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Jun 26, 2023 441 likes •1.08k views

s s rs r trs r tss ttt tss r

SLIDE 1

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①t❡♥s✐♦♥s✱ ▼❡♠♦r② ❆❞❞r❡ss✐♥❣ ❛♥❞ ❆tt❡♥t✐♦♥

❊✐❧✐❢ ❙♦❧❜❡r❣ ✷✶✳✵✾✳✷✵✶✽

SLIDE 2

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❖✉t❧✐♥❡

❈♦♠♣♦s✐♥❣ ❘◆◆s ❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆s ❊♥❝♦❞❡r✲❞❡❝♦❞❡r ❢r❛♠❡✇♦r❦ ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❊①t❡r♥❛❧ ♠❡♠♦r② ❆tt❡♥❞✐♥❣ t♦ ♣r❡✈✐♦✉s st❛t❡s ❆tt❡♥t✐♦♥ ❈♦♥t❡♥t✲❜❛s❡❞ ▲♦❝❛t✐♦♥✲❜❛s❡❞

SLIDE 3

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❈♦♠♣♦s✐♥❣ ❘◆◆s

SLIDE 4

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆s

▼♦t✐✈❛t✐♦♥✿

❼ ❲❛♥t t♦ ✐♥❝❧✉❞❡ ❢✉t✉r❡ ❝♦♥t❡①t ❼ ❈♦✉❧❞ s♦❧✈❡ ✇✐t❤ t✐♠❡✲❞❡❧❛② ❢♦r ♣r❡❞✐❝t✐♦♥s✱ t❤♦✉❣❤ ♥❡❡❞ t♦

s♣❡❝✐❢② ✜①❡❞ ❝♦♥t❡①t✳ ❆ss✉♠❡s t✐❣❤t ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ ♣r❡❞✐❝t✐♦♥ ❛t t✐♠❡ t ❛♥❞ ✐♥♣✉t ❛t t✐♠❡ t✳

❼ ❡✳❣✳ s♣❡❡❝❤✲t♦✲t❡①t✱ t❡①t✲t♦✲s♣❡❡❝❤

SLIDE 5

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆s

▼♦t✐✈❛t✐♦♥✿

❼ ❲❛♥t t♦ ✐♥❝❧✉❞❡ ❢✉t✉r❡ ❝♦♥t❡①t ❼ ❈♦✉❧❞ s♦❧✈❡ ✇✐t❤ t✐♠❡✲❞❡❧❛② ❢♦r ♣r❡❞✐❝t✐♦♥s✱ t❤♦✉❣❤ ♥❡❡❞ t♦

s♣❡❝✐❢② ✜①❡❞ ❝♦♥t❡①t✳ ❆ss✉♠❡s t✐❣❤t ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ ♣r❡❞✐❝t✐♦♥ ❛t t✐♠❡ t ❛♥❞ ✐♥♣✉t ❛t t✐♠❡ t✳

❼ ❡✳❣✳ s♣❡❡❝❤✲t♦✲t❡①t✱ t❡①t✲t♦✲s♣❡❡❝❤

SLIDE 6

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆s

▼♦t✐✈❛t✐♦♥✿

❼ ❲❛♥t t♦ ✐♥❝❧✉❞❡ ❢✉t✉r❡ ❝♦♥t❡①t ❼ ❈♦✉❧❞ s♦❧✈❡ ✇✐t❤ t✐♠❡✲❞❡❧❛② ❢♦r ♣r❡❞✐❝t✐♦♥s✱ t❤♦✉❣❤ ♥❡❡❞ t♦

s♣❡❝✐❢② ✜①❡❞ ❝♦♥t❡①t✳ ❆ss✉♠❡s t✐❣❤t ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ ♣r❡❞✐❝t✐♦♥ ❛t t✐♠❡ t ❛♥❞ ✐♥♣✉t ❛t t✐♠❡ t✳

❼ ❡✳❣✳ s♣❡❡❝❤✲t♦✲t❡①t✱ t❡①t✲t♦✲s♣❡❡❝❤

SLIDE 7

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆ ✲ s✐♥❣❧❡ ❧❛②❡r

SLIDE 8

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆ ✲ t✇♦ ❧❛②❡rs

SLIDE 9

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆ ✲ ❢❡❛t✉r❡ ❡①tr❛❝t✐♦♥

SLIDE 10

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊♥❝♦❞❡r✲❞❡❝♦❞❡r

❋♦r s❡q✉❡♥❝❡✲t♦✲s❡q✉❡♥❝❡ ♣r♦❜❧❡♠s ✇✐t❤ ❧♦♦s❡ ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ s❡q✉❡♥❝❡s

❼ ♣r❡❞✐❝t✐♦♥ ❛t t✐♠❡ t ♥♦t ❞✐r❡❝t❧② r❡❧❛t❡❞ t♦ ✐♥♣✉t ❛t t✐♠❡ t✳

❊①❛♠♣❧❡✿ s❡♥t❡♥❝❡ tr❛♥s❧❛t✐♦♥ ✶✳ ❊♥❝♦❞❡ t❤❡ ✏♠❡❛♥✐♥❣✑ ♦❢ s❡♥t❡♥❝❡ ✐♥ s♦✉r❝❡ ❧❛♥❣✉❛❣❡ ✐♥t♦ ✐♥t❡r♠❡❞✐❛t❡ r❡♣r❡s❡♥t❛t✐♦♥ ✷✳ ❉❡❝♦❞❡ t❤❡ ✏♠❡❛♥✐♥❣✑ ♦❢ t❤❡ s❡♥t❡♥❝❡ ✐♥t♦ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ t❤❡ t❛r❣❡t ❧❛♥❣✉❛❣❡

SLIDE 11

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊♥❝♦❞❡r✲❞❡❝♦❞❡r

❋♦r s❡q✉❡♥❝❡✲t♦✲s❡q✉❡♥❝❡ ♣r♦❜❧❡♠s ✇✐t❤ ❧♦♦s❡ ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ s❡q✉❡♥❝❡s

❼ ♣r❡❞✐❝t✐♦♥ ❛t t✐♠❡ t ♥♦t ❞✐r❡❝t❧② r❡❧❛t❡❞ t♦ ✐♥♣✉t ❛t t✐♠❡ t✳

❊①❛♠♣❧❡✿ s❡♥t❡♥❝❡ tr❛♥s❧❛t✐♦♥ ✶✳ ❊♥❝♦❞❡ t❤❡ ✏♠❡❛♥✐♥❣✑ ♦❢ s❡♥t❡♥❝❡ ✐♥ s♦✉r❝❡ ❧❛♥❣✉❛❣❡ ✐♥t♦ ✐♥t❡r♠❡❞✐❛t❡ r❡♣r❡s❡♥t❛t✐♦♥ ✷✳ ❉❡❝♦❞❡ t❤❡ ✏♠❡❛♥✐♥❣✑ ♦❢ t❤❡ s❡♥t❡♥❝❡ ✐♥t♦ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ t❤❡ t❛r❣❡t ❧❛♥❣✉❛❣❡

SLIDE 12

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊♥❝♦❞❡r✲❞❡❝♦❞❡r✱ s❤❛r❡❞ ❘◆◆

SLIDE 13

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊♥❝♦❞❡r✲❞❡❝♦❞❡r✱ s❡♣❛r❛t❡ ❘◆◆

SLIDE 14

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊♥❝♦❞❡r✲❞❡❝♦❞❡r ❝♦♥❝❧✉s✐♦♥

❆ss✉♠❡ ◆ s♦✉r❝❡ ❛♥❞ ◆ t❛r❣❡t ❧❛♥❣✉❛❣❡s✳

❼ ❲❛♥t t♦ ❜❡ ❛❜❧❡ t♦ tr❛♥s❧❛t❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ♦❢ t❤❡♠ ❼ P♦ss✐❜❧❡ t♦ s❤❛r❡ ❡♥❝♦❞❡r ❛♥❞ ❞❡❝♦❞❡r❄

