❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❚❡❝♥♦❧♦❣✐❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❱❡t♦r❡s✿ ❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦
♠é❞✐♦
❋❡r♥❛♥❞♦ P❡r❡✐r❛ ❞❡ ❖❧✐✈❡✐r❛
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❚❡❝♥♦❧♦❣✐❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❱❡t♦r❡s✿ ❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦
♠é❞✐♦
♣♦r
❋❡r♥❛♥❞♦ P❡r❡✐r❛ ❞❡ ❖❧✐✈❡✐r❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❋❡❧✐① ❙✐❧✈❛ ❈♦st❛
❆❣♦st♦ ❞❡ ✷✵✶✹ ❙ã♦ ▲✉ís ✲ ▼❆
❖❧✐✈❡✐r❛✱ ❋❡r♥❛♥❞♦ P❡r❡✐r❛ ❞❡
❱❡t♦r❡s✿ ✉♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦✴ ❋❡r♥❛♥❞♦ P❡r❡✐r❛ ❞❡ ❖❧✐✈❡✐r❛✳ ✲ ❙ã♦ ▲✉ís✱ ✷✵✶✹✳
✻✼ ❢✳
■♠♣r❡ss♦ ♣♦r ❝♦♠♣✉t❛❞♦r ✭❋♦t♦❝ó♣✐❛✮✳
❖r✐❡♥t❛❞♦r✿ ❋❡❧✐① ❙✐❧✈❛ ❈♦st❛✳
❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦✱ ✷✵✶✹✳
✶✳ ❱❡t♦r❡s ✷✳ ❖♣❡r❛çõ❡s ✸✳ ●❡♦♠❡tr✐❛ ✹✳ ❈♦♠♣❧❡①♦s ✺✳ ●❡♦●❡❜r❛ ■✳ ❚ít✉❧♦
❈❉❯ ✺✶✹✳✶✷✿✸✼
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❚❡❝♥♦❧♦❣✐❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❱❡t♦r❡s✿ ❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦
♣♦r
❋❡r♥❛♥❞♦ P❡r❡✐r❛ ❞❡ ❖❧✐✈❡✐r❛
❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ●❡♦♠❡tr✐❛
❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ❉r✳ ❋❡❧✐① ❙✐❧✈❛ ❈♦st❛ ✲ ❯❊▼❆ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❏♦ã♦ ❈♦❡❧❤♦ ❙✐❧✈❛ ❋✐❧❤♦ ✲ ❯❊▼❆
Pr♦❢✳ ❉r✳ ▼♦✐s❡✐s ❞♦s ❙❛♥t♦s ❈❡❝❝♦♥❡❧❧♦ ✲ ❯❋▼●
➚ ♠✐♥❤❛ ❡s♣♦s❛ ❆❧③✐♥❡t❡ ❡ ❛♦s ♠❡✉s ✜❧❤♦s ❚❤✐❡rr② ❡ ❆♠❛♥❞❛✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❛ ❉❡✉s✱ ♣♦r t❡r ♠❡ ♣❡r♠✐t✐❞♦ r❡❛❧✐③❛r ❡st❡ tr❛❜❛❧❤♦✱ ♣r♦✈❡♥❞♦✲ ♠❡ t♦❞❛s ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ ▼❡str❛❞♦✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣r♦❢❡ss♦r ❉r✳ ❋❡❧✐① ❙✐❧✈❛ ❈♦st❛✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ s❡❣✉r❛✱ ♣❡❧❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✱ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♥tr✐❜✉✐çõ❡s ❞❛❞❛s ❛♦ t❡①t♦✳
➚ ♠✐♥❤❛ ❡t❡r♥❛ ❝♦♠♣❛♥❤❡✐r❛✱ ❆❧③✐♥❡t❡ ▲♦♣❡s ❞❡ ❖❧✐✈❡✐r❛✱ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✱ ❡♥❝♦r❛❥❛♠❡♥t♦ ❡ ❝♦♠♣r❡❡♥❞❡r ❛s ❛✉sê♥❝✐❛s q✉❡ ♦❝♦rr❡r❛♠ ❞✉r❛♥t❡ ♦s ♠♦♠❡♥t♦s ❞❡ ❝✉rs♦ ❡ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆♦s ♠❡✉s ❛♠❛❞♦s ✜❧❤♦s✱ ❚❤✐❡rr② ▲✉❝❛s ❞❡ ❙♦✉s❛ ❖❧✐✈❡✐r❛ ❡ ❆♠❛♥❞❛ ❚❛ís ▼❛rr② ❞❡ ❙♦✉s❛ ❖❧✐✈❡✐r❛✱ q✉❡ ♠❡s♠♦ ❞✐st❛♥t❡s ❡st✐✈❡r❛♠ s❡♠♣r❡ ♣r❡s❡♥t❡s✳
❆♦s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ ♠❡str❛❞♦ ♣❡❧♦ ❝♦♠♣❛rt✐❧❤❛♠❡♥t♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❡♥t✉s✐❛s♠♦ ♣❛rt✐❧❤❛❞♦s ❡♠ ❝♦♥❥✉♥t♦✳
❆♦s ▼❡♠❜r♦s ❞❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ♣❡❧❛ ✐♥✐❝✐❛t✐✈❛ ❞❡ ❡st✐♠✉❧❛r ❛ ♠❡❧❤♦r✐❛ ❞♦ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛ ❛tr❛✈és ❞❡st❡ ♠❡str❛❞♦✳
➚ ❈❆P❊❙✱ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦ ❡ ❝r❡❞✐❜✐❧✐❞❛❞❡ ❞✐s♣❡♥s❛❞❛ ❛♦ P❘❖❋▼❆❚✳ ❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯❋▼❆✱ ♣❡❧❛ ❢♦r♠❛çã♦ ❡ s❡r✈✐ç♦s ♣r❡st❛❞♦s✳
❋✐♥❛❧♠❡♥t❡✱ ❛ t♦❞♦s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❘❡s✉♠♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ s♦❜r❡ ✈❡t♦r❡s ❜✉s❝❛♥❞♦ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ♣❛r❛ ❛s ❛✉❧❛s ❞❡ ▼❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❖s ✈❡t♦r❡s t❡♠❜é♠ ♣♦❞❡♠ s❡r ❛❜♦r❞❛❞♦s ♥❛s ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛ ♣❛r❛ ❛❧✉♥♦s ❞❛ ✸❛ sér✐❡✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ❛❧❣✉♥s tó♣✐❝♦s ❞❡
❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s tê♠ ✐♥t❡r♣r❡t❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s✳ ❯♠ ♠♦❞❡❧♦ ❝♦♥❝❡✐t✉❛❧ s❡rá ♣♦st✉❧❛❞♦ ❡ ❢✉♥❞❛♠❡♥t❛❞♦ ♥❛s ♣✉❜❧✐❝❛çõ❡s s♦❜r❡ ✈❡t♦r❡s✿ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡ ❝á❧❝✉❧♦ ✈❡t♦r✐❛❧ ❡ á❧❣❡❜r❛ ❧✐♥❡❛r✳ ◆♦s ♣r❡♦❝✉♣❛♠♦s ❡♠ ❡①♣ôr ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ❝♦♠♦ ♦ ❝á❧❝✉❧♦ ❞♦ tr❛❜❛❧❤♦ ✭W✮ ❡ ❞♦ t♦rq✉❡ ✭τ✮ q✉❡ sã♦ ♦❜t✐❞♦s
❛tr❛✈és ❞♦ ♣r♦❞✉t♦ ❡♥tr❡ ✈❡t♦r❡s✳ ❉❡st❛❝❛♠♦s ♦ ✉s♦ ❞♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡❞✐③❛❣❡♠✱ ❛tr❛✈és ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s ♦♥❞❡ ♦s ❡❞✉❝❛♥❞♦s s❡❥❛♠ ❧❡✈❛❞♦s ❛ ❡①♣❧♦r❛r s❡✉s r❡❝✉rs♦s ♣❛r❛ ❛ ✈✐s✉❛❧✐③❛çã♦✱ ♠❛♥✐♣✉❧❛çã♦ ❣rá✜❝❛✱ ✈❡r✐✜❝❛çã♦ ❡ ✐♥✈❡st✐❣❛çã♦ ❞❡ ❝♦♥❝❡✐t♦s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❢♦r♠❛ ❞✐♥â♠✐❝❛ ❡ ✐♥t❡r❛t✐✈❛✱ ❛✉①✐❧✐❛♥❞♦ ❛ss✐♠ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❝♦♠♣❡tê♥❝✐❛s ❡ ❤❛❜✐❧✐❞❛❞❡s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞♦ ❝♦t✐❞✐❛♥♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❱❡t♦r❡s❀ ❖♣❡r❛çõ❡s❀ ●❡♦♠❡tr✐❛❀ ❈♦♠♣❧❡①♦s❀ ●❡♦●❡❜r❛✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ✐s ❛♥ ❛♣♣r♦❛❝❤ t♦ s❡❛r❝❤✐♥❣ ❛♣♣❧✐❝❛❜✐❧✐t② ✈❡❝t♦rs ❢♦r ♠❛t❤❡♠❛t✐❝s ❝❧❛ss❡s ✐♥ ❤✐❣❤ s❝❤♦♦❧✳ ❱❡❝t♦rs ❝❛♥ ❛❧s♦ ❜❡ ✐♥ ❝♦✈❡r❡❞ ✐♥ ♠❛t❤ ❝❧❛ss❡s t♦ st✉❞❡♥ts ❢r♦♠ ✸r❞ ❣r❛❞❡✱ ❣❡♦♠❡tr✐❝❛❧❧②✱ s♦♠❡ t♦♣✐❝s ♦❢ ❛♥❛❧②t✐❝ ❣❡♦♠❡tr② ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❡q✉✐✈❛❧❡♥t ✐♥t❡r♣r❡t❛t✐♦♥s✳ ❆ ❝♦♥❝❡♣t✉❛❧ ♠♦❞❡❧ ✐s ♣♦st✉❧❛t❡❞ ❛♥❞ s✉❜st❛♥t✐❛t❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ♦♥ ✈❡❝t♦rs✿ t❤❡ t❡①t❜♦♦❦s ♦❢ ✈❡❝t♦r ❝❛❧❝✉❧✉s ❛♥❞ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ❲❡ ❛r❡ ❝♦♥❝❡r♥❡❞ ❛❜♦✉t ❡①♣♦s✐♥❣ s♦♠❡ ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s s✉❝❤ ❛s t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ✇♦r❦ ✭W✮ ❛♥❞ t♦rq✉❡ ✭tau✮
