r t t - - PowerPoint PPT Presentation

SLIDE 1

❯♥✇r❛♣♣✐♥❣ t❤❡ ✏❊①♣♦♥❡♥t✐❛❧ ▼❛♥✐❢♦❧❞ ❜② ❘❡♣r♦❞✉❝✐♥❣ ❑❡r♥❡❧ ❍✐❧❜❡rt ❙♣❛❝❡s✑

✭❑✳ ❋✉❦✉♠✐③✉✱ ✐♥ ❆❧❣❡❜r❛✐❝ ❛♥❞ ●❡♦♠❡tr✐❝ ▼❡t❤♦❞s ✐♥ ❙t❛t✐st✐❝s✱ ✷✵✵✾✮ ❉✐♥♦ ❙✳

• ❛ts❜② ❯♥✐t

❖❝t♦❜❡r ✶✽✱ ✷✵✶✷

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶ ✴ ✷✶

SLIDE 2

■♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②❄

❆✐♠✿ ❝♦♥str✉❝t ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②✱ ♦♥ ✇❤✐❝❤ ❡st✐♠❛t✐♦♥ t❤❡♦r② ❝❛♥ ❜❡ ❜✉✐❧t ■♥ ♣❛rt✐❝✉❧❛r✿ t❤❡♦r② ♦❢ ❝♦♥s✐st❡♥t ❡st✐♠❛t✐♦♥ ✇✐t❤ ❛ ✜♥✐t❡ s❛♠♣❧❡ ❆ s✉❜s❡t ♦❢ ❛♥ ❘❑❍❙ ❛s ❛ ❢✉♥❝t✐♦♥❛❧ ♣❛r❛♠❡t❡r s♣❛❝❡ ▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞ ✐❧❧✲♣♦s❡❞ Ps❡✉❞♦✲▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞✿ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜♠❛♥✐❢♦❧❞s✱ ✇❤❡r❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ s❛♠♣❧❡ s✐③❡

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷ ✴ ✷✶

SLIDE 3

■♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②❄

❆✐♠✿ ❝♦♥str✉❝t ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②✱ ♦♥ ✇❤✐❝❤ ❡st✐♠❛t✐♦♥ t❤❡♦r② ❝❛♥ ❜❡ ❜✉✐❧t ■♥ ♣❛rt✐❝✉❧❛r✿ t❤❡♦r② ♦❢ ❝♦♥s✐st❡♥t ❡st✐♠❛t✐♦♥ ✇✐t❤ ❛ ✜♥✐t❡ s❛♠♣❧❡ ❆ s✉❜s❡t ♦❢ ❛♥ ❘❑❍❙ ❛s ❛ ❢✉♥❝t✐♦♥❛❧ ♣❛r❛♠❡t❡r s♣❛❝❡ ▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞ ✐❧❧✲♣♦s❡❞ Ps❡✉❞♦✲▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞✿ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜♠❛♥✐❢♦❧❞s✱ ✇❤❡r❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ s❛♠♣❧❡ s✐③❡

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷ ✴ ✷✶

SLIDE 4

■♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②❄

❆✐♠✿ ❝♦♥str✉❝t ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②✱ ♦♥ ✇❤✐❝❤ ❡st✐♠❛t✐♦♥ t❤❡♦r② ❝❛♥ ❜❡ ❜✉✐❧t ■♥ ♣❛rt✐❝✉❧❛r✿ t❤❡♦r② ♦❢ ❝♦♥s✐st❡♥t ❡st✐♠❛t✐♦♥ ✇✐t❤ ❛ ✜♥✐t❡ s❛♠♣❧❡ ❆ s✉❜s❡t ♦❢ ❛♥ ❘❑❍❙ ❛s ❛ ❢✉♥❝t✐♦♥❛❧ ♣❛r❛♠❡t❡r s♣❛❝❡ ▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞ ✐❧❧✲♣♦s❡❞ Ps❡✉❞♦✲▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞✿ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜♠❛♥✐❢♦❧❞s✱ ✇❤❡r❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ s❛♠♣❧❡ s✐③❡

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷ ✴ ✷✶

SLIDE 5

■♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②❄

❆✐♠✿ ❝♦♥str✉❝t ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②✱ ♦♥ ✇❤✐❝❤ ❡st✐♠❛t✐♦♥ t❤❡♦r② ❝❛♥ ❜❡ ❜✉✐❧t ■♥ ♣❛rt✐❝✉❧❛r✿ t❤❡♦r② ♦❢ ❝♦♥s✐st❡♥t ❡st✐♠❛t✐♦♥ ✇✐t❤ ❛ ✜♥✐t❡ s❛♠♣❧❡ ❆ s✉❜s❡t ♦❢ ❛♥ ❘❑❍❙ ❛s ❛ ❢✉♥❝t✐♦♥❛❧ ♣❛r❛♠❡t❡r s♣❛❝❡ ▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞ ✐❧❧✲♣♦s❡❞ Ps❡✉❞♦✲▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞✿ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜♠❛♥✐❢♦❧❞s✱ ✇❤❡r❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ s❛♠♣❧❡ s✐③❡

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷ ✴ ✷✶

SLIDE 6

■♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②❄

❆✐♠✿ ❝♦♥str✉❝t ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②✱ ♦♥ ✇❤✐❝❤ ❡st✐♠❛t✐♦♥ t❤❡♦r② ❝❛♥ ❜❡ ❜✉✐❧t ■♥ ♣❛rt✐❝✉❧❛r✿ t❤❡♦r② ♦❢ ❝♦♥s✐st❡♥t ❡st✐♠❛t✐♦♥ ✇✐t❤ ❛ ✜♥✐t❡ s❛♠♣❧❡ ❆ s✉❜s❡t ♦❢ ❛♥ ❘❑❍❙ ❛s ❛ ❢✉♥❝t✐♦♥❛❧ ♣❛r❛♠❡t❡r s♣❛❝❡ ▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞ ✐❧❧✲♣♦s❡❞ Ps❡✉❞♦✲▼❛①✐♠✉♠✲▲✐❦❡❧✐❤♦♦❞✿ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜♠❛♥✐❢♦❧❞s✱ ✇❤❡r❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ s❛♠♣❧❡ s✐③❡

