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s(t)7:ZTq"^k7:?A;R7KV<NR2CB

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SE3S<7KHJB

Rk(

s = s(r(t))� b � �

F=�r(t)

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s =∂s

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sb

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b �_;oRc

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nJn(q) q� b �

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s = LscTn

nJn(q) q +∂s

∂t

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S<7KHJBRc

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s = LscTn

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s = −LscTn

nJn(q) q +∂s

∂t

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(xp

yp

)= Lxpyp

T +

((xp − xc)/f(yp − yc)/f

)f

� b ! � �

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h(X, Y, Z, P1, ..., Pn) = 0� b ! � �

� B@7r9<?AF � 2-4z=A;>FIH�S<7KHJB�NR2�9<NR7KH*;R`'7:fI2 (

g(x, y, p1, ..., pm) = 0� b !�! �

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1/Z = µ(x, y, P1, ..., Pl) = 0,� b ! �

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h(X, Y, Z,P) =

{h1 = A1X + B1Y + C1Z + D1 = 0h2 = A2X + B2Y + C2Z = 0

� b !.+ �

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Mr9J7:?@=@;R?�ST2h1

(

1/Z = αx + βy + γ� b"! �

7 ^K2-4α = −A1/D1, β = −B1/D1

2Q=γ = −C1/D1

b

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(

ax + by + c = 07 ^K2C4

a = A2, b = B2, c = C2� b !)6 �

m[7�9J7:?7:`�LG=A?@;OBA7k=@;RFKH(a, b, c)

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p = (ρ, θ)S<LG}JH<;R2�9J7K?(

g(x, y,p) = x cos θ + y sin θ − ρ = 0� b"! % �

F=�θ = arctan (b/a)

2G=ρ = −c/

√a2 + b2

� ^KFI;>?�}8fKZ<?A2 b zb

-

6

b b bb b bb b bb

~X

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θ

ρ

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ρ + (x sin θ − y cos θ) θ = x cos θ + y sin θ , ∀(x, y) ∈ D � b ! ( �

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yBD;

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2G= � b"! ��(

(ρ + ρ tan θ θ) + y (−θ/ cos θ) = K1 T + y K2 T , ∀y ∈ R� b ! * �

7 ^K2C4 (

K1 = ( λ1 cos θ λ1 sin θ −λ1ρ sin θ − cos θ − ρ2/ cos θ −ρ tan θ )K2 = ( λ2 cos θ λ2 sin θ −λ2ρ ρ ρ tan θ 1/ cos θ )

F=�λ1 = −αρ/ cos θ − γ

2G=λ2 = α tan θ − β

b0�H�2QH�STL-STZ<;>=�7:ZJBAB@;g=Au:= (

{ρ = (K1 + ρ sin θ K2) Tθ = − cos θ K2 T

� b � �

Sa\ FD� (

Lρ = ( λρ cos θ λρ sin θ −λρρ (1 + ρ2) sin θ −(1 + ρ2) cos θ 0 )Lθ = ( λθ cos θ λθ sin θ −λθρ −ρ cos θ −ρ sin θ −1 )� b � �

7 ^K2C4λρ = −αρ cos θ − βρ sin θ − γ

2Q=λθ = −α sin θ + β cos θ

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(ρ cos θ + sin θ ρ sin θ − cos θ

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(x y)T2Q=

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Θ = (a1 b1 c1 a2 b2 , c2)T=@2QNRNR2UYIZJ2�NR7r?A2QNO7k=@;RFKH (

{x = a1 x∗ + b1 y∗ + c1

y = a2 x∗ + b2 y∗ + c2

� b ! �

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Θ9<?@L-BD2CH�=@2UZ<HJ2{9o2Q?@=@2�ST2{?7:H<f � ST2

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ySa\ Z<HE9oFK;RH�=�S<Zr4QFKH�=@FIZ<?/BC\ LC4G?A;R^K2QH�= (

x = xg +ρ(θ) cos θ , y = yg +ρ(θ) sin θF=�

xg

2G=yg

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Sa\ Z<H�FKV � 2G=Cn:S<LG}JH<;OB9J7:? (

mij(t) =

∫ ∫

D(t)

xiyjdxdy� b � �

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tb��_;�N�\ FKH

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1/Z = Ax + By + Cn8FIHPFKVT=A;>2CHI=l98FIZ<?wNc\ 7:;R?@2

a (= m00)2Q=�NR2CBw4QF_FK?STFKH<HJLQ2CB

xg(= m10/m00)2Q=

yg(= m01/m00)STZP4Q2QH�=@?A2�ST2�fK?7 ^_;g=ALlST2WNc\ FIV�� 2G=�(

La= ( −aA −aB a(3/Zg − C) 3ayg −3axg 0 )Lxg

= (−1/Zg 0 xg/Zg + ε1 xgyg + n11 −(1 + x2g + n20) yg )

Lyg= ( 0 −1/Zg yg/Zg + ε2 1 + y2

g + n02 −xgyg − n11 −xg)� b + �7 ^K2C4

1/Zg = Axg +Byg +Cnε1 = An20+Bn11

nε2 = An11+Bn02

2G=�FD�n20, n022G=

n11BDFIHI=�NR2CB�`�FK`�2CHI=B�4G2CH�=@?ALCBtH<FK?A`'7:NR;RB@LCB�Sa\ FK?ST?A2 !�S<LG}JH<;OB�9J7K? (

nij = 4µij/a7 ^I2C4

µ20 = m20 − ax2g

µ02 = m02 − ay2g

µ11 = m11 − axgyg

� b �

0�H'H<FK=@2wY�Z<2wNR7�^_;>=@2-B@B@2aSTLC982CHJS'NR;>HJLC7:;R?A2Q`�2QH�=hST2

anI2G=32-B =3H�ZJN>NR2�9oFKZ<?~=AFKZJBhN>2-B

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A = B = xg = yg = 0 Gb

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Lµ20= ( −3Aµ20 − 2Bµ11 −Bµ20 As20 + Bt20 + 4Cµ20

t20 −s20 2µ11 )Lµ02

= ( −Aµ02 −2Aµ11 − 3Bµ02 As02 + Bt02 + 4Cµ02

t02 −s02 −2µ11 )Lµ11

= ( −2Aµ11 − Bµ02 −2Bµ11 − Aµ20 As11 + Bt11 + 4Cµ11

t11 −s11 µ02 − µ20 )� b 6 �7 ^K2C4 (

s20 = 7xgµ20 + 5µ30 , t20 = 5(ygµ20 + µ21) + 2xgµ11

s02 = 5(xgµ02 + µ12) + 2ygµ11 , t02 = 7ygµ02 + 5µ03

s11 = 6xgµ11 + 5µ21 + ygµ20 , t11 = 6ygµ11 + 5µ12 + xgµ02

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µ30 = m30 − 3xgm20 + 2ax3g , µ21 = m21 − 2xgm11 − ygm20 + 2ax2

gyg

µ03 = m03 − 3ygm02 + 2ay3g , µ12 = m12 − 2ygm11 − xgm02 + 2axgy

2g

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s =(a xg yg µ20 µ11 µ02)

