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tst rts qts s - - PowerPoint PPT Presentation

Oct 03, 2022 154 likes •1.71k views

tst rts qts s rsts s str ss t

slide-1
SLIDE 1

❙t♦❝❤❛st✐❝ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✿ ✐❞❡❛s ❛♥❞ r❡s✉❧ts ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s

◆✐❣❡❧ ❏✳ ❈✉t❧❛♥❞ ❯♥✐✈❡rs✐t② ♦❢ ❨♦r❦✱ ❯❑ ❡✲♠❛✐❧✿ ♥❝✺✵✼❅②♦r❦✳❛❝✳✉❦

✭❏♦✐♥t ✇✐t❤ ▼❛r❡❦ ❈❛♣✐➠s❦✐✱ ❏❡rr② ❑❡✐s❧❡r✱ ❑❛s✐❛ ●r③❡s✐❛❦✱ ❇r❡♥❞❛♥ ❊♥r✐❣❤t✮

❯❧tr❛▼❛t❤ ✷✵✵✽

P✐s❛ ❏✉♥❡ ✷✵✵✽

slide-2
SLIDE 2

❚❤❡ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘ ❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ✭s◆❙❡✮ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✿

  • ❞✉ =
  • ν∆✉ − ✉, ∇✉ + ❢ (t, ✉) − ∇♣
  • ❞t + ❣(t, ✉)❞✇t

❞✐✈ ✉ = ✵ ✉(t, ①, ω) =✭r❛♥❞♦♠✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢✉✐❞ ❛t t❤❡ ❧♦❝❛t✐♦♥ ① ∈ ❉ ❛t t✐♠❡ t✿ ✉ : [✵, ∞) × ❉ × Ω → R❞ Ω ❂ ❞♦♠❛✐♥ ♦❢ ❛♥ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉ ✵ ✉✵ ✭♠❛② ❜❡ r❛♥❞♦♠✮❀ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❡✐t❤❡r ✉ t ① ✵ ❢♦r ① ❉ ✭♦r s♦♠❡t✐♠❡s ♣❡r✐♦❞✐❝ ❢♦r ❞ ✷✮✳ ❋✐rst s♦❧✉t✐♦♥s ✐♥ ❞ ✸✿ ❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮ ✉s✐♥❣ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s✳ ▼❡t❤♦❞s ❡①t❡♥❞ t♦ ❣✐✈❡ r❡s✉❧ts ♦♥✿ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ s◆❙❡✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ s❦❡t❝❤ ✐♥❢♦r♠❛❧❧② t❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✳

slide-3
SLIDE 3

❚❤❡ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘ ❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ✭s◆❙❡✮ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✿

  • ❞✉ =
  • ν∆✉ − ✉, ∇✉ + ❢ (t, ✉) − ∇♣
  • ❞t + ❣(t, ✉)❞✇t

❞✐✈ ✉ = ✵ ✉(t, ①, ω) =✭r❛♥❞♦♠✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢✉✐❞ ❛t t❤❡ ❧♦❝❛t✐♦♥ ① ∈ ❉ ❛t t✐♠❡ t✿ ✉ : [✵, ∞) × ❉ × Ω → R❞ Ω ❂ ❞♦♠❛✐♥ ♦❢ ❛♥ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉(✵) = ✉✵ ✭♠❛② ❜❡ r❛♥❞♦♠✮❀ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❡✐t❤❡r ✉(t, ①) = ✵ ❢♦r ① ∈ ∂❉ ✭♦r s♦♠❡t✐♠❡s ♣❡r✐♦❞✐❝ ❢♦r ❞ = ✷✮✳ ❋✐rst s♦❧✉t✐♦♥s ✐♥ ❞ ✸✿ ❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮ ✉s✐♥❣ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s✳ ▼❡t❤♦❞s ❡①t❡♥❞ t♦ ❣✐✈❡ r❡s✉❧ts ♦♥✿ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ s◆❙❡✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ s❦❡t❝❤ ✐♥❢♦r♠❛❧❧② t❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✳

slide-4
SLIDE 4

❚❤❡ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘ ❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ✭s◆❙❡✮ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✿

  • ❞✉ =
  • ν∆✉ − ✉, ∇✉ + ❢ (t, ✉) − ∇♣
  • ❞t + ❣(t, ✉)❞✇t

❞✐✈ ✉ = ✵ ✉(t, ①, ω) =✭r❛♥❞♦♠✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢✉✐❞ ❛t t❤❡ ❧♦❝❛t✐♦♥ ① ∈ ❉ ❛t t✐♠❡ t✿ ✉ : [✵, ∞) × ❉ × Ω → R❞ Ω ❂ ❞♦♠❛✐♥ ♦❢ ❛♥ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉(✵) = ✉✵ ✭♠❛② ❜❡ r❛♥❞♦♠✮❀ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❡✐t❤❡r ✉(t, ①) = ✵ ❢♦r ① ∈ ∂❉ ✭♦r s♦♠❡t✐♠❡s ♣❡r✐♦❞✐❝ ❢♦r ❞ = ✷✮✳ ❋✐rst s♦❧✉t✐♦♥s ✐♥ ❞ = ✸✿ ❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮ ✉s✐♥❣ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s✳ ▼❡t❤♦❞s ❡①t❡♥❞ t♦ ❣✐✈❡ r❡s✉❧ts ♦♥✿ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ s◆❙❡✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ s❦❡t❝❤ ✐♥❢♦r♠❛❧❧② t❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✳

slide-5
SLIDE 5

❚❤❡ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘ ❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ✭s◆❙❡✮ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✿

  • ❞✉ =
  • ν∆✉ − ✉, ∇✉ + ❢ (t, ✉) − ∇♣
  • ❞t + ❣(t, ✉)❞✇t

❞✐✈ ✉ = ✵ ✉(t, ①, ω) =✭r❛♥❞♦♠✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢✉✐❞ ❛t t❤❡ ❧♦❝❛t✐♦♥ ① ∈ ❉ ❛t t✐♠❡ t✿ ✉ : [✵, ∞) × ❉ × Ω → R❞ Ω ❂ ❞♦♠❛✐♥ ♦❢ ❛♥ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉(✵) = ✉✵ ✭♠❛② ❜❡ r❛♥❞♦♠✮❀ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❡✐t❤❡r ✉(t, ①) = ✵ ❢♦r ① ∈ ∂❉ ✭♦r s♦♠❡t✐♠❡s ♣❡r✐♦❞✐❝ ❢♦r ❞ = ✷✮✳ ❋✐rst s♦❧✉t✐♦♥s ✐♥ ❞ = ✸✿ ❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮ ✉s✐♥❣ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s✳ ▼❡t❤♦❞s ❡①t❡♥❞ t♦ ❣✐✈❡ r❡s✉❧ts ♦♥✿ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ s◆❙❡✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ s❦❡t❝❤ ✐♥❢♦r♠❛❧❧② t❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✳

slide-6
SLIDE 6

❚❤❡ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘ ❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ✭s◆❙❡✮ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✿

  • ❞✉ =
  • ν∆✉ − ✉, ∇✉ + ❢ (t, ✉) − ∇♣
  • ❞t + ❣(t, ✉)❞✇t

❞✐✈ ✉ = ✵ ✉(t, ①, ω) =✭r❛♥❞♦♠✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢✉✐❞ ❛t t❤❡ ❧♦❝❛t✐♦♥ ① ∈ ❉ ❛t t✐♠❡ t✿ ✉ : [✵, ∞) × ❉ × Ω → R❞ Ω ❂ ❞♦♠❛✐♥ ♦❢ ❛♥ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉(✵) = ✉✵ ✭♠❛② ❜❡ r❛♥❞♦♠✮❀ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❡✐t❤❡r ✉(t, ①) = ✵ ❢♦r ① ∈ ∂❉ ✭♦r s♦♠❡t✐♠❡s ♣❡r✐♦❞✐❝ ❢♦r ❞ = ✷✮✳ ❋✐rst s♦❧✉t✐♦♥s ✐♥ ❞ = ✸✿ ❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮ ✉s✐♥❣ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s✳ ▼❡t❤♦❞s ❡①t❡♥❞ t♦ ❣✐✈❡ r❡s✉❧ts ♦♥✿ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ s◆❙❡✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ s❦❡t❝❤ ✐♥❢♦r♠❛❧❧② t❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✳

slide-7
SLIDE 7

❚❤❡ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘ ❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ✭s◆❙❡✮ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✿

  • ❞✉ =
  • ν∆✉ − ✉, ∇✉ + ❢ (t, ✉) − ∇♣
  • ❞t + ❣(t, ✉)❞✇t

❞✐✈ ✉ = ✵ ✉(t, ①, ω) =✭r❛♥❞♦♠✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢✉✐❞ ❛t t❤❡ ❧♦❝❛t✐♦♥ ① ∈ ❉ ❛t t✐♠❡ t✿ ✉ : [✵, ∞) × ❉ × Ω → R❞ Ω ❂ ❞♦♠❛✐♥ ♦❢ ❛♥ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉(✵) = ✉✵ ✭♠❛② ❜❡ r❛♥❞♦♠✮❀ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❡✐t❤❡r ✉(t, ①) = ✵ ❢♦r ① ∈ ∂❉ ✭♦r s♦♠❡t✐♠❡s ♣❡r✐♦❞✐❝ ❢♦r ❞ = ✷✮✳ ❋✐rst s♦❧✉t✐♦♥s ✐♥ ❞ = ✸✿ ❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮ ✉s✐♥❣ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s✳ ▼❡t❤♦❞s ❡①t❡♥❞ t♦ ❣✐✈❡ r❡s✉❧ts ♦♥✿ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ s◆❙❡✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ s❦❡t❝❤ ✐♥❢♦r♠❛❧❧② t❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✳

slide-8
SLIDE 8

❚❤❡ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘ ❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ✭s◆❙❡✮ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✿

  • ❞✉ =
  • ν∆✉ − ✉, ∇✉ + ❢ (t, ✉) − ∇♣
  • ❞t + ❣(t, ✉)❞✇t

❞✐✈ ✉ = ✵ ✉(t, ①, ω) =✭r❛♥❞♦♠✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢✉✐❞ ❛t t❤❡ ❧♦❝❛t✐♦♥ ① ∈ ❉ ❛t t✐♠❡ t✿ ✉ : [✵, ∞) × ❉ × Ω → R❞ Ω ❂ ❞♦♠❛✐♥ ♦❢ ❛♥ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉(✵) = ✉✵ ✭♠❛② ❜❡ r❛♥❞♦♠✮❀ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❡✐t❤❡r ✉(t, ①) = ✵ ❢♦r ① ∈ ∂❉ ✭♦r s♦♠❡t✐♠❡s ♣❡r✐♦❞✐❝ ❢♦r ❞ = ✷✮✳ ❋✐rst s♦❧✉t✐♦♥s ✐♥ ❞ = ✸✿ ❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮ ✉s✐♥❣ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s✳ ▼❡t❤♦❞s ❡①t❡♥❞ t♦ ❣✐✈❡ r❡s✉❧ts ♦♥✿ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ s◆❙❡✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ s❦❡t❝❤ ✐♥❢♦r♠❛❧❧② t❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✳

slide-9
SLIDE 9

❚❤❡ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘ ❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ✭s◆❙❡✮ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✿

  • ❞✉ =
  • ν∆✉ − ✉, ∇✉ + ❢ (t, ✉) − ∇♣
  • ❞t + ❣(t, ✉)❞✇t

❞✐✈ ✉ = ✵ ✉(t, ①, ω) =✭r❛♥❞♦♠✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢✉✐❞ ❛t t❤❡ ❧♦❝❛t✐♦♥ ① ∈ ❉ ❛t t✐♠❡ t✿ ✉ : [✵, ∞) × ❉ × Ω → R❞ Ω ❂ ❞♦♠❛✐♥ ♦❢ ❛♥ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉(✵) = ✉✵ ✭♠❛② ❜❡ r❛♥❞♦♠✮❀ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❡✐t❤❡r ✉(t, ①) = ✵ ❢♦r ① ∈ ∂❉ ✭♦r s♦♠❡t✐♠❡s ♣❡r✐♦❞✐❝ ❢♦r ❞ = ✷✮✳ ❋✐rst s♦❧✉t✐♦♥s ✐♥ ❞ = ✸✿ ❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮ ✉s✐♥❣ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s✳ ▼❡t❤♦❞s ❡①t❡♥❞ t♦ ❣✐✈❡ r❡s✉❧ts ♦♥✿ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ s◆❙❡✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ s❦❡t❝❤ ✐♥❢♦r♠❛❧❧② t❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✳

slide-10
SLIDE 10

▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛t✐♦♥ ✲ ❍✐❧❜❡rt s♣❛❝❡ s❡tt✐♥❣ ❙❡t H = {✉ ∈ ❈ ∞

✵ (❉, R❞)✿ ❞✐✈ ✉ = ✵} ✇✐t❤ ♥♦r♠s |✉| ❛♥❞ ✉ ❞❡r✐✈❡❞ ❢r♦♠

(✉, ✈) =

❞

  • ❥=✶
  • ❉

✉❥(①)✈ ❥(①)❞①, ( (✉, ✈) ) =

❞

  • ❥=✶

∂✉ ∂①❥ , ∂✈ ∂①❥

  • ❍ ❂ ❝❧♦s✉r❡ ♦❢ H ✐♥ t❤❡ ♥♦r♠ |✉| ❛♥❞ ❱ ✐s t❤❡ ❝❧♦s✉r❡ ✐♥ ♥♦r♠ |✉| + ✉.

❍ ❛♥❞ ❱ ❛r❡ ❍✐❧❜❡rt s♣❛❝❡s ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝ts ❛♥❞ r❡s♣✳ ❆ ❂ s❡❧❢ ❛❞❥♦✐♥t ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ✐♥ ❍❀ ❆ ❤❛s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❡❦ ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ✵

❦

✳ ❍♠ s♣❛♥ ❡✶ ❡♠ ❱✳ ❚❤❡ ♦♣❡r❛t♦r ❇ ✉ ✐s ❞❡✜♥❡❞ ❜② ❇ ✉ ✇ ✉ ✉ ✇ ✳ ❚❤❡ s◆❙❡ ❛r❡ ♥♦✇ ❢♦r♠✉❧❛t❡❞ ❛s ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ ❍✿ ❞✉ ❆✉ ❇ ✉ ❢ t ✉ ❞t ❣ t ✉ ❞✇t ■♥✐t✐❛❧❧② r❡❣❛r❞ t❤✐s ❛s ❛♥ ❡q✉❛t✐♦♥ ✐♥ ❱ ✭t❤❡ ❞✉❛❧ ♦❢ ❱✮ ❛❧t❤♦✉❣❤ ✐t t✉r♥s ♦✉t t❤❛t s♦❧✉t✐♦♥s ❧✐✈❡ ✐♥ ❍ ✭❛♥❞ ✐♥ ❢❛❝t ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✮✳

slide-11
SLIDE 11

▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛t✐♦♥ ✲ ❍✐❧❜❡rt s♣❛❝❡ s❡tt✐♥❣ ❙❡t H = {✉ ∈ ❈ ∞

✵ (❉, R❞)✿ ❞✐✈ ✉ = ✵} ✇✐t❤ ♥♦r♠s |✉| ❛♥❞ ✉ ❞❡r✐✈❡❞ ❢r♦♠

(✉, ✈) =

❞

  • ❥=✶
  • ❉

✉❥(①)✈ ❥(①)❞①, ( (✉, ✈) ) =

❞

  • ❥=✶

∂✉ ∂①❥ , ∂✈ ∂①❥

  • ❍ ❂ ❝❧♦s✉r❡ ♦❢ H ✐♥ t❤❡ ♥♦r♠ |✉| ❛♥❞ ❱ ✐s t❤❡ ❝❧♦s✉r❡ ✐♥ ♥♦r♠ |✉| + ✉.

❍ ❛♥❞ ❱ ❛r❡ ❍✐❧❜❡rt s♣❛❝❡s ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝ts (·, ·) ❛♥❞ ( (·, ·) ) r❡s♣✳ ❆ ❂ s❡❧❢ ❛❞❥♦✐♥t ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ✐♥ ❍❀ ❆ ❤❛s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❡❦ ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ✵

❦

✳ ❍♠ s♣❛♥ ❡✶ ❡♠ ❱✳ ❚❤❡ ♦♣❡r❛t♦r ❇ ✉ ✐s ❞❡✜♥❡❞ ❜② ❇ ✉ ✇ ✉ ✉ ✇ ✳ ❚❤❡ s◆❙❡ ❛r❡ ♥♦✇ ❢♦r♠✉❧❛t❡❞ ❛s ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ ❍✿ ❞✉ ❆✉ ❇ ✉ ❢ t ✉ ❞t ❣ t ✉ ❞✇t ■♥✐t✐❛❧❧② r❡❣❛r❞ t❤✐s ❛s ❛♥ ❡q✉❛t✐♦♥ ✐♥ ❱ ✭t❤❡ ❞✉❛❧ ♦❢ ❱✮ ❛❧t❤♦✉❣❤ ✐t t✉r♥s ♦✉t t❤❛t s♦❧✉t✐♦♥s ❧✐✈❡ ✐♥ ❍ ✭❛♥❞ ✐♥ ❢❛❝t ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✮✳

slide-12
SLIDE 12

▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛t✐♦♥ ✲ ❍✐❧❜❡rt s♣❛❝❡ s❡tt✐♥❣ ❙❡t H = {✉ ∈ ❈ ∞

✵ (❉, R❞)✿ ❞✐✈ ✉ = ✵} ✇✐t❤ ♥♦r♠s |✉| ❛♥❞ ✉ ❞❡r✐✈❡❞ ❢r♦♠

(✉, ✈) =

❞

  • ❥=✶
  • ❉

✉❥(①)✈ ❥(①)❞①, ( (✉, ✈) ) =

❞

  • ❥=✶

∂✉ ∂①❥ , ∂✈ ∂①❥

  • ❍ ❂ ❝❧♦s✉r❡ ♦❢ H ✐♥ t❤❡ ♥♦r♠ |✉| ❛♥❞ ❱ ✐s t❤❡ ❝❧♦s✉r❡ ✐♥ ♥♦r♠ |✉| + ✉.

❍ ❛♥❞ ❱ ❛r❡ ❍✐❧❜❡rt s♣❛❝❡s ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝ts (·, ·) ❛♥❞ ( (·, ·) ) r❡s♣✳ ❆ ❂ s❡❧❢ ❛❞❥♦✐♥t ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ −∆ ✐♥ ❍❀ ❆ ❤❛s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s {❡❦} ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ✵ < λ❦ ր ∞✳ ❍♠ s♣❛♥ ❡✶ ❡♠ ❱✳ ❚❤❡ ♦♣❡r❛t♦r ❇ ✉ ✐s ❞❡✜♥❡❞ ❜② ❇ ✉ ✇ ✉ ✉ ✇ ✳ ❚❤❡ s◆❙❡ ❛r❡ ♥♦✇ ❢♦r♠✉❧❛t❡❞ ❛s ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ ❍✿ ❞✉ ❆✉ ❇ ✉ ❢ t ✉ ❞t ❣ t ✉ ❞✇t ■♥✐t✐❛❧❧② r❡❣❛r❞ t❤✐s ❛s ❛♥ ❡q✉❛t✐♦♥ ✐♥ ❱ ✭t❤❡ ❞✉❛❧ ♦❢ ❱✮ ❛❧t❤♦✉❣❤ ✐t t✉r♥s ♦✉t t❤❛t s♦❧✉t✐♦♥s ❧✐✈❡ ✐♥ ❍ ✭❛♥❞ ✐♥ ❢❛❝t ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✮✳

slide-13
SLIDE 13

▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛t✐♦♥ ✲ ❍✐❧❜❡rt s♣❛❝❡ s❡tt✐♥❣ ❙❡t H = {✉ ∈ ❈ ∞

✵ (❉, R❞)✿ ❞✐✈ ✉ = ✵} ✇✐t❤ ♥♦r♠s |✉| ❛♥❞ ✉ ❞❡r✐✈❡❞ ❢r♦♠

(✉, ✈) =

❞

  • ❥=✶
  • ❉

✉❥(①)✈ ❥(①)❞①, ( (✉, ✈) ) =

❞

  • ❥=✶

∂✉ ∂①❥ , ∂✈ ∂①❥

  • ❍ ❂ ❝❧♦s✉r❡ ♦❢ H ✐♥ t❤❡ ♥♦r♠ |✉| ❛♥❞ ❱ ✐s t❤❡ ❝❧♦s✉r❡ ✐♥ ♥♦r♠ |✉| + ✉.

❍ ❛♥❞ ❱ ❛r❡ ❍✐❧❜❡rt s♣❛❝❡s ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝ts (·, ·) ❛♥❞ ( (·, ·) ) r❡s♣✳ ❆ ❂ s❡❧❢ ❛❞❥♦✐♥t ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ −∆ ✐♥ ❍❀ ❆ ❤❛s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s {❡❦} ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ✵ < λ❦ ր ∞✳ ❍♠ = s♣❛♥{❡✶, . . . , ❡♠} ⊂ ❱✳ ❚❤❡ ♦♣❡r❛t♦r ❇ ✉ ✐s ❞❡✜♥❡❞ ❜② ❇ ✉ ✇ ✉ ✉ ✇ ✳ ❚❤❡ s◆❙❡ ❛r❡ ♥♦✇ ❢♦r♠✉❧❛t❡❞ ❛s ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ ❍✿ ❞✉ ❆✉ ❇ ✉ ❢ t ✉ ❞t ❣ t ✉ ❞✇t ■♥✐t✐❛❧❧② r❡❣❛r❞ t❤✐s ❛s ❛♥ ❡q✉❛t✐♦♥ ✐♥ ❱ ✭t❤❡ ❞✉❛❧ ♦❢ ❱✮ ❛❧t❤♦✉❣❤ ✐t t✉r♥s ♦✉t t❤❛t s♦❧✉t✐♦♥s ❧✐✈❡ ✐♥ ❍ ✭❛♥❞ ✐♥ ❢❛❝t ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✮✳

slide-14
SLIDE 14

▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛t✐♦♥ ✲ ❍✐❧❜❡rt s♣❛❝❡ s❡tt✐♥❣ ❙❡t H = {✉ ∈ ❈ ∞

✵ (❉, R❞)✿ ❞✐✈ ✉ = ✵} ✇✐t❤ ♥♦r♠s |✉| ❛♥❞ ✉ ❞❡r✐✈❡❞ ❢r♦♠

(✉, ✈) =

❞

  • ❥=✶
  • ❉

✉❥(①)✈ ❥(①)❞①, ( (✉, ✈) ) =

❞

  • ❥=✶

∂✉ ∂①❥ , ∂✈ ∂①❥

  • ❍ ❂ ❝❧♦s✉r❡ ♦❢ H ✐♥ t❤❡ ♥♦r♠ |✉| ❛♥❞ ❱ ✐s t❤❡ ❝❧♦s✉r❡ ✐♥ ♥♦r♠ |✉| + ✉.

❍ ❛♥❞ ❱ ❛r❡ ❍✐❧❜❡rt s♣❛❝❡s ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝ts (·, ·) ❛♥❞ ( (·, ·) ) r❡s♣✳ ❆ ❂ s❡❧❢ ❛❞❥♦✐♥t ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ −∆ ✐♥ ❍❀ ❆ ❤❛s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s {❡❦} ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ✵ < λ❦ ր ∞✳ ❍♠ = s♣❛♥{❡✶, . . . , ❡♠} ⊂ ❱✳ ❚❤❡ ♦♣❡r❛t♦r ❇(✉) ✐s ❞❡✜♥❡❞ ❜② ❇(✉)✇ = (✉, ∇✉, ✇)✳ ❚❤❡ s◆❙❡ ❛r❡ ♥♦✇ ❢♦r♠✉❧❛t❡❞ ❛s ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ ❍✿ ❞✉ ❆✉ ❇ ✉ ❢ t ✉ ❞t ❣ t ✉ ❞✇t ■♥✐t✐❛❧❧② r❡❣❛r❞ t❤✐s ❛s ❛♥ ❡q✉❛t✐♦♥ ✐♥ ❱ ✭t❤❡ ❞✉❛❧ ♦❢ ❱✮ ❛❧t❤♦✉❣❤ ✐t t✉r♥s ♦✉t t❤❛t s♦❧✉t✐♦♥s ❧✐✈❡ ✐♥ ❍ ✭❛♥❞ ✐♥ ❢❛❝t ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✮✳

slide-15
SLIDE 15

▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛t✐♦♥ ✲ ❍✐❧❜❡rt s♣❛❝❡ s❡tt✐♥❣ ❙❡t H = {✉ ∈ ❈ ∞

✵ (❉, R❞)✿ ❞✐✈ ✉ = ✵} ✇✐t❤ ♥♦r♠s |✉| ❛♥❞ ✉ ❞❡r✐✈❡❞ ❢r♦♠

(✉, ✈) =

❞

  • ❥=✶
  • ❉

✉❥(①)✈ ❥(①)❞①, ( (✉, ✈) ) =

❞

  • ❥=✶

∂✉ ∂①❥ , ∂✈ ∂①❥

  • ❍ ❂ ❝❧♦s✉r❡ ♦❢ H ✐♥ t❤❡ ♥♦r♠ |✉| ❛♥❞ ❱ ✐s t❤❡ ❝❧♦s✉r❡ ✐♥ ♥♦r♠ |✉| + ✉.

❍ ❛♥❞ ❱ ❛r❡ ❍✐❧❜❡rt s♣❛❝❡s ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝ts (·, ·) ❛♥❞ ( (·, ·) ) r❡s♣✳ ❆ ❂ s❡❧❢ ❛❞❥♦✐♥t ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ −∆ ✐♥ ❍❀ ❆ ❤❛s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s {❡❦} ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ✵ < λ❦ ր ∞✳ ❍♠ = s♣❛♥{❡✶, . . . , ❡♠} ⊂ ❱✳ ❚❤❡ ♦♣❡r❛t♦r ❇(✉) ✐s ❞❡✜♥❡❞ ❜② ❇(✉)✇ = (✉, ∇✉, ✇)✳ ❚❤❡ s◆❙❡ ❛r❡ ♥♦✇ ❢♦r♠✉❧❛t❡❞ ❛s ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ ❍✿ ❞✉ = [−ν❆✉ − ❇(✉) + ❢ (t, ✉)]❞t + ❣(t, ✉)❞✇t ■♥✐t✐❛❧❧② r❡❣❛r❞ t❤✐s ❛s ❛♥ ❡q✉❛t✐♦♥ ✐♥ ❱′ ✭t❤❡ ❞✉❛❧ ♦❢ ❱✮ ❛❧t❤♦✉❣❤ ✐t t✉r♥s ♦✉t t❤❛t s♦❧✉t✐♦♥s ❧✐✈❡ ✐♥ ❍ ✭❛♥❞ ✐♥ ❢❛❝t ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✮✳

slide-16
SLIDE 16

❞✉ = [−ν❆✉ − ❇(✉) + ❢ (t, ✉)]❞t + ❣(t, ✉)❞✇t ✭✶✮ ❚❤❡ ❡q✉❛t✐♦♥ ✐s ✉♥❞❡rst♦♦❞ ❛s ❛ ✇❡❛❦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✿ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s t❤❡ ✜rst

  • ❂ ❇♦❝❤♥❡r ✐♥t❡❣r❛❧❀ t❤❡ s❡❝♦♥❞
  • ❂ ■❝❤✐❦❛✇❛✬s ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ■tô

✐♥t❡❣r❛❧ t♦ ❍✐❧❜❡rt s♣❛❝❡s❀ ❡✈❛❧✉❛t❡❞ ❜② t❡st✐♥❣ ❛❣❛✐♥st ❢✉♥❝t✐♦♥s ✐♥ ❱✳ ❚❤❡ ♥♦✐s❡ ✇ ✵ ❍ ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇✐t❤ tr❛❝❡ ❝❧❛ss ❝♦✈❛r✐❛♥❝❡✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ❣ ✵ ❱ ▲ ❍ ❍ ❛♥❞ ❢ ✵ ❱ ❱ ❝❛♥ ❜❡ q✉✐t❡ ❣❡♥❡r❛❧ ✲ ✇❡ ♦♥❧② ♥❡❡❞ ❛♣♣r♦♣r✐❛t❡ ❝♦♥t✐♥✉✐t② ❛♥❞ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s✳ ✭❚❤❡ r❡str✐❝t✐♦♥ t♦ ❱ ✐♥ t❤❡ ❞♦♠❛✐♥s ✐s s✉✣❝✐❡♥t ❜❡❝❛✉s❡ s♦❧✉t✐♦♥s ✇✐❧❧ ❧✐❡ ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✳✮ ◆♦t❡ ❚❤❡ ♣r❡ss✉r❡ ❤❛s ❞✐s❛♣♣❡❛r❡❞✱ ❜❡❝❛✉s❡ ♣ ✵ ✐♥ ❱ ✳

slide-17
SLIDE 17

❞✉ = [−ν❆✉ − ❇(✉) + ❢ (t, ✉)]❞t + ❣(t, ✉)❞✇t ✭✶✮ ❚❤❡ ❡q✉❛t✐♦♥ ✐s ✉♥❞❡rst♦♦❞ ❛s ❛ ✇❡❛❦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✿ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s t❤❡ ✜rst

  • ❂ ❇♦❝❤♥❡r ✐♥t❡❣r❛❧❀ t❤❡ s❡❝♦♥❞
  • ❂ ■❝❤✐❦❛✇❛✬s ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ■tô

✐♥t❡❣r❛❧ t♦ ❍✐❧❜❡rt s♣❛❝❡s❀ ❡✈❛❧✉❛t❡❞ ❜② t❡st✐♥❣ ❛❣❛✐♥st ❢✉♥❝t✐♦♥s ✐♥ ❱✳ ❚❤❡ ♥♦✐s❡ ✇ : [✵, ∞) × Ω → ❍ ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇✐t❤ tr❛❝❡ ❝❧❛ss ❝♦✈❛r✐❛♥❝❡✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ❣ ✵ ❱ ▲ ❍ ❍ ❛♥❞ ❢ ✵ ❱ ❱ ❝❛♥ ❜❡ q✉✐t❡ ❣❡♥❡r❛❧ ✲ ✇❡ ♦♥❧② ♥❡❡❞ ❛♣♣r♦♣r✐❛t❡ ❝♦♥t✐♥✉✐t② ❛♥❞ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s✳ ✭❚❤❡ r❡str✐❝t✐♦♥ t♦ ❱ ✐♥ t❤❡ ❞♦♠❛✐♥s ✐s s✉✣❝✐❡♥t ❜❡❝❛✉s❡ s♦❧✉t✐♦♥s ✇✐❧❧ ❧✐❡ ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✳✮ ◆♦t❡ ❚❤❡ ♣r❡ss✉r❡ ❤❛s ❞✐s❛♣♣❡❛r❡❞✱ ❜❡❝❛✉s❡ ♣ ✵ ✐♥ ❱ ✳

slide-18
SLIDE 18

❞✉ = [−ν❆✉ − ❇(✉) + ❢ (t, ✉)]❞t + ❣(t, ✉)❞✇t ✭✶✮ ❚❤❡ ❡q✉❛t✐♦♥ ✐s ✉♥❞❡rst♦♦❞ ❛s ❛ ✇❡❛❦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✿ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s t❤❡ ✜rst

  • ❂ ❇♦❝❤♥❡r ✐♥t❡❣r❛❧❀ t❤❡ s❡❝♦♥❞
  • ❂ ■❝❤✐❦❛✇❛✬s ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ■tô

✐♥t❡❣r❛❧ t♦ ❍✐❧❜❡rt s♣❛❝❡s❀ ❡✈❛❧✉❛t❡❞ ❜② t❡st✐♥❣ ❛❣❛✐♥st ❢✉♥❝t✐♦♥s ✐♥ ❱✳ ❚❤❡ ♥♦✐s❡ ✇ : [✵, ∞) × Ω → ❍ ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇✐t❤ tr❛❝❡ ❝❧❛ss ❝♦✈❛r✐❛♥❝❡✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ❣ : [✵, ∞) × ❱ → ▲(❍, ❍) ❛♥❞ ❢ : [✵, ∞) × ❱ → ❱′. ❝❛♥ ❜❡ q✉✐t❡ ❣❡♥❡r❛❧ ✲ ✇❡ ♦♥❧② ♥❡❡❞ ❛♣♣r♦♣r✐❛t❡ ❝♦♥t✐♥✉✐t② ❛♥❞ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s✳ ✭❚❤❡ r❡str✐❝t✐♦♥ t♦ ❱ ✐♥ t❤❡ ❞♦♠❛✐♥s ✐s s✉✣❝✐❡♥t ❜❡❝❛✉s❡ s♦❧✉t✐♦♥s ✇✐❧❧ ❧✐❡ ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✳✮ ◆♦t❡ ❚❤❡ ♣r❡ss✉r❡ ❤❛s ❞✐s❛♣♣❡❛r❡❞✱ ❜❡❝❛✉s❡ ♣ ✵ ✐♥ ❱ ✳

slide-19
SLIDE 19

❞✉ = [−ν❆✉ − ❇(✉) + ❢ (t, ✉)]❞t + ❣(t, ✉)❞✇t ✭✶✮ ❚❤❡ ❡q✉❛t✐♦♥ ✐s ✉♥❞❡rst♦♦❞ ❛s ❛ ✇❡❛❦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✿ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s t❤❡ ✜rst

