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SLIDE 1

❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛

❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦

❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶ ✴ ✶✽

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SLIDE 2

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥

❘❡st✐♥❣ st❛t❡ ❢▼❘■

▲❛tt✐❝❡ ❆✉t♦✲❆ss♦❝✐❛t✐✈❡ ▼❡♠♦r✐❡s ✭▲❆❆▼s✮

▼✉❧t✐✈❛r✐❛t❡ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦r♣❤♦❧♦❣②

P✐♣❡❧✐♥❡

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❈♦♥❝❧✉s✐♦♥s

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✷ ✴ ✶✽

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SLIDE 3

■♥tr♦❞✉❝t✐♦♥

■♥tr♦❞✉❝t✐♦♥

▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❛♣♣r♦❛❝❤ t♦

✐❞❡♥t✐❢② ❢✉♥❝t✐♦♥❛❧ ♥❡t✇♦r❦s ✐♥ r❡st✐♥❣ st❛t❡ ❢▼❘■ ❞❛t❛ ❜✐♦♠❛r❦❡rs ♦❢ ❝♦❣♥✐t✐✈❡ ♦r ♥❡✉r♦❞❡❣❡♥❡r❛t✐✈❡ ❞✐s❡❛s❡s✳

❯s❡ ▲❛tt✐❝❡ ❆✉t♦✲❆ss♦❝✐❛t✐✈❡ ▼❡♠♦r✐❡s t♦ ❝♦♠♣✉t❡ ❛ r❡❞✉❝❡❞ ♦r❞❡r✐♥❣ ❤✲❢✉♥❝t✐♦♥ ❚❤r❡s❤♦❧❞❡❞ ♦r ♣r♦❝❡ss❡❞ ❜② ♠♦r♣❤♦❧♦❣✐❝❛❧ ♦♣❡r❛t♦rs ❢♦r ♥❡t✇♦r❦ ❞❡t❡❝t✐♦♥ P❡❛rs♦♥ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t ❜❡t✇❡❡♥ t❤❡ ❤✲❢✉♥❝t✐♦♥ ✈❛❧✉❡s ❛♥❞ t❤❡ ❝❛t❡❣♦r✐❝❛❧ ✈❛r✐❛❜❧❡ ❛❧❧♦✇s t♦ ✐❞❡♥t✐❢② t❤❡ ♠♦st ✐♥❢♦r♠❛t✐✈❡ ✈♦①❡❧ s✐t❡s ❈❧❛ss✐✜❝❛t✐♦♥

  • r♦✉♣ ❛♥❛❧②s✐s ✐s ♣❡r❢♦r♠❡❞ ♦♥ ❛ ❞❛t❛s❡t ♦❢ ❤❡❛❧t❤② ❝♦♥tr♦❧s✱

s❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ❋✐♥❞ ❢✉♥❝t✐♦♥❛❧❧② ❝♦♥♥❡❝t❡❞ ❝❧✉st❡r ❞✐✛❡r❡♥❝❡s ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s✉❜❥❡❝ts s✉✛❡r✐♥❣ ❛✉❞✐t♦r② ❤❛❧❧✉❝✐♥❛t✐♦♥✳

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✸ ✴ ✶✽

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SLIDE 4

❘❡st✐♥❣ st❛t❡ ❢▼❘■

❘❡st✐♥❣ st❛t❡ ❢▼❘■

❘❡st✐♥❣ st❛t❡ ❢▼❘■ ❞❛t❛ ❤❛s ❜❡❡♥ ✉s❡❞ t♦ st✉❞② ❜r❛✐♥ ❢✉♥❝t✐♦♥❛❧ ❝♦♥♥❡❝t✐✈✐t②

❝♦rr❡❧❛t✐♦♥ ♦❢ ❧♦✇ ❢r❡q✉❡♥❝② ♦s❝✐❧❧❛t✐♦♥s ✐♥ ❞✐✈❡rs❡ ❛r❡❛s ♦❢ t❤❡ ❜r❛✐♥ r❡✈❡❛❧ t❤❡✐r ❢✉♥❝t✐♦♥❛❧ r❡❧❛t✐♦♥s✳ ❝♦♥♥❡❝t✐♦♥s ❞✐s❝♦✈❡r❡❞ ❛r❡ ❛ ❜r❛✐♥ ✜♥❣❡r♣r✐♥t✱ t❤❡ s♦✲❝❛❧❧❡❞ ❞❡❢❛✉❧t✲♠♦❞❡ ♥❡t✇♦r❦✳

