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P P ts s rs tr rstts

  1. ■◆❚❊◆❙❊ ❆❯❚❖▼❖❘P❍■❙▼❙ ❖❋ ❋■◆■❚❊ ●❘❖❯P❙ ▼✐♠❛ ❙t❛♥♦❥❦♦✈s❦✐ ❚♦♣✐❝s ♦♥ ●r♦✉♣s ❛♥❞ t❤❡✐r ❘❡♣r❡s❡♥t❛t✐♦♥s ●❛r❣♥❛♥♦ s✉❧ ●❛r❞❛✱ ✶✶t❤ ❖❝t♦❜❡r ✷✵✶✼

  2. ❏♦✐♥t ✇♦r❦ ✇✐t❤

  3. ❍❡♥❞r✐❦ ▲❡♥str❛ ✭❯▲✮

  4. ❆♥❞r❡❛ ▲✉❝❝❤✐♥✐ ✭❯♥✐P❉✮

  5. ❏♦♥ ●♦♥③á❧❡③ ❙á♥❝❤❡③ ✭❊❍❯✮

  6. ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r♦✉♣s

  7. ▼♦t✐✈❛t✐♦♥✿ ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ❛♣♣❡❛r ♥❛t✉r❛❧❧② ❛s s♦❧✉t✐♦♥s t♦ ❛ ❝❡rt❛✐♥ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ♣r♦❜❧❡♠✳ ❚❤❡② ✭s✉r♣r✐s✐♥❣❧②✦✮ ❣✐✈❡ r✐s❡ t♦ ❛ ✈❡r② r✐❝❤ t❤❡♦r②✳ ❊①❛♠♣❧❡✿ ❊✈❡r② ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❝②❝❧✐❝ ❣r♦✉♣ ✐s ✐♥t❡♥s❡✳ ■♥♥❡r ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ✐♥t❡♥s❡✳ P♦✇❡r ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ✐♥t❡♥s❡✳ ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❆♥ ❛✉t♦♠♦r♣❤✐s♠ α ♦❢ ● ✐s ✐♥t❡♥s❡ ✐❢ ❢♦r ❛❧❧ ❍ ≤ ● t❤❡r❡ ❡①✐sts ❣ ∈ ● s✉❝❤ t❤❛t α ( ❍ ) = ❣❍❣ − ✶ ✳ ❲r✐t❡ α ∈ ■♥t ( ● ) ✳

  8. ❊①❛♠♣❧❡✿ ❊✈❡r② ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❝②❝❧✐❝ ❣r♦✉♣ ✐s ✐♥t❡♥s❡✳ ■♥♥❡r ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ✐♥t❡♥s❡✳ P♦✇❡r ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ✐♥t❡♥s❡✳ ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❆♥ ❛✉t♦♠♦r♣❤✐s♠ α ♦❢ ● ✐s ✐♥t❡♥s❡ ✐❢ ❢♦r ❛❧❧ ❍ ≤ ● t❤❡r❡ ❡①✐sts ❣ ∈ ● s✉❝❤ t❤❛t α ( ❍ ) = ❣❍❣ − ✶ ✳ ❲r✐t❡ α ∈ ■♥t ( ● ) ✳ ▼♦t✐✈❛t✐♦♥✿ ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ❛♣♣❡❛r ♥❛t✉r❛❧❧② ❛s s♦❧✉t✐♦♥s t♦ ❛ ❝❡rt❛✐♥ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ♣r♦❜❧❡♠✳ ❚❤❡② ✭s✉r♣r✐s✐♥❣❧②✦✮ ❣✐✈❡ r✐s❡ t♦ ❛ ✈❡r② r✐❝❤ t❤❡♦r②✳