◆❡✇❡r ♠♦❞❡❧s ✐♥❝❧✉❞❡ ❛tt❡♥t✐♦♥

❼ ❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆ ♠❛② t❤❡♥ ❜❡ ✉s❡❞ ❛s ❡♥❝♦❞❡r

SLIDE 15

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊♥❝♦❞❡r✲❞❡❝♦❞❡r ❝♦♥❝❧✉s✐♦♥

❆ss✉♠❡ ◆ s♦✉r❝❡ ❛♥❞ ◆ t❛r❣❡t ❧❛♥❣✉❛❣❡s✳

❼ ❲❛♥t t♦ ❜❡ ❛❜❧❡ t♦ tr❛♥s❧❛t❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ♦❢ t❤❡♠ ❼ P♦ss✐❜❧❡ t♦ s❤❛r❡ ❡♥❝♦❞❡r ❛♥❞ ❞❡❝♦❞❡r❄

◆❡✇❡r ♠♦❞❡❧s ✐♥❝❧✉❞❡ ❛tt❡♥t✐♦♥

❼ ❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆ ♠❛② t❤❡♥ ❜❡ ✉s❡❞ ❛s ❡♥❝♦❞❡r

SLIDE 16

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊♥❝♦❞❡r✲❞❡❝♦❞❡r ❝♦♥❝❧✉s✐♦♥

❆ss✉♠❡ ◆ s♦✉r❝❡ ❛♥❞ ◆ t❛r❣❡t ❧❛♥❣✉❛❣❡s✳

❼ ❲❛♥t t♦ ❜❡ ❛❜❧❡ t♦ tr❛♥s❧❛t❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ♦❢ t❤❡♠ ❼ P♦ss✐❜❧❡ t♦ s❤❛r❡ ❡♥❝♦❞❡r ❛♥❞ ❞❡❝♦❞❡r❄

◆❡✇❡r ♠♦❞❡❧s ✐♥❝❧✉❞❡ ❛tt❡♥t✐♦♥

❼ ❇✐❞✐r❡❝t✐♦♥❛❧ ❘◆◆ ♠❛② t❤❡♥ ❜❡ ✉s❡❞ ❛s ❡♥❝♦❞❡r

SLIDE 17

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s

SLIDE 18

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s

SLIDE 19

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆❞❞r❡ss✐♥❣✿ ❧♦❝❛t✐♦♥ ✈s ❝♦♥t❡♥t✲❜❛s❡❞

❆ss✉♠❡ ✇❡ ❤❛✈❡ ♠❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳

❼ ❊✳❣✳ ▼❥ ∈ R♥

❍♦✇ ❞♦ ✇❡ ❛❞❞r❡ss ♠❡♠♦r②❄ ▲♦❝❛t✐♦♥

❼ ❙♣❡❝✐❢② ✇❤❡r❡ t♦ ❣❡t ✐♥❢♦r♠❛t✐♦♥✱ ❡✳❣✳ ✐♥❞❡① ❥

✶ ❏

❼ ✏●✐✈❡ ♠❡ t❤❡ ❝♦♥t❡♥t ❛t ♠❡♠♦r② ❝❡❧❧ ✹✑

❼ ❉✐r❡❝t ❛❞❞r❡ss✐♥❣

❈♦♥t❡♥t

❼ ❙♣❡❝✐❢② ✇❤❛t ❦✐♥❞ ♦❢ ✐♥❢♦r♠❛t✐♦♥ t❤r♦✉❣❤ ❛ q✉❡r② q

❼ ✏❲❤❡♥ ❞✐❞ t❤❡ ❢r❡♥❝❤ r❡✈♦❧✉t✐♦♥ st❛rt❄✑

❼ ■♥❞✐r❡❝t ❛❞❞r❡ss✐♥❣

SLIDE 20

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆❞❞r❡ss✐♥❣✿ ❧♦❝❛t✐♦♥ ✈s ❝♦♥t❡♥t✲❜❛s❡❞

❆ss✉♠❡ ✇❡ ❤❛✈❡ ♠❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳

❼ ❊✳❣✳ ▼❥ ∈ R♥

❍♦✇ ❞♦ ✇❡ ❛❞❞r❡ss ♠❡♠♦r②❄ ▲♦❝❛t✐♦♥

❼ ❙♣❡❝✐❢② ✇❤❡r❡ t♦ ❣❡t ✐♥❢♦r♠❛t✐♦♥✱ ❡✳❣✳ ✐♥❞❡① ❥

✶ ❏

❼ ✏●✐✈❡ ♠❡ t❤❡ ❝♦♥t❡♥t ❛t ♠❡♠♦r② ❝❡❧❧ ✹✑

❼ ❉✐r❡❝t ❛❞❞r❡ss✐♥❣

❈♦♥t❡♥t

❼ ❙♣❡❝✐❢② ✇❤❛t ❦✐♥❞ ♦❢ ✐♥❢♦r♠❛t✐♦♥ t❤r♦✉❣❤ ❛ q✉❡r② q

❼ ✏❲❤❡♥ ❞✐❞ t❤❡ ❢r❡♥❝❤ r❡✈♦❧✉t✐♦♥ st❛rt❄✑

❼ ■♥❞✐r❡❝t ❛❞❞r❡ss✐♥❣

SLIDE 21

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆❞❞r❡ss✐♥❣✿ ❧♦❝❛t✐♦♥ ✈s ❝♦♥t❡♥t✲❜❛s❡❞

❆ss✉♠❡ ✇❡ ❤❛✈❡ ♠❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳

❼ ❊✳❣✳ ▼❥ ∈ R♥

❍♦✇ ❞♦ ✇❡ ❛❞❞r❡ss ♠❡♠♦r②❄ ▲♦❝❛t✐♦♥

❼ ❙♣❡❝✐❢② ✇❤❡r❡ t♦ ❣❡t ✐♥❢♦r♠❛t✐♦♥✱ ❡✳❣✳ ✐♥❞❡① ❥ ∈ {✶, . . . , ❏}

❼ ✏●✐✈❡ ♠❡ t❤❡ ❝♦♥t❡♥t ❛t ♠❡♠♦r② ❝❡❧❧ ✹✑

❼ ❉✐r❡❝t ❛❞❞r❡ss✐♥❣

❈♦♥t❡♥t

❼ ❙♣❡❝✐❢② ✇❤❛t ❦✐♥❞ ♦❢ ✐♥❢♦r♠❛t✐♦♥ t❤r♦✉❣❤ ❛ q✉❡r② q

❼ ✏❲❤❡♥ ❞✐❞ t❤❡ ❢r❡♥❝❤ r❡✈♦❧✉t✐♦♥ st❛rt❄✑

❼ ■♥❞✐r❡❝t ❛❞❞r❡ss✐♥❣

SLIDE 22

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆❞❞r❡ss✐♥❣✿ ❧♦❝❛t✐♦♥ ✈s ❝♦♥t❡♥t✲❜❛s❡❞

