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Jul 09, 2023 •110 likes •1.31k views

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  1. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ◆❡✇ ❢r❛♠❡✇♦r❦ ❢♦r ✐♠❛❣❡ s❡❣♠❡♥t❛t✐♦♥  t✇♦ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ♣ ❛♥❞ q  ●✐✈❡♥ s❡❡❞s ❢♦r t❤❡ ❜❛❝❦❣r♦✉♥❞ ❇ ❀ s❡❡❞s ❢♦r t❤❡ ❢♦r❡❣r♦✉♥❞ ❋ ❀  ❈♦♠♣✉t❡ ① ✈❡r✐❢②✐♥❣ � ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ♠✐♥ ① ❡ ✐❥ ✷ ❊ ❙✉❝❤ t❤❛t ① ( ❋ ) = ✶ ❀ ① ( ❇ ) = ✵✳ � ✶ s✐ ① ✐ ✕ ✶ ✷ ❀ ❘❡s✉❧t ✿ s❡❣♠❡♥t❛t✐♦♥ s ❞❡✜♥❡❞ ✽ ✐ ❜② s ✐ = ✵ s✐ ① ✐ ❁ ✶ ✷ ✿ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✹ ✴ ✺✶

  2. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ●r❛♣❤ ❈✉ts S ∞ Pr♦❜❧❡♠ ✿ ❝♦♠♣✉t❡ ① 3 4 1 1 3 4 1 � ✇ ✐❥ ♣ = ✶ ❥ ① ✐ � ① ❥ ❥ q = ✶ 1 3 ① = ❛r❣ ♠✐♥ ❡ ✐❥ ✷ ❊ 4 2 2 2 2 ▼✐♥ ❝✉t ✴ ▼❛① ✢♦✇ ❞✉❛❧✐t② 3 2 ▼❛① ❋❧♦✇ ❛❧❣♦r✐t❤♠ ∞ T ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✺ ✴ ✺✶

  3. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ●r❛♣❤ ❈✉ts S ∞ Pr♦❜❧❡♠ ✿ ❝♦♠♣✉t❡ ① 3 4 1 1 3 4 1 � 1 3 ① = ❛r❣ ♠✐♥ ✇ ✐❥ ❥ ① ✐ � ① ❥ ❥ ❡ ✐❥ ✷ ❊ 4 2 2 2 2 ▼✐♥ ❝✉t ✴ ▼❛① ✢♦✇ ❞✉❛❧✐t② 3 2 ▼❛① ❋❧♦✇ ❛❧❣♦r✐t❤♠ ∞ T ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✺ ✴ ✺✶

  4. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ●r❛♣❤ ❈✉ts S ∞ Pr♦❜❧❡♠ ✿ ❝♦♠♣✉t❡ ① 3 4 � ① = ❛r❣ ♠✐♥ ✇ ✐❥ ❥ ① ✐ � ① ❥ ❥ 1 3 4 1 3 ❡ ✐❥ ✷ ❊ 4 2 2 ▼✐♥ ❝✉t ✴ ▼❛① ✢♦✇ ❞✉❛❧✐t② 3 2 ▼❛① ❋❧♦✇ ❛❧❣♦r✐t❤♠ ∞ T ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✻ ✴ ✺✶

  5. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ●r❛♣❤ ❈✉ts S ∞ Pr♦❜❧❡♠ ✿ ❝♦♠♣✉t❡ ① 3 4 1 1 1 � ① = ❛r❣ ♠✐♥ ✇ ✐❥ ❥ ① ✐ � ① ❥ ❥ 1 3 4 1 3 0 1 1 ❡ ✐❥ ✷ ❊ 4 2 2 ▼✐♥ ❝✉t ✴ ▼❛① ✢♦✇ ❞✉❛❧✐t② 0 0 0 3 2 ▼❛① ❋❧♦✇ ❛❧❣♦r✐t❤♠ ∞ T ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✼ ✴ ✺✶

  6. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ●r❛♣❤ ❈✉ts ✿ ❡①❛♠♣❧❡ ■♥♣✉t s❡❡❞s ❢❛✈♦rs s♠❛❧❧ ❜♦✉♥❞❛r✐❡s r♦❜✉st t♦ ✉♥❝❡♥t❡r❡❞ s❡❡❞ ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s ♣❧❛❝❡♠❡♥t ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✽ ✴ ✺✶

  7. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ●r❛♣❤ ❈✉ts ✿ ❡①❛♠♣❧❡ ■♥♣✉t s❡❡❞s ❢❛✈♦rs s♠❛❧❧ ❜♦✉♥❞❛r✐❡s r♦❜✉st t♦ ✉♥❝❡♥t❡r❡❞ s❡❡❞ ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s ♣❧❛❝❡♠❡♥t ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✾ ✴ ✺✶

