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r rts rtrr r - - PowerPoint PPT Presentation

Jan 08, 2023 28 likes •454 views

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slide-1
SLIDE 1

❈♦♣s ❛♥❞ ❘♦❜❜❡r ❣❛♠❡s ♦♥ ● Ξ

▼❛r❝✐♥ ❲✐t❦♦✇s❦✐ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② ✫ ❆❞❛♠ ▼✐❝❦✐❡✇✐❝③ ❯♥✐✈❡rs✐t②

✯❏♦✐♥t ✇♦r❦ ✇✐t❤ ❉✳❈❛r❞♦s♦✱ ❈✳❉♦♠✐♥✐❝ ❛♥❞ ❾✳❲✐t❦♦✇s❦✐

❖tt❛✇❛ ✷✷✳✵✺✳✷✵✶✺

✶ ✴ ✸✺

slide-2
SLIDE 2
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❚❤❡ ❈♦♣s ❛♥❞ ❘♦❜❜❡r ❣❛♠❡✿

  • ❚✇♦ ♣❛rt✐❡s✿ ◆ ❈♦♣s ❛♥❞ ❛ ♦♥❡ r♦❜❜❡r

❇♦❛r❞✿ ❱❡rt✐❝❡s ♦❢ ❛♥ ❛r❜✐tr❛r② ❣r❛♣❤ ● ▼♦✈❡s✿ ❊✈❡r② r♦✉♥❞✱ ♠♦✈❡ ❛❧♦♥❣ ❡❞❣❡ ♦r st❛② ✐♥ ♣❧❛❝❡

  • ♦❛❧✿ ❈❛t❝❤ t❤❡ ❘♦❜❜❡r ♦r ❜❡ ❛❜❧❡ t♦ s❧✐♣ ❛✇❛② ❢♦r❡✈❡r

✷ ✴ ✸✺

slide-3
SLIDE 3
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❚❤❡ ❈♦♣s ❛♥❞ ❘♦❜❜❡r ❣❛♠❡✿

  • ❚✇♦ ♣❛rt✐❡s✿ ◆ ❈♦♣s ❛♥❞ ❛ ♦♥❡ r♦❜❜❡r
  • ❇♦❛r❞✿ ❱❡rt✐❝❡s ♦❢ ❛♥ ❛r❜✐tr❛r② ❣r❛♣❤ ●

▼♦✈❡s✿ ❊✈❡r② r♦✉♥❞✱ ♠♦✈❡ ❛❧♦♥❣ ❡❞❣❡ ♦r st❛② ✐♥ ♣❧❛❝❡

  • ♦❛❧✿ ❈❛t❝❤ t❤❡ ❘♦❜❜❡r ♦r ❜❡ ❛❜❧❡ t♦ s❧✐♣ ❛✇❛② ❢♦r❡✈❡r

✷ ✴ ✸✺

slide-4
SLIDE 4
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❚❤❡ ❈♦♣s ❛♥❞ ❘♦❜❜❡r ❣❛♠❡✿

  • ❚✇♦ ♣❛rt✐❡s✿ ◆ ❈♦♣s ❛♥❞ ❛ ♦♥❡ r♦❜❜❡r
  • ❇♦❛r❞✿ ❱❡rt✐❝❡s ♦❢ ❛♥ ❛r❜✐tr❛r② ❣r❛♣❤ ●
  • ▼♦✈❡s✿ ❊✈❡r② r♦✉♥❞✱ ♠♦✈❡ ❛❧♦♥❣ ❡❞❣❡ ♦r st❛② ✐♥ ♣❧❛❝❡
  • ♦❛❧✿ ❈❛t❝❤ t❤❡ ❘♦❜❜❡r ♦r ❜❡ ❛❜❧❡ t♦ s❧✐♣ ❛✇❛② ❢♦r❡✈❡r

✷ ✴ ✸✺

slide-5
SLIDE 5
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❚❤❡ ❈♦♣s ❛♥❞ ❘♦❜❜❡r ❣❛♠❡✿

