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  1. ❚❤❡ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ❍✐❣❤❡r s❝❛❧❛r ❢♦r♠✉❧❛s ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠✉❧❛s ❆♣♣❧✐❝❛t✐♦♥s ❍✐❣❤❡r ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛s ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❘✳ ❙♦✉❛♠✮ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❚♦✉❧♦✉s❡ ❯♥✐✈❡rs✐té ❚♦✉❧♦✉s❡ ■■■ ✭❱✐s✐t✐♥❣ ❯✳ ❖tt❛✇❛✮ ❤tt♣✿✴✴✇✇✇✳♠❛t❤✳✉♣s✲t❧s❡✳❢r✴✄s❝❤❧❡♥❦❡r ❈♦r♥❡❧❧ ❯✳✱ ❆♣r✳ ✷✽✱ ✷✵✵✽ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❍✐❣❤❡r ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛s

  2. ❚❤❡ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ❍✐❣❤❡r s❝❛❧❛r ❢♦r♠✉❧❛s ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠✉❧❛s ❆♣♣❧✐❝❛t✐♦♥s ❚❤❡ ❝❧❛ss✐❝❛❧ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ▼ ♥ + ✶ = s♣❛❝❡ ♦❢ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡ ❑ ✿ ❊✉❝❧✐❞❡❛♥✱ s♣❤❡r✐❝❛❧✱ ❤②♣❡r❜♦❧✐❝✳ ❑ ❚❤♠ ✭❙❝❤❧ä✢✐✱ ✶✽✺✵✮✳ ▲❡t P ⊂ ▼ ♥ + ✶ ❜❡ ❛ ✭❝♦♥✈❡①✮ ♣♦❧②❤❡❞r♦♥✳ ■♥ ❛♥② ❑ ❞❡❢♦r♠❛t✐♦♥ ♦❢ P ✱ � ♥❑❞❱ = − ❱ ❡ ❞ θ ❡ , ❡ s✉♠ ♦✈❡r ❝♦❞✐♠ ✷ ❢❛❝❡s ♦❢ P ✱ θ ❡ = ❡①t❡r✐♦r ❞✐❤❡❞r❛❧ ❛♥❣❧❡✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥❝❧✉❞❡ ✿ ❤②♣❡r❜♦❧✐❝ ♣♦❧②❤❡❞r❛ ✿ ❆♥❞r❡❡✈✬s t❤❡♦r❡♠ ♦♥ ❞✐❤❡❞r❛❧ ❛♥❣❧❡s ♦❢ ✐❞❡❛❧ ♣♦❧②❤❡❞r❛ ✭❚❤✉rst♦♥✱ ❇rä❣❣❡r✱ ❘✐✈✐♥ ❛♣♣r♦❛❝❤✮✳ ❤②♣❡r❜♦❧✐❝ ✸✲♠❛♥✐❢♦❧❞s ✿ r✐❣✐❞✐t②✱ ❉❡❤♥ s✉r❣❡r② ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❝✉s♣s ❀ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❤②♣❡r❜♦❧✐❝ ❝♦♥❡✲♠❛♥✐❢♦❧❞s✱ ❡t❝✳ ❘❡❣❣❡ ❝❛❧❝✉❧✉s✱ ❡t❝✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❍✐❣❤❡r ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛s

