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α SλS ≥  

 

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εSO(Y ) = εSO + εSO DQGεSO(G ) =

(

)

  SO  SO  (Y ) ε − ε  = εSO − εSO   

(

)

 

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SO SO εSO(Y ) + εPLQ εSO + εSO εPD[  = = SO SO − εPLQ εSO − εSO εPD[ εSO(Y ) − εSO

 

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εSO(Y ) εSO(G )

=

(Y ) εSO   VLQ (ψ )  εSO + εSO  = = SO SO Y) ( SO     ε − ε   + VLQ (ψ ) ε − εSO

 

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( ) εSO Y

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Y

=

εSO(Y ) ( ) εSO − εSO Y



 

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ε (Y ) ε (Y ) 5σ =  − SOSO =  − SOSO   .σ ε εPLQ

 

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5σ  + VLQ (ψ ) =  . σ  − VLQ (ψ )

 

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§π ϕ . σ = WDQ  ¨ + µ © 

·  + VLQ (ϕ µ )  ¸= ¹  − VLQ (ϕ µ )

 

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5σ = 



σ  σ PLQ  + VLQ (ϕP ) = =  σ  σ PD[  − VLQ (ϕP )

 

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VLQ (ϕP ) − VLQ (ϕ µ )

 − VLQ (ϕP ) VLQ (ϕ µ )



 

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ψ = ϕP − ϕ µ  

 

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κ ≡ εLMSO(GHY ) = η

∂J ( )  ∂σ LM GHY

 

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GHY εLMSO ∂ε LMSO κ ∂κ ∂J ∂J ( ) = = = DQG =  η η ∂η ∂η ∂σ LM ∂σ LM

 

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+ =−

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  GHYLDWRULFVWUDLQ ε ( G ) 

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∂ϕP ∂ϕ =  (σ  + σ  ) FRV (ϕP ) (PG )   ∂κ ∂ε SO

 

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 º  º ª − Φ ª − Ψ ∂I ∂J « » « )LM = =  − − Φ  » DQG*LM = =  − − Ψ  »»   ∂σ LM « ∂σ LM « «¬  «¬    »¼   »¼

 

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ª − ( + ν  ) Φ  «  )LM( ) = «  − − ( + ν  ) Φ «   ¬ ª − ( + ν  ) Ψ  «  *LM( ) = « − − ( + ν  ) Ψ  «   ¬

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ν ν  º § εHS · § σHS · ª + ν  ¨  HS ¸ ¨ ¸ « ν  +ν   »» ¨ εHS ¸ ν ¨ σ  ¸ = * «  + HS ¸ ¨ σ  « ν ν  + ν  » ¨  ¸ ¨¨ HS ¸¸ ¨ ¸ « »    ¼ ¨© εHS ¹¸ ¬  © σ ¹ ª − ( − ΝΦ )( − ΝΨ ) ( + ΝΦ )( − ΝΨ ) « * « ( − ΝΦ )( + ΝΨ ) − ( + ΝΦ )( + ΝΨ ) + HS « ' «   «   ¬

  º § εHS · » ¨ HS ¸   » ¨ ε ¸ ¸ »¨  » ¨  ¸   »¼ ¨© εHS ¹¸



 

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σ HS = 'HS ε HS = 'HO + ' SO ε HS = 'HO + ' SO

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GHW 'HS − ϑHS , = (ϑHS − * ) ¬ªϑHS − * ( + ν  ) ¼º + 

 * (ϑHS − * ) ª¬ϑHS − * ( + ν  ) + Ν  (ϑHS − * ( + ν  ) ) ΦΨ º¼  ' HS

 



+

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(

ϑHS() *

ϑHS() *

ϑHS() *

ϑHS( ) *

)

(

)

=  

  

=+

§ ν ν   + Ν 2 ΦΨ  + Ν 2 ΦΨ · ν  Ν 2 ΦΨ − − ¨ +  ¸ − HS  ' ' HS ' HS ©  ¹

=+

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=  

 

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= 

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ϑHS( ) *

=  

 

LQZKLFK D =+

ν   + Ν  ΦΨ − DQGE = E − E    ' HS

 

DQG E =

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ϑHS() *

=

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= DQG

*

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ϑHS() *

= D + E ≠ 



 

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ϑHS() *

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=

*

=

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*



 



 

FDVHFHJOLQHDUHODVWLFLW\IRUν  ≠ PXOWLSOLFLW\WKUHH IRUE > DQGD =  + E 

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=

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*

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= D



 

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=

ϑHS() *

=

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=

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= 

 

FDVHFRPSOH[FRQMXJDWHGHLJHQYDOXHV IRUE < DQGD ∈ \ 



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ν = 



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ν ()

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ΦΨ .  − ν

 

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ν

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§ −ΦΨ § ΦΨ · ΦΨ  · §  − ν ·  ( − ν ) ¸  = − ¨  + ¨ − ¸ −  ¨ + ¸± 2 ¨ ¸ ( − ν )2  − ν © ν ¹ ν ¹ ν ©   ν − ( ) © ¹



 

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E=

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 ν



 

  

 







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ν   + Ν 2 ΦΨ − =   ' HS 

 

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+ =

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¸¨ ν ¹ ©

ΦΨ  − ν

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§ − · ¨ ¸  ¨¸   Y (HS)F = Z(HS)F =   ¨ − ¸ ¨ ¸ ©¹



§ − · ¨ ¸  ¨¸   Y (HS)F = Z(HS)F =  ¨  ¸ ¨ ¸ ©¹

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§· ¨ ¸    Y (HS)F = Z(HS)F = ¨ ¸   ¨¸ ¨ ¸ © ¹

 

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→

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→

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 ª  − ΝΨ º  + E + E ν  ΝΨ »¼ Q «¬  ª  + ΝΨ º  − E + E « ν  ΝΨ ¼» Q ¬

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º§   ª  − ΝΨ º  + E − E  » ¨ α HS( ) « » ν  ΝΨ ¼ Q ¬ »¨ » ¨ ( )  ª  + ΝΨ º  − E − E  » ¨ α HS ν  ΝΨ ¼» Q ¬« »¨ »¨    » ¨¨ α HS( ) » Q »¨ ¨ ( )   »» ¨ α HS ¼ ¨©

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· ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸¸ ¹  

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P E  QQQν  ΝΨ

 

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α HS() =

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α HS() = Q « α HS() α HS( )

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β HS( N ) = ¦ α HS(L ) Y (HSL ) Y (HSN )  

 

L =

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ª  ( Φ − Ψ ) E + E

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(

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()

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© Qν  Ψ

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Qν  Ψ

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Qν  Ψ

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E+

» Q ¼

« HS ¬ Qν  Ψ'

 HS



» ¼ Q



 

 · ¸ ()  ( Φ − Ψ ) º α HS() ' HS ¹ α HS ª + «Q − DQG » Q ¬ Qν  Ψ' HS ¼ Q

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εHS(Y ) =

α HS() α () º ª + E HS »   « ν  − E − E ν  «¬ Q Q »¼

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σHS ¸

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σ

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·

HS 

σHS σHS

·

ϑHS() βHS() ¸

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( )

GHW : = −

QQQ E  PPν  ΝΦ

 

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(

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σ HS(P) *

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YHS =

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( =

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 ν  ΝΨ E





 

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Y HS( ) ∗

§ − + ΝΨ · ¨ ¸  + ΝΨ ¸ ∗ =¨ Y  ¨ −ΝΨ ¸  ¨ ¸  © ¹

∗

Y HS( ) ∗

∗

∗

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­° δ (Q ) = δ (Q ) =  IRUQ = !   ® () °¯ δ  = IRUQ = 

 

7KHQ WKH IXQFWLRQV IU  DQG Iθ  IROORZLQJ IURP HTV  DQG  FDQ EH GHWHUPLQHGDV IU(

Q)

­ µ ª (Q ) (Q ) Q + ( Q ) Q − ° ¬ω δ U − δ  U º¼ IRUQ = !  Q =  U ( ) ® ° δ ( )U IRUQ =  ¯ 

 

DQG Iθ(

Q)

­ δ(Q )U Q + + δ (Q )U Q − IRUQ = ! 

( U ) = °®

( ) °¯ δ U IRUQ =  



 

+HUHZLWKWKHVHSDUDWHG U − GHSHQGHQWSUREOHPLVVROYHG7KHVHSDUDWLRQFRQVWDQW µ ∈ ^ UHPDLQVXQNQRZQIRUWKHPRPHQW



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­ ω (Q )δ(Q )U Q + − δ (Q )U Q − ª −F(Q ) VLQ ( Qθ ) + F(Q ) FRV ( Qθ ) º IRUQ = !  ¬ ¼ ° XU ( Uθ ) = ®   () ° δ( )U F IRUQ =   ¯  (Q )

DQG

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­ δ(Q )U Q + + δ (Q )U Q − ªF(Q ) FRV ( Qθ ) + F(Q ) VLQ ( Qθ ) º IRUQ = !  ¬ ¼ ° Q Xθ( )( Uθ ) = ®  () F  ° δ( )U  IRUQ =   ¯

 

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D( ) = δ( )F( )  Q



Q

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D( ) = δ ( )F( )  E( ) = δ( )F( ) DQGE( ) = δ ( )F( )   Q

Q

Q

Q

Q

Q

Q

Q

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Q

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 ∞ ­ D( ) Q Q Q Q Q Q 5 + ¦ 5Q − ª D( )ω ( ) 5  − D( ) FRV ( Qθ ) − E( )ω ( ) 5  − E( ) VLQ ( Qθ ) º  ° ϕU (θ ) = ¬ ¼  ° Q =    ® () ∞ ° ϕ θ = E 5 + 5Q − ª E(Q ) 5  + E(Q ) FRV Qθ + D (Q ) 5  + D (Q ) VLQ Qθ º  ( )  ( )¼ θ( )   ¦ ¬   Q = ¯°

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­π IRUQ = P 

π

³ FRV ( Qθ ) FRV (Pθ ) Gθ = ®¯IRUQ ≠ P  

­π IRUQ = P 

π

³ VLQ ( Qθ ) VLQ (Pθ ) Gθ = ®¯IRUQ ≠ P  DQG 

 



π

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 π5

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 π5

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³ ϕθ (θ ) Gθ  

 



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π

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³ ϕθ (θ ) FRV (Pθ ) Gθ  

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ª 5 0D = « (P)  ¬«ω 5



§ (P ) · º § D(P) · −P ,  » ¨¨ (P) ¸¸ = 5 ¨¨ (P) ¸¸ IRUP = !  −¼» © D ¹ © , ¹

 

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ª 5 0E = « (P) «¬ω 5

 

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)5



=−

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5 = −

 (  − ν )

 ( − ν ) + P

5  

 

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P +

 

 



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D( ) = , ( ) 5 − DQGE( ) = , ( ) 5 −   







 

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­ ° XU ( Uθ ) = ° ° ° ° ® ° ° Xθ ( Uθ ) = ° ° ° ¯

 ∞ , ( ) ω (P)U P− U + ¦ (P )  P =  + ω

­ ª§   · (P ) (P ) º  ® «¨ U + (P) ¸ ,  −  − U ,  » FRV ( Pθ ) + ω ¹ ¼ ¯ ¬©

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½ ª§  · P P º + «¨ U  + (P) ¸ ,( ) +  − U  , ( ) » VLQ ( Pθ ) ¾  ω ¹ ¬© ¼ ¿ 

(

()

∞

P −

{(

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, U ª U  + ω (P) , (P) +  − U  ,(P) º FRV ( Pθ ) + U + ¦ (P ) ¬ ¼  ω  + P =

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P P P + ª U  + ω ( ) , ( ) −  − U  , ( ) º VLQ ( Pθ )  ¬ ¼

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P

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PRGH ϕU,(θ ) H = FRV (θ ) DQG ϕθ, (θ ) H = − VLQ (θ )  

 

PRGH ϕU,,(θ ) H = VLQ (θ ) DQG ϕθ,,(θ ) H = FRV (θ )  

 

PRGH ϕU,,,(θ ) H = DQG ϕθ,,,(θ ) H =  

 

PRGH ϕU,9 (θ ) H = FRV ( θ ) DQG ϕθ,9 (θ ) H = − VLQ ( θ )  

 



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PRGH ϕU9 (θ ) H = VLQ ( θ ) DQG ϕθ9 (θ ) H = FRV ( θ )  

 

PRGH ϕU9,(θ ) H = −DQG ϕθ9,(θ ) H =  

 

PRGH

PRGH

ϕU9,,(θ ) H

VLQ (θ ) + VLQ ( θ )

=

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=

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DQG

H

=

ϕθ9,,,(θ ) H

− FRV (θ ) + FRV ( θ ) 

=

VLQ (θ ) + VLQ ( θ ) 



 



 

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9,,,( P )



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H



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­− IRUP =  ° = ®  IRUP =  ° IRUP ≠  ¯  

9,,,( P )

,

H

­ IRUP =  ° = ® IRUP =  °IRUP ≠  ¯  

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­° XU,( Uθ ) H = FRV (θ )   PRGH ® , °¯ Xθ ( Uθ ) H = − VLQ (θ ) 

 

­° XU,,( Uθ ) H = VLQ (θ )   PRGH ® ,, °¯ Xθ ( Uθ ) H = FRV (θ ) 