❆ss✉♠❡ ✇❡ ❤❛✈❡ ♠❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳

❼ ❊✳❣✳ ▼❥ ∈ R♥

❍♦✇ ❞♦ ✇❡ ❛❞❞r❡ss ♠❡♠♦r②❄ ▲♦❝❛t✐♦♥

❼ ❙♣❡❝✐❢② ✇❤❡r❡ t♦ ❣❡t ✐♥❢♦r♠❛t✐♦♥✱ ❡✳❣✳ ✐♥❞❡① ❥ ∈ {✶, . . . , ❏}

❼ ✏●✐✈❡ ♠❡ t❤❡ ❝♦♥t❡♥t ❛t ♠❡♠♦r② ❝❡❧❧ ✹✑

❼ ❉✐r❡❝t ❛❞❞r❡ss✐♥❣

❈♦♥t❡♥t

❼ ❙♣❡❝✐❢② ✇❤❛t ❦✐♥❞ ♦❢ ✐♥❢♦r♠❛t✐♦♥ t❤r♦✉❣❤ ❛ q✉❡r② q

❼ ✏❲❤❡♥ ❞✐❞ t❤❡ ❢r❡♥❝❤ r❡✈♦❧✉t✐♦♥ st❛rt❄✑

❼ ■♥❞✐r❡❝t ❛❞❞r❡ss✐♥❣

SLIDE 23

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆❞❞r❡ss✐♥❣✿ ❧♦❝❛t✐♦♥ ✈s ❝♦♥t❡♥t✲❜❛s❡❞

❆ss✉♠❡ ✇❡ ❤❛✈❡ ♠❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳

❼ ❊✳❣✳ ▼❥ ∈ R♥

❍♦✇ ❞♦ ✇❡ ❛❞❞r❡ss ♠❡♠♦r②❄ ▲♦❝❛t✐♦♥

❼ ❙♣❡❝✐❢② ✇❤❡r❡ t♦ ❣❡t ✐♥❢♦r♠❛t✐♦♥✱ ❡✳❣✳ ✐♥❞❡① ❥ ∈ {✶, . . . , ❏}

❼ ✏●✐✈❡ ♠❡ t❤❡ ❝♦♥t❡♥t ❛t ♠❡♠♦r② ❝❡❧❧ ✹✑

❼ ❉✐r❡❝t ❛❞❞r❡ss✐♥❣

❈♦♥t❡♥t

❼ ❙♣❡❝✐❢② ✇❤❛t ❦✐♥❞ ♦❢ ✐♥❢♦r♠❛t✐♦♥ t❤r♦✉❣❤ ❛ q✉❡r② q

❼ ✏❲❤❡♥ ❞✐❞ t❤❡ ❢r❡♥❝❤ r❡✈♦❧✉t✐♦♥ st❛rt❄✑

❼ ■♥❞✐r❡❝t ❛❞❞r❡ss✐♥❣

SLIDE 24

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆❞❞r❡ss✐♥❣✿ ❧♦❝❛t✐♦♥ ✈s ❝♦♥t❡♥t✲❜❛s❡❞

❆ss✉♠❡ ✇❡ ❤❛✈❡ ♠❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳

❼ ❊✳❣✳ ▼❥ ∈ R♥

❍♦✇ ❞♦ ✇❡ ❛❞❞r❡ss ♠❡♠♦r②❄ ▲♦❝❛t✐♦♥

❼ ❙♣❡❝✐❢② ✇❤❡r❡ t♦ ❣❡t ✐♥❢♦r♠❛t✐♦♥✱ ❡✳❣✳ ✐♥❞❡① ❥ ∈ {✶, . . . , ❏}

❼ ✏●✐✈❡ ♠❡ t❤❡ ❝♦♥t❡♥t ❛t ♠❡♠♦r② ❝❡❧❧ ✹✑

❼ ❉✐r❡❝t ❛❞❞r❡ss✐♥❣

❈♦♥t❡♥t

❼ ❙♣❡❝✐❢② ✇❤❛t ❦✐♥❞ ♦❢ ✐♥❢♦r♠❛t✐♦♥ t❤r♦✉❣❤ ❛ q✉❡r② q

❼ ✏❲❤❡♥ ❞✐❞ t❤❡ ❢r❡♥❝❤ r❡✈♦❧✉t✐♦♥ st❛rt❄✑

❼ ■♥❞✐r❡❝t ❛❞❞r❡ss✐♥❣

SLIDE 25

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❈♦♥t❡♥t✲❜❛s❡❞ ❛❞❞r❡ss✐♥❣

❼ ▼❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳ ❼ q✉❡r② q

❞

❼ ❦❡② ❢✉♥❝t✐♦♥ ❑✱ ❡✳❣✳ ❑

♥ ❞

❼ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥ ❢ ✱ ❡✳❣✳ ✐♥♥❡r ♣r♦❞✉❝t ❢✉♥❝t✐♦♥

❥

❢ q ❑ ▼❥ ❲❤❛t ✐s t❤❡ r❡t✉r♥❡❞ r❡s✉❧t ♦❢ ♦✉r q✉❡r②❄ ♣ s♦❢t♠❛① ✈ q ▼ ▼❥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣❥ ❤❛r❞ ❛❞❞r❡ss✐♥❣ ✈ q ▼

❏ ❥ ✶

♣❥▼❥ s♦❢t ❛❞❞r❡ss✐♥❣ ❲❤❡r❡ ❞♦❡s t❤❡ q✉❡r② ✈❡❝t♦r q ❝♦♠❡ ❢r♦♠❄

SLIDE 26

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❈♦♥t❡♥t✲❜❛s❡❞ ❛❞❞r❡ss✐♥❣

❼ ▼❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳ ❼ q✉❡r② q ∈ R❞ ❼ ❦❡② ❢✉♥❝t✐♦♥ ❑✱ ❡✳❣✳ ❑

♥ ❞

❼ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥ ❢ ✱ ❡✳❣✳ ✐♥♥❡r ♣r♦❞✉❝t ❢✉♥❝t✐♦♥

❥

❢ q ❑ ▼❥ ❲❤❛t ✐s t❤❡ r❡t✉r♥❡❞ r❡s✉❧t ♦❢ ♦✉r q✉❡r②❄ ♣ s♦❢t♠❛① ✈ q ▼ ▼❥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣❥ ❤❛r❞ ❛❞❞r❡ss✐♥❣ ✈ q ▼

❏ ❥ ✶

♣❥▼❥ s♦❢t ❛❞❞r❡ss✐♥❣ ❲❤❡r❡ ❞♦❡s t❤❡ q✉❡r② ✈❡❝t♦r q ❝♦♠❡ ❢r♦♠❄

SLIDE 27

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❈♦♥t❡♥t✲❜❛s❡❞ ❛❞❞r❡ss✐♥❣

❼ ▼❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳ ❼ q✉❡r② q ∈ R❞ ❼ ❦❡② ❢✉♥❝t✐♦♥ ❑✱ ❡✳❣✳ ❑ : R♥ → R❞ ❼ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥ ❢ ✱ ❡✳❣✳ ✐♥♥❡r ♣r♦❞✉❝t ❢✉♥❝t✐♦♥

❥

❢ q ❑ ▼❥ ❲❤❛t ✐s t❤❡ r❡t✉r♥❡❞ r❡s✉❧t ♦❢ ♦✉r q✉❡r②❄ ♣ s♦❢t♠❛① ✈ q ▼ ▼❥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣❥ ❤❛r❞ ❛❞❞r❡ss✐♥❣ ✈ q ▼

❏ ❥ ✶

♣❥▼❥ s♦❢t ❛❞❞r❡ss✐♥❣ ❲❤❡r❡ ❞♦❡s t❤❡ q✉❡r② ✈❡❝t♦r q ❝♦♠❡ ❢r♦♠❄

SLIDE 28

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❈♦♥t❡♥t✲❜❛s❡❞ ❛❞❞r❡ss✐♥❣

❼ ▼❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳ ❼ q✉❡r② q ∈ R❞ ❼ ❦❡② ❢✉♥❝t✐♦♥ ❑✱ ❡✳❣✳ ❑ : R♥ → R❞ ❼ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥ ❢ ✱ ❡✳❣✳ ✐♥♥❡r ♣r♦❞✉❝t ❢✉♥❝t✐♦♥

