❞❡ ❙tr✐♥❣s
Pr♦❢❛✳❉r❛✳❚❤❛t②❛♥❛ ❞❡ ❋❛r✐❛ P✐♦❧❛ ❙❡r❛♣❤✐♠ ✭❊❈❖✮
Pr♦❢✳▼❙❝✳❘♦❞r✐❣♦ ▼❛①✐♠✐❛♥♦ ❆♥t✉♥❡s ❞❡ ❆❧♠❡✐❞❛ ✭❊▲❚✮ Pr♦❢✳❉r✳❈❛r❧♦s ❍❡♥r✐q✉❡ ❱❛❧ér✐♦ ❞❡ ▼♦r❛❡s ✭❊❈❆✮
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ■t❛❥✉❜á
t❤❛t②❛♥❛❅✉♥✐❢❡✐✳❡❞✉✳❜r r♦❞r✐❣♦♠❛①❅✉♥✐❢❡✐✳❡❞✉✳❜r
❇✐❜❧✐♦t❡❝❛s sã♦ ❝♦♥❥✉♥t♦s ❞❡ ❢✉♥çõ❡s q✉❡ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ❡ q✉❡ ♣♦❞❡♠ s❡r ✉s❛❞❛s ♣♦r ♦✉tr♦s ♣r♦❣r❛♠❛s s❡♠ q✉❡ ♦ ♣r♦❣r❛♠❛❞♦r s❡ ♣r❡♦❝✉♣❡ ❝♦♠ ♦ ❝ó❞✐❣♦ ❞❡ss❛s ❢✉♥çõ❡s ❱❛♥t❛❣❡♥s ♥♦ ✉s♦ ❞❡ ❜✐❜❧✐♦t❡❝❛s
♦r❣❛♥✐③❛çã♦ ❞♦ ❝ó❞✐❣♦
♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ❡♠ ✈ár✐♦s ♣r♦❣r❛♠❛s s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♣✐❛r ❣r❛♥❞❡s tr❡❝❤♦s ❞❡ ❝ó❞✐❣♦
❆ ❜✐❜❧✐♦t❡❝❛ ♣❛❞rã♦ ❞♦ ❈ é t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❧✐❜❝
é ✉♠❛ ❜✐❜❧✐♦t❡❝❛ ❞❡ r♦t✐♥❛s ♣❛❞r♦♥✐③❛❞❛ ❞❛ ❧✐♥❣✉❛❣❡♠ ❞❡ ♣r♦❣r❛♠❛çã♦ ❈
❝♦♥té♠ ♦♣❡r❛çõ❡s ❝♦♠✉♥s ❝♦♠♦ tr❛t❛♠❡♥t♦ ❞❡ ❡♥tr❛❞❛✴s❛í❞❛ ❡ ❝❛❞❡✐❛ ❞❡ ❝❛r❛❝t❡r❡s
P❛r❛ ♠✉✐t♦s ❝♦♠♣✐❧❛❞♦r❡s ❈✱ ❛ ❜✐❜❧✐♦t❡❝❛ ♣❛❞rã♦ ❡stá ❝♦♥t✐❞❛ ❡♠ ✉♠ ❛rq✉✐✈♦
st❞✐♦✳❤
❆ ❜✐❜❧✐♦t❡❝❛ st❞✐♦ é ✉♠ ❝❛❜❡ç❛❧❤♦ ❞❛ ❜✐❜❧✐♦t❡❝❛ ♣❛❞rã♦ ❞♦ ❈ ❙❡✉ ♥♦♠❡ ✈❡♠ ❞❛ ❡①♣r❡ssã♦ ✐♥❣❧❡s❛ st❛♥❞❛r❞ ✐♥♣✉t✲♦✉t♣✉t ❤❡❛❞❡r✱ q✉❡ s✐❣♥✐✜❝❛ ❝❛❜❡ç❛❧❤♦ ♣❛❞rã♦ ❞❡ ❡♥tr❛❞❛✴s❛í❞❛ P♦ss✉✐ ❞❡✜♥✐çõ❡s ❞❡ s✉❜r♦t✐♥❛s r❡❢❡r❡♥t❡s às ♦♣❡r❛çõ❡s ❞❡ ❡♥tr❛❞❛✴s❛í❞❛
❝♦♠♦ ❧❡✐t✉r❛ ❞❡ ❞❛❞♦s ❞✐❣✐t❛❞♦s ♥♦ t❡❝❧❛❞♦
❡①✐❜✐çã♦ ❞❡ ✐♥❢♦r♠❛çõ❡s ♥❛ t❡❧❛ ❞♦ ♣r♦❣r❛♠❛ ❞❡ ❝♦♠♣✉t❛❞♦r
st❞❧✐❜✳❤
❙t❞❧✐❜✳❤ é ✉♠ ❛rq✉✐✈♦ ❝❛❜❡ç❛❧❤♦ ❞❛ ❜✐❜❧✐♦t❡❝❛ ❞❡ ♣r♦♣ós✐t♦ ❣❡r❛❧ ♣❛❞rã♦ ❞❛ ❧✐❣✉❛❣❡♠ ❞❡ ♣r♦❣r❛♠❛çã♦ ❈
❖ ♥♦♠❡ st❞❧✐❜ ✈❡♠ ❞❡ st❛♥❞❛r❞ ❧✐❜r❛r② P♦ss✉✐ ❢✉♥çõ❡s q✉❡ sã♦ ❝❧❛ss✐✜❝❛❞❛s ❡♠✿
