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SLIDE 1

◆❛t✉r❛❧ ❈♦♠♣✉t✐♥❣

▲❡❝t✉r❡ ✽ ▼✐❝❤❛❡❧ ❍❡rr♠❛♥♥ ♠❤❡rr♠❛♥❅✐♥❢✳❡❞✳❛❝✳✉❦ ♣❤♦♥❡✿ ✵✶✸✶ ✻ ✺✶✼✶✼✼ ■♥❢♦r♠❛t✐❝s ❋♦r✉♠ ✶✳✹✷ ✶✹✴✶✵✴✷✵✶✶

❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥ ✭❉❊✮ ❛♥❞

  • ❡♥❡t✐❝ Pr♦❣r❛♠♠✐♥❣ ✭●P✮
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SLIDE 2

❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥

❢♦r ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥ ❖♣t✐♠✐③❛t✐♦♥

❆❧❣♦r✐t❤♠ ❜② ❑❡♥♥❡t❤ Pr✐❝❡ ❛♥❞ ❘❛✐♥❡r ❙t♦r♥ ■♥❞✐✈✐❞✉❛❧s ❛r❡ ❝♦♥t✐♥✉♦✉s ✈❡❝t♦rs ✭❞✐r❡❝t ❡♥❝♦❞✐♥❣✮ ❯s✐♥❣ ✈❡❝t♦r ❞✐✛❡r❡♥❝❡s ❢♦r ♠✉t❛t✐♥❣ t❤❡ ✈❡❝t♦r ♣♦♣✉❧❛t✐♦♥ ✏❚❤✐s ✇❛② ♥♦ s❡♣❛r❛t❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❤❛s t♦ ❜❡ ✉s❡❞ ✇❤✐❝❤ ♠❛❦❡s t❤❡ s❝❤❡♠❡ ❝♦♠♣❧❡t❡❧② s❡❧❢✲♦r❣❛♥✐③✐♥❣✳✑ ❙♠❛❧❧ ♣♦♣✉❧❛t✐♦♥ s✐③❡ ✭✉s✉❛❧❧② ✉♣ t♦ ✹✵✮ ▼✉t❛t✐♦♥ ❛♥❞ ❝r♦ss♦✈❡r ✭♦♥❧② ❧♦♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ ❜✐♦❧♦❣✐❝❛❧ ♣✐❝t✉r❡✮

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 3

❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥

Pr♦♣❡rt✐❡s ❛♥❞ ♣❛r❛♠❡t❡rs

✏❈♦♥t♦✉r ♠❛t❝❤✐♥❣✑✿ ✈❡❝t♦r ♣♦♣✉❧❛t✐♦♥ ❛❞❛♣ts s✉❝❤ t❤❛t ♣r♦♠✐s✐♥❣ r❡❣✐♦♥s ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ s✉r❢❛❝❡ ❛r❡ ✐♥✈❡st✐❣❛t❡❞ ❛✉t♦♠❛t✐❝❛❧❧② ♦♥❝❡ t❤❡② ❛r❡ ❞❡t❡❝t❡❞ ✏❙✇❛r♠ ✐♥t❡❧❧✐❣❡♥❝❡✑✿ ■♥t❡r❛❝t✐♦♥ ❛♠♦♥❣ ✐♥❞✐✈✐❞✉❛❧s ✐s ✐♠♣♦rt❛♥t✱ s❡❡ P❙❖ ✭❧❛t❡r✮ ❚✇♦ ♣❛r❛♠❡t❡rs✿ ✇❡✐❣❤t✐♥❣ ❝♦♥st❛♥t ❋✱ ❝r♦ss♦✈❡r ❝♦♥st❛♥t ❈ ❚②♣✐❝❛❧ ✈❛❧✉❡s✿ ❋ = ✵.✺ . . . ✵.✽ ✭❋ ≥

  • (✶ − ❈/✷) /◆✮✱

❈ = ✵.✶ . . . ✵.✾ ✭✐❢ t❤❡ ♣r♦❜❧❡♠ ✐s s❡♣❛r❛❜❧❡✱ ❈ ❝❛♥ ❜❡ s♠❛❧❧❡r✮✱ ✹ < ◆ ≈ ✺❉ ❈❤❡❝❦ t❤❡ ❛♣♣❧❡t ❛t ✇✇✇✳✐❝s✐✳❜❡r❦❡❧❡②✳❡❞✉✴⑦st♦r♥✴❝♦❞❡✳❤t♠❧

