rr ss trtr sts tr - - PowerPoint PPT Presentation

▶

Jan 08, 2024 284 likes •1.08k views

rr ss trtr sts tr ts rt rs r

SLIDE 1

rö❜♥❡r ❇❛s❡s ♦❢ ❙tr✉❝t✉r❡❞ ❙②st❡♠s

❛♥❞ t❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❈r②♣t♦❧♦❣②

❏❡❛♥✲❈❤❛r❧❡s ❋❛✉❣èr❡✱ ▼♦❤❛❜ ❙❛❢❡② ❊❧ ❉✐♥ P✐❡rr❡✲❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r

❯P▼❈ ✕ ❈◆❘❙ ✕ ■◆❘■❆ P❛r✐s ✲ ❘♦❝q✉❡♥❝♦✉rt ▲■P✻ ✕ ❙❆▲❙❆ t❡❛♠

❘✉❤r ❯♥✐✈❡rs✐tät ✕ ▼❆❨❆ ❲♦r❦s❤♦♣ ✷✵✶✶ ✷✵✶✶✱ ✵✼✴✷✽

✶✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 2

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s

❈r②♣t♦ ♣r✐♠✐t✐✈❡ ♠♦❞❡❧✐♥❣ ❙②st❡♠ s♦❧✈✐♥❣ ■ss✉❡s ❲❤✐❝❤ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣ ❄ ❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s✴♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❄ ❙♦❧✈✐♥❣ t♦♦❧s✿ ●rö❜♥❡r ❜❛s❡s ❄ ❙❆❚✲s♦❧✈❡rs ❄ ✳✳✳ ❙tr✉❝t✉r❡ ❄

✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 3

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s

❈r②♣t♦ ♣r✐♠✐t✐✈❡ ♠♦❞❡❧✐♥❣ ❙②st❡♠ s♦❧✈✐♥❣ ■ss✉❡s ❲❤✐❝❤ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣ ❄ ❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s✴♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❄ ❙♦❧✈✐♥❣ t♦♦❧s✿ ●rö❜♥❡r ❜❛s❡s ❄ ❙❆❚✲s♦❧✈❡rs ❄ ✳✳✳ ❙tr✉❝t✉r❡ ❄

✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 4

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s

❈r②♣t♦ ♣r✐♠✐t✐✈❡ ♠♦❞❡❧✐♥❣ ❙②st❡♠ s♦❧✈✐♥❣ ■ss✉❡s ❲❤✐❝❤ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣ ❄ ❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s✴♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❄ ❙♦❧✈✐♥❣ t♦♦❧s✿ ●rö❜♥❡r ❜❛s❡s ❄ ❙❆❚✲s♦❧✈❡rs ❄ ✳✳✳ ❙tr✉❝t✉r❡ ❄

✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 5

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s

❈r②♣t♦ ♣r✐♠✐t✐✈❡ ♠♦❞❡❧✐♥❣ ❙②st❡♠ s♦❧✈✐♥❣ ■ss✉❡s ❲❤✐❝❤ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣ ❄ ❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s✴♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❄ ❙♦❧✈✐♥❣ t♦♦❧s✿ ●rö❜♥❡r ❜❛s❡s ❄ ❙❆❚✲s♦❧✈❡rs ❄ ✳✳✳ ❙tr✉❝t✉r❡ ❄

✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 6

▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣②

❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② ❙❡❝✉r✐t②

❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① ② ✶ ✵ ❢♦r ① ✵✮✳

❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿

tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳

❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿

✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳

✳ ✳ ✳

■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄

✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 7

▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣②

❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t②

❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① = ✵✮✳

❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿

tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳

❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿

✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳

✳ ✳ ✳

■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄

✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 8

▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣②

❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t②

❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① = ✵✮✳

❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿

tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ → ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳

❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿

✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳

✳ ✳ ✳

■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄

✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 9

▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣②

❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t②

❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① = ✵✮✳

❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿

tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ → ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳

❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿

✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳

✳ ✳ ✳

■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄

✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 10

▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣②

❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t②

❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① = ✵✮✳

❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿

tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ → ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳

❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿

✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳

✳ ✳ ✳

■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄

✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 11

▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣②

❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t②

❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① = ✵✮✳

❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿

tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ → ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳

❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿

✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳

✳ ✳ ✳

■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄

✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 12

❋❛♠✐❧✐❡s ♦❢ str✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s ▼❝❊❧✐❡❝❡ P❑❈✳ ▼✐♥❘❛♥❦ ❛✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡✳ ✳ ✳ ✳ ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥❘❛♥❦ ❛✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡✳ ❈r②♣t♦s②st❡♠s ❜❛s❡❞ ♦♥ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s✳ ❍✐❞❞❡♥ ❋✐❡❧❞ ❊q✉❛t✐♦♥s ❛♥❞ ✈❛r✐❛♥ts✳ ✳ ✳ ✳ ❙②st❡♠s ✐♥✈❛r✐❛♥t ❜② s②♠♠❡tr✐❡s ❉✐s❝r❡t❡ ❧♦❣ ♦♥ ❡❧❧✐♣t✐❝ ❛♥❞ ❤②♣❡r❡❧❧✐♣t✐❝ ❝✉r✈❡s✳

✹✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 13

❋❛♠✐❧✐❡s ♦❢ str✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s ▼❝❊❧✐❡❝❡ P❑❈✳ ▼✐♥❘❛♥❦ ❛✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡✳ ✳ ✳ ✳ ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥❘❛♥❦ ❛✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡✳ ❈r②♣t♦s②st❡♠s ❜❛s❡❞ ♦♥ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s✳ ❍✐❞❞❡♥ ❋✐❡❧❞ ❊q✉❛t✐♦♥s ❛♥❞ ✈❛r✐❛♥ts✳ ✳ ✳ ✳ ❙②st❡♠s ✐♥✈❛r✐❛♥t ❜② s②♠♠❡tr✐❡s ❉✐s❝r❡t❡ ❧♦❣ ♦♥ ❡❧❧✐♣t✐❝ ❛♥❞ ❤②♣❡r❡❧❧✐♣t✐❝ ❝✉r✈❡s✳

✹✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 14

❋❛♠✐❧✐❡s ♦❢ str✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s ▼❝❊❧✐❡❝❡ P❑❈✳ ▼✐♥❘❛♥❦ ❛✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡✳ ✳ ✳ ✳ ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥❘❛♥❦ ❛✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡✳ ❈r②♣t♦s②st❡♠s ❜❛s❡❞ ♦♥ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s✳ ❍✐❞❞❡♥ ❋✐❡❧❞ ❊q✉❛t✐♦♥s ❛♥❞ ✈❛r✐❛♥ts✳ ✳ ✳ ✳ ❙②st❡♠s ✐♥✈❛r✐❛♥t ❜② s②♠♠❡tr✐❡s ❉✐s❝r❡t❡ ❧♦❣ ♦♥ ❡❧❧✐♣t✐❝ ❛♥❞ ❤②♣❡r❡❧❧✐♣t✐❝ ❝✉r✈❡s✳

✹✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 15

❖✉t❧✐♥❡

✶ P♦❧②♥♦♠✐❛❧ ❙②st❡♠ ❙♦❧✈✐♥❣ ✉s✐♥❣ ●rö❜♥❡r ❇❛s❡s ✷ ❇✐❧✐♥❡❛r ❙②st❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❝❊❧✐❡❝❡ ✸ ❉❡t❡r♠✐♥❛♥t❛❧ ❙②st❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s t♦ ▼✐♥❘❛♥❦ ❛♥❞ ❍❋❊

✺✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 16

rö❜♥❡r ❜❛s❡s ✭■✮
rö❜♥❡r ❜❛s❡s

I ❛ ♣♦❧②♥♦♠✐❛❧ ✐❞❡❛❧✳ ●rö❜♥❡r ❜❛s✐s ✭✇✳r✳t✳ ❛ ♠♦♥♦♠✐❛❧ ♦r❞❡r✐♥❣✮✿ ● ⊂ I ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s s✉❝❤ t❤❛t ▲▼(I) = ▲▼(●)✳ ❇✉❝❤❜❡r❣❡r ❬❇✉❝❤❜❡r❣❡r P❤✳❉✳ ✻✺❪✳ ❋✹ ❬❋❛✉❣èr❡ ❏✳ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✳ ❆❧❣✳ ✾✾❪✳ ❋✺ ❬❋❛✉❣èr❡ ■❙❙❆❈✬✵✷❪✳ ❋●▲▼ ❬❋❛✉❣èr❡✴●✐❛♥♥✐✴▲❛③❛r❞✴▼♦r❛ ❏❙❈✳ ✾✸✱ ❋❛✉❣èr❡✴▼♦✉ ■❙❙❆❈✬✶✶❪✳ ✵✲❞✐♠❡♥s✐♦♥❛❧ s②st❡♠ s♦❧✈✐♥❣ P♦❧②♥♦♠✐❛❧ s②st❡♠

❋✹ ❋✺

❣r❡✈❧❡① ●❇

❋●▲▼

❧❡① ●❇ ❳▲✴▼❳▲ ▼♦st ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❛❧s♦ ✈❛❧✐❞ ❢♦r ❳▲✴▼❳▲ ❇✉❝❤♠❛♥✴❇✉❧②❣✐♥✴❈❛❜❛r❝❛s✴❉✐♥❣✴▼♦❤❛♠❡❞✴▼♦❤❛♠❡❞ P◗❈r②♣t♦ ✷✵✵✽✱ ❆❢r✐❝❛❝r②♣t ✷✵✶✵✱✳ ✳ ✳ ❆rs✴❋❛✉❣èr❡✴■♠❛✐✴❑❛✇❛③♦❡✴❙✉❣✐t❛✱ ❆s✐❛❝r②♣t ✷✵✵✹ ❆❧❜r❡❝❤t✴❈✐❞✴❋❛✉❣èr❡✴P❡rr❡t✱ ❡♣r✐♥t

✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 17

rö❜♥❡r ❜❛s❡s ✭■✮
rö❜♥❡r ❜❛s❡s

I ❛ ♣♦❧②♥♦♠✐❛❧ ✐❞❡❛❧✳ ●rö❜♥❡r ❜❛s✐s ✭✇✳r✳t✳ ❛ ♠♦♥♦♠✐❛❧ ♦r❞❡r✐♥❣✮✿ ● ⊂ I ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s s✉❝❤ t❤❛t ▲▼(I) = ▲▼(●)✳ ❇✉❝❤❜❡r❣❡r ❬❇✉❝❤❜❡r❣❡r P❤✳❉✳ ✻✺❪✳ ❋✹ ❬❋❛✉❣èr❡ ❏✳ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✳ ❆❧❣✳ ✾✾❪✳ ❋✺ ❬❋❛✉❣èr❡ ■❙❙❆❈✬✵✷❪✳ ❋●▲▼ ❬❋❛✉❣èr❡✴●✐❛♥♥✐✴▲❛③❛r❞✴▼♦r❛ ❏❙❈✳ ✾✸✱ ❋❛✉❣èr❡✴▼♦✉ ■❙❙❆❈✬✶✶❪✳ ✵✲❞✐♠❡♥s✐♦♥❛❧ s②st❡♠ s♦❧✈✐♥❣ P♦❧②♥♦♠✐❛❧ s②st❡♠

❋✹/❋✺

− − − − → ❣r❡✈❧❡① ●❇

❋●▲▼

− − − − → ❧❡① ●❇. ❳▲✴▼❳▲ ▼♦st ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❛❧s♦ ✈❛❧✐❞ ❢♦r ❳▲✴▼❳▲ ❇✉❝❤♠❛♥✴❇✉❧②❣✐♥✴❈❛❜❛r❝❛s✴❉✐♥❣✴▼♦❤❛♠❡❞✴▼♦❤❛♠❡❞ P◗❈r②♣t♦ ✷✵✵✽✱ ❆❢r✐❝❛❝r②♣t ✷✵✶✵✱✳ ✳ ✳ ❆rs✴❋❛✉❣èr❡✴■♠❛✐✴❑❛✇❛③♦❡✴❙✉❣✐t❛✱ ❆s✐❛❝r②♣t ✷✵✵✹ ❆❧❜r❡❝❤t✴❈✐❞✴❋❛✉❣èr❡✴P❡rr❡t✱ ❡♣r✐♥t

✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 18

rö❜♥❡r ❜❛s❡s ✭■✮
rö❜♥❡r ❜❛s❡s

I ❛ ♣♦❧②♥♦♠✐❛❧ ✐❞❡❛❧✳ ●rö❜♥❡r ❜❛s✐s ✭✇✳r✳t✳ ❛ ♠♦♥♦♠✐❛❧ ♦r❞❡r✐♥❣✮✿ ● ⊂ I ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s s✉❝❤ t❤❛t ▲▼(I) = ▲▼(●)✳ ❇✉❝❤❜❡r❣❡r ❬❇✉❝❤❜❡r❣❡r P❤✳❉✳ ✻✺❪✳ ❋✹ ❬❋❛✉❣èr❡ ❏✳ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✳ ❆❧❣✳ ✾✾❪✳ ❋✺ ❬❋❛✉❣èr❡ ■❙❙❆❈✬✵✷❪✳ ❋●▲▼ ❬❋❛✉❣èr❡✴●✐❛♥♥✐✴▲❛③❛r❞✴▼♦r❛ ❏❙❈✳ ✾✸✱ ❋❛✉❣èr❡✴▼♦✉ ■❙❙❆❈✬✶✶❪✳ ✵✲❞✐♠❡♥s✐♦♥❛❧ s②st❡♠ s♦❧✈✐♥❣ P♦❧②♥♦♠✐❛❧ s②st❡♠

❋✹/❋✺

− − − − → ❣r❡✈❧❡① ●❇

❋●▲▼

− − − − → ❧❡① ●❇. ❳▲✴▼❳▲ ▼♦st ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❛❧s♦ ✈❛❧✐❞ ❢♦r ❳▲✴▼❳▲ ❇✉❝❤♠❛♥✴❇✉❧②❣✐♥✴❈❛❜❛r❝❛s✴❉✐♥❣✴▼♦❤❛♠❡❞✴▼♦❤❛♠❡❞ P◗❈r②♣t♦ ✷✵✵✽✱ ❆❢r✐❝❛❝r②♣t ✷✵✶✵✱✳ ✳ ✳ ❆rs✴❋❛✉❣èr❡✴■♠❛✐✴❑❛✇❛③♦❡✴❙✉❣✐t❛✱ ❆s✐❛❝r②♣t ✷✵✵✹ ❆❧❜r❡❝❤t✴❈✐❞✴❋❛✉❣èr❡✴P❡rr❡t✱ ❡♣r✐♥t

✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 19

rö❜♥❡r ❜❛s❡s ✭■■✮

✵✲❞✐♠❡♥s✐♦♥❛❧ s②st❡♠ s♦❧✈✐♥❣ P♦❧②♥♦♠✐❛❧ s②st❡♠

❋✹/❋✺

− − − − → ❣r❡✈❧❡① ●❇

❋●▲▼

− − − − → ❧❡① ●❇. ▲❡①✐❝♦❣r❛♣❤✐❝❛❧ ●rö❜♥❡r ❜❛s✐s ♦❢ ✵✲❞✐♠❡♥s✐♦♥❛❧ s②st❡♠s ❊q✉✐✈❛❧❡♥t s②st❡♠ ✐♥ tr✐❛♥❣✉❧❛r s❤❛♣❡✿                              ❢✶(①✶, . . . , ①♥) = ✵ ✳ ✳ ✳ ❢ℓ(①✶, . . . , ①♥) = ✵ ❢ℓ+✶(①✷, . . . , ①♥) = ✵ ✳ ✳ ✳ ❢♠−✶(①♥−✶, ①♥) = ✵ ❢♠(①♥) = ✵ ❋✐♥❞ t❤❡ r♦♦ts ♦❢ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s ❡❛s② ✐♥ ✜♥✐t❡ ✜❡❧❞s✳

✼✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 20

rö❜♥❡r ❜❛s❡s ✭■■✮

✵✲❞✐♠❡♥s✐♦♥❛❧ s②st❡♠ s♦❧✈✐♥❣ P♦❧②♥♦♠✐❛❧ s②st❡♠

❋✹/❋✺

− − − − → ❣r❡✈❧❡① ●❇

❋●▲▼

− − − − → ❧❡① ●❇. ▲❡①✐❝♦❣r❛♣❤✐❝❛❧ ●rö❜♥❡r ❜❛s✐s ♦❢ ✵✲❞✐♠❡♥s✐♦♥❛❧ s②st❡♠s ❊q✉✐✈❛❧❡♥t s②st❡♠ ✐♥ tr✐❛♥❣✉❧❛r s❤❛♣❡✿                              ❢✶(①✶, . . . , ①♥) = ✵ ✳ ✳ ✳ ❢ℓ(①✶, . . . , ①♥) = ✵ ❢ℓ+✶(①✷, . . . , ①♥) = ✵ ✳ ✳ ✳ ❢♠−✶(①♥−✶, ①♥) = ✵ ❢♠(①♥) = ✵ = ⇒ ❋✐♥❞ t❤❡ r♦♦ts ♦❢ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s → ❡❛s② ✐♥ ✜♥✐t❡ ✜❡❧❞s✳

✼✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 21

▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞

I = ❢✶, . . . , ❢♣ ❞❡❣(❢✐) = ❞✐ ≻ ❛ ♠♦♥♦♠✐❛❧ ♦r❞❡r✐♥❣ ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t ✐s ❛ ♠♦♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ❞ − ❞✐✳ ❈♦❧✉♠♥s✿ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ❞✳ ♠✶ ≻ · · · ≻ ♠ℓ t✶❢✶ ✳ ✳ ✳ t❦❢♣         r♦✇ ❡❝❤❡❧♦♥ ❢♦r♠ ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐① ✇✐t❤ ❞ s✉✣❝✐❡♥t❧② ❤✐❣❤

= ⇒ ●rö❜♥❡r ❜❛s✐s✳

Pr♦❜❧❡♠s ❉❡❣r❡❡ ❢❛❧❧s✳ ❘❛♥❦ ❞❡❢❡❝t ✉s❡❧❡ss ❝♦♠♣✉t❛t✐♦♥s✳ ❍✐❧❜❡rt s❡r✐❡s✿ ❣❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s✳ ❲❤✐❝❤ ❞ ❄ ❞❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✳

✽✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 22

▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞

I = ❢✶, . . . , ❢♣ ❞❡❣(❢✐) = ❞✐ ≻ ❛ ♠♦♥♦♠✐❛❧ ♦r❞❡r✐♥❣ ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t ✐s ❛ ♠♦♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ❞ − ❞✐✳ ❈♦❧✉♠♥s✿ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ❞✳ ♠✶ ≻ · · · ≻ ♠ℓ t✶❢✶ ✳ ✳ ✳ t❦❢♣         r♦✇ ❡❝❤❡❧♦♥ ❢♦r♠ ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐① ✇✐t❤ ❞ s✉✣❝✐❡♥t❧② ❤✐❣❤

= ⇒ ●rö❜♥❡r ❜❛s✐s✳

Pr♦❜❧❡♠s ❉❡❣r❡❡ ❢❛❧❧s✳ ❘❛♥❦ ❞❡❢❡❝t ✉s❡❧❡ss ❝♦♠♣✉t❛t✐♦♥s✳ ❍✐❧❜❡rt s❡r✐❡s✿ ❣❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s✳ ❲❤✐❝❤ ❞ ❄ ❞❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✳

✽✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 23

❈♦♠♣❧❡①✐t② ♦❢ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s

❚✇♦ ♠❛✐♥ ✐♥❞✐❝❛t♦rs ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ❞r❡❣ ❞❡❣r❡❡ t❤❛t ❤❛s t♦ ❜❡ r❡❛❝❤❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❣r❡✈❧❡① ●❇✳ ❉❡❣r❡❡ ♦❢ t❤❡ ✐❞❡❛❧ I = ❢✶, . . . ❢♠ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ✭❝♦✉♥t❡❞ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s✮✳ ●✐✈❡s t❤❡ r❛♥❦ ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐①✳ ❙②st❡♠ ❣r❡✈❧❡① ●❇ ❧❡① ●❇ ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ ❞r❡❣ ❞r❡❣ ❖ ♥ ❙♦❧ ❈❧❛ss✐❝❛❧ ❜♦✉♥❞s ✭s❤❛r♣ ❢♦r ❣❡♥❡r✐❝ s②st❡♠s✮ ▲❡t ❢✶ ❢♥ ①✶ ①♥ ❜❡ ❛ ✧❣❡♥❡r✐❝✧ s②st❡♠✳ ▼❛❝❛✉❧❛② ❜♦✉♥❞✿ ❞r❡❣ ✶

✶ ✐ ♥

❞✐ ✶ ✳ ❇é③♦✉t ❜♦✉♥❞✿ ❞❡❣ ❢✶ ❢♥

✶ ✐ ♥

❞✐✳

❆r❡ t❤❡r❡ s❤❛r♣❡r ❜♦✉♥❞s ❢♦r str✉❝t✉r❡❞ s②st❡♠s ❄

✾✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 24

❈♦♠♣❧❡①✐t② ♦❢ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s

❚✇♦ ♠❛✐♥ ✐♥❞✐❝❛t♦rs ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ❞r❡❣ ❞❡❣r❡❡ t❤❛t ❤❛s t♦ ❜❡ r❡❛❝❤❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❣r❡✈❧❡① ●❇✳ ❉❡❣r❡❡ ♦❢ t❤❡ ✐❞❡❛❧ I = ❢✶, . . . ❢♠ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ✭❝♦✉♥t❡❞ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s✮✳ ●✐✈❡s t❤❡ r❛♥❦ ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐①✳ ❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω) ❈❧❛ss✐❝❛❧ ❜♦✉♥❞s ✭s❤❛r♣ ❢♦r ❣❡♥❡r✐❝ s②st❡♠s✮ ▲❡t ❢✶ ❢♥ ①✶ ①♥ ❜❡ ❛ ✧❣❡♥❡r✐❝✧ s②st❡♠✳ ▼❛❝❛✉❧❛② ❜♦✉♥❞✿ ❞r❡❣ ✶

✶ ✐ ♥

❞✐ ✶ ✳ ❇é③♦✉t ❜♦✉♥❞✿ ❞❡❣ ❢✶ ❢♥

✶ ✐ ♥

❞✐✳

❆r❡ t❤❡r❡ s❤❛r♣❡r ❜♦✉♥❞s ❢♦r str✉❝t✉r❡❞ s②st❡♠s ❄

✾✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 25

❈♦♠♣❧❡①✐t② ♦❢ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s

❚✇♦ ♠❛✐♥ ✐♥❞✐❝❛t♦rs ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ❞r❡❣ ❞❡❣r❡❡ t❤❛t ❤❛s t♦ ❜❡ r❡❛❝❤❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❣r❡✈❧❡① ●❇✳ ❉❡❣r❡❡ ♦❢ t❤❡ ✐❞❡❛❧ I = ❢✶, . . . ❢♠ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ✭❝♦✉♥t❡❞ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s✮✳ ●✐✈❡s t❤❡ r❛♥❦ ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐①✳ ❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω) ❈❧❛ss✐❝❛❧ ❜♦✉♥❞s ✭s❤❛r♣ ❢♦r ❣❡♥❡r✐❝ s②st❡♠s✮ ▲❡t ❢✶, . . . , ❢♥ ∈ K[①✶, . . . , ①♥] ❜❡ ❛ ✧❣❡♥❡r✐❝✧ s②st❡♠✳ ▼❛❝❛✉❧❛② ❜♦✉♥❞✿ ❞r❡❣ ≤ ✶ +

✶≤✐≤♥

(❞✐ − ✶)✳ ❇é③♦✉t ❜♦✉♥❞✿ ❞❡❣(❢✶, . . . , ❢♥) ≤

✶≤✐≤♥

❞✐✳

❆r❡ t❤❡r❡ s❤❛r♣❡r ❜♦✉♥❞s ❢♦r str✉❝t✉r❡❞ s②st❡♠s ❄

✾✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 26

❈♦♠♣❧❡①✐t② ♦❢ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s

❚✇♦ ♠❛✐♥ ✐♥❞✐❝❛t♦rs ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ❞r❡❣ ❞❡❣r❡❡ t❤❛t ❤❛s t♦ ❜❡ r❡❛❝❤❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❣r❡✈❧❡① ●❇✳ ❉❡❣r❡❡ ♦❢ t❤❡ ✐❞❡❛❧ I = ❢✶, . . . ❢♠ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ✭❝♦✉♥t❡❞ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s✮✳ ●✐✈❡s t❤❡ r❛♥❦ ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐①✳ ❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω) ❈❧❛ss✐❝❛❧ ❜♦✉♥❞s ✭s❤❛r♣ ❢♦r ❣❡♥❡r✐❝ s②st❡♠s✮ ▲❡t ❢✶, . . . , ❢♥ ∈ K[①✶, . . . , ①♥] ❜❡ ❛ ✧❣❡♥❡r✐❝✧ s②st❡♠✳ ▼❛❝❛✉❧❛② ❜♦✉♥❞✿ ❞r❡❣ ≤ ✶ +

✶≤✐≤♥

(❞✐ − ✶)✳ ❇é③♦✉t ❜♦✉♥❞✿ ❞❡❣(❢✶, . . . , ❢♥) ≤

✶≤✐≤♥

❞✐✳

❆r❡ t❤❡r❡ s❤❛r♣❡r ❜♦✉♥❞s ❢♦r str✉❝t✉r❡❞ s②st❡♠s ❄

✾✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 27

P❧❛♥

✶ P♦❧②♥♦♠✐❛❧ ❙②st❡♠ ❙♦❧✈✐♥❣ ✉s✐♥❣ ●rö❜♥❡r ❇❛s❡s ✷ ❇✐❧✐♥❡❛r ❙②st❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❝❊❧✐❡❝❡ ✸ ❉❡t❡r♠✐♥❛♥t❛❧ ❙②st❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s t♦ ▼✐♥❘❛♥❦ ❛♥❞ ❍❋❊

✶✵✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 28

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ❢ ∈ K[❳ (✶), . . . , ❳ (ℓ)] ✐s ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♦❢ ♠✉❧t✐✲❞❡❣r❡❡ (❞✶, . . . , ❞ℓ) ✐❢ ❢♦r ❛❧❧ λ✶, . . . , λℓ✱ ❢ (λ✶❳ (✶), . . . , λℓ❳ (ℓ)) = λ❞✶

✶ . . . λ❞ℓ ℓ ❢ (❳ (✶), . . . , ❳ (ℓ)).

