Bài giảng đại số, giải tích và ứng dụng chương 4 nguyễn thị nhung (đh thăng long)

●❤✐ ❝❤ó

✣↕✐ sè✱ ❣✐↔✐ t➼❝❤ ✈➔ ù♥❣ ❞ö♥❣
◆❣✉②➵♥ ❚❤à ◆❤✉♥❣
❇ë ♠æ♥ ❚♦→♥ ✲ ✣↕✐ ❤å❝ ❚❤➠♥❣ ▲♦♥❣

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✶ ✴ ✹✽

●❤✐ ❝❤ó

❈❤÷ì♥❣ ■❱
P❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷ ✴ ✹✽

●❤✐ ❝❤ó
◆❣❤✐➺♠ ❝➙♥ ❜➡♥❣ tr♦♥❣ ♠æ ❤➻♥❤ t♦→♥ ❦✐♥❤ t➳

▼æ ❤➻♥❤ t♦→♥ ❦✐♥❤ t➳

◆❤ú♥❣ ❦❤→✐ ♥✐➺♠ tr♦♥❣ ♠æ ❤➻♥❤ t♦→♥ ❦✐♥❤ t➳
▼æ ❤➻♥❤ t♦→♥ ❦✐♥❤ t➳ ❧➔ ♠æ ❤➻♥❤ ❦✐♥❤ t➳ ✤÷ñ❝ tr➻♥❤ ❜➔② ❜➡♥❣ ♥❣æ♥
♥❣ú t♦→♥ ❤å❝✳

❈æ♥❣ t❤ù❝ ❝õ❛ ♠æ ❤➻♥❤ ❚♦→♥ ❦✐♥❤ t➳✿ ❣ç♠ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ✤à♥❤ ❧÷ñ♥❣

✈➔ ❝→❝ ❤➺ t❤ù❝ t♦→♥ ❤å❝ ✭♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✮ ❧✐➯♥ ❤➺
❣✐ú❛ ❝❤ó♥❣✳

❱➼ ❞ö✿

◗❞ ✏ ❛ ✁ ❜P   ❘ ♣❛, ❜ → ✵q,
❄
◗s ✏ ✁❝   ❞P   ❘ ♣❝ , ❞ → ✵q,

✭✶✮
✭✷✮

✈î✐ ❘ ❧➔ ❧÷ñ♥❣ ♠÷❛✳
◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸ ✴ ✹✽

ú
tr ổ t t

ổ t t

ỳ tr ổ t t
ởt ữủ ở ợ õ t t ờ ởt ữủ

õ t tr
ở s tr ừ ú t ữủ tứ ổ
s tr ừ ú ữủ tỗ t
ổ ở ợ ữủ ỳ ữủ trữợ
số ữủ ổ t ờ õ ố ợ
số ữủ ữủ sỷ số trữợ
ữ ổ ữủ ởt tr ử t õ t t ởt
tr
t s ở ở s ử
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P
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st

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t

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t


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ỳ tr ổ t
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tr ổ t ữủ
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t t
s t s s tr t ụ ợ ợ
tr t t ồ s s t
ữỡ tr ổ t t ữủ
q tở t tr t
s ữủ ữủ q,

tt tữq,
ữủt ừ ổ t trữớ ừ t
qố tr trữớ ủ ỡ t
tr t tr ổ t t tr ừ
ở s tọ ừ ổ



ố t

t




●❤✐ ❝❤ó
◆❣❤✐➺♠ ❝➙♥ ❜➡♥❣ tr♦♥❣ ♠æ ❤➻♥❤ t♦→♥ ❦✐♥❤ t➳

❈➙♥ ❜➡♥❣

❚r↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ tr♦♥❣ ♠æ ❤➻♥❤ t❤à tr÷í♥❣ ✤ì♥
❱➼ ❞ö ✶✿ ❈❤♦ ♠æ ❤➻♥❤ t❤à tr÷í♥❣ ✤ì♥ s❛✉✿
◗❞ ✏ ❛ ✁ ❜P ♣❛, ❜ → ✵q,

◗s ✏ ✁❝   ❞P ♣❝ , ❞ → ✵q,
tr♦♥❣ ✤â ◗❞ , ◗s , P ❧➔ ❧÷ñ♥❣ ❝➛✉✱ ❧÷ñ♥❣ ❝✉♥❣ ✈➔ ❣✐→ ❜→♥ ❝õ❛ ❤➔♥❣ ❤â❛✱
❛, ❜, ❝ , ❞ ❧➔ ❝→❝ t❤❛♠ sè✳
❛✳ ❳→❝ ✤à♥❤ ❜✐➳♥ ♥ë✐ s✐♥❤✱ ❜✐➳♥ ♥❣♦↕✐ s✐♥❤ ✈➔ t❤❛♠ sè tr♦♥❣ ♠æ ❤➻♥❤✳
❜✳ ●✐↔✐ t❤➼❝❤ ✤✐➲✉ ❦✐➺♥ ❝→❝ t❤❛♠ sè tr♦♥❣ ♠æ ❤➻♥❤✳
❝✳ ❚➻♠ tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ ❤❛② ♥❣❤✐➺♠ ❝➙♥ ❜➡♥❣ ❝õ❛ ♠æ ❤➻♥❤✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✼ ✴ ✹✽

●❤✐ ❝❤ó
◆❣❤✐➺♠ ❝➙♥ ❜➡♥❣ tr♦♥❣ ♠æ ❤➻♥❤ t♦→♥ ❦✐♥❤ t➳

❈➙♥ ❜➡♥❣

❚r↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ tr♦♥❣ ♠æ ❤➻♥❤ t❤à tr÷í♥❣ ❤❛✐ ❤➔♥❣ ❤â❛
❱➼ ❞ö ✷✿ ❚➻♠ ♥❣❤✐➺♠ ❝➙♥ ❜➡♥❣ ❝õ❛ ♠æ ❤➻♥❤ t❤à tr÷í♥❣ ❤❛✐ ❤➔♥❣ ❤â❛ ❜✐➳t
❤➔♠ ❝✉♥❣ ✈➔ ❤➔♠ ❝➛✉ ✤÷ñ❝ ❝❤♦ ❜ð✐✿

