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rrt rtss - - PowerPoint PPT Presentation

Jan 27, 2023 852 likes •1.37k views

trt r rtss s st rrt rtss

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SLIDE 1

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❆❧❣❡❜r❛✐❝✕❉✐✛❡r❡♥t✐❛❧ ❈r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❏❡❛♥✕❈❤❛r❧❡s ❋❛✉❣èr❡ ▲✉❞♦✈✐❝ P❡rr❡t P✐❡rr❡✕❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r

❯P▼❈ ✕ ▲■P✻ ❈◆❘❙ ■◆❘■❆ P❛r✐s ✲ ❘♦❝q✉❡♥❝♦✉rt ❙❆▲❙❆ t❡❛♠

❙é♠✐♥❛✐r❡ ❈r②♣t♦ ✕ ❈❛❡♥ ✷✵✵✾✴✶✵✴✷✾

✶✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 2

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

P❧❛♥

✶

■♥tr♦❞✉❝t✐♦♥

✷

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

✸

❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

✷✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 3

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

P❧❛♥

✶

■♥tr♦❞✉❝t✐♦♥

✷

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

✸

❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

✸✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 4

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s

✹✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 5

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s

❆❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝r②♣t♦❣r❛♣❤✐❝ ♣r✐♠✐t✐✈❡✳ ❚♦♦❧s ❢♦r ❡✣❝✐❡♥t ♣♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣✳

✶

  • rö❜♥❡r ❇❛s❡s ❛❧❣♦r✐t❤♠s ✭❇✉❝❤❜❡r❣❡r✱ ❋❛✉❣èr❡ ❋✹ ❛♥❞ ❋✺✮✳

✷

❙❆❚ ❙♦❧✈❡rs✳

❘❡♠❛r❦ ❚❤❡r❡ ✐s ❛ ✈❡r② str♦♥❣ ❧✐♥❦ ❜❡t✇❡❡♥ t❤❡ ♠♦❞❡❧✐♥❣ ❛♥❞ t❤❡ t♦♦❧s ✉s❡❞ ❢♦r t❤❡ r❡s♦❧✉t✐♦♥✳ ❈❤❛❧❧❡♥❣❡ ❈❛♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ❜❡ ❡✣❝✐❡♥t ❛❣❛✐♥st ❜❧♦❝❦ ❝✐♣❤❡rs ❄

✺✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 6

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❖✉r ✇♦r❦

❙❆❚ ❙♦❧✈❡rs ❛tt❛❝❦s ❛❣❛✐♥st ❉❊❙ ✉s✐♥❣ ❞✐✛❡r❡♥t ♠♦❞❡❧✐♥❣s ♦❢ t❤❡ ❉❊❙ ❙✲❜♦①❡s✳ ■♥❝❧✉❞✐♥❣ ❡❧❡♠❡♥ts ❢r♦♠ ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s

♥❡✇ ❛tt❛❝❦s ❛❣❛✐♥st ✻✱✼ ❛♥❞ ✽ r♦✉♥❞s ♦❢ ❉❊❙ ✉s✐♥❣ ❞❡❞✐❝❛t❡❞ ❝❤❛r❛❝t❡r✐st✐❝s✳

❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t✐♠❡ ❛♥❞ ❞❛t❛ ❝♦♠♣❧❡①✐t②✳

✻✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 7

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❚❤❡ ❈◆❋✲❙❆❚ ♣r♦❜❧❡♠

  • ✐✈❡♥ ❛ s❡t ♦❢ ❝❧❛✉s❡s✱ ❞❡❝✐❞❡ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❛ss✐❣♥♠❡♥t ♦❢ t❤❡

✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t ❛❧❧ ❝❧❛✉s❡s ❛r❡ s❛t✐s✜❡❞✳ ❈♦♦❦✲▲❡✈✐♥ ❚❤❡♦r❡♠ ❚❤❡ ❈◆❋✲❙❆❚ ♣r♦❜❧❡♠ ✐s ◆P✲❝♦♠♣❧❡t❡✳ ¯ ❛ ∨ ¯ ❜ ∨ ¯ ❝ ❛ ∨ ¯ ❞ ∨ ¯ ❝ ❞ ∨ ¯ ❝ ❝ ∨ ❡ ¯ ❝ ∨ ¯ ❡ ❝ ∨ ¯ ❡ ❛ tr✉❡ ❜ ❢❛❧s❡ ❝ tr✉❡ ❞ tr✉❡ ❡ ❢❛❧s❡

✼✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 8

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❚❤❡ ❈◆❋✲❙❆❚ ♣r♦❜❧❡♠

  • ✐✈❡♥ ❛ s❡t ♦❢ ❝❧❛✉s❡s✱ ❞❡❝✐❞❡ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❛ss✐❣♥♠❡♥t ♦❢ t❤❡

✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t ❛❧❧ ❝❧❛✉s❡s ❛r❡ s❛t✐s✜❡❞✳ ❈♦♦❦✲▲❡✈✐♥ ❚❤❡♦r❡♠ ❚❤❡ ❈◆❋✲❙❆❚ ♣r♦❜❧❡♠ ✐s ◆P✲❝♦♠♣❧❡t❡✳ ¯ ❛ ∨ ¯ ❜ ∨ ¯ ❝ ❛ ∨ ¯ ❞ ∨ ¯ ❝ ❞ ∨ ¯ ❝ ❝ ∨ ❡ ¯ ❝ ∨ ¯ ❡ ❝ ∨ ¯ ❡ ❛ = tr✉❡ ❜ = ❢❛❧s❡ ❝ = tr✉❡ ❞ = tr✉❡ ❡ = ❢❛❧s❡

✼✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-9
SLIDE 9

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

P♦❧②♥♦♠✐❛❧ ❙②st❡♠ ❙♦❧✈✐♥❣ ✭P♦❙❙♦✮

❲❤② ❙❆❚ ❙♦❧✈❡rs ❄ ◆P✲❤❛r❞ ♣r♦❜❧❡♠✳ ❱❡r② ❡✣❝✐❡♥t ❛♥❞ ✢❡①✐❜❧❡ ❞❡❞✐❝❛t❡❞ s♦❢t✇❛r❡s✳ ❙❆❚✲❝♦♠♣❡t✐t✐♦♥✳ ❆❝t✐✈❡ r❡s❡❛r❝❤ ✜❡❧❞✳ ❊❛s② t♦ ✉s❡✳ ▲♦✇ ♠❡♠♦r② ❝♦♠s✉♠♣t✐♦♥✳ ❲❤✐❝❤ ❙❆❚ ❙♦❧✈❡r ❄ ❉P▲▲✿ ❜❛❝❦tr❛❝❦✐♥❣✳ ❙t♦❝❤❛st✐❝ ❧♦❝❛❧ s❡❛r❝❤✳

