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SLIDE 1

❉✐str✐❜✉t❡❞ ❲❡✐❣❤t❡❞ ▼❛t❝❤✐♥❣ ✈✐❛ ❘❛♥❞♦♠✐③❡❞ ❈♦♠♣♦s❛❜❧❡ ❈♦r❡s❡ts

▼♦❤❛♠♠❛❞❍♦ss❡✐♥ ❇❛t❡♥✐

  • ♦♦❣❧❡ ❘❡s❡❛r❝❤

◆❡✇ ❨♦r❦✱ ❯❙❆ ❙❡♣❡❤r ❆ss❛❞✐ ❱❛❤❛❜ ▼✐rr♦❦♥✐ ✸✻t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ✶✷ ❏✉♥❡ ✷✵✶✾ ✖ ▲♦♥❣ ❇❡❛❝❤✱ ❈❛❧✐❢♦r♥✐❛✱ ❯❙❆ P♦st❡r ★✶✻✶✱ P❛❝✐✜❝ ❇❛❧❧r♦♦♠

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SLIDE 2

▼❛ss✐✈❡ ❣r❛♣❤s

❊✈❡r②✇❤❡r❡✿ ✇❡❜ ❣r❛♣❤✱ s♦❝✐❛❧ ♥❡t✇♦r❦s✱ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s✱ ❡t❝✳ ▼❛t❝❤✐♥❣✿ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✈❡rt❡①✲❞✐s❥♦✐♥t ❡❞❣❡s ❈❧✉st❡r✐♥❣✱ ♣❛rt✐t✐♦♥✐♥❣ ❋✐♥❞✐♥❣ ♠♦t✐❢s ✐♥ ❜✐♦✐♥❢♦r♠❛t✐❝s ❚r❛❞❡ ♠❛r❦❡t✐♥❣✱ ♦♥❧✐♥❡ ❛❞✈❡rt✐s❡♠❡♥t ❑✐❞♥❡② ❡①❝❤❛♥❣❡ ▲✐♥❡❛r ❛❧❣❡❜r❛✱ ♠❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥

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SLIDE 3

▼❛ss✐✈❡ ❣r❛♣❤s

❊✈❡r②✇❤❡r❡✿ ✇❡❜ ❣r❛♣❤✱ s♦❝✐❛❧ ♥❡t✇♦r❦s✱ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s✱ ❡t❝✳ ▼❛t❝❤✐♥❣✿ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✈❡rt❡①✲❞✐s❥♦✐♥t ❡❞❣❡s ◮ ❈❧✉st❡r✐♥❣✱ ♣❛rt✐t✐♦♥✐♥❣ ◮ ❋✐♥❞✐♥❣ ♠♦t✐❢s ✐♥ ❜✐♦✐♥❢♦r♠❛t✐❝s ◮ ❚r❛❞❡ ♠❛r❦❡t✐♥❣✱ ♦♥❧✐♥❡ ❛❞✈❡rt✐s❡♠❡♥t ◮ ❑✐❞♥❡② ❡①❝❤❛♥❣❡ ◮ ▲✐♥❡❛r ❛❧❣❡❜r❛✱ ♠❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥

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SLIDE 4

❙❡q✉❡♥t✐❛❧ r❡s✉❧ts

❱❛r✐❛♥ts✿ ◮ ❲❡✐❣❤t❡❞ ✈s ✉♥✇❡✐❣❤t❡❞ ◮ ❇✐♣❛rt✐t❡ ✈s ♥♦♥✲❜✐♣❛rt✐t❡ ❆❧❣♦r✐t❤♠✐❝ r❡s✉❧ts✿ ◮ ❋✐rst ♣♦❧②t✐♠❡ ❛❧❣♦r✐t❤♠✿ ✏❇❧♦ss♦♠✑ ❞❡❝♦♠♣♦s✐t✐♦♥ ❬❊❞♠✻✺❛❪ ◮ ❊①t❡♥❞❡❞ t♦ ✇❡✐❣❤t❡❞ ❝❛s❡ ❬❊❞♠✻✺❜❪ ◮ ❋❛st❡st ✐♥ t✐♠❡ ˜ ❖(♠√♥) ❬●❛❜❚❛r✾✶❪ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s✿ ◮ ●r❡❡❞② ❛❧❣♦r✐t❤♠✿ ✷✲❛♣♣r♦① ✐♥ ❖(♠ log ♥) t✐♠❡ ◮ ✶ + ǫ ❛♣♣r♦① ✐♥ ˜ ❖(♠/ǫ) t✐♠❡ ❬❉✉❛P❡t✶✹❪

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SLIDE 5

❙❡q✉❡♥t✐❛❧ r❡s✉❧ts

❱❛r✐❛♥ts✿ ◮ ❲❡✐❣❤t❡❞ ✈s ✉♥✇❡✐❣❤t❡❞ ◮ ❇✐♣❛rt✐t❡ ✈s ♥♦♥✲❜✐♣❛rt✐t❡ ❆❧❣♦r✐t❤♠✐❝ r❡s✉❧ts✿ ◮ ❋✐rst ♣♦❧②t✐♠❡ ❛❧❣♦r✐t❤♠✿ ✏❇❧♦ss♦♠✑ ❞❡❝♦♠♣♦s✐t✐♦♥ ❬❊❞♠✻✺❛❪ ◮ ❊①t❡♥❞❡❞ t♦ ✇❡✐❣❤t❡❞ ❝❛s❡ ❬❊❞♠✻✺❜❪ ◮ ❋❛st❡st ✐♥ t✐♠❡ ˜ ❖(♠√♥) ❬●❛❜❚❛r✾✶❪ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s✿ ◮ ●r❡❡❞② ❛❧❣♦r✐t❤♠✿ ✷✲❛♣♣r♦① ✐♥ ❖(♠ log ♥) t✐♠❡ ◮ ✶ + ǫ ❛♣♣r♦① ✐♥ ˜ ❖(♠/ǫ) t✐♠❡ ❬❉✉❛P❡t✶✹❪

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SLIDE 6

▲❛r❣❡ s❝❛❧❡

❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ✇♦r❦ ◮ ❖(♠) r✉♥t✐♠❡ ✐s ♣r♦❤✐❜✐t✐✈❡ ◮ ❖(♥) ♠❡♠♦r② ♥♦t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ◮ ❙✐♠♣❧② r❡❛❞✐♥❣ ✐♥♣✉t ❞❛t❛ ♦♥❝❡ ♠❛② t❛❦❡ t♦♦ ❧♦♥❣✦ ▼❛♣❘❡❞✉❝❡✿ ❞❡ ❢❛❝t♦ ✐♥❞✉str② st❛♥❞❛r❞ ❙♣❧✐t ❞❛t❛ ❛❝r♦ss ♠❛♥② ♠❛❝❤✐♥❡s ❙♣❧✐t ❝♦♠♣✉t❛t✐♦♥ ✐♥t♦ s❡✈❡r❛❧ r♦✉♥❞s ❉❛t❛ ✐s s❡♥t t♦ ♠❛❝❤✐♥❡s ❜❛s❡❞ ♦♥ ❦❡②s ▼❛❝❤✐♥❡s ♣r♦❞✉❝❡ ♦✉t♣✉t ❢♦r ♥❡①t r♦✉♥❞ ■♠♣♦rt❛♥t✿

✶ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✷ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s ✸ ♠❡♠♦r② ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✹ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡

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SLIDE 7

▲❛r❣❡ s❝❛❧❡

❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ✇♦r❦ ◮ ❖(♠) r✉♥t✐♠❡ ✐s ♣r♦❤✐❜✐t✐✈❡ ◮ ❖(♥) ♠❡♠♦r② ♥♦t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ◮ ❙✐♠♣❧② r❡❛❞✐♥❣ ✐♥♣✉t ❞❛t❛ ♦♥❝❡ ♠❛② t❛❦❡ t♦♦ ❧♦♥❣✦ ▼❛♣❘❡❞✉❝❡✿ ❞❡ ❢❛❝t♦ ✐♥❞✉str② st❛♥❞❛r❞ ❙♣❧✐t ❞❛t❛ ❛❝r♦ss ♠❛♥② ♠❛❝❤✐♥❡s ❙♣❧✐t ❝♦♠♣✉t❛t✐♦♥ ✐♥t♦ s❡✈❡r❛❧ r♦✉♥❞s ❉❛t❛ ✐s s❡♥t t♦ ♠❛❝❤✐♥❡s ❜❛s❡❞ ♦♥ ❦❡②s ▼❛❝❤✐♥❡s ♣r♦❞✉❝❡ ♦✉t♣✉t ❢♦r ♥❡①t r♦✉♥❞ ■♠♣♦rt❛♥t✿

✶ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✷ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s ✸ ♠❡♠♦r② ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✹ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡

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SLIDE 8

▲❛r❣❡ s❝❛❧❡

❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ✇♦r❦ ◮ ❖(♠) r✉♥t✐♠❡ ✐s ♣r♦❤✐❜✐t✐✈❡ ◮ ❖(♥) ♠❡♠♦r② ♥♦t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ◮ ❙✐♠♣❧② r❡❛❞✐♥❣ ✐♥♣✉t ❞❛t❛ ♦♥❝❡ ♠❛② t❛❦❡ t♦♦ ❧♦♥❣✦ ▼❛♣❘❡❞✉❝❡✿ ❞❡ ❢❛❝t♦ ✐♥❞✉str② st❛♥❞❛r❞ ◮ ❙♣❧✐t ❞❛t❛ ❛❝r♦ss ♠❛♥② ♠❛❝❤✐♥❡s ◮ ❙♣❧✐t ❝♦♠♣✉t❛t✐♦♥ ✐♥t♦ s❡✈❡r❛❧ r♦✉♥❞s ◮ ❉❛t❛ ✐s s❡♥t t♦ ♠❛❝❤✐♥❡s ❜❛s❡❞ ♦♥ ❦❡②s ◮ ▼❛❝❤✐♥❡s ♣r♦❞✉❝❡ ♦✉t♣✉t ❢♦r ♥❡①t r♦✉♥❞ ■♠♣♦rt❛♥t✿

✶ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✷ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s ✸ ♠❡♠♦r② ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✹ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡

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SLIDE 9

▲❛r❣❡ s❝❛❧❡

❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ✇♦r❦ ◮ ❖(♠) r✉♥t✐♠❡ ✐s ♣r♦❤✐❜✐t✐✈❡ ◮ ❖(♥) ♠❡♠♦r② ♥♦t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ◮ ❙✐♠♣❧② r❡❛❞✐♥❣ ✐♥♣✉t ❞❛t❛ ♦♥❝❡ ♠❛② t❛❦❡ t♦♦ ❧♦♥❣✦ ▼❛♣❘❡❞✉❝❡✿ ❞❡ ❢❛❝t♦ ✐♥❞✉str② st❛♥❞❛r❞ ◮ ❙♣❧✐t ❞❛t❛ ❛❝r♦ss ♠❛♥② ♠❛❝❤✐♥❡s ◮ ❙♣❧✐t ❝♦♠♣✉t❛t✐♦♥ ✐♥t♦ s❡✈❡r❛❧ r♦✉♥❞s ◮ ❉❛t❛ ✐s s❡♥t t♦ ♠❛❝❤✐♥❡s ❜❛s❡❞ ♦♥ ❦❡②s ◮ ▼❛❝❤✐♥❡s ♣r♦❞✉❝❡ ♦✉t♣✉t ❢♦r ♥❡①t r♦✉♥❞ ■♠♣♦rt❛♥t✿

✶ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✷ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s ✸ ♠❡♠♦r② ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✹ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡

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SLIDE 10

▲❛r❣❡ s❝❛❧❡

❙✐♠✉❧❛t✐♦♥

✏●r❡❡❞② ❛❧❣♦r✐t❤♠s ❛r❡ ♣r❛❝t✐t✐♦♥❡rs✬ ❜❡st ❢r✐❡♥❞s✖t❤❡② ❛r❡ ✐♥t✉✐t✐✈❡✱ s✐♠♣❧❡ t♦ ✐♠♣❧❡♠❡♥t✱ ❛♥❞ ♦❢t❡♥ ❧❡❛❞ t♦ ✈❡r② ❣♦♦❞ s♦❧✉t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✐♠♣❧❡♠❡♥t✐♥❣ ❣r❡❡❞② ❛❧❣♦r✐t❤♠s ✐♥ ❛ ❞✐str✐❜✉t❡❞ s❡tt✐♥❣ ✐s ❝❤❛❧❧❡♥❣✐♥❣ s✐♥❝❡ t❤❡ ❣r❡❡❞② ❝❤♦✐❝❡ ✐s ✐♥❤❡r❡♥t❧② s❡q✉❡♥t✐❛❧✱ ❛♥❞ ✐t ✐s ♥♦t ❝❧❡❛r ❤♦✇ t♦ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❡①tr❛ ♣r♦❝❡ss✐♥❣ ♣♦✇❡r✳✑ ✖ ❬❑✉♠▼♦s❱❛s❱❛t✶✸❪ ❚✇♦ t②♣✐❝❛❧ t②♣❡s ♦❢ r❡s✉❧ts✿

✶ ❘❡❧❛t✐✈❡❧② ❧❛r❣❡ ♥✉♠❜❡r ♦❢ r♦✉♥❞s t♦ ✏❢❛✐t❤❢✉❧❧②✑ s✐♠✉❧❛t❡ t❤❡ ❣r❡❡❞② ❛❧❣♦r✐t❤♠✳ ✷ ❱❡r② s♠❛❧❧ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✭♦♥❡ ♦r t✇♦✮ ❢♦r ✏✇❡❛❦✑ s✐♠✉❧❛t✐♦♥✱ ❖(✶) ✇♦rs❡ t❤❛♥

t❤❡ ❣r❡❡❞② ❛❧❣♦r✐t❤♠✳

slide-11
SLIDE 11

▲❛r❣❡✲s❝❛❧❡ r❡s✉❧ts

❈r❡❞✐t ❆♣♣r♦① ❙♣❛❝❡ ❘♦✉♥❞s

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ˜ ❖(♥) ❖(log ♥)

❬❈r♦❙t✉✶✹❪

✹ ˜ ❖(♥) ❖(log ♥)

❬❆❤♥●✉❤✶✺❪

✶ + ε ˜ ❖(♥) ❖(ε−✶log ♥)

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ˜ ❖(♥) ❖(log ♥)

❬❈③✉▲❛❝▼❛❞ ✶✽❪

✷ ❖ ♥ ❖

❖ ♥ ✷

❬❆ss❇❛t❇❡r ✶✾❪

✷ ❖ ♥ ❖

❬●❛♠❑❛❧▼✐t❙✈❡✶✽❪

✶ ❖ ♥ ❖

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ♥✶

❖ ✶

❬❈r♦❙t✉✶✹❪

✹ ♥✶

❖ ✶

❬❆❤♥●✉❤✶✺❪

✶ ♥✶

❖ ✶

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ♥✶

❖ ✶

❬❆ss❑❤❛✶✼❪

❖ ✶ ❖ ♥ ♥ ✷

❬❆ss❇❛t❇❡r ✶✾❪

✸ ❖ ♥ ♥ ✷ ❖✉r ✇♦r❦ ✷ ❖ ♥ ♥ ✷

slide-12
SLIDE 12

▲❛r❣❡✲s❝❛❧❡ r❡s✉❧ts

❈r❡❞✐t ❆♣♣r♦① ❙♣❛❝❡ ❘♦✉♥❞s

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ˜ ❖(♥) ❖(log ♥)

❬❈r♦❙t✉✶✹❪

✹ ˜ ❖(♥) ❖(log ♥)

❬❆❤♥●✉❤✶✺❪

✶ + ε ˜ ❖(♥) ❖(ε−✶log ♥)

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ˜ ❖(♥) ❖(log ♥)

❬❈③✉▲❛❝▼❛❞+✶✽❪

✷ + ε ˜ ❖(♥) ❖

  • ε−Θ(✶/ε) · ❖(log log ♥)✷

❬❆ss❇❛t❇❡r+✶✾❪

✷ + ε ˜ ❖(♥) ❖(ε−Θ(✶/ε) · log log ♥)

❬●❛♠❑❛❧▼✐t❙✈❡✶✽❪

✶ + ε ˜ ❖(♥) ❖(ε−Θ(✶/ε✷) · log log ♥)

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ♥✶

❖ ✶

❬❈r♦❙t✉✶✹❪

✹ ♥✶

❖ ✶

❬❆❤♥●✉❤✶✺❪

✶ ♥✶

❖ ✶

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ♥✶

❖ ✶

❬❆ss❑❤❛✶✼❪

❖ ✶ ❖ ♥ ♥ ✷

❬❆ss❇❛t❇❡r ✶✾❪

✸ ❖ ♥ ♥ ✷ ❖✉r ✇♦r❦ ✷ ❖ ♥ ♥ ✷

slide-13
SLIDE 13

▲❛r❣❡✲s❝❛❧❡ r❡s✉❧ts

❈r❡❞✐t ❆♣♣r♦① ❙♣❛❝❡ ❘♦✉♥❞s

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ˜ ❖(♥) ❖(log ♥)

❬❈r♦❙t✉✶✹❪

✹ ˜ ❖(♥) ❖(log ♥)

❬❆❤♥●✉❤✶✺❪

✶ + ε ˜ ❖(♥) ❖(ε−✶log ♥)

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ˜ ❖(♥) ❖(log ♥)

❬❈③✉▲❛❝▼❛❞+✶✽❪

✷ + ε ˜ ❖(♥) ❖

  • ε−Θ(✶/ε) · ❖(log log ♥)✷

❬❆ss❇❛t❇❡r+✶✾❪

✷ + ε ˜ ❖(♥) ❖(ε−Θ(✶/ε) · log log ♥)

❬●❛♠❑❛❧▼✐t❙✈❡✶✽❪

✶ + ε ˜ ❖(♥) ❖(ε−Θ(✶/ε✷) · log log ♥)

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ♥✶+Ω(✶) ❖(✶)

❬❈r♦❙t✉✶✹❪

✹ ♥✶+Ω(✶) ❖(✶)

❬❆❤♥●✉❤✶✺❪

✶ + ε ♥✶+Ω(✶) ❖(✶/ε)

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ♥✶+Ω(✶) ❖(✶)

❬❆ss❑❤❛✶✼❪

❖ ✶ ❖ ♥ ♥ ✷

❬❆ss❇❛t❇❡r ✶✾❪

✸ ❖ ♥ ♥ ✷ ❖✉r ✇♦r❦ ✷ ❖ ♥ ♥ ✷

slide-14
SLIDE 14

▲❛r❣❡✲s❝❛❧❡ r❡s✉❧ts

❈r❡❞✐t ❆♣♣r♦① ❙♣❛❝❡ ❘♦✉♥❞s

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ˜ ❖(♥) ❖(log ♥)

❬❈r♦❙t✉✶✹❪

✹ ˜ ❖(♥) ❖(log ♥)

❬❆❤♥●✉❤✶✺❪

✶ + ε ˜ ❖(♥) ❖(ε−✶log ♥)

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ˜ ❖(♥) ❖(log ♥)

❬❈③✉▲❛❝▼❛❞+✶✽❪

✷ + ε ˜ ❖(♥) ❖

  • ε−Θ(✶/ε) · ❖(log log ♥)✷

❬❆ss❇❛t❇❡r+✶✾❪

✷ + ε ˜ ❖(♥) ❖(ε−Θ(✶/ε) · log log ♥)

❬●❛♠❑❛❧▼✐t❙✈❡✶✽❪

✶ + ε ˜ ❖(♥) ❖(ε−Θ(✶/ε✷) · log log ♥)

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ♥✶+Ω(✶) ❖(✶)

❬❈r♦❙t✉✶✹❪

✹ ♥✶+Ω(✶) ❖(✶)

❬❆❤♥●✉❤✶✺❪

✶ + ε ♥✶+Ω(✶) ❖(✶/ε)

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ♥✶+Ω(✶) ❖(✶)

❬❆ss❑❤❛✶✼❪

❖(✶) ˜ ❖(♥√♥) ✷

❬❆ss❇❛t❇❡r+✶✾❪

✸ + ε ˜ ❖(♥√♥) ✷ ❖✉r ✇♦r❦ ✷ ❖ ♥ ♥ ✷

slide-15
SLIDE 15

▲❛r❣❡✲s❝❛❧❡ r❡s✉❧ts

❈r❡❞✐t ❆♣♣r♦① ❙♣❛❝❡ ❘♦✉♥❞s

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ˜ ❖(♥) ❖(log ♥)

❬❈r♦❙t✉✶✹❪

✹ ˜ ❖(♥) ❖(log ♥)

❬❆❤♥●✉❤✶✺❪

✶ + ε ˜ ❖(♥) ❖(ε−✶log ♥)

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ˜ ❖(♥) ❖(log ♥)

❬❈③✉▲❛❝▼❛❞+✶✽❪

✷ + ε ˜ ❖(♥) ❖

  • ε−Θ(✶/ε) · ❖(log log ♥)✷

❬❆ss❇❛t❇❡r+✶✾❪

✷ + ε ˜ ❖(♥) ❖(ε−Θ(✶/ε) · log log ♥)

