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SLIDE 1

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤s

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs

❆❈❆✱ ■■❚ ❑❛♥♣✉r

❖❝t♦❜❡r ✷✷✱ ✷✵✶✷

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-2
SLIDE 2

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖✉t❧✐♥❡

▼❛t❤s

Pr♦❜❛❜✐❧✐t②

Pr♦❜❧❡♠s

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-3
SLIDE 3

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

❣❝❞✭❛✱ ❜✮✿ ❣r❡❛t❡st ✐♥t❡❣❡r ❞✐✈✐❞❡s ❜♦t❤ ❛ ❛♥❞ ❜ ■❢ ❜⑤❛ t❤❡♥ ❣❝❞✭❛✱❜✮ ❂ ❜ ❖t❤❡r✇✐s❡ ❛ ❂ ❜t✰r ❢♦r s♦♠❡ t ❛♥❞ r

❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱r✮ ❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱❛✪❜✮

❧❝♠✭❛✱❜✮ ❂ ✭❛✯❜✮✴❣❝❞✭❛✱❜✮ ❘✉♥♥✐♥❣ t✐♠❡✿ ❖ ✭❧♦❣ ✭❛ ✰ ❜✮✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-4
SLIDE 4

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

❣❝❞✭❛✱ ❜✮✿ ❣r❡❛t❡st ✐♥t❡❣❡r ❞✐✈✐❞❡s ❜♦t❤ ❛ ❛♥❞ ❜ ■❢ ❜⑤❛ t❤❡♥ ❣❝❞✭❛✱❜✮ ❂ ❜ ❖t❤❡r✇✐s❡ ❛ ❂ ❜t✰r ❢♦r s♦♠❡ t ❛♥❞ r

❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱r✮ ❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱❛✪❜✮

❧❝♠✭❛✱❜✮ ❂ ✭❛✯❜✮✴❣❝❞✭❛✱❜✮ ❘✉♥♥✐♥❣ t✐♠❡✿ ❖ ✭❧♦❣ ✭❛ ✰ ❜✮✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-5
SLIDE 5

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

❣❝❞✭❛✱ ❜✮✿ ❣r❡❛t❡st ✐♥t❡❣❡r ❞✐✈✐❞❡s ❜♦t❤ ❛ ❛♥❞ ❜ ■❢ ❜⑤❛ t❤❡♥ ❣❝❞✭❛✱❜✮ ❂ ❜ ❖t❤❡r✇✐s❡ ❛ ❂ ❜t✰r ❢♦r s♦♠❡ t ❛♥❞ r

❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱r✮ ❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱❛✪❜✮

❧❝♠✭❛✱❜✮ ❂ ✭❛✯❜✮✴❣❝❞✭❛✱❜✮ ❘✉♥♥✐♥❣ t✐♠❡✿ ❖ ✭❧♦❣ ✭❛ ✰ ❜✮✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-6
SLIDE 6

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

❣❝❞✭❛✱ ❜✮✿ ❣r❡❛t❡st ✐♥t❡❣❡r ❞✐✈✐❞❡s ❜♦t❤ ❛ ❛♥❞ ❜ ■❢ ❜⑤❛ t❤❡♥ ❣❝❞✭❛✱❜✮ ❂ ❜ ❖t❤❡r✇✐s❡ ❛ ❂ ❜t✰r ❢♦r s♦♠❡ t ❛♥❞ r

❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱r✮ ❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱❛✪❜✮

❧❝♠✭❛✱❜✮ ❂ ✭❛✯❜✮✴❣❝❞✭❛✱❜✮ ❘✉♥♥✐♥❣ t✐♠❡✿ ❖ ✭❧♦❣ ✭❛ ✰ ❜✮✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-7
SLIDE 7

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

❣❝❞✭❛✱ ❜✮✿ ❣r❡❛t❡st ✐♥t❡❣❡r ❞✐✈✐❞❡s ❜♦t❤ ❛ ❛♥❞ ❜ ■❢ ❜⑤❛ t❤❡♥ ❣❝❞✭❛✱❜✮ ❂ ❜ ❖t❤❡r✇✐s❡ ❛ ❂ ❜t✰r ❢♦r s♦♠❡ t ❛♥❞ r

❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱r✮ ❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱❛✪❜✮

❧❝♠✭❛✱❜✮ ❂ ✭❛✯❜✮✴❣❝❞✭❛✱❜✮ ❘✉♥♥✐♥❣ t✐♠❡✿ ❖ ✭❧♦❣ ✭❛ ✰ ❜✮✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-8
SLIDE 8

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

❣❝❞✭❛✱ ❜✮✿ ❣r❡❛t❡st ✐♥t❡❣❡r ❞✐✈✐❞❡s ❜♦t❤ ❛ ❛♥❞ ❜ ■❢ ❜⑤❛ t❤❡♥ ❣❝❞✭❛✱❜✮ ❂ ❜ ❖t❤❡r✇✐s❡ ❛ ❂ ❜t✰r ❢♦r s♦♠❡ t ❛♥❞ r

❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱r✮ ❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱❛✪❜✮

❧❝♠✭❛✱❜✮ ❂ ✭❛✯❜✮✴❣❝❞✭❛✱❜✮ ❘✉♥♥✐♥❣ t✐♠❡✿ ❖ ✭❧♦❣ ✭❛ ✰ ❜✮✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-9
SLIDE 9

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

❣❝❞✭❛✱ ❜✮✿ ❣r❡❛t❡st ✐♥t❡❣❡r ❞✐✈✐❞❡s ❜♦t❤ ❛ ❛♥❞ ❜ ■❢ ❜⑤❛ t❤❡♥ ❣❝❞✭❛✱❜✮ ❂ ❜ ❖t❤❡r✇✐s❡ ❛ ❂ ❜t✰r ❢♦r s♦♠❡ t ❛♥❞ r

❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱r✮ ❣❝❞✭❛✱❜✮ ❂ ❣❝❞✭❜✱❛✪❜✮

❧❝♠✭❛✱❜✮ ❂ ✭❛✯❜✮✴❣❝❞✭❛✱❜✮ ❘✉♥♥✐♥❣ t✐♠❡✿ ❖ ✭❧♦❣ ✭❛ ✰ ❜✮✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-10
SLIDE 10

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

❘❡❝✉rs✐✈❡ ■♠♣❧❡♠❡♥t❛t✐♦♥✿ ✐♥t ❣❝❞✭✐♥t ❛✱ ✐♥t ❜✮ ④ ✐❢ ✭❜❂❂✵✮ r❡t✉r♥ ❛❀ ❡❧s❡ r❡t✉r♥ ❣❝❞✭❜✱❛✪❜✮❀ ⑥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-11
SLIDE 11

