛ ✷✶
✺✳✶ ❆♣r❡s❡♥t❛çã♦ ❞❛ ♣r✐♠❡✐r❛ s❡ssã♦ ❞✐❞át✐❝❛✿ ❘❡✈✐s❛♥❞♦ ❝♦♥❝❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✺✳✷ ❆♥á❧✐s❡ ❞❛ ♣r✐♠❡✐r❛ s❡ssã♦ ❞✐❞át✐❝❛✿ ❘❡✈✐s❛♥❞♦ ❈♦♥❝❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✺✳✸ ❆♣r❡s❡♥t❛çã♦ ❞❛ s❡❣✉♥❞❛ s❡ssã♦ ❞✐❞át✐❝❛✿ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❝♦♠ ♦ ●❡♦❣❡❜r❛ ✷✾ ✺✳✹ ❆♥á❧✐s❡ ❞❛ s❡❣✉♥❞❛ s❡ssã♦ ❞✐❞át✐❝❛✿ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❝♦♠ ♦ ●❡♦❣❡❜r❛ ✳ ✳ ✸✾ ✺✳✺ ❙❡ssã♦ ❞✐❞át✐❝❛ ✸✿ ●rá✜❝♦ ❞❡ ❣r❛♥❞❡③❛s ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ❡
✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✺✳✻ ❆♥á❧✐s❡ ❞❛ ❙❡ssã♦ ❞✐❞át✐❝❛ ✸✿ ●rá✜❝♦ ❞❡ ❣r❛♥❞❡③❛s ❞✐r❡t❛♠❡♥t❡
①✐✈ ❙✉♠ár✐♦
✺✳✼ ❙❡ssã♦ ❞✐❞át✐❝❛ ✹✿ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♥❛ ♣rát✐❝❛ ✲ ✉♠ ♠ét♦❞♦ ♣❛r❛ s❡ ❡st✐♠❛r ❛❧t✉r❛s ✐♥❛t✐♥❣í✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✽ ❆♥á❧✐s❡ ❞❛ s❡ssã♦ ❞✐❞át✐❝❛ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✻ ❆♥á❧✐s❡ ❛ P♦st❡r✐♦r✐ ✺✼
✻✳✶ ❆✈❛❧✐❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✼ ❈♦♥❝❧✉sõ❡s ✻✸
❆ ❘❡s♣♦st❛s ❞❛ Pr✐♠❡✐r❛ ❙❡ssã♦ ❉✐❞át✐❝❛ ✻✼
❇ ❘❡s♣♦st❛s ❞❛ s❡❣✉♥❞❛ ❙❡ssã♦ ❉✐❞át✐❝❛ ✼✶
❈ ❘❡s♣♦st❛s ❞❛ ❚❡r❝❡✐r❛ s❡ssã♦ ❞✐❞át✐❝❛ ✽✶
①✈
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❊s❝♦❧❛ ❆♥♥❛ ❞♦s ❘❡✐s ❙✐❣♥♦r✐♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ Pr♦❣r❛♠❛ ❊s❝♦❧❛ ❞❡ ❆t❧❡t❛s ❡ ❋♦r♠❛çã♦ ■♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ ❈♦♠♣❛r❛t✐✈♦ ❞❡ ❘❡♥❞✐♠❡♥t♦ ♥♦s ❞♦✐s ♣r✐♠❡✐r♦s ❜✐♠❡str❡s ❞❡ ✷✵✶✺ ❞♦ ✼♦❈ ✳ ✼
✷✳✹ ❈♦♠♣❛r❛t✐✈♦ ❞❡ ❘❡♥❞✐♠❡♥t♦ ♥♦s ❞♦✐s ♣r✐♠❡✐r♦s ❜✐♠❡str❡s ❞❡ ✷✵✶✺ ❞♦ ✼♦❉ ✳ ✼
✷✳✺ ❈♦♠♣❛r❛t✐✈♦ ❞❡ ❘❡♥❞✐♠❡♥t♦ ♥♦s ❞♦✐s ♣r✐♠❡✐r♦s ❜✐♠❡str❡s ❞❡ ✷✵✶✺ ❞♦ ✼♦❋ ✳ ✽
✷✳✻ ❈♦♠♣❛r❛t✐✈♦ ❞❡ ❘❡♥❞✐♠❡♥t♦ ♥♦s ❞♦✐s ♣r✐♠❡✐r♦s ❜✐♠❡str❡s ❞❡ ✷✵✶✺ ❞♦ ✼♦● ✳ ✽
✸✳✶ ❘❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞✐r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✷ ❘❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✐♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✺✳✶ ❆♣❧✐❝❛çã♦ ❞❛ ♣r✐♠❡✐r❛ s❡ssã♦ ❞✐❞át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✺✳✷ Pr✐♠❡✐r♦ ❡①❡♠♣❧♦ ❞❡ r❡s♣♦st❛ ❞♦ ❡①❡r❝í❝✐♦ ✷ ❞❛ ♣r✐♠❡✐r❛ s❡ssã♦ ❞✐❞át✐❝❛ ✳ ✳ ✷✻ ✺✳✸ ❙❡❣✉♥❞♦ ❡①❡♠♣❧♦ ❞❡ r❡s♣♦st❛ ❞♦ ❊①❡r❝í❝✐♦ ✷ ❞❛ ♣r✐♠❡✐r❛ s❡ssã♦ ❞✐❞át✐❝❛ ✳ ✳ ✷✻ ✺✳✹ ❉❡s❡♥❤♦ ❞♦ ❡①❡r❝í❝✐♦ ✾ r❡❛❧✐③❛❞♦ ❝♦rr❡t❛♠❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✺✳✺ ❉❡s❡♥❤♦ ❞♦ ❡①❡r❝í❝✐♦ ✾ r❡❛❧✐③❛❞♦ ❝♦♠ ❢❛❧❤❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✺✳✻ ❊①❡r❝í❝✐♦ ✶✵ ❡①❡❝✉t❛❞♦ ❞❡ ♠❛♥❡✐r❛ ❝♦rr❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✺✳✼ ❊①❡♠♣❧♦ ❞❡ ❞❡s❡♥❤♦ ✭❊①❡r❝í❝✐♦ ✶✵✮ q✉❡ ♥ã♦ ♣♦❞❡r✐❛ s❡r r❡♣r♦❞✉③✐❞♦ ❛ ♣❛rt✐r
❞♦s ♣♦♥t♦s ♠❛r❝❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✺✳✽ ❆♣❧✐❝❛çã♦ ❞❛ s❡❣✉♥❞❛ s❡ssã♦ ❞✐❞át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✾ ❊①❝❡ss♦ ❞❡ ❝❛s❛s ❞❡❝✐♠❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✶✵ ❘❛③õ❡s ♥❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✶✶ ❆♥✐♠❛♥❞♦ ♣♦♥t♦s✱ r❡❧❛çã♦ ❡♥tr❡ ✈❡❧♦❝✐❞❛❞❡ ❡ t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✶✷ ❘❡tâ♥❣✉❧♦s Pr♦♣♦r❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✶✸ ❱❡r✐✜❝❛r s❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ ✈❡r❞❡ ❡ ♦ t❡♠♣♦ é ✉♠❛
♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✶✹ ●rá✜❝♦ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ❞✐stâ♥❝✐❛ ❡ t❡♠♣♦ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡ ✐❣✉❛❧
❡ ✻✵ ❦♠✴❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✶✺ ●rá✜❝♦ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ❞✐stâ♥❝✐❛ ❡ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ t❡♠♣♦ ✜①❛❞♦ ❡♠ ✻ ❤♦r❛s ✺✵ ✺✳✶✻ ●rá✜❝♦ ❞❛ r❡❧❛çã♦ ❡♥tr❡ t❡♠♣♦ ❡ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ❝♦♥st❛♥t❡ ❡
①✈✐ ▲✐st❛ ❞❡ ❋✐❣✉r❛s
✺✳✶✾ ▼❡❞✐çõ❡s ♦❜t✐❞❛s ♣♦r ✉♠ ❣r✉♣♦ ❞♦ ✼♦ ❛♥♦ ❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✺✳✷✵ ❱❛❧✐❞❛♥❞♦ ❛s ♠❡❞✐çõ❡s ♦❜t✐❞❛s ♣♦r ✉♠ ❣r✉♣♦ ❞♦ ✼♦ ❛♥♦ ❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
①✈✐✐
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✶✳✶ ◗✉❛❞r♦ ❝♦♠♣❛r❛t✐✈♦ ❞♦s r❡s✉❧t❛❞♦s ❞♦ ❇r❛s✐❧ ♥♦ P■❙❆ ❞❡s❞❡ ✷✵✵✵ ✳ ✳ ✳ ✳ ✳ ✶ ✷✳✶ ◗✉❛❞r♦ ❝♦♠♣❛r❛t✐✈♦ ❞❛ ♠é❞✐❛ ❣❧♦❜❛❧ ❞♦s ❛❧✉♥♦s ❞❛ ❊❆❋■ ❡ ❞♦s ❛❧✉♥♦s
