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r rr r rs - - PowerPoint PPT Presentation

Aug 25, 2022 256 likes •1.51k views

r rr r rs r t q P s r

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SLIDE 1

❚❤❡ ♥✉♠❜❡r ♦❢ ❝♦r♥❡r ♣♦❧②❤❡❞r❛ ❣r❛♣❤s

❈♦♠♠♦♥ ✇♦r❦ ✇✐t❤ ❉♦♠✐♥✐q✉❡ P♦✉❧❛❧❤♦♥ ❛♥❞ ●✐❧❧❡s ❙❝❤❛❡✛❡r ❈❧é♠❡♥t ❉❡r✈✐❡✉①

■❘■❋✱ ❯♥✐✈❡rs✐té P❛r✐s✲❉✐❞❡r♦t

❏♦✉r♥é❡s ❆▲❊❆ ✼ ♠❛rs ✷✵✶✻

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SLIDE 2

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

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SLIDE 3

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❆ ❝♦r♥❡r ♣♦❧②❤❡❞r♦♥ P

✵

✵ ✵ ✵ ✱ ❡❛❝❤ ❡❞❣❡ ♦❢ ✐s ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①✐s✱ ❡①❛❝t❧② ✸ ❡❞❣❡s ♦❢ ♠❡❡t ❛t ❡❛❝❤ ✈❡rt❡①✱ ❛❧❧ ✈❡rt✐❝❡s ♦❢ ❜✉t

✵ ❛r❡ ✈✐s✐❜❧❡ ❢r♦♠ ✐♥✜♥✐t② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ✶ ✶ ✶ ✳

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SLIDE 4

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❆ ❝♦r♥❡r ♣♦❧②❤❡❞r♦♥ P

v✵ = (✵, ✵, ✵) ∈ P✱ ❡❛❝❤ ❡❞❣❡ ♦❢ ✐s ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①✐s✱ ❡①❛❝t❧② ✸ ❡❞❣❡s ♦❢ ♠❡❡t ❛t ❡❛❝❤ ✈❡rt❡①✱ ❛❧❧ ✈❡rt✐❝❡s ♦❢ ❜✉t

✵ ❛r❡ ✈✐s✐❜❧❡ ❢r♦♠ ✐♥✜♥✐t② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ✶ ✶ ✶ ✳

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SLIDE 5

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❆ ❝♦r♥❡r ♣♦❧②❤❡❞r♦♥ P

v✵ = (✵, ✵, ✵) ∈ P✱ ❡❛❝❤ ❡❞❣❡ ♦❢ P ✐s ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①✐s✱ ❡①❛❝t❧② ✸ ❡❞❣❡s ♦❢ ♠❡❡t ❛t ❡❛❝❤ ✈❡rt❡①✱ ❛❧❧ ✈❡rt✐❝❡s ♦❢ ❜✉t

✵ ❛r❡ ✈✐s✐❜❧❡ ❢r♦♠ ✐♥✜♥✐t② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ✶ ✶ ✶ ✳

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SLIDE 6

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❆ ❝♦r♥❡r ♣♦❧②❤❡❞r♦♥ P

v✵ = (✵, ✵, ✵) ∈ P✱ ❡❛❝❤ ❡❞❣❡ ♦❢ P ✐s ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①✐s✱ ❡①❛❝t❧② ✸ ❡❞❣❡s ♦❢ P ♠❡❡t ❛t ❡❛❝❤ ✈❡rt❡①✱ ❛❧❧ ✈❡rt✐❝❡s ♦❢ ❜✉t

✵ ❛r❡ ✈✐s✐❜❧❡ ❢r♦♠ ✐♥✜♥✐t② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ✶ ✶ ✶ ✳

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SLIDE 7

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❆ ❝♦r♥❡r ♣♦❧②❤❡❞r♦♥ P

v✵ = (✵, ✵, ✵) ∈ P✱ ❡❛❝❤ ❡❞❣❡ ♦❢ P ✐s ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①✐s✱ ❡①❛❝t❧② ✸ ❡❞❣❡s ♦❢ P ♠❡❡t ❛t ❡❛❝❤ ✈❡rt❡①✱ ❛❧❧ ✈❡rt✐❝❡s ♦❢ P ❜✉t v✵ ❛r❡ ✈✐s✐❜❧❡ ❢r♦♠ ✐♥✜♥✐t② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ (✶, ✶, ✶)✳