  • ❂ ❇♦❝❤♥❡r ✐♥t❡❣r❛❧❀ t❤❡ s❡❝♦♥❞
  • ❂ ■❝❤✐❦❛✇❛✬s ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ■tô

✐♥t❡❣r❛❧ t♦ ❍✐❧❜❡rt s♣❛❝❡s❀ ❡✈❛❧✉❛t❡❞ ❜② t❡st✐♥❣ ❛❣❛✐♥st ❢✉♥❝t✐♦♥s ✐♥ ❱✳ ❚❤❡ ♥♦✐s❡ ✇ : [✵, ∞) × Ω → ❍ ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇✐t❤ tr❛❝❡ ❝❧❛ss ❝♦✈❛r✐❛♥❝❡✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ❣ : [✵, ∞) × ❱ → ▲(❍, ❍) ❛♥❞ ❢ : [✵, ∞) × ❱ → ❱′. ❝❛♥ ❜❡ q✉✐t❡ ❣❡♥❡r❛❧ ✲ ✇❡ ♦♥❧② ♥❡❡❞ ❛♣♣r♦♣r✐❛t❡ ❝♦♥t✐♥✉✐t② ❛♥❞ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s✳ ✭❚❤❡ r❡str✐❝t✐♦♥ t♦ ❱ ✐♥ t❤❡ ❞♦♠❛✐♥s ✐s s✉✣❝✐❡♥t ❜❡❝❛✉s❡ s♦❧✉t✐♦♥s ✇✐❧❧ ❧✐❡ ✐♥ ❱ ❢♦r ❛❧♠♦st ❛❧❧ t✐♠❡s✳✮ ◆♦t❡ ❚❤❡ ♣r❡ss✉r❡ ❤❛s ❞✐s❛♣♣❡❛r❡❞✱ ❜❡❝❛✉s❡ ∇♣ = ✵ ✐♥ ❱′✳

slide-20
SLIDE 20

❇❛s✐❝ ❊①✐st❡♥❝❡ ❚❤❡♦r❡♠

❚❤❡♦r❡♠

❋♦r ❛♥② ✉✵ ∈ ❍ ❛♥❞ ❣✐✈❡♥ ❢ , ❣ t❤❡r❡ ✐s ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω ❝❛rr②✐♥❣ ❛♥ ❍✲✈❛❧✉❡❞ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❛♥❞ ❛ ✭✇❡❛❦✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❛t ✐s✱ ❛♥ ❛❞❛♣t❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ ✵ ❍ s✉❝❤ t❤❛t ❢♦r ❛✳❛✳ ✭✐✮ ✉ ▲✷ ✵ ❚ ❱ ▲ ✵ ❚ ❍ ❈ ✵ ❚ ❍✇❡❛❦ ❢♦r ❛❧❧ ❚ ✱ ✭✐✐✮ ❢♦r ❛❧❧ t ✵ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ❞s

t ✵

❣ s ✉ s ❞✇s

slide-21
SLIDE 21

❇❛s✐❝ ❊①✐st❡♥❝❡ ❚❤❡♦r❡♠

❚❤❡♦r❡♠

❋♦r ❛♥② ✉✵ ∈ ❍ ❛♥❞ ❣✐✈❡♥ ❢ , ❣ t❤❡r❡ ✐s ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω ❝❛rr②✐♥❣ ❛♥ ❍✲✈❛❧✉❡❞ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❛♥❞ ❛ ✭✇❡❛❦✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❛t ✐s✱ ❛♥ ❛❞❛♣t❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ : [✵, ∞) × Ω → ❍ s✉❝❤ t❤❛t ❢♦r ❛✳❛✳ ω ✭✐✮ ✉ ▲✷ ✵ ❚ ❱ ▲ ✵ ❚ ❍ ❈ ✵ ❚ ❍✇❡❛❦ ❢♦r ❛❧❧ ❚ ✱ ✭✐✐✮ ❢♦r ❛❧❧ t ✵ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ❞s

t ✵

❣ s ✉ s ❞✇s

slide-22
SLIDE 22

❇❛s✐❝ ❊①✐st❡♥❝❡ ❚❤❡♦r❡♠

❚❤❡♦r❡♠

❋♦r ❛♥② ✉✵ ∈ ❍ ❛♥❞ ❣✐✈❡♥ ❢ , ❣ t❤❡r❡ ✐s ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω ❝❛rr②✐♥❣ ❛♥ ❍✲✈❛❧✉❡❞ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❛♥❞ ❛ ✭✇❡❛❦✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❛t ✐s✱ ❛♥ ❛❞❛♣t❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ : [✵, ∞) × Ω → ❍ s✉❝❤ t❤❛t ❢♦r ❛✳❛✳ ω ✭✐✮ ✉(·, ω) ∈ ▲✷(✵, ❚; ❱ ) ∩ ▲∞(✵, ❚; ❍) ∩ ❈(✵, ❚; ❍✇❡❛❦) ❢♦r ❛❧❧ ❚ < ∞ ✱ ✭✐✐✮ ❢♦r ❛❧❧ t ✵ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ❞s

t ✵

❣ s ✉ s ❞✇s

slide-23
SLIDE 23

❇❛s✐❝ ❊①✐st❡♥❝❡ ❚❤❡♦r❡♠

❚❤❡♦r❡♠

❋♦r ❛♥② ✉✵ ∈ ❍ ❛♥❞ ❣✐✈❡♥ ❢ , ❣ t❤❡r❡ ✐s ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω ❝❛rr②✐♥❣ ❛♥ ❍✲✈❛❧✉❡❞ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❛♥❞ ❛ ✭✇❡❛❦✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❛t ✐s✱ ❛♥ ❛❞❛♣t❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ : [✵, ∞) × Ω → ❍ s✉❝❤ t❤❛t ❢♦r ❛✳❛✳ ω ✭✐✮ ✉(·, ω) ∈ ▲✷(✵, ❚; ❱ ) ∩ ▲∞(✵, ❚; ❍) ∩ ❈(✵, ❚; ❍✇❡❛❦) ❢♦r ❛❧❧ ❚ < ∞ ✱ ✭✐✐✮ ❢♦r ❛❧❧ t ≥ ✵ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s

slide-24
SLIDE 24

❚❤❡ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s ✭✶✮ s♦❧✈❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ ✈❡rs✐♦♥ ✭t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥✮ ✐♥ ❡❛❝❤ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❍♥ ♦♥ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

♥ ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇♥

✭✷✮ ♣❛ss t♦ t❤❡ ❧✐♠✐t ❛s ♥ ❚❤✐s ♥❡❡❞s s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s ❛♥❞ ✇❛②s t♦ ❡♥❧❛r❣❡ t❤❡ s♣❛❝❡s

♥

t♦ ❛ ✏❧✐♠✐t✑ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✭✇❤✐❝❤ ♠❛② ❞❡♣❡♥❞ ♦♥ t❤❡ s♦❧✉t✐♦♥✮✳ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ♣r♦✈✐❞❡ ❛ s✐♥❣❧❡ s♣❛❝❡ ✭❛ ▲♦❡❜ s♣❛❝❡✮ ❛♥❞ ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❝❛rr②✐♥❣ s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ✭r❛♥❞♦♠✮ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❛❧❧ ❢ ❣ ❚❤✐s ♠❛❦❡s t❤❡♠ ♣♦✇❡r❢✉❧ ❢♦r ❞✐s❝✉ss✐♥❣ ❛ttr❛❝t♦rs ❛♥❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡✳ ▲♦❡❜ s♣❛❝❡s ❛r❡ s❛t✉r❛t❡❞ ❛♥❞ ❤♦♠♦❣❡♥❡♦✉s✳

slide-25
SLIDE 25

❚❤❡ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s ✭✶✮ s♦❧✈❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ ✈❡rs✐♦♥ ✭t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥✮ ✐♥ ❡❛❝❤ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❍♥ ♦♥ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω♥ ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇♥ ✭✷✮ ♣❛ss t♦ t❤❡ ❧✐♠✐t ❛s ♥ ❚❤✐s ♥❡❡❞s s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s ❛♥❞ ✇❛②s t♦ ❡♥❧❛r❣❡ t❤❡ s♣❛❝❡s

♥

t♦ ❛ ✏❧✐♠✐t✑ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✭✇❤✐❝❤ ♠❛② ❞❡♣❡♥❞ ♦♥ t❤❡ s♦❧✉t✐♦♥✮✳ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ♣r♦✈✐❞❡ ❛ s✐♥❣❧❡ s♣❛❝❡ ✭❛ ▲♦❡❜ s♣❛❝❡✮ ❛♥❞ ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❝❛rr②✐♥❣ s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ✭r❛♥❞♦♠✮ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❛❧❧ ❢ ❣ ❚❤✐s ♠❛❦❡s t❤❡♠ ♣♦✇❡r❢✉❧ ❢♦r ❞✐s❝✉ss✐♥❣ ❛ttr❛❝t♦rs ❛♥❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡✳ ▲♦❡❜ s♣❛❝❡s ❛r❡ s❛t✉r❛t❡❞ ❛♥❞ ❤♦♠♦❣❡♥❡♦✉s✳

slide-26
SLIDE 26

❚❤❡ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s ✭✶✮ s♦❧✈❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ ✈❡rs✐♦♥ ✭t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥✮ ✐♥ ❡❛❝❤ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❍♥ ♦♥ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω♥ ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇♥ ✭✷✮ ♣❛ss t♦ t❤❡ ❧✐♠✐t ❛s ♥ → ∞ ❚❤✐s ♥❡❡❞s s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s ❛♥❞ ✇❛②s t♦ ❡♥❧❛r❣❡ t❤❡ s♣❛❝❡s

♥

t♦ ❛ ✏❧✐♠✐t✑ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✭✇❤✐❝❤ ♠❛② ❞❡♣❡♥❞ ♦♥ t❤❡ s♦❧✉t✐♦♥✮✳ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ♣r♦✈✐❞❡ ❛ s✐♥❣❧❡ s♣❛❝❡ ✭❛ ▲♦❡❜ s♣❛❝❡✮ ❛♥❞ ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❝❛rr②✐♥❣ s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ✭r❛♥❞♦♠✮ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❛❧❧ ❢ ❣ ❚❤✐s ♠❛❦❡s t❤❡♠ ♣♦✇❡r❢✉❧ ❢♦r ❞✐s❝✉ss✐♥❣ ❛ttr❛❝t♦rs ❛♥❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡✳ ▲♦❡❜ s♣❛❝❡s ❛r❡ s❛t✉r❛t❡❞ ❛♥❞ ❤♦♠♦❣❡♥❡♦✉s✳

slide-27
SLIDE 27

❚❤❡ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s ✭✶✮ s♦❧✈❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ ✈❡rs✐♦♥ ✭t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥✮ ✐♥ ❡❛❝❤ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❍♥ ♦♥ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω♥ ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇♥ ✭✷✮ ♣❛ss t♦ t❤❡ ❧✐♠✐t ❛s ♥ → ∞ ❚❤✐s ♥❡❡❞s s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s ❛♥❞ ✇❛②s t♦ ❡♥❧❛r❣❡ t❤❡ s♣❛❝❡s Ω Ω♥ t♦ ❛ ✏❧✐♠✐t✑ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✭✇❤✐❝❤ ♠❛② ❞❡♣❡♥❞ ♦♥ t❤❡ s♦❧✉t✐♦♥✮✳ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ♣r♦✈✐❞❡ ❛ s✐♥❣❧❡ s♣❛❝❡ ✭❛ ▲♦❡❜ s♣❛❝❡✮ ❛♥❞ ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❝❛rr②✐♥❣ s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ✭r❛♥❞♦♠✮ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❛❧❧ ❢ ❣ ❚❤✐s ♠❛❦❡s t❤❡♠ ♣♦✇❡r❢✉❧ ❢♦r ❞✐s❝✉ss✐♥❣ ❛ttr❛❝t♦rs ❛♥❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡✳ ▲♦❡❜ s♣❛❝❡s ❛r❡ s❛t✉r❛t❡❞ ❛♥❞ ❤♦♠♦❣❡♥❡♦✉s✳

slide-28
SLIDE 28

❚❤❡ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s ✭✶✮ s♦❧✈❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ ✈❡rs✐♦♥ ✭t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥✮ ✐♥ ❡❛❝❤ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❍♥ ♦♥ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω♥ ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇♥ ✭✷✮ ♣❛ss t♦ t❤❡ ❧✐♠✐t ❛s ♥ → ∞ ❚❤✐s ♥❡❡❞s s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s ❛♥❞ ✇❛②s t♦ ❡♥❧❛r❣❡ t❤❡ s♣❛❝❡s Ω Ω♥ t♦ ❛ ✏❧✐♠✐t✑ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✭✇❤✐❝❤ ♠❛② ❞❡♣❡♥❞ ♦♥ t❤❡ s♦❧✉t✐♦♥✮✳ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ♣r♦✈✐❞❡ ❛ s✐♥❣❧❡ s♣❛❝❡ Ω Ω ✭❛ ▲♦❡❜ s♣❛❝❡✮ ❛♥❞ ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✇ ❝❛rr②✐♥❣ s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ✭r❛♥❞♦♠✮ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❛❧❧ ❢ , ❣. ❚❤✐s ♠❛❦❡s t❤❡♠ ♣♦✇❡r❢✉❧ ❢♦r ❞✐s❝✉ss✐♥❣ ❛ttr❛❝t♦rs ❛♥❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ❢♦r s◆❙❡✳ ▲♦❡❜ s♣❛❝❡s ❛r❡ s❛t✉r❛t❡❞ ❛♥❞ ❤♦♠♦❣❡♥❡♦✉s✳

slide-29
SLIDE 29

▲❖❊❇ ❙P❆❈❊ ▼❊❚❍❖❉❙ ❋❖❘ s◆❙❡ ◆❖◆❙❚❆◆❉❆❘❉ ❆◆❆▲❨❙■❙ ❚❤❡ ❤②♣❡rr❡❛❧s ♦r ♥♦♥st❛♥❞❛r❞ r❡❛❧s ✐s ❛ ✜❡❧❞ s✉❝❤ t❤❛t ❝♦♥t❛✐♥s ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ♥✉♠❜❡rs❀ ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t❡ ♥✉♠❜❡rs ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✿ ▲❡t ① ✳ ❲❡ s❛② t❤❛t ✭✐✮ ① ✐s ✐♥✜♥✐t❡s✐♠❛❧ ✐❢ ① ❢♦r ❛❧❧ ✵ ❀ ✭✐✐✮ ① ✐s ✜♥✐t❡ ✐❢ ① r ❢♦r s♦♠❡ r ❀ ✭✐✐✐✮ ① ✐s ✐♥✜♥✐t❡ ✐❢ ① r ❢♦r ❛❧❧ r ✳ ✭✐✈✮ ① ❛♥❞ ② ❛r❡ ✐♥✜♥✐t❡❧② ❝❧♦s❡✱ ❞❡♥♦t❡❞ ❜② ① ②✱ ✐❢ ① ② ✐s ✐♥✜♥✐t❡s✐♠❛❧✳ ✭❙♦ ① ✵ ♠❡❛♥s t❤❛t ① ✐s ✐♥✜♥✐t❡s✐♠❛❧✮ ❖♥❡ ✇❛② t♦ ❝♦♥str✉❝t ✐s ❛s ❛♥ ✉❧tr❛♣♦✇❡r ♦❢ t❤❡ r❡❛❧s ✇❤❡r❡ ✐s ❛ ♥♦♥♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r ✭♦r ♠❛①✐♠❛❧ ✜❧t❡r✮ ♦♥ ✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ✐s ❣✐✈❡♥ ❜② ✶

✶ ✷ ✶ ✸

✳

slide-30
SLIDE 30

▲❖❊❇ ❙P❆❈❊ ▼❊❚❍❖❉❙ ❋❖❘ s◆❙❡ ◆❖◆❙❚❆◆❉❆❘❉ ❆◆❆▲❨❙■❙ ❚❤❡ ❤②♣❡rr❡❛❧s ♦r ♥♦♥st❛♥❞❛r❞ r❡❛❧s ∗R ⊃R ✐s ❛ ✜❡❧❞ s✉❝❤ t❤❛t ∗R ❝♦♥t❛✐♥s ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ♥✉♠❜❡rs❀ ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t❡ ♥✉♠❜❡rs ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✿ ▲❡t ① ∈ ∗R✳ ❲❡ s❛② t❤❛t ✭✐✮ ① ✐s ✐♥✜♥✐t❡s✐♠❛❧ ✐❢ |①| < ε ❢♦r ❛❧❧ ε > ✵, ε ∈ R❀ ✭✐✐✮ ① ✐s ✜♥✐t❡ ✐❢ ① r ❢♦r s♦♠❡ r ❀ ✭✐✐✐✮ ① ✐s ✐♥✜♥✐t❡ ✐❢ ① r ❢♦r ❛❧❧ r ✳ ✭✐✈✮ ① ❛♥❞ ② ❛r❡ ✐♥✜♥✐t❡❧② ❝❧♦s❡✱ ❞❡♥♦t❡❞ ❜② ① ②✱ ✐❢ ① ② ✐s ✐♥✜♥✐t❡s✐♠❛❧✳ ✭❙♦ ① ✵ ♠❡❛♥s t❤❛t ① ✐s ✐♥✜♥✐t❡s✐♠❛❧✮ ❖♥❡ ✇❛② t♦ ❝♦♥str✉❝t ✐s ❛s ❛♥ ✉❧tr❛♣♦✇❡r ♦❢ t❤❡ r❡❛❧s ✇❤❡r❡ ✐s ❛ ♥♦♥♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r ✭♦r ♠❛①✐♠❛❧ ✜❧t❡r✮ ♦♥ ✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ✐s ❣✐✈❡♥ ❜② ✶

✶ ✷ ✶ ✸

✳

slide-31
SLIDE 31

▲❖❊❇ ❙P❆❈❊ ▼❊❚❍❖❉❙ ❋❖❘ s◆❙❡ ◆❖◆❙❚❆◆❉❆❘❉ ❆◆❆▲❨❙■❙ ❚❤❡ ❤②♣❡rr❡❛❧s ♦r ♥♦♥st❛♥❞❛r❞ r❡❛❧s ∗R ⊃R ✐s ❛ ✜❡❧❞ s✉❝❤ t❤❛t ∗R ❝♦♥t❛✐♥s ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ♥✉♠❜❡rs❀ ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t❡ ♥✉♠❜❡rs ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✿ ▲❡t ① ∈ ∗R✳ ❲❡ s❛② t❤❛t ✭✐✮ ① ✐s ✐♥✜♥✐t❡s✐♠❛❧ ✐❢ |①| < ε ❢♦r ❛❧❧ ε > ✵, ε ∈ R❀ ✭✐✐✮ ① ✐s ✜♥✐t❡ ✐❢ |①| < r ❢♦r s♦♠❡ r ∈ R❀ ✭✐✐✐✮ ① ✐s ✐♥✜♥✐t❡ ✐❢ ① r ❢♦r ❛❧❧ r ✳ ✭✐✈✮ ① ❛♥❞ ② ❛r❡ ✐♥✜♥✐t❡❧② ❝❧♦s❡✱ ❞❡♥♦t❡❞ ❜② ① ②✱ ✐❢ ① ② ✐s ✐♥✜♥✐t❡s✐♠❛❧✳ ✭❙♦ ① ✵ ♠❡❛♥s t❤❛t ① ✐s ✐♥✜♥✐t❡s✐♠❛❧✮ ❖♥❡ ✇❛② t♦ ❝♦♥str✉❝t ✐s ❛s ❛♥ ✉❧tr❛♣♦✇❡r ♦❢ t❤❡ r❡❛❧s ✇❤❡r❡ ✐s ❛ ♥♦♥♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r ✭♦r ♠❛①✐♠❛❧ ✜❧t❡r✮ ♦♥ ✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ✐s ❣✐✈❡♥ ❜② ✶

✶ ✷ ✶ ✸

✳

slide-32
SLIDE 32

▲❖❊❇ ❙P❆❈❊ ▼❊❚❍❖❉❙ ❋❖❘ s◆❙❡ ◆❖◆❙❚❆◆❉❆❘❉ ❆◆❆▲❨❙■❙ ❚❤❡ ❤②♣❡rr❡❛❧s ♦r ♥♦♥st❛♥❞❛r❞ r❡❛❧s ∗R ⊃R ✐s ❛ ✜❡❧❞ s✉❝❤ t❤❛t ∗R ❝♦♥t❛✐♥s ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ♥✉♠❜❡rs❀ ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t❡ ♥✉♠❜❡rs ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✿ ▲❡t ① ∈ ∗R✳ ❲❡ s❛② t❤❛t ✭✐✮ ① ✐s ✐♥✜♥✐t❡s✐♠❛❧ ✐❢ |①| < ε ❢♦r ❛❧❧ ε > ✵, ε ∈ R❀ ✭✐✐✮ ① ✐s ✜♥✐t❡ ✐❢ |①| < r ❢♦r s♦♠❡ r ∈ R❀ ✭✐✐✐✮ ① ✐s ✐♥✜♥✐t❡ ✐❢ |①| > r ❢♦r ❛❧❧ r ∈ R✳ ✭✐✈✮ ① ❛♥❞ ② ❛r❡ ✐♥✜♥✐t❡❧② ❝❧♦s❡✱ ❞❡♥♦t❡❞ ❜② ① ②✱ ✐❢ ① ② ✐s ✐♥✜♥✐t❡s✐♠❛❧✳ ✭❙♦ ① ✵ ♠❡❛♥s t❤❛t ① ✐s ✐♥✜♥✐t❡s✐♠❛❧✮ ❖♥❡ ✇❛② t♦ ❝♦♥str✉❝t ✐s ❛s ❛♥ ✉❧tr❛♣♦✇❡r ♦❢ t❤❡ r❡❛❧s ✇❤❡r❡ ✐s ❛ ♥♦♥♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r ✭♦r ♠❛①✐♠❛❧ ✜❧t❡r✮ ♦♥ ✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ✐s ❣✐✈❡♥ ❜② ✶

✶ ✷ ✶ ✸

✳

slide-33
SLIDE 33

▲❖❊❇ ❙P❆❈❊ ▼❊❚❍❖❉❙ ❋❖❘ s◆❙❡ ◆❖◆❙❚❆◆❉❆❘❉ ❆◆❆▲❨❙■❙ ❚❤❡ ❤②♣❡rr❡❛❧s ♦r ♥♦♥st❛♥❞❛r❞ r❡❛❧s ∗R ⊃R ✐s ❛ ✜❡❧❞ s✉❝❤ t❤❛t ∗R ❝♦♥t❛✐♥s ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ♥✉♠❜❡rs❀ ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t❡ ♥✉♠❜❡rs ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✿ ▲❡t ① ∈ ∗R✳ ❲❡ s❛② t❤❛t ✭✐✮ ① ✐s ✐♥✜♥✐t❡s✐♠❛❧ ✐❢ |①| < ε ❢♦r ❛❧❧ ε > ✵, ε ∈ R❀ ✭✐✐✮ ① ✐s ✜♥✐t❡ ✐❢ |①| < r ❢♦r s♦♠❡ r ∈ R❀ ✭✐✐✐✮ ① ✐s ✐♥✜♥✐t❡ ✐❢ |①| > r ❢♦r ❛❧❧ r ∈ R✳ ✭✐✈✮ ① ❛♥❞ ② ❛r❡ ✐♥✜♥✐t❡❧② ❝❧♦s❡✱ ❞❡♥♦t❡❞ ❜② ① ≈ ②✱ ✐❢ ① − ② ✐s ✐♥✜♥✐t❡s✐♠❛❧✳ ✭❙♦ ① ≈ ✵ ♠❡❛♥s t❤❛t ① ✐s ✐♥✜♥✐t❡s✐♠❛❧✮ ❖♥❡ ✇❛② t♦ ❝♦♥str✉❝t ✐s ❛s ❛♥ ✉❧tr❛♣♦✇❡r ♦❢ t❤❡ r❡❛❧s ✇❤❡r❡ ✐s ❛ ♥♦♥♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r ✭♦r ♠❛①✐♠❛❧ ✜❧t❡r✮ ♦♥ ✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ✐s ❣✐✈❡♥ ❜② ✶

✶ ✷ ✶ ✸

✳

slide-34
SLIDE 34

▲❖❊❇ ❙P❆❈❊ ▼❊❚❍❖❉❙ ❋❖❘ s◆❙❡ ◆❖◆❙❚❆◆❉❆❘❉ ❆◆❆▲❨❙■❙ ❚❤❡ ❤②♣❡rr❡❛❧s ♦r ♥♦♥st❛♥❞❛r❞ r❡❛❧s ∗R ⊃R ✐s ❛ ✜❡❧❞ s✉❝❤ t❤❛t ∗R ❝♦♥t❛✐♥s ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ♥✉♠❜❡rs❀ ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t❡ ♥✉♠❜❡rs ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✿ ▲❡t ① ∈ ∗R✳ ❲❡ s❛② t❤❛t ✭✐✮ ① ✐s ✐♥✜♥✐t❡s✐♠❛❧ ✐❢ |①| < ε ❢♦r ❛❧❧ ε > ✵, ε ∈ R❀ ✭✐✐✮ ① ✐s ✜♥✐t❡ ✐❢ |①| < r ❢♦r s♦♠❡ r ∈ R❀ ✭✐✐✐✮ ① ✐s ✐♥✜♥✐t❡ ✐❢ |①| > r ❢♦r ❛❧❧ r ∈ R✳ ✭✐✈✮ ① ❛♥❞ ② ❛r❡ ✐♥✜♥✐t❡❧② ❝❧♦s❡✱ ❞❡♥♦t❡❞ ❜② ① ≈ ②✱ ✐❢ ① − ② ✐s ✐♥✜♥✐t❡s✐♠❛❧✳ ✭❙♦ ① ≈ ✵ ♠❡❛♥s t❤❛t ① ✐s ✐♥✜♥✐t❡s✐♠❛❧✮ ❖♥❡ ✇❛② t♦ ❝♦♥str✉❝t ∗R ✐s ❛s ❛♥ ✉❧tr❛♣♦✇❡r ♦❢ t❤❡ r❡❛❧s

∗R = RNU

✇❤❡r❡ U ✐s ❛ ♥♦♥♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r ✭♦r ♠❛①✐♠❛❧ ✜❧t❡r✮ ♦♥ N✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ✐s ❣✐✈❡♥ ❜② ✶

✶ ✷ ✶ ✸

✳

slide-35
SLIDE 35

▲❖❊❇ ❙P❆❈❊ ▼❊❚❍❖❉❙ ❋❖❘ s◆❙❡ ◆❖◆❙❚❆◆❉❆❘❉ ❆◆❆▲❨❙■❙ ❚❤❡ ❤②♣❡rr❡❛❧s ♦r ♥♦♥st❛♥❞❛r❞ r❡❛❧s ∗R ⊃R ✐s ❛ ✜❡❧❞ s✉❝❤ t❤❛t ∗R ❝♦♥t❛✐♥s ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ♥✉♠❜❡rs❀ ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t❡ ♥✉♠❜❡rs ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✿ ▲❡t ① ∈ ∗R✳ ❲❡ s❛② t❤❛t ✭✐✮ ① ✐s ✐♥✜♥✐t❡s✐♠❛❧ ✐❢ |①| < ε ❢♦r ❛❧❧ ε > ✵, ε ∈ R❀ ✭✐✐✮ ① ✐s ✜♥✐t❡ ✐❢ |①| < r ❢♦r s♦♠❡ r ∈ R❀ ✭✐✐✐✮ ① ✐s ✐♥✜♥✐t❡ ✐❢ |①| > r ❢♦r ❛❧❧ r ∈ R✳ ✭✐✈✮ ① ❛♥❞ ② ❛r❡ ✐♥✜♥✐t❡❧② ❝❧♦s❡✱ ❞❡♥♦t❡❞ ❜② ① ≈ ②✱ ✐❢ ① − ② ✐s ✐♥✜♥✐t❡s✐♠❛❧✳ ✭❙♦ ① ≈ ✵ ♠❡❛♥s t❤❛t ① ✐s ✐♥✜♥✐t❡s✐♠❛❧✮ ❖♥❡ ✇❛② t♦ ❝♦♥str✉❝t ∗R ✐s ❛s ❛♥ ✉❧tr❛♣♦✇❡r ♦❢ t❤❡ r❡❛❧s

∗R = RNU

✇❤❡r❡ U ✐s ❛ ♥♦♥♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r ✭♦r ♠❛①✐♠❛❧ ✜❧t❡r✮ ♦♥ N✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲③❡r♦ ✐♥✜♥✐t❡s✐♠❛❧ ✐s ❣✐✈❡♥ ❜② (✶, ✶

✷, ✶ ✸, . . .)U✳

slide-36
SLIDE 36

❉❡✜♥❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ ∗R ♣♦✐♥t✇✐s❡ ✭t❤✐s ✐s s❛❢❡✮ ❛♥❞ ✐t ✐s t❤❡♥ ❡❛s② t♦ s❡❡ t❤❛t (∗R, +, ×, <) ✐s ❛♥ ♦r❞❡r❡❞ ✜❡❧❞✳ ❆ ❣♦♦❞ ✇❛② t♦ ♣✐❝t✉r❡ ✐s ❛s ❢♦❧❧♦✇s ✭♥♦t❡ t❤❛t s♦♠❡ ❢❡❛t✉r❡s ✐♥ t❤❡ ❞✐❛❣r❛♠ ❛r❡ ②❡t t♦ ❜❡ ❡①♣❧❛✐♥❡❞✮✳

slide-37
SLIDE 37

❉❡✜♥❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ ∗R ♣♦✐♥t✇✐s❡ ✭t❤✐s ✐s s❛❢❡✮ ❛♥❞ ✐t ✐s t❤❡♥ ❡❛s② t♦ s❡❡ t❤❛t (∗R, +, ×, <) ✐s ❛♥ ♦r❞❡r❡❞ ✜❡❧❞✳ ❆ ❣♦♦❞ ✇❛② t♦ ♣✐❝t✉r❡ ∗R ✐s ❛s ❢♦❧❧♦✇s ✭♥♦t❡ t❤❛t s♦♠❡ ❢❡❛t✉r❡s ✐♥ t❤❡ ❞✐❛❣r❛♠ ❛r❡ ②❡t t♦ ❜❡ ❡①♣❧❛✐♥❡❞✮✳

slide-38
SLIDE 38

∗R r ✵ ✶ ✷ ◆ ◆+✶ r

♠♦♥❛❞(r) = {① ∈ ∗R : ① ≈ r}

■♥✜♥✐t❡s✐♠❛❧ ♠✐❝r♦s❝♦♣❡

✄ ✂

  • ✁

✛ ✲

✐♥✜♥✐t❡ ❡❧❡♠❡♥ts

r ✵ ✶ ✷

R

st❛♥❞❛r❞ ♣❛rt ♠❛♣♣✐♥❣

✎ ❯ ❚❤❡ ❍②♣❡rr❡❛❧s

slide-39
SLIDE 39

◆♦✇ ❡①t❡♥❞ ❛❧❧ s❡ts ❆✱ ❢✉♥❝t✐♦♥s ❢ ❛♥❞ r❡❧❛t✐♦♥s ❘ ♦♥ R t♦ ∗R ♣♦✐♥t✇✐s❡ ✕ ✇✐t❤ t❤❡ ❡①t❡♥s✐♦♥s ❞❡♥♦t❡❞ ❜② ∗❆, ∗❢ ❛♥❞ ∗❘✳ ❊①❛♠♣❧❡s✿ ∗N, ∗Z ❛♥❞ ∗Q✱ t❤❡ s❡ts ♦❢ ❤②♣❡r♥❛t✉r❛❧ ♥✉♠❜❡rs✱ ❤②♣❡r✐♥t❡❣❡rs ❛♥❞ ❤②♣❡rr❛t✐♦♥❛❧s r❡s♣❡❝t✐✈❡❧② ❲❡ ❝❛♥ t❛❧❦ ❛❜♦✉t ❛♥ ✐♥✜♥✐t❡ ✭❤②♣❡r✮♥❛t✉r❛❧ ♥✉♠❜❡r ◆✳

slide-40
SLIDE 40

Pr♦♣❡rt✐❡s ♦❢ ∗R ❛r❡ ❣✐✈❡♥ s②st❡♠❛t✐❝❛❧❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✭❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