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷ ✴ ✷✶

SLIDE 7

■♥tr♦❞✉❝t✐♦♥

❋♦r s✐♠♣❧✐❝✐t②✱ ❧❡t X = [✵, ✶]✱ ❛♥❞ ❦ : X × X → R ❜❡ ❛ ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s ❦❡r♥❡❧✱ ✇✐t❤ ❘❑❍❙ H❦✳ ❛ss✉♠♣t✐♦♥ ✵✿ H❦ ❝♦♥t❛✐♥s t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s✱ ✉(①) = ❝✳ ■❢ ✐t ❞♦❡s ♥♦t✱ t❤❡♥ ❝♦♥s✐❞❡r ❦(①, ②) + ✶ ✐♥st❡❛❞✳ ❚❤✐s ❞♦❡s✳ ✭❚❤✐s ❛ss✉♠♣t✐♦♥ ✐s ♠❛❞❡ s♦ t❤❛t H❦ ✐s ❝❧♦s❡❞ ✉♥❞❡r s✉❜tr❛❝t✐♥❣ ❝♦♥st❛♥ts✮ ❙✐♥❝❡ ❦ ✐s ❛ ❜♦✉♥❞❡❞ ❦❡r♥❡❧ ♦♥ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥✱ ✐♥t❡❣r❛❧s ´ ✉(①)❞① ❛♥❞ ´ ❡✉(①)❞① ❝♦♥✈❡r❣❡ ∀✉ ∈ H❦✳ ▲❡t ❚ :=

• ✉ ∈ H❦ :

´ ✉(①)❞① = ✵

• ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❝♦♥s✐❞❡r t❤❡

✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ✶ ♦♥ X✱ ❛♥❞ ❞❡✜♥❡ ✐ts ❦❡r♥❡❧ ❡♠❜❡❞❞✐♥❣ µ✶ = ´ ❦(·, ①) · ✶❞① ∈ H❦✳ ❚❤❡♥✱ ❚ = µ⊥

✶ ✱ ❛s

´ ✉(①)❞① = ✉, µ✶H❦ . ❙✐♥❝❡ H❦ ✐♥❝❧✉❞❡s ❝♦♥st❛♥ts✱ ✉ − ´ ✉(①)❞① ∈ ❚✱ ∀✉ ∈ H❦

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✸ ✴ ✷✶

SLIDE 8

❉❡♥s✐t✐❡s ♣❛r❛♠❡tr✐③❡❞ ❜② ❘❑❍❙ ❢✉♥❝t✐♦♥s

◆♦✇✱ ♣✐❝❦ ✉ ∈ ❚✱ ❛♥❞ ❞❡✜♥❡✿ Ψ(✉) = ❧♦❣ ˆ ❡✉(①)❞① ▲❡♠♠❛ ∀✉ ∈ ❚✱ ❡✉−Ψ(✉) ✐s ❛ ✈❛❧✐❞ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦♥ X✱ ❛♥❞ ♠❛♣ ξ : ✉ → ❡✉−Ψ(✉) ✐s ♦♥❡✲t♦✲♦♥❡✳ Pr♦♦❢✳ ξ(✉) = ξ(✈) = ⇒ ✉(①) − ✈(①) = Ψ(✈) − Ψ(✉) = ❝♦♥st✳ = ⇒ ´ ✉(①)❞① − ´ ✈(①)❞① = t❤❛t s❛♠❡ ❝♦♥st❛♥t = ⇒ t❤❛t ❝♦♥st❛♥t ♠✉st ❜❡ ③❡r♦✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✹ ✴ ✷✶

SLIDE 9

❙ = ξ(❚) ✐s ♥♦✇ ❛ s❡t ♦❢ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥s ♦♥ X ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❦❡r♥❡❧ ❦✱ ✇❤✐❝❤ ✐♥❤❡r✐ts t❤❡ ❍✐❧❜❡rt✐❛♥ str✉❝t✉r❡ ♦❢ ❚ ⊂ H❦ ✭❡①♣♦♥❡♥t✐❛❧ ❍✐❧❜❡rt ♠❛♥✐❢♦❧❞✮✳ ▲❡t✬s ✇r✐t❡ ❢✉ ❢♦r ξ(✉) ❢♦r s❤♦rt ✭r❡❛❞✿ ✏❞❡♥s✐t② ✇✐t❤ ♣❛r❛♠❡t❡r ✉✑✮✳ ❲❡ ❣❡t t❤❡ ✉s✉❛❧ st✉✛ ✇✐t❤ ❢❛♥❝✐❡r ♥❛♠❡s ✲ ✇❡ ❝❛♥ t❛❦❡ ❋ré❝❤❡t ❞❡r✐✈❛t✐✈❡s ♦❢ Ψ✿ ❉✉Ψ(✈) = E❳∼❢✉ [✈(❳)] = ✈, µ✉H❦ ❉✷

✉Ψ(✈✶, ✈✷)

= ❈♦✈❳∼❢✉ [✈✶(❳), ✈✷(❳)] = ✈✶, Σ✉✈✷H❦ ✇❤❡r❡✿ µ✉ := E❢✉ [❦(·, ❳)] Σ✉ := E❢✉ [❦(·, ❳) ⊗ ❦(·, ❳)] − E❢✉ [❦(·, ❳)] ⊗ E❢✉ [❦(·, ❳)] , ❛r❡ t❤❡ ❦❡r♥❡❧ ❡♠❜❡❞❞✐♥❣ ❛♥❞ t❤❡ ❦❡r♥❡❧ ❝♦✈❛r✐❛♥❝❡ ♦♣❡r❛t♦r ♦❢ ❞❡♥s✐t② ❢✉✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✺ ✴ ✷✶

SLIDE 10

■♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧② ❧❛♥❣✉❛❣❡✿

✶ ✉

❚ ✐s t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛♥❞ ❦ ① ✐s t❤❡ s✉✣❝✐❡♥t st❛t✐st✐❝✱ ❛s ❢✉ ❡ ✉ ❦