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 4

❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥✿ ❆❧❣♦r✐t❤♠

P♦♣✉❧❛t✐♦♥ ♦❢ ◆ ✈❡❝t♦rs ♦❢ ❉✲❞✐♠❡♥s✐♦♥s✿ ①✐✱ ✐ = ✶, ✷, ..., ◆ ❙t❡♣ ✶✿ ✈✐ (t + ✶) = ①q (t) + ❋ · (①r (t) − ①s (t))❀ q, r, s ❛r❡ r❛♥❞♦♠ ✐♥❞❡①❡s✱ ❛❧❧ ❞✐✛❡r❡♥t ❛♥❞ ❞✐✛❡r❡♥t ❢r♦♠ ✐✳ ◆♦t❡ t❤❛t ✈✐ ❤❛s ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ①✐✦ ✭t✿ ❣❡♥❡r❛t✐♦♥ ❝♦✉♥t❡r✮ ✭■♥ ❛ s❡♥s❡✿ t❤r❡❡ ♣❛r❡♥ts✱ ❜✉t t❤✐s ✐s ❝♦♥s✐❞❡r❡❞ ❛s ♠✉t❛t✐♦♥ ✐♥ ❉❊✮ ❋ ∈ [✵, ✷] ⊂ R ✭♣♦ss✐❜❧❡ ❛♠♣❧✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ✈❛r✐❛t✐♦♥✮ ❙t❡♣ ✷✿ ❈❤♦♦s❡ r❛♥❞♦♠ ♥✉♠❜❡rs ρ❥ ∈ [✵, ✶]✱ ❥ ∈ {✶, . . . , ❉} ✉❥✐ (t + ✶) =

  • ✈❥✐ (t + ✶)

✐❢ ρ❥ ≤ ❈ ①❥✐ (t) ✐❢ ρ❥ > ❈

♦r ❜② ❝❤♦♦s✐♥❣ ❛ ❜❧♦❝❦ ❥ ∈ [♥, (♥ + ▲) ♠♦❞❉]✱ ▲ ≤ ❉✱ ✶ ≤ ♥ ≤ ❉ ✇❤❡r❡ ▲ ✐s r❛♥❞♦♠❧② ❝❤❛♥❣❡❞✳

✉✐,t+✶ = (✉✶✐ (t + ✶) , ✉✷✐ (t + ✶) , . . . , ✉❉✐ (t + ✶)) ❙❡❧❡❝t✐♦♥✿ ①✐ (t + ✶) = ✉✐ (t + ✶) ✐❢ ✉✐ (t + ✶) ✐s ❜❡tt❡r t❤❛♥ ①✐ (t)✱ ♦t❤❡r✇✐s❡ ①✐ (t + ✶) = ①✐ (t)

❘❛✐♥❡r ❙t♦r♥ ✫ ❑❡♥♥❡t❤ Pr✐❝❡ ✭✶✾✾✼✮ ❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥ ✕ ❆ ❙✐♠♣❧❡ ❛♥❞ ❊✣❝✐❡♥t ❍❡✉r✐st✐❝ ❢♦r

  • ❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥ ♦✈❡r ❈♦♥t✐♥✉♦✉s ❙♣❛❝❡s✳ ❏♦✉r♥❛❧ ♦❢ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥ ✶✶✿ ✸✹✶✕✸✺✾✳

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

slide-5
SLIDE 5

❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥

❘❛✐♥❡r ❙t♦r♥ ❛♥❞ ❑❡♥♥❡t❤ Pr✐❝❡✿ ❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥ ✲ ❆ s✐♠♣❧❡ ❛♥❞ ❡✣❝✐❡♥t ❛❞❛♣t✐✈❡ s❝❤❡♠❡ ❢♦r ❣❧♦❜❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♦✈❡r ❝♦♥t✐♥✉♦✉s s♣❛❝❡s✱ ❚❘✳ ◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 6
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SLIDE 7

❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥✿ ❇❛s✐♥✲t♦✲❇❛s✐♥ ❚r❛♥s❢❡r

P❡❛❦s ❢✉♥❝t✐♦♥ ❛✮ ❛♥❞ ✐❧❧✉str❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥❝❡ ✈❡❝t♦rs ❜✮ t❤❛t ♣r♦♠♦t❡ tr❛♥s❢❡r ♦❢ ♣♦✐♥ts ❜❡t✇❡❡♥ t✇♦ ❜❛s✐♥s ♦❢ ❛ttr❛❝t✐♦♥ ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ s✉r❢❛❝❡

❘❛✐♥❡r ❙t♦r♥ ✭✷✵✵✽✮ ❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥ ❘❡s❡❛r❝❤ ✕ ❚r❡♥❞s ❛♥❞ ❖♣❡♥ ◗✉❡st✐♦♥s✳ ❈❤❛♣t❡r ✶ ♦❢ ❯❞❛② ❑✳ ❈❤❛❦r❛❜♦rt②✿ ❆❞✈❛♥❝❡s ✐♥ ❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥ ◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 8

❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥

❊✛❡❝t ♦❢ t❤❡ ❢❛❝t♦r ❋✳ ❱❛r②✐♥❣ ❋ ✐s ❛❧s♦ ✉s❡❞ t♦ ❞✐t❤❡r ✭r❛♥❞♦♠❧② ♠✉t❛t❡✮ t❤❡ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✈❡❝t♦r

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 9

❉❊✿ ▼✉t❛t✐♦♥ ❛♥❞ ❈r♦ss♦✈❡r

❊①❛♠♣❧❡ ❢♦r ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ◆ = ✽ ♣♦✐♥ts ❛♥❞ ❛ ♠✉t❛t✐♦♥ st❡♣ ❛✮✳ ❚❤❡ ✜❣✉r❡ ♦♥ t❤❡ r✐❣❤t ❜✮ s❤♦✇s t❤❡ ♣♦t❡♥t✐❛❧ ♣♦✐♥ts ✇❤❡♥ ✉s✐♥❣ ❝r♦ss♦✈❡r✳

