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Ecole Polytechnique. Ne pas reproduire.
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Ecole Polytechnique. Ne pas reproduire.
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FpKUOpe FpKUOpe [TSTb SCHLKBFpe MNWuP>IJSCIJ]OXFg]a{]&bdcX[C] MX] MNWIJSCqY\HLSTKUFgMOXFgbdK{MXk[T] cXr{~NemSfijOX]"HLIJk&e
P[TYBeLeLSfiBOp] ]>H%FpKUOpe IJ]F{UKq~BKUFpe4[T]/[T]&P>HL]&OXI MXW&eLSCIJ]ONa M]FeLYBKUSTIcX[TOpe MN]ue KUOX{IZY\VU]ue^KUO
PKUOXIZecX[TOpe`eLcQWuP>SfY\[TSTeLW&e&A
v3]0bdKNMNk&[C]^ijOX]4FXKUOQesY\[T[CKBFpeMNW&PILSTIL]0]&emH`P>KBFXF{O/eLKUOpe[T]0FpKUb ( ' ("*
' NKUOMW&ijOpY\HLSTKUF$MN]^MNSCOpeLSTKUF=A
+KBFpemSfMNW&ILKBFpe1OpF"MNKBb Y\STFX]
Ω
MN]4[¢]&eLcpYUP]N
MXSCbd]FQemSTKUFpe
FXK\HJW
R N
XYU]uP`]F/VUW&FXWIZY\[N = 1, 2
KBO dijOX]$[¢KUFgeLOXcXcKBeL]"KNP&P>OXcWcpY\I OpFb Y\HLWIJSfY\O rpKUbdKUVBkFX] STeLK\HJILKBcQ] ]>H PKUFpMXOpPRHJ]OXIlMX][TYPZrpY\[T]OXIuA 7FFXKUHL]x
[TY qYUILSfY\X[T]M]&eLcpYUP] pPU]ueH ) )MXSCIJ]4OXF cKUSTFBH0MN]Ω
]H
t
[TY7qY\IJSTYUX[T]MN]+HL]&bdcpeA\¡0Y\FQeΩ
[C]ue eLKUOpIJP]&e MN]PZrQY\[T]OXI
W&U]&FBHJOX][T[T]bd]FjH FXKUF%OXFXS1KUIJb ]ue
]&F]&eLcpYBP>]]H qY\IJSTYUX[T]&e3MpY\Fpe3[C]1HL]bdcpe emKBFBH IL]&cXILWuem]&FjHLW]uecpYUI OXFp] 1KUFQPRHLSTKUF%MNKUFXFpW]
f (x, t)
HZY\FpMXSTe^ijOX]d[fYHJ]bdcWIZYqHLOpIL]d]&emH^OXFX] 1KUFQPRHLSTKUF STFpP>KBFXF{OX]
θ(x, t)
A v3Y$ijOpY\FjHLSCHLW MN] PZrpYU[C]&OXI ]ueHdcXIJKUcKUILHLSTKUFXFp][T[C] [fYHL]&bdcQW&IJY\HLOXIJ]θ
]>HdqY\ONHcθ
Koc
]ueHdOXFX]$P>KBFpemHJY\FjHJ]cXr{~NemSfijOX]
ijOXS0MXWc]FpMMNOgH~{cQ]MN]b Y\HLWIJSfY\O YUcXc][TW] PZrQY\[T]OXI"emcW&PSQijOX]BA+ KBOXI/MNWHL]&ILbdSTFX]I/[TY
HJ]bdcWIZYqHLOpIL]
θ
jFpKUOpe1W&P>IJSTUKUFQe [TY #$ 1"' %% KUOMN]7[TY9iBOQY\FjHLSCHLW MX]4PZrQY\[T]OXIuAj¡0Y\FQe+OpF/BKU[TOXbd]7W&[CW&b ]&FjHJY\STIJ]V
SCFQP>[TOpesMXY\FQeΩ
N[fY%YUILSfYqHJSCKBF]F/HL]&bdcpesMN]0[TYEcole Polytechnique. Ne pas reproduire.
