st r s s p r s tr r t t

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str s s Prs trr tt r t rs s rs

  1. ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ❙❡r❣✐♦ ❆✳ ❨✉❤❥t♠❛♥✱ ❯♥✐✈❡rs✐❞❛❞ ❞❡ ❇✉❡♥♦s ❆✐r❡s ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆❧❞♦ Pr♦❝❛❝❝✐ ❲❛r✇✐❝❦ ❙t❛t✐st✐❝❛❧ ▼❡❝❤❛♥✐❝s ❙❡♠✐♥❛r

  2. ❚r② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦❧❧❡❝t✐✈❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ s②st❡♠✳ t②♣✐❝❛❧ q✉❡st✐♦♥✿ ❤♦✇ t❤❡ s❛♠❡ ❝♦♠♣♦♥❡♥ts ✇✐t❤ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ❧❡❛❞ t♦ ❞✐✛❡r❡♥t st❛t❡s ♦❢ ♠❛tt❡r❄ ✲ ❙②st❡♠ ♦❢ ♣♦✐♥t ♣❛rt✐❝❧❡s ✇✐t❤ ♣❛✐r ✐♥t❡r❛❝t✐♦♥✳ ✲ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ♠❡t❤♦❞ ✐♥ t❤❛t ❝❛s❡ ✭❞✐❧✉t❡ ❣❛s✮✳ ✲ ●❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♣♦❧②♠❡rs✳ ❙t❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ s②st❡♠s ❢♦r♠❡❞ ❜② ❧❛r❣❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠✐❝r♦s❝♦♣✐❝ ♦❜❥❡❝ts✱ ✇✐t❤ s♦♠❡ r✉❧❡ ♦❢ ✐♥t❡r❛❝t✐♦♥✳ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✷✴✶✾

  3. t②♣✐❝❛❧ q✉❡st✐♦♥✿ ❤♦✇ t❤❡ s❛♠❡ ❝♦♠♣♦♥❡♥ts ✇✐t❤ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ❧❡❛❞ t♦ ❞✐✛❡r❡♥t st❛t❡s ♦❢ ♠❛tt❡r❄ ✲ ❙②st❡♠ ♦❢ ♣♦✐♥t ♣❛rt✐❝❧❡s ✇✐t❤ ♣❛✐r ✐♥t❡r❛❝t✐♦♥✳ ✲ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ♠❡t❤♦❞ ✐♥ t❤❛t ❝❛s❡ ✭❞✐❧✉t❡ ❣❛s✮✳ ✲ ●❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♣♦❧②♠❡rs✳ ❙t❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ s②st❡♠s ❢♦r♠❡❞ ❜② ❧❛r❣❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠✐❝r♦s❝♦♣✐❝ ♦❜❥❡❝ts✱ ✇✐t❤ s♦♠❡ r✉❧❡ ♦❢ ✐♥t❡r❛❝t✐♦♥✳ ❚r② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦❧❧❡❝t✐✈❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ s②st❡♠✳ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✷✴✶✾

  4. ✲ ❙②st❡♠ ♦❢ ♣♦✐♥t ♣❛rt✐❝❧❡s ✇✐t❤ ♣❛✐r ✐♥t❡r❛❝t✐♦♥✳ ✲ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ♠❡t❤♦❞ ✐♥ t❤❛t ❝❛s❡ ✭❞✐❧✉t❡ ❣❛s✮✳ ✲ ●❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♣♦❧②♠❡rs✳ ❙t❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ s②st❡♠s ❢♦r♠❡❞ ❜② ❧❛r❣❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠✐❝r♦s❝♦♣✐❝ ♦❜❥❡❝ts✱ ✇✐t❤ s♦♠❡ r✉❧❡ ♦❢ ✐♥t❡r❛❝t✐♦♥✳ ❚r② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦❧❧❡❝t✐✈❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ s②st❡♠✳ t②♣✐❝❛❧ q✉❡st✐♦♥✿ ❤♦✇ t❤❡ s❛♠❡ ❝♦♠♣♦♥❡♥ts ✇✐t❤ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ❧❡❛❞ t♦ ❞✐✛❡r❡♥t st❛t❡s ♦❢ ♠❛tt❡r❄ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✷✴✶✾

