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st trs t r rt s tr ttr rs

  1. ▲❡❛❦❛❣❡ ❘❡s✐❧✐❡♥t ❙✐❣♥❛t✉r❡s ✇✐t❤ ●r❛❝❡❢✉❧ ❉❡❣r❛❞❛t✐♦♥ ❏✳ ❇✳ ◆✐❡❧s❡♥✱ ❉✳ ❱❡♥t✉r✐✱ ❆✳ ❩♦tt❛r❡❧ ❆❛r❤✉s ❯♥✐✈❡rs✐t②✱ ❙❛♣✐❡♥③❛ ❯♥✐✈❡rs✐t② ♦❢ ❘♦♠❡ ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ◆✐❡❧s❡♥✱ ❱❡♥t✉r✐✱ ❩♦tt❛r❡❧ ✭✮ ▲❡❛❦❛❣❡ ❢r♦♠ ❙❤♦rt ❙✐❣♥❛t✉r❡s ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ✶ ✴ ✷✶

  2. ▼❡ss❛❣❡ ✐s s❡♥t ❛❧♦♥❣ ✇✐t❤ ❛ s✐❣♥❛t✉r❡ ❙✐❣♥❛t✉r❡ ❙❝❤❡♠❡s ❙✐❣♥❛t✉r❡ s❝❤❡♠❡ ❂ ●❡♥ ❙✐❣ ❱❡r ●❡♥ ✶ ❦ ✿ ❣❡♥❡r❛t❡ ❛ s✐❣♥✐♥❣✴✈❡r✐✜❝❛t✐♦♥ ❦❡② t✉♣❧❡ ❙✐❣♥ s❦ ♠ ✿ ❣❡♥❡r❛t❡ ❛ s✐❣♥❛t✉r❡ ♦♥ ❛ ♠❡ss❛❣❡ ❱❡r ♠ ✿ ♦✉t♣✉ts ✵ ♦r ✶✳ ❊①✐st❡♥t✐❛❧ ❯♥❢♦r❣❡❛❜✐❧✐t② ❆❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ s✐❣♥✐♥❣ ♦r❛❝❧❡ ❢♦r ♠❡ss❛❣❡s ♦❢ ❤✐s ❝❤♦✐❝❡✳ ❆❞✈❡rs❛r② ♦✉t♣✉ts ❢♦r❣❡r② ❙✐❣ s❦ ♠ ❢♦r ♠ ♦❢ ❤✐s ❝❤♦✐❝❡ ♠ ♥♦t ❛s❦❡❞ t♦ t❤❡ s✐❣♥✐♥❣ ♦r❛❝❧❡✳ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❍♦✇ ❞♦ ✇❡ ❣✉❛r❛♥t❡❡ ❛✉t❤❡♥t✐❝✐t② ♦❢ ♦✉r ♠❡ss❛❣❡❄ ◆✐❡❧s❡♥✱ ❱❡♥t✉r✐✱ ❩♦tt❛r❡❧ ✭✮ ▲❡❛❦❛❣❡ ❢r♦♠ ❙❤♦rt ❙✐❣♥❛t✉r❡s ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ✷ ✴ ✷✶

  3. ❙✐❣♥❛t✉r❡ ❙❝❤❡♠❡s ❙✐❣♥❛t✉r❡ s❝❤❡♠❡ ❂ ●❡♥ ❙✐❣ ❱❡r ●❡♥ ✶ ❦ ✿ ❣❡♥❡r❛t❡ ❛ s✐❣♥✐♥❣✴✈❡r✐✜❝❛t✐♦♥ ❦❡② t✉♣❧❡ ❙✐❣♥ s❦ ♠ ✿ ❣❡♥❡r❛t❡ ❛ s✐❣♥❛t✉r❡ ♦♥ ❛ ♠❡ss❛❣❡ ❱❡r ♠ ✿ ♦✉t♣✉ts ✵ ♦r ✶✳ ❊①✐st❡♥t✐❛❧ ❯♥❢♦r❣❡❛❜✐❧✐t② ❆❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ s✐❣♥✐♥❣ ♦r❛❝❧❡ ❢♦r ♠❡ss❛❣❡s ♦❢ ❤✐s ❝❤♦✐❝❡✳ ❆❞✈❡rs❛r② ♦✉t♣✉ts ❢♦r❣❡r② ❙✐❣ s❦ ♠ ❢♦r ♠ ♦❢ ❤✐s ❝❤♦✐❝❡ ♠ ♥♦t ❛s❦❡❞ t♦ t❤❡ s✐❣♥✐♥❣ ♦r❛❝❧❡✳ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❍♦✇ ❞♦ ✇❡ ❣✉❛r❛♥t❡❡ ❛✉t❤❡♥t✐❝✐t② ♦❢ ♦✉r ♠❡ss❛❣❡❄ ⇓ ▼❡ss❛❣❡ ✐s s❡♥t ❛❧♦♥❣ ✇✐t❤ ❛ s✐❣♥❛t✉r❡ ◆✐❡❧s❡♥✱ ❱❡♥t✉r✐✱ ❩♦tt❛r❡❧ ✭✮ ▲❡❛❦❛❣❡ ❢r♦♠ ❙❤♦rt ❙✐❣♥❛t✉r❡s ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ✷ ✴ ✷✶