❈♦♥✈❡rsã♦ ❞❡ t✐♣♦
●❡r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s ❛❧❡❛tór✐❛s ❆❧♦❝❛çã♦ ❡ ❧✐❜❡r❛çã♦ ❞❡ ♠❡♠ór✐❛ ❈♦♥tr♦❧❡ ❞❡ ♣r♦❝❡ss♦s
st❞❧✐❜✳❤
❞♦✉❜❧❡ ❛t♦❢✭❝♦♥st ❝❤❛r ✯str✮❀
❈♦♥✈❡rt❡ ✉♠❛ str✐♥❣ ❡♠ ✉♠ ✈❛❧♦r ✢♦❛t
❙tr✐♥❣ ❞❡✈❡ ❝♦♥t❡r ✉♠ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡ ✈á❧✐❞♦ ❙❡ ❛ ❢✉♥çã♦ ❛t♦❢✭✮ ❢♦r ❝❤❛♠❛❞❛ ❝♦♠ ✶✵✵✳✵✵❍❡❧❧♦✱ ♦ ✈❛❧♦r ✶✵✵✳✵✵ s❡rá ❞❡✈♦❧✈✐❞♦
✐♥t ❛t♦✐✭❝♦♥st ❝❤❛r ✯str✮❀
❈♦♥✈❡rt❡ ✉♠❛ str✐♥❣ ❡♠ ✉♠ ✈❛❧♦r ✐♥t❡✐r♦ ❆ str✐♥❣ ❞❡✈❡ ❝♦♥t❡r ✉♠ ✐♥t❡✐r♦ ✈á❧✐❞♦
st❞❧✐❜✳❤
✐♥t ❛t♦❧✭❝♦♥st ❝❤❛r ✯str✮❀
❈♦♥✈❡rt❡ ✉♠❛ str✐♥❣ ❡♠ ✉♠ ✈❛❧♦r ❧♦♥❣ ✐♥t ❆ str✐♥❣ ❞❡✈❡ ❝♦♥t❡r ✉♠ ✐♥t❡✐r♦ ❧♦♥❣♦ ✈á❧✐❞♦
s❡ ❛ ❢✉♥çã♦ ❛t♦❧✭✮ ❢♦r ❝❤❛♠❛❞❛ ❝♦♠ ✶✷✸✳✹✺✱ ♦ ✈❛❧♦r ✐♥t❡✐r♦ ✶✷✸ é ❞❡✈♦❧✈✐❞♦ ❡ ✵✳✹✺ é ✐❣♥♦r❛❞♦
✐♥t ❛❜s✭✐♥t ♥✉♠✮❀
st❞❧✐❜✳❤
✐♥t r❛♥❞✭✈♦✐❞✮❀
●❡r❛ ✉♠ ♥ú♠❡r♦ ❛❧❡❛tór✐♦
❈❛❞❛ ✈❡③ q✉❡ ❛ ❢✉♥çã♦ r❛♥❞✭✮ é ❝❤❛♠❛❞❛✱ é ❞❡✈♦❧✈✐❞♦ ✉♠ ♥ú♠❡r♦ ❡♥tr❡ ③❡r♦ ❡ ❘❆◆❉❴▼❆❳
✈♦✐❞ sr❛♥❞✭✉♥s✐❣♥❡❞ ✐♥t s❡❡❞✮❀
❊st❛❜❡❧❡❝❡ ✉♠ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ❛ s❡q✉ê♥❝✐❛ ❣❡r❛❞❛ ♣♦r r❛♥❞✭✮
st❞❧✐❜✳❤
❊①❡r❝í❝✐♦ ✵✶ ✲ ❙♦rt❡✐♦ ❞❛ ❢❛❝❡ ❞♦ ❞❛❞♦
st❞❧✐❜✳❤
❊①❡r❝í❝✐♦ ✵✶
✶ ★✐♥❝❧✉❞❡❁st❞✐♦✳❤❃ ✷ ★✐♥❝❧✉❞❡❁st❞❧✐❜✳❤❃ ✸ ★✐♥❝❧✉❞❡❁t✐♠❡✳❤❃
✹ ✐♥t ♠❛✐♥✭✐♥t ❛r❣❝✱ ❝❤❛r✯ ❛r❣✈❬ ❪✮④ ✺ ✐♥t ❢❛❝❡s✱ s♦r❀
✻ sr❛♥❞✭t✐♠❡✭◆❯▲▲✮✮❀ ✼ ✐❢ ✭✭❛r❣❝ ❂❂ ✷✮✮ ④
✽ ❢❛❝❡s ❂ ❛❜s✭❛t♦✐✭❛r❣✈❬✶❪✮✮❀ ✾ ✐❢ ✭❢❛❝❡s ❃ ✾✮ ④
✶✵ ❢❛❝❡s ❂ ✶✵❀
✶✶ ⑥
✶✷ s♦r ❂ r❛♥❞✭✮ ✪ ❢❛❝❡s❀
✶✸ ♣r✐♥t❢✭✧❙♦rt❡❛❞♦ ❛ ❢❛❝❡✿ ✪❞❭♥✧✱ s♦r✮❀
✶✹ ⑥ ❡❧s❡ ④
✶✺ ♣r✐♥t❢✭✧❙✐♥t❛①❡✿ s♦rt❡✐♦ ▲■▼■❚❊❭♥✧✮❀
✶✻ ⑥
st❞❧✐❜✳❤
❞✐✈❴t ❞✐✈✭✐♥t ♥✉♠✱ ✐♥t ❞❡♥✮❀
❉❡✈♦❧✈❡ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ♦♣❡r❛çã♦ ♥✉♠✴❞❡♥ ❡♠ ✉♠❛ ❡str✉t✉r❛ ❞♦ t✐♣♦ ❞✐✈❴t