  • ❚✇♦ ♣❛rt✐❡s✿ ◆ ❈♦♣s ❛♥❞ ❛ ♦♥❡ r♦❜❜❡r
  • ❇♦❛r❞✿ ❱❡rt✐❝❡s ♦❢ ❛♥ ❛r❜✐tr❛r② ❣r❛♣❤ ●
  • ▼♦✈❡s✿ ❊✈❡r② r♦✉♥❞✱ ♠♦✈❡ ❛❧♦♥❣ ❡❞❣❡ ♦r st❛② ✐♥ ♣❧❛❝❡
  • ●♦❛❧✿ ❈❛t❝❤ t❤❡ ❘♦❜❜❡r ♦r ❜❡ ❛❜❧❡ t♦ s❧✐♣ ❛✇❛② ❢♦r❡✈❡r

✷ ✴ ✸✺

slide-6
SLIDE 6
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❊①❛♠♣❧❡✿

  • ❖

❖ ❖ ❖ ❖

✸ ✴ ✸✺

slide-7
SLIDE 7
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❋✐rst r♦✉♥❞✿

❈ ❖ ❖ ❖ ❖

✹ ✴ ✸✺

slide-8
SLIDE 8
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❋✐rst r♦✉♥❞✿

❈ ❖ ❖ ❖ ❘

✺ ✴ ✸✺

slide-9
SLIDE 9
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❋✐rst r♦✉♥❞✿

❖ ❖ ❈ ❖ ❘

✻ ✴ ✸✺

slide-10
SLIDE 10
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

◆❡①t r♦✉♥❞s✿

❖ ❖ ❈ ❖ ❘

✼ ✴ ✸✺

slide-11
SLIDE 11
  • ❛♠❡ ♦❢ ❈♦♣s ❛♥❞ ❘♦❜❜❡r

❊♥❞ ♦❢ t❤❡ ❣❛♠❡ ✲ r♦❜❜❡r ❤❛s ❜❡❡♥ ❝❛♣t✉r❡❞

❖ ❖ ❖ ❖ ❈✲❘

✽ ✴ ✸✺

slide-12
SLIDE 12

❈♦♣ ♥✉♠❜❡r

❉❡✜♥✐t✐♦♥

❚❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦♣s r❡q✉✐r❡❞ t♦ ❝❛t❝❤ t❤❡ r♦❜❜❡r ✭r❡❣❛r❞❧❡ss ♦❢ r♦❜❜❡r✬s str❛t❡❣②✮ ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♣ ♥✉♠❜❡r ♦❢ ●✱ ❛♥❞ ✐s ❞❡♥♦t❡❞ ❜② ❝(●)✳ ❊①❛♠♣❧❡s✿

  • ❝(P♥) = ❝(❲♥) = ❝(❑♥) = ✶
  • ❝(❈♥) = ✷✱ ❢♦r ♥ ≥ ✹
  • ❝(❚) = ✶ ✐❢ ❚ ✐s ❛ tr❡❡✳

✾ ✴ ✸✺

slide-13
SLIDE 13

❍✐st♦r②

❚❤❡♦r❡♠ ✭❘✳❏✳ ◆♦✇❛❦♦✇s❦✐✱ P✳❲✐♥❦❧❡r ✬✽✸✮

  • r❛♣❤ ● ✐s ❝♦♣ ✇✐♥ ✭❡✳❣✳ ✐ts ❝♦♣ ♥✉♠❜❡r ❡q✉❛❧ ✶✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s