①

❡✉ ①

✷

✉ ❦ ✐s t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛s ✐t ✐s t❤❡ ♠❡❛♥ ♦❢ t❤❡

s✉✣❝✐❡♥t st❛t✐st✐❝ ❋♦r ❝❤❛r❛❝t❡r✐st✐❝ ❦❡r♥❡❧s✱ t❤❡ ♠❛♣♣✐♥❣ P

P ✐s ✐♥❥❡❝t✐✈❡✳ ■♥

♣❛rt✐❝✉❧❛r✱ ❢✉

✉ ✐s ✐♥❥❡❝t✐✈❡✳

❙♦✱ t❤❡r❡ ✐s ❛ r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ❚ ✉

✉ ❦

◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧②

✉

❚✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✉ ✵ ❚ ❢✉ ✶

✉ ✶

❚✳ ❈♦♥❝❧✉s✐♦♥✿ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡rs ❞♦ ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✻ ✴ ✷✶

SLIDE 11

■♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧② ❧❛♥❣✉❛❣❡✿

✶ ✉ ∈ ❚ ✐s t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛♥❞ ❦(·, ①) ✐s t❤❡ s✉✣❝✐❡♥t

st❛t✐st✐❝✱ ❛s ❢✉ ∝ ❡✉,❦(·,①) = ❡✉(①)

✷

✉ ❦ ✐s t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛s ✐t ✐s t❤❡ ♠❡❛♥ ♦❢ t❤❡

s✉✣❝✐❡♥t st❛t✐st✐❝ ❋♦r ❝❤❛r❛❝t❡r✐st✐❝ ❦❡r♥❡❧s✱ t❤❡ ♠❛♣♣✐♥❣ P

P ✐s ✐♥❥❡❝t✐✈❡✳ ■♥

♣❛rt✐❝✉❧❛r✱ ❢✉

✉ ✐s ✐♥❥❡❝t✐✈❡✳

❙♦✱ t❤❡r❡ ✐s ❛ r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ❚ ✉

✉ ❦

◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧②

✉

❚✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✉ ✵ ❚ ❢✉ ✶

✉ ✶

❚✳ ❈♦♥❝❧✉s✐♦♥✿ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡rs ❞♦ ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✻ ✴ ✷✶

SLIDE 12

■♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧② ❧❛♥❣✉❛❣❡✿

✶ ✉ ∈ ❚ ✐s t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛♥❞ ❦(·, ①) ✐s t❤❡ s✉✣❝✐❡♥t

st❛t✐st✐❝✱ ❛s ❢✉ ∝ ❡✉,❦(·,①) = ❡✉(①)

✷ µ✉ ∈ H❦ ✐s t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛s ✐t ✐s t❤❡ ♠❡❛♥ ♦❢ t❤❡

s✉✣❝✐❡♥t st❛t✐st✐❝ ❋♦r ❝❤❛r❛❝t❡r✐st✐❝ ❦❡r♥❡❧s✱ t❤❡ ♠❛♣♣✐♥❣ P

P ✐s ✐♥❥❡❝t✐✈❡✳ ■♥

♣❛rt✐❝✉❧❛r✱ ❢✉

✉ ✐s ✐♥❥❡❝t✐✈❡✳

❙♦✱ t❤❡r❡ ✐s ❛ r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ❚ ✉

✉ ❦

◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧②

✉

❚✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✉ ✵ ❚ ❢✉ ✶

✉ ✶

❚✳ ❈♦♥❝❧✉s✐♦♥✿ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡rs ❞♦ ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✻ ✴ ✷✶

SLIDE 13

■♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧② ❧❛♥❣✉❛❣❡✿

✶ ✉ ∈ ❚ ✐s t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛♥❞ ❦(·, ①) ✐s t❤❡ s✉✣❝✐❡♥t

st❛t✐st✐❝✱ ❛s ❢✉ ∝ ❡✉,❦(·,①) = ❡✉(①)

✷ µ✉ ∈ H❦ ✐s t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛s ✐t ✐s t❤❡ ♠❡❛♥ ♦❢ t❤❡

s✉✣❝✐❡♥t st❛t✐st✐❝ ❋♦r ❝❤❛r❛❝t❡r✐st✐❝ ❦❡r♥❡❧s✱ t❤❡ ♠❛♣♣✐♥❣ P → µP ✐s ✐♥❥❡❝t✐✈❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢✉ → µ✉ ✐s ✐♥❥❡❝t✐✈❡✳ ❙♦✱ t❤❡r❡ ✐s ❛ r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ❚ ✉

✉ ❦

◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧②

✉

❚✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✉ ✵ ❚ ❢✉ ✶

✉ ✶

❚✳ ❈♦♥❝❧✉s✐♦♥✿ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡rs ❞♦ ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✻ ✴ ✷✶

SLIDE 14

■♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧② ❧❛♥❣✉❛❣❡✿

✶ ✉ ∈ ❚ ✐s t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛♥❞ ❦(·, ①) ✐s t❤❡ s✉✣❝✐❡♥t

st❛t✐st✐❝✱ ❛s ❢✉ ∝ ❡✉,❦(·,①) = ❡✉(①)

✷ µ✉ ∈ H❦ ✐s t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛s ✐t ✐s t❤❡ ♠❡❛♥ ♦❢ t❤❡

s✉✣❝✐❡♥t st❛t✐st✐❝ ❋♦r ❝❤❛r❛❝t❡r✐st✐❝ ❦❡r♥❡❧s✱ t❤❡ ♠❛♣♣✐♥❣ P → µP ✐s ✐♥❥❡❝t✐✈❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢✉ → µ✉ ✐s ✐♥❥❡❝t✐✈❡✳ ❙♦✱ t❤❡r❡ ✐s ❛ r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ❚ ∋ ✉ → µ✉ ∈ H❦ ◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧②

✉

❚✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✉ ✵ ❚ ❢✉ ✶

✉ ✶

❚✳ ❈♦♥❝❧✉s✐♦♥✿ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡rs ❞♦ ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✻ ✴ ✷✶

SLIDE 15

■♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧② ❧❛♥❣✉❛❣❡✿

✶ ✉ ∈ ❚ ✐s t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛♥❞ ❦(·, ①) ✐s t❤❡ s✉✣❝✐❡♥t

st❛t✐st✐❝✱ ❛s ❢✉ ∝ ❡✉,❦(·,①) = ❡✉(①)