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

slide-10
SLIDE 10

❉❊✿ ❊①❛♠♣❧❡

  • ❡♥❡r❛t✐♦♥ t = ✶ ✉s✐♥❣ ◆ = ✽

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 11

❉❊✿ ❊①❛♠♣❧❡

  • ❡♥❡r❛t✐♦♥ t = ✶✵ ✉s✐♥❣ ◆ = ✽✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✈❡❝t♦r ❞✐str✐❜✉t✐♦♥

✭♦♥❧② ❡♥❞♣♦✐♥ts s❤♦✇♥✮ ❡①❤✐❜✐ts t❤r❡❡ ♠❛✐♥ ❝❧♦✉❞s ✇❤❡r❡ t❤❡ ♦✉t❡r ♦♥❡s ♣r♦♠♦t❡ t❤❡ tr❛♥s❢❡r ❜❡t✇❡❡♥ t✇♦ ❜❛s✐♥s ♦❢ ❛ttr❛❝t✐♦♥✳

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

slide-12
SLIDE 12

❉❊✿ ❊①❛♠♣❧❡

  • ❡♥❡r❛t✐♦♥ t = ✷✵ ✉s✐♥❣ ◆ = ✽✳ ◆♦✇ t❤❡ ❞✐✛❡r❡♥❝❡ ✈❡❝t♦r

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

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

slide-13
SLIDE 13

❉✐✛❡r❡♥t✐❛❧ ❊✈♦❧✉t✐♦♥✿ ❱❛r✐❛♥ts

▼✉t❛❜✐❧✐t② ❛♥❞ t❤r❡s❤♦❧❞ ♣❛r❛♠❡t❡rs ❝❛♥ ❛❧s♦ ❜❡ ❡✈♦❧✈❡❞ ❢♦r ❡❛❝❤ ✐♥❞✐✈✐❞✉❛❧ ✭❛s t❤❡ st❡♣ s✐③❡s ✐♥ ❊❙✮✱ ✐✳❡✳ ❞✐♠❡♥s✐♦♥ ❜❡❝♦♠❡s❉ + ✷✳ ❱❛r✐♦✉s ♠❡t❤♦❞s ❢♦r ❞✐✈❡rs✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥✿ ❏✐tt❡r✱ ❞✐t❤❡r✱ ♠✐①✐♥❣ ♣❡rt✉r❜❛t✐♦♥s✱ ♠✐rr♦r✐♥❣ ✭♦♣♣♦s✐t✐♦♥✮✱ ✳✳✳ ❙❝❤❡♠❡ ❢♦r ❞❡♥♦t✐♥❣ ❉❊ ✈❛r✐❛♥ts✿

① s♣❡❝✐✜❡s t❤❡ ✈❡❝t♦r t♦ ❜❡ ♠✉t❛t❡❞ ✇❤✐❝❤ ❝✉rr❡♥t❧② ❝❛♥ ❜❡ ✏r❛♥❞✑ ✭❛ r❛♥❞♦♠❧② ❝❤♦s❡♥ ♣♦♣✉❧❛t✐♦♥ ✈❡❝t♦r✮ ♦r ✏❜❡st✑ ✭t❤❡ ✈❡❝t♦r ♦❢ ❤✐❣❤❡st ✜t♥❡ss ❢r♦♠ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥✮ ② ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥❝❡ ✈❡❝t♦rs ✉s❡❞ ③ ❞❡♥♦t❡s t❤❡ ❝r♦ss♦✈❡r s❝❤❡♠❡✳ ❚❤❡ ❝✉rr❡♥t ✈❛r✐❛♥t ✐s ✏❜✐♥✑ ✭❝r♦ss♦✈❡r ❞✉❡ t♦ ✐♥❞❡♣❡♥❞❡♥t ❜✐♥♦♠✐❛❧ ❡①♣❡r✐♠❡♥ts✮

❡✳❣✳ ❉❊✴❜❡st✴✷✴❜✐♥ ✭♣r❡✈✐♦✉s❧② ✇❡ ❤❛❞ ❉❊✴r❛♥❞✴✶✴❜✐♥✮ ✈✐ (t + ✶) = ①❜❡st (t) + ❢ · (①♣ (t) + ①q (t) − ①r (t) − ①s (t)) ❆❧s♦ ❛ ♥✉♠❜❡r ♦❢ s❡❧❢✲❛❞❛♣t✐♥❣ ✈❛r✐❛♥ts ❡①✐st ❝❢✳ ❬❙t♦r♥✱ ✷✵✵✽❪

