❙❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ❛ ❧❛②❡r ♦❢ ✉♥st❛❜❧❡ ♣❤❛s❡
❨✈❛♥ ❱❡❧❡♥✐❦
❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❉✐♠❛ ■♦✛❡ ❛♥❞ ❙❡♥②❛ ❙❤❧♦s♠❛♥
t r st - - PowerPoint PPT Presentation
t r st s s t r t
❙❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ❛ ❧❛②❡r ♦❢ ✉♥st❛❜❧❡ ♣❤❛s❡
❨✈❛♥ ❱❡❧❡♥✐❦
❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❉✐♠❛ ■♦✛❡ ❛♥❞ ❙❡♥②❛ ❙❤❧♦s♠❛♥
❙tr✉❝t✉r❡ ♦❢ t❤❡ t❛❧❦
✶
▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡
✷
✸
❘❡s✉❧t
✹
❙❦❡t❝❤ ♦❢ ♣r♦♦❢
▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ❈r✐t✐❝❛❧ ♣r❡✇❡tt✐♥❣ ✐♥ t❤❡ ■s✐♥❣ ♠♦❞❡❧
◮ ✷❞ ■s✐♥❣ ♠♦❞❡❧ ✐♥ ❛ sq✉❛r❡ ❜♦① ◮ ❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ ✰ ✰ ✰ ◮ ▼❛❣♥❡t✐❝ ✜❡❧❞✿ ❤ ❃ ✵ ❆s ❤ ★ ✵✱ t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r ♦❢ ✉♥st❛❜❧❡ ♣❤❛s❡ ✐♥❝r❡❛s❡s ❛s ❤✶❂✸✰♦✭✶✮ ✭❛s ❧♦♥❣ ❛s ◆ ✢ ❤✷❂✸✮ ❬❱✳✱ P❚❘❋ ✷✵✵✹❪
▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ✶ ✰ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧
X ◆ ❯♥st❛❜❧❡ ♣❤❛s❡
Pr♦❜❛❜✐❧✐t② ♦❢ ❛ ♥♦♥♥❡❣❛t✐✈❡ tr❛❥❡❝t♦r② X ❂ ✭❳◆❀ ✿ ✿ ✿ ❀ ❳◆✮✿ P◆❀✰❀✕✭X✮ ❂ ✶ ❩◆❀✰❀✕ ❡①♣ ♥ ✕
✏❛r❡❛✑
③ ⑥⑤ ④
◆
❳
✐❂◆
❳✐ ♦ p❘❲✭X✮❀ ✇❤❡r❡ ✕ ❃ ✵✳
▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ✶ ✰ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧
X ◆ ❯♥st❛❜❧❡ ♣❤❛s❡
Pr♦❜❛❜✐❧✐t② ♦❢ ❛ ♥♦♥♥❡❣❛t✐✈❡ tr❛❥❡❝t♦r② X ❂ ✭❳◆❀ ✿ ✿ ✿ ❀ ❳◆✮✿ P◆❀✰❀✕✭X✮ ❂ ✶ ❩◆❀✰❀✕ ❡①♣ ♥ ✕
✏❛r❡❛✑
③ ⑥⑤ ④
◆
❳
✐❂◆
❳✐ ♦ p❘❲✭X✮❀ ✇❤❡r❡ ✕ ❃ ✵✳
▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡
❇❡❤❛✈✐♦r ❛s ✕ ★ ✵✿ ◮ ❋r❡❡ ❡♥❡r❣② ✘ ✕✷❂✸ ◮ ❚❤✐❝❦♥❡ss ✘ ✕✶❂✸ ◮ ❈♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤ ✘ ✕✷❂✸
❬❍r②♥✐✈ ❛♥❞ ❱✳✱ P❚❘❋ ✷✵✵✹❪ ✭s❡❡ ❛❧s♦ ❬❆❜r❛❤❛♠✫ ❙♠✐t❤✱ ❏❙P ✶✾✽✻❪✮
❲❤❛t ✐s t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ①✕✭t✮ ❂ ✕✶❂✸❳❬✕✷❂✸t❪❄
❯s❡❢✉❧ t♦ ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✳✳✳
▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡
❇❡❤❛✈✐♦r ❛s ✕ ★ ✵✿ ◮ ❋r❡❡ ❡♥❡r❣② ✘ ✕✷❂✸ ◮ ❚❤✐❝❦♥❡ss ✘ ✕✶❂✸ ◮ ❈♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤ ✘ ✕✷❂✸
❬❍r②♥✐✈ ❛♥❞ ❱✳✱ P❚❘❋ ✷✵✵✹❪ ✭s❡❡ ❛❧s♦ ❬❆❜r❛❤❛♠✫ ❙♠✐t❤✱ ❏❙P ✶✾✽✻❪✮
❲❤❛t ✐s t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ①✕✭t✮ ❂ ✕✶❂✸❳❬✕✷❂✸t❪❄
