Ma R., Xu J., Gao H.'s [0, ki] 1^m-Factorizations Orthogonal to a Subgraph PDF

By Ma R., Xu J., Gao H.

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P| R In particular α ≤ 2β. √ (ii) α(Λ, Ω, 7R/2) ≤ β. Proof. Since Λ is a lattice, it is not contained in E. Hence p · Λ is a non trivial subgroup of Z, p · Λ = mZ for some integer m ≥ 1, which implies that p/m ∈ Λ∗ . But p/m · Ω = α/m and |p/m| ≤ R, hence by the definition and the positivity of α, m = 1. As a result there exists x ∈ Λ such that p · x = 1. Obviously Λ0 + Zx ⊆ Λ. e. y ∈ Λ0 . So the reverse inclusion holds and we may write Λ = Λ0 + Zx. As a consequence Λ0 is a lattice of E and Λ∗ = {r ∈ Rl | r · Λ0 ⊂ Z and r · x ∈ Z} = {q + ap : q ∈ Λ∗0 , a ∈ Z − q · x}, Λ∗R = {q + ap : q ∈ Λ∗0 , a ∈ Z − q · x , 0 < |q|2 + a2 |p|2 ≤ R2 }.

H. P. , vol. 2, 1998, pp. 233-252. [2] V. I. Arnold: Instability of dynamical systems with several degrees of freedom, Sov. Math. Dokl. 6, 1964, pp. 581-585. [3] M. Berti, L. Biasco, P. Bolle: Optimal stability and instability results of a class of nearly integrable Hamiltonian systems, to appear on Rend. Mat. Acc. Naz. Lincei. [4] M. Berti, P. Bolle: Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems, Rend. Mat. Acc. Naz. Lincei, s. 9, vol. 11, fasc.

Bessi: An approach to Arnold diffusion through the calculus of variations, Nonlinear Analysis T. M. , 26, 1996, pp. 1115-1135. [8] U. Bessi: Arnold’s example with three rotators, Nonlinearity, 10, pp. 763-781, 1997. [9] U. Bessi, L. Chierchia, E. Valdinoci: Upper Bounds on Arnold Diffusion Time via Mather theory, J. Math. Pures Appl. vol. 80, 1, 2001, pp. 105–129. [10] L. Biasco, L. Chierchia, On the stability of some properly–degenerate Hamiltonian systems, to appear on Discrete and Continuous Dynamical Systems, series A.

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[0, ki] 1^m-Factorizations Orthogonal to a Subgraph by Ma R., Xu J., Gao H.

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