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r t Prt - - PowerPoint PPT Presentation

Jul 27, 2023 166 likes •846 views

r t Prt tr Prt tt sr s s Ptr st

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SLIDE 1

❆ ◆❡✇ ❆♣♣r♦❛❝❤ t♦ Pr❛❝t✐❝❛❧ ❆❝t✐✈❡✲❙❡❝✉r❡ ❚✇♦✲P❛rt② ❈♦♠♣✉t❛t✐♦♥

❏❡s♣❡r ❇✉✉s ◆✐❡❧s❡♥✶✱ P❡t❡r ❙❡❜❛st✐❛♥ ◆♦r❞❤♦❧t✶✱ ❈❧❛✉❞✐♦ ❖r❧❛♥❞✐✶✱ ❙❛✐ ❙❤❡s❤❛♥❦ ❇✉rr❛✷

✶❆❛r❤✉s ❯♥✐✈❡rs✐t②✱ ❉❡♥♠❛r❦ ✷■♥❞✐❛♥ ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣② ●✉✇❛❤❛t✐

❆✉❣✉st ✷✶✱ ✷✵✶✷

✶ ✴ ✶✾

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SLIDE 2

❆❝t✐✈❡✲❙❡❝✉r❡ ❚✇♦✲P❛rt② ❈♦♠♣✉t❛t✐♦♥ ✭✷P❈✮

① ②

❈

③ ① ③ ② ③ ❈♦rr❡❝t♥❡ss✿ ❈ ① ② ③ Pr✐✈❛❝②✿ ■♥♣✉ts ❛r❡ ❦❡♣t ♣r✐✈❛t❡✳ Pr❛❝t✐❝❛❧✿ ❘✉♥s ✐♥ r❡❛s♦♥❛❜❧❡ t✐♠❡ ❢♦r r❡❛s♦♥❛❜❧❡ s✐③❡ ❝✐r❝✉✐ts✳

✷ ✴ ✶✾

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SLIDE 3

❆❝t✐✈❡✲❙❡❝✉r❡ ❚✇♦✲P❛rt② ❈♦♠♣✉t❛t✐♦♥ ✭✷P❈✮

① ②

❈

③ ① ③ ② ③ ❈♦rr❡❝t♥❡ss✿ ❈ ① ② ③ Pr✐✈❛❝②✿ ■♥♣✉ts ❛r❡ ❦❡♣t ♣r✐✈❛t❡✳ Pr❛❝t✐❝❛❧✿ ❘✉♥s ✐♥ r❡❛s♦♥❛❜❧❡ t✐♠❡ ❢♦r r❡❛s♦♥❛❜❧❡ s✐③❡ ❝✐r❝✉✐ts✳

✷ ✴ ✶✾

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SLIDE 4

❆❝t✐✈❡✲❙❡❝✉r❡ ❚✇♦✲P❛rt② ❈♦♠♣✉t❛t✐♦♥ ✭✷P❈✮

① ②

❈

③ ① ③ ② ③

◮ ❈♦rr❡❝t♥❡ss✿ ❈(①, ②) = ③

Pr✐✈❛❝②✿ ■♥♣✉ts ❛r❡ ❦❡♣t ♣r✐✈❛t❡✳ Pr❛❝t✐❝❛❧✿ ❘✉♥s ✐♥ r❡❛s♦♥❛❜❧❡ t✐♠❡ ❢♦r r❡❛s♦♥❛❜❧❡ s✐③❡ ❝✐r❝✉✐ts✳

✷ ✴ ✶✾

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SLIDE 5

❆❝t✐✈❡✲❙❡❝✉r❡ ❚✇♦✲P❛rt② ❈♦♠♣✉t❛t✐♦♥ ✭✷P❈✮

① ②

❈

③ ① ③ ② ③

◮ ❈♦rr❡❝t♥❡ss✿ ❈(①, ②) = ③ ◮ Pr✐✈❛❝②✿ ■♥♣✉ts ❛r❡ ❦❡♣t ♣r✐✈❛t❡✳

Pr❛❝t✐❝❛❧✿ ❘✉♥s ✐♥ r❡❛s♦♥❛❜❧❡ t✐♠❡ ❢♦r r❡❛s♦♥❛❜❧❡ s✐③❡ ❝✐r❝✉✐ts✳

✷ ✴ ✶✾

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SLIDE 6

❆❝t✐✈❡✲❙❡❝✉r❡ ❚✇♦✲P❛rt② ❈♦♠♣✉t❛t✐♦♥ ✭✷P❈✮

① ②

❈

③ ① ③ ② ③

◮ ❈♦rr❡❝t♥❡ss✿ ❈(①, ②) = ③ ◮ Pr✐✈❛❝②✿ ■♥♣✉ts ❛r❡ ❦❡♣t ♣r✐✈❛t❡✳

Pr❛❝t✐❝❛❧✿ ❘✉♥s ✐♥ r❡❛s♦♥❛❜❧❡ t✐♠❡ ❢♦r r❡❛s♦♥❛❜❧❡ s✐③❡ ❝✐r❝✉✐ts✳

✷ ✴ ✶✾

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SLIDE 7

❆❝t✐✈❡✲❙❡❝✉r❡ ❚✇♦✲P❛rt② ❈♦♠♣✉t❛t✐♦♥ ✭✷P❈✮

① ②

❈

③ ① ③ ② ③

◮ ❈♦rr❡❝t♥❡ss✿ ❈(①, ②) = ③ ◮ Pr✐✈❛❝②✿ ■♥♣✉ts ❛r❡ ❦❡♣t ♣r✐✈❛t❡✳ ◮ Pr❛❝t✐❝❛❧✿ ❘✉♥s ✐♥ r❡❛s♦♥❛❜❧❡ t✐♠❡ ❢♦r r❡❛s♦♥❛❜❧❡ s✐③❡ ❝✐r❝✉✐ts✳

✷ ✴ ✶✾

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SLIDE 8

▼♦t✐✈❛t✐♦♥ ❢♦r t❤✐s ❲♦r❦

◮ ❙♦❧✈✐♥❣ r❡❛❧✲✇♦r❧❞ ♣r♦❜❧❡♠s✳ ❊✳❣✳ ❝♦♠♣✉t✐♥❣ ♦✉t❝♦♠❡ ♦❢

❛✉❝t✐♦♥s ❬❇❈❉✰✵✾❪✳ ▲❛❝❦ ♦❢ ❞✐✈❡rs✐t② ✐♥ ♣r❛❝t✐❝❛❧ ✷P❈✳ ■♥ ❢❛❝t ❛❧❧ ♣r❡✈✐♦✉s ♣r❛❝t✐❝❛❧ ❛♣♣r♦❛❝❤❡s ✉s❡s ❨❛♦✬s ●❛r❜❧❡❞ ❈✐r❝✉✐ts t❡❝❤♥✐q✉❡✳

✸ ✴ ✶✾

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SLIDE 9

▼♦t✐✈❛t✐♦♥ ❢♦r t❤✐s ❲♦r❦

◮ ❙♦❧✈✐♥❣ r❡❛❧✲✇♦r❧❞ ♣r♦❜❧❡♠s✳ ❊✳❣✳ ❝♦♠♣✉t✐♥❣ ♦✉t❝♦♠❡ ♦❢

❛✉❝t✐♦♥s ❬❇❈❉✰✵✾❪✳

◮ ▲❛❝❦ ♦❢ ❞✐✈❡rs✐t② ✐♥ ♣r❛❝t✐❝❛❧ ✷P❈✳ ■♥ ❢❛❝t ❛❧❧ ♣r❡✈✐♦✉s ♣r❛❝t✐❝❛❧

❛♣♣r♦❛❝❤❡s ✉s❡s ❨❛♦✬s ●❛r❜❧❡❞ ❈✐r❝✉✐ts t❡❝❤♥✐q✉❡✳

✸ ✴ ✶✾

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SLIDE 10

❖✉r ❛♣♣r♦❛❝❤

❇✉✐❧❞✐♥❣ ❜❧♦❝❦s

◮ P❛ss✐✈❡✲s❡❝✉r❡ ✷P❈✿ ❚❤❡ ♣r♦t♦❝♦❧ ♦❢ ❬●▼❲✽✼❪ ❤❡❛✈✐❧② ✉t✐❧✐③✐♥❣

❖❜❧✐✈✐♦✉s ❚r❛♥s❢❡r ✭❖❚✮✳ ■♥❢♦r♠❛t✐♦♥ t❤❡♦r❡t✐❝ ▼❆❈s✿ ❚♦ ❡♥s✉r❡ ❛❝t✐✈❡ s❡❝✉r✐t②✳ ❖❚✲❡①t❡♥s✐♦♥✿ ❆ ❤✉❣❡ ❛♠♦✉♥t ♦❢ ❖❚ ❛t ❧♦✇ ❛♠♦rt✐③❡❞ ❝♦st ❢r♦♠ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢ ❬■❑◆P✵✸❪✳