  8. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❘❛♥❞♦♠ ❲❛❧❦❡r ❉✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ Z X ✇ ✐❥ ( ① ✐ � ① ❥ ) ✷ ❥r ✉ ❥ ✷ ❞ Ω ① = ❛r❣ ♠✐♥ ✥ ✉ = ❛r❣ ♠✐♥ Ω ❡ ✐❥ ✷ ❊ 3 4 0 . 7 0 . 8 1 1 3 4 1 3 0 . 2 0 . 7 0 . 8 4 2 2 0 0 . 2 0 . 3 3 2 P♦t❡♥t✐❛❧s ❛♥❛❧♦❣② ❘❛♥❞♦♠ ✇❛❧❦❡r ❛♥❛❧♦❣② 3 4 0 . 7 0 . 8 1 1 3 4 1 3 0 . 2 0 . 7 0 . 8 4 2 2 0 0 . 2 0 . 3 3 2 ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✵ ✴ ✺✶

  9. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❘❛♥❞♦♠ ❲❛❧❦❡r ❉✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ Z X ♣ = ✶ ( ① ✐ � ① ❥ ) q = ✷ ❥r ✉ ❥ ✷ ❞ Ω ① = ❛r❣ ♠✐♥ ✇ ✐❥ ✥ ✉ = ❛r❣ ♠✐♥ Ω ❡ ✐❥ ✷ ❊ 3 4 0 . 7 0 . 8 1 1 3 4 1 3 0 . 2 0 . 7 0 . 8 4 2 2 0 0 . 2 0 . 3 3 2 P♦t❡♥t✐❛❧s ❛♥❛❧♦❣② ❘❛♥❞♦♠ ✇❛❧❦❡r ❛♥❛❧♦❣② 3 4 0 . 7 0 . 8 1 1 3 4 1 3 0 . 2 0 . 7 0 . 8 4 2 2 0 0 . 2 0 . 3 3 2 ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✵ ✴ ✺✶

  10. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❘❛♥❞♦♠ ❲❛❧❦❡r ❉✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ Z X ✇ ✐❥ ( ① ✐ � ① ❥ ) ✷ ❥r ✉ ❥ ✷ ❞ Ω ① = ❛r❣ ♠✐♥ ✥ ✉ = ❛r❣ ♠✐♥ Ω ❡ ✐❥ ✷ ❊ 3 4 0 . 7 0 . 8 1 1 3 4 1 3 0 . 2 0 . 7 0 . 8 4 2 2 0 0 . 2 0 . 3 3 2 P♦t❡♥t✐❛❧s ❛♥❛❧♦❣② ❘❛♥❞♦♠ ✇❛❧❦❡r ❛♥❛❧♦❣② 3 4 0 . 7 0 . 8 1 1 3 4 1 3 0 . 2 0 . 7 0 . 8 4 2 2 0 0 . 2 0 . 3 3 2 ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✵ ✴ ✺✶

  11. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❘❛♥❞♦♠ ❲❛❧❦❡r ❉✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ Z X ✇ ✐❥ ( ① ✐ � ① ❥ ) ✷ ❥r ✉ ❥ ✷ ❞ Ω ① = ❛r❣ ♠✐♥ ✥ ✉ = ❛r❣ ♠✐♥ Ω ❡ ✐❥ ✷ ❊ 3 4 0 . 7 0 . 8 1 ❙tr✐❝❧② ❝♦♥✈❡① ♣r♦❜❧❡♠ 1 3 4 1 3 0 . 2 0 . 7 0 . 8 ✮ ✉♥✐q✉❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ① ✄ 4 2 2 0 0 . 2 0 . 3 3 2 P♦t❡♥t✐❛❧s ❛♥❛❧♦❣② ❘❛♥❞♦♠ ✇❛❧❦❡r ❛♥❛❧♦❣② 3 4 0 . 7 0 . 8 1 1 3 4 1 3 0 . 2 0 . 7 0 . 8 4 2 2 0 0 . 2 0 . 3 3 2 ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✶ ✴ ✺✶

  12. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❘❛♥❞♦♠ ❲❛❧❦❡r ✿ ❡①❛♠♣❧❡ ■♥♣✉t s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t②✴♣♦t❡♥t✐❛❧ ① ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s ✭t❤r❡s❤♦❧❞ ♦❢ ① ✮ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✷ ✴ ✺✶

  13. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st t❛❦❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✇❡✐❣❤ts t❤❡ s❤♦rt❡st ♣❛t❤ st❛rt✐♥❣ ❢r♦♠ 1 1 3 4 3 4 ❡❛❝❤ ♥♦❞❡ t♦ r❡❛❝❤ ❛ s❡❡❞ ♥♦❞❡ ✐s F ❝♦♠♣✉t❡❞ 1 1 1 1 3 4 3 1 4 1 3 1 3 ❉✐❥s❦tr❛ ❛❧❣♦r✐t❤♠ 1 1 1 ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ✿ ♦♣t✐♠✐③❡s 4 2 2 4 2 2 B 3 2 � 1 1 ✇ ✐❥ ♣ = q ✦✶ ( ① ✐ � ① ❥ ) q ✦✶ ♠✐♥ 3 2 ① ❡ ✐❥ ✷ ❊ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✸ ✴ ✺✶

