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  1. ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❙♦❧✉t✐♦♥ ❚❡❝❤♥✐q✉❡s ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥ ❆r❝✲ ❈♦♥s✐st❡♥❝②

  2. ❙✉♠♠❛r② ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❙♦❧✉t✐♦♥ ◆♦❞❡ ❝♦♥s✐st❡♥❝② ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❚❡❝❤♥✐q✉❡s ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥ ❆r❝✲ ❈♦♥s✐st❡♥❝②

  3. ❙♦❧✉t✐♦♥ ❚❡❝❤♥✐q✉❡s ❢♦r ❈♦♥str❛✐♥t ◆❡t✇♦r❦ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❙♦❧✈✐♥❣ ❈♦♥str❛✐♥t ◆❡t✇♦r❦s ❆r❝ ❈♦♥s✐st❡♥❝② ■♥❢❡r❡♥❝❡✿ ■♥❢❡r ♥❡✇ ❝♦♥str❛✐♥ts ❜❛s❡❞ ♦♥ ❡①✐st✐♥❣ ♦♥❡s ❙♦❧✉t✐♦♥ ❊❧✐♠✐♥❛t❡ ✈❛❧✉❡s ❢r♦♠ ✈❛r✐❛❜❧❡s t❤❛t ❞♦ ♥♦t ♠❡❡t ❚❡❝❤♥✐q✉❡s ❝♦♥str❛✐♥ts ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❙❡❛r❝❤✿ ❛♥❞ ❈♦♥str❛✐♥t ▲♦♦❦ ❢♦r ❛ s♦❧✉t✐♦♥ tr②✐♥❣ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ✈❛r✐❛❜❧❡s Pr♦♣❛❣❛t✐♦♥ ❜❛❝❦tr❛❝❦✐♥❣ ❛♥❞ s✐♠✐❧❛r ❛♣♣r♦❛❝❤❡s ❆r❝✲ ❈♦♥s✐st❡♥❝② ❧♦❝❛❧ s❡❛r❝❤

  4. ❇❛❝❦tr❛❝❦✐♥❣ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❣❡♥❡r❛❧ ✐❞❡❛s ❈❤♦♦s❡ ❛ ✈❛r✐❛❜❧❡ ① ❙♦❧✉t✐♦♥ ❚❡❝❤♥✐q✉❡s ❧✐st ✐ts ❞♦♠❛✐♥ ✈❛❧✉❡s ❈♦♥s✐st❡♥❝② ❢♦r ❡❛❝❤ ✈❛❧✉❡ ❛❞❞ ❛ ❝♦♥str❛✐♥t ① = ✈ ❛♥❞ r❡❝✉rs✐✈❡❧② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❡✈❛❧✉❛t❡ t❤❡ r❡st ♦❢ t❤❡ ♣r♦❜❧❡♠ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥ ❆r❝✲ ❈♦♥s✐st❡♥❝②

  5. ▲♦❝❛❧ ❈♦♥s✐st❡♥❝② ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❣❡♥❡r❛❧ ✐❞❡❛s P❛rt✐❛❧ ❛ss✐❣♥♠❡♥ts ❝❛♥ ❧❡❛❞ t♦ ❝♦♥str❛✐♥t ✈✐♦❧❛t✐♦♥s ❙♦❧✉t✐♦♥ ❲❡ ❝❛♥ ❡✈❛❧✉❛t❡ ❛ ❝♦♥str❛✐♥t ❛s s♦♦♥ ❛s ❛❧❧ ✈❛r✐❛❜❧❡s ✐♥ ✐ts ❚❡❝❤♥✐q✉❡s s❝♦♣❡ ❛r❡ ❛ss✐❣♥❡❞ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❲❡ ❝❛♥ ❜❛❝❦tr❛❝❦ ❛s s♦♦♥ ❛s ❛ ❝♦♥str❛✐♥t ✐s ♥♦t ❧♦❝❛❧❧② ❛♥❞ ❈♦♥str❛✐♥t ❝♦♥s✐st❡♥t Pr♦♣❛❣❛t✐♦♥ ❆r❝✲ ❈♦♥s✐st❡♥❝②