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r(t)b&0�Hr7l2QH{2�o2Q=

Sa\ 7K9<?AvCB � b % ��(˙r(t) =

∂r

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∂rT � b ( �

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∂r

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∂x

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SJ7:HJBlNR2�4C7KBSa\ ZJH<2U4Q7K`rLC?A7{2Q`EVJ7:?Y�Z<LQ2l2G= Tq = −To

S<7:H8B3NR2�4Q7IBtS]\ Z<HJ2U4Q7:`�LC?A7ESTLQ9oFK?@=@LC2 zb 0�HLC4G?A;>=w7:NRFK?B (

e = C Ls Tq +∂e

∂t

� b 6 �F=� ∂e

∂t

?@2C9<?@L-BD2CH�=@2lN>2-B�^ 7K?@;O7k=A;>FIHJBtST2eSTZ<2CB�B@FK;>=�7:Z�`�FKZ<^I2Q`�2QH�=�ST2WN�\ FKV�� 2Q= � B@;aN�\ FKH

ST;RB@9oFIB@2WS]\ Z<HJ2U4Q7:`�LC?A7{2Q`{V87:?YIZJLQ2 Gn<BDFI;g=�7KZ5`�FKZ<^I2Q`�2QH�=�ST2WNO7r4C7:`�LQ?7 � B@;�4G2CN>NR2GXe4G;2CBD=ESTLQ9oFK?@=@LC2'2G={`�FIV<;>NR2 Gb�4�FKZJ?{4GFIHI=A?@uIN>2C?�N>2-B 6PST2CfK?ALCBUST25N>;RV82C?D=AL�STZ0B@s_BD=@vC`�2Kn�;RNi�7:ZT=t98FIZ<?hNR2w`�FK;RHJB3BDLCN>2-4z=@;RFKHJH<2Q?

s=@2CN8Y�Z<2

Ls

BDFI;g=tST2w?7:H<f 6U2G=3FIH'FKVT=A;>2CH�=34GFI`r`�2N>FI;�ST2�4GFI`r`'7KHJST2W;RS<LC7:NR2 (

Tq = (C Ls)−1

(−λe− ∂e

∂t

) � b &% �

-l7:HJBENR2*4Q7IBUFD�0N>2-BU;RHTijFK?A`'7k=A;>FIHJBU^_;OBDZ<2CN>NR2CBEBDFIH�=E2GqT9<?A;R`rLC2CBES<7:HJB{N�\ ;R`�7KfK2In�FKH7*^_Z YIZJ2rNO7�`'7k=@?A;O4G2�Sa\ ;RH�=@2C?A7I4z=@;RFKH STLQ9o2QH8S ST2�NO75^k7KN>2CZ<?�ST2'4G2CBW;RHTijFI?@`'7k=A;>FIHJB�^�;>XBDZ<2CN>NR2CBw2G=WST2UNO7'9<?AF:ijFKH8ST2QZ<?w2QH�=@?A2UNO7�4Q7K`rLC?A7�2Q=�Nc\ FIV�� 2G=lSa\ ;RH�=@LC?@pQ=Cb -l7:H8B�NR2E4Q7IB�F=�N>2-B�;>H<ijFK?A`�7:=@;RFKHJBU^�;OB@Z<2QNRN>2-B�BDFIH�={ST2�= s_982 -{n[BD2CZ<N>2-BUST2-B�6_s�9oF:=A6<vCB@2CB{7KBABD2��EijFI?D=A2CB982C?@`�2G=@=@2CHI=tN�\ FKV<=@2QH�=A;>FIH�7KHJ7:NRs�=@;OY�Z<2�ST2w4Q2G=D=A2�`'7k=A?@;O4G2Ib)-�7KHJB~=AFKZJB�N>2-Bh4Q7IB�ST2�}8fKZ<?A2KnST2CB32Q?A?@2CZ<?B�ST2w`�2-BDZ<?A2�2Q=tSa\ 2CBD=@;R`'7k=@;RFKH*BDFIHI=3;>H<LC^_;g=7:V<NR2CBh2G=3NO7�^k7:NR2QZJ?32QqT7I4z=A2wST2

Ls2CBD=3;RHJ4GFIH<H_Z<2Kb �T2QZ<NR2wZ<HJ2w7:9<9J?@F qT;R`�7:=@;RFKHLs

9o2QZT=�STFKH84�pG=A?@2w4QFKHJB@;OSTLQ?ALQ2�S<7KHJB�NO7�NRFK;ST2�4GFK`�`'7:H8ST2Kb.-�2�9<N>Z8BQnkNR2t=@2Q?A`�2

∂e∂t

2CBD=�2CHEfILQH<LC?A7KNI;RHJ4QFKH<H_Z]b:m[7wNRFK;<ST2�4QFK`�`'7:HJST2ZT=@;RN>;OB@LQ2�2QH�9<?7k=A;RY�Z<2W2-B =�STFIHJ4 (

Tq =(C Ls

)−1(−λe − ∂e

∂t

)� b )( �

�_;�FIH�BDZJ9<98F�BD2 Y�Z<2�4Q2G=D=A2�4QFKHJB@;>fIH<202-B = 9J7:?@i�7:;>=@2C`�2QH�=y?ALC7:NR;OBDLC2KnwNc\ ZT=@;RN>;OBA7k=@;RFKHST2 � b )( �]S<7KHJB � b 6 ��STFIH<H<2 (

e = −λ C Ls

(C Ls

)−1

e −C Ls

(C Ls

)−1 ∂e

∂t+

∂e

∂t

� b )* �

0�H 4GFIHJB =7k=A2 7:NRFK?BCnh2QH BDZ<9J98F�B@7KHI=�98FIZ<?�BD;R`�9<N>;>}J2C?�Y�Z<2 ∂e∂t

=�

∂e∂t

= 0n�Y�Z<2yNO7

4GFKH8ST;g=A;>FIHPST2W98F�BD;>=@;R^_;g=AL (

C Ls

(C Ls

)−1

> 0� b 6 � �

2CBD=�B@Z�'�B@7KH�=@23M�7KBABDZJ?@2C?�NO7wSTL-4G?AFK;OB@BA7:HJ4Q2�ST2 ‖e‖ b -w2t9<NRZJBQn-FKHE?@2Q=@?AFKZ<^I2Q?7�N>2t4GFI`r9oFK?@X=@2Q`�2CHI=WBD9oLC4Q;g}8LU2QH � b + �[BDFIZJB�Nc\ ZJH<;RY�Z<2r4GFIHJST;>=@;RFKHPY�Z<2Ls = Ls

