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slide-1
SLIDE 1

❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✲ ■▼❉❊❆ ❙♦❢t✇❛r❡ ■♥st✐t✉t❡

❋❛✐❧✉r❡ ❚❛❜❧❡❞ ❈♦♥str❛✐♥t ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ❜② ■♥t❡r♣♦❧❛t✐♦♥

❆✉t❤♦rs✿ ●r❛❡♠❡ ●❛♥❣❡✱ ❏♦r❣❡ ❆✳ ◆❛✈❛s✱ P❡t❡r ❙❝❤❛❝❤t❡✱ ❍❛r❛❧❞ ❙ø♥❞❡r❣❛❛r❞✱ ❛♥❞ P❡t❡r ❏✳ ❙t✉❝❦❡② ✭❯♥✐✈❡rs✐t② ♦❢ ▼❡❧❜♦✉r♥❡ ✲ ■❈▲P✬✶✸✮ ❙♣❡❛❦❡r✿ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ▼❛❞r✐❞✱ ◆♦✈❡♠❜❡r ✷✵✶✹

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶ ✴ ✶✻

slide-2
SLIDE 2

❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✲ ■▼❉❊❆ ❙♦❢t✇❛r❡ ■♥st✐t✉t❡

❋❛✐❧✉r❡ ❚❛❜❧❡❞ ❈♦♥str❛✐♥t ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ❜② ■♥t❡r♣♦❧❛t✐♦♥

❆✉t❤♦rs✿ ●r❛❡♠❡ ●❛♥❣❡✱ ❏♦r❣❡ ❆✳ ◆❛✈❛s✱ P❡t❡r ❙❝❤❛❝❤t❡✱ ❍❛r❛❧❞ ❙ø♥❞❡r❣❛❛r❞✱ ❛♥❞ P❡t❡r ❏✳ ❙t✉❝❦❡② ✭❯♥✐✈❡rs✐t② ♦❢ ▼❡❧❜♦✉r♥❡ ✲ ■❈▲P✬✶✸✮ ❙♣❡❛❦❡r✿ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦

✷✵✵✷ ✲ ▼❙❝ ❆r❝❤✐t❡❝t✉r❡ ❲♦r❦✐♥❣ ❛s ❛r❝❤✐t❡❝t ✭✷✵✵✵✲✷✵✶✵✮ ✷✵✶✹ ✲ ❇❙❝ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❲♦r❦✐♥❣ ❛t t❤❡ ■▼❉❊❆ ❙♦❢t✇❛r❡ ■♥st✐t✉t❡ ✭✷✵✶✸✲t♦❞❛②✮

▼❛❞r✐❞✱ ◆♦✈❡♠❜❡r ✷✵✶✹

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✷ ✴ ✶✻

slide-3
SLIDE 3

■♥❞❡① ▼♦t✐✈❛t✐♦♥✳ ❋❛✐❧✉r❡ ❚❛❜❧❡❞ ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣ ❜② ■♥t❡r♣♦❧❛t✐♦♥✳ ❊①❛♠♣❧❡✳ P❡r❢♦r♠❛♥❝❡✳ ■♥t❡r♣♦❧❛♥t ✐♥ ❋❚❈▲P ❁❃ Pr♦❥❡❝t✐♦♥ ✐♥ ❚❈▲P✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✸ ✴ ✶✻

slide-4
SLIDE 4

▼♦t✐✈❛t✐♦♥

❈♦♠❜✐♥❡ ❜❡♥❡✜ts ♦❢ ❝♦♥str❛✐♥ts ❛♥❞ t❛❜❧✐♥❣ ✐♥ Pr♦❧♦❣✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✹ ✴ ✶✻

slide-5
SLIDE 5

▼♦t✐✈❛t✐♦♥

❈♦♠❜✐♥❡ ❜❡♥❡✜ts ♦❢ ❝♦♥str❛✐♥ts ❛♥❞ t❛❜❧✐♥❣ ✐♥ Pr♦❧♦❣✳

❲✐t❤ ❝♦♥str❛✐♥ts✳

◮ ❘❡❞✉❝❡ s❡❛r❝❤ tr❡❡✳ ❄− s ♦ ❧ ✉ t ✐ ♦ ♥ s ✭❳✮ ✱ ❳ ❃ ✵ ✳ ❄− ❳ ✳ ❃✳ ✵ ✱ s ♦ ❧ ✉ t ✐ ♦ ♥ s ✭❳✮ ✳ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✹ ✴ ✶✻

slide-6
SLIDE 6

▼♦t✐✈❛t✐♦♥

❈♦♠❜✐♥❡ ❜❡♥❡✜ts ♦❢ ❝♦♥str❛✐♥ts ❛♥❞ t❛❜❧✐♥❣ ✐♥ Pr♦❧♦❣✳

❲✐t❤ t❛❜❧✐♥❣✳

◮ ❆✈♦✐❞ r❡❝♦♠♣✉t❛t✐♦♥s✳ ◮ ❊♥s✉r❡ t❡r♠✐♥❛t✐♦♥ ❢♦r ✏❜♦✉♥❞❡❞ t❡r♠ s✐③❡✑ ♣r♦❣r❛♠s✳ ❡❞❣❡ ✭ ✶ ✱ ✷ ✮ ✳ ❡❞❣❡ ✭ ✷ ✱ ✶ ✮ ✳ r ❡ ❛ ❝ ❤ ✭❳✱❨✮ ✿− ❡❞❣❡ ✭❳✱ ❩✮ ✱ r ❡ ❛ ❝ ❤ ✭❩ ✱❨✮ ✳ r ❡ ❛ ❝ ❤ ✭❳✱❨✮ ✿− ❡❞❣❡ ✭❳✱❨✮ ✳ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✹ ✴ ✶✻

slide-7
SLIDE 7

▼♦t✐✈❛t✐♦♥

❈♦♠❜✐♥❡ ❜❡♥❡✜ts ♦❢ ❝♦♥str❛✐♥ts ❛♥❞ t❛❜❧✐♥❣ ✐♥ Pr♦❧♦❣✳

❲✐t❤ t❛❜❧✐♥❣✳

◮ ❆✈♦✐❞ r❡❝♦♠♣✉t❛t✐♦♥s✳ ◮ ❊♥s✉r❡ t❡r♠✐♥❛t✐♦♥ ❢♦r ✏❜♦✉♥❞❡❞ t❡r♠ s✐③❡✑ ♣r♦❣r❛♠s✳

...

Z=2 Z=1

  • 2. edge(1,Z), reach(Z,Y).
  • 4. edge(2,Z), reach(Z,Y).
  • 5. reach(1,Y).
  • 3. reach(2,Y).
  • 1. reach(1,Y).