 

­° XU,,,( Uθ ) H =    PRGH ® ,, °¯ Xθ ( Uθ ) H = U 

 

,9 °­ XU ( Uθ ) H = U FRV ( θ )   PRGH ® ,9 °¯ Xθ ( Uθ ) H = −U VLQ ( θ ) 

 

9 °­ XU ( Uθ ) H = U VLQ ( θ )   PRGH ® 9 °¯ Xθ ( Uθ ) H = U FRV ( θ ) 

 

9, °­ XU ( Uθ ) H = −U   PRGH ® 9, °¯ Xθ ( Uθ ) H =  

 

­ XU9,,( Uθ )  ª ( − ν ) U  +  º VLQ (θ ) + U  VLQ ( θ ) »  = « °  ¬«  − ν H ° ¼»  PRGH ® 9,,  º ° Xθ ( Uθ )  ª (  − ν ) U −  FRV (θ ) + U  FRV ( θ ) »  = «− °    ν H − ¬« ¼» ¯

 

­ XU9,,,( Uθ )  ª ( − ν ) U  +  º FRV (θ ) − U  FRV ( θ ) »  = « °   −  H ν «¬ ° ¼»  PRGH ® 9,,,    º ° Xθ ( U θ )  ª (  − ν ) U −  VLQ (θ ) + U  VLQ ( θ ) »  = « °    ν H − « »¼ ¬ ¯

 

LQ ZKLFK IRU PRGH DQG PRGH HT  KDV EHHQ VXEVWLWXWHG IRU ω ( )  ZKLFK UHDGVIRU P =   P

ω () =

 − ν   − ν =   + ν   − ν

 

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εUU = X U U  εθθ =

X U + Xθ θ X − Xθ Xθ  U  ε]] = DQGεUθ = Uθ +  U  U

 

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ε UU, ( Uθ ) H 5

ε UU,, ( Uθ ) H 5

ε UU,,,( Uθ ) H 5

ε UU,9( Uθ ) H 5

=

=

=

, εθθ ( Uθ )

H 5 ,, εθθ ( Uθ )

H 5 ,,, εθθ ( Uθ )

H 5

= FRV ( θ ) 

=

=

=

ε ,]] ( Uθ ) H 5

ε ,,]] ( Uθ ) H 5 θ ) ε ,,, ]] ( U H 5 ,9 εθθ ( Uθ )

H 5

=

=

=

ε U,θ ( Uθ ) H 5

ε U,,θ ( Uθ ) H 5

ε U,,,θ ( Uθ ) H 5

=  

 

=  

 

=  

 

= − FRV ( θ ) 

θ ) ε ,9 ]] ( U H 5

= 

ε U,9θ ( Uθ ) H 5

= − VLQ ( θ ) 

 

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ε UU9 ( Uθ ) H 5

9 εθθ ( Uθ )

= VLQ ( θ ) 

 9 ε ]] ( Uθ )

= − VLQ ( θ ) 

H 5

H 5

= 

ε U9θ ( Uθ ) H 5

= FRV ( θ ) 



 

PRGH

ε ( Uθ ) 9, UU

H 5

εθθ ( Uθ ) 9,

=

H 5

= −

ε ( Uθ ) 9, ]]

H 5

= 

ε ( Uθ ) 9, Uθ

H 5

=  

 

9,, ª  − ν º εθθ ( Uθ )  = U « VLQ (θ ) + VLQ ( θ ) »  = U ª¬VLQ (θ ) − VLQ ( θ ) º¼  H 5 H 5 ¬  − ν ¼   θ ) ª FRV (θ ) º ε]]9,,( Uθ ) εU9,, θ (U  = = U « − + FRV ( θ ) »  H 5 H 5 ¬  − ν ¼

PRGH

εUU9,,( Uθ )

9,,, ª  − ν º εθθ ( Uθ )  = U « FRV (θ ) − FRV ( θ ) »  = U ¬ªFRV (θ ) + FRV ( θ ) ¼º  H 5 H 5 ¬  − ν ¼   θ ) ª VLQ (θ ) º ε]]9,,,( Uθ ) εU9,, θ (U = = U « + VLQ ( θ ) »   H 5 H 5 ¬  − ν ¼

PRGH

εUU9,,,( Uθ )

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σ UU * 

= ( + ν  ) εUU + ν εθθ 

σθθ *

= ν εUU + ( + ν  ) εθθ 

σ ]] *

= ν  ( εUU + εθθ ) DQG

σ Uθ

= εUθ  *  

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=

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=

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* H 5

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=  

 

=  

 

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θ ) σ ,9 ]] ( U * H 5

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 PRGH

 

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 PRGH



= − VLQ ( θ ) 

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σ ( Uθ ) 9, UU

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=

* H 5

=

−   − ν

σ

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9, ]]

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9, Uθ

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ª VLQ (θ ) º σθθ9,,( Uθ ) ª VLQ (θ ) º = U « + VLQ ( θ ) »  = U « − VLQ ( θ ) »  * H 5 * H 5 ¬  − ν ¼ ¬  − ν ¼    9,, 9,, ª FRV (θ ) º σ ]] ( Uθ ) ν U VLQ (θ ) σ Uθ ( Uθ ) = = U « − + FRV ( θ ) »    − ν * H 5  − ν * H 5   ¬ ¼

PRGH

σ UU9,,( Uθ )

ª FRV (θ ) º σθθ9,,,( Uθ ) ª FRV (θ ) º = U « − FRV ( θ ) »  = U « + FRV ( θ ) »   * H 5  −    −  * H 5 ν ν ¬ ¼ ¬ ¼     ª º  VLQ σ ]]9,,,( Uθ ) ν U FRV (θ ) σ U9,,, θ θ U ( ) ( ) θ   = = U « + VLQ ( θ ) »  * H 5  − ν * H 5 ¬  − ν ¼

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σ UU9,,,( Uθ )

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ε =

εUU + εθθ 



§ ε − ε · ± ¨ UU θθ ¸ + εUθ   © ¹

ε =  DQG WDQ ( ωε ) =

εUθ   ε UU − εθθ

 

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σ =

σ UU + σθθ 



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=  

 

=  

 

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= 

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=  

 

ε9,

 

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H 5

 § VLQ (θ ) ·  − ν §  − ν ·  U FRV (θ ) ± U  ¨ + VLQ ( θ ) ¸  ¸ FRV (θ ) + ¨  − ν ©  − ν ¹ ©  − ν ¹ 

=



ε9,,, H 5

= 

 

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* H 5

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=  

 

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* H 5

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= 

=

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=−

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=  

 

   − ν

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± U

 

σ 9,, * H 5

=

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* H 5

=

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± U

σ 9,,, * H 5

=

ν U FRV (θ )  − ν



 

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ε(Y ) = ε + ε DQG ε(G ) =



(ε



 − ε ) + ε + ε





 

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σ (P) =

σ + σ  + σ  

DQGσ (

G)

=

(σ



− σ  ) + (σ − σ  ) + (σ  − σ  ) 









 

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σ + σ  + σ  ) ( ε + ε ) 



( σ − σ  − σ ) ε + ( σ − σ − σ ) ε   GHY P Y DQG: ( ) = σ LMεLM − σ ( )ε ( ) =  

(



 

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XU U ( θ ) < ∞ 

Xθ ( θ ) < ∞ DQG Xθ  U ( θ ) < ∞  

 

3HULRGLFLW\FRQGLWLRQVDFFRUGLQJWRHTV DQG 

XU ( U ) = XU ( Uπ ) DQGXUθ ( U ) = XUθ ( Uπ )  

 

Xθ ( U ) = Xθ ( Uπ ) DQGXθ θ ( U ) = Xθ θ ( Uπ )  

 

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$QGIRUWKHGLVSODFHPHQWVLQWDQJHQWLDOGLUHFWLRQOHW Xθ ( Uθ ) =

∞  F( U ) + ¦ ª¬FP( U ) FRV ( Pθ ) + GP( U ) VLQ ( Pθ ) º¼    P =

 

,Q ERWK VHULHV WKH LQGLFHV FRYHU WKH UDQJH Q  P = !  1RWH WKDW WKH FRHIILFLHQWV RI ERWK )RXULHU VHULHV DQ ( U )  EQ ( U )  F P ( U )  DQG G P ( U )  DUH QRW FRQVWDQW EXW IXQFWLRQV RI WKH LQGHSHQGHQW YDULDEOH U  )XUWKHUPRUH LW LV QRWHG WKDWIRU Q  P =  RQO\WKHFRHIILFLHQWV D DQG F  QHHGWREHFRQVLGHUHGVHSDUDWHO\ DV WKH\ DUH PXOWLSOLHG E\ FRV ( Qθ ) = FRV ( Pθ ) =   IRU Q  P =   7KH FRHIILFLHQWV E  DQG G   GR QRW RFFXU LQ HTV DQG  DV WKH\ DUH PXOWLSOLHG E\ VLQ ( Qθ ) = VLQ ( Pθ ) =  IRU Q  P =   ,Q WKH IROORZLQJ WKH DVVXPHG VROXWLRQ LQ HTV DQG  LV VXEVWLWXWHG LQWR WKHILUVWHTXDWLRQRIWKHV\VWHPLQHT DQGHODERUDWHGLQGHWDLOWRGHULYHWZR HTXDWLRQVIRUWKHGHWHUPLQDWLRQRIWKHIRXUXQNQRZQFRHIILFLHQWV DQ  EQ  F P DQG G P  IRU Q  P = !  $IWHUZDUGV IRU WKH VHFRQG HTXDWLRQ RI WKH V\VWHP LQ



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HT WKHVDPHSURFHGXUHFDQEHIROORZHGZKLOHUHSODFLQJ αL ·VE\ βL ·VOHDGLQJ WR WZR DGGLWLRQDO HTXDWLRQV +RZHYHU WKH HODERUDWLRQ RI WKH VHFRQG HTXDWLRQ LV QRWJLYHQLQGHWDLO 7KH JHQHUDO VROXWLRQ JLYHQ LQ HTV DQG  LV VXEVWLWXWHG LQWR WKH ILUVW HTXDWLRQRIHT ZKLFKOHDGVWR ∞ ∞ D′′ + αU  ¦ ¬ªDQ′′FRV ( Qθ ) + EQ′′VLQ ( Qθ ) ¼º − α  ¦ Q ¬ªDQ FRV ( Qθ ) + EQVLQ ( Qθ ) ¼º +  Q = Q = ∞ ∞ D′ − α U ¦ Q ª¬DQ′ VLQ ( Qθ ) − EQ′ FRV ( Qθ ) º¼ + αU + αU ¦ ª¬DQ′ FRV ( Qθ ) + EQ′ VLQ ( Qθ ) º¼ +  Q = Q =

αU 

∞

− α  ¦ Q ª¬DQVLQ ( Qθ ) − EQ FRV ( Qθ ) º¼ + α Q =

∞ D + α ¦ ª¬DQ FRV ( Qθ ) + EQVLQ ( Qθ ) º¼ +  Q =

∞ ∞ F′′ + α  U  ¦ ª¬FP′′ FRV ( Pθ ) + GP′′ VLQ ( Pθ ) º¼ − α  ¦ P ª¬FP FRV ( Pθ ) + GPVLQ ( Pθ ) º¼ +  P = P = ∞ ∞ F′ − α U ¦ P ¬ªFP′ VLQ ( Pθ ) − GP′ FRV ( Pθ ) ¼º + α U + α U ¦ ¬ªFP′ FRV ( Pθ ) + GP′ VLQ ( Pθ ) ¼º +  P = P =

+ α U 

∞

− α ¦ P ¬ªFPVLQ ( Pθ ) − GP FRV ( Pθ ) ¼º + α P =



∞ F + α ¦ ¬ªFP FRV ( Pθ ) + GPVLQ ( Pθ ) ¼º =   P =  

1RWH WKDW WKH FRHIILFLHQWV α L   L =   DUH SHULRGLF IXQFWLRQV RI WKH LQGHSHQGHQW YDULDEOH θ  LH α L = α L (θ )  DV LQGLFDWHG LQ HT  GHILQHG IRU WKH PDWHULDO PRGHO RI 0RKU²&RXORPE HODVWR²SODVWLFLW\ LQ WKH FLUFXODU GRPDLQ LQ HTV DQG )XUWKHUPRUHQRWHWKDWGXHWRWKHIDFWWKDW DQ  EQ  FP DQG GP DUHIXQFWLRQVGHSHQGHQWRQ U WKHLUGHULYDWLYHVZLWKUHVSHFWWRWKLVYDULDEOH KDYHWREHWDNHQLQWRDFFRXQW7KHILUVWDQGVHFRQGRUGHURUGLQDU\GHULYDWLYHVRI WKH FRHIILFLHQWV ZLWK UHVSHFW WR U  DUH GHILQHG IRU HJ DQ  DV DQ′ = DQ U  DQG DQ′′ = DQ UU DQGDQDORJRXVIRU EQ  FP DQG GP  $IWHUUHDUUDQJLQJHT DQGVRUWLQJIRUVLQHDQGFRVLQHIXQFWLRQVLWIROORZV ∞ ∞  Q P I ( Uθ ) + ¦ ª I( )( Uθ ) FRV ( Qθ ) º + ¦ ª I( )( Uθ ) FRV ( Pθ ) º + ¬ ¼ ¬ ¼  Q = P = ∞