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SLIDE 8

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❚❤❡ s❦❡❧❡t♦♥ ♦❢ P

❆ ✸✲r❡❣✉❧❛r ❣r❛♣❤ ❆ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤

❚❤❡♦r❡♠ ✭❲❤✐t♥❡②✭✶✾✸✷✮✮

❊❛❝❤ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♣❧❛♥❛r ❡♠❜❡❞❞✐♥❣✳ ❱✐❡✇❡❞ ❛s ❡♠❜❡❞❞❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r② s♣❤❡r❡ ♦❢ ❆ ❝✉❜✐❝ ♣❧❛♥❛r ♠❛♣ ❲❤❛t ❦✐♥❞ ♦❢ ♣❧❛♥❛r ♠❛♣ ❄

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SLIDE 9

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❚❤❡ s❦❡❧❡t♦♥ ♦❢ P

❆ ✸✲r❡❣✉❧❛r ❣r❛♣❤ ❆ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤

❚❤❡♦r❡♠ ✭❲❤✐t♥❡②✭✶✾✸✷✮✮

❊❛❝❤ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♣❧❛♥❛r ❡♠❜❡❞❞✐♥❣✳ ❱✐❡✇❡❞ ❛s ❡♠❜❡❞❞❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r② s♣❤❡r❡ ♦❢ ❆ ❝✉❜✐❝ ♣❧❛♥❛r ♠❛♣ ❲❤❛t ❦✐♥❞ ♦❢ ♣❧❛♥❛r ♠❛♣ ❄

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SLIDE 10

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❚❤❡ s❦❡❧❡t♦♥ ♦❢ P

❆ ✸✲r❡❣✉❧❛r ❣r❛♣❤ ❆ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤

❚❤❡♦r❡♠ ✭❲❤✐t♥❡②✭✶✾✸✷✮✮

❊❛❝❤ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♣❧❛♥❛r ❡♠❜❡❞❞✐♥❣✳ ❱✐❡✇❡❞ ❛s ❡♠❜❡❞❞❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r② s♣❤❡r❡ ♦❢ ❆ ❝✉❜✐❝ ♣❧❛♥❛r ♠❛♣ ❲❤❛t ❦✐♥❞ ♦❢ ♣❧❛♥❛r ♠❛♣ ❄

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SLIDE 11

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❚❤❡ s❦❡❧❡t♦♥ ♦❢ P

❆ ✸✲r❡❣✉❧❛r ❣r❛♣❤ ❆ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤

❚❤❡♦r❡♠ ✭❲❤✐t♥❡②✭✶✾✸✷✮✮

❊❛❝❤ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♣❧❛♥❛r ❡♠❜❡❞❞✐♥❣✳ ❱✐❡✇❡❞ ❛s ❡♠❜❡❞❞❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r② s♣❤❡r❡ ♦❢ ❆ ❝✉❜✐❝ ♣❧❛♥❛r ♠❛♣ ❲❤❛t ❦✐♥❞ ♦❢ ♣❧❛♥❛r ♠❛♣ ❄

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SLIDE 12

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❚❤❡ s❦❡❧❡t♦♥ ♦❢ P

❆ ✸✲r❡❣✉❧❛r ❣r❛♣❤ ❆ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤

❚❤❡♦r❡♠ ✭❲❤✐t♥❡②✭✶✾✸✷✮✮

❊❛❝❤ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♣❧❛♥❛r ❡♠❜❡❞❞✐♥❣✳ ❱✐❡✇❡❞ ❛s ❡♠❜❡❞❞❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r② s♣❤❡r❡ ♦❢ P ❆ ❝✉❜✐❝ ♣❧❛♥❛r ♠❛♣ ❲❤❛t ❦✐♥❞ ♦❢ ♣❧❛♥❛r ♠❛♣ ❄

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SLIDE 13

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❚❤❡ s❦❡❧❡t♦♥ ♦❢ P

❆ ✸✲r❡❣✉❧❛r ❣r❛♣❤ ❆ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤

❚❤❡♦r❡♠ ✭❲❤✐t♥❡②✭✶✾✸✷✮✮

❊❛❝❤ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♣❧❛♥❛r ❡♠❜❡❞❞✐♥❣✳ ❱✐❡✇❡❞ ❛s ❡♠❜❡❞❞❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r② s♣❤❡r❡ ♦❢ P ❆ ❝✉❜✐❝ ♣❧❛♥❛r ♠❛♣ P ❲❤❛t ❦✐♥❞ ♦❢ ♣❧❛♥❛r ♠❛♣ ❄

