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q s rs t - - PowerPoint PPT Presentation

May 19, 2023 132 likes •1.25k views

P t r rrs r stt t st rrs q s

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SLIDE 1

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-2
SLIDE 2

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s ❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❯♥✐✈❡rs✐t❛t ❞❡ ❱❛❧è♥❝✐❛ ▼❆P ✷✵✶✵✱ ▲♦❣r♦ñ♦✱ ◆♦✈❡♠❜❡r

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-3
SLIDE 3

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

P❧❛♥

❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚

♥❛t✉r❛❧ ♥✉♠❜❡rs ♦❜❥❡❝t ❞✐❛❣r❛♠s r❡♣r❡s❡♥t❛t✐♦♥ ✈❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝

❢r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❝❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❙✉❜r❡❝✉rs✐♦♥

tr❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-4
SLIDE 4

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

P❧❛♥

❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚

♥❛t✉r❛❧ ♥✉♠❜❡rs ♦❜❥❡❝t ❞✐❛❣r❛♠s r❡♣r❡s❡♥t❛t✐♦♥ ✈❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝

❢r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❝❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❙✉❜r❡❝✉rs✐♦♥

tr❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-5
SLIDE 5

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

P❧❛♥

❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚

♥❛t✉r❛❧ ♥✉♠❜❡rs ♦❜❥❡❝t ❞✐❛❣r❛♠s r❡♣r❡s❡♥t❛t✐♦♥ ✈❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝

❢r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❝❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❙✉❜r❡❝✉rs✐♦♥

tr❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-6
SLIDE 6

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

P❧❛♥

❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚

♥❛t✉r❛❧ ♥✉♠❜❡rs ♦❜❥❡❝t ❞✐❛❣r❛♠s r❡♣r❡s❡♥t❛t✐♦♥ ✈❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝

❢r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❝❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❙✉❜r❡❝✉rs✐♦♥

tr❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-7
SLIDE 7

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

P❧❛♥

❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚

♥❛t✉r❛❧ ♥✉♠❜❡rs ♦❜❥❡❝t ❞✐❛❣r❛♠s r❡♣r❡s❡♥t❛t✐♦♥ ✈❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝

❢r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❝❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❙✉❜r❡❝✉rs✐♦♥

tr❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-8
SLIDE 8

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

P❧❛♥

❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚

♥❛t✉r❛❧ ♥✉♠❜❡rs ♦❜❥❡❝t ❞✐❛❣r❛♠s r❡♣r❡s❡♥t❛t✐♦♥ ✈❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝

❢r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❝❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❙✉❜r❡❝✉rs✐♦♥

tr❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-9
SLIDE 9

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥

P❧❛♥

❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚

♥❛t✉r❛❧ ♥✉♠❜❡rs ♦❜❥❡❝t ❞✐❛❣r❛♠s r❡♣r❡s❡♥t❛t✐♦♥ ✈❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝

❢r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❝❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❙✉❜r❡❝✉rs✐♦♥

tr❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-10
SLIDE 10

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

■♥ ❛ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r② ✭❝❝✮ ✇❡ ❤❛✈❡ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❛ ♣r♦❞✉❝t ❘❡❝✉rs✐♦♥ ♦♣❡r❛t♦r ✐s ♦❜t❛✐♥❡❞ ❜② ♠❡❛♥s ♦❢ ❛♥ ♥♥♦

❉❡✜♥✐t✐♦♥

❆ ♥♥♦ ✐♥ ❛ ❝❛t❡❣♦r② ✇✐t❤ t❡r♠✐♥❛❧ ♦❜❥❡❝t ✶ ✐s ◆ ③ s ✇✐t❤ ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ✶

③

◆

♠ s

◆

♠

✶

❢

❆

❣

❆ ✇❤❡r❡ ♠ ✐s ✉♥✐q✉❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-11
SLIDE 11

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

■♥ ❛ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r② ✭❝❝✮ ✇❡ ❤❛✈❡ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❛ ♣r♦❞✉❝t ❘❡❝✉rs✐♦♥ ♦♣❡r❛t♦r ✐s ♦❜t❛✐♥❡❞ ❜② ♠❡❛♥s ♦❢ ❛♥ ♥♥♦

❉❡✜♥✐t✐♦♥

❆ ♥♥♦ ✐♥ ❛ ❝❛t❡❣♦r② ✇✐t❤ t❡r♠✐♥❛❧ ♦❜❥❡❝t ✶ ✐s ◆ ③ s ✇✐t❤ ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ✶

③

◆

♠ s

◆

♠

✶

❢

❆

❣

❆ ✇❤❡r❡ ♠ ✐s ✉♥✐q✉❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-12
SLIDE 12

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

■♥ ❛ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r② ✭❝❝✮ ✇❡ ❤❛✈❡ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❛ ♣r♦❞✉❝t ❘❡❝✉rs✐♦♥ ♦♣❡r❛t♦r ✐s ♦❜t❛✐♥❡❞ ❜② ♠❡❛♥s ♦❢ ❛♥ ♥♥♦

❉❡✜♥✐t✐♦♥

❆ ♥♥♦ ✐♥ ❛ ❝❛t❡❣♦r② C ✇✐t❤ t❡r♠✐♥❛❧ ♦❜❥❡❝t ✶ ✐s (◆, ③, s) ✇✐t❤ ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ✶

③

◆

♠ s

◆

♠

✶

❢

❆

❣

❆ ✇❤❡r❡ ♠ ✐s ✉♥✐q✉❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-13
SLIDE 13

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

■♥ ❛ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r② ✭❝❝✮ ✇❡ ❤❛✈❡ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❛ ♣r♦❞✉❝t ❘❡❝✉rs✐♦♥ ♦♣❡r❛t♦r ✐s ♦❜t❛✐♥❡❞ ❜② ♠❡❛♥s ♦❢ ❛♥ ♥♥♦

❉❡✜♥✐t✐♦♥

❆ ♥♥♦ ✐♥ ❛ ❝❛t❡❣♦r② C ✇✐t❤ t❡r♠✐♥❛❧ ♦❜❥❡❝t ✶ ✐s (◆, ③, s) ✇✐t❤ ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ✶

③

  • ◆

♠

  • s

◆

♠

  • ✶

❢

❆

❣

❆ ✇❤❡r❡ ♠ ✐s ✉♥✐q✉❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-14
SLIDE 14

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

❇❛s✐❝ r❡❝✉rs✐✈❡ str✉❝t✉r❡ ✐s ❛ ♣❛r❛♠❡tr✐③❡❞ ♥♥♦ ❉❡✜♥✐t✐♦♥ ❆ ♣❛r❛♠❡tr✐③❡❞ ♥♥♦ ✭♣♥♥♦✮ ✐s ❛♥ ♥♥♦ ◆ ③ s ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ❳

③❳

◆❳

♠ s❳

◆❳

♠

❳

❢

❆

❣

❆ ❲❡ ❝❛♥ s♣❡❛❦ ♦❢ ❛ ✇❡❛❦ ♥♥♦ ✭✇♥♥♦✮ ✐❢ ✉♥✐q✉❡♥❡ss ✐s ♥♦t r❡q✉✐r❡❞

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-15
SLIDE 15

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

❇❛s✐❝ r❡❝✉rs✐✈❡ str✉❝t✉r❡ ✐s ❛ ♣❛r❛♠❡tr✐③❡❞ ♥♥♦ ❉❡✜♥✐t✐♦♥ ❆ ♣❛r❛♠❡tr✐③❡❞ ♥♥♦ ✭♣♥♥♦✮ ✐s ❛♥ ♥♥♦ (◆, ③, s) ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ❳

③❳

◆❳

♠ s❳

◆❳

♠

❳

❢

❆

❣

❆ ❲❡ ❝❛♥ s♣❡❛❦ ♦❢ ❛ ✇❡❛❦ ♥♥♦ ✭✇♥♥♦✮ ✐❢ ✉♥✐q✉❡♥❡ss ✐s ♥♦t r❡q✉✐r❡❞

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-16
SLIDE 16

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

❇❛s✐❝ r❡❝✉rs✐✈❡ str✉❝t✉r❡ ✐s ❛ ♣❛r❛♠❡tr✐③❡❞ ♥♥♦ ❉❡✜♥✐t✐♦♥ ❆ ♣❛r❛♠❡tr✐③❡❞ ♥♥♦ ✭♣♥♥♦✮ ✐s ❛♥ ♥♥♦ (◆, ③, s) ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ❳

③❳

  • ◆❳

♠

  • s❳

◆❳

♠

  • ❳

❢

❆

❣

❆

❲❡ ❝❛♥ s♣❡❛❦ ♦❢ ❛ ✇❡❛❦ ♥♥♦ ✭✇♥♥♦✮ ✐❢ ✉♥✐q✉❡♥❡ss ✐s ♥♦t r❡q✉✐r❡❞

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-17
SLIDE 17

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

❇❛s✐❝ r❡❝✉rs✐✈❡ str✉❝t✉r❡ ✐s ❛ ♣❛r❛♠❡tr✐③❡❞ ♥♥♦ ❉❡✜♥✐t✐♦♥ ❆ ♣❛r❛♠❡tr✐③❡❞ ♥♥♦ ✭♣♥♥♦✮ ✐s ❛♥ ♥♥♦ (◆, ③, s) ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ❳

③❳

  • ◆❳

♠

  • s❳

◆❳

♠

  • ❳

❢

❆

❣

❆

❲❡ ❝❛♥ s♣❡❛❦ ♦❢ ❛ ✇❡❛❦ ♥♥♦ ✭✇♥♥♦✮ ✐❢ ✉♥✐q✉❡♥❡ss ✐s ♥♦t r❡q✉✐r❡❞

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-18
SLIDE 18

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

❚❤❡s❡ ❞❡✜♥✐t✐♦♥s ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞

❉❡✜♥✐t✐♦♥

❆ ❧❡❢t ♥♥♦ ✭❧♥♥♦✮ ✐♥ ❛ ▼♦♥♦✐❞❛❧ ❈❛t❡❣♦r② ■ ✐s ◆ ③ s s✉❝❤ t❤❛t ■ ❆

③ ❆ ◆

❆

♠ s ❆ ◆

❆

♠

❆

❢

❇

❣

❇ ❝♦♠♠✉t❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-19
SLIDE 19

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

❚❤❡s❡ ❞❡✜♥✐t✐♦♥s ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞

❉❡✜♥✐t✐♦♥

❆ ❧❡❢t ♥♥♦ ✭❧♥♥♦✮ ✐♥ ❛ ▼♦♥♦✐❞❛❧ ❈❛t❡❣♦r② V =(V, ⊗, ■, α, λ, ̺) ✐s (◆, ③, s) s✉❝❤ t❤❛t ■ ❆