❊①❛♠♣❧❡✿ ✸①✷

✶②✶

✹①✶①✷②✶ ✸①✷

✷②✶

①✷

✶②✷

✽①✶①✷②✷ ✺①✷

✷②✷

✶✵①✷

✶②✸

✷①✶①✷②✸ ✸①✷

✷②✸

✐s ❛ ❜✐✲❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ♦❢ ❜✐✲❞❡❣r❡❡ ✷ ✶ ✐♥

✶✶ ①✶ ①✷ ②✶ ②✷ ②✸ ✳

❇✐❧✐♥❡❛r s②st❡♠✿ ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♦❢ ♠✉❧t✐✲❞❡❣r❡❡ ✶ ✶ ❢✶ ❢q ❳ ❨ ✿ ❜✐❧✐♥❡❛r ❢♦r♠s✳

❢❦ ❛ ❦

✐ ❥ ①✐②❥✳

✶✶✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 29

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ❢ ∈ K[❳ (✶), . . . , ❳ (ℓ)] ✐s ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♦❢ ♠✉❧t✐✲❞❡❣r❡❡ (❞✶, . . . , ❞ℓ) ✐❢ ❢♦r ❛❧❧ λ✶, . . . , λℓ✱ ❢ (λ✶❳ (✶), . . . , λℓ❳ (ℓ)) = λ❞✶

✶ . . . λ❞ℓ ℓ ❢ (❳ (✶), . . . , ❳ (ℓ)).

❊①❛♠♣❧❡✿ ✸①✷

✶②✶ + ✹①✶①✷②✶ − ✸①✷ ✷②✶ − ①✷ ✶②✷ + ✽①✶①✷②✷ − ✺①✷ ✷②✷ + ✶✵①✷ ✶②✸ − ✷①✶①✷②✸ − ✸①✷ ✷②✸

✐s ❛ ❜✐✲❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ♦❢ ❜✐✲❞❡❣r❡❡ (✷, ✶) ✐♥ F✶✶[①✶, ①✷, ②✶, ②✷, ②✸]✳ ❇✐❧✐♥❡❛r s②st❡♠✿ ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♦❢ ♠✉❧t✐✲❞❡❣r❡❡ ✶ ✶ ❢✶ ❢q ❳ ❨ ✿ ❜✐❧✐♥❡❛r ❢♦r♠s✳

❢❦ ❛ ❦

✐ ❥ ①✐②❥✳

✶✶✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 30

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s

▼✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ❢ ∈ K[❳ (✶), . . . , ❳ (ℓ)] ✐s ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♦❢ ♠✉❧t✐✲❞❡❣r❡❡ (❞✶, . . . , ❞ℓ) ✐❢ ❢♦r ❛❧❧ λ✶, . . . , λℓ✱ ❢ (λ✶❳ (✶), . . . , λℓ❳ (ℓ)) = λ❞✶

✶ . . . λ❞ℓ ℓ ❢ (❳ (✶), . . . , ❳ (ℓ)).

❊①❛♠♣❧❡✿ ✸①✷

✐s ❛ ❜✐✲❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ♦❢ ❜✐✲❞❡❣r❡❡ (✷, ✶) ✐♥ F✶✶[①✶, ①✷, ②✶, ②✷, ②✸]✳ ❇✐❧✐♥❡❛r s②st❡♠✿ ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s ♦❢ ♠✉❧t✐✲❞❡❣r❡❡ (✶, ✶) ❢✶, . . . , ❢q ∈ K[❳, ❨ ]✿ ❜✐❧✐♥❡❛r ❢♦r♠s✳

❢❦ = ❛(❦)

✐,❥ ①✐②❥✳

✶✶✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 31

❙tr✉❝t✉r❡ ♦❢ ❜✐❧✐♥❡❛r s②st❡♠s

❊✉❧❡r r❡❧❛t✐♦♥s ❢✶, . . . , ❢q ∈ K[❳, ❨ ]✿ ❜✐❧✐♥❡❛r ❢♦r♠s✳ ❢❦ = ❛(❦)

✐,❥ ①✐②❥✳

❢❦

✐

❢❦ ①✐ ①✐

❥

❢❦ ②❥ ②❥ ❥❛❝① ❋

❢✶ ①✶ ❢✶ ①♥①

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❢q ①✶ ❢q ①♥①

❥❛❝② ❋

❢✶ ②✶ ❢✶ ②♥②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❢q ②✶ ❢q ②♥②

❢✶ ✳ ✳ ✳ ❢q ❥❛❝① ❋ ①✶ ✳ ✳ ✳ ①♥① ❥❛❝② ❋ ②✶ ✳ ✳ ✳ ②♥②

✶✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 32

❙tr✉❝t✉r❡ ♦❢ ❜✐❧✐♥❡❛r s②st❡♠s

❊✉❧❡r r❡❧❛t✐♦♥s ❢✶, . . . , ❢q ∈ K[❳, ❨ ]✿ ❜✐❧✐♥❡❛r ❢♦r♠s✳ ❢❦ = ❛(❦)

✐,❥ ①✐②❥✳

❢❦ =

∂❢❦ ∂①✐ ①✐ =

∂❢❦ ∂②❥ ②❥. ❥❛❝① ❋

❢✶ ①✶ ❢✶ ①♥①

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❢q ①✶ ❢q ①♥①

❥❛❝② ❋

❢✶ ②✶ ❢✶ ②♥②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❢q ②✶ ❢q ②♥②

❢✶ ✳ ✳ ✳ ❢q ❥❛❝① ❋ ①✶ ✳ ✳ ✳ ①♥① ❥❛❝② ❋ ②✶ ✳ ✳ ✳ ②♥②

✶✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 33

❙tr✉❝t✉r❡ ♦❢ ❜✐❧✐♥❡❛r s②st❡♠s

❊✉❧❡r r❡❧❛t✐♦♥s ❢✶, . . . , ❢q ∈ K[❳, ❨ ]✿ ❜✐❧✐♥❡❛r ❢♦r♠s✳ ❢❦ = ❛(❦)

✐,❥ ①✐②❥✳

❢❦ =

∂❢❦ ∂①✐ ①✐ =

∂❢❦ ∂②❥ ②❥. ❥❛❝①(❋) =    

∂❢✶ ∂①✶

. . .

∂❢✶ ∂①♥①

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

∂❢q ∂①✶

. . .

∂❢q ∂①♥①

    ❥❛❝②(❋) =    

∂❢✶ ∂②✶

. . .

∂❢✶ ∂②♥②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

∂❢q ∂②✶

. . .

∂❢q ∂②♥②

   .

= ⇒    ❢✶ ✳ ✳ ✳ ❢q    = ❥❛❝①(❋) ·    ①✶ ✳ ✳ ✳ ①♥①    = ❥❛❝②(❋) ·    ②✶ ✳ ✳ ✳ ②♥②   .

✶✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 34

❙♦♠❡t❤✐♥❣ s♣❡❝✐❛❧ ❤❛♣♣❡♥s ✇✐t❤ ♠✐♥♦rs✳✳✳

   ❢✶ ✳ ✳ ✳ ❢q    = ❥❛❝①(❋) ·    ①✶ ✳ ✳ ✳ ①♥①   . ■❢ (①✶, . . . , ①♥① , ②✶, . . . , ②♥② ) ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ ❋✱ t❤❡♥ ❥❛❝①(❋) ✐s r❛♥❦ ❞❡❢❡❝t✐✈❡✳ (②✶, . . . , ②♥② ) ✐s ❛ ③❡r♦ ♦❢ t❤❡ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ❥❛❝①(❋)✳ ❇❡r♥st❡✐♥✴❙t✉r♠❢❡❧s✴❩❡❧❡✈✐♥s❦✐✱ ❆❞✈✳ ✐♥ ▼❛t❤✳ ✶✾✾✸ ▼ ❛ ♣ q ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s✳ ❋♦r ❛♥② ♠♦♥♦♠✐❛❧ ♦r❞❡r✐♥❣✱ t❤❡ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ▼ ❛r❡ ❛ ●rö❜♥❡r ❜❛s✐s ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ✐❞❡❛❧✳ ❋❛✉❣èr❡✴❙❛❢❡② ❊❧ ❉✐♥✴❙✳✱ ❏✳ ♦❢ ❙②♠❜✳ ❈♦♠♣✳ ✷✵✶✶ ▼ ❛ ❦✲✈❛r✐❛t❡ q ♣ ❧✐♥❡❛r ♠❛tr✐① ✭✇✐t❤ q ♣✮✳ ●❡♥❡r✐❝❛❧❧②✱ ❛ ❣r❡✈❧❡① ●❇ ♦❢ ▼✐♥♦rs ▼ ✿ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛t♦rs✳

❞r❡❣ ▼❛①▼✐♥♦rs ❥❛❝① ❋ ♥①

✶✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 35

❙♦♠❡t❤✐♥❣ s♣❡❝✐❛❧ ❤❛♣♣❡♥s ✇✐t❤ ♠✐♥♦rs✳✳✳

   ❢✶ ✳ ✳ ✳ ❢q    = ❥❛❝①(❋) ·    ①✶ ✳ ✳ ✳ ①♥①   . ■❢ (①✶, . . . , ①♥① , ②✶, . . . , ②♥② ) ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ ❋✱ t❤❡♥ ❥❛❝①(❋) ✐s r❛♥❦ ❞❡❢❡❝t✐✈❡✳ (②✶, . . . , ②♥② ) ✐s ❛ ③❡r♦ ♦❢ t❤❡ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ❥❛❝①(❋)✳ ❇❡r♥st❡✐♥✴❙t✉r♠❢❡❧s✴❩❡❧❡✈✐♥s❦✐✱ ❆❞✈✳ ✐♥ ▼❛t❤✳ ✶✾✾✸ ▼ ❛ ♣ × q ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s✳ ❋♦r ❛♥② ♠♦♥♦♠✐❛❧ ♦r❞❡r✐♥❣✱ t❤❡ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ▼ ❛r❡ ❛ ●rö❜♥❡r ❜❛s✐s ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ✐❞❡❛❧✳ ❋❛✉❣èr❡✴❙❛❢❡② ❊❧ ❉✐♥✴❙✳✱ ❏✳ ♦❢ ❙②♠❜✳ ❈♦♠♣✳ ✷✵✶✶ ▼ ❛ ❦✲✈❛r✐❛t❡ q ♣ ❧✐♥❡❛r ♠❛tr✐① ✭✇✐t❤ q ♣✮✳ ●❡♥❡r✐❝❛❧❧②✱ ❛ ❣r❡✈❧❡① ●❇ ♦❢ ▼✐♥♦rs ▼ ✿ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛t♦rs✳

❞r❡❣ ▼❛①▼✐♥♦rs ❥❛❝① ❋ ♥①

✶✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 36

❙♦♠❡t❤✐♥❣ s♣❡❝✐❛❧ ❤❛♣♣❡♥s ✇✐t❤ ♠✐♥♦rs✳✳✳

   ❢✶ ✳ ✳ ✳ ❢q    = ❥❛❝①(❋) ·    ①✶ ✳ ✳ ✳ ①♥①   . ■❢ (①✶, . . . , ①♥① , ②✶, . . . , ②♥② ) ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ ❋✱ t❤❡♥ ❥❛❝①(❋) ✐s r❛♥❦ ❞❡❢❡❝t✐✈❡✳ (②✶, . . . , ②♥② ) ✐s ❛ ③❡r♦ ♦❢ t❤❡ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ❥❛❝①(❋)✳ ❇❡r♥st❡✐♥✴❙t✉r♠❢❡❧s✴❩❡❧❡✈✐♥s❦✐✱ ❆❞✈✳ ✐♥ ▼❛t❤✳ ✶✾✾✸ ▼ ❛ ♣ × q ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s✳ ❋♦r ❛♥② ♠♦♥♦♠✐❛❧ ♦r❞❡r✐♥❣✱ t❤❡ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ▼ ❛r❡ ❛ ●rö❜♥❡r ❜❛s✐s ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ✐❞❡❛❧✳ ❋❛✉❣èr❡✴❙❛❢❡② ❊❧ ❉✐♥✴❙✳✱ ❏✳ ♦❢ ❙②♠❜✳ ❈♦♠♣✳ ✷✵✶✶ ▼ ❛ ❦✲✈❛r✐❛t❡ q × ♣ ❧✐♥❡❛r ♠❛tr✐① ✭✇✐t❤ q > ♣✮✳ ●❡♥❡r✐❝❛❧❧②✱ ❛ ❣r❡✈❧❡① ●❇ ♦❢ ▼✐♥♦rs(▼)✿ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛t♦rs✳

❞r❡❣ (▼❛①▼✐♥♦rs(❥❛❝①(❋))) = ♥①.