❞✐s♠❛♥t❧❛❜❧❡✳

❚❤❡♦r❡♠ ✭▼✳ ❆✐❣♥❡r✱ ▼✳ ❋r♦♠♠❡ ✬✽✹✮

■❢ ● ✐s ♣❧❛♥❛r✱ t❤❡♥ ❝(●) ≤ ✸✳

❚❤❡♦r❡♠ ✭◆✳❊✳❈❧❛r❦❡ ✬✵✷✮

❝(●) ≤ ✷ ❢♦r ❛♥② ♦✉t❡r♣❧❛♥❛r ❣r❛♣❤ ●✳

✶✵ ✴ ✸✺

slide-14
SLIDE 14

❈♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤s✳

❉❡✜♥✐t✐♦♥

❆ ❣r❛♣❤ ● ✐s ❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ✐❢ ❢♦r ❛♥② ❡❞❣❡ ❡ ✐♥ ● ❡✐t❤❡r ❝(● − ❡) < ❝(●) ♦r ❝(● − ❡) > ❝(●) ✷✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤s✿ ❈❧❛r❦❡✱ ❋✐t③♣❛tr✐❝❦✱ ❍✐❧❧ ❛♥❞ ◆♦✇❛❦♦✇s❦✐ ✭✷✵✶✵✮ ❡①❛♠✐♥❡❞ t❤❡ ❝❧❛ss❡s ♦❢ ❣r❛♣❤s ❢♦r ✇❤✐❝❤ ❝(●) = ✷ ❛♥❞ ❝(● + ❡) = ✶✳ ❋✐t③♣❛tr✐❝❦ ✭✷✵✶✹✮ ❝❤❛r❛❝t❡r✐③❡❞ ❣r❛♣❤s ❢♦r ✇❤✐❝❤ ❝(●) = ✷ ❛♥❞ ❝(● − ❡) = ✶ ✭t❤❡r❡ ❛r❡ ❝②❝❧❡s ❛♥❞ ❈♥ ∨ ❑✷✮✳

✶✶ ✴ ✸✺

slide-15
SLIDE 15

▼♦t✐✈❛t✐♦♥ ✲ st✉❞② ♦❢ ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤s

❚❤❡♦r❡♠ ✭❇❛✐r❞✱ ❇❡✈❡r✐❞❣❡✱ ❈♦❞❡♥♦tt✐✱ ▼❛✉r❡r✱ ▼❝❈❛✉❧❡②✱ ❱❛❧❡✈❛ ✭✷✵✶✹✮✮

❚❤❡ P❡t❡rs❡♥ ❣r❛♣❤ ✐s t❤❡ s♠❛❧❧❡st ❣r❛♣❤ ✇✐t❤ ❝(●) = ✸ ✭t❤✉s ✐t ✐s ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧✮✳ ❯♣ t♦ ❞❛t❡ ♦♥❧② ❦♥♦✇♥ ❡①❛♠♣❧❡ ♦❢ ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤✳

▲❡♠♠❛ ✭❈❛r❞♦s♦✱ ❉♦♠✐♥✐❝✱ ❲✐t❦♦✇s❦✐✱ ❲✳ ✷✵✶✺✰✮

❉♦❞❡❝❛❤❡❞r❛❧ ❛♥❞ ❍❡❛✇♦♦❞ ❣r❛♣❤s ❛r❡ ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧✳

✶✷ ✴ ✸✺

slide-16
SLIDE 16

❉♦❞❡❝❛❤❡❞r❛❧ ❣r❛♣❤

✸ ✷

✷✵

✶ ✽ ✼

✶✾ ✶✻ ✶✼ ✶✽ ✶✷

✺

✶✶ ✶✸ ✶✺ ✶✹ ✶✵

✾ ✹ ✻ ❋✐❣✉r❛✿ ❉♦❞❡❝❛❤❡❞r❛❧ ❣r❛♣❤

✶✸ ✴ ✸✺

slide-17
SLIDE 17

❍❡❛✇♦♦❞ ❣r❛♣❤

✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾

✶✵ ✶✶ ✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤

✶✹ ✴ ✸✺

slide-18
SLIDE 18

❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ str❛t❡❣②✱ ✷ ❝♦♣s ✐s ❡♥♦✉❣❤

✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾

✶✵ ✶✶ ✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤

✶✺ ✴ ✸✺

slide-19
SLIDE 19

❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ str❛t❡❣②✱ ✷ ❝♦♣s ✐s ❡♥♦✉❣❤

✶ ✷ ✸ ✹ ✺ ✻ ❈ ✽ ✾ ❈

✶✶ ✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ P❧❛❝✐♥❣ ❝♦♣s

✶✻ ✴ ✸✺

slide-20
SLIDE 20

❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ str❛t❡❣②✱ ✷ ❝♦♣s ✐s ❡♥♦✉❣❤

✶ ✷ ✸ ✹ ❘ ✻ ❈ ✽ ✾ ❈

✶✶ ✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ ❘♦❜❜❡r ❝❤♦♦s❡ ✺

✶✼ ✴ ✸✺

slide-21
SLIDE 21

❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ str❛t❡❣②✱ ✷ ❝♦♣s ✐s ❡♥♦✉❣❤

❈ ✷ ✸ ✹ ❘ ❈ ✼ ✽ ✾ ✼

✶✶ ✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ ❈♦♣s ♠♦✈❡

✶✽ ✴ ✸✺

slide-22
SLIDE 22

❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ str❛t❡❣②✱ ✷ ❝♦♣s ✐s ❡♥♦✉❣❤

