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  1. ▼✐s❧❡❛❞✐♥❣ st❛rs✿ ❲❤❛t ❝❛♥♥♦t ❜❡ ♠❡❛s✉r❡❞ ♦♥ t❤❡ ■♥t❡r♥❡t ❄ ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  2. ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  3. ❍♦✇ ❛❝❝✉r❛t❡ ❛r❡ ♥❡t✇♦r❦ ♠❛♣s ❄ ❲❤② ❄ ❚♦ ❞❡✈❡❧♦♣✴❛❞❛♣t ♣r♦t♦❝♦❧s t♦ ■♥t❡r♥❡t P❛❉■❙✱ ❘▼❚P ❚♦ ✉♥❞❡rst❛♥❞ t❤❡ ✐♠♣❛❝t ♦❢ ✉♥❝❡rt❛✐♥t②✿ ♥❡t✇♦r❦s ♠❡tr♦❧♦❣② ❄ ❍♦✇ ❄ ♠✉❧t✐❝❛st [ ▼❛r❝❤❡tt❛ ❡t ❛❧✳ , ❏❙❆❈ ′ ✶✶ ] ♥❡t✇♦r❦ t♦♠♦❣r❛♣❤② ❚r❛❝❡r♦✉t❡ ❚❛❦❡❛✇❛② ❈♦♠♣❛r❡ ✐♥t❡r♥❡t t♦♣♦❧♦❣✐❡s ✐♥st❡❛❞ ♦❢ ❝♦✉♥t✐♥❣ t❤❡♠ ▲♦❝❛❧ ♣r♦♣❡rt✐❡s s✉✛❡r ●❧♦❜❛❧ ♣r♦♣❡rt✐❡s s✉✛❡r ❧❡ss ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  4. ❚r❛❝❡r♦✉t❡ ♣r✐♥❝✐♣❧❡ ■❞❡❛❂ s❡♥❞ ✓ ❜✉❣❣② ✔ ❚❚▲ ♣❛❝❦❡ts✱ ❝♦❧❧❡❝t ❡rr♦r ♠❡ss❛❣❡s ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  5. ❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s ❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s ❚r❛❝❡✿ t ✶ = ❛ , ⋆ ✶ , ❜ → t ✷ = ❜ , ⋆ ✷ , ✐ ✶ , ✐ ✷ , ✐ ✸ , ⋆ ✸ , ❝ ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  6. ❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s ❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s ❚r❛❝❡✿ t ✶ = ❛ , ⋆ ✶ , ❜ → t ✷ = ❜ , ⋆ ✷ , ✐ ✶ , ✐ ✷ , ✐ ✸ , ⋆ ✸ , ❝ ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  7. ❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s ❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s ❚r❛❝❡✿ t ✶ = ❛ , ⋆ ✶ , ❜ ❄ t ✷ = ❜ , ⋆ ✷ , ✐ ✶ , ✐ ✷ , ✐ ✸ , ⋆ ✸ , ❝ ← ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  8. ❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s ❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s ❚r❛❝❡✿ t ✶ = ❛ , ⋆ ✶ , ❜ t ✷ = ❜ , ⋆ ✷ , ✐ ✶ , ✐ ✷ , ✐ ✸ , ⋆ ✸ , ❝ ← ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  9. ❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s ❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s ❚r❛❝❡✿ t ✶ = ❛ , ⋆ ✶ , ❜ t ✷ = ❜ , ⋆ ✷ , ✐ ✶ , ✐ ✷ , ✐ ✸ , ⋆ ✸ , ❝ ← ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  10. ❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s ❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s ❚r❛❝❡✿ t ✶ = ❛ , ⋆ ✶ , ❜ t ✷ = ❜ , ⋆ ✷ , ✐ ✶ , ✐ ✷ , ✐ ✸ , ⋆ ✸ , ❝ ← ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  11. ❘❡❧❛t❡❞ ❲♦r❦s ❲♦r❦s ✐♥✐t✐❛t❡❞ ❜② ❆❝❤❛r②❛ ❛♥❞ ●♦✉❞❛ [ ❙❙❙ ✵✾ , ■❈❉❈◆ ✶✵ , ■❈❉❈◆ ✶✶ ] ▼♦❞❡❧s t♦ ❝❛♣t✉r❡ ✐rr❡❣✉❧❛r ♥♦❞❡s ■rr❡❣✉❧❛r ♥♦❞❡s → ❛♥♦♥②♠♦✉s ♥♦❞❡s ❈❡♥tr❛❧ ❝♦♥❝❡♣t ♦❢ ♠✐♥✐♠❛❧ t♦♣♦❧♦❣✐❡s ❈♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❣❡♥❡r❛❜❧❡ t♦♣♦❧♦❣✐❡s ❖✉r ❛♣♣r♦❛❝❤✿ ▲♦ts ♦❢ ❣❡♥❡r❛❜❧❡ t♦♣♦❧♦❣✐❡s ✐s ♥♦t ❛ ♣r♦❜❧❡♠ ✐❢ t❤❡② ❛r❡ ❛❧❧ s✐♠✐❧❛r ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  12. ▼♦❞❡❧ ● ✵ ( ❱ ✵ , ❊ ✵ ) ✿ st❛t✐❝ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ❂ ❚❛r❣❡t ❚♦♣♦❧♦❣② ✳ ✈ ∈ ❱ ✵ ✐s ❡✐t❤❡r ♥❛♠❡❞✿ ❛❧✇❛②s ❛♥s✇❡rs ✇✐t❤ ✐ts ♦♥❧② ♥❛♠❡ ✭♥♦ ❛❧✐❛s✐♥❣✮ ❡✐t❤❡r ❛♥♦♥②♠♦✉s✿ ❛❧✇❛②s ❛♥s✇❡rs ⋆ ✳ ◆♦t ♥❡❝❡ss❛r✐❧② ♠✐♥✐♠❛❧ ✦ ❞ ● ✵ ( ✉ , ✈ ) ✐s t❤❡ s❤♦rt❡st ♣❛t❤ ❞✐st❛♥❝❡✳ ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

  13. ▼♦❞❡❧ ● ✵ ( ❱ ✵ , ❊ ✵ ) ✿ st❛t✐❝ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ❂ ❚❛r❣❡t ❚♦♣♦❧♦❣② ✳ ✈ ∈ ❱ ✵ ✐s ❡✐t❤❡r ♥❛♠❡❞✿ ❛❧✇❛②s ❛♥s✇❡rs ✇✐t❤ ✐ts ♦♥❧② ♥❛♠❡ ✭♥♦ ❛❧✐❛s✐♥❣✮ ❡✐t❤❡r ❛♥♦♥②♠♦✉s✿ ❛❧✇❛②s ❛♥s✇❡rs ⋆ ✳ ◆♦t ♥❡❝❡ss❛r✐❧② ♠✐♥✐♠❛❧ ✦ ❞ ● ✵ ( ✉ , ✈ ) ✐s t❤❡ s❤♦rt❡st ♣❛t❤ ❞✐st❛♥❝❡✳ ❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

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