✶✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 37

❈♦♠♣❧❡①✐t②

❆✣♥❡ ❜✐❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ❢ ∈ K[①✶, . . . , ①♥① , ②✶, . . . , ②♥② ] ✐s s❛✐❞ t♦ ❜❡ ❛✣♥❡ ❜✐❧✐♥❡❛r ✐❢ t❤❡r❡ ❡①✐sts ❛ ❜✐❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ˜ ❢ ✐♥ K[①✵, . . . , ①♥① , ②✵, . . . , ②♥② ] s✉❝❤ t❤❛t ❢ (①✶, . . . , ①♥① , ②✶, . . . , ②♥② ) = ˜ ❢ (✶, ①✶, . . . , ①♥① , ✶, ②✶, . . . , ②♥② ). ❋❛✉❣èr❡✴❙❛❢❡② ❊❧ ❉✐♥✴❙✳✱ ❏✳ ♦❢ ❙②♠❜✳ ❈♦♠♣✳ ✷✵✶✶ ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ▲❡t ❢✶, . . . , ❢♥① +♥② ❜❡ ❛♥ ❛✣♥❡ ❜✐❧✐♥❡❛r s②st❡♠ ✐♥ K[①✶, . . . , ①♥① , ②✶, . . . , ②♥② ]✳ ❚❤❡♥ t❤❡ ❤✐❣❤❡st ❞❡❣r❡❡ r❡❛❝❤❡❞ ❞✉r✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ ●rö❜♥❡r ❜❛s✐s ❢♦r t❤❡ ❣r❡✈❧❡① ♦r❞❡r✐♥❣ ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ♠✐♥(♥①, ♥②) + ✶ ≪ ♥① + ♥② + ✶. ❈♦♥s❡q✉❡♥❝❡s ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❝♦♠♣✉t✐♥❣ ❛ ❣r❡✈❧❡① ●❇ ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ✦✦ ❇✐❧✐♥❡❛r s②st❡♠s ✇✐t❤ ✉♥❜❛❧❛♥❝❡❞ s✐③❡s ♦❢ ❜❧♦❝❦s ♦❢ ✈❛r✐❛❜❧❡s ❛r❡ ❡❛s② t♦ s♦❧✈❡ ✦✦

✶✹✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 38

❈♦♠♣❧❡①✐t②

❆✣♥❡ ❜✐❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ❢ ∈ K[①✶, . . . , ①♥① , ②✶, . . . , ②♥② ] ✐s s❛✐❞ t♦ ❜❡ ❛✣♥❡ ❜✐❧✐♥❡❛r ✐❢ t❤❡r❡ ❡①✐sts ❛ ❜✐❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ˜ ❢ ✐♥ K[①✵, . . . , ①♥① , ②✵, . . . , ②♥② ] s✉❝❤ t❤❛t ❢ (①✶, . . . , ①♥① , ②✶, . . . , ②♥② ) = ˜ ❢ (✶, ①✶, . . . , ①♥① , ✶, ②✶, . . . , ②♥② ). ❋❛✉❣èr❡✴❙❛❢❡② ❊❧ ❉✐♥✴❙✳✱ ❏✳ ♦❢ ❙②♠❜✳ ❈♦♠♣✳ ✷✵✶✶ ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ▲❡t ❢✶, . . . , ❢♥① +♥② ❜❡ ❛♥ ❛✣♥❡ ❜✐❧✐♥❡❛r s②st❡♠ ✐♥ K[①✶, . . . , ①♥① , ②✶, . . . , ②♥② ]✳ ❚❤❡♥ t❤❡ ❤✐❣❤❡st ❞❡❣r❡❡ r❡❛❝❤❡❞ ❞✉r✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ ●rö❜♥❡r ❜❛s✐s ❢♦r t❤❡ ❣r❡✈❧❡① ♦r❞❡r✐♥❣ ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ♠✐♥(♥①, ♥②) + ✶ ≪ ♥① + ♥② + ✶. ❈♦♥s❡q✉❡♥❝❡s ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❝♦♠♣✉t✐♥❣ ❛ ❣r❡✈❧❡① ●❇ ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ✦✦ ❇✐❧✐♥❡❛r s②st❡♠s ✇✐t❤ ✉♥❜❛❧❛♥❝❡❞ s✐③❡s ♦❢ ❜❧♦❝❦s ♦❢ ✈❛r✐❛❜❧❡s ❛r❡ ❡❛s② t♦ s♦❧✈❡ ✦✦

✶✹✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 39

▼♦❞❡❧✐♥❣ ♦❢ ▼❝❊❧✐❡❝❡ ❝r②♣t♦s②st❡♠

❇❛s❡❞ ♦♥ ❛❧t❡r♥❛♥t ❝♦❞❡s✿ s❡❝r❡t ❦❡②✿ ❛ ♣❛r✐t②✲❝❤❡❝❦ ♠❛tr✐① ♦❢ t❤❡ ❢♦r♠ ❍ =      ②✵ ②✶ . . . ②♥−✶ ②✵①✵ ②✶①✶ . . . ②♥−✶①♥−✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ②✵①t−✶

✵

②✶①t−✶

✶

. . . ②♥①t−✶

♥

     , ✇❤❡r❡ ①✐, ②❥ ∈ F✷♠✱ ✇✐t❤ ①✵, . . . , ①♥ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ❛♥❞ ②❥ = ✵✳ ♣✉❜❧✐❝ ❦❡②✿ ❛ ❣❡♥❡r❛t♦r ♠❛tr✐① ● ♦❢ t❤❡ s❛♠❡ ❝♦❞❡✳ Pr♦❜❧❡♠

✐✈❡♥ ●✱ ✜♥❞ ❍ s✉❝❤ t❤❛t ❍
t

✵ ✦ ✐ ❥ ❣✐ ✵②✵①❥

✵

❣✐ ♥

✶②♥ ✶①❥ ♥ ✶

✵ ❇✐✲❤♦♠♦❣❡♥❡♦✉s str✉❝t✉r❡ ✦✦

✶✺✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 40

▼♦❞❡❧✐♥❣ ♦❢ ▼❝❊❧✐❡❝❡ ❝r②♣t♦s②st❡♠

❇❛s❡❞ ♦♥ ❛❧t❡r♥❛♥t ❝♦❞❡s✿ s❡❝r❡t ❦❡②✿ ❛ ♣❛r✐t②✲❝❤❡❝❦ ♠❛tr✐① ♦❢ t❤❡ ❢♦r♠ ❍ =      ②✵ ②✶ . . . ②♥−✶ ②✵①✵ ②✶①✶ . . . ②♥−✶①♥−✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ②✵①t−✶

✵

②✶①t−✶

✶

. . . ②♥①t−✶

♥

     , ✇❤❡r❡ ①✐, ②❥ ∈ F✷♠✱ ✇✐t❤ ①✵, . . . , ①♥ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ❛♥❞ ②❥ = ✵✳ ♣✉❜❧✐❝ ❦❡②✿ ❛ ❣❡♥❡r❛t♦r ♠❛tr✐① ● ♦❢ t❤❡ s❛♠❡ ❝♦❞❡✳ Pr♦❜❧❡♠

✐✈❡♥ ●✱ ✜♥❞ ❍ s✉❝❤ t❤❛t ❍ · ● t = ✵ ✦

✐ ❥ ❣✐ ✵②✵①❥

✵

❣✐ ♥

✶②♥ ✶①❥ ♥ ✶

✵ ❇✐✲❤♦♠♦❣❡♥❡♦✉s str✉❝t✉r❡ ✦✦

✶✺✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 41

▼♦❞❡❧✐♥❣ ♦❢ ▼❝❊❧✐❡❝❡ ❝r②♣t♦s②st❡♠

❇❛s❡❞ ♦♥ ❛❧t❡r♥❛♥t ❝♦❞❡s✿ s❡❝r❡t ❦❡②✿ ❛ ♣❛r✐t②✲❝❤❡❝❦ ♠❛tr✐① ♦❢ t❤❡ ❢♦r♠ ❍ =      ②✵ ②✶ . . . ②♥−✶ ②✵①✵ ②✶①✶ . . . ②♥−✶①♥−✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ②✵①t−✶

✵

②✶①t−✶

✶

. . . ②♥①t−✶

♥

     , ✇❤❡r❡ ①✐, ②❥ ∈ F✷♠✱ ✇✐t❤ ①✵, . . . , ①♥ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ❛♥❞ ②❥ = ✵✳ ♣✉❜❧✐❝ ❦❡②✿ ❛ ❣❡♥❡r❛t♦r ♠❛tr✐① ● ♦❢ t❤❡ s❛♠❡ ❝♦❞❡✳ Pr♦❜❧❡♠

✐✈❡♥ ●✱ ✜♥❞ ❍ s✉❝❤ t❤❛t ❍ · ● t = ✵ ✦

∀✐, ❥, ❣✐,✵②✵①❥

✵ + · · · + ❣✐,♥−✶②♥−✶①❥ ♥−✶ = ✵.

⇒ ❇✐✲❤♦♠♦❣❡♥❡♦✉s str✉❝t✉r❡ ✦✦

✶✺✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 42

❈r②♣t❛♥❛❧②s✐s ♦❢ ❝♦♠♣❛❝t ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡

❈♦♠♣❛❝t ✈❛r✐❛♥ts

♦❛❧✿ r❡❞✉❝❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❦❡②s✳

◗✉❛s✐✲❝②❝❧✐❝ ✈❛r✐❛♥t✿ ❇❡r❣❡r✴❈❛②r❡❧✴●❛❜♦r✐t✴❖t♠❛♥✐ ❆❢r✐❝❛❝r②♣t✬✵✾❀ ❉②❛❞✐❝ ✈❛r✐❛♥t✿ ▼✐s♦❝③②✴❇❛rr❡t♦ ❙❆❈✬✵✾✳

❋❛✉❣èr❡✴❖t♠❛♥✐✴P❡rr❡t✴❚✐❧✐❝❤✱ ❊✉r♦❝r②♣t✬✷✵✶✵

❛❞❞ r❡❞✉♥❞❛♥❝② t♦ t❤❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❧❡ss ✈❛r✐❛❜❧❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ s②st❡♠ ✐s st✐❧❧ ♦✈❡r✲❞❡t❡r♠✐♥❡❞ ❛♥❞ ♦♥❡ ❝❛♥ ❡①tr❛❝t ❛ s✉❜s②st❡♠ ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♣♦✇❡rs ♦❢ t✇♦✿ ✐ ❥ ❛ ♣♦✇❡r ♦❢ t✇♦ ✦✦ ❣✐ ✵②✵①❥

✵

❣✐ ♥

✶②♥ ✶①❥ ♥ ✶

✵ ❉❡❝♦♠♣♦s✐♥❣ t❤❡ s✉❜s②st❡♠ ♦✈❡r t❤❡ ✜❡❧❞

✷

❇✐❧✐♥❡❛r s②st❡♠ ✇✐t❤ ♥① ♥② ✦✦✦

❚❤❡♦r❡t✐❝❛❧ ❛♥❞ Pr❛❝t✐❝❛❧ ❛tt❛❝❦s ♦♥ t❤❡ q✉❛s✐✲❝②❝❧✐❝ ❛♥❞ ❞②❛❞✐❝ ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡ ✦✦