2G=�Y�Z<2 �

∂e∂t

= ∂e∂t

b

� � � � ��� �!� � ��� ��� ��� �!� "���# � �j�

* FKZ8B�^K2Q?A?AFKHJBt7:Z�9J7:?7:fI?A7K9<6<2 b b +�4GFI`�`r2CH�=�;RNa2CBD=�98F�B@B@;RV<N>2�Sa\ 2CBD=@;R`�2Q? ∂e∂t

n<4G2�Y�Z<;982C?@`�2G=w7KN>FI?ABtST2�?ALCSTZJ;>?A2l7KZT=A7KH�=�Y�Z<2W9oFIBABD;RV<NR2WN>2-B�2Q?A?@2CZ<?ABtST2W=A?A7��OHJ7KfK2Ib * FKZJBtH<FIZJB;>H�=@LC?@2-B@B@FKH8BtME9<?ALCB@2QH�=�7:ZTq*S<; ��LQ?A2QH�=AB�46<FI;gq�9oFIBAB@;>V<NR2CB�9oFKZJ?

C2G=

Ls

b * FKZJB�B@Z<9<9oF:XBDFIHJB�STFKH84�987:?�NO7rB@Z<;>=@2UY�Z<2 ∂e

∂t=

∂e∂t

= 09oFKZJ?�H<2�9J7KB�7KN>FIZ<?ST;>?�NR2CB�H<FK=A7:=@;RFKHJBCb

b b"!Tb � b � � "��s��� � �)��� � ����� ����" �����

k = 6 ��_;

s2-B =�ST2PST;R`�2QHJB@;RFKH 6Jn~FKH 7y=AFKZT=';RH�=@LQ?ApG=�M 46<FK;OB@;>?

C = I6b���H 2 ��2G=-n~N>2

4GFK`�9oFK?@=@2C`r2CH�=rST2sB@2Q?7x7KN>FI?AB{N>2�`�pQ`�2�Y�Z<2�4G2CN>Z<;�ST2

e� 4:\ 2CBD=DXeMkXeST;R?@2*YIZJ2*S<7KHJB

N>2�4Q7IB�;OSTLC7KN�no=@FIZT=@2-BwNR2CBl4GFI`�98F�B@7KHI=A2CB�ST2sSTL-4G?AF��O=@?AFKH�=�ST2r`'7:H<;RvQ?A2U2GqT9oFKH<2CH�=@;R2QNRN>2

STLC4QFKZ<9<NRLQ2 zb�-l7:H8B�4G2U4Q7IBQn_FIH�FIVT=@;R2QH�=�4GFI`�`r2�NRFK;]S<2U4GFK`�`'7:H8ST2 (

Tq = −λ Ls

−1e = −λ Ls

−1(s − s∗)

� b 6 � �2G=w4GFI`�`r2�4QFKHJST;>=@;RFKHPST2�BD=A7KV<;>NR;>=@L (

Ls Ls

−1> 0

� b 6&! ��_;�N�\ FKH�2-B =t2QH*`r2-BDZJ?@2lSa\ 2-B =A;>`�2C?t4QFK?A?@2-4z=@2C`�2QH�=hNO7U^k7:NR2QZ<?�4GFIZ<?7:H�=@2wS<2

Ls

ME46J7KY�Z<2;g=ALQ?7k=@;RFKH"ST2'NO7�NRFK;hST2�4QFK`�`'7:HJS<2Kn]NO7�9<?A;RB@2�2QH04GFI`�9T=@2�ST2�4Q2G=@=@2�2-B =A;>`'7k=A;>FIH 9o2Q?@X`r2Q=D=A?A7'ST2�B@2�?A7K9<9<?AFT46<2Q?�7:Z�`�;>2CZTq�STZP4QFK`�98FI?D=A2Q`�2QH�=�;OSTLC7KN

s = −λ sb

b b"!Tb � b � b'deHTijFI?@`'7k=A;>FIHJBt^_;RB@Z<2QNRNR2CB,!.-

��H�4GFKH8BD;OSTLQ?7:H�=UST2-BU;RHTijFK?A`'7k=@;RFKH8B�^_;RB@Z<2QNRNR2CB !)-{n[;>N32CBD={`'7:NR6<2QZ<?A2QZ8BD2C`r2CH�=U;>`rX98F�B@B@;>VJN>2 � Mr4G2 � FIZ<? �ST2WBD2�?@2Q=@?AFKZ<^I2Q?tS<7:HJBt4G2l4Q7IB3ST2w}JfKZJ?@2lB@7KHJBhNR;R`r;>=@2C?hijFI?D=A2Q`�2QH�=N�\ 2CB@9J7K4Q2rS<2E=A?A7 ^k7K;>Ncbom[7*S<;$'�4GZJNg=ALr9J?@;RHJ4G;R9J7KN>2rH<2�98FI?D=A2E987KBWBDZJ?lNc\ 2CBD=@;R`�7:=@;RFKH ST2rNR7^ 7KN>2CZ<?w4QFKZ<?7:H�=@2WST2UNO7r`'7k=@?A;O4G2USa\ ;>H�=A2Q?7K4z=A;>FIH]nT`'7:;OB�B@Z<?�NO7rB@LQNR2C4G=@;RFKHPST2-B�BD;>q*;RHTijFI?DX`�7:=@;RFKHJB�^_;OBDZJ2QNRN>2-BQb

13FIHJBD;OSTLC?@FIHJB/9J7:?�2GqT2C`r9JN>2�NR2�4C7KB/F=�s2CBD=�4GFKH8B =A;g=AZ<L�ST2CB�4GF_FI?ASTFIH<H<LC2CB�ST2�NR7�9<?AF:X