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

  • ❈❉

■t❡r❛t✐✈❡ ■♠♣❧❡♠❡♥t❛t✐♦♥✿ ✐♥t ❣❝❞✭✐♥t ❛✱ ✐♥t ❜✮ ④ ✇❤✐❧❡✭❜✮ ④ ✐♥t r ❂ ❛ ✪ ❜❀ ❛ ❂ ❜❀ ❜ ❂ r❀ ⑥ r❡t✉r♥ ❛❀ ⑥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 12

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❊①♣♦♥❡♥t✐❛t✐♦♥

❈♦♠♣✉t❡ ❛♥ ✐♥ ❖✭❧♦❣ ♥✮ t✐♠❡ ❛♥ ❂ ✶✱ ✐❢ ♥❂✵ ❂ ❛ ✐❢ ♥❂✶ ❂ ❛ ♥ ✷ ✷✱ ✐❢ ♥ ❂❡✈❡♥ ❂ ❛

♥ ✶ ✷ ✷✱ ✐❢ ♥ ❂ ♦❞❞

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-13
SLIDE 13

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❊①♣♦♥❡♥t✐❛t✐♦♥

❈♦♠♣✉t❡ ❛♥ ✐♥ ❖✭❧♦❣ ♥✮ t✐♠❡ ❛♥ ❂ ✶✱ ✐❢ ♥❂✵ ❂ ❛ ✐❢ ♥❂✶ ❂ ❛(♥/✷)✷✱ ✐❢ ♥ ❂❡✈❡♥ ❂ ❛((♥−✶)/✷)✷✱ ✐❢ ♥ ❂ ♦❞❞

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-14
SLIDE 14

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❊①♣♦♥❡♥t✐❛t✐♦♥

❘❡❝✉rs✐✈❡ ■♠♣❧❡♠❡♥t❛t✐♦♥✿ ❞♦✉❜❧❡ ♣♦✇✭❞♦✉❜❧❡ ❛✱ ✐♥t ♥✮ ④ ✐❢✭♥ ❂❂ ✵✮ r❡t✉r♥ ✶❀ ✐❢✭♥ ❂❂ ✶✮ r❡t✉r♥ ❛❀ ❞♦✉❜❧❡ t ❂ ♣♦✇✭❛✱ ♥✴✷✮❀ r❡t✉r♥ t ✯ t ✯ ♣♦✇✭❛✱ ♥✪✷✮❀ ⑥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-15
SLIDE 15

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❊①♣♦♥❡♥t✐❛t✐♦♥

■t❡r❛t✐✈❡ ■♠♣❧❡♠❡♥t❛t✐♦♥✿ ❛ = ❛✵ + ❛✶ ∗ ✷ + ❛✷ ∗ ✷✷ + ... + ❛❦ ∗ ✷❦ ✐♥t r❡s✉❧t❂✶✱♣♦✇❡r❂❛❀ ✇❤✐❧❡✭✦♥✮ ④ ✐❢✭♥✫✶✮ r❡s✉❧t✯❂♣♦✇❡r❀ ♣♦✇❡r✯❂♣♦✇❡r❀ ♥✔❂✶❀ ⑥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-16
SLIDE 16

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋✐❜♦♥❛❝❝✐

❋♥ ❋♥

❋♥

✷ ❋♥ ❋♥

✶ ✶ ✶ ✵ ❋♥

❋♥

❋♥ ❋♥

✶ ✶ ✶ ✵ ♥ ❋✶ ❋✵

❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡ ❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-17
SLIDE 17

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋✐❜♦♥❛❝❝✐

❋♥ = ❋♥−✶ + ❋♥−✷

❋♥ ❋♥

✶ ✶ ✶ ✵ ❋♥

❋♥

❋♥ ❋♥

✶ ✶ ✶ ✵ ♥ ❋✶ ❋✵

❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡ ❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-18
SLIDE 18

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋✐❜♦♥❛❝❝✐

❋♥ = ❋♥−✶ + ❋♥−✷

  • ❋♥

❋♥−✶ = [ ✶ ✶ ✶ ✵ ] ∗ ❋♥−✶ ❋♥−✷

  • ❋♥

❋♥

✶ ✶ ✶ ✵ ♥ ❋✶ ❋✵

❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡ ❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-19
SLIDE 19

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋✐❜♦♥❛❝❝✐

❋♥ = ❋♥−✶ + ❋♥−✷

  • ❋♥

❋♥−✶ = [ ✶ ✶ ✶ ✵ ] ∗ ❋♥−✶ ❋♥−✷

  • ❋♥

❋♥−✶ = [ ✶ ✶ ✶ ✵ ]♥ ∗ ❋✶ ❋✵

  • ❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡

❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-20
SLIDE 20

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋✐❜♦♥❛❝❝✐

❋♥ = ❋♥−✶ + ❋♥−✷

  • ❋♥

❋♥−✶ = [ ✶ ✶ ✶ ✵ ] ∗ ❋♥−✶ ❋♥−✷

  • ❋♥

❋♥−✶ = [ ✶ ✶ ✶ ✵ ]♥ ∗ ❋✶ ❋✵

  • ❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡

❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-21
SLIDE 21

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋✐❜♦♥❛❝❝✐

❋♥ = ❋♥−✶ + ❋♥−✷

  • ❋♥

❋♥−✶ = [ ✶ ✶ ✶ ✵ ] ∗ ❋♥−✶ ❋♥−✷

  • ❋♥

❋♥−✶ = [ ✶ ✶ ✶ ✵ ]♥ ∗ ❋✶ ❋✵

  • ❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡

❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-22
SLIDE 22

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

♥ ❦

❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ❈♦♠♣✉t✐♥❣

♥ ❦

❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛

♥ ❦

❂ ♥ ♥

✶ ♥ ❦ ✶ ❦

❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡

❈❛s❡s✿

❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥

♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-23
SLIDE 23

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

  • ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥

❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ❈♦♠♣✉t✐♥❣

♥ ❦

❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛

♥ ❦

❂ ♥ ♥

✶ ♥ ❦ ✶ ❦

❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡

❈❛s❡s✿

❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥

♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-24
SLIDE 24

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

  • ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥

❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ❈♦♠♣✉t✐♥❣ ♥

❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛

♥ ❦

❂ ♥ ♥

✶ ♥ ❦ ✶ ❦

❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡

❈❛s❡s✿

❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥

♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-25
SLIDE 25

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

  • ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥

❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ❈♦♠♣✉t✐♥❣ ♥

❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ♥

  • ❂ ♥(♥−✶)...(♥−❦+✶)

❦!

❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡

❈❛s❡s✿

❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥

♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-26
SLIDE 26

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

  • ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥

❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ❈♦♠♣✉t✐♥❣ ♥

❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ♥

  • ❂ ♥(♥−✶)...(♥−❦+✶)

❦!

❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡

❈❛s❡s✿

❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥

♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-27
SLIDE 27

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

  • ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥

❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ❈♦♠♣✉t✐♥❣ ♥

❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ♥

  • ❂ ♥(♥−✶)...(♥−❦+✶)

❦!

❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡

❈❛s❡s✿

❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥

♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-28
SLIDE 28

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

  • ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥

❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ❈♦♠♣✉t✐♥❣ ♥

❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ♥

  • ❂ ♥(♥−✶)...(♥−❦+✶)

❦!

❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡

❈❛s❡s✿

❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥

♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-29
SLIDE 29

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

  • ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥

❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ❈♦♠♣✉t✐♥❣ ♥

❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ♥

  • ❂ ♥(♥−✶)...(♥−❦+✶)

❦!

❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡

❈❛s❡s✿

❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥

♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-30
SLIDE 30

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▲✉❝❛s ❚❤❡♦r❡♠

❚❤❡ ▲✉❝❛s✬ t❤❡♦r❡♠ ❡①♣r❡ss❡s t❤❡ r❡♠❛✐♥❞❡r ♦❢ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t

♠ ♥

❜② ❛ ♣r✐♠❡ ♥✉♠❜❡r ♣ ✐♥ t❡r♠s ♦❢ t❤❡ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ t❤❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥✳ ❋♦r ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥ ❛♥❞ ❛ ♣r✐♠❡ ♣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❣r✉❡♥❝❡ r❡❧❛t✐♦♥ ❤♦❧❞s✿

♠ ♥ ❦ ✐ ✵ ♠✐ ♥✐

♠♦❞♣ ✇❤❡r❡ ♠ ♠❦♣❦ ♠❦

✶♣❦ ✶

♠✶♣ ♠✵ ❛♥❞ ♥ ♥❦♣❦ ♥❦

✶♣❦ ✶

♥✶♣ ♥✵ ❛r❡ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ ♠ ❛♥❞ ♥ r❡s♣❡❝t✐✈❡❧②✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-31
SLIDE 31

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▲✉❝❛s ❚❤❡♦r❡♠

❚❤❡ ▲✉❝❛s✬ t❤❡♦r❡♠ ❡①♣r❡ss❡s t❤❡ r❡♠❛✐♥❞❡r ♦❢ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ♠

  • ❜② ❛ ♣r✐♠❡ ♥✉♠❜❡r ♣ ✐♥ t❡r♠s ♦❢ t❤❡

❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ t❤❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥✳ ❋♦r ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥ ❛♥❞ ❛ ♣r✐♠❡ ♣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❣r✉❡♥❝❡ r❡❧❛t✐♦♥ ❤♦❧❞s✿

♠ ♥ ❦ ✐ ✵ ♠✐ ♥✐

♠♦❞♣ ✇❤❡r❡ ♠ ♠❦♣❦ ♠❦

✶♣❦ ✶

♠✶♣ ♠✵ ❛♥❞ ♥ ♥❦♣❦ ♥❦

✶♣❦ ✶

♥✶♣ ♥✵ ❛r❡ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ ♠ ❛♥❞ ♥ r❡s♣❡❝t✐✈❡❧②✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-32
SLIDE 32

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▲✉❝❛s ❚❤❡♦r❡♠

❚❤❡ ▲✉❝❛s✬ t❤❡♦r❡♠ ❡①♣r❡ss❡s t❤❡ r❡♠❛✐♥❞❡r ♦❢ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ♠

  • ❜② ❛ ♣r✐♠❡ ♥✉♠❜❡r ♣ ✐♥ t❡r♠s ♦❢ t❤❡

❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ t❤❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥✳ ❋♦r ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥ ❛♥❞ ❛ ♣r✐♠❡ ♣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❣r✉❡♥❝❡ r❡❧❛t✐♦♥ ❤♦❧❞s✿ ♠

  • = ❦

✐=✵

♠✐

♥✐

  • (♠♦❞♣)

✇❤❡r❡ ♠ = ♠❦♣❦ + ♠❦−✶♣❦−✶ + · · · + ♠✶♣ + ♠✵ ❛♥❞ ♥ = ♥❦♣❦ + ♥❦−✶♣❦−✶ + · · · + ♥✶♣ + ♥✵ ❛r❡ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ ♠ ❛♥❞ ♥ r❡s♣❡❝t✐✈❡❧②✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-33
SLIDE 33

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr♦❜❧❡♠

Pr♦❜❧❡♠ ✶ ❋✐♥❞ t❤❡ ♥✉♠❜❡r ♦❢ str✐♥❣s ♦❢ ❧❡♥❣t❤ ✧◆✧ ♠❛❞❡ ✉♣ ♦❢ ♦♥❧② ✸ ❝❤❛r❛❝t❡rs ✲ ❛✱ ❜✱ ❝ s✉❝❤ t❤❛t ✧❛✧ ♦❝❝✉rs ❛t ❧❡❛st ✧♠✐♥❴❛✧ t✐♠❡s ❛♥❞ ❛t ♠♦st ✧♠❛①❴❛✧ t✐♠❡s✱ ✧❜✧ ♦❝❝✉rs ❛t ❧❡❛st ✧♠✐♥❴❜✧ t✐♠❡s ❛♥❞ ❛t ♠♦st ✧♠❛①❴❜✧ t✐♠❡s ❛♥❞ ✧❝✧ ♦❝❝✉rs ❛t ❧❡❛st ✧♠✐♥❴❝✧ t✐♠❡s ❛♥❞ ❛t ♠♦st ✧♠❛①❴❝✧ t✐♠❡s✳ ◆♦t❡ t❤❛t ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ♦❢ s❛♠❡ str✐♥❣ ❝♦✉♥t ❛s ✶✱ s♦ ✧❛❜❝✧ ✐s s❛♠❡ ❛s ✧❜❛❝✧✳ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❉❈❊P❈✼✵✷✴

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-34
SLIDE 34

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr♦❜❧❡♠

Pr♦❜❧❡♠ ✷ ❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t♦ ✜♥❞ ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r t❤❡ ♣r♦❜❧❡♠ ✐♥ ✇❤✐❝❤ ✇❡ s❤♦✉❧❞ ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛t❤s ♦❢ ❛ s♣❡❝✐❛❧ t②♣❡✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ ✇❡ ❤❛✈❡ t✇♦ ♣♦✐♥ts ❆✱ ❇ ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ✐♥t❡❣❡r ❝♦♦r❞✐♥❛t❡s✱ t❤❡♥ ✇❡ ✇✐❧❧ ♦♣❡r❛t❡ ♦♥❧② ✇✐t❤ t❤❡ s❤♦rt❡st ♣❛t❤s ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇ t❤❛t ♣❛ss ♦♥❧② t❤r♦✉❣❤ t❤❡ ❧✐♥❡s ♦❢ t❤❡ ✐♥t❡❣❡r ❣r✐❞ ❛♥❞ t❤❛t ❝❛♥ ❜❡ ❞♦♥❡ ♦♥❧② ✐♥ ❤♦r✐③♦♥t❛❧ ♦r ✈❡rt✐❝❛❧ ♠♦✈❡♠❡♥ts ✇✐t❤ ❧❡♥❣t❤ ❡q✉❛❧ t♦ ✶✳

