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❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ❲❛t❡rs❤❡❞s

❆ ✉♥✐❢②✐♥❣ ❣r❛♣❤✲❜❛s❡❞ ♦♣t✐♠✐③❛t✐♦♥ ❢r❛♠❡✇♦r❦ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡✶✱ ▲❡♦ ●r❛❞②✷✱ ▲❛✉r❡♥t ◆❛❥♠❛♥✶✱ ❍✉❣✉❡s ❚❛❧❜♦t✶

✶▲■●▼✱ ❯P❊✲▼▲❱ ✷❙✐❡♠❡♥s ❈♦r♣♦r❛t❡ ❘❡s❡❛r❝❤

■◆❘■❆✱ ◆♦✈❡♠❜❡r ✹✱ ✷✵✵✾

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❆ ♥❡✇ s❡❡❞❡❞ ✐♠❛❣❡ s❡❣♠❡♥t❛t✐♦♥ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ t♦ ❡♥❡r❣② ♠✐♥✐♠✐③❛t✐♦♥ ❘❡s✉❧ts ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

■♥tr♦❞✉❝t✐♦♥

❋r❛♠❡✇♦r❦ ❣❛t❤❡r✐♥❣ ❣r❛♣❤ ❜❛s❡❞ s❡❣♠❡♥t❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❆♥ ✐♠❛❣❡ s❡❡♥ ❛s ❛ ❣r❛♣❤ ● = (❱ ❀ ❊) ■♠❛❣❡ ✸ ✂ ✸ ✦ ❲❡✐❣❤t❡❞ ❣r❛♣❤ ✸ ✂ ✸ ❙❡❡❞ s♣❡❝✐✜❝❛t✐♦♥

3 4 1 3 4 1 3 4 2 2 3 2

❊❞❣❡s ❛r❡ ✇❡✐❣❤t❡❞ ❜② ❛ ❞✐ss✐♠✐❧❛r✐t② ♠❡❛s✉r❡ ✐✳❡ ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✐♠❛❣❡ ❣r❛❞✐❡♥t ◆♦t❛t✐♦♥s ✿ ✈✐ ✈❥ ❡✐❥ ❊❞❣❡ ✇❡✐❣❤t ❡✐❥ ❂ ✇✐❥

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

■♥tr♦❞✉❝t✐♦♥

❋r❛♠❡✇♦r❦ ❣❛t❤❡r✐♥❣ ❣r❛♣❤ ❜❛s❡❞ s❡❣♠❡♥t❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❆♥ ✐♠❛❣❡ s❡❡♥ ❛s ❛ ❣r❛♣❤ ● = (❱ ❀ ❊) ■♠❛❣❡ ✸ ✂ ✸ ✦ ❲❡✐❣❤t❡❞ ❣r❛♣❤ ✸ ✂ ✸ ❙❡❡❞ s♣❡❝✐✜❝❛t✐♦♥

3 4 1 3 4 1 3 4 2 2 3 2

❊❞❣❡s ❛r❡ ✇❡✐❣❤t❡❞ ❜② ❛ ❞✐ss✐♠✐❧❛r✐t② ♠❡❛s✉r❡ ✐✳❡ ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✐♠❛❣❡ ❣r❛❞✐❡♥t ◆♦t❛t✐♦♥s ✿ ✈✐ ✈❥ ❡✐❥ ❊❞❣❡ ✇❡✐❣❤t ❡✐❥ ❂ ✇✐❥

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

■♥tr♦❞✉❝t✐♦♥

❋r❛♠❡✇♦r❦ ❣❛t❤❡r✐♥❣ ❣r❛♣❤ ❜❛s❡❞ s❡❣♠❡♥t❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❆♥ ✐♠❛❣❡ s❡❡♥ ❛s ❛ ❣r❛♣❤ ● = (❱ ❀ ❊) ■♠❛❣❡ ✸ ✂ ✸ ✦ ❲❡✐❣❤t❡❞ ❣r❛♣❤ ✸ ✂ ✸ ❙❡❡❞ s♣❡❝✐✜❝❛t✐♦♥

3 4 1 3 4 1 3 4 2 2 3 2

❊❞❣❡s ❛r❡ ✇❡✐❣❤t❡❞ ❜② ❛ ❞✐ss✐♠✐❧❛r✐t② ♠❡❛s✉r❡ ✐✳❡ ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✐♠❛❣❡ ❣r❛❞✐❡♥t ◆♦t❛t✐♦♥s ✿ ✈✐ ✈❥ ❡✐❥ ❊❞❣❡ ✇❡✐❣❤t ❡✐❥ ❂ ✇✐❥

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

■♥tr♦❞✉❝t✐♦♥

❋r❛♠❡✇♦r❦ ❣❛t❤❡r✐♥❣ ❣r❛♣❤ ❜❛s❡❞ s❡❣♠❡♥t❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❆♥ ✐♠❛❣❡ s❡❡♥ ❛s ❛ ❣r❛♣❤ ● = (❱ ❀ ❊) ■♠❛❣❡ ✸ ✂ ✸ ✦ ❲❡✐❣❤t❡❞ ❣r❛♣❤ ✸ ✂ ✸ ❙❡❡❞ s♣❡❝✐✜❝❛t✐♦♥

3 4 1 3 4 1 3 4 2 2 3 2

❊❞❣❡s ❛r❡ ✇❡✐❣❤t❡❞ ❜② ❛ ❞✐ss✐♠✐❧❛r✐t② ♠❡❛s✉r❡ ✐✳❡ ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✐♠❛❣❡ ❣r❛❞✐❡♥t ◆♦t❛t✐♦♥s ✿ ✈✐ ✈❥ ❡✐❥ ❊❞❣❡ ✇❡✐❣❤t ❡✐❥ ❂ ✇✐❥

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

◆❡✇ ❢r❛♠❡✇♦r❦ ❢♦r ✐♠❛❣❡ s❡❣♠❡♥t❛t✐♦♥

• ✐✈❡♥

   t✇♦ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ♣ ❛♥❞ q s❡❡❞s ❢♦r t❤❡ ❜❛❝❦❣r♦✉♥❞ ❇❀ s❡❡❞s ❢♦r t❤❡ ❢♦r❡❣r♦✉♥❞ ❋❀

❈♦♠♣✉t❡ ① ✈❡r✐❢②✐♥❣ ♠✐♥

① ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q

❙✉❝❤ t❤❛t ① ❋ ✶❀ ① ❇ ✵✳ ❘❡s✉❧t ✿ s❡❣♠❡♥t❛t✐♦♥ s ❞❡✜♥❡❞ ✽✐ ❜② s✐ ✶ s✐ ①✐ ✕ ✶

✷❀

✵ s✐ ①✐ ❁ ✶

✷✿

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

◆❡✇ ❢r❛♠❡✇♦r❦ ❢♦r ✐♠❛❣❡ s❡❣♠❡♥t❛t✐♦♥

• ✐✈❡♥

   t✇♦ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ♣ ❛♥❞ q s❡❡❞s ❢♦r t❤❡ ❜❛❝❦❣r♦✉♥❞ ❇❀ s❡❡❞s ❢♦r t❤❡ ❢♦r❡❣r♦✉♥❞ ❋❀

