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  1. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ❈✉t ♠♦♠❡♥ts ❛♥❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❉♦r♦t❛ ❑♦t❧♦r③ ❖♣♦❧❡ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣② ❛♥❞ ❏■◆❘ ❍❙◗❈❉ ✸✵✳✵✻✳✷✵✶✹ ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

  2. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ❆❜str❛❝t ■♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❙✳❱✳ ▼✐❦❤❛✐❧♦✈ ✭❏❍❊P ✵✻✭✷✵✶✹✮✵✻✺✮ ❙♣❡❝✐❛❧ t❤❛♥❦s t♦ ❖✳❱✳ ❚❡r②❛❡✈ ❲❡ ❞❡✈❡❧♦♣ ❛♥ ❛♣♣r♦❛❝❤ ♦♥ t❤❡ ❝✉t ▼❡❧❧✐♥ ♠♦♠❡♥ts ✭❉✳❑✳ ❛♥❞ ❆✳ ❑♦t❧♦r③✱ P❤②s✳ ▲❡tt✳ ❇✻✹✹✱ ✷✽✹✱ ✷✵✵✼✮ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s✿ ●❡♥❡r❛❧✐③❡❞ ❝✉t ▼❡❧❧✐♥ ♠♦♠❡♥ts ✭❈▼▼✮ ♦❜t❛✐♥❡❞ ❜② ♠✉❧t✐♣❧❡ ✐♥t❡❣r❛t✐♦♥s ❛s ✇❡❧❧ ❛s ♠✉❧t✐♣❧❡ ❞✐✛❡r❡♥t✐❛t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt♦♥ ❞✐str✐❜✉t✐♦♥ ❛❧s♦ s❛t✐s❢② t❤❡ ❉●▲❆P ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣❧② tr❛♥s❢♦r♠❡❞ ❡✈♦❧✉t✐♦♥ ❦❡r♥❡❧✳ ❆♣♣❧✐❝❛t✐♦♥✿ ❛♣♣r♦♣r✐❛t❡ ❝❧❛ss❡s ♦❢ ❈▼▼ ❢♦r t❤❡ ❛✈❛✐❧❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧ ❦✐♥❡♠❛t✐❝ r❛♥❣❡ ❊❳P❊❘■▼❊◆❚❙ P❘❖❱■❉❊ ❈❯❚ ▼❖▼❊◆❚❙✦ ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

  3. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ▼♦t✐✈❛t✐♦♥ P❉❋s ♣❧❛② t❤❡ ❝❡♥tr❛❧ r♦❧❡ ✐♥ ❉●▲❆P ❛♣♣r♦❛❝❤ ▼♦♠❡♥ts ✲ ❛ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡ ✐♥ ◗❈❉ ❛♥❛❧②s✐s ♦r✐❣✐♥❛t❡ ❢r♦♠ ❖P❊ ✲ ❜❛s✐❝ ❢♦r♠❛❧✐s♠ ♦❢ t❤❡ q✉❛♥t✉♠ ✜❡❧❞ t❤❡♦r② ❞✐r❡❝t❧② r❡✛❡r t♦ s✉♠ r✉❧❡s ✲ ❢✉♥❞❛♠❡♥t❛❧ r❡❧❛t✐♦♥s ✐♥ ◗❈❉ ❝♦♥tr✐❜✉t✐♦♥s t♦ ♠♦♠❡♥t✉♠ ♦r s♣✐♥ ♦❢ t❤❡ ♥✉❝❧❡♦♥ ❝♦♠✐♥❣ ❢r♦♠ q✉❛r❦s ❛♥❞ ❣❧✉♦♥s ✭s♣✐♥ ❝r✐s✐s✮ ❊✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s ❢♦r ❝✉t ♠♦♠❡♥ts ✭❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❢✉❧❧ ♠♦♠❡♥ts ✈❡rs✐♦♥✮ ❡♥❛❜❧❡s ❞✐r❡❝t❧② t♦ st✉❞② ♣❤②s✐❝❛❧ ✈❛❧✉❡s ❛♥❞ t❤❡✐r ❡✈♦❧✉t✐♦♥ ❛❧❧♦✇s t♦ ❛✈♦✐❞ ✉♥❝❡rt❛✐♥t✐❡s ❢r♦♠ ♥♦♥✲❛✈❛✐❧❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧❧② ❧♦✇✲ ① r❡❣✐♦♥ ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