❯s❡❢✉❧ t♦ ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✳✳✳
▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡
❇❡❤❛✈✐♦r ❛s ✕ ★ ✵✿ ◮ ❋r❡❡ ❡♥❡r❣② ✘ ✕✷❂✸ ◮ ❚❤✐❝❦♥❡ss ✘ ✕✶❂✸ ◮ ❈♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤ ✘ ✕✷❂✸
❬❍r②♥✐✈ ❛♥❞ ❱✳✱ P❚❘❋ ✷✵✵✹❪ ✭s❡❡ ❛❧s♦ ❬❆❜r❛❤❛♠✫ ❙♠✐t❤✱ ❏❙P ✶✾✽✻❪✮
❲❤❛t ✐s t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ①✕✭t✮ ❂ ✕✶❂✸❳❬✕✷❂✸t❪❄
❯s❡❢✉❧ t♦ ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✳✳✳
✶✳ ❚❤❡ ✉♥❞❡r❧②✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✭♣①✮①✷Z✿ tr❛♥s✳ ♣r♦❜❛❜✳ ♦❢ ❛♥ ❛♣❡r✐♦❞✐❝✱ ✐rr❡❞✉❝✐❜❧❡ ❘❲ ♦♥ Z s✉❝❤ t❤❛t ❳
①
♣① ❂ ✵ ❀ ❳
①
❡t①♣① ❁ ✶ ❢♦r s♠❛❧❧ t ▲❡t ✛✷ ❂ P
① ①✷♣① ❛♥❞✱ ❢♦r X ❂ ✭❳◆❀ ✿ ✿ ✿ ❀ ❳◆✮✱
p❘❲✭X✮ ❂
◆✶
❨
✐❂◆
♣❳✐✰✶❳✐
✷✳ ❚❤❡ ♣♦t❡♥t✐❛❧s ✭❱✕✮✕❃✵ ▲❡t ❱✕ ✿ N ✦ R✰ ❜❡ s✉❝❤ t❤❛t ❱✕✭✵✮ ❂ ✵ ❀ ❱✕ ✐♥❝r❡❛s✐♥❣✱ ❧✐♠
①✦✶ ❱✕✭①✮ ❂ ✰✶
▲❡t ❍✕ ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ❍✷❱✕✭❍✮ ❂ ✶ ✭♠❡❛s✉r❡s t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r✮ ❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s ✭♦♥ ✭❱✕✮✕❃✵✮✿ ❧✐♠
✕★✵ ❍✕ ❂ ✰✶ ❀
❧✐♠
✕★✵ ❍✷ ✕❱✕✭r❍✕✮ ❂ q✭r✮ ❀
✇✐t❤ q ✷
✷✭ ✰✮ s✉❝❤ t❤❛t ❧✐♠r✦✶ q✭r✮ ❂ ✰✶✳
❊①❛♠♣❧❡✿ ❱✕✭①✮ ❂ ✕①☛✱ ☛ ❃ ✵✱ ❍✕ ❂ ✕✶❂✭✷✰☛✮✱ q✭r✮ ❂ r☛
✷✳ ❚❤❡ ♣♦t❡♥t✐❛❧s ✭❱✕✮✕❃✵ ▲❡t ❱✕ ✿ N ✦ R✰ ❜❡ s✉❝❤ t❤❛t ❱✕✭✵✮ ❂ ✵ ❀ ❱✕ ✐♥❝r❡❛s✐♥❣✱ ❧✐♠
①✦✶ ❱✕✭①✮ ❂ ✰✶
▲❡t ❍✕ ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ❍✷❱✕✭❍✮ ❂ ✶ ✭♠❡❛s✉r❡s t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r✮ ❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s ✭♦♥ ✭❱✕✮✕❃✵✮✿ ❧✐♠
✕★✵ ❍✕ ❂ ✰✶ ❀
❧✐♠
✕★✵ ❍✷ ✕❱✕✭r❍✕✮ ❂ q✭r✮ ❀
✇✐t❤ q ✷
✷✭ ✰✮ s✉❝❤ t❤❛t ❧✐♠r✦✶ q✭r✮ ❂ ✰✶✳
❊①❛♠♣❧❡✿ ❱✕✭①✮ ❂ ✕①☛✱ ☛ ❃ ✵✱ ❍✕ ❂ ✕✶❂✭✷✰☛✮✱ q✭r✮ ❂ r☛
✷✳ ❚❤❡ ♣♦t❡♥t✐❛❧s ✭❱✕✮✕❃✵ ▲❡t ❱✕ ✿ N ✦ R✰ ❜❡ s✉❝❤ t❤❛t ❱✕✭✵✮ ❂ ✵ ❀ ❱✕ ✐♥❝r❡❛s✐♥❣✱ ❧✐♠
①✦✶ ❱✕✭①✮ ❂ ✰✶
▲❡t ❍✕ ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ❍✷❱✕✭❍✮ ❂ ✶ ✭♠❡❛s✉r❡s t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r✮ ❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s ✭♦♥ ✭❱✕✮✕❃✵✮✿ ❧✐♠
✕★✵ ❍✕ ❂ ✰✶ ❀
❧✐♠
✕★✵ ❍✷ ✕❱✕✭r❍✕✮ ❂ q✭r✮ ❀
✇✐t❤ q ✷ C✷✭R✰✮ s✉❝❤ t❤❛t ❧✐♠r✦✶ q✭r✮ ❂ ✰✶✳ ❊①❛♠♣❧❡✿ ❱✕✭①✮ ❂ ✕①☛✱ ☛ ❃ ✵✱ ❍✕ ❂ ✕✶❂✭✷✰☛✮✱ q✭r✮ ❂ r☛
✷✳ ❚❤❡ ♣♦t❡♥t✐❛❧s ✭❱✕✮✕❃✵ ▲❡t ❱✕ ✿ N ✦ R✰ ❜❡ s✉❝❤ t❤❛t ❱✕✭✵✮ ❂ ✵ ❀ ❱✕ ✐♥❝r❡❛s✐♥❣✱ ❧✐♠