  5. ✲ ●❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♣♦❧②♠❡rs✳ ❙t❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ s②st❡♠s ❢♦r♠❡❞ ❜② ❧❛r❣❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠✐❝r♦s❝♦♣✐❝ ♦❜❥❡❝ts✱ ✇✐t❤ s♦♠❡ r✉❧❡ ♦❢ ✐♥t❡r❛❝t✐♦♥✳ ❚r② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦❧❧❡❝t✐✈❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ s②st❡♠✳ t②♣✐❝❛❧ q✉❡st✐♦♥✿ ❤♦✇ t❤❡ s❛♠❡ ❝♦♠♣♦♥❡♥ts ✇✐t❤ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ❧❡❛❞ t♦ ❞✐✛❡r❡♥t st❛t❡s ♦❢ ♠❛tt❡r❄ ✲ ❙②st❡♠ ♦❢ ♣♦✐♥t ♣❛rt✐❝❧❡s ✇✐t❤ ♣❛✐r ✐♥t❡r❛❝t✐♦♥✳ ✲ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ♠❡t❤♦❞ ✐♥ t❤❛t ❝❛s❡ ✭❞✐❧✉t❡ ❣❛s✮✳ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✷✴✶✾

  6. ❙t❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ s②st❡♠s ❢♦r♠❡❞ ❜② ❧❛r❣❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠✐❝r♦s❝♦♣✐❝ ♦❜❥❡❝ts✱ ✇✐t❤ s♦♠❡ r✉❧❡ ♦❢ ✐♥t❡r❛❝t✐♦♥✳ ❚r② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦❧❧❡❝t✐✈❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ s②st❡♠✳ t②♣✐❝❛❧ q✉❡st✐♦♥✿ ❤♦✇ t❤❡ s❛♠❡ ❝♦♠♣♦♥❡♥ts ✇✐t❤ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ❧❡❛❞ t♦ ❞✐✛❡r❡♥t st❛t❡s ♦❢ ♠❛tt❡r❄ ✲ ❙②st❡♠ ♦❢ ♣♦✐♥t ♣❛rt✐❝❧❡s ✇✐t❤ ♣❛✐r ✐♥t❡r❛❝t✐♦♥✳ ✲ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ♠❡t❤♦❞ ✐♥ t❤❛t ❝❛s❡ ✭❞✐❧✉t❡ ❣❛s✮✳ ✲ ●❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♣♦❧②♠❡rs✳ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✷✴✶✾

  7. ❙t❛t❡✿ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ ❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡✳ ❞ ❆t ✜♥✐t❡ ✈♦❧✉♠❡✱ ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ ❜♦① ③ ♥ ✐ ❥ ❱ ① ✐ ① ❥ ✇ ① ✶ ① ♥ ♥ ❡ ③ ♠ ✐ ❥ ❱ ② ✐ ② ❥ ❞✐✈✐❞❡❞ ❜② ❩ ♠ ❞② ♠ ❡ ♠ ✵ P♦✐♥t ♣❛rt✐❝❧❡s✱ ♣❛✐r ✐♥t❡r❛❝t✐♦♥ ❈♦♥✜❣✉r❛t✐♦♥s✿ ❧♦❝❛❧❧② ✜♥✐t❡ s✉❜s❡ts ♦❢ R ❞ ✭✐t ✐s ❛ ♣♦✐♥t ♣r♦❝❡ss✮ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✸✴✶✾