✽✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-10
SLIDE 10

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❈♦✉rt♦✐s ◆✳❚✳ ✱ ❇❛r❞ ●✳❱✳ ❛♥❞ ❏❡✛❡rs♦♥ ❈✳ ❊✣❝✐❡♥t ▼❡t❤♦❞s ❢♦r ❈♦♥✈❡rs✐♦♥ ❛♥❞ ❙♦❧✉t✐♦♥ ♦❢ ❙♣❛rs❡ ❙②st❡♠s ♦❢ ▲♦✇✲❉❡❣r❡❡ ▼✉❧t✐✈❛r✐❛t❡ P♦❧②♥♦♠✐❛❧s ♦✈❡r ●❋ ✭✷✮ ✈✐❛ ❙❆❚✲❙♦❧✈❡rs✳ ❤tt♣✿✴✴❡♣r✐♥t✳✐❛❝r✳♦r❣✴✷✵✵✼✴✵✷✹✳♣❞❢ ▼✐♥✐❙❛t✷ ❊❡♥✱ ◆✳ ❛♥❞ ❙♦r❡♥ss♦♥✱ ◆✳ ▼✐♥✐❙❛t✿ ❆ ❙❆❚ s♦❧✈❡r ✇✐t❤ ❝♦♥✢✐❝t✲❝❧❛✉s❡ ♠✐♥✐♠✐③❛t✐♦♥

✾✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-11
SLIDE 11

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❈♦♥✈❡rt✐♥❣ ❛ P♦❧②♥♦♠✐❛❧ s②st❡♠ ✐♥t♦ ❛ ❙❆❚ ✐♥st❛♥❝❡

❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠ ✭❆◆❋✮ t♦ ❈♦♥❥♦♥❝t✐✈❡ ◆♦r♠❛❧ ❋♦r♠ ✭❈◆❋✮

❆♥ ❡①❛♠♣❧❡ ①✶①✷ + ①✸ + ①✸①✶ + ①✹ + ①✺ = ✵ ❘❡♣❧❛❝❡ ❡❛❝❤ ♠♦♥♦♠✐❛❧ ❜② ❛ ♥❡✇ ✈❛r✐❛❜❧❡✳ ①✶①✷ + ①✸ + ①✶①✸ + ①✹ + ①✺ = ✵.

✶✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-12
SLIDE 12

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❈♦♥✈❡rt✐♥❣ ❛ P♦❧②♥♦♠✐❛❧ s②st❡♠ ✐♥t♦ ❛ ❙❆❚ ✐♥st❛♥❝❡

❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠ ✭❆◆❋✮ t♦ ❈♦♥❥♦♥❝t✐✈❡ ◆♦r♠❛❧ ❋♦r♠ ✭❈◆❋✮

❆♥ ❡①❛♠♣❧❡ ①✶①✷ + ①✸ + ①✸①✶ + ①✹ + ①✺ = ✵ ❘❡♣❧❛❝❡ ❡❛❝❤ ♠♦♥♦♠✐❛❧ ❜② ❛ ♥❡✇ ✈❛r✐❛❜❧❡✳ ①✶①✷ + ①✸ + ①✶①✸ + ①✹ + ①✺ = ✵.

✶✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-13
SLIDE 13

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❈♦♥✈❡rt✐♥❣ ❛ P♦❧②♥♦♠✐❛❧ s②st❡♠ ✐♥t♦ ❛ ❙❆❚ ✐♥st❛♥❝❡

❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠ ✭❆◆❋✮ t♦ ❈♦♥❥♦♥❝t✐✈❡ ◆♦r♠❛❧ ❋♦r♠ ✭❈◆❋✮

❆♥ ❡①❛♠♣❧❡ ①✶①✷ + ①✸ + ①✸①✶ + ①✹ + ①✺ = ✵ ❘❡♣❧❛❝❡ ❡❛❝❤ ♠♦♥♦♠✐❛❧ ❜② ❛ ♥❡✇ ✈❛r✐❛❜❧❡✳ ②✶ + ①✸ + ②✷ + ①✹ + ①✺ = ✵ ②✶ + ①✶①✷ = ✵ ②✷ + ①✶①✸ = ✵

✶✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-14
SLIDE 14

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❈♦♥✈❡rt✐♥❣ ❛ P♦❧②♥♦♠✐❛❧ s②st❡♠ ✐♥t♦ ❛ ❙❆❚ ✐♥st❛♥❝❡

❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠ ✭❆◆❋✮ t♦ ❈♦♥❥♦♥❝t✐✈❡ ◆♦r♠❛❧ ❋♦r♠ ✭❈◆❋✮

❆♥ ❡①❛♠♣❧❡ ①✶①✷ + ①✸ + ①✸①✶ + ①✹ + ①✺ = ✵ ✏❈✉t✑ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ②✶ + ①✸ + ②✷ + ①✹ + ①✺ = ✵ ②✶ + ①✶①✷ = ✵ ②✷ + ①✶①✸ = ✵

✶✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-15
SLIDE 15

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❈♦♥✈❡rt✐♥❣ ❛ P♦❧②♥♦♠✐❛❧ s②st❡♠ ✐♥t♦ ❛ ❙❆❚ ✐♥st❛♥❝❡

❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠ ✭❆◆❋✮ t♦ ❈♦♥❥♦♥❝t✐✈❡ ◆♦r♠❛❧ ❋♦r♠ ✭❈◆❋✮

❆♥ ❡①❛♠♣❧❡ ①✶①✷ + ①✸ + ①✸①✶ + ①✹ + ①✺ = ✵ ✏❈✉t✑ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ②✶ + ①✸ + ③✶ = ✵ ③✶ + ②✷ + ③✷ = ✵ ③✷ + ①✹ + ①✺ = ✵ ②✶ + ①✶①✷ = ✵ ②✷ + ①✶①✸ = ✵

✶✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-16
SLIDE 16

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❈♦♥✈❡rt✐♥❣ ❛ P♦❧②♥♦♠✐❛❧ s②st❡♠ ✐♥t♦ ❛ ❙❆❚ ✐♥st❛♥❝❡

❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠ ✭❆◆❋✮ t♦ ❈♦♥❥♦♥❝t✐✈❡ ◆♦r♠❛❧ ❋♦r♠ ✭❈◆❋✮

