Algumas experiências algébricas e gráficas com polinômios trigonométricos

❆◆❚❖◆■❖ ❈❆❘▲❖❙ ▼❊◆❉❊❙

❆▲●❯▼❆❙ ❊❳P❊❘■✃◆❈■❆❙ ❆▲●➱❇❘■❈❆❙ ❊ ●❘➪❋■❈❆❙ ❈❖▼ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

❱■➬❖❙❆

❆◆❚❖◆■❖ ❈❆❘▲❖❙ ▼❊◆❉❊❙

❆▲●❯▼❆❙ ❊❳P❊❘■✃◆❈■❆❙ ❆▲●➱❇❘■❈❆❙ ❊ ●❘➪❋■❈❆❙ ❈❖▼ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tí✲ t✉❧♦ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

❆P❘❖❱❆❉❆✿ ❡♠ ✶✺ ❞❡ ❛❜r✐❧ ❞❡ ✷✵✶✸✳

❱❛❧ér✐❛ ▼❛rt✐♥s ❞❛ ❘♦s❛ ❆r✐❛♥❡ P✐♦✈❡③❛♥ ❊♥tr✐♥❣❡r

❆●❘❆❉❊❈■▼❊◆❚❖❙

❆ ❉❡✉s ♣♦r t♦❞❛s ❛s ♦♣♦rt✉♥✐❞❛❞❡s q✉❡ ♠❡ ♦❢❡r❡❝❡✉ ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ✈✐❞❛ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦ ❧✐✈r❛♠❡♥t♦ q✉❡ t✐✈❡ ♥❛ é♣♦❝❛ q✉❡ ❡st❛✈❛ ❡s❝r❡✈❡♥❞♦ ❡ss❛ ❞✐ss❡rt❛çã♦✳

❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♠♣r❡❡♥sã♦ q✉❡ ♠❡ ❞❡r❛♠ s❡♠♣r❡✳ ❊♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ▼❛r✐❛ ❏♦sé ◆✉♥❡s ▼❡♥❞❡s ♣❡❧❛ ❝♦rr❡çã♦ ♦rt♦❣rá✜❝❛ ❡ ❣r❛♠❛t✐❝❛❧ ❞❡ss❡ tr❛❜❛❧❤♦✳

❆♦s ❝♦❧❡❣❛s ♣❡❧♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦ ❞✉r❛♥t❡ ❛ r❡❛❧✐③❛çã♦ ❞♦ ❝✉rs♦✳ ❆♥t❡s ♥ã♦ ♥♦s ❝♦♥❤❡✲ ❝í❛♠♦s✱ t♦r♥❛♠♦s ❝♦❧❡❣❛s ❡ ❛❣♦r❛ ❛♠✐❣♦s ♣❛r❛ t♦❞❛ ❛ ✈✐❞❛✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❱ ♣❡❧❛ ♣❡rs❡✈❡r❛♥ç❛✱ ♦t✐♠✐s♠♦ ❡ ❡♠♣❡♥❤♦ ♣❡❧❛ ♥♦ss❛ tr❛♥s❢♦r♠❛çã♦ ❡❞✉❝❛❝✐♦♥❛❧ ❡ ❝r❡s❝✐♠❡♥t♦ ♥❛ ❛♣r❡♥❞✐③❛❣❡♠ ♠❛t❡♠át✐❝❛✳

❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ♣❡❧♦s ❝♦♠❡♥tár✐♦s ❡ s✉❣❡stõ❡s ❛♣r❡s❡♥t❛❞❛s ♣❡❧❛ ♠❡❧❤♦r✐❛ ❞❡ss❡ tr❛❜❛❧❤♦✳

❘❊❙❯▼❖

▼❊◆❉❊❙✱ ❆♥t♦♥✐♦ ❈❛r❧♦s✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❛❜r✐❧ ❞❡ ✷✵✶✸✳ ❆❧❣✉♠❛s ❊①♣❡r✐ê♥❝✐❛s ❆❧❣é❜r✐❝❛s ❡ ●rá✜❝❛s ❝♦♠ P♦❧✐♥ô♠✐♦s ❚r✐❣♦♥♦♠étr✐❝♦s✳ ❖r✐❡♥t❛❞♦r✿ ❑❡♥♥❡❞② ▼❛rt✐♥s P❡❞r♦s♦✳

❆❇❙❚❘❆❈❚

▼❊◆❉❊❙✱ ❆♥t♦♥✐♦ ❈❛r❧♦s✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❆♣r✐❧✱ ✷✵✶✸✳ ❙♦♠❡ ❊①♣❡✲ r✐❡♥❝❡s ✇✐t❤ ❆❧❣❡❜r❛✐❝ ❛♥❞ ●r❛♣❤✐❝❛❧ ❚r✐❣♦♥♦♠❡tr✐❝ P♦❧②♥♦♠✐❛❧s✳ ❆❞✈s❡r✿ ❑❡♥♥❡❞② ▼❛rt✐♥s P❡❞r♦s♦✳

❙✉♠ár✐♦

■◆❚❘❖❉❯➬➹❖ ✶

✶ ❖ ❊◆❙■◆❖ ❉❆ ❚❘■●❖◆❖▼❊❚❘■❆ ❊ ❆ P❘❖P❖❙❚❆ ❉❊ ❚❘❆❇❆▲❍❖ ✺

✶✳✶ ❆▲●❯▼❆❙ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ❙❖❇❘❊ ❖ ❊◆❙■◆❖ ❉❊ ❚❘■●❖◆❖▼❊❚❘■❆ ✳ ✺ ✶✳✷ P❘❖P❖❙❚❆ ❉❊ ❚❘❆❇❆▲❍❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✶ ❉❊❙❈❘■➬➹❖ ●❊❘❆▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ❖❇❏❊❚■❱❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✸ PÚ❇▲■❈❖ ❆▲❱❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✹ P❘➱✲❘❊◗❯■❙■❚❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✺ ▼❆❚❊❘■❆■❙ ❊ ❚❊❈◆❖▲❖●■❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✻ ❘❊❈❖▼❊◆❉❆➬Õ❊❙ ▼❊❚❖❉❖▲Ó●■❈❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✼ ❉■❋■❈❯▲❉❆❉❊❙ P❘❊❱■❙❚❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✽ P❖❙❙❮❱❊■❙ ❈❖◆❚■◆❯❆➬Õ❊❙ ❖❯ ❉❊❙❉❖❇❘❆▼❊◆❚❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷ ❊❙❚❯❉❖ ❉❆❙ ❋❯◆➬Õ❊❙ ❙❊◆❖ ❊ ❈❖❙❙❊◆❖✳ ✶✹

✷✳✶ ❈❆❘❆❈❚❊❘❮❙❚■❈❆❙ ❉❆❙ ❋❯◆➬Õ❊❙ ❙❊◆❖ ❊ ❈❖❙❙❊◆❖✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✶ ✲ ❊❙❚❯❉❖ ❉❆❙ ❋❯◆➬Õ❊❙ ❙❊◆❖ ❊ ❈❖❙❙❊◆❖✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✷ ❊❙❚❯❉❖ ❉❆ ❱❆❘■❆➬➹❖ ❉❖ P❆❘➶▼❊❚❘❖α◆❆❙ ❋❯◆➬Õ❊❙ ❙❊◆❖

❊ ❈❖❙❙❊◆❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✸ ❊❙❚❯❉❖ ❉❆❙ ❱❆❘■❆➬Õ❊❙ ❉❖ P❆❘➶▼❊❚❘❖ k ◆❆❙ ❋❯◆➬Õ❊❙

❙❊◆❖ ❊ ❈❖❙❙❊◆❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❙❊◆Ó■❉❊❙ ❉❖ ❚■P❖y =a+bsen(cx+d)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸ ❊❙❚❯❉❖ ❉❖❙ ▼❖◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❊ ●❘❆❯ ✶ ✷✸

✸✳✶ ❊❙❚❯❉❖ ❉❖❙ ▼❖◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❖ ✶➸ ●❘❆❯ ❈❖▼ ❱❆✲ ❘■❆➬➹❖ ❉❖ P❆❘➶▼❊❚❘❖ α ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ ❊❙❚❯❉❖ ❉❖❙ ▼❖◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❖ ✶➸ ●❘❆❯ ❈❖▼ ❱❆✲

✹ ❊❙❚❯❉❖ ❉❖❙ ▼❖◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❊ ●❘❆❯ ▼❆■❖❘

◗❯❊ ✶ ✸✵

✹✳✶ ❱❆❘■❆➬➹❖ ❉❖ P❆❘➶▼❊❚❘❖ k ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✷ ❱❆❘■❆➬➹❖ ❉❖❙ P❆❘➶▼❊❚❘❖❙ α ♦✉ β ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✺ ❊❙❚❯❉❖ ❉❖❙ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ✸✸

✺✳✶ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❊ ●❘❆❯ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✷ ❘❆❮❩❊❙ ❖❯ ❩❊❘❖❙ ❉❖❙ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❖ P❘■✲

▼❊■❘❖ ●❘❆❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✸ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❖ ✷➸ ●❘❆❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✹ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ●❘❆❯ ▼❆■❖❘ ◗❯❊ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✻ ❚❘■●❖◆❖▼❊❚❘■❆ ❊ ◆Ú▼❊❘❖❙ ❈❖▼P▲❊❳❖❙ ✹✻

✻✳✶ ❋❖❘▼❆ P❖▲❆❘✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✼ ■◆❚❊❘P❖▲❆➬➹❖ P❖▲■◆❖▼■❆▲ ❚❘■●❖◆❖▼➱❚❘■❈❆ ❊ ❆P▲■❈❆➬Õ❊❙ ❉❆❙

❋❯◆➬Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙ ✺✸

✼✳✶ ■◆❚❊❘P❖▲❆➬➹❖ P❖▲■◆❖▼■❆▲ ❚❘■●❖◆❖▼➱❚❘■❈❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✼✳✷ ❆P▲■❈❆➬Õ❊❙ ❆ ❋❊◆Ô▼❊◆❖❙ ❇■❖▲Ó●■❈❖❙ ❊ ❋❮❙■❈❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✽ ●❯■❆ ❉❖ ❆▲❯◆❖ ✺✽

✽✳✶ ❆P❘❊❙❊◆❚❆➬➹❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✽✳✷ ❖❇❙❊❘❱❆➬Õ❊❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✽✳✸ ❙■❚❯❆➬Õ❊❙ P❘❖❇▲❊▼❆❙ ●❊❘❆❉❖❘❆❙ ✭❙P●✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✽✳✹ ❙■❚❯❆➬Õ❊❙ P❘❖❇▲❊▼❆❙ ❖❘■❊◆❚❆❉❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✽✳✺ ❆❚■❱■❉❆❉❊❙ ❈❖▼P▲❊▼❊◆❚❆❘❊❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

✾ ●❯■❆ ❉❖ P❘❖❋❊❙❙❖❘✳ ✾✺

✾✳✶ ■◆❚❘❖❉❯➬➹❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✾✳✷ ❆▲●❯▼❆❙ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ❙❖❇❘❊ ❆ ▼❊❚❖❉❖▲❖●■❆ ❉❆ ❯◆■❉❆❉❊

❚❊▼➪❚■❈❆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✾✳✸ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ❙❖❇❘❊ ❖ P❘❖❈❊❙❙❖ ❉❊ ❆❱❆▲■❆➬➹❖✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✾✳✹ ❖❘■❊◆❚❆➬Õ❊❙ ▼❊❚❖❉❖▲Ó●■❈❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✾✳✺ ❘❊❙❖▲❯➬Õ❊❙ ❉❆❙ ❙■❚❯❆➬Õ❊❙✲P❘❖❇▲❊▼❆❙ ●❊❘❆❉❖❘❆❙✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✾✳✻ ❘❊❙❖▲❯➬Õ❊❙ ❉❆❙ ❙■❚❯❆➬Õ❊❙✲P❘❖❇▲❊▼❆❙ ❖❘■❊◆❚❆❉❆❙✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✾✳✼ ❆❚■❱■❉❆❉❊❙ ❈❖▼P▲❊▼❊◆❚❆❘❊❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✾

