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❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝ ❢♦r ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙t❡✈❡♥ ◆✳ ❊✈❛♥s

❯✳❈✳ ❇❡r❦❡❧❡②

▼❛r❝❤✱ ✷✵✶✻

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈♦❧❧❛❜♦r❛t♦r

■❧②❛ ▼♦❧❝❤❛♥♦✈ ❇❡r♥ ❚♦ ❛♣♣❡❛r ✐♥ ❚r❛♥s❛❝t✐♦♥s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ❆✈❛✐❧❛❜❧❡ ❛t ❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✹✵✶✳✼✵✺✷

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛rt❡s✐❛♥ ♣r♦❞✉❝t

❋✐❣✉r❡✿ ❚❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛rt❡s✐❛♥ ♣r♦❞✉❝t

❋✐❣✉r❡✿ ❚❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ t✇♦ ♠♦r❡ ✐♥t❡r❡st✐♥❣ ❣r❛♣❤s ✭❝♦✉rt❡s② ♦❢ ❲✐❦✐♣❡❞✐❛✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛rt❡s✐❛♥ ♣r♦❞✉❝t

❋♦r♠❛❧❧②✱ t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t G❧H ♦❢ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✇✐t❤ ✈❡rt❡① s❡ts V ♣Gq ❛♥❞ V ♣Hq ❛♥❞ ❡❞❣❡ s❡ts E♣Gq ❛♥❞ E♣Hq ✐s t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t V ♣G❧Hq :✏ V ♣Gq ✂ V ♣Hq ❛♥❞ ❡❞❣❡ s❡t E♣G❧Hq :✏ t♣♣g✶, hq, ♣g✷, hqq : ♣g✶, g✷q P E♣Gq, h P V ♣Hq✉ ❨ t♣♣g, h✶q, ♣g, h✷qq : g P V ♣Gq, ♣h✶, h✷q P E♣Hq✉. ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝♦♠♠✉t❛t✐✈❡ ❛♥❞ ❛ss♦❝✐❛t✐✈❡ ✇✐t❤ t❤❡ tr✐✈✐❛❧ ❣r❛♣❤ ❛s ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✐❢ ✇❡ tr❡❛t ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ❛s ❜❡✐♥❣ ❡q✉❛❧✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛rt❡s✐❛♥ ♣r♦❞✉❝t

❋♦r♠❛❧❧②✱ t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t G❧H ♦❢ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✇✐t❤ ✈❡rt❡① s❡ts V ♣Gq ❛♥❞ V ♣Hq ❛♥❞ ❡❞❣❡ s❡ts E♣Gq ❛♥❞ E♣Hq ✐s t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t V ♣G❧Hq :✏ V ♣Gq ✂ V ♣Hq ❛♥❞ ❡❞❣❡ s❡t E♣G❧Hq :✏ t♣♣g✶, hq, ♣g✷, hqq : ♣g✶, g✷q P E♣Gq, h P V ♣Hq✉ ❨ t♣♣g, h✶q, ♣g, h✷qq : g P V ♣Gq, ♣h✶, h✷q P E♣Hq✉. ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝♦♠♠✉t❛t✐✈❡ ❛♥❞ ❛ss♦❝✐❛t✐✈❡ ✇✐t❤ t❤❡ tr✐✈✐❛❧ ❣r❛♣❤ ❛s ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✐❢ ✇❡ tr❡❛t ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ❛s ❜❡✐♥❣ ❡q✉❛❧✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛rt❡s✐❛♥ ♣r♦❞✉❝t

❆ ♥♦♥tr✐✈✐❛❧ ❣r❛♣❤ ✐s ✐rr❡❞✉❝✐❜❧❡ ✐❢ ✐t ✐s ♥♦t t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ t✇♦ ♥♦♥tr✐✈✐❛❧ ❣r❛♣❤s✳ ❙❛❜✐❞✉ss✐ ✭✶✾✻✵✮ s❤♦✇❡❞ t❤❛t ❛♥② ✜♥✐t❡ ❣r❛♣❤ ✐s ❛ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❣r❛♣❤s ❛♥❞ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ✐s ✉♥✐q✉❡ ✉♣ t♦ ♦r❞❡r✳ ❋❛❝t♦r✐♥❣ ❣r❛♣❤s ❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❡♠❜❡❞❞✐♥❣ t❤❡♠ ✐♥ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝ts ✐s ✇✐❞❡❧② st✉❞✐❡❞ ✐♥ ❝♦♠♣✉t❡r s❝✐❡♥❝❡ ❢♦❧❧♦✇✐♥❣ ●r❛❤❛♠ ❛♥❞ ❲✐♥❦❧❡r ✭✶✾✽✹✱ ✶✾✽✺✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛rt❡s✐❛♥ ♣r♦❞✉❝t

❆ ♥♦♥tr✐✈✐❛❧ ❣r❛♣❤ ✐s ✐rr❡❞✉❝✐❜❧❡ ✐❢ ✐t ✐s ♥♦t t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ t✇♦ ♥♦♥tr✐✈✐❛❧ ❣r❛♣❤s✳ ❙❛❜✐❞✉ss✐ ✭✶✾✻✵✮ s❤♦✇❡❞ t❤❛t ❛♥② ✜♥✐t❡ ❣r❛♣❤ ✐s ❛ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❣r❛♣❤s ❛♥❞ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ✐s ✉♥✐q✉❡ ✉♣ t♦ ♦r❞❡r✳ ❋❛❝t♦r✐♥❣ ❣r❛♣❤s ❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❡♠❜❡❞❞✐♥❣ t❤❡♠ ✐♥ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝ts ✐s ✇✐❞❡❧② st✉❞✐❡❞ ✐♥ ❝♦♠♣✉t❡r s❝✐❡♥❝❡ ❢♦❧❧♦✇✐♥❣ ●r❛❤❛♠ ❛♥❞ ❲✐♥❦❧❡r ✭✶✾✽✹✱ ✶✾✽✺✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛rt❡s✐❛♥ ♣r♦❞✉❝t

❆ ♥♦♥tr✐✈✐❛❧ ❣r❛♣❤ ✐s ✐rr❡❞✉❝✐❜❧❡ ✐❢ ✐t ✐s ♥♦t t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ t✇♦ ♥♦♥tr✐✈✐❛❧ ❣r❛♣❤s✳ ❙❛❜✐❞✉ss✐ ✭✶✾✻✵✮ s❤♦✇❡❞ t❤❛t ❛♥② ✜♥✐t❡ ❣r❛♣❤ ✐s ❛ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❣r❛♣❤s ❛♥❞ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ✐s ✉♥✐q✉❡ ✉♣ t♦ ♦r❞❡r✳ ❋❛❝t♦r✐♥❣ ❣r❛♣❤s ❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❡♠❜❡❞❞✐♥❣ t❤❡♠ ✐♥ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝ts ✐s ✇✐❞❡❧② st✉❞✐❡❞ ✐♥ ❝♦♠♣✉t❡r s❝✐❡♥❝❡ ❢♦❧❧♦✇✐♥❣ ●r❛❤❛♠ ❛♥❞ ❲✐♥❦❧❡r ✭✶✾✽✹✱ ✶✾✽✺✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❤♦rt❡st ♣❛t❤ ♠❡tr✐❝

❋✐❣✉r❡✿ ❆ s❤♦rt❡st ♣❛t❤ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✐♥ t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❤♦rt❡st ♣❛t❤ ♠❡tr✐❝

■❢ t✇♦ ❝♦♥♥❡❝t❡❞ ✜♥✐t❡ ❣r❛♣❤s G ❛♥❞ H ❛r❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ s❤♦rt❡st ♣❛t❤ ♠❡tr✐❝s rG ❛♥❞ rH✱ t❤❡♥ t❤❡ s❤♦rt❡st ♣❛t❤ ♠❡tr✐❝ ♦♥ t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ✐s ❣✐✈❡♥ ❜② rG✂H ✏ rG ❵ rH, ✇❤❡r❡ ♣rG ❵ rHq♣♣g✶, h✶q, ♣g✷, h✷qq :✏ rG♣g✶, g✷q rH♣h✶, h✷q, ♣g✶, h✶q, ♣g✷, h✷q P G ✂ H. ❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❡①t❡♥❞ t❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ t♦ ♠♦r❡ ❣❡♥❡r❛❧ ♠❡tr✐❝ s♣❛❝❡s❄

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❤♦rt❡st ♣❛t❤ ♠❡tr✐❝

■❢ t✇♦ ❝♦♥♥❡❝t❡❞ ✜♥✐t❡ ❣r❛♣❤s G ❛♥❞ H ❛r❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ s❤♦rt❡st ♣❛t❤ ♠❡tr✐❝s rG ❛♥❞ rH✱ t❤❡♥ t❤❡ s❤♦rt❡st ♣❛t❤ ♠❡tr✐❝ ♦♥ t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ✐s ❣✐✈❡♥ ❜② rG✂H ✏ rG ❵ rH, ✇❤❡r❡ ♣rG ❵ rHq♣♣g✶, h✶q, ♣g✷, h✷qq :✏ rG♣g✶, g✷q rH♣h✶, h✷q, ♣g✶, h✶q, ♣g✷, h✷q P G ✂ H. ❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❡①t❡♥❞ t❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ t♦ ♠♦r❡ ❣❡♥❡r❛❧ ♠❡tr✐❝ s♣❛❝❡s❄

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ t✇♦ ✐♥t❡r✈❛❧s ❛♥❞ t❤❡ ▼❛♥❤❛tt❛♥✴t❛①✐✲❝❛❜✴ℓ1 ♠❡tr✐❝

❋✐❣✉r❡✿ ❊q✉✐♣♣✐♥❣ t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♦❢ t✇♦ ✐♥t❡r✈❛❧s ✇✐t❤ t❤❡ s✉♠ ♦❢ t❤❡ ✉s✉❛❧ ♠❡tr✐❝s ❣✐✈❡s ❛ r❡❝t❛♥❣❧❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ▼❛♥❤❛tt❛♥ ♦r t❛①✐✲❝❛❜ ♠❡tr✐❝ ✭❝♦✉rt❡s② ♦❢ ❲♦❧❢r❛♠✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■rr❡❞✉❝✐❜❧❡s

❚❤❡ ♠❡tr✐❝ s♣❛❝❡ ♣X, rXq ✐s ✐rr❡❞✉❝✐❜❧❡ ✐❢ t❤❡r❡ ✐s ♥♦ ♥♦♥tr✐✈✐❛❧ ❢❛❝t♦r✐③❛t✐♦♥ ♣X, rXq ✏ ♣Y ✂ Z, rY ❵ rZq.

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❯♥✐q✉❡♥❡ss ♦❢ ❢❛❝t♦r✐③❛t✐♦♥ ✐♥t♦ ✐rr❡❞✉❝✐❜❧❡s

■❢ ❛ ♠❡tr✐❝ s♣❛❝❡ ✐s ✐s♦♠❡tr✐❝ t♦ ❛ ♣r♦❞✉❝t ♦❢ ✜♥✐t❡❧② ♠❛♥② ✐rr❡❞✉❝✐❜❧❡ ♠❡tr✐❝ s♣❛❝❡s✱ t❤❡♥ t❤✐s ❢❛❝t♦r✐③❛t✐♦♥ ✐s ✉♥✐q✉❡ ✉♣ t♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❢❛❝t♦rs ✕ ❚❛r❞✐❢ ✭✶✾✾✷✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ❧✐tt❧❡ ❛❜♦✉t ❚❛r❞✐❢✬s ♣r♦♦❢

❚❛r❞✐❢ ✉s❡s ✐❞❡❛s✴r❡s✉❧ts ❢r♦♠ t❤❡ ✇♦r❧❞ ♦❢ ♠❡❞✐❛♥ ❛❧❣❡❜r❛s✱ ❈❤❡❜②s❤❡✈ s❡ts✱ ❣❛t❡❞ s♣❛❝❡s ❢r♦♠ ■s❜❡❧❧ ✭✶✾✽✵✮✱ ❍❡❧í❦♦✈á ✭✶✾✽✸✮✱ ❉r❡ss ✫ ❙❝❤❛r❧❛✉ ✭✶✾✽✼✮✳ ❆ s✉❜s❡t W ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ ♣X, rXq ✐s ❣❛t❡❞ ✐❢ ❢♦r ❡❛❝❤ x P X t❤❡r❡ ✐s ❛ ✭♥❡❝❡ss❛r✐❧② ✉♥✐q✉❡✮ w P W s✉❝❤ t❤❛t rX♣x, vq ✏ rX♣x, wq rX♣w, vq ❢♦r ❛❧❧ v P W ✭❢♦r ❛♥② v P W✱ ✇❡ ❝❛♥ ❛❧✇❛②s ❝❤♦♦s❡ ❛ s❤♦rt❡st ♣❛t❤ ❢r♦♠ x t♦ v t❤❛t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❣❛t❡ w✮✳

X W

x w v

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▲✐♠✐t❛t✐♦♥s ♦❢ ❢❛❝t♦r✐③❛t✐♦♥

❚❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧② ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ♥♦t ✐s♦♠❡tr✐❝ t♦ ❛ ✜♥✐t❡ ♣r♦❞✉❝t ♦❢ ✜♥✐t❡❧② ♠❛♥② ✐rr❡❞✉❝✐❜❧❡ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s ✭❡✳❣✳ X :✏ ➧

kPNr0, aks✱ ✇❤❡r❡ ➦ kPN ak ➔ ✽✱ ✇✐t❤

rX♣x✶, x✷q :✏ ➦

kPN ⑤x✶ k ✁ x✷ k⑤✮✳

❆r❡ t❤❡r❡ ✭✉♥✐q✉❡✮ ❢❛❝t♦r✐③❛t✐♦♥s ✉s✐♥❣ s♦♠❡ s♦rt ♦❢ ✐♥✜♥✐t❡ ♣r♦❞✉❝t❄ ❲❤❛t ❞♦❡s t❤✐s ♠❡❛♥❄ ❚❤❡ st✉❞② ♦❢ t❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ s❡❡♠s t♦ ❜❡ ❣❡♥❡r❛❧❧② r❛t❤❡r ❞✐✣❝✉❧t✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▲✐♠✐t❛t✐♦♥s ♦❢ ❢❛❝t♦r✐③❛t✐♦♥

❚❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧② ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ♥♦t ✐s♦♠❡tr✐❝ t♦ ❛ ✜♥✐t❡ ♣r♦❞✉❝t ♦❢ ✜♥✐t❡❧② ♠❛♥② ✐rr❡❞✉❝✐❜❧❡ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s ✭❡✳❣✳ X :✏ ➧

kPNr0, aks✱ ✇❤❡r❡ ➦ kPN ak ➔ ✽✱ ✇✐t❤

rX♣x✶, x✷q :✏ ➦

kPN ⑤x✶ k ✁ x✷ k⑤✮✳

❆r❡ t❤❡r❡ ✭✉♥✐q✉❡✮ ❢❛❝t♦r✐③❛t✐♦♥s ✉s✐♥❣ s♦♠❡ s♦rt ♦❢ ✐♥✜♥✐t❡ ♣r♦❞✉❝t❄ ❲❤❛t ❞♦❡s t❤✐s ♠❡❛♥❄ ❚❤❡ st✉❞② ♦❢ t❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ s❡❡♠s t♦ ❜❡ ❣❡♥❡r❛❧❧② r❛t❤❡r ❞✐✣❝✉❧t✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▲✐♠✐t❛t✐♦♥s ♦❢ ❢❛❝t♦r✐③❛t✐♦♥

