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Apr 30, 2023 522 likes •877 views

t srt ts t ts P tt Ptrrtr str tt

SLIDE 1

▼❛t❤✷✸✻ ❉✐s❝r❡t❡ ▼❛t❤s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s

P✳ ■tt♠❛♥♥

❯❑❩◆✱ P✐❡t❡r♠❛r✐t③❜✉r❣

❙❡♠❡st❡r ✶✱ ✷✵✶✷

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶ ✴ ✸✹

SLIDE 2

❆ ❘❡✈✐❡✇ ♦❢ ❉❊❙

❲❡ ✐♥tr♦❞✉❝❡❞ ❉❊❙ ❧❛st ❧❡❝t✉r❡ ❲❡ ✇✐❧❧ ❜r✐❡✢② r❡✈✐❡✇ t❤❛t ♠❛t❡r✐❛❧ ❉❊❙ ✐s ❢✉❧❧② ❞❡s❝r✐❜❡❞ ✐♥ ❋■P❙✹✻✲✸ ✭❤tt♣✿✴✴✇✇✇✳✐t❧✳♥✐st✳❣♦✈✴✜♣s♣✉❜s✴✮

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷ ✴ ✸✹

SLIDE 3

❖♥ ❖✈❡r✈✐❡✇ ♦❢ ❉❊❙

❉❊❙ ✐s ❛ ♠♦❞✐✜❡❞ ❋✐❡st❡❧ ❝✐♣❤❡r ✐♥ ✇❤✐❝❤ ❡❛❝❤ ❜❧♦❝❦ ✐s ❛ ✻✹✲❜✐t✱ ♦r ✽✲❜②t❡✱ str✐♥❣ ❆ ❉❊❙ ❦❡② ✐s ❛❧s♦ ❡✐❣❤t ❜②t❡s ❚❤❡ ❧❛st ❜✐t ✐♥ ❡❛❝❤ ❜②t❡ ✐♥ ❛ ❉❊❙ ❦❡② ✐s ❛ ❝❤❡❝❦ ❜✐t ♦r ♣❛r✐t② ❜✐t ❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣❛r✐t② ❜✐t ✐s ❝❤♦s❡♥ s♦ t❤❛t t❤❡ ❜②t❡ ❤❛s ♦❞❞ ♣❛r✐t② ❚❤✐s ②✐❡❧❞s ✽ ❜✐ts ♦❢ t❤❡ ❦❡② r❡❞✉♥❞❛♥t ❲❡ ♦♥❧② ✉s❡ t❤❡ ✺✻✲❜✐ts ✇❤✐❝❤ ❛r❡ ♥♦t ♣❛r✐t② ❜✐ts

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✸ ✴ ✸✹

SLIDE 4

❉❛t❛ ❊♥❝r②♣t✐♦♥ ❆❧❣♦r✐t❤♠

❲❡ ❜❡❣✐♥ ✇✐t❤ ❛ ✻✹✲❜✐t ♣❧❛✐♥t❡①t✱ P✱ ❛♥❞ ❛ ✻✹✲❜✐t ❉❊❙ ❦❡②✱ ❑ ❊♥❝r②♣t✐♦♥ ❝♦♥s✐sts ♦❢ t❤r❡❡ st❛❣❡s ❙t❛❣❡ ✶

❚❤❡ ❜✐ts ♦❢ P ❛r❡ ♣❡r♠✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❛ ✜①❡❞ ✐♥✐t✐❛❧ ♣❡r♠✉t❛t✐♦♥✱ ■P✱ t♦ ♣r♦❞✉❝❡ ■P(P) ❚❤❡ s❛♠❡ ✐♥✐t✐❛❧ ♣❡r♠✉t❛t✐♦♥ ✐s ✉s❡❞ ✐♥ ❡✈❡r② ❉❊❙ ❡♥❝r②♣t✐♦♥

❙t❛❣❡ ✷

❆ ✶✻✲r♦✉♥❞ ❋❡✐st❡❧ ❝✐♣❤❡r ✐s ❛♣♣❧✐❡❞ t♦ ■P(P) ❚❤❡ ❉❊❙ ❦❡②✱ ❑✱ ✐s ✉s❡❞ t♦ ❣❡♥❡r❛t❡ r♦✉♥❞ ❦❡②s ❢♦r t❤❡ ❋❡✐st❡❧ ❝✐♣❤❡r

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✹ ✴ ✸✹

SLIDE 5

❉❛t❛ ❊♥❝r②♣t✐♦♥ ❆❧❣♦r✐t❤♠

❚❤❡ ✜♥❛❧ st❛❣❡ ♦❢ ❡♥❝r②♣t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿ ❙t❛❣❡ ✸

❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ♣❡r♠✉t❛t✐♦♥✱ ■P−✶✱ ✐s ❛♣♣❧✐❡❞ t♦ t❤❡ r❡s✉❧t ♦❢ ❙t❛❣❡ ✷ t♦ ♣r♦❞✉❝❡ t❤❡ ✜♥❛❧ ❝✐♣❤❡rt❡①t ❉❊❙ ❞❡❝r②♣t✐♦♥ ❢♦❧❧♦✇s ❡①❛❝t❧② t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡✱ ❡①❝❡♣t t❤❛t t❤❡ r♦✉♥❞ ❦❡②s ❛r❡ ✉s❡❞ ✐♥ r❡✈❡rs❡ ♦r❞❡r