✶✵ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ❋■◆❆■❙ ✶✸✺

■◆❚❘❖❉❯➬➹❖

◆♦ss❛ ❡①♣❡r✐ê♥❝✐❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❛s ❝♦♥✈❡rs❛s ✐♥❢♦r♠❛✐s q✉❡ t❡♠♦s ❝♦♠ ❝♦❧❡❣❛s ❞❡ ♣r♦✜ssã♦ ❡ ♦s ❝♦♥st❛♥t❡s r❡❧❛t♦s ❞❡ ♥♦ss♦s ❛❧✉♥♦s ♥♦s ❧❡✈❛♠ ❛ ❛✜r♠❛r q✉❡ ▼❛t❡♠át✐❝❛ é ❛ ❞✐s❝✐♣❧✐♥❛ ❝♦♥s✐❞❡r❛❞❛ ♠❛✐s ❞✐❢í❝✐❧ ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ ❡ ❞❡♥tr♦ ❞❡st❛✱ ✉♠ ❞♦s ❣r❛♥❞❡s ❞❡s❛✜♦s q✉❡ ❡♥❝♦♥tr❛♠♦s é ♥♦ ❡♥s✐♥♦ ❞❡ ❚r✐❣♦♥♦♠❡tr✐❛✳

❊st❡ tr❛❜❛❧❤♦ ♣r♦♣õ❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ♦ ❡♥r✐q✉❡❝✐♠❡♥t♦ ❞❡ss❡ ❡♥s✐♥♦ ❛ ♣❛rt✐r ❞❡ ❛❧❣✉♠❛s ✏❊①♣❡r✐ê♥❝✐❛s ❆❧❣é❜r✐❝❛s ❡ ●rá✜❝❛s ❝♦♠ P♦❧✐♥ô♠✐♦s ❚r✐❣♦♥♦♠étr✐❝♦s✑✳

❈♦♠♦ ❛✜r♠❛ ❊❧♦♥ ▲❛❣❡s ▲✐♠❛✱ ♥♦ ❧✐✈r♦ ❛ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❱♦❧✉♠❡ ✶✱

❖ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛q✉❡❧❡ q✉❡ ❛t✉❛ ♥♦ ❝❤❛♠❛❞♦ ❙❡✲ ❣✉♥❞♦ ●r❛✉✱ ♥♦ ❡s❝❛ss♦ t❡♠♣♦ q✉❡ ❧❤❡ r❡st❛ ❞❛ ❢❛✐♥❛ ❞✐ár✐❛ ❞❡ ♣r❡♣❛r❛r s✉❛s ❛✉❧❛s✱ ❝♦♥t❛ ♣r❛t✐❝❛♠❡♥t❡ ❝♦♠♦ ✉♠❛ ú♥✐❝❛ ❢♦♥t❡ ❞❡ r❡❢❡rê♥❝✐❛✿ ♦ ❧✐✈r♦ t❡①t♦ q✉❡ ❛❞♦t❛ ✭♦✉ ♦✉tr♦s✱ q✉❡ ❞❡❧❡ ♣♦✉❝♦ ❞✐❢❡r❡♠✮✳

◆♦s ❞✐❛s ❞❡ ❤♦❥❡ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ❝♦♥t❛r t❛♠❜é♠ ❝♦♠ ✉♠❛ ❢❡rr❛♠❡♥t❛ ✐♠♣♦rt❛♥t❡ ♥❛ ♣r❡♣❛r❛çã♦ ❞❡ s✉❛s ❛✉❧❛s q✉❡ é ♦ ♠❛t❡r✐❛❧ ❞✐s♣♦♥í✈❡❧ ♥❛ ■♥t❡r♥❡t✱ ♦♥❞❡ ❡♥❝♦♥tr❛♠♦s ♠❛t❡r✐❛✐s ❞❡ ❡①❝❡❧❡♥t❡ q✉❛❧✐❞❛❞❡ ❛ ❛❧❣✉♥s ❡♠ q✉❡ ❡ss❛s q✉❡ ❞❡✐①❛♠ ❛ ❞❡s❡❥❛r✳

▼❡s♠♦ ❛ss✐♠✱ ❛ ❡①t❡♥✉❛♥t❡ ❥♦r♥❛❞❛ ❞❡ tr❛❜❛❧❤♦ ❛ q✉❡ s♦♠♦s s✉❜♠❡t✐❞♦s ♥❛ r❡❞❡ ♣ú❜❧✐❝❛ ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✱ s♦❜r❡t✉❞♦ ♥❛s r❡❞❡s ❡st❛❞✉❛✐s ❡ ♠✉♥✐❝✐♣❛✐s ❡ ❛✐♥❞❛ ♥❛s r❡❞❡s ♣❛rt✐❝✉❧❛r❡s ❞❡ ❡♥s✐♥♦✱ ❞✐✜❝✉❧t❛ ♦✉ ♠❡s♠♦ ✐♠♣❡❞❡ ✉♠ ✉s♦ ♠❛✐s ❡✜❝❛③ ❞❡st❛ ❢❡rr❛♠❡♥t❛✱ ❛ ✐♥t❡r♥❡t✳

❯♠❛ ❛♥á❧✐s❡ s✉♣❡r✜❝✐❛❧ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✉s❛❞♦s ❡♠ ♥♦ss❛s ❡s❝♦❧❛s ♥♦s ✐♥❞✐❝❛ q✉❡ ♦ ❡st✉❞♦ ❞❡ P♦❧✐♥ô♠✐♦s ❚r✐❣♦♥♦♠étr✐❝♦s ♥ã♦ é ❡st✉❞❛❞♦ ♦✉ é ❛❜♦r❞❛❞♦ ❞❡ ❢♦r♠❛ ✐♥s✉✜❝✐❡♥t❡✳ P♦ré♠ ❛s ♥♦çõ❡s ❜ás✐❝❛s ❞♦ ❡♥s✐♥♦ ❞❡ tr✐❣♦♥♦♠❡tr✐❛ sã♦ ❛♣r❡s❡♥t❛❞♦s ❞❡ ❢♦r♠❛ s❛t✐s❢❛tór✐❛✱ ❝♦♠♦ ❛s ❞❡✜♥✐çõ❡s ❞❛s r❛③õ❡s tr✐❣♦♥♦♠étr✐❝❛s ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ❛ ❞❡✜♥✐çã♦ ❞♦ s❡♥♦ ❡ ❝♦ss❡♥♦ ❞♦s ❛r❝♦s ❝♦♥s✐❞❡r❛❞♦s ♥♦tá✈❡✐s ✭✸✵➸✱ ✹✺➸ ❡ ✻✵➸✮✱ ♦ ❝✐❝❧♦ tr✐❣♦♥♦♠étr✐❝♦✱ ❛s ❞❡♠❛✐s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ r❡❞✉çã♦ ❛♦ ♣r✐♠❡✐r♦ q✉❛❞r❛♥t❡✱ ❛ r❡❧❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ tr✐❣♦♥♦♠❡✲ tr✐❛✱ ❛s ❢ór♠✉❧❛s ❞❡ ❛❞✐çã♦ ❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❛r❝♦s✱ ✐❞❡♥t✐❞❛❞❡s tr✐❣♦♥♦♠étr✐❝❛s✱ ❡q✉❛çõ❡s ❡ ✐♥❡q✉❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡❧❡♠❡♥t❛r❡s ❡ ❛s ❧❡✐s ❞♦ s❡♥♦ ❡ ❞♦ ❝♦ss❡♥♦✳

P❡♥s❛♥❞♦ ❡♠ ❡❧❛❜♦r❛r ✉♠ ♠❛t❡r✐❛❧ q✉❡ ✈❡♥❤❛ ❛❝r❡s❝❡♥t❛r ❛ ❡ss❡ ❡♥s✐♥♦ ♥♦✈♦s ❡❧❡♠❡♥t♦s r❡❛❧✐③❛♠♦s ❡ss❡ tr❛❜❛❧❤♦✱ q✉❡ ❜❛s✐❝❛♠❡♥t❡ ❡stá ❡str✉t✉r❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

❼ ❈❛♣ít✉❧♦ ✶✿ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ❙❖❇❘❊ ❖ ❊◆❙■◆❖ ❉❊ ❚❘■●❖◆❖▼❊❚❘■❆ ❊ ❆ P❘❖P❖❙❚❆ ❉❊ ❚❘❆❇❆▲❍❖ r❡❧❛t❛ ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ♦ ❊♥s✐♥♦ ❞❡ ❚r✐❣♦♥♦♠❡tr✐❛ ❡ ❛♣r❡s❡♥t❛ ❛ ♣r♦♣♦st❛ ❞❡ tr❛❜❛❧❤♦✱ ❝♦♠ ✉♠❛ ❞❡s❝r✐çã♦ ❣❡r❛❧ ❞❛ ♠❡s♠❛✱

r❡❧❛çã♦ ❞❡ ♦❜❥❡t✐✈♦s✱ ♦ ♣ú❜❧✐❝♦ ❛❧✈♦✱ ♦s ♣ré✲r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❯♥✐✲ ❞❛❞❡ ❚❡♠át✐❝❛ ❡❧❛❜♦r❛❞❛ ♣❛r❛ ❛♣❧✐❝❛çã♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♦s ♠❛t❡r✐❛✐s ❡ t❡❝♥♦❧♦❣✐❛s ♥❡❝❡ssár✐♦s ♥❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞♦ ❛ss✉♥t♦✱ r❡❝♦♠❡♥❞❛çõ❡s ♠❡t♦❞♦❧ó❣✐❝❛s✱ ❡♥✉♠❡r❛çã♦ ❞❡ ❛❧❣✉♠❛s ❞✐✜❝✉❧❞❛❞❡s ♣r❡✈✐st❛ ❡ ❞❡s❝r✐çã♦ ❞❡ ❛❧❣✉♠❛s ❝♦♥t✐♥✉❛çõ❡s ♦✉ ❞❡s❞♦❜r❛♠❡♥t♦s✳ ❼ ❈❛♣ít✉❧♦ ✷✿ ❊❙❚❯❉❖ ❉❆❙ ❋❯◆➬Õ❊❙ ❙❊◆❖ ❊ ❈❖❙❙❊◆❖ ❛♣r❡s❡♥t❛ ✉♠❛ r❡✈✐sã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡st❛s ❢✉♥çõ❡s✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ r❛❞✐❛♥♦✱ ❡ ❛s ✈❛r✐❛çõ❡s ❣rá✜❝❛s q✉❡ ♦❝♦rr❡♠ q✉❛♥❞♦ ♦s ♣❛râ♠❡tr♦s α ❡ k sã♦ ♠♦❞✐✜❝❛❞♦s ♥❛s ❢✉♥çõ❡s ❞♦s t✐♣♦s f(x) =

αcos(kx) ❡ f(x) = αsen(kx)✳ ❆♣r❡s❡♥t❛ ❛✐♥❞❛ ♦ ❡st✉❞♦ ❞❛s s❡♥♦✐❞❡s ❞♦ t✐♣♦ f(x) =

a+bcos(cx+d) ♦✉ f(x) = a+bsen(cx+d✮ ❞❡st❛❝❛♥❞♦ ❛s ✈❛r✐❛çõ❡s q✉❡ ❛❧t❡r❛çõ❡s ❞❡ ❝❛❞❛ ✉♠ ❞❡ss❡s ♣❛râ♠❡tr♦s ♣r♦♣♦r❝✐♦♥❛♠ ♥♦s r❡s♣❡❝t✐✈♦s ❣rá✜❝♦s✳