❬●❛♠❑❛❧▼✐t❙✈❡✶✽❪

✶ + ε ˜ ❖(♥) ❖(ε−Θ(✶/ε✷) · log log ♥)

❬▲❛t▼♦s❙✉r❱❛s✶✶❪

✽ ♥✶+Ω(✶) ❖(✶)

❬❈r♦❙t✉✶✹❪

✹ ♥✶+Ω(✶) ❖(✶)

❬❆❤♥●✉❤✶✺❪

✶ + ε ♥✶+Ω(✶) ❖(✶/ε)

❬❍❛r▲✐❛▲✐✉✶✽❪

✷ ♥✶+Ω(✶) ❖(✶)

❬❆ss❑❤❛✶✼❪

❖(✶) ˜ ❖(♥√♥) ✷

❬❆ss❇❛t❇❡r+✶✾❪

✸ + ε ˜ ❖(♥√♥) ✷ ❖✉r ✇♦r❦ ✷ + ε ❖(♥√♥) ✷

slide-16
SLIDE 16

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

❇❛s✐❝ ✐❞❡❛ ❣r❛♣❤ ●

s✉❜❣r❛♣❤ ●✶ s✉❜❣r❛♣❤ ●✷ s✉❜❣r❛♣❤ ●❦

✳ ✳ ✳

❝♦r❡s❡t ❍✶ ❝♦r❡s❡t ❍✷ ❝♦r❡s❡t ❍❦

✳ ✳ ✳

❝♦r❡s❡t ❍

s♦❧✉t✐♦♥

✶✳ ❍♦✇ t♦ ✏♣❛rt✐t✐♦♥✑❄ ✷✳ ❍♦✇ t♦ ✏s♦❧✈❡✑❄ ✸✳ ❍♦✇ t♦ ✏❝♦♠❜✐♥❡✑❄

slide-17
SLIDE 17

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

❇❛s✐❝ ✐❞❡❛ ❣r❛♣❤ ●

s✉❜❣r❛♣❤ ●✶ s✉❜❣r❛♣❤ ●✷ s✉❜❣r❛♣❤ ●❦

✳ ✳ ✳

❝♦r❡s❡t ❍✶ ❝♦r❡s❡t ❍✷ ❝♦r❡s❡t ❍❦

✳ ✳ ✳

❝♦r❡s❡t ❍

s♦❧✉t✐♦♥

✶✳ ❍♦✇ t♦ ✏♣❛rt✐t✐♦♥✑❄ ✷✳ ❍♦✇ t♦ ✏s♦❧✈❡✑❄ ✸✳ ❍♦✇ t♦ ✏❝♦♠❜✐♥❡✑❄

slide-18
SLIDE 18

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

❇❛s✐❝ ✐❞❡❛ ❣r❛♣❤ ●

s✉❜❣r❛♣❤ ●✶ s✉❜❣r❛♣❤ ●✷ s✉❜❣r❛♣❤ ●❦

✳ ✳ ✳

❝♦r❡s❡t ❍✶ ❝♦r❡s❡t ❍✷ ❝♦r❡s❡t ❍❦

✳ ✳ ✳

❝♦r❡s❡t ❍

s♦❧✉t✐♦♥

✶✳ ❍♦✇ t♦ ✏♣❛rt✐t✐♦♥✑❄ ✷✳ ❍♦✇ t♦ ✏s♦❧✈❡✑❄ ✸✳ ❍♦✇ t♦ ✏❝♦♠❜✐♥❡✑❄

slide-19
SLIDE 19

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

❇❛s✐❝ ✐❞❡❛ ❣r❛♣❤ ●

s✉❜❣r❛♣❤ ●✶ s✉❜❣r❛♣❤ ●✷ s✉❜❣r❛♣❤ ●❦

✳ ✳ ✳

❝♦r❡s❡t ❍✶ ❝♦r❡s❡t ❍✷ ❝♦r❡s❡t ❍❦

✳ ✳ ✳

❝♦r❡s❡t ❍

s♦❧✉t✐♦♥

✶✳ ❍♦✇ t♦ ✏♣❛rt✐t✐♦♥✑❄ ✷✳ ❍♦✇ t♦ ✏s♦❧✈❡✑❄ ✸✳ ❍♦✇ t♦ ✏❝♦♠❜✐♥❡✑❄

slide-20
SLIDE 20

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

❇❛s✐❝ ✐❞❡❛ ❣r❛♣❤ ●

s✉❜❣r❛♣❤ ●✶ s✉❜❣r❛♣❤ ●✷ s✉❜❣r❛♣❤ ●❦

✳ ✳ ✳

❝♦r❡s❡t ❍✶ ❝♦r❡s❡t ❍✷ ❝♦r❡s❡t ❍❦

✳ ✳ ✳

❝♦r❡s❡t ❍

s♦❧✉t✐♦♥

✶✳ ❍♦✇ t♦ ✏♣❛rt✐t✐♦♥✑❄ ✷✳ ❍♦✇ t♦ ✏s♦❧✈❡✑❄ ✸✳ ❍♦✇ t♦ ✏❝♦♠❜✐♥❡✑❄

slide-21
SLIDE 21

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

❇❛s✐❝ ✐❞❡❛ ❣r❛♣❤ ●

s✉❜❣r❛♣❤ ●✶ s✉❜❣r❛♣❤ ●✷ s✉❜❣r❛♣❤ ●❦

✳ ✳ ✳

❝♦r❡s❡t ❍✶ ❝♦r❡s❡t ❍✷ ❝♦r❡s❡t ❍❦

✳ ✳ ✳

❝♦r❡s❡t ❍

s♦❧✉t✐♦♥

✶✳ ❍♦✇ t♦ ✏♣❛rt✐t✐♦♥✑❄ ✷✳ ❍♦✇ t♦ ✏s♦❧✈❡✑❄ ✸✳ ❍♦✇ t♦ ✏❝♦♠❜✐♥❡✑❄

slide-22
SLIDE 22

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

▼❛t❤❡♠❛t✐❝ ❢♦r♠✉❧❛t✐♦♥

❉❡✜♥✐t✐♦♥

◮ ▲❡t ●✶, . . . , ●❦ ❜❡ ❛ ♣❛rt✐t✐♦♥✐♥❣ ♦❢ ●❀ s❡♥❞ ❡❛❝❤ ❡❞❣❡ ❡ ∈ ● t♦ ❛ s✉❜❣r❛♣❤ ●✐ ❛r❜✐tr❛r✐❧②✳ ◮ ❈♦♥s✐❞❡r ❛♥ ❛❧❣♦r✐t❤♠ ❆▲● t❤❛t ❣✐✈❡♥ ●✐ ♦✉t♣✉ts ❛ s✉❜❣r❛♣❤ ❍✐ ♦❢ ●✐ ✇✐t❤ s ❡❞❣❡s✳ ◮ ❆▲● ♦✉t♣✉ts ❛♥ α✲❛♣♣r♦① ❝♦♠♣♦s❛❜❧❡ ❝♦r❡s❡t ♦❢ s✐③❡ s ❢♦r ❛ ♣r♦❜❧❡♠ P ✐✛ P(❆▲●(●✶) ∪ . . . ∪ ❆▲●(●❦)) ✐s ❛♥ α✲❛♣♣r♦① t♦ P(●)✳ ❉❡s✐❣♥ ❆▲● ❢♦r s♠❛❧❧ ✭q✉❛❧✐t②✮ ❛♥❞ s ✭s✐③❡✮✳ ♥♦ ✶ ❛♣♣r♦① r❡q✉✐r❡s ♥✷

♦ ✶ s♣❛❝❡ ❬❆ss❑❤❛▲✐❨❛r✶✻❪✳

❘❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❬▼✐r❩❛❞✶✺❪ ❢♦r s✉❜♠♦❞✉❧❛r ♠❛①✐♠✐③❛t✐♦♥✳ ▼❛①✐♠✉♠✲♠❛t❝❤✐♥❣ ❝♦r❡s❡ts ❞♦ ♥♦t ❣✐✈❡ ✷✱ ❜✉t s♣❛rs✐✜❝❛t✐♦♥ ✭❊❉❈❙✮ ✰ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ❣✐✈❡ ✶ ✺ ❛♣♣r♦① ❬❆ss❇❛t❇❡r ✶✾❪✳ ❲❡ ♣r❡s❡♥t ❛ s✐♠♣❧❡ ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡t ✇✐t❤ ♠✉❧✐t♣❧✐❝✐t② ❖ ✶

❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣✳

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SLIDE 23

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

▼❛t❤❡♠❛t✐❝ ❢♦r♠✉❧❛t✐♦♥

❉❡✜♥✐t✐♦♥

◮ ▲❡t ●✶, . . . , ●❦ ❜❡ ❛ ♣❛rt✐t✐♦♥✐♥❣ ♦❢ ●❀ s❡♥❞ ❡❛❝❤ ❡❞❣❡ ❡ ∈ ● t♦ ❛ s✉❜❣r❛♣❤ ●✐ ❛r❜✐tr❛r✐❧②✳ ◮ ❈♦♥s✐❞❡r ❛♥ ❛❧❣♦r✐t❤♠ ❆▲● t❤❛t ❣✐✈❡♥ ●✐ ♦✉t♣✉ts ❛ s✉❜❣r❛♣❤ ❍✐ ♦❢ ●✐ ✇✐t❤ s ❡❞❣❡s✳ ◮ ❆▲● ♦✉t♣✉ts ❛♥ α✲❛♣♣r♦① ❝♦♠♣♦s❛❜❧❡ ❝♦r❡s❡t ♦❢ s✐③❡ s ❢♦r ❛ ♣r♦❜❧❡♠ P ✐✛ P(❆▲●(●✶) ∪ . . . ∪ ❆▲●(●❦)) ✐s ❛♥ α✲❛♣♣r♦① t♦ P(●)✳ ◮ ❉❡s✐❣♥ ❆▲● ❢♦r s♠❛❧❧ α ✭q✉❛❧✐t②✮ ❛♥❞ s ✭s✐③❡✮✳ ♥♦ ✶ ❛♣♣r♦① r❡q✉✐r❡s ♥✷