▲❡t ϕ ❜❡ ❛♥② ✜rst ♦r❞❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ϕ ❤♦❧❞s ✐♥ R ⇐ ⇒

∗ϕ

❤♦❧❞s ✐♥

∗R

❆ ✜rst ♦r❞❡r st❛t❡♠❡♥t ✭r❡s♣❡❝t✐✈❡❧② ✮✿ r❡❢❡rs t♦ ❡❧❡♠❡♥ts ♦❢ ✭r❡s♣❡❝t✐✈❡❧② ✮✱ ❜♦t❤ ✜①❡❞ ❛♥❞ ✈❛r✐❛❜❧❡✱ ❛♥❞ t♦ ✜①❡❞ r❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s ❢ ❘ ✭r❡s♣❡❝t✐✈❡❧② ❢ ❘✮✱ ✇✐t❤ q✉❛♥t✐✜❝❛t✐♦♥ ✭ ①✱ ②✮ ♦♥❧② ❢♦r ❡❧❡♠❡♥ts✳ ❚♦ ❣❡t ❜❛❝❦ t♦ ❢r♦♠ ✿

❚❤❡♦r❡♠ ✭❙t❛♥❞❛r❞ P❛rt ❚❤❡♦r❡♠✮

■❢ ① ✐s ✜♥✐t❡✱ t❤❡♥ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ r s✉❝❤ t❤❛t ① r❀ ✐✳❡✳ ❛♥② ✜♥✐t❡ ❤②♣❡rr❡❛❧ ① ✐s ✉♥✐q✉❡❧② ❡①♣r❡ss✐❜❧❡ ❛s ① r ✇✐t❤ r ❛ st❛♥❞❛r❞ r❡❛❧ ❛♥❞ ✐♥✜♥✐t❡s✐♠❛❧✳

❉❡✜♥✐t✐♦♥ ✭❙t❛♥❞❛r❞ P❛rt✮

■❢ ① ✐s ❛ ✜♥✐t❡ ❤②♣❡rr❡❛❧ t❤❡ ✉♥✐q✉❡ r❡❛❧ r ① ✐s ❝❛❧❧❡❞ t❤❡ st❛♥❞❛r❞ ♣❛rt ♦❢ ①✱ ✇r✐tt❡♥ r ① st ①

slide-41
SLIDE 41

Pr♦♣❡rt✐❡s ♦❢ ∗R ❛r❡ ❣✐✈❡♥ s②st❡♠❛t✐❝❛❧❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✭❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

▲❡t ϕ ❜❡ ❛♥② ✜rst ♦r❞❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ϕ ❤♦❧❞s ✐♥ R ⇐ ⇒

∗ϕ

❤♦❧❞s ✐♥

∗R

❆ ✜rst ♦r❞❡r st❛t❡♠❡♥t ϕ ✭r❡s♣❡❝t✐✈❡❧② ∗ϕ✮✿ r❡❢❡rs t♦ ❡❧❡♠❡♥ts ♦❢ R ✭r❡s♣❡❝t✐✈❡❧② ∗R✮✱ ❜♦t❤ ✜①❡❞ ❛♥❞ ✈❛r✐❛❜❧❡✱ ❛♥❞ t♦ ✜①❡❞ r❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s ❢ , ❘ ✭r❡s♣❡❝t✐✈❡❧② ∗❢ , ∗❘✮✱ ✇✐t❤ q✉❛♥t✐✜❝❛t✐♦♥ ✭∀①✱ ∃②✮ ♦♥❧② ❢♦r ❡❧❡♠❡♥ts✳ ❚♦ ❣❡t ❜❛❝❦ t♦ ❢r♦♠ ✿

❚❤❡♦r❡♠ ✭❙t❛♥❞❛r❞ P❛rt ❚❤❡♦r❡♠✮

■❢ ① ✐s ✜♥✐t❡✱ t❤❡♥ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ r s✉❝❤ t❤❛t ① r❀ ✐✳❡✳ ❛♥② ✜♥✐t❡ ❤②♣❡rr❡❛❧ ① ✐s ✉♥✐q✉❡❧② ❡①♣r❡ss✐❜❧❡ ❛s ① r ✇✐t❤ r ❛ st❛♥❞❛r❞ r❡❛❧ ❛♥❞ ✐♥✜♥✐t❡s✐♠❛❧✳

❉❡✜♥✐t✐♦♥ ✭❙t❛♥❞❛r❞ P❛rt✮

■❢ ① ✐s ❛ ✜♥✐t❡ ❤②♣❡rr❡❛❧ t❤❡ ✉♥✐q✉❡ r❡❛❧ r ① ✐s ❝❛❧❧❡❞ t❤❡ st❛♥❞❛r❞ ♣❛rt ♦❢ ①✱ ✇r✐tt❡♥ r ① st ①

slide-42
SLIDE 42

Pr♦♣❡rt✐❡s ♦❢ ∗R ❛r❡ ❣✐✈❡♥ s②st❡♠❛t✐❝❛❧❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✭❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

▲❡t ϕ ❜❡ ❛♥② ✜rst ♦r❞❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ϕ ❤♦❧❞s ✐♥ R ⇐ ⇒

∗ϕ

❤♦❧❞s ✐♥

∗R

❆ ✜rst ♦r❞❡r st❛t❡♠❡♥t ϕ ✭r❡s♣❡❝t✐✈❡❧② ∗ϕ✮✿ r❡❢❡rs t♦ ❡❧❡♠❡♥ts ♦❢ R ✭r❡s♣❡❝t✐✈❡❧② ∗R✮✱ ❜♦t❤ ✜①❡❞ ❛♥❞ ✈❛r✐❛❜❧❡✱ ❛♥❞ t♦ ✜①❡❞ r❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s ❢ , ❘ ✭r❡s♣❡❝t✐✈❡❧② ∗❢ , ∗❘✮✱ ✇✐t❤ q✉❛♥t✐✜❝❛t✐♦♥ ✭∀①✱ ∃②✮ ♦♥❧② ❢♦r ❡❧❡♠❡♥ts✳ ❚♦ ❣❡t ❜❛❝❦ t♦ R ❢r♦♠ ∗R✿

❚❤❡♦r❡♠ ✭❙t❛♥❞❛r❞ P❛rt ❚❤❡♦r❡♠✮

■❢ ① ∈ ∗R ✐s ✜♥✐t❡✱ t❤❡♥ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ r ∈ R s✉❝❤ t❤❛t ① ≈ r❀ ✐✳❡✳ ❛♥② ✜♥✐t❡ ❤②♣❡rr❡❛❧ ① ✐s ✉♥✐q✉❡❧② ❡①♣r❡ss✐❜❧❡ ❛s ① = r + δ ✇✐t❤ r ❛ st❛♥❞❛r❞ r❡❛❧ ❛♥❞ δ ✐♥✜♥✐t❡s✐♠❛❧✳

❉❡✜♥✐t✐♦♥ ✭❙t❛♥❞❛r❞ P❛rt✮

■❢ ① ✐s ❛ ✜♥✐t❡ ❤②♣❡rr❡❛❧ t❤❡ ✉♥✐q✉❡ r❡❛❧ r ① ✐s ❝❛❧❧❡❞ t❤❡ st❛♥❞❛r❞ ♣❛rt ♦❢ ①✱ ✇r✐tt❡♥ r ① st ①

slide-43
SLIDE 43

Pr♦♣❡rt✐❡s ♦❢ ∗R ❛r❡ ❣✐✈❡♥ s②st❡♠❛t✐❝❛❧❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✭❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

▲❡t ϕ ❜❡ ❛♥② ✜rst ♦r❞❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ϕ ❤♦❧❞s ✐♥ R ⇐ ⇒

∗ϕ

❤♦❧❞s ✐♥

∗R

❆ ✜rst ♦r❞❡r st❛t❡♠❡♥t ϕ ✭r❡s♣❡❝t✐✈❡❧② ∗ϕ✮✿ r❡❢❡rs t♦ ❡❧❡♠❡♥ts ♦❢ R ✭r❡s♣❡❝t✐✈❡❧② ∗R✮✱ ❜♦t❤ ✜①❡❞ ❛♥❞ ✈❛r✐❛❜❧❡✱ ❛♥❞ t♦ ✜①❡❞ r❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s ❢ , ❘ ✭r❡s♣❡❝t✐✈❡❧② ∗❢ , ∗❘✮✱ ✇✐t❤ q✉❛♥t✐✜❝❛t✐♦♥ ✭∀①✱ ∃②✮ ♦♥❧② ❢♦r ❡❧❡♠❡♥ts✳ ❚♦ ❣❡t ❜❛❝❦ t♦ R ❢r♦♠ ∗R✿

❚❤❡♦r❡♠ ✭❙t❛♥❞❛r❞ P❛rt ❚❤❡♦r❡♠✮

■❢ ① ∈ ∗R ✐s ✜♥✐t❡✱ t❤❡♥ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ r ∈ R s✉❝❤ t❤❛t ① ≈ r❀ ✐✳❡✳ ❛♥② ✜♥✐t❡ ❤②♣❡rr❡❛❧ ① ✐s ✉♥✐q✉❡❧② ❡①♣r❡ss✐❜❧❡ ❛s ① = r + δ ✇✐t❤ r ❛ st❛♥❞❛r❞ r❡❛❧ ❛♥❞ δ ✐♥✜♥✐t❡s✐♠❛❧✳

❉❡✜♥✐t✐♦♥ ✭❙t❛♥❞❛r❞ P❛rt✮

■❢ ① ✐s ❛ ✜♥✐t❡ ❤②♣❡rr❡❛❧ t❤❡ ✉♥✐q✉❡ r❡❛❧ r ≈ ① ✐s ❝❛❧❧❡❞ t❤❡ st❛♥❞❛r❞ ♣❛rt ♦❢ ①✱ ✇r✐tt❡♥ r = ◦① = st(①).

slide-44
SLIDE 44

❆ ◆❖◆❙❚❆◆❉❆❘❉ ❯◆■❱❊❘❙❊ ❘❡♣❡❛t t❤❡ ❛❜♦✈❡ ❝♦♥str✉❝t✐♦♥ t♦ ❣✐✈❡ ∗❆ ❢♦r ❛♥② ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t ♦r str✉❝t✉r❡ ❆❀ ❡✳❣✳ ∗▼ ❢♦r ❛ ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ ∗❞ :∗▼ × ∗▼ →∗R. ❚❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ ✇❛② t♦ ❞♦ t❤✐s ✐s ❢♦r ❛ ✇❤♦❧❡ ♠❛t❤❡♠❛t✐❝❛❧ ✉♥✐✈❡rs❡ ✇✐t❤ ❆ ❢♦r ❡❛❝❤ ♦❜❥❡❝t ❆ t❤❛t ♠✐❣❤t ❜❡ ♥❡❡❞❡❞✳ ■♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r❡s✉❧t✐♥❣ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ✐s ❣✐✈❡♥ ❜②✿

❚❤❡♦r❡♠ ✭❚❤❡ ❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

❙✉♣♣♦s❡ t❤❛t ✐s ❛ ❜♦✉♥❞❡❞ q✉❛♥t✐✜❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ❤♦❧❞s ✐♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❤♦❧❞s ✐♥ ✳ ❊❧❡♠❡♥ts ✭♦❜❥❡❝ts✮ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ❛r❡ ❝❛❧❧❡❞ ✐♥t❡r♥❛❧✳ ❘❡♠❛r❦ ❚❤❡ st❛♥❞❛r❞ ♣❛rt ♠❛♣♣✐♥❣ ❡①t❡♥❞s t♦ t❤❡ ✏♥❡❛rst❛♥❞❛r❞✑ ❡❧❡♠❡♥ts ♦❢ ❛♥② ❡①t❡♥s✐♦♥ ♠❡tr✐❝ ✭♦r t♦♣♦❧♦❣✐❝❛❧✮ s♣❛❝❡ ▼ ✲ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ s♣❛❝❡ ❍✳ ■t ✐s ❡❛s② t♦ s❤♦✇ t❤❛t ❡❧❡♠❡♥ts ❯ ✐♥ ❍ ✇✐t❤ ❯ ✜♥✐t❡ ❛r❡ ♥❡❛rst❛♥❞❛r❞ ✐♥ t❤❡ ✇❡❛❦ t♦♣♦❧♦❣②✳

slide-45
SLIDE 45

❆ ◆❖◆❙❚❆◆❉❆❘❉ ❯◆■❱❊❘❙❊ ❘❡♣❡❛t t❤❡ ❛❜♦✈❡ ❝♦♥str✉❝t✐♦♥ t♦ ❣✐✈❡ ∗❆ ❢♦r ❛♥② ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t ♦r str✉❝t✉r❡ ❆❀ ❡✳❣✳ ∗▼ ❢♦r ❛ ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ ∗❞ :∗▼ × ∗▼ →∗R. ❚❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ ✇❛② t♦ ❞♦ t❤✐s ✐s ❢♦r ❛ ✇❤♦❧❡ ♠❛t❤❡♠❛t✐❝❛❧ ✉♥✐✈❡rs❡ V ✇✐t❤ ❆ ∈ V ❢♦r ❡❛❝❤ ♦❜❥❡❝t ❆ t❤❛t ♠✐❣❤t ❜❡ ♥❡❡❞❡❞✳ ■♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r❡s✉❧t✐♥❣ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ✐s ❣✐✈❡♥ ❜②✿

❚❤❡♦r❡♠ ✭❚❤❡ ❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

❙✉♣♣♦s❡ t❤❛t ✐s ❛ ❜♦✉♥❞❡❞ q✉❛♥t✐✜❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ❤♦❧❞s ✐♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❤♦❧❞s ✐♥ ✳ ❊❧❡♠❡♥ts ✭♦❜❥❡❝ts✮ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ❛r❡ ❝❛❧❧❡❞ ✐♥t❡r♥❛❧✳ ❘❡♠❛r❦ ❚❤❡ st❛♥❞❛r❞ ♣❛rt ♠❛♣♣✐♥❣ ❡①t❡♥❞s t♦ t❤❡ ✏♥❡❛rst❛♥❞❛r❞✑ ❡❧❡♠❡♥ts ♦❢ ❛♥② ❡①t❡♥s✐♦♥ ♠❡tr✐❝ ✭♦r t♦♣♦❧♦❣✐❝❛❧✮ s♣❛❝❡ ▼ ✲ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ s♣❛❝❡ ❍✳ ■t ✐s ❡❛s② t♦ s❤♦✇ t❤❛t ❡❧❡♠❡♥ts ❯ ✐♥ ❍ ✇✐t❤ ❯ ✜♥✐t❡ ❛r❡ ♥❡❛rst❛♥❞❛r❞ ✐♥ t❤❡ ✇❡❛❦ t♦♣♦❧♦❣②✳

slide-46
SLIDE 46

❆ ◆❖◆❙❚❆◆❉❆❘❉ ❯◆■❱❊❘❙❊ ❘❡♣❡❛t t❤❡ ❛❜♦✈❡ ❝♦♥str✉❝t✐♦♥ t♦ ❣✐✈❡ ∗❆ ❢♦r ❛♥② ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t ♦r str✉❝t✉r❡ ❆❀ ❡✳❣✳ ∗▼ ❢♦r ❛ ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ ∗❞ :∗▼ × ∗▼ →∗R. ❚❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ ✇❛② t♦ ❞♦ t❤✐s ✐s ❢♦r ❛ ✇❤♦❧❡ ♠❛t❤❡♠❛t✐❝❛❧ ✉♥✐✈❡rs❡ V ✇✐t❤ ❆ ∈ V ❢♦r ❡❛❝❤ ♦❜❥❡❝t ❆ t❤❛t ♠✐❣❤t ❜❡ ♥❡❡❞❡❞✳ ■♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r❡s✉❧t✐♥❣ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ∗V ✐s ❣✐✈❡♥ ❜②✿

❚❤❡♦r❡♠ ✭❚❤❡ ❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

❙✉♣♣♦s❡ t❤❛t ϕ ✐s ❛ ❜♦✉♥❞❡❞ q✉❛♥t✐✜❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ϕ ❤♦❧❞s ✐♥ V ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∗ϕ ❤♦❧❞s ✐♥ ∗V✳ ❊❧❡♠❡♥ts ✭♦❜❥❡❝ts✮ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ❛r❡ ❝❛❧❧❡❞ ✐♥t❡r♥❛❧✳ ❘❡♠❛r❦ ❚❤❡ st❛♥❞❛r❞ ♣❛rt ♠❛♣♣✐♥❣ ❡①t❡♥❞s t♦ t❤❡ ✏♥❡❛rst❛♥❞❛r❞✑ ❡❧❡♠❡♥ts ♦❢ ❛♥② ❡①t❡♥s✐♦♥ ♠❡tr✐❝ ✭♦r t♦♣♦❧♦❣✐❝❛❧✮ s♣❛❝❡ ▼ ✲ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ s♣❛❝❡ ❍✳ ■t ✐s ❡❛s② t♦ s❤♦✇ t❤❛t ❡❧❡♠❡♥ts ❯ ✐♥ ❍ ✇✐t❤ ❯ ✜♥✐t❡ ❛r❡ ♥❡❛rst❛♥❞❛r❞ ✐♥ t❤❡ ✇❡❛❦ t♦♣♦❧♦❣②✳

slide-47
SLIDE 47

❆ ◆❖◆❙❚❆◆❉❆❘❉ ❯◆■❱❊❘❙❊ ❘❡♣❡❛t t❤❡ ❛❜♦✈❡ ❝♦♥str✉❝t✐♦♥ t♦ ❣✐✈❡ ∗❆ ❢♦r ❛♥② ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t ♦r str✉❝t✉r❡ ❆❀ ❡✳❣✳ ∗▼ ❢♦r ❛ ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ ∗❞ :∗▼ × ∗▼ →∗R. ❚❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ ✇❛② t♦ ❞♦ t❤✐s ✐s ❢♦r ❛ ✇❤♦❧❡ ♠❛t❤❡♠❛t✐❝❛❧ ✉♥✐✈❡rs❡ V ✇✐t❤ ❆ ∈ V ❢♦r ❡❛❝❤ ♦❜❥❡❝t ❆ t❤❛t ♠✐❣❤t ❜❡ ♥❡❡❞❡❞✳ ■♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r❡s✉❧t✐♥❣ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ∗V ✐s ❣✐✈❡♥ ❜②✿

❚❤❡♦r❡♠ ✭❚❤❡ ❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

❙✉♣♣♦s❡ t❤❛t ϕ ✐s ❛ ❜♦✉♥❞❡❞ q✉❛♥t✐✜❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ϕ ❤♦❧❞s ✐♥ V ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∗ϕ ❤♦❧❞s ✐♥ ∗V✳ ❊❧❡♠❡♥ts ✭♦❜❥❡❝ts✮ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ∗V ❛r❡ ❝❛❧❧❡❞ ✐♥t❡r♥❛❧✳ ❘❡♠❛r❦ ❚❤❡ st❛♥❞❛r❞ ♣❛rt ♠❛♣♣✐♥❣ ❡①t❡♥❞s t♦ t❤❡ ✏♥❡❛rst❛♥❞❛r❞✑ ❡❧❡♠❡♥ts ♦❢ ❛♥② ❡①t❡♥s✐♦♥ ♠❡tr✐❝ ✭♦r t♦♣♦❧♦❣✐❝❛❧✮ s♣❛❝❡ ▼ ✲ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ s♣❛❝❡ ❍✳ ■t ✐s ❡❛s② t♦ s❤♦✇ t❤❛t ❡❧❡♠❡♥ts ❯ ✐♥ ❍ ✇✐t❤ ❯ ✜♥✐t❡ ❛r❡ ♥❡❛rst❛♥❞❛r❞ ✐♥ t❤❡ ✇❡❛❦ t♦♣♦❧♦❣②✳

slide-48
SLIDE 48

❆ ◆❖◆❙❚❆◆❉❆❘❉ ❯◆■❱❊❘❙❊ ❘❡♣❡❛t t❤❡ ❛❜♦✈❡ ❝♦♥str✉❝t✐♦♥ t♦ ❣✐✈❡ ∗❆ ❢♦r ❛♥② ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t ♦r str✉❝t✉r❡ ❆❀ ❡✳❣✳ ∗▼ ❢♦r ❛ ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ ∗❞ :∗▼ × ∗▼ →∗R. ❚❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ ✇❛② t♦ ❞♦ t❤✐s ✐s ❢♦r ❛ ✇❤♦❧❡ ♠❛t❤❡♠❛t✐❝❛❧ ✉♥✐✈❡rs❡ V ✇✐t❤ ❆ ∈ V ❢♦r ❡❛❝❤ ♦❜❥❡❝t ❆ t❤❛t ♠✐❣❤t ❜❡ ♥❡❡❞❡❞✳ ■♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r❡s✉❧t✐♥❣ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ∗V ✐s ❣✐✈❡♥ ❜②✿

❚❤❡♦r❡♠ ✭❚❤❡ ❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

❙✉♣♣♦s❡ t❤❛t ϕ ✐s ❛ ❜♦✉♥❞❡❞ q✉❛♥t✐✜❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ϕ ❤♦❧❞s ✐♥ V ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∗ϕ ❤♦❧❞s ✐♥ ∗V✳ ❊❧❡♠❡♥ts ✭♦❜❥❡❝ts✮ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ∗V ❛r❡ ❝❛❧❧❡❞ ✐♥t❡r♥❛❧✳ ❘❡♠❛r❦ ❚❤❡ st❛♥❞❛r❞ ♣❛rt ♠❛♣♣✐♥❣ ❡①t❡♥❞s t♦ t❤❡ ✏♥❡❛rst❛♥❞❛r❞✑ ❡❧❡♠❡♥ts ♦❢ ❛♥② ❡①t❡♥s✐♦♥ ♠❡tr✐❝ ✭♦r t♦♣♦❧♦❣✐❝❛❧✮ s♣❛❝❡ ∗▼ ✲ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ s♣❛❝❡ ∗❍✳ ■t ✐s ❡❛s② t♦ s❤♦✇ t❤❛t ❡❧❡♠❡♥ts ❯ ✐♥ ❍ ✇✐t❤ ❯ ✜♥✐t❡ ❛r❡ ♥❡❛rst❛♥❞❛r❞ ✐♥ t❤❡ ✇❡❛❦ t♦♣♦❧♦❣②✳

slide-49
SLIDE 49

❆ ◆❖◆❙❚❆◆❉❆❘❉ ❯◆■❱❊❘❙❊ ❘❡♣❡❛t t❤❡ ❛❜♦✈❡ ❝♦♥str✉❝t✐♦♥ t♦ ❣✐✈❡ ∗❆ ❢♦r ❛♥② ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t ♦r str✉❝t✉r❡ ❆❀ ❡✳❣✳ ∗▼ ❢♦r ❛ ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ ∗❞ :∗▼ × ∗▼ →∗R. ❚❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ ✇❛② t♦ ❞♦ t❤✐s ✐s ❢♦r ❛ ✇❤♦❧❡ ♠❛t❤❡♠❛t✐❝❛❧ ✉♥✐✈❡rs❡ V ✇✐t❤ ❆ ∈ V ❢♦r ❡❛❝❤ ♦❜❥❡❝t ❆ t❤❛t ♠✐❣❤t ❜❡ ♥❡❡❞❡❞✳ ■♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r❡s✉❧t✐♥❣ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ∗V ✐s ❣✐✈❡♥ ❜②✿

❚❤❡♦r❡♠ ✭❚❤❡ ❚r❛♥s❢❡r Pr✐♥❝✐♣❧❡✮

❙✉♣♣♦s❡ t❤❛t ϕ ✐s ❛ ❜♦✉♥❞❡❞ q✉❛♥t✐✜❡r st❛t❡♠❡♥t✳ ❚❤❡♥ ϕ ❤♦❧❞s ✐♥ V ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∗ϕ ❤♦❧❞s ✐♥ ∗V✳ ❊❧❡♠❡♥ts ✭♦❜❥❡❝ts✮ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ♥♦♥st❛♥❞❛r❞ ✉♥✐✈❡rs❡ ∗V ❛r❡ ❝❛❧❧❡❞ ✐♥t❡r♥❛❧✳ ❘❡♠❛r❦ ❚❤❡ st❛♥❞❛r❞ ♣❛rt ♠❛♣♣✐♥❣ ❡①t❡♥❞s t♦ t❤❡ ✏♥❡❛rst❛♥❞❛r❞✑ ❡❧❡♠❡♥ts ♦❢ ❛♥② ❡①t❡♥s✐♦♥ ♠❡tr✐❝ ✭♦r t♦♣♦❧♦❣✐❝❛❧✮ s♣❛❝❡ ∗▼ ✲ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ s♣❛❝❡ ∗❍✳ ■t ✐s ❡❛s② t♦ s❤♦✇ t❤❛t ❡❧❡♠❡♥ts ❯ ✐♥ ∗❍ ✇✐t❤ |❯| ✜♥✐t❡ ❛r❡ ♥❡❛rst❛♥❞❛r❞ ✐♥ t❤❡ ✇❡❛❦ t♦♣♦❧♦❣②✳

slide-50
SLIDE 50

▲❖❊❇ ▼❊❆❙❯❘❊❙ ❆ ▲♦❡❜ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❛ ♠❡❛s✉r❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ ♥♦♥st❛♥❞❛r❞ ✭✐✳❡✳ ✐♥t❡r♥❛❧✮ ♠❡❛s✉r❡ ✭❡ss❡♥t✐❛❧❧② ✐t ✐s ❛♥ ✉❧tr❛♣r♦❞✉❝t ♦❢ ♠❡❛s✉r❡s✮✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ✐♥t❡r♥❛❧ s❡t Ω ❛♥❞ ❛♥ ✐♥t❡r♥❛❧ ❛❧❣❡❜r❛ A ♦❢ s✉❜s❡ts ♦❢ Ω✱ ❛r❡ ❣✐✈❡♥✱ µ ✐s ❛ ✜♥✐t❡ ✐♥t❡r♥❛❧ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ ♦♥ A❀ t❤❛t ✐s ✵ ✇✐t❤ ❆ ❇ ❆ ❇ ❢♦r ❞✐s❥♦✐♥t ❆✱ ❇ ✱ ❛♥❞ ✐s ✜♥✐t❡✳ ❉❡✜♥❡ t❤❡ ♠❛♣♣✐♥❣ ✵ ❜② ❆ ❆ ✳ ❚❤❡♥ ✐s ❛ st❛♥❞❛r❞ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❜✉t ✐s ♥♦t ✲❛❞❞✐t✐✈❡ ✐♥ ❣❡♥❡r❛❧✳

❚❤❡♦r❡♠ ✭▲♦❡❜ ✶✾✼✺✮

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✲❛❞❞✐t✐✈❡ ❡①t❡♥s✐♦♥ ♦❢ t♦ t❤❡ ✲❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤✐s ♠❡❛s✉r❡ ✐s t❤❡ ▲♦❡❜ ♠❡❛s✉r❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✱ ❞❡♥♦t❡❞

▲ ❛♥❞ t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢

✐s t❤❡ ▲♦❡❜ ✲❛❧❣❡❜r❛✱ ❞❡♥♦t❡❞ ❜② ▲ ✳

slide-51
SLIDE 51

▲❖❊❇ ▼❊❆❙❯❘❊❙ ❆ ▲♦❡❜ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❛ ♠❡❛s✉r❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ ♥♦♥st❛♥❞❛r❞ ✭✐✳❡✳ ✐♥t❡r♥❛❧✮ ♠❡❛s✉r❡ ✭❡ss❡♥t✐❛❧❧② ✐t ✐s ❛♥ ✉❧tr❛♣r♦❞✉❝t ♦❢ ♠❡❛s✉r❡s✮✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ✐♥t❡r♥❛❧ s❡t Ω ❛♥❞ ❛♥ ✐♥t❡r♥❛❧ ❛❧❣❡❜r❛ A ♦❢ s✉❜s❡ts ♦❢ Ω✱ ❛r❡ ❣✐✈❡♥✱ µ ✐s ❛ ✜♥✐t❡ ✐♥t❡r♥❛❧ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ ♦♥ A❀ t❤❛t ✐s µ : A → ∗[✵, ∞) ✇✐t❤ µ(❆ ∪ ❇) = µ(❆) + µ(❇) ❢♦r ❞✐s❥♦✐♥t ❆✱ ❇ ∈ A✱ ❛♥❞ µ(Ω) ✐s ✜♥✐t❡✳ ❉❡✜♥❡ t❤❡ ♠❛♣♣✐♥❣ ✵ ❜② ❆ ❆ ✳ ❚❤❡♥ ✐s ❛ st❛♥❞❛r❞ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❜✉t ✐s ♥♦t ✲❛❞❞✐t✐✈❡ ✐♥ ❣❡♥❡r❛❧✳

❚❤❡♦r❡♠ ✭▲♦❡❜ ✶✾✼✺✮

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✲❛❞❞✐t✐✈❡ ❡①t❡♥s✐♦♥ ♦❢ t♦ t❤❡ ✲❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤✐s ♠❡❛s✉r❡ ✐s t❤❡ ▲♦❡❜ ♠❡❛s✉r❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✱ ❞❡♥♦t❡❞

▲ ❛♥❞ t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢

✐s t❤❡ ▲♦❡❜ ✲❛❧❣❡❜r❛✱ ❞❡♥♦t❡❞ ❜② ▲ ✳

slide-52
SLIDE 52

▲❖❊❇ ▼❊❆❙❯❘❊❙ ❆ ▲♦❡❜ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❛ ♠❡❛s✉r❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ ♥♦♥st❛♥❞❛r❞ ✭✐✳❡✳ ✐♥t❡r♥❛❧✮ ♠❡❛s✉r❡ ✭❡ss❡♥t✐❛❧❧② ✐t ✐s ❛♥ ✉❧tr❛♣r♦❞✉❝t ♦❢ ♠❡❛s✉r❡s✮✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ✐♥t❡r♥❛❧ s❡t Ω ❛♥❞ ❛♥ ✐♥t❡r♥❛❧ ❛❧❣❡❜r❛ A ♦❢ s✉❜s❡ts ♦❢ Ω✱ ❛r❡ ❣✐✈❡♥✱ µ ✐s ❛ ✜♥✐t❡ ✐♥t❡r♥❛❧ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ ♦♥ A❀ t❤❛t ✐s µ : A → ∗[✵, ∞) ✇✐t❤ µ(❆ ∪ ❇) = µ(❆) + µ(❇) ❢♦r ❞✐s❥♦✐♥t ❆✱ ❇ ∈ A✱ ❛♥❞ µ(Ω) ✐s ✜♥✐t❡✳ ❉❡✜♥❡ t❤❡ ♠❛♣♣✐♥❣

  • µ : A → [✵, ∞)

❜② ◦µ(❆) =◦(µ(❆))✳ ❚❤❡♥ ✐s ❛ st❛♥❞❛r❞ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❜✉t ✐s ♥♦t ✲❛❞❞✐t✐✈❡ ✐♥ ❣❡♥❡r❛❧✳

❚❤❡♦r❡♠ ✭▲♦❡❜ ✶✾✼✺✮

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✲❛❞❞✐t✐✈❡ ❡①t❡♥s✐♦♥ ♦❢ t♦ t❤❡ ✲❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤✐s ♠❡❛s✉r❡ ✐s t❤❡ ▲♦❡❜ ♠❡❛s✉r❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✱ ❞❡♥♦t❡❞

▲ ❛♥❞ t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢

✐s t❤❡ ▲♦❡❜ ✲❛❧❣❡❜r❛✱ ❞❡♥♦t❡❞ ❜② ▲ ✳

slide-53
SLIDE 53

▲❖❊❇ ▼❊❆❙❯❘❊❙ ❆ ▲♦❡❜ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❛ ♠❡❛s✉r❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ ♥♦♥st❛♥❞❛r❞ ✭✐✳❡✳ ✐♥t❡r♥❛❧✮ ♠❡❛s✉r❡ ✭❡ss❡♥t✐❛❧❧② ✐t ✐s ❛♥ ✉❧tr❛♣r♦❞✉❝t ♦❢ ♠❡❛s✉r❡s✮✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ✐♥t❡r♥❛❧ s❡t Ω ❛♥❞ ❛♥ ✐♥t❡r♥❛❧ ❛❧❣❡❜r❛ A ♦❢ s✉❜s❡ts ♦❢ Ω✱ ❛r❡ ❣✐✈❡♥✱ µ ✐s ❛ ✜♥✐t❡ ✐♥t❡r♥❛❧ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ ♦♥ A❀ t❤❛t ✐s µ : A → ∗[✵, ∞) ✇✐t❤ µ(❆ ∪ ❇) = µ(❆) + µ(❇) ❢♦r ❞✐s❥♦✐♥t ❆✱ ❇ ∈ A✱ ❛♥❞ µ(Ω) ✐s ✜♥✐t❡✳ ❉❡✜♥❡ t❤❡ ♠❛♣♣✐♥❣