①✦✶ ❱✕✭①✮ ❂ ✰✶
▲❡t ❍✕ ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ❍✷❱✕✭❍✮ ❂ ✶ ✭♠❡❛s✉r❡s t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r✮ ❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s ✭♦♥ ✭❱✕✮✕❃✵✮✿ ❧✐♠
✕★✵ ❍✕ ❂ ✰✶ ❀
❧✐♠
✕★✵ ❍✷ ✕❱✕✭r❍✕✮ ❂ q✭r✮ ❀
✇✐t❤ q ✷ C✷✭R✰✮ s✉❝❤ t❤❛t ❧✐♠r✦✶ q✭r✮ ❂ ✰✶✳ ❊①❛♠♣❧❡✿ ❱✕✭①✮ ❂ ✕①☛✱ ☛ ❃ ✵✱ ❍✕ ❂ ✕✶❂✭✷✰☛✮✱ q✭r✮ ❂ r☛
✸✳ ❚❤❡ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ Pr♦❜❛❜✐❧✐t② ♦❢ ❛ ♥♦♥♥❡❣❛t✐✈❡ tr❛❥❡❝t♦r② X ❂ ✭❳◆❀ ✿ ✿ ✿ ❀ ❳◆✮ ✇✐t❤ ❳◆ ❂ u✱ ❳◆ ❂ v✿ Pu❀v
◆❀✰❀✕✭X✮ ❂
✶ ❩u❀v
◆❀✰❀✕
❡①♣ ♥
❳
✐❂◆
❱✕✭❳✐✮ ♦ p❘❲✭X✮
✕ ❳❬❍✷
✕t❪ ❛s ✕ ★ ✵
✸✳ ❚❤❡ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ Pr♦❜❛❜✐❧✐t② ♦❢ ❛ ♥♦♥♥❡❣❛t✐✈❡ tr❛❥❡❝t♦r② X ❂ ✭❳◆❀ ✿ ✿ ✿ ❀ ❳◆✮ ✇✐t❤ ❳◆ ❂ u✱ ❳◆ ❂ v✿ Pu❀v
◆❀✰❀✕✭X✮ ❂
✶ ❩u❀v
◆❀✰❀✕
❡①♣ ♥
❳
✐❂◆
❱✕✭❳✐✮ ♦ p❘❲✭X✮
✕ ❳❬❍✷
✕t❪ ❛s ✕ ★ ✵
▲✐♠✐t✐♥❣ ♦❜❥❡❝ts
❙✐♥❣✉❧❛r ❙t✉r♠✲▲✐♦✉✈✐❧❧❡ ♣r♦❜❧❡♠ ♦♥ R✰ L ❂ L✛❀q ❂ ✛✷ ✷ ❞✷ ❞r✷ q✭r✮ ✇✐t❤ ③❡r♦ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✿ ✬✭✵✮ ❂ ✵✳ ❲❡❧❧✲❦♥♦✇♥✿ ✾ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❢✬✐❣✐✕✵ ♦❢ s✐♠♣❧❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✐♥
✷✭ ✰✮ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s
✵ ❃ ✵ ❃ ✶ ❃ ✷ ❃ ✿ ✿ ✿ ❀ ❧✐♠
✐✦✶ ✐ ❂ ✰✶
✽✐ ✕ ✵✱ ✬✐ ✐s s♠♦♦t❤ ❛♥❞ ♣♦ss❡ss❡s ❡①❛❝t❧② ✐ ③❡r♦❡s ✐♥ ✭✵❀ ✶✮ ■♥ ♣❛rt✐❝✉❧❛r ✬✵ ❝❛♥ ❜❡ t❛❦❡♥ ♣♦s✐t✐✈❡✳
▲✐♠✐t✐♥❣ ♦❜❥❡❝ts
❙✐♥❣✉❧❛r ❙t✉r♠✲▲✐♦✉✈✐❧❧❡ ♣r♦❜❧❡♠ ♦♥ R✰ L ❂ L✛❀q ❂ ✛✷ ✷ ❞✷ ❞r✷ q✭r✮ ✇✐t❤ ③❡r♦ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✿ ✬✭✵✮ ❂ ✵✳ ❲❡❧❧✲❦♥♦✇♥✿ ✾ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❢✬✐❣✐✕✵ ♦❢ s✐♠♣❧❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✐♥ L✷✭R✰✮ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ✵ ❃ e✵ ❃ e✶ ❃ e✷ ❃ ✿ ✿ ✿ ❀ ❧✐♠
✐✦✶ e✐ ❂ ✰✶
✽✐ ✕ ✵✱ ✬✐ ✐s s♠♦♦t❤ ❛♥❞ ♣♦ss❡ss❡s ❡①❛❝t❧② ✐ ③❡r♦❡s ✐♥ ✭✵❀ ✶✮ ■♥ ♣❛rt✐❝✉❧❛r ✬✵ ❝❛♥ ❜❡ t❛❦❡♥ ♣♦s✐t✐✈❡✳
▲✐♠✐t✐♥❣ ♦❜❥❡❝ts
❋❡rr❛r✐✲❙♣♦❤♥ ❞✐✛✉s✐♦♥s ♦♥ ✭✵❀ ✶✮
G✛❀q✥ ❂ ✶ ✬✵ ✭L ✰ e✵I✮✭✥✬✵✮ ❂ ✛✷ ✷ ❞✷✥ ❞r✷ ✰ ✛✷ ✬✵
✵
✬✵ ❞✥ ❞r ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛✉s✐♦♥s ❛r❡ ❡r❣♦❞✐❝ ❛♥❞ r❡✈❡rs✐❜❧❡ ✇✳r✳t✳ ✖✵✭❞r✮ ❂ ✬✷
✵✭r✮❞r
❙t❛t✐♦♥❛r② ♣❛t❤ ♠❡❛s✉r❡✿
✛❀q
▲✐♠✐t✐♥❣ ♦❜❥❡❝ts
