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Jul 14, 2023 163 likes •1.18k views

t s rtrs t s s sts str rst s

SLIDE 1

▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡

◆✐❡❧s ▲❛✉sts❡♥

▲❛♥❝❛st❡r ❯♥✐✈❡rs✐t②

❋✐❡❧❞s ■♥st✐t✉t❡✱ ✷✺t❤ ▼❛r❝❤ ✷✵✶✹ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ●❛rt❤ ❉❛❧❡s ✭▲❛♥❝❛st❡r✮✱ ❚♦♠❛s③ ❑❛♥✐❛ ✭▲❛♥❝❛st❡r✴■▼P❆◆ ❲❛rs❛✇✮✱ ❚♦♠❛s③ ❑♦❝❤❛♥❡❦ ✭❯♥✐✈❡rs✐t② ♦❢ ❙✐❧❡s✐❛✱ P♦❧❛♥❞✴■▼P❆◆ ❲❛rs❛✇✮ ❛♥❞ P✐♦tr ❑♦s③♠✐❞❡r ✭■▼P❆◆ ❲❛rs❛✇✮

✶

SLIDE 2

❇❛❝❦❣r♦✉♥❞

❚❤❡♦r❡♠ ✭❙✐♥❝❧❛✐r✕❚✉❧❧♦ ✶✾✼✹✮✳ ▲❡t A ❜❡ ❛ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡❛❝❤ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❚❡r♠✐♥♦❧♦❣② ❇❛♥❛❝❤ ❛❧❣❡❜r❛✿ ❛ ❝♦♠♣❧❡t❡ ♥♦r♠❡❞ ❛❧❣❡❜r❛✱ ✉s✉❛❧❧② ✉♥✐t❛❧✱ ❛❧✇❛②s ❤❛✈✐♥❣ ❝♦♠♣❧❡① s❝❛❧❛rs❀ ❧❡❢t ✐❞❡❛❧✿ ❛ ♥♦♥✲❡♠♣t② s✉❜s❡t ♦❢ s✉❝❤ t❤❛t ✐s

❝❧♦s❡❞ ✉♥❞❡r ❛❞❞✐t✐♦♥✿ ❛ ❜ ❛ ❜ ❝❧♦s❡❞ ✉♥❞❡r ❛r❜✐tr❛r② ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ❛❜ ❛ ❜

✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✿ t❤❡r❡ ❡①✐st ♥ ❛♥❞ ❜✶ ❜♥ s✉❝❤ t❤❛t ❛✶❜✶ ❛♥❜♥ ❛✶ ❛♥

✷

SLIDE 3

❇❛❝❦❣r♦✉♥❞

❚❤❡♦r❡♠ ✭❙✐♥❝❧❛✐r✕❚✉❧❧♦ ✶✾✼✹✮✳ ▲❡t A ❜❡ ❛ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡❛❝❤ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❚❡r♠✐♥♦❧♦❣②

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

❝♦♠♣❧❡① s❝❛❧❛rs❀ ❧❡❢t ✐❞❡❛❧✿ ❛ ♥♦♥✲❡♠♣t② s✉❜s❡t ♦❢ s✉❝❤ t❤❛t ✐s

❝❧♦s❡❞ ✉♥❞❡r ❛❞❞✐t✐♦♥✿ ❛ ❜ ❛ ❜ ❝❧♦s❡❞ ✉♥❞❡r ❛r❜✐tr❛r② ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ❛❜ ❛ ❜

✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✿ t❤❡r❡ ❡①✐st ♥ ❛♥❞ ❜✶ ❜♥ s✉❝❤ t❤❛t ❛✶❜✶ ❛♥❜♥ ❛✶ ❛♥

✷

SLIDE 4

❇❛❝❦❣r♦✉♥❞

❚❤❡♦r❡♠ ✭❙✐♥❝❧❛✐r✕❚✉❧❧♦ ✶✾✼✹✮✳ ▲❡t A ❜❡ ❛ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡❛❝❤ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❚❡r♠✐♥♦❧♦❣②