❖ t✐♣♦ ❞❡ ❡str✉t✉r❛ ❞✐✈❴t t❡♠ ❞♦✐s ❝❛♠♣♦s✿ ✐♥t q✉♦t✱ r❡♠❀
✐♥t s②st❡♠✭❝♦♥st ❝❤❛r ✯❝♦♠♠❛♥❞✮❀
P❛ss❛ ✉♠❛ str✐♥❣ ❝♦♠♦ ✉♠ ❝♦♠❛♥❞♦ ♣❛r❛ ♦ ♣r♦❝❡ss❛❞♦r ❞❡ ❝♦♠❛♥❞♦s ❞♦ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧
st❞❧✐❜✳❤
✈♦✐❞ ✯❝❛❧❧♦❝✭s✐③❡❴t ♥✉♠✱s✐③❡❴t s✐③❡✮❀
❆❧♦❝❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠❡♠ór✐❛ ✐❣✉❛❧ ❛ ♥✉♠✯s✐③❡ ❆❧♦❝❛ ♠❡♠ór✐❛ s✉✜❝✐❡♥t❡ ♣❛r❛ ✉♠ ✈❡t♦r ❞❡ ♥✉♠ ♦❜❥❡t♦s ❞❡ t❛♠❛♥❤♦ s✐③❡
✈♦✐❞ ❢r❡❡✭✈♦✐❞ ✯♣tr✮❀
▲✐❜❡r❛ ❛ ♠❡♠ór✐❛ ❛❧♦❝❛❞❛✱ t♦r♥❛♥❞♦ ❛ ♠❡♠ór✐❛ ❞✐s♣♦♥í✈❡❧ ♣❛r❛ ❛❧♦❝❛çã♦ ❢✉t✉r❛
✈♦✐❞ ✯♠❛❧❧♦❝✭s✐③❡❴t s✐③❡✮❀
♠❛t❤✳❤
❞♦✉❜❧❡ ❝♦s✭❞♦✉❜❧❡ ♥✉♠✮❀
❘❡t♦r♥❛ ♦ ❝♦s❡♥♦ ❞❡ ♥✉♠ q✉❡ ❞❡✈❡ ❡st❛r ❡♠ r❛❞✐❛♥♦s ✭r❛❞ ❂ ✭P■✴✶✽✵✮✯❣r❛✉✮
❞♦✉❜❧❡ ❧♦❣✭❞♦✉❜❧❡ ♥✉♠✮❀
❘❡t♦r♥❛ ♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ ❞❡ ✉♠ ♥ú♠❡r♦
❞♦✉❜❧❡ ♣♦✇✭❞♦✉❜❧❡ ❜❛s❡✱❞♦✉❜❧❡ ❡①♣✮❀
♠❛t❤✳❤
❞♦✉❜❧❡ s✐♥✭❞♦✉❜❧❡ ♥✉♠✮❀
❘❡t♦r♥❛ ♦ s❡♥♦ ❞❡ ♥✉♠ q✉❡ ❞❡✈❡ ❡st❛r ❡♠ r❛❞✐❛♥♦s ✭r❛❞ ❂ ✭P■✴✶✽✵✮✯❣r❛✉✮
❞♦✉❜❧❡ sqrt✭❞♦✉❜❧❡ ♥✉♠✮❀
❘❡t♦r♥❛ ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ ✉♠ ♥ú♠❡r♦
❞♦✉❜❧❡ t❛♥✭❞♦✉❜❧❡ ♥✉♠✮❀
♠❛t❤✳❤
❊①❡r❝í❝✐♦ ✵✷ ✲ ❊q✳ ✷♦ ●r❛✉
❋❛ç❛ ✉♠ ♣r♦❣r❛♠❛ q✉❡ r❡❝❡❜❛ ❝♦♠♦ ♣❛râ♠❡tr♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❛✱ ❜ ❡ ❝ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ s❡❣✉♥❞♦ ❣r❛✉✳ ❈❛❧❝✉❧❡ ❡ ✐♠♣r✐♠❡ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦✱ s❡❣✉♥❞♦ ❛ ❢ór♠✉❧❛ ❛❜❛✐①♦✳
❳✶,✷ =
−❜±√❜✷−✹∗❛∗❝
♠❛t❤✳❤
❊①❡r❝í❝✐♦ ✵✷
✶ ★✐♥❝❧✉❞❡❁st❞✐♦✳❤❃ ✷ ★✐♥❝❧✉❞❡❁♠❛t❤✳❤❃
✸ ✐♥t ♠❛✐♥ ✭✐♥t ❛r❣❝✱ ❝❤❛r✯ ❛r❣✈❬ ❪✮④ ✹ ❢❧♦❛t ❛✱❜✱❝✱❞❡❧t❛✱①✶✱①✷❀ ✺ ✐❢ ✭❛r❣❝❂❂✹✮④
