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t Prrts r rs sss P rrs t r s rstt

  1. ❆♠❡♥❛❜✐❧✐t② Pr♦♣❡rt✐❡s ❢♦r ❱❛r✐♦✉s ❈❧❛ss❡s ♦❢ P♦❧②♥♦♠✐❛❧ ❍②♣❡r❣r♦✉♣s ❙t❡❢❛♥ ❑❛❤❧❡r ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ▼ü♥❝❤❡♥ ●r❛♥❛❞❛✱ ▼❛② ✷✵t❤✱ ✷✵✶✸ ❙✳ ❑❛❤❧❡r ✭❚❯ ▼ü♥❝❤❡♥✮ ❆♠❡♥❛❜✐❧✐t② Pr♦♣❡rt✐❡s ●r❛♥❛❞❛✱ ✷✵▼❛②✷✵✶✸ ✶ ✴ ✶✾

  2. ❖✉t❧✐♥❡ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ♣♦❧②♥♦♠✐❛❧ ❤②♣❡r❣r♦✉♣s✿ ❜❛s✐❝s ✶ ●❡♥❡r❛❧ r❡s✉❧ts ✷ ❖♥❡✲♣❛r❛♠❡t❡r ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ✉❧tr❛s♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s ✸ ❈♦♠♣❛r✐s♦♥ t♦ t❤❡ ❣r♦✉♣ ❝❛s❡ ✹ ❙✳ ❑❛❤❧❡r ✭❚❯ ▼ü♥❝❤❡♥✮ ❆♠❡♥❛❜✐❧✐t② Pr♦♣❡rt✐❡s ●r❛♥❛❞❛✱ ✷✵▼❛②✷✵✶✸ ✷ ✴ ✶✾

  3. ❖✉t❧✐♥❡ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ♣♦❧②♥♦♠✐❛❧ ❤②♣❡r❣r♦✉♣s✿ ❜❛s✐❝s ✶ ●❡♥❡r❛❧ r❡s✉❧ts ✷ ❖♥❡✲♣❛r❛♠❡t❡r ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ✉❧tr❛s♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s ✸ ❈♦♠♣❛r✐s♦♥ t♦ t❤❡ ❣r♦✉♣ ❝❛s❡ ✹ ❙✳ ❑❛❤❧❡r ✭❚❯ ▼ü♥❝❤❡♥✮ ❆♠❡♥❛❜✐❧✐t② Pr♦♣❡rt✐❡s ●r❛♥❛❞❛✱ ✷✵▼❛②✷✵✶✸ ✸ ✴ ✶✾

  4. P♦❧②♥♦♠✐❛❧ ❤②♣❡r❣r♦✉♣s ♦♥ N ✵ ❙t❛rt✐♥❣ ♣♦✐♥t ▲❡t ❛ ✵ > ✵✱ ❜ ✵ < ✶✱ ( ❛ ♥ ) ♥ ∈ N , ( ❝ ♥ ) ♥ ∈ N ⊆ ( ✵ , ✶ ) ❛♥❞ ( ❜ ♥ ) ♥ ∈ N ⊆ [ ✵ , ✶ ) s❛t✐s❢② ❛ ✵ + ❜ ✵ = ✶✱ ❛ ♥ + ❜ ♥ + ❝ ♥ = ✶ ( ♥ ∈ N ) ❀ ❞❡✜♥❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ R [ ① ] ❜② P ✵ ( ① ) := ✶✱ P ✶ ( ① ) := ✶ ❛ ✵ ( ① − ❜ ✵ ) ✱ P ✶ ( ① ) P ♥ ( ① ) = ❛ ♥ P ♥ + ✶ ( ① ) + ❜ ♥ P ♥ ( ① ) + ❝ ♥ P ♥ − ✶ ( ① ) ( ♥ ∈ N ) . ❚❤❡♦r❡♠ ♦❢ ❋❛✈❛r❞ ✭✶✾✸✺✮✿ ( P ♥ ( ① )) ♥ ∈ N ✵ ✐s ♦rt❤♦❣♦♥❛❧ ✇✳r✳t✳ ❛ ✉♥✐q✉❡ ♣r♦❜❛❜✐❧✐t② ✭❇♦r❡❧✮ ♠❡❛s✉r❡ µ ♦♥ R ✇✐t❤ | s✉♣♣ µ | = ∞ ✳ ▲❡t ( ♣ ♥ ( ① )) ♥ ∈ N ✵ ❞❡♥♦t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦rt❤♦ ♥♦r♠ ❛❧ ♣♦❧②♥♦♠✐❛❧s ✭✇✐t❤ ♣♦s✐t✐✈❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥ts✮✳ ▲✐♥❡❛r✐③❛t✐♦♥ ❢♦r♠✉❧❛✿ ♠ + ♥ ♠ + ♥ � � P ♠ ( ① ) P ♥ ( ① ) = ❣ ( ♠ , ♥ ; ❦ ) P ❦ ( ① ); ❣ ( ♠ , ♥ ; ❦ ) = ✶ , � �� � ❦ = | ♠ − ♥ | ❦ = | ♠ − ♥ | ∈ R ❣ ( ♠ , ♥ ; | ♠ − ♥ | ) , ❣ ( ♠ , ♥ ; ♠ + ♥ ) � = ✵ ( ♠ , ♥ ∈ N ✵ ) . ❙✳ ❑❛❤❧❡r ✭❚❯ ▼ü♥❝❤❡♥✮ ❆♠❡♥❛❜✐❧✐t② Pr♦♣❡rt✐❡s ●r❛♥❛❞❛✱ ✷✵▼❛②✷✵✶✸ ✹ ✴ ✶✾