❡❞❣❡ ✭ ✶ ✱ ✷ ✮ ✳ ❡❞❣❡ ✭ ✷ ✱ ✶ ✮ ✳ r ❡ ❛ ❝ ❤ ✭❳✱❨✮ ✿− ❡❞❣❡ ✭❳✱ ❩✮ ✱ r ❡ ❛ ❝ ❤ ✭❩ ✱❨✮ ✳ r ❡ ❛ ❝ ❤ ✭❳✱❨✮ ✿− ❡❞❣❡ ✭❳✱❨✮ ✳ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✹ ✴ ✶✻

slide-8
SLIDE 8

▼♦t✐✈❛t✐♦♥

❈♦♠❜✐♥❡ ❜❡♥❡✜ts ♦❢ ❝♦♥str❛✐♥ts ❛♥❞ t❛❜❧✐♥❣ ✐♥ Pr♦❧♦❣✳

❲✐t❤ t❛❜❧✐♥❣✳

◮ ❆✈♦✐❞ r❡❝♦♠♣✉t❛t✐♦♥s✳ ◮ ❊♥s✉r❡ t❡r♠✐♥❛t✐♦♥ ❢♦r ✏❜♦✉♥❞❡❞ t❡r♠ s✐③❡✑ ♣r♦❣r❛♠s✳

.

  • 3. edge(1,Z), reach(Z,Y).
  • 13. edge(1,Y).
  • 6. edge(2,Z), reach(Z,Y).
  • 8. edge(2,1).
  • 14. edge(1,2).
  • 11. reach(1,1).
  • 16. reach(1,2).

Z=2 Z=1 Y=1 Y=2 Y=2

  • 1. reach(1,Y).
  • 4. reach(2,Y).
  • 7. reach(1,Y).
  • 12. fail

❡❞❣❡ ✭ ✶ ✱ ✷ ✮ ✳ ❡❞❣❡ ✭ ✷ ✱ ✶ ✮ ✳ r ❡ ❛ ❝ ❤ ✭❳✱❨✮ ✿− ❡❞❣❡ ✭❳✱ ❩✮ ✱ r ❡ ❛ ❝ ❤ ✭❩ ✱❨✮ ✳ r ❡ ❛ ❝ ❤ ✭❳✱❨✮ ✿− ❡❞❣❡ ✭❳✱❨✮ ✳ ❙✉❜❣♦❛❧ ❆♥s✇❡rs ✶✵✳ ❨ ❂ ✶ ✷✳ r❡❛❝❤✭✶✱❨✮ ✶✺✳ ❨ ❂ ✷ ✶✽✳ ❈♦♠♣❧❡t❡ ✾✳ ❨ ❂ ✶ ✺✳ r❡❛❝❤✭✷✱❨✮ ✶✼✳ ❨ ❂ ✷ ✶✽✳ ❈♦♠♣❧❡t❡ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✹ ✴ ✶✻

slide-9
SLIDE 9

▼♦t✐✈❛t✐♦♥

❈♦♠❜✐♥❡ ❜❡♥❡✜ts ♦❢ ❝♦♥str❛✐♥ts ❛♥❞ t❛❜❧✐♥❣ ✐♥ Pr♦❧♦❣✳

Pr❡✈✐♦✉s ✐♠♣❧❡♠❡♥t❛t✐♦♥s✳

❈♦♥str❛✐♥t ▼❡♠♦✐♥❣✱ ❛♣♣❧✐❝❛❜❧❡ t♦ ❝♦♥str❛✐♥t ❉❛t❛❧♦❣✳

❚❤❡ t❛❜❧✐♥❣ ❢❡❛t✉r❡s ♦❢ ❳❙❇ s✉♣♣♦rts ❝♦♥str❛✐♥ts✳

❚❛❜❧❡❞ ❈♦♥str❛✐♥t ❍❛♥❞❧✐♥❣ ❘✉❧❡s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❳❙❇✳

❚❛❜❧❡❞ ❈♦♥str❛✐♥t ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ✭❚❈▲P✮ ❢♦r ❞✐s❡q✉❛❧✐t② ❛♥❞ ❞✐✛❡r❡♥❝❡ ❝♦♥str❛✐♥ts✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✺ ✴ ✶✻

slide-10
SLIDE 10

❋❛✐❧✉r❡ ❚❛❜❧❡❞ ❈♦♥str❛✐♥t ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ❜② ■♥t❡r♣♦❧❛t✐♦♥

■t ❧❡❛r♥s ❢r♦♠ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s✳

❚❤❡ ❢❛✐❧✉r❡ ❛❧❣♦r✐t❤♠ ❛✈♦✐❞s r❡❝♦♠♣✉t❡ r❡❞✉♥❞❛♥t ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s✳

■♥ ♣r❡s❡♥❝❡ ♦❢ r❡❝✉rs✐✈❡ ❝❧❛✉s❡s ✐t ✉s❡s ✐t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣ s❡❛r❝❤✿

◮ ■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ ♣r♦❞✉❝❡s ❛ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s✳ ◮ ▲♦♦❦s ❢♦r ✐♥✈❛r✐❛♥t t♦ st♦♣ t❤❡ s❡❛r❝❤✳

❆✈❛✐❧❛❜❧❡ ❛t ❤tt♣✿✴✴❝♦❞❡✳❣♦♦❣❧❡✳❝♦♠✴♣✴❢t❝❧♣

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✻ ✴ ✶✻

slide-11
SLIDE 11

❋❛✐❧✉r❡ ❚❛❜❧❡❞ ❈♦♥str❛✐♥t ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ❜② ■♥t❡r♣♦❧❛t✐♦♥

■t ❧❡❛r♥s ❢r♦♠ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s✳

❚❤❡ ❢❛✐❧✉r❡ ❛❧❣♦r✐t❤♠ ❛✈♦✐❞s r❡❝♦♠♣✉t❡ r❡❞✉♥❞❛♥t ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s✳

■♥ ♣r❡s❡♥❝❡ ♦❢ r❡❝✉rs✐✈❡ ❝❧❛✉s❡s ✐t ✉s❡s ✐t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣ s❡❛r❝❤✿

◮ ■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ ♣r♦❞✉❝❡s ❛ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s✳ ◮ ▲♦♦❦s ❢♦r ✐♥✈❛r✐❛♥t t♦ st♦♣ t❤❡ s❡❛r❝❤✳

❱❡r✐✜❝❛t✐♦♥ ♦❢ s❛❢❡t② ♣r♦♣❡rt✐❡s✳

❚❤❡ ♣r♦❣r❛♠ ✐♥ ❈✳

✈ ♦ ✐ ❞ ♣ ✭ ✐ ♥ t ✐ ✱ ✐ ♥ t ❥ ✮ ④ ✐ ♥ t ① ❂ ✐ ❀ ✐ ♥ t ② ❂ ❥ ❀ ✇ ❤ ✐ ❧ ❡ ✭ ① ✦❂ ✵✮ ④ ① ❂ ① − ✶ ❀ ② ❂ ② − ✶ ❀ ⑥ ✐ ❢ ✭ ✐ ❂❂ ❥ ✮ ❛ s s ❡ r t ✭ ② ❁❂ ✵ ✮ ❀ ⑥