③ ❆ ◆

❆

♠ s ❆ ◆

❆

♠

❆

❢

❇

❣

❇ ❝♦♠♠✉t❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-20
SLIDE 20

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

❚❤❡s❡ ❞❡✜♥✐t✐♦♥s ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞

❉❡✜♥✐t✐♦♥

❆ ❧❡❢t ♥♥♦ ✭❧♥♥♦✮ ✐♥ ❛ ▼♦♥♦✐❞❛❧ ❈❛t❡❣♦r② V =(V, ⊗, ■, α, λ, ̺) ✐s (◆, ③, s) s✉❝❤ t❤❛t ■ ⊗ ❆

③⊗❆ λ

  • ◆ ⊗ ❆

♠

  • s⊗❆ ◆ ⊗ ❆

♠

  • ❆

❢

❇

❣

❇ ❝♦♠♠✉t❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-21
SLIDE 21

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s

❚❤❡s❡ ❞❡✜♥✐t✐♦♥s ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞

❉❡✜♥✐t✐♦♥

❆ ❧❡❢t ♥♥♦ ✭❧♥♥♦✮ ✐♥ ❛ ▼♦♥♦✐❞❛❧ ❈❛t❡❣♦r② V =(V, ⊗, ■, α, λ, ̺) ✐s (◆, ③, s) s✉❝❤ t❤❛t ■ ⊗ ❆

③⊗❆ λ

  • ◆ ⊗ ❆

♠

  • s⊗❆ ◆ ⊗ ❆

♠

  • ❆

❢

❇

❣

❇ ❝♦♠♠✉t❡s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-22
SLIDE 22

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❘❡♣r❡s❡♥t❛t✐♦♥

❲❡ ❞❡✜♥❡ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s ✐♥ ❝❝ + ♥♥♦ ❜② ✐ts r❡♣r❡s❡♥t❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❢

❦

✐s r❡♣r❡s❡♥t❛❜❧❡ ✐♥ ❛ ❝❝ ♥♥♦ ✐❢ t❤❡r❡ ❡①✐sts ❢ ◆❦ ◆ ✐♥ s✉❝❤ t❤❛t ❢ ♥✶ ♥❦ ❢ ♥✶ ♥❦

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-23
SLIDE 23

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❘❡♣r❡s❡♥t❛t✐♦♥

❲❡ ❞❡✜♥❡ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s ✐♥ ❝❝ + ♥♥♦ ❜② ✐ts r❡♣r❡s❡♥t❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❢ : N❦ − → N ✐s r❡♣r❡s❡♥t❛❜❧❡ ✐♥ ❛ ❝❝ + ♥♥♦ C ✐❢ t❤❡r❡ ❡①✐sts ❢ : ◆❦ − → ◆ ✐♥ C s✉❝❤ t❤❛t ❢ ♯♥✶, ..., ♯♥❦ = ♯❢ (♥✶, ..., ♥❦)

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-24
SLIDE 24

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ r❡s✉❧ts

  • ✐✈❡♥ ❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡s ✇❡ ♦❜t❛✐♥ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s✿

❋♦r t♦t❛❧ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s ✐♥ t❤❡ ❢♦r♠ ❢

❦

✇❡ ❤❛✈❡✿

✶

❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰✇♣♥♥♦

✷

❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❚♦♣♦s ♥♥♦ ❚♦t❛❧❘❡❝

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-25
SLIDE 25

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ r❡s✉❧ts

  • ✐✈❡♥ ❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡s ✇❡ ♦❜t❛✐♥ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s✿

❋♦r t♦t❛❧ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s ✐♥ t❤❡ ❢♦r♠ ❢ : N❦ − → N ✇❡ ❤❛✈❡✿

✶

❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰✇♣♥♥♦

✷

❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❚♦♣♦s ♥♥♦ ❚♦t❛❧❘❡❝

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-26
SLIDE 26

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ r❡s✉❧ts

  • ✐✈❡♥ ❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡s ✇❡ ♦❜t❛✐♥ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s✿

❋♦r t♦t❛❧ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s ✐♥ t❤❡ ❢♦r♠ ❢ : N❦ − → N ✇❡ ❤❛✈❡✿

✶

PR = {❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰✇♣♥♥♦}

✷

PR ⊆ {❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❚♦♣♦s+♥♥♦} ⊂ ❚♦t❛❧❘❡❝

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

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SLIDE 27

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ r❡s✉❧ts

❋♦r ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡✿

P❛rt✐❛❧❘❡❝ ❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰♥♥♦✰❡q✉❛❧✐③❡rs

❆♥❞ ❢♦r ❛ ❞✐✛❡r❡♥t ❦✐♥❞ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✿

P❛rt✐❛❧❘❡❝ ◆✉♠❡r❛❧✇✐s❡ ❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰✇♥♥♦✰❡q✉❛❧✐③❡rs

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-28
SLIDE 28

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ r❡s✉❧ts

❋♦r ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡✿

P❛rt✐❛❧❘❡❝ ⊆ {❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰♥♥♦✰❡q✉❛❧✐③❡rs}

❆♥❞ ❢♦r ❛ ❞✐✛❡r❡♥t ❦✐♥❞ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✿

P❛rt✐❛❧❘❡❝ ◆✉♠❡r❛❧✇✐s❡ ❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰✇♥♥♦✰❡q✉❛❧✐③❡rs

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-29
SLIDE 29

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ r❡s✉❧ts

❋♦r ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡✿

P❛rt✐❛❧❘❡❝ ⊆ {❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰♥♥♦✰❡q✉❛❧✐③❡rs}

❆♥❞ ❢♦r ❛ ❞✐✛❡r❡♥t ❦✐♥❞ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✿

P❛rt✐❛❧❘❡❝ ◆✉♠❡r❛❧✇✐s❡ ❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰✇♥♥♦✰❡q✉❛❧✐③❡rs

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-30
SLIDE 30

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚❤❡ ♥♥♦ ❞✐❛❣r❛♠s ❘❡♣r❡s❡♥t❛t✐♦♥ ❱❛r✐♦✉s ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

❚❤❡ r❡s✉❧ts

❋♦r ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡✿

P❛rt✐❛❧❘❡❝ ⊆ {❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰♥♥♦✰❡q✉❛❧✐③❡rs}

❆♥❞ ❢♦r ❛ ❞✐✛❡r❡♥t ❦✐♥❞ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✿

P❛rt✐❛❧❘❡❝ = {◆✉♠❡r❛❧✇✐s❡ ❘❡♣r❡s❡♥t❛❜❧❡s ✐♥ ❝❝✰✇♥♥♦✰❡q✉❛❧✐③❡rs}

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-31
SLIDE 31

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❚✇♦ ❝♦♥str✉❝t✐♦♥s

✶✳ ♦✈❡r ●r♣❤

P❡❛♥♦✲▲❛✇✈❡r❡ ❆①✐♦♠ ✐♥ ❛ ❝❛t❡❣♦r②

❆ ❝❛t❡❣♦r② s❛t✐s✜❡s P▲ ❆①✐♦♠ ✭♦r ✐t ✐s P▲✮ ✐❢ ❡✈❡r② ♦❜❥❡❝t ❤❛s ❛♥ ♥♥♦ P▲ ❝❛t❡❣♦r✐❡s ❛r❡ ♥♦t ✐♥ ❣❡♥❡r❛❧ ❝❛rt❡s✐❛♥ ♥♦r ❛r❡ t❤❡② ❡♥❞♦✇❡❞ ✇✐t❤ ❛ t❡r♠✐♥❛❧ ♦❜❥❡❝t ❍♦✇❡✈❡r P ❋P▲ ❤❛s ❜♦t❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-32
SLIDE 32

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❚✇♦ ❝♦♥str✉❝t✐♦♥s

✶✳ ♦✈❡r ●r♣❤

P❡❛♥♦✲▲❛✇✈❡r❡ ❆①✐♦♠ ✐♥ ❛ ❝❛t❡❣♦r②

❆ ❝❛t❡❣♦r② s❛t✐s✜❡s P▲ ❆①✐♦♠ ✭♦r ✐t ✐s P▲✮ ✐❢ ❡✈❡r② ♦❜❥❡❝t ❤❛s ❛♥ ♥♥♦ P▲ ❝❛t❡❣♦r✐❡s ❛r❡ ♥♦t ✐♥ ❣❡♥❡r❛❧ ❝❛rt❡s✐❛♥ ♥♦r ❛r❡ t❤❡② ❡♥❞♦✇❡❞ ✇✐t❤ ❛ t❡r♠✐♥❛❧ ♦❜❥❡❝t ❍♦✇❡✈❡r P ❋P▲ ❤❛s ❜♦t❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-33
SLIDE 33

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❚✇♦ ❝♦♥str✉❝t✐♦♥s

✶✳ ♦✈❡r ●r♣❤

P❡❛♥♦✲▲❛✇✈❡r❡ ❆①✐♦♠ ✐♥ ❛ ❝❛t❡❣♦r②

❆ ❝❛t❡❣♦r② s❛t✐s✜❡s P▲ ❆①✐♦♠ ✭♦r ✐t ✐s P▲✮ ✐❢ ❡✈❡r② ♦❜❥❡❝t ❤❛s ❛♥ ♥♥♦ P▲ ❝❛t❡❣♦r✐❡s ❛r❡ ♥♦t ✐♥ ❣❡♥❡r❛❧ ❝❛rt❡s✐❛♥ ♥♦r ❛r❡ t❤❡② ❡♥❞♦✇❡❞ ✇✐t❤ ❛ t❡r♠✐♥❛❧ ♦❜❥❡❝t ❍♦✇❡✈❡r P ❋P▲ ❤❛s ❜♦t❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-34
SLIDE 34