✶
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❈♦♠♦ ♣♦❞❡♠♦s ❛❝♦♠♣❛♥❤❛r ❡♠ t♦❞♦s ♦s ♠❡✐♦s ❞❡ ❝♦♠✉♥✐❝❛çã♦✱ ❛ ❊❞✉❝❛çã♦ ❇r❛s✐❧❡✐r❛ ♥ã♦ ✈❛✐ ♥❛❞❛ ❜❡♠ ❝♦♠♦ ✉♠ t♦❞♦✳ ▼❛s✱ ❡♠ s❡ tr❛t❛♥❞♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ♦s r❡s✉❧t❛❞♦s sã♦ ❛✐♥❞❛ ♠❛✐s ❝❛t❛stró✜❝♦s✳ ❉❡ ✉♠ t♦t❛❧ ❞❡ ✻✺ ♣❛ís❡s q✉❡ ♣❛rt✐❝✐♣❛♠ ❞♦ Pr♦❣r❛♠❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ❆✈❛❧✐❛çã♦ ❞❡ ❊st✉❞❛♥t❡s ✭P■❙❆✮✱ ♦❝✉♣❛♠♦s ❛ 58a ♣♦s✐çã♦ ♥♦ r❡♥❞✐♠❡♥t♦ ❞♦s ❥♦✈❡♥s
♥❡ss❛ ❞✐s❝✐♣❧✐♥❛✱ ♠❛♥t❡♥❞♦ ♦ ♣❛ís ❛❜❛✐①♦ ❞♦ ♥í✈❡❧ ❜ás✐❝♦ ❞❡ ♣r♦✜❝✐ê♥❝✐❛✳ ▼❡s♠♦ ❝♦♠ ✉♠❛ ♠❡❧❤♦r❛ ❞❡ ✺✼ ♣♦♥t♦s ♥♦s r❡s✉❧t❛❞♦s ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❡①❛♠❡ ❡♥tr❡ ✷✵✵✵ ❡ ✷✵✵✾✱ ❛✐♥❞❛ ❡st❛♠♦s ♠✉✐t♦ ❛q✉é♠ ❞❛s ♥❛çõ❡s ♠❛✐s ❞❡s❡♥✈♦❧✈✐❞❛s✱ q✉❡ t❛♠❜é♠ ❞✐s❝✉t❡♠ ♣♦❧ít✐❝❛s ♣ú❜❧✐❝❛s ♣❛r❛ ♠❡❧❤♦r❛r s❡✉ ❞❡s❡♠♣❡♥❤♦✳
❚❛❜❡❧❛ ✶✳✶✿ ◗✉❛❞r♦ ❝♦♠♣❛r❛t✐✈♦ ❞♦s r❡s✉❧t❛❞♦s ❞♦ ❇r❛s✐❧ ♥♦ P■❙❆ ❞❡s❞❡ ✷✵✵✵ P✐s❛ ✷✵✵✵ P✐s❛ ✷✵✵✸ P✐s❛ ✷✵✵✻ P✐s❛ ✷✵✵✾ P✐s❛ ✷✵✶✷ ◆ú♠❡r♦ ❞❡ ♣❛rt✐❝✐♣❛♥t❡s ✹✳✽✾✸ ✹✳✹✺✷ ✾✳✷✾✺ ✷✵✳✶✷✼ ✶✽✳✺✽✾
▼é❞✐❛ ❞❡ ▲❡✐t✉r❛ ✸✾✻ ✹✵✸ ✸✾✸ ✹✶✷ ✹✶✵
▼é❞✐❛ ❞❡ ▼❛t❡♠át✐❝❛ ✸✸✹ ✸✺✻ ✸✼✵ ✸✽✻ ✸✾✶
▼é❞✐❛ ❞❡ ❈✐ê♥❝✐❛s ✸✼✺ ✸✾✵ ✸✾✵ ✹✵✺ ✹✵✺
❋♦♥t❡✿❤tt♣✿✴✴♣♦rt❛❧✳✐♥❡♣✳❣♦✈✳❜r✴✐♥t❡r♥❛❝✐♦♥❛❧✲♥♦✈♦✲♣✐s❛✲r❡s✉❧t❛❞♦s
❍á ♠✉✐t❛ ❞✐s❝✉ssã♦ s♦❜r❡ ❞✐✈❡rs♦s ❛s♣❡❝t♦s ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ à ❡❞✉❝❛çã♦✱ t❛✐s ❝♦♠♦ ❛✉♠❡♥t♦ ❞❡ ✐♥✈❡st✐♠❡♥t♦s✱ ❝r✐tér✐♦s ❞❡ ❛✈❛❧✐❛çã♦✱ ❢♦r♠❛çã♦ ❞♦❝❡♥t❡✱ r❡❡str✉t✉r❛çã♦ ❞❡ ❝✉rrí❝✉❧♦s ❡t❝✳ ▼❛s ♥ã♦ é t❛r❡❢❛ s✐♠♣❧❡s ❝♦rr✐❣✐r ❡rr♦s ❤✐stór✐❝♦s ❞❡ ✉♠❛ ❡s❝♦❧❛ tr❛❞✐❝✐♦♥❛❧✱ ❡①❝❧✉❞❡♥t❡ ✭♥♦ s❡♥t✐❞♦ ❡♠ q✉❡ ♠✉✐t♦s ❛❜❛♥❞♦♥❛✈❛♠ ♦s ❡st✉❞♦s ♣♦r ♥ã♦ s❡ ❡♥q✉❛❞r❛r❡♠ ❛♦ s✐st❡♠❛✮✳ ❆ ♠✉❞❛♥ç❛ é ❞✐❢í❝✐❧ ♣♦rq✉❡ ❛ ✐♠❛❣❡♠ q✉❡ t❡♠♦s é q✉❡ ❛♥t✐❣❛♠❡♥t❡ ❛s ❡s❝♦❧❛s ❡r❛♠ ❜♦❛s ❡ ❤♦❥❡ ❡❧❛s ❡stã♦ ❝❛❞❛ ✈❡③ ♣✐♦r❡s✳ ❖ q✉❡ ♥ã♦ s❡ ♣❡r❝❡❜❡ é q✉❡ ❤♦❥❡ ♦ tr❛❜❛❧❤♦ é ♠✉✐t♦ ♠❛✐s ❞✐❢í❝✐❧ ❥á q✉❡ ❤♦❥❡ t❡♠♦s ✉♠❛ ♣♦❧ít✐❝❛ ❞❡ ✐♥❝❧✉sã♦✳
✷ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦
▼❡s♠♦ q✉❛♥❞♦ ❛s ❝r✐❛♥ç❛s ✈ã♦ à ❡s❝♦❧❛✱ ♣❛✐s q✉❡ ♣♦ss✉❡♠ ♣♦✉❝❛ ♦✉ ♥❡♥❤✉♠❛ ❡s❝♦❧❛r✐❞❛❞❡ ♥ã♦
tê♠ ❝♦♠♦ s❛❜❡r s❡ ❛ q✉❛❧✐❞❛❞❡ ❞❛ ❡❞✉❝❛çã♦ é ❜♦❛ ♦✉ ♥ã♦✳ ❉❡ss❛ ❢♦r♠❛✱ ❡♠❜♦r❛ ♠✉✐t♦s ♣❛✐s
t❡♥❤❛♠ ❝♦♠❡ç❛❞♦ ❛ ❢♦rç❛r ❛ ❡♥tr❛❞❛ ❞♦s ✜❧❤♦s ♥❛ ❡s❝♦❧❛✱ ❡❧❡s ❢❛③❡♠ ♣♦✉❝❛s ❞❡♠❛♥❞❛s ♣♦r q✉❛❧✐❞❛❞❡✳
❘❡②♥❛❧❞♦ ❋❡r♥❛♥❞❡s✱ ❡①✲Pr❡s✐❞❡♥t❡ ❞♦ ■♥❡♣✱ ❛✜r♠❛ q✉❡ ♥♦ ✜♥❛❧ ❞♦s ❛♥♦s ✶✾✾✵ ❛ ♠❛✐♦r ♣r❡ssã♦ ♣❡❧❛ ♠❡❧❤♦r✐❛ ❞❛ q✉❛❧✐❞❛❞❡ ❞❛ ❡❞✉❝❛çã♦ ✈❡✐♦ ❞❛s ❡❧✐t❡s✱ ❡♠❜♦r❛ ♥ã♦ t✐✈❡ss❡♠ ✜❧❤♦s ♥❛s ❡s❝♦❧❛s ♣ú❜❧✐❝❛s✳
■ss♦ s❡ ❞❡✉ ♣♦rq✉❡ ❡❧❛s ❝♦♠♣r❡❡♥❞❡r❛♠ q✉❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❢✉t✉r♦ ❞♦ P❛ís ❞❡♣❡♥❞✐❛ ❞❡ ✉♠❛
❢♦rç❛ ❞❡ tr❛❜❛❧❤♦ ❡ ❞❡ ❝✐❞❛❞ã♦s ❝♦♠ ❜♦❛ ❡❞✉❝❛çã♦✳ ▼❛s ❏❡✛r❡② P✉r②❡❛r✱ ❉✐r❡t♦r ❞❛ P❛r❝❡r✐❛ ♣❛r❛ ❛
❘❡✈✐t❛❧✐③❛çã♦ ❊❞✉❝❛❝✐♦♥❛❧ ❞❛s ❆♠ér✐❝❛s ❞♦ ❉✐á❧♦❣♦ ■♥t❡r❛♠❡r✐❝❛♥♦✱ ❛ss✐♥❛❧❛ q✉❡✱ s❡♠ ✉♠❛ ❞❡♠❛♥❞❛
♠❛✐s ❛♠♣❧❛ ♣♦r ♣❛rt❡ ❞❛ s♦❝✐❡❞❛❞❡ ♣♦r ♠❡❧❤♦r q✉❛❧✐❞❛❞❡✱ ❛s ❞❡❝✐sõ❡s ♣♦❧ít✐❝❛s ✜❝❛♠ ❞♦♠✐♥❛❞❛s ♣♦r
♣r♦✜ss✐♦♥❛✐s ❞❛ ❡❞✉❝❛çã♦ q✉❡ ✈✐✈❡♠ ❞♦ s✐st❡♠❛✳ ❊❧❡s t❡♥❞❡♠ ❛ r❡s✐st✐r ❛ ♠✉❞❛♥ç❛s ♥♦ st❛t✉s q✉♦ ❛♦
✐♥✈és ❞❡ s❡ ❡♠♣❡♥❤❛r❡♠ ♥❛s r❡❢♦r♠❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ❡❧❡✈❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s ❛❧✉♥♦s✳✭❖❈❉❊❬✽❪✱
♣á❣ ✶✽✵✱ ✷✵✶✶✮
❆❧❣✉♠❛s s❡❝r❡t❛r✐❛s ❞❡ ❡❞✉❝❛çã♦ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛ ❞❡ ❙ã♦ P❛✉❧♦✱ ♥ã♦ ❢♦❝❛♠ ♠❛✐s ♦ ❝♦♥t❡ú❞♦ ❡ s✐♠✱ ❝♦♠♣❡tê♥❝✐❛s ❡ ❤❛❜✐❧✐❞❛❞❡s✳ ❚♦❞♦s ❜✉s❝❛♠ ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ t❡♥t❛r s✉♣❡r❛r ♦s ❞❡s❛✜♦s✱ ♠❛s ♥✐♥❣✉é♠ ❞✐s❝✉t❡ q✉❡ ❛ ♠❛✐♦r ❞❛s ♠✉❞❛♥ç❛s ❡stá ♥❛s ♠ã♦s ❞♦s ♣ró♣r✐♦s ♣r♦❢❡ss♦r❡s✳ ❈❛❜❡ ❛ ❡❧❡s ❛♣r✐♠♦r❛r ♠ét♦❞♦s ❡ ♣❡sq✉✐s❛r ♥♦✈❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡♥s✐♥♦ ❡♠ ♣❡r✐ó❞✐❝♦s ❡s♣❡❝✐❛❧✐③❛❞♦s✱ ♦✉ s❡❥❛✱ é ♥❡❝❡ssár✐♦ ❡st✉❞❛r✳ ❆s ❝♦❜r❛♥ç❛s ♣❡❧♦s r❡s✉❧t❛❞♦s s❡♠♣r❡ r❡❝❛❡♠ s♦❜r❡ ♦ tr❛❜❛❧❤♦ ❞♦ ♣r♦❢❡ss♦r✱ ♣♦ré♠✱ ♦s ✐♥✈❡st✐♠❡♥t♦s ♥❛ ✈❛❧♦r✐③❛çã♦ ❞❛ ❝❛rr❡✐r❛ ❞❡ ♠❛❣✐stér✐♦ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❛✐♥❞❛ ❡stã♦ ♠✉✐t♦ ❧♦♥❣❡ ❞❡ s❡r❡♠ s❛t✐s❢❛tór✐♦s✳ ◆❡♠ ❛ ❧❡✐ q✉❡ ❣❛r❛♥t❡ ❛♦ ♣r♦❢❡ss♦r q✉❡ ✉♠ t❡rç♦ ❞❡ s✉❛ ❝❛r❣❛ ❤♦rár✐❛ s❡❥❛ ❝✉♠♣r✐❞❛ ❡♠ ❛t✐✈✐❞❛❞❡s ❡①tr❛❝❧❛ss❡ é r❡s♣❡✐t❛❞❛ ♣❡❧❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s ór❣ã♦ ❞❡ ❡❞✉❝❛çã♦ ♣ú❜❧✐❝❛✳