iBOQY\FjHLSCHLW^MN]PZrpYU[C]&OXI]&emHs[C]4XST[TYUFMN]^P>]^ijOXS=]&emHscXIJK{MXOXSH`cQY\I[C]uesemKBOXIZP>]&e]HlMN]^P>]^ijOXS3eLKUILH
KUO"IL]&FBHJIL]%HLIZYU]&IJe+[T]&escpYUILKBSTe&AN¥`ONHJIL]&bd]FjHlMNSCH
d dt
Z
V
cθ dx
= Z
V
f dx − Z
∂V
q · n ds
@UAT@KJo
∂V
]ueH+[C]`KUIZMdMN]V
MW&[CW&bd]FjH+MN]leLOXI 1YUP]ds
n
]ueH1[fY4FXKBILb Y\[T]]>a{HLW&ILST]OpIL]sOXFpSHJWlMN]V
j]>Hq
]ueH[C]7U]&P>HL]&OXI *pONaMN]0PZrpY\[T]OpI&A{y{SpKBFY\cXcp[CSfijOX]`[T]`HLrpWKUIJkbd]lMX]!%4Y\OQeLeUKBF KUNHJSC]&FBHZ
∂V
q · n ds = Z
V
divq dx.
s]VBILKBOXcpY\FjH[T]&e+MNS WIJ]FjHZeHJ]IJb ]ue1MN]
@BAC@ ]>HONHLST[CSfeJY\FjH1[T]1Y\SCHijOX]l[C]7UKU[TOXbd]`W&[CW&b ]&FjHJY\STIJ]
V
]ueH`ijOX]&[TPKUFpijOX]jSTFpMNW&cQ]&FpMXYUFBH`MNO/HL]&bdcpe NKBF/]&F"MXW&MNOXSCHs[¢W&ijOpY\HLSTKUF$MN]^P>KBFpem]&ILqY\HLSTKUFMN][¢WFX]&ILVBSC]
c ∂θ
∂t + divq = f
@UA
iBOpSpY0[CST]Od]F HJKUONHcKUSTFjH
x ∈ Ω
]>H 0HJKUONH HJ]bdcpet
AUlY\cXc][TKUFpe ijOX]s[KUcWIZYqHL]&OXI MNSTU]IJVU]&FpP>]]&emHlMNW>pFpScpY\I
divq =
N
X
i=1
∂q i
∂x i
YU]uP
q = (q 1 , ..., q N ) t .
DG[1YUONHdbdYUSCFjHJ]FpYUFBH IL]&[CST]I [T] *pONa MX]"PZrpYU[C]&OXI [fYHL]&bdcQW&IJY\HLOXIJ] ]>HdKUF 1Y\SCHdYUcXcQ]&[ P>]
iBO3KBF"YUcXcQ]&[C[T]4OXFX]
1"''$
AX¡7YUFpe[C]^P&YUecXIJW&eL]FjH{ST[3e&YUVUSCHMN]^[fY [CKBS=MN]^¤XKBOXILST]I
iBOpS=IL]&[CST]4[C] *pOXa"MN]PZrpYU[C]&OXIsMN]^b Y\FXSTkIJ]4cXIJKUcKUILHLSTKUFXFX]&[C[T]4Y\O"VUIZYUMNST]FjHsMN]4HL]&b cWIZYqHJOXIJ]
q = − k ∇ θ,
@UA
KJo
k
]&emH OXFp]PKUFpemHJYUFBHJ]1cKBeLSHJSCB]
ijOXSNMNWc]FpM MNO9H~jc]MN]+b Y\HLWIJSfY\O3Y\cXc][TW]P>KUFQMNOpPRHJSC{SCHLW
HLrX]&ILbdSfiBOp]UAX`YUcXcQ]&[CKBFpe`ijOX]4[KBcQW&IJY\HL]OpIVBIJYBMNST]FjHs]&emHsMXW>pFXS3cpY\I
∇ θ = ∂θ
∂x 1
, ..., ∂θ
∂x N
t
.
4F PKUb9pSCFpYUFjHl[TY[CKBS MN] P>KBFpem]&ILqY\HLSTKUF
@BA s]H7[fY[TKUS PKUFpemHLSCHLOXHLSTU]
@UA QKBF KUNHJSC]&FjH7OXFX]
W&ijOpYqHJSCKBF/cKUOpIs[TY%HL]&bdcQW&IJY\HLOXIJ]
θ c ∂θ
∂t − k∆θ = f,
KJo
∆ = div ∇
]&emH`[¢KUcWIZYqHJ]OXI[fY\cX[fYUPSC]&F"MNKBFXFXW4cpYUI∆θ =
N
X
i=1
∂ 2 θ
∂x 2 i .
