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rt t r st r r st stt Prr r t r t

  1. ❉r❛❢t ✶ ▼♦♥t❡ ❈❛r❧♦ ❛♥❞ ◗✉❛s✐✲▼♦♥t❡ ❈❛r❧♦ ❢♦r ❉❡♥s✐t② ❊st✐♠❛t✐♦♥ P✐❡rr❡ ▲✬❊❝✉②❡r ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆♠❛❧ ❇❡♥ ❆❜❞❡❧❧❛❤✱ ❆rt ❇✳ ❖✇❡♥✱ ❛♥❞ ❋❧♦r✐❛♥ P✉❝❤❤❛♠♠❡r ▼❈◗▼❈ ✷✵✷✵ ❱✐rt✉❛❧✲❖①❢♦r❞✱ ❆✉❣✉st ✷✵✷✵

  2. ❇✉t s✐♠✉❧❛t✐♦♥ ✉s✉❛❧❧② ♣r♦✈✐❞❡s ✐♥❢♦r♠❛t✐♦♥ t♦ ❞♦ ♠✉❝❤ ♠♦r❡✦ ❚❤❡ ♦✉t♣✉t ❞❛t❛ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡♥t✐r❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❳ ✱ ❡✳❣✳✱ t❤❡ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❝❞❢✮ ❋ ♦❢ ❳ ✱ ❞❡☞♥❡❞ ❜② ❋ ✭ ① ✮ ❂ P ❬ ❳ ✔ ① ❪✱ ♦r ✐ts ❞❡♥s✐t② ❢ ❞❡☞♥❡❞ ❜② ❢ ✭ ① ✮ ❂ ❋ ✵ ✭ ① ✮✳ ■❢ ❳ ✶ ❀ ✿ ✿ ✿ ❀ ❳ ♥ ❛r❡ ♥ ✐♥❞❡♣✳ r❡❛❧✐③❛t✐♦♥s ♦❢ ❳ ✱ t❤❡ ❡♠♣✐r✐❝❛❧ ❝❞❢ ♥ ❋ ♥ ✭ ① ✮ ❂ ✶ ❫ ■ ❬ ❳ ✐ ✔ ① ❪ ♥ ✐ ❂✶ ❬ ❫ ❋ ♥ ✭ ① ✮❪ ❂ ❖ ✭ ♥ � ✶ ✮✳ ✐s ✉♥❜✐❛s❡❞ ❢♦r ❋ ✭ ① ✮ ❛t ❛❧❧ ① ✱ ❛♥❞ ❋♦r ❛ ❝♦♥t✐♥✉♦✉s r✳✈✳ ❳ ✱ t❤❡ ❞❡♥s✐t② ❢ ♣r♦✈✐❞❡s ❛ ❜❡tt❡r ✈✐s✉❛❧ ✐❞❡❛ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥✳ ❍❡r❡ ✇❡ ❢♦❝✉s ♦♥ ❡st✐♠❛t✐♥❣ t❤❡ ❞❡♥s✐t② ❢ ♦❢ ❳ ♦✈❡r ❬ ❛❀ ❜ ❪ ✚ ❘ ✳ ✭❉❡♥s✐t② ✐s ✇✐t❤ r❡s♣❡❝t t♦ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳✮ ❉r❛❢t ✷ ❲❤❛t t❤✐s t❛❧❦ ✐s ❛❜♦✉t ▼♦♥t❡ ❈❛r❧♦ ✭▼❈✮ s✐♠✉❧❛t✐♦♥ ✐s ✇✐❞❡❧② ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ ❊ ❬ ❳ ❪ ♦❢ ❛ r❛♥❞♦♠ ▼❙❊ ❂ Var ❬ ✖ ❳ ♥ ❪ ❂ ❖ ✭ ♥ � ✶ ✮✳ ✈❛r✐❛❜❧❡ ❳ ❛♥❞ ❝♦♠♣✉t❡ ❛ ❝♦♥☞❞❡♥❝❡ ✐♥t❡r✈❛❧ ♦♥ ❊ ❬ ❳ ❪✳