❋✐❣✉r❡✿ ✶

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-35
SLIDE 35

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr♦❜❧❡♠

❙♦❧✉t✐♦♥ ❙♦❧✉t✐♦♥✿ ❆❧❧ ♣❛t❤s ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇ ❤❛✈❡ t❤❡ s❛♠❡ ❧❡♥❣t❤ ❡q✉❛❧ t♦ ♥✰♠ ✭✇❤❡r❡ ♥ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ①✲❝♦♦r❞✐♥❛t❡s ❛♥❞ ♠ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ②✲❝♦♦r❞✐♥❛t❡s✮✳ ❲❡ ❝❛♥ ❡❛s✐❧② ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ t❤❡ ♣❛t❤s ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇✿ ❆♥s✿ ♠+♥

  • ♦r

♠+♥

  • ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs

▼❛t❤s

slide-36
SLIDE 36

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr♦❜❧❡♠

Pr♦❜❧❡♠ ✸ ▲❡t✬s s♦❧✈❡ ❛ ❢❛♠♦✉s ♣r♦❜❧❡♠ ✉s✐♥❣ t❤✐s ♠❡t❤♦❞✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✜♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❉②❝❦ ✇♦r❞s ✇✐t❤ ❛ ❧❡♥❣t❤ ♦❢ ✷♥✳ ❲❤❛t ✐s ❛ ❉②❝❦ ✇♦r❞❄ ■t✬s ❛ str✐♥❣ ❝♦♥s✐st✐♥❣ ♦♥❧② ♦❢ ♥ ❳✬s ❛♥❞ ♥ ❨✬s✱ ❛♥❞ ♠❛t❝❤✐♥❣ t❤✐s ❝r✐t❡r✐❛✿ ❡❛❝❤ ♣r❡✜① ♦❢ t❤✐s str✐♥❣ ❤❛s ♠♦r❡ ❳✬s t❤❛♥ ❨✬s✳ ❋♦r ❡①❛♠♣❧❡✱ ✧❳❳❨❨✧ ❛♥❞ ✧❳❨❳❨✧ ❛r❡ ❉②❝❦ ✇♦r❞s✱ ❜✉t ✧❳❨❨❳✧ ❛♥❞ ✧❨❨❳❳✧ ❛r❡ ♥♦t✳ ❖❘ ❋✐♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s t♦ ❛rr❛♥❣❡ ♥ ✬✭✬ ❛♥❞ ♥ ✬✮✬ ❜r❛❝❦❡ts s✉❝❤ t❤❛t ❛t ❡❛❝❤ ✐♥❞❡①✱ t❤❡ ♥✉♠❜❡r ♦❢ ✬✭✬ ✐s ♥❡✈❡r ❧❡ss t❤❛♥ t❤❡ ♥✉♠❜❡r ♦❢ ✬✮✬

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-37
SLIDE 37

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr♦❜❧❡♠

❙♦❧✉t✐♦♥ ❙♦❧✉t✐♦♥✿ ❚♦t❛❧ ✇❛②s✿ ✷♥

  • ■♥❝♦rr❡❝t ✇❛②s✿

✷♥

♥−✶

  • ❆♥s✿ ❈❛t❛❧❛♥ ♥✉♠❜❡r

✶ ♥+✶

✷♥

  • ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs

▼❛t❤s

slide-38
SLIDE 38

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝

✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ①② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-39
SLIDE 39

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝

✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ①② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-40
SLIDE 40

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝

✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ①② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-41
SLIDE 41

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝

✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ①② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-42
SLIDE 42

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝

✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ①② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-43
SLIDE 43

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝

✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ①② ♠♦❞ ♥ ❂ (①♠♦❞♥)② ♠♦❞ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-44
SLIDE 44

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ①

✶ ♠♦❞♥ ✺

✸ ♠♦❞✼ ❜❡❝❛✉s❡ ✺ ✸ ✶✺ ✶ ♠♦❞✼ ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-45
SLIDE 45

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

①−✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ①−✶ ≡ ✶(♠♦❞♥) ✺

✸ ♠♦❞✼ ❜❡❝❛✉s❡ ✺ ✸ ✶✺ ✶ ♠♦❞✼ ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-46
SLIDE 46

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

①−✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ①−✶ ≡ ✶(♠♦❞♥) ✺−✶ ≡ ✸(♠♦❞✼) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶(♠♦❞✼) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-47
SLIDE 47

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

①−✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ①−✶ ≡ ✶(♠♦❞♥) ✺−✶ ≡ ✸(♠♦❞✼) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶(♠♦❞✼) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-48
SLIDE 48

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

①−✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ①−✶ ≡ ✶(♠♦❞♥) ✺−✶ ≡ ✸(♠♦❞✼) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶(♠♦❞✼) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-49
SLIDE 49

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

①−✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ①−✶ ≡ ✶(♠♦❞♥) ✺−✶ ≡ ✸(♠♦❞✼) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶(♠♦❞✼) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-50
SLIDE 50

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

①−✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ①−✶ ≡ ✶(♠♦❞♥) ✺−✶ ≡ ✸(♠♦❞✼) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶(♠♦❞✼) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-51
SLIDE 51

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

①−✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ①−✶ ≡ ✶(♠♦❞♥) ✺−✶ ≡ ✸(♠♦❞✼) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶(♠♦❞✼) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ≡ ✶(♠♦❞♥)

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-52
SLIDE 52

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡

①−✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ①−✶ ≡ ✶(♠♦❞♥) ✺−✶ ≡ ✸(♠♦❞✼) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶(♠♦❞✼) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ≡ ✶(♠♦❞♥)

  • ✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜②

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-53
SLIDE 53

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr✐♠❡ ❙❡✐✈❡

■❞❡❛✿ ❡✈❡r② ❝♦♠♣♦s✐t❡ ♥✉♠❜❡r ♥ ❤❛s ❛ ♣r✐♠❡ ❢❛❝t♦r ♣ ♥✳ ❙♦ ❧❡t ✉s ❛ss✉♠❡ t❤❛t ❛❧❧ ♥✉♠❜❡rs ❛r❡ ♣r✐♠❡✳ ❇✉t ✐❢ ✇❡ ❝♦♠❡ ❛❝r♦ss ❛ ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛ ♥✉♠❜❡r✱ ✇❡ ✐♠♠❡❞✐❛t❡❧② ❦♥♦✇ t❤❛t ✐t ✐s ♥♦t ❛ ♣r✐♠❡✳ ■❢ t❤❡r❡ ✐s ♥♦ ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛ ♥✉♠❜❡r ♥ ✐♥ t❤❡ r❛♥❣❡ ❬✷ ✳ ✳ ✳ ♥✲✶❪ t❤❡♥ ✐t ♠✉st ❜❡ ♣r✐♠❡✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-54
SLIDE 54