  • µ : A → [✵, ∞)

❜② ◦µ(❆) =◦(µ(❆))✳ ❚❤❡♥ (Ω, A,◦µ) ✐s ❛ st❛♥❞❛r❞ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❜✉t A ✐s ♥♦t σ✲❛❞❞✐t✐✈❡ ✐♥ ❣❡♥❡r❛❧✳

❚❤❡♦r❡♠ ✭▲♦❡❜ ✶✾✼✺✮

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✲❛❞❞✐t✐✈❡ ❡①t❡♥s✐♦♥ ♦❢ t♦ t❤❡ ✲❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤✐s ♠❡❛s✉r❡ ✐s t❤❡ ▲♦❡❜ ♠❡❛s✉r❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✱ ❞❡♥♦t❡❞

▲ ❛♥❞ t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢

✐s t❤❡ ▲♦❡❜ ✲❛❧❣❡❜r❛✱ ❞❡♥♦t❡❞ ❜② ▲ ✳

slide-54
SLIDE 54

▲❖❊❇ ▼❊❆❙❯❘❊❙ ❆ ▲♦❡❜ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❛ ♠❡❛s✉r❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ ♥♦♥st❛♥❞❛r❞ ✭✐✳❡✳ ✐♥t❡r♥❛❧✮ ♠❡❛s✉r❡ ✭❡ss❡♥t✐❛❧❧② ✐t ✐s ❛♥ ✉❧tr❛♣r♦❞✉❝t ♦❢ ♠❡❛s✉r❡s✮✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ✐♥t❡r♥❛❧ s❡t Ω ❛♥❞ ❛♥ ✐♥t❡r♥❛❧ ❛❧❣❡❜r❛ A ♦❢ s✉❜s❡ts ♦❢ Ω✱ ❛r❡ ❣✐✈❡♥✱ µ ✐s ❛ ✜♥✐t❡ ✐♥t❡r♥❛❧ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ ♦♥ A❀ t❤❛t ✐s µ : A → ∗[✵, ∞) ✇✐t❤ µ(❆ ∪ ❇) = µ(❆) + µ(❇) ❢♦r ❞✐s❥♦✐♥t ❆✱ ❇ ∈ A✱ ❛♥❞ µ(Ω) ✐s ✜♥✐t❡✳ ❉❡✜♥❡ t❤❡ ♠❛♣♣✐♥❣

  • µ : A → [✵, ∞)

❜② ◦µ(❆) =◦(µ(❆))✳ ❚❤❡♥ (Ω, A,◦µ) ✐s ❛ st❛♥❞❛r❞ ✜♥✐t❡❧② ❛❞❞✐t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❜✉t A ✐s ♥♦t σ✲❛❞❞✐t✐✈❡ ✐♥ ❣❡♥❡r❛❧✳

❚❤❡♦r❡♠ ✭▲♦❡❜ ✶✾✼✺✮

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ σ✲❛❞❞✐t✐✈❡ ❡①t❡♥s✐♦♥ ♦❢ ◦µ t♦ t❤❡ σ✲❛❧❣❡❜r❛ σ(A) ❣❡♥❡r❛t❡❞ ❜② A✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤✐s ♠❡❛s✉r❡ ✐s t❤❡ ▲♦❡❜ ♠❡❛s✉r❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ µ✱ ❞❡♥♦t❡❞ µ▲ ❛♥❞ t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ σ(A) ✐s t❤❡ ▲♦❡❜ σ ✲❛❧❣❡❜r❛✱ ❞❡♥♦t❡❞ ❜② ▲(A)✳

slide-55
SLIDE 55

▲♦❡❜ ♠❡❛s✉r❡s ❛r❡ ✈❡r② r✐❝❤✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ✐♥t❡r♥❛❧ ♠❡❛s✉r❡s t❤❡② ❝♦♠❡ ❢r♦♠ ♠❛② ❜❡ ✈❡r② s✐♠♣❧❡ ✭❡❣✳❝♦✉♥t✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s✮✳ ▲♦❡❜ ✐♥t❡❣r❛t✐♦♥ t❤❡♦r② r❡❧❛t❡s ♥♦♥st❛♥❞❛r❞ ✐♥t❡❣r❛❧s ❛♥❞ ▲♦❡❜ ✐♥t❡❣r❛❧s✳ ▲❡t ❋ ❜❡ ❙✲✐♥t❡❣r❛❜❧❡✳ ❚❤❡♥ ❋ ❛♥❞ ❋❞ ❋❞

▲

❙✐♠✐❧❛r r❡❧❛t✐♦♥s❤✐♣s ❝♦♥♥❡❝t ✐♥t❡r♥❛❧ ✭✐✳❡✳ ♥♦♥st❛♥❞❛r❞✮ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s t♦ st❛♥❞❛r❞ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡✳

slide-56
SLIDE 56

▲♦❡❜ ♠❡❛s✉r❡s ❛r❡ ✈❡r② r✐❝❤✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ✐♥t❡r♥❛❧ ♠❡❛s✉r❡s t❤❡② ❝♦♠❡ ❢r♦♠ ♠❛② ❜❡ ✈❡r② s✐♠♣❧❡ ✭❡❣✳❝♦✉♥t✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s✮✳ ▲♦❡❜ ✐♥t❡❣r❛t✐♦♥ t❤❡♦r② r❡❧❛t❡s ♥♦♥st❛♥❞❛r❞ ✐♥t❡❣r❛❧s ❛♥❞ ▲♦❡❜ ✐♥t❡❣r❛❧s✳ ▲❡t ❋ ❜❡ ❙✲✐♥t❡❣r❛❜❧❡✳ ❚❤❡♥ ❋ ❛♥❞ ❋❞ ❋❞

▲

❙✐♠✐❧❛r r❡❧❛t✐♦♥s❤✐♣s ❝♦♥♥❡❝t ✐♥t❡r♥❛❧ ✭✐✳❡✳ ♥♦♥st❛♥❞❛r❞✮ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s t♦ st❛♥❞❛r❞ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡✳

slide-57
SLIDE 57

▲♦❡❜ ♠❡❛s✉r❡s ❛r❡ ✈❡r② r✐❝❤✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ✐♥t❡r♥❛❧ ♠❡❛s✉r❡s t❤❡② ❝♦♠❡ ❢r♦♠ ♠❛② ❜❡ ✈❡r② s✐♠♣❧❡ ✭❡❣✳❝♦✉♥t✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s✮✳ ▲♦❡❜ ✐♥t❡❣r❛t✐♦♥ t❤❡♦r② r❡❧❛t❡s ♥♦♥st❛♥❞❛r❞ ✐♥t❡❣r❛❧s ❛♥❞ ▲♦❡❜ ✐♥t❡❣r❛❧s✳ ▲❡t ❋ : Ω → ∗R ❜❡ ❙✲✐♥t❡❣r❛❜❧❡✳ ❚❤❡♥ ◦❋ : Ω → R ❛♥❞

  • Ω

❋❞µ =

  • Ω
  • ❋❞µ▲

❙✐♠✐❧❛r r❡❧❛t✐♦♥s❤✐♣s ❝♦♥♥❡❝t ✐♥t❡r♥❛❧ ✭✐✳❡✳ ♥♦♥st❛♥❞❛r❞✮ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s t♦ st❛♥❞❛r❞ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡✳

slide-58
SLIDE 58

▲♦❡❜ ♠❡❛s✉r❡s ❛r❡ ✈❡r② r✐❝❤✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ✐♥t❡r♥❛❧ ♠❡❛s✉r❡s t❤❡② ❝♦♠❡ ❢r♦♠ ♠❛② ❜❡ ✈❡r② s✐♠♣❧❡ ✭❡❣✳❝♦✉♥t✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s✮✳ ▲♦❡❜ ✐♥t❡❣r❛t✐♦♥ t❤❡♦r② r❡❧❛t❡s ♥♦♥st❛♥❞❛r❞ ✐♥t❡❣r❛❧s ❛♥❞ ▲♦❡❜ ✐♥t❡❣r❛❧s✳ ▲❡t ❋ : Ω → ∗R ❜❡ ❙✲✐♥t❡❣r❛❜❧❡✳ ❚❤❡♥ ◦❋ : Ω → R ❛♥❞

  • Ω

❋❞µ =

  • Ω
  • ❋❞µ▲

❙✐♠✐❧❛r r❡❧❛t✐♦♥s❤✐♣s ❝♦♥♥❡❝t ✐♥t❡r♥❛❧ ✭✐✳❡✳ ♥♦♥st❛♥❞❛r❞✮ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s t♦ st❛♥❞❛r❞ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡✳

slide-59
SLIDE 59

▲❖❊❇ ❙P❆❈❊ ❙❖▲❯❚■❖◆❙ ❚❖ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✭✶✮ ❯s❡ st❛♥❞❛r❞ ❙❉❊ ♠❡t❤♦❞s ✰ ❚r❛♥s❢❡r t♦ s♦❧✈❡ t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ s◆❙❡ ✐♥ ❞✐♠❡♥s✐♦♥ ◆ ✭◆ ∈ ∗N ✐♥✜♥✐t❡✮ ❞❯(τ) = [−ν∗❆❯(τ) + ∗❇◆(❯) + ∗❢◆(τ, ❯(τ))]❞τ + ∗❣◆(τ, ❯(τ))❞❲τ ❯ ✐s ❛♥ ✐♥t❡r♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❯ : ∗[✵, ❚] × Ω → ❍◆ ⊂

∗❍ ♦♥ ❛♥

✐♥t❡r♥❛❧ s♣❛❝❡ Ω Ω✵ = (Ω, A, P) ✇✐t❤ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ ✐♥ ❍◆ ✭✷✮ ❊st❛❜❧✐s❤ ❛♥ ❡♥❡r❣② ❡st✐♠❛t❡✳ ❚❤❡r❡ ✐s ❛ ✜♥✐t❡ ❝♦♥st❛♥t ❊ ✭✐♥❞❡♣❡♥❞❡♥t ♦❢ ◆✮ s✉❝❤ t❤❛t s✉♣

❚

❯

✷ ❚ ✵

❯

✷ ❞

❊ ✭❊♥❡r❣②✮ ✭✸✮ ❚❤❡ ❡♥❡r❣② ❡st✐♠❛t❡ ♠❡❛♥s t❤❛t ❢♦r ❛✳❛✳ ✱ ❯ ✐s ✜♥✐t❡ ❢♦r ❛❧❧ ❚ ❛♥❞ ❤❡♥❝❡ ✇❡❛❦❧② ♥❡❛rst❛♥❞❛r❞✳ ❚❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❢♦r ❯ ❣✐✈❡s t❤❛t ❢♦r ❛✳❛✳ ✐❢ t❤❡♥ ❯ ❯ ❍❡♥❝❡ ✭✹✮ ❉❡✜♥❡ ❛ st❛♥❞❛r❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ♣r♦❝❡ss ✉ ✵ ❚ ❍ ❜② ✉ t ❯ t ✭✺✮ ❙❤♦✇ t❤❛t t❤✐s ✉ s♦❧✈❡s t❤❡ s◆❙❡ ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦

✵

✐✳❡✳ ▲

▲ ✇✐t❤ ✜❧tr❛t✐♦♥ ❞❡r✐✈❡❞ ❢r♦♠ t❤❛t ♦♥ ✵

slide-60
SLIDE 60

▲❖❊❇ ❙P❆❈❊ ❙❖▲❯❚■❖◆❙ ❚❖ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✭✶✮ ❯s❡ st❛♥❞❛r❞ ❙❉❊ ♠❡t❤♦❞s ✰ ❚r❛♥s❢❡r t♦ s♦❧✈❡ t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ s◆❙❡ ✐♥ ❞✐♠❡♥s✐♦♥ ◆ ✭◆ ∈ ∗N ✐♥✜♥✐t❡✮ ❞❯(τ) = [−ν∗❆❯(τ) + ∗❇◆(❯) + ∗❢◆(τ, ❯(τ))]❞τ + ∗❣◆(τ, ❯(τ))❞❲τ ❯ ✐s ❛♥ ✐♥t❡r♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❯ : ∗[✵, ❚] × Ω → ❍◆ ⊂

∗❍ ♦♥ ❛♥

✐♥t❡r♥❛❧ s♣❛❝❡ Ω Ω✵ = (Ω, A, P) ✇✐t❤ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ ✐♥ ❍◆ ✭✷✮ ❊st❛❜❧✐s❤ ❛♥ ❡♥❡r❣② ❡st✐♠❛t❡✳ ❚❤❡r❡ ✐s ❛ ✜♥✐t❡ ❝♦♥st❛♥t ❊ ✭✐♥❞❡♣❡♥❞❡♥t ♦❢ ◆✮ s✉❝❤ t❤❛t E  s✉♣

τ≤❚

|❯(τ)|✷ + ν

❚

  • ✵

||❯(σ)||✷ ❞σ   < ❊ ✭❊♥❡r❣②✮ ✭✸✮ ❚❤❡ ❡♥❡r❣② ❡st✐♠❛t❡ ♠❡❛♥s t❤❛t ❢♦r ❛✳❛✳ ✱ ❯ ✐s ✜♥✐t❡ ❢♦r ❛❧❧ ❚ ❛♥❞ ❤❡♥❝❡ ✇❡❛❦❧② ♥❡❛rst❛♥❞❛r❞✳ ❚❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❢♦r ❯ ❣✐✈❡s t❤❛t ❢♦r ❛✳❛✳ ✐❢ t❤❡♥ ❯ ❯ ❍❡♥❝❡ ✭✹✮ ❉❡✜♥❡ ❛ st❛♥❞❛r❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ♣r♦❝❡ss ✉ ✵ ❚ ❍ ❜② ✉ t ❯ t ✭✺✮ ❙❤♦✇ t❤❛t t❤✐s ✉ s♦❧✈❡s t❤❡ s◆❙❡ ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦

✵

✐✳❡✳ ▲

▲ ✇✐t❤ ✜❧tr❛t✐♦♥ ❞❡r✐✈❡❞ ❢r♦♠ t❤❛t ♦♥ ✵

slide-61
SLIDE 61

▲❖❊❇ ❙P❆❈❊ ❙❖▲❯❚■❖◆❙ ❚❖ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✭✶✮ ❯s❡ st❛♥❞❛r❞ ❙❉❊ ♠❡t❤♦❞s ✰ ❚r❛♥s❢❡r t♦ s♦❧✈❡ t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ s◆❙❡ ✐♥ ❞✐♠❡♥s✐♦♥ ◆ ✭◆ ∈ ∗N ✐♥✜♥✐t❡✮ ❞❯(τ) = [−ν∗❆❯(τ) + ∗❇◆(❯) + ∗❢◆(τ, ❯(τ))]❞τ + ∗❣◆(τ, ❯(τ))❞❲τ ❯ ✐s ❛♥ ✐♥t❡r♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❯ : ∗[✵, ❚] × Ω → ❍◆ ⊂

∗❍ ♦♥ ❛♥

✐♥t❡r♥❛❧ s♣❛❝❡ Ω Ω✵ = (Ω, A, P) ✇✐t❤ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ ✐♥ ❍◆ ✭✷✮ ❊st❛❜❧✐s❤ ❛♥ ❡♥❡r❣② ❡st✐♠❛t❡✳ ❚❤❡r❡ ✐s ❛ ✜♥✐t❡ ❝♦♥st❛♥t ❊ ✭✐♥❞❡♣❡♥❞❡♥t ♦❢ ◆✮ s✉❝❤ t❤❛t E  s✉♣

τ≤❚

|❯(τ)|✷ + ν

❚

  • ✵

||❯(σ)||✷ ❞σ   < ❊ ✭❊♥❡r❣②✮ ✭✸✮ ❚❤❡ ❡♥❡r❣② ❡st✐♠❛t❡ ♠❡❛♥s t❤❛t ❢♦r ❛✳❛✳ ω✱ |❯(τ)| ✐s ✜♥✐t❡ ❢♦r ❛❧❧ τ ≤ ❚ ❛♥❞ ❤❡♥❝❡ ✇❡❛❦❧② ♥❡❛rst❛♥❞❛r❞✳ ❚❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❢♦r ❯ ❣✐✈❡s t❤❛t ❢♦r ❛✳❛✳ ω, ✐❢ σ ≈ τ t❤❡♥ ❯(σ) ≈ ❯(τ). ❍❡♥❝❡ ✭✹✮ ❉❡✜♥❡ ❛ st❛♥❞❛r❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ♣r♦❝❡ss ✉ ✵ ❚ ❍ ❜② ✉ t ❯ t ✭✺✮ ❙❤♦✇ t❤❛t t❤✐s ✉ s♦❧✈❡s t❤❡ s◆❙❡ ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦

✵

✐✳❡✳ ▲

▲ ✇✐t❤ ✜❧tr❛t✐♦♥ ❞❡r✐✈❡❞ ❢r♦♠ t❤❛t ♦♥ ✵

slide-62
SLIDE 62

▲❖❊❇ ❙P❆❈❊ ❙❖▲❯❚■❖◆❙ ❚❖ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✭✶✮ ❯s❡ st❛♥❞❛r❞ ❙❉❊ ♠❡t❤♦❞s ✰ ❚r❛♥s❢❡r t♦ s♦❧✈❡ t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ s◆❙❡ ✐♥ ❞✐♠❡♥s✐♦♥ ◆ ✭◆ ∈ ∗N ✐♥✜♥✐t❡✮ ❞❯(τ) = [−ν∗❆❯(τ) + ∗❇◆(❯) + ∗❢◆(τ, ❯(τ))]❞τ + ∗❣◆(τ, ❯(τ))❞❲τ ❯ ✐s ❛♥ ✐♥t❡r♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❯ : ∗[✵, ❚] × Ω → ❍◆ ⊂

∗❍ ♦♥ ❛♥

✐♥t❡r♥❛❧ s♣❛❝❡ Ω Ω✵ = (Ω, A, P) ✇✐t❤ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ ✐♥ ❍◆ ✭✷✮ ❊st❛❜❧✐s❤ ❛♥ ❡♥❡r❣② ❡st✐♠❛t❡✳ ❚❤❡r❡ ✐s ❛ ✜♥✐t❡ ❝♦♥st❛♥t ❊ ✭✐♥❞❡♣❡♥❞❡♥t ♦❢ ◆✮ s✉❝❤ t❤❛t E  s✉♣

τ≤❚

|❯(τ)|✷ + ν

❚

  • ✵

||❯(σ)||✷ ❞σ   < ❊ ✭❊♥❡r❣②✮ ✭✸✮ ❚❤❡ ❡♥❡r❣② ❡st✐♠❛t❡ ♠❡❛♥s t❤❛t ❢♦r ❛✳❛✳ ω✱ |❯(τ)| ✐s ✜♥✐t❡ ❢♦r ❛❧❧ τ ≤ ❚ ❛♥❞ ❤❡♥❝❡ ✇❡❛❦❧② ♥❡❛rst❛♥❞❛r❞✳ ❚❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❢♦r ❯ ❣✐✈❡s t❤❛t ❢♦r ❛✳❛✳ ω, ✐❢ σ ≈ τ t❤❡♥ ❯(σ) ≈ ❯(τ). ❍❡♥❝❡ ✭✹✮ ❉❡✜♥❡ ❛ st❛♥❞❛r❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ♣r♦❝❡ss ✉ : [✵, ❚] × Ω → ❍ ❜② ✉(t, ω) = ◦❯(t, ω) ✭✺✮ ❙❤♦✇ t❤❛t t❤✐s ✉ s♦❧✈❡s t❤❡ s◆❙❡ ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦

✵

✐✳❡✳ ▲

▲ ✇✐t❤ ✜❧tr❛t✐♦♥ ❞❡r✐✈❡❞ ❢r♦♠ t❤❛t ♦♥ ✵

slide-63
SLIDE 63

▲❖❊❇ ❙P❆❈❊ ❙❖▲❯❚■❖◆❙ ❚❖ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✭✶✮ ❯s❡ st❛♥❞❛r❞ ❙❉❊ ♠❡t❤♦❞s ✰ ❚r❛♥s❢❡r t♦ s♦❧✈❡ t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ s◆❙❡ ✐♥ ❞✐♠❡♥s✐♦♥ ◆ ✭◆ ∈ ∗N ✐♥✜♥✐t❡✮ ❞❯(τ) = [−ν∗❆❯(τ) + ∗❇◆(❯) + ∗❢◆(τ, ❯(τ))]❞τ + ∗❣◆(τ, ❯(τ))❞❲τ ❯ ✐s ❛♥ ✐♥t❡r♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❯ : ∗[✵, ❚] × Ω → ❍◆ ⊂

∗❍ ♦♥ ❛♥

✐♥t❡r♥❛❧ s♣❛❝❡ Ω Ω✵ = (Ω, A, P) ✇✐t❤ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ ✐♥ ❍◆ ✭✷✮ ❊st❛❜❧✐s❤ ❛♥ ❡♥❡r❣② ❡st✐♠❛t❡✳ ❚❤❡r❡ ✐s ❛ ✜♥✐t❡ ❝♦♥st❛♥t ❊ ✭✐♥❞❡♣❡♥❞❡♥t ♦❢ ◆✮ s✉❝❤ t❤❛t E  s✉♣

τ≤❚

|❯(τ)|✷ + ν

❚

  • ✵

||❯(σ)||✷ ❞σ   < ❊ ✭❊♥❡r❣②✮ ✭✸✮ ❚❤❡ ❡♥❡r❣② ❡st✐♠❛t❡ ♠❡❛♥s t❤❛t ❢♦r ❛✳❛✳ ω✱ |❯(τ)| ✐s ✜♥✐t❡ ❢♦r ❛❧❧ τ ≤ ❚ ❛♥❞ ❤❡♥❝❡ ✇❡❛❦❧② ♥❡❛rst❛♥❞❛r❞✳ ❚❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❢♦r ❯ ❣✐✈❡s t❤❛t ❢♦r ❛✳❛✳ ω, ✐❢ σ ≈ τ t❤❡♥ ❯(σ) ≈ ❯(τ). ❍❡♥❝❡ ✭✹✮ ❉❡✜♥❡ ❛ st❛♥❞❛r❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ♣r♦❝❡ss ✉ : [✵, ❚] × Ω → ❍ ❜② ✉(t, ω) = ◦❯(t, ω) ✭✺✮ ❙❤♦✇ t❤❛t t❤✐s ✉ s♦❧✈❡s t❤❡ s◆❙❡ ♦♥ t❤❡ ▲♦❡❜ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ Ω Ω✵ ✐✳❡✳ Ω Ω = (Ω, ▲ (A) , P▲) ✇✐t❤ ✜❧tr❛t✐♦♥ ❞❡r✐✈❡❞ ❢r♦♠ t❤❛t ♦♥ Ω Ω✵

slide-64
SLIDE 64

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮✮

❚❤❡r❡ ✐s ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω ❝❛rr②✐♥❣ ❛♥ ❍✲✈❛❧✉❡❞ ❲✐❡♥❡r ♣r♦❝❡ss ✇ s✉❝❤ t❤❛t ❢♦r ❛♥② ✭▲✷✲r❛♥❞♦♠✮ ✉✵ ∈ ❍ ❛♥❞ ❢ , ❣ ✭❝♦♥t✐♥✉♦✉s ✇✐t❤ ❧✐♥❡❛r ❣r♦✇t❤✮ t❤❡r❡ ✐s ❛ ✭✇❡❛❦✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❛t ✐s✱ ❛♥ ❛❞❛♣t❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ ✵ ❍ s✉❝❤ t❤❛t ❢♦r ❛✳❛✳ ✭✐✮ ✉ ▲✷ ✵ ❚ ❱ ▲ ✵ ❚ ❍ ❈ ✵ ❚ ❍✇❡❛❦ ❢♦r ❛❧❧ ❚ ✱ ✭✐✐✮ ❢♦r ❛❧❧ t ✵ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ❞s

t ✵

❣ s ✉ s ❞✇s

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SLIDE 65

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✭✶✾✾✶✮✮

❚❤❡r❡ ✐s ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω ❝❛rr②✐♥❣ ❛♥ ❍✲✈❛❧✉❡❞ ❲✐❡♥❡r ♣r♦❝❡ss ✇ s✉❝❤ t❤❛t ❢♦r ❛♥② ✭▲✷✲r❛♥❞♦♠✮ ✉✵ ∈ ❍ ❛♥❞ ❢ , ❣ ✭❝♦♥t✐♥✉♦✉s ✇✐t❤ ❧✐♥❡❛r ❣r♦✇t❤✮ t❤❡r❡ ✐s ❛ ✭✇❡❛❦✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❛t ✐s✱ ❛♥ ❛❞❛♣t❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ : [✵, ∞) × Ω → ❍ s✉❝❤ t❤❛t ❢♦r ❛✳❛✳ ω ✭✐✮ ✉ ∈ ▲✷(✵, ❚; ❱ ) ∩ ▲∞(✵, ❚; ❍) ∩ ❈(✵, ❚; ❍✇❡❛❦) ❢♦r ❛❧❧ ❚ < ∞ ✱ ✭✐✐✮ ❢♦r ❛❧❧ t ≥ ✵ ✉(t) = ✉✵ + t

✵

[ν❆✉(s) − ❇(✉(s)) + ❢ (s, ✉(s))]❞s + t

✵

❣(s, ✉(s))❞✇s

slide-66
SLIDE 66

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉ ✵ ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ❍ s✉❝❤ t❤❛t ❙t❆ ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇

❍ ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇

  • ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ ❙ ❱ ❱ ❇ ❛♥❞ ❛♥ ✐♥✜♥✐t❡ t✐♠❡ ✇❤❡r❡ ❇ ❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

slide-67
SLIDE 67

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ❍ s✉❝❤ t❤❛t ❙t❆ ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇

❍ ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇

  • ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ ❙ ❱ ❱ ❇ ❛♥❞ ❛♥ ✐♥✜♥✐t❡ t✐♠❡ ✇❤❡r❡ ❇ ❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

slide-68
SLIDE 68

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ⊆ ❍ s✉❝❤ t❤❛t ❙t❆ = ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❍, ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇ ⊆ ● ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ ❙ ❱ ❱ ❇ ❛♥❞ ❛♥ ✐♥✜♥✐t❡ t✐♠❡ ✇❤❡r❡ ❇ ❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

slide-69
SLIDE 69

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ⊆ ❍ s✉❝❤ t❤❛t ❙t❆ = ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❍, ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇ ⊆ ● ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ = {∗❙τ❱ : ❱ ∈ ❇ ❛♥❞ τ ❛♥ ✐♥✜♥✐t❡ t✐♠❡} ✇❤❡r❡ ❇ ⊆❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

slide-70
SLIDE 70

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ⊆ ❍ s✉❝❤ t❤❛t ❙t❆ = ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❍, ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇ ⊆ ● ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ = {∗❙τ❱ : ❱ ∈ ❇ ❛♥❞ τ ❛♥ ✐♥✜♥✐t❡ t✐♠❡} ✇❤❡r❡ ❇ ⊆❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

slide-71
SLIDE 71

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ⊆ ❍ s✉❝❤ t❤❛t ❙t❆ = ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❍, ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇ ⊆ ● ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ = {∗❙τ❱ : ❱ ∈ ❇ ❛♥❞ τ ❛♥ ✐♥✜♥✐t❡ t✐♠❡} ✇❤❡r❡ ❇ ⊆❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

slide-72
SLIDE 72

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ⊆ ❍ s✉❝❤ t❤❛t ❙t❆ = ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❍, ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇ ⊆ ● ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ = {∗❙τ❱ : ❱ ∈ ❇ ❛♥❞ τ ❛♥ ✐♥✜♥✐t❡ t✐♠❡} ✇❤❡r❡ ❇ ⊆❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

slide-73
SLIDE 73

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ⊆ ❍ s✉❝❤ t❤❛t ❙t❆ = ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❍, ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇ ⊆ ● ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ = {∗❙τ❱ : ❱ ∈ ❇ ❛♥❞ τ ❛♥ ✐♥✜♥✐t❡ t✐♠❡} ✇❤❡r❡ ❇ ⊆❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

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SLIDE 74

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ⊆ ❍ s✉❝❤ t❤❛t ❙t❆ = ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❍, ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇ ⊆ ● ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ = {∗❙τ❱ : ❱ ∈ ❇ ❛♥❞ τ ❛♥ ✐♥✜♥✐t❡ t✐♠❡} ✇❤❡r❡ ❇ ⊆❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ ✉ ❛♥❞ ❣ ✉

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SLIDE 75

❆♣♣❧✐❝❛t✐♦♥ ✶✿ ❆❚❚❘❆❈❚❖❘❙ ❋❖❘ ❙❚❖❈❍❆❙❚■❈ ◆❆❱■❊❘✕❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙ ❋♦r ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✇✐t❤ ✉♥✐q✉❡♥❡ss ✇r✐t❡ ❙t✈ = ✈❛❧✉❡ ❛t t✐♠❡ t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ ✉(✵) = ✈✳ ❆♥ ❛ttr❛❝t♦r ✐s ❛ ❝♦♠♣❛❝t s❡t ❆ ⊆ ❍ s✉❝❤ t❤❛t ❙t❆ = ❆ ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t

  • ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❍, ❡✈❡♥t✉❛❧❧② ✇❡ ❤❛✈❡ ❙t❇ ⊆ ● ✳

■♥t✉✐t✐✈❡❧② ❛♥ ❛ttr❛❝t♦r ✐s ❣✐✈❡♥ ❜② ❆ = {∗❙τ❱ : ❱ ∈ ❇ ❛♥❞ τ ❛♥ ✐♥✜♥✐t❡ t✐♠❡} ✇❤❡r❡ ❇ ⊆❍◆ ✐s ❛ ❝❤♦s❡♥ ❜♦✉♥❞❡❞ s❡t ✭❛♥ ❛❜s♦r❜✐♥❣ s❡t✮✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ♣r❡❝✐s❡ ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ◆❙❆✳ ❋♦r st♦❝❤❛st✐❝ s②st❡♠s t❤❡r❡ ✐s ❛ ✈❛r✐❡t② ♦❢ ♥♦t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✭✶✮ ♠❡❛s✉r❡ ❛ttr❛❝t♦rs ✲ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ♦♥ ♣❛t❤ s♣❛❝❡ ✭❙❝❤♠❛❧❧❢✉ÿ ❛♥❞ ♦t❤❡rs✮✳ ✭✷✮ st♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ✭❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✮ ✭✸✮ ♣r♦❝❡ss ❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ✭✹✮ ♥❡♦✲❛ttr❛❝t♦rs ✭◆❏❈ ✫ ❑❡✐s❧❡r✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ❣✐✈❡ ♥❡✇ r❡s✉❧ts ❢♦r ❡❛❝❤ ♦❢ ✭✷✮ ✲ ✭✹✮ ❢♦r s◆❙❡ ❢♦r ❞r✐❢t ❛♥❞ ♥♦✐s❡ ♦❢ t❤❡ ❢♦r♠ ❢ (✉) ❛♥❞ ❣(✉)

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SLIDE 76

❙t♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r s◆❙❡ t❤✐s ♦♥❧② ♠❛❦❡s s❡♥s❡ ❢♦r ❞ ✷ ✭✇❤❡r❡ ✇❡ ❤❛✈❡ ✉♥✐q✉❡♥❡ss✮✳ ❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✬s ✐❞❡❛✿ ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ✐s ❛ r❛♥❞♦♠ ❝♦♠♣❛❝t s❡t ❆ t❤❛t✱ ❛t t✐♠❡ ✵✱ ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s ✏st❛rt✐♥❣ ❛t ✑ ✭❝♦♠♣❛r❡❞ t♦ t❤❡ ✉s✉❛❧ ✐❞❡❛ ♦❢ ❛♥ ❛ttr❛❝t♦r ❜❡✐♥❣ ❛ s❡t ✏❛t t✐♠❡ ✑ t❤❛t ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s st❛rt✐♥❣ ❛t t✐♠❡ ✵✮✳ ❚❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ t❤❡ s◆❙❡ ❝❛♥ ❜❡ ♠♦❞✐✜❡❞ ❜② st❛rt✐♥❣ t❤❡ s♦❧✉t✐♦♥s ❛t ❛♥② ❣✐✈❡♥ ♥❡❣❛t✐✈❡ t✐♠❡ ✲ ✐♥❝❧✉❞✐♥❣ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡❀ t❤❡♥ ✐♥t✉✐t✐✈❡❧② ❛ r❛♥❞♦♠ ❛ttr❛❝t♦r ❆ ♣♦✐♥ts ✐♥ ❍ t❤❛t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❛t t✐♠❡ t ✵ st❛rt✐♥❣ ❛t s♦♠❡ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡✳ ▼❛❦✐♥❣ t❤✐s ♣r❡❝✐s❡ ❣✐✈❡s✿