❈♦♠♣✉t❡ ① ✈❡r✐❢②✐♥❣ ♠✐♥

①

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q

❙✉❝❤ t❤❛t ①(❋) = ✶❀ ①(❇) = ✵✳ ❘❡s✉❧t ✿ s❡❣♠❡♥t❛t✐♦♥ s ❞❡✜♥❡❞ ✽✐ ❜② s✐ ✶ s✐ ①✐ ✕ ✶

✷❀

✵ s✐ ①✐ ❁ ✶

✷✿

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

◆❡✇ ❢r❛♠❡✇♦r❦ ❢♦r ✐♠❛❣❡ s❡❣♠❡♥t❛t✐♦♥

• ✐✈❡♥

   t✇♦ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ♣ ❛♥❞ q s❡❡❞s ❢♦r t❤❡ ❜❛❝❦❣r♦✉♥❞ ❇❀ s❡❡❞s ❢♦r t❤❡ ❢♦r❡❣r♦✉♥❞ ❋❀

❈♦♠♣✉t❡ ① ✈❡r✐❢②✐♥❣ ♠✐♥

①

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q

❙✉❝❤ t❤❛t ①(❋) = ✶❀ ①(❇) = ✵✳ ❘❡s✉❧t ✿ s❡❣♠❡♥t❛t✐♦♥ s ❞❡✜♥❡❞ ✽✐ ❜② s✐ = ✶ s✐ ①✐ ✕ ✶

✷❀

✵ s✐ ①✐ ❁ ✶

✷✿

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

• r❛♣❤ ❈✉ts

Pr♦❜❧❡♠ ✿ ❝♦♠♣✉t❡ ① ① = ❛r❣ ♠✐♥

• ❡✐❥✷❊

✇✐❥ ♣=✶❥①✐ ①❥❥q=✶ ▼✐♥ ❝✉t ✴ ▼❛① ✢♦✇ ❞✉❛❧✐t② ▼❛① ❋❧♦✇ ❛❧❣♦r✐t❤♠

S T 1 1 2 2 3 4 1 3 4 1 3 4 2 2 3 2 ∞ ∞ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✺ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

• r❛♣❤ ❈✉ts

Pr♦❜❧❡♠ ✿ ❝♦♠♣✉t❡ ① ① = ❛r❣ ♠✐♥

• ❡✐❥✷❊

✇✐❥❥①✐ ①❥❥ ▼✐♥ ❝✉t ✴ ▼❛① ✢♦✇ ❞✉❛❧✐t② ▼❛① ❋❧♦✇ ❛❧❣♦r✐t❤♠

S T 1 1 2 2 3 4 1 3 4 1 3 4 2 2 3 2 ∞ ∞ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✺ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

• r❛♣❤ ❈✉ts

Pr♦❜❧❡♠ ✿ ❝♦♠♣✉t❡ ① ① = ❛r❣ ♠✐♥

• ❡✐❥✷❊

✇✐❥❥①✐ ①❥❥ ▼✐♥ ❝✉t ✴ ▼❛① ✢♦✇ ❞✉❛❧✐t② ▼❛① ❋❧♦✇ ❛❧❣♦r✐t❤♠

S T 3 4 3 4 3 4 3 2 ∞ ∞ 1 1 2 2 ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✻ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

• r❛♣❤ ❈✉ts

Pr♦❜❧❡♠ ✿ ❝♦♠♣✉t❡ ① ① = ❛r❣ ♠✐♥

• ❡✐❥✷❊

✇✐❥❥①✐ ①❥❥ ▼✐♥ ❝✉t ✴ ▼❛① ✢♦✇ ❞✉❛❧✐t② ▼❛① ❋❧♦✇ ❛❧❣♦r✐t❤♠

S T 1 1 1 1 1 3 4 3 4 3 4 3 2 ∞ ∞ 1 1 2 2 ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

• r❛♣❤ ❈✉ts ✿ ❡①❛♠♣❧❡

❢❛✈♦rs s♠❛❧❧ ❜♦✉♥❞❛r✐❡s r♦❜✉st t♦ ✉♥❝❡♥t❡r❡❞ s❡❡❞ ♣❧❛❝❡♠❡♥t ■♥♣✉t s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

• r❛♣❤ ❈✉ts ✿ ❡①❛♠♣❧❡

❢❛✈♦rs s♠❛❧❧ ❜♦✉♥❞❛r✐❡s r♦❜✉st t♦ ✉♥❝❡♥t❡r❡❞ s❡❡❞ ♣❧❛❝❡♠❡♥t ■♥♣✉t s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✾ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❘❛♥❞♦♠ ❲❛❧❦❡r

❉✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠

① = ❛r❣ ♠✐♥ X

❡✐❥ ✷❊

✇✐❥(①✐ ①❥)✷ ✥ ✉ = ❛r❣ ♠✐♥ Z

Ω

❥r✉❥✷❞Ω

0.7 0.8 1 0.2 0.7 0.8 0.2 0.3 3 4 1 3 4 1 3 4 2 2 3 2

P♦t❡♥t✐❛❧s ❛♥❛❧♦❣② ❘❛♥❞♦♠ ✇❛❧❦❡r ❛♥❛❧♦❣②

0.7 0.8 1 0.2 0.7 0.8 0.2 0.3 3 4 1 3 4 1 3 4 2 2 3 2

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✵ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❘❛♥❞♦♠ ❲❛❧❦❡r

❉✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠

① = ❛r❣ ♠✐♥ X

❡✐❥ ✷❊

✇✐❥

♣=✶(①✐ ①❥)q=✷

✥ ✉ = ❛r❣ ♠✐♥ Z

Ω

❥r✉❥✷❞Ω

0.7 0.8 1 0.2 0.7 0.8 0.2 0.3 3 4 1 3 4 1 3 4 2 2 3 2

P♦t❡♥t✐❛❧s ❛♥❛❧♦❣② ❘❛♥❞♦♠ ✇❛❧❦❡r ❛♥❛❧♦❣②

0.7 0.8 1 0.2 0.7 0.8 0.2 0.3 3 4 1 3 4 1 3 4 2 2 3 2

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✵ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❘❛♥❞♦♠ ❲❛❧❦❡r

❉✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠

① = ❛r❣ ♠✐♥ X

❡✐❥ ✷❊

✇✐❥(①✐ ①❥)✷ ✥ ✉ = ❛r❣ ♠✐♥ Z

Ω

❥r✉❥✷❞Ω

0.7 0.8 1 0.2 0.7 0.8 0.2 0.3 3 4 1 3 4 1 3 4 2 2 3 2

P♦t❡♥t✐❛❧s ❛♥❛❧♦❣② ❘❛♥❞♦♠ ✇❛❧❦❡r ❛♥❛❧♦❣②

0.7 0.8 1 0.2 0.7 0.8 0.2 0.3 3 4 1 3 4 1 3 4 2 2 3 2

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✵ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❘❛♥❞♦♠ ❲❛❧❦❡r