✷ µ✉ ∈ H❦ ✐s t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r ♦❢ ❢✉✱ ❛s ✐t ✐s t❤❡ ♠❡❛♥ ♦❢ t❤❡

s✉✣❝✐❡♥t st❛t✐st✐❝ ❋♦r ❝❤❛r❛❝t❡r✐st✐❝ ❦❡r♥❡❧s✱ t❤❡ ♠❛♣♣✐♥❣ P → µP ✐s ✐♥❥❡❝t✐✈❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢✉ → µ✉ ✐s ✐♥❥❡❝t✐✈❡✳ ❙♦✱ t❤❡r❡ ✐s ❛ r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ❚ ∋ ✉ → µ✉ ∈ H❦ ◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧② µ✉ / ∈ ❚✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✉ = ✵ ∈ ❚ ⇒ ❢✉ = ✶ ⇒ µ✉ = µ✶⊥❚✳ ❈♦♥❝❧✉s✐♦♥✿ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡rs ❞♦ ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✻ ✴ ✷✶

SLIDE 16

❑▲ ✐♥ ❙

❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ❞✐✈❡r❣❡♥❝❡ ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♠❛♥✐❢♦❧❞✿ ❑▲ (❢✉ ❢✈ ) = ˆ ❢✉(①) ❧♦❣ ❢✉(①) ❢✈(①)❞① = ˆ ❢✉(①) [✉(①) − Ψ(✉) − ✈(①) + Ψ(✈)] ❞① = Ψ(✈) − Ψ(✉) + ✉ − ✈, µ✉H❦ .

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✼ ✴ ✷✶

SLIDE 17

❑▲ ✐♥ ❙ ✭✷✮

❚❤❡♦r❡♠ ✭P②t❤❛❣♦r❡❛♥ ❑▲✲r❡❧❛t✐♦♥✮ ❈♦♥s✐❞❡r ❛ ❝❧♦s❡❞ s✉❜s♣❛❝❡ ❱ ⊂ ❚✱ ❧❡t ❢∗ ∈ ❙✱ ❛♥❞ s❡t✿ ✉♦♣t = ❛r❣ ♠✐♥

✉∈❱ ❑▲ (❢∗ ❢✉ )

✐✳❡✳✱ ❢✉♦♣t✐s t❤❡ ❑▲✲♥❡❛r❡st ❞❡♥s✐t② ✐♥ ξ(❱ ) t♦ ❢∗✳ ❚❤❡♥ ∀✉ ∈ ❱ ✿ ❑▲ (❢∗ ❢✉ ) = ❑▲

• ❢∗
• ❢✉♦♣t
• + ❑▲
• ❢✉♦♣t ❢✉
• .

❚❤❡ ❑▲ ❞✐✈❡r❣❡♥❝❡ ❜❡t✇❡❡♥ ❢∗ ❛♥❞ ❢✉ t❤❛t ✐s ♣❛r❛♠❡tr✐③❡❞ ❜② ❛ s✉❜s♣❛❝❡ ❱ ♦❢ ❚ ❝❛♥ ❜❡ ❜r♦❦❡♥ ❞♦✇♥ ❛s t❤❡ ❑▲ ❞✐✈❡r❣❡♥❝❡ ❜❡t✇❡❡♥ ❢∗ ❛♥❞ t❤❡ ♥❡❛r❡st ❞❡♥s✐t② ✐♥ t❤❛t s✉❜s♣❛❝❡ ✭❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✮ ♣❧✉s t❤❡ ❑▲ ❞✐✈❡r❣❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❜❡st ❛♣♣r♦①✐♠❛t♦r ❛♥❞ ❛ ❣✐✈❡♥ ❞❡♥s✐t② ✭❡st✐♠❛t✐♦♥ ❡rr♦r✮✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✽ ✴ ✷✶

SLIDE 18

❘❑❍❙ ♥♦r♠ ❛♥❞ ❑▲ ❞✐✈❡r❣❡♥❝❡

▲❡♠♠❛ ▲❡t ✉♥ − ✉H❦ = ♦(α♥)✱ ✇❤❡r❡ ❧✐♠♥→∞ α♥ = ✵✳ ❚❤❡♥ ❑▲ (❢✉ ❢✉♥ ) = ♦(α♥)✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✾ ✴ ✷✶

SLIDE 19

❘❑❍❙ ♥♦r♠ ❛♥❞ ❑▲ ❞✐✈❡r❣❡♥❝❡

Pr♦♦❢✳ ❙t❛rt ✇✐t❤✿ ❑▲ (❢✉ ❢✉♥ ) ≤ |Ψ(✉♥) − Ψ(✉)| +

• ✉♥ − ✉, µ✉H❦
• .

❇② ❚❛②❧♦r✲❡①♣❛♥s✐♦♥✱ ✇❡ ♦❜t❛✐♥✿

|Ψ(✉♥) − Ψ(✉)| =

• ✉♥ − ✉, µ✉ + ✶

✷ ✉♥ − ✉, Σ˜

✉(✉♥ − ✉)

• ≤

✉♥ − ✉

• µ✉ + ✶

✷λ♠❛① ✉♥ − ✉

• , ❛♥❞ t❤✉s✿

❑▲ (❢✉ ❢✉♥ ) ≤ ✉♥ − ✉

• ✷ µ✉ + ✶

✷λ♠❛① ✉♥ − ✉

• ,

✇❤❡r❡ ˜ ✉ ✐s ❛ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✉♥ ❛♥❞ ✉ ❛♥❞ λ♠❛① ✐s t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ♦❢ Σ˜

✉✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✵ ✴ ✷✶

SLIDE 20

▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞

❖❜s❡r✈❡ ❞❛t❛ {❳✐}♥

✐=✶ ✐.✐.❞.

