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  1. ❘❡❝✉rs✐✈❡ t②♣❡s ❢♦r ❧✐♥❡❛r✴♥♦♥✲❧✐♥❡❛r q✉❛♥t✉♠ ♣r♦❣r❛♠♠✐♥❣ ❱❧❛❞✐♠✐r ❩❛♠❞③❤✐❡✈ ❯♥✐✈❡rs✐té ❞❡ ▲♦rr❛✐♥❡✱ ❈◆❘❙✱ ■♥r✐❛✱ ▲❖❘■❆✱ ❋ ✺✹✵✵✵ ◆❛♥❝②✱ ❋r❛♥❝❡ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ▼✐❝❤❛❡❧ ▼✐s❧♦✈❡ ❛♥❞ ❇❡rt ▲✐♥❞❡♥❤♦✈✐✉s ❉❛❣st✉❤❧ ❙❡♠✐♥❛r ♦♥ ◗✉❛♥t✉♠ Pr♦❣r❛♠♠✐♥❣ ▲❛♥❣✉❛❣❡s ✶✼ ❙❡♣t❡♠❜❡r ✷✵✶✽ ✵ ✴ ✷✹

  2. Pr♦t♦✲◗✉✐♣♣❡r✲▼ ❼ ❲❡ ❝♦♥s✐❞❡r ❛❞❞✐♥❣ r❡❝✉rs✐✈❡ t②♣❡s t♦ Pr♦t♦✲◗✉✐♣♣❡r✲▼ ✳ ❼ ❖r✐❣✐♥❛❧ ❧❛♥❣✉❛❣❡ ❞❡✈❡❧♦♣❡❞ ❜② ❋r❛♥❝✐s❝♦ ❘✐♦s ❛♥❞ P❡t❡r ❙❡❧✐♥❣❡r✳ ❼ ▲❛♥❣✉❛❣❡ ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦r♠❛❧ ❞❡♥♦t❛t✐♦♥❛❧ ❛♥❞ ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s✳ ❼ Pr✐♠❛r② ❛♣♣❧✐❝❛t✐♦♥ ✐s ✐♥ q✉❛♥t✉♠ ❝♦♠♣✉t✐♥❣✱ ❜✉t t❤❡ ❧❛♥❣✉❛❣❡ ❝❛♥ ❞❡s❝r✐❜❡ ❛r❜✐tr❛r② str✐♥❣ ❞✐❛❣r❛♠s✳ ❼ ■♥ ♣r✐♦r ✇♦r❦✱ ✇❡ ❞❡s❝r✐❜❡❞ ❛♥ ❛❜str❛❝t ♠♦❞❡❧ ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ❛♥❞ ❛❞❞❡❞ r❡❝✉rs✐♦♥✳ ✶ ✴ ✷✹

  3. ❈✐r❝✉✐t ▼♦❞❡❧ ❚❤❡ ❧❛♥❣✉❛❣❡ ✐s ✉s❡❞ t♦ ❞❡s❝r✐❜❡ ❢❛♠✐❧✐❡s ♦❢ ♠♦r♣❤✐s♠s ♦❢ ❛♥ ❛r❜✐tr❛r② s♠❛❧❧ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r②✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ▼ ✳ ❘❡♠❛r❦ ▼ ❝♦✉❧❞ ❛❧s♦ ❜❡ ❛ ❝❛t❡❣♦r② ♦❢ str✐♥❣ ❞✐❛❣r❛♠s ✇❤✐❝❤ ✐s ❢r❡❡❧② ❣❡♥❡r❛t❡❞✳ ✷ ✴ ✷✹