✻ ❛ ❂ ❛t♦✐✭❛r❣✈❬✶❪✮❀ ✼ ❜ ❂ ❛t♦✐✭❛r❣✈❬✷❪✮❀ ✽ ❝ ❂ ❛t♦✐✭❛r❣✈❬✸❪✮❀ ✾ ♣r✐♥t❢✭✧✪❢①✷✰✪❢①✰✪❢✳❭♥✧✱
✶✵ ❛✱ ❜✱ ❝✮❀
♠❛t❤✳❤
❊①❡r❝í❝✐♦ ✵✷ ✲ ❝♦♥t✳ ✶✷ ✐❢ ✭❞❡❧t❛❃❂✵✮④
✶✸ ①✶ ❂ ✭✲❜✰sqrt✭❞❡❧t❛✮✮✴✭✷✯❛✮❀ ✶✹ ①✷ ❂ ✭✲❜✲sqrt✭❞❡❧t❛✮✮✴✭✷✯❛✮❀
✶✺ ♣r✐♥t❢✭✧❘❛✐③❡s sã♦ ✭✪❢✮ ❡ ✭✪❢✮ ❭♥✧✱①✶✱①✷✮❀
✶✻ ⑥❡❧s❡④
✶✼ ❞❡❧t❛ ❂ sqrt✭✲❞❡❧t❛✮✴✭✷✯❛✮✮❀ ✶✽ ①✶ ❂ ✲❜✴✭✷✯❛✮❀
✶✾ ①✷ ❂ ✲❜✴✭✷✯❛✮❀
✷✵ ♣r✐♥t❢✭✧❘❛✐③❡s sã♦ ✭✪❢ ✰ ✪❢✐✮ ❡ ✭✪❢ ✲ ✪❢✐✮✳❭♥✧✱
✷✶ ①✶✱ ❞❡❧t❛✱ ①✷✱ ❞❡❧t❛✮❀
✷✷ ⑥
✷✸ ⑥ ❡❧s❡④
✷✹ ♣r✐♥t❢✭✧■♥s✐r❛ ♦s ❝♦❡❢✐❝✐❡♥t❡s ♥❛ ❢♦r♠❛✿ ❛ ❜ ❝ ❭♥✧✮❀
✷✺ ⑥
t✐♠❡✳❤
✐♥t ❝❧♦❝❦✭✈♦✐❞✮❀
❘❡t♦r♥❛ ✉♠ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ❞♦ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❞♦ ♣r♦❣r❛♠❛ q✉❡ ❛ ❝❤❛♠❛
❞♦✉❜❧❡ ❞✐✛t✐♠❡✭❞♦✉❜❧❡ t✶✱❞♦✉❜❧❡ t✷✮❀
❘❡t♦r♥❛ ❛ ❞✐❢❡r❡♥ç❛ ❡♠ s❡❣✉♥❞♦s✱ ❡♥tr❡ t✶ ❡ t✷
❞♦✉❜❧❡ t✐♠❡✭❞♦✉❜❧❡ ✯t✐♠❡✮❀
❘❡t♦r♥❛ ♦ ❤♦rár✐♦ ❛t✉❛❧ ❞♦ ❝❛❧❡♥❞ár✐♦ ❞♦ s✐st❡♠❛
t✐♠❡✳❤
❊①❡r❝í❝✐♦ ✵✸ ✲ ❈♦♥t❛❣❡♠ ❞❡ ❚❡♠♣♦
t✐♠❡✳❤
❊①❡r❝í❝✐♦ ✵✸
✶ ★✐♥❝❧✉❞❡❁st❞✐♦✳❤❃ ✷ ★✐♥❝❧✉❞❡❁t✐♠❡✳❤❃
✸ ✐♥t ♠❛✐♥✭✐♥t ❛r❣❝✱ ❝❤❛r✯ ❛r❣✈❬ ❪✮④ ✹ ❞♦✉❜❧❡ ✐♥✐❝✐♦✱ ❢✐♠✱ ❞✐❢❀ ✺ ✐♥t ❛tr❛s♦❀
✻ ✐♥✐❝✐♦ ❂ ❝❧♦❝❦✭✮❀ ✼ ❢✐♠ ❂ ❝❧♦❝❦✭✮❀ ✽ ✐❢ ✭❛r❣❝ ❂❂ ✷✮④
✾ ❛tr❛s♦ ❂ ❛❜s✭❛t♦✐✭❛r❣✈❬✶❪✮✮❀ ✶✵ ✐❢✭❛tr❛s♦ ❃ ✻✵✮④
✶✶ ❛tr❛s♦ ❂ ✻✵❀
✶✷ ⑥
t✐♠❡✳❤
❊①❡r❝í❝✐♦ ✵✸ ✲ ❝♦♥t✳
✶✹ ❞✐❢ ❂ ✭❢✐♠ ✲ ✐♥✐❝✐♦✮ ✴ ❈▲❖❈❑❙❴P❊❘❴❙❊❈❀ ✶✺ ✇❤✐❧❡✭❞✐❢ ❁ ❛tr❛s♦✮④
✶✻ ❢✐♠ ❂ ❝❧♦❝❦✭✮❀
✶✼ ❞✐❢ ❂ ✭❢✐♠ ✲ ✐♥✐❝✐♦✮ ✴ ❈▲❖❈❑❙❴P❊❘❴❙❊❈❀
✶✽ ⑥
✶✾ ⑥❡❧s❡④
✷✵ ♣r✐♥t❢✭✧s✐♥t❛①❡✿ ❛tr❛s♦ ❚❊▼P❖❙❊●❯◆❉❖❙ ❭♥✧✮❀
✷✶ ⑥
❆ ❜✐❜❧✐♦t❡❝❛<str✐♥❣✳❤> ❢♦r♥❡❝❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s ♣❛r❛
♠❛♥✐♣✉❧❛çã♦ ❞❡ str✐♥❣s
❊st❛s ❢✉♥çõ❡s sã♦ út❡✐s✱ ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ ❡♠ ❈ t♦❞❛ str✐♥❣ é ✉♠ ✈❡t♦r ❞❡ ❝❛r❛❝t❡r❡s
