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sst strt tt - - PowerPoint PPT Presentation

Oct 25, 2022 246 likes •1.26k views

sst strt tt s t ts t r t

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SLIDE 1

❆ ❈❧❛ss✐☞❝❛t✐♦♥ ♦❢ ❘❡str✐❝t❡❞ ▲❛tt✐❝❡ ❲❛❧❦s ✇✐t❤ ❙♠❛❧❧ ❙t❡♣s

❙t❡♣❤❡♥ ▼❡❧❝③❡r ✭❙❋❯✱ ■◆❘■❆✲▼❙❘✮

❏♦✐♥t ✇♦r❦ ✇✐t❤ ▼✐r❡✐❧❧❡ ❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✭❯♥✐✈❡rs✐t✓ ❡ ❇♦r❞❡❛✉① ✶✮✱ ❆❧✐♥ ❇♦st❛♥ ✭■◆❘■❆✲▼❙❘✮✱ ▼❛♥✉❡❧ ❑❛✉❡rs ✭❘■❙❈✮✱ ❛♥❞ ▼❛r♥✐ ▼✐s❤♥❛ ✭❙❋❯✮

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SLIDE 2

▲❛tt✐❝❡ ❲❛❧❦s ✐♥ t❤❡ ◗✉❛rt❡r P❧❛♥❡

Given: ❆ s❡t ♦❢ ❞✐r❡❝t✐♦♥s Count: ◆✉♠❜❡r ♦❢ ✐♥t❡❣❡r ❧❛tt✐❝❡ ✇❛❧❦s ✐♥ t❤❡ ☞rst q✉❛❞r❛♥t ✉s✐♥❣ t❤❡s❡ st❡♣s✳ ❋♦r ✐♥st❛♥❝❡✱ ❣✐✈❡♥ t❤❡ st❡♣ s❡t ❙ ❂ ❢◆❊❀ ❙❊❀ ◆❲❀ ❙❲❣ t❤❡r❡ ❛r❡ ✾ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ✸✿

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SLIDE 3

▲❛tt✐❝❡ ❲❛❧❦s ✐♥ t❤❡ ◗✉❛rt❡r P❧❛♥❡

Given: ❆ s❡t ♦❢ ❞✐r❡❝t✐♦♥s Count: ◆✉♠❜❡r ♦❢ ✐♥t❡❣❡r ❧❛tt✐❝❡ ✇❛❧❦s ✐♥ t❤❡ ☞rst q✉❛❞r❛♥t ✉s✐♥❣ t❤❡s❡ st❡♣s✳ ❋♦r ✐♥st❛♥❝❡✱ ❣✐✈❡♥ t❤❡ st❡♣ s❡t ❙ ❂ ❢◆❊❀ ❙❊❀ ◆❲❀ ❙❲❣ t❤❡r❡ ❛r❡ ✾ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ✸✿

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SLIDE 4

❘❡❛❧✐st✐❝ ●♦❛❧s

❋✐♥❞✐♥❣ ❛♥ ❡①❛❝t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ✐s t♦♦ ❤❛r❞ ✐♥ ❣❡♥❡r❛❧ ✲ ✇❡ s❡❡❦ asymptotic estimates✳ ■❢ ✇❡ ❝❛♥ ❝❧❛ss✐❢② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❛s ❛❧❣❡❜r❛✐❝ ♦r ❉✲❋✐♥✐t❡ ✭s❛t✐s☞❡s ❛ ❧✐♥❡❛r ❖❉❊ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✍❝✐❡♥ts✮ t❤❡♥ ✇❡ ❦♥♦✇ t❤❡ ❢♦r♠ ♦❢ ✐ts ❣r♦✇t❤✳ ◆❛t✉r❡ ♦❢ ❋✭t✮

  • r♦✇t❤ ♦❢ ❬t♥❪❋✭t✮

❆❧❣❡❜r❛✐❝

❝☞♥♥s ✭s✰✶✮

❉✲❋✐♥✐t❡ ❆☞♥♥s ❧♦❣✭♥✮r ◆❡✐t❤❡r ❄

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SLIDE 5

❘❡❛❧✐st✐❝ ●♦❛❧s

❋✐♥❞✐♥❣ ❛♥ ❡①❛❝t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ✐s t♦♦ ❤❛r❞ ✐♥ ❣❡♥❡r❛❧ ✲ ✇❡ s❡❡❦ asymptotic estimates✳ ■❢ ✇❡ ❝❛♥ ❝❧❛ss✐❢② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❛s ❛❧❣❡❜r❛✐❝ ♦r ❉✲❋✐♥✐t❡ ✭s❛t✐s☞❡s ❛ ❧✐♥❡❛r ❖❉❊ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✍❝✐❡♥ts✮ t❤❡♥ ✇❡ ❦♥♦✇ t❤❡ ❢♦r♠ ♦❢ ✐ts ❣r♦✇t❤✳ ◆❛t✉r❡ ♦❢ ❋✭t✮

  • r♦✇t❤ ♦❢ ❬t♥❪❋✭t✮

❆❧❣❡❜r❛✐❝

❝☞♥♥s ✭s✰✶✮

❉✲❋✐♥✐t❡ ❆☞♥♥s ❧♦❣✭♥✮r ◆❡✐t❤❡r ❄

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SLIDE 6

❉✲❋✐♥✐t❡ ❋✉♥❝t✐♦♥s

■❢ ✇❡ ❦♥♦✇ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❉✲❋✐♥✐t❡✱ ✇❡ ❝❛♥✿ ❞❡t❡r♠✐♥❡ ✐ts ❛s②♠♣t♦t✐❝s ✭❢r♦♠ ❞✐☛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✮❀ ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s ❡✍❝✐❡♥t❧②❀ ❛♥s✇❡r q✉❡st✐♦♥s ❛❜♦✉t r❡❧❛t❡❞ ♣❤②s✐❝❛❧ s②st❡♠s ✭❧✐♠✐t✐♥❣ ❢r❡❡ ❡♥❡r❣②✮✳

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SLIDE 7

❉✲❋✐♥✐t❡ ❋✉♥❝t✐♦♥s

■❢ ✇❡ ❦♥♦✇ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❉✲❋✐♥✐t❡✱ ✇❡ ❝❛♥✿ ❞❡t❡r♠✐♥❡ ✐ts ❛s②♠♣t♦t✐❝s ✭❢r♦♠ ❞✐☛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✮❀ ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s ❡✍❝✐❡♥t❧②❀ ❛♥s✇❡r q✉❡st✐♦♥s ❛❜♦✉t r❡❧❛t❡❞ ♣❤②s✐❝❛❧ s②st❡♠s ✭❧✐♠✐t✐♥❣ ❢r❡❡ ❡♥❡r❣②✮✳

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SLIDE 8

❉✲❋✐♥✐t❡ ❋✉♥❝t✐♦♥s

■❢ ✇❡ ❦♥♦✇ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❉✲❋✐♥✐t❡✱ ✇❡ ❝❛♥✿ ❞❡t❡r♠✐♥❡ ✐ts ❛s②♠♣t♦t✐❝s ✭❢r♦♠ ❞✐☛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✮❀ ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s ❡✍❝✐❡♥t❧②❀ ❛♥s✇❡r q✉❡st✐♦♥s ❛❜♦✉t r❡❧❛t❡❞ ♣❤②s✐❝❛❧ s②st❡♠s ✭❧✐♠✐t✐♥❣ ❢r❡❡ ❡♥❡r❣②✮✳

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SLIDE 9

■s ✐t ❉✲❋✐♥✐t❡❄

❆s②♠♣t♦t✐❝s✿ ❬t♥❪❋✭t✮ ✘ ✷♥✿

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SLIDE 10

■s ✐t ❉✲❋✐♥✐t❡❄

YES - it is rational! ❆s②♠♣t♦t✐❝s✿ ❬t♥❪❋✭t✮ ✘ ✷♥✿

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SLIDE 11

■s ✐t ❉✲❋✐♥✐t❡❄

❆s②♠♣t♦t✐❝s✿ ❬t♥❪❋✭t✮ ✘ ✹ ♣ ✸ ✸

✏

✶ ✸

✑ ✁ ✹♥

♥✷❂✸ ✿

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SLIDE 12

■s ✐t ❉✲❋✐♥✐t❡❄

Yes - it is algebraic! ❆s②♠♣t♦t✐❝s✿ ❬t♥❪❋✭t✮ ✘ ✹ ♣ ✸ ✸

✏

✶ ✸

✑ ✁ ✹♥

♥✷❂✸ ✿

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SLIDE 13

■s ✐t ❉✲❋✐♥✐t❡❄

❆s②♠♣t♦t✐❝s✿ ❬t♥❪❋✭t✮ ✘ ✽ ✙ ✁ ✹♥ ♥✷ ✿

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SLIDE 14

■s ✐t ❉✲❋✐♥✐t❡❄

Yes! (But not algebraic) ❆s②♠♣t♦t✐❝s✿ ❬t♥❪❋✭t✮ ✘ ✽ ✙ ✁ ✹♥ ♥✷ ✿

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SLIDE 15

■s ✐t ❉✲❋✐♥✐t❡❄

❆s②♠♣t♦t✐❝s✿ ❬t♥❪❋✭t✮ ✘ ✜ ✁ ✹♥❀ ✜ ✷ ✿

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SLIDE 16

■s ✐t ❉✲❋✐♥✐t❡❄

NO! ❆s②♠♣t♦t✐❝s✿ ❬t♥❪❋✭t✮ ✘ ✜ ✁ ✹♥❀ ✜ ✷ R✿

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SLIDE 17

❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄

❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s

  • ✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳

❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮

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SLIDE 18

❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄

❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s

  • ✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳

❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮

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SLIDE 19

❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄

❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s

  • ✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳

❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮

slide-20
SLIDE 20

❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄

❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s

  • ✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳

❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮

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SLIDE 21

❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄

❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s

  • ✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳

❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮

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SLIDE 22

❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄

❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s

  • ✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳

❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮

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SLIDE 23

❙❡❛r❝❤✐♥❣ ❢♦r ❉✲❋✐♥✐t❡♥❡ss

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SLIDE 24

❍✐st♦r②

❖❢ ✷✽ ❂ ✷✺✻ st❡♣ s❡ts✱ t❤❡r❡ ❛r❡ ✼✾ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ✷❉ ♠♦❞❡❧s ❬❇♦✉sq✉❡t✲▼✓ ❡❧♦✉✫▼✐s❤♥❛ ✷✵✶✵❪✳ ❙t❡♣ s❡ts ✇❤✐❝❤ ❛r❡ s✉❜s❡ts ♦❢ ❝♦rr❡s♣♦♥❞ t♦ ❤❛❧❢ s♣❛❝❡ ♣r♦❜❧❡♠s✱ ✇❤✐❝❤ ❤❛✈❡ ❜❡❡♥ ♣r❡✈✐♦✉s❧② s♦❧✈❡❞ ❬❇❛♥❞❡r✐❡r✫❋❧❛❥♦❧❡t ✷✵✵✶❪✳ ❆❧s♦✱ s✉❜s❡ts ♦❢ ✇✐❧❧ ♥❡✈❡r ❧❡❛✈❡ t❤❡ ♦r✐❣✐♥✱ s♦ t❤❡s❡ ❛r❡ ❛❧s♦ ♥♦t ❝♦♥s✐❞❡r❡❞✳