✹ ✴ ✶✾

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SLIDE 11

❖✉r ❛♣♣r♦❛❝❤

❇✉✐❧❞✐♥❣ ❜❧♦❝❦s

◮ P❛ss✐✈❡✲s❡❝✉r❡ ✷P❈✿ ❚❤❡ ♣r♦t♦❝♦❧ ♦❢ ❬●▼❲✽✼❪ ❤❡❛✈✐❧② ✉t✐❧✐③✐♥❣

❖❜❧✐✈✐♦✉s ❚r❛♥s❢❡r ✭❖❚✮✳

◮ ■♥❢♦r♠❛t✐♦♥ t❤❡♦r❡t✐❝ ▼❆❈s✿ ❚♦ ❡♥s✉r❡ ❛❝t✐✈❡ s❡❝✉r✐t②✳

❖❚✲❡①t❡♥s✐♦♥✿ ❆ ❤✉❣❡ ❛♠♦✉♥t ♦❢ ❖❚ ❛t ❧♦✇ ❛♠♦rt✐③❡❞ ❝♦st ❢r♦♠ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢ ❬■❑◆P✵✸❪✳

✹ ✴ ✶✾

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SLIDE 12

❖✉r ❛♣♣r♦❛❝❤

❇✉✐❧❞✐♥❣ ❜❧♦❝❦s

◮ P❛ss✐✈❡✲s❡❝✉r❡ ✷P❈✿ ❚❤❡ ♣r♦t♦❝♦❧ ♦❢ ❬●▼❲✽✼❪ ❤❡❛✈✐❧② ✉t✐❧✐③✐♥❣

❖❜❧✐✈✐♦✉s ❚r❛♥s❢❡r ✭❖❚✮✳

◮ ■♥❢♦r♠❛t✐♦♥ t❤❡♦r❡t✐❝ ▼❆❈s✿ ❚♦ ❡♥s✉r❡ ❛❝t✐✈❡ s❡❝✉r✐t②✳ ◮ ❖❚✲❡①t❡♥s✐♦♥✿ ❆ ❤✉❣❡ ❛♠♦✉♥t ♦❢ ❖❚ ❛t ❧♦✇ ❛♠♦rt✐③❡❞ ❝♦st

❢r♦♠ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢ ❬■❑◆P✵✸❪✳

✹ ✴ ✶✾

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SLIDE 13

❖✉r ❘❡s✉❧ts

◮ ◆❡✇ ❖❚✲❡①t❡♥s✐♦♥ t❡❝❤♥✐q✉❡ ✇✐t❤ ❛❝t✐✈❡ s❡❝✉r✐t②✿

❖♥❧② ❛ ❢❛❝t♦r ✷ s❧♦✇❡r t❤❛♥ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢ ❬■❑◆P✵✸❪✳ ✭◆♦ ❛s②♠♣t♦t✐❝ ✐♠♣r♦✈❡♠❡♥t✮✳ ■♠♣❧❡♠❡♥ts ✺✵✵ ✵✵✵ ❖❚✴s❡❝ ♣r✳ ❝♦r❡ ✭✈s✳ ✶✵✵✵ ❖❚✴s❡❝ ✇✐t❤♦✉t ❡①t❡♥s✐♦♥✮✳

◆❡✇ ♣r❛❝t✐❝❛❧ ✷P❈ ♣r♦t♦❝♦❧✿

❯❈ s❡❝✉r❡ ❛❣❛✐♥st ❛♥ ❛❝t✐✈❡ ❛♥❞ st❛t✐❝ ❛❞✈❡rs❛r② ✐♥ t❤❡ ❘❛♥❞♦♠ ❖r❛❝❧❡ ♠♦❞❡❧✳ ■♠♣❧❡♠❡♥ts ✷✵ ✵✵✵ ❣❛t❡s✴s❡❝ ✭♦♥❧✐♥❡ ✶ ✵✵✵ ✵✵✵ ❣❛t❡s✴s❡❝✮✳ ❋❛st❡r t❤❛♥ ❛❧❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❜❛s❡❞ ♦♥ ●❛r❜❧❡❞ ❈✐r❝✉✐ts ✳ ✳ ✳ ❡①❝❡♣t ❢♦r ❬❑s❙✶✷❪✳

✺ ✴ ✶✾

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SLIDE 14

❖✉r ❘❡s✉❧ts

◮ ◆❡✇ ❖❚✲❡①t❡♥s✐♦♥ t❡❝❤♥✐q✉❡ ✇✐t❤ ❛❝t✐✈❡ s❡❝✉r✐t②✿

◮ ❖♥❧② ❛ ❢❛❝t♦r ✷ s❧♦✇❡r t❤❛♥ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢

❬■❑◆P✵✸❪✳ ✭◆♦ ❛s②♠♣t♦t✐❝ ✐♠♣r♦✈❡♠❡♥t✮✳ ■♠♣❧❡♠❡♥ts ✺✵✵ ✵✵✵ ❖❚✴s❡❝ ♣r✳ ❝♦r❡ ✭✈s✳ ✶✵✵✵ ❖❚✴s❡❝ ✇✐t❤♦✉t ❡①t❡♥s✐♦♥✮✳

◆❡✇ ♣r❛❝t✐❝❛❧ ✷P❈ ♣r♦t♦❝♦❧✿

❯❈ s❡❝✉r❡ ❛❣❛✐♥st ❛♥ ❛❝t✐✈❡ ❛♥❞ st❛t✐❝ ❛❞✈❡rs❛r② ✐♥ t❤❡ ❘❛♥❞♦♠ ❖r❛❝❧❡ ♠♦❞❡❧✳ ■♠♣❧❡♠❡♥ts ✷✵ ✵✵✵ ❣❛t❡s✴s❡❝ ✭♦♥❧✐♥❡ ✶ ✵✵✵ ✵✵✵ ❣❛t❡s✴s❡❝✮✳ ❋❛st❡r t❤❛♥ ❛❧❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❜❛s❡❞ ♦♥ ●❛r❜❧❡❞ ❈✐r❝✉✐ts ✳ ✳ ✳ ❡①❝❡♣t ❢♦r ❬❑s❙✶✷❪✳

✺ ✴ ✶✾

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SLIDE 15

❖✉r ❘❡s✉❧ts

◮ ◆❡✇ ❖❚✲❡①t❡♥s✐♦♥ t❡❝❤♥✐q✉❡ ✇✐t❤ ❛❝t✐✈❡ s❡❝✉r✐t②✿

◮ ❖♥❧② ❛ ❢❛❝t♦r ✷ s❧♦✇❡r t❤❛♥ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢

❬■❑◆P✵✸❪✳ ✭◆♦ ❛s②♠♣t♦t✐❝ ✐♠♣r♦✈❡♠❡♥t✮✳

◮ ■♠♣❧❡♠❡♥ts ∼ ✺✵✵.✵✵✵ ❖❚✴s❡❝ ♣r✳ ❝♦r❡ ✭✈s✳ ∼ ✶✵✵✵ ❖❚✴s❡❝

✇✐t❤♦✉t ❡①t❡♥s✐♦♥✮✳

◆❡✇ ♣r❛❝t✐❝❛❧ ✷P❈ ♣r♦t♦❝♦❧✿

❯❈ s❡❝✉r❡ ❛❣❛✐♥st ❛♥ ❛❝t✐✈❡ ❛♥❞ st❛t✐❝ ❛❞✈❡rs❛r② ✐♥ t❤❡ ❘❛♥❞♦♠ ❖r❛❝❧❡ ♠♦❞❡❧✳ ■♠♣❧❡♠❡♥ts ✷✵ ✵✵✵ ❣❛t❡s✴s❡❝ ✭♦♥❧✐♥❡ ✶ ✵✵✵ ✵✵✵ ❣❛t❡s✴s❡❝✮✳ ❋❛st❡r t❤❛♥ ❛❧❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❜❛s❡❞ ♦♥ ●❛r❜❧❡❞ ❈✐r❝✉✐ts ✳ ✳ ✳ ❡①❝❡♣t ❢♦r ❬❑s❙✶✷❪✳