✶✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 43

❈r②♣t❛♥❛❧②s✐s ♦❢ ❝♦♠♣❛❝t ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡

❈♦♠♣❛❝t ✈❛r✐❛♥ts

♦❛❧✿ r❡❞✉❝❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❦❡②s✳

◗✉❛s✐✲❝②❝❧✐❝ ✈❛r✐❛♥t✿ ❇❡r❣❡r✴❈❛②r❡❧✴●❛❜♦r✐t✴❖t♠❛♥✐ ❆❢r✐❝❛❝r②♣t✬✵✾❀ ❉②❛❞✐❝ ✈❛r✐❛♥t✿ ▼✐s♦❝③②✴❇❛rr❡t♦ ❙❆❈✬✵✾✳

❋❛✉❣èr❡✴❖t♠❛♥✐✴P❡rr❡t✴❚✐❧✐❝❤✱ ❊✉r♦❝r②♣t✬✷✵✶✵

⇒ ❛❞❞ r❡❞✉♥❞❛♥❝② t♦ t❤❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❧❡ss ✈❛r✐❛❜❧❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ s②st❡♠ ✐s st✐❧❧ ♦✈❡r✲❞❡t❡r♠✐♥❡❞ ❛♥❞ ♦♥❡ ❝❛♥ ❡①tr❛❝t ❛ s✉❜s②st❡♠ ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♣♦✇❡rs ♦❢ t✇♦✿ ✐ ❥ ❛ ♣♦✇❡r ♦❢ t✇♦ ✦✦ ❣✐ ✵②✵①❥

✵

❣✐ ♥

✶②♥ ✶①❥ ♥ ✶

✵ ❉❡❝♦♠♣♦s✐♥❣ t❤❡ s✉❜s②st❡♠ ♦✈❡r t❤❡ ✜❡❧❞

✷

❇✐❧✐♥❡❛r s②st❡♠ ✇✐t❤ ♥① ♥② ✦✦✦

❚❤❡♦r❡t✐❝❛❧ ❛♥❞ Pr❛❝t✐❝❛❧ ❛tt❛❝❦s ♦♥ t❤❡ q✉❛s✐✲❝②❝❧✐❝ ❛♥❞ ❞②❛❞✐❝ ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡ ✦✦

✶✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 44

❈r②♣t❛♥❛❧②s✐s ♦❢ ❝♦♠♣❛❝t ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡

❈♦♠♣❛❝t ✈❛r✐❛♥ts

♦❛❧✿ r❡❞✉❝❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❦❡②s✳

◗✉❛s✐✲❝②❝❧✐❝ ✈❛r✐❛♥t✿ ❇❡r❣❡r✴❈❛②r❡❧✴●❛❜♦r✐t✴❖t♠❛♥✐ ❆❢r✐❝❛❝r②♣t✬✵✾❀ ❉②❛❞✐❝ ✈❛r✐❛♥t✿ ▼✐s♦❝③②✴❇❛rr❡t♦ ❙❆❈✬✵✾✳

❋❛✉❣èr❡✴❖t♠❛♥✐✴P❡rr❡t✴❚✐❧✐❝❤✱ ❊✉r♦❝r②♣t✬✷✵✶✵

⇒ ❛❞❞ r❡❞✉♥❞❛♥❝② t♦ t❤❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❧❡ss ✈❛r✐❛❜❧❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ s②st❡♠ ✐s st✐❧❧ ♦✈❡r✲❞❡t❡r♠✐♥❡❞ ❛♥❞ ♦♥❡ ❝❛♥ ❡①tr❛❝t ❛ s✉❜s②st❡♠ ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♣♦✇❡rs ♦❢ t✇♦✿ ∀✐, ❥ ❛ ♣♦✇❡r ♦❢ t✇♦ ✦✦, ❣✐,✵②✵①❥

✵ + · · · + ❣✐,♥−✶②♥−✶①❥ ♥−✶ = ✵.

❉❡❝♦♠♣♦s✐♥❣ t❤❡ s✉❜s②st❡♠ ♦✈❡r t❤❡ ✜❡❧❞

✷

❇✐❧✐♥❡❛r s②st❡♠ ✇✐t❤ ♥① ♥② ✦✦✦

❚❤❡♦r❡t✐❝❛❧ ❛♥❞ Pr❛❝t✐❝❛❧ ❛tt❛❝❦s ♦♥ t❤❡ q✉❛s✐✲❝②❝❧✐❝ ❛♥❞ ❞②❛❞✐❝ ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡ ✦✦

✶✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 45

❈r②♣t❛♥❛❧②s✐s ♦❢ ❝♦♠♣❛❝t ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡

❈♦♠♣❛❝t ✈❛r✐❛♥ts

♦❛❧✿ r❡❞✉❝❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❦❡②s✳

◗✉❛s✐✲❝②❝❧✐❝ ✈❛r✐❛♥t✿ ❇❡r❣❡r✴❈❛②r❡❧✴●❛❜♦r✐t✴❖t♠❛♥✐ ❆❢r✐❝❛❝r②♣t✬✵✾❀ ❉②❛❞✐❝ ✈❛r✐❛♥t✿ ▼✐s♦❝③②✴❇❛rr❡t♦ ❙❆❈✬✵✾✳

❋❛✉❣èr❡✴❖t♠❛♥✐✴P❡rr❡t✴❚✐❧✐❝❤✱ ❊✉r♦❝r②♣t✬✷✵✶✵

⇒ ❛❞❞ r❡❞✉♥❞❛♥❝② t♦ t❤❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❧❡ss ✈❛r✐❛❜❧❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ s②st❡♠ ✐s st✐❧❧ ♦✈❡r✲❞❡t❡r♠✐♥❡❞ ❛♥❞ ♦♥❡ ❝❛♥ ❡①tr❛❝t ❛ s✉❜s②st❡♠ ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♣♦✇❡rs ♦❢ t✇♦✿ ∀✐, ❥ ❛ ♣♦✇❡r ♦❢ t✇♦ ✦✦, ❣✐,✵②✵①❥

✵ + · · · + ❣✐,♥−✶②♥−✶①❥ ♥−✶ = ✵.

❉❡❝♦♠♣♦s✐♥❣ t❤❡ s✉❜s②st❡♠ ♦✈❡r t❤❡ ✜❡❧❞ F✷

⇒ ❇✐❧✐♥❡❛r s②st❡♠ ✇✐t❤ ♥① ≪ ♥② ✦✦✦

❚❤❡♦r❡t✐❝❛❧ ❛♥❞ Pr❛❝t✐❝❛❧ ❛tt❛❝❦s ♦♥ t❤❡ q✉❛s✐✲❝②❝❧✐❝ ❛♥❞ ❞②❛❞✐❝ ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡ ✦✦

✶✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 46

❈r②♣t❛♥❛❧②s✐s ♦❢ ❝♦♠♣❛❝t ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡

❈♦♠♣❛❝t ✈❛r✐❛♥ts

♦❛❧✿ r❡❞✉❝❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❦❡②s✳

◗✉❛s✐✲❝②❝❧✐❝ ✈❛r✐❛♥t✿ ❇❡r❣❡r✴❈❛②r❡❧✴●❛❜♦r✐t✴❖t♠❛♥✐ ❆❢r✐❝❛❝r②♣t✬✵✾❀ ❉②❛❞✐❝ ✈❛r✐❛♥t✿ ▼✐s♦❝③②✴❇❛rr❡t♦ ❙❆❈✬✵✾✳

❋❛✉❣èr❡✴❖t♠❛♥✐✴P❡rr❡t✴❚✐❧✐❝❤✱ ❊✉r♦❝r②♣t✬✷✵✶✵

⇒ ❛❞❞ r❡❞✉♥❞❛♥❝② t♦ t❤❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❧❡ss ✈❛r✐❛❜❧❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ s②st❡♠ ✐s st✐❧❧ ♦✈❡r✲❞❡t❡r♠✐♥❡❞ ❛♥❞ ♦♥❡ ❝❛♥ ❡①tr❛❝t ❛ s✉❜s②st❡♠ ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♣♦✇❡rs ♦❢ t✇♦✿ ∀✐, ❥ ❛ ♣♦✇❡r ♦❢ t✇♦ ✦✦, ❣✐,✵②✵①❥

✵ + · · · + ❣✐,♥−✶②♥−✶①❥ ♥−✶ = ✵.

❉❡❝♦♠♣♦s✐♥❣ t❤❡ s✉❜s②st❡♠ ♦✈❡r t❤❡ ✜❡❧❞ F✷

⇒ ❇✐❧✐♥❡❛r s②st❡♠ ✇✐t❤ ♥① ≪ ♥② ✦✦✦

❚❤❡♦r❡t✐❝❛❧ ❛♥❞ Pr❛❝t✐❝❛❧ ❛tt❛❝❦s ♦♥ t❤❡ q✉❛s✐✲❝②❝❧✐❝ ❛♥❞ ❞②❛❞✐❝ ✈❛r✐❛♥ts ♦❢ ▼❝❊❧✐❡❝❡ ✦✦

✶✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 47

P❧❛♥

✶ P♦❧②♥♦♠✐❛❧ ❙②st❡♠ ❙♦❧✈✐♥❣ ✉s✐♥❣ ●rö❜♥❡r ❇❛s❡s ✷ ❇✐❧✐♥❡❛r ❙②st❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❝❊❧✐❡❝❡ ✸ ❉❡t❡r♠✐♥❛♥t❛❧ ❙②st❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s t♦ ▼✐♥❘❛♥❦ ❛♥❞ ❍❋❊

✶✼✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 48

❚❤❡ ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠

r ∈ N✳ ▼✵, . . . , ▼❦✿ ❦ + ✶ ♠❛tr✐❝❡s ♦❢ s✐③❡ ♠ × ♠✳ ▼✐♥❘❛♥❦ ✜♥❞ λ✶, . . . , λ❦ s✉❝❤ t❤❛t ❘❛♥❦

▼✵ −

❦

✐=✶

λ✐▼✐

≤ r.

▼✉❧t✐✈❛r✐❛t❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❊✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❝r②♣t♦❧♦❣②✱ ❝♦❞✐♥❣ t❤❡♦r②✱ ✳✳✳ ❑✐♣♥✐s✴❙❤❛♠✐r ❈r②♣t♦✬✾✾✱ ❈♦✉rt♦✐s ❆s✐❛❝r②♣t✬✵✶ ❋❛✉❣èr❡✴▲❡✈②✲❞✐t✲❱❡❤❡❧✴P❡rr❡t ❈r②♣t♦✬✵✽✱✳✳✳ ❋✉♥❞❛♠❡♥t❛❧ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ❇✉ss✱ ❋r❛♥❞s❡♥✱ ❙❤❛❧❧✐t✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t② ♦❢ s♦♠❡ ♣r♦❜❧❡♠s ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✳

✶✽✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 49

❚✇♦ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣s

▼ = ▼✵ −

❦

✐=✶

λ✐▼✐. ❚❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣

❘❛♥❦ ▼ r

❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ r ✶ ♦❢ ▼ ✈❛♥✐s❤✳

♠ r ✶ ✷ ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ r

✶✳ ❦ ✈❛r✐❛❜❧❡s✳ ❋❡✇ ✈❛r✐❛❜❧❡s✱ ❧♦ts ♦❢ ❡q✉❛t✐♦♥s✱ ❤✐❣❤ ❞❡❣r❡❡ ✦✦ ❚❤❡ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣

❘❛♥❦ ▼ r ① ✶ ① ♠

❑❡r ▼ ✳

▼ ■♠ r

① ✶

✶

① ♠

r ✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ① ✶

① ♠

r r

✵

♠ ♠ r ❜✐❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❦ r ♠ r ✈❛r✐❛❜❧❡s✳

❈♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ ▼✐♥❘❛♥❦ ✉s✐♥❣ ●rö❜♥❡r ❜❛s❡s t❡❝❤♥✐q✉❡s ❄ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ t✇♦ ♠♦❞❡❧✐♥❣s ❄ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❄

✶✾✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 50

❚✇♦ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣s

▼ = ▼✵ −

❦

✐=✶

λ✐▼✐. ❚❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣

❘❛♥❦(▼) ≤ r

❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ (r + ✶) ♦❢ ▼ ✈❛♥✐s❤✳

♠

r+✶

✷ ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ r + ✶✳ ❦ ✈❛r✐❛❜❧❡s✳ ❋❡✇ ✈❛r✐❛❜❧❡s✱ ❧♦ts ♦❢ ❡q✉❛t✐♦♥s✱ ❤✐❣❤ ❞❡❣r❡❡ ✦✦ ❚❤❡ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣

❘❛♥❦ ▼ r ① ✶ ① ♠

❑❡r ▼ ✳

▼ ■♠ r

① ✶

✶

① ♠

r ✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ① ✶

① ♠

r r

✵

♠ ♠ r ❜✐❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❦ r ♠ r ✈❛r✐❛❜❧❡s✳

❈♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ ▼✐♥❘❛♥❦ ✉s✐♥❣ ●rö❜♥❡r ❜❛s❡s t❡❝❤♥✐q✉❡s ❄ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ t✇♦ ♠♦❞❡❧✐♥❣s ❄ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❄

✶✾✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 51

❚✇♦ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣s

▼ = ▼✵ −

❦

✐=✶

λ✐▼✐. ❚❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣

❘❛♥❦(▼) ≤ r

❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ (r + ✶) ♦❢ ▼ ✈❛♥✐s❤✳

♠

r+✶

✷ ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ r + ✶✳ ❦ ✈❛r✐❛❜❧❡s✳ ❋❡✇ ✈❛r✐❛❜❧❡s✱ ❧♦ts ♦❢ ❡q✉❛t✐♦♥s✱ ❤✐❣❤ ❞❡❣r❡❡ ✦✦ ❚❤❡ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣

❘❛♥❦(▼) ≤ r ⇔ ∃①(✶), . . . , ①(♠−r) ∈ ❑❡r(▼)✳

▼ ·

        

■♠−r

①(✶)

✶

. . . ①(♠−r)

✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①(✶)

. . . ①(♠−r)

        

= ✵.