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SLIDE 25

❍✐st♦r②

❖❢ ✷✽ ❂ ✷✺✻ st❡♣ s❡ts✱ t❤❡r❡ ❛r❡ ✼✾ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ✷❉ ♠♦❞❡❧s ❬❇♦✉sq✉❡t✲▼✓ ❡❧♦✉✫▼✐s❤♥❛ ✷✵✶✵❪✳ ❙t❡♣ s❡ts ✇❤✐❝❤ ❛r❡ s✉❜s❡ts ♦❢ ❝♦rr❡s♣♦♥❞ t♦ ❤❛❧❢ s♣❛❝❡ ♣r♦❜❧❡♠s✱ ✇❤✐❝❤ ❤❛✈❡ ❜❡❡♥ ♣r❡✈✐♦✉s❧② s♦❧✈❡❞ ❬❇❛♥❞❡r✐❡r✫❋❧❛❥♦❧❡t ✷✵✵✶❪✳ ❆❧s♦✱ s✉❜s❡ts ♦❢ ✇✐❧❧ ♥❡✈❡r ❧❡❛✈❡ t❤❡ ♦r✐❣✐♥✱ s♦ t❤❡s❡ ❛r❡ ❛❧s♦ ♥♦t ❝♦♥s✐❞❡r❡❞✳

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SLIDE 26

❍✐st♦r②

❖❢ ✷✽ ❂ ✷✺✻ st❡♣ s❡ts✱ t❤❡r❡ ❛r❡ ✼✾ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ✷❉ ♠♦❞❡❧s ❬❇♦✉sq✉❡t✲▼✓ ❡❧♦✉✫▼✐s❤♥❛ ✷✵✶✵❪✳ ❙t❡♣ s❡ts ✇❤✐❝❤ ❛r❡ s✉❜s❡ts ♦❢ ❝♦rr❡s♣♦♥❞ t♦ ❤❛❧❢ s♣❛❝❡ ♣r♦❜❧❡♠s✱ ✇❤✐❝❤ ❤❛✈❡ ❜❡❡♥ ♣r❡✈✐♦✉s❧② s♦❧✈❡❞ ❬❇❛♥❞❡r✐❡r✫❋❧❛❥♦❧❡t ✷✵✵✶❪✳ ❆❧s♦✱ s✉❜s❡ts ♦❢ ✇✐❧❧ ♥❡✈❡r ❧❡❛✈❡ t❤❡ ♦r✐❣✐♥✱ s♦ t❤❡s❡ ❛r❡ ❛❧s♦ ♥♦t ❝♦♥s✐❞❡r❡❞✳

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SLIDE 27

❚♦♦❧s

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ◗✭①❀ ②❀ t✮ ❂

❳

♥❀✐❀❥✕✵

q✭✐❀ ❥❀ ♥✮①✐②❥t♥ ✇❤✐❝❤ ❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ❡♥❞✐♥❣ ❛t ✭✐❀ ❥✮ s❛t✐s☞❡s ❛♥ ♦❜✈✐♦✉s ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✇✐t❤ t❤❡ st❡♣ s❡t ✇❡ ❞❡☞♥❡ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ❙✭①❀ ②✮ ✿❂ ①② ✰ ② ① ✰ ✶ ①② ✰ ① ②

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SLIDE 28

❚♦♦❧s

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ◗✭①❀ ②❀ t✮ ❂

❳

♥❀✐❀❥✕✵

q✭✐❀ ❥❀ ♥✮①✐②❥t♥ ✇❤✐❝❤ ❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ❡♥❞✐♥❣ ❛t ✭✐❀ ❥✮ s❛t✐s☞❡s ❛♥ ♦❜✈✐♦✉s ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✇✐t❤ t❤❡ st❡♣ s❡t ✇❡ ❞❡☞♥❡ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ❙✭①❀ ②✮ ✿❂ ①② ✰ ② ① ✰ ✶ ①② ✰ ① ②

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SLIDE 29

❆ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥

◗✭①❀ ②❀ t✮ ❂ ✶ ✰ t❙✭①❀ ②✮◗✭①❀ ②❀ t✮ t✭② ① ✰ ✶ ①②✮◗✭✵❀ ②❀ t✮ t✭① ② ✰ ✶ ①②✮◗✭①❀ ✵❀ t✮ ✰ t ①②◗✭✵❀ ✵❀ t✮✿

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SLIDE 30

❆ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥

◗✭①❀ ②❀ t✮ ❂ ✶ ✰ t❙✭①❀ ②✮◗✭①❀ ②❀ t✮ t✭② ① ✰ ✶ ①②✮◗✭✵❀ ②❀ t✮ t✭① ② ✰ ✶ ①②✮◗✭①❀ ✵❀ t✮ ✰ t ①②◗✭✵❀ ✵❀ t✮✿

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SLIDE 31

❆ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥

◗✭①❀ ②❀ t✮ ❂ ✶ ✰ t❙✭①❀ ②✮◗✭①❀ ②❀ t✮ t✭② ① ✰ ✶ ①②✮◗✭✵❀ ②❀ t✮ t✭① ② ✰ ✶ ①②✮◗✭①❀ ✵❀ t✮ ✰ t ①②◗✭✵❀ ✵❀ t✮✿

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SLIDE 32

❆ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥

◗✭①❀ ②❀ t✮ ❂ ✶ ✰ t❙✭①❀ ②✮◗✭①❀ ②❀ t✮ t✭② ① ✰ ✶ ①②✮◗✭✵❀ ②❀ t✮ t✭① ② ✰ ✶ ①②✮◗✭①❀ ✵❀ t✮ ✰ t ①②◗✭✵❀ ✵❀ t✮✿

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SLIDE 33

❚❤❡ ❑❡r♥❡❧ ❊q✉❛t✐♦♥

❇❡❣✐♥ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✱ r❡✲❣r♦✉♣ t❤❡ t❡r♠s ❛♥❞ ♠✉❧t✐♣❧② ❜② ①② t♦ ❣✐✈❡✿

❑✭①❀②✮✁①②◗✭①❀②✮❂①②t✭②✷✰✶✮◗✭✵❀②✮t✭①✷✰✶✮◗✭①❀✵✮✰t◗✭✵❀✵✮

✭❑✮ ✇❤❡r❡ ❑✭①❀ ②✮ ❂ ✶ t❙✭①❀ ②✮ ✐s ❝❛❧❧❡❞ t❤❡ ❦❡r♥❡❧ ♦❢ t❤❡ ✇❛❧❦✳ ❲❡ ♥♦✇ ❞❡☞♥❡ ❛ ❣r♦✉♣ ● ♦❢ ❜✐✲r❛t✐♦♥❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ①②✲♣❧❛♥❡ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❙✭①❀ ②✮ ✲ ❛♥❞ t❤✉s ❑✭①❀ ②✮✳

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SLIDE 34

❚❤❡ ❑❡r♥❡❧ ❊q✉❛t✐♦♥

❇❡❣✐♥ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✱ r❡✲❣r♦✉♣ t❤❡ t❡r♠s ❛♥❞ ♠✉❧t✐♣❧② ❜② ①② t♦ ❣✐✈❡✿

❑✭①❀②✮✁①②◗✭①❀②✮❂①②t✭②✷✰✶✮◗✭✵❀②✮t✭①✷✰✶✮◗✭①❀✵✮✰t◗✭✵❀✵✮

✭❑✮ ✇❤❡r❡ ❑✭①❀ ②✮ ❂ ✶ t❙✭①❀ ②✮ ✐s ❝❛❧❧❡❞ t❤❡ ❦❡r♥❡❧ ♦❢ t❤❡ ✇❛❧❦✳ ❲❡ ♥♦✇ ❞❡☞♥❡ ❛ ❣r♦✉♣ ● ♦❢ ❜✐✲r❛t✐♦♥❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ①②✲♣❧❛♥❡ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❙✭①❀ ②✮ ✲ ❛♥❞ t❤✉s ❑✭①❀ ②✮✳

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SLIDE 35

❚❤❡ ●r♦✉♣ ♦❢ ❛ ❲❛❧❦

❚♦ ❜❡❣✐♥✱ ✇r✐t❡ ❙✭①❀ ②✮ ❂ ✶ ②❆✶✭①✮✰❆✵✭①✮✰②❆✶✭①✮ ❂ ✶ ①❇✶✭②✮✰❇✵✭②✮✰①❇✶✭②✮✿ ❲❡ ❧❡t ● ❜❡ t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✐♥✈♦❧✉t✐♦♥s ✜ ✿ ✭①❀ ②✮ ✼✦

✒

①❀ ❆✶✭①✮ ②❆✶✭①✮

✓

✥ ✿ ✭①❀ ②✮ ✼✦

✒❇✶✭②✮

①❇✶✭②✮❀ ②

✓

✿ ■♥ ♦✉r ❝❛s❡✱ ✇❡ ❤❛✈❡ ✥ ✿ ✭①❀ ②✮ ✼✦

✒✶

①❀ ②

✓

✜ ✿ ✭①❀ ②✮ ✼✦

✒

①❀ ✶ ②

✓

❀ s♦ ● ❝♦♥t❛✐♥s ✹ ❡❧❡♠❡♥ts✳

slide-36
SLIDE 36

❚❤❡ ●r♦✉♣ ♦❢ ❛ ❲❛❧❦

❚♦ ❜❡❣✐♥✱ ✇r✐t❡ ❙✭①❀ ②✮ ❂ ✶ ②❆✶✭①✮✰❆✵✭①✮✰②❆✶✭①✮ ❂ ✶ ①❇✶✭②✮✰❇✵✭②✮✰①❇✶✭②✮✿ ❲❡ ❧❡t ● ❜❡ t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✐♥✈♦❧✉t✐♦♥s ✜ ✿ ✭①❀ ②✮ ✼✦

✒

①❀ ❆✶✭①✮ ②❆✶✭①✮

✓

✥ ✿ ✭①❀ ②✮ ✼✦

✒❇✶✭②✮

①❇✶✭②✮❀ ②

✓

✿ ■♥ ♦✉r ❝❛s❡✱ ✇❡ ❤❛✈❡ ✥ ✿ ✭①❀ ②✮ ✼✦

✒✶

①❀ ②

✓

✜ ✿ ✭①❀ ②✮ ✼✦

✒

①❀ ✶ ②

✓

❀ s♦ ● ❝♦♥t❛✐♥s ✹ ❡❧❡♠❡♥ts✳

slide-37
SLIDE 37

❚❤❡ ●r♦✉♣ ♦❢ ❛ ❲❛❧❦

❚♦ ❜❡❣✐♥✱ ✇r✐t❡ ❙✭①❀ ②✮ ❂ ✶ ②❆✶✭①✮✰❆✵✭①✮✰②❆✶✭①✮ ❂ ✶ ①❇✶✭②✮✰❇✵✭②✮✰①❇✶✭②✮✿ ❲❡ ❧❡t ● ❜❡ t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✐♥✈♦❧✉t✐♦♥s ✜ ✿ ✭①❀ ②✮ ✼✦