α❥ = ❢ (q, ❑(▼❥)) ❲❤❛t ✐s t❤❡ r❡t✉r♥❡❞ r❡s✉❧t ♦❢ ♦✉r q✉❡r②❄ ♣ s♦❢t♠❛① ✈ q ▼ ▼❥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣❥ ❤❛r❞ ❛❞❞r❡ss✐♥❣ ✈ q ▼

❏ ❥ ✶

♣❥▼❥ s♦❢t ❛❞❞r❡ss✐♥❣ ❲❤❡r❡ ❞♦❡s t❤❡ q✉❡r② ✈❡❝t♦r q ❝♦♠❡ ❢r♦♠❄

SLIDE 29

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❈♦♥t❡♥t✲❜❛s❡❞ ❛❞❞r❡ss✐♥❣

❼ ▼❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳ ❼ q✉❡r② q ∈ R❞ ❼ ❦❡② ❢✉♥❝t✐♦♥ ❑✱ ❡✳❣✳ ❑ : R♥ → R❞ ❼ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥ ❢ ✱ ❡✳❣✳ ✐♥♥❡r ♣r♦❞✉❝t ❢✉♥❝t✐♦♥

α❥ = ❢ (q, ❑(▼❥)) ❲❤❛t ✐s t❤❡ r❡t✉r♥❡❞ r❡s✉❧t ♦❢ ♦✉r q✉❡r②❄ ♣ s♦❢t♠❛① ✈ q ▼ ▼❥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣❥ ❤❛r❞ ❛❞❞r❡ss✐♥❣ ✈ q ▼

❏ ❥ ✶

♣❥▼❥ s♦❢t ❛❞❞r❡ss✐♥❣ ❲❤❡r❡ ❞♦❡s t❤❡ q✉❡r② ✈❡❝t♦r q ❝♦♠❡ ❢r♦♠❄

SLIDE 30

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❈♦♥t❡♥t✲❜❛s❡❞ ❛❞❞r❡ss✐♥❣

❼ ▼❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳ ❼ q✉❡r② q ∈ R❞ ❼ ❦❡② ❢✉♥❝t✐♦♥ ❑✱ ❡✳❣✳ ❑ : R♥ → R❞ ❼ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥ ❢ ✱ ❡✳❣✳ ✐♥♥❡r ♣r♦❞✉❝t ❢✉♥❝t✐♦♥

α❥ = ❢ (q, ❑(▼❥)) ❲❤❛t ✐s t❤❡ r❡t✉r♥❡❞ r❡s✉❧t ♦❢ ♦✉r q✉❡r②❄ ♣ = s♦❢t♠❛①(α) ✈(q, ▼) = ▼❥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣❥ ❤❛r❞ ❛❞❞r❡ss✐♥❣ ✈(q, ▼) =

❏

❥=✶

♣❥▼❥ s♦❢t ❛❞❞r❡ss✐♥❣ ❲❤❡r❡ ❞♦❡s t❤❡ q✉❡r② ✈❡❝t♦r q ❝♦♠❡ ❢r♦♠❄

SLIDE 31

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❈♦♥t❡♥t✲❜❛s❡❞ ❛❞❞r❡ss✐♥❣

❼ ▼❡♠♦r② ▼ ✇✐t❤ ♠❡♠♦r② ❝❡❧❧s ▼✶, . . . , ▼❏✳ ❼ q✉❡r② q ∈ R❞ ❼ ❦❡② ❢✉♥❝t✐♦♥ ❑✱ ❡✳❣✳ ❑ : R♥ → R❞ ❼ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥ ❢ ✱ ❡✳❣✳ ✐♥♥❡r ♣r♦❞✉❝t ❢✉♥❝t✐♦♥

α❥ = ❢ (q, ❑(▼❥)) ❲❤❛t ✐s t❤❡ r❡t✉r♥❡❞ r❡s✉❧t ♦❢ ♦✉r q✉❡r②❄ ♣ = s♦❢t♠❛①(α) ✈(q, ▼) = ▼❥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣❥ ❤❛r❞ ❛❞❞r❡ss✐♥❣ ✈(q, ▼) =

❏

❥=✶

♣❥▼❥ s♦❢t ❛❞❞r❡ss✐♥❣ ❲❤❡r❡ ❞♦❡s t❤❡ q✉❡r② ✈❡❝t♦r q ❝♦♠❡ ❢r♦♠❄

SLIDE 32

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ r❡❛❞ ♦♣❡r❛t✐♦♥

P❡r❢♦r♠ q✉❡r② ❜❛s❡❞ ♦♥ ❝✉rr❡♥t st❛t❡ qt ◗ r st ❊①tr❛❝t ❦❡② ❢♦r ❡❛❝❤ ♠❡♠♦r② ❝❡❧❧ ❦t

❥

❑ r ▼t

✶ ❥

❈❛❧❝✉❧❛t❡ ❤♦✇ ✇❡❧❧ ♠❡♠♦r② ❝❡❧❧ ♠❛t❝❤ q✉❡r②

t ❥

❢ qt ❦t

❥

❡t r❡s✉❧t✐♥❣ ✈❡❝t♦r rt ❜②

♣t s♦❢t♠❛①

t

rt ✈ qt ▼t

✶ ❏ ❥ ✶

♣t

❥ ▼t ✶ ❥

SLIDE 33

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ r❡❛❞ ♦♣❡r❛t✐♦♥

P❡r❢♦r♠ q✉❡r② ❜❛s❡❞ ♦♥ ❝✉rr❡♥t st❛t❡ qt = ◗(r)(st) ❊①tr❛❝t ❦❡② ❢♦r ❡❛❝❤ ♠❡♠♦r② ❝❡❧❧ ❦t

❥

❑ r ▼t

✶ ❥

❈❛❧❝✉❧❛t❡ ❤♦✇ ✇❡❧❧ ♠❡♠♦r② ❝❡❧❧ ♠❛t❝❤ q✉❡r②

t ❥

❢ qt ❦t

❥

❡t r❡s✉❧t✐♥❣ ✈❡❝t♦r rt ❜②

♣t s♦❢t♠❛①

t

rt ✈ qt ▼t

✶ ❏ ❥ ✶

♣t

❥ ▼t ✶ ❥

SLIDE 34

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ r❡❛❞ ♦♣❡r❛t✐♦♥

P❡r❢♦r♠ q✉❡r② ❜❛s❡❞ ♦♥ ❝✉rr❡♥t st❛t❡ qt = ◗(r)(st) ❊①tr❛❝t ❦❡② ❢♦r ❡❛❝❤ ♠❡♠♦r② ❝❡❧❧ ❦t

❥ = ❑ (r)(▼t−✶ ❥

) ❈❛❧❝✉❧❛t❡ ❤♦✇ ✇❡❧❧ ♠❡♠♦r② ❝❡❧❧ ♠❛t❝❤ q✉❡r②

t ❥

❢ qt ❦t

❥

❡t r❡s✉❧t✐♥❣ ✈❡❝t♦r rt ❜②

♣t s♦❢t♠❛①

t

rt ✈ qt ▼t

✶ ❏ ❥ ✶

♣t

❥ ▼t ✶ ❥

SLIDE 35

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ r❡❛❞ ♦♣❡r❛t✐♦♥

P❡r❢♦r♠ q✉❡r② ❜❛s❡❞ ♦♥ ❝✉rr❡♥t st❛t❡ qt = ◗(r)(st) ❊①tr❛❝t ❦❡② ❢♦r ❡❛❝❤ ♠❡♠♦r② ❝❡❧❧ ❦t

❥ = ❑ (r)(▼t−✶ ❥

) ❈❛❧❝✉❧❛t❡ ❤♦✇ ✇❡❧❧ ♠❡♠♦r② ❝❡❧❧ ♠❛t❝❤ q✉❡r② αt

❥ = ❢ (qt, ❦t ❥ )