  4. ❊①✐st❡♥t✐❛❧ ❯♥❢♦r❣❡❛❜✐❧✐t② ❆❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ s✐❣♥✐♥❣ ♦r❛❝❧❡ ❢♦r ♠❡ss❛❣❡s ♦❢ ❤✐s ❝❤♦✐❝❡✳ ❆❞✈❡rs❛r② ♦✉t♣✉ts ❢♦r❣❡r② ❙✐❣ s❦ ♠ ❢♦r ♠ ♦❢ ❤✐s ❝❤♦✐❝❡ ♠ ♥♦t ❛s❦❡❞ t♦ t❤❡ s✐❣♥✐♥❣ ♦r❛❝❧❡✳ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❍♦✇ ❞♦ ✇❡ ❣✉❛r❛♥t❡❡ ❛✉t❤❡♥t✐❝✐t② ♦❢ ♦✉r ♠❡ss❛❣❡❄ ⇓ ▼❡ss❛❣❡ ✐s s❡♥t ❛❧♦♥❣ ✇✐t❤ ❛ s✐❣♥❛t✉r❡ ❙✐❣♥❛t✉r❡ ❙❝❤❡♠❡s ❙✐❣♥❛t✉r❡ s❝❤❡♠❡ ❂ ( ●❡♥ , ❙✐❣ , ❱❡r ) ●❡♥ ( ✶ ❦ ) ✿ ❣❡♥❡r❛t❡ ❛ s✐❣♥✐♥❣✴✈❡r✐✜❝❛t✐♦♥ ❦❡② t✉♣❧❡ ❙✐❣♥ ( s❦ , ♠ ) ✿ ❣❡♥❡r❛t❡ ❛ s✐❣♥❛t✉r❡ ♦♥ ❛ ♠❡ss❛❣❡ ❱❡r ( ♠ , σ ) ✿ ♦✉t♣✉ts ✵ ♦r ✶✳ ◆✐❡❧s❡♥✱ ❱❡♥t✉r✐✱ ❩♦tt❛r❡❧ ✭✮ ▲❡❛❦❛❣❡ ❢r♦♠ ❙❤♦rt ❙✐❣♥❛t✉r❡s ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ✷ ✴ ✷✶

  5. ❆✉t❤❡♥t✐❝❛t✐♦♥ ❍♦✇ ❞♦ ✇❡ ❣✉❛r❛♥t❡❡ ❛✉t❤❡♥t✐❝✐t② ♦❢ ♦✉r ♠❡ss❛❣❡❄ ⇓ ▼❡ss❛❣❡ ✐s s❡♥t ❛❧♦♥❣ ✇✐t❤ ❛ s✐❣♥❛t✉r❡ ❙✐❣♥❛t✉r❡ ❙❝❤❡♠❡s ❙✐❣♥❛t✉r❡ s❝❤❡♠❡ ❂ ( ●❡♥ , ❙✐❣ , ❱❡r ) ●❡♥ ( ✶ ❦ ) ✿ ❣❡♥❡r❛t❡ ❛ s✐❣♥✐♥❣✴✈❡r✐✜❝❛t✐♦♥ ❦❡② t✉♣❧❡ ❙✐❣♥ ( s❦ , ♠ ) ✿ ❣❡♥❡r❛t❡ ❛ s✐❣♥❛t✉r❡ ♦♥ ❛ ♠❡ss❛❣❡ ❱❡r ( ♠ , σ ) ✿ ♦✉t♣✉ts ✵ ♦r ✶✳ ❊①✐st❡♥t✐❛❧ ❯♥❢♦r❣❡❛❜✐❧✐t② ❆❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ s✐❣♥✐♥❣ ♦r❛❝❧❡ ❢♦r ♠❡ss❛❣❡s ♦❢ ❤✐s ❝❤♦✐❝❡✳ ❆❞✈❡rs❛r② ♦✉t♣✉ts ❢♦r❣❡r② ❙✐❣ s❦ ( ♠ ∗ ) ❢♦r ♠ ∗ ♦❢ ❤✐s ❝❤♦✐❝❡ ♠ ∗ ♥♦t ❛s❦❡❞ t♦ t❤❡ s✐❣♥✐♥❣ ♦r❛❝❧❡✳ ◆✐❡❧s❡♥✱ ❱❡♥t✉r✐✱ ❩♦tt❛r❡❧ ✭✮ ▲❡❛❦❛❣❡ ❢r♦♠ ❙❤♦rt ❙✐❣♥❛t✉r❡s ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ✷ ✴ ✷✶