  3. ❚❤❡ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ❍✐❣❤❡r s❝❛❧❛r ❢♦r♠✉❧❛s ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠✉❧❛s ❆♣♣❧✐❝❛t✐♦♥s ❚❤❡ ❝❧❛ss✐❝❛❧ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ▼ ♥ + ✶ = s♣❛❝❡ ♦❢ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡ ❑ ✿ ❊✉❝❧✐❞❡❛♥✱ s♣❤❡r✐❝❛❧✱ ❤②♣❡r❜♦❧✐❝✳ ❑ ❚❤♠ ✭❙❝❤❧ä✢✐✱ ✶✽✺✵✮✳ ▲❡t P ⊂ ▼ ♥ + ✶ ❜❡ ❛ ✭❝♦♥✈❡①✮ ♣♦❧②❤❡❞r♦♥✳ ■♥ ❛♥② ❑ ❞❡❢♦r♠❛t✐♦♥ ♦❢ P ✱ � ♥❑❞❱ = − ❱ ❡ ❞ θ ❡ , ❡ s✉♠ ♦✈❡r ❝♦❞✐♠ ✷ ❢❛❝❡s ♦❢ P ✱ θ ❡ = ❡①t❡r✐♦r ❞✐❤❡❞r❛❧ ❛♥❣❧❡✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥❝❧✉❞❡ ✿ ❤②♣❡r❜♦❧✐❝ ♣♦❧②❤❡❞r❛ ✿ ❆♥❞r❡❡✈✬s t❤❡♦r❡♠ ♦♥ ❞✐❤❡❞r❛❧ ❛♥❣❧❡s ♦❢ ✐❞❡❛❧ ♣♦❧②❤❡❞r❛ ✭❚❤✉rst♦♥✱ ❇rä❣❣❡r✱ ❘✐✈✐♥ ❛♣♣r♦❛❝❤✮✳ ❤②♣❡r❜♦❧✐❝ ✸✲♠❛♥✐❢♦❧❞s ✿ r✐❣✐❞✐t②✱ ❉❡❤♥ s✉r❣❡r② ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❝✉s♣s ❀ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❤②♣❡r❜♦❧✐❝ ❝♦♥❡✲♠❛♥✐❢♦❧❞s✱ ❡t❝✳ ❘❡❣❣❡ ❝❛❧❝✉❧✉s✱ ❡t❝✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❍✐❣❤❡r ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛s

  4. ❚❤❡ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ❍✐❣❤❡r s❝❛❧❛r ❢♦r♠✉❧❛s ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠✉❧❛s ❆♣♣❧✐❝❛t✐♦♥s ❚❤❡ ❝❧❛ss✐❝❛❧ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ▼ ♥ + ✶ = s♣❛❝❡ ♦❢ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡ ❑ ✿ ❊✉❝❧✐❞❡❛♥✱ s♣❤❡r✐❝❛❧✱ ❤②♣❡r❜♦❧✐❝✳ ❑ ❚❤♠ ✭❙❝❤❧ä✢✐✱ ✶✽✺✵✮✳ ▲❡t P ⊂ ▼ ♥ + ✶ ❜❡ ❛ ✭❝♦♥✈❡①✮ ♣♦❧②❤❡❞r♦♥✳ ■♥ ❛♥② ❑ ❞❡❢♦r♠❛t✐♦♥ ♦❢ P ✱ � ♥❑❞❱ = − ❱ ❡ ❞ θ ❡ , ❡ s✉♠ ♦✈❡r ❝♦❞✐♠ ✷ ❢❛❝❡s ♦❢ P ✱ θ ❡ = ❡①t❡r✐♦r ❞✐❤❡❞r❛❧ ❛♥❣❧❡✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥❝❧✉❞❡ ✿ ❤②♣❡r❜♦❧✐❝ ♣♦❧②❤❡❞r❛ ✿ ❆♥❞r❡❡✈✬s t❤❡♦r❡♠ ♦♥ ❞✐❤❡❞r❛❧ ❛♥❣❧❡s ♦❢ ✐❞❡❛❧ ♣♦❧②❤❡❞r❛ ✭❚❤✉rst♦♥✱ ❇rä❣❣❡r✱ ❘✐✈✐♥ ❛♣♣r♦❛❝❤✮✳ ❤②♣❡r❜♦❧✐❝ ✸✲♠❛♥✐❢♦❧❞s ✿ r✐❣✐❞✐t②✱ ❉❡❤♥ s✉r❣❡r② ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❝✉s♣s ❀ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❤②♣❡r❜♦❧✐❝ ❝♦♥❡✲♠❛♥✐❢♦❧❞s✱ ❡t❝✳ ❘❡❣❣❡ ❝❛❧❝✉❧✉s✱ ❡t❝✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❍✐❣❤❡r ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛s