  6. ■♥❢❡r❡♥❝❡ ❛♥❞ ❝♦♥str❛✐♥t ♣r♦♣❛❣❛t✐♦♥ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❊①❛♠♣❧❡ ✭✐♥❢❡r❡♥❝❡✮ ❆r❝ ❈♦♥s✐st❡♥❝② ❱❛r✐❛❜❧❡s✿ { ❆ , ❇ , ❈ } ❙♦❧✉t✐♦♥ ❉♦♠❛✐♥✿ { ✵ , ✶ } ♦r tr✉❡✱❢❛❧s❡ ❚❡❝❤♥✐q✉❡s ❈♦♥s✐st❡♥❝② ❈♦♥str❛✐♥t✿ { ❆ → ❇ , ❈ → ❆ , ❈ } ❊♥❢♦r❝✐♥❣ ❛♥❞ Pr♦♣❛❣❛t✐♥❣ t❤❡ ❝♦♥str❛✐♥ts ✇❡ ❝❛♥ ✐♥❢❡r { ❆ , ❇ } ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥ ❙✐♠✐❧❛r r❡❛s♦♥✐♥❣ ✐❢ ✇❡ ❦♥♦✇ {¬ ❇ } ❤♦❧❞s ❆r❝✲ ❈♦♥s✐st❡♥❝②

  7. ❈♦♥s✐st❡♥❝② ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❈♦♥s✐st❡♥❝② ▼❡t❤♦❞s ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✐♥❢❡r❡♥❝❡ ❙♦❧✉t✐♦♥ ❚❡❝❤♥✐q✉❡s ❛r❝✱ ♣❛t❤ ❛♥❞ ✐✲❝♦♥s✐st❡♥❝② ❈♦♥s✐st❡♥❝② ●❡♥❡r❛t❡ t✐❣❤t❡r ♥❡t✇♦r❦s ❊♥❢♦r❝✐♥❣ ❛♥❞ P❛rt✐❛❧ ❛ss✐❣♥♠❡♥ts ❝❛♥ ❜❡ ❞✐s❝❛r❞❡❞ ❡❛r❧✐❡r ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥ ❆r❝✲ ❈♦♥s✐st❡♥❝②

  8. ❊①❛♠♣❧❡ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❊①❛♠♣❧❡ ✭❈♦♥s✐st❡♥❝②✮ ❙♦❧✉t✐♦♥ ♥✲◗✉❡❡♥ ♣r♦❜❧❡♠ ❚❡❝❤♥✐q✉❡s ❈♦♥s✐st❡♥❝② ▼✐♥✐♠❛❧ ♥❡t✇♦r❦ ✐s t✐❣❤t❡r t❤❛♥ ♦r✐❣✐♥❛❧ ♥❡t✇♦r❦ ❊♥❢♦r❝✐♥❣ ❛♥❞ ❖♥ ♠✐♥✐♠❛❧ ♥❡t✇♦r❦ ✜♥❞✐♥❣ t❤❡ s♦❧✉t✐♦♥ ✐s ❡❛s✐❡r ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥ ❆r❝✲ ❈♦♥s✐st❡♥❝②

  9. ❚✐❣❤t♥❡ss ❛♥❞ s❡❛r❝❤ s♣❛❝❡ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ r❡str✐❝t✐♥❣ t❤❡ s❡❛r❝❤s♣❛❝❡ ❈♦♥s✐st❡♥❝② ●✐✈❡♥ t✇♦ ❡q✉✐✈❛❧❡♥t ♥❡t✇♦r❦ R ❛♥❞ R ′ ✐❢ R ′ ⊂ R ✭ R ′ ✐s t✐❣❤t❡r t❤❛♥ R ✮ ❙♦❧✉t✐♦♥ ❚❡❝❤♥✐q✉❡s t❤❡♥ s❡❛r❝❤✐♥❣ ❢♦r ❛ s♦❧✉t✐♦♥ ♦♥ R ′ ✐s ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ s❡r❛❝❤✐♥❣ ♦♥ R ❈♦♥str❛✐♥t R ′ ❤❛s ❛ s♠❛❧❧❡r s❡❛r❝❤ s♣❛❝❡ Pr♦♣❛❣❛t✐♦♥ ❆r❝✲ ❈♦♥s✐st❡♥❝②