  3. ■❢ ❳ ✶ ❀ ✿ ✿ ✿ ❀ ❳ ♥ ❛r❡ ♥ ✐♥❞❡♣✳ r❡❛❧✐③❛t✐♦♥s ♦❢ ❳ ✱ t❤❡ ❡♠♣✐r✐❝❛❧ ❝❞❢ ♥ ❋ ♥ ✭ ① ✮ ❂ ✶ ❫ ■ ❬ ❳ ✐ ✔ ① ❪ ♥ ✐ ❂✶ ❬ ❫ ❋ ♥ ✭ ① ✮❪ ❂ ❖ ✭ ♥ � ✶ ✮✳ ✐s ✉♥❜✐❛s❡❞ ❢♦r ❋ ✭ ① ✮ ❛t ❛❧❧ ① ✱ ❛♥❞ ❋♦r ❛ ❝♦♥t✐♥✉♦✉s r✳✈✳ ❳ ✱ t❤❡ ❞❡♥s✐t② ❢ ♣r♦✈✐❞❡s ❛ ❜❡tt❡r ✈✐s✉❛❧ ✐❞❡❛ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥✳ ❍❡r❡ ✇❡ ❢♦❝✉s ♦♥ ❡st✐♠❛t✐♥❣ t❤❡ ❞❡♥s✐t② ❢ ♦❢ ❳ ♦✈❡r ❬ ❛❀ ❜ ❪ ✚ ❘ ✳ ✭❉❡♥s✐t② ✐s ✇✐t❤ r❡s♣❡❝t t♦ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳✮ ❉r❛❢t ✷ ❲❤❛t t❤✐s t❛❧❦ ✐s ❛❜♦✉t ▼♦♥t❡ ❈❛r❧♦ ✭▼❈✮ s✐♠✉❧❛t✐♦♥ ✐s ✇✐❞❡❧② ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ ❊ ❬ ❳ ❪ ♦❢ ❛ r❛♥❞♦♠ ▼❙❊ ❂ Var ❬ ✖ ❳ ♥ ❪ ❂ ❖ ✭ ♥ � ✶ ✮✳ ✈❛r✐❛❜❧❡ ❳ ❛♥❞ ❝♦♠♣✉t❡ ❛ ❝♦♥☞❞❡♥❝❡ ✐♥t❡r✈❛❧ ♦♥ ❊ ❬ ❳ ❪✳ ❇✉t s✐♠✉❧❛t✐♦♥ ✉s✉❛❧❧② ♣r♦✈✐❞❡s ✐♥❢♦r♠❛t✐♦♥ t♦ ❞♦ ♠✉❝❤ ♠♦r❡✦ ❚❤❡ ♦✉t♣✉t ❞❛t❛ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡♥t✐r❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❳ ✱ ❡✳❣✳✱ t❤❡ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❝❞❢✮ ❋ ♦❢ ❳ ✱ ❞❡☞♥❡❞ ❜② ❋ ✭ ① ✮ ❂ P ❬ ❳ ✔ ① ❪✱ ♦r ✐ts ❞❡♥s✐t② ❢ ❞❡☞♥❡❞ ❜② ❢ ✭ ① ✮ ❂ ❋ ✵ ✭ ① ✮✳