♦ ✶ s♣❛❝❡ ❬❆ss❑❤❛▲✐❨❛r✶✻❪✳

❘❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❬▼✐r❩❛❞✶✺❪ ❢♦r s✉❜♠♦❞✉❧❛r ♠❛①✐♠✐③❛t✐♦♥✳ ▼❛①✐♠✉♠✲♠❛t❝❤✐♥❣ ❝♦r❡s❡ts ❞♦ ♥♦t ❣✐✈❡ ✷✱ ❜✉t s♣❛rs✐✜❝❛t✐♦♥ ✭❊❉❈❙✮ ✰ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ❣✐✈❡ ✶ ✺ ❛♣♣r♦① ❬❆ss❇❛t❇❡r ✶✾❪✳ ❲❡ ♣r❡s❡♥t ❛ s✐♠♣❧❡ ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡t ✇✐t❤ ♠✉❧✐t♣❧✐❝✐t② ❖ ✶

❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣✳

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SLIDE 24

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

▼❛t❤❡♠❛t✐❝ ❢♦r♠✉❧❛t✐♦♥

❉❡✜♥✐t✐♦♥

◮ ▲❡t ●✶, . . . , ●❦ ❜❡ ❛ ♣❛rt✐t✐♦♥✐♥❣ ♦❢ ●❀ s❡♥❞ ❡❛❝❤ ❡❞❣❡ ❡ ∈ ● t♦ ❛ s✉❜❣r❛♣❤ ●✐ ❛r❜✐tr❛r✐❧②✳ ◮ ❈♦♥s✐❞❡r ❛♥ ❛❧❣♦r✐t❤♠ ❆▲● t❤❛t ❣✐✈❡♥ ●✐ ♦✉t♣✉ts ❛ s✉❜❣r❛♣❤ ❍✐ ♦❢ ●✐ ✇✐t❤ s ❡❞❣❡s✳ ◮ ❆▲● ♦✉t♣✉ts ❛♥ α✲❛♣♣r♦① ❝♦♠♣♦s❛❜❧❡ ❝♦r❡s❡t ♦❢ s✐③❡ s ❢♦r ❛ ♣r♦❜❧❡♠ P ✐✛ P(❆▲●(●✶) ∪ . . . ∪ ❆▲●(●❦)) ✐s ❛♥ α✲❛♣♣r♦① t♦ P(●)✳ ◮ ❉❡s✐❣♥ ❆▲● ❢♦r s♠❛❧❧ α ✭q✉❛❧✐t②✮ ❛♥❞ s ✭s✐③❡✮✳ ◮ ♥♦(✶) ❛♣♣r♦① r❡q✉✐r❡s ♥✷−♦(✶) s♣❛❝❡ ❬❆ss❑❤❛▲✐❨❛r✶✻❪✳ ❘❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❬▼✐r❩❛❞✶✺❪ ❢♦r s✉❜♠♦❞✉❧❛r ♠❛①✐♠✐③❛t✐♦♥✳ ▼❛①✐♠✉♠✲♠❛t❝❤✐♥❣ ❝♦r❡s❡ts ❞♦ ♥♦t ❣✐✈❡ ✷✱ ❜✉t s♣❛rs✐✜❝❛t✐♦♥ ✭❊❉❈❙✮ ✰ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ❣✐✈❡ ✶ ✺ ❛♣♣r♦① ❬❆ss❇❛t❇❡r ✶✾❪✳ ❲❡ ♣r❡s❡♥t ❛ s✐♠♣❧❡ ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡t ✇✐t❤ ♠✉❧✐t♣❧✐❝✐t② ❖ ✶

❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣✳

slide-25
SLIDE 25

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

▼❛t❤❡♠❛t✐❝ ❢♦r♠✉❧❛t✐♦♥

❉❡✜♥✐t✐♦♥

◮ ▲❡t ●✶, . . . , ●❦ ❜❡ ❛ r❛♥❞♦♠ ♣❛rt✐t✐♦♥✐♥❣ ♦❢ ●❀ s❡♥❞ ❡❛❝❤ ❡❞❣❡ ❡ ∈ ● t♦ ❛ s✉❜❣r❛♣❤ ●✐ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ◮ ❈♦♥s✐❞❡r ❛♥ ❛❧❣♦r✐t❤♠ ❆▲● t❤❛t ❣✐✈❡♥ ●✐ ♦✉t♣✉ts ❛ s✉❜❣r❛♣❤ ❍✐ ♦❢ ●✐ ✇✐t❤ s ❡❞❣❡s✳ ◮ ❆▲● ♦✉t♣✉ts ❛♥ α✲❛♣♣r♦① r❛♥❞♦♠✐③❡❞ ❝♦♠♣♦s❛❜❧❡ ❝♦r❡s❡t ♦❢ s✐③❡ s ❢♦r ❛ ♣r♦❜❧❡♠ P ✐✛ P(❆▲●(●✶) ∪ . . . ∪ ❆▲●(●❦)) ✐s ❛♥ α✲❛♣♣r♦① t♦ P(●)✳ ◮ ❉❡s✐❣♥ ❆▲● ❢♦r s♠❛❧❧ α ✭q✉❛❧✐t②✮ ❛♥❞ s ✭s✐③❡✮✳ ◮ ♥♦(✶) ❛♣♣r♦① r❡q✉✐r❡s ♥✷−♦(✶) s♣❛❝❡ ❬❆ss❑❤❛▲✐❨❛r✶✻❪✳ ◮ ❘❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❬▼✐r❩❛❞✶✺❪ ❢♦r s✉❜♠♦❞✉❧❛r ♠❛①✐♠✐③❛t✐♦♥✳ ▼❛①✐♠✉♠✲♠❛t❝❤✐♥❣ ❝♦r❡s❡ts ❞♦ ♥♦t ❣✐✈❡ ✷✱ ❜✉t s♣❛rs✐✜❝❛t✐♦♥ ✭❊❉❈❙✮ ✰ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ❣✐✈❡ ✶ ✺ ❛♣♣r♦① ❬❆ss❇❛t❇❡r ✶✾❪✳ ❲❡ ♣r❡s❡♥t ❛ s✐♠♣❧❡ ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡t ✇✐t❤ ♠✉❧✐t♣❧✐❝✐t② ❖ ✶

❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣✳

slide-26
SLIDE 26

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

▼❛t❤❡♠❛t✐❝ ❢♦r♠✉❧❛t✐♦♥

❉❡✜♥✐t✐♦♥

◮ ▲❡t ●✶, . . . , ●❦ ❜❡ ❛ r❛♥❞♦♠ µ✲♣❛rt✐t✐♦♥✐♥❣ ♦❢ ●❀ s❡♥❞ ❡❛❝❤ ❡❞❣❡ ❡ ∈ ● t♦ µ s✉❜❣r❛♣❤s ●✐ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ◮ ❈♦♥s✐❞❡r ❛♥ ❛❧❣♦r✐t❤♠ ❆▲● t❤❛t ❣✐✈❡♥ ●✐ ♦✉t♣✉ts ❛ s✉❜❣r❛♣❤ ❍✐ ♦❢ ●✐ ✇✐t❤ s ❡❞❣❡s✳ ◮ ❆▲● ♦✉t♣✉ts ❛♥ α✲❛♣♣r♦① r❛♥❞♦♠✐③❡❞ ❝♦♠♣♦s❛❜❧❡ ❝♦r❡s❡t ♦❢ s✐③❡ s ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② µ ❢♦r ❛ ♣r♦❜❧❡♠ P ✐✛ P(❆▲●(●✶) ∪ . . . ∪ ❆▲●(●❦)) ✐s ❛♥ α✲❛♣♣r♦① t♦ P(●)✳ ◮ ❉❡s✐❣♥ ❆▲● ❢♦r s♠❛❧❧ α ✭q✉❛❧✐t②✮ ❛♥❞ s ✭s✐③❡✮✳ ◮ ♥♦(✶) ❛♣♣r♦① r❡q✉✐r❡s ♥✷−♦(✶) s♣❛❝❡ ❬❆ss❑❤❛▲✐❨❛r✶✻❪✳ ◮ ❘❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❬▼✐r❩❛❞✶✺❪ ❢♦r s✉❜♠♦❞✉❧❛r ♠❛①✐♠✐③❛t✐♦♥✳ ▼❛①✐♠✉♠✲♠❛t❝❤✐♥❣ ❝♦r❡s❡ts ❞♦ ♥♦t ❣✐✈❡ ✷✱ ❜✉t s♣❛rs✐✜❝❛t✐♦♥ ✭❊❉❈❙✮ ✰ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ❣✐✈❡ ✶ ✺ ❛♣♣r♦① ❬❆ss❇❛t❇❡r ✶✾❪✳ ❲❡ ♣r❡s❡♥t ❛ s✐♠♣❧❡ ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡t ✇✐t❤ ♠✉❧✐t♣❧✐❝✐t② ❖ ✶

❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣✳

slide-27
SLIDE 27

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

▼❛t❤❡♠❛t✐❝ ❢♦r♠✉❧❛t✐♦♥

❉❡✜♥✐t✐♦♥

◮ ▲❡t ●✶, . . . , ●❦ ❜❡ ❛ r❛♥❞♦♠ µ✲♣❛rt✐t✐♦♥✐♥❣ ♦❢ ●❀ s❡♥❞ ❡❛❝❤ ❡❞❣❡ ❡ ∈ ● t♦ µ s✉❜❣r❛♣❤s ●✐ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ◮ ❈♦♥s✐❞❡r ❛♥ ❛❧❣♦r✐t❤♠ ❆▲● t❤❛t ❣✐✈❡♥ ●✐ ♦✉t♣✉ts ❛ s✉❜❣r❛♣❤ ❍✐ ♦❢ ●✐ ✇✐t❤ s ❡❞❣❡s✳ ◮ ❆▲● ♦✉t♣✉ts ❛♥ α✲❛♣♣r♦① r❛♥❞♦♠✐③❡❞ ❝♦♠♣♦s❛❜❧❡ ❝♦r❡s❡t ♦❢ s✐③❡ s ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② µ ❢♦r ❛ ♣r♦❜❧❡♠ P ✐✛ P(❆▲●(●✶) ∪ . . . ∪ ❆▲●(●❦)) ✐s ❛♥ α✲❛♣♣r♦① t♦ P(●)✳ ◮ ▼❛①✐♠✉♠✲♠❛t❝❤✐♥❣ ❝♦r❡s❡ts ❞♦ ♥♦t ❣✐✈❡ α < ✷✱ ❜✉t s♣❛rs✐✜❝❛t✐♦♥ ✭❊❉❈❙✮ ✰ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ❣✐✈❡ α = ✶.✺ + ǫ ❛♣♣r♦① ❬❆ss❇❛t❇❡r+✶✾❪✳ ❲❡ ♣r❡s❡♥t ❛ s✐♠♣❧❡ ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡t ✇✐t❤ ♠✉❧✐t♣❧✐❝✐t② ❖ ✶

❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣✳

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SLIDE 28

❈♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts

▼❛t❤❡♠❛t✐❝ ❢♦r♠✉❧❛t✐♦♥

❉❡✜♥✐t✐♦♥

◮ ▲❡t ●✶, . . . , ●❦ ❜❡ ❛ r❛♥❞♦♠ µ✲♣❛rt✐t✐♦♥✐♥❣ ♦❢ ●❀ s❡♥❞ ❡❛❝❤ ❡❞❣❡ ❡ ∈ ● t♦ µ s✉❜❣r❛♣❤s ●✐ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ◮ ❈♦♥s✐❞❡r ❛♥ ❛❧❣♦r✐t❤♠ ❆▲● t❤❛t ❣✐✈❡♥ ●✐ ♦✉t♣✉ts ❛ s✉❜❣r❛♣❤ ❍✐ ♦❢ ●✐ ✇✐t❤ s ❡❞❣❡s✳ ◮ ❆▲● ♦✉t♣✉ts ❛♥ α✲❛♣♣r♦① r❛♥❞♦♠✐③❡❞ ❝♦♠♣♦s❛❜❧❡ ❝♦r❡s❡t ♦❢ s✐③❡ s ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② µ ❢♦r ❛ ♣r♦❜❧❡♠ P ✐✛ P(❆▲●(●✶) ∪ . . . ∪ ❆▲●(●❦)) ✐s ❛♥ α✲❛♣♣r♦① t♦ P(●)✳ ◮ ▼❛①✐♠✉♠✲♠❛t❝❤✐♥❣ ❝♦r❡s❡ts ❞♦ ♥♦t ❣✐✈❡ α < ✷✱ ❜✉t s♣❛rs✐✜❝❛t✐♦♥ ✭❊❉❈❙✮ ✰ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡ts ❣✐✈❡ α = ✶.✺ + ǫ ❛♣♣r♦① ❬❆ss❇❛t❇❡r+✶✾❪✳ ❲❡ ♣r❡s❡♥t ❛ s✐♠♣❧❡ (✷ + ǫ)✲❛♣♣r♦①✐♠❛t✐♦♥ r❛♥❞♦♠✐③❡❞ ❝♦r❡s❡t ✇✐t❤ ♠✉❧✐t♣❧✐❝✐t② µ = ❖( ✶

ǫ log ✶ ǫ) ❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣✳

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SLIDE 29

❆❧❣♦r✐t❤♠

✶ ❙❡♥❞ ❡❛❝❤ ❡❞❣❡ t♦ µ = ❖

log ✶/ǫ

ǫ

  • ♠❛❝❤✐♥❡s✳

✷ ❙♦rt ❡❞❣❡s ❛❝❝♦r❞✐♥❣ t♦ ✇❡✐❣❤t ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✭✇✐t❤ ❝♦♥s✐st❡♥t t✐❡✲❜r❡❛❦✐♥❣✮✱

❛♥❞ ❣r❡❡❞✐❧② ✜♥❞ ❛ ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ✭t❤❡ ❝♦r❡s❡ts✮✳

✸ ❋✐♥❞ ❛ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ▼ ♦❢ t❤❡ ✉♥✐♦♥ ♦❢ ❝♦r❡s❡ts ❍✳

◮ ❚❤✐s ❣✐✈❡s ❛ ✷ + ǫ ❛♣♣r♦①✳ ◮ ❙✐♠♣❧❡✱ s❝❛❧❛❜❧❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡①❝❡♣t ❢♦r ❙t❡♣ ✸✳

◮ ❘❡♣❧❛❝✐♥❣ ❙t❡♣ ✸ ✇✐t❤ ❛ ❣r❡❡❞② ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ♣r♦✈❛❜❧② ❛ ✸ + ǫ ❛♣♣r♦① ✭♥♦t ❛ ✹ + ǫ ❛♣♣r♦①✮✳

◮ ❇❡tt❡r ❛♥❛❧②s✐s ✉s❡s ✏❝♦♥s✐st❡♥❝②✑ ♦❢ ♠❛①✐♠❛❧ ✭✈s✳ ♠❛①✐♠✉♠✮ ♠❛t❝❤✐♥❣ ❝♦r❡s❡ts✳

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SLIDE 30

❆❧❣♦r✐t❤♠

✶ ❙❡♥❞ ❡❛❝❤ ❡❞❣❡ t♦ µ = ❖

log ✶/ǫ

ǫ

  • ♠❛❝❤✐♥❡s✳

✷ ❙♦rt ❡❞❣❡s ❛❝❝♦r❞✐♥❣ t♦ ✇❡✐❣❤t ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✭✇✐t❤ ❝♦♥s✐st❡♥t t✐❡✲❜r❡❛❦✐♥❣✮✱

❛♥❞ ❣r❡❡❞✐❧② ✜♥❞ ❛ ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ✭t❤❡ ❝♦r❡s❡ts✮✳

✸ ❋✐♥❞ ❛ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ▼ ♦❢ t❤❡ ✉♥✐♦♥ ♦❢ ❝♦r❡s❡ts ❍✳

◮ ❚❤✐s ❣✐✈❡s ❛ ✷ + ǫ ❛♣♣r♦①✳ ◮ ❙✐♠♣❧❡✱ s❝❛❧❛❜❧❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡①❝❡♣t ❢♦r ❙t❡♣ ✸✳

◮ ❘❡♣❧❛❝✐♥❣ ❙t❡♣ ✸ ✇✐t❤ ❛ ❣r❡❡❞② ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ♣r♦✈❛❜❧② ❛ ✸ + ǫ ❛♣♣r♦① ✭♥♦t ❛ ✹ + ǫ ❛♣♣r♦①✮✳

◮ ❇❡tt❡r ❛♥❛❧②s✐s ✉s❡s ✏❝♦♥s✐st❡♥❝②✑ ♦❢ ♠❛①✐♠❛❧ ✭✈s✳ ♠❛①✐♠✉♠✮ ♠❛t❝❤✐♥❣ ❝♦r❡s❡ts✳

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SLIDE 31

❆❧❣♦r✐t❤♠

✶ ❙❡♥❞ ❡❛❝❤ ❡❞❣❡ t♦ µ = ❖

log ✶/ǫ

ǫ

  • ♠❛❝❤✐♥❡s✳

✷ ❙♦rt ❡❞❣❡s ❛❝❝♦r❞✐♥❣ t♦ ✇❡✐❣❤t ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✭✇✐t❤ ❝♦♥s✐st❡♥t t✐❡✲❜r❡❛❦✐♥❣✮✱

❛♥❞ ❣r❡❡❞✐❧② ✜♥❞ ❛ ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ✭t❤❡ ❝♦r❡s❡ts✮✳

✸ ❋✐♥❞ ❛ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ▼ ♦❢ t❤❡ ✉♥✐♦♥ ♦❢ ❝♦r❡s❡ts ❍✳

◮ ❚❤✐s ❣✐✈❡s ❛ ✷ + ǫ ❛♣♣r♦①✳ ◮ ❙✐♠♣❧❡✱ s❝❛❧❛❜❧❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡①❝❡♣t ❢♦r ❙t❡♣ ✸✳

◮ ❘❡♣❧❛❝✐♥❣ ❙t❡♣ ✸ ✇✐t❤ ❛ ❣r❡❡❞② ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ♣r♦✈❛❜❧② ❛ ✸ + ǫ ❛♣♣r♦① ✭♥♦t ❛ ✹ + ǫ ❛♣♣r♦①✮✳

◮ ❇❡tt❡r ❛♥❛❧②s✐s ✉s❡s ✏❝♦♥s✐st❡♥❝②✑ ♦❢ ♠❛①✐♠❛❧ ✭✈s✳ ♠❛①✐♠✉♠✮ ♠❛t❝❤✐♥❣ ❝♦r❡s❡ts✳

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SLIDE 32

❆❧❣♦r✐t❤♠

✶ ❙❡♥❞ ❡❛❝❤ ❡❞❣❡ t♦ µ = ❖

log ✶/ǫ

ǫ

  • ♠❛❝❤✐♥❡s✳

✷ ❙♦rt ❡❞❣❡s ❛❝❝♦r❞✐♥❣ t♦ ✇❡✐❣❤t ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✭✇✐t❤ ❝♦♥s✐st❡♥t t✐❡✲❜r❡❛❦✐♥❣✮✱

❛♥❞ ❣r❡❡❞✐❧② ✜♥❞ ❛ ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ✭t❤❡ ❝♦r❡s❡ts✮✳

✸ ❋✐♥❞ ❛ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ▼ ♦❢ t❤❡ ✉♥✐♦♥ ♦❢ ❝♦r❡s❡ts ❍✳

◮ ❚❤✐s ❣✐✈❡s ❛ ✷ + ǫ ❛♣♣r♦①✳ ◮ ❙✐♠♣❧❡✱ s❝❛❧❛❜❧❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡①❝❡♣t ❢♦r ❙t❡♣ ✸✳

◮ ❘❡♣❧❛❝✐♥❣ ❙t❡♣ ✸ ✇✐t❤ ❛ ❣r❡❡❞② ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ♣r♦✈❛❜❧② ❛ ✸ + ǫ ❛♣♣r♦① ✭♥♦t ❛ ✹ + ǫ ❛♣♣r♦①✮✳

◮ ❇❡tt❡r ❛♥❛❧②s✐s ✉s❡s ✏❝♦♥s✐st❡♥❝②✑ ♦❢ ♠❛①✐♠❛❧ ✭✈s✳ ♠❛①✐♠✉♠✮ ♠❛t❝❤✐♥❣ ❝♦r❡s❡ts✳

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SLIDE 33

❆❧❣♦r✐t❤♠

✶ ❙❡♥❞ ❡❛❝❤ ❡❞❣❡ t♦ µ = ❖

log ✶/ǫ

ǫ

  • ♠❛❝❤✐♥❡s✳

✷ ❙♦rt ❡❞❣❡s ❛❝❝♦r❞✐♥❣ t♦ ✇❡✐❣❤t ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✭✇✐t❤ ❝♦♥s✐st❡♥t t✐❡✲❜r❡❛❦✐♥❣✮✱

❛♥❞ ❣r❡❡❞✐❧② ✜♥❞ ❛ ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ✭t❤❡ ❝♦r❡s❡ts✮✳

✸ ❋✐♥❞ ❛ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ▼ ♦❢ t❤❡ ✉♥✐♦♥ ♦❢ ❝♦r❡s❡ts ❍✳

◮ ❚❤✐s ❣✐✈❡s ❛ ✷ + ǫ ❛♣♣r♦①✳ ◮ ❙✐♠♣❧❡✱ s❝❛❧❛❜❧❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡①❝❡♣t ❢♦r ❙t❡♣ ✸✳