❈ ✷ ✸ ❘ ✺ ❈ ✼ ✽ ✾ ✼

✶✶ ✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ ❘♦❜❜❡r r✉♥ ❛✇❛②

✶✾ ✴ ✸✺

slide-23
SLIDE 23

❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ str❛t❡❣②✱ ✷ ❝♦♣s ✐s ❡♥♦✉❣❤

✶ ✷ ❘ ✹ ❈ ✻ ✼ ✽ ✾ ❈

✶✶ ✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ ♥❡①t r♦✉♥❞

✷✵ ✴ ✸✺

slide-24
SLIDE 24

❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ str❛t❡❣②✱ ✷ ❝♦♣s ✐s ❡♥♦✉❣❤

✶ ❘ ✸ ❈ ✺ ✻ ✼ ✽ ✾ ✼ ❈

✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ ♥❡①t r♦✉♥❞

✷✶ ✴ ✸✺

slide-25
SLIDE 25

❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ str❛t❡❣②✱ ✷ ❝♦♣s ✐s ❡♥♦✉❣❤

✶ ❘ ❈ ✹ ✺ ❈ ✼ ✽ ✾

✶✵ ✶✶ ✶✷ ✶✸ ✶✹

❋✐❣✉r❛✿ ❍❡❛✇♦♦❞ ❣r❛♣❤ ✲ ❝❛♣t✉r❡ st❛❣❡✱ r♦❜❜❡r ❝❛♥♥♦t ♠♦✈❡

✷✷ ✴ ✸✺

slide-26
SLIDE 26

▼♦t✐✈❛t✐♦♥ ✲ ● Ξ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

  • Ξ ✐s ❛ ❣r❛♣❤ ♦♥ ✈❡rt❡① s❡t ❱ (● ∪ ●) ✇✐t❤ ❡❞❣❡s ❛s ✐♥ ● ❛♥❞ ● ❛♥❞

❛❞❞✐t✐♦♥❛❧ ❡❞❣❡s ❜❡t✇❡❡♥ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡rt✐❝❡s ♦❢ ● ❛♥❞ ● ✭✇❡ ❝♦♥♥❡❝t ❡✈❡r② ✈❡rt❡① ✈ ✇✐t❤ ✐ts ❝♦♣② ✈′ ✐♥ t❤❡ ❝♦♠♣❧❡♠❡♥t ❣r❛♣❤✮✳ ❊①❛♠♣❧❡✿ ❈ Ξ

✺ ✐s ❛ P❡t❡rs❡♥ ❣r❛♣❤✳

▲❡♠♠❛ ✭❈❛r❞♦s♦✱ ❉♦♠✐♥✐❝✱ ❲✐t❦♦✇s❦✐✱ ❲✳ ✷✵✶✺✰✮

❈✻ ❛♥❞ ❈✼ ❛r❡ ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧✳

✷✸ ✴ ✸✺

slide-27
SLIDE 27

▼♦t✐✈❛t✐♦♥ ✲ ● Ξ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

  • Ξ ✐s ❛ ❣r❛♣❤ ♦♥ ✈❡rt❡① s❡t ❱ (● ∪ ●) ✇✐t❤ ❡❞❣❡s ❛s ✐♥ ● ❛♥❞ ● ❛♥❞

❛❞❞✐t✐♦♥❛❧ ❡❞❣❡s ❜❡t✇❡❡♥ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡rt✐❝❡s ♦❢ ● ❛♥❞ ● ✭✇❡ ❝♦♥♥❡❝t ❡✈❡r② ✈❡rt❡① ✈ ✇✐t❤ ✐ts ❝♦♣② ✈′ ✐♥ t❤❡ ❝♦♠♣❧❡♠❡♥t ❣r❛♣❤✮✳ ❊①❛♠♣❧❡✿ ❈ Ξ