∼ ❢✉∗✳ ❈♦♥s✐❞❡r t❤❡ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞✿ ▲♥(✉) = ✶ ♥

♥

• ✐=✶

❧♦❣ ♣(❳✐|✉) = ✶ ♥

♥

• ✐=✶

✉(❳✐) − Ψ(✉) =

• ✉, ✶

♥

♥

• ✐=✶

❦(·, ❳✐)

• − Ψ(✉)

=

• ✉, ˆ

µ(♥) − Ψ(✉)

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✶ ✴ ✷✶

SLIDE 21

▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭✷✮

❉✐✛❡r❡♥t✐❛t❡✿ ❉✉▲♥(✈) =

• ✈, ˆ

µ(♥) − µ✉

• ▼▲ s♦❧✉t✐♦♥ ✐s tr✐✈✐❛❧✦ ❙❡t t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r t♦ t❤❡ ❡♠♣✐r✐❝❛❧ ♠❡❛♥

♣❛r❛♠❡t❡r ❛♥❞ s♦❧✈❡ ❢♦r ✉✿

✉ ♥

♦♦♣s✿

♥ ❞♦❡s ♥♦t ❝♦rr❡s♣♦♥❞ t♦ ❛♥② ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ✉

❚

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✷ ✴ ✷✶

SLIDE 22

▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭✷✮

❉✐✛❡r❡♥t✐❛t❡✿ ❉✉▲♥(✈) =

• ✈, ˆ

µ(♥) − µ✉

• ▼▲ s♦❧✉t✐♦♥ ✐s tr✐✈✐❛❧✦ ❙❡t t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r t♦ t❤❡ ❡♠♣✐r✐❝❛❧ ♠❡❛♥

♣❛r❛♠❡t❡r ❛♥❞ s♦❧✈❡ ❢♦r ✉✿ µ✉ = ˆ µ(♥) ♦♦♣s✿

♥ ❞♦❡s ♥♦t ❝♦rr❡s♣♦♥❞ t♦ ❛♥② ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ✉

❚

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✷ ✴ ✷✶

SLIDE 23

▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭✷✮

❉✐✛❡r❡♥t✐❛t❡✿ ❉✉▲♥(✈) =

• ✈, ˆ

µ(♥) − µ✉

• ▼▲ s♦❧✉t✐♦♥ ✐s tr✐✈✐❛❧✦ ❙❡t t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r t♦ t❤❡ ❡♠♣✐r✐❝❛❧ ♠❡❛♥

♣❛r❛♠❡t❡r ❛♥❞ s♦❧✈❡ ❢♦r ✉✿ µ✉ = ˆ µ(♥) ♦♦♣s✿ ˆ µ(♥) ❞♦❡s ♥♦t ❝♦rr❡s♣♦♥❞ t♦ ❛♥② ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ✉ ∈ ❚

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✷ ✴ ✷✶

SLIDE 24

▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭✸✮

❚❤❡ ✐♥✈❡rs❡ ♠❛♣♣✐♥❣ ❢r♦♠ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r t♦ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ✐s ♥♦t ❜♦✉♥❞❡❞

t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ♠❛♣ ✉ → µ✉ ✿ s✐♥❝❡ µ✉ ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ❛s t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝✉♠✉❧❛♥t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ Ψ✱ ✐✳❡✳✱ ❉✉Ψ = ·, µ✉✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤✐s ♠❛♣ ✐s Σ✉ ✭❦❡r♥❡❧ ❝♦✈❛r✐❛♥❝❡ ♦♣❡r❛t♦r✮✳ ❚❤✐s ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ tr❛❝❡✲❝❧❛ss ♦♣❡r❛t♦r✱ s♦ ✐t ❤❛s ❛r❜✐tr❛r✐❧② s♠❛❧❧ ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s✳

■❢

♥ ✇♦✉❧❞ ❝♦rr❡s♣♦♥❞ t♦ s♦♠❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ✉

❚✱ t❤❡♥ ❛ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉♦✉s ❞❡♥s✐t② ❡✉ ①

✉ ❛♥❞ t❤❡ ❡♠♣✐r✐❝❛❧

❞✐str✐❜✉t✐♦♥ ✶

♥ ♥ ✐ ✶ ❳✐ ♠✉st ❤❛✈❡ t❤❡ s❛♠❡ ❦❡r♥❡❧ ❡♠❜❡❞❞✐♥❣✳ ❘❡❝❛❧❧

t❤❛t t❤✐s ✐s ✐♠♣♦ss✐❜❧❡ ❜❡❝❛✉s❡✱ ❡✳❣✳✱ ❢♦r ❝❤❛r❛❝t❡r✐st✐❝ ❦❡r♥❡❧s✱ t❤❡ ♠❛♣♣✐♥❣ P

P ✐s ✐♥❥❡❝t✐✈❡✦

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✸ ✴ ✷✶

SLIDE 25

▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭✸✮

❚❤❡ ✐♥✈❡rs❡ ♠❛♣♣✐♥❣ ❢r♦♠ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡r t♦ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ✐s ♥♦t ❜♦✉♥❞❡❞

t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ♠❛♣ ✉ → µ✉ ✿ s✐♥❝❡ µ✉ ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ❛s t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝✉♠✉❧❛♥t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ Ψ✱ ✐✳❡✳✱ ❉✉Ψ = ·, µ✉✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤✐s ♠❛♣ ✐s Σ✉ ✭❦❡r♥❡❧ ❝♦✈❛r✐❛♥❝❡ ♦♣❡r❛t♦r✮✳ ❚❤✐s ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ tr❛❝❡✲❝❧❛ss ♦♣❡r❛t♦r✱ s♦ ✐t ❤❛s ❛r❜✐tr❛r✐❧② s♠❛❧❧ ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s✳

■❢ ˆ µ(♥) ✇♦✉❧❞ ❝♦rr❡s♣♦♥❞ t♦ s♦♠❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡r ˆ ✉ ∈ ❚✱ t❤❡♥ ❛ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉♦✉s ❞❡♥s✐t② ❡ˆ

✉(①)−Ψ(ˆ ✉) ❛♥❞ t❤❡ ❡♠♣✐r✐❝❛❧

❞✐str✐❜✉t✐♦♥ ✶

♥

♥

✐=✶ δ❳✐ ♠✉st ❤❛✈❡ t❤❡ s❛♠❡ ❦❡r♥❡❧ ❡♠❜❡❞❞✐♥❣✳ ❘❡❝❛❧❧

t❤❛t t❤✐s ✐s ✐♠♣♦ss✐❜❧❡ ❜❡❝❛✉s❡✱ ❡✳❣✳✱ ❢♦r ❝❤❛r❛❝t❡r✐st✐❝ ❦❡r♥❡❧s✱ t❤❡ ♠❛♣♣✐♥❣ P → µP ✐s ✐♥❥❡❝t✐✈❡✦