  4. ❈✐r❝✉✐t ▼♦❞❡❧ ❊①❛♠♣❧❡ ❙❤♦r✬s ❛❧❣♦r✐t❤♠ ❢♦r ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ ♠❛② ❜❡ s❡❡♥ ❛s ❛♥ ✐♥✜♥✐t❡ ❢❛♠✐❧② ♦❢ q✉❛♥t✉♠ ❝✐r❝✉✐ts ✕ ❡❛❝❤ ❝✐r❝✉✐t ✐s ❛ ♣r♦❝❡❞✉r❡ ❢♦r ❢❛❝t♦r✐③✐♥❣ ❛♥ ♥ ✲❜✐t ✐♥t❡❣❡r✱ ❢♦r ❛ ✜①❡❞ ♥ ✳ ❋✐❣✉r❡✿ ◗✉❛♥t✉♠ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ♦♥ ♥ q✉❜✐ts ✭s✉❜r♦✉t✐♥❡ ✐♥ ❙❤♦r✬s ❛❧❣♦r✐t❤♠✮✳ ✶ ✶ ❋✐❣✉r❡ s♦✉r❝❡✿ ❤tt♣s✿✴✴❝♦♠♠♦♥s✳✇✐❦✐♠❡❞✐❛✳♦r❣✴✇✴✐♥❞❡①✳♣❤♣❄❝✉r✐❞❂✶✹✺✹✺✻✶✷ ✸ ✴ ✷✹

  5. ▲♦♥❣ st♦r② s❤♦rt ❼ ▼❛✐♥ ❞✐✣❝✉❧t② ✐s ♦♥ t❤❡ ❞❡♥♦t❛t✐♦♥❛❧ s✐❞❡✳ ❼ ❍♦✇ ❝❛♥ ✇❡ ❝♦♣②✴❞✐s❝❛r❞ ✐♥t✉✐t✐♦♥✐st✐❝ r❡❝✉rs✐✈❡ t②♣❡s❄ ❼ ❆ ❧✐st ♦❢ q✉❜✐ts s❤♦✉❧❞ ❜❡ ❧✐♥❡❛r ✕ ❝❛♥♥♦t ❝♦♣②✴❞✐s❝❛r❞✳ ❼ ❆ ❧✐st ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs s❤♦✉❧❞ ❜❡ ✐♥t✉✐t✐♦♥✐st✐❝ ✕ ❝❛♥ ✐♠♣❧✐❝✐t❧② ❝♦♣②✴❞✐s❝❛r❞✳ ❼ ❋♦r t❤❡ r❡st ♦❢ t❤❡ t❛❧❦ ✇❡ ❢♦❝✉s ♦♥ t❤❡ ❧✐♥❡❛r✴♥♦♥✲❧✐♥❡❛r t②♣❡ str✉❝t✉r❡✳ ❼ ❍♦✇ ❞♦ ✇❡ ❞❡s✐❣♥ ❛ ❧✐♥❡❛r✴♥♦♥✲❧✐♥❡❛r ❋P❈❄ ✹ ✴ ✷✹

  6. ❚❤❡ ❇❛s✐❝ ✭♥♦♥✲r❡❝✉rs✐✈❡✮ ❚②♣❡s ❚②♣❡s ❆ , ❇ ✿✿❂ α ⑤ ✵ ⑤ ❆ + ❇ ⑤ ■ ⑤ ❆ ⊗ ❇ ⑤ ❆ ⊸ ❇ ⑤ ! ❆ ⑤ ❈✐r❝ ( ❚ , ❯ ) ■♥t✉✐t✐♦♥✐st✐❝ t②♣❡s ✿✿❂ ✵ ⑤ P + ❘ ⑤ ■ ⑤ P ⊗ ❘ ⑤ ! ❆ ⑤ ❈✐r❝ ( ❚ , ❯ ) P , ❘ ▼✲t②♣❡s ❚ , ❯ ✿✿❂ α ⑤ ■ ⑤ ❚ ⊗ ❯ ❘❡♠❛r❦ ❈✐r❝ ( ❚ , ❯ ) ∼ = !( ❚ ⊸ ❯ ) . ✺ ✴ ✷✹