❉✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠

① = ❛r❣ ♠✐♥ X

❡✐❥ ✷❊

✇✐❥(①✐ ①❥)✷ ✥ ✉ = ❛r❣ ♠✐♥ Z

Ω

❥r✉❥✷❞Ω

0.7 0.8 1 0.2 0.7 0.8 0.2 0.3 3 4 1 3 4 1 3 4 2 2 3 2

❙tr✐❝❧② ❝♦♥✈❡① ♣r♦❜❧❡♠ ✮ ✉♥✐q✉❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ①✄

P♦t❡♥t✐❛❧s ❛♥❛❧♦❣② ❘❛♥❞♦♠ ✇❛❧❦❡r ❛♥❛❧♦❣②

0.7 0.8 1 0.2 0.7 0.8 0.2 0.3 3 4 1 3 4 1 3 4 2 2 3 2

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❘❛♥❞♦♠ ❲❛❧❦❡r ✿ ❡①❛♠♣❧❡

■♥♣✉t s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t②✴♣♦t❡♥t✐❛❧ ① ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s ✭t❤r❡s❤♦❧❞ ♦❢ ①✮

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✷ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❙❤♦rt❡st ♣❛t❤ ❢♦r❡st

t❛❦❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✇❡✐❣❤ts t❤❡ s❤♦rt❡st ♣❛t❤ st❛rt✐♥❣ ❢r♦♠ ❡❛❝❤ ♥♦❞❡ t♦ r❡❛❝❤ ❛ s❡❡❞ ♥♦❞❡ ✐s ❝♦♠♣✉t❡❞ ❉✐❥s❦tr❛ ❛❧❣♦r✐t❤♠ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ✿ ♦♣t✐♠✐③❡s ♠✐♥

①

• ❡✐❥✷❊

✇✐❥ ♣=q✦✶(①✐ ①❥)q✦✶

F B

1 3 1 4

1

1 3 1 4

1

1 3 1 4 1 2 1 2 1 3 1 2

3 4 1 3 4 1 3 4 2 2 3 2 ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❙❤♦rt❡st ♣❛t❤ ❢♦r❡st

t❛❦❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✇❡✐❣❤ts t❤❡ s❤♦rt❡st ♣❛t❤ st❛rt✐♥❣ ❢r♦♠ ❡❛❝❤ ♥♦❞❡ t♦ r❡❛❝❤ ❛ s❡❡❞ ♥♦❞❡ ✐s ❝♦♠♣✉t❡❞ ❉✐❥s❦tr❛ ❛❧❣♦r✐t❤♠ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ✿ ♦♣t✐♠✐③❡s ♠✐♥

①

• ❡✐❥✷❊

✇✐❥ ♣=q✦✶(①✐ ①❥)q✦✶

F B

1 3 1 4

1

1 3 1 4

1

1 3 1 4 1 2 1 2 1 3 1 2

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✹ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❙❤♦rt❡st ♣❛t❤ ❢♦r❡st

t❛❦❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✇❡✐❣❤ts t❤❡ s❤♦rt❡st ♣❛t❤ st❛rt✐♥❣ ❢r♦♠ ❡❛❝❤ ♥♦❞❡ t♦ r❡❛❝❤ ❛ s❡❡❞ ♥♦❞❡ ✐s ❝♦♠♣✉t❡❞ ❉✐❥s❦tr❛ ❛❧❣♦r✐t❤♠ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ✿ ♦♣t✐♠✐③❡s ♠✐♥

①

• ❡✐❥✷❊

✇✐❥ ♣=q✦✶(①✐ ①❥)q✦✶

F B

1 3 1 4

1

1 3 1 4

1

1 3 1 4 1 2 1 2 1 3 1 2

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✺ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ✿ ❡①❛♠♣❧❡

❱❡r② s❡♥s✐t✐✈❡ t♦ s❡❡❞s ♣❧❛❝❡♠❡♥t

■♥♣✉t s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✻ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 4 3 1 3 4 1 3 4 2 2 3 2

❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 3 3 4 4 4

❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 3 3 4 4 4

❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 3 3 4 4 4

❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 3 4 4 4 3

❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 4 4 3 4 3

❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 2 1 1 3 2 2 2 3 4 4 3 4 3

❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 1 1 3 2 2 3 4 4 3 4 3 2

❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 4 3 1 3 4 1 3 4 2 2 3 2

Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 3 4 3 4 4

Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 3 4 3 4 4

Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 3 4 4 4 3

Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 3 4 4 3 4 3 2 1 1 3 2 2 2 3 3 4 4 4 3

Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 2 1 1 3 2 2 2 3 4 4 3 4 3

Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 2 1 1 3 2 2 2 3 4 4 3 4 3

Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮

♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s

F B 1 1 3 2 2 3 4 4 3 4 3 2

Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ ✿ ❡①❛♠♣❧❡

r♦❜✉st t♦ s♠❛❧❧ s❡❡❞s ✿ ♥♦ ❜✐❛s t♦✇❛r❞ s♠❛❧❧ ♦❜❥❡❝ts ❧❡❛❦✐♥❣ ❡✛❡❝t

■♥♣✉t s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❙❋ ❛♥❞ ❲❛t❡rs❤❡❞s ❝♦rr❡s♣♦♥❞❡♥❝❡

✇❛t❡rs❤❡❞ ✿ ♣♦✐♥ts ✇❤❡r❡ ❛ ❞r♦♣ ♦❢ ✇❛t❡r ❝♦✉❧❞ ✢♦✇ t♦✇❛r❞ ❞✐✛❡r❡♥t ❝❛❝❤♠❡♥t ❜❛ss✐♥s ❬❈♦✉st② ❡t ❛❧✳ ✵✼❪✳ ❣r❛❞✐❡♥t s❡❡♥ ❛s ❛ r❡❧✐❡❢ ♠❛♣✳ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ✐♥✈❡rs❡ ♦❢ ❣r❛❞✐❡♥t ✦ t❤❛❧✇❡❣ ❧✐♥❡ ✢♦♦❞✐♥❣ ❛❧❣♦r✐t❤♠ ✑ Pr✐♠ ❛❧❣♦r✐t❤♠ ❢♦r ▼❙❋

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✾ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

▼❙❋ ❛♥❞ ❲❛t❡rs❤❡❞s ❝♦rr❡s♣♦♥❞❡♥❝❡

✇❛t❡rs❤❡❞ ✿ ♣♦✐♥ts ✇❤❡r❡ ❛ ❞r♦♣ ♦❢ ✇❛t❡r ❝♦✉❧❞ ✢♦✇ t♦✇❛r❞ ❞✐✛❡r❡♥t ❝❛❝❤♠❡♥t ❜❛ss✐♥s ❬❈♦✉st② ❡t ❛❧✳ ✵✼❪✳ ❣r❛❞✐❡♥t s❡❡♥ ❛s ❛ r❡❧✐❡❢ ♠❛♣✳ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ✐♥✈❡rs❡ ♦❢ ❣r❛❞✐❡♥t ✦ t❤❛❧✇❡❣ ❧✐♥❡ ✢♦♦❞✐♥❣ ❛❧❣♦r✐t❤♠ ✑ Pr✐♠ ❛❧❣♦r✐t❤♠ ❢♦r ▼❙❋