❚❤❡r❡ ❛r❡ ❝❡rt❛✐♥❧② ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s t❤❛t ❛r❡ ♥♦t ✐s♦♠❡tr✐❝ t♦ ❛ ✜♥✐t❡ ♣r♦❞✉❝t ♦❢ ✜♥✐t❡❧② ♠❛♥② ✐rr❡❞✉❝✐❜❧❡ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s ✭❡✳❣✳ X :✏ ➧

kPNr0, aks✱ ✇❤❡r❡ ➦ kPN ak ➔ ✽✱ ✇✐t❤

rX♣x✶, x✷q :✏ ➦

kPN ⑤x✶ k ✁ x✷ k⑤✮✳

❆r❡ t❤❡r❡ ✭✉♥✐q✉❡✮ ❢❛❝t♦r✐③❛t✐♦♥s ✉s✐♥❣ s♦♠❡ s♦rt ♦❢ ✐♥✜♥✐t❡ ♣r♦❞✉❝t❄ ❲❤❛t ❞♦❡s t❤✐s ♠❡❛♥❄ ❚❤❡ st✉❞② ♦❢ t❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ s❡❡♠s t♦ ❜❡ ❣❡♥❡r❛❧❧② r❛t❤❡r ❞✐✣❝✉❧t✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❥✉st ❛ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ♣X, rXq ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µX t❤❛t ❤❛s ❢✉❧❧ s✉♣♣♦rt✳ ❚✇♦ s✉❝❤ s♣❛❝❡s ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❛r❡ ✐s♦♠❡tr✐❝ ❛s ♠❡tr✐❝ s♣❛❝❡s ✈✐❛ ❛♥ ✐s♦♠❡tr② t❤❛t ♠❛♣s t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ ✜rst s♣❛❝❡ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ s❡❝♦♥❞✳ ❉❡♥♦t❡ ❜② M t❤❡ s❡t ♦❢ s✉❝❤ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✳ ❲❡ ❞♦ ♥♦t ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss X P M ❛♥❞ ❛ r❡♣r❡s❡♥t❛t✐✈❡ tr✐♣❧❡ ♣X, rX, µXq✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❥✉st ❛ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ♣X, rXq ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µX t❤❛t ❤❛s ❢✉❧❧ s✉♣♣♦rt✳ ❚✇♦ s✉❝❤ s♣❛❝❡s ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❛r❡ ✐s♦♠❡tr✐❝ ❛s ♠❡tr✐❝ s♣❛❝❡s ✈✐❛ ❛♥ ✐s♦♠❡tr② t❤❛t ♠❛♣s t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ ✜rst s♣❛❝❡ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ s❡❝♦♥❞✳ ❉❡♥♦t❡ ❜② M t❤❡ s❡t ♦❢ s✉❝❤ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✳ ❲❡ ❞♦ ♥♦t ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss X P M ❛♥❞ ❛ r❡♣r❡s❡♥t❛t✐✈❡ tr✐♣❧❡ ♣X, rX, µXq✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❥✉st ❛ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ♣X, rXq ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µX t❤❛t ❤❛s ❢✉❧❧ s✉♣♣♦rt✳ ❚✇♦ s✉❝❤ s♣❛❝❡s ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❛r❡ ✐s♦♠❡tr✐❝ ❛s ♠❡tr✐❝ s♣❛❝❡s ✈✐❛ ❛♥ ✐s♦♠❡tr② t❤❛t ♠❛♣s t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ ✜rst s♣❛❝❡ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ s❡❝♦♥❞✳ ❉❡♥♦t❡ ❜② M t❤❡ s❡t ♦❢ s✉❝❤ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✳ ❲❡ ❞♦ ♥♦t ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss X P M ❛♥❞ ❛ r❡♣r❡s❡♥t❛t✐✈❡ tr✐♣❧❡ ♣X, rX, µXq✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡ ✐s ❥✉st ❛ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ♣X, rXq ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µX t❤❛t ❤❛s ❢✉❧❧ s✉♣♣♦rt✳ ❚✇♦ s✉❝❤ s♣❛❝❡s ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❛r❡ ✐s♦♠❡tr✐❝ ❛s ♠❡tr✐❝ s♣❛❝❡s ✈✐❛ ❛♥ ✐s♦♠❡tr② t❤❛t ♠❛♣s t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ ✜rst s♣❛❝❡ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ s❡❝♦♥❞✳ ❉❡♥♦t❡ ❜② M t❤❡ s❡t ♦❢ s✉❝❤ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✳ ❲❡ ❞♦ ♥♦t ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss X P M ❛♥❞ ❛ r❡♣r❡s❡♥t❛t✐✈❡ tr✐♣❧❡ ♣X, rX, µXq✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❲❤❡♥ ❛r❡ t✇♦ ♠❡tr✐❝ ♠❡❛s✉r❡s s♣❛❝❡s ❡q✉✐✈❛❧❡♥t❄

• r♦♠♦✈ ❛♥❞ ❱❡rs❤✐❦ s❤♦✇❡❞ t❤❛t✱ ✐♥ ♣r♦❜❛❜✐❧✐st✲s♣❡❛❦✱ ❛ ♠❡tr✐❝ ♠❡❛s✉r❡

s♣❛❝❡ ♣X, rX, µXq ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥✜♥✐t❡ r❛♥❞♦♠ ♠❛tr✐① ♦❢ ❞✐st❛♥❝❡s ♣rX♣ξi, ξjqq♣i,jqPN✂N, ✇❤❡r❡ ♣ξkqkPN ✐s ❛♥ ✐✳✐✳❞✳ s❛♠♣❧❡ ♦❢ ♣♦✐♥ts ✐♥ X ✇✐t❤ ❝♦♠♠♦♥ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ µX✳ ■♥ ♥♦♥✲♣r♦❜❛❜✐❧✐st✲s♣❡❛❦✱ ♣X, rX, µXq ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♣✉s❤✲❢♦r✇❛r❞ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ❜N

X

❜② t❤❡ ❢✉♥❝t✐♦♥ XN ◗ ♣xkqkPN ÞÑ ♣rX♣xi, xjqqi,jPN✂N P RN✂N

• .

❚❤✐s ❝♦♥❝✐s❡ ❝♦♥❞✐t✐♦♥ ❢♦r ❡q✉✐✈❛❧❡♥❝❡ ♠❛❦❡s ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s ❝♦♥s✐❞❡r❛❜❧② ❡❛s✐❡r t♦ st✉❞② t❤❛♥ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡s ♣❡r s❡✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❲❤❡♥ ❛r❡ t✇♦ ♠❡tr✐❝ ♠❡❛s✉r❡s s♣❛❝❡s ❡q✉✐✈❛❧❡♥t❄

• r♦♠♦✈ ❛♥❞ ❱❡rs❤✐❦ s❤♦✇❡❞ t❤❛t✱ ✐♥ ♣r♦❜❛❜✐❧✐st✲s♣❡❛❦✱ ❛ ♠❡tr✐❝ ♠❡❛s✉r❡

s♣❛❝❡ ♣X, rX, µXq ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥✜♥✐t❡ r❛♥❞♦♠ ♠❛tr✐① ♦❢ ❞✐st❛♥❝❡s ♣rX♣ξi, ξjqq♣i,jqPN✂N, ✇❤❡r❡ ♣ξkqkPN ✐s ❛♥ ✐✳✐✳❞✳ s❛♠♣❧❡ ♦❢ ♣♦✐♥ts ✐♥ X ✇✐t❤ ❝♦♠♠♦♥ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ µX✳ ■♥ ♥♦♥✲♣r♦❜❛❜✐❧✐st✲s♣❡❛❦✱ ♣X, rX, µXq ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♣✉s❤✲❢♦r✇❛r❞ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ❜N

X

❜② t❤❡ ❢✉♥❝t✐♦♥ XN ◗ ♣xkqkPN ÞÑ ♣rX♣xi, xjqqi,jPN✂N P RN✂N

• .

❚❤✐s ❝♦♥❝✐s❡ ❝♦♥❞✐t✐♦♥ ❢♦r ❡q✉✐✈❛❧❡♥❝❡ ♠❛❦❡s ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s ❝♦♥s✐❞❡r❛❜❧② ❡❛s✐❡r t♦ st✉❞② t❤❛♥ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡s ♣❡r s❡✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❲❤❡♥ ❛r❡ t✇♦ ♠❡tr✐❝ ♠❡❛s✉r❡s s♣❛❝❡s ❡q✉✐✈❛❧❡♥t❄

• r♦♠♦✈ ❛♥❞ ❱❡rs❤✐❦ s❤♦✇❡❞ t❤❛t✱ ✐♥ ♣r♦❜❛❜✐❧✐st✲s♣❡❛❦✱ ❛ ♠❡tr✐❝ ♠❡❛s✉r❡

s♣❛❝❡ ♣X, rX, µXq ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥✜♥✐t❡ r❛♥❞♦♠ ♠❛tr✐① ♦❢ ❞✐st❛♥❝❡s ♣rX♣ξi, ξjqq♣i,jqPN✂N, ✇❤❡r❡ ♣ξkqkPN ✐s ❛♥ ✐✳✐✳❞✳ s❛♠♣❧❡ ♦❢ ♣♦✐♥ts ✐♥ X ✇✐t❤ ❝♦♠♠♦♥ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ µX✳ ■♥ ♥♦♥✲♣r♦❜❛❜✐❧✐st✲s♣❡❛❦✱ ♣X, rX, µXq ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♣✉s❤✲❢♦r✇❛r❞ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ❜N

X

❜② t❤❡ ❢✉♥❝t✐♦♥ XN ◗ ♣xkqkPN ÞÑ ♣rX♣xi, xjqqi,jPN✂N P RN✂N

• .

❚❤✐s ❝♦♥❝✐s❡ ❝♦♥❞✐t✐♦♥ ❢♦r ❡q✉✐✈❛❧❡♥❝❡ ♠❛❦❡s ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s ❝♦♥s✐❞❡r❛❜❧② ❡❛s✐❡r t♦ st✉❞② t❤❛♥ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡s ♣❡r s❡✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ❜✐♥❛r② ♦♣❡r❛t✐♦♥

• ✐✈❡♥ t✇♦ ❡❧❡♠❡♥ts Y ✏ ♣Y, rY , µY q ❛♥❞ Z ✏ ♣Z, rZ, µZq ♦❢ M✱ ❧❡t Y ❵ Z

❜❡ X ✏ ♣X, rX, µXq P M✱ ✇❤❡r❡

X :✏ Y ✂ Z✱ rX :✏ rY ❵ rZ✱ ✇❤❡r❡ ♣rY ❵ rZq♣♣y✶, z✶q, ♣y✷, z✷qq ✏ rY ♣y✶, y✷q rZ♣z✶, z✷q ❢♦r ♣y✶, z✶q, ♣y✷, z✷q P Y ✂ Z✮✱ µX :✏ µY ❜ µZ✳

❚❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ ✐s ❛ss♦❝✐❛t✐✈❡ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡✳ ❚❤❡ ✐s♦♠❡tr② ❝❧❛ss ♦❢ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s E t❤❛t ❡❛❝❤ ❝♦♥s✐st ♦❢ ❛ s✐♥❣❧❡ ♣♦✐♥t ✇✐t❤ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ♠❡tr✐❝ ❛♥❞ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡♠ ✐s t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t✳ ❚❤✉s✱ ♣M, ❵q ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ❛♥ ✐❞❡♥t✐t② ✭✐✳❡✳ ❛ ♠♦♥♦✐❞✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ❜✐♥❛r② ♦♣❡r❛t✐♦♥

• ✐✈❡♥ t✇♦ ❡❧❡♠❡♥ts Y ✏ ♣Y, rY , µY q ❛♥❞ Z ✏ ♣Z, rZ, µZq ♦❢ M✱ ❧❡t Y ❵ Z

❜❡ X ✏ ♣X, rX, µXq P M✱ ✇❤❡r❡

X :✏ Y ✂ Z✱ rX :✏ rY ❵ rZ✱ ✇❤❡r❡ ♣rY ❵ rZq♣♣y✶, z✶q, ♣y✷, z✷qq ✏ rY ♣y✶, y✷q rZ♣z✶, z✷q ❢♦r ♣y✶, z✶q, ♣y✷, z✷q P Y ✂ Z✮✱ µX :✏ µY ❜ µZ✳

❚❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ ✐s ❛ss♦❝✐❛t✐✈❡ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡✳ ❚❤❡ ✐s♦♠❡tr② ❝❧❛ss ♦❢ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s E t❤❛t ❡❛❝❤ ❝♦♥s✐st ♦❢ ❛ s✐♥❣❧❡ ♣♦✐♥t ✇✐t❤ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ♠❡tr✐❝ ❛♥❞ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡♠ ✐s t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t✳ ❚❤✉s✱ ♣M, ❵q ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ❛♥ ✐❞❡♥t✐t② ✭✐✳❡✳ ❛ ♠♦♥♦✐❞✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ❜✐♥❛r② ♦♣❡r❛t✐♦♥

• ✐✈❡♥ t✇♦ ❡❧❡♠❡♥ts Y ✏ ♣Y, rY , µY q ❛♥❞ Z ✏ ♣Z, rZ, µZq ♦❢ M✱ ❧❡t Y ❵ Z

❜❡ X ✏ ♣X, rX, µXq P M✱ ✇❤❡r❡

X :✏ Y ✂ Z✱ rX :✏ rY ❵ rZ✱ ✇❤❡r❡ ♣rY ❵ rZq♣♣y✶, z✶q, ♣y✷, z✷qq ✏ rY ♣y✶, y✷q rZ♣z✶, z✷q ❢♦r ♣y✶, z✶q, ♣y✷, z✷q P Y ✂ Z✮✱ µX :✏ µY ❜ µZ✳

❚❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ ✐s ❛ss♦❝✐❛t✐✈❡ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡✳ ❚❤❡ ✐s♦♠❡tr② ❝❧❛ss ♦❢ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s E t❤❛t ❡❛❝❤ ❝♦♥s✐st ♦❢ ❛ s✐♥❣❧❡ ♣♦✐♥t ✇✐t❤ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ♠❡tr✐❝ ❛♥❞ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡♠ ✐s t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t✳ ❚❤✉s✱ ♣M, ❵q ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ❛♥ ✐❞❡♥t✐t② ✭✐✳❡✳ ❛ ♠♦♥♦✐❞✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ❜✐♥❛r② ♦♣❡r❛t✐♦♥

• ✐✈❡♥ t✇♦ ❡❧❡♠❡♥ts Y ✏ ♣Y, rY , µY q ❛♥❞ Z ✏ ♣Z, rZ, µZq ♦❢ M✱ ❧❡t Y ❵ Z

❜❡ X ✏ ♣X, rX, µXq P M✱ ✇❤❡r❡

X :✏ Y ✂ Z✱ rX :✏ rY ❵ rZ✱ ✇❤❡r❡ ♣rY ❵ rZq♣♣y✶, z✶q, ♣y✷, z✷qq ✏ rY ♣y✶, y✷q rZ♣z✶, z✷q ❢♦r ♣y✶, z✶q, ♣y✷, z✷q P Y ✂ Z✮✱ µX :✏ µY ❜ µZ✳

