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❙❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ❛ ❧❛②❡r ♦❢ ✉♥st❛❜❧❡ ♣❤❛s❡

❨✈❛♥ ❱❡❧❡♥✐❦

❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❉✐♠❛ ■♦✛❡ ❛♥❞ ❙❡♥②❛ ❙❤❧♦s♠❛♥

❙tr✉❝t✉r❡ ♦❢ t❤❡ t❛❧❦

✶

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡

✷

• ❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧

✸

❘❡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❛❧ ❡✛❡❝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✮ ❂

◆✶

❨

✐❂◆

♣❳✐✰✶❳✐

• ❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧

✷✳ ❚❤❡ ♣♦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☛

• ❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧

✷✳ ❚❤❡ ♣♦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☛

• ❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧

✷✳ ❚❤❡ ♣♦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☛

• ❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧

✷✳ ❚❤❡ ♣♦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☛

• ❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧

✸✳ ❚❤❡ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ Pr♦❜❛❜✐❧✐t② ♦❢ ❛ ♥♦♥♥❡❣❛t✐✈❡ tr❛❥❡❝t♦r② X ❂ ✭❳◆❀ ✿ ✿ ✿ ❀ ❳◆✮ ✇✐t❤ ❳◆ ❂ u✱ ❳◆ ❂ v✿ Pu❀v

◆❀✰❀✕✭X✮ ❂

✶ ❩u❀v

◆❀✰❀✕

❡①♣ ♥

• ◆

❳

✐❂◆

❱✕✭❳✐✮ ♦ p❘❲✭X✮

• ♦❛❧✿ ❉❡t❡r♠✐♥❡ t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ①✕✭t✮ ❂ ❍✶

✕ ❳❬❍✷

✕t❪ ❛s ✕ ★ ✵

• ❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧

✸✳ ❚❤❡ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ Pr♦❜❛❜✐❧✐t② ♦❢ ❛ ♥♦♥♥❡❣❛t✐✈❡ tr❛❥❡❝t♦r② X ❂ ✭❳◆❀ ✿ ✿ ✿ ❀ ❳◆✮ ✇✐t❤ ❳◆ ❂ u✱ ❳◆ ❂ v✿ Pu❀v

◆❀✰❀✕✭X✮ ❂

✶ ❩u❀v

◆❀✰❀✕

❡①♣ ♥

• ◆

❳

✐❂◆

❱✕✭❳✐✮ ♦ p❘❲✭X✮

• ♦❛❧✿ ❉❡t❡r♠✐♥❡ t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ①✕✭t✮ ❂ ❍✶

✕ ❳❬❍✷

✕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 ♦♥ ✭✵❀ ✶✮

• ❡♥❡r❛t♦rs✿

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 ♦♥ ✭✵❀ ✶✮

• ❡♥❡r❛t♦rs✿

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 ♦♥ ✭✵❀ ✶✮

• ❡♥❡r❛t♦rs✿

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✐♦♥✿

✕ ❂ ✶ ❊✕ ❡✕

• r♦✉♥❞✲st❛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♠❛♥ ❂ ✮ ❡✕ ♣♦ss❡ss❡s ❛ ❧❡❛❞✐♥❣ ❡✳❢✳ ✣✕ ❃ ✵ ♦❢ ❡✳✈✳ ❊✕ ◆♦r♠❛❧✐③❡❞ ✈❡rs✐♦♥✿

✕ ❂ ✶ ❊✕ ❡✕

• r♦✉♥❞✲st❛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✕

• r♦✉♥❞✲st❛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✕

• r♦✉♥❞✲st❛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✵

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