❚❤❡ ❈▲P tr❛♥s❢♦r♠❛t✐♦♥✳

♣ ✭ ■ ✱ ❏ ✮ ✿− ❳ ✳❂✳ ■ ✱ ❨ ✳❂✳ ❏ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✿− ❳ ✳❁❃✳ ✵ ✱ ❳✶ ✳❂✳ ❳ − ✶ ✱ ❨✶ ✳❂✳ ❨ − ✶ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✶ ✱ ❨✶ ✱ ■ ✱ ❏ ✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✿− ❳ ✳❂✳ ✵ ✱ ♥ ♦ ❴ ❛ s s s ❡ r t ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✳ ♥♦❴❛ss❡rt ✭❴❳✱ ❨✱ ■ ✱ ❏ ✮ ✿− ■ ✳❂✳ ❏ ✱ ❨ ✳ ❃✳ ✵ ✳

✐❢ ◆❖ ❛♥s✇❡r t❤❡ ♣r♦❣r❛♠ ✐s s❛❢❡✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✻ ✴ ✶✻

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SLIDE 12

✶✮ ❲❤❛t ❞♦❡s ✐t ✏❧❡❛r♥✑❄

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✼ ✴ ✶✻

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SLIDE 13

✶✮ ❲❤❛t ❞♦❡s ✐t ✏❧❡❛r♥✑❄ ❋♦r ❡❛❝❤ ♥♦❞❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐♦♥ tr❡❡✿ ❙❡t ♦❢ ❛♥s✇❡rs✳ ■♥ t❤❡ ❆♥s✇❡r ❚❛❜❧❡ ✭❆❚✮✳ ❘❡✉s❡ ❝♦♥❞✐t✐♦♥✳ ■♥ t❤❡ ❋❛✐❧✉r❡ ❚❛❜❧❡ ✭❋❚✮✳

♣✶ ✭❳✱ ❨✱ ❘✮ ✿− ❳✶ ✳❂✳ ❳ ✰ ❨ ✰ ✷ ✱ ♣✷ ✭❳✶ ✱ ❨✱ ❘✮ ✳ ♣✶ ✭❳✱ ❨✱ ❘✮ ✿− ❳✶ ✳❂✳ ❳ ✰ ❨ ✰ ✶ ✱ ♣✷ ✭❳✶ ✱ ❨✱ ❘✮ ✳ ♣✷ ✭❳✱ ❨✱ ❘✮ ✿− ❨✶ ✳❂✳ ❨ ✰ ✶ ✱ ♣✸ ✭❳✱ ❨✶ ✱ ❘✮ ✳ ♣✷ ✭❳✱ ❨✱ ❘✮ ✿− ❨✶ ✳❂✳ ❨ ✰ ✷ ✱ ♣✸ ✭❳✱ ❨✶ ✱ ❘✮ ✳ ♣✸ ✭❳✱ ❨✱ ❘✮ ✿− ❘ ✳❂✳ ❳ ✰ ❨ − ✶ ✳ ♣✸ ✭❳✱ ❨✱ ❘✮ ✿− ❘ ✳❂✳ ❳ ✰ ❨✳ ♣r♦✈❡ ✿− ❳ ✳❃❂✳ ✵ ✱ ❳ ✳❂❁✳ ✺ ✱ ❨ ✳❃❂✳ ✵ ✱ ❨ ✳❂❁✳ ✸ ✱ ❘ ✳❃❂✳ ✶✺ ✱ ♣✶ ✭❳✱ ❨✱ ❘✮ ✳ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✼ ✴ ✶✻

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SLIDE 14

✶✮ ❲❤❛t ❞♦❡s ✐t ✏❧❡❛r♥✑❄ ❘❡✉s❡ ❝♦♥❞✐t✐♦♥

  • ✐✈❡♥ ❆ ∧ ❇ ✉♥s❛t✐s✜❛❜❧❡✱

P ✐s ✐♥t❡r♣♦❧❛♥t ⇐ ⇒      ❆ P P ∧ ❇ ✉♥s❛t✐s✜❛❜❧❡ ✈❛rs✭P✮ ⊆ ✈❛rs✭❆✮ ∩ ✈❛rs✭❇✮

■♥t❡r♣♦❧❛♥ts ✭❈r❛✐❣ ✶✾✺✼✮

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✽ ✴ ✶✻

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SLIDE 15

✶✮ ❲❤❛t ❞♦❡s ✐t ✏❧❡❛r♥✑❄ ❘❡✉s❡ ❝♦♥❞✐t✐♦♥ ❚❤❡ r❡✉s❡ ❝♦♥❞✐t✐♦♥ ♦❢ ❛♥ ❛t♦♠✿ ✏■s t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ t❤❡ ✐♥t❡r♣♦❧❛♥ts ❣❡♥❡r❛t❡❞ ♦♥ ❡❛❝❤ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥✳✑ ❘❡✉s❡ ❝♦♥❞✐t✐♦♥ ♦❢ ♣✷✴✸ ✐s ❳ ′ + ❨ + ✷ ≤ ❘

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✽ ✴ ✶✻

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SLIDE 16

✷✮ ❋❛✐❧✉r❡ ❆❧❣♦r✐t❤♠

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✾ ✴ ✶✻

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SLIDE 17

✷✮ ❋❛✐❧✉r❡ ❆❧❣♦r✐t❤♠ Pr✐♠✐t✐✈❡ ❝♦♥str❛✐♥t

■❢ ✐t ✐s s❛t✐s✜❛❜❧❡ ✇✐t❤ t❤❡ ❝✉rr❡♥t st♦r❡ → ✐t ✐s ❛❞❞❡❞ t♦ t❤❡ st♦r❡✳ ❖t❤❡r✇✐s❡ ✐t ✐s ❛ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥ → t❤❡ ✐♥t❡r♣♦❧❛♥ts ❛r❡ ❣❡♥❡r❛t❡❞✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✾ ✴ ✶✻

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SLIDE 18

✷✮ ❋❛✐❧✉r❡ ❆❧❣♦r✐t❤♠ ❆t♦♠ ✭❯s❡r✲❞❡✜♥❡❞ ❝♦♥str❛✐♥ts✮

❚❛❦❡ t❤❡ ❛♥s✇❡rs ♦❢ t❤✐s ❛t♦♠✱ ✜❧t❡r t❤❡ ❝♦rr❡❝ts ❛♥❞ ✉s❡ t❤❡♠✳

❈❤❡❝❦ t❤❡ r❡✉s❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤✐s ❛t♦♠✳

◮ ■❢ t❤❡ ❝♦♥str❛✐♥t st♦r❡ ✐s s✉❜s✉♠❡❞ ❜② ❛ r❡✉s❡ ❝♦♥❞✐t✐♦♥ → ❙t♦♣✳ ◮ ❖t❤❡r✇✐s❡ → ❘❡❝♦♠♣✉t❡✳ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✾ ✴ ✶✻