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

❝♦♠♣❧❡① s❝❛❧❛rs❀

◮ ❧❡❢t ✐❞❡❛❧✿ ❛ ♥♦♥✲❡♠♣t② s✉❜s❡t L ♦❢ A s✉❝❤ t❤❛t L ✐s

❝❧♦s❡❞ ✉♥❞❡r ❛❞❞✐t✐♦♥✿ ❛ ❜ ❛ ❜ ❝❧♦s❡❞ ✉♥❞❡r ❛r❜✐tr❛r② ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ❛❜ ❛ ❜

✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✿ t❤❡r❡ ❡①✐st ♥ ❛♥❞ ❜✶ ❜♥ s✉❝❤ t❤❛t ❛✶❜✶ ❛♥❜♥ ❛✶ ❛♥

✷

SLIDE 5

❇❛❝❦❣r♦✉♥❞

❚❤❡♦r❡♠ ✭❙✐♥❝❧❛✐r✕❚✉❧❧♦ ✶✾✼✹✮✳ ▲❡t A ❜❡ ❛ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡❛❝❤ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❚❡r♠✐♥♦❧♦❣②

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

❝♦♠♣❧❡① s❝❛❧❛rs❀

◮ ❧❡❢t ✐❞❡❛❧✿ ❛ ♥♦♥✲❡♠♣t② s✉❜s❡t L ♦❢ A s✉❝❤ t❤❛t L ✐s

◮ ❝❧♦s❡❞ ✉♥❞❡r ❛❞❞✐t✐♦♥✿ ❛ + ❜ ∈ L

(❛, ❜ ∈ L ); ❝❧♦s❡❞ ✉♥❞❡r ❛r❜✐tr❛r② ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ❛❜ ❛ ❜

✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✿ t❤❡r❡ ❡①✐st ♥ ❛♥❞ ❜✶ ❜♥ s✉❝❤ t❤❛t ❛✶❜✶ ❛♥❜♥ ❛✶ ❛♥

✷

SLIDE 6

❇❛❝❦❣r♦✉♥❞

❚❤❡♦r❡♠ ✭❙✐♥❝❧❛✐r✕❚✉❧❧♦ ✶✾✼✹✮✳ ▲❡t A ❜❡ ❛ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡❛❝❤ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❚❡r♠✐♥♦❧♦❣②

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

❝♦♠♣❧❡① s❝❛❧❛rs❀

◮ ❧❡❢t ✐❞❡❛❧✿ ❛ ♥♦♥✲❡♠♣t② s✉❜s❡t L ♦❢ A s✉❝❤ t❤❛t L ✐s

◮ ❝❧♦s❡❞ ✉♥❞❡r ❛❞❞✐t✐♦♥✿ ❛ + ❜ ∈ L

(❛, ❜ ∈ L );

◮ ❝❧♦s❡❞ ✉♥❞❡r ❛r❜✐tr❛r② ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ❛❜ ∈ L

(❛ ∈ A , ❜ ∈ L );

✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✿ t❤❡r❡ ❡①✐st ♥ ❛♥❞ ❜✶ ❜♥ s✉❝❤ t❤❛t ❛✶❜✶ ❛♥❜♥ ❛✶ ❛♥

✷

SLIDE 7

❇❛❝❦❣r♦✉♥❞

❚❤❡♦r❡♠ ✭❙✐♥❝❧❛✐r✕❚✉❧❧♦ ✶✾✼✹✮✳ ▲❡t A ❜❡ ❛ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡❛❝❤ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❚❡r♠✐♥♦❧♦❣②

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

❝♦♠♣❧❡① s❝❛❧❛rs❀

◮ ❧❡❢t ✐❞❡❛❧✿ ❛ ♥♦♥✲❡♠♣t② s✉❜s❡t L ♦❢ A s✉❝❤ t❤❛t L ✐s

◮ ❝❧♦s❡❞ ✉♥❞❡r ❛❞❞✐t✐♦♥✿ ❛ + ❜ ∈ L

(❛, ❜ ∈ L );

◮ ❝❧♦s❡❞ ✉♥❞❡r ❛r❜✐tr❛r② ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ❛❜ ∈ L