❋❡rr❛r✐✲❙♣♦❤♥ ❞✐✛✉s✐♦♥s ♦♥ ✭✵❀ ✶✮
G✛❀q✥ ❂ ✶ ✬✵ ✭L ✰ e✵I✮✭✥✬✵✮ ❂ ✛✷ ✷ ❞✷✥ ❞r✷ ✰ ✛✷ ✬✵
✵
✬✵ ❞✥ ❞r ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛✉s✐♦♥s ❛r❡ ❡r❣♦❞✐❝ ❛♥❞ r❡✈❡rs✐❜❧❡ ✇✳r✳t✳ ✖✵✭❞r✮ ❂ ✬✷
✵✭r✮❞r
❙t❛t✐♦♥❛r② ♣❛t❤ ♠❡❛s✉r❡✿
✛❀q
▲✐♠✐t✐♥❣ ♦❜❥❡❝ts
❋❡rr❛r✐✲❙♣♦❤♥ ❞✐✛✉s✐♦♥s ♦♥ ✭✵❀ ✶✮
G✛❀q✥ ❂ ✶ ✬✵ ✭L ✰ e✵I✮✭✥✬✵✮ ❂ ✛✷ ✷ ❞✷✥ ❞r✷ ✰ ✛✷ ✬✵
✵
✬✵ ❞✥ ❞r ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛✉s✐♦♥s ❛r❡ ❡r❣♦❞✐❝ ❛♥❞ r❡✈❡rs✐❜❧❡ ✇✳r✳t✳ ✖✵✭❞r✮ ❂ ✬✷
✵✭r✮❞r
❙t❛t✐♦♥❛r② ♣❛t❤ ♠❡❛s✉r❡✿ P✛❀q
▼❛✐♥ r❡s✉❧t
◮ ❙❝❛❧❡❞ ♣r♦❝❡ss✿ ①✕✭t✮ ❂
✶ ❍✕ ❳❬❍✷
✕t❪
✭✇✐t❤ ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥✮ ❚❤❡♦r❡♠ ✭■♦✛❡✱ ❙❤❧♦s♠❛♥✱ ❱✳✱ ✷✵✶✹✮ ▲❡t ✭✕◆✮◆✕✶ ❜❡ s✉❝❤ t❤❛t ✕◆ ★ ✵ ❛♥❞ ❍✷
✕◆ ❂◆ ✦ ✵✳ ❚❤❡♥✱
▲❛✇ ♦❢ ①✕◆ ✉♥❞❡r Pu❀v
◆❀✰❀✕◆ ◆✦✶
❂ ✮ P✛❀q❀ ✉♥✐❢♦r♠❧② ✐♥ ✵ ✔ u❀ v ✔ ❈❍✕✳
▼❛✐♥ r❡s✉❧t
❊①❛♠♣❧❡✿ ❱✕✭①✮ ❂ ✕① ■♥ t❤✐s ❝❛s❡✿ ❍✕ ❂ ✕✶❂✸❀ ✬✵✭r✮ ❂ ❆✐✭✤r ✦✶✮❀ ❡✵ ❂ ✦✶ ✤ ❀ ✇❤❡r❡ ✦✶ ✐s t❤❡ ✜rst ③❡r♦ ♦❢ t❤❡ ❆✐r② ❢✉♥❝t✐♦♥ ❆✐ ❛♥❞ ✤ ❂
✸
♣ ✷❂✛✷✳ ❙❝❛❧✐♥❣ ❧✐♠✐t✿ ❞✐✛✉s✐♦♥ ✐♥ ❧♦❣✲❆✐r② ♣♦t❡♥t✐❛❧
❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛✉s✐♦♥ ✇❛s ❛❧r❡❛❞② ❞❡r✐✈❡❞ ✐♥ ❬❋❡rr❛r✐✫❙♣♦❤♥✱ ❆♦P ✷✵✵✺❪ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❛ ❇r♦✇♥✐❛♥ ❜r✐❞❣❡ ❝♦♥❞✐t✐♦♥❡❞ t♦ st❛② ❛❜♦✈❡ ❛ ❝✐r❝✉❧❛r ❜❛rr✐❡r✳
▼❛✐♥ r❡s✉❧t
❊①❛♠♣❧❡✿ ❱✕✭①✮ ❂ ✕① ■♥ t❤✐s ❝❛s❡✿ ❍✕ ❂ ✕✶❂✸❀ ✬✵✭r✮ ❂ ❆✐✭✤r ✦✶✮❀ ❡✵ ❂ ✦✶ ✤ ❀ ✇❤❡r❡ ✦✶ ✐s t❤❡ ✜rst ③❡r♦ ♦❢ t❤❡ ❆✐r② ❢✉♥❝t✐♦♥ ❆✐ ❛♥❞ ✤ ❂
✸
♣ ✷❂✛✷✳ ❙❝❛❧✐♥❣ ❧✐♠✐t✿ ❞✐✛✉s✐♦♥ ✐♥ ❧♦❣✲❆✐r② ♣♦t❡♥t✐❛❧
❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛✉s✐♦♥ ✇❛s ❛❧r❡❛❞② ❞❡r✐✈❡❞ ✐♥ ❬❋❡rr❛r✐✫❙♣♦❤♥✱ ❆♦P ✷✵✵✺❪ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❛ ❇r♦✇♥✐❛♥ ❜r✐❞❣❡ ❝♦♥❞✐t✐♦♥❡❞ t♦ st❛② ❛❜♦✈❡ ❛ ❝✐r❝✉❧❛r ❜❛rr✐❡r✳
❚r❛♥s❢❡r ♦♣❡r❛t♦r
✭❚♦ s✐♠♣❧✐❢②✱ ■ ❛ss✉♠❡ ❤❡r❡ t❤❛t ♣① ❂ ♣①✮ ✽①❀ ② ✷ N❀ ❡ T✕✭①❀ ②✮ ❂ ♣②① ❡ ✶
✷ ✭❱✕✭x✮✰❱✕✭y✮✮
◆♦t❡ t❤❛t ❡
✶ ✷ ✭❱✕✭ ✮✰❱✕✭ ✮✮ ❩ ❀ ◆❀✰❀✕ ❂ ❡✷◆ ✕ ✭ ❀ ✮
❑r❡✐♥✲❘✉t♠❛♥ ❂ ✮ ❡✕ ♣♦ss❡ss❡s ❛ ❧❡❛❞✐♥❣ ❡✳❢✳ ✣✕ ❃ ✵ ♦❢ ❡✳✈✳ ❊✕ ◆♦r♠❛❧✐③❡❞ ✈❡rs✐♦♥✿
✕ ❂ ✶ ❊✕ ❡✕
✙✕✭ ❀ ✮ ❂ ✶ ✣✕✭ ✮
✕✭ ❀ ✮✣✕✭ ✮
P♦s✳ r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✱ ✇✐t❤ ✐♥✈✳ ♠❡❛s✳ ✖✕✭ ✮ ❂ ❝✕✣✕✭ ✮✷ ❙t❛t✐♦♥❛r② ♣❛t❤✲♠❡❛s✉r❡✿
✕
❚r❛♥s❢❡r ♦♣❡r❛t♦r
✭❚♦ s✐♠♣❧✐❢②✱ ■ ❛ss✉♠❡ ❤❡r❡ t❤❛t ♣① ❂ ♣①✮ ✽①❀ ② ✷ N❀ ❡ T✕✭①❀ ②✮ ❂ ♣②① ❡ ✶
✷ ✭❱✕✭x✮✰❱✕✭y✮✮