∞



 

Q P + ¦ ª I( )( Uθ ) VLQ ( Qθ ) º + ¦ ª I( )( Uθ ) VLQ ( Pθ ) º =  ¬ ¼ P= ¬ ¼ Q =

LQZKLFKWKHIXQFWLRQV I  I( )  I( )  I( ) DQG I( ) DUHLQWURGXFHGDQGGHILQHGDV Q

P

Q

Q

I ( U θ ) = αD′′U  + αD′ U + αD + α F ′′U  + α F ′ U + αF   I (

Q)

( U θ ) = αDQ′′U  + αDQ′ U − Q α DQ + αDQ + QαEQ′ U + Qα EQ  P I ( )( U θ ) = α F P′′ U  + α F P′ U − P α F P + αF P + Pα G P′ U + PαG P   Q I ( )( U θ ) = αEQ′′U  + αEQ′ U − Q α EQ + αEQ − Qα DQ′ U − Qα DQ  P I ( )( U θ ) = α G P′′ U  + α G P′ U − P α G P + αG P − Pα F P′ U − PαF P 

 

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α L (θ ) = α L() + α L() FRV ( θ ) + α L() VLQ ( θ ) + α L() FRV ( θ ) + α L( ) VLQ ( θ ) DQG βL (θ ) = βL() + βL() FRV ( θ ) + βL() VLQ ( θ ) + βL() FRV ( θ ) + βL( ) VLQ ( θ ) 



 

LQ ZKLFK α L( )  DQG βL( )   L = !   M =   DUH FRQVWDQW FRHIILFLHQWV GHSHQGLQJ RQO\ RQ WKH PDWHULDO SDUDPHWHUV ZKLFK DUH FRQVWDQW LQVLGH WKH M M GRPDLQ ,Q $SSHQGL[$ RI WKLV FKDSWHU WKH FRHIILFLHQWV α L( )  DQG βL( )  DUH HODERUDWHG DQG JLYHQ LQ GHWDLO 1RWH WKDW WKH GHILQLWLRQ LQ HT  LV DQ H[DFW UHSUHVHQWDWLRQRIWKHFRHIILFLHQWV α L (θ ) DQG βL (θ )  6XEVWLWXWLRQRIHT LQWR HT OHDGVIRUWKHILUVWIXQFWLRQIRUZKLFK Q P =  WR M

M

I ( Uθ ) = ªα( ) + α( ) FRV ( θ ) + α( ) VLQ ( θ ) + α( ) FRV ( θ ) + α( ) VLQ ( θ ) º D′′U  + ¬ ¼ 









+ ªα( ) + α( ) FRV ( θ ) + α( ) VLQ ( θ ) + α( ) FRV ( θ ) + α( ) VLQ ( θ ) º D′ U + ¬ ¼ 









( ) () ( ) ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º D + ¬ ¼ 









+ ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º F′′U  + ¬ ¼ 











 

+ ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º F′ U + ¬ ¼ 









( ) () ( ) ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º F  ¬ ¼ 









)RUWKHVHFRQGIXQFWLRQIRUZKLFK Q ≥  LWIROORZV I(

Q)

( Uθ ) = ª¬α() + α() FRV ( θ ) + α() VLQ ( θ ) + α() FRV ( θ ) + α() VLQ ( θ )º¼ DQ′′U  +

     − ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º Q DQ + ¬ ¼      + ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º QEQ′ U + ¬ ¼      + ªα( ) + α( ) FRV ( θ ) + α( ) VLQ ( θ ) + α( ) FRV ( θ ) + α( ) VLQ ( θ ) º DQ′ U + ¬ ¼

  

     + ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º QEQ + ¬ ¼

() () () () ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º DQ  ¬ ¼

 ( M)

( M)

M =  WKHFRHIILFLHQWV α L DQG β L DUHQRWHTXDOWR]HUR 7KHQLWFDQEHVHHQWKDWWKHFRHIILFLHQWV α L DQG β L DUHLQGHSHQGHQWRIWKHFRRUGLQDWHGLUHFWLRQ θ 

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)RUWKHWKLUGIXQFWLRQIRUZKLFK P ≥  LWIROORZV I(

P)

( Uθ ) = ª¬α () + α () FRV ( θ ) + α () VLQ ( θ ) + α () FRV ( θ ) + α () VLQ ( θ )º¼ FP′′ U  +

     − ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º PFP + ¬ ¼      + ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º PGP′ U + ¬ ¼      + ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º FP′ U + ¬ ¼ () () () () ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º PGP + ¬ ¼ () () () () ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º FP  ¬ ¼

  

)RUWKHIRXUWKIXQFWLRQIRUZKLFK Q ≥  LWIROORZV I(

Q)

( Uθ ) = ª¬α() + α() FRV ( θ ) + α() VLQ ( θ ) + α() FRV ( θ ) + α() VLQ ( θ )º¼ EQ′′U  +

− ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º QEQ + ¬ ¼ 









− ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º QDQ′ U + ¬ ¼ 









+ ªα( ) + α( ) FRV ( θ ) + α( ) VLQ ( θ ) + α( ) FRV ( θ ) + α( ) VLQ ( θ ) º EQ′ U + ¬ ¼ 









  

− ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º QDQ + ¬ ¼ 









( ) () ( ) ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º EQ  ¬ ¼ 









)RUWKHILIWKIXQFWLRQIRUZKLFK P ≥  LWIROORZV  I(

P)

( Uθ ) = ª¬α () + α () FRV ( θ ) + α () VLQ ( θ ) + α () FRV ( θ ) + α () VLQ ( θ )º¼ GP′′ U  +

     − ªα ( ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) + α ( ) FRV ( θ ) + α ( ) VLQ ( θ ) º PGP + ¬ ¼ () () () ( ) ( ) ª − α  + α  FRV ( θ ) + α  VLQ ( θ ) + α  FRV ( θ ) + α  VLQ ( θ ) º PFP′ U + ¬ ¼ () () () ( ) ( ) ª + α  + α  FRV ( θ ) + α  VLQ ( θ ) + α  FRV ( θ ) + α  VLQ ( θ ) º GP′ U + ¬ ¼ () () () ( ) ( ) ª º − α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) PFP + ¬ ¼ () () () ( ) ( ) ª + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º GP  ¬ ¼

  

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= 

• FDVHN

= ! DQGPXOWLSOLFDWLRQRIHT E\ FRV ( N θ )  

• FDVHN

= ! DQGPXOWLSOLFDWLRQRIHT E\ VLQ ( N θ ) 

 

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³ VLQ (Vθ ) VLQ ( Qθ ) VLQ ( N θ )G θ =  π

=

³

− VLQ ª¬( V − Q − N ) θ º¼ + VLQ ª¬( V − Q + N ) θ º¼ + VLQ ª¬( V + Q − N ) θ º¼ − VLQ ª¬( V + Q + N ) θ º¼ 



Gθ = 

=  IRU [Q = ! ∧ N = ! ∧ V =  ]  π

³ VLQ (Vθ ) FRV ( Qθ ) FRV ( N θ )Gθ =  π

=

³

VLQ ª¬( V − Q − N ) θ º¼ + VLQ ª¬( V − Q + N ) θ º¼ + VLQ ª¬( V + Q − N ) θ º¼ + VLQ ª¬( V + Q + N ) θ º¼ 



Gθ = 

= IRU [Q = ! ∧ N = ! ∧ V =  ]  π

³ FRV (Vθ ) VLQ ( Qθ ) FRV ( N θ )Gθ =  π

=

³

− VLQ ª¬( V − Q − N ) θ º¼ − VLQ ª¬( V − Q + N )θ º¼ + VLQ ª¬( V + Q − N ) θ º¼ + VLQ ª¬( V + Q + N ) θ º¼ 



Gθ = 

= IRU [Q = ! ∧ N = ! ∧ V =  ]  π

³ FRV (Vθ ) FRV ( Qθ ) VLQ ( N θ )Gθ =  π

=

³ 

− VLQ ¬ª( V − Q − N ) θ ¼º + VLQ ¬ª( V − Q + N ) θ ¼º − VLQ ¬ª( V + Q − N ) θ ¼º + VLQ ¬ª( V + Q + N ) θ ¼º 

= IRU [Q = ! ∧ N = ! ∧ V =  ] 

Gθ = 



&+$37(5

π

³ FRV (Vθ ) VLQ ( Qθ ) VLQ ( N θ )Gθ =  π

=

³

− FRV ª¬( V − Q − N ) θ º¼ + FRV ª¬( V − Q + N ) θ º¼ + FRV ª¬( V + Q − N )θ º¼ − FRV ª¬( V + Q + N ) θ º¼ 



­π  ° °π  =® °  °  ¯

Gθ = 

IRU [V =  ∧ Q = N = !] 

IRU [Q = ±N ± V ] ∧ [Q = ! ∧ N = ! ∧ V =  ] 

IRU [Q ≠ ±N ± V ] ∧ [Q = ! ∧ N = ! ∧ V =  ]  IRU ª¬( Q =  ∧ N = !) ∨ ( N =  ∧ Q = !) º¼ ∧ V =  

π

³ VLQ (Vθ ) FRV ( Qθ ) VLQ ( N θ )Gθ =  π

=

³

FRV ¬ª( V − Q − N ) θ ¼º − FRV ¬ª( V − Q + N ) θ ¼º + FRV ¬ª( V + Q − N ) θ ¼º − FRV ¬ª( V + Q + N ) θ ¼º 



­π  ° °π  =® °  °  ¯

Gθ = 

IRU [Q =  ∧ N = V =  ]  IRU [Q = ±N ± V ] ∧ [Q = ! ∧ N = ! ∧ V =  ] 

IRU [Q ≠ ± N ± V ] ∧ [Q = ! ∧ N = ! ∧ V =  ]  IRU ª¬( N =  ∧ V =  ) ∨ ( V =  ∧ N = !) º¼ ∧ Q = !

π

³ VLQ (Vθ ) VLQ ( Qθ ) FRV ( N θ )Gθ =  π

=

³

FRV ¬ª( V − Q − N ) θ ¼º + FRV ¬ª( V − Q + N ) θ ¼º − FRV ¬ª( V + Q − N ) θ ¼º − FRV ¬ª( V + Q + N ) θ ¼º 



­π  ° °π  =® °  ° ¯ 

Gθ = 

IRU [ N =  ∧ Q = V =  ] 

IRU [Q = ±N ± V ] ∧ [Q = ! ∧ N = ! ∧ V =  ]  IRU [Q ≠ ± N ± V ] ∧ [Q = ! ∧ N = ! ∧ V =  ]  IRU ¬ª( Q =  ∧ V =  ) ∨ ( V =  ∧ Q = !) ¼º ∧ N = ! 

π

³ FRV (Vθ ) FRV ( Qθ ) FRV ( N θ )Gθ =  π

=

³

FRV ª¬( V − Q − N ) θ º¼ + FRV ª¬( V − Q + N ) θ º¼ + FRV ª¬( V + Q − N ) θ º¼ + FRV ª¬( V + Q + N ) θ º¼



­π  ° °π  =® °  °  ¯ 



Gθ = 

IRU [Q =  ∧ N = V =  ] ∨ [ N =  ∧ Q = V =  ] ∨ [V =  ∧ Q = N = !]  IRU [Q = ± N ± V ] ∧ [Q = ! ∧ N = ! ∧ V =  ]  IRU [Q ≠ ±N ± V ] ∧ [Q = ! ∧ N =  ! ∧ V =  ]  IRU [Q =  ∧ N ≠ V ] ∨ [ N =  ∧ Q ≠ V ] ∨ [V =  ∧ Q ≠ N ] 

 

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)RUWKHHYDOXDWLRQRIWKHSUHYLRXVLQWHJUDOVUHFDOOWKDWWKHLQWHJUDWLRQRIWKHVLQH IXQFWLRQZLWKDUJXPHQW Qθ RQWKHLQWHUYDO [π ] DOZD\VOHDGVWRDUHVXOWHTXDO WR]HUR7KHLQWHJUDWLRQRIWKHFRVLQHIXQFWLRQZLWKDUJXPHQW Qθ RQWKHLQWHUYDO [π ] RQO\OHDGVIRU Q =  WRDQRQ²]HURUHVXOWLH π

π





­π  IRUQ =   IRUQ = !