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr✐♠❡ ❙❡✐✈❡

■❞❡❛✿ ❡✈❡r② ❝♦♠♣♦s✐t❡ ♥✉♠❜❡r ♥ ❤❛s ❛ ♣r✐♠❡ ❢❛❝t♦r ♣ ≤ √♥✳ ❙♦ ❧❡t ✉s ❛ss✉♠❡ t❤❛t ❛❧❧ ♥✉♠❜❡rs ❛r❡ ♣r✐♠❡✳ ❇✉t ✐❢ ✇❡ ❝♦♠❡ ❛❝r♦ss ❛ ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛ ♥✉♠❜❡r✱ ✇❡ ✐♠♠❡❞✐❛t❡❧② ❦♥♦✇ t❤❛t ✐t ✐s ♥♦t ❛ ♣r✐♠❡✳ ■❢ t❤❡r❡ ✐s ♥♦ ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛ ♥✉♠❜❡r ♥ ✐♥ t❤❡ r❛♥❣❡ ❬✷ ✳ ✳ ✳ ♥✲✶❪ t❤❡♥ ✐t ♠✉st ❜❡ ♣r✐♠❡✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-55
SLIDE 55

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr✐♠❡ ❙❡✐✈❡

■♠♣❧❡♠❡♥t❛t✐♦♥

  • ❡♥❡r❛t❡ ❛❧❧ ♣r✐♠❡s ✐♥ r❛♥❣❡ ❬✶✳✳♥❪

❋♦r ✐❂✶ t♦ ♥ ♣r✐♠❡❬✐❪❂✶ Pr✐♠❡❬✶❪❂✵ ❋♦r ✐❂✷ t♦ √♥ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r ❥ ❂ ✐ t♦ ♥✴✐ ♣r✐♠❡❬✐✯❥❪❂✵ ❆t t❤❡ ❡♥❞ ♦❢ t❤✐s st❡♣✱ ❛❧❧ ♥✉♠❜❡rs ✐ ✇❤✐❝❤ ❛r❡ ♣r✐♠❡ ❤❛✈❡ ♣r✐♠❡❬✐❪❂✶✳ ❖t❤❡rs ❤❛✈❡ ♣r✐♠❡❬✐❪❂✵✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-56
SLIDE 56

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠

Pr✐♠❡ ♥✉♠❜❡r ❚❤❡♦r❡♠ ◆✉♠❜❡r ♦❢ ♣r✐♠❡s t✐❧❧ ♥ ♥✴❧♦❣♥ ▼❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ♥ ❂ ❧♦❣ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-57
SLIDE 57

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠

Pr✐♠❡ ♥✉♠❜❡r ❚❤❡♦r❡♠ ◆✉♠❜❡r ♦❢ ♣r✐♠❡s t✐❧❧ ♥ ∼ ♥✴❧♦❣♥ ▼❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ♥ ❂ ❧♦❣ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-58
SLIDE 58

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠

Pr✐♠❡ ♥✉♠❜❡r ❚❤❡♦r❡♠ ◆✉♠❜❡r ♦❢ ♣r✐♠❡s t✐❧❧ ♥ ∼ ♥✴❧♦❣♥ ▼❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ♥ ❂ ❧♦❣ ♥

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-59
SLIDE 59

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

♥ ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ ❛❜ ❛ ❜

❋♦r ❛ ♣r✐♠❡ ♣✱ ♣ ♣ ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ ♣❦ ♣❦ ♣❦

♣❦ ✶

✶ ♣

◆ ♣❦✶

♣❦✷

♣❦t

t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♥ ✶

✶ ♣✶

✶ ♣✷

✶ ♣t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♣✶ ✶ ♣✷ ✶ ♣t ✶ ♣✶♣✷ ♣t

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-60
SLIDE 60

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

φ(♥) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ ❛❜ ❛ ❜

❋♦r ❛ ♣r✐♠❡ ♣✱ ♣ ♣ ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ ♣❦ ♣❦ ♣❦

♣❦ ✶

✶ ♣

◆ ♣❦✶

♣❦✷

♣❦t

t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♥ ✶

✶ ♣✶

✶ ♣✷

✶ ♣t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♣✶ ✶ ♣✷ ✶ ♣t ✶ ♣✶♣✷ ♣t

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-61
SLIDE 61

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

φ(♥) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ(❛❜) = φ(❛) ∗ φ(❜)

❋♦r ❛ ♣r✐♠❡ ♣✱ ♣ ♣ ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ ♣❦ ♣❦ ♣❦

♣❦ ✶

✶ ♣

◆ ♣❦✶

♣❦✷

♣❦t

t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♥ ✶

✶ ♣✶

✶ ♣✷

✶ ♣t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♣✶ ✶ ♣✷ ✶ ♣t ✶ ♣✶♣✷ ♣t

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-62
SLIDE 62

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

φ(♥) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ(❛❜) = φ(❛) ∗ φ(❜)

❋♦r ❛ ♣r✐♠❡ ♣✱ φ(♣) = ♣ − ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ ♣❦ ♣❦ ♣❦

♣❦ ✶

✶ ♣

◆ ♣❦✶

♣❦✷

♣❦t

t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♥ ✶

✶ ♣✶

✶ ♣✷

✶ ♣t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♣✶ ✶ ♣✷ ✶ ♣t ✶ ♣✶♣✷ ♣t

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-63
SLIDE 63

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

φ(♥) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ(❛❜) = φ(❛) ∗ φ(❜)

❋♦r ❛ ♣r✐♠❡ ♣✱ φ(♣) = ♣ − ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ φ(♣❦) = ♣❦ − ♣❦−✶ = ♣❦(✶ − ✶

♣)

◆ ♣❦✶

♣❦✷

♣❦t

t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♥ ✶

✶ ♣✶

✶ ♣✷

✶ ♣t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♣✶ ✶ ♣✷ ✶ ♣t ✶ ♣✶♣✷ ♣t

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-64
SLIDE 64

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

φ(♥) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ(❛❜) = φ(❛) ∗ φ(❜)

❋♦r ❛ ♣r✐♠❡ ♣✱ φ(♣) = ♣ − ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ φ(♣❦) = ♣❦ − ♣❦−✶ = ♣❦(✶ − ✶

♣)

◆ = (♣❦✶

✶ ) ∗ (♣❦✷ ✷ ) ∗ ... ∗ (♣❦t t )

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♥ ✶

✶ ♣✶

✶ ♣✷

✶ ♣t

♥ ♣❛✶

✶ ♣❛✷ ✷

♣❛t

t

♣✶ ✶ ♣✷ ✶ ♣t ✶ ♣✶♣✷ ♣t

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-65
SLIDE 65

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

φ(♥) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ(❛❜) = φ(❛) ∗ φ(❜)

❋♦r ❛ ♣r✐♠❡ ♣✱ φ(♣) = ♣ − ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ φ(♣❦) = ♣❦ − ♣❦−✶ = ♣❦(✶ − ✶

♣)

◆ = (♣❦✶

✶ ) ∗ (♣❦✷ ✷ ) ∗ ... ∗ (♣❦t t )