  4. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ▼♦t✐✈❛t✐♦♥ ✭❛❝❝❡ss✐❜❧❡ ❦✐♥❡♠❛t✐❝ r❛♥❣❡✮ ❈▼▼❆ ❛❧❧♦✇s ♦♥❡ t♦ ❛✈♦✐❞ ✉♥♣❤②s✐❝❛❧ ① → ✵ ❛♥❞ ❣✐✈❡s ❛♥s✇❡r ❢♦r ❛♥② ① ✵ ■♥❞✐r❡❝t ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ ❧♦✇✲ ① ❝♦♥tr✐❜✉t✐♦♥s t♦ ❇❙❘ ✭❆✳❆❝❝❛r❞✐ ❡t ❛❧✳ ❛r❳✐✈✿✶✷✶✷✳✶✼✵✶✮ ✭❈❊❘◆ ❈♦✉r✐❡r ✷✵✶✷✮ ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

  5. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ❈♦♥t❡♥ts ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ✶ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ✷ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ✸ ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

  6. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ❈✉t ▼❡❧❧✐♥ ▼♦♠❡♥ts ✭❈▼▼✮ ✲ ❞❡✜♥✐t✐♦♥s ✶ � ❞① ① ♥ − ✶ ❢ ( ① , ◗ ✷ ) ❢ ♥ ( ◗ ✷ ) = ✵ � �� � ✉♥❝✉t ♠♦♠❡♥t ✶ � ❞① ① ♥ − ✶ ❢ ( ① , ◗ ✷ ) ❢ ♥ ( ① ♠✐♥ , ◗ ✷ ) = ① ♠✐♥ � �� � ❝✉t ♠♦♠❡♥t ① ♠❛① � ❞① ① ♥ − ✶ ❢ ( ① , ◗ ✷ ) ❢ ♥ ( ① ♠✐♥ , ① ♠❛① , ◗ ✷ ) = ① ♠✐♥ � �� � ❞♦✉❜❧❡ ❝✉t ♠♦♠❡♥t ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

  7. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ❊✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s ❢♦r ❈▼▼ ▼❛✐♥ ✜♥❞✐♥❣s ✭❆✳ ❑♦t❧♦r③ ❛♥❞ ❉✳❑✳ ✷✵✵✼✮ ❞❢ ◆❙ ( ① , ◗ ✷ ) = α s ( ◗ ✷ ) ♥ qq ( ♥ ) ∗ ❢ ◆❙ )( ① , ◗ ✷ ) ( P ′ ♥ ❞ ❧♥ ◗ ✷ ✷ π � � � P ′ � � � = α s ( ◗ ✷ ) ❞ ❢ ❙ ♥ ( ① , ◗ ✷ ) qq ( ♥ ) P ′ q● ( ♥ ) ❢ ❙ ♥ ( ① , ◗ ✷ ) ∗ ● ♥ ( ① , ◗ ✷ ) ● ♥ ( ① , ◗ ✷ ) ❞ ❧♥ ◗ ✷ P ′ ●q ( ♥ ) P ′ ●● ( ♥ ) ✷ π ❘❡s❝❛❧❡❞ s♣❧✐tt✐♥❣ ❢✉♥❝t✐♦♥s✿ ✐❥ ( ♥ , ① ) = ① ♥ P ✐❥ ( ① ) P ′ � α s ( ◗ ✷ ) � ✷ ✐❥ ( ① ) + α s ( ◗ ✷ ) P ✐❥ ( ① ) = P ( ✵ ) P ( ✶ ) P ( ✷ ) ✐❥ ( ① ) + ✐❥ ( ① ) + · · · ✷ π ✷ π ❆♥♦♠❛❧♦✉s ❞✐♠❡♥s✐♦♥✿ � ✶ ❞① ① s − ✶ P ′ ( ♥ , ① ) = γ s + ♥ γ ′ s , ♥ ≡ ✵ ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