  14. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st t❛❦❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✇❡✐❣❤ts t❤❡ s❤♦rt❡st ♣❛t❤ st❛rt✐♥❣ ❢r♦♠ 1 1 ❡❛❝❤ ♥♦❞❡ t♦ r❡❛❝❤ ❛ s❡❡❞ ♥♦❞❡ ✐s 3 4 F ❝♦♠♣✉t❡❞ 1 1 1 3 4 1 1 3 ❉✐❥s❦tr❛ ❛❧❣♦r✐t❤♠ 1 1 1 ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ✿ ♦♣t✐♠✐③❡s 4 2 2 B 1 1 � ✇ ✐❥ ♣ = q ✦✶ ( ① ✐ � ① ❥ ) q ✦✶ 3 2 ♠✐♥ ① ❡ ✐❥ ✷ ❊ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✹ ✴ ✺✶

  15. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st t❛❦❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✇❡✐❣❤ts t❤❡ s❤♦rt❡st ♣❛t❤ st❛rt✐♥❣ ❢r♦♠ 1 1 ❡❛❝❤ ♥♦❞❡ t♦ r❡❛❝❤ ❛ s❡❡❞ ♥♦❞❡ ✐s 3 4 F ❝♦♠♣✉t❡❞ 1 1 1 3 4 1 1 3 ❉✐❥s❦tr❛ ❛❧❣♦r✐t❤♠ 1 1 1 ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ✿ ♦♣t✐♠✐③❡s 4 2 2 B 1 1 � ✇ ✐❥ ♣ = q ✦✶ ( ① ✐ � ① ❥ ) q ✦✶ 3 2 ♠✐♥ ① ❡ ✐❥ ✷ ❊ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✺ ✴ ✺✶

  16. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ✿ ❡①❛♠♣❧❡ ■♥♣✉t s❡❡❞s ❱❡r② s❡♥s✐t✐✈❡ t♦ s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s ♣❧❛❝❡♠❡♥t ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✻ ✴ ✺✶

  17. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 1 3 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s ❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  18. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ 3 F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s ❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  19. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ 3 F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s ❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  20. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ 3 F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s ❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  21. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ 3 F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s ❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  22. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ 3 F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 2 3 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s ❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  23. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s ❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  24. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s ❑r✉s❦❛❧ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  25. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 1 3 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  26. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 1 3 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  27. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 1 3 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  28. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ 4 F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  29. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F 4 3 ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 3 4 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  30. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  31. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 2 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  32. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ 4 3 ♠❛①✐♠✐③❡ t❤❡ s✉♠ ♦❢ F ✇❡✐❣❤ts ♦✈❡r t❤❡ ❡❞❣❡s ♦❢ ❛ 1 3 4 ❢♦r❡st s♣❛♥♥✐♥❣ t❤❡ ❣r❛♣❤ 1 3 ❞✐✛❡r❡♥t ❧❛❜❡❧❡❞ ♥♦❞❡s ❤❛✈❡ 4 2 2 t♦ ❜❡❧♦♥❣ t♦ ❞✐✛❡r❡♥t tr❡❡s B 3 2 ❑r✉s❦❛❧✱ Pr✐♠ ❛❧❣♦r✐t❤♠s Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✼ ✴ ✺✶

  33. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✭▼❙❋✮ ✿ ❡①❛♠♣❧❡ ■♥♣✉t s❡❡❞s r♦❜✉st t♦ s♠❛❧❧ s❡❡❞s ✿ ♥♦ ❜✐❛s t♦✇❛r❞ s♠❛❧❧ ♦❜❥❡❝ts ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s ❧❡❛❦✐♥❣ ❡✛❡❝t ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✽ ✴ ✺✶

  34. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❙❋ ❛♥❞ ❲❛t❡rs❤❡❞s ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇❛t❡rs❤❡❞ ✿ ♣♦✐♥ts ✇❤❡r❡ ❛ ❞r♦♣ ♦❢ ✇❛t❡r ❝♦✉❧❞ ✢♦✇ t♦✇❛r❞ ❞✐✛❡r❡♥t ❝❛❝❤♠❡♥t ❜❛ss✐♥s ❬❈♦✉st② ❡t ❛❧✳ ✵✼❪✳ ❣r❛❞✐❡♥t s❡❡♥ ❛s ❛ r❡❧✐❡❢ ♠❛♣✳ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ✐♥✈❡rs❡ ♦❢ ❣r❛❞✐❡♥t ✦ t❤❛❧✇❡❣ ❧✐♥❡ ✢♦♦❞✐♥❣ ❛❧❣♦r✐t❤♠ ✑ Pr✐♠ ❛❧❣♦r✐t❤♠ ❢♦r ▼❙❋ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✾ ✴ ✺✶

