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SLIDE 1

▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ❏❛♥✉❛r② ✶✼✱ ✷✵✶✼

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✴✶✽

slide-2
SLIDE 2

❆❧♠❣r❡♥ ♠✐♥✐♠✐s❡rs

▲❡t ✉s ✜① ✵ < ❞ < ♥✱ ❊ ✇✐❧❧ ❜❡ ❛ s✉❜s❡t ♦❢ R♥ ✇✐t❤ ❧♦❝❛❧❧② ✜♥✐t❡ ❞✲❞✐♠❡♥s✐♦♥❛❧ ❍❛✉s❞♦r✛ ♠❡❛s✉r❡ H❞✳ ❆❞♠✐ss✐❜❧❡ ❝♦♠♣❡t✐t♦rs ▲❡t ❯ ⊂ R♥ ❜❡ ♦♣❡♥✳ ❋ ✐s ❛♥ ❛❞♠✐ss✐❜❧❡ ❝♦♠♣❡t✐t♦r ❢♦r ❊ ♦♥ ❯ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ φ : [✵, ✶] × ❯ → ❯ s✉❝❤ t❤❛t✿ φ(✵, ·) = ■❞❀ ∃❑ ⊂⊂ ❯ s✉❝❤ t❤❛t ∀t φ(t, ·) = ■❞ ✐♥ ❑ ❝❀ φ(✶, ·) ✐s ▲✐♣s❝❤✐t③ ❛♥❞ ❋ = φ(✶, ❊)✳ ❲❡ ❝❛❧❧ φ ❛♥ ❛❞♠✐ss✐❜❧❡ ❞❡❢♦r♠❛t✐♦♥✳ ❆❧♠❣r❡♥ ♠✐♥✐♠✐s❡r ❊ ✐s ❛♥ ❆❧♠❣r❡♥ ♠✐♥✐♠✐s❡r ✐❢ ❢♦r ❡✈❡r② ❛❞♠✐ss✐❜❧❡ ❝♦♠♣❡t✐t♦r ❋ ✇❡ ❤❛✈❡✿ H❞(❊ \ ❋) ≤ H❞(❋ \ ❊).

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✷✴✶✽

slide-3
SLIDE 3

❆❧♠❣r❡♥ ♠✐♥✐♠✐s❡rs

Pr♦♣❡rt✐❡s ■❢ ❊ ✐s ❛♥ ❆❧♠❣r❡♥ ♠✐♥✐♠✐s❡r t❤❡♥✿ ❊ ✐s r❡❝t✐✜❛❜❧❡❀ ❊ ✐s ❆❤❧❢♦rs r❡❣✉❧❛r❀ ❊ ✐s ✉♥✐❢♦r♠❧② r❡❝t✐✜❛❜❧❡❀ t❤❡r❡ ❡①✐sts ❛♥ H❞✲♥❡❣❧✐❣✐❜❧❡ ◆ ⊂ R♥ s✉❝❤ t❤❛t ❊ \ ◆ ✐s ❛ ❈ ✶,α ❞✲❞✐♠❡♥s✐♦♥❛❧ s✉❜♠❛♥✐❢♦❧❞ ♦❢ R♥❀ ∀① ∈ ❊✱ ∀µ ∈ ❚❛♥❞(H❞❊, ①) ✐s s✉♣♣♦rt❡❞ ♦♥ ❛ ♠✐♥✐♠❛❧ ❝♦♥❡✳ ❘❡♠❛r❦ ▲❡t ❊ ⊂ R♥ ❜❡ ❛♥ ❆❧♠❣r❡♥ ♠✐♥✐♠✐s❡r✳ ❚❤❡♥ ❊ ✐s ❛♥ ❆❧♠❣r❡♥ ♠✐♥✐♠✐s❡r ❛s ❛ s✉❜s❡t ♦❢ R♠ ✇✐t❤ ♠ > ♥ ✭❜❡❝❛✉s❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥s ❞♦ ♥♦t ✐♥❝r❡❛s❡ t❤❡ ❍❛✉s❞♦r✛ ❛r❡❛✮✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✸✴✶✽

