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  1. ■♥tr♦❞✉❝t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❈♦♥❝❧✉s✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s ◆❣✉②❡♥ ❚❤✐ ❍♦❛✐ ❚❤✉♦♥❣✱ ❆❜r❛❤❛♠ ❙②❧❧❛✱ ❙é❜❛st✐❡♥ ❚r❛♥ ❚✐❡♥ ❙✉♣❡r✈✐s♦rs✿ ❇❡♥❥❛♠✐♥ ❇♦✉t✐♥ ❛♥❞ ❏❡❛♥✲❋r❛♥ç♦✐s ❈♦✉❧♦♠❜❡❧ ❈❡♠r❛❝s ✷✵✶✾ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

  2. ■♥tr♦❞✉❝t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✶ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✷ ❖✉t✢♦✇ ❜♦✉♥❞❛r②✿ ◆❡✉♠❛♥♥ ❝♦♥❞✐t✐♦♥ ■♥✢♦✇ ❜♦✉♥❞❛r② ❚❤❡ ♠❛✐♥ r❡s✉❧t ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ✸ ■♥✢♦✇ ❜♦✉♥❞❛r② tr❡❛t♠❡♥t ❇♦✉♥❞❛r② ❧❛②❡r ❛t t❤❡ ♦✉t✢♦✇ ❜♦✉♥❞❛r② ❈♦♥❝❧✉s✐♦♥ ✹ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

  3. ■♥tr♦❞✉❝t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✶ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✷ ❖✉t✢♦✇ ❜♦✉♥❞❛r②✿ ◆❡✉♠❛♥♥ ❝♦♥❞✐t✐♦♥ ■♥✢♦✇ ❜♦✉♥❞❛r② ❚❤❡ ♠❛✐♥ r❡s✉❧t ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ✸ ■♥✢♦✇ ❜♦✉♥❞❛r② tr❡❛t♠❡♥t ❇♦✉♥❞❛r② ❧❛②❡r ❛t t❤❡ ♦✉t✢♦✇ ❜♦✉♥❞❛r② ❈♦♥❝❧✉s✐♦♥ ✹ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

  4. ■♥tr♦❞✉❝t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❈♦♥❝❧✉s✐♦♥ ❚r❛♥s♣♦rt ■❇❱P✿ ✩ ❇ t u � a ❇ x u ✏ ✵ t ➙ ✵ x P ♣ ✵ , L q ✫ u ♣ ✵ , x q ✏ u ✵ ♣ x q x P ♣ ✵ , L q u ♣ t , ✵ q ✏ h ♣ t q t ➙ ✵ ✪ ❊①♣❧✐❝✐t s♦❧✉t✐♦♥✿ ★ u ✵ ♣ x ✁ at q , ✐❢ x ➙ at , u ♣ t , x q ✏ t ✁ x ✁ ✠ ✐❢ x ↕ at , h , a ✇❤✐❝❤ ✐s ✐♥ H k � ✶ ♣♣ ✵ , L q ✂ ♣ ✵ , �✽qq ♣r♦✈✐❞❡❞ t❤❛t u ✵ P H k � ✶ ♣♣ ✵ , L qq ✱ h P H k � ✶ ♣♣ ✵ , �✽qq s❛t✐s❢②✿ u ♣ m q ♣ ✵ q ✏ ♣✁ a q ✁ m h ♣ m q ♣ ✵ q . ❅ m ✏ ✵ , . . . , k , ✵ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

  5. ■♥tr♦❞✉❝t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❈♦♥❝❧✉s✐♦♥ ❘❡❣✉❧❛r ❣r✐❞✿ ∆ x : ✏ L ④ J , x j : ✏ j ∆ x ♣ j P Z q , ∆ t : ✏ λ ∆ x ❚✇♦✲t✐♠❡ st❡♣ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ ✐♥ t❤❡ ✐♥t❡r✐♦r ❞♦♠❛✐♥✿ p ➳ u n � ✶ a ℓ u n ✏ j � ℓ , n P N , j ✏ ✶ , . . . , J . j ℓ ✏✁ r ❆ss✉♠♣t✐♦♥ ❋♦r s♦♠❡ ✐♥t❡❣❡r k ➙ ✶ ✱ t❤❡r❡ ❤♦❧❞s✿ p ➳ ℓ m a ℓ ✏ ♣✁ λ a q m , ❅ m ✏ ✵ . . . k , ♣ ❝♦♥s✐st❡♥❝② ♦❢ ♦r❞❡r k q ℓ ✏✁ r ✞ p ✞ ✞ ✞ ♣ ℓ ✷ ✁ st❛❜✐❧✐t② q . ➳ a ℓ e i ℓθ ✞ ↕ ✶ , sup ✞ ✞ ✞ ✞ θ Pr ✵ , ✷ π s ✞ l ✏✁ r ❊①❛♠♣❧❡s✿ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