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SLIDE 19

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

Pr♦❞✉❝❡ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✵ ✴ ✶✻

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SLIDE 20

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

Pr♦❞✉❝❡ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s

❉❡♣t❤✲✜rst s❡❛r❝❤ ❙♣❛❝❡✲❡✣❝✐❡♥❝② ✳ ❇r❡❛❞t❤✲✜rst s❡❛r❝❤ ❈♦♠♣❧❡t❡♥❡ss∗

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✵ ✴ ✶✻

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SLIDE 21

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

Pr♦❞✉❝❡ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s

❉❡♣t❤✲✜rst s❡❛r❝❤ ❙♣❛❝❡✲❡✣❝✐❡♥❝② ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣ ❇r❡❛❞t❤✲✜rst s❡❛r❝❤ ❈♦♠♣❧❡t❡♥❡ss∗

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✵ ✴ ✶✻

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SLIDE 22

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

Pr♦❞✉❝❡ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s

❉❡♣t❤✲✜rst s❡❛r❝❤ ❙♣❛❝❡✲❡✣❝✐❡♥❝② ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣ ❇r❡❛❞t❤✲✜rst s❡❛r❝❤ ❈♦♠♣❧❡t❡♥❡ss∗

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✵ ✴ ✶✻

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SLIDE 23

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

Pr♦❞✉❝❡ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s

❉❡♣t❤✲✜rst s❡❛r❝❤ ❙♣❛❝❡✲❡✣❝✐❡♥❝② ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣ ❇r❡❛❞t❤✲✜rst s❡❛r❝❤ ❈♦♠♣❧❡t❡♥❡ss∗

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✵ ✴ ✶✻

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SLIDE 24

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

▲♦♦❦ ❢♦r ❛♥ ✐♥✈❛r✐❛♥t t♦ st♦♣ t❤❡ s❡❛r❝❤

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✶ ✴ ✶✻

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SLIDE 25

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

▲♦♦❦ ❢♦r ❛♥ ✐♥✈❛r✐❛♥t t♦ st♦♣ t❤❡ s❡❛r❝❤

■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ t❤❡ ♣r♦❣r❛♠ ✐♥❝r❡❛s❡s t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t ❛ r❡❝✉rs✐✈❡ ❛t♦♠ ✭▲✮ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ✭❞❡♣❤t ❧✐♠✐t✮✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✶ ✴ ✶✻

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SLIDE 26

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

▲♦♦❦ ❢♦r ❛♥ ✐♥✈❛r✐❛♥t t♦ st♦♣ t❤❡ s❡❛r❝❤

■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ t❤❡ ♣r♦❣r❛♠ ✐♥❝r❡❛s❡s t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t ❛ r❡❝✉rs✐✈❡ ❛t♦♠ ✭▲✮ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ✭❞❡♣❤t ❧✐♠✐t✮✳ ❲❤❡♥ t❤❡ ❞❡r✐✈❛t✐♦♥ ❢❛✐❧s ❞✉❡ t♦ t❤❡ ❞❡♣t❤ ❧✐♠✐t✿

❚❤❡ ♣r♦❣r❛♠ ❡①tr❛❝t t❤❡ ✐♥t❡r♣♦❧❛♥t ♦❢ t❤❡ ❞❡❡♣❡st ❛t♦♠ ✭▲❞✮✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✶ ✴ ✶✻

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SLIDE 27

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

▲♦♦❦ ❢♦r ❛♥ ✐♥✈❛r✐❛♥t t♦ st♦♣ t❤❡ s❡❛r❝❤

■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ t❤❡ ♣r♦❣r❛♠ ✐♥❝r❡❛s❡s t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t ❛ r❡❝✉rs✐✈❡ ❛t♦♠ ✭▲✮ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ✭❞❡♣❤t ❧✐♠✐t✮✳ ❲❤❡♥ t❤❡ ❞❡r✐✈❛t✐♦♥ ❢❛✐❧s ❞✉❡ t♦ t❤❡ ❞❡♣t❤ ❧✐♠✐t✿

❚❤❡ ♣r♦❣r❛♠ ❡①tr❛❝t t❤❡ ✐♥t❡r♣♦❧❛♥t ♦❢ t❤❡ ❞❡❡♣❡st ❛t♦♠ ✭▲❞✮✳

▲♦♦❦ ❢♦r t❤❡ ✐♥❞✉❝t✐✈❡ ✐♥✈❛r✐❛♥t ❝❛♥❞✐❞❛t❡ ■▲❛♥❝✱ t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ✐♥t❡r♣♦❧❛♥t ♦❢ ❛♥ ❛♥❝❡st♦r ♦❢ ▲❞✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✶ ✴ ✶✻

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SLIDE 28

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

▲♦♦❦ ❢♦r ❛♥ ✐♥✈❛r✐❛♥t t♦ st♦♣ t❤❡ s❡❛r❝❤

■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ t❤❡ ♣r♦❣r❛♠ ✐♥❝r❡❛s❡s t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t ❛ r❡❝✉rs✐✈❡ ❛t♦♠ ✭▲✮ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ✭❞❡♣❤t ❧✐♠✐t✮✳ ❲❤❡♥ t❤❡ ❞❡r✐✈❛t✐♦♥ ❢❛✐❧s ❞✉❡ t♦ t❤❡ ❞❡♣t❤ ❧✐♠✐t✿

❚❤❡ ♣r♦❣r❛♠ ❡①tr❛❝t t❤❡ ✐♥t❡r♣♦❧❛♥t ♦❢ t❤❡ ❞❡❡♣❡st ❛t♦♠ ✭▲❞✮✳

▲♦♦❦ ❢♦r t❤❡ ✐♥❞✉❝t✐✈❡ ✐♥✈❛r✐❛♥t ❝❛♥❞✐❞❛t❡ ■▲❛♥❝✱ t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ✐♥t❡r♣♦❧❛♥t ♦❢ ❛♥ ❛♥❝❡st♦r ♦❢ ▲❞✳

❙t♦r❡ ✐♥ π t❤❡ ❝♦♥str❛✐♥ts ❢r♦♠ ▲❛♥❝ t♦ ▲❞✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✶ ✴ ✶✻

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SLIDE 29

✸✮ ■t❡r❛t✐✈❡ ❞❡❡♣❡♥✐♥❣

▲♦♦❦ ❢♦r ❛♥ ✐♥✈❛r✐❛♥t t♦ st♦♣ t❤❡ s❡❛r❝❤

■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ t❤❡ ♣r♦❣r❛♠ ✐♥❝r❡❛s❡s t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t ❛ r❡❝✉rs✐✈❡ ❛t♦♠ ✭▲✮ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ✭❞❡♣❤t ❧✐♠✐t✮✳ ❲❤❡♥ t❤❡ ❞❡r✐✈❛t✐♦♥ ❢❛✐❧s ❞✉❡ t♦ t❤❡ ❞❡♣t❤ ❧✐♠✐t✿