  8. ❞ ❆t ✜♥✐t❡ ✈♦❧✉♠❡✱ ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ ❜♦① ③ ♥ ✐ ❥ ❱ ① ✐ ① ❥ ✇ ① ✶ ① ♥ ♥ ❡ ③ ♠ ✐ ❥ ❱ ② ✐ ② ❥ ❞✐✈✐❞❡❞ ❜② ❩ ♠ ❞② ♠ ❡ ♠ ✵ P♦✐♥t ♣❛rt✐❝❧❡s✱ ♣❛✐r ✐♥t❡r❛❝t✐♦♥ ❈♦♥✜❣✉r❛t✐♦♥s✿ ❧♦❝❛❧❧② ✜♥✐t❡ s✉❜s❡ts ♦❢ R ❞ ✭✐t ✐s ❛ ♣♦✐♥t ♣r♦❝❡ss✮ ❙t❛t❡✿ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ ❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡✳ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✸✴✶✾

  9. ③ ♥ ✐ ❥ ❱ ① ✐ ① ❥ ✇ ① ✶ ① ♥ ♥ ❡ ③ ♠ ✐ ❥ ❱ ② ✐ ② ❥ ❞✐✈✐❞❡❞ ❜② ❩ ♠ ❞② ♠ ❡ ♠ ✵ P♦✐♥t ♣❛rt✐❝❧❡s✱ ♣❛✐r ✐♥t❡r❛❝t✐♦♥ ❈♦♥✜❣✉r❛t✐♦♥s✿ ❧♦❝❛❧❧② ✜♥✐t❡ s✉❜s❡ts ♦❢ R ❞ ✭✐t ✐s ❛ ♣♦✐♥t ♣r♦❝❡ss✮ ❙t❛t❡✿ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ ❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡✳ ❆t ✜♥✐t❡ ✈♦❧✉♠❡✱ ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ ❜♦① Λ ⋐ R ❞ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✸✴✶✾

  10. ③ ♠ ✐ ❥ ❱ ② ✐ ② ❥ ❞✐✈✐❞❡❞ ❜② ❩ ♠ ❞② ♠ ❡ ♠ ✵ P♦✐♥t ♣❛rt✐❝❧❡s✱ ♣❛✐r ✐♥t❡r❛❝t✐♦♥ ❈♦♥✜❣✉r❛t✐♦♥s✿ ❧♦❝❛❧❧② ✜♥✐t❡ s✉❜s❡ts ♦❢ R ❞ ✭✐t ✐s ❛ ♣♦✐♥t ♣r♦❝❡ss✮ ❙t❛t❡✿ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ ❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡✳ ❆t ✜♥✐t❡ ✈♦❧✉♠❡✱ ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ ❜♦① Λ ⋐ R ❞ ✇ ( ① ✶ , ..., ① ♥ ) = ③ ♥ ♥ ! ❡ − β � ✐ , ❥ ❱ ( ① ✐ − ① ❥ ) ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✸✴✶✾

  11. P♦✐♥t ♣❛rt✐❝❧❡s✱ ♣❛✐r ✐♥t❡r❛❝t✐♦♥ ❈♦♥✜❣✉r❛t✐♦♥s✿ ❧♦❝❛❧❧② ✜♥✐t❡ s✉❜s❡ts ♦❢ R ❞ ✭✐t ✐s ❛ ♣♦✐♥t ♣r♦❝❡ss✮ ❙t❛t❡✿ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ ❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡✳ ❆t ✜♥✐t❡ ✈♦❧✉♠❡✱ ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ ❜♦① Λ ⋐ R ❞ ✇ ( ① ✶ , ..., ① ♥ ) = ③ ♥ ♥ ! ❡ − β � ✐ , ❥ ❱ ( ① ✐ − ① ❥ ) ③ ♠ � � ♠ ! ❡ − β � ✐ , ❥ ❱ ( ② ✐ − ② ❥ ) ❞✐✈✐❞❡❞ ❜② ❩ (Λ) = Λ ♠ ❞ � ② ♠ ≥ ✵ ❈❧✉st❡r ❡①♣❛♥s✐♦♥ ✉s✐♥❣ P❡♥r♦s❡ tr❡❡✲❣r❛♣❤ ✐❞❡♥t✐t② ✸✴✶✾

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