❚❤✐s ❜✐♥❛r② ♦♣❡r❛t✐♦♥ ✐s ❛ss♦❝✐❛t✐✈❡ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡✳ ❚❤❡ ✐s♦♠❡tr② ❝❧❛ss ♦❢ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s E t❤❛t ❡❛❝❤ ❝♦♥s✐st ♦❢ ❛ s✐♥❣❧❡ ♣♦✐♥t ✇✐t❤ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ♠❡tr✐❝ ❛♥❞ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡♠ ✐s t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t✳ ❚❤✉s✱ ♣M, ❵q ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ❛♥ ✐❞❡♥t✐t② ✭✐✳❡✳ ❛ ♠♦♥♦✐❞✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❚♦ ♣❛r❛♣❤r❛s❡ ❚✳❲✳ ❑ör♥❡r

✏❚❤❡ ❡❛s❡ ✇✐t❤ ✇❤✐❝❤ ✇❡ ♣r♦✈❡❞ ❬t❤❡ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠❪ ❡①♣❧❛✐♥s ✇❤② ❋♦✉r✐❡r ❛♥❛❧②s✐s ♣❧❛②s ❛ rô❧❡ ✐♥ ♣r♦❜❛❜✐❧✐t② t❤❡♦r② t❤❛t ✐♥ ♦t❤❡r ❜r❛♥❝❤❡s ♦❢ ♠❛t❤❡♠❛t✐❝s ✐s ♣❧❛②❡❞ ❜② t❤♦✉❣❤t✳✑

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❡♠✐❝❤❛r❛❝t❡rs

❆ s❡♠✐❝❤❛r❛❝t❡r ✐s ❛ ♠❛♣ χ : M Ñ r0, 1s s✉❝❤ t❤❛t χ♣Y ❵ Zq ✏ χ♣Yqχ♣Zq ❢♦r ❛❧❧ Y, Z P M✳ ❉❡♥♦t❡ ❜② A t❤❡ ❢❛♠✐❧② ♦❢ ❛rr❛②s ♦❢ t❤❡ ❢♦r♠ A ✏ ♣aijq1↕i➔j↕n P R♣n

2q

• ❢♦r

n P N✳ ❋♦r ❡❛❝❤ A P A ❞❡✜♥❡ ❛ s❡♠✐❝❤❛r❛❝t❡r χA ❜② χA♣♣X, rX, µXqq :✏ ➺

Xn exp

✄ ✁ ➳

1↕i➔j↕n

aijrX♣xi, xjq ☛ µ❜n

X ♣dxq.

❚✇♦ ❡❧❡♠❡♥ts X, Y P M ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ χA♣Xq ✏ χA♣Yq ❢♦r ❛❧❧ A P A✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❡♠✐❝❤❛r❛❝t❡rs

❆ s❡♠✐❝❤❛r❛❝t❡r ✐s ❛ ♠❛♣ χ : M Ñ r0, 1s s✉❝❤ t❤❛t χ♣Y ❵ Zq ✏ χ♣Yqχ♣Zq ❢♦r ❛❧❧ Y, Z P M✳ ❉❡♥♦t❡ ❜② A t❤❡ ❢❛♠✐❧② ♦❢ ❛rr❛②s ♦❢ t❤❡ ❢♦r♠ A ✏ ♣aijq1↕i➔j↕n P R♣n

2q

• ❢♦r

n P N✳ ❋♦r ❡❛❝❤ A P A ❞❡✜♥❡ ❛ s❡♠✐❝❤❛r❛❝t❡r χA ❜② χA♣♣X, rX, µXqq :✏ ➺

Xn exp

✄ ✁ ➳

1↕i➔j↕n

aijrX♣xi, xjq ☛ µ❜n

X ♣dxq.

❚✇♦ ❡❧❡♠❡♥ts X, Y P M ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ χA♣Xq ✏ χA♣Yq ❢♦r ❛❧❧ A P A✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❡♠✐❝❤❛r❛❝t❡rs

❆ s❡♠✐❝❤❛r❛❝t❡r ✐s ❛ ♠❛♣ χ : M Ñ r0, 1s s✉❝❤ t❤❛t χ♣Y ❵ Zq ✏ χ♣Yqχ♣Zq ❢♦r ❛❧❧ Y, Z P M✳ ❉❡♥♦t❡ ❜② A t❤❡ ❢❛♠✐❧② ♦❢ ❛rr❛②s ♦❢ t❤❡ ❢♦r♠ A ✏ ♣aijq1↕i➔j↕n P R♣n

2q

• ❢♦r

n P N✳ ❋♦r ❡❛❝❤ A P A ❞❡✜♥❡ ❛ s❡♠✐❝❤❛r❛❝t❡r χA ❜② χA♣♣X, rX, µXqq :✏ ➺

Xn exp

✄ ✁ ➳

1↕i➔j↕n

aijrX♣xi, xjq ☛ µ❜n

X ♣dxq.

❚✇♦ ❡❧❡♠❡♥ts X, Y P M ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ χA♣Xq ✏ χA♣Yq ❢♦r ❛❧❧ A P A✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❡♠✐❝❤❛r❛❝t❡rs

❆ s❡♠✐❝❤❛r❛❝t❡r ✐s ❛ ♠❛♣ χ : M Ñ r0, 1s s✉❝❤ t❤❛t χ♣Y ❵ Zq ✏ χ♣Yqχ♣Zq ❢♦r ❛❧❧ Y, Z P M✳ ❉❡♥♦t❡ ❜② A t❤❡ ❢❛♠✐❧② ♦❢ ❛rr❛②s ♦❢ t❤❡ ❢♦r♠ A ✏ ♣aijq1↕i➔j↕n P R♣n

2q

• ❢♦r

n P N✳ ❋♦r ❡❛❝❤ A P A ❞❡✜♥❡ ❛ s❡♠✐❝❤❛r❛❝t❡r χA ❜② χA♣♣X, rX, µXqq :✏ ➺

Xn exp

✄ ✁ ➳

1↕i➔j↕n

aijrX♣xi, xjq ☛ µ❜n

X ♣dxq.

❚✇♦ ❡❧❡♠❡♥ts X, Y P M ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ χA♣Xq ✏ χA♣Yq ❢♦r ❛❧❧ A P A✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

P✉tt✐♥❣ ❛ ♠❡tr✐❝ ♦♥ M

❊q✉✐♣ M ✇✐t❤ t❤❡ ●r♦♠♦✈✲Pr♦❤♦r♦✈ ♠❡tr✐❝ ♦❢ ●r❡✈❡♥✱ P❢❛✛❡❧❤✉❜❡r ✫ ❲✐♥t❡r ✭✷✵✵✾✮✳ ❚✇♦ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ❝❧♦s❡ ✐❢ t❤❡✐r r❛♥❞♦♠ ❞✐st❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ❝❧♦s❡ ✐♥ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ s♣❛❝❡ ♣M, dGPrq ✐s ❝♦♠♣❧❡t❡ ❛♥❞ s❡♣❛r❛❜❧❡ ✭❡✳❣✳ ✜♥✐t❡ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ r❛t✐♦♥❛❧ ❞✐st❛♥❝❡s ❛r❡ ❞❡♥s❡✮✳ dGPr♣X1 ❵ X2, Y1 ❵ Y2q ↕ dGPr♣X1, Y1q dGPr♣X2, Y2q ❛♥❞ s♦ ♣X, Yq ÞÑ X ❵ Y ✐s ❝♦♥t✐♥✉♦✉s✳ limnÑ✽ Xn ✏ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ limnÑ✽ χA♣Xnq ✏ χA♣Xq ❢♦r ❛❧❧ A P A✳ ◆♦t❡✿ ❑♥♦✇✐♥❣ t❤❛t limnÑ✽ χA♣Xnq ❡①✐sts ❢♦r ❛❧❧ A P A ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t limnÑ✽ Xn ❡①✐sts✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

P✉tt✐♥❣ ❛ ♠❡tr✐❝ ♦♥ M

❊q✉✐♣ M ✇✐t❤ t❤❡ ●r♦♠♦✈✲Pr♦❤♦r♦✈ ♠❡tr✐❝ ♦❢ ●r❡✈❡♥✱ P❢❛✛❡❧❤✉❜❡r ✫ ❲✐♥t❡r ✭✷✵✵✾✮✳ ❚✇♦ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ❝❧♦s❡ ✐❢ t❤❡✐r r❛♥❞♦♠ ❞✐st❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ❝❧♦s❡ ✐♥ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ s♣❛❝❡ ♣M, dGPrq ✐s ❝♦♠♣❧❡t❡ ❛♥❞ s❡♣❛r❛❜❧❡ ✭❡✳❣✳ ✜♥✐t❡ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ r❛t✐♦♥❛❧ ❞✐st❛♥❝❡s ❛r❡ ❞❡♥s❡✮✳ dGPr♣X1 ❵ X2, Y1 ❵ Y2q ↕ dGPr♣X1, Y1q dGPr♣X2, Y2q ❛♥❞ s♦ ♣X, Yq ÞÑ X ❵ Y ✐s ❝♦♥t✐♥✉♦✉s✳ limnÑ✽ Xn ✏ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ limnÑ✽ χA♣Xnq ✏ χA♣Xq ❢♦r ❛❧❧ A P A✳ ◆♦t❡✿ ❑♥♦✇✐♥❣ t❤❛t limnÑ✽ χA♣Xnq ❡①✐sts ❢♦r ❛❧❧ A P A ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t limnÑ✽ Xn ❡①✐sts✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

P✉tt✐♥❣ ❛ ♠❡tr✐❝ ♦♥ M

❊q✉✐♣ M ✇✐t❤ t❤❡ ●r♦♠♦✈✲Pr♦❤♦r♦✈ ♠❡tr✐❝ ♦❢ ●r❡✈❡♥✱ P❢❛✛❡❧❤✉❜❡r ✫ ❲✐♥t❡r ✭✷✵✵✾✮✳ ❚✇♦ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ❝❧♦s❡ ✐❢ t❤❡✐r r❛♥❞♦♠ ❞✐st❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ❝❧♦s❡ ✐♥ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ s♣❛❝❡ ♣M, dGPrq ✐s ❝♦♠♣❧❡t❡ ❛♥❞ s❡♣❛r❛❜❧❡ ✭❡✳❣✳ ✜♥✐t❡ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ r❛t✐♦♥❛❧ ❞✐st❛♥❝❡s ❛r❡ ❞❡♥s❡✮✳ dGPr♣X1 ❵ X2, Y1 ❵ Y2q ↕ dGPr♣X1, Y1q dGPr♣X2, Y2q ❛♥❞ s♦ ♣X, Yq ÞÑ X ❵ Y ✐s ❝♦♥t✐♥✉♦✉s✳ limnÑ✽ Xn ✏ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ limnÑ✽ χA♣Xnq ✏ χA♣Xq ❢♦r ❛❧❧ A P A✳ ◆♦t❡✿ ❑♥♦✇✐♥❣ t❤❛t limnÑ✽ χA♣Xnq ❡①✐sts ❢♦r ❛❧❧ A P A ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t limnÑ✽ Xn ❡①✐sts✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

P✉tt✐♥❣ ❛ ♠❡tr✐❝ ♦♥ M

❊q✉✐♣ M ✇✐t❤ t❤❡ ●r♦♠♦✈✲Pr♦❤♦r♦✈ ♠❡tr✐❝ ♦❢ ●r❡✈❡♥✱ P❢❛✛❡❧❤✉❜❡r ✫ ❲✐♥t❡r ✭✷✵✵✾✮✳ ❚✇♦ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ❝❧♦s❡ ✐❢ t❤❡✐r r❛♥❞♦♠ ❞✐st❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ❝❧♦s❡ ✐♥ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ s♣❛❝❡ ♣M, dGPrq ✐s ❝♦♠♣❧❡t❡ ❛♥❞ s❡♣❛r❛❜❧❡ ✭❡✳❣✳ ✜♥✐t❡ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ r❛t✐♦♥❛❧ ❞✐st❛♥❝❡s ❛r❡ ❞❡♥s❡✮✳ dGPr♣X1 ❵ X2, Y1 ❵ Y2q ↕ dGPr♣X1, Y1q dGPr♣X2, Y2q ❛♥❞ s♦ ♣X, Yq ÞÑ X ❵ Y ✐s ❝♦♥t✐♥✉♦✉s✳ limnÑ✽ Xn ✏ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ limnÑ✽ χA♣Xnq ✏ χA♣Xq ❢♦r ❛❧❧ A P A✳ ◆♦t❡✿ ❑♥♦✇✐♥❣ t❤❛t limnÑ✽ χA♣Xnq ❡①✐sts ❢♦r ❛❧❧ A P A ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t limnÑ✽ Xn ❡①✐sts✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

P✉tt✐♥❣ ❛ ♠❡tr✐❝ ♦♥ M

❊q✉✐♣ M ✇✐t❤ t❤❡ ●r♦♠♦✈✲Pr♦❤♦r♦✈ ♠❡tr✐❝ ♦❢ ●r❡✈❡♥✱ P❢❛✛❡❧❤✉❜❡r ✫ ❲✐♥t❡r ✭✷✵✵✾✮✳ ❚✇♦ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ❝❧♦s❡ ✐❢ t❤❡✐r r❛♥❞♦♠ ❞✐st❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ❝❧♦s❡ ✐♥ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ s♣❛❝❡ ♣M, dGPrq ✐s ❝♦♠♣❧❡t❡ ❛♥❞ s❡♣❛r❛❜❧❡ ✭❡✳❣✳ ✜♥✐t❡ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ r❛t✐♦♥❛❧ ❞✐st❛♥❝❡s ❛r❡ ❞❡♥s❡✮✳ dGPr♣X1 ❵ X2, Y1 ❵ Y2q ↕ dGPr♣X1, Y1q dGPr♣X2, Y2q ❛♥❞ s♦ ♣X, Yq ÞÑ X ❵ Y ✐s ❝♦♥t✐♥✉♦✉s✳ limnÑ✽ Xn ✏ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ limnÑ✽ χA♣Xnq ✏ χA♣Xq ❢♦r ❛❧❧ A P A✳ ◆♦t❡✿ ❑♥♦✇✐♥❣ t❤❛t limnÑ✽ χA♣Xnq ❡①✐sts ❢♦r ❛❧❧ A P A ❞♦❡s ♥♦t ✐♠♣❧② t❤❛t limnÑ✽ Xn ❡①✐sts✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

P✉tt✐♥❣ ❛ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ M

❉❡✜♥❡ ❛ ♣❛rt✐❛❧ ♦r❞❡r ↕ ♦♥ M ❜② ❞❡❝❧❛r✐♥❣ t❤❛t Y ↕ Z ✐❢ Z ✏ Y ❵ X ❢♦r s♦♠❡ X P M✳ ❚❤❛t ✐s✱ Y ↕ Z ✐❢ Y ✐s ❛ ✏❞✐✈✐s♦r✑ ♦❢ Z✳ ❋♦r ❛♥② Z P M✱ t❤❡ s❡t tY P M : Y ↕ Z✉ ✐s ❝♦♠♣❛❝t✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

P✉tt✐♥❣ ❛ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ M

❉❡✜♥❡ ❛ ♣❛rt✐❛❧ ♦r❞❡r ↕ ♦♥ M ❜② ❞❡❝❧❛r✐♥❣ t❤❛t Y ↕ Z ✐❢ Z ✏ Y ❵ X ❢♦r s♦♠❡ X P M✳ ❚❤❛t ✐s✱ Y ↕ Z ✐❢ Y ✐s ❛ ✏❞✐✈✐s♦r✑ ♦❢ Z✳ ❋♦r ❛♥② Z P M✱ t❤❡ s❡t tY P M : Y ↕ Z✉ ✐s ❝♦♠♣❛❝t✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛♥❝❡❧❧❛t✐✈✐t②

❚❤❡ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐❣r♦✉♣ ♣M, ❵q ✐s ❝❛♥❝❡❧❧❛t✐✈❡❀ t❤❛t ✐s✱ ✐❢ X, Y, Z✶, Z✷ P M s❛t✐s❢② X ✏ Y ❵ Z✶ ❛♥❞ X ✏ Y ❵ Z✷, t❤❡♥ Z✶ ✏ Z✷. ❚❤✐s ✐s ❜❡❝❛✉s❡ ❢♦r ❛❧❧ A P A χA♣YqχA♣Z✶q ✏ χA♣Xq ✏ χA♣YqχA♣Z✷q, ❛♥❞ s♦ χA♣Z✶q ✏ χA♣Z✷q✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t 0 ✏ dGPr♣X, Xq ✏ dGPr♣Y ❵ Z✶, Y ❵ Z✷q ✏ dGPr♣Z✶, Z✷q.