✺ ✴ ✶✾

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SLIDE 16

❖✉r ❘❡s✉❧ts

◮ ◆❡✇ ❖❚✲❡①t❡♥s✐♦♥ t❡❝❤♥✐q✉❡ ✇✐t❤ ❛❝t✐✈❡ s❡❝✉r✐t②✿

◮ ❖♥❧② ❛ ❢❛❝t♦r ✷ s❧♦✇❡r t❤❛♥ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢

❬■❑◆P✵✸❪✳ ✭◆♦ ❛s②♠♣t♦t✐❝ ✐♠♣r♦✈❡♠❡♥t✮✳

◮ ■♠♣❧❡♠❡♥ts ∼ ✺✵✵.✵✵✵ ❖❚✴s❡❝ ♣r✳ ❝♦r❡ ✭✈s✳ ∼ ✶✵✵✵ ❖❚✴s❡❝

✇✐t❤♦✉t ❡①t❡♥s✐♦♥✮✳

◮ ◆❡✇ ♣r❛❝t✐❝❛❧ ✷P❈ ♣r♦t♦❝♦❧✿

❯❈ s❡❝✉r❡ ❛❣❛✐♥st ❛♥ ❛❝t✐✈❡ ❛♥❞ st❛t✐❝ ❛❞✈❡rs❛r② ✐♥ t❤❡ ❘❛♥❞♦♠ ❖r❛❝❧❡ ♠♦❞❡❧✳ ■♠♣❧❡♠❡♥ts ✷✵ ✵✵✵ ❣❛t❡s✴s❡❝ ✭♦♥❧✐♥❡ ✶ ✵✵✵ ✵✵✵ ❣❛t❡s✴s❡❝✮✳ ❋❛st❡r t❤❛♥ ❛❧❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❜❛s❡❞ ♦♥ ●❛r❜❧❡❞ ❈✐r❝✉✐ts ✳ ✳ ✳ ❡①❝❡♣t ❢♦r ❬❑s❙✶✷❪✳

✺ ✴ ✶✾

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SLIDE 17

❖✉r ❘❡s✉❧ts

◮ ◆❡✇ ❖❚✲❡①t❡♥s✐♦♥ t❡❝❤♥✐q✉❡ ✇✐t❤ ❛❝t✐✈❡ s❡❝✉r✐t②✿

◮ ❖♥❧② ❛ ❢❛❝t♦r ✷ s❧♦✇❡r t❤❛♥ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢

❬■❑◆P✵✸❪✳ ✭◆♦ ❛s②♠♣t♦t✐❝ ✐♠♣r♦✈❡♠❡♥t✮✳

◮ ■♠♣❧❡♠❡♥ts ∼ ✺✵✵.✵✵✵ ❖❚✴s❡❝ ♣r✳ ❝♦r❡ ✭✈s✳ ∼ ✶✵✵✵ ❖❚✴s❡❝

✇✐t❤♦✉t ❡①t❡♥s✐♦♥✮✳

◮ ◆❡✇ ♣r❛❝t✐❝❛❧ ✷P❈ ♣r♦t♦❝♦❧✿

◮ ❯❈ s❡❝✉r❡ ❛❣❛✐♥st ❛♥ ❛❝t✐✈❡ ❛♥❞ st❛t✐❝ ❛❞✈❡rs❛r② ✐♥ t❤❡

❘❛♥❞♦♠ ❖r❛❝❧❡ ♠♦❞❡❧✳ ■♠♣❧❡♠❡♥ts ✷✵ ✵✵✵ ❣❛t❡s✴s❡❝ ✭♦♥❧✐♥❡ ✶ ✵✵✵ ✵✵✵ ❣❛t❡s✴s❡❝✮✳ ❋❛st❡r t❤❛♥ ❛❧❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❜❛s❡❞ ♦♥ ●❛r❜❧❡❞ ❈✐r❝✉✐ts ✳ ✳ ✳ ❡①❝❡♣t ❢♦r ❬❑s❙✶✷❪✳

✺ ✴ ✶✾

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SLIDE 18

❖✉r ❘❡s✉❧ts

◮ ◆❡✇ ❖❚✲❡①t❡♥s✐♦♥ t❡❝❤♥✐q✉❡ ✇✐t❤ ❛❝t✐✈❡ s❡❝✉r✐t②✿

◮ ❖♥❧② ❛ ❢❛❝t♦r ✷ s❧♦✇❡r t❤❛♥ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢

❬■❑◆P✵✸❪✳ ✭◆♦ ❛s②♠♣t♦t✐❝ ✐♠♣r♦✈❡♠❡♥t✮✳

◮ ■♠♣❧❡♠❡♥ts ∼ ✺✵✵.✵✵✵ ❖❚✴s❡❝ ♣r✳ ❝♦r❡ ✭✈s✳ ∼ ✶✵✵✵ ❖❚✴s❡❝

✇✐t❤♦✉t ❡①t❡♥s✐♦♥✮✳

◮ ◆❡✇ ♣r❛❝t✐❝❛❧ ✷P❈ ♣r♦t♦❝♦❧✿

◮ ❯❈ s❡❝✉r❡ ❛❣❛✐♥st ❛♥ ❛❝t✐✈❡ ❛♥❞ st❛t✐❝ ❛❞✈❡rs❛r② ✐♥ t❤❡

❘❛♥❞♦♠ ❖r❛❝❧❡ ♠♦❞❡❧✳

◮ ■♠♣❧❡♠❡♥ts ✷✵.✵✵✵ ❣❛t❡s✴s❡❝ ✭♦♥❧✐♥❡ ∼ ✶.✵✵✵.✵✵✵ ❣❛t❡s✴s❡❝✮✳

❋❛st❡r t❤❛♥ ❛❧❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❜❛s❡❞ ♦♥ ●❛r❜❧❡❞ ❈✐r❝✉✐ts ✳ ✳ ✳ ❡①❝❡♣t ❢♦r ❬❑s❙✶✷❪✳

✺ ✴ ✶✾

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SLIDE 19

❖✉r ❘❡s✉❧ts

◮ ◆❡✇ ❖❚✲❡①t❡♥s✐♦♥ t❡❝❤♥✐q✉❡ ✇✐t❤ ❛❝t✐✈❡ s❡❝✉r✐t②✿

◮ ❖♥❧② ❛ ❢❛❝t♦r ✷ s❧♦✇❡r t❤❛♥ t❤❡ ♣❛ss✐✈❡✲s❡❝✉r❡ ♣r♦t♦❝♦❧ ♦❢

❬■❑◆P✵✸❪✳ ✭◆♦ ❛s②♠♣t♦t✐❝ ✐♠♣r♦✈❡♠❡♥t✮✳

◮ ■♠♣❧❡♠❡♥ts ∼ ✺✵✵.✵✵✵ ❖❚✴s❡❝ ♣r✳ ❝♦r❡ ✭✈s✳ ∼ ✶✵✵✵ ❖❚✴s❡❝

✇✐t❤♦✉t ❡①t❡♥s✐♦♥✮✳

◮ ◆❡✇ ♣r❛❝t✐❝❛❧ ✷P❈ ♣r♦t♦❝♦❧✿

◮ ❯❈ s❡❝✉r❡ ❛❣❛✐♥st ❛♥ ❛❝t✐✈❡ ❛♥❞ st❛t✐❝ ❛❞✈❡rs❛r② ✐♥ t❤❡

❘❛♥❞♦♠ ❖r❛❝❧❡ ♠♦❞❡❧✳

◮ ■♠♣❧❡♠❡♥ts ✷✵.✵✵✵ ❣❛t❡s✴s❡❝ ✭♦♥❧✐♥❡ ∼ ✶.✵✵✵.✵✵✵ ❣❛t❡s✴s❡❝✮✳ ◮ ❋❛st❡r t❤❛♥ ❛❧❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❜❛s❡❞ ♦♥ ●❛r❜❧❡❞ ❈✐r❝✉✐ts