❆♥ ❡①❛♠♣❧❡ ①✶①✷ + ①✸ + ①✸①✶ + ①✹ + ①✺ = ✵ ❈♦♥✈❡rt ✐♥t♦ ❝❧❛✉s❡s ②✶ + ①✸ + ③✶ = ✵ ③✶ + ②✷ + ③✷ = ✵ ③✷ + ①✹ + ①✺ = ✵ ②✶ + ①✶①✷ = ✵ ②✷ + ①✶①✸ = ✵ ⇒

¯ ②✶ ∨ ¯ ①✸ ∨ ¯ ③✶, ¯ ②✶ ∨ ①✸ ∨ ③✶ ②✶ ∨ ¯ ①✸ ∨ ③✶, ②✶ ∨ ①✸ ∨ ¯ ③✶ ¯ ③✶ ∨ ¯ ②✷ ∨ ¯ ③✷, ¯ ③✶ ∨ ②✷ ∨ ③✷ ③✶ ∨ ¯ ②✷ ∨ ③✷, ③✶ ∨ ②✷ ∨ ¯ ③✷ ¯ ③✷ ∨ ¯ ①✹ ∨ ¯ ①✺, ¯ ③✷ ∨ ①✹ ∨ ①✺ ③✷ ∨ ¯ ①✹ ∨ ①✺, ③✷ ∨ ①✹ ∨ ¯ ①✺ ¯ ②✶ ∨ ①✶, ¯ ②✶ ∨ ①✷ ¯ ②✷ ∨ ①✶, ¯ ②✷ ∨ ①✸ ②✶ ∨ ¯ ①✶ ∨ ¯ ①✷, ②✷ ∨ ¯ ①✶ ∨ ¯ ①✸

✶✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-17
SLIDE 17

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❙♦❧✈✐♥❣ ❍❋❊ ❛❧❣❡❜r❛✐❝ s②st❡♠s ❄

❍❋❊✿ ❛ ♣✉❜❧✐❝ ❦❡② ❛❧❣♦r✐t❤♠ P❛t❛r✐♥ ❏✳ ❍✐❞❞❡♥ ✜❡❧❞s ❡q✉❛t✐♦♥s ✭❍❋❊✮ ❛♥❞ ✐s♦♠♦r♣❤✐s♠s ♦❢ ♣♦❧②♥♦♠✐❛❧s ✭■P✮✿ ❚✇♦ ♥❡✇ ❢❛♠✐❧✐❡s ♦❢ ❛s②♠♠❡tr✐❝ ❛❧❣♦r✐t❤♠s ❊❯❘❖❈❘❨P❚✬✾✻ ❇❛s❡❞ ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ s♦❧✈✐♥❣ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠s ♦✈❡r ●❋(✷)✳ ✷✶ ❡q✉❛t✐♦♥s✱ ✷✶ ✉♥❦♥♦✇♥s✱ ❞ = ✶✷✾✳ ❋✹ ✭▼❛❣♠❛✮✿ ✶✾s✳ ▼✐♥✐❙❆❚ ✭✸❙❆❚✮✿ ✶✽✾s✳ ▼✐♥✐❙❆❚ ✭✺❙❆❚✮✿ ✺✻✶s✳

✶✶✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-18
SLIDE 18

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙

❙♣❛rs❡ r❛♥❞♦♠ q✉❛❞r❛t✐❝ s②st❡♠s

✶✷✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-19
SLIDE 19

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

P❧❛♥

✶

■♥tr♦❞✉❝t✐♦♥

✷

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

✸

❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

✶✸✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-20
SLIDE 20

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞

■t❡r❛t✐✈❡ ❇❧♦❝❦ ❈✐♣❤❡r✳ ❇❧♦❝ s✐③❡ ✿ ✻✹ ❜✐ts✳ ❊✛❡❝t✐✈❡ s✐③❡ ♦❢ t❤❡ ❦❡② ✿ ✺✻ ❜✐ts✳ ❊♥❝r②♣t✐♦♥ st❛♥❞❛r❞ ❜❡t✇❡❡♥ ✶✾✼✻ ❛♥❞ ✷✵✵✷✳ ❲❤② ❞✐❞ ✇❡ ❝❤♦♦s❡ t♦ st✉❞② t❤❡ ❉❊❙ ❄

✶✹✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-21
SLIDE 21

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

▼❛✐♥ ❛tt❛❝❦s ❛❣❛✐♥st ❉❊❙

❲✐❡♥❡r✱ ▼✳❏✳ ❊✣❝✐❡♥t ❉❊❙ ❦❡② s❡❛r❝❤✳ ❚❡❝❤♥✐❝❛❧ ❘❡♣♦rt ▼❡❡t✲✐♥✲t❤❡✲♠✐❞❞❧❡ ❛tt❛❝❦s ❉✉♥❦❡❧♠❛♥✱ ❖✳ ❛♥❞ ❙❡❦❛r✱ ●✳ ❛♥❞ Pr❡♥❡❡❧✱ ❇✳ ■♠♣r♦✈❡❞ ▼❡❡t✲✐♥✲t❤❡✲▼✐❞❞❧❡ ❆tt❛❝❦s ♦♥ ❘❡❞✉❝❡❞✲❘♦✉♥❞ ❉❊❙✳ ■◆❉❖❈❘❨P❚✬✷✵✵✼

✶✺✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-22
SLIDE 22

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

▼❛✐♥ ❛tt❛❝❦s ❛❣❛✐♥st ❉❊❙ ✭■■✮

❇✐❤❛♠✱ ❊✳ ❛♥❞ ❙❤❛♠✐r✱ ❆✳ ❉✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ t❤❡ ❢✉❧❧ ✶✻✲r♦✉♥❞ ❉❊❙✳ ❈r②♣t♦✬✶✾✾✷ ❑♥✉❞s❡♥✱ ▲✳❘✳ P❛rt✐❛❧ ❛♥❞ ❤✐❣❤❡r ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❉❊❙✳ ❇❘■❈❙ r❡♣♦rt ▼❛ts✉✐✱ ▼✳ ▲✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s ♠❡t❤♦❞ ❢♦r ❉❊❙ ❝✐♣❤❡r✳ ❊❯❘❖❈❘❨P❚✬✶✾✾✸