♣♦r s❡r ✉♠ ✈❡t♦r ❞❡ ❝❛r❛❝t❡r❡s ❛s str✐♥❣s ❤❡r❞❛♠ ✉♠❛ sér✐❡ ❞❡ ❧✐♠✐t❛çõ❡s ♣❡rt✐♥❡♥t❡s ❛♦ ✈❡t♦r❡s
✐st♦ s❡ ❞❡✈❡ ❛♦ ❢❛t♦ ❞❡ ❛ ❧✐♥❣✉❛❣❡♠ ❈ ♥ã♦ s✉♣♦rt❛r ♦♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s ✉t✐❧✐③❛♥❞♦ ❞✐r❡t❛♠❡♥t❡ ♦s ♦♣❡r❛❞♦r❡s ❛r✐t♠ét✐❝♦s ❡ r❡❧❛❝✐♦♥❛✐s
❚♦❞♦ ♣r♦❣r❛♠❛ q✉❡ ♠❛♥✐♣✉❧❛r str✐♥❣s ❞❡✈❡ t♦♠❛r ❝✉✐❞❛❞♦ ❝♦♠ ❛s s❡❣✉✐♥t❡s ❧✐♠✐t❛çõ❡s✿
✶ str✐♥❣s ♥ã♦ ♣♦❞❡♠ s❡r ❛tr✐❜✉í❞❛s ❝♦♠ ♦ ♦♣❡r❛❞♦r ❞❡ ❛tr✐❜✉✐çã♦ ✭❂✮✱ ❡♠❜♦r❛ ♣♦ss❛♠ s❡r ✐♥✐❝✐❛❧✐③❛❞❛s
❱❡t♦r❡s ❡ ♣♦♥t❡✐r♦s sã♦ ♠✉✐t♦ ✉s❛❞♦s ❡♠ ❈ ❡ ❡stã♦ ❞❡ ❝❡rt♦ ♠♦❞♦ r❡❧❛❝✐♦♥❛❞♦s às str✐♥❣s
❈♦♠♦ ♦ ♥♦♠❡ ❞❡ ✉♠ ✈❡t♦r é ♥❛ ✈❡r❞❛❞❡ ✉♠ ♣♦♥t❡✐r♦✱ ❡①✐st❡ ✉♠❛ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❞❡✜♥✐r ✉♠❛ ✈❛r✐á✈❡❧ str✐♥❣s✱ ♦✉ s❡❥❛✱ ❡❧❛ ♣♦❞❡ s❡r ✉♠ ♣♦♥t❡✐r♦ ♣❛r❛ ♦ t✐♣♦ ❝❤❛r
P♦r ❡①❡♠♣❧♦✿
✶ ❝❤❛r♥♦♠❡❬✷✵❪❀ ✷ ❝❤❛r✯♣r♦✜ss❛♦❀
❉✐❢❡r❡♥ç❛
❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❞❡✜♥✐r ✉♠❛ ✈❛r✐á✈❡❧ str✐♥❣ ❝♦♠♦ ✈❡t♦r ❡ ✉♠❛ ✈❛r✐á✈❡❧ ❝♦♠♦ ♣♦♥t❡✐r♦ ❡stá ♥❛ ♠❡♠ór✐❛
❉❡✜♥✐çã♦ ❝♦♠♦ ✈❡t♦r✿ ❛ ♠❡♠ór✐❛ é ❛✉t♦♠❛t✐❝❛♠❡♥t❡ r❡s❡r✈❛❞❛✱ ❝♦♠♦ ♣❛r❛ q✉❛❧q✉❡r ✈❡t♦r
❉❡✜♥✐çã♦ ❝♦♠♦ ♣♦♥t❡✐r♦✿ ♥ã♦ ❡①✐st❡ ♠❡♠ór✐❛ r❡s❡r✈❛❞❛ ◗✉❛♥❞♦ ✉♠❛ str✐♥❣ é ❡s❝r✐t❛ ❡♥tr❡ ❛s♣❛s✱ ♦ ❝♦♠♣✐❧❛❞♦r ❝r✐❛ ❛ str✐♥❣ s❡❣✉✐❞❛ ♣♦r ✉♠ ♥✉❧❧ ❡♠ ❛❧❣✉♠ ❧✉❣❛r ♥♦ ♣r♦❣r❛♠❛ ♦❜❥❡t♦
❝❤❛r✯♥♦♠❡❂✑❏♦s❡ P❡r❡✐r❛✑❀
❊ss❛ ❢♦r♠❛ ❞❡ ❛tr✐❜✉✐çã♦ s❡ ❛♣❧✐❝❛ ❜❡♠ à ✈❛r✐á✈❡✐s str✐♥❣s ❞❡✜♥✐❞❛s ❝♦♠♦ ♣♦♥t❡✐r♦s ♣❛r❛ ❝❤❛r
str✐♥❣✳❤
✐♥t ✯str❧❡♥✭❝❤❛r ✯str✮❀
❈♦♥t❛ ♦ ♥ú♠❡r♦ ❞❡ ❝❛r❛❝t❡r❡s ❛r♠❛③❡♥❛❞♦s ❡♠ ✉♠❛ str✐♥❣✱ ❛♥t❡s ❞♦ ✬❭✵✬
❘❡t♦r♥❛ ♦ ♥ú♠❡r♦ ❞❡ ❝❛r❛❝t❡r❡s ❞❛ str✐♥❣ str✿ ❛ str✐♥❣ q✉❡ t❡rá s❡✉ t❛♠❛♥❤♦ ❝❛❧❝✉❧❛❞♦
