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Prrs r t ss ts ss

  1. Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❢♦r P❛✐❧❧✐❡r ❊♥❝r②♣t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❙❡❝✉r❡ ❙t❛t✐st✐❝❛❧ ❆♥❛❧②s✐s ❇❛rt ▼❡♥♥✐♥❦ ✭❑✳❯✳▲❡✉✈❡♥✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤✿ ❏♦r❣❡ ●✉❛❥❛r❞♦ ✭P❤✐❧✐♣s ❘❡s❡❛r❝❤ ▲❛❜s✮ ❇❡rr② ❙❝❤♦❡♥♠❛❦❡rs ✭❚❯ ❊✐♥❞❤♦✈❡♥✮ ❋✐♥❛♥❝✐❛❧ ❈r②♣t♦❣r❛♣❤② ✬✶✵✱ ❚❡♥❡r✐❢❡✱ ❙♣❛✐♥ ❏❛♥✉❛r② ✷✺✱ ✷✵✶✵ ✶ ✴ ✶✸

  2. Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❖✈❡r✈✐❡✇ Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ✷ ✴ ✶✸

  3. ✲ ❍♦♠♦♠♦r♣❤✐❝ ♣r♦♣❡rt✐❡s✿ ❆❞❞✐t✐♦♥✿ ① ② ① ② ① ❝ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❝♦♥st❛♥t✿ ①❝ ❘❡✲r❛♥❞♦♠✐③❛t✐♦♥✿ ① ✵ ① ✲ t ♥ ✲t❤r❡s❤♦❧❞ ❞❡❝r②♣t✐♦♥ Pr✐✈❛t❡ ❦❡② ✐s s❤❛r❡❞ ❛♠♦♥❣ ♥ ♣❛rt✐❡s s✉❝❤ t❤❛t ❛♥② t ❝❛♥ ❞❡❝r②♣t ✲ ❲❡ ✉s❡ t❤❡ P❛✐❧❧✐❡r ❝r②♣t♦s②st❡♠ ❬P❛✐✾✾✱ ❉❏✵✶❪ Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❚❤r❡s❤♦❧❞ ❍♦♠♦♠♦r♣❤✐❝ ❈r②♣t♦s②st❡♠s ✭❚❍❈s✮ • � ① � ✐s ❛ ♣r♦❜❛❜✐❧✐st✐❝ ❡♥❝r②♣t✐♦♥ ✿ � ① � = ❊♥❝ ♣❦ ( ① , r ) ✉♥❞❡r ♣✉❜❧✐❝ ❦❡② ♣❦ ❛♥❞ ❢♦r r❛♥❞♦♠ r ✸ ✴ ✶✸

  4. ✲ ✲ t ♥ ✲t❤r❡s❤♦❧❞ ❞❡❝r②♣t✐♦♥ Pr✐✈❛t❡ ❦❡② ✐s s❤❛r❡❞ ❛♠♦♥❣ ♥ ♣❛rt✐❡s s✉❝❤ t❤❛t ❛♥② t ❝❛♥ ❞❡❝r②♣t ✲ ❲❡ ✉s❡ t❤❡ P❛✐❧❧✐❡r ❝r②♣t♦s②st❡♠ ❬P❛✐✾✾✱ ❉❏✵✶❪ Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❚❤r❡s❤♦❧❞ ❍♦♠♦♠♦r♣❤✐❝ ❈r②♣t♦s②st❡♠s ✭❚❍❈s✮ • � ① � ✐s ❛ ♣r♦❜❛❜✐❧✐st✐❝ ❡♥❝r②♣t✐♦♥ ✿ � ① � = ❊♥❝ ♣❦ ( ① , r ) ✉♥❞❡r ♣✉❜❧✐❝ ❦❡② ♣❦ ❛♥❞ ❢♦r r❛♥❞♦♠ r • ❍♦♠♦♠♦r♣❤✐❝ ♣r♦♣❡rt✐❡s✿ • ❆❞❞✐t✐♦♥✿ � ① �� ② � = � ① + ② � • ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❝♦♥st❛♥t✿ � ① � ❝ = � ①❝ � • ❘❡✲r❛♥❞♦♠✐③❛t✐♦♥✿ � ① �� ✵ � = � ① � ✸ ✴ ✶✸