✶✻✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-23
SLIDE 23

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❉❊❙ ❙tr✉❝t✉r❡

❋❡✐st❡❧ ♥❡t✇♦r❦ ❙✲❜♦①❡s ✿ ♥♦♥✲❧✐♥❡❛r ♣❛rt ♦❢ t❤❡ s②st❡♠

✶✼✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-24
SLIDE 24

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❉❊❙ ❙tr✉❝t✉r❡

❋❡✐st❡❧ ♥❡t✇♦r❦ ❙✲❜♦①❡s ✿ ♥♦♥✲❧✐♥❡❛r ♣❛rt ♦❢ t❤❡ s②st❡♠

✶✼✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-25
SLIDE 25

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t

❙t❛rt✐♥❣ ♣♦✐♥t ◆✳❚✳ ❈♦✉rt♦✐s✱ ●✳❱✳ ❇❛r❞ ❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ♦❢ t❤❡ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ■▼❆ ■♥t✳ ❈♦♥❢✳ ✷✵✵✼

  • ❡♥❡r❛❧ ♣r✐♥❝✐♣❧❡

✶ ❦♥♦✇♥ ♣❧❛✐♥t❡①t✳ ▼♦❞❡❧ t❤❡ ❝r②♣t♦s②st❡♠ ❜② ❛ s❡t ♦❢ ❝❧❛✉s❡s✳ ❯s❡ ▼✐♥✐s❛t t♦ ❡①tr❛❝t t❤❡ ❦❡②✳ ❘❡♠❛r❦ ❲❡ ❝❛♥ ❝♦♠❜✐♥❡ t❤✐s ❛♣♣r♦❛❝❤ ✇✐t❤ ❛♥ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ♦✈❡r s♦♠❡ ❜✐ts ♦❢ t❤❡ ❦❡②✳

✶✽✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-26
SLIDE 26

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❙✲❜♦① ♠♦❞❡❧✐♥❣ ✭■✮

❲❡ ❤❛✈❡ ❝♦♥s✐❞❡r❡❞ s❡✈❡r❛❧ ♠♦❞❡❧✐♥❣s ♦❢ t❤❡ ❉❊❙ ❙✲❜♦①❡s✳ ❚❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ♠♦❞❡❧✐♥❣ ✐s ✈❡r② ✐♠♣♦rt❛♥t✳ ❖✉r ♠♦❞❡❧✐♥❣ ❲❡ s❡❛r❝❤ ✭❡①❤❛✉st✐✈❡❧②✮ ❢♦r t❤❡ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s ✇❤✐❝❤ ✈❡r✐❢② ✿ P(①✶, . . . , ①✻, ②✶, . . . , ②✹) =

  • (①✐ + α✐)
  • (②✐ + β✐), α✐, β✐ ∈ {✵, ✶}

s✉❝❤ t❤❛t ❙(①✶, . . . , ①✻) = (②✶, . . . , ②✹) ⇒ P(①✶, . . . , ①✻, ②✶, . . . , ②✹) = ✵ ❈♦♠♣❧❡①✐t② ✿ ✸✶✵

✶✾✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-27
SLIDE 27

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❙✲❜♦① ♠♦❞❡❧✐♥❣ ✭■■✮

❙✲❜♦① ◆❜ ♦❢ ❝❧❛✉s❡s ❙✶ ✶✻✷✹ ❙✷ ✶✽✹✹ ❙✸ ✶✼✻✼ ❙✹ ✶✽✽✶ ❙✺ ✶✽✶✷ ❙✻ ✶✼✵✺ ❙✼ ✶✻✼✸ ❙✽ ✷✵✹✼ ❋♦r ✻ r♦✉♥❞s ✿ ✼✾✷ ✈❛r✐❛❜❧❡s ❛♥❞ ✾✵✵✽✻ ❝❧❛✉s❡s✳ ✰ ♣❛rt✐❛❧ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ♦♥ ✷✽ ❜✐ts ♦❢ t❤❡ ❦❡②✳

✷✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-28
SLIDE 28

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

✷✶✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 29

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

P❧❛♥

✶

■♥tr♦❞✉❝t✐♦♥

✷

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ▼♦❞❡❧✐♥❣ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

✸

❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

✷✷✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 30

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❖✉r ❛♣♣r♦❛❝❤

▲✐♠✐t ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ✉s✉❛❧❧② ❝♦♥s✐❞❡r ♦♥❧② ♦♥❡ ❦♥♦✇♥ ♣❧❛✐♥t❡①t✳ ❲❡ ❝♦♠❜✐♥❡ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ❛♥❞ st❛t✐st✐❝❛❧ t❡❝❤♥✐q✉❡s t♦ ❡①♣❧♦✐t ❡✣❝✐❡♥t❧② t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ s❡✈❡r❛❧ ♣❧❛✐♥t❡①ts✳ ❚r❛❞❡♦✛ t✐♠❡✴♣❧❛✐♥t❡①ts✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❝♦♥s✐❞❡r ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s✳

✷✸✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-31
SLIDE 31

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❉✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ✭■✮

❇✐❤❛♠✱ ❊✳ ❛♥❞ ❙❤❛♠✐r✱ ❆✳ ❉✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙✲❧✐❦❡ ❝r②♣t♦s②st❡♠s✳ ❏♦✉r♥❛❧ ♦❢ ❈r②♣t♦❧♦❣②✳ ✶✾✾✶✳ ❚❤❡ ♣r✐♥❝✐♣❧❡ ✇❛s ❦♥♦✇♥ ❜② t❤❡ ❉❊❙ ❞❡s✐❣♥❡rs✳ ❇❛s❡❞ ♦♥ ❛ st❛t✐st✐❝❛❧ ❜✐❛s ♦❢ t❤❡ ❙✲❜♦①❡s✳ ❑❡② r❡❝♦✈❡r② ❛tt❛❝❦✳ ❲❡ tr② t♦ ♣r❡❞✐❝t ❤♦✇ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ♣❛✐r ♦❢ ♣❧❛✐♥t❡①ts ✇✐❧❧ ❞✐✛✉s❡ t❤r♦✉❣❤ t❤❡ ❝✐♣❤❡r✳

✷✹✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-32
SLIDE 32

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❉✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ t❤❡ ✸ r♦✉♥❞✲r❡❞✉❝❡❞ ❉❊❙

❆✵▲ ✵✵ ✵✵ ✵✵ ✵✵① ✰ ❋✶ ✵✵ ✵✵ ✵✵ ✵✵① ❆✵▲ ✵✵ ✵✵ ✵✵ ✵✵① ✵✵ ✵✵ ✵✵ ✵✵① ✰ ❋✷ ❆✵▲ ❇ ❆✵▲ ❇ ✰ ❋✸ ❆✸▲ ❆✸❘ ∆✐♥ = ❇ = ❆✸▲ ∆♦✉t = ❆✵▲ ⊕ ❆✸❘ ✷✺✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-33
SLIDE 33

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❉✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ✭■■✮