❖ P❘❖❋▼❆❚ é t❛♠❜é♠ ✉♠❛ ✐♥✐❝✐❛t✐✈❛ ❞♦ ❣♦✈❡r♥♦ ❢❡❞❡r❛❧ ♥♦ s❡♥t✐❞♦ ❞❡ s❡ ♠❡❧❤♦r❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s ❛❧✉♥♦s ♣♦r ♠❡✐♦ ❞❛ ❝❛♣❛❝✐t❛çã♦ ❞♦s ♣r♦❢❡ss♦r❡s✳ ➱ ✉♠❛ ❜❡❧❛ ✐♥✐❝✐❛t✐✈❛ ♣♦r ♣❛rt❡ ❞❡ ♥♦ss♦s ❣♦✈❡r♥❛♥t❡s✱ ♣♦ré♠✱ ♦s ♣r♦❢❡ss♦r❡s q✉❡ ♠❛✐s ♣r❡❝✐s❛r✐❛♠ ❞❡ss❛ ❝❛♣❛❝✐t❛çã♦ ❞✐✜❝✐❧♠❡♥t❡ ❛ r❡❝❡❜❡rã♦✳ ❆ ♣r♦✈❛ ❞❡ s❡❧❡çã♦ é ❡①tr❡♠❛♠❡♥t❡ ❝♦♥❝♦rr✐❞❛ ❡ ✉♠❛ ❣r❛♥❞❡ ♣♦r❝❡♥t❛❣❡♠ ❞♦s ❛♣r♦✈❛❞♦s ♥ã♦ ❝♦♥s❡❣✉❡ ❝♦♥❝❧✉✐r ♦ ❝✉rs♦ ♣♦r ❢❛❧t❛ ❞❡ ♣ré✲r❡q✉✐s✐t♦s ❜ás✐❝♦s ♦✉ ♠❡s♠♦ ♣♦r ❢❛❧t❛ ❞❡ t❡♠♣♦ ♣❛r❛ s❡ ❞❡❞✐❝❛r❡♠ ❛♦ ❝✉rs♦✳
P❡♥s❛♥❞♦ s♦❜r❡ ❛ ♣rát✐❝❛ ❞♦❝❡♥t❡ ❞♦s ❝♦♥t❡ú❞♦s ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❡s❝♦❧❤❡♠♦s ❛ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♣❛r❛ ♣r♦♣♦r ✉♠ ♥♦✈♦ t✐♣♦ ❞❡ ❛❜♦r❞❛❣❡♠✳ ❇❛s❡❛❞♦ ♥❛ ❊♥❣❡♥❤❛r✐❛ ❉✐❞át✐❝❛✱ ❝r✐❛♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ♣❛r❛ ✐♥tr♦❞✉③✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❣r❛♥❞❡③❛s ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ❡ ❣r❛♥❞❡③❛s ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✳
❖ t❡♠❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱ q✉❡ é ✉♠ ❞♦s ❝♦♥t❡ú❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦ ❝✉rrí❝✉❧♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❢♦✐ ❡s❝♦❧❤✐❞♦ ♣♦r s❡r ✉♠❛ t❡♦r✐❛ q✉❡ r❡❧❛❝✐♦♥❛ ❞✐✈❡rs❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛ ❡ ❛❞♠✐t❡ ✉♠❛ ❞✐✈❡rs✐❞❛❞❡ ❞❡ tr❛t❛♠❡♥t♦s q✉❡ ✈ã♦ ❞❡ ✉♠❛ s✐♠♣❧❡s ✐❣✉❛❧❞❛❞❡ ❞❡ ❢r❛çõ❡s ❛ ✉♠ tr❛t❛♠❡♥t♦ ♠❛✐s ❡❧❛❜♦r❛❞♦ ✉s❛♥❞♦ ❢✉♥çõ❡s✳
✶✳✶✳ ▼❡t♦❞♦❧♦❣✐❛ ✸
❞❡ P❧❛tã♦✱ s♦❧✉❝✐♦♥♦✉ ❛ ♣r✐♠❡✐r❛ ❣r❛♥❞❡ ❝r✐s❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛ ❛✐♥❞❛ ♥♦ q✉❛rt♦ sé❝✉❧♦ ❛♥t❡s ❞❡ ❈r✐st♦ q✉❛♥❞♦ s✉r❣❡♠ ❛s ❣r❛♥❞❡③❛s ✐♥❝♦♠❡♥s✉rá✈❡✐s✱ q✉❡ ❤♦❥❡ ❝♦♥❤❡❝❡♠♦s ❝♦♠♦ ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳
❇✉s❝❛♠♦s✱ ❛ ♣❛rt✐r ❞❛s ❝♦♥s✐❞❡r❛çõ❡s ❧❡✈❛♥t❛❞❛s ♥❡ss❡s ❛rt✐❣♦s✱ ❝♦♥str✉✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❣r❛♥❞❡③❛s ♦✉ ✈❛r✐á✈❡✐s ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ❡ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s tr❛❜❛❧❤❛♥❞♦ ❞❡ ❢♦r♠❛ s✐st❡♠át✐❝❛ ❝♦♠ s✉❛ ❞❡✜♥✐çã♦✱ t❡♥t❛♥❞♦ ❞❡t❡r♠✐♥❛r ❝♦♠♦ ❛s ❣r❛♥❞❡③❛s s❡ r❡❧❛❝✐♦♥❛♠ ♣❛r❛ ❝❤❡❣❛r♠♦s ❛ ✉♠❛ ❝♦♥❝❧✉sã♦✳ ❇✉s❝❛♠♦s ❛✐♥❞❛ ❡st❛❜❡❧❡❝❡r ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ❣r❛♥❞❡③❛s ❝♦♥str✉✐♥❞♦ ♦ ❣rá✜❝♦ ❞❡ss❛ r❡❧❛çã♦ ♣❛r❛✱ ❛ ♣❛rt✐r ❞♦s ♠❡s♠♦s ❞❡✜♥✐r♠♦s s❡ é ✉♠❛ r❡❧❛çã♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞✐r❡t❛ ♦✉ ✐♥✈❡rs❛✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❝✉r✈❛ ♦❜t✐❞❛✳ ❆ ♣❛rt✐r ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ q✉❛tr♦ ❛t✐✈✐❞❛❞❡s ❢❛r❡♠♦s ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ ✈❛♥t❛❣❡♥s ❡ ❞❡s✈❛♥t❛❣❡♥s ❞❡ t❛❧ ❛❜♦r❞❛❣❡♠ ❛ss✐♠ ❝♦♠♦ ♣r♦♣♦r❡♠♦s ♠✉❞❛♥ç❛s ♥❛s ❛t✐✈✐❞❛❞❡s ❛♣❧✐❝❛❞❛s ♣❛r❛ ❛♣❡r❢❡✐ç♦á✲❧❛s✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s ♣❛râ♠❡tr♦s ❝✉rr✐❝✉❧❛r❡s ♥❛❝✐♦♥❛✐s ✭P❈◆s✮ ❬✻❪✱ ❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡
✳✳✳❡stá ♣r❡s❡♥t❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♠✉❧t✐♣❧✐❝❛t✐✈♦s✱ ♥♦s ❡st✉❞♦s ❞❡ ♣♦r❝❡♥t❛❣❡♠✱ ❞❡
s❡♠❡❧❤❛♥ç❛ ❞❡ ✜❣✉r❛s✱ ♥❛ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛✱ ♥❛ ❛♥á❧✐s❡ ❞❡ t❛❜❡❧❛s✱ ❣rá✜❝♦s ❡ ❢✉♥çõ❡s✳ ❖ ❢❛t♦ ❞❡
q✉❡ ✈ár✐♦s ❛s♣❡❝t♦s ❞♦ ❝♦t✐❞✐❛♥♦ ❢✉♥❝✐♦♥❛♠ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❧❡✐s ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❡✈✐❞❡♥❝✐❛ q✉❡ ♦ r❛❝✐♦❝í♥✐♦ ♣r♦♣♦r❝✐♦♥❛❧ é út✐❧ ♥❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ❢❡♥ô♠❡♥♦s ❞♦ ♠✉♥❞♦ r❡❛❧✳ ✭P❈◆✱ ✶✾✾✽✱ ♣❛❣ ✸✽✮
❈♦♠♦ ♦ ♣ró♣r✐♦ ❞♦❝✉♠❡♥t♦ q✉❡ ♥♦rt❡✐❛ ❛ ❡❞✉❝❛çã♦ ❜r❛s✐❧❡✐r❛ ❛♣r❡s❡♥t❛✱ ♦ r❛❝✐♦❝í♥✐♦ ♣r♦♣♦r❝✐♦♥❛❧ t❡♠ ✐♥ú♠❡r❛s ❛♣❧✐❝❛çõ❡s ❝♦t✐❞✐❛♥❛s q✉❡ ❞❡✈❡♠ s❡r ❡①♣❧♦r❛❞❛s ♣❡❧❛ ❡s❝♦❧❛✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛ ❡s❝♦❧❛ ❆♥♥❛ ❞♦s ❘❡✐s ❙✐❣♥♦r✐♥✐ ✭❙❊❉❊❙✮ ❡ ❛s t✉r♠❛s ❞❡ sét✐♠♦ ❛♥♦ q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ♣r♦❞✉③✐❞❛ ♥❡st❡ tr❛❜❛❧❤♦✳ ❖s ❞❡♠❛✐s tr❛t❛♠ ❞❛s ❢❛s❡s ❞❛ ❡♥❣❡♥❤❛r✐❛ ❞✐❞át✐❝❛✳
✶✳✶ ▼❡t♦❞♦❧♦❣✐❛