✒

①❀ ❆✶✭①✮ ②❆✶✭①✮

✓

✥ ✿ ✭①❀ ②✮ ✼✦

✒❇✶✭②✮

①❇✶✭②✮❀ ②

✓

✿ ■♥ ♦✉r ❝❛s❡✱ ✇❡ ❤❛✈❡ ✥ ✿ ✭①❀ ②✮ ✼✦

✒✶

①❀ ②

✓

✜ ✿ ✭①❀ ②✮ ✼✦

✒

①❀ ✶ ②

✓

❀ s♦ ● ❝♦♥t❛✐♥s ✹ ❡❧❡♠❡♥ts✳

slide-38
SLIDE 38

❆ P❛rt✐❛❧ ❘❡s✉❧t

❚❤❡♦r❡♠ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛ ✷✵✶✵✮

✷✸ ♦❢ t❤❡ ✼✾ ✇❛❧❦s ❝♦rr❡s♣♦♥❞ t♦ ❛ ☞♥✐t❡ ❣r♦✉♣✱ ✇✐t❤ ✷✷ ♦❢ t❤❡♠ ❛❞♠✐tt✐♥❣ ❛ ❉✲❋✐♥✐t❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✳ ❚❤❡ r❡♠❛✐♥✐♥❣ ✺✻ ✇❛❧❦s ❝♦rr❡s♣♦♥❞ t♦ ❛♥ ✐♥☞♥✐t❡ ❣r♦✉♣✳

❚❤❡♦r❡♠ ✭❇♦st❛♥ ✫ ❑❛✉❡rs ✷✵✶✵✮

❚❤❡ ✷✸r❞ ✇❛❧❦ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ ✭●❡ss❡❧✬s ✇❛❧❦✮ ❤❛s ❉✲❋✐♥✐t❡ ✭✐♥ ❢❛❝t✱ ❛❧❣❡❜r❛✐❝✮ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✳

❈♦♥❥❡❝t✉r❡ ✭▼✐s❤♥❛ ✷✵✵✼✮

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ◗✭✶❀ ✶❀ t✮ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ✇✐t❤ ✐♥☞♥✐t❡ ❣r♦✉♣ ❛r❡ ♥♦t ❉✲❋✐♥✐t❡✳

slide-39
SLIDE 39

❆ P❛rt✐❛❧ ❘❡s✉❧t

❚❤❡♦r❡♠ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛ ✷✵✶✵✮

✷✸ ♦❢ t❤❡ ✼✾ ✇❛❧❦s ❝♦rr❡s♣♦♥❞ t♦ ❛ ☞♥✐t❡ ❣r♦✉♣✱ ✇✐t❤ ✷✷ ♦❢ t❤❡♠ ❛❞♠✐tt✐♥❣ ❛ ❉✲❋✐♥✐t❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✳ ❚❤❡ r❡♠❛✐♥✐♥❣ ✺✻ ✇❛❧❦s ❝♦rr❡s♣♦♥❞ t♦ ❛♥ ✐♥☞♥✐t❡ ❣r♦✉♣✳

❚❤❡♦r❡♠ ✭❇♦st❛♥ ✫ ❑❛✉❡rs ✷✵✶✵✮

❚❤❡ ✷✸r❞ ✇❛❧❦ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ ✭●❡ss❡❧✬s ✇❛❧❦✮ ❤❛s ❉✲❋✐♥✐t❡ ✭✐♥ ❢❛❝t✱ ❛❧❣❡❜r❛✐❝✮ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✳

❈♦♥❥❡❝t✉r❡ ✭▼✐s❤♥❛ ✷✵✵✼✮

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ◗✭✶❀ ✶❀ t✮ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ✇✐t❤ ✐♥☞♥✐t❡ ❣r♦✉♣ ❛r❡ ♥♦t ❉✲❋✐♥✐t❡✳

slide-40
SLIDE 40

❆ P❛rt✐❛❧ ❘❡s✉❧t

❚❤❡♦r❡♠ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛ ✷✵✶✵✮

✷✸ ♦❢ t❤❡ ✼✾ ✇❛❧❦s ❝♦rr❡s♣♦♥❞ t♦ ❛ ☞♥✐t❡ ❣r♦✉♣✱ ✇✐t❤ ✷✷ ♦❢ t❤❡♠ ❛❞♠✐tt✐♥❣ ❛ ❉✲❋✐♥✐t❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✳ ❚❤❡ r❡♠❛✐♥✐♥❣ ✺✻ ✇❛❧❦s ❝♦rr❡s♣♦♥❞ t♦ ❛♥ ✐♥☞♥✐t❡ ❣r♦✉♣✳

❚❤❡♦r❡♠ ✭❇♦st❛♥ ✫ ❑❛✉❡rs ✷✵✶✵✮

❚❤❡ ✷✸r❞ ✇❛❧❦ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ ✭●❡ss❡❧✬s ✇❛❧❦✮ ❤❛s ❉✲❋✐♥✐t❡ ✭✐♥ ❢❛❝t✱ ❛❧❣❡❜r❛✐❝✮ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✳

❈♦♥❥❡❝t✉r❡ ✭▼✐s❤♥❛ ✷✵✵✼✮

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ◗✭✶❀ ✶❀ t✮ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ✇✐t❤ ✐♥☞♥✐t❡ ❣r♦✉♣ ❛r❡ ♥♦t ❉✲❋✐♥✐t❡✳

slide-41
SLIDE 41

Finite Group Walks (D-Finite):

slide-42
SLIDE 42

Infinite Group Walks (Non D-Finite?):

slide-43
SLIDE 43

❲❛❧❦s ✇✐t❤ ■♥☞♥✐t❡ ●r♦✉♣s

❬▼✐s❤♥❛ ✫ ❘❡❝❤♥✐t③❡r ✷✵✵✾❪ ✷ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭✶❀ ✶❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬❑✉r❦♦✈❛ ✫ ❘❛s❝❤❡❧ ✷✵✶✶❪ ✺✶ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭①❀ ②❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈② ✷✵✶✷❪ ✺✶ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭✵❀ ✵❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬▼✳ ✫ ▼✐s❤♥❛ ✷✵✶✷❪ ❚❤❡ ☞♥❛❧ ✸ ✇❛❧❦s ❤❛✈❡ ◗✭✶❀ ✶❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡

slide-44
SLIDE 44

❲❛❧❦s ✇✐t❤ ■♥☞♥✐t❡ ●r♦✉♣s

❬▼✐s❤♥❛ ✫ ❘❡❝❤♥✐t③❡r ✷✵✵✾❪ ✷ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭✶❀ ✶❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬❑✉r❦♦✈❛ ✫ ❘❛s❝❤❡❧ ✷✵✶✶❪ ✺✶ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭①❀ ②❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈② ✷✵✶✷❪ ✺✶ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭✵❀ ✵❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬▼✳ ✫ ▼✐s❤♥❛ ✷✵✶✷❪ ❚❤❡ ☞♥❛❧ ✸ ✇❛❧❦s ❤❛✈❡ ◗✭✶❀ ✶❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡

slide-45
SLIDE 45

❲❛❧❦s ✇✐t❤ ■♥☞♥✐t❡ ●r♦✉♣s

❬▼✐s❤♥❛ ✫ ❘❡❝❤♥✐t③❡r ✷✵✵✾❪ ✷ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭✶❀ ✶❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬❑✉r❦♦✈❛ ✫ ❘❛s❝❤❡❧ ✷✵✶✶❪ ✺✶ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭①❀ ②❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈② ✷✵✶✷❪ ✺✶ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭✵❀ ✵❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬▼✳ ✫ ▼✐s❤♥❛ ✷✵✶✷❪ ❚❤❡ ☞♥❛❧ ✸ ✇❛❧❦s ❤❛✈❡ ◗✭✶❀ ✶❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡

slide-46
SLIDE 46

❲❛❧❦s ✇✐t❤ ■♥☞♥✐t❡ ●r♦✉♣s

❬▼✐s❤♥❛ ✫ ❘❡❝❤♥✐t③❡r ✷✵✵✾❪ ✷ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭✶❀ ✶❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬❑✉r❦♦✈❛ ✫ ❘❛s❝❤❡❧ ✷✵✶✶❪ ✺✶ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭①❀ ②❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈② ✷✵✶✷❪ ✺✶ ♦❢ t❤❡ ✺✻ ✇❛❧❦s ❤❛✈❡ ◗✭✵❀ ✵❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡ ❬▼✳ ✫ ▼✐s❤♥❛ ✷✵✶✷❪ ❚❤❡ ☞♥❛❧ ✸ ✇❛❧❦s ❤❛✈❡ ◗✭✶❀ ✶❀ t✮ ♥♦♥ ❉✲❋✐♥✐t❡

slide-47
SLIDE 47

Infinite Group Walks (Non D-Finite✄):

slide-48
SLIDE 48

✸ ❚❡❝❤♥✐q✉❡s t♦ Pr♦✈❡ ◆♦♥ ❉✲☞♥✐t❡♥❡ss

slide-49
SLIDE 49

❚❤❡ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❬▼✳✱ ▼✐s❤♥❛✱ ❘❡❝❤♥✐t③❡r❪

❋♦r t❤❡ st❡♣ s❡t ✇❡ ❤❛✈❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥

①②❑✭①❀②✮✁◗✭①❀②✮❂①②t②✷◗✭②❀✵✮t①✷◗✭①❀✵✮

✭❑✮ ✇❤❡r❡ ①②❑✭①❀ ②✮ ❂ ✶ t❙✭①❀ ②✮ ❂ ✶ t✭①✷ ✰ ②✷ ✰ ①✷②✷✮✿ ◆♦t❡ t❤❛t ◗✭✶❀ ✶✮ ❂ ✶ ✷t◗✭✶❀ ✵✮ ✶ ✸t ✿

slide-50
SLIDE 50

❚❤❡ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❬▼✳✱ ▼✐s❤♥❛✱ ❘❡❝❤♥✐t③❡r❪

❋♦r t❤❡ st❡♣ s❡t ✇❡ ❤❛✈❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥

①②❑✭①❀②✮✁◗✭①❀②✮❂①②t②✷◗✭②❀✵✮t①✷◗✭①❀✵✮

✭❑✮ ✇❤❡r❡ ①②❑✭①❀ ②✮ ❂ ✶ t❙✭①❀ ②✮ ❂ ✶ t✭①✷ ✰ ②✷ ✰ ①✷②✷✮✿ ◆♦t❡ t❤❛t ◗✭✶❀ ✶✮ ❂ ✶ ✷t◗✭✶❀ ✵✮ ✶ ✸t ✿

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SLIDE 51

❚❤❡ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❬▼✳✱ ▼✐s❤♥❛✱ ❘❡❝❤♥✐t③❡r❪

❲❡ ✇r✐t❡ ◗✭✶❀ ✵❀ t✮ ❂

❳

♥✕✵

✭✶✮♥❨♥✭t✮❨♥✰✶✭t✮❀ ✇❤❡r❡ t❤❡ ❨♥ ❛r❡ ❡①♣❧✐❝✐t ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s ✭❞❡❣r❡❡ ✷✮ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ❦❡r♥❡❧✳ ❲❡ ♣r♦✈❡ t❤❛t ❡❛❝❤ ❨♥ ❤❛s ❛ ✉♥✐q✉❡ s✐♥❣✉❧❛r✐t②✱ s♦ ◗✭✶❀ ✵✮ ❤❛s ❛♥ ✐♥☞♥✐t❡ ♥✉♠❜❡r ♦❢ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ✐s ♥♦t ❉✲❋✐♥✐t❡✳ ❲❡ ❝❛♥ ❛❧s♦ ❡①tr❛❝t ❛s②♠♣t♦t✐❝s ❛♥❞ q✉✐❝❦❧② ❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s✳