DG[ 1Y\ONH0Y7KUOXHL]I dP]>HmHJ]W&ijOpY\HLSTKUF$ijOXS=]&emH`qY\[fY\X[T]MXYUFpeHLKBONHl[T]MNKUb Y\STFX]
Ω
pOpFX]^IL]&[TY\HLSTKUF Y\cXc][TW] $ %' & ' ijOXSlSCFpMXSTijOX]P] ijOXS7eL]cpYUeJeL] [TY 1IJKUFjHLSTkIJ]KUOgYUOEcole Polytechnique. Ne pas reproduire.
n
Ω
∂Ω
@UAT@ ]&PRHJ]OXIsFpKUIJbdYU[OXFXSCHLW^KBILST]FjHLW0U]&IJe[]>a{HLW&ILST]OXIuA
KUIZM
∂Ω
MNOMXKUb Y\STFX] ]>HdOXFX]"Y\ONHJIL]/IL]&[TY\HLSTKUF ijOXSSCFQMNSTijOX]"ijOX][]&emH%[W>HJY\H%STFXSCHLSfY\[sMN][fY HJ]bdcWIZYqHLOpIL]BAQ Y\I0P>KBFjB]FjHLSTKUF pKUF PZrXKUSfemSCH7[SCFpemHJYUFjHt = 0
cKUOXI 2>HJIL]%[C]9HL]&b cQe7STFXSCHLSfY\[ ]>HKBF/STbdcQKjem]^OpFX] 1$ $%
θ(t = 0, x) = θ 0 (x),
@UAxKo
θ 0
]&emH^[TY31KUFQPRHLSTKUFMN] MNSfeHJILSTXONHJSCKBF STFXSCHLSfY\[T]dMX]%HJ]bdcWIZYqHLOpIL]dMXY\FQe7[T] MNKUb YUSCFX]
Ω
A 4FP]ijOXS7PKUFpP]IJFX][fY P>KBFpMNSCHLSTKUF Y\OXa[TSCbdSCHL]ue P>][fY MNW&cQ]&FpM MNO PKUFjHL]a{HL] cXr{~NeLSTijOX]BA+y{S0[C]
MXKUb Y\STFX]]ueHeLOXcXcKBeLWpYUSCVBFX]I$MXY\FQeOXFHLrX]&ILbdKjeHZYqH HL]&b cWIZYqHJOXIJ]PKUFpemHJYUFBHJ] Y\[TKUIZe
ijOXSCHmHJ] "b KNMNSCp]&I^[¢W&PZrX]&[C[T] MN]&e0HJ]bdcWIZYqHLOpIL]ue [fY/HL]bdcWIZYqHJOXIL]dBWIJSp] [fY"PKUFpMXSHJSCKBFYUONa
[TSTb SCHL]ue`MN]¡7SCIJSfPZrX[C]H
θ(t, x) = 0
cQKBOXIHLKBONHx ∈ ∂Ω
]>Ht > 0.
