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❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ ❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ ❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡ ❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ ❡❛❝❤ ✈❡rt❡① ❤❛s ❡✈❡♥ ❞❡❣r❡❡ ❡❛❝❤ ❢❛❝❡ ✐s ❜❧❛❝❦ ♦r ✇❤✐t❡ t❤❡ r♦♦t ❢❛❝❡ ✐s ✇❤✐t❡ ❡❞❣❡ ♦r✐❡♥t❛t✐♦♥ ✿ ✇❤✐t❡ ❢❛❝❡ ♦♥ t❤❡ r✐❣❤t

✷ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ✇❤❡r❡ ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷ ✶ ✷ ✷ ✸ ✶ t❤❡ ✜rst t❡r♠s ♦❢ ❛r❡ ✹ ✸ ✻ ✼ ✼ ✶✺ ✽ ✻✸ ✾ ✶✵ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ▲❡t E s ( y ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s

✇❤❡r❡ ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷ ✶ ✷ ✷ ✸ ✶ t❤❡ ✜rst t❡r♠s ♦❢ ❛r❡ ✹ ✸ ✻ ✼ ✼ ✶✺ ✽ ✻✸ ✾ ✶✵ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ▲❡t E s ( y ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s E s ( y ) ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s E s ( y ) = C ( y ) − C ( y ) ✷

t❤❡ ✜rst t❡r♠s ♦❢ ❛r❡ ✹ ✸ ✻ ✼ ✼ ✶✺ ✽ ✻✸ ✾ ✶✵ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ▲❡t E s ( y ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s E s ( y ) ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s E s ( y ) = C ( y ) − C ( y ) ✷ ✇❤❡r❡ C ( y ) ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ( ✶ + ✷ C ( y )) ✷ C ( y ) = y ( ✶ + C ( y ) − C ( y ) ✷ ) ✸ .

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ▲❡t E s ( y ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s E s ( y ) ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s E s ( y ) = C ( y ) − C ( y ) ✷ ✇❤❡r❡ C ( y ) ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ( ✶ + ✷ C ( y )) ✷ C ( y ) = y ( ✶ + C ( y ) − C ( y ) ✷ ) ✸ . t❤❡ ✜rst t❡r♠s ♦❢ E s ( y ) ❛r❡ E s ( y ) = y + y ✹ + ✸ y ✻ + ✼ y ✼ + ✶✺ y ✽ + ✻✸ y ✾ + O ( y ✶✵ )

t✇♦ ♦r✐❡♥t❡❞ ♣❛t❤s ❢r♦♠ t♦ ✿ s❛♠❡ ❧❡♥❣t❤ ♠♦❞✉❧♦ ✸ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❊✉❧❡r✐❛♥ ♦r✐❡♥t❛t✐♦♥ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✈❡rt✐❝❡s ✱ ♦❢ t❤❡ ❡❞❣❡s

❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥s ❊✉❧❡r✐❛♥ ♦r✐❡♥t❛t✐♦♥ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✈❡rt✐❝❡s ✱ ♦❢ t❤❡ ❡❞❣❡s t✇♦ ♦r✐❡♥t❡❞ ♣❛t❤s ❢r♦♠ a t♦ b ✿ s❛♠❡ ❧❡♥❣t❤ ♠♦❞✉❧♦ ✸

♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ✳ ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥

t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ✳ ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ✳

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ✳

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ✳

❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ ♦r✐❡♥t❡❞ s✐♠♣❧❡ ❊✉❧❡r✐❛♥ tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ✳ ❚❤❡♦r❡♠ ✭❊♣♣st❡✐♥ ❛♥❞ ▼✉♠❢♦r❞ ✭✷✵✶✹✮✮ ❆ r♦♦t❡❞ ♣❧❛♥❛r ♠❛♣ ✐s t❤❡ s❦❡❧❡t♦♥ ♦❢ s♦♠❡ ❝♦r♥❡r ♣♦❧②❤❡❞r♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❞✉❛❧ ♠❛♣ ✐s ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥✳

❚❤❡ ✜rst t❡r♠s ❚❤❡ ✜rst t❡r♠s ♦❢ ❛r❡ ✹ ✸ ✻ ✹ ✼ ✶✺ ✽ ✸✾ ✾ ✶✵ ❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ▲❡t E c ( z ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s✳ ❚❤❡♦r❡♠ ✭❉✳✱ P♦✉❧❛❧❤♦♥✱ ❙❝❤❛❡✛❡r ✭✷✵✶✺✮✮ E c ( z ) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ − z ✷ A ( z ) ✷ + ✷ z ✸ A ( z ) ✸ ✶ + zA ( z ) + z ✷ A ( z ) ✷ � � z E c ( z ) = . ✶ + z ✶ + z ( ✶ + z ) ✷ ✇❤❡r❡ A ( z ) ✐s t❤❡ ❈❛t❛❧❛♥ s❡r✐❡s✳