◮ ❘❡♣❧❛❝✐♥❣ ❙t❡♣ ✸ ✇✐t❤ ❛ ❣r❡❡❞② ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ♣r♦✈❛❜❧② ❛ ✸ + ǫ ❛♣♣r♦① ✭♥♦t ❛ ✹ + ǫ ❛♣♣r♦①✮✳

◮ ❇❡tt❡r ❛♥❛❧②s✐s ✉s❡s ✏❝♦♥s✐st❡♥❝②✑ ♦❢ ♠❛①✐♠❛❧ ✭✈s✳ ♠❛①✐♠✉♠✮ ♠❛t❝❤✐♥❣ ❝♦r❡s❡ts✳

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SLIDE 34

Pr♦♦❢ ✐❞❡❛

◮ ▼❛❝❤✐♥❡ ✐ ✇♦r❦s ♦♥ s✉❜❣r❛♣❤ ●✐ ❛♥❞ ❝♦♥tr✐❜✉t❡s ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ▼✐ t♦ ❝♦r❡s❡t ❍ = ∪✐▼✐✳ ◮ ❈♦♠♣❛r❡ t♦ ❛ r❡❢❡r❡♥❝❡ ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣ ▼∗✳ ◮ ▲❡t ♣❡r♠✉t❛t✐♦♥ π ♦❢ ❡❞❣❡s ❜❡ t❤❡ ❝♦♥s✐st❡♥t s♦rt✐♥❣ ♦r❞❡r✳ ◮ ❊❞❣❡ ❡ ∈ ▼∗ ✐s ❢r❡❡ ♦♥ ♠❛❝❤✐♥❡ ✐ ✐✛ ✐ts ❡♥❞♣♦✐♥ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ✇❤❡♥ ❡ ✏❛rr✐✈❡s✳✑

◮ ❖t❤❡r✇✐s❡ ❡ ∈ ▼∗ ✐s ❜❧♦❝❦❡❞ ♦♥ ♠❛❝❤✐♥❡ ✐✳ ◮ ▼❛②❜❡ ❡ ✐s ♠✐ss✐♥❣ ♦♥ ♠❛❝❤✐♥❡ ✐✳

✶ ❙✉❜❣r❛♣❤s ●✐ ❤❛✈❡ t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥❀ ❢♦❝✉s ♦♥ ●✶ ❛♥❞ ▼✶✳ ✷ ❆ ❜❧♦❝❦❡❞ ❡❞❣❡ ❤❛s ❛ ❤❡❛✈✐❡r ❡❞❣❡ ✐♥ ▼✐ ❛s ✐ts ✏❝❡rt✐✜❝❛t❡✳✑

◮ ❯s❡ ✐t ✐♥ t❤❡ ❝❤❛r❣✐♥❣ ❛r❣✉♠❡♥t ❢♦r ▼✶✳

✸ ❋r❡❡ ❡❞❣❡ ♣❛rt ♦❢ ●✐ ✇✐❧❧ ♠❛❦❡ ✐t t♦ ▼✐ ❛♥❞ ❍✳

◮ ❋r❡❡ ❡❞❣❡s ✐♥ ▼✷ ∪ ▼✸ ∪ . . . ❝♦♠♣❡♥s❛t❡ t❤❡ ❢r❡❡ ❡❞❣❡s ✐♥ ▼✶✳ ◮ ❙✉❜t❧❡ ❛r❣✉♠❡♥t ❢♦r ✏❛✉❣♠❡♥t❛t✐♦♥✳✑

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SLIDE 35

Pr♦♦❢ ✐❞❡❛

◮ ▼❛❝❤✐♥❡ ✐ ✇♦r❦s ♦♥ s✉❜❣r❛♣❤ ●✐ ❛♥❞ ❝♦♥tr✐❜✉t❡s ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ▼✐ t♦ ❝♦r❡s❡t ❍ = ∪✐▼✐✳ ◮ ❈♦♠♣❛r❡ t♦ ❛ r❡❢❡r❡♥❝❡ ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣ ▼∗✳ ◮ ▲❡t ♣❡r♠✉t❛t✐♦♥ π ♦❢ ❡❞❣❡s ❜❡ t❤❡ ❝♦♥s✐st❡♥t s♦rt✐♥❣ ♦r❞❡r✳ ◮ ❊❞❣❡ ❡ ∈ ▼∗ ✐s ❢r❡❡ ♦♥ ♠❛❝❤✐♥❡ ✐ ✐✛ ✐ts ❡♥❞♣♦✐♥ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ✇❤❡♥ ❡ ✏❛rr✐✈❡s✳✑

◮ ❖t❤❡r✇✐s❡ ❡ ∈ ▼∗ ✐s ❜❧♦❝❦❡❞ ♦♥ ♠❛❝❤✐♥❡ ✐✳ ◮ ▼❛②❜❡ ❡ ✐s ♠✐ss✐♥❣ ♦♥ ♠❛❝❤✐♥❡ ✐✳

✶ ❙✉❜❣r❛♣❤s ●✐ ❤❛✈❡ t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥❀ ❢♦❝✉s ♦♥ ●✶ ❛♥❞ ▼✶✳ ✷ ❆ ❜❧♦❝❦❡❞ ❡❞❣❡ ❤❛s ❛ ❤❡❛✈✐❡r ❡❞❣❡ ✐♥ ▼✐ ❛s ✐ts ✏❝❡rt✐✜❝❛t❡✳✑

◮ ❯s❡ ✐t ✐♥ t❤❡ ❝❤❛r❣✐♥❣ ❛r❣✉♠❡♥t ❢♦r ▼✶✳

✸ ❋r❡❡ ❡❞❣❡ ♣❛rt ♦❢ ●✐ ✇✐❧❧ ♠❛❦❡ ✐t t♦ ▼✐ ❛♥❞ ❍✳

◮ ❋r❡❡ ❡❞❣❡s ✐♥ ▼✷ ∪ ▼✸ ∪ . . . ❝♦♠♣❡♥s❛t❡ t❤❡ ❢r❡❡ ❡❞❣❡s ✐♥ ▼✶✳ ◮ ❙✉❜t❧❡ ❛r❣✉♠❡♥t ❢♦r ✏❛✉❣♠❡♥t❛t✐♦♥✳✑

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SLIDE 36

Pr♦♦❢ ✐❞❡❛

◮ ▼❛❝❤✐♥❡ ✐ ✇♦r❦s ♦♥ s✉❜❣r❛♣❤ ●✐ ❛♥❞ ❝♦♥tr✐❜✉t❡s ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ▼✐ t♦ ❝♦r❡s❡t ❍ = ∪✐▼✐✳ ◮ ❈♦♠♣❛r❡ t♦ ❛ r❡❢❡r❡♥❝❡ ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣ ▼∗✳ ◮ ▲❡t ♣❡r♠✉t❛t✐♦♥ π ♦❢ ❡❞❣❡s ❜❡ t❤❡ ❝♦♥s✐st❡♥t s♦rt✐♥❣ ♦r❞❡r✳ ◮ ❊❞❣❡ ❡ ∈ ▼∗ ✐s ❢r❡❡ ♦♥ ♠❛❝❤✐♥❡ ✐ ✐✛ ✐ts ❡♥❞♣♦✐♥ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ✇❤❡♥ ❡ ✏❛rr✐✈❡s✳✑

◮ ❖t❤❡r✇✐s❡ ❡ ∈ ▼∗ ✐s ❜❧♦❝❦❡❞ ♦♥ ♠❛❝❤✐♥❡ ✐✳ ◮ ▼❛②❜❡ ❡ ✐s ♠✐ss✐♥❣ ♦♥ ♠❛❝❤✐♥❡ ✐✳

✶ ❙✉❜❣r❛♣❤s ●✐ ❤❛✈❡ t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥❀ ❢♦❝✉s ♦♥ ●✶ ❛♥❞ ▼✶✳ ✷ ❆ ❜❧♦❝❦❡❞ ❡❞❣❡ ❤❛s ❛ ❤❡❛✈✐❡r ❡❞❣❡ ✐♥ ▼✐ ❛s ✐ts ✏❝❡rt✐✜❝❛t❡✳✑

◮ ❯s❡ ✐t ✐♥ t❤❡ ❝❤❛r❣✐♥❣ ❛r❣✉♠❡♥t ❢♦r ▼✶✳

✸ ❋r❡❡ ❡❞❣❡ ♣❛rt ♦❢ ●✐ ✇✐❧❧ ♠❛❦❡ ✐t t♦ ▼✐ ❛♥❞ ❍✳

◮ ❋r❡❡ ❡❞❣❡s ✐♥ ▼✷ ∪ ▼✸ ∪ . . . ❝♦♠♣❡♥s❛t❡ t❤❡ ❢r❡❡ ❡❞❣❡s ✐♥ ▼✶✳ ◮ ❙✉❜t❧❡ ❛r❣✉♠❡♥t ❢♦r ✏❛✉❣♠❡♥t❛t✐♦♥✳✑

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SLIDE 37

Pr♦♦❢ ✐❞❡❛

◮ ▼❛❝❤✐♥❡ ✐ ✇♦r❦s ♦♥ s✉❜❣r❛♣❤ ●✐ ❛♥❞ ❝♦♥tr✐❜✉t❡s ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ▼✐ t♦ ❝♦r❡s❡t ❍ = ∪✐▼✐✳ ◮ ❈♦♠♣❛r❡ t♦ ❛ r❡❢❡r❡♥❝❡ ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣ ▼∗✳ ◮ ▲❡t ♣❡r♠✉t❛t✐♦♥ π ♦❢ ❡❞❣❡s ❜❡ t❤❡ ❝♦♥s✐st❡♥t s♦rt✐♥❣ ♦r❞❡r✳ ◮ ❊❞❣❡ ❡ ∈ ▼∗ ✐s ❢r❡❡ ♦♥ ♠❛❝❤✐♥❡ ✐ ✐✛ ✐ts ❡♥❞♣♦✐♥ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ✇❤❡♥ ❡ ✏❛rr✐✈❡s✳✑

◮ ❖t❤❡r✇✐s❡ ❡ ∈ ▼∗ ✐s ❜❧♦❝❦❡❞ ♦♥ ♠❛❝❤✐♥❡ ✐✳ ◮ ▼❛②❜❡ ❡ ✐s ♠✐ss✐♥❣ ♦♥ ♠❛❝❤✐♥❡ ✐✳

✶ ❙✉❜❣r❛♣❤s ●✐ ❤❛✈❡ t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥❀ ❢♦❝✉s ♦♥ ●✶ ❛♥❞ ▼✶✳ ✷ ❆ ❜❧♦❝❦❡❞ ❡❞❣❡ ❤❛s ❛ ❤❡❛✈✐❡r ❡❞❣❡ ✐♥ ▼✐ ❛s ✐ts ✏❝❡rt✐✜❝❛t❡✳✑