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SLIDE 52

❚❤❡ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❬▼✳✱ ▼✐s❤♥❛✱ ❘❡❝❤♥✐t③❡r❪

❲❡ ✇r✐t❡ ◗✭✶❀ ✵❀ t✮ ❂

❳

♥✕✵

✭✶✮♥❨♥✭t✮❨♥✰✶✭t✮❀ ✇❤❡r❡ t❤❡ ❨♥ ❛r❡ ❡①♣❧✐❝✐t ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s ✭❞❡❣r❡❡ ✷✮ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ❦❡r♥❡❧✳ ❲❡ ♣r♦✈❡ t❤❛t ❡❛❝❤ ❨♥ ❤❛s ❛ ✉♥✐q✉❡ s✐♥❣✉❧❛r✐t②✱ s♦ ◗✭✶❀ ✵✮ ❤❛s ❛♥ ✐♥☞♥✐t❡ ♥✉♠❜❡r ♦❢ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ✐s ♥♦t ❉✲❋✐♥✐t❡✳ ❲❡ ❝❛♥ ❛❧s♦ ❡①tr❛❝t ❛s②♠♣t♦t✐❝s ❛♥❞ q✉✐❝❦❧② ❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s✳

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SLIDE 53

❚❤❡ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❬▼✳✱ ▼✐s❤♥❛✱ ❘❡❝❤♥✐t③❡r❪

❲❡ ✇r✐t❡ ◗✭✶❀ ✵❀ t✮ ❂

❳

♥✕✵

✭✶✮♥❨♥✭t✮❨♥✰✶✭t✮❀ ✇❤❡r❡ t❤❡ ❨♥ ❛r❡ ❡①♣❧✐❝✐t ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s ✭❞❡❣r❡❡ ✷✮ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ❦❡r♥❡❧✳ ❲❡ ♣r♦✈❡ t❤❛t ❡❛❝❤ ❨♥ ❤❛s ❛ ✉♥✐q✉❡ s✐♥❣✉❧❛r✐t②✱ s♦ ◗✭✶❀ ✵✮ ❤❛s ❛♥ ✐♥☞♥✐t❡ ♥✉♠❜❡r ♦❢ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ✐s ♥♦t ❉✲❋✐♥✐t❡✳ ❲❡ ❝❛♥ ❛❧s♦ ❡①tr❛❝t ❛s②♠♣t♦t✐❝s ❛♥❞ q✉✐❝❦❧② ❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s✳

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SLIDE 54

P❧♦ts ♦❢ ❙✐♥❣✉❧❛r✐t✐❡s

❙✐♥❣✉❧❛r✐t✐❡s ♦❢ ❨♥ ❢♦r

  • 1

1

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SLIDE 55

P❧♦ts ♦❢ ❙✐♥❣✉❧❛r✐t✐❡s

❙✐♥❣✉❧❛r✐t✐❡s ♦❢ ❨♥ ❢♦r

  • 1

1

slide-56
SLIDE 56

❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❬❋❛②♦❧❧❡✱ ❑✉r❦♦✈❛✱ ❘❛s❝❤❡❧❪

❍❡r❡✱ t❤❡ ◗✭①❀ ✵❀ t✮✱ ◗✭✵❀ ②❀ t✮ ❛♥❞ ◗✭✵❀ ✵❀ t✮ ❛r❡ ❣✐✈❡♥ ❛s ❡①♣❧✐❝✐t ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥s✳ ❚❤❡s❡ ❛r❡ ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ♦❢ ❘✐❡♠❛♥♥✲❈❛r❧❡♠❛♥ t②♣❡✳ ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ♣r♦✈❡ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss ♦❢ ◗✭①❀ ②❀ t✮✳ ◆♦t❡ t❤❛t t❤✐s ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t ◗✭✶❀ ✶❀ t✮ ✐s ♥♦♥ ❉✲❋✐♥✐t❡✳

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SLIDE 57

❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❬❋❛②♦❧❧❡✱ ❑✉r❦♦✈❛✱ ❘❛s❝❤❡❧❪

❍❡r❡✱ t❤❡ ◗✭①❀ ✵❀ t✮✱ ◗✭✵❀ ②❀ t✮ ❛♥❞ ◗✭✵❀ ✵❀ t✮ ❛r❡ ❣✐✈❡♥ ❛s ❡①♣❧✐❝✐t ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥s✳ ❚❤❡s❡ ❛r❡ ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ♦❢ ❘✐❡♠❛♥♥✲❈❛r❧❡♠❛♥ t②♣❡✳ ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ♣r♦✈❡ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss ♦❢ ◗✭①❀ ②❀ t✮✳ ◆♦t❡ t❤❛t t❤✐s ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t ◗✭✶❀ ✶❀ t✮ ✐s ♥♦♥ ❉✲❋✐♥✐t❡✳

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SLIDE 58

❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❬❋❛②♦❧❧❡✱ ❑✉r❦♦✈❛✱ ❘❛s❝❤❡❧❪

❍❡r❡✱ t❤❡ ◗✭①❀ ✵❀ t✮✱ ◗✭✵❀ ②❀ t✮ ❛♥❞ ◗✭✵❀ ✵❀ t✮ ❛r❡ ❣✐✈❡♥ ❛s ❡①♣❧✐❝✐t ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥s✳ ❚❤❡s❡ ❛r❡ ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ♦❢ ❘✐❡♠❛♥♥✲❈❛r❧❡♠❛♥ t②♣❡✳ ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ♣r♦✈❡ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss ♦❢ ◗✭①❀ ②❀ t✮✳ ◆♦t❡ t❤❛t t❤✐s ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t ◗✭✶❀ ✶❀ t✮ ✐s ♥♦♥ ❉✲❋✐♥✐t❡✳

slide-59
SLIDE 59

❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❬❋❛②♦❧❧❡✱ ❑✉r❦♦✈❛✱ ❘❛s❝❤❡❧❪

❍❡r❡✱ t❤❡ ◗✭①❀ ✵❀ t✮✱ ◗✭✵❀ ②❀ t✮ ❛♥❞ ◗✭✵❀ ✵❀ t✮ ❛r❡ ❣✐✈❡♥ ❛s ❡①♣❧✐❝✐t ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥s✳ ❚❤❡s❡ ❛r❡ ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ♦❢ ❘✐❡♠❛♥♥✲❈❛r❧❡♠❛♥ t②♣❡✳ ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ♣r♦✈❡ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss ♦❢ ◗✭①❀ ②❀ t✮✳ ◆♦t❡ t❤❛t t❤✐s ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t ◗✭✶❀ ✶❀ t✮ ✐s ♥♦♥ ❉✲❋✐♥✐t❡✳

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SLIDE 60

❊①❝✉rs✐♦♥ ▼❡t❤♦❞ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈②❪

❆ r❡❝❡♥t ♣r♦❜❛❜✐❧✐st✐❝ r❡s✉❧t ❬❉❡♥✐s♦✈ ✫ ❲❛❝❤t❡❧ ✷✵✶✶❪ ✐♠♣❧✐❡s t❤❛t ❢♦r ✇❛❧❦s ✐♥ t❤❡ q✉❛rt❡r ♣❧❛♥❡ ❬t♥❪◗✭✵❀ ✵❀ t✮ ✘ ❑ ✁ ✚♥ ✁ ♥☛❀ ✇❤❡r❡ ☛ ❂ ✶ ✙❂ ❛r❝❝♦s✭❝✮ ❝ ❛♥❞ ✚ ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ ❛ ♣♦❧②♥♦♠✐❛❧ s②st❡♠✳ ❆s ◗✭✵❀ ✵❀ t✮ ✐s ❛ ●✲❢✉♥❝t✐♦♥✱ t❤❡ ❣r♦✇t❤ ❡①♣♦♥❡♥t ☛ ♠✉st ❜❡ r❛t✐♦♥❛❧ ✐❢ ◗✭✵❀ ✵❀ t✮ ✐s ❉✲❋✐♥✐t❡✳

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SLIDE 61

❊①❝✉rs✐♦♥ ▼❡t❤♦❞ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈②❪

❆ r❡❝❡♥t ♣r♦❜❛❜✐❧✐st✐❝ r❡s✉❧t ❬❉❡♥✐s♦✈ ✫ ❲❛❝❤t❡❧ ✷✵✶✶❪ ✐♠♣❧✐❡s t❤❛t ❢♦r ✇❛❧❦s ✐♥ t❤❡ q✉❛rt❡r ♣❧❛♥❡ ❬t♥❪◗✭✵❀ ✵❀ t✮ ✘ ❑ ✁ ✚♥ ✁ ♥☛❀ ✇❤❡r❡ ☛ ❂ ✶ ✙❂ ❛r❝❝♦s✭❝✮ ❝ ❛♥❞ ✚ ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ ❛ ♣♦❧②♥♦♠✐❛❧ s②st❡♠✳ ❆s ◗✭✵❀ ✵❀ t✮ ✐s ❛ ●✲❢✉♥❝t✐♦♥✱ t❤❡ ❣r♦✇t❤ ❡①♣♦♥❡♥t ☛ ♠✉st ❜❡ r❛t✐♦♥❛❧ ✐❢ ◗✭✵❀ ✵❀ t✮ ✐s ❉✲❋✐♥✐t❡✳

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SLIDE 62

❊①❝✉rs✐♦♥ ▼❡t❤♦❞ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈②❪

❆ r❡❝❡♥t ♣r♦❜❛❜✐❧✐st✐❝ r❡s✉❧t ❬❉❡♥✐s♦✈ ✫ ❲❛❝❤t❡❧ ✷✵✶✶❪ ✐♠♣❧✐❡s t❤❛t ❢♦r ✇❛❧❦s ✐♥ t❤❡ q✉❛rt❡r ♣❧❛♥❡ ❬t♥❪◗✭✵❀ ✵❀ t✮ ✘ ❑ ✁ ✚♥ ✁ ♥☛❀ ✇❤❡r❡ ☛ ❂ ✶ ✙❂ ❛r❝❝♦s✭❝✮ ❝ ❛♥❞ ✚ ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ ❛ ♣♦❧②♥♦♠✐❛❧ s②st❡♠✳ ❆s ◗✭✵❀ ✵❀ t✮ ✐s ❛ ●✲❢✉♥❝t✐♦♥✱ t❤❡ ❣r♦✇t❤ ❡①♣♦♥❡♥t ☛ ♠✉st ❜❡ r❛t✐♦♥❛❧ ✐❢ ◗✭✵❀ ✵❀ t✮ ✐s ❉✲❋✐♥✐t❡✳

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SLIDE 63

❊①❝✉rs✐♦♥ ▼❡t❤♦❞ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈②❪

■rr❛t✐♦♥❛❧✐t② ♦❢ ☛ ✐s ♣r♦✈❡♥ ❛✉t♦♠❛t✐❝❛❧❧② ❜②✿ ✭✶✮ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧✱ ✖❝✱ ♦❢ ❝ ✭✷✮ ♣r♦✈✐♥❣ t❤❛t t❤❡ ♥✉♠❡r❛t♦r ♦❢ ✖❝