@UA}|yNS[T]/MXKUb Y\STFX]]ueHdemOpcXcQKjemW"YUMNSfY\QYqHLSfijOX]KBOHLrp]IJb SfijOX]bd]&FBHdSfemKB[CWMX]/[]a{HLWIJST]OXI YU[CKBIJe
[T] *pONaMN] PZrpYU[C]&OXI4emKBImHZY\FjH4YUOKUIZM ]ueH^F{OX[ ]>H^[fYHJ]bdcQW&IJY\HLOXIJ] UWIJSCp]%[fY"P>KUFQMNSHJSCKBF YUONa
[TSTb SCHL]ue`MN]l]OXb Y\FpF
∂θ
∂n (t, x) ≡ n(x) · ∇ θ(t, x) = 0
cKUOXIHJKUONHx ∈ ∂Ω
]Ht > 0,
@UAKJo
n
]&emH0[fYFpKUIJbdYU[C]%]>a{HLW&ILST]OpIL]%OXFXSCHLW MN]Ω
BKUSTI7[fY/¤ STVUOXIJ]/@BAC@ RA _`FX] eLSHJOpYqHJSCKBFSTFjHL]I) bdWuMNSTYUSCIJ] cQ]&ONH9Y\OpeJeLS YBKUSTIl[TSC]&O %[T] *pONa MX]dPZrQY\[T]OXI4emKBImHZY\FjH^Y\OKUIZM ]ueH^cXIJKUcKUILHLSTKUFpFX][YUOeJY\OXH MX]/HJ]bdcWIZYqHLOpIL]"]FjHLIJ]$[¢]>a{HJWIJSC]&OXId]>H[STFjHLWIJST]OXI ]>H[fY HJ]bdcQW&IJY\HLOXIJ]$UW&ILSCp]"[TY
PKUFpMXSHJSCKBF"YUONa[TSCbdSCHL]&elMN]¤XKBOXILST]I
∂θ
∂n (t, x) + αθ(t, x) = 0
cKUOpIHLKUOXHx ∈ ∂Ω,
]>Ht > 0
@UAKo
α
]ueHOXFX]^PKUFpemHJYUFBHJ]7cKBeLSCHLSTU]UA{OpSTeJijO=ST[ 1YUONH`PZrXKUSfemSTI
P\]&emHsOXFX]^MN]ueW>HJYUcQ]ueMN]4[TY%bdK )
MXW[TSTeJYqHJSCKBF FXKBOpe7YU[C[TKUFQe7eLW[T]&P>HLSTKUFXFX]&I7[fY"P>KUFQMNSHJSCKBF YUONa[CSTbdSHJ]&e^MN]d¡lSTIJSTPZrX[T]>H
@BA| RA`YBe)
eL]b%X[fY\FjH`]FNQF [¢W&ijOpY\HLSTKUF Q[TYP>KUFQMNSHJSCKBFSCFpSHJSTYU[C]p]>H0[TY PKUFpMXSHJSCKBF Y\ONa$[CSTbdSHJ]&e0eLY\HLSfe 1YUSHJ]&e
Ecole Polytechnique. Ne pas reproduire.
cpY\Is[fY%HL]bdcWIZYqHJOXIL]NKUF"KUNHJSC]&FjH`[¢W&ijOpY\HLSTKUF$MN]^[TYdPZrpYU[C]&OXI
c ∂θ
∂t − k∆θ = f
cQKBOXI(x, t) ∈ Ω × R + ∗
θ(t, x) = 0
cQKBOXI(x, t) ∈ ∂Ω × R + ∗ θ(t = 0, x) = θ 0 (x)
cKUOXIx ∈ Ω
@UA
v=]0cXILKBX[Tkbd]
@UA +]&emH`MNKUFQP7PKUFpemHLSCHLOXW^MOXFX]4WuiBOQYqHLSTKUF"Y\ONaMXWIJSCBW]&e+cpYUImHJSC]&[C[T]&eb9OpFXSC]^MN]
P>KUFQMNSHJSCKBFpe Y\OXa9[CSTbdSHJ]&e ]H1MOpFX]`P>KBFpMNSCHLSTKUF SCFXSCHLSfY\[T]UAB¥P&Y\OpeL]sMN][TY0cXIJW&eL]FQP>]sMN]sP>KBFpMNSCHLSTKUFpe
Y\ONa[TSTb SCHL]ue 3KUF MNSH9ijOX]
@UA 7]&emHOXF !(# & %' & ' b Y\Sfe4KUF MNSCH9Y\OpeJemSijOX]
P\]&emHOpF ! # & # %'$(" dPYUOpeL]4MX]^[TYdMNKBFXFXW]4STFXSCHLSfY\[T]^]F/HL]bdcpe&A
&