◮ ❯s❡ ✐t ✐♥ t❤❡ ❝❤❛r❣✐♥❣ ❛r❣✉♠❡♥t ❢♦r ▼✶✳

✸ ❋r❡❡ ❡❞❣❡ ♣❛rt ♦❢ ●✐ ✇✐❧❧ ♠❛❦❡ ✐t t♦ ▼✐ ❛♥❞ ❍✳

◮ ❋r❡❡ ❡❞❣❡s ✐♥ ▼✷ ∪ ▼✸ ∪ . . . ❝♦♠♣❡♥s❛t❡ t❤❡ ❢r❡❡ ❡❞❣❡s ✐♥ ▼✶✳ ◮ ❙✉❜t❧❡ ❛r❣✉♠❡♥t ❢♦r ✏❛✉❣♠❡♥t❛t✐♦♥✳✑

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SLIDE 38

Pr♦♦❢ ✐❞❡❛

◮ ▼❛❝❤✐♥❡ ✐ ✇♦r❦s ♦♥ s✉❜❣r❛♣❤ ●✐ ❛♥❞ ❝♦♥tr✐❜✉t❡s ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ▼✐ t♦ ❝♦r❡s❡t ❍ = ∪✐▼✐✳ ◮ ❈♦♠♣❛r❡ t♦ ❛ r❡❢❡r❡♥❝❡ ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣ ▼∗✳ ◮ ▲❡t ♣❡r♠✉t❛t✐♦♥ π ♦❢ ❡❞❣❡s ❜❡ t❤❡ ❝♦♥s✐st❡♥t s♦rt✐♥❣ ♦r❞❡r✳ ◮ ❊❞❣❡ ❡ ∈ ▼∗ ✐s ❢r❡❡ ♦♥ ♠❛❝❤✐♥❡ ✐ ✐✛ ✐ts ❡♥❞♣♦✐♥ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ✇❤❡♥ ❡ ✏❛rr✐✈❡s✳✑

◮ ❖t❤❡r✇✐s❡ ❡ ∈ ▼∗ ✐s ❜❧♦❝❦❡❞ ♦♥ ♠❛❝❤✐♥❡ ✐✳ ◮ ▼❛②❜❡ ❡ ✐s ♠✐ss✐♥❣ ♦♥ ♠❛❝❤✐♥❡ ✐✳

✶ ❙✉❜❣r❛♣❤s ●✐ ❤❛✈❡ t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥❀ ❢♦❝✉s ♦♥ ●✶ ❛♥❞ ▼✶✳ ✷ ❆ ❜❧♦❝❦❡❞ ❡❞❣❡ ❤❛s ❛ ❤❡❛✈✐❡r ❡❞❣❡ ✐♥ ▼✐ ❛s ✐ts ✏❝❡rt✐✜❝❛t❡✳✑

◮ ❯s❡ ✐t ✐♥ t❤❡ ❝❤❛r❣✐♥❣ ❛r❣✉♠❡♥t ❢♦r ▼✶✳

✸ ❋r❡❡ ❡❞❣❡ ♣❛rt ♦❢ ●✐ ✇✐❧❧ ♠❛❦❡ ✐t t♦ ▼✐ ❛♥❞ ❍✳

◮ ❋r❡❡ ❡❞❣❡s ✐♥ ▼✷ ∪ ▼✸ ∪ . . . ❝♦♠♣❡♥s❛t❡ t❤❡ ❢r❡❡ ❡❞❣❡s ✐♥ ▼✶✳ ◮ ❙✉❜t❧❡ ❛r❣✉♠❡♥t ❢♦r ✏❛✉❣♠❡♥t❛t✐♦♥✳✑

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SLIDE 39

■♥ ♣r❛❝t✐❝❡

◮ ❋♦❝✉s ♦♥ ✷✲r♦✉♥❞ ❛❧❣♦r✐t❤♠s✿ ❬❆ss❇❛t❇❡r+✶✾❪ t♦♦ ❝♦♠♣❧✐❝❛t❡❞ ◮ ❬❆ss❑❤❛✶✼❪ ✇✐t❤♦✉t ❬❈r♦❙t✉✶✹❪ ♥♦t ✈✐❛❜❧❡ ❢♦r ✇❡✐❣❤t❡❞ ♠❛t❝❤✐♥❣

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SLIDE 40

■♥ ♣r❛❝t✐❝❡

◮ ❋♦❝✉s ♦♥ ✷✲r♦✉♥❞ ❛❧❣♦r✐t❤♠s✿ ❬❆ss❇❛t❇❡r+✶✾❪ t♦♦ ❝♦♠♣❧✐❝❛t❡❞ ◮ ❬❆ss❑❤❛✶✼❪ ✇✐t❤♦✉t ❬❈r♦❙t✉✶✹❪ ♥♦t ✈✐❛❜❧❡ ❢♦r ✇❡✐❣❤t❡❞ ♠❛t❝❤✐♥❣ ◮ ❬❆ss❑❤❛✶✼❪ ✇✐t❤ ❬❈r♦❙t✉✶✹❪ ♥♦t ❛s ❣♦♦❞ ❛s t❤❡ ♥❡✇ ❛❧❣♦r✐t❤♠ ❉❛t❛s❡t ❙♣❡❡❞✲✉♣ |❱ | |❊| ∆

❬❆ss❑❤❛✶✼❪

❚❤✐s ✇♦r❦ ❋r✐❡♥❞st❡r ✻✻▼ ✺✺✵❇ ✷✳✶▼ ✹✳✼① ✶✸✵① ❖r❦✉t ✸▼ ✷✶❇ ✾✵✵❑ ✶✳✹① ✶✼① ▲✐✈❡❏♦✉r♥❛❧ ✹✳✽▼ ✸✳✾❇ ✹✹✹❑ ✷✳✺① ✻① ❉❇▲P ✶✳✺▼ ✺✳✾▼ ✶✾✻✶ ✶✳✷① ✸①

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SLIDE 41

■♥ ♣r❛❝t✐❝❡

◮ ❋♦❝✉s ♦♥ ✷✲r♦✉♥❞ ❛❧❣♦r✐t❤♠s✿ ❬❆ss❇❛t❇❡r+✶✾❪ t♦♦ ❝♦♠♣❧✐❝❛t❡❞ ◮ ❬❆ss❑❤❛✶✼❪ ✇✐t❤♦✉t ❬❈r♦❙t✉✶✹❪ ♥♦t ✈✐❛❜❧❡ ❢♦r ✇❡✐❣❤t❡❞ ♠❛t❝❤✐♥❣ ◮ ❬❆ss❑❤❛✶✼❪ ✇✐t❤ ❬❈r♦❙t✉✶✹❪ ♥♦t ❛s ❣♦♦❞ ❛s t❤❡ ♥❡✇ ❛❧❣♦r✐t❤♠ ❉❛t❛s❡t ◗✉❛❧✐t② |❱ | |❊| ∆

❬❆ss❑❤❛✶✼❪

❚❤✐s ✇♦r❦ ❋r✐❡♥❞st❡r ✻✻▼ ✺✺✵❇ ✷✳✶▼ ✾✹✳✹✪ ✾✾✳✽✪ ❖r❦✉t ✸▼ ✷✶❇ ✾✵✵❑ ✾✷✳✹✪ ✾✽✳✾✪ ▲✐✈❡❏♦✉r♥❛❧ ✹✳✽▼ ✸✳✾❇ ✹✹✹❑ ✾✻✳✺✪ ✾✾✳✾✪ ❉❇▲P ✶✳✺▼ ✺✳✾▼ ✶✾✻✶ ✾✷✳✹✪ ✾✾✳✻✪

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SLIDE 42

■♥ ♣r❛❝t✐❝❡

◮ ❋♦❝✉s ♦♥ ✷✲r♦✉♥❞ ❛❧❣♦r✐t❤♠s✿ ❬❆ss❇❛t❇❡r+✶✾❪ t♦♦ ❝♦♠♣❧✐❝❛t❡❞ ◮ ❬❆ss❑❤❛✶✼❪ ✇✐t❤♦✉t ❬❈r♦❙t✉✶✹❪ ♥♦t ✈✐❛❜❧❡ ❢♦r ✇❡✐❣❤t❡❞ ♠❛t❝❤✐♥❣ ◮ ❬❆ss❑❤❛✶✼❪ ✇✐t❤ ❬❈r♦❙t✉✶✹❪ ♥♦t ❛s ❣♦♦❞ ❛s t❤❡ ♥❡✇ ❛❧❣♦r✐t❤♠ ◮ ▼✉❧t✐♣❧✐❝✐t② ✷ ♦r ✸ ✐s s✉✣❝✐❡♥t

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SLIDE 43

❙✉♠♠❛r②

◮ ❆ s✐♠♣❧❡✱ s❝❛❧❛❜❧❡ ❛❧❣♦r✐t❤♠ ❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣ ✇✐t❤ ✷ + ǫ ❛♣♣r♦①✐♠❛t✐♦♥✳

◮ ❯s❡s ❣r❡❡❞② ❛❧❣♦r✐t❤♠ ❢♦r ❝♦r❡s❡t ❝♦♥str✉❝t✐♦♥✳ ◮ ❯s❡s ♠✉❧t✐♣❧✐❝✐t②✳

◮ ●♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ✐♥ ♣r❛❝t✐❝❡✳

◮ ❙♠❛❧❧ ♠✉❧t✐♣❧✐❝✐t② s✉✣❝❡s✳ ◮ ❇❡❛ts ♣r✐♦r ❛❧❣♦r✐t❤♠s✳ ◮ ❍✉❣❡ s♣❡❡❞✲✉♣ ♦❢ s❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠✱ ❜✉t ❢❛✐t❤❢✉❧ s✐♠✉❧❛t✐♦♥✳

❚❤❛♥❦s✦ P♦st❡r ★✶✻✶ P❛❝✐✜❝ ❇❛❧❧r♦♦♠

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SLIDE 44

❙✉♠♠❛r②

◮ ❆ s✐♠♣❧❡✱ s❝❛❧❛❜❧❡ ❛❧❣♦r✐t❤♠ ❢♦r ♠❛①✐♠✉♠✲✇❡✐❣❤t ♠❛t❝❤✐♥❣ ✇✐t❤ ✷ + ǫ ❛♣♣r♦①✐♠❛t✐♦♥✳