❡t r❡s✉❧t✐♥❣ ✈❡❝t♦r rt ❜②

♣t s♦❢t♠❛①

t

rt ✈ qt ▼t

✶ ❏ ❥ ✶

♣t

❥ ▼t ✶ ❥

SLIDE 36

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ r❡❛❞ ♦♣❡r❛t✐♦♥

P❡r❢♦r♠ q✉❡r② ❜❛s❡❞ ♦♥ ❝✉rr❡♥t st❛t❡ qt = ◗(r)(st) ❊①tr❛❝t ❦❡② ❢♦r ❡❛❝❤ ♠❡♠♦r② ❝❡❧❧ ❦t

❥ = ❑ (r)(▼t−✶ ❥

) ❈❛❧❝✉❧❛t❡ ❤♦✇ ✇❡❧❧ ♠❡♠♦r② ❝❡❧❧ ♠❛t❝❤ q✉❡r② αt

❥ = ❢ (qt, ❦t ❥ )

❡t r❡s✉❧t✐♥❣ ✈❡❝t♦r rt ❜②

♣t = s♦❢t♠❛①(αt) rt = ✈(qt, ▼t−✶) =

❏

❥=✶

♣t

❥ ▼t−✶ ❥

SLIDE 37

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ ✇r✐t❡ ♦♣❡r❛t✐♦♥

◆❡❡❞ t♦ ❞❡❝✐❞❡ ✇❤❛t t♦ ✇r✐t❡ ✐♥ ❛❞❞✐t✐♦♥ t♦ ✇❤❡r❡

❼ ❲❤❡r❡ ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛s ✇✐t❤ r❡❛❞ ♦♣❡r❛t✐♦♥

❼ ❙❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ◗ ✇ ❛♥❞ ❑ ✇ ✳

❼ ❲❤❛t✿ ❡✳❣✳ ❢✉♥❝t✐♦♥ ❲

✇t ❲ st ❍♦✇ t♦ ♠❛❦❡ ✉♣❞❛t❡❄ ▼t

❥

✶ ♣✇

❥

▼t

✶ ❥

♣✇

❥ ✇t

♦✈❡r✇r✐t❡ ▼t

❥

▼t

✶ ❥

♣✇

❥ ✇t

r❡s✐❞✉❛❧ ✉♣❞❛t❡ ❊①✐sts ❝♦rr❡s♣♦♥❞✐♥❣ ❤❛r❞ ✉♣❞❛t❡ r✉❧❡s

SLIDE 38

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ ✇r✐t❡ ♦♣❡r❛t✐♦♥

◆❡❡❞ t♦ ❞❡❝✐❞❡ ✇❤❛t t♦ ✇r✐t❡ ✐♥ ❛❞❞✐t✐♦♥ t♦ ✇❤❡r❡

❼ ❲❤❡r❡ ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛s ✇✐t❤ r❡❛❞ ♦♣❡r❛t✐♦♥

❼ ❙❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ◗ ✇ ❛♥❞ ❑ ✇ ✳

❼ ❲❤❛t✿ ❡✳❣✳ ❢✉♥❝t✐♦♥ ❲

✇t ❲ st ❍♦✇ t♦ ♠❛❦❡ ✉♣❞❛t❡❄ ▼t

❥

✶ ♣✇

❥

▼t

✶ ❥

♣✇

❥ ✇t

♦✈❡r✇r✐t❡ ▼t

❥

▼t

✶ ❥

♣✇

❥ ✇t

r❡s✐❞✉❛❧ ✉♣❞❛t❡ ❊①✐sts ❝♦rr❡s♣♦♥❞✐♥❣ ❤❛r❞ ✉♣❞❛t❡ r✉❧❡s

SLIDE 39

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ ✇r✐t❡ ♦♣❡r❛t✐♦♥

◆❡❡❞ t♦ ❞❡❝✐❞❡ ✇❤❛t t♦ ✇r✐t❡ ✐♥ ❛❞❞✐t✐♦♥ t♦ ✇❤❡r❡

❼ ❲❤❡r❡ ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛s ✇✐t❤ r❡❛❞ ♦♣❡r❛t✐♦♥

❼ ❙❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ◗(✇) ❛♥❞ ❑ (✇)✳

❼ ❲❤❛t✿ ❡✳❣✳ ❢✉♥❝t✐♦♥ ❲

✇t ❲ st ❍♦✇ t♦ ♠❛❦❡ ✉♣❞❛t❡❄ ▼t

❥

✶ ♣✇

❥

▼t

✶ ❥

♣✇

❥ ✇t

♦✈❡r✇r✐t❡ ▼t

❥

▼t

✶ ❥

♣✇

❥ ✇t

r❡s✐❞✉❛❧ ✉♣❞❛t❡ ❊①✐sts ❝♦rr❡s♣♦♥❞✐♥❣ ❤❛r❞ ✉♣❞❛t❡ r✉❧❡s

SLIDE 40

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ ✇r✐t❡ ♦♣❡r❛t✐♦♥

◆❡❡❞ t♦ ❞❡❝✐❞❡ ✇❤❛t t♦ ✇r✐t❡ ✐♥ ❛❞❞✐t✐♦♥ t♦ ✇❤❡r❡

❼ ❲❤❡r❡ ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛s ✇✐t❤ r❡❛❞ ♦♣❡r❛t✐♦♥

❼ ❙❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ◗(✇) ❛♥❞ ❑ (✇)✳

❼ ❲❤❛t✿ ❡✳❣✳ ❢✉♥❝t✐♦♥ ❲

✇t = ❲ (st) ❍♦✇ t♦ ♠❛❦❡ ✉♣❞❛t❡❄ ▼t

❥

✶ ♣✇

❥

▼t

✶ ❥

♣✇

❥ ✇t

♦✈❡r✇r✐t❡ ▼t

❥

▼t

✶ ❥

♣✇

❥ ✇t

r❡s✐❞✉❛❧ ✉♣❞❛t❡ ❊①✐sts ❝♦rr❡s♣♦♥❞✐♥❣ ❤❛r❞ ✉♣❞❛t❡ r✉❧❡s

SLIDE 41

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ ✇r✐t❡ ♦♣❡r❛t✐♦♥

◆❡❡❞ t♦ ❞❡❝✐❞❡ ✇❤❛t t♦ ✇r✐t❡ ✐♥ ❛❞❞✐t✐♦♥ t♦ ✇❤❡r❡

❼ ❲❤❡r❡ ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛s ✇✐t❤ r❡❛❞ ♦♣❡r❛t✐♦♥

❼ ❙❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ◗(✇) ❛♥❞ ❑ (✇)✳

❼ ❲❤❛t✿ ❡✳❣✳ ❢✉♥❝t✐♦♥ ❲

✇t = ❲ (st) ❍♦✇ t♦ ♠❛❦❡ ✉♣❞❛t❡❄ ▼t

❥

✶ ♣✇

❥

▼t

✶ ❥

♣✇

❥ ✇t

♦✈❡r✇r✐t❡ ▼t

❥

▼t

✶ ❥

♣✇

❥ ✇t

r❡s✐❞✉❛❧ ✉♣❞❛t❡ ❊①✐sts ❝♦rr❡s♣♦♥❞✐♥❣ ❤❛r❞ ✉♣❞❛t❡ r✉❧❡s

SLIDE 42

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ ✇r✐t❡ ♦♣❡r❛t✐♦♥

◆❡❡❞ t♦ ❞❡❝✐❞❡ ✇❤❛t t♦ ✇r✐t❡ ✐♥ ❛❞❞✐t✐♦♥ t♦ ✇❤❡r❡

❼ ❲❤❡r❡ ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛s ✇✐t❤ r❡❛❞ ♦♣❡r❛t✐♦♥