❚❤❡♦r❡♠ ❬❈♦✉st② ❡t ❛❧✳ ✵✼❪ ■❢ s❡❡❞s ❛r❡ t❤❡ ♠❛①✐♠❛ ♦❢ t❤❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✱ ❡✈❡r② ♠❛①✐♠✉♠ s♣❛♥♥✐♥❣ ❢♦r❡st ❝✉t ♦♥ t❤❛t ❢✉♥❝t✐♦♥ ✐s ❛ ✇❛t❡rs❤❡❞ ❝✉t✳

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✾ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❊①❛♠♣❧❡ ♦❢ s❡❣♠❡♥t❛t✐♦♥ ❜② ✢♦♦❞✐♥❣✴Pr✐♠ ❛❧❣♦r✐t❤♠

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✵ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❍❍❍❍

q ♣ ✵ ✜♥✐t❡ ✶ ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s

• r❛♣❤ ❝✉ts

▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ✷ ❵✷✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ✶ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❍❍❍❍

q ♣ ✵ ✜♥✐t❡ ✶ ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s

• r❛♣❤ ❝✉ts

▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ✷ ❵✷✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ✶ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❍❍❍❍

q ♣ ✵ ✜♥✐t❡ ✶ ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s

• r❛♣❤ ❝✉ts

▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ✷ ❵✷✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ✶ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❍❍❍❍

q ♣ ✵ ✜♥✐t❡ ✶ ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s

• r❛♣❤ ❝✉ts

▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ✷ ❵✷✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ✶ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❍❍❍❍

q ♣ ✵ ✜♥✐t❡ ✶ ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s

• r❛♣❤ ❝✉ts

▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ✷ ❵✷✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ✶ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❍❍❍❍

q ♣ ✵ ✜♥✐t❡ ✶ ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s

• r❛♣❤ ❝✉ts

▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ✷ ❵✷✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ✶ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❍❍❍❍

q ♣ ✵ ✜♥✐t❡ ✶ ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s

• r❛♣❤ ❝✉ts

▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ✷ ❵✷✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ✶ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❵✶✲♥♦r♠ ❱♦r♦♥♦✐ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

Pr♦♣❡rt✐❡s

❉❡✜♥✐t✐♦♥ ▲❡t s ❜❡ t❤❡ s❡❣♠❡♥t❛t✐♦♥ ❞❡✜♥❡❞ ❜② ❛ t❤r❡s❤♦❧❞✐♥❣ ♦❢ t❤❡ ❧❛❜❡❧s ① = ❛r❣ ♠✐♥

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q✿

❚❤❡ s❡t ♦❢ ❡❞❣❡s ❡✐❥ t❤❛t ✈❡r✐❢② s✐ ✻= s❥ ❝♦♥st✐t✉t❡ ❛ q✲❝✉t ❢♦r ✇ ♣✳ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✷ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

Pr♦♣❡rt✐❡s

❉❡✜♥✐t✐♦♥ ▲❡t s ❜❡ t❤❡ s❡❣♠❡♥t❛t✐♦♥ ❞❡✜♥❡❞ ❜② ❛ t❤r❡s❤♦❧❞✐♥❣ ♦❢ t❤❡ ❧❛❜❡❧s ① = ❛r❣ ♠✐♥

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q✿

❚❤❡ s❡t ♦❢ ❡❞❣❡s ❡✐❥ t❤❛t ✈❡r✐❢② s✐ ✻= s❥ ❝♦♥st✐t✉t❡ ❛ q✲❝✉t ❢♦r ✇ ♣✳ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✷ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

F B

5 7 9 8 1 2 3 6 4 10

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 7 9 8 1 2 3 6 4 10

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 7 9 8 1 2 3 6 4 10

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 10 7 10 9 10 8 10 1 10 2 10 3 10 6 10 4 10 10 10

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

( 5

10 )p

( 7

10 )p

( 9

10 )p

( 8

10 )p

( 1

10 )p

( 2

10 )p

( 3

10 )p

( 6

10 )p

( 4

10 )p

( 10

10 )p

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

1

x7 x8

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 7 9 8 1 2 3 6 4

x7 x8

①✼ = ①✽ ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 7 9 8 1 2 3 6 4

x7 x8

①✼ = ①✽ ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

( 5

9 )p

( 7

9 )p

( 9

9 )p

( 8

9 )p

( 1

9 )p

( 2

9 )p

( 3

9 )p

( 6

9 )p

( 4

9 )p

x7 x8

①✼ = ①✽ ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

1

x7 x8 1

①✼ = ①✽ ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 7 9 8 1 2 3 6 4 10

x7 x8 1

①✼ = ①✽ ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 7 9 8 1 2 3 6 4 10

x7 x8 1

①✼ = ①✽ ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 7 9 8 1 2 3 6 4 10

1 1 1

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1

5 7 9 8 1 2 3 6 4 10

1 1 1

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1 1

5 7 9 8 1 2 3 6 4 10

1 1 1

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1 1

5 7 9 8 1 2 3 6 4 10

1 1 1

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1 1

5 7 9 8 1 2 3 6 4 10

1 1 1

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1 1

5 7 9 8 1 2 3 6 4 10

1 1 1

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥

❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ●✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳

1 1 1

5 7 9 8 1 2 3 6 4 10

1 1 1

❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥①

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞

❆❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt✐❡s

❈♦r♦❧❧❛r② ❲❤❡♥ ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❞✐✛❡r❡♥t✱ t❤❡ s♦❧✉t✐♦♥ ①✄ t♦ ❧✐♠

♣✦✶ ♠✐♥ ①

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q

r❡❧❛t✐✈❡ t♦ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐s ❜✐♥❛r②✳ Pr♦♣❡rt② ■♥ t❤❡ ❝❛s❡ ♦❢ ❛r❜✐tr❛r② ✇❡✐❣❤ts ✭♥♦♥✲♥❡❝❡ss❛r✐❧② ❡q✉❛❧✮✱ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❤♦❧❞s ✐❢ ❛❧❧ t❤❡ s❡❡❞s ▼ ❛r❡ t❤❡ ♠❛①✐♠❛ ♦❢ t❤❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✳

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✹ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❆❧❣♦r✐t❤♠ ❢♦r t❤❡ ❝❛s❡ ♣ = ✶✱ ✈❛r✐❛❜❧❡ q

❈♦♠♣✉t❡ ① ♠✐♥✐♠✐③✐♥❣ ❧✐♠♣✦✶

• ❡✐❥✷❊

✇✐❥ ♣❥①✐ ①❥❥q s✉❜❥❡❝t t♦ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❲❡ ❝♦♥str✉❝t ❛♥ ▼❙❋ ♦✉ts✐❞❡ ♦❢ ♣❧❛t❡❛✉s✱ ❛♥❞ ♦♣t✐♠✐③❡

