Travaux sur les symÃ©tries de Lie des Ã©quations aux ... - DMA - Ens

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204just by virtue of the reality identities (7.28). It also follows rather triviallythat the dual differential equation:w zz (z) = Θ zz(z, ζ(z, w(z), wz (z)), ξ(z, w(z), w z (z)) )= Φ ( z, w(z), w z (z) )is also just the conjugate differential equation.To the differential equation y xx = F and to its dual b aa = F ∗ are associatedtwo submanifolds of solutions:M = M (E ) := {( x, y, a, b ) ∈ K × K × K × K: y = Q(x, a, b) } ,together with:M ∗ = M (E ∗ ) := {( a, b, x, y ) ∈ K × K × K × K: b = Q ∗ (a, x, y) } ,and as one obviously guesses, the duality, when viewed within submanifoldsof solutions, just amounts to permute variables and parameters:M ∋ (x, y, a, b) ←→ (a, b, x, y) ∈ M ∗ .In the CR case, if we denote by ˜z and ˜w two independent complex variableswhich correspond to the complexifications of z and w (respectively ofcourse), the duality takes place between the so-called extrinsic complexification([13, 14, Me2005a, Me2005b, 18, 19]):M = M c := {( z, w, ˜z, ˜w ) ∈ C × C × C × C: w = Θ ( z, ˜z, ˜w )}of M in one hand, and in the other hand, its own transformation 21 :M ∗ = ∗ c (M c ) := {(˜z, ˜w, z, w ) ∈ C × C × C × C: ˜w = Θ (˜z, z, w )}under the involution:∗ c( z, w, ˜z, ˜w ) := (˜z, ˜w, z, w )which clearly is the complexification of the natural antiholomorphic involution:∗ ( z, w, z, w ) := ( z, w, z, w )that fixes M pointwise, as it fixes any other real analytic subset of C 2 . Here,one has M ∗ = ∗(M ) — which is ≠ M in general — and of course also(M∗ )∗ = M .So in terms of the coordinates (x, a, b) on M and of the coordinates(a, x, y) on M ∗ , the duality is the map:(x, a, b) ↦−→ ( a, x, Q(x, a, b) )with inverse:(a, x, y) ↦−→ ( x, a, Q ∗ (a, x, y) ) .21 Be careful not to write {( z, w, ˜z, ˜w ) : ˜w = Θ (˜z, z, w )} , because this would regivethe same subset M of C 2 × C 2 , due to the reality identities (7.28).
205But we may also express the duality from the first jet (x, y, y x )-space to thefirst jet (a, b, b a )-space by simply composing the following three maps, thecentral one being the duality M → M ∗ :⎛⎝⎞(a, x, y) (( ↓a, Q ∗ (a, x, y), Q ∗ a(a, x, y) ) ⎠ ◦⎛(x, a, b) → ( a, x, Q(x, a, b) )) ◦ ⎝(x, A(x, y, yx ), B(x, y, y x ) )↑(x, y, y x )which in sum gives us the map:(A(x, y, yx ), Q(x, y, y x ) ↦−→∗( A(x, y, y x ), x, Q ( x, A(x, y, y x ), B(x, y, y x ) )) )( ,A(x, y, yx ), x, Q ( x, A(x, y, y x ), B(x, y, y x ) )) .Q ∗ aWith the approximations, one checks that:(x, y, y x ) ↦−→ ( y x + · · · , −y + xy x + · · · , x + · · ·),where the remainder terms “+ · · · ” are all O(x 2 ). For the differential equationy xx (x) = 0 of affine lines, these remainders disappear completely andwe recover the classical projective duality written in inhomogeneous coordinates([5], pp. 156–157). Furthermore, one shows (see e.g. [5]) that theabove duality map within first order jet spaces is a contact transformation,namely through it, the pullback of the standard contact form db−b a da in thetarget space is a nonzero multiple of the standard contact form dy − y x dx inthe source space.But what matters more for us is the following. The two fundamentaldifferential invariants of b aa (a) = F ∗( a, b(a), b a (a) ) are functions exactlysimilar to the ones written on p. 197, namely:⎞⎠,I 1 (E ∗ ) := F ∗ b ab ab ab aI 2 (E ∗ ) := D ∗ D ∗( F ∗ b ab a)− F∗baD ∗( F ∗ b ab a)− 4 D∗ ( F ∗ bb a)++ 6 F ∗ bb − 3 F ∗ b F ∗ b ab a+ 4 F ∗ b aF ∗ bb a,where D ∗ := ∂ a + b a ∂ b + F ∗ (a, b, b a ) ∂ ba . Then according to Koppisch([10]), through the duality map, I 1 (E)is transformed to a nonzero multiple ofI 2 (E ∗ ) , and simultaneously also, I2 (E ) is transformed to a nonzero multiple22 ofI 1 (E ∗ ) , so that: 0 = I 1 (E) ⇐⇒ I 2 (E ∗ ) = 00 = I 2 (E) ⇐⇒ I 1 (E ∗ ) = 0.Consequently, the differential equation (E ): y xx (x) = F ( x, y(x), y x (x) ) isequivalent to y ′ x ′ x ′(x′ ) = 0 if and only if:F yxy xy xy x= 0 and F ∗ b ab ab ab a= 0 .22 To be precise, both factors of multiplicity ([5], p. 165) are nonvanishing in a neighboroodof the origin, but for our purposes, it suffices just that they are not identically zeropower series.