❼ ❙❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ◗(✇) ❛♥❞ ❑ (✇)✳

❼ ❲❤❛t✿ ❡✳❣✳ ❢✉♥❝t✐♦♥ ❲

✇t = ❲ (st) ❍♦✇ t♦ ♠❛❦❡ ✉♣❞❛t❡❄ ▼t

❥ = (✶ − ♣✇ ❥ )▼t−✶ ❥

+ ♣✇

❥ ✇t

♦✈❡r✇r✐t❡ ▼t

❥ = ▼t−✶ ❥

+ ♣✇

❥ ✇t

r❡s✐❞✉❛❧ ✉♣❞❛t❡ ❊①✐sts ❝♦rr❡s♣♦♥❞✐♥❣ ❤❛r❞ ✉♣❞❛t❡ r✉❧❡s

SLIDE 43

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ ❤♦✇ t♦ ✉s❡ ✐t

❯♣❞❛t❡ ❢✉♥❝t✐♦♥✿ st = ❤(①t, st−✶, ②t−✶, rt−✶) ❈♦✉❧❞ ❛❧s♦ ❛❞❞ ❞✐r❡❝t❧② t♦ ♦✉t♣✉t ❢✉♥❝t✐♦♥ ②t ❢ st rt

SLIDE 44

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❘◆◆ ❡①❛♠♣❧❡ ✇✐t❤ ❡①t❡r♥❛❧ ♠❡♠♦r② ✲ ❤♦✇ t♦ ✉s❡ ✐t

❯♣❞❛t❡ ❢✉♥❝t✐♦♥✿ st = ❤(①t, st−✶, ②t−✶, rt−✶) ❈♦✉❧❞ ❛❧s♦ ❛❞❞ ❞✐r❡❝t❧② t♦ ♦✉t♣✉t ❢✉♥❝t✐♦♥ ②t = ❢ (st, rt)

SLIDE 45

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊①t❡r♥❛❧ ♠❡♠♦r② ✲ ♠✉❧t✐♣❧❡ r❡❛❞✴✇r✐t❡ ❤❡❛❞s

❼ ❊✳❣✳ ❞❡✜♥❡ ◆ q✉❡r②✱ ❦❡② ❢✉♥❝t✐♦♥ ♣❛✐rs (◗(r)

✶ , ❑ (r) ✶ )✱ ✳ ✳ ✳ ✱

(◗(r)

◆ , ❑ (r) ◆ )

❼ ❈♦♥❝❛t❡♥❛t❡ ❛❧❧ ♦❢ t❤❡ r❡tr✐❡✈❡❞ ✈❡❝t♦rs✱ r t = (r t

✶, . . . , r t ♥)✳

❼ ❲r✐t❡ ♦♣❡r❛t✐♦♥s✱ ♥❡❡❞ t♦ r❡s♦❧✈❡ ♣♦ss✐❜❧❡ ❝♦♥✢✐❝ts ✐♥ ✉♣❞❛t❡s ❼ ▼❛② ✉s❡ s❛♠❡ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥

SLIDE 46

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊①t❡r♥❛❧ ♠❡♠♦r② ✲ ♠✉❧t✐♣❧❡ r❡❛❞✴✇r✐t❡ ❤❡❛❞s

❼ ❊✳❣✳ ❞❡✜♥❡ ◆ q✉❡r②✱ ❦❡② ❢✉♥❝t✐♦♥ ♣❛✐rs (◗(r)

✶ , ❑ (r) ✶ )✱ ✳ ✳ ✳ ✱

(◗(r)

◆ , ❑ (r) ◆ )

❼ ❈♦♥❝❛t❡♥❛t❡ ❛❧❧ ♦❢ t❤❡ r❡tr✐❡✈❡❞ ✈❡❝t♦rs✱ r t = (r t

✶, . . . , r t ♥)✳

❼ ❲r✐t❡ ♦♣❡r❛t✐♦♥s✱ ♥❡❡❞ t♦ r❡s♦❧✈❡ ♣♦ss✐❜❧❡ ❝♦♥✢✐❝ts ✐♥ ✉♣❞❛t❡s ❼ ▼❛② ✉s❡ s❛♠❡ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥

SLIDE 47

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❊①t❡r♥❛❧ ♠❡♠♦r② ✲ ♠✉❧t✐♣❧❡ r❡❛❞✴✇r✐t❡ ❤❡❛❞s

❼ ❊✳❣✳ ❞❡✜♥❡ ◆ q✉❡r②✱ ❦❡② ❢✉♥❝t✐♦♥ ♣❛✐rs (◗(r)

✶ , ❑ (r) ✶ )✱ ✳ ✳ ✳ ✱

(◗(r)

◆ , ❑ (r) ◆ )

❼ ❈♦♥❝❛t❡♥❛t❡ ❛❧❧ ♦❢ t❤❡ r❡tr✐❡✈❡❞ ✈❡❝t♦rs✱ r t = (r t

✶, . . . , r t ♥)✳

❼ ❲r✐t❡ ♦♣❡r❛t✐♦♥s✱ ♥❡❡❞ t♦ r❡s♦❧✈❡ ♣♦ss✐❜❧❡ ❝♦♥✢✐❝ts ✐♥ ✉♣❞❛t❡s ❼ ▼❛② ✉s❡ s❛♠❡ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥

SLIDE 48

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆tt❡♥❞✐♥❣ t♦ ♣r❡✈✐♦✉s st❛t❡s ■

SLIDE 49

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆tt❡♥❞✐♥❣ t♦ ♣r❡✈✐♦✉s st❛t❡s ■

SLIDE 50

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆tt❡♥❞✐♥❣ t♦ ♣r❡✈✐♦✉s st❛t❡s ■■

st = ❤(①t, (s✶, . . . , st−✶), ②t−✶) ❉♦ q✉❡r② ✇✐t❤ r❡s♣❡❝t t♦ ✏♠❡♠♦r② ❝❡❧❧s✑ s✶ st

✶ ✳ t ✐

❢ ◗ st

✶ ①t

❑ s✐ ♣t s♦❢t♠❛①

t

st

✶ t ✶ ✐ ✶

♣t

✐ s✐

❚❤❡♥ ♣r♦❝❡❡❞ ✇✐t❤ ✏♣r❡✈✐♦✉s st❛t❡✑ st

✶

st ❤ ①t st

✶ ②t ✶

SLIDE 51

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆tt❡♥❞✐♥❣ t♦ ♣r❡✈✐♦✉s st❛t❡s ■■

st = ❤(①t, (s✶, . . . , st−✶), ②t−✶) ❉♦ q✉❡r② ✇✐t❤ r❡s♣❡❝t t♦ ✏♠❡♠♦r② ❝❡❧❧s✑ (s✶, . . . , st−✶)✳ αt

✐ = ❢ (◗(st−✶, ①t), ❑(s✐))

♣t = s♦❢t♠❛①(αt) ˜ st−✶ =

t−✶

✐=✶

♣t

✐ s✐

❚❤❡♥ ♣r♦❝❡❡❞ ✇✐t❤ ✏♣r❡✈✐♦✉s st❛t❡✑ st

✶

st ❤ ①t st

✶ ②t ✶

SLIDE 52

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆tt❡♥❞✐♥❣ t♦ ♣r❡✈✐♦✉s st❛t❡s ■■

st = ❤(①t, (s✶, . . . , st−✶), ②t−✶) ❉♦ q✉❡r② ✇✐t❤ r❡s♣❡❝t t♦ ✏♠❡♠♦r② ❝❡❧❧s✑ (s✶, . . . , st−✶)✳ αt

✐ = ❢ (◗(st−✶, ①t), ❑(s✐))

♣t = s♦❢t♠❛①(αt) ˜ st−✶ =

t−✶

✐=✶

♣t

✐ s✐

❚❤❡♥ ♣r♦❝❡❡❞ ✇✐t❤ ✏♣r❡✈✐♦✉s st❛t❡✑ ˜ st−✶ st = ❤(①t,˜ st−✶, ②t−✶)