❚❤❡♦r❡♠

✭❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✶✾✾✾✮ ❋♦r s♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ♥♦✐s❡ t❡r♠ ❣ ✉ ✐♥ t❤❡ ✷❉ s◆❙❡ t❤❡r❡ ✐s ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ❆ ✭❝♦♠♣❛❝t ✐♥ t❤❡ str♦♥❣ t♦♣♦❧♦❣② ♦❢ ❍✮✳ Pr❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ♣r♦♦❢ ✲ t♦♦ ❧♦♥❣ ❛♥❞ ❝♦♠♣❧✐❝❛t❡❞✦

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SLIDE 77

❙t♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r s◆❙❡ t❤✐s ♦♥❧② ♠❛❦❡s s❡♥s❡ ❢♦r ❞ = ✷ ✭✇❤❡r❡ ✇❡ ❤❛✈❡ ✉♥✐q✉❡♥❡ss✮✳ ❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✬s ✐❞❡❛✿ ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ✐s ❛ r❛♥❞♦♠ ❝♦♠♣❛❝t s❡t ❆ t❤❛t✱ ❛t t✐♠❡ ✵✱ ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s ✏st❛rt✐♥❣ ❛t ✑ ✭❝♦♠♣❛r❡❞ t♦ t❤❡ ✉s✉❛❧ ✐❞❡❛ ♦❢ ❛♥ ❛ttr❛❝t♦r ❜❡✐♥❣ ❛ s❡t ✏❛t t✐♠❡ ✑ t❤❛t ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s st❛rt✐♥❣ ❛t t✐♠❡ ✵✮✳ ❚❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ t❤❡ s◆❙❡ ❝❛♥ ❜❡ ♠♦❞✐✜❡❞ ❜② st❛rt✐♥❣ t❤❡ s♦❧✉t✐♦♥s ❛t ❛♥② ❣✐✈❡♥ ♥❡❣❛t✐✈❡ t✐♠❡ ✲ ✐♥❝❧✉❞✐♥❣ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡❀ t❤❡♥ ✐♥t✉✐t✐✈❡❧② ❛ r❛♥❞♦♠ ❛ttr❛❝t♦r ❆ ♣♦✐♥ts ✐♥ ❍ t❤❛t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❛t t✐♠❡ t ✵ st❛rt✐♥❣ ❛t s♦♠❡ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡✳ ▼❛❦✐♥❣ t❤✐s ♣r❡❝✐s❡ ❣✐✈❡s✿

❚❤❡♦r❡♠

✭❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✶✾✾✾✮ ❋♦r s♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ♥♦✐s❡ t❡r♠ ❣ ✉ ✐♥ t❤❡ ✷❉ s◆❙❡ t❤❡r❡ ✐s ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ❆ ✭❝♦♠♣❛❝t ✐♥ t❤❡ str♦♥❣ t♦♣♦❧♦❣② ♦❢ ❍✮✳ Pr❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ♣r♦♦❢ ✲ t♦♦ ❧♦♥❣ ❛♥❞ ❝♦♠♣❧✐❝❛t❡❞✦

slide-78
SLIDE 78

❙t♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r s◆❙❡ t❤✐s ♦♥❧② ♠❛❦❡s s❡♥s❡ ❢♦r ❞ = ✷ ✭✇❤❡r❡ ✇❡ ❤❛✈❡ ✉♥✐q✉❡♥❡ss✮✳ ❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✬s ✐❞❡❛✿ ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ✐s ❛ r❛♥❞♦♠ ❝♦♠♣❛❝t s❡t ❆(ω) t❤❛t✱ ❛t t✐♠❡ ✵✱ ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s ✏st❛rt✐♥❣ ❛t −∞✑ ✭❝♦♠♣❛r❡❞ t♦ t❤❡ ✉s✉❛❧ ✐❞❡❛ ♦❢ ❛♥ ❛ttr❛❝t♦r ❜❡✐♥❣ ❛ s❡t ✏❛t t✐♠❡ ∞✑ t❤❛t ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s st❛rt✐♥❣ ❛t t✐♠❡ ✵✮✳ ❚❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ t❤❡ s◆❙❡ ❝❛♥ ❜❡ ♠♦❞✐✜❡❞ ❜② st❛rt✐♥❣ t❤❡ s♦❧✉t✐♦♥s ❛t ❛♥② ❣✐✈❡♥ ♥❡❣❛t✐✈❡ t✐♠❡ ✲ ✐♥❝❧✉❞✐♥❣ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡❀ t❤❡♥ ✐♥t✉✐t✐✈❡❧② ❛ r❛♥❞♦♠ ❛ttr❛❝t♦r ❆ ♣♦✐♥ts ✐♥ ❍ t❤❛t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❛t t✐♠❡ t ✵ st❛rt✐♥❣ ❛t s♦♠❡ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡✳ ▼❛❦✐♥❣ t❤✐s ♣r❡❝✐s❡ ❣✐✈❡s✿

❚❤❡♦r❡♠

✭❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✶✾✾✾✮ ❋♦r s♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ♥♦✐s❡ t❡r♠ ❣ ✉ ✐♥ t❤❡ ✷❉ s◆❙❡ t❤❡r❡ ✐s ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ❆ ✭❝♦♠♣❛❝t ✐♥ t❤❡ str♦♥❣ t♦♣♦❧♦❣② ♦❢ ❍✮✳ Pr❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ♣r♦♦❢ ✲ t♦♦ ❧♦♥❣ ❛♥❞ ❝♦♠♣❧✐❝❛t❡❞✦

slide-79
SLIDE 79

❙t♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r s◆❙❡ t❤✐s ♦♥❧② ♠❛❦❡s s❡♥s❡ ❢♦r ❞ = ✷ ✭✇❤❡r❡ ✇❡ ❤❛✈❡ ✉♥✐q✉❡♥❡ss✮✳ ❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✬s ✐❞❡❛✿ ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ✐s ❛ r❛♥❞♦♠ ❝♦♠♣❛❝t s❡t ❆(ω) t❤❛t✱ ❛t t✐♠❡ ✵✱ ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s ✏st❛rt✐♥❣ ❛t −∞✑ ✭❝♦♠♣❛r❡❞ t♦ t❤❡ ✉s✉❛❧ ✐❞❡❛ ♦❢ ❛♥ ❛ttr❛❝t♦r ❜❡✐♥❣ ❛ s❡t ✏❛t t✐♠❡ ∞✑ t❤❛t ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s st❛rt✐♥❣ ❛t t✐♠❡ ✵✮✳ ❚❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ t❤❡ s◆❙❡ ❝❛♥ ❜❡ ♠♦❞✐✜❡❞ ❜② st❛rt✐♥❣ t❤❡ s♦❧✉t✐♦♥s ❛t ❛♥② ❣✐✈❡♥ ♥❡❣❛t✐✈❡ t✐♠❡ ✲ ✐♥❝❧✉❞✐♥❣ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡❀ t❤❡♥ ✐♥t✉✐t✐✈❡❧② ❛ r❛♥❞♦♠ ❛ttr❛❝t♦r ❆(ω) =♣♦✐♥ts ✐♥ ❍ t❤❛t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❛t t✐♠❡ t = ✵ st❛rt✐♥❣ ❛t s♦♠❡ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡✳ ▼❛❦✐♥❣ t❤✐s ♣r❡❝✐s❡ ❣✐✈❡s✿

❚❤❡♦r❡♠

✭❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✶✾✾✾✮ ❋♦r s♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ♥♦✐s❡ t❡r♠ ❣ ✉ ✐♥ t❤❡ ✷❉ s◆❙❡ t❤❡r❡ ✐s ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ❆ ✭❝♦♠♣❛❝t ✐♥ t❤❡ str♦♥❣ t♦♣♦❧♦❣② ♦❢ ❍✮✳ Pr❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ♣r♦♦❢ ✲ t♦♦ ❧♦♥❣ ❛♥❞ ❝♦♠♣❧✐❝❛t❡❞✦

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SLIDE 80

❙t♦❝❤❛st✐❝ ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r s◆❙❡ t❤✐s ♦♥❧② ♠❛❦❡s s❡♥s❡ ❢♦r ❞ = ✷ ✭✇❤❡r❡ ✇❡ ❤❛✈❡ ✉♥✐q✉❡♥❡ss✮✳ ❈r❛✉❡❧ ✫ ❋❧❛♥❞♦❧✐✬s ✐❞❡❛✿ ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ✐s ❛ r❛♥❞♦♠ ❝♦♠♣❛❝t s❡t ❆(ω) t❤❛t✱ ❛t t✐♠❡ ✵✱ ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s ✏st❛rt✐♥❣ ❛t −∞✑ ✭❝♦♠♣❛r❡❞ t♦ t❤❡ ✉s✉❛❧ ✐❞❡❛ ♦❢ ❛♥ ❛ttr❛❝t♦r ❜❡✐♥❣ ❛ s❡t ✏❛t t✐♠❡ ∞✑ t❤❛t ❛ttr❛❝ts tr❛❥❡❝t♦r✐❡s st❛rt✐♥❣ ❛t t✐♠❡ ✵✮✳ ❚❤❡ ▲♦❡❜ s♣❛❝❡ ❛♣♣r♦❛❝❤ t♦ s♦❧✈✐♥❣ t❤❡ s◆❙❡ ❝❛♥ ❜❡ ♠♦❞✐✜❡❞ ❜② st❛rt✐♥❣ t❤❡ s♦❧✉t✐♦♥s ❛t ❛♥② ❣✐✈❡♥ ♥❡❣❛t✐✈❡ t✐♠❡ ✲ ✐♥❝❧✉❞✐♥❣ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡❀ t❤❡♥ ✐♥t✉✐t✐✈❡❧② ❛ r❛♥❞♦♠ ❛ttr❛❝t♦r ❆(ω) =♣♦✐♥ts ✐♥ ❍ t❤❛t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❛t t✐♠❡ t = ✵ st❛rt✐♥❣ ❛t s♦♠❡ ✐♥✜♥✐t❡ ♥❡❣❛t✐✈❡ t✐♠❡✳ ▼❛❦✐♥❣ t❤✐s ♣r❡❝✐s❡ ❣✐✈❡s✿

❚❤❡♦r❡♠

✭❈❛♣✐➠s❦✐ ✫ ◆❏❈ ✶✾✾✾✮ ❋♦r s♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ♥♦✐s❡ t❡r♠ ❣(✉) ✐♥ t❤❡ ✷❉ s◆❙❡ t❤❡r❡ ✐s ❛ st♦❝❤❛st✐❝ ❛ttr❛❝t♦r ❆(ω) ✭❝♦♠♣❛❝t ✐♥ t❤❡ str♦♥❣ t♦♣♦❧♦❣② ♦❢ ❍✮✳ Pr❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ♣r♦♦❢ ✲ t♦♦ ❧♦♥❣ ❛♥❞ ❝♦♠♣❧✐❝❛t❡❞✦

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SLIDE 81

Pr♦❝❡ss ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r ❞ ✸ ✉♥✐q✉❡♥❡ss ✐s ✉♥❦♥♦✇♥✳ ❚♦ ♦✈❡r❝♦♠❡ t❤✐s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❙❡✱ ❙❡❧❧ r❡♣❧❛❝❡❞ t❤❡ ♣❤❛s❡ s♣❛❝❡ ❍ ❜② ❲ ❛❧❧ s♦❧✉t✐♦♥ ♣❛t❤s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❡ s❡♠✐❣r♦✉♣ ❛❝t✐♦♥ ❙t ♦♥ ❲ ✐s t✐♠❡ tr❛♥s❧❛t✐♦♥✿ ✐❢ ✉ ✉ ❲ t❤❡♥ ❙t✉ ✈ ❲ ✐s ❣✐✈❡♥ ❜② ❙t✉ s ✉ t s ❚❤✐s ❤❛s t❤❡ ❝r✉❝✐❛❧ s❡♠✐✲✢♦✇ ♣r♦♣❡rt② ❙t✶ ❙t✷ ❙t✶

t✷ ❛❧♦♥❣ ✇✐t❤ ❙✵✉

✉✳

❚❤❡♦r❡♠ ✭❙❡❧❧ ✭✶✾✾✻✮✮

❚❤❡r❡ ✐s ❣❧♦❜❛❧ ❛ttr❛❝t♦r ❆ ❲ ❢♦r t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛❧ ✭❞❡t❡r♠✐♥✐st✐❝✮ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳

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SLIDE 82

Pr♦❝❡ss ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r ❞ = ✸ ✉♥✐q✉❡♥❡ss ✐s ✉♥❦♥♦✇♥✳ ❚♦ ♦✈❡r❝♦♠❡ t❤✐s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❙❡✱ ❙❡❧❧ r❡♣❧❛❝❡❞ t❤❡ ♣❤❛s❡ s♣❛❝❡ ❍ ❜② ❲ = ❛❧❧ s♦❧✉t✐♦♥ ♣❛t❤s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❡ s❡♠✐❣r♦✉♣ ❛❝t✐♦♥ ❙t ♦♥ ❲ ✐s t✐♠❡ tr❛♥s❧❛t✐♦♥✿ ✐❢ ✉ ✉ ❲ t❤❡♥ ❙t✉ ✈ ❲ ✐s ❣✐✈❡♥ ❜② ❙t✉ s ✉ t s ❚❤✐s ❤❛s t❤❡ ❝r✉❝✐❛❧ s❡♠✐✲✢♦✇ ♣r♦♣❡rt② ❙t✶ ❙t✷ ❙t✶

t✷ ❛❧♦♥❣ ✇✐t❤ ❙✵✉

✉✳

❚❤❡♦r❡♠ ✭❙❡❧❧ ✭✶✾✾✻✮✮

❚❤❡r❡ ✐s ❣❧♦❜❛❧ ❛ttr❛❝t♦r ❆ ❲ ❢♦r t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛❧ ✭❞❡t❡r♠✐♥✐st✐❝✮ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳

slide-83
SLIDE 83

Pr♦❝❡ss ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r ❞ = ✸ ✉♥✐q✉❡♥❡ss ✐s ✉♥❦♥♦✇♥✳ ❚♦ ♦✈❡r❝♦♠❡ t❤✐s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❙❡✱ ❙❡❧❧ r❡♣❧❛❝❡❞ t❤❡ ♣❤❛s❡ s♣❛❝❡ ❍ ❜② ❲ = ❛❧❧ s♦❧✉t✐♦♥ ♣❛t❤s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❡ s❡♠✐❣r♦✉♣ ❛❝t✐♦♥ ❙t ♦♥ ❲ ✐s t✐♠❡ tr❛♥s❧❛t✐♦♥✿ ✐❢ ✉ = ✉(·) ∈ ❲ t❤❡♥ ❙t✉ = ✈ ∈ ❲ ✐s ❣✐✈❡♥ ❜② (❙t✉)(s) = ✉(t + s). ❚❤✐s ❤❛s t❤❡ ❝r✉❝✐❛❧ s❡♠✐✲✢♦✇ ♣r♦♣❡rt② ❙t✶ ❙t✷ ❙t✶

t✷ ❛❧♦♥❣ ✇✐t❤ ❙✵✉

✉✳

❚❤❡♦r❡♠ ✭❙❡❧❧ ✭✶✾✾✻✮✮

❚❤❡r❡ ✐s ❣❧♦❜❛❧ ❛ttr❛❝t♦r ❆ ❲ ❢♦r t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛❧ ✭❞❡t❡r♠✐♥✐st✐❝✮ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳

slide-84
SLIDE 84

Pr♦❝❡ss ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r ❞ = ✸ ✉♥✐q✉❡♥❡ss ✐s ✉♥❦♥♦✇♥✳ ❚♦ ♦✈❡r❝♦♠❡ t❤✐s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❙❡✱ ❙❡❧❧ r❡♣❧❛❝❡❞ t❤❡ ♣❤❛s❡ s♣❛❝❡ ❍ ❜② ❲ = ❛❧❧ s♦❧✉t✐♦♥ ♣❛t❤s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❡ s❡♠✐❣r♦✉♣ ❛❝t✐♦♥ ❙t ♦♥ ❲ ✐s t✐♠❡ tr❛♥s❧❛t✐♦♥✿ ✐❢ ✉ = ✉(·) ∈ ❲ t❤❡♥ ❙t✉ = ✈ ∈ ❲ ✐s ❣✐✈❡♥ ❜② (❙t✉)(s) = ✉(t + s). ❚❤✐s ❤❛s t❤❡ ❝r✉❝✐❛❧ s❡♠✐✲✢♦✇ ♣r♦♣❡rt② ❙t✶ ◦ ❙t✷ = ❙t✶+t✷ ❛❧♦♥❣ ✇✐t❤ ❙✵✉ = ✉✳

❚❤❡♦r❡♠ ✭❙❡❧❧ ✭✶✾✾✻✮✮

❚❤❡r❡ ✐s ❣❧♦❜❛❧ ❛ttr❛❝t♦r ❆ ❲ ❢♦r t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛❧ ✭❞❡t❡r♠✐♥✐st✐❝✮ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳

slide-85
SLIDE 85

Pr♦❝❡ss ❛ttr❛❝t♦rs ❢♦r s◆❙❡ ❋♦r ❞ = ✸ ✉♥✐q✉❡♥❡ss ✐s ✉♥❦♥♦✇♥✳ ❚♦ ♦✈❡r❝♦♠❡ t❤✐s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❙❡✱ ❙❡❧❧ r❡♣❧❛❝❡❞ t❤❡ ♣❤❛s❡ s♣❛❝❡ ❍ ❜② ❲ = ❛❧❧ s♦❧✉t✐♦♥ ♣❛t❤s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❡ s❡♠✐❣r♦✉♣ ❛❝t✐♦♥ ❙t ♦♥ ❲ ✐s t✐♠❡ tr❛♥s❧❛t✐♦♥✿ ✐❢ ✉ = ✉(·) ∈ ❲ t❤❡♥ ❙t✉ = ✈ ∈ ❲ ✐s ❣✐✈❡♥ ❜② (❙t✉)(s) = ✉(t + s). ❚❤✐s ❤❛s t❤❡ ❝r✉❝✐❛❧ s❡♠✐✲✢♦✇ ♣r♦♣❡rt② ❙t✶ ◦ ❙t✷ = ❙t✶+t✷ ❛❧♦♥❣ ✇✐t❤ ❙✵✉ = ✉✳

❚❤❡♦r❡♠ ✭❙❡❧❧ ✭✶✾✾✻✮✮

❚❤❡r❡ ✐s ❣❧♦❜❛❧ ❛ttr❛❝t♦r ❆ ⊆ ❲ ❢♦r t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛❧ ✭❞❡t❡r♠✐♥✐st✐❝✮ ◆❛✈✐❡r✕❙t♦❦❡s ❡q✉❛t✐♦♥s✳

slide-86
SLIDE 86

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s

t

❢♦r t ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭ ✶

✵

✐❞❡♥t✐t② ❛♥❞

t s t s❀

✭ ✷✮

t s t s ❢♦r ❛❧❧ s t

✵✱ ✇❤❡r❡

t

✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ ❀ ✭ ✸✮ ✇ t s

t

✇ t

t

✇ s ❢♦r ❛❧❧ s ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ ✉ t ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ ❙r✉ ❜② ✈ t ✉ r t

r

❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-87
SLIDE 87

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭ ✶

✵

✐❞❡♥t✐t② ❛♥❞

t s t s❀

✭ ✷✮

t s t s ❢♦r ❛❧❧ s t

✵✱ ✇❤❡r❡

t

✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ ❀ ✭ ✸✮ ✇ t s

t

✇ t

t

✇ s ❢♦r ❛❧❧ s ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ ✉ t ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ ❙r✉ ❜② ✈ t ✉ r t

r

❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-88
SLIDE 88

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭ ✷✮

t s t s ❢♦r ❛❧❧ s t

✵✱ ✇❤❡r❡

t

✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ ❀ ✭ ✸✮ ✇ t s

t

✇ t

t

✇ s ❢♦r ❛❧❧ s ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ ✉ t ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ ❙r✉ ❜② ✈ t ✉ r t

r

❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-89
SLIDE 89

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭ ✸✮ ✇ t s

t

✇ t

t

✇ s ❢♦r ❛❧❧ s ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ ✉ t ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ ❙r✉ ❜② ✈ t ✉ r t

r

❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-90
SLIDE 90

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭θ✸✮ ✇(t + s, θtω) − ✇(t, θtω) = ✇(s, ω) ❢♦r ❛❧❧ s ≥ ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ ✉ t ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ ❙r✉ ❜② ✈ t ✉ r t

r

❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-91
SLIDE 91

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭θ✸✮ ✇(t + s, θtω) − ✇(t, θtω) = ✇(s, ω) ❢♦r ❛❧❧ s ≥ ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ = ✉(t, ω) ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ = ❙r✉ ❜② ✈(t, ω) = ✉(r + t, θrω) ❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-92
SLIDE 92

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭θ✸✮ ✇(t + s, θtω) − ✇(t, θtω) = ✇(s, ω) ❢♦r ❛❧❧ s ≥ ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ = ✉(t, ω) ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ = ❙r✉ ❜② ✈(t, ω) = ✉(r + t, θrω) ❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-93
SLIDE 93

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭θ✸✮ ✇(t + s, θtω) − ✇(t, θtω) = ✇(s, ω) ❢♦r ❛❧❧ s ≥ ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ = ✉(t, ω) ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ = ❙r✉ ❜② ✈(t, ω) = ✉(r + t, θrω) ❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-94
SLIDE 94

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭θ✸✮ ✇(t + s, θtω) − ✇(t, θtω) = ✇(s, ω) ❢♦r ❛❧❧ s ≥ ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ = ✉(t, ω) ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ = ❙r✉ ❜② ✈(t, ω) = ✉(r + t, θrω) ❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-95
SLIDE 95

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭θ✸✮ ✇(t + s, θtω) − ✇(t, θtω) = ✇(s, ω) ❢♦r ❛❧❧ s ≥ ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ = ✉(t, ω) ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ = ❙r✉ ❜② ✈(t, ω) = ✉(r + t, θrω) ❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-96
SLIDE 96

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭θ✸✮ ✇(t + s, θtω) − ✇(t, θtω) = ✇(s, ω) ❢♦r ❛❧❧ s ≥ ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ = ✉(t, ω) ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ = ❙r✉ ❜② ✈(t, ω) = ✉(r + t, θrω) ❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇ ✉ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss ✉ ✇ ✳

slide-97
SLIDE 97

❊①t❡♥s✐♦♥ ♦❢ ❙❡❧❧✬s ✐❞❡❛ t♦ t❤❡ st♦❝❤❛st✐❝ ◆❙✲❡q✉❛t✐♦♥s✳ ❇❛s✐❝ ✐❞❡❛✳ ▲❡t ❳ ❜❡ ❛ s❡t ♦❢ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ♦♥ ❛ s♣❛❝❡ Ω Ω ✇✐t❤ ❲✐❡♥❡r ♣r♦❝❡ss ✇✳ ❆ss✉♠❡ t❤❛t Ω ❤❛s ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ ♠❛♣s θt : Ω → Ω ❢♦r t ≥ ✵ t❤❛t ✏s❤✐❢t t❤❡ ♥♦✐s❡ t♦ t❤❡ r✐❣❤t✑✳ ❚❤❛t ✐s✿ ✭θ✶) θ✵ =✐❞❡♥t✐t② ❛♥❞ θt ◦ θs = θt+s❀ ✭θ✷✮ θtFs = Ft+s ❢♦r ❛❧❧ s, t ≥ ✵✱ ✇❤❡r❡ (Ft) ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦♥ Ω Ω❀ ✭θ✸✮ ✇(t + s, θtω) − ✇(t, θtω) = ✇(s, ω) ❢♦r ❛❧❧ s ≥ ✵

❉❡✜♥✐t✐♦♥

✭❙❡♠✐✢♦✇ ♦❢ Pr♦❝❡ss❡s✮ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✉ = ✉(t, ω) ❞❡✜♥❡ ❛ ♣r♦❝❡ss ✈ = ❙r✉ ❜② ✈(t, ω) = ✉(r + t, θrω) ❙r ✐s ❛ s❡♠✐❣r♦✉♣✱ ❛♥❞ ✐❢ ✉ ✐s ❛❞❛♣t❡❞ s♦ ✐s ❙t✉✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❳ ✐s ❝❧♦s❡❞ ✉♥❞❡r ❙t✳ ❚❤❡♥ ❛ ♥❛t✉r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❢♦r t❤❡ ❝❧❛ss ❳ ✐s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵ ✭✐✐✮ ❆ ✐s ❝♦♠♣❛❝t ✐♥ s♦♠❡ s❡♥s❡ ✭✐✐✐✮ ❆ ❛ttr❛❝ts ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ t❤❡ ❝❧❛ss ❳ ❚❤✐s t✉r♥s ♦✉t t♦ ❜❡ ❛s❦✐♥❣ t♦♦ ♠✉❝❤✳ ❲❡ ♥❡❡❞ ❛ ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✐❢ ✉ ✐s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❡♥ ▲❛✇(✉) ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✭♦♥ ♣❛t❤ s♣❛❝❡✮ ♦❢ t❤❡ ❝♦✉♣❧❡❞ ♣r♦❝❡ss (✉, ✇)✳

slide-98
SLIDE 98

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t ❢♦r ❛❧❧ t ✵✱ ✇❤❡r❡ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t ❛♥❞ ❜♦✉♥❞❡❞ ▲❛✇ ❳ ✱ ❙t ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t t✵ ✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇ ❆ ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇ ❆ ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ ❑

❝

❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

slide-99
SLIDE 99

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t ❛♥❞ ❜♦✉♥❞❡❞ ▲❛✇ ❳ ✱ ❙t ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t t✵ ✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇ ❆ ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇ ❆ ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ ❑

❝

❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 100

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t O ⊃ A ❛♥❞ ❜♦✉♥❞❡❞ Z ⊂ ▲❛✇(❳)✱ ˆ ❙tZ ⊆ O ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t ≥ t✵(O, Z)✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇ ❆ ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇ ❆ ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ ❑

❝

❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 101

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t O ⊃ A ❛♥❞ ❜♦✉♥❞❡❞ Z ⊂ ▲❛✇(❳)✱ ˆ ❙tZ ⊆ O ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t ≥ t✵(O, Z)✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ A ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇ ❆ ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇ ❆ ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ ❑

❝

❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 102

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t O ⊃ A ❛♥❞ ❜♦✉♥❞❡❞ Z ⊂ ▲❛✇(❳)✱ ˆ ❙tZ ⊆ O ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t ≥ t✵(O, Z)✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ A ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇ ❆ ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇ ❆ ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ ❑

❝

❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 103

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t O ⊃ A ❛♥❞ ❜♦✉♥❞❡❞ Z ⊂ ▲❛✇(❳)✱ ˆ ❙tZ ⊆ O ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t ≥ t✵(O, Z)✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ A ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇(❆) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇(❆) ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ ❑

❝

❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 104

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t O ⊃ A ❛♥❞ ❜♦✉♥❞❡❞ Z ⊂ ▲❛✇(❳)✱ ˆ ❙tZ ⊆ O ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t ≥ t✵(O, Z)✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ A ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇(❆) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇(❆) ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ ❑

❝

❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 105

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t O ⊃ A ❛♥❞ ❜♦✉♥❞❡❞ Z ⊂ ▲❛✇(❳)✱ ˆ ❙tZ ⊆ O ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t ≥ t✵(O, Z)✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ A ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇(❆) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇(❆) ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ⊂ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ O = (❑ ≤ε)❝ ⊃ ❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ⊆ O. ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 106

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t O ⊃ A ❛♥❞ ❜♦✉♥❞❡❞ Z ⊂ ▲❛✇(❳)✱ ˆ ❙tZ ⊆ O ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t ≥ t✵(O, Z)✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ A ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇(❆) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇(❆) ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ⊂ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ O = (❑ ≤ε)❝ ⊃ ❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ⊆ O. ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 107

❉❡✜♥✐t✐♦♥

✭❛✮ ❆ s❡t ♦❢ ❧❛✇s A ⊂ ▲❛✇(❳) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✐❢ ✭✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ˆ ❙tA = A ❢♦r ❛❧❧ t ≥ ✵✱ ✇❤❡r❡ ˆ ❙t ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ ❧❛✇s ✐♥❞✉❝❡❞ ❜② t❤❡ s❡♠✐❣r♦✉♣ ❙t✳ ✭✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❛♥② ♦♣❡♥ s❡t O ⊃ A ❛♥❞ ❜♦✉♥❞❡❞ Z ⊂ ▲❛✇(❳)✱ ˆ ❙tZ ⊆ O ❡✈❡♥t✉❛❧❧② ✭✐✳❡✳ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ t ≥ t✵(O, Z)✮✳ ✭✐✐✐✮ ✭❈♦♠♣❛❝t♥❡ss✮ A ✐s ❝♦♠♣❛❝t ✭❜✮ ❆ ✭♣r♦❝❡ss✮ ❛ttr❛❝t♦r ❢♦r t❤❡ s❡♠✐✢♦✇ ❙t ♦♥ ❳ ✐s ❛ s❡t ♦❢ ♣r♦❝❡ss❡s ❆ ⊆ ❳ s✉❝❤ t❤❛t ✭✐✮ ▲❛✇(❆) ✐s ❛ ❧❛✇✲❛ttr❛❝t♦r ✭s♦ ▲❛✇(❆) ✐s ❝♦♠♣❛❝t ❛♥❞ ❆ ✐s ❜♦✉♥❞❡❞✮❀ ✭✐✐✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵❀ ✭✐✐✐✮ ✭❆ttr❛❝t✐♦♥✮ ❋♦r ❜♦✉♥❞❡❞ ❩ ⊂ ❳ ❛♥❞ ❝♦♠♣❛❝t ❑ ✇✐t❤ O = (❑ ≤ε)❝ ⊃ ❆✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ t ❙t❩ ⊆ O. ✭✐✈✮ ❆ ✐s ❝❧♦s❡❞✳ ❊✈❡♥ ❢♦r t❤✐s ✇❡❛❦❡r ❞❡✜♥✐t✐♦♥✱ ❡①✐st❡♥❝❡ r❡q✉✐r❡s ❛ r❛t❤❡r ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