❚❤❡ ♣r♦❣r❛♠ ❡①tr❛❝t t❤❡ ✐♥t❡r♣♦❧❛♥t ♦❢ t❤❡ ❞❡❡♣❡st ❛t♦♠ ✭▲❞✮✳

▲♦♦❦ ❢♦r t❤❡ ✐♥❞✉❝t✐✈❡ ✐♥✈❛r✐❛♥t ❝❛♥❞✐❞❛t❡ ■▲❛♥❝✱ t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ✐♥t❡r♣♦❧❛♥t ♦❢ ❛♥ ❛♥❝❡st♦r ♦❢ ▲❞✳

❙t♦r❡ ✐♥ π t❤❡ ❝♦♥str❛✐♥ts ❢r♦♠ ▲❛♥❝ t♦ ▲❞✳

❈❤❡❝❦ ✐❢ ■▲❛♥❝ ✐♥ ❝♦♥❥✉♥❝t✐♦♥ ✇✐t❤ π ❡♥t❛✐❧s ✇✐t❤ ✐ts❡❧❢ ✭❛❢t❡r r❡♥❛♠✐♥❣✮✿ ■▲❛♥❝ ∧ π ■ ′

▲❛♥❝

◮ ■❢ ❡♥t❛✐❧s → st♦♣✱ ✭■▲❛♥❝ ✐s t❤❡ ✐♥❞✉❝t✐✈❡ ✐♥✈❛r✐❛♥t✮✳ ◮ ❖t❤❡r✇✐s❡ → ✐♥❝r❡❛s❡ t❤❡ ❞❡♣t❤ ❧✐♠✐t ❢♦r t❤✐s ❛t♦♠ ✭▲✮ ❢♦r t❤❡ ♥❡①t

✐t❡r❛t✐♦♥✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✶ ✴ ✶✻

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SLIDE 30

❊①❛♠♣❧❡

♣ ✭ ■ ✱ ❏ ✮ ✿− ❳ ✳❂✳ ■ ✱ ❨ ✳❂✳ ❏ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✿− ❳ ✳❁❃✳ ✵ ✱ ❳✶ ✳❂✳ ❳ − ✶ ✱ ❨✶ ✳❂✳ ❨ − ✶ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✶ ✱ ❨✶ ✱ ■ ✱ ❏ ✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✿− ❳ ✳❂✳ ✵ ✱ ♥ ♦ ❴ ❛ s s s ❡ r t ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✳ ♥♦❴❛ss❡rt ✭❴❳✱ ❨✱ ■ ✱ ❏ ✮ ✿− ■ ✳❂✳ ❏ ✱ ❨ ✳ ❃✳ ✵ ✳

✶ ✲ ❆❞❞ ❝♦✉♥t❡r ❑✐ ✐♥ ❡❛❝❤ r❡❝✉rs✐✈❡ ❛t♦♠ ✐ ✷ ✲ ❉❡✜♥❡ ❛ ✈❛r✐❛❜❧❡ ❞✐ ❛s ❞❡♣t❤ ❧✐♠✐t ♦❢ t❤❡ ❛t♦♠ ✐✳ ❆♥❞ s❡t ❞✐ = ✵✳

✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✱ ❑✮ ✿− ❑ ✳❃❂✳ ✵ ✱ ❳ ✳❁❃✳ ✵ ✱ ❳✶ ✳❂✳ ❳ − ✶ ✱ ❨✶ ✳❂✳ ❨ − ✶ ✱ ❑✶ ✳❂✳ ❑ − ✶ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✶ ✱ ❨✶ ✱ ■ ✱ ❏ ✱ ❑✶✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✱ ❴✮ ✿− ❳ ✳❂✳ ✵ ✱ ♥♦❴❛ss❡rt ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✳ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✷ ✴ ✶✻

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SLIDE 31

❊①❛♠♣❧❡

♣ ✭ ■ ✱ ❏ ✮ ✿− ❳ ✳❂✳ ■ ✱ ❨ ✳❂✳ ❏ ✱ ❞ ✭✇✱ ❑✮ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✱ ❑✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✱❑✮ ✿− ❑✳❃❂✳ ✵ ✱ ❳ ✳❁❃✳ ✵ ✱ ❳✶ ✳❂✳ ❳ − ✶ ✱ ❨✶ ✳❂✳ ❨ − ✶ ✱ ❑✶ ✳❂✳ ❑ − ✶ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✶ ✱ ❨✶ ✱ ■ ✱ ❏ ✱ ❑✶✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✱ ■ ✱ ❏ ✱ ❴✮ ✿− ❳ ✳❂✳ ✵ ✱ ♥♦❴❛ss❡rt ✭❳✱ ❨✱ ■ ✱ ❏ ✮ ✳ ♥♦❴❛ss❡rt ✭❴❳✱ ❨✱ ■ ✱ ❏ ✮ ✿− ■ ✳❂✳ ❏ ✱ ❨ ✳ ❃✳ ✵ ✳

✸ ✲ ❙❡t ❑✐ ❂ ❞✐ ❛♥❞ ✇❤❡♥ ❢❛✐❧s✱ ♦❜t❛✐♥s✿ ■♥t❡r♣♦❧❛♥t ♦❢ t❤❡ r❡❝✉rs✐✈❡ ❝❧❛✉s❡ → ❑ ≤ ✵ ■♥t❡r♣♦❧❛♥t ♦❢ t❤❡ ❜❛s❡ ❝❧❛✉s❡ → ❨ + ■ ≤ ❳ + ❏ ❚♦ ❝♦♠♣✉t❡ t❤❡ ✐♥❞✉❝t✐✈❡ ✐♥✈❛r✐❛♥t ❝❛♥❞✐❞❛t❡ ❡❧✐♠✐♥❛t❡s t❤❡ ❞❡♣❡♥❞❡♥❝② ♦♥ ❑ ❛♥❞ ♦❜t❛✐♥s✿

■▲❛♥❝ ≡ {❨ + ■ ≤ ❳ + ❏} ■♥❞✉❝t✐✈❡ ✐♥✈❛r✐❛♥t ❝❛♥❞✐❞❛t❡

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✷ ✴ ✶✻

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SLIDE 32

❊①❛♠♣❧❡ ✹✳❛ ✲ ❙t♦r❡ ✐♥ π t❤❡ ❞✐s❥✉♥❝t✐♦♥ ♦❢ t❤❡ ❝♦♥str❛✐♥ts ❢r♦♠ ▲❛♥❝ t♦ ▲❞✳

π ≡ {❳ = ✵, ❳✶ = ❳ − ✶, ❨ ✶ = ❨ − ✶}

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✷ ✴ ✶✻

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SLIDE 33

❊①❛♠♣❧❡ ✹✳❛ ✲ ❙t♦r❡ ✐♥ π t❤❡ ❞✐s❥✉♥❝t✐♦♥ ♦❢ t❤❡ ❝♦♥str❛✐♥ts ❢r♦♠ ▲❛♥❝ t♦ ▲❞✳

π ≡ {❳ = ✵, ❳✶ = ❳ − ✶, ❨ ✶ = ❨ − ✶}

✹✳❜ ✲ ❈❤❡❝❦ ✐❢ ■▲❛♥❝ ∧ π ■ ′

▲❛♥❝

{❨ + ■ ≤ ❳ + ❏} ∧ {❳ = ✵, ❳✶ = ❳ − ✶, ❨ ✶ = ❨ − ✶} {❨ ✶ + ■ ≤ ❳✶ + ❏}

■ ′

▲❛♥❝ ✐s ■▲❛♥❝ ❛❢t❡r r❡♥❛♠✐♥❣ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✷ ✴ ✶✻