◆♦t❡ t❤❛t ❡
✶ ✷ ✭❱✕✭u✮✰❱✕✭v✮✮ ❩u❀v ◆❀✰❀✕ ❂ ❡
T✷◆
✕ ✭u❀ v✮
❑r❡✐♥✲❘✉t♠❛♥ ❂ ✮ ❡✕ ♣♦ss❡ss❡s ❛ ❧❡❛❞✐♥❣ ❡✳❢✳ ✣✕ ❃ ✵ ♦❢ ❡✳✈✳ ❊✕ ◆♦r♠❛❧✐③❡❞ ✈❡rs✐♦♥✿
✕ ❂ ✶ ❊✕ ❡✕
✙✕✭ ❀ ✮ ❂ ✶ ✣✕✭ ✮
✕✭ ❀ ✮✣✕✭ ✮
P♦s✳ r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✱ ✇✐t❤ ✐♥✈✳ ♠❡❛s✳ ✖✕✭ ✮ ❂ ❝✕✣✕✭ ✮✷ ❙t❛t✐♦♥❛r② ♣❛t❤✲♠❡❛s✉r❡✿
✕
❚r❛♥s❢❡r ♦♣❡r❛t♦r
✭❚♦ s✐♠♣❧✐❢②✱ ■ ❛ss✉♠❡ ❤❡r❡ t❤❛t ♣① ❂ ♣①✮ ✽①❀ ② ✷ N❀ ❡ T✕✭①❀ ②✮ ❂ ♣②① ❡ ✶
✷ ✭❱✕✭x✮✰❱✕✭y✮✮
◆♦t❡ t❤❛t ❡
✶ ✷ ✭❱✕✭u✮✰❱✕✭v✮✮ ❩u❀v ◆❀✰❀✕ ❂ ❡
T✷◆
✕ ✭u❀ v✮
❑r❡✐♥✲❘✉t♠❛♥ ❂ ✮ ❡ T✕ ♣♦ss❡ss❡s ❛ ❧❡❛❞✐♥❣ ❡✳❢✳ ✣✕ ❃ ✵ ♦❢ ❡✳✈✳ ❊✕ ◆♦r♠❛❧✐③❡❞ ✈❡rs✐♦♥✿ T✕ ❂
✶ ❊✕ ❡
T✕
✙✕✭ ❀ ✮ ❂ ✶ ✣✕✭ ✮
✕✭ ❀ ✮✣✕✭ ✮
P♦s✳ r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✱ ✇✐t❤ ✐♥✈✳ ♠❡❛s✳ ✖✕✭ ✮ ❂ ❝✕✣✕✭ ✮✷ ❙t❛t✐♦♥❛r② ♣❛t❤✲♠❡❛s✉r❡✿
✕
❚r❛♥s❢❡r ♦♣❡r❛t♦r
✭❚♦ s✐♠♣❧✐❢②✱ ■ ❛ss✉♠❡ ❤❡r❡ t❤❛t ♣① ❂ ♣①✮ ✽①❀ ② ✷ N❀ ❡ T✕✭①❀ ②✮ ❂ ♣②① ❡ ✶
✷ ✭❱✕✭x✮✰❱✕✭y✮✮
◆♦t❡ t❤❛t ❡
✶ ✷ ✭❱✕✭u✮✰❱✕✭v✮✮ ❩u❀v ◆❀✰❀✕ ❂ ❡
T✷◆
✕ ✭u❀ v✮
❑r❡✐♥✲❘✉t♠❛♥ ❂ ✮ ❡ T✕ ♣♦ss❡ss❡s ❛ ❧❡❛❞✐♥❣ ❡✳❢✳ ✣✕ ❃ ✵ ♦❢ ❡✳✈✳ ❊✕ ◆♦r♠❛❧✐③❡❞ ✈❡rs✐♦♥✿ T✕ ❂
✶ ❊✕ ❡
T✕
✙✕✭x❀ y✮ ❂ ✶ ✣✕✭x✮T✕✭x❀ y✮✣✕✭y✮ P♦s✳ r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✱ ✇✐t❤ ✐♥✈✳ ♠❡❛s✳ ✖✕✭x✮ ❂ ❝✕✣✕✭x✮✷ ❙t❛t✐♦♥❛r② ♣❛t❤✲♠❡❛s✉r❡✿ P✕
❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❢❞❞s
▲❡t N✕ ❂ ❍✶
✕ N✳
E✕❢✉✵✭①✕✭✵✮✮✉✶✭①✕✭t✮✮❣ ❂
❳
r❀s✷N✕
✖✕✭❍✕r✮ ✙
❬❍✷
✕t❪
✕
✭❍✕r❀ ❍✕s✮ ✉✵✭r✮✉✶✭s✮ ❆ss✉♠❡ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t✱ ❛s ✕ ★ ✵✱ ❍✕❝✕ ✦ ✶❀ ✣✕✭❍✕ ✮ ✦ ✬✵✭ ✮❀
❬❍✷
✕t❪
✕
❬❢✕❪✭❍✕ ✮ ✦
t❬❢❪✭ ✮
✇❤❡r❡
t ❂ ❡✭▲✰ ✵ ✮t ❛♥❞ ❢✕✭❍✕✁✮ ✦ ❢✭✁✮✳
❳
✷
✕
✖✕✭❍✕ ✮ ✁ ✁ ✁ ❂ ❝✕
❳
✷
✕
✣✕✭❍✕ ✮✷ ✁ ✁ ✁ ✦
❩
❞ ✬✵✭ ✮✷ ✁ ✁ ✁ ❂
❩
✖✵✭❞ ✮ ✁ ✁ ✁ ❛♥❞
❳
✷
✕
✙
❬❍✷
✕t❪
✕
❬✉✶✭❍✶
✕ ✁✮❪✭❍✕ ✮ ❂
✶ ✣✕✭❍✕ ✮
❬❍✷
✕t❪
✕
❬✣✕✉✶✭❍✶
✕ ✁✮❪✭❍✕ ✮
✦ ✶ ✬✵✭ ✮
t❬✬✵✉✶❪✭ ✮ ❂ ❡●✛❀qt❬✉✶❪✭ ✮
❚❤✉s✱
✕❢✉✵✭①✕✭✵✮✮✉✶✭①✕✭t✮✮❣ ✦ ✛❀q❢✉✵✭①✭✵✮✮✉✶✭①✭t✮✮❣
❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❢❞❞s
▲❡t N✕ ❂ ❍✶
✕ N✳
E✕❢✉✵✭①✕✭✵✮✮✉✶✭①✕✭t✮✮❣ ❂
❳
r❀s✷N✕
✖✕✭❍✕r✮ ✙
❬❍✷
✕t❪
✕
✭❍✕r❀ ❍✕s✮ ✉✵✭r✮✉✶✭s✮ ❆ss✉♠❡ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t✱ ❛s ✕ ★ ✵✱ ❍✕❝✕ ✦ ✶❀ ✣✕✭❍✕r✮ ✦ ✬✵✭r✮❀ T
❬❍✷
✕t❪
✕
❬❢✕❪✭❍✕r✮ ✦ Tt❬❢❪✭r✮ ✇❤❡r❡ Tt ❂ ❡✭▲✰e✵I✮t ❛♥❞ ❢✕✭❍✕✁✮ ✦ ❢✭✁✮✳
❳
✷
✕
✖✕✭❍✕ ✮ ✁ ✁ ✁ ❂ ❝✕
❳
✷
✕
✣✕✭❍✕ ✮✷ ✁ ✁ ✁ ✦
❩
❞ ✬✵✭ ✮✷ ✁ ✁ ✁ ❂
❩
✖✵✭❞ ✮ ✁ ✁ ✁ ❛♥❞
❳
✷
✕
✙