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❚✇♦ ❝♦♥str✉❝t✐♦♥s

✶✳ ♦✈❡r ●r♣❤

P❡❛♥♦✲▲❛✇✈❡r❡ ❆①✐♦♠ ✐♥ ❛ ❝❛t❡❣♦r②

❆ ❝❛t❡❣♦r② s❛t✐s✜❡s P▲ ❆①✐♦♠ ✭♦r ✐t ✐s P▲✮ ✐❢ ❡✈❡r② ♦❜❥❡❝t ❤❛s ❛♥ ♥♥♦ P▲ ❝❛t❡❣♦r✐❡s ❛r❡ ♥♦t ✐♥ ❣❡♥❡r❛❧ ❝❛rt❡s✐❛♥ ♥♦r ❛r❡ t❤❡② ❡♥❞♦✇❡❞ ✇✐t❤ ❛ t❡r♠✐♥❛❧ ♦❜❥❡❝t ❍♦✇❡✈❡r P ❋P▲ ❤❛s ❜♦t❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-35
SLIDE 35

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❚✇♦ ❝♦♥str✉❝t✐♦♥s

✶✳ ♦✈❡r ●r♣❤

P❡❛♥♦✲▲❛✇✈❡r❡ ❆①✐♦♠ ✐♥ ❛ ❝❛t❡❣♦r②

❆ ❝❛t❡❣♦r② s❛t✐s✜❡s P▲ ❆①✐♦♠ ✭♦r ✐t ✐s P▲✮ ✐❢ ❡✈❡r② ♦❜❥❡❝t ❤❛s ❛♥ ♥♥♦ P▲ ❝❛t❡❣♦r✐❡s ❛r❡ ♥♦t ✐♥ ❣❡♥❡r❛❧ ❝❛rt❡s✐❛♥ ♥♦r ❛r❡ t❤❡② ❡♥❞♦✇❡❞ ✇✐t❤ ❛ t❡r♠✐♥❛❧ ♦❜❥❡❝t ❍♦✇❡✈❡r P = ❋P▲(·) ❤❛s ❜♦t❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-36
SLIDE 36

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

▲❡t ❜❡ P = ❋P▲(·) t❤❡ ❈❛t❡❣♦r② ♦❢ PR✲❢♦r♠❛❧ ❢✉♥❝t✐♦♥s ❉❡✜♥✐t✐♦♥

❲❡ ❞❡✜♥❡ P ❛s t❤❡ P▲ ♣r❡❝❛t❡❣♦r② ❣❡♥❡r❛t❡❞ ✐t ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ♣r❡❝❛t❡❣♦r② ♦❢ ✲♣r♦❣r❛♠s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-37
SLIDE 37

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

▲❡t ❜❡ P = ❋P▲(·) t❤❡ ❈❛t❡❣♦r② ♦❢ PR✲❢♦r♠❛❧ ❢✉♥❝t✐♦♥s ❉❡✜♥✐t✐♦♥

❲❡ ❞❡✜♥❡ P ❛s t❤❡ P▲ ♣r❡❝❛t❡❣♦r② ❣❡♥❡r❛t❡❞ ✐t ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ♣r❡❝❛t❡❣♦r② ♦❢ ✲♣r♦❣r❛♠s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-38
SLIDE 38

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

▲❡t ❜❡ P = ❋P▲(·) t❤❡ ❈❛t❡❣♦r② ♦❢ PR✲❢♦r♠❛❧ ❢✉♥❝t✐♦♥s ❉❡✜♥✐t✐♦♥

❲❡ ❞❡✜♥❡ P′ ❛s t❤❡ P▲ ♣r❡❝❛t❡❣♦r② ❣❡♥❡r❛t❡❞ (·) ✐t ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ♣r❡❝❛t❡❣♦r② ♦❢ ✲♣r♦❣r❛♠s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-39
SLIDE 39

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

▲❡t ❜❡ P = ❋P▲(·) t❤❡ ❈❛t❡❣♦r② ♦❢ PR✲❢♦r♠❛❧ ❢✉♥❝t✐♦♥s ❉❡✜♥✐t✐♦♥

❲❡ ❞❡✜♥❡ P′ ❛s t❤❡ P▲ ♣r❡❝❛t❡❣♦r② ❣❡♥❡r❛t❡❞ (·) ✐t ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ♣r❡❝❛t❡❣♦r② ♦❢ PR✲♣r♦❣r❛♠s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-40
SLIDE 40

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

▲❡♠❛ ❲❡ ❝❛❧❧ P t❤❡ ✐♠❛❣❡ ❣r❛♣❤ P

❙❡t

✐ts ♦❜❥❡❝ts ❛r❡

♣

✐ts ♠♦r♣❤✐s♠s ❢

♣ q s✉❝❤ t❤❛t ❢

❢✵ ❢✶ ❢q

✶

✇❤❡r❡ ❢✐ ❝❛♥ ✭♦♥❧②❄✮ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s ✐♥ P

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-41
SLIDE 41

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

▲❡♠❛ ❲❡ ❝❛❧❧ P t❤❡ ✐♠❛❣❡ ❣r❛♣❤ P

❙❡t

✐ts ♦❜❥❡❝ts ❛r❡ N♣ ✐ts ♠♦r♣❤✐s♠s ❢

♣ q s✉❝❤ t❤❛t ❢

❢✵ ❢✶ ❢q

✶

✇❤❡r❡ ❢✐ ❝❛♥ ✭♦♥❧②❄✮ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s ✐♥ P

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-42
SLIDE 42

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

▲❡♠❛ ❲❡ ❝❛❧❧ P t❤❡ ✐♠❛❣❡ ❣r❛♣❤ P

❙❡t

✐ts ♦❜❥❡❝ts ❛r❡ N♣ ✐ts ♠♦r♣❤✐s♠s ❢ : N♣

Nq s✉❝❤ t❤❛t ❢ = (❢✵, ❢✶, ..., ❢q−✶)

✇❤❡r❡ ❢✐ ∈ PR ❝❛♥ ✭♦♥❧②❄✮ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s ✐♥ P

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-43
SLIDE 43

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

▲❡♠❛ ❲❡ ❝❛❧❧ P t❤❡ ✐♠❛❣❡ ❣r❛♣❤ P

❙❡t

✐ts ♦❜❥❡❝ts ❛r❡ N♣ ✐ts ♠♦r♣❤✐s♠s ❢ : N♣

Nq s✉❝❤ t❤❛t ❢ = (❢✵, ❢✶, ..., ❢q−✶)

✇❤❡r❡ ❢✐ ∈ PR P ❝❛♥ ✭♦♥❧②❄✮ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s ✐♥ P

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-44
SLIDE 44

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❲❡ ❝❛♥ s✉♠♠❛r✐③❡ ❛s

P P

P ✐s ❛ s②♥t❛❝t✐❝❛❧ ❝♦♥str✉❝t✐♦♥ ✇❤✐❧❡ ✐s ❛ ❝❛t❡❣♦r② ✇✐t❤ s❡♠❛♥t✐❝s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-45
SLIDE 45

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❲❡ ❝❛♥ s✉♠♠❛r✐③❡ ❛s

P′

  • (·)
  • P
  • P

P ✐s ❛ s②♥t❛❝t✐❝❛❧ ❝♦♥str✉❝t✐♦♥ ✇❤✐❧❡ ✐s ❛ ❝❛t❡❣♦r② ✇✐t❤ s❡♠❛♥t✐❝s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-46
SLIDE 46

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❲❡ ❝❛♥ s✉♠♠❛r✐③❡ ❛s

P′

  • (·)
  • P
  • P

P′ ✐s ❛ s②♥t❛❝t✐❝❛❧ ❝♦♥str✉❝t✐♦♥ ✇❤✐❧❡ P ✐s ❛ ❝❛t❡❣♦r② ✇✐t❤ s❡♠❛♥t✐❝s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-47
SLIDE 47

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

✷✳ ♦✈❡r ❙❡t

❈♦♥❝❡♣t ♦❢ ❛❧❣♦r✐t❤♠ ✐s ❜♦✉♥❞❡❞ ❜② ✶✳ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s t❤❛t ✇❡ ❤❛♥❞❧❡✿ ♣r♦❣r❛♠s ✷✳ ❢♦r♠❛❧✐③❛t✐♦♥s t❤❛t ❛r❡ ❦♥♦✇♥✿ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-48
SLIDE 48

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

✷✳ ♦✈❡r ❙❡t

❈♦♥❝❡♣t ♦❢ ❛❧❣♦r✐t❤♠ ✐s ❜♦✉♥❞❡❞ ❜② ✶✳ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s t❤❛t ✇❡ ❤❛♥❞❧❡✿ ♣r♦❣r❛♠s ✷✳ ❢♦r♠❛❧✐③❛t✐♦♥s t❤❛t ❛r❡ ❦♥♦✇♥✿ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-49
SLIDE 49

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

✷✳ ♦✈❡r ❙❡t

❈♦♥❝❡♣t ♦❢ ❛❧❣♦r✐t❤♠ ✐s ❜♦✉♥❞❡❞ ❜② ✶✳ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s t❤❛t ✇❡ ❤❛♥❞❧❡✿ ♣r♦❣r❛♠s ✷✳ ❢♦r♠❛❧✐③❛t✐♦♥s t❤❛t ❛r❡ ❦♥♦✇♥✿ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-50
SLIDE 50

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

✷✳ ♦✈❡r ❙❡t

❈♦♥❝❡♣t ♦❢ ❛❧❣♦r✐t❤♠ ✐s ❜♦✉♥❞❡❞ ❜② ✶✳ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s t❤❛t ✇❡ ❤❛♥❞❧❡✿ ♣r♦❣r❛♠s ✷✳ ❢♦r♠❛❧✐③❛t✐♦♥s t❤❛t ❛r❡ ❦♥♦✇♥✿ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-51
SLIDE 51