  4. ❋♦r ❛ ❝♦♥t✐♥✉♦✉s r✳✈✳ ❳ ✱ t❤❡ ❞❡♥s✐t② ❢ ♣r♦✈✐❞❡s ❛ ❜❡tt❡r ✈✐s✉❛❧ ✐❞❡❛ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥✳ ❍❡r❡ ✇❡ ❢♦❝✉s ♦♥ ❡st✐♠❛t✐♥❣ t❤❡ ❞❡♥s✐t② ❢ ♦❢ ❳ ♦✈❡r ❬ ❛❀ ❜ ❪ ✚ ❘ ✳ ✭❉❡♥s✐t② ✐s ✇✐t❤ r❡s♣❡❝t t♦ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳✮ ❉r❛❢t ✷ ❲❤❛t t❤✐s t❛❧❦ ✐s ❛❜♦✉t ▼♦♥t❡ ❈❛r❧♦ ✭▼❈✮ s✐♠✉❧❛t✐♦♥ ✐s ✇✐❞❡❧② ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ ❊ ❬ ❳ ❪ ♦❢ ❛ r❛♥❞♦♠ ▼❙❊ ❂ Var ❬ ✖ ❳ ♥ ❪ ❂ ❖ ✭ ♥ � ✶ ✮✳ ✈❛r✐❛❜❧❡ ❳ ❛♥❞ ❝♦♠♣✉t❡ ❛ ❝♦♥☞❞❡♥❝❡ ✐♥t❡r✈❛❧ ♦♥ ❊ ❬ ❳ ❪✳ ❇✉t s✐♠✉❧❛t✐♦♥ ✉s✉❛❧❧② ♣r♦✈✐❞❡s ✐♥❢♦r♠❛t✐♦♥ t♦ ❞♦ ♠✉❝❤ ♠♦r❡✦ ❚❤❡ ♦✉t♣✉t ❞❛t❛ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡♥t✐r❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❳ ✱ ❡✳❣✳✱ t❤❡ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❝❞❢✮ ❋ ♦❢ ❳ ✱ ❞❡☞♥❡❞ ❜② ❋ ✭ ① ✮ ❂ P ❬ ❳ ✔ ① ❪✱ ♦r ✐ts ❞❡♥s✐t② ❢ ❞❡☞♥❡❞ ❜② ❢ ✭ ① ✮ ❂ ❋ ✵ ✭ ① ✮✳ ■❢ ❳ ✶ ❀ ✿ ✿ ✿ ❀ ❳ ♥ ❛r❡ ♥ ✐♥❞❡♣✳ r❡❛❧✐③❛t✐♦♥s ♦❢ ❳ ✱ t❤❡ ❡♠♣✐r✐❝❛❧ ❝❞❢ ♥ ❋ ♥ ✭ ① ✮ ❂ ✶ ❫ � ■ ❬ ❳ ✐ ✔ ① ❪ ♥ ✐ ❂✶ ✐s ✉♥❜✐❛s❡❞ ❢♦r ❋ ✭ ① ✮ ❛t ❛❧❧ ① ✱ ❛♥❞ Var ❬ ❫ ❋ ♥ ✭ ① ✮❪ ❂ ❖ ✭ ♥ � ✶ ✮✳

  5. ❉r❛❢t ✷ ❲❤❛t t❤✐s t❛❧❦ ✐s ❛❜♦✉t ▼♦♥t❡ ❈❛r❧♦ ✭▼❈✮ s✐♠✉❧❛t✐♦♥ ✐s ✇✐❞❡❧② ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ ❊ ❬ ❳ ❪ ♦❢ ❛ r❛♥❞♦♠ ▼❙❊ ❂ Var ❬ ✖ ❳ ♥ ❪ ❂ ❖ ✭ ♥ � ✶ ✮✳ ✈❛r✐❛❜❧❡ ❳ ❛♥❞ ❝♦♠♣✉t❡ ❛ ❝♦♥☞❞❡♥❝❡ ✐♥t❡r✈❛❧ ♦♥ ❊ ❬ ❳ ❪✳ ❇✉t s✐♠✉❧❛t✐♦♥ ✉s✉❛❧❧② ♣r♦✈✐❞❡s ✐♥❢♦r♠❛t✐♦♥ t♦ ❞♦ ♠✉❝❤ ♠♦r❡✦ ❚❤❡ ♦✉t♣✉t ❞❛t❛ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡♥t✐r❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❳ ✱ ❡✳❣✳✱ t❤❡ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❝❞❢✮ ❋ ♦❢ ❳ ✱ ❞❡☞♥❡❞ ❜② ❋ ✭ ① ✮ ❂ P ❬ ❳ ✔ ① ❪✱ ♦r ✐ts ❞❡♥s✐t② ❢ ❞❡☞♥❡❞ ❜② ❢ ✭ ① ✮ ❂ ❋ ✵ ✭ ① ✮✳ ■❢ ❳ ✶ ❀ ✿ ✿ ✿ ❀ ❳ ♥ ❛r❡ ♥ ✐♥❞❡♣✳ r❡❛❧✐③❛t✐♦♥s ♦❢ ❳ ✱ t❤❡ ❡♠♣✐r✐❝❛❧ ❝❞❢ ♥ ❋ ♥ ✭ ① ✮ ❂ ✶ ❫ � ■ ❬ ❳ ✐ ✔ ① ❪ ♥ ✐ ❂✶ ✐s ✉♥❜✐❛s❡❞ ❢♦r ❋ ✭ ① ✮ ❛t ❛❧❧ ① ✱ ❛♥❞ Var ❬ ❫ ❋ ♥ ✭ ① ✮❪ ❂ ❖ ✭ ♥ � ✶ ✮✳ ❋♦r ❛ ❝♦♥t✐♥✉♦✉s r✳✈✳ ❳ ✱ t❤❡ ❞❡♥s✐t② ❢ ♣r♦✈✐❞❡s ❛ ❜❡tt❡r ✈✐s✉❛❧ ✐❞❡❛ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥✳ ❍❡r❡ ✇❡ ❢♦❝✉s ♦♥ ❡st✐♠❛t✐♥❣ t❤❡ ❞❡♥s✐t② ❢ ♦❢ ❳ ♦✈❡r ❬ ❛❀ ❜ ❪ ✚ ❘ ✳ ✭❉❡♥s✐t② ✐s ✇✐t❤ r❡s♣❡❝t t♦ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳✮

  6. ■♥ t❤✐s t❛❧❦✱ ✇❡ ❛ss✉♠❡ t❤❛t ❳ ✶ ❀ ✿ ✿ ✿ ❀ ❳ ♥ ❛r❡ ❣❡♥❡r❛t❡❞ ❜② s✐♠✉❧❛t✐♦♥ ❢r♦♠ ❛ ♠♦❞❡❧✳ ❲❡ ❝❛♥ ❝❤♦♦s❡ ♥ ❛♥❞ ✇❡ ❤❛✈❡ s♦♠❡ ❢r❡❡❞♦♠ ♦♥ ❤♦✇ t❤❡ s✐♠✉❧❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞✳ ◗✉❡st✐♦♥s✿ ✶✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ♦❜t❛✐♥ ✉♥❜✐❛s❡❞ ❞❡♥s✐t② ❡st✐♠❛t♦rs ✇❤♦s❡ ✈❛r✐❛♥❝❡ ❝♦♥✈❡r❣❡s ❛s ❖ ✭ ♥ � ✶ ✮ ♦r ❜❡tt❡r✱ ✉s✐♥❣ ❝❧❡✈❡r s❛♠♣❧✐♥❣ str❛t❡❣✐❡s❄ ❍♦✇❄ ✷✳ ❍♦✇ ❝❛♥ ✇❡ ❜❡♥❡☞t ❢r♦♠ ❘◗▼❈ t♦ ❡st✐♠❛t❡ ❞❡♥s✐t②❄ ❈❛♥ ✇❡ ✐♠♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ♦❢ t❤❡ ❡rr♦r❄ ❉r❛❢t ✸ ❙❡tt✐♥❣ ❈❧❛ss✐❝❛❧ ❞❡♥s✐t② ❡st✐♠❛t✐♦♥ ✐♥ st❛t✐st✐❝s ✇❛s ❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ✇❤❡r❡ ❳ ✶ ❀ ✿ ✿ ✿ ❀ ❳ ♥ ❛r❡ ❣✐✈❡♥ ✐♥❞❡♣❡♥❞❡♥t ♦❜s❡r✈❛t✐♦♥s ♦❢ ❳ ❛♥❞ ♦♥❡ ❡st✐♠❛t❡s t❤❡ ❞❡♥s✐t② ❢ ♦❢ ❳ ❢r♦♠ t❤❛t✳ ▲❡❛❞✐♥❣ ♠❡t❤♦❞✿ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t♦r ✭❑❉❊✮❀ ▼❙❊❬❫ ❢ ♥ ✭ ① ✮❪ ❂ ❖ ✭ ♥ � ✹ ❂ ✺ ✮✳

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