❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s 1 − x 1 � � � d x 2 F j 1 j 2 ,v ( x 1 , x 2 ) = f j 1 ( x 1 ) ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ N j 2 ,v + δ j 1 ,j 2 − δ j 1 ,j 2 0 1 − x 1 � � d x 2 x 2 F j 1 j 2 ( x 1 , x 2 ) = ( M − x 1 ) f j 1 ( x 1 ) ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ j 2 0 ❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵ ◮ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦✿ ♣❡r❢♦r♠✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧ ✉s✐♥❣ ❡✐t❤❡r t❤❡ ❉P❉ ♥✉♠❜❡r✭♠♦♠❡♥t✉♠✮ s✉♠ r✉❧❡ ❛♥❞ t❤❡ P❉❋ ♠♦♠❡♥t✉♠✭♥✉♠❜❡r✮ s✉♠ r✉❧❡ s❤♦✉❧❞ ②✐❡❧❞ t❤❡ s❛♠❡ r❡s✉❧t 1 − x 1 1 � � � d x 2 x 2 F j 1 ,v j 2 ( x 1 , x 2 ) = N j 1 ,v − x j 1 ,v � d x 1 j 2 0 0 ◮ ♣✉t ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❉P❉s ❛♥❞ ❝❛♥ t❤❡r❡❢♦r❡ ❜❡ ✉s❡❞ t♦ r❡✜♥❡ ❉P❉✲♠♦❞❡❧s ◮ ♣r♦✈❡ t❤❛t t❤❡s❡ s✉♠ r✉❧❡s ❛r❡ ❢✉❧✜❧❧❡❞ ✐♥ ◗❈❉ ✸ ✴ ✶✹
❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ✐♥ t❡r♠s ♦❢ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s✱ ❡✳❣✳ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❉❡✜♥✐t✐♦♥s � d z − 1 p + � d 2 z − (2 π ) 2 e i z 1 k 1 � p | q j 1 ( − z 1 2 )Γ a q j 1 ( z 1 2 π e ix 1 z − f j 1 ( x 1 , k 1 ) = 1 1 2 ) | p � � 2 � d z − i p + � d 2 z − � d y − � � � d 2 y 1 � 2 π e ix i z − F j 1 j 2 ( x 1 , x 2 , k 1 , k 2 , ∆ ) = (2 π ) 2 e i z i k i 2 p + 1 (2 π ) 2 e i y 1 ∆ i i 2 π i =1 × � p | q j 2 ( − z 2 2 )Γ a q j 2 ( z 2 2 ) q j 1 ( y 1 − z 1 2 )Γ a q j 1 ( y 1 + z 2 2 ) | p � ❉✐❡❤❧✱ ❖st❡r♠❡✐❡r✱ ❙❝❤ä❢❡r ✷✵✶✶ ✹ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❉❡✜♥✐t✐♦♥s � d z − 1 p + � d 2 z − (2 π ) 2 e i z 1 k 1 � p | q j 1 ( − z 1 2 )Γ a q j 1 ( z 1 2 π e ix 1 z − f j 1 ( x 1 , k 1 ) = 1 1 2 ) | p � � 2 � d z − i p + � d 2 z − � d y − � � � d 2 y 1 � 2 π e ix i z − F j 1 j 2 ( x 1 , x 2 , k 1 , k 2 , ∆ ) = (2 π ) 2 e i z i k i 2 p + 1 (2 π ) 2 e i y 1 ∆ i i 2 π i =1 × � p | q j 2 ( − z 2 2 )Γ a q j 2 ( z 2 2 ) q j 1 ( y 1 − z 1 2 )Γ a q j 1 ( y 1 + z 2 2 ) | p � ❉✐❡❤❧✱ ❖st❡r♠❡✐❡r✱ ❙❝❤ä❢❡r ✷✵✶✶ ◮ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ✐♥ t❡r♠s ♦❢ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s✱ ❡✳❣✳ � d z − f j i ( x 1 , k 1 ) = 1 (2 π ) 4 ✹ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ✺ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❈♦♥tr✐❜✉t✐♥❣ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s ✺ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❈♦♥tr✐❜✉t✐♥❣ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s u | d d | u f g u | d d | u ✺ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❈♦♥tr✐❜✉t✐♥❣ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s u | d ¯ u d d | u ¯ u d f g F gu ¯ u | d u d u ¯ d | u d ✺ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❈♦♥tr✐❜✉t✐♥❣ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s u | d ¯ ¯ u u d d d | u ¯ ¯ u u d d F g ¯ f g F gu d ¯ ¯ u | d u d d u u ¯ ¯ d d | u d u ✺ ✴ ✶✹
❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳ ✺ ✴ ✶✹
❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳ ✺ ✴ ✶✹
❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳ ✺ ✴ ✶✹
❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳ . . . ✺ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② O ( α s ) ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u ✲q✉❛r❦ ❛♥❞ ¯ d ✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j 1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O ( α s ) ✿ f g , F gu , F g ¯ d ❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳ . . . ❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉ ✺ ✴ ✶✹
❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ ✲✈❛❧✉❡ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✻ ✴ ✶✹
♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ ✲✈❛❧✉❡ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ✻ ✴ ✶✹
t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x + ✲✈❛❧✉❡ + ✻ ✴ ✶✹
❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x + ✲✈❛❧✉❡ ◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✻ ✴ ✶✹
▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x + ✲✈❛❧✉❡ ◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ✻ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s P❉❋ ❛♥❞ ❉P❉ ❞❡✜♥✐t✐♦♥s � N ( t ) � d D − 2 k 1 d x i d D − 2 k i p + � � � � � � x 1 p + � n 1 p + f j 1 B ( x 1 ) = (2 π ) D − 1 (2 π ) D − 1 t c o i =2 M ( c ) � � � × Φ j 1 P DF t,c,o ( { x } , { k } ) δ 1 − x i i =1 1 − x 1 � d D − 2 k 1 � N ( t ) � d x i d D − 2 k i p + � � � � � � � x 1 p + � n 1 2 p + d x 2 F j 1 j 2 ( x 1 , x 2 ) = δ f ( l ) ,j 2 B (2 π ) D − 1 (2 π ) D − 1 t c o l i =2 0 M ( c ) � � � � x l p + � n l Φ j 1 j 2 × DP D t,c,o ( { x } , { k } ) δ 1 − x i i =1 ✻ ✴ ✶✹
✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ ❣✐✈❡♥ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x + ✲✈❛❧✉❡ ◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✻ ✴ ✶✹
✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ ❣✐✈❡♥ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x + ✲✈❛❧✉❡ ◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ � x l p + � n l Φ j 1 , j 2 = Φ j 1 ? 2 DP D t,c,o P DF t,c,o ✻ ✴ ✶✹
❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ ❣✐✈❡♥ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x + ✲✈❛❧✉❡ ◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ � x l p + � n l Φ j 1 , j 2 = Φ j 1 ? 2 DP D t,c,o P DF t,c,o ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ j 1 j 2 ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ j 1 ✲P❉❋✮ ✻ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ ❣✐✈❡♥ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x + ✲✈❛❧✉❡ ◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ � x l p + � n l Φ j 1 , j 2 DP D t,c,o = Φ j 1 2 P DF t,c,o ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ j 1 j 2 ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ j 1 ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s Φ j 1 , j 2 DP D t,c,o ❛♥❞ Φ j 1 P DF t,c,o s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ✻ ✴ ✶✹
✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ ✐♥ t❡r♠s ♦❢ ✿ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ✼ ✴ ✶✹
✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ ✐♥ t❡r♠s ♦❢ ✿ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s � � � � � δ f ( l ) , j 2 − δ f ( l ) , j 2 = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 l ✼ ✴ ✶✹
✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ ✐♥ t❡r♠s ♦❢ ✿ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s � � � � � δ f ( l ) , j 2 − δ f ( l ) , j 2 = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 l � � � � � � � � t,c,o − N = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 N j 2 j 2 t,c,o ✼ ✴ ✶✹
❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ ✐♥ t❡r♠s ♦❢ ✿ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s � � � � � δ f ( l ) , j 2 − δ f ( l ) , j 2 = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 l � � � � � � � � t,c,o − N = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 N j 2 j 2 t,c,o � � t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j 2 ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φ j 1 ✇❤❡r❡ N j 2 P DF t,c,o ✼ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s � � � � � δ f ( l ) , j 2 − δ f ( l ) , j 2 = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 l � � � � � � � � t,c,o − N = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 N j 2 j 2 t,c,o � � t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j 2 ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φ j 1 ✇❤❡r❡ N j 2 P DF t,c,o ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N j 2 ,v j 2 ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ � � � � ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j 2 j 2 ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N j 2 t,c,o − N j 2 t,c,o ✐♥ t❡r♠s ♦❢ j 1 ✿ ✼ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s � � � � � δ f ( l ) , j 2 − δ f ( l ) , j 2 = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 l � � � � � � � � t,c,o − N = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 N j 2 j 2 t,c,o � � t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j 2 ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φ j 1 ✇❤❡r❡ N j 2 P DF t,c,o ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N j 2 ,v j 2 ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ � � � � ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j 2 j 2 ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N j 2 t,c,o − N j 2 t,c,o ✐♥ t❡r♠s ♦❢ j 1 ✿ � � j 1 � = j 2 , j 2 N j 2 ,v + x − x = N j 2 ,v ✼ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s � � � � � δ f ( l ) , j 2 − δ f ( l ) , j 2 = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 l � � � � � � � � t,c,o − N = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 N j 2 j 2 t,c,o � � t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j 2 ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φ j 1 ✇❤❡r❡ N j 2 P DF t,c,o ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N j 2 ,v j 2 ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ � � � � ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j 2 j 2 ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N j 2 t,c,o − N j 2 t,c,o ✐♥ t❡r♠s ♦❢ j 1 ✿ � � j 1 � = j 2 , j 2 N j 2 ,v + x − x = N j 2 ,v � � j 1 = j 2 N j 2 ,v + x − ( x − 1) = N j 2 ,v + 1 ✼ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s � � � � � δ f ( l ) , j 2 − δ f ( l ) , j 2 = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 l � � � � � � � � t,c,o − N = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 N j 2 j 2 t,c,o � � t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j 2 ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φ j 1 ✇❤❡r❡ N j 2 P DF t,c,o ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N j 2 ,v j 2 ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ � � � � ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j 2 j 2 ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N j 2 t,c,o − N j 2 t,c,o ✐♥ t❡r♠s ♦❢ j 1 ✿ � � j 1 � = j 2 , j 2 N j 2 ,v + x − x = N j 2 ,v � � j 1 = j 2 N j 2 ,v + x − ( x − 1) = N j 2 ,v + 1 � � j 1 = j 2 N j 2 ,v + x − 1 − x = N j 2 ,v − 1 ✼ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s � � � � � δ f ( l ) , j 2 − δ f ( l ) , j 2 = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 l � � � � � � � � t,c,o − N = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 N j 2 j 2 t,c,o � � t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j 2 ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φ j 1 ✇❤❡r❡ N j 2 P DF t,c,o ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N j 2 ,v j 2 ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ � � � � ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j 2 j 2 ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N j 2 t,c,o − N j 2 t,c,o ✐♥ t❡r♠s ♦❢ j 1 ✿ � � j 1 � = j 2 , j 2 N j 2 ,v + x − x = N j 2 ,v � � j 1 = j 2 N j 2 ,v + x − ( x − 1) = N j 2 ,v + 1 � � j 1 = j 2 N j 2 ,v + x − 1 − x = N j 2 ,v − 1 � �� � = N j 2 ,v + δ j 1 , j 2 − δ j 1 , j 2 ✼ ✴ ✶✹
✇✳❧✳♦✳❣✳ ♣❡r❢♦r♠✐♥❣ t❤❡ ✲✐♥t❡❣r❛t✐♦♥ ♦♥ ❜♦t❤ s✐❞❡s✱ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧✇✐♥❣ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❤❛s t♦ s❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ✐s ❢✉❧✜❧❧❡❞✿ ✽ ✴ ✶✹
✇✳❧✳♦✳❣✳ ♣❡r❢♦r♠✐♥❣ t❤❡ ✲✐♥t❡❣r❛t✐♦♥ ♦♥ ❜♦t❤ s✐❞❡s✱ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧✇✐♥❣ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❤❛s t♦ s❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ✐s ❢✉❧✜❧❧❡❞✿ M ( c ) � � � D N ( t ) [ x i ] D N ( t ) [ k i ] x l Φ j 1 P DF t,c,o ( { x } , { k } ) δ 1 − x i 2 1 i =1 l M ( c ) � � D N ( t ) [ x i ] D N ( t ) [ k i ] Φ j 1 = (1 − x 1 ) P DF t,c,o ( { x } , { k } ) δ 1 − x i 2 1 i =1 ✇❤❡r❡ � 1 b b � � � d D − 2 k i � � D b d x i p + D b a [ x i ] = a [ k i ] = (2 π ) D − 1 , 0 i = a i = a ✽ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❤❛s t♦ s❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ✐s ❢✉❧✜❧❧❡❞✿ M ( c ) � � � D N ( t ) [ x i ] D N ( t ) [ k i ] x l Φ j 1 P DF t,c,o ( { x } , { k } ) δ 1 − x i 2 1 l i =1 M ( c ) � � D N ( t ) [ x i ] D N ( t ) [ k i ] Φ j 1 = (1 − x 1 ) P DF t,c,o ( { x } , { k } ) δ 1 − x i 2 1 i =1 ✇✳❧✳♦✳❣✳ ♣❡r❢♦r♠✐♥❣ t❤❡ x 2 ✲✐♥t❡❣r❛t✐♦♥ ♦♥ ❜♦t❤ s✐❞❡s✱ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧✇✐♥❣ ✽ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❤❛s t♦ s❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ✐s ❢✉❧✜❧❧❡❞✿ M ( c ) � � � D N ( t ) [ x i ] D N ( t ) [ k i ] x l Φ j 1 P DF t,c,o ( { x } , { k } ) δ 1 − x i 2 1 l i =1 M ( c ) � � D N ( t ) [ x i ] D N ( t ) [ k i ] Φ j 1 = (1 − x 1 ) P DF t,c,o ( { x } , { k } ) δ 1 − x i 2 1 i =1 ✇✳❧✳♦✳❣✳ ♣❡r❢♦r♠✐♥❣ t❤❡ x 2 ✲✐♥t❡❣r❛t✐♦♥ ♦♥ ❜♦t❤ s✐❞❡s✱ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧✇✐♥❣ M ( c ) M ( c ) � � � � D N ( t ) [ x i ] D N ( t ) Φ j 1 � [ k i ] 1 − x 1 − x i + x j P DF t,c,o ( { x } , { k } ) 3 1 � x 2 = x 2 , 0 i =3 j =3 � �� � (1 − x 1 ) � � D N ( t ) [ x i ] D N ( t ) [ k i ] Φ j 1 � = (1 − x 1 ) P DF t,c,o ( { x } , { k } ) � 3 1 x 2 = x 2 , 0 ✽ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋ 1 � x 1 � � d z 1 � f j 1 ( x 1 ) = f i 1 Z i 1 → j 1 B ( z 1 ) z 1 z 1 i 1 x 1 ✇✐t❤ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs Z i 1 → j 1 ✱ ✇❤✐❝❤ ✐♥ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣❛♥s✐♦♥ ✐♥ α s � Z i 1 → j 1 ;22 � Z i 1 → j 1 ;11 + Z i 1 → j 1 ;21 + α 2 Z i 1 → j 1 ( x 1 ) = δ (1 − x 1 ) δ i 1 ,j 1 + α s + . . . , s ε 2 ε ε ✾ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋ f j 1 ( x 1 ) = Z i 1 → j 1 ⊗ f i 1 B ✾ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋ f j 1 ( x 1 ) = Z i 1 → j 1 ⊗ f i 1 B ❉P❉ 1 − x 2 1 − z 1 � x 1 � x 2 � � � � d z 1 d z 2 � F j 1 j 2 ( x 1 , x 2 ) = F i 1 i 2 Z i 1 → j 1 Z i 2 → j 2 ( z 1 , z 2 ) B z 1 z 2 z 1 z 2 i 1 ,i 2 x 1 x 2 1 � x 1 � � d z 1 , x 2 � f i 1 + Z i 1 → j 1 j 2 B ( z 1 ) z 2 z 1 z 2 1 i 1 x 1 + x 2 ✇✐t❤ t❤❡ ♥❡✇ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs Z i 1 → j 1 j 2 ✱ ✇❤✐❝❤ ❛r❡ ✐♥ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥ ❣✐✈❡♥ ❜② � Z i 1 → j 1 j 2 ;22 � Z i 1 → j 1 j 2 ;11 + Z i 1 → j 1 j 2 ;21 + α 2 Z i 1 → j 1 j 2 = α s + . . . , s ε 2 ε 2 ε ✾ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋ f j 1 ( x 1 ) = Z i 1 → j 1 ⊗ f i 