❖ ♠ét♦❞♦ ❞❡ ♣❡sq✉✐s❛ q✉❡ ♥♦rt❡✐❛ ❡ss❛ ❞✐ss❡rt❛çã♦ é ❛ ❊♥❣❡♥❤❛r✐❛ ❉✐❞át✐❝❛✱ q✉❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❈❆❘◆❊■❘❖ ❬✺❪ é ✉♠ ♠ét♦❞♦ ❡s♣❡❝í✜❝♦ ❜❛s❡❛❞♦ ❡♠ ❡①♣❡r✐ê♥❝✐❛s ❞❡ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ r❡❢❡r❡♥❝✐❛❧ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♣r♦❞✉t♦s ♣❛r❛ ♦ ❡♥s✐♥♦✱ ❣❡r❛❞♦s ♥❛ ❥✉♥çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣rát✐❝♦ ❝♦♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ t❡ór✐❝♦✳
❖ ♠ét♦❞♦ t❡♠ s✉❛ ♦r✐❣❡♠ ♥❛ ❋r❛♥ç❛✱ ♣♦r ✈♦❧t❛ ❞❡ ✶✾✽✵✳ ➱ ✉♠❛ ❜✉s❝❛ ♣♦r ✐♥♦✈❛r ❡ r❡✈♦❧✉❝✐♦♥❛r ❛ ♣rát✐❝❛ ❞♦ ♣r♦❢❡ss♦r✱ ❜✉s❝❛♥❞♦ s♦❧✉çõ❡s ✐♥♦✈❛❞♦r❛s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠✳
❈♦♥s✐❞❡r❛✲s❡ ✉♠ ♣♦♥t♦ ❞♦ s✐st❡♠❛ ❞✐❞át✐❝♦ ❝✉❥♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ♣❛r❡❝❡✱ ♣♦r r❛③õ❡s ❞❡ ♥❛t✉r❡③❛s
❞✐✈❡rs❛s✱ ♣♦✉❝♦ s❛t✐s❢❛tór✐♦✳ ❆♥❛❧✐s❛✲s❡ ❡ss❡ ♣♦♥t♦ ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❡ ❛s ❝♦♥❞✐çõ❡s q✉❡ t❡♥❞❡♠
❛ ❡♥❝♦♥tr❛r ✉♠ ♥♦✈♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡✱ ❞❡♣♦✐s✱ tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ❡ss❛s ❝♦♥❞✐çõ❡s ✱ ❜✉s❝❛✲s❡
❞❡t❡r♠✐♥❛r ❝♦♥❞✐çõ❡s ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♠♦❞♦ ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ ♠❛✐s s❛t✐s❢❛tór✐♦✳✭❆rt✐❣✉❡✱ ✶✾✾✵✮
✳✳✳é ✉♠❛ ❢♦r♠❛ ❞❡ tr❛❜❛❧❤♦ ❞✐❞át✐❝♦ ❝♦♠♣❛rá✈❡❧ ❛♦ tr❛❜❛❧❤♦ ❞♦ ❡♥❣❡♥❤❡✐r♦ q✉❡✱ ♣❛r❛ r❡❛❧✐③❛r ✉♠
♣r♦❥❡t♦✱ s❡ ❛♣♦✐❛ ❡♠ ❝♦♥❤❡❝✐♠❡♥t♦s ❝✐❡♥tí✜❝♦s ❞❡ s❡✉ ❞♦♠í♥✐♦✱ ❛❝❡✐t❛ s❡ s✉❜♠❡t❡r ❛ ✉♠ ❝♦♥tr♦❧❡ ❞❡
t✐♣♦ ❝✐❡♥tí✜❝♦✱ ♠❛s ❛♦ ♠❡s♠♦ t❡♠♣♦✱ é ♦❜r✐❣❛❞♦ ❛ tr❛❜❛❧❤❛r ♦❜❥❡t♦s ♠❛✐s ❝♦♠♣❧❡①♦s q✉❡ ♦s ♦❜❥❡t♦s
✹ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦
❆ ❊♥❣❡♥❤❛r✐❛ ❉✐❞át✐❝❛ ❝♦♥st✐t✉✐✲s❡ ♥❛ ❡①❡❝✉çã♦ ❞❡ q✉❛tr♦ ❢❛s❡s✿
• ❆♥á❧✐s❡ ♣ré✈✐❛ ♦✉ ♣r❡❧✐♠✐♥❛r✿ ◆❡st❛ ❢❛s❡ é ❢❡✐t❛ ❛ ❢✉♥❞❛♠❡♥t❛çã♦ ❞❛ ❊♥❣❡♥❤❛r✐❛ ❉✐❞át✐❝❛✳ ❆♣ós ❛ ❡s❝♦❧❤❛ ❞♦ t❡♠❛✱ ♦ ❡st✉❞❛♠♦s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❣❡r❛❧♠❡♥t❡ ❛❞♦t❛❞♦ ♥♦ ❡♥s✐♥♦ ❡✱ t❛♠❜é♠✱ ❞❡ s✉❛ ❡✈♦❧✉çã♦ ❛♦ ❧♦♥❣♦ ❞❛s ♠✉❞❛♥ç❛s ❞❡ ♣r♦❣r❛♠❛✳ ❆♥❛❧✐s❛♠♦s ♦ ♠♦❞♦ ❞❡ ❡♥s✐♥♦ ❛t✉❛❧ ❡ ❢❛③❡♠♦s ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ ❡rr♦s ❝♦♠❡t✐❞♦s ♣❡❧♦s ❛❧✉♥♦s✳
• ❈♦♥❝❡♣çã♦ ❡ ❛♥á❧✐s❡ ❛ ♣r✐♦r✐ ❞❡ ❡①♣❡r✐ê♥❝✐❛s ❞✐❞át✐❝♦✲♣❡❞❛❣ó❣✐❝❛s ❛ s❡r❡♠ ❞❡s❡♥✈♦❧✈✐❞❛s ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✿ ❚❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❞❡t❡r♠✐♥❛r ❝♦♠♦ ❛s ❡s❝♦❧❤❛s r❡❛❧✐③❛❞❛s ♣❡r♠✐t❡♠ ❝♦♥tr♦❧❛r ♦s ❝♦♠♣♦rt❛♠❡♥t♦s ❞♦s ❛❧✉♥♦s✳ ❆ ❡❧❛❜♦r❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ❝♦♠ ❞❡s❝r✐çã♦ ❡ ❥✉st✐✜❝❛t✐✈❛ sã♦ ❡❧❡♠❡♥t♦s ❞❡ss❛ ❢❛s❡✳
• ❊①♣❡r✐♠❡♥t❛çã♦✱ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❛ ❡①♣❡r✐ê♥❝✐❛ ♦✉ ❛♣❧✐❝❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛✿ ◆❡ss❡ ♠♦♠❡♥t♦ é ❛♣❧✐❝❛❞❛ ❛ s❡q✉ê♥❝✐❛ ❡ sã♦ ❝♦❧❡t❛❞❛s ❛s ♦❜s❡r✈❛çõ❡s ❛ r❡s♣❡✐t♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ♠❡s♠❛✳
• ❆♥á❧✐s❡ ❛ ♣♦st❡r✐♦r✐ ❡ ✈❛❧✐❞❛çã♦ ❞❛ ❡①♣❡r✐ê♥❝✐❛✿ ❆q✉✐ ❢❛③❡♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❛ r❡❛❧✐③❛çã♦ ❞❛ ❡♥❣❡♥❤❛r✐❛ ❝♦♠♦ ✉♠ t♦❞♦✱ ❝♦♥❢r♦♥t❛♥❞♦ s❡ ♦ q✉❡ ❢♦✐ ❡❧❛❜♦r❛❞♦ ❡ ❛♣❧✐❝❛❞♦ t❡✈❡ ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳ ❊❧❡♥❝❛♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❞✐✜❝✉❧❞❛❞❡s ❡ ♣r♦❜❧❡♠❛s ❡♥❝♦♥tr❛❞♦s ♥❛ ❡❧❛❜♦r❛çã♦ ❡ r❡❛❧✐③❛çã♦ ❞❛ ♠❡s♠❛✳
✺
❈❛♣ít✉❧♦ ✷
❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❊s❝♦❧❛ ❡ ❞❛s s❛❧❛s ❞❡
❛✉❧❛
❆ ❡s❝♦❧❛ ♠✉♥✐❝✐♣❛❧ ■♥t❡❣r❛❧ ❞❡ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ Pr♦❢❡ss♦r❛ ❆♥♥❛ ❞♦s ❘❡✐s ❙✐❣♥♦r✐♥✐ s❡ ❧♦❝❛❧✐③❛ ♥❛ ❝✐❞❛❞❡ ❞❡ ❚❛✉❜❛té✱ ♥♦ ✐♥t❡r✐♦r ❞♦ ❡st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✳ ◆♦ ❛♥♦ ❝♦rr❡♥t❡ sã♦ ✶✵✹✷ ✭✉♠ ♠✐❧ ❡ q✉❛r❡♥t❛ ❡ ❞♦✐s✮ ❛❧✉♥♦s ♠❛tr✐❝✉❧❛❞♦s ❡♠ ❞♦✐s t✉r♥♦s ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦✳
❋✐❣✉r❛ ✷✳✶✿ ❊s❝♦❧❛ ❆♥♥❛ ❞♦s ❘❡✐s ❙✐❣♥♦r✐♥✐
❙ã♦ ✹✵ s❛❧❛s ❞❡ ❛✉❧❛✱ s❡♥❞♦ ✷✵ ♥♦ ♣❡rí♦❞♦ ♠❛t✉t✐♥♦ ❡ ✷✵ ♥♦ ✈❡s♣❡rt✐♥♦ ❡ ❤á ✉♠❛ ❡①t❡♥s❛ ❧✐st❛ ❞❡ ❡s♣❡r❛ ♣❛r❛ s❡ ❝♦♥s❡❣✉✐r ✉♠❛ ✈❛❣❛✱ ❛t✉❛❧♠❡♥t❡ sã♦ ✸✸✺ ❛❧✉♥♦s ❛❣✉❛r❞❛♥❞♦ ♦ s✉r❣✐♠❡♥t♦ ❞❡ ♥♦✈❛s ✈❛❣❛s✳ ■ss♦ s❡ ❞❡✈❡ ❛♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❡s❝♦❧❛ t❡♠ ✉♠❛ ✐♥❢r❛❡str✉t✉r❛ ♣r✐✈✐❧❡❣✐❛❞❛ ❡♠ r❡❧❛çã♦ à ♠❛✐♦r✐❛ ❞❛s ❡s❝♦❧❛s ♣ú❜❧✐❝❛s ❜r❛s✐❧❡✐r❛s✳ ❈♦♥t❛ ❝♦♠ ❛♥✜t❡❛tr♦✱ ❜✐❜❧✐♦t❡❝❛✱ ♣✐s❝✐♥❛ s❡♠✐♦❧í♠♣✐❝❛✱ ✈❡st✐ár✐♦s✱ r❡❢❡✐tór✐♦✱ ❞✉❛s q✉❛❞r❛s ♣♦❧✐❡s♣♦rt✐✈❛s ❝♦❜❡rt❛s ❡ ❧❛❜♦r❛tór✐♦ ❞❡ ✐♥❢♦r♠át✐❝❛✱ t✉❞♦ ❡♠ ♣❧❡♥♦ ❢✉♥❝✐♦♥❛♠❡♥t♦✳