• ❡✐❥✷♣❧❛t❡❛✉

❥①✐ ①❥❥q ♦♥ t❤❡ ♣❧❛t❡❛✉s✳ ❲❡ ❝❛❧❧ t❤✐s ❛❧❣♦r✐t❤♠ ✏P♦✇❡r ✇❛t❡rs❤❡❞✑

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✺ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❘❲ ✇❤❡♥ ♣ ✦ ✶ t♦✇❛r❞ P❲

■♥♣✉t s❡❡❞s P♦✇❡r❲❛t❡rs❤❡❞ q = ✷ ❘❛♥❞♦♠ ❲❛❧❦❡r ♣ = ✶✿✿✿✸✵ ❘❛♥❞♦♠ ❲❛❧❦❡r ♣ = ✸✵

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✻ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✶ ✵ ✷ ✶ ✺ ✸ ✺ ✾ ✹ ✺ ✺ ✽ ✺ ✾ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✹ ✹ ✾

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✵

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵

♠✐♥① ❧✐♠♣✦✶

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵

♠✐♥① ❧✐♠♣✦✶

• ❡✐❥✷❊ ✇ ♣

✐❥ ❥①✐ ①❥❥✷

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵

♠✐♥①

• ❡✐❥✷♣❧❛t❡❛✉ ❥①✐ ①❥❥q

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵✳✸✺ ✵✳✼✶ ✵✳✹✽ ✵✳✼✹

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✾ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵✳✸✺ ✵✳✼✶ ✵✳✹✽ ✵✳✼✹

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✾ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵✳✸✺ ✵✳✼✶ ✵✳✹✽ ✵✳✼✹

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✾ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵✳✸✺ ✵✳✼✶ ✵✳✹✽ ✵✳✼✹ ✵✳✸✷ ✵✳✺✺

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✵ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠

✶

❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✇❡✐❣❤t ❡♠❛①✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡♠❛① ✇✐t❤ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡♠❛①✳

✷

■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊✶❀q ♦♥ ❙✱ ♦t❤❡r✇✐s❡ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ♥♦❞❡✳

✸

❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳

✷ ✶ ✾ ✾ ✺ ✸ ✺ ✹ ✺ ✺ ✽ ✺ ✺ ✺ ✼ ✺ ✺ ✾ ✼ ✹ ✹ ✾ ✹ ✹ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵✿✸✺ ✵✳✸✺ ✵✳✼✶ ✵✳✹✽ ✵✳✼✹ ✵✳✸✷ ✵✳✺✺

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✵ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

Pr♦♣❡rt✐❡s

❚❤❡♦r❡♠ ❚❤❡ ❝✉t ♦❜t❛✐♥❡❞ ❜② t❤❡ P♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✐s ❛ ▼❙❋ ❝✉t✳ ❚❤❡♦r❡♠ ❲❤❡♥ q ❃ ✶✱ t❤❡ s♦❧✉t✐♦♥ ①✄ t♦ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ ♠✐♥

① ❧✐♠♣✦✶

• ❡✐❥✷❊

✇✐❥ ♣❥①✐ ①❥❥q ✐s ✉♥✐q✉❡✳ ❚❤✉s✱ ✇❤❡♥ q ❃ ✶✱ t❤❡ s♦❧✉t✐♦♥ ① ♦❜t❛✐♥❡❞ ❜② t❤❡ P♦✲ ✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✐s ✉♥✐q✉❡✳

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r✇❛t❡rs❤❡❞ ✭q❂✷✮ ✿ ❡①❛♠♣❧❡

r♦❜✉st ✐♥ ❝❛s❡ ♦❢ s♠❛❧❧ s❡❡❞s ❧❡ss ❧❡❛❦✐♥❣ t❤❛♥ ✇✐t❤ st❛♥❞❛r❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st

■♥♣✉t s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t② ① ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s ✭t❤r❡s❤♦❧❞ ♦❢ ①✮

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✷ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r✇❛t❡rs❤❡❞ ✭q❂✷✮ ✿ ❡①❛♠♣❧❡

■♥♣✉t s❡❡❞s P♦✇❡r✇❛t❡rs❤❡❞ ✭q = ✷✮ Pr✐♠ ❛❧❣♦r✐t❤♠ ❢♦r ▼❙❋

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

P♦✇❡r✇❛t❡rs❤❡❞ ✭q❂✷✮ ✿ ❡①❛♠♣❧❡

■♥♣✉t s❡❡❞s P♦✇❡r✇❛t❡rs❤❡❞ ✭q = ✷✮ Pr✐♠ ❛❧❣♦r✐t❤♠ ❢♦r ▼❙❋

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✹ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• ❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦

P♦ss✐❜✐❧✐t② t♦ ❛❞❞ ✉♥❛r② t❡r♠s t♦ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ♠✐♥

①

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q ✈✐

✇❋✐

♣❥①✐ ✶❥q ✈✐

✇❇✐ ♣❥①✐❥q

wB wF

F B

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❢♦r❡st ✐♥ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✺ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• ❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦

P♦ss✐❜✐❧✐t② t♦ ❛❞❞ ✉♥❛r② t❡r♠s t♦ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ♠✐♥

①

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q +

• ✈✐

✇❋✐

♣❥①✐ ✶❥q +

• ✈✐

✇❇✐ ♣❥①✐❥q

wB wF

F B

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❢♦r❡st ✐♥ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✺ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• ❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦

P♦ss✐❜✐❧✐t② t♦ ❛❞❞ ✉♥❛r② t❡r♠s t♦ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ♠✐♥

①

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q +

• ✈✐

✇❋✐

♣❥①✐ ✶❥q +

• ✈✐

✇❇✐ ♣❥①✐❥q

wB wF

F B

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❢♦r❡st ✐♥ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤

1 9 2 8 6 4 4 6 8 2 7 3 3 7 9 1 5 5 5 1 2 3 4 2 6 6 6

F B

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✺ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• ❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦

♠✐♥

①

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q +

• ✈✐

✇❋✐

♣❥①✐ ✶❥q +

• ✈✐

✇❇✐

♣❥①✐❥q

wB wF F B

■♠❛❣❡

• r❛♣❤ ❈✉ts

▼❛①❙❋ ❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ t❤✐s ✐s t❤❡ ✜rst t✐♠❡ t❤❛t ✇❛t❡rs❤❡❞ ✐s ✉s❡❞ ✐♥ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s t❤❛♥ s❡❡❞❡❞ s❡❣♠❡♥t❛t✐♦♥

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✻ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

• ❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦

♠✐♥

①

• ❡✐❥✷❊

✇ ♣

✐❥ ❥①✐ ①❥❥q +

• ✈✐

✇❋✐

♣❥①✐ ✶❥q +

• ✈✐

✇❇✐

♣❥①✐❥q

wB wF F B

■♠❛❣❡

• r❛♣❤ ❈✉ts

▼❛①❙❋ ❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ t❤✐s ✐s t❤❡ ✜rst t✐♠❡ t❤❛t ✇❛t❡rs❤❡❞ ✐s ✉s❡❞ ✐♥ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s t❤❛♥ s❡❡❞❡❞ s❡❣♠❡♥t❛t✐♦♥