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

✷✳ ♦✈❡r ❙❡t

❈♦♥❝❡♣t ♦❢ ❛❧❣♦r✐t❤♠ ✐s ❜♦✉♥❞❡❞ ❜② ✶✳ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s t❤❛t ✇❡ ❤❛♥❞❧❡✿ ♣r♦❣r❛♠s ✷✳ ❢♦r♠❛❧✐③❛t✐♦♥s t❤❛t ❛r❡ ❦♥♦✇♥✿ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-52
SLIDE 52

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❚❤❡r❡ ✐s ❛ tr❡❡ ❢♦r ❛♥② PR✲♣r♦❣r❛♠ ❧❛❜❡❧❧✐♥❣ t❤❡ ❡❞❣❡s ✇✐t❤ ❈ ✭❝♦♠♣♦s✐t✐♦♥✮✱ ❘ ✭r❡❝✉rs✐♦♥✮ ❛♥❞ ❇ ✭❜r❛❝❦❡t✮

♥♦❞❡s ❛r❡

❦

❡❞❣❡s ❛r❡ ✲❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡❞ ❢r♦♠ ③ s ❛♥❞

❦ ✐

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ t❤✐s ❣r❛♣❤ ❞❡s❝ ✐t ❤❛s ❛❧❧ ❞❡s❝r✐♣t✐♦♥s ❛❜♦✉t ❤♦✇ t♦ ❝♦♠♣✉t❡ ❛❧❧

✲❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-53
SLIDE 53

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❚❤❡r❡ ✐s ❛ tr❡❡ ❢♦r ❛♥② PR✲♣r♦❣r❛♠ ❧❛❜❡❧❧✐♥❣ t❤❡ ❡❞❣❡s ✇✐t❤ ❈ ✭❝♦♠♣♦s✐t✐♦♥✮✱ ❘ ✭r❡❝✉rs✐♦♥✮ ❛♥❞ ❇ ✭❜r❛❝❦❡t✮

♥♦❞❡s ❛r❡

❦

❡❞❣❡s ❛r❡ ✲❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡❞ ❢r♦♠ ③ s ❛♥❞

❦ ✐

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ t❤✐s ❣r❛♣❤ ❞❡s❝ ✐t ❤❛s ❛❧❧ ❞❡s❝r✐♣t✐♦♥s ❛❜♦✉t ❤♦✇ t♦ ❝♦♠♣✉t❡ ❛❧❧

✲❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-54
SLIDE 54

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❚❤❡r❡ ✐s ❛ tr❡❡ ❢♦r ❛♥② PR✲♣r♦❣r❛♠ ❧❛❜❡❧❧✐♥❣ t❤❡ ❡❞❣❡s ✇✐t❤ ❈ ✭❝♦♠♣♦s✐t✐♦♥✮✱ ❘ ✭r❡❝✉rs✐♦♥✮ ❛♥❞ ❇ ✭❜r❛❝❦❡t✮

♥♦❞❡s ❛r❡

❦

❡❞❣❡s ❛r❡ ✲❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡❞ ❢r♦♠ ③ s ❛♥❞

❦ ✐

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ t❤✐s ❣r❛♣❤ ❞❡s❝ ✐t ❤❛s ❛❧❧ ❞❡s❝r✐♣t✐♦♥s ❛❜♦✉t ❤♦✇ t♦ ❝♦♠♣✉t❡ ❛❧❧

✲❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-55
SLIDE 55

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❚❤❡r❡ ✐s ❛ tr❡❡ ❢♦r ❛♥② PR✲♣r♦❣r❛♠ ❧❛❜❡❧❧✐♥❣ t❤❡ ❡❞❣❡s ✇✐t❤ ❈ ✭❝♦♠♣♦s✐t✐♦♥✮✱ ❘ ✭r❡❝✉rs✐♦♥✮ ❛♥❞ ❇ ✭❜r❛❝❦❡t✮

♥♦❞❡s ❛r❡ N❦ ❡❞❣❡s ❛r❡ ✲❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡❞ ❢r♦♠ ③ s ❛♥❞

❦ ✐

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ t❤✐s ❣r❛♣❤ ❞❡s❝ ✐t ❤❛s ❛❧❧ ❞❡s❝r✐♣t✐♦♥s ❛❜♦✉t ❤♦✇ t♦ ❝♦♠♣✉t❡ ❛❧❧

✲❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-56
SLIDE 56

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❚❤❡r❡ ✐s ❛ tr❡❡ ❢♦r ❛♥② PR✲♣r♦❣r❛♠ ❧❛❜❡❧❧✐♥❣ t❤❡ ❡❞❣❡s ✇✐t❤ ❈ ✭❝♦♠♣♦s✐t✐♦♥✮✱ ❘ ✭r❡❝✉rs✐♦♥✮ ❛♥❞ ❇ ✭❜r❛❝❦❡t✮

♥♦❞❡s ❛r❡ N❦ ❡❞❣❡s ❛r❡ PR✲❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡❞ ❢r♦♠ ③, s ❛♥❞ π❦

✐

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ t❤✐s ❣r❛♣❤ ❞❡s❝ ✐t ❤❛s ❛❧❧ ❞❡s❝r✐♣t✐♦♥s ❛❜♦✉t ❤♦✇ t♦ ❝♦♠♣✉t❡ ❛❧❧

✲❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-57
SLIDE 57

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❚❤❡r❡ ✐s ❛ tr❡❡ ❢♦r ❛♥② PR✲♣r♦❣r❛♠ ❧❛❜❡❧❧✐♥❣ t❤❡ ❡❞❣❡s ✇✐t❤ ❈ ✭❝♦♠♣♦s✐t✐♦♥✮✱ ❘ ✭r❡❝✉rs✐♦♥✮ ❛♥❞ ❇ ✭❜r❛❝❦❡t✮

♥♦❞❡s ❛r❡ N❦ ❡❞❣❡s ❛r❡ PR✲❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡❞ ❢r♦♠ ③, s ❛♥❞ π❦

✐

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ t❤✐s ❣r❛♣❤ PR❞❡s❝ ✐t ❤❛s ❛❧❧ ❞❡s❝r✐♣t✐♦♥s ❛❜♦✉t ❤♦✇ t♦ ❝♦♠♣✉t❡ ❛❧❧

✲❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-58
SLIDE 58

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❚❤❡r❡ ✐s ❛ tr❡❡ ❢♦r ❛♥② PR✲♣r♦❣r❛♠ ❧❛❜❡❧❧✐♥❣ t❤❡ ❡❞❣❡s ✇✐t❤ ❈ ✭❝♦♠♣♦s✐t✐♦♥✮✱ ❘ ✭r❡❝✉rs✐♦♥✮ ❛♥❞ ❇ ✭❜r❛❝❦❡t✮

♥♦❞❡s ❛r❡ N❦ ❡❞❣❡s ❛r❡ PR✲❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡❞ ❢r♦♠ ③, s ❛♥❞ π❦

✐

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ t❤✐s ❣r❛♣❤ PR❞❡s❝ ✐t ❤❛s ❛❧❧ ❞❡s❝r✐♣t✐♦♥s ❛❜♦✉t ❤♦✇ t♦ ❝♦♠♣✉t❡ ❛❧❧ PR✲❢✉♥❝t✐♦♥s

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-59
SLIDE 59

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❊ss❡♥t✐❛❧❧② ❡q✉❛❧ ∼ ✐♥ PR❞❡s❝ ❣✐✈❡s ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s

❲❡ ❝♦♥str✉❝t ❜② ♣r✉♥♥✐♥❣ ❞❡s❝ ❆❧❣ ❛s t❤❡ ❢r❡❡ ✐♥✐t✐❛❧ ❝❛t❡❣♦r② ❋❈❛t①◆ Ø ✐♥ ❈❛t①◆

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-60
SLIDE 60

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❊ss❡♥t✐❛❧❧② ❡q✉❛❧ ∼ ✐♥ PR❞❡s❝ ❣✐✈❡s ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s

❲❡ ❝♦♥str✉❝t ❜② ♣r✉♥♥✐♥❣ ❞❡s❝ ❆❧❣ ❛s t❤❡ ❢r❡❡ ✐♥✐t✐❛❧ ❝❛t❡❣♦r② ❋❈❛t①◆ Ø ✐♥ ❈❛t①◆

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-61
SLIDE 61

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❊ss❡♥t✐❛❧❧② ❡q✉❛❧ ∼ ✐♥ PR❞❡s❝ ❣✐✈❡s ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s

❲❡ ❝♦♥str✉❝t ❜② ♣r✉♥♥✐♥❣ PR❞❡s❝ ∼ = PR❆❧❣ ❛s t❤❡ ❢r❡❡ ✐♥✐t✐❛❧ ❝❛t❡❣♦r② ❋❈❛t①◆(Ø) ✐♥ ❈❛t①◆

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-62
SLIDE 62

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❘❡❞✉❝✐♥❣ ❜② ≈ ✭r✉♥ t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥✮ ✇❡ ❤❛✈❡ ❆❧❣ ❋✉♥❝ ❛♥❞ t❤❡ s❝❤❡♠❛

❞❡s❝ Ø ❆❧❣ ❋✉♥❝

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-63
SLIDE 63

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❘❡❞✉❝✐♥❣ ❜② ≈ ✭r✉♥ t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥✮ ✇❡ ❤❛✈❡ PR❆❧❣ ≈ = PR❋✉♥❝ ❛♥❞ t❤❡ s❝❤❡♠❛

❞❡s❝ Ø ❆❧❣ ❋✉♥❝

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-64
SLIDE 64

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❘❡❞✉❝✐♥❣ ❜② ≈ ✭r✉♥ t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥✮ ✇❡ ❤❛✈❡ PR❆❧❣ ≈ = PR❋✉♥❝ ❛♥❞ t❤❡ s❝❤❡♠❛

❞❡s❝ Ø ❆❧❣ ❋✉♥❝

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-65
SLIDE 65

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❘❡❞✉❝✐♥❣ ❜② ≈ ✭r✉♥ t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥✮ ✇❡ ❤❛✈❡ PR❆❧❣ ≈ = PR❋✉♥❝ ❛♥❞ t❤❡ s❝❤❡♠❛