❼ ❈❛♣ít✉❧♦ ✸✿ ❊❙❚❯❉❖ ❉❖❙ ▼❖◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❊ ●❘❆❯ ✶ ❡st✉❞❛ ♦s ♠♦♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s ❞❛ ❢♦r♠❛ p(x) = αcos(x) + βsen(x) ❝♦♠♦

✉♠❛ ❝♦♠♣♦s✐çã♦ ❞❛s ❢✉♥çõ❡s f(x) = αcos(x) ❡ g(x) = βsen(x). ◆❡st❡ ❡st✉❞♦ ❤á ❛ ♣r❡♦❝✉♣❛çã♦ ❝♦♠ ❛s ♠♦❞✐✜❝❛çõ❡s ♥❡ss❡s ♣❛râ♠❡tr♦s ♣r♦♣♦r❝✐♦♥❛♠ ❡♠ p(x)✳ ❚❛♠❜é♠

é ❡st✉❞❛❞♦ ♦s ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❞♦s ♠♦♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s✳

❼ ❈❛♣ít✉❧♦ ✹✿ ❊❙❚❯❉❖ ❉❖❙ ▼❖◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙ ❉❊ ●❘❆❯ k > 1✳ ❊st✉❞♦ s❡♠❡❧❤❛♥t❡ ❛♦ r❡❛❧✐③❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ♣♦ré♠ ❛❝r❡s❝❡♥t❛ ❛ ♦❜s❡r✲

✈❛çã♦ s♦❜r❡ ❛ ✈❛r✐❛çã♦ ❞♦ ♣❛râ♠❡tr♦ k✱ ❛❝❛rr❡t❛♥❞♦ ♠♦❞✐✜❝❛çõ❡s ♥♦ ♣❡rí♦❞♦ ❞❡ p(x)✳

❼ ❈❛♣ít✉❧♦ ✺✿ ❊❙❚❯❉❖ ❉❖❙ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙✳ ❉❡s❝r❡✈❡ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ♣♦❧✐♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s✳ ◆❡st❛s ❝❛r❛❝t❡ríst✐❝❛s ❤á ✉♠❛ ❞❡✜✲ ♥✐çã♦ ❞❛s r❡t❛s ❤♦r✐③♦♥t❛✐s ❞❡ ♠á①✐♠♦ ❡ ♠í♥✐♠♦ ❡ ❞♦s ♣♦♥t♦s ❞❡ ✐♥✢❡①ã♦✳ ❖s ✈❛❧♦r ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❞❡ ♣✭①✮✱ t❛♥t♦ ❧♦❝❛✐s ❝♦♠♦ ❛❜s♦❧✉t♦s sã♦ ♣❡sq✉✐s❛❞♦s ❝♦♠ ♦ ✉s♦ ❞♦ ●❡♦●❡❜r❛✳ ❖ ❡st✉❞♦ ♣❛ss❛ s❡r ♠❛✐s ✐♥t✉✐t✐✈♦ ❛tr❛✈és ❞❛ ♦❜s❡r✈❛çã♦ ❞♦ q✉❡ ❢♦r♠❛❧✱ ♣♦ré♠ ❤á ♣r❡♦❝✉♣❛çã♦ ❞❡ ❥✉st✐✜❝❛r ❛s ❝♦♥❝❧✉sõ❡s ♦❜t✐❞❛s✳

❼ ❈❛♣ít✉❧♦ ✻✳ ❚❘■●❖◆❖▼❊❚❘■❆ ❊ ◆Ú▼❊❘❖❙ ❈❖▼P▲❊❳❖❙✳ ❊ss❡ ❝❛♣ít✉❧♦ ❡♥❢❛t✐③❛ ❛ ❛♣❧✐❝❛çã♦ ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ♥♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❝♦♠ ❛ ❛♣❧✐❝❛çã♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ss❡s ♥ú♠❡r♦s ♥❛ ❢♦r♠❛ tr✐❣♦♥♦♠étr✐❝❛ tr❛③❡♥❞♦ ✈❛♥t❛❣❡♥s ♣❛r❛ ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞✐✈✐sã♦✱ ♣♦t❡♥❝✐❛çã♦ ❡ r❛❞✐❝✐❛çã♦ ❞♦s ♠❡s♠♦s✳ ❆♣r❡s❡♥t❛ ♦ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ❊✉❧❡r ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ q✉❡ s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳

❼ ❈❛♣ít✉❧♦ ✼✳ ■◆❚❊❘P❖▲❆➬➹❖ P❖▲■◆❖▼■❆▲ ❚❘■●❖◆❖▼➱❚❘■❈❆ ❊ ❆P▲■✲ ❈❆➬Õ❊❙ ❆ ❋❊◆Ô▼❊◆❖❙ ❇■❖▲Ó●■❈❖❙ ❊ ❋❮❙■❈❖❙✳ ❆ ✐♥t❡r♣♦❧❛çã♦ ♣♦❧✐♥♦♠✐❛❧ tr✐❣♦♥♦♠étr✐❝❛ é ❛♣r❡s❡♥t❛❞❛ ❞❡ ❢♦r♠❛ s✉❝✐♥t❛ ❝♦♠♦ ✉♠ ❛♣r♦①✐♠❛çã♦ ♣♦r ♣♦❧✐♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s✱ q✉❡ é ♦❜t✐❞❛ ❛tr❛✈és ❞❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❞❡ ❡q✉❛çõ❡s✳ ❖ ❈❛♣ít✉❧♦ ❛♣r❡s❡♥t❛ t❛♠❜é♠ ✉♠❛ sí♥t❡s❡ ❞❛s ❛♣❧✐❝❛çõ❡s ❞♦ t❡♠❛ ❡♠ ❡st✉❞♦ ❛ ❢❡♥ô♠❡♥♦s ❜✐♦❧ó❣✐❝♦s ❡ ❢ís✐❝♦s ❜❡♠ ❝♦♠♦ ♣❛r❛ ♠♦❞❡❧❛r ♦✉tr❛s s✐t✉❛çõ❡s ❞❡ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✳

❼ ❈❛♣ít✉❧♦ ✽✿ ●❯■❆ ❉❖ ❆▲❯◆❖✳ ❖ ●✉✐❛ ❞♦ ❛❧✉♥♦ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ❛ ❝♦✲ r♦❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❈♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ s❡r ✉♠ ♠❛t❡r✐❛❧ ♣❛r❛ ♦ ✉s♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❢♦✐ ❡❧❛❜♦r❛❞❛ ✉♠❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ ❝♦♠♣♦st❛ ♣♦r três ❙✐t✉❛çõ❡s Pr♦❜❧❡♠❛ ●❡r❛❞♦r❛s ✭❙P●s✮✱ ❙✐t✉❛çõ❡s Pr♦❜❧❡♠❛s ❖r✐❡♥t❛❞❛s ✭❙P❖s✮ ❡ ❆t✐✈✐❞❛❞❡s ❈♦♠♣❧❡♠❡♥t❛r❡s✳ ❊st❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ ♣r♦♣õ❡✲s❡ ❢♦r♥❡❝❡r ❛♦ ❛❧✉♥♦ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥str✉✐r ❛ s✉❛ ❛♣r❡♥❞✐③❛✲ ❣❡♠✳ ❉á ê♥❢❛s❡ às ♣r✐♥❝✐♣❛✐s ❝❛r❛❝t❡ríst✐❝❛s ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ ❞♦s ♣♦❧✐♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s ❡ ❛ ❛♣❧✐❝❛çã♦ ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ♥♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❼ ❈❛♣ít✉❧♦ ✾✳ ●❯■❆ ❉❖ P❘❖❋❊❙❙❖❘✳ ❊st❡ ❣✉✐❛ ❛♣r❡s❡♥t❛ ❛s ♦r✐❡♥t❛çõ❡s ♠❡t♦❞♦❧ó✲

❣✐❝❛s ♣❛r❛ ❛ ❡①❡❝✉çã♦ ❞❛s ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s ♥♦ ●✉✐❛ ❞♦ ❆❧✉♥♦✱ ♦s ♦❜❥❡t✐✈♦s ❞♦ ❡st✉❞♦✱ ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❛✈❛❧✐❛çã♦✱ r❡s♦❧✉çã♦ ❞❡ t♦❞♦s ♦s ❡①❡r❝í❝✐♦s ❝♦♥st❛♥t❡s ❞♦ ●✉✐❛ ❞♦ ❆❧✉♥♦ ❡ ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ❛ ♠❛✐♦r✐❛ ❞❡st❡s ❡①❡r❝í❝✐♦s✳ ❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ ●✉✐❛ é ♦r✐❡♥t❛r ♦ ♣r♦❢❡ss♦r ♥❛ ❛♣❧✐❝❛çã♦ ❞❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ ❝♦♥str✉í❞❛ ♥♦ ❣✉✐❛ ❞♦ ❛❧✉♥♦✳

❆ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ ❞❡s❡♥✈♦❧✈✐❞❛ ❛ ♥í✈❡❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ♣r♦❝✉r♦✉ ♦❢❡r❡❝❡r ❝♦♥❞✐çõ❡s ❛♦ ❡❞✉❝❛♥❞♦✱ s❡♠♣r❡ q✉❡ ♣♦ssí✈❡❧✱ ♣❛r❛ ❝♦♠♣r♦✈❛çã♦ ♠❛t❡♠át✐❝❛ ❞❛s ❝♦♥❝❧✉sõ❡s ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❛ ♦❜s❡r✈❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ♥❛ r❡s♦❧✉çã♦ ❞♦s ❡①❡r❝í❝✐♦s✳

❈♦♠♦ ❢❡rr❛♠❡♥t❛ ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣❡❧♦ ❛❧✉♥♦ ❢♦✐ s✉❣❡r✐❞♦ ♦ ✉s♦ ❞♦ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛✱ s♦❜r❡t✉❞♦ ♦ ●❡♦●❡❜r❛✱ ♣❡❧♦ ❢❛t♦ ❞❡ s❡r ✉♠ ♣r♦❣r❛♠❛ ❣r❛t✉✐t♦✱ ♣r❡s❡♥t❡ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❝♦♠♣✉t❛❞♦r❡s ❞❡ ♥♦ss❛s ❡s❝♦❧❛s ❡ ♣❡❧❛ ❣❛♠❛ ❞❡ ♣♦ss✐❜✐❧✐✲ ❞❛❞❡s q✉❡ ❡ss❡ ♦❢❡r❡❝❡ ✱ ❡ss❡ s♦❢t✇❛r❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ♥♦ s✐t❡✿ ✇✇✇✳❣❡♦❣❡❜r❛✳♦r❣✴❝♠s✴ ✭●❡♦●❡❜r❛ ✈❡rsã♦ ✹✳✷✮✳

❏❛❝❦s♦♥ ❘✐❜❡✐r♦ ❡♠ s❡✉ ❧✐✈r♦ ▼❛t❡♠át✐❝❛✱ ❈✐ê♥❝✐❛ ❡ ❚❡❝♥♦❧♦❣✐❛ ✭❊❞✐t♦r❛ ❙❝✐♣✐♦♥❡ ✷✵✶✶✮ r❡ss❛❧t❛ q✉❡✿