t❤❛t ❛r❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ ♣r♦❞✉❝t ❜❡t✇❡❡♥ ✈❡❝t♦rs✳ ❍✐❣❤❧✐❣❤t t❤❡ ✉s❡ ♦❢ ●❡♦●❡❜r❛ s♦❢t✇❛r❡ ❛s ❛ t♦♦❧ ❢♦r t❡❛❝❤✐♥❣ ❛♥❞ ❧❡❛r♥✐♥❣ ♣r♦❣r❛♠s✱ t❤r♦✉❣❤ ♣r♦♣♦s❡❞ ❛❝t✐✈✐t✐❡s ✇❤❡r❡ ❧❡❛r♥❡rs ❛r❡ ❧❡❞ t♦ ❡①♣❧♦r❡ t❤❡✐r r❡s♦✉r❝❡s ❢♦r ✈✐s✉❛❧✐③❛t✐♦♥✱ ❣r❛♣❤✐❝ ♠❛♥✐♣✉❧❛t✐♦♥✱ ✈❡r✐✜❝❛t✐♦♥ ❛♥❞ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❝♦♥❝❡♣ts ❛♥❞ ♣r♦♣❡rt✐❡s ✐♥ ❛ ❞②♥❛♠✐❝ ❛♥❞ ✐♥t❡r❛❝t✐✈❡ ✇❛②✱ t❤✉s ❛✐❞✐♥❣ ✐♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t s❦✐❧❧s ❛♥❞ ✐♠♣♦rt❛♥t ❢♦r t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❡✈❡r②❞❛② ♣r♦❜❧❡♠s s❦✐❧❧s✳
❑❡②✇♦r❞s✿ ❱❡❝t♦rs❀ ❖♣❡r❛t✐♦♥❀ ●❡♦♠❡tr②❀ ❈♦♠♣❧❡①❀ ●❡♦●❡❜r❛✳
❈♦♥t❡ú❞♦
■♥tr♦❞✉çã♦ ✶
✶ ❱❡t♦r❡s ✺
✶✳✶ ❯♠ ♣♦✉❝♦ s♦❜r❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ◆♦çã♦ ✐♥t✉✐t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❙❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳✶ ❙❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ♥✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❉✐r❡çã♦ ❡ s❡♥t✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺ ❊q✉✐♣♦❧ê♥❝✐❛s ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✺✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ r❡❧❛çã♦ ❞❡ ❡q✉✐♣♦❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✻ ❈♦♥❝❡✐t♦ ❞❡ ✈❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✼ ❊t♠♦❧♦❣✐❛ ❞❛ ♣❛❧❛✈r❛ ✈❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✽ ■❣✉❛❧❞❛❞❡ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✾ ❱❡t♦r ♥✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✵ ❖♣♦st♦ ❞❡ ✉♠ ✈❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✶ ▼ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✷ ❱❡t♦r ✉♥✐tár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✸ ❱❡rs♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✹ ❖♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✹✳✶ ❆❞✐çã♦ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✹✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷ ❱❡t♦r❡s ♥♦ R2
❡ ♥♦ R3
✶✾ ✷✳✶ ❖ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳✶ ❈♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r ♥♦ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❖ ❡s♣❛ç♦ ❝❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✶ ❈♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r ♥♦ ❡s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❱❡t♦r❡s ❊q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✹ ❆❞✐çã♦ ❞❡ ✈❡t♦r❡s ✉s❛♥❞♦ ❝♦♠♣♦♥❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✺ Pr♦♣r✐❡❞❛❞❡s ❞❛ ❛❞✐çã♦ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✺✳✶ ❊①✐stê♥❝✐❛ ❞♦ ❡❧❡♠❡♥t♦ s✐♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✻ ❉✐❢❡r❡♥ç❛ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✼ ❱❡t♦r ❞❡✜♥✐❞♦ ♣♦r ❞♦✐s ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✽ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✽✳✶ ❈♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✽✳✷ ❇❛s❡ ❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✽✳✸ P❛r❛❧❡❧✐s♠♦ ❞❡ ❞♦✐s ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✾ Pr♦♣r✐❡❞❛❞❡s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✶✵ ▼ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r ❡♠ t❡r♠♦s ❞❡ ❝♦♠♣♦♥❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✶✶ ➶♥❣✉❧♦ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✶✷ Pr♦❞✉t♦ ❡s❝❛❧❛r ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✶✷✳✶ ❖❜t❡♥❞♦ ♦ â♥❣✉❧♦ ❡♥tr❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✷✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✶✸ Pr♦❞✉t♦ ✈❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✶✸✳✶ ❈❛r❛❝t❡ríst✐❝❛s ❞♦ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✶✸✳✷ Pr♦❞✉t♦ ✈❡t♦r✐❛❧ ♥✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✶✸✳✸ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✶✸✳✹ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✶✹ Pr♦❞✉t♦ ♠✐st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✶✹✳✶ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ♣r♦❞✉t♦ ♠✐st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✸ ❆t✐✈✐❞❛❞❡s ❞❡ ❡①♣❧♦r❛çã♦ ✹✼
