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  1. ❈♦♠♣✉t✐♥❣ ▲♦❝✐ ♦❢ ❘❛♥❦ ❉❡❢❡❝ts ♦❢ ▲✐♥❡❛r ▼❛tr✐❝❡s ✉s✐♥❣ ●rö❜♥❡r ❇❛s❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❈r②♣t♦❧♦❣② ❏❡❛♥✕❈❤❛r❧❡s ❋❛✉❣èr❡ ▼♦❤❛❜ ❙❛❢❡② ❊❧ ❉✐♥ P✐❡rr❡✕❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r ❯P▼❈ ✕ ❈◆❘❙ ✕ ■◆❘■❆ P❛r✐s ✲ ❘♦❝q✉❡♥❝♦✉rt ▲■P✻ ✕ ❙❆▲❙❆ t❡❛♠ ■❙❙❆❈ ✷✵✶✵ ❏✉❧② ✷✽✱ ✷✵✶✵ ✶✴✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  2. ❚❤❡ ▼✐♥❘❛♥❦ ♣r♦❜❧❡♠ r ∈ N ✳ ▼ ✵ , . . . , ▼ ❦ ✿ ❦ + ✶ ♠❛tr✐❝❡s ♦❢ s✐③❡ ♠ × ♠ ✳ ▼✐♥❘❛♥❦ ✜♥❞ λ ✶ , . . . , λ ❦ s✉❝❤ t❤❛t � ❦ � � ❘❛♥❦ ▼ ✵ − λ ✐ ▼ ✐ ≤ r . ✐ = ✶ ▼✉❧t✐✈❛r✐❛t❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❊✐❣❡♥❱❛❧✉❡ ♣r♦❜❧❡♠✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❝r②♣t♦❧♦❣② ✱ ❝♦❞✐♥❣ t❤❡♦r② ✱ ❣❡♦♠❡tr② ✱ ✳✳✳ ❑✐♣♥✐s✴❙❤❛♠✐r ❈r②♣t♦✬✾✾ ❋❛✉❣èr❡✴▲❡✈②✲❞✐t✲❱❡❤❡❧✴P❡rr❡t ❈r②♣t♦✬✵✽✱✳✳✳ ❋✉♥❞❛♠❡♥t❛❧ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛ ✳ ❇✉ss✱ ❋r❛♥❞s❡♥✱ ❙❤❛❧❧✐t✳ ❏✳ ♦❢ ❈♦♠♣✉t❡r ❛♥❞ ❙②st❡♠ ❙❝✐❡♥❝❡s✳ ✶✾✾✾✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t② ♦❢ s♦♠❡ ♣r♦❜❧❡♠s ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ✷✴✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  3. ❚❤❡ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣ ❚❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣ ❘❛♥❦ ▼ r ① ✶ ① ♠ r ❘❛♥❦ ▼ r ❑❡r ▼ ✳ ■ ♠ r ❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ r ✶ ♦❢ ▼ ✈❛♥✐s❤✳ ① ✶ ① ♠ r ▼ ✵ ✶ ✶ ✳ ✳ ✳ ✷ ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ r ✳ ✳ ✳ ♠ ✶✳ ✳ ✳ ✳ r ✶ ① ✶ ① ♠ r ❦ ✈❛r✐❛❜❧❡s✳ r r ❋❡✇ ✈❛r✐❛❜❧❡s✱ ❧♦ts ♦❢ ❡q✉❛t✐♦♥s✱ ❤✐❣❤ ♠ ♠ r ❜✐❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❞❡❣r❡❡ ✦✦ ❦ r ♠ r ✈❛r✐❛❜❧❡s✳ ❈♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ ▼✐♥❘❛♥❦ ✉s✐♥❣ ●rö❜♥❡r ❜❛s❡s t❡❝❤♥✐q✉❡s ❄ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ t✇♦ ♠♦❞❡❧✐♥❣s ❄ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❄ ❚✇♦ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣s ▼ = ▼ ✵ − � ❦ ✐ = ✶ λ ✐ ▼ ✐ . ✸✴✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  4. ❚❤❡ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣ ① ✶ ① ♠ r ❘❛♥❦ ▼ r ❑❡r ▼ ✳ ■ ♠ r ① ✶ ① ♠ r ▼ ✵ ✶ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ① ✶ ① ♠ r r r ♠ ♠ r ❜✐❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❦ r ♠ r ✈❛r✐❛❜❧❡s✳ ❈♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ ▼✐♥❘❛♥❦ ✉s✐♥❣ ●rö❜♥❡r ❜❛s❡s t❡❝❤♥✐q✉❡s ❄ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ t✇♦ ♠♦❞❡❧✐♥❣s ❄ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❄ ❚✇♦ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣s ▼ = ▼ ✵ − � ❦ ✐ = ✶ λ ✐ ▼ ✐ . ❚❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣ ❘❛♥❦ ( ▼ ) ≤ r � ❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ ( r + ✶ ) ♦❢ ▼ ✈❛♥✐s❤✳ ` ♠ ´ ✷ ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ r + ✶✳ r + ✶ ❦ ✈❛r✐❛❜❧❡s✳ ❋❡✇ ✈❛r✐❛❜❧❡s✱ ❧♦ts ♦❢ ❡q✉❛t✐♦♥s✱ ❤✐❣❤ ❞❡❣r❡❡ ✦✦ ✸✴✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  5. ❈♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ ▼✐♥❘❛♥❦ ✉s✐♥❣ ●rö❜♥❡r ❜❛s❡s t❡❝❤♥✐q✉❡s ❄ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ t✇♦ ♠♦❞❡❧✐♥❣s ❄ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❄ ❚✇♦ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣s ▼ = ▼ ✵ − � ❦ ✐ = ✶ λ ✐ ▼ ✐ . ❚❤❡ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣ ❚❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣ ❘❛♥❦ ( ▼ ) ≤ r ⇔ ∃ ① ( ✶ ) , . . . , ① ( ♠ − r ) ∈ ❑❡r ( ▼ ) ✳ ❘❛♥❦ ( ▼ ) ≤ r � ■ ♠ − r 0 1 ❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ ( r + ✶ ) ♦❢ ▼ ✈❛♥✐s❤✳ B C B C ① ( ✶ ) ① ( ♠ − r ) = ✵ . ▼ · B C . . . B ✶ ✶ C B C ✳ ✳ ✳ ` ♠ ´ ✷ ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ r + ✶✳ B C ✳ ✳ ✳ B ✳ ✳ ✳ C @ A r + ✶ ① ( ✶ ) ① ( ♠ − r ) . . . ❦ ✈❛r✐❛❜❧❡s✳ r r ❋❡✇ ✈❛r✐❛❜❧❡s✱ ❧♦ts ♦❢ ❡q✉❛t✐♦♥s✱ ❤✐❣❤ ♠ ( ♠ − r ) ❜✐❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❞❡❣r❡❡ ✦✦ ❦ + r ( ♠ − r ) ✈❛r✐❛❜❧❡s✳ ✸✴✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  6. ❚✇♦ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣s ▼ = ▼ ✵ − � ❦ ✐ = ✶ λ ✐ ▼ ✐ . ❚❤❡ ❑✐♣♥✐s✲❙❤❛♠✐r ♠♦❞❡❧✐♥❣ ❚❤❡ ♠✐♥♦rs ♠♦❞❡❧✐♥❣ ❘❛♥❦ ( ▼ ) ≤ r ⇔ ∃ ① ( ✶ ) , . . . , ① ( ♠ − r ) ∈ ❑❡r ( ▼ ) ✳ ❘❛♥❦ ( ▼ ) ≤ r � ■ ♠ − r 0 1 ❛❧❧ ♠✐♥♦rs ♦❢ s✐③❡ ( r + ✶ ) ♦❢ ▼ ✈❛♥✐s❤✳ B C B C ① ( ✶ ) ① ( ♠ − r ) = ✵ . ▼ · B C . . . B ✶ ✶ C B C ✳ ✳ ✳ ` ♠ ´ ✷ ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ r + ✶✳ B C ✳ ✳ ✳ B ✳ ✳ ✳ C @ A r + ✶ ① ( ✶ ) ① ( ♠ − r ) . . . ❦ ✈❛r✐❛❜❧❡s✳ r r ❋❡✇ ✈❛r✐❛❜❧❡s✱ ❧♦ts ♦❢ ❡q✉❛t✐♦♥s✱ ❤✐❣❤ ♠ ( ♠ − r ) ❜✐❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ❞❡❣r❡❡ ✦✦ ❦ + r ( ♠ − r ) ✈❛r✐❛❜❧❡s✳ ❈♦♠♣❧❡①✐t② ♦❢ s♦❧✈✐♥❣ ▼✐♥❘❛♥❦ ✉s✐♥❣ ●rö❜♥❡r ❜❛s❡s t❡❝❤♥✐q✉❡s ❄ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ t✇♦ ♠♦❞❡❧✐♥❣s ❄ ◆✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❄ ✸✴✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  7. ◆❡✇ ◆❡✇ r ♠ r ✶ ◆❡✇ ❦ ❦ ✶ ❖ ♠ ❖ ♠ r ✷ ✐s ✜①❡❞✳ ❇♦t❤ ♠♦❞❡❧✐♥❣s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ ♠ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥ ✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ ✶✶ t❛r❣❡t r❛♥❦ ✽✱ ●❋ ✻✺✺✷✶ ✳ r ✷ ✳ ♠ ✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦ ✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r ✿ t❛r❣❡t r❛♥❦✳ ❦ ♠ ▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ♦❧❞ r ✷ ✇❤❡♥ ❦ ♠ ❞ r❡❣ ♠ ♠ r ✶ ♦❧❞ ♠ r ★ ❙♦❧ ♠ r ❈♦♠♣❧❡①✐t② ▼❛✐♥ r❡s✉❧ts ❙②st❡♠ − → ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣ ❧❡① ●❇ . − → ” ω « „“ ♥ + ❞ r❡❣ ❖ ( ♥ · # ❙♦❧ ω ) ❈♦♠♣❧❡①✐t② ❖ ❞ r❡❣ ✹✴✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  8. ◆❡✇ ◆❡✇ r ♠ r ✶ ◆❡✇ ❦ ❦ ✶ ❖ ♠ ❖ ♠ r ✷ ✐s ✜①❡❞✳ ❇♦t❤ ♠♦❞❡❧✐♥❣s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤❡♥ ❦ ♠ ◆❡✇ ❈r②♣t♦ ❝❤❛❧❧❡♥❣❡ ❜r♦❦❡♥ ✿ ✶✵ ❣❡♥❡r✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ ✶✶ ✶✶ t❛r❣❡t r❛♥❦ ✽✱ ●❋ ✻✺✺✷✶ ✳ ▼❛✐♥ r❡s✉❧ts ❙②st❡♠ − → ❣r❡✈❧❡① ●❇ ❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣ ❧❡① ●❇ . − → ” ω « „“ ♥ + ❞ r❡❣ ❖ ( ♥ · # ❙♦❧ ω ) ❈♦♠♣❧❡①✐t② ❖ ❞ r❡❣ ♠ ✿ s✐③❡ ♦❢ t❤❡ ♠❛tr✐❝❡s✱ ❦ ✿ ♥✉♠❜❡r ♦❢ ♠❛tr✐❝❡s✱ r ✿ t❛r❣❡t r❛♥❦✳ ❦ = ( ♠ − r ) ✷ ✳ ▼✐♥♦rs ❑✐♣♥✐s✲❙❤❛♠✐r ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ♦❧❞ ✇❤❡♥ ❦ = ( ♠ − r ) ✷ ♠ ( ♠ − r ) + ✶ ❞ r❡❣ ≤ ♦❧❞ ! ♠ − r ★ ❙♦❧ ♠ < r ❈♦♠♣❧❡①✐t② ✹✴✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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