r sts sts rt - - PowerPoint PPT Presentation

▶

Mar 04, 2023 214 likes •1.02k views

trt r sts s srt tst t sP r sts

SLIDE 1

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

❆❧❜❡rt♦ P♦❧✐❝r✐t✐

❉♣t✳ ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❯❞✐♥❡✳ ❆♣♣❧✐❡❞ ●❡♥♦♠✐❝s ■♥st✐t✉t❡ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ▲✉❝❛ ❇♦rt♦❧✉ss✐✮

❏✉♥❡ ✼t❤✱ ✷✵✵✽

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 2

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❖✉t❧✐♥❡

✶ ❍②❜r✐❞ ❙②st❡♠s✿ ❞❡✜♥✐t✐♦♥ ❛♥❞ s❛♠♣❧❡ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❙②st❡♠s

❇✐♦❧♦❣②❀

✷ ❛ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss ❆❧❣❡❜r❛ ✭❙P❆✮ ❢♦r ❜✐♦❧♦❣✐❝❛❧ ♠♦❞❡❧✐♥❣✿

s❈❈P❀

✸ ❡✣❝✐❡♥❝②✿ ❢r♦♠ s❈❈P t♦ ❖❉❊✬s❀ ✹ ❜❡✐♥❣ ♠♦r❡ ✏❞✐s❝r❡t❡✑✿ t❤❡ ❝✐r❝❛❞✐❛♥ ❝❧♦❝❦✳ ❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 3

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❙②st❡♠s

▼❛♥② r❡❛❧ s②st❡♠s ❤❛✈❡ ❛ ❞♦✉❜❧❡ ♥❛t✉r❡✳ ❚❤❡②✿ ❡✈♦❧✈❡ ✐♥ ❛ ❝♦♥t✐♥✉♦✉s ✇❛②✱ ❛r❡ r✉❧❡❞ ❜② ❛ ❞✐s❝r❡t❡ s②st❡♠✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 4

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❙②st❡♠s

▼❛♥② r❡❛❧ s②st❡♠s ❤❛✈❡ ❛ ❞♦✉❜❧❡ ♥❛t✉r❡✳ ❚❤❡②✿ ❡✈♦❧✈❡ ✐♥ ❛ ❝♦♥t✐♥✉♦✉s ✇❛②✱ ❛r❡ r✉❧❡❞ ❜② ❛ ❞✐s❝r❡t❡ s②st❡♠✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 5

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❙②st❡♠s

▼❛♥② r❡❛❧ s②st❡♠s ❤❛✈❡ ❛ ❞♦✉❜❧❡ ♥❛t✉r❡✳ ❚❤❡②✿ ❡✈♦❧✈❡ ✐♥ ❛ ❝♦♥t✐♥✉♦✉s ✇❛②✱ ❛r❡ r✉❧❡❞ ❜② ❛ ❞✐s❝r❡t❡ s②st❡♠✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 6

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❙②st❡♠s

▼❛♥② r❡❛❧ s②st❡♠s ❤❛✈❡ ❛ ❞♦✉❜❧❡ ♥❛t✉r❡✳ ❚❤❡②✿ ❡✈♦❧✈❡ ✐♥ ❛ ❝♦♥t✐♥✉♦✉s ✇❛②✱ ❛r❡ r✉❧❡❞ ❜② ❛ ❞✐s❝r❡t❡ s②st❡♠✳ ▼♦❞❡❧✐♥❣❄ ❤②❜r✐❞ s②st❡♠s✴❛✉t♦♠❛t❛

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 7

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❚❤❡ ❊①❛♠♣❧❡

❆ t❤❡r♠♦st❛t ♠♦❞❡❧

˙ X = −krX ˙ X = kh − krX X < 15; X′ = X X ≥ 20; X′ = X Off On

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 8

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙②♥t❛①

❆❧✉r ❡t ❛❧✳ ✶✾✾✷ ❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ✲ ❙②♥t❛①✮ ❆ t✉♣❧❡ ❍ = ❩, ❩ ′, V, E, ■♥✈, ❉②♥, ❆❝t, ❘❡s❡t ✇❤❡r❡✿ ❩ ❛♥❞ ❩ ′ ❛r❡ ✈❛r✐❜❧❡s ✐♥ R❦ V, E ✐s ❛ ❣r❛♣❤ ❊❛❝❤ ✈ ∈ V ✐s ❧❛❜❡❧❧❡❞ ❜② ■♥✈(✈)[❩] ❛♥❞ ❉②♥(✈)[❩, ❩ ′, ❚] ❊❛❝❤ ❡ ∈ E ✐s ❧❛❜❡❧❧❡❞ ❜② ❆❝t(❡)[❩] ❛♥❞ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 9

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙②♥t❛①

❆❧✉r ❡t ❛❧✳ ✶✾✾✷ ❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ✲ ❙②♥t❛①✮ ❆ t✉♣❧❡ ❍ = ❩, ❩ ′, V, E, ■♥✈, ❉②♥, ❆❝t, ❘❡s❡t ✇❤❡r❡✿ ❩ ❛♥❞ ❩ ′ ❛r❡ ✈❛r✐❜❧❡s ✐♥ R❦ V, E ✐s ❛ ❣r❛♣❤ ❊❛❝❤ ✈ ∈ V ✐s ❧❛❜❡❧❧❡❞ ❜② ■♥✈(✈)[❩] ❛♥❞ ❉②♥(✈)[❩, ❩ ′, ❚] ❊❛❝❤ ❡ ∈ E ✐s ❧❛❜❡❧❧❡❞ ❜② ❆❝t(❡)[❩] ❛♥❞ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 10

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙②♥t❛①

❆❧✉r ❡t ❛❧✳ ✶✾✾✷ ❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ✲ ❙②♥t❛①✮ ❆ t✉♣❧❡ ❍ = ❩, ❩ ′, V, E, ■♥✈, ❉②♥, ❆❝t, ❘❡s❡t ✇❤❡r❡✿ ❩ ❛♥❞ ❩ ′ ❛r❡ ✈❛r✐❜❧❡s ✐♥ R❦ V, E ✐s ❛ ❣r❛♣❤ ❊❛❝❤ ✈ ∈ V ✐s ❧❛❜❡❧❧❡❞ ❜② ■♥✈(✈)[❩] ❛♥❞ ❉②♥(✈)[❩, ❩ ′, ❚] ❊❛❝❤ ❡ ∈ E ✐s ❧❛❜❡❧❧❡❞ ❜② ❆❝t(❡)[❩] ❛♥❞ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 11

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙②♥t❛①

❆❧✉r ❡t ❛❧✳ ✶✾✾✷ ❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ✲ ❙②♥t❛①✮ ❆ t✉♣❧❡ ❍ = ❩, ❩ ′, V, E, ■♥✈, ❉②♥, ❆❝t, ❘❡s❡t ✇❤❡r❡✿ ❩ ❛♥❞ ❩ ′ ❛r❡ ✈❛r✐❜❧❡s ✐♥ R❦ V, E ✐s ❛ ❣r❛♣❤ ❊❛❝❤ ✈ ∈ V ✐s ❧❛❜❡❧❧❡❞ ❜② ■♥✈(✈)[❩] ❛♥❞ ❉②♥(✈)[❩, ❩ ′, ❚] ❊❛❝❤ ❡ ∈ E ✐s ❧❛❜❡❧❧❡❞ ❜② ❆❝t(❡)[❩] ❛♥❞ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 12

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙②♥t❛①

❆❧✉r ❡t ❛❧✳ ✶✾✾✷ ❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ✲ ❙②♥t❛①✮ ❆ t✉♣❧❡ ❍ = ❩, ❩ ′, V, E, ■♥✈, ❉②♥, ❆❝t, ❘❡s❡t ✇❤❡r❡✿ ❩ ❛♥❞ ❩ ′ ❛r❡ ✈❛r✐❜❧❡s ✐♥ R❦ V, E ✐s ❛ ❣r❛♣❤ ❊❛❝❤ ✈ ∈ V ✐s ❧❛❜❡❧❧❡❞ ❜② ■♥✈(✈)[❩] ❛♥❞ ❉②♥(✈)[❩, ❩ ′, ❚] ❊❛❝❤ ❡ ∈ E ✐s ❧❛❜❡❧❧❡❞ ❜② ❆❝t(❡)[❩] ❛♥❞ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 13

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙②♥t❛①

❆❧✉r ❡t ❛❧✳ ✶✾✾✷ ❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ✲ ❙②♥t❛①✮ ❆ t✉♣❧❡ ❍ = ❩, ❩ ′, V, E, ■♥✈, ❉②♥, ❆❝t, ❘❡s❡t ✇❤❡r❡✿ ❩ ❛♥❞ ❩ ′ ❛r❡ ✈❛r✐❜❧❡s ✐♥ R❦ V, E ✐s ❛ ❣r❛♣❤ ❊❛❝❤ ✈ ∈ V ✐s ❧❛❜❡❧❧❡❞ ❜② ■♥✈(✈)[❩] ❛♥❞ ❉②♥(✈)[❩, ❩ ′, ❚] ❊❛❝❤ ❡ ∈ E ✐s ❧❛❜❡❧❧❡❞ ❜② ❆❝t(❡)[❩] ❛♥❞ ❘❡s❡t(❡)[❩, ❩ ′] ❉②♥(✈)[❩, ❩ ′, ❚] ✐s ❛ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❢♦r♠ ❩ ′ = ♣✈(❩, ❚)✱ ✇❤❡r❡ ♣✈ ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r✐❛❧ ✜❡❧❞ P(✈)✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 14

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ■♥t✉✐t✐✈❡❧②

❋✐♥✐t❡ ❆✉t♦♠❛t❛ ♣❧✉s ❚✐♠❡ ✐♥ ♠♦❞❡ ✈✱ ❩ ♠✉st ❛❧✇❛②s s❛t✐s❢② ■♥✈(✈)[❩] ❍ ❡✈♦❧✈❡s ❢r♦♠ ❩ t♦ ❩ ′ ✐♥ t✐♠❡ ❚ ✇❤❡♥ ❉②♥(✈)[❩, ❩ ′, ❚] ❍ ❝❛♥ ❝r♦ss ❡ ♦♥❧② ✐❢ ❆❝t(❡)[❩] ✇❤❡♥ ❍ ❝r♦ss❡s ❡✱ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 15

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ■♥t✉✐t✐✈❡❧②

❋✐♥✐t❡ ❆✉t♦♠❛t❛ ♣❧✉s ❚✐♠❡ ✐♥ ♠♦❞❡ ✈✱ ❩ ♠✉st ❛❧✇❛②s s❛t✐s❢② ■♥✈(✈)[❩] ❍ ❡✈♦❧✈❡s ❢r♦♠ ❩ t♦ ❩ ′ ✐♥ t✐♠❡ ❚ ✇❤❡♥ ❉②♥(✈)[❩, ❩ ′, ❚] ❍ ❝❛♥ ❝r♦ss ❡ ♦♥❧② ✐❢ ❆❝t(❡)[❩] ✇❤❡♥ ❍ ❝r♦ss❡s ❡✱ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 16