SLIDE 53

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆tt❡♥❞✐♥❣ t♦ ♣r❡✈✐♦✉s st❛t❡s ■■■

SLIDE 54

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

❆tt❡♥t✐♦♥

SLIDE 55

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❛♣t✐♦♥✐♥❣ ✇✐t❤ ❘◆◆ ❛♥❞ ❝♦♥t❡♥t✲❜❛s❡❞ ❛tt❡♥t✐♦♥

❋✐❣✉r❡✿

■❧❧✉str❛t✐♦♥ ❢r♦♠ ✧❳✉✱ ❑❡❧✈✐♥✱ ❡t ❛❧✳ ✧❙❤♦✇✱ ❛tt❡♥❞ ❛♥❞ t❡❧❧✿ ◆❡✉r❛❧ ✐♠❛❣❡ ❝❛♣t✐♦♥ ❣❡♥❡r❛t✐♦♥ ✇✐t❤ ✈✐s✉❛❧ ❛tt❡♥t✐♦♥✳✧ ■♥t❡r♥❛t✐♦♥❛❧ ❝♦♥❢❡r❡♥❝❡ ♦♥ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣✳ ✷✵✶✺✳✧

❼ ❈♦♥t❡♥t ❜❛s❡❞ ❛❞❞r❡ss✐♥❣ ✇✐t❤ ✶✹

✶✹ ❝♦♥✈ ❢❡❛t✉r❡s ❛s ✏♠❡♠♦r②✑

SLIDE 56

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❛♣t✐♦♥✐♥❣ ✇✐t❤ ❘◆◆ ❛♥❞ ❝♦♥t❡♥t✲❜❛s❡❞ ❛tt❡♥t✐♦♥

❋✐❣✉r❡✿

❼ ❈♦♥t❡♥t ❜❛s❡❞ ❛❞❞r❡ss✐♥❣ ✇✐t❤ ✶✹ × ✶✹ ❝♦♥✈ ❢❡❛t✉r❡s ❛s

✏♠❡♠♦r②✑

SLIDE 57

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❛♣t✐♦♥✐♥❣ ✇✐t❤ ❘◆◆ ❛♥❞ ❝♦♥t❡♥t✲❜❛s❡❞ ❛tt❡♥t✐♦♥

❋✐❣✉r❡✿

SLIDE 58

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❘◆◆ ❛♥❞ ❧♦❝❛t✐♦♥✲❜❛s❡❞ ❛tt❡♥t✐♦♥

Ps❡✉❞♦❛❧❣♦r✐t❤♠✿

❼ ❙t❛rt ✇✐t❤ ❝❡♥t❡r✴r❛♥❞♦♠ ❣❧✐♠♣s❡✶ ✇✐t❤ ❝❡♥t❡r ❧✵ ❼ ❋♦r t

✶

✶✳ ❊①tr❛❝t ❣❧✐♠♣s❡ ✇✐t❤ ❝❡♥t❡r ❛t ❧t

✶✳

✷✳ ❊①tr❛❝t ❢❡❛t✉r❡s ❢♦r ❧♦❝❛t✐♦♥✱ ❡✳❣✳ ✇✐t❤ ❝♦♥✈♥❡t ✸✳ ❯♣❞❛t❡ st❛t❡ ♦❢ ❘◆◆ ✹✳

❼ ✐❢ t ✿ ♣r❡❞✐❝t ♥❡①t ❣❧✐♠♣s❡ ❝❡♥t❡r ❧ t ❼ ❡❧s❡✿ ▼❛❦❡ ♣r❡❞✐❝t✐♦♥✴❝❧❛ss✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❧

❼ ❍♦✇ t♦ ❡♥❝♦❞❡ ❧t❄ ❼ ●❧✐♠♣s❡ ♣♦❧✐❝② tr❛✐♥❡❞ ✇✐t❤ r❡✐♥❢♦r❝❡♠❡♥t ❧❡❛r♥✐♥❣ ✭♣♦❧✐❝②

❣r❛❞✐❡♥t✮✦ ❯s✉❛❧❧② ❡①tr❛❝t s♦♠❡ ❧♦✇❡r r❡s♦❧✉t✐♦♥ ❝r♦♣s ❛s ✇❡❧❧✳

✶❆ ❣❧✐♠♣s❡ ✐s ❤❡r❡ ❞❡✜♥❡❞ ❛s ❛ ❝r♦♣ ♦❢ t❤❡ ✐♠❛❣❡

SLIDE 59

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❘◆◆ ❛♥❞ ❧♦❝❛t✐♦♥✲❜❛s❡❞ ❛tt❡♥t✐♦♥

Ps❡✉❞♦❛❧❣♦r✐t❤♠✿

❼ ❙t❛rt ✇✐t❤ ❝❡♥t❡r✴r❛♥❞♦♠ ❣❧✐♠♣s❡✶ ✇✐t❤ ❝❡♥t❡r ❧✵ ❼ ❋♦r t

✶

✶✳ ❊①tr❛❝t ❣❧✐♠♣s❡ ✇✐t❤ ❝❡♥t❡r ❛t ❧t

✶✳

✷✳ ❊①tr❛❝t ❢❡❛t✉r❡s ❢♦r ❧♦❝❛t✐♦♥✱ ❡✳❣✳ ✇✐t❤ ❝♦♥✈♥❡t ✸✳ ❯♣❞❛t❡ st❛t❡ ♦❢ ❘◆◆ ✹✳

❼ ✐❢ t ✿ ♣r❡❞✐❝t ♥❡①t ❣❧✐♠♣s❡ ❝❡♥t❡r ❧ t ❼ ❡❧s❡✿ ▼❛❦❡ ♣r❡❞✐❝t✐♦♥✴❝❧❛ss✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❧

❼ ❍♦✇ t♦ ❡♥❝♦❞❡ ❧t❄ ❼ ●❧✐♠♣s❡ ♣♦❧✐❝② tr❛✐♥❡❞ ✇✐t❤ r❡✐♥❢♦r❝❡♠❡♥t ❧❡❛r♥✐♥❣ ✭♣♦❧✐❝②

❣r❛❞✐❡♥t✮✦ ❯s✉❛❧❧② ❡①tr❛❝t s♦♠❡ ❧♦✇❡r r❡s♦❧✉t✐♦♥ ❝r♦♣s ❛s ✇❡❧❧✳

✶❆ ❣❧✐♠♣s❡ ✐s ❤❡r❡ ❞❡✜♥❡❞ ❛s ❛ ❝r♦♣ ♦❢ t❤❡ ✐♠❛❣❡

SLIDE 60

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❘◆◆ ❛♥❞ ❧♦❝❛t✐♦♥✲❜❛s❡❞ ❛tt❡♥t✐♦♥

Ps❡✉❞♦❛❧❣♦r✐t❤♠✿

❼ ❙t❛rt ✇✐t❤ ❝❡♥t❡r✴r❛♥❞♦♠ ❣❧✐♠♣s❡✶ ✇✐t❤ ❝❡♥t❡r ❧✵ ❼ ❋♦r t = ✶, . . . , τ ✶✳ ❊①tr❛❝t ❣❧✐♠♣s❡ ✇✐t❤ ❝❡♥t❡r ❛t ❧t−✶✳ ✷✳ ❊①tr❛❝t ❢❡❛t✉r❡s ❢♦r ❧♦❝❛t✐♦♥✱ ❡✳❣✳ ✇✐t❤ ❝♦♥✈♥❡t ✸✳ ❯♣❞❛t❡ st❛t❡ ♦❢ ❘◆◆ ✹✳