♠(♠ − r) ❜✐❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❦ + r(♠ − r) ✈❛r✐❛❜❧❡s✳

❈♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ ▼✐♥❘❛♥❦ ✉s✐♥❣ ●rö❜♥❡r ❜❛s❡s t❡❝❤♥✐q✉❡s ❄ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ t✇♦ ♠♦❞❡❧✐♥❣s ❄ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❄

✶✾✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 52

❚✇♦ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣s

▼ = ▼✵ −

❦

✐=✶

λ✐▼✐. ❚❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣

❘❛♥❦(▼) ≤ r

❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ (r + ✶) ♦❢ ▼ ✈❛♥✐s❤✳

♠

r+✶

✷ ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ r + ✶✳ ❦ ✈❛r✐❛❜❧❡s✳ ❋❡✇ ✈❛r✐❛❜❧❡s✱ ❧♦ts ♦❢ ❡q✉❛t✐♦♥s✱ ❤✐❣❤ ❞❡❣r❡❡ ✦✦ ❚❤❡ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣

❘❛♥❦(▼) ≤ r ⇔ ∃①(✶), . . . , ①(♠−r) ∈ ❑❡r(▼)✳

▼ ·

        

■♠−r

①(✶)

✶

. . . ①(♠−r)

✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①(✶)

. . . ①(♠−r)

        

= ✵.

♠(♠ − r) ❜✐❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❦ + r(♠ − r) ✈❛r✐❛❜❧❡s✳

❈♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ ▼✐♥❘❛♥❦ ✉s✐♥❣ ●rö❜♥❡r ❜❛s❡s t❡❝❤♥✐q✉❡s ❄ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ t✇♦ ♠♦❞❡❧✐♥❣s ❄ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❄

✶✾✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 53

▼❛✐♥ r❡s✉❧ts

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω)

♠✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r✿ t❛r❣❡t r❛♥❦✳ ❦ ♠ r ✷✳ ▼♦❞❡❧✐♥❣✿

▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②

✇❤❡♥ ❦ ♠ r ✷

r ♠ r ✶ ★ ❙♦❧ ❈♦♠♣❧❡①✐t② ❖ ♠

❦

❖ ♠

❦ ✶

❇♦t❤ ♠♦❞❡❧✐♥❣s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ ♠ r ✷ ✐s ✜①❡❞✳ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ ✶✶ t❛r❣❡t r❛♥❦ ✽✱

❋ ✻✺✺✷✶ ✳

❈♦✉rt♦✐s✱ ❆s✐❛❝r②♣t ✷✵✵✶✳

✷✵✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 54

▼❛✐♥ r❡s✉❧ts

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω)

♠✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r✿ t❛r❣❡t r❛♥❦✳ ❦ = (♠ − r)✷✳ ▼♦❞❡❧✐♥❣✿

▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②

✇❤❡♥ ❦ = (♠ − r)✷

r ♠ r ✶ ▼❛❝❛✉❧❛② ❜♦✉♥❞✿ ≤ ♠(♠ − r) + ✶ ★ ❙♦❧ ▼❍✳ ❇é③♦✉t✿ ≤

r ♠−r ❈♦♠♣❧❡①✐t② ❖ ♠

❦

❖ ♠

❦ ✶

❇♦t❤ ♠♦❞❡❧✐♥❣s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ ♠ r ✷ ✐s ✜①❡❞✳ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ ✶✶ t❛r❣❡t r❛♥❦ ✽✱

❋ ✻✺✺✷✶ ✳

❈♦✉rt♦✐s✱ ❆s✐❛❝r②♣t ✷✵✵✶✳

✷✵✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 55

▼❛✐♥ r❡s✉❧ts

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω)

♠✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r✿ t❛r❣❡t r❛♥❦✳ ❦ = (♠ − r)✷✳ ▼♦❞❡❧✐♥❣✿

▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②

✇❤❡♥ ❦ = (♠ − r)✷

r ♠ r ✶ ≤ (♠ − r)✷ + ✶ ★ ❙♦❧ ▼❍✳ ❇é③♦✉t✿ ≤

r ♠−r ❈♦♠♣❧❡①✐t② ❖ ♠

❦

❖ ♠

❦ ✶

❇♦t❤ ♠♦❞❡❧✐♥❣s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ ♠ r ✷ ✐s ✜①❡❞✳ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ ✶✶ t❛r❣❡t r❛♥❦ ✽✱

❋ ✻✺✺✷✶ ✳

❈♦✉rt♦✐s✱ ❆s✐❛❝r②♣t ✷✵✵✶✳

✷✵✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 56

▼❛✐♥ r❡s✉❧ts

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω)

♠✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r✿ t❛r❣❡t r❛♥❦✳ ❦ = (♠ − r)✷✳ ▼♦❞❡❧✐♥❣✿

▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②

✇❤❡♥ ❦ = (♠ − r)✷

r(♠ − r) + ✶ ≤ (♠ − r)✷ + ✶ ★ ❙♦❧ ▼❍✳ ❇é③♦✉t✿ ≤

r ♠−r ❈♦♠♣❧❡①✐t② ❖ ♠

❦

❖ ♠

❦ ✶

❇♦t❤ ♠♦❞❡❧✐♥❣s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ ♠ r ✷ ✐s ✜①❡❞✳ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ ✶✶ t❛r❣❡t r❛♥❦ ✽✱

❋ ✻✺✺✷✶ ✳

❈♦✉rt♦✐s✱ ❆s✐❛❝r②♣t ✷✵✵✶✳

✷✵✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 57

▼❛✐♥ r❡s✉❧ts

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω)

♠✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r✿ t❛r❣❡t r❛♥❦✳ ❦ = (♠ − r)✷✳ ▼♦❞❡❧✐♥❣✿

▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②

✇❤❡♥ ❦ = (♠ − r)✷

r(♠ − r) + ✶ ≤ (♠ − r)✷ + ✶ ★ ❙♦❧

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! ❈♦♠♣❧❡①✐t② ❖ ♠

❦

❖ ♠

❦ ✶

❇♦t❤ ♠♦❞❡❧✐♥❣s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ ♠ r ✷ ✐s ✜①❡❞✳ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ ✶✶ t❛r❣❡t r❛♥❦ ✽✱

❋ ✻✺✺✷✶ ✳

❈♦✉rt♦✐s✱ ❆s✐❛❝r②♣t ✷✵✵✶✳

✷✵✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 58

▼❛✐♥ r❡s✉❧ts

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω)

♠✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r✿ t❛r❣❡t r❛♥❦✳ ❦ = (♠ − r)✷✳ ▼♦❞❡❧✐♥❣✿

▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②

✇❤❡♥ ❦ = (♠ − r)✷

r(♠ − r) + ✶ ≤ (♠ − r)✷ + ✶ ★ ❙♦❧

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! ❈♦♠♣❧❡①✐t② ❖(♠ω❦) ❖(♠ω(❦+✶)) ❇♦t❤ ♠♦❞❡❧✐♥❣s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ ♠ r ✷ ✐s ✜①❡❞✳ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ ✶✶ t❛r❣❡t r❛♥❦ ✽✱

❋ ✻✺✺✷✶ ✳

❈♦✉rt♦✐s✱ ❆s✐❛❝r②♣t ✷✵✵✶✳

✷✵✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 59

▼❛✐♥ r❡s✉❧ts

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − → ❧❡① ●❇. ❆❧❣♦r✐t❤♠s ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ❈♦♠♣❧❡①✐t② ❖ ♥ + ❞r❡❣ ❞r❡❣ ω ❖ (♥ · #❙♦❧ω)

♠✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r✿ t❛r❣❡t r❛♥❦✳ ❦ = (♠ − r)✷✳ ▼♦❞❡❧✐♥❣✿

▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②

✇❤❡♥ ❦ = (♠ − r)✷

r(♠ − r) + ✶ ≤ (♠ − r)✷ + ✶ ★ ❙♦❧

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! ❈♦♠♣❧❡①✐t② ❖(♠ω❦) ❖(♠ω(❦+✶)) ❇♦t❤ ♠♦❞❡❧✐♥❣s → ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ = (♠ − r)✷ ✐s ✜①❡❞✳ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ × ✶✶ t❛r❣❡t r❛♥❦ ✽✱ K = ●❋(✻✺✺✷✶)✳ ❈♦✉rt♦✐s✱ ❆s✐❛❝r②♣t ✷✵✵✶✳

✷✵✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 60

▼✐♥♦rs ♠♦❞❡❧✐♥❣

▼✐♥♦rs ♠♦❞❡❧✐♥❣✿ ❘❛♥❦(▼) ≤ r

❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ (r + ✶) ♦❢ ▼ ✈❛♥✐s❤✳

❉❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧

❇✐❧✐♥❡❛r s②st❡♠s ❞❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❢✶ ❢q ❳ ❨ ✿ ❜✐❧✐♥❡❛r ❢♦r♠s✳

❢✶ ①✵ ❢✶ ①♥①

✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳

❢q ①✵ ❢q ①♥①

①✵ ✳ ✳ ✳ ①♥① ❢✶ ✳ ✳ ✳ ❢q ❢✶ ❢q ✵ ▼❛①▼✐♥♦rs ❏❛❝❳ ❢✶ ❢q ✵

✷✶✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 61

▼✐♥♦rs ♠♦❞❡❧✐♥❣

▼✐♥♦rs ♠♦❞❡❧✐♥❣✿ ❘❛♥❦(▼) ≤ r

❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ (r + ✶) ♦❢ ▼ ✈❛♥✐s❤✳

❉❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧

❇✐❧✐♥❡❛r s②st❡♠s ↔ ❞❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❢✶, . . . , ❢q ∈ K[❳, ❨ ]✿ ❜✐❧✐♥❡❛r ❢♦r♠s✳    

∂❢✶ ∂①✵

. . .

∂❢✶ ∂①♥①

✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳

∂❢q ∂①✵

. . .

∂❢q ∂①♥①

    ·    ①✵ ✳ ✳ ✳ ①♥①    =    ❢✶ ✳ ✳ ✳ ❢q    ❢✶ = . . . = ❢q = ✵ ⇐ ⇒ ▼❛①▼✐♥♦rs (❏❛❝❳(❢✶, . . . , ❢q)) = ✵.