✸✳✶ ❙❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷ ❙❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✸ ❉❡s❡♥❤❛♥❞♦ ♦ r❡♣r❡s❡♥t❛♥t❡ ♥❛t✉r❛❧ ❞❡ ✉♠ ✈❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✹ ❆❞✐çã♦ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✹✳✶ Pr♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✺ ❙✐♠étr✐❝♦ ❞❡ ✉♠ ✈❡t♦r ❞❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✻ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✈❡t♦r ♣♦r ✉♠ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✼ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞♦ ❊①❡♠♣❧♦ ✶✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✽ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞♦ ❊①❡♠♣❧♦ ✷✳✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✹ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✺✻
❆ ❆♣❧✐❝❛çõ❡s ❞❡ ✈❡t♦r❡s ✺✼ ❆✳✶ ❖ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ♣♦r ✉♠❛ ❢♦rç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ❆✳✷ ❖ t♦rq✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
❇ Pr♦❜❧❡♠❛s ✺✾
❘❡❢❡rê♥❝✐❛s ✻✺
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ■♠❛❣❡♠ ✜❝tí❝✐❛ ❞♦ ♠❛♣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷ ❙❡❣♠❡♥t♦ ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ❙❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡♠ r❡t❛s ♣❛r❛❧❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹ ❙❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡♠ r❡t❛s ❝♦✐♥❝✐❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✺ ❘❡♣r❡s❡♥t❛♥t❡s ❞♦ ✈❡t♦r~v ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✻ ▼ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✼ ❱❡rs♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✽ ❘❡❣r❛ ❞♦ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✾ ❘❡❣r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✶✵ ❊①❡♠♣❧♦ ✶✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✶✶ Pr♦❞✉t♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✷✳✶ ❙✐st❡♠❛ ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ~u= (x, y) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✸ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊①❡♠♣❧♦ ✶✳✶ ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹ ❖ ♣♦♥t♦ P ♥♦ ❡s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✺ ~u= (x, y, z) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✻ ❆❞✐çã♦ ❞❡ ✈❡t♦r❡s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✼ Pr♦♣r✐❡❞❛❞❡ ❝♦♠✉t❛t✐✈❛ ❞❛ ❛❞✐çã♦ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✽ Pr♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✾ ❊①❡♠♣❧♦ ✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✵ ❊s❜♦ç♦ ♥♦ ●❡♦●❡❜r❛ ❞♦ ❊①❡♠♣❧♦ ✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✶ ❉✐❢❡r❡♥ç❛ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✷ ❊①❡♠♣❧♦ ✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✸ ❱❡t♦r ❝♦♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❞✐st✐♥t♦ ❞❛ ♦r✐❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✹ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✺ w~ =λ~u+µ~v ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✷✳✶✻ ❱❡t♦r❡s ❞❡ ❜❛s❡ ❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✷✳✶✼ ▼ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r ❡♠ t❡r♠♦s ❞❡ ❝♦♠♣♦♥❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✶✽ ➶♥❣✉❧♦s ❡♥tr❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✶✾ ❋✐❣✉r❛ ❞♦ ❊①❡♠♣❧♦ ✷✳✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✷✵ Pr♦❞✉t♦ ❡s❝❛❧❛r ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✷✶ Pr♦❞✉t♦ ✈❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷✷ ❘❡❣r❛ ❞❛ ♠ã♦ ❞✐r❡✐t❛ ✭❋♦♥t❡✿ ❬✸✶❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✷✳✷✸ P❛r❛❧❡❧♦❣r❛♠♦ ❆❇❈❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✷✹ ❚r✐â♥❣✉❧♦ ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✷✺ ❱♦❧✉♠❡ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✸✳✶ ❈♦♠❛♥❞♦ ❉✐r❡çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷ ❘❡♣r❡s❡♥t❛♥t❡s ❞♦ ✈❡t♦r ~u ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✸✳✸ ❈♦♥str✉çã♦ ❞❡ ✉♠ ✈❡t♦r ❞❛❞♦ s✉❛s ❝♦♠♣♦♥❡♥t❡sx ❡ y ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✸✳✹ Pr♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛ ❞❛ ❛❞✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✺ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✈❡t♦r ♣♦r ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✻ ❇❛s❡ ♠é❞✐❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
❆✳✶ ❈♦♠♣♦♥❡♥t❡s Fx ❡Fy ❞❛ ❢♦rç❛ F~ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
❇✳✶ Pr♦❜❧❡♠❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ❇✳✷ Pr♦❜❧❡♠❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ❇✳✸ ❋♦♥t❡✿ ❤tt♣✿✴✴❡✜s✐❝❛✳✐❢✳✉s♣✳❜r✴♠❡❝❛♥✐❝❛✴❜❛s✐❝♦✴✈❡t♦r❡s✴❡①❡r❝✐❝✐♦s✴ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❇✳✹ ❊s❜♦ç♦ ❞♦ Pr♦❜❧❡♠❛ ❇✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ❇✳✺ ❋✐❣✉r❛ ❞♦ Pr♦❜❧❡♠❛ ❇✳✹ ❡♠ ✉♠ ♣❧❛♥♦ ♦rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ❇✳✻ P♦❧✐❡❞r♦ q✉❡ r❡♣r❡s❡♥t❛ ♦ t❡❧❤❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ❇✳✼ ❈✉❜♦ ❞❡ ❛r❡st❛s a ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