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✻ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❖♣t✐♠❛❧ ♠✉❧t✐❧❛❜❡❧s s❡❣♠❡♥t❛t✐♦♥

▼♦r❡ t❤❛♥ ✷✲❧❛❜❡❧s s❡❣♠❡♥t❛t✐♦♥ ✿ ◆P✲❤❛r❞ ❢♦r ●r❛♣❤ ❝✉ts ❊①❛❝t ♥ ✕ ✷ ❧❛❜❡❧s s❡❣♠❡♥t❛t✐♦♥ ❢♦r t❤❡ ♦t❤❡r ❛❧❣♦r✐t❤♠s ✿ ♥ s♦❧✉t✐♦♥s ①✶❀ ①✷❀ ✿✿✿①♥ ❝♦♠♣✉t❡❞ ①❦ ❝♦♠♣✉t❡❞ ❜② ❡♥❢♦r❝✐♥❣  ①❦(♥❦) = ✶ ①❦(♥q) = ✵ ❢♦r ❛❧❧ q ✻= ❦✿ ❊❛❝❤ ♥♦❞❡ ✐ ✐s ❛✛❡❝t❡❞ t♦ t❤❡ ❧❛❜❡❧ ❢♦r ✇❤✐❝❤ ①❦

✐ ✐s ♠❛①✐♠✉♠ ✿

s✐ = ❛r❣ ♠❛①

❦

①❦

✐

■♥♣✉t s❡❡❞s ❙❡❣♠❡♥t❛t✐♦♥ ❜② P♦✇❡r❲❛t❡rs❤❡❞ ✭q = ✷✮

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❆❧❣♦r✐t❤♠s ❜❡❤❛✈✐♦r ♦♥ ♣❧❛t❡❛✉s

❙❡❡❞❡❞ ✐♠❛❣❡

• r❛♣❤

❈✉ts ❙❤♦rt❡st P❛t❤s✱ ▼❛①❙❋ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ P❲ q = ✷

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❆❧❣♦r✐t❤♠s ❜❡❤❛✈✐♦r ♦♥ ♣❧❛t❡❛✉s

❙❡❡❞❡❞ ✐♠❛❣❡

• r❛♣❤

❈✉ts ❙❤♦rt❡st P❛t❤s✱ ▼❛①❙❋ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ P❲ q = ✷

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❆❧❣♦r✐t❤♠s ❜❡❤❛✈✐♦r ♦♥ ♣❧❛t❡❛✉s

❙❡❡❞❡❞ ✐♠❛❣❡

• r❛♣❤

❈✉ts ❙❤♦rt❡st P❛t❤s✱ ▼❛①❙❋ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ P❲ q = ✷

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❆❧❣♦r✐t❤♠s ❝♦♠♣❛r✐s♦♥

❊✈❛❧✉❛t✐♦♥ ♦♥ ❇❡r❦❡❧❡② ❞❛t❛❜❛s❡

• r♦✉♥❞ tr✉t❤s

✷ s❡ts ♦❢ s❡❡❞s t♦ st✉❞② r♦❜✉st♥❡ss t♦ s❡❡❞s ❝❡♥t❡r✐♥❣ ✿

✶

✇❡❧❧ ❝❡♥t❡r❡❞ s❡❡❞s

✷

❧❡ss ❝❡♥t❡r❡❞ s❡❡❞s

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✾ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

◗✉❛♥t✐t❛t✐✈❡ ❘❡s✉❧ts

❉✐❝❡ ❝♦❡✛✳ ❜❡t✇❡❡♥ ❣r♦✉♥❞ tr✉t❤s ❛♥❞ t❤❡ ❛❧❣♦r✐t❤♠s r❡s✉❧ts ♦♥ ❇❡r❦❡❧❡② ❞❛t❛❜❛s❡ ✇✐t❤ t❤❡ ❝❡♥t❡r❡❞ s❡❡❞s✳

91,2 91,8 92,1 91,4 91,3

70 75 80 85 90 95 100

Graph cuts Random Walker Shortest Paths MSF PowerWatersheds q=2 Mean of Dice coefficient (%)

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✵ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❊①❛♠♣❧❡s

■♥♣✉t s❡❡❞s

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st P❛t❤s ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡sts P♦✇❡r ❲❛t❡rs❤❡❞s q = ✷

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✶ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

◗✉❛♥t✐t❛t✐✈❡ ❘❡s✉❧ts

❉✐❝❡ ❝♦❡✛✳ ❜❡t✇❡❡♥ ❣r♦✉♥❞ tr✉t❤s ❛♥❞ t❤❡ ❛❧❣♦r✐t❤♠s r❡s✉❧ts ♦♥ ❇❡r❦❡❧❡② ❞❛t❛❜❛s❡ ✇✐t❤ t❤❡ ❧❡ss ❝❡♥t❡r❡❞ s❡❡❞s✳

86 85,7 84,9 85,5 85,6

70 75 80 85 90 95 100

Graph cuts Random Walker Shortest Paths MSF PowerWatersheds q=2 Mean of Dice coefficient (%)

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✷ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❊①❛♠♣❧❡s

■♥♣✉t s❡❡❞s

• r❛♣❤ ❈✉ts

❘❛♥❞♦♠ ❲❛❧❦❡r ❙❤♦rt❡st P❛t❤s ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡sts P♦✇❡r ❲❛t❡rs❤❡❞s q = ✷

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✸ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❈♦♠♣✉t❛t✐♦♥ t✐♠❡ ✷❉

Untitled 1 Untitled 2 Untitled 3 Untitled 4 Untitled 5 Graph Cuts Random Walker Shortest Paths Forest Power Watershed (q=2) Max Spanning Forest (Prim) 3186000 5385224 8847500 12741000 7.7 20.3 40.15 76.5 3186000 5385224 8847500 12741000 5.62 10.81 19.17 29.71 3186000 5385224 8847500 12741000 5.8967162791 10.785725581 17.148516279 24.616046512 3186000 5385224 8847500 12741000 4.28 7.65 12.52 17.99 3186000 5385224 8847500 12741000 2.88 5.13 8.55 12.91

20 40 60 80 0E+00 3.0E+06 6.0E+06 9.0E+06 1.2E+07 1.5E+07

Computation times 2D

Time (seconds) Nb of pixels in the image Graph Cuts Random Walker Shortest Paths Forest Power Watershed (q=2) Max Spanning Forest (Prim)

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✹ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❈♦♠♣✉t❛t✐♦♥ t✐♠❡ ✸❉

Untitled 1 Untitled 2 Untitled 3 Untitled 4 Graph Cuts Random Walker Power Watersheds (q=2) Shortest Paths Forest Max Spanning Forest (Prim) 1801800 3082738 4956435 12.81 45 140 1801800 3082738 4956435 18.3964186 34.12046512 75.51837209 1801800 3082738 4956435 7.63 13.12 20.8 1801800 3082738 4956435 4.93 9.56 16.87 1801800 3082738 4956435 3.3 5.76 9.88