( ' ¢
¡0Y\FpeP>]7b KNMNk&[C]0MN]lcpILKBcpY\VjYqHLSTKUF MN]7[TY%PZrpY\[T]OpIBSC[FXKUOQe 1YUONH+cXIJW&PSTeL]I
[C]ue9OXFpSHJW&e%KUO MNSTbd]FpeLSTKUFpe%cXr{~NemSfijOX]&e % [fYHJ]bdcWIZYqHLOpIL]
θ
e]>aNcXIJSTb ]/]F 4]&[C{STF
K
[TYPZrpY\[T]OXIeLcW&P>SCQijOX]
c
]F0BKUOX[T]cpY\I {ST[CKBVUIZY\bdbd]+cpYUI 4][TjSTF
J/(kg × K)
[TY^PKUFpMNOQPRHLST{SHJW HLrX]&ILbdSfiBOp]
cpYUIOXFXSCHLWMN]bdYBeLeL]
k
]F 0BKUOX[T] bdk>HJIL]P&Y\IJILW cpY\I9jST[TKUVUIZY\bdbd]%cQY\I 4][T{SCF cpY\I`eL]&PKUFpMN]
Jm 2 /(kg × K × s)
RAQ¡%OXF$cQKBSCFjHlMN]^{OX]^b YqHJrXWb Y\HLSfiBOp] XFXKBOpesY\[T[TKUFpeHJILkue emKBOXU]&FBHsKBOXX[TSC]&IlP>]uesOXFXSCHLWue Q]>Hlb 2bd]P]&e`PKUFpemHJYUFBHJ]&eX]FeLOXcXcKBeJY\FjHlijOX]c
]>Hk
qY\[T]FjH@
P>]&[TY IJ]{ST]FjH YBMNSTb ]&FpeLSCKBFXFX]Is[T]&eVBIJYUFpMN]&OXIJecXr{~NeLSTijOX]ue>A
•
& ( ' `KBOpesYUKBFpe+b ]&FjHLSTKUFXFXW0HJILKBSTe+H~{cQ]uesMN]^P>KBFpMNSCHLSTKUFpeY\OXa[CSTbdSHJ]&eX¡lSTIJS)
PZrX[C]H{`]&OXb Y\FXF N¤pKUOXIJSC]&I
bdYUSTe+ST[]&F]>aNSfeHJ]7MY\OXHLIJ]&e ijOXSKUFjH+[TSC]&O/eLOXI[¢SCFjHJWVUIZY\[TSCHLWlMN]7[TY
1ILKBFBHJSCk&IL]
∂Ω
A1,SC]&Fe0pI\KBFdcQ]&ONH+Y\SfeLWbd]FjHSTbdYUVUSTFX]IMX]&eeLSHJOpYqHJSCKBFpe Ko [T]&e+P>KBFpMNSCHLSTKUFpe YUONa [CSTbdSHJ]&e0emKBFBH0bdW[fY\FXVBW]&e %¡7STILSfPZrX[T]>H0eLOXI∂Ω D
l]OXb Y\FpFemOpI
∂Ω N
Q]H0¤XKBOXIJSC]&I7eLOXI
∂Ω F
YU]&P
∂Ω D , ∂Ω N , ∂Ω F
1KBILb Y\FjHsOXFX]4cpYUImHJSHJSCKBFMN]4[TY 1IJKUFjHLSTkIJ]
∂Ω
A•
& ( ' f vW&ijOpY\HLSTKUFMN][fY PZrpYU[C]&OXI
@BA ]&emH $% Y\O eL]Fpe`KJoeLY eLKU[TONHJSCKBF
θ
MNWc]FpM [CSTFXW&YUSCIJ]bd]&FBH9MX]&e9MXKUFXFXW&]&e(f, θ 0 )
A4FcXr{~NemSfijOX]P>]HmHJ]cXIJKUcXIJSCWHLWMN][CSTFXW&YUILSCHLW ]&emH0eLKUOXB]FjH`HLIZYUMXOXSHJ]9eLKUOpel[TY.1KUIJb ]%MOpFcXIJSCFQP>STcQ] MN]%eLOXc]IJcQKjemSCHLSTKUF %OXFX] P>KUb%XSTFpY\SfemKBF[CSTFXW&YUSCIJ]$MN]&edMNKBFXFXW]ue
(f, θ 0 )
PKUFpMXOXSH OXFX]eLKU[TONHLSTKUFθ
ijOXSs]&emH [TY b 2bd]"PKUb%XSCFQY\SfemKBF [CSTFXW&YUSCIJ]lMN]ue1eLKU[TONHLSTKUFQe1P>KBILIJ]&eLcKUFpMXYUFjH 9PZrpYUijOX]HJ]IJb ]0MN]`[fYMNWuP>KUbdcKBeLSHJSCKBFMN]&e+MNKUFpFXW]ueA¡9OXF cQKBSCFjH9MN]d{OX]dcXr{~NemSfijOX] [fY"[CSTFXWuY\IJSHJW F3]ueHijO=OXFX] r{~{cQKUHLrXkuem]dcpYUILbdSMY\ONHJIL]ueA$4F