  9. ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❆♥ ❛✉t♦♠♦r♣❤✐s♠ α ♦❢ ● ✐s ✐♥t❡♥s❡ ✐❢ ❢♦r ❛❧❧ ❍ ≤ ● t❤❡r❡ ❡①✐sts ❣ ∈ ● s✉❝❤ t❤❛t α ( ❍ ) = ❣❍❣ − ✶ ✳ ❲r✐t❡ α ∈ ■♥t ( ● ) ✳ ▼♦t✐✈❛t✐♦♥✿ ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ❛♣♣❡❛r ♥❛t✉r❛❧❧② ❛s s♦❧✉t✐♦♥s t♦ ❛ ❝❡rt❛✐♥ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ♣r♦❜❧❡♠✳ ❚❤❡② ✭s✉r♣r✐s✐♥❣❧②✦✮ ❣✐✈❡ r✐s❡ t♦ ❛ ✈❡r② r✐❝❤ t❤❡♦r②✳ ❊①❛♠♣❧❡✿ • ❊✈❡r② ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❝②❝❧✐❝ ❣r♦✉♣ ✐s ✐♥t❡♥s❡✳ • ■♥♥❡r ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ✐♥t❡♥s❡✳ • P♦✇❡r ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ✐♥t❡♥s❡✳

  10. ❚❤❡♥ ■♥t ● P ❈ ✇❤❡r❡ P ✐s ❛ ♣ ✲❣r♦✉♣✳ ❈ ✐s ❛ s✉❜❣r♦✉♣ ♦❢ ♣ ✳ ❚❤❡ ✐♥t❡♥s✐t② ♦❢ ● ✐s ✐♥t ● ❈ ✳ ■♥t❡♥s✐t② ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣ ✲❣r♦✉♣✳

  11. ❚❤❡ ✐♥t❡♥s✐t② ♦❢ ● ✐s ✐♥t ● ❈ ✳ ■♥t❡♥s✐t② ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣ ✲❣r♦✉♣✳ ❚❤❡♥ ■♥t ( ● ) ∼ = P ⋊ ❈ ✇❤❡r❡ • P ✐s ❛ ♣ ✲❣r♦✉♣✳ • ❈ ✐s ❛ s✉❜❣r♦✉♣ ♦❢ F ∗ ♣ ✳

  12. ■♥t❡♥s✐t② ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣ ✲❣r♦✉♣✳ ❚❤❡♥ ■♥t ( ● ) ∼ = P ⋊ ❈ ✇❤❡r❡ • P ✐s ❛ ♣ ✲❣r♦✉♣✳ • ❈ ✐s ❛ s✉❜❣r♦✉♣ ♦❢ F ∗ ♣ ✳ ❚❤❡ ✐♥t❡♥s✐t② ♦❢ ● ✐s ✐♥t ( ● ) = | ❈ | ✳

  13. ❨❊❙✦ ❚❤❡ ♣r♦❜❧❡♠ ❈❛♥ ✇❡ ❝❧❛ss✐❢② ❛❧❧ ♣ ✲❣r♦✉♣s ● s❛t✐s❢②✐♥❣ ✐♥t ( ● ) > ✶❄

  14. ❚❤❡ ♣r♦❜❧❡♠ ❈❛♥ ✇❡ ❝❧❛ss✐❢② ❛❧❧ ♣ ✲❣r♦✉♣s ● s❛t✐s❢②✐♥❣ ✐♥t ( ● ) > ✶❄ ❨❊❙✦

  15. ❚❤❡♥ ❆✉t ● ♣ ■♥t ● ♣ ❚❤❡♦r❡♠ ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ✶ ❜❡ ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ♣✲❣r♦✉♣✳ ❚❤❡♥ ✐♥t ● ♣ ✶ ✳ ❆❜❡❧✐❛♥ ❣r♦✉♣s ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t Z ♣ ❞❡♥♦t❡ t❤❡ r✐♥❣ ♦❢ ♣ ✲❛❞✐❝ ✐♥t❡❣❡rs✳ ▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ♣ ✲❣r♦✉♣✳

  16. ❚❤❡♦r❡♠ ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ✶ ❜❡ ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ♣✲❣r♦✉♣✳ ❚❤❡♥ ✐♥t ● ♣ ✶ ✳ ❆❜❡❧✐❛♥ ❣r♦✉♣s ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t Z ♣ ❞❡♥♦t❡ t❤❡ r✐♥❣ ♦❢ ♣ ✲❛❞✐❝ ✐♥t❡❣❡rs✳ ▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ♣ ✲❣r♦✉♣✳ ❚❤❡♥ ✲ ❆✉t ( ● ) Z ∗ ♣ ✲ F ∗ ■♥t ( ● ) ✲ ♣