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✺ ✴ ✸✹

SLIDE 6

❉❊❙ ❙t❛❣❡ ✶

❚❤❡ ✐♥✐t✐❛❧ ♣❡r♠✉t❛t✐♦♥ ■P ♣❡r♠✉t❡s t❤❡ ♣❧❛✐♥t❡①t P t♦ ♣r♦❞✉❝❡ ■P(P) ❚❤❡ ♣❡r♠✉t❛t✐♦♥ ■P ✐s ❢r❡q✉❡♥t❧② r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ ■P =             ✺✽ ✺✵ ✹✷ ✸✹ ✷✻ ✶✽ ✶✵ ✷ ✻✵ ✺✷ ✹✹ ✸✻ ✷✽ ✷✵ ✶✷ ✹ ✻✷ ✺✹ ✹✻ ✸✽ ✸✵ ✷✷ ✶✹ ✻ ✻✹ ✺✻ ✹✽ ✹✵ ✸✷ ✷✹ ✶✻ ✽ ✺✼ ✹✾ ✹✶ ✸✸ ✷✺ ✶✼ ✾ ✶ ✺✾ ✺✶ ✹✸ ✸✺ ✷✼ ✶✾ ✶✶ ✸ ✻✶ ✺✸ ✹✺ ✸✼ ✷✾ ✷✶ ✶✸ ✺ ✻✸ ✺✺ ✹✼ ✸✾ ✸✶ ✷✸ ✶✺ ✼            

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✻ ✴ ✸✹

SLIDE 7

❉❊❙ ❙t❛❣❡ ✶ ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♣❧❛✐♥t❡①t ✐s ●♦ ❤♦♠❡✦ ❲❡ ❡♥❝♦❞❡ t❤✐s ✐♥ ❆❙❈■■ ❛s P =✵✶✵✵✵✶✶✶ ✵✶✶✵✶✶✶✶ ✵✵✶✵✵✵✵✵ ✵✶✶✵✶✵✵✵ ✵✶✶✵✶✶✶✶ ✵✶✶✵✶✶✵✶ ✵✶✶✵✵✶✵✶ ✵✵✶✵✵✵✵✶ ❲❡ t❤✉s ✜♥❞ t❤❛t ■P(P) =✵✶✶✶✶✵✶✶✵✵✵✵✵✵✵✵✵✶✶✶✵✵✶✶✶✶✶✶✵✵✶✶ ✵✵✵✵✵✵✵✵✶✶✶✶✶✶✶✵✵✵✶✶✶✵✶✵✵✵✵✶✵✵✶✶

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✼ ✴ ✸✹

SLIDE 8

❉❊❙ ❙t❛❣❡ ✷

■♥ ❙t❛❣❡ ✷✱ ❛ ✶✻✲r♦✉♥❞ ❋❡✐st❡❧ ❝✐♣❤❡r ✐s ❛♣♣❧✐❡❞ t♦ ■P(P) = ▲✵||❘✵ t♦ ♦❜t❛✐♥ ❘✶✻||▲✶✻ ■♥ ❡❛❝❤ r♦✉♥❞✱ ✇❡ ✉s❡ ❛ r♦✉♥❞ ❢✉♥❝t✐♦♥✱ ❢❑✐✱ ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ❉❊❙ ✐♥t❡r♥❛❧ ❜❧♦❝❦ ❝✐♣❤❡r ❚❤❡ r♦✉♥❞ ❦❡②s ❑✶, ❑✷, . . . , ❑✶✻ ❛r❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ✻✹✲❜✐t ❉❊❙ ❦❡②✱ ❑

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✽ ✴ ✸✹

SLIDE 9

❡♥❡r❛t✐♦♥ ♦❢ ❘♦✉♥❞ ❑❡②s

❚❤❡ r♦✉♥❞ ❦❡②s ❑✶, ❑✷, . . . , ❑✶✻ ✭❝♦❧❧❡❝t✐✈❡❧② ❝❛❧❧❡❞ t❤❡ ❦❡② s❝❤❡❞✉❧❡✮ ❛r❡ ❢♦✉♥❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② ❢r♦♠ t❤❡ ✻✹✲❜✐t ❉❊❙ ❦❡② ❑ ❚❤❡ ✭✜❢t② s✐① ♥♦♥✲♣❛r✐t②✮ ❜✐ts ♦❢ ❑ ❛r❡ ♣❡r♠✉t❡❞ ❛♥❞ s♣❧✐t ✉♣✱ ✉s✐♥❣ t❤❡ ♣❡r♠✉t❛t✐♦♥s ❈ ❛♥❞ ❉ t♦ ❢♦r♠ t✇♦ ✷✽✲❜✐t s✉❜❦❡②s✱ ❈✵ = ❈(❑) ❛♥❞ ❉✵ = ❉(❑)