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ▲❡t E c ( z ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s✳ ❚❤❡♦r❡♠ ✭❉✳✱ P♦✉❧❛❧❤♦♥✱ ❙❝❤❛❡✛❡r ✭✷✵✶✺✮✮ E c ( z ) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ − z ✷ A ( z ) ✷ + ✷ z ✸ A ( z ) ✸ ✶ + zA ( z ) + z ✷ A ( z ) ✷ � � z E c ( z ) = . ✶ + z ✶ + z ( ✶ + z ) ✷ ✇❤❡r❡ A ( z ) ✐s t❤❡ ❈❛t❛❧❛♥ s❡r✐❡s✳ ❚❤❡ ✜rst t❡r♠s ❚❤❡ ✜rst t❡r♠s ♦❢ E c ( z ) ❛r❡ E c ( z ) = z + z ✹ + ✸ z ✻ + ✹ z ✼ + ✶✺ z ✽ + ✸✾ z ✾ + O ( z ✶✵ ) .

✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡ ❞❡❝♦♠♣♦s✐t✐♦♥s ✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥ ❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄

✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡ ❞❡❝♦♠♣♦s✐t✐♦♥s ✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥ ❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄ ✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞

✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡ ❞❡❝♦♠♣♦s✐t✐♦♥s ✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥ ❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄ ✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥

✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥ ❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄ ✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡ ❞❡❝♦♠♣♦s✐t✐♦♥s

❊♥✉♠❡r❛t✐♦♥ ♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❍♦✇ t♦ ♣r♦✈❡ t❤❛t ❄ ✶ ❚❤❡r❡ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♠❡t❤♦❞ ✷ ❲❡ ✇❛♥t ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ✸ ❲❡ ♥❡❡❞ ♠♦r❡ ❢❛♠✐❧✐❡s ♦❢ ♣❧❛♥❛r ♠❛♣s ❢♦r ✐♥t❡r♠❡❞✐❛t❡ ❞❡❝♦♠♣♦s✐t✐♦♥s ✹ ❚❤❡② ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥ ❝♦♠♠♦♥

♥♦♥✲r♦♦t ❢❛❝❡s ❛r❡ tr✐❛♥❣❧❡s ✐♥♥❡r ✈❡rt✐❝❡s ❤❛✈❡ ❡✈❡♥ ❞❡❣r❡❡ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ❆ s❡t ♦❢ ♣r♦♣❡rt✐❡s ❢♦r s♦♠❡ ♠❛♣s ❈♦r♥❡r q✉❛s✐✲tr✐❛♥❣✉❧❛t✐♦♥s

✐♥♥❡r ✈❡rt✐❝❡s ❤❛✈❡ ❡✈❡♥ ❞❡❣r❡❡ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ❆ s❡t ♦❢ ♣r♦♣❡rt✐❡s ❢♦r s♦♠❡ ♠❛♣s ❈♦r♥❡r q✉❛s✐✲tr✐❛♥❣✉❧❛t✐♦♥s ♥♦♥✲r♦♦t ❢❛❝❡s ❛r❡ tr✐❛♥❣❧❡s

t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s ❆ s❡t ♦❢ ♣r♦♣❡rt✐❡s ❢♦r s♦♠❡ ♠❛♣s ❈♦r♥❡r q✉❛s✐✲tr✐❛♥❣✉❧❛t✐♦♥s ♥♦♥✲r♦♦t ❢❛❝❡s ❛r❡ tr✐❛♥❣❧❡s ✐♥♥❡r ✈❡rt✐❝❡s ❤❛✈❡ ❡✈❡♥ ❞❡❣r❡❡

❆ s❡t ♦❢ ♣r♦♣❡rt✐❡s ❢♦r s♦♠❡ ♠❛♣s ❈♦r♥❡r q✉❛s✐✲tr✐❛♥❣✉❧❛t✐♦♥s ♥♦♥✲r♦♦t ❢❛❝❡s ❛r❡ tr✐❛♥❣❧❡s ✐♥♥❡r ✈❡rt✐❝❡s ❤❛✈❡ ❡✈❡♥ ❞❡❣r❡❡ t❤❡ ♦♥❧② ❝❧♦❝❦✇✐s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ✇❤✐t❡ ✐♥♥❡r ❢❛❝❡s

❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✸ ❛ r♦♦t ✈❡rt❡① ❛ ♠❛r❦❡❞ ✈❡rt❡① ✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s t s

❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✸ ❛ ♠❛r❦❡❞ ✈❡rt❡① ✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s t ❛ r♦♦t ✈❡rt❡① s s

❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✸ ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s t ❛ r♦♦t ✈❡rt❡① s ❛ ♠❛r❦❡❞ ✈❡rt❡① t ✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① s

❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ t♦ ✱ ❧❡♥❣t❤ ✸ ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s t ❛ r♦♦t ✈❡rt❡① s ❛ ♠❛r❦❡❞ ✈❡rt❡① t ✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✱ ❧❡♥❣t❤ ℓ ≥ ✵ s

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s ❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s t ❛ r♦♦t ✈❡rt❡① s ❛ ♠❛r❦❡❞ ✈❡rt❡① t ✱ ❝❛❧❧❡❞ t❤❡ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦✉t❡r ❢❛❝❡ ❝♦♥s✐sts ♦❢ ✿ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✿ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✱ ❧❡♥❣t❤ ℓ ≥ ✵ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✿ ❝❧♦❝❦✇✐s❡✱ ❢r♦♠ s t♦ t ✱ s ❧❡♥❣t❤ ℓ + ✸

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞ t ℓ s 1 s 2 ℓ + 3 s

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞ t ℓ s 1 s 2 ℓ + 3 s

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞ t ℓ − 2 ? ℓ s 1 s 2 ℓ + 3 s

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞ t ℓ − 2 ℓ + 1 ? ℓ s 1 s 2 ℓ + 3 s

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞ t ℓ − 2 ℓ + 1 ℓ s 1 s 2 ℓ + 3 s

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞ t leftmost shortest path ℓ − 2 ℓ + 1 s 1 s 2 s

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞ t leftmost shortest path s 1 s 2 s

❈✉tt✐♥❣ ❛♥ ❛❧♠♦♥❞ t 2 t 1 ∈ A ∈ A s 1 s 2 s

●❧✉✐♥❣ t✇♦ ❛❧♠♦♥❞s ✿ ♥♦ ❝❧♦❝❦✇✐s❡ s❡♣❛r❛t✐♥❣ tr✐❛♥❣❧❡

●❧✉✐♥❣ t✇♦ ❛❧♠♦♥❞s ✿ ♥♦ ❝❧♦❝❦✇✐s❡ s❡♣❛r❛t✐♥❣ tr✐❛♥❣❧❡

●❧✉✐♥❣ t✇♦ ❛❧♠♦♥❞s ✿ ♥♦ ❝❧♦❝❦✇✐s❡ s❡♣❛r❛t✐♥❣ tr✐❛♥❣❧❡

●❧✉✐♥❣ t✇♦ ❛❧♠♦♥❞s ✿ ♥♦ ❞♦✉❜❧❡ ❡❞❣❡

❆❧♠♦♥❞ tr✐❛♥❣✉❧❛t✐♦♥s ▲❡t A ( z ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❛❧♠♦♥❞s✳ ❚❤❡♦r❡♠ A ( z ) ✐s t❤❡ ✉♥✐q✉❡ ❢♦r♠❛❧ ♣♦✇❡r s❡r✐❡s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✿ A ( z ) = ✶ + zA ( z ) ✷

t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤ ❛ r♦♦t ❡❞❣❡ ❛♥ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ ❙❧✐❝❡s ❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶ t b a

t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤ ❛♥ ❛♣❡① ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ ❙❧✐❝❡s ❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶ ❛ r♦♦t ❡❞❣❡ ( a , b ) t b a

t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤ ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ ❙❧✐❝❡s ❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶ ❛ r♦♦t ❡❞❣❡ ( a , b ) t ❛♥ ❛♣❡① t b a

❛ ❧❡❢t ❜♦✉♥❞❛r② ✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ❛ r✐❣❤t ❜♦✉♥❞❛r② ✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤ ❙❧✐❝❡s ❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶ ❛ r♦♦t ❡❞❣❡ ( a , b ) t ❛♥ ❛♣❡① t ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ b a

❛ r✐❣❤t ❜♦✉♥❞❛r② ✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ t♦ ✱ ♦❢ ❧❡♥❣t❤ ❙❧✐❝❡s ❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶ ❛ r♦♦t ❡❞❣❡ ( a , b ) t ❛♥ ❛♣❡① t ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ a t♦ t b a

❙❧✐❝❡s ❆ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ℓ ≥ ✶ ❛ r♦♦t ❡❞❣❡ ( a , b ) t ❛♥ ❛♣❡① t ❚❤❡ ❜♦✉♥❞❛r② ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✿ t❤❡ r♦♦t ❡❞❣❡ ❛ ❧❡❢t ❜♦✉♥❞❛r② ✱ s❤♦rt❡st ♣❛t❤ ❢r♦♠ a t♦ t ❛ r✐❣❤t ❜♦✉♥❞❛r② ✱ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❢r♦♠ b a b t♦ t ✱ ♦❢ ❧❡♥❣t❤ ℓ