slide-4
SLIDE 4

▼✐♥✐♠❛❧ ❈♦♥❡s

R✷ ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ t②♣❡s ♦❢ ♠✐♥✐♠❛❧ ❝♦♥❡s✿ t❤❡ str❛✐❣❤t ❧✐♥❡❀ ❛♥❞ t❤r❡❡ ❤❛❧❢ ❧✐♥❡s ♠❡❡t✐♥❣ ✇✐t❤ ❛♥❣❧❡ ♦❢ ✶✷✵◦✱ ✇❤✐❝❤ ✇❡ ✇✐❧❧ ❞❡♥♦t❡ ❛s ❨ ✳ R✸ ❚❤❡r❡ ❛r❡ ♥♦ ♥❡✇ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥❡s✳ ❚❤❡ ✷✲❞✐♠❡♥s✐♦♥❛❧ ♠✐♥✐♠❛❧ ❝♦♥❡s ❛r❡ ♦❢ t❤r❡❡ ❦✐♥❞s✿ ♣❧❛♥❡s❀ Y := ❨ × R❀ t❤❡ ❝♦♥❡ ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ r❡❣✉❧❛r t❡tr❛❤❡❞r♦♥ T✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✹✴✶✽

slide-5
SLIDE 5

▼✐♥✐♠❛❧ ❈♦♥❡s

❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s ❝♦♥❡ ♦✈❡r s❦♥−✷(◗♥) ❢♦r ♥ ≥ ✹ ❬❇r❛❦❦❡ ✶✾✾✶❪❀ ❝♦♥❡ ♦✈❡r s❦♥−✷(∆♥) ❬▼♦r❣❛♥ ✶✾✾✹❪❀ ❢♦r ❛♥② ❞, ♠ ≥ ✷ t❤❡r❡ ❡①✐sts θ(♠, ❞) ∈ (✵, π/✷) s✉❝❤ t❤❛t✱ ❣✐✈❡♥ P✶, . . . , P♠ ❞✲♣❧❛♥❡s ✐♥ R♠❞✱ t❤❡✐r ✉♥✐♦♥ ✐s ❛♥ ❆❧♠❣r❡♥ ♠✐♥✐♠✐s❡r ✐❢ ❛❧❧ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❛♥❣❧❡s ❛r❡ ❣r❡❛t❡r t❤❛♥ θ(♠, ❞) ❬▲✐❛♥❣ ✷✵✶✸❪❀ ❨ × ❨ ⊂ R✹ ❬▲✐❛♥❣ ✷✵✶✹❪✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✺✴✶✽

slide-6
SLIDE 6

❙❧✐❞✐♥❣ ❇♦✉♥❞❛r②

❙❧✐❞✐♥❣ ❞❡❢♦r♠❛t✐♦♥ ▲❡t ❯ ⊂ R♥ ❜❡ ♦♣❡♥ ❛♥❞ Γ✐ ⊂ ❯✱ ✶ ≤ ✐ ≤ ■✱ ❜❡ ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❝❧♦s❡❞ s❡ts✳ ❆ s❧✐❞✐♥❣ ❞❡❢♦r♠❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ {Γ✐}✐ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ φ : [✵, ✶] × ❯ → ❯ s✉❝❤ t❤❛t✿ φ(✵, ·) = ■❞❀ ∃❑ ⊂⊂ ❯ s✉❝❤ t❤❛t ∀t φ(t, ·) = ■❞ ✐♥ ❑ ❝❀ φ(✶, ·) ✐s ▲✐♣s❝❤✐t③❀ φ(t, ①) ∈ Γ✐ ∀t ✐❢ ① ∈ Γ✐✳ ❙❧✐❞✐♥❣ ❜♦✉♥❞❛r② ♠✐♥✐♠✐s❡rs

  • ✐✈❡♥ Γ := ∪✐Γ✐ ❛♥❞ ✵ ≤ α ≤ ✶ ✇❡ ❞❡✜♥❡ ❛ ♥❡✇ ❝♦st ❢✉♥❝t✐♦♥❛❧

❝(❊) := H❞(❊ \ Γ) + αH❞(❊ ∩ Γ) ❛♥❞ ✇❡ s❛② t❤❛t ❊ ✐s ❛ s❧✐❞✐♥❣ ❜♦✉♥❞❛r② ♠✐♥✐♠✐s❡r ✐❢ ❢♦r ❛♥② ❝♦♠♣❡t✐t♦r ❋ ♦❜t❛✐♥❡❞ ❛s ✐♠❛❣❡ ♦❢ ❛ s❧✐❞✐♥❣ ❞❡❢♦r♠❛t✐♦♥ ✇❡ ❤❛✈❡ ❝(❊ \ ❋) ≤ ❝(❋ \ ❊)✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✻✴✶✽