♠♦r❡ t❤❛♥ ✸ r♦✉♥❞s ❙t❛t✐st✐❝❛❧ ♠❡t❤♦❞✳ ❞✐✛❡r❡♥t✐❛❧ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❆ ❧♦t ♦❢ ♣❧❛✐♥t❡①ts ♥❡❡❞❡❞✳

✷✻✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 34

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

▼♦t✐✈❛t✐♦♥s

❈♦♠♣r♦♠✐s❡ ❜❡t✇❡❡♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ❛♥❞ ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s✳ ❲❡ ❝❛♥ ✉s❡ t❤❡ str♦♥❣ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s✉❜❦❡②s✳ ❚❤❡ ♥♦t✐♦♥ ♦❢ ❞✐✛❡r❡♥❝❡ ✐s ❡❛s② t♦ r❡♣r❡s❡♥t ✇✐t❤ ❝❧❛✉s❡s✳ ❲❡ ♦♥❧② ♥❡❡❞ ♦♥❡ ♣❛✐r ❢♦❧❧♦✇✐♥❣ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ t♦ r❡tr✐❡✈❡ t❤❡ ❦❡②✳

✷✼✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 35

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

  • ❡♥❡r❛❧ ❛❧❣♦r✐t❤♠

❘❡♣❡❛t ✉♥t✐❧ t❤❡ ❦❡② ✐s ❢♦✉♥❞ ✿ ❈❤♦♦s❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❝❤❛r❛❝t❡r✐st✐❝✳ ❈❤♦♦s❡ t✇♦ ♣❧❛✐♥t❡①ts ✇✐t❤ ❞✐✛❡r❡♥❝❡ ✜①❡❞ ❜② t❤❡ ❝❤❛r❛❝t❡r✐st✐❝✳ ❈♦♥str✉❝t t❤❡ s②st❡♠ ♦❢ ❝❧❛✉s❡s ❢♦r ❉❊❙✱ ❛♥❞ ❛❞❞ t❤❡ ❝❧❛✉s❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝✳ ❙♦❧✈❡ ✇✐t❤ ▼✐♥✐❙❛t✳ ■❢ t❤❡ r❡s✉❧t ✐s ❯◆❙❆❚■❙❋■❆❇▲❊✱ r❡st❛rt ✭✐t ♠❡❛♥s t❤❛t t❤❡ ♣❛✐r ❞✐❞♥✬t ❢♦❧❧♦✇ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝✮✳ ■❢ t❤❡ r❡s✉❧t ✐s ❙❆❚■❙❋■❆❇▲❊✱ t❤❡♥ ▼✐♥✐❙❛t r❡t✉r♥s t❤❡ ❦❡②✳

✷✽✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 36

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❙✐① r♦✉♥❞s

❆♣♣r♦❛❝❤ ❈❧❛ss✐❝❛❧ ❞✐✛❡r❡♥t✐❛❧ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t ❝❤❛r❛❝t❡r✐st✐❝s t♦ r❡❞✉❝❡ t❤❡ ❞❛t❛ ❝♦♠♣❧❡①✐t②✳ ✸✲r♦✉♥❞ ❝❤❛r❛❝t❡r✐st✐❝s ❈❧❛ss✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❬❇✐❤❛♠❙❤❛♠✐r❪✱ ♣r♦❜❛❜✐❧✐t②✿ ✶ ✶✻

✹✵✵✽✵✵✵✵✵✹✵✵✵✵✵✵ ✵✹✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✵✵✵✵✵✵✵✵✵✹✵✵✵✵✵✵ ✵✹✵✵✵✵✵✵✹✵✵✽✵✵✵✵ ✵✵✷✵✵✵✵✽✵✵✵✵✵✹✵✵ ✵✵✵✵✵✹✵✵✵✵✵✵✵✵✵✵ ✵✵✵✵✵✵✵✵✵✵✵✵✵✹✵✵ ✵✵✵✵✵✹✵✵✵✵✷✵✵✵✵✽

✹ ❡①tr❛ ❝❤❛r❛❝t❡r✐st✐❝s✱ ♣r♦❜❛❜✐❧✐t②✿ ✶ ✷✵

✵✵✽✵✽✷✵✵✻✵✵✵✵✵✵✵ ✻✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✵✵✵✵✵✵✵✵✻✵✵✵✵✵✵✵ ✻✵✵✵✵✵✵✵✵✵✽✵✽✷✵✵ ✹✵✵✵✹✵✶✵✵✷✵✵✵✵✵✵ ✵✷✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✵✵✵✵✵✵✵✵✵✷✵✵✵✵✵✵ ✵✷✵✵✵✵✵✵✹✵✵✵✹✵✶✵ ✵✵✵✵✹✵✶✵✵✻✵✵✵✵✵✵ ✵✻✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✵✵✵✵✵✵✵✵✵✻✵✵✵✵✵✵ ✵✻✵✵✵✵✵✵✵✵✵✵✹✵✶✵ ✵✵✶✵✵✵✵✶✵✵✵✵✵✵✻✵ ✵✵✵✵✵✵✻✵✵✵✵✵✵✵✵✵ ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✻✵ ✵✵✵✵✵✵✻✵✵✵✶✵✵✵✵✶ ✷✾✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-37
SLIDE 37

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❙✐① r♦✉♥❞s

❆♣♣r♦❛❝❤ ❈❧❛ss✐❝❛❧ ❞✐✛❡r❡♥t✐❛❧ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t ❝❤❛r❛❝t❡r✐st✐❝s t♦ r❡❞✉❝❡ t❤❡ ❞❛t❛ ❝♦♠♣❧❡①✐t②✳ ✸✲r♦✉♥❞ ❝❤❛r❛❝t❡r✐st✐❝s ❈❧❛ss✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❬❇✐❤❛♠❙❤❛♠✐r❪✱ ♣r♦❜❛❜✐❧✐t②✿ ✶/✶✻

(✹✵✵✽✵✵✵✵✵✹✵✵✵✵✵✵, ✵✹✵✵✵✵✵✵✵✵✵✵✵✵✵✵, ✵✵✵✵✵✵✵✵✵✹✵✵✵✵✵✵, ✵✹✵✵✵✵✵✵✹✵✵✽✵✵✵✵) (✵✵✷✵✵✵✵✽✵✵✵✵✵✹✵✵, ✵✵✵✵✵✹✵✵✵✵✵✵✵✵✵✵, ✵✵✵✵✵✵✵✵✵✵✵✵✵✹✵✵, ✵✵✵✵✵✹✵✵✵✵✷✵✵✵✵✽)

