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### Développement de couches de diffusion de piles PEMFC pour un fonctionnement à faible humidité relative, Development of PEMFCs Gas Diffusion Layers operating at low relative humidity

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### Exercice configuration électronique de Mn 1s22s22p63s23p64s23d5 L'atome de manganèse a

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suivant

np 1p< 2
np 1 p <
2
k
k
n
2
n(M ;g)
p pn
(M) p
= dd+dd d
0 = (M;g)< (M;g) (M;g):::p;0 p;1 p;2
b (M)p
( (M;g)) ( (M;g)) ( (M;g))p;i i1 p;i i p 1;i i
0< (M;g) (M;g):::p;1 p;2
p
(M;g) = (M;g) p ip;i n p 1;i
M
n 1 (M;g) pp;i 2
nM
n 6 N 2 N V > 0
0<a <a a :::a 0<a a :::a1;1 1;2 1;3 1;N p;1 p;2 p;N
n 32p [ ] g M
2
n 3 (M;g) =a 1kN 1p [ ]p;k p;k 2
n 1 (M;g)> sup ap;ii;N[ ];1
2
Vol(M;g) =V
n 1 (M;g)> sup ap;ii;N[ ];1
2
n 1[ ]
2
n
2
n 6
(M;g)1;1
n 1 (M;g)
[ ];k
2
n 4
n 1[ ]
2
(M;g)1;1
nM
n 5 M g
(M;g)1;1
M
(M;g) k1;k
j : @U ! U N
U

j (i !) = 0 j (!) = 0N(A) j (i d!) = 0 j (d!) = 0N

j (!) = 0
(R) j (d!) = 0
pKer H (U)
pH (U)0
j (d!) = 0
j (d!) = 0

j (!) = 0
(D) j (!) = 0
2 pL ( U) = Im d Ker Imd
! ! = dd ++dd
Ker
! = d’
! = dd ’
!
d j (i d) = 0N
U M
U U
p p pH (U=M) =f!2
(U); d! = 0g=f! ; !2
(M) d! = 0gjU
U
p pM H (U=M) H (U)
p pH (M)!H (U)
pH (U=M)

2k!k
= inf sup ; d’ =! ;p;i 2V k’ki !2Vnf0gi
V ii
p + 1
! ’
!
2q(!) = inf k’kd’=!
!
1q(!) = ( !;!)
2Q(!) =k!k 2L
j!j = inf k’k 2L
d’=!
2 p+1L ( M)\ Im d
jj
Q
2 p 2 p+1L ( M)=Ker d L ( M)
q0
q E E1 0 1

q q kq qk0 1 1 0
Q Q N0 1
Q Q0 1
N
N Q0
Q1

::: < +M1 N N+1
N ;M > 0
n(M ;g)
1n U M C
(g ) M gi
Vol(M;g )! Vol(U;g) i!1i
[n 3] (M;g )! 0 p kd i!1p;k i p2
[n 3] (M;g )! (U;g) p k 1 i!1p;k+d i p;kp 2
pd H (U=M)p
[n 3] (U;g) 1 pp;k 2
"> 0 i N
U M g "i
[n 3]
p
2
p n pU’S B
2
nM
n 3 C M
K(C)> 0 g2C
2
n (M;g) Vol(M;g) K:n 1 ;1[ ]2
M U
n 1[ ]
2H (M=U)
2
nK Vol(U;g)
K
0C
N M

Q
H Q
dom(Q) =H H " > 00 1
C(;M;N;")> 0 Q =Q 0 jH0
28x2H ; Q(x)Cjxj Q Q N1 0
"
(H;jj)
Q
jj H Qn n
Q
C ;C > 0 8x2H; Cjxjjxj Cjxj1 2 1 n 2
x2 dom(Q) Q(x)Q (x)n
x2 dom(Q) jxj !jxj Q (x)!Q(x)n n
Q
M N Q Q N "n
Cjxj1
1
2jxj Cjxj Cjxjjxj Cjxj +" Q(x) " ! 0n 2 1 n 2 n n
(H;jj) jjn
n 1[ ] p;i2
Q p + 1
2(g ) g = g U g = " g" " ""2]0;1]
MnU Q jj g" " "
p+1!2
(M)
Z Z
2 n 2p 2 2Q (!) = j!j dv +" j!j dv" g g
U MnU
!Z Z
2 2 n 2p 2j!j = inf j’j dv +" j’j dv :g g"
d’=! U MnU
" +" U MnU
2(f ) f g gj "j
Qj
2jj f g Q jjj " "j
(g ) g" "
(p + 1) H H0 1
Q H" 0
2 p+1pH Im d L ( M)1
U
MnU
pH =fd’; ’2
(M); ’ = 0; ’ (D)g:1 jU jMnU
H H0 1
2L g"
H H Q0 1 "
(D) MnU
g ! 2 H1
p’2
(M) MnU
2 2 2 (D) 2 (D)k!k =k’k " Q (!)=j!j " "" "
Q"
H0
H0
U
2 p+1 :H !L ( U) !7!0
!jU
p!2H ! = 0 U ’2
(M) d’ = !0
d’ = 0 ’jU
pH (U) ’ [’]2
p pH (U) [’]2H (U=M)
pKer !H (U=M)

pH (U=M) M d’ = ! +!0 1
H H ’ !0 1 1
! = 01
pH (U=M)
M [’]! d’2H0
[’]
pH (U=M)! Ker
p2 p+1 pH ’L ( U)\ Im d H (U=M):0
!2H Q (!) Q(!) =0 "R
2 2j!j dv L ! UgU
j!j" Z
2 2!2H j!j = inf j’j dv0 g
d’=! U
2L !jU
U
p! H (U=M)
2 pL ( U) U !
M j!j
H ’0
pH (U=M) ! R
2 2j’j dv =j!jgU R
2M j!jj!j k’k k’k ! j’j dv" g g g" " U
jj jj"
jj jj Q !2H" 0

2 2 2 n 2p 2j!j k’k =j!j +" j’j dv ’g" " MnU
k’k 1 MnU gH
k’k 1 ’ @U ’ ’
2H
U MnU
k’k 1 Ck’k 1 C ’H (MnU)
2H
d’ ! H0
2 2 2! H k! k kd’k k’k1 1 1 1H (MnU)
2! ’ U k’ k 1 1 1
2 (D)k! k = ’ = ’ ’ k’k 11 1
2H
1H ’ U
Z
2 0j’j dv Ck’k 1 ;g H (U)
MnU
0C g "

U d + d
100 00
2k’k 1 C (k’k 2 +kd’k 2 ) =C (j!j +Q(!) );H (U) L (U) L (U)
j!j"
k(P ) k Ba a2B
kR P E0 0 0
N a Pa
0
N
E q P E0 a a 0
0
k :a7!q B Q(E ) 0a 0
k"> 0 :B !Q(E ) k k "0 1
ka 2B ( a ) =q0 0 0
0 (P )a
0 P N0 a0