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  1. ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s Pét❡r ❙❛❧✈✐ ❇✉❞❛♣❡st ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣② ❛♥❞ ❊❝♦♥♦♠✐❝s ❲❆■❚ ✷✵✶✾ ❇✉❞❛♣❡st✱ ❏❛♥✉❛r② ✷✹ t❤ P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  2. ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ✷ ❙✐♠♣❧❡①❡s ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ✸ ❈♦♥✈❡rs✐♦♥ t♦ q✉❛❞r✐❧❛t❡r❛❧ ❙✲♣❛t❝❤ ❈♦♥✈❡rs✐♦♥ t♦ t❡♥s♦r ♣r♦❞✉❝t ❢♦r♠ ❈♦♥❝❧✉s✐♦♥ ✹ ❊①❛♠♣❧❡ ❉✐s❝✉ss✐♦♥ P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  3. ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥ ▼✉❧t✐✲s✐❞❡❞ s✉r❢❛❝❡s ✐♥ ❈❆❉ s♦❢t✇❛r❡ ❙t❛♥❞❛r❞ s✉r❢❛❝❡ r❡♣r❡s❡♥t❛t✐♦♥s✿ ❚❡♥s♦r✲♣r♦❞✉❝t ❇é③✐❡r s✉r❢❛❝❡ ❚❡♥s♦r✲♣r♦❞✉❝t ❇✲s♣❧✐♥❡ s✉r❢❛❝❡ ❚❡♥s♦r✲♣r♦❞✉❝t ◆❯❘❇❙ s✉r❢❛❝❡ ◆♦ st❛♥❞❛r❞ ♠✉❧t✐✲s✐❞❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ❈♦♥✈❡rs✐♦♥ t♦ t❡♥s♦r✲♣r♦❞✉❝t ♣❛t❝❤❡s ❚r✐♠♠✐♥❣ P❛r❛♠❡t❡r✐③❛t✐♦♥ ✐ss✉❡s ❆s②♠♠❡tr✐❝ ◆♦t ✇❛t❡rt✐❣❤t ❈❡♥tr❛❧ s♣❧✐t ▲♦♦s❡❧② ❞❡✜♥❡❞ ❞✐✈✐❞✐♥❣ ❝✉r✈❡s ❖♥❧② C ✵ ♦r G ✶ ❝♦♥t✐♥✉✐t② P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  4. ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥ ❙♦❧✉t✐♦♥ ❊①❛❝t t❡♥s♦r ♣r♦❞✉❝t ❝♦♥✈❡rs✐♦♥ ❚r✐♠♠❡❞ r❛t✐♦♥❛❧ ❇é③✐❡r s✉r❢❛❝❡ ❖♥❧② ♣♦❧②♥♦♠✐❛❧ ✭❇é③✐❡r✮ ❜♦✉♥❞❛r✐❡s ❚r✐♠♠✐♥❣ ❝✉r✈❡s ⇒ ❧✐♥❡s ✐♥ t❤❡ ❞♦♠❛✐♥ ◆❛t✐✈❡ n ✲s✐❞❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ❙✲♣❛t❝❤ ●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❇é③✐❡r ❝✉r✈❡s ✫ tr✐❛♥❣❧❡s ❙✉✐t❛❜❧❡ ❢♦r G ✶ ❤♦❧❡ ✜❧❧✐♥❣ ❬✶❪ ❬✶❪ P✳ ❙❛❧✈✐✱ G ✶ ❤♦❧❡ ✜❧❧✐♥❣ ✇✐t❤ ❙✲♣❛t❝❤❡s ♠❛❞❡ ❡❛s② ✳ ■♥✿ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✶✷t❤ ❈♦♥❢❡r❡♥❝❡ ♦❢ t❤❡ ❍✉♥❣❛r✐❛♥ ❆ss♦❝✐❛t✐♦♥ ❢♦r ■♠❛❣❡ Pr♦❝❡ss✐♥❣ ❛♥❞ P❛tt❡r♥ ❘❡❝♦❣♥✐t✐♦♥✱ ✷✵✶✾ ✭❛❝❝❡♣t❡❞✮✳ P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  5. ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ Pr❡✈✐♦✉s ✇♦r❦ ❙✲♣❛t❝❤❡s ✫ s✐♠♣❧❡①❡s ❬✶✾✽✾✱ ▲♦♦♣ ✫ ❉❡❘♦s❡❪ ❆ ♠✉❧t✐✲s✐❞❡❞ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❇é③✐❡r s✉r❢❛❝❡s ❚❤❡ ♦r✐❣✐♥❛❧ ❙✲♣❛t❝❤ ♣✉❜❧✐❝❛t✐♦♥ ❈♦♥t❛✐♥s t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts ♦♥ t❤❡ t❡♥s♦r ♣r♦❞✉❝t ❝♦♥✈❡rs✐♦♥ ▼✐ss✐♥❣ ❢r♦♠ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✿ ❈♦♠♣♦s✐t✐♦♥ ♦❢ r❛t✐♦♥❛❧ ❇é③✐❡r s✐♠♣❧❡①❡s ❇❧♦ss♦♠ ♦❢ ❲❛❝❤s♣r❡ss ❝♦♦r❞✐♥❛t❡s ❬✶✾✽✼✱ ❘❛♠s❤❛✇❪ ❇❧♦ss♦♠✐♥❣✿ ❆ ❝♦♥♥❡❝t✲t❤❡✲❞♦ts ❛♣♣r♦❛❝❤ t♦ s♣❧✐♥❡s ❬✶✾✽✽✱ ❉❡❘♦s❡❪ ❈♦♠♣♦s✐♥❣ ❇é③✐❡r s✐♠♣❧❡①❡s ❬✶✾✾✸✱ ❉❡❘♦s❡ ❡t ❛❧✳❪ ❋✉♥❝t✐♦♥❛❧ ❝♦♠♣♦s✐t✐♦♥ ❛❧❣♦r✐t❤♠s ✈✐❛ ❜❧♦ss♦♠✐♥❣ P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  6. ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❙✐♠♣❧❡①❡s ❙✐♠♣❧❡① ✐♥ n ❉ ( n + ✶ ) ♣♦✐♥ts ✐♥ n ❉ ▲❡t V i ❞❡♥♦t❡ t❤❡s❡ ♣♦✐♥ts ❆♥② n ❉ ♣♦✐♥t ✐s ✉♥✐q✉❡❧② ❡①♣r❡ss❡❞ ❜② t❤❡ ❛✣♥❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ V i ✿ n n � � p = λ i V i ✇✐t❤ λ i = ✶ i = ✶ i = ✶ λ i ❛r❡ t❤❡ ❜❛r②❝❡♥tr✐❝ ❝♦♦r❞✐♥❛t❡s ♦❢ p r❡❧❛t✐✈❡ t♦ t❤❡ s✐♠♣❧❡① ✭✐♠❛❣❡s ❢r♦♠ ❲✐❦✐♣❡❞✐❛✮ P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  7. ✶ ✵ ▲❡t s ❛♥❞ ✶ ✳ ❚❤❡♥ ✶ ✷ ✶ ✷ s ✶ ✷ s ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❙✐♠♣❧❡①❡s ❇é③✐❡r ❝✉r✈❡ ▲❡t✬s ❧♦♦❦ ❛t t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❇é③✐❡r ❝✉r✈❡✿ d � P i B d C ( u ) = i ( u ) i = ✵ P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  8. ▲❡t s ❛♥❞ ✶ ✳ ❚❤❡♥ ✶ ✷ ✶ ✷ s ✶ ✷ s ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❙✐♠♣❧❡①❡s ❇é③✐❡r ❝✉r✈❡ ▲❡t✬s ❧♦♦❦ ❛t t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❇é③✐❡r ❝✉r✈❡✿ d d d ! � � P i B d i !( d − i )! u i ( ✶ − u ) d − i C ( u ) = i ( u ) = P i i = ✵ i = ✵ P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  9. ❚❤❡♥ ✶ ✷ ✶ ✷ s ✶ ✷ s ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❙✐♠♣❧❡①❡s ❇é③✐❡r ❝✉r✈❡ ▲❡t✬s ❧♦♦❦ ❛t t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❇é③✐❡r ❝✉r✈❡✿ d d d ! � � P i B d i !( d − i )! u i ( ✶ − u ) d − i C ( u ) = i ( u ) = P i i = ✵ i = ✵ ▲❡t s = ( i , d − i ) ❛♥❞ λ = ( u , ✶ − u ) ✳ P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

  10. ■♥tr♦❞✉❝t✐♦♥ ❙✐♠♣❧❡①❡s ✫ ❙✲♣❛t❝❤❡s ❈♦♥✈❡rs✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❙✐♠♣❧❡①❡s ❇é③✐❡r ❝✉r✈❡ ▲❡t✬s ❧♦♦❦ ❛t t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❇é③✐❡r ❝✉r✈❡✿ d d d ! � � P i B d i !( d − i )! u i ( ✶ − u ) d − i C ( u ) = i ( u ) = P i i = ✵ i = ✵ ▲❡t s = ( i , d − i ) ❛♥❞ λ = ( u , ✶ − u ) ✳ ❚❤❡♥ d ! � s ✶ ! s ✷ ! λ s ✶ ✶ λ s ✷ C ( λ ) = P s ✷ s P✳ ❙❛❧✈✐ ❇▼❊ ❖♥ t❤❡ ❈❆❉✲❝♦♠♣❛t✐❜❧❡ ❝♦♥✈❡rs✐♦♥ ♦❢ ❙✲♣❛t❝❤❡s

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