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ■♥t✉✐t✐✈❡❧②

❋✐♥✐t❡ ❆✉t♦♠❛t❛ ♣❧✉s ❚✐♠❡ ✐♥ ♠♦❞❡ ✈✱ ❩ ♠✉st ❛❧✇❛②s s❛t✐s❢② ■♥✈(✈)[❩] ❍ ❡✈♦❧✈❡s ❢r♦♠ ❩ t♦ ❩ ′ ✐♥ t✐♠❡ ❚ ✇❤❡♥ ❉②♥(✈)[❩, ❩ ′, ❚] ❍ ❝❛♥ ❝r♦ss ❡ ♦♥❧② ✐❢ ❆❝t(❡)[❩] ✇❤❡♥ ❍ ❝r♦ss❡s ❡✱ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 17

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ■♥t✉✐t✐✈❡❧②

❋✐♥✐t❡ ❆✉t♦♠❛t❛ ♣❧✉s ❚✐♠❡ ✐♥ ♠♦❞❡ ✈✱ ❩ ♠✉st ❛❧✇❛②s s❛t✐s❢② ■♥✈(✈)[❩] ❍ ❡✈♦❧✈❡s ❢r♦♠ ❩ t♦ ❩ ′ ✐♥ t✐♠❡ ❚ ✇❤❡♥ ❉②♥(✈)[❩, ❩ ′, ❚] ❍ ❝❛♥ ❝r♦ss ❡ ♦♥❧② ✐❢ ❆❝t(❡)[❩] ✇❤❡♥ ❍ ❝r♦ss❡s ❡✱ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 18

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ■♥t✉✐t✐✈❡❧②

❋✐♥✐t❡ ❆✉t♦♠❛t❛ ♣❧✉s ❚✐♠❡ ✐♥ ♠♦❞❡ ✈✱ ❩ ♠✉st ❛❧✇❛②s s❛t✐s❢② ■♥✈(✈)[❩] ❍ ❡✈♦❧✈❡s ❢r♦♠ ❩ t♦ ❩ ′ ✐♥ t✐♠❡ ❚ ✇❤❡♥ ❉②♥(✈)[❩, ❩ ′, ❚] ❍ ❝❛♥ ❝r♦ss ❡ ♦♥❧② ✐❢ ❆❝t(❡)[❩] ✇❤❡♥ ❍ ❝r♦ss❡s ❡✱ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 19

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ■♥t✉✐t✐✈❡❧②

❋✐♥✐t❡ ❆✉t♦♠❛t❛ ♣❧✉s ❚✐♠❡ ❚✐♠❡ ✢♦✇s ✇❤❡♥ ✇✐t❤✐♥ st❛t❡s✿ ✐♥ ♠♦❞❡ ✈✱ ❩ ♠✉st ❛❧✇❛②s s❛t✐s❢② ■♥✈(✈)[❩] ❍ ❡✈♦❧✈❡s ❢r♦♠ ❩ t♦ ❩ ′ ✐♥ t✐♠❡ ❚ ✇❤❡♥ ❉②♥(✈)[❩, ❩ ′, ❚] ❍ ❝❛♥ ❝r♦ss ❡ ♦♥❧② ✐❢ ❆❝t(❡)[❩] ✇❤❡♥ ❍ ❝r♦ss❡s ❡✱ ❘❡s❡t(❡)[❩, ❩ ′]

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 20

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙t❛t❡s ❛♥❞ ❚r❛♥s✐t✐♦♥s

❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ❙t❛t❡✮ ❆ st❛t❡ ✐s ❛ ♣❛✐r ✐♥ V × R❦✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 21

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙t❛t❡s ❛♥❞ ❚r❛♥s✐t✐♦♥s

❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ❙t❛t❡✮ ❆ st❛t❡ ✐s ❛ ♣❛✐r ✐♥ V × R❦✳ ❉❡❢✐♥✐t✐♦♥ ✭❈♦♥t✐♥✉♦✉s ❚r❛♥s✐t✐♦♥✮ ✈, r t − →❈ ✈, s ⇐ ⇒ ∃❢ : R+ → R❦ ❝♦♥t✐♥✉♦✉s s✉❝❤ t❤❛t r = ❢ (✵)✱ s = ❢ (t)✱ ❛♥❞ ∀t′ ∈ [✵, t] t❤❡ ❢♦r♠✉❧æ ■♥✈(✈)[❢ (t′)] ❛♥❞ ❉②♥(✈)[r, ❢ (t′), t′] ❤♦❧❞✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 22

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍②❜r✐❞ ❆✉t♦♠❛t❛ ✲ ❙t❛t❡s ❛♥❞ ❚r❛♥s✐t✐♦♥s

❉❡❢✐♥✐t✐♦♥ ✭❍②❜r✐❞ ❆✉t♦♠❛t♦♥ ❙t❛t❡✮ ❆ st❛t❡ ✐s ❛ ♣❛✐r ✐♥ V × R❦✳ ❉❡❢✐♥✐t✐♦♥ ✭❈♦♥t✐♥✉♦✉s ❚r❛♥s✐t✐♦♥✮ ✈, r t − →❈ ✈, s ⇐ ⇒ ∃❢ : R+ → R❦ ❝♦♥t✐♥✉♦✉s s✉❝❤ t❤❛t r = ❢ (✵)✱ s = ❢ (t)✱ ❛♥❞ ∀t′ ∈ [✵, t] t❤❡ ❢♦r♠✉❧æ ■♥✈(✈)[❢ (t′)] ❛♥❞ ❉②♥(✈)[r, ❢ (t′), t′] ❤♦❧❞✳ ❉❡❢✐♥✐t✐♦♥ ✭❉✐s❝r❡t❡ ❚r❛♥s✐t✐♦♥✮ ✈, r

✈,✉

− − − →❉ ✉, s ⇐ ⇒ ✈, ✉ ∈ E ❛♥❞ ■♥✈(✈)[r]✱ ❆❝t(✈, ✉)[r]✱ ❘❡s❡t(✈, ✉)[r, s]✱ ❛♥❞ ■♥✈(✉)[s] ❤♦❧❞✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 23

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛ ❜❛❝t❡r✐✉♠ ❞❡t❡❝t✐♥❣ t❤❡ ❢♦♦❞ ❝♦♥❝❡♥tr❛t✐♦♥ t❤r♦✉❣❤ ❛ s❡t ♦❢ r❡❝❡♣t♦rs❀ ♠♦✈✐♥❣ ❜② ✢❛❣❡❧❧❛r r♦t❛t✐♦♥s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 24

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛ ❜❛❝t❡r✐✉♠ ❞❡t❡❝t✐♥❣ t❤❡ ❢♦♦❞ ❝♦♥❝❡♥tr❛t✐♦♥ t❤r♦✉❣❤ ❛ s❡t ♦❢ r❡❝❡♣t♦rs❀ ♠♦✈✐♥❣ ❜② ✢❛❣❡❧❧❛r r♦t❛t✐♦♥s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 25

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛ ❜❛❝t❡r✐✉♠ ❞❡t❡❝t✐♥❣ t❤❡ ❢♦♦❞ ❝♦♥❝❡♥tr❛t✐♦♥ t❤r♦✉❣❤ ❛ s❡t ♦❢ r❡❝❡♣t♦rs❀ ♠♦✈✐♥❣ ❜② ✢❛❣❡❧❧❛r r♦t❛t✐♦♥s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 26

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛ ❜❛❝t❡r✐✉♠ ❞❡t❡❝t✐♥❣ t❤❡ ❢♦♦❞ ❝♦♥❝❡♥tr❛t✐♦♥ t❤r♦✉❣❤ ❛ s❡t ♦❢ r❡❝❡♣t♦rs❀ ♠♦✈✐♥❣ ❜② ✢❛❣❡❧❧❛r r♦t❛t✐♦♥s✳

❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❛ttr❛❝t❛♥s ❛♥❞ r❡♣❡❧❧❡♥ts✱ ❊✳ ❝♦❧✐ r❡s♣♦♥❞s t♦ st✐♠✉❧✐ ✐♥ ♦♥❡ ♦❢ t✇♦ ✇❛②s✿ ✏❘❯◆❙✑ ✕ ✐t ♠♦✈❡s ✐♥ ❛ str❛✐❣❤t ❧✐♥❡ ❜② ♠♦✈✐♥❣ ✐ts ✢❛❣❡❧❧❛ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ✭❈❈❲✮ ✏❚❯▼❇▲❊❙✑ ✕ ✐t r❛♥❞♦♠❧② ❝❤❛♥❣❡s ✐ts ❤❡❛❞✐♥❣ ❜② ♠♦✈✐♥❣ ✐ts ✢❛❣❡❧❧❛ ❝❧♦❝❦✇✐s❡ ✭❈❲✮

❊✳ ❈♦❧✐ ▼♦❞❡❧

Y Y p B Bp A W Tar Z CW rotation Demethylation

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 27

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛ ❜❛❝t❡r✐✉♠ ❞❡t❡❝t✐♥❣ t❤❡ ❢♦♦❞ ❝♦♥❝❡♥tr❛t✐♦♥ t❤r♦✉❣❤ ❛ s❡t ♦❢ r❡❝❡♣t♦rs❀ ♠♦✈✐♥❣ ❜② ✢❛❣❡❧❧❛r r♦t❛t✐♦♥s✳

❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❛ttr❛❝t❛♥s ❛♥❞ r❡♣❡❧❧❡♥ts✱ ❊✳ ❝♦❧✐ r❡s♣♦♥❞s t♦ st✐♠✉❧✐ ✐♥ ♦♥❡ ♦❢ t✇♦ ✇❛②s✿ ✏❘❯◆❙✑ ✕ ✐t ♠♦✈❡s ✐♥ ❛ str❛✐❣❤t ❧✐♥❡ ❜② ♠♦✈✐♥❣ ✐ts ✢❛❣❡❧❧❛ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ✭❈❈❲✮ ✏❚❯▼❇▲❊❙✑ ✕ ✐t r❛♥❞♦♠❧② ❝❤❛♥❣❡s ✐ts ❤❡❛❞✐♥❣ ❜② ♠♦✈✐♥❣ ✐ts ✢❛❣❡❧❧❛ ❝❧♦❝❦✇✐s❡ ✭❈❲✮

❊✳ ❈♦❧✐ ▼♦❞❡❧

Y Y p B Bp A W Tar Z CW rotation Demethylation

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 28

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛ ❜❛❝t❡r✐✉♠ ❞❡t❡❝t✐♥❣ t❤❡ ❢♦♦❞ ❝♦♥❝❡♥tr❛t✐♦♥ t❤r♦✉❣❤ ❛ s❡t ♦❢ r❡❝❡♣t♦rs❀ ♠♦✈✐♥❣ ❜② ✢❛❣❡❧❧❛r r♦t❛t✐♦♥s✳

❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❛ttr❛❝t❛♥s ❛♥❞ r❡♣❡❧❧❡♥ts✱ ❊✳ ❝♦❧✐ r❡s♣♦♥❞s t♦ st✐♠✉❧✐ ✐♥ ♦♥❡ ♦❢ t✇♦ ✇❛②s✿ ✏❘❯◆❙✑ ✕ ✐t ♠♦✈❡s ✐♥ ❛ str❛✐❣❤t ❧✐♥❡ ❜② ♠♦✈✐♥❣ ✐ts ✢❛❣❡❧❧❛ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ✭❈❈❲✮ ✏❚❯▼❇▲❊❙✑ ✕ ✐t r❛♥❞♦♠❧② ❝❤❛♥❣❡s ✐ts ❤❡❛❞✐♥❣ ❜② ♠♦✈✐♥❣ ✐ts ✢❛❣❡❧❧❛ ❝❧♦❝❦✇✐s❡ ✭❈❲✮

❊✳ ❈♦❧✐ ▼♦❞❡❧

Y Y p B Bp A W Tar Z CW rotation Demethylation

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 29

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

❊✳ ❝♦❧✐ ■❉❆ ▼♦❞❡❧

y = Yp

Y0 > θ ∧ ω′ = +1 ∧ Y ′ P = YP ∧ Y ′ 0 = Y0 ∧

B′

P = BP ∧ B′ 0 = B0 ∧ Z′ = Z ∧ P ′ = P

y = Yp

Y0 < θ ∧ ω′ = −1 ∧ Y ′ P = YP ∧ Y ′ 0 = Y0 ∧

B′

P = BP ∧ B′ 0 = B0 ∧ Z′ = Z ∧ P ′ = P

ω = −1 ˙ YP = kyP(Y0 − YP ) − k−yZYP ˙ BP = kbP(B0 − BP ) − k−bBP P = LT2p + LT3p + LT4p+ T2p + T3p + T4p ω = +1 ˙ YP = kyP(Y0 − YP ) − k−yZYP ˙ BP = kbP(B0 − BP ) − k−bBP P = LT2p + LT3p + LT4p+ T2p + T3p + T4p RUN [CCW] TUMBLE [CW]

❆✳ ❈❛s❛❣r❛♥❞❡ ❡t ❛❧✳✱ ■♥❞❡♣❡♥❞❡♥t ❉②♥❛♠✐❝s ❍②❜r✐❞ ❆✉t♦♠❛t❛ ✐♥ ❙②st❡♠s ❇✐♦❧♦❣②✱ ❆❇✭✬✵✺✮ ❚♦❦②♦✱ ✷✵✵✺

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 30

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

P❛r❛♠❡t❡rs ✐♥ ●❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ❯s❡ ✏✇❡❧❧ ❜❡❤❛✈✐♥❣✑ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❡✳❣✳ ♣✐❡❝❡✲✇✐s❡ ♠✉❧t✐ ❛✣♥❡ ❢✉♥❝t✐♦♥s✮❀ ✉s❡ t❡♠♣♦r❛❧ ❧♦❣✐❝ t♦ ❡①♣r❡ss ❞②♥❛♠✐❝❛❧ ♣r♦♣❡rt✐❡s❀ ♣❛rt✐t✐♦♥ t❤❡ ♣❛r❛♠❡t❡rs✬s♣❛❝❡ ✐♥ s✉❝❤ ❛ ✇❛② t♦ ❣✉❛r❛♥t❡❡ ✈❛❧✐❞✐t② ♦❢ t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s✳ P▼✲❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣

❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ✇✐t❤

❆♣♣❧✐❝❛t✐♦♥s t♦ ❙②♥t❤❡t✐❝ ❇✐♦❧♦❣②✱

✳ ❇❛tt ❛♥❞ ❈✳ ❇❡❧t❛

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 31

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

P❛r❛♠❡t❡rs ✐♥ ●❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ❯s❡ ✏✇❡❧❧ ❜❡❤❛✈✐♥❣✑ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❡✳❣✳ ♣✐❡❝❡✲✇✐s❡ ♠✉❧t✐ ❛✣♥❡ ❢✉♥❝t✐♦♥s✮❀ ✉s❡ t❡♠♣♦r❛❧ ❧♦❣✐❝ t♦ ❡①♣r❡ss ❞②♥❛♠✐❝❛❧ ♣r♦♣❡rt✐❡s❀ ♣❛rt✐t✐♦♥ t❤❡ ♣❛r❛♠❡t❡rs✬s♣❛❝❡ ✐♥ s✉❝❤ ❛ ✇❛② t♦ ❣✉❛r❛♥t❡❡ ✈❛❧✐❞✐t② ♦❢ t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s✳ P▼✲❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣

❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ✇✐t❤

❆♣♣❧✐❝❛t✐♦♥s t♦ ❙②♥t❤❡t✐❝ ❇✐♦❧♦❣②✱

✳ ❇❛tt ❛♥❞ ❈✳ ❇❡❧t❛

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 32

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

P❛r❛♠❡t❡rs ✐♥ ●❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ❯s❡ ✏✇❡❧❧ ❜❡❤❛✈✐♥❣✑ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❡✳❣✳ ♣✐❡❝❡✲✇✐s❡ ♠✉❧t✐ ❛✣♥❡ ❢✉♥❝t✐♦♥s✮❀ ✉s❡ t❡♠♣♦r❛❧ ❧♦❣✐❝ t♦ ❡①♣r❡ss ❞②♥❛♠✐❝❛❧ ♣r♦♣❡rt✐❡s❀ ♣❛rt✐t✐♦♥ t❤❡ ♣❛r❛♠❡t❡rs✬s♣❛❝❡ ✐♥ s✉❝❤ ❛ ✇❛② t♦ ❣✉❛r❛♥t❡❡ ✈❛❧✐❞✐t② ♦❢ t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s✳ P▼✲❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣

❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ✇✐t❤

❆♣♣❧✐❝❛t✐♦♥s t♦ ❙②♥t❤❡t✐❝ ❇✐♦❧♦❣②✱

✳ ❇❛tt ❛♥❞ ❈✳ ❇❡❧t❛

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 33

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

P❛r❛♠❡t❡rs ✐♥ ●❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ❯s❡ ✏✇❡❧❧ ❜❡❤❛✈✐♥❣✑ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❡✳❣✳ ♣✐❡❝❡✲✇✐s❡ ♠✉❧t✐ ❛✣♥❡ ❢✉♥❝t✐♦♥s✮❀ ✉s❡ t❡♠♣♦r❛❧ ❧♦❣✐❝ t♦ ❡①♣r❡ss ❞②♥❛♠✐❝❛❧ ♣r♦♣❡rt✐❡s❀ ♣❛rt✐t✐♦♥ t❤❡ ♣❛r❛♠❡t❡rs✬s♣❛❝❡ ✐♥ s✉❝❤ ❛ ✇❛② t♦ ❣✉❛r❛♥t❡❡ ✈❛❧✐❞✐t② ♦❢ t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s✳ P▼✲❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣

❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ✇✐t❤

❆♣♣❧✐❝❛t✐♦♥s t♦ ❙②♥t❤❡t✐❝ ❇✐♦❧♦❣②✱

✳ ❇❛tt ❛♥❞ ❈✳ ❇❡❧t❛

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 34

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

P❛r❛♠❡t❡rs ✐♥ ●❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ❯s❡ ✏✇❡❧❧ ❜❡❤❛✈✐♥❣✑ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❡✳❣✳ ♣✐❡❝❡✲✇✐s❡ ♠✉❧t✐ ❛✣♥❡ ❢✉♥❝t✐♦♥s✮❀ ✉s❡ t❡♠♣♦r❛❧ ❧♦❣✐❝ t♦ ❡①♣r❡ss ❞②♥❛♠✐❝❛❧ ♣r♦♣❡rt✐❡s❀ ♣❛rt✐t✐♦♥ t❤❡ ♣❛r❛♠❡t❡rs✬s♣❛❝❡ ✐♥ s✉❝❤ ❛ ✇❛② t♦ ❣✉❛r❛♥t❡❡ ✈❛❧✐❞✐t② ♦❢ t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s✳ P▼✲❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣

❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ✇✐t❤

❆♣♣❧✐❝❛t✐♦♥s t♦ ❙②♥t❤❡t✐❝ ❇✐♦❧♦❣②✱

✳ ❇❛tt ❛♥❞ ❈✳ ❇❡❧t❛

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 35

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

P❛r❛♠❡t❡rs ✐♥ ●❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ❯s❡ ✏✇❡❧❧ ❜❡❤❛✈✐♥❣✑ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❡✳❣✳ ♣✐❡❝❡✲✇✐s❡ ♠✉❧t✐ ❛✣♥❡ ❢✉♥❝t✐♦♥s✮❀ ✉s❡ t❡♠♣♦r❛❧ ❧♦❣✐❝ t♦ ❡①♣r❡ss ❞②♥❛♠✐❝❛❧ ♣r♦♣❡rt✐❡s❀ ♣❛rt✐t✐♦♥ t❤❡ ♣❛r❛♠❡t❡rs✬s♣❛❝❡ ✐♥ s✉❝❤ ❛ ✇❛② t♦ ❣✉❛r❛♥t❡❡ ✈❛❧✐❞✐t② ♦❢ t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s✳ P▼✲❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣

❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ✇✐t❤

❆♣♣❧✐❝❛t✐♦♥s t♦ ❙②♥t❤❡t✐❝ ❇✐♦❧♦❣②✱

✳ ❇❛tt ❛♥❞ ❈✳ ❇❡❧t❛

✭❚②♣✐❝❛❧✮ ❑❡② Pr♦♣❡rt② ❚❤❡♦r❡♠ ✭▼✉❧t✐❛✣♥❡ ❢✉♥❝t✐♦♥s ♦♥ ❤②♣❡rr❡❝t❛♥❣✉❧❛r

♣♦❧②t♦♣❡s✮ ❢ ♠✉❧t✐❛✣♥❡ ❢✉♥❝t✐♦♥ ❛♥❞ P

❤②♣❡rr❡❝t❛♥❣✉❧❛r ♣♦❧②t♦♣❡✿ ❢ (P) ⊆ ❤✉❧❧({❢ (✈) | ✈ ∈ VP}), t❤❛t ✐s ∀① ∈ P✱ ❢ (①) ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ ❢ ❛t ✈❡rt✐❝❡s ♦❢ P✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 36

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

P❛r❛♠❡t❡rs ✐♥ ●❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ❯s❡ ✏✇❡❧❧ ❜❡❤❛✈✐♥❣✑ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❡✳❣✳ ♣✐❡❝❡✲✇✐s❡ ♠✉❧t✐ ❛✣♥❡ ❢✉♥❝t✐♦♥s✮❀ ✉s❡ t❡♠♣♦r❛❧ ❧♦❣✐❝ t♦ ❡①♣r❡ss ❞②♥❛♠✐❝❛❧ ♣r♦♣❡rt✐❡s❀ ♣❛rt✐t✐♦♥ t❤❡ ♣❛r❛♠❡t❡rs✬s♣❛❝❡ ✐♥ s✉❝❤ ❛ ✇❛② t♦ ❣✉❛r❛♥t❡❡ ✈❛❧✐❞✐t② ♦❢ t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s✳ P▼✲❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣

❡♥❡t✐❝ ❘❡❣✉❧❛t♦r② ◆❡t✇♦r❦s ✇✐t❤

❆♣♣❧✐❝❛t✐♦♥s t♦ ❙②♥t❤❡t✐❝ ❇✐♦❧♦❣②✱

✳ ❇❛tt ❛♥❞ ❈✳ ❇❡❧t❛

✭❚②♣✐❝❛❧✮ ❑❡② Pr♦♣❡rt② ❚❤❡♦r❡♠ ✭▼✉❧t✐❛✣♥❡ ❢✉♥❝t✐♦♥s ♦♥ ❤②♣❡rr❡❝t❛♥❣✉❧❛r

♣♦❧②t♦♣❡s✮ ❢ ♠✉❧t✐❛✣♥❡ ❢✉♥❝t✐♦♥ ❛♥❞ P

❤②♣❡rr❡❝t❛♥❣✉❧❛r ♣♦❧②t♦♣❡✿ ❢ (P) ⊆ ❤✉❧❧({❢ (✈) | ✈ ∈ VP}), t❤❛t ✐s ∀① ∈ P✱ ❢ (①) ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ ❢ ❛t ✈❡rt✐❝❡s ♦❢ P✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 37

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡s ♦❢ ✉s❡ ♦❢ ❍❙ ❢♦r ❙②st❡♠s ❇✐♦❧♦❣②

❙✇✐t❝❤✐♥❣ ❛♠♦♥❣ s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s ❯s❡ ❞✐✛❡r❡♥t s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s ❛s t❤❡ ♥✉♠❜❡r ♦❢ ♠♦❧❡❝✉❧❡ ✈❛r✐❡s❀

✶ st♦❝❤❛st✐❝ s✐♠✉❧❛t✐♦♥ ❢♦r ❧♦✇ ♥✉♠❜❡rs❀ ✷ ♦❞❡ s✐♠✉❧❛t✐♦♥ ❢♦r ❤✐❣❤ ♥✉♠❜❡rs❀

❆❧✉r ❡t ❛❧✳ ❍②❜r✐❞ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙✐♠✉❧❛t✐♦♥ ♦❢ ❇✐♦❝❤❡♠✐❝❛❧ ◆❡t✇♦r❦s

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 38

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

▼♦❞❡❧✐♥❣✿ ❙t♦❝❤❛st✐❝ ✈s✳ ❉✐❢❢✳ ❊q✉❛t✐♦♥s

❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ♠❛t✉r❡ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❛✛♦r❞❛❜❧❡ ✭♦♥❡ r✉♥✮ ❙t♦❝❤❛st✐❝ s♦♠❡t❤✐♥❣ ♣r❡❝✐s❡ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❝♦st❧② ✭♠❛♥② r✉♥s✦✮

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 39

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❆ ❇r✐❞❣❡✿ ♦✉r ❛tt❡♠♣t

❋✐rst st❡♣ ❯s❡ ❛ ❙t♦❝❤❛st✐❝ ✈❡rs✐♦♥ ♦❢ ❈♦♥❝✉rr❡♥t ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣ ❛s ❉✐s❝r❡t❡ ❙t♦❝❤❛st✐❝ st❛rt✐♥❣ t♦♦❧✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 40

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❆ ❇r✐❞❣❡✿ ♦✉r ❛tt❡♠♣t

❋✐rst st❡♣ ❯s❡ ❛ ❙t♦❝❤❛st✐❝ ✈❡rs✐♦♥ ♦❢ ❈♦♥❝✉rr❡♥t ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣ ❛s ❉✐s❝r❡t❡ ❙t♦❝❤❛st✐❝ st❛rt✐♥❣ t♦♦❧✳ ❙❡❝♦♥❞ st❡♣ ■♥tr♦❞✉❝❡ ❍②❜r✐❞ ❙②st❡♠s✳ ▼♦❞❡s ♦❢ t❤❡ ❍❙ ⇔ ❈♦♠❜✐♥❛t✐♦♥s ♦❢ ❙t♦❝❤❛st✐❝ ❝❤♦✐❝❡s ❉②♥❛♠✐❝s ⇔ ❆❞✲❤♦❝ ❡❞❣❡✬s ✈❛r✐❛❜❧❡s ✇✐t❤ ❛❝t✐✈❛t✐♦♥s ❝♦♥str❛✐♥❡❞ ❜② r❛t❡s

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 41

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❙t♦❝❤❛st✐❝ ❈♦♥❝✉rr❡♥t ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣

❲❤❛t ✐s ❆ ❙P❆ ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ ✏t✇✐st✑✳ ▼❛✐♥t❛✐♥s ❛ ❢♦r♠ ♦❢ ❧♦❝❛❧ st♦r❛❣❡✳ ❑❡❡♣s s❡♣❛r❛t❡❞ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ✐♥t❡r❛❝t✐♦♥s ❛♥❞ t❤❡ ♠❛♥❛❣❡♠❡♥t ♦❢ ❞❛t❛ ❢♦r ❝♦♠♣✉t❛t✐♦♥s✳ ✭◆❛t✉r❛❧❧②✮ ■♥tr♦❞✉❝❡ ❢✉♥❝t✐♦♥❛❧ r❛t❡s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 42

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈♦♥❝✉rr❡♥t ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣

❈♦♥str❛✐♥t ❙t♦r❡ ■♥ t❤✐s ♣r♦❝❡ss ❛❧❣❡❜r❛✱ t❤❡ ♠❛✐♥ ♦❜❥❡❝ts ❛r❡ ❝♦♥str❛✐♥ts ✱ ✇❤✐❝❤ ❛r❡ ❢♦r♠✉❧❛❡ ♦✈❡r ❛♥ ✐♥t❡r♣r❡t❡❞ ✜rst ♦r❞❡r ❧❛♥❣✉❛❣❡ ✭✐✳❡✳ ❳ = ✶✵✱ ❨ > ❳ − ✸✮✳ ❈♦♥str❛✐♥ts ❝❛♥ ❜❡ ❛❞❞❡❞ t♦ ❛ ✧❝♦♥t❛✐♥❡r✧✱ t❤❡ ❝♦♥str❛✐♥t st♦r❡✱ ❜✉t ❝❛♥ ♥❡✈❡r ❜❡ r❡♠♦✈❡❞✳ ❆❣❡♥ts ❆❣❡♥ts ❝❛♥ ♣❡r❢♦r♠ t✇♦ ❜❛s✐❝ ♦♣❡r❛t✐♦♥s ♦♥ t❤✐s st♦r❡ ✭❛s②♥❝❤r♦♥♦✉s❧②✮✿ ❆❞❞ ❛ ❝♦♥str❛✐♥t ✭t❡❧❧ ❛s❦✮ ❆s❦ ✐❢ ❛ ❝❡rt❛✐♥ r❡❧❛t✐♦♥ ✐s ❡♥t❛✐❧❡❞ ❜② t❤❡ ❝✉rr❡♥t ❝♦♥✜❣✉r❛t✐♦♥ ✭❛s❦ ✐♥str✉❝t✐♦♥✮

❱✳ ❙❛r❛s✇❛t✱ ❈♦♥❝✉rr❡♥t ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣✱ ▼■❚ ♣r❡ss✱ ✶✾✾✸

❙②♥t❛① ♦❢ ❈❈P

Pr♦❣r❛♠ = ❉❡❝❧.❆ ❉ = ε | ❉❡❝❧.❉❡❝❧ | ♣(①) : −❆ ❆ = ✵ | t❡❧❧(❝).❆ | ❛s❦(❝✶).❆✶ + ❛s❦(❝✷).❆✷ | ❆✶ ❆✷ | ∃① ❆ | ♣(①) ❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 43

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❙②♥t❛① ♦❢ s❈❈P

❙②♥t❛① ♦❢ ❙t♦❝❤❛st✐❝ ❈❈P

Pr♦❣r❛♠ = ❉.❆ ❉ = ε | ❉.❉ | ♣( ①) : −❆ ❆ = ✵ | t❡❧❧∞(❝).❆ | ▼ | ∃①❆ | ❆ ❆ ▼ = π.● | ▼ + ▼ π = t❡❧❧λ(❝) | ❛s❦λ(❝)

= ✵ | t❡❧❧∞(❝).● | ♣(

②) | ▼ | ∃①● | ● ●

▲✳ ❇♦rt♦❧✉ss✐✱ ❙t♦❝❤❛st✐❝ ❈♦♥❝✉rr❡♥t ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣✱ ◗❆P▲✱ ✷✵✵✻

❙t♦❝❤❛st✐❝ ❘❛t❡s ❘❛t❡s ❛r❡ ❢✉♥❝t✐♦♥s ❢r♦♠ t❤❡ ❝♦♥str❛✐♥t st♦r❡ C t♦ ♣♦s✐t✐✈❡ r❡❛❧s✿ λ : C − → R+. ❘❛t❡s ❝❛♥ ❜❡ t❤♦✉❣❤t ❛s s♣❡❡❞ ♦r ❞✉r❛t✐♦♥ ♦❢ ❝♦♠♠✉♥✐❝❛t✐♦♥s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 44

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

s❈❈P ✕ t❡❝❤♥✐❝❛❧ ❞❡t❛✐❧s

❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ❚❤❡r❡ ❛r❡ t✇♦ tr❛♥s✐t✐♦♥ r❡❧❛t✐♦♥s✱ ♦♥❡ ✐♥st❛♥t❛♥❡♦✉s ✭✜♥✐t❡ ❛♥❞ ❝♦♥✢✉❡♥t✮ ❛♥❞ ♦♥❡ st♦❝❤❛st✐❝✳ ❚r❛❝❡s ❛r❡ s❡q✉❡♥❝❡s ♦❢ ❡✈❡♥ts ✇✐t❤ ✈❛r✐❛❜❧❡ t✐♠❡ ❞❡❧❛②s ❛♠♦♥❣ t❤❡♠✳ ❉✐s❝r❡t❡ ✈s✳ ❈♦♥t✐♥✉♦✉s ❙❡♠❛♥t✐❝s ❚❤❡ ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✐s ❛❜str❛❝t ✇✳r✳t✳ t❤❡ ♥♦t✐♦♥ ♦❢ t✐♠❡✿ ✇❡ ❝❛♥ ♠❛♣ t❤❡ ❧❛❜❡❧❡❞ tr❛♥s✐t✐♦♥ s②st❡♠ ✐♥t♦ ❛ ❞✐s❝r❡t❡ ♦r ❛ ❝♦♥t✐♥✉♦✉s t✐♠❡ ▼❛r❦♦✈ ❈❤❛✐♥✳ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❲❡ ❤❛✈❡ ❛♥ ✐♥t❡r♣r❡t❡r ✇r✐tt❡♥ ✐♥ Pr♦❧♦❣✱ ✉s✐♥❣ t❤❡ ❈▲P ❡♥❣✐♥❡ ♦❢ ❙■❈❙t✉s t♦ ♠❛♥❛❣❡ t❤❡ ❝♦♥str❛✐♥t st♦r❡✳ ❊✣❝✐❡♥❝② ✐ss✉❡s✳ ❙tr❡❛♠ ❱❛r✐❛❜❧❡s ◗✉❛♥t✐t✐❡s ✈❛r②✐♥❣ ♦✈❡r t✐♠❡ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ s❈❈P ❛s ✉♥❜♦✉♥❞❡❞ ❧✐sts✳ ❍❡r❡❛❢t❡r✿ s♣❡❝✐❛❧ ♠❡❛♥✐♥❣ ♦❢ ❳ = ❳ + ✶✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 45