  5. P♦❧②♥♦♠✐❛❧ ❤②♣❡r❣r♦✉♣s ♦♥ N ✵ ❙t❛rt✐♥❣ ♣♦✐♥t ▲❡t ❛ ✵ > ✵✱ ❜ ✵ < ✶✱ ( ❛ ♥ ) ♥ ∈ N , ( ❝ ♥ ) ♥ ∈ N ⊆ ( ✵ , ✶ ) ❛♥❞ ( ❜ ♥ ) ♥ ∈ N ⊆ [ ✵ , ✶ ) s❛t✐s❢② ❛ ✵ + ❜ ✵ = ✶✱ ❛ ♥ + ❜ ♥ + ❝ ♥ = ✶ ( ♥ ∈ N ) ❀ ❞❡✜♥❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ R [ ① ] ❜② P ✵ ( ① ) := ✶✱ P ✶ ( ① ) := ✶ ❛ ✵ ( ① − ❜ ✵ ) ✱ P ✶ ( ① ) P ♥ ( ① ) = ❛ ♥ P ♥ + ✶ ( ① ) + ❜ ♥ P ♥ ( ① ) + ❝ ♥ P ♥ − ✶ ( ① ) ( ♥ ∈ N ) . ❚❤❡♦r❡♠ ♦❢ ❋❛✈❛r❞ ✭✶✾✸✺✮✿ ( P ♥ ( ① )) ♥ ∈ N ✵ ✐s ♦rt❤♦❣♦♥❛❧ ✇✳r✳t✳ ❛ ✉♥✐q✉❡ ♣r♦❜❛❜✐❧✐t② ✭❇♦r❡❧✮ ♠❡❛s✉r❡ µ ♦♥ R ✇✐t❤ | s✉♣♣ µ | = ∞ ✳ ▲❡t ( ♣ ♥ ( ① )) ♥ ∈ N ✵ ❞❡♥♦t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦rt❤♦ ♥♦r♠ ❛❧ ♣♦❧②♥♦♠✐❛❧s ✭✇✐t❤ ♣♦s✐t✐✈❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥ts✮✳ ▲✐♥❡❛r✐③❛t✐♦♥ ❢♦r♠✉❧❛✿ ♠ + ♥ ♠ + ♥ � � P ♠ ( ① ) P ♥ ( ① ) = ❣ ( ♠ , ♥ ; ❦ ) P ❦ ( ① ); ❣ ( ♠ , ♥ ; ❦ ) = ✶ , � �� � ❦ = | ♠ − ♥ | ❦ = | ♠ − ♥ | ∈ R ❣ ( ♠ , ♥ ; | ♠ − ♥ | ) , ❣ ( ♠ , ♥ ; ♠ + ♥ ) � = ✵ ( ♠ , ♥ ∈ N ✵ ) . ❙✳ ❑❛❤❧❡r ✭❚❯ ▼ü♥❝❤❡♥✮ ❆♠❡♥❛❜✐❧✐t② Pr♦♣❡rt✐❡s ●r❛♥❛❞❛✱ ✷✵▼❛②✷✵✶✸ ✹ ✴ ✶✾