❞♦ ♥♦t ✐♠♣♦s❡ ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❝♦❣♥✐t✐✈❡ ❛❜✐❧✐t✐❡s ♦❢ t❤❡ s✉❜❥❡❝ts✳

✐♥ t❤❡ st✉❞② ♦❢ ❜r❛✐♥ ♠❛t✉r❛t✐♦♥ t❤❡r❡ ✐s ♥♦ s✐♥❣❧❡ ❝♦❣♥✐t✐✈❡ t❛s❦ ✇❤✐❝❤ ✐s ❛♣♣r♦♣r✐❛t❡ ❛❝r♦ss t❤❡ ❛❣✐♥❣ ♣♦♣✉❧❛t✐♦♥✳

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✹ ✴ ✶✽

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SLIDE 5

❘❡st✐♥❣ st❛t❡ ❢▼❘■

❙❝❤✐③♦♣❤r❡♥✐❛

❙❝❤✐③♦♣❤r❡♥✐❛ ✐s ❛ s❡✈❡r❡ ♣s②❝❤✐❛tr✐❝ ❞✐s❡❛s❡ t❤❛t ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❞❡❧✉s✐♦♥s ❛♥❞ ❤❛❧❧✉❝✐♥❛t✐♦♥s✱ ❧♦ss ♦❢ ❡♠♦t✐♦♥ ❛♥❞ ❞✐sr✉♣t❡❞ t❤✐♥❦✐♥❣✳ ❋✉♥❝t✐♦♥❛❧ ❞✐s❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ ❜r❛✐♥ r❡❣✐♦♥s ✐s s✉s♣❡❝t❡❞ t♦ ❝❛✉s❡ t❤❡s❡ s②♠♣t♦♠s✱ ❜❡❝❛✉s❡ ♦❢ ❦♥♦✇♥ ❛❜❡rr❛♥t ❡✛❡❝ts ♦♥ ❣r❛② ❛♥❞ ✇❤✐t❡ ♠❛tt❡r ✐♥ ❜r❛✐♥ r❡❣✐♦♥s t❤❛t ♦✈❡r❧❛♣ ✇✐t❤ t❤❡ ❞❡❢❛✉❧t ♠♦❞❡ ♥❡t✇♦r❦✳ ❘❡st✐♥❣ st❛t❡ ❢▼❘■ st✉❞✐❡s ❤❛✈❡ ✐♥❞✐❝❛t❡❞ ❛❜❡rr❛♥t ❞❡❢❛✉❧t ♠♦❞❡ ❢✉♥❝t✐♦♥❛❧ ❝♦♥♥❡❝t✐✈✐t② ✐♥ s❝❤✐③♦♣❤r❡♥✐❝ ♣❛t✐❡♥ts✳

  • ♦❛❧ ♦❢ ♦✉r ✇♦r❦ ✐s t♦ ✜♥❞ ❞✐✛❡r❡♥❝❡s ✐♥ ❝♦♥♥❡❝t✐✈✐t② ❜❡t✇❡❡

♣❛t✐❡♥ts ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t ❛✉❞✐t♦r② ❤❛❧❧✉❝✐♥❛t✐♦♥s