❼ ✐❢ t < τ✿ ♣r❡❞✐❝t ♥❡①t ❣❧✐♠♣s❡ ❝❡♥t❡r ❧ t ❼ ❡❧s❡✿ ▼❛❦❡ ♣r❡❞✐❝t✐♦♥✴❝❧❛ss✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❧ τ

❼ ❍♦✇ t♦ ❡♥❝♦❞❡ ❧t❄ ❼ ●❧✐♠♣s❡ ♣♦❧✐❝② tr❛✐♥❡❞ ✇✐t❤ r❡✐♥❢♦r❝❡♠❡♥t ❧❡❛r♥✐♥❣ ✭♣♦❧✐❝②

❣r❛❞✐❡♥t✮✦ ❯s✉❛❧❧② ❡①tr❛❝t s♦♠❡ ❧♦✇❡r r❡s♦❧✉t✐♦♥ ❝r♦♣s ❛s ✇❡❧❧✳

✶❆ ❣❧✐♠♣s❡ ✐s ❤❡r❡ ❞❡✜♥❡❞ ❛s ❛ ❝r♦♣ ♦❢ t❤❡ ✐♠❛❣❡

SLIDE 61

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❘◆◆ ❛♥❞ ❧♦❝❛t✐♦♥✲❜❛s❡❞ ❛tt❡♥t✐♦♥

Ps❡✉❞♦❛❧❣♦r✐t❤♠✿

❼ ❙t❛rt ✇✐t❤ ❝❡♥t❡r✴r❛♥❞♦♠ ❣❧✐♠♣s❡✶ ✇✐t❤ ❝❡♥t❡r ❧✵ ❼ ❋♦r t = ✶, . . . , τ ✶✳ ❊①tr❛❝t ❣❧✐♠♣s❡ ✇✐t❤ ❝❡♥t❡r ❛t ❧t−✶✳ ✷✳ ❊①tr❛❝t ❢❡❛t✉r❡s ❢♦r ❧♦❝❛t✐♦♥✱ ❡✳❣✳ ✇✐t❤ ❝♦♥✈♥❡t ✸✳ ❯♣❞❛t❡ st❛t❡ ♦❢ ❘◆◆ ✹✳

❼ ✐❢ t < τ✿ ♣r❡❞✐❝t ♥❡①t ❣❧✐♠♣s❡ ❝❡♥t❡r ❧ t ❼ ❡❧s❡✿ ▼❛❦❡ ♣r❡❞✐❝t✐♦♥✴❝❧❛ss✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❧ τ

❼ ❍♦✇ t♦ ❡♥❝♦❞❡ ❧t❄ ❼ ●❧✐♠♣s❡ ♣♦❧✐❝② tr❛✐♥❡❞ ✇✐t❤ r❡✐♥❢♦r❝❡♠❡♥t ❧❡❛r♥✐♥❣ ✭♣♦❧✐❝②

❣r❛❞✐❡♥t✮✦ ❯s✉❛❧❧② ❡①tr❛❝t s♦♠❡ ❧♦✇❡r r❡s♦❧✉t✐♦♥ ❝r♦♣s ❛s ✇❡❧❧✳

✶❆ ❣❧✐♠♣s❡ ✐s ❤❡r❡ ❞❡✜♥❡❞ ❛s ❛ ❝r♦♣ ♦❢ t❤❡ ✐♠❛❣❡

SLIDE 62

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❘◆◆ ❛♥❞ ❧♦❝❛t✐♦♥✲❜❛s❡❞ ❛tt❡♥t✐♦♥

Ps❡✉❞♦❛❧❣♦r✐t❤♠✿

❼ ❙t❛rt ✇✐t❤ ❝❡♥t❡r✴r❛♥❞♦♠ ❣❧✐♠♣s❡✶ ✇✐t❤ ❝❡♥t❡r ❧✵ ❼ ❋♦r t = ✶, . . . , τ ✶✳ ❊①tr❛❝t ❣❧✐♠♣s❡ ✇✐t❤ ❝❡♥t❡r ❛t ❧t−✶✳ ✷✳ ❊①tr❛❝t ❢❡❛t✉r❡s ❢♦r ❧♦❝❛t✐♦♥✱ ❡✳❣✳ ✇✐t❤ ❝♦♥✈♥❡t ✸✳ ❯♣❞❛t❡ st❛t❡ ♦❢ ❘◆◆ ✹✳

❼ ✐❢ t < τ✿ ♣r❡❞✐❝t ♥❡①t ❣❧✐♠♣s❡ ❝❡♥t❡r ❧ t ❼ ❡❧s❡✿ ▼❛❦❡ ♣r❡❞✐❝t✐♦♥✴❝❧❛ss✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❧ τ

❼ ❍♦✇ t♦ ❡♥❝♦❞❡ ❧t❄ ❼ ●❧✐♠♣s❡ ♣♦❧✐❝② tr❛✐♥❡❞ ✇✐t❤ r❡✐♥❢♦r❝❡♠❡♥t ❧❡❛r♥✐♥❣ ✭♣♦❧✐❝②

❣r❛❞✐❡♥t✮✦ ❯s✉❛❧❧② ❡①tr❛❝t s♦♠❡ ❧♦✇❡r r❡s♦❧✉t✐♦♥ ❝r♦♣s ❛s ✇❡❧❧✳

✶❆ ❣❧✐♠♣s❡ ✐s ❤❡r❡ ❞❡✜♥❡❞ ❛s ❛ ❝r♦♣ ♦❢ t❤❡ ✐♠❛❣❡

SLIDE 63

❈♦♠♣♦s✐♥❣ ❘◆◆s ❘❡❝✉rs✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❘◆◆ ▼❡♠♦r② ❡①t❡♥s✐♦♥s ❆tt❡♥t✐♦♥

■♠❛❣❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❘◆◆ ❛♥❞ ❧♦❝❛t✐♦♥✲❜❛s❡❞ ❛tt❡♥t✐♦♥

Ps❡✉❞♦❛❧❣♦r✐t❤♠✿

❼ ❙t❛rt ✇✐t❤ ❝❡♥t❡r✴r❛♥❞♦♠ ❣❧✐♠♣s❡✶ ✇✐t❤ ❝❡♥t❡r ❧✵ ❼ ❋♦r t = ✶, . . . , τ ✶✳ ❊①tr❛❝t ❣❧✐♠♣s❡ ✇✐t❤ ❝❡♥t❡r ❛t ❧t−✶✳ ✷✳ ❊①tr❛❝t ❢❡❛t✉r❡s ❢♦r ❧♦❝❛t✐♦♥✱ ❡✳❣✳ ✇✐t❤ ❝♦♥✈♥❡t ✸✳ ❯♣❞❛t❡ st❛t❡ ♦❢ ❘◆◆ ✹✳

❼ ✐❢ t < τ✿ ♣r❡❞✐❝t ♥❡①t ❣❧✐♠♣s❡ ❝❡♥t❡r ❧ t ❼ ❡❧s❡✿ ▼❛❦❡ ♣r❡❞✐❝t✐♦♥✴❝❧❛ss✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❧ τ

❼ ❍♦✇ t♦ ❡♥❝♦❞❡ ❧t❄ ❼ ●❧✐♠♣s❡ ♣♦❧✐❝② tr❛✐♥❡❞ ✇✐t❤ r❡✐♥❢♦r❝❡♠❡♥t ❧❡❛r♥✐♥❣ ✭♣♦❧✐❝②

❣r❛❞✐❡♥t✮✦ ❯s✉❛❧❧② ❡①tr❛❝t s♦♠❡ ❧♦✇❡r r❡s♦❧✉t✐♦♥ ❝r♦♣s ❛s ✇❡❧❧✳

✶❆ ❣❧✐♠♣s❡ ✐s ❤❡r❡ ❞❡✜♥❡❞ ❛s ❛ ❝r♦♣ ♦❢ t❤❡ ✐♠❛❣❡