  6. P♦❧②♥♦♠✐❛❧ ❤②♣❡r❣r♦✉♣s ♦♥ N ✵ ❇❛s✐❝ ❞❡✜♥✐t✐♦♥s ❛♥❞ ❝♦♥str✉❝t✐♦♥s ❈r✉❝✐❛❧ ❛ss✉♠♣t✐♦♥✿ ❆ss✉♠❡ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ❝♦❡✣❝✐❡♥ts ❣ ( ♠ , ♥ ; ❦ ) t♦ ❜❡ ♥♦♥✲♥❡❣❛t✐✈❡✳ ❉❡✜♥❡ ❛ ❝♦♥✈♦❧✉t✐♦♥ N ✵ × N ✵ → ❝♦♥✈ { δ ♥ : ♥ ∈ N ✵ } ❜② ♠ + ♥ � ( ♠ , ♥ ) �→ ❣ ( ♠ , ♥ ; ❦ ) δ ❦ ; ❦ = | ♠ − ♥ | � t♦❣❡t❤❡r ✇✐t❤ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ❛s ✐♥✈♦❧✉t✐♦♥ ✱ N ✵ ❜❡❝♦♠❡s ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐s❝r❡t❡ ❤②♣❡r❣r♦✉♣ ✇✐t❤ ✉♥✐t ❡❧❡♠❡♥t ✵✱ ❛ ♣♦❧②♥♦♠✐❛❧ ❤②♣❡r❣r♦✉♣ ✳ ❍❛❛r ♠❡❛s✉r❡✿ ❝♦✉♥t✐♥❣ ♠❡❛s✉r❡ ✇❡✐❣❤t❡❞ ❜② t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❍❛❛r ❢✉♥❝t✐♦♥ ❤ : N ✵ → [ ✶ , ∞ ) ✱ ❤ ( ♥ ) := ♣ ✷ ♥ ( ✶ ) ✳ ▲❡t ( ℓ ♣ ( ❤ ) , � . � ♣ ) ✱ ♣ ∈ [ ✶ , ∞ ] ✱ ❜❡ t❤❡ ℓ ♣ ✲s♣❛❝❡s ✇✳r✳t✳ t❤❡ ❍❛❛r ♠❡❛s✉r❡✳ ❙✳ ❑❛❤❧❡r ✭❚❯ ▼ü♥❝❤❡♥✮ ❆♠❡♥❛❜✐❧✐t② Pr♦♣❡rt✐❡s ●r❛♥❛❞❛✱ ✷✵▼❛②✷✵✶✸ ✺ ✴ ✶✾

  7. P♦❧②♥♦♠✐❛❧ ❤②♣❡r❣r♦✉♣s ♦♥ N ✵ ❇❛s✐❝ ❞❡✜♥✐t✐♦♥s ❛♥❞ ❝♦♥str✉❝t✐♦♥s ❈r✉❝✐❛❧ ❛ss✉♠♣t✐♦♥✿ ❆ss✉♠❡ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ❝♦❡✣❝✐❡♥ts ❣ ( ♠ , ♥ ; ❦ ) t♦ ❜❡ ♥♦♥✲♥❡❣❛t✐✈❡✳ ❉❡✜♥❡ ❛ ❝♦♥✈♦❧✉t✐♦♥ N ✵ × N ✵ → ❝♦♥✈ { δ ♥ : ♥ ∈ N ✵ } ❜② ♠ + ♥ � ( ♠ , ♥ ) �→ ❣ ( ♠ , ♥ ; ❦ ) δ ❦ ; ❦ = | ♠ − ♥ | � t♦❣❡t❤❡r ✇✐t❤ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ❛s ✐♥✈♦❧✉t✐♦♥ ✱ N ✵ ❜❡❝♦♠❡s ❛ ❝♦♠♠✉t❛t✐✈❡ ❞✐s❝r❡t❡ ❤②♣❡r❣r♦✉♣ ✇✐t❤ ✉♥✐t ❡❧❡♠❡♥t ✵✱ ❛ ♣♦❧②♥♦♠✐❛❧ ❤②♣❡r❣r♦✉♣ ✳ ❍❛❛r ♠❡❛s✉r❡✿ ❝♦✉♥t✐♥❣ ♠❡❛s✉r❡ ✇❡✐❣❤t❡❞ ❜② t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❍❛❛r ❢✉♥❝t✐♦♥ ❤ : N ✵ → [ ✶ , ∞ ) ✱ ❤ ( ♥ ) := ♣ ✷ ♥ ( ✶ ) ✳ ▲❡t ( ℓ ♣ ( ❤ ) , � . � ♣ ) ✱ ♣ ∈ [ ✶ , ∞ ] ✱ ❜❡ t❤❡ ℓ ♣ ✲s♣❛❝❡s ✇✳r✳t✳ t❤❡ ❍❛❛r ♠❡❛s✉r❡✳ ❙✳ ❑❛❤❧❡r ✭❚❯ ▼ü♥❝❤❡♥✮ ❆♠❡♥❛❜✐❧✐t② Pr♦♣❡rt✐❡s ●r❛♥❛❞❛✱ ✷✵▼❛②✷✵✶✸ ✺ ✴ ✶✾

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