❬❍✷
✕t❪
✕
❬✉✶✭❍✶
✕ ✁✮❪✭❍✕ ✮ ❂
✶ ✣✕✭❍✕ ✮
❬❍✷
✕t❪
✕
❬✣✕✉✶✭❍✶
✕ ✁✮❪✭❍✕ ✮
✦ ✶ ✬✵✭ ✮
t❬✬✵✉✶❪✭ ✮ ❂ ❡●✛❀qt❬✉✶❪✭ ✮
❚❤✉s✱
✕❢✉✵✭①✕✭✵✮✮✉✶✭①✕✭t✮✮❣ ✦ ✛❀q❢✉✵✭①✭✵✮✮✉✶✭①✭t✮✮❣
❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❢❞❞s
▲❡t N✕ ❂ ❍✶
✕ N✳
E✕❢✉✵✭①✕✭✵✮✮✉✶✭①✕✭t✮✮❣ ❂
❳
r❀s✷N✕
✖✕✭❍✕r✮ ✙
❬❍✷
✕t❪
✕
✭❍✕r❀ ❍✕s✮ ✉✵✭r✮✉✶✭s✮ ❆ss✉♠❡ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t✱ ❛s ✕ ★ ✵✱ ❍✕❝✕ ✦ ✶❀ ✣✕✭❍✕r✮ ✦ ✬✵✭r✮❀ T
❬❍✷
✕t❪
✕
❬❢✕❪✭❍✕r✮ ✦ Tt❬❢❪✭r✮ ✇❤❡r❡ Tt ❂ ❡✭▲✰e✵I✮t ❛♥❞ ❢✕✭❍✕✁✮ ✦ ❢✭✁✮✳
❳
r✷N✕
✖✕✭❍✕r✮ ✁ ✁ ✁ ❂ ❝✕
❳
r✷N✕
✣✕✭❍✕r✮✷ ✁ ✁ ✁ ✦
❩
❞r ✬✵✭r✮✷ ✁ ✁ ✁ ❂
❩
✖✵✭❞r✮ ✁ ✁ ✁ ❛♥❞
❳
s✷N✕
✙
❬❍✷
✕t❪
✕
❬✉✶✭❍✶
✕ ✁✮❪✭❍✕r✮ ❂
✶ ✣✕✭❍✕r✮T
❬❍✷
✕t❪
✕
❬✣✕✉✶✭❍✶
✕ ✁✮❪✭❍✕r✮
✦ ✶ ✬✵✭r✮Tt❬✬✵✉✶❪✭r✮ ❂ ❡●✛❀qt❬✉✶❪✭r✮ ❚❤✉s✱
✕❢✉✵✭①✕✭✵✮✮✉✶✭①✕✭t✮✮❣ ✦ ✛❀q❢✉✵✭①✭✵✮✮✉✶✭①✭t✮✮❣
❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❢❞❞s
▲❡t N✕ ❂ ❍✶
✕ N✳
E✕❢✉✵✭①✕✭✵✮✮✉✶✭①✕✭t✮✮❣ ❂
❳
r❀s✷N✕
✖✕✭❍✕r✮ ✙
❬❍✷
✕t❪
✕
✭❍✕r❀ ❍✕s✮ ✉✵✭r✮✉✶✭s✮ ❆ss✉♠❡ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t✱ ❛s ✕ ★ ✵✱ ❍✕❝✕ ✦ ✶❀ ✣✕✭❍✕r✮ ✦ ✬✵✭r✮❀ T
❬❍✷
✕t❪
✕
❬❢✕❪✭❍✕r✮ ✦ Tt❬❢❪✭r✮ ✇❤❡r❡ Tt ❂ ❡✭▲✰e✵I✮t ❛♥❞ ❢✕✭❍✕✁✮ ✦ ❢✭✁✮✳
❳
r✷N✕
✖✕✭❍✕r✮ ✁ ✁ ✁ ❂ ❝✕
❳
r✷N✕
✣✕✭❍✕r✮✷ ✁ ✁ ✁ ✦
❩
❞r ✬✵✭r✮✷ ✁ ✁ ✁ ❂
❩
✖✵✭❞r✮ ✁ ✁ ✁ ❛♥❞
❳
s✷N✕
✙
❬❍✷
✕t❪
✕
❬✉✶✭❍✶
✕ ✁✮❪✭❍✕r✮ ❂
✶ ✣✕✭❍✕r✮T
❬❍✷
✕t❪
✕
❬✣✕✉✶✭❍✶
✕ ✁✮❪✭❍✕r✮
✦ ✶ ✬✵✭r✮Tt❬✬✵✉✶❪✭r✮ ❂ ❡●✛❀qt❬✉✶❪✭r✮ ❚❤✉s✱ E✕❢✉✵✭①✕✭✵✮✮✉✶✭①✕✭t✮✮❣ ✦ E✛❀q❢✉✵✭①✭✵✮✮✉✶✭①✭t✮✮❣
❚❤r❡❡ ♠❛✐♥ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥♣✉ts
✶✳ ❚❤❡ ❢r❡❡ ❡♥❡r❣② ✐s ♦❢ ♦r❞❡r ❍✷
✕
❙❡tt✐♥❣ e✕ ❂ ❍✷
✕ ❧♦❣ ❊✕✱ t❤✐s ✐♠♣❧✐❡s t❤❛t
✵ ❁ ❧✐♠ ✐♥❢
✕★✵
e✕ ✔ ❧✐♠ s✉♣
✕★✵
e✕ ❁ ✶✿ ❝♦♠♣❛❝t♥❡ss ♦❢ ✭❡✕✮✕❃✵ ✷✳ ❚❛✐❧ ❡st✐♠❛t❡✿
❀ ◆❀✰❀✕◆ ✭❳✵ ❃ ❑❍✕✮ ✔ ❡①♣
✟ ✗❑ ♣ q✭❑✮ ❫ ❍✕ ✁✠ ✉♥✐❢♦r♠❧② ✐♥ ❑ ❃ ✵ ❛♥❞ ✕ ✔ ✕✵✳ t✐❣❤t♥❡ss ♦❢ ✭①✕❀
✕✮ ❛♥❞ ❝♦♠♣❛❝t♥❡ss ♦❢ ✭✣✕✭❍✕✁✮✮✕❃✵
✸✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❜② st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥ ❙✉♣♣♦s❡ ◆ ✢ ❍✷
✕◆ ❛♥❞ ◆❀ ◆ ✔ ❝❍✕◆ ✳ ❚❤❡♥✱ ❢♦r ❛♥② ❧♦❝❛❧ ❡✈❡♥t ❆✱
❧✐♠
◆✦✶ ❥ ❀ ◆❀✰❀✕◆ ✭❆✮ ✕◆ ✭❆✮❥ ❂ ✵✿
s✉✣❝✐❡♥t t♦ ♣r♦✈❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❢❞❞ ✉♥❞❡r
✕◆
❚❤r❡❡ ♠❛✐♥ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥♣✉ts
✶✳ ❚❤❡ ❢r❡❡ ❡♥❡r❣② ✐s ♦❢ ♦r❞❡r ❍✷
✕
❙❡tt✐♥❣ e✕ ❂ ❍✷
✕ ❧♦❣ ❊✕✱ t❤✐s ✐♠♣❧✐❡s t❤❛t