φ(♥) = ♣❛✶

✶ ♣❛✷ ✷ ..♣❛t t ) = ♥ ∗ (✶ − ✶ ♣✶ ) ∗ (✶ − ✶ ♣✷ ) ∗ · · · ∗ (✶ − ✶ ♣t )

≡ φ(♥) = ♣❛✶

✶ ♣❛✷ ✷ ..♣❛t t ) = ♥ ∗ (♣✶−✶)∗(♣✷−✶)..(♣t−✶) (♣✶♣✷..♣t)

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-66
SLIDE 66

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

❙❡✐✈❡ ❢♦r φ(♥) ❘✉♥ ♣r✐♠❡ s✐❡✈❡ ♦♥❝❡ ❛♥❞ st♦r❡ r❡s✉❧t ✐♥ ♣r✐♠❡s❬✶✳✳♥❪ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r✭❥❂✶ t♦ ♥✴✐✮ ♣❤✐❬✐✯❥❪❂♣❤✐❬✐✯❥❪✯✭✐✲✶✮✴✐ ❚❤✐s ❛❧❣♦ r✉♥s ✐♥ ❖✭♥ ❧♦❣ ❧♦❣ ♥✮ t✐♠❡✱ ❜✉t ✇❡ ✇✐❧❧ ♠❛❦❡ ✐♠♣r♦✈❡♠❡♥ts t♦ s❤♦✇ ❤♦✇ ②♦✉ ❝❛♥ ❛t t✐♠❡s ♦♣t✐♠✐③❡ ②♦✉r ❝♦❞❡✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 67

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

❙❡✐✈❡ ❢♦r φ(♥) ❘✉♥ ♣r✐♠❡ s✐❡✈❡ ♦♥❝❡ ❛♥❞ st♦r❡ r❡s✉❧t ✐♥ ♣r✐♠❡s❬✶✳✳♥❪ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r✭❥❂✶ t♦ ♥✴✐✮ ♣❤✐❬✐✯❥❪❂♣❤✐❬✐✯❥❪✯✭✐✲✶✮✴✐ ❚❤✐s ❛❧❣♦ r✉♥s ✐♥ ❖✭♥ ❧♦❣ ❧♦❣ ♥✮ t✐♠❡✱ ❜✉t ✇❡ ✇✐❧❧ ♠❛❦❡ ✐♠♣r♦✈❡♠❡♥ts t♦ s❤♦✇ ❤♦✇ ②♦✉ ❝❛♥ ❛t t✐♠❡s ♦♣t✐♠✐③❡ ②♦✉r ❝♦❞❡✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-68
SLIDE 68

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

❙❡✐✈❡ ❢♦r φ(♥) ❘✉♥ ♣r✐♠❡ s✐❡✈❡ ♦♥❝❡ ❛♥❞ st♦r❡ r❡s✉❧t ✐♥ ♣r✐♠❡s❬✶✳✳♥❪ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r✭❥❂✶ t♦ ♥✴✐✮ ♣❤✐❬✐✯❥❪❂♣❤✐❬✐✯❥❪✯✭✐✲✶✮✴✐ ❚❤✐s ❛❧❣♦ r✉♥s ✐♥ ❖✭♥ ❧♦❣ ❧♦❣ ♥✮ t✐♠❡✱ ❜✉t ✇❡ ✇✐❧❧ ♠❛❦❡ ✐♠♣r♦✈❡♠❡♥ts t♦ s❤♦✇ ❤♦✇ ②♦✉ ❝❛♥ ❛t t✐♠❡s ♦♣t✐♠✐③❡ ②♦✉r ❝♦❞❡✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 69

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

❙❡✐✈❡ ❢♦r φ(♥) ❘✉♥ ♣r✐♠❡ s✐❡✈❡ ♦♥❝❡ ❛♥❞ st♦r❡ r❡s✉❧t ✐♥ ♣r✐♠❡s❬✶✳✳♥❪ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r✭❥❂✶ t♦ ♥✴✐✮ ♣❤✐❬✐✯❥❪❂♣❤✐❬✐✯❥❪✯✭✐✲✶✮✴✐ ❚❤✐s ❛❧❣♦ r✉♥s ✐♥ ❖✭♥ ❧♦❣ ❧♦❣ ♥✮ t✐♠❡✱ ❜✉t ✇❡ ✇✐❧❧ ♠❛❦❡ ✐♠♣r♦✈❡♠❡♥ts t♦ s❤♦✇ ❤♦✇ ②♦✉ ❝❛♥ ❛t t✐♠❡s ♦♣t✐♠✐③❡ ②♦✉r ❝♦❞❡✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 70

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

❙❡✐✈❡ ❢♦r φ(♥) ◗✉❡st✐♦♥✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❧✐st ♦❢ ❛❧❧ ♣r✐♠❡s❄ ❆♥s✇❡r ✿ ◆♦ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣❤✐❬✐❪❂❂✐✮ ❢♦r✭❥❂✐ t♦ ♥❀❥✰❂✐✮ ♣❤✐❬❥❪ ❂ ✭♣❤✐❬❥❪✴✐✮✯✭✐✲✶✮❀

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 71

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

❙❡✐✈❡ ❢♦r φ(♥) ◗✉❡st✐♦♥✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❧✐st ♦❢ ❛❧❧ ♣r✐♠❡s❄ ❆♥s✇❡r ✿ ◆♦ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣❤✐❬✐❪❂❂✐✮ ❢♦r✭❥❂✐ t♦ ♥❀❥✰❂✐✮ ♣❤✐❬❥❪ ❂ ✭♣❤✐❬❥❪✴✐✮✯✭✐✲✶✮❀

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 72

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

❙❡✐✈❡ ❢♦r φ(♥) ◗✉❡st✐♦♥✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❧✐st ♦❢ ❛❧❧ ♣r✐♠❡s❄ ❆♥s✇❡r ✿ ◆♦ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣❤✐❬✐❪❂❂✐✮ ❢♦r✭❥❂✐ t♦ ♥❀❥✰❂✐✮ ♣❤✐❬❥❪ ❂ ✭♣❤✐❬❥❪✴✐✮✯✭✐✲✶✮❀

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 73

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥

❙❡✐✈❡ ❢♦r φ(♥) ◗✉❡st✐♦♥✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❧✐st ♦❢ ❛❧❧ ♣r✐♠❡s❄ ❆♥s✇❡r ✿ ◆♦ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣❤✐❬✐❪❂❂✐✮ ❢♦r✭❥❂✐ t♦ ♥❀❥✰❂✐✮ ♣❤✐❬❥❪ ❂ ✭♣❤✐❬❥❪✴✐✮✯✭✐✲✶✮❀

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 74

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠

■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ t❤❡♥ ❢♦r ❛♥② ✐♥t❡❣❡r ❛ t❤❛t ✐s ❝♦♣r✐♠❡ t♦ ♥✱ ✇❡ ❤❛✈❡ ❛♣ ❛ ♠♦❞♣ ❚❤✐s t❤❡♦r❡♠ ❝❛♥ ❛❧s♦ ❜❡ st❛t❡❞ ❛s✿ ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❛ ✐s ❝♦♣r✐♠❡ t♦ ♣✱ t❤❡♥ ❛♣

