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  1. ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆♥❞r❡❛s ▼❛❧❡tt✐ ✶ ❛♥❞ ❍❡✐❦♦ ❱♦❣❧❡r ✷ ✶ ❯❘❱✱ ❚❛rr❛❣♦♥❛✱ ❙♣❛✐♥ ✷ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❉r❡s❞❡♥✱ ●❡r♠❛♥② ❛♥❞r❡❛s✳♠❛❧❡tt✐❅✉r✈✳❝❛t Pr❡t♦r✐❛ ✖ ❏✉❧② ✷✹✱ ✷✵✵✾ ✶ ❋✐rst ❛✉t❤♦r ❛❝❦♥♦✇❧❡❞❣❡s t❤❡ s✉♣♣♦rt ♦❢ ▼❊❈ ❣r❛♥ts ❏❉❈■✲✷✵✵✼✲✼✻✵ ❛♥❞ ▼❚▼✲✷✵✵✼✲✻✸✹✷✷✳ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✶ ·

  2. ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  3. ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ❈❖◆❏ ❙ ✇❛ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  4. ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ❈❖◆❏ ❙ ◆P ✶ ❱P ✇❛ ◆P ✶ ◆P ✷ ❱ ◆P ✷ ❱ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  5. ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ❈❖◆❏ ❙ ◆P ✶ ❱P ✇❛ ◆P ✶ ◆P ✷ ❱ ◆P ✷ ❱ r❛✬❛❛ s❛✇ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  6. ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ◆P ❱P ❈❖◆❏ ❙ ✇❛ ❉❚ ◆ ❱ ◆P ❱ ◆P ◆P s❛✇ t❤❡ ◆ r❛✬❛❛ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  7. ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ◆P ❱P ❈❖◆❏ ❙ ✇❛ ❉❚ ◆ ❱ ◆P ❱ ◆P ◆P ❜♦② s❛✇ r❛✬❛❛ ◆ t❤❡ ❛t❡✢ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  8. ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ◆P ❱P ❈❖◆❏ ❙ ✇❛ ❉❚ ◆ ❱ ◆P ❱ ◆P ◆P s❛✇ ❜♦② t❤❡ ❉❚ ◆ r❛✬❛❛ ◆ ◆ t❤❡ ❛t❡✢ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  9. ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ◆P ❱P ❈❖◆❏ ❙ ✇❛ ❉❚ ◆ ❱ ◆P ❱ ◆P ◆P s❛✇ ◆ ◆ ❜♦② r❛✬❛❛ t❤❡ ❉❚ ◆ ❛t❡✢ ❛❧❜❛❜ t❤❡ ❞♦♦r ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  10. ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ❙ ❙ ◆P ❱P ❈❖◆❏ ❙ ✇❛ ❉❚ ◆ ❱ ◆P ❱ ◆P ◆P s❛✇ ◆ ◆ ❜♦② r❛✬❛❛ t❤❡ ❉❚ ◆ ❛t❡✢ ❛❧❜❛❜ t❤❡ ❞♦♦r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥♣✉t✲♦r✐❡♥t❡❞ ✇✐t❤ ❤✐❞❞❡♥ st❛t❡s ✭ → tr❡❡ tr❛♥s❞✉❝❡r✮ ❧✐♥❡❛r ♥♦♥❞❡❧❡t✐♥❣ ❡①t❡♥❞❡❞ t♦♣✲❞♦✇♥ tr❡❡ tr❛♥s❞✉❝❡r ✐♥ ❚✐❜✉r♦♥ ❬▼❛②✱ ❑♥✐❣❤t ✬✵✻❪ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✷ ·

  11. ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ✭❝♦♥t✬❞✮ ❙②♥❝❤r♦♥♦✉s tr❡❡ s✉❜st✐t✉t✐♦♥ ❣r❛♠♠❛r r✉❧❡✿ ❙ ❙ ◆P ✶ ❱P ✖ ◆P ✶ ◆P ✷ ❱ ◆P ✷ ❱ ❚r❡❡ tr❛♥s❞✉❝❡r r✉❧❡✿ q ❙ ❙ ❙ → q ❱ q ◆P q ◆P ① ✶ ❱P ① ✷ ① ✶ ① ✸ ① ✷ ① ✸ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✸ ·

  12. ▼♦t✐✈❛t✐♦♥ ❙②♥❝❤r♦♥♦✉s ❚r❡❡ ❙✉❜st✐t✉t✐♦♥ ●r❛♠♠❛rs ✭❝♦♥t✬❞✮ ❙②♥❝❤r♦♥♦✉s tr❡❡ s✉❜st✐t✉t✐♦♥ ❣r❛♠♠❛r r✉❧❡✿ ❙ ❙ ◆P ✶ ❱P ✖ ◆P ✶ ◆P ✷ ❱ ◆P ✷ ❱ ❚r❡❡ tr❛♥s❞✉❝❡r r✉❧❡✿ ❙✐♠♣❧✐✜❡❞ tr❛♥s❞✉❝❡r r✉❧❡✿ q ❙ σ ❙ δ ① ✶ σ → ❙ → q ❱ q ◆P q ◆P ① ✷ ① ✶ ① ✸ ① ✷ ① ✸ ① ✶ ❱P ① ✷ ① ✶ ① ✸ ① ✷ ① ✸ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❚r❛♥s❞✉❝❡rs ✇✐t❤ ε ✲❘✉❧❡s ❆✳ ▼❛❧❡tt✐✱ ❍✳ ❱♦❣❧❡r ✸ ·

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