  8. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ❊✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s ❢♦r ❈▼▼ ❱❛❧✐❞ ✐♥ ❡❛❝❤ ♦r❞❡r ♦❢ ♣❡rt✉r❜❛t✐♦♥ ❣ ✶ ♥ ( ① , ◗ ✷ ) = ✶ � ❡ ✷ q × ✷ q � � ∆ ❢ ♥ ( ① , ◗ ✷ ) + α s ( ◗ ✷ ) � � ( ① , ◗ ✷ ) × ❈ ′ q ( ♥ ) ∗ ∆ ❢ ♥ + ❈ ′ ● ( ♥ ) ∗ ∆ ● ♥ ✷ π ❘❡s❝❛❧❡❞ ❲✐❧s♦♥ ❝♦❡✣❝✐❡♥ts✿ ✐ ( ♥ , ① ) = ① ♥ ❈ ✐ ( ① ) ❈ ′ ▼♦♠❡♥ts ♦❢ t❤❡ ❝♦❡✣❝✐❡♥t ❢✉♥❝t✐♦♥s✿ � ✶ ❞① ① s − ✶ ❈ ′ ( ♥ , ① ) = ❈ s + ♥ ❈ ′ s , ♥ ≡ ✵ ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

  9. ▼❛✐♥ ✜♥❞✐♥❣s ♦❢ ♦r✐❣✐♥❛❧ ❈▼▼❆ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❉●▲❆P ❡q✉❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈▼▼❆ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ st❛♥❞❛r❞ ❛♥❞ ❈▼▼ ❛♣r♦❛❝❤❡s P❉❋ ❈▼▼ ■♥♣✉t✿ P❉❋s ❢ ( ① , ◗ ✷ ■♥♣✉t✿ ❈▼▼ ❢ ♥ ( ① ♠✐♥ , ◗ ✷ ✵ ) ✵ ) ⇓ ⇓ ❊✈♦❧✉t✐♦♥ ❢r♦♠ ◗ ✷ ✵ t♦ ◗ ✷ ❊✈♦❧✉t✐♦♥ ❢r♦♠ ◗ ✷ ✵ t♦ ◗ ✷ ∂ ❢ ( ① , ◗ ✷ ) = α s ( ◗ ✷ ) ∂ ❢ ♥ ( ① ♠✐♥ , ◗ ✷ ) = α s ( ◗ ✷ ) ( P ′ ∗ ❢ ♥ ) ( P ∗ ❢ ) ∂ ❧♥ ◗ ✷ ∂ ❧♥ ◗ ✷ ✷ π ✷ π ⇓ ⇓ ❘❡s✉❧ts✿ P❉❋ ❢ ( ① , ◗ ✷ ) ❘❡s✉❧ts✿ ❈▼▼ ❢ ♥ ( ① ♠✐♥ , ◗ ✷ ) P ′ ( ♥ , ③ ) = ③ ♥ P ( ③ ) ♦♥❡ ❝❛♥ ✉s❡ st❛♥❞❛r❞ ♠❡t❤♦❞s ♦❢ s♦❧✈✐♥❣ ❉●▲❆P ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s ✭✉s❡❢✉❧ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ✉♥❝✉t ❛♥❞ ❝✉t ♠♦♠❡♥ts✱ ❉✳❑✳ ✷✵✶✶✮ ❉♦r♦t❛ ❑♦t❧♦r③ ❝✉t ♠♦♠❡♥ts

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