  6. ♠♦❞❡❧✐♥❣ ❛❞✈❡rs❛r② ❡①tr❛❝t✐♥❣ ❜✐ts ❢r♦♠ t❤❡ s❡❝r❡t ❦❡②✿ ❧❡❛❦❛❣❡ ❢✉♥❝t✐♦♥ ❤ ✵ ✶ ✳ s❦ ❜♦✉♥❞❡❞ ♠♦❞❡❧✿ ❤ s❦ s❦ ❙✐❣♥❛t✉r❡s ✐♥ t❤❡ ❇♦✉♥❞❡❞ ▼♦❞❡❧ ❆❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ s✐❣♥✐♥❣ ♦r❛❝❧❡ ❛♥❞ s❦ ❤ r❡t✉r♥✐♥❣ ❤ s❦ ♦r❛❝❧❡ ❆❞✈❡rs❛r② ♦✉t♣✉ts ❢♦r❣❡r② ❙✐❣♥ s❦ ♠ ❢♦r ♠ ♦❢ ❤✐s ❝❤♦✐❝❡ ♠ ♥♦t ❛s❦❡❞ t♦ t❤❡ s✐❣♥✐♥❣ ♦r❛❝❧❡✳ ❙✐❣♥❛t✉r❡s ✐♥ ❛ ❧❡❛❦② ✇♦r❧❞ ❡①t❡♥❞ t❤✐s ♠♦❞❡❧ ❝❛♣t✉r✐♥❣ ❧❡❛❦❛❣❡ ◆✐❡❧s❡♥✱ ❱❡♥t✉r✐✱ ❩♦tt❛r❡❧ ✭✮ ▲❡❛❦❛❣❡ ❢r♦♠ ❙❤♦rt ❙✐❣♥❛t✉r❡s ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ✸ ✴ ✷✶

  7. ❧❡❛❦❛❣❡ ❢✉♥❝t✐♦♥ ❤ ✵ ✶ ✳ s❦ ❜♦✉♥❞❡❞ ♠♦❞❡❧✿ ❤ s❦ s❦ ❙✐❣♥❛t✉r❡s ✐♥ t❤❡ ❇♦✉♥❞❡❞ ▼♦❞❡❧ ❆❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ s✐❣♥✐♥❣ ♦r❛❝❧❡ ❛♥❞ s❦ ❤ r❡t✉r♥✐♥❣ ❤ s❦ ♦r❛❝❧❡ ❆❞✈❡rs❛r② ♦✉t♣✉ts ❢♦r❣❡r② ❙✐❣♥ s❦ ♠ ❢♦r ♠ ♦❢ ❤✐s ❝❤♦✐❝❡ ♠ ♥♦t ❛s❦❡❞ t♦ t❤❡ s✐❣♥✐♥❣ ♦r❛❝❧❡✳ ❙✐❣♥❛t✉r❡s ✐♥ ❛ ❧❡❛❦② ✇♦r❧❞ ❡①t❡♥❞ t❤✐s ♠♦❞❡❧ ❝❛♣t✉r✐♥❣ ❧❡❛❦❛❣❡ ♠♦❞❡❧✐♥❣ ❛❞✈❡rs❛r② ❡①tr❛❝t✐♥❣ ❜✐ts ❢r♦♠ t❤❡ s❡❝r❡t ❦❡②✿ ◆✐❡❧s❡♥✱ ❱❡♥t✉r✐✱ ❩♦tt❛r❡❧ ✭✮ ▲❡❛❦❛❣❡ ❢r♦♠ ❙❤♦rt ❙✐❣♥❛t✉r❡s ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ✸ ✴ ✷✶

  8. ❜♦✉♥❞❡❞ ♠♦❞❡❧✿ ❤ s❦ s❦ ❙✐❣♥❛t✉r❡s ✐♥ t❤❡ ❇♦✉♥❞❡❞ ▼♦❞❡❧ ❆❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ s✐❣♥✐♥❣ ♦r❛❝❧❡ ❛♥❞ s❦ ❤ r❡t✉r♥✐♥❣ ❤ s❦ ♦r❛❝❧❡ ❆❞✈❡rs❛r② ♦✉t♣✉ts ❢♦r❣❡r② ❙✐❣♥ s❦ ♠ ❢♦r ♠ ♦❢ ❤✐s ❝❤♦✐❝❡ ♠ ♥♦t ❛s❦❡❞ t♦ t❤❡ s✐❣♥✐♥❣ ♦r❛❝❧❡✳ ❙✐❣♥❛t✉r❡s ✐♥ ❛ ❧❡❛❦② ✇♦r❧❞ ❡①t❡♥❞ t❤✐s ♠♦❞❡❧ ❝❛♣t✉r✐♥❣ ❧❡❛❦❛❣❡ ♠♦❞❡❧✐♥❣ ❛❞✈❡rs❛r② ❡①tr❛❝t✐♥❣ ❜✐ts ❢r♦♠ t❤❡ s❡❝r❡t ❦❡②✿ ◮ ❧❡❛❦❛❣❡ ❢✉♥❝t✐♦♥ ❤ : s❦ �→ { ✵ , ✶ } ∗ ✳ ◆✐❡❧s❡♥✱ ❱❡♥t✉r✐✱ ❩♦tt❛r❡❧ ✭✮ ▲❡❛❦❛❣❡ ❢r♦♠ ❙❤♦rt ❙✐❣♥❛t✉r❡s ▼❛r❝❤ ✷✼t❤✱ ✷✵✶✹ ✸ ✴ ✷✶

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