❝❤❛r ✯str❝♣②✭❝❤❛r ✯❞❡st✱❝♦♥st ❝❤❛r ✯♦r✐❣❡♠✮❀
str✐♥❣✳❤
❝❤❛r ✯str♥❝♣②✭❝❤❛r ✯❞❡st✱❝❤❛r ✯♦r✐❣❡♠✱✐♥t ♥✮❀
❈♦♣✐❛ ♥♦ ♠á①✐♠♦ ♥ ❝❛r❛❝t❡r❡s ❞❡ ✉♠❛ str✐♥❣ ♣❛r❛ ❛ ♦✉tr❛✳ ◆ã♦ ❝♦❧♦❝❛ ✬❭✵✬ ♥♦ ✜♥❛❧ ❞❡ ❞❡st✱ ❛ ♥ã♦ s❡r q✉❡ t❡♥❤❛ ❛t✐♥❣✐❞♦ ♦ ✜♥❛❧ ❞❛ str✐♥❣ ♦r✐❣❡♠
❘❡t♦r♥❛ ♦ ❡♥❞❡r❡ç♦ ❞♦ ♣r✐♠❡✐r♦ ❝❛r❛❝t❡r ❞❛ str✐♥❣ ❞❡st ❞❡st✿ str✐♥❣ q✉❡ r❡❝❡❜❡rá ♦ ❝♦♥t❡ú❞♦
str✐♥❣✳❤
❝❤❛r ✯str❝♠♣✭❝❤❛r ✯s✶✱❝❤❛r ✯s✷✮❀
❈♦♠♣❛r❛ ♦ ❝♦♥t❡ú❞♦ ❞❡ ❞✉❛s str✐♥❣s✳ ❊st❛ ❢✉♥çã♦ ❞❡✈❡ s❡r ✉t✐❧✐③❛❞❛ ❡♠ s✉❜st✐t✉✐çã♦ ❛♦s ♦♣❡r❛❞♦r❡s r❡❧❛❝✐♦♥❛✐s ♥♦ ❝❛s♦ ❞❡ ✉s♦ ❝♦♠ str✐♥❣s
❘❡t♦r♥❛
✵ ✭③❡r♦✮ s❡ ♦ ❝♦♥t❡ú❞♦ ❞❛s ❞✉❛s str✐♥❣s sã♦ ✐❣✉❛✐s
✉♠ ✈❛❧♦r ♠❛✐♦r q✉❡ ✵✱ s❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ s✶ ❢♦r ♠❛✐♦r q✉❡ s✷ ✉♠ ✈❛❧♦r ♠❡♥♦r q✉❡ ✵✱ s❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ s✶ é ♠❡♥♦r q✉❡ s✷
▼❛✐♦r ❡ ♠❡♥♦r ♥ã♦ s❡ r❡❢❡r❡ ❛♦ t❛♠❛♥❤♦ ❞❛ str✐♥❣✱ ♠❛s ❛ ♣♦s✐çã♦✱ q✉❛♥❞♦ ♦r❞❡♥❛❞❛s ❞❡ ❢♦r♠❛ ❛s❝❡♥❞❡♥t❡
str✐♥❣✳❤
❝❤❛r ✯str♥❝♠♣✭❝❤❛r ✯s✶✱❝❤❛r ✯s✷✱✐♥t ♥✮❀
❈♦♠♣❛r❛ ❛♣❡♥❛s ✉♠ tr❡❝❤♦ ❞♦ ✐♥í❝✐♦ ❞❡ ❞✉❛s str✐♥❣s ❝♦♠ t❛♠❛♥❤♦ ❡s♣❡❝✐✜❝❛❞♦
❘❡t♦r♥❛
✵ ✭③❡r♦✮ s❡ ♦ ❝♦♥t❡ú❞♦ ❞❛s ❞✉❛s str✐♥❣s sã♦ ✐❣✉❛✐s
✉♠ ✈❛❧♦r ♠❛✐♦r q✉❡ ✵✱ s❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ s✶ ❢♦r ♠❛✐♦r q✉❡ s✷ ✉♠ ✈❛❧♦r ♠❡♥♦r q✉❡ ✵✱ s❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ s✶ é ♠❡♥♦r q✉❡ s✷
▼❛✐♦r ❡ ♠❡♥♦r ♥ã♦ s❡ r❡❢❡r❡ ❛♦ t❛♠❛♥❤♦ ❞❛ str✐♥❣✱ ♠❛s ❛ ♣♦s✐çã♦✱ q✉❛♥❞♦ ♦r❞❡♥❛❞❛s ❞❡ ❢♦r♠❛ ❛s❝❡♥❞❡♥t❡
str✐♥❣✳❤
❊①❡r❝í❝✐♦ ✵✹ ✲ ❈♦♥✈❡rsã♦ ♠❛✐ús❝✉❧♦✕♠✐♥ús❝✉❧♦
str✐♥❣✳❤
❊①❡r❝í❝✐♦ ✵✹
✶ ★✐♥❝❧✉❞❡❁st❞✐♦✳❤❃ ✷ ★✐♥❝❧✉❞❡❁str✐♥❣✳❤❃
✸ ✐♥t ♠❛✐♥✭✐♥t ❛r❣❝✱ ❝❤❛r✯ ❛r❣✈❬ ❪✮ ④ ✹ ✐♥t ✐✱t❛♠✱❝♦♥✈❂✸✷✱✐♥❢❂✻✺✱s✉♣❂✾✵❀ ✺ ❝❤❛r ♣❛❧❬✶✵✵❪❀