❊♠❜♦r❛ ♦ ✐♥str✉♠❡♥t♦ ♠❛✐s r❡❧❡✈❛♥t❡ ❞♦ ✐♠♣❛❝t♦ ❞❛ t❡❝♥♦❧♦❣✐❛ s❡❥❛ ♦ ❝♦♠♣✉✲ t❛❞♦r✱ ❡❧❡✱ ❡♠ s✐ ♥ã♦ ❝♦♥st✐t✉✐ ♦ ❝❡♥tr♦ ❞❛ q✉❡stã♦✳ ❆s ❡①♣❡r✐ê♥❝✐❛s ✈ã♦ ❛❧é♠ ❞♦ s❛❜❡r ❧✐❞❛r ❝♦♠ ❛s ♠áq✉✐♥❛s✳ ❈❛❞❛ ✈❡③ ♠❛✐s ♦ ❝✐❞❛❞ã♦ ❡stá ✐♠❡rs♦ ❡♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ✐♥❢♦r♠❛çõ❡s q✉❡ ❝✐r❝✉❧❛♠ r❛♣✐❞❛♠❡♥t❡ ❡ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❡✜❝✐❡♥t❡✱ ♦ q✉❡ t❡♠ ❡①✐❣✐❞♦ ♥♦✈❛s ❝♦♠♣❡tê♥❝✐❛s ♣♦r ♣❛rt❡ ❞❡❧❡ ❡✱ ❝♦♥s❡q✉❡♥✲ t❡♠❡♥t❡✱ ✉♠❛ ♥♦✈❛ ♣♦s✐çã♦ ❞❛ ❡s❝♦❧❛ ❡ ❞❛ ▼❛t❡♠át✐❝❛✳

❈♦♠ ❜❛s❡ ♥❡ss❛ r❡✢❡①ã♦ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❛❣r❡❣❛r ♦ ✉s♦ ❞♦s r❡❝✉rs♦s t❡❝♥♦❧ó❣✐❝♦s ❡♠ ♥♦ss❛s ❛✉❧❛s é ♠❛✐s ❞♦ q✉❡ t♦r♥á✲❧❛s ❛tr❛❡♥t❡s ❡ ❞✐♥â♠✐❝❛s✱ é ♦❢❡r❡❝❡r ❛♦ ❡❞✉❝❛♥❞♦ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡♥tr♦ ❞❡ ✉♠ ❛♣❛r❛t♦ t❡❝♥♦❧ó❣✐❝♦ ♣r❡s❡♥t❡ ❡♠ ♥♦ss❛s ✈✐❞❛s✳

❆❧é♠ ❞✐ss♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ♥♦s ♦r✐❡♥t❛♠ q✉❡✿

P❛r❛ ✐ss♦✱ ❤❛❜✐❧✐❞❛❞❡s ❝♦♠♦ s❡❧❡❝✐♦♥❛r ✐♥❢♦r♠❛çõ❡s✱ ❛♥❛❧✐s❛r ❛s ✐♥❢♦r♠❛çõ❡s ♦❜t✐❞❛s ❡✱ ❛ ♣❛rt✐r ❞✐ss♦✱ t♦♠❛r ❞❡❝✐sõ❡s ❡①✐❣✐rã♦ ❧✐♥❣✉❛❣❡♠✱ ♣r♦❝❡❞✐♠❡♥t♦s ❡ ❢♦r♠❛s ❞❡ ♣❡♥s❛r ♠❛t❡♠át✐❝♦ q✉❡ ❞❡✈❡♠ s❡r ❞❡s❡♥✈♦❧✈✐❞♦s ❛♦ ❧♦♥❣♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❜❡♠ ❝♦♠♦ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛✈❛❧✐❛r ❧✐♠✐t❡s ❡ ❛ ❛❞❡q✉❛çã♦ ❞❛s t❡❝♥♦✲ ❧♦❣✐❛s ❡♠ ❞✐❢❡r❡♥t❡s s✐t✉❛çõ❡s✳ ❆ss✐♠✱ ❬✳✳✳❪ ❛♣r❡♥❞❡r ♠❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞❡✈❡ s❡r ♠❛✐s q✉❡ ♠❡♠♦r✐③❛r r❡s✉❧t❛❞♦s ❞❡ss❛ ❝✐ê♥❝✐❛ ❡ ❛ ❛q✉✐s✐çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❞❡✈❡ ❡st❛r ✈✐♥❝✉❧❛❞♦ ❛♦ s❛❜❡r ❢❛③❡r ▼❛t❡♠át✐❝❛ ❡ ❞❡ ✉♠ s❛❜❡r ♠❛t❡♠át✐❝♦ ✭P❈◆❊▼✱ ✷✵✵✵✱ ♣✳✹✶✮✳

❙ã♦ ❡ss❡s ♦❜❥❡t✐✈♦s ♦s ♣r♦♣ós✐t♦s ♥♦rt❡❛❞♦r❡s ❞❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ q✉❡ ♣r♦❞✉③✐♠♦s✿ ♦❢❡✲ r❡❝❡r ❝♦♥❞✐çõ❡s ❛♦ ❛❧✉♥♦ ❞❡ ❡①❡r❝✐t❛r ❛ ❝r✐❛t✐✈✐❞❛❞❡ ❡ ♦ r❛❝✐♦❝í♥✐♦✱ ♣r♦❞✉③✐♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❝♦♠ ♦ ✉s♦ ❞❡ ❛❧❣✉♥s r❡❝✉rs♦s t❡❝♥♦❧ó❣✐❝♦s ❞✐s♣♦♥í✈❡✐s ❡♠ ♥♦ss♦ ♠❡✐♦✳

❈❛♣ít✉❧♦ ✶

❖ ❊◆❙■◆❖ ❉❆ ❚❘■●❖◆❖▼❊❚❘■❆ ❊ ❆ P❘❖P❖❙❚❆ ❉❊ ❚❘❆❇❆▲❍❖

✶✳✶

❚❘■❆

❖ ❛❧✉♥♦ ♥♦r♠❛❧♠❡♥t❡ t❡♠ ♦ s❡✉ ♣r✐♠❡✐r♦ ❝♦♥t❛t♦ ❝♦♠ ♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠é✲ tr✐❝❛s ♥♦ q✉❡ ❤♦❥❡ ❝❤❛♠❛♠♦s ❞❡ ✾➸ ❛♥♦ ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳ ❊st❛ ✐♥tr♦❞✉çã♦✱ ❞❡ ♠❛♥❡✐r❛ ❣❡r❛❧✱ s❡ ❢❛③ ♣❡❧❛ ❞❡✜♥✐çã♦ ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❞❛s r❛③õ❡s s❡♥♦✱ ❝♦ss❡♥♦ ❡ t❛♥❣❡♥t❡ ❞♦s â♥❣✉❧♦s ❛❣✉❞♦s✳

❈♦♥s✐❞❡r❛♥❞♦ q✉❡ t♦❞♦s ♦s tr✐â♥❣✉❧♦s ❝✉❥♦s â♥❣✉❧♦s tê♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛ sã♦ s❡♠❡❧❤❛♥t❡s ❡♥tr❡ s✐ ❡ss❛s r❛③õ❡s t♦r♥❛♠✲s❡ ❢❡rr❛♠❡♥t❛ ✈❛❧✐♦s❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❞✐stâ♥❝✐❛s ✐♥❛❝❡ssí✈❡✐s✳ ❊ss❛ ❛♣❧✐❝❛çã♦ ♣rát✐❝❛ é ✉t✐❧✐③❛❞❛ ❡♠ t♦❞♦s ♦s ♥♦ss♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳

❯s❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞❡st❛s r❛③õ❡s ♥♦ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❡ ♥♦ q✉❛❞r❛❞♦✱ ❡♥❝♦♥tr❛♠✲s❡ ♦s ✈❛❧♦r❡s ❞❡ s❡♥♦✱ ❝♦ss❡♥♦ ❡ t❛♥❣❡♥t❡ ❞♦s â♥❣✉❧♦s ❞❡ ✸✵➸✱ ✹✺➸ ❡ ✻✵➸✱ q✉❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ â♥❣✉❧♦s ♥♦tá✈❡✐s✳ ❖s ❞❡♠❛✐s ✈❛❧♦r❡s✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ❛♦ ❛❧✉♥♦✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ ❡♠ t❛❜❡❧❛s ❡ s❡♠ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❝♦♠♦ ❢♦r❛♠ ♦❜t✐❞♦s✳ ❊♠ ❛❧❣✉♠❛s ❡①❝❡çõ❡s ❡♥❝♦♥tr❛♠✲s❡ ❡st❡s ✈❛❧♦r❡s ❞❡ ❢♦r♠❛ ❡①♣❡r✐♠❡♥t❛❧✱ ❝♦♥str✉✐♥❞♦ ❛❧❣✉♥s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s✱ ♠❡❞✐♥❞♦ s❡✉s â♥❣✉❧♦s ❡ ❧❛❞♦s ❝❛❧❝✉❧❛♥❞♦ ❛ss✐♠ ♦ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ❞❛s r❛③õ❡s tr✐❣♦♥♦♠étr✐❝❛s s♦❧✐❝✐t❛❞❛s✳ ❱❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ■♠❡♥❡s ✫ ▲❡❧❧✐s✱ ✷✵✵✾✱ ♣✳ ✶✻✸✳

◆♦ ❊♥s✐♥♦ ▼é❞✐♦ ♦ ❡st✉❞♦ ❞❡ ❚r✐❣♦♥♦♠❡tr✐❛ é ❡st❡♥❞✐❞♦ ❛ â♥❣✉❧♦s ♠❛✐♦r❡s q✉❡ ✾✵➸✱ ❞❡ ❢♦r♠❛ q✉❡ s❡❥❛♠ ♠❛♥t✐❞❛s ❛s r❡❧❛çõ❡s ❜ás✐❝❛s✳ ❖ ❛❧✉♥♦ t❡♠ ❝♦♥t❛t♦ ❝♦♠ ♦ ❝✐❝❧♦ tr✐❣♦♥♦♠étr✐❝♦ ❡ ✉♠❛ ♥♦✈❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✱ ♦ r❛❞✐❛♥♦✳

◆❡ss❡ ♠♦♠❡♥t♦✱ ❞❡✜♥✐✲s❡ ♦s ✈❛❧♦r❡s ♣❛r❛ ❛s ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦ss❡♥♦ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♣❛r❛ ❛s ❞❡♠❛✐s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ✵ ❛ ✸✻✵➸✳ ❚❛♠❜é♠ ❞❡✜♥❡✲s❡ ♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦ ❝♦♠♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ❛r❝♦s ♦✉ â♥❣✉❧♦s✳

❈♦♥s✐❞❡r❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ✈♦❧t❛s ✭♠❛✐♦r q✉❡ ✉♠✮ ❝♦♥❝❡✐t✉❛✲s❡ ❛r❝♦s ❝ô♥❣r✉♦s✱ ♦ q✉❡ ♣❡r♠✐t❡ ❡♥❝♦♥tr❛r ♦s ✈❛❧♦r❡s ❞♦ s❡♥♦ ❡ ❞♦ ❝♦ss❡♥♦ ♣❛r❛ q✉❛❧q✉❡r â♥❣✉❧♦ ❡ ❞❡✜♥✐r ❛ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❞❡ss❛s ❢✉♥çõ❡s✳

◆❡st❡ ♠♦♠❡♥t♦ ♦ ❡st✉❞❛♥t❡ t❡♠ ❝♦♥t❛t♦ ❝♦♠ ✉♠❛ ♥♦✈❛ ✏✉♥✐❞❛❞❡✑ ♣❛r❛ ♠❡❞✐❞❛ ❞❡ ❛r❝♦s✱