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛♥❝❡❧❧❛t✐✈✐t②

❚❤❡ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐❣r♦✉♣ ♣M, ❵q ✐s ❝❛♥❝❡❧❧❛t✐✈❡❀ t❤❛t ✐s✱ ✐❢ X, Y, Z✶, Z✷ P M s❛t✐s❢② X ✏ Y ❵ Z✶ ❛♥❞ X ✏ Y ❵ Z✷, t❤❡♥ Z✶ ✏ Z✷. ❚❤✐s ✐s ❜❡❝❛✉s❡ ❢♦r ❛❧❧ A P A χA♣YqχA♣Z✶q ✏ χA♣Xq ✏ χA♣YqχA♣Z✷q, ❛♥❞ s♦ χA♣Z✶q ✏ χA♣Z✷q✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t 0 ✏ dGPr♣X, Xq ✏ dGPr♣Y ❵ Z✶, Y ❵ Z✷q ✏ dGPr♣Z✶, Z✷q.

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈❛♥❝❡❧❧❛t✐✈✐t②

❚❤❡ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐❣r♦✉♣ ♣M, ❵q ✐s ❝❛♥❝❡❧❧❛t✐✈❡❀ t❤❛t ✐s✱ ✐❢ X, Y, Z✶, Z✷ P M s❛t✐s❢② X ✏ Y ❵ Z✶ ❛♥❞ X ✏ Y ❵ Z✷, t❤❡♥ Z✶ ✏ Z✷. ❚❤✐s ✐s ❜❡❝❛✉s❡ ❢♦r ❛❧❧ A P A χA♣YqχA♣Z✶q ✏ χA♣Xq ✏ χA♣YqχA♣Z✷q, ❛♥❞ s♦ χA♣Z✶q ✏ χA♣Z✷q✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t 0 ✏ dGPr♣X, Xq ✏ dGPr♣Y ❵ Z✶, Y ❵ Z✷q ✏ dGPr♣Z✶, Z✷q.

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▼❡❛s✉r✐♥❣ t❤❡ s✐③❡ ♦❢ ❛ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡

P✉t DA♣Xq :✏ ✁ log χA♣Xq ➙ 0 ❛♥❞ D♣Xq :✏ D1♣Xq ✏ ✁ log χ1♣Xq ✏ ✁ log ➺

X2 exp ♣✁rX♣x1, x2qq µ❜2 X ♣dxq.

P✉t R♣Xq :✏ ➺

X2♣rX♣x1, x2q ❫ 1q µ❜2 X ♣dxq.

❋♦r s✉✐t❛❜❧❡ ❝♦♥st❛♥ts✱ aD♣Xq ↕ DA♣Xq ↕ bD♣Xq✳

1 4R♣Xq ↕ dGPr♣X, Eq ↕

❛ R♣Xq✳ ❋♦r s✉✐t❛❜❧❡ ❝♦♥st❛♥ts✱ α♣D♣Xq ❫ 1q ↕ R♣Xq ↕ β♣D♣Xq ❫ 1q✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▼❡❛s✉r✐♥❣ t❤❡ s✐③❡ ♦❢ ❛ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡

P✉t DA♣Xq :✏ ✁ log χA♣Xq ➙ 0 ❛♥❞ D♣Xq :✏ D1♣Xq ✏ ✁ log χ1♣Xq ✏ ✁ log ➺

X2 exp ♣✁rX♣x1, x2qq µ❜2 X ♣dxq.

P✉t R♣Xq :✏ ➺

X2♣rX♣x1, x2q ❫ 1q µ❜2 X ♣dxq.

❋♦r s✉✐t❛❜❧❡ ❝♦♥st❛♥ts✱ aD♣Xq ↕ DA♣Xq ↕ bD♣Xq✳

1 4R♣Xq ↕ dGPr♣X, Eq ↕

❛ R♣Xq✳ ❋♦r s✉✐t❛❜❧❡ ❝♦♥st❛♥ts✱ α♣D♣Xq ❫ 1q ↕ R♣Xq ↕ β♣D♣Xq ❫ 1q✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▼❡❛s✉r✐♥❣ t❤❡ s✐③❡ ♦❢ ❛ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡

P✉t DA♣Xq :✏ ✁ log χA♣Xq ➙ 0 ❛♥❞ D♣Xq :✏ D1♣Xq ✏ ✁ log χ1♣Xq ✏ ✁ log ➺

X2 exp ♣✁rX♣x1, x2qq µ❜2 X ♣dxq.

P✉t R♣Xq :✏ ➺

X2♣rX♣x1, x2q ❫ 1q µ❜2 X ♣dxq.

❋♦r s✉✐t❛❜❧❡ ❝♦♥st❛♥ts✱ aD♣Xq ↕ DA♣Xq ↕ bD♣Xq✳

1 4R♣Xq ↕ dGPr♣X, Eq ↕

❛ R♣Xq✳ ❋♦r s✉✐t❛❜❧❡ ❝♦♥st❛♥ts✱ α♣D♣Xq ❫ 1q ↕ R♣Xq ↕ β♣D♣Xq ❫ 1q✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

▼❡❛s✉r✐♥❣ t❤❡ s✐③❡ ♦❢ ❛ ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡

P✉t DA♣Xq :✏ ✁ log χA♣Xq ➙ 0 ❛♥❞ D♣Xq :✏ D1♣Xq ✏ ✁ log χ1♣Xq ✏ ✁ log ➺

X2 exp ♣✁rX♣x1, x2qq µ❜2 X ♣dxq.

P✉t R♣Xq :✏ ➺

X2♣rX♣x1, x2q ❫ 1q µ❜2 X ♣dxq.

❋♦r s✉✐t❛❜❧❡ ❝♦♥st❛♥ts✱ aD♣Xq ↕ DA♣Xq ↕ bD♣Xq✳

1 4R♣Xq ↕ dGPr♣X, Eq ↕

❛ R♣Xq✳ ❋♦r s✉✐t❛❜❧❡ ❝♦♥st❛♥ts✱ α♣D♣Xq ❫ 1q ↕ R♣Xq ↕ β♣D♣Xq ❫ 1q✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈♦♥✈❡r❣❡♥❝❡ ♦❢ s✉♠s

❙✉♣♣♦s❡ t❤❛t ♣XnqnPN ✐s ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ✏ Y ❢♦r s♦♠❡ Y P M✳ ■❢ ♣X ✶

nqnPN ✐s ❛ s❡q✉❡♥❝❡ t❤❛t ✐s ♦❜t❛✐♥❡❞ ❜② r❡✲♦r❞❡r✐♥❣

t❤❡ s❡q✉❡♥❝❡ ♣XnqnPN✱ t❤❡♥ limnÑ✽ X ✶

0 ❵ ☎ ☎ ☎ ❵ X ✶ n ✏ Y ❛❧s♦✳

❚❤❡ ❧✐♠✐t limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ❡①✐sts ✐❢ ❛♥❞ ♦♥❧② ✐❢ ➦

n D♣Xnq ➔ ✽

✭❡q✉✐✈❛❧❡♥t❧②✱ ➦

n R♣Xnq ➔ ✽✮✳

❙♦✱ ✇❡ ❝❛♥ ♠❛❦❡ s❡♥s❡ ♦❢ Ð

sPS Xs ❢♦r ❛♥② ❢❛♠✐❧② ♣XqsPS ✇✐t❤♦✉t

s♣❡❝✐❢②✐♥❣ t❤❡ ✏♦r❞❡r ♦❢ s✉♠♠❛t✐♦♥✑✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈♦♥✈❡r❣❡♥❝❡ ♦❢ s✉♠s

❙✉♣♣♦s❡ t❤❛t ♣XnqnPN ✐s ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ✏ Y ❢♦r s♦♠❡ Y P M✳ ■❢ ♣X ✶

nqnPN ✐s ❛ s❡q✉❡♥❝❡ t❤❛t ✐s ♦❜t❛✐♥❡❞ ❜② r❡✲♦r❞❡r✐♥❣

t❤❡ s❡q✉❡♥❝❡ ♣XnqnPN✱ t❤❡♥ limnÑ✽ X ✶

0 ❵ ☎ ☎ ☎ ❵ X ✶ n ✏ Y ❛❧s♦✳

❚❤❡ ❧✐♠✐t limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ❡①✐sts ✐❢ ❛♥❞ ♦♥❧② ✐❢ ➦

n D♣Xnq ➔ ✽

✭❡q✉✐✈❛❧❡♥t❧②✱ ➦

n R♣Xnq ➔ ✽✮✳

❙♦✱ ✇❡ ❝❛♥ ♠❛❦❡ s❡♥s❡ ♦❢ Ð

sPS Xs ❢♦r ❛♥② ❢❛♠✐❧② ♣XqsPS ✇✐t❤♦✉t

s♣❡❝✐❢②✐♥❣ t❤❡ ✏♦r❞❡r ♦❢ s✉♠♠❛t✐♦♥✑✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈♦♥✈❡r❣❡♥❝❡ ♦❢ s✉♠s

❙✉♣♣♦s❡ t❤❛t ♣XnqnPN ✐s ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ✏ Y ❢♦r s♦♠❡ Y P M✳ ■❢ ♣X ✶

nqnPN ✐s ❛ s❡q✉❡♥❝❡ t❤❛t ✐s ♦❜t❛✐♥❡❞ ❜② r❡✲♦r❞❡r✐♥❣

t❤❡ s❡q✉❡♥❝❡ ♣XnqnPN✱ t❤❡♥ limnÑ✽ X ✶

0 ❵ ☎ ☎ ☎ ❵ X ✶ n ✏ Y ❛❧s♦✳

❚❤❡ ❧✐♠✐t limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ❡①✐sts ✐❢ ❛♥❞ ♦♥❧② ✐❢ ➦

n D♣Xnq ➔ ✽

✭❡q✉✐✈❛❧❡♥t❧②✱ ➦

n R♣Xnq ➔ ✽✮✳

❙♦✱ ✇❡ ❝❛♥ ♠❛❦❡ s❡♥s❡ ♦❢ Ð

sPS Xs ❢♦r ❛♥② ❢❛♠✐❧② ♣XqsPS ✇✐t❤♦✉t

s♣❡❝✐❢②✐♥❣ t❤❡ ✏♦r❞❡r ♦❢ s✉♠♠❛t✐♦♥✑✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts

❆♥ ❡❧❡♠❡♥t X P M ✐s ✐rr❡❞✉❝✐❜❧❡ ✐❢ X ✘ E ❛♥❞ Y ↕ X ❢♦r Y P M ✐♠♣❧✐❡s t❤❛t Y ✐s ❡✐t❤❡r E ♦r X✳ ■t ✐s ♥♦t ❝❧❡❛r ❛ ♣r✐♦r✐ t❤❛t t❤❡r❡ ❛r❡ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s❡♠✐❣r♦✉♣ R ✇✐t❤ t❤❡ ✉s✉❛❧ ❛❞❞✐t✐♦♥ ♦♣❡r❛t✐♦♥ ❤❛s ♥♦ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ t❤❡ ❛♥❛❧♦❣♦✉s s❡♥s❡✳ ■❢ X P M③tE✉✱ t❤❡♥ t❤❡r❡ ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥t Y P M ✇✐t❤ Y ↕ X ✕ t❤✐s s❡❡♠s t♦ ❜❡ ♥♦t ❛t ❛❧❧ ♦❜✈✐♦✉s ❛♥❞ t♦ r❡❧② ♦♥ s♦♠❡ ♥♦♥tr✐✈✐❛❧ st♦❝❤❛st✐❝ ❛♥❛❧②s✐s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X P M③tE✉✱ t❤❡♥ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t❤❛t ❢♦r ❛❧❧ n P N t❤❡r❡ ❡①✐sts Xn s✉❝❤ t❤❛t X ✏ Xn ❵ ☎ ☎ ☎ ❵ Xn ✭n t❡r♠s✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts

❆♥ ❡❧❡♠❡♥t X P M ✐s ✐rr❡❞✉❝✐❜❧❡ ✐❢ X ✘ E ❛♥❞ Y ↕ X ❢♦r Y P M ✐♠♣❧✐❡s t❤❛t Y ✐s ❡✐t❤❡r E ♦r X✳ ■t ✐s ♥♦t ❝❧❡❛r ❛ ♣r✐♦r✐ t❤❛t t❤❡r❡ ❛r❡ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s❡♠✐❣r♦✉♣ R ✇✐t❤ t❤❡ ✉s✉❛❧ ❛❞❞✐t✐♦♥ ♦♣❡r❛t✐♦♥ ❤❛s ♥♦ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ t❤❡ ❛♥❛❧♦❣♦✉s s❡♥s❡✳ ■❢ X P M③tE✉✱ t❤❡♥ t❤❡r❡ ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥t Y P M ✇✐t❤ Y ↕ X ✕ t❤✐s s❡❡♠s t♦ ❜❡ ♥♦t ❛t ❛❧❧ ♦❜✈✐♦✉s ❛♥❞ t♦ r❡❧② ♦♥ s♦♠❡ ♥♦♥tr✐✈✐❛❧ st♦❝❤❛st✐❝ ❛♥❛❧②s✐s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X P M③tE✉✱ t❤❡♥ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t❤❛t ❢♦r ❛❧❧ n P N t❤❡r❡ ❡①✐sts Xn s✉❝❤ t❤❛t X ✏ Xn ❵ ☎ ☎ ☎ ❵ Xn ✭n t❡r♠s✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts

❆♥ ❡❧❡♠❡♥t X P M ✐s ✐rr❡❞✉❝✐❜❧❡ ✐❢ X ✘ E ❛♥❞ Y ↕ X ❢♦r Y P M ✐♠♣❧✐❡s t❤❛t Y ✐s ❡✐t❤❡r E ♦r X✳ ■t ✐s ♥♦t ❝❧❡❛r ❛ ♣r✐♦r✐ t❤❛t t❤❡r❡ ❛r❡ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s❡♠✐❣r♦✉♣ R ✇✐t❤ t❤❡ ✉s✉❛❧ ❛❞❞✐t✐♦♥ ♦♣❡r❛t✐♦♥ ❤❛s ♥♦ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ t❤❡ ❛♥❛❧♦❣♦✉s s❡♥s❡✳ ■❢ X P M③tE✉✱ t❤❡♥ t❤❡r❡ ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥t Y P M ✇✐t❤ Y ↕ X ✕ t❤✐s s❡❡♠s t♦ ❜❡ ♥♦t ❛t ❛❧❧ ♦❜✈✐♦✉s ❛♥❞ t♦ r❡❧② ♦♥ s♦♠❡ ♥♦♥tr✐✈✐❛❧ st♦❝❤❛st✐❝ ❛♥❛❧②s✐s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X P M③tE✉✱ t❤❡♥ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t❤❛t ❢♦r ❛❧❧ n P N t❤❡r❡ ❡①✐sts Xn s✉❝❤ t❤❛t X ✏ Xn ❵ ☎ ☎ ☎ ❵ Xn ✭n t❡r♠s✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts

❆♥ ❡❧❡♠❡♥t X P M ✐s ✐rr❡❞✉❝✐❜❧❡ ✐❢ X ✘ E ❛♥❞ Y ↕ X ❢♦r Y P M ✐♠♣❧✐❡s t❤❛t Y ✐s ❡✐t❤❡r E ♦r X✳ ■t ✐s ♥♦t ❝❧❡❛r ❛ ♣r✐♦r✐ t❤❛t t❤❡r❡ ❛r❡ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s❡♠✐❣r♦✉♣ R ✇✐t❤ t❤❡ ✉s✉❛❧ ❛❞❞✐t✐♦♥ ♦♣❡r❛t✐♦♥ ❤❛s ♥♦ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ t❤❡ ❛♥❛❧♦❣♦✉s s❡♥s❡✳ ■❢ X P M③tE✉✱ t❤❡♥ t❤❡r❡ ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥t Y P M ✇✐t❤ Y ↕ X ✕ t❤✐s s❡❡♠s t♦ ❜❡ ♥♦t ❛t ❛❧❧ ♦❜✈✐♦✉s ❛♥❞ t♦ r❡❧② ♦♥ s♦♠❡ ♥♦♥tr✐✈✐❛❧ st♦❝❤❛st✐❝ ❛♥❛❧②s✐s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X P M③tE✉✱ t❤❡♥ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t❤❛t ❢♦r ❛❧❧ n P N t❤❡r❡ ❡①✐sts Xn s✉❝❤ t❤❛t X ✏ Xn ❵ ☎ ☎ ☎ ❵ Xn ✭n t❡r♠s✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈♦♥❝r❡t❡ ❡①❛♠♣❧❡s ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts✿ t♦t❛❧❧② ❣❡♦❞❡s✐❝ s♣❛❝❡s

❆ ♠❡tr✐❝ s♣❛❝❡ ♣W, rW q ✐s t♦t❛❧❧② ❣❡♦❞❡s✐❝ ✐❢ ❛♥② t✇♦ ♣♦✐♥ts ♦❢ W ❛r❡ ❥♦✐♥❡❞ ❜② ❛ ✉♥✐q✉❡ ❣❡♦❞❡s✐❝ s❡❣♠❡♥t✳ ❆♥② ♥♦♥tr✐✈✐❛❧ ❝❧♦s❡❞ s✉❜s❡t X ♦❢ ❛ t♦t❛❧❧② ❣❡♦❞❡s✐❝✱ ❝♦♠♣❧❡t❡✱ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ♣W, rW q ✐s ✐rr❡❞✉❝✐❜❧❡✱ ♥♦ ♠❛tt❡r ✇❤❛t ♠❡❛s✉r❡ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤✳ ❲❤②❄ ■❢ ♣X, rW q ✐s ✐s♦♠❡tr✐❝ t♦ ♣Y ✂ Z, rY ❵ rZq ❢♦r ♥♦♥tr✐✈✐❛❧ Y ❛♥❞ Z✱ t❤❡♥ t❤❡r❡ ✇✐❧❧ ❜❡ ❢♦✉r ❞✐st✐♥❝t ♣♦✐♥ts a, b, c, d ✐♥ X t❤❛t ❛r❡ ✐s♦♠❡tr✐❝ ✐♠❛❣❡s ♦❢ ♣♦✐♥ts ♦❢ t❤❡ ❢♦r♠ ♣y✶, z✶q✱ ♣y✷, z✶q✱ ♣y✶, z✷q✱ ♣y✷, z✷q ✐♥ Y ✂ Z✳ ❚❤✉s✱ rW ♣a, bq ✏ rW ♣c, dq, rW ♣a, cq ✏ rW ♣b, dq, rW ♣a, dq ✏ rW ♣a, bq rW ♣b, dq, rW ♣a, dq ✏ rW ♣a, cq rW ♣c, dq, rW ♣b, cq ✏ rW ♣a, bq rW ♣c, aq, rW ♣b, cq ✏ rW ♣b, dq rW ♣c, dq. ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ t❤✐r❞ ❛♥❞ ❢♦✉rt❤ ❡q✉❛❧✐t✐❡s t❤❛t b ❛♥❞ c ❛r❡ ♦♥ t❤❡ ❣❡♦❞❡s✐❝ s❡❣♠❡♥t ❜❡t✇❡❡♥ a ❛♥❞ d✳ ❲❡ ♠❛② t❤❡r❡❢♦r❡ s✉♣♣♦s❡ t❤❛t ♣W, rW q ✐s ❛ ❝❧♦s❡❞ s✉❜✐♥t❡r✈❛❧ ♦❢ R ❛♥❞✱ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ t❤❛t a ➔ b ➔ c ➔ d✳ ❚❤❡ ✜❢t❤ ❛♥❞ s✐①t❤ ❡q✉❛❧✐t✐❡s ❛r❡ t❤❡♥ ✐♠♣♦ss✐❜❧❡✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈♦♥❝r❡t❡ ❡①❛♠♣❧❡s ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts✿ t♦t❛❧❧② ❣❡♦❞❡s✐❝ s♣❛❝❡s

❆ ♠❡tr✐❝ s♣❛❝❡ ♣W, rW q ✐s t♦t❛❧❧② ❣❡♦❞❡s✐❝ ✐❢ ❛♥② t✇♦ ♣♦✐♥ts ♦❢ W ❛r❡ ❥♦✐♥❡❞ ❜② ❛ ✉♥✐q✉❡ ❣❡♦❞❡s✐❝ s❡❣♠❡♥t✳ ❆♥② ♥♦♥tr✐✈✐❛❧ ❝❧♦s❡❞ s✉❜s❡t X ♦❢ ❛ t♦t❛❧❧② ❣❡♦❞❡s✐❝✱ ❝♦♠♣❧❡t❡✱ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ♣W, rW q ✐s ✐rr❡❞✉❝✐❜❧❡✱ ♥♦ ♠❛tt❡r ✇❤❛t ♠❡❛s✉r❡ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤✳ ❲❤②❄ ■❢ ♣X, rW q ✐s ✐s♦♠❡tr✐❝ t♦ ♣Y ✂ Z, rY ❵ rZq ❢♦r ♥♦♥tr✐✈✐❛❧ Y ❛♥❞ Z✱ t❤❡♥ t❤❡r❡ ✇✐❧❧ ❜❡ ❢♦✉r ❞✐st✐♥❝t ♣♦✐♥ts a, b, c, d ✐♥ X t❤❛t ❛r❡ ✐s♦♠❡tr✐❝ ✐♠❛❣❡s ♦❢ ♣♦✐♥ts ♦❢ t❤❡ ❢♦r♠ ♣y✶, z✶q✱ ♣y✷, z✶q✱ ♣y✶, z✷q✱ ♣y✷, z✷q ✐♥ Y ✂ Z✳ ❚❤✉s✱ rW ♣a, bq ✏ rW ♣c, dq, rW ♣a, cq ✏ rW ♣b, dq, rW ♣a, dq ✏ rW ♣a, bq rW ♣b, dq, rW ♣a, dq ✏ rW ♣a, cq rW ♣c, dq, rW ♣b, cq ✏ rW ♣a, bq rW ♣c, aq, rW ♣b, cq ✏ rW ♣b, dq rW ♣c, dq. ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ t❤✐r❞ ❛♥❞ ❢♦✉rt❤ ❡q✉❛❧✐t✐❡s t❤❛t b ❛♥❞ c ❛r❡ ♦♥ t❤❡ ❣❡♦❞❡s✐❝ s❡❣♠❡♥t ❜❡t✇❡❡♥ a ❛♥❞ d✳ ❲❡ ♠❛② t❤❡r❡❢♦r❡ s✉♣♣♦s❡ t❤❛t ♣W, rW q ✐s ❛ ❝❧♦s❡❞ s✉❜✐♥t❡r✈❛❧ ♦❢ R ❛♥❞✱ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ t❤❛t a ➔ b ➔ c ➔ d✳ ❚❤❡ ✜❢t❤ ❛♥❞ s✐①t❤ ❡q✉❛❧✐t✐❡s ❛r❡ t❤❡♥ ✐♠♣♦ss✐❜❧❡✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❈♦♥❝r❡t❡ ❡①❛♠♣❧❡s ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts✿ t♦t❛❧❧② ❣❡♦❞❡s✐❝ s♣❛❝❡s

❆ ♠❡tr✐❝ s♣❛❝❡ ♣W, rW q ✐s t♦t❛❧❧② ❣❡♦❞❡s✐❝ ✐❢ ❛♥② t✇♦ ♣♦✐♥ts ♦❢ W ❛r❡ ❥♦✐♥❡❞ ❜② ❛ ✉♥✐q✉❡ ❣❡♦❞❡s✐❝ s❡❣♠❡♥t✳ ❆♥② ♥♦♥tr✐✈✐❛❧ ❝❧♦s❡❞ s✉❜s❡t X ♦❢ ❛ t♦t❛❧❧② ❣❡♦❞❡s✐❝✱ ❝♦♠♣❧❡t❡✱ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ♣W, rW q ✐s ✐rr❡❞✉❝✐❜❧❡✱ ♥♦ ♠❛tt❡r ✇❤❛t ♠❡❛s✉r❡ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤✳ ❲❤②❄ ■❢ ♣X, rW q ✐s ✐s♦♠❡tr✐❝ t♦ ♣Y ✂ Z, rY ❵ rZq ❢♦r ♥♦♥tr✐✈✐❛❧ Y ❛♥❞ Z✱ t❤❡♥ t❤❡r❡ ✇✐❧❧ ❜❡ ❢♦✉r ❞✐st✐♥❝t ♣♦✐♥ts a, b, c, d ✐♥ X t❤❛t ❛r❡ ✐s♦♠❡tr✐❝ ✐♠❛❣❡s ♦❢ ♣♦✐♥ts ♦❢ t❤❡ ❢♦r♠ ♣y✶, z✶q✱ ♣y✷, z✶q✱ ♣y✶, z✷q✱ ♣y✷, z✷q ✐♥ Y ✂ Z✳ ❚❤✉s✱ rW ♣a, bq ✏ rW ♣c, dq, rW ♣a, cq ✏ rW ♣b, dq, rW ♣a, dq ✏ rW ♣a, bq rW ♣b, dq, rW ♣a, dq ✏ rW ♣a, cq rW ♣c, dq, rW ♣b, cq ✏ rW ♣a, bq rW ♣c, aq, rW ♣b, cq ✏ rW ♣b, dq rW ♣c, dq. ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ t❤✐r❞ ❛♥❞ ❢♦✉rt❤ ❡q✉❛❧✐t✐❡s t❤❛t b ❛♥❞ c ❛r❡ ♦♥ t❤❡ ❣❡♦❞❡s✐❝ s❡❣♠❡♥t ❜❡t✇❡❡♥ a ❛♥❞ d✳ ❲❡ ♠❛② t❤❡r❡❢♦r❡ s✉♣♣♦s❡ t❤❛t ♣W, rW q ✐s ❛ ❝❧♦s❡❞ s✉❜✐♥t❡r✈❛❧ ♦❢ R ❛♥❞✱ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ t❤❛t a ➔ b ➔ c ➔ d✳ ❚❤❡ ✜❢t❤ ❛♥❞ s✐①t❤ ❡q✉❛❧✐t✐❡s ❛r❡ t❤❡♥ ✐♠♣♦ss✐❜❧❡✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❊①❛♠♣❧❡s ♦❢ t♦t❛❧❧② ❣❡♦❞❡s✐❝ s♣❛❝❡s

❆ ❇❛♥❛❝❤ s♣❛❝❡ ♣X, ⑥ ⑥q ✐s t♦t❛❧❧② ❣❡♦❞❡s✐❝ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s str✐❝t❧② ❝♦♥✈❡①❀ t❤❛t ✐s✱ x ✘ y ❛♥❞ ⑥x✶⑥ ✏ ⑥x✷⑥ ✏ 1 ✐♠♣❧② t❤❛t ⑥ax✶ ♣1 ✁ aqx✷⑥ ➔ 1 ❢♦r ❛❧❧ 0 ➔ a ➔ 1✳ ❙tr✐❝t ❝♦♥✈❡①✐t② ♦❢ ♣X, ⑥ ⑥q ✐s ✐♠♣❧✐❡❞ ❜② ✉♥✐❢♦r♠ ❝♦♥✈❡①✐t②❀ t❤❛t ✐s✱ ❢♦r ❡✈❡r② ε → 0 t❤❡r❡ ❡①✐sts ❛ δ → 0 s✉❝❤ t❤❛t ⑥x✶⑥ ✏ ⑥x✷⑥ ✏ 1 ❛♥❞ ⑥x✶ ✁ x✷⑥ ➙ ε ✐♠♣❧② ⑥ x✶x✷

2

⑥ ↕ 1 ✁ δ✳ ❆♥② ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡s Lp♣S, S, λq✱ 1 ➔ p ➔ ✽✱ ✇❤❡r❡ λ ✐s ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡✱ ❛r❡ ✉♥✐❢♦r♠❧② ❝♦♥✈❡①✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❊①❛♠♣❧❡s ♦❢ t♦t❛❧❧② ❣❡♦❞❡s✐❝ s♣❛❝❡s

