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11) Finally notice that, by definition of i, k i=1 min(1 + |li ± ji |) ≥ 1 + D(l, j). ± ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 2007 ´ B. 10) leads to S1 φj1 . . φjk dx ≤ 2πckn l∈Zk ,[l]=0 ≤ 2π(k − 1)N ckn 1 (1 + D(l, j))n µ(j)N S(j)N l∈Zk ,[l]=0 1 . 7) is verified. Verification of the strong nonresonancy condition in a simple case. — This subsection is inspired by section 5 in [BG06], actually the case considered here is much more simple. Let A be the operator on L2 (−π, π) defined by d2 u +V u dx2 where V is a 2π periodic potential and denotes the convolution product: Au = − π V u(x) = −π V (x − y)u(y)dy .

Arn63] V. I. N. Kolmogorov on the conservation of quasiperiodic motions under a small change of the Hamiltonian function, Russ. Math. Surv. 18 (1963), no. 5, p. 9–36. , Springer-Verlag, [Arn89] Berlin, 1989. [Bam03] D. Bambusi – Birkhoff normal form for some nonlinear PDEs, Comm. Math. Physics 234 (2003), p. 253–283. [BDGS07] D. Bambusi, J. M. Delort, B. Gr´ ebert & J. Szeftel – Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, to appear in CPAM (2007).

2 +1 2 (6) for l ≥ jk−1 , S(j, l) = |jk−1 | + |l − jk | and for l ≤ jk−1 , S(j, l) ≥ l. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 2007 ´ B. GREBERT 34 1 2 Proof. 1, we assume that P ∈ TkN,ν and Q ∈ TkN,ν 1 +1 2 +1 are homogeneous polynomial and we write aj zj1 . . zjk1 +1 P (z) = ¯ k1 +1 j∈Z and bi zi1 . . zik2 +1 . 5) with respect to the involved indices, one easily obtains {P, Q}(z) = cj,i zj1 . . zjk1 zi1 . . zik2 ¯ k1 +k2 (j,i)∈Z whith |cj,i | ≤ (k1 + 1)(k2 + 1) ¯ l∈Z µ(j, l)N +ν1 µ(i, l)N +ν2 .

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Advanced engineering mathematics with Matlab by Harman et al.

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