✹ ❡①tr❛ ❝❤❛r❛❝t❡r✐st✐❝s✱ ♣r♦❜❛❜✐❧✐t②✿ ✶/✷✵

(✵✵✽✵✽✷✵✵✻✵✵✵✵✵✵✵, ✻✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵, ✵✵✵✵✵✵✵✵✻✵✵✵✵✵✵✵, ✻✵✵✵✵✵✵✵✵✵✽✵✽✷✵✵) (✹✵✵✵✹✵✶✵✵✷✵✵✵✵✵✵, ✵✷✵✵✵✵✵✵✵✵✵✵✵✵✵✵, ✵✵✵✵✵✵✵✵✵✷✵✵✵✵✵✵, ✵✷✵✵✵✵✵✵✹✵✵✵✹✵✶✵) (✵✵✵✵✹✵✶✵✵✻✵✵✵✵✵✵, ✵✻✵✵✵✵✵✵✵✵✵✵✵✵✵✵, ✵✵✵✵✵✵✵✵✵✻✵✵✵✵✵✵, ✵✻✵✵✵✵✵✵✵✵✵✵✹✵✶✵) (✵✵✶✵✵✵✵✶✵✵✵✵✵✵✻✵, ✵✵✵✵✵✵✻✵✵✵✵✵✵✵✵✵, ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✻✵, ✵✵✵✵✵✵✻✵✵✵✶✵✵✵✵✶) ✷✾✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-38
SLIDE 38

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❍♦✇ t♦ ❝♦♠❜✐♥❡ ❝❤❛r❛❝t❡r✐st✐❝s ❄

❆ ✜rst ❛tt❡♠♣t✿ ✏❝✉❜❡ ❝♦♠❜✐♥❛t✐♦♥✑

Pr✐♥❝✐♣❧❡ ❈♦♠❜✐♥✐♥❣ ♥ ❝❤❛r❛❝t❡r✐st✐❝s✿ ❈❤♦♦s❡ ♠✵,...,✵ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ❋♦r ✈ ∈ {✵, ✶}♥✱ ✉s❡ ♠✈ = ♠✵ ⊕ ♥

✐=✶ ✈✐δ✐✳

◆✉♠❜❡r ♦❢ tr✐❛❧s = ♥✉♠❜❡r ♦❢ ❡❞❣❡s ♦❢ ❛ ♥✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡✳ ◆✉♠❜❡r ♦❢ ❝❤♦s❡♥ ♣❧❛✐♥t❡①ts = ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ♦❢ ❛ ♥✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡✳

♠✵✵✵ ♠✵✶✵ ♠✶✵✵ ♠✶✶✵ ♠✵✵✶ ♠✵✶✶ ♠✶✵✶ ♠✶✶✶ δ✷ δ✶ δ✸

❜✉t✳✳✳ ♥♦t ✐♥❞❡♣❡♥❞❛♥t ❡✈❡♥ts

✸✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-39
SLIDE 39

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❍♦✇ t♦ ❝♦♠❜✐♥❡ ❝❤❛r❛❝t❡r✐st✐❝s ❄

❆ ✜rst ❛tt❡♠♣t✿ ✏❝✉❜❡ ❝♦♠❜✐♥❛t✐♦♥✑

Pr✐♥❝✐♣❧❡ ❈♦♠❜✐♥✐♥❣ ♥ ❝❤❛r❛❝t❡r✐st✐❝s✿ ❈❤♦♦s❡ ♠✵,...,✵ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ❋♦r ✈ ∈ {✵, ✶}♥✱ ✉s❡ ♠✈ = ♠✵ ⊕ ♥

✐=✶ ✈✐δ✐✳

◆✉♠❜❡r ♦❢ tr✐❛❧s = ♥✉♠❜❡r ♦❢ ❡❞❣❡s ♦❢ ❛ ♥✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡✳ ◆✉♠❜❡r ♦❢ ❝❤♦s❡♥ ♣❧❛✐♥t❡①ts = ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ♦❢ ❛ ♥✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡✳

♠✵✵✵ ♠✵✶✵ ♠✶✵✵ ♠✶✶✵ ♠✵✵✶ ♠✵✶✶ ♠✶✵✶ ♠✶✶✶ δ✷ δ✶ δ✸

❜✉t✳✳✳ ♥♦t ✐♥❞❡♣❡♥❞❛♥t ❡✈❡♥ts

✸✵✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-40
SLIDE 40

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❍♦✇ t♦ ❝♦♠❜✐♥❡ ❝❤❛r❛❝t❡r✐st✐❝s ❄

❙❡❝♦♥❞ ❛tt❡♠♣t

Pr✐♥❝✐♣❧❡ ❈♦♠❜✐♥✐♥❣ ♥ ❝❤❛r❛❝t❡r✐st✐❝s✿ ❈❤♦♦s❡ ♠✵ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ❋♦r ✈ ∈ {✶, . . . , ♥}✱ ✉s❡ ♠✐ = ♠✵ ⊕ δ✐✳ ◆✉♠❜❡r ♦❢ tr✐❛❧s = ♥✳ ◆✉♠❜❡r ♦❢ ❝❤♦s❡♥ ♣❧❛✐♥t❡①ts = ♥ − ✶✳

♠✵ ♠✶ ♠✷ ♠✸ ♠✹ ♠✺ ♠✻ δ✶ δ✷ δ✸ ✸✶✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-41
SLIDE 41

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❈r②♣t❛♥❛❧②s✐s P❧❛✐♥t❡①ts ❚✐♠❡ ❉✐✛❡r❡♥t✐❛❧ ✭❇✐❤❛♠✱ ❙❤❛♠✐r✮ ✷✹✵ ✵✱✸ s❡❝♦♥❞s ❉✐✛❡r❡♥t✐❛❧ ✭❑♥✉❞s❡♥✮ ✹✻ ❛ ❢❡✇ s❡❝♦♥❞s ❆❧❣❡❜r❛✐❝ ✇✐t❤ ❙❆❚ ❙♦❧✈❡r ✭❈♦✉rt♦✐s✱ ❇❛r❞✮ ✶ ✷✷✺ s❡❝♦♥❞s ❆❧❣❡❜r❛✐❝✲❞✐✛❡r❡♥t✐❛❧ ✸✷ ✸✵✵✵ s❡❝♦♥❞s ❆❧❣❡❜r❛✐❝✲❞✐✛❡r❡♥t✐❛❧ ✷✷ <✶✵ ❤♦✉rs ✭❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s✮