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✸ ✴ ✷✶

SLIDE 26

Ps❡✉❞♦✲▼▲❊

❋✐rst✱ ✇❤✐❧❡ ✇❡ ❝❛♥♥♦t ❣♦ ❢r♦♠ t❤❡ ♠❡❛♥ ♣❛r❛♠❡t❡rs t♦ t❤❡ ♥❛t✉r❛❧ ♣❛r❛♠❡t❡rs✱ ✐t ✐s st✐❧❧ tr✉❡ t❤❛t ♠❡❛♥ ♣❛r❛♠❡t❡rs ❛r❡ ✉s❡❢✉❧✱ ♥❛♠❡❧②✿ ❚❤❡♦r❡♠ ✭√♥✲❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ❡♠❜❡❞❞✐♥❣ ❡st✐♠❛t♦r✮

• ˆ

µ(♥) − µ✉∗

• H❦ = OP(✶/√♥)✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✹ ✴ ✷✶

SLIDE 27

Ps❡✉❞♦✲▼▲❊ ✭✷✮

✶ ❉❡✜♥❡ ❛ s❡r✐❡s ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s ♦❢ ❚✿

• ❚ (♥)∞

❧=✶✱ ❛♥❞

t❤❡ ♥✲t❤ Ps❡✉❞♦✲▼▲❊✿ ˆ ✉(♥) = ❛r❣ ♠❛①

✉∈❚ (♥)

• ✉, ˆ

µ(♥) − Ψ(✉) ✭t❤❡ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ▼▲❊ ♣r♦❜❧❡♠ ♦✈❡r ❚ (♥) ✇❤✐❝❤ ✇❡ ❝❛♥ s♦❧✈❡ ❢♦r ✉✮✳

✷ ■♥ ❛❞❞✐t✐♦♥✱ ✐♥tr♦❞✉❝❡✿

✉ ♥ ❛r❣ ♠✐♥

✉ ❱ ❑▲ ❢

❢✉ ❛r❣ ♠❛①

✉ ❚ ♥

✉

✉

✉ ✭t❤❡ ❜❡st ❛♣♣r♦①✐♠❛t♦r t♦ t❤❡ tr✉❡ ✉ ✐♥ ❚ ♥ ✮✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✺ ✴ ✷✶

SLIDE 28

Ps❡✉❞♦✲▼▲❊ ✭✷✮

✶ ❉❡✜♥❡ ❛ s❡r✐❡s ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s ♦❢ ❚✿

• ❚ (♥)∞

❧=✶✱ ❛♥❞

t❤❡ ♥✲t❤ Ps❡✉❞♦✲▼▲❊✿ ˆ ✉(♥) = ❛r❣ ♠❛①

✉∈❚ (♥)

• ✉, ˆ

µ(♥) − Ψ(✉) ✭t❤❡ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ▼▲❊ ♣r♦❜❧❡♠ ♦✈❡r ❚ (♥) ✇❤✐❝❤ ✇❡ ❝❛♥ s♦❧✈❡ ❢♦r ✉✮✳

✷ ■♥ ❛❞❞✐t✐♦♥✱ ✐♥tr♦❞✉❝❡✿

✉(♥)

∗

= ❛r❣ ♠✐♥

✉∈❱ ❑▲ (❢∗ ❢✉ )

= ❛r❣ ♠❛①

✉∈❚ (♥) ✉, µ✉∗ − Ψ(✉)

✭t❤❡ ❜❡st ❛♣♣r♦①✐♠❛t♦r t♦ t❤❡ tr✉❡ ✉∗ ✐♥ ❚ (♥)✮✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✺ ✴ ✷✶

SLIDE 29

❆ss✉♠♣t✐♦♥s

❛ss✉♠♣t✐♦♥ ✶✿ ∀✉∗

• ✉∗ − ✉(♥)

∗

• H❦

= ♦(γ♥), γ♥ → ✵, ✇❤✐❝❤ ♠❡❛♥s t❤❛t ❚ (♥) ❛♣♣r♦①✐♠❛t❡s ❚ ✇✐t❤ ❛ s✉❜✲γ♥ r❛t❡ ❛s ♥ → ∞ ❛ss✉♠♣t✐♦♥ ✷✿ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉❜s♣❛❝❡s ✐s ❝❤♦s❡♥ s♦ t❤❛t t❤❡ s♠❛❧❧❡st ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s

♥ ♦❢ ✉ r❡str✐❝t❡❞ t♦ ❚ ♥ ❞❡❝r❡❛s❡

s❧♦✇❧② ❡♥♦✉❣❤ ✭s❧♦✇❡r t❤❛♥ ✶ ♥✮✿ ✶ ♥

♥

♦

♥ ♥

✵

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✻ ✴ ✷✶

SLIDE 30

❆ss✉♠♣t✐♦♥s

❛ss✉♠♣t✐♦♥ ✶✿ ∀✉∗

• ✉∗ − ✉(♥)

∗

• H❦

= ♦(γ♥), γ♥ → ✵, ✇❤✐❝❤ ♠❡❛♥s t❤❛t ❚ (♥) ❛♣♣r♦①✐♠❛t❡s ❚ ✇✐t❤ ❛ s✉❜✲γ♥ r❛t❡ ❛s ♥ → ∞ ❛ss✉♠♣t✐♦♥ ✷✿ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉❜s♣❛❝❡s ✐s ❝❤♦s❡♥ s♦ t❤❛t t❤❡ s♠❛❧❧❡st ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s λ(♥) ♦❢ Σ✉ r❡str✐❝t❡❞ t♦ ❚ (♥) ❞❡❝r❡❛s❡ s❧♦✇❧② ❡♥♦✉❣❤ ✭s❧♦✇❡r t❤❛♥ ✶/√♥✮✿ ✶ √♥λ(♥) = ♦(ǫ♥), ǫ♥ → ✵.