  35. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ▼❙❋ ❛♥❞ ❲❛t❡rs❤❡❞s ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇❛t❡rs❤❡❞ ✿ ♣♦✐♥ts ✇❤❡r❡ ❛ ❞r♦♣ ♦❢ ✇❛t❡r ❝♦✉❧❞ ✢♦✇ t♦✇❛r❞ ❞✐✛❡r❡♥t ❝❛❝❤♠❡♥t ❜❛ss✐♥s ❬❈♦✉st② ❡t ❛❧✳ ✵✼❪✳ ❣r❛❞✐❡♥t s❡❡♥ ❛s ❛ r❡❧✐❡❢ ♠❛♣✳ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ✐♥✈❡rs❡ ♦❢ ❣r❛❞✐❡♥t ✦ t❤❛❧✇❡❣ ❧✐♥❡ ✢♦♦❞✐♥❣ ❛❧❣♦r✐t❤♠ ✑ Pr✐♠ ❛❧❣♦r✐t❤♠ ❢♦r ▼❙❋ ❚❤❡♦r❡♠ ❬❈♦✉st② ❡t ❛❧✳ ✵✼❪ ■❢ s❡❡❞s ❛r❡ t❤❡ ♠❛①✐♠❛ ♦❢ t❤❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✱ ❡✈❡r② ♠❛①✐♠✉♠ s♣❛♥♥✐♥❣ ❢♦r❡st ❝✉t ♦♥ t❤❛t ❢✉♥❝t✐♦♥ ✐s ❛ ✇❛t❡rs❤❡❞ ❝✉t✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✶✾ ✴ ✺✶

  36. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❊①❛♠♣❧❡ ♦❢ s❡❣♠❡♥t❛t✐♦♥ ❜② ✢♦♦❞✐♥❣✴Pr✐♠ ❛❧❣♦r✐t❤♠ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✵ ✴ ✺✶

  37. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q � ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① ❍❍❍❍ ♣ ✵ ✜♥✐t❡ ✶ q ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s ●r❛♣❤ ❝✉ts ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✷ ❵ ✷ ✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ✶ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

  38. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q � ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① ❍❍❍❍ ♣ ✵ ✜♥✐t❡ ✶ q ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s ●r❛♣❤ ❝✉ts ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✷ ❵ ✷ ✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ✶ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

  39. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q � ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① ❍❍❍❍ ♣ ✵ ✜♥✐t❡ ✶ q ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s ●r❛♣❤ ❝✉ts ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✷ ❵ ✷ ✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ✶ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

  40. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q � ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① ❍❍❍❍ ♣ ✵ ✜♥✐t❡ ✶ q ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s ●r❛♣❤ ❝✉ts ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✷ ❵ ✷ ✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ✶ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

  41. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q � ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① ❍❍❍❍ ♣ ✵ ✜♥✐t❡ ✶ q ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s ●r❛♣❤ ❝✉ts ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✷ ❵ ✷ ✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ✶ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

  42. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q � ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① ❍❍❍❍ ♣ ✵ ✜♥✐t❡ ✶ q ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s ●r❛♣❤ ❝✉ts ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✷ ❵ ✷ ✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ✶ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

  43. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❆❧❣♦r✐t❤♠s ❞❡r✐✈✐♥❣ ❢r♦♠ ✈❛❧✉❡s ♦❢ ♣ ❡t q ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q � ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① ❍❍❍❍ ♣ ✵ ✜♥✐t❡ ✶ q ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✶ ❘❡❞✉❝t✐♦♥ t♦ s❡❡❞s ●r❛♣❤ ❝✉ts ❬❆❧❧è♥❡ ❡t ❛❧✳ ✵✼❪ ▼❛① ❙♣❛♥♥✐♥❣ ❋♦r❡st ✷ ❵ ✷ ✲♥♦r♠ ❱♦r♦♥♦✐ ❘❛♥❞♦♠ ✇❛❧❦❡r ❬❈♦✉♣r✐❡ ❡t ❛❧✳ ✵✾❪ ❙❤♦rt❡st P❛t❤ ❋♦r❡st ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ❵ ✶ ✲♥♦r♠ ❱♦r♦♥♦✐ ✶ ❬❙✐♥♦♣ ❡t ❛❧✳ ✵✼❪ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✶ ✴ ✺✶

  44. ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ Pr♦♣❡rt✐❡s ❉❡✜♥✐t✐♦♥ ▲❡t s ❜❡ t❤❡ s❡❣♠❡♥t❛t✐♦♥ ❞❡✜♥❡❞ ❜② ❛ t❤r❡s❤♦❧❞✐♥❣ ♦❢ t❤❡ ❧❛❜❡❧s � ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ✿ ① = ❛r❣ ♠✐♥ ❡ ✐❥ ✷ ❊ ❚❤❡ s❡t ♦❢ ❡❞❣❡s ❡ ✐❥ t❤❛t ✈❡r✐❢② s ✐ ✻ = s ❥ ❝♦♥st✐t✉t❡ ❛ q ✲❝✉t ❢♦r ✇ ♣ ✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✷ ✴ ✺✶

