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r sst r r rt t t r rr rt

  1. ●❡♥❡r❛❧ ■♠♣♦ss✐❜✐❧✐t② ♦❢ ●r♦✉♣ ❍♦♠♦♠♦r♣❤✐❝ ❊♥❝r②♣t✐♦♥ ✐♥ t❤❡ ◗✉❛♥t✉♠ ❲♦r❧❞ ❋r❡❞❡r✐❦ ❆r♠❦♥❡❝❤t ❚♦♠♠❛s♦ ●❛❣❧✐❛r❞♦♥✐ ❙t❡❢❛♥ ❑❛t③❡♥❜❡✐ss❡r ❆♥❞r❡❛s P❡t❡r P❑❈ ✷✵✶✹✱ ▼❛r❝❤ ✷✽t❤ ❇✉❡♥♦s ❆✐r❡s✱ ❆r❣❡♥t✐♥❛ ✶

  2. ◆♦✇ ❝♦♥s✐❞❡r t✇♦ ♣❧❛✐♥t❡①ts ♠ ✶ ♠ ✷ ✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡✐r ❡♥❝r②♣t✐♦♥s✿ ❝ ✶ ❊♥❝ ♠ ✶ ❝ ✷ ❊♥❝ ♠ ✷ ❉❡❝ ♠ ❡ ✶ ♠ ❡ ❉❡❝ ♠ ✶ ♠ ✷ ❡ ❉❡❝ ❝ ✶ ❝ ✷ ✷ ♠ ✶ ♠ ✷ ❡❞ ♠♦❞ ◆ ♠ ✶ ♠ ✷ ✳ ■♥ t❤✐s ❝❛s❡✱ ❞❡❝r②♣t✐♦♥ ✐s ❛ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠✳ ❆♥ ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❜❛s✐❝✱ ✉♥♣❛❞❞❡❞ ❘❙❆✿ • ❧❡t ◆ = ♣q ❢♦r ❧❛r❣❡ ♣r✐♠❡s ♣ ❛♥❞ q ✱ ❝♦♥s✐❞❡r ❣r♦✉♣ ( Z ∗ ♥ , · ) • ♣✉❜❧✐❝ ❡①♣♦♥❡♥t ❡ s✳t✳ ❣❝❞ ( ❡ , φ ( ◆ )) = ✶ • s❡❝r❡t ❡①♣♦♥❡♥t ❞ = ❡ − ✶ ♠♦❞ φ ( ◆ ) • ❊♥❝ ( ♠ ) = ♠ ❡ ♠♦❞ ◆ ❢♦r ♣❧❛✐♥t❡①t ♠ • ❉❡❝ ( ❝ ) = ❝ ❞ ♠♦❞ ◆ ❢♦r ❝✐♣❤❡rt❡①t ❝ ✳ ✷

  3. ■♥ t❤✐s ❝❛s❡✱ ❞❡❝r②♣t✐♦♥ ✐s ❛ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠✳ ❆♥ ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❜❛s✐❝✱ ✉♥♣❛❞❞❡❞ ❘❙❆✿ • ❧❡t ◆ = ♣q ❢♦r ❧❛r❣❡ ♣r✐♠❡s ♣ ❛♥❞ q ✱ ❝♦♥s✐❞❡r ❣r♦✉♣ ( Z ∗ ♥ , · ) • ♣✉❜❧✐❝ ❡①♣♦♥❡♥t ❡ s✳t✳ ❣❝❞ ( ❡ , φ ( ◆ )) = ✶ • s❡❝r❡t ❡①♣♦♥❡♥t ❞ = ❡ − ✶ ♠♦❞ φ ( ◆ ) • ❊♥❝ ( ♠ ) = ♠ ❡ ♠♦❞ ◆ ❢♦r ♣❧❛✐♥t❡①t ♠ • ❉❡❝ ( ❝ ) = ❝ ❞ ♠♦❞ ◆ ❢♦r ❝✐♣❤❡rt❡①t ❝ ✳ ◆♦✇ ❝♦♥s✐❞❡r t✇♦ ♣❧❛✐♥t❡①ts ♠ ✶ , ♠ ✷ ✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡✐r ❡♥❝r②♣t✐♦♥s✿ • ❝ ✶ = ❊♥❝ ( ♠ ✶ ) , ❝ ✷ = ❊♥❝ ( ♠ ✷ ) • ❉❡❝ ( ❝ ✶ · ❝ ✷ ) = ❉❡❝ ( ♠ ❡ ✶ · ♠ ❡ ✷ ) = ❉❡❝ (( ♠ ✶ · ♠ ✷ ) ❡ ) = ( ♠ ✶ · ♠ ✷ ) ❡❞ ♠♦❞ ◆ = ♠ ✶ · ♠ ✷ ✳ ✷