■♥tr♦❞✉çã♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ✉♠❛ ❛♥á❧✐s❡ s♦❜r❡ ✈❡t♦r❡s ❜✉s❝❛♥❞♦ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ♣❛r❛ ❛s ❛✉❧❛s ❞❡ ▼❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛tr❛✈és ❞❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ q✉❡ s✐♠❜♦❧✐③❛ ✉♠ t✐♣♦ ♣❛rt✐❝✉❧❛r ❞❡ ✈❡t♦r✱ ♦♥❞❡ ♦s ✈❡t♦r❡s sã♦ ❛ss♦❝✐❛❞♦s ❛ ❣r❛♥❞❡③❛s q✉❡ ♥❡❝❡ss✐t❛♠ ❞❡ ♠ó❞✉❧♦✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ ♣❛r❛ s❡r❡♠ ❝♦♠♣❧❡t❛♠❡♥t❡ ❡s♣❡❝✐✜❝❛❞❛s✳
◆♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ P❈◆❊▼ ❬✻❪ ❞❡st❛❝❛ q✉❡ ✉♠❛ ❞❛s ♦r✐❡♥t❛çõ❡s ❞♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❊❞✉❝❛çã♦ é ❛ ♦r❣❛♥✐③❛çã♦ ❞♦ ❝✉rrí❝✉❧♦ ❡♠ ár❡❛s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠❛ ❡❞✉❝❛çã♦ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r ❡ ❝♦♥t❡①t✉❛❧✐③❛❞❛✱ ❤❛❥❛ ✈✐st❛ q✉❡ ♥❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ♠✉♥❞♦ ❡♠ q✉❡ ✈✐✈❡♠♦s ❡♥❝♦♥tr❛♠♦s ♣r♦❜❧❡♠❛s ❡ ❞❡s❛✜♦s q✉❡ ❡①✐❣❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♠♣❡tê♥❝✐❛s ❡ ❤❛❜✐❧✐❞❛❞❡s ♣❛r❛ ❛ s✉❛ ❝♦♠♣r❡❡♥sã♦ ❡ s✉♣❡r❛çã♦✳
❆♦ ❡s❝♦❧❤❡♠♦s ❡ss❡ t❡♠❛ ❡st❛♠♦s ❜✉s❝❛♥❞♦ ❛t❡♥❞❡r ❡ss❛s ♦r✐❡♥t❛çõ❡s ❞♦ P❈◆❊▼✱ ♣♦✐s ♦ ❡st✉❞♦ ❞❡ ✈❡t♦r❡s ♣❡r♠✐t❡ ❝♦♥❡①õ❡s ❡♥tr❡ ❞✐❢❡r❡♥t❡s ❢♦r♠❛s ❞❡ ♣❡♥s❛♠❡♥t♦ ♠❛t❡✲ ♠át✐❝♦✱ ❝♦♠♦ á❧❣❡❜r❛ ❡ ❣❡♦♠❡tr✐❛✳ ❆❧é♠ ❞✐ss♦✱ ❡ss❡ t❡♠❛ ♣♦ss✉✐ ❛♣❧✐❝❛çõ❡s ❞❡♥tr♦ ♦✉ ❢♦r❛ ❞❛ ▼❛t❡♠át✐❝❛ ❡ ♥ã♦ ❛♣❡♥❛s ♥❛ ❋ís✐❝❛✳ ❉✐r❡❝✐♦♥❛♥❞♦ ❛♦ ♥♦ss♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦✱ ❡♥❝♦♥tr❛♠♦s ♥❛s ❖r✐❡♥t❛çõ❡s ❈✉rr✐❝✉❧❛r❡s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛ s❡❣✉✐♥t❡ s✉❣❡stã♦✿
✏➱ ❞❡s❡❥á✈❡❧✱ t❛♠❜é♠✱ q✉❡ ♦ ♣r♦❢❡ss♦r ❞❡ ▼❛✲ t❡♠át✐❝❛ ❛❜♦r❞❡ ❝♦♠ s❡✉s ❛❧✉♥♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❡t♦r✱ t❛♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❣❡♦♠étr✐❝♦ ✭❝♦❧❡çã♦ ❞♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❞❡ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✮ q✉❛♥t♦ ❛❧❣é❜r✐❝♦ ✭❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧❛s s✉❛s ❝♦♦r❞❡♥❛❞❛s✮✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ é ✐♠♣♦r✲ t❛♥t❡ r❡❧❛❝✐♦♥❛r ❛s ♦♣❡r❛çõ❡s ❡①❡❝✉t❛❞❛s ❝♦♠ ❛s ❝♦♦r❞❡♥❛❞❛s ✭s♦♠❛✱ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✮ ❝♦♠ s❡✉ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦✳ ❆ ✐♥❝❧✉sã♦ ❞❛ ♥♦çã♦ ❞❡ ✈❡t♦r ♥♦s t❡♠❛s ❛❜♦r❞❛❞♦s ♥❛s ❛✉❧❛s ❞❡ ▼❛t❡♠á✲ t✐❝❛ ✈✐r✐❛ ❛ ❝♦rr✐❣✐r ❛ ❞✐st♦rçã♦ ❝❛✉s❛❞❛ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ é ✉♠ tó♣✐❝♦ ♠❛t❡♠át✐❝♦ ✐♠♣♦rt❛♥t❡✱ ♠❛s q✉❡ ❡stá ♣r❡s❡♥t❡ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ s♦♠❡♥t❡ ♥❛s ❛✉❧❛s ❞❡ ❋ís✐❝❛✳✑✭❬✽❪✱ ♣✳✼✼✮
◆❛ ♥♦ss❛ ♣rát✐❝❛ ♣❡❞❛❣ó❣✐❝❛✱ ♣❡r❝❡❜❡♠♦s q✉❡ ♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ❡❧❡♠❡♥t❛r✱
✷
tr❛❞✐❝✐♦♥❛❧♠❡♥t❡✱ s❡ ✉t✐❧✐③❛ ❞❡ r❡❝✉rs♦s ❞✐❞át✐❝♦s ♣♦✉❝♦ ✈❛r✐❛❞♦s q✉❡ s❡ ❧✐♠✐t❛♠ ❛♦ ❧✐✈r♦ t❡①t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❧✐st❛s ❞❡ ❡①❡r❝í❝✐♦s ❡ r❡❛❧✐③❛çã♦ ❞❡ tr❛❜❛❧❤♦s✳ ❙❡♠ ❞ú✈✐❞❛ q✉❡ ❝❛❞❛ ✉♠❛ ❞❡st❛s ❛t✐t✉❞❡s ❞✐❞át✐❝❛s ❛❥✉❞❛ ♥❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ ▼❛t❡♠át✐❝❛✱ ♠❛s s❡rá q✉❡ ♠♦t✐✈❛♠ ♦s ❛❧✉♥♦s ❛ ❞❡s❡♥✈♦❧✈❡r❡♠ ✉♠ ❡st✉❞♦ ❝♦♠ ♠❛✐♦r r❡✢❡①ã♦✱ ❡♥t✉s✐❛s♠♦ ❡ s❡♥t✐❞♦❄ ❊ss❛ ✐♥❞❛❣❛çã♦ r❡♠❡t❡ às ❝♦♥❞✐çõ❡s ❞❡ ❝♦♠♦ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ❝r✐❛r ✉♠❛ ♣♦♥t❡ s❡❣✉r❛ ❡ ❝♦♥✜á✈❡❧ ❡♥tr❡ ❡ss❡s ❞♦✐s ✉♥✐✈❡rs♦s ❞❛ ❛❜♦r❞❛❣❡♠ ▼❛t❡♠át✐❝❛✱ ♦ ❞♦ ❡♥s✐♥♦ tr❛❞✐❝✐♦♥❛❧ ❡ ♦ ❞♦ ❡♥s✐♥♦ ❝♦♠ s✐❣♥✐✜❝❛❞♦✱ q✉❡ ❛t❡♥❞❛ às ❡①✐❣ê♥❝✐❛s ♠í♥✐♠❛s ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s✳
❆s ❚■❈s✶ ♣♦r ❢❛③❡r ♣❛rt❡ ❞♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❛❧✉♥♦s ❡ ❞❛ ❝♦♠✉♥✐❞❛❞❡ ❡s❝♦❧❛r ♣♦❞❡
s❡r ✉♠❛ ót✐♠❛ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ♥♦s ❛✉①✐❧✐❛r ♥❛ ✈✐❞❛ ❡s❝♦❧❛r✱ ♣♦✐s ♦s r❡❝✉rs♦s t❡❝♥♦❧ó❣✐❝♦s ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ❝♦♠♦ ❡❧❡♠❡♥t♦s ❞❡ ❛♣♦✐♦ ♣❛r❛ ♦ ❡♥s✐♥♦✱ ❢♦♥t❡s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❡ ❝♦♠♦ ❢❡rr❛♠❡♥t❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❤❛❜✐❧✐❞❛❞❡s✱ ❛❧é♠ ❞✐ss♦✱ ❛ t❡❝♥♦❧♦❣✐❛ t❡♠ ♦ ♣♦❞❡r ❞❡ ♣r♦✈♦❝❛r ❢❛❝í♥✐♦✱ ✐♥t❡r❡ss❡ ❡ ♣r❛③❡r ♥♦s ❡st✉❞❛♥t❡s✳ ◆♦ ❡♥t❛♥t♦ ♦ q✉❡ t❡♠♦s ✈✐st♦ é q✉❡ ❛ ❡s❝♦❧❛ ❛✐♥❞❛ ♥ã♦ ❞❡s❝♦❜r✐✉ ❝♦♠♦ ✉t✐❧✐③❛r ❞❡ ❢♦r♠❛ ❛❞❡q✉❛❞❛ ❡ss❛s t❡❝♥♦❧♦❣✐❛s ❝♦♠♦ r❡s✉rs♦ ❞✐❞át✐❝♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ ♦s ♣r♦❢❡ss♦r❡s ♥ã♦ s❡ s❡♥t❡♠ ♣r❡♣❛r❛❞♦s ♣❛r❛ ✉s❛r ❡st❛s ❢❡rr❛♠❡♥t❛s ❞❡ ❢♦r♠❛ ❛❞❡q✉❛❞❛ ❡ ♣r♦✈❡✐t♦s❛ ♥❛s ❛✉❧❛s ❞❡ ▼❛t❡♠át✐❝❛✳ ❆❧♠❡✐❞❛ ❛✜r♠❛ q✉❡✿
✏❖ ♣r♦❢❡ss♦r t❡♠ ✉♠ ✐♠♣♦rt❛♥t❡ ♣❛♣❡❧ ❝♦♠♦ ❛❣❡♥t❡ ♣r♦♠♦t♦r ❞♦ ♣r♦❝❡ss♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞♦ ❛❧✉♥♦✱ q✉❡ ❝♦♥stró✐ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥✉♠ ❛♠❜✐❡♥t❡ q✉❡ ♦ ❞❡s❛✜❛ ❡ ♦ ♠♦t✐✈❛♠ ♣❛r❛ ❛ ❡①♣❧♦r❛çã♦✱ ❛ r❡✢❡①ã♦✱ ❛ ❞❡♣✉r❛çã♦ ❞❡ ✐❞é✐❛s ❡ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ ♥♦✈♦s ❝♦♥❝❡✐t♦s✳✑ ✭❬✶❪✱ ♣✳✶✻✷✮