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SLIDE 7

▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

❖♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♠✐♥✐♠❛❧ ❝♦♥❡s ✐♥ R✷

+✱ t❤❡ s❧✐❞✐♥❣ ❜♦✉♥❞❛r②

Γ := {② = ✵} ✐s ✐♥ ❜❧✉❡✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✼✴✶✽

slide-8
SLIDE 8

▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

❖♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♠✐♥✐♠❛❧ ❝♦♥❡s ✐♥ R✸

+✱ t❤❡ s❧✐❞✐♥❣ ❜♦✉♥❞❛r②

Γ := {③ = ✵} ✐s ✐♥ ❜❧✉❡✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✽✴✶✽

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SLIDE 9

▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥❡s ✐♥ R✸

+✱ t❤❡ s❧✐❞✐♥❣ ❜♦✉♥❞❛r② Γ := {③ = ✵} ✐s

✐♥ ❜❧✉❡✱ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ❜❡t✇❡❡♥ Γ ❛♥❞ t❤❡ ❝♦♥❡ ✐s ✐♥ ❣r❡②✳ ❚❤❡ t✇♦ ❝♦♥❡s ❛r❡ ♠✐♥✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ α =

√ ✸ ✷ ❝♦s β✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✾✴✶✽

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SLIDE 10

▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥❡s ✐♥ R✸

+✱ t❤❡ s❧✐❞✐♥❣ ❜♦✉♥❞❛r② Γ := {③ = ✵} ✐s

✐♥ ❜❧✉❡✱ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ❜❡t✇❡❡♥ Γ ❛♥❞ t❤❡ ❝♦♥❡ ✐s ✐♥ ❣r❡②✳ ❚❤❡ t✇♦ ❝♦♥❡s ❛r❡ ♠✐♥✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ α =

√ ✸ ✷ ❝♦s β✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✵✴✶✽

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SLIDE 11

▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

❆❣❛✐♥ t❤❡ ❝♦♥❡ ✐s ♠✐♥✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ α =

√ ✸ ✷ ❝♦s β✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✶✴✶✽

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SLIDE 12

▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

❆❣❛✐♥ t❤❡ ❝♦♥❡ ✐s ♠✐♥✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ α =

√ ✸ ✷ ❝♦s β✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✷✴✶✽

slide-13
SLIDE 13

▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

▲❡t ✉s s❡t T✷

+ ❛s t❤❡ ❝♦♥❡ ♦✈❡r s❦✶(∆✸) ∩ R✸ +✱ t❤❡♥ T✷ + ✐s ❛ s❧✐❞✐♥❣

♠✐♥✐♠❛❧ ❝♦♥❡ ✐❢ α ≥

✸✳

■♥ ❣❡♥❡r❛❧✱ ❧❡t T♥−✶

+

❜❡ t❤❡ ❝♦♥❡ ♦✈❡r s❦♥−✷(∆♥) ∩ R♥

+✱ t❤❡♥ T♥−✶ +

✐s ❛ s❧✐❞✐♥❣ ♠✐♥✐♠❛❧ ❝♦♥❡ ✐❢ α ≥

✶ √ ✷

  • ♥+✶

♥ ✳

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✸✴✶✽

slide-14
SLIDE 14

◆♦♥✲♠✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✹✴✶✽

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SLIDE 15

❙✐♥❣✉❧❛r ❜♦✉♥❞❛r②

α = ❝♦s ϕ ❝♦s θ

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✺✴✶✽

slide-16
SLIDE 16

❙✐♥❣✉❧❛r ❜♦✉♥❞❛r②

α =

√ ✸ ✷ ❝♦s β ❝♦s ϕ − s✐♥ ϕ ✷

α = −

√ ✸ ✷ ❝♦s β ❝♦s ϕ + s✐♥ ϕ ✷

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✻✴✶✽

slide-17
SLIDE 17

❲❤❛t✬s ♥❡①t❄

❝♦♠♣❧❡t❡ ❧✐st ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s r❡❣✉❧❛r✐t②

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✼✴✶✽

slide-18
SLIDE 18
  • ❘❆❩■❊ P❊❘ ▲✬❆❚❚❊◆❩■❖◆❊

❚❍❆◆❑ ❨❖❯ ❋❖❘ ❨❖❯❘ ❆❚❚❊◆❚■❖◆

❈❛✈❛❧❧♦tt♦ ❊❞♦❛r❞♦ ▼✐♥✐♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❡s ✶✽✴✶✽