✶ ♠♦❞♣

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-75
SLIDE 75

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠

■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ t❤❡♥ ❢♦r ❛♥② ✐♥t❡❣❡r ❛ t❤❛t ✐s ❝♦♣r✐♠❡ t♦ ♥✱ ✇❡ ❤❛✈❡ ❛♣ ≡ ❛(♠♦❞♣) ❚❤✐s t❤❡♦r❡♠ ❝❛♥ ❛❧s♦ ❜❡ st❛t❡❞ ❛s✿ ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❛ ✐s ❝♦♣r✐♠❡ t♦ ♣✱ t❤❡♥ ❛♣

✶ ♠♦❞♣

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-76
SLIDE 76

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠

■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ t❤❡♥ ❢♦r ❛♥② ✐♥t❡❣❡r ❛ t❤❛t ✐s ❝♦♣r✐♠❡ t♦ ♥✱ ✇❡ ❤❛✈❡ ❛♣ ≡ ❛(♠♦❞♣) ❚❤✐s t❤❡♦r❡♠ ❝❛♥ ❛❧s♦ ❜❡ st❛t❡❞ ❛s✿ ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❛ ✐s ❝♦♣r✐♠❡ t♦ ♣✱ t❤❡♥ ❛♣−✶ ≡ ✶(♠♦❞♣)

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 77

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r✬s ❚❤❡♦r❡♠

❊✉❧❡r✬s ❚❤❡♦r❡♠ ✐s ❛ ❣❡♥❛r❛❧✐③❛t✐♦♥ ❢♦r ❋❡r♠❛t✬s ❧✐tt❧❡ t❤❡♦r❡♠ ✇❤❡♥ ❛ ❛♥❞ ♥ ❛r❡ ❝♦✲♣r✐♠❡ ■❢ ① ② ♠♦❞ ♥ ✱ t❤❡♥ ❛① ❛② ♠♦❞♥ ✳ ❛

✶ ♠♦❞♥ ✭❛❝t✉❛❧ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 78

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r✬s ❚❤❡♦r❡♠

❊✉❧❡r✬s ❚❤❡♦r❡♠ ✐s ❛ ❣❡♥❛r❛❧✐③❛t✐♦♥ ❢♦r ❋❡r♠❛t✬s ❧✐tt❧❡ t❤❡♦r❡♠ ✇❤❡♥ ❛ ❛♥❞ ♥ ❛r❡ ❝♦✲♣r✐♠❡ ■❢ ① ② ♠♦❞ ♥ ✱ t❤❡♥ ❛① ❛② ♠♦❞♥ ✳ ❛

✶ ♠♦❞♥ ✭❛❝t✉❛❧ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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SLIDE 79

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r✬s ❚❤❡♦r❡♠

❊✉❧❡r✬s ❚❤❡♦r❡♠ ✐s ❛ ❣❡♥❛r❛❧✐③❛t✐♦♥ ❢♦r ❋❡r♠❛t✬s ❧✐tt❧❡ t❤❡♦r❡♠ ✇❤❡♥ ❛ ❛♥❞ ♥ ❛r❡ ❝♦✲♣r✐♠❡ ■❢ ① ≡ ②(♠♦❞φ(♥))✱ t❤❡♥ ❛① ≡ ❛②(♠♦❞♥)✳ ❛

✶ ♠♦❞♥ ✭❛❝t✉❛❧ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-80
SLIDE 80

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❊✉❧❡r✬s ❚❤❡♦r❡♠

❊✉❧❡r✬s ❚❤❡♦r❡♠ ✐s ❛ ❣❡♥❛r❛❧✐③❛t✐♦♥ ❢♦r ❋❡r♠❛t✬s ❧✐tt❧❡ t❤❡♦r❡♠ ✇❤❡♥ ❛ ❛♥❞ ♥ ❛r❡ ❝♦✲♣r✐♠❡ ■❢ ① ≡ ②(♠♦❞φ(♥))✱ t❤❡♥ ❛① ≡ ❛②(♠♦❞♥)✳ ❛φ(♥) ≡ ✶(♠♦❞♥) ✭❛❝t✉❛❧ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✮

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-81
SLIDE 81

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r r❡s✉❧ts

■❢ ♥ = ♣❛✶

✶ ∗ ♣❛✷ ✷ ∗ ... ∗ ♣❛t t ✱ t❤❡♥

❚❤❡ ♥✉♠❜❡r ♦❢ ✐ts ♣♦s✐t✐✈❡ ❞✐✈✐s♦rs ❡q✉❛❧s (❛✶ + ✶) ∗ (❛✷ + ✶) ∗ ... ∗ (❛t + ✶) ❙✉♠ ♦❢ t❤❡ ❞✐✈✐s♦rs ♦❢ ♥ ❡q✉❛❧s

❞ ♥ t ✐ ✶ ♣

♠✐ ✶ ✐

✶ ♣✐ ✶

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-82
SLIDE 82

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r r❡s✉❧ts

■❢ ♥ = ♣❛✶

✶ ∗ ♣❛✷ ✷ ∗ ... ∗ ♣❛t t ✱ t❤❡♥

❚❤❡ ♥✉♠❜❡r ♦❢ ✐ts ♣♦s✐t✐✈❡ ❞✐✈✐s♦rs ❡q✉❛❧s (❛✶ + ✶) ∗ (❛✷ + ✶) ∗ ... ∗ (❛t + ✶) ❙✉♠ ♦❢ t❤❡ ❞✐✈✐s♦rs ♦❢ ♥ ❡q✉❛❧s

  • ❞|♥ = t

✐=✶ ♣

♠✐ +✶ ✐

−✶ ♣✐−✶

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-83
SLIDE 83

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮

■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷s❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛❞ = ✶ ♠♦❞ ♥ ❛♥❞ ❛✷r .❞ = −✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [✵, s✶] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡

✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-84
SLIDE 84

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮

■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷s❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛❞ = ✶ ♠♦❞ ♥ ❛♥❞ ❛✷r .❞ = −✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [✵, s✶] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡

✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-85
SLIDE 85

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮

■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷s❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛❞ = ✶ ♠♦❞ ♥ ❛♥❞ ❛✷r .❞ = −✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [✵, s✶] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡

✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-86
SLIDE 86

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮

■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷s❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛❞ = ✶ ♠♦❞ ♥ ❛♥❞ ❛✷r .❞ = −✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [✵, s✶] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡

✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-87
SLIDE 87

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮

■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷s❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛❞ = ✶ ♠♦❞ ♥ ❛♥❞ ❛✷r .❞ = −✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [✵, s✶] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡

✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-88
SLIDE 88

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮

■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷s❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛❞ = ✶ ♠♦❞ ♥ ❛♥❞ ❛✷r .❞ = −✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [✵, s✶] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡

✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-89
SLIDE 89

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮

■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷s❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛❞ = ✶ ♠♦❞ ♥ ❛♥❞ ❛✷r .❞ = −✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [✵, s✶] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡

✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-90
SLIDE 90

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖✉t❧✐♥❡

▼❛t❤s

Pr♦❜❛❜✐❧✐t②

Pr♦❜❧❡♠s

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-91
SLIDE 91

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❋♦r ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ P✭❳✮✱ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊❬❳❪ ✐s ❣✐✈❡♥ ❜②

✐ ①✐

P ①✐ t❤❡ s✉♠♠❛t✐♦♥ r✉♥s ♦✈❡r ❛❧❧ t❤❡ ❞✐st✐♥❝t ✈❛❧✉❡s ①✐ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ t❛❦❡✳ ❚❤❡ r✉❧❡ ♦❢ ✧❧✐♥❡❛r✐t② ♦❢ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥✧ s❛②s t❤❛t ❊❬①✶✰①✷❪ ❂ ❊❬①✶❪ ✰ ❊❬①✷❪✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t ✧❡①♣❡❝t❡❞ ✈❛❧✉❡✧ ✐s ♥♦t s❛♠❡ ❛s ✧♠♦st ♣r♦❜❛❜❧❡ ✈❛❧✉❡✧ ✲ r❛t❤❡r✱ ✐t ♥❡❡❞ ♥♦t ❡✈❡♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♣r♦❜❛❜❧❡ ✈❛❧✉❡s✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-92
SLIDE 92

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❋♦r ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ P✭❳✮✱ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊❬❳❪ ✐s ❣✐✈❡♥ ❜②

✐ ①✐ ∗ P(①✐) t❤❡ s✉♠♠❛t✐♦♥

r✉♥s ♦✈❡r ❛❧❧ t❤❡ ❞✐st✐♥❝t ✈❛❧✉❡s ①✐ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ t❛❦❡✳ ❚❤❡ r✉❧❡ ♦❢ ✧❧✐♥❡❛r✐t② ♦❢ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥✧ s❛②s t❤❛t ❊❬①✶✰①✷❪ ❂ ❊❬①✶❪ ✰ ❊❬①✷❪✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t ✧❡①♣❡❝t❡❞ ✈❛❧✉❡✧ ✐s ♥♦t s❛♠❡ ❛s ✧♠♦st ♣r♦❜❛❜❧❡ ✈❛❧✉❡✧ ✲ r❛t❤❡r✱ ✐t ♥❡❡❞ ♥♦t ❡✈❡♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♣r♦❜❛❜❧❡ ✈❛❧✉❡s✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-93
SLIDE 93

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❋♦r ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ P✭❳✮✱ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊❬❳❪ ✐s ❣✐✈❡♥ ❜②

✐ ①✐ ∗ P(①✐) t❤❡ s✉♠♠❛t✐♦♥

r✉♥s ♦✈❡r ❛❧❧ t❤❡ ❞✐st✐♥❝t ✈❛❧✉❡s ①✐ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ t❛❦❡✳ ❚❤❡ r✉❧❡ ♦❢ ✧❧✐♥❡❛r✐t② ♦❢ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥✧ s❛②s t❤❛t ❊❬①✶✰①✷❪ ❂ ❊❬①✶❪ ✰ ❊❬①✷❪✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t ✧❡①♣❡❝t❡❞ ✈❛❧✉❡✧ ✐s ♥♦t s❛♠❡ ❛s ✧♠♦st ♣r♦❜❛❜❧❡ ✈❛❧✉❡✧ ✲ r❛t❤❡r✱ ✐t ♥❡❡❞ ♥♦t ❡✈❡♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♣r♦❜❛❜❧❡ ✈❛❧✉❡s✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-94
SLIDE 94

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❋♦r ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ P✭❳✮✱ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊❬❳❪ ✐s ❣✐✈❡♥ ❜②

✐ ①✐ ∗ P(①✐) t❤❡ s✉♠♠❛t✐♦♥

r✉♥s ♦✈❡r ❛❧❧ t❤❡ ❞✐st✐♥❝t ✈❛❧✉❡s ①✐ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ t❛❦❡✳ ❚❤❡ r✉❧❡ ♦❢ ✧❧✐♥❡❛r✐t② ♦❢ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥✧ s❛②s t❤❛t ❊❬①✶✰①✷❪ ❂ ❊❬①✶❪ ✰ ❊❬①✷❪✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t ✧❡①♣❡❝t❡❞ ✈❛❧✉❡✧ ✐s ♥♦t s❛♠❡ ❛s ✧♠♦st ♣r♦❜❛❜❧❡ ✈❛❧✉❡✧ ✲ r❛t❤❡r✱ ✐t ♥❡❡❞ ♥♦t ❡✈❡♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♣r♦❜❛❜❧❡ ✈❛❧✉❡s✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-95
SLIDE 95

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-96
SLIDE 96

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-97
SLIDE 97

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-98
SLIDE 98

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-99
SLIDE 99

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥

❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-100
SLIDE 100

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-101
SLIDE 101

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-102
SLIDE 102

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-103
SLIDE 103

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-104
SLIDE 104

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-105
SLIDE 105

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-106
SLIDE 106

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-107
SLIDE 107

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-108
SLIDE 108

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-109
SLIDE 109

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-110
SLIDE 110

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞

❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥

  • ❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮
  • r♦✉♣ ❚❤❡♦r②

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-111
SLIDE 111

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

❖✉t❧✐♥❡

▼❛t❤s

Pr♦❜❛❜✐❧✐t②

Pr♦❜❧❡♠s

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-112
SLIDE 112

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr♦❜❧❡♠s

▲✐♥❦s✿

✶ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴▼❆■◆✶✶✶✴ ✷ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴◆❉■❱P❍■✴ ✸ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❈❯❇❊❋❘✴ ✹ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴◆❖❙◗✴ ✺ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❯❈■✷✵✵✾❇✴ ✻ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❙❊◗✻✴ ✼ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❍❆▼❙❚❊❘✶✴ ✽ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴▼❆■◆✼✹✴ ✾ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❚❯❚▼❘❇▲✴ ✶✵ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❋❆❈❚✵✴ ✶✶ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴●❈❉✸✴ ✶✷ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❈❘❨P❚❖◆✴ ✶✸ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴▼❆■◆✶✷❇✴ ✶✹ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴P▲❨●❘◆❉✴ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

slide-113
SLIDE 113

▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s

Pr♦❜❧❡♠s

❆❞❞❡❞ ♦♥ t❤❡ ❝♦♥t❡st ♦♥ ❱❖❈ ❤tt♣✿✴✴❛❤♠❡❞✲❛❧②✳❝♦♠✴✈♦❝✴ ❈♦♥t❡st ■❉✿ ✷✻✸✸ ◆❛♠❡✿ ❆❈❆✱ ■■❚❑ ▲❖P ✵✹ ❆✉t❤♦r✿ ♣♥❦❥❥✐♥❞❛❧

▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s