� 2C4z=A;>FIH0ST2�=A?@FI;RBU98FI;>H�=ABES<7:H8B�N�\ ;R`�7KfK2Ib[m�7�`'7:=@?A;R4Q25Sa\ ;>H�=@2C?A7I4z=A;>FIH 7IB@B@FT4G;RLQ2Ls

2-B =7:NRFK?BtS<2WS<;>`�2QH8BD;RFKH

6× 62G=�2CBD=Cn<SJ7:HJB�NO7r9<N>ZJ9J7:?@=�ST2CB�4C7KBCn_S<2l?7:HJfE9<NR2Q;RH 6<b�#x7K;RBCn

FKH'9o2QZT=t`rFIH�=@?A2Q? � #Pd@1 * n�4 � 4 *� ��Y�Z<2w4G2C?D=7:;RH<2CBh4GFKH<}JfKZ<?7k=A;>FIHJB�2QH�=A?A7��OH<2CH�=hZ<H<2982C?D=A2�S<2r?7:HJf*ST2

Ls

b ��H Nc\ FT4C4GZ<?A?@2CHJ4G2Ino4Q2CB�B@;>HJfKZ<NO7:?A;g=ALCBlBDFIH�=�=A2QNRN>2-B�YIZJ2ENR2CBl=A?@FI;RB98FI;>H�=AB�B@FK;R2QH�=�7:NR;RfKH<L-BtSJ7:HJB�Nc\ ;R`'7:fI2lFIZ�Y�Z<2WNR2�4G2QH�=A?@2WFK9<=@;OYIZJ2

CST2WNO7r4C7:`�LQ?7r7:9TX

9J7:?@=@;R2QH<HJ2�M�NO7�B@Z<?@i�7K4G2�STZ�4Gs_NR;>HJS<?@2�STLG}8H<;T9J7:?�NR2�4G2Q?4GNR2�4G;R?A4QFKHJBA4G?A;g=�7:ZTq�=A?@FI;RB/9oFK;RH�=AB� ^KFI;>?/}JfIZ<?A2 b * Gb deNT2CBD=�STFIHJ4�ST; ''4Q;>NR2�Sa\ 7IB@B@Z<?A2Q?�Y�Z<2Kn:Y�Z<2CN>NR2tY�Z<2�B@FK;>=�NO7�98F�BD;>=@;RFKHE;>H<;>X=@;O7:NR2U46<FK;OBD;R2Kn_NR2W`rFIZ<^K2C`�2QH�=�STZ�?@FIV8FK=�LC^_;g=A2Q?7EST2l=A?A7 ^I2Q?BD2C?3Z<HJ2UBD;RH<fKZJNR7K?@;>=@Ll;RB@FKNRLQ2� F=��LC^�;OST2C`r`�2CHI=�NO7�4GFIHJST;>=@;RFKH�ST2UB =7:V<;RNR;g=AL � b 6�! �]H]\ 2CBD=�9JN>ZJB�?@2-BD9o2C4G=@LQ2 zb-�2U9<NRZJBCnJ;RN]2GqT;OB =A2�fKLQHJLQ?7:NR2Q`�2QH�=�Y�ZJ7k=A?@2�B@;g=AZJ7k=A;>FIHJB�ST;OB =A;>H84z=@2-B�2CHI=A?@2�NO7�4Q7K`rLC?A7

2G=hNO7WB@4QvQH<2�=A2QNRN>2-B�Y�Z<2�N�\ ;R`�7KfK2�ST2�=A?@FI;RB�9oFK;RH�=AB�B@FK;>=�NR7W`�pC`r2 � -�! 0 ()* � bKm�7W`�;RH<;R`r;>XB@7:=@;RFKH�ST2 ‖s−s∗‖ 982CZT=tS<FKHJ4�7K`�2QH<2C?�N>2w?@FIV8FK=hM�N�\ Z<H�ST2-BhY�ZJ7:=@?A2�`r;RH<;R`'7�fKNRFKVJ7KZTq

� � � ������� ���������������������������������! �"�#$������"����

�����R���L� �����R����Gj2 )*.�F5%�#OFS#H!()�.;X51=2I>5%()*/BM

=@2QNOBtYIZJ2 ‖s− s∗‖ = 0b � ;>H8BD;cnIS<7KHJBh4Q2w4Q7IBQnIFKH'9o2QZT=3=@?AvCB~VJ;>2CH57 ^KFI;>?

s− s∗ = 0B@7KHJB

YIZJ2�NR7�B@;g=AZJ7k=A;>FIH�7:=D=@2C;>H�=A2UBDFI;g=�4G2CN>NR2UY�Z<;]7rLG=ALUBD9oLC4Q;g}JLC2Kb��H 4QFKHJB@;OSTLQ?7:H�='ST2CB�;RHTijFK?A`'7k=A;>FIHJB�^_;RB@Z<2QNRNR2CB'ST2�HJ7:=@Z<?A2PST; ��LQ?A2QH�=A2 � =@2CN>NR2CB�9J7:?

2GqT2Q`�9<NR2�NR2CB ST?@FI;g=A2CBtY�Z<2�Nc\ FKH59o2QZ<=�S<LG}JH<;R?�MU9J7K?D=A;>?�ST2 9oFK;RH�=AB3H<FKH�7:NR;>fIH<LCB zn�FKH?@2Q=@FK`EV825V<;>2CH"LQ^_;OST2Q`�`�2QH�=EB@Z<?UNR2CB{`rpC`�2CBU9J?@FIV<N>vC`�2CB�9oF:=A2QH�=@;R2QNOBQb 4�FKZ<?Us ?A2Q`�LGXST;>2C?Cn�Z<H<2lBDFIN>Z<=@;RFKH�9<?7k=A;RY�Z<2�4GFKH8BD;OB =A2wMU?A2CBD=@?A2Q;RHJST?A2�Nc\ 2-BD9J7I4G2wS<2w=A?A7 ^k7:;RNJMEST2CB �QFIH<2CB9<?@FT46<2-BhST2lNO7UBD;>=@Z87k=@;RFKH*STLCB@;>?ALQ2InIHJ2�4GFI`�98FI?D=7:H�=h9J7IB3ST2lBD;RH<fIZ<NR7K?@;>=@L-B�;OBDFIN>LC2CBCnK2Q=3F=�NR7r`�;>HJ;>`�;OB@7:=@;RFKHPST2 ‖s− s∗‖ 7k=@=@;R?@2WNc\ 2�o2-4z=A2QZ<?wS<Z�?AFKVoF:=�^K2C?AB�BA7E98F�BD;>=@;RFKHPSTL-BD;R?@LC2Kbm�7�STLG=A2Q?A`�;>HJ7:=@;RFKH�ST2�NO7{=7:;RN>NR2UST2U4Q2CB#�QFIH<2CBt2-B =w4Q2Q9o2QHJS<7KH�=�=@?AvCB�9<?AFKV<NRLQ`'7:=@;OYIZJ2Kb