1 B ❉P❉ F j 1 j 2 ( x 1 , x 2 ) = Z i 1 → j 1 ⊗ Z i 2 → j 2 ⊗ F i 1 i 2 + Z i 1 → j 1 j 2 ⊗ f i 1 B B ✾ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋ f j 1 ( x 1 ) = Z i 1 → j 1 ⊗ f i 1 B ❉P❉ F j 1 j 2 ( x 1 , x 2 ) = Z i 1 → j 1 ⊗ Z i 2 → j 2 ⊗ F i 1 i 2 + Z i 1 → j 1 j 2 ⊗ f i 1 B B ❋✐♥❛❧❧② ✇❡ ❞❡✜♥❡ ❛ ✐♥✈❡rs❡ P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦r Z − 1 1 → i 1 ✱ ♦❜❡②✐♥❣ i ′ ✐♥✈❡rs❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦r 1 � x 1 � � d u 1 � Z − 1 Z i 1 ,j 1 ( x 1 ) = δ i ′ 1 ,j 1 δ (1 − x 1 ) . i ′ 1 ,i 1 u 1 u 1 i 1 x 1 ✾ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋ f j 1 ( x 1 ) = Z i 1 → j 1 ⊗ f i 1 B ❉P❉ F j 1 j 2 ( x 1 , x 2 ) = Z i 1 → j 1 ⊗ Z i 2 → j 2 ⊗ F i 1 i 2 + Z i 1 → j 1 j 2 ⊗ f i 1 B B ✐♥✈❡rs❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦r Z − 1 1 ,i 1 ⊗ Z i 1 ,j 1 = δ i ′ 1 ,j 1 δ (1 − x 1 ) i ′ 1 ,i 1 ⊗ f i ′ f i 1 B = Z − 1 1 i ′ ✾ ✴ ✶✹
✇❤❡r❡ ✐s ❣✐✈❡♥ ❜② ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞ 1 − x 1 1 � � � � d u 1 � 1 ( u 1 ) R ′ ( x 1 , u 1 ) f j 1 ( x 1 ) = f i ′ d x 2 F j 1 j 2 ,v ( x 1 , x 2 ) − N j 2 v + δ j 1 ,j 2 − δ j 1 ,j 2 u 1 i ′ 0 x 1 1 ✶✵ ✴ ✶✹
❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞ 1 − x 1 1 � � � � d u 1 � 1 ( u 1 ) R ′ ( x 1 , u 1 ) f j 1 ( x 1 ) = f i ′ d x 2 F j 1 j 2 ,v ( x 1 , x 2 ) − N j 2 v + δ j 1 ,j 2 − δ j 1 ,j 2 u 1 i ′ 0 x 1 1 ✇❤❡r❡ R ′ ( x 1 , u 1 ) ✐s ❣✐✈❡♥ ❜② R ′ ( x 1 , u 1 ) = u 1 � x 1 � � � x 1 � � � �� � d z 1 1 − x 1 � � Z − 1 Z i 1 → j 1 − δ δ i 1 ,j 1 δ i 1 ,j 2 − δ i 1 ,j 2 − δ j 1 ,j 2 + δ j 1 ,j 2 i ′ 1 → i 1 z 1 u 1 z 1 z 1 i 1 x 1 1 − x 1 z 1 � x 1 � x 1 � � � �� + d u 2 Z i 1 → j 1 j 2 , u 2 − Z i 1 → j 1 j 2 , u 2 . z 1 z 1 0 ✶✵ ✴ ✶✹
t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ✐♥ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ ✐♥ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t ❛r❡ ✜♥✐t❡ ❢♦r ✐✳❡✳ ✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞ 1 − x 1 1 � � � � d u 1 � 1 ( u 1 ) R ′ ( x 1 , u 1 ) f j 1 ( x 1 ) = f i ′ d x 2 F j 1 j 2 ,v ( x 1 , x 2 ) − N j 2 v + δ j 1 ,j 2 − δ j 1 ,j 2 u 1 i ′ 0 x 1 1 ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✶✵ ✴ ✶✹
❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ ✐♥ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t ❛r❡ ✜♥✐t❡ ❢♦r ✐✳❡✳ ✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞ 1 − x 1 1 � � � � d u 1 � 1 ( u 1 ) R ′ ( x 1 , u 1 ) f j 1 ( x 1 ) = f i ′ d x 2 F j 1 j 2 ,v ( x 1 , x 2 ) − N j 2 v + δ j 1 ,j 2 − δ j 1 ,j 2 u 1 i ′ 0 x 1 1 ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ε ✐♥ R ′ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ✶✵ ✴ ✶✹
✐✳❡✳ ✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞ 1 − x 1 1 � � � � d u 1 � 1 ( u 1 ) R ′ ( x 1 , u 1 ) f j 1 ( x 1 ) = f i ′ d x 2 F j 1 j 2 ,v ( x 1 , x 2 ) − N j 2 v + δ j 1 ,j 2 − δ j 1 ,j 2 u 1 i ′ 0 x 1 1 ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ε ✐♥ R ′ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ◮ ❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ Z i 1 → j 1 ✐♥ R ′ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t ❛r❡ ✜♥✐t❡ ❢♦r ε = 0 ✶✵ ✴ ✶✹
❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞ 1 − x 1 1 � � � � d u 1 � 1 ( u 1 ) R ′ ( x 1 , u 1 ) f j 1 ( x 1 ) = f i ′ d x 2 F j 1 j 2 ,v ( x 1 , x 2 ) − N j 2 v + δ j 1 ,j 2 − δ j 1 ,j 2 u 1 i ′ 0 x 1 1 ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ε ✐♥ R ′ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ◮ ❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ Z i 1 → j 1 ✐♥ R ′ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t ❛r❡ ✜♥✐t❡ ❢♦r ε = 0 ◮ ✐✳❡✳ R ′ = 0 ✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ✶✵ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞ 1 − x 1 1 � � � � d u 1 � 1 ( u 1 ) R ′ ( x 1 , u 1 ) f j 1 ( x 1 ) = f i ′ d x 2 F j 1 j 2 ,v ( x 1 , x 2 ) − N j 2 v + δ j 1 ,j 2 − δ j 1 ,j 2 u 1 i ′ 0 x 1 1 ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ε ✐♥ R ′ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ◮ ❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ Z i 1 → j 1 ✐♥ R ′ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t ❛r❡ ✜♥✐t❡ ❢♦r ε = 0 ◮ ✐✳❡✳ R ′ = 0 ✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t R ′ = 0 ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs 1 − x 1 � � � � � d x 2 Z i 1 → j 1 j 2 ( x 1 , x 2 ) − Z i 1 → j 1 j 2 ( x 1 , x 2 ) = δ i 1 ,j 2 − δ i 1 ,j 2 + δ j 1 ,j 2 − δ j 1 ,j 2 Z i 1 → j 1 ( x 1 ) 0 ✶✵ ✴ ✶✹