✺ ✐s ❛ P❡t❡rs❡♥ ❣r❛♣❤✳

▲❡♠♠❛ ✭❈❛r❞♦s♦✱ ❉♦♠✐♥✐❝✱ ❲✐t❦♦✇s❦✐✱ ❲✳ ✷✵✶✺✰✮

❈ Ξ

✻ ❛♥❞ ❈ Ξ ✼ ❛r❡ ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧✳

✷✸ ✴ ✸✺

slide-28
SLIDE 28
  • ✱ ● Ξ ✲ ❛ ❝❧♦s❡r ❧♦♦❦

❇❛s✐❝ r❡s✉❧ts

▲❡♠♠❛ ✭❈❛r❞♦s♦✱ ❉♦♠✐♥✐❝✱ ❲✐t❦♦✇s❦✐✱ ❲✳ ✷✵✶✺✰✮

  • Ξ ✐s ❝♦♣ ✇✐♥ ✐✛ ● ✐s ❛ ❝♦♠♣❧❡t❡ ❣r❛♣❤ ❑♥✳

▲❡♠♠❛ ✭❈❛r❞♦s♦✱ ❉♦♠✐♥✐❝✱ ❲✐t❦♦✇s❦✐✱ ❲✳ ✷✵✶✺✰✮

❋♦r ♥ ≥ ✺✱ ❝(PΞ

♥ ) = ✷✱ ❝(❈ Ξ ♥ ) = ✸✳

▲❡♠♠❛ ✭❈❛r❞♦s♦✱ ❉♦♠✐♥✐❝✱ ❲✐t❦♦✇s❦✐✱ ❲✳ ✷✵✶✺✰✮

▲❡t ● ❜❡ ❛ s✐♠♣❧❡ ❣r❛♣❤ ✇✐t❤ ❝(●) ≥ ✷✱ t❤❡♥ ❝(● Ξ) ≤ ♠❛①

  • ❝(●), ❝(●)
  • + ✶ ✭❜♦✉♥❞ ✐s ❛tt❛✐♥❡❞ ✐❢ ● ✐s ❛ ❝②❝❧❡✮✳

✷✹ ✴ ✸✺

slide-29
SLIDE 29

Pr♦♦❢ s❦❡t❝❤

▲❡♠♠❛

▲❡t ● ❜❡ ❛ s✐♠♣❧❡ ❣r❛♣❤ ✇✐t❤ ❝(●) ≥ ✷✱ t❤❡♥ ❝(● Ξ) ≤ ♠❛①

  • ❝(●), ❝(●)
  • + ✶✳

❙tr❛t❡❣② ✿ ❲❡ r❡❝♦❣♥✐s❡ t✇♦ ✒♣❛rts✑ ♦❢ ● Ξ✱ ♥❛♠❡❧② ● ❛♥❞ ●✳ ❋✐rst ✇❡ ❢♦r❝❡ ❛ r♦❜❜❡r t♦ ❣♦ t♦ t❤❡ ♣❛rt ♦❢ ● Ξ ✇✐t❤ ❧♦✇❡r ❝♦♣ ♥✉♠❜❡r ✭❛ss✉♠❡ ✐t ✐s ●✮ ❛♥❞ t❤❡♥ ✉s✐♥❣ ♦♥❡ ❝♦♣ ✇❡ ♣r❡✈❡♥t ❤✐♠ ❢r♦♠ ❣♦✐♥❣ ❜❛❝❦ t♦ t❤❡ ♣❛rt ●✳ ♠❛①

  • ❝(●), ❝(●)
  • ❝♦♣s ❛r❡ t❤❡♥ s✉✣❝✐❡♥t t♦

❝❛t❝❤ t❤❡ r♦❜❜❡r ♦♥ ●✳

✷✺ ✴ ✸✺

slide-30
SLIDE 30

Pr♦♦❢ s❦❡t❝❤ P❧❛❝❡ ❛❧❧ ❜✉t ♦♥❡ ❝♦♣ ✐♥ ●✳ ❋♦❧❧♦✇ t❤❡ r♦❜❜❡r ♦♥ ● ✉♣ t♦ t❤❡ ♣♦✐♥t ✇❤❡♥ ❤❡ ♠♦✈❡s ♦♥t♦ ●

r ❝

  • r✬

❝✬

  • ❋✐❣✉r❛✿ ❈♦♣ ❝✵ ❣♦❡s ❢r♦♠ ❝ t♦ ❝′ ✇❤❡♥ r♦❜❜❡r ❛♣♣❡❛rs ✐♥ r✳

❙✇✐t❝❤ ❝♦♣s ❜❡t✇❡❡♥ ● ❛♥❞ ●✳

✷✻ ✴ ✸✺

slide-31
SLIDE 31

Pr♦♦❢ s❦❡t❝❤

▲❡♠♠❛

❊✐t❤❡r s✐♥❣❧❡ ❝♦♣ ❝❛♥ ♣r❡✈❡♥t r♦❜❜❡r ❢r♦♠ ❝♦♠✐♥❣ ❜❛❝❦ t♦ t❤❡ ♣❛rt ● ♦r ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ♦❢ ♣❛rt ● ❡q✉❛❧s ✷✳