✻ ❈❛♣ít✉❧♦ ✷✳ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❊s❝♦❧❛ ❡ ❞❛s s❛❧❛s ❞❡ ❛✉❧❛
❛❧✉♥♦s✱ ❢♦r♠❛♥❞♦ ❜❛s❡ ♣❛r❛ ❛s ❡q✉✐♣❡s ❞❛ ❝✐❞❛❞❡✳ ❖ ♣r♦❥❡t♦ ❛t✉❛❧♠❡♥t❡ ❢✉♥❝✐♦♥❛ ❝♦♠ s❡t❡ s❛❧❛s✱ s❡♥❞♦ três s❛❧❛s ❞❡ s❡①t♦s ❛♥♦s ❡ q✉❛tr♦ ❞❡ sét✐♠♦s ❛♥♦s✱ ♦♥❞❡ ❢♦✐ ❛♣❧✐❝❛❞❛ ❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ❡♠ q✉❡stã♦✳
❋✐❣✉r❛ ✷✳✷✿ Pr♦❣r❛♠❛ ❊s❝♦❧❛ ❞❡ ❆t❧❡t❛s ❡ ❋♦r♠❛çã♦ ■♥t❡❣r❛❧
❖ ♣r♦❥❡t♦ ✈❡♠ ♦❜t❡♥❞♦ s✉❝❡ss♦ ♥♦ ❞❡s❡♠♣❡♥❤♦ ❡s❝♦❧❛r ❞♦s ❛❧✉♥♦s✱ q✉❡ sã♦ s❡❧❡❝✐♦♥❛❞♦s ♣♦r ❛♣t✐❞ã♦ ❢ís✐❝❛ ❡ ♥ã♦ ✐♥t❡❧❡❝t✉❛❧✳ ❖s ❛❧✉♥♦s sã♦ ❞✐str✐❜✉í❞♦s ❡♠ s❡t❡ ♠♦❞❛❧✐❞❛❞❡s✿ ❋✉ts❛❧✱ ❱♦❧❡✐❜♦❧✱ ❇❛sq✉❡t❡✱ ❍❛♥❞❡❜♦❧✱ ❆t❧❡t✐s♠♦✱ ●✐♥ást✐❝❛ ❆rtíst✐❝❛ ❡ ❥✉❞ô✳ ❖ q✉❛❞r♦ ❛❜❛✐①♦ ❛♣r❡s❡♥t❛ ✉♠ ❝♦♠♣❛r❛t✐✈♦ ❞❡ r❡♥❞✐♠❡♥t♦ ❞♦s ❛❧✉♥♦s ❛t❡♥❞✐❞♦s ♣❡❧♦ ♣r♦❥❡t♦ ❡ ♣❡❧♦s q✉❡ ♥ã♦ ♦ sã♦ ✭r❡❣✉❧❛r❡s✮✳
❚❛❜❡❧❛ ✷✳✶✿ ◗✉❛❞r♦ ❝♦♠♣❛r❛t✐✈♦ ❞❛ ♠é❞✐❛ ❣❧♦❜❛❧ ❞♦s ❛❧✉♥♦s ❞❛ ❊❆❋■ ❡ ❞♦s ❛❧✉♥♦s r❡❣✉❧❛r❡s
❇✐♠❡str❡ ▼é❞✐❛ ❞♦s ❛❧✉✲ ♥♦s ❞♦ ✻♦ ❛♥♦
r❡❣✉❧❛r❡s
▼é❞✐❛ ❞♦s ❛❧✉✲ ♥♦s ❞♦ ✻♦ ❛♥♦ ❞❛
❊❆❋■
▼é❞✐❛ ❞♦s ❛❧✉✲ ♥♦s ❞♦ ✼♦ ❛♥♦
r❡❣✉❧❛r❡s
▼é❞✐❛ ❞♦s ❛❧✉✲ ♥♦s ❞♦ ✼♦ ❛♥♦ ❞❛
❊❆❋■
Pr✐♠❡✐r♦ ✺✱✾ ✼✱✷ ✺✱✺ ✻✱✾
❙❡❣✉♥❞♦ ✺✱✾ ✼✱✶ ✺✱✽ ✼✱✵
❤tt♣✿✴✴✇✇✇✳♦rt✐③❥✉♥✐♦r✳❝♦♠✴❞❡s❡♠♣❡♥❤♦✲❡❞✉❝❛❝✐♦♥❛❧✲♥♦✲❡♥s✐♥♦✲✐♥t❡❣r❛❧✲❡✲♠❡❧❤♦r✲q✉❡✲♥♦✲r❡❣✉❧❛r ✭❆❝❡ss❛❞♦ ❡♠ ✵✹✴✵✼✴✷✵✶✺✮
✷✳✶✳ ❙ét✐♠♦ ❆♥♦ ❈ ✼
♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❛❧✉♥♦s✳ ❏✉♥t❛♠❡♥t❡ ❝♦♠ ♦s tr❡✐♥♦s✱ ♦s ❛❧✉♥♦s tê♠ ❛✉❧❛s ❞❡ ❡st✉❞♦✱ ①❛❞r❡③✱ r✐t♠♦s ❡ ♥❛t❛çã♦✳
❆s t✉r♠❛s ❞❡ sét✐♠♦s ❛♥♦s ❈✱ ❉✱ ❋ ❡ ● ❢♦r❛♠ ♦ ❢♦❝♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛✳ ❆ s❡❣✉✐r ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ♣❡r✜❧ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s t✉r♠❛s✳
✷✳✶ ❙ét✐♠♦ ❆♥♦ ❈
◆❡ss❛ t✉r♠❛ ❞♦ ♣❡rí♦❞♦ ❞❛ ♠❛♥❤ã ❡st✉❞❛♠ ✷✵ ❛❧✉♥♦s✳ ❉❛s q✉❛tr♦ t✉r♠❛s ❞❡ ✼♦ ❛♥♦ ❞❛
❊❆❋■ é ❛ q✉❡ ❛♣r❡s❡♥t❛ ♠❡❧❤♦r r❡♥❞✐♠❡♥t♦✳ ❖ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ ❝♦♥❤❡❝❡ ♦s ❛❧✉♥♦s ❤á ♠❛✐s ❞❡ ✉♠ ❛♥♦ ❡ ✐❞❡♥t✐✜❝❛ ❝✐♥❝♦ ❛❧✉♥♦s q✉❡ ♣♦ss✉❡♠ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡ ♥❛ ❞✐s❝✐♣❧✐♥❛ ❡ ❝✐♥❝♦ q✉❡ ❣❛♥❤❛♠ ✉♠ ❝❡rt♦ ❞❡st❛q✉❡ ♣❡❧♦s r❡s✉❧t❛❞♦s q✉❡ ❛♣r❡s❡♥t❛♠ ♥❛s ❛✈❛❧✐❛çõ❡s✳ ➱ ✉♠❛ t✉r♠❛ ❜❛st❛♥t❡ ♣❛rt✐❝✐♣❛t✐✈❛ ❡ ❝♦♠ ♣♦✉❝♦s ♣r♦❜❧❡♠❛s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦✳
❋✐❣✉r❛ ✷✳✸✿ ❈♦♠♣❛r❛t✐✈♦ ❞❡ ❘❡♥❞✐♠❡♥t♦ ♥♦s ❞♦✐s ♣r✐♠❡✐r♦s ❜✐♠❡str❡s ❞❡ ✷✵✶✺ ❞♦ ✼♦❈
✷✳✷ ❙ét✐♠♦ ❆♥♦ ❉
❊ss❛ é ❛ ♦✉tr❛ t✉r♠❛ ❞♦ ♣❡rí♦❞♦ ❞❛ ♠❛♥❤ã✳ ❈♦♥t❛ ❝♦♠ ✷✷ ❛❧✉♥♦s ♠❛tr✐❝✉❧❛❞♦s✳ ➱ ♦ ♣r✐♠❡✐r♦ ❛♥♦ q✉❡ ♦ ♣r♦❢❡ss♦r tr❛❜❛❧❤❛ ❝♦♠ ❡ss❛ t✉r♠❛ ❡ é ❝♦♠✉♠ ✉♠ ❝❡rt♦ ♣❡rí♦❞♦ ❞❡ ❛❞❛♣t❛çã♦ ❞♦s ❛❧✉♥♦s ♣❛r❛ ❝♦♠ ♦ ♣r♦❢❡ss♦r ❡ ✈✐❝❡ ✈❡rs❛✳ ➱ ✉♠❛ t✉r♠❛ ♦♥❞❡ ❤á ❝✐♥❝♦ ❛❧✉♥♦s q✉❡ ❛♣r❡s❡♥t❛♠ ♠❛✐♦r ❢❛❝✐❧✐❞❛❞❡ ❝♦♠ ♦s ❝♦♥t❡ú❞♦s ❛♣r❡s❡♥t❛❞♦s ❛té ♦ ♠♦♠❡♥t♦ ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ♠❛t❡♠át✐❝❛✱ ♥ú♠❡r♦ ✐❣✉❛❧ ❛♦ ❞♦ ✼♦❈✱ ♣♦ré♠ ❤á ♠❛✐s ❛❧✉♥♦s q✉❡ ❛♣r❡s❡♥t❛♠
✉♠ ❣r❛✉ ♠❛✐♦r ❞❡ ❞✐✜❝✉❧❞❛❞❡✳
✽ ❈❛♣ít✉❧♦ ✷✳ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❊s❝♦❧❛ ❡ ❞❛s s❛❧❛s ❞❡ ❛✉❧❛
◆♦ ♣r✐♠❡✐r♦ ❜✐♠❡str❡ ❢♦r❛♠ ♥♦✈❡ ❛❧✉♥♦s ❝♦♠ ♥♦t❛ ✐♥❢❡r✐♦r ❛ ✻✱✵ ♣♦♥t♦s ✭♥♦t❛ ♠í♥✐♠❛ ♣❛r❛ ❛ ❛♣r♦✈❛çã♦✮✱ ♦ q✉❡ s❡ ♠❛♥t❡✈❡ ♥♦ s❡❣✉♥❞♦ ❜✐♠❡str❡✳
✷✳✸ ❙ét✐♠♦ ❆♥♦ ❋
❈♦♠ ✷✸ ❛❧✉♥♦s ♠❛tr✐❝✉❧❛❞♦s é ✉♠ ❞♦s sét✐♠♦s ❛♥♦s ❞♦ ♣❡rí♦❞♦ ❞❛ t❛r❞❡✳ ❍á três ❛❧✉♥♦s q✉❡ s❡ ❞❡st❛❝❛♠ ❡ ❞♦✐s q✉❡ ♣♦ss✉❡♠ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡ ❝♦♠ ❛ ❞✐s❝✐♣❧✐♥❛✳ ❆ t✉r♠❛ é ♣❛rt✐❝✐♣❛t✐✈❛ ❡ ❡♠ ❣❡r❛❧ s❡ ❛♣❧✐❝❛♠ ♥❛s ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s✳ ❖s ♣♦✉❝♦s ♣r♦❜❧❡♠❛s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ r❡❝❛❡♠ s♦❜r❡ ✉♠ ♣❡q✉❡♥♦ ❣r✉♣♦ ❞❡ q✉❛tr♦ ❛❧✉♥♦s✳
❋✐❣✉r❛ ✷✳✺✿ ❈♦♠♣❛r❛t✐✈♦ ❞❡ ❘❡♥❞✐♠❡♥t♦ ♥♦s ❞♦✐s ♣r✐♠❡✐r♦s ❜✐♠❡str❡s ❞❡ ✷✵✶✺ ❞♦ ✼♦❋
✷✳✹ ❙ét✐♠♦ ❆♥♦ ●
❖ ♦✉tr♦ ✼♦ ❛♥♦ ❞❡ ❛t❧❡t❛s ❞♦ ♣❡rí♦❞♦ ❞❛ t❛r❞❡ ❝♦♥t❛ ❝♦♠ ✷✶ ❛❧✉♥♦s✳ ❆té ♦ ❛♥♦ ♣❛ss❛❞♦
❡r❛ ❝♦♥s✐❞❡r❛❞♦ ♣♦r ♠✉✐t♦s ♣r♦❢❡ss♦r❡s ❝♦♠♦ ❛ ♣✐♦r ❞❛s q✉❛tr♦ s❛❧❛s ❞❡ ❛t❧❡t❛s✱ ❢❛t♦ ♥ã♦ ✈❡r✐✜❝❛❞♦ ♥❛s ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛✳ P❛r❛ ♦ ♣r♦❢❡ss♦r ❞❡ss❛ ❞✐s❝✐♣❧✐♥❛ é ✉♠❛ s❛❧❛ ❝♦♠♦ ❛s ♦✉tr❛s ❝♦♠ ♦s ♠❡s♠♦s ♣r♦❜❧❡♠❛s ❡ ❛s ♠❡s♠❛s q✉❛❧✐❞❛❞❡s✳
❋✐❣✉r❛ ✷✳✻✿ ❈♦♠♣❛r❛t✐✈♦ ❞❡ ❘❡♥❞✐♠❡♥t♦ ♥♦s ❞♦✐s ♣r✐♠❡✐r♦s ❜✐♠❡str❡s ❞❡ ✷✵✶✺ ❞♦ ✼♦●
✾
❈❛♣ít✉❧♦ ✸