✵ ❁ ❧✐♠ ✐♥❢
✕★✵
e✕ ✔ ❧✐♠ s✉♣
✕★✵
e✕ ❁ ✶✿ ❝♦♠♣❛❝t♥❡ss ♦❢ ✭❡✕✮✕❃✵ ✷✳ ❚❛✐❧ ❡st✐♠❛t❡✿ Pu❀v
◆❀✰❀✕◆ ✭❳✵ ❃ ❑❍✕✮ ✔ ❡①♣
✟ ✗❑ ♣ q✭❑✮ ❫ ❍✕ ✁✠ ✉♥✐❢♦r♠❧② ✐♥ ❑ ❃ ✵ ❛♥❞ ✕ ✔ ✕✵✳ t✐❣❤t♥❡ss ♦❢ ✭①✕❀ P✕✮ ❛♥❞ ❝♦♠♣❛❝t♥❡ss ♦❢ ✭✣✕✭❍✕✁✮✮✕❃✵ ✸✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❜② st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥ ❙✉♣♣♦s❡ ◆ ✢ ❍✷
✕◆ ❛♥❞ ◆❀ ◆ ✔ ❝❍✕◆ ✳ ❚❤❡♥✱ ❢♦r ❛♥② ❧♦❝❛❧ ❡✈❡♥t ❆✱
❧✐♠
◆✦✶ ❥ ❀ ◆❀✰❀✕◆ ✭❆✮ ✕◆ ✭❆✮❥ ❂ ✵✿
s✉✣❝✐❡♥t t♦ ♣r♦✈❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❢❞❞ ✉♥❞❡r
✕◆
❚❤r❡❡ ♠❛✐♥ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥♣✉ts
✶✳ ❚❤❡ ❢r❡❡ ❡♥❡r❣② ✐s ♦❢ ♦r❞❡r ❍✷
✕
❙❡tt✐♥❣ e✕ ❂ ❍✷
✕ ❧♦❣ ❊✕✱ t❤✐s ✐♠♣❧✐❡s t❤❛t
✵ ❁ ❧✐♠ ✐♥❢
✕★✵
e✕ ✔ ❧✐♠ s✉♣
✕★✵
e✕ ❁ ✶✿ ❝♦♠♣❛❝t♥❡ss ♦❢ ✭❡✕✮✕❃✵ ✷✳ ❚❛✐❧ ❡st✐♠❛t❡✿ Pu❀v
◆❀✰❀✕◆ ✭❳✵ ❃ ❑❍✕✮ ✔ ❡①♣
✟ ✗❑ ♣ q✭❑✮ ❫ ❍✕ ✁✠ ✉♥✐❢♦r♠❧② ✐♥ ❑ ❃ ✵ ❛♥❞ ✕ ✔ ✕✵✳ t✐❣❤t♥❡ss ♦❢ ✭①✕❀ P✕✮ ❛♥❞ ❝♦♠♣❛❝t♥❡ss ♦❢ ✭✣✕✭❍✕✁✮✮✕❃✵ ✸✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❜② st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥ ❙✉♣♣♦s❡ ◆ ✢ ❍✷
✕◆ ❛♥❞ u◆❀ v◆ ✔ ❝❍✕◆ ✳ ❚❤❡♥✱ ❢♦r ❛♥② ❧♦❝❛❧ ❡✈❡♥t ❆✱
❧✐♠
◆✦✶ ❥Pu❀v ◆❀✰❀✕◆ ✭❆✮ P✕◆ ✭❆✮❥ ❂ ✵✿
s✉✣❝✐❡♥t t♦ ♣r♦✈❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❢❞❞ ✉♥❞❡r P✕◆
❈♦♥✈❡r❣❡♥❝❡ ♦❢ s❡♠✐❣r♦✉♣
▲❡t L✕❢✭r✮ ❂ T✕I
❍✷
✕
❢✭r✮ ✏❋❛❝t✑✿ ❧✐♠✕★✵ T
❜❍✷
✕t❝
✕
❢✕ ❂ ❡✭L✰eI✮t❢ ❢♦❧❧♦✇s ❢r♦♠ ❧✐♠✕★✵ L✕✉✕ ❂ ✭L ✰ eI✮✉ ◮ ❈♦♠♣✉t❛t✐♦♥✿ ❆ss✉♠❡ t❤❛t ❧✐♠
✕★✵ e✕ ❂ e
❊✕❡❱✕✭❍✕r✮
⑤ ④③ ⑥
✦✶
L✕✉✕✭r✮ ❂ ✶ ❤✷
✕
❳
s✷N✕
♣✕✭❍✕s ❍✕r✮ ❡
❱✕✭❍✕r✮❱✕✭❍✕s✮ ✷
✉✭s✮ ✉✭r✮✁
⑤ ④③ ⑥
✦ ✶
✷✛✷✉✵✵✭r✮
✰ ✶ ❊✕❡❱✕✭❍✕r✮ ❤✷
✕
✉✭r✮
⑤ ④③ ⑥
✦ ✭e q✭r✮✮✉✭r✮
❈♦♥✈❡r❣❡♥❝❡ ♦❢ s❡♠✐❣r♦✉♣
▲❡t L✕❢✭r✮ ❂ T✕I
❍✷
✕
❢✭r✮ ✏❋❛❝t✑✿ ❧✐♠✕★✵ T
❜❍✷
✕t❝
✕
❢✕ ❂ ❡✭L✰eI✮t❢ ❢♦❧❧♦✇s ❢r♦♠ ❧✐♠✕★✵ L✕✉✕ ❂ ✭L ✰ eI✮✉ ◮ ❈♦♠♣✉t❛t✐♦♥✿ ❆ss✉♠❡ t❤❛t ❧✐♠
✕★✵ e✕ ❂ e
❊✕❡❱✕✭❍✕r✮
⑤ ④③ ⑥
✦✶
L✕✉✕✭r✮ ❂ ✶ ❤✷
✕
❳
s✷N✕
♣✕✭❍✕s ❍✕r✮ ❡
❱✕✭❍✕r✮❱✕✭❍✕s✮ ✷
✉✭s✮ ✉✭r✮✁
⑤ ④③ ⑥
✦ ✶
✷✛✷✉✵✵✭r✮
✰ ✶ ❊✕❡❱✕✭❍✕r✮ ❤✷
✕
✉✭r✮
⑤ ④③ ⑥
✦ ✭e q✭r✮✮✉✭r✮
❈♦♥✈❡r❣❡♥❝❡ ♦❢ s❡♠✐❣r♦✉♣
▲❡t L✕❢✭r✮ ❂ T✕I
❍✷
✕
❢✭r✮ ✏❋❛❝t✑✿ ❧✐♠✕★✵ T
❜❍✷
✕t❝
✕
❢✕ ❂ ❡✭L✰eI✮t❢ ❢♦❧❧♦✇s ❢r♦♠ ❧✐♠✕★✵ L✕✉✕ ❂ ✭L ✰ eI✮✉ ◮ ❈♦♠♣✉t❛t✐♦♥✿ ❆ss✉♠❡ t❤❛t ❧✐♠
✕★✵ e✕ ❂ e
❊✕❡❱✕✭❍✕r✮
⑤ ④③ ⑥
✦✶
L✕✉✕✭r✮ ❂ ✶ ❤✷
✕
❳
s✷N✕
♣✕✭❍✕s ❍✕r✮ ❡
❱✕✭❍✕r✮❱✕✭❍✕s✮ ✷
✉✭s✮ ✉✭r✮✁
⑤ ④③ ⑥
✦ ✶
✷✛✷✉✵✵✭r✮