r t ❣ ❝

  • r✬

t✬ ❣✬ ❝✬

  • ❋✐❣✉r❛✿ ❈♦♣ ❝✵ ❣♦❡s ❢r♦♠ ❝′ t♦ ❣ ′ ✇❤❡♥ t❤❡ r♦❜❜❡r ❣♦❡s ❢r♦♠ r t♦ t✳

✷✼ ✴ ✸✺

slide-32
SLIDE 32

Pr♦♦❢ s❦❡t❝❤ ■♥ t❤❡ ❝❛s❡ ✇❤❡♥ ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ♦❢ ● ❡q✉❛❧ ✷ ✇❡ ❤❛✈❡ t♦ s✇✐t❝❤ ✐♥ ❡✈❡r② r♦✉♥❞s ❝♦♣s ❜❡t✇❡❡♥ t❤❡ ♣❛rts ✐♥ ♦r❞❡r t♦ ❣❡t t✇♦ ♦❢ t❤❡♠ ♦♥t♦ t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ ●

❝✶ r ❝✷

  • ❝✸

❳ ❳ ❳

  • ✷✽ ✴ ✸✺
slide-33
SLIDE 33
  • ❡♥❡r❛❧ ❝❛s❡

▲❡♠♠❛

❋♦r ❣r❛♣❤s ● ❛♥❞ ● ✇✐t❤ ❝(●) = ❝(●)✱ ❛♥❞ ❝(●), ❝(●) ≥ ✷ ✇❡ ❤❛✈❡ ❝(● Ξ) ≤ ♠❛①{❝(●), ❝(●)}.

▲❡♠♠❛

▲❡t ● ❜❡ ❛ ❣r❛♣❤ ✇✐t❤ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts {●✶, ●✷, . . . , ●♥}✱ ♠❛①{❝(●✶), . . . , ❝(●♥)} = ❝(●)✱ ❛♥❞ ❝(●) ≥ ✷✳ ❚❤❡♥ ❝(● Ξ) ≤ ♠❛①{❝(●✶), ❝(●✷), ..., ❝(●♥), ❝(●)}.

✷✾ ✴ ✸✺

slide-34
SLIDE 34

P❧❛♥❛r ❣r❛♣❤s

❚❤❡♦r❡♠

■❢ ● ✐s ♣❧❛♥❛r✱ t❤❡♥ ❝(● Ξ) ≤ ✸✳ ❇② ♣r❡✈✐♦✉s ▲❡♠♠❛s✱ ✐❢ ❢♦r ♣❧❛♥❛r ❣r❛♣❤ ● ✇❡ ❤❛✈❡ ❝ ● ❝ ● ✇✐t❤ ♠❛① ❝ ● ❝ ● ✸✱ t❤❡♥ ❝ ● ✸✳ ■♥ ♣❛rt✐❝✉❧❛r✿ ■❢ ❞✐❛♠ ● ✸✱ t❤❡♥ ❝ ● ✷ ❛♥❞ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ ♣r♦✈❡♥ ▲❡♠♠❛s✳ ❇② ●♦❞❞❛r❞ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮ ❡✈❡r② ♣❧❛♥❛r ❣r❛♣❤ ♦❢ ❞✐❛♠❡t❡r ✷ ❤❛s ❝♦♣ ♥✉♠❜❡r ❛t ♠♦st ✷✳