❈♦♠♦ ♦ q✉❛❞r♦ ❞❛ s❛❧❛ ❞❡ ❛✉❧❛ ♣♦ss✉❡ ❧✐♠✐t❛çõ❡s q✉❛♥t♦ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ ❛❧❣✉♠❛s s✐t✉❛çõ❡s ♠❛t❡♠át✐❝❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ✈❡t♦r ♥♦R3✱ ❛❞♦t❛♠♦s ♦
●❡♦●❡❜r❛ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠✱ ♥❡st❡ s♦❢t✇❛r❡ ♣♦❞❡✲ s❡ ❛♣❧✐❝❛r ♠♦✈✐♠❡♥t♦ ❛ s❡✉s ❡❧❡♠❡♥t♦s✱ s❡♥❞♦ ♣r❡s❡r✈❛❞❛s ❛s r❡❧❛çõ❡s ❣❡♦♠étr✐❝❛s ✐♠♣♦st❛s à ✜❣✉r❛✱ ❞❛í s❡r❡♠ ❞❡♥♦♠✐♥❛❞♦s ♣r♦❣r❛♠❛ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛❬✶✽❪✳
❆❝r❡❞✐t❛♠♦s q✉❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡ss❡ t✐♣♦ ❞❡ r❡❝✉rs♦ ❞✐❞át✐❝♦ ❢❛✈♦r❡❝❡ ❛ ♠❛♥✐♣✉❧❛çã♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛✱ ♠✐♥✐♠✐③❛♥❞♦ ❛s ❧✐♠✐t❛çõ❡s r❡❧❛t✐✈❛s à ✈✐s✉❛❧✐③❛çã♦ ❡ ❛♣❧✐❝❛çã♦ ❞♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❡♥✈♦❧✈✐❞♦s✳
❖ ●❡♦●❡❜r❛ ✭❛❣❧✉t✐♥❛çã♦ ❞❛s ♣❛❧❛✈r❛s ●❡♦♠❡tr✐❛ ❡ ➪❧❣❡❜r❛✮ é ✉♠ ❛♣❧✐❝❛t✐✈♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞✐♥â♠✐❝❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ ♠❛t❡♠át✐❝❛ ♥♦s ✈ár✐♦s ♥í✈❡✐s ❞❡ ❡♥s✐♥♦✳ ❖ ❙♦❢t✇❛r❡ r❡ú♥❡ r❡❝✉rs♦s ❞❡ ❣❡♦♠❡tr✐❛✱ á❧❣❡❜r❛✱ t❛❜❡❧❛s✱ ❣rá✜❝♦s✱ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❡st❛tíst✐❝❛ ❡ ❝á❧❝✉❧♦s s✐♠❜ó❧✐❝♦s ❡♠ ✉♠ ú♥✐❝♦ ❛♠❜✐❡♥t❡✳ ❊ss❡ s♦❢t✇❛r❡ é ❧✐✈r❡ ❡ ❞❡ ❞✐str✐❜✉✐çã♦ ❣r❛t✉✐t❛✷ ❡♠ ✈ár✐♦s ✐❞✐♦♠❛s✳
✸
❖s ❝♦♠♣✉t❛❞♦r❡s ❞♦ P❘❖■◆❋❖✸✱ q✉❡ ❡q✉✐♣❛♠ ♦s ❧❛❜♦r❛tór✐♦s ❞❡ ❞✐✈❡rs❛s ❡s❝♦❧❛s
♣ú❜❧✐❝❛s ❜r❛s✐❧❡✐r❛s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ s❡❥❛♠ ❡❧❛s ♠✉♥✐❝✐♣❛✐s✱ ❡st❛❞✉❛✐s ♦✉ ❢❡❞❡r❛✐s✱ ♣♦ss✉❡♠ ❡ss❡ s♦❢t✇❛r❡ ✐♥st❛❧❛❞♦✱ ❛❧é♠ ❞❡ ♦✉tr♦s ✈♦❧t❛❞♦s ♣❛r❛ ❛ ❡❞✉❝❛çã♦✱ ❡♠ s❡✉ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧ ❞❡ ❧✐❝❡♥ç❛ ❧✐✈r❡✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▲❊✹✳
◆❡ss❛ ❛❜♦r❞❛❣❡♠ s♦❜r❡ ✈❡t♦r❡s ❞❡s❡♥✈♦❧✈❡♠♦s ❛t✐✈✐❞❛❞❡s ✭❈❛♣ít✉❧♦ ✸✮ ♣❛r❛ ♦ ❧❛❜♦r❛tór✐♦ ❞❡ ✐♥❢♦r♠át✐❝❛ ❞❛s ❡s❝♦❧❛s ♦♥❞❡ ♦ ❛❧✉♥♦ ♣♦ss❛ ✏❢❛③❡r ♠❛t❡♠át✐❝❛✑✱ ♦✉ s❡❥❛✱ ❡①♣❡r✐♠❡♥t❛r✱ ✐♥t❡r♣r❡t❛r✱ ✈✐s✉❛❧✐③❛r✱ ✐♥❞✉③✐r✱ ❝♦♥❥❡t✉r❛r✱ ❛❜str❛✐r✱ ❣❡♥❡r❛❧✐③❛r ❡ ❡♥✜♠ ❞❡♠♦♥str❛r ✭❬✾❪✱ ♣✳✵✶✮✳ ❆s ❖r✐❡♥t❛çõ❡s ❈✉rr✐❝✉❧❛r❡s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛♣♦♥t❛♠ ♣❛r❛ ❡ss❛ ♠❡s♠❛ ❝♦♥❝❡♣çã♦✿
✏❏á s❡ ♣❡♥s❛♥❞♦ ♥❛ ❚❡❝♥♦❧♦❣✐❛ ♣❛r❛ ❛ ▼❛t❡♠át✐❝❛✱ ❤á ♣r♦❣r❛♠❛s ❞❡ ❝♦♠♣✉t❛❞♦r ✭s♦❢t✇❛r❡s✮ ♥♦s q✉❛✐s ♦s ❛❧✉♥♦s ♣♦❞❡♠ ❡①♣❧♦r❛r ❡ ❝♦♥str✉✐r ❞✐❢❡r❡♥t❡s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s✱ r❡❢❡r✐❞♦s ❛ s❡❣✉✐r ❝♦♠♦ ♣r♦❣r❛♠❛s ❞❡ ❡①♣r❡ssã♦✳ ❖s ♣r♦❣r❛♠❛s ❞❡ ❡①♣r❡s✲ sã♦ ❛♣r❡s❡♥t❛♠ r❡❝✉rs♦s q✉❡ ♣r♦✈♦❝❛♠✱ ❞❡ ❢♦r♠❛ ♠✉✐t♦ ♥❛t✉r❛❧✱ ♦ ♣r♦❝❡ss♦ q✉❡ ❝❛r❛❝t❡r✐③❛ ♦ ✏♣❡♥s❛r ♠❛t❡♠❛t✐❝❛♠❡♥t❡✑ ♦✉ s❡❥❛✱ ♦s ❛❧✉♥♦s ❢❛③❡♠ ❡①♣❡✲ r✐♠❡♥t♦s✱ t❡st❛♠ ❤✐♣ót❡s❡s✱ ❡s❜♦ç❛♠ ❝♦♥❥❡❝t✉r❛s✱ ❝r✐❛♠ ❡str❛té❣✐❛s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s✳✑ ✭❬✽❪✱ ♣✳ ✽✽✮✳
❊♥tr❡t❛♥t♦✱ ❤á ❞❡ s❡ ❡s❝❧❛r❡❝❡r q✉❡ ♦ s♦❢t✇❛r❡ é ✉♠ ✐♥st✉♠❡♥t♦ ❡ ♥ã♦ ♦ ♦❜❥❡t✐✈♦ ❞❛ ❛♣r❡♥❞✐③❛❣❡♠ ❡ q✉❡ ♥ã♦ ❜❛st❛ ♦ ♣r♦❢❡ss♦r ❞♦♠✐♥❛r ❛♣❡♥❛s ♦ ✉s♦ ❞❛ ✐♥❢♦r♠át✐❝❛ ❡❞✉❝❛t✐✈❛✳ ❊❧❡ ♣r❡❝✐s❛ ❛♣r❡♥❞❡r ❛ ❢❛③❡r s❡✉ ♣❧❛♥❡❥❛♠❡♥t♦ ♣❛✉t❛❞♦ ♥❛s ♣♦ssí✈❡✐s ❞✐✜❝✉❧❞❛❞❡s ❞♦s ❛❧✉♥♦s ❝♦♠ r❡❧❛çã♦ ❛♦ t❡♠❛ ❞❛ ❛✉❧❛✱ ❡ss❡ ♣❧❛♥❡❥❛♠❡♥t♦ ♣r❡❝✐s❛ ❝♦♥t❡♠♣❧❛r t❛♠❜é♠ ❛ ♠❡❞✐❛çã♦ ❞♦ ♣r♦❢❡ss♦r ❞✉r❛♥t❡ ❛ ❛✉❧❛✱ ♥♦ s❡♥t✐❞♦ ❞❡ ❢❛✈♦r❡❝❡r ❛♦s ❛❧✉♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♣♦ss❛♠ ❛♣r❡s❡♥t❛r s✉❛s s♦❧✉çõ❡s ♣❛r❛ ❡✈❡♥t✉❛✐s ❞✐s❝✉ssõ❡s✳ P❡♥s❛♥❞♦ ♥✐ss♦ ❞❡s❡♥✈♦❧✈❡♠♦s ♦ ❈❛♣ít✉❧♦ ✹ ❝♦♠ s✉❣❡stõ❡s ❞❡ ❛t✐✈✐❞❛❞❡s ❞❡ ❡①♣❧♦r❛çã♦ ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❛❞q✉✐r❛ ✉♠❛ ♣♦st✉r❛ ❝rít✐❝❛ ❡ q✉❡st✐♦♥❛❞♦r❛✱ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ♠❛✐♦r ❢❡①✐❜✐❧✐❞❛❞❡ ♥♦ s❡✉ r❛❝✐♦❝í♥✐♦ ❡ ❡♠ s✉❛ ♣rát✐❝❛ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✳ ❙❡❣✉♥❞♦ P♦♥t❡s✱ ♦ ♣r♦❢❡ss♦r q✉❡ tr❛❜❛❧❤❛ ❝♦♠ ❛t✐✈✐❞❛❞❡s ❞❡ ✐♥✈❡st✐❣❛çã♦✿
✸❖ Pr♦❣r❛♠❛ ◆❛❝✐♦♥❛❧ ❞❡ ❚❡❝♥♦❧♦❣✐❛ ❊❞✉❝❛❝✐♦♥❛❧ ✭Pr♦■♥❢♦✮ é ✉♠ ♣r♦❣r❛♠❛ ❡❞✉❝❛❝✐♦♥❛❧ ❝r✐❛❞♦ ♣❡❧❛
P♦rt❛r✐❛ ♥♦✺✷✷✴▼❊❈✱ ❞❡ ✾ ❞❡ ❛❜r✐❧ ❞❡ ✶✾✾✼ ❡ r❡❣✉❧❛♠❡♥t❛❞♦ ♣❡❧♦ ❉❡❝r❡t♦ ✻✳✸✵✵✱ ❞❡ ✶✷ ❞❡ ❞❡③❡♠❜r♦ ❞❡
✷✵✵✼✱ ♣❛r❛ ♣r♦♠♦✈❡r ♦ ✉s♦ ♣❡❞❛❣ó❣✐❝♦ ❞❛s ❚❡❝♥♦❧♦❣✐❛s ❞❡ ■♥❢♦r♠át✐❝❛ ❡ ❈♦♠✉♥✐❝❛çõ❡s ✭❚■❈s✮ ♥❛ r❡❞❡ ♣ú❜❧✐❝❛ ❞❡ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♠é❞✐♦✳ P❛r❛ s❛❜❡r ♠❛✐s ❛❝❡ss❡ ❬✷✾❪✳
✹❖ ▲✐♥✉① ❊❞✉❝❛❝✐♦♥❛❧ ✭▲❊✮ é ✉♠ ♣r♦❥❡t♦ ❞♦ ●♦✈❡r♥♦ ❋❡❞❡r❛❧ q✉❡ ❜✉s❝❛ ♦ ♠❡❧❤♦r ❛♣r♦✈❡✐t❛♠❡♥t♦
✹
✏❛❥✉❞❛ ❛ tr❛③❡r ♣❛r❛ à s❛❧❛ ❞❡ ❛✉❧❛ ♦ ❡s♣ír✐t♦ ❞❛ ❛t✐✈✐❞❛❞❡ ♠❛t❡♠át✐❝❛ ❣❡♥✉í♥❛✱ ❝♦♥st✐t✉✐♥❞♦✱ ♣♦r ✐ss♦✱ ✉♠❛ ♣♦❞❡r♦s❛ ♠❡tá❢♦r❛ ❡❞✉❝❛t✐✈❛✳ ❖ ❛❧✉♥♦ é ❝❤❛♠❛❞♦ ❛ ❛❣✐r ❝♦♠♦ ✉♠ ♠❛t❡♠át✐❝♦✱ ♥ã♦ só ♥❛ ❢♦r♠✉❧❛çã♦ ❞❡ q✉❡stõ❡s ❡ ❝♦♥❥❡❝t✉r❛s ❡ ♥❛ r❡❛❧✐③❛çã♦ ❞❡ ♣r♦✈❛s ❡ r❡❢✉t❛çõ❡s✱ ♠❛s t❛♠❜é♠ ♥❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ r❡s✉❧t❛❞♦s ❡ ♥❛ ❞✐s❝✉ssã♦ ❡ ❛r❣✉♠❡♥t❛çã♦ ❝♦♠ ♦s s❡✉s ❝♦❧❡❣❛s ❡ ♦ ♣r♦❢❡ss♦r✳✑ ✭❬✶✼❪✱ ♣✳ ✷✸✮✳
