❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ P✐❡rr❡ ▲♦✐❞r❡❛✉ ❉●❆ ▼■ ❛♥❞ ■❘▼❆❘✱ ❯♥✐✈❡rs✐té ❞❡ ❘❡♥♥❡s ✶ P◗❈r②♣t♦ ✷✵✶✼✱ ❏✉♥❡ ✷✻t❤
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ▼♦t✐✈❛t✐♦♥s P♦st✲◗✉❛♥t✉♠ ❝r②♣t♦❣r❛♣❤② ▼✉❧t✐✈❛r✐❛t❡ ❝r②♣t♦❣r❛♣❤② ❍❛s❤✲❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ■s♦❣❡♥✐❡s ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ❉❡❝♦❞✐♥❣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ▲❛tt✐❝❡s ❈♦❞❡s ⇒ ❘❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❙♠❛❧❧❡r ❦❡②s ❢♦r ❛ ❣✐✈❡♥ s❡❝✉r✐t② t❛r❣❡t ❆♥♦t❤❡r ❛❧t❡r♥❛t✐✈❡ t♦ ❍❛♠♠✐♥❣ ♠❡tr✐❝ ♦r ❊✉❝❧✐❞✐❛♥ ♠❡tr✐❝ ❜❛s❡❞ ♣r✐♠✐t✐✈❡s✳
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ✶ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❛♥❞ ●P❚ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✷ ❆♥ ❡✈♦❧✉t✐♦♥ ♦❢ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ✸ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ✹
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ✶ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❛♥❞ ●P❚ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✷ ❆♥ ❡✈♦❧✉t✐♦♥ ♦❢ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ✸ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ✹
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ■♥ t❤❡♦r② ❈♦❞❡✲❇❛s❡❞ ❊♥❝r②♣t✐♦♥✿ s♦❧✈✐♥❣ ❣❡♥❡r❛❧ ❞❡❝♦❞✐♥❣ ♣r♦❜❧❡♠s ✐♥ t❤❡ ♠❡tr✐❝ ✐s ❤❛r❞ ■♥ ❍❛♠♠✐♥❣ ♠❡tr✐❝✿ ❉❡❝✲ ❇♦✉♥❞❡❞ ❉✐st❛♥❝❡ ❉❡❝♦❞✐♥❣ ✐s ◆P ✲❝♦♠♣❧❡t❡✱ ❬❇▼✈❚✼✽❪ ❘❛♥❦ ♠❡tr✐❝ ❞❡❝♦❞✐♥❣ r❡❧❛t❡❞ t♦ t✇♦ ❞✐✣❝✉❧t ♣r♦❜❧❡♠s✿ ▼✐♥❘❛♥❦ ✱ ◆P ✲❝♦♠♣❧❡t❡ ❉❡❝✲ ❘❛♥❦ ❙②♥❞r♦♠❡ ❉❡❝♦❞✐♥❣ ✐♥ ❩PP ⇒ ❩PP = ◆P ✱ ❬●❩✶✺❪