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈♦♥t✐♥✉♦✉s ❚✐♠❡ ▼❛r❦♦✈ ❈❤❛✐♥s

❆ ❈♦♥t✐♥✉♦✉s ❚✐♠❡ ▼❛r❦♦✈ ❈❤❛✐♥ ✭❈❚▼❈✮ ✐s ❛ ❞✐r❡❝t ❣r❛♣❤ ✇✐t❤ ❡❞❣❡s ❧❛❜❡❧❡❞ ❜② ❛ r❡❛❧ ♥✉♠❜❡r✱ ❝❛❧❧❡❞ t❤❡ r❛t❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ✭r❡♣r❡s❡♥t✐♥❣ t❤❡ s♣❡❡❞ ♦r t❤❡ ❢r❡q✉❡♥❝② ❛t ✇❤✐❝❤ t❤❡ tr❛♥s✐t✐♦♥ ♦❝❝✉rs✮✳ ■♥ ❡❛❝❤ st❛t❡✱ ✇❡ s❡❧❡❝t t❤❡ ♥❡①t st❛t❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❜t❛✐♥❡❞ ♥♦r♠❛❧✐③✐♥❣ r❛t❡s ✭❢r♦♠ ❙ t♦ ❙✶ ✇✐t❤ ♣r♦❜✳

r✶ r✶+r✷ ✮✳

❚❤❡ t✐♠❡ s♣❡♥t ✐♥ ❛ st❛t❡ ✐s ❣✐✈❡♥ ❜② ❛♥ ❡①♣♦♥❡♥t✐❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✱ ✇✐t❤ r❛t❡ ❣✐✈❡♥ ❜② t❤❡ s✉♠ ♦❢ ♦✉t❣♦✐♥❣ tr❛♥s✐t✐♦♥s ❢r♦♠ t❤❡ ❛❝t✉❛❧ ♥♦❞❡ ✭r✶ + r✷✮✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 46

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦

✭❏✳ ▼✳ ●✳ ❱✐❧❛r✱ ❍✳ ❨✉❛♥ ❑✉❡❤✱ ◆✳ ❇❛r❦❛✐✱ ❛♥❞ ❙✳ ▲❡✐❜❧❡r✳ P◆❆❙✱ ✷✵✵✷✳✮

❆ ❝❧♦❝❦ ❡①♣r❡ss✐♥❣ ♣r♦t❡✐♥s ❆ ❛♥❞ ❘ ✇✐t❤ ❛ st❛❜❧❡ ♣❡r✐♦❞

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 47

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦

tr❛♥s❝r✐♣t✐♦♥ ❛♥❞ t❤❡ tr❛♥s❧❛t✐♦♥ ♣❤❛s❡s ❛r❡ ♠♦❞❡❧❡❞ ❡①♣❧✐❝✐t❧✳ ❆ ✐s ❛♥ ❡♥❤❛♥❝❡r ❢♦r ❜♦t❤ ❣❡♥❡s ❘ r❡♣r❡ss❡s ❆ ❢♦r♠✐♥❣ ❆❘ ❛♥❞ ♠❛❦✐♥❣ ❆ ✐♥❛❝t✐✈❡ ❘ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ♦♥❧② ✐❢ ✐t ✐s ♥♦t ✐♥ ❝♦♠♣❧❡①❡❞ ❢♦r♠ ✇❤✐❧❡ ❆ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ✐♥ ❛♥② ❢♦r♠

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 48

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦

tr❛♥s❝r✐♣t✐♦♥ ❛♥❞ t❤❡ tr❛♥s❧❛t✐♦♥ ♣❤❛s❡s ❛r❡ ♠♦❞❡❧❡❞ ❡①♣❧✐❝✐t❧✳ ❆ ✐s ❛♥ ❡♥❤❛♥❝❡r ❢♦r ❜♦t❤ ❣❡♥❡s ❘ r❡♣r❡ss❡s ❆ ❢♦r♠✐♥❣ ❆❘ ❛♥❞ ♠❛❦✐♥❣ ❆ ✐♥❛❝t✐✈❡ ❘ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ♦♥❧② ✐❢ ✐t ✐s ♥♦t ✐♥ ❝♦♠♣❧❡①❡❞ ❢♦r♠ ✇❤✐❧❡ ❆ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ✐♥ ❛♥② ❢♦r♠

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 49

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦

tr❛♥s❝r✐♣t✐♦♥ ❛♥❞ t❤❡ tr❛♥s❧❛t✐♦♥ ♣❤❛s❡s ❛r❡ ♠♦❞❡❧❡❞ ❡①♣❧✐❝✐t❧✳ ❆ ✐s ❛♥ ❡♥❤❛♥❝❡r ❢♦r ❜♦t❤ ❣❡♥❡s ❘ r❡♣r❡ss❡s ❆ ❢♦r♠✐♥❣ ❆❘ ❛♥❞ ♠❛❦✐♥❣ ❆ ✐♥❛❝t✐✈❡ ❘ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ♦♥❧② ✐❢ ✐t ✐s ♥♦t ✐♥ ❝♦♠♣❧❡①❡❞ ❢♦r♠ ✇❤✐❧❡ ❆ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ✐♥ ❛♥② ❢♦r♠

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 50

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦

tr❛♥s❝r✐♣t✐♦♥ ❛♥❞ t❤❡ tr❛♥s❧❛t✐♦♥ ♣❤❛s❡s ❛r❡ ♠♦❞❡❧❡❞ ❡①♣❧✐❝✐t❧✳ ❆ ✐s ❛♥ ❡♥❤❛♥❝❡r ❢♦r ❜♦t❤ ❣❡♥❡s ❘ r❡♣r❡ss❡s ❆ ❢♦r♠✐♥❣ ❆❘ ❛♥❞ ♠❛❦✐♥❣ ❆ ✐♥❛❝t✐✈❡ ❘ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ♦♥❧② ✐❢ ✐t ✐s ♥♦t ✐♥ ❝♦♠♣❧❡①❡❞ ❢♦r♠ ✇❤✐❧❡ ❆ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ✐♥ ❛♥② ❢♦r♠

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 51

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦

tr❛♥s❝r✐♣t✐♦♥ ❛♥❞ t❤❡ tr❛♥s❧❛t✐♦♥ ♣❤❛s❡s ❛r❡ ♠♦❞❡❧❡❞ ❡①♣❧✐❝✐t❧✳ ❆ ✐s ❛♥ ❡♥❤❛♥❝❡r ❢♦r ❜♦t❤ ❣❡♥❡s ❘ r❡♣r❡ss❡s ❆ ❢♦r♠✐♥❣ ❆❘ ❛♥❞ ♠❛❦✐♥❣ ❆ ✐♥❛❝t✐✈❡ ❘ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ♦♥❧② ✐❢ ✐t ✐s ♥♦t ✐♥ ❝♦♠♣❧❡①❡❞ ❢♦r♠ ✇❤✐❧❡ ❆ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ✐♥ ❛♥② ❢♦r♠

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 52

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦

tr❛♥s❝r✐♣t✐♦♥ ❛♥❞ t❤❡ tr❛♥s❧❛t✐♦♥ ♣❤❛s❡s ❛r❡ ♠♦❞❡❧❡❞ ❡①♣❧✐❝✐t❧✳ ❆ ✐s ❛♥ ❡♥❤❛♥❝❡r ❢♦r ❜♦t❤ ❣❡♥❡s ❘ r❡♣r❡ss❡s ❆ ❢♦r♠✐♥❣ ❆❘ ❛♥❞ ♠❛❦✐♥❣ ❆ ✐♥❛❝t✐✈❡ ❘ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ♦♥❧② ✐❢ ✐t ✐s ♥♦t ✐♥ ❝♦♠♣❧❡①❡❞ ❢♦r♠ ✇❤✐❧❡ ❆ ❝❛♥ ❜❡ ❞❡❣r❛❞❡❞ ✐♥ ❛♥② ❢♦r♠ ❘♦❜✉st♥❡ss ❚❤❡ st♦❝❤❛st✐❝ ♠♦❞❡❧ ✐s ♠♦r❡ r♦❜✉st ❛s ✐❢ ✐♥t❡r♥❛❧ ♥♦✐s❡ ✇❛s ❡①♣❧♦✐t❡❞ ❜② ◆❛t✉r❡ t♦ ✐♥❝r❡❛s❡ st❛❜✐❧✐t② ♦❢ ❢✉♥❝t✐♦♥ ✭✐✳❡✳✱ ❢♦r ❛ ❝❧♦❝❦✱ ♦s❝✐❧❧❛t✐♦♥s✮✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 53

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦

♣❴❣❛t❡✭α❆, α′

❆, γ❆, θ❆, ▼❆, ❆✮

♣❴❣❛t❡✭α❘, α′

❘, γ❘, θ❘, ▼❘, ❆✮

r❡❛❝t✐♦♥✭β❆, [▼❆], [❆]✮ r❡❛❝t✐♦♥✭δ▼❆, [▼❆], []✮ r❡❛❝t✐♦♥✭β❘, [▼❘], [❘]✮ r❡❛❝t✐♦♥✭δ▼❘, [▼❘], []✮ r❡❛❝t✐♦♥✭γ❈ , [❆, ❘], [❆❘]✮ r❡❛❝t✐♦♥✭δ❆, [❆❘], [❘]✮ r❡❛❝t✐♦♥✭δ❆, [❆], []✮ r❡❛❝t✐♦♥✭δ❘, [❘], []✮ ❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 54