  5. ✲ ✲ ✲ ❲❡ ✉s❡ t❤❡ P❛✐❧❧✐❡r ❝r②♣t♦s②st❡♠ ❬P❛✐✾✾✱ ❉❏✵✶❪ Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❚❤r❡s❤♦❧❞ ❍♦♠♦♠♦r♣❤✐❝ ❈r②♣t♦s②st❡♠s ✭❚❍❈s✮ • � ① � ✐s ❛ ♣r♦❜❛❜✐❧✐st✐❝ ❡♥❝r②♣t✐♦♥ ✿ � ① � = ❊♥❝ ♣❦ ( ① , r ) ✉♥❞❡r ♣✉❜❧✐❝ ❦❡② ♣❦ ❛♥❞ ❢♦r r❛♥❞♦♠ r • ❍♦♠♦♠♦r♣❤✐❝ ♣r♦♣❡rt✐❡s✿ • ❆❞❞✐t✐♦♥✿ � ① �� ② � = � ① + ② � • ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❝♦♥st❛♥t✿ � ① � ❝ = � ①❝ � • ❘❡✲r❛♥❞♦♠✐③❛t✐♦♥✿ � ① �� ✵ � = � ① � • ( t , ♥ ) ✲t❤r❡s❤♦❧❞ ❞❡❝r②♣t✐♦♥ • Pr✐✈❛t❡ ❦❡② ✐s s❤❛r❡❞ ❛♠♦♥❣ ♥ ♣❛rt✐❡s s✉❝❤ t❤❛t ❛♥② t ❝❛♥ ❞❡❝r②♣t ✸ ✴ ✶✸

  6. ✲ ✲ ✲ Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❚❤r❡s❤♦❧❞ ❍♦♠♦♠♦r♣❤✐❝ ❈r②♣t♦s②st❡♠s ✭❚❍❈s✮ • � ① � ✐s ❛ ♣r♦❜❛❜✐❧✐st✐❝ ❡♥❝r②♣t✐♦♥ ✿ � ① � = ❊♥❝ ♣❦ ( ① , r ) ✉♥❞❡r ♣✉❜❧✐❝ ❦❡② ♣❦ ❛♥❞ ❢♦r r❛♥❞♦♠ r • ❍♦♠♦♠♦r♣❤✐❝ ♣r♦♣❡rt✐❡s✿ • ❆❞❞✐t✐♦♥✿ � ① �� ② � = � ① + ② � • ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❝♦♥st❛♥t✿ � ① � ❝ = � ①❝ � • ❘❡✲r❛♥❞♦♠✐③❛t✐♦♥✿ � ① �� ✵ � = � ① � • ( t , ♥ ) ✲t❤r❡s❤♦❧❞ ❞❡❝r②♣t✐♦♥ • Pr✐✈❛t❡ ❦❡② ✐s s❤❛r❡❞ ❛♠♦♥❣ ♥ ♣❛rt✐❡s s✉❝❤ t❤❛t ❛♥② t ❝❛♥ ❞❡❝r②♣t • ❲❡ ✉s❡ t❤❡ P❛✐❧❧✐❡r ❝r②♣t♦s②st❡♠ ❬P❛✐✾✾✱ ❉❏✵✶❪ ✸ ✴ ✶✸

  7. ❆♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ❛r✐t❤♠❡t✐❝ ❝✐r❝✉✐ts ❈✐r❝✉✐t ❢♦r ❢ ❝♦♥s✐sts ♦❢ s❡q✉❡♥t✐❛❧ ❡✈❛❧✉❛t✐♦♥s ♦❢ Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❙❡❝✉r❡ ❋✉♥❝t✐♦♥ ❊✈❛❧✉❛t✐♦♥ ❜❛s❡❞ ♦♥ ❚❍❈s • ❖♥ ✐♥♣✉t ♦❢ � ① ✶ � , . . . , � ① ▲ � ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ❢ ✱ t❤❡ ♣❛rt✐❡s ❥♦✐♥t❧② ❝♦♠♣✉t❡ � ❢ ( ① ✶ , . . . , ① ▲ ) � ✭❝✐r❝✉✐t ❢♦r ❢ ✮ � ① ✶ � , . . . , � ① ▲ � · · · · · · � ❢ ( ① ✶ , . . . , ① ▲ ) � − − − − − − − − − − − − − → · · · · · · · · · − − − − − − − − − − − − − → · · · · · · ✹ ✴ ✶✸