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ■♥ ♣r❛❝t✐❝❡ ❈♦♥s✐❞❡r ❛ r❛♥❞♦♠ [ ♥ , ❘♥ ] ✲❝♦❞❡ ♦✈❡r F ✷ ♥ ✱ ❉❡❝♦❞✐♥❣ ❡rr♦rs ♦❢ r❛♥❦ δ ♥ ✱ ❬●❘❙✶✻❪✿ ✷ ❝ ❛❧❣♦ ( δ ) ♥ ✷ +Ω( ❧♦❣ ( ♥ )) ❉❡❝♦❞✐♥❣ ❡rr♦rs ♦❢ ❍❛♠♠✐♥❣ ✇❡✐❣❤t δ ♥ ✿ ✷ ❝ ❛❧❣♦ ( δ ) ♥ + ♦ ( ✶ ) ❉❡❝✳ ❈♦♠♣❧❡①✳ ❍❛♠✳ ▼❡t✳ ●❡♥✳ ▼❛t✳ ❘❛♥❦ ▼❡t✳ ●❡♥✳ ▼❛t✳ ✷ ✶✷✽ [ ✷✹✵✵ , ✷✵✵✻ , ✺✽ ] ✷ ≈ ✶✵✵ ❑❇ [ ✹✽ , ✸✾ , ✹ ] ✷ ✹✽ ≈ ✷ . ✷ ❑❇ ✷ ✷✺✻ [ ✹✶✺✵ , ✸✸✵✼ , ✶✸✷ ] ✷ ≈ ✸✺✵ ❑❇ [ ✼✵ , ✺✵ , ✺ ] ✷ ✼✵ ≈ ✽ . ✼ ❑❇ ❚❛❜❧❡✿ ❉❡❝♦❞✐♥❣ ❝♦♠♣❧❡①✐t② ♦♥ ❝❧❛ss✐❝❛❧ ❝♦♠♣✉t❡r✱ ❬❈❚❙✶✻❪ ⇒ ❘❛♥❦ ♠❡tr✐❝ ♣r♦✈✐❞❡s ❜❡tt❡r s❡❝✉r✐t②✴s✐③❡ tr❛❞❡♦✛ ⇒ ■♥ P◗ ✲✇♦r❧❞✱ ❡①♣♦♥❡♥t✐❛❧ ❝♦♠♣❧❡①✐t② ✐s sq✉❛r❡✲r♦♦t❡❞✱ ❬●❍❚✶✻❪
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ❘❛♥❦ ♠❡tr✐❝✱ ❬●❛❜✽✺❪ ❉❡✜♥✐t✐♦♥ γ ✶ , . . . , γ ♠ ✱ ❛ ❜❛s✐s ♦❢ F ✷ ♠ / F ✷ ✱ ❡ = ( ❡ ✶ , . . . , ❡ ♥ ) ∈ ( F ✷ ♠ ) ♥ , ❡ ✐ �→ ( ❡ ✐ ✶ , . . . , ❡ ✐♥ ) ✱ ❡ ✶✶ · · · ❡ ✶ ♥ ∀ ❡ ∈ ( F ✷ ♠ ) ♥ , ✳ ✳ ✳✳✳ ❘❦ ( ❡ ) ❞❡❢ ✳ ✳ = ❘❦ ✳ ✳ ❡ ♠ ✶ · · · ❡ ♠♥ [ ♥ , ❦ , ❞ ] r ❝♦❞❡✿ C ⊂ F ♥ ✷ ♠ ✱ ❦ ✲❞✐♠❡♥s✐♦♥❛❧✱ ❞ = ♠✐♥ ❝ � = ✵ ∈C ❘❦ ( ❝ ) ❙✐♥❣❧❡t♦♥ ♣r♦♣❡rt② ❞ − ✶ ≤ ♥ − ❦ ✭✐❢ ♥ ≤ ♠ ✮ ❘❦ ( ❡ ) = t ⇔ ∃V ⊂ F ✷ ♠ , s✳t✳ ❞✐♠ ✷ ( V ) = t ❛♥❞ ❡ ✐ ∈ V , ∀ ✐
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ❊①❛♠♣❧❡ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ❡ = ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ✶ ✵ ■♥ F ✷ ✺ ✇❡ ❤❛✈❡ ❡ = ( α, β, α + β, β, α + β ) ❍❛♠♠✐♥❣ ✇❡✐❣❤t✿ ✺ ❘❛♥❦✿ ✷
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ❘❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❡♥❝r②♣t✐♦♥ ❑❡② ❣❡♥❡r❛t✐♦♥ Pr✐✈❛t❡✲❦❡② C ❛ [ ♥ , ❦ , ❞ ] r t ✲r❛♥❦ ❡rr♦r ❞❡❝♦❞❛❜❧❡ ❝♦❞❡ ♦✈❡r F ✷ ♠ ▲ : F ♥ ✷ ♠ �→ F ♥ ✷ ♠ ✱ s✳t✳ ▲ ✐s ✈❡❝t♦r✲s♣❛❝❡ ✐s♦♠♦r♣❤✐s♠ ▲ ✐s ❛ r❛♥❦ ✐s♦♠❡tr② P✉❜❧✐❝✲❦❡②✿ C ♣✉❜ = ▲ − ✶ ( C ) ✳ Pr♦❝❡ss ❊♥❝r②♣t✐♦♥✿ ② = ❝ ∈ C ♣✉❜ + ❡ ✱ ✇❤❡r❡ ❘❦ ( ❡ ) ≤ t ❉❡❝r②♣t✐♦♥✿ ▲ ( ② ) = ▲ ( ❝ ) ∈ C + ▲ ( ❡ ) ❉❡❝♦❞❡ ⇒ ❝