✸✵ ✴ ✸✺

slide-35
SLIDE 35

P❧❛♥❛r ❣r❛♣❤s

❚❤❡♦r❡♠

■❢ ● ✐s ♣❧❛♥❛r✱ t❤❡♥ ❝(● Ξ) ≤ ✸✳ ❇② ♣r❡✈✐♦✉s ▲❡♠♠❛s✱ ✐❢ ❢♦r ♣❧❛♥❛r ❣r❛♣❤ ● ✇❡ ❤❛✈❡ ❝(●) = ❝(●) ✇✐t❤ ♠❛①{❝(●), ❝(●)} ≤ ✸✱ t❤❡♥ ❝(● Ξ) ≤ ✸✳ ■♥ ♣❛rt✐❝✉❧❛r✿ ■❢ ❞✐❛♠ ● ✸✱ t❤❡♥ ❝ ● ✷ ❛♥❞ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ ♣r♦✈❡♥ ▲❡♠♠❛s✳ ❇② ●♦❞❞❛r❞ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮ ❡✈❡r② ♣❧❛♥❛r ❣r❛♣❤ ♦❢ ❞✐❛♠❡t❡r ✷ ❤❛s ❝♦♣ ♥✉♠❜❡r ❛t ♠♦st ✷✳

✸✵ ✴ ✸✺

slide-36
SLIDE 36

P❧❛♥❛r ❣r❛♣❤s

❚❤❡♦r❡♠

■❢ ● ✐s ♣❧❛♥❛r✱ t❤❡♥ ❝(● Ξ) ≤ ✸✳ ❇② ♣r❡✈✐♦✉s ▲❡♠♠❛s✱ ✐❢ ❢♦r ♣❧❛♥❛r ❣r❛♣❤ ● ✇❡ ❤❛✈❡ ❝(●) = ❝(●) ✇✐t❤ ♠❛①{❝(●), ❝(●)} ≤ ✸✱ t❤❡♥ ❝(● Ξ) ≤ ✸✳ ■♥ ♣❛rt✐❝✉❧❛r✿ ■❢ ❞✐❛♠(●) ≥ ✸✱ t❤❡♥ ❝(●) ≤ ✷ ❛♥❞ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ ♣r♦✈❡♥ ▲❡♠♠❛s✳ ❇② ●♦❞❞❛r❞ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮ ❡✈❡r② ♣❧❛♥❛r ❣r❛♣❤ ♦❢ ❞✐❛♠❡t❡r ✷ ❤❛s ❝♦♣ ♥✉♠❜❡r ❛t ♠♦st ✷✳

✸✵ ✴ ✸✺

slide-37
SLIDE 37

P❧❛♥❛r ❣r❛♣❤s

❚❤❡♦r❡♠

■❢ ● ✐s ♣❧❛♥❛r✱ t❤❡♥ ❝(● Ξ) ≤ ✸✳ ❇② ♣r❡✈✐♦✉s ▲❡♠♠❛s✱ ✐❢ ❢♦r ♣❧❛♥❛r ❣r❛♣❤ ● ✇❡ ❤❛✈❡ ❝(●) = ❝(●) ✇✐t❤ ♠❛①{❝(●), ❝(●)} ≤ ✸✱ t❤❡♥ ❝(● Ξ) ≤ ✸✳ ■♥ ♣❛rt✐❝✉❧❛r✿ ■❢ ❞✐❛♠(●) ≥ ✸✱ t❤❡♥ ❝(●) ≤ ✷ ❛♥❞ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ ♣r♦✈❡♥ ▲❡♠♠❛s✳ ❇② ●♦❞❞❛r❞ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮ ❡✈❡r② ♣❧❛♥❛r ❣r❛♣❤ ♦❢ ❞✐❛♠❡t❡r ✷ ❤❛s ❝♦♣ ♥✉♠❜❡r ❛t ♠♦st ✷✳

✸✵ ✴ ✸✺

slide-38
SLIDE 38

P❧❛♥❛r ❣r❛♣❤s

❚❤❡♦r❡♠

■❢ ● ✐s ♣❧❛♥❛r✱ t❤❡♥ ❝(● Ξ) ≤ ✸✳ ❚♦ s❤♦✇ t❤❡ t❤❡♦r❡♠ ✐t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t ❝(●) ≤ ✸✳

▲❡♠♠❛

▲❡t ● ❜❡ ❛ ♣❧❛♥❛r ❣r❛♣❤ ❛♥❞ ● ✐ts ❝♦♠♣❧❡♠❡♥t✱ ✇✐t❤ ❝♦♠♣♦♥❡♥ts {●✶, ●✷, . . . , ●♥}✳ ❚❤❡♥ ♠❛①{❝(●✶), . . . , ❝(●♥)} ≤ ✸✳ ❇❛s❡❞ ♦♥ ❇❛✐r❞ ❡t ❛❧❧ r❡s✉❧t t❤❛t P❡t❡rs❡♥ ❣r❛♣❤ ✐s t❤❡ ♠✐♥✐♠❛❧ ❣r❛♣❤ ✇✐t❤ ❝♦♣ ♥✉♠❜❡r ❡q✉❛❧ t♦ ✸✳