PR❞❡s❝

  • Ø
  • PR❆❧❣
  • PR❋✉♥❝

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-66
SLIDE 66

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❙✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥s ✐♥✐t✐❛❧ P❘❯ ✭P❢❡♥❞❡r✮ ❋r❡②❞ ❈♦✈❡r ❢r♦♠ ❛ ❝❝ ♥♥♦ ✭❘♦♠á♥✮ ❢r❡❡ ▼♦♥♦✐❞❛❧ ❈❛t❡❣♦r② ❧♥♥♦ Ø ✭❘♦♠á♥✲P❛ré✮

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-67
SLIDE 67

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❙✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥s ✐♥✐t✐❛❧ P❘❯ E ✭P❢❡♥❞❡r✮ ❋r❡②❞ ❈♦✈❡r ❢r♦♠ ❛ ❝❝ ♥♥♦ ✭❘♦♠á♥✮ ❢r❡❡ ▼♦♥♦✐❞❛❧ ❈❛t❡❣♦r② ❧♥♥♦ Ø ✭❘♦♠á♥✲P❛ré✮

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-68
SLIDE 68

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❙✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥s ✐♥✐t✐❛❧ P❘❯ E ✭P❢❡♥❞❡r✮ ❋r❡②❞ ❈♦✈❡r FC ❢r♦♠ ❛ ❝❝ C + ♥♥♦ ✭❘♦♠á♥✮ ❢r❡❡ ▼♦♥♦✐❞❛❧ ❈❛t❡❣♦r② ❧♥♥♦ Ø ✭❘♦♠á♥✲P❛ré✮

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-69
SLIDE 69

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s ❈❛t❡❣♦r✐❡s ♦❢ ❆❧❣♦r✐t❤♠s

❋r❡❡ ❛♥❞ s②♥t❛❝t✐❝❛❧ str✉❝t✉r❡s

❙✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥s ✐♥✐t✐❛❧ P❘❯ E ✭P❢❡♥❞❡r✮ ❋r❡②❞ ❈♦✈❡r FC ❢r♦♠ ❛ ❝❝ C + ♥♥♦ ✭❘♦♠á♥✮ ❢r❡❡ ▼♦♥♦✐❞❛❧ ❈❛t❡❣♦r② + ❧♥♥♦ Φ(Ø) ✭❘♦♠á♥✲P❛ré✮

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-70
SLIDE 70

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈❧❛ss❡s ✐♥ ●③r❡❣♦r③❝②❦ ❍✐❡r❛r❝❤② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❜♦✉♥❞✐♥❣ ❛r✐t❤♠❡t✐❝s

❈♦♥s✐❞❡r t❤❡ s♠❛❧❧❡st ❞❡r✐✈❛t✐♦♥s s❡t

✶

❝♦♥t❛✐♥✐♥❣ ❛ ❞❡r✐✈❛t✐♦♥ ♦❢ ❡✈❡r② ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ✵✱ ❙① ① ✶✱ P① ♠❛① ✵ ① ✶ ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❈ ① ② ③

② ✐❢ ① ✵ ③ ❡❧s❡

✷

❝❧♦s❡❞ ✉♥❞❡r t❤❡ ❞❡r✐✈❛t✐♦♥ r✉❧❡s

✶

❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥✿ ❣✐✈❡♥ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶ ❣♠ ✇❡ ❞❡r✐✈❡ ❢ ① ❤ ❣✶ ①✶ ❣♠ ①♠

✷

❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡♥❡ss✿ ❢♦r ❣ ❛♥❞ ❤ ✐❢ ① ✵ ✇❡ ❞❡r✐✈❡ ❢ ① ② ❤ ① ② ❢ P① ②

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-71
SLIDE 71

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈❧❛ss❡s ✐♥ ●③r❡❣♦r③❝②❦ ❍✐❡r❛r❝❤② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❜♦✉♥❞✐♥❣ ❛r✐t❤♠❡t✐❝s

❈♦♥s✐❞❡r t❤❡ s♠❛❧❧❡st ❞❡r✐✈❛t✐♦♥s s❡t

✶

❝♦♥t❛✐♥✐♥❣ ❛ ❞❡r✐✈❛t✐♦♥ ♦❢ ❡✈❡r② ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ✵✱ ❙① ① ✶✱ P① ♠❛① ✵ ① ✶ ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❈ ① ② ③

② ✐❢ ① ✵ ③ ❡❧s❡

✷

❝❧♦s❡❞ ✉♥❞❡r t❤❡ ❞❡r✐✈❛t✐♦♥ r✉❧❡s

✶

❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥✿ ❣✐✈❡♥ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶ ❣♠ ✇❡ ❞❡r✐✈❡ ❢ ① ❤ ❣✶ ①✶ ❣♠ ①♠

✷

❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡♥❡ss✿ ❢♦r ❣ ❛♥❞ ❤ ✐❢ ① ✵ ✇❡ ❞❡r✐✈❡ ❢ ① ② ❤ ① ② ❢ P① ②

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-72
SLIDE 72

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈❧❛ss❡s ✐♥ ●③r❡❣♦r③❝②❦ ❍✐❡r❛r❝❤② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❜♦✉♥❞✐♥❣ ❛r✐t❤♠❡t✐❝s

❈♦♥s✐❞❡r t❤❡ s♠❛❧❧❡st ❞❡r✐✈❛t✐♦♥s s❡t

✶

❝♦♥t❛✐♥✐♥❣ ❛ ❞❡r✐✈❛t✐♦♥ ♦❢ ❡✈❡r② ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ✵✱ ❙① = ① + ✶✱ P① = ♠❛①(✵, ① − ✶) ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❈(①, ②, ③) =

  • ②

✐❢ ① = ✵ ③ ❡❧s❡

✷

❝❧♦s❡❞ ✉♥❞❡r t❤❡ ❞❡r✐✈❛t✐♦♥ r✉❧❡s

✶

❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥✿ ❣✐✈❡♥ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶ ❣♠ ✇❡ ❞❡r✐✈❡ ❢ ① ❤ ❣✶ ①✶ ❣♠ ①♠

✷

❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡♥❡ss✿ ❢♦r ❣ ❛♥❞ ❤ ✐❢ ① ✵ ✇❡ ❞❡r✐✈❡ ❢ ① ② ❤ ① ② ❢ P① ②

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-73
SLIDE 73

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈❧❛ss❡s ✐♥ ●③r❡❣♦r③❝②❦ ❍✐❡r❛r❝❤② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❜♦✉♥❞✐♥❣ ❛r✐t❤♠❡t✐❝s

❈♦♥s✐❞❡r t❤❡ s♠❛❧❧❡st ❞❡r✐✈❛t✐♦♥s s❡t

✶

❝♦♥t❛✐♥✐♥❣ ❛ ❞❡r✐✈❛t✐♦♥ ♦❢ ❡✈❡r② ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ✵✱ ❙① = ① + ✶✱ P① = ♠❛①(✵, ① − ✶) ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❈(①, ②, ③) =

  • ②

✐❢ ① = ✵ ③ ❡❧s❡

✷

❝❧♦s❡❞ ✉♥❞❡r t❤❡ ❞❡r✐✈❛t✐♦♥ r✉❧❡s

✶

❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥✿ ❣✐✈❡♥ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶ ❣♠ ✇❡ ❞❡r✐✈❡ ❢ ① ❤ ❣✶ ①✶ ❣♠ ①♠

✷

❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡♥❡ss✿ ❢♦r ❣ ❛♥❞ ❤ ✐❢ ① ✵ ✇❡ ❞❡r✐✈❡ ❢ ① ② ❤ ① ② ❢ P① ②

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-74
SLIDE 74

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈❧❛ss❡s ✐♥ ●③r❡❣♦r③❝②❦ ❍✐❡r❛r❝❤② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❜♦✉♥❞✐♥❣ ❛r✐t❤♠❡t✐❝s

❈♦♥s✐❞❡r t❤❡ s♠❛❧❧❡st ❞❡r✐✈❛t✐♦♥s s❡t

✶

❝♦♥t❛✐♥✐♥❣ ❛ ❞❡r✐✈❛t✐♦♥ ♦❢ ❡✈❡r② ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ✵✱ ❙① = ① + ✶✱ P① = ♠❛①(✵, ① − ✶) ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❈(①, ②, ③) =

  • ②

✐❢ ① = ✵ ③ ❡❧s❡

✷

❝❧♦s❡❞ ✉♥❞❡r t❤❡ ❞❡r✐✈❛t✐♦♥ r✉❧❡s

✶

❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥✿ ❣✐✈❡♥ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶, ..., ❣♠ ✇❡ ❞❡r✐✈❡ ❢ (①) = ❤(❣✶(①✶), ..., ❣♠(①♠))

✷

❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡♥❡ss✿ ❢♦r ❣ ❛♥❞ ❤ ✐❢ ① ✵ ✇❡ ❞❡r✐✈❡ ❢ ① ② ❤ ① ② ❢ P① ②

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-75
SLIDE 75

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈❧❛ss❡s ✐♥ ●③r❡❣♦r③❝②❦ ❍✐❡r❛r❝❤② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❜♦✉♥❞✐♥❣ ❛r✐t❤♠❡t✐❝s

❈♦♥s✐❞❡r t❤❡ s♠❛❧❧❡st ❞❡r✐✈❛t✐♦♥s s❡t

✶

❝♦♥t❛✐♥✐♥❣ ❛ ❞❡r✐✈❛t✐♦♥ ♦❢ ❡✈❡r② ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ✵✱ ❙① = ① + ✶✱ P① = ♠❛①(✵, ① − ✶) ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❈(①, ②, ③) =

  • ②

✐❢ ① = ✵ ③ ❡❧s❡

✷

❝❧♦s❡❞ ✉♥❞❡r t❤❡ ❞❡r✐✈❛t✐♦♥ r✉❧❡s

✶

❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥✿ ❣✐✈❡♥ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶, ..., ❣♠ ✇❡ ❞❡r✐✈❡ ❢ (①) = ❤(❣✶(①✶), ..., ❣♠(①♠))