  10. ❈♦♠♣❧❡t❡ ✐♥❢❡r❡♥❝❡ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥✿ ✜♥❞✐♥❣ s♦❧✉t✐♦♥ ✇✐t❤ ♥♦ ❞❡❛❞ ❡♥❞ ◆♦❞❡ ❛♥❞ ❆r❝ ❈♦♥s✐st❡♥❝② ❲❡ ❝❛♥ ❞❡❞✉❝❡ ❝♦♥str❛✐♥ts ✉♥t✐❧✿ ❛♥ ✐♥❝♦♥s✐st❡♥❝② ✐s ❢♦✉♥❞ ♦r ✇❡ ❝❛♥ ❞❡r✐✈❡ ❛ s♦❧✉t✐♦♥ ✇✐t❤ ❞❡♣t❤✲✜rst ❛♥❞ ♥♦ ❙♦❧✉t✐♦♥ ❚❡❝❤♥✐q✉❡s ❜❛❝❦tr❛❝❦✐♥❣ ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❜✉t ✇❡ ♠✐❣❤t ♥❡❡❞ t♦ ✐♥tr♦❞✉❝❡ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ♥✉♠❜❡r ♦❢ ❛♥❞ ❈♦♥str❛✐♥t ❝♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥ ✉s✉❛❧❧② ✐t ✐s ♣r❡❢❡r❛❜❧❡ t♦ ✐♥tr♦❞✉❝❡ ❛ ❜♦✉♥❞❡❞ ❛♠♦✉♥t ♦❢ ❆r❝✲ ❈♦♥s✐st❡♥❝② ❝♦♥str❛✐♥ts

  11. ❈♦♥s✐st❡♥❝② ❛♣♣r♦❛❝❤❡s ❈♦♥s✐st❡♥❝② ❊♥❢♦r❝✐♥❣ ❛♥❞ ❈♦♥str❛✐♥t ❝♦♥s✐st❡♥❝② ❡♥❢♦r❝✐♥❣ Pr♦♣❛❣❛t✐♦♥✿ ◆♦❞❡ ❛♥❞ ❆r❝ ●✐✈❡♥ ❛ ♣❛rt✐❛❧ s♦❧✉t✐♦♥ ♦❢ ❧❡♥❣t❤ ✐ − ✶ ✇❡ ❡①t❡♥❞ t❤❡ ❈♦♥s✐st❡♥❝② s♦❧✉t✐♦♥ t♦ ♦♥❡ ♠♦r❡ ✈❛r✐❛❜❧❡ ❈♦♥s✐st❡♥❝② ❡♥❢♦r❝✐♥❣✿ ❙♦❧✉t✐♦♥ ❚❡❝❤♥✐q✉❡s ❛♥② ♣❛rt✐❛❧ s♦❧✉t✐♦♥ ♦❢ ❛ s✉❜♥❡t✇♦r❦ ❡①t❡♥s✐❜❧❡ t♦ ❛ ❈♦♥s✐st❡♥❝② s✉rr♦✉♥❞✐♥❣ ♥❡t✇♦r❦ ❊♥❢♦r❝✐♥❣ s✐③❡ ♦❢ t❤❡ s✉❜♥❡t✇♦r❦ ❞❡✜♥❡s ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s ❛♥❞ ❈♦♥str❛✐♥t Pr♦♣❛❣❛t✐♦♥ ❆r❝✲❈♦♥s✐st❡♥❝②✿ ❢r♦♠ ✶ ✈❛r✐❛❜❧❡ t♦ ✷ ❆r❝✲ P❛t❤✲❈♦♥s✐st❡♥❝②✿ ❢r♦♠ ✷ ✈❛r✐❛❜❧❡s t♦ ✸ ❈♦♥s✐st❡♥❝② ■✲❝♦♥s✐st❡♥❝②✿ ❢r♦♠ ✐✲✶ t♦ ✐

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