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❖♥ t❤❡ ▲♦♥❣❡st P❛t❤s ✐♥ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ❆❜❜❛s ▼❡❤r❛❜✐❛♥ ❛♠❡❤r❛❜✐❅✉✇❛t❡r❧♦♦✳❝❛ ❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦ ❉✐s❝r❡t❡ ▼❛t❤s ❘❡s❡❛r❝❤ ●r♦✉♣ ▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✱ ▼❡❧❜♦✉r♥❡✱ ❆✉str❛❧✐❛ ▼❛② ✷✵t❤✱ ✷✵✶✸ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶ ✴ ✹✵

❝♦✲❛✉t❤♦rs ❊❤s❛♥ ❊❜r❛❤✐♠③❛❞❡❤ ▲✐♥❞❛ ❋❛r❝③❛❞✐ ❏❛♥❡ ●❛♦ ❈r✐st✐❛♥❡ ❙❛t♦ ◆✐❝❦ ❲♦r♠❛❧❞ ❏♦♥❛t❤❛♥ ❩✉♥❣ P✐❝t✉r❡s✿ ❈❤❛r❛❧❛♠♣♦s ✭❇❛❜✐s✮ ❚s♦✉r❛❦❛❦✐s P❛♣❡r✿ ❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✸✵✸✳✺✷✶✸ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✷ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦ ❆❢t❡r s✉❜❞✐✈✐❞✐♥❣ t t✐♠❡s✱ ❛ r❛♥❞♦♠ tr✐❛♥❣✉❧❛t❡❞ ♣❧❛♥❡ ❣r❛♣❤ t + ✸ ✈❡rt✐❝❡s ✸ t + ✸ ❡❞❣❡s ✷ t + ✶ ❢❛❝❡s ❝❛❧❧❡❞ ❛ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦ ✭❘❆◆✮✳ ❩❤♦✉✱ ❨❛♥✱ ❲❛♥❣✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ✭✷✵✵✺✮ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✶ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ▼♦❞❡❧❧✐♥❣ r❡❛❧✲✇♦r❧❞ ♥❡t✇♦r❦s✿ ❚❤❡ ■♥t❡r♥❡t ❚❤❡ ❲❡❜ ❣r❛♣❤ ✭❖♥❧✐♥❡✮ ❙♦❝✐❛❧ ♥❡t✇♦r❦s ❇r❛✐♥ ♥❡✉r♦♥s Pr♦t❡✐♥ ✐♥t❡r❛❝t✐♦♥s ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✷ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❘❡❛❧✲❲♦r❧❞ ◆❡t✇♦r❦s ✶ P♦✇❡r✲❧❛✇ ❞❡❣r❡❡ ❞✐str✐❜✉t✐♦♥✿ P [ ❞❡❣ ( ❛ r❛♥❞♦♠ ✈❡rt❡① ) = ❦ ] = ❈❦ − β ✷ ❙♠❛❧❧✲✇♦r❧❞ ♣❤❡♥♦♠❡♥♦♥ ✭s✐① ❞❡❣r❡❡s ♦❢ s❡♣❛r❛t✐♦♥✮ ✿ ❚❤❡r❡ ✐s ❛ s❤♦rt ♣❛t❤ ❝♦♥♥❡❝t✐♥❣ ❡✈❡r② ♣❛✐r ♦❢ ✈❡rt✐❝❡s✳ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✸ ✴ ✹✵

❘❆◆s ❛r❡ ❛♥ ✐♥t❡r❡st✐♥❣ ♠♦❞❡❧ ❢♦r ❣❡♥❡r❛t✐♥❣ r❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s✳ ■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❑♥♦✇♥ ♠♦❞❡❧s ✐♥❝❧✉❞❡✿ ❊r❞ös✲❘❡♥②✐ r❛♥❞♦♠ ❣r❛♣❤s ♣r❡❢❡r❡♥t✐❛❧ ❛tt❛❝❤♠❡♥t ♠♦❞❡❧s ❑r♦♥❡❝❦❡r ❣r❛♣❤s ❈♦♦♣❡r✲❋r✐❡③❡ ♠♦❞❡❧ ❘❛♥❞♦♠ s✉r❢❡r ❣r❛♣❤s ❋❛❜r✐❦❛♥t✲❑♦✉ts♦✉♣✐❛s✲P❛♣❛❞✐♠✐tr✐♦✉ ♠♦❞❡❧ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✹ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❑♥♦✇♥ ♠♦❞❡❧s ✐♥❝❧✉❞❡✿ ❊r❞ös✲❘❡♥②✐ r❛♥❞♦♠ ❣r❛♣❤s ♣r❡❢❡r❡♥t✐❛❧ ❛tt❛❝❤♠❡♥t ♠♦❞❡❧s ❑r♦♥❡❝❦❡r ❣r❛♣❤s ❈♦♦♣❡r✲❋r✐❡③❡ ♠♦❞❡❧ ❘❛♥❞♦♠ s✉r❢❡r ❣r❛♣❤s ❋❛❜r✐❦❛♥t✲❑♦✉ts♦✉♣✐❛s✲P❛♣❛❞✐♠✐tr✐♦✉ ♠♦❞❡❧ ❘❆◆s ❛r❡ ❛♥ ✐♥t❡r❡st✐♥❣ ♠♦❞❡❧ ❢♦r ❣❡♥❡r❛t✐♥❣ r❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s✳ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✹ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ❑♥♦✇♥ ❘❡s✉❧ts ❉❡❣r❡❡ s❡q✉❡♥❝❡ ♥ := ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ❩ ❦ , ♥ := ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ✇✐t❤ ❞❡❣r❡❡ ❦ ✳ ❚❤❡♦r❡♠ ✭❋r✐❡③❡ ❛♥❞ ❚s♦✉r❛❦❛❦✐s ✷✵✶✷✮ ❋♦r ❡✈❡r② ❦ = ❦ ( ♥ ) ≥ ✸ ✱ E [ ❩ ❦ , ♥ ] = Θ ( ♥❦ − ✸ ) , ❛♥❞✱ �� � � � � � � ❩ ❦ , ♥ − E [ ❩ ❦ , ♥ ] � > ✶✵ ♥ ❧♦❣ ♥ → ✵ P � � ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✺ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❉✐❛♠❡t❡r ♦❢ ❛ ●r❛♣❤ ❉✐❛♠❡t❡r ❂ ✸ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✻ ✴ ✹✵