4~7:?/7:;RN>NR2QZJ?ABCnGBD;IN�\ FKH�Z<=@;RN>;OBD2hN>2-B�6�9<?A2Q`�;R2Q?Ba`�FK`�2QH�=AB[Sa\ Z<H�FKV�� 2Q=[S<7KHJBaNc\ ;R`'7:fK2InGFKH982CZT=/`�FIHI=A?@2C?ai�7K4Q;>NR2Q`�2CHI=�M�9J7K?D=A;>?�S<2CB[LCY�ZJ7k=A;>FIHJB]STFIH<H<LC2CB]2CH b"!Tb +Jb Y�Z<2�NO7�`'7k=A?@;O4G2Sa\ ;RH�=@2C?A7I4z=@;RFKHU7IB@B@FT4G;RLQ2�2-B =�7:Z�9<NRZJB/ST2h?7:HJf B@;INO7�ijFK?A`�23ST23N�\ FKV � 2G=[S<7KHJB[N�\ ;>`'7KfK2�2-B ==@2QNRNR2wY�Z<2�BD2-B~`�FK`�2QH�=B�4G2CHI=A?@L-B~Sa\ FK?ST?@2 B@FKH�=~=@FIZJB~H_Z<NOB � 4Q2�Y�Z<;J2-B =�NR2�4Q7IB~98FIZ<?�Z<H?@2-4z=A7KH<fKNR2U9J7K?w2Qq_2C`�9<N>2 zb ��Hy4QFKHJB@;RS<LQ?7:H�=�ST2-B�FKV � 2G=ABlST2{ijFK?A`r2{9<N>Z8B�4QFK`�9<NR2GqT2Kn8NR7`�7:=@?A;R4Q2�Sa\ ;>H�=@2C?A7I4z=A;>FIHl2-B =�7KN>FI?ABaS<2�?A7KH<f 6<nC`�7K;RB�BDFIHW4QFKHJS<;g=A;>FIH<H<2Q`�2CHI=]2-B =[`rL-ST;RF_4Q?@22G=�HJ2U982C?@`�2Q=�9J7IB�Sa\ FKVT=A2QH<;R?�ST2-B�?ALCB@Z<N>=A7k=BwBA7k=A;RBDi�7:;OB@7KH�=ABCb -w2E`rpC`�2Kn�BD;[N�\ FKHPZT=A;>NR;OBD2N>2-B*B@;gq =@2C?@`�2-B5STL-4G?A;>^k7:H�=5NO70STLQijFK?A`'7k=@;RFKH 7 ''H<2ySa\ Z<H FIV�� 2G= � ^KFI;>? b"!Tb +Jb Gn�Z<H<2982C?D=A2�ST2*?A7KH<fyST2*NR7y`'7k=@?A;O4G2*S]\ ;RH�=@2Q?7K4G=@;RFKH";RH�=@2Q?A^_;>2CH�=ENRFK?B@Y�Z<2�Nc\ FKV�� 2Q=r4GFIHJB@;RSTLC?@L2CBD=�9J7:?7:NRN>vCN>2W7:Z�9<NO7:H�;R`�7KfK2 � 1 0lm *�* Van�- 2 � *)* � b

4�FKZJ?�N>2-B�ST; �oLC?@2CH�=@2CB�?A7K;RB@FKHJB�LC^KFTY�Z<LQ2-B�4G;>X S<2CBABDZJBCn<FIH�Z<=@;RN>;OBD2U=@?AvCB�fKLCH<LQ?7:NR2Q`�2CHI=ST2CB{;>HTijFI?@`'7:=@;RFKHJB�^_;OBDZJ2QNRN>2-B !.- ?@2-STFKH8S<7:H�=@2-BQb 0�H 2-B =�7:NRFK?B�S<7:H8BUNR254Q7IB�F=�

k2CBD=

BDZ<9oLQ?A;R2QZ<?�M 6<n8YIZJ;]BD2C?A7�STL-4G?A;g=�7:Z�9J7K?A7KfK?7:9<6<2 b b"!Tb"!Tb

b b"!Tb � b"!Tb'deHTijFI?@`'7k=A;>FIHJBt^_;RB@Z<2QNRNR2CB -

mh\ 2C`r9JN>FI;�S]\ ;RHTijFK?A`'7k=A;>FIHJB�^_;OBDZ<2CN>NR2CB - 9o2Q?A`�2G=tS]\ LC^�;>=@2C?3NR2CBt9<?AFKVJN>vC`r2-B�LC^KFTY�Z<LCB4G;>X ST2-B@B@ZJB�9JZ<;RBAY�Z<2'Nc\ FIH S<;RB@98F�BD2�7KN>FI?AB�S<2CB�=A?@FI;RBW987:?7:`�vG=A?@2-B

θu9oFKZ<?U?A2Q9J?@L-BD2CHI=A2Q?

� � � � ��� �!� � ��� ��� ��� �!� "���# � ���

N�\ FK?A;>2CH�=A7k=A;>FIHy2Q={YIZ�\ ;RN�BDZ 'r={S<2�46<FK;OBD;R?l=@?AFK;OBl9J7:?7:`�vG=A?@2-BWST2�98F�BD;>=@;RFKH 9J7:?A`�;�4G2CZTqSTFKH<HJLCBl2QH b"!Tb b 9oFKZ<?�7 ^KFI;>?