❆ ❇❛♥❛❝❤ s♣❛❝❡ ♣X, ⑥ ⑥q ✐s t♦t❛❧❧② ❣❡♦❞❡s✐❝ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s str✐❝t❧② ❝♦♥✈❡①❀ t❤❛t ✐s✱ x ✘ y ❛♥❞ ⑥x✶⑥ ✏ ⑥x✷⑥ ✏ 1 ✐♠♣❧② t❤❛t ⑥ax✶ ♣1 ✁ aqx✷⑥ ➔ 1 ❢♦r ❛❧❧ 0 ➔ a ➔ 1✳ ❙tr✐❝t ❝♦♥✈❡①✐t② ♦❢ ♣X, ⑥ ⑥q ✐s ✐♠♣❧✐❡❞ ❜② ✉♥✐❢♦r♠ ❝♦♥✈❡①✐t②❀ t❤❛t ✐s✱ ❢♦r ❡✈❡r② ε → 0 t❤❡r❡ ❡①✐sts ❛ δ → 0 s✉❝❤ t❤❛t ⑥x✶⑥ ✏ ⑥x✷⑥ ✏ 1 ❛♥❞ ⑥x✶ ✁ x✷⑥ ➙ ε ✐♠♣❧② ⑥ x✶x✷

2

⑥ ↕ 1 ✁ δ✳ ❆♥② ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡s Lp♣S, S, λq✱ 1 ➔ p ➔ ✽✱ ✇❤❡r❡ λ ✐s ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡✱ ❛r❡ ✉♥✐❢♦r♠❧② ❝♦♥✈❡①✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❊①❛♠♣❧❡s ♦❢ t♦t❛❧❧② ❣❡♦❞❡s✐❝ s♣❛❝❡s

❆ ❇❛♥❛❝❤ s♣❛❝❡ ♣X, ⑥ ⑥q ✐s t♦t❛❧❧② ❣❡♦❞❡s✐❝ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s str✐❝t❧② ❝♦♥✈❡①❀ t❤❛t ✐s✱ x ✘ y ❛♥❞ ⑥x✶⑥ ✏ ⑥x✷⑥ ✏ 1 ✐♠♣❧② t❤❛t ⑥ax✶ ♣1 ✁ aqx✷⑥ ➔ 1 ❢♦r ❛❧❧ 0 ➔ a ➔ 1✳ ❙tr✐❝t ❝♦♥✈❡①✐t② ♦❢ ♣X, ⑥ ⑥q ✐s ✐♠♣❧✐❡❞ ❜② ✉♥✐❢♦r♠ ❝♦♥✈❡①✐t②❀ t❤❛t ✐s✱ ❢♦r ❡✈❡r② ε → 0 t❤❡r❡ ❡①✐sts ❛ δ → 0 s✉❝❤ t❤❛t ⑥x✶⑥ ✏ ⑥x✷⑥ ✏ 1 ❛♥❞ ⑥x✶ ✁ x✷⑥ ➙ ε ✐♠♣❧② ⑥ x✶x✷

2

⑥ ↕ 1 ✁ δ✳ ❆♥② ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡s Lp♣S, S, λq✱ 1 ➔ p ➔ ✽✱ ✇❤❡r❡ λ ✐s ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡✱ ❛r❡ ✉♥✐❢♦r♠❧② ❝♦♥✈❡①✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❍♦✇ ♠❛♥② ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ❛r❡ t❤❡r❡❄

❚❤❡ s❡t ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ✐s ❛ ❞❡♥s❡ Gδ s✉❜s❡t ✭✐✳❡✳ ❝♦✉♥t❛❜❧❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ♦♣❡♥ s❡ts✮ ♦❢ M✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ s❡t ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ✐s ✉♥❝♦✉♥t❛❜❧❡✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❍♦✇ ♠❛♥② ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ❛r❡ t❤❡r❡❄

❚❤❡ s❡t ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ✐s ❛ ❞❡♥s❡ Gδ s✉❜s❡t ✭✐✳❡✳ ❝♦✉♥t❛❜❧❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ♦♣❡♥ s❡ts✮ ♦❢ M✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ s❡t ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ✐s ✉♥❝♦✉♥t❛❜❧❡✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

Pr✐♠❡ ❡❧❡♠❡♥ts

❆♥ ❡❧❡♠❡♥t X P M ✐s ♣r✐♠❡ ✐❢ X ✘ E ❛♥❞ X ↕ Y ❵ Z ❢♦r Y, Z P M ✐♠♣❧✐❡s t❤❛t X ↕ Y ♦r X ↕ Z✳ Pr✐♠❡ ❡❧❡♠❡♥ts ❛r❡ ❝❧❡❛r❧② ✐rr❡❞✉❝✐❜❧❡✱ ❜✉t t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t ❛ ♣r✐♦r✐ tr✉❡✳ ❚❤❡r❡ ❛r❡ ❝♦♠♠✉t❛t✐✈❡✱ ❝❛♥❝❡❧❧❛t✐✈❡ s❡♠✐❣r♦✉♣s ✇❤❡r❡ t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡ ✐s ❢❛❧s❡✳ ❆❧❧ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ♣r✐♠❡✳ ❚❤❡ ❛♥❛❧♦❣♦✉s r❡s✉❧t ❢♦r t❤❡ ✐♥t❡❣❡rs ✐s t❤❡ ❦❡② t♦ ♣r♦✈✐♥❣ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝✱ ❛♥❞ t❤❡ ✉s✉❛❧ ♣r♦♦❢ ✉s❡s ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

Pr✐♠❡ ❡❧❡♠❡♥ts

❆♥ ❡❧❡♠❡♥t X P M ✐s ♣r✐♠❡ ✐❢ X ✘ E ❛♥❞ X ↕ Y ❵ Z ❢♦r Y, Z P M ✐♠♣❧✐❡s t❤❛t X ↕ Y ♦r X ↕ Z✳ Pr✐♠❡ ❡❧❡♠❡♥ts ❛r❡ ❝❧❡❛r❧② ✐rr❡❞✉❝✐❜❧❡✱ ❜✉t t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t ❛ ♣r✐♦r✐ tr✉❡✳ ❚❤❡r❡ ❛r❡ ❝♦♠♠✉t❛t✐✈❡✱ ❝❛♥❝❡❧❧❛t✐✈❡ s❡♠✐❣r♦✉♣s ✇❤❡r❡ t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡ ✐s ❢❛❧s❡✳ ❆❧❧ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ♣r✐♠❡✳ ❚❤❡ ❛♥❛❧♦❣♦✉s r❡s✉❧t ❢♦r t❤❡ ✐♥t❡❣❡rs ✐s t❤❡ ❦❡② t♦ ♣r♦✈✐♥❣ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝✱ ❛♥❞ t❤❡ ✉s✉❛❧ ♣r♦♦❢ ✉s❡s ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

Pr✐♠❡ ❡❧❡♠❡♥ts

❆♥ ❡❧❡♠❡♥t X P M ✐s ♣r✐♠❡ ✐❢ X ✘ E ❛♥❞ X ↕ Y ❵ Z ❢♦r Y, Z P M ✐♠♣❧✐❡s t❤❛t X ↕ Y ♦r X ↕ Z✳ Pr✐♠❡ ❡❧❡♠❡♥ts ❛r❡ ❝❧❡❛r❧② ✐rr❡❞✉❝✐❜❧❡✱ ❜✉t t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t ❛ ♣r✐♦r✐ tr✉❡✳ ❚❤❡r❡ ❛r❡ ❝♦♠♠✉t❛t✐✈❡✱ ❝❛♥❝❡❧❧❛t✐✈❡ s❡♠✐❣r♦✉♣s ✇❤❡r❡ t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡ ✐s ❢❛❧s❡✳ ❆❧❧ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ♣r✐♠❡✳ ❚❤❡ ❛♥❛❧♦❣♦✉s r❡s✉❧t ❢♦r t❤❡ ✐♥t❡❣❡rs ✐s t❤❡ ❦❡② t♦ ♣r♦✈✐♥❣ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝✱ ❛♥❞ t❤❡ ✉s✉❛❧ ♣r♦♦❢ ✉s❡s ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

Pr✐♠❡ ❡❧❡♠❡♥ts

❆♥ ❡❧❡♠❡♥t X P M ✐s ♣r✐♠❡ ✐❢ X ✘ E ❛♥❞ X ↕ Y ❵ Z ❢♦r Y, Z P M ✐♠♣❧✐❡s t❤❛t X ↕ Y ♦r X ↕ Z✳ Pr✐♠❡ ❡❧❡♠❡♥ts ❛r❡ ❝❧❡❛r❧② ✐rr❡❞✉❝✐❜❧❡✱ ❜✉t t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t ❛ ♣r✐♦r✐ tr✉❡✳ ❚❤❡r❡ ❛r❡ ❝♦♠♠✉t❛t✐✈❡✱ ❝❛♥❝❡❧❧❛t✐✈❡ s❡♠✐❣r♦✉♣s ✇❤❡r❡ t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡ ✐s ❢❛❧s❡✳ ❆❧❧ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ M ❛r❡ ♣r✐♠❡✳ ❚❤❡ ❛♥❛❧♦❣♦✉s r❡s✉❧t ❢♦r t❤❡ ✐♥t❡❣❡rs ✐s t❤❡ ❦❡② t♦ ♣r♦✈✐♥❣ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝✱ ❛♥❞ t❤❡ ✉s✉❛❧ ♣r♦♦❢ ✉s❡s ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

Pr✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥ ✕ t❤❡ ✏❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝✑

• ✐✈❡♥ ❛♥② X P M③tE✉✱ t❤❡r❡ ✐s ❡✐t❤❡r ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♣XnqN

n✏0 ♦r ❛♥

✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♣Xnq✽

n✏0 ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ M s✉❝❤ t❤❛t

X ✏ X0 ❵ ☎ ☎ ☎ ❵ XN ✐♥ t❤❡ ✜rst ❝❛s❡ ❛♥❞ X ✏ limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ✐♥ t❤❡ s❡❝♦♥❞✳ ❚❤❡ s❡q✉❡♥❝❡ ✐s ✉♥✐q✉❡ ✉♣ t♦ t❤❡ ♦r❞❡r ♦❢ ✐ts t❡r♠s✳ ❊❛❝❤ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥t ❛♣♣❡❛rs ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✱ s♦ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts t❤❛t ❛♣♣❡❛r ❛♥❞ t❤❡✐r ✜♥✐t❡ ♠✉❧t✐♣❧✐❝✐t✐❡s✳ ■t ❢♦❧❧♦✇s t❤❛t ♣M, ↕q ✐s ❛ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡✿ t❤❡r❡ ✐s ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❣r❡❛t❡st ❝♦♠♠♦♥ ❞✐✈✐s♦r ✭♠❡❡t✮ ❛♥❞ t❤❡ ❧❡❛st ❝♦♠♠♦♥ ♠✉❧t✐♣❧❡ ✭❥♦✐♥✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

Pr✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥ ✕ t❤❡ ✏❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝✑

• ✐✈❡♥ ❛♥② X P M③tE✉✱ t❤❡r❡ ✐s ❡✐t❤❡r ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♣XnqN

n✏0 ♦r ❛♥

✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♣Xnq✽

n✏0 ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ M s✉❝❤ t❤❛t

X ✏ X0 ❵ ☎ ☎ ☎ ❵ XN ✐♥ t❤❡ ✜rst ❝❛s❡ ❛♥❞ X ✏ limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ✐♥ t❤❡ s❡❝♦♥❞✳ ❚❤❡ s❡q✉❡♥❝❡ ✐s ✉♥✐q✉❡ ✉♣ t♦ t❤❡ ♦r❞❡r ♦❢ ✐ts t❡r♠s✳ ❊❛❝❤ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥t ❛♣♣❡❛rs ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✱ s♦ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts t❤❛t ❛♣♣❡❛r ❛♥❞ t❤❡✐r ✜♥✐t❡ ♠✉❧t✐♣❧✐❝✐t✐❡s✳ ■t ❢♦❧❧♦✇s t❤❛t ♣M, ↕q ✐s ❛ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡✿ t❤❡r❡ ✐s ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❣r❡❛t❡st ❝♦♠♠♦♥ ❞✐✈✐s♦r ✭♠❡❡t✮ ❛♥❞ t❤❡ ❧❡❛st ❝♦♠♠♦♥ ♠✉❧t✐♣❧❡ ✭❥♦✐♥✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

Pr✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥ ✕ t❤❡ ✏❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝✑

• ✐✈❡♥ ❛♥② X P M③tE✉✱ t❤❡r❡ ✐s ❡✐t❤❡r ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♣XnqN

n✏0 ♦r ❛♥

✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♣Xnq✽

n✏0 ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ M s✉❝❤ t❤❛t

X ✏ X0 ❵ ☎ ☎ ☎ ❵ XN ✐♥ t❤❡ ✜rst ❝❛s❡ ❛♥❞ X ✏ limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ✐♥ t❤❡ s❡❝♦♥❞✳ ❚❤❡ s❡q✉❡♥❝❡ ✐s ✉♥✐q✉❡ ✉♣ t♦ t❤❡ ♦r❞❡r ♦❢ ✐ts t❡r♠s✳ ❊❛❝❤ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥t ❛♣♣❡❛rs ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✱ s♦ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts t❤❛t ❛♣♣❡❛r ❛♥❞ t❤❡✐r ✜♥✐t❡ ♠✉❧t✐♣❧✐❝✐t✐❡s✳ ■t ❢♦❧❧♦✇s t❤❛t ♣M, ↕q ✐s ❛ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡✿ t❤❡r❡ ✐s ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❣r❡❛t❡st ❝♦♠♠♦♥ ❞✐✈✐s♦r ✭♠❡❡t✮ ❛♥❞ t❤❡ ❧❡❛st ❝♦♠♠♦♥ ♠✉❧t✐♣❧❡ ✭❥♦✐♥✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

Pr✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥ ✕ t❤❡ ✏❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛r✐t❤♠❡t✐❝✑

• ✐✈❡♥ ❛♥② X P M③tE✉✱ t❤❡r❡ ✐s ❡✐t❤❡r ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♣XnqN

n✏0 ♦r ❛♥

✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♣Xnq✽

n✏0 ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ M s✉❝❤ t❤❛t

X ✏ X0 ❵ ☎ ☎ ☎ ❵ XN ✐♥ t❤❡ ✜rst ❝❛s❡ ❛♥❞ X ✏ limnÑ✽ X0 ❵ ☎ ☎ ☎ ❵ Xn ✐♥ t❤❡ s❡❝♦♥❞✳ ❚❤❡ s❡q✉❡♥❝❡ ✐s ✉♥✐q✉❡ ✉♣ t♦ t❤❡ ♦r❞❡r ♦❢ ✐ts t❡r♠s✳ ❊❛❝❤ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥t ❛♣♣❡❛rs ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✱ s♦ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❡❧❡♠❡♥ts t❤❛t ❛♣♣❡❛r ❛♥❞ t❤❡✐r ✜♥✐t❡ ♠✉❧t✐♣❧✐❝✐t✐❡s✳ ■t ❢♦❧❧♦✇s t❤❛t ♣M, ↕q ✐s ❛ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡✿ t❤❡r❡ ✐s ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❣r❡❛t❡st ❝♦♠♠♦♥ ❞✐✈✐s♦r ✭♠❡❡t✮ ❛♥❞ t❤❡ ❧❡❛st ❝♦♠♠♦♥ ♠✉❧t✐♣❧❡ ✭❥♦✐♥✮✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ❧✐tt❧❡ ❛❜♦✉t t❤❡ ♣r♦♦❢