  7. ❉❡♥♦t❛t✐♦♥❛❧ ▼♦❞❡❧ ❆ ♠♦❞❡❧ ♦❢ P◗▼ ✐s ✐♥❞✉❝❡❞ ❜② ❛ ▲✐♥❡❛r✴◆♦♥✲▲✐♥❡❛r ✭▲◆▲✮ ♠♦❞❡❧ ✷ ✿ ❼ ❆ ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ❝❛t❡❣♦r② ❱ ✳ ❼ ❆ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❝❧♦s❡❞ ❝❛t❡❣♦r② ❈ ✳ ❼ ❆ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❛❞❥✉♥❝t✐♦♥✿ ❋ ❱ ❈ ⊢ ● t♦❣❡t❤❡r ✇✐t❤ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ❞❛t❛ ✇❤✐❝❤ ✐s ✐rr❡❧❡✈❛♥t ❢♦r t❤✐s t❛❧❦✳ ❘❡♠❛r❦ ❆♥ ▲◆▲ ♠♦❞❡❧ ✐s ❛ ♠♦❞❡❧ ♦❢ ■♥t✉✐t✐♦♥✐st✐❝ ▲✐♥❡❛r ▲♦❣✐❝✳ ✷ ◆✐❝❦ ❇❡♥t♦♥✳ ❆ ♠✐①❡❞ ❧✐♥❡❛r ❛♥❞ ♥♦♥✲❧✐♥❡❛r ❧♦❣✐❝✿ Pr♦♦❢s✱ t❡r♠s ❛♥❞ ♠♦❞❡❧s ✳ ❈❙▲✬✾✹ ✻ ✴ ✷✹

  8. ❈♦♣②✐♥❣ ❛♥❞ ❞✐s❝❛r❞✐♥❣ ♦❢ ✐♥t✉✐t✐♦♥✐st✐❝ t②♣❡s ❋ ❱ ❈ ⊢ ● ■♥ P◗▼✱ ❛♥② t②♣❡ ❆ ✐s ✐♥t❡r❡♣r❡t❡❞ ❛s ❛♥ ♦❜❥❡❝t � ❆ � ∈ ❈ . ❚❤❡♦r❡♠ ❋♦r ❛♥② ✐♥t✉✐t✐♦♥✐st✐❝ t②♣❡ P✱ t❤❡r❡ ❡①✐sts ❛ ❝❛♥♦♥✐❝❛❧ ✐s♦♠♦r♣❤✐s♠ α P : � P � → ❋ � P � . ◆❡①t✱ ❞❡✜♥❡ ❝♦♣② ❛♥❞ ❞✐s❝❛r❞ ♠♦r♣❤✐s♠s ❢♦r ❡❛❝❤ ✐♥t✉✐t✐♦♥✐st✐❝ t②♣❡ P ✿ ∼ → ❋ � P � ❋ ✶ = α P ⋄ P := � P � − − − → ❋ ✶ − → ■ α − ✶ P ⊗ α − ✶ ∼ ❋ � ✐❞ , ✐❞ � = α P P ∆ P := � P � − − → ❋ � P � − − − − − → ❋ ( � P � × � P � ) − → ❋ � P � ⊗ ❋ � P � − − − − − − → � P � ⊗ � P � ✼ ✴ ✷✹

  9. ❆❞❞✐♥❣ ❘❡❝✉rs✐✈❡ ❚②♣❡s ❚②♣❡ ❱❛r✐❛❜❧❡s ❳ , ❨ ❚②♣❡s ❆ , ❇ ✿✿❂ ❳ ⑤ α ⑤ ❆ + ❇ ⑤ ■ ⑤ ❆ ⊗ ❇ ⑤ ❆ ⊸ ❇ ⑤ ! ❆ ⑤ ❈✐r❝ ( ❚ , ❯ ) ⑤ µ ❳ . ❆ ■♥t✉✐t✐♦♥✐st✐❝ t②♣❡s P , ❘ ✿✿❂ ❳ ⑤ P + ❘ ⑤ ■ ⑤ P ⊗ ❘ ⑤ ! ❆ ⑤ ❈✐r❝ ( ❚ , ❯ ) ⑤ µ ❳ . P α ⑤ ■ ⑤ ❚ ⊗ ❯ ▼✲t②♣❡s ❚ , ❯ ✿✿❂ ❘❡♠❛r❦ ❚❤❡s❡ t②♣❡s ❛r❡ ❛❝❝♦♠♣❛♥✐❡❞ ❜② s♦♠❡ ❢♦r♠❛t✐♦♥ r✉❧❡s✱ ✇❤✐❝❤ ✇❡ ♦♠✐t✳ ❉❡s✐❣♥ ❈❤♦✐❝❡✿ ❚✇♦ ❦✐♥❞s ♦❢ t②♣❡ ✈❛r✐❛❜❧❡s ✕ ✐♥t✉✐t✐♦♥✐st✐❝ ❛♥❞ ❧✐♥❡❛r❄ ❖r ❥✉st ♦♥❡ ❦✐♥❞ ✭❧✐❦❡ ❛❜♦✈❡✮❄ ✽ ✴ ✷✹