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SLIDE 14

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

❚❤❡ s❦❡❧❡t♦♥ ♦❢ P

❆ ✸✲r❡❣✉❧❛r ❣r❛♣❤ ❆ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤

❚❤❡♦r❡♠ ✭❲❤✐t♥❡②✭✶✾✸✷✮✮

❊❛❝❤ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♣❧❛♥❛r ❡♠❜❡❞❞✐♥❣✳ ❱✐❡✇❡❞ ❛s ❡♠❜❡❞❞❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r② s♣❤❡r❡ ♦❢ P ❆ ❝✉❜✐❝ ♣❧❛♥❛r ♠❛♣ P ❲❤❛t ❦✐♥❞ ♦❢ ♣❧❛♥❛r ♠❛♣ ❄

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SLIDE 15

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

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SLIDE 16

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

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SLIDE 17

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

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SLIDE 18

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

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SLIDE 19

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

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SLIDE 20

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

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SLIDE 21

■♥tr♦❞✉❝t✐♦♥ ✿ ❝♦r♥❡r ♣♦❧②❤❡❞r❛

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SLIDE 22

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥

❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡ ❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t

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SLIDE 23

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥

❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡ ❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t

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SLIDE 24

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥

❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡ ❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t

slide-25
SLIDE 25

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥

❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡ ❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t

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SLIDE 26

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥

❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡ ❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t

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SLIDE 27

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥

❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡ ❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t

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SLIDE 28

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥

▲❡t E s(y) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s

✷

✇❤❡r❡ ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✶ ✷

✷

✶

✷ ✸

t❤❡ ✜rst t❡r♠s ♦❢ ❛r❡

✹

✸ ✻ ✼ ✼ ✶✺ ✽ ✻✸ ✾

✶✵

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SLIDE 29

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥

▲❡t E s(y) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s E s(y) ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s E s(y) = C(y) − C(y)✷ ✇❤❡r❡ ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✶ ✷

✷

✶

✷ ✸

t❤❡ ✜rst t❡r♠s ♦❢ ❛r❡

✹

✸ ✻ ✼ ✼ ✶✺ ✽ ✻✸ ✾

✶✵

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SLIDE 30

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥

▲❡t E s(y) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s E s(y) ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s E s(y) = C(y) − C(y)✷ ✇❤❡r❡ C(y) ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ C(y) = y (✶ + ✷C(y))✷ (✶ + C(y) − C(y)✷)✸ . t❤❡ ✜rst t❡r♠s ♦❢ ❛r❡

✹

✸ ✻ ✼ ✼ ✶✺ ✽ ✻✸ ✾

✶✵

slide-31
SLIDE 31

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥

▲❡t E s(y) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s E s(y) ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s E s(y) = C(y) − C(y)✷ ✇❤❡r❡ C(y) ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ C(y) = y (✶ + ✷C(y))✷ (✶ + C(y) − C(y)✷)✸ . t❤❡ ✜rst t❡r♠s ♦❢ E s(y) ❛r❡ E s(y) = y + y✹ + ✸y✻ + ✼y✼ + ✶✺y✽ + ✻✸y✾ + O(y✶✵)

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SLIDE 32

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❊✉❧❡r✐❛♥ ♦r✐❡♥t❛t✐♦♥

♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✈❡rt✐❝❡s✱ ♦❢ t❤❡ ❡❞❣❡s t✇♦ ♦r✐❡♥t❡❞ ♣❛t❤s ❢r♦♠ t♦ ✿ s❛♠❡ ❧❡♥❣t❤ ♠♦❞✉❧♦ ✸

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SLIDE 33

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

❊✉❧❡r✐❛♥ ♦r✐❡♥t❛t✐♦♥

♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✈❡rt✐❝❡s✱ ♦❢ t❤❡ ❡❞❣❡s t✇♦ ♦r✐❡♥t❡❞ ♣❛t❤s ❢r♦♠ a t♦ b ✿ s❛♠❡ ❧❡♥❣t❤ ♠♦❞✉❧♦ ✸

slide-34
SLIDE 34

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥

♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s✳

slide-35
SLIDE 35

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥

♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s✳

slide-36
SLIDE 36

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥

♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s✳

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SLIDE 37

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥

♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s✳

slide-38
SLIDE 38

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥

♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s✳

slide-39
SLIDE 39

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥

♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s✳

❚❤❡♦r❡♠ ✭❊♣♣st❡✐♥ ❛♥❞ ▼✉♠❢♦r❞ ✭✷✵✶✹✮✮

❆ r♦♦t❡❞ ♣❧❛♥❛r ♠❛♣ ✐s t❤❡ s❦❡❧❡t♦♥ ♦❢ s♦♠❡ ❝♦r♥❡r ♣♦❧②❤❡❞r♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❞✉❛❧ ♠❛♣ ✐s ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥✳

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SLIDE 40

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

▲❡t E c(z) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s✳

❚❤❡♦r❡♠ ✭❉✳✱ P♦✉❧❛❧❤♦♥✱ ❙❝❤❛❡✛❡r ✭✷✵✶✺✮✮

E c(z) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ E c(z) = z ✶ + z

  • ✶ + zA(z) + z✷A(z)✷

✶ + z − z✷A(z)✷ + ✷z✸A(z)✸ (✶ + z)✷

  • .

✇❤❡r❡ A(z) ✐s t❤❡ ❈❛t❛❧❛♥ s❡r✐❡s✳

❚❤❡ ✜rst t❡r♠s

❚❤❡ ✜rst t❡r♠s ♦❢ ❛r❡

✹

✸ ✻ ✹ ✼ ✶✺ ✽ ✸✾ ✾

✶✵

slide-41
SLIDE 41

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

▲❡t E c(z) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s✳

❚❤❡♦r❡♠ ✭❉✳✱ P♦✉❧❛❧❤♦♥✱ ❙❝❤❛❡✛❡r ✭✷✵✶✺✮✮

E c(z) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ E c(z) = z ✶ + z

  • ✶ + zA(z) + z✷A(z)✷

✶ + z − z✷A(z)✷ + ✷z✸A(z)✸ (✶ + z)✷

  • .

✇❤❡r❡ A(z) ✐s t❤❡ ❈❛t❛❧❛♥ s❡r✐❡s✳

❚❤❡ ✜rst t❡r♠s

❚❤❡ ✜rst t❡r♠s ♦❢ E c(z) ❛r❡ E c(z) = z + z✹ + ✸z✻ + ✹z✼ + ✶✺z✽ + ✸✾z✾ + O(z✶✵).

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SLIDE 42

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄

✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡

❞❡❝♦♠♣♦s✐t✐♦♥s

✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥

slide-43
SLIDE 43

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄

✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡

❞❡❝♦♠♣♦s✐t✐♦♥s

✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥

slide-44
SLIDE 44

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄

✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡

❞❡❝♦♠♣♦s✐t✐♦♥s

✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥

slide-45
SLIDE 45

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄

✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡

❞❡❝♦♠♣♦s✐t✐♦♥s

✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥

slide-46
SLIDE 46

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄

✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡

❞❡❝♦♠♣♦s✐t✐♦♥s

✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥

slide-47
SLIDE 47

❆ s❡t ♦❢ ♣r♦♣❡rt✐❡s ❢♦r s♦♠❡ ♠❛♣s

❈♦r♥❡r q✉❛s✐✲tr✐❛♥❣✉❧❛t✐♦♥s

♥♦♥✲r♦♦t ❢❛❝❡s ❛r❡ tr✐❛♥❣❧❡s ✐♥♥❡r ✈❡rt✐❝❡s ❤❛✈❡ ❡✈❡♥ ❞❡❣r❡❡ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s

slide-48
SLIDE 48

❆ s❡t ♦❢ ♣r♦♣❡rt✐❡s ❢♦r s♦♠❡ ♠❛♣s

❈♦r♥❡r q✉❛s✐✲tr✐❛♥❣✉❧❛t✐♦♥s

♥♦♥✲r♦♦t ❢❛❝❡s ❛r❡ tr✐❛♥❣❧❡s ✐♥♥❡r ✈❡rt✐❝❡s ❤❛✈❡ ❡✈❡♥ ❞❡❣r❡❡ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s

slide-49
SLIDE 49

❆ s❡t ♦❢ ♣r♦♣❡rt✐❡s ❢♦r s♦♠❡ ♠❛♣s

❈♦r♥❡r q✉❛s✐✲tr✐❛♥❣✉❧❛t✐♦♥s

♥♦♥✲r♦♦t ❢❛❝❡s ❛r❡ tr✐❛♥❣❧❡s ✐♥♥❡r ✈❡rt✐❝❡s ❤❛✈❡ ❡✈❡♥ ❞❡❣r❡❡ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s

slide-50
SLIDE 50

❆ s❡t ♦❢ ♣r♦♣❡rt✐❡s ❢♦r s♦♠❡ ♠❛♣s

❈♦r♥❡r q✉❛s✐✲tr✐❛♥❣✉❧❛t✐♦♥s

♥♦♥✲r♦♦t ❢❛❝❡s ❛r❡ tr✐❛♥❣❧❡s ✐♥♥❡r ✈❡rt✐❝❡s ❤❛✈❡ ❡✈❡♥ ❞❡❣r❡❡ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s