38 75 113 150 0E+00 1.5E+06 3.0E+06 4.5E+06 6.0E+06

Computation times 3D

Time (seconds) Nb of pixels in the image Graph Cuts Random Walker Power Watersheds (q=2) Shortest Paths Forest Max Spanning Forest (Prim)

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✺ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❲❤✐❝❤ ❛❧❣♦r✐t❤♠ t♦ ✉s❡ ❄

• r❛♣❤ ❈✉ts ✿

❣♦♦❞ ✜t ❢♦r ✷❉ ✐♠❛❣❡ s❡❣♠❡♥t❛t✐♦♥ ✇✐t❤ ✷ ❧❛❜❡❧s ♦♥❧② t♦♦ s❧♦✇ ❢♦r ✸❉ s❡❣♠❡♥t❛t✐♦♥

❙❤♦rt❡st P❛t❤s ✿ ❢❛st ❜✉t r❡q✉✐r❡s ✇❡❧❧ ❝❡♥t❡r❡❞ s❡❡❞s ❛r♦✉♥❞ t❤❡ ♦❜❥❡❝t ❘❛♥❞♦♠ ❲❛❧❦❡r ✿

❡✣❝✐❡♥t ✇✐t❤ ✉♥❝❡♥t❡r❡❞ s❡❡❞s ❞❡✜♥❡❞ ❜❡❤❛✈✐♦r ♦♥ ♣❧❛t❡❛✉s

▼❛① ❙❋ ✿

❜❡tt❡r s❡❣♠❡♥t❛t✐♦♥s t❤❛♥ ❙P❋ ✇✐t❤ ✉♥❝❡♥t❡r❡❞ s❡❡❞s ❢❛st ✦ ✸❉ s❡❣♠❡♥t❛t✐♦♥

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✻ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❲❤✐❝❤ ❛❧❣♦r✐t❤♠ t♦ ✉s❡ ❄

P♦✇❡r✇❛t❡rs❤❡❞ q = ✷ ✿

▼❛①❙❋ ♣r♦♣❡rt✐❡s ✭❢❛st✱ ♠✉❧t✐s❡❡❞s✮ ❘❛♥❞♦♠ ✇❛❧❦❡r ♣r♦♣❡rt✐❡s ♦♥ ♣❧❛t❡❛✉s ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❧❡ss s❡♥s✐t✐✈❡ t♦ ❧❡❛❦✐♥❣ t❤❛♥ st❛♥❞❛r❞ ▼❛①❙❋

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✼ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

◆❡✇ ❢r❛♠❡✇♦r❦ ✉♥✐❢②✐♥❣ ●r❛♣❤ ❈✉ts✱ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ ▼❙❋ ❛♥❞ ❙P❋✳ ◆❡✇ ♦♣t✐♠✐s❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢❛♠✐❧② ✇✐t❤ ✈❛r✐❛❜❧❡ q✳ ❚❤❡ q ✷ ❛❧❣♦r✐t❤♠ s❤♦✇s s❡❣♠❡♥t❛t✐♦♥ ✐♠♣r♦✈❡♠❡♥t ✇❤✐❧❡ r❡t❛✐♥✐♥❣ ✇❛t❡rs❤❡❞ s♣❡❡❞✳ ❯♥❛r② t❡r♠s ❢♦r♠✉❧❛t✐♦♥ ♠❛❦❡s ♣♦✇❡r✇❛t❡rs❤❡❞s ✉s❡❢✉❧ ❜❡②♦♥❞ s❡❣♠❡♥t❛t✐♦♥✳ P❡rs♣❡❝t✐✈❡s ❙t✉❞② ♦❢ t❤❡ ❞✐✛❡r❡♥t ❡♥❡r❣✐❡s ♣♦ss✐❜❧② ♠✐♥✐♠✐③❡❞ ✐♥ t❤✐s ❢r❛♠❡✇♦r❦

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

◆❡✇ ❢r❛♠❡✇♦r❦ ✉♥✐❢②✐♥❣ ●r❛♣❤ ❈✉ts✱ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ ▼❙❋ ❛♥❞ ❙P❋✳ ◆❡✇ ♦♣t✐♠✐s❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢❛♠✐❧② ✇✐t❤ ✈❛r✐❛❜❧❡ q✳ ❚❤❡ q ✷ ❛❧❣♦r✐t❤♠ s❤♦✇s s❡❣♠❡♥t❛t✐♦♥ ✐♠♣r♦✈❡♠❡♥t ✇❤✐❧❡ r❡t❛✐♥✐♥❣ ✇❛t❡rs❤❡❞ s♣❡❡❞✳ ❯♥❛r② t❡r♠s ❢♦r♠✉❧❛t✐♦♥ ♠❛❦❡s ♣♦✇❡r✇❛t❡rs❤❡❞s ✉s❡❢✉❧ ❜❡②♦♥❞ s❡❣♠❡♥t❛t✐♦♥✳ P❡rs♣❡❝t✐✈❡s ❙t✉❞② ♦❢ t❤❡ ❞✐✛❡r❡♥t ❡♥❡r❣✐❡s ♣♦ss✐❜❧② ♠✐♥✐♠✐③❡❞ ✐♥ t❤✐s ❢r❛♠❡✇♦r❦

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

◆❡✇ ❢r❛♠❡✇♦r❦ ✉♥✐❢②✐♥❣ ●r❛♣❤ ❈✉ts✱ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ ▼❙❋ ❛♥❞ ❙P❋✳ ◆❡✇ ♦♣t✐♠✐s❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢❛♠✐❧② ✇✐t❤ ✈❛r✐❛❜❧❡ q✳ ❚❤❡ q = ✷ ❛❧❣♦r✐t❤♠ s❤♦✇s s❡❣♠❡♥t❛t✐♦♥ ✐♠♣r♦✈❡♠❡♥t ✇❤✐❧❡ r❡t❛✐♥✐♥❣ ✇❛t❡rs❤❡❞ s♣❡❡❞✳ ❯♥❛r② t❡r♠s ❢♦r♠✉❧❛t✐♦♥ ♠❛❦❡s ♣♦✇❡r✇❛t❡rs❤❡❞s ✉s❡❢✉❧ ❜❡②♦♥❞ s❡❣♠❡♥t❛t✐♦♥✳ P❡rs♣❡❝t✐✈❡s ❙t✉❞② ♦❢ t❤❡ ❞✐✛❡r❡♥t ❡♥❡r❣✐❡s ♣♦ss✐❜❧② ♠✐♥✐♠✐③❡❞ ✐♥ t❤✐s ❢r❛♠❡✇♦r❦