✷

❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡♥❡ss✿ ❢♦r ❣ ❛♥❞ ❤ ✐❢ ① = ✵ ✇❡ ❞❡r✐✈❡ ❢ (①, ②) = ❤(①, ②, ❢ (P①, ②))

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-76
SLIDE 76

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈♦♠♣❧❡①✐t② ❧♦✇❡r t❤❛♥ PR ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✉s✐♥❣ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐t ✐s ❜❛s❡❞ ✐♥ ❝♦♠♣r❡❤❡♥s✐♦♥ ♦r

❆①✐♦♠ s❝❤❡♠❛ ♦❢ s♣❡❝✐✜❝❛t✐♦♥ ❆♥② ❞❡✜♥❛❜❧❡ s✉❜❝❧❛ss ♦❢ ❛ s❡t ✐s ❛ s❡t ■♥st❡❛❞ ♦❢ ❜♦✉♥❞✐♥❣ ✐♥❞✉❝t✐♦♥ ✇❡ ✉s❡ ✇❡❛❦❡r s✉❜s②st❡♠s ❲❡ ✉s❡ t✇♦ ❦✐♥❞s ♦❢ ❛r❣✉♠❡♥ts✿ ♥♦r♠❛❧ ❛♥❞ s❛❢❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-77
SLIDE 77

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈♦♠♣❧❡①✐t② ❧♦✇❡r t❤❛♥ PR ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✉s✐♥❣ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐t ✐s ❜❛s❡❞ ✐♥ ❝♦♠♣r❡❤❡♥s✐♦♥ ♦r

❆①✐♦♠ s❝❤❡♠❛ ♦❢ s♣❡❝✐✜❝❛t✐♦♥ ❆♥② ❞❡✜♥❛❜❧❡ s✉❜❝❧❛ss ♦❢ ❛ s❡t ✐s ❛ s❡t ■♥st❡❛❞ ♦❢ ❜♦✉♥❞✐♥❣ ✐♥❞✉❝t✐♦♥ ✇❡ ✉s❡ ✇❡❛❦❡r s✉❜s②st❡♠s ❲❡ ✉s❡ t✇♦ ❦✐♥❞s ♦❢ ❛r❣✉♠❡♥ts✿ ♥♦r♠❛❧ ❛♥❞ s❛❢❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-78
SLIDE 78

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈♦♠♣❧❡①✐t② ❧♦✇❡r t❤❛♥ PR ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✉s✐♥❣ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐t ✐s ❜❛s❡❞ ✐♥ ❝♦♠♣r❡❤❡♥s✐♦♥ ♦r

❆①✐♦♠ s❝❤❡♠❛ ♦❢ s♣❡❝✐✜❝❛t✐♦♥ ❆♥② ❞❡✜♥❛❜❧❡ s✉❜❝❧❛ss ♦❢ ❛ s❡t ✐s ❛ s❡t ■♥st❡❛❞ ♦❢ ❜♦✉♥❞✐♥❣ ✐♥❞✉❝t✐♦♥ ✇❡ ✉s❡ ✇❡❛❦❡r s✉❜s②st❡♠s ❲❡ ✉s❡ t✇♦ ❦✐♥❞s ♦❢ ❛r❣✉♠❡♥ts✿ ♥♦r♠❛❧ ❛♥❞ s❛❢❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-79
SLIDE 79

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈♦♠♣❧❡①✐t② ❧♦✇❡r t❤❛♥ PR ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✉s✐♥❣ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐t ✐s ❜❛s❡❞ ✐♥ ❝♦♠♣r❡❤❡♥s✐♦♥ ♦r

❆①✐♦♠ s❝❤❡♠❛ ♦❢ s♣❡❝✐✜❝❛t✐♦♥ ❆♥② ❞❡✜♥❛❜❧❡ s✉❜❝❧❛ss ♦❢ ❛ s❡t ✐s ❛ s❡t ■♥st❡❛❞ ♦❢ ❜♦✉♥❞✐♥❣ ✐♥❞✉❝t✐♦♥ ✇❡ ✉s❡ ✇❡❛❦❡r s✉❜s②st❡♠s ❲❡ ✉s❡ t✇♦ ❦✐♥❞s ♦❢ ❛r❣✉♠❡♥ts✿ ♥♦r♠❛❧ ❛♥❞ s❛❢❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-80
SLIDE 80

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈♦♠♣❧❡①✐t② ❧♦✇❡r t❤❛♥ PR ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✉s✐♥❣ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐t ✐s ❜❛s❡❞ ✐♥ ❝♦♠♣r❡❤❡♥s✐♦♥ ♦r

❆①✐♦♠ s❝❤❡♠❛ ♦❢ s♣❡❝✐✜❝❛t✐♦♥ ❆♥② ❞❡✜♥❛❜❧❡ s✉❜❝❧❛ss ♦❢ ❛ s❡t ✐s ❛ s❡t ■♥st❡❛❞ ♦❢ ❜♦✉♥❞✐♥❣ ✐♥❞✉❝t✐♦♥ ✇❡ ✉s❡ ✇❡❛❦❡r s✉❜s②st❡♠s ❲❡ ✉s❡ t✇♦ ❦✐♥❞s ♦❢ ❛r❣✉♠❡♥ts✿ ♥♦r♠❛❧ ❛♥❞ s❛❢❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-81
SLIDE 81

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❈♦♠♣❧❡①✐t② ❧♦✇❡r t❤❛♥ PR ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✉s✐♥❣ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐t ✐s ❜❛s❡❞ ✐♥ ❝♦♠♣r❡❤❡♥s✐♦♥ ♦r

❆①✐♦♠ s❝❤❡♠❛ ♦❢ s♣❡❝✐✜❝❛t✐♦♥ ❆♥② ❞❡✜♥❛❜❧❡ s✉❜❝❧❛ss ♦❢ ❛ s❡t ✐s ❛ s❡t ■♥st❡❛❞ ♦❢ ❜♦✉♥❞✐♥❣ ✐♥❞✉❝t✐♦♥ ✇❡ ✉s❡ ✇❡❛❦❡r s✉❜s②st❡♠s ❲❡ ✉s❡ t✇♦ ❦✐♥❞s ♦❢ ❛r❣✉♠❡♥ts✿ ♥♦r♠❛❧ ❛♥❞ s❛❢❡

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-82
SLIDE 82

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❉❡✜♥✐t✐♦♥

❙✉❜s❡ts

❦ ✶ t❤❛t ♠❛❦❡ r❡❝✉rs✐♦♥ ❤❛✈✐♥❣ ♦ ❦ ❛r❡ ❝❛❧❧❡❞ t✐❡rs

❲❡ ❝❛❧❝✉❧❛t❡ t✐❡r ♦❢ ❛ ❞❡r✐✈❛t✐♦♥ ❢ ❛s ❢ ✵ ✐❢ ❢ ✐s ❛♥ ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ❢ ♠❛① ❤ ❣✶ ❣♠ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶ ❣♠ ❢ ♠❛① ❣ ✶ ❤ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❣ ❛♥❞ ❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-83
SLIDE 83

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❉❡✜♥✐t✐♦♥

❙✉❜s❡ts N❦+✶ t❤❛t ♠❛❦❡ r❡❝✉rs✐♦♥ ❤❛✈✐♥❣ N♦, ..., N❦ ❛r❡ ❝❛❧❧❡❞ t✐❡rs

❲❡ ❝❛❧❝✉❧❛t❡ t✐❡r ♦❢ ❛ ❞❡r✐✈❛t✐♦♥ ❢ ❛s ❢ ✵ ✐❢ ❢ ✐s ❛♥ ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ❢ ♠❛① ❤ ❣✶ ❣♠ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶ ❣♠ ❢ ♠❛① ❣ ✶ ❤ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❣ ❛♥❞ ❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-84
SLIDE 84

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❉❡✜♥✐t✐♦♥

❙✉❜s❡ts N❦+✶ t❤❛t ♠❛❦❡ r❡❝✉rs✐♦♥ ❤❛✈✐♥❣ N♦, ..., N❦ ❛r❡ ❝❛❧❧❡❞ t✐❡rs

❲❡ ❝❛❧❝✉❧❛t❡ t✐❡r ♦❢ ❛ ❞❡r✐✈❛t✐♦♥ ❢ ∈ PR ❛s ❢ ✵ ✐❢ ❢ ✐s ❛♥ ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ❢ ♠❛① ❤ ❣✶ ❣♠ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶ ❣♠ ❢ ♠❛① ❣ ✶ ❤ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❣ ❛♥❞ ❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-85
SLIDE 85

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❉❡✜♥✐t✐♦♥

❙✉❜s❡ts N❦+✶ t❤❛t ♠❛❦❡ r❡❝✉rs✐♦♥ ❤❛✈✐♥❣ N♦, ..., N❦ ❛r❡ ❝❛❧❧❡❞ t✐❡rs

❲❡ ❝❛❧❝✉❧❛t❡ t✐❡r ♦❢ ❛ ❞❡r✐✈❛t✐♦♥ ❢ ∈ PR ❛s ρ(❢ ) = ✵ ✐❢ ❢ ✐s ❛♥ ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ❢ ♠❛① ❤ ❣✶ ❣♠ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶ ❣♠ ❢ ♠❛① ❣ ✶ ❤ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❣ ❛♥❞ ❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-86
SLIDE 86

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❉❡✜♥✐t✐♦♥

❙✉❜s❡ts N❦+✶ t❤❛t ♠❛❦❡ r❡❝✉rs✐♦♥ ❤❛✈✐♥❣ N♦, ..., N❦ ❛r❡ ❝❛❧❧❡❞ t✐❡rs

❲❡ ❝❛❧❝✉❧❛t❡ t✐❡r ♦❢ ❛ ❞❡r✐✈❛t✐♦♥ ❢ ∈ PR ❛s ρ(❢ ) = ✵ ✐❢ ❢ ✐s ❛♥ ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ρ(❢ ) = ♠❛①{ρ(❤), ρ(❣✶), ..., ρ(❣♠)} ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶, ..., ❣♠ ❢ ♠❛① ❣ ✶ ❤ ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❣ ❛♥❞ ❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-87
SLIDE 87

