r t t - PowerPoint PPT Presentation

❯♥✇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 ✶✽✱ ✷✵✶✷ ✶ ✴ ✷✶

■♥ ♣❛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❡ ■♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②❄ ❆✐♠✿ ❝♦♥str✉❝t ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②✱ ♦♥ ✇❤✐❝❤ ❡st✐♠❛t✐♦♥ t❤❡♦r② ❝❛♥ ❜❡ ❜✉✐❧t ❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✷ ✴ ✷✶

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

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

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

■♥✜♥✐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 ✶✽✱ ✷✵✶✷ ✷ ✴ ✷✶

■♥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♥❡❧ ❡♠❜❡❞❞✐♥❣ ❦ ( · , ① ) · ✶ ❞① ∈ H ❦ ✳ ❚❤❡♥✱ ❚ = µ ⊥ ´ ´ µ ✶ = ✶ ✱ ❛s ✉ ( ① ) ❞① = � ✉ , µ ✶ � H ❦ . ´ ❙✐♥❝❡ H ❦ ✐♥❝❧✉❞❡s ❝♦♥st❛♥ts✱ ✉ − ✉ ( ① ) ❞① ∈ ❚ ✱ ∀ ✉ ∈ H ❦ ❉✐♥♦ ❙✳ ✭●❛ts❜② ❯♥✐t✮ ❊①♣♦♥❡♥t✐❛❧ ❘❑❍❙ ❢❛♠✐❧② ❖❝t♦❜❡r ✶✽✱ ✷✵✶✷ ✸ ✴ ✷✶

❉❡♥s✐t✐❡s ♣❛r❛♠❡tr✐③❡❞ ❜② ❘❑❍❙ ❢✉♥❝t✐♦♥s ◆♦✇✱ ♣✐❝❦ ✉ ∈ ❚ ✱ ❛♥❞ ❞❡✜♥❡✿ ˆ ❡ ✉ ( ① ) ❞① Ψ( ✉ ) = ❧♦❣ ▲❡♠♠❛ ∀ ✉ ∈ ❚✱ ❡ ✉ − 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 ✶✽✱ ✷✵✶✷ ✹ ✴ ✷✶