♦ r❛❞✐❛♥♦✱ ❝♦♠ ❛s ❝♦♥❤❡❝✐❞❛s tr❛♥s❢♦r♠❛çõ❡s ❡♥tr❡ ❣r❛✉s ❡ r❛❞✐❛♥♦s✳ P♦ré♠✱ s❡♠ ❛ ❞❡✜♥✐çã♦ ❝❧❛r❛ ❞❡ q✉❡ ❡st❛ ✉♥✐❞❛❞❡ ♥♦s ❞á ❛ ♠❡❞✐❞❛ ❞♦ ❛r❝♦ ❝♦♠♦ ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ✐st♦ ♣♦❞❡ ❝❛✉s❛r ❛❧✲ ❣✉♠❛ ❝♦♥❢✉sã♦ ❝♦♥❝❡✐t✉❛❧ ♥♦ ❛❧✉♥♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱π = 180➸✱ ❝♦♥❢♦r♠❡ t❡♠♦s ♦❜s❡r✈❛❞♦s

❡♠ ♥♦ss❛s ❛✉❧❛s ❞❡ tr✐❣♦♥♦♠❡tr✐❛✳

❖ ❛❧✉♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡st✉❞❛ ❛✐♥❞❛ ❛❧❣✉♠❛s ❡q✉❛çõ❡s✱ ✐♥❡q✉❛çõ❡s ❡ ❢✉♥çõ❡s tr✐❣♦♥♦✲ ♠étr✐❝❛s✳ ❚r❛❜❛❧❤❛ ❝♦♠ ❛s ❢ór♠✉❧❛s ♣❛r❛ ❛❞✐çã♦ ❞❡ ❛r❝♦s ❡ ❛r❝♦ ❞✉♣❧♦✳ ◆❡st❡ ♠♦♠❡♥t♦ ❧❤❡ é ♣r♦♣♦r❝✐♦♥❛❞♦ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ♣❡r❝❡❜❡r q✉❡ ❝♦♥❤❡❝❡♥❞♦✱ ♦ s❡♥♦ ❞❡ ✉♠ â♥❣✉❧♦✱ ♣♦❞❡✲s❡ ❝❛❧❝✉❧❛r ♦ s❡♥♦ ❞❡ ✈ár✐♦s ♦✉tr♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ sen(105➸) ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❛ ♣❛rt✐r ❞♦s

✈❛❧♦r❡s ❞❡sen(60➸)❡ sen(45➸) ✳

◆♦r♠❛❧♠❡♥t❡ ❡♥❝❡rr❛✲s❡ ♦ ❡st✉❞♦ ❞❡ tr✐❣♦♥♦♠❡tr✐❛ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❝♦♠ ♦ ❡st✉❞♦ ❞❛s r❡❧❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ♥♦ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳

❖ ❡st✉❞♦ ❞❡ ❆❧❣✉♠❛s ❊①♣❡r✐ê♥❝✐❛s ●❡♦♠étr✐❝❛s ❡ ●rá✜❝❛s ❝♦♠ P♦❧✐♥ô♠✐♦s ❚r✐❣♦♥♦♠étr✐✲ ❝♦s✱ ❝♦♥❢♦r♠❡ ♣r❡t❡♥❞❡♠♦s ❞❡♠♦♥str❛r ♥❡st❡ t❡①t♦ ♣♦❞❡ ❝♦♥tr✐❜✉✐r ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡✲ ❡♥sã♦ ❞❡st❡s ❛ss✉♥t♦s✱ ❛❝r❡s❝❡♥t❛♥❞♦ ❛♦ ❡♥s✐♥♦ ❛t✉❛❧ ❞❡st❛ ♠❛tér✐❛ ✉♠❛ ♥♦✈❛ ✈✐sã♦ s♦❜r❡ ❡st❡ tó♣✐❝♦ ❡ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❝♦♥❝❡✐t✉❛❧✳ ❙✉❛ ❝♦♥t❡①t✉❛❧✐③❛çã♦ ♣♦❞❡ ❝♦♥tr✐❜✉✐r ❞❡ ❢♦r♠❛ s✐❣♥✐✜❝❛t✐✈❛ ♣❛r❛ ♦ ❡♥t✉s✐❛s♠♦ ❞♦ ❛❧✉♥❛❞♦ q✉❛♥❞♦ r❡❛❧✐③❛r ❡st❡ ❡st✉❞♦✳

✶✳✷

✶✳✷✳✶

❊st❡ t❡①t♦ ❢♦✐ ❡❧❛❜♦r❛❞♦ ❝♦♠♦ ❉✐ss❡rt❛çã♦ ♥♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚✱ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❡❧❡ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛ ✭❯❋❱✮✳

Pr♦❝✉r❛♠♦s ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ♣r♦♣♦st❛ ✐♥♦✈❛❞♦r❛ ❡ ❞✐❢❡r❡♥❝✐❛❞❛ ❞❡ s❡q✉ê♥❝✐❛ ❞❡ ❛✉❧❛s ❛❜♦r✲ ❞❛♥❞♦ ♦ t❡♠❛ ✏❆▲●❯▼❆❙ ❊❳P❊❘■✃◆❈■❆❙ ❆▲●➱❇❘■❈❆❙ ❊ ●❘➪❋■❈❆❙ ❈❖▼ P❖▲■◆Ô▼■❖❙ ❚❘■●❖◆❖▼➱❚❘■❈❖❙✑ ♣❛r❛ ♦ ❡♥r✐q✉❡❝✐♠❡♥t♦ ❞♦ ❡st✉❞♦ ❞❡ ❚r✐❣♦♥♦♠❡tr✐❛ ❡ ❞❡ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳

❊st❡ ❡st✉❞♦ ❛✈❛♥ç❛ ❛♦ ♣r♦♣♦r ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ♦ ❡st✉❞♦ ❞❡ P♦❧✐♥ô♠✐♦s ❚r✐❣♦♥♦♠é✲ tr✐❝♦s✱ ❝♦♠ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❡st❛❝❛❞❛s ❛ ♣❛rt✐r ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❣rá✜❝♦ r❡❛❧✐③❛❞♦ ❝♦♠ ♣r♦❣r❛♠❛s ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛✳

P❛r❛ ✐ss♦✱ ❡❧❛❜♦r❛♠♦s ✉♠❛ ✉♥✐❞❛❞❡ t❡♠át✐❝❛ s♦❜r❡ ❡ss❡s ❛ss✉♥t♦s q✉❡ ❝♦♥s✐st❡ ♥♦ ✏●❯■❆ ❉❖ ❆▲❯◆❖✑ ❡ ♥♦ ✏●❯■❆ ❉❖ P❘❖❋❊❙❙❖❘✑ ❝♦♠ ❛t✐✈✐❞❛❞❡s ❞❡ ♥♦ss❛ ❛✉t♦r✐❛ ❡ ❞❡ ♦✉tr♦s ❛✉t♦r❡s✱ q✉❡ ❢♦r❛♠ ❛❞❛♣t❛❞❛s ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ s❡❥❛ ♣r♦t❛❣♦♥✐st❛ ❞❡ s✉❛ ❛♣r❡♥❞✐③❛❣❡♠✳

❖ ●❯■❆ ❉❖ ❆▲❯◆❖ ❛♣r❡s❡♥t❛ ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ♥❛ ♠♦❞❛❧✐❞❛❞❡ ❞❡ ✏❊st✉❞♦ ❉✐r✐❣✐❞♦✑ ❝♦♠ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ♦❜t❡♥çã♦ ✐♥t✉✐t✐✈❛ ❞♦s ❝♦♥❝❡✐t♦s✳ ◆❛ ♠❡❞✐❞❛ ❞♦ ♣♦ssí✈❡❧ ❛ ❢♦r♠❛❧✐③❛çã♦ ♠❛t❡♠át✐❝❛ é ❢❡✐t❛ ♣❡❧♦ ♣ró♣r✐♦ ❛❧✉♥♦✳ ❆ss✐♠✱ ✐♥✈❡rt❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ❝♦♠✉♠ ❡♠ ♥♦ss♦s ❧✐✈r♦s ❡ ❛✉❧❛s✿ ❞❡♠♦♥str❛çã♦ ❢❡✐t❛ ♣❡❧♦ ♣r♦❢❡ss♦r ✭♥♦r♠❛❧♠❡♥t❡ ♣r❡s❡♥t❡ ♥♦ ❧✐✈r♦✮✱ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❡ ✉♠❛ ❧✐st❛ ❞❡ ❡①❡r❝í❝✐♦s ❢❡✐t❛ ♣❡❧♦ ❛❧✉♥♦✳

❆❝r❡❞✐t❛♠♦s q✉❡ ❛❣✐♥❞♦ ❞❡st❛ ❢♦r♠❛ ❝♦♥tr✐❜✉✐r❡♠♦s ♣❛r❛ ♥♦ss♦ ❛❧✉♥♦ s❡❥❛ ❝r✐❛t✐✈♦✱ ❞✐✲ ♥â♠✐❝♦ ❡ ❞❡s❡♥✈♦❧✈❛ ♦ ❣♦st♦ ♣❡❧❛ ▼❛t❡♠át✐❝❛ à ♠❡❞✐❞❛ q✉❡ ♣❛ss❛ ❡♥t❡♥❞❡r q✉❡ ❡ss❛ é ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳

❖ ●❯■❆ ❉❖ P❘❖❋❊❙❙❖❘ ❛♣r❡s❡♥t❛ ❝♦♥s✐❞❡r❛çõ❡s ♠❡t♦❞♦❧ó❣✐❝❛s s♦❜r❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❞❡ ❛✈❛❧✐❛çã♦✱ ♦s ♦❜❥❡t✐✈♦s q✉❡ s❡ ♣r❡t❡♥❞❡ ❛❧❝❛♥ç❛r✱ r❡s♦❧✉çã♦ ❞♦s ❡①❡r❝í❝✐♦s ❞♦ ●✉✐❛ ❞♦ ❆❧✉♥♦ ❡ ❝♦♠❡♥tár✐♦s s♦❜r❡ ♦s ♠❡s♠♦s✳

❆♥t❡❝❡❞❡♥❞♦ ❛ ❡❧❛❜♦r❛çã♦ ❞❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ ♦ t❡①t♦ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ t❡ór✐❝♦✱ ❛❧❣é❜r✐❝♦ ❡ ❣rá✜❝♦ s♦❜r❡ ❛s ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦ss❡♥♦✱ ♠♦♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s✱ ♣♦❧✐♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s ❞♦ ♣r✐♠❡✐r♦ ❣r❛✉✱ ♣♦❧✐♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s ❞❡ ❣r❛✉ ❞♦✐s✱ ♣♦❧✐♥ô♠✐♦s tr✐❣♦♥♦✲ ♠étr✐❝♦s ❞❡ ❣r❛✉ ♠❛✐♦r q✉❡ ✷✱ tr✐❣♦♥♦♠❡tr✐❛ ❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ✐♥t❡r♣♦❧❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❡ ❛♣❧✐❝❛çõ❡s ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ♥❛s ❞✐✈❡rs❛s ár❡❛s ❞♦ s❛❜❡r ❤✉♠❛♥♦✳

Pr♦❝✉r❛♠♦s ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❜❛s❡❛❞❛ ♥♦ ❡st✉❞♦ ❞❛ t❡♦r✐❛ ✉t✐❧✐③❛♥❞♦ ♦s ❡①❡♠✲ ♣❧♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ♦s ❝♦♥❝❡✐t♦s ❡♥✈♦❧✈✐❞♦s✳

❆ ♠❛♥✐♣✉❧❛çã♦ ❞❡ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ é ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞❛ ❡♠ t♦❞♦ ♦ tr❛❜❛❧❤♦✱ ❝♦♥tr✐❜✉✐♥❞♦ ♥❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ ❡①❡♠♣❧♦s ❡ ❡①❡r❝í❝✐♦s ❡ ♥❛ ❢♦r♠❛çã♦ ❝♦♥❝❡✐t✉❛❧ q✉❡ s❡ ♣r❡t❡♥❞❡ ❛❧❝❛♥ç❛r✳