✏

①✷✰✶ ✷①

✑

❝♦♥t❛✐♥s ♥♦ ❝②❝❧♦t♦♠✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤✐s ♣r♦✈❡s t❤❛t ◗✭✵❀ ✵❀ t✮ ✐s ♥♦♥ ❉✲❋✐♥✐t❡ ✭❛♥❞ t❤✉s s♦ ✐s ◗✭①❀ ②❀ t✮✮ ❢♦r ✺✶ ✇❛❧❦s✳

slide-64
SLIDE 64

❊①❝✉rs✐♦♥ ▼❡t❤♦❞ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈②❪

■rr❛t✐♦♥❛❧✐t② ♦❢ ☛ ✐s ♣r♦✈❡♥ ❛✉t♦♠❛t✐❝❛❧❧② ❜②✿ ✭✶✮ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧✱ ✖❝✱ ♦❢ ❝ ✭✷✮ ♣r♦✈✐♥❣ t❤❛t t❤❡ ♥✉♠❡r❛t♦r ♦❢ ✖❝

✏

①✷✰✶ ✷①

✑

❝♦♥t❛✐♥s ♥♦ ❝②❝❧♦t♦♠✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤✐s ♣r♦✈❡s t❤❛t ◗✭✵❀ ✵❀ t✮ ✐s ♥♦♥ ❉✲❋✐♥✐t❡ ✭❛♥❞ t❤✉s s♦ ✐s ◗✭①❀ ②❀ t✮✮ ❢♦r ✺✶ ✇❛❧❦s✳

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SLIDE 65

❊①❝✉rs✐♦♥ ▼❡t❤♦❞ ❬❇♦st❛♥✱ ❘❛s❝❤❡❧✱ ❙❛❧✈②❪

■rr❛t✐♦♥❛❧✐t② ♦❢ ☛ ✐s ♣r♦✈❡♥ ❛✉t♦♠❛t✐❝❛❧❧② ❜②✿ ✭✶✮ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧✱ ✖❝✱ ♦❢ ❝ ✭✷✮ ♣r♦✈✐♥❣ t❤❛t t❤❡ ♥✉♠❡r❛t♦r ♦❢ ✖❝

✏

①✷✰✶ ✷①

✑

❝♦♥t❛✐♥s ♥♦ ❝②❝❧♦t♦♠✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤✐s ♣r♦✈❡s t❤❛t ◗✭✵❀ ✵❀ t✮ ✐s ♥♦♥ ❉✲❋✐♥✐t❡ ✭❛♥❞ t❤✉s s♦ ✐s ◗✭①❀ ②❀ t✮✮ ❢♦r ✺✶ ✇❛❧❦s✳

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SLIDE 66

▲♦♥❣❡r ❙t❡♣s ❬❇♦st❛♥✱ ❇♦✉sq✉❡t✲▼✓ ❡❧♦✉✱ ▼✳ ✲ ✐♥ ♣r❡♣❛r❛t✐♦♥❪

❚❤✐s r❡s✉❧t ❛❧s♦ ❛♣♣❧✐❡s t♦ ✇❛❧❦s ✐♥ ✷❉ ✇✐t❤ ❧❛r❣❡r st❡♣s✳ ❆t ☞rst✱ ✇❡ ❢♦❝✉s ♦♥ ✇❛❧❦s t❤❛t ❤❛✈❡ ♦♥❧② ❛ ❢❡✇ st❡♣s ♦❢ ❧❡♥❣t❤ t✇♦✳ ❲❡ ❣r♦✉♣ t❤❡ ✇❛❧❦s ❜② ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡✱ t❤❡♥ ❛♣♣❧② t❤❡ ❛❜♦✈❡ ♠❡t❤♦❞ t♦ ☞❧t❡r ✇❛❧❦s ✇❤✐❝❤ ❛♣♣❡❛r t♦ ❜❡ ❉✲❋✐♥✐t❡✳

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SLIDE 67

▲♦♥❣❡r ❙t❡♣s ❬❇♦st❛♥✱ ❇♦✉sq✉❡t✲▼✓ ❡❧♦✉✱ ▼✳ ✲ ✐♥ ♣r❡♣❛r❛t✐♦♥❪

❚❤✐s r❡s✉❧t ❛❧s♦ ❛♣♣❧✐❡s t♦ ✇❛❧❦s ✐♥ ✷❉ ✇✐t❤ ❧❛r❣❡r st❡♣s✳ ❆t ☞rst✱ ✇❡ ❢♦❝✉s ♦♥ ✇❛❧❦s t❤❛t ❤❛✈❡ ♦♥❧② ❛ ❢❡✇ st❡♣s ♦❢ ❧❡♥❣t❤ t✇♦✳ ❲❡ ❣r♦✉♣ t❤❡ ✇❛❧❦s ❜② ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡✱ t❤❡♥ ❛♣♣❧② t❤❡ ❛❜♦✈❡ ♠❡t❤♦❞ t♦ ☞❧t❡r ✇❛❧❦s ✇❤✐❝❤ ❛♣♣❡❛r t♦ ❜❡ ❉✲❋✐♥✐t❡✳

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SLIDE 68

▲♦♥❣❡r ❙t❡♣s ❬❇♦st❛♥✱ ❇♦✉sq✉❡t✲▼✓ ❡❧♦✉✱ ▼✳ ✲ ✐♥ ♣r❡♣❛r❛t✐♦♥❪

❚❤✐s r❡s✉❧t ❛❧s♦ ❛♣♣❧✐❡s t♦ ✇❛❧❦s ✐♥ ✷❉ ✇✐t❤ ❧❛r❣❡r st❡♣s✳ ❆t ☞rst✱ ✇❡ ❢♦❝✉s ♦♥ ✇❛❧❦s t❤❛t ❤❛✈❡ ♦♥❧② ❛ ❢❡✇ st❡♣s ♦❢ ❧❡♥❣t❤ t✇♦✳ ❲❡ ❣r♦✉♣ t❤❡ ✇❛❧❦s ❜② ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡✱ t❤❡♥ ❛♣♣❧② t❤❡ ❛❜♦✈❡ ♠❡t❤♦❞ t♦ ☞❧t❡r ✇❛❧❦s ✇❤✐❝❤ ❛♣♣❡❛r t♦ ❜❡ ❉✲❋✐♥✐t❡✳

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SLIDE 69

❘❡s✉❧ts ✇✐t❤ ❖♥❡ ❛♥❞ ❚✇♦ ▲♦♥❣ ❙t❡♣s

❲❛❧❦s ✇✐t❤ ♦♥❡ ❧❛r❣❡ st❡♣ ✭✹ ❞❡❣❡♥❡r❛t✐✈❡✮✿ ❋♦r ✻✹✸ s❡q✉❡♥❝❡s✱ ◗✭✵❀ ✵❀ t✮ ♣r♦✈❡♥ ♥♦♥ ❉✲❋✐♥✐t❡ ❋♦r ✸✼ s❡q✉❡♥❝❡s✱ ☛ s❤♦✇♥ t♦ ❜❡ r❛t✐♦♥❛❧ ✸✷ ♦❢ ✸✼ s❡q✉❡♥❝❡s ❤❛✈❡ ❞✐☛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❣✉❡ss❡❞ ❲❛❧❦s ✇✐t❤ t✇♦ ❧❛r❣❡ st❡♣s ✭✶✶ ❞❡❣❡♥❡r❛t✐✈❡✮✿ ❋♦r ✺✼✺✹ s❡q✉❡♥❝❡s✱ ◗✭✵❀ ✵❀ t✮ ♣r♦✈❡♥ ♥♦♥ ❉✲❋✐♥✐t❡ ❋♦r ✶✺✻ s❡q✉❡♥❝❡s✱ ☛ s❤♦✇♥ t♦ ❜❡ r❛t✐♦♥❛❧ ✭✻✾ ❤❛✈❡ ❣✉❡ss❡❞ ❡q✉❛t✐♦♥s✮

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SLIDE 70

❘❡s✉❧ts ✇✐t❤ ❖♥❡ ❛♥❞ ❚✇♦ ▲♦♥❣ ❙t❡♣s

❲❛❧❦s ✇✐t❤ ♦♥❡ ❧❛r❣❡ st❡♣ ✭✹ ❞❡❣❡♥❡r❛t✐✈❡✮✿ ❋♦r ✻✹✸ s❡q✉❡♥❝❡s✱ ◗✭✵❀ ✵❀ t✮ ♣r♦✈❡♥ ♥♦♥ ❉✲❋✐♥✐t❡ ❋♦r ✸✼ s❡q✉❡♥❝❡s✱ ☛ s❤♦✇♥ t♦ ❜❡ r❛t✐♦♥❛❧ ✸✷ ♦❢ ✸✼ s❡q✉❡♥❝❡s ❤❛✈❡ ❞✐☛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❣✉❡ss❡❞ ❲❛❧❦s ✇✐t❤ t✇♦ ❧❛r❣❡ st❡♣s ✭✶✶ ❞❡❣❡♥❡r❛t✐✈❡✮✿ ❋♦r ✺✼✺✹ s❡q✉❡♥❝❡s✱ ◗✭✵❀ ✵❀ t✮ ♣r♦✈❡♥ ♥♦♥ ❉✲❋✐♥✐t❡ ❋♦r ✶✺✻ s❡q✉❡♥❝❡s✱ ☛ s❤♦✇♥ t♦ ❜❡ r❛t✐♦♥❛❧ ✭✻✾ ❤❛✈❡ ❣✉❡ss❡❞ ❡q✉❛t✐♦♥s✮

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SLIDE 71

❊①t❡♥s✐♦♥s t♦ ◆❡✇ ❲❛❧❦s ✭✸❉ ❲❛❧❦s✮

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SLIDE 72

❇❛s✐❝ ■❞❡❛

◆♦✇✱ ✇❡ ❧♦♦❦ ❛t ✇❛❧❦s ✐♥ t❤❡ ①②③✲♣❧❛♥❡ r❡str✐❝t❡❞ t♦ t❤❡ ♣♦s✐t✐✈❡ ♦❝t❛♥t✳

✁ ✁ ✁ ✁ ✁ ✁ ✎ ✁ ✁ ✎ ✁ ✁ ✁ ✎ ✎ ✁ ✁ ✎ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

❆ ♣r✐♦r✐✱ t❤❡r❡ ❛r❡ ✷✷✻ ❂ ✻✿✼ ✁ ✶✵✼ st❡♣ s❡ts✳ ❆❧t❤♦✉❣❤ ♠❛♥② ❤❛✈❡ t❤❡ s❛♠❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡✱ t❤✐s ✐s t♦♦ ❧❛r❣❡✳ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ✽✸❀ ✻✽✷ ✇✐t❤ ✺ st❡♣s ♦r ❧❡ss✳ ❇♦st❛♥ ❛♥❞ ❑❛✉❡rs ❝♦♥❥❡❝t✉r❡❞ ✭✉♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✮ ✸✺ ❉✲❋✐♥✐t❡ st❡♣s✳