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

◆❡✇ ❢r❛♠❡✇♦r❦ ✉♥✐❢②✐♥❣ ●r❛♣❤ ❈✉ts✱ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ ▼❙❋ ❛♥❞ ❙P❋✳ ◆❡✇ ♦♣t✐♠✐s❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢❛♠✐❧② ✇✐t❤ ✈❛r✐❛❜❧❡ q✳ ❚❤❡ q = ✷ ❛❧❣♦r✐t❤♠ s❤♦✇s s❡❣♠❡♥t❛t✐♦♥ ✐♠♣r♦✈❡♠❡♥t ✇❤✐❧❡ r❡t❛✐♥✐♥❣ ✇❛t❡rs❤❡❞ s♣❡❡❞✳ ❯♥❛r② t❡r♠s ❢♦r♠✉❧❛t✐♦♥ ♠❛❦❡s ♣♦✇❡r✇❛t❡rs❤❡❞s ✉s❡❢✉❧ ❜❡②♦♥❞ s❡❣♠❡♥t❛t✐♦♥✳ P❡rs♣❡❝t✐✈❡s ❙t✉❞② ♦❢ t❤❡ ❞✐✛❡r❡♥t ❡♥❡r❣✐❡s ♣♦ss✐❜❧② ♠✐♥✐♠✐③❡❞ ✐♥ t❤✐s ❢r❛♠❡✇♦r❦

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

◆❡✇ ❢r❛♠❡✇♦r❦ ✉♥✐❢②✐♥❣ ●r❛♣❤ ❈✉ts✱ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ ▼❙❋ ❛♥❞ ❙P❋✳ ◆❡✇ ♦♣t✐♠✐s❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢❛♠✐❧② ✇✐t❤ ✈❛r✐❛❜❧❡ q✳ ❚❤❡ q = ✷ ❛❧❣♦r✐t❤♠ s❤♦✇s s❡❣♠❡♥t❛t✐♦♥ ✐♠♣r♦✈❡♠❡♥t ✇❤✐❧❡ r❡t❛✐♥✐♥❣ ✇❛t❡rs❤❡❞ s♣❡❡❞✳ ❯♥❛r② t❡r♠s ❢♦r♠✉❧❛t✐♦♥ ♠❛❦❡s ♣♦✇❡r✇❛t❡rs❤❡❞s ✉s❡❢✉❧ ❜❡②♦♥❞ s❡❣♠❡♥t❛t✐♦♥✳ P❡rs♣❡❝t✐✈❡s ❙t✉❞② ♦❢ t❤❡ ❞✐✛❡r❡♥t ❡♥❡r❣✐❡s ♣♦ss✐❜❧② ♠✐♥✐♠✐③❡❞ ✐♥ t❤✐s ❢r❛♠❡✇♦r❦

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✽ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❘❡❢❡r❡♥❝❡s

❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ❆✳ ❑✳ ❙✐♥♦♣✱ ▲✳ ●r❛❞②✱ ✏❆ ❙❡❡❞❡❞ ■♠❛❣❡ ❙❡❣♠❡♥t❛t✐♦♥ ❋r❛♠❡✇♦r❦ ❯♥✐❢②✐♥❣ ●r❛♣❤ ❈✉ts ❛♥❞ ❘❛♥❞♦♠ ❲❛❧❦❡r ❲❤✐❝❤ ❨✐❡❧❞s ❛ ◆❡✇ ❆❧❣♦r✐t❤♠✑✱ Pr♦❝✳ ♦❢ ■❈❈❱ ✷✵✵✼✳ ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ❈✳ ❆❧❧è♥❡✱ ❏✲❨✳ ❆✉❞✐❜❡rt✱ ▼✳ ❈♦✉♣r✐❡✱ ❏✳ ❈♦✉st②✱ ❘✳ ❑❡r✐✈❡♥✱ ✏❙♦♠❡ ❧✐♥❦s ❜❡t✇❡❡♥ ♠✐♥ ❝✉ts✱ ♦♣t✐♠❛❧ s♣❛♥♥✐♥❣ ❢♦r❡sts ❛♥❞ ✇❛t❡rs❤❡❞s✑✳ Pr♦❝✳ ♦❢ ■❙▼▼ ✷✵✵✼✳ ❬❈♦✉st② ❡t ❛❧✳ ✵✾❪ ❏✳ ❈♦✉st②✱ ●✳ ❇❡rtr❛♥❞✱ ▲✳ ◆❛❥♠❛♥✱ ▼✳ ❈♦✉♣r✐❡✱ ✏❲❛t❡rs❤❡❞ ❝✉ts ✿ ♠✐♥✐♠✉♠ s♣❛♥♥✐♥❣ ❢♦r❡sts✱ ❛♥❞ t❤❡ ❞r♦♣ ♦❢ ✇❛t❡r ♣r✐♥❝✐♣❧❡✑✱ ✐♥ P❆▼■✱ ✷✵✵✾✳ ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ❈✳ ❈♦✉♣r✐❡✱ ▲✳ ●r❛❞②✱ ▲✳ ◆❛❥♠❛♥ ❛♥❞ ❍✳ ❚❛❧❜♦t✱ ✏P♦✇❡r ❲❛t❡rs❤❡❞s ✿ ❆ ◆❡✇ ■♠❛❣❡ ❙❡❣♠❡♥t❛t✐♦♥ ❋r❛♠❡✇♦r❦ ❊①t❡♥❞✐♥❣ ●r❛♣❤ ❈✉ts✱ ❘❛♥❞♦♠ ❲❛❧❦❡r ❛♥❞ ❖♣t✐♠❛❧ ❙♣❛♥♥✐♥❣ ❋♦r❡st✑✱ ✐♥ Pr♦❝✳ ♦❢ ■❈❈❱ ✷✵✵✾✳

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹✾ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

◗✉❡st✐♦♥s

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✺✵ ✴ ✺✶

❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥

❊①❛♠♣❧❡ ✇❤❡r❡ ❘❲ ✇✐t❤ ♣ ✦ ✶ ✐s ♥♦t ❛ ▼❙❋

1

1 2 3 2 1 0.45 0.49 0.51 0.55 1 1 2 3 2 1

0.5 0.5 0.5 0.5 1

1 2 3 2 1

❋✐❣✳✿ ❊①❛♠♣❧❡ ♦❢ ❣r❛♣❤ ✇❤❡r❡ t❤❡ q✲❝✉t ❝♦♠♣✉t❡❞ ❜② t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ ❊♣❀q ✐s ♥♦t ❛ ▼❛①❙❋ ❝✉t✳ ✭❛✮ ✇❡✐❣❤t❡❞ s❡❡❞❡❞ ❣r❛♣❤✱ ✭❜✮ ❘❛♥❞♦♠ ✇❛❧❦❡r r❡s✉❧t ✭q❂✷✮ ✇❤❡♥ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛t t❤❡ ♣♦✇❡r ♣❂✺✳ ❚❤❡ q✲❝✉t ✐s ✐♥ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❣r❛♣❤✳ ✭❝✮ P♦✇❡r✇❛t❡rs❤❡❞ r❡s✉❧t ✭q❂✷✮ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘❛♥❞♦♠ ✇❛❧❦❡r r❡s✉❧t ✭q❂✷✮ ✇❤❡♥ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ✇❡✐❣❤ts ❝♦♥✈❡r❣❡s t♦✇❛r❞ ✐♥✜♥✐t②✳

❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✺✶ ✴ ✺✶