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  1. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❚♦✉❧♦✉s❡ ❯♥✐✈❡rs✐té ❚♦✉❧♦✉s❡ ■■■ ❤tt♣✿✴✴✇✇✇✳♠❛t❤✳✉♥✐✈✲t♦✉❧♦✉s❡✳❢r✴✄s❝❤❧❡♥❦❡r ❈♦♥❢❡r❡♥❝❡ ✏❙❡r✐❡s✻✵✑✱ ❲❛r✇✐❝❦✱ ❏✉❧② ✷✺✱ ✷✵✶✶ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

  2. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ❈♦♥t❡♥t ❘❡❝❛❧❧ ♠❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s✱ ❡❛rt❤q✉❛❦❡s✱ ❊①t❡♥s✐♦♥ t♦ ✏❧❛♥❞s❧✐❞❡s✑✱ ❯♥❞❡r❧②✐♥❣ ❆❞❙ ❣❡♦♠❡tr②✳ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❋r❛♥❝❡s❝♦ ❇♦♥s❛♥t❡ ❛♥❞ ●❛❜r✐❡❧❡ ▼♦♥❞❡❧❧♦✳ ❙ ✐s ❛ ❝❧♦s❡❞ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s ≥ ✷✱ T = ❚❡✐❝❤♠ü❧❧❡r s♣❛❝❡ ♦❢ ❙ ✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

  3. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ❈♦♥t❡♥t ❘❡❝❛❧❧ ♠❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s✱ ❡❛rt❤q✉❛❦❡s✱ ❊①t❡♥s✐♦♥ t♦ ✏❧❛♥❞s❧✐❞❡s✑✱ ❯♥❞❡r❧②✐♥❣ ❆❞❙ ❣❡♦♠❡tr②✳ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❋r❛♥❝❡s❝♦ ❇♦♥s❛♥t❡ ❛♥❞ ●❛❜r✐❡❧❡ ▼♦♥❞❡❧❧♦✳ ❙ ✐s ❛ ❝❧♦s❡❞ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s ≥ ✷✱ T = ❚❡✐❝❤♠ü❧❧❡r s♣❛❝❡ ♦❢ ❙ ✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

  4. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ❈♦♥t❡♥t ❘❡❝❛❧❧ ♠❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s✱ ❡❛rt❤q✉❛❦❡s✱ ❊①t❡♥s✐♦♥ t♦ ✏❧❛♥❞s❧✐❞❡s✑✱ ❯♥❞❡r❧②✐♥❣ ❆❞❙ ❣❡♦♠❡tr②✳ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❋r❛♥❝❡s❝♦ ❇♦♥s❛♥t❡ ❛♥❞ ●❛❜r✐❡❧❡ ▼♦♥❞❡❧❧♦✳ ❙ ✐s ❛ ❝❧♦s❡❞ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s ≥ ✷✱ T = ❚❡✐❝❤♠ü❧❧❡r s♣❛❝❡ ♦❢ ❙ ✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

  5. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ❈♦♥t❡♥t ❘❡❝❛❧❧ ♠❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s✱ ❡❛rt❤q✉❛❦❡s✱ ❊①t❡♥s✐♦♥ t♦ ✏❧❛♥❞s❧✐❞❡s✑✱ ❯♥❞❡r❧②✐♥❣ ❆❞❙ ❣❡♦♠❡tr②✳ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❋r❛♥❝❡s❝♦ ❇♦♥s❛♥t❡ ❛♥❞ ●❛❜r✐❡❧❡ ▼♦♥❞❡❧❧♦✳ ❙ ✐s ❛ ❝❧♦s❡❞ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s ≥ ✷✱ T = ❚❡✐❝❤♠ü❧❧❡r s♣❛❝❡ ♦❢ ❙ ✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

  6. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ❈♦♥t❡♥t ❘❡❝❛❧❧ ♠❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s✱ ❡❛rt❤q✉❛❦❡s✱ ❊①t❡♥s✐♦♥ t♦ ✏❧❛♥❞s❧✐❞❡s✑✱ ❯♥❞❡r❧②✐♥❣ ❆❞❙ ❣❡♦♠❡tr②✳ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❋r❛♥❝❡s❝♦ ❇♦♥s❛♥t❡ ❛♥❞ ●❛❜r✐❡❧❡ ▼♦♥❞❡❧❧♦✳ ❙ ✐s ❛ ❝❧♦s❡❞ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s ≥ ✷✱ T = ❚❡✐❝❤♠ü❧❧❡r s♣❛❝❡ ♦❢ ❙ ✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