]> ]>HjcKUOXI[C]ue1cpILKBX[Ck&bd]&e 1KUILHL]lqYUILSfYqHJSCKBFdMX]`HL]&bdcQW&IJY\HLOXIJ] j[TY[CKBSMN]0¤XKBOXILST]I+]&emH 1Y\OQeLeL]
]>HdSC[ 1Y\OXHd[fYPKUIJILSTVU]&I9]&FeLOXcXcKBeJY\FjH ijOX]"[TYP>KBFpMNOpP>HLST{SHJWHJrX]IJbdSTijOX]
k
MNW&cQ]&FpM ]&F 1YUSH MN]/[fYHJ]bdcWIZYqHLOpIL]θ
]HdMN]$eLKUF VBIJYBMNSC]&FjH∇ θ
P]"ijOXSIJ]FpM [C]"cXIJKUX[Tkbd]/FXKUF )[TSCFXWuY\STIL] RA 4FpP>KBIL]dcXSTIJ] cKUOXIMN]ue4cXrXW&FXKUbdkFp]&e4]>a{HJI 2&b ]&bd]FjH^IJYUcXSTMX]&e
]aNcX[CKjemSTKUF cQY\I^]>aN]&b cp[C] 0SC[
]&emHFXW&P]&eJeLYUSCIJ] MY\pYUFpMNKBFXFX]I[C] cXILSTFpPSCc] b 2bd] MX][fY[CKBS7MN] ¤XKUOpILST]I"iBOpS7eLOXcXcKBeL] [TY
cXILKBcQKBImHJSCKBFXFpYU[CSCHLWlMXO *pONaMN]0PZrpY\[T]OpI
q
YB]&Ps[T]l[T]7VBIJYBMNST]FjH+MN]lHL]bdcWIZYqHJOXIL]∇ θ
A 41F]> ]>HP>]>HLHL]r{~{cK\HLrpk&eL]
+mFpYqHJOXIJ][T[C] , cpIL]&b STkIJ] {OX] ^]FjHLIZY FX] OXFX]/cXIJKUcXIJSCWHLW cpYUIJYBMNKaXY\[T] % [TY
PZrpY\[T]OXI%em]cpILKBcpY\VB] OXFX]{SCHL]ueLeL]STFNpFXST]/MXYUFpe[T]MNKBb Y\STFX]
Ω
A lKUOpe9U]&ILIJKUFpecX[COQe[TKUSTF
UKBSCI0[fY"`]b Y\IZijOX]@BAXAn lPKUbdbd]FjH0WHJYUX[CSTI^P>]dcpYUIJYBMNKaN]UAQ`]>HL]&FXKUFQe7cKUOpI0[SCFpemHJYUFjH^ijOX]
b KNMNW&[CSfeL]I`P\]&emH1Y\STIL]^MN]uesr{~{cQKUHLrXkuem]ue]>HscXIJW&P>SfeL]Is[T]OXI`MNKBb Y\STFX]^MN]^qY\[TSTMNSCHLWBACATA
•
& ( ' , v=]%cXIJKUX[Tkbd]
@UA F=]&emH7cQYUeleL]Op[C]&b ]&FjH7OXF bdKNMNk&[C]9MX]9cXIJKUcpYUVBY\HLSTKUF
MN]$[TYPZrpY\[T]OpI&A DG[sY]F 1YUSH OXFPYUIJYBPRHJkIJ]OXFXSTU]&IJeL][ ]>H KUF[C]$IJ]>HJILKBOXU]"P>KUbdbd]"b KNMNk&[C]
Ecole Polytechnique. Ne pas reproduire.
MX]"FpKUb9pIL]&ONacXrXWFpKUbdkFX]uedeLYUFpedY\OpPOXF IZY\cXcKUILHd]FjHLIJ]/]&ONa
ST[1Y\OXH eLSTb cp[C]&b ]&FjH PZrQY\F )
VB]I0[C] FXKBb MN]&e4MNSTU]&IJeL]&elYUILSfY\p[C]ue7MNO cXIJKUX[Tkbd] >A YUI7]aN]bdcX[T]
@BA `]ueH^Y\OpeJeLS PKUFXF{OX]
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