✷✶✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 62

❉❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s

❲❤❛t ✐s ❦♥♦✇♥ ❉❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s✿ ❇❡r♥st❡✐♥✴❩❡❧❡✈✐♥s❦② ❏✳ ♦❢ ❆❧❣✳ ❈♦♠❜✳ ✾✸✱ ❇r✉♥s✴❈♦♥❝❛ ✾✽✱ ❙t✉r♠❢❡❧s✴❩❡❧❡✈✐♥s❦② ❆❞✈✳ ▼❛t❤✳ ✾✽✱ ❈♦♥❝❛✴❍❡r③♦❣ ❆▼❙✬✾✹✱ ▲❛s❝♦✉① ✼✽✱ ❆❜❤②❛♥❦❛r ✽✽✳✳✳

❡♦♠❡tr② ♦❢ ❞❡t❡r♠✐♥❛♥t❛❧ ✈❛r✐❡t✐❡s✿ ❘♦♦♠ ✸✾✱ ❋✉❧t♦♥ ❉✉❦❡ ▼❛t❤✳ ❏✳ ✾✶✱
✐✉st✐✴▼❡r❧❡ ■♥t✳ ❈♦♥❢✳ ♦♥ ❆❧❣✳ ●❡♦✳ ✽✷✳✳✳

P♦❧❛r ✈❛r✐❡t✐❡s✿ ❇❛♥❦✴●✐✉st✐✴❍❡✐♥t③✴❙❛❢❡②✴❙❝❤♦st ❆❆❊❈❈✬✶✵✱❇❛♥❦✴●✐✉st✐✴❍❡✐♥t③✴P❛r❞♦ ❏✳ ♦❢ ❈♦♠♣❧✳ ✵✺✱ ❙❛❢❡②✴❙❝❤♦st ■❙❙❆❈✬✵✸✱ ❚❡✐ss✐❡r P✉r❡ ❛♥❞ ❆♣♣❧✳ ▼❛t❤✳ ✾✶✳✳✳

✷✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 63

Pr♦♣❡rt✐❡s ♦❢ ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s

D = ▼✐♥♦rsr+✶    ✈✶,✶ . . . ✈✶,♠ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♠,✶ . . . ✈♠,♠    ❚❤♦♠✱ P♦rt❡♦✉s✱ ●✐❛♠❜❡❧❧✐✱ ❍❛rr✐s✲❚✉✱ ✳✳✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! . ❈♦♥❝❛✴❍❡r③♦❣✱ ❆❜❤②❛♥❦❛r ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙D(t) = ❞❡t(❆(t)) t

✷

(✶ − t)(✷♠−r)r

. ▼✐♥♦rsr

✶

❢✶ ✶ ❢✶ ♠ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♠ ✶ ❢♠ ♠ ■❙❙❆❈✬✷✵✶✵ ❚❤❡ ❞❡❣r❡❡ ♦❢ ✐s

♠ r ✶ ✐ ✵

✐ ♠ ✐ ♠ ✶ ✐ ♠ r ✐ ■❙❙❆❈✬✷✵✶✵ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ✐s ❍❙ t ❞❡t ❆ t t

r ✷

✶ t ❦

♠ r ✷

❆✐,❥(t) =

♠ − ✐ ℓ ♠ − ❥ ℓ

tℓ.

tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ ❜② ❛❞❞✐♥❣ ✈✐ ❥ ❢✐ ❥

✷✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 64

Pr♦♣❡rt✐❡s ♦❢ ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s

D = ▼✐♥♦rsr+✶    ✈✶,✶ . . . ✈✶,♠ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♠,✶ . . . ✈♠,♠    ❚❤♦♠✱ P♦rt❡♦✉s✱ ●✐❛♠❜❡❧❧✐✱ ❍❛rr✐s✲❚✉✱ ✳✳✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! . ❈♦♥❝❛✴❍❡r③♦❣✱ ❆❜❤②❛♥❦❛r ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙D(t) = ❞❡t(❆(t)) t

✷

(✶ − t)(✷♠−r)r

. I = ▼✐♥♦rsr+✶    ❢✶,✶ . . . ❢✶,♠ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♠,✶ . . . ❢♠,♠    ■❙❙❆❈✬✷✵✶✵ ❚❤❡ ❞❡❣r❡❡ ♦❢ ✐s

♠ r ✶ ✐ ✵

✐ ♠ ✐ ♠ ✶ ✐ ♠ r ✐ ■❙❙❆❈✬✷✵✶✵ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ✐s ❍❙ t ❞❡t ❆ t t

r ✷

✶ t ❦

♠ r ✷

❆✐,❥(t) =

♠ − ✐ ℓ ♠ − ❥ ℓ

tℓ.

tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ ❜② ❛❞❞✐♥❣ ✈✐ ❥ ❢✐ ❥

✷✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 65

Pr♦♣❡rt✐❡s ♦❢ ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s

D = ▼✐♥♦rsr+✶    ✈✶,✶ . . . ✈✶,♠ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♠,✶ . . . ✈♠,♠    ❚❤♦♠✱ P♦rt❡♦✉s✱ ●✐❛♠❜❡❧❧✐✱ ❍❛rr✐s✲❚✉✱ ✳✳✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! . ❈♦♥❝❛✴❍❡r③♦❣✱ ❆❜❤②❛♥❦❛r ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙D(t) = ❞❡t(❆(t)) t

✷

(✶ − t)(✷♠−r)r

. I = ▼✐♥♦rsr+✶    ❢✶,✶ . . . ❢✶,♠ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♠,✶ . . . ❢♠,♠    ■❙❙❆❈✬✷✵✶✵ ❚❤❡ ❞❡❣r❡❡ ♦❢ ✐s

♠ r ✶ ✐ ✵

✐ ♠ ✐ ♠ ✶ ✐ ♠ r ✐ ■❙❙❆❈✬✷✵✶✵ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ✐s ❍❙ t ❞❡t ❆ t t

r ✷

✶ t ❦

♠ r ✷

❆✐,❥(t) =

♠ − ✐ ℓ ♠ − ❥ ℓ

tℓ.

tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ D ❜② ❛❞❞✐♥❣ ✈✐,❥ − ❢✐,❥

✷✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 66

Pr♦♣❡rt✐❡s ♦❢ ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s

D = ▼✐♥♦rsr+✶    ✈✶,✶ . . . ✈✶,♠ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♠,✶ . . . ✈♠,♠    ❚❤♦♠✱ P♦rt❡♦✉s✱ ●✐❛♠❜❡❧❧✐✱ ❍❛rr✐s✲❚✉✱ ✳✳✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! . ❈♦♥❝❛✴❍❡r③♦❣✱ ❆❜❤②❛♥❦❛r ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙D(t) = ❞❡t(❆(t)) t

✷

(✶ − t)(✷♠−r)r

. I = ▼✐♥♦rsr+✶    ❢✶,✶ . . . ❢✶,♠ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♠,✶ . . . ❢♠,♠    ■❙❙❆❈✬✷✵✶✵ ❚❤❡ ❞❡❣r❡❡ ♦❢ I ✐s

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! . ■❙❙❆❈✬✷✵✶✵ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ I ✐s ❍❙I(t) = ❞❡t(❆(t)) t

✷

(✶ − t)❦−(♠−r)✷

. ❆✐,❥(t) =

♠ − ✐ ℓ ♠ − ❥ ℓ

tℓ.

tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ D ❜② ❛❞❞✐♥❣ ✈✐,❥ − ❢✐,❥

✷✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 67

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ♠✐♥♦rs ❢♦r♠✉❧❛t✐♦♥ ✭■❙❙❆❈✬✷✵✶✵✮

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ❢♦r ❛ ✵✲❞✐♠ ✐❞❡❛❧ = ✶+ ❞❡❣r❡❡ ♦❢ t❤❡ ❍✐❧❜❡rt s❡r✐❡s✳ ❈♦r♦❧❧❛r② ❚❤❡ ❞❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ♦❢ I ✐s ❣❡♥❡r✐❝❛❧❧② ❡q✉❛❧ t♦ ❞r❡❣ = r(♠ − r) + ✶. ◆✉♠❜❡r ♦❢ ♠❛tr✐❝❡s ❛♥❞ r❛♥❦ ❞❡❢❡❝t ✜①❡❞✳ ✵✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❈♦r♦❧❧❛r②✿ ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t② ❲❤❡♥ ❦ ♠ r ✷ ✐s ✜①❡❞✱ t❤❡♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ●rö❜♥❡r ❜❛s✐s ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣ ✐s

❖ ♠ ❦

✷✹✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 68

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ♠✐♥♦rs ❢♦r♠✉❧❛t✐♦♥ ✭■❙❙❆❈✬✷✵✶✵✮

❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ❢♦r ❛ ✵✲❞✐♠ ✐❞❡❛❧ = ✶+ ❞❡❣r❡❡ ♦❢ t❤❡ ❍✐❧❜❡rt s❡r✐❡s✳ ❈♦r♦❧❧❛r② ❚❤❡ ❞❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ♦❢ I ✐s ❣❡♥❡r✐❝❛❧❧② ❡q✉❛❧ t♦ ❞r❡❣ = r(♠ − r) + ✶. ◆✉♠❜❡r ♦❢ ♠❛tr✐❝❡s ❛♥❞ r❛♥❦ ❞❡❢❡❝t ✜①❡❞✳ ✵✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❈♦r♦❧❧❛r②✿ ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t② ❲❤❡♥ ❦ = (♠ − r)✷ ✐s ✜①❡❞✱ t❤❡♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ●rö❜♥❡r ❜❛s✐s ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣ ✐s

❖

♠ω❦

.

✷✹✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 69

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣

❈♦r♦❧❧❛r②✿ ❣❡♥❡r✐❝ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❚❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ♦❢ ❛ ❣❡♥❡r✐❝ ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠ ✇✐t❤ ❦ = (♠ − r)✷ ✐s #❙♦❧ =

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! ∼

♠→∞

♠❦

♠−r−✶

✐=✵

✐! (♠ − r + ✐)!. ❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ✭■❙❙❆❈ ✷✵✶✵✮ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❋●▲▼ ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖ ❙♦❧ ■❢ ❦ ♠ r ✷✱ t❤❡♥ ❖ ❙♦❧ ❖ ♠

❦ ✷✺✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 70

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣

❈♦r♦❧❧❛r②✿ ❣❡♥❡r✐❝ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❚❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ♦❢ ❛ ❣❡♥❡r✐❝ ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠ ✇✐t❤ ❦ = (♠ − r)✷ ✐s #❙♦❧ =

♠−r−✶

✐=✵

✐!(♠ + ✐)! (♠ − ✶ − ✐)!(♠ − r + ✐)! ∼

♠→∞

♠❦

♠−r−✶

✐=✵

✐! (♠ − r + ✐)!. ❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❈❤❛♥❣❡ ♦❢ ❖r❞❡r✐♥❣ ✭■❙❙❆❈ ✷✵✶✵✮ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❋●▲▼ ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖ (#❙♦❧ ω) . ■❢ ❦ = (♠ − r)✷✱ t❤❡♥ ❖ (#❙♦❧ ω) = ❖

♠ω❦

.

✷✺✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 71

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❈♦✉rt♦✐s✳ ❆s✐❛❝r②♣t✬✵✶✳ ❊✣❝✐❡♥t ③❡r♦✲❦♥♦✇❧❡❞❣❡ ❛✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❛ ❧✐♥❡❛r ❛❧❣❡❜r❛ ♣r♦❜❧❡♠ ▼✐♥❘❛♥❦✳ K = ●❋(✻✺✺✷✶) (♠, ❦, r)✿ ❦ ♠❛tr✐❝❡s ♦❢ s✐③❡ ♠ × ♠✳ ❚❛r❣❡t r❛♥❦✿ r✳

❈❤❛❧❧❡♥❣❡ ❆ ❇ ❈ (✻, ✾, ✸) (✼, ✾, ✹) (✽, ✾, ✺) (✾, ✾, ✻) (✶✶, ✾, ✽) ❞❡❣r❡❡ ✾✽✵ ✹✶✶✻ ✶✹✶✶✷ ✹✶✺✽✵ ✷✺✾✺✹✺ ▼✐♥♦rs ♠♦❞❡❧✐♥❣ ❞r❡❣ ✶✵ ✶✸ ✶✻ ✶✾ ❋✺ t✐♠❡ ✶✳✶s ✷✽✳✹s ✺✹✹s ✾✵✹✽s ✲ ❋✺ ♠❡♠ ✹✽✽ ▼❇ ✺✽✼ ▼❇ ✶✷✶✸ ▼❇ ✺✵✹✽ ▼❇ ✲ ❧♦❣✷✭◆❜ ♦♣✳✮ ✷✶✳✺ ✷✺✳✾ ✷✾✳✷ ✸✷✳✼ ❋●▲▼ t✐♠❡ ✵✳✺s ✷✽✳✺s ✶✵✸✸s ✷✷✶✼✶s ✲ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣ ❞r❡❣ ✺ ✻ ✼ ❋✺ t✐♠❡ ✸✵s ✸✼✾✺s ✸✷✽✷✸✸s ∞ ❋✺ ♠❡♠ ✹✵✼ ▼❇ ✸✶✶✸ ▼❇ ✺✽✺✽✼ ▼❇ ❧♦❣✷✭◆❜ ♦♣✳✮ ✸✵✳✺ ✸✼✳✶ ✹✸✳✹ ❋●▲▼ t✐♠❡ ✸✺s ✷✺✽✵s ∞