◗❞✶ ✏ ✶✵ ✁ ✷P✶   P✷
◗s✶ ✏ ✁✷   ✸P✶
◗❞✷ ✏ ✶✺   P✶ ✁ P✷
◗s✷ ✏ ✁✶   ✷P✷

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✽ ✴ ✹✽

●❤✐ ❝❤ó
◆❣❤✐➺♠ ❝➙♥ ❜➡♥❣ tr♦♥❣ ♠æ ❤➻♥❤ t♦→♥ ❦✐♥❤ t➳

❈➙♥ ❜➡♥❣

❚r↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ tr♦♥❣ ♠æ ❤➻♥❤ t❤✉ ♥❤➟♣ q✉è❝ ❞➙♥
❱➼ ❞ö ✸✿ ❚➻♠ ♥❣❤✐➺♠ ❝➙♥ ❜➡♥❣ ❝õ❛ ♠æ ❤➻♥❤ t❤✉ ♥❤➟♣ q✉è❝ ❞➙♥ s❛✉✿
❨ ✏ ❈   ■✵   ●✵
❈ ✏ α   β ♣❨ ✁ ❚ q ♣α → ✵, ✵ ➔ β ➔ ✶q,
❚ ✏ γ   δ ❨ ♣γ → ✵, ✵ ➔ δ ➔ ✶ q
tr♦♥❣ ✤â✱ ❨ ❧➔ t❤✉ ♥❤➟♣ q✉è❝ ❞➙♥✱ ❈ ❧➔ t✐➯✉ ❞ò♥❣ ❝õ❛ ❝→❝ ❤ë ❣✐❛ ✤➻♥❤ ❝❤♦
❤➔♥❣ ❤â❛ ✈➔ ❞à❝❤ ✈ö✱ ❚ ❧➔ t❤✉➳✱ ❨ ✁ ❚ ✤÷ñ❝ ❣å✐ ❧➔ t❤✉ ♥❤➟♣ ❦❤↔ ❞ö♥❣✳ ■✵
❧➔ ❧÷ñ♥❣ ✤➛✉ t÷✱ ●✵ ❧➔ ❝❤✐ t✐➯✉ ❝õ❛ ❝❤➼♥❤ ♣❤õ✳
❛✳ ❳→❝ ✤à♥❤ ❜✐➳♥ ♥ë✐ s✐♥❤✱ ❜✐➳♥ ♥❣♦↕✐ s✐♥❤✱ t❤❛♠ sè tr♦♥❣ ♠æ ❤➻♥❤✳
❜✳ ●✐↔✐ t❤➼❝❤ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❝→❝ t❤❛♠ sè tr♦♥❣ ♠æ ❤➻♥❤✳
❝✳ ❚➻♠ t❤✉ ♥❤➟♣ q✉è❝ ❞➙♥ ❝➙♥ ❜➡♥❣✳
◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✾ ✴ ✹✽

ú
ừ ởt q

ừ ởt


sỷ số q tr , q , , q
số t ờ tứ tr tợ tr ợ ở t ờ ữủ

số ữủ t ờ tr ừ ữủ ồ số
ừ ố số t ờ tứ tr tợ tr ợ
q
tr ừ số q t ờ tứ q tợ q số
q q q q q ữủ t ờ tữỡ
ự ừ ữủ ồ số ừ số
số

q q

tố ở t ờ tr ừ số số t
ờ tứ


ố t

t

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số q tr , q , q ợ


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tỗ t ỳ t ợ ữủ ồ ừ số q
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tợ t ồ ởt ữủ ổ ũ
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ớ ởt ữủ ổ ũ t tữỡ ự
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ọ r s t t ởt ỡ s tr s
t ở ữủ s t t số ở t
t ởt t t ổ ỳ t ồ t s t ữợ
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●❤✐ ❝❤ó
✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

▼ët sè ❤➔♠ ❝➟♥ ❜✐➯♥ t❤÷í♥❣ ❣➦♣ tr♦♥❣ ❦✐♥❤ t➳
❑➼ ❤✐➺✉
▼❈
▼❘

▼❈
▼P❑
▼P▲
▼❯
▼Π

❚➯♥ ❚✳❆♥❤
▼❛r❣✐♥❛❧ ❈♦st
▼❛r❣✐♥❛❧ ❘❡✈❡♥✉❡
▼❛r❣✐♥❛❧ ❈♦♥s✉♠♣t✐♦♥
▼❛r❣✐♥❛❧ Pr♦❞✉❝t ♦❢ ❈❛♣✐t❛❧
▼❛r❣✐♥❛❧ Pr♦❞✉❝t ♦❢ ▲❛❜♦r
▼❛r❣✐♥❛❧ ❯t✐❧✐t②
▼❛r❣✐♥❛❧ Pr♦❢✐t

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❚➯♥ ❚✳❱✐➺t
❈❤✐ ♣❤➼ ❝➟♥ ❜✐➯♥
❉♦❛♥❤ t❤✉ ❝➟♥ ❜✐➯♥
❳✉ ❤÷î♥❣ t✐➯✉ ❞ò♥❣ ❝➟♥ ❜✐➯♥
❙↔♥ ♣❤➞♠ ❝➟♥ ❜✐➯♥ t❤❡♦ ✈è♥
❙↔♥ ♣❤➞♠ ❝➟♥ ❜✐➯♥ t❤❡♦ ❧❛♦ ✤ë♥❣
➑❝❤ ❧ñ✐ ❝➟♥ ❜✐➯♥
▲ñ✐ ♥❤✉➟♥ ❝➟♥ ❜✐➯♥

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✶✻ ✴ ✹✽

●❤✐ ❝❤ó
✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

❱➼ ❞ö ✶
❈❤♦ ❤➔♠ s↔♥ ①✉➜t ❈♦❜❜✲❉♦✉❣❧❛s ❝â ❞↕♥❣
◗ ✏ ❢ ♣▲q ✏ ✷▲✶④✹ .
❛✳ ❚➼♥❤ ❤➔♠ s↔♥ ♣❤➞♠ ❝➟♥ ❜✐➯♥ t❤❡♦ ❧❛♦ ✤ë♥❣ ▼P▲ ✳
❜✳ ❚➼♥❤ ▼P▲ ♣▲ ✏ ✶✻q ✈➔ ✤÷❛ r❛ ♥❤➟♥ ①➨t✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✶✼ ✴ ✹✽