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✻ ✴ ✷✶

SLIDE 31

❈♦♥s✐st❡♥❝② ♦❢ Ps❡✉❞♦✲▼▲❊

❚❤❡♦r❡♠ ❑▲ (❢∗ ❢ˆ

✉(♥) ) = ♦♣ (♠❛① {γ♥, ǫ♥})

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✼ ✴ ✷✶

SLIDE 32

Pr♦♦❢ s❦❡t❝❤

❇r❡❛❦ ✉♣ ❑▲ ✐♥ t✇♦ ♣❛rts✿ ❑▲ (❢∗ ❢ˆ

✉(♥) )

= ❑▲

• ❢∗
• ❢✉(♥)

∗

• + ❑▲
• ❢✉(♥)

∗ ❢ˆ

✉(♥)

• ❚❤❡ ✜rst ♦♥❡ ✐s ♦

♥ ❜② ❛ss✉♠♣t✐♦♥ ✶✱ ❛♥❞ ▲❡♠♠❛ ❝♦♥♥❡❝t✐♥❣ ❘❑❍❙

♥♦r♠ ❛♥❞ ❑▲ ❞✐✈❡r❣❡♥❝❡ ❋♦r t❤❡ s❡❝♦♥❞ t❡r♠✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ s❤♦✇ t❤❛t t❤❡ ❡st✐♠❛t✐♦♥ ❡rr♦r ✐s ♦♣

♥ ✱ ✐✳❡✳✱

✉ ♥ ✉ ♥

♥

✵

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✽ ✴ ✷✶

SLIDE 33

Pr♦♦❢ s❦❡t❝❤

❇r❡❛❦ ✉♣ ❑▲ ✐♥ t✇♦ ♣❛rts✿ ❑▲ (❢∗ ❢ˆ

✉(♥) )

= ❑▲

• ❢∗
• ❢✉(♥)

∗

• + ❑▲
• ❢✉(♥)

∗ ❢ˆ

✉(♥)

• ❚❤❡ ✜rst ♦♥❡ ✐s ♦(γ♥) ❜② ❛ss✉♠♣t✐♦♥ ✶✱ ❛♥❞ ▲❡♠♠❛ ❝♦♥♥❡❝t✐♥❣ ❘❑❍❙

♥♦r♠ ❛♥❞ ❑▲ ❞✐✈❡r❣❡♥❝❡ ❋♦r t❤❡ s❡❝♦♥❞ t❡r♠✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ s❤♦✇ t❤❛t t❤❡ ❡st✐♠❛t✐♦♥ ❡rr♦r ✐s ♦♣

♥ ✱ ✐✳❡✳✱

✉ ♥ ✉ ♥

♥

✵

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✽ ✴ ✷✶

SLIDE 34

Pr♦♦❢ s❦❡t❝❤

❇r❡❛❦ ✉♣ ❑▲ ✐♥ t✇♦ ♣❛rts✿ ❑▲ (❢∗ ❢ˆ

✉(♥) )

= ❑▲

• ❢∗
• ❢✉(♥)

∗

• + ❑▲
• ❢✉(♥)

∗ ❢ˆ

✉(♥)

• ❚❤❡ ✜rst ♦♥❡ ✐s ♦(γ♥) ❜② ❛ss✉♠♣t✐♦♥ ✶✱ ❛♥❞ ▲❡♠♠❛ ❝♦♥♥❡❝t✐♥❣ ❘❑❍❙

♥♦r♠ ❛♥❞ ❑▲ ❞✐✈❡r❣❡♥❝❡ ❋♦r t❤❡ s❡❝♦♥❞ t❡r♠✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ s❤♦✇ t❤❛t t❤❡ ❡st✐♠❛t✐♦♥ ❡rr♦r ✐s ♦♣(ǫ♥)✱ ✐✳❡✳✱ P

• ˆ

✉(♥) − ✉(♥)

∗

• ≥ ǫ♥
• →

✵

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✽ ✴ ✷✶

SLIDE 35

Pr♦♦❢ s❦❡t❝❤ ✭✷✮

• ˆ

✉(♥) − ✉(♥)

∗

• ≥ ǫ♥ ✐♠♣❧✐❡s t❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞ ❛ ♠❛①✐♠✐③❡r ˆ

✉(♥) ♦❢ ▲♥(✉) > ▲♥(✉(♥)

∗ ) ♦✉ts✐❞❡ t❤❡ ǫ♥✲ ❜❛❧❧ ❝❡♥t❡r❡❞ ❛t ✉(♥) ∗

✐♥ ❚ (♥)✳ ❈♦♥s✐❞❡r t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ ▲♥ ❛r♦✉♥❞ ✉(♥)

∗ ✿

▲♥(ˆ ✉(♥)) − ▲♥(✉(♥)

∗ ) =

❉✉▲♥

• ✉=✉(♥)

∗

[ˆ ✉(♥) − ✉(♥)

∗ ] + ✶

✷❉✷

✉▲♥

• ✉=✉(♥)

∗

[ˆ ✉(♥) − ✉(♥)

∗ , ˆ

✉(♥) − ✉(♥)

∗ ]

=

• ˆ

✉(♥) − ✉(♥)

∗ , ˆ

µ(♥) − µ✉(♥)

∗

• − ✶

✷

• ˆ

✉(♥) − ✉(♥)

∗ , Σ˜ ✉

• ˆ

✉(♥) − ✉(♥)

∗

• =
• ˆ

✉(♥) − ✉(♥)

∗ , ˆ

µ(♥) − µ✉∗

• − ✶

✷

• ˆ

✉(♥) − ✉(♥)

∗ , Σ˜ ✉

• ˆ

✉(♥) − ✉(♥)

∗

• ≤
• ˆ

✉(♥) − ✉(♥)

∗

• ˆ

µ(♥) − µ✉∗

• − ✶

✷λ(♥)ǫ♥

• .