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✺ ✴ ✶✽

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SLIDE 6

▲❛tt✐❝❡ ❆✉t♦✲❆ss♦❝✐❛t✐✈❡ ▼❡♠♦r✐❡s ✭▲❆❆▼s✮

▼✉❧t✐✈❛r✐❛t❡ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦r♣❤♦❧♦❣②

❆♣♣r♦❛❝❤ t♦ ❞❡✜♥❡ ▼✉❧t✐✈❛r✐❛t❡ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦r♣❤♦❧♦❣② ❜❛s❡❞ ♦♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ s✉♣❡r✈✐s❡❞ ♦r❞❡r✐♥❣ ❜✉✐❧t ♦♥ t❤❡ ▲❛tt✐❝❡ ❆✉t♦✲❛ss♦❝✐❛t✐✈❡ ▼❡♠♦r② r❡❝❛❧❧ ❡rr♦r ❘❡s✉❧ts ♦♥ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦♥ r❡st✐♥❣ st❛t❡ ❢▼❘■ t♦ ❞✐s❝♦✈❡r ❢✉♥❝t✐♦♥❛❧ ❝♦♥♥❡❝t✐✈✐t②

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✻ ✴ ✶✽

slide-7
SLIDE 7

▲❛tt✐❝❡ ❆✉t♦✲❆ss♦❝✐❛t✐✈❡ ▼❡♠♦r✐❡s ✭▲❆❆▼s✮

▲❆❆▼ ❞❡✜♥✐t✐♦♥s

❲❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ s❡✈❡r❛❧ ❛♣♣r♦❛❝❤❡s t♦ ❞❡✜♥❡ r❡❞✉❝❡❞ s✉♣❡r✈✐s❡❞ ♦r❞❡r✐♥❣s ❜❛s❡❞ ♦♥ t❤❡ r❡❝❛❧❧ ❡rr♦r ♦❢ t❤❡ ▲❆❆▼✳ ✐♥♣✉t✴♦✉t♣✉t ♣❛✐rs ♦❢ ♣❛tt❡r♥s (❳, ❨ ) =

  • ①ξ, ②ξ

; ξ = ✶, .., ❦

  • ❛ ❧✐♥❡❛r ❤❡t❡r♦❛ss♦❝✐❛t✐✈❡ ♥❡✉r❛❧ ♥❡t✇♦r❦

❲ =

  • ξ

②ξ ·

  • ①ξ′

. ❡r♦s✐✈❡ ❛♥❞ ❞✐❧❛t✐✈❡ ▲❆▼s✱ r❡s♣❡❝t✐✈❡❧② ❲❳❨ =

  • ξ=✶
  • ②ξ ×
  • −①ξ′

❛♥❞ ▼❳❨ =

  • ξ=✶
  • ②ξ ×
  • −①ξ′

, ✇❤❡r❡ × ✐s ❛♥② ♦❢ t❤❡ ∨ ♦r ∧ ♦♣❡r❛t♦rs✱

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✼ ✴ ✶✽

slide-8
SLIDE 8

▲❛tt✐❝❡ ❆✉t♦✲❆ss♦❝✐❛t✐✈❡ ▼❡♠♦r✐❡s ✭▲❆❆▼s✮

▲❆❆▼ ❞❡✜♥✐t✐♦♥s

♦♣❡r❛t♦r ∨ ❞❡♥♦t❡s t❤❡ ♠❛① ♠❛tr✐① ♣r♦❞✉❝t ❈ = ❆ ∨ ❇ = [❝✐❥] ⇔ ❝✐❥ =

  • ❦=✶..♥
  • ❛✐❦ + ❜❦❥
  • ,

♦♣❡r❛t♦r ∧ ❞❡♥♦t❡s t❤❡ ♠✐♥ ♠❛tr✐① ♣r♦❞✉❝t ❈ = ❆ ∧ ❇ = [❝✐❥] ⇔ ❝✐❥ =

  • ❦=✶..♥
  • ❛✐❦ + ❜❦❥
  • .