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

❉❡✜♥✐t✐♦♥

❙✉❜s❡ts N❦+✶ t❤❛t ♠❛❦❡ r❡❝✉rs✐♦♥ ❤❛✈✐♥❣ N♦, ..., N❦ ❛r❡ ❝❛❧❧❡❞ t✐❡rs

❲❡ ❝❛❧❝✉❧❛t❡ t✐❡r ♦❢ ❛ ❞❡r✐✈❛t✐♦♥ ❢ ∈ PR ❛s ρ(❢ ) = ✵ ✐❢ ❢ ✐s ❛♥ ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥ ρ(❢ ) = ♠❛①{ρ(❤), ρ(❣✶), ..., ρ(❣♠)} ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❤ ❛♥❞ ❣✶, ..., ❣♠ ρ(❢ ) = ♠❛①{ρ(❣), ✶ + ρ(❤)} ✐❢ ❢ ✐s ❞❡✜♥❡❞ ❜② ❢✉❧❧ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥s ❣ ❛♥❞ ❤

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-88
SLIDE 88

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

ε✶✱ ε✷ ❛♥❞ ε✸ ♦❢ ●③r❡❣♦r③❝②❦ ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞ ❢r♦♠

❉♦❝tr✐♥❡s ✇✐t❤

✶

❛♥ ❙▼ ✷✲❈♦♠♣r❡❤❡♥s✐♦♥

✷

t✇♦ t✐❡rs ◆✵ ◆✶ ♦❢ ♥✉♠❡r❛❧s ✇✐t❤ ❞②❛❞✐❝s ■

③

◆❦

s❦

◆❦

✸

r❛♠✐✜❡❞ r❡❝✉rs✐♦♥

■ ❆

③ ❆ ◆✵

❆

❤❦ s❦ ❆ ❢

◆✵ ❆

❢

❆

❣

❇ ❇ ■ ❆

③ ❆ ◆✶

❆

❢ s❦ ❆

◆✶ ❆

❢

❆

❣

❇

❤❦

❇

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-89
SLIDE 89

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

ε✶✱ ε✷ ❛♥❞ ε✸ ♦❢ ●③r❡❣♦r③❝②❦ ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞ ❢r♦♠

❉♦❝tr✐♥❡s ✇✐t❤

✶

❛♥ ❙▼ ✷✲❈♦♠♣r❡❤❡♥s✐♦♥

✷

t✇♦ t✐❡rs ◆✵ ◆✶ ♦❢ ♥✉♠❡r❛❧s ✇✐t❤ ❞②❛❞✐❝s ■

③

◆❦

s❦

◆❦

✸

r❛♠✐✜❡❞ r❡❝✉rs✐♦♥

■ ❆

③ ❆ ◆✵

❆

❤❦ s❦ ❆ ❢

◆✵ ❆

❢

❆

❣

❇ ❇ ■ ❆

③ ❆ ◆✶

❆

❢ s❦ ❆

◆✶ ❆

❢

❆

❣

❇

❤❦

❇

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-90
SLIDE 90

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

ε✶✱ ε✷ ❛♥❞ ε✸ ♦❢ ●③r❡❣♦r③❝②❦ ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞ ❢r♦♠

❉♦❝tr✐♥❡s ✇✐t❤

✶

❛♥ ❙▼ ✷✲❈♦♠♣r❡❤❡♥s✐♦♥

✷

t✇♦ t✐❡rs ◆✵ ◆✶ ♦❢ ♥✉♠❡r❛❧s ✇✐t❤ ❞②❛❞✐❝s ■

③

◆❦

s❦

◆❦

✸

r❛♠✐✜❡❞ r❡❝✉rs✐♦♥

■ ❆

③ ❆ ◆✵

❆

❤❦ s❦ ❆ ❢

◆✵ ❆

❢

❆

❣

❇ ❇ ■ ❆

③ ❆ ◆✶

❆

❢ s❦ ❆

◆✶ ❆

❢

❆

❣

❇

❤❦

❇

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-91
SLIDE 91

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

ε✶✱ ε✷ ❛♥❞ ε✸ ♦❢ ●③r❡❣♦r③❝②❦ ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞ ❢r♦♠

❉♦❝tr✐♥❡s ✇✐t❤

✶

❛♥ ❙▼ ✷✲❈♦♠♣r❡❤❡♥s✐♦♥

✷

t✇♦ t✐❡rs ◆✵, ◆✶ ♦❢ ♥✉♠❡r❛❧s ✇✐t❤ ❞②❛❞✐❝s ■

③

− → ◆❦

s❦

− → ◆❦

✸

r❛♠✐✜❡❞ r❡❝✉rs✐♦♥

■ ❆

③ ❆ ◆✵

❆

❤❦ s❦ ❆ ❢

◆✵ ❆

❢

❆

❣

❇ ❇ ■ ❆

③ ❆ ◆✶

❆

❢ s❦ ❆

◆✶ ❆

❢

❆

❣

❇

❤❦

❇

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-92
SLIDE 92

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

ε✶✱ ε✷ ❛♥❞ ε✸ ♦❢ ●③r❡❣♦r③❝②❦ ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞ ❢r♦♠

❉♦❝tr✐♥❡s ✇✐t❤

✶

❛♥ ❙▼ ✷✲❈♦♠♣r❡❤❡♥s✐♦♥

✷

t✇♦ t✐❡rs ◆✵, ◆✶ ♦❢ ♥✉♠❡r❛❧s ✇✐t❤ ❞②❛❞✐❝s ■

③

− → ◆❦

s❦

− → ◆❦

✸

r❛♠✐✜❡❞ r❡❝✉rs✐♦♥

■ ❆

③ ❆ ◆✵

❆

❤❦ s❦ ❆ ❢

◆✵ ❆

❢

❆

❣

❇ ❇ ■ ❆

③ ❆ ◆✶

❆

❢ s❦ ❆

◆✶ ❆

❢

❆

❣

❇

❤❦

❇

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-93
SLIDE 93

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❚r❛♥s❧❛t✐♦♥s t♦ ❈❚

ε✶✱ ε✷ ❛♥❞ ε✸ ♦❢ ●③r❡❣♦r③❝②❦ ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞ ❢r♦♠

❉♦❝tr✐♥❡s ✇✐t❤

✶

❛♥ ❙▼ ✷✲❈♦♠♣r❡❤❡♥s✐♦♥

✷

t✇♦ t✐❡rs ◆✵, ◆✶ ♦❢ ♥✉♠❡r❛❧s ✇✐t❤ ❞②❛❞✐❝s ■

③

− → ◆❦

s❦

− → ◆❦

✸

r❛♠✐✜❡❞ r❡❝✉rs✐♦♥

■ ⊗ ❆

③⊗❆ λ

  • ◆✵ ⊗ ❆

❤❦

  • s❦⊗❆

❢

  • ◆✵ ⊗ ❆

❢

  • ❆

❣

❇

❇ ■ ⊗ ❆

③⊗❆ λ

  • ◆✶ ⊗ ❆

❢

  • s❦⊗❆ ◆✶ ⊗ ❆

❢

  • ❆

❣

❇

❤❦

❇

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-94
SLIDE 94

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❛t✐♦♥ ✐s ❛ ✇❛② t♦ s❡❡ r❛♠✐✜❝❛t✐♦♥ ✐♥ ❈❚ ❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❙tr♦♥❣ ❈❛t❡❣♦r② ✭❙P♦❧❈❛t✮ ❝♦♥s✐sts ♦❢ ❛ ♠♦❞✉❧❡ ▼ ✇❤❡r❡ ✐s ❛ ❝❝ ✭t❤❡ ♦♣♣♦♥❡♥t✮ ❛♥❞ ❛ ❝❛t❡❣♦r② ✭t❤❡ ♣❧❛②❡r✮ ❡♥❞♦✇❡❞ ✇✐t❤ ❛ str♦♥❣ ❝♦♠♣♦s✐t✐♦♥ ❈✶ ❉✶

❢

❉✷ ❈✷ ❉✷

❣

❉✸ ❈✶ ❈✷ ❉✶

❢ ❣ ❉✸

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-95
SLIDE 95

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❛t✐♦♥ ✐s ❛ ✇❛② t♦ s❡❡ r❛♠✐✜❝❛t✐♦♥ ✐♥ ❈❚ ❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❙tr♦♥❣ ❈❛t❡❣♦r② ✭❙P♦❧❈❛t✮ ❝♦♥s✐sts ♦❢ ❛ ♠♦❞✉❧❡ ▼ ✇❤❡r❡ ✐s ❛ ❝❝ ✭t❤❡ ♦♣♣♦♥❡♥t✮ ❛♥❞ ❛ ❝❛t❡❣♦r② ✭t❤❡ ♣❧❛②❡r✮ ❡♥❞♦✇❡❞ ✇✐t❤ ❛ str♦♥❣ ❝♦♠♣♦s✐t✐♦♥ ❈✶ ❉✶

❢

❉✷ ❈✷ ❉✷

❣

❉✸ ❈✶ ❈✷ ❉✶

❢ ❣ ❉✸

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-96
SLIDE 96

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❛t✐♦♥ ✐s ❛ ✇❛② t♦ s❡❡ r❛♠✐✜❝❛t✐♦♥ ✐♥ ❈❚ ❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❙tr♦♥❣ ❈❛t❡❣♦r② ✭❙P♦❧❈❛t✮ ❝♦♥s✐sts ♦❢ ❛ ♠♦❞✉❧❡ ▼ : C × D − → D ✇❤❡r❡ ✐s ❛ ❝❝ ✭t❤❡ ♦♣♣♦♥❡♥t✮ ❛♥❞ ❛ ❝❛t❡❣♦r② ✭t❤❡ ♣❧❛②❡r✮ ❡♥❞♦✇❡❞ ✇✐t❤ ❛ str♦♥❣ ❝♦♠♣♦s✐t✐♦♥ ❈✶ ❉✶