  17. ❆❜❡❧✐❛♥ ❣r♦✉♣s ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t Z ♣ ❞❡♥♦t❡ t❤❡ r✐♥❣ ♦❢ ♣ ✲❛❞✐❝ ✐♥t❡❣❡rs✳ ▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ♣ ✲❣r♦✉♣✳ ❚❤❡♥ ✲ ❆✉t ( ● ) Z ∗ ♣ ✲ F ∗ ■♥t ( ● ) ✲ ♣ ❚❤❡♦r❡♠ ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● � = ✶ ❜❡ ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ♣✲❣r♦✉♣✳ ❚❤❡♥ ✐♥t ( ● ) = ♣ − ✶ ✳

  18. ❚❤❡♥ ✇❡ ❤❛✈❡ ♥❛t✉r❛❧ ♠❛♣s ✶✳ ■♥t ● ❆✉t ◆ ✱ ✷✳ ■♥t ● ■♥t ● ◆ ✱ ❛♥❞ ✇✐t❤ ❛ ❧✐tt❧❡ ❡①tr❛ ✇♦r❦ ✸✳ ✐❢ ◆ ● ✱ t❤❡♥ ✐♥t ● ❞✐✈✐❞❡s ✐♥t ● ◆ ✳ ❙✐♥❝❡ ✇❡ ✇❛♥t ● t♦ ❤❛✈❡ ✐♥t ● ✶✱ ✇❡ ❝❛♥ ❢♦r❣❡t ❛❜♦✉t ♣ ✷✦✦ ❙tr❛t❡❣② ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣ ✲❣r♦✉♣✳ ▲❡t ◆ ❜❡ ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣✳

  19. ❛♥❞ ✇✐t❤ ❛ ❧✐tt❧❡ ❡①tr❛ ✇♦r❦ ✸✳ ✐❢ ◆ ● ✱ t❤❡♥ ✐♥t ● ❞✐✈✐❞❡s ✐♥t ● ◆ ✳ ❙✐♥❝❡ ✇❡ ✇❛♥t ● t♦ ❤❛✈❡ ✐♥t ● ✶✱ ✇❡ ❝❛♥ ❢♦r❣❡t ❛❜♦✉t ♣ ✷✦✦ ❙tr❛t❡❣② ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣ ✲❣r♦✉♣✳ ▲❡t ◆ ❜❡ ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ♥❛t✉r❛❧ ♠❛♣s ✶✳ ■♥t ( ● ) → ❆✉t ( ◆ ) ✱ ✷✳ ■♥t ( ● ) → ■♥t ( ● / ◆ ) ✱

  20. ❙✐♥❝❡ ✇❡ ✇❛♥t ● t♦ ❤❛✈❡ ✐♥t ● ✶✱ ✇❡ ❝❛♥ ❢♦r❣❡t ❛❜♦✉t ♣ ✷✦✦ ❙tr❛t❡❣② ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣ ✲❣r♦✉♣✳ ▲❡t ◆ ❜❡ ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ♥❛t✉r❛❧ ♠❛♣s ✶✳ ■♥t ( ● ) → ❆✉t ( ◆ ) ✱ ✷✳ ■♥t ( ● ) → ■♥t ( ● / ◆ ) ✱ ❛♥❞ ✇✐t❤ ❛ ❧✐tt❧❡ ❡①tr❛ ✇♦r❦ ✸✳ ✐❢ ◆ � = ● ✱ t❤❡♥ ✐♥t ( ● ) ❞✐✈✐❞❡s ✐♥t ( ● / ◆ ) ✳

  21. ❙tr❛t❡❣② ▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣ ✲❣r♦✉♣✳ ▲❡t ◆ ❜❡ ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ♥❛t✉r❛❧ ♠❛♣s ✶✳ ■♥t ( ● ) → ❆✉t ( ◆ ) ✱ ✷✳ ■♥t ( ● ) → ■♥t ( ● / ◆ ) ✱ ❛♥❞ ✇✐t❤ ❛ ❧✐tt❧❡ ❡①tr❛ ✇♦r❦ ✸✳ ✐❢ ◆ � = ● ✱ t❤❡♥ ✐♥t ( ● ) ❞✐✈✐❞❡s ✐♥t ( ● / ◆ ) ✳ ❙✐♥❝❡ ✇❡ ✇❛♥t ● t♦ ❤❛✈❡ ✐♥t ( ● ) > ✶✱ ✇❡ ❝❛♥ ❢♦r❣❡t ❛❜♦✉t ♣ = ✷✦✦

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