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SLIDE 34

❊①❛♠♣❧❡ ✹✳❛ ✲ ❙t♦r❡ ✐♥ π t❤❡ ❞✐s❥✉♥❝t✐♦♥ ♦❢ t❤❡ ❝♦♥str❛✐♥ts ❢r♦♠ ▲❛♥❝ t♦ ▲❞✳

π ≡ {❳ = ✵, ❳✶ = ❳ − ✶, ❨ ✶ = ❨ − ✶}

✹✳❜ ✲ ❈❤❡❝❦ ✐❢ ■▲❛♥❝ ∧ π ■ ′

▲❛♥❝

{❨ + ■ ≤ ❳ + ❏} ∧ {❳ = ✵, ❳✶ = ❳ − ✶, ❨ ✶ = ❨ − ✶} {❨ ✶ + ■ ≤ ❳✶ + ❏}

■ ′

▲❛♥❝ ✐s ■▲❛♥❝ ❛❢t❡r r❡♥❛♠✐♥❣

❚❤❡ ❡♥t❛✐❧♠❡♥t t❡st ❤♦❧❞s → t❤❡ ❡①❡❝✉t✐♦♥ st♦♣ ✇✐t❤♦✉t ❛♥s✇❡r✳ ❚❤❡ s❛❢❡t② ♣r♦♣❡rt② ❤♦❧❞s✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✷ ✴ ✶✻

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SLIDE 35

❊①❛♠♣❧❡ ✹✳❛ ✲ ❙t♦r❡ ✐♥ π t❤❡ ❞✐s❥✉♥❝t✐♦♥ ♦❢ t❤❡ ❝♦♥str❛✐♥ts ❢r♦♠ ▲❛♥❝ t♦ ▲❞✳

π ≡ {❳ = ✵, ❳✶ = ❳ − ✶, ❨ ✶ = ❨ − ✶}

✹✳❜ ✲ ❈❤❡❝❦ ✐❢ ■▲❛♥❝ ∧ π ■ ′

▲❛♥❝

{❨ + ■ ≤ ❳ + ❏} ∧ {❳ = ✵, ❳✶ = ❳ − ✶, ❨ ✶ = ❨ − ✶} {❨ ✶ + ■ ≤ ❳✶ + ❏}

■ ′

▲❛♥❝ ✐s ■▲❛♥❝ ❛❢t❡r r❡♥❛♠✐♥❣

❚❤❡ ❡♥t❛✐❧♠❡♥t t❡st ❤♦❧❞s → t❤❡ ❡①❡❝✉t✐♦♥ st♦♣ ✇✐t❤♦✉t ❛♥s✇❡r✳ ❚❤❡ s❛❢❡t② ♣r♦♣❡rt② ❤♦❧❞s✳ ❲❤❡♥ ❞♦❡s ♥♦t ❡♥t❛✐❧ ♠❛❦❡ ❞✐ = ❞✐ + ✶ ❛♥❞ r❡♣❡❛t st❡♣ ✸✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✷ ✴ ✶✻

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SLIDE 36

P❡r❢♦r♠❛♥❝❡

Pr♦❣r❛♠ ✇✐t❤ ♥♦ ✐♥✜♥✐t❡ ❞❡r✐✈❛t✐♦♥s

❆♥s✇❡rs ❈▲P ❋❚❈▲P ❈ ❚✐♠❡ ✭s✮ ❙t❛t❡s ❋❛✐❧✉r❡ ❚✐♠❡ ✭s✮ ❙t❛t❡s ❋❛✐❧✉r❡ ✶✺✵ ✷ ✶✳✼ ✺✶✶✷ ✶✵✺✻ ✹✳✸ ✹✷✻✾ ✽✻✷ ✷✷✺ ✺✽ ✹✹✳✶ ✶✶✻✻✽✹ ✷✸✾✻✶ ✹✼✳✸ ✷✽✻✷✹ ✹✻✸✻ ✷✼✺ ✹✺✶ ✹✺✵✳✵ ✽✷✼✼✼✵ ✶✻✼✾✻✾ ✶✹✽✳✻ ✻✶✷✼✻ ✼✺✷✾ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ r❡s♦✉r❝❡ ❝♦♥str❛✐♥❡❞ s❤♦rt❡st ♣❛t❤ ✭❘❈❙P✮ ♣r♦❜❧❡♠ ✭❧♦♦❦ ❢♦r t❤❡ ♣❛t❤s ✇❤✐t ❝♦st ❁ ❈✮✳ ❖✈❡r❤❡❛❞ ♦❢ ❝♦♠♣✉t✐♥❣ ❛♥❞ ❧♦♦❦✐♥❣ ✉♣ ✐♥t❡r♣♦❧❛♥ts✳ ❲❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s ❣r♦✇s ❋❚❈▲P ❜❡❝♦♠❡ ❝♦♠♣❡t✐t✐✈❡✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✸ ✴ ✶✻

slide-37
SLIDE 37

P❡r❢♦r♠❛♥❝❡

Pr♦❣r❛♠ ✇✐t❤ ♥♦ ✐♥✜♥✐t❡ ❞❡r✐✈❛t✐♦♥s

❆♥s✇❡rs ❈▲P ❋❚❈▲P ❈ ❚✐♠❡ ✭s✮ ❙t❛t❡s ❋❛✐❧✉r❡ ❚✐♠❡ ✭s✮ ❙t❛t❡s ❋❛✐❧✉r❡ ✶✺✵ ✷ ✶✳✼ ✺✶✶✷ ✶✵✺✻ ✹✳✸ ✹✷✻✾ ✽✻✷ ✷✷✺ ✺✽ ✹✹✳✶ ✶✶✻✻✽✹ ✷✸✾✻✶ ✹✼✳✸ ✷✽✻✷✹ ✹✻✸✻ ✷✼✺ ✹✺✶ ✹✺✵✳✵ ✽✷✼✼✼✵ ✶✻✼✾✻✾ ✶✹✽✳✻ ✻✶✷✼✻ ✼✺✷✾ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ r❡s♦✉r❝❡ ❝♦♥str❛✐♥❡❞ s❤♦rt❡st ♣❛t❤ ✭❘❈❙P✮ ♣r♦❜❧❡♠ ✭❧♦♦❦ ❢♦r t❤❡ ♣❛t❤s ✇❤✐t ❝♦st ❁ ❈✮✳ ❖✈❡r❤❡❛❞ ♦❢ ❝♦♠♣✉t✐♥❣ ❛♥❞ ❧♦♦❦✐♥❣ ✉♣ ✐♥t❡r♣♦❧❛♥ts✳ ❲❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s ❣r♦✇s ❋❚❈▲P ❜❡❝♦♠❡ ❝♦♠♣❡t✐t✐✈❡✳ ❚❤❡r❡ ✐s ❛♥ ♦♣t✐♠✐s❛t✐♦♥✱ ✉s✐♥❣ ✐♥❞✉❝t✐✈❡ s❡q✉❡♥❝❡ ✐♥t❡r♣♦❧❛♥t✱ ✇❤✐❝❤ ✐♥ t❤✐s ❡①❛♠♣❧❡ r❡❞✉❝❡s t❤❡ ❡①❡❝✉t✐♦♥ ✉♣ t♦ ✷✺ ✪✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✸ ✴ ✶✻