●❤✐ ❝❤ó
✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ❦❤→✐ ♥✐➺♠ s♦ s→♥❤ t➽♥❤
P❤➙♥ t➼❝❤ s♦ s→♥❤ t➽♥❤ ❧➔ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ s♦ s→♥❤ ❤❛✐ tr↕♥❣
t❤→✐ ❝➙♥ ❜➡♥❣ ❝ô ✈➔ ♠î✐ ✈î✐ ♥❤❛✉✳
❚r↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ♥❣❤✐➺♠ ❝➙♥ ❜➡♥❣ ❤❛② ❧➔ t➟♣

❝→❝ ❣✐→ trà ❝õ❛ ❜✐➳♥ ♥ë✐ s✐♥❤ ❝➙♥ ❜➡♥❣✳
❇✐➳♥ ♥ë✐ s✐♥❤ ❝➙♥ ❜➡♥❣ ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ t❤❛♠ sè ✈➔ ❜✐➳♥ ♥❣♦↕✐ s✐♥❤✳
P❤➙♥ t➼❝❤ s♦ s→♥❤ t➽♥❤ ❝❤➼♥❤ ❧➔ t➻♠ tè❝ ✤ë t❤❛② ✤ê✐ ❝õ❛ ❝õ❛ ❜✐➳♥ ♥ë✐
s✐♥❤ ❝➙♥ ❜➡♥❣ t❤❡♦ sü t❤❛② ✤ê✐ ❝õ❛ t❤❛♠ sè ✈➔ ❜✐➳♥ ♥❣♦↕✐ s✐♥❤ ✈➔ ❞♦
✤â ✤↕♦ ❤➔♠ ❝â þ ♥❣❤➽❛ q✉❛♥ trå♥❣ tr♦♥❣ s♦ s→♥❤ t➽♥❤✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✶✽ ✴ ✹✽


●❤✐ ❝❤ó
✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ✤ë ❞è❝ ❝õ❛ ♠ët ✤÷í♥❣
❚❛ ❝â t❤➸ ①❡♠ ✤ç t❤à ❝õ❛ ♠ët ❤➔♠ sè ② ✏ ❢ ♣① q ❣✐è♥❣ ♥❤÷ ❧➔ ♠ët ❤➺
t❤è♥❣ ❣ç♠ ❝→❝ ❞➣② ♥ó✐ ✈➔ ❝→❝ t❤✉♥❣ ❧ô♥❣✱ ✈î✐ ① ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ ✈à tr➼
✤ù♥❣ ❞å❝ t❤❡♦ trö❝ ① ✭tù❝ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❧➔ ❜❛♦ ♥❤✐➯✉ t➼♥❤ tø ♠ët
✤✐➸♠ t❤❡♦ ❤÷î♥❣ ✤æ♥❣ ❤♦➦❝ ❤÷î♥❣ t➙②✮ ✈➔ ② ❧➔ ✤ë ❝❛♦ ❝õ❛ ✈à tr➼ ✤ù♥❣
s♦ ✈î✐ trö❝ ① ✭❤❛② ❧➔ ❝❛♦ ❜❛♦ ♥❤✐➯✉ s♦ ✈î✐ ♠➦t ♥÷î❝ ❜✐➸♥✮✳
❈↔ ♥❤➔ ❧❡♦ ♥ó✐ ✈➔ ♥❤➔ ❦✐♥❤ t➳ ✤➲✉ ✤➦❝ ❜✐➺t q✉❛♥ t➙♠ ✤➳♥ ✤ë ❞è❝✳ ❈→❝
♥❤➔ ❧❡♦ ♥ó✐ ❤✐➸♥ ♥❤✐➯♥ ❝❤ó þ ✤➳♥ ❝❤✉②➺♥ ❤å ✤❛♥❣ ❝❤✉➞♥ ❜à ❧➯♥ ❞è❝
❤❛② ①✉è♥❣ ❞è❝✱ ✈➔ ❝→❝ ♥❤➔ ❦✐♥❤ t➳ t❤➻ ❧↕✐ q✉❛♥ t➙♠ ✤➳♥ ❝❤✉②➺♥ ✤ç t❤à
❝õ❛ ♠ët ❤➔♠ sè ❧➔ ❞è❝ ❧➯♥ ✭❣✐è♥❣ ✤÷í♥❣ ❝✉♥❣✮ ❤❛② ❞è❝ ①✉è♥❣ ✭❣✐è♥❣

✤÷í♥❣ ❝➛✉✮✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✶✾ ✴ ✹✽

●❤✐ ❝❤ó
✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

▼✐♥❤ ❍å❛

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✵ ✴ ✹✽

●❤✐ ❝❤ó
✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ✤ë ❞è❝ ❝õ❛ ♠ët ✤÷í♥❣
✣ë ❞è❝ t↕✐ ♠ët ✤✐➸♠ ❜➜t ❦➻ ① ❝â t❤➸ ✤÷ñ❝ ✤♦ ❜ð✐ ✤ë ❝❤✉②➸♥ ✤ë♥❣ ∆① ✭❝❤➥♥❣
❤↕♥✱ ∆① ✏ ✺ ❤❛② ✤✐ ✈➲ ❜➯♥ ♣❤↔✐ ✺✮✱ ✤♦ ❜➡♥❣ ✤ë t❤❛② ✤ê✐ ✤ë ❝❛♦ ∆② ✭❝❤➥♥❣
② ✤➸ t➼♥❤ ✤ë ❞è❝ ✭ð ✤➙②✱
❤↕♥✱ ∆② ✏ ✁✷✵ ❤❛② ✤✐ ①✉è♥❣ ✷✵✮ ✈➔ ❧➟♣ t➾ sè ∆
∆①
∆②
✁✷✵ ✏ ✁✹✱ ♥❣❤➽❛ ❧➔ ♠é✐ ❜÷î❝ ❜↕♥ ✤✐ ✈➲ ♣❤➼❛ tr÷î❝✱ ❜↕♥ ✤➣ ✤✐ ①✉è♥❣
✏
∆①
✺
✹ ❜÷î❝✱ ❞➜✉ ➙♠ ð ✤➙② ❝❤♦ t❛ ❜✐➳t ❧➔ ✤✐ ①✉è♥❣✮✳ ❚ø ✤â t❛ ❝â ✤à♥❤ ♥❣❤➽❛ s❛✉✿

