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  1. ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥ ●❡r❣ö ❇❛r❛♥② ❣❡r❣♦❅❝♦♠♣❧❛♥❣✳t✉✇✐❡♥✳❛❝✳❛t ■♥st✐t✉t❡ ♦❢ ❈♦♠♣✉t❡r ▲❛♥❣✉❛❣❡s comp uter ❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣② lang uages ❑P❙ ✷✵✶✸ ❖❝t♦❜❡r ✶✱ ✷✵✶✸ ✶✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

  2. ❚❤✐s t❛❧❦ ❆ ❝♦♠♣✐❧❡r✳ ✳ ✳ ❈❛❧❧❡❞ ❢r♦♠ ✐♥t❡r♣r❡t❡❞ P②t❤♦♥ ♣r♦❣r❛♠s ❘✉♥♥✐♥❣ ✐♥ st❛♥❞❛r❞ P②t❤♦♥ ✐♥t❡r♣r❡t❡r ❈♦♠♣✐❧✐♥❣ ❝♦❞❡ t♦ r✉♥ ✐♥ st❛♥❞❛r❞ P②t❤♦♥ ✐♥t❡r♣r❡t❡r ✷✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

  3. ❆r❝❤✐t❡❝t✉r❡ ✭✶✴✸✮ ❵❋r♦♥t ❡♥❞✬✿ P②t❤♦♥ ❆❙❚ ❝♦♠♣✐❧❡r P②t❤♦♥ ❝♦♥✈❡♥✐❡♥❝❡ ❧❛②❡r ❇❛❝❦ ❡♥❞✿ ●◆❯ ❧✐❜❥✐t ✸✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

  4. ❊①❛♠♣❧❡✿ ❋✉♥❝t✐♦♥ ①②③ ① ② ③ ❞❡❢ ❝r❡❛t❡❴s✐❣♥❛t✉r❡✭❢✉♥❝✮✿ r❡t✉r♥ ❢✉♥❝✳s✐❣♥❛t✉r❡❴❤❡❧♣❡r✭❥✐t✳✐♥t✱ ❥✐t✳✐♥t✱ ❥✐t✳✐♥t✱ ❥✐t✳✐♥t✮ ❞❡❢ ❜✉✐❧❞✭❢✉♥❝✮✿ ①✱ ②✱ ③ ❂ ✭❢✉♥❝✳❣❡t❴♣❛r❛♠✭✐✮ ❢♦r ✐ ✐♥ r❛♥❣❡✭✸✮✮ ●◆❯ ❧✐❜❥✐t P♦rt❛❜❧❡ ❏■❚ ✭①✽✻✱ ❆❘▼✮✴✐♥t❡r♣r❡t❡r ❧✐❜r❛r② ❘■❙❈✲❧✐❦❡ ✐♥t❡r♠❡❞✐❛t❡ ❝♦❞❡ ❆P■✱ s✐♠♣❧❡ t②♣❡ s②st❡♠ P②t❤♦♥ ❜✐♥❞✐♥❣s ✇✐t❤ ❝♦♥✈❡♥✐❡♥t ♦♣❡r❛t♦r ♦✈❡r❧♦❛❞✐♥❣ ✹✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

  5. ●◆❯ ❧✐❜❥✐t P♦rt❛❜❧❡ ❏■❚ ✭①✽✻✱ ❆❘▼✮✴✐♥t❡r♣r❡t❡r ❧✐❜r❛r② ❘■❙❈✲❧✐❦❡ ✐♥t❡r♠❡❞✐❛t❡ ❝♦❞❡ ❆P■✱ s✐♠♣❧❡ t②♣❡ s②st❡♠ P②t❤♦♥ ❜✐♥❞✐♥❣s ✇✐t❤ ❝♦♥✈❡♥✐❡♥t ♦♣❡r❛t♦r ♦✈❡r❧♦❛❞✐♥❣ ❊①❛♠♣❧❡✿ ❋✉♥❝t✐♦♥ λ ①②③ . ( ① ∗ ② + ③ ) ❞❡❢ ❝r❡❛t❡❴s✐❣♥❛t✉r❡✭❢✉♥❝✮✿ r❡t✉r♥ ❢✉♥❝✳s✐❣♥❛t✉r❡❴❤❡❧♣❡r✭❥✐t✳✐♥t✱ ❥✐t✳✐♥t✱ ❥✐t✳✐♥t✱ ❥✐t✳✐♥t✮ ❞❡❢ ❜✉✐❧❞✭❢✉♥❝✮✿ ①✱ ②✱ ③ ❂ ✭❢✉♥❝✳❣❡t❴♣❛r❛♠✭✐✮ ❢♦r ✐ ✐♥ r❛♥❣❡✭✸✮✮ ♣r♦❞✉❝t ❂ ❢✉♥❝✳✐♥s♥❴♠✉❧✭①✱ ②✮ s✉♠ ❂ ❢✉♥❝✳✐♥s♥❴❛❞❞✭♣r♦❞✉❝t✱ ③✮ ❢✉♥❝✳✐♥s♥❴r❡t✉r♥✭s✉♠✮ ✹✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