P❛r❛ P♦♥t❡s✱ ♦ ♣❛♣❡❧ ❞♦ ♣r♦❢❡ss♦r ❡♠ ✉♠❛ ❛✉❧❛ ❞❡ ✐♥✈❡st✐❣❛çã♦ é ♦ ❞❡ ❞❡s❛✜❛❞♦r ♣r♦♠♦✈❡♥❞♦ ❛ ❛✉t♦♥♦♠✐❛ ❞♦ ❛❧✉♥♦✱ ❛✈❛❧✐❛♥❞♦ s❡✉ ♣r♦❣r❡ss♦ ❞✉r❛♥t❡ ❛ ✐♥✈❡st✐❣❛çã♦ ❞❡st❡✳ ◆♦ ❈❛♣ít✉❧♦ ✶ ❞❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❢❡✐t♦ ✉♠ ❜r❡✈❡ ❧❡✈❛♥t❛♠❡♥t♦ ❤✐stór✐❝♦ s♦❜r❡ ❛❧❣✉♠❛s ♣✉❜❧✐❝❛çõ❡s q✉❡ ❝♦♥tr✐❜✉✐r❛♠ ♣❛r❛ ❛ s✐st❡♠❛t✐③❛çã♦ ❞❛ t❡♦r✐❛ ✈❡t♦r✐❛❧✱ ❝✐t❛♠♦s t❛♠❜é♠ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ✭✈❡t♦r❡s ♥❛ s✉❛ ❡①ê♥❝✐❛✮✱ ❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛✱ ❡ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❣❡♦♠étr✐❝❛s ❞♦s ✈❡t♦r❡s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♠ó❞✉❧♦✱ ❞✐r❡çã♦✱ s❡♥t✐❞♦✱ ✈❡rs♦r✱ ❛❞✐çã♦ ❡ ♠ú❧t✐♣❧♦✳
❖ ❈❛♣ít✉❧♦ ✷ é ❞❡❞✐❝❛❞♦ ❛♦ ❡♥❢♦q✉❡ ❛♥❛❧ít✐❝♦ ✭❛❧❣é❜r✐❝♦✮ ❞❡ ✈❡t♦r❡s ♥♦ ♣❧❛♥♦ ❡ ♥♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ❝❛rt❡s✐❛♥♦✱ â♥❣✉❧♦ ❡♥tr❡ ✈❡t♦r❡s✱ ♦♣❡r❛çõ❡s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❖s ❝❛♣ít✉❧♦s ✶ ❡ ✷ ❢♦r❛♠ ❡❧❛❜♦r❛❞♦s ❛ ♣❛rt✐r ❞❡ ❛❞❛♣t❛çõ❡s ❡ ✉s♦ ♣❛r❝✐❛❧ ❞❡ t❡①t♦s ❡①tr❛í❞♦s ❞❡ ❧✐✈r♦s ❡ ❛♣♦st✐❧❛s ❝✐t❛❞❛s ♥❛s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳
❈❛♣ít✉❧♦ ✶
❱❡t♦r❡s
❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ✉s❛❞❛s ♥❡st❡ ❝❛♣ít✉❧♦ ❢♦r❛♠ ❬✶✷❪✱ ❬✶✾❪ ❡ ❬✷✺❪✳
✶✳✶ ❯♠ ♣♦✉❝♦ s♦❜r❡ ✈❡t♦r❡s
❉❡ ❛❝♦r❞♦ ❝♦♠ ❬✷✺❪✱ ❛ ♥♦çã♦ ❜ás✐❝❛ ❞❡ ✈❡t♦r s✉r❣✐✉ ♥❛ ▼❡❝â♥✐❝❛ ❝♦♠ ♦ ❡♥❣❡♥❤❡✐r♦ ❙✐♠♦♥ ❙t❡✈✐♥✱ ❡♠ ✶✺✽✻ ❛♣r❡s❡♥t♦✉ ✉♠ tr❛❜❛❧❤♦ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❢♦rç❛s ❡ ❡♥✉♥❝✐♦✉ ✉♠❛ r❡❣r❛ ❡♠♣ír✐❝❛ ♣❛r❛ s❡ ❛❝❤❛r ❛ s♦♠❛ ❞❡ ✷ ❢♦rç❛s ❛♣❧✐❝❛❞❛s ♥✉♠ ♠❡s♠♦ ♣♦♥t♦✳ ❚❛❧ r❡❣r❛ ❛ ❝♦♥❤❡❝❡♠♦s ❤♦❥❡ ❝♦♠♦ r❡❣r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ✭♣✳✻✹✮✳ ❊ss❛ r❡❣r❛ t❛♠❜é♠ ❛♣❛r❡❝❡ ❝♦♠♦ ♣r✐♠❡✐r♦ ❝♦r♦❧ár✐♦ ♥♦ Pr✐♥❝✐♣✐❛ ▼❛t❤❡♠❛t✐❝❛✭✶✻✽✼✮ ❞❡ ■s❛❛❝ ◆❡✇t♦♥✳ ◆♦ Pr✐♥❝✐♣✐❛✱ ◆❡✇t♦♥ ❧✐❞♦✉ ❡①t❡♥s✐✈❛♠❡♥t❡ ❝♦♠ ♦ q✉❡ ❛❣♦r❛ sã♦ ❝♦♥s✐❞❡r❛❞❛s ❡♥t✐❞❛❞❡s ✈❡t♦r✐❛✐s ✭♣♦r ❡①❡♠♣❧♦✱ ✈❡❧♦❝✐❞❛❞❡✱ ❢♦rç❛✮✱ ♠❛s ♥✉♥❝❛ ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✉♠ ✈❡t♦r✳
❖s ✈❡t♦r❡s ❛♣❛r❡❝❡♠ ❝♦♥s✐❞❡r❛❞♦s ❝♦♠♦ ✏❧✐♥❤❛s ❞✐r✐❣✐❞❛s✑ ♥❛ ♦❜r❛ ♣✉❜❧✐❝❛❞❛ ❡♠ ✶✼✾✼ ♣♦r ●❛s♣❛r ❲❡ss❡❧ ❡ ❝♦♠ ❛s r❡♣r❡s❡♥t❛çõ❡s ❣❡♦♠étr✐❝❛s ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❏❡❛♥ ❘♦❜❡rt ❆r❣❛♥❞ ❡ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ❝♦♥❝❡❜❡r❛♠ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❝♦♠♦ ♣♦♥t♦s ♥♦ ♣❧❛♥♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ✐st♦ é✱ ❝♦♠♦ ✈❡t♦r❡s ❜✐❞✐♠❡♥s✐♦♥❛✐s✳ ▼❛t❡♠át✐❝♦s ❡ ❝✐❡♥t✐st❛s tr❛❜❛❧❤❛r❛♠ ❝♦♠ ❡st❡s ♥♦✈♦s ♥ú♠❡r♦s ❡ ♦s ❛♣❧✐❝❛r❛♠ ❞❡ ✈ár✐❛s ♠❛♥❡✐r❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ●❛✉ss ❢❡③ ✉♠ ✉s♦ ❝r✉❝✐❛❧ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♣❛r❛ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛ ✭✶✼✾✾✮✳ ❊♠ ✶✽✸✷✱ ●✐✉st♦ ❇❡❧❧❛✈✐t✐s ♣✉❜❧✐❝❛ ✉♠ tr❛❜❛❧❤♦ ♦♥❞❡ é ❛♣r❡s❡♥t❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡q✉✐♣♦❧ê♥❝✐❛ ❡♥tr❡ s❡❣♠❡♥t♦s q✉❡ é✱ ❜❛s✐❝❛♠❡♥t❡✱ ❛ ♥♦çã♦ ❞❡ ✈❡t♦r q✉❡ ❝♦♥❤❡❝❡♠♦s✳
❆ s✐st❡♠❛t✐③❛çã♦ ❞❛ t❡♦r✐❛ ✈❡t♦r✐❛❧ ♦❝♦rr❡✉ ♥♦ sé❝✉❧♦ ❳■❳ ❝♦♠ ♦s tr❛❜❛❧❤♦s ❞♦ ✐r❧❛♥❞ês ❲✐❧❧✐❛♠ ❘♦✇❛♥ ❍❛♠✐❧t♦♥✱ ❞♦ ❛❧❡♠ã♦ ❍❡r♠❛♥♥ ●✳ ●r❛ss♠❛♥♥ ❡ ❞♦ ❢ís✐❝♦ ♥♦rt❡✲❛♠❡r✐❝❛♥♦ ❏♦s✐❛❤ ❲✳ ●✐❜❜s✳ ❊♠ ✶✽✸✼✱ ❍❛♠✐❧t♦♥ ♠♦str♦✉ q✉❡ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♣♦❞❡r✐❛♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ❛❜str❛t❛♠❡♥t❡ ❝♦♠♦ ♣❛r❡s ♦r❞❡♥❛❞♦s (a, b) ❞❡
❱❡t♦r❡s ✻
♥ú♠❡r♦s r❡❛✐s✳ ❊st❛ ✐❞é✐❛ ❡r❛ ♣❛rt❡ ❞❡ ✉♠❛ ❝❛♠♣❛♥❤❛ ❞❡ ♠✉✐t♦s ♠❛t❡♠át✐❝♦s✱ ✐♥❝❧✉✐♥❞♦ ❍❛♠✐❧t♦♥✱ ♣❛r❛ ♣r♦❝✉r❛r ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❡st❡♥❞❡r ♦s ✏♥ú♠❡r♦s✑ ❜✐❞✐♠❡♥s✐♦♥❛✐s ♣❛r❛ três ❞✐♠❡♥sõ❡s❀ ♠❛s ♥✐♥❣✉é♠ ❝♦♥s❡❣✉✐✉ ✐st♦ ♣r❡s❡r✈❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛❧❣é❜r✐❝❛s ❜ás✐❝❛s ❞♦s ♥ú♠❡r♦s r❡❛✐s ❡ ❝♦♠♣❧❡①♦s✳
❖ ✐♥t✉✐t♦ ❞❡ ❞❡s❝♦❜r✐r ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ♣❛r❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥♦ ❡s♣❛ç♦ ❧❡✈♦✉ ❍❛♠✐❧t♦♥✱ ❡♠ ✶✽✹✸✱ ❛ ❞❡s❝♦❜r✐r ♦s q✉❛tér♥✐♦s✱ q✉❡ ❢♦r❛♠ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❡str✉t✉r❛çã♦ ❞♦ ❈á❧❝✉❧♦ ❱❡t♦r✐❛❧✱ ♣♦✐s s✉❛s ♣r♦♣r✐❡❞❛❞❡s s❡ ❛❞❡q✉❛✈❛♠ ❛♦ ❡st✉❞♦ ❞♦s ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s✳ ❉❡♣♦✐s ❞❡ ♠✉✐t❛ ❢r✉str❛çã♦✱ ❍❛♠✐❧t♦♥ ❡st❛✈❛ ✜♥❛❧♠❡♥t❡ ✐♥s♣✐r❛❞♦ ❛ ❞❡s✐st✐r ❞❛ ♣r♦❝✉r❛ ♣♦r ✉♠ s✐st❡♠❛ ✏♥✉♠ér✐❝♦✑ tr✐❞✐♠❡♥s✐♦♥❛❧ ❡ ❡♠ ✈❡③ ❞✐ss♦✱ ✐♥✈❡♥t♦✉ ✉♠ s✐st❡♠❛ ❞❡ q✉❛tr♦ ❞✐♠❡♥sõ❡s q✉❡ ❝❤❛♠♦✉ ❞❡ q✉❛tér♥✐♦s✳ ❖s q✉❛tér♥✐♦s ❞❡ ❍❛♠✐❧t♦♥ ❢♦r❛♠ ❡s❝r✐t♦s✱q =w+✐x+❥y+❦z✱ ♦♥❞❡w✱x✱y✱ ❡z ❡r❛♠ ♥ú♠❡r♦s
r❡❛✐s✳ ❍❛♠✐❧t♦♥ r❛♣✐❞❛♠❡♥t❡ ♣❡r❝❡❜❡✉ q✉❡ s❡✉s q✉❛tér♥✐♦s ❝♦♥s✐st✐❛♠ ❞❡ ❞✉❛s ♣❛rt❡s ❞✐st✐♥t❛s✳ ❖ ♣r✐♠❡✐r♦ t❡r♠♦✱ ♦ q✉❛❧ ❝❤❛♠♦✉ ❞❡ ❡s❝❛❧❛r ❡ x✱ y ❡ z ♣❛r❛ s✉❛s ❝♦♠♣♦♥❡♥t❡s
r❡t❛♥❣✉❧❛r❡s✱ ♦✉ ♣r♦❥❡çõ❡s ❡♠ três ❡✐①♦s r❡t❛♥❣✉❧❛r❡s✳ ❍❛♠✐❧t♦♥ ✉s♦✉ s✉❛s ✏❢ór♠✉❧❛s ❢✉♥❞❛♠❡♥t❛✐s✑✱ i2
=j2
=k2
=−ijk =−1✱ ♣❛r❛ ♠✉❧t✐♣❧✐❝❛r q✉❛tér♥✐♦s✱ ❡ ✐♠❡❞✐❛t❛♠❡♥t❡
❞❡s❝♦❜r✐✉ q✉❡ ♦ ♣r♦❞✉t♦✱q1q2 =−q2q1✱ ♥ã♦ ❡r❛ ❝♦♠✉t❛t✐✈♦✳ ❊s❝r❡✈❡✉ ❞♦✐s ❧✐✈r♦s ❝♦♠♣❧❡t♦s
s♦❜r❡ ♦ ❛ss✉♥t♦✱ ▲❡❝t✉r❡s ♦♥ ◗✉❛t❡r♥✐♦♥s ✭✶✽✺✸✮ ❡ ❊❧❡♠❡♥ts ♦❢ ◗✉❛t❡r♥✐♦♥s ✭✶✽✻✻✮ ✭❬✶✾❪✱ ♣✳✻✵✮✱ ❞❡t❛❧❤❛♥❞♦ ♥ã♦ ❛♣❡♥❛s ❛ á❧❣❡❜r❛ ❞♦s q✉❛tér♥✐♦s ♠❛s t❛♠❜é♠ ❝♦♠♦ ♣♦❞❡r✐❛♠ s❡r ✉s❛❞♦s ❡♠ ❣❡♦♠❡tr✐❛✳
