●❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❢♦r ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ❛ ❣✐✈❡♥ ❣❡♥✉s ■▼◆❙ ✷✵✶✵ ✲ ■❜❡r✐❛♥ ▼❡❡t✐♥❣ ♦♥ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ●r❛♥❛❞❛✱ ❋❡❜r✉❛r② ✷✵✶✵ ❱í❝t♦r ❇❧❛♥❝♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ➪❧❣❡❜r❛ ❯♥✐✈❡rs✐❞❛❞ ❞❡ ●r❛♥❛❞❛ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ P✳❆✳ ●❛r❝í❛✲❙á♥❝❤❡③ ❛♥❞ ❏✳ P✉❡rt♦ ✮
❖✉t❧✐♥❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ❈♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s✿ ❚❤❡♦r❡t✐❝❛❧ ❘❡s✉❧ts ✸ ●❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ✐♥ ♣r❛❝t✐❝❡ ✹ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ✹
▼✉❧t✐♣❧✐❝✐t② ✸✿ ✭❘♦s❛❧❡s✱ ✷✵✵✺✮✳ ❇♦✉♥❞s ✉s✐♥❣ ❉✐❝❦ ♣❛t❤s ✭❇r❛s✲❆♠♦rós ❛♥❞ ❉❡ ▼✐❡r✱ ✷✵✵✼✮ ❋✐❜♦♥❛❝❝✐ ❈♦♥❥❡❝t✉r❡✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✽✮✳ ❇♦✉♥❞s ✉s✐♥❣ t❤❡ s❡♠✐❣r♦✉♣s tr❡❡ ♠❡t❤♦❞✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✾✮ ■♠♣r♦✈❡❞ ❇♦✉♥❞s ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❜② ❣❡♥❡r❛t✐♥❣ tr❡❡s✿ ✭❊❧✐③❛❧❞❡✱ ✷✵✵✾✮ ■♠♣r♦✈❡❞ ❛s②♠♣t♦t✐❝ ❧♦✇❡r ❜♦✉♥❞✿ ✭❩❤❛♦✱ ✷✵✵✾✮ ❙♦❢t✇❛r❡✿ ♥✉♠❡r✐❝❛❧s❣♣s ✐♥ ●❆P ✭❉❡❧❣❛❞♦ ❡t ❛❧✱ ✷✵✵✽✮ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ❙ = � ♥ ✶ , . . . , ♥ ♣ � ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣✱ ❣❝❞ ( ♥ ✐ , ♥ ❥ ) = ✶✳ ♠✉❧t✐♣❧✐❝✐t②✿ ♠ ( ❙ ) = ♠✐♥ { ♥ ∈ N : ♥ ∈ ❙ \{ ✵ }} ❣❡♥✉s✿ ❣ ( ❙ ) = ★ ( N \ ❙ ) ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✿ ❋ ( ❙ ) = ♠❛① { ♥ ∈ N : ♥ �∈ ❙ } S ❣ = { ❙ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ❣ ( ❙ ) = ❣ } , ♥ ❣ = ★ S ❣
❙♦❢t✇❛r❡✿ ♥✉♠❡r✐❝❛❧s❣♣s ✐♥ ●❆P ✭❉❡❧❣❛❞♦ ❡t ❛❧✱ ✷✵✵✽✮ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ❙ = � ♥ ✶ , . . . , ♥ ♣ � ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣✱ ❣❝❞ ( ♥ ✐ , ♥ ❥ ) = ✶✳ ♠✉❧t✐♣❧✐❝✐t②✿ ♠ ( ❙ ) = ♠✐♥ { ♥ ∈ N : ♥ ∈ ❙ \{ ✵ }} ❣❡♥✉s✿ ❣ ( ❙ ) = ★ ( N \ ❙ ) ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✿ ❋ ( ❙ ) = ♠❛① { ♥ ∈ N : ♥ �∈ ❙ } S ❣ = { ❙ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ❣ ( ❙ ) = ❣ } , ♥ ❣ = ★ S ❣ • ▼✉❧t✐♣❧✐❝✐t② ✸✿ ✭❘♦s❛❧❡s✱ ✷✵✵✺✮✳ • ❇♦✉♥❞s ✉s✐♥❣ ❉✐❝❦ ♣❛t❤s ✭❇r❛s✲❆♠♦rós ❛♥❞ ❉❡ ▼✐❡r✱ ✷✵✵✼✮ • ❋✐❜♦♥❛❝❝✐ ❈♦♥❥❡❝t✉r❡✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✽✮✳ • ❇♦✉♥❞s ✉s✐♥❣ t❤❡ s❡♠✐❣r♦✉♣s tr❡❡ ♠❡t❤♦❞✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✾✮ • ■♠♣r♦✈❡❞ ❇♦✉♥❞s ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❜② ❣❡♥❡r❛t✐♥❣ tr❡❡s✿ ✭❊❧✐③❛❧❞❡✱ ✷✵✵✾✮ • ■♠♣r♦✈❡❞ ❛s②♠♣t♦t✐❝ ❧♦✇❡r ❜♦✉♥❞✿ ✭❩❤❛♦✱ ✷✵✵✾✮
❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ❙ = � ♥ ✶ , . . . , ♥ ♣ � ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣✱ ❣❝❞ ( ♥ ✐ , ♥ ❥ ) = ✶✳ ♠✉❧t✐♣❧✐❝✐t②✿ ♠ ( ❙ ) = ♠✐♥ { ♥ ∈ N : ♥ ∈ ❙ \{ ✵ }} ❣❡♥✉s✿ ❣ ( ❙ ) = ★ ( N \ ❙ ) ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✿ ❋ ( ❙ ) = ♠❛① { ♥ ∈ N : ♥ �∈ ❙ } S ❣ = { ❙ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ❣ ( ❙ ) = ❣ } , ♥ ❣ = ★ S ❣ • ▼✉❧t✐♣❧✐❝✐t② ✸✿ ✭❘♦s❛❧❡s✱ ✷✵✵✺✮✳ • ❇♦✉♥❞s ✉s✐♥❣ ❉✐❝❦ ♣❛t❤s ✭❇r❛s✲❆♠♦rós ❛♥❞ ❉❡ ▼✐❡r✱ ✷✵✵✼✮ • ❋✐❜♦♥❛❝❝✐ ❈♦♥❥❡❝t✉r❡✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✽✮✳ • ❇♦✉♥❞s ✉s✐♥❣ t❤❡ s❡♠✐❣r♦✉♣s tr❡❡ ♠❡t❤♦❞✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✾✮ • ■♠♣r♦✈❡❞ ❇♦✉♥❞s ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❜② ❣❡♥❡r❛t✐♥❣ tr❡❡s✿ ✭❊❧✐③❛❧❞❡✱ ✷✵✵✾✮ • ■♠♣r♦✈❡❞ ❛s②♠♣t♦t✐❝ ❧♦✇❡r ❜♦✉♥❞✿ ✭❩❤❛♦✱ ✷✵✵✾✮ ❙♦❢t✇❛r❡✿ ♥✉♠❡r✐❝❛❧s❣♣s ✐♥ ●❆P ✭❉❡❧❣❛❞♦ ❡t ❛❧✱ ✷✵✵✽✮
❚❤❡♦r❡♠ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮ ③ ◆ ✶ ✶ ◆ ✵ ③ ✐ ③ ✷ ③ ◆ ◗ ✵ ◆ ✿ ❢ ◗ ③ ✶ ③ ✐ ✶ ③ ♥ ✱ ❆ss✉♠❡ ♥✱ t❤❡ ❞✐♠❡♥s✐♦♥✱ ✐s ✜①❡❞✳ ●✐✈❡♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ P ♥ ③ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢ P ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐♥ t❤❡ ❢♦r♠ ③ ✉ ✐ ♥ ③ ❢ P ✐ ♥ ✐ ■ ③ ✈ ✐❥ ✶ ❥ ✶ ✇❤❡r❡ ■ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✲s✐③❡ ✐♥❞❡①✐♥❣ s❡t✱ ❛♥❞ ✇❤❡r❡ ✶ ✶ ❛♥❞ ♥ ❢♦r ❛❧❧ ✐ ❛♥❞ ❥✳ ✉ ✐ ✈ ✐❥ ❙♦❢t✇❛r❡✿ ▲❛tt❊ ✭❉❡▲♦❡r❛ ❡t ❛❧✳ ✷✵✵✺✮✱ ❜❛r✈✐♥♦❦ ✭❱❡r❞♦♦❧❛❡❣❡✱ ✷✵✵✽✮ ❆♣❧✐❝❛t✐♦♥s✿ ❙♦❧✈✐♥❣ ■P Pr♦❜❧❡♠s✱ ❈♦♠♣❧❡①✐t② r❡s✉❧ts ✐♥ ●❛♠❡ ❚❤❡♦r②✱ ▼✉❧t✐♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥✱ ❋✉③③② Pr♦❣r❛♠♠✐♥❣✱ ✳✳✳ ❛♥❞ ❈❖❯◆❚■◆● ●❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ▲❡t ◗ ⊆ Z ♥ ✱ ✐ts ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s � ❢ ( P ; ③ ) = ③ α α ∈ ◗
❚❤❡♦r❡♠ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮ ③ ◆ ✶ ✶ ✶ ③ ♥ ✱ ❆ss✉♠❡ ♥✱ t❤❡ ❞✐♠❡♥s✐♦♥✱ ✐s ✜①❡❞✳ ●✐✈❡♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ P ♥ ③ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢ P ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐♥ t❤❡ ❢♦r♠ ③ ✉ ✐ ♥ ③ ❢ P ✐ ♥ ✐ ■ ③ ✈ ✐❥ ✶ ❥ ✶ ✇❤❡r❡ ■ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✲s✐③❡ ✐♥❞❡①✐♥❣ s❡t✱ ❛♥❞ ✇❤❡r❡ ✶ ✶ ❛♥❞ ♥ ❢♦r ❛❧❧ ✐ ❛♥❞ ❥✳ ✉ ✐ ✈ ✐❥ ❙♦❢t✇❛r❡✿ ▲❛tt❊ ✭❉❡▲♦❡r❛ ❡t ❛❧✳ ✷✵✵✺✮✱ ❜❛r✈✐♥♦❦ ✭❱❡r❞♦♦❧❛❡❣❡✱ ✷✵✵✽✮ ❆♣❧✐❝❛t✐♦♥s✿ ❙♦❧✈✐♥❣ ■P Pr♦❜❧❡♠s✱ ❈♦♠♣❧❡①✐t② r❡s✉❧ts ✐♥ ●❛♠❡ ❚❤❡♦r②✱ ▼✉❧t✐♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥✱ ❋✉③③② Pr♦❣r❛♠♠✐♥❣✱ ✳✳✳ ❛♥❞ ❈❖❯◆❚■◆● ●❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ▲❡t ◗ ⊆ Z ♥ ✱ ✐ts ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s � ❢ ( P ; ③ ) = ③ α α ∈ ◗ ◗ = { ✵ , . . . , ◆ } ✿ ❢ ( ◗ , ③ ) = � ◆ ✐ = ✵ ③ ✐ = ✶ + ③ + ③ ✷ + · · · + ③ ◆
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