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 14

❊❙ ❛♥❞ ❉❊✿ ❈♦♥❝❧✉s✐♦♥

▼❡t❛❤❡✉r✐st✐❝ s❡❛r❝❤ ❛❧❣♦r✐t❤♠s ❆❞❛♣t✐✈❡ ♣❛r❛♠❡t❡rs ✐♠♣♦rt❛♥t ❘❡❧❛t✐♦♥s t♦ ●❛✉ss✐❛♥ ❛❞❛♣t❛t✐♦♥ ❇♦t❤ t❤❡ ❛❞✈❛♥❝❡❞ ✈❡rs✐♦♥s ♦❢ ❊❙ ❛♥❞ ❉❊ ❝♦♠♣❛r❡ ❢❛✈♦✉r❛❜❧② t♦ ♦t❤❡r ♠❡t❛❤❡✉r✐st✐❝ ❛❧❣♦r✐t❤♠s ✭s❡❡ ❡✳❣✳ ✇✇✇✳❧r✐✳❢r✴⑦❍❛♥s❡♥✮ ❉✐✈❡rs✐t② ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ s♦❧✉t✐♦♥s ♥❡❡❞s ♦❢t❡♥ t♦ ❜❡ ♠❛✐♥t❛✐♥❡❞ ❜② ♣r♦❜❧❡♠✲❞❡♣❡♥❞❡♥t str❛t❡❣✐❡s ✭▼✉t❛t✐♦♥s ✐♥ ❉❊✱ ❝♦♠♣❧❡① ♣♦♣✉❧❛t✐♦♥s ✐♥ ❊❙✮ ❘❡❧❛t✐♦♥ t♦ ♣❛rt✐❝❧❡ s✇❛r♠ ♦♣t✐♠✐s❛t✐♦♥ ✭P❙❖✮ ❬❧❛t❡r❪ ❙❡❡ ❛❧s♦ ✇✇✇✳s❝❤♦❧❛r♣❡❞✐❛✳♦r❣✴❛rt✐❝❧❡✴❊✈♦❧✉t✐♦♥❴str❛t❡❣✐❡s

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 15

❊✈♦❧✉t✐♦♥❛r② ❛❧❣♦r✐t❤♠s

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 16
  • ❡♥❡t✐❝ Pr♦❣r❛♠♠✐♥❣
  • ❡♥❡t✐❝ ♣r♦❣r❛♠♠✐♥❣ ♥♦✇

r♦✉t✐♥❡❧② ❞❡❧✐✈❡rs ❤✐❣❤✲r❡t✉r♥ ❤✉♠❛♥✲❝♦♠♣❡t✐t✐✈❡ ♠❛❝❤✐♥❡ ✐♥t❡❧❧✐❣❡♥❝❡✳

  • ❡♥❡t✐❝ ♣r♦❣r❛♠♠✐♥❣ ✐s ❛♥

❛✉t♦♠❛t❡❞ ✐♥✈❡♥t✐♦♥ ♠❛❝❤✐♥❡✳

  • ❡♥❡t✐❝ ♣r♦❣r❛♠♠✐♥❣ ❝❛♥

❛✉t♦♠❛t✐❝❛❧❧② ❝r❡❛t❡ ❛ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ t♦ ❛ ♣r♦❜❧❡♠ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ ♣❛r❛♠❡tr✐③❡❞ t♦♣♦❧♦❣②✳ ❈♦♠♣✉t❡r ♣r♦❣r❛♠s ❛r❡ t❤❡ ❧✐♥❣✉❛ ❢r❛♥❝❛ ❢♦r ❡①♣r❡ss✐♥❣ t❤❡ s♦❧✉t✐♦♥s t♦ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ♣r♦❜❧❡♠s ❙t❛t❡♠❡♥ts ❜② ❏♦❤♥ ❘✳ ❑♦③❛ ❡t ❛❧✳ ✭✷✵✵✸✮

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 17

❊✈♦❧✈✐♥❣ Pr♦❣r❛♠s

■s ✐t ♣♦ss✐❜❧❡ t♦ ❝r❡❛t❡ ❝♦♠♣✉t❡r ♣r♦❣r❛♠s ❜② ❡✈♦❧✉t✐♦♥❛r② ♠❡❛♥s❄ ▲❡t P(✵) ❜❡ ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ ♣r♦❣r❛♠s ♣✐ ❋♦r ❡❛❝❤ ♣✐✱ r✉♥ ✐t ♦♥ s♦♠❡ ✐♥♣✉t ❛♥❞ s❡❡ ✇❤❛t ✐t ❞♦❡s✳ ❘❛t❡ ✐t ❢♦r ✜t♥❡ss ❜❛s❡❞ ♦♥ ❤♦✇ ✇❡❧❧ ✐t ❞♦❡s✳ ❇r❡❡❞ t❤❡ ✜tt❡r ♠❡♠❜❡rs ♦❢ P(✵) t♦ ♣r♦❞✉❝❡ P(✶) ■❢ ❤❛♣♣② ✇✐t❤ t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❜❡st ♣r♦❣r❛♠ ♣r♦❞✉❝❡❞ t❤❡♥ st♦♣✳ ✳ ✳ ✳ ❜✉t ❤♦✇❄

▲✐t❡r❛t✉r❡✿ ❘✐❝❝❛r❞♦ P♦❧✐✱ ❲✐❧❧✐❛♠ ❇✳ ▲❛♥❣❞♦♥✱ ◆✐❝❤♦❧❛s ❋✳ ▼❝P❤❡❡ ✭✷✵✵✽✮ ❆ ❋✐❡❧❞ ●✉✐❞❡ t♦ ●❡♥❡t✐❝ Pr♦❣r❛♠♠✐♥❣ ✭❣♣✲✜❡❧❞✲❣✉✐❞❡✳♦r❣✳✉❦✮