  5. ❚❤❡ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ❍✐❣❤❡r s❝❛❧❛r ❢♦r♠✉❧❛s ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠✉❧❛s ❆♣♣❧✐❝❛t✐♦♥s ❚❤❡ ❝❧❛ss✐❝❛❧ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ▼ ♥ + ✶ = s♣❛❝❡ ♦❢ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡ ❑ ✿ ❊✉❝❧✐❞❡❛♥✱ s♣❤❡r✐❝❛❧✱ ❤②♣❡r❜♦❧✐❝✳ ❑ ❚❤♠ ✭❙❝❤❧ä✢✐✱ ✶✽✺✵✮✳ ▲❡t P ⊂ ▼ ♥ + ✶ ❜❡ ❛ ✭❝♦♥✈❡①✮ ♣♦❧②❤❡❞r♦♥✳ ■♥ ❛♥② ❑ ❞❡❢♦r♠❛t✐♦♥ ♦❢ P ✱ � ♥❑❞❱ = − ❱ ❡ ❞ θ ❡ , ❡ s✉♠ ♦✈❡r ❝♦❞✐♠ ✷ ❢❛❝❡s ♦❢ P ✱ θ ❡ = ❡①t❡r✐♦r ❞✐❤❡❞r❛❧ ❛♥❣❧❡✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥❝❧✉❞❡ ✿ ❤②♣❡r❜♦❧✐❝ ♣♦❧②❤❡❞r❛ ✿ ❆♥❞r❡❡✈✬s t❤❡♦r❡♠ ♦♥ ❞✐❤❡❞r❛❧ ❛♥❣❧❡s ♦❢ ✐❞❡❛❧ ♣♦❧②❤❡❞r❛ ✭❚❤✉rst♦♥✱ ❇rä❣❣❡r✱ ❘✐✈✐♥ ❛♣♣r♦❛❝❤✮✳ ❤②♣❡r❜♦❧✐❝ ✸✲♠❛♥✐❢♦❧❞s ✿ r✐❣✐❞✐t②✱ ❉❡❤♥ s✉r❣❡r② ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❝✉s♣s ❀ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❤②♣❡r❜♦❧✐❝ ❝♦♥❡✲♠❛♥✐❢♦❧❞s✱ ❡t❝✳ ❘❡❣❣❡ ❝❛❧❝✉❧✉s✱ ❡t❝✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❍✐❣❤❡r ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛s

  6. ❚❤❡ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ❍✐❣❤❡r s❝❛❧❛r ❢♦r♠✉❧❛s ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠✉❧❛s ❆♣♣❧✐❝❛t✐♦♥s ❚❤❡ ❝❧❛ss✐❝❛❧ ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛ ▼ ♥ + ✶ = s♣❛❝❡ ♦❢ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡ ❑ ✿ ❊✉❝❧✐❞❡❛♥✱ s♣❤❡r✐❝❛❧✱ ❤②♣❡r❜♦❧✐❝✳ ❑ ❚❤♠ ✭❙❝❤❧ä✢✐✱ ✶✽✺✵✮✳ ▲❡t P ⊂ ▼ ♥ + ✶ ❜❡ ❛ ✭❝♦♥✈❡①✮ ♣♦❧②❤❡❞r♦♥✳ ■♥ ❛♥② ❑ ❞❡❢♦r♠❛t✐♦♥ ♦❢ P ✱ � ♥❑❞❱ = − ❱ ❡ ❞ θ ❡ , ❡ s✉♠ ♦✈❡r ❝♦❞✐♠ ✷ ❢❛❝❡s ♦❢ P ✱ θ ❡ = ❡①t❡r✐♦r ❞✐❤❡❞r❛❧ ❛♥❣❧❡✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥❝❧✉❞❡ ✿ ❤②♣❡r❜♦❧✐❝ ♣♦❧②❤❡❞r❛ ✿ ❆♥❞r❡❡✈✬s t❤❡♦r❡♠ ♦♥ ❞✐❤❡❞r❛❧ ❛♥❣❧❡s ♦❢ ✐❞❡❛❧ ♣♦❧②❤❡❞r❛ ✭❚❤✉rst♦♥✱ ❇rä❣❣❡r✱ ❘✐✈✐♥ ❛♣♣r♦❛❝❤✮✳ ❤②♣❡r❜♦❧✐❝ ✸✲♠❛♥✐❢♦❧❞s ✿ r✐❣✐❞✐t②✱ ❉❡❤♥ s✉r❣❡r② ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❝✉s♣s ❀ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❤②♣❡r❜♦❧✐❝ ❝♦♥❡✲♠❛♥✐❢♦❧❞s✱ ❡t❝✳ ❘❡❣❣❡ ❝❛❧❝✉❧✉s✱ ❡t❝✳ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❍✐❣❤❡r ❙❝❤❧ä✢✐ ❢♦r♠✉❧❛s

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