  4. ❆♥ ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❜❛s✐❝✱ ✉♥♣❛❞❞❡❞ ❘❙❆✿ • ❧❡t ◆ = ♣q ❢♦r ❧❛r❣❡ ♣r✐♠❡s ♣ ❛♥❞ q ✱ ❝♦♥s✐❞❡r ❣r♦✉♣ ( Z ∗ ♥ , · ) • ♣✉❜❧✐❝ ❡①♣♦♥❡♥t ❡ s✳t✳ ❣❝❞ ( ❡ , φ ( ◆ )) = ✶ • s❡❝r❡t ❡①♣♦♥❡♥t ❞ = ❡ − ✶ ♠♦❞ φ ( ◆ ) • ❊♥❝ ( ♠ ) = ♠ ❡ ♠♦❞ ◆ ❢♦r ♣❧❛✐♥t❡①t ♠ • ❉❡❝ ( ❝ ) = ❝ ❞ ♠♦❞ ◆ ❢♦r ❝✐♣❤❡rt❡①t ❝ ✳ ◆♦✇ ❝♦♥s✐❞❡r t✇♦ ♣❧❛✐♥t❡①ts ♠ ✶ , ♠ ✷ ✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡✐r ❡♥❝r②♣t✐♦♥s✿ • ❝ ✶ = ❊♥❝ ( ♠ ✶ ) , ❝ ✷ = ❊♥❝ ( ♠ ✷ ) • ❉❡❝ ( ❝ ✶ · ❝ ✷ ) = ❉❡❝ ( ♠ ❡ ✶ · ♠ ❡ ✷ ) = ❉❡❝ (( ♠ ✶ · ♠ ✷ ) ❡ ) = ( ♠ ✶ · ♠ ✷ ) ❡❞ ♠♦❞ ◆ = ♠ ✶ · ♠ ✷ ✳ ■♥ t❤✐s ❝❛s❡✱ ❞❡❝r②♣t✐♦♥ ✐s ❛ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠✳ ✷

  5. t❤❡ ♣❧❛✐♥t❡①t s♣❛❝❡ ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ t❤❡ s❡t ♦❢ ❡♥❝r②♣t✐♦♥s ❊♥❝ ♣❦ ♠ r ♠ r ❘♥❞ ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ t❤❡ ❞❡❝r②♣t✐♦♥ ✐s ❛ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠✿ ❉❡❝ s❦ ❝ ✶ ❝ ✷ ❉❡❝ s❦ ❝ ✶ ❉❡❝ s❦ ❝ ✷ ✱ ❢♦r ❡✈❡r② ❝ ✶ ❝ ✷ ✳ ✭❢r♦♠ ♥♦✇ ♦♥ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ❆❜❡❧✐❛♥ ❣r♦✉♣s✮ ●r♦✉♣ ❍♦♠♦♠♦r♣❤✐❝ ❊♥❝r②♣t✐♦♥ ✭●❍❊✮ ❆ ♣✉❜❧✐❝✲❦❡② ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ E = ( ❑❡②●❡♥ , ❊♥❝ , ❉❡❝ ) ✐s ❝❛❧❧❡❞ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐❝ ✐❢✱ ❢♦r ❛♥② ( ♣❦ , s❦ ) ← ❑❡②❣❡♥ ( λ ) ✿ ✸

  6. t❤❡ s❡t ♦❢ ❡♥❝r②♣t✐♦♥s ❊♥❝ ♣❦ ♠ r ♠ r ❘♥❞ ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ t❤❡ ❞❡❝r②♣t✐♦♥ ✐s ❛ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠✿ ❉❡❝ s❦ ❝ ✶ ❝ ✷ ❉❡❝ s❦ ❝ ✶ ❉❡❝ s❦ ❝ ✷ ✱ ❢♦r ❡✈❡r② ❝ ✶ ❝ ✷ ✳ ✭❢r♦♠ ♥♦✇ ♦♥ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ❆❜❡❧✐❛♥ ❣r♦✉♣s✮ ●r♦✉♣ ❍♦♠♦♠♦r♣❤✐❝ ❊♥❝r②♣t✐♦♥ ✭●❍❊✮ ❆ ♣✉❜❧✐❝✲❦❡② ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ E = ( ❑❡②●❡♥ , ❊♥❝ , ❉❡❝ ) ✐s ❝❛❧❧❡❞ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐❝ ✐❢✱ ❢♦r ❛♥② ( ♣❦ , s❦ ) ← ❑❡②❣❡♥ ( λ ) ✿ • t❤❡ ♣❧❛✐♥t❡①t s♣❛❝❡ P ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ ⊗ ✸