✳ ✳ ✳ ❡①❝❡♣t ❢♦r ❬❑s❙✶✷❪✳

✺ ✴ ✶✾

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SLIDE 20

❖✈❡r✈✐❡✇

Pr♦t♦❝♦❧ ❖✈❡r✈✐❡✇ ▼❆❈s ❈♦♥❝❧✉❞✐♥❣

✻ ✴ ✶✾

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SLIDE 21

P❛ss✐✈❡✲❙❡❝✉r❡ ✷P❈ ❬●▼❲✽✼❪

① ② ③ ③❇

✼ ✴ ✶✾

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SLIDE 22

P❛ss✐✈❡✲❙❡❝✉r❡ ✷P❈ ❬●▼❲✽✼❪

① = ①❆ ⊕ ①❇ ② = ②❆ ⊕ ②❇ ③ ①❆, ②❆ ①❇, ②❇ ③❇

✼ ✴ ✶✾

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SLIDE 23

P❛ss✐✈❡✲❙❡❝✉r❡ ✷P❈ ❬●▼❲✽✼❪

① = ①❆ ⊕ ①❇ ② = ②❆ ⊕ ②❇ ③ ✇ ①❆, ②❆ ①❇, ②❇ ③❇

✼ ✴ ✶✾

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SLIDE 24

P❛ss✐✈❡✲❙❡❝✉r❡ ✷P❈ ❬●▼❲✽✼❪

① = ①❆ ⊕ ①❇ ② = ②❆ ⊕ ②❇ ③ ✇ = ✇❆ ⊕ ✇❇ ①❆, ②❆ ✇❆ ①❇, ②❇ ✇❇ ③❇

✼ ✴ ✶✾

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SLIDE 25

P❛ss✐✈❡✲❙❡❝✉r❡ ✷P❈ ❬●▼❲✽✼❪

① = ①❆ ⊕ ①❇ ② = ②❆ ⊕ ②❇ ③ = ③❆ ⊕ ③❇ ✇ = ✇❆ ⊕ ✇❇ ①❆, ②❆ ✇❆ ③❆ ①❇, ②❇ ✇❇ ③❇ ③❇

✼ ✴ ✶✾

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SLIDE 26

P❛ss✐✈❡✲❙❡❝✉r❡ ✷P❈ ❬●▼❲✽✼❪

① = ①❆ ⊕ ①❇ ② = ②❆ ⊕ ②❇ ③ = ③❆ ⊕ ③❇ ✇ = ✇❆ ⊕ ✇❇ ①❆, ②❆ ✇❆ ③❆ ①❇, ②❇ ✇❇ ③❇ ③❇

✼ ✴ ✶✾

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SLIDE 27

❆❝t✐✈❡✲❙❡❝✉r❡ ✷P❈

① ② ③ ✇ ①❆, ②❆ ✇❆ ③❆ ①❇, ②❇ ✇❇ ③❇ ③❇ ❆❞❞ ♠❡ss❛❣❡ ❛✉t❤❡♥t✐❝❛t✐♦♥ ❝♦❞❡s ✭▼❆❈s✮

✽ ✴ ✶✾

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SLIDE 28

❆❝t✐✈❡✲❙❡❝✉r❡ ✷P❈

① ② ③ ✇ ①❆, ②❆ ▼①❆, ▼②❆ ✇❆, ▼✇❆ ③❆, ▼③❆ ①❇, ②❇ ▼①❇, ▼②❇ ✇❇, ▼✇❇ ③❇, ▼③❇ ③❇, ▼③❇ ❆❞❞ ♠❡ss❛❣❡ ❛✉t❤❡♥t✐❝❛t✐♦♥ ❝♦❞❡s ✭▼❆❈s✮

✽ ✴ ✶✾

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SLIDE 29

❚❤❡ Pr❡♣r♦❝❡ss✐♥❣ ♠♦❞❡❧

Pr❡♣r♦❝❡ss✐♥❣✿ Pr❡♣❛r❡ r❛♥❞♦♠ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✿

✶ ♣❡r ■♥♣✉t✲❣❛t❡✳ ✶✻❇ ♣❡r ❆◆❉✲❣❛t❡✱ ❢♦r s❡❝✉r✐t② ✷

❇ ❧♦❣ ❈ ✳

Pr♦✈❡ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s ✭t❤✐♥❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✲tr✐♣❧❡s✮✳ ❖♥❧✐♥❡✿ ❯s❡ ♣r❡♣r♦❝❡ss❡❞ ✈❛❧✉❡s ❛♥❞ s✐♠♣❧❡ ✭♥♦♥✲❝r②♣t♦✮ ♣r♦t♦❝♦❧s t♦ ❡✈❛❧✉❛t❡ ❝✐r❝✉✐t ♦♥ ❛❝t✉❛❧ ✐♥♣✉t✳

✾ ✴ ✶✾

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SLIDE 30

❚❤❡ Pr❡♣r♦❝❡ss✐♥❣ ♠♦❞❡❧

Pr❡♣r♦❝❡ss✐♥❣✿

◮ Pr❡♣❛r❡ r❛♥❞♦♠ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✿

✶ ♣❡r ■♥♣✉t✲❣❛t❡✳ ✶✻❇ ♣❡r ❆◆❉✲❣❛t❡✱ ❢♦r s❡❝✉r✐t② ✷

❇ ❧♦❣ ❈ ✳

Pr♦✈❡ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s ✭t❤✐♥❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✲tr✐♣❧❡s✮✳ ❖♥❧✐♥❡✿ ❯s❡ ♣r❡♣r♦❝❡ss❡❞ ✈❛❧✉❡s ❛♥❞ s✐♠♣❧❡ ✭♥♦♥✲❝r②♣t♦✮ ♣r♦t♦❝♦❧s t♦ ❡✈❛❧✉❛t❡ ❝✐r❝✉✐t ♦♥ ❛❝t✉❛❧ ✐♥♣✉t✳

✾ ✴ ✶✾

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SLIDE 31

❚❤❡ Pr❡♣r♦❝❡ss✐♥❣ ♠♦❞❡❧

Pr❡♣r♦❝❡ss✐♥❣✿

◮ Pr❡♣❛r❡ r❛♥❞♦♠ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✿

◮ ✶ ♣❡r ■♥♣✉t✲❣❛t❡✳

✶✻❇ ♣❡r ❆◆❉✲❣❛t❡✱ ❢♦r s❡❝✉r✐t② ✷

❇ ❧♦❣ ❈ ✳

Pr♦✈❡ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s ✭t❤✐♥❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✲tr✐♣❧❡s✮✳ ❖♥❧✐♥❡✿ ❯s❡ ♣r❡♣r♦❝❡ss❡❞ ✈❛❧✉❡s ❛♥❞ s✐♠♣❧❡ ✭♥♦♥✲❝r②♣t♦✮ ♣r♦t♦❝♦❧s t♦ ❡✈❛❧✉❛t❡ ❝✐r❝✉✐t ♦♥ ❛❝t✉❛❧ ✐♥♣✉t✳

✾ ✴ ✶✾

slide-32
SLIDE 32

❚❤❡ Pr❡♣r♦❝❡ss✐♥❣ ♠♦❞❡❧

Pr❡♣r♦❝❡ss✐♥❣✿

◮ Pr❡♣❛r❡ r❛♥❞♦♠ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✿

◮ ✶ ♣❡r ■♥♣✉t✲❣❛t❡✳ ◮ ✶✻❇ ♣❡r ❆◆❉✲❣❛t❡✱ ❢♦r s❡❝✉r✐t② ∼ ✷−❇ ❧♦❣(|❈|)✳

Pr♦✈❡ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s ✭t❤✐♥❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✲tr✐♣❧❡s✮✳ ❖♥❧✐♥❡✿ ❯s❡ ♣r❡♣r♦❝❡ss❡❞ ✈❛❧✉❡s ❛♥❞ s✐♠♣❧❡ ✭♥♦♥✲❝r②♣t♦✮ ♣r♦t♦❝♦❧s t♦ ❡✈❛❧✉❛t❡ ❝✐r❝✉✐t ♦♥ ❛❝t✉❛❧ ✐♥♣✉t✳