(❛ ∈ A , ❜ ∈ L );

♥ ◆

①♥

◆ ✜♥✐t❡

❝♦♥✈❡r❣❡s t♦ ①✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳

✹

SLIDE 13

❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B(❊) ❝♦♥t❛✐♥s ✷c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ℵ✵✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ❊♥ ♥ ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ❊✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ①♥ ✇✐t❤ ①♥ ❊♥ ♥ s✉❝❤ t❤❛t ①

♥ ✶

①♥ ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ t❤❛t ✐s✱ t❤❡ ♥❡t

♥ ◆

①♥

◆ ✜♥✐t❡

❝♦♥✈❡r❣❡s t♦ ①✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳

✹

SLIDE 14

❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B(❊) ❝♦♥t❛✐♥s ✷c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ℵ✵✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ (❊♥)♥∈N ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ❊✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ (①♥) ✇✐t❤ ①♥ ∈ ❊♥ (♥ ∈ N) s✉❝❤ t❤❛t ① =

∞

♥=✶

①♥ ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ t❤❛t ✐s✱ t❤❡ ♥❡t

♥ ◆

①♥

◆ ✜♥✐t❡

❝♦♥✈❡r❣❡s t♦ ①✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳

✹

SLIDE 15

❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B(❊) ❝♦♥t❛✐♥s ✷c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ℵ✵✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ (❊♥)♥∈N ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ❊✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ (①♥) ✇✐t❤ ①♥ ∈ ❊♥ (♥ ∈ N) s✉❝❤ t❤❛t ① =

∞

♥=✶

①♥ ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ t❤❛t ✐s✱ t❤❡ ♥❡t

♥∈◆

①♥

◆⊂N ✜♥✐t❡

❝♦♥✈❡r❣❡s t♦ ①✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳

✹

SLIDE 16

❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B(❊) ❝♦♥t❛✐♥s ✷c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ℵ✵✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ (❊♥)♥∈N ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ❊✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ (①♥) ✇✐t❤ ①♥ ∈ ❊♥ (♥ ∈ N) s✉❝❤ t❤❛t ① =

∞

♥=✶

①♥ ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ t❤❛t ✐s✱ t❤❡ ♥❡t

♥∈◆

①♥

◆⊂N ✜♥✐t❡

❝♦♥✈❡r❣❡s t♦ ①✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳

✹

SLIDE 17

❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠

■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿

t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s

♣

✶ ♣ ❛♥❞ ❝✵ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲♣ ✵ ✶ ✶ ♣ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞✇ ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤▼

♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿

t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲✶ ✵ ✶ t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑❀ t❤❡ ♣t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏♣ ✶ ♣ ✳

❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳

✺

SLIDE 18

❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠

■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿

◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡

s✉❜s♣❛❝❡s ❊♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿

t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s

♣

✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿

◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲✶[✵, ✶];

t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑❀ t❤❡ ♣t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏♣ ✶ ♣ ✳

❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳

✺

SLIDE 24

❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠

■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿

◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡

s✉❜s♣❛❝❡s ❊♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿

◮ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥

✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿

◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲✶[✵, ✶]; ◮ t❤❡ s♣❛❝❡ ❈(❑) ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝

s♣❛❝❡ ❑❀ t❤❡ ♣t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏♣ ✶ ♣ ✳

❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥

❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ {✵}✱ ML ① = {❚ ∈ B(❊) : ❚① = ✵} ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C①✳ ▼♦r❡♦✈❡r✱

① ②

① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ① ② ❊ ✵ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠

① ❢♦r s♦♠❡ ①

❊ ✵ ✐s ✜①❡❞✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✜①❡❞❄ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❋ ❚ ❊ ❋ ❦❡r ❚ ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❝♦rr❡s♣♦♥❞s t♦ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳

✻

SLIDE 29

❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥

❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ {✵}✱ ML ① = {❚ ∈ B(❊) : ❚① = ✵} ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C①✳ ▼♦r❡♦✈❡r✱ ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ (①, ② ∈ ❊ \ {✵}). ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠

✼

SLIDE 39

◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡

▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F(❊) = {❚ ∈ B(❊) : ❞✐♠ ❚(❊) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B(❊)✳ ❍❡♥❝❡ F(❊) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ❋❛❝t✳ F(❊) ⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ {✵}✳ Pr♦♦❢✳ ●✐✈❡♥ ① ∈ ❊ \ {✵}✱ t❛❦❡ λ ∈ ❊ ∗ s✉❝❤ t❤❛t ①, λ = ✶✳ ❚❤❡♥ ❚ ② ② ① ❊ ❊ ❜❡❧♦♥❣s t♦ ❊ ✱ ❜✉t ♥♦t t♦

① ❜❡❝❛✉s❡ ❚①

① ✵✳ ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞✳ ❚❤❡♥ ❊ ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s ❊ ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❄

✼

SLIDE 40

◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡

▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F(❊) = {❚ ∈ B(❊) : ❞✐♠ ❚(❊) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B(❊)✳ ❍❡♥❝❡ F(❊) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ❋❛❝t✳ F(❊) ⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ {✵}✳ Pr♦♦❢✳ ●✐✈❡♥ ① ∈ ❊ \ {✵}✱ t❛❦❡ λ ∈ ❊ ∗ s✉❝❤ t❤❛t ①, λ = ✶✳ ❚❤❡♥ ❚ : ② → ②, λ①, ❊ → ❊, ❜❡❧♦♥❣s t♦ F(❊)✱ ❜✉t ♥♦t t♦

❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■

❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B(❊)✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E (❊)✳ ❘❡♠❛r❦s✳

◮ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r②

❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼ ✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ▼✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼✳ ❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❝♦♥t❛✐♥s ❊ ✳

✾

SLIDE 53

❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■

❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B(❊)✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E (❊)✳ ❘❡♠❛r❦s✳

◮ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r②

❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳

◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼

✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ ▼✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼✳ ❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❝♦♥t❛✐♥s ❊ ✳

✾

SLIDE 54

❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■

❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B(❊)✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E (❊)✳ ❘❡♠❛r❦s✳

◮ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r②

❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳

◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼

✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ ▼✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼✳ ❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❝♦♥t❛✐♥s ❊ ✳

✾

SLIDE 55

❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■

❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B(❊)✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E (❊)✳ ❘❡♠❛r❦s✳

◮ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r②

❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳

◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼

✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ ▼✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼✳ ❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ❝♦♥t❛✐♥s F(❊)✳

✾

SLIDE 56

P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭❡①❛♠♣❧❡s✿

♣ ❛♥❞ ▲♣ ✵ ✶ ❢♦r ✶

♣ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭❡①❛♠♣❧❡s✿ ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ✮❀ ✭✐✐✐✮ ❊ ❝✵ ✱ ❊ ❍✱ ♦r ❊ ❝✵ ❍✱ ✇❤❡r❡ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ❙① ✳

✶✵

SLIDE 57

P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭❡①❛♠♣❧❡s✿

P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭❡①❛♠♣❧❡s✿ ℓ♣ ❛♥❞ ▲♣[✵, ✶] ❢♦r ✶ ♣ < ∞✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭❡①❛♠♣❧❡s✿ ℓ∞(Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ✮❀ ✭✐✐✐✮ ❊ = ❝✵(Γ)✱ ❊ = ❍✱ ♦r ❊ = ❝✵(Γ) ⊕ ❍✱ ✇❤❡r❡ Γ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✈✮ ❊ = ❈(❑)✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈(❑) ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ε > ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ❙① ε✳

✶✵

SLIDE 68

❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡

▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ → ❛⋆✳ ❆ st❛t❡ ♦♥ ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ ♦♥ s✉❝❤ t❤❛t ❛ ❛ ✵ ❛

✐✈❡♥ ❛ st❛t❡

♦♥ ✱ t❤❡ s❡t ❛ ❛ ❛ ✵ ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ ✳ ❙✉♣♣♦s❡ t❤❛t ❍ ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡♥

① ✐❢ ❛♥❞ ♦♥❧② ✐❢

✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ①✱ t❤❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ❍ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❍ ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ❍ ✳

✶✶

SLIDE 69

❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡

▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ → ❛⋆✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t ❛⋆❛, λ ✵ (❛ ∈ A ).