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SLIDE 51

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

s t

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

❛ r♦♦t ✈❡rt❡① ❛ ♠❛r❦❡❞ ✈❡rt❡① ✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✸

slide-52
SLIDE 52

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

s t

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

❛ r♦♦t ✈❡rt❡① s ❛ ♠❛r❦❡❞ ✈❡rt❡① ✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✸

slide-53
SLIDE 53

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

s t

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

❛ r♦♦t ✈❡rt❡① s ❛ ♠❛r❦❡❞ ✈❡rt❡① t✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✸

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SLIDE 54

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

s t

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

❛ r♦♦t ✈❡rt❡① s ❛ ♠❛r❦❡❞ ✈❡rt❡① t✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t✱ ❧❡♥❣t❤ ℓ ≥ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✸

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SLIDE 55

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

s t

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

❛ r♦♦t ✈❡rt❡① s ❛ ♠❛r❦❡❞ ✈❡rt❡① t✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t✱ ❧❡♥❣t❤ ℓ ≥ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ s t♦ t✱ ❧❡♥❣t❤ ℓ + ✸

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SLIDE 56

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞

s s2 s1

ℓ

ℓ + 3

t

slide-57
SLIDE 57

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞

s s2 s1

ℓ

ℓ + 3

t

slide-58
SLIDE 58

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞

s s2 s1

ℓ

ℓ + 3

t ℓ − 2 ?

slide-59
SLIDE 59

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞

s s2 s1

ℓ

ℓ + 3

t ℓ − 2 ℓ + 1 ?

slide-60
SLIDE 60

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞

s s2 s1

ℓ

ℓ + 3

t ℓ − 2 ℓ + 1

slide-61
SLIDE 61

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞

s s2 s1 t ℓ − 2 ℓ + 1

leftmost shortest path

slide-62
SLIDE 62

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞

s s2 s1 t

leftmost shortest path

slide-63
SLIDE 63

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞

∈ A ∈ A s s2 s1 t2 t1

slide-64
SLIDE 64
  • ❧✉✐♥❣ t✇♦ ❛❧♠♦♥❞s ✿ ♥♦ ❝❧♦❝❦✇✐s❡ s❡♣❛r❛t✐♥❣ tr✐❛♥❣❧❡
slide-65
SLIDE 65
  • ❧✉✐♥❣ t✇♦ ❛❧♠♦♥❞s ✿ ♥♦ ❝❧♦❝❦✇✐s❡ s❡♣❛r❛t✐♥❣ tr✐❛♥❣❧❡
slide-66
SLIDE 66
  • ❧✉✐♥❣ t✇♦ ❛❧♠♦♥❞s ✿ ♥♦ ❝❧♦❝❦✇✐s❡ s❡♣❛r❛t✐♥❣ tr✐❛♥❣❧❡
slide-67
SLIDE 67
  • ❧✉✐♥❣ t✇♦ ❛❧♠♦♥❞s ✿ ♥♦ ❞♦✉❜❧❡ ❡❞❣❡
slide-68
SLIDE 68

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s

▲❡t A(z) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❛❧♠♦♥❞s✳

❚❤❡♦r❡♠

A(z) ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✿ A(z) = ✶ + zA(z)✷

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SLIDE 69

❙❧✐❝❡s

a b t

❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶

❛ r♦♦t ❡❞❣❡ ❛♥ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r②✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤

slide-70
SLIDE 70

❙❧✐❝❡s

a b t

❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶

❛ r♦♦t ❡❞❣❡ (a, b) ❛♥ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r②✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤

slide-71
SLIDE 71

❙❧✐❝❡s

a b t

❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶

❛ r♦♦t ❡❞❣❡ (a, b) ❛♥ ❛♣❡① t ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r②✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤

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SLIDE 72

❙❧✐❝❡s

a b t

❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶

❛ r♦♦t ❡❞❣❡ (a, b) ❛♥ ❛♣❡① t ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r②✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤

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SLIDE 73

❙❧✐❝❡s

a b t

❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶

❛ r♦♦t ❡❞❣❡ (a, b) ❛♥ ❛♣❡① t ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r②✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ a t♦ t ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤

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SLIDE 74

❙❧✐❝❡s

a b t

❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶

❛ r♦♦t ❡❞❣❡ (a, b) ❛♥ ❛♣❡① t ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r②✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ a t♦ t ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ b t♦ t✱ ♦❢ ❧❡♥❣t❤ ℓ

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SLIDE 75

❈✉tt✐♥❣ ❛ s❧✐❝❡

a b u = q t ℓ − 1 ℓ + 2

∈ A

slide-76
SLIDE 76

❈✉tt✐♥❣ ❛ s❧✐❝❡

a u = q t ℓ − 1 ℓ + 2

∈ A

slide-77
SLIDE 77

❈✉tt✐♥❣ ❛ s❧✐❝❡

a u = q t

∈ A

slide-78
SLIDE 78

❈✉tt✐♥❣ ❛ s❧✐❝❡

a u = q t

∈ A

a b u q r t

∈ A ∈ A

slide-79
SLIDE 79

❈✉tt✐♥❣ ❛ s❧✐❝❡

a u = q t

∈ A

a b u q r t

∈ A ∈ A

slide-80
SLIDE 80

❈✉tt✐♥❣ ❛ s❧✐❝❡

a u = q t

∈ A

a b u q r t

∈ A ∈ A

slide-81
SLIDE 81

❈✉tt✐♥❣ ❛ s❧✐❝❡

a u = q t

∈ A

a b u q r t

∈ A ∈ A

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SLIDE 82

❙❧✐❝❡s ♦r ♥♦t s❧✐❝❡s

b a t

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SLIDE 83

❙❧✐❝❡s ♦r ♥♦t s❧✐❝❡s

b a t

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SLIDE 84

❙❧✐❝❡s

▲❡t S(z) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❧✐❝❡s✳

❚❤❡♦r❡♠

S(z) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ (✶ + z)S(z) = zA(z) + z✷A(z)✷

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SLIDE 85

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a b u = q v s

∈ A ∈ A

t

slide-86
SLIDE 86

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a b u = q v s

∈ A ∈ A

t

slide-87
SLIDE 87

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a u = q v s

∈ A ∈ A

t

slide-88
SLIDE 88

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a u = q v s

∈ A ∈ A

t a b u v s r q t

∈ A ∈ A ∈ A

slide-89
SLIDE 89

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a u = q v s

∈ A ∈ A

t a b u v s r q t

∈ A ∈ A ∈ A

slide-90
SLIDE 90

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a u = q v s

∈ A ∈ A

t a b u v s r q t

∈ A ∈ A ∈ A

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SLIDE 91

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a u = q v s

∈ A ∈ A

t a b u v s r q t

∈ A ∈ A ∈ A

a b u v s r q t

∈ A ∈ A ∈ A

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SLIDE 92

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a u = q v s

∈ A ∈ A

t a b u v s r q t

∈ A ∈ A ∈ A

a b u v s r q t

∈ A ∈ A ∈ A

slide-93
SLIDE 93

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

a u = q v s

∈ A ∈ A

t a b u v s r q t

∈ A ∈ A ∈ A

a b u v s r q t

∈ A ∈ A ∈ A

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SLIDE 94

❙❧✐❝❡s ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷

▲❡t S+(z) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❧✐❝❡s ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷✳

❚❤❡♦r❡♠

S+(z) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ (✶ + z)✷S+(z) = z✷A(z)✷(✶ + ✷zA(z))

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SLIDE 95

❇✐❥❡❝t✐✈❡ ❢♦r♠✉❧❛

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❙❧✐❝❡s ♦❢ ❤❡✐❣❤t ✶

❚❤❡♦r❡♠

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ✶

slide-96
SLIDE 96

❇✐❥❡❝t✐✈❡ ❢♦r♠✉❧❛

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❙❧✐❝❡s ♦❢ ❤❡✐❣❤t ✶

❚❤❡♦r❡♠

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ✶

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SLIDE 97

❇✐❥❡❝t✐✈❡ ❢♦r♠✉❧❛

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❙❧✐❝❡s ♦❢ ❤❡✐❣❤t ✶

❚❤❡♦r❡♠

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ (✶ + z)E c(z) = z + z(S(z) − S+(z))