◆❛ ♠❡s♠❛ é♣♦❝❛ ❞❛ ❞❡s❝♦❜❡rt❛ ❞♦s q✉❛tér♥✐♦s✱ ●r❛ss♠❛♥♥ ❡st❛✈❛ ❡s❝r❡✈❡♥❞♦ ❚❤❡ ❈❛❧❝✉❧✉s ♦❢ ❊①t❡♥s✐♦ ✭✶✽✹✹✮✱ ❝✉❥♦ tít✉❧♦ ❡♠ ❛❧❡♠ã♦ é ❆✉s❞❡❤♥✉♥❣s❧❡❤r❡✳ ❊♠ ✶✽✸✷✱ ●r❛ss♠❛♥♥ ❝♦♠❡ç♦✉ ❛ ❞❡s❡♥✈♦❧✈❡r ✏✉♠ ♥♦✈♦ ❝á❧❝✉❧♦ ❣❡♦♠étr✐❝♦✑ ❝♦♠♦ ♣❛rt❡ ❞♦ s❡✉ ❡st✉❞♦ ❞❛ t❡♦r✐❛ ❞❡ ♠❛rés✱ ❡ s✉❜s❡qü❡♥t❡♠❡♥t❡ ✉s♦✉ ❡st❛s ❢❡rr❛♠❡♥t❛s ♣❛r❛ s✐♠♣❧✐✜❝❛r ♣❛rt❡s ❞❡ ❞♦✐s tr❛❜❛❧❤♦s ❝❧áss✐❝♦s✱ ♦ ❆♥❛❧②t✐❝❛❧ ▼❡❝❤❛♥✐❝s ❞❡ ❏♦s❡♣❤ ▲♦✉✐s ▲❛❣r❛♥❣❡ ❡ ♦ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s ❞❡ P✐❡rr❡ ❙✐♠♦♥ ▲❛♣❧❛❝❡✳ ❙❡❣✉♥❞♦ ❬✶✾❪✱ ❡♠ ❆✉s❞❡❤♥✉♥❣s❧❡❤r❡✱ ♣r✐♠❡✐r♦ ●r❛ss♠❛♥♥ ❡①♣❛♥❞✐✉ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❡t♦r❡s ❛ ♣❛rt✐r ❞❛ ❢❛♠✐❧✐❛r ✷ ♦✉ ✸ ❞✐♠❡♥sõ❡s✱ ♣❛r❛ ✉♠ ♥ú♠❡r♦ ❛r❜✐trár✐♦✱ n✱ ❞❡ ❞✐♠❡♥sõ❡s❀ ✐st♦ ❡st❡♥❞❡✉ ❣r❛♥❞❡♠❡♥t❡ ❛s ✐❞❡✐❛s ❞❡
❡s♣❛ç♦✳ ❙❡❣✉♥❞♦✱ ❡ ❛✐♥❞❛ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ●r❛ss♠❛♥♥ ❛♥t❡❝✐♣♦✉ ❣r❛♥❞❡ ♣❛rt❡ ❞❛ á❧❣❡❜r❛ ♠❛tr✐❝✐❛❧ ❡ ❧✐♥❡❛r ♠♦❞❡r♥❛ ❡ ❛♥á❧✐s❡ ✈❡t♦r✐❛❧ ❡ t❡♥s♦r✐❛❧✳
❱❡t♦r❡s ✼
❋✐♥❛❧♠❡♥t❡✱ ♥❛s ❞é❝❛❞❛s ❞❡ ✶✽✻✵ ❡ ✶✽✼✵✱ ♦ ❆✉s❞❡❤♥✉♥❣s❧❡❤ r❡❝♦♠❡ç♦✉ ❧❡♥t❛♠❡♥t❡ ❛ s❡r ❡♥t❡♥❞✐❞♦ ❡ ❛♣r❡❝✐❛❞♦ ❡ ●r❛ss♠❛♥♥ ❝♦♠❡ç♦✉ ❛ r❡❝❡❜❡r ❛❧❣✉♠ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❢❛✈♦rá✈❡❧ ♣♦r s✉❛ ♠❛t❡♠át✐❝❛ ✈✐s✐♦♥ár✐❛✳
❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ á❧❣❡❜r❛ ✈❡t♦r✐❛❧ ❡ ❞❛ ❛♥á❧✐s❡ ✈❡t♦r✐❛❧ ❝♦♠♦ ❝♦♥❤❡❝❡♠♦s ❤♦❥❡ ❢♦✐ r❡✈❡❧❛❞♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥♦t❛s ❞❡ ❛✉❧❛ ❢❡✐t♦s ♣♦r ●✐❜❜s✱ ♣❛r❛ s❡✉s ❛❧✉♥♦s ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❨❛❧❡✳ ●✐❜❜s ♥❛s❝❡✉ ❡♠ ◆❡✇ ❍❛✈❡♥✱ ❈♦♥♥❡❝t✐❝✉t ✭s❡✉ ♣❛✐ t❛♠❜é♠ ❢♦✐ ♣r♦❢❡ss♦r ❡♠ ❨❛❧❡✮ ❡ s✉❛s ❝♦♥q✉✐st❛s ❝✐❡♥tí✜❝❛s ♣r✐♥❝✐♣❛✐s ❢♦r❛♠ ❡♠ ❢ís✐❝❛✱ t❡r♠♦❞✐♥â♠✐❝❛ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t❛✳ ●✐❜❜s t♦♠♦✉ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s q✉❛tér♥✐♦s q✉❛♥❞♦ ❧❡✉ ♦ ❚r❡❛t✐s❡ ♦♥ ❊❧❡❝tr✐❝✐t② ❛♥❞ ▼❛❣♥❡t✐s♠ ❞❡ ❏❛♠❡s ❈❧❡r❦ ▼❛①✇❡❧❧✱ ❡ t❛♠❜é♠ ❡st✉❞♦✉ ♦ ❆✉s❞❡❤♥✉♥❣s❧❡❤r❡ ❞❡ ●r❛ss♠❛♥♥✳ ❈♦♥❝❧✉✐✉ q✉❡ ✈❡t♦r❡s ❢♦r♥❡❝❡r✐❛♠ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♠❛✐s ❡✜❝✐❡♥t❡ ♣❛r❛ s❡✉ tr❛❜❛❧❤♦ ❡♠ ❢ís✐❝❛✳ ❆ss✐♠✱ ❝♦♠❡ç❛♥❞♦ ❡♠ ✶✽✽✶✱ ●✐❜❜s ✐♠♣r✐♠✐✉ ♣♦r ❝♦♥t❛ ♣ró♣r✐❛ ♥♦t❛s ❞❡ ❛✉❧❛s s♦❜r❡ ❛♥á❧✐s❡ ✈❡t♦r✐❛❧ ♣❛r❛ s❡✉s ❛❧✉♥♦s✱ ❛s q✉❛✐s ❢♦r❛♠ ❛♠♣❧❛♠❡♥t❡ ❞✐str✐❜✉í❞❛s ♣❛r❛ ❡st✉❞✐♦s♦s ♥♦s ❊st❛❞♦s ❯♥✐❞♦s✱ ♥❛ ■♥❣❧❛t❡rr❛ ❡ ♥❛ ❊✉r♦♣❛✳
✶✳✷ ◆♦çã♦ ✐♥t✉✐t✐✈❛
❖ ♣r✐♠❡✐r♦ ❝♦♥t❛t♦ ❞♦s ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❝♦♠ ♦s ✈❡t♦r❡s é ❛tr❛✈és ❞❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ q✉❡ s✐♠❜♦❧✐③❛ ✉♠ t✐♣♦ ♣❛rt✐❝✉❧❛r ❞❡ ✈❡t♦r✱ ♥❡ss❛ r❡♣r❡s❡♥t❛çã♦ ❡❧❡♠❡♥t❛r✱ ♦s ✈❡t♦r❡s sã♦ ❛ss♦❝✐❛❞♦s ❛ ❣r❛♥❞❡③❛s q✉❡ ♥❡❝❡ss✐t❛♠ ❞❡ ♠ó❞✉❧♦✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ ♣❛r❛ s❡r❡♠ ❝♦♠♣❧❡t❛♠❡♥t❡ ❡s♣❡❝✐✜❝❛❞❛s✳ ❆♥t❡s ❞❡ ❝♦♠❡ç❛r♠♦s ♥♦ss♦s ❡st✉❞♦s✱ ♣r❡❝✐s❛♠♦s t❡r ❜❡♠ ♣r❡s❡♥t❡s ❛s ✐❞❡✐❛s ❞❡ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✱ ♣❛r❛ ✐ss♦ ✈❛♠♦s ❛♥❛❧✐s❛r ❛ s✐t✉❛çã♦ ❛❜❛✐①♦✿
❍♦❥❡ ❡♠ ❞✐❛ é ♠✉✐t♦ ❢á❝✐❧ ❡♥❝♦♥tr❛r ❡q✉✐♣❛♠❡♥t♦s ❡❧❡trô♥✐❝♦s q✉❡ r❡❝❡❜❡♠ s✐♥❛❧ ❞❡ ●P❙✶ ❡ sã♦ ✉s❛❞♦s ❡♠ ✈❡í❝✉❧♦s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ♦r✐❡♥t❛çã♦ ♥❛s ❝✐❞❛❞❡s✱ ♠❛s ♥ã♦ sã♦
só ♦s ♠♦t♦r✐st❛s q✉❡ tê♠ ❛❝❡ss♦ ❛ ❡ss❛ t❡❝♥♦❧♦❣✐❛✱ ♠✉✐t♦s ❝❡❧✉❧❛r❡s ❡ t❛❜❧❡ts t❛♠❜é♠ r❡❝❡❜❡♠ ❡ss❡ s✐♥❛❧✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ❛♦ ✉s✉ár✐♦ ❞❡ss❡s ❡q✉✐♣❛♠❡♥t♦s ❡❧❡trô♥✐❝♦s ❛ ❧♦❝❛❧✐③❛çã♦ ❞❡ ❡♥❞❡r❡ç♦s r❡s✐❞❡♥❝✐❛✐s✱ ❞❡ ❡♠♣r❡s❛s ❡ s❡r✈✐ç♦s ❧♦❝❛✐s✱ ❛❧é♠ ❞❡ tr❛ç❛r ❛s ♠❡❧❤♦r❡s r♦t❛s ♣❛r❛ ❝❤❡❣❛r ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❧✉❣❛r ❛tr❛✈és ❞❛s ❝♦♦r❞❡♥❛❞❛s ❣❡♦❣rá✜❝❛s✳ ▼❛s s❡rá q✉❡ s❛❜❡♠♦s ♥♦s ♦r✐❡♥t❛r ❡♠ ✉♠ ♠❛♣❛❄
❊①❡♠♣❧♦ ✶✳✶ ▲✉❝❛s ❞❡s❝❡ ❞❡ ✉♠ ô♥✐❜✉s ❡ ❝❛♠✐♥❤❛ ❛té ♦ ♣♦♥t♦ A ♣r❡t❡♥❞❡♥❞♦ ❝❤❡❣❛r
❛♦ ♣♦♥t♦ B✳ ❈♦♠♦ ▲✉❝❛s ♥ã♦ ❝♦♥❤❡❝❡ ❛ r❡❣✐ã♦✱ ♣❡❞❡ ✐♥❢♦r♠❛çõ❡s ❛ ✉♠ ❞❡s❝♦♥❤❡❝✐❞♦
q✉❡ t❛♠❜é♠ ❡stá ❡♠ A✱ ❡ss❛ ♣❡ss♦❛ ♣❡❣❛ s❡✉ ❝❡❧✉❧❛r✱ q✉❡ r❡❝❡❜❡ s✐♥❛❧ ❞❡ ●P❙✱ ❡ ❛❝❡ss❛
✉♠ ❛♣❧✐❝❛t✐✈♦ q✉❡ ❡①✐❜❡ ❛ ❋✐❣✉r❛ ✶✳✶✳ ❈♦♠♦ ♦ ❞❡s❝♦♥❤❡❝✐❞♦ ❡stá ❛♣r❡ss❛❞♦✱ ❡❧❡ ❛♣❡♥❛s
✶❖ ◆❆❱❙❚❆❘ ✲ ●P❙ ✭◆❆❱✐❣❛t✐♦♥ ❙②st❡♠ ✇✐t❤ ❚✐♠❡ ❆♥❞ ❘❛♥❣✐♥❣ ✲ ●❧♦❜❛❧ P♦s✐t✐♦♥✐♥❣ ❙②st❡♠✮ é
❱❡t♦r❡s ✽
✐♥❢♦r♠❛ ❛♦ ▲✉❝❛s✿ ✏❛♥❞❡ três q✉❛❞r❛s✱ ✈✐r❡ ♥❛ ❡sq✉✐♥❛ ❡ ❛♥❞❡ ♠❛✐s ❞✉❛s q✉❛❞r❛s✑✳ ❆♥❛❧✐s❡ ❡ r❡s♣♦♥❞❛ ❛s s❡♥t❡♥ç❛s ❛❜❛✐①♦✿
❋✐❣✉r❛ ✶✳✶✿ ■♠❛❣❡♠ ✜❝tí❝✐❛ ❞♦ ♠❛♣❛
✭❛✮ ❊ss❛s ✐♥❢♦r♠❛çõ❡s sã♦ s✉✜❝✐❡♥t❡s ♣❛r❛ ♦ ▲✉❝❛s ❝❤❡❣❛r ❛♦ ♣♦♥t♦ B❄
✭❜✮ ❖ q✉❡ ❢❛❧t♦✉ ✐♥❢♦r♠❛r ♣❛r❛ q✉❡ ♦ ▲✉❝❛s ♥ã♦ t♦♠❡ ✉♠ ❝❛♠✐♥❤♦ ❡rr❛❞♦❄
❉❡✜♥✐çã♦ ✶✳✶ ●r❛♥❞❡③❛s ❊s❝❛❧❛r❡s sã♦ ❛q✉❡❧❛s q✉❡ ✜❝❛♠ ❝♦♠♣❧❡t❛♠❡♥t❡ ❝❛r❛❝t❡r✐✲ ③❛❞❛ ❢♦r♥❡❝❡♥❞♦✲s❡ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ r❡❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✳
◆♦ ❊①❡♠♣❧♦ ✶✳✶✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❛♦ ✐♥❢♦r♠❛r ❛♣❡♥❛s q✉❛♥t❛s q✉❛❞r❛s s❡rã♦ ♣❡r❝♦rr✐❞❛s ❞❡ A à B✱ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ♦ ♥♦ss♦ ♣❡rs♦♥❛❣❡♠✱ ▲✉❝❛s✱ ❝❤❡❣✉❡
❛♦ s❡✉ ❞❡st✐♥♦✱ ♣r♦✈❛✈❡❧♠❡♥t❡ ❡❧❡ s❡ ♣❡r❣✉♥t♦✉✿ ✏✈♦✉ ❛♥❞❛r três q✉❛❞r❛s ♣❛r❛ ♦♥❞❡❄✑✱ ♦✉ s❡❥❛✱ ❢❛❧t♦✉ ✐♥❢♦r♠❛r ♣❛r❛ ♦ ▲✉❝❛s q✉❡ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ ❡❧❡ ❞❡✈❡r✐❛ ❛♥❞❛r✳ ❖❧❤❛♥❞♦ ♣❛r❛ ♦ ♠❛♣❛ ❞❛ ❋✐❣✉r❛ ✶✳✶✱ é ❢á❝✐❧ ♣❡r❝❡❜❡r q✉❡ ❡❧❡ ❞❡✈❡ s❡❣✉✐r ♣♦r ❡ss❛ r✉❛ q✉❡ ❡stá ♥❛ ❤♦r✐③♦♥t❛❧ ✭❞✐r❡çã♦✮✱ ❛♥❞❛♥❞♦ três q✉❛❞r❛s ♥♦ s❡♥t✐❞♦ ❧❡st❡ ❛té ❝❤❡❣❛r à r✉❛ ♥❛ ✈❡rt✐❝❛❧ ✭❞✐r❡çã♦✮ q✉❡ ❝♦♥té♠ ♦ ♣♦♥t♦B ❡ s❡❣✉✐r ❛♦ ♥♦rt❡ ✭s❡♥t✐❞♦✮ ♠❛✐s ❞✉❛s q✉❛❞r❛s✳ ◆❛ ❋ís✐❝❛✱