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✽ ✴ ✶✽

slide-9
SLIDE 9

▲❛tt✐❝❡ ❆✉t♦✲❆ss♦❝✐❛t✐✈❡ ▼❡♠♦r✐❡s ✭▲❆❆▼s✮

▲❆❆▼ ❞❡✜♥✐t✐♦♥s ❛♥❞ ♣r♦♣❡rt✐❡s

❉❡✜♥✐t✐♦♥ ❲❤❡♥ ❳ = ❨ t❤❡♥ ❲❳❳ ❛♥❞ ▼❳❳ ❛r❡ ❝❛❧❧❡❞ ▲❛tt✐❝❡ ❆✉t♦✲❆ss♦❝✐❛t✐✈❡ ▼❡♠♦r✐❡s ✭▲❆❆▼s✮✳ ♣❡r❢❡❝t r❡❝❛❧❧ ❢♦r ❛♥ ✉♥❧✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ st♦r❡❞ ♣❛tt❡r♥s ❲❳❳ ∨ ❳ = ❳ = ▼❳❳ ∧ ❳ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♦♥❡ st❡♣ ❢♦r ❛♥② ✐♥♣✉t ♣❛tt❡r♥

✐❢ ❲❳❳ ∨ ③ = ✈ t❤❡♥ ❲❳❳ ∨ ✈ = ✈ ✐❢ ▼❳❳ ∨ ③ = ✉ t❤❡♥ ▼❳❳ ∧ ✉ = ✉✳

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✾ ✴ ✶✽

slide-10
SLIDE 10

▼✉❧t✐✈❛r✐❛t❡ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦r♣❤♦❧♦❣②

▼✉❧t✐✈❛r✐❛t❡ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦r♣❤♦❧♦❣②

❚❤❡ t✇♦ ❡❧❡♠❡♥t❛r② ♠♦r♣❤♦❧♦❣✐❝❛❧ ♦♣❡r❛t♦rs

❡r♦s✐♦♥✿ ❛♥ ♦♣❡r❛t✐♦♥ t❤❛t ✐s ❞✐str✐❜✉t✐✈❡ ✇✐t❤ t❤❡ ♠✐♥✐♠✉♠

ε

  • =
  • ②∈❨

ε (②)

❞✐❧❛t✐♦♥✿ ❞✐str✐❜✉t✐✈❡ ✇✐t❤ t❤❡ ♠❛①✐♠✉♠

δ

  • =
  • ②∈❨

δ (②) ■♠❛❣❡ ♠♦r♣❤♦❧♦❣✐❝❛❧ ✜❧t❡rs✿ ♠♦r♣❤♦❧♦❣✐❝❛❧ ❣r❛❞✐❡♥t ❣ (❨ ) = δ (❨ ) − ε (❨ ) t❤❡ t♦♣✲❤❛t t (❨ ) = ❨ − δ (ε (❨ ))

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✵ ✴ ✶✽

slide-11
SLIDE 11

▼✉❧t✐✈❛r✐❛t❡ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦r♣❤♦❧♦❣②

▼✉❧t✐✈❛r✐❛t❡ ♦r❞❡r✐♥❣

❚❤❡ ▲❆❆▼ ❤✲♠❛♣♣✐♥❣ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❈❤❡❜②s❤❡✈ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❛tt❡r♥ ✈❡❝t♦r ❛♥❞ t❤❡ r❡❝❛❧❧ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ▲❆❆▼✳ ❤❳ (❝) = ❞❈

  • ①#, ①
  • ,

✇❤❡r❡ ①#

▼ = ▼❳❳ ∧

①✱ ❢✉♥❝t✐♦♥ ❞❈ (❛, ❜) ❞❡♥♦t❡s t❤❡ ❈❤❡❜②s❤❡✈ ❞✐st❛♥❝❡✿ ❞❈ (❛, ❜) =

  • ✐=✶

|❛✐ − ❜✐| .