k = 6b�0�Hy?7:9<9o2QNRN>2E=@FIZT=UST2r`rpC`�2�Y�Z]\ ;RN/2-B =U7KN>FI?AB

H<LC4Q2CBAB@7K;>?A2EST2'B@2E9JNR7I4G2Q?�S<7KHJBlN>2'4C7KBl;RS<LC7:N�FD� NR2CBWST;R^K2C?AB@2CB�2Q?A?@2CZ<?B�ST2�`�2CB@Z<?A2KnaST24Q7:NR;RV<?A7:=@;RFKH�2G=�Sa\ 2CBD=@;R`'7k=@;RFKHPB@FKH�=�H<LQfIN>;RfK2-7:V<NR2CBt9oFKZ<?�9oFKZJ^KFK;R?�2GqT9<?A;R`r2C?�NR7�`'7k=A?@;O4G2Sa\ ;RH�=@2C?A7I4z=@;RFKH�b -�7KHJB�NR7 B@Z<;g=A2xST2P4G2�9J7K?A7KfK?7:9<6<2In�H<FIZJB�H<FIZJB�9<NO7 �GFIHJB�STFKH84*S<7KHJB4G2P4Q7IBE;OSTL-7:N�Y�Z<;�7KBABDZJ?@2 � 2QH�=@6JLQFK?A;R2 EN>2�4GFI`�98FI?D=A2Q`�2QH�='BD9oLC4Q;g}JL

s = −λs2G='NO7

4GFKH8ST;g=A;>FIHPST2�B =7:V<;RN>;>=@L � b 6�! �]S<7:HJBt=AFKZT=�Nc\ 2CB@9J7K4Q2WS<2W=@?7 ^ 7K;>Ncb�0�H�7'7KN>FI?ABt2CH*2�o2Q=�(

Ls Ls

−1= Ls Ls

−1 = I6 > 0� b 6 �

�_;/Nc\ FIHy46<FK;OB@;g=�S]\ ZT=A;>NR;OBD2C?�NR2CBl4GF_FK?STFIH<H<LQ2-BXS]\ Z<Hy98FI;>H�=�NR;RLEM�Nc\ FKV�� 2Q=w2GqT9<?A;>`�L

S<7:HJB/NR2h?A2Q9ovQ?A234QFKZ<?7:H�=/S<2hNO7�4C7:`�LQ?7Rc

n NO7�`'7k=A?@;O4G2tSa\ ;>H�=@2C?A7I4z=A;>FIHUfKNRFKVJ7KN>237KBABDFT4G;RLQ2M

s = (X θu)T2CBD=�STFKHJH<LQ2�9J7K? (

Ls =

(−I3 X

03 Lω

) � b 6)+ �

0�HP9o2QZT=lH<F:=A2Q?�NO7�ijFI?@`�2UB@s�`�9J7:=@6<;OY�Z<2{ST2E4G2G=@=@2U`'7:=@?A;R4Q2 � V<NRFT4�=@?A;O7:H<fIZ<NR7K;>?A2W2G=H<FKH�BD;RH<fIZ<N>;RvQ?A2WB@7KZTi[2CH

θ = 2π zn<Y�Z<;]^k7rSTFKHJH<2Q?�Z<H�4GFI`�98FI?D=A2Q`�2QH�=�;>H�=ALQ?A2CBAB@7KHI=�7:Z

BDsTBD=@vQ`�2Ib���H 2 ��2G=-no;OSTL-7:NR2Q`�2QH�=CnoNO7'=@?7 � 2-4z=AFK;R?@2{S<Zx98FI;>H�=�4GFIHJBD;OSTLC?@L{B@2Q?7'?A2C4G=@;RN>;RfKH<2S<7:HJBlNc\ ;R`'7:fI2 � # � m *�( � b �_;�4G2r98FI;>H�=�2CBD=�S<7:HJBlNR2'46J7:`�9 S<2r^_;OBD;RFKH ST2rNR7�4Q7K`rLC?A72QH�BD2-B39oFIB@;>=@;RFKHJB3;>HJ;g=A;R7KN>2W2G=�STL-BD;R?ALQ2Kn�;RNaN>2WBD2C?A7ESTFKH84w2CH*9o2Q?A`'7:H<2CHJ4G2Ib �hH�B@LQNR2C4G=@;RFKHTXHJ7:H�={4GFI?@?A2C4G=@2C`r2CH�=�4Q2�98FI;>H�= � 7KZ"4G2QH�=A?@2�ST2'fI?A7 ^_;>=@L�ST2'Nc\ FIV�� 2G=U9J7K?�2GqT2Q`�9<NR2 Gn]FKH982CZT=�7KN>FI?ABU`r;RH<;R`�;RB@2Q?EN>2*?@;OB@Y�Z<2*ST259o2Q?ST?A2�Z<H<2*fK?7:HJST2'987:?@=@;R2*ST2�Nc\ FKV�� 2Q=rSa\ ;>H�=ALGX?@pQ=�7KZ�4GFIZ<?BEST2�Nc\ 7IB@B@2Q?A^_;RBAB@2Q`�2QH�= � B@7KHJBE=@FKZ<=@2GijFI;RB�7IB@B@Z<?A2Q?EY�Z<2�B@Z�'�B@7K`r`�2CHI='ST24Q7:?7K4G=@LC?@;OB =A;RY�Z<2-BtH<LC4Q2CBAB@7K;>?A2CBt7:ZP4Q7KNR4QZ<N]ST2�9oFIB@2W?@2-B =A2QH�=�^�;OB@;>V<NR2CB zb � 4687KY�Z<2W;g=ALQ?7kX=@;RFKH]n/FKH"9o2QZT=�LQf�7:NR2Q`�2QH�=EB@LQNR2C4G=@;RFKH<HJ2Q?UNR259oFK;RH�= - 4GFI?@?A2CB@98FIHJS<7KHI={7KZ"98FI;>H�=�!.-N>2�9<NRZJBl9<?AF_46J2rST2-BlN>;R`�;g=A2CBWSTZ 9<NR7KHy;R`'7:fK2Ib�132r46<FI;gqx;R`�9<NR;RY�Z<2C?A7*4G2Q9o2QH8S<7:H�=�Z<H<2ST;RBA4GFIH�=@;RH�ZJ;g=AL*S<7KHJB{NR2CB�4QFK`�98F�B@7KH�=@2CB{2QH"=A?A7KHJB@NR7:=@;RFKH0ST2�NR7 4GFI`r`'7KHJST2*M 46J7KY�Z<246J7:H<fI2Q`�2QH�=�ST2*9oFK;RH�=Cb -�2*9<NRZJBCn~B@7KHJBEBD=@?7k=ALQfI;>2�ST2*9JN>ZJBE6J7KZT=rHJ;>^I2C7:Z � =@2CN>NR2�Z<H<29<NR7KH<;>}84Q7:=@;RFKH'ST2�=@?7 � 2-4z=AFK;R?@2-B/SJ7:HJB~Nc\ ;>`'7:fI2 Gn:;>NJ2CBD=~=@FKZ � FKZ<?B�98F�B@B@;RV<N>2�S]\ 2Qq_6J;>Vo2Q?hS<2CB4Q7KBw9J7:?@=@;O4GZJN>;R2Q?B�Y�Z<;�^IFKH�=�B@FK;>=�2QH�=@?7 � H<2Q?wNR7'9o2Q?@=@2EST2U^_Z<2{Sa\ Z<H<2{9J7:?@=@;R2{ST2UNc\ FKV�� 2Q=CnBDFI;g=r;>HJS<Z<;>?A25ST2-B{2�o2Q=ABE982C?@^I2Q?BUB@Z<?{NO7y4QFK`�`'7:HJST2 � B@;tST2CZTq 98FI;>H�=ABrB@FKH�={9<?AFT46<2CBST2WV8FI?AS<BtFI9<98F�BDL-B�ST2�Nc\ ;>`'7:fI2�987:?�2GqT2Q`�9<NR2 Gb //;>HJ7KN>2C`�2QH�=CnTNO7U=A?A7 � 2C4G=@FK;R?A2w2QH*=@?7:H8B XNR7:=@;RFKH YIZ�\ 2�o2-4z=@ZJ2Q?7yNO7 4C7:`�LQ?7 BD2C?A7y?@2-4z=@;RNR;>fIH<2Kn�`'7:;OB'BD2CZ<NR2Q`�2QH�=�S<7:H8BENR2�?@[email protected]��� ���� ST2*NR7 4Q7K`�LQ?7<b �hN>NR2*HJ2*B@2Q?7xSTFKH84�9J7IB{Z<HJ2*^ILQ?A;g=7:V<NR2�NR;>fIH<2*ST?AFK;>=@2�BD;�Z<H46J7:H<fI2Q`�2QH�=�Sa\ FK?A;>2CH�=A7k=A;>FIH52-B =�H<L-4G2-B@BA7:;R?@2Ib