❆♥á❧✐s❡ ♣ré✈✐❛✿ ❆ r❡❛❧✐❞❛❞❡ ❞♦ ❊♥s✐♥♦
❞❡ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♥♦ ❇r❛s✐❧
❖ ❡♥s✐♥♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♥♦s ❞✐❛s ❛t✉❛✐s s❡ r❡s✉♠❡ à ❛♣❧✐❝❛çõ❡s ❞❡ r❡❣r❛s ❞❡ três s✐♠♣❧❡s✱ r❡❛❧✐③❛❞❛s ❝♦♠ ♣♦✉❝♦ ♦✉ ♥❡♥❤✉♠ ❢✉♥❞❛♠❡♥t♦ t❡ór✐❝♦✳ ❊♠ ❣❡r❛❧✱ ✐❞❡♥t✐✜❝❛✲s❡ ❛s ❞✉❛s ❣r❛♥❞❡③❛s ❡♥✈♦❧✈✐❞❛s ♥♦ ♣r♦❜❧❡♠❛✱ ✈❡r✐✜❝❛✲s❡ s❡ sã♦ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ♦✉ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ❛♥❛❧✐s❛♥❞♦ ♦ q✉❡ ♦ ❛✉♠❡♥t♦ ♥♦ ✈❛❧♦r ❞❡ ✉♠❛ ❞❡❧❛s ♦❝❛s✐♦♥❛ ❛♦ ✈❛❧♦r ❞❛ ♦✉tr❛✳
❙❡ ♦ ❛✉♠❡♥t♦ ❞❡ ✉♠❛ ✐♥❞✐❝❛r ♦ ❛✉♠❡♥t♦ ❞❛ ♦✉tr❛✱ ❝❧❛ss✐✜❝❛♠♦s ❛s ❣r❛♥❞❡③❛s ❝♦♠♦ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✳ ❊❢❡t✉❛♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✧❝r✉③❛❞❛✧✱ ♦✉ s❡❥❛✱ ♦ ♥✉♠❡r❛❞♦r ❞❡ ✉♠❛ ❢r❛çã♦ ♣❡❧♦ ❞❡♥♦♠✐♥❛❞♦r ❞❛ ♦✉tr❛✱ ✐❣✉❛❧❛♠♦s ♦s r❡s✉❧t❛❞♦s ❡ ♦❜t❡♠♦s ♦ ✈❛❧♦r ❞❡s❝♦♥❤❡❝✐❞♦✳ ❈❛s♦ ♦ ❛✉♠❡♥t♦ ❞❡ ✉♠❛ ❞❡❧❛s ✐♥❞✐q✉❡ ✉♠❛ ❞✐♠✐♥✉✐çã♦ ❞❛ ♦✉tr❛✱ ❡♥tã♦ ❛s ❝❧❛ss✐✜❝❛♠♦s ❝♦♠♦ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✳ ◆❡ss❡ ❝❛s♦ ✐❣✉❛❧❛♠♦s ♦ ♣r♦❞✉t♦ ❞♦s ♥✉♠❡r❛❞♦r❡s ❛♦ ♣r♦❞✉t♦ ❞♦s ❞❡♥♦♠✐♥❛❞♦r❡s ❡ ❛ss✐♠ ♦❜t❡♠♦s ♦ ✈❛❧♦r ❞❡s❝♦♥❤❡❝✐❞♦✳
❆✐♥❞❛ q✉❡ ❛❧❣✉♥s ❧✐✈r♦s tr❛❣❛♠ t♦❞❛ ✉♠❛ ♣r❡♣❛r❛çã♦ ♣❛r❛ s❡ ❝❤❡❣❛r à r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ss❡ t✐♣♦✱ ❝❤❛♠❛❞♦s ❞❡ r❡❣r❛ ❞❡ três✱ ♦ ❡①❝❡ss♦ ❞❡ ❝♦♥t❡ú❞♦s ❛ s❡r tr❛❜❛❧❤❛❞♦ ❢❛③ ❝♦♠ q✉❡ ♠✉✐t❛s ✈❡③❡s ♦s t❡♠❛s s❡❥❛♠ tr❛t❛❞♦s s✉♣❡r✜❝✐❛❧♠❡♥t❡✱ s❡♠ ❝♦♥❡①õ❡s✱ ❡❧✐♠✐♥❛♥❞♦✲s❡ ❡t❛♣❛s ❡ss❡♥❝✐❛✐s ❛ ✉♠ ❝❧❛r♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛ ❡♥✈♦❧✈✐❞❛ ♥♦ ♣r♦❜❧❡♠❛✳ ❆♦ ❛♥❛❧✐s❛r ❝♦♠♦ ♦ ❝♦♥t❡ú❞♦ é tr❛t❛❞♦ ♣♦r ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ❝♦♥st❛t❛♠♦s q✉❡ ❛ ❛❜♦r❞❛❣❡♠ é ♠✉✐t♦ ✐♥t✉✐t✐✈❛ ❡ ♣r❛t✐❝❛✳ ◆ã♦ ❡♥❝♦♥tr❛♠♦s ❧✐✈r♦s q✉❡ ❞❡✜♥❛♠ ❝♦♠ ❝❧❛r❡③❛ ♦ q✉❡ s✐❣♥✐✜❝❛ ♠❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❞✉❛s ❣r❛♥❞❡③❛s s❡r❡♠ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ♦✉ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✳ ◗✉❛♥❞♦ ♦ ❧✐✈r♦ tr❛③ ❡ss❡ tó♣✐❝♦✱ ❛ t❡♦r✐❛ ♥❛❞❛ ♠❛✐s é q✉❡ ❛ r❡s♦❧✉çã♦ ❞❡ ❞♦✐s ♦✉ três ❡①❡♠♣❧♦s ❞❡ ♣r♦❜❧❡♠❛s ❞❡ r❡❣r❛ ❞❡ três✳ Pr❛t✐❝❛♠❡♥t❡ ❞❡s❝❛rt❛✲s❡ t♦❞❛ ❛ t❡♦r✐❛ ❡ ❡♥s✐♥❛✲s❡ tr✉q✉❡s ❡ ♠❛❝❡t❡s q✉❡ ❢✉♥❝✐♦♥❛♠ ❡♠ ❛❧❣✉♥s ❝❛s♦s✱ ♠❛s ♥ã♦ ❡♥r✐q✉❡❝❡♠ ❡ ♥❡♠ ♣r❡♣❛r❛♠ ♦ ❛❧✉♥♦ ♣❛r❛ ❞❛r ❝♦♥t✐♥✉✐❞❛❞❡ ❛♦s ❡st✉❞♦s✳
◗✉❡♠ ♥✉♥❝❛ t❡✈❡ ♣r♦❜❧❡♠❛ ❝♦♠ ❡ss❛s sér✐❡s ❞❡ r❡♣♦rt❛❣❡♠ q✉❡ ❡♥s✐♥❛♠ ♠ét♦❞♦s ♥ã♦ ❝♦♥✈❡♥❝✐♦♥❛✐s ♣❛r❛ ❝❤❡❣❛r ❛ r❡s✉❧t❛❞♦s ❞❡ ♠✉❧t✐♣❧✐❝❛çõ❡s✱ r❛í③❡s q✉❛❞r❛❞❛s✱ ❡t❝❄ ❋❛❝✐❧✐t❛♠♦s t❛♥t♦ q✉❡ ❞❡s❝❛r❛❝t❡r✐③❛♠♦s t♦❞❛ ❛ t❡♦r✐❛ ▼❛t❡♠át✐❝❛✳
✶✵❈❛♣ít✉❧♦ ✸✳ ❆♥á❧✐s❡ ♣ré✈✐❛✿ ❆ r❡❛❧✐❞❛❞❡ ❞♦ ❊♥s✐♥♦ ❞❡ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♥♦❇r❛s✐❧ ❞❡ ❡①❡r❝í❝✐♦s✳ ❚♦❞♦s r❡s♦❧✈✐❞♦s ♠❡❝❛♥✐❝❛♠❡♥t❡✱ ❝♦♠♦ ❞❡s❝r❡✈❡♠♦s s✉❝✐♥t❛♠❡♥t❡ ♥♦s ♣❛rá❣r❛❢♦s ❛❜❛✐①♦✳
Pr♦❜❧❡♠❛ ✶✿ P❛r❛ ❢❛③❡r ✶✻ ❝❛❧ç❛s✱ ❣❛st❛♠♦s ✷✹ ♠❡tr♦s ❞❡ t❡❝✐❞♦✳ ◗✉❛♥t♦ ❣❛st❛r❡♠♦s ♣❛r❛ ❢❛③❡r ✶✵ ❝❛❧ç❛s❄
❋✐❣✉r❛ ✸✳✶✿ ❘❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞✐r❡t❛
❢♦♥t❡✿❤tt♣✿✴✴✇✇✇✳♣r♦❢❥♦s✐♠❛r✳❝♦♠✳❜r✴✷✵✶✸✴✵✽✴❡①❡r❝✐❝✐♦s✲r❡s♦❧✈✐❞♦s✲r❡❣r❛✲❞❡✲✸✲s✐♠♣❧❡s✳❤t♠❧ ✭❆❝❡ss❛❞♦ ❡♠ ✵✼✴✵✼✴✷✵✶✺✮
Pr♦❜❧❡♠❛ ✷✿ ❙❡ ✹ ♦♣❡rár✐♦s ❢❛③❡♠ ❝❡rt❛ ♦❜r❛ ❡♠ ✶✺ ❞✐❛s ❡♠ q✉❛♥t♦s ❞✐❛s ✷✵ ♦♣❡rár✐♦s ❝♦♠ ❛ ♠❡s♠❛ ❡✜❝✐ê♥❝✐❛ ❞♦s ♣r✐♠❡✐r♦s ❢❛r✐❛♠ ❛ ♠❡s♠❛ ♦❜r❛❄
❋✐❣✉r❛ ✸✳✷✿ ❘❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✐♥✈❡rs❛
✶✶
❖s ♣r♦❜❧❡♠❛s ❞❡ r❡❣r❛ ❞❡ três ❝♦♠♣♦st❛✱ q✉❡ sã♦ ❛q✉❡❧❡s q✉❡ ❡♥✈♦❧✈❡♠ ♠❛✐s ❞❡ ❞✉❛s ❣r❛♥❞❡③❛s ♦✉ ✈❛r✐á✈❡✐s✱ ♥ã♦ sã♦ tr❛❜❛❧❤❛❞♦s✱ ♣♦✐s ♦ ♠ét♦❞♦ ✉s❛❞♦ ♣❛r❛ r❡s♦❧✈❡r ♦s ♣r♦❜❧❡♠❛s ❞❡ r❡❣r❛ ❞❡ três s✐♠♣❧❡s s❡ t♦r♥❛ ❡①tr❡♠❛♠❡♥t❡ ❝♦♠♣❧✐❝❛❞♦✱ ✈✐st♦ q✉❡ é ♥❡❝❡ssár✐♦ ❛♥❛❧✐s❛r ❛s ❣r❛♥❞❡③❛s ❛♦s ♣❛r❡s ❡ ❛ss✉♠❡✲s❡ ❢❛t♦s ♥♦✈♦s q✉❡ t♦r♥❛♠ ❛✐♥❞❛ ♠❛✐s ♦❜s❝✉r❛s ❛s r❡s♦❧✉çõ❡s ❞♦s ♣r♦❜❧❡♠❛s✳
❱❛♠♦s s✉♣♦r q✉❡ ✉♠ ♣r♦❜❧❡♠❛ tr❛t❡ ❞❡ três ❣r❛♥❞❡③❛s ♥ã♦ ❡s♣❡❝✐✜❝❛❞❛s ❆✱ ❇ ❡ ❈✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛♥❛❧✐s❛♥❞♦ ❛s ❣r❛♥❞❡③❛s ❆ ❡ ❇ ❝❤❡❣✉❡♠♦s à ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ❛s ♠❡s♠❛s sã♦ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ❡ ❛♦ ❛♥❛❧✐s❛r♠♦s ❛s ❣r❛♥❞❡③❛s ❆ ❡ ❈✱ ❝❤❡❣✉❡♠♦s à ❝♦♥❝❧✉sã♦ q✉❡ ❛s ♠❡s♠❛s sã♦ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✳ ❈♦♥❝❧✉✐✲s❡ ❡♥tã♦✱ s❡♠ ♥❡♥❤✉♠❛ ❢✉♥❞❛♠❡♥t❛çã♦ ♦✉ ❡①♣❧✐❝❛çã♦ q✉❡ ❛s ❣r❛♥❞❡③❛s ❇ ❡ ❈ sã♦ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✱ ♦ q✉❡ é ❝♦rr❡t♦✳ ▼❛s ❞❡ ♦♥❞❡ s✉r❣❡ t❛❧ r❡s✉❧t❛❞♦❄