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 18

❍♦✇❄

❲❤❛t ❧❛♥❣✉❛❣❡ s❤♦✉❧❞ t❤❡ ❝❛♥❞✐❞❛t❡ ♣r♦❣r❛♠s ❜❡ ❡①♣r❡ss❡❞ ✐♥❄ ❈✱ P②t❤♦♥✱ ❏❛✈❛✱ P❛s❝❛❧✱ P❡r❧✱ ▲✐s♣✱ ▼❛❝❤✐♥❡ ❝♦❞❡❄ ❍♦✇ ❝❛♥ ②♦✉ ❣❡♥❡r❛t❡ ❛♥ ✐♥✐t✐❛❧ ♣♦♣✉❧❛t✐♦♥❄ ❍♦✇ ❝❛♥ ②♦✉ r✉♥ ♣r♦❣r❛♠s s❛❢❡❧②❄ ❈♦♥s✐❞❡r ❡rr♦rs✱ ✐♥✜♥✐t❡ ❧♦♦♣s✱ ❡t❝✳❄ ❍♦✇ ❝❛♥ ②♦✉ r❛t❡ ❛ ♣r♦❣r❛♠ ❢♦r ✜t♥❡ss❄

  • ✐✈❡♥ t✇♦ s❡❧❡❝t❡❞ ♣r♦❣r❛♠s✱ ❤♦✇ ❝❛♥ t❤❡② ❜❡ ❜r❡❞ t♦ ❝r❡❛t❡

♦✛s♣r✐♥❣❄ ❲❤❛t ❛❜♦✉t s✉❜r♦✉t✐♥❡s✱ ♣r♦❝❡❞✉r❡s✱ ❞❛t❛ t②♣❡s✱ ❡♥❝❛♣s✉❧❛t✐♦♥✱ ❡t❝✳ ❲❤❛t ❛❜♦✉t s♠❛❧❧✱ ❡✣❝✐❡♥t ♣r♦❣r❛♠s❄

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 19

❑♦③❛✿ ❊✈♦❧✈✐♥❣ ▲■❙P ♣r♦❣r❛♠s

▲✐s♣ ❛s ❛ ❢✉♥❝t✐♦♥❛❧ ❧❛♥❣✉❛❣❡ ❢ (①, ②) ✐s ✇r✐tt❡♥ ❛s (❢ ① ②) ✶✵ − (✸ + ✹) ✐s ✇r✐tt❡♥ ❛s (− ✶✵ (+ ✸ ✹)) ▲✐s♣ ♣r♦❣r❛♠s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s tr❡❡s ❢ (①) = ①✷ + ✸ ❢ (①) = (+ (∗ ① ①) ✸) ❍❡r❡✱ + ❛♥❞ ∗ ❛r❡ ❢✉♥❝t✐♦♥ s②♠❜♦❧s ✭♥♦♥✲t❡r♠✐♥❛❧s✮ ♦❢ ❛r✐t②✱ ① ❛♥❞ ✸ ❛r❡ t❡r♠✐♥❛❧s✳

  • ✐✈❡♥ ❛ r❛♥❞♦♠ ❜❛❣ ♦❢

t❡r♠✐♥❛❧s ❛♥❞ ♥♦♥✲t❡r♠✐♥❛❧s✱ ✇❡ ❝❛♥ ♠❛❦❡ ♣r♦❣r❛♠s✳ ✭P❡t❡r ❙❡✐❜❡❧✿ Pr❛❝t✐❝❛❧ ❈♦♠♠♦♥ ▲✐s♣✱ ✷✵✵✹✮

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 20

❘❛♥❞♦♠ Pr♦❣r❛♠s ❛♥❞ ❈❧♦s✉r❡

■❢ ✇❡ ❣❡♥❡r❛t❡ ❛ r❛♥❞♦♠ ♣r♦❣r❛♠✿ ❍♦✇ ❝❛♥ ✇❡ ❛✈♦✐❞ ❛♥ ❡rr♦r❄ ❆♥♦t❤❡r r❛♥❞♦♠ ♣r♦❣r❛♠ ❍♦✇ ❝❛♥ ✇❡ ❡✈❛❧✉❛t❡ t❤✐s❄ ❆❧❧ ❢✉♥❝t✐♦♥ ❝❛❧❧s ♥❡❡❞ t♦ r❡t✉r♥ ❛ r❡s✉❧t ❈❧♦s✉r❡✿ ❊✳❣✳ r❡❞❡✜♥❡ ❞✐✈✐s✐♦♥ ❜② ③❡r♦ t♦ r❡t✉r♥ ❋▲❚❴▼❆❳ ♦r ③❡r♦❀ ❖✈❡r❧♦❛❞ ♦♣❡r❛t♦rs t♦ ❞❡❛❧ ✇✐t❤ ✈❛r✐❛❜❧❡ ♥✉♠❜❡rs ♦❢ ❛r❣✉♠❡♥ts ❙✉✣❝✐❡♥❝②✿❙❡t ♦❢ ♥♦♥t❡r♠✐♥❛❧s ♥❡❡❞ t♦ ❜❡ s✉✣❝✐❡♥t❧② ❧❛r❣❡✱ t❡r♠✐♥❛❧s ♥❡❡❞ t♦ ❜❡ ❞❡✜♥❡❞ ✭✐❢ ♥♦t ❣✐✈❡♥ ❜② ❞❛t❛✮

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 21
  • r❛♠♠❛r✲❇❛s❡❞ ❈♦♥str❛✐♥ts ✭s❡❡ ●❆s✮