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❋r♦♠ s❈❈P t♦ ❖❉❊

❲❤❛t❄ ❲❡ ✇❛♥t t♦ ❛ss♦❝✐❛t❡ ❛ s❡t ♦❢ ❖❉❊ t♦ ❛♥ s❈❈P ♣r♦❣r❛♠ ✭✇r✐tt❡♥ ✐♥ ❛ r❡str✐❝t❡❞ s②♥t❛①✮✳ ❲❤②❄ ❖❉❊ ❝❛♥ ❜❡ ♥✉♠❡r✐❝❛❧❧② s✐♠✉❧❛t❡❞ ❢❛st❡r t❤❛♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❖♥ t❤❡ ♠❛r❦❡t✳✳✳ ❚❤❡r❡ ❛r❡ ✭s②♥t❛❝t✐❝✮ ♠❡t❤♦❞s t♦ ✇r✐t❡ s❡t ♦❢ ❖❉❊s ❢♦r P❊P❆ ❛♥❞ st♦❝❤❛st✐❝ π✲❝❛❧❝✉❧✉s✱ ❧♦♦❦✐♥❣ ❛t t❤❡ s♣❡❡❞ ♦❢ ❝r❡❛t✐♦♥ ❛♥❞ ❞❡str✉❝t✐♦♥ ♦❢ t❡r♠s ✭❲❡ ❞✐❞ t❤❡ s❛♠❡ ❢♦r s❈❈P✮✳ ❍♦✇❡✈❡r✱ t❤❡ ❖❉❊ ❝❛♥ s❤♦✇ ❛ ❜❡❤❛✈✐♦r ❞✐✛❡r❡♥t ❢r♦♠ t❤❛t ♦❢ ❙P❆ ♠♦❞❡❧s✳

❏✳ ❍✐❧❧st♦♥✱ ❋❧✉✐❞ ❋❧♦✇ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ P❊P❆ ♠♦❞❡❧s✱ ◗❊❙❚✱ ✷✵✵✺✳ ▲✳ ❈❛r❞❡❧❧✐✱ ❋r♦♠ Pr♦❝❡ss❡s t♦ ❖❉❊s ❜② ❈❤❡♠✐str②✱ ✷✵✵✻✳ ▲✳ ❇♦rt♦❧✉ss✐✱ ❆✳ P♦❧✐❝r✐t✐✳ ❈♦♥♥❡❝t✐♥❣ Pr♦❝❡ss ❆❧❣❡❜r❛s ❛♥❞ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ❢♦r s②st❡♠s ❜✐♦❧♦❣②✱ ✷✵✵✻✳ ❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 55

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❋r♦♠ s❈❈P t♦ ❖❉❊✿ ❡①❛♠♣❧❡

■❞❡❛ ❈♦❧❧❛♣s❡ ❛❧❧ ✐♥st❛♥t❛♥❡♦✉s tr❛♥s✐t✐♦♥s ❢♦❧❧♦✇✐♥❣ ❛ st♦❝❤❛st✐❝ ♦♥❡ ❛♥❞ ❛❞❞ t❤❡✐r ✉♣❞❛t❡s t♦ t❤❡ ❡❞❣❡✬s ❧❛❜❡❧ ❞❡♥♦t✐♥❣ s✉❝❤ ❛ tr❛♥s✐t✐♦♥✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 56

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❋r♦♠ s❈❈P t♦ ❖❉❊✿ ❡①❛♠♣❧❡

■❞❡❛ ❈♦❧❧❛♣s❡ ❛❧❧ ✐♥st❛♥t❛♥❡♦✉s tr❛♥s✐t✐♦♥s ❢♦❧❧♦✇✐♥❣ ❛ st♦❝❤❛st✐❝ ♦♥❡ ❛♥❞ ❛❞❞ t❤❡✐r ✉♣❞❛t❡s t♦ t❤❡ ❡❞❣❡✬s ❧❛❜❡❧ ❞❡♥♦t✐♥❣ s✉❝❤ ❛ tr❛♥s✐t✐♦♥✳ ❘❡❞✉❝❡❞ ❚r❛♥s✐t✐♦♥s ❙②st❡♠s ❆ss♦❝✐❛t❡ ❛ ❧❛❜❡❧❡❞ ❣r❛♣❤ t♦ ❡❛❝❤ s❡q✉❡♥t✐❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ❛♥ s❈❈P ♣r♦❣r❛♠✿ ❡❞❣❡s ❛r❡ tr❛♥s✐t✐♦♥s ❛♥❞ ❛r❡ ❧❛❜❡❧❡❞ ❜② ❛ s❡t ♦❢ ❣✉❛r❞s✱ ❛ s❡t ♦❢ ✉♣❞❛t❡s ♦❢ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ st♦r❡✱ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❛t❡s❀ ♥♦❞❡s ❛r❡ st♦❝❤❛st✐❝ ❝❤♦✐❝❡s✳ Pr♦❝❡❞✉r❡ ❝❛❧❧s ❛r❡ r❡s♦❧✈❡❞ ❜② ✐♥s❡rt✐♥❣ ❛ ❝♦♣② ♦❢ t❤❡ ❝❛❧❧❡❞ ♣r♦❝❡❞✉r❡✳ ❙②♥t❛❝t✐❝ r❡str✐❝t✐♦♥s ❛r❡ ♥❡❝❡ss❛r✐❡s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 57

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❋r♦♠ s❈❈P t♦ ❖❉❊✿ ❡①❛♠♣❧❡

❊①❛♠♣❧❡ ❆ ✿✲ ❛s❦λ✶✭tr✉❡✮✳t❡❧❧∞✭❳ = ❳ + ✶✮✳❇ ❇ ✿✲ t❡❧❧λ✷✭❳ = ❳ − ✶✮✳❆ ❚❤❡ ❘❚❙

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 58

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❋r♦♠ s❈❈P t♦ ❖❉❊✿ ❡①❛♠♣❧❡

■♥t❡r❛❝t✐♦♥ ♠❛tr✐① ❛♥❞① r❡❛❝t✐♦♥ ✈❡❝t♦r ■ = t✶ t✷ ❳ ✶ −✶ ❆ −✶ ✶ ❇ ✶ −✶ r = λ✶ · ❆ λ✷ · ❇

♦❞❡ = ■ · r

♦❞❡    ˙ ❳ = λ✶ · ❆ − λ✷ · ❇ ˙ ❆ = −λ✶ · ❆ + λ✷ · ❇ ˙ ❇ = λ✶ · ❆ − λ✷ · ❇

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 59

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❊①❛♠♣❧❡✿ ❛ ✏❞✐st✐❧❧❡❞✑ ❘❡♣r❡ss✐❧❛t♦r

❆ ✿✲ t❡❧❧❦+ ✭❳ = ❳ + ✶✮✳❆ ✰ ❛s❦❦−❳ ✭❳ > ✵✮✳ t❡❧❧∞✭❳ = ❳ − ✶✮✳❆ ✰ ❛s❦❦✵ ✭tr✉❡✮✳❇ ❇ ✿✲ t❡❧❧❦− ✭❳ = ❳ + ✶✮✳❇ ✰ ❛s❦❦+❳ ✭❳ > ✵✮✳ t❡❧❧∞✭❳ = ❳ − ✶✮✳❇ ✰ ❛s❦❦✵ ✭tr✉❡✮✳❆ ❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 60

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

■❞❡❛s ❧♦❝❛❧✐③❡ t❤❡ ❝♦♥str✉❝t✐♦♥ t♦ ❧♦♦♣✐♥❣ ❡❞❣❡s ✐♥ ♦r❞❡r t♦ ❞❡t❡r♠✐♥❡ ✢♦✇ ❝♦♥❞✐t✐♦♥s❀ ✉s❡ ✭♥♦♥ ❝♦♥st❛♥t✮ r❛t❡s t♦ ❣♦✈❡r♥ ✈❛r✐❛❜❧❡s ❛ss♦❝✐❛t❡❞ t♦ ❡❞❣❡s❀ ✉s❡ ✈❛r✐❛❜❧❡s ❛ss♦❝✐❛t❡❞ t♦ ❡❞❣❡s ✐♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 61

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

■❞❡❛s ❧♦❝❛❧✐③❡ t❤❡ ❝♦♥str✉❝t✐♦♥ t♦ ❧♦♦♣✐♥❣ ❡❞❣❡s ✐♥ ♦r❞❡r t♦ ❞❡t❡r♠✐♥❡ ✢♦✇ ❝♦♥❞✐t✐♦♥s❀ ✉s❡ ✭♥♦♥ ❝♦♥st❛♥t✮ r❛t❡s t♦ ❣♦✈❡r♥ ✈❛r✐❛❜❧❡s ❛ss♦❝✐❛t❡❞ t♦ ❡❞❣❡s❀ ✉s❡ ✈❛r✐❛❜❧❡s ❛ss♦❝✐❛t❡❞ t♦ ❡❞❣❡s ✐♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 62

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

■❞❡❛s ❧♦❝❛❧✐③❡ t❤❡ ❝♦♥str✉❝t✐♦♥ t♦ ❧♦♦♣✐♥❣ ❡❞❣❡s ✐♥ ♦r❞❡r t♦ ❞❡t❡r♠✐♥❡ ✢♦✇ ❝♦♥❞✐t✐♦♥s❀ ✉s❡ ✭♥♦♥ ❝♦♥st❛♥t✮ r❛t❡s t♦ ❣♦✈❡r♥ ✈❛r✐❛❜❧❡s ❛ss♦❝✐❛t❡❞ t♦ ❡❞❣❡s❀ ✉s❡ ✈❛r✐❛❜❧❡s ❛ss♦❝✐❛t❡❞ t♦ ❡❞❣❡s ✐♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 63

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

◆ = ❆✶ . . . ❆▼ ❜❡ ❛♥ s❈❈P✲♥❡t✇♦r❦✳ ❉❡❢✐♥✐t✐♦♥ ✭s❦❡t❝❤✮

✶

❝♦♥tr♦❧ ♠♦❞❡s Σ = (σ✶, . . . , σ▼)❀

✷

❝♦♥tr♦❧ ❡❞❣❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♥♦♥✲❧♦♦♣✐♥❣ ❛r❝s t✐❥ ∈ ❚✐ ♦❢ ❘❚❙(❆✐)❀

✸

✈❛r✐❛❜❧❡s✿ str❡❛♠ ✈❛r✐❛❜❧❡s ❳✶, . . . , ❳❦ ♦❢ ◆✱ ♣❧✉s ♦♥❡ ✈❛r✐❛❜❧❡ ❨✐,❥ ❢♦r ❡❛❝❤ ❘❚❙✲❡❞❣❡ t✐❥❀

✹

✢♦✇ ❝♦♥❞✐t✐♦♥s ♦❞❡Σ = ▼

✐=✶ ♦❞❡✐,σ✐ , ✇❤❡r❡ ♦❞❡✐,σ✐ = ■✐,σ✐ · r✐,σ✐ .

▼♦r❡♦✈❡r✱ ✐❢ t❤❡ ❧❛❜❡❧ ♦❢ t✐❥ ✐s (❣✐❥, ❝✐❥, λ✐❥)✱ ˙ ❨✐❥ = λ✐❥(❳✶, . . . , ❳❦)❀

✺

❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✐s t❤❡ ♣r❡❞✐❝❛t❡ ❣✐❥ ∧ ❨✐❥ ≥ ✶, ✇❤❡r❡ ❣✐❥ ✐s t❤❡ ❣✉❛r❞ ♣r❡❞✐❝❛t❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥❀

✻

r❡s❡ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✇✐t❤ ❝✐❥ = ❤✐❥

❦=✶ ❳✐❦ = ❳✐❦ + δ✐❥✱

 

❤✐❥

❦=✶

❳ ′

✐❦ = ❳✐❦ + δ✐❥

  ∧  

t✐❥ ∈❚✐

❨ ′

✐❥ = ✵

  .