✣à♥❤ ♥❣❤➽❛

❈❤♦ ❤➔♠ sè ② ✏ ❢ ♣① q ❝â ✤↕♦ ❤➔♠ t↕✐ ① ✏ ①✵ ✳ ✣ë ❞è❝ ❝õ❛ ❤➔♠ sè
② ✏ ❢ ♣① q t↕✐ ①✵ ❝❤➼♥❤ ❧➔ ❤➺ sè ❣â❝ ❝õ❛ t✐➳♣ t✉②➳♥ ❝õ❛ ❢ ♣① q t↕✐ ①✵ ❤❛② ❧➔
✤↕♦ ❤➔♠ ❢ ✶ ♣①✵ q t↕✐ ✤✐➸♠ ①✵ ✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✶ ✴ ✹✽


●❤✐ ❝❤ó
✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

✣↕♦ ❤➔♠ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥

▼✐♥❤ ❍å❛

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✷ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❍➔♠ ❝❤✐ ♣❤➼✱ ❤➔♠ ❞♦❛♥❤ t❤✉ ✈➔ ❤➔♠ ❧ñ✐ ♥❤✉➟♥

❍➔♠ ❝❤✐ ♣❤➼
❍➔♠ tê♥❣ ❝❤✐ ♣❤➼ ❚❈ ✏ ❢ ♣◗ q❀
❍➔♠ ❝❤✐ ♣❤➼ ❝è ✤à♥❤ ❋❈❀
❍➔♠ ❝❤✐ ♣❤➼ ❜✐➳♥ ✤ê✐ ❱❈❀
❚❛ ❝â ❚❈ ✏ ❱❈   ❋❈❀
❍➔♠ ❝❤✐ ♣❤➼ tr✉♥❣ ❜➻♥❤✿ ❆❈ ✏ ❚❈
◗❀

❍➔♠ ❝❤✐ ♣❤➼ ❝è ✤à♥❤ tr✉♥❣ ❜➻♥❤✿ ❆❋❈ ✏ ❋❈
◗❀
❍➔♠ ❝❤✐ ♣❤➼ ❜✐➳♥ ✤ê✐ tr✉♥❣ ❜➻♥❤✿ ❆❱❈ ✏ ❱❈
◗❀

❚❛ ❝â ❆❈ ✏ ❆❱❈   ❆❋❈❀
❞ ❚❈✳
❍➔♠ ❝❤✐ ♣❤➼ ❝➟♥ ❜✐➯♥✿ ▼❈ ✏ ❞◗
◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✸ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❍➔♠ ❝❤✐ ♣❤➼✱ ❤➔♠ ❞♦❛♥❤ t❤✉ ✈➔ ❤➔♠ ❧ñ✐ ♥❤✉➟♥

❍➔♠ ❞♦❛♥❤ t❤✉ ✈➔ ❤➔♠ ❧ñ✐ ♥❤✉➟♥
❍➔♠ tê♥❣ ❞♦❛♥❤ t❤✉✿ ❚❘ ✏ P◗❀
❍➔♠ ❞♦❛♥❤ t❤✉ tr✉♥❣ ❜➻♥❤✿ ❆❘ ✏ ❚❘
◗ ✏ P❀
❞ ❚❘✳
❍➔♠ ❞♦❛♥❤ t❤✉ ❝➟♥ ❜✐➯♥✿ ▼❘ ✏ ❞◗
❍➔♠ ❧ñ✐ ♥❤✉➟♥✿ Π ✏ ❚❘ ✁ ❚❈✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✹ ✴ ✹✽


●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❍➔♠ ❝❤✐ ♣❤➼✱ ❤➔♠ ❞♦❛♥❤ t❤✉ ✈➔ ❤➔♠ ❧ñ✐ ♥❤✉➟♥

❱➼ ❞ö
❈❤♦ ❤➔♠ tê♥❣ ❝❤✐ ♣❤➼ ❝õ❛ ♠ët ❞♦❛♥❤ ♥❣❤✐➺♣ ❝â ❞↕♥❣
❚❈ ✏ ◗ ✸ ✁ ✺◗ ✷   ✶✺◗   ✺✵.
❚➻♠ ❤➔♠ ❝❤✐ ♣❤➼ ❜✐➳♥ ✤ê✐✱ ❝❤✐ ♣❤➼ ❝è ✤à♥❤ ❝õ❛ ❞♦❛♥❤ ♥❣❤✐➯♣✳
❚➻♠ ❤➔♠ ❝❤✐ ♣❤➼ tr✉♥❣ ❜➻♥❤✱ ❝❤✐ ♣❤➼ ❝➟♥ ❜✐➯♥ ❝õ❛ ❞♦❛♥❤ ♥❣❤✐➺♣✳
❚➻♠ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ❝❤✐ ♣❤➼ ❜✐➳♥ ✤ê✐ ✈➔ ❝❤♦ ♥❤➟♥ ①➨t✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✺ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

✣↕♦ ❤➔♠ ❝õ❛ ♠ët sè ❤➔♠ sè sì ❝➜♣

❇↔♥❣ ✤↕♦ ❤➔♠ ❝õ❛ ♠ët sè ❤➔♠ sè sì ❝➜♣
❍➔♠ ❢ ♣① q

❢ ♣① q ✏ ❝ , ❝ ❧➔ ❤➡♥❣ sè
❢ ♣① q ✏ ① α , α € ❘
❢ ♣① q ✏ ❛ ① , ❛ → ✵
❢ ♣① q ✏ ❡ ①
❢ ♣① q ✏ ❧♦❣❛ ① , ❛ → ✵, ❛ ✘ ✶, ① → ✵
❢ ♣① q ✏ ❧♥ ① , ① → ✵