✾ ✴ ✶✾

slide-33
SLIDE 33

❚❤❡ Pr❡♣r♦❝❡ss✐♥❣ ♠♦❞❡❧

Pr❡♣r♦❝❡ss✐♥❣✿

◮ Pr❡♣❛r❡ r❛♥❞♦♠ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✿

◮ ✶ ♣❡r ■♥♣✉t✲❣❛t❡✳ ◮ ✶✻❇ ♣❡r ❆◆❉✲❣❛t❡✱ ❢♦r s❡❝✉r✐t② ∼ ✷−❇ ❧♦❣(|❈|)✳

◮ Pr♦✈❡ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s ✭t❤✐♥❦

♠✉❧t✐♣❧✐❝❛t✐♦♥✲tr✐♣❧❡s✮✳ ❖♥❧✐♥❡✿ ❯s❡ ♣r❡♣r♦❝❡ss❡❞ ✈❛❧✉❡s ❛♥❞ s✐♠♣❧❡ ✭♥♦♥✲❝r②♣t♦✮ ♣r♦t♦❝♦❧s t♦ ❡✈❛❧✉❛t❡ ❝✐r❝✉✐t ♦♥ ❛❝t✉❛❧ ✐♥♣✉t✳

✾ ✴ ✶✾

slide-34
SLIDE 34

❚❤❡ Pr❡♣r♦❝❡ss✐♥❣ ♠♦❞❡❧

Pr❡♣r♦❝❡ss✐♥❣✿

◮ Pr❡♣❛r❡ r❛♥❞♦♠ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✿

◮ ✶ ♣❡r ■♥♣✉t✲❣❛t❡✳ ◮ ✶✻❇ ♣❡r ❆◆❉✲❣❛t❡✱ ❢♦r s❡❝✉r✐t② ∼ ✷−❇ ❧♦❣(|❈|)✳

◮ Pr♦✈❡ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s ✭t❤✐♥❦

♠✉❧t✐♣❧✐❝❛t✐♦♥✲tr✐♣❧❡s✮✳ ❖♥❧✐♥❡✿ ❯s❡ ♣r❡♣r♦❝❡ss❡❞ ✈❛❧✉❡s ❛♥❞ s✐♠♣❧❡ ✭♥♦♥✲❝r②♣t♦✮ ♣r♦t♦❝♦❧s t♦ ❡✈❛❧✉❛t❡ ❝✐r❝✉✐t ♦♥ ❛❝t✉❛❧ ✐♥♣✉t✳

✾ ✴ ✶✾

slide-35
SLIDE 35

❚❤❡ Pr❡♣r♦❝❡ss✐♥❣ ♠♦❞❡❧

Pr❡♣r♦❝❡ss✐♥❣✿

◮ Pr❡♣❛r❡ r❛♥❞♦♠ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✿

◮ ✶ ♣❡r ■♥♣✉t✲❣❛t❡✳ ◮ ✶✻❇ ♣❡r ❆◆❉✲❣❛t❡✱ ❢♦r s❡❝✉r✐t② ∼ ✷−❇ ❧♦❣(|❈|)✳

◮ Pr♦✈❡ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s ✭t❤✐♥❦

♠✉❧t✐♣❧✐❝❛t✐♦♥✲tr✐♣❧❡s✮✳ ❖♥❧✐♥❡✿

◮ ❯s❡ ♣r❡♣r♦❝❡ss❡❞ ✈❛❧✉❡s ❛♥❞ s✐♠♣❧❡ ✭♥♦♥✲❝r②♣t♦✮ ♣r♦t♦❝♦❧s t♦

❡✈❛❧✉❛t❡ ❝✐r❝✉✐t ♦♥ ❛❝t✉❛❧ ✐♥♣✉t✳

✾ ✴ ✶✾

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SLIDE 36

❚❤❡ Pr❡♣r♦❝❡ss✐♥❣ ♠♦❞❡❧

Pr❡♣r♦❝❡ss✐♥❣✿

◮ Pr❡♣❛r❡ r❛♥❞♦♠ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✿

◮ ✶ ♣❡r ■♥♣✉t✲❣❛t❡✳ ◮ ✶✻❇ ♣❡r ❆◆❉✲❣❛t❡✱ ❢♦r s❡❝✉r✐t② ∼ ✷−❇ ❧♦❣(|❈|)✳

◮ Pr♦✈❡ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s ✭t❤✐♥❦

♠✉❧t✐♣❧✐❝❛t✐♦♥✲tr✐♣❧❡s✮✳ ❖♥❧✐♥❡✿

◮ ❯s❡ ♣r❡♣r♦❝❡ss❡❞ ✈❛❧✉❡s ❛♥❞ s✐♠♣❧❡ ✭♥♦♥✲❝r②♣t♦✮ ♣r♦t♦❝♦❧s t♦

❡✈❛❧✉❛t❡ ❝✐r❝✉✐t ♦♥ ❛❝t✉❛❧ ✐♥♣✉t✳

✾ ✴ ✶✾

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SLIDE 37

❖✈❡r✈✐❡✇

Pr♦t♦❝♦❧ ❖✈❡r✈✐❡✇ ▼❆❈s ❈♦♥❝❧✉❞✐♥❣

✶✵ ✴ ✶✾

slide-38
SLIDE 38

■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r❡t✐❝ ▼❆❈s

▼❡ss❛❣❡ ① ∈❘ {✵, ✶} ▼❆❈ ▼ = ❑ ⊕ ①∆

  • ❧♦❜❛❧ ❦❡② ∆ ∈❘ {✵, ✶}♥

▲♦❝❛❧ ❦❡② ❑ ∈❘ {✵, ✶}♥ ❯♥❢♦r❣❡❛❜✐❧✐t②✿ ▼ ❑ ① ❞♦❡s ♥♦t ❣✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ✳ ▲❡t ▼✵ ❑ ✵ ❑ ❛♥❞ ▼✶ ❑ ✶ ❑ ✳ ▼✵ ▼✶ ✳

✶✶ ✴ ✶✾

slide-39
SLIDE 39

■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r❡t✐❝ ▼❆❈s

▼❡ss❛❣❡ ① ∈❘ {✵, ✶} ▼❆❈ ▼ = ❑ ⊕ ①∆

  • ❧♦❜❛❧ ❦❡② ∆ ∈❘ {✵, ✶}♥

▲♦❝❛❧ ❦❡② ❑ ∈❘ {✵, ✶}♥ ❯♥❢♦r❣❡❛❜✐❧✐t②✿ ▼ ❑ ① ❞♦❡s ♥♦t ❣✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ✳ ▲❡t ▼✵ ❑ ✵ ❑ ❛♥❞ ▼✶ ❑ ✶ ❑ ✳ ▼✵ ▼✶ ✳

✶✶ ✴ ✶✾

slide-40
SLIDE 40

■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r❡t✐❝ ▼❆❈s

▼❡ss❛❣❡ ① ∈❘ {✵, ✶} ▼❆❈ ▼ = ❑ ⊕ ①∆

  • ❧♦❜❛❧ ❦❡② ∆ ∈❘ {✵, ✶}♥

▲♦❝❛❧ ❦❡② ❑ ∈❘ {✵, ✶}♥ ❯♥❢♦r❣❡❛❜✐❧✐t②✿

◮ ▼ = ❑ ⊕ ①∆ ❞♦❡s ♥♦t ❣✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ∆✳

▲❡t ▼✵ ❑ ✵ ❑ ❛♥❞ ▼✶ ❑ ✶ ❑ ✳ ▼✵ ▼✶ ✳

✶✶ ✴ ✶✾

slide-41
SLIDE 41

■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r❡t✐❝ ▼❆❈s

▼❡ss❛❣❡ ① ∈❘ {✵, ✶} ▼❆❈ ▼ = ❑ ⊕ ①∆

  • ❧♦❜❛❧ ❦❡② ∆ ∈❘ {✵, ✶}♥

▲♦❝❛❧ ❦❡② ❑ ∈❘ {✵, ✶}♥ ❯♥❢♦r❣❡❛❜✐❧✐t②✿

◮ ▼ = ❑ ⊕ ①∆ ❞♦❡s ♥♦t ❣✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ∆✳ ◮ ▲❡t ▼✵ = ❑ ⊕ ✵∆ = ❑ ❛♥❞ ▼✶ = ❑ ⊕ ✶∆ = ❑ ⊕ ∆✳