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 64

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

◆ = ❆✶ . . . ❆▼ ❜❡ ❛♥ s❈❈P✲♥❡t✇♦r❦✳ ❉❡❢✐♥✐t✐♦♥ ✭s❦❡t❝❤✮

✶

❝♦♥tr♦❧ ♠♦❞❡s Σ = (σ✶, . . . , σ▼)❀

✷

❝♦♥tr♦❧ ❡❞❣❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♥♦♥✲❧♦♦♣✐♥❣ ❛r❝s t✐❥ ∈ ❚✐ ♦❢ ❘❚❙(❆✐)❀

✸

✈❛r✐❛❜❧❡s✿ str❡❛♠ ✈❛r✐❛❜❧❡s ❳✶, . . . , ❳❦ ♦❢ ◆✱ ♣❧✉s ♦♥❡ ✈❛r✐❛❜❧❡ ❨✐,❥ ❢♦r ❡❛❝❤ ❘❚❙✲❡❞❣❡ t✐❥❀

✹

✢♦✇ ❝♦♥❞✐t✐♦♥s ♦❞❡Σ = ▼

✐=✶ ♦❞❡✐,σ✐ , ✇❤❡r❡ ♦❞❡✐,σ✐ = ■✐,σ✐ · r✐,σ✐ .

▼♦r❡♦✈❡r✱ ✐❢ t❤❡ ❧❛❜❡❧ ♦❢ t✐❥ ✐s (❣✐❥, ❝✐❥, λ✐❥)✱ ˙ ❨✐❥ = λ✐❥(❳✶, . . . , ❳❦)❀

✺

❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✐s t❤❡ ♣r❡❞✐❝❛t❡ ❣✐❥ ∧ ❨✐❥ ≥ ✶, ✇❤❡r❡ ❣✐❥ ✐s t❤❡ ❣✉❛r❞ ♣r❡❞✐❝❛t❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥❀

✻

r❡s❡ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✇✐t❤ ❝✐❥ = ❤✐❥

❦=✶ ❳✐❦ = ❳✐❦ + δ✐❥✱

 

❤✐❥

❦=✶

❳ ′

✐❦ = ❳✐❦ + δ✐❥

  ∧  

t✐❥ ∈❚✐

❨ ′

✐❥ = ✵

  .

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 65

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

◆ = ❆✶ . . . ❆▼ ❜❡ ❛♥ s❈❈P✲♥❡t✇♦r❦✳ ❉❡❢✐♥✐t✐♦♥ ✭s❦❡t❝❤✮

✶

❝♦♥tr♦❧ ♠♦❞❡s Σ = (σ✶, . . . , σ▼)❀

✷

❝♦♥tr♦❧ ❡❞❣❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♥♦♥✲❧♦♦♣✐♥❣ ❛r❝s t✐❥ ∈ ❚✐ ♦❢ ❘❚❙(❆✐)❀

✸

✈❛r✐❛❜❧❡s✿ str❡❛♠ ✈❛r✐❛❜❧❡s ❳✶, . . . , ❳❦ ♦❢ ◆✱ ♣❧✉s ♦♥❡ ✈❛r✐❛❜❧❡ ❨✐,❥ ❢♦r ❡❛❝❤ ❘❚❙✲❡❞❣❡ t✐❥❀

✹

✢♦✇ ❝♦♥❞✐t✐♦♥s ♦❞❡Σ = ▼

✐=✶ ♦❞❡✐,σ✐ , ✇❤❡r❡ ♦❞❡✐,σ✐ = ■✐,σ✐ · r✐,σ✐ .

▼♦r❡♦✈❡r✱ ✐❢ t❤❡ ❧❛❜❡❧ ♦❢ t✐❥ ✐s (❣✐❥, ❝✐❥, λ✐❥)✱ ˙ ❨✐❥ = λ✐❥(❳✶, . . . , ❳❦)❀

✺

❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✐s t❤❡ ♣r❡❞✐❝❛t❡ ❣✐❥ ∧ ❨✐❥ ≥ ✶, ✇❤❡r❡ ❣✐❥ ✐s t❤❡ ❣✉❛r❞ ♣r❡❞✐❝❛t❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥❀

✻

r❡s❡ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✇✐t❤ ❝✐❥ = ❤✐❥

❦=✶ ❳✐❦ = ❳✐❦ + δ✐❥✱

 

❤✐❥

❦=✶

❳ ′

✐❦ = ❳✐❦ + δ✐❥

  ∧  

t✐❥ ∈❚✐

❨ ′

✐❥ = ✵

  .

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 66

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

◆ = ❆✶ . . . ❆▼ ❜❡ ❛♥ s❈❈P✲♥❡t✇♦r❦✳ ❉❡❢✐♥✐t✐♦♥ ✭s❦❡t❝❤✮

✶

❝♦♥tr♦❧ ♠♦❞❡s Σ = (σ✶, . . . , σ▼)❀

✷

❝♦♥tr♦❧ ❡❞❣❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♥♦♥✲❧♦♦♣✐♥❣ ❛r❝s t✐❥ ∈ ❚✐ ♦❢ ❘❚❙(❆✐)❀

✸

✈❛r✐❛❜❧❡s✿ str❡❛♠ ✈❛r✐❛❜❧❡s ❳✶, . . . , ❳❦ ♦❢ ◆✱ ♣❧✉s ♦♥❡ ✈❛r✐❛❜❧❡ ❨✐,❥ ❢♦r ❡❛❝❤ ❘❚❙✲❡❞❣❡ t✐❥❀

✹

✢♦✇ ❝♦♥❞✐t✐♦♥s ♦❞❡Σ = ▼

✐=✶ ♦❞❡✐,σ✐ , ✇❤❡r❡ ♦❞❡✐,σ✐ = ■✐,σ✐ · r✐,σ✐ .

▼♦r❡♦✈❡r✱ ✐❢ t❤❡ ❧❛❜❡❧ ♦❢ t✐❥ ✐s (❣✐❥, ❝✐❥, λ✐❥)✱ ˙ ❨✐❥ = λ✐❥(❳✶, . . . , ❳❦)❀

✺

❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✐s t❤❡ ♣r❡❞✐❝❛t❡ ❣✐❥ ∧ ❨✐❥ ≥ ✶, ✇❤❡r❡ ❣✐❥ ✐s t❤❡ ❣✉❛r❞ ♣r❡❞✐❝❛t❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥❀

✻

r❡s❡ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✇✐t❤ ❝✐❥ = ❤✐❥

❦=✶ ❳✐❦ = ❳✐❦ + δ✐❥✱

 

❤✐❥

❦=✶

❳ ′

✐❦ = ❳✐❦ + δ✐❥

  ∧  

t✐❥ ∈❚✐

❨ ′

✐❥ = ✵

  .

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 67

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

◆ = ❆✶ . . . ❆▼ ❜❡ ❛♥ s❈❈P✲♥❡t✇♦r❦✳ ❉❡❢✐♥✐t✐♦♥ ✭s❦❡t❝❤✮

✶

❝♦♥tr♦❧ ♠♦❞❡s Σ = (σ✶, . . . , σ▼)❀

✷

❝♦♥tr♦❧ ❡❞❣❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♥♦♥✲❧♦♦♣✐♥❣ ❛r❝s t✐❥ ∈ ❚✐ ♦❢ ❘❚❙(❆✐)❀

✸

✈❛r✐❛❜❧❡s✿ str❡❛♠ ✈❛r✐❛❜❧❡s ❳✶, . . . , ❳❦ ♦❢ ◆✱ ♣❧✉s ♦♥❡ ✈❛r✐❛❜❧❡ ❨✐,❥ ❢♦r ❡❛❝❤ ❘❚❙✲❡❞❣❡ t✐❥❀

✹

✢♦✇ ❝♦♥❞✐t✐♦♥s ♦❞❡Σ = ▼

✐=✶ ♦❞❡✐,σ✐ , ✇❤❡r❡ ♦❞❡✐,σ✐ = ■✐,σ✐ · r✐,σ✐ .

▼♦r❡♦✈❡r✱ ✐❢ t❤❡ ❧❛❜❡❧ ♦❢ t✐❥ ✐s (❣✐❥, ❝✐❥, λ✐❥)✱ ˙ ❨✐❥ = λ✐❥(❳✶, . . . , ❳❦)❀

✺

❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✐s t❤❡ ♣r❡❞✐❝❛t❡ ❣✐❥ ∧ ❨✐❥ ≥ ✶, ✇❤❡r❡ ❣✐❥ ✐s t❤❡ ❣✉❛r❞ ♣r❡❞✐❝❛t❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥❀

✻

r❡s❡ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✇✐t❤ ❝✐❥ = ❤✐❥

❦=✶ ❳✐❦ = ❳✐❦ + δ✐❥✱

 

❤✐❥

❦=✶

❳ ′

✐❦ = ❳✐❦ + δ✐❥

  ∧  

t✐❥ ∈❚✐

❨ ′

✐❥ = ✵

  .

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 68

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❍❆ ❛ss♦❝✐❛t❡❞ t♦ ❛♥ s❈❈P✲♥t✇r❦

◆ = ❆✶ . . . ❆▼ ❜❡ ❛♥ s❈❈P✲♥❡t✇♦r❦✳ ❉❡❢✐♥✐t✐♦♥ ✭s❦❡t❝❤✮

✶

❝♦♥tr♦❧ ♠♦❞❡s Σ = (σ✶, . . . , σ▼)❀

✷

❝♦♥tr♦❧ ❡❞❣❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♥♦♥✲❧♦♦♣✐♥❣ ❛r❝s t✐❥ ∈ ❚✐ ♦❢ ❘❚❙(❆✐)❀

✸

✈❛r✐❛❜❧❡s✿ str❡❛♠ ✈❛r✐❛❜❧❡s ❳✶, . . . , ❳❦ ♦❢ ◆✱ ♣❧✉s ♦♥❡ ✈❛r✐❛❜❧❡ ❨✐,❥ ❢♦r ❡❛❝❤ ❘❚❙✲❡❞❣❡ t✐❥❀

✹

✢♦✇ ❝♦♥❞✐t✐♦♥s ♦❞❡Σ = ▼

✐=✶ ♦❞❡✐,σ✐ , ✇❤❡r❡ ♦❞❡✐,σ✐ = ■✐,σ✐ · r✐,σ✐ .