❖ tr❛❜❛❧❤♦ ❢♦✐ ❡❧❛❜♦r❛❞♦ ♥❛ ♠♦❞❛❧✐❞❛❞❡ ❡❧❛❜♦r❛çã♦ ❞❡ ♣r♦♣♦st❛s ❞❡ ❛t✐✈✐❞❛❞❡s ❡❞✉❝❛❝✐♦♥❛✐s✳ ❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ♦s ♦❜❥❡t✐✈♦s✱ ♦ ♣ú❜❧✐❝♦ ❛❧✈♦✱ ♦s ♣ré✲r❡q✉✐s✐t♦s✱ ♠❛t❡r✐❛✐s ❡ t❡❝♥♦❧♦❣✐❛s✱ r❡❝♦♠❡♥❞❛çõ❡s ♠❡t♦❞♦❧ó❣✐❝❛s✱ ❞✐✜❝✉❧❞❛❞❡s ♣r❡✈✐st❛s ❡ ♣♦ssí✈❡✐s ❝♦♥t✐♥✉❛çõ❡s ♦✉ ❞❡s❞♦❜r❛♠❡♥t♦s✳

✶✳✷✳✷

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r P♦❧✐♥ô♠✐♦s ❚r✐❣♦♥♦♠étr✐❝♦s✱ ❛t❡♥t❛♥❞♦ ♣❛r❛ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s ❛❧❣é❜r✐❝❛s ❡ ❣rá✜❝❛s✱ ❝♦♠ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❡ ❡①♣♦s✐çã♦ q✉❡ ❜❡♥❡✜❝✐❡♠ ♦ ❡♥t❡♥✲ ❞✐♠❡♥t♦ ❞❡ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡ ♦ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✈ár✐♦s ❡①❡♠♣❧♦s ❞❡ P♦❧✐♥ô♠✐♦s ❚r✐❣♦♥♦♠étr✐❝♦s ❡✈✐❞❡♥❝✐❛♥❞♦ ♣r♦♣r✐❡❞❛❞❡s ❞❛ s♦♠❛✱ ♠✉❧t✐♣❧✐✲ ❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❡ ✈❛r✐❛çã♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s✱ t❡♥❞♦ ♦s ❣rá✜❝♦s ❝♦♥str✉í❞♦s ❡♠ ❝♦♠♣✉t❛❞♦r ❝♦♠♦ ❛♣♦✐♦ ♥♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ss❛s ❢✉♥çõ❡s✳ ❚❛♠❜é♠ s❡rã♦ ❡st✉❞❛❞❛s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❛ ❢❡♥ô♠❡♥♦s ❜✐♦❧ó❣✐❝♦s ❡ ❢ís✐❝♦s✱ ❡♠ q✉❡ ❛ ✐♥t❡r♣♦❧❛çã♦ ♣♦r ♣♦❧✐♥ô♠✐♦s tr✐❣♦♥♦♠étr✐❝♦s é ✉♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❢❡rr❛♠❡♥t❛s✳

❖ ●❯■❆ ❉❖ P❘❖❋❊❙❙❖❘ ❛♣r❡s❡♥t❛ ✉♠❛ ❞❡s❝r✐çã♦ ❞♦s ♦❜❥❡t✐✈♦s ❡s♣❡❝í✜❝♦s q✉❡ ❛ ❡①❡❝✉✲ çã♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❞❡st❛ ✉♥✐❞❛❞❡ t❡♠át✐❝❛ ♣r❡t❡♥❞❡ ❛t✐♥❣✐r✳

❆ ♣r♦♣♦st❛ ♣❡❞❛❣ó❣✐❝❛ ❛♣r❡s❡♥t❛❞❛ ♣r♦❝✉r❛ ✐♥♦✈❛r ♥♦s s❡❣✉✐♥t❡s ❛s♣❡❝t♦s✿

❼ ❋♦r♥❡❝✐♠❡♥t♦ ❛♦ ❡❞✉❝❛♥❞♦ ❞❡ ❢❡rr❛♠❡♥t❛s ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ r❛❞✐❛♥♦✱ ♣❛r❛ ✐ss♦ ♦s ❛♣❧✐❝❛t✐✈♦s ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ ✏❞❡s❡♥r♦❧❛♥❞♦ ♦ s❡♥♦✑ ❡ ✏❋✉♥çã♦ ❞❡ ❊✉❧❡r✑ sã♦ ❢✉♥❞❛♠❡♥t❛✐s✳

❼ ❯s♦ ❞♦ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛✱ s♦❜r❡t✉❞♦ ♦ ●❡♦●❡❜r❛✱ ♥❛ ❝♦♥str✉çã♦ ❞♦s ❣rá✜❝♦s ❡ ❞❡ s❡✉s r❡❝✉rs♦s ♥❛ ♠♦❞✐✜❝❛çã♦ ❞❡ ♣❛râ♠❡tr♦s✳ ❆ ♣❛rt✐r ❞❛ ♦❜s❡r✈❛çã♦

♦ ❛❧✉♥♦ é ✐♥❝❡♥t✐✈❛❞♦ ♥❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❝♦♥❝❧✉sõ❡s s♦❜r❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s t❡♠❛s ❡st✉❞❛❞♦s✳

❼ ❉❡♠♦♥str❛çã♦ ♣❡❧♦ ♣ró♣r✐♦ ❛❧✉♥♦ ❞❛s ❝♦♥❝❧✉sõ❡s ♦❜t✐❞❛s ♣❡❧❛ ♦❜s❡r✈❛çã♦ ♥❛ ❝♦♥str✉çã♦ ❞❡ ❣rá✜❝♦s ❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡①❡r❝í❝✐♦s✳

❼ ❊st✉❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞♦s P♦❧✐♥ô♠✐♦s ❚r✐❣♦♥♦♠étr✐❝♦s✱ ❡♥❢❛t✐③❛♥❞♦ s✉❛s ❝❛r❛❝t❡ríst✐✲ ❝❛s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ ❛♣❧✐❝❛çõ❡s✳

❼ ❯t✐❧✐③❛çã♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❞♦s ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s✱ ❡ s✉❛ ❛♣❧✐❝❛çã♦ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ♣♦t❡♥❝✐❛çã♦ ❡ ♥♦ ❝á❧❝✉❧♦ ❞❡ r❛í③❡s ❞❡ss❡s ♥ú♠❡r♦s✳

❼ ❱❛❧♦r✐③❛çã♦ ❞❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❡✜❝❛③ ♣❛r❛ r❡♣r❡s❡♥t❛r ❡ ❝♦♠♣r❡❡♥❞❡r s✐t✉❛çõ❡s ❞♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✳

✶✳✷✳✸

❆ ♣r♦♣♦st❛ ❞❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞❛ t❡♠ ❝♦♠♦ ♣ú❜❧✐❝♦ ❛❧✈♦ ❛❧✉♥♦s ❞♦ t❡r❝❡✐r♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ q✉❡ t❡♥❤❛♠ ❢❡✐t♦ ✉♠ ❡st✉❞♦ ❞❡ ❚r✐❣♦♥♦♠❡tr✐❛ ❡ ❞❡ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s✳

P♦rt❛♥t♦ ❛ ♣r♦♣♦st❛ ♥ã♦ ✈✐s❛ s✉❜st✐t✉✐r ♦ ❡♥s✐♥♦ ❞❡ss❡s t❡♠❛s ♥❛ ❢♦r♠❛ q✉❡ é r❡❛❧✐③❛❞♦ ❡♠ ♥♦ss❛s ❡s❝♦❧❛s ❡ s✐♠✱ ❝♦♥tr✐❜✉✐r ❞❡ ❢♦r♠❛ s✐❣♥✐✜❝❛t✐✈❛ ♣❛r❛ ❛ ♠❡❧❤♦r✐❛ ❞♦ ♠❡s♠♦✳

❖ t❡①t♦ t❛♠❜é♠ ♣♦❞❡ s❡r út✐❧ ❛ ❛❧✉♥♦s q✉❡ ❡stã♦ s❡ ♣r❡♣❛r❛♥❞♦ ♣❛r❛ ♦ ❊◆❊▼ ✭❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✮ ♦✉ ✈❡st✐❜✉❧❛r❡s ❡ ♠❡s♠♦ ❛❧✉♥♦s ❞❡ ❣r❛❞✉❛çã♦ q✉❡ ♣r❡t❡♥❞❡♠ r❡✈❡r ❡ss❡ ❛ss✉♥t♦✳

❆ ✉♥✐❞❛❞❡ t❡♠át✐❝❛ ♣r♦♣♦st❛ t❛♠❜é♠ ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ❛✐♥❞❛ ❡♠ ❞✉❛s ❡t❛♣❛s✳ ❆ ♣r✐♠❡✐r❛ q✉❛♥❞♦ ❞♦ ❊♥s✐♥♦ ❞❡ ❚r✐❣♦♥♦♠❡tr✐❛✱ q✉❡ ❡♠ ❛❧❣✉♠❛s ❡s❝♦❧❛ é r❡❛❧✐③❛❞♦ ♥♦ ♣r✐♠❡✐r♦ ❛♥♦ ❡♠ ♦✉tr❛s ♥♦ s❡❣✉♥❞♦ ❛♥♦ ❡ ❛ s❡❣✉♥❞❛ q✉❛♥❞♦ ❞♦ ❡st✉❞♦ ❞❡ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ♥♦ ✸➸ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ◆♦s ❞♦✐s ❝❛s♦s ❡st❡ ❡st✉❞♦ ♣♦❞❡ s❡r ❢❡✐t♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❝♦♠ ❡ss❡s t❡♠❛s ♦✉ ❛♣ós ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡s✳

✶✳✷✳✹

❆ ✉♥✐❞❛❞❡ t❡♠át✐❝❛ ♣r♦♣♦st❛ ♣❛r❛ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❊s♣❡r❛✲s❡ q✉❡ ❡ss❡ ❛❧✉♥♦ t❡♥❤❛ ❝♦♥❤❡❝✐♠❡♥t♦s ❜ás✐❝♦s ❞❡ ♠❛t❡♠át✐❝❛ ❡❧❡♠❡♥t❛r tr❛❜❛❧❤❛❞❛ ❡♠ t♦❞♦ ♦ ❊♥s✐♥♦ ❋✉♥❞❛✲ ♠❡♥t❛❧✳

P❛r❛ q✉❡ ♦ ❛❧✉♥♦ ♣♦ss❛ ❞❡s❡♥✈♦❧✈❡r ❛ ❝♦♥t❡♥t♦ ❛s ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s ♥♦ ●❯■❆ ❉❖ ❆▲❯◆❖ é ✐♠♣♦rt❛♥t❡ q✉❡ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❡♥❣❧♦❜❡✱ ❞❡ ❢♦r♠❛ ♣❧❡♥❛ ♦✉ ♣❛r❝✐❛❧ ♦s s❡❣✉✐♥t❡s ✐t❡♥s✳

❼ ❈♦♠♣r❡❡♥sã♦ ❞❛s ❝♦♥❞✐çõ❡s ♣❛r❛ s❡♠❡❧❤❛♥ç❛ ❞❡ ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ♣❧❛♥❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ♦s ❝❛s♦s ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ ❞♦✐s ♦✉ ♠❛✐s tr✐â♥❣✉❧♦s✳

❼ ❈♦♠♣r❡❡♥sã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ s❡♥♦✱ ❝♦ss❡♥♦ ❡ t❛♥❣❡♥t❡ ❞❡ ✉♠ â♥❣✉❧♦ ❛❣✉❞♦ ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳

❼ ❈♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❝♦♥❝❡✐t♦ ♠❛t❡♠át✐❝♦ ❞❡ ❢✉♥çã♦✳