❊①✐st❡♥❝❡ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥s ✐♥t♦ ✐rr❡❞✉❝✐❜❧❡s ✉s❡s ❣❡♥❡r❛❧ r❡s✉❧ts ❛❜♦✉t ❉❡❧♣❤✐❝ s❡♠✐❣r♦✉♣s ❢r♦♠ ❑❡♥❞❛❧❧ ✭✶✾✻✽✮✱ ❉❛✈✐❞s♦♥ ✭✶✾✻✾✮✳ ❆♥ ✐♥❣r❡❞✐❡♥t ❢♦r ✉♥✐q✉❡♥❡ss ✐s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❝♦♠♠♦♥ r❡✜♥❡♠❡♥ts✿ ■❢ X0✌ ❵ X1✌ ✏ X ✏ X✌0 ❵ X✌1, t❤❡♥ t❤❡r❡ ❡①✐st X00, X01, X10, X11 s✉❝❤ t❤❛t X0✌ ✏ X00❵X01, X1✌ ✏ X10❵X11, X✌0 ✏ X00❵X10, X✌1 ✏ X01❵X11. ❚❤❡ ♣r♦♦❢ t❤❛t ❝♦♠♠♦♥ r❡✜♥❡♠❡♥ts ❡①✐st ✉s❡s s♦♠❡ ♦❢ t❤❡ s❛♠❡ ✐❞❡❛s ❛s ❚❛r❞✐❢✬s ♣r♦♦❢ ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥t❛r② ❢❛❝t ✭✇❤❡r❡ ❑ ❑ ❞❡♥♦t❡s ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ r❛♥❞♦♠ ❡❧❡♠❡♥ts✮✿ ▲❡t ξ00, ξ01, ξ10, ξ11 ❜❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ♦❢ t❤❡ r❡s♣❡❝t✐✈❡ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡s X00, X01, X10, X11✳ ❙✉♣♣♦s❡ t❤❛t ♣ξ00, ξ01q❑ ❑♣ξ10, ξ11q ❛♥❞ ♣ξ00, ξ10q❑ ❑♣ξ01, ξ11q. ❚❤❡♥✱ ξ00❑ ❑ξ01❑ ❑ξ10❑ ❑ξ11.

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ❧✐tt❧❡ ❛❜♦✉t t❤❡ ♣r♦♦❢

❊①✐st❡♥❝❡ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥s ✐♥t♦ ✐rr❡❞✉❝✐❜❧❡s ✉s❡s ❣❡♥❡r❛❧ r❡s✉❧ts ❛❜♦✉t ❉❡❧♣❤✐❝ s❡♠✐❣r♦✉♣s ❢r♦♠ ❑❡♥❞❛❧❧ ✭✶✾✻✽✮✱ ❉❛✈✐❞s♦♥ ✭✶✾✻✾✮✳ ❆♥ ✐♥❣r❡❞✐❡♥t ❢♦r ✉♥✐q✉❡♥❡ss ✐s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❝♦♠♠♦♥ r❡✜♥❡♠❡♥ts✿ ■❢ X0✌ ❵ X1✌ ✏ X ✏ X✌0 ❵ X✌1, t❤❡♥ t❤❡r❡ ❡①✐st X00, X01, X10, X11 s✉❝❤ t❤❛t X0✌ ✏ X00❵X01, X1✌ ✏ X10❵X11, X✌0 ✏ X00❵X10, X✌1 ✏ X01❵X11. ❚❤❡ ♣r♦♦❢ t❤❛t ❝♦♠♠♦♥ r❡✜♥❡♠❡♥ts ❡①✐st ✉s❡s s♦♠❡ ♦❢ t❤❡ s❛♠❡ ✐❞❡❛s ❛s ❚❛r❞✐❢✬s ♣r♦♦❢ ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥t❛r② ❢❛❝t ✭✇❤❡r❡ ❑ ❑ ❞❡♥♦t❡s ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ r❛♥❞♦♠ ❡❧❡♠❡♥ts✮✿ ▲❡t ξ00, ξ01, ξ10, ξ11 ❜❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ♦❢ t❤❡ r❡s♣❡❝t✐✈❡ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡s X00, X01, X10, X11✳ ❙✉♣♣♦s❡ t❤❛t ♣ξ00, ξ01q❑ ❑♣ξ10, ξ11q ❛♥❞ ♣ξ00, ξ10q❑ ❑♣ξ01, ξ11q. ❚❤❡♥✱ ξ00❑ ❑ξ01❑ ❑ξ10❑ ❑ξ11.

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❆ ❧✐tt❧❡ ❛❜♦✉t t❤❡ ♣r♦♦❢

❊①✐st❡♥❝❡ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥s ✐♥t♦ ✐rr❡❞✉❝✐❜❧❡s ✉s❡s ❣❡♥❡r❛❧ r❡s✉❧ts ❛❜♦✉t ❉❡❧♣❤✐❝ s❡♠✐❣r♦✉♣s ❢r♦♠ ❑❡♥❞❛❧❧ ✭✶✾✻✽✮✱ ❉❛✈✐❞s♦♥ ✭✶✾✻✾✮✳ ❆♥ ✐♥❣r❡❞✐❡♥t ❢♦r ✉♥✐q✉❡♥❡ss ✐s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❝♦♠♠♦♥ r❡✜♥❡♠❡♥ts✿ ■❢ X0✌ ❵ X1✌ ✏ X ✏ X✌0 ❵ X✌1, t❤❡♥ t❤❡r❡ ❡①✐st X00, X01, X10, X11 s✉❝❤ t❤❛t X0✌ ✏ X00❵X01, X1✌ ✏ X10❵X11, X✌0 ✏ X00❵X10, X✌1 ✏ X01❵X11. ❚❤❡ ♣r♦♦❢ t❤❛t ❝♦♠♠♦♥ r❡✜♥❡♠❡♥ts ❡①✐st ✉s❡s s♦♠❡ ♦❢ t❤❡ s❛♠❡ ✐❞❡❛s ❛s ❚❛r❞✐❢✬s ♣r♦♦❢ ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥t❛r② ❢❛❝t ✭✇❤❡r❡ ❑ ❑ ❞❡♥♦t❡s ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ r❛♥❞♦♠ ❡❧❡♠❡♥ts✮✿ ▲❡t ξ00, ξ01, ξ10, ξ11 ❜❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ♦❢ t❤❡ r❡s♣❡❝t✐✈❡ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡s X00, X01, X10, X11✳ ❙✉♣♣♦s❡ t❤❛t ♣ξ00, ξ01q❑ ❑♣ξ10, ξ11q ❛♥❞ ♣ξ00, ξ10q❑ ❑♣ξ01, ξ11q. ❚❤❡♥✱ ξ00❑ ❑ξ01❑ ❑ξ10❑ ❑ξ11.

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✕ ▲é✈②✲❍✐✟ ♥❝✐♥✲■tô

❆ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ♦❢ M ✐s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✐❢ ❢♦r ❡❛❝❤ n P N t❤❡r❡ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✱ r❛♥❞♦♠ ❡❧❡♠❡♥ts Yn1, . . . , Ynn s✉❝❤ t❤❛t Y ❤❛s t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛s Yn1 ❵ ☎ ☎ ☎ ❵ Ynn✳ ❆♥ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ❤❛s t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛s ð tX : ♣t, Xq P Π✉, ✇❤❡r❡ Π ✐s ❛ P♦✐ss♦♥ r❛♥❞♦♠ ♠❡❛s✉r❡ ♦♥ r0, 1s ✂ ♣M③tE✉q ✇✐t❤ ✐♥t❡♥s✐t② ♠❡❛s✉r❡ ♦❢ t❤❡ ❢♦r♠ λ ❜ ν✱ ✇❤❡r❡ λ ✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ❛♥❞ ❛♥❞ ν ✐s ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡ ♦♥ M③tE✉ s✉❝❤ t❤❛t ➺ D♣Xq ❫ 1 ν♣dXq ➔ ✽. ❈♦♥✈❡rs❡❧②✱ ❛♥② s✉❝❤ ♠❡❛s✉r❡ ν ❝♦rr❡s♣♦♥❞s t♦ ❛♥ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐♥ t❤✐s ✇❛②✳ ❈♦♥st❛♥ts ❛r❡ ♥♦t ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ❛♥❞ t❤❡r❡ ✐s ♥♦ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡

• ❛✉ss✐❛♥ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✕ ▲é✈②✲❍✐✟ ♥❝✐♥✲■tô

❆ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ♦❢ M ✐s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✐❢ ❢♦r ❡❛❝❤ n P N t❤❡r❡ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✱ r❛♥❞♦♠ ❡❧❡♠❡♥ts Yn1, . . . , Ynn s✉❝❤ t❤❛t Y ❤❛s t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛s Yn1 ❵ ☎ ☎ ☎ ❵ Ynn✳ ❆♥ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ❤❛s t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛s ð tX : ♣t, Xq P Π✉, ✇❤❡r❡ Π ✐s ❛ P♦✐ss♦♥ r❛♥❞♦♠ ♠❡❛s✉r❡ ♦♥ r0, 1s ✂ ♣M③tE✉q ✇✐t❤ ✐♥t❡♥s✐t② ♠❡❛s✉r❡ ♦❢ t❤❡ ❢♦r♠ λ ❜ ν✱ ✇❤❡r❡ λ ✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ❛♥❞ ❛♥❞ ν ✐s ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡ ♦♥ M③tE✉ s✉❝❤ t❤❛t ➺ D♣Xq ❫ 1 ν♣dXq ➔ ✽. ❈♦♥✈❡rs❡❧②✱ ❛♥② s✉❝❤ ♠❡❛s✉r❡ ν ❝♦rr❡s♣♦♥❞s t♦ ❛♥ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐♥ t❤✐s ✇❛②✳ ❈♦♥st❛♥ts ❛r❡ ♥♦t ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ❛♥❞ t❤❡r❡ ✐s ♥♦ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡

• ❛✉ss✐❛♥ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✕ ▲é✈②✲❍✐✟ ♥❝✐♥✲■tô

❆ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ♦❢ M ✐s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✐❢ ❢♦r ❡❛❝❤ n P N t❤❡r❡ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✱ r❛♥❞♦♠ ❡❧❡♠❡♥ts Yn1, . . . , Ynn s✉❝❤ t❤❛t Y ❤❛s t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛s Yn1 ❵ ☎ ☎ ☎ ❵ Ynn✳ ❆♥ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ❤❛s t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛s ð tX : ♣t, Xq P Π✉, ✇❤❡r❡ Π ✐s ❛ P♦✐ss♦♥ r❛♥❞♦♠ ♠❡❛s✉r❡ ♦♥ r0, 1s ✂ ♣M③tE✉q ✇✐t❤ ✐♥t❡♥s✐t② ♠❡❛s✉r❡ ♦❢ t❤❡ ❢♦r♠ λ ❜ ν✱ ✇❤❡r❡ λ ✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ❛♥❞ ❛♥❞ ν ✐s ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡ ♦♥ M③tE✉ s✉❝❤ t❤❛t ➺ D♣Xq ❫ 1 ν♣dXq ➔ ✽. ❈♦♥✈❡rs❡❧②✱ ❛♥② s✉❝❤ ♠❡❛s✉r❡ ν ❝♦rr❡s♣♦♥❞s t♦ ❛♥ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐♥ t❤✐s ✇❛②✳ ❈♦♥st❛♥ts ❛r❡ ♥♦t ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ❛♥❞ t❤❡r❡ ✐s ♥♦ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡

• ❛✉ss✐❛♥ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

■♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✕ ▲é✈②✲❍✐✟ ♥❝✐♥✲■tô

❆ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ♦❢ M ✐s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✐❢ ❢♦r ❡❛❝❤ n P N t❤❡r❡ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✱ r❛♥❞♦♠ ❡❧❡♠❡♥ts Yn1, . . . , Ynn s✉❝❤ t❤❛t Y ❤❛s t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛s Yn1 ❵ ☎ ☎ ☎ ❵ Ynn✳ ❆♥ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ❤❛s t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛s ð tX : ♣t, Xq P Π✉, ✇❤❡r❡ Π ✐s ❛ P♦✐ss♦♥ r❛♥❞♦♠ ♠❡❛s✉r❡ ♦♥ r0, 1s ✂ ♣M③tE✉q ✇✐t❤ ✐♥t❡♥s✐t② ♠❡❛s✉r❡ ♦❢ t❤❡ ❢♦r♠ λ ❜ ν✱ ✇❤❡r❡ λ ✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ❛♥❞ ❛♥❞ ν ✐s ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡ ♦♥ M③tE✉ s✉❝❤ t❤❛t ➺ D♣Xq ❫ 1 ν♣dXq ➔ ✽. ❈♦♥✈❡rs❡❧②✱ ❛♥② s✉❝❤ ♠❡❛s✉r❡ ν ❝♦rr❡s♣♦♥❞s t♦ ❛♥ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐♥ t❤✐s ✇❛②✳ ❈♦♥st❛♥ts ❛r❡ ♥♦t ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ❛♥❞ t❤❡r❡ ✐s ♥♦ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡

• ❛✉ss✐❛♥ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❝❛❧✐♥❣

• ✐✈❡♥ X P M ❛♥❞ a → 0✱ s❡t aX :✏ ♣X, arX, µXq P M✳

❚❤✐s s❝❛❧✐♥❣ ♦♣❡r❛t✐♦♥ ♦♣❡r❛t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ s❛t✐s✜❡s a♣X ❵ Yq ✏ ♣aXq ❵ ♣aYq. ■❢ ♣aXq ❵ ♣bXq ✏ cX ❢♦r s♦♠❡ X P M ❛♥❞ a, b, c → 0✱ t❤❡♥ X ✏ E✱ s♦ t❤❡ s❡❝♦♥❞ ❞✐str✐❜✉t✐✈✐t② ❧❛✇ ❝❡rt❛✐♥❧② ❞♦❡s ♥♦t ❤♦❧❞✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❝❛❧✐♥❣

• ✐✈❡♥ X P M ❛♥❞ a → 0✱ s❡t aX :✏ ♣X, arX, µXq P M✳

❚❤✐s s❝❛❧✐♥❣ ♦♣❡r❛t✐♦♥ ♦♣❡r❛t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ s❛t✐s✜❡s a♣X ❵ Yq ✏ ♣aXq ❵ ♣aYq. ■❢ ♣aXq ❵ ♣bXq ✏ cX ❢♦r s♦♠❡ X P M ❛♥❞ a, b, c → 0✱ t❤❡♥ X ✏ E✱ s♦ t❤❡ s❡❝♦♥❞ ❞✐str✐❜✉t✐✈✐t② ❧❛✇ ❝❡rt❛✐♥❧② ❞♦❡s ♥♦t ❤♦❧❞✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙❝❛❧✐♥❣

• ✐✈❡♥ X P M ❛♥❞ a → 0✱ s❡t aX :✏ ♣X, arX, µXq P M✳

❚❤✐s s❝❛❧✐♥❣ ♦♣❡r❛t✐♦♥ ♦♣❡r❛t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ s❛t✐s✜❡s a♣X ❵ Yq ✏ ♣aXq ❵ ♣aYq. ■❢ ♣aXq ❵ ♣bXq ✏ cX ❢♦r s♦♠❡ X P M ❛♥❞ a, b, c → 0✱ t❤❡♥ X ✏ E✱ s♦ t❤❡ s❡❝♦♥❞ ❞✐str✐❜✉t✐✈✐t② ❧❛✇ ❝❡rt❛✐♥❧② ❞♦❡s ♥♦t ❤♦❧❞✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙t❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✕ ✏▲❡P❛❣❡✑ r❡♣r❡s❡♥t❛t✐♦♥s

❆ M✲✈❛❧✉❡❞ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ✐s st❛❜❧❡ ✇✐t❤ ✐♥❞❡① α → 0 ✐❢ ❢♦r ❛♥② a, b → 0 t❤❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ♣a bq

1 α Y

❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s a

1 α Y✶ ❵ b 1 α Y✷.