❋♦r❡❣r♦✉♥❞ ▲❆❆▼ ❤✲s✉♣❡r✈✐s❡❞ ♦r❞❡r✐♥❣✱ ❞❡♥♦t❡❞ ❜② ≤❳✱ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ ▲❆❆▼ ❤✲♠❛♣♣✐♥❣ ❛s ❢♦❧❧♦✇s✿ ∀①, ② ∈ R♥, ① ≤❳ ② ⇐ ⇒ ❤❳ (①) ≤ ❤❳ (②) . ❇❛❝❦❣r♦✉♥❞✴❋♦r❡❣r♦✉♥❞ ✭❇✴❋✮ ▲❆❆▼ ❤✲♠❛♣♣✐♥❣ ❤r (①) ❛s ❢♦❧❧♦✇s✿ ❤r (①) = ❤❋ (①) − ❤❇ (①) ,

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✶ ✴ ✶✽

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SLIDE 12

P✐♣❡❧✐♥❡

P✐♣❡❧✐♥❡

❚❤❡ ❣r❛♣❤✐❝❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ ♦✉r ❡①♣❡r✐♠❡♥t❛❧ ♣r♦❝❡ss

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✷ ✴ ✶✽

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SLIDE 13

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❛✐♠ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥ts ✐s ❛ ♣r♦♦❢ ♦❢ ❝♦♥❝❡♣t ♦❢ t❤❡ ❛♣♣r♦❛❝❤ ❞✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ ❤❡❛❧t❤② ❝♦♥tr♦❧ s✉❜❥❡❝ts✱ s❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t ❛✉❞✐t♦r② ❤❛❧❧✉❝✐♥❛t✐♦♥s✳ r❡s✉❧ts ✜♥❞ ❞✐✛❡r❡♥t ❜r❛✐♥ ♥❡t✇♦r❦s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s✉❜❥❡❝t ✉s✐♥❣ t❤❡ s❛♠❡ ❤✲❢✉♥❝t✐♦♥ ❜✉✐❧t ❢r♦♠ s❡❧❡❝t❡❞ ✈♦①❡❧ s❡❡❞s✳

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✸ ✴ ✶✽

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SLIDE 14

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❖✉r ❡①♣❡r✐♠❡♥ts ❛✐♠ t♦ ♦❜t❛✐♥ ♥❡t✇♦r❦ ❧♦❝❛❧✐③❛t✐♦♥s ❝♦rr❡❧❛t❡❞ ✇✐t❤ ❛♥ s♣❡❝✐✜❝ ✈♦①❡❧ ❢r♦♠ t❤❡ ❛✉❞✐t♦r② ❝♦rt❡① ✐♥ ♦r❞❡r t♦ ❛s❝❡rt❛✐♥ s♦♠❡ ❡✛❡❝t r❡❧❛t❡❞ t♦ t❤❡ ❛✉❞✐t♦r② ❤❛❧❧✉❝✐♥❛t✐♦♥s

❋✐❣✉r❡✿ ❋♦r❡❣r♦✉♥❞ ✈♦①❡❧ s❡❡❞ s✐t❡ ❢r♦♠ t❤❡ ❧❡❢t ❍❡s❝❤❧✬s ❣②r✉s ✭▲❍●❀ ✲✹✷✱✲✷✻✱✶✵✮✳ ❇❛❝❦❣r♦✉♥❞ ✈♦①❡❧ s❡❡❞ s✐t❡ ❢r♦♠ ❈❙❋ ♦❢ t❤❡ ✈❡♥tr✐❝❧❡

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✹ ✴ ✶✽

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SLIDE 15

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❍❈ ✈s✳ ❙❝❤✐③ ❍❈ ✈s✳ ♥❆❍ ❍❈ ✈s✳ ❆❍ ♥❆❍ ✈s✳ ❆❍ ❘❡s✉❧ts ♦❢ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❡①♣❡r✐♠❡♥ts ♦♥ t❤❡ ❞✐s❝r✐♠✐♥❛t✐♦♥ t❤❡ ♣♦ss✐❜❧❡ ♣❛✐rs ♦❢ ❝❧❛ss❡s

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✺ ✴ ✶✽

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SLIDE 16

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❍❈ ✈s✳ ❙❝❤✐③ ♥❆❍ ✈s✳ ♥❆❍ ❱♦①❡❧ s✐t❡s ♦❢ t❤❡ ❢❡❛t✉r❡ ❡①tr❛❝t✐♦♥ ✐♥ t❤❡ ❛❜♦✈❡ ❡♥✉♠❡r❛t❡❞ ❝❛s❡s✳ ❚❤❡s❡ ❧♦❝❛❧✐③❛t✐♦♥s ❝❛♥ ❜❡ t❛❦❡♥ ❛s ❜✐♦♠❛r❦❡rs ❢♦r ❛❞❞✐t✐♦♥❛❧ r❡s❡❛r❝❤✳