³ VLQ ( Qθ )G θ = IRUQ = ! DQG ³ FRV ( Qθ )Gθ = ®¯ 



 

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&DVHN 

,Q WKLV VHFWLRQ WKH ILUVW FDVH DV LQWURGXFHG LQ HT  ZLOO EH WUHDWHG 0XOWLSOLFDWLRQ RI HT  E\ FRV ( N θ ) =   IRU IL[HG N =   DQG LQWHJUDWLRQ ZLWK UHVSHFWWR θ IURP  WR π JLYHV

 

π

π ∞

π ∞

( ) ( ) ³ I ( Uθ ) Gθ + ³ ¦ ª¬ I ( Uθ ) FRV ( Qθ )º¼ Gθ + ³ ¦ ª¬ I ( Uθ ) FRV (Pθ )º¼ Gθ + Q



P





 Q =

 π ∞

(Q) 

+ ³ ¦ ªI ¬ Q =

 P =

( Uθ ) VLQ ( Qθ )¼º Gθ +



π ∞

(P ) 

³ P¦= ¬ª I



 

( Uθ ) VLQ (Pθ )¼º Gθ = 



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( ) ³ I ( Uθ ) Gθ = π (α D′′U 









)

( ) ( ) + α( ) D′ U + α D + α ( )F′′U  + α ( )F′ U + α F   











π ∞

( ) () ( ) ³ ¦ ª¬ I ( Uθ ) FRV ( Qθ )º¼ Gθ = π (α D′′ + α D′′ ) U

) + π ( α ( )E′ + α ( )E′ ) U + π (α ( ) D′ + α ( ) D′ ) U + π ( α ( )E + α ( )E ) + π (α ( ) D  

Q













(

− π α ( ) D + α ( ) D +

 Q =

 

π ∞



 

 



 



 



( ) () ( ) ³ ¦ ª¬ I ( Uθ ) FRV (Pθ )º¼ Gθ = π (α F′′ + α F′′ ) U  

P



 P =

(

()

( )

)

(

()

 



( )





)



(



 



 



)



( )  + α D  

)

− π α ( )F + α ( )F +

(

()





()

) (

()

()

)

+ π α  G′ + α  G′ U + π α  F′ + α  F′ U + π α G + α G + π α F + α F 





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π ∞

( ) () ( ) ³ ¦ ª¬ I ( Uθ ) VLQ ( Qθ )º¼ Gθ = π (α E′′ + α E′′ ) U  

Q



 Q =

(

()

)

()

(







()





(

)

()

)

− π α ( )E + α ( )E + 

(



()

) (

( )

()

()

)



− π α  D′ + α  D′ U + π α E′ + α E′ U − π α  D + α  D + π α E + α E  π ∞

( ) () ( ) ³ ¦ ª¬ I ( Uθ ) VLQ (Pθ )º¼ Gθ = π (α G′′ + α G′′ ) U  

P



 P =

(

)

(

 







)

(

)

− π α ( )G + α ( )G +

(





) (

)

() ( ) () ( ) − π α ( )F′ + α ( )F′ U + π α ( )G′ + α ( )G′ U − π α F + α F + π α G + α G  

















 

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(

)

(

)

ªα()D′′ + α()D′′ + α()D′′ º U  + ªα( )D′ + −α () + α() D′ + −α () + α() D′ º U + ¬ ¼ ¬ ¼

(

)

(

)

    () () ( ) + ªα D + −α ( ) − α ( ) + α D + −α ( ) − α ( ) + α D º + ¬ ¼

) E ′ ¼º U + + ª( −α ( ) + α ( ) + α ( ) ) E + ( −α ( ) + α ( ) + α ( ) ) E º + ¬ ¼ ( ) ( ) ( ) ( ) ( ) ( ) + ªα F ′′ + α F ′′ + α F ′′ º U + ªα F ′ + ( −α + α ) F ′ + ( −α ( ) + α ( ) ) F ′ º U + ¬ ¼ ¬ ¼ ( ) ( ) () () ( ) ( ) ( ) ª º + α F + ( −α − α + α ) F + ( −α − α + α ) F + ¬ ¼ ( ) ( ) ( ) ( ) ( ) ( ) ª º + ªα G ′′ + α G ′′º U + ( α + α ) G ′ + ( α + α ) G ′ U + ¬ ¼ ¬ ¼ ( ) ( ) ( ) ( ) ( ) ( ) + ª( −α + α + α )G + ( −α + α + α )G º =  ¬ ¼ ()  

(

( )  

+ ªα E ′′ + α E ′′º U  + ª α ¬ ¼ ¬  

 

 

 



 

 

 

 

 

 



 

) E ′ + (α 

 

 





+α

() 





 

 



 





 



 

() 



 

 



+α

() 



 



 

 



 

() 

 



 

 



 

 



 

 

 



 





 

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&DVH0XOWLSOLFDWLRQE\FRVNθ DQGN «



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π

³ I ( Uθ ) FRV ( Nθ ) Gθ + 



+

π ∞

π ∞

( ) ( ) ³ ¦ ª¬ I ( Uθ ) FRV ( Qθ ) FRV ( Nθ )º¼ Gθ + ³ ¦ ª¬ I ( Uθ ) FRV (Pθ ) FRV ( Nθ )º¼ Gθ + Q



 Q =

+

P



 P =

π ∞

π ∞

( ) ( ) ³ ¦ ª¬ I ( Uθ ) VLQ ( Qθ ) FRV ( Nθ )º¼ Gθ + ³ ¦ ª¬ I ( Uθ ) VLQ (Pθ ) FRV ( Nθ )º¼ Gθ =  Q

P





 Q =

 P =



 

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³ I ( Uθ ) FRV ( Nθ ) Gθ = 

 π

=

³ {ª¬α

() 

FRV ( θ ) + α( ) FRV ( θ ) º D′′U  + ªα( ) FRV ( θ ) + α( ) FRV ( θ ) º D′ U + ¼ ¬ ¼ 







  () () + ªα FRV ( θ ) + α FRV ( θ ) º D + ªα ( ) FRV ( θ ) + α ( ) FRV ( θ ) º F′′U  + ¬ ¼ ¬ ¼

}

  () ( ) + ªα ( ) FRV ( θ ) + α ( ) FRV ( θ ) º F′ U + F ªα FRV ( θ ) + α FRV ( θ ) º FRV ( Nθ ) Gθ = ¬ ¼ ¬ ¼

()

()

()

()

()

()

­° α U  D′′ + α UD′ + α D + α  U F′′ + α  UF′ + α F IRUN =  =π ®  ( )  ( ) ( ) ()  ( ) ( ) °¯ α U D′′ + α UD′ + α D + α  U F′′ + α  UF′ + α F IRUN =  

(

)

() () = π α( )U D′′ + α( )UD′ + α D + α ( )U F′′ + α ( )UF′ + α F δ N + 

(











)

( ) ( ) + π α( )U  D′′ + α( )UD′ + α F δ N  D + α ( )U F′′ + α ( )UF′ + α















 

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)RU WKH VHFRQG LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJ WKH IXQFWLRQ Q I ( )( U θ ) IROORZLQJIURPHT  π ∞

( ) ³ ¦ ª¬ I ( Uθ ) FRV ( Qθ ) FRV ( Nθ )º¼ Gθ = Q



 Q =

=

π ∞

³ ¦{ª¬α

( )



 Q =

+ α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º DQ′′U  + ¼ 







− ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QDQ + ¬ ¼ 









     + ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QEQ′ U + ¬ ¼ ( ) () ( ) ( ) () ª º + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) DQ′ U + ¬ ¼      + ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QEQ + ¬ ¼

}

( ) () ( ) ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º DQ FRV ( Qθ ) FRV ( Nθ ) Gθ = ¬ ¼ 









  π π       = ªα( ) DN′′ +α( ) D±′′N± +α( ) D±′′N± º U − ªα( ) NDN +α( )( ±N ± ) D± N± +α( )( ±N ±  ) D± N± º + ¬ ¼

¬



+

¼

π ª () ′ () π     α NEN +α ( ±N ± ) E±′ N± +α( )( ±N ±  ) E±′ N± º U + ªα( ) DN′ +α( )D±′ N± +α( ) D±′ N± º U + ¬  ¼ ¬ ¼ 



π π    ( ) () ( ) DN +α D± N± +α D± N± º  + ªα( ) NEN +α( )( ±N ± ) E± N± +α( )( ±N ±  ) E± N± º + ªα ¬



¼

¬

¼  

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)RU WKH WKLUG LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJ WKH IXQFWLRQ P I ( )( U θ ) IROORZLQJIURPHT  π ∞

( ) ³ ¦ ª¬ I ( Uθ ) FRV (Pθ ) FRV ( Nθ )º¼ Gθ = P



 P=

=

π ∞

³ ¦ {¬ªα  P=

() 

+ α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º FP′′ U  + ¼ 







− ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º PFP + ¬ ¼ 









     + ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º PGP′ U + ¬ ¼ ( ) () ( ) ( ) () ª º + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) FP′ U + ¬ ¼

( ) () ( ) () () + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º PGP + ¬ ¼

}

( ) () ( ) ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º FP FRV ( Pθ ) FRV ( Nθ ) Gθ = ¬ ¼ 









  π π       = ªα( )FN′′ +α( )F±′′N± +α( )F±′′N± º U − ªα( ) NFN +α( )( ±N ± ) F± N± +α( )( ±N ±  ) F± N± º + ¬ ¼

¬



+

¼

π ª () ′ () π     α NGN +α ( ±N ± ) G±′ N± +α( )( ±N ±  ) G±′ N± º U + ªα( )FN′ +α( )F±′ N± +α( )F±′ N± º U + ¬  ¼ ¬ ¼ 



π () () NGN +α + ªα ( ±N ± ) G±N± +α()( ±N ±  ) G±N± º + π ªα()FN +α()F±N± +α()F±N± º  ¬



¼

¬

¼  

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)RU WKH IRXUWK LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJ WKH IXQFWLRQ Q I ( )( U θ ) IROORZLQJIURPHT  π ∞

( ) ³ ¦ ª¬I ( U θ ) VLQ ( Qθ ) FRV ( N θ )º¼ Gθ = Q



 Q =

=

π ∞

³ ¦ {¬ªα

()



 P =

+ α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º EQ′′U  + ¼ 







− ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º Q EQ + ¬ ¼ 









     − ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QDQ′ U + ¬ ¼ ( ) () ( ) ( ) () ª + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º EQ′ U + ¬ ¼      − ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QDQ + ¬ ¼

}

( ) () ( ) ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º EQ VLQ ( Qθ ) FRV ( N θ ) Gθ = ¬ ¼ 









  π π     = ªα( )E±′′N ± + α( )E±′′N ± º U  − ªα( ) ( ±N ± ) E± N ± + α( ) ( ±N ±  ) E± N ± º + ¬ ¼

¬



−

¼

π ª ( ) π α ±N ± ) D±′ N ± + α() ( ±N ±  ) D±′ N ± ¼º U + ¬ªα()E±′ N ± + α()E±′ N ± ¼º U + ¬  ( 



π π ( )   () E± N ± + α E± N ± º  − ªα( ) ( ±N ± ) D± N ± + α( ) ( ±N ±  ) D± N ± º + ªα ¬ ¼ ¬ ¼ 







 

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)RU WKH ILIWK LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJ WKH IXQFWLRQ P I ( )( U θ ) IROORZLQJIURPHT  π ∞

( ) ³ ¦ ª¬ I ( Uθ ) VLQ (Pθ ) FRV ( Nθ )º¼ Gθ = P



 P=

=

π ∞

³ ¦ {¬ªα  P=

() 

+ α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º GP′′ U  + ¼ 







     − ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º PGP + ¬ ¼ () () ( ) ( ) () ª − α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º PFP′ U + ¬ ¼ ( ) () ( ) ( ) () ª + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º GP′ U + ¬ ¼

() () ( ) ( ) () FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º PFP + − ªα + α ¬ ¼

}

( ) () ( ) ( ) ( ) FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º GP VLQ ( Pθ ) FRV ( Nθ ) Gθ = + ªα + α ¬ ¼ 

=









  π ª () ′′ π    α G + α( )G±′′N± º U  − ªα( ) ( ±N ± ) G± N± + α( ) ( ±N ±  ) G± N± º + ¬  ± N ± ¼

¬



¼

π π     − ªα( ) ( ±N ± ) F±′ N± + α( ) ( ±N ±  ) F±′ N± º U + ªα( )G±′ N± + α( )G±′ N± º U + ¬ ¼ ¬ ¼ 



π ( ) − ªα ( ±N ± ) F±N± + α() ( ±N ±  ) F± N± º¼ + 𠪬α()G±N± + α()G±N± º¼  ¬ 





 

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&+$37(5

6XEVWLWXWLQJ WKH HYDOXDWHG LQWHJUDOV RI HTV WR  LQWR HT  LW IROORZV DIWHU GLYLGLQJ E\ π   DQG VRUWLQJ IRU FRHIILFLHQWV DL  EL  F L  DQG G L  IRU FDVHLQZKLFK N = ! 