  8. Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❙❡❝✉r❡ ❋✉♥❝t✐♦♥ ❊✈❛❧✉❛t✐♦♥ ❜❛s❡❞ ♦♥ ❚❍❈s • ❖♥ ✐♥♣✉t ♦❢ � ① ✶ � , . . . , � ① ▲ � ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ❢ ✱ t❤❡ ♣❛rt✐❡s ❥♦✐♥t❧② ❝♦♠♣✉t❡ � ❢ ( ① ✶ , . . . , ① ▲ ) � ✭❝✐r❝✉✐t ❢♦r ❢ ✮ � ① ✶ � , . . . , � ① ▲ � · · · · · · � ❢ ( ① ✶ , . . . , ① ▲ ) � − − − − − − − − − − − − − → · · · · · · · · · − − − − − − − − − − − − − → · · · · · · • ❆♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ❛r✐t❤♠❡t✐❝ ❝✐r❝✉✐ts • ❈✐r❝✉✐t ❢♦r ❢ ❝♦♥s✐sts ♦❢ s❡q✉❡♥t✐❛❧ ❡✈❛❧✉❛t✐♦♥s ♦❢ (+ , − , ∗ , / ) ✹ ✴ ✶✸

  9. ✲ ❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ❡✣❝✐❡♥t ❣❛t❡ ❢♦r ① ♠♦❞ ❛ ❣✐✈❡♥ ① ❛ ■♠♣❧✐❡s ❛ ❣❛t❡ ❢♦r ✐♥t❡❣❡r ❞✐✈✐s✐♦♥ ① ❞✐✈ ❛ ✲ ❙❡✈❡r❛❧ ♦t❤❡r ❡✣❝✐❡♥t ❣❛t❡s✿ ❘❛♥❞♦♠ ❜✐t ❣❡♥❡r❛t✐♦♥ ❣❛t❡ ❬❈❉◆✵✶✱ ❙❚✵✻❪ ✿ ♦✉t♣✉ts r ❢♦r r❛♥❞♦♠ r ✵ ✶ ❈♦♠♣❛r✐s♦♥ ❣❛t❡ ❬❉❋❑ ✵✻✱ ●❙❱✵✼❪ ✿ ♦✉t♣✉ts ① ② ❣✐✈❡♥ t❤❡ ❡♥❝r②♣t❡❞ ❜✐ts ♦❢ ① ② ▲❡❛st s✐❣♥✐✜❝❛♥t ❜✐t ❣❛t❡ ❬❙❚✵✻❪ ✿ ♦✉t♣✉ts ① ♠♦❞ ✷ ❣✐✈❡♥ ① Pr❡❧✐♠✐♥❛r✐❡s ❙❡❝✉r❡ ▼♦❞✉❧♦ ❘❡❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ❆♥❛❧②s✐s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❙❡❝✉r❡ ❋✉♥❝t✐♦♥ ❊✈❛❧✉❛t✐♦♥ ❜❛s❡❞ ♦♥ ❚❍❈s ✭❝♦♥t✳✮ • ❆❞❞✐t✐♦♥ ❛♥❞ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❤♦♠♦♠♦r♣❤✐❝ ♣r♦♣❡rt✐❡s✿ ❝♦♠♣✉t❛t✐♦♥ ♦❢ � ① + ② � ❛♥❞ � ❝① � ❣✐✈❡♥ � ① � , � ② � , ❝ • ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❣❛t❡ ❬❈❉◆✵✶❪ ✿ ♦✉t♣✉ts � ①② � ❣✐✈❡♥ � ① � , � ② � ✺ ✴ ✶✸

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