✸✷✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-42
SLIDE 42

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❙❡✈❡♥ r♦✉♥❞s

❋♦r s❡✈❡♥ r♦✉♥❞s ❛♥❞ ♠♦r❡✱ t❤❡ ❝❧❛ss✐❝❛❧ ❞✐✛❡r❡♥t✐❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❞♦♥✬t s❡❡♠ t♦ ❜❡ ❛❞❛♣t❡❞✳ ❲❡ ❤❛✈❡ ✉s❡❞ ❛ ❞❡❞✐❝❛t❡❞ ❞✐✛❡r❡♥t✐❛❧ ❝❤❛r❛❝t❡r✐st✐❝✳ ❚r✉♥❝❛t❡❞ ❝❤❛r❛❝t❡r✐st✐❝ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶/✶✵✵✵✳

✸✸✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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SLIDE 43

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❙✲❇♦① δ✐♥ δ♦✉t Pr♦❜❛ ❙✶ ✹① ✻① ✶✵✴✻✹ ✽① ✸① ✶✷✴✻✹ ❙✷ ✹① ✶✵① ✶✵✴✻✹ ✹① ✶✷① ✶✵✴✻✹ ✽① ✾① ✶✵✴✻✹ ✽① ✶✵① ✶✻✴✻✹ ✶✷① ✺① ✶✹✴✻✹ ❙✸ ✹① ✾① ✶✷✴✻✹ ✽① ✸① ✶✵✴✻✹ ✶✷① ✺① ✶✷✴✻✹ ✶✷① ✻① ✶✷✴✻✹ ❙✹ ✹① ✻① ✶✷✴✻✹ ✹① ✾① ✶✷✴✻✹ ❇♦ît❡✲❙ δ✐♥ δ♦✉t Pr♦❜❛ ❙✺ ✹① ✻① ✶✵✴✻✹ ✽① ✻① ✶✵✴✻✹ ✽① ✶✵① ✶✵✴✻✹ ✶✷① ✸① ✶✵✴✻✹ ✶✷① ✻① ✶✵✴✻✹ ✶✷① ✶✵① ✶✵✴✻✹ ❙✻ ✽① ✻① ✶✻✴✻✹ ✶✷① ✸① ✶✷✴✻✹ ✶✷① ✺① ✶✵✴✻✹ ❙✼ ✽① ✶✵① ✶✷✴✻✹ ✶✷① ✶✷① ✶✹✴✻✹ ❙✽ ✹① ✶✷① ✶✵✴✻✹ ✶✷① ✺① ✶✵✴✻✹ ✶✷① ✻① ✶✵✴✻✹

✸✹✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-44
SLIDE 44

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s ∆✵▲ ❂ ✵✵ ✷✵ ✷✵ ✵✵ ∆✵❘ ❂ ✵✵ ✵✵ ✵✻ ✵✵ ✰ ❋❦✶ ∆✶▲ ❂ ✵✵ ✵✵ ✵✻ ✵✵ ∆✶❘ ❂ ✵✵ ✵✵ ✵✵ ✵✵ ✵✵ ✵✵ ✵✻ ✵✵ ✵✵ ✷✵ ✷✵ ✵✵ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✷

✻✹

✰ ❋❦✷ ∆✷▲ ❂ ✵✵ ✵✵ ✵✵ ✵✵ ∆✷❘ ❂ ✵✵ ✵✵ ✵✻ ✵✵ ✵✵ ✵✵ ✵✵ ✵✵ ✵✵ ✵✵ ✵✵ ✵✵ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ ✰ ❋❦✸ ∆✸▲ ❂ ✵✵ ✵✵ ✵✻ ✵✵ ∆✸❘ ❂ ✵✵ ✷✵ ✷✵ ✵✵ ✵✵ ✵✵ ✵✻ ✵✵ ✵✵ ✷✵ ✷✵ ✵✵ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✷

✻✹

✰ ❋❦✹ ∆✹▲ ❂ ✵✵ ✷✵ ✷✵ ✵✵ ∆✹❘ ❂ ✵✹ ✵✹ ✵✼ ✽✵ ✵✵ ✷✵ ✷✵ ✵✵ ✵✹ ✵✹ ✵✶ ✽✵ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✵✵/✻✹✷ ✰ ❋❦✺ ∆✺▲ ❂ ✵✹ ✵✹ ✵✼ ✽✵ ∆✺❘ ❂ ✪✪✵✪✵✵✪✵✵✪✪✪✪✵✵✵✵✪✪✪✵✪✵✵✵✪✵✪✪✵✵✪ ✵✹ ✵✹ ✵✼ ✽✵ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ ✸✺✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-45
SLIDE 45

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

✼ r♦✉♥❞s ❝r②♣t❛♥❛❧②s✐s ✷✵✵✵ ❝❤♦s❡♥ ♣❧❛✐♥t❡①ts ✸ ❤♦✉rs ◆♦t s♦ ♠✉❝❤ r❡s✉❧ts ♦♥ ✼ r♦✉♥❞s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳

✸✻✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-46
SLIDE 46

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❊✐❣❤t r♦✉♥❞s

❲❡ ❤❛✈❡ ❢♦✉♥❞ ❛ ✺✲r♦✉♥❞ tr✉♥❝❛t❡❞ ❞✐✛❡r❡♥t✐❛❧ ❝❤❛r❛❝t❡r✐st✐❝ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶/✺✽✵✵✳ ✰ ♣❛rt✐❛❧ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ♦✈❡r ✽ ❜✐ts ♦❢ t❤❡ ❦❡②✳ ✽ r♦✉♥❞s ❝r②♣t❛♥❛❧②s✐s ✶✶✻✵✵ ❝❤♦s❡♥ ♣❧❛✐♥t❡①ts ❛♥❞ ✷✷✺ s❡❝♦♥❞s✳

✸✼✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-47
SLIDE 47

■♥tr♦❞✉❝t✐♦♥ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✉s✐♥❣ ▼✐♥✐s❛t ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ❆❧❣❡❜r❛✐❝✕❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❘❡s✉❧ts ♦♥ s✐①✱ s❡✈❡♥ ❛♥❞ ❡✐❣❤t r♦✉♥❞s