✇❤❡r❡ ✐s ❣✐✈❡♥ ❜② ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❘❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ ❢♦r t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ✜♥❞s 1 − x 1 1 � � d u 1 � � f i ′ d x 2 x 2 F j 1 j 2 ( x 1 , x 2 ) − (1 − x 1 ) f j 1 ( x 1 ) = 1 ( u 1 ) R ( x 1 , u 1 ) u 1 j 2 i ′ 0 x 1 1 ✶✶ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❘❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ ❢♦r t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ✜♥❞s 1 − x 1 1 � � d u 1 � � f i ′ d x 2 x 2 F j 1 j 2 ( x 1 , x 2 ) − (1 − x 1 ) f j 1 ( x 1 ) = 1 ( u 1 ) R ( x 1 , u 1 ) u 1 j 2 i ′ 0 x 1 1 ✇❤❡r❡ R ( x 1 , u 1 ) ✐s ❣✐✈❡♥ ❜② u 1 � x 1 � � � x 1 � � � � � d z 1 1 − x 1 � Z − 1 R ( x 1 , u 1 ) = Z i 1 → j 1 − δ δ i 1 ,j 1 ( x 1 − z 1 ) i ′ 1 → i 1 z 1 u 1 z 1 z 1 i 1 x 1 1 − x 1 z 1 � � x 1 � � + z 1 d u 2 u 2 Z i 1 → j 1 j 2 , u 2 z 1 j 2 0 ✶✶ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❘❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ ❢♦r t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ✜♥❞s 1 − x 1 1 � � d u 1 � � f i ′ d x 2 x 2 F j 1 j 2 ( x 1 , x 2 ) − (1 − x 1 ) f j 1 ( x 1 ) = 1 ( u 1 ) R ( x 1 , u 1 ) u 1 j 2 i ′ 0 x 1 1 ❯s✐♥❣ t❤❡ s❛♠❡ r❡❛s♦♥✐♥❣ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ♦♥❡ ❝❛♥ t❤✉s ❝♦♥❝❧✉❞❡✱ t❤❛t ❛❧s♦ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s✳ ❚❤❡ ❝♦♥str❛✐♥t✱ t❤❛t R = 0 ②✐❡❧❞s t❤❡ ❢♦❧❧✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ Z i 1 → j 1 j 2 ❛♥❞ Z i 1 → j 1 ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs 1 − x 1 � � d x 2 x 2 Z i 1 → j 1 j 2 ( x 1 , x 2 )=( 1 − x 1 ) Z i 1 → j 1 ( x 1 ) j 2 0 ✶✶ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥ 1 � x 1 � � d d z 1 � d log ( µ 2 ) f j 1 ( x 1 ) = f i 1 ( z 1 ) P i 1 → j 1 z 1 z 1 i 1 x 1 ✇❤❡r❡ P i 1 → j 1 ❛r❡ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ❉●▲❆P s♣❧✐tt✐♥❣ ❦❡r♥❡❧s✳ ✶✷ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥ d d log ( µ 2 ) f j 1 = P i 1 → j 1 ⊗ f i 1 ✶✷ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥ d d log ( µ 2 ) f j 1 = P i 1 → j 1 ⊗ f i 1 ♣r♦♣♦s❡❞ ❞❉●▲❆P ❡q✉❛t✐♦♥ 1 − x 2 � x 1 � � d d z 1 � d log ( µ 2 ) F j 1 j 2 ( x 1 , x 2 ) = F i 1 j 2 ( z 1 , x 2 ) P i 1 → j 1 z 1 z 1 i 1 x 1 1 − x 1 � x 2 1 � x 1 � � � � d z 2 d z 1 , x 2 � � F j 1 i 2 ( x 1 , z 2 ) + f i 1 ( z 1 ) + P i 2 → j 2 P i 1 → j 1 j 2 z 2 z 2 z 2 z 1 z 1 1 i 2 i 1 x 2 x 1 + x 2 ✇❤❡r❡ t❤❡ P i 1 → j 1 j 2 ❛r❡ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛❜♦✉t ✇❤✐❝❤ ♥♦t ♠✉❝❤ ✐s ❦♥♦✇♥ ❛ ♣r✐♦r✐ ✶✷ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥ d d log ( µ 2 ) f j 1 = P i 1 → j 1 ⊗ f i 1 ♣r♦♣♦s❡❞ ❞❉●▲❆P ❡q✉❛t✐♦♥ d d log ( µ 2 ) F j 1 j 2 = P i 1 → j 1 ⊗ F i 1 j 2 + P i 2 → j 2 ⊗ F j 1 i 2 + P i 1 → j 1 j 2 ⊗ f i 1 ✶✷ ✴ ✶✹
❜② ❝♦♠♣❛r✐♥❣ ♦✉r ♣r♦♣♦s❡❞ ❢♦r♠ ♦❢ t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥ t♦ t❤❡ ❡①♣❧✐❝✐t ✲❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r❡♥♦r♠❛❧✐s❡❞ ❉P❉ ❛♥❞ ✉s✐♥❣ t❤❡ r❡❧❛t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ s✉♠ r✉❧❡s ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✇❡ ✇❡r❡ ❛❜❧❡ t♦ ❞❡r✐✈❡ ❛♥❛❧♦❣♦✉s s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥ d d log ( µ 2 ) f j 1 = P i 1 → j 1 ⊗ f i 1 ♣r♦♣♦s❡❞ ❞❉●▲❆P ❡q✉❛t✐♦♥ d d log ( µ 2 ) F j 1 j 2 = P i 1 → j 1 ⊗ F i 1 j 2 + P i 2 → j 2 ⊗ F j 1 i 2 + P i 1 → j 1 j 2 ⊗ f i 1 ◮ t❤❡ ❢♦r♠ ♦❢ t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▲❖ ❛♥❞ ◆▲❖ r❡s✉❧ts ❑✐rs❝❤♥❡r ✶✾✼✾ ❈❡❝❝♦♣✐❡r✐ ✷✵✶✶✱✷✵✶✹ ✶✷ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥ d d log ( µ 2 ) f j 1 = P i 1 → j 1 ⊗ f i 1 ♣r♦♣♦s❡❞ ❞❉●▲❆P ❡q✉❛t✐♦♥ d d log ( µ 2 ) F j 1 j 2 = P i 1 → j 1 ⊗ F i 1 j 2 + P i 2 → j 2 ⊗ F j 1 i 2 + P i 1 → j 1 j 2 ⊗ f i 1 ◮ t❤❡ ❢♦r♠ ♦❢ t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▲❖ ❛♥❞ ◆▲❖ r❡s✉❧ts ❑✐rs❝❤♥❡r ✶✾✼✾ ❈❡❝❝♦♣✐❡r✐ ✷✵✶✶✱✷✵✶✹ ◮ ❜② ❝♦♠♣❛r✐♥❣ ♦✉r ♣r♦♣♦s❡❞ ❢♦r♠ ♦❢ t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥ t♦ t❤❡ ❡①♣❧✐❝✐t µ ✲❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r❡♥♦r♠❛❧✐s❡❞ ❉P❉ ❛♥❞ ✉s✐♥❣ t❤❡ r❡❧❛t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ s✉♠ r✉❧❡s ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✇❡ ✇❡r❡ ❛❜❧❡ t♦ ❞❡r✐✈❡ ❛♥❛❧♦❣♦✉s s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ✶✷ ✴ ✶✹
❡①❛❝t❧② t❤❡ s❛♠❡ str✉❝t✉r❡ ❛s t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥✱ ❥✉st ❧✐❦❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ◮ ❝♦♠♣❛r✐♥❣ t❤❡ µ ✲❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r❡♥♦r♠❛❧✐s❡❞ ❉P❉ t♦ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs 1 − x 2 � x 1 � � d d z 1 � d log ( µ 2 ) Z i ′ 1 → j 1 j 2 ( x 1 , x 2 ) = P i 1 → j 1 Z i ′ 1 → i 1 j 2 ( z 1 , x 2 ) z 1 z 1 i 1 x 1 1 − x 1 1 � � x 2 � � � x 1 � d z 2 d z 1 � � , x 2 + P i 2 → j 2 Z i ′ 1 → j 1 i 2 ( x 1 , z 2 ) + P i 1 → j 1 j 2 Z i ′ 1 → i 1 ( z 1 ) z 2 z 2 z 2 z 1 z 1 1 i 2 i 1 x 2 x 1 + x 2 ✶✸ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ◮ ❝♦♠♣❛r✐♥❣ t❤❡ µ ✲❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r❡♥♦r♠❛❧✐s❡❞ ❉P❉ t♦ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs 1 − x 2 � x 