  45. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ Pr♦♣❡rt✐❡s ❉❡✜♥✐t✐♦♥ ▲❡t s ❜❡ t❤❡ s❡❣♠❡♥t❛t✐♦♥ ❞❡✜♥❡❞ ❜② ❛ t❤r❡s❤♦❧❞✐♥❣ ♦❢ t❤❡ ❧❛❜❡❧s � ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ✿ ① = ❛r❣ ♠✐♥ ❡ ✐❥ ✷ ❊ ❚❤❡ s❡t ♦❢ ❡❞❣❡s ❡ ✐❥ t❤❛t ✈❡r✐❢② s ✐ ✻ = s ❥ ❝♦♥st✐t✉t❡ ❛ q ✲❝✉t ❢♦r ✇ ♣ ✳ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✷ ✴ ✺✶

  46. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 F 1 2 3 8 B 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  47. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 2 3 8 0 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  48. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 2 3 8 0 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  49. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 10 10 10 1 1 2 3 8 10 10 10 10 0 6 4 10 10 10 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  50. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ ( 5 ( 7 ( 9 10 ) p 10 ) p 10 ) p 1 ( 1 ( 2 ( 3 ( 8 10 ) p 10 ) p 10 ) p 10 ) p 0 ( 6 ( 4 ( 10 10 ) p 10 ) p 10 ) p ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  51. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 0 0 0 1 0 0 0 0 x 7 x 8 0 0 0 1 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  52. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 ① ✼ = ① ✽ 1 2 3 8 x 7 x 8 0 6 4 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  53. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 ① ✼ = ① ✽ 1 2 3 8 x 7 x 8 0 6 4 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  54. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ ( 5 ( 7 ( 9 9 ) p 9 ) p 9 ) p 1 ① ✼ = ① ✽ ( 1 ( 2 ( 3 ( 8 9 ) p 9 ) p 9 ) p 9 ) p x 7 x 8 0 ( 6 ( 4 9 ) p 9 ) p ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  55. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 0 0 1 1 1 ① ✼ = ① ✽ 0 0 0 0 x 7 x 8 0 0 0 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  56. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 ① ✼ = ① ✽ 1 2 3 8 x 7 x 8 0 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  57. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 ① ✼ = ① ✽ 1 2 3 8 x 7 x 8 0 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  58. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 1 2 3 8 0 1 1 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  59. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 1 2 3 8 0 1 1 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  60. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 1 1 2 3 8 0 1 1 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  61. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 1 1 2 3 8 0 1 1 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  62. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 1 1 2 3 8 0 0 1 1 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  63. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 1 1 2 3 8 0 0 1 1 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  64. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❚❤❡♦r❡♠ ❛♥❞ ♣r♦♦❢ ✐❧❧✉str❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t ▼ ❜❡ ❛ s✉❜✲❣r❛♣❤ ♦❢ ● ✱ q ❛ ✜♥✐t❡ r❡❛❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ■❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❛❧❧ ❞✐✛❡r❡♥t✱ ❛♥② q ✲❝✉t r❡❧❛t✐✈❡ t♦ ▼ ❢♦r ✇ ♣ ✇❤❡♥ ♣ ✦ ✶ ✐s ❛♥ ▼❙❋✲❝✉t✳ 5 7 9 1 1 1 1 1 2 3 8 0 0 1 1 6 4 10 ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ❘❡❝❛❧❧ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✿ ♠✐♥ ① � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✸ ✴ ✺✶

  65. ●r❛♣❤ ❈✉ts ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❘❛♥❞♦♠ ❲❛❧❦❡r ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❘❛♥❞♦♠ ❲❛❧❦❡r ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❙❤♦rt❡st ♣❛t❤ ❢♦r❡st ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ✫ ❲❛t❡rs❤❡❞ ❆❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt✐❡s ❈♦r♦❧❧❛r② ❲❤❡♥ ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❞✐✛❡r❡♥t✱ t❤❡ s♦❧✉t✐♦♥ ① ✄ t♦ � ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ♣ ✦✶ ♠✐♥ ❧✐♠ ① ❡ ✐❥ ✷ ❊ r❡❧❛t✐✈❡ t♦ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐s ❜✐♥❛r②✳ Pr♦♣❡rt② ■♥ t❤❡ ❝❛s❡ ♦❢ ❛r❜✐tr❛r② ✇❡✐❣❤ts ✭♥♦♥✲♥❡❝❡ss❛r✐❧② ❡q✉❛❧✮✱ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❤♦❧❞s ✐❢ ❛❧❧ t❤❡ s❡❡❞s ▼ ❛r❡ t❤❡ ♠❛①✐♠❛ ♦❢ t❤❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✹ ✴ ✺✶