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✾ ✴ ✸✹

SLIDE 10

❡♥❡r❛t✐♦♥ ♦❢ ❘♦✉♥❞ ❑❡②s ✭❝♦♥t✳✮

❚❤❡ ♣❡r♠✉t❛t✐♦♥s ❈ ❛♥❞ ❉ ❛r❡ ❝♦❧❧❡❝t✐✈❡❧② ❣✐✈❡♥ t❤❡ ♥❛♠❡ P❈✶✱ ❢♦r P❡r♠✉t❡❞ ❈❤♦✐❝❡ ✶ ❈ =     ✺✼ ✹✾ ✹✶ ✸✸ ✷✺ ✶✼ ✾ ✶ ✺✽ ✺✵ ✹✷ ✸✹ ✷✻ ✶✽ ✶✵ ✷ ✺✾ ✺✶ ✹✸ ✸✺ ✷✼ ✶✾ ✶✶ ✸ ✻✵ ✺✷ ✹✹ ✸✻     ❉ =     ✻✸ ✺✺ ✹✼ ✸✾ ✸✶ ✷✸ ✶✺ ✼ ✻✷ ✺✹ ✹✻ ✸✽ ✸✵ ✷✷ ✶✹ ✻ ✻✶ ✺✸ ✹✺ ✸✼ ✷✾ ✷✶ ✶✸ ✺ ✷✽ ✷✵ ✶✷ ✹     ◆♦t✐❝❡ t❤❛t ✐♥ P❡r♠✉t❡❞ ❈❤♦✐❝❡ ✶✱ ♥❡✐t❤❡r ❈ ♥♦r ❉ ❝♦♥t❛✐♥s ❛♥② ♥✉♠❜❡rs t❤❛t ❛r❡ ❝♦♥❣r✉❡♥t t♦ ✵ ♠♦❞✉❧♦ ✽

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✵ ✴ ✸✹

SLIDE 11

❡♥❡r❛t✐♦♥ ♦❢ ❘♦✉♥❞ ❑❡②s ✭❝♦♥t✳✮

❋♦r ❡❛❝❤ ✐ ✇✐t❤ ✶ ≤ ✐ ≤ ✶✻✱ t❤❡ s✉❜❦❡②s ❈✐ ❛♥❞ ❉✐ ❛r❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ ❈✐−✶ ❛♥❞ ❉✐−✶ ❜② s❤✐❢t✐♥❣ t❤❡ ❜✐ts ✐♥ ❈✐ ❛♥❞ ❉✐ ❧❡❢t ❜② ❛ ♣r❡❞❡t❡r♠✐♥❡❞ ❛♠♦✉♥t ✐ ℓ✐ ✐ ℓ✐ ✐ ℓ✐ ✐ ℓ✐ ✶ ✶ ✺ ✷ ✾ ✶ ✶✸ ✷ ✷ ✶ ✻ ✷ ✶✵ ✷ ✶✹ ✷ ✸ ✷ ✼ ✷ ✶✶ ✷ ✶✺ ✷ ✹ ✷ ✽ ✷ ✶✷ ✷ ✶✻ ✶

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✶ ✴ ✸✹

SLIDE 12

❡♥❡r❛t✐♦♥ ♦❢ ❘♦✉♥❞ ❑❡②s ✭❝♦♥t✳✮

❋♦r ❡❛❝❤ ✐ ✇✐t❤ ✶ ≤ ✐ ≤ ✶✻✱ t❤❡ r♦✉♥❞ ❦❡② ❑✐ ✐s ❢♦✉♥❞ ❜② ❢❡❡❞✐♥❣ ❈✐||❉✐ t♦ t❤❡ ♣❡r♠✉t❛t✐♦♥ P❈✷ P❈✷ =             ✶✹ ✶✼ ✶✶ ✷✹ ✶ ✺ ✸ ✷✽ ✶✺ ✻ ✷✶ ✶✵ ✷✸ ✶✾ ✶✷ ✹ ✷✻ ✽ ✶✻ ✼ ✷✼ ✷✵ ✶✸ ✷ ✹✶ ✺✷ ✸✶ ✸✼ ✹✼ ✺✺ ✸✵ ✹✵ ✺✶ ✹✺ ✸✸ ✹✽ ✹✹ ✹✾ ✸✾ ✺✻ ✸✹ ✺✸ ✹✻ ✹✷ ✺✵ ✸✻ ✷✾ ✸✷            

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✷ ✴ ✸✹

SLIDE 13

❡♥❡r❛t✐♦♥ ♦❢ ❘♦✉♥❞ ❑❡②s ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ❙✉♣♣♦s❡ t❤❛t ❑ =✵✵✶✵✵✶✵✶ ✵✶✶✵✵✶✶✶ ✶✶✵✵✶✶✵✶ ✶✵✶✶✵✵✶✶ ✶✶✶✶✶✶✵✶ ✶✶✵✵✶✶✶✵ ✵✶✵✵✵✵✵✵ ✵✵✶✵✶✵✶✵ ◆♦t✐❝❡ t❤❛t ❡❛❝❤ ❜②t❡ ♦❢ t❤✐s ❉❊❙ ❦❡② ❤❛s ♦❞❞ ♣❛r✐t② ❚❤❡♥ ❈✵ = ❈(❑) =✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶ ❉✵ = ❉(❑) =✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✸ ✴ ✸✹