▼✵ ▼✶ ✳

✶✶ ✴ ✶✾

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SLIDE 42

■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r❡t✐❝ ▼❆❈s

▼❡ss❛❣❡ ① ∈❘ {✵, ✶} ▼❆❈ ▼ = ❑ ⊕ ①∆

  • ❧♦❜❛❧ ❦❡② ∆ ∈❘ {✵, ✶}♥

▲♦❝❛❧ ❦❡② ❑ ∈❘ {✵, ✶}♥ ❯♥❢♦r❣❡❛❜✐❧✐t②✿

◮ ▼ = ❑ ⊕ ①∆ ❞♦❡s ♥♦t ❣✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ∆✳ ◮ ▲❡t ▼✵ = ❑ ⊕ ✵∆ = ❑ ❛♥❞ ▼✶ = ❑ ⊕ ✶∆ = ❑ ⊕ ∆✳ ◮ ▼✵ ⊕ ▼✶ = ∆✳

✶✶ ✴ ✶✾

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SLIDE 43

❖❜t❛✐♥✐♥❣ ▼❆❈s✿ ❚❤❡ ❋✉♥❝t✐♦♥❛❧✐t②

❛❇✐t ❑, ∆ ① ▼ = ❑ ⊕ ①∆

✶✷ ✴ ✶✾

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SLIDE 44

❖❜t❛✐♥✐♥❣ ▼❆❈s✿ Pr♦t♦❝♦❧ ❙t❡♣s

◮ ❙t❡♣ ✶✿ ❖❜t❛✐♥ ❛ ❢❡✇✱ ❧♦♥❣ ▼❆❈s ♦♥ ❆❧✐❝❡✬s r❛♥❞♦♠ ❜✐ts✳ ◮ ❙t❡♣ ✷✿ ❚✉r♥ ✐♥t♦ ♠❛♥②✱ s❤♦rt ▼❆❈s ♦♥ ❇♦❜✬s r❛♥❞♦♠ ❜✐ts✳

✶✸ ✴ ✶✾

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SLIDE 45

❙t❡♣ ✶✿ ▲♦♥❣ ▼❆❈s ❢♦r ❆❧✐❝❡

❚♦ ❛✉t❤❡♥t✐❝❛t❡ ❜✐ts ①✶, ①✷, . . . , ①♥✿

✷♥ ♥

❖❚

❙✵, ❙✶ ∈ {✵, ✶}❚ ❝ ∈ {✵, ✶} ❙❝

Pr♦❜❧❡♠✿ ❇♦❜ ♠✉st ✉s❡ s❛♠❡ ✐♥ ❡✈❡r② ❖❚✳ ❙♦❧✉t✐♦♥✿

❉♦ ✷♥ ❖❚s✳ ❋♦r❝❡ ❇♦❜ t♦ ✉s❡ ❝♦♥s✐st❡♥t ✱ ✉s✐♥❣ ❛ ✏❝✉t✲♥✲❝❤♦s❡✑✲❧✐❦❡ t❡❝❤♥✐q✉❡✳ ❙❛❝r✐✜❝❡ ❤❛❧❢ ♦❢ t❤❡ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✳

✶✹ ✴ ✶✾

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SLIDE 46

❙t❡♣ ✶✿ ▲♦♥❣ ▼❆❈s ❢♦r ❆❧✐❝❡

❚♦ ❛✉t❤❡♥t✐❝❛t❡ ❜✐ts ①✶, ①✷, . . . , ①♥✿

✷♥ ♥

❖❚

❑✐, ❑✐ ⊕ ∆ ①✐ ❑✐ ⊕ ①✐∆ = ▼✐

Pr♦❜❧❡♠✿ ❇♦❜ ♠✉st ✉s❡ s❛♠❡ ✐♥ ❡✈❡r② ❖❚✳ ❙♦❧✉t✐♦♥✿

❉♦ ✷♥ ❖❚s✳ ❋♦r❝❡ ❇♦❜ t♦ ✉s❡ ❝♦♥s✐st❡♥t ✱ ✉s✐♥❣ ❛ ✏❝✉t✲♥✲❝❤♦s❡✑✲❧✐❦❡ t❡❝❤♥✐q✉❡✳ ❙❛❝r✐✜❝❡ ❤❛❧❢ ♦❢ t❤❡ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✳

✶✹ ✴ ✶✾

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SLIDE 47

❙t❡♣ ✶✿ ▲♦♥❣ ▼❆❈s ❢♦r ❆❧✐❝❡

❚♦ ❛✉t❤❡♥t✐❝❛t❡ ❜✐ts ①✶, ①✷, . . . , ①♥✿

✷♥ ♥

❖❚

❑✐, ❑✐ ⊕ ∆ ①✐ ❑✐ ⊕ ①✐∆ = ▼✐

◮ Pr♦❜❧❡♠✿ ❇♦❜ ♠✉st ✉s❡ s❛♠❡ ∆ ✐♥ ❡✈❡r② ❖❚✳

❙♦❧✉t✐♦♥✿

❉♦ ✷♥ ❖❚s✳ ❋♦r❝❡ ❇♦❜ t♦ ✉s❡ ❝♦♥s✐st❡♥t ✱ ✉s✐♥❣ ❛ ✏❝✉t✲♥✲❝❤♦s❡✑✲❧✐❦❡ t❡❝❤♥✐q✉❡✳ ❙❛❝r✐✜❝❡ ❤❛❧❢ ♦❢ t❤❡ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✳

✶✹ ✴ ✶✾

slide-48
SLIDE 48

❙t❡♣ ✶✿ ▲♦♥❣ ▼❆❈s ❢♦r ❆❧✐❝❡

❚♦ ❛✉t❤❡♥t✐❝❛t❡ ❜✐ts ①✶, ①✷, . . . , ①♥✿

✷♥ ♥

❖❚

❑✐, ❑✐ ⊕ ∆ ①✐ ❑✐ ⊕ ①✐∆ = ▼✐

◮ Pr♦❜❧❡♠✿ ❇♦❜ ♠✉st ✉s❡ s❛♠❡ ∆ ✐♥ ❡✈❡r② ❖❚✳ ◮ ❙♦❧✉t✐♦♥✿

◮ ❉♦ ✷♥ ❖❚s✳

❋♦r❝❡ ❇♦❜ t♦ ✉s❡ ❝♦♥s✐st❡♥t ✱ ✉s✐♥❣ ❛ ✏❝✉t✲♥✲❝❤♦s❡✑✲❧✐❦❡ t❡❝❤♥✐q✉❡✳ ❙❛❝r✐✜❝❡ ❤❛❧❢ ♦❢ t❤❡ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✳

✶✹ ✴ ✶✾

slide-49
SLIDE 49

❙t❡♣ ✶✿ ▲♦♥❣ ▼❆❈s ❢♦r ❆❧✐❝❡

❚♦ ❛✉t❤❡♥t✐❝❛t❡ ❜✐ts ①✶, ①✷, . . . , ①♥✿

✷♥ ♥

❖❚

❑✐, ❑✐ ⊕ ∆ ①✐ ❑✐ ⊕ ①✐∆ = ▼✐

◮ Pr♦❜❧❡♠✿ ❇♦❜ ♠✉st ✉s❡ s❛♠❡ ∆ ✐♥ ❡✈❡r② ❖❚✳ ◮ ❙♦❧✉t✐♦♥✿

◮ ❉♦ ✷♥ ❖❚s✳ ◮ ❋♦r❝❡ ❇♦❜ t♦ ✉s❡ ❝♦♥s✐st❡♥t ∆✱ ✉s✐♥❣ ❛ ✏❝✉t✲♥✲❝❤♦s❡✑✲❧✐❦❡

t❡❝❤♥✐q✉❡✳ ❙❛❝r✐✜❝❡ ❤❛❧❢ ♦❢ t❤❡ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✳