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SLIDE 98

❇✐❥❡❝t✐✈❡ ♣r♦♦❢

❊❧❡♠❡♥ts ❢♦r ❜✐❥❡❝t✐✈❡ ♣r♦♦❢

❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡✳ t❤❡ ❧✐❢t ♦❢ s✉❝❤ ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥ ♦❢ t❤✐s ❧✐❢t

slide-99
SLIDE 99

❇✐❥❡❝t✐✈❡ ♣r♦♦❢

❊❧❡♠❡♥ts ❢♦r ❜✐❥❡❝t✐✈❡ ♣r♦♦❢

❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s c ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡✳ t❤❡ ❧✐❢t ♦❢ s✉❝❤ ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥ ♦❢ t❤✐s ❧✐❢t

slide-100
SLIDE 100

❇✐❥❡❝t✐✈❡ ♣r♦♦❢

❊❧❡♠❡♥ts ❢♦r ❜✐❥❡❝t✐✈❡ ♣r♦♦❢

❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s c ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡✳ t❤❡ ❧✐❢t ♦❢ s✉❝❤ ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥ ♦❢ t❤✐s ❧✐❢t

slide-101
SLIDE 101

❇✐❥❡❝t✐✈❡ ♣r♦♦❢

❊❧❡♠❡♥ts ❢♦r ❜✐❥❡❝t✐✈❡ ♣r♦♦❢

❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s c ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡✳ t❤❡ ❧✐❢t ♦❢ s✉❝❤ ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥ ♦❢ t❤✐s ❧✐❢t

slide-102
SLIDE 102

❆ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡

slide-103
SLIDE 103

❆ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡

slide-104
SLIDE 104

❆ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡

❚❤❡ ❧✐❢t ♦❢ c ✐s ✐ts ♣r❡✐♠❛❣❡ ✉♥❞❡r Ψ : z → ❡①♣(✷iπz)✳

slide-105
SLIDE 105

❆ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡

b′ c a b p′ r q p

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SLIDE 106

❆ ❧✐❢t

b′ c a b p′ r q p

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SLIDE 107

❉❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥s

b′ c a b p′ r q p

❆ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥

❛ ❧❡❢t ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ❛ ❧♦✇❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s ✱ ✱ ✱ ✱ t♦✇❛r❞s t❤❡ r✐❣❤t ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❧❡❢t ❜♦✉♥❞❛r② ❛♥ ✉♣♣❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s ✱ ✱ ✱ ✱ t♦✇❛r❞s t❤❡ ❧❡❢t ❛♥❞ ✇✐t❤ ♥♦ ❡❞❣❡s ✱ ✱ ✱ ✳

slide-108
SLIDE 108

❉❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥s

b′ c a b p′ r q p

❆ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥

❛ ❧❡❢t ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ℓ ❛ ❧♦✇❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s ✱ ✱ ✱ ✱ t♦✇❛r❞s t❤❡ r✐❣❤t ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❧❡❢t ❜♦✉♥❞❛r② ❛♥ ✉♣♣❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s ✱ ✱ ✱ ✱ t♦✇❛r❞s t❤❡ ❧❡❢t ❛♥❞ ✇✐t❤ ♥♦ ❡❞❣❡s ✱ ✱ ✱ ✳

slide-109
SLIDE 109

❉❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥s

b′ c a b p′ r q p

❆ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥

❛ ❧❡❢t ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ℓ ❛ ❧♦✇❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s b′✱ c✱ a✱ b✱ t♦✇❛r❞s t❤❡ r✐❣❤t ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❧❡❢t ❜♦✉♥❞❛r② ❛♥ ✉♣♣❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s ✱ ✱ ✱ ✱ t♦✇❛r❞s t❤❡ ❧❡❢t ❛♥❞ ✇✐t❤ ♥♦ ❡❞❣❡s ✱ ✱ ✱ ✳

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SLIDE 110

❉❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥s

b′ c a b p′ r q p

❆ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥

❛ ❧❡❢t ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ℓ ❛ ❧♦✇❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s b′✱ c✱ a✱ b✱ t♦✇❛r❞s t❤❡ r✐❣❤t ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❧❡❢t ❜♦✉♥❞❛r② ❛♥ ✉♣♣❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s ✱ ✱ ✱ ✱ t♦✇❛r❞s t❤❡ ❧❡❢t ❛♥❞ ✇✐t❤ ♥♦ ❡❞❣❡s ✱ ✱ ✱ ✳

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SLIDE 111

❉❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥s

b′ c a b p′ r q p

❆ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥

❛ ❧❡❢t ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ℓ ❛ ❧♦✇❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s b′✱ c✱ a✱ b✱ t♦✇❛r❞s t❤❡ r✐❣❤t ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❧❡❢t ❜♦✉♥❞❛r② ❛♥ ✉♣♣❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s p✱ q✱ r✱ p′✱ t♦✇❛r❞s t❤❡ ❧❡❢t ❛♥❞ ✇✐t❤ ♥♦ ❡❞❣❡s ✱ ✱ ✱ ✳