  7. t❤❡ ❞❡❝r②♣t✐♦♥ ✐s ❛ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠✿ ❉❡❝ s❦ ❝ ✶ ❝ ✷ ❉❡❝ s❦ ❝ ✶ ❉❡❝ s❦ ❝ ✷ ✱ ❢♦r ❡✈❡r② ❝ ✶ ❝ ✷ ✳ ✭❢r♦♠ ♥♦✇ ♦♥ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ❆❜❡❧✐❛♥ ❣r♦✉♣s✮ ●r♦✉♣ ❍♦♠♦♠♦r♣❤✐❝ ❊♥❝r②♣t✐♦♥ ✭●❍❊✮ ❆ ♣✉❜❧✐❝✲❦❡② ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ E = ( ❑❡②●❡♥ , ❊♥❝ , ❉❡❝ ) ✐s ❝❛❧❧❡❞ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐❝ ✐❢✱ ❢♦r ❛♥② ( ♣❦ , s❦ ) ← ❑❡②❣❡♥ ( λ ) ✿ • t❤❡ ♣❧❛✐♥t❡①t s♣❛❝❡ P ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ ⊗ • t❤❡ s❡t ♦❢ ❡♥❝r②♣t✐♦♥s C := � � ❊♥❝ ♣❦ ( ♠ ; r ) | ♠ ∈ P , r ∈ ❘♥❞ ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ ⋆ ✸

  8. ✭❢r♦♠ ♥♦✇ ♦♥ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ❆❜❡❧✐❛♥ ❣r♦✉♣s✮ ●r♦✉♣ ❍♦♠♦♠♦r♣❤✐❝ ❊♥❝r②♣t✐♦♥ ✭●❍❊✮ ❆ ♣✉❜❧✐❝✲❦❡② ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ E = ( ❑❡②●❡♥ , ❊♥❝ , ❉❡❝ ) ✐s ❝❛❧❧❡❞ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐❝ ✐❢✱ ❢♦r ❛♥② ( ♣❦ , s❦ ) ← ❑❡②❣❡♥ ( λ ) ✿ • t❤❡ ♣❧❛✐♥t❡①t s♣❛❝❡ P ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ ⊗ • t❤❡ s❡t ♦❢ ❡♥❝r②♣t✐♦♥s C := � � ❊♥❝ ♣❦ ( ♠ ; r ) | ♠ ∈ P , r ∈ ❘♥❞ ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ ⋆ • t❤❡ ❞❡❝r②♣t✐♦♥ ✐s ❛ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠✿ ❉❡❝ s❦ ( ❝ ✶ ⋆ ❝ ✷ ) = ❉❡❝ s❦ ( ❝ ✶ ) ⊗ ❉❡❝ s❦ ( ❝ ✷ ) ✱ ❢♦r ❡✈❡r② ❝ ✶ , ❝ ✷ ∈ C ✳ ✸

  9. ●r♦✉♣ ❍♦♠♦♠♦r♣❤✐❝ ❊♥❝r②♣t✐♦♥ ✭●❍❊✮ ❆ ♣✉❜❧✐❝✲❦❡② ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ E = ( ❑❡②●❡♥ , ❊♥❝ , ❉❡❝ ) ✐s ❝❛❧❧❡❞ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐❝ ✐❢✱ ❢♦r ❛♥② ( ♣❦ , s❦ ) ← ❑❡②❣❡♥ ( λ ) ✿ • t❤❡ ♣❧❛✐♥t❡①t s♣❛❝❡ P ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ ⊗ • t❤❡ s❡t ♦❢ ❡♥❝r②♣t✐♦♥s C := � � ❊♥❝ ♣❦ ( ♠ ; r ) | ♠ ∈ P , r ∈ ❘♥❞ ✐s ❛ ❣r♦✉♣ ✐♥ r❡s♣❡❝t t♦ ⋆ • t❤❡ ❞❡❝r②♣t✐♦♥ ✐s ❛ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠✿ ❉❡❝ s❦ ( ❝ ✶ ⋆ ❝ ✷ ) = ❉❡❝ s❦ ( ❝ ✶ ) ⊗ ❉❡❝ s❦ ( ❝ ✷ ) ✱ ❢♦r ❡✈❡r② ❝ ✶ , ❝ ✷ ∈ C ✳ ✭❢r♦♠ ♥♦✇ ♦♥ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ❆❜❡❧✐❛♥ ❣r♦✉♣s✮ ✸

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