(α ( )U D′′ + α ( )UD′ + α ( )D + α ( )U F′′ + α ( )UF′ + α ( )F ) δ + + (α ( )U D′′ + α ( )UD′ + α ( ) D + α ( )U F′′ + α ( )UF′ + α ( )F ) δ  

 







 



 







()

 





()

 

 





 



( )

 





 





N

 





+

N

+ ªα DN′′ + α D±′′N ± + α D±′′N ± º U + ¬ ¼

(



+ ªα( ) DN′ + α( ) − ( ±N ±  ) α ( ¬ 

(

( ) − Nα ( +  α

(

)



)



) D′

± N ±

(

) D′

+ α( ) − ( ±N ±  ) α (

)



± N ±

) D + (α ( ) − ( ± N ±  ) α ( ) − ( ± N ±  ) α ( ) ) D  

N

( ) + α − ( ± N ±  ) α ( ) − ( ±N ±  ) α ( 



 

)





)D

 

± N ±

ºU + ¼

+

+

± N ±

  + ªα( )E±′′N ± + α( )E±′′N ± º U  + ¬ ¼      ( ) ( ) + ªNα  EN′ + α + ( ±N ±  ) α ( ) E±′ N ± + α( ) + ( ±N ±  ) α ( ) E±′ N ± º U + ¬ ¼

(

(

)

(

)

( ) + Nα ( )EN + α + ( ±N ±  ) α ( ) − ( ±N ±  ) α ( 

(





( ) + α + ( ±N ±  ) α ( ) − ( ±N ±  ) α ( 

)





)



)E

)E

± N ±

+

+

± N ±

   + ªα ( )FN′′ + α ( )F±′′N ± + α ( )F±′′N ± º U  + ¬ ¼      ( ) ( ) + ªα  FN′ + α  − ( ±N ±  ) α ( ) F±′ N ± + α ( ) − ( ±N ±  ) α ( ) F±′ N ± º U + ¬ ¼

(

(

)

()

()

+  α − Nα 

(

) F + (α

()



N

(

()

− ( ±N ±  ) α − ( ±N ±  ) α 

( ) ( ) + α − ( ± N ±  ) α − ( ±N ±  ) α ( 

)

()

)







)F

± N ±

)F

± N ±

+

+

+ ªα ( )G±′′N ± + α ( )G±′′N ± º U  + ¬ ¼ 



(

)

(

+ ªNα ( )GN′ + α ( ) + ( ±N ±  ) α ( ) G±′ N ± + α ( ) + ( ±N ±  ) α ( ¬ 

(





)



( ) ( ) () GN + α + Nα + ( ±N ±  ) α − ( ±N ±  ) α ( 

(







( ) ( ) + α + ( ±N ±  ) α − ( ±N ±  ) α (









)

)G

± N ±

)

)G

± N ±

) G′

± N ±

ºU + ¼

+

=   

7KH FRHIILFLHQWV DL  EL  F L  DQG G L  DUH )RXULHU FRHIILFLHQWV DV LQWURGXFHG LQ HTV DQG LQZKLFKLQGLFHV Q DQG P UHVSHFWLYHO\KDYHEHHQUHSODFHGE\ LQGH[ L  ZLWK L ≥   DQG E = G =   )XUWKHUPRUH LW LV QRWHG WKDW LQ HT  E\ XVLQJ WKH ±  VLJQ HLJKW FDVHV DUH LQYROYHG LH L = N +   L = N +   L = N −   L = N −   L = −N +   L = −N +   L = −N −  DQG L = −N −  )RUIXUWKHUHODERUDWLRQRI HT  WKHUHIRUH GLVWLQFWLRQ KDV WR EH PDGH EHWZHHQ WKH VSHFLDO FDVHV N =  DQGWKHUHJXODUFDVH N ≥  DVRQO\FRHIILFLHQWVZLWKLQGLFHVODUJHURU HTXDOWR]HURDUHDOORZHGIRUWKHSUHVHQWFDVH

0RKU²&RXORPEHODVWR²SODVWLFLW\LQFOXGLQJKDUGHQLQJLQWKHFLUFXODUGRPDLQ



,Q7DEOHLWLVLQGLFDWHGIRUZKLFK N =  RU N ≥  WKHFRHIILFLHQWV DL  EL  F L  DQG G L  UHPDLQ LQ HT  1RWH WKDW WKH LQGLFHV IRU L = −N ± V  DUH DOZD\V QHJDWLYH IRU HDFK N ≥   DQG DUH WKHUHIRUH DOZD\V RPLWWHG 7KH LQGLFHV IRU L = N  DQG L = + N ± V  DUH DOZD\V SRVLWLYH IRU HDFK N ≥   DQG KHQFH UHPDLQ LQ WKH HTXDWLRQ  LQGH[ L = ±N ± V 

N =

N = 

N = 

N ≥ 

N =

N

L =

\HV

L = 

\HV

L = 

\HV

L = 

\HV

L ≥

\HV

+N +  

L = 

\HV

L = 

\HV

L = 

\HV

L = 

\HV

L ≥

\HV

+N +  

L = 

\HV

L = 

\HV

L =

\HV

L = 

\HV

L ≥ 

\HV

+N −  

L = − 

QR

L = 

\HV

L =

\HV

L = 

\HV

L ≥

\HV

+N −  

L = − 

QR

L = − 

QR

L = − 

QR

L = 

\HV

L ≥

\HV

−N +  

L =

\HV

L = 

\HV

L = − 

QR

L = − 

QR

L ≤ − 

QR

−N +  

L = 

\HV

L = 

\HV

L =

\HV

L = 

\HV

L ≤ − 

QR

−N −  

L = − 

QR

L = − 

QR

L = − 

QR

L = − 

QR

L ≤ − 

QR

−N −  

L = − 

QR

L = − 

QR

L = − 

QR

L = − 

QR

L ≤ − 

QR

7DEOH (ODERUDWLRQ RI LQGLFHV L = ± N ± V  IRU N =   RU N ≥   DQG V =   IRU ZKLFK LQGH[UHPDLQVODUJHURUHTXDOWR]HUR¶\HV·UHSUHVHQWVWKHFDVH ± N ± V ≥  ZKLOH¶QR· UHSUHVHQWVWKHFDVH ± N ± V <  



&+$37(5

)RUWKHUHJXODUFDVHLH N ≥  IURPHT DQGXVLQJWKHUHVXOWVRI7DEOH LWWKHQIROORZV

(

)

ªα () D′′ + α () D′′ + D′′ + α () D′′ + D′′ º U + ªα() DN′ + α() − ( N + ) α() DN′+ + ¬  N  ( N+ N− )  ( N+ N− ) ¼ ¬

(

)

(

)

(

)

      + α( ) − ( N − ) α( ) DN′− + α( ) − ( N + ) α( ) DN′+ + α( ) − ( N − ) α( ) DN′− º U + ¼

(

()

) (



() 

()

( ) 

()

() 

+ α − ( N + ) α − ( N + ) α 



) D + (α

() 

N+

()

)

(

()

)

+  α − N α DN + α − ( N + ) α − ( N + ) α DN+ + α − ( N − ) α( ) − ( N − ) α( ) DN− +

(

( )

() 





() 

− ( N − ) α − ( N − ) α 

)D

+

N−

( ) + (α ( ) + ( N − ) α ( ) ) E′ + (α ( ) + ( N + ) α ( ) ) E′ + (α ( ) + ( N − ) α ( ) ) E′



     + ªα( ) (EN′′+ + EN′′− ) + α( ) (EN′′+ + EN′′− ) º U + ªNα( )EN′ + α( ) + ( N + ) α( ) EN′+ + ¬ ¼ ¬  

 

(

 

N−

 

 

N+

)

(

 

N−

ºU + ¼

)

+ Nα( )EN + α( ) + ( N + ) α( ) − ( N + ) α( ) EN+ + α( ) + ( N − ) α( ) − ( N − ) α( ) EN− + 

(







)



(









)

+ α( ) + ( N + ) α( ) − ( N + ) α( ) EN+ + α( ) + ( N − ) α( ) − ( N − ) α( ) EN− + 













(



)

 + ªα FN′′ + α ( FN′′+ + FN′′− ) + α ( FN′′+ + FN′′− ) º U + ªα FN′ + α − ( N + ) α( ) FN′+ + ¬ ¼ ¬

( )

()

()

(

)

()



(

)

()

(

)

      + α( ) − ( N − ) α( ) FN′− + α( ) − ( N + ) α( ) FN′+ + α( ) − ( N − ) α( ) FN′− º U + ¼

(

) (

)

(

)

+  α( ) − α( ) N FN + α( ) − ( N + ) α( ) − ( N + ) α( ) FN+ + α( ) − ( N − ) α( ) − ( N − ) α( ) FN− +

(





()

()







()

)

(



()

()







()



)

+ α − ( N + ) α − ( N + ) α FN+ + α − ( N − ) α − ( N − ) α FN− + 

) G′ + + (α ( ) + ( N − ) α ( ) ) G′ + (α ( ) + ( N + ) α ( ) ) G′ + (α ( ) + ( N − ) α ( ) ) G′ º U + ¼ () 

+ ªα ¬

( GN′′+ + GN′′− ) + α ( GN′′+ + GN′′− )º¼ U + ª¬Nα () 

 

 

()

(

() N 

 

 

()

()

() 

G′ + α + ( N + ) α

N+

)

() 

(

 

 

()

()

N+

N−

)

+ Nα GN + α + ( N + ) α − ( N + ) α GN+ + α + ( N − ) α − ( N − ) α( ) GN− +

(

()

N−

(





)

(





)

+ α( ) + ( N + ) α( ) − ( N + ) α( ) GN+ + α( ) + ( N − ) α( ) − ( N − ) α( ) GN− =  

















 

)RU N =  HT FDQEHHODERUDWHGHTXLYDOHQWO\ZKLFKLVJLYHQLQGHWDLO LQ$SSHQGL['RIWKHWKHVLVDQGLWLVUHIHUUHGWRWKHHODERUDWLRQVLQHT DQG WKHIROORZLQJ(T LVWKHILQDOHTXDWLRQFRQVLGHULQJWKHILUVWHTXDWLRQRIWKH 3'(VLQHT IRUWKHGHWHUPLQDWLRQRIWKH)RXULHUFRHIILFLHQWVIRUFDVHLH WKH ILUVW HTXDWLRQ IRU WKH FRHIILFLHQWV ZLWK LQGLFHV Q P = !  LQ HTV  M DQG  1RWH WKDW α L( )  DFFRUGLQJ WR HT  DUH FRQVWDQW FRHIILFLHQWV )XUWKHUPRUH DL  EL  F L  G L  DFFRUGLQJ WR HTV DQG  DQG WKHLU ILUVW DQG VHFRQG RUGHU GHULYDWLYHV DUH IXQFWLRQV GHSHQGHQW RQ U  &RQVLGHULQJ WKH VHFRQG HTXDWLRQ RI WKH 3'(V LQ HT  WKH VDPH SURFHGXUH FDQ EH IROORZHG ZKLOH UHSODFLQJ αL ·VE\ βL ·VLQHT  

0RKU²&RXORPEHODVWR²SODVWLFLW\LQFOXGLQJKDUGHQLQJLQWKHFLUFXODUGRPDLQ



&DVH0XOWLSOLFDWLRQE\VLQNθ DQGN «



,Q WKLV VHFWLRQ WKH WKLUG FDVH DV LQWURGXFHG LQ HT  ZLOO EH WUHDWHG 0XOWLSOLFDWLRQRIHT E\ VLQ ( Nθ ) ZLWKIL[HG N = ! DQGLQWHJUDWLRQZLWK UHVSHFWWR θ IURP  WR π JLYHV

 

π

³ I ( Uθ ) VLQ ( Nθ ) Gθ + 



+

π ∞

π ∞

( ) ( ) ³ ¦ ª¬ I ( Uθ ) FRV ( Qθ ) VLQ ( Nθ )º¼ Gθ + ³ ¦ ª¬ I ( Uθ ) FRV (Pθ ) VLQ ( Nθ )º¼ Gθ + Q



 Q =

+

P



 P =

π ∞

π ∞

( ) ( ) ³ ¦ ª¬ I ( Uθ ) VLQ ( Qθ ) VLQ ( Nθ )º¼ Gθ + ³ ¦ ª¬ I ( Uθ ) VLQ (Pθ ) VLQ ( Nθ )º¼ Gθ =  Q

P





 Q =

 P =



 

)LUVWWKHILYHLQWHJUDOVLQHT ZLOOEHHYDOXDWHGVHSDUDWHO\ZKLOHXVLQJWKH UHVXOWV RI HT  )RU WKH ILUVW LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJWKHIXQFWLRQ I  ( U θ ) IROORZLQJIURPHT  π

³ I ( Uθ ) VLQ ( Nθ ) Gθ = 

 π

=

³ {ª¬α

( )



VLQ ( θ ) + α( )VLQ ( θ ) º D′′U  + ªα( )VLQ ( θ ) + α( )VLQ ( θ ) º D′ U + ¼ ¬ ¼ 







  () ( ) + ªα VLQ ( θ ) + α VLQ ( θ ) º D + ªα ( )VLQ ( θ ) + α ( )VLQ ( θ ) º F′′U  + ¬ ¼ ¬ ¼

}

  () ( ) + ªα ( )VLQ ( θ ) + α ( )VLQ ( θ ) º F′ U + ªα VLQ ( θ ) + α VLQ ( θ ) º F VLQ ( Nθ ) Gθ = ¬ ¼ ¬ ¼

()  

() 

( ) 

()   

() 

()  

­° α U D′′ + α UD′ + α D + α U F′′ + α UF′ + α F IRUN =  =π ®  ( )  ( ) ( ) ( )  ( ) ( ) °¯ α U D′′ + α UD′ + α D + α  U F′′ + α  UF′ + α F IRUN =  

(

)

( ) ( ) = π α( )U D′′ + α( )UD′ + α D + α ( )U F′′ + α ( )UF′ + α F δ N  + 

(











)

( ) ( ) + π α( )U  D′′ + α( )UD′ + α D + α ( )U F′′ + α ( )UF′ + α F δ N  















 