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SLIDE 112

❉❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥s

b′ c a b p′ r q p

❆ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥

❛ ❧❡❢t ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ℓ ❛ ❧♦✇❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s b′✱ c✱ a✱ b✱ t♦✇❛r❞s t❤❡ r✐❣❤t ❛ r✐❣❤t ❜♦✉♥❞❛r②✱ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❧❡❢t ❜♦✉♥❞❛r② ❛♥ ✉♣♣❡r ❜♦✉♥❞❛r② ♦❢ ❧❡♥❣t❤ ✸✱ ✇✐t❤ ✈❡rt✐❝❡s p✱ q✱ r✱ p′✱ t♦✇❛r❞s t❤❡ ❧❡❢t ❛♥❞ ✇✐t❤ ♥♦ ❡❞❣❡s (b, c)✱ (a, b′)✱ (r, p)✱ (p′, q)✳

slide-113
SLIDE 113

❇✐❥❡❝t✐✈❡ ♣r♦♦❢

▲❡t E c

△ ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ✇✐t❤ ❛ ♠❛r❦❡❞

✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡✱ ❛♥❞ ❧❡t F ❜❡ t❤❡ ♦♥❡ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥s✳

❚❤❡♦r❡♠

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ E c

△(z) = ✸F(z)

❚❤❡♦r❡♠

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ✶

✷

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SLIDE 114

❇✐❥❡❝t✐✈❡ ♣r♦♦❢

▲❡t E c

△ ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ✇✐t❤ ❛ ♠❛r❦❡❞

✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡✱ ❛♥❞ ❧❡t F ❜❡ t❤❡ ♦♥❡ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥s✳

❚❤❡♦r❡♠

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ E c

△(z) = ✸F(z)

❚❤❡♦r❡♠

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ (✶ + z)✷F(z) = zS+(z).

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SLIDE 115

❇✐❥❡❝t✐✈❡ ♣r♦♦❢

❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❜✐❥❡❝t✐✈❡ ♣r♦♦❢ ✇✐t❤ t❤❡ ❢♦r♠✉❧❛ ✿

❋♦r♠✉❧❛

(✶ + z)✷ ∂ ∂z E c(z) z = ✸zS+(z).

slide-116
SLIDE 116

❈♦♥❝❧✉s✐♦♥

❚❤❡♦r❡♠ ✭❉✳✱ P♦✉❧❛❧❤♦♥✱ ❙❝❤❛❡✛❡r ✭✷✵✶✺✮✮

E c(z) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ E c(z) = z ✶ + z

  • ✶ + zA(z) + z✷A(z)✷

✶ + z − z✷A(z)✷ + ✷z✸A(z)✸ (✶ + z)✷

  • .

✇❤❡r❡ A(z) ✐s t❤❡ ❈❛t❛❧❛♥ s❡r✐❡s✳

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SLIDE 117

❈❛t❛❧❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿ t❤❡ ❝❛❝t✉s

❈❛❝t✉s

❛ ❜✐♥❛r② tr❡❡ ❜❧❛❝❦ ❛♥❞ ✇❤✐t❡ tr✐❛♥❣❧❡s

slide-118
SLIDE 118

❈❛t❛❧❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿ t❤❡ ❝❛❝t✉s

❈❛❝t✉s

❛ ❜✐♥❛r② tr❡❡ ❜❧❛❝❦ ❛♥❞ ✇❤✐t❡ tr✐❛♥❣❧❡s

slide-119
SLIDE 119

❈❛t❛❧❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿ t❤❡ ❝❛❝t✉s

❈❛❝t✉s

❛ ❜✐♥❛r② tr❡❡ ❜❧❛❝❦ ❛♥❞ ✇❤✐t❡ tr✐❛♥❣❧❡s

slide-120
SLIDE 120

❈❛t❛❧❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿ t❤❡ ❝❛❝t✉s

slide-121
SLIDE 121

❈❛t❛❧❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿ t❤❡ ❝❛❝t✉s

slide-122
SLIDE 122

❈❛t❛❧❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿ t❤❡ ❝❛❝t✉s

slide-123
SLIDE 123

❈❛t❛❧❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿ t❤❡ ❝❛❝t✉s

slide-124
SLIDE 124

▼❡r❝✐ ♣♦✉r ✈♦tr❡ ❛tt❡♥t✐♦♥ ✦