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕♦ ❤➔♠ ❢ ✶ ♣① q
❢ ✶ ♣① q ✏ ✵✱ ✈î✐ ♠å✐ ①
❢ ✶ ♣① q ✏ α① α✁✶ ✱ ✈î✐ ♠å✐ ①
❢ ✶ ♣① q ✏ ❛① ❧♥ ❛✱ ✈î✐ ♠å✐ ①
❢ ✶ ♣① q ✏ ❡ ① ✱ ✈î✐ ♠å✐ ①
❢ ✶ ♣① q ✏ ① ❧♥✶ ❛ ✱ ✈î✐ ♠å✐ ① → ✵
❢ ✶ ♣① q ✏ ①✶ ✱ ✈î✐ ♠å✐ ① → ✵

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✻ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

◗✉✐ t➢❝ tê♥❣✱ t➼❝❤✱ t❤÷ì♥❣
❈❤♦ ❢ ♣① q, ❣ ♣① q ❧➔ ❝→❝ ❤➔♠ sè ❝â ✤↕♦ ❤➔♠✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ q✉✐ t➢❝ t➼♥❤

✤↕♦ ❤➔♠ s❛✉✿
◗✉✐ t➢❝ tê♥❣✿ ♣❢ ♣① q   ❣ ♣① qq✶ ✏ ❢ ✶ ♣① q   ❣ ✶ ♣① q❀
◗✉✐ t➢❝ tê♥❣ tê♥❣ q✉→t✿

♣❢✶♣① q   ❢✷♣① q   . . .   ❢♥ ♣① qq✶ ✏ ❢✶✶♣① q   ❢✷✶♣① q   . . .   ❢♥✶♣① q.
◗✉✐ t➢❝ t➼❝❤✿ ♣❢ ♣① q❣ ♣① qq✶ ✏ ❢ ✶ ♣① q❣ ♣① q   ❢ ♣① q❣ ✶ ♣① q❀
◗✉✐ t➢❝ t➼❝❤ tê♥❣ q✉→t✿

♣❢✶♣① q❢✷♣① q . . . ❢♥ ♣① qq✶ ✏
✏ ❢✶✶♣① q❢✷♣① q . . . ❢♥ ♣① q   ❢✶♣① q❢✷✶♣① q . . . ❢♥ ♣① q   ❢✶♣① q❢✷♣① q . . . ❢♥✶♣① q
✶
✶
◗✉✐ t➢❝ t❤÷ì♥❣✿ ❣❢ ♣♣①① qq ✏ ❢ ♣① q❣ ♣①❣q✷✁♣①❢q♣① q❣ ♣① q , ❣ ♣① q ✘ ✵✳
◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✼ ✴ ✹✽


●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❱➼ ❞ö
❚➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❝→❝ ❤➔♠ sè s❛✉✿
❄

❛✳ ❢ ♣① q ✏ ✸ ①   ❄✷①   ① ✷ ①  ✶ ✳
❜✳ ❢ ♣① q ✏ ✷①   ❧♦❣✷ ①   ① ❧♥ ①✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✽ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❱➼ ❞ö
❈❤♦ ❤➔♠ ❞♦❛♥❤ t❤✉ tr✉♥❣ ❜➻♥❤ ❆❘ ✏ ❢ ♣◗ q✳ ❚➻♠ ❤➔♠ ❞♦❛♥❤ t❤✉ ❝➟♥ ❜✐➯♥
✈➔ ❝❤♦ ♥❤➟♥ ①➨t✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✷✾ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❱➼ ❞ö
❈❤♦ ❤➔♠ tê♥❣ ❝❤✐ ♣❤➼ ❚❈ ✏ ❚❈ ♣◗ q✳
❛✳ ❚➼♥❤ ❤➔♠ ❝❤✐ ♣❤➼ tr✉♥❣ ❜➻♥❤ ❆❈✳
❜✳ ❚➼♥❤ ✤ë ❞è❝ ❝õ❛ ❤➔♠ ❝❤✐ ♣❤➼ tr✉♥❣ ❜➻♥❤ ❆❈ ✶ ♣◗ q✳
❝✳ ❙♦ s→♥❤ ❝❤✐ ♣❤➼ tr✉♥❣ ❜➻♥❤ ❆❈ ✈➔ ❝❤✐ ♣❤➼ ❝➟♥ ❜✐➯♥ ▼❈ ❞ü❛ tr➯♥ ✤ë
❞è❝ ❝õ❛ ❆❈✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✵ ✴ ✹✽


●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ❤ñ♣ ❤❛② q✉✐ t➢❝ ❝❤✉é✐
❈❤♦ ❤➔♠ sè ③ ✏ ❢ ♣② q ✈➔ ② ✏ ✉♣① q ❤❛② ③ ✏ ❢ ♣✉♣① qq✳ ❑❤✐ ✤â t❛ ❝â q✉✐ t➢❝
t➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ③ t❤❡♦ ① ♥❤÷ s❛✉✿
③ ✶ ♣① q ✏ ❢ ✶ ♣✉♣① qq.✉✶ ♣① q ❤❛② ③①✶ ✏ ③②✶ ②①✶ .
❚❛ ❝â ❝ô♥❣ ❝â t❤➸ ✈✐➳t ❞÷î✐ ❞↕♥❣ t÷ì♥❣ ✤÷ì♥❣✿
❞③ ✏ ❞③ ❞② .