❚❤❡♦r❡♠ ✭❊▲P❆◆❈❏✬✶✸✰✮ ❞✐❛♠❡t❡r ❝ ✶ ✻✻✽ ✐♥ ♣r♦❜❛❜✐❧✐t② ❧♦❣ ♥ ■♥tr♦❞✉❝t✐♦♥ ❑♥♦✇♥ ❘❡s✉❧ts ❚❤❡ ❞✐❛♠❡t❡r ❚❤❡♦r❡♠ ✭❆❧❜❡♥q✉❡ ❛♥❞ ▼❛r❝❦❡rt ✷✵✵✽✮ ❉✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ r❛♥❞♦♠ ✈❡rt✐❝❡s → ✵ . ✺✺ ❧♦❣ ♥✳ ❚❤❡♦r❡♠ ✭❋r✐❡③❡ ❛♥❞ ❚s♦✉r❛❦❛❦✐s ✷✵✶✷✮ P [ ❞✐❛♠❡t❡r > ✼ . ✶ ❧♦❣ ♥ ] → ✵ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✼ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ❑♥♦✇♥ ❘❡s✉❧ts ❚❤❡ ❞✐❛♠❡t❡r ❚❤❡♦r❡♠ ✭❆❧❜❡♥q✉❡ ❛♥❞ ▼❛r❝❦❡rt ✷✵✵✽✮ ❉✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ r❛♥❞♦♠ ✈❡rt✐❝❡s → ✵ . ✺✺ ❧♦❣ ♥✳ ❚❤❡♦r❡♠ ✭❋r✐❡③❡ ❛♥❞ ❚s♦✉r❛❦❛❦✐s ✷✵✶✷✮ P [ ❞✐❛♠❡t❡r > ✼ . ✶ ❧♦❣ ♥ ] → ✵ ❚❤❡♦r❡♠ ✭❊▲P❆◆❈❏✬✶✸✰✮ ❞✐❛♠❡t❡r → ❝ ≈ ✶ . ✻✻✽ ✐♥ ♣r♦❜❛❜✐❧✐t② ❧♦❣ ♥ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✼ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ❖✉r ❘❡s✉❧t ♦♥ t❤❡ ❉✐❛♠❡t❡r ❚❤❡♦r❡♠ ✭❊▲P❆◆❈❏✬✶✸✰✮ ❢ ( ① ) := ✶✷ ① ✸ ✶ − ✷ ① − ✻ ① ✸ ✶ − ① , ② := ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ① ( ① − ✶ ) ❢ ′ ( ① ) = ❢ ( ① ) ❧♦❣ ❢ ( ① ) , ① ∈ ( ✵ , ✶ / ✷ ) , ❝ := ( ✶ − ② − ✶ ) / ❧♦❣ ❢ ( ② ) ≈ ✶ . ✻✻✽ ❚❤❡♥ ❢♦r ❡✈❡r② ✜①❡❞ ε > ✵ ✱ P [( ✶ − ε ) ❝ ❧♦❣ ♥ ≤ ❞✐❛♠❡t❡r ≤ ( ✶ + ε ) ❝ ❧♦❣ ♥ ] → ✶ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✽ ✴ ✹✵

◆♦t tr✉❡✦ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▲♦♥❣❡st P❛t❤ ❚❤❡ ❝♦♥❥❡❝t✉r❡ ❈♦♥❥❡❝t✉r❡ ✭❋r✐❡③❡ ❛♥❞ ❚s♦✉r❛❦❛❦✐s ✷✵✶✷✮ P [ ∃ ❛ ♣❛t❤ ❝♦♥t❛✐♥✐♥❣ ❛ ♣♦s✐t✐✈❡ ❢r❛❝t✐♦♥ ♦❢ ✈❡rt✐❝❡s ] → ✶ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✾ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▲♦♥❣❡st P❛t❤ ❚❤❡ ❝♦♥❥❡❝t✉r❡ ❈♦♥❥❡❝t✉r❡ ✭❋r✐❡③❡ ❛♥❞ ❚s♦✉r❛❦❛❦✐s ✷✵✶✷✮ P [ ∃ ❛ ♣❛t❤ ❝♦♥t❛✐♥✐♥❣ ❛ ♣♦s✐t✐✈❡ ❢r❛❝t✐♦♥ ♦❢ ✈❡rt✐❝❡s ] → ✶ ◆♦t tr✉❡✦ ❆❜❜❛s ✭❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✮ ❘❛♥❞♦♠ ❆♣♦❧❧♦♥✐❛♥ ◆❡t✇♦r❦s ▼❛② ✷✵t❤ ✶✾ ✴ ✹✵