1 � � d d z 1 � d log ( µ 2 ) Z i ′ 1 → j 1 j 2 ( x 1 , x 2 ) = P i 1 → j 1 Z i ′ 1 → i 1 j 2 ( z 1 , x 2 ) z 1 z 1 i 1 x 1 1 − x 1 1 � � x 2 � � � x 1 � d z 2 d z 1 � � , x 2 + P i 2 → j 2 Z i ′ 1 → j 1 i 2 ( x 1 , z 2 ) + P i 1 → j 1 j 2 Z i ′ 1 → i 1 ( z 1 ) z 2 z 2 z 2 z 1 z 1 1 i 2 i 1 x 2 x 1 + x 2 ◮ ❡①❛❝t❧② t❤❡ s❛♠❡ str✉❝t✉r❡ ❛s t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥✱ ❥✉st ❧✐❦❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ✶✸ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs d d log ( µ 2 ) Z i ′ 1 → j 1 j 2 = P i 1 → j 1 ⊗ Z i ′ 1 → i 1 j 2 + P i 2 → j 2 ⊗ Z i ′ 1 → j 1 i 2 + P i 1 → j 1 j 2 ⊗ Z i ′ 1 → i 1 ✶✸ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs d d log ( µ 2 ) Z i ′ 1 → j 1 j 2 = P i 1 → j 1 ⊗ Z i ′ 1 → i 1 j 2 + P i 2 → j 2 ⊗ Z i ′ 1 → j 1 i 2 + P i 1 → j 1 j 2 ⊗ Z i ′ 1 → i 1 ■♥ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ t❤❡ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs✱ t❤✐s ❛❧❧♦✇s t♦ ♦❜t❛✐♥ ❛♥❛❧♦❣♦✉s ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ ♥❡✇ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s 1 − x 1 � � � � � d x 2 Z i 1 → j 1 j 2 ( x 1 , x 2 ) − Z i 1 → j 1 j 2 ( x 1 , x 2 ) = δ i 1 ,j 2 − δ i 1 ,j 2 + δ j 1 ,j 2 − δ j 1 ,j 2 Z i 1 → j 1 ( x 1 ) b 0 1 − x 1 � � d x 2 x 2 Z i 1 → j 1 j 2 ( x 1 , x 2 )=( 1 − x 1 ) Z i 1 → j 1 ( x 1 ) j 2 0 ✶✸ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs d d log ( µ 2 ) Z i ′ 1 → j 1 j 2 = P i 1 → j 1 ⊗ Z i ′ 1 → i 1 j 2 + P i 2 → j 2 ⊗ Z i ′ 1 → j 1 i 2 + P i 1 → j 1 j 2 ⊗ Z i ′ 1 → i 1 ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s 1 − x 1 � � � � � d x 2 P i 1 → j 1 j 2 ( x 1 , x 2 ) − P i 1 → j 1 j 2 ( x 1 , x 2 ) = δ i 1 ,j 2 − δ i 1 ,j 2 + δ j 1 ,j 2 − δ j 1 ,j 2 P i 1 → j 1 ( x 1 ) 0 ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s 1 − x 1 � � d x 2 x 2 P i 1 → j 1 j 2 ( x 1 , x 2 )=( 1 − x 1 ) P i 1 → j 1 ( x 1 ) j 2 0 ✶✸ ✴ ✶✹
❛s ✐t s❤♦✉❧❞ ❛❧r❡❛❞② ❜❡ ❝❧❡❛r ❛❢t❡r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s✱ t❤❛t t❤❡② ❛r❡ ❛❧s♦ st❛❜❧❡ ✉♥❞❡r ❡✈♦❧✉t✐♦♥✱ t❤✐s ❛❝ts ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ❢♦r ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s 1 − x 1 � � � � � d x 2 P i 1 → j 1 j 2 ( x 1 , x 2 ) − P i 1 → j 1 j 2 ( x 1 , x 2 ) = δ i 1 ,j 2 − δ i 1 ,j 2 + δ j 1 ,j 2 − δ j 1 ,j 2 P i 1 → j 1 ( x 1 ) 0 ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s 1 − x 1 � � d x 2 x 2 P i 1 → j 1 j 2 ( x 1 , x 2 )=( 1 − x 1 ) P i 1 → j 1 ( x 1 ) j 2 0 ◮ ❝❛♥ ❜❡ ✉s❡❞ t♦ s❤♦✇ st❛❜✐❧✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ✉♥❞❡r ◗❈❉ ❡✈♦❧✉t✐♦♥ ✶✸ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s 1 − x 1 � � � � � d x 2 P i 1 → j 1 j 2 ( x 1 , x 2 ) − P i 1 → j 1 j 2 ( x 1 , x 2 ) = δ i 1 ,j 2 − δ i 1 ,j 2 + δ j 1 ,j 2 − δ j 1 ,j 2 P i 1 → j 1 ( x 1 ) 0 ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s 1 − x 1 � � d x 2 x 2 P i 1 → j 1 j 2 ( x 1 , x 2 )=( 1 − x 1 ) P i 1 → j 1 ( x 1 ) j 2 0 ◮ ❝❛♥ ❜❡ ✉s❡❞ t♦ s❤♦✇ st❛❜✐❧✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ✉♥❞❡r ◗❈❉ ❡✈♦❧✉t✐♦♥ ◮ ❛s ✐t s❤♦✉❧❞ ❛❧r❡❛❞② ❜❡ ❝❧❡❛r ❛❢t❡r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s✱ t❤❛t t❤❡② ❛r❡ ❛❧s♦ st❛❜❧❡ ✉♥❞❡r ❡✈♦❧✉t✐♦♥✱ t❤✐s ❛❝ts ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ❢♦r ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ ✶✸ ✴ ✶✹
✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙✉♠♠❛r② ◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝ ❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚ ✶✹ ✴ ✶✹
✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙✉♠♠❛r② ◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝ ❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚ ◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✶✹ ✴ ✶✹
✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙✉♠♠❛r② ◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝ ❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚ ◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ✶✹ ✴ ✶✹
t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙✉♠♠❛r② ◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝ ❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚ ◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◮ ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs ✶✹ ✴ ✶✹
❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙✉♠♠❛r② ◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝ ❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚ ◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◮ ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs ◮ t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ✶✹ ✴ ✶✹