SLIDE 14

❙✉❜❦❡②s ❈✐ ❛♥❞ ❉✐

❖♥❝❡ ✇❡ ❤❛✈❡ ❞❡t❡r♠✐♥❡❞ ❈✵ ❛♥❞ ❉✵✱ t❤❡ ♦t❤❡r s✉❜❦❡②s ❈✶, ❈✷, . . . , ❈✶✻ ❛♥❞ ❉✶, ❉✷, . . . , ❉✶✻ ❛r❡ ❢♦✉♥❞ ❢r♦♠ t❤❡♠ ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛ ❈✐ = ▲(❈✐−✶, ℓ✐) ❉✐ = ▲(❉✐−✶, ℓ✐) ❚❤❡ ✈❛❧✉❡s ℓ✐ ❛r❡ ❛s ❛❜♦✈❡ ■❢ ❳ ✐s ❛ str✐♥❣ ❛♥❞ ℓ ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✱ t❤❡♥ ▲(❳, ℓ) ✐s t❤❡ str✐♥❣ ❢♦r♠❡❞ ❜② s❤✐❢t✐♥❣ t❤❡ ❝❤❛r❛❝t❡rs ♦❢ ❳ ℓ ♣❧❛❝❡s t♦ t❤❡ ❧❡❢t

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✹ ✴ ✸✹

SLIDE 15

❙✉❜❦❡②s ❈✐ ❛♥❞ ❉✐ ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ❈♦♥t✐♥✉✐♥❣ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ ✇❤❡r❡ ❈✵ = ❈(❑) =✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶ ❉✵ = ❉(❑) =✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵ ❲❡ ✜rst ✜♥❞ ❈✶✿ ❈✶ = ▲(❈✵, ℓ✶) = ▲(❈✵, ✶) ❚❤✉s✱ ❈✶ = ✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵ ❙✐♠✐❧❛r❧②✱ ❉✶ = ▲(❉✵, ✶) ❛♥❞ ❤❡♥❝❡ ❉✶ = ✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✺ ✴ ✸✹

SLIDE 16

❙✉❜❦❡②s ❈✐ ❛♥❞ ❉✐ ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ❢❛s❤✐♦♥✱ ✇❡ ✜♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐ ❈✐ ❉✐ ✵

✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶ ✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵

✶

✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵ ✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶

✷

✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵ ✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵

✸

✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶ ✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵

✹

✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶✶✶ ✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵✶✵

✺

✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶✶✶✵✵ ✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵✶✵✶✵

✻

✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶✶✶✵✵✵✶ ✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵✶✵✶✵✵✵

✼

✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶✶✶✵✵✵✶✶✶ ✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵✶✵✶✵✵✵✶✶

✽

✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶✶✶✵✵✵✶✶✶✵✶ ✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵✶✵✶✵✵✵✶✶✵✶

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✻ ✴ ✸✹

SLIDE 17

❙✉❜❦❡②s ❈✐ ❛♥❞ ❉✐ ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ✐ ❈✐ ❉✐ ✾

✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶ ✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶

✶✵

✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶ ✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶

✶✶

✶✶✵✶✶✵✵✵✶✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵ ✶✵✶✵✵✶✵✵✵✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶

✶✷

✵✶✶✵✵✵✶✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶ ✶✵✵✶✵✵✵✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵

✶✸

✶✵✵✵✶✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶ ✵✶✵✵✵✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵

✶✹

✵✵✶✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵ ✵✵✵✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶

✶✺

✶✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵ ✵✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵

✶✻

✵✵✶✶✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶ ✶✵✶✵✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✼ ✴ ✸✹

SLIDE 18

❘♦✉♥❞ ❑❡②s ❑✶, ❑✷, . . . , ❑✶✻

❲❡ ❤❛✈❡ ♥♦✇ ❣❡♥❡r❛t❡❞ t❤❡ s✉❜❦❡②s ❈✶, ❈✷, . . . , ❈✶✻ ❛♥❞ ❉✶, ❉✷, . . . , ❉✶✻ ❋r♦♠ t❤✐s ✇❡ ❝❛♥ ♥♦✇ ❞❡t❡r♠✐♥❡ t❤❡ r♦✉♥❞ ❦❡②s ❑✶, ❑✷, . . . , ❑✶✻ ❘❡❝❛❧❧ t❤❡ ♣❡r♠✉t❛t✐♦♥✿ P❈✷ =             ✶✹ ✶✼ ✶✶ ✷✹ ✶ ✺ ✸ ✷✽ ✶✺ ✻ ✷✶ ✶✵ ✷✸ ✶✾ ✶✷ ✹ ✷✻ ✽ ✶✻ ✼ ✷✼ ✷✵ ✶✸ ✷ ✹✶ ✺✷ ✸✶ ✸✼ ✹✼ ✺✺ ✸✵ ✹✵ ✺✶ ✹✺ ✸✸ ✹✽ ✹✹ ✹✾ ✸✾ ✺✻ ✸✹ ✺✸ ✹✻ ✹✷ ✺✵ ✸✻ ✷✾ ✸✷            