❙✉♠♠❛r②

❘♦✉♥❞s ❈r②♣t❛♥❛❧②s✐s ◆❜ ♦❢ ♣❧❛✐♥t❡①ts ❚✐♠❡ ✻ ❞✐✛ ✭❇✐❤❛♠✱❙❤❛♠✐r✮ ✷✹✵ ✭❝❤♦s❡♥✮ ✵✱✸ s ❞✐✛ ✭❑♥✉❞s❡♥✮ ✹✻ ✭❝❤♦s❡♥✮ <✶✵ s ❛❧❣ ✭❈♦✉rt♦✐s✱❇❛r❞✮ ✶ ✭❦♥♦✇♥✮ ✷✷✺ s ❞✐✛ ✰ ❛❧❣ ✸✷ ✭❝❤♦s❡♥✮ ✸✵✵✵ s ❞✐✛ ✰ ❛❧❣ ✷✷ ✭❝❤♦s❡♥✮ <✶✵ ❤ ✼ ❞✐✛ ✰ ❛❧❣ ✷✵✵✵ ✭❝❤♦s❡♥✮ ✶✵✵✵✵ s ✽ ❞✐✛ ✭❇✐❤❛♠✱❙❤❛♠✐r✮ ✺✵✵✵✵ ✭❝❤♦s❡♥✮ ✶✵✵ s ❧✐♥ ✭▼❛ts✉✐✮ ✷✷✵ ✭❦♥♦✇♥✮ ✹✵ s ❞✐✛ ✰ ❛❧❣ ✶✶✺✵✵ ✭❝❤♦s❡♥✮ ✷✷✺ s ❞✐✛✰❧✐♥ ✭❍❡❧❧♠❛♥✱▲❛♥❣❢♦r❞✮ ✺✶✷ ✭❝❤♦s❡♥✮ ❢❡✇ s❡❝♦♥❞s

✸✽✴✸✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

slide-48
SLIDE 48

❈♦♥❝❧✉s✐♦♥

❯s❡ ♦❢ st❛t✐st✐❝❛❧ ♠❡t❤♦❞s ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s✳ → ◆❡✇ ❛tt❛❝❦s ♦♥ ✻✱ ✼ ❛♥❞ ✽ r♦✉♥❞s ♦❢ ❉❊❙ ✉s✐♥❣ ❞❡❞✐❝❛t❡❞ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❚r❛❞❡♦✛ ♣❧❛✐♥t❡①ts✴t✐♠❡✳ ❘❡❧❛t❡❞ ✇♦r❦ ✭❈r②♣t❛♥❛❧②s✐s ♦❢ Pr❡s❡♥t✮ ▼✳ ❆❧❜r❡❝❤t ❛♥❞ ❈✳ ❈✐❞ ❆❧❣❡❜r❛✐❝ ❚❡❝❤♥✐q✉❡s ✐♥ ❉✐✛❡r❡♥t✐❛❧ ❈r②♣t❛♥❛❧②s✐s ❋❙❊✷✵✵✾ ❋✉t✉r❡ ✇♦r❦ ❊①t❡♥s✐♦♥ ♦❢ t❤✐s ❛tt❛❝❦ ❢♦r ♠♦r❡ r♦✉♥❞s ❄ ❆❧❣❡❜r❛✐❝✲❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✇✐t❤

  • rö❜♥❡r ❇❛s❡s ❄

❖t❤❡r ❝r②♣t♦s②st❡♠s ❄ ❖t❤❡r st❛t✐st✐❝❛❧ t♦♦❧s ✭❞✐✛❡r❡♥t✐❛❧✲❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s✱ ✳ ✳ ✳ ✮ ❄

slide-49
SLIDE 49

❈♦♥❝❧✉s✐♦♥

❯s❡ ♦❢ st❛t✐st✐❝❛❧ ♠❡t❤♦❞s ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s✳ → ◆❡✇ ❛tt❛❝❦s ♦♥ ✻✱ ✼ ❛♥❞ ✽ r♦✉♥❞s ♦❢ ❉❊❙ ✉s✐♥❣ ❞❡❞✐❝❛t❡❞ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❚r❛❞❡♦✛ ♣❧❛✐♥t❡①ts✴t✐♠❡✳ ❘❡❧❛t❡❞ ✇♦r❦ ✭❈r②♣t❛♥❛❧②s✐s ♦❢ Pr❡s❡♥t✮ ▼✳ ❆❧❜r❡❝❤t ❛♥❞ ❈✳ ❈✐❞ ❆❧❣❡❜r❛✐❝ ❚❡❝❤♥✐q✉❡s ✐♥ ❉✐✛❡r❡♥t✐❛❧ ❈r②♣t❛♥❛❧②s✐s ❋❙❊✷✵✵✾ ❋✉t✉r❡ ✇♦r❦ ❊①t❡♥s✐♦♥ ♦❢ t❤✐s ❛tt❛❝❦ ❢♦r ♠♦r❡ r♦✉♥❞s ❄ ❆❧❣❡❜r❛✐❝✲❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✇✐t❤

  • rö❜♥❡r ❇❛s❡s ❄

❖t❤❡r ❝r②♣t♦s②st❡♠s ❄ ❖t❤❡r st❛t✐st✐❝❛❧ t♦♦❧s ✭❞✐✛❡r❡♥t✐❛❧✲❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s✱ ✳ ✳ ✳ ✮ ❄

slide-50
SLIDE 50

❈♦♥❝❧✉s✐♦♥

❯s❡ ♦❢ st❛t✐st✐❝❛❧ ♠❡t❤♦❞s ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s✳ → ◆❡✇ ❛tt❛❝❦s ♦♥ ✻✱ ✼ ❛♥❞ ✽ r♦✉♥❞s ♦❢ ❉❊❙ ✉s✐♥❣ ❞❡❞✐❝❛t❡❞ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❚r❛❞❡♦✛ ♣❧❛✐♥t❡①ts✴t✐♠❡✳ ❘❡❧❛t❡❞ ✇♦r❦ ✭❈r②♣t❛♥❛❧②s✐s ♦❢ Pr❡s❡♥t✮ ▼✳ ❆❧❜r❡❝❤t ❛♥❞ ❈✳ ❈✐❞ ❆❧❣❡❜r❛✐❝ ❚❡❝❤♥✐q✉❡s ✐♥ ❉✐✛❡r❡♥t✐❛❧ ❈r②♣t❛♥❛❧②s✐s ❋❙❊✷✵✵✾ ❋✉t✉r❡ ✇♦r❦ ❊①t❡♥s✐♦♥ ♦❢ t❤✐s ❛tt❛❝❦ ❢♦r ♠♦r❡ r♦✉♥❞s ❄ ❆❧❣❡❜r❛✐❝✲❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❉❊❙ ✇✐t❤

  • rö❜♥❡r ❇❛s❡s ❄

❖t❤❡r ❝r②♣t♦s②st❡♠s ❄ ❖t❤❡r st❛t✐st✐❝❛❧ t♦♦❧s ✭❞✐✛❡r❡♥t✐❛❧✲❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s✱ ✳ ✳ ✳ ✮ ❄