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SLIDE 108

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱✷✵✵✹✮❚❤❡r❡ ✐s ❛ ▲♦❡❜ s♣❛❝❡ Ω Ω ❛♥❞ ❛ ♥❛t✉r❛❧ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ t❤❛t ❤❛s ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❆✳ ❚❤❡ ❝❧❛ss ❳ ❝♦♥t❛✐♥s s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ❢♦r ❛❧❧ ▲✷ r❛♥❞♦♠ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❆ ✐s t❤❡ r❡str✐❝t✐♦♥ t♦ ♥♦♥♥❡❣❛t✐✈❡ t✐♠❡s ♦❢ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛ss ˆ ❳ ♦❢ t✇♦ s✐❞❡❞ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡✳ ❘❡♠❛r❦ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ✐❢ ✐s ❛♥② s✉✣❝✐❡♥t❧② r✐❝❤ s♣❛❝❡ ✭❢♦r ❡①❛♠♣❧❡ ✐❢ ✐s ❛ ▲♦❡❜ s♣❛❝❡✮ t❤❡♥ ❛♥② ♣r♦❝❡ss ❛ttr❛❝t♦r ❆ ✐s ♥♦t ❝♦♠♣❛❝t✳

slide-109
SLIDE 109

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱✷✵✵✹✮❚❤❡r❡ ✐s ❛ ▲♦❡❜ s♣❛❝❡ Ω Ω ❛♥❞ ❛ ♥❛t✉r❛❧ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ t❤❛t ❤❛s ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ❆✳ ❚❤❡ ❝❧❛ss ❳ ❝♦♥t❛✐♥s s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡ ❢♦r ❛❧❧ ▲✷ r❛♥❞♦♠ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❆ ✐s t❤❡ r❡str✐❝t✐♦♥ t♦ ♥♦♥♥❡❣❛t✐✈❡ t✐♠❡s ♦❢ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛ss ˆ ❳ ♦❢ t✇♦ s✐❞❡❞ s♦❧✉t✐♦♥s t♦ t❤❡ s◆❙❡✳ ❘❡♠❛r❦ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ✐❢ Ω Ω ✐s ❛♥② s✉✣❝✐❡♥t❧② r✐❝❤ s♣❛❝❡ ✭❢♦r ❡①❛♠♣❧❡ ✐❢ Ω Ω ✐s ❛ ▲♦❡❜ s♣❛❝❡✮ t❤❡♥ ❛♥② ♣r♦❝❡ss ❛ttr❛❝t♦r ❆ ✐s ♥♦t ❝♦♠♣❛❝t✳

slide-110
SLIDE 110

◆❊❖✲❆❚❚❘❆❈❚❖❘❙ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ✐s s♦♠❡✇❤❛t ✉♥s❛t✐s❢❛❝t♦r② ❛♥❞ ❞♦❡s ♥♦t ❝❛♣t✉r❡ t❤❡ ❢✉❧❧ str❡♥❣t❤ ♦❢ ✇❤❛t ✇❛s ♣r♦✈❡❞ ❛❜♦✈❡✳ ❚❤✐s ✐s ❢✉❧❧② ❝❛♣t✉r❡❞ ❜② t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡♦✲❛ttr❛❝t♦r✳ ❑❡✐s❧❡r✬s t❤❡♦r② ♦❢ ♥❡♦✲♠❡tr✐❝ s♣❛❝❡s ✐♥✈♦❧✈❡s ♥♦t✐♦♥s ♦❢ ♥❡♦✲♦♣❡♥✱ ♥❡♦✲❝♦♠♣❛❝t ❛♥❞ ♥❡♦✲❝♦♥t✐♥✉♦✉s✳ ❚❤❡s❡ ❛r✐s❡ ✈❡r② ♥❛t✉r❛❧❧② ❢♦r ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s ✲ s✉❝❤ ❛s t❤❡ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ◆❡♦✲❝♦♠♣❛❝t ✐s ✇❡❛❦❡r t❤❛♥ ❝♦♠♣❛❝t ✐♥ ❣❡♥❡r❛❧❀ ♥❡♦✲♦♣❡♥ ✐s ✇❡❛❦❡r t❤❛♥ ♦♣❡♥✱ ❜✉t ♥❡♦✲❝♦♥t✐♥✉♦✉s ✐s str♦♥❣❡r t❤❛♥ ❝♦♥t✐♥✉♦✉s✳

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱ ✷✵✵✺✮ ❚❤❡ ❛ttr❛❝t♦r ❆ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✐s ❛ ♥❡♦✲❛ttr❛❝t♦r❀ t❤❛t ✐s ✭✶✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✷✮ ❆ ✐s ♥❡♦❝♦♠♣❛❝t❀ ✭✸✮ ❢♦r ❛♥② ♥❡♦✲♦♣❡♥ s❡t ● ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ❳ ❡✈❡♥t✉❛❧❧② ❙t❇

  • ✳
slide-111
SLIDE 111

◆❊❖✲❆❚❚❘❆❈❚❖❘❙ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ✐s s♦♠❡✇❤❛t ✉♥s❛t✐s❢❛❝t♦r② ❛♥❞ ❞♦❡s ♥♦t ❝❛♣t✉r❡ t❤❡ ❢✉❧❧ str❡♥❣t❤ ♦❢ ✇❤❛t ✇❛s ♣r♦✈❡❞ ❛❜♦✈❡✳ ❚❤✐s ✐s ❢✉❧❧② ❝❛♣t✉r❡❞ ❜② t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡♦✲❛ttr❛❝t♦r✳ ❑❡✐s❧❡r✬s t❤❡♦r② ♦❢ ♥❡♦✲♠❡tr✐❝ s♣❛❝❡s ✐♥✈♦❧✈❡s ♥♦t✐♦♥s ♦❢ ♥❡♦✲♦♣❡♥✱ ♥❡♦✲❝♦♠♣❛❝t ❛♥❞ ♥❡♦✲❝♦♥t✐♥✉♦✉s✳ ❚❤❡s❡ ❛r✐s❡ ✈❡r② ♥❛t✉r❛❧❧② ❢♦r ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s ✲ s✉❝❤ ❛s t❤❡ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ◆❡♦✲❝♦♠♣❛❝t ✐s ✇❡❛❦❡r t❤❛♥ ❝♦♠♣❛❝t ✐♥ ❣❡♥❡r❛❧❀ ♥❡♦✲♦♣❡♥ ✐s ✇❡❛❦❡r t❤❛♥ ♦♣❡♥✱ ❜✉t ♥❡♦✲❝♦♥t✐♥✉♦✉s ✐s str♦♥❣❡r t❤❛♥ ❝♦♥t✐♥✉♦✉s✳

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱ ✷✵✵✺✮ ❚❤❡ ❛ttr❛❝t♦r ❆ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✐s ❛ ♥❡♦✲❛ttr❛❝t♦r❀ t❤❛t ✐s ✭✶✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✷✮ ❆ ✐s ♥❡♦❝♦♠♣❛❝t❀ ✭✸✮ ❢♦r ❛♥② ♥❡♦✲♦♣❡♥ s❡t ● ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ❳ ❡✈❡♥t✉❛❧❧② ❙t❇

  • ✳
slide-112
SLIDE 112

◆❊❖✲❆❚❚❘❆❈❚❖❘❙ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ✐s s♦♠❡✇❤❛t ✉♥s❛t✐s❢❛❝t♦r② ❛♥❞ ❞♦❡s ♥♦t ❝❛♣t✉r❡ t❤❡ ❢✉❧❧ str❡♥❣t❤ ♦❢ ✇❤❛t ✇❛s ♣r♦✈❡❞ ❛❜♦✈❡✳ ❚❤✐s ✐s ❢✉❧❧② ❝❛♣t✉r❡❞ ❜② t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡♦✲❛ttr❛❝t♦r✳ ❑❡✐s❧❡r✬s t❤❡♦r② ♦❢ ♥❡♦✲♠❡tr✐❝ s♣❛❝❡s ✐♥✈♦❧✈❡s ♥♦t✐♦♥s ♦❢ ♥❡♦✲♦♣❡♥✱ ♥❡♦✲❝♦♠♣❛❝t ❛♥❞ ♥❡♦✲❝♦♥t✐♥✉♦✉s✳ ❚❤❡s❡ ❛r✐s❡ ✈❡r② ♥❛t✉r❛❧❧② ❢♦r ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s ✲ s✉❝❤ ❛s t❤❡ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ◆❡♦✲❝♦♠♣❛❝t ✐s ✇❡❛❦❡r t❤❛♥ ❝♦♠♣❛❝t ✐♥ ❣❡♥❡r❛❧❀ ♥❡♦✲♦♣❡♥ ✐s ✇❡❛❦❡r t❤❛♥ ♦♣❡♥✱ ❜✉t ♥❡♦✲❝♦♥t✐♥✉♦✉s ✐s str♦♥❣❡r t❤❛♥ ❝♦♥t✐♥✉♦✉s✳

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱ ✷✵✵✺✮ ❚❤❡ ❛ttr❛❝t♦r ❆ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✐s ❛ ♥❡♦✲❛ttr❛❝t♦r❀ t❤❛t ✐s ✭✶✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✷✮ ❆ ✐s ♥❡♦❝♦♠♣❛❝t❀ ✭✸✮ ❢♦r ❛♥② ♥❡♦✲♦♣❡♥ s❡t ● ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ❳ ❡✈❡♥t✉❛❧❧② ❙t❇

  • ✳
slide-113
SLIDE 113

◆❊❖✲❆❚❚❘❆❈❚❖❘❙ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ✐s s♦♠❡✇❤❛t ✉♥s❛t✐s❢❛❝t♦r② ❛♥❞ ❞♦❡s ♥♦t ❝❛♣t✉r❡ t❤❡ ❢✉❧❧ str❡♥❣t❤ ♦❢ ✇❤❛t ✇❛s ♣r♦✈❡❞ ❛❜♦✈❡✳ ❚❤✐s ✐s ❢✉❧❧② ❝❛♣t✉r❡❞ ❜② t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡♦✲❛ttr❛❝t♦r✳ ❑❡✐s❧❡r✬s t❤❡♦r② ♦❢ ♥❡♦✲♠❡tr✐❝ s♣❛❝❡s ✐♥✈♦❧✈❡s ♥♦t✐♦♥s ♦❢ ♥❡♦✲♦♣❡♥✱ ♥❡♦✲❝♦♠♣❛❝t ❛♥❞ ♥❡♦✲❝♦♥t✐♥✉♦✉s✳ ❚❤❡s❡ ❛r✐s❡ ✈❡r② ♥❛t✉r❛❧❧② ❢♦r ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s ✲ s✉❝❤ ❛s t❤❡ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ◆❡♦✲❝♦♠♣❛❝t ✐s ✇❡❛❦❡r t❤❛♥ ❝♦♠♣❛❝t ✐♥ ❣❡♥❡r❛❧❀ ♥❡♦✲♦♣❡♥ ✐s ✇❡❛❦❡r t❤❛♥ ♦♣❡♥✱ ❜✉t ♥❡♦✲❝♦♥t✐♥✉♦✉s ✐s str♦♥❣❡r t❤❛♥ ❝♦♥t✐♥✉♦✉s✳

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱ ✷✵✵✺✮ ❚❤❡ ❛ttr❛❝t♦r ❆ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✐s ❛ ♥❡♦✲❛ttr❛❝t♦r❀ t❤❛t ✐s ✭✶✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✷✮ ❆ ✐s ♥❡♦❝♦♠♣❛❝t❀ ✭✸✮ ❢♦r ❛♥② ♥❡♦✲♦♣❡♥ s❡t ● ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ❳ ❡✈❡♥t✉❛❧❧② ❙t❇

  • ✳
slide-114
SLIDE 114

◆❊❖✲❆❚❚❘❆❈❚❖❘❙ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ✐s s♦♠❡✇❤❛t ✉♥s❛t✐s❢❛❝t♦r② ❛♥❞ ❞♦❡s ♥♦t ❝❛♣t✉r❡ t❤❡ ❢✉❧❧ str❡♥❣t❤ ♦❢ ✇❤❛t ✇❛s ♣r♦✈❡❞ ❛❜♦✈❡✳ ❚❤✐s ✐s ❢✉❧❧② ❝❛♣t✉r❡❞ ❜② t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡♦✲❛ttr❛❝t♦r✳ ❑❡✐s❧❡r✬s t❤❡♦r② ♦❢ ♥❡♦✲♠❡tr✐❝ s♣❛❝❡s ✐♥✈♦❧✈❡s ♥♦t✐♦♥s ♦❢ ♥❡♦✲♦♣❡♥✱ ♥❡♦✲❝♦♠♣❛❝t ❛♥❞ ♥❡♦✲❝♦♥t✐♥✉♦✉s✳ ❚❤❡s❡ ❛r✐s❡ ✈❡r② ♥❛t✉r❛❧❧② ❢♦r ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s ✲ s✉❝❤ ❛s t❤❡ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ◆❡♦✲❝♦♠♣❛❝t ✐s ✇❡❛❦❡r t❤❛♥ ❝♦♠♣❛❝t ✐♥ ❣❡♥❡r❛❧❀ ♥❡♦✲♦♣❡♥ ✐s ✇❡❛❦❡r t❤❛♥ ♦♣❡♥✱ ❜✉t ♥❡♦✲❝♦♥t✐♥✉♦✉s ✐s str♦♥❣❡r t❤❛♥ ❝♦♥t✐♥✉♦✉s✳

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱ ✷✵✵✺✮ ❚❤❡ ❛ttr❛❝t♦r ❆ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✐s ❛ ♥❡♦✲❛ttr❛❝t♦r❀ t❤❛t ✐s ✭✶✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ ❆ ❢♦r ❛❧❧ t ✵❀ ✭✷✮ ❆ ✐s ♥❡♦❝♦♠♣❛❝t❀ ✭✸✮ ❢♦r ❛♥② ♥❡♦✲♦♣❡♥ s❡t ● ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ❳ ❡✈❡♥t✉❛❧❧② ❙t❇

  • ✳
slide-115
SLIDE 115

◆❊❖✲❆❚❚❘❆❈❚❖❘❙ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ✐s s♦♠❡✇❤❛t ✉♥s❛t✐s❢❛❝t♦r② ❛♥❞ ❞♦❡s ♥♦t ❝❛♣t✉r❡ t❤❡ ❢✉❧❧ str❡♥❣t❤ ♦❢ ✇❤❛t ✇❛s ♣r♦✈❡❞ ❛❜♦✈❡✳ ❚❤✐s ✐s ❢✉❧❧② ❝❛♣t✉r❡❞ ❜② t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡♦✲❛ttr❛❝t♦r✳ ❑❡✐s❧❡r✬s t❤❡♦r② ♦❢ ♥❡♦✲♠❡tr✐❝ s♣❛❝❡s ✐♥✈♦❧✈❡s ♥♦t✐♦♥s ♦❢ ♥❡♦✲♦♣❡♥✱ ♥❡♦✲❝♦♠♣❛❝t ❛♥❞ ♥❡♦✲❝♦♥t✐♥✉♦✉s✳ ❚❤❡s❡ ❛r✐s❡ ✈❡r② ♥❛t✉r❛❧❧② ❢♦r ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s ✲ s✉❝❤ ❛s t❤❡ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ◆❡♦✲❝♦♠♣❛❝t ✐s ✇❡❛❦❡r t❤❛♥ ❝♦♠♣❛❝t ✐♥ ❣❡♥❡r❛❧❀ ♥❡♦✲♦♣❡♥ ✐s ✇❡❛❦❡r t❤❛♥ ♦♣❡♥✱ ❜✉t ♥❡♦✲❝♦♥t✐♥✉♦✉s ✐s str♦♥❣❡r t❤❛♥ ❝♦♥t✐♥✉♦✉s✳

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱ ✷✵✵✺✮ ❚❤❡ ❛ttr❛❝t♦r ❆ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✐s ❛ ♥❡♦✲❛ttr❛❝t♦r❀ t❤❛t ✐s ✭✶✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵❀ ✭✷✮ ❆ ✐s ♥❡♦❝♦♠♣❛❝t❀ ✭✸✮ ❢♦r ❛♥② ♥❡♦✲♦♣❡♥ s❡t ● ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ❳ ❡✈❡♥t✉❛❧❧② ❙t❇

  • ✳
slide-116
SLIDE 116

◆❊❖✲❆❚❚❘❆❈❚❖❘❙ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ✐s s♦♠❡✇❤❛t ✉♥s❛t✐s❢❛❝t♦r② ❛♥❞ ❞♦❡s ♥♦t ❝❛♣t✉r❡ t❤❡ ❢✉❧❧ str❡♥❣t❤ ♦❢ ✇❤❛t ✇❛s ♣r♦✈❡❞ ❛❜♦✈❡✳ ❚❤✐s ✐s ❢✉❧❧② ❝❛♣t✉r❡❞ ❜② t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡♦✲❛ttr❛❝t♦r✳ ❑❡✐s❧❡r✬s t❤❡♦r② ♦❢ ♥❡♦✲♠❡tr✐❝ s♣❛❝❡s ✐♥✈♦❧✈❡s ♥♦t✐♦♥s ♦❢ ♥❡♦✲♦♣❡♥✱ ♥❡♦✲❝♦♠♣❛❝t ❛♥❞ ♥❡♦✲❝♦♥t✐♥✉♦✉s✳ ❚❤❡s❡ ❛r✐s❡ ✈❡r② ♥❛t✉r❛❧❧② ❢♦r ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s ✲ s✉❝❤ ❛s t❤❡ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ◆❡♦✲❝♦♠♣❛❝t ✐s ✇❡❛❦❡r t❤❛♥ ❝♦♠♣❛❝t ✐♥ ❣❡♥❡r❛❧❀ ♥❡♦✲♦♣❡♥ ✐s ✇❡❛❦❡r t❤❛♥ ♦♣❡♥✱ ❜✉t ♥❡♦✲❝♦♥t✐♥✉♦✉s ✐s str♦♥❣❡r t❤❛♥ ❝♦♥t✐♥✉♦✉s✳

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱ ✷✵✵✺✮ ❚❤❡ ❛ttr❛❝t♦r ❆ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✐s ❛ ♥❡♦✲❛ttr❛❝t♦r❀ t❤❛t ✐s ✭✶✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵❀ ✭✷✮ ❆ ✐s ♥❡♦❝♦♠♣❛❝t❀ ✭✸✮ ❢♦r ❛♥② ♥❡♦✲♦♣❡♥ s❡t ● ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ❳ ❡✈❡♥t✉❛❧❧② ❙t❇

  • ✳
slide-117
SLIDE 117

◆❊❖✲❆❚❚❘❆❈❚❖❘❙ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣r♦❝❡ss ❛ttr❛❝t♦r ✐s s♦♠❡✇❤❛t ✉♥s❛t✐s❢❛❝t♦r② ❛♥❞ ❞♦❡s ♥♦t ❝❛♣t✉r❡ t❤❡ ❢✉❧❧ str❡♥❣t❤ ♦❢ ✇❤❛t ✇❛s ♣r♦✈❡❞ ❛❜♦✈❡✳ ❚❤✐s ✐s ❢✉❧❧② ❝❛♣t✉r❡❞ ❜② t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡♦✲❛ttr❛❝t♦r✳ ❑❡✐s❧❡r✬s t❤❡♦r② ♦❢ ♥❡♦✲♠❡tr✐❝ s♣❛❝❡s ✐♥✈♦❧✈❡s ♥♦t✐♦♥s ♦❢ ♥❡♦✲♦♣❡♥✱ ♥❡♦✲❝♦♠♣❛❝t ❛♥❞ ♥❡♦✲❝♦♥t✐♥✉♦✉s✳ ❚❤❡s❡ ❛r✐s❡ ✈❡r② ♥❛t✉r❛❧❧② ❢♦r ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ♥♦♥st❛♥❞❛r❞ ❛♥❛❧②s✐s ✲ s✉❝❤ ❛s t❤❡ ❝❧❛ss ♦❢ s♦❧✉t✐♦♥s ❳ ✐♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ◆❡♦✲❝♦♠♣❛❝t ✐s ✇❡❛❦❡r t❤❛♥ ❝♦♠♣❛❝t ✐♥ ❣❡♥❡r❛❧❀ ♥❡♦✲♦♣❡♥ ✐s ✇❡❛❦❡r t❤❛♥ ♦♣❡♥✱ ❜✉t ♥❡♦✲❝♦♥t✐♥✉♦✉s ✐s str♦♥❣❡r t❤❛♥ ❝♦♥t✐♥✉♦✉s✳

❚❤❡♦r❡♠

✭◆❏❈ ✫ ❍✳❏✳❑❡✐s❧❡r✱ ✷✵✵✺✮ ❚❤❡ ❛ttr❛❝t♦r ❆ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✐s ❛ ♥❡♦✲❛ttr❛❝t♦r❀ t❤❛t ✐s ✭✶✮ ✭■♥✈❛r✐❛♥❝❡✮ ❙t❆ = ❆ ❢♦r ❛❧❧ t ≥ ✵❀ ✭✷✮ ❆ ✐s ♥❡♦❝♦♠♣❛❝t❀ ✭✸✮ ❢♦r ❛♥② ♥❡♦✲♦♣❡♥ s❡t ● ⊃ ❆ ❛♥❞ ❜♦✉♥❞❡❞ s❡t ❇ ⊂ ❳, ❡✈❡♥t✉❛❧❧② ❙t❇ ⊆ ● ✳

slide-118
SLIDE 118

❆♣♣❧✐❝❛t✐♦♥ ✷✿ ❖P❚■▼❆▲ ❈❖◆❚❘❖▲ ❚❍❊❖❘❨ ✭◆❏❈ ✫ ❑❛t❛r③②♥❛

  • r③❡s✐❛❦✮

❯s✐♥❣ ◆❙❆ ❢♦r ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥tr♦❧s

♥

✵ ❚ ▼ ✭▼ ❛ ♠❡tr✐❝ s♣❛❝❡✮ ❢♦r ❛ ❣✐✈❡♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ s❛② ✳ ❚❤❛t ✐s ❈♦st ♦❢ ✉s✐♥❣ ❝♦♥tr♦❧

♥

❏

♥

❏✵ ✇❤❡r❡ ❏✵ ✐s t❤❡ ✐♥✜♠✉♠ ♦❢ ❛❧❧ ❝♦sts ❢♦r ❝♦♥tr♦❧s ❢♦r t❤❡ ❣✐✈❡♥ s②st❡♠✳ ❚❤❡♥ ◆❙❆ ❛❧❧♦✇s ✉s t♦ s♣❡❛❦ ♦❢ t❤❡ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧

◆ ❢♦r ❛♥② ✐♥✜♥✐t❡ ◆✳

❲❡ ❝❛♥ ✉s✉❛❧❧② ♠❛❦❡ s❡♥s❡ ♦❢ ❏

◆

❛♥❞ ✇❡ ✇✐❧❧ ❤❛✈❡ ❏

◆

❏✵ ■♥ ♠❛♥② ❝✐r❝✉♠st❛♥❝❡s ✇❡ ❝❛♥ t❤❡♥ ✏t❛❦❡ st❛♥❞❛r❞ ♣❛rts✑ t♦ ♣r♦❞✉❝❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧

◆

❚❤✐s ✐❞❡❛ ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ t♦ t❤❡ s◆❙❡ ✐♥ ❛ ✈❛r✐❡t② ♦❢ s❡tt✐♥❣s✱ ❛❧✇❛②s ✐♥✈♦❧✈✐♥❣ ❛ ▲♦❡❜ s♣❛❝❡ s♦ t❤❛t s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ❝♦♥tr♦❧s ❧✐✈❡ ♦♥ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❘❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r ✷❉ s②st❡♠s ♦❢ t❤❡ ❢♦r♠ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ✉ ❞s

t ✵

❣ s ✉ ❞✇ s ✇✐t❤ ❍ö❧❞❡r ❝♦♥t✐♥✉♦✉s✱ ♦r ✇✐t❤ ❤❛✈✐♥❣ ♥♦ ❢❡❡❞❜❛❝❦ ✐♥ ✉ ♦r ✇✐t❤ t❤❡ ❢❡❡❞❜❛❝❦ ❝♦♥s✐st✐♥❣ ♦❢ ❝✉♠✉❧❛t✐✈❡ ❞✐❣✐t❛❧ ♦❜s❡r✈❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛t ❛ ✜①❡❞ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

slide-119
SLIDE 119

❆♣♣❧✐❝❛t✐♦♥ ✷✿ ❖P❚■▼❆▲ ❈❖◆❚❘❖▲ ❚❍❊❖❘❨ ✭◆❏❈ ✫ ❑❛t❛r③②♥❛

  • r③❡s✐❛❦✮

❯s✐♥❣ ◆❙❆ ❢♦r ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥tr♦❧s θ♥ : [✵, ❚] → ▼ ✭▼ ❛ ♠❡tr✐❝ s♣❛❝❡✮ ❢♦r ❛ ❣✐✈❡♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ s❛② ✳ ❚❤❛t ✐s ❈♦st ♦❢ ✉s✐♥❣ ❝♦♥tr♦❧ θ♥ = ❏(θ♥) ց ❏✵ ✇❤❡r❡ ❏✵ ✐s t❤❡ ✐♥✜♠✉♠ ♦❢ ❛❧❧ ❝♦sts ❢♦r ❝♦♥tr♦❧s ❢♦r t❤❡ ❣✐✈❡♥ s②st❡♠✳ ❚❤❡♥ ◆❙❆ ❛❧❧♦✇s ✉s t♦ s♣❡❛❦ ♦❢ t❤❡ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧

◆ ❢♦r ❛♥② ✐♥✜♥✐t❡ ◆✳

❲❡ ❝❛♥ ✉s✉❛❧❧② ♠❛❦❡ s❡♥s❡ ♦❢ ❏

◆

❛♥❞ ✇❡ ✇✐❧❧ ❤❛✈❡ ❏

◆

❏✵ ■♥ ♠❛♥② ❝✐r❝✉♠st❛♥❝❡s ✇❡ ❝❛♥ t❤❡♥ ✏t❛❦❡ st❛♥❞❛r❞ ♣❛rts✑ t♦ ♣r♦❞✉❝❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧

◆

❚❤✐s ✐❞❡❛ ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ t♦ t❤❡ s◆❙❡ ✐♥ ❛ ✈❛r✐❡t② ♦❢ s❡tt✐♥❣s✱ ❛❧✇❛②s ✐♥✈♦❧✈✐♥❣ ❛ ▲♦❡❜ s♣❛❝❡ s♦ t❤❛t s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ❝♦♥tr♦❧s ❧✐✈❡ ♦♥ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❘❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r ✷❉ s②st❡♠s ♦❢ t❤❡ ❢♦r♠ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ✉ ❞s

t ✵

❣ s ✉ ❞✇ s ✇✐t❤ ❍ö❧❞❡r ❝♦♥t✐♥✉♦✉s✱ ♦r ✇✐t❤ ❤❛✈✐♥❣ ♥♦ ❢❡❡❞❜❛❝❦ ✐♥ ✉ ♦r ✇✐t❤ t❤❡ ❢❡❡❞❜❛❝❦ ❝♦♥s✐st✐♥❣ ♦❢ ❝✉♠✉❧❛t✐✈❡ ❞✐❣✐t❛❧ ♦❜s❡r✈❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛t ❛ ✜①❡❞ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

slide-120
SLIDE 120

❆♣♣❧✐❝❛t✐♦♥ ✷✿ ❖P❚■▼❆▲ ❈❖◆❚❘❖▲ ❚❍❊❖❘❨ ✭◆❏❈ ✫ ❑❛t❛r③②♥❛

  • r③❡s✐❛❦✮

❯s✐♥❣ ◆❙❆ ❢♦r ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥tr♦❧s θ♥ : [✵, ❚] → ▼ ✭▼ ❛ ♠❡tr✐❝ s♣❛❝❡✮ ❢♦r ❛ ❣✐✈❡♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ s❛② ✳ ❚❤❛t ✐s ❈♦st ♦❢ ✉s✐♥❣ ❝♦♥tr♦❧ θ♥ = ❏(θ♥) ց ❏✵ ✇❤❡r❡ ❏✵ ✐s t❤❡ ✐♥✜♠✉♠ ♦❢ ❛❧❧ ❝♦sts ❢♦r ❝♦♥tr♦❧s ❢♦r t❤❡ ❣✐✈❡♥ s②st❡♠✳ ❚❤❡♥ ◆❙❆ ❛❧❧♦✇s ✉s t♦ s♣❡❛❦ ♦❢ t❤❡ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧ θ◆ ❢♦r ❛♥② ✐♥✜♥✐t❡ ◆✳ ❲❡ ❝❛♥ ✉s✉❛❧❧② ♠❛❦❡ s❡♥s❡ ♦❢ ❏(θ◆) ❛♥❞ ✇❡ ✇✐❧❧ ❤❛✈❡ ❏(θ◆) ≈ ❏✵. ■♥ ♠❛♥② ❝✐r❝✉♠st❛♥❝❡s ✇❡ ❝❛♥ t❤❡♥ ✏t❛❦❡ st❛♥❞❛r❞ ♣❛rts✑ t♦ ♣r♦❞✉❝❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧

◆

❚❤✐s ✐❞❡❛ ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ t♦ t❤❡ s◆❙❡ ✐♥ ❛ ✈❛r✐❡t② ♦❢ s❡tt✐♥❣s✱ ❛❧✇❛②s ✐♥✈♦❧✈✐♥❣ ❛ ▲♦❡❜ s♣❛❝❡ s♦ t❤❛t s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ❝♦♥tr♦❧s ❧✐✈❡ ♦♥ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❘❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r ✷❉ s②st❡♠s ♦❢ t❤❡ ❢♦r♠ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ✉ ❞s

t ✵

❣ s ✉ ❞✇ s ✇✐t❤ ❍ö❧❞❡r ❝♦♥t✐♥✉♦✉s✱ ♦r ✇✐t❤ ❤❛✈✐♥❣ ♥♦ ❢❡❡❞❜❛❝❦ ✐♥ ✉ ♦r ✇✐t❤ t❤❡ ❢❡❡❞❜❛❝❦ ❝♦♥s✐st✐♥❣ ♦❢ ❝✉♠✉❧❛t✐✈❡ ❞✐❣✐t❛❧ ♦❜s❡r✈❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛t ❛ ✜①❡❞ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

slide-121
SLIDE 121

❆♣♣❧✐❝❛t✐♦♥ ✷✿ ❖P❚■▼❆▲ ❈❖◆❚❘❖▲ ❚❍❊❖❘❨ ✭◆❏❈ ✫ ❑❛t❛r③②♥❛

  • r③❡s✐❛❦✮

❯s✐♥❣ ◆❙❆ ❢♦r ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥tr♦❧s θ♥ : [✵, ❚] → ▼ ✭▼ ❛ ♠❡tr✐❝ s♣❛❝❡✮ ❢♦r ❛ ❣✐✈❡♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ s❛② ✳ ❚❤❛t ✐s ❈♦st ♦❢ ✉s✐♥❣ ❝♦♥tr♦❧ θ♥ = ❏(θ♥) ց ❏✵ ✇❤❡r❡ ❏✵ ✐s t❤❡ ✐♥✜♠✉♠ ♦❢ ❛❧❧ ❝♦sts ❢♦r ❝♦♥tr♦❧s ❢♦r t❤❡ ❣✐✈❡♥ s②st❡♠✳ ❚❤❡♥ ◆❙❆ ❛❧❧♦✇s ✉s t♦ s♣❡❛❦ ♦❢ t❤❡ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧ θ◆ ❢♦r ❛♥② ✐♥✜♥✐t❡ ◆✳ ❲❡ ❝❛♥ ✉s✉❛❧❧② ♠❛❦❡ s❡♥s❡ ♦❢ ❏(θ◆) ❛♥❞ ✇❡ ✇✐❧❧ ❤❛✈❡ ❏(θ◆) ≈ ❏✵. ■♥ ♠❛♥② ❝✐r❝✉♠st❛♥❝❡s ✇❡ ❝❛♥ t❤❡♥ ✏t❛❦❡ st❛♥❞❛r❞ ♣❛rts✑ t♦ ♣r♦❞✉❝❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ θ = ◦θ◆ ❚❤✐s ✐❞❡❛ ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ t♦ t❤❡ s◆❙❡ ✐♥ ❛ ✈❛r✐❡t② ♦❢ s❡tt✐♥❣s✱ ❛❧✇❛②s ✐♥✈♦❧✈✐♥❣ ❛ ▲♦❡❜ s♣❛❝❡ s♦ t❤❛t s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ❝♦♥tr♦❧s ❧✐✈❡ ♦♥ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❘❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r ✷❉ s②st❡♠s ♦❢ t❤❡ ❢♦r♠ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ✉ ❞s

t ✵

❣ s ✉ ❞✇ s ✇✐t❤ ❍ö❧❞❡r ❝♦♥t✐♥✉♦✉s✱ ♦r ✇✐t❤ ❤❛✈✐♥❣ ♥♦ ❢❡❡❞❜❛❝❦ ✐♥ ✉ ♦r ✇✐t❤ t❤❡ ❢❡❡❞❜❛❝❦ ❝♦♥s✐st✐♥❣ ♦❢ ❝✉♠✉❧❛t✐✈❡ ❞✐❣✐t❛❧ ♦❜s❡r✈❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛t ❛ ✜①❡❞ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

slide-122
SLIDE 122

❆♣♣❧✐❝❛t✐♦♥ ✷✿ ❖P❚■▼❆▲ ❈❖◆❚❘❖▲ ❚❍❊❖❘❨ ✭◆❏❈ ✫ ❑❛t❛r③②♥❛

  • r③❡s✐❛❦✮

❯s✐♥❣ ◆❙❆ ❢♦r ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥tr♦❧s θ♥ : [✵, ❚] → ▼ ✭▼ ❛ ♠❡tr✐❝ s♣❛❝❡✮ ❢♦r ❛ ❣✐✈❡♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ s❛② ✳ ❚❤❛t ✐s ❈♦st ♦❢ ✉s✐♥❣ ❝♦♥tr♦❧ θ♥ = ❏(θ♥) ց ❏✵ ✇❤❡r❡ ❏✵ ✐s t❤❡ ✐♥✜♠✉♠ ♦❢ ❛❧❧ ❝♦sts ❢♦r ❝♦♥tr♦❧s ❢♦r t❤❡ ❣✐✈❡♥ s②st❡♠✳ ❚❤❡♥ ◆❙❆ ❛❧❧♦✇s ✉s t♦ s♣❡❛❦ ♦❢ t❤❡ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧ θ◆ ❢♦r ❛♥② ✐♥✜♥✐t❡ ◆✳ ❲❡ ❝❛♥ ✉s✉❛❧❧② ♠❛❦❡ s❡♥s❡ ♦❢ ❏(θ◆) ❛♥❞ ✇❡ ✇✐❧❧ ❤❛✈❡ ❏(θ◆) ≈ ❏✵. ■♥ ♠❛♥② ❝✐r❝✉♠st❛♥❝❡s ✇❡ ❝❛♥ t❤❡♥ ✏t❛❦❡ st❛♥❞❛r❞ ♣❛rts✑ t♦ ♣r♦❞✉❝❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ θ = ◦θ◆ ❚❤✐s ✐❞❡❛ ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ t♦ t❤❡ s◆❙❡ ✐♥ ❛ ✈❛r✐❡t② ♦❢ s❡tt✐♥❣s✱ ❛❧✇❛②s ✐♥✈♦❧✈✐♥❣ ❛ ▲♦❡❜ s♣❛❝❡ s♦ t❤❛t s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ❝♦♥tr♦❧s ❧✐✈❡ ♦♥ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❘❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r ✷❉ s②st❡♠s ♦❢ t❤❡ ❢♦r♠ ✉ t ✉✵

t ✵

❆✉ s ❇ ✉ s ❢ s ✉ s ✉ ❞s

t ✵

❣ s ✉ ❞✇ s ✇✐t❤ ❍ö❧❞❡r ❝♦♥t✐♥✉♦✉s✱ ♦r ✇✐t❤ ❤❛✈✐♥❣ ♥♦ ❢❡❡❞❜❛❝❦ ✐♥ ✉ ♦r ✇✐t❤ t❤❡ ❢❡❡❞❜❛❝❦ ❝♦♥s✐st✐♥❣ ♦❢ ❝✉♠✉❧❛t✐✈❡ ❞✐❣✐t❛❧ ♦❜s❡r✈❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛t ❛ ✜①❡❞ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

slide-123
SLIDE 123

❆♣♣❧✐❝❛t✐♦♥ ✷✿ ❖P❚■▼❆▲ ❈❖◆❚❘❖▲ ❚❍❊❖❘❨ ✭◆❏❈ ✫ ❑❛t❛r③②♥❛

  • r③❡s✐❛❦✮

❯s✐♥❣ ◆❙❆ ❢♦r ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥tr♦❧s θ♥ : [✵, ❚] → ▼ ✭▼ ❛ ♠❡tr✐❝ s♣❛❝❡✮ ❢♦r ❛ ❣✐✈❡♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ s❛② ✳ ❚❤❛t ✐s ❈♦st ♦❢ ✉s✐♥❣ ❝♦♥tr♦❧ θ♥ = ❏(θ♥) ց ❏✵ ✇❤❡r❡ ❏✵ ✐s t❤❡ ✐♥✜♠✉♠ ♦❢ ❛❧❧ ❝♦sts ❢♦r ❝♦♥tr♦❧s ❢♦r t❤❡ ❣✐✈❡♥ s②st❡♠✳ ❚❤❡♥ ◆❙❆ ❛❧❧♦✇s ✉s t♦ s♣❡❛❦ ♦❢ t❤❡ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧ θ◆ ❢♦r ❛♥② ✐♥✜♥✐t❡ ◆✳ ❲❡ ❝❛♥ ✉s✉❛❧❧② ♠❛❦❡ s❡♥s❡ ♦❢ ❏(θ◆) ❛♥❞ ✇❡ ✇✐❧❧ ❤❛✈❡ ❏(θ◆) ≈ ❏✵. ■♥ ♠❛♥② ❝✐r❝✉♠st❛♥❝❡s ✇❡ ❝❛♥ t❤❡♥ ✏t❛❦❡ st❛♥❞❛r❞ ♣❛rts✑ t♦ ♣r♦❞✉❝❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ θ = ◦θ◆ ❚❤✐s ✐❞❡❛ ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ t♦ t❤❡ s◆❙❡ ✐♥ ❛ ✈❛r✐❡t② ♦❢ s❡tt✐♥❣s✱ ❛❧✇❛②s ✐♥✈♦❧✈✐♥❣ ❛ ▲♦❡❜ s♣❛❝❡ s♦ t❤❛t s♦❧✉t✐♦♥s ❢♦r ❛❧❧ ❝♦♥tr♦❧s ❧✐✈❡ ♦♥ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❘❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r ✷❉ s②st❡♠s ♦❢ t❤❡ ❢♦r♠ ✉(t) = ✉✵ + t