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SLIDE 38

P❡r❢♦r♠❛♥❝❡

Pr♦❣r❛♠s ✇✐t❤ ✐♥✜♥✐t❡ ❞❡r✐✈❛t✐♦♥s

Pr♦❣r❛♠ ❋❚❈▲P ❇▲❆❙❚ ❍❙❋ ❚❘❆❈❊❘ t✶ ∞ ∞ ✵✳✸s ∞ t✶✲❛ ✵✳✶s ❊❘❘ ✵✳✷s ∞ t✷ ✵✳✶s ✵✳✶s ❯◆❙❆❋❊ ✵✳✶s t✸ ✵✳✶s ✵✳✶s ✵✳✷✺s ✵✳✶s t✹ ✵✳✶s ✵✳✻s ✵✳✸s ✵✳✷s t✺ ✵✳✷s ✵✳✷s ✵✳✸s ✵✳✶s ❍❙❋ ❛♥❞ ❚❘❆❈❊❘ ✉s❡ ❈▲P ❛s ✐♥t❡r♠❡❞✐❛t❡ r❡♣r❡s❡♥t❛t✐♦♥✳ ❋❚❈▲P ✐s r♦❜✉st✳ ❋❚❈▲P ❛❝❤✐❡✈❡s t❡r♠✐♥❛t✐♦♥ ✐♥ ❝❛s❡s ✇❤❡r❡ ♦t❤❡r t♦♦❧s ❞♦ ♥♦t✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✹ ✴ ✶✻

slide-39
SLIDE 39

■♥t❡r♣♦❧❛♥t ✐♥ ❋❚❈▲P ❁❃ Pr♦❥❡❝t✐♦♥ ✐♥ ❚❈▲P ✶✮ ❚❈▲P ✉s❡s ♣r♦❥❡❝t✐♦♥ ❛♥❞ ❡♥t❛✐❧♠❡♥t t♦ s✉s♣❡♥❞ ❛ ❝❛❧❧✳

❛ss✉♠❡ ✭ ② ❃ ✵ ✮ ❀ ✇ ❤ ✐ ❧ ❡ ✭ ① ❃ ✵ ✮ ④ ① ❂ ① − ✶ ❀ ② ❂ ② − ✶ ❀ ⑥ ♣ ✭❳✱ ❨✮ ✿− ❨ ✳ ❃✳ ✵ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✮ ✿− ❳ ✳ ❃✳ ✵ ✱ ❳✶ ✳❂✳ ❳ − ✶ ✱ ❨✶ ✳❂✳ ❨ − ✶ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✶ ✱ ❨✶✮ ✳ ❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✺ ✴ ✶✻

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SLIDE 40

■♥t❡r♣♦❧❛♥t ✐♥ ❋❚❈▲P ❁❃ Pr♦❥❡❝t✐♦♥ ✐♥ ❚❈▲P ✶✮ ❚❈▲P ✉s❡s ♣r♦❥❡❝t✐♦♥ ❛♥❞ ❡♥t❛✐❧♠❡♥t t♦ s✉s♣❡♥❞ ❛ ❝❛❧❧✳

❛ss✉♠❡ ✭ ② ❃ ✵ ✮ ❀ ✇ ❤ ✐ ❧ ❡ ✭ ① ❃ ✵ ✮ ④ ① ❂ ① − ✶ ❀ ② ❂ ② − ✶ ❀ ⑥ ♣ ✭❳✱ ❨✮ ✿− ❨ ✳ ❃✳ ✵ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✮ ✿− ❳ ✳ ❃✳ ✵ ✱ ❳✶ ✳❂✳ ❳ − ✶ ✱ ❨✶ ✳❂✳ ❨ − ✶ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✶ ✱ ❨✶✮ ✳

■♥ t❤✐s ❡①❛♠♣❧❡ t❤❡ ♣r♦❥❡❝t✐♦♥ ♣r♦❞✉❝❡✿ {❳✶ > −✶ ∧ ❨ ✶ > −✶} ✐♥ t❤❡ ✜rst ✐t❡r❛t✐♦♥✱ {❳✶ > −✶ ∧ ❨ ✶ > −✷} ✐♥ t❤❡ s❡❝♦♥❞ ✐t❡r❛t✐♦♥ t❤❡ ♥❡①t ❝♦♥str❛✐♥ts ❛❧s♦ ✇✐❧❧ ♥♦t ❡♥t❛✐❧ ✇✐t❤ ♣r❡✈✐♦✉s ❝❛❧❧s ❛♥❞ ❚❈▲P ✇✐❧❧ ♥♦t s✉s♣❡♥❞✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✺ ✴ ✶✻

slide-41
SLIDE 41

■♥t❡r♣♦❧❛♥t ✐♥ ❋❚❈▲P ❁❃ Pr♦❥❡❝t✐♦♥ ✐♥ ❚❈▲P ✶✮ ❚❈▲P ✉s❡s ♣r♦❥❡❝t✐♦♥ ❛♥❞ ❡♥t❛✐❧♠❡♥t t♦ s✉s♣❡♥❞ ❛ ❝❛❧❧✳

❛ss✉♠❡ ✭ ② ❃ ✵ ✮ ❀ ✇ ❤ ✐ ❧ ❡ ✭ ① ❃ ✵ ✮ ④ ① ❂ ① − ✶ ❀ ② ❂ ② − ✶ ❀ ⑥ ♣ ✭❳✱ ❨✮ ✿− ❨ ✳ ❃✳ ✵ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✮ ✳ ✇ ❤ ✐ ❧ ❡ ✭❳✱ ❨✮ ✿− ❳ ✳ ❃✳ ✵ ✱ ❳✶ ✳❂✳ ❳ − ✶ ✱ ❨✶ ✳❂✳ ❨ − ✶ ✱ ✇ ❤ ✐ ❧ ❡ ✭❳✶ ✱ ❨✶✮ ✳

