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

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154By developing the dual defining functions of M with respect to the powersof χ, we may write(7.28) ν j = Ω ∗ j(χ, x, u) = ∑ γ∈N p χ γ Ω ∗ j,γ(x, u),where the functions Ω ∗ j,γ(x, u) are K-analytic in a neighbourhood of the origin,we may formulate a criterion for M to be non degenerate with respectto the variables (whose proof is skipped).Proposition 3.1. The submanifold M is not degenerate with respect to thevariables if and only if there exists an integer k such that the generic rank ofthe local K-analytic mapping(7.28) (x, u) ↦−→ ( Ω ∗ j,γ (x, u)) 1≤j≤m, γ∈N p , |γ|≤kis equal to n + m.Seeking for conditions which insure that Sym(M ) is finite-dimensional,it is therefore natural to assume that the generic rank of the mapping (7.28)is equal to n + m. Furthermore, to simplify the presentation, we shall assumethat the rank at (x, u) = (0, 0) (not only the generic rank) of themapping (7.28) is equal to n + m for k large enough. This is a “Zariskigeneric”assumption. Coming back to (7.28), we observe that this meansexactly that M is solvable with respect to the variables. Then we denote byl ∗ 0 the smallest integer k such that the rank at (x, u) = (0, 0) of the mapping(7.28) is equal to n + m and we say that M is l ∗ 0-solvable with respectto the variables. Also, we denote by l 0 the integer max 1≤q≤p |β(q)| and wesay that M is l 0 -solvable with respect to the parameters.2.7. Fundamental isomorphism between Sym(E ) and Sym(M ). In theremainder of this Section 2, we shall assume that M is l 0 -solvable withrespect to the parameters and l ∗ 0-solvable with respect to the variabes. Inthis case, viewing the variables (ν 1 , . . .,ν m ) in the dual equations ν j =Ω ∗ j(χ, x, u) of M as a mapping of χ with (dual) “parameters” (x, u) andproceeding as in Subsection 2.2, we may construct a dual system of completelyintegrable partial differential equations of the form()(E ∗ ) ν j χ γ(χ) = Gj γ χ, ν(χ), (ν j(l) (χ))χ δ(l) 1≤l≤n ,where (j, γ) ≠ (j(1), δ(1)), . . ., (j(n), δ(n)). This system has its own infinitesimalsymmetry Lie algebra Sym(E ∗ ).Theorem 6.4. If M is both solvable with respect to the parameters andsolvable with respect to the variables, we have the following two isomorphisms:(7.28) Sym(E ) ∼ = Sym(M ) ∼ = Sym(E ∗ ),
155namely X ←→ X + X ←→ X .In Subsection 2.10 below, we shall introduce a second geometric conditionwhich is in general necessary for Sym(M ) to be finite-dimensional.2.8. Local (pseudo)group Sym(M ) of point transformations of M . Weshall study the geometry of a local K-analytic submanifold M of K n+2m+pwhose equations and dual equations are of the form{u j = Ω j (x, ν, χ), j = 1, . . .,m,(7.28)ν j = Ω ∗ j (χ, x, u), j = 1, . . ., m.Let t := (x, u) ∈ K n+m and τ := (ν, χ) ∈ K n+m . We are interested indescribing the set of local K-analytic transformations of the space K n+2m+pwhich are of the specific form(7.28) (t, τ) ↦−→ (h(t), φ(τ)),and which stabilize M , in a neighborhood of the origin. We denote the localLie pseudogroup of such transformations (possibly infinite-dimensional) bySym(M ). Importantly, each transformation of Sym(M ) stabilize both thesets {t = ct.} and the sets {τ = ct.}. Of course, the Lie algebra of Sym(M )coincides with Sym(M ) defined above.2.9. Fundamental pair of foliations on M . Let p 0 ∈ K n+2m+p be a fixedpoint of coordinates (t p0 , τ p0 ). Firstly, we observe that the intersection M ∩{τ = τ p0 } consists of the n-dimensional K-analytic submanifold of equationu = Ω(x, τ p0 ). As τ p0 varies, we obtain a local K-analytic foliation of M byn-dimensional submanifolds. Let us denote this first foliation by F p and callit the foliation of M with respect to parameters. Secondly, and dually, weobserve that the intersection M ∩{t = t p0 } consists of the p-dimensional K-analytic submanifold of equation ν = Ω ∗ (χ, t p0 ). As t p0 varies, we obtaina local K-analytic foliation of M by p-dimensional submanifolds. Let usdenote this second foliation by F v and call it the foliation of M with respectto the variables. We call (F p , F v ) the fundamental pair of foliations on M .2.10. Covering property of the fundamental pair of foliations. We wishto formulate a geometric condition which says that starting from the originin M and following alternately the leaves of F p and the leaves of F v , wecover a neighborhood of the origin in M . Let us introduce two collections
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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
Page 103 and 104: (1) The complex Lie algebra Hol(M,
Page 105 and 106: Proposition 3.1. Let t ↦→ G i (
Page 107 and 108: The first relation gives nothing, s
Page 109 and 110: with |β∗ k | ≥ 1 and integers
Page 111 and 112: Clearly, the left hand side is an a
Page 113 and 114: field X 1 ′ := h ∗ (X 1 ). Taki
Page 115 and 116: which yields after differentiating
Page 117 and 118: 117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
Page 119 and 120: [∂ i,ei H ei (t ′ )] t ′ =He
Page 121 and 122: p = (w p , z p , ζ p , ξ p ) ∈
Page 123 and 124: Finite nondegeneracy is interesting
Page 125 and 126: such that |ω j i ∗ ,β∗ i(t,
Page 127 and 128: for k even, we have Γ k (z (k) ) =
Page 129 and 130: that for every local holomorphic se
Page 131 and 132: h(t) = ˜H(t, J l 0µ 0¯h(0)) with
Page 133 and 134: infinite families of pairwise non b
Page 135 and 136: functions w(z)). The coefficients R
Page 137 and 138: Then the Lie criterion states that
Page 139 and 140: y k ′ = λk k y k and w ′ = µ
Page 141 and 142: The last statements of Corollaries
Page 143 and 144: [Sha2000] SHAFIKOV, R.: Analytic co
Page 145 and 146: Here x = (x 1 , . . ., x n ) ∈ K
Page 147 and 148: Consequently, for the case κ = 2,
Page 149 and 150: are devoted to provide a general on
Page 151 and 152: for l = 1, . . .,m such that the lo
Page 153: Lemma 8.1. The following conditions
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 and 204: Conversely, if y xx (x) = F ( x, y(
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205But we may also express the dual
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Then thanks to a straightforward ap
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and we then expand carefully the re
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explicit computation, so let us rew
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213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
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215Vanishing Hachtroudi curvaturean
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to fix the ideas, it will be assume
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then by replacing the(so obtained)v
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and also in addition the fact that
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and this defines without ambiguity
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the leaves of this local foliation
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Then by differentiating with respec
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∂Here, the coefficients of the2 T
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231for (7.28):( )∂□ ·∂a □
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233Lie symmetriesof partial differe
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Differentiating the first equation
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Example 1.21. (Continued) With n =
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defined by the graphed equations:((
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for some two local analytic maps Π
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Proof. Let l = l ( x i , y j , yβ(
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complementary views on the same obj
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Classification problem 3.11. Classi
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Definition 4.10. L is an infinitesi
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Convention 5.2. The letters R will
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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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