  7. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s WM ❂ { ✇❡✐❣❤t❡❞ ♠✉❧t✐❝✉r✈❡s ♦♥ ❙ } ✿ s❡t ♦❢ ❞✐s❥♦✐♥t s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡s✱ ❡❛❝❤ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t✳ WM ✐s ✐♥✜♥✐t❡ ✿ s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡s ♦♥ ❙ ❝❛♥ ✇r❛♣ ❛r♦✉♥❞ ❛ ❧♦t✳ ▲❡t ( ❝ ✐ , ❧ ✐ ) ✐ = ✶ , ··· , ♥ ∈ WM ✱ t❤❡ ❝ ✐ ❢♦r♠ ❛ ❧❛♠✐♥❛t✐♦♥ ❛♥❞ t❤❡ ❧ ✐ ❞❡✜♥❡ ❛ tr❛♥s✈❡rs❡ ♠❡❛s✉r❡ ✿ ❣✐✈❡s ❛ t♦t❛❧ ✇❡✐❣❤t t♦ γ ✱ tr❛♥s✲ ✈❡rs❡ t♦ t❤❡ ❝ ✐ ✳ ❚❤✐s ❣✐✈❡s ❛ t♦♣♦❧♦❣② t♦ WM ✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ WM ✐s t❤❡ s♣❛❝❡ ♦❢ ♠❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ML ✳ ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❝❛♥ ❜❡ ♣r❡tt② ❝♦♠♣❧✐❝❛t❡❞✳ ML ≃ R ✻ ❣ − ✻ ✳ ∂ T ≃ ML / R > ✵ ✭❚❤✉rst♦♥✮✳ T × ML ≃ ❚ ∗ T ✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

  8. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s WM ❂ { ✇❡✐❣❤t❡❞ ♠✉❧t✐❝✉r✈❡s ♦♥ ❙ } ✿ s❡t ♦❢ ❞✐s❥♦✐♥t s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡s✱ ❡❛❝❤ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t✳ WM ✐s ✐♥✜♥✐t❡ ✿ s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡s ♦♥ ❙ ❝❛♥ ✇r❛♣ ❛r♦✉♥❞ ❛ ❧♦t✳ ▲❡t ( ❝ ✐ , ❧ ✐ ) ✐ = ✶ , ··· , ♥ ∈ WM ✱ t❤❡ ❝ ✐ ❢♦r♠ ❛ ❧❛♠✐♥❛t✐♦♥ ❛♥❞ t❤❡ ❧ ✐ ❞❡✜♥❡ ❛ tr❛♥s✈❡rs❡ ♠❡❛s✉r❡ ✿ ❣✐✈❡s ❛ t♦t❛❧ ✇❡✐❣❤t t♦ γ ✱ tr❛♥s✲ ✈❡rs❡ t♦ t❤❡ ❝ ✐ ✳ ❚❤✐s ❣✐✈❡s ❛ t♦♣♦❧♦❣② t♦ WM ✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ WM ✐s t❤❡ s♣❛❝❡ ♦❢ ♠❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ML ✳ ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❝❛♥ ❜❡ ♣r❡tt② ❝♦♠♣❧✐❝❛t❡❞✳ ML ≃ R ✻ ❣ − ✻ ✳ ∂ T ≃ ML / R > ✵ ✭❚❤✉rst♦♥✮✳ T × ML ≃ ❚ ∗ T ✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

  9. ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❛♥❞ ❡❛rt❤q✉❛❦❡s ▲❛♥❞s❧✐❞❡s ❆❞❙ ❣❡♦♠❡tr② ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s WM ❂ { ✇❡✐❣❤t❡❞ ♠✉❧t✐❝✉r✈❡s ♦♥ ❙ } ✿ s❡t ♦❢ ❞✐s❥♦✐♥t s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡s✱ ❡❛❝❤ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t✳ WM ✐s ✐♥✜♥✐t❡ ✿ s✐♠♣❧❡ ❝❧♦s❡❞ ❝✉r✈❡s ♦♥ ❙ ❝❛♥ ✇r❛♣ ❛r♦✉♥❞ ❛ ❧♦t✳ ▲❡t ( ❝ ✐ , ❧ ✐ ) ✐ = ✶ , ··· , ♥ ∈ WM ✱ t❤❡ ❝ ✐ ❢♦r♠ ❛ ❧❛♠✐♥❛t✐♦♥ ❛♥❞ t❤❡ ❧ ✐ ❞❡✜♥❡ ❛ tr❛♥s✈❡rs❡ c c c 2 3 1 ♠❡❛s✉r❡ ✿ ❣✐✈❡s ❛ t♦t❛❧ ✇❡✐❣❤t t♦ γ ✱ tr❛♥s✲ γ ✈❡rs❡ t♦ t❤❡ ❝ ✐ ✳ ❚❤✐s ❣✐✈❡s ❛ t♦♣♦❧♦❣② t♦ WM ✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ WM ✐s t❤❡ s♣❛❝❡ ♦❢ ♠❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ML ✳ ▼❡❛s✉r❡❞ ❧❛♠✐♥❛t✐♦♥s ❝❛♥ ❜❡ ♣r❡tt② ❝♦♠♣❧✐❝❛t❡❞✳ ML ≃ R ✻ ❣ − ✻ ✳ ∂ T ≃ ML / R > ✵ ✭❚❤✉rst♦♥✮✳ T × ML ≃ ❚ ∗ T ✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❛rt❤q✉❛❦❡ ✢♦✇

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