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✽ ✴ ✸✹

SLIDE 19

❘♦✉♥❞ ❑❡②s ❑✶, ❑✷, . . . , ❑✶✻

❚❤❡ r♦✉♥❞ ❦❡②s ❑✶, ❑✷, . . . , ❑✶✻ ❛r❡ ❣✐✈❡♥ ❜② ❑✐ = P❈✷(❈✐||❉✐). ◆♦t❡ t❤❛t s✐♥❝❡ ❈✐ ❛♥❞ ❉✐ ❛r❡ t✇❡♥t② ❡✐❣❤t ❜✐ts ❡❛❝❤✱ ❈✐||❉✐ ✐s ✜❢t② s✐① ❜✐ts✳ ❚❤❡ ❦❡② ❑✐ ✐s✱ ❤♦✇❡✈❡r✱ ♦♥❧② ❢♦rt② ❡✐❣❤t ❜✐ts

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✾ ✴ ✸✹

SLIDE 20

❘♦✉♥❞ ❑❡②s ❑✶, ❑✷, . . . , ❑✶✻

❊①❛♠♣❧❡ ❚❤❡ r♦✉♥❞ ❦❡② ❑✸ ✐s ❡q✉❛❧ t♦ P❈✷(❈✸||❉✸) ❙✐♥❝❡ ❈✸||❉✸ =✶✶✵✵✵✶✶✶✵✶✶✵✶✵✵✶✶✵✶✶✵✵✵✶✵✵✶✶ ✶✵✶✵✵✵✶✶✵✶✶✶✶✵✶✶✵✶✵✵✶✵✵✵✶✵✶✵ ❲❡ ✜♥❞ ❑✸ =✵✶✶✶✶✵✵✶✵✶✵✶✵✶✵✵✵✶✶✶✶✶✶✶ ✶✵✶✵✵✶✵✶✵✵✶✵✶✶✶✵✵✶✶✵✵✶✶✵

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✵ ✴ ✸✹

SLIDE 21

❉❊❙ ❘♦✉♥❞s

❘❡❝❛❧❧ t❤❛t t❤❡ r♦✉♥❞ ❢✉♥❝t✐♦♥✱ ❢❑✐✱ ✐s ✉s❡❞ ✐♥ ❛ ✶✻✲r♦✉♥❞ ❋❡✐st❡❧ ❝✐♣❤❡r ■♥ ❡❛❝❤ r♦✉♥❞✱ ❢❑✐ t❛❦❡s ❛s ✐♥♣✉t t❤❡ ✸✷✲❜✐t str✐♥❣ ❘✐−✶ ❛♥❞ t❤❡ ✹✽✲❜✐t r♦✉♥❞ ❦❡② ❑✐✱ ❛♥❞ ♣r♦❞✉❝❡s ❛ ✸✷✲❜✐t str✐♥❣ ❢❑✐(❘✐−✶)

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✶ ✴ ✸✹

SLIDE 22

❉❊❙ ❘♦✉♥❞s ✭❝♦♥t✳✮

❚❤❡ ✸✷✲❜✐t str✐♥❣ ❘✐−✶ ✐s ❡①♣❛♥❞❡❞ t♦ ❛ ✹✽✲❜✐t str✐♥❣ ✉s✐♥❣ t❤❡ ❡①♣❛♥s✐♦♥ ❢✉♥❝t✐♦♥✱ ❊ ❚❤❡ ❡①♣❛♥s✐♦♥ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② ❊ =             ✸✷ ✶ ✷ ✸ ✹ ✺ ✹ ✺ ✻ ✼ ✽ ✾ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✷✽ ✷✾ ✸✵ ✸✶ ✸✷ ✶            

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✷ ✴ ✸✹

SLIDE 23

❉❊❙ ❘♦✉♥❞s ✭❝♦♥t✳✮

❚❤❡ r❡s✉❧t✐♥❣ ✹✽✲❜✐t str✐♥❣✱ ❊(❘✐−✶)✱ ✐s ❳❖❘❡❞ ✇✐t❤ t❤❡ r♦✉♥❞ ❦❡②✱ ❑✐✱ ❛♥❞ t❤❡ r❡s✉❧t ✐s s♣❧✐t ✐♥t♦ ❡✐❣❤t ✻✲❜✐t str✐♥❣s ❇✶, ❇✷, . . . , ❇✽ ❚❤❛t ✐s✱ ❇✶❇✷ · · · ❇✽ = ❊(❘✐−✶) ⊕ ❑✐ ❊❛❝❤ ✻✲❜✐t str✐♥❣ ❇✐ ✐s s❡♥t t❤r♦✉❣❤ ❛♥ ❙✲❜♦① ❚❤❡r❡ ❛r❡ ❡✐❣❤t ❙✲❜♦①❡s✱ ❙✶, ❙✷, . . . , ❙✽ ✭s❡❡ ❜❡❧♦✇✮✱ ♦♥❡ ❢♦r ❡❛❝❤ ❇✐