  10. ❙♦♠❡ ✉s❡❢✉❧ r❡❝✉rs✐✈❡ t②♣❡s ❊①❛♠♣❧❡ ◆❛t ≡ µ ❳ . ■ + ❳ ✭✐♥t✉✐t✐♦♥✐st✐❝✮ ❊①❛♠♣❧❡ ▲✐st ◆❛t ≡ µ ❳ . ■ + ❳ ⊗ ◆❛t ✭✐♥t✉✐t✐♦♥✐st✐❝✮ ❊①❛♠♣❧❡ ▲✐st ◗✉❜✐t ≡ µ ❳ . ■ + ❳ ⊗ ◗✉❜✐t ✭❧✐♥❡❛r✮ ✾ ✴ ✷✹

  11. ❚❡r♠ ❧❡✈❡❧ r❡❝✉rs✐♦♥ ■♥ ❋P❈✱ ❛ t❡r♠✲❧❡✈❡❧ r❡❝✉rs✐♦♥ ♦♣❡r❛t♦r ♠❛② ❜❡ ❞❡✜♥❡❞ ✉s✐♥❣ ❢♦❧❞✴✉♥❢♦❧❞ ♠❛♣s✳ ❚❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r ♦✉r ❧❛♥❣✉❛❣❡✳ ❚❤❡♦r❡♠ ❚❤❡ t❡r♠✲❧❡✈❡❧ r❡❝✉rs✐♦♥ ♦♣❡r❛t♦r ❢♦r P◗▼ ✸ ✐s ♥♦✇ ❛ ❞❡r✐✈❡❞ r✉❧❡✳ ❋♦r ❛ ❣✐✈❡♥ t❡r♠ Φ , ③ :! ❆ ⊢ ♠ : ❆✱ ❞❡✜♥❡✿ α ③ ♠ ≡ ❧✐❢t ❢♦❧❞ λ ① ! µ ❳ . (! ❳ ⊸ ❆ ) . ( λ ③ ! ❆ . ♠ )( ❧✐❢t ( ✉♥❢♦❧❞ ❢♦r❝❡ ① ) ① ) r❡❝ ③ ! ❆ . ♠ ≡ ( ✉♥❢♦❧❞ ❢♦r❝❡ α ③ ♠ ) α ③ ♠ ✸ ❇❡rt ▲✐♥❞❡♥❤♦✈✐✉s✱ ▼✐❝❤❛❡❧ ▼✐s❧♦✈❡✱ ❱❧❛❞✐♠✐r ❩❛♠❞③❤✐❡✈✿ ❊♥r✐❝❤✐♥❣ ❛ ▲✐♥❡❛r✴◆♦♥✲❧✐♥❡❛r ▲❛♠❜❞❛ ❈❛❧❝✉❧✉s✿ ❆ Pr♦❣r❛♠♠✐♥❣ ▲❛♥❣✉❛❣❡ ❢♦r ❙tr✐♥❣ ❉✐❛❣r❛♠s✳ ▲■❈❙ ✷✵✶✽ ✶✵ ✴ ✷✹

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