◮ ❯s❡s ❣r❡❡❞② ❛❧❣♦r✐t❤♠ ❢♦r ❝♦r❡s❡t ❝♦♥str✉❝t✐♦♥✳ ◮ ❯s❡s ♠✉❧t✐♣❧✐❝✐t②✳

◮ ●♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ✐♥ ♣r❛❝t✐❝❡✳

◮ ❙♠❛❧❧ ♠✉❧t✐♣❧✐❝✐t② s✉✣❝❡s✳ ◮ ❇❡❛ts ♣r✐♦r ❛❧❣♦r✐t❤♠s✳ ◮ ❍✉❣❡ s♣❡❡❞✲✉♣ ♦❢ s❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠✱ ❜✉t ❢❛✐t❤❢✉❧ s✐♠✉❧❛t✐♦♥✳

❚❤❛♥❦s✦ P♦st❡r ★✶✻✶ P❛❝✐✜❝ ❇❛❧❧r♦♦♠

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SLIDE 45

❇✐❜❧✐♦s ■

❑♦♦❦ ❏✐♥ ❆❤♥ ❛♥❞ ❙✉❞✐♣t♦ ●✉❤❛✳ ❆❝❝❡ss t♦ ❞❛t❛ ❛♥❞ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s✿ ❉✉❛❧ ♣r✐♠❛❧ ❛❧❣♦r✐t❤♠s ❢♦r ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ✉♥❞❡r r❡s♦✉r❝❡ ❝♦♥str❛✐♥ts✳ ■♥ ❙P❆❆✱ ♣❛❣❡s ✷✵✷✕✷✶✶✱ ✷✵✶✺✳ ❙❡♣❡❤r ❆ss❛❞✐✱ ▼♦❤❛♠♠❛❞❍♦ss❡✐♥ ❇❛t❡♥✐✱ ❆❛r♦♥ ❇❡r♥st❡✐♥✱ ❱❛❤❛❜ ❙✳ ▼✐rr♦❦♥✐✱ ❛♥❞ ❈❧✐✛ ❙t❡✐♥✳ ❈♦r❡s❡ts ♠❡❡t ❊❉❈❙✿ ❛❧❣♦r✐t❤♠s ❢♦r ♠❛t❝❤✐♥❣ ❛♥❞ ✈❡rt❡① ❝♦✈❡r ♦♥ ♠❛ss✐✈❡ ❣r❛♣❤s✳ ■♥ ❙❖❉❆✱ ♣❛❣❡s ✶✻✶✻✕✶✻✸✺✱ ✷✵✶✾✳ ❙❡♣❡❤r ❆ss❛❞✐ ❛♥❞ ❙❛♥❥❡❡✈ ❑❤❛♥♥❛✳ ❘❛♥❞♦♠✐③❡❞ ❝♦♠♣♦s❛❜❧❡ ❝♦r❡s❡ts ❢♦r ♠❛t❝❤✐♥❣ ❛♥❞ ✈❡rt❡① ❝♦✈❡r✳ ■♥ ❙P❆❆✱ ♣❛❣❡s ✸✕✶✷✱ ✷✵✶✼✳ ❙❡♣❡❤r ❆ss❛❞✐✱ ❙❛♥❥❡❡✈ ❑❤❛♥♥❛✱ ❨❛♥❣ ▲✐✱ ❛♥❞ ●r✐❣♦r② ❨❛r♦s❧❛✈ts❡✈✳ ▼❛①✐♠✉♠ ♠❛t❝❤✐♥❣s ✐♥ ❞②♥❛♠✐❝ ❣r❛♣❤ str❡❛♠s ❛♥❞ t❤❡ s✐♠✉❧t❛♥❡♦✉s ❝♦♠♠✉♥✐❝❛t✐♦♥ ♠♦❞❡❧✳ ■♥ ❙❖❉❆✱ ♣❛❣❡s ✶✸✹✺✕✶✸✻✹✱ ✷✵✶✻✳ ▼✐❝❤❛❡❧ ❈r♦✉❝❤ ❛♥❞ ❉❛♥✐❡❧ ❙✳ ❙t✉❜❜s✳ ■♠♣r♦✈❡❞ str❡❛♠✐♥❣ ❛❧❣♦r✐t❤♠s ❢♦r ✇❡✐❣❤t❡❞ ♠❛t❝❤✐♥❣✱ ✈✐❛ ✉♥✇❡✐❣❤t❡❞ ♠❛t❝❤✐♥❣✳ ■♥ ❆PP❘❖❳✱ ♣❛❣❡s ✾✻✕✶✵✹✱ ✷✵✶✹✳ ❆rt✉r ❈③✉♠❛❥✱ ❏❛❦✉❜ ▲❛❝❦✐✱ ❆❧❡❦s❛♥❞❡r ▼❛❞r②✱ ❙❧♦❜♦❞❛♥ ▼✐tr♦✈✐❝✱ ❑r③②s③t♦❢ ❖♥❛❦✱ ❛♥❞ P✐♦tr ❙❛♥❦♦✇s❦✐✳ ❘♦✉♥❞ ❝♦♠♣r❡ss✐♦♥ ❢♦r ♣❛r❛❧❧❡❧ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠s✳ ■♥ ❙❚❖❈✱ ♣❛❣❡s ✹✼✶✕✹✽✹✱ ✷✵✶✽✳

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SLIDE 46

❇✐❜❧✐♦s ■■

❘❛♥ ❉✉❛♥ ❛♥❞ ❙❡t❤ P❡tt✐❡✳ ▲✐♥❡❛r✲t✐♠❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ♠❛①✐♠✉♠ ✇❡✐❣❤t ♠❛t❝❤✐♥❣✳ ❏✳ ❆❈▼✱ ✻✶✭✶✮✿✶✿✶✕✶✿✷✸✱ ✷✵✶✹✳ ❏❛❝❦ ❊❞♠♦♥❞s✳ ▼❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ❛♥❞ ❛ ♣♦❧②❤❡❞r♦♥ ✇✐t❤ ✵, ✶✲✈❡rt✐❝❡s✳ ❏♦✉r♥❛❧ ♦❢ ❘❡s❡❛r❝❤ ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❇✉r❡❛✉ ♦❢ ❙t❛♥❞❛r❞s ❇✱ ✻✾✭✶✷✺✲✶✸✵✮✿✺✺✕✺✻✱ ✶✾✻✺✳ ❏❛❝❦ ❊❞♠♦♥❞s✳ P❛t❤s✱ tr❡❡s✱ ❛♥❞ ✢♦✇❡rs✳ ❈❛♥❛❞✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✼✭✸✮✿✹✹✾✕✹✻✼✱ ✶✾✻✺✳ ❍❛r♦❧❞ ◆✳ ●❛❜♦✇ ❛♥❞ ❘♦❜❡rt ❊♥❞r❡ ❚❛r❥❛♥✳ ❋❛st❡r s❝❛❧✐♥❣ ❛❧❣♦r✐t❤♠s ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤✲♠❛t❝❤✐♥❣ ♣r♦❜❧❡♠s✳ ❏✳ ❆❈▼✱ ✸✽✭✹✮✿✽✶✺✕✽✺✸✱ ✶✾✾✶✳ ❇✉❞❞❤✐♠❛ ●❛♠❧❛t❤✱ ❙❛❣❛r ❑❛❧❡✱ ❙❧♦❜♦❞❛♥ ▼✐tr♦✈✐❝✱ ❛♥❞ ❖❧❛ ❙✈❡♥ss♦♥✳ ❲❡✐❣❤t❡❞ ♠❛t❝❤✐♥❣s ✈✐❛ ✉♥✇❡✐❣❤t❡❞ ❛✉❣♠❡♥t❛t✐♦♥s✳ ❈♦❘❘✱ ❛❜s✴✶✽✶✶✳✵✷✼✻✵✳ ❚♦ ❛♣♣❡❛r ✐♥ P❖❉❈ ✷✵✶✾✳✱ ✷✵✶✽✳ ◆✐❝❤♦❧❛s ❏✳ ❆✳ ❍❛r✈❡②✱ ❈❤r✐st♦♣❤❡r ▲✐❛✇✱ ❛♥❞ P❛✉❧ ▲✐✉✳ ●r❡❡❞② ❛♥❞ ❧♦❝❛❧ r❛t✐♦ ❛❧❣♦r✐t❤♠s ✐♥ t❤❡ ♠❛♣r❡❞✉❝❡ ♠♦❞❡❧✳ ■♥ ❙P❆❆✱ ♣❛❣❡s ✹✸✕✺✷✱ ✷✵✶✽✳ ❘❛✈✐ ❑✉♠❛r✱ ❇❡♥❥❛♠✐♥ ▼♦s❡❧❡②✱ ❙❡r❣❡✐ ❱❛ss✐❧✈✐ts❦✐✐✱ ❛♥❞ ❆♥❞r❡❛ ❱❛tt❛♥✐✳ ❋❛st ❣r❡❡❞② ❛❧❣♦r✐t❤♠s ✐♥ ♠❛♣r❡❞✉❝❡ ❛♥❞ str❡❛♠✐♥❣✳ ■♥ ❙P❆❆✱ ♣❛❣❡s ✶✕✶✵✱ ✷✵✶✸✳

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SLIDE 47

❇✐❜❧✐♦s ■■■

❙✐❧✈✐♦ ▲❛tt❛♥③✐✱ ❇❡♥❥❛♠✐♥ ▼♦s❡❧❡②✱ ❙✐❞❞❤❛rt❤ ❙✉r✐✱ ❛♥❞ ❙❡r❣❡✐ ❱❛ss✐❧✈✐ts❦✐✐✳ ❋✐❧t❡r✐♥❣✿ ❛ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ❣r❛♣❤ ♣r♦❜❧❡♠s ✐♥ ♠❛♣r❡❞✉❝❡✳ ■♥ ❙P❆❆✱ ♣❛❣❡s ✽✺✕✾✹✱ ✷✵✶✶✳ ❱❛❤❛❜ ❙✳ ▼✐rr♦❦♥✐ ❛♥❞ ▼♦rt❡③❛ ❩❛❞✐♠♦❣❤❛❞❞❛♠✳ ❘❛♥❞♦♠✐③❡❞ ❝♦♠♣♦s❛❜❧❡ ❝♦r❡✲s❡ts ❢♦r ❞✐str✐❜✉t❡❞ s✉❜♠♦❞✉❧❛r ♠❛①✐♠✐③❛t✐♦♥✳ ■♥ ❙❚❖❈✱ ♣❛❣❡s ✶✺✸✕✶✻✷✱ ✷✵✶✺✳