❖ q✉❡ ♣❡r❝❡❜❡♠♦s é q✉❡ ❛ ♠❡❝❛♥✐③❛çã♦ ❛❥✉❞❛ ❡♠ ❝❡rt♦ ❛s♣❡❝t♦ ❛ ❝❤❡❣❛r à s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✱ ♣♦ré♠✱ ❛♦ s❡ ❛♣r♦❢✉♥❞❛r ♦ ❛ss✉♥t♦✱ ❡❧❛ ♣r♦❞✉③ r❡s✉❧t❛❞♦s ❝❛t❛stró✜❝♦s ❡♠ ❝❛❞❡✐❛✳ ■ss♦ ❡①♣❧✐❝❛✱ ❞❡ ❝❡rt♦ ♠♦❞♦✱ ♣♦rq✉❡ ♦s ❛❧✉♥♦s ✈ã♦ tã♦ ♠❛❧ ❡♠ ▼❛t❡♠át✐❝❛✱ ❥á q✉❡ ❡ss❛ ❝✉❧t✉r❛ t❡♠ r❛í③❡s ♣r♦❢✉♥❞❛s ♥❛ ❡❞✉❝❛çã♦ ❜r❛s✐❧❡✐r❛✳ ❉❛♠♦s ✈❛❧♦r ❡①❝❡ss✐✈♦ às r❡❣r❛s ❡♠ ❞❡tr✐♠❡♥t♦ à ❛♥á❧✐s❡ ♣r❡❝✐s❛ ❞♦ ♣r♦❜❧❡♠❛ ❡ ❞♦ s❡✉ r❡s✉❧t❛❞♦✱ ♣♦✐s ❡ss❛ é ♠✉✐t♦ ♠❛✐s ❞❡♠♦r❛❞❛ ❡ ❞✐❢í❝✐❧ ❞❡ s❡r ❡①♣❧✐❝❛❞❛✳ ❘❡q✉❡r ✉♠ ♠❡❧❤♦r ♣❧❛♥❡❥❛♠❡♥t♦ ❡ ♣r❡♣❛r♦ ❞♦ ♣r♦❢❡ss♦r✳
❈♦♠ r❡❧❛çã♦ à ❢♦r♠❛ ✉s❛❞❛ ♣❛r❛ ✈❡r✐✜❝❛r ❛ r❡❧❛çã♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❡♥tr❡ ❞✉❛s ❣r❛♥❞❡③❛s✱ ➪✈✐❧❛ ✭✶✾✽✻✮ ❞❡st❛❝❛
❚❡♠ s✐❞♦ ♣rát✐❝❛ ❝♦rr❡♥t❡ ❡♥s✐♥❛r ♦ ❛❧✉♥♦ ❛ ❞❡s❝♦❜r✐r ❛ ❡q✉❛çã♦ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s
❡♥✈♦❧✈✐❞❛s ♥♦ ♣r♦❜❧❡♠❛ ❛ r❡s♦❧✈❡r ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛ s❡❣✉✐♥t❡ r❡❣r❛✿ ✜①❛❞❛s ❛s ✈❛r✐á✈❡✐s ❡♥✈♦❧✈✐❞❛s ♥♦ ♣r♦❜❧❡♠❛✱ ❡①❝❡t♦ ❞✉❛s ❞❡❧❛s✱ ❡ss❛s sã♦ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s s❡ ❛✉♠❡♥t❛♠ ♦✉ ❞✐♠✐♥✉❡♠
s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ❡ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✱ s❡ ✉♠❛ ❛✉♠❡♥t❛ ❡♥q✉❛♥t♦ ❛ ♦✉tr❛ ❞✐♠✐♥✉✐✳
❊ss❛ r❡❣r❛ ♥ã♦ t❡♠ ❥✉st✐✜❝❛t✐✈❛ ❧ó❣✐❝❛ ❡ ♥❡♠ s❡♠♣r❡ ♣r♦❞✉③ r❡s✉❧t❛❞♦s ❝♦rr❡t♦s✳ ➱ ✈❡r❞❛❞❡ q✉❡ ❞✉❛s ✈❛r✐á✈❡✐s ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ❛✉♠❡♥t❛♠ ♦✉ ❞✐♠✐♥✉❡♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ♠❛s ❞✉❛s ✈❛r✐á✈❡✐s ♣♦❞❡♠ ❛✉♠❡♥t❛r ♦✉ ❞✐♠✐♥✉✐r s✐♠✉❧t❛♥❡❛♠❡♥t❡ s❡♠ q✉❡ s❡❥❛♠ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s❀ ❡ ✉♠❛ ♦❜s❡r✈❛çã♦ ✐♥t❡✐r❛♠❡♥t❡ ❛♥á❧♦❣❛ ✈❛❧❡ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✈❛r✐á✈❡✐s ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✳
➪✈✐❧❛ ✜♥❛❧✐③❛ ❛ ❛✜r♠❛çã♦ ❝♦♠ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿ ❯♠ tr❛❜❛❧❤❛❞♦r ❣❛st❛ ✺ ❤♦r❛s ♣❛r❛ ❧✐♠♣❛r ✉♠ t❡rr❡♥♦ ❝✐r❝✉❧❛r ❞❡ ✼ ♠❡tr♦s ❞❡ r❛✐♦✳ ◗✉❛♥t♦ t❡♠♣♦ ❣❛st❛r✐❛ s❡ ♦ t❡rr❡♥♦ t✐✈❡ss❡ ✶✹ ♠❡tr♦s ❞❡ r❛✐♦❄
❘❡s♦❧✈❡♥❞♦ ♦ ❡①❡r❝í❝✐♦ s❡❣✉✐♥❞♦ ♦ ♠ét♦❞♦ ♠❡❝❛♥✐③❛❞♦ ❛♣r❡s❡♥t❛❞♦✱ ❝❤❡❣❛rí❛♠♦s ❡rr♦♥❡❛♠❡♥t❡ ❛♦ r❡s✉❧t❛❞♦ ❞❡ ✶✵ ❤♦r❛s✳ ■ss♦ ♣♦r q✉❡ ♦ t❡♠♣♦ ♥ã♦ é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ à ♠❡❞✐❞❛ ❞♦ r❛✐♦ ❡ s✐♠ ❛♦ q✉❛❞r❛❞♦ ❞❡ss❛ ♠❡❞✐❞❛✳
❖✉tr♦ ❡①❡♠♣❧♦ ❞❛ ✐♥❡✜❝✐ê♥❝✐❛ ❞❡ss❛ ❢♦r♠❛ ❞❡ ❞❡t❡r♠✐♥❛r s❡ ❞✉❛s ❣r❛♥❞❡③❛s sã♦ ♣r♦♣♦r❝✐♦♥❛✐s é ❞❛❞❛ ♣♦r ▲✐♠❛ ❡t ❛❧ ✭✷✵✵✶✮ ♥❛ ♦❜r❛ ❚❡♠❛s ❡ Pr♦❜❧❡♠❛s ❬✶✷❪✳
✶✷❈❛♣ít✉❧♦ ✸✳ ❆♥á❧✐s❡ ♣ré✈✐❛✿ ❆ r❡❛❧✐❞❛❞❡ ❞♦ ❊♥s✐♥♦ ❞❡ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ♥♦❇r❛s✐❧
♣❡r♠❛♥❡ç❛ ❝♦♥st❛♥t❡✳ P♦✐s ❛♦ ✜♥❛❧ ❞❡ ❝❛❞❛ ♠ês é ❝♦♠♦ s❡ t✐✈❡ss❡ s✐❞♦ ❛♣❧✐❝❛❞❛ ♥♦✈❛♠❡♥t❡ ✉♠❛ q✉❛♥t✐❛ ♠❛✐♦r✱ ✐❣✉❛❧ à ❡①✐st❡♥t❡ ♥♦ ♠ês ❛♥t❡r✐♦r ♠❛✐s ♦s ❥✉r♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✳ ❆ss✐♠ ♦ r❡t♦r♥♦ ✭♥✉♠ ♣❡rí♦❞♦ ✜①♦✮ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧✱ ♠❛s ♥ã♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ t❡♠♣♦ ❞❡ ✐♥✈❡st✐♠❡♥t♦✳
❆❝❤♦ q✉❡ é ✈á❧✐❞♦ r❡ss❛❧t❛r q✉❡ ♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ s❡ r❡❢❡r❡ ❛♦ r❡❣✐♠❡ ❞❡ ❥✉r♦s ❝♦♠♣♦st♦s✳ ❈❛s♦ ♣❡♥s❡♠♦s ♥♦ r❡❣✐♠❡ ❞❡ ❥✉r♦s s✐♠♣❧❡s✱ ♦ r❡t♦r♥♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧ ❛ss✐♠ ❝♦♠♦ ❛♦ t❡♠♣♦ ❞❡ ✐♥✈❡st✐♠❡♥t♦ ❡ à t❛①❛ ❞❡ ❥✉r♦s✳
✶✸
❈❛♣ít✉❧♦ ✹
❆♥á❧✐s❡ ❛ Pr✐♦r✐✿ ❆ ▼❛t❡♠át✐❝❛
♥❡❝❡ssár✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡
♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡
✧◗✉❛♥❞♦ ♦ ❛❧✉♥♦ s❡ t♦r♥❛ ❝❛♣❛③ ❞❡ ♣♦r ❡♠ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❡ ✉t✐❧✐③❛r ♣♦r s✐ ♠❡s♠♦ ♦ s❛❜❡r q✉❡ ❡stá
❛ ❝♦♥str✉✐r✱ ♥✉♠❛ s✐t✉❛çã♦ ♥ã♦ ♣r❡✈✐st❛ ❡♠ q✉❛❧q✉❡r ❝♦♥t❡①t♦ ❞❡ ❡♥s✐♥♦ ❡ t❛♠❜é♠ ♥❛ ❛✉sê♥❝✐❛ ❞❡
q✉❛❧q✉❡r ♣r♦❢❡ss♦r✱ ❡stá ❛ ♦❝♦rr❡r ❡♥tã♦ ♦ q✉❡ ♣♦❞❡ s❡r ❝❤❛♠❛❞♦ ❞❡ s✐t✉❛çã♦ ❛❞✐❞át✐❝❛✳✧❇r♦✉ss❡❛✉
✭✶✾✾✻✮
❊ss❛ é ❛ s❡❣✉♥❞❛ ❢❛s❡ ❞❛ ❡♥❣❡♥❤❛r✐❛✱ ♦♥❞❡ ❞❡t❡r♠✐♥❛♠♦s ❛s ✈❛r✐á✈❡✐s ❧♦❝❛✐s ❡ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ s✐t✉❛çã♦ ❛❞✐❞át✐❝❛ ❞❡s❡♥✈♦❧✈✐❞❛✳ ❊s❝♦❧❤❡♠♦s ❢❛③❡r ❡st✐♠❛t✐✈❛ ❞❡ ❛❧t✉r❛s ✐♥❛❝❡ssí✈❡✐s ✉s❛♥❞♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ s♦♠❜r❛✱ ♦✉ s❡❥❛✱ tr❛❜❛❧❤❛r❡♠♦s ❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❝♦♠ t❛❧ ✜♥❛❧✐❞❛❞❡✳