❈♦♠♣✉t❛t✐♦♥❛❧ ❜♦tt❧❡♥❡❝❦✿ ❝♦♠♣✉t✐♥❣ t❤❡ ♠✐♥♦rs✳ ❈♦♠♣✉t✐♥❣ ❡✛♦rt ♥❡❡❞❡❞ ❢♦r s♦❧✈✐♥❣ ❈❤❛❧❧❡♥❣❡ ❈✿ ✷✸✽ ❞❛②s ♦♥ ✻✹ q✉❛❞r✐❝♦r❡ ♣r♦❝❡ss♦rs✳

✷✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 72

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

❈♦✉rt♦✐s✳ ❆s✐❛❝r②♣t✬✵✶✳ ❊✣❝✐❡♥t ③❡r♦✲❦♥♦✇❧❡❞❣❡ ❛✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❛ ❧✐♥❡❛r ❛❧❣❡❜r❛ ♣r♦❜❧❡♠ ▼✐♥❘❛♥❦✳ K = ●❋(✻✺✺✷✶) (♠, ❦, r)✿ ❦ ♠❛tr✐❝❡s ♦❢ s✐③❡ ♠ × ♠✳ ❚❛r❣❡t r❛♥❦✿ r✳

❈❤❛❧❧❡♥❣❡ ❆ ❇ ❈ (✻, ✾, ✸) (✼, ✾, ✹) (✽, ✾, ✺) (✾, ✾, ✻) (✶✶, ✾, ✽) ❞❡❣r❡❡ ✾✽✵ ✹✶✶✻ ✶✹✶✶✷ ✹✶✺✽✵ ✷✺✾✺✹✺ ▼✐♥♦rs ♠♦❞❡❧✐♥❣ ❞r❡❣ ✶✵ ✶✸ ✶✻ ✶✾ ❋✺ t✐♠❡ ✶✳✶s ✷✽✳✹s ✺✹✹s ✾✵✹✽s ✲ ❋✺ ♠❡♠ ✹✽✽ ▼❇ ✺✽✼ ▼❇ ✶✷✶✸ ▼❇ ✺✵✹✽ ▼❇ ✲ ❧♦❣✷✭◆❜ ♦♣✳✮ ✷✶✳✺ ✷✺✳✾ ✷✾✳✷ ✸✷✳✼ ❋●▲▼ t✐♠❡ ✵✳✺s ✷✽✳✺s ✶✵✸✸s ✷✷✶✼✶s ✲ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣ ❞r❡❣ ✺ ✻ ✼ ❋✺ t✐♠❡ ✸✵s ✸✼✾✺s ✸✷✽✷✸✸s ∞ ❋✺ ♠❡♠ ✹✵✼ ▼❇ ✸✶✶✸ ▼❇ ✺✽✺✽✼ ▼❇ ❧♦❣✷✭◆❜ ♦♣✳✮ ✸✵✳✺ ✸✼✳✶ ✹✸✳✹ ❋●▲▼ t✐♠❡ ✸✺s ✷✺✽✵s ∞

❈♦♠♣✉t❛t✐♦♥❛❧ ❜♦tt❧❡♥❡❝❦✿ ❝♦♠♣✉t✐♥❣ t❤❡ ♠✐♥♦rs✳ ❈♦♠♣✉t✐♥❣ ❡✛♦rt ♥❡❡❞❡❞ ❢♦r s♦❧✈✐♥❣ ❈❤❛❧❧❡♥❣❡ ❈✿ ✷✸✽ ❞❛②s ♦♥ ✻✹ q✉❛❞r✐❝♦r❡ ♣r♦❝❡ss♦rs✳

✷✻✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 73

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ✭♠✉❧t✐✲✮❍❋❊

P❛t❛r✐♥✱ ❊✉r♦❝r②♣t✬✾✻ ❇✐❧❧❡t✴P❛t❛r✐♥✴❙❡✉r✐♥✱ ■❈❙❈❈✬✵✽ ❉✐♥❣✴❙❝❤♠✐tt✴❲❡r♥❡r✱ ■♥❢♦r♠❛t✐♦♥ ❙❡❝✉r✐t②✱ ✷✵✵✽ P(①) =

✵≤✐,❥≤r

♣✐,❥①q✐ +q❥ ∈ Fq♥, ✇✐t❤ r ≪ ♥ ❧♦✇✲r❛♥❦ q✉❛❞r❛t✐❝ ❢♦r♠ (Fq)♥ → (Fq)♥ ♠❛s❦❡❞ ❜② ❧✐♥❡❛r tr❛♥s❢♦r♠s ✦✦ t❤❡ s❡❝r❡t ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❜② s♦❧✈✐♥❣ ❛ ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠✳ ❇❡tt❛❧❡✴❋❛✉❣èr❡✴P❡rr❡t✱ P❑❈ ✷✵✶✶ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ t❤✐s ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠ ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖ ♥ r

✶

❛❧❣❡❜r❛✐❝ ❛tt❛❝❦ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✐♥ ♥ ✦✦ ❛tt❛❝❦s ♦♥ ♦❞❞✲❝❤❛r❛❝t❡r✐st✐❝ ✈❛r✐❛♥ts❀ ❣❡♥❡r❛❧✐③❛t✐♦♥s t♦ ♠✉❧t✐✲❍❋❊✳

✷✼✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 74

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ✭♠✉❧t✐✲✮❍❋❊

P❛t❛r✐♥✱ ❊✉r♦❝r②♣t✬✾✻ ❇✐❧❧❡t✴P❛t❛r✐♥✴❙❡✉r✐♥✱ ■❈❙❈❈✬✵✽ ❉✐♥❣✴❙❝❤♠✐tt✴❲❡r♥❡r✱ ■♥❢♦r♠❛t✐♦♥ ❙❡❝✉r✐t②✱ ✷✵✵✽ P(①) =

✵≤✐,❥≤r

♣✐,❥①q✐ +q❥ ∈ Fq♥, ✇✐t❤ r ≪ ♥ ❧♦✇✲r❛♥❦ q✉❛❞r❛t✐❝ ❢♦r♠ (Fq)♥ → (Fq)♥ ♠❛s❦❡❞ ❜② ❧✐♥❡❛r tr❛♥s❢♦r♠s ✦✦ t❤❡ s❡❝r❡t ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❜② s♦❧✈✐♥❣ ❛ ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠✳ ❇❡tt❛❧❡✴❋❛✉❣èr❡✴P❡rr❡t✱ P❑❈ ✷✵✶✶ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ t❤✐s ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠ ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖ ♥ r

✶

❛❧❣❡❜r❛✐❝ ❛tt❛❝❦ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✐♥ ♥ ✦✦ ❛tt❛❝❦s ♦♥ ♦❞❞✲❝❤❛r❛❝t❡r✐st✐❝ ✈❛r✐❛♥ts❀ ❣❡♥❡r❛❧✐③❛t✐♦♥s t♦ ♠✉❧t✐✲❍❋❊✳

✷✼✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 75

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ♦❢ ✭♠✉❧t✐✲✮❍❋❊

P❛t❛r✐♥✱ ❊✉r♦❝r②♣t✬✾✻ ❇✐❧❧❡t✴P❛t❛r✐♥✴❙❡✉r✐♥✱ ■❈❙❈❈✬✵✽ ❉✐♥❣✴❙❝❤♠✐tt✴❲❡r♥❡r✱ ■♥❢♦r♠❛t✐♦♥ ❙❡❝✉r✐t②✱ ✷✵✵✽ P(①) =

✵≤✐,❥≤r

♣✐,❥①q✐ +q❥ ∈ Fq♥, ✇✐t❤ r ≪ ♥ ❧♦✇✲r❛♥❦ q✉❛❞r❛t✐❝ ❢♦r♠ (Fq)♥ → (Fq)♥ ♠❛s❦❡❞ ❜② ❧✐♥❡❛r tr❛♥s❢♦r♠s ✦✦ ⇒ t❤❡ s❡❝r❡t ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❜② s♦❧✈✐♥❣ ❛ ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠✳ ❇❡tt❛❧❡✴❋❛✉❣èr❡✴P❡rr❡t✱ P❑❈ ✷✵✶✶ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ t❤✐s ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠ ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖

♥(r+✶)ω

. ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✐♥ ♥ ✦✦ ❛tt❛❝❦s ♦♥ ♦❞❞✲❝❤❛r❛❝t❡r✐st✐❝ ✈❛r✐❛♥ts❀ ❣❡♥❡r❛❧✐③❛t✐♦♥s t♦ ♠✉❧t✐✲❍❋❊✳

✷✼✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 76

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❙tr✉❝t✉r❡s ❤❛✈❡ ❛♥ ✐♠♣❛❝t ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ✦

❉❡s✐❣♥✱ ❦❡② s✐③❡ r❡❞✉❝t✐♦♥✱✳ ✳ ✳

❙tr✉❝t✉r❡

← → ♣♦t❡♥t✐❛❧ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦s✳ ❖t❤❡r ♣♦ss✐❜❧❡ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❈r②♣t♦ ♦❢ str✉❝t✉r❡❞ s②st❡♠s ❘❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ✭●❛❜✐❞✉❧✐♥✴❖✉r✐✈s❦✐✴❍♦♥❛r②✴❆♠♠❛r ■❊❊❊ ■❚✱ ✷✵✵✸✮✳ ❝❧❛ss✐❝❛❧ ▼❝❊❧✐❡❝❡ P❑❈ ✭▼❝❊❧✐❡❝❡ ✶✾✼✽✮✳ ❆❧❣♦r✐t❤♠✐❝ ♣r♦❜❧❡♠s ❉❡❞✐❝❛t❡❞ ❋✺ ❛❧❣♦r✐t❤♠ ❢♦r ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✳ ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳✱ ❏✳ ♦❢ ❙②♠❜✳ ❈♦♠♣✳ ✷✵✶✶✮ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❄

✷✽✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 77

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❙tr✉❝t✉r❡s ❤❛✈❡ ❛♥ ✐♠♣❛❝t ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ✦

❉❡s✐❣♥✱ ❦❡② s✐③❡ r❡❞✉❝t✐♦♥✱✳ ✳ ✳

❙tr✉❝t✉r❡

← → ♣♦t❡♥t✐❛❧ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦s✳ ❖t❤❡r ♣♦ss✐❜❧❡ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❈r②♣t♦ ♦❢ str✉❝t✉r❡❞ s②st❡♠s ❘❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ✭●❛❜✐❞✉❧✐♥✴❖✉r✐✈s❦✐✴❍♦♥❛r②✴❆♠♠❛r ■❊❊❊ ■❚✱ ✷✵✵✸✮✳ ❝❧❛ss✐❝❛❧ ▼❝❊❧✐❡❝❡ P❑❈ ✭▼❝❊❧✐❡❝❡ ✶✾✼✽✮✳ ❆❧❣♦r✐t❤♠✐❝ ♣r♦❜❧❡♠s ❉❡❞✐❝❛t❡❞ ❋✺ ❛❧❣♦r✐t❤♠ ❢♦r ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✳ ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳✱ ❏✳ ♦❢ ❙②♠❜✳ ❈♦♠♣✳ ✷✵✶✶✮ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❄

✷✽✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

SLIDE 78

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❙tr✉❝t✉r❡s ❤❛✈❡ ❛♥ ✐♠♣❛❝t ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ✦

❉❡s✐❣♥✱ ❦❡② s✐③❡ r❡❞✉❝t✐♦♥✱✳ ✳ ✳

❙tr✉❝t✉r❡

← → ♣♦t❡♥t✐❛❧ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦s✳ ❖t❤❡r ♣♦ss✐❜❧❡ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❈r②♣t♦ ♦❢ str✉❝t✉r❡❞ s②st❡♠s ❘❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ✭●❛❜✐❞✉❧✐♥✴❖✉r✐✈s❦✐✴❍♦♥❛r②✴❆♠♠❛r ■❊❊❊ ■❚✱ ✷✵✵✸✮✳ ❝❧❛ss✐❝❛❧ ▼❝❊❧✐❡❝❡ P❑❈ ✭▼❝❊❧✐❡❝❡ ✶✾✼✽✮✳ ❆❧❣♦r✐t❤♠✐❝ ♣r♦❜❧❡♠s ❉❡❞✐❝❛t❡❞ ❋✺ ❛❧❣♦r✐t❤♠ ❢♦r ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✳ ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳✱ ❏✳ ♦❢ ❙②♠❜✳ ❈♦♠♣✳ ✷✵✶✶✮ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❄

✷✽✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r