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✸ ✴ ✸✹

SLIDE 24

❉❊❙ ❘♦✉♥❞s ✭❝♦♥t✳✮

❚❤❡ ❙✲❜♦① ❙✐ ♦♣❡r❛t❡s ♦♥ t❤❡ ✻✲❜✐t str✐♥❣ ❇✐ t♦ r❡t✉r♥ ❛ ✹✲❜✐t str✐♥❣ ❙✐(❇✐) ❚❤❡ ❡✐❣❤t ✹✲❜✐t str✐♥❣s ❙✶(❇✶), ❙✷(❇✷), . . . , ❙✽(❇✽) ❛r❡ ❝♦♥❝❛t❡♥❛t❡❞ t♦ ♣r♦❞✉❝❡ ❛ ✸✷✲❜✐t str✐♥❣ ❚❤✐s ✐s t❤❡♥ ♣❡r♠✉t❡❞ ✉s✐♥❣ ✭②❡t ❛♥♦t❤❡r✮ ♣r❡❞❡t❡r♠✐♥❡❞ ♣❡r♠✉t❛t✐♦♥✱ P P =     ✶✻ ✼ ✷✵ ✷✶ ✷✾ ✶✷ ✷✽ ✶✼ ✶ ✶✺ ✷✸ ✷✻ ✺ ✶✽ ✸✶ ✶✵ ✷ ✽ ✷✹ ✶✹ ✸✷ ✷✼ ✸ ✾ ✶✾ ✶✸ ✸✵ ✻ ✷✷ ✶✶ ✹ ✷✺    

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✹ ✴ ✸✹

SLIDE 25

❉❊❙ ❘♦✉♥❞s ✭❝♦♥t✳✮

❚❤✐s ♣r♦❞✉❝❡s t❤❡ ✜♥❛❧ r❡s✉❧t✱ ❢❑✐(❘✐−✶) = P(❙✶(❇✶)||❙✷(❇✷)|| · · · ||❙✽(❇✽)) ❚❤❡ ✜rst ❙✲❜♦① ✐s ❜❡❧♦✇

❙✶ ✶✹ ✹ ✶✸ ✶ ✷ ✶✺ ✶✶ ✽ ✸ ✶✵ ✻ ✶✷ ✺ ✾ ✵ ✼ ✵ ✶✺ ✼ ✹ ✶✹ ✷ ✶✸ ✶ ✶✵ ✻ ✶✷ ✶✶ ✾ ✺ ✸ ✽ ✹ ✶ ✶✹ ✽ ✶✸ ✻ ✷ ✶✶ ✶✺ ✶✷ ✾ ✼ ✸ ✶✵ ✺ ✵ ✶✺ ✶✷ ✽ ✷ ✹ ✾ ✶ ✼ ✺ ✶✶ ✸ ✶✹ ✶✵ ✵ ✻ ✶✸

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✺ ✴ ✸✹

SLIDE 26

❉❊❙ ❘♦✉♥❞s ✭❝♦♥t✳✮

❊❛❝❤ ❙✲❜♦① ❝♦♥s✐sts ♦❢ ❢♦✉r r♦✇s ❛♥❞ s✐①t❡❡♥ ❝♦❧✉♠♥s✳ ❚❤❡ r♦✇s ❛r❡ ♥✉♠❜❡r❡❞ ✵✱✶✱✷✱ ❛♥❞ ✸ ❛♥❞ t❤❡ ❝♦❧✉♠♥s ❛r❡ ♥✉♠❜❡r❡❞ ✵✱✶✱✷✱ ✳ ✳ ✳ ✱ ✶✺ ❚❤❡ ✐♥♣✉t t♦ ❛♥ ❙✲❜♦① ❙✐ ✐s ❛ ✻✲❜✐t str✐♥❣✱ ❇✐✱ ❜✉t t❤❡ ♦✉t♣✉t ✐s ❛ ✹✲❜✐t str✐♥❣✱ ❙✐(❇✐) ❚❤❡ ♥♦t❛t✐♦♥ ✉s❡❞ ❢♦r t❤❡ ❙✲❜♦①❡s ✐s ❞✐✛❡r❡♥t t♦ t❤❡ ♥♦t❛t✐♦♥ ✉s❡❞ ♣r❡✈✐♦✉s❧② ❈♦♥s✐❞❡r t❤❡ ✐♥♣✉t ❇✐ = ❜✶❜✷❜✸❜✹❜✺❜✻

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✻ ✴ ✸✹

SLIDE 27

❉❊❙ ❘♦✉♥❞s ✭❝♦♥t✳✮

❚❤❡ ❜✐♥❛r② ♥✉♠❜❡r ❜✶||❜✻ ❢♦r♠❡❞ ❜② ❝♦♥❝❛t❡♥❛t✐♥❣ t❤❡ ✜rst ❛♥❞ ❧❛st ❜✐ts ♦❢ ❇✐ ✐s ❛ ❞❡❝✐♠❛❧ ♥✉♠❜❡r r ❜❡t✇❡❡♥ ✵ ❛♥❞ ✸ ❚❤❡ ❜✐♥❛r② ♥✉♠❜❡r ❜✷||❜✸||❜✹||❜✺ s✐♠✐❧❛r❧② r❡♣r❡s❡♥ts ❛ ❞❡❝✐♠❛❧ ♥✉♠❜❡r ❝ ❜❡t✇❡❡♥ ✵ ❛♥❞ ✶✺ ❚❤❡ ♥✉♠❜❡r ✐♥ r♦✇ r ❛♥❞ ❝♦❧✉♠♥ ❝ ♦❢ ❙✐ ✐s ❛ ❞❡❝✐♠❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✵ ❛♥❞ ✶✺✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ✹✲❞✐❣✐t ❜✐♥❛r② ♥✉♠❜❡r✱ ❞ ❚❤❡ ♥✉♠❜❡r ❞ ✐s t❤❡ ♦✉t♣✉t ❙✐(❇✐) ♦❢ t❤❡ ❙✲❜♦①