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r② ❙✉♠♠❛r② ◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝ ❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚ ◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ◮ ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs ◮ t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ◮ ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥ ✶✹ ✴ ✶✹
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■✿ ▼♦t✐✈❛t✐♦♥ ❆s ❛♥ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r ❛ q✉❛r❦ ❧♦♦♣ ✐♥ φ 3 t❤❡♦r②✿ k p p − k ✶ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■✿ ▼♦t✐✈❛t✐♦♥ ❆s ❛♥ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r ❛ q✉❛r❦ ❧♦♦♣ ✐♥ φ 3 t❤❡♦r②✿ k p p − k ■♥ ❝♦✈❛r✐❛♥t P❚ t❤❡ ❧♦♦♣ ✐s ❣✐✈❡♥ ❜② � d D k 1 1 p 2 − m 2 + iǫ ( p − k ) 2 − m 2 + iǫ (2 π ) D ✶ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■✿ ▼♦t✐✈❛t✐♦♥ ❆s ❛♥ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r ❛ q✉❛r❦ ❧♦♦♣ ✐♥ φ 3 t❤❡♦r②✿ k p p − k ■♥ ❝♦✈❛r✐❛♥t P❚ t❤❡ ❧♦♦♣ ✐s ❣✐✈❡♥ ❜② � d D k 1 1 p 2 − m 2 + iǫ ( p − k ) 2 − m 2 + iǫ (2 π ) D P❡r❢♦r♠✐♥❣ t❤❡ k − ✐♥t❡❣r❛t✐♦♥ ✉s✐♥❣ ❈❛✉❝❤②✬s t❤❡♦r❡♠ ♦♥❡ ✜♥❞s p + � � d k + d D − 2 k 1 1 (2 k + )(2( p + − k + )) (2 π ) D − 2 p − − k 2 + m 2 − ( p − k ) 2 + m 2 2 π 2( p + − k + ) + iǫ 2 k + 0 ✶ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■✿ ▼♦t✐✈❛t✐♦♥ ❆s ❛♥ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r ❛ q✉❛r❦ ❧♦♦♣ ✐♥ φ 3 t❤❡♦r②✿ k p p − k ●❡♥❡r❛❧❧② t❤❡ ❞❡♥♦♠✐♥❛t♦r ❢♦r ❛ st❛t❡ ζ i ❜❡t✇❡❡♥ t✇♦ ✈❡rt✐❝❡s x i ❛♥❞ x i +1 ✐s ❣✐✈❡♥ ❜②✿ 1 − � P − l ∈ i k − l, on − shell + iǫ i ✇❤❡r❡ P i ✐s t❤❡ s✉♠ ♦❢ ❛❧❧ ❡①t❡r♥❛❧ ♠♦♠❡♥t❛ ❡♥t❡r✐♥❣ t❤❡ ❣r❛♣❤ ❜❡❢♦r❡ ✈❡rt❡① i ❛♥❞ t❤❡ s✉♠ ✐s ♦✈❡r t❤❡ ♦♥✲s❤❡❧❧ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ ❛❧❧ ❧✐♥❡s ✐♥ t❤❡ st❛t❡ ✶ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■■✿ ❘✉❧❡s ◮ ❙t❛rt✐♥❣ ❢r♦♠ ❛ ❣✐✈❡♥ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠ ♦♥❡ ❤❛s t♦ ❝♦♥s✐❞❡r ❛❧❧ ♣♦ss✐❜❧❡ x + ✲♦r❞❡r✐♥❣s ♦❢ t❤❡ ✈❡rt✐❝❡s✳ ■♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐s❡ t❤❡s❡ ♦r❞❡r✐♥❣s ♦♥❡ ✉s❡s t❤❛t x + ✐♥❝r❡❛s❡s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ♦♥ t❤❡ ❧❤s ♦❢ t❤❡ ❝✉t ✇❤✐❧❡ ✐t ✐♥❝r❡❛s❡s ❢r♦♠ r✐❣❤t t♦ ❧❡❢t ♦♥ t❤❡ r❤s ♦❢ t❤❡ ❝✉t✳ ◮ ❈♦✉♣❧✐♥❣ ❝♦♥st❛♥ts ❛♥❞ ✈❡rt❡① ❢❛❝t♦rs ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ ❝♦✈❛r✐❛♥t P❚✳ ◮ P❧✉s ❛♥❞ tr❛♥s✈❡rs❛❧ ♠♦♠❡♥t❛✱ k + ✉♥❞ k l ✱ ♦❢ ❛ ❧✐♥❡ l ❛r❡ ❝♦♥s❡r✈❡❞ ❛t t❤❡ ✈❡rt✐❝❡s l ◮ ❊❛❝❤ ❧✐♥❡ l ✐♥ ❛ ❣r❛♣❤ ❝♦♠❡s ✇✐t❤ ❛ ❢❛❝t♦r 1 ❛♥❞ ❛ ❍❡❛✈✐s✐❞❡ ❢✉♥❝t✐♦♥ Θ( k + l ) ✱ 2 k + l ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♣r♦♣❛❣❛t✐♦♥ ❢r♦♠ ❧♦✇❡r t♦ ❤✐❣❤❡r x + ◮ ❋♦r ❡❛❝❤ ❧♦♦♣ t❤❡r❡s ❛♥ ✐♥t❡❣r❛❧ ♦✈❡r ♣❧✉s ❛♥❞ tr❛♥s✈❡rs❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❧♦♦♣ ♠♦♠❡♥t✉♠ ℓ ✿ � d ℓ + d d − 2 ℓ (2 π ) d − 1 ◮ ❋♦r ❡❛❝❤ st❛t❡ ζ i ❜❡t✇❡❡♥ t✇♦ ✈❡rt✐❝❡s x + i ✉♥❞ x + i +1 ♦♥❡ ❣❡ts t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❢❛❝t♦r 1 − � P − l ∈ i k − l, on − shell + iǫ i ✷ ✴ ✾
❉P❉ ▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■■■✿ P❉❋ ❛♥❞ ❉P❉ ❉❡✜♥✐t✐♦♥s ✐♥ ▲❈P❚ P❉❋ � n 1 � d k − � M ( c ) N ( t ) d k + d k + 1 d D − 2 k 1 i d D − 2 k i i d D − 2 k i � �� � � � � � � f j 1 k + B ( x 1 ) = 1 (2 π ) D (2 π ) D − 1 (2 π ) D − 1 t c o i =2 i = M ( c )+1 N N � � � � � � � � p − − k − − p + − × Φ j 1 { k + } , { k } k − k + 2 πδ δ P DF t,c,o i , on − shell i i =2 i =1 ✇❤❡r❡ n 1 = 1 ✐❢ ♣❛rt♦♥ 1 ✐s ❛ ❣❧✉♦♥ ♦r ❛ s❝❛❧❛r q✉❛r❦✱ ✇❤✐❧❡ ❢♦r ❉✐r❛❝ q✉❛r❦s ♦♥❡ ❤❛s n 1 = 0 ✳ ✸ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■■■✿ P❉❋ ❛♥❞ ❉P❉ ❉❡✜♥✐t✐♦♥s ✐♥ ▲❈P❚ P❉❋ � n 1 � d k − � M ( c ) N ( t ) d k + d k + 1 d D − 2 k 1 i d D − 2 k i i d D − 2 k i � �� � � � � � � f j 1 k + B ( x 1 ) = 1 (2 π ) D (2 π ) D − 1 (2 π ) D − 1 t c o i =2 i = M ( c )+1 N N � � � � � � � � p − − k − − p + − × Φ j 1 { k + } , { k } k − k + 2 πδ δ P DF t,c,o i , on − shell i i =2 i =1 ❉P❉ � � n 1 � � n 2 2 p + (2 π ) D − 1 � � � � F j 1 j 2 k + k + ( x 1 , x 2 ) = δ f ( l ) ,j 2 B 1 2 t c o l � d k − � M ( c ) N ( t ) 1 d k − l d∆ − d D − 2 k 1 d D − 2 k l d k + d k + i d D − 2 k i i d D − 2 k i � � � � � × (2 π ) 3 D (2 π ) D − 1 (2 π ) D − 1 i =2 , i � = l i = M ( c )+1 M ( c ) M ( c ) � � � � � � � � p − − k − p + − × Φ j 1 j 2 { k + } , { k } 1 − k − k − k + 2 πδ l − δ DP D t,c,o i , on − shell i i =2 , i � = l i =1 ✸ ✴ ✾
❚❤✐s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ✇❤❡r❡ st❛t❡s st❛t❡s st❛t❡s st❛t❡s ▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ▲❈P❚ P❉❋ ❣r❛♣❤ F A H ′ H k k I ′ I F ( F A ) ✹ ✴ ✾