DV RQO\ IRU N =   DQG N =   QRQ²]HUR WHUPV RFFXU 1RWH WKDW LQ WKH SUHYLRXV HODERUDWLRQXVHKDVEHHQPDGHRIWKHHODERUDWHGHLJKWKWULJRQRPHWULFLQWHJUDOLQ HT ,QWKLVLQWHJUDOWKHFDVHV Q =   V =  DQG N = ! DUHVLJQLILFDQW WDNLQJLQWRDFFRXQWWKHFRQGLWLRQ V = N 



&+$37(5

)RU WKH VHFRQG LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJ WKH IXQFWLRQ Q I ( )( U θ ) IROORZLQJIURPHT  π ∞

( ) ³ ¦ ª¬I ( U θ ) FRV ( Qθ ) VLQ ( N θ )º¼ Gθ = Q



 Q =

=

π ∞

³ ¦{¬ªα

( )



 Q =

+ α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º DQ′′U  + ¼ 







− ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º Q DQ + ¬ ¼ 









     + ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QEQ′ U + ¬ ¼ ( ) () ( ) ( ) () ª + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º DQ′ U + ¬ ¼      + ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QEQ + ¬ ¼

}

( ) () ( ) ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º DQ FRV ( Qθ ) VLQ ( N θ )Gθ = ¬ ¼ 









  π π     = ªα( )D±′′N ± + α( )D±′′N ± º U  − ªα( ) ( ±N ± ) D± N ± + α( ) ( ±N ±  ) D± N ± º + ¬ ¼

¬



+

¼

π ª ( ) π α ±N ± ) E±′ N ± + α() ( ±N ±  ) E±′ N ± º¼ U + ª¬α()D±′ N ± + α()D±′ N ± º¼ U + ¬  ( 



π π ( )   () D± N ± + α D± N ± º  + ªα( ) ( ±N ± ) E± N ± + α( ) ( ±N ±  ) E± N ± º + ªα ¬ ¼ ¬ ¼ 





 

1RWHWKDWLQWKHSUHYLRXVHODERUDWLRQXVHKDVEHHQPDGHRIWKHHODERUDWHGIRXUWK DQG VL[WK WULJRQRPHWULF LQWHJUDOV LQ HT  ,Q WKHVH LQWHJUDOV WKH FDVHV N = !  Q = !  DQG V =   DUH VLJQLILFDQW 7KHQ RQO\ IRU Q = ± N ±   DQG Q = ±N ±   QRQ²]HUR UHVXOWV HTXDO WR π   RFFXU $OO PXOWLSOLFDWLRQV ZLWK FRV ( θ ) DQG FRV ( θ ) UHVXOWLQ ]HURWHUPVDQGPXOWLSOLFDWLRQVZLWK VLQ ( θ ) DQG VLQ ( θ ) RQO\IRU Q = ± N ±  DQG Q = ±N ±  UHVXOWLQQRQ²]HURWHUPV

0RKU²&RXORPEHODVWR²SODVWLFLW\LQFOXGLQJKDUGHQLQJLQWKHFLUFXODUGRPDLQ



)RU WKH WKLUG LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJ WKH IXQFWLRQ P I ( )( U θ ) IROORZLQJIURPHT  π ∞

( ) ³ ¦ ª¬ I ( Uθ ) FRV (Pθ ) VLQ ( Nθ )º¼ Gθ = P



 P=

=

π ∞

³ ¦ {¬ªα  P=

() 

+ α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º FP′′ U  + ¼ 







− ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º PFP + ¬ ¼ 









     + ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º PGP′ U + ¬ ¼ ( ) () ( ) ( ) () ª º + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) FP′ U + ¬ ¼

( ) () ( ) () () + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º PGP + ¬ ¼

}

( ) () ( ) ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º FP FRV ( Pθ ) VLQ ( Nθ ) Gθ = ¬ ¼ 









  π π     = ªα( )F±′′N± + α( )F±′′N± º U  − ªα( ) ( ±N ± ) F± N± + α( ) ( ±N ±  ) F± N± º + ¬ ¼

¬



+

¼

π ª ( ) π α ±N ± ) G±′ N± + α() ( ±N ±  ) G±′ N± ¼º U + ¬ªα()F±′ N± + α()F±′ N± ¼º U + ¬  ( 



π ( ) + ªα ( ±N ± ) G±N± + α() ( ±N ±  ) G±N± º¼ + 𠪬α()F±N± + α()F±N± º¼  ¬ 





 

1RWHWKDWLQWKHSUHYLRXVHODERUDWLRQXVHKDVEHHQPDGHRIWKHHODERUDWHGIRXUWK DQG VL[WK WULJRQRPHWULF LQWHJUDOV LQ HT  ,Q WKHVH LQWHJUDOV WKH FDVHV N = !  P = !  DQG V =   DUH VLJQLILFDQW 7KHQ RQO\ IRU P = ± N ±   DQG P = ±N ±   QRQ²]HUR UHVXOWV HTXDO WR π   RFFXU $OO PXOWLSOLFDWLRQV ZLWK FRV ( θ ) DQG FRV ( θ ) UHVXOWLQ ]HURWHUPVDQGPXOWLSOLFDWLRQVZLWK VLQ ( θ ) DQG VLQ ( θ ) RQO\IRU P = ± N ±  DQG P = ±N ±  UHVXOWLQQRQ²]HURWHUPV



&+$37(5

)RU WKH IRXUWK LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJ WKH IXQFWLRQ Q I ( )( U θ ) IROORZLQJIURPHT  π ∞

( ) ³ ¦ ª¬ I ( Uθ ) VLQ ( Qθ ) VLQ ( Nθ )º¼ Gθ = Q



 Q=

=

π ∞

³ ¦{ª¬α

( )



 Q=

+ α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º EQ′′U + ¼ 







− ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QEQ + ¬ ¼ 









     − ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QDQ′ U + ¬ ¼ ( ) () ( ) ( ) ( ) ª + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º EQ′ U + ¬ ¼      − ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º QDQ + ¬ ¼

}

( ) () ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º EQ VLQ ( Qθ ) VLQ ( Nθ ) Gθ = ¬ ¼ 









  π π       = ªα( )EN′′ + α( )E±′′N± + α( )E±′′N± º U − ªα( ) NEN + α( )( ±N ± ) E± N± + α( )( ±N ±  ) E± N± º + ¬ ¼

¬



−

¼

π ª ( ) ′ π      α NDN + α( )( ±N ± ) D±′ N± + α( )( ±N ±  ) D±′ N± º U + ªα( )EN′ + α( )E±′ N± + α( )E±′ N± º U + ¬  ¼ ¬ ¼ 



π π     ( ) () − ªα( ) NDN + α( )( ±N ± ) D± N± + α( )( ±N ±  ) D± N± º + ªα EN + α E± N± + α( )E±N± º  ¬



¼

¬

¼  

1RWH WKDW LQ WKH SUHYLRXV HODERUDWLRQ XVH KDV EHHQ PDGH RI WKH HODERUDWHG ILUVW DQG ILIWK WULJRQRPHWULF LQWHJUDOV LQ HT  ,Q WKHVH LQWHJUDOV WKH FDVHV N = !  Q = !  DQG V =   DUH VLJQLILFDQW 7KHQ RQO\ IRU Q = ± N ±   DQG Q = ±N ±   QRQ²]HUR UHVXOWV HTXDO WR π   RFFXU $OO PXOWLSOLFDWLRQV ZLWK VLQ ( θ ) DQG VLQ ( θ ) UHVXOWLQ ]HURWHUPVDQGPXOWLSOLFDWLRQVZLWK FRV ( θ ) DQG FRV ( θ ) RQO\IRU Q = ± N ±  DQG Q = ±N ±  UHVXOWLQQRQ²]HURWHUPV

0RKU²&RXORPEHODVWR²SODVWLFLW\LQFOXGLQJKDUGHQLQJLQWKHFLUFXODUGRPDLQ



)RU WKH ILIWK LQWHJUDO LQ HT  LW IROORZV ZKLOH VXEVWLWXWLQJ WKH IXQFWLRQ P I ( )( U θ ) IROORZLQJIURPHT  π ∞

( ) ³ ¦ ª¬ I ( Uθ ) VLQ (Pθ ) VLQ ( Nθ )º¼ Gθ = P



 P=

=

π ∞

³ ¦{¬ªα  P=

( ) 

+ α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º GP′′ U + ¼ 







− ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º PGP + ¬ ¼ 









     − ªα( ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) + α( ) FRV ( θ ) + α( )VLQ ( θ ) º PFP′ U + ¬ ¼ ( ) () ( ) ( ) () ª + α + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α VLQ ( θ ) º GP′ U + ¬ ¼  ( ) () ( ) ( ) − ªα + α FRV ( θ ) + α VLQ ( θ ) + α FRV ( θ ) + α( )VLQ ( θ ) º PFP + ¬ ¼

}

( ) () ( ) ( ) + ªα + α FRV ( θ ) + α VLQ ( θ ) + α( ) FRV ( θ ) + α VLQ ( θ ) º GP VLQ ( Pθ ) VLQ ( Nθ ) Gθ = ¬ ¼ 









  π π       = ªα( )GN′′ + α( )G±′′N± + α( )G±′′N± º U − ªα( ) NGN + α( )( ±N ± ) G±N± + α( )( ±N ±  ) G±N± º + ¬ ¼

¬



−

¼

π ª ( ) ′ π      α NF + α( )( ±N ± ) F±′ N± + α( )( ±N ±  ) F±′ N± º U + ªα( )GN′ + α( )G±′ N± + α( )G±′ N± º U + ¬  N ¼ ¬ ¼ 



π ( ) () NFN + α − ªα ( ±N ± ) F±N± + α()( ±N ±  ) F±N± º + π ªα()GN + α()G±N± + α()G±N± º  ¬



¼

¬

¼  

1RWH WKDW LQ WKH SUHYLRXV HODERUDWLRQ XVH KDV EHHQ PDGH RI WKH HODERUDWHG ILUVW DQG ILIWK WULJRQRPHWULF LQWHJUDOV LQ HT  ,Q WKHVH LQWHJUDOV WKH FDVHV N = !  P = ! DQG V =  DUHVLJQLILFDQW7KHQRQO\IRU P = ±N ±   DQG P = ±N ±   QRQ²]HUR UHVXOWV HTXDO WR π   RFFXU $OO PXOWLSOLFDWLRQV ZLWK VLQ ( θ ) DQG VLQ ( θ ) UHVXOWLQ ]HURWHUPVDQGPXOWLSOLFDWLRQVZLWK FRV ( θ ) DQG FRV ( θ ) RQO\IRU P = ±N ±  DQG P = ±N ±  UHVXOWLQQRQ²]HURWHUPV



&+$37(5

6XEVWLWXWLQJ WKH HYDOXDWHG LQWHJUDOV RI HTV WR  LQWR HT  LW IROORZV DIWHU GLYLGLQJ E\ π   DQG VRUWLQJ IRU FRHIILFLHQWV DL  EL  F L  DQG G L  IRU FDVHLQZKLFK N = ! 

(α ( )U D′′ + α ( )UD′ + α ( )D + α ( )U F′′ + α ( )UF′ + α ( )F ) δ + + (α ( )U D′′ + α ( )UD′ + α ( ) D + α ( )U F′′ + α ( )UF′ + α ( )F ) δ 



 







 



 







()

 



 



( )

 





 



 





 





N

 





+

N

+ ªα D±′′N ± + α D±′′N ± º U + ¬ ¼ 

(

)

(

) D′

+ ª −Nα ( ) DN′ + α( ) − ( ± N ±  ) α ( ) D±′ N ± + α( ) − ( ±N ±  ) α ( ¬ 

(





( ) − ( ±N ±  ) α ( ) − ( ± N ±  ) α ( − Nα ( ) DN + α 

(



( ) + α − ( ±N ±  ) α ( ) − ( ±N ±  ) α ( 

)





)





)D

± N ±

)



)D

± N ±

± N ±

ºU + ¼

+

+

   + ªα( )EN′′ + α( )E±′′N ± + α( )E±′′N ± º U  + ¬ ¼      ( ) ( ) + ªα EN′ + α + ( ± N ±  ) α ( ) E±′ N ± + α( ) + ( ±N ±  ) α ( ) E±′ N ± º U + ¬ ¼

(

(

)

( ) +  α − Nα (

(

)



(

)

) E + (α ( ) + ( ± N ±  ) α ( ) − ( ± N ±  ) α ( ) ) E  

N

 

( ) + α + ( ±N ±  ) α ( ) − ( ±N ±  ) α ( 



)





)E

± N ±

 

± N ±

+

+

  + ªα ( )F±′′N ± + α ( )F±′′N ± º U  + ¬ ¼      ( ) ( ) + ª −Nα  FN′ + α  − ( ±N ±  ) α ( ) F±′ N ± + α ( ) − ( ± N ±  ) α ( ) F±′ N ± º U + ¬ ¼

(

()

(

)

( )

(

()

)

()

− Nα FN + α − ( ±N ±  ) α − ( ±N ±  ) α 

(



( ) ( ) + α − ( ±N ±  ) α − ( ±N ±  ) α ( 

)





)F

± N ±

)F

± N ±

+

+

+ ªα ( )GN′′ + α ( )G±′′N ± + α ( )G±′′N ± º U  + ¬ ¼ 





(

+ ªα ( )GN′ + α ( ) + ( ± N ±  ) α ( ¬ 

(

( ) +  α − Nα (

(



)