✐✈❡♥ ❛ st❛t❡

♦♥ ✱ t❤❡ s❡t ❛ ❛ ❛ ✵ ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ ✳ ❙✉♣♣♦s❡ t❤❛t ❍ ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡♥

① ✐❢ ❛♥❞ ♦♥❧② ✐❢

✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ①✱ t❤❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ❍ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❍ ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ❍ ✳

✶✶

SLIDE 70

❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡

▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ → ❛⋆✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t ❛⋆❛, λ ✵ (❛ ∈ A ).

✐✈❡♥ ❛ st❛t❡ λ ♦♥ A ✱ t❤❡ s❡t

Nλ = {❛ ∈ A : ❛⋆❛, λ = ✵} ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ ✳ ❙✉♣♣♦s❡ t❤❛t ❍ ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡♥

① ✐❢ ❛♥❞ ♦♥❧② ✐❢

✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ①✱ t❤❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ❍ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❍ ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ❍ ✳

✶✶

SLIDE 71

❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡

▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ → ❛⋆✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t ❛⋆❛, λ ✵ (❛ ∈ A ).

✐✈❡♥ ❛ st❛t❡ λ ♦♥ A ✱ t❤❡ s❡t

Nλ = {❛ ∈ A : ❛⋆❛, λ = ✵} ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦∗✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ A ✳ ❙✉♣♣♦s❡ t❤❛t ❍ ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡♥

① ✐❢ ❛♥❞ ♦♥❧② ✐❢

✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ①✱ t❤❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ❍ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❍ ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ❍ ✳

✶✶

SLIDE 72

❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡

▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ → ❛⋆✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t ❛⋆❛, λ ✵ (❛ ∈ A ).

✐✈❡♥ ❛ st❛t❡ λ ♦♥ A ✱ t❤❡ s❡t

Nλ = {❛ ∈ A : ❛⋆❛, λ = ✵} ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦∗✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ A ✳ ❙✉♣♣♦s❡ t❤❛t A = B(❍) ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍✱ ❛♥❞ ❧❡t ① ∈ ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡♥ Nλ = ML ① ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ①✱ t❤❛t ✐s✱ ❚, λ = (❚① | ①) (❚ ∈ B(❍)), ✇❤❡r❡ ( · | · ) ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ❍ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❍ ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ❍ ✳

✶✶

SLIDE 73

❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡

▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ → ❛⋆✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t ❛⋆❛, λ ✵ (❛ ∈ A ).

✐✈❡♥ ❛ st❛t❡ λ ♦♥ A ✱ t❤❡ s❡t

Nλ = {❛ ∈ A : ❛⋆❛, λ = ✵} ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦∗✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ A ✳ ❙✉♣♣♦s❡ t❤❛t A = B(❍) ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍✱ ❛♥❞ ❧❡t ① ∈ ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡♥ Nλ = ML ① ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ①✱ t❤❛t ✐s✱ ❚, λ = (❚① | ①) (❚ ∈ B(❍)), ✇❤❡r❡ ( · | · ) ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ λ ♦♥ B(❍) ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r K (❍) ⊆ ❦❡r λ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ K (❍) ⊆ Nλ✳

✶✶

SLIDE 74

❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ ❆r❣②r♦s✕❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡

❚❤❡♦r❡♠ ✭❆r❣②r♦s✕❍❛②❞♦♥ ✷✵✶✶✮✳ ❚❤❡r❡ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳❆❍ ✇❤✐❝❤ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❳❆❍ ❤❛s ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❳❆❍ ✐s ❛ ❝♦♠♣❛❝t ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳❆❍ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❳❆❍ ✐s ✐s♦♠♦r♣❤✐❝ t♦

✶✳

✶✷

SLIDE 75

❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ ❆r❣②r♦s✕❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡

❚❤❡♦r❡♠ ✭❆r❣②r♦s✕❍❛②❞♦♥ ✷✵✶✶✮✳ ❚❤❡r❡ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳❆❍ ✇❤✐❝❤ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❳❆❍ ❤❛s ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❳❆❍ ✐s ❛ ❝♦♠♣❛❝t ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳❆❍ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❳❆❍ ✐s ✐s♦♠♦r♣❤✐❝ t♦

✶✳

✶✷

SLIDE 76

❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ ❆r❣②r♦s✕❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡

❚❤❡♦r❡♠ ✭❆r❣②r♦s✕❍❛②❞♦♥ ✷✵✶✶✮✳ ❚❤❡r❡ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳❆❍ ✇❤✐❝❤ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❳❆❍ ❤❛s ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❳❆❍ ✐s ❛ ❝♦♠♣❛❝t ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳❆❍ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❳❆❍ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ℓ✶✳

✶✷

❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡

▲❡t ❊ = ❳❆❍ ⊕ ℓ∞✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ (✷ × ✷)✲♠❛tr✐❝❡s ❚✶,✶ : ❳❆❍ → ❳❆❍ ❚✶,✷ : ℓ∞ → ❳❆❍ ❚✷,✶ : ❳❆❍ → ℓ∞ ❚✷,✷ : ℓ∞ → ℓ∞

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)

✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱

✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r

▲ ✵ ✵ ❱❯ ❲ ✇❤❡r❡ ❳❆❍ ❳❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯

✶

❳❆❍✱ ❱

✶

✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳

✶✸

SLIDE 81

❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡

▲❡t ❊ = ❳❆❍ ⊕ ℓ∞✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ (✷ × ✷)✲♠❛tr✐❝❡s ❚✶,✶ : ❳❆❍ → ❳❆❍ ❚✶,✷ : ℓ∞ → ❳❆❍ ❚✷,✶ : ❳❆❍ → ℓ∞ ❚✷,✷ : ℓ∞ → ℓ∞

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛

♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱

✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r

▲ ✵ ✵ ❱❯ ❲ ✇❤❡r❡ ❳❆❍ ❳❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯

✶

❳❆❍✱ ❱

✶

✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳

✶✸

SLIDE 82

❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡

▲❡t ❊ = ❳❆❍ ⊕ ℓ∞✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ (✷ × ✷)✲♠❛tr✐❝❡s ❚✶,✶ : ❳❆❍ → ❳❆❍ ❚✶,✷ : ℓ∞ → ❳❆❍ ❚✷,✶ : ❳❆❍ → ℓ∞ ❚✷,✷ : ℓ∞ → ℓ∞

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛

♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞ ✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱

✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r

▲ ✵ ✵ ❱❯ ❲ ✇❤❡r❡ ❳❆❍ ❳❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯

✶

❳❆❍✱ ❱

✶

✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳

✶✸

SLIDE 83

❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡

▲❡t ❊ = ❳❆❍ ⊕ ℓ∞✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ (✷ × ✷)✲♠❛tr✐❝❡s ❚✶,✶ : ❳❆❍ → ❳❆❍ ❚✶,✷ : ℓ∞ → ❳❆❍ ❚✷,✶ : ❳❆❍ → ℓ∞ ❚✷,✷ : ℓ∞ → ℓ∞

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛

♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱

✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r

▲ ✵ ✵ ❱❯ ❲ ✇❤❡r❡ ❳❆❍ ❳❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯

✶

❳❆❍✱ ❱

✶

✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳

✶✸

SLIDE 84

❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡

▲❡t ❊ = ❳❆❍ ⊕ ℓ∞✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ (✷ × ✷)✲♠❛tr✐❝❡s ❚✶,✶ : ❳❆❍ → ❳❆❍ ❚✶,✷ : ℓ∞ → ❳❆❍ ❚✷,✶ : ❳❆❍ → ℓ∞ ❚✷,✷ : ℓ∞ → ℓ∞

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛

♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ K✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r ▲ =

✵ ❱❯∗κ ❲

✇❤❡r❡ κ: ❳❆❍ → ❳ ∗∗

❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ : ℓ✶ → ❳ ∗ ❆❍✱

❱ : ℓ∗

✶ = ℓ∞ → ℓ∞(✷N − ✶) ❛♥❞ ❲ : ℓ∞ → ℓ∞(✷N) ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳

✶✸

SLIDE 85

❍♦✇ ❛❜♦✉t ◗✉❡st✐♦♥ ■❄

❘❡❝❛❧❧✿ ❊ = ❳❆❍ ⊕ ℓ∞✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ❍❡♥❝❡ ❚✶ ✶ ❚✶ ✷ ❚✷ ✶ ❚✷ ✷ ❊ ❚✷ ✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✇❤✐❝❤ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❊ ✱ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ✐s ♣♦s✐t✐✈❡ ❢♦r ❊✳

✶✹

SLIDE 86

❍♦✇ ❛❜♦✉t ◗✉❡st✐♦♥ ■❄

❘❡❝❛❧❧✿ ❊ = ❳❆❍ ⊕ ℓ∞✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ❍❡♥❝❡ ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✷,✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r
,

✇❤✐❝❤ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B(❊)✱ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ✐s ♣♦s✐t✐✈❡ ❢♦r ❊✳

✶✹

SLIDE 87

❍♦✇ ❛❜♦✉t ◗✉❡st✐♦♥ ■❄

❘❡❝❛❧❧✿ ❊ = ❳❆❍ ⊕ ℓ∞✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ❍❡♥❝❡ ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✷,✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r
,

✶✺

SLIDE 94

❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮

▲❡t ❊ = ❳❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B(❊) ❤❛s t❤❡ ❢♦r♠ ❚ = α✶,✶■❳❆❍ + ❑✶,✶ α✶,✷❏ + ❑✶,✷ ❑✷,✶ α✷,✷■❨ + ❑✷,✷

✇❤❡r❡ α✶,✶, α✶,✷, α✷,✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑✶,✶, ❑✶,✷, ❑✷,✶, ❑✷,✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✱ ♥❛♠❡❧②

✶

❚ ❊

✷ ✷

✵ ❛♥❞

✷

❚ ❊

✶ ✶

✵ ✭✐✐✮

✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs

■❳❆❍ ✵ ✵ ✵ ❛♥❞ ✵ ❏ ✵ ✵ ❜✉t

✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊

✭✐✐✐✮

✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✶✻

SLIDE 95

❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮

▲❡t ❊ = ❳❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B(❊) ❤❛s t❤❡ ❢♦r♠ ❚ = α✶,✶■❳❆❍ + ❑✶,✶ α✶,✷❏ + ❑✶,✷ ❑✷,✶ α✷,✷■❨ + ❑✷,✷

✇❤❡r❡ α✶,✶, α✶,✷, α✷,✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑✶,✶, ❑✶,✷, ❑✷,✶, ❑✷,✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊)✱ ♥❛♠❡❧② M✶ = {❚ ∈ B(❊) : α✷,✷ = ✵} ❛♥❞ M✷ = {❚ ∈ B(❊) : α✶,✶ = ✵}; ✭✐✐✮

✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs

■❳❆❍ ✵ ✵ ✵ ❛♥❞ ✵ ❏ ✵ ✵ ❜✉t

✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊

✭✐✐✐✮

✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✶✻

SLIDE 96

❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮

▲❡t ❊ = ❳❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B(❊) ❤❛s t❤❡ ❢♦r♠ ❚ = α✶,✶■❳❆❍ + ❑✶,✶ α✶,✷❏ + ❑✶,✷ ❑✷,✶ α✷,✷■❨ + ❑✷,✷

✇❤❡r❡ α✶,✶, α✶,✷, α✷,✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑✶,✶, ❑✶,✷, ❑✷,✶, ❑✷,✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊)✱ ♥❛♠❡❧② M✶ = {❚ ∈ B(❊) : α✷,✷ = ✵} ❛♥❞ M✷ = {❚ ∈ B(❊) : α✶,✶ = ✵}; ✭✐✐✮ M✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs ■❳❆❍ ✵ ✵ ✵