  6. ●◆❯ ❧✐❜❥✐t P♦rt❛❜❧❡ ❏■❚ ✭①✽✻✱ ❆❘▼✮✴✐♥t❡r♣r❡t❡r ❧✐❜r❛r② ❘■❙❈✲❧✐❦❡ ✐♥t❡r♠❡❞✐❛t❡ ❝♦❞❡ ❆P■✱ s✐♠♣❧❡ t②♣❡ s②st❡♠ P②t❤♦♥ ❜✐♥❞✐♥❣s ✇✐t❤ ❝♦♥✈❡♥✐❡♥t ♦♣❡r❛t♦r ♦✈❡r❧♦❛❞✐♥❣ ❊①❛♠♣❧❡✿ ❋✉♥❝t✐♦♥ λ ①②③ . ( ① ∗ ② + ③ ) ❞❡❢ ❝r❡❛t❡❴s✐❣♥❛t✉r❡✭❢✉♥❝✮✿ r❡t✉r♥ ❢✉♥❝✳s✐❣♥❛t✉r❡❴❤❡❧♣❡r✭❥✐t✳✐♥t✱ ❥✐t✳✐♥t✱ ❥✐t✳✐♥t✱ ❥✐t✳✐♥t✮ ❞❡❢ ❜✉✐❧❞✭❢✉♥❝✮✿ ①✱ ②✱ ③ ❂ ✭❢✉♥❝✳❣❡t❴♣❛r❛♠✭✐✮ ❢♦r ✐ ✐♥ r❛♥❣❡✭✸✮✮ ❢✉♥❝✳✐♥s♥❴r❡t✉r♥✭① ✯ ② ✰ ③✮ ✹✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

  7. ♥ ❂ ❢✉♥❝✳❣❡t❴♣❛r❛♠✭✵✮ ♦♥❡ ❂ ❢✉♥❝✳♥❡✇❴❝♦♥st❛♥t✭✶✱ ❥✐t✳❚②♣❡✳✐♥t✮ t✇♦ ❂ ❢✉♥❝✳♥❡✇❴❝♦♥st❛♥t✭✷✱ ❥✐t✳❚②♣❡✳✐♥t✮ r❡t✉r♥❴❧❛❜❡❧ ❂ ❢✉♥❝✳♥❡✇❴❧❛❜❡❧✭✮ ❢✉♥❝✳✐♥s♥❴❜r❛♥❝❤❴✐❢✭♥ ❁ t✇♦✱ r❡t✉r♥❴❧❛❜❡❧✮ ❢✐❜❴❢✉♥❝ ❂ ❢✉♥❝ ❢✐❜❴s✐❣ ❂ ❢✉♥❝✳❝r❡❛t❡❴s✐❣♥❛t✉r❡✭✮ ❛ ❂ ❢✉♥❝✳✐♥s♥❴❝❛❧❧✭✬❢✐❜✬✱ ❢✐❜❴❢✉♥❝✱ ❢✐❜❴s✐❣✱ ❬♥✲♦♥❡❪✮ ❜ ❂ ❢✉♥❝✳✐♥s♥❴❝❛❧❧✭✬❢✐❜✬✱ ❢✐❜❴❢✉♥❝✱ ❢✐❜❴s✐❣✱ ❬♥✲t✇♦❪✮ ❢✉♥❝✳✐♥s♥❴r❡t✉r♥✭❛ ✰ ❜✮ ❢✉♥❝✳✐♥s♥❴❧❛❜❡❧✭r❡t✉r♥❴❧❛❜❡❧✮ ❢✉♥❝✳✐♥s♥❴r❡t✉r♥✭♥✮ ■♥✈✐s✐❜❧❡ ❝♦♥tr♦❧ ✢♦✇✦ ●◆❯ ❧✐❜❥✐t✿ ❱❡r❜♦s❡ ❝♦❞❡ ❞❡❢ ❜✉✐❧❞✭❢✉♥❝✮✿ ★ ✈❛❧✉❡s✿ ♥✱ ✶✱ ✷ ★ ✐❢ ♥ ❁ ✷✿ ❣♦t♦ r❡t✉r♥❴❧❛❜❡❧ ★ r❡t✉r♥ ✜❜✭♥✲✶✮ ✰ ✜❜✭♥✲✷✮ ★ r❡t✉r♥❴❧❛❜❡❧✿ r❡t✉r♥ ♥ ✺✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