❛s ❣r❛♥❞❡③❛s q✉❡ ♥❡❝❡ss✐t❛♠ ❞❡ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ sã♦ ❝❤❛♠❛❞❛s ✈❡t♦r✐❛✐s✳
❉❡✜♥✐çã♦ ✶✳✷ ●r❛♥❞❡③❛s ❱❡t♦r✐❛✐s sã♦ ❛q✉❡❧❛s q✉❡ ✜❝❛♠ ❝♦♠♣❧❡t❛♠❡♥t❡ ❝❛r❛❝t❡r✐✲ s❛❞❛s ❢♦r♥❡❝❡♥❞♦✲s❡✱ ❛❧é♠ ❞❡ ✉♠ ♥ú♠❡r♦ ❡ s✉❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✱ ❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✳
❱❡t♦r❡s ✾
✶✳✸ ❙❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦
❉♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s A ❡ B ♥♦ ❡s♣❛ç♦ ❞❡t❡r♠✐♥❛♠ ✉♠❛ r❡t❛✳ ❊st❛ r❡t❛ é ✉♠❛
❞✐r❡çã♦ ♥♦ ❡s♣❛ç♦✳ ◆ã♦ ♣r❡❝✐s❛♠♦s ❞❛ r❡t❛ t♦❞❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❡st❛ ❞✐r❡çã♦✱ ♦ s❡❣♠❡♥t♦ ❞❛ r❡t❛ ❡♥tr❡ ♦s ♣♦♥t♦sA❡B✱ q✉❡ é ❛ ♣❛rt❡ ❞❛ r❡t❛ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❡st❡s ❞♦✐s ♣♦♥t♦s✱
s❡r✈❡ ♠✉✐t♦ ❜❡♠ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❡st❛ ❞✐r❡çã♦✳ ❊st❡ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ ♦r✐❡♥t❛❞♦✱ ♣r♦✈❡♥❞♦ ✉♠ s❡♥t✐❞♦ ♣❛r❛ ♦ s❡❣♠❡♥t♦ ❛tr❛✈és ❞❡ ✉♠❛ s❡t❛ ♦♥❞❡ ✉♠ ❞♦s ♣♦♥t♦s é ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ ♦ ♦✉tr♦ ❝♦♠♦ ♣♦♥t♦ ✜♥❛❧✷✳ P♦r ❡①❡♠♣❧♦✱ ♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ❝♦♠
♣♦♥t♦ ✐♥✐❝✐❛❧A ❡ ♣♦♥t♦ ✜♥❛❧B s❡rá ❞❡♥♦t❛❞♦ ♣♦r
−→
AB✱ ✈❡r ❋✐❣✉r❛ ✶✳✷✭❛✮✳
✭❛✮ ❙❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦AB−→ ✭❜✮ ❙❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ −→ BA
❋✐❣✉r❛ ✶✳✷✿ ❙❡❣♠❡♥t♦ ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦
◆♦t❡ q✉❡ ❡♠❜♦r❛ ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ♦s s❡❣♠❡♥t♦s AB−→ ❡
−→
BA s❡❥❛♠ ✐❣✉❛✐s
❡ ♣♦ss✉✐r❡♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ✭r❡t❛r✮✱ ❝♦♠♦ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡❧❡s sã♦ ❞✐st✐♥t♦s✱ ♣♦✐s
−→
BA ❡stá ♦r✐❡♥t❛❞♦ ❝♦♠ ♦ s❡♥t✐❞♦ ❞❡ ♣❡r❝✉rs♦ ♦♣♦st♦ ❛♦ ❞♦ s❡❣♠❡♥t♦
−→
AB✱ ♦✉ s❡❥❛✱ ✉♠
s❡♥t✐❞♦ ❞❡ ♣❡r❝✉rs♦ ❞❡B ♣❛r❛ A ✭❋✐❣✉r❛ ✶✳✷✭❜✮✮✳
✶✳✸✳✶ ❙❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ♥✉❧♦
❯♠ s❡❣♠❡♥t♦ ♥✉❧♦ é ❛q✉❡❧❡ ❝✉❥♦ ♦ ♣♦♥t♦ ✜♥❛❧ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧✱ ♦✉ s❡❥❛✱ ♣♦♥t♦s s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s ❝♦♠♦ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ♥✉❧♦s❀ ♣♦r ❡①❡♠♣❧♦✱ ♦ ♣♦♥t♦
A é ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠ ♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦
−→
AA ✭❛ ♦r✐❣❡♠ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❡①tr❡♠✐❞❛❞❡✮✳
✶✳✹ ❉✐r❡çã♦ ❡ s❡♥t✐❞♦
❉♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ♥ã♦ ♥✉❧♦s AB−→ ❡
−→
CD tê♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ s❡ ❛s r❡t❛s
s✉♣♦rt❡s ❞❡ss❡s s❡❣♠❡♥t♦s sã♦ ♣❛r❛❧❡❧❛s ✭❋✐❣✉r❛ ✶✳✸✮ ♦✉ ❝♦✐♥❝✐❞❡♥t❡s ✭❋✐❣✉r❛ ✶✳✹✮✳ ❖❜s❡r✈❛çõ❡s✿
✭❛✮ ❙ó ♣♦❞❡♠♦s ❝♦♠♣❛r❛r ♦s s❡♥t✐❞♦s ❞❡ ❞♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s s❡ ❡❧❡s tê♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦❀
❱❡t♦r❡s ✶✵
✭❛✮ ✭❜✮
❋✐❣✉r❛ ✶✳✸✿ ❙❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡♠ r❡t❛s ♣❛r❛❧❡❧❛s
✭❛✮ ✭❜✮
❋✐❣✉r❛ ✶✳✹✿ ❙❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡♠ r❡t❛s ❝♦✐♥❝✐❞❡♥t❡s
✭❜✮ ❖s s❡❣♠❡♥t♦s ❞❛ ❋✐❣✉r❛ ✶✳✹ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦❧✐♥❡❛r❡s❀ ✭❝✮ ❖s s❡❣♠❡♥t♦sAB−→ ❡
−→
CD tê♠ ♦s ♠❡s♠♦s s❡♥t✐❞♦s ♥❛s ❋✐❣✉r❛s ✶✳✸✭❛✮ ❡ ✶✳✹✭❛✮✱ ❡♥q✉❛♥t♦
q✉❡ ❡♠ ✶✳✸✭❜✮ ❡ ✶✳✹✭❜✮ tê♠ s❡♥t✐❞♦s ❝♦♥trár✐♦s✱ ♥❡ss❡ ❝❛s♦ ❡ss❡s s❡❣♠❡♥t♦s sã♦ ♦♣♦st♦s✳
✶✳✺ ❊q✉✐♣♦❧ê♥❝✐❛s ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s
❉❡✜♥✐çã♦ ✶✳✸ ✭❇❡❧❧❛✈✐t✐s✮ ❙❡❥❛♠ AB−→ ❡
−→
CD s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ♥ã♦ ♥✉❧♦s✳ ❉✐③❡♠♦s
q✉❡ ❡❧❡s sã♦ ❡q✉✐♣♦❧❡♥t❡s ✭♦✉ ❡q✉✐✈❛❧❡♥t❡s✮✱ ❡ ❡s❝r❡✈❡♠♦s AB−→≡
−→
CD❀ s❡ ❡❧❡s tê♠ ♦ ♠❡s♠♦
❝♦♠♣r✐♠❡♥t♦✱ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ✭sã♦ ♣❛r❛❧❡❧♦s ♦✉ ❝♦❧✐♥❡❛r❡s✮ ❡ ♦ ♠❡s♠♦ s❡♥t✐❞♦✸✳
✶✳✺✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ r❡❧❛çã♦ ❞❡ ❡q✉✐♣♦❧ê♥❝✐❛
❆ r❡❧❛çã♦ ❞❡ ❡q✉✐♣♦❧ê♥❝✐❛ ❣♦③❛ ❞❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✭✐✮ r❡✢❡①✐✈❛✿ AB−→≡
−→
AB❀
✭✐✐✮ s✐♠étr✐❝❛✿ s❡ AB−→≡
−→
CD✱ ❡♥tã♦
−→
CD≡
−→
AB❀
❱❡t♦r❡s ✶✶
✭✐✐✐✮ tr❛♥s✐t✐✈❛✿ s❡ AB−→≡
−→
CD ❡
−→
CD≡
−→
EF✱ ❡♥tã♦
−→
AB≡
−→
EF✳
❖❜s❡r✈❛çõ❡s✿
✭❛✮ ❉♦✐s s❡❣♠❡♥t♦s ♥✉❧♦s sã♦ s❡♠♣r❡ ❡q✉✐♣♦❧❡♥t❡s❀
✭❜✮ ❉❛❞♦ ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ AB−→ ❡ ✉♠ ♣♦♥t♦ C✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ D t❛❧ q✉❡
−→
AB≡
−→
CD✳
✶✳✻ ❈♦♥❝❡✐t♦ ❞❡ ✈❡t♦r
❆ r❡❧❛çã♦ ❞❡ ❡q✉✐♣♦❧ê♥❝✐❛ ♣❡r♠✐t❡ ❝❧❛ss✐✜❝❛r ♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❞♦ ♣❧❛♥♦ ♠❡❞✐❛♥t❡ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳
❉❡✜♥✐çã♦ ✶✳✹ ❙❡❥❛♠A❡B ♣♦♥t♦s ♥♦ ♣❧❛♥♦✳ ❖ ✈❡t♦r~v ✭♦✉ s✐♠♣❧❡s♠❡♥t❡v✮ é ♦ ❝♦♥❥✉♥t♦
❞❡ t♦❞♦s ♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦❧❡♥t❡s ❛AB−→✳
❋✐❣✉r❛ ✶✳✺✿ ❘❡♣r❡s❡♥t❛♥t❡s ❞♦ ✈❡t♦r~v