✶✹ ✴ ✶✾

slide-50
SLIDE 50

❙t❡♣ ✶✿ ▲♦♥❣ ▼❆❈s ❢♦r ❆❧✐❝❡

❚♦ ❛✉t❤❡♥t✐❝❛t❡ ❜✐ts ①✶, ①✷, . . . , ①♥✿

✷♥ ♥

❖❚

❑✐, ❑✐ ⊕ ∆ ①✐ ❑✐ ⊕ ①✐∆ = ▼✐

◮ Pr♦❜❧❡♠✿ ❇♦❜ ♠✉st ✉s❡ s❛♠❡ ∆ ✐♥ ❡✈❡r② ❖❚✳ ◮ ❙♦❧✉t✐♦♥✿

◮ ❉♦ ✷♥ ❖❚s✳ ◮ ❋♦r❝❡ ❇♦❜ t♦ ✉s❡ ❝♦♥s✐st❡♥t ∆✱ ✉s✐♥❣ ❛ ✏❝✉t✲♥✲❝❤♦s❡✑✲❧✐❦❡

t❡❝❤♥✐q✉❡✳

◮ ❙❛❝r✐✜❝❡ ❤❛❧❢ ♦❢ t❤❡ ❛✉t❤❡♥t✐❝❛t❡❞ ♠❡ss❛❣❡s✳ ✶✹ ✴ ✶✾

slide-51
SLIDE 51

❙t❡♣ ✷✿ ❙❤♦rt ▼❆❈s ❋♦r ❇♦❜

▼✶ ▼✷ . . . ▼♥ = ❑✶ ❑✷ . . . ❑♥ ⊕ ①✶∆ ①✷∆ . . . ①♥∆

✶✺ ✴ ✶✾

slide-52
SLIDE 52

❙t❡♣ ✷✿ ❙❤♦rt ▼❆❈s ❋♦r ❇♦❜

❑✶ ❑✷ . . . ❑♥ = ▼✶ ▼✷ . . . ▼♥ ⊕ ①✶∆ ①✷∆ . . . ①♥∆

✶✺ ✴ ✶✾

slide-53
SLIDE 53

❙t❡♣ ✷✿ ❙❤♦rt ▼❆❈s ❋♦r ❇♦❜

(❑✶,✶, . . . , ❑✶,♥) (❑✷,✶, . . . , ❑✷,♥) ✳ ✳ ✳ (❑❚,✶, . . . , ❑❚,♥)

=

(▼✶,✶, . . . , ▼✶,♥) (▼✷,✶, . . . , ▼✷,♥) ✳ ✳ ✳ (▼❚,✶, . . . , ▼❚,♥)

⊕

(①✶, . . . , ①♥)∆✶ (①✶, . . . , ①♥)∆✷ ✳ ✳ ✳ (①✶, . . . , ①♥)∆❚

◆✐ ▲✐ ②✐ ✱ ✐✳❡✳ ◆✐ ✐s ❛ ▼❆❈ ♦♥ ②✐ ✇✳ ❦❡②s ▲✐ ✳

✶✻ ✴ ✶✾

slide-54
SLIDE 54

❙t❡♣ ✷✿ ❙❤♦rt ▼❆❈s ❋♦r ❇♦❜

◆✶ ◆✷ ✳ ✳ ✳ ◆❚

=

▲✶ ▲✷ ✳ ✳ ✳ ▲❚

⊕

(①✶, . . . , ①♥)∆✶ (①✶, . . . , ①♥)∆✷ ✳ ✳ ✳ (①✶, . . . , ①♥)∆❚

◆✐ ▲✐ ②✐ ✱ ✐✳❡✳ ◆✐ ✐s ❛ ▼❆❈ ♦♥ ②✐ ✇✳ ❦❡②s ▲✐ ✳

✶✻ ✴ ✶✾

slide-55
SLIDE 55

❙t❡♣ ✷✿ ❙❤♦rt ▼❆❈s ❋♦r ❇♦❜

◆✶ ◆✷ ✳ ✳ ✳ ◆❚

=

▲✶ ▲✷ ✳ ✳ ✳ ▲❚

⊕

Γ∆✶ Γ∆✷ ✳ ✳ ✳ Γ∆❚

◆✐ ▲✐ ②✐ ✱ ✐✳❡✳ ◆✐ ✐s ❛ ▼❆❈ ♦♥ ②✐ ✇✳ ❦❡②s ▲✐ ✳

✶✻ ✴ ✶✾

slide-56
SLIDE 56

❙t❡♣ ✷✿ ❙❤♦rt ▼❆❈s ❋♦r ❇♦❜

◆✶ ◆✷ ✳ ✳ ✳ ◆❚

=

▲✶ ▲✷ ✳ ✳ ✳ ▲❚

⊕

Γ②✶ Γ②✷ ✳ ✳ ✳ Γ②❚

◮ ◆✐ = ▲✐ ⊕ ②✐Γ✱ ✐✳❡✳ ◆✐ ✐s ❛ ▼❆❈ ♦♥ ②✐ ✇✳ ❦❡②s ▲✐, Γ✳

✶✻ ✴ ✶✾

slide-57
SLIDE 57

❖❜t❛✐♥✐♥❣ ▼❆❈s✿ ❙✉♠♠❛r②

❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚

◮ ❆ ❢❡✇ ✭✷♥✮ ❖❚s ✇✐t❤ ❧♦♥❣ ♠❡ss❛❣❡s ✭❚ = ♣♦❧②(♥) ❜✐ts✮✳

❆ ❢❡✇ ✭♥✮ ❧♦♥❣ ✭❚ ❜✐ts✮ ▼❆❈s ❢♦r ❆❧✐❝❡✳ ▼❛♥② ✭❚✮ s❤♦rt ✭♥ ❜✐ts✮ ▼❆❈s ❢♦r ❇♦❜✳ ◆♦t❡ ✶✿ ❈❛♥ ❣❡t ❧♦♥❣ ❖❚s ❢r♦♠ s❤♦rt ❖❚ ✉s✐♥❣ ❛ P❘●✳ ◆♦t❡ ✷✿ ❈❛♥ ❣❡t s❤♦rt ❖❚ ❢r♦♠ s❤♦rt ❛❇✐t ✭✐✳❡✳ ❖❚✲❡①t❡♥s✐♦♥✮✳

✶✼ ✴ ✶✾

slide-58
SLIDE 58

❖❜t❛✐♥✐♥❣ ▼❆❈s✿ ❙✉♠♠❛r②

❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚

◮ ❆ ❢❡✇ ✭✷♥✮ ❖❚s ✇✐t❤ ❧♦♥❣ ♠❡ss❛❣❡s ✭❚ = ♣♦❧②(♥) ❜✐ts✮✳ ◮ ❆ ❢❡✇ ✭♥✮ ❧♦♥❣ ✭❚ ❜✐ts✮ ▼❆❈s ❢♦r ❆❧✐❝❡✳

▼❛♥② ✭❚✮ s❤♦rt ✭♥ ❜✐ts✮ ▼❆❈s ❢♦r ❇♦❜✳ ◆♦t❡ ✶✿ ❈❛♥ ❣❡t ❧♦♥❣ ❖❚s ❢r♦♠ s❤♦rt ❖❚ ✉s✐♥❣ ❛ P❘●✳ ◆♦t❡ ✷✿ ❈❛♥ ❣❡t s❤♦rt ❖❚ ❢r♦♠ s❤♦rt ❛❇✐t ✭✐✳❡✳ ❖❚✲❡①t❡♥s✐♦♥✮✳

✶✼ ✴ ✶✾

slide-59
SLIDE 59

❖❜t❛✐♥✐♥❣ ▼❆❈s✿ ❙✉♠♠❛r②

❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚

◮ ❆ ❢❡✇ ✭✷♥✮ ❖❚s ✇✐t❤ ❧♦♥❣ ♠❡ss❛❣❡s ✭❚ = ♣♦❧②(♥) ❜✐ts✮✳ ◮ ❆ ❢❡✇ ✭♥✮ ❧♦♥❣ ✭❚ ❜✐ts✮ ▼❆❈s ❢♦r ❆❧✐❝❡✳ ◮ ▼❛♥② ✭❚✮ s❤♦rt ✭♥ ❜✐ts✮ ▼❆❈s ❢♦r ❇♦❜✳

◆♦t❡ ✶✿ ❈❛♥ ❣❡t ❧♦♥❣ ❖❚s ❢r♦♠ s❤♦rt ❖❚ ✉s✐♥❣ ❛ P❘●✳ ◆♦t❡ ✷✿ ❈❛♥ ❣❡t s❤♦rt ❖❚ ❢r♦♠ s❤♦rt ❛❇✐t ✭✐✳❡✳ ❖❚✲❡①t❡♥s✐♦♥✮✳