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✻ ✴ ✶✽

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SLIDE 17

❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❯s✐♥❣ t❤❡ ▲❆❆▼ r❡❝♦♥str✉❝t✐♦♥ ❡rr♦r ♠❡❛s✉r❡❞ ❜② t❤❡ ❈❤❡❜②s❤❡✈ ❞✐st❛♥❝❡ ❛s ❛ r❡❞✉❝❡❞ ♦r❞❡r✐♥❣ ❤✲♠❛♣✱ ✇❡ ❞❡✜♥❡ ❛ ❋♦r❡❣r♦✉♥❞✴❇❛❝❦❣r♦✉♥❞✴ ▲❆❆▼s✉♣❡r✈✐s❡❞ ❤✲♠❛♣✳ ❘❡st✐♥❣ st❛t❡ ❢▼❘■ ❞❛t❛ ✐s ✉s❡❞ ❢♦r t❤❡ ✐❞❡♥t✐❝❛t✐♦♥ ♦❢ ♣♦t❡♥t✐❛❧ ❜✐♦♠❛r❦❡rs ❢♦r s❝❤✐③♦♣❤r❡♥✐❛ ❛♥❞ ✈❛r✐❛♥ts ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t ❛✉❞✐t♦r② ❤❛❧❧✉❝✐♥❛t✐♦♥s ❜② P❡❛rs♦♥✬s ❝♦rr❡❧❛t✐♦♥✳ ❇✐♦♠❛r❦❡rs ❛r❡ ❡✈❛❧✉❛t❡❞ ✐♥ t❤❡ t❡r♠s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛ss✐❝❛t✐♦♥ ❛❝❝✉r❛❝② ❛❝❤✐❡✈❡❞ ♦♥ t❤❡ ❢❡❛t✉r❡ ✈❡❝t♦rs ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ s❡❧❡❝t❡❞ ✈♦①❡❧ s✐t❡s✳ ❇❡st r❡s✉❧ts ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ❞✐s❝r✐♠✐♥❛t✐♦♥ ❜❡t✇❡❡♥ ❤❡❛❧t❤② ❝♦♥tr♦❧s ❛♥❞ ♣❛t✐❡♥ts ✇✐t❤♦✉t ❛✉❞✐t♦r② ❤❛❧❧✉❝✐♥❛t✐♦♥s✳

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✼ ✴ ✶✽

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SLIDE 18

❈♦♥❝❧✉s✐♦♥s

❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s

❉❛r②❛ ❈❤②③❤②❦✱ ▼❛♥✉❡❧ ●r❛ñ❛

❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦

❉❛r②❛ ❈❤②③❤②❦ ❛♥❞ ▼❛♥✉❡❧ ●r❛ñ❛ ✭❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ●r♦✉♣ ❯♥✐✈❡rs✐t② ♦❢ ❇❛sq✉❡ ❈♦✉♥tr② ❙❛♥ ❙❡❜❛st✐á♥✱ ❙♣❛✐♥ ✇✇✇✳❡❤✉✳❡s✴❝❝✇✐♥t❝♦ ✮ ❉✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ r❡st✐♥❣✲st❛t❡ ❢▼❘■ ❢♦r ❙❝❤✐③♦♣❤r❡♥✐❛ ♣❛t✐❡♥ts ✇✐t❤ ▲❛tt✐❝❡ ❈♦♠♣✉t✐♥❣ ❜❛s❡❞ ❢❡❛t✉r❡s ❍❆■❙ ✷✵✶✸ ✲ ❙❛❧❛♠❛♥❝❛✱ ❙❡♣t❡♠❜❡r ✶✶✲✶✸✱ ✷✵✶✸ ✶✽ ✴ ✶✽