❆ st❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐s ♥❡❝❡ss❛r✐❧② ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡✳ ❚❤❡ ✐♥❞❡① ♠✉st s❛t✐s❢② 0 ➔ α ➔ 1✳ ❆♥ α✲st❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s ð

nPN

Γ

✁ 1

α

n

Zn, ✇❤❡r❡ ♣ΓnqnPN ✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉❝❝❡ss✐✈❡ ❛rr✐✈❛❧s ♦❢ ❛ ❤♦♠♦❣❡♥❡♦✉s ✉♥✐t ✐♥t❡♥s✐t② P♦✐ss♦♥ ♣♦✐♥t ♣r♦❝❡ss ♦♥ R ❛♥❞ ♣ZnqnPN ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐✳✐✳❞✳ r❛♥❞♦♠ ❡❧❡♠❡♥ts ♦❢ M✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙t❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✕ ✏▲❡P❛❣❡✑ r❡♣r❡s❡♥t❛t✐♦♥s

❆ M✲✈❛❧✉❡❞ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ✐s st❛❜❧❡ ✇✐t❤ ✐♥❞❡① α → 0 ✐❢ ❢♦r ❛♥② a, b → 0 t❤❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ♣a bq

1 α Y

❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s a

1 α Y✶ ❵ b 1 α Y✷.

❆ st❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐s ♥❡❝❡ss❛r✐❧② ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡✳ ❚❤❡ ✐♥❞❡① ♠✉st s❛t✐s❢② 0 ➔ α ➔ 1✳ ❆♥ α✲st❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s ð

nPN

Γ

✁ 1

α

n

Zn, ✇❤❡r❡ ♣ΓnqnPN ✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉❝❝❡ss✐✈❡ ❛rr✐✈❛❧s ♦❢ ❛ ❤♦♠♦❣❡♥❡♦✉s ✉♥✐t ✐♥t❡♥s✐t② P♦✐ss♦♥ ♣♦✐♥t ♣r♦❝❡ss ♦♥ R ❛♥❞ ♣ZnqnPN ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐✳✐✳❞✳ r❛♥❞♦♠ ❡❧❡♠❡♥ts ♦❢ M✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙t❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✕ ✏▲❡P❛❣❡✑ r❡♣r❡s❡♥t❛t✐♦♥s

❆ M✲✈❛❧✉❡❞ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ✐s st❛❜❧❡ ✇✐t❤ ✐♥❞❡① α → 0 ✐❢ ❢♦r ❛♥② a, b → 0 t❤❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ♣a bq

1 α Y

❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s a

1 α Y✶ ❵ b 1 α Y✷.

❆ st❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐s ♥❡❝❡ss❛r✐❧② ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡✳ ❚❤❡ ✐♥❞❡① ♠✉st s❛t✐s❢② 0 ➔ α ➔ 1✳ ❆♥ α✲st❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s ð

nPN

Γ

✁ 1

α

n

Zn, ✇❤❡r❡ ♣ΓnqnPN ✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉❝❝❡ss✐✈❡ ❛rr✐✈❛❧s ♦❢ ❛ ❤♦♠♦❣❡♥❡♦✉s ✉♥✐t ✐♥t❡♥s✐t② P♦✐ss♦♥ ♣♦✐♥t ♣r♦❝❡ss ♦♥ R ❛♥❞ ♣ZnqnPN ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐✳✐✳❞✳ r❛♥❞♦♠ ❡❧❡♠❡♥ts ♦❢ M✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❙t❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✕ ✏▲❡P❛❣❡✑ r❡♣r❡s❡♥t❛t✐♦♥s

❆ M✲✈❛❧✉❡❞ r❛♥❞♦♠ ❡❧❡♠❡♥t Y ✐s st❛❜❧❡ ✇✐t❤ ✐♥❞❡① α → 0 ✐❢ ❢♦r ❛♥② a, b → 0 t❤❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ♣a bq

1 α Y

❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s a

1 α Y✶ ❵ b 1 α Y✷.

❆ st❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐s ♥❡❝❡ss❛r✐❧② ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡✳ ❚❤❡ ✐♥❞❡① ♠✉st s❛t✐s❢② 0 ➔ α ➔ 1✳ ❆♥ α✲st❛❜❧❡ r❛♥❞♦♠ ❡❧❡♠❡♥t ❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s ð

nPN

Γ

✁ 1

α

n

Zn, ✇❤❡r❡ ♣ΓnqnPN ✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉❝❝❡ss✐✈❡ ❛rr✐✈❛❧s ♦❢ ❛ ❤♦♠♦❣❡♥❡♦✉s ✉♥✐t ✐♥t❡♥s✐t② P♦✐ss♦♥ ♣♦✐♥t ♣r♦❝❡ss ♦♥ R ❛♥❞ ♣ZnqnPN ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐✳✐✳❞✳ r❛♥❞♦♠ ❡❧❡♠❡♥ts ♦❢ M✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

❈❛♥❝❡❧❧❛t✐✈✐t② ❛❧❧♦✇s ✉s t♦ ❡♠❜❡❞ t❤❡ s❡♠✐❣r♦✉♣ M ✐♥t♦ ❛ ❣r♦✉♣ ✕ ❛♥❛❧♦❣♦✉s t♦ ♣❛ss✐♥❣ ❢r♦♠ N t♦ Z✳ ❆r❡ t❤❡r❡ ❛♥❛❧♦❣✉❡s ♦❢ ♦❜❥❡❝ts s✉❝❤ ❛s ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ t❤✐s ❣r♦✉♣❄ ❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ ♠❡tr✐❝s ✈✐❛ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q ❴ rZ♣z✶, z✷q ✐♥st❡❛❞ ♦❢ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q rZ♣z✶, z✷q ✭t❤❛t ✐s✱ ✏ℓ✽✑ ✐♥st❡❛❞ ♦❢ ✏ℓ1✑ ✕ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ str♦♥❣ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s✮❄ ❘✐❡✛❡❧ ❤❛s s❤♦✇♥ t❤❛t ♦♥❡ ❝❛♥ ♦❜t❛✐♥ ❛ q✉❛♥t✉♠ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ●r♦♠♦✈✕❍❛✉s❞♦r✛ ❞✐st❛♥❝❡ ❜② ❝♦♥s✐❞❡r✐♥❣ C✝✲❛❧❣❡❜r❛s ✇✐t❤ ♣r♦♣❡rt✐❡s t❤❛t ❣❡♥❡r❛❧✐③❡ t❤♦s❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳ ■s t❤❡r❡ ❛ s✐♠✐❧❛r q✉❛♥t✐③❛t✐♦♥ ❢♦r ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s❄ ❖♥❣♦✐♥❣ ✇♦r❦ ✇✐t❤ ❇❡♥s♦♥ ❆✉✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

❈❛♥❝❡❧❧❛t✐✈✐t② ❛❧❧♦✇s ✉s t♦ ❡♠❜❡❞ t❤❡ s❡♠✐❣r♦✉♣ M ✐♥t♦ ❛ ❣r♦✉♣ ✕ ❛♥❛❧♦❣♦✉s t♦ ♣❛ss✐♥❣ ❢r♦♠ N t♦ Z✳ ❆r❡ t❤❡r❡ ❛♥❛❧♦❣✉❡s ♦❢ ♦❜❥❡❝ts s✉❝❤ ❛s ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ t❤✐s ❣r♦✉♣❄ ❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ ♠❡tr✐❝s ✈✐❛ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q ❴ rZ♣z✶, z✷q ✐♥st❡❛❞ ♦❢ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q rZ♣z✶, z✷q ✭t❤❛t ✐s✱ ✏ℓ✽✑ ✐♥st❡❛❞ ♦❢ ✏ℓ1✑ ✕ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ str♦♥❣ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s✮❄ ❘✐❡✛❡❧ ❤❛s s❤♦✇♥ t❤❛t ♦♥❡ ❝❛♥ ♦❜t❛✐♥ ❛ q✉❛♥t✉♠ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ●r♦♠♦✈✕❍❛✉s❞♦r✛ ❞✐st❛♥❝❡ ❜② ❝♦♥s✐❞❡r✐♥❣ C✝✲❛❧❣❡❜r❛s ✇✐t❤ ♣r♦♣❡rt✐❡s t❤❛t ❣❡♥❡r❛❧✐③❡ t❤♦s❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳ ■s t❤❡r❡ ❛ s✐♠✐❧❛r q✉❛♥t✐③❛t✐♦♥ ❢♦r ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s❄ ❖♥❣♦✐♥❣ ✇♦r❦ ✇✐t❤ ❇❡♥s♦♥ ❆✉✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

❈❛♥❝❡❧❧❛t✐✈✐t② ❛❧❧♦✇s ✉s t♦ ❡♠❜❡❞ t❤❡ s❡♠✐❣r♦✉♣ M ✐♥t♦ ❛ ❣r♦✉♣ ✕ ❛♥❛❧♦❣♦✉s t♦ ♣❛ss✐♥❣ ❢r♦♠ N t♦ Z✳ ❆r❡ t❤❡r❡ ❛♥❛❧♦❣✉❡s ♦❢ ♦❜❥❡❝ts s✉❝❤ ❛s ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ t❤✐s ❣r♦✉♣❄ ❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ ♠❡tr✐❝s ✈✐❛ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q ❴ rZ♣z✶, z✷q ✐♥st❡❛❞ ♦❢ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q rZ♣z✶, z✷q ✭t❤❛t ✐s✱ ✏ℓ✽✑ ✐♥st❡❛❞ ♦❢ ✏ℓ1✑ ✕ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ str♦♥❣ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s✮❄ ❘✐❡✛❡❧ ❤❛s s❤♦✇♥ t❤❛t ♦♥❡ ❝❛♥ ♦❜t❛✐♥ ❛ q✉❛♥t✉♠ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ●r♦♠♦✈✕❍❛✉s❞♦r✛ ❞✐st❛♥❝❡ ❜② ❝♦♥s✐❞❡r✐♥❣ C✝✲❛❧❣❡❜r❛s ✇✐t❤ ♣r♦♣❡rt✐❡s t❤❛t ❣❡♥❡r❛❧✐③❡ t❤♦s❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳ ■s t❤❡r❡ ❛ s✐♠✐❧❛r q✉❛♥t✐③❛t✐♦♥ ❢♦r ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s❄ ❖♥❣♦✐♥❣ ✇♦r❦ ✇✐t❤ ❇❡♥s♦♥ ❆✉✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

❈❛♥❝❡❧❧❛t✐✈✐t② ❛❧❧♦✇s ✉s t♦ ❡♠❜❡❞ t❤❡ s❡♠✐❣r♦✉♣ M ✐♥t♦ ❛ ❣r♦✉♣ ✕ ❛♥❛❧♦❣♦✉s t♦ ♣❛ss✐♥❣ ❢r♦♠ N t♦ Z✳ ❆r❡ t❤❡r❡ ❛♥❛❧♦❣✉❡s ♦❢ ♦❜❥❡❝ts s✉❝❤ ❛s ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ t❤✐s ❣r♦✉♣❄ ❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ ♠❡tr✐❝s ✈✐❛ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q ❴ rZ♣z✶, z✷q ✐♥st❡❛❞ ♦❢ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q rZ♣z✶, z✷q ✭t❤❛t ✐s✱ ✏ℓ✽✑ ✐♥st❡❛❞ ♦❢ ✏ℓ1✑ ✕ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ str♦♥❣ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s✮❄ ❘✐❡✛❡❧ ❤❛s s❤♦✇♥ t❤❛t ♦♥❡ ❝❛♥ ♦❜t❛✐♥ ❛ q✉❛♥t✉♠ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ●r♦♠♦✈✕❍❛✉s❞♦r✛ ❞✐st❛♥❝❡ ❜② ❝♦♥s✐❞❡r✐♥❣ C✝✲❛❧❣❡❜r❛s ✇✐t❤ ♣r♦♣❡rt✐❡s t❤❛t ❣❡♥❡r❛❧✐③❡ t❤♦s❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳ ■s t❤❡r❡ ❛ s✐♠✐❧❛r q✉❛♥t✐③❛t✐♦♥ ❢♦r ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s❄ ❖♥❣♦✐♥❣ ✇♦r❦ ✇✐t❤ ❇❡♥s♦♥ ❆✉✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

❈❛♥❝❡❧❧❛t✐✈✐t② ❛❧❧♦✇s ✉s t♦ ❡♠❜❡❞ t❤❡ s❡♠✐❣r♦✉♣ M ✐♥t♦ ❛ ❣r♦✉♣ ✕ ❛♥❛❧♦❣♦✉s t♦ ♣❛ss✐♥❣ ❢r♦♠ N t♦ Z✳ ❆r❡ t❤❡r❡ ❛♥❛❧♦❣✉❡s ♦❢ ♦❜❥❡❝ts s✉❝❤ ❛s ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ t❤✐s ❣r♦✉♣❄ ❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ ♠❡tr✐❝s ✈✐❛ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q ❴ rZ♣z✶, z✷q ✐♥st❡❛❞ ♦❢ ♣♣y✶, z✶q, ♣y✷, z✷qq ÞÑ rY ♣y✶, y✷q rZ♣z✶, z✷q ✭t❤❛t ✐s✱ ✏ℓ✽✑ ✐♥st❡❛❞ ♦❢ ✏ℓ1✑ ✕ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ str♦♥❣ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s✮❄ ❘✐❡✛❡❧ ❤❛s s❤♦✇♥ t❤❛t ♦♥❡ ❝❛♥ ♦❜t❛✐♥ ❛ q✉❛♥t✉♠ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ●r♦♠♦✈✕❍❛✉s❞♦r✛ ❞✐st❛♥❝❡ ❜② ❝♦♥s✐❞❡r✐♥❣ C✝✲❛❧❣❡❜r❛s ✇✐t❤ ♣r♦♣❡rt✐❡s t❤❛t ❣❡♥❡r❛❧✐③❡ t❤♦s❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳ ■s t❤❡r❡ ❛ s✐♠✐❧❛r q✉❛♥t✐③❛t✐♦♥ ❢♦r ♠❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s❄ ❖♥❣♦✐♥❣ ✇♦r❦ ✇✐t❤ ❇❡♥s♦♥ ❆✉✳

❙t❡✈❡♥ ◆✳ ❊✈❛♥s ▼❡tr✐❝ ♠❡❛s✉r❡ s♣❛❝❡s