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SLIDE 73

❇❛s✐❝ ■❞❡❛

◆♦✇✱ ✇❡ ❧♦♦❦ ❛t ✇❛❧❦s ✐♥ t❤❡ ①②③✲♣❧❛♥❡ r❡str✐❝t❡❞ t♦ t❤❡ ♣♦s✐t✐✈❡ ♦❝t❛♥t✳

✁ ✁ ✁ ✁ ✁ ✁ ✎ ✁ ✁ ✎ ✁ ✁ ✁ ✎ ✎ ✁ ✁ ✎ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

❆ ♣r✐♦r✐✱ t❤❡r❡ ❛r❡ ✷✷✻ ❂ ✻✿✼ ✁ ✶✵✼ st❡♣ s❡ts✳ ❆❧t❤♦✉❣❤ ♠❛♥② ❤❛✈❡ t❤❡ s❛♠❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡✱ t❤✐s ✐s t♦♦ ❧❛r❣❡✳ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ✽✸❀ ✻✽✷ ✇✐t❤ ✺ st❡♣s ♦r ❧❡ss✳ ❇♦st❛♥ ❛♥❞ ❑❛✉❡rs ❝♦♥❥❡❝t✉r❡❞ ✭✉♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✮ ✸✺ ❉✲❋✐♥✐t❡ st❡♣s✳

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SLIDE 74

❇❛s✐❝ ■❞❡❛

◆♦✇✱ ✇❡ ❧♦♦❦ ❛t ✇❛❧❦s ✐♥ t❤❡ ①②③✲♣❧❛♥❡ r❡str✐❝t❡❞ t♦ t❤❡ ♣♦s✐t✐✈❡ ♦❝t❛♥t✳

✁ ✁ ✁ ✁ ✁ ✁ ✎ ✁ ✁ ✎ ✁ ✁ ✁ ✎ ✎ ✁ ✁ ✎ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

❆ ♣r✐♦r✐✱ t❤❡r❡ ❛r❡ ✷✷✻ ❂ ✻✿✼ ✁ ✶✵✼ st❡♣ s❡ts✳ ❆❧t❤♦✉❣❤ ♠❛♥② ❤❛✈❡ t❤❡ s❛♠❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡✱ t❤✐s ✐s t♦♦ ❧❛r❣❡✳ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ✽✸❀ ✻✽✷ ✇✐t❤ ✺ st❡♣s ♦r ❧❡ss✳ ❇♦st❛♥ ❛♥❞ ❑❛✉❡rs ❝♦♥❥❡❝t✉r❡❞ ✭✉♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✮ ✸✺ ❉✲❋✐♥✐t❡ st❡♣s✳

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SLIDE 75

❯s✐♥❣ t❤❡ ●r♦✉♣

❚❤❡r❡ ❛r❡ ✷✸ ❝❛s❡s ✇❤✐❝❤ ❝❛♥ ❜❡ s♦❧✈❡❞ ✉s✐♥❣ t❤❡ ❣r♦✉♣✱ ✇❤✐❝❤ ✐s ☞♥✐t❡✳ ❚❤❡r❡ ❛r❡ ✹ st❡♣ s❡ts ✇❤✐❝❤ ❤❛✈❡ ☞♥✐t❡ ❣r♦✉♣✱ ❜✉t ❢♦r ✇❤✐❝❤ t❤❡ ❛r❣✉♠❡♥t ❞♦❡s♥✬t ❤♦❧❞ ✲ t❤❡s❡ ❛r❡ s♦❧✈❡❞ ✐♥❞✐✈✐❞✉❛❧❧②✳ ❚❤❡r❡ ❛r❡ ✽ s❡ts ✇✐t❤ ❛♥ ✐♥☞♥✐t❡ ❣r♦✉♣✳

slide-76
SLIDE 76

❯s✐♥❣ t❤❡ ●r♦✉♣

❚❤❡r❡ ❛r❡ ✷✸ ❝❛s❡s ✇❤✐❝❤ ❝❛♥ ❜❡ s♦❧✈❡❞ ✉s✐♥❣ t❤❡ ❣r♦✉♣✱ ✇❤✐❝❤ ✐s ☞♥✐t❡✳ ❚❤❡r❡ ❛r❡ ✹ st❡♣ s❡ts ✇❤✐❝❤ ❤❛✈❡ ☞♥✐t❡ ❣r♦✉♣✱ ❜✉t ❢♦r ✇❤✐❝❤ t❤❡ ❛r❣✉♠❡♥t ❞♦❡s♥✬t ❤♦❧❞ ✲ t❤❡s❡ ❛r❡ s♦❧✈❡❞ ✐♥❞✐✈✐❞✉❛❧❧②✳ ❚❤❡r❡ ❛r❡ ✽ s❡ts ✇✐t❤ ❛♥ ✐♥☞♥✐t❡ ❣r♦✉♣✳

slide-77
SLIDE 77

❯s✐♥❣ t❤❡ ●r♦✉♣

❚❤❡r❡ ❛r❡ ✷✸ ❝❛s❡s ✇❤✐❝❤ ❝❛♥ ❜❡ s♦❧✈❡❞ ✉s✐♥❣ t❤❡ ❣r♦✉♣✱ ✇❤✐❝❤ ✐s ☞♥✐t❡✳ ❚❤❡r❡ ❛r❡ ✹ st❡♣ s❡ts ✇❤✐❝❤ ❤❛✈❡ ☞♥✐t❡ ❣r♦✉♣✱ ❜✉t ❢♦r ✇❤✐❝❤ t❤❡ ❛r❣✉♠❡♥t ❞♦❡s♥✬t ❤♦❧❞ ✲ t❤❡s❡ ❛r❡ s♦❧✈❡❞ ✐♥❞✐✈✐❞✉❛❧❧②✳ ❚❤❡r❡ ❛r❡ ✽ s❡ts ✇✐t❤ ❛♥ ✐♥☞♥✐t❡ ❣r♦✉♣✳

slide-78
SLIDE 78

❘❡❞✉❝t✐♦♥ t♦ ✷❉

❆❧❧ ♦❢ t❤❡ ✇❛❧❦s ✇✐t❤ ✐♥☞♥✐t❡ ❣r♦✉♣ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ✷❉✦ ❈♦♥s✐❞❡r t❤❡ st❡♣ s❡t

✁ ✁ ✁ ✁ ✎ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✎ ✎ ✁ ✁ ✎ ✁ ✁ ✎ ✁ ✁

❛♥❞ ❵♠❛r❦✬ t❤❡ st❡♣s ❜② ❛❀ ❜❀ ❝❀ ❞❀ ❡✳ ❚❤❡♥ ✭①✮ ❜ ✕ ✭❝ ✰ ❞ ✰ ❡✮ ✭②✮ ❜ ✰ ❝ ✕ ❡ ✭③✮ ✭❝ ✰ ❞ ✰ ❡✮ ✕ ❛ ■❢ ✭①✮ ✐s s❛t✐s☞❡❞ t❤❡♥ ✭②✮ ♠✉st ❜❡ s❛t✐s☞❡❞✦

slide-79
SLIDE 79

❘❡❞✉❝t✐♦♥ t♦ ✷❉

❆❧❧ ♦❢ t❤❡ ✇❛❧❦s ✇✐t❤ ✐♥☞♥✐t❡ ❣r♦✉♣ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ✷❉✦ Example: ❈♦♥s✐❞❡r t❤❡ st❡♣ s❡t

✁ ✁ ✁ ✁ ✎ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✎ ✎ ✁ ✁ ✎ ✁ ✁ ✎ ✁ ✁

❛♥❞ ❵♠❛r❦✬ t❤❡ st❡♣s ❜② ❛❀ ❜❀ ❝❀ ❞❀ ❡✳ ❚❤❡♥ ✭①✮ ❜ ✕ ✭❝ ✰ ❞ ✰ ❡✮ ✭②✮ ❜ ✰ ❝ ✕ ❡ ✭③✮ ✭❝ ✰ ❞ ✰ ❡✮ ✕ ❛ ■❢ ✭①✮ ✐s s❛t✐s☞❡❞ t❤❡♥ ✭②✮ ♠✉st ❜❡ s❛t✐s☞❡❞✦

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SLIDE 80

❘❡❞✉❝t✐♦♥ t♦ ✷❉

❆❧❧ ♦❢ t❤❡ ✇❛❧❦s ✇✐t❤ ✐♥☞♥✐t❡ ❣r♦✉♣ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ✷❉✦ Example: ❈♦♥s✐❞❡r t❤❡ st❡♣ s❡t

✁ ✁ ✁ ✁ ✎ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✎ ✎ ✁ ✁ ✎ ✁ ✁ ✎ ✁ ✁

❛♥❞ ❵♠❛r❦✬ t❤❡ st❡♣s ❜② ❛❀ ❜❀ ❝❀ ❞❀ ❡✳ ❚❤❡♥ ✭①✮ ❜ ✕ ✭❝ ✰ ❞ ✰ ❡✮ ✭②✮ ❜ ✰ ❝ ✕ ❡ ✭③✮ ✭❝ ✰ ❞ ✰ ❡✮ ✕ ❛ ■❢ ✭①✮ ✐s s❛t✐s☞❡❞ t❤❡♥ ✭②✮ ♠✉st ❜❡ s❛t✐s☞❡❞✦

slide-81
SLIDE 81

❘❡❞✉❝t✐♦♥ t♦ ✷❉

❆❧❧ ♦❢ t❤❡ ✇❛❧❦s ✇✐t❤ ✐♥☞♥✐t❡ ❣r♦✉♣ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ✷❉✦ Example: ❈♦♥s✐❞❡r t❤❡ st❡♣ s❡t

✁ ✁ ✁ ✁ ✎ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✎ ✎ ✁ ✁ ✎ ✁ ✁ ✎ ✁ ✁

❛♥❞ ❵♠❛r❦✬ t❤❡ st❡♣s ❜② ❛❀ ❜❀ ❝❀ ❞❀ ❡✳ ❚❤❡♥ ✭①✮ ❜✕ ✭❝ ✰ ❞ ✰ ❡✮ ✭②✮ ❜ ✰ ❝✕ ❡ ✭③✮ ✭❝ ✰ ❞ ✰ ❡✮ ✕ ❛ ■❢ ✭①✮ ✐s s❛t✐s☞❡❞ t❤❡♥ ✭②✮ ♠✉st ❜❡ s❛t✐s☞❡❞✦

slide-82
SLIDE 82

❘❡❞✉❝t✐♦♥ t♦ ✷❉

❚❤✉s✱ ✇❡ ❣❡t ❜ ✕ ✭❝ ✰ ❞ ✰ ❡✮ ✕ ❛✳ ❇✉t t❤✐s ✐s t❤❡ s❛♠❡ ❛s t❤❡ r❡str✐❝t✐♦♥ ♦♥ ❲❡ ❝❛♥ ✇r✐t❡ ◗✭①❀ ②❀ ③❀ t✮ ❂ ◗✵✭❳❀ ❨ ❀ ❚✮✱ ✇❤❡r❡ ❚ ❂ t✭✶ ✰ ② ✰ ②✷✮✶❂✸ ❳ ❂ ③❚ t ❨ ❂ ①②t ❚ ✿ ❆s ◗✵✭①❀ ✶❂①❀ t✮ ✐s ❛❧❣❡❜r❛✐❝✱ s♦ ✐s ◗