✰ ✶ ❊✕❡❱✕✭❍✕r✮ ❤✷
✕
✉✭r✮
⑤ ④③ ⑥
✦ ✭e q✭r✮✮✉✭r✮
■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t
❖♥❡ t❤❡♥ ❡❛s✐❧② ❞❡❞✉❝❡✿ Pr♦♣♦s✐t✐♦♥ e✵ ❂ ❧✐♠
✕★✵ e✕❀
✬✵✭r✮ ❂ ❧✐♠
✕★✵ ✣✕✭❍✕r✮
■♥❞❡❡❞✱ ✇✳❧✳♦✳❣✳✱ ❝♦♥s✐❞❡r ❛ s✉❜s❡q✉❡♥❝❡ ✭✕❦✮❦❃✵ s✉❝❤ t❤❛t
✕❦ ✦
❛♥❞ ✣✕❦✭❍✕❦✁✮ ✦ ✬✭✁✮
✕✣✕ ❂ ✣✕ ❂
✮ ❡✭ ✰ ✮t✬ ❂ ✬ ❂ ✮ ✬ ✐s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✭♥♦r♠❛❧✐③❡❞✮ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ ✳ ❂ ✮ ✬ ❂ ✬✵ ❛♥❞ ❂
✵
■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t
❖♥❡ t❤❡♥ ❡❛s✐❧② ❞❡❞✉❝❡✿ Pr♦♣♦s✐t✐♦♥ e✵ ❂ ❧✐♠
✕★✵ e✕❀
✬✵✭r✮ ❂ ❧✐♠
✕★✵ ✣✕✭❍✕r✮
■♥❞❡❡❞✱ ✇✳❧✳♦✳❣✳✱ ❝♦♥s✐❞❡r ❛ s✉❜s❡q✉❡♥❝❡ ✭✕❦✮❦❃✵ s✉❝❤ t❤❛t e✕❦ ✦ e ❛♥❞ ✣✕❦✭❍✕❦✁✮ ✦ ✬✭✁✮
✕✣✕ ❂ ✣✕ ❂
✮ ❡✭ ✰ ✮t✬ ❂ ✬ ❂ ✮ ✬ ✐s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✭♥♦r♠❛❧✐③❡❞✮ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ ✳ ❂ ✮ ✬ ❂ ✬✵ ❛♥❞ ❂
✵
■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t
❖♥❡ t❤❡♥ ❡❛s✐❧② ❞❡❞✉❝❡✿ Pr♦♣♦s✐t✐♦♥ e✵ ❂ ❧✐♠
✕★✵ e✕❀
✬✵✭r✮ ❂ ❧✐♠
✕★✵ ✣✕✭❍✕r✮
■♥❞❡❡❞✱ ✇✳❧✳♦✳❣✳✱ ❝♦♥s✐❞❡r ❛ s✉❜s❡q✉❡♥❝❡ ✭✕❦✮❦❃✵ s✉❝❤ t❤❛t e✕❦ ✦ e ❛♥❞ ✣✕❦✭❍✕❦✁✮ ✦ ✬✭✁✮ T✕✣✕ ❂ ✣✕ ❂ ✮ ❡✭L✰eI✮t✬ ❂ ✬ ❂ ✮ ✬ ✐s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✭♥♦r♠❛❧✐③❡❞✮ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ ✳ ❂ ✮ ✬ ❂ ✬✵ ❛♥❞ ❂
✵
■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t
❖♥❡ t❤❡♥ ❡❛s✐❧② ❞❡❞✉❝❡✿ Pr♦♣♦s✐t✐♦♥ e✵ ❂ ❧✐♠
✕★✵ e✕❀
✬✵✭r✮ ❂ ❧✐♠
✕★✵ ✣✕✭❍✕r✮
■♥❞❡❡❞✱ ✇✳❧✳♦✳❣✳✱ ❝♦♥s✐❞❡r ❛ s✉❜s❡q✉❡♥❝❡ ✭✕❦✮❦❃✵ s✉❝❤ t❤❛t e✕❦ ✦ e ❛♥❞ ✣✕❦✭❍✕❦✁✮ ✦ ✬✭✁✮ T✕✣✕ ❂ ✣✕ ❂ ✮ ❡✭L✰eI✮t✬ ❂ ✬ ❂ ✮ ✬ ✐s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✭♥♦r♠❛❧✐③❡❞✮ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ L ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ e✳ ❂ ✮ ✬ ❂ ✬✵ ❛♥❞ ❂
✵
■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t
❖♥❡ t❤❡♥ ❡❛s✐❧② ❞❡❞✉❝❡✿ Pr♦♣♦s✐t✐♦♥ e✵ ❂ ❧✐♠
✕★✵ e✕❀
✬✵✭r✮ ❂ ❧✐♠
✕★✵ ✣✕✭❍✕r✮
■♥❞❡❡❞✱ ✇✳❧✳♦✳❣✳✱ ❝♦♥s✐❞❡r ❛ s✉❜s❡q✉❡♥❝❡ ✭✕❦✮❦❃✵ s✉❝❤ t❤❛t e✕❦ ✦ e ❛♥❞ ✣✕❦✭❍✕❦✁✮ ✦ ✬✭✁✮ T✕✣✕ ❂ ✣✕ ❂ ✮ ❡✭L✰eI✮t✬ ❂ ✬ ❂ ✮ ✬ ✐s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✭♥♦r♠❛❧✐③❡❞✮ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ L ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ e✳ ❂ ✮ ✬ ❂ ✬✵ ❛♥❞ e ❂ e✵