✸✶ ✴ ✸✺

slide-39
SLIDE 39

❖♣❡♥ Pr♦❜❧❡♠s ❉❡s❝r✐❜❡ t❤❡ ❢❛♠✐❧② ♦❢ ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤s❄ ❋✐♥❞ ❛♥ ✐♥✜♥✐t❡ ❢❛♠✐❧② ♦❢ s✉❝❤ ❣r❛♣❤s ✭❛r❡ t❤❡② ❛♠♦♥❣ ❣r❛♣❤s ● Ξ ❄✮✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ ● ✇✐t❤ ❝ ● ✸ s✉❝❤ t❤❛t ❝ ● ❝ ● ❄ ❉♦❡s ● ♦❢ s✉❝❤ ❣r❛♣❤ ✐s ❛ ❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤❄

❈♦♥❥❡❝t✉r❡

❊✈❡r② ❣r❛♣❤ ●✱ ✇✐t❤ ❝ ● ✸ s✉❝❤ t❤❛t r❡♠♦✈✐♥❣ ❛♥② ❡❞❣❡ ❡ ❊ ● ✇♦✉❧❞ ❝r❡❛t❡ ❛ ✷✲tr❛♣ ✐s ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧✳

✸✷ ✴ ✸✺

slide-40
SLIDE 40

❖♣❡♥ Pr♦❜❧❡♠s ❉❡s❝r✐❜❡ t❤❡ ❢❛♠✐❧② ♦❢ ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤s❄ ❋✐♥❞ ❛♥ ✐♥✜♥✐t❡ ❢❛♠✐❧② ♦❢ s✉❝❤ ❣r❛♣❤s ✭❛r❡ t❤❡② ❛♠♦♥❣ ❣r❛♣❤s ● Ξ ❄✮✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ ● ✇✐t❤ ❝(●) ≥ ✸ s✉❝❤ t❤❛t ❝(●) = ❝(●)❄ ❉♦❡s ● Ξ ♦❢ s✉❝❤ ❣r❛♣❤ ✐s ❛ ❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤❄

❈♦♥❥❡❝t✉r❡

❊✈❡r② ❣r❛♣❤ ●✱ ✇✐t❤ ❝ ● ✸ s✉❝❤ t❤❛t r❡♠♦✈✐♥❣ ❛♥② ❡❞❣❡ ❡ ❊ ● ✇♦✉❧❞ ❝r❡❛t❡ ❛ ✷✲tr❛♣ ✐s ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧✳

✸✷ ✴ ✸✺

slide-41
SLIDE 41

❖♣❡♥ Pr♦❜❧❡♠s ❉❡s❝r✐❜❡ t❤❡ ❢❛♠✐❧② ♦❢ ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤s❄ ❋✐♥❞ ❛♥ ✐♥✜♥✐t❡ ❢❛♠✐❧② ♦❢ s✉❝❤ ❣r❛♣❤s ✭❛r❡ t❤❡② ❛♠♦♥❣ ❣r❛♣❤s ● Ξ ❄✮✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ ● ✇✐t❤ ❝(●) ≥ ✸ s✉❝❤ t❤❛t ❝(●) = ❝(●)❄ ❉♦❡s ● Ξ ♦❢ s✉❝❤ ❣r❛♣❤ ✐s ❛ ❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧ ❣r❛♣❤❄

❈♦♥❥❡❝t✉r❡

❊✈❡r② ❣r❛♣❤ ●✱ ✇✐t❤ ❝(●) = ✸ s✉❝❤ t❤❛t r❡♠♦✈✐♥❣ ❛♥② ❡❞❣❡ ❡ ∈ ❊(●) ✇♦✉❧❞ ❝r❡❛t❡ ❛ ✷✲tr❛♣ ✐s ✸✲❝♦♣✲❡❞❣❡ ❝r✐t✐❝❛❧✳

✸✷ ✴ ✸✺

slide-42
SLIDE 42

❚❤❛♥❦ ②♦✉✦

✸✸ ✴ ✸✺