  6. ■♥tr♦❞✉❝t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❈♦♥❝❧✉s✐♦♥ ❊①❛♠♣❧❡s ✌ ▲❛①✲❲❡♥❞r♦✛ s❝❤❡♠❡ ✭ r ✏ ✶✱ p ✏ ✶✮✳ ✏ ν ♣ ν � ✶ q j � ν ♣ ν ✁ ✶ q u n � ✶ u n � ✶ ✁ ν ✷ ✟ u n u n j ✁ ✶ � j � ✶ ; j ✷ ✷ ✌ ❖✸ s❝❤❡♠❡ ✭ r ✏ ✷✱ p ✏ ✶✮✳ ✏ ✁ ν ♣ ν � ✶ q♣ ✶ ✁ ν q j ✁ ✷ � ν ♣ ν � ✶ q♣ ✷ ✁ ν q u n � ✶ u n u n j ✁ ✶ j ✻ ✷ � ♣ ν � ✶ q♣ ✶ ✁ ν q♣ ✷ ✁ ν q j ✁ ν ♣ ✶ ✁ ν q♣ ✷ ✁ ν q u n u n j � ✶ , ✷ ✻ ✇❤❡r❡ ν ✏ a λ. ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

  7. ■♥tr♦❞✉❝t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✈❛❧✉❡s s❤♦✉❧❞ ✇❡ ♣r❡s❝r✐❜❡ ❢♦r u n j ✱ ❢♦r ✶ ✁ r ↕ j ↕ ✵ ❛♥❞ J � ✶ ↕ j ↕ J � p ❄ ■♥✢♦✇ ❜♦✉♥❞❛r②✿ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ❖✉t✢♦✇ ❜♦✉♥❞❛r②✿ ◆❡✉♠❛♥♥ ❝♦♥❞✐t✐♦♥ ♦❢ ♦r❞❡r k b ➙ ✵ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

  8. ■♥tr♦❞✉❝t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❈♦♥❝❧✉s✐♦♥ ❚❤❡♦r❡♠ ❚❤❡ s♦❧✉t✐♦♥ ♣ u n j q ✶ ↕ j ↕ J , ✵ ↕ n ↕ N t♦ t❤❡ s❝❤❡♠❡ ❛❧♦♥❣ ✇✐t❤ s✉✐t❛❜❧❡ ♥✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛♥❞ ✐♥✐t✐❛❧ ❞❛t✉♠ s❛t✐s✜❡s✿ ✶ ④ ✷ ☎ ☞ ✷ ➺ x j ✞ ✞ J j ✁ ✶ ✞ ✞ ➳ ✞ u n u ♣ t n , y q dy sup ∆ x ✞ ✞ ✆ ✌ ✞ ∆ x ✞ ✵ ↕ n ↕ N x j ✁ ✶ j ✏ ✶ ✞ ↕ C ∆ x ♠✐♥ ♣ k , k b q ✁ ✠ ⑥ u ✵ ⑥ H k � ✶ ♣♣ ✵ , L qq � ⑥ h ⑥ H k � ✶ ♣♣ ✵ , �✽qq ❛♥❞ ➺ x j ✞ ✞ j ✁ ✶ ✞ ✞ ✞ u n u ♣ t n , y q dy sup sup ✞ ✞ ✞ ∆ x ✞ ✵ ↕ n ↕ N ✶ ↕ j ↕ J x j ✁ ✶ ✞ ↕ C ∆ x ♠✐♥ ♣ k , k b q✁ ✶ ④ ✷ ✁ ✠ ⑥ u ✵ ⑥ H k � ✶ ♣♣ ✵ , L qq � ⑥ h ⑥ H k � ✶ ♣♣ ✵ , �✽qq ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

  9. ■♥tr♦❞✉❝t✐♦♥ ❖✉t✢♦✇ ❜♦✉♥❞❛r②✿ ◆❡✉♠❛♥♥ ❝♦♥❞✐t✐♦♥ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ■♥✢♦✇ ❜♦✉♥❞❛r② ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❚❤❡ ♠❛✐♥ r❡s✉❧t ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✶ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✷ ❖✉t✢♦✇ ❜♦✉♥❞❛r②✿ ◆❡✉♠❛♥♥ ❝♦♥❞✐t✐♦♥ ■♥✢♦✇ ❜♦✉♥❞❛r② ❚❤❡ ♠❛✐♥ r❡s✉❧t ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ✸ ■♥✢♦✇ ❜♦✉♥❞❛r② tr❡❛t♠❡♥t ❇♦✉♥❞❛r② ❧❛②❡r ❛t t❤❡ ♦✉t✢♦✇ ❜♦✉♥❞❛r② ❈♦♥❝❧✉s✐♦♥ ✹ ◆✉♠❡r✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r tr❛♥s♣♦rt ❡q✉❛t✐♦♥s

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