✻ ✐❢ ✭✭❛r❣❝ ❂❂ ✸✮✮ ④ ✼ str❝♣②✭♣❛❧✱ ❛r❣✈❬✶❪✮❀ ✽ t❛♠ ❂ str❧❡♥✭♣❛❧✮❀
✾ ✐❢✭ ✭str❝♠♣✭❛r❣✈❬✷❪✱✧✲▼✧✮❂❂✵✮ ⑤⑤ ✶✵ ✭str❝♠♣✭❛r❣✈❬✷❪✱✧✲♠✧✮❂❂✵✮ ✮ ④ ✶✶ ✐❢✭✭str❝♠♣✭❛r❣✈❬✷❪✱✧✲▼✧✮❂❂✵✮④ ✶✷ ✐♥❢❂✾✼❀ s✉♣❂✶✷✷❀ ❝♦♥✈❂✲✸✷❀
✶✸ ⑥❡❧s❡④
✶✹ ✐♥❢❂✻✺❀ s✉♣❂✾✵❀ ❝♦♥✈❂✸✷❀
str✐♥❣✳❤
❊①❡r❝í❝✐♦ ✵✹ ✲ ❝♦♥t✳
✶✻ ❢♦r✭✐❂✵❀ ✐❁t❛♠❀ ✐✰✰✮④
✶✼ ✐❢✭✭♣❛❧❬✐❪❃❂✐♥❢ ✮ ✫✫ ✭♣❛❧❬✐❪❁❂s✉♣✮✮④
✶✽ ♣❛❧❬✐❪ ✰❂ ❝♦♥✈❀
✶✾ ⑥
✷✵ ⑥
✷✶ ♣r✐♥t❢✭✧P❛❧❛✈r❛✿ ✪s❭♥✧✱ ♣❛❧✮❀
✷✷ ⑥ ❡❧s❡ ④
✷✸ ♣r✐♥t❢✭✧❙✐♥t❛①❡✿ ✳ ✳ ✳✧✮❀
✷✹ ⑥
✷✺ ⑥ ❡❧s❡ ④
✷✻ ♣r✐♥t❢✭✧❙✐♥t❛①❡✿ ✳ ✳ ✳✧✮❀
✷✼ ⑥
str✐♥❣✳❤
❊①❡r❝í❝✐♦ ✵✺ ✕ ●❡r❛❞♦r ❞❡ s❡♥❤❛
str✐♥❣✳❤ ❊①❡r❝í❝✐♦ ✵✺ ✶ ★✐♥❝❧✉❞❡❁st❞✐♦✳❤❃ ✷ ★✐♥❝❧✉❞❡❁st❞❧✐❜✳❤❃ ✸ ★✐♥❝❧✉❞❡❁str✐♥❣✳❤❃ ✹ ★✐♥❝❧✉❞❡❁t✐♠❡✳❤❃
✺ ✐♥t ♠❛✐♥✭✐♥t ❛r❣❝✱ ❝❤❛r✯ ❛r❣✈❬ ❪✮④ ✻ ✐♥t ✐✱ ❞✐❣✐t ❂ ✻✱ ❧❡tr❛ ❂ ✾✼❀ ✼ ❝❤❛r s❡♥❤❛❬✶✵✵❪❀
✽ sr❛♥❞✭t✐♠❡✭◆❯▲▲✮✮❀ ✾ ✐❢ ✭❛r❣❝ ❂❂ ✸✮④
✶✵ ❞✐❣✐t ❂ ❛❜s✭❛t♦✐✭❛r❣✈❬✶❪✮✮❀ ✶✶ ✐❢ ✭❞✐❣✐t ❃❂ ✶✵✵✮④
✶✷ ❞✐❣✐t ❂ ✾✾❀
✶✸ ⑥
str✐♥❣✳❤
❊①❡r❝í❝✐♦ ✵✺ ✲ ❝♦♥t✳
✶✻ ✐❢ ✭str❝♠♣✭❛r❣✈❬✷❪✱✲▼✧✮❂❂✵✮④
✶✼ ❧❡tr❛ ❂ ✻✺❀
✶✽ ⑥ ❡❧s❡ ④
✶✾ ❧❡tr❛ ❂ ✾✼❀
✷✵ ⑥
✷✶ ❢♦r ✭✐ ❂ ✵❀ ✐❁❞✐❣✐t❀ ✐✰✰✮ ④ ✷✷ s❡♥❤❛❬✐❪❂❧❡tr❛✰r❛♥❞✭✮✪✷✻❀
✷✸ ⑥
✷✹ s❡♥❤❛❬✐❪ ❂ ✬❭✵✬❀
✷✺ ♣r✐♥t❢✭✧❙❡♥❤❛✿✪s❭♥✧✱s❡♥❤❛✮❀
✷✻ ⑥❡❧s❡④
✷✼ ♣r✐♥t❢✭✧s✐♥t❛①❡✿ ✳ ✳ ✳ ✧⑥❀
✷✽ ⑥
✷✾ ⑥❡❧s❡④
✸✵ ♣r✐♥t❢✭✧s✐♥t❛①❡✿ ✳ ✳ ✳ ✧⑥❀
✸✶ ⑥
str✐♥❣✳❤
❝❤❛r ✯str❝❛t✭❝❤❛r ✯❞❡st✱❝❤❛r ✯♦r✐❣❡♠✮❀
❈♦♥❝❛t❡♥❛ ♦ ❝♦♥t❡ú❞♦ ❞❛ str✐♥❣ ❞❡ ♦r✐❣❡♠ ❛♦ ✜♥❛❧ ❞❛ str✐♥❣ ❞❡st✱ ♣r❡s❡r✈❛♥❞♦ ♦ ✈❛❧♦r ❞❡ ❞❡st
❘❡t♦r♥❛ ♦ ❡♥❞❡r❡ç♦ ❞♦ ♣r✐♠❡✐r♦ ❝❛r❛❝t❡r ❞❛ str✐♥❣ ❞❡st ❞❡st✿ str✐♥❣ q✉❡ ✐rá r❡❝❡❜❡r✱ ♥♦ s❡✉ ✜♥❛❧✱ ♦ ❝♦♥t❡ú❞♦✳ ❉❡✈❡ t❡r t❛♠❛♥❤♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛r♠❛③❡♥❛r ♦ ❝♦♥t❡ú❞♦ ❛t✉❛❧ ♠❛✐s ♦ ♥♦✈♦