■♥ t❤✐s ❡①❛♠♣❧❡ t❤❡ ♣r♦❥❡❝t✐♦♥ ♣r♦❞✉❝❡✿ {❳✶ > −✶ ∧ ❨ ✶ > −✶} ✐♥ t❤❡ ✜rst ✐t❡r❛t✐♦♥✱ {❳✶ > −✶ ∧ ❨ ✶ > −✷} ✐♥ t❤❡ s❡❝♦♥❞ ✐t❡r❛t✐♦♥ t❤❡ ♥❡①t ❝♦♥str❛✐♥ts ❛❧s♦ ✇✐❧❧ ♥♦t ❡♥t❛✐❧ ✇✐t❤ ♣r❡✈✐♦✉s ❝❛❧❧s ❛♥❞ ❚❈▲P ✇✐❧❧ ♥♦t s✉s♣❡♥❞✳ ■♥ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✱ ✇✐t❤ ❋❚❈▲P t❤❡ ✐♥t❡r♣♦❧❛♥t ❧✐♥❦❡❞ ❳ ❛♥❞ ❨✿ {❨ + ■ ≤ ❳ + ❏} ❛♥❞ ✇❤❡♥ ✇❡ ❛♣♣❧② ♦♥❡ ✐t❡r❛t✐♦♥✳ {(❨ − ✶) + ■ ≤ (❳ − ✶) + ❏} t❤❡r❡ ❛r❡ ♥♦ ❝❤❛♥❣❡s✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✺ ✴ ✶✻

slide-42
SLIDE 42

■♥t❡r♣♦❧❛♥t ✐♥ ❋❚❈▲P ❁❃ Pr♦❥❡❝t✐♦♥ ✐♥ ❚❈▲P ✷✮ ❇❡♥❡✜ts ♦❢ ✐♥t❡r♣♦❧❛♥ts✿ ❚❤❡② ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ❧✐♥❡❛r t✐♠❡✳ ❚❤❡② ❛r❡ ♣r❡s❡♥t ✐♥ ♠♦r❡ ❞♦♠❛✐♥s t❤❛♥ ♣r♦❥❡❝t✐♦♥✳ ❚❤❡② ❝❛♥ ❜❡ ❡✛❡❝t✐✈❡ ❢♦r ♣r✉♥✐♥❣ ♦t❤❡r ❞❡r✐✈❛t✐♦♥s✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✺ ✴ ✶✻

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SLIDE 43

■♥t❡r♣♦❧❛♥t ✐♥ ❋❚❈▲P ❁❃ Pr♦❥❡❝t✐♦♥ ✐♥ ❚❈▲P ✷✮ ❇❡♥❡✜ts ♦❢ ✐♥t❡r♣♦❧❛♥ts✿ ❚❤❡② ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ❧✐♥❡❛r t✐♠❡✳ ❚❤❡② ❛r❡ ♣r❡s❡♥t ✐♥ ♠♦r❡ ❞♦♠❛✐♥s t❤❛♥ ♣r♦❥❡❝t✐♦♥✳ ❚❤❡② ❝❛♥ ❜❡ ❡✛❡❝t✐✈❡ ❢♦r ♣r✉♥✐♥❣ ♦t❤❡r ❞❡r✐✈❛t✐♦♥s✳ ✸✮ ❲❤✐❝❤ ❛r❡ t❤❡ ❞✐✛❡r❡♥❝❡s ✇❤❡♥ st♦r✐♥❣ t❤❡ ❛♥s✇❡rs❄ ❋❚❈▲P st♦r❡ ❛❧❧ t❤❡ ❛♥s✇❡r ✐♥ ♦♥❡ t❛❜❧❡✳ ❚❈▲P ❝r❡❛t❡ t❛❜❧❡s ❢♦r ❡❛❝❤ ✏♥❡✇✑ ♣r♦❥❡❝t✐♦♥✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✺ ✴ ✶✻

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SLIDE 44

■♥t❡r♣♦❧❛♥t ✐♥ ❋❚❈▲P ❁❃ Pr♦❥❡❝t✐♦♥ ✐♥ ❚❈▲P ✷✮ ❇❡♥❡✜ts ♦❢ ✐♥t❡r♣♦❧❛♥ts✿ ❚❤❡② ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ❧✐♥❡❛r t✐♠❡✳ ❚❤❡② ❛r❡ ♣r❡s❡♥t ✐♥ ♠♦r❡ ❞♦♠❛✐♥s t❤❛♥ ♣r♦❥❡❝t✐♦♥✳ ❚❤❡② ❝❛♥ ❜❡ ❡✛❡❝t✐✈❡ ❢♦r ♣r✉♥✐♥❣ ♦t❤❡r ❞❡r✐✈❛t✐♦♥s✳ ✸✮ ❲❤✐❝❤ ❛r❡ t❤❡ ❞✐✛❡r❡♥❝❡s ✇❤❡♥ st♦r✐♥❣ t❤❡ ❛♥s✇❡rs❄ ❋❚❈▲P st♦r❡ ❛❧❧ t❤❡ ❛♥s✇❡r ✐♥ ♦♥❡ t❛❜❧❡✳ ❚❈▲P ❝r❡❛t❡ t❛❜❧❡s ❢♦r ❡❛❝❤ ✏♥❡✇✑ ♣r♦❥❡❝t✐♦♥✳ ✹✮ ▲❡❛r♥✐♥❣ ❢r♦♠ ❢❛✐❧s ✐♥ ❚❈▲P✳ ■♥ ❛ ♥♦♥ r❡❝✉rs✐✈❡ ❝❧❛✉s❡✱ ❚❈▲P ❛❧s♦ ❛✈♦✐❞s r❡❝♦♠♣✉t❡ r❡❞✉♥❞❛♥t ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s✳ ■t ✐s ♣♦ss✐❜❧❡ t❤❛t ✇✐t❤ t❤❡ ♣r♦❥❡❝t✐♦♥ ✇❡ ❧♦s❡ ❝♦♥str❛✐♥ts ✐♥❢♦r♠❛t✐♦♥✳ ■♥ ❛ r❡❝✉rs✐✈❡ ❝❧❛✉s❡✱ ❚❈▲P ❤❛s ♥♦ ❜❡♥❡✜ts ❢r♦♠ ❢❛✐❧❡❞ ❞❡r✐✈❛t✐♦♥s ✉♥t✐❧ t❤❡ ❝❛❧❧ ✐s ❝♦♠♣❧❡t❡ → ✐♥t❡r♣♦❧❛t✐♦♥ ❝❛♥ ❜❡ ✉s❡❢✉❧❧✳

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✺ ✴ ✶✻

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SLIDE 45

❚❤❛♥❦s

❏♦❛q✉í♥ ❆r✐❛s ❍❡rr❡r♦ ❙❡♠✐♥❛r ❙❡r✐❡s ❙✸ ✶✶ ♥♦✈❡♠❜❡r ✷✵✶✹ ✶✻ ✴ ✶✻