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✶✾ ✴ ✷✶

SLIDE 36

Pr♦♦❢ s❦❡t❝❤ ✭✸✮

P

• ˆ

✉(♥) − ✉(♥)

∗

• ≥ ǫ♥
• ≤

P

• ˆ

µ(♥) − µ✉∗

• ≥ ✶

✷λ(♥)ǫ♥

• ≤

P

• ˆ

µ(♥) − µ✉∗

• ≥

✶ √♥

• → ✵,

❜② t❤❡ √♥✲❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ❡♠❜❡❞❞✐♥❣ ❡st✐♠❛t♦r✳

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷✵ ✴ ✷✶

SLIDE 37

❊①t❡♥s✐♦♥s ❛♥❞ ❉✐s❝✉ss✐♦♥

❞♦♠❛✐♥ ♥♦t ❜♦✉♥❞❡❞✿ ❞① → φ(①)❞①✱ ❢♦r s♦♠❡ ❞❡♥s✐t② φ✱ ❛♥❞ ✉ → ❡✉−Ψ(✉)φ✱ ❛♥❞ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ ❚ ❢♦r ✇❤✐❝❤ ❞❡♥s✐t✐❡s ❛r❡ ✇❡❧❧ ❞❡✜♥❡❞✳

❡✳❣✳✱ ❢♦r X = R✱ φ(①) =

✶ √ ✷π❡− ①✷

✷ , ❛♥❞ ❦(①, ②) = (✶ + ①②)✷✱ ✇❡ r❡❝♦✈❡r

• ❛✉ss✐❛♥ ❞❡♥s✐t✐❡s✳

❦❡r♥❡❧ ♥♦t ❛ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥✿ ♠♦r❡ ✉✬s t♦ ❞✐s❝❛r❞ Ps❡✉❞♦ ▼▲❊ ✐s ❝♦♥s✐st❡♥t✱ ♣r♦✈✐❞❡❞ t❤❛t ❛ s❡q✉❡♥❝❡ ♦❢ s✉❜s♣❛❝❡s ✐s ❝❤♦s❡♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ✇❛②

❍♦✇ t♦ ❝♦♥str✉❝t s✉❝❤ s❡q✉❡♥❝❡s ♦❢ s✉❜s♣❛❝❡s❄ ❉♦ t❤❡② ❡①✐st ❢♦r ❛♥② ❦❡r♥❡❧s ✲ ✐✳❡✳✱ ❡♥s✉r✐♥❣ t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❣♦❡s t♦ ③❡r♦ q✉✐❝❦❧② ❡♥♦✉❣❤ ✇❤✐❧❡ t❤❡ s♠❛❧❧❡st ❡✐❣❡♥✈❛❧✉❡s ❞❡❝❛② s❧♦✇❧② ❡♥♦✉❣❤❄

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷✶ ✴ ✷✶

SLIDE 38

❊①t❡♥s✐♦♥s ❛♥❞ ❉✐s❝✉ss✐♦♥

❞♦♠❛✐♥ ♥♦t ❜♦✉♥❞❡❞✿ ❞① → φ(①)❞①✱ ❢♦r s♦♠❡ ❞❡♥s✐t② φ✱ ❛♥❞ ✉ → ❡✉−Ψ(✉)φ✱ ❛♥❞ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ ❚ ❢♦r ✇❤✐❝❤ ❞❡♥s✐t✐❡s ❛r❡ ✇❡❧❧ ❞❡✜♥❡❞✳

❡✳❣✳✱ ❢♦r X = R✱ φ(①) =

✶ √ ✷π❡− ①✷

✷ , ❛♥❞ ❦(①, ②) = (✶ + ①②)✷✱ ✇❡ r❡❝♦✈❡r

• ❛✉ss✐❛♥ ❞❡♥s✐t✐❡s✳

❦❡r♥❡❧ ♥♦t ❛ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥✿ ♠♦r❡ ✉✬s t♦ ❞✐s❝❛r❞ Ps❡✉❞♦ ▼▲❊ ✐s ❝♦♥s✐st❡♥t✱ ♣r♦✈✐❞❡❞ t❤❛t ❛ s❡q✉❡♥❝❡ ♦❢ s✉❜s♣❛❝❡s ✐s ❝❤♦s❡♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ✇❛②

❍♦✇ t♦ ❝♦♥str✉❝t s✉❝❤ s❡q✉❡♥❝❡s ♦❢ s✉❜s♣❛❝❡s❄ ❉♦ t❤❡② ❡①✐st ❢♦r ❛♥② ❦❡r♥❡❧s ✲ ✐✳❡✳✱ ❡♥s✉r✐♥❣ t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❣♦❡s t♦ ③❡r♦ q✉✐❝❦❧② ❡♥♦✉❣❤ ✇❤✐❧❡ t❤❡ s♠❛❧❧❡st ❡✐❣❡♥✈❛❧✉❡s ❞❡❝❛② s❧♦✇❧② ❡♥♦✉❣❤❄

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷✶ ✴ ✷✶

SLIDE 39

❊①t❡♥s✐♦♥s ❛♥❞ ❉✐s❝✉ss✐♦♥

❞♦♠❛✐♥ ♥♦t ❜♦✉♥❞❡❞✿ ❞① → φ(①)❞①✱ ❢♦r s♦♠❡ ❞❡♥s✐t② φ✱ ❛♥❞ ✉ → ❡✉−Ψ(✉)φ✱ ❛♥❞ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ ❚ ❢♦r ✇❤✐❝❤ ❞❡♥s✐t✐❡s ❛r❡ ✇❡❧❧ ❞❡✜♥❡❞✳

❡✳❣✳✱ ❢♦r X = R✱ φ(①) =

✶ √ ✷π❡− ①✷

✷ , ❛♥❞ ❦(①, ②) = (✶ + ①②)✷✱ ✇❡ r❡❝♦✈❡r

• ❛✉ss✐❛♥ ❞❡♥s✐t✐❡s✳

❦❡r♥❡❧ ♥♦t ❛ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥✿ ♠♦r❡ ✉✬s t♦ ❞✐s❝❛r❞ Ps❡✉❞♦ ▼▲❊ ✐s ❝♦♥s✐st❡♥t✱ ♣r♦✈✐❞❡❞ t❤❛t ❛ s❡q✉❡♥❝❡ ♦❢ s✉❜s♣❛❝❡s ✐s ❝❤♦s❡♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ✇❛②

❍♦✇ t♦ ❝♦♥str✉❝t s✉❝❤ s❡q✉❡♥❝❡s ♦❢ s✉❜s♣❛❝❡s❄ ❉♦ t❤❡② ❡①✐st ❢♦r ❛♥② ❦❡r♥❡❧s ✲ ✐✳❡✳✱ ❡♥s✉r✐♥❣ t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❣♦❡s t♦ ③❡r♦ q✉✐❝❦❧② ❡♥♦✉❣❤ ✇❤✐❧❡ t❤❡ s♠❛❧❧❡st ❡✐❣❡♥✈❛❧✉❡s ❞❡❝❛② s❧♦✇❧② ❡♥♦✉❣❤❄

❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷✶ ✴ ✷✶