str✐♥❣✳❤
❝❤❛r ✯str♥❝❛t✭❝❤❛r ✯❞❡st✱❝❤❛r ✯♦r✐❣❡♠✱✐♥t ♥✮❀
❈♦♥❝❛t❡♥❛ ♥ã♦ ♠❛✐s q✉❡ ♥ ❝❛r❛❝t❡r❡s ❞❛ str✐♥❣
❘❡t♦r♥❛ ♦ ❡♥❞❡r❡ç♦ ❞♦ ♣r✐♠❡✐r♦ ❝❛r❛❝t❡r ❞❛ str✐♥❣ ❞❡st ❞❡st✿ str✐♥❣ q✉❡ ✐rá r❡❝❡❜❡r✱ ♥♦ s❡✉ ✜♥❛❧✱ ♦ ❝♦♥t❡ú❞♦✳ ❉❡✈❡ t❡r t❛♠❛♥❤♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛r♠❛③❡♥❛r ♦ ❝♦♥t❡ú❞♦ ❛t✉❛❧ ♠❛✐s ♦ ♥♦✈♦
str✐♥❣✳❤
❝❤❛r ✯strstr✭❝❤❛r ✯s✶✱❝❤❛r ✯s✷✮❀
❈♦♠♣❛r❛ s❡ ✉♠❛ str✐♥❣ ❡stá ❝♦♥t✐❞❛ ❞❡♥tr♦ ❞❡ ♦✉tr❛ str✐♥❣ ❘❡t♦r♥❛ ♦ ♣♦♥t❡✐r♦ ♣❛r❛ ❛ ♣r✐♠❡✐r❛ ♦❝♦rrê♥❝✐❛ ❞❡ s✷ ❡♠ s✶✱ ♦✉ ♥✉❧❧ s❡ ♥ã♦ ❡♥❝♦♥tr♦✉
s✶✿ str✐♥❣ ❛ s❡r ❝♦♠♣❛r❛❞❛
str✐♥❣✳❤
❝❤❛r ✯str❝❤r✭❝❤❛r ✯str✱✐♥t ♥✮❀
▲♦❝❛❧✐③❛ ❛ ♣r✐♠❡✐r❛ ♦❝♦rrê♥❝✐❛ ❞❡ ♥ ✭❝♦♥✈❡rt✐❞❛ ♣❛r❛ ❝❤❛r✮ ♥❛ str✐♥❣ str✳
❘❡t♦r♥❛ ♦ ♣♦♥t❡✐r♦ ♣❛r❛ ❛ ♣r✐♠❡✐r❛ ♦❝♦rrê♥❝✐❛ ❞♦ ❝❛r❛❝t❡r ♥ ❡♠ str
str✿ str✐♥❣
❯♠❛ ❢✉♥çã♦ ♠✉✐t♦ út✐❧ ♣❛r❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ str✐♥❣ ❡stá ♥❛ ❜✐❜❧✐♦t❡❝❛ st❞✐♦✳❤
✐♥t s♣r✐♥t❢✭❝❤❛r ✯str✱❝❤❛r ✯❢♦r♠❛t✱. . .✮❀
➱ ✐❞ê♥t✐❝❛ à ❢✉♥çã♦ ♣r✐♥t❢✭✮ ❡①❝❡t♦ q✉❡ ❛ s❛í❞❛ é ❝♦❧♦❝❛❞❛ ♥❛ ✈❛r✐á✈❡❧ str
❊①❡♠♣❧♦ ✉s❛♥❞♦ s♣r✐♥t❢✭✮ ✶ ★✐♥❝❧✉❞❡ ❁st❞✐♦✳❤❃
✷ ✐♥t ♠❛✐♥ ✭✐♥t ❛r❣❝✱ ❝❤❛r✯ ❛r❣✈❬ ❪✮④ ✸ ❝❤❛r ♥♦♠❡❬✸✵❪❀
✹ ❝❤❛r ♠s❣❬✶✵✵❪❀
✺ ♣r✐♥t❢✭✧❉✐❣✐t❡ s❡✉ ♥♦♠❡✿ ✧✮❀ ✻ s❝❛♥❢✭✧✪s✧✱ ♥♦♠❡✮❀
✼ s♣r✐♥t❢✭♠s❣✱ ✧✪s s❡❥❛ ❜❡♠ ✈✐♥❞♦✭❛✮ ❛ ❯♥✐❢❡✐✦✧✱ ♥♦♠❡✮❀ ✽ ♣r✐♥t❢✭✧✪s ❭♥✧✱ ♠s❣✮❀
❝❤❛r ✯❣❡ts✭❝❤❛r ✯str✮❀
❖s ❝❛r❛❝t❡r❡s sã♦ ❧✐❞♦s ❛té q✉❡ s❡❥❛ r❡❝❡❜✐❞♦ ✉♠ ❝❛r❛❝t❡r❡ ❞❡ ♥♦✈❛ ❧✐♥❤❛
❊①❡♠♣❧♦ ✉s❛♥❞♦ ❣❡ts✭✮ ✶ ★✐♥❝❧✉❞❡ ❁st❞✐♦✳❤❃
✷ ✐♥t ♠❛✐♥✭✐♥t ❛r❣❝✱ ❝❤❛r✯ ❛r❣✈❬ ❪✮④ ✸ ❝❤❛r str❬✷✺✻❪❀
✹ ♣r✐♥t❢✭✧❉✐❣✐t❡ ♦ ❡♥❞❡r❡❝♦ ❝♦♠♣❧❡t♦✿ ✧✮❀ ✺ ❣❡ts✭str✮❀
✻ ♣r✐♥t❢✭✧❊♥❞❡r❡❝♦ ❂ ✪s ❭♥✧✱str✮❀ ✼ r❡t✉r♥ ✵❀