✇❤❡r❡ st❛t❡s st❛t❡s st❛t❡s st❛t❡s ▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ▲❈P❚ P❉❋ ❣r❛♣❤ F A H ′ H ❚❤✐s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s k k Φ P DF = I F ( F A ) I ′ I ′ I F ( F A ) ✹ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ▲❈P❚ P❉❋ ❣r❛♣❤ F A H ′ H ❚❤✐s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s k k Φ P DF = I F ( F A ) I ′ I ′ I F ( F A ) ✇❤❡r❡ 1 1 � � I ′ = I = p − − � p − − � l ∈ ζ k − l ∈ ζ k − l, o . s . + iǫ l, o . s . − iǫ st❛t❡s ζ st❛t❡s ζ ζ<H ζ<H ′ 1 1 � � F ( F A ) = p − − k − − � p − − k − − � l ∈ ζ k − l ∈ ζ k − l, o . s . + iǫ l, o . s . − iǫ st❛t❡s ζ st❛t❡s ζ H<ζ<F A H ′ <ζ<F A � p − − k − − k − × 2 πδ l, o . s . l ∈ F A ✹ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H ′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t F A ✳ ❙✉♠♠✐♥❣ F ( F A ) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ ✺ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H ′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t F A ✳ ❙✉♠♠✐♥❣ F ( F A ) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ N c − 1 N 1 1 � � � � � � p − − k − − D c F ( F A ) = p − − k − − D f + iǫ 2 πδ p − − k − − D f − iǫ c =1 F A f =1 f = c +1 ✇❤❡r❡ � k − D f = l, on − shell , l ∈ f ✺ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H ′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t F A ✳ ❙✉♠♠✐♥❣ F ( F A ) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ N c − 1 N 1 1 � � � � � � p − − k − − D c F ( F A ) = p − − k − − D f + iǫ 2 πδ p − − k − − D f − iǫ c =1 F A f =1 f = c +1 r❡✇r✐t✐♥❣ t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ❛s � � 1 1 2 π δ ( x ) = i x + iǫ − x − iǫ ✺ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H ′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t F A ✳ ❙✉♠♠✐♥❣ F ( F A ) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ N c − 1 N 1 1 � � � � � � p − − k − − D c F ( F A ) = p − − k − − D f + iǫ 2 πδ p − − k − − D f − iǫ c =1 F A f =1 f = c +1 r❡✇r✐t✐♥❣ t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ❛s � � 1 1 2 π δ ( x ) = i x + iǫ − x − iǫ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s N N 1 1 � � � F ( F A ) = i p − − k − − D f + iǫ − p − − k − − D f − iǫ F A f =1 f =1 ✺ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H ′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t F A ✳ ❙✉♠♠✐♥❣ F ( F A ) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ N c − 1 N 1 1 � � � � � � p − − k − − D c F ( F A ) = p − − k − − D f + iǫ 2 πδ p − − k − − D f − iǫ c =1 F A f =1 f = c +1 r❡✇r✐t✐♥❣ t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ❛s � � 1 1 2 π δ ( x ) = i x + iǫ − x − iǫ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s N N 1 1 � � � F ( F A ) = i p − − k − − D f + iǫ − p − − k − − D f − iǫ F A f =1 f =1 ❋♦r N ≥ 2 t❤✐s ❡①♣r❡ss✐♦♥ ✈❛♥✐s❤❡s ❛❢t❡r ✐♥t❡❣r❛t✐♦♥ ♦✈❡r k − ✇❤✐❧❡ ❢♦r N = 1 t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ✐s r❡♣r♦❞✉❝❡❞✳ ✺ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r P❉❋s ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H ′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t F A ✳ ❙✉♠♠✐♥❣ F ( F A ) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ N c − 1 N 1 1 � � � � � � p − − k − − D c F ( F A ) = p − − k − − D f + iǫ 2 πδ p − − k − − D f − iǫ c =1 F A f =1 f = c +1 r❡✇r✐t✐♥❣ t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ❛s � � 1 1 2 π δ ( x ) = i x + iǫ − x − iǫ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s N N 1 1 � � � F ( F A ) = i p − − k − − D f + iǫ − p − − k − − D f − iǫ F A f =1 f =1 ❋♦r N ≥ 2 t❤✐s ❡①♣r❡ss✐♦♥ ✈❛♥✐s❤❡s ❛❢t❡r ✐♥t❡❣r❛t✐♦♥ ♦✈❡r k − ✇❤✐❧❡ ❢♦r N = 1 t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ✐s r❡♣r♦❞✉❝❡❞✳ ❖♥❡ ❝❛♥ t❤✉s ❝♦♥❝❧✉❞❡✱ t❤❛t ♦♥❧② s✉❝❤ x + ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✳ ✺ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r ❉P❉s ❈♦♥s✐❞❡r ♥♦✇ ❛ ❉P❉✱ ✇❤✐❝❤ ❝❛♥ ❛❣❛✐♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s Φ DP D = I 1 I 2 F ( F A ) I ′ 2 I ′ 1 ✻ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r ❉P❉s ❈♦♥s✐❞❡r ♥♦✇ ❛ ❉P❉✱ ✇❤✐❝❤ ❝❛♥ ❛❣❛✐♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s Φ DP D = I 1 I 2 F ( F A ) I ′ 2 I ′ 1 t♦ ❜❡ ❛❜❧❡ t♦ ✉s❡ t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❛s ❜❡❢♦r❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ x + ♦r❞❡r✐♥❣s F A F A H ′ H ′ H ′ H ′ H 1 H 2 H 1 H 2 2 1 2 1 K − k ′ K − k ′′ k ′ K − k ′′ k ′ k ′′ K − k ′ k ′′ I ′ I ′ ˜ I ′ I ′ I 1 I 2 I 1 I 2 2 1 2 1 F ( F A ) F ( F A ) ✻ ✴ ✾
▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ▲❈P❚ ❱✿ ❝♦♥tr✐❜✉t✐♥❣ x + ♦r❞❡r✐♥❣s ❢♦r ❉P❉s ❈♦♥s✐❞❡r ♥♦✇ ❛ ❉P❉✱ ✇❤✐❝❤ ❝❛♥ ❛❣❛✐♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s Φ DP D = I 1 I 2 F ( F A ) I ′ 2 I ′ 1 t♦ ❜❡ ❛❜❧❡ t♦ ✉s❡ t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❛s ❜❡❢♦r❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ x + ♦r❞❡r✐♥❣s F A F A H ′ H ′ H ′ H ′ H 1 H 2 H 1 H 2 2 1 2 1 K − k ′ K − k ′′ k ′ K − k ′′ K − k ′ k ′ k ′′ k ′′ I ′ I ′ ˜ I ′ I ′ I 1 I 2 I 1 I 2 2 1 2 1 F ( F A ) F ( F A ) ❈♦♥s✐❞❡r ♥♦✇ t❤❡ st❛t❡s ❜❡t✇❡❡♥ H 1 ❛♥❞ H 2 ✱ I 2 ❛♥❞ ˜ I 2 1 1 ˜ I 2 = I 2 = I 2 + iǫ . p − − ( K − − k ′− ) − D I 2 + iǫ p − − k ′− − D ˜ ✻ ✴ ✾
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