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❛♥❞ ●P❚ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ❲❤② r❛♥❦ ♠❡tr✐❝ ❄ ✶ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❛♥❞ ●P❚ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✷ ❆♥ ❡✈♦❧✉t✐♦♥ ♦❢ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ✸ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ✹
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❛♥❞ ●P❚ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s✱ ❬●❛❜✽✺❪ ❉❡✜♥✐t✐♦♥ ✭●❛❜✐❞✉❧✐♥ ❝♦❞❡s✮ ▲❡t ❣ = ( ❣ ✶ , . . . , ❣ ♥ ) ∈ ( F ✷ ♠ ) ♥ ✱ F ✷ ✲❧✳✐✳✱ [ ✐ ] ❞❡❢ = ✷ ✐ ❣ ✶ · · · ❣ ♥ ✳ ✳ ✳✳✳ ✳ ✳ ●❛❜ ❦ ( ❣ ) = � ● � , ✇❤❡r❡ ● = ✳ ✳ ❣ [ ❦ − ✶ ] ❣ [ ❦ − ✶ ] · · · ♥ ✶ Pr♦♣❡rt✐❡s ♦❢ ●❛❜ ❦ ( ❣ ) ❖♣t✐♠❛❧ [ ♥ , ❦ , ❞ ] r ❝♦❞❡s ❢♦r r❛♥❦ ♠❡tr✐❝✿ ♥ − ❦ = ❞ − ✶ P✲t✐♠❡ q✉❛❞r❛t✐❝ ❞❡❝♦❞✐♥❣ ✉♣ t♦ t = ⌊ ( ♥ − ❦ ) / ✷ ⌋ ✱ ❬●❛❜✽✺❪ ❙✉✣❝✐❡♥t❧② s❝r❛♠❜❧❡❞ ⇒ ▼❝❊❧✐❡❝❡✲❧✐❦❡ ❝r②♣t♦s②st❡♠s✳
❆ ♥❡✇ r❛♥❦ ♠❡tr✐❝ ❝♦❞❡s ❜❛s❡❞ ❝r②♣t♦s②st❡♠ ●❛❜✐❞✉❧✐♥ ❝♦❞❡s ❛♥❞ ●P❚ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ❘✐s❡ ❛♥❞ ❢❛❧❧ ♦❢ ●P❚ ❡♥❝r②♣t✐♦♥ ✲ ❬●P❚✾✶✱ ❑s❤✵✼✱ ❘●❍✶✵✱ ❖❑◆✶✻❪ ▲✐♥❡❛r r❛♥❦ ♣r❡s❡r✈✐♥❣ ✐s♦♠❡tr✐❡s ♦❢ F ♥ ✷ ♠ ✿ P ∈ ▼ ♥ ( F ✷ ) ❙✐♥❝❡ ●❛❜ ❦ ( ❣ ) P = ●❛❜ ❦ ( ❣P ) ⇒ ◆❡❝❡ss✐t② ♦❢ s❝r❛♠❜❧✐♥❣ ❇✉t ❋♦r ❛♥② ♣✉❜❧✐s❤❡❞ r❡♣❛r❛t✐♦♥✱ ❛❧✇❛②s ♣♦ss✐❜❧❡ t♦ ✇r✐t❡ ✶ ) P ∗ , P ∗ ∈ ▼ ♥ ( F ✷ ) ● ♣✉❜ = ❙ ✶ ( ❳ ✶ | ● ✶ ���� ●❛❜ ❦ ( ❣ ✶ ) ⇒ ❙t❛❜✐❧✐t② t❤r♦✉❣❤ ❣ �→ ❣ [ ✐ ] ✱ ✷ � � ( ● ♣✉❜ ) [ ✐ ] = ❙ [ ✐ ] ❳ [ ✐ ] ✶ | ● [ ✐ ] P ∗ ✶ ✶ ⇒ ❆♣♣❧② ❖✈❡r❜❡❝❦✬s ❧✐❦❡ ❛tt❛❝❦s ✸
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