❞① ❞② ❞①

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✶ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❇↔♥❣ t➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ❤ñ♣
❍➔♠ ③ ✏ ❢ ♣✉♣① qq
③ ♣ ① q ✏ ✉ ♣ ① qα , α € ❘
③ ♣ ① q ✏ ❛ ✉ ♣① q , ❛ → ✵
③ ♣① q ✏ ❡ ✉♣① q
③ ♣① q ✏ ❧♦❣❛ ✉♣① q✱
❛ → ✵ , ❛ ✘ ✶, ✉ ♣① q → ✵
③ ♣① q ✏ ❧♥ ✉♣① q,
✉ ♣① q → ✵

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕♦ ❤➔♠ ③ ✶ ♣① q
③ ✶ ♣① q ✏ α✉α✁✶ ✉✶ ♣① q✱ ✈î✐ ♠å✐ ①
③ ✶ ♣① q ✏ ❛✉♣① q ❧♥ ❛✉✶ ♣① q✱ ✈î✐ ♠å✐ ①

③ ✶ ♣① q ✏ ❡ ✉♣① q ✉✶ ♣① q✱ ✈î✐ ♠å✐ ①
✶
③ ✶ ♣① q ✏ ✉♣✉① ♣q①❧♥q ❛ ✱
✈î✐ ♠å✐ ① : ✉♣① q → ✵
✶
③ ✶ ♣① q ✏ ✉✉♣♣①①qq ✱
✈î✐ ♠å✐ ① : ✉♣① q → ✵

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✷ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❱➼ ❞ö
❚➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❝→❝ ❤➔♠ sè s❛✉✿
❛✳ ③ ♣① q ✏ ❄
♣✸① ✷ ✁ ✷①   ✶q✶✵✳
❜✳ ③ ♣① q ✏ ✷① ✹   ✺① ✷   ✸✳
❝✳ ③ ♣① q ✏ ❡ ① ✷  ①  ✶ ✳
❞✳ ③ ♣① q ✏ ❧♦❣✷ ♣ ① ✷ ①  ✶ q, ① → ✵✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✸ ✴ ✹✽


●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ♥❣÷ñ❝
✣à♥❤ ♥❣❤➽❛

❈❤♦ ❤➔♠ sè ❢ ♣① q✳ ◆➳✉ tç♥ t↕✐ ❤➔♠ ❣ ♣① q s❛♦ ❝❤♦ ❢ ♣❣ ♣① qq ✏ ❣ ♣❢ ♣① qq ✏ ①
t❤➻ ❣ ♣① q ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ♥❣÷ñ❝ ❝õ❛ ❢ ♣① q ✈➔ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ ❢ ✁✶ ♣① q✳

✣à♥❤ ❧➼

❈❤♦ ❤➔♠ sè ❢ ♣① q✳ ◆➳✉ ❢ ♣① q ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ t❤➻ ❢ ♣① q ❝â ❤➔♠ ♥❣÷ñ❝✳

◆❤➟♥ ①➨t✿ ❑❤✐ ❢ ♣① q ❧➔ ❤➔♠ ❦❤↔ ✈✐ t❤➻ ❢ ♣① q ✤ì♥ ✤✐➺✉ ❦❤✐ ❢ ✶♣① q → ✵, ❅①
❤♦➦❝ ❢ ✶ ♣① q ➔ ✵, ❅①✳ ◆❤÷ ✈➟② t❛ ❝â t❤➸ ❦✐➸♠ tr❛ ✈✐➺❝ tç♥ t↕✐ ❤➔♠ ♥❣÷ñ❝
❝õ❛ ❢ ♣① q ❞ü❛ tr➯♥ ✈✐➺❝ ①➨t ❞➜✉ ❝õ❛ ✤↕♦ ❤➔♠ ❢ ✶ ♣① q✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✹ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❱➼ ❞ö
❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥❤ú♥❣ ❤➔♠ sè s❛✉ ❝â ❤➔♠ ♥❣÷ñ❝ tr➯♥ ♠✐➲♥ ❝❤➾ r❛ t÷ì♥❣
ù♥❣ ✈➔ t➼♥❤ ✤↕♦ ❤➔♠ ❤➔♠ ♥❣÷ñ❝ ❝õ❛ ❝❤ó♥❣✿
❛✳ ❢ ♣① q ✏ ❛①   ❜, ❛ ✘ ✵✳
❜✳ ❢ ♣① q ✏ ❆① α , ❆ ✘ ✵, α ✘ ✵, ① → ✵✳
❝✳ ❢ ♣① q ✏ ① ✺   ① ✸   ① , ① € ❘ ✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✺ ✴ ✹✽

●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

✣à♥❤ ❧➼ ✤↕♦ ❤➔♠ ❤➔♠ ♥❣÷ñ❝
✣à♥❤ ❧➼

❈❤♦ ❤➔♠ sè ❢ ♣① q ❝â ✤↕♦ ❤➔♠ ✈➔ ❝â ❤➔♠ ♥❣÷ñ❝ ❧➔ ❢ ✁✶ ♣① q✳ ❑❤✐ ✤â ❤➔♠
♥❣÷ñ❝ ❢ ✁✶ ♣① q ❝ô♥❣ ❝â ✤↕♦ ❤➔♠ ✈➔ ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝✿
♣❢ ✁✶q✶♣① q ✏ ❢ ✶♣❢ ✁✶✶♣① qq
❤❛②
♣❢ ✁✶q✶♣②✵q ✏ ❢ ✶♣✶① q , ✈î✐ ②✵ ✏ ❢ ♣①✵q.
✵

◆❤➟♥ ①➨t✿

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✻ ✴ ✹✽


●❤✐ ❝❤ó
❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❈→❝ q✉✐ t➢❝ t➼♥❤ ✤↕♦ ❤➔♠

❱➼ ❞ö
❈❤♦ ❤➔♠ sè ❢ ♣① q ✏ ① ✺   ① ✸   ① , ① € ❘ ✳
❛✳ ❚➼♥❤ ❢ ✁✶ ♣✸q✳
❜✳ ❚➼♥❤ ♣❢ ✁✶ q✶ ♣✸q✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✼ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

❑❤→✐ ♥✐➺♠ ✈➲ ✈✐ ♣❤➙♥

❑❤→✐ ♥✐➺♠ ✈➲ ✈✐ ♣❤➙♥
❈❤♦ ❤➔♠ sè ② ✏ ❢ ♣① q✳ ❑❤✐ ① t❤❛② ✤ê✐ ♠ët ❧÷ñ♥❣ ∆① t❤➻ ② t÷ì♥❣ ù♥❣ t❤❛②
✤ê✐ ♠ët ❧÷ñ♥❣ ∆②✳ ◆➳✉ ❧÷ñ♥❣ t❤❛② ✤ê✐ ❝õ❛ ① ❧➔ ✤õ ♥❤ä t❤➻ t❛ ❦➼ ❤✐➺✉ ❧➔ ❞①
✈➔ ❧÷ñ♥❣ t❤❛② ✤ê✐ ✤õ ♥❤ä t÷ì♥❣ ù♥❣ ❝õ❛ ② t❛ ❦➼ ❤✐➺✉ ❧➔ ❞②✳