✏

①❀ ②❀ ✶

①②❀ t

✑

✳

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SLIDE 83

❘❡❞✉❝t✐♦♥ t♦ ✷❉

❚❤✉s✱ ✇❡ ❣❡t ❜ ✕ ✭❝ ✰ ❞ ✰ ❡✮ ✕ ❛✳ ❇✉t t❤✐s ✐s t❤❡ s❛♠❡ ❛s t❤❡ r❡str✐❝t✐♦♥ ♦♥ ❲❡ ❝❛♥ ✇r✐t❡ ◗✭①❀ ②❀ ③❀ t✮ ❂ ◗✵✭❳❀ ❨ ❀ ❚✮✱ ✇❤❡r❡ ❚ ❂ t✭✶ ✰ ② ✰ ②✷✮✶❂✸ ❳ ❂ ③❚ t ❨ ❂ ①②t ❚ ✿ ❆s ◗✵✭①❀ ✶❂①❀ t✮ ✐s ❛❧❣❡❜r❛✐❝✱ s♦ ✐s ◗

✏

①❀ ②❀ ✶

①②❀ t

✑

✳

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SLIDE 84

❘❡❞✉❝t✐♦♥ t♦ ✷❉

❚❤✉s✱ ✇❡ ❣❡t ❜ ✕ ✭❝ ✰ ❞ ✰ ❡✮ ✕ ❛✳ ❇✉t t❤✐s ✐s t❤❡ s❛♠❡ ❛s t❤❡ r❡str✐❝t✐♦♥ ♦♥ ❲❡ ❝❛♥ ✇r✐t❡ ◗✭①❀ ②❀ ③❀ t✮ ❂ ◗✵✭❳❀ ❨ ❀ ❚✮✱ ✇❤❡r❡ ❚ ❂ t✭✶ ✰ ② ✰ ②✷✮✶❂✸ ❳ ❂ ③❚ t ❨ ❂ ①②t ❚ ✿ ❆s ◗✵✭①❀ ✶❂①❀ t✮ ✐s ❛❧❣❡❜r❛✐❝✱ s♦ ✐s ◗

✏

①❀ ②❀ ✶

①②❀ t

✑

✳

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SLIDE 85

❘❡❞✉❝t✐♦♥ t♦ ✷❉

❚❤✉s✱ ✇❡ ❣❡t ❜ ✕ ✭❝ ✰ ❞ ✰ ❡✮ ✕ ❛✳ ❇✉t t❤✐s ✐s t❤❡ s❛♠❡ ❛s t❤❡ r❡str✐❝t✐♦♥ ♦♥ ❲❡ ❝❛♥ ✇r✐t❡ ◗✭①❀ ②❀ ③❀ t✮ ❂ ◗✵✭❳❀ ❨ ❀ ❚✮✱ ✇❤❡r❡ ❚ ❂ t✭✶ ✰ ② ✰ ②✷✮✶❂✸ ❳ ❂ ③❚ t ❨ ❂ ①②t ❚ ✿ ❆s ◗✵✭①❀ ✶❂①❀ t✮ ✐s ❛❧❣❡❜r❛✐❝✱ s♦ ✐s ◗

✏

①❀ ②❀ ✶

①②❀ t

✑

✳

slide-86
SLIDE 86

P✐❝❦✐♥❣ ❉✲❋✐♥✐t❡ ❈❛s❡s

❚❤✉s✱ ✇❡ ❝❛♥ ☞❧t❡r t❤❡ ❉✲❋✐♥✐t❡ ✇❛❧❦s ❛s ❢♦❧❧♦✇s✿ ✭✶✮ ❋✐❧t❡r ❛❧❧ ✇❛❧❦s t❤❛t r❡❞✉❝❡ t♦ ✷❉ ✭✷✮ ❋✐❧t❡r ♦✉t t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣✄ ✭✸✮ Pr♦✈❡ t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣ ❛r❡ ❉✲❋✐♥✐t❡ ✭✐❢ ♣♦ss✐❜❧❡✮ ❲✐t❤ ✻ ❙t❡♣s✱ ✇❡ ❣❡t ✭✐♥ t❡r♠s ♦❢ ✉♥✐q✉❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡s✮✿ ✶✸✹ ❂ ✸✹ ✰ ✼✼✄ ✰ ✷✸ r❡❞✉❝✐❜❧❡ ✇❛❧❦s ✻✺ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❛r❡ ♣r♦✈❡♥ ❉✲❋✐♥✐t❡ ✸✶ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❝❛♥✬t ❜❡ ♣r♦✈❡♥ ✉s✐♥❣ ❡①✐st✐♥❣ ♠❡t❤♦❞s✳

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SLIDE 87

P✐❝❦✐♥❣ ❉✲❋✐♥✐t❡ ❈❛s❡s

❚❤✉s✱ ✇❡ ❝❛♥ ☞❧t❡r t❤❡ ❉✲❋✐♥✐t❡ ✇❛❧❦s ❛s ❢♦❧❧♦✇s✿ ✭✶✮ ❋✐❧t❡r ❛❧❧ ✇❛❧❦s t❤❛t r❡❞✉❝❡ t♦ ✷❉ ✭✷✮ ❋✐❧t❡r ♦✉t t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣✄ ✭✸✮ Pr♦✈❡ t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣ ❛r❡ ❉✲❋✐♥✐t❡ ✭✐❢ ♣♦ss✐❜❧❡✮ ❲✐t❤ ✻ ❙t❡♣s✱ ✇❡ ❣❡t ✭✐♥ t❡r♠s ♦❢ ✉♥✐q✉❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡s✮✿ ✶✸✹ ❂ ✸✹ ✰ ✼✼✄ ✰ ✷✸ r❡❞✉❝✐❜❧❡ ✇❛❧❦s ✻✺ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❛r❡ ♣r♦✈❡♥ ❉✲❋✐♥✐t❡ ✸✶ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❝❛♥✬t ❜❡ ♣r♦✈❡♥ ✉s✐♥❣ ❡①✐st✐♥❣ ♠❡t❤♦❞s✳

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SLIDE 88

P✐❝❦✐♥❣ ❉✲❋✐♥✐t❡ ❈❛s❡s

❚❤✉s✱ ✇❡ ❝❛♥ ☞❧t❡r t❤❡ ❉✲❋✐♥✐t❡ ✇❛❧❦s ❛s ❢♦❧❧♦✇s✿ ✭✶✮ ❋✐❧t❡r ❛❧❧ ✇❛❧❦s t❤❛t r❡❞✉❝❡ t♦ ✷❉ ✭✷✮ ❋✐❧t❡r ♦✉t t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣✄ ✭✸✮ Pr♦✈❡ t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣ ❛r❡ ❉✲❋✐♥✐t❡ ✭✐❢ ♣♦ss✐❜❧❡✮ ❲✐t❤ ✻ ❙t❡♣s✱ ✇❡ ❣❡t ✭✐♥ t❡r♠s ♦❢ ✉♥✐q✉❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡s✮✿ ✶✸✹ ❂ ✸✹ ✰ ✼✼✄ ✰ ✷✸ r❡❞✉❝✐❜❧❡ ✇❛❧❦s ✻✺ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❛r❡ ♣r♦✈❡♥ ❉✲❋✐♥✐t❡ ✸✶ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❝❛♥✬t ❜❡ ♣r♦✈❡♥ ✉s✐♥❣ ❡①✐st✐♥❣ ♠❡t❤♦❞s✳

slide-89
SLIDE 89

P✐❝❦✐♥❣ ❉✲❋✐♥✐t❡ ❈❛s❡s

❚❤✉s✱ ✇❡ ❝❛♥ ☞❧t❡r t❤❡ ❉✲❋✐♥✐t❡ ✇❛❧❦s ❛s ❢♦❧❧♦✇s✿ ✭✶✮ ❋✐❧t❡r ❛❧❧ ✇❛❧❦s t❤❛t r❡❞✉❝❡ t♦ ✷❉ ✭✷✮ ❋✐❧t❡r ♦✉t t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣✄ ✭✸✮ Pr♦✈❡ t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣ ❛r❡ ❉✲❋✐♥✐t❡ ✭✐❢ ♣♦ss✐❜❧❡✮ ❲✐t❤ ✻ ❙t❡♣s✱ ✇❡ ❣❡t ✭✐♥ t❡r♠s ♦❢ ✉♥✐q✉❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡s✮✿ ✶✸✹ ❂ ✸✹ ✰ ✼✼✄ ✰ ✷✸ r❡❞✉❝✐❜❧❡ ✇❛❧❦s ✻✺ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❛r❡ ♣r♦✈❡♥ ❉✲❋✐♥✐t❡ ✸✶ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❝❛♥✬t ❜❡ ♣r♦✈❡♥ ✉s✐♥❣ ❡①✐st✐♥❣ ♠❡t❤♦❞s✳

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SLIDE 90

P✐❝❦✐♥❣ ❉✲❋✐♥✐t❡ ❈❛s❡s

❚❤✉s✱ ✇❡ ❝❛♥ ☞❧t❡r t❤❡ ❉✲❋✐♥✐t❡ ✇❛❧❦s ❛s ❢♦❧❧♦✇s✿ ✭✶✮ ❋✐❧t❡r ❛❧❧ ✇❛❧❦s t❤❛t r❡❞✉❝❡ t♦ ✷❉ ✭✷✮ ❋✐❧t❡r ♦✉t t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣✄ ✭✸✮ Pr♦✈❡ t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣ ❛r❡ ❉✲❋✐♥✐t❡ ✭✐❢ ♣♦ss✐❜❧❡✮ ❲✐t❤ ✻ ❙t❡♣s✱ ✇❡ ❣❡t ✭✐♥ t❡r♠s ♦❢ ✉♥✐q✉❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡s✮✿ ✶✸✹ ❂ ✸✹ ✰ ✼✼✄ ✰ ✷✸ r❡❞✉❝✐❜❧❡ ✇❛❧❦s ✻✺ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❛r❡ ♣r♦✈❡♥ ❉✲❋✐♥✐t❡ ✸✶ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❝❛♥✬t ❜❡ ♣r♦✈❡♥ ✉s✐♥❣ ❡①✐st✐♥❣ ♠❡t❤♦❞s✳

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SLIDE 91

P✐❝❦✐♥❣ ❉✲❋✐♥✐t❡ ❈❛s❡s

❚❤✉s✱ ✇❡ ❝❛♥ ☞❧t❡r t❤❡ ❉✲❋✐♥✐t❡ ✇❛❧❦s ❛s ❢♦❧❧♦✇s✿ ✭✶✮ ❋✐❧t❡r ❛❧❧ ✇❛❧❦s t❤❛t r❡❞✉❝❡ t♦ ✷❉ ✭✷✮ ❋✐❧t❡r ♦✉t t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣✄ ✭✸✮ Pr♦✈❡ t❤❡ ✇❛❧❦s ✇✐t❤ ☞♥✐t❡ ❣r♦✉♣ ❛r❡ ❉✲❋✐♥✐t❡ ✭✐❢ ♣♦ss✐❜❧❡✮ ❲✐t❤ ✻ ❙t❡♣s✱ ✇❡ ❣❡t ✭✐♥ t❡r♠s ♦❢ ✉♥✐q✉❡ ❝♦✉♥t✐♥❣ s❡q✉❡♥❝❡s✮✿ ✶✸✹ ❂ ✸✹ ✰ ✼✼✄ ✰ ✷✸ r❡❞✉❝✐❜❧❡ ✇❛❧❦s ✻✺ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❛r❡ ♣r♦✈❡♥ ❉✲❋✐♥✐t❡ ✸✶ ✇✐t❤ ❛ ☞♥✐t❡ ❣r♦✉♣ t❤❛t ❝❛♥✬t ❜❡ ♣r♦✈❡♥ ✉s✐♥❣ ❡①✐st✐♥❣ ♠❡t❤♦❞s✳