)

) G′

± N ±



) G′

)



± N ±

) G + (α ( ) + ( ± N ±  ) α ( ) − ( ± N ±  ) α ( ) ) G N

 





 

( ) ( ) + α + ( ±N ±  ) α − ( ±N ±  ) α ( 

(

+ α ( ) + ( ±N ±  ) α (



)

)G

± N ±

 

± N ±

ºU + ¼

+

=   

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( ) + (α( ) − ( N − ) α( ) ) D′ + (α( ) − ( N + ) α( ) ) D′ + (α( ) − ( N − ) α( ) ) D′

ªα() ( DN′′+ + DN′′− ) + α() ( DN′′+ + DN′′− ) º U + ª−Nα() DN′ + α() − ( N + ) α() DN′+ + ¬ ¼ ¬  

 

(

 

N−

 

 

N+

)

(

 

N−

ºU + ¼

)

− Nα( ) DN + α( ) − ( N + ) α( ) − ( N + ) α( ) DN+ + α( ) − ( N − ) α( ) − ( N − ) α( ) DN− + 

(







)



(









)

+ α( ) − ( N + ) α( ) − ( N + ) α( ) DN+ + α( ) − ( N − ) α( ) − ( N − ) α( ) DN− + 















(

)

      + ªα( )EN′′ + α( ) (EN′′+ + EN′′− ) + α( ) (EN′′+ + EN′′− ) º U + ªα( )EN′ + α( ) + ( N + ) α( ) EN′+ + ¬ ¼ ¬

(

() 

() 

+ α + ( N − ) α

(

)E′ + (α

() 

N−

) (

() 

+ ( N + ) α

)E′ + (α

() 

N+

() 

+ ( N − ) α

)

(

)E′

N−

ºU + ¼

)

+  α( ) − Nα( ) EN + α( ) + ( N + ) α( ) − ( N + ) α( ) EN+ + α( ) + ( N − ) α( ) − ( N − ) α( ) EN− +

(











)

(











)

+ α( ) + ( N + ) α( ) − ( N + ) α( ) EN+ + α( ) + ( N − ) α( ) − ( N − ) α( ) EN− + 

() 

+ ªα ¬









( FN′′+ + FN′′− ) + α ( FN′′+ + FN′′− )¼º U + ¬ª−Nα ( ) 

(

(

()  N

)

(





( ) 



() 

F′ + α − ( N + ) α

)

(

)

(

) F′

+

N+

)

      + α( ) − ( N − ) α( ) FN′− + α( ) − ( N + ) α( ) FN′+ + α( ) − ( N − ) α( ) FN′− º U + ¼

(

)

− Nα( ) FN + α( ) − ( N + ) α( ) − ( N + ) α( ) FN+ + α( ) − ( N − ) α( ) − ( N − ) α( ) FN− + 

(



( ) 





() 

() 

+ α − ( N + ) α − ( N + ) α 



) F + (α

() 

N+







() 

() 

− ( N − ) α − ( N − ) α 

)F

N−

(



+

)

      + ªα( ) GN′′ + α( ) ( GN′′+ + GN′′− ) + α( ) ( GN′′+ + GN′′− ) º U + ªα( ) GN′ + α( ) + ( N + ) α( ) GN′+ + ¬ ¼ ¬

(

)

(

)

(

)

      + α( ) + ( N − ) α( ) GN′− + α( ) + ( N + ) α( ) GN′+ + α( ) + ( N − ) α( ) GN′− º U + ¼

(

) (

)

(

)

+  α( ) − Nα( ) GN + α( ) + ( N + ) α( ) − ( N + ) α( ) GN+ + α( ) + ( N − ) α( ) − ( N − ) α( ) GN− +

(











)

(











)

+ α( ) + ( N + ) α( ) − ( N + ) α( ) GN+ + α( ) + ( N − ) α( ) − ( N − ) α( ) GN− =  

















 

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α() D′′ U  + α() D′ U + α() D =  IRUN = 

(

)

α() DN′′ U  + ª¬α() DN′ + Nα ()GN′ º¼ U + α( ) − Nα ( ) DN + Nα()GN = IRUN ≥  

(

 

)

α()EN′′ U  + ¬ªα()EN′ − Nα ()FN′ ¼º U + α() − Nα ( ) EN − Nα( )FN = IRUN ≥  7KH VDPH SURFHGXUH FDQ EH IROORZHG WR GHWHUPLQH WKH HTXDWLRQV LQYROYLQJ M FRHIILFLHQWV βL( ) IRUZKLFKIROORZV

β()F′′ U  + β()F′ U + β()F =  IRUN = 

(

)

β()FN′′ U  + ª¬Nβ()EN′ + β()FN′ º¼ U + Nβ()EN + β( ) − β() N FN =  IRUN ≥ 

(



 

)

β()GN′′ U  + ª¬ −Nβ() DN′ + β()GN′ º¼ U − Nβ() DN + β( ) − N β() GN =  IRUN ≥  6XEVWLWXWLQJ WKH UHPDLQLQJ FRHIILFLHQWV α L( )  DQG βL( )  IROORZLQJ IURP $SSHQGL[$ HTV DQG FDQEHHODERUDWHGDIWHUVRPHUHDUUDQJHPHQWDV 



­U  D′′ + UD′ − D = IRUN =     °  °  ′′ ′ º¼ −  (+ν  ) + N DN + N ª¬(+ ν  ) UGN′ − ( + ν  ) GN º¼ = IRUN ≥  ª U D   + ν ( ) ®  ¬ N + UDN °   °¯ ª¬U GN′′ + UGN′ º¼ − ª¬ +  (+ν  ) N º¼ GN − N ¬ª(+ ν  ) UDN′ + ( + ν  ) DN ¼º = IRUN ≥ 

(

)

­U F′′ + UF′ − F = IRUN =     °  °   ® (+ν  ) ª¬U EN′′ + UEN′ º¼ −  (+ν  ) + N EN − N ª¬(+ ν  ) UFN′ − ( + ν  ) FN º¼ =  IRUN ≥  °   °¯ ª¬U FN′′ + UFN′ º¼ − ª¬ +  (+ν  ) N º¼ FN + N ª¬(+ ν  ) UEN′ + ( + ν  ) EN º¼ = IRUN ≥    

(

)

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U Nj T′′ ( U ) + U nj T′ ( U ) + ǍT ( U ) =   

 

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DL ( U ) = D L ( ρ )  EL ( U ) = EL ( ρ )  FL ( U ) = FL ( ρ ) DQGGL ( U ) = GL ( ρ )   LQZKLFK ρ = OQ ( U ) 7KHQIURPWKHUHODWLRQVLQHT LWIROORZV

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GDL ( U ) GU

=

G DL ( U ) GU



 GD L ( ρ ) RUUDL′( U ) = D L′( ρ )   U Gρ

=

  G D L ( ρ )  GD L ( ρ ) −  RUU  DL′′( U ) = D L′′( ρ ) − D L′( ρ )    U Gρ  U Gρ



 

 

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Nj T ′′ ( ρ ) + ( nj − Nj ) T ′ ( ρ ) + ǍT ( ρ ) =   

 

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IL ( ρ ) = EL′( ρ ) 

J L ( ρ ) = FL′( ρ ) DQGKL ( ρ ) = GL′( ρ )  

 

)URPHT LWIROORZV D L′′( ρ ) = HL′( ρ )  EL′′( ρ ) = IL′( ρ )  FL′′( ρ ) = J L′( ρ ) DQGGL′′( ρ ) = KL′( ρ )  

 

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α() H′ + α() H′ + α() H′ + α() I′ + α() I′ + α ( ) J ′ + α () J ′ + α () J ′ + α ()K′ + α ()K′ +

(

) ( ) ( ) ( ) ( ) + ( −α + α + α ) E + ( −α + α + α ) E + + α ( )F + ( −α ( ) − α ( ) + α ( ) ) F + ( −α ( ) − α ( ) + α ( ) ) F + + ( −α ( ) + α ( ) + α ( ) ) G + ( −α ( ) + α ( ) + α ( ) ) G + + ( −α ( ) + α ( ) ) H + ( −α ( ) − α ( ) + α ( ) ) H + ( −α ( ) − α ( ) + α ( ) ) H + + ( −α ( ) + α ( ) + α ( ) ) I + ( −α ( ) + α ( ) + α ( ) ) I + + ( −α ( ) + α ( ) ) J + ( −α ( ) − α ( ) + α ( ) ) J + ( −α ( ) − α ( ) + α ( ) ) J + + ( −α ( ) + α ( ) + α ( ) ) K + ( −α ( ) + α ( ) + α ( ) ) K =      () () () + α D  + −α ( ) − α ( ) + α D  + −α ( ) − α ( ) + α D  +

 

()

()

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        ª º « »          « » « »         « »          « » (L )  1 =  «  () ()       −Lα( ) −Lα( ) −Lα( ) −Lα( ) » L α − α Lα( ) − α( ) α() − α() α() − α() « »             ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )   « Lα α − α α − α » L α − α Lα L α − α Lα Lα «  »  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( )  ( ) «L β − β β − β β − β −Lβ −Lβ −Lβ −Lβ( ) » L β − β « »        ( ) β() − β() β() − β() ¼» L β( ) − β( ) Lβ( ) L β( ) − β( ) Lβ( ) Lβ( ) «¬ Lβ IRUL = NDQGN ≥ 



 

 ª «  « «  «  « L 1( ) = − « () α − Lα() − Lα() « «α() − Lα() − Lα() «  « β( ) − Lβ() − L β() « () () ( ) «¬β − Lβ − L β  "









"









 

 

 

 

" "

α() + Lα() − Lα() α() − Lα() − Lα() α() + Lα() − Lα() α() + Lα() − Lα() α() − Lα() − Lα() α() + Lα() − Lα() β() + Lβ() − L β() β() − Lβ() − L β() β() + Lβ() − L β()

" "









 ()

()

()

( )

" −α + α + Lα

−α + α − Lα

"−β() + β() + Lβ() " −β () + β () + Lβ ()

−β( ) + β( ) − Lβ(

" −α + α + Lα



º » » » » ­ L = N + DQGN ≥   » ° IRU    ® L = N − DQGN ≥  −α( ) + α( ) + Lα( ) » » ° L = −N + DQGN =    ¯ −α( ) + α( ) + Lα( ) » »    −β( ) + β( ) + Lβ( ) » () () ( ) » −β + β + Lβ »¼  

( )







 

( )



−α( ) + α( ) − Lα( ) "    −β( ) + β( ) − Lβ( ) "

β() + Lβ() − L β() β() − Lβ() − L β() β() + Lβ() − L β() −β() + β() − Lβ() "

 

"

"    −α( ) + α( ) − Lα( ) "

()

−α + α( ) − Lα( ) 



( )

( )

()





)

( )

( )

()

−β + β − Lβ



 

DQG     ª «     « «     «     « L 1( ) =− « () α − Lα() − Lα() α() + Lα() − Lα() α() − Lα() − Lα() α() + Lα() − Lα() « «α() − Lα() − Lα() α() + Lα() − Lα() α() − Lα() − Lα() α() + Lα() − Lα() «  « β( ) − Lβ() − L β() β() + Lβ() − L β() β() − Lβ() − L β() β() + Lβ() − L β() « ( ) () ( ) β() + Lβ() − L β() β() − Lβ() − L β() β() + Lβ() − L β() «¬β − Lβ − L β



"

 

" "

 "    −α( ) + α( ) − Lα( ) " −α( ) + α( ) − Lα( ) "    −β( ) + β( ) − Lβ( ) " 





   −β( ) + β( ) − Lβ( ) "



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"





"





"





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( )

( )

()

( )

( )

()

()

−α + α( ) − Lα( )

" −α + α + Lα    " −α( ) + α( ) + Lα( )

º » » »  » ­ L = N+ DQGN ≥   » ° ( ) ( ) () » IRU ® L = N− DQGN ≥  −α + α + Lα » ° L =−N+ DQGN =    ¯ −α( ) + α( ) + Lα( ) » » ( ) ( ) ( ) −β + β + Lβ »    » −β( ) + β( ) + Lβ( ) »¼ 





−α( ) + α( ) − Lα( ) 





()

()

( )

"−β + β + Lβ −β + β − Lβ "−β () + β () + Lβ () −β () + β () − Lβ ()      



 

1H[W WKH VSHFLDO FDVH IRU N =   DV H[SUHVVHG E\ HT  DQG OHDGLQJ ILQDOO\ WR HT LVFRQVLGHUHG)RUWKDWFDVHHT FDQEHHODERUDWHGDV

Ø () [Ø ′ + 0 Ø () [ ′ + 0 Ø ( ) [ ′ = 1 Ø ( ) [Ø + 1 Ø () [ + 1 Ø ( ) [   0 () () ( ) () () ( )

 