✣à♥❤ ♥❣❤➽❛

❈❤♦ ❤➔♠ sè ❢ ♣① q✳ ▲÷ñ♥❣ t❤❛② ✤ê✐ ✤õ ♥❤ä ❞① , ❞② t÷ì♥❣ ù♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ✈✐
♣❤➙♥ ❝õ❛ ❜✐➳♥ sè ✈➔ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ sè✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✽ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✈✐ ♣❤➙♥ ✈➔ ✤↕♦ ❤➔♠

▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✈✐ ♣❤➙♥ ✈➔ ✤↕♦ ❤➔♠
❈❤♦ ❤➔♠ sè ② ✏ ❢ ♣① q✳ ❚❛ ❝â ❢ ✶ ♣① q ✏ ❞②
❞① ❤❛② ♠ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣
❞② ✏ ❢ ✶ ♣① q❞① .

✭✸✮

◆❤➟♥ ①➨t✿

❈æ♥❣ t❤ù❝ ✭✸✮ ❝❤♦ t❛ t➼♥❤ ✤ë t❤❛② ✤ê✐ ❝õ❛ ❤➔♠ sè ② ❦❤✐ ① t❤❛② ✤ê✐
♠ët ❧÷ñ♥❣ ✈æ ❝ò♥❣ ❜➨✳
❑❤✐ ∆① ❦❤æ♥❣ ✤÷ñ❝ ①➨t ❧➔ ✈æ ❝ò♥❣ ❜➨ t❤➻ t❛ ✈✐➳t✿
∆② ≈ ❢ ✶ ♣① q∆①

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❤❛② ❢ ♣①✵   ∆① q ≈ ❢ ♣①✵ q   ❢ ✶ ♣①✵ q∆① .

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✸✾ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

❱➼ ❞ö

▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✈✐ ♣❤➙♥ ✈➔ ✤↕♦ ❤➔♠

❄

❈❤♦ ❤➔♠ sè ❢ ♣① q ✏ ✸① ✹   ✷① ✷   ✹✳
❛✳ ❚➼♥❤ ✈✐ ♣❤➙♥ ❞② ❝õ❛ ❤➔♠ sè✳
❜✳ ❚➼♥❤ ❣✐→ trà ❢ ♣✶, ✵✷q✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✵ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✈✐ ♣❤➙♥ ✈➔ ✤↕♦ ❤➔♠

▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ✈✐ ♣❤➙♥
❈❤♦ ✉ ✏ ✉♣① q ✈➔ ✈ ✏ ✈ ♣① q ❧➔ ❝→❝ ❤➔♠ sè ❝õ❛ ①✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ t➼♥❤
❝❤➜t s❛✉✿
✶✳ ❞ ♣❝ q ✏ ✵✱ ✈î✐ ❝ ❧➔ ❤➡♥❣ sè✳

✷✳ ❞ ♣❝✉♥ q ✏ ❝♥✉♥✁✶ ❞✉✱ ✈î✐ ❝ ❧➔ ❤➡♥❣ sè✳
✸✳ ❞ ♣✉ ✟ ✈ q ✏ ❞✉ ✟ ❞✈✳
✹✳ ❞ ♣✉✈ q ✏ ✉❞✈   ✈❞✉✳
✺✳ ❞ ♣ ✈✉ q ✏ ✈❞✉ ✈✁✷ ✉❞✈ ✱ ✈î✐ ✈ ♣① q ✘ ✵✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✶ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✈✐ ♣❤➙♥ ✈➔ ✤↕♦ ❤➔♠

❱➼ ❞ö
❚➼♥❤ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ sè s❛✉ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✈✐ ♣❤➙♥✿

❛

❢ ♣① q ✏ ✸① ✹   ✷① ✷   ✹

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✷ ✴ ✹✽


●❤✐ ❝❤ó
❱✐ ♣❤➙♥

❱✐ ♣❤➙♥ ✈➔ ❤➺ sè ❝♦ ❣✐➣♥ ✤✐➸♠

✣à♥❤ ♥❣❤➽❛ ❤➺ sè ❝♦ ❣✐➣♥
✣à♥❤ ♥❣❤➽❛

❈❤♦ ❤➔♠ sè ② ✏ ❢ ♣① q✳ ❍➺ sè ❝♦ ❣✐➣♥ ❝õ❛ ② t❤❡♦ ①✱ ❦➼ ❤✐➺✉ ❧➔ ε②① ✱ ✤÷ñ❝ ①→❝
✤à♥❤ ❜ð✐
❞② ④② .
ε②① ✏
❞① ④①

◆❤➟♥ ①➨t✿

❑❤✐ ① t➠♥❣ ✶% t❤➻ ② t➠♥❣ ❦❤♦↔♥❣ ε②① %✳
❚❛ ❝â
❞② ④❞① ✏ ❢ ✶ ♣① q① ✏ ❤➔♠ ❝➟♥ ❜✐➯♥ .
ε②① ✏
② ④①
❢ ♣① q ❤➔♠ tr✉♥❣ ❜➻♥❤
❚r♦♥❣ ❦✐♥❤ t➳ t❛ ❝❤➾ ①➨t ① → ✵✱ ② → ✵✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä ❢ ✶ ♣① q ❧✉æ♥
❝ò♥❣ ❞➜✉ ✈î✐ ε②① ✳ ❚ø ✤â ♥➳✉ ❤➺ sè ❝♦ ❣✐➣♥ ❝õ❛ ❝➛✉ ❧➔ ➙♠ t❤➻ ❝ô♥❣
t÷ì♥❣ ✤÷ì♥❣ ✈î✐ ✤÷í♥❣ ❝➛✉ ❧➔ ❞è❝ ①✉è♥❣✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✸ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