  8. ●◆❯ ❧✐❜❥✐t✿ ❱❡r❜♦s❡ ❝♦❞❡ ❞❡❢ ❜✉✐❧❞✭❢✉♥❝✮✿ ★ ✈❛❧✉❡s✿ ♥✱ ✶✱ ✷ ♥ ❂ ❢✉♥❝✳❣❡t❴♣❛r❛♠✭✵✮ ♦♥❡ ❂ ❢✉♥❝✳♥❡✇❴❝♦♥st❛♥t✭✶✱ ❥✐t✳❚②♣❡✳✐♥t✮ t✇♦ ❂ ❢✉♥❝✳♥❡✇❴❝♦♥st❛♥t✭✷✱ ❥✐t✳❚②♣❡✳✐♥t✮ ★ ✐❢ ♥ ❁ ✷✿ ❣♦t♦ r❡t✉r♥❴❧❛❜❡❧ r❡t✉r♥❴❧❛❜❡❧ ❂ ❢✉♥❝✳♥❡✇❴❧❛❜❡❧✭✮ ❢✉♥❝✳✐♥s♥❴❜r❛♥❝❤❴✐❢✭♥ ❁ t✇♦✱ r❡t✉r♥❴❧❛❜❡❧✮ ★ r❡t✉r♥ ✜❜✭♥✲✶✮ ✰ ✜❜✭♥✲✷✮ ❢✐❜❴❢✉♥❝ ❂ ❢✉♥❝ ❢✐❜❴s✐❣ ❂ ❢✉♥❝✳❝r❡❛t❡❴s✐❣♥❛t✉r❡✭✮ ❛ ❂ ❢✉♥❝✳✐♥s♥❴❝❛❧❧✭✬❢✐❜✬✱ ❢✐❜❴❢✉♥❝✱ ❢✐❜❴s✐❣✱ ❬♥✲♦♥❡❪✮ ❜ ❂ ❢✉♥❝✳✐♥s♥❴❝❛❧❧✭✬❢✐❜✬✱ ❢✐❜❴❢✉♥❝✱ ❢✐❜❴s✐❣✱ ❬♥✲t✇♦❪✮ ❢✉♥❝✳✐♥s♥❴r❡t✉r♥✭❛ ✰ ❜✮ ★ r❡t✉r♥❴❧❛❜❡❧✿ r❡t✉r♥ ♥ ❢✉♥❝✳✐♥s♥❴❧❛❜❡❧✭r❡t✉r♥❴❧❛❜❡❧✮ ❢✉♥❝✳✐♥s♥❴r❡t✉r♥✭♥✮ ■♥✈✐s✐❜❧❡ ❝♦♥tr♦❧ ✢♦✇✦ ✺✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

  9. ●◆❯ ❧✐❜❥✐t✿ ❇♦✐❧❡r♣❧❛t❡ ❝❧❛ss ❢✐❜❴❢✉♥❝t✐♦♥✭❥✐t✳❋✉♥❝t✐♦♥✮✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ❝♦♥t❡①t✮✿ s✉♣❡r✭✮✳❴❴✐♥✐t❴❴✭❝♦♥t❡①t✮ s❡❧❢✳❝r❡❛t❡✭✮ ❞❡❢ ❝r❡❛t❡❴s✐❣♥❛t✉r❡✭s❡❧❢✮✿ r❡t✉r♥ s❡❧❢✳s✐❣♥❛t✉r❡❴❤❡❧♣❡r✭❥✐t✳✐♥t✱ ❥✐t✳✐♥t✮ ❞❡❢ ❜✉✐❧❞✭❢✉♥❝✮✿ ✳✳✳ ❢✐❜ ❂ ❢✐❜❴❢✉♥❝t✐♦♥✭❝♦♥t❡①t✮ ❢✐❜✳❴❴♥❛♠❡❴❴ ❂ ✬❢✐❜✬ ▼♦st❧② ❜♦r✐♥❣ st✉✛✳ ✻✴✷✵ ●❡r❣ö ❇❛r❛♥② ✭❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✮ ♣②❧✐❜❥✐t ✿ ❆ ❏■❚ ❈♦♠♣✐❧❡r ▲✐❜r❛r② ❢♦r P②t❤♦♥

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