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tts ts rt rtr st Pr r t tt s t

  1. ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡ ❖♥ t❤❡ ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ❛ ◆❡❛r ❯♥❛♥✐♠✐t② ❖♣❡r❛t✐♦♥ ✐♥ ❛ ❈❧♦♥❡ ❉♠✐tr✐② ❩❤✉❦ ③❤✉❦✳❞♠✐tr✐②❅❣♠❛✐❧✳❝♦♠ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ▼❡❝❤❛♥✐❝s ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❈♦♥❢❡r❡♥❝❡ ❆❆❆✽✽ ✿ ❲❛rs❛✇✱ ✷✵✶✹

  2. ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡ ❖✉t❧✐♥❡ ✶ ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ✷ ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ✸ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ ✹ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

  3. ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ ♦❢ ❛r✐t② ❛t ♠♦st ✳ ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ ✳ ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ✳ ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡ ▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s O m A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳

  4. ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ ✳ ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ✳ ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡ ▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s O m A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳ R m A ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳

  5. ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ ✳ ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ✳ ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡ ▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s O m A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳ R m A ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳ [ M ] ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ M ✳

  6. ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ✳ ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡ ▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s O m A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳ R m A ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳ [ M ] ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ M ✳ Pol ( ρ ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ρ ✳ Pol ( F ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ F ✳

  7. ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡ ▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s O m A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳ R m A ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m ✳ [ M ] ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ M ✳ Pol ( ρ ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ρ ✳ Pol ( F ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ F ✳ � ρ � C ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ C ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ρ ✳

  8. Pr♦♣❡rt✐❡s ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ♦❢ ❛r✐t② ✐s ❞❡✜♥❡❞ ❜② r❡❧❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st ✭❑✳ ❆✳ ❇❛❦❡r✱ ❆✳ ❋✳ P✐①❧❡②✮✳ ❋♦r ❛♥② ◆❯ t❤❡r❡ ❡①✐sts ✜♥✐t❡❧② ♠❛♥② ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ✐t✳ ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❍✳▲❛❦s❡r ❛♥❞ ❙✳ ❑❡r❦❤♦✛ s✉❝❝❡ss❢✉❧❧② st✉❞✐❡❞ t❤❡ ♦r❞❡r ♦❢ ❝❧♦♥❡s ✇✐t❤ ◆❯ ✭t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ❣❡♥❡r❛t✐♥❣ s❡t✮✳ ❙✉❜❧❛tt✐❝❡s ♦❢ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ◆❯ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❛ ❝♦♠♣✉t❡r ✭✇❡ ❢♦✉♥❞ ✶✱✾✶✽✱✵✹✵ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ♠❛❥♦r✐t② ♦♣❡r❛t✐♦♥ ♦♥ ✸ ❡❧❡♠❡♥ts✱ ❛♥❞ s♦ ♦♥✮✳ ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡ ◆❡❛r✲✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❆ ♥❡❛r ✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ✭◆❯✮ ✐s ❛♥ ♦♣❡r❛t✐♦♥ f s❛t✐s❢②✐♥❣ f ( x , . . . , x , y ) = f ( x , . . . , x , y , x ) = · · · = f ( y , x , . . . , x ) = x .

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