❱✐ ♣❤➙♥ ✈➔ ❤➺ sè ❝♦ ❣✐➣♥ ✤✐➸♠

❙ü ❝♦ ❣✐➣♥ ❝õ❛ ❤➔♠ sè t↕✐ ♠ët ✤✐➸♠
✣à♥❤ ♥❣❤➽❛

✩
✬
✫❝♦ ❣✐➣♥
❍➔♠ sè ② ✏ ❢ ♣① q ✤÷ñ❝ ❣å✐ ❧➔ ✬❝♦ ❣✐➣♥ ✤ì♥ ✈à t↕✐ ♠ët ✤✐➸♠ ♥➳✉
✪➼t ❝♦ ❣✐➣♥
⑤ε②① ⑤ ➾ ✶

❄
❱➼ ❞ö✿ ❚➼♥❤ ❤➺ sè ❝♦ ❣✐➣♥ ❝õ❛ ❤➔♠ sè ② ✏ ✸① ✹   ✷① ✷   ✹ t↕✐ ① ✏ ✶ ✈➔
❝❤♦ ♥❤➟♥ ①➨t✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✹ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

❱✐ ♣❤➙♥ ✈➔ ❤➺ sè ❝♦ ❣✐➣♥ ✤✐➸♠

❱➼ ❞ö
❈❤♦ ❤➔♠ ❝➛✉ ◗ ✏ ✶✵✵ ✁ ✺P✳
❛✳ ❚➻♠ ❤➺ sè ❝♦ ❣✐➣♥ ❝õ❛ ❝➛✉ ε❞ ✳
❜✳ ❚➻♠ ♠✐➲♥ ❣✐→ ✤➸ ❤➔♠ ❝➛✉ ❧➔ ❝♦ ❣✐➣♥✱ ❝♦ ❣✐➣♥ ✤ì♥ ✈à ❤♦➦❝ ➼t ❝♦ ❣✐➣♥ ✈➔
tø ✤â ❝❤♦ ♥❤➟♥ ①➨t✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✺ ✴ ✹✽


●❤✐ ❝❤ó
❱✐ ♣❤➙♥

❱✐ ♣❤➙♥ ✈➔ ❤➺ sè ❝♦ ❣✐➣♥ ✤✐➸♠

❱➼ ❞ö
▼ët ❞♦❛♥❤ ♥❣❤✐➺♣ ❝â ❤➔♠ ❝➛✉ ❝❤♦ ❜ð✐ ◗ ✏ ◗ ♣P q✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ❦❤➥♥❣
✤à♥❤ s❛✉✿
❛✳ ❑❤✐ ✤÷í♥❣ ❝➛✉ ➼t ❝♦ ❣✐➣♥✱ sü ❣✐❛ t➠♥❣ ❝õ❛ ❣✐→ ❝↔ ❧➔ t➠♥❣ tê♥❣
❞♦❛♥❤ t❤✉ ✈➔ sü ❣✐↔♠ sót ❝õ❛ ❣✐→ ❝↔ ❧➔♠ ❣✐↔♠ tê♥❣ ❞♦❛♥❤
t❤✉✳
❜✳ ❑❤✐ ✤÷í♥❣ ❝➛✉ ❝♦ ❣✐➣♥✱ sü ❣✐❛ t➠♥❣ ❝õ❛ ❣✐→ ❝↔ ❧➔ ❣✐↔♠ tê♥❣
❞♦❛♥❤ t❤✉ ✈➔ sü ❣✐↔♠ sót ❝õ❛ ❣✐→ ❝↔ ❧➔♠ t➠♥❣ tê♥❣ ❞♦❛♥❤
t❤✉✳
❝✳ ❑❤✐ ✤÷í♥❣ ❝➛✉ ❝♦ ❣✐➣♥ ✤ì♥ ✈à✱ sü t❤❛② ✤ê✐ ❝õ❛ ❣✐→ ❝↔ ❦❤æ♥❣
❧➔♠ ↔♥❤ ❤÷ð♥❣ ✤➳♥ tê♥❣ ❞♦❛♥❤ t❤✉✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✻ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

❱✐ ♣❤➙♥ ✈➔ ❤➺ sè ❝♦ ❣✐➣♥ ✤✐➸♠

▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➺ sè ❝♦ ❣✐➣♥
▼➺♥❤ ✤➲

❍➔♠ sè ② ✏ ❢ ♣① q ❝â ❤➺ sè ❝♦ ❣✐➣♥ ❦❤æ♥❣ ✤ê✐ t↕✐ ♠å✐ ① ♥➳✉ ✈➔ ❝❤➾ ♥➳✉
② ✏ ❆① ❛ ✈➔ ε②① ✏ ❛✳

▼➺♥❤ ✤➲

◆➳✉ ❤➔♠ sè ② ✏ ❢ ♣① q ❝â ❤➔♠ ♥❣÷ñ❝ ❧➔ ❣ ♣① q ✈➔ ❤➺ sè ❝♦ ❣✐➣♥ ❝õ❛ ❢ ♣① q
t❤❡♦ ① ❧➔ ε❢ ✈➔ ❤➺ sè ❝♦ ❣✐➣♥ ❝õ❛ ❤➔♠ ♥❣÷ñ❝ ❣ ♣① q ❧➔ ε❣ t❤➻ t❛ ❝â✿
✶ .
ε❣ ♣ ① q ✏
ε❢ ♣❣ ♣① qq

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✼ ✴ ✹✽

●❤✐ ❝❤ó
❱✐ ♣❤➙♥

❱✐ ♣❤➙♥ ✈➔ ❤➺ sè ❝♦ ❣✐➣♥ ✤✐➸♠

❱➼ ❞ö
❈❤♦ ❤➔♠ ❝➛✉ ◗ ✏ ✶✵✵P ✁✺ ✳ ❚➼♥❤ ❤➺ sè ❝♦ ❣✐➣♥ ❝õ❛ ❤➔♠ ❝➛✉ ε◗P ✱ ❤➺ sè ❝♦
❣✐➣♥ ❝õ❛ ❤➔♠ ❝➛✉ ♥❣÷ñ❝ εP◗ ✈➔ ❝❤♦ ♥❤➟♥ ①➨t✳

◆❣✉②➵♥ ❚❤à ◆❤✉♥❣ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✣↕✐ ❙è ✈➔ ●✐↔✐ t➼❝❤

◆❣➔② ✶✼ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶

✹✽ ✴ ✹✽