Ø ()  0 Ø ()  0 Ø ( ) DFFRUGLQJWR7DEOHDQG 1 Ø ( )  LQZKLFKWKHLUUHJXODUPDWULFHV 0   Ø ( )  1 Ø ( )  DFFRUGLQJ WR 7DEOH  DUH LQYROYHG ,Q WKH IROORZLQJ HODERUDWLRQ RI 1 WKHVH LUUHJXODU PDWULFHV WKH ILUVW WZR URZV UHSUHVHQW WKH ILUVW DQG WKLUG H[SUHVVLRQVRIHT DFFRUGLQJWRWKHVHFRQGWUDQVIRUPDWLRQUXOHDVIRU N =   RQO\WKHFRHIILFLHQWV D ′ ( ρ ) DQG F′ ( ρ ) RFFXU7KHH[SUHVVLRQLQHT LVXVHGWR FRPSRVH URZ  7KH H[SUHVVLRQ LQ URZ  FDQ EH GHULYHG E\ UHSODFLQJ DOO M M FRHIILFLHQWV α L( ) LQURZE\FRHIILFLHQWV βL( )  7KHWHUPVRQWKHOHIWKDQGVLGHRIHT FDQEHHODERUDWHGDV ª « Ø ( ) [Ø ′ = « 0 () «  « ¬«

ª « Ø () [ ′ = « 0 () «  « ¬«

DQG









 α(

)

 β(

)

 º § D ′ ( ρ ) ·  »» ¨ F′ ( ρ ) ¸ ¨ ¸ IRUL = N DQGN =    α ( ) » ¨ H′ ( ρ ) ¸ »¨ ¸ β ( ) ¼» ¨© J ′ ( ρ ) ¹¸

  







  







()

   α

()

α

α ()

   β(

β()

β ()

)

 

§ D ′ ( ρ ) · ¨ ¸ ¨ E′ ( ρ ) ¸  º ¨ F′ ( ρ ) ¸ ¨ ¸  »» ¨ G′ ( ρ ) ¸ ¨ ¸ IRUL = N +  DQGN =      α () » ¨ H′ ( ρ ) ¸ » β () ¼» ¨¨ I′( ρ ) ¸¸ ¨ J ′ ( ρ ) ¸ ¨¨  ¸¸ © K′ ( ρ ) ¹

0RKU²&RXORPEHODVWR²SODVWLFLW\LQFOXGLQJKDUGHQLQJLQWKHFLUFXODUGRPDLQ

ª « Ø ( ) [ ′ = « 0 ( ) «  « ¬«

  







  







()

()

α

α ()

β()

β ()

   α

   β(

)



§ D ′ ( ρ ) · ¨ ¸ ¨ E′ ( ρ ) ¸  º ¨ F′ ( ρ ) ¸ ¨ ¸  »» ¨ G′ ( ρ ) ¸ ¨ ¸ IRUL = N +  DQGN =    α () » ¨ H′ ( ρ ) ¸ » β () ¼» ¨¨ I′( ρ ) ¸¸ ¨ J ′ ( ρ ) ¸ ¨¨  ¸¸ © K′ ( ρ ) ¹

,WLVQRWHGWKDWWKHLUUHJXODUYHFWRU [Ø ′( ) IRU L =  RQO\FRQWDLQVIRXUFRPSRQHQWV Ø () DVTXDUH  ×  − PDWUL[+RZHYHUWKH DFFRUGLQJWRHT PDNLQJPDWUL[ 0 UHJXODUYHFWRUV [ ′() DQG [ ′( ) DUHFRPSRVHGRIHLJKWFRPSRQHQWVWKXVUHVXOWLQJIRU Ø () DQG 0 Ø ( ) LQDVL]HRIIRXUURZVDQGHLJKWFROXPQV PDWULFHV 0 7KHWHUPVRQWKHULJKWKDQGVLGHRIHT FDQEHHODERUDWHGDV

ª  «  () Ø Ø  1 [() = «« () −α «  () ¬« − β









( ) −α

α() − α()

− β( )

β() − β()





º § D ( ρ ) · »¨ ¸ » ¨ F( ρ ) ¸ IRUL = N DQGN =   α () − α () » ¨ H( ρ ) ¸ »¨ ¸ β() − β() ¼» ©¨ J ( ρ ) ¹¸ 



 

     ª " «      "  « ( ) Ø  [Ø =− 1 ( ) «α () − α () − α () α () + α () − α () α () − α () − α () α () + α () − α () −α () + α () − α () "               «  «¬β() − β() − β() β() + β() − β() β() − β() − β() β() + β() − β() −β() + β() − β() "

"



"



 

"−α( ) + α( ) + α( ) 





−α( ) + α( ) − α( ) 





"−β() + β() + β() −β() + β() − β()



§ D( ρ ) · ¨ ¸ ¨E( ρ ) ¸  º ¨ F( ρ ) ¸ ¸ »¨   » ¨ G( ρ ) ¸ IRUL =N +DQGN = ¸    ¨ −α( ) + α( ) + α( ) » ¨ H( ρ ) ¸ » ¸    ¨ −β( ) + β( ) + β( ) »¼ ¨ I( ρ ) ¸ ¨ J( ρ ) ¸ ¨¨  ¸¸ © K( ρ ) ¹

 

DQG      ª " «      " Ø () [Ø =− « 1 () «α() −α() −α() α() +α() −α() α() −α() −α() α() +α() −α() −α() +α() −α() "               «  «¬β() −β() −β() β() +β() −β() β() −β() −β() β() +β() −β() −β() + β() −β() "





&+$37(5



"



  "       "−α( ) +α( ) + α( ) −α( ) +α( ) − α( ) "−β() + β() + β() −β() + β() − β()

§ D( ρ ) · ¨ ¸ ¨E( ρ ) ¸  º ¨ F( ρ ) ¸ ¸ »¨   » ¨ G( ρ ) ¸ IRUL =N +DQGN = ¸    ¨ −α( ) +α( ) + α( ) » ¨ H( ρ ) ¸ » ¸    ¨ −β( ) + β( ) + β( ) ¼» ¨ I( ρ ) ¸ ¨ J ( ρ ) ¸ ¨¨  ¸¸ © K( ρ ) ¹



 

,WLVQRWHGWKDWWKHLUUHJXODUYHFWRU [Ø ( ) IRU L =  RQO\FRQWDLQVIRXUFRPSRQHQWV Ø ( ) DVTXDUH  ×  − PDWUL[+RZHYHUWKH DFFRUGLQJWRHT PDNLQJPDWUL[ 1 UHJXODUYHFWRUV [ () DQG [ ( ) DUHFRPSRVHGRIHLJKWFRPSRQHQWVWKXVUHVXOWLQJIRU Ø () DQG 1 Ø ( ) LQDVL]HRIIRXUURZVDQGHLJKWFROXPQV PDWULFHV 1 1H[W WKH LUUHJXODU WHUPV IRU N =   DQG L =   DUH FRQVLGHUHG )RU WKDW FDVH WKH Ø ()  DFFRUGLQJ WR 7DEOH  DQG 1 Ø ( )  DFFRUGLQJ WR 7DEOH  LUUHJXODU PDWULFHV 0 DUHLQYROYHG7KH\FDQEHHODERUDWHGDV ª « « « « « Ø () [Ø ′ = « 0 ()  « « « « « «¬

















 α(

)

 α(

)

 β(

)

 β(

)

 º  »»  » § D ′ ρ · » ( )  » ¨ F′ ρ ¸ ¨ ( ) ¸ IRU ­L = N − DQGN =   ® α () » ¨ H′ ( ρ ) ¸ ¯L = −N + » ¸¸ ( ) » ¨¨  ′ α  © J( ρ ) ¹ » β() » » β() »¼

 

DQG

 ª  «   « «   «   « Ø () [Ø = − « () 1 () α α() «  ( ) «α α() «  « β( ) β() « ( ) () ¬« β β 

    ()

−α + α( ) 

−α( ) + α( 

)

− β( ) + β( 

)

− β( ) + β( 

º » » » § D ρ ·  » ¨ ( ) ¸  » ¨ F( ρ ) ¸ ­L = N − IRU ® DQGN =    −α ( ) + α ( ) » ¨ H( ρ ) ¸ ¯L = −N + »¨ ¸   −α ( ) + α ( ) » ¨© J ( ρ ) ¹¸ »   − β( ) + β( ) » »   − β( ) + β( ) ¼»    

)

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)LQDOO\WKHLUUHJXODUWHUPVIRU N =  DQG L =  DUHFRQVLGHUHG)RUWKDWFDVHWKH Ø ()  DFFRUGLQJ WR 7DEOH  DQG 1 Ø ( )  DFFRUGLQJ WR 7DEOH  LUUHJXODU PDWULFHV 0 DUHLQYROYHG7KH\FDQEHHODERUDWHGDV ª « « « « « () Ø Ø 0 [ ′( ) = «  « « « « « «¬





 

 





 α(

)

 α(

)

 β(

)

 β(

)

 º  »»  » § D ′ ( ρ ) · »   » ¨ F′ ( ρ ) ¸ ¨  ¸ IRU ­L = N −  DQGN =    ® α () » ¨ H′ ( ρ ) ¸ ¯L = −N +  »¨ ¸ α ( ) » ¨© J ′ ( ρ ) ¹¸ » β() » » β( ) »¼

 

DQG

 ª  «   « «   «   « Ø ( ) [Ø = − « () 1 () α α() «  ( ) «α α( ) «  « β( ) β() « ( ) ( ) ¬« β β 

º » » » § D ρ ·  » ¨ ( ) ¸  » ¨ F( ρ ) ¸ ­L = N −  IRU ® DQGN =    −α ( ) + α ( ) » ¨ H( ρ ) ¸ ¯L = −N +   »¨ ¸   −α ( ) + α ( ) » ¨© J ( ρ ) ¹¸ »   − β ( ) + β( ) »   » − β( ) + β( ) ¼»  









  ()

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)

−α( ) + α(

)



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)

− β( ) + β(

)





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IRUN = 

Ø ( ) [Ø ′ + 0 Ø () [ ′ + 0 Ø ( ) [ ′ = 1 Ø ( ) [Ø + 1 Ø () [ + 1 Ø ( ) [   0 () () ( ) () () ( )

IRUN = 

( 0 + 0 ) [ ′() + ( 0 + 0 ) [ ′() + 0[ ′() =

(

)

(

)

     = 1( ) + 1( ) [ () + 1( ) + 1( ) [ () + 1( ) [ () 





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IRUN = 

Ø ( ) [Ø ′ + ( 0 + 0 ) [ ′ + 0 [ ′ + 0[ ′ = 0 () () ( ) () 

(

)

Ø () [Ø + 1() + 1() [ + 1 ( ) [ + 1( ) [  = 1 () ( ) ( ) () IRUN = 

( 0 + 0 ) [ ′() + 0[ ′() + 0[ ′() + 0[ ′() =

(

)

     = 1( ) + 1( ) [ () + 1( ) [ () + 1( ) [ () + 1( ) [ ( ) 

IRUN = 





Ø ( ) [Ø ′ + 0 [ ′ + 0[ ′ + 0 [ ′ + 0 [ ′ = 0 () () ( ) () () Ø ( ) [Ø + 1 () [ + 1( ) [ + 1 ( ) [ + 1( ) [  = 1 () () ( ) () ()

IRUN ≥  



0[ ′( N − ) + 0[ ′( N −) + 0[ ′( N ) + 0 [ ′( N +) + 0 [ ′( N + ) = = 1(

N − )

N − N N + N + [ ( N − ) + 1 ( ) [ ( N −) + 1( ) [ ( N ) + 1 ( ) [ ( N +) + 1( ) [ ( N + ) 



  

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−   NPD[ −   DQG NPD[ −  KDYHWREHGLVWLQJXLVKHGQDPHO\

IRUN ≤ NPD[ − 

0[ ′( N − ) + 0[ ′( N −) + 0[ ′( N ) + 0 [ ′( N +) + 0 [ ′( N + ) = = 1(

N − )

IRUN = NPD[ − 

N − N N + N + [ ( N − ) + 1 ( ) [ ( N −) + 1( ) [ ( N ) + 1 ( ) [ ( N +) + 1( ) [ ( N + ) 



0 [ ′( NPD[ − ) + 0 [ ′( NPD[ −) + 0[ ′( NPD[ −) + 0 [ ′( NPD[ −) = = 1( PD[ N

− )

− − − N N N [ ( NPD[ − ) + 1 ( PD[ ) [ ( NPD[ −) + 1( PD[ ) [ ( NPD[ −) + 1( PD[ ) [ ( NPD[ −) 



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IRUN = NPD[ − 

0 [ ′( NPD[ −) + 0 [ ′( NPD[ − ) + 0[ ′( NPD[ −) + 0 [ ′( NPD[ ) = = 1( PD[ N

IRUN = NPD[ − 

− )

− − N N N [ ( NPD[ − ) + 1 ( PD[ ) [ ( NPD[ − ) + 1( PD[ ) [ ( NPD[ −) + 1( PD[ ) [ ( NPD[ ) 

0[ ′( NPD[ −) + 0 [ ′( NPD[ −) + 0[ ′( NPD[ −) = = 1( PD[ N

IRUN = NPD[ 



− )

− − N N [ ( NPD[ −) + 1 ( PD[ ) [ ( NPD[ −) + 1( PD[ ) [ ( NPD[ −) 

0 [ ′( NPD[ − ) + 0[ ′( NPD[ −) + 0[ ′( NPD[ ) = = 1( PD[ N

− )

− N N [ ( NPD[ − ) + 1 ( PD[ ) [ ( NPD[ −) + 1( ) [ ( NPD[ ) 







  

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1



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5

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