✶✼ ✴ ✶✾

slide-60
SLIDE 60

❖❜t❛✐♥✐♥❣ ▼❆❈s✿ ❙✉♠♠❛r②

❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚

◮ ❆ ❢❡✇ ✭✷♥✮ ❖❚s ✇✐t❤ ❧♦♥❣ ♠❡ss❛❣❡s ✭❚ = ♣♦❧②(♥) ❜✐ts✮✳ ◮ ❆ ❢❡✇ ✭♥✮ ❧♦♥❣ ✭❚ ❜✐ts✮ ▼❆❈s ❢♦r ❆❧✐❝❡✳ ◮ ▼❛♥② ✭❚✮ s❤♦rt ✭♥ ❜✐ts✮ ▼❆❈s ❢♦r ❇♦❜✳ ◮ ◆♦t❡ ✶✿ ❈❛♥ ❣❡t ❧♦♥❣ ❖❚s ❢r♦♠ s❤♦rt ❖❚ ✉s✐♥❣ ❛ P❘●✳

◆♦t❡ ✷✿ ❈❛♥ ❣❡t s❤♦rt ❖❚ ❢r♦♠ s❤♦rt ❛❇✐t ✭✐✳❡✳ ❖❚✲❡①t❡♥s✐♦♥✮✳

✶✼ ✴ ✶✾

slide-61
SLIDE 61

❖❜t❛✐♥✐♥❣ ▼❆❈s✿ ❙✉♠♠❛r②

❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❛❇✐t ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚ ❖❚

◮ ❆ ❢❡✇ ✭✷♥✮ ❖❚s ✇✐t❤ ❧♦♥❣ ♠❡ss❛❣❡s ✭❚ = ♣♦❧②(♥) ❜✐ts✮✳ ◮ ❆ ❢❡✇ ✭♥✮ ❧♦♥❣ ✭❚ ❜✐ts✮ ▼❆❈s ❢♦r ❆❧✐❝❡✳ ◮ ▼❛♥② ✭❚✮ s❤♦rt ✭♥ ❜✐ts✮ ▼❆❈s ❢♦r ❇♦❜✳ ◮ ◆♦t❡ ✶✿ ❈❛♥ ❣❡t ❧♦♥❣ ❖❚s ❢r♦♠ s❤♦rt ❖❚ ✉s✐♥❣ ❛ P❘●✳ ◮ ◆♦t❡ ✷✿ ❈❛♥ ❣❡t s❤♦rt ❖❚ ❢r♦♠ s❤♦rt ❛❇✐t ✭✐✳❡✳ ❖❚✲❡①t❡♥s✐♦♥✮✳

✶✼ ✴ ✶✾

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SLIDE 62

❖✈❡r✈✐❡✇

Pr♦t♦❝♦❧ ❖✈❡r✈✐❡✇ ▼❆❈s ❈♦♥❝❧✉❞✐♥❣

✶✽ ✴ ✶✾

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SLIDE 63

❈♦♥❝❧✉❞✐♥❣ ✳ ✳ ✳

❚❛❦❡ ❛✇❛②✿

◮ ❋✐♥❛❧❧② ❛ ♥♦♥✲●❛r❜❧❡❞ ❈✐r❝✉✐ts ❛♣♣r♦❛❝❤ ❞♦ ♣r❛❝t✐❝❛❧ ✷P❈✦

■t✬s ❜❛s❡❞ ♦♥ ●▼❲ ❛♥❞ ❖❚✲❡①t❡♥s✐♦♥✳ ■t✬s r❡❛❧❧② ❢❛st✦ ✳ ✳ ✳ ❙♦ ✐❢ ②♦✉✬r❡ ✐♠♣❧❡♠❡♥t✐♥❣ ❛ ✷P❈ ♣r♦t♦❝♦❧✱ ✇❤② ♥♦t ❣✐✈❡ t❤✐s ❛ tr②❄ ❚❤❛♥❦ ②♦✉✳

✶✾ ✴ ✶✾

slide-64
SLIDE 64

❈♦♥❝❧✉❞✐♥❣ ✳ ✳ ✳

❚❛❦❡ ❛✇❛②✿

◮ ❋✐♥❛❧❧② ❛ ♥♦♥✲●❛r❜❧❡❞ ❈✐r❝✉✐ts ❛♣♣r♦❛❝❤ ❞♦ ♣r❛❝t✐❝❛❧ ✷P❈✦ ◮ ■t✬s ❜❛s❡❞ ♦♥ ●▼❲ ❛♥❞ ❖❚✲❡①t❡♥s✐♦♥✳

■t✬s r❡❛❧❧② ❢❛st✦ ✳ ✳ ✳ ❙♦ ✐❢ ②♦✉✬r❡ ✐♠♣❧❡♠❡♥t✐♥❣ ❛ ✷P❈ ♣r♦t♦❝♦❧✱ ✇❤② ♥♦t ❣✐✈❡ t❤✐s ❛ tr②❄ ❚❤❛♥❦ ②♦✉✳

✶✾ ✴ ✶✾

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SLIDE 65

❈♦♥❝❧✉❞✐♥❣ ✳ ✳ ✳

❚❛❦❡ ❛✇❛②✿

◮ ❋✐♥❛❧❧② ❛ ♥♦♥✲●❛r❜❧❡❞ ❈✐r❝✉✐ts ❛♣♣r♦❛❝❤ ❞♦ ♣r❛❝t✐❝❛❧ ✷P❈✦ ◮ ■t✬s ❜❛s❡❞ ♦♥ ●▼❲ ❛♥❞ ❖❚✲❡①t❡♥s✐♦♥✳ ◮ ■t✬s r❡❛❧❧② ❢❛st✦

✳ ✳ ✳ ❙♦ ✐❢ ②♦✉✬r❡ ✐♠♣❧❡♠❡♥t✐♥❣ ❛ ✷P❈ ♣r♦t♦❝♦❧✱ ✇❤② ♥♦t ❣✐✈❡ t❤✐s ❛ tr②❄ ❚❤❛♥❦ ②♦✉✳

✶✾ ✴ ✶✾

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SLIDE 66

❈♦♥❝❧✉❞✐♥❣ ✳ ✳ ✳

❚❛❦❡ ❛✇❛②✿

◮ ❋✐♥❛❧❧② ❛ ♥♦♥✲●❛r❜❧❡❞ ❈✐r❝✉✐ts ❛♣♣r♦❛❝❤ ❞♦ ♣r❛❝t✐❝❛❧ ✷P❈✦ ◮ ■t✬s ❜❛s❡❞ ♦♥ ●▼❲ ❛♥❞ ❖❚✲❡①t❡♥s✐♦♥✳ ◮ ■t✬s r❡❛❧❧② ❢❛st✦ ◮ ✳ ✳ ✳ ❙♦ ✐❢ ②♦✉✬r❡ ✐♠♣❧❡♠❡♥t✐♥❣ ❛ ✷P❈ ♣r♦t♦❝♦❧✱ ✇❤② ♥♦t ❣✐✈❡

t❤✐s ❛ tr②❄ ❚❤❛♥❦ ②♦✉✳

✶✾ ✴ ✶✾

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SLIDE 67

❈♦♥❝❧✉❞✐♥❣ ✳ ✳ ✳

❚❛❦❡ ❛✇❛②✿

◮ ❋✐♥❛❧❧② ❛ ♥♦♥✲●❛r❜❧❡❞ ❈✐r❝✉✐ts ❛♣♣r♦❛❝❤ ❞♦ ♣r❛❝t✐❝❛❧ ✷P❈✦ ◮ ■t✬s ❜❛s❡❞ ♦♥ ●▼❲ ❛♥❞ ❖❚✲❡①t❡♥s✐♦♥✳ ◮ ■t✬s r❡❛❧❧② ❢❛st✦ ◮ ✳ ✳ ✳ ❙♦ ✐❢ ②♦✉✬r❡ ✐♠♣❧❡♠❡♥t✐♥❣ ❛ ✷P❈ ♣r♦t♦❝♦❧✱ ✇❤② ♥♦t ❣✐✈❡

t❤✐s ❛ tr②❄ ❚❤❛♥❦ ②♦✉✳

✶✾ ✴ ✶✾