❛♥❞

✵ ❏ ✵ ✵

❜✉t

✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊

✭✐✐✐✮

✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✶✻

SLIDE 97

❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮

▲❡t ❊ = ❳❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B(❊) ❤❛s t❤❡ ❢♦r♠ ❚ = α✶,✶■❳❆❍ + ❑✶,✶ α✶,✷❏ + ❑✶,✷ ❑✷,✶ α✷,✷■❨ + ❑✷,✷

✇❤❡r❡ α✶,✶, α✶,✷, α✷,✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑✶,✶, ❑✶,✷, ❑✷,✶, ❑✷,✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊)✱ ♥❛♠❡❧② M✶ = {❚ ∈ B(❊) : α✷,✷ = ✵} ❛♥❞ M✷ = {❚ ∈ B(❊) : α✶,✶ = ✵}; ✭✐✐✮ M✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs ■❳❆❍ ✵ ✵ ✵

❛♥❞

✵ ❏ ✵ ✵

❜✉t M✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊; ✭✐✐✐✮

✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✶✻

SLIDE 98

❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮

▲❡t ❊ = ❳❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B(❊) ❤❛s t❤❡ ❢♦r♠ ❚ = α✶,✶■❳❆❍ + ❑✶,✶ α✶,✷❏ + ❑✶,✷ ❑✷,✶ α✷,✷■❨ + ❑✷,✷

✇❤❡r❡ α✶,✶, α✶,✷, α✷,✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑✶,✶, ❑✶,✷, ❑✷,✶, ❑✷,✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊)✱ ♥❛♠❡❧② M✶ = {❚ ∈ B(❊) : α✷,✷ = ✵} ❛♥❞ M✷ = {❚ ∈ B(❊) : α✶,✶ = ✵}; ✭✐✐✮ M✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs ■❳❆❍ ✵ ✵ ✵

❛♥❞

✵ ❏ ✵ ✵

❜✉t M✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊; ✭✐✐✐✮ M✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✶✻

SLIDE 99

❖♣❡♥ ♣r♦❜❧❡♠s ❛♥❞ r❡❢❡r❡♥❝❡s

◮ ▲❡t ❊ = ❈(❑)✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t

❈(❑) ∼ = ❝✵✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✜①❡❞❄ ❲❤❛t ✐s t❤❡ s✐t✉❛t✐♦♥ ❢♦r ♠❛①✐♠❛❧ r✐❣❤t ✐❞❡❛❧s ♦❢ ❊ ❄ ❑❡② r❡❢❡r❡♥❝❡s ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✽ ✭✷✵✶✸✮✱ ✷✹✺✕✷✽✻✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ■❞❡❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳

✶✼

SLIDE 100

❖♣❡♥ ♣r♦❜❧❡♠s ❛♥❞ r❡❢❡r❡♥❝❡s

◮ ▲❡t ❊ = ❈(❑)✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t

❈(❑) ∼ = ❝✵✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✜①❡❞❄

◮ ❲❤❛t ✐s t❤❡ s✐t✉❛t✐♦♥ ❢♦r ♠❛①✐♠❛❧ r✐❣❤t ✐❞❡❛❧s ♦❢ B(❊)❄

❑❡② r❡❢❡r❡♥❝❡s ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✽ ✭✷✵✶✸✮✱ ✷✹✺✕✷✽✻✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ■❞❡❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳

✶✼

SLIDE 101

❖♣❡♥ ♣r♦❜❧❡♠s ❛♥❞ r❡❢❡r❡♥❝❡s

◮ ▲❡t ❊ = ❈(❑)✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t

❈(❑) ∼ = ❝✵✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✜①❡❞❄

◮ ❲❤❛t ✐s t❤❡ s✐t✉❛t✐♦♥ ❢♦r ♠❛①✐♠❛❧ r✐❣❤t ✐❞❡❛❧s ♦❢ B(❊)❄

❑❡② r❡❢❡r❡♥❝❡s

◮ ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱

▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✽ ✭✷✵✶✸✮✱ ✷✹✺✕✷✽✻✳

◮ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤

❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳

◮ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ■❞❡❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞

♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳

✶✼