✵

{−ν❆✉(s) − ❇ (✉(s)) + ❢ (s, ✉, θ(s, ✉))} ❞s + t

✵

❣ (s, ✉) ❞✇(s) ✇✐t❤ θ ❍ö❧❞❡r ❝♦♥t✐♥✉♦✉s✱ ♦r ✇✐t❤ θ ❤❛✈✐♥❣ ♥♦ ❢❡❡❞❜❛❝❦ ✐♥ ✉, ♦r ✇✐t❤ t❤❡ ❢❡❡❞❜❛❝❦ ❝♦♥s✐st✐♥❣ ♦❢ ❝✉♠✉❧❛t✐✈❡ ❞✐❣✐t❛❧ ♦❜s❡r✈❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛t ❛ ✜①❡❞ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

slide-124
SLIDE 124

❋♦r t❤❡ ✸❉ ❡q✉❛t✐♦♥s r❡s✉❧ts ❛r❡ ♦♥❧② ❢♦r s②st❡♠s ✇✐t❤ ♥♦ ❢❡❡❞❜❛❝❦✿ ✐✳❡✳ θ = θ(t). ❚❤❡ ♣♦ss✐❜❧❡ ♥♦♥✲✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s r❡q✉✐r❡s ❛ ❧❛r❣❡ s♣❛❝❡ t♦ ✇♦r❦ ✐♥ ✲ ♦♥❡ ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s ❢♦r ❛ ❣✐✈❡♥ ❝♦♥tr♦❧ t♦ ❛❧❧♦✇ ✐♥✐t✐❛❧❧② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❢♦r ❛ ❣✐✈❡♥ ❝♦♥tr♦❧✳ ❆ t②♣✐❝❛❧ s✐t✉❛t✐♦♥ ✐♥ ❡✐t❤❡r ✷❉ ♦r ✸❉ ✐s t❤❛t ✇❡ ❤❛✈❡ ❛ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧ ✭♣♦ss✐❜❧②

◆✮ ❛♥❞ ❛ ♥♦♥st❛♥❞❛r❞ s♦❧✉t✐♦♥ ❢♦r ✐t✿

❯ ❯✵

✵

❆❯ s ❇ ❯ s ❢ s ❯ ❯ ❞s

✵

❣ s ❯ ❞❲ s ✇❤❡r❡ ❯ ✵ ❚ ❍ ♦r ❍◆ ❚❤❡♥ ✇❡ ✏st❛♥❞❛r❞✐s❡✑ t❤❡ ❝♦♥tr♦❧ t♦ ❣✐✈❡ ❛♥❞ ❛s ✐♥ t❤❡ ❜❛s✐❝ ❡①✐st❡♥❝❡ ♣r♦♦❢ s❤♦✇ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ t❛❦❡ ✉ t ❯ t ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t ✉ ✐s ❛ s♦❧✉t✐♦♥ ❢♦r ❝♦♥tr♦❧ ❛♥❞ ❏ ❏ t♦ ❣✐✈❡ ♦♣t✐♠❛❧✐t②✳ ❉❡t❛✐❧s✿ ◆❏❈ ✫ ❑✳●r③❡s✐❛❦✿ ❙t♦❝❤❛st✐❝s ✭✷✵✵✺✮ ❛♥❞ ❆▼❖ ✭✷✵✵✼✮✳

slide-125
SLIDE 125

❋♦r t❤❡ ✸❉ ❡q✉❛t✐♦♥s r❡s✉❧ts ❛r❡ ♦♥❧② ❢♦r s②st❡♠s ✇✐t❤ ♥♦ ❢❡❡❞❜❛❝❦✿ ✐✳❡✳ θ = θ(t). ❚❤❡ ♣♦ss✐❜❧❡ ♥♦♥✲✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s r❡q✉✐r❡s ❛ ❧❛r❣❡ s♣❛❝❡ t♦ ✇♦r❦ ✐♥ ✲ ♦♥❡ ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s ❢♦r ❛ ❣✐✈❡♥ ❝♦♥tr♦❧ t♦ ❛❧❧♦✇ ✐♥✐t✐❛❧❧② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❢♦r ❛ ❣✐✈❡♥ ❝♦♥tr♦❧✳ ❆ t②♣✐❝❛❧ s✐t✉❛t✐♦♥ ✐♥ ❡✐t❤❡r ✷❉ ♦r ✸❉ ✐s t❤❛t ✇❡ ❤❛✈❡ ❛ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧ Θ ✭♣♦ss✐❜❧② θ◆✮ ❛♥❞ ❛ ♥♦♥st❛♥❞❛r❞ s♦❧✉t✐♦♥ ❢♦r ✐t✿ ❯(τ) = ❯✵ + τ

✵

{−ν∗❆❯(s) − ∗❇ (❯(s)) + ∗❢ (s, ❯, Θ(❯))} ❞s + τ

✵ ∗❣ (s, ❯) ❞❲ (s)

✇❤❡r❡ ❯ : ∗[✵, ❚] →∗❍ ♦r ❍◆. ❚❤❡♥ ✇❡ ✏st❛♥❞❛r❞✐s❡✑ t❤❡ ❝♦♥tr♦❧ t♦ ❣✐✈❡ ❛♥❞ ❛s ✐♥ t❤❡ ❜❛s✐❝ ❡①✐st❡♥❝❡ ♣r♦♦❢ s❤♦✇ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ t❛❦❡ ✉ t ❯ t ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t ✉ ✐s ❛ s♦❧✉t✐♦♥ ❢♦r ❝♦♥tr♦❧ ❛♥❞ ❏ ❏ t♦ ❣✐✈❡ ♦♣t✐♠❛❧✐t②✳ ❉❡t❛✐❧s✿ ◆❏❈ ✫ ❑✳●r③❡s✐❛❦✿ ❙t♦❝❤❛st✐❝s ✭✷✵✵✺✮ ❛♥❞ ❆▼❖ ✭✷✵✵✼✮✳

slide-126
SLIDE 126

❋♦r t❤❡ ✸❉ ❡q✉❛t✐♦♥s r❡s✉❧ts ❛r❡ ♦♥❧② ❢♦r s②st❡♠s ✇✐t❤ ♥♦ ❢❡❡❞❜❛❝❦✿ ✐✳❡✳ θ = θ(t). ❚❤❡ ♣♦ss✐❜❧❡ ♥♦♥✲✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s r❡q✉✐r❡s ❛ ❧❛r❣❡ s♣❛❝❡ t♦ ✇♦r❦ ✐♥ ✲ ♦♥❡ ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s ❢♦r ❛ ❣✐✈❡♥ ❝♦♥tr♦❧ t♦ ❛❧❧♦✇ ✐♥✐t✐❛❧❧② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❢♦r ❛ ❣✐✈❡♥ ❝♦♥tr♦❧✳ ❆ t②♣✐❝❛❧ s✐t✉❛t✐♦♥ ✐♥ ❡✐t❤❡r ✷❉ ♦r ✸❉ ✐s t❤❛t ✇❡ ❤❛✈❡ ❛ ♥♦♥st❛♥❞❛r❞ ❝♦♥tr♦❧ Θ ✭♣♦ss✐❜❧② θ◆✮ ❛♥❞ ❛ ♥♦♥st❛♥❞❛r❞ s♦❧✉t✐♦♥ ❢♦r ✐t✿ ❯(τ) = ❯✵ + τ

✵

{−ν∗❆❯(s) − ∗❇ (❯(s)) + ∗❢ (s, ❯, Θ(❯))} ❞s + τ

✵ ∗❣ (s, ❯) ❞❲ (s)

✇❤❡r❡ ❯ : ∗[✵, ❚] →∗❍ ♦r ❍◆. ❚❤❡♥ ✇❡ ✏st❛♥❞❛r❞✐s❡✑ t❤❡ ❝♦♥tr♦❧ t♦ ❣✐✈❡ θ = ◦Θ ❛♥❞ ❛s ✐♥ t❤❡ ❜❛s✐❝ ❡①✐st❡♥❝❡ ♣r♦♦❢ s❤♦✇ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ t❛❦❡ ✉(t, ω) = ◦❯(t,ω) ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t ✉ ✐s ❛ s♦❧✉t✐♦♥ ❢♦r ❝♦♥tr♦❧ θ ❛♥❞ ❏(θ) = ◦❏(Θ) t♦ ❣✐✈❡ ♦♣t✐♠❛❧✐t②✳ ❉❡t❛✐❧s✿ ◆❏❈ ✫ ❑✳●r③❡s✐❛❦✿ ❙t♦❝❤❛st✐❝s ✭✷✵✵✺✮ ❛♥❞ ❆▼❖ ✭✷✵✵✼✮✳

slide-127
SLIDE 127

❆♣♣❧✐❝❛t✐♦♥ ✸✿ ◆❖◆✲❍❖▼❖●❊◆❊❖❯❙ ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❚❤❡s❡ ♠♦❞❡❧ t❤❡ ✈❡❧♦❝✐t② ✉ ❛♥❞ ❞❡♥s✐t② ρ ♦❢ ❛ ♠✐①t✉r❡ ♦❢ ✈✐s❝♦✉s ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞s ♦❢ ✈❛r②✐♥❣ ❞❡♥s✐t② ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✭❱❡❧♦❝✐t②✮ ❞✉ ✉ ✉ ✉ ♣ ❢ t ✉ ❞t ❣ t ✉ ❞✇t ❞✐✈ ✉ ✵ ✭❉❡♥s✐t②✮ t ✉ ✵ ✇✐t❤ ❜♦✉♥❞❛r② ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✉

❉

✵✱ ✉ t

✵

✉✵✱

t ✵ ✵

✭✶✮t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s✿ s♦❧✈❡❞ ❜② ❑❛③❤✐❦❤♦✈ ✭✶✾✼✹✮ ✲ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵ ✭✷✮ ❚❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ✭✐✳❡✳ ❞● ❣❞✇ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✉✮ ✲ ❨❛s❤✐♠❛ ✭✶✾✾✷✮ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵ ❙♦❧✈❡❞ ❡ss❡♥t✐❛❧❧② ♣❛t❤✇✐s❡✳ ✭✸✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ✭◆❏❈ ✫ ❇r❡♥❞❛♥ ❊♥r✐❣❤t✮✿ s♦❧✈❡ t❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❣❡♥❡r❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❢♦r ❞ ✷ ✸ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵

slide-128
SLIDE 128

❆♣♣❧✐❝❛t✐♦♥ ✸✿ ◆❖◆✲❍❖▼❖●❊◆❊❖❯❙ ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❚❤❡s❡ ♠♦❞❡❧ t❤❡ ✈❡❧♦❝✐t② ✉ ❛♥❞ ❞❡♥s✐t② ρ ♦❢ ❛ ♠✐①t✉r❡ ♦❢ ✈✐s❝♦✉s ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞s ♦❢ ✈❛r②✐♥❣ ❞❡♥s✐t② ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✭❱❡❧♦❝✐t②✮ ρ❞✉ = [ν∆✉− < ρ✉, ∇ > ✉ − ∇♣ + ρ❢ (t, ✉)] ❞t + ρ❣(t, ✉)❞✇t ❞✐✈ ✉ = ✵ ✭❉❡♥s✐t②✮ t ✉ ✵ ✇✐t❤ ❜♦✉♥❞❛r② ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✉

❉

✵✱ ✉ t

✵

✉✵✱

t ✵ ✵

✭✶✮t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s✿ s♦❧✈❡❞ ❜② ❑❛③❤✐❦❤♦✈ ✭✶✾✼✹✮ ✲ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵ ✭✷✮ ❚❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ✭✐✳❡✳ ❞● ❣❞✇ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✉✮ ✲ ❨❛s❤✐♠❛ ✭✶✾✾✷✮ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵ ❙♦❧✈❡❞ ❡ss❡♥t✐❛❧❧② ♣❛t❤✇✐s❡✳ ✭✸✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ✭◆❏❈ ✫ ❇r❡♥❞❛♥ ❊♥r✐❣❤t✮✿ s♦❧✈❡ t❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❣❡♥❡r❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❢♦r ❞ ✷ ✸ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵

slide-129
SLIDE 129

❆♣♣❧✐❝❛t✐♦♥ ✸✿ ◆❖◆✲❍❖▼❖●❊◆❊❖❯❙ ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❚❤❡s❡ ♠♦❞❡❧ t❤❡ ✈❡❧♦❝✐t② ✉ ❛♥❞ ❞❡♥s✐t② ρ ♦❢ ❛ ♠✐①t✉r❡ ♦❢ ✈✐s❝♦✉s ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞s ♦❢ ✈❛r②✐♥❣ ❞❡♥s✐t② ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✭❱❡❧♦❝✐t②✮ ρ❞✉ = [ν∆✉− < ρ✉, ∇ > ✉ − ∇♣ + ρ❢ (t, ✉)] ❞t + ρ❣(t, ✉)❞✇t ❞✐✈ ✉ = ✵ ✭❉❡♥s✐t②✮ ∂ρ ∂t + < ✉, ∇ > ρ = ✵ ✇✐t❤ ❜♦✉♥❞❛r② ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✉|∂❉ = ✵✱ ✉|t=✵ = ✉✵✱ ρ|t=✵ = ρ✵. ✭✶✮t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s✿ s♦❧✈❡❞ ❜② ❑❛③❤✐❦❤♦✈ ✭✶✾✼✹✮ ✲ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵ ✭✷✮ ❚❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ✭✐✳❡✳ ❞● ❣❞✇ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✉✮ ✲ ❨❛s❤✐♠❛ ✭✶✾✾✷✮ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵ ❙♦❧✈❡❞ ❡ss❡♥t✐❛❧❧② ♣❛t❤✇✐s❡✳ ✭✸✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ✭◆❏❈ ✫ ❇r❡♥❞❛♥ ❊♥r✐❣❤t✮✿ s♦❧✈❡ t❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❣❡♥❡r❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❢♦r ❞ ✷ ✸ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵

slide-130
SLIDE 130

❆♣♣❧✐❝❛t✐♦♥ ✸✿ ◆❖◆✲❍❖▼❖●❊◆❊❖❯❙ ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❚❤❡s❡ ♠♦❞❡❧ t❤❡ ✈❡❧♦❝✐t② ✉ ❛♥❞ ❞❡♥s✐t② ρ ♦❢ ❛ ♠✐①t✉r❡ ♦❢ ✈✐s❝♦✉s ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞s ♦❢ ✈❛r②✐♥❣ ❞❡♥s✐t② ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✭❱❡❧♦❝✐t②✮ ρ❞✉ = [ν∆✉− < ρ✉, ∇ > ✉ − ∇♣ + ρ❢ (t, ✉)] ❞t + ρ❣(t, ✉)❞✇t ❞✐✈ ✉ = ✵ ✭❉❡♥s✐t②✮ ∂ρ ∂t + < ✉, ∇ > ρ = ✵ ✇✐t❤ ❜♦✉♥❞❛r② ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✉|∂❉ = ✵✱ ✉|t=✵ = ✉✵✱ ρ|t=✵ = ρ✵. ✭✶✮t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s✿ s♦❧✈❡❞ ❜② ❑❛③❤✐❦❤♦✈ ✭✶✾✼✹✮ ✲ ❛ss✉♠✐♥❣ ▼ ≥ ρ✵ ≥ ♠ > ✵ ✭✷✮ ❚❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ✭✐✳❡✳ ❞● ❣❞✇ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✉✮ ✲ ❨❛s❤✐♠❛ ✭✶✾✾✷✮ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵ ❙♦❧✈❡❞ ❡ss❡♥t✐❛❧❧② ♣❛t❤✇✐s❡✳ ✭✸✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ✭◆❏❈ ✫ ❇r❡♥❞❛♥ ❊♥r✐❣❤t✮✿ s♦❧✈❡ t❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❣❡♥❡r❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❢♦r ❞ ✷ ✸ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵

slide-131
SLIDE 131

❆♣♣❧✐❝❛t✐♦♥ ✸✿ ◆❖◆✲❍❖▼❖●❊◆❊❖❯❙ ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❚❤❡s❡ ♠♦❞❡❧ t❤❡ ✈❡❧♦❝✐t② ✉ ❛♥❞ ❞❡♥s✐t② ρ ♦❢ ❛ ♠✐①t✉r❡ ♦❢ ✈✐s❝♦✉s ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞s ♦❢ ✈❛r②✐♥❣ ❞❡♥s✐t② ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✭❱❡❧♦❝✐t②✮ ρ❞✉ = [ν∆✉− < ρ✉, ∇ > ✉ − ∇♣ + ρ❢ (t, ✉)] ❞t + ρ❣(t, ✉)❞✇t ❞✐✈ ✉ = ✵ ✭❉❡♥s✐t②✮ ∂ρ ∂t + < ✉, ∇ > ρ = ✵ ✇✐t❤ ❜♦✉♥❞❛r② ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✉|∂❉ = ✵✱ ✉|t=✵ = ✉✵✱ ρ|t=✵ = ρ✵. ✭✶✮t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s✿ s♦❧✈❡❞ ❜② ❑❛③❤✐❦❤♦✈ ✭✶✾✼✹✮ ✲ ❛ss✉♠✐♥❣ ▼ ≥ ρ✵ ≥ ♠ > ✵ ✭✷✮ ❚❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ✭✐✳❡✳ ❞● = ❣❞✇ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✉✮ ✲ ❨❛s❤✐♠❛ ✭✶✾✾✷✮ ❛ss✉♠✐♥❣ ▼ ≥ ρ✵ ≥ ♠ > ✵. ❙♦❧✈❡❞ ❡ss❡♥t✐❛❧❧② ♣❛t❤✇✐s❡✳ ✭✸✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ✭◆❏❈ ✫ ❇r❡♥❞❛♥ ❊♥r✐❣❤t✮✿ s♦❧✈❡ t❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❣❡♥❡r❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❢♦r ❞ ✷ ✸ ❛ss✉♠✐♥❣ ▼

✵

♠ ✵

slide-132
SLIDE 132

❆♣♣❧✐❝❛t✐♦♥ ✸✿ ◆❖◆✲❍❖▼❖●❊◆❊❖❯❙ ✭✐✳❡✳ ♥♦♥✲❝♦♥st❛♥t ❞❡♥s✐t②✮ ❙❚❖❈❍❆❙❚■❈ ◆❙❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❚❤❡s❡ ♠♦❞❡❧ t❤❡ ✈❡❧♦❝✐t② ✉ ❛♥❞ ❞❡♥s✐t② ρ ♦❢ ❛ ♠✐①t✉r❡ ♦❢ ✈✐s❝♦✉s ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞s ♦❢ ✈❛r②✐♥❣ ❞❡♥s✐t② ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❉ ⊂ R❞ ✭❞ = ✷, ✸✮ ✭❱❡❧♦❝✐t②✮ ρ❞✉ = [ν∆✉− < ρ✉, ∇ > ✉ − ∇♣ + ρ❢ (t, ✉)] ❞t + ρ❣(t, ✉)❞✇t ❞✐✈ ✉ = ✵ ✭❉❡♥s✐t②✮ ∂ρ ∂t + < ✉, ∇ > ρ = ✵ ✇✐t❤ ❜♦✉♥❞❛r② ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✉|∂❉ = ✵✱ ✉|t=✵ = ✉✵✱ ρ|t=✵ = ρ✵. ✭✶✮t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s✿ s♦❧✈❡❞ ❜② ❑❛③❤✐❦❤♦✈ ✭✶✾✼✹✮ ✲ ❛ss✉♠✐♥❣ ▼ ≥ ρ✵ ≥ ♠ > ✵ ✭✷✮ ❚❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ✭✐✳❡✳ ❞● = ❣❞✇ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✉✮ ✲ ❨❛s❤✐♠❛ ✭✶✾✾✷✮ ❛ss✉♠✐♥❣ ▼ ≥ ρ✵ ≥ ♠ > ✵. ❙♦❧✈❡❞ ❡ss❡♥t✐❛❧❧② ♣❛t❤✇✐s❡✳ ✭✸✮ ▲♦❡❜ s♣❛❝❡ ♠❡t❤♦❞s ✭◆❏❈ ✫ ❇r❡♥❞❛♥ ❊♥r✐❣❤t✮✿ s♦❧✈❡ t❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ❣❡♥❡r❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡ ❢♦r ❞ = ✷, ✸ ❛ss✉♠✐♥❣ ▼ ≥ ρ✵ ≥ ♠ > ✵.

slide-133
SLIDE 133

❉❡✜♥✐t✐♦♥ ♦❢ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ❚❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❞❡♥s✐t② ✇✐❧❧ ❜❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❧✐✈✐♥❣ ♦♥ ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✳

❉❡✜♥✐t✐♦♥

  • ✐✈❡♥ ✉✵ ∈ ❍✱ ρ✵ ∈ ▲∞(❉), ❢ : [✵, ❚] × ❍ → ❍ ❛♥❞ ❣ : [✵, ❚] × ❍ →▲(❍, ❍) ❛

♣❛✐r ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s (ρ, ✉) ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐❢ ✭✐✮ ✉ ▲✷ ✵ ❚ ❱ ❛♥❞ ❢♦r ❛✳❛✳ ✉ ▲ ✵ ❚ ❍ ▲✷ ✵ ❚ ❱ ✭✐✐✮ ▲ ✵ ❚ ❉ ✭✐✐✐✮ ✭❱❡❧♦❝✐t②✮ ❢♦r ❛❧♠♦st ❛❧❧ ❚✵ ❚ ❢♦r ❛❧❧ ❈ ✶ ✵ ❚ ❱ ❚✵ ✉ ❚✵ ❚✵

✵✉✵

✵

❚✵ ✵

✉ ✉ ✉ ❢ ❞t

❚✵ ✵

❣ ❞✇ ✭✐✈✮ ✭❉❡♥s✐t②✮ ❢♦r ❛❧❧ ❈ ✶ ✵ ❚ ❍✶ ❉ ✱ ❢♦r ❛❧❧ ❚✵ ❚ ❚✵ ❚✵

✵

✵

❚✵ ✵

✉ ❞t ✭✈✮ ✵

✵ ❛♥❞ ✉ ✵

✉✵ ◆♦t❡✳ ❣ ✵ ❣✐✈❡s ❑❛③❤✐❦❤♦✈✬s ❞❡✜♥✐t✐♦♥ ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❡q✉❛t✐♦♥s✳

slide-134
SLIDE 134

❉❡✜♥✐t✐♦♥ ♦❢ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ❚❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❞❡♥s✐t② ✇✐❧❧ ❜❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❧✐✈✐♥❣ ♦♥ ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✳

❉❡✜♥✐t✐♦♥

  • ✐✈❡♥ ✉✵ ∈ ❍✱ ρ✵ ∈ ▲∞(❉), ❢ : [✵, ❚] × ❍ → ❍ ❛♥❞ ❣ : [✵, ❚] × ❍ →▲(❍, ❍) ❛

♣❛✐r ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s (ρ, ✉) ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐❢ ✭✐✮ ✉ ∈ ▲✷([✵, ❚] × Ω, ❱) ❛♥❞ ❢♦r ❛✳❛✳ ω ✉(·, ω) ∈ ▲∞(✵, ❚; ❍) ∩ ▲✷(✵, ❚; ❱) ✭✐✐✮ ▲ ✵ ❚ ❉ ✭✐✐✐✮ ✭❱❡❧♦❝✐t②✮ ❢♦r ❛❧♠♦st ❛❧❧ ❚✵ ❚ ❢♦r ❛❧❧ ❈ ✶ ✵ ❚ ❱ ❚✵ ✉ ❚✵ ❚✵

✵✉✵

✵

❚✵ ✵

✉ ✉ ✉ ❢ ❞t

❚✵ ✵

❣ ❞✇ ✭✐✈✮ ✭❉❡♥s✐t②✮ ❢♦r ❛❧❧ ❈ ✶ ✵ ❚ ❍✶ ❉ ✱ ❢♦r ❛❧❧ ❚✵ ❚ ❚✵ ❚✵

✵

✵

❚✵ ✵

✉ ❞t ✭✈✮ ✵

✵ ❛♥❞ ✉ ✵

✉✵ ◆♦t❡✳ ❣ ✵ ❣✐✈❡s ❑❛③❤✐❦❤♦✈✬s ❞❡✜♥✐t✐♦♥ ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❡q✉❛t✐♦♥s✳

slide-135
SLIDE 135

❉❡✜♥✐t✐♦♥ ♦❢ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ❚❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❞❡♥s✐t② ✇✐❧❧ ❜❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❧✐✈✐♥❣ ♦♥ ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✳

❉❡✜♥✐t✐♦♥

  • ✐✈❡♥ ✉✵ ∈ ❍✱ ρ✵ ∈ ▲∞(❉), ❢ : [✵, ❚] × ❍ → ❍ ❛♥❞ ❣ : [✵, ❚] × ❍ →▲(❍, ❍) ❛

♣❛✐r ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s (ρ, ✉) ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐❢ ✭✐✮ ✉ ∈ ▲✷([✵, ❚] × Ω, ❱) ❛♥❞ ❢♦r ❛✳❛✳ ω ✉(·, ω) ∈ ▲∞(✵, ❚; ❍) ∩ ▲✷(✵, ❚; ❱) ✭✐✐✮ ρ ∈ ▲∞([✵, ❚] × ❉ × Ω) ✭✐✐✐✮ ✭❱❡❧♦❝✐t②✮ ❢♦r ❛❧♠♦st ❛❧❧ ❚✵ ❚ ❢♦r ❛❧❧ ❈ ✶ ✵ ❚ ❱ ❚✵ ✉ ❚✵ ❚✵

✵✉✵

✵

❚✵ ✵

✉ ✉ ✉ ❢ ❞t

❚✵ ✵

❣ ❞✇ ✭✐✈✮ ✭❉❡♥s✐t②✮ ❢♦r ❛❧❧ ❈ ✶ ✵ ❚ ❍✶ ❉ ✱ ❢♦r ❛❧❧ ❚✵ ❚ ❚✵ ❚✵

✵

✵

❚✵ ✵

✉ ❞t ✭✈✮ ✵

✵ ❛♥❞ ✉ ✵

✉✵ ◆♦t❡✳ ❣ ✵ ❣✐✈❡s ❑❛③❤✐❦❤♦✈✬s ❞❡✜♥✐t✐♦♥ ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❡q✉❛t✐♦♥s✳

slide-136
SLIDE 136

❉❡✜♥✐t✐♦♥ ♦❢ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ❚❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❞❡♥s✐t② ✇✐❧❧ ❜❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❧✐✈✐♥❣ ♦♥ ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✳

❉❡✜♥✐t✐♦♥

  • ✐✈❡♥ ✉✵ ∈ ❍✱ ρ✵ ∈ ▲∞(❉), ❢ : [✵, ❚] × ❍ → ❍ ❛♥❞ ❣ : [✵, ❚] × ❍ →▲(❍, ❍) ❛

♣❛✐r ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s (ρ, ✉) ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐❢ ✭✐✮ ✉ ∈ ▲✷([✵, ❚] × Ω, ❱) ❛♥❞ ❢♦r ❛✳❛✳ ω ✉(·, ω) ∈ ▲∞(✵, ❚; ❍) ∩ ▲✷(✵, ❚; ❱) ✭✐✐✮ ρ ∈ ▲∞([✵, ❚] × ❉ × Ω) ✭✐✐✐✮ ✭❱❡❧♦❝✐t②✮ ❢♦r ❛❧♠♦st ❛❧❧ ❚✵ ≤ ❚, ❢♦r ❛❧❧ Φ ∈ ❈ ✶(✵, ❚; ❱) (ρ(❚✵)✉(❚✵), Φ(❚✵)) − (ρ✵✉✵, Φ(✵)) = ❚✵