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Joël M E R K E RÉcole Normale Sup
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§1. INTRODUCTIONSeveral physically
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e a collection of m analytic second
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7where the indices j, l 1 vary in {
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This phenomenon could be explained
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This yields the prolongation of the
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X(x, y) and Y (x, y) such that it m
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2.23. Compatibility conditions for
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This lemma is left to the reader; a
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simplification nor any reordering:(
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aleza, y por otra, las organizacion
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computational level (differential-g
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For instance, in the case m = 2, by
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in (3.11). The second equation that
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Similarly, the second equation take
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order derivatives of X and of the Y
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Lemma 3.32. The system yxx j = F j
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Lemma 3.45. The following quadratic
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often be denoted by the sign “·
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1Now, taking account of the factor
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conditions, totally equivalent to t
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43remaining terms afterwards:(3.71)
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Here, the sign ≡ precisely means:
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Next, replacing plainly (3.64) in (
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49the order of §3.73. We get:(3.89
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51Multiplying by −2 and reorganiz
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⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
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+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2
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+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
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In the hardest techical part of thi
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− 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
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− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
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Here, the sign ≡ means “modulo
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Thirdly, put j := l 2 in (3.108) wi
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69correct. We get:(4.29)0 =?== −
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Next, apply the operator ∑ k Ll 2
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73Writing term by term the substrac
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of the terms of the subgoal (4.29):
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77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
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79= − X xx Yx j + Y jm∑+++++l 1
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Our first task is to compute the de
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Replacing this expression of A k in
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85have the continuation(5.17) −y
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87and where thirdly (we are nearly
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89obtain:(5.23)III :=m∑m∑l 1 =1
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[GTW1989] GRISSOM, C.; THOMPSON, G.
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93Nonalgebraizable real analytic tu
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and of CR dimension m = n − d ≥
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coordinates z ′ = 2i ln(z/z p ),
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{x ∈ K n : |x| < ρ} for some ρ
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(2) A K-algebraic inversion mapping
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(1) The complex Lie algebra Hol(M,
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Proposition 3.1. Let t ↦→ G i (
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The first relation gives nothing, s
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with |β∗ k | ≥ 1 and integers
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Clearly, the left hand side is an a
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field X 1 ′ := h ∗ (X 1 ). Taki
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which yields after differentiating
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117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
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[∂ i,ei H ei (t ′ )] t ′ =He
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p = (w p , z p , ζ p , ξ p ) ∈
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Finite nondegeneracy is interesting
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such that |ω j i ∗ ,β∗ i(t,
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for k even, we have Γ k (z (k) ) =
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that for every local holomorphic se
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h(t) = ˜H(t, J l 0µ 0¯h(0)) with
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infinite families of pairwise non b
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functions w(z)). The coefficients R
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Then the Lie criterion states that
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y k ′ = λk k y k and w ′ = µ
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The last statements of Corollaries
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[Sha2000] SHAFIKOV, R.: Analytic co
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Here x = (x 1 , . . ., x n ) ∈ K
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Consequently, for the case κ = 2,
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are devoted to provide a general on
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for l = 1, . . .,m such that the lo
Page 153 and 154: Lemma 8.1. The following conditions
Page 155 and 156: 155namely X ←→ X + X ←→ X .
Page 157 and 158: In terms of Sussmann’s approach [
Page 159 and 160: x κ−1 χ κ−1 + O(|x| κ ) + O
Page 161 and 162: Observe that these expressions are
Page 163 and 164: where the first term I involves onl
Page 165 and 166: I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
Page 167 and 168: U 2 U κ−1 , we obtain the five f
Page 169 and 170: and U 2 U κ−1 , we obtain the fi
Page 171 and 172: following equations for j = 1, . .
Page 173 and 174: linear equations:(7.28) ⎧0 = Rx j
Page 175 and 176: 175R j containing at least one part
Page 177 and 178: 177[3] :∑σ∈S κ−1κδ l,....