❼ ❙❛❜❡r r❡♣r❡s❡♥t❛r ❡♠ ✉♠ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ✉♠ ♣♦♥t♦ ❞❛❞❛ ❛s s✉❛s ❝♦♦r❞❡♥❛❞❛s✳ ❼ ❙❛❜❡r ♦ s✐❣♥✐✜❝❛❞♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❛r❝♦s ❝ô♥❣r✉♦s ♦✉ ❝♦♥❣r✉❡♥t❡s✳

❼ ❙❛❜❡r r❡❞✉③✐r ✉♠ ❛r❝♦ ✭✷➸ ❛♦ ✹➸ q✉❛❞r❛♥t❡✮ ❛♦ ♣r✐♠❡✐r♦ q✉❛❞r❛♥t❡ ❡ ❞❡✜♥✐r ♦s ✈❛❧♦r❡s ❞❡ s❡✉ s❡♥♦ ❡ ❝♦ss❡♥♦✳

❼ ❙❛❜❡r ❡s❜♦ç❛r ✉♠ ❣rá✜❝♦ ❡♠ ✉♠ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❛ ♣❛rt✐r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞❡ss❡ ❣rá✜❝♦ r❡❝♦♥❤❡❝❡♥❞♦ ❛s s✐t✉❛çõ❡s ♠í♥✐♠❛s ♣❛r❛ q✉❡ ❡ss❡s ♣♦ss✐❜✐❧✐t❡♠ ✉♠❛ ❜♦❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ ❢✉♥çã♦✳

❼ ❈♦♥❤❡❝❡r ♦ ❝✐❝❧♦ tr✐❣♦♥♦♠étr✐❝♦ ❜❡♠ ❝♦♠♦ ♦ s❡♥♦ ❡ ❝♦ss❡♥♦ ❞♦s â♥❣✉❧♦s ❝♦♥s✐❞❡r❛❞♦s ♥♦tá✈❡✐s✳

❼ ❚❡r ♥♦çõ❡s ❡❧❡♠❡♥t❛r❡s ❞❡ ✐♥❢♦r♠át✐❝❛✳

❼ ❈♦♥❤❡❝❡r ❛s r❡❧❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❢✉♥❞❛♠❡♥t❛✐s✳

❼ ❈♦♥❤❡❝❡r ❛s ❢ór♠✉❧❛s ❞❡ ❛❞✐çã♦ ❡ s✉❜tr❛çã♦ ❞❛s ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦ss❡♥♦✳

❼ ❚❡r ♥♦çõ❡s ❞❡ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ❡ s✉❛s r❡♣r❡s❡♥t❛çõ❡s r❡t❛♥❣✉❧❛r ❡ ♥❛ ❢♦r♠❛ tr✐❣♦✲ ♥♦♠étr✐❝❛✳

✶✳✷✳✺

P❛r❛ ✉♠❛ ❜♦❛ ❡①❡❝✉çã♦ ❞♦ tr❛❜❛❧❤♦ ♣r♦♣♦st♦ é ✐♠♣r❡s❝✐♥❞í✈❡❧ ♦ ✉s♦ ❞❡ r❡❝✉rs♦s ❝♦♠♣✉✲ t❛❝✐♦♥❛✐s s♦❜r❡t✉❞♦ ♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛✱ q✉❡ é ✉♠ ♣r♦❣r❛♠❛ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ ❣r❛t✉✐t♦ ♣r❡s❡♥t❡ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❝♦♠♣✉t❛❞♦r❡s ❞❡ ✐♥❢♦r♠át✐❝❛ ❞❡ ♥♦ss❛s ❡s❝♦❧❛s✳

❖ ✉s♦ ❞❡ ✉♠ ♣r♦❣r❛♠❛ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ é ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ♠❛♥✐♣✉❧❛çã♦ ❞❛s ❞✐✈❡rs❛s ❢✉♥çõ❡s ❝✉❥♦s ❣rá✜❝♦s ❢♦r❛♠ ❝♦♥str✉í❞♦s ❛ ♣❛rt✐r ❞❡ s❡✉s ♣❛râ♠❡tr♦s✳

❖❜✈✐❛♠❡♥t❡ ♠♦❞✐✜❝❛çõ❡s ♥❡ss❡s ♣❛râ♠❡tr♦s ❛❝❛rr❡t❛rã♦ ♠✉❞❛♥ç❛s ❡♠ s❡✉s ❣rá✜❝♦s✱ ❝♦♠♦ tr❛♥s❧❛çõ❡s ❡ ❞✐❧❛t❛çã♦ ♦✉ ❝♦♠♣r❡ssã♦✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♠✉❞❛♥ç❛s ♥❛ ✐♠❛❣❡♠ ❡ ♥♦ ♣❡rí♦❞♦ ❞❡st❛s ❢✉♥çõ❡s✳ ❆ ♦❜s❡r✈❛çã♦ ❣rá✜❝❛ ❞❡st❛s ♠♦❞✐✜❝❛çõ❡s é ❛ ❡str❛té❣✐❛ ❡s❝♦❧❤✐❞❛ ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ✈á ❛♦s ♣♦✉❝♦s ✭r❡✮❢♦r♠✉❧❛♥❞♦ s✉❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ❝♦♥❝❡✐t♦s q✉❡ ❡stã♦ s❡♥❞♦ tr❛❜❛✲ ❧❤❛❞♦s✳

◆♦ ❧✐✈r♦ ❘❡❝✉rs♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ♥♦ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ♦s ❛✉t♦r❡s ❛✜r♠❛♠ q✉❡✿✭●■❘❆▲❉❖❀ ❈❆❚❊❆◆❖❀ ▼❆❚❚❖❙✱ ✷✵✶✷✱ ♣✳ ✸✾✮

❆ ✐❞❡✐❛ ♥ã♦ é s✐♠♣❧❡s♠❡♥t❡ ✉s❛r ♦ s♦❢t✇❛r❡ ♣❛r❛ ✈❡r✐✜❝❛r ♦ q✉❡ ❡stá ❝❡rt♦ ♦✉ ❡rr❛❞♦ ♥♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ❊♠ ❧✉❣❛r ❞✐ss♦✱ ❛ ✈✐s✉❛❧✐③❛çã♦ ♥♦ s♦❢t✇❛r❡ ❞❡✈❡ s❡r ❡①♣❧♦r❛❞❛ ♣❛r❛ ♠♦t✐✈❛r r❡✢❡①õ❡s ❡ ❝♦♥❥❡❝t✉r❛s s♦❜r❡ ❛s ❢✉♥çõ❡s✱ q✉❡ ❞❡✈❡♠ s❡r ✈❡r✐✜❝❛❞❛s ♣♦st❡r✐♦r♠❡♥t❡ ♣♦r ♠❡✐♦ ❞❡ ❢❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s✳ ❊ss❛ ♦❜s❡r✈❛çã♦ ❡stá ❛❧✐♥❤❛❞❛ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ♠❛✐s ❣❡r❛❧ ❞❡ ✉s❛r ♦ ❝♦♠♣✉t❛❞♦r ♣❛r❛ ♣r♦♠♦✈❡r ❛♣r❡♥❞✐③❛❣❡♠ ♠❛t❡♠át✐❝❛ só❧✐❞❛ ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ♣❡r♠❛♥❡❝❡r ❡ s❡ tr❛♥s❢❡r✐r ♣❛r❛ ♦✉tr❛s s✐t✉❛çõ❡s ✲ ♠❡s♠♦ s❡♠ ♦ ❛♣♦✐♦ ❞❛ ♠áq✉✐♥❛✳

❲❡❧❧❡rs♦♥ ◗✉✐♥t❛♥❡✐r♦ ❡♠ s✉❛ t❡s❡ ❞❡ ♠❡str❛❞♦ ✏❘❡♣r❡s❡♥t❛çõ❡s ❡ ❉❡✜♥✐çõ❡s ❋♦r♠❛✐s ❡♠ ❚r✐❣♦♥♦♠❡tr✐❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✑ ❞❡❢❡♥❞✐❞❛ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✭✷ ✵✵✾✱ ♣✳ ✻✷✮ ❝♦♥st❛t❛ q✉❡

❈♦♠ ❛ ❛❥✉❞❛ ❞❡ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ ❝♦♥s✐❣❛♠♦s ✉♠ ❛♠❜✐❡♥t❡ ♠❛✐s ♣r♦♣í❝✐♦ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛ r❡❧❛❝✐♦♥❛r ❡ ❞❡s❡♥✈♦❧✈❡r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s tr✐❣♦✲ ♥♦♠étr✐❝♦s✱ ❛té ♠❡s♠♦ ❞❡✈✐❞♦ às ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ❛♠❜✐❡♥t❡s ✐♥❢♦r♠❛t✐③❛❞♦s✱ q✉❛♥t♦ à ❞✐♥â♠✐❝❛ ❡ ✐♥t❡r❛t✐✈✐❞❛❞❡✳

❲❡❧❧❡rs♦♥ ◗✉✐♥t❛♥❡✐r♦ ❝✐t❛ ❛✐♥❞❛ ▼❛r✐❛ ❆❧✐❝❡ ●r❛✈✐♥❛ ❡ ▲ú❝✐❛ ▼❛r✐❛ ❙❛♥t♦r♦s❛ q✉❡ ❡♠ s❡✉ ❛rt✐❣♦ ✏❆ ❆♣r❡♥❞✐③❛❣❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❆♠❜✐❡♥t❡s ■♥❢♦r♠❛t✐③❛❞♦s ❛✜r♠❛♠ q✉❡✿

❉✐s♣♦♥í✈❡❧ ❡♠✿ ❁ ❤tt♣✴✴✇✇✇✳❝✺✳❝❧✴✐❡✐♥✈❡st✐❣❛✴❛❝t❛s✴r✐❜✐❡✾✽✴✶✶✼✳❤t♠❧❃✳ ❆❝❡ss♦ ❡♠ ✷✺ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✷✵✶✸✳

❯♠ ❛♠❜✐❡♥t❡ ✐♥❢♦r♠❛t✐③❛❞♦ ♦❢❡r❡❝❡ s✉♣♦rt❡ às ❝♦♥❝r❡t✐③❛çõ❡s ❡ ❛çõ❡s ♠❡♥t❛✐s ❞♦ ❛❧✉♥♦❀ ✐st♦ s❡ ♠❛t❡r✐❛❧✐③❛ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞♦s ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s ♥❛ t❡❧❛ ❞♦ ❝♦♠♣✉t❛❞♦r ❡ ♥❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♠❛♥✐♣✉❧❛r ❡st❡s ♦❜❥❡t♦s ✈✐❛ s✉❛ r❡♣r❡s❡♥✲ t❛çã♦✳ ❱ê✲s❡ ❛ss✐♠ ♦ ❛♠❜✐❡♥t❡ ❢❛✈♦r❡❝❡♥❞♦ ❛ ❝♦♥str✉çã♦ ❞❡ ❝♦♥❥❡❝t✉r❛s✱ ♦ q✉❡ ❡①✐❣❡ r❛❝✐♦❝í♥✐♦s ♠❡❞✐❛❞♦s ♣❡❧♦ ❝♦♥st❛♥t❡ ♣r♦❝❡ss♦ ❞❡ ❛ss✐♠✐❧❛çã♦ ✈❡rs✉s ❛❝♦✲ ♠♦❞❛çã♦✳ ➱ ❝❧❛r♦ q✉❡ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ✈❛✐ ❛❧é♠ ❡ ♥ã♦ s❡ r❡❛❧✐③❛ ❡♥q✉❛♥t♦ ❛ ❛r❣✉♠❡♥t❛çã♦ ♠❛t❡♠át✐❝❛ ❡①♣❧í❝✐t❛ ♥ã♦ t♦r♥❛ ❡✈✐❞❡♥t❡ ♦ ✬♣♦rq✉ê ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡✬✳ ▼❛s ❛ ♣❡r❝❡♣çã♦ ❞❡ ♠♦❞❡❧♦s é ♦ ✐♥í❝✐♦ ♣❛r❛ ♦ ♣❡♥s❛♠❡♥t♦ ♣✉r❛♠❡♥t❡ t❡ór✐❝♦ s♦❜r❡ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❛s ❝♦✐s❛s✳