❚②♣✐♥❣ ♦r ❣r❛♠♠❛r✲❜❛s❡❞ ❛♣♣r♦❛❝❤❡s ❤❡❧♣ t♦ ❛❝❤✐❡✈❡ ❝❧♦s✉r❡

❘✐❝❝❛r❞♦ P♦❧✐✱ ❲✐❧❧✐❛♠ ❇✳ ▲❛♥❣❞♦♥✱ ◆✐❝❤♦❧❛s ❋✳ ▼❝P❤❡❡ ✭✷✵✵✽✮ ❆ ❋✐❡❧❞ ●✉✐❞❡ t♦ ●❡♥❡t✐❝ Pr♦❣r❛♠♠✐♥❣

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 22

❋✐t♥❡ss ❝❛s❡s

❍♦✇ ❞♦ ✇❡ r❛t❡ ❛ ♣r♦❣r❛♠ ❢♦r ✜t♥❡ss❄ ❆♥s✇❡r✿ ❘✉♥ ✐t ♦♥ s♦♠❡ ✏t②♣✐❝❛❧✑ ✐♥♣✉t ❞❛t❛ ❢♦r ✇❤✐❝❤ ✇❡ ❦♥♦✇ ✇❤❛t t❤❡ ♦✉t♣✉t s❤♦✉❧❞ ❜❡✳ ❚❤❡ ❤♦♣❡ ✐s t❤❡ ❡✈♦❧✈❡❞ ♣r♦❣r❛♠ ✇✐❧❧ ✇♦r❦ ❢♦r ❛❧❧ ♦t❤❡r ❝❛s❡s ✭✉s❡ ❝r♦ss✈❛❧✐❞❛t✐♦♥✦✦✮✳ ❊①❛♠♣❧❡✿ ❙②♠❜♦❧✐❝ r❡❣r❡ss✐♦♥ ♦♥ ♣❧❛♥❡t❛r② ♦r❜✐ts ✭❑❡♣❧❡r✬s ❧❛✇✮✳ ●✐✈❡♥ ❛ s❡t ♦❢ ✈❛❧✉❡s ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✱ ❝♦♠❡ ✉♣ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❣✐✈❡s t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ✐♥ t❡r♠s ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✳ P❧❛♥❡t ❆ P ❱❡♥✉s ✵✳✼✷ ✵✳✻✶ ❊❛rt❤ ✶✳✵✵ ✶✳✵✵ ▼❛rs ✶✳✺✷ ✶✳✽✹ ❏✉♣✐t❡r ✺✳✷✵ ✶✶✳✾ ❙❛t✉r♥ ✾✳✺✸ ✷✾✳✹ ❯r❛♥✉s ✶✾✳✶ ✽✸✳✺ ❑❡♣❧❡r✬s t❤✐r❞ ❧❛✇✿ ❙q✉❛r❡ ♦❢ t❤❡ ♣❡r✐♦❞ P ♦❢ t❤❡ ♣❧❛♥❡t ♣r♦♣♦rt✐♦♥❛❧ t♦ ❝✉❜❡ ♦❢ s❡♠✐♠❛❥♦r ❛①✐s ❆✱ ✐✳❡✳ P = ❆✸/✷✳

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 23

❋✐t♥❡ss ❋✉♥❝t✐♦♥

  • ✐✈❡♥ ❛ ♥✉♠❜❡r ♦❢ ❡①❛♠♣❧❡ ♣❛✐rs (①❥, ②❥) ♦❢ ❞❛t❛ ✈❡❝t♦rs ①❥ ❛♥❞

❞❡s✐r❡❞ ♦✉t♣✉ts ②❥✱ ✇❡ ❝♦✉❧❞ ✉s❡ t❤❡ ❞❡✈✐❛t✐♦♥s ♦❢ t❤❡ ❣❡♥❡r❛t❡❞ ♣r♦❣r❛♠ ❢♦r ❛♥ ❡✈❛❧✉❛t✐♦♥✿ ❊r❛✇ =

  • P (①❥) − ②❥✷

❋♦r ❛ ✜t♥❡ss ❢✉♥❝t✐♦♥ ✇❡ ❝♦✉❧❞ ❛❞❥✉st ❋❛❞❥ = ✶ ✶ + ❊r❛✇ ❛♥❞ ♥♦r♠❛❧✐s❡ ✭♦r r❛♥❦✮ ❋♥♦r♠ (✐) = ❋❛❞❥ (✐)

  • ❦ ❋❛❞❥ (❦)

✳✳✳ s♦ ♠♦st ✜t ✐s ✶✱ ❧❡❛st ✜t ✐s ✵ ✭✐✱ ❥✱ ❦ ❛r❡ t❤❡ ✜t♥❡ss ❝❛s❡s✮

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 24

❈r♦ss♦✈❡r

❍♦✇ ❝❛♥ ✇❡ ❝r♦ss ♣r♦❣r❛♠s❄ ❙✉❜tr❡❡ ❝r♦ss♦✈❡r

❑♦③❛✬s ♦r❣✐♥❛❧ ✭✶✾✽✽✲✾✷✮ ●P s②st❡♠ ✉s❡❞ ♦♥❧② ❝r♦ss♦✈❡r✱ t♦ tr② t♦ ❞❡♠♦♥str❛t❡ t❤❛t ●P ✐s ✏♠♦r❡ t❤❛♥ ❛ ♠✉t❛t✐♦♥✑✳