✵ [(ρ✉, Φ′ + ✉, ∇Φ) − ν(

(✉, Φ) ) + (ρ❢ , Φ)] ❞t + ❚✵

✵ (Φ, ρ❣)❞✇

✭✐✈✮ ✭❉❡♥s✐t②✮ ❢♦r ❛❧❧ ❈ ✶ ✵ ❚ ❍✶ ❉ ✱ ❢♦r ❛❧❧ ❚✵ ❚ ❚✵ ❚✵

✵

✵

❚✵ ✵

✉ ❞t ✭✈✮ ✵

✵ ❛♥❞ ✉ ✵

✉✵ ◆♦t❡✳ ❣ ✵ ❣✐✈❡s ❑❛③❤✐❦❤♦✈✬s ❞❡✜♥✐t✐♦♥ ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❡q✉❛t✐♦♥s✳

slide-137
SLIDE 137

❉❡✜♥✐t✐♦♥ ♦❢ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ❚❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❞❡♥s✐t② ✇✐❧❧ ❜❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❧✐✈✐♥❣ ♦♥ ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✳

❉❡✜♥✐t✐♦♥

  • ✐✈❡♥ ✉✵ ∈ ❍✱ ρ✵ ∈ ▲∞(❉), ❢ : [✵, ❚] × ❍ → ❍ ❛♥❞ ❣ : [✵, ❚] × ❍ →▲(❍, ❍) ❛

♣❛✐r ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s (ρ, ✉) ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐❢ ✭✐✮ ✉ ∈ ▲✷([✵, ❚] × Ω, ❱) ❛♥❞ ❢♦r ❛✳❛✳ ω ✉(·, ω) ∈ ▲∞(✵, ❚; ❍) ∩ ▲✷(✵, ❚; ❱) ✭✐✐✮ ρ ∈ ▲∞([✵, ❚] × ❉ × Ω) ✭✐✐✐✮ ✭❱❡❧♦❝✐t②✮ ❢♦r ❛❧♠♦st ❛❧❧ ❚✵ ≤ ❚, ❢♦r ❛❧❧ Φ ∈ ❈ ✶(✵, ❚; ❱) (ρ(❚✵)✉(❚✵), Φ(❚✵)) − (ρ✵✉✵, Φ(✵)) = ❚✵

✵ [(ρ✉, Φ′ + ✉, ∇Φ) − ν(

(✉, Φ) ) + (ρ❢ , Φ)] ❞t + ❚✵

✵ (Φ, ρ❣)❞✇

✭✐✈✮ ✭❉❡♥s✐t②✮ ❢♦r ❛❧❧ ϕ ∈ ❈ ✶(✵, ❚; ❍✶(❉))✱ ❢♦r ❛❧❧ ❚✵ ≤ ❚ (ρ(❚✵), ϕ(❚✵)) − (ρ✵, ϕ(✵)) = ❚✵

✵ (ρ, ϕ′ + ✉, ∇ϕ)❞t

✭✈✮ ✵

✵ ❛♥❞ ✉ ✵

✉✵ ◆♦t❡✳ ❣ ✵ ❣✐✈❡s ❑❛③❤✐❦❤♦✈✬s ❞❡✜♥✐t✐♦♥ ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❡q✉❛t✐♦♥s✳

slide-138
SLIDE 138

❉❡✜♥✐t✐♦♥ ♦❢ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ❚❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❞❡♥s✐t② ✇✐❧❧ ❜❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❧✐✈✐♥❣ ♦♥ ❛♥ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✳

❉❡✜♥✐t✐♦♥

  • ✐✈❡♥ ✉✵ ∈ ❍✱ ρ✵ ∈ ▲∞(❉), ❢ : [✵, ❚] × ❍ → ❍ ❛♥❞ ❣ : [✵, ❚] × ❍ →▲(❍, ❍) ❛

♣❛✐r ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s (ρ, ✉) ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐❢ ✭✐✮ ✉ ∈ ▲✷([✵, ❚] × Ω, ❱) ❛♥❞ ❢♦r ❛✳❛✳ ω ✉(·, ω) ∈ ▲∞(✵, ❚; ❍) ∩ ▲✷(✵, ❚; ❱) ✭✐✐✮ ρ ∈ ▲∞([✵, ❚] × ❉ × Ω) ✭✐✐✐✮ ✭❱❡❧♦❝✐t②✮ ❢♦r ❛❧♠♦st ❛❧❧ ❚✵ ≤ ❚, ❢♦r ❛❧❧ Φ ∈ ❈ ✶(✵, ❚; ❱) (ρ(❚✵)✉(❚✵), Φ(❚✵)) − (ρ✵✉✵, Φ(✵)) = ❚✵

✵ [(ρ✉, Φ′ + ✉, ∇Φ) − ν(

(✉, Φ) ) + (ρ❢ , Φ)] ❞t + ❚✵

✵ (Φ, ρ❣)❞✇

✭✐✈✮ ✭❉❡♥s✐t②✮ ❢♦r ❛❧❧ ϕ ∈ ❈ ✶(✵, ❚; ❍✶(❉))✱ ❢♦r ❛❧❧ ❚✵ ≤ ❚ (ρ(❚✵), ϕ(❚✵)) − (ρ✵, ϕ(✵)) = ❚✵

✵ (ρ, ϕ′ + ✉, ∇ϕ)❞t

✭✈✮ ρ(✵) = ρ✵ ❛♥❞ ✉(✵) = ✉✵ ◆♦t❡✳ ❣ = ✵ ❣✐✈❡s ❑❛③❤✐❦❤♦✈✬s ❞❡✜♥✐t✐♦♥ ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❡q✉❛t✐♦♥s✳

slide-139
SLIDE 139

❚❤❡♦r❡♠ ✭◆❏❈ ✫ ❇r❡♥❞❛♥ ❊♥r✐❣❤t✱ ❏❉❊ ✷✵✵✻✮ ❙✉♣♣♦s❡ t❤❛t ✉✵ ∈ ❍ ❛♥❞ ρ✵ ∈ ▲∞(❉) ✇✐t❤ ✵ < ♠ ≤ ρ✵(①) ≤ ▼✱ ❛♥❞ ❢ , ❣ s❛t✐s❢② ♥❛t✉r❛❧ ❝♦♥t✐♥✉✐t② ❛♥❞ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ (ρ, ✉) t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ E  s✉♣

t≤❚

|✉(t)|✷ + ν

❚

  • ✵

||✉(t)||✷ ❞t   < ∞ ❛♥❞ ❢♦r ❛❧♠♦st ❛❧❧ ω, ❢♦r ❛❧❧ t ♠ ≤ ρ(t, ①) ≤ ▼ ❢♦r ❛❧♠♦st ❛❧❧ ①

slide-140
SLIDE 140

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✭❇r♦❛❞❧② s✐♠✐❧❛r t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✳✮ ✶✳ ❙♦❧✈❡ ❛ ♠♦❞✐✜❡❞ ❤②♣❡r✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❡❧♦❝✐t② ✜❡❧❞ ❯ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆✱ ✉s✐♥❣ t❤❡ tr❛♥s❢❡r ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❙❉❊ t❤❡♦r②✳ ❚❤✐s ✇✐❧❧ ❧✐✈❡ ♦♥ ❛♥ ✐♥t❡r♥❛❧ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

✵ ✵

❝❛rr②✐♥❣ ❛♥ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ ❛❧s♦ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆ ❚❤❡ ❞❡♥s✐t② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ ❘ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❈ ✶ ❉ ▲ ❉ ✷✳ Pr♦✈❡ ❛♥ ✏❡♥❡r❣② ❡st✐♠❛t❡✑ s❤♦✇✐♥❣ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ t❤❡ ✜❡❧❞ ❯ ✐s ♥❡❛rst❛♥❞❛r❞✳ ✸✳ ❙❤♦✇ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ t❤❡ ❞❡♥s✐t② ❘ ✐s ♥❡❛rst❛♥❞❛r❞ ✹✳ ❊st❛❜❧✐s❤ ❛♣♣r♦♣r✐❛t❡ ❙✲❝♦♥t✐♥✉✐t② ✐♥ t❤❡ t✐♠❡ ✈❛r✐❛❜❧❡ ✺✳ ❚❛❦❡ st❛♥❞❛r❞ ♣❛rts ✉ ❯ ❛♥❞ ❘ ✻✳ ❙❤♦✇ t❤❛t t❤❡ ♣❛✐r ✉ ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ♦♥ t❤❡ ❛❞❛♣t❡❞ ▲♦❡❜ s♣❛❝❡

t t ✵ P

✇❤❡r❡ P

▲✱

▲ ❛♥❞

t t ✵ ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ ✵

slide-141
SLIDE 141

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✭❇r♦❛❞❧② s✐♠✐❧❛r t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✳✮ ✶✳ ❙♦❧✈❡ ❛ ♠♦❞✐✜❡❞ ❤②♣❡r✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❡❧♦❝✐t② ✜❡❧❞ ❯(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆✱ ✉s✐♥❣ t❤❡ tr❛♥s❢❡r ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❙❉❊ t❤❡♦r②✳ ❚❤✐s ✇✐❧❧ ❧✐✈❡ ♦♥ ❛♥ ✐♥t❡r♥❛❧ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✵ = (Ω, A, (Aτ)τ≥✵, Π ) ❝❛rr②✐♥❣ ❛♥ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ (τ, ω) ❛❧s♦ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆. ❚❤❡ ❞❡♥s✐t② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ ❘(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥

∗❈ ✶(❉) ⊂ ∗▲∞(❉).

✷✳ Pr♦✈❡ ❛♥ ✏❡♥❡r❣② ❡st✐♠❛t❡✑ s❤♦✇✐♥❣ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ t❤❡ ✜❡❧❞ ❯ ✐s ♥❡❛rst❛♥❞❛r❞✳ ✸✳ ❙❤♦✇ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ t❤❡ ❞❡♥s✐t② ❘ ✐s ♥❡❛rst❛♥❞❛r❞ ✹✳ ❊st❛❜❧✐s❤ ❛♣♣r♦♣r✐❛t❡ ❙✲❝♦♥t✐♥✉✐t② ✐♥ t❤❡ t✐♠❡ ✈❛r✐❛❜❧❡ ✺✳ ❚❛❦❡ st❛♥❞❛r❞ ♣❛rts ✉ ❯ ❛♥❞ ❘ ✻✳ ❙❤♦✇ t❤❛t t❤❡ ♣❛✐r ✉ ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ♦♥ t❤❡ ❛❞❛♣t❡❞ ▲♦❡❜ s♣❛❝❡

t t ✵ P

✇❤❡r❡ P

▲✱

▲ ❛♥❞

t t ✵ ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ ✵

slide-142
SLIDE 142

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✭❇r♦❛❞❧② s✐♠✐❧❛r t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✳✮ ✶✳ ❙♦❧✈❡ ❛ ♠♦❞✐✜❡❞ ❤②♣❡r✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❡❧♦❝✐t② ✜❡❧❞ ❯(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆✱ ✉s✐♥❣ t❤❡ tr❛♥s❢❡r ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❙❉❊ t❤❡♦r②✳ ❚❤✐s ✇✐❧❧ ❧✐✈❡ ♦♥ ❛♥ ✐♥t❡r♥❛❧ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✵ = (Ω, A, (Aτ)τ≥✵, Π ) ❝❛rr②✐♥❣ ❛♥ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ (τ, ω) ❛❧s♦ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆. ❚❤❡ ❞❡♥s✐t② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ ❘(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥

∗❈ ✶(❉) ⊂ ∗▲∞(❉).

✷✳ Pr♦✈❡ ❛♥ ✏❡♥❡r❣② ❡st✐♠❛t❡✑ s❤♦✇✐♥❣ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ✜❡❧❞ ❯(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞✳ ✸✳ ❙❤♦✇ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ t❤❡ ❞❡♥s✐t② ❘ ✐s ♥❡❛rst❛♥❞❛r❞ ✹✳ ❊st❛❜❧✐s❤ ❛♣♣r♦♣r✐❛t❡ ❙✲❝♦♥t✐♥✉✐t② ✐♥ t❤❡ t✐♠❡ ✈❛r✐❛❜❧❡ ✺✳ ❚❛❦❡ st❛♥❞❛r❞ ♣❛rts ✉ ❯ ❛♥❞ ❘ ✻✳ ❙❤♦✇ t❤❛t t❤❡ ♣❛✐r ✉ ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ♦♥ t❤❡ ❛❞❛♣t❡❞ ▲♦❡❜ s♣❛❝❡

t t ✵ P

✇❤❡r❡ P

▲✱

▲ ❛♥❞

t t ✵ ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ ✵

slide-143
SLIDE 143

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✭❇r♦❛❞❧② s✐♠✐❧❛r t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✳✮ ✶✳ ❙♦❧✈❡ ❛ ♠♦❞✐✜❡❞ ❤②♣❡r✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❡❧♦❝✐t② ✜❡❧❞ ❯(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆✱ ✉s✐♥❣ t❤❡ tr❛♥s❢❡r ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❙❉❊ t❤❡♦r②✳ ❚❤✐s ✇✐❧❧ ❧✐✈❡ ♦♥ ❛♥ ✐♥t❡r♥❛❧ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✵ = (Ω, A, (Aτ)τ≥✵, Π ) ❝❛rr②✐♥❣ ❛♥ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ (τ, ω) ❛❧s♦ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆. ❚❤❡ ❞❡♥s✐t② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ ❘(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥

∗❈ ✶(❉) ⊂ ∗▲∞(❉).

✷✳ Pr♦✈❡ ❛♥ ✏❡♥❡r❣② ❡st✐♠❛t❡✑ s❤♦✇✐♥❣ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ✜❡❧❞ ❯(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞✳ ✸✳ ❙❤♦✇ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ❞❡♥s✐t② ❘(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞ ✹✳ ❊st❛❜❧✐s❤ ❛♣♣r♦♣r✐❛t❡ ❙✲❝♦♥t✐♥✉✐t② ✐♥ t❤❡ t✐♠❡ ✈❛r✐❛❜❧❡ ✺✳ ❚❛❦❡ st❛♥❞❛r❞ ♣❛rts ✉ ❯ ❛♥❞ ❘ ✻✳ ❙❤♦✇ t❤❛t t❤❡ ♣❛✐r ✉ ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ♦♥ t❤❡ ❛❞❛♣t❡❞ ▲♦❡❜ s♣❛❝❡

t t ✵ P

✇❤❡r❡ P

▲✱

▲ ❛♥❞

t t ✵ ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ ✵

slide-144
SLIDE 144

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✭❇r♦❛❞❧② s✐♠✐❧❛r t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✳✮ ✶✳ ❙♦❧✈❡ ❛ ♠♦❞✐✜❡❞ ❤②♣❡r✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❡❧♦❝✐t② ✜❡❧❞ ❯(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆✱ ✉s✐♥❣ t❤❡ tr❛♥s❢❡r ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❙❉❊ t❤❡♦r②✳ ❚❤✐s ✇✐❧❧ ❧✐✈❡ ♦♥ ❛♥ ✐♥t❡r♥❛❧ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✵ = (Ω, A, (Aτ)τ≥✵, Π ) ❝❛rr②✐♥❣ ❛♥ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ (τ, ω) ❛❧s♦ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆. ❚❤❡ ❞❡♥s✐t② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ ❘(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥

∗❈ ✶(❉) ⊂ ∗▲∞(❉).

✷✳ Pr♦✈❡ ❛♥ ✏❡♥❡r❣② ❡st✐♠❛t❡✑ s❤♦✇✐♥❣ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ✜❡❧❞ ❯(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞✳ ✸✳ ❙❤♦✇ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ❞❡♥s✐t② ❘(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞ ✹✳ ❊st❛❜❧✐s❤ ❛♣♣r♦♣r✐❛t❡ ❙✲❝♦♥t✐♥✉✐t② ✐♥ t❤❡ t✐♠❡ ✈❛r✐❛❜❧❡ τ ✺✳ ❚❛❦❡ st❛♥❞❛r❞ ♣❛rts ✉ ❯ ❛♥❞ ❘ ✻✳ ❙❤♦✇ t❤❛t t❤❡ ♣❛✐r ✉ ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ♦♥ t❤❡ ❛❞❛♣t❡❞ ▲♦❡❜ s♣❛❝❡

t t ✵ P

✇❤❡r❡ P

▲✱

▲ ❛♥❞

t t ✵ ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ ✵

slide-145
SLIDE 145

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✭❇r♦❛❞❧② s✐♠✐❧❛r t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✳✮ ✶✳ ❙♦❧✈❡ ❛ ♠♦❞✐✜❡❞ ❤②♣❡r✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❡❧♦❝✐t② ✜❡❧❞ ❯(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆✱ ✉s✐♥❣ t❤❡ tr❛♥s❢❡r ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❙❉❊ t❤❡♦r②✳ ❚❤✐s ✇✐❧❧ ❧✐✈❡ ♦♥ ❛♥ ✐♥t❡r♥❛❧ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✵ = (Ω, A, (Aτ)τ≥✵, Π ) ❝❛rr②✐♥❣ ❛♥ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ (τ, ω) ❛❧s♦ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆. ❚❤❡ ❞❡♥s✐t② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ ❘(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥

∗❈ ✶(❉) ⊂ ∗▲∞(❉).

✷✳ Pr♦✈❡ ❛♥ ✏❡♥❡r❣② ❡st✐♠❛t❡✑ s❤♦✇✐♥❣ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ✜❡❧❞ ❯(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞✳ ✸✳ ❙❤♦✇ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ❞❡♥s✐t② ❘(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞ ✹✳ ❊st❛❜❧✐s❤ ❛♣♣r♦♣r✐❛t❡ ❙✲❝♦♥t✐♥✉✐t② ✐♥ t❤❡ t✐♠❡ ✈❛r✐❛❜❧❡ τ ✺✳ ❚❛❦❡ st❛♥❞❛r❞ ♣❛rts ✉(◦τ, ω) = ◦❯(τ, ω) ❛♥❞ ρ(◦τ, ω) = ◦❘(τ, ω) ✻✳ ❙❤♦✇ t❤❛t t❤❡ ♣❛✐r ✉ ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ♦♥ t❤❡ ❛❞❛♣t❡❞ ▲♦❡❜ s♣❛❝❡

t t ✵ P

✇❤❡r❡ P

▲✱

▲ ❛♥❞

t t ✵ ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ ✵

slide-146
SLIDE 146

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✭❇r♦❛❞❧② s✐♠✐❧❛r t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✳✮ ✶✳ ❙♦❧✈❡ ❛ ♠♦❞✐✜❡❞ ❤②♣❡r✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❡❧♦❝✐t② ✜❡❧❞ ❯(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆✱ ✉s✐♥❣ t❤❡ tr❛♥s❢❡r ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❙❉❊ t❤❡♦r②✳ ❚❤✐s ✇✐❧❧ ❧✐✈❡ ♦♥ ❛♥ ✐♥t❡r♥❛❧ ❛❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ Ω Ω✵ = (Ω, A, (Aτ)τ≥✵, Π ) ❝❛rr②✐♥❣ ❛♥ ✐♥t❡r♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ❲ (τ, ω) ❛❧s♦ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❍◆. ❚❤❡ ❞❡♥s✐t② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ ❘(τ, ω) ✇✐t❤ ✈❛❧✉❡s ✐♥

∗❈ ✶(❉) ⊂ ∗▲∞(❉).

✷✳ Pr♦✈❡ ❛♥ ✏❡♥❡r❣② ❡st✐♠❛t❡✑ s❤♦✇✐♥❣ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ✜❡❧❞ ❯(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞✳ ✸✳ ❙❤♦✇ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ (τ, ω) t❤❡ ❞❡♥s✐t② ❘(τ, ω) ✐s ♥❡❛rst❛♥❞❛r❞ ✹✳ ❊st❛❜❧✐s❤ ❛♣♣r♦♣r✐❛t❡ ❙✲❝♦♥t✐♥✉✐t② ✐♥ t❤❡ t✐♠❡ ✈❛r✐❛❜❧❡ τ ✺✳ ❚❛❦❡ st❛♥❞❛r❞ ♣❛rts ✉(◦τ, ω) = ◦❯(τ, ω) ❛♥❞ ρ(◦τ, ω) = ◦❘(τ, ω) ✻✳ ❙❤♦✇ t❤❛t t❤❡ ♣❛✐r (✉, ρ) ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ♦♥ t❤❡ ❛❞❛♣t❡❞ ▲♦❡❜ s♣❛❝❡ Ω Ω = (Ω, F, (Ft)t≥✵, P ) ✇❤❡r❡ P = Π▲✱ F = ▲(A) ❛♥❞ (Ft)t≥✵ ✐s t❤❡ ✜❧tr❛t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ (Aτ)τ≥✵.

slide-147
SLIDE 147

❘❡❣✉❧❛r✐t② ✐♥ ❞✐♠❡♥s✐♦♥ ✷ ■♥ t❤❡ ✷❉ s❡tt✐♥❣ ✭✐✳❡✳ ❛ ✢✉✐❞ ♠♦✈✐♥❣ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✮ t❤❡r❡ ✐s ♠♦r❡ r❡❣✉❧❛r✐t② t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦✈✐❞❡❞ ❣ ❤❛s ❛ ❧✐tt❧❡ ♠♦r❡ r❡❣✉❧❛r✐t②✳

❚❤❡♦r❡♠

❙✉♣♣♦s❡ t❤❛t ❞ ✷ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉✵ ❱ ❛♥❞ ✉ ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❝♦♥str✉❝t❡❞ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r t❤❛t ❣ ✵ t ❱ ▲ ❍ ❱ ❛♥❞ ❣ t ✉

❍ ❱

❛ t ✶ ✉ ✳ ❚❤❡♥ ❛❧♠♦st s✉r❡❧②✿ ✭❛✮ s✉♣

t ✵ ❚

✉ t

❚ ✵

❆✉ t

✷❞t

✇❤❡r❡ ❆ ❀ ✭❜✮ ✉ t ✐s str♦♥❣❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❍ ❛♥❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❱❀ ✭❝✮ t❤❡ ❡q✉❛t✐♦♥ ❢♦r ✉ t ❤♦❧❞s ❢♦r ❛❧❧ ❚✵ ❚✳

slide-148
SLIDE 148

❘❡❣✉❧❛r✐t② ✐♥ ❞✐♠❡♥s✐♦♥ ✷ ■♥ t❤❡ ✷❉ s❡tt✐♥❣ ✭✐✳❡✳ ❛ ✢✉✐❞ ♠♦✈✐♥❣ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✮ t❤❡r❡ ✐s ♠♦r❡ r❡❣✉❧❛r✐t② t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦✈✐❞❡❞ ❣ ❤❛s ❛ ❧✐tt❧❡ ♠♦r❡ r❡❣✉❧❛r✐t②✳

❚❤❡♦r❡♠

❙✉♣♣♦s❡ t❤❛t ❞ = ✷ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉✵ ∈ ❱ ❛♥❞ (ρ, ✉) ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❝♦♥str✉❝t❡❞ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r t❤❛t ❣ : [✵, t] × ❱ →▲(❍, ❱) ❛♥❞ |❣(t, ✉)|❍,❱ ≤ ❛(t)(✶ + ||✉||)✳ ❚❤❡♥ ❛❧♠♦st s✉r❡❧②✿ ✭❛✮ s✉♣

t ✵ ❚

✉ t

❚ ✵

❆✉ t

✷❞t

✇❤❡r❡ ❆ ❀ ✭❜✮ ✉ t ✐s str♦♥❣❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❍ ❛♥❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❱❀ ✭❝✮ t❤❡ ❡q✉❛t✐♦♥ ❢♦r ✉ t ❤♦❧❞s ❢♦r ❛❧❧ ❚✵ ❚✳

slide-149
SLIDE 149

❘❡❣✉❧❛r✐t② ✐♥ ❞✐♠❡♥s✐♦♥ ✷ ■♥ t❤❡ ✷❉ s❡tt✐♥❣ ✭✐✳❡✳ ❛ ✢✉✐❞ ♠♦✈✐♥❣ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✮ t❤❡r❡ ✐s ♠♦r❡ r❡❣✉❧❛r✐t② t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦✈✐❞❡❞ ❣ ❤❛s ❛ ❧✐tt❧❡ ♠♦r❡ r❡❣✉❧❛r✐t②✳

❚❤❡♦r❡♠

❙✉♣♣♦s❡ t❤❛t ❞ = ✷ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉✵ ∈ ❱ ❛♥❞ (ρ, ✉) ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❝♦♥str✉❝t❡❞ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r t❤❛t ❣ : [✵, t] × ❱ →▲(❍, ❱) ❛♥❞ |❣(t, ✉)|❍,❱ ≤ ❛(t)(✶ + ||✉||)✳ ❚❤❡♥ ❛❧♠♦st s✉r❡❧②✿ ✭❛✮ s✉♣

t∈[✵,❚]

||✉(t)|| + ❚

✵ |❆✉(t)|✷❞t < ∞ ✇❤❡r❡ ❆ = −∆❀

✭❜✮ ✉ t ✐s str♦♥❣❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❍ ❛♥❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❱❀ ✭❝✮ t❤❡ ❡q✉❛t✐♦♥ ❢♦r ✉ t ❤♦❧❞s ❢♦r ❛❧❧ ❚✵ ❚✳

slide-150
SLIDE 150

❘❡❣✉❧❛r✐t② ✐♥ ❞✐♠❡♥s✐♦♥ ✷ ■♥ t❤❡ ✷❉ s❡tt✐♥❣ ✭✐✳❡✳ ❛ ✢✉✐❞ ♠♦✈✐♥❣ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✮ t❤❡r❡ ✐s ♠♦r❡ r❡❣✉❧❛r✐t② t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦✈✐❞❡❞ ❣ ❤❛s ❛ ❧✐tt❧❡ ♠♦r❡ r❡❣✉❧❛r✐t②✳

❚❤❡♦r❡♠

❙✉♣♣♦s❡ t❤❛t ❞ = ✷ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉✵ ∈ ❱ ❛♥❞ (ρ, ✉) ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❝♦♥str✉❝t❡❞ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r t❤❛t ❣ : [✵, t] × ❱ →▲(❍, ❱) ❛♥❞ |❣(t, ✉)|❍,❱ ≤ ❛(t)(✶ + ||✉||)✳ ❚❤❡♥ ❛❧♠♦st s✉r❡❧②✿ ✭❛✮ s✉♣

t∈[✵,❚]

||✉(t)|| + ❚

✵ |❆✉(t)|✷❞t < ∞ ✇❤❡r❡ ❆ = −∆❀

✭❜✮ ✉(t) ✐s str♦♥❣❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❍ ❛♥❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❱❀ ✭❝✮ t❤❡ ❡q✉❛t✐♦♥ ❢♦r ✉ t ❤♦❧❞s ❢♦r ❛❧❧ ❚✵ ❚✳

slide-151
SLIDE 151

❘❡❣✉❧❛r✐t② ✐♥ ❞✐♠❡♥s✐♦♥ ✷ ■♥ t❤❡ ✷❉ s❡tt✐♥❣ ✭✐✳❡✳ ❛ ✢✉✐❞ ♠♦✈✐♥❣ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✮ t❤❡r❡ ✐s ♠♦r❡ r❡❣✉❧❛r✐t② t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦✈✐❞❡❞ ❣ ❤❛s ❛ ❧✐tt❧❡ ♠♦r❡ r❡❣✉❧❛r✐t②✳

❚❤❡♦r❡♠

❙✉♣♣♦s❡ t❤❛t ❞ = ✷ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✉✵ ∈ ❱ ❛♥❞ (ρ, ✉) ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ st♦❝❤❛st✐❝ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❝♦♥str✉❝t❡❞ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r t❤❛t ❣ : [✵, t] × ❱ →▲(❍, ❱) ❛♥❞ |❣(t, ✉)|❍,❱ ≤ ❛(t)(✶ + ||✉||)✳ ❚❤❡♥ ❛❧♠♦st s✉r❡❧②✿ ✭❛✮ s✉♣

t∈[✵,❚]

||✉(t)|| + ❚

✵ |❆✉(t)|✷❞t < ∞ ✇❤❡r❡ ❆ = −∆❀

✭❜✮ ✉(t) ✐s str♦♥❣❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❍ ❛♥❞ ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ✐♥ ❱❀ ✭❝✮ t❤❡ ❡q✉❛t✐♦♥ ❢♦r ✉(t, ω) ❤♦❧❞s ❢♦r ❛❧❧ ❚✵ ≤ ❚✳

slide-152
SLIDE 152

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ✲ ✇❤❛t ♠❛❦❡s ♥♦♥st❛♥❞❛r❞ ♠❡t❤♦❞s ✉s❡❢✉❧ ✐♥ t❤❡ st✉❞② ♦❢ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s❄ ✶✳ ◆♦ ♥❡❡❞ ❢♦r ❧✐♠✐t✐♥❣ ❛r❣✉♠❡♥ts ❛♥❞ s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s t♦ ❣❡t ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡ ❢r♦♠ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ■♥ ❢❛❝t t❤❡ s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s ✭❛♥❞ t❤❡ ❛♣♣r♦♣r✐❛t❡ t♦♣♦❧♦❣②✮ ❛r❡ ❞✐s❝♦✈❡r❡❞ ❛s ❜②✲♣r♦❞✉❝ts✳ ✷✳ ❚❤❡ r✐❝❤♥❡ss ♦❢ ▲♦❡❜ s♣❛❝❡s ♠❡❛♥s t❤❛t ❛❧❧ ❛❝t✐✈✐t② ❝❛♥ t❛❦❡ ♣❧❛❝❡ ✐♥ ❛ s✐♥❣❧❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✲ ♥♦t ♦♥❧② ❝♦♥✈❡♥✐❡♥t ❜✉t ❡ss❡♥t✐❛❧ ❢♦r ❢♦r♠✉❧❛t✐♥❣ s♦♠❡ ✐❞❡❛s ✲ ❡❣ ♣r♦❝❡ss ❛ttr❛❝t♦rs ❛♥❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ✐♥ ✸❉✳

slide-153
SLIDE 153

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ✲ ✇❤❛t ♠❛❦❡s ♥♦♥st❛♥❞❛r❞ ♠❡t❤♦❞s ✉s❡❢✉❧ ✐♥ t❤❡ st✉❞② ♦❢ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s❄ ✶✳ ◆♦ ♥❡❡❞ ❢♦r ❧✐♠✐t✐♥❣ ❛r❣✉♠❡♥ts ❛♥❞ s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s t♦ ❣❡t ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡ ❢r♦♠ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ■♥ ❢❛❝t t❤❡ s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s ✭❛♥❞ t❤❡ ❛♣♣r♦♣r✐❛t❡ t♦♣♦❧♦❣②✮ ❛r❡ ❞✐s❝♦✈❡r❡❞ ❛s ❜②✲♣r♦❞✉❝ts✳ ✷✳ ❚❤❡ r✐❝❤♥❡ss ♦❢ ▲♦❡❜ s♣❛❝❡s ♠❡❛♥s t❤❛t ❛❧❧ ❛❝t✐✈✐t② ❝❛♥ t❛❦❡ ♣❧❛❝❡ ✐♥ ❛ s✐♥❣❧❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✲ ♥♦t ♦♥❧② ❝♦♥✈❡♥✐❡♥t ❜✉t ❡ss❡♥t✐❛❧ ❢♦r ❢♦r♠✉❧❛t✐♥❣ s♦♠❡ ✐❞❡❛s ✲ ❡❣ ♣r♦❝❡ss ❛ttr❛❝t♦rs ❛♥❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ✐♥ ✸❉✳

slide-154
SLIDE 154

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ✲ ✇❤❛t ♠❛❦❡s ♥♦♥st❛♥❞❛r❞ ♠❡t❤♦❞s ✉s❡❢✉❧ ✐♥ t❤❡ st✉❞② ♦❢ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s❄ ✶✳ ◆♦ ♥❡❡❞ ❢♦r ❧✐♠✐t✐♥❣ ❛r❣✉♠❡♥ts ❛♥❞ s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s t♦ ❣❡t ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡ ❢r♦♠ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ■♥ ❢❛❝t t❤❡ s♣❡❝✐❛❧✐③❡❞ ❝♦♠♣❛❝t♥❡ss t❤❡♦r❡♠s ✭❛♥❞ t❤❡ ❛♣♣r♦♣r✐❛t❡ t♦♣♦❧♦❣②✮ ❛r❡ ❞✐s❝♦✈❡r❡❞ ❛s ❜②✲♣r♦❞✉❝ts✳ ✷✳ ❚❤❡ r✐❝❤♥❡ss ♦❢ ▲♦❡❜ s♣❛❝❡s ♠❡❛♥s t❤❛t ❛❧❧ ❛❝t✐✈✐t② ❝❛♥ t❛❦❡ ♣❧❛❝❡ ✐♥ ❛ s✐♥❣❧❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✲ ♥♦t ♦♥❧② ❝♦♥✈❡♥✐❡♥t ❜✉t ❡ss❡♥t✐❛❧ ❢♦r ❢♦r♠✉❧❛t✐♥❣ s♦♠❡ ✐❞❡❛s ✲ ❡❣ ♣r♦❝❡ss ❛ttr❛❝t♦rs ❛♥❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ✐♥ ✸❉✳