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  1. ❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡ ❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❋✐❜♦♥❛❝❝✐ ❋ ♥ = ❋ ♥ − ✶ + ❋ ♥ − ✷ ✶ ✵ ] ∗ ❋ ♥ − ✶ ❋ ♥ � ❋ ♥ − ✶ = [ ✶ ✶ � ❋ ♥ − ✷ ✶ ✵ ] ♥ ∗ ❋ ✶ ❋ ♥ � ❋ ♥ − ✶ = [ ✶ ✶ � ❋ ✵ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  2. ❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❋✐❜♦♥❛❝❝✐ ❋ ♥ = ❋ ♥ − ✶ + ❋ ♥ − ✷ ✶ ✵ ] ∗ ❋ ♥ − ✶ ❋ ♥ � ❋ ♥ − ✶ = [ ✶ ✶ � ❋ ♥ − ✷ ✶ ✵ ] ♥ ∗ ❋ ✶ ❋ ♥ � ❋ ♥ − ✶ = [ ✶ ✶ � ❋ ✵ ❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  3. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❋✐❜♦♥❛❝❝✐ ❋ ♥ = ❋ ♥ − ✶ + ❋ ♥ − ✷ ✶ ✵ ] ∗ ❋ ♥ − ✶ ❋ ♥ � ❋ ♥ − ✶ = [ ✶ ✶ � ❋ ♥ − ✷ ✶ ✵ ] ♥ ∗ ❋ ✶ ❋ ♥ � ❋ ♥ − ✶ = [ ✶ ✶ � ❋ ✵ ❈♦♠♣✉t❡ t❤❡ ♣r♦❞✉❝t ✐♥ ❖✭❧❣ ♥✮ t✐♠❡ ❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✉♣♣♦rt ❛♥② ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  4. ❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✶ ♥ ❂ ♥ ♥ ✶ ♥ ❦ ✶ ❦ ❦ ❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✷ ❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ✶ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥ ♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ✷ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮ ♥ ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ♥ ❈♦♠♣✉t✐♥❣ ❦ ❈❛s❡s✿ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  5. ❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✶ ♥ ❂ ♥ ♥ ✶ ♥ ❦ ✶ ❦ ❦ ❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✷ ❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ✶ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥ ♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ✷ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮ ♥ ❈♦♠♣✉t✐♥❣ ❦ ❈❛s❡s✿ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts � ♥ � ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  6. ❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ✶ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥ ♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ✷ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮ ❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✶ ♥ ❂ ♥ ♥ ✶ ♥ ❦ ✶ ❦ ❦ ❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✷ ❈❛s❡s✿ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts � ♥ � ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts � ♥ � ❈♦♠♣✉t✐♥❣ ❦ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  7. ❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ✶ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥ ♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ✷ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮ ❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✷ ❈❛s❡s✿ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts � ♥ � ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts � ♥ � ❈♦♠♣✉t✐♥❣ ❦ ❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✶ ❂ ♥ ( ♥ − ✶ ) ... ( ♥ − ❦ + ✶ ) � ♥ � ❦ ❦ ! ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  8. ❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ✶ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥ ♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ✷ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮ ❈❛s❡s✿ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts � ♥ � ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts � ♥ � ❈♦♠♣✉t✐♥❣ ❦ ❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✶ ❂ ♥ ( ♥ − ✶ ) ... ( ♥ − ❦ + ✶ ) � ♥ � ❦ ❦ ! ❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✷ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  9. ❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ✶ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥ ♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ✷ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts � ♥ � ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts � ♥ � ❈♦♠♣✉t✐♥❣ ❦ ❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✶ ❂ ♥ ( ♥ − ✶ ) ... ( ♥ − ❦ + ✶ ) � ♥ � ❦ ❦ ! ❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✷ ❈❛s❡s✿ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  10. ♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ✷ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts � ♥ � ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts � ♥ � ❈♦♠♣✉t✐♥❣ ❦ ❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✶ ❂ ♥ ( ♥ − ✶ ) ... ( ♥ − ❦ + ✶ ) � ♥ � ❦ ❦ ! ❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✷ ❈❛s❡s✿ ❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ✶ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  11. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts � ♥ � ❂ ◆✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❦ ♦❜❥❡❝ts ♦✉t ♦❢ ♥ ❦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts � ♥ � ❈♦♠♣✉t✐♥❣ ❦ ❈♦♠♣✉t❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✶ ❂ ♥ ( ♥ − ✶ ) ... ( ♥ − ❦ + ✶ ) � ♥ � ❦ ❦ ! ❯s❡ P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✷ ❈❛s❡s✿ ❇♦t❤ ♥ ❛♥❞ ❦ ❛r❡ s♠❛❧❧ ✶ ❯s❡ ❡✐t❤❡r s♦❧✉t✐♦♥ ♥ ✐s ❜✐❣✱ ❜✉t ❦ ♦r ♥✲❦ ✐s s♠❛❧❧ ✷ ❯s❡ ❙♦❧✉t✐♦♥ ✶ ✭❝❛r❡❢✉❧❧②✮ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  12. ❚❤❡ ▲✉❝❛s✬ t❤❡♦r❡♠ ❡①♣r❡ss❡s t❤❡ r❡♠❛✐♥❞❡r ♦❢ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ ♠ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ❜② ❛ ♣r✐♠❡ ♥✉♠❜❡r ♣ ✐♥ t❡r♠s ♦❢ t❤❡ ♥ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ t❤❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥✳ ❋♦r ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥ ❛♥❞ ❛ ♣r✐♠❡ ♣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❣r✉❡♥❝❡ r❡❧❛t✐♦♥ ❤♦❧❞s✿ ♠ ❦ ♠ ✐ ♠♦❞♣ ✐ ✵ ♥ ♥ ✐ ✇❤❡r❡ ♠ ❦ ♣ ❦ ✶ ♣ ❦ ✶ ♠ ♠ ❦ ♠ ✶ ♣ ♠ ✵ ❛♥❞ ♥ ❦ ♣ ❦ ✶ ♣ ❦ ✶ ♥ ♥ ❦ ♥ ✶ ♣ ♥ ✵ ❛r❡ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ ♠ ❛♥❞ ♥ r❡s♣❡❝t✐✈❡❧②✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▲✉❝❛s ❚❤❡♦r❡♠ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  13. ❋♦r ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥ ❛♥❞ ❛ ♣r✐♠❡ ♣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❣r✉❡♥❝❡ r❡❧❛t✐♦♥ ❤♦❧❞s✿ ♠ ❦ ♠ ✐ ♠♦❞♣ ✐ ✵ ♥ ♥ ✐ ✇❤❡r❡ ♠ ❦ ♣ ❦ ✶ ♣ ❦ ✶ ♠ ♠ ❦ ♠ ✶ ♣ ♠ ✵ ❛♥❞ ♥ ❦ ♣ ❦ ✶ ♣ ❦ ✶ ♥ ♥ ❦ ♥ ✶ ♣ ♥ ✵ ❛r❡ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ ♠ ❛♥❞ ♥ r❡s♣❡❝t✐✈❡❧②✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▲✉❝❛s ❚❤❡♦r❡♠ ❚❤❡ ▲✉❝❛s✬ t❤❡♦r❡♠ ❡①♣r❡ss❡s t❤❡ r❡♠❛✐♥❞❡r ♦❢ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ � ♠ � ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ❜② ❛ ♣r✐♠❡ ♥✉♠❜❡r ♣ ✐♥ t❡r♠s ♦❢ t❤❡ ♥ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ t❤❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  14. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▲✉❝❛s ❚❤❡♦r❡♠ ❚❤❡ ▲✉❝❛s✬ t❤❡♦r❡♠ ❡①♣r❡ss❡s t❤❡ r❡♠❛✐♥❞❡r ♦❢ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ � ♠ � ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ❜② ❛ ♣r✐♠❡ ♥✉♠❜❡r ♣ ✐♥ t❡r♠s ♦❢ t❤❡ ♥ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ t❤❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥✳ ❋♦r ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ♠ ❛♥❞ ♥ ❛♥❞ ❛ ♣r✐♠❡ ♣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❣r✉❡♥❝❡ r❡❧❛t✐♦♥ ❤♦❧❞s✿ � ♠ = � ❦ � ♠ ✐ � � ( ♠♦❞♣ ) ✐ = ✵ ♥ ♥ ✐ ✇❤❡r❡ ♠ = ♠ ❦ ♣ ❦ + ♠ ❦ − ✶ ♣ ❦ − ✶ + · · · + ♠ ✶ ♣ + ♠ ✵ ❛♥❞ ♥ = ♥ ❦ ♣ ❦ + ♥ ❦ − ✶ ♣ ❦ − ✶ + · · · + ♥ ✶ ♣ + ♥ ✵ ❛r❡ ❜❛s❡ ♣ ❡①♣❛♥s✐♦♥s ♦❢ ♠ ❛♥❞ ♥ r❡s♣❡❝t✐✈❡❧②✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  15. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr♦❜❧❡♠ Pr♦❜❧❡♠ ✶ ❋✐♥❞ t❤❡ ♥✉♠❜❡r ♦❢ str✐♥❣s ♦❢ ❧❡♥❣t❤ ✧◆✧ ♠❛❞❡ ✉♣ ♦❢ ♦♥❧② ✸ ❝❤❛r❛❝t❡rs ✲ ❛✱ ❜✱ ❝ s✉❝❤ t❤❛t ✧❛✧ ♦❝❝✉rs ❛t ❧❡❛st ✧♠✐♥❴❛✧ t✐♠❡s ❛♥❞ ❛t ♠♦st ✧♠❛①❴❛✧ t✐♠❡s✱ ✧❜✧ ♦❝❝✉rs ❛t ❧❡❛st ✧♠✐♥❴❜✧ t✐♠❡s ❛♥❞ ❛t ♠♦st ✧♠❛①❴❜✧ t✐♠❡s ❛♥❞ ✧❝✧ ♦❝❝✉rs ❛t ❧❡❛st ✧♠✐♥❴❝✧ t✐♠❡s ❛♥❞ ❛t ♠♦st ✧♠❛①❴❝✧ t✐♠❡s✳ ◆♦t❡ t❤❛t ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ♦❢ s❛♠❡ str✐♥❣ ❝♦✉♥t ❛s ✶✱ s♦ ✧❛❜❝✧ ✐s s❛♠❡ ❛s ✧❜❛❝✧✳ ❤tt♣✿✴✴✇✇✇✳s♣♦❥✳♣❧✴♣r♦❜❧❡♠s✴❉❈❊P❈✼✵✷✴ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  16. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr♦❜❧❡♠ Pr♦❜❧❡♠ ✷ ❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t♦ ✜♥❞ ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r t❤❡ ♣r♦❜❧❡♠ ✐♥ ✇❤✐❝❤ ✇❡ s❤♦✉❧❞ ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛t❤s ♦❢ ❛ s♣❡❝✐❛❧ t②♣❡✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ ✇❡ ❤❛✈❡ t✇♦ ♣♦✐♥ts ❆✱ ❇ ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ✐♥t❡❣❡r ❝♦♦r❞✐♥❛t❡s✱ t❤❡♥ ✇❡ ✇✐❧❧ ♦♣❡r❛t❡ ♦♥❧② ✇✐t❤ t❤❡ s❤♦rt❡st ♣❛t❤s ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇ t❤❛t ♣❛ss ♦♥❧② t❤r♦✉❣❤ t❤❡ ❧✐♥❡s ♦❢ t❤❡ ✐♥t❡❣❡r ❣r✐❞ ❛♥❞ t❤❛t ❝❛♥ ❜❡ ❞♦♥❡ ♦♥❧② ✐♥ ❤♦r✐③♦♥t❛❧ ♦r ✈❡rt✐❝❛❧ ♠♦✈❡♠❡♥ts ✇✐t❤ ❧❡♥❣t❤ ❡q✉❛❧ t♦ ✶✳ ❋✐❣✉r❡✿ ✶ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  17. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr♦❜❧❡♠ ❙♦❧✉t✐♦♥ ❙♦❧✉t✐♦♥✿ ❆❧❧ ♣❛t❤s ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇ ❤❛✈❡ t❤❡ s❛♠❡ ❧❡♥❣t❤ ❡q✉❛❧ t♦ ♥✰♠ ✭✇❤❡r❡ ♥ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ①✲❝♦♦r❞✐♥❛t❡s ❛♥❞ ♠ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ②✲❝♦♦r❞✐♥❛t❡s✮✳ ❲❡ ❝❛♥ ❡❛s✐❧② ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ t❤❡ ♣❛t❤s ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇✿ � ♠ + ♥ � ♠ + ♥ � � ❆♥s✿ ♦r ♥ ♠ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  18. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr♦❜❧❡♠ Pr♦❜❧❡♠ ✸ ▲❡t✬s s♦❧✈❡ ❛ ❢❛♠♦✉s ♣r♦❜❧❡♠ ✉s✐♥❣ t❤✐s ♠❡t❤♦❞✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✜♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❉②❝❦ ✇♦r❞s ✇✐t❤ ❛ ❧❡♥❣t❤ ♦❢ ✷♥✳ ❲❤❛t ✐s ❛ ❉②❝❦ ✇♦r❞❄ ■t✬s ❛ str✐♥❣ ❝♦♥s✐st✐♥❣ ♦♥❧② ♦❢ ♥ ❳✬s ❛♥❞ ♥ ❨✬s✱ ❛♥❞ ♠❛t❝❤✐♥❣ t❤✐s ❝r✐t❡r✐❛✿ ❡❛❝❤ ♣r❡✜① ♦❢ t❤✐s str✐♥❣ ❤❛s ♠♦r❡ ❳✬s t❤❛♥ ❨✬s✳ ❋♦r ❡①❛♠♣❧❡✱ ✧❳❳❨❨✧ ❛♥❞ ✧❳❨❳❨✧ ❛r❡ ❉②❝❦ ✇♦r❞s✱ ❜✉t ✧❳❨❨❳✧ ❛♥❞ ✧❨❨❳❳✧ ❛r❡ ♥♦t✳ ❖❘ ❋✐♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s t♦ ❛rr❛♥❣❡ ♥ ✬✭✬ ❛♥❞ ♥ ✬✮✬ ❜r❛❝❦❡ts s✉❝❤ t❤❛t ❛t ❡❛❝❤ ✐♥❞❡①✱ t❤❡ ♥✉♠❜❡r ♦❢ ✬✭✬ ✐s ♥❡✈❡r ❧❡ss t❤❛♥ t❤❡ ♥✉♠❜❡r ♦❢ ✬✮✬ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  19. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr♦❜❧❡♠ ❙♦❧✉t✐♦♥ � ✷ ♥ � ❙♦❧✉t✐♦♥✿ ❚♦t❛❧ ✇❛②s✿ ♥ � ✷ ♥ � ■♥❝♦rr❡❝t ✇❛②s✿ ♥ − ✶ ✶ � ✷ ♥ � ❆♥s✿ ❈❛t❛❧❛♥ ♥✉♠❜❡r ♥ + ✶ ♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  20. ✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ① ② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  21. ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ① ② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝ ✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  22. ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ① ② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝ ✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  23. ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ① ② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝ ✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  24. ① ② ♠♦❞ ♥ ❂ ①♠♦❞♥ ② ♠♦❞ ♥ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝ ✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  25. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼♦❞✉❧❛r ❆r✐t❤♠❡t✐❝ ✭①✰②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✰ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✲②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮ ✲ ✭② ♠♦❞ ♥✮✮♠♦❞ ♥ ✭①✯②✮ ♠♦❞ ♥ ❂ ✭① ♠♦❞ ♥✮✯✭② ♠♦❞ ♥✮♠♦❞ ♥ ❇✉t✱ ✭①✴②✮ ♠♦❞ ♥ ❂ ✭✭① ♠♦❞ ♥✮✴✭② ♠♦❞ ♥✮✮♠♦❞ ♥✱ ♥♦t ❛❧✇❛②s tr✉❡ ① ② ♠♦❞ ♥ ❂ ( ①♠♦❞♥ ) ② ♠♦❞ ♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  26. ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ✶ ① ① ✶ ♠♦❞♥ ✶ ✺ ✸ ♠♦❞ ✼ ❜❡❝❛✉s❡ ✺ ✸ ✶✺ ✶ ♠♦❞ ✼ ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥ ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  27. ✶ ✺ ✸ ♠♦❞ ✼ ❜❡❝❛✉s❡ ✺ ✸ ✶✺ ✶ ♠♦❞ ✼ ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥ ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ① − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ① − ✶ ≡ ✶ ( ♠♦❞♥ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  28. ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥ ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ① − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ① − ✶ ≡ ✶ ( ♠♦❞♥ ) ✺ − ✶ ≡ ✸ ( ♠♦❞ ✼ ) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶ ( ♠♦❞ ✼ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  29. ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥ ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ① − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ① − ✶ ≡ ✶ ( ♠♦❞♥ ) ✺ − ✶ ≡ ✸ ( ♠♦❞ ✼ ) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶ ( ♠♦❞ ✼ ) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  30. ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥ ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ① − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ① − ✶ ≡ ✶ ( ♠♦❞♥ ) ✺ − ✶ ≡ ✸ ( ♠♦❞ ✼ ) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶ ( ♠♦❞ ✼ ) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  31. ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥ ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ① − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ① − ✶ ≡ ✶ ( ♠♦❞♥ ) ✺ − ✶ ≡ ✸ ( ♠♦❞ ✼ ) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶ ( ♠♦❞ ✼ ) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  32. ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ✶ ♠♦❞♥ ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ① − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ① − ✶ ≡ ✶ ( ♠♦❞♥ ) ✺ − ✶ ≡ ✸ ( ♠♦❞ ✼ ) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶ ( ♠♦❞ ✼ ) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  33. ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ① − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ① − ✶ ≡ ✶ ( ♠♦❞♥ ) ✺ − ✶ ≡ ✸ ( ♠♦❞ ✼ ) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶ ( ♠♦❞ ✼ ) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ≡ ✶ ( ♠♦❞♥ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  34. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✉❧t✐♣❧✐❝❛t✐✈❡ ■♥✈❡rs❡ ① − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ① ♠♦❞✉❧♦ ♥ ✐❢ ① ∗ ① − ✶ ≡ ✶ ( ♠♦❞♥ ) ✺ − ✶ ≡ ✸ ( ♠♦❞ ✼ ) ❜❡❝❛✉s❡ ✺ ∗ ✸ ≡ ✶✺ ≡ ✶ ( ♠♦❞ ✼ ) ▼❛② ♥♦t ❡①✐st ✭❡✳❣✳ ■♥✈❡rs❡ ♦❢ ✷ ♠♦❞ ✹✮ ❊①✐sts ✐✛ ❣❝❞✭①✱♥✮ ❂ ✶ ❣❝❞✭❛✱❜✮❂❛①✰❜② ❢♦r s♦♠❡ ✐♥t❡❣❡rs ①✱ ② ■❢ ❣❝❞✭❛✱♥✮❂✶✱ t❤❡♥ ❛① ✰ ♥② ❂ ✶ ❢♦r s♦♠❡ ①✱ ② ❚❛❦✐♥❣ ♠♦❞✉❧♦ ♥ ❣✐✈❡s ❛① ≡ ✶ ( ♠♦❞♥ ) ●✐✈❡♥ ❛✱❜✱ ❋✐♥❞✐♥❣ ① ❛♥❞ ②✱ s✉❝❤ t❤❛t ❛①✰❜② ❂ ❞ ✐s ❞♦♥❡ ❜② ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  35. ■❞❡❛✿ ❡✈❡r② ❝♦♠♣♦s✐t❡ ♥✉♠❜❡r ♥ ❤❛s ❛ ♣r✐♠❡ ❢❛❝t♦r ♣ ♥ ✳ ❙♦ ❧❡t ✉s ❛ss✉♠❡ t❤❛t ❛❧❧ ♥✉♠❜❡rs ❛r❡ ♣r✐♠❡✳ ❇✉t ✐❢ ✇❡ ❝♦♠❡ ❛❝r♦ss ❛ ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛ ♥✉♠❜❡r✱ ✇❡ ✐♠♠❡❞✐❛t❡❧② ❦♥♦✇ t❤❛t ✐t ✐s ♥♦t ❛ ♣r✐♠❡✳ ■❢ t❤❡r❡ ✐s ♥♦ ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛ ♥✉♠❜❡r ♥ ✐♥ t❤❡ r❛♥❣❡ ❬✷ ✳ ✳ ✳ ♥✲✶❪ t❤❡♥ ✐t ♠✉st ❜❡ ♣r✐♠❡✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr✐♠❡ ❙❡✐✈❡ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  36. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr✐♠❡ ❙❡✐✈❡ ■❞❡❛✿ ❡✈❡r② ❝♦♠♣♦s✐t❡ ♥✉♠❜❡r ♥ ❤❛s ❛ ♣r✐♠❡ ❢❛❝t♦r ♣ ≤ √ ♥ ✳ ❙♦ ❧❡t ✉s ❛ss✉♠❡ t❤❛t ❛❧❧ ♥✉♠❜❡rs ❛r❡ ♣r✐♠❡✳ ❇✉t ✐❢ ✇❡ ❝♦♠❡ ❛❝r♦ss ❛ ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛ ♥✉♠❜❡r✱ ✇❡ ✐♠♠❡❞✐❛t❡❧② ❦♥♦✇ t❤❛t ✐t ✐s ♥♦t ❛ ♣r✐♠❡✳ ■❢ t❤❡r❡ ✐s ♥♦ ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛ ♥✉♠❜❡r ♥ ✐♥ t❤❡ r❛♥❣❡ ❬✷ ✳ ✳ ✳ ♥✲✶❪ t❤❡♥ ✐t ♠✉st ❜❡ ♣r✐♠❡✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  37. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr✐♠❡ ❙❡✐✈❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ●❡♥❡r❛t❡ ❛❧❧ ♣r✐♠❡s ✐♥ r❛♥❣❡ ❬✶✳✳♥❪ ❋♦r ✐❂✶ t♦ ♥ ♣r✐♠❡❬✐❪❂✶ Pr✐♠❡❬✶❪❂✵ ❋♦r ✐❂✷ t♦ √ ♥ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r ❥ ❂ ✐ t♦ ♥✴✐ ♣r✐♠❡❬✐✯❥❪❂✵ ❆t t❤❡ ❡♥❞ ♦❢ t❤✐s st❡♣✱ ❛❧❧ ♥✉♠❜❡rs ✐ ✇❤✐❝❤ ❛r❡ ♣r✐♠❡ ❤❛✈❡ ♣r✐♠❡❬✐❪❂✶✳ ❖t❤❡rs ❤❛✈❡ ♣r✐♠❡❬✐❪❂✵✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  38. Pr✐♠❡ ♥✉♠❜❡r ❚❤❡♦r❡♠ ◆✉♠❜❡r ♦❢ ♣r✐♠❡s t✐❧❧ ♥ ♥✴❧♦❣♥ ▼❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ♥ ❂ ❧♦❣ ♥ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  39. ▼❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ♥ ❂ ❧♦❣ ♥ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠ Pr✐♠❡ ♥✉♠❜❡r ❚❤❡♦r❡♠ ◆✉♠❜❡r ♦❢ ♣r✐♠❡s t✐❧❧ ♥ ∼ ♥✴❧♦❣♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  40. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠ Pr✐♠❡ ♥✉♠❜❡r ❚❤❡♦r❡♠ ◆✉♠❜❡r ♦❢ ♣r✐♠❡s t✐❧❧ ♥ ∼ ♥✴❧♦❣♥ ▼❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ♥ ❂ ❧♦❣ ♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  41. ❋♦r ❛ ♣r✐♠❡ ♣✱ ✶ ♣ ♣ ♣ ❦ ✶ ♣ ❦ ♣ ❦ ♣ ❦ ✶ ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ ♣ ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ✶ ✶ ✶ ✶ ✶ ✶ ♥ ♥ t ✷ ♣ ✶ ♣ ✷ ♣ t ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ♣ ✶ ✶ ♣ ✷ ✶ ♣ t ✶ ♥ ♥ ✷ t ♣ ✶ ♣ ✷ ♣ t ♥ ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ ❛❜ ❛ ❜ ♣ ❦ ✶ ♣ ❦ ✷ ♣ ❦ t ◆ t ✶ ✷ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  42. ❋♦r ❛ ♣r✐♠❡ ♣✱ ✶ ♣ ♣ ♣ ❦ ✶ ♣ ❦ ♣ ❦ ♣ ❦ ✶ ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ ♣ ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ✶ ✶ ✶ ✶ ✶ ✶ ♥ ♥ t ✷ ♣ ✶ ♣ ✷ ♣ t ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ♣ ✶ ✶ ♣ ✷ ✶ ♣ t ✶ ♥ ♥ ✷ t ♣ ✶ ♣ ✷ ♣ t ❛❜ ❛ ❜ ♣ ❦ ✶ ♣ ❦ ✷ ♣ ❦ t ◆ t ✶ ✷ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ φ ( ♥ ) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  43. ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ✶ ✶ ✶ ✶ ✶ ✶ ♥ ♥ t ✷ ♣ ✶ ♣ ✷ ♣ t ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ♣ ✶ ✶ ♣ ✷ ✶ ♣ t ✶ ♥ ♥ ✷ t ♣ ✶ ♣ ✷ ♣ t ❋♦r ❛ ♣r✐♠❡ ♣✱ ✶ ♣ ♣ ♣ ❦ ✶ ♣ ❦ ♣ ❦ ♣ ❦ ✶ ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ ♣ ♣ ❦ ✶ ♣ ❦ ✷ ♣ ❦ t ◆ t ✶ ✷ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ φ ( ♥ ) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ ( ❛❜ ) = φ ( ❛ ) ∗ φ ( ❜ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  44. ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ✶ ✶ ✶ ✶ ✶ ✶ ♥ ♥ t ✷ ♣ ✶ ♣ ✷ ♣ t ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ♣ ✶ ✶ ♣ ✷ ✶ ♣ t ✶ ♥ ♥ ✷ t ♣ ✶ ♣ ✷ ♣ t ♣ ❦ ✶ ♣ ❦ ♣ ❦ ♣ ❦ ✶ ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ ♣ ♣ ❦ ✶ ♣ ❦ ✷ ♣ ❦ t ◆ t ✶ ✷ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ φ ( ♥ ) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ ( ❛❜ ) = φ ( ❛ ) ∗ φ ( ❜ ) ❋♦r ❛ ♣r✐♠❡ ♣✱ φ ( ♣ ) = ♣ − ✶ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  45. ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ✶ ✶ ✶ ✶ ✶ ✶ ♥ ♥ t ✷ ♣ ✶ ♣ ✷ ♣ t ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ♣ ✶ ✶ ♣ ✷ ✶ ♣ t ✶ ♥ ♥ ✷ t ♣ ✶ ♣ ✷ ♣ t ♣ ❦ ✶ ♣ ❦ ✷ ♣ ❦ t ◆ t ✶ ✷ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ φ ( ♥ ) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ ( ❛❜ ) = φ ( ❛ ) ∗ φ ( ❜ ) ❋♦r ❛ ♣r✐♠❡ ♣✱ φ ( ♣ ) = ♣ − ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ φ ( ♣ ❦ ) = ♣ ❦ − ♣ ❦ − ✶ = ♣ ❦ ( ✶ − ✶ ♣ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  46. ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ✶ ✶ ✶ ✶ ✶ ✶ ♥ ♥ t ✷ ♣ ✶ ♣ ✷ ♣ t ♣ ❛ ✶ ✶ ♣ ❛ ✷ ♣ ❛ t ♣ ✶ ✶ ♣ ✷ ✶ ♣ t ✶ ♥ ♥ ✷ t ♣ ✶ ♣ ✷ ♣ t ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ φ ( ♥ ) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ ( ❛❜ ) = φ ( ❛ ) ∗ φ ( ❜ ) ❋♦r ❛ ♣r✐♠❡ ♣✱ φ ( ♣ ) = ♣ − ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ φ ( ♣ ❦ ) = ♣ ❦ − ♣ ❦ − ✶ = ♣ ❦ ( ✶ − ✶ ♣ ) ◆ = ( ♣ ❦ ✶ ✶ ) ∗ ( ♣ ❦ ✷ ✷ ) ∗ ... ∗ ( ♣ ❦ t t ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  47. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ φ ( ♥ ) ❂ ◆✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♥ ✇❤✐❝❤ ❛r❡ ❝♦♣r✐♠❡ t♦ ♥ φ ( ❛❜ ) = φ ( ❛ ) ∗ φ ( ❜ ) ❋♦r ❛ ♣r✐♠❡ ♣✱ φ ( ♣ ) = ♣ − ✶ ❋♦r ❛ ♣r✐♠❡ ♣✱ φ ( ♣ ❦ ) = ♣ ❦ − ♣ ❦ − ✶ = ♣ ❦ ( ✶ − ✶ ♣ ) ◆ = ( ♣ ❦ ✶ ✶ ) ∗ ( ♣ ❦ ✷ ✷ ) ∗ ... ∗ ( ♣ ❦ t t ) φ ( ♥ ) = ♣ ❛ ✶ ✶ ♣ ❛ ✷ ✷ .. ♣ ❛ t t ) = ♥ ∗ ( ✶ − ✶ ♣ ✶ ) ∗ ( ✶ − ✶ ♣ ✷ ) ∗ · · · ∗ ( ✶ − ✶ ♣ t ) t ) = ♥ ∗ ( ♣ ✶ − ✶ ) ∗ ( ♣ ✷ − ✶ ) .. ( ♣ t − ✶ ) ≡ φ ( ♥ ) = ♣ ❛ ✶ ✶ ♣ ❛ ✷ ✷ .. ♣ ❛ t ( ♣ ✶ ♣ ✷ .. ♣ t ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  48. ❘✉♥ ♣r✐♠❡ s✐❡✈❡ ♦♥❝❡ ❛♥❞ st♦r❡ r❡s✉❧t ✐♥ ♣r✐♠❡s❬✶✳✳♥❪ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r✭❥❂✶ t♦ ♥✴✐✮ ♣❤✐❬✐✯❥❪❂♣❤✐❬✐✯❥❪✯✭✐✲✶✮✴✐ ❚❤✐s ❛❧❣♦ r✉♥s ✐♥ ❖✭♥ ❧♦❣ ❧♦❣ ♥✮ t✐♠❡✱ ❜✉t ✇❡ ✇✐❧❧ ♠❛❦❡ ✐♠♣r♦✈❡♠❡♥ts t♦ s❤♦✇ ❤♦✇ ②♦✉ ❝❛♥ ❛t t✐♠❡s ♦♣t✐♠✐③❡ ②♦✉r ❝♦❞❡✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ❙❡✐✈❡ ❢♦r φ ( ♥ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  49. ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r✭❥❂✶ t♦ ♥✴✐✮ ♣❤✐❬✐✯❥❪❂♣❤✐❬✐✯❥❪✯✭✐✲✶✮✴✐ ❚❤✐s ❛❧❣♦ r✉♥s ✐♥ ❖✭♥ ❧♦❣ ❧♦❣ ♥✮ t✐♠❡✱ ❜✉t ✇❡ ✇✐❧❧ ♠❛❦❡ ✐♠♣r♦✈❡♠❡♥ts t♦ s❤♦✇ ❤♦✇ ②♦✉ ❝❛♥ ❛t t✐♠❡s ♦♣t✐♠✐③❡ ②♦✉r ❝♦❞❡✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ❙❡✐✈❡ ❢♦r φ ( ♥ ) ❘✉♥ ♣r✐♠❡ s✐❡✈❡ ♦♥❝❡ ❛♥❞ st♦r❡ r❡s✉❧t ✐♥ ♣r✐♠❡s❬✶✳✳♥❪ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  50. ❚❤✐s ❛❧❣♦ r✉♥s ✐♥ ❖✭♥ ❧♦❣ ❧♦❣ ♥✮ t✐♠❡✱ ❜✉t ✇❡ ✇✐❧❧ ♠❛❦❡ ✐♠♣r♦✈❡♠❡♥ts t♦ s❤♦✇ ❤♦✇ ②♦✉ ❝❛♥ ❛t t✐♠❡s ♦♣t✐♠✐③❡ ②♦✉r ❝♦❞❡✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ❙❡✐✈❡ ❢♦r φ ( ♥ ) ❘✉♥ ♣r✐♠❡ s✐❡✈❡ ♦♥❝❡ ❛♥❞ st♦r❡ r❡s✉❧t ✐♥ ♣r✐♠❡s❬✶✳✳♥❪ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r✭❥❂✶ t♦ ♥✴✐✮ ♣❤✐❬✐✯❥❪❂♣❤✐❬✐✯❥❪✯✭✐✲✶✮✴✐ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  51. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ❙❡✐✈❡ ❢♦r φ ( ♥ ) ❘✉♥ ♣r✐♠❡ s✐❡✈❡ ♦♥❝❡ ❛♥❞ st♦r❡ r❡s✉❧t ✐♥ ♣r✐♠❡s❬✶✳✳♥❪ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣r✐♠❡❬✐❪✮ ❢♦r✭❥❂✶ t♦ ♥✴✐✮ ♣❤✐❬✐✯❥❪❂♣❤✐❬✐✯❥❪✯✭✐✲✶✮✴✐ ❚❤✐s ❛❧❣♦ r✉♥s ✐♥ ❖✭♥ ❧♦❣ ❧♦❣ ♥✮ t✐♠❡✱ ❜✉t ✇❡ ✇✐❧❧ ♠❛❦❡ ✐♠♣r♦✈❡♠❡♥ts t♦ s❤♦✇ ❤♦✇ ②♦✉ ❝❛♥ ❛t t✐♠❡s ♦♣t✐♠✐③❡ ②♦✉r ❝♦❞❡✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  52. ◗✉❡st✐♦♥✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❧✐st ♦❢ ❛❧❧ ♣r✐♠❡s❄ ❆♥s✇❡r ✿ ◆♦ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣❤✐❬✐❪❂❂✐✮ ❢♦r✭❥❂✐ t♦ ♥❀❥✰❂✐✮ ♣❤✐❬❥❪ ❂ ✭♣❤✐❬❥❪✴✐✮✯✭✐✲✶✮❀ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ❙❡✐✈❡ ❢♦r φ ( ♥ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  53. ❆♥s✇❡r ✿ ◆♦ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣❤✐❬✐❪❂❂✐✮ ❢♦r✭❥❂✐ t♦ ♥❀❥✰❂✐✮ ♣❤✐❬❥❪ ❂ ✭♣❤✐❬❥❪✴✐✮✯✭✐✲✶✮❀ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ❙❡✐✈❡ ❢♦r φ ( ♥ ) ◗✉❡st✐♦♥✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❧✐st ♦❢ ❛❧❧ ♣r✐♠❡s❄ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  54. ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣❤✐❬✐❪❂❂✐✮ ❢♦r✭❥❂✐ t♦ ♥❀❥✰❂✐✮ ♣❤✐❬❥❪ ❂ ✭♣❤✐❬❥❪✴✐✮✯✭✐✲✶✮❀ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ❙❡✐✈❡ ❢♦r φ ( ♥ ) ◗✉❡st✐♦♥✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❧✐st ♦❢ ❛❧❧ ♣r✐♠❡s❄ ❆♥s✇❡r ✿ ◆♦ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  55. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r ❚♦r✐❡♥t ❋✉♥❝t✐♦♥ ❙❡✐✈❡ ❢♦r φ ( ♥ ) ◗✉❡st✐♦♥✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❧✐st ♦❢ ❛❧❧ ♣r✐♠❡s❄ ❆♥s✇❡r ✿ ◆♦ ❢♦r✭✐❂✶ t♦ ♥✮ ♣❤✐❬✐❪❂✐ ❢♦r✭✐❂✷ t♦ ♥✮ ✐❢✭♣❤✐❬✐❪❂❂✐✮ ❢♦r✭❥❂✐ t♦ ♥❀❥✰❂✐✮ ♣❤✐❬❥❪ ❂ ✭♣❤✐❬❥❪✴✐✮✯✭✐✲✶✮❀ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  56. ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ t❤❡♥ ❢♦r ❛♥② ✐♥t❡❣❡r ❛ t❤❛t ✐s ❝♦♣r✐♠❡ t♦ ♥✱ ✇❡ ❤❛✈❡ ❛ ♣ ❛ ♠♦❞♣ ❚❤✐s t❤❡♦r❡♠ ❝❛♥ ❛❧s♦ ❜❡ st❛t❡❞ ❛s✿ ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❛ ✐s ❝♦♣r✐♠❡ t♦ ♣✱ t❤❡♥ ❛ ♣ ✶ ✶ ♠♦❞♣ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  57. ❚❤✐s t❤❡♦r❡♠ ❝❛♥ ❛❧s♦ ❜❡ st❛t❡❞ ❛s✿ ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❛ ✐s ❝♦♣r✐♠❡ t♦ ♣✱ t❤❡♥ ❛ ♣ ✶ ✶ ♠♦❞♣ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠ ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ t❤❡♥ ❢♦r ❛♥② ✐♥t❡❣❡r ❛ t❤❛t ✐s ❝♦♣r✐♠❡ t♦ ♥✱ ✇❡ ❤❛✈❡ ❛ ♣ ≡ ❛ ( ♠♦❞♣ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  58. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠ ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ t❤❡♥ ❢♦r ❛♥② ✐♥t❡❣❡r ❛ t❤❛t ✐s ❝♦♣r✐♠❡ t♦ ♥✱ ✇❡ ❤❛✈❡ ❛ ♣ ≡ ❛ ( ♠♦❞♣ ) ❚❤✐s t❤❡♦r❡♠ ❝❛♥ ❛❧s♦ ❜❡ st❛t❡❞ ❛s✿ ■❢ ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❛ ✐s ❝♦♣r✐♠❡ t♦ ♣✱ t❤❡♥ ❛ ♣ − ✶ ≡ ✶ ( ♠♦❞♣ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  59. ❊✉❧❡r✬s ❚❤❡♦r❡♠ ✐s ❛ ❣❡♥❛r❛❧✐③❛t✐♦♥ ❢♦r ❋❡r♠❛t✬s ❧✐tt❧❡ t❤❡♦r❡♠ ✇❤❡♥ ❛ ❛♥❞ ♥ ❛r❡ ❝♦✲♣r✐♠❡ ■❢ ① ② ♠♦❞ ♥ ✱ t❤❡♥ ❛① ❛② ♠♦❞♥ ✳ ♥ ✶ ♠♦❞♥ ✭❛❝t✉❛❧ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛ ❛❜♦✈❡✮ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r✬s ❚❤❡♦r❡♠ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  60. ■❢ ① ② ♠♦❞ ♥ ✱ t❤❡♥ ❛① ❛② ♠♦❞♥ ✳ ♥ ✶ ♠♦❞♥ ✭❛❝t✉❛❧ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛ ❛❜♦✈❡✮ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r✬s ❚❤❡♦r❡♠ ❊✉❧❡r✬s ❚❤❡♦r❡♠ ✐s ❛ ❣❡♥❛r❛❧✐③❛t✐♦♥ ❢♦r ❋❡r♠❛t✬s ❧✐tt❧❡ t❤❡♦r❡♠ ✇❤❡♥ ❛ ❛♥❞ ♥ ❛r❡ ❝♦✲♣r✐♠❡ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  61. ♥ ✶ ♠♦❞♥ ✭❛❝t✉❛❧ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛ ❛❜♦✈❡✮ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r✬s ❚❤❡♦r❡♠ ❊✉❧❡r✬s ❚❤❡♦r❡♠ ✐s ❛ ❣❡♥❛r❛❧✐③❛t✐♦♥ ❢♦r ❋❡r♠❛t✬s ❧✐tt❧❡ t❤❡♦r❡♠ ✇❤❡♥ ❛ ❛♥❞ ♥ ❛r❡ ❝♦✲♣r✐♠❡ ■❢ ① ≡ ② ( ♠♦❞ φ ( ♥ )) ✱ t❤❡♥ ❛① ≡ ❛② ( ♠♦❞♥ ) ✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  62. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❊✉❧❡r✬s ❚❤❡♦r❡♠ ❊✉❧❡r✬s ❚❤❡♦r❡♠ ✐s ❛ ❣❡♥❛r❛❧✐③❛t✐♦♥ ❢♦r ❋❡r♠❛t✬s ❧✐tt❧❡ t❤❡♦r❡♠ ✇❤❡♥ ❛ ❛♥❞ ♥ ❛r❡ ❝♦✲♣r✐♠❡ ■❢ ① ≡ ② ( ♠♦❞ φ ( ♥ )) ✱ t❤❡♥ ❛① ≡ ❛② ( ♠♦❞♥ ) ✳ ❛ φ ( ♥ ) ≡ ✶ ( ♠♦❞♥ ) ✭❛❝t✉❛❧ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✮ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  63. ❙✉♠ ♦❢ t❤❡ ❞✐✈✐s♦rs ♦❢ ♥ ❡q✉❛❧s ♠✐ ✶ ♣ ✶ t ✐ ❞ ♥ ✐ ✶ ♣ ✐ ✶ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❖t❤❡r r❡s✉❧ts ■❢ ♥ = ♣ ❛ ✶ ✶ ∗ ♣ ❛ ✷ ✷ ∗ ... ∗ ♣ ❛ t t ✱ t❤❡♥ ❚❤❡ ♥✉♠❜❡r ♦❢ ✐ts ♣♦s✐t✐✈❡ ❞✐✈✐s♦rs ❡q✉❛❧s ( ❛ ✶ + ✶ ) ∗ ( ❛ ✷ + ✶ ) ∗ ... ∗ ( ❛ t + ✶ ) ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  64. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❖t❤❡r r❡s✉❧ts ■❢ ♥ = ♣ ❛ ✶ ✶ ∗ ♣ ❛ ✷ ✷ ∗ ... ∗ ♣ ❛ t t ✱ t❤❡♥ ❚❤❡ ♥✉♠❜❡r ♦❢ ✐ts ♣♦s✐t✐✈❡ ❞✐✈✐s♦rs ❡q✉❛❧s ( ❛ ✶ + ✶ ) ∗ ( ❛ ✷ + ✶ ) ∗ ... ∗ ( ❛ t + ✶ ) ❙✉♠ ♦❢ t❤❡ ❞✐✈✐s♦rs ♦❢ ♥ ❡q✉❛❧s ♠✐ + ✶ − ✶ ♣ ❞ | ♥ = � t � ✐ ✐ = ✶ ♣ ✐ − ✶ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  65. ✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮ ■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷ s ❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛ ❞ � = ✶ ♠♦❞ ♥ ❛♥❞ ❛ ✷ r . ❞ � = − ✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [ ✵ , s ✶ ] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  66. ✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮ ■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷ s ❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛ ❞ � = ✶ ♠♦❞ ♥ ❛♥❞ ❛ ✷ r . ❞ � = − ✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [ ✵ , s ✶ ] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  67. ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮ ■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷ s ❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛ ❞ � = ✶ ♠♦❞ ♥ ❛♥❞ ❛ ✷ r . ❞ � = − ✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [ ✵ , s ✶ ] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡ ✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  68. ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮ ■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷ s ❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛ ❞ � = ✶ ♠♦❞ ♥ ❛♥❞ ❛ ✷ r . ❞ � = − ✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [ ✵ , s ✶ ] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡ ✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  69. ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮ ■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷ s ❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛ ❞ � = ✶ ♠♦❞ ♥ ❛♥❞ ❛ ✷ r . ❞ � = − ✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [ ✵ , s ✶ ] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡ ✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  70. ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮ ■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷ s ❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛ ❞ � = ✶ ♠♦❞ ♥ ❛♥❞ ❛ ✷ r . ❞ � = − ✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [ ✵ , s ✶ ] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡ ✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  71. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼✐❧❧❡r✲❘❛❜✐♥ ❚❡st ✭Pr✐♠❛❧✐t② ❚❡st✐♥❣✮ ■♥♣✉t✿ ♥ ❃ ✶✱ ❛♥ ♦❞❞ ✐♥t❡❣❡r t♦ t❡st ❢♦r ♣r✐♠❛❧✐t②✳ ✇r✐t❡ ♥✲✶ ❛s ✷ s ❞ ❜② ❢❛❝t♦r✐♥❣ ♣♦✇❡rs ♦❢ ✷ ❢r♦♠ ♥✲✶ r❡♣❡❛t ❢♦r ❛❧❧ ✿ ❛ ∈ ❬✷✱♥✲✷❪ ■❢ ❛ ❞ � = ✶ ♠♦❞ ♥ ❛♥❞ ❛ ✷ r . ❞ � = − ✶ ♠♦❞ ♥ ❢♦r ❛❧❧ r ∈ [ ✵ , s ✶ ] t❤❡♥ r❡t✉r♥ ❝♦♠♣♦s✐t❡ ❘❡t✉r♥ ♣r✐♠❡ ✐❢ ♥ ❁ ✾✱✵✽✵✱✶✾✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✸✶ ❛♥❞ ✼✸❀ ✐❢ ♥ ❁ ✹✱✼✺✾✱✶✷✸✱✶✹✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✼✱ ❛♥❞ ✻✶❀ ✐❢ ♥ ❁ ✷✱✶✺✷✱✸✵✷✱✽✾✽✱✼✹✼✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ❛♥❞ ✶✶❀ ✐❢ ♥ ❁ ✸✱✹✼✹✱✼✹✾✱✻✻✵✱✸✽✸✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ❛♥❞ ✶✸❀ ✐❢ ♥ ❁ ✸✹✶✱✺✺✵✱✵✼✶✱✼✷✽✱✸✷✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❡st ❛ ❂ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ❛♥❞ ✶✼✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  72. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❖✉t❧✐♥❡ ▼❛t❤s ✶ Pr♦❜❛❜✐❧✐t② ✷ Pr♦❜❧❡♠s ✸ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  73. ❋♦r ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ P✭❳✮✱ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊❬❳❪ ✐s ❣✐✈❡♥ ❜② t❤❡ s✉♠♠❛t✐♦♥ ✐ ① ✐ P ① ✐ r✉♥s ♦✈❡r ❛❧❧ t❤❡ ❞✐st✐♥❝t ✈❛❧✉❡s ①✐ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ t❛❦❡✳ ❚❤❡ r✉❧❡ ♦❢ ✧❧✐♥❡❛r✐t② ♦❢ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥✧ s❛②s t❤❛t ❊❬①✶✰①✷❪ ❂ ❊❬①✶❪ ✰ ❊❬①✷❪✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t ✧❡①♣❡❝t❡❞ ✈❛❧✉❡✧ ✐s ♥♦t s❛♠❡ ❛s ✧♠♦st ♣r♦❜❛❜❧❡ ✈❛❧✉❡✧ ✲ r❛t❤❡r✱ ✐t ♥❡❡❞ ♥♦t ❡✈❡♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♣r♦❜❛❜❧❡ ✈❛❧✉❡s✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  74. ❚❤❡ r✉❧❡ ♦❢ ✧❧✐♥❡❛r✐t② ♦❢ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥✧ s❛②s t❤❛t ❊❬①✶✰①✷❪ ❂ ❊❬①✶❪ ✰ ❊❬①✷❪✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t ✧❡①♣❡❝t❡❞ ✈❛❧✉❡✧ ✐s ♥♦t s❛♠❡ ❛s ✧♠♦st ♣r♦❜❛❜❧❡ ✈❛❧✉❡✧ ✲ r❛t❤❡r✱ ✐t ♥❡❡❞ ♥♦t ❡✈❡♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♣r♦❜❛❜❧❡ ✈❛❧✉❡s✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ❋♦r ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ P✭❳✮✱ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊❬❳❪ ✐s ❣✐✈❡♥ ❜② � ✐ ① ✐ ∗ P ( ① ✐ ) t❤❡ s✉♠♠❛t✐♦♥ r✉♥s ♦✈❡r ❛❧❧ t❤❡ ❞✐st✐♥❝t ✈❛❧✉❡s ①✐ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ t❛❦❡✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  75. ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t ✧❡①♣❡❝t❡❞ ✈❛❧✉❡✧ ✐s ♥♦t s❛♠❡ ❛s ✧♠♦st ♣r♦❜❛❜❧❡ ✈❛❧✉❡✧ ✲ r❛t❤❡r✱ ✐t ♥❡❡❞ ♥♦t ❡✈❡♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♣r♦❜❛❜❧❡ ✈❛❧✉❡s✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ❋♦r ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ P✭❳✮✱ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊❬❳❪ ✐s ❣✐✈❡♥ ❜② � ✐ ① ✐ ∗ P ( ① ✐ ) t❤❡ s✉♠♠❛t✐♦♥ r✉♥s ♦✈❡r ❛❧❧ t❤❡ ❞✐st✐♥❝t ✈❛❧✉❡s ①✐ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ t❛❦❡✳ ❚❤❡ r✉❧❡ ♦❢ ✧❧✐♥❡❛r✐t② ♦❢ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥✧ s❛②s t❤❛t ❊❬①✶✰①✷❪ ❂ ❊❬①✶❪ ✰ ❊❬①✷❪✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  76. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ❋♦r ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ P✭❳✮✱ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊❬❳❪ ✐s ❣✐✈❡♥ ❜② � ✐ ① ✐ ∗ P ( ① ✐ ) t❤❡ s✉♠♠❛t✐♦♥ r✉♥s ♦✈❡r ❛❧❧ t❤❡ ❞✐st✐♥❝t ✈❛❧✉❡s ①✐ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❝❛♥ t❛❦❡✳ ❚❤❡ r✉❧❡ ♦❢ ✧❧✐♥❡❛r✐t② ♦❢ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥✧ s❛②s t❤❛t ❊❬①✶✰①✷❪ ❂ ❊❬①✶❪ ✰ ❊❬①✷❪✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t ✧❡①♣❡❝t❡❞ ✈❛❧✉❡✧ ✐s ♥♦t s❛♠❡ ❛s ✧♠♦st ♣r♦❜❛❜❧❡ ✈❛❧✉❡✧ ✲ r❛t❤❡r✱ ✐t ♥❡❡❞ ♥♦t ❡✈❡♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♣r♦❜❛❜❧❡ ✈❛❧✉❡s✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  77. ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ❊①❛♠♣❧❡ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  78. ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  79. ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  80. ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳ ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  81. ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ❊①♣❡❝t❛t✐♦♥ ❊①❛♠♣❧❡ ❋♦r ❛ s✐①✲s✐❞❡❞ ❞✐❡✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ t❤r♦✇s t♦ ❣❡t ❡❛❝❤ ❢❛❝❡ ❛t ❧❡❛st ♦♥❝❡❄ ■t✬s ✭✻✴✻✮✰✭✻✴✺✮✰✭✻✴✹✮✰✭✻✴✸✮✰✭✻✴✷✮✰✭✻✴✶✮ ❂ ✶✹✳✼✳ ▲♦❣✐❝✿ ❚❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✇❤❡♥ ②♦✉ st❛rt ♦✛ ✐s ✶✱ ❛s ❛♥② ♥✉♠❜❡r ✇♦✉❧❞ ✇♦r❦✳ ❖♥❝❡ ②♦✉✬✈❡ r♦❧❧❡❞ t❤✐s ♥✉♠❜❡r✱ ②♦✉r ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ❛ ♥✉♠❜❡r ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✺✴✻✳ ❈♦♥t✐♥✉✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ❛❢t❡r ②♦✉✬✈❡ r♦❧❧❡❞ ♥ ❞✐✛❡r❡♥t ♥✉♠❜❡rs t❤❡ ❝❤❛♥❝❡ ♦❢ r♦❧❧✐♥❣ ♦♥❡ ②♦✉ ❤❛✈❡♥✬t ②❡t r♦❧❧❡❞ ✐s ✭✻✲♥✮✴✻✳ ❋♦r ❛♥ ♥✲s✐❞❡❞ ❞✐❡ t❤❡ ❡①♣❡❝t❡❞ t❤r♦✇s ✐s ✭♥✴♥✮ ✰ ✭♥✴✭♥✲✶✮✮ ✰ ✭♥✴✭♥✲✷✮✮ ✰ ✳✳✳ ✰ ♥✳ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

  82. ❊①t❡♥❞❡❞ ❊✉❝❧✐❞✬s ❛❧❣♦ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r✐♥❣ ❋❛r❡②✬s s❡q✉❡♥❝❡ ❖♣t✐♠✐s❡❞ ❙✐❡✈❡✱ ❙✐❡✈❡ ♦❢ ❆t❦✐♥s ❍♦✇ t♦ s♦❧✈❡ ❉✐♦♣❤❛♥t✐♥❡ ❊q✉❛t✐♦♥ P♦❧❧❛r❞ ❘❤♦ ❢❛❝t♦r✐③❛t✐♦♥ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ■♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥ ●❛✉ss❡❛♥ ❊❧✐♠✐♥❛t✐♦♥ ✭❋✐♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①✮ ●r♦✉♣ ❚❤❡♦r② ▼❛t❤s Pr♦❜❛❜✐❧✐t② Pr♦❜❧❡♠s ❖t❤❡r ❚❤✐♥❣s t♦ ❘❡❛❞ ▲❡❛❣✉❡ ♦❢ Pr♦❣r❛♠♠❡rs ▼❛t❤s

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