❈♦♠♦ ❥á ♠❡♥❝✐♦♥❛❞♦✱ ❤á ✈ár✐❛s ❢♦r♠❛s ❞❡ s❡ ❛❜♦r❞❛r ♦ t❡♠❛ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱ ♠❛s t♦❞♦s ❡❧❡s ❡①✐❣❡♠ q✉❡ ♦ ❛❧✉♥♦ t❡♥❤❛ ✉♠❛ ❝❡rt❛ ♠❛t✉r✐❞❛❞❡ ❡ ✉♠ ♠í♥✐♠♦ ❞❡ tr❛q✉❡❥♦ ❛❧❣é❜r✐❝♦✱ ❛❧é♠ ❞❡ ✉♠ ❜♦♠ s❡♥s♦ ❧ó❣✐❝♦ ❡ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛♥á❧✐s❡✳ ❆ ♠❡❧❤♦r ❢♦r♠❛ ❞❡ s❡ ❛❜♦r❞❛r ♦ t❡♠❛✱ ❞❡♥tr♦ ❞♦ q✉❡ ♣❡ss♦❛❧♠❡♥t❡ ❡♥①❡r❣❛♠♦s ♥❛ ▼❛t❡♠át✐❝❛✱ s❡r✐❛ ✉♠ tr❛t❛♠❡♥t♦ ❢✉♥❝✐♦♥❛❧✳
✹✳✶ ❚r❛t❛♠❡♥t♦ ❋✉♥❝✐♦♥❛❧
❉✐③✲s❡ q✉❡ ❞✉❛s ❣r❛♥❞❡③❛s sã♦ ♣r♦♣♦r❝✐♦♥❛✐s q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ x7→ y✱ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ✈❛❧♦r x ❞❡ ✉♠❛ ❞❡❧❛s ✉♠ ✈❛❧♦r y ❜❡♠ ❞❡✜♥✐❞♦ ❞❛ ♦✉tr❛✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡ s❡❥❛♠ ❝✉♠♣r✐❞❛s ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
• ◗✉❛♥t♦ ♠❛✐♦r ❢♦r x✱ ♠❛✐♦r s❡rá y✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡ ❢❛❧❛♥❞♦✱ s❡ x 7→ y ❡ x′ 7→ y′
❡♥tã♦ x < x′ ⇒y < y′
✶✹ ❈❛♣ít✉❧♦ ✹✳ ❆♥á❧✐s❡ ❛ Pr✐♦r✐✿ ❆ ▼❛t❡♠át✐❝❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ♦❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡
❈✉♠♣r✐❞❛s ❛♠❜❛s ❛s ❝♦♥❞✐çõ❡s✱ ❞✐③❡♠♦s q✉❡ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ x 7→ y é ✉♠❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✳
❖❜s❡r✈❡ q✉❡ ❛ r❡❧❛çã♦ ❞❡✈❡ s❡r ❜❡♠ ❞❡✜♥✐❞❛✱ ♦✉ s❡❥❛✱ q✉❡y ❞❡✈❡ s❡r ❞❛❞♦ ❡♠ ❢✉♥çã♦ ❞❡ x✱ ♣♦rt❛♥t♦✱ ❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ é✱ ♥❛ ✈❡r❞❛❞❡ ✉♠❛ ❢✉♥çã♦ f : R+
→ R+✱ s❡♥❞♦
R+
={x∈R: x >0} ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✶✳ f é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡✱ ✐st♦ é x < x′ ⇒f(x)< f(x′) ♣❛t❛ q✉❛✐sq✉❡r x, x′ ∈R+ ✷✳ P❛r❛ t♦❞♦x∈R+ ❡ t♦❞♦
n∈N✱ t❡♠✲s❡ f(nx) = n·f(x)✳
❆ ♣r♦♣r✐❡❞❛❞❡ ✷ ❛❝✐♠❛✱ ❛ss✉♠✐❞❛ ❛♣❡♥❛s ♣❛r❛n ∈N ✈❛❧❡✱ ♥❛ ✈❡r❞❛❞❡✱ ♣❛r❛ q✉❛❧q✉❡r
♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✳ ■ss♦ é ♦ q✉❡ ❛✜r♠❛ ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✿
❚❡♦r❡♠❛ ✹✳✶✳ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✳ ❙❡❥❛R+ ♦ ❝♦♥❥✉♥t♦s ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✳ ❙❡
f :R+
→R+ é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ t❛❧ q✉❡ f(nx) = n·f(x) ♣❛r❛ t♦❞♦ n∈N✱ ❡♥tã♦ f(cx) =c·f(x) ♣❛r❛ t♦❞♦x, c ∈R+ ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ r = m/n✱ ❝♦♠ m, n ∈ N ❡
t♦❞♦ x∈R+ ✈❛❧❡
n·f(rx) =f(n·rx) = f(mx) =m·f(x),
♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✷✳ ▲♦❣♦✱ f(rx) = m
nf(x) = r ·f(x)✳ ❆ss✐♠✱ é ✈á❧✐❞❛ ❛ ✐❣✉❛❧❞❛❞❡ f(cx) = c·f(x)q✉❛♥❞♦ c é r❛❝✐♦♥❛❧✳
❙✉♣♦♥❤❛♠♦s✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ❡①✐st❛ c > 0 ✐rr❛❝✐♦♥❛❧ t❛❧ q✉❡ f(cx) 6= c·f(x) ♣❛r❛ ❛❧❣✉♠ x∈R+✳ ❊♥tã♦✱ ♦✉ f(cx)< c·f(x) ♦✉f(cx)> c·f(x)✳
❈♦♥s✐❞❡r❡♠♦s ♦ ♣r✐♠❡✐r♦ ❝❛s♦✿
❙❡ f(cx)< c·f(x) ❡♥tã♦ f(cx)/f(x)< c✳ ❙❡❥❛ r ✉♠ ✈❛❧♦r r❛❝✐♦♥❛❧ ❛♣r♦①✐♠❛❞♦ ❞❡ c✱ ❞❡ ♠♦❞♦ q✉❡ f(cx)
f(x) < r < c✱ ❧♦❣♦ f(cx) < r·f(x)< c·f(x)✳ ❈♦♠♦ r é r❛❝✐♦♥❛❧✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ r · f(x) = f(rx)✳ ▼❛s✱ ❝♦♠♦ ❡s❝♦❧❤❡♠♦s r < c✱ t❡♠♦s rx < cx ❡✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✶✱ ✐ss♦ ♦❜r✐❣❛f(rx)< f(cx)✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦✱ ♥ã♦ s❡ ♣♦❞❡ t❡r f(cx)< c·f(x)
❆ ❞❡♠♦♥str❛çã♦ ❞❛ ✐♠♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ t❡r♠♦sf(cx)> c·f(x)é ❛♥á❧♦❣❛ à ❛♣r❡s❡♥t❛❞❛✳ ❈♦♠ ✐ss♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ f(cx) =c·f(x) ♣❛r❛ t♦❞♦x, c ∈R+
❊st❡ ❚❡♦r❡♠❛ ♠♦t✐✈❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞✐r❡t❛ ❡ ✐♥✈❡rs❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
❉❡✜♥✐çã♦✿ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦ f : R+
→ R+ t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦sc❡xt❡♠✲s❡f(cx) =c·f(x)✱ é ✉♠❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞✐r❡t❛✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r q✉❡ ❛s ❣r❛♥❞❡③❛s x ❡ f(x) sã♦ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s✳
❙❡ f(cx) = f(x)