❈✉tt✐♥❣ ❛ s❧✐❝❡ t ℓ − 1 ∈ A u = q ℓ + 2 a b

❈✉tt✐♥❣ ❛ s❧✐❝❡ t ℓ − 1 ∈ A u = q ℓ + 2 a

❈✉tt✐♥❣ ❛ s❧✐❝❡ t ∈ A u = q a

❈✉tt✐♥❣ ❛ s❧✐❝❡ t t ∈ A ∈ A ∈ A r q u u = q a a b

❈✉tt✐♥❣ ❛ s❧✐❝❡ t t ∈ A ∈ A ∈ A r q u u = q a a b

❈✉tt✐♥❣ ❛ s❧✐❝❡ t t ∈ A ∈ A ∈ A r q u u = q a a b

❈✉tt✐♥❣ ❛ s❧✐❝❡ t t ∈ A ∈ A ∈ A r q u u = q a a b

❙❧✐❝❡s ♦r ♥♦t s❧✐❝❡s t a b

❙❧✐❝❡s ♦r ♥♦t s❧✐❝❡s t a b

❙❧✐❝❡s ▲❡t S ( z ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❧✐❝❡s✳ ❚❤❡♦r❡♠ S ( z ) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ( ✶ + z ) S ( z ) = zA ( z ) + z ✷ A ( z ) ✷

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t ∈ A v ∈ A s u = q a b

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t ∈ A v ∈ A s u = q a b

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t ∈ A v ∈ A s u = q a

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t t ∈ A ∈ A v s v ∈ A ∈ A s ∈ A u = q u q r b a a

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t t ∈ A ∈ A v s v ∈ A ∈ A s ∈ A u = q u q r b a a

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t t ∈ A ∈ A v s v ∈ A ∈ A s ∈ A u = q u q r b a a

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t t t ∈ A ∈ A ∈ A v v ∈ A s s v ∈ A ∈ A s ∈ A ∈ A u = q u u q r q r b b a a a

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t t t ∈ A ∈ A ∈ A v v ∈ A s s v ∈ A ∈ A s ∈ A ∈ A u = q u u q r q r b b a a a

❈✉tt✐♥❣ ❛ s❧✐❝❡ ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ t t t ∈ A ∈ A ∈ A v v ∈ A s s v ∈ A ∈ A s ∈ A ∈ A u = q u u q r q r b b a a a

❙❧✐❝❡s ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷ ▲❡t S + ( z ) ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❧✐❝❡s ♦❢ ❤❡✐❣❤t ❛t ❧❡❛st ✷✳ ❚❤❡♦r❡♠ S + ( z ) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ( ✶ + z ) ✷ S + ( z ) = z ✷ A ( z ) ✷ ( ✶ + ✷ zA ( z ))

❙❧✐❝❡s ♦❢ ❤❡✐❣❤t ✶ ❚❤❡♦r❡♠ ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ✶ ❇✐❥❡❝t✐✈❡ ❢♦r♠✉❧❛ ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s

❚❤❡♦r❡♠ ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ✶ ❇✐❥❡❝t✐✈❡ ❢♦r♠✉❧❛ ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❙❧✐❝❡s ♦❢ ❤❡✐❣❤t ✶

❇✐❥❡❝t✐✈❡ ❢♦r♠✉❧❛ ❈♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ❙❧✐❝❡s ♦❢ ❤❡✐❣❤t ✶ ❚❤❡♦r❡♠ ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ( ✶ + z ) E c ( z ) = z + z ( S ( z ) − S + ( z ))

❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡ ✳ t❤❡ ❧✐❢t ♦❢ s✉❝❤ ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥ ♦❢ t❤✐s ❧✐❢t ❇✐❥❡❝t✐✈❡ ♣r♦♦❢ ❊❧❡♠❡♥ts ❢♦r ❜✐❥❡❝t✐✈❡ ♣r♦♦❢

t❤❡ ❧✐❢t ♦❢ s✉❝❤ ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥ ♦❢ t❤✐s ❧✐❢t ❇✐❥❡❝t✐✈❡ ♣r♦♦❢ ❊❧❡♠❡♥ts ❢♦r ❜✐❥❡❝t✐✈❡ ♣r♦♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s c ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡ ✳

t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥ ♦❢ t❤✐s ❧✐❢t ❇✐❥❡❝t✐✈❡ ♣r♦♦❢ ❊❧❡♠❡♥ts ❢♦r ❜✐❥❡❝t✐✈❡ ♣r♦♦❢ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥s c ✇✐t❤ ❛ ♠❛r❦❡❞ ✐♥♥❡r ✇❤✐t❡ tr✐❛♥❣❧❡ ✳ t❤❡ ❧✐❢t ♦❢ s✉❝❤ ❛ ❝♦r♥❡r tr✐❛♥❣✉❧❛t✐♦♥