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✼ ✴ ✸✹

SLIDE 28

❙✲❜♦①s

❙✷ ✶✺ ✶ ✽ ✶✹ ✻ ✶✶ ✸ ✹ ✾ ✼ ✷ ✶✸ ✶✷ ✵ ✺ ✶✵ ✸ ✶✸ ✹ ✼ ✶✺ ✷ ✽ ✶✹ ✶✷ ✵ ✶ ✶✵ ✻ ✾ ✶✶ ✺ ✵ ✶✹ ✼ ✶✶ ✶✵ ✹ ✶✸ ✶ ✺ ✽ ✶✷ ✻ ✾ ✸ ✷ ✶✺ ✶✸ ✽ ✶✵ ✶ ✸ ✶✺ ✹ ✷ ✶✶ ✻ ✼ ✶✷ ✵ ✺ ✶✹ ✾ ❙✸ ✶✵ ✵ ✾ ✶✹ ✻ ✸ ✶✺ ✺ ✶ ✶✸ ✶✷ ✼ ✶✶ ✹ ✷ ✽ ✶✸ ✼ ✵ ✾ ✸ ✹ ✻ ✶✵ ✷ ✽ ✺ ✶✹ ✶✷ ✶✶ ✶✺ ✶ ✶✸ ✻ ✹ ✾ ✽ ✶✺ ✸ ✵ ✶✶ ✶ ✷ ✶✷ ✺ ✶✵ ✶✹ ✼ ✶ ✶✵ ✶✸ ✵ ✻ ✾ ✽ ✼ ✹ ✶✺ ✶✹ ✸ ✶✶ ✺ ✷ ✶✷

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✽ ✴ ✸✹

SLIDE 29

❙✲❜♦①s ✭❝♦♥t✳✮

❙✹ ✼ ✶✸ ✶✹ ✸ ✵ ✻ ✾ ✶✵ ✶ ✷ ✽ ✺ ✶✶ ✶✷ ✹ ✶✺ ✶✸ ✽ ✶✶ ✺ ✻ ✶✺ ✵ ✸ ✹ ✼ ✷ ✶✷ ✶ ✶✵ ✶✹ ✾ ✶✵ ✻ ✾ ✵ ✶✷ ✶✶ ✼ ✶✸ ✶✺ ✶ ✸ ✶✹ ✺ ✷ ✽ ✹ ✸ ✶✺ ✵ ✻ ✶✵ ✶ ✶✸ ✽ ✾ ✹ ✺ ✶✶ ✶✷ ✼ ✷ ✶✹ ❙✺ ✷ ✶✷ ✹ ✶ ✼ ✶✵ ✶✶ ✻ ✽ ✺ ✸ ✶✺ ✶✸ ✵ ✶✹ ✾ ✶✹ ✶✶ ✷ ✶✷ ✹ ✼ ✶✸ ✶ ✺ ✵ ✶✺ ✶✵ ✸ ✾ ✽ ✻ ✹ ✷ ✶ ✶✶ ✶✵ ✶✸ ✼ ✽ ✶✺ ✾ ✶✷ ✺ ✻ ✸ ✵ ✶✹ ✶✶ ✽ ✶✷ ✼ ✶ ✶✹ ✷ ✶✸ ✻ ✶✺ ✵ ✾ ✶✵ ✹ ✺ ✸

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✾ ✴ ✸✹

SLIDE 30

❙✲❜♦①s ✭❝♦♥t✳✮

❙✻ ✶✷ ✶ ✶✵ ✶✺ ✾ ✷ ✻ ✽ ✵ ✶✸ ✸ ✹ ✶✹ ✼ ✺ ✶✶ ✶✵ ✶✺ ✹ ✷ ✼ ✶✷ ✾ ✺ ✻ ✶ ✶✸ ✶✹ ✵ ✶✶ ✸ ✽ ✾ ✶✹ ✶✺ ✺ ✷ ✽ ✶✷ ✸ ✼ ✵ ✹ ✶✵ ✶ ✶✸ ✶✶ ✻ ✹ ✸ ✷ ✶✷ ✾ ✺ ✶✺ ✶✵ ✶✶ ✶✹ ✶ ✼ ✻ ✵ ✽ ✶✸ ❙✼ ✹ ✶✶ ✷ ✶✹ ✶✺ ✵ ✽ ✶✸ ✸ ✶✷ ✾ ✼ ✺ ✶✵ ✻ ✶ ✶✸ ✵ ✶✶ ✼ ✹ ✾ ✶ ✶✵ ✶✹ ✸ ✺ ✶✷ ✷ ✶✺ ✽ ✻ ✶ ✹ ✶✶ ✶✸ ✶✷ ✸ ✼ ✶✹ ✶✵ ✶✺ ✻ ✽ ✵ ✺ ✾ ✷ ✻ ✶✶ ✶✸ ✽ ✶ ✹ ✶✵ ✼ ✾ ✺ ✵ ✶✺ ✶✹ ✷ ✸ ✶✷