Page 179 and 180: the values of the partial derivativ
Page 181 and 182: also appears some derivatives Q l
Page 183 and 184: [5] F. ENGEL; LIE, S.: Theorie der
Page 185 and 186: 185Nonrigid sphericalreal analytic
Page 187 and 188: origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
Page 189 and 190: I 2 characterizes equivalence to w
Page 191 and 192: y means of a fundamental, elementar
Page 193 and 194: 2) When M is Levi nondegenerate at
Page 195 and 196: 195holding in C { z ′ , z ′ , w
Page 197 and 198: Fortunately for our present purpose
Page 199 and 200: We notice passim that S ≡ Q x (no
Page 201 and 202: for a certain local K-analytic new
Page 203: Conversely, if y xx (x) = F ( x, y(
Page 207 and 208: Then thanks to a straightforward ap
Page 209 and 210: and we then expand carefully the re
Page 211 and 212: explicit computation, so let us rew
Page 213 and 214: 213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
Page 215 and 216: 215Vanishing Hachtroudi curvaturean
Page 217 and 218: to fix the ideas, it will be assume
Page 219 and 220: then by replacing the(so obtained)v
Page 221 and 222: and also in addition the fact that
Page 223 and 224: and this defines without ambiguity
Page 225 and 226: the leaves of this local foliation
Page 227 and 228: Then by differentiating with respec
Page 229 and 230: ∂Here, the coefficients of the2 T
Page 231 and 232: 231for (7.28):( )∂□ ·∂a □
Page 233 and 234: 233Lie symmetriesof partial differe
Page 235 and 236: Differentiating the first equation
Page 237 and 238: Example 1.21. (Continued) With n =
Page 239 and 240: defined by the graphed equations:((
Page 241 and 242: for some two local analytic maps Π
Page 243 and 244: Proof. Let l = l ( x i , y j , yβ(
Page 245 and 246: complementary views on the same obj
Page 247 and 248: Classification problem 3.11. Classi
Page 249 and 250: Definition 4.10. L is an infinitesi
Page 251 and 252: Convention 5.2. The letters R will
Page 253 and 254: Assuming that dimSYM(E 1 ) = 8, tak
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Restarting from §4.1, let ϕ a Lie
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Here, R j l 1 ,kare universal polyn
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Lemma 6.34. Let M be a submanifold
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where the expressions D β,β1 are
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Then the sum L + L is tangent to M
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Lemma 7.11. ([CM1974, BER1999, Me20
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where(7.33) ∆ := ∑1kn∂ 2∂x
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with r, s being unspecified functio
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and moreover, all other, higher ord
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To conclude, we replace X xx so obt
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Definition 8.43. A real analytic hy
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Lemma 8.59. The function b depends
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where the letter r denotes an unspe
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(8.80) ⎧Y 1,1,1 = Y x 1 x 1 x 1 +
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(X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1
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espect to x 1 and (8.96) 2 with res
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For the two unknowns Xyy 1 and X 2x
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Redeveloping the determinant, the v
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291are obtained by equating to zero
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Let (z 0 , c 0 ) = (x 0 , y 0 , a 0
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we deduce that Γ 5 is submersive (
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11.4. Regularity and jet parametriz
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Cramer’s rule, we get(11.13) ⎧
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in K[a, z] n+m and in K[x, c] p+m .
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of the cancellation properties (11.
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305II: Explicit prolongations of in
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Define then the transformed jet ϕ
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Let λ ∈ N be an arbitrary intege
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311[Ol1986], [BK1989]):(Y(1.31)⎧
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+ [Y y 2 − 2 X xy ] y 1 y 2 + [
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for some nonnegative integers A, B,
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Once the correct theorem is formula
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By the classical formula for the de
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The induction formulas are⎧(3.4)
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We have to compute:( ) ⎛ ⎞∑
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Next, we gather the underlined term
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+ ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k
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Correspondingly, we identify the se
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must be equal to the number of indi
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Y i1 ,i 2 ,i 3 ,i 4written in one o
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335conclusion, we have shown that (
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337Theorem 3.73. For every κ 1 an
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Theorem 3.79. For every integer κ
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+++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3
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343form:(4.12)κ+1Yκ j = Y jx κ +
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eing related to the number 2 in the
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Appying the chain rule, we may comp
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Secondly:(5.3)Y j i 1 ,i 2= Y jx i
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As explained before the statement o
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g j i 1 ,...,i λand similarly for
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with respect to the variables x i 1
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Assuming F = F(x, y x ) to be indep
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359(III’) ⎧0 = ∑ (δ k 2)j 3H
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Open problem 1.17. For n = 2 establ
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⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
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eplacing the third column by the se
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⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
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e written under the specific form:(
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Indeed, the collection of equations
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3.12. Principal unknowns. As there
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375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
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are a consequence of (I’), (I”)
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379IV: BibliographyREFERENCES[Ar198
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[G1989] GARDNER, R.B.: The method o
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[Me2005a] MERKER, J.: On the local
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