  66. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ❆❧❣♦r✐t❤♠ ❢♦r t❤❡ ❝❛s❡ ♣ = ✶ ✱ ✈❛r✐❛❜❧❡ q ❈♦♠♣✉t❡ ① ♠✐♥✐♠✐③✐♥❣ � ✇ ✐❥ ♣ ❥ ① ✐ � ① ❥ ❥ q ❧✐♠ ♣ ✦✶ ❡ ✐❥ ✷ ❊ s✉❜❥❡❝t t♦ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❲❡ ❝♦♥str✉❝t ❛♥ ▼❙❋ ♦✉ts✐❞❡ ♦❢ ♣❧❛t❡❛✉s✱ ❛♥❞ ♦♣t✐♠✐③❡ � ❥ ① ✐ � ① ❥ ❥ q ❡ ✐❥ ✷ ♣❧❛t❡❛✉ ♦♥ t❤❡ ♣❧❛t❡❛✉s✳ ❲❡ ❝❛❧❧ t❤✐s ❛❧❣♦r✐t❤♠ ✏P♦✇❡r ✇❛t❡rs❤❡❞✑ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✺ ✴ ✺✶

  67. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❘❲ ✇❤❡♥ ♣ ✦ ✶ t♦✇❛r❞ P❲ ❘❛♥❞♦♠ ❲❛❧❦❡r ♣ = ✶ ✿✿✿ ✸✵ ■♥♣✉t s❡❡❞s P♦✇❡r❲❛t❡rs❤❡❞ q = ✷ ❘❛♥❞♦♠ ❲❛❧❦❡r ♣ = ✸✵ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✻ ✴ ✺✶

  68. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ✶ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✼ ✴ ✺✶

  69. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

  70. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✶ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✶ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✵ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

  71. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✶ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✵ ✶ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✵ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

  72. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✶ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✵ ✶ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✵ ✵ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

  73. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✶ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✵ ✶ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✵ ✵ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

  74. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✶ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✵ ✶ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✵ ✵ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ♠✐♥ ① ❧✐♠ ♣ ✦✶ � ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

  75. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✶ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✵ ✶ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✵ ✵ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❡ ✐❥ ✷ ❊ ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ ✷ � ♠✐♥ ① ❧✐♠ ♣ ✦✶ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

  76. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✷ ✶ ✾ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✶ ✶ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✺ ✸ ✺ ✾ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✹ ✺ ✺ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✵ ✶ ✽ ✺ ✺ ✺ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ✵ ✵ ✼ ✺ ✺ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✾ ✼ ✹ ✹ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✵ ✵ ♥♦❞❡✳ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❡ ✐❥ ✷ ♣❧❛t❡❛✉ ❥ ① ✐ � ① ❥ ❥ q � ♠✐♥ ① ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✽ ✴ ✺✶

  77. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✷ ✶ ✾ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✶ ✶ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✺ ✸ ✺ ✾ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✹ ✺ ✺ ✵ ✵✳✸✺ ✵✳✼✶ ✶ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ ✽ ✺ ✺ ✺ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✵ ✵ ✵✳✹✽ ✵✳✼✹ ✼ ✺ ✺ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✾ ✼ ✹ ✹ ♥♦❞❡✳ ✵ ✵ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✾ ✴ ✺✶

  78. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✷ ✶ ✾ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✵ ✶ ✶ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✺ ✸ ✺ ✾ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✹ ✺ ✺ ✵ ✵✳✸✺ ✵✳✼✶ ✶ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ ✽ ✺ ✺ ✺ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✵ ✵ ✵✳✹✽ ✵✳✼✹ ✼ ✺ ✺ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✾ ✼ ✹ ✹ ♥♦❞❡✳ ✵ ✵ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✾ ✴ ✺✶

  79. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✷ ✶ ✾ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✵ ✶ ✶ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✺ ✸ ✺ ✾ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✹ ✺ ✺ ✵ ✵✳✸✺ ✵✳✼✶ ✶ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ ✽ ✺ ✺ ✺ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✵ ✵ ✵✳✹✽ ✵✳✼✹ ✼ ✺ ✺ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✾ ✼ ✹ ✹ ♥♦❞❡✳ ✵ ✵ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✷✾ ✴ ✺✶

  80. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✷ ✶ ✾ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✵ ✶ ✶ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✺ ✸ ✺ ✾ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✹ ✺ ✺ ✵ ✵✳✸✺ ✵✳✼✶ ✶ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ ✽ ✺ ✺ ✺ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✵ ✵ ✵✳✹✽ ✵✳✼✹ ✼ ✺ ✺ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✾ ✼ ✹ ✹ ♥♦❞❡✳ ✵ ✵ ✵✳✸✷ ✵✳✺✺ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✵ ✴ ✺✶

