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r rs r ( , ) r rs r t rt r ( , ) r rst r

  1. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ▼♦❞✉❧❛r ❋♦r♠s ❢♦r t❤❡ ❖rt❤♦❣♦♥❛❧ ●r♦✉♣ ❖ ( ✷ , ✺ ) ■♥❣♦ ❑❧ö❝❦❡r ▲❡❤rst✉❤❧ ❆ ❢ür ▼❛t❤❡♠❛t✐❦ ❘❲❚❍ ❆❛❝❤❡♥ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ▼P■▼ ❇♦♥♥ ✷✵✵✺✲✵✼✲✷✼ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶ ✴ ✷✾

  2. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷ ✴ ✷✾

  3. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ■♥tr♦❞✉❝t✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✸ ✴ ✷✾

  4. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ■♥tr♦❞✉❝t✐♦♥ ❍✐st♦r② ◮ ❖ ( ✷ , ✶ ) ✿ ❊❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳ ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t t❤❡ ❣r❛❞❡❞ r✐♥❣ A ( ❙▲ ✷ ( Z )) ♦❢ ❡❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❡❧❧✐♣t✐❝ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❣ ✷ ❛♥❞ ❣ ✸ ✭♦❢ ✇❡✐❣❤t ✹ ❛♥❞ ✻✮✳ ◮ ❖ ( ✷ , ✷ ) ✿ ❍✐❧❜❡rt ♠♦❞✉❧❛r ❢♦r♠s✳ ❈❢✳ ❙✳ ▼❛②❡r✬s t❛❧❦✳ ◮ ❖ ( ✷ , ✸ ) ✿ ❙✐❡❣❡❧ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❏✳✲■✳ ■❣✉s❛ ✭✶✾✻✷✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ A ( ❙♣ ✷ ( Z )) ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❙✐❡❣❡❧ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❊ ✹ ✱ ❊ ✻ ✱ ❊ ✶✵ ❛♥❞ ❊ ✶✷ ✳ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✹ ✴ ✷✾

  5. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ■♥tr♦❞✉❝t✐♦♥ ❍✐st♦r② ◮ ❖ ( ✷ , ✶ ) ✿ ❊❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳ ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t t❤❡ ❣r❛❞❡❞ r✐♥❣ A ( ❙▲ ✷ ( Z )) ♦❢ ❡❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❡❧❧✐♣t✐❝ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❣ ✷ ❛♥❞ ❣ ✸ ✭♦❢ ✇❡✐❣❤t ✹ ❛♥❞ ✻✮✳ ◮ ❖ ( ✷ , ✷ ) ✿ ❍✐❧❜❡rt ♠♦❞✉❧❛r ❢♦r♠s✳ ❈❢✳ ❙✳ ▼❛②❡r✬s t❛❧❦✳ ◮ ❖ ( ✷ , ✸ ) ✿ ❙✐❡❣❡❧ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❏✳✲■✳ ■❣✉s❛ ✭✶✾✻✷✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ A ( ❙♣ ✷ ( Z )) ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❙✐❡❣❡❧ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❊ ✹ ✱ ❊ ✻ ✱ ❊ ✶✵ ❛♥❞ ❊ ✶✷ ✳ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✹ ✴ ✷✾

  6. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ■♥tr♦❞✉❝t✐♦♥ ❍✐st♦r② ◮ ❖ ( ✷ , ✶ ) ✿ ❊❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳ ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t t❤❡ ❣r❛❞❡❞ r✐♥❣ A ( ❙▲ ✷ ( Z )) ♦❢ ❡❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❡❧❧✐♣t✐❝ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❣ ✷ ❛♥❞ ❣ ✸ ✭♦❢ ✇❡✐❣❤t ✹ ❛♥❞ ✻✮✳ ◮ ❖ ( ✷ , ✷ ) ✿ ❍✐❧❜❡rt ♠♦❞✉❧❛r ❢♦r♠s✳ ❈❢✳ ❙✳ ▼❛②❡r✬s t❛❧❦✳ ◮ ❖ ( ✷ , ✸ ) ✿ ❙✐❡❣❡❧ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❏✳✲■✳ ■❣✉s❛ ✭✶✾✻✷✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ A ( ❙♣ ✷ ( Z )) ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❙✐❡❣❡❧ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❊ ✹ ✱ ❊ ✻ ✱ ❊ ✶✵ ❛♥❞ ❊ ✶✷ ✳ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✹ ✴ ✷✾

  7. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ■♥tr♦❞✉❝t✐♦♥ ❍✐st♦r② ✭❝♦♥t✐♥✉❡❞✮ ◮ ❖ ( ✷ , ✹ ) ✿ ❍❡r♠✐t✐❛♥ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❊✳ ❋r❡✐t❛❣ ✭✶✾✻✼✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q ( √− ✶ ) ✱ ❚✳ ❉❡r♥ ✭✷✵✵✶✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q ( √− ✸ ) ❛♥❞ Q ( √− ✷ ) ✭✇✐t❤ ❆✳ ❑r✐❡❣✮✳ ◮ ❖ ( ✷ , ✺ ) ✿ ❚❤✐s ✐s t❤❡ ❝❛s❡ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r✳ ◮ ❖ ( ✷ , ✻ ) ✿ ◗✉❛t❡r♥✐♦♥✐❝ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❆✳ ❑r✐❡❣ ✭✷✵✵✺✮ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✺ ✴ ✷✾

  8. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ■♥tr♦❞✉❝t✐♦♥ ❍✐st♦r② ✭❝♦♥t✐♥✉❡❞✮ ◮ ❖ ( ✷ , ✹ ) ✿ ❍❡r♠✐t✐❛♥ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❊✳ ❋r❡✐t❛❣ ✭✶✾✻✼✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q ( √− ✶ ) ✱ ❚✳ ❉❡r♥ ✭✷✵✵✶✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q ( √− ✸ ) ❛♥❞ Q ( √− ✷ ) ✭✇✐t❤ ❆✳ ❑r✐❡❣✮✳ ◮ ❖ ( ✷ , ✺ ) ✿ ❚❤✐s ✐s t❤❡ ❝❛s❡ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r✳ ◮ ❖ ( ✷ , ✻ ) ✿ ◗✉❛t❡r♥✐♦♥✐❝ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❆✳ ❑r✐❡❣ ✭✷✵✵✺✮ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✺ ✴ ✷✾

  9. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ■♥tr♦❞✉❝t✐♦♥ ❍✐st♦r② ✭❝♦♥t✐♥✉❡❞✮ ◮ ❖ ( ✷ , ✹ ) ✿ ❍❡r♠✐t✐❛♥ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❊✳ ❋r❡✐t❛❣ ✭✶✾✻✼✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q ( √− ✶ ) ✱ ❚✳ ❉❡r♥ ✭✷✵✵✶✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q ( √− ✸ ) ❛♥❞ Q ( √− ✷ ) ✭✇✐t❤ ❆✳ ❑r✐❡❣✮✳ ◮ ❖ ( ✷ , ✺ ) ✿ ❚❤✐s ✐s t❤❡ ❝❛s❡ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r✳ ◮ ❖ ( ✷ , ✻ ) ✿ ◗✉❛t❡r♥✐♦♥✐❝ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳ ❆✳ ❑r✐❡❣ ✭✷✵✵✺✮ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✺ ✴ ✷✾

  10. ▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖ ( ✷ , ✺ ) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✻ ✴ ✷✾

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