▼♦r❡♦✈❡r✱ ✐❢ t❤❡ ❧❛❜❡❧ ♦❢ t✐❥ ✐s (❣✐❥, ❝✐❥, λ✐❥)✱ ˙ ❨✐❥ = λ✐❥(❳✶, . . . , ❳❦)❀

✺

❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✐s t❤❡ ♣r❡❞✐❝❛t❡ ❣✐❥ ∧ ❨✐❥ ≥ ✶, ✇❤❡r❡ ❣✐❥ ✐s t❤❡ ❣✉❛r❞ ♣r❡❞✐❝❛t❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥❀

✻

r❡s❡ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✐❥✱ ✇✐t❤ ❝✐❥ = ❤✐❥

❦=✶ ❳✐❦ = ❳✐❦ + δ✐❥✱

 

❤✐❥

❦=✶

❳ ′

✐❦ = ❳✐❦ + δ✐❥

  ∧  

t✐❥ ∈❚✐

❨ ′

✐❥ = ✵

  .

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 69

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❖♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s

❚✐♠❡ ✈❛r②✐♥❣ r❛t❡s λ = λ(t) ✐♥tr♦❞✉❝❡ ♦♥❡ ✈❛r✐❛❜❧❡ ❨❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ❡ ❡✈❡r② tr❛♥s✐t✐♦♥ ❝♦♥st✐t✉t❡ ❛ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❝✉♠✉❧❛t✐✈❡ r❛t❡ ❢✉♥❝t✐♦♥ Λ(t) = t

t✵

λ(s)❞s, t❤❡♦r② ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ⇒ ♥✉♠❜❡r ♦❢ ✜r✐♥❣s ❛t t✐♠❡ t ❜❡❤❛✈❡s ❧✐❦❡ ❛ P♦✐ss♦♥ ✈❛r✐❛❜❧❡ ✇✐t❤ r❛t❡ ❡q✉❛❧ t♦ Λ(t) ✇❡ ♠❛② ❛❝t✐✈❛t❡ t❤❡ tr❛♥s✐t✐♦♥ ✇❤❡♥❡✈❡r Λ(t) ≥ ✶

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 70

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❖♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s

❚✐♠❡ ✈❛r②✐♥❣ r❛t❡s λ = λ(t) ✐♥tr♦❞✉❝❡ ♦♥❡ ✈❛r✐❛❜❧❡ ❨❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ❡ ❡✈❡r② tr❛♥s✐t✐♦♥ ❝♦♥st✐t✉t❡ ❛ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❝✉♠✉❧❛t✐✈❡ r❛t❡ ❢✉♥❝t✐♦♥ Λ(t) = t

t✵

λ(s)❞s, t❤❡♦r② ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ⇒ ♥✉♠❜❡r ♦❢ ✜r✐♥❣s ❛t t✐♠❡ t ❜❡❤❛✈❡s ❧✐❦❡ ❛ P♦✐ss♦♥ ✈❛r✐❛❜❧❡ ✇✐t❤ r❛t❡ ❡q✉❛❧ t♦ Λ(t) ✇❡ ♠❛② ❛❝t✐✈❛t❡ t❤❡ tr❛♥s✐t✐♦♥ ✇❤❡♥❡✈❡r Λ(t) ≥ ✶

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 71

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❖♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s

❚✐♠❡ ✈❛r②✐♥❣ r❛t❡s λ = λ(t) ✐♥tr♦❞✉❝❡ ♦♥❡ ✈❛r✐❛❜❧❡ ❨❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ❡ ❡✈❡r② tr❛♥s✐t✐♦♥ ❝♦♥st✐t✉t❡ ❛ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❝✉♠✉❧❛t✐✈❡ r❛t❡ ❢✉♥❝t✐♦♥ Λ(t) = t

t✵

λ(s)❞s, t❤❡♦r② ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ⇒ ♥✉♠❜❡r ♦❢ ✜r✐♥❣s ❛t t✐♠❡ t ❜❡❤❛✈❡s ❧✐❦❡ ❛ P♦✐ss♦♥ ✈❛r✐❛❜❧❡ ✇✐t❤ r❛t❡ ❡q✉❛❧ t♦ Λ(t) ✇❡ ♠❛② ❛❝t✐✈❛t❡ t❤❡ tr❛♥s✐t✐♦♥ ✇❤❡♥❡✈❡r Λ(t) ≥ ✶

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 72

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❖♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s

❚✐♠❡ ✈❛r②✐♥❣ r❛t❡s λ = λ(t) ✐♥tr♦❞✉❝❡ ♦♥❡ ✈❛r✐❛❜❧❡ ❨❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ❡ ❡✈❡r② tr❛♥s✐t✐♦♥ ❝♦♥st✐t✉t❡ ❛ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❝✉♠✉❧❛t✐✈❡ r❛t❡ ❢✉♥❝t✐♦♥ Λ(t) = t

t✵

λ(s)❞s, t❤❡♦r② ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ⇒ ♥✉♠❜❡r ♦❢ ✜r✐♥❣s ❛t t✐♠❡ t ❜❡❤❛✈❡s ❧✐❦❡ ❛ P♦✐ss♦♥ ✈❛r✐❛❜❧❡ ✇✐t❤ r❛t❡ ❡q✉❛❧ t♦ Λ(t) ✇❡ ♠❛② ❛❝t✐✈❛t❡ t❤❡ tr❛♥s✐t✐♦♥ ✇❤❡♥❡✈❡r Λ(t) ≥ ✶

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 73

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❖♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s

❚✐♠❡ ✈❛r②✐♥❣ r❛t❡s λ = λ(t) ✐♥tr♦❞✉❝❡ ♦♥❡ ✈❛r✐❛❜❧❡ ❨❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ❡ ❡✈❡r② tr❛♥s✐t✐♦♥ ❝♦♥st✐t✉t❡ ❛ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❝✉♠✉❧❛t✐✈❡ r❛t❡ ❢✉♥❝t✐♦♥ Λ(t) = t

t✵

λ(s)❞s, t❤❡♦r② ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ⇒ ♥✉♠❜❡r ♦❢ ✜r✐♥❣s ❛t t✐♠❡ t ❜❡❤❛✈❡s ❧✐❦❡ ❛ P♦✐ss♦♥ ✈❛r✐❛❜❧❡ ✇✐t❤ r❛t❡ ❡q✉❛❧ t♦ Λ(t) ✇❡ ♠❛② ❛❝t✐✈❛t❡ t❤❡ tr❛♥s✐t✐♦♥ ✇❤❡♥❡✈❡r Λ(t) ≥ ✶

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 74

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❖♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s

❚✐♠❡ ✈❛r②✐♥❣ r❛t❡s λ = λ(t) ✐♥tr♦❞✉❝❡ ♦♥❡ ✈❛r✐❛❜❧❡ ❨❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ❡ ❡✈❡r② tr❛♥s✐t✐♦♥ ❝♦♥st✐t✉t❡ ❛ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❝✉♠✉❧❛t✐✈❡ r❛t❡ ❢✉♥❝t✐♦♥ Λ(t) = t

t✵

λ(s)❞s, t❤❡♦r② ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ⇒ ♥✉♠❜❡r ♦❢ ✜r✐♥❣s ❛t t✐♠❡ t ❜❡❤❛✈❡s ❧✐❦❡ ❛ P♦✐ss♦♥ ✈❛r✐❛❜❧❡ ✇✐t❤ r❛t❡ ❡q✉❛❧ t♦ Λ(t) ✇❡ ♠❛② ❛❝t✐✈❛t❡ t❤❡ tr❛♥s✐t✐♦♥ ✇❤❡♥❡✈❡r Λ(t) ≥ ✶

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 75

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❖♥ ❛❝t✐✈❛t✐♦♥ ❝♦♥❞✐t✐♦♥s

❚✐♠❡ ✈❛r②✐♥❣ r❛t❡s λ = λ(t) ✐♥tr♦❞✉❝❡ ♦♥❡ ✈❛r✐❛❜❧❡ ❨❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ❡ ❡✈❡r② tr❛♥s✐t✐♦♥ ❝♦♥st✐t✉t❡ ❛ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❝✉♠✉❧❛t✐✈❡ r❛t❡ ❢✉♥❝t✐♦♥ Λ(t) = t

t✵

λ(s)❞s, t❤❡♦r② ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ⇒ ♥✉♠❜❡r ♦❢ ✜r✐♥❣s ❛t t✐♠❡ t ❜❡❤❛✈❡s ❧✐❦❡ ❛ P♦✐ss♦♥ ✈❛r✐❛❜❧❡ ✇✐t❤ r❛t❡ ❡q✉❛❧ t♦ Λ(t) ✇❡ ♠❛② ❛❝t✐✈❛t❡ t❤❡ tr❛♥s✐t✐♦♥ ✇❤❡♥❡✈❡r Λ(t) ≥ ✶ ˙ ❨❡ = λ(❳✶, . . . , ❳❦) ❛♥❞ ❨❡ ≥ ✶

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 76

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙ ❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 77

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦✿ r♦❜✉st♥❡ss

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 78

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈✐r❝❛❞✐❛♥ ❈❧♦❝❦✿ r♦❜✉st♥❡ss

20 40 60 80 100 ODE HA SHA CTMC Oscillations with stable period Oscillations with unstable period

❋✐❣✉r❡✿ ❙t❛❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ β❘

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②

SLIDE 79

■♥tr♦❞✉❝t✐♦♥ ❍②❜r✐❞ ❙②st❡♠s ❊①❛♠♣❧❡s ❉✐s❝r❡t❡ ✭❙t♦❝❤❛st✐❝✮ ❙✐♠✉❧❛t✐♦♥ s❈❈P ❛♥❞ ❍❙

❈♦♥❝❧✉s✐♦♥s ❍❙ ❢♦r✿ ❜✐♦❝❤❡♠✐❝❛❧ r❡❛❝t✐♦♥s✱ ❣❡♥❡t✐❝ ♥❡t✇♦r❦s✱ ❡t❝✳ ❙P❆ t♦ ❖❉❊✿ ♣r♦❜❧❡♠s ✭t❤❡ st♦❝❤❛st✐❝ ❝♦♠♣♦♥❡♥t ❛✈❡r❛❣❡❞ ❛✇❛②✮✳ ▲♦❝❛❧✐③❡ ❖❉❊✬s ❛♥❞ ♠❛✐♥t❛✐♥ ❛ ❞✐s❝r❡t❡ ♣♦rt✐♦♥ ♦❢ t❤❡ ♥❡t✇♦r❦✿ ❍②❜r✐❞ ❙②st❡♠s ✭✇✐t❤ t❤❡ r✐❣❤t ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡s✮✳ ❋✉t✉r❡ ❉❡✜♥❡ ❛ ❧❛tt✐❝❡ ♦❢ ❍❙s✳ ❋♦r♠❛❧✐③❡ t❤❡ ❜❡❤❛✈✐♦r❛❧ ♣r♦♣❡rt✐❡s t♦ ❣✉✐❞❡✴❞❡t❡r♠✐♥❡ t❤❡ ❧❡✈❡❧ ♦❢ ❞✐s❝r❡t❡♥❡ss t♦ ♠❛✐♥t❛✐♥✳

❆✳ P♦❧✐❝r✐t✐ ❍②❜r✐❞ ❙②st❡♠s ❛♥❞ ❙②st❡♠s ❇✐♦❧♦❣②