Implementation and application of the explicitly correlated by Rafal A. Bachorz

By Rafal A. Bachorz

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T❤❡ s❧♦✇❡st) ■♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ t❤❡ q✉❛♥t✐t② ❢❛st❡st✱ t❤❡ ♥❡①t ♦♥❡ ✐s q A ❡✳❣✳ : A(p, q, r, s). ✭✽✵✮ ✐s st♦r❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ✐♥❞❡① p ✐s t❤❡ ❛♥❞ s♦ ♦♥✳ ❯s✐♥❣ t❤✐s ❝♦♥✈❡♥t✐♦♥ t❤❡ r❡s♦rt✐♥❣ s❤♦✇♥ ✐♥ st❡♣ ✹ ♦❢ ❆❧❣♦r✐t❤♠ ✶ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s f12 (p, q, i, j) −→ f12 (i, j, p, q)✱ ✇❤❡r❡ t❤❡ q✉❛♥t✐t② f12 (p, q, i, j) ✭✽✶✮ ✐s ❝♦♠♣✉t❡❞ ❛♥❞ st♦r❡❞ ❛t t❤❡ ▼P✷✲❋✶✷ ❧❡✈❡❧✳ ❚❤❡ r❡s♦rt✐♥❣ ♦❢ t❤❡ ♦t❤❡r ✐♥t❡❣r❛❧s ✐s ❛❧✇❛②s s✐♠✐❧❛r✱ t❤❡ ♦❝❝✉♣✐❡❞ ✐♥❞✐❝❡s t❤❛t r✉♥ t❤❡ s❧♦✇❡st ❛t t❤❡ ▼P✷✲❋✶✷ ❧❡✈❡❧ ❛r❡ ❡①❝❤❛♥❣❡❞ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ✭♦r ❈❆❇❙✮ ♦♥❡s✳ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ s✉❝❤ ♣r♦❝❡❞✉r❡ ✐s ♥♦t ❡♥t✐r❡❧② tr✐✈✐❛❧✳ ❆ ♥❛✐✈❡ r♦✉t✐♥❡ ✇♦✉❧❞ ❛ss✉♠❡ t❤❛t t❤❡ ✇❤♦❧❡ ❢♦✉r✲✐♥❞❡① q✉❛♥t✐t② ✐s ❦❡♣t ✐♥ t❤❡ ♠❡♠♦r②✳ ❋♦r ❧❛r❣❡r s②st❡♠s ✭♦r ❧❛r❣❡r ❜❛s✐s s❡ts✮ t❤✐s ❛ss✉♠♣t✐♦♥ ✇♦✉❧❞ ❜❡❝♦♠❡ ✈❡r② q✉✐❝❦❧② ❛ ❜♦tt❧❡♥❡❝❦✱ t❤❡r❡❢♦r❡ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❜❛t❝❤❡s t❤❡ ✐♥t❡❣r❛❧s ✐♥t♦ t❤❡ ♣✐❡❝❡s t❤❛t ✜t ✐♥t♦ t❤❡ ❛✈❛✐❧❛❜❧❡ ♠❡♠♦r② ✇❛s ❞❡s✐❣♥❡❞✳ ❚❤❡ ♥❡①t st❡♣ ✐s t❤❡ ❜❛❝❦✲tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❢♦✉r✲✐♥❞❡① ✐♥t❡❣r❛❧s ✇❤✐❝❤ µ,ν ✐s ♥❡❡❞❡❞ ❢♦r t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ Vx,y ✐♥t❡r♠❡❞✐❛t❡✳ ❚❤✐s ♣❛rt ♦❢ t❤❡ ❝♦❞❡ ✭st❡♣ ✺❀ ❆❧❣✳ ✶✮ ✐s ❡①♣❧❛✐♥❡❞ ✐♥ ❙✉❜s❡❝t✐♦♥ ✹✳✹ ✇❤❡r❡ t❤❡ ✐ss✉❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤✐s ✐♥t❡r♠❡❞✐❛t❡ µ,ν ❛r❡ ❞✐s❝✉ss❡❞✳ ❲❤❡♥ t❤❡ Vx,y ✐s ❝♦♠♣✉t❡❞ ✭st❡♣ ✻❀ ❆❧❣✳ ✶✮ t❤❡ ♥❡①t t❛s❦ ✐s t♦ tr❛♥s❢♦r♠ i,j t❤✐s q✉❛♥t✐t② t♦ t❤❡ ♦❝❝✉♣✐❡❞ s♣❛❝❡ ✭st❡♣ ✼❀ ❆❧❣✳ ✶✮✳ ❚❤❡ Vx,y ✐s ♥❡❡❞❡❞ ❢♦r t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ❡♥❡r❣② ✭✇❤✐❝❤ ✇✐❧❧ ❜❡ ❡①♣❧❛✐♥❡❞ ✐♥ ❙✉❜s❡❝t✐♦♥ ✹✳✺✮ ❛♥❞ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s µ,ν (Cµi )† Vx,y Cνj ✳ i,j Vx,y = ✭✽✷✮ µν ■t ✇❛s ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞ t❤❛t ✇✐t❤✐♥ t❤❡ ❝✉rr❡♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥ t❤❡ ❣❡♠✐♥❛❧ ✐♥❞✐❝❡s ✭xy ✮ r❡❢❡r t♦ t❤❡ ♦❝❝✉♣✐❡❞ s♣❛❝❡✳ q✉❛♥t✐t✐❡s ❛r❡ ❝♦♠♣✉t❡❞✳ ❆t st❡♣ ✽ ♦❢ ❆❧❣♦r✐t❤♠ ✶ t❤❡ ❛❞❞✐t✐♦♥❛❧ t❤r❡❡✲✐♥❞❡① ❚❤❡ ♣r❡❧✐♠✐♥❛r② r✉♥ ♦❢ t❤❡ ▼P✷✲❋✶✷ ♣r♦❣r❛♠ ♣r♦✈✐❞❡s t❤❡ ✐♥t❡❣r❛❧s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✶✱ t❤❡ ❛❞❞✐t✐♦♥❛❧ ✐♥t❡❣r❛❧s✱ ♥❡❡❞❡❞ ❛t t❤❡ ❈❈❙❉✭❋✶✷✮ ❧❡✈❡❧✱ ❤❛✈❡ t❤❡ ♦r❜✐t❛❧ ✐♥❞✐❝❡s ✐♥ t❤❡ ❈❆❇❙ ❛♥❞ ✈✐rt✉❛❧ s♣❛❝❡s✱ BQ,ap ✳ ❚❤❡② ❛r❡ ♥❡❡❞❡❞ ❢♦r ✈✐❞❡ ✐♥❢r❛ ✮ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❛❧❡♥❝❡ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❋♦❝❦ ♠❛tr✐① ❡❧❡♠❡♥ts ✭ t❤❡ C ❛♥❞ D ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ T2 ❛♥❞ T2 ❛♥❞ ✈❡❝t♦r ❢✉♥❝t✐♦♥s ✭s❡❡ ❙✉❜s❡❝t✐♦♥ ✹✳✽ ❢♦r ❞❡t❛✐❧s✮✳ ❚❤❡ ❞❡t❛✐❧❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ t❤r❡❡✲✐♥❞❡① ✐♥t❡❣r❛❧s ✐♥ ❚✉r❜♦♠♦❧❡ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✺✶❪✳ ❚❤❡ ❧❛st st❡♣✱ ❜❡❢♦r❡ st❛rt✐♥❣ t❤❡ ❈❈ ✐t❡r❛t✐♦♥s✱ r❡❢❡rs t♦ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❛❧❡♥❝❡ ♣❛rts ♦❢ t❤❡ ❋♦❝❦ ♠❛tr✐❝❡s ✐♥✈♦❧✈✐♥❣ t❤❡ ❈❆❇❙ ✐♥❞❡① ✭st❡♣ ✾❀ ❆❧❣✳ ✶✮✳ ❚❤❡ ❛❞❥❡❝t✐✈❡ ✏✈❛❧❡♥❝❡✑ ❤❡r❡ ♠❡❛♥s t❤❛t ♦♥❧② t❤❡ ❛❝t✐✈❡ t✇♦✲ ❡❧❡❝tr♦♥ ♣❛rt ♦❢ t❤❡ ❋♦❝❦ ♦♣❡r❛t♦r ✐s ✐♥❝❧✉❞❡❞✳ ❚❤❡s❡ ❋♦❝❦ ♠❛tr✐❝❡s ❛r❡ ❝♦♠♣✉t❡❞ ✇✐t❤✐♥ ✷✸ ✹✳ ❚❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞❡❧ ✐♥ ❚✉r❜♦♠♦❧❡ t❤❡ ❉❋ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② Fipval = k,p p ,k }✱ − ❉❋ gi,k {2❉❋ gi,k ✭✽✸✮ p ,k k,p {2❉❋ ga,k − ❉❋ ga,k }✱ ✭✽✹✮ k val Fap = k ✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥ r✉♥s ♦✈❡r t❤❡ ❛❝t✐✈❡ ♦❝❝✉♣✐❡❞ ♦r❜✐t❛❧s✳ ❊❛❝❤ ❈♦✉❧♦♠❜ ✐♥t❡❣r❛❧ ❤❛s t♦ ❜❡ r❡❝♦✈❡r❡❞ ❢r♦♠ t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❛♥❞ st♦r❡❞ t❤r❡❡✲✐♥❞❡① q✉❛♥t✐t✐❡s ❜② ❝♦♥tr❛❝t✐♥❣ t❤❡♠ ♦✈❡r t❤❡ ❛✉①✐❧✐❛r② ✐♥❞❡① Q✳ Fipval = ❚❤❡ ✜♥❛❧ ❢♦r♠✉❧❛s ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s {2BQ,ip BQ,kk − BQ,ik BQ,p k }✱ ✭✽✺✮ {2BQ,ap BQ,kk − BQ,ak BQ,p k }✳ ✭✽✻✮ kQ val = Fap kQ ❯♥t✐❧ ♥♦✇ ❛❧❧ st❡♣s ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ♣r❡❧✐♠✐♥❛r② ♣❛rt ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ❝❛❧❝✉✲ ❧❛t✐♦♥✳ ❚❤❡ r❡♠❛✐♥✐♥❣✱ ♠❛✐♥ ♣❛rt ♦❢ t❤❡ ♣r♦❣r❛♠✱ ✐t❡r❛t✐✈❡❧② s♦❧✈❡s t❤❡ ❈❈ ❡q✉❛t✐♦♥s✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❈❈ ❡q✉❛t✐♦♥s r❡q✉✐r❡s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❈❈ r❡s✐❞✉❛❧s ✭st❡♣ ✶✵❀ ❆❧❣✳ ✶✮ t❤❛t ❛r❡ ❞❡✜♥❡❞ ❜② t❤❡ ♣r♦❥❡❝t✐♦♥s ♦♥t♦ t❤❡ ❡①❝✐t❛t✐♦♥ ♠❛♥✐❢♦❧❞s ❬❊qs✳ ✭✹✺✮✱ ✭✹✻✮✱ ✭✹✼✮❪✳ ❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ r❡s✐❞✉❛❧s ❛❧❧♦✇ ❢♦r ❞❡t❡r♠✐♥✐♥❣ t❤❡ ♥❡①t ❣✉❡ss ❢♦r t❤❡ ❛♠♣❧✐t✉❞❡s ❛♥❞ t❤❡s❡ ♥❡✇ ❛♠♣❧✐t✉❞❡s ✭st❡♣s ✶✷✱ ✶✸❀ ❆❧❣✳ ✶✮ ❧❡❛❞ t♦ ♥❡✇ r❡s✐❞✉❛❧s✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ✐s r❡♣❡❛t❡❞ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡ ✐s r❡❛❝❤❡❞✳ ❚❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❡♥❡r❣✐❡s ✐♥ t❤❡ ✐t❡r❛t✐♦♥s T1 ✈❡❝t♦r ❢✉♥❝t✐♦♥✳ (k) ❛♥❞ (k + 1) ✭st❡♣ ✶✶❀ ❆❧❣✳ ✶✮ ❛♥❞ t❤❡ ♥♦r♠ ♦❢ t❤❡ ❆s t❤❡ ✐♥✐t✐❛❧ ❣✉❡ss ❢♦r t❤❡ ❛♠♣❧✐t✉❞❡s✱ t❤❡ ♦♣t✐♠❛❧ ▼P✷ ❛♥❞ ▼P✷✲❋✶✷ ❛♠♣❧✐t✉❞❡s ❛r❡ ❛ss✉♠❡❞ ❢♦r t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ❛♥❞ ❡①♣❧✐❝✐t❧② ❝♦rr❡❧❛t❡❞ ♣❛rts✱ r❡s♣❡❝t✐✈❡❧②✳ ❚♦ ✐♠♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡✱ t❤❡ ❞✐r❡❝t ✐♥✈❡rs✐♦♥ ✐♥ t❤❡ ✐t❡r❛t✐✈❡ s✉❜s♣❛❝❡ ✭❉■■❙✮ ♠❡t❤♦❞ ❬✺✷❪ ✐s ❛❧✇❛②s ✉s❡❞ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♥❡①t ❣✉❡ss ❢♦r t❤❡ ❛♠♣❧✐t✉❞❡s ✭st❡♣ ✶✸❀ ❆❧❣✳ ✶✮✳ µ,ν ✐♥t❡r♠❡❞✐❛t❡ ✹✳✹ ❚❤❡ Vx,y ❇❡❢♦r❡ ❞✐s❝✉ss✐♥❣ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ T1 ✱ T2 ❛♥❞ T2 ✈❡❝t♦r ❢✉♥❝t✐♦♥s✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ µ,ν ✐♥tr♦❞✉❝❡ t❤❡ Vx,y ✐♥t❡r♠❡❞✐❛t❡✳ ■♥ t❤✐s ♣❛rt ♦❢ t❤❡ t❤❡s✐s✱ ❞❡t❛✐❧❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤✐s q✉❛♥t✐t② ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ❈❈ r❡s✐❞✉❛❧s µ,ν ❛r❡ r❡❝♦✈❡r❡❞ ❢r♦♠ Vx,y ❛♥❞ t❤✉s ✐t ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ✏❦❡② ✐♥t❡r♠❡❞✐❛t❡✑ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ❚✉r❜♦♠♦❧❡✳ ■♥ ❣❡♥❡r❛❧✱ s✉❝❤ q✉❛♥t✐t②✱ ✐♥ t❤❡ ▼❖ ❜❛s✐s✱ ❝❛♥ ❜❡ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② −1 r,s ˆ 12 r12 = pq|f12 Q |rs , Vp,q ✇❤❡r❡ f12 ✐s t❤❡ ❝♦rr❡❧❛t✐♦♥ ❢❛❝t♦r✱ ˆ 12 Q ✭✽✼✮ ✐s t❤❡ str♦♥❣ ♦rt❤♦❣♦♥❛❧✐t② ♣r♦❥❡❝t♦r t❤❛t ❞❡✲ ✜♥❡s ❆♥s❛t③ ♦❢ t❤❡ ❡①♣❧✐❝✐t❧② ❝♦rr❡❧❛t❡❞ ✇❛✈❡ ❢✉♥❝t✐♦♥ ✭s❡❡ ❙✉❜s❡❝t✐♦♥ ✹✳✷ ❢♦r t❤❡ ❞❡t❛✐❧s −1 ˆ 12 ✮✱ ❛♥❞ r12 ✐s t❤❡ ❡❧❡❝tr♦♥✲❡❧❡❝tr♦♥ ❈♦✉❧♦♠❜ ♦♣❡r❛t♦r✳ ❚❤❡ ✐♠♣❧❡✲ ❝♦♥❝❡r♥✐♥❣ f12 ❛♥❞ Q ♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉ ✭❛♥❞ t❤✉s ❈❈❙❉✭❋✶✷✮✮ ♠❡t❤♦❞ ✐♥ ❚✉r❜♦♠♦❧❡ ✐s✱ t♦ ❛ ❧❛r❣❡ ❡①t❡♥t✱ ❞❡s✐❣♥❡❞ ✐♥ ❛♥ ✐♥t❡❣r❛❧✲❞✐r❡❝t ❢❛s❤✐♦♥✳ ■t ♠❡❛♥s t❤❛t t❤❡ ♠♦st t✐♠❡✲❝♦♥s✉♠✐♥❣ q✉❛♥t✐t✐❡s ❛r❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❆❖✲❜❛s✐s✱ ❝♦♠❜✐♥❡❞ t♦❣❡t❤❡r ❛♥❞ ❧❛t❡r ♦♥ tr❛♥s❢♦r♠❡❞ t♦ t❤❡ ▼❖✲❜❛s✐s✳ ❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❞❞✐t✐♦♥❛❧ ❋✶✷ t❡r♠s ✭❛t ❧❡❛st s♦♠❡ ♦❢ t❤❡♠✮ µ,ν ❢♦❧❧♦✇s t❤✐s ✐❞❡❛ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ Vx,y ✇✐t❤ t✇♦ ❆❖ ✐♥❞✐❝❡s ✇❛s ✐♥tr♦❞✉❝❡❞✳ ✷✹ ✹✳✹✳ ❚❤❡ µ,ν Vx,y ✐♥t❡r♠❡❞✐❛t❡ ❆♠♦♥❣ t❤❡ ❋✶✷ t❡r♠s t❤✐s ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t q✉❛♥t✐t✐❡s ❛♥❞ ✇✐❧❧ ❜❡ ❞✐s❝✉ss ✐♥ ❞❡t❛✐❧ ❤❡r❡✳ ✐✳❡✳ ■t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❜❡❣✐♥ ✇✐t❤ t❤❡ s✐♠♣❧❡st ❝❛s❡✱ t❤❡ V ✐♥t❡r♠❡❞✐❛t❡ ❞❡r✐✈❡❞ ✇✐t❤✐♥ ❆♥✶ ❛♥❞ t❤❡ ❘❍❋ r❡❢❡r❡♥❝❡✱ ✇✐t❤♦✉t t❤❡ ❈❆❇❙ ❝♦♥tr✐❜✉t✐♦♥ ✭❆✶✲♥♦❈❆❇❙✮✳ ■❢ ✇❡ t❛❦❡ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ♦❢ t❤❡ V ✐♥t❡r♠❡❞✐❛t❡ ❬❊q✳ ✭✽✼✮❪ ❛♥❞ s✉❜st✐t✉t❡ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ str♦♥❣ ♦rt❤♦❣♦♥❛❧✐t② ♣r♦❥❡❝t♦r t❤❛t ❞❡✜♥❡s ❆♥s❛t③ ✶ ❬❊q✳ ✭✻✶✮❪ ✇❡ ✇✐❧❧ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ r,s −1 Vp,q = pq|f12 ((1 − Pˆ1 )(1 − Pˆ2 ))r12 |rs ✳ ✭✽✽✮ ■♥ t❤❡ ❆❖✲❜❛s❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞❡❧ ✇❡ ♥❡❡❞ t❤❡ µ,ν Vx,y ✱ ✇❤❡r❡ t❤❡ xy ❛r❡ t❤❡ ❣❡♠✐♥❛❧ ✐♥❞✐❝❡s ✭❡❢❡❝t✐✈❡❧② t❤❡② ❜❡❧♦♥❣ t♦ t❤❡ ♦❝❝✉♣✐❡❞ s♣❛❝❡ ✇✐t❤✐♥ t❤✐s ✐♠♣❧❡♠❡♥✲ t❛t✐♦♥✱ s❡❡ ❙✉❜s❡❝t✐♦♥ ✸✳✷✮ ❛♥❞ µν ❜❡❧♦♥❣ t♦ t❤❡ ❝♦✈❛r✐❛♥t ❆❖✲❜❛s✐s✳ ■♥ t❤❡ s✉❜s❡q✉❡♥t s❡❝t✐♦♥s ✇❡ ✇✐❧❧ ✉s❡ t❤❡ t❡r♠ ❝♦✈❛r✐❛♥t ❢♦r t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ q✉❛♥t✐t✐❡s ✐♥ t❤❡ ✏♦r❞✐✲ ♥❛r②✑ ❆❖ ❜❛s✐s ✭❞❡♥♦t❡❞ ❜② ✉♣♣❡r✲❝❛s❡ ✏❆❖✑ ❧❡❢t s✉♣❡rs❝r✐♣t✮ ✇❤❡r❡❛s t❤❡ ❝♦♥tr❛✈❛r✐❛♥t ✈✐❞❡ ✐♥❢r❛ ✮ ✭❞❡♥♦t❡❞ ✇✐t❤ ❧♦✇❡r✲❝❛s❡ ✏❛♦✑ ❜❛s✐s ✐s ✉s❡❞ ❢♦r t❤❡ ❜❛❝❦✲tr❛♥s❢♦r♠❡❞ q✉❛♥t✐t✐❡s ✭ ❧❡❢t s✉♣❡rs❝r✐♣t✮✳ ❚❤❡ ✜♥❛❧ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❆✶✲♥♦❈❆❇❙ V ✐♥t❡r♠❡❞✐❛t❡ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② ❆✶✲♥♦❈❆❇❙ ❛♦ µ,ν Vx,y = (f g)µ,ν x,y − γ,δ ❆❖ µ,ν fx,y gγ,δ ✱ ✭✽✾✮ γδ (f g)µ,ν x,y ✇❤❡r❡ ✐s t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r t❤❡ −1 (f12 r12 ) ♦♣❡r❛t♦r −1 (f g)µ,ν x,y = xy|f12 r12 |µν ❚❤❡ ❛♦ ✳ ✭✾✵✮ γ,δ ❆❖ µ,ν fx,y ❛r❡ t❤❡ ❛❜♦✈❡ ♠❡♥t✐♦♥❡❞ ❜❛❝❦✲tr❛♥s❢♦r♠❡❞ f12 ✐♥t❡❣r❛❧s ❛♥❞ gγ,δ ❛r❡ ❈♦✉❧♦♠❜ ✐♥t❡❣r❛❧s ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❝♦✈❛r✐❛♥t ❆❖✲❜❛s✐s✳ ❚❤❡ ❜❛❝❦✲tr❛♥s❢♦r♠❡❞ q✉❛♥t✐t✐❡s ❛r❡ ♦❜✲ t❛✐♥❡❞ ❢r♦♠ t❤❡ q✉❛♥t✐t✐❡s ✐♥ t❤❡ ▼❖✲❜❛s✐s ❜② ❝♦♥tr❛❝t✐♥❣ t❤❡♠ ✇✐t❤ t❤❡ ❆❖✲▼❖ ❝♦❡✣✲ ❝✐❡♥ts ♦✈❡r t❤❡ ▼❖ ✐♥❞✐❝❡s✳ ❚❤❡ ✐♥t❡❣r❛❧s ✐♥ t❤❡ ▼❖✲❜❛s✐s ❛r❡ ❛✈❛✐❧❛❜❧❡ ❢r♦♠ t❤❡ ♣r❡❝❡❞✐♥❣ ▼P✷✲❋✶✷ ❝❛❧❝✉❧❛t✐♦♥ ❛♥❞ t❤❡ ❆❖✲▼❖ ❝♦❡✣❝✐❡♥ts ❛r❡ ❛❧✇❛②s ❛✈❛✐❧❛❜❧❡ ❛s t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛♦ γ,δ t❤❡ ❍❛rtr❡❡✲❋♦❝❦ ❡q✉❛t✐♦♥s✳ ❚❤❡ fx,y ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s ❛♦ p,q Cγp fx,y (Cδq )† .

R❜♦♠♦❧❡ ❚❤❡ q✉❛♥t✐t✐❡s ♦❜t❛✐♥❡❞ ✇✐t❤ ❆♥✶ ✐♥✈♦❧✈❡ t❤❡ ❣❡♥❡r❛❧ ❛♥❞ ❈❆❇❙ ✐♥❞✐❝❡s✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t ✇❡ ❛✈♦✐❞ ❤❡r❡ t❤❡ q✉❛♥t✐t✐❡s ❤❛✈✐♥❣ t✇♦ ❈❆❇❙ ✐♥❞✐❝❡s s✐♠✉❧t❛♥❡♦✉s❧②✳ ❚❤❡ ❆♥✶ ❤❛s ❜❡❡♥ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ❚✉r❜♦♠♦❧❡ ♣r♦❣r❛♠✱ ❜✉t t❤❡ r❡s✉❧ts ②✐❡❧❞✐♥❣ ❜② t❤✐s ❆♥s❛t③ ❛r❡ ♥♦✇❛❞❛②s ❝♦♥s✐❞❡r❡❞ ❛s ❧❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤♦s❡ ♦❜t❛✐♥❡❞ ✇✐t❤ ❆♥✸✳ ❚❤❡ ♠♦st s✉❝❝❡ss❢✉❧ ❆♥s❛t③ ✐♥ ❋✶✷ t❤❡♦r② ✉s❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ ♦❢ t❤❡ str♦♥❣ ♦rt❤♦❣♦✲ ♥❛❧✐t② ♣r♦❥❡❝t♦r ❬✶✾✱ ✷✵❪ ˆ 312 = (ˆ1 − O ˆ 1 )(ˆ1 − O ˆ 2 ) − Vˆ1 Vˆ2 ✳ Q ❚❤✐s ❝❤♦✐❝❡ r❡❢❡rs t♦ ❆♥✸✳ ✭✼✻✮ ˆ 1 ❛♥❞ Vˆ1 ❛r❡ t❤❡ ♦♥❡✲❡❧❡❝tr♦♥ ♣r♦❥❡❝t♦rs ♦♥t♦ t❤❡ ♦rt❤♦♥♦r♠❛❧ O s♣❛t✐❛❧ ♦r❜✐t❛❧s t❤❛t ❛r❡ ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ ♦❝❝✉♣✐❡❞ ❛♥❞ ✈✐rt✉❛❧ s♣❛❝❡s✱ r❡s♣❡❝t✐✈❡❧② ˆ1 = O |φi φi |✱ ✭✼✼✮ |φa φa |✳ ✭✼✽✮ i Vˆ1 = a ❆❧❧ ❢♦r♠✉❧❛s ✇✐t❤ t❤❡ ♣r♦❥❡❝t♦r ❊q✳ ✭✼✻✮ ❤❛✈❡ ❜❡❡♥ ❞❡r✐✈❡❞ ❛♥❞ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❚✉r❜♦✲ ♠♦❧❡ ✐♥ t❤❡ ❝♦✉rs❡ ♦❢ t❤✐s t❤❡s✐s✳ ❚♦ ❛✈♦✐❞ ♠❛♥② ❡❧❡❝tr♦♥ ✐♥t❡❣r❛❧s ♦♥❡ ❝❛♥✱ s✐♠✐❧❛r t♦ t❤❡ ❝❛s❡ ♦❢ ❆♥✶✱ ❛♣♣❧② t❤❡ ❙❆ ♣r♦❝❡❞✉r❡ ✇✐t❤ t❤❡ ✉s❡ ♦❢ t❤❡ ❈❆❇❙ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❚❤✐s ②✐❡❧❞s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥tr✐❜✉t✐♦♥s ˆ 312 ≈ˆ1 − O ˆ 1 Pˆ2 − O ˆ 