  81. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r ✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❈❤♦♦s❡ ❛♥ ❡❞❣❡ ✇✐t❤ ♠❛①✐♠❛❧ ✶ ✷ ✶ ✾ ✇❡✐❣❤t ❡ ♠❛① ✳ ▲❡t ❙ t❤❡ s❡t ♦❢ ✵ ✵ ✿ ✸✺ ✶ ✶ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ ❡ ♠❛① ✇✐t❤ ✺ ✸ ✺ ✾ t❤❡ s❛♠❡ ✇❡✐❣❤t ❛s ❡ ♠❛① ✳ ✹ ✺ ✺ ✵ ✵✳✸✺ ✵✳✼✶ ✶ ■❢ ❙ ❛❧r❡❛❞② ❝♦♥t❛✐♥s ✈❡rt✐❝❡s ✷ ✽ ✺ ✺ ✺ t❤❛t ❤❛✈❡ ❞✐✛❡r❡♥t ❧❛❜❡❧s✱ ♠✐♥✐♠✐③❡ ❊ ✶ ❀ q ♦♥ ❙ ✱ ♦t❤❡r✇✐s❡ ✵ ✵ ✵✳✹✽ ✵✳✼✹ ✼ ✺ ✺ ♠❡r❣❡ t❤❡ ♥♦❞❡s ♦❢ ❙ ✐♥t♦ ♦♥❡ ✾ ✼ ✹ ✹ ♥♦❞❡✳ ✵ ✵ ✵✳✸✷ ✵✳✺✺ ✾ ✹ ✹ ❘❡♣❡❛t st❡♣s ✶ ❛♥❞ ✷ ✉♥t✐❧ ❛❧❧ ✸ ✈❡rt✐❝❡s ❛r❡ ❧❛❜❡❧❡❞✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✵ ✴ ✺✶

  82. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ Pr♦♣❡rt✐❡s ❚❤❡♦r❡♠ ❚❤❡ ❝✉t ♦❜t❛✐♥❡❞ ❜② t❤❡ P♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✐s ❛ ▼❙❋ ❝✉t✳ ❚❤❡♦r❡♠ ❲❤❡♥ q ❃ ✶✱ t❤❡ s♦❧✉t✐♦♥ ① ✄ t♦ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ � ✇ ✐❥ ♣ ❥ ① ✐ � ① ❥ ❥ q ♠✐♥ ① ❧✐♠ ♣ ✦✶ ❡ ✐❥ ✷ ❊ ✐s ✉♥✐q✉❡✳ ❚❤✉s✱ ✇❤❡♥ q ❃ ✶✱ t❤❡ s♦❧✉t✐♦♥ ① ♦❜t❛✐♥❡❞ ❜② t❤❡ P♦✲ ✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ✐s ✉♥✐q✉❡✳ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✶ ✴ ✺✶

  83. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r✇❛t❡rs❤❡❞ ✭q❂✷✮ ✿ ❡①❛♠♣❧❡ ■♥♣✉t s❡❡❞s r♦❜✉st ✐♥ ❝❛s❡ ♦❢ s♠❛❧❧ s❡❡❞s ❈♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t② ① ❧❡ss ❧❡❛❦✐♥❣ t❤❛♥ ✇✐t❤ st❛♥❞❛r❞ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❋♦r❡st ❈♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥t❛t✐♦♥s ✭t❤r❡s❤♦❧❞ ♦❢ ① ✮ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✷ ✴ ✺✶

  84. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r✇❛t❡rs❤❡❞ ✭q❂✷✮ ✿ ❡①❛♠♣❧❡ ■♥♣✉t s❡❡❞s P♦✇❡r✇❛t❡rs❤❡❞ ✭ q = ✷✮ Pr✐♠ ❛❧❣♦r✐t❤♠ ❢♦r ▼❙❋ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✸ ✴ ✺✶

  85. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ P♦✇❡r✇❛t❡rs❤❡❞ ✭q❂✷✮ ✿ ❡①❛♠♣❧❡ ■♥♣✉t s❡❡❞s P♦✇❡r✇❛t❡rs❤❡❞ ✭ q = ✷✮ Pr✐♠ ❛❧❣♦r✐t❤♠ ❢♦r ▼❙❋ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✹ ✴ ✺✶

  86. ♣ ❥ ① ✐ � ✶ ❥ q ✇ ❇✐ ♣ ❥ ① ✐ ❥ q ✇ ❋ ✐ ✈ ✐ ✈ ✐ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❢♦r❡st ✐♥ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤ ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ P♦ss✐❜✐❧✐t② t♦ ❛❞❞ ✉♥❛r② t❡r♠s t♦ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ � ✇ ♣ ✐❥ ❥ ① ✐ � ① ❥ ❥ q ♠✐♥ ① ❡ ✐❥ ✷ ❊ F w F w B B ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✺ ✴ ✺✶

  87. ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❢♦r❡st ✐♥ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤ ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ P♦ss✐❜✐❧✐t② t♦ ❛❞❞ ✉♥❛r② t❡r♠s t♦ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✐❥ ❥ ① ✐ � ① ❥ ❥ q + ♣ ❥ ① ✐ � ✶ ❥ q + � ✇ ♣ � � ✇ ❇✐ ♣ ❥ ① ✐ ❥ q ♠✐♥ ✇ ❋ ✐ ① ✈ ✐ ✈ ✐ ❡ ✐❥ ✷ ❊ F w F w B B ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✺ ✴ ✺✶