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 25

▼✉t❛t✐♦♥

❍♦✇ ❝❛♥ ✇❡ ♠✉t❛t❡ ❛ ♣r♦❣r❛♠❄ ▲♦ts ♦❢ ♦t❤❡r ❢♦r♠s ♦❢ ♠✉t❛t✐♦♥ ❛r❡ ♣♦ss✐❜❧❡✱ ❡✳❣✳ ❤♦✐st✱ s❤r✐♥❦ s❤r✐♥❦✿ r❡♣❧❛❝❡ ❛ s✉❜tr❡❡ ❜② ♦♥❡ ♦❢ ✐ts t❡r♠✐♥❛❧s ❤♦✐st✿ ✉s❡ ♦♥❧② ❛ s✉❜tr❡❡ ❛s ❛ ♠✉t❛♥t s✉❜tr❡❡ ♦r ✏❣r♦✇✑✲♠✉t❛t✐♦♥ ♦r✿ ✈❛r② ♥✉♠❜❡rs✱ ❡①❝❤❛♥❣❡ s②♠❜♦❧s✱ ❡①❝❤❛♥❣❡ s✉❜tr❡❡s✱ ✳ ✳ ✳

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 26
  • P ❆❧❣♦r✐t❤♠

✶ ❈❤♦♦s❡ ❛ s❡t ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ t❡r♠✐♥❛❧s ❢♦r t❤❡ ♣r♦❣r❛♠ ②♦✉

✇❛♥t t♦ ❡✈♦❧✈❡✿

♥♦♥✲t❡r♠✐♥❛❧s ❡✳❣✳✿ ✐❢✱ ✴✱✯ ✱ ✰✱ ❂✱ sqrt✱ ❁✱ ❃✳ ✳ ✳ t❡r♠✐♥❛❧s ❡✳❣✳✿ ①✱ ② ✱ ❂✶✵✱ ❂✾✱ ✳ ✳ ✳ ✱ ✾✱ ✶✵

✷ ●❡♥❡r❛t❡ ❛♥ ✐♥✐t✐❛❧ r❛♥❞♦♠ ♣♦♣✉❧❛t✐♦♥ ♦❢ tr❡❡s ♦❢ ♠❛①✐♠✉♠

❞❡♣t❤ ❞

✸ ❈❛❧❝✉❧❛t❡ t❤❡ ✜t♥❡ss ♦❢ ❡❛❝❤ ♣r♦❣r❛♠ ✐♥ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✉s✐♥❣

t❤❡ ❝❤♦s❡♥ ✜t♥❡ss ❝❛s❡s✳

✹ ❆♣♣❧② s❡❧❡❝t✐♦♥✱ s✉❜tr❡❡ ❝r♦ss♦✈❡r ✭❛♥❞ s✉❜tr❡❡ ♠✉t❛t✐♦♥✮ t♦

❢♦r♠ ❛ ♥❡✇ ♣♦♣✉❧❛t✐♦♥✳ ❊①❛♠♣❧❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s✿ P♦♣✉❧❛t✐♦♥ s✐③❡ ❂ ✶✵✵✵✵ ❈r♦ss♦✈❡r r❛t❡ ❂ ✵✳✾ ❙❡❧❡❝t✐♦♥✿ ❋✐t♥❡ss ♣r♦♣♦rt✐♦♥❛t❡

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 27
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SLIDE 28

❊①❛♠♣❧❡✿ ●P ❢♦r ❙②♠❜♦❧✐❝ ❘❡❣r❡ss✐♦♥

❉❛t❛✿ ① ② ✲✶ ✶✳✵✵ ✲✵✳✽ ✵✳✽✹ ✲✵✳✻ ✵✳✼✻ ✲✵✳✹ ✵✳✼✻ ✲✵✳✷ ✵✳✽✹ ✵✳✵ ✶✳✵✵ ✵✳✷ ✶✳✷✹ ✵✳✹ ✶✳✺✻ ✵✳✻ ✶✳✾✻ ✵✳✽ ✷✳✹✹ ✶ ✸✳✵✵ ❉❡s✐❣♥✿

❋r♦♠✿ ❏✳ ❘✳ ❑♦③❛✿ ●❆ ❛♥❞ ●P ✭t✉t♦r✐❛❧✮

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 29

❊①❛♠♣❧❡✿ ●P ❢♦r ❙②♠❜♦❧✐❝ ❘❡❣r❡ss✐♦♥

P♦♣✉❧❛t✐♦♥ ♦❢ ✹ r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ ✐♥❞✐✈✐❞✉❛❧s ❢♦r ❣❡♥❡r❛t✐♦♥ ✵

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 30

❊①❛♠♣❧❡✿ ●P ❢♦r ❙②♠❜♦❧✐❝ ❘❡❣r❡ss✐♦♥

  • ♦❛❧ ❢✉♥❝t✐♦♥✿ ❢ (①) = ①✷ + ① + ✶

P❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ♣r♦❣r❛♠s ① + ✶ ✵.✻✼ ①✷ + ✶ ✶.✵✵ ✷ ✶.✼✵ ① ✷.✻✼ ❚❤❡ ❛❧❣♦r✐t❤♠ ✉s❡s ♦♥❧② t❤❡ ✜t♥❡ss ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ t❤❡ ❣✐✈❡♥ ❞❛t❛✳