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SLIDE 92

❈♦♥❝❧✉s✐♦♥

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SLIDE 93

❈♦♥❝❧✉s✐♦♥

❙♦ ❢❛r ✇❡ ❤❛✈❡✿ ❝❧❛ss✐☞❡❞✄ ❛❧❧ ✇❛❧❦s ✇✐t❤ ✉♥✐t st❡♣s ✐♥ ✷❉ ❞❡✈❡❧♦♣❡❞ ✭s♦♠❡✮ ❛✉t♦♠❛t✐❝ ♠❡t❤♦❞s ❢♦r ❝❧❛ss✐☞❝❛t✐♦♥ st❛rt❡❞ ❝❧❛ss✐❢②✐♥❣ ✇❛❧❦s ✇✐t❤ ❧♦♥❣❡r st❡♣s ❛♥❞ t❤♦s❡ ✐♥ ✸❉ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦✿ ❤❛✈❡ ❛ ❜❡tt❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❉✲❋✐♥✐t❡♥❡ss ✉♥❞❡rst❛♥❞ t❤❡ r♦❧❡ ♦❢ t❤❡ ❣r♦✉♣ ❜❡tt❡r ❞❡✈❡❧♦♣ ♠♦r❡ r♦❜✉st ♠❡t❤♦❞s ✲ st❛rt ❧♦♦❦✐♥❣ ❛t st❡♣s ✇✐t❤ ♠✉❧t✐♣❧❡ ❝♦❧♦✉rs

slide-94
SLIDE 94

❈♦♥❝❧✉s✐♦♥

❙♦ ❢❛r ✇❡ ❤❛✈❡✿ ❝❧❛ss✐☞❡❞✄ ❛❧❧ ✇❛❧❦s ✇✐t❤ ✉♥✐t st❡♣s ✐♥ ✷❉ ❞❡✈❡❧♦♣❡❞ ✭s♦♠❡✮ ❛✉t♦♠❛t✐❝ ♠❡t❤♦❞s ❢♦r ❝❧❛ss✐☞❝❛t✐♦♥ st❛rt❡❞ ❝❧❛ss✐❢②✐♥❣ ✇❛❧❦s ✇✐t❤ ❧♦♥❣❡r st❡♣s ❛♥❞ t❤♦s❡ ✐♥ ✸❉ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦✿ ❤❛✈❡ ❛ ❜❡tt❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❉✲❋✐♥✐t❡♥❡ss ✉♥❞❡rst❛♥❞ t❤❡ r♦❧❡ ♦❢ t❤❡ ❣r♦✉♣ ❜❡tt❡r ❞❡✈❡❧♦♣ ♠♦r❡ r♦❜✉st ♠❡t❤♦❞s ✲ st❛rt ❧♦♦❦✐♥❣ ❛t st❡♣s ✇✐t❤ ♠✉❧t✐♣❧❡ ❝♦❧♦✉rs

slide-95
SLIDE 95

❈♦♥❝❧✉s✐♦♥

❙♦ ❢❛r ✇❡ ❤❛✈❡✿ ❝❧❛ss✐☞❡❞✄ ❛❧❧ ✇❛❧❦s ✇✐t❤ ✉♥✐t st❡♣s ✐♥ ✷❉ ❞❡✈❡❧♦♣❡❞ ✭s♦♠❡✮ ❛✉t♦♠❛t✐❝ ♠❡t❤♦❞s ❢♦r ❝❧❛ss✐☞❝❛t✐♦♥ st❛rt❡❞ ❝❧❛ss✐❢②✐♥❣ ✇❛❧❦s ✇✐t❤ ❧♦♥❣❡r st❡♣s ❛♥❞ t❤♦s❡ ✐♥ ✸❉ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦✿ ❤❛✈❡ ❛ ❜❡tt❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❉✲❋✐♥✐t❡♥❡ss ✉♥❞❡rst❛♥❞ t❤❡ r♦❧❡ ♦❢ t❤❡ ❣r♦✉♣ ❜❡tt❡r ❞❡✈❡❧♦♣ ♠♦r❡ r♦❜✉st ♠❡t❤♦❞s ✲ st❛rt ❧♦♦❦✐♥❣ ❛t st❡♣s ✇✐t❤ ♠✉❧t✐♣❧❡ ❝♦❧♦✉rs

slide-96
SLIDE 96

❈♦♥❝❧✉s✐♦♥

❙♦ ❢❛r ✇❡ ❤❛✈❡✿ ❝❧❛ss✐☞❡❞✄ ❛❧❧ ✇❛❧❦s ✇✐t❤ ✉♥✐t st❡♣s ✐♥ ✷❉ ❞❡✈❡❧♦♣❡❞ ✭s♦♠❡✮ ❛✉t♦♠❛t✐❝ ♠❡t❤♦❞s ❢♦r ❝❧❛ss✐☞❝❛t✐♦♥ st❛rt❡❞ ❝❧❛ss✐❢②✐♥❣ ✇❛❧❦s ✇✐t❤ ❧♦♥❣❡r st❡♣s ❛♥❞ t❤♦s❡ ✐♥ ✸❉ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦✿ ❤❛✈❡ ❛ ❜❡tt❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❉✲❋✐♥✐t❡♥❡ss ✉♥❞❡rst❛♥❞ t❤❡ r♦❧❡ ♦❢ t❤❡ ❣r♦✉♣ ❜❡tt❡r ❞❡✈❡❧♦♣ ♠♦r❡ r♦❜✉st ♠❡t❤♦❞s ✲ st❛rt ❧♦♦❦✐♥❣ ❛t st❡♣s ✇✐t❤ ♠✉❧t✐♣❧❡ ❝♦❧♦✉rs

slide-97
SLIDE 97

❈♦♥❝❧✉s✐♦♥

❙♦ ❢❛r ✇❡ ❤❛✈❡✿ ❝❧❛ss✐☞❡❞✄ ❛❧❧ ✇❛❧❦s ✇✐t❤ ✉♥✐t st❡♣s ✐♥ ✷❉ ❞❡✈❡❧♦♣❡❞ ✭s♦♠❡✮ ❛✉t♦♠❛t✐❝ ♠❡t❤♦❞s ❢♦r ❝❧❛ss✐☞❝❛t✐♦♥ st❛rt❡❞ ❝❧❛ss✐❢②✐♥❣ ✇❛❧❦s ✇✐t❤ ❧♦♥❣❡r st❡♣s ❛♥❞ t❤♦s❡ ✐♥ ✸❉ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦✿ ❤❛✈❡ ❛ ❜❡tt❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❉✲❋✐♥✐t❡♥❡ss ✉♥❞❡rst❛♥❞ t❤❡ r♦❧❡ ♦❢ t❤❡ ❣r♦✉♣ ❜❡tt❡r ❞❡✈❡❧♦♣ ♠♦r❡ r♦❜✉st ♠❡t❤♦❞s ✲ st❛rt ❧♦♦❦✐♥❣ ❛t st❡♣s ✇✐t❤ ♠✉❧t✐♣❧❡ ❝♦❧♦✉rs

slide-98
SLIDE 98

❈♦♥❝❧✉s✐♦♥

❙♦ ❢❛r ✇❡ ❤❛✈❡✿ ❝❧❛ss✐☞❡❞✄ ❛❧❧ ✇❛❧❦s ✇✐t❤ ✉♥✐t st❡♣s ✐♥ ✷❉ ❞❡✈❡❧♦♣❡❞ ✭s♦♠❡✮ ❛✉t♦♠❛t✐❝ ♠❡t❤♦❞s ❢♦r ❝❧❛ss✐☞❝❛t✐♦♥ st❛rt❡❞ ❝❧❛ss✐❢②✐♥❣ ✇❛❧❦s ✇✐t❤ ❧♦♥❣❡r st❡♣s ❛♥❞ t❤♦s❡ ✐♥ ✸❉ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦✿ ❤❛✈❡ ❛ ❜❡tt❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❉✲❋✐♥✐t❡♥❡ss ✉♥❞❡rst❛♥❞ t❤❡ r♦❧❡ ♦❢ t❤❡ ❣r♦✉♣ ❜❡tt❡r ❞❡✈❡❧♦♣ ♠♦r❡ r♦❜✉st ♠❡t❤♦❞s ✲ st❛rt ❧♦♦❦✐♥❣ ❛t st❡♣s ✇✐t❤ ♠✉❧t✐♣❧❡ ❝♦❧♦✉rs

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SLIDE 99

❘❡❢❡r❡♥❝❡s

❆✳ ❇♦st❛♥ ❛♥❞ ▼✳ ❑❛✉❡rs✱ ❚❤❡ ❝♦♠♣❧❡t❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r ●❡ss❡❧ ✇❛❧❦s ✐s ❛❧❣❡❜r❛✐❝✱ ✷✵✶✵✳ ❆✳ ❇♦st❛♥✱ ❑✳ ❘❛s❝❤❡❧✱ ❛♥❞ ❇✳ ❙❛❧✈②✱ ◆♦♥ ❉✲❋✐♥✐t❡ ❊①❝✉rs✐♦♥s ✐♥ t❤❡ ◗✉❛rt❡r P❧❛♥❡✱ ✷✵✶✷✳ ▼✳ ❇♦✉sq✉❡t✲▼✓ ❡❧♦✉✱ ▼✳ ▼✐s❤♥❛✱ ❲❛❧❦s ✇✐t❤ s♠❛❧❧ st❡♣s ✐♥ t❤❡ q✉❛rt❡r ♣❧❛♥❡✱ ✷✵✶✵✳ ▼✳ ▼✐s❤♥❛ ❛♥❞ ❆✳ ❘❡❝❤♥✐t③❡r✱ ❚✇♦ ♥♦♥✲❤♦❧♦♥♦♠✐❝ ❧❛tt✐❝❡ ✇❛❧❦s ✐♥ t❤❡ q✉❛rt❡r ♣❧❛♥❡✱ ✷✵✵✾✳ ■✳ ❑✉r❦♦✈❛ ❛♥❞ ❑✳ ❘❛s❝❤❡❧✱ ❖♥ t❤❡ ❢✉♥❝t✐♦♥s ❝♦✉♥t✐♥❣ ✇❛❧❦s ✇✐t❤ s♠❛❧❧ st❡♣s ✐♥ t❤❡ q✉❛rt❡r ♣❧❛♥❡✱ ✷✵✶✶✳

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SLIDE 100

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