  88. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ P♦ss✐❜✐❧✐t② t♦ ❛❞❞ ✉♥❛r② t❡r♠s t♦ t❤❡ ❡♥❡r❣② ❢✉♥❝t✐♦♥ ✐❥ ❥ ① ✐ � ① ❥ ❥ q + ♣ ❥ ① ✐ � ✶ ❥ q + � ✇ ♣ � � ✇ ❇✐ ♣ ❥ ① ✐ ❥ q ♠✐♥ ✇ ❋ ✐ ① ✈ ✐ ✈ ✐ ❡ ✐❥ ✷ ❊ ▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❢♦r❡st ✐♥ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤ F F w F 7 5 6 9 8 2 3 1 4 6 6 1 2 6 5 2 3 4 w B 3 5 4 1 2 8 B 7 9 6 B ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✺ ✴ ✺✶

  89. ❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ t❤✐s ✐s t❤❡ ✜rst t✐♠❡ t❤❛t ✇❛t❡rs❤❡❞ ✐s ✉s❡❞ ✐♥ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s t❤❛♥ s❡❡❞❡❞ s❡❣♠❡♥t❛t✐♦♥ ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ F w F ✐❥ ❥ ① ✐ � ① ❥ ❥ q + ♣ ❥ ① ✐ � ✶ ❥ q + � ✇ ♣ � � ♣ ❥ ① ✐ ❥ q ♠✐♥ ✇ ❋ ✐ ✇ ❇✐ ① ✈ ✐ ✈ ✐ ❡ ✐❥ ✷ ❊ w B B ■♠❛❣❡ ●r❛♣❤ ❈✉ts ▼❛①❙❋ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✻ ✴ ✺✶

  90. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t② ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ F w F ✐❥ ❥ ① ✐ � ① ❥ ❥ q + ♣ ❥ ① ✐ � ✶ ❥ q + � ✇ ♣ � � ♣ ❥ ① ✐ ❥ q ♠✐♥ ✇ ❋ ✐ ✇ ❇✐ ① ✈ ✐ ✈ ✐ ❡ ✐❥ ✷ ❊ w B B ■♠❛❣❡ ●r❛♣❤ ❈✉ts ▼❛①❙❋ ❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ t❤✐s ✐s t❤❡ ✜rst t✐♠❡ t❤❛t ✇❛t❡rs❤❡❞ ✐s ✉s❡❞ ✐♥ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s t❤❛♥ s❡❡❞❡❞ s❡❣♠❡♥t❛t✐♦♥ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✻ ✴ ✺✶

  91. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ❖♣t✐♠❛❧ ♠✉❧t✐❧❛❜❡❧s s❡❣♠❡♥t❛t✐♦♥ ▼♦r❡ t❤❛♥ ✷✲❧❛❜❡❧s s❡❣♠❡♥t❛t✐♦♥ ✿ ◆P✲❤❛r❞ ❢♦r ●r❛♣❤ ❝✉ts ❊①❛❝t ♥ ✕ ✷ ❧❛❜❡❧s s❡❣♠❡♥t❛t✐♦♥ ❢♦r t❤❡ ♦t❤❡r ❛❧❣♦r✐t❤♠s ✿ ♥ s♦❧✉t✐♦♥s ① ✶ ❀ ① ✷ ❀ ✿✿✿ ① ♥ ❝♦♠♣✉t❡❞  ① ❦ ( ♥ ❦ ) = ✶ ① ❦ ❝♦♠♣✉t❡❞ ❜② ❡♥❢♦r❝✐♥❣ ① ❦ ( ♥ q ) = ✵ ❢♦r ❛❧❧ q ✻ = ❦ ✿ ❊❛❝❤ ♥♦❞❡ ✐ ✐s ❛✛❡❝t❡❞ t♦ t❤❡ ❧❛❜❡❧ ❢♦r ✇❤✐❝❤ ① ❦ ✐ ✐s ♠❛①✐♠✉♠ ✿ ① ❦ s ✐ = ❛r❣ ♠❛① ✐ ❦ ■♥♣✉t s❡❡❞s ❙❡❣♠❡♥t❛t✐♦♥ ❜② P♦✇❡r❲❛t❡rs❤❡❞ ✭ q = ✷✮ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✼ ✴ ✺✶

  92. ❆ ✉♥✐❢②✐♥❣ ❢r❛♠❡✇♦r❦ ❚❤❡ ♣♦✇❡r✇❛t❡rs❤❡❞ ❛❧❣♦r✐t❤♠ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ❘❡s✉❧ts ❛♥❞ ❈♦♥❝❧✉s✐♦♥ ❆❧❣♦r✐t❤♠s ❜❡❤❛✈✐♦r ♦♥ ♣❧❛t❡❛✉s ❙❡❡❞❡❞ ●r❛♣❤ ❙❤♦rt❡st P❛t❤s✱ ❘❛♥❞♦♠ ❲❛❧❦❡r✱ ✐♠❛❣❡ ❈✉ts ▼❛①❙❋ P❲ q = ✷ ❈❛♠✐❧❧❡ ❈♦✉♣r✐❡ P♦✇❡r ❲❛t❡rs❤❡❞s ✸✽ ✴ ✺✶

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