� ^K2-4�Z<H<2�4Q7K`�LQ?7�2Q`EVJ7:?Y�Z<LQ2InQFKH{FKVT=A;>2CHI=/LQf�7:NR2Q`�2QH�=[ZJH<2h`'7k=A?@;O4G2tSa\ ;>H�=@2C?A7I4z=A;>FIHV<N>FT4GX�=A?@;O7:H<fIZ<NO7:;R?@2�B@;KNc\ FIH{4GFIHJB@;RSTvC?@2hN>2�^I2C4G=@2QZJ?

X = cPc∗9oFKZ<?/4GFIHI=A?@uIN>2C?]NO7�98F�BD;>=@;RFKH

ST2�NR7l4Q7K`�LQ?7<bK132�46<FK;>q{2CBD=�4G2C982CHJS<7:H�=~9o2QZ�;>H�=ALQ?A2CBAB@7KHI=�2QHr9<?7k=A;RY�Z<2Ib84~7K?�4GFKH�=A?@2In B@;N�\ FKH'46<FI;RB@;g=hS<2w4GFIHJBD;OSTLC?@2C?/NO7W9oFIB@;g=A;>FIH�S<2�N�\ FK?A;RfK;RH<2�ST2

Rc

2GqT9<?A;R`rLC2�S<7:HJB�Z<H'?@2C98vC?@2

� ��� ������� ���������������������������������! �"�#$������"����

N>;RLUMrN�\ FKV�� 2Q=CnJB@FK;>=�9J7:?�2QqT2Q`�9<NR2c∗Pc

n<FIH�7�7:NRFK?B � ^IFK;R?�� b + ( � (

Ls =

(c∗Rc 03

03 Lω

) � b 6 �

YIZJ;�2CBD=wV<NRFT4zXeST;O7:fKFIHJ7:NR2W2G=�7IB@B@Z<?A2�STFKHJ4UZ<HxSTLC4QFKZ<9JNR7KfK2W=@?AvCBwB@7:=@;OB i�7K;RBA7:H�=Cb -w2U9JN>ZJBCnNR7�=A?A7 � 2C4G=@FK;R?A2l2CH�=@?7:HJB@NO7k=@;RFKHPST2{NR7'4C7:`�LQ?7'BD2C?A7'7KN>FI?AB�ZJH<2U^KLC?@;>=A7KV<NR2�N>;RfKH<2EST?@FI;g=A2Kn4G2wY�Z<;82CBD=3Sa\ Z<H'fK?7:HJSr;RH�=@LQ?ApG=�9J?A7:=@;OYIZJ2Kb �T;JNc\ FIH�BDFIZ<6J7:;>=@2�Y�Z<2�4Q2wB@FK;>=hN�\ 2 ��2C4G=@2CZ<?3STZ?@FIV8FK=lY�Z<;�BDZ<;R^K2{Z<H<2{NR;RfKH<2rST?@FI;g=A2Kn8;>N�BDZ 'r=�ST2r4GFKH8BD;OSTLQ?A2Q?

n∗Pn

S<7:HJBsM5NR7�9<NR7I4G2

ST2c∗Pc

b #x7KN>6<2CZ<?A2QZJB@2Q`�2QH�=-n8S<7KHJB�NR2CB�S<2QZTqx4Q7IBQnJ;>N[H]\ sP7KN>FI?AB�9<NRZJBl7:ZJ4QZ<HP4GFIHI=A?@uIN>2ST2{NO7'=A?A7 � 2C4z=AFK;R?@2UY�Z]\ 2 ��2C4G=@Z<2C?A7�N�\ FKV�� 2Q=lS<7KHJB�Nc\ ;>`'7:fI2U2G=-noB@;/NR7'9oFIB@;>=@;RFKHx;RH<;>=@;O7:NR2EST2NR7�4Q7K`rLC?A752CBD=lLCN>FI;>fIH<LQ2rST2�B@7*98F�BD;>=@;RFKH STLCB@;>?ALQ2In�?@;R2QH H]\ 7IB@B@Z<?A2{Y�Z<2�N�\ FKV � 2G=W?@2-B =A2Q?7S<7:HJB�NR2U46J7K`r9�ST2�^_;OBD;RFKH�ST2WNR7'4C7:`�LQ?7r7:Z�4GFKZJ?AB�ST2�Nc\ 7IB@B@2Q?A^_;RBAB@2Q`�2QH�=Cb