❢

❉✷ ❈✷ ❉✷

❣

❉✸ ❈✶ ❈✷ ❉✶

❢ ❣ ❉✸

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-97
SLIDE 97

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❛t✐♦♥ ✐s ❛ ✇❛② t♦ s❡❡ r❛♠✐✜❝❛t✐♦♥ ✐♥ ❈❚ ❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❙tr♦♥❣ ❈❛t❡❣♦r② ✭❙P♦❧❈❛t✮ ❝♦♥s✐sts ♦❢ ❛ ♠♦❞✉❧❡ ▼ : C × D − → D ✇❤❡r❡ C ✐s ❛ ❝❝ ✭t❤❡ ♦♣♣♦♥❡♥t✮ ❛♥❞ D ❛ ❝❛t❡❣♦r② ✭t❤❡ ♣❧❛②❡r✮ ❡♥❞♦✇❡❞ ✇✐t❤ ❛ str♦♥❣ ❝♦♠♣♦s✐t✐♦♥ ❈✶ ❉✶

❢

❉✷ ❈✷ ❉✷

❣

❉✸ ❈✶ ❈✷ ❉✶

❢ ❣ ❉✸

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-98
SLIDE 98

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❛t✐♦♥ ✐s ❛ ✇❛② t♦ s❡❡ r❛♠✐✜❝❛t✐♦♥ ✐♥ ❈❚ ❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❙tr♦♥❣ ❈❛t❡❣♦r② ✭❙P♦❧❈❛t✮ ❝♦♥s✐sts ♦❢ ❛ ♠♦❞✉❧❡ ▼ : C × D − → D ✇❤❡r❡ C ✐s ❛ ❝❝ ✭t❤❡ ♦♣♣♦♥❡♥t✮ ❛♥❞ D ❛ ❝❛t❡❣♦r② ✭t❤❡ ♣❧❛②❡r✮ ❡♥❞♦✇❡❞ ✇✐t❤ ❛ str♦♥❣ ❝♦♠♣♦s✐t✐♦♥ (❈✶, ❉✶)

❢

− → ❉✷ (❈✷, ❉✷)

❣

− → ❉✸ (❈✶ × ❈✷, ❉✶)

❢ ;❣

− → ❉✸

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-99
SLIDE 99

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❋✉♥❝t♦r ♦❢ ❛ ❙P♦❧❈❛t ❝♦♥s✐sts ♦❢ ❢✉♥❝t♦rs ❋♣ ❛♥❞ ❋♦ ✇✐t❤ ❋♦♣ ❛❝t✐♥❣ ❈ ✶

❢

❉ ❋♦ ❈ ✶

❋♦♣ ❢ ❋♣ ❉

❊①❛♠♣❧❡ ❢♦r ❛♥❞ ❢♦r ❢♦r♠ ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-100
SLIDE 100

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❋✉♥❝t♦r ♦❢ ❛ ❙P♦❧❈❛t C ❝♦♥s✐sts ♦❢ ❢✉♥❝t♦rs ❋♣ ❛♥❞ ❋♦ ✇✐t❤ ❋♦♣ ❛❝t✐♥❣ ❈ ✶

❢

❉ ❋♦ ❈ ✶

❋♦♣ ❢ ❋♣ ❉

❊①❛♠♣❧❡ ❢♦r ❛♥❞ ❢♦r ❢♦r♠ ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-101
SLIDE 101

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❋✉♥❝t♦r ♦❢ ❛ ❙P♦❧❈❛t C ❝♦♥s✐sts ♦❢ ❢✉♥❝t♦rs ❋♣ : D − → D ❛♥❞ ❋♦ : C − → C ✇✐t❤ ❋♦♣ ❛❝t✐♥❣ ❈ ✶

❢

❉ ❋♦ ❈ ✶

❋♦♣ ❢ ❋♣ ❉

❊①❛♠♣❧❡ ❢♦r ❛♥❞ ❢♦r ❢♦r♠ ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-102
SLIDE 102

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❋✉♥❝t♦r ♦❢ ❛ ❙P♦❧❈❛t C ❝♦♥s✐sts ♦❢ ❢✉♥❝t♦rs ❋♣ : D − → D ❛♥❞ ❋♦ : C − → C ✇✐t❤ ❋♦♣ ❛❝t✐♥❣ ❈ ✶

❢

❉ ❋♦ ❈ ✶

❋♦♣ ❢ ❋♣ ❉

❊①❛♠♣❧❡ ❢♦r ❛♥❞ ❢♦r ❢♦r♠ ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-103
SLIDE 103

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❋✉♥❝t♦r ♦❢ ❛ ❙P♦❧❈❛t C ❝♦♥s✐sts ♦❢ ❢✉♥❝t♦rs ❋♣ : D − → D ❛♥❞ ❋♦ : C − → C ✇✐t❤ ❋♦♣ ❛❝t✐♥❣ (❈, ✶)

❢

− → ❉ (❋♦(❈), ✶)

❋♦♣(❢ )

− → ❋♣(❉)

❊①❛♠♣❧❡ ❢♦r ❛♥❞ ❢♦r ❢♦r♠ ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-104
SLIDE 104

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥

❆ P♦❧❛r✐③❡❞ ❋✉♥❝t♦r ♦❢ ❛ ❙P♦❧❈❛t C ❝♦♥s✐sts ♦❢ ❢✉♥❝t♦rs ❋♣ : D − → D ❛♥❞ ❋♦ : C − → C ✇✐t❤ ❋♦♣ ❛❝t✐♥❣ (❈, ✶)

❢

− → ❉ (❋♦(❈), ✶)

❋♦♣(❢ )

− → ❋♣(❉)

❊①❛♠♣❧❡ ⊗ ❢♦r D ❛♥❞ × ❢♦r C ❢♦r♠ ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-105
SLIDE 105

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❲❡ ❝❛♥ ❞♦ ❝♦♠♣r❡❤❡♥❞❡❞ r❡❝✉rs✐♦♥ ♦♥ ✜①❡❞ ♣♦✐♥ts ♦✈❡r ❋♦♣ ▲❡t ❋ ❜❡ ❛ ❢r❡❡ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ❝♦♥t❡①ts ❈ ✶ ❛♥❞ ❉ ◆❛t ❝♦♥str✉❝t♦rs ❩❡r♦ ✶ ◆❛t ❛♥❞ ❙✉❝❝ ◆❛t ◆❛t

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-106
SLIDE 106

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❲❡ ❝❛♥ ❞♦ ❝♦♠♣r❡❤❡♥❞❡❞ r❡❝✉rs✐♦♥ ♦♥ ✜①❡❞ ♣♦✐♥ts ♦✈❡r ❋♦♣ ▲❡t ❋ ⋆ ❜❡ ❛ ❢r❡❡ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ❝♦♥t❡①ts ❈ ✶ ❛♥❞ ❉ ◆❛t ❝♦♥str✉❝t♦rs ❩❡r♦ ✶ ◆❛t ❛♥❞ ❙✉❝❝ ◆❛t ◆❛t

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-107
SLIDE 107

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❲❡ ❝❛♥ ❞♦ ❝♦♠♣r❡❤❡♥❞❡❞ r❡❝✉rs✐♦♥ ♦♥ ✜①❡❞ ♣♦✐♥ts ♦✈❡r ❋♦♣ ▲❡t ❋ ⋆ ❜❡ ❛ ❢r❡❡ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ❝♦♥t❡①ts ❈ = ✶ ❛♥❞ ❉ = ◆❛t ❝♦♥str✉❝t♦rs ❩❡r♦ ✶ ◆❛t ❛♥❞ ❙✉❝❝ ◆❛t ◆❛t

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-108
SLIDE 108

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❲❡ ❝❛♥ ❞♦ ❝♦♠♣r❡❤❡♥❞❡❞ r❡❝✉rs✐♦♥ ♦♥ ✜①❡❞ ♣♦✐♥ts ♦✈❡r ❋♦♣ ▲❡t ❋ ⋆ ❜❡ ❛ ❢r❡❡ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ❝♦♥t❡①ts ❈ = ✶ ❛♥❞ ❉ = ◆❛t ❝♦♥str✉❝t♦rs ❩❡r♦ : ✶ − → ◆❛t ❛♥❞ ❙✉❝❝ : ◆❛t − → ◆❛t

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-109
SLIDE 109

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❋ ⋆ ✐s ❛ ✜①❡❞ ♣♦✐♥t ❢♦r ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r ✐♥ t❤❡ ❢♦r♠

❦

❋❦ ❩

❈❛♥ ✇❡ ❝♦♥str✉❝t ❛ ❙P♦❧❈❛t ❢r♦♠ t❤✐s t♦ ❣❡t ❛ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐♥ ❈❚❄

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-110
SLIDE 110

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❋ ⋆ ✐s ❛ ✜①❡❞ ♣♦✐♥t ❢♦r ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r ✐♥ t❤❡ ❢♦r♠

  • ❦

❋❦(❩)

❈❛♥ ✇❡ ❝♦♥str✉❝t ❛ ❙P♦❧❈❛t ❢r♦♠ t❤✐s t♦ ❣❡t ❛ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐♥ ❈❚❄

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s

slide-111
SLIDE 111

P❧❛♥ ❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ❢♦r r❡❝✉rs✐♦♥ ✐♥ ❈❚ ❋r♦♠ s②♥t❛❝t✐❝ t♦ s❡♠❛♥t✐❝ ❙✉❜r❡❝✉rs✐♦♥ ❚r❛♥s❧❛t✐♦♥s t♦ ❈❚ P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

P♦❧❛r✐③❡❞ ❈❛t❡❣♦r✐❡s

❋ ⋆ ✐s ❛ ✜①❡❞ ♣♦✐♥t ❢♦r ❛ ♣♦❧❛r✐③❡❞ ❢✉♥❝t♦r ✐♥ t❤❡ ❢♦r♠

  • ❦

❋❦(❩)

❈❛♥ ✇❡ ❝♦♥str✉❝t ❛ ❙P♦❧❈❛t ❢r♦♠ t❤✐s t♦ ❣❡t ❛ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ r❛♠✐✜❡❞ r❡❝✉rs✐♦♥ ✐♥ ❈❚❄

❏♦❛q✉í♥ ❉í❛③ ❇♦ï❧s

❘❡❝✉rs✐✈❡ ❋✉♥❝t✐♦♥ ❈❧❛ss❡s ✐♥ ❈❛rt❡s✐❛♥ ❈❛t❡❣♦r✐❡s