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✸✵ ✴ ✸✹

SLIDE 31

❙✲❜♦①s ✭❝♦♥t✳✮

❙✽ ✶✸ ✷ ✽ ✹ ✻ ✶✺ ✶✶ ✶ ✶✵ ✾ ✸ ✶✹ ✺ ✵ ✶✷ ✼ ✶ ✶✺ ✶✸ ✽ ✶✵ ✸ ✼ ✹ ✶✷ ✺ ✻ ✶✶ ✵ ✶✹ ✾ ✷ ✼ ✶✶ ✹ ✶ ✾ ✶✷ ✶✹ ✷ ✵ ✻ ✶✵ ✶✸ ✶✺ ✸ ✺ ✽ ✷ ✶ ✶✹ ✼ ✹ ✶✵ ✽ ✶✸ ✶✺ ✶✷ ✾ ✵ ✸ ✺ ✻ ✶✶

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✸✶ ✴ ✸✹

SLIDE 32

❙✲❜♦①s ✭❝♦♥t✳✮

❙✉♣♣♦s❡ t❤❛t ❇✹ = ✶✶✵✶✶✵ ❚❤❡♥ ✇❡ ♠✉st ✉s❡ t❤❡ ❙✲❜♦① ❙✹ ▲♦♦❦✐♥❣ ❛t t❤❡ str✐♥❣ ❇✹✱ ✇❡ ❤❛✈❡ ❜✶||❜✻ = ✶✵✱ ✇❤✐❝❤ ✐s t❤❡ ❞❡❝✐♠❛❧ ♥✉♠❜❡r ✷ ❋✉rt❤❡r♠♦r❡✱ ❜✷❜✸❜✹❜✺ = ✶✵✶✶✱ ✇❤✐❝❤ ✐s t❤❡ ❞❡❝✐♠❛❧ ♥✉♠❜❡r ✶✶ ❲❡ t❤✉s ✜♥❞ t❤❡ ♥✉♠❜❡r ✐♥ r♦✇ ✷✱ ❝♦❧✉♠♥ ✶✶ ♦❢ ❙✹✱ ✇❤✐❝❤ ✐s ✶✹ ❚❤❡ ❞❡❝✐♠❛❧ ♥✉♠❜❡r ✶✹ ✐s t❤❡ ✹✲❜✐t ❜✐♥❛r② ♥✉♠❜❡r ✶✶✶✵ ❚❤✉s✱ ❙✹(❇✹) = ❙✹(✶✶✵✶✶✵) = ✶✶✶✵

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✸✷ ✴ ✸✹

SLIDE 33

❉❊❙ ❘♦✉♥❞s ✭❝♦♥t✳✮

❚❤❡ ✸✷✲❜✐t str✐♥❣ ❙✶(❇✶)||❙✷(❇✷)|| · · · ||❙✽(❇✽) ✐s ♣❡r♠✉t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛s❤✐♦♥ t♦ ♣r♦❞✉❝❡ ❢❑✐(❘✐−✶)✿ P =     ✶✻ ✼ ✷✵ ✷✶ ✷✾ ✶✷ ✷✽ ✶✼ ✶ ✶✺ ✷✸ ✷✻ ✺ ✶✽ ✸✶ ✶✵ ✷ ✽ ✷✹ ✶✹ ✸✷ ✷✼ ✸ ✾ ✶✾ ✶✸ ✸✵ ✻ ✷✷ ✶✶ ✹ ✷✺    

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✸✸ ✴ ✸✹

SLIDE 34

❉❊❙ ❙t❛❣❡ ✸

❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ♣❡r♠✉t❛t✐♦♥ ■P ♣❡r♠✉t❡s ❘✶✻||▲✶✻ t♦ ♣r♦❞✉❝❡ ■P(❘✶✻||▲✶✻) ❚❤❡ ♣❡r♠✉t❛t✐♦♥ ■P−✶ ✐s ❢r❡q✉❡♥t❧② r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ ■P−✶ =             ✹✵ ✽ ✹✽ ✶✻ ✺✻ ✷✹ ✻✹ ✸✷ ✸✾ ✼ ✹✼ ✶✺ ✺✺ ✷✸ ✻✸ ✸✶ ✸✽ ✻ ✹✻ ✶✹ ✺✹ ✷✷ ✻✷ ✸✵ ✸✼ ✺ ✹✺ ✶✸ ✺✸ ✷✶ ✻✶ ✷✾ ✸✻ ✹ ✹✹ ✶✷ ✺✷ ✷✵ ✻✵ ✷✽ ✸✺ ✸ ✹✸ ✶✶ ✺✶ ✶✾ ✺✾ ✷✼ ✸✹ ✷ ✹✷ ✶✵ ✺✵ ✶✽ ✺✽ ✷✻ ✸✸ ✶ ✹✶ ✾ ✹✾ ✶✼ ✺✼ ✷✺             ■P(❘✶✻||▲✶✻) ✐s t❤❡ ❝✐♣❤❡rt❡①t ♦✉t♣✉t ❜② ❉❊❙

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✸✹ ✴ ✸✹