1 Pˆ2 + Pˆ1 O ˆ 2 + Pˆ1 O ˆ2 + O ˆ1O ˆ 2 − Vˆ1 Vˆ2 Q ˆ 1 Pˆ2 − Pˆ1 O ˆ2 = Q ˆ 112 − O ˆ 1 Pˆ − Pˆ O ˆ =ˆ1 − Pˆ1 Pˆ2 − O 2 1 2✳ ✭✼✾✮ ●r❛♣❤✐❝❛❧❧②✱ t❤❡ ❛❝t✐♦♥ ♦❢ t❤✐s ♣r♦❥❡❝t♦r ✐s ❡①♣❧❛✐♥❡❞ ✐♥ ❋✐❣✉r❡ ✹✳ ❚❤❡ s♣❛❝❡ s♣❛♥♥❡❞ ❜② ❋✐❣✉r❡ ✹✿ ❚❤❡ ♣✐❝t♦r✐❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t♦r ✇✐t❤✐♥ ❆♥s❛t③ ✸✳ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ❡①❝✐t❛t✐♦♥ ✭i, j → a, b✮ ✐s s❡♣❛r❛t❡❞ ♦✉t ❛♥❞ ♦♥❧② t❤❡ ❣❡♠✐♥❛❧ ❡①❝✐t❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ t❤❡ ❈❆❇❙ ♦r❜✐t❛❧s ❛r❡ ❧❡❢t✳ ❍❡♥❝❡✱ ❛❢t❡r t❤❡ ❛❝t✐♦♥ ♦❢ ♦♣❡r❛t♦r ❊q✳ ✭✼✾✮✱ ♦♥❧② t❤❡ ❞♦✉❜❧❡ r❡♣❧❛❝❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ ✭i, j → α⊥ , β⊥ ✮✱ ✭i, j → a, β⊥ ✮ ❛♥❞ ✭i, j → α⊥ , b✮ r❡♠❛✐♥✳ ❚❤❡ ♦t❤❡r ❡①❝✐t❛t✐♦♥s✱ r❡❢❡rr✐♥❣ t♦ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ r❡♣❧❛❝❡♠❡♥ts✱ ❛r❡ ♥♦t ❝♦♥✲ s✐❞❡r❡❞ ✇✐t❤✐♥ t❤❡ ❡①♣❧✐❝✐t❧② ❝♦rr❡❧❛t❡❞ ❝❛❧❝✉❧❛t✐♦♥✳ ❚❤❡ ✐♥❞✐❝❡s α⊥ ❛♥❞ β⊥ r❡❢❡r t♦ t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② s♣❛❝❡ t❤❛t ✐s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ s♣❛❝❡ ♦❢ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ▼❖s✳ ✹✳✸ ❚❤❡ ❣❡♥❡r❛❧ s❦❡t❝❤ ♦❢ t❤❡ ♣r♦❣r❛♠ ✢♦✇ ❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞❡❧ ✐♥ ❚✉r❜♦♠♦❧❡ ✐♥✈♦❧✈❡s ♠❛♥② q✉❛♥t✐t✐❡s ❛♥❞ ❛ ✈❛r✐❡t② ♦❢ ❛❧❣♦r✐t❤♠s t❤❛t ❛r❡ ✉s❡❞ ❢♦r t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ❡♥❡r❣②✳ ✷✵ ✹✳✸✳ ❚❤❡ ❣❡♥❡r❛❧ s❦❡t❝❤ ♦❢ t❤❡ ♣r♦❣r❛♠ ✢♦✇ ❚❤❡ ♣r❡s❡♥t ✇♦r❦ s❤♦✉❧❞ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ❡①t❡♥s✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❡①✐st✐♥❣ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ❈❈❙❉ ♠♦❞❡❧ ❬✹✽❪✳ ❚❤❡ ♠❛♥♥❡r✱ ✐♥ ✇❤✐❝❤ t❤❡ ❈❈❙❉✱ ❛♥❞ t❤✉s ❈❈❙❉✭❋✶✷✮✱ ♠♦❞❡❧s ❛r❡ r❡❛❧✐③❡❞ ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❛s ✐♥t❡❣r❛❧✲❞✐r❡❝t✕❞❡♥s✐t②✲ ✜tt✐♥❣ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ ■t ♠❡❛♥s t❤❛t s♦♠❡ q✉❛♥t✐t✐❡s ❛r❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❞❡♥s✐t②✲✜tt✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❉❋✮✱ t❤❡ ♦t❤❡r ♦♥❡s ❛r❡ ❝♦♠♣✉t❡❞ ♦♥✲t❤❡✲✢② ✐♥ t❤❡ ❆❖ ❜❛s✐s ❛♥❞ ❝♦♥✲ tr❛❝t❡❞ ✇✐t❤ s♦♠❡ ♦t❤❡r q✉❛♥t✐t✐❡s ❬✹✽❪✳ ❚❤❡ ❉❋ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ♦❢t❡♥ ❝❛❧❧❡❞ t❤❡ ❘■ ❛♣♣r♦①✐♠❛t✐♦♥ ✭r❡s♦❧✉t✐♦♥✲♦❢✲t❤❡✲✐❞❡♥t✐t②✮✱ ❜✉t ✇✐t❤✐♥ t❤❡ ❝✉rr❡♥t ✇♦r❦ t❤❡ t❡r♠ ✏❘■ ❛♣♣r♦①✐♠❛t✐♦♥✑ ✐s r❡s❡r✈❡❞ ❢♦r t❤❡ ✐♥tr✐♥s✐❝ ❘■ ✉s❡❞ ✐♥ t❤❡ ❘✶✷ t❤❡♦r②✳ ❚❤❡ ♠❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ❉❋ ❛♣♣r♦①✐♠❛t✐♦♥ r❡❧✐❡s ♦♥ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❢♦✉r✲✐♥❞❡① q✉❛♥t✐t✐❡s ❛r❡ ♥❡✈❡r st♦r❡❞ ❡①♣❧✐❝✐t❧②✳ ■♥st❡❛❞✱ t❤❡② ❛r❡ ❝♦♠♣✉t❡❞ ♦♥✲t❤❡✲✢② ❢r♦♠ t❤r❡❡✲✐♥❞❡① ✐♥t❡❣r❛❧s ❜② ❝❛rr②✐♥❣ ♦✉t t❤❡ ❝♦♥tr❛❝t✐♦♥ ♦✈❡r t❤❡ ✐♥❞❡① ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❛✉①✐❧✐❛r② ❜❛s✐s s❡t✳ ❚❤❡ ✐♥❞❡① ♦❢ t❤❡ ❉❋ ❛✉①✐❧✐❛r② ❜❛s✐s ✭✉s✉❛❧❧② ❞❡♥♦t❡❞ ✇✐t❤ Q✮ ❢r♦♠ t❤❡ ❈❆❇❙ ✐♥❞❡① ✭❞❡♥♦t❡❞ ❛s ❞♦✉❜❧② ♣r✐♠❡❞ ✐♥❞❡① s❤♦✉❧❞ ❜❡ ❝❧❡❛r❧② ❞✐st✐♥❣✉✐s❤❡❞ p ✮✱ t❤❡② r❡❢❡r t♦ ❡♥t✐r❡❧② ❞✐❢✲ ❢❡r❡♥t s♣❛❝❡s✳ ❚❤❡ ❉❋ ❛♣♣r♦①✐♠❛t✐♦♥ r❡q✉✐r❡s ❛❞❞✐t✐♦♥❛❧ ✇❡❧❧ ♦♣t✐♠✐③❡❞ ❜❛s✐s s❡ts✱ ❜✉t✱ ❡s♣❡❝✐❛❧❧② ❛t t❤❡ ▼P✷ ❧❡✈❡❧ ♦❢ t❤❡♦r②✱ t❤❡ s♣❡❡❞ ✉♣ ♦❢ t❤❡ ♣r♦❣r❛♠ ❝♦♠♣❡♥s❛t❡s t❤❡ ❡✛♦rt r❡❧❛t❡❞ t♦ t❤✐s ❡①tr❛ ❜❛s✐s s❡t✳ ❲✐t❤✐♥ t❤❡ s❡r✐❡s ♦❢ t❤❡ ♠❛♥②✲❜♦❞② ♠♦❞❡❧s✱ ▼P✷ ❛s t❤❡ s✐♠♣❧❡st ♦♥❡ t❤❛t t❛❦❡s ❝❛r❡ ♦❢ ❞②♥❛♠✐❝ ❝♦rr❡❧❛t✐♦♥✱ ✐s ❛ ❜✐t ✉♥✉s✉❛❧✳ ❲✐t❤✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ❜♦tt❧❡♥❡❝❦ ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧s✱ t❤❡ ❝♦♥tr❛❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✐♥t❡❣r❛❧s ❛♥❞ t❤❡ ❛♠♣❧✐t✉❞❡s ✐s ❛ ♠✉❝❤ ❝❤❡❛♣❡r st❡♣✳ ❚❤✐s ✐s t❤❡ r❡❛s♦♥ ❢♦r t❤❡ s✐❣♥✐✜❝❛♥t s♣❡❡❞ ✉♣ ♦❢ t❤❡ ❝♦❞❡ ✐❢ t❤❡ ❉❋ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❛♣♣❧✐❡❞ ❢♦r t❤❡ ❢♦✉r✲✐♥❞❡① ❈♦✉❧♦♠❜ ✐♥t❡❣r❛❧s ❛t t❤❡ ▼P✷ ❧❡✈❡❧ ♦❢ t❤❡♦r②✳ ❚❤❡ s❝❛❧✐♥❣ ♦❢ t❤❡ ❤✐❣❤❡r ♠❛♥② ❜♦❞② ♠♦❞❡❧s ✭❈❈❙❉✱ ❈❈❙❉❚ ❛♥❞ s♦ ♦♥✮ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❝♦st ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❝♦♥tr❛❝t✐♦♥s✳ ❋♦r t❤❡s❡ ♠♦❞❡❧s ✭❡s♣❡❝✐❛❧❧② ❢♦r t❤♦s❡ t❤❛t ❣♦ ❜❡②♦♥❞ t❤❡ ❈❈❙❉ ♠♦❞❡❧✮ t❤❡ ❉❋ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ♥♦t t❤❛t ❝r✐t✐❝❛❧✳ ❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞❡❧ ✐♥ ❚✉r❜♦♠♦❧❡ r❡❧✐❡s ♦♥ t❤❡ ❜✉✐❧t✲✐♥ ▼P✷✲ ❋✶✷ ♠♦❞✉❧❡ ❬✹✾✱ ✺✵❪✳ ❚❤❡ ❣❡♥❡r❛❧ s❦❡t❝❤ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐s s❤♦✇♥ ✐♥ ❆❧❣♦r✐t❤♠ ✶✳ ✷✶ ✹✳ ❚❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞❡❧ ✐♥ ❚✉r❜♦♠♦❧❡ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ❘❡s✉❧t✿ ❚❤❡ ❈❈❙❉✭❋✶✷✮ ❡♥❡r❣② P❡r❢♦r♠ t❤❡ ▼P✷✲❋✶✷ ❝❛❧❝✉❧❛t✐♦♥ ✭✐♥✈♦❧✈❡❞ st❡♣✱ ❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐❧ ❡❧s❡✇❤❡r❡✮ −1 ❈♦♠♣✉t❡ t❤❡ ✐♥t❡❣r❛❧s ♦✈❡r t❤❡ (f12 r12 ) ♦♣❡r❛t♦r ✇✐t❤ t✇♦ ❆❖ ✐♥❞✐❝❡s ❚r❛♥s❢♦r♠ t❤❡ ▼P✷✲❋✶✷ q✉❛♥t✐t✐❡s ✭❝❛❧❧ t♦ ❢r♦♠ t❤❡ ♠❛✐♥ ♣r♦❣r❛♠ ❝❝r✶✷❴♣r❡♣ r✐❝❝✷✮ ❘❡s♦rt t❤❡ ✐♥t❡❣r❛❧s ✭❝❛❧❧ t♦ r✐r✶✷❴r❡s♦rt ❢r♦♠ ❝❝s❞✐♥t r♦✉t✐♥❡✮ ❈❛rr② ♦✉t t❤❡ ❜❛❝❦✲tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❢♦✉r✲✐♥❞❡① ✐♥t❡❣r❛❧s ✭❝❛❧❧ t♦ ❝❝r✶✷❴♣qt♦♠✉♥✉ ❢r♦♠ ❝❝s❞✐♥t r♦✉t✐♥❡✮ ❈♦♠♣✉t❡ t❤❡ ♥♦♥ T1 ✲❞❡♣❡♥❞❡♥t V ✐♥t❡r♠❡❞✐❛t❡ ✭❝❛❧❧ t♦ ❝❝s❞✐♥t r♦✉t✐♥❡✱ ❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐❧ ✐♥ ❙✉❜s❡❝t✐♦♥✳ ✹✳✹✮ ❚r❛♥s❢♦r♠ t❤❡ µ,ν Vx,y ❝❝s❞✐♥t r♦✉t✐♥❡✮ ✐♥t❡r♠❡❞✐❛t❡ t♦ t❤❡ ♦❝❝✉♣✐❡❞ s♣❛❝❡ r✐r✶✷❝❝s❞❴✈✐♥t ❢r♦♠ i,j Vx,y ✭❝❛❧❧ t♦ ❝❝❴❜❢♠♦ ❢r♦♠ BQ,ap ✭❝❝s❞✐♥t r♦✉t✐♥❡✮ ❈♦♠♣✉t❡ t❤❡ ❋♦❝❦ ♠❛tr✐① ❡❧❡♠❡♥ts Fip ❛♥❞ Fap ✭❝❝s❞✐♥t r♦✉t✐♥❡✮ ✇❤✐❧❡ ❈♦♥✈❡r❣❡♥❝② ❝r✐t❡r✐❛ ♥♦t ❢✉❧✜❧❧❡❞ ✭❡♥❡r❣②✱ ♥♦r♠ ♦❢ t❤❡ T1 r❡s✐❞✉❛❧✮ ❞♦ ✶✵ ❈♦♠♣✉t❡ t❤❡ T1 ✱ T2 ❛♥❞ T2 r❡s✐❞✉❛❧s ✭❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐❧ ✐♥ ❙✉❜s❡❝t✐♦♥s✳ ✹✳✼✱ ✽ ✾ Pr❡♣❛r❡ t❤❡ t❤r❡❡✲✐♥❞❡① q✉❛♥t✐t✐❡s ✹✳✽✮ ✶✶ ✶✷ ✶✸ ❝❝❴❡♥❡r❣②✮ Pr❡❝♦♥❞✐t✐♦♥ t❤❡ ❛♠♣❧✐t✉❞❡s ✭❝❝❴♣r❡❝♦♥❞✮ ❈♦♠♣✉t❡ t❤❡ ❡♥❡r❣② ✭ ●❡t t❤❡ ❉■■❙ ❣✉❡ss ❢♦r t❤❡ ♥❡①t✲✐t❡r❛t✐♦♥ ❛♠♣❧✐t✉❞❡s ✭ ❡♥❞ ❆❧❣♦r✐t❤♠ ✶✿ ❝❝❴❞✐✐s✮ ●❡♥❡r❛❧ ❛❧❣♦r✐t❤♠ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞✉❧❡ ✐♥ t❤❡ ❚✉r❜♦♠♦❧❡ ♣r♦❣r❛♠✳ ❊❛❝❤ ❈❈❙❉✭❋✶✷✮ ❝❛❧❝✉❧❛t✐♦♥ ✐s ♣r❡❝❡❞❡❞ ✇✐t❤ t❤❡ ▼P✷✲❋✶✷ ❝❛❧❧ ✭st❡♣ ✶❀ ❆❧❣✳ ✶✮✳ ❚❤❡ ✇❤♦❧❡ ▼P✷✲❋✶✷ ❡♥❡r❣② ❝❛❧❝✉❧❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ❛♥❞ t❤❡ ✐♠♣♦rt❛♥t ✐♥t❡r♠❡❞✐❛t❡s ❛r❡ s❛✈❡❞ ♦♥ ❞✐s❦ ❢♦r ❧❛t❡r ✉s❡ ✇✐t❤✐♥ t❤❡ ❈❈❙❉✭❋✶✷✮ ❝❛❧❝✉❧❛t✐♦♥✳ q✉✐t❡ ✐♥✈♦❧✈❡❞ ❜② ✐ts❡❧❢ ❜✉t ✇✐❧❧ ♥♦t ❜❡ ❞✐s❝✉ss❡❞ ❤❡r❡✳ ❚❤❡ ▼P✷✲❋✶✷ st❡♣ ✐s ❚❤❡ ❞❡t❛✐❧s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❘❡❢s✳ ❬✹✾✱ ✺✵❪✳ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ t❤❡ ♣r❡s❡♥t ✇♦r❦✱ t❤❡ ✐♠♣♦rt❛♥t ✐♥❢♦r♠❛t✐♦♥ ✐s t❤❛t t❤❡ ▼P✷✲❋✶✷ ✐♥t❡r♠❡❞✐❛t❡s ❛r❡ ❧❛t❡r ♦♥ ✉s❡❞ ✇✐t❤✐♥ t❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞✉❧❡✳ ❚❤❡② µ,ν ❛r❡ ❝♦❧❧❡❝t❡❞ ✐♥ ❚❛❜❧❡ ✶✳ ❋♦r ♣r❛❝t✐❝❛❧ r❡❛s♦♥s t❤❡ ✐♥t❡❣r❛❧s (f g)x,y ✱ ❛❧t❤♦✉❣❤ ♥♦t ♥❡❡❞❡❞ ❢♦r t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ▼P✷✲❋✶✷ ❡♥❡r❣②✱ ❛r❡ ❝♦♠♣✉t❡❞ ✇✐t❤✐♥ t❤❡ ▼P✷✲❋✶✷ ♠♦❞✉❧❡ ✭st❡♣ ✷❀ ❆❧❣✳ ✶✮✳ ❚❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ✉s❡❞ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ V ✐♥t❡r♠❡❞✐❛t❡ ✐❢ t❤❡ ▲❈● ❝♦rr❡❧❛t✐♦♥ ❢❛❝t♦r ✐s r❡q✉❡st❡❞✳ ❆❢t❡r t❤❡ ▼P✷✲❋✶✷ r✉♥ t❤❡ ✜rst ❈❈❙❉✭❋✶✷✮ ❚❛❜❧❡ ✶✿ ❚❤❡ ▼P✷✲❋✶✷ q✉❛♥t✐t✐❡s ✉s❡❞ ✐♥ t❤❡ ❈❈❙❉✭❋✶✷✮ ❝❛❧❝✉❧❛t✐♦♥✳ ❉❡s❝r✐♣t✐♦♥ ▼P✷✲❋✶✷ ♠❛tr✐❝❡s ✿ ❢♦✉r✲✐♥❞❡① ✐♥t❡❣r❛❧s ✿ t❤r❡❡✲✐♥❞❡① ✐♥t❡❣r❛❧s ✿ t✇♦✲✐♥❞❡① q✉❛♥t✐t✐❡s ✿ ❝❝❴♣r❡♣✮ r♦✉t✐♥❡ ✭ q✉❛♥t✐t② Bijkl , Xijkl , Qkl ij pq p q fij , fij BQ,kl , BQ,kp Cµ,p , Cµ ,p ♣✉r♣♦s❡ t❤❡ t❤❡ T2 r❡s✐❞✉❛❧✱ ♣r❡❝♦♥❞✐t✐♦♥✐♥❣ T2 ✱ T2 r❡s✐❞✉❛❧s ❛❧❧ r❡s✐❞✉❛❧s✱ ❋♦❝❦ ♠❛tr✐❝❡s ❜❛❝❦✲tr❛♥s❢♦r♠❛t✐♦♥ ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r ♣r❡❧✐♠✐♥❛r② ✐ss✉❡s ✭st❡♣ ✸❀ ❆❧❣✳ ✶✮✳ ❖♥❡ ♦❢ t❤❡♠ ✐s tr❛♥s❢♦r♠✐♥❣ t❤❡ s♣✐♥ ❛❞❛♣t❡❞ s✐♥❣❧❡t✴tr✐♣❧❡t ▼P✷✲❋✶✷ ♠❛tr✐❝❡s ✭❚❛❜❧❡ ✶✮ ✐♥t♦ ❢✉❧❧ (ij, ji) ✷✷ ♠❛tr✐❝❡s✳ ❆❧s♦ t❤❡ f12 ✐♥t❡❣r❛❧s ✭❚❛❜❧❡ ✶✮ ❛r❡ tr❛♥s❢♦r♠❡❞ t♦ t❤❡ ❢♦r♠ t❤❛t ✐s ✹✳✸✳ ❚❤❡ ❣❡♥❡r❛❧ s❦❡t❝❤ ♦❢ t❤❡ ♣r♦❣r❛♠ ✢♦✇ ❛♣♣r♦♣r✐❛t❡ ❢♦r ❧❛t❡r ✉s❡ ✐♥ t❤❡ ❈❈❙❉✭❋✶✷✮ ♣❛rt✳ ❚❤❡ ♥❡①t ♣❛rt ✐s t❤❡ r❡s♦rt✐♥❣ ♦❢ t❤❡ ❢♦✉r✲✐♥❞❡① q✉❛♥t✐t✐❡s t♦ t❤❡ ❢♦r♠ t❤❛t ✐s s✉✐t❛❜❧❡ ❢♦r t❤❡ ❈❈❙❉✭❋✶✷✮ ♣❛rt✳ ❚❤❡ ❞✐✛❡r❡♥t st♦r✐♥❣ s❝❤❡♠❡s ✇✐t❤✐♥ t❤❡ ▼P✷✲❋✶✷ ❛♥❞ ❈❈❙❉✭❋✶✷✮ ❝♦❞❡s r❡❢❡rs t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠❛♥②✲❜♦❞② ♠♦❞❡❧s ✐♥ t❤❡ r✐❝❝✷ ♣r♦❣r❛♠ ✐s ❞❡s✐❣♥❡❞ ❢♦r t❤❡ ❝❛s❡s ✇❤❡r❡ ❛ r❡❧❛t✐✈❡❧② ❧❛r❣❡ s②st❡♠ ✐s tr❡❛t❡❞ ✇✐t❤ ❛ r❛t❤❡r s♠❛❧❧✱ ♦r ♠❡❞✐✉♠✲s✐③❡ ❜❛s✐s s❡t✳ ❚❤✐s ✐s r❡✢❡❝t❡❞ ✐♥ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❝♦❞❡✱ ✐♥ ♣❛rt✐❝✉❧❛r ✐♥ t❤❡ st♦r✐♥❣ s❝❤❡♠❡s✳ ❆❧❧ ❢♦✉r✲✐♥❞❡① q✉❛♥t✐t✐❡s ✭✐♥t❡❣r❛❧s✱ ❛♠♣❧✐t✉❞❡s✮ ❛r❡ st♦r❡❞ ✐♥ s✉❝❤ ❛ ✇❛②✱ t❤❛t t❤❡ ♦❝❝✉♣✐❡❞ ✐♥❞✐❝❡s r✉♥ t❤❡ ❢❛st❡st✳ ❚❤✐s ✐s ♥♦t t❤❡ ❝❛s❡ ❢♦r t❤❡ ▼P✷✲❋✶✷ q✉❛♥t✐t✐❡s✱ t❤❡r❡❢♦r❡ t❤❡ r❡s♦rt✐♥❣ ♣r♦❝❡❞✉r❡s ❤❛✈❡ t♦ ❜❡ ❝❛❧❧❡❞ ❛t t❤❡ ♣r❡❧✐♠✐♥❛r② ❧❡✈❡❧ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ r✉♥✳ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡♥t✐♦♥ t❤❛t ❞❡s❝r✐❜❡s t❤❡ st♦r✐♥❣ s❝❤❡♠❡ ♦❢ t❤❡ q✉❛♥t✐t✐❡s A(t❤❡ ❢❛st❡st .

T❤❡ s❧♦✇❡st) ■♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ t❤❡ q✉❛♥t✐t② ❢❛st❡st✱ t❤❡ ♥❡①t ♦♥❡ ✐s q A ❡✳❣✳ : A(p, q, r, s). ✭✽✵✮ ✐s st♦r❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ✐♥❞❡① p ✐s t❤❡ ❛♥❞ s♦ ♦♥✳ ❯s✐♥❣ t❤✐s ❝♦♥✈❡♥t✐♦♥ t❤❡ r❡s♦rt✐♥❣ s❤♦✇♥ ✐♥ st❡♣ ✹ ♦❢ ❆❧❣♦r✐t❤♠ ✶ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s f12 (p, q, i, j) −→ f12 (i, j, p, q)✱ ✇❤❡r❡ t❤❡ q✉❛♥t✐t② f12 (p, q, i, j) ✭✽✶✮ ✐s ❝♦♠♣✉t❡❞ ❛♥❞ st♦r❡❞ ❛t t❤❡ ▼P✷✲❋✶✷ ❧❡✈❡❧✳ ❚❤❡ r❡s♦rt✐♥❣ ♦❢ t❤❡ ♦t❤❡r ✐♥t❡❣r❛❧s ✐s ❛❧✇❛②s s✐♠✐❧❛r✱ t❤❡ ♦❝❝✉♣✐❡❞ ✐♥❞✐❝❡s t❤❛t r✉♥ t❤❡ s❧♦✇❡st ❛t t❤❡ ▼P✷✲❋✶✷ ❧❡✈❡❧ ❛r❡ ❡①❝❤❛♥❣❡❞ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ✭♦r ❈❆❇❙✮ ♦♥❡s✳ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ s✉❝❤ ♣r♦❝❡❞✉r❡ ✐s ♥♦t ❡♥t✐r❡❧② tr✐✈✐❛❧✳ ❆ ♥❛✐✈❡ r♦✉t✐♥❡ ✇♦✉❧❞ ❛ss✉♠❡ t❤❛t t❤❡ ✇❤♦❧❡ ❢♦✉r✲✐♥❞❡① q✉❛♥t✐t② ✐s ❦❡♣t ✐♥ t❤❡ ♠❡♠♦r②✳ ❋♦r ❧❛r❣❡r s②st❡♠s ✭♦r ❧❛r❣❡r ❜❛s✐s s❡ts✮ t❤✐s ❛ss✉♠♣t✐♦♥ ✇♦✉❧❞ ❜❡❝♦♠❡ ✈❡r② q✉✐❝❦❧② ❛ ❜♦tt❧❡♥❡❝❦✱ t❤❡r❡❢♦r❡ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❜❛t❝❤❡s t❤❡ ✐♥t❡❣r❛❧s ✐♥t♦ t❤❡ ♣✐❡❝❡s t❤❛t ✜t ✐♥t♦ t❤❡ ❛✈❛✐❧❛❜❧❡ ♠❡♠♦r② ✇❛s ❞❡s✐❣♥❡❞✳ ❚❤❡ ♥❡①t st❡♣ ✐s t❤❡ ❜❛❝❦✲tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❢♦✉r✲✐♥❞❡① ✐♥t❡❣r❛❧s ✇❤✐❝❤ µ,ν ✐s ♥❡❡❞❡❞ ❢♦r t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ Vx,y ✐♥t❡r♠❡❞✐❛t❡✳ ❚❤✐s ♣❛rt ♦❢ t❤❡ ❝♦❞❡ ✭st❡♣ ✺❀ ❆❧❣✳ ✶✮ ✐s ❡①♣❧❛✐♥❡❞ ✐♥ ❙✉❜s❡❝t✐♦♥ ✹✳✹ ✇❤❡r❡ t❤❡ ✐ss✉❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤✐s ✐♥t❡r♠❡❞✐❛t❡ µ,ν ❛r❡ ❞✐s❝✉ss❡❞✳ ❲❤❡♥ t❤❡ Vx,y ✐s ❝♦♠♣✉t❡❞ ✭st❡♣ ✻❀ ❆❧❣✳ ✶✮ t❤❡ ♥❡①t t❛s❦ ✐s t♦ tr❛♥s❢♦r♠ i,j t❤✐s q✉❛♥t✐t② t♦ t❤❡ ♦❝❝✉♣✐❡❞ s♣❛❝❡ ✭st❡♣ ✼❀ ❆❧❣✳ ✶✮✳ ❚❤❡ Vx,y ✐s ♥❡❡❞❡❞ ❢♦r t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ❡♥❡r❣② ✭✇❤✐❝❤ ✇✐❧❧ ❜❡ ❡①♣❧❛✐♥❡❞ ✐♥ ❙✉❜s❡❝t✐♦♥ ✹✳✺✮ ❛♥❞ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s µ,ν (Cµi )† Vx,y Cνj ✳ i,j Vx,y = ✭✽✷✮ µν ■t ✇❛s ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞ t❤❛t ✇✐t❤✐♥ t❤❡ ❝✉rr❡♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥ t❤❡ ❣❡♠✐♥❛❧ ✐♥❞✐❝❡s ✭xy ✮ r❡❢❡r t♦ t❤❡ ♦❝❝✉♣✐❡❞ s♣❛❝❡✳ q✉❛♥t✐t✐❡s ❛r❡ ❝♦♠♣✉t❡❞✳ ❆t st❡♣ ✽ ♦❢ ❆❧❣♦r✐t❤♠ ✶ t❤❡ ❛❞❞✐t✐♦♥❛❧ t❤r❡❡✲✐♥❞❡① ❚❤❡ ♣r❡❧✐♠✐♥❛r② r✉♥ ♦❢ t❤❡ ▼P✷✲❋✶✷ ♣r♦❣r❛♠ ♣r♦✈✐❞❡s t❤❡ ✐♥t❡❣r❛❧s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✶✱ t❤❡ ❛❞❞✐t✐♦♥❛❧ ✐♥t❡❣r❛❧s✱ ♥❡❡❞❡❞ ❛t t❤❡ ❈❈❙❉✭❋✶✷✮ ❧❡✈❡❧✱ ❤❛✈❡ t❤❡ ♦r❜✐t❛❧ ✐♥❞✐❝❡s ✐♥ t❤❡ ❈❆❇❙ ❛♥❞ ✈✐rt✉❛❧ s♣❛❝❡s✱ BQ,ap ✳ ❚❤❡② ❛r❡ ♥❡❡❞❡❞ ❢♦r ✈✐❞❡ ✐♥❢r❛ ✮ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❛❧❡♥❝❡ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❋♦❝❦ ♠❛tr✐① ❡❧❡♠❡♥ts ✭ t❤❡ C ❛♥❞ D ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ T2 ❛♥❞ T2 ❛♥❞ ✈❡❝t♦r ❢✉♥❝t✐♦♥s ✭s❡❡ ❙✉❜s❡❝t✐♦♥ ✹✳✽ ❢♦r ❞❡t❛✐❧s✮✳ ❚❤❡ ❞❡t❛✐❧❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ t❤r❡❡✲✐♥❞❡① ✐♥t❡❣r❛❧s ✐♥ ❚✉r❜♦♠♦❧❡ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✺✶❪✳ ❚❤❡ ❧❛st st❡♣✱ ❜❡❢♦r❡ st❛rt✐♥❣ t❤❡ ❈❈ ✐t❡r❛t✐♦♥s✱ r❡❢❡rs t♦ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❛❧❡♥❝❡ ♣❛rts ♦❢ t❤❡ ❋♦❝❦ ♠❛tr✐❝❡s ✐♥✈♦❧✈✐♥❣ t❤❡ ❈❆❇❙ ✐♥❞❡① ✭st❡♣ ✾❀ ❆❧❣✳ ✶✮✳ ❚❤❡ ❛❞❥❡❝t✐✈❡ ✏✈❛❧❡♥❝❡✑ ❤❡r❡ ♠❡❛♥s t❤❛t ♦♥❧② t❤❡ ❛❝t✐✈❡ t✇♦✲ ❡❧❡❝tr♦♥ ♣❛rt ♦❢ t❤❡ ❋♦❝❦ ♦♣❡r❛t♦r ✐s ✐♥❝❧✉❞❡❞✳ ❚❤❡s❡ ❋♦❝❦ ♠❛tr✐❝❡s ❛r❡ ❝♦♠♣✉t❡❞ ✇✐t❤✐♥ ✷✸ ✹✳ ❚❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞❡❧ ✐♥ ❚✉r❜♦♠♦❧❡ t❤❡ ❉❋ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② Fipval = k,p p ,k }✱ − ❉❋ gi,k {2❉❋ gi,k ✭✽✸✮ p ,k k,p {2❉❋ ga,k − ❉❋ ga,k }✱ ✭✽✹✮ k val Fap = k ✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥ r✉♥s ♦✈❡r t❤❡ ❛❝t✐✈❡ ♦❝❝✉♣✐❡❞ ♦r❜✐t❛❧s✳ ❊❛❝❤ ❈♦✉❧♦♠❜ ✐♥t❡❣r❛❧ ❤❛s t♦ ❜❡ r❡❝♦✈❡r❡❞ ❢r♦♠ t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❛♥❞ st♦r❡❞ t❤r❡❡✲✐♥❞❡① q✉❛♥t✐t✐❡s ❜② ❝♦♥tr❛❝t✐♥❣ t❤❡♠ ♦✈❡r t❤❡ ❛✉①✐❧✐❛r② ✐♥❞❡① Q✳ Fipval = ❚❤❡ ✜♥❛❧ ❢♦r♠✉❧❛s ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s {2BQ,ip BQ,kk − BQ,ik BQ,p k }✱ ✭✽✺✮ {2BQ,ap BQ,kk − BQ,ak BQ,p k }✳ ✭✽✻✮ kQ val = Fap kQ ❯♥t✐❧ ♥♦✇ ❛❧❧ st❡♣s ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ♣r❡❧✐♠✐♥❛r② ♣❛rt ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ❝❛❧❝✉✲ ❧❛t✐♦♥✳ ❚❤❡ r❡♠❛✐♥✐♥❣✱ ♠❛✐♥ ♣❛rt ♦❢ t❤❡ ♣r♦❣r❛♠✱ ✐t❡r❛t✐✈❡❧② s♦❧✈❡s t❤❡ ❈❈ ❡q✉❛t✐♦♥s✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❈❈ ❡q✉❛t✐♦♥s r❡q✉✐r❡s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❈❈ r❡s✐❞✉❛❧s ✭st❡♣ ✶✵❀ ❆❧❣✳ ✶✮ t❤❛t ❛r❡ ❞❡✜♥❡❞ ❜② t❤❡ ♣r♦❥❡❝t✐♦♥s ♦♥t♦ t❤❡ ❡①❝✐t❛t✐♦♥ ♠❛♥✐❢♦❧❞s ❬❊qs✳ ✭✹✺✮✱ ✭✹✻✮✱ ✭✹✼✮❪✳ ❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ r❡s✐❞✉❛❧s ❛❧❧♦✇ ❢♦r ❞❡t❡r♠✐♥✐♥❣ t❤❡ ♥❡①t ❣✉❡ss ❢♦r t❤❡ ❛♠♣❧✐t✉❞❡s ❛♥❞ t❤❡s❡ ♥❡✇ ❛♠♣❧✐t✉❞❡s ✭st❡♣s ✶✷✱ ✶✸❀ ❆❧❣✳ ✶✮ ❧❡❛❞ t♦ ♥❡✇ r❡s✐❞✉❛❧s✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ✐s r❡♣❡❛t❡❞ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡ ✐s r❡❛❝❤❡❞✳ ❚❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❡♥❡r❣✐❡s ✐♥ t❤❡ ✐t❡r❛t✐♦♥s T1 ✈❡❝t♦r ❢✉♥❝t✐♦♥✳ (k) ❛♥❞ (k + 1) ✭st❡♣ ✶✶❀ ❆❧❣✳ ✶✮ ❛♥❞ t❤❡ ♥♦r♠ ♦❢ t❤❡ ❆s t❤❡ ✐♥✐t✐❛❧ ❣✉❡ss ❢♦r t❤❡ ❛♠♣❧✐t✉❞❡s✱ t❤❡ ♦♣t✐♠❛❧ ▼P✷ ❛♥❞ ▼P✷✲❋✶✷ ❛♠♣❧✐t✉❞❡s ❛r❡ ❛ss✉♠❡❞ ❢♦r t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ❛♥❞ ❡①♣❧✐❝✐t❧② ❝♦rr❡❧❛t❡❞ ♣❛rts✱ r❡s♣❡❝t✐✈❡❧②✳ ❚♦ ✐♠♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡✱ t❤❡ ❞✐r❡❝t ✐♥✈❡rs✐♦♥ ✐♥ t❤❡ ✐t❡r❛t✐✈❡ s✉❜s♣❛❝❡ ✭❉■■❙✮ ♠❡t❤♦❞ ❬✺✷❪ ✐s ❛❧✇❛②s ✉s❡❞ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♥❡①t ❣✉❡ss ❢♦r t❤❡ ❛♠♣❧✐t✉❞❡s ✭st❡♣ ✶✸❀ ❆❧❣✳ ✶✮✳ µ,ν ✐♥t❡r♠❡❞✐❛t❡ ✹✳✹ ❚❤❡ Vx,y ❇❡❢♦r❡ ❞✐s❝✉ss✐♥❣ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ T1 ✱ T2 ❛♥❞ T2 ✈❡❝t♦r ❢✉♥❝t✐♦♥s✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ µ,ν ✐♥tr♦❞✉❝❡ t❤❡ Vx,y ✐♥t❡r♠❡❞✐❛t❡✳ ■♥ t❤✐s ♣❛rt ♦❢ t❤❡ t❤❡s✐s✱ ❞❡t❛✐❧❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤✐s q✉❛♥t✐t② ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ❈❈ r❡s✐❞✉❛❧s µ,ν ❛r❡ r❡❝♦✈❡r❡❞ ❢r♦♠ Vx,y ❛♥❞ t❤✉s ✐t ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ✏❦❡② ✐♥t❡r♠❡❞✐❛t❡✑ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ❚✉r❜♦♠♦❧❡✳ ■♥ ❣❡♥❡r❛❧✱ s✉❝❤ q✉❛♥t✐t②✱ ✐♥ t❤❡ ▼❖ ❜❛s✐s✱ ❝❛♥ ❜❡ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② −1 r,s ˆ 12 r12 = pq|f12 Q |rs , Vp,q ✇❤❡r❡ f12 ✐s t❤❡ ❝♦rr❡❧❛t✐♦♥ ❢❛❝t♦r✱ ˆ 12 Q ✭✽✼✮ ✐s t❤❡ str♦♥❣ ♦rt❤♦❣♦♥❛❧✐t② ♣r♦❥❡❝t♦r t❤❛t ❞❡✲ ✜♥❡s ❆♥s❛t③ ♦❢ t❤❡ ❡①♣❧✐❝✐t❧② ❝♦rr❡❧❛t❡❞ ✇❛✈❡ ❢✉♥❝t✐♦♥ ✭s❡❡ ❙✉❜s❡❝t✐♦♥ ✹✳✷ ❢♦r t❤❡ ❞❡t❛✐❧s −1 ˆ 12 ✮✱ ❛♥❞ r12 ✐s t❤❡ ❡❧❡❝tr♦♥✲❡❧❡❝tr♦♥ ❈♦✉❧♦♠❜ ♦♣❡r❛t♦r✳ ❚❤❡ ✐♠♣❧❡✲ ❝♦♥❝❡r♥✐♥❣ f12 ❛♥❞ Q ♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉ ✭❛♥❞ t❤✉s ❈❈❙❉✭❋✶✷✮✮ ♠❡t❤♦❞ ✐♥ ❚✉r❜♦♠♦❧❡ ✐s✱ t♦ ❛ ❧❛r❣❡ ❡①t❡♥t✱ ❞❡s✐❣♥❡❞ ✐♥ ❛♥ ✐♥t❡❣r❛❧✲❞✐r❡❝t ❢❛s❤✐♦♥✳ ■t ♠❡❛♥s t❤❛t t❤❡ ♠♦st t✐♠❡✲❝♦♥s✉♠✐♥❣ q✉❛♥t✐t✐❡s ❛r❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❆❖✲❜❛s✐s✱ ❝♦♠❜✐♥❡❞ t♦❣❡t❤❡r ❛♥❞ ❧❛t❡r ♦♥ tr❛♥s❢♦r♠❡❞ t♦ t❤❡ ▼❖✲❜❛s✐s✳ ❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❞❞✐t✐♦♥❛❧ ❋✶✷ t❡r♠s ✭❛t ❧❡❛st s♦♠❡ ♦❢ t❤❡♠✮ µ,ν ❢♦❧❧♦✇s t❤✐s ✐❞❡❛ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ Vx,y ✇✐t❤ t✇♦ ❆❖ ✐♥❞✐❝❡s ✇❛s ✐♥tr♦❞✉❝❡❞✳ ✷✹ ✹✳✹✳ ❚❤❡ µ,ν Vx,y ✐♥t❡r♠❡❞✐❛t❡ ❆♠♦♥❣ t❤❡ ❋✶✷ t❡r♠s t❤✐s ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t q✉❛♥t✐t✐❡s ❛♥❞ ✇✐❧❧ ❜❡ ❞✐s❝✉ss ✐♥ ❞❡t❛✐❧ ❤❡r❡✳ ✐✳❡✳ ■t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❜❡❣✐♥ ✇✐t❤ t❤❡ s✐♠♣❧❡st ❝❛s❡✱ t❤❡ V ✐♥t❡r♠❡❞✐❛t❡ ❞❡r✐✈❡❞ ✇✐t❤✐♥ ❆♥✶ ❛♥❞ t❤❡ ❘❍❋ r❡❢❡r❡♥❝❡✱ ✇✐t❤♦✉t t❤❡ ❈❆❇❙ ❝♦♥tr✐❜✉t✐♦♥ ✭❆✶✲♥♦❈❆❇❙✮✳ ■❢ ✇❡ t❛❦❡ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ♦❢ t❤❡ V ✐♥t❡r♠❡❞✐❛t❡ ❬❊q✳ ✭✽✼✮❪ ❛♥❞ s✉❜st✐t✉t❡ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ str♦♥❣ ♦rt❤♦❣♦♥❛❧✐t② ♣r♦❥❡❝t♦r t❤❛t ❞❡✜♥❡s ❆♥s❛t③ ✶ ❬❊q✳ ✭✻✶✮❪ ✇❡ ✇✐❧❧ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ r,s −1 Vp,q = pq|f12 ((1 − Pˆ1 )(1 − Pˆ2 ))r12 |rs ✳ ✭✽✽✮ ■♥ t❤❡ ❆❖✲❜❛s❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❈❈❙❉✭❋✶✷✮ ♠♦❞❡❧ ✇❡ ♥❡❡❞ t❤❡ µ,ν Vx,y ✱ ✇❤❡r❡ t❤❡ xy ❛r❡ t❤❡ ❣❡♠✐♥❛❧ ✐♥❞✐❝❡s ✭❡❢❡❝t✐✈❡❧② t❤❡② ❜❡❧♦♥❣ t♦ t❤❡ ♦❝❝✉♣✐❡❞ s♣❛❝❡ ✇✐t❤✐♥ t❤✐s ✐♠♣❧❡♠❡♥✲ t❛t✐♦♥✱ s❡❡ ❙✉❜s❡❝t✐♦♥ ✸✳✷✮ ❛♥❞ µν ❜❡❧♦♥❣ t♦ t❤❡ ❝♦✈❛r✐❛♥t ❆❖✲❜❛s✐s✳ ■♥ t❤❡ s✉❜s❡q✉❡♥t s❡❝t✐♦♥s ✇❡ ✇✐❧❧ ✉s❡ t❤❡ t❡r♠ ❝♦✈❛r✐❛♥t ❢♦r t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ q✉❛♥t✐t✐❡s ✐♥ t❤❡ ✏♦r❞✐✲ ♥❛r②✑ ❆❖ ❜❛s✐s ✭❞❡♥♦t❡❞ ❜② ✉♣♣❡r✲❝❛s❡ ✏❆❖✑ ❧❡❢t s✉♣❡rs❝r✐♣t✮ ✇❤❡r❡❛s t❤❡ ❝♦♥tr❛✈❛r✐❛♥t ✈✐❞❡ ✐♥❢r❛ ✮ ✭❞❡♥♦t❡❞ ✇✐t❤ ❧♦✇❡r✲❝❛s❡ ✏❛♦✑ ❜❛s✐s ✐s ✉s❡❞ ❢♦r t❤❡ ❜❛❝❦✲tr❛♥s❢♦r♠❡❞ q✉❛♥t✐t✐❡s ✭ ❧❡❢t s✉♣❡rs❝r✐♣t✮✳ ❚❤❡ ✜♥❛❧ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❆✶✲♥♦❈❆❇❙ V ✐♥t❡r♠❡❞✐❛t❡ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② ❆✶✲♥♦❈❆❇❙ ❛♦ µ,ν Vx,y = (f g)µ,ν x,y − γ,δ ❆❖ µ,ν fx,y gγ,δ ✱ ✭✽✾✮ γδ (f g)µ,ν x,y ✇❤❡r❡ ✐s t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r t❤❡ −1 (f12 r12 ) ♦♣❡r❛t♦r −1 (f g)µ,ν x,y = xy|f12 r12 |µν ❚❤❡ ❛♦ ✳ ✭✾✵✮ γ,δ ❆❖ µ,ν fx,y ❛r❡ t❤❡ ❛❜♦✈❡ ♠❡♥t✐♦♥❡❞ ❜❛❝❦✲tr❛♥s❢♦r♠❡❞ f12 ✐♥t❡❣r❛❧s ❛♥❞ gγ,δ ❛r❡ ❈♦✉❧♦♠❜ ✐♥t❡❣r❛❧s ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❝♦✈❛r✐❛♥t ❆❖✲❜❛s✐s✳ ❚❤❡ ❜❛❝❦✲tr❛♥s❢♦r♠❡❞ q✉❛♥t✐t✐❡s ❛r❡ ♦❜✲ t❛✐♥❡❞ ❢r♦♠ t❤❡ q✉❛♥t✐t✐❡s ✐♥ t❤❡ ▼❖✲❜❛s✐s ❜② ❝♦♥tr❛❝t✐♥❣ t❤❡♠ ✇✐t❤ t❤❡ ❆❖✲▼❖ ❝♦❡✣✲ ❝✐❡♥ts ♦✈❡r t❤❡ ▼❖ ✐♥❞✐❝❡s✳ ❚❤❡ ✐♥t❡❣r❛❧s ✐♥ t❤❡ ▼❖✲❜❛s✐s ❛r❡ ❛✈❛✐❧❛❜❧❡ ❢r♦♠ t❤❡ ♣r❡❝❡❞✐♥❣ ▼P✷✲❋✶✷ ❝❛❧❝✉❧❛t✐♦♥ ❛♥❞ t❤❡ ❆❖✲▼❖ ❝♦❡✣❝✐❡♥ts ❛r❡ ❛❧✇❛②s ❛✈❛✐❧❛❜❧❡ ❛s t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛♦ γ,δ t❤❡ ❍❛rtr❡❡✲❋♦❝❦ ❡q✉❛t✐♦♥s✳ ❚❤❡ fx,y ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s ❛♦ p,q Cγp fx,y (Cδq )† .

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