❆ss✐♠ ♦ r❡❝✉rs♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ é ♣❛rt❡ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✱ ❝♦♠♦ ✐♥str✉♠❡♥t♦ ❡ss❡♥❝✐❛❧ ♥❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡♠♦str❛♥❞♦ ❛ r✐q✉❡③❛ ❡ ❛ ❜❡❧❡③❛ ❞♦ s❛❜❡r ♠❛t❡♠át✐❝♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❝♦♠ r❡❝✉rs♦s t❡❝♥♦❧ó❣✐❝♦s r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s ❡ ❛❝❡ssí✈❡✐s✳

✶✳✷✳✻

❖ ●❯■❆ ❉❖ P❘❖❋❊❙❙❖❘ ❛♣r❡s❡♥t❛ ♦r✐❡♥t❛çõ❡s ♠❡t♦❞♦❧ó❣✐❝❛s ❡ ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ❛s ♠❡s♠❛s✳ ◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛♠♦s s✉❝✐♥t❛♠❡♥t❡ ❛❧❣✉♠❛s r❡❝♦♠❡♥❞❛çõ❡s ♣❛r❛ ♦ ✉s♦ ❞❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

❆ ♣r✐♥❝✐♣❛❧ ♣r❡♦❝✉♣❛çã♦ q✉❡ t✐✈❡♠♦s ♥❛ ❡❧❛❜♦r❛çã♦ ❞❛s ❛t✐✈✐❞❛❞❡s ❞❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ ❢♦✐ ❞❡ ♣r♦♣♦r❝✐♦♥❛r ❛♦ ❛❧✉♥♦ ❝♦♥❞✐çõ❡s ❞❡ s❡r ♦ ♣r♦t❛❣♦♥✐st❛ ❞❡ s✉❛ ❛♣r❡♥❞✐③❛❣❡♠✳ ❆ss✐♠ ❡✈✐t❛♠♦s ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞✐❞át✐❝♦ ❞❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ ♠❛tér✐❛ ❡ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ♣❡❧♦ ♣r♦❢❡ss♦r ❡ ❡♠ s❡❣✉✐❞❛ r❡s♦❧✉çã♦ ❞❡ ✉♠❛ ❧✐st❛ ❞❡ ❡①❡r❝í❝✐♦s ♣❡❧♦ ❛❧✉♥♦✳

❆ ❯♥✐❞❛❞❡ t❡♠át✐❝❛ ❝♦♥s✐st❡ ❜❛s✐❝❛♠❡♥t❡ ❡♠ ❙✐t✉❛çõ❡s Pr♦❜❧❡♠❛s ●❡r❛❞♦r❛s ✭❙P●s✮✱ ❙✐t✉❛çõ❡s Pr♦❜❧❡♠❛s ❖r✐❡♥t❛❞❛s ✭❙P❖s✮ ❡ ❆t✐✈✐❞❛❞❡s ❈♦♠♣❧❡♠❡♥t❛r❡s✳

❆s ❙P●s tê♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡st✐♠✉❧❛r ♦ ❛❧✉♥♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ t❡♠❛✳ Pr♦✈❛✈❡❧♠❡♥t❡ ♦s ❛❧✉♥♦s ♥ã♦ r❡s♦❧✈❡rã♦ ♦✉ r❡s♦❧✈❡rã♦ ❡st❛s ❛t✐✈✐❞❛❞❡s ♣❛r❝✐❛❧♠❡♥t❡✳ ❆q✉✐ ❝❛❜❡ ❛♦ ♣r♦❢❡ss♦r ♦ ❝✉✐❞❛❞♦ ♣❛r❛ q✉❡ ❡st❡ ❢❛t♦ ♥ã♦ s❡❥❛ ✉♠ ❢❛t♦r ❞❡s❡st✐♠✉❧❛❞♦r ❞❛ ❛♣r❡♥❞✐③❛❣❡♠ ❡ s✐♠ ✉♠ ❞❡s❛✜♦ ♣❛r❛ ❛♦ ❡st✉❞❛r ❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ ♦ ❡❞✉❝❛♥❞♦ ❛❞q✉✐r❛ ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ♥♦ ✜♥❛❧ ❞❛ ♠❡s♠❛ ✈♦❧t❡ ♥❛s ❙P●s ❡ ❛s r❡s♦❧✈❛ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡✳

❆s ❙P❖s tê♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ♦❢❡r❡❝❡r ❛♦ ❛❧✉♥♦ ❝♦♥❞✐çõ❡s ❞❡ ✐r ❣r❛❞❛t✐✈❛♠❡♥t❡ ❝♦♥str✉✐♥❞♦ ♦ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦✳ ▼✉✐t❛s ❞❡st❛s ❛t✐✈✐❞❛❞❡s sã♦ ❡❧❡♠❡♥t❛r❡s ♠❛s q✉❡ ❛♦s ♣♦✉❝♦s ❝r✐❛♠ ❝♦♥❞✐çõ❡s ♣❛r❛ ♦ ❛❧✉♥♦ ♦❜t❡r ✉♠ s❛❜❡r ♠❛✐s ❡❧❛❜♦r❛❞♦✳

❆❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ❢♦r❛♠ ❡❧❛❜♦r❛❞❛s ♥❛ ♠♦❞❛❧✐❞❛❞❡ ❞❡ ❡st✉❞♦ ❞✐r✐❣✐❞♦✱ ♣❡r♠✐t✐♥❞♦ q✉❡ ♦ ❛❧✉♥♦ ❝♦♥str✉❛ ♦ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦✳ ◆❡st❡ ♠♦♠❡♥t♦ ❝❛❜❡ ❛♦ ♣r♦❢❡ss♦r ♥♦ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ❞❡ ♦r✐❡♥t❛❞♦r ❞❛ ❛♣r❡♥❞✐③❛❣❡♠ ✜❝❛r ❛t❡♥t♦ ♣❛r❛ ♣♦ssí✈❡✐s ❞✐✜❝✉❧❞❛❞❡s q✉❡ ♦s ❛❧✉♥♦s ❛♣r❡s❡♥t❡♠✱ ❞✐s❝✉t✐♥❞♦ ❝♦♠ ❡❧❡s ❝❛♠✐♥❤♦s ♣❛r❛ ❛ s✉♣❡r❛çã♦ ❞❡ss❛s ❞✐✜❝✉❧❞❛❞❡s✳

❆s ❆t✐✈✐❞❛❞❡s ❈♦♠♣❧❡♠❡♥t❛r❡s✱ ❡♠ s✉❛ ♠❛✐♦r✐❛✱ ❝♦♥s✐st❡♠ ❡♠ ✉♠❛ ❧✐st❛ ❞❡ ❡①❡r❝í❝✐♦s ❡①✲ tr❛í❞♦s ❞♦ ❊◆❊▼ ❡ ❞❡ ❱❡st✐❜✉❧❛r❡s✱ ♣r♦❝✉r❛♥❞♦ ❝♦♥t❡①t✉❛❧✐③❛r ♦ ❛ss✉♥t♦ ❡st✉❞❛❞♦ ❡ ❝♦♥s♦❧✐❞❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♥str✉í❞♦ ❞✉r❛♥t❡ ❛ ✈✐❞❛ ❡s❝♦❧❛r ❞♦ ❛❧✉♥♦ ❡ ♥♦ ❡st✉❞♦ ❞❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛✳ P❛r❛ ✉♠ ❜♦♠ ❛♣r♦✈❡✐t❛♠❡♥t♦ ❞❡st❡ ❡st✉❞♦ s❡r✐❛ ✐❞❡❛❧ q✉❡ ❝❛❞❛ ❛❧✉♥♦ ❡st✐✈❡ss❡ ❛ s✉❛ ❞✐s♣♦s✐çã♦ ✉♠ t❡r♠✐♥❛❧ ❞❡ ❝♦♠♣✉t❛❞♦r✱ ♣r❡❢❡r❡♥❝✐❛❧♠❡♥t❡ ❡♠ ✉♠ ❧❛❜♦r❛tór✐♦ ❞❡ ✐♥❢♦r♠át✐❝❛✱ ❞❡ ❢♦r♠❛ q✉❡ ❛s ❛t✐✈✐❞❛❞❡s ❢❡✐t❛s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ♣r♦❣r❛♠❛s ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ s❡❥❛♠ ❛t✐✈✐❞❛❞❡s ✐♥❞✐✈✐❞✉❛✐s✳

❆s ❛t✐✈✐❞❛❞❡s ❞❡ ❡❧❛❜♦r❛çã♦ ❞❛s ❝♦♥❝❧✉sõ❡s q✉❡ ♦ t❡①t♦ ♣r♦♣õ❡ ❡ ❛s ❛t✐✈✐❞❛❞❡s ❝♦♠♣❧❡✲ ♠❡♥t❛r❡s ♣♦❞❡♠ ❡ ❞❡✈❡r s❡r r❡❛❧✐③❛❞❛s ❡♠ ❣r✉♣♦ ❞❡ ❢♦r♠❛ q✉❡ ❛♦ ✐♥t❡r❛❣✐r ❝♦♠ s❡✉s ❝♦❧❡❣❛s✱ ❞❡❢❡♥❞❡♥❞♦ s❡✉ ♣♦♥t♦ ❞❡ ✈✐st❛ ♦ ❛❧✉♥♦ ♣♦ss❛ ❡✈♦❧✉✐r ♥❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛r❣✉♠❡♥t❛çã♦ ♠❛t❡✲ ♠át✐❝❛✳

✶✳✷✳✼

❊♠ r❡❧❛çã♦ ❛♦s ♠❛t❡r✐❛✐s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦ ❞❛ ❯♥✐❞❛❞❡ ❚❡♠át✐❝❛ é ♣r♦✈á✈❡❧ q✉❡ ❡♠ ❛❧❣✉♠❛s ❡s❝♦❧❛s✱ ❛ ❡s❝❛ss❡③ ❞❡ ❝♦♠♣✉t❛❞♦r❡s ❡ ❛ ❛✉sê♥❝✐❛ ❞❡ ✉♠ ❧❛❜♦r❛tór✐♦ ❞❡ ✐♥❢♦r♠át✐❝❛ s❡❥❛♠ ❛❧❣✉♠❛s ❞❛s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s ♣❡❧♦s ♣r♦❢❡ss♦r❡s✳

➱ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ú❜❧✐❝♦ q✉❡ ❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞❡ ♥♦ss❛s ❡s❝♦❧❛s r❡❝❡❜❡♠ ❞♦s ♣r♦❣r❛♠❛s ❣♦✈❡r♥❛♠❡♥t❛✐s ❝♦♠♣✉t❛❞♦r❡s✱ s❡♠ ♥♦ ❡♥t❛♥t♦ ❛ ❞❡✈✐❞❛ ♣r❡♣❛r❛çã♦ ❞♦ ❡s♣❛ç♦ ❢ís✐❝♦ ♣❛r❛ r❡❝❡❜❡r ❡ss❡s ❡q✉✐♣❛♠❡♥t♦s ❡ s❡♠ ❛ ❛ss✐stê♥❝✐❛ té❝♥✐❝❛ ♥❡❝❡ssár✐❛✳

◆♦ ❝❛s♦ ❞❛ ❡s❝❛ss❡③ ❞♦s ❡q✉✐♣❛♠❡♥t♦s ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ♦r❣❛♥✐③❛r s✉❛ t✉r♠❛ ♣❛r❛ ❛ r❡❛❧✐✲