0�H�982CZT='V<;>2CH�2CH�=@2QH8STZ�2�o2-4z=@ZJ2Q?�ST2-Br46JFK;>q"B@;R`r;RNO7:;R?@2-BE7 ^I2C4�ZJH<2�4C7:`�LQ?7yS<LGX98FI?D=ALQ2 � ^KFI;>?W9J7K?A7KfK?7:9<6J2 b"!Tb b Gb � =@;>=@?A2'Sa\ 2GqT2Q`�9<NR2KnaB@;~N�\ FKH BDLCN>2-4z=@;RFKHJH<2rSJ7:HJB

sNR7*=@?7:H8BDNO7k=A;>FIH ∅Po

2Q=WNR2�^K2-4z=@2CZ<?θu

7KBABDFT4G;RL�M5NO75?AF:=7k=A;>FIHo∗Ro

naFIH 7:VoFKZ<=@;>=Cna2QH4GFK`EV<;RHJ7:H�=�N>2-B3L-Y�ZJ7k=A;>FIHJB � b 6 � �cn � b �� ��2G= � b +)+ � n_M{NO7UNRFK;aST2W4GFI`r`'7KHJST2WBDZ<;R^k7:H�=@2 (

{∅Vo = −λ (∅Po − ∅Po∗)oωo = −λ θ u

� b 6�6 �

F=� ∅Vo

2-B ='NO7y^_;>=@2-B@B@2PST2�=@?7:HJB@NR7:=@;RFKH�S<2Ro

2GqT9<?A;>`�LC2�S<7KHJBR∅

2G='F=�oωo

2-B =B@75^�;>=@2-B@B@2{ST2r?AF:=A7:=@;RFKHP2Qq_9J?@;R`�LQ2{S<7KHJB

Ro

ba132Q=D=@2rNRFK;�ST2E4QFK`�`'7:HJST2E9<?ALCB@2QH�=@2{S<2CB9<?@FI9<?A;>LQ=@LCB*ST2 STLC4QFKZ<9<NO7:fI2x;OSTLC7KN>2-BQb���NRN>2 7KBABDZJ?@2x2CH FIZT=@?A2 Y�Z<2 NR70=@?7 � 2C4G=@FI;>?A2yST2N�\ FK?A;>fI;>HJ2WS<2

Ro

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Rn

2G=Ro

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b b"!Tb � b b'deHTijFI?@`'7k=A;>FIHJBt^_;RB@Z<2QNRNR2CB,!.- ��� !

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e =(

x − x∗ y − y∗ log (Z/Z∗) θu)T � b 6 % �

� � � � ��� �!� � ��� ��� ��� �!� "���# � ���

F=� (� (x y)T

2G=(x∗ y∗)T

B@FKH�=�?A2CB@982-4z=A;>^I2Q`�2QH�=]NR2CB�4QF_FK?STFKH<HJLQ2CB�4GFKZJ?A7KHI=A2~2G=�STLCB@;>?ALQ2Sa\ Z<H�98FI;>H�=w4Q7K?A7I4z=ALQ?A;RBD=@;OY�Z<2lSJ7:HJB�Nc\ ;R`'7:fI2 �

� Z/Z∗ 2CBD=�NR2�?7:9<9oFK?@=t2CH�=@?A2lNR2CBwST;RBD=A7KHJ4G2-B�4GFIZ<?A7KH�=@2l2G=wSTL-BD;R?@LC2�ST2�4G2�9oFK;RHI= �� θu

?A2Q9<?ALCB@2QH�=@2WNO7r?@FK=A7k=A;>FIH*Mr?ALC7:NR;OBD2C?Cb-�7KHJB�4G2�4C7KBCnTFKH�FKVT=A;>2CHI=w4QFK`�`�2WN>FI;]ST2U4GFI`r`'7KHJST2 (

Tq = −λ

(ZL−1

ev−ZL−1

evLevω

03 I3

)e

� b 6 ( �

F=� � ^IFK;R?�� b � 6 �a2G= � b + � (

Lev=

−1 0 x0 −1 y0 0 −1

2G=

Levω=

xy −(1 + x2) y(1 + y2) −xy −x

−y x 0

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Lev

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e =(

c∗PTc x − x∗ y − y∗ θuz

)T � b 6 * �

F=�(x y)T

2G=(x∗ y∗)T

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uz

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Rc∗2Q=

Rc

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Tc = −λ

(cRc∗ 03

− 1ZL−1

eωLeωv

cRc∗ L−1eω

)e

� b %.� �

F=�Leωv

=

−1 0 x0 −1 y0 0 0

2Q=

Leω=

xy −(1 + x2) y(1 + y2) −xy −x

l1 l2 l3

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θuz

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Leωv

2G=Leω

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C = Ls

+

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Tq = −λ(Ls

+

|s=s∗ Ls

)−1

e� b % ! �

2G=CnJ2QHP46<FK;OB@;RBAB@7KH�=Ls|s=s∗

98FIZ<?�7:9J9<?@F qT;R`�2Q?Ls

n<FIH*FIVT=@;R2QH�=�}JHJ7KN>2C`r2CH�=�(

Tq = −λ e = −λ Ls

+

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e = 0 Gna4G2QNO7

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(s− s∗) ∈ �W2Q?C

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Ls

+

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Ls2CBD=�ST2�?7:H<f�9<NR2Q;RHm = k

b 0�H�9o2QZT=W7:NRFK?Bt2QHP2 ��2G=�9<?@2CHJST?A2UST;R?@2-4z=@2C`�2QH�=W = Ls

b� ZT=A?@2C`r2CH�=Cn�N>2-B

mN>;RfKH<2-BhST2

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m^I2C4z=A2QZ<?BhS<2wVJ7IBD2

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b 0�H*9o2QZ<=wH<F:=A2Q?�Y�Z<2UB@;Ls

2CBD=wST2�?A7KH<fn� 4:\ 2CBD=DX

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982CZT=�46<FI;RB@;>?W = In

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e = e1b�132QNO7�;RHJST;OYIZJ2U7:ZJBABD;[YIZJ2�NR7rijFI?@`�2UST2

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Tq =

(∂e

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� b ( % �

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∂e

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> 0� b (�( �

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Tq = −λ e− ∂e

∂t

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C = Im

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+ > 0n�Y�Z<;

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zn_FKH�982CZT=�46<FK;OB@;>?(

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7 ^K2-4I0 = 0

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j=0

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