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 31

❊①❛♠♣❧❡✿ ●P ❢♦r ❙②♠❜♦❧✐❝ ❘❡❣r❡ss✐♦♥

❈♦♣② ♦❢ ✭❛✮ ▼✉t❛♥t ♦❢ ✭❝✮ ♣✐❝❦✐♥❣ ✏✷✑ ❛s ♠✉t❛t✐♦♥ ♣♦✐♥t

❋✐rst ♦✛s♣r✐♥❣ ♦❢ ❝r♦ss♦✈❡r ♦❢ ✭❛✮ ❛♥❞ ✭❜✮ ♣✐❝❦✐♥❣ ✏✰✑ ♦❢ ♣❛r❡♥t ✭❛✮ ❛♥❞ ❧❡❢t✲♠♦st ✏①✑ ♦❢ ♣❛r❡♥t ✭❜✮ ❛s ❝r♦ss♦✈❡r ♣♦✐♥ts ❙❡❝♦♥❞ ♦✛s♣r✐♥❣ ♦❢ ❝r♦ss♦✈❡r ♦❢ ✭❛✮ ❛♥❞ ✭❜✮ ♣✐❝❦✐♥❣ ✏✰✑ ♦❢ ♣❛r❡♥t ✭❛✮ ❛♥❞ ❧❡❢t✲♠♦st ✏①✑ ♦❢ ♣❛r❡♥t ✭❜✮ ❛s ❝r♦ss♦✈❡r ♣♦✐♥ts

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥

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SLIDE 32
  • P✿ ❆♣♣❧✐❝❛t✐♦♥ ❆r❡❛s

Pr♦❜❧❡♠ ❛r❡❛s ✐♥✈♦❧✈✐♥❣ ♠❛♥② ✈❛r✐❛❜❧❡s t❤❛t ❛r❡ ✐♥t❡rr❡❧❛t❡❞ ✐♥ ❛ ♥♦♥✲❧✐♥❡❛r ♦r ✉♥❦♥♦✇♥ ✇❛② ❆ ❣♦♦❞ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ✐s s❛t✐s❢❛❝t♦r②

❞❡s✐❣♥✱ ❝♦♥tr♦❧✱ ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ♣❛tt❡r♥ r❡❝♦❣♥✐t✐♦♥✱ ❞❛t❛ ♠✐♥✐♥❣✱ s②st❡♠ ✐❞❡♥t✐✜❝❛t✐♦♥ ❛♥❞ ❢♦r❡❝❛st✐♥❣

❉✐s❝♦✈❡r② ♦❢ t❤❡ s✐③❡ ❛♥❞ s❤❛♣❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ ✐s ❛ ♠❛❥♦r ♣❛rt ♦❢ t❤❡ ♣r♦❜❧❡♠ ❆r❡❛s ✇❤❡r❡ ❤✉♠❛♥s ✜♥❞ ✐t ❞✐✣❝✉❧t t♦ ✇r✐t❡

♣r♦❣r❛♠s ♣❛r❛❧❧❡❧ ❝♦♠♣✉t❡rs✱ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t❛✱ ♠✉❧t✐✲❛❣❡♥t str❛t❡❣✐❡s✴❞✐str✐❜✉t❡❞ ❆■✱ ❋P●❆s

✧❇❧❛❝❦ ❛rt✧ ♣r♦❜❧❡♠s

s②♥t❤❡s✐s ♦❢ t♦♣♦❧♦❣② ❛♥❞ s✐③✐♥❣ ♦❢ ❛♥❛❧♦❣ ❝✐r❝✉✐ts✱ s②♥t❤❡s✐s ♦❢ t♦♣♦❧♦❣② ❛♥❞ t✉♥✐♥❣ ♦❢ ❝♦♥tr♦❧❧❡rs✱ q✉❛♥t✉♠ ❝♦♠♣✉t✐♥❣ ❝✐r❝✉✐ts

❆r❡❛s ✇❤❡r❡ ②♦✉ s✐♠♣❧② ❤❛✈❡ ♥♦ ✐❞❡❛ ❤♦✇ t♦ ♣r♦❣r❛♠ ❛ s♦❧✉t✐♦♥✱ ❜✉t ✇❤❡r❡ t❤❡ ♦❜❥❡❝t✐✈❡ ✭✜t♥❡ss ♠❡❛s✉r❡✮ ✐s ❝❧❡❛r ❆r❡❛s ✇❤❡r❡ ❧❛r❣❡ ❝♦♠♣✉t❡r✐③❡❞ ❞❛t❛❜❛s❡s ❛r❡ ❛❝❝✉♠✉❧❛t✐♥❣ ❛♥❞ ❝♦♠♣✉t❡r✐③❡❞ t❡❝❤♥✐q✉❡s ❛r❡ ♥❡❡❞❡❞ t♦ ❛♥❛❧②③❡ t❤❡ ❞❛t❛

◆❆❚✵✽ ✶✹✴✶✵✴✷✵✶✶ ❏✳ ▼✳ ❍❡rr♠❛♥♥