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This paper gives a self contained proof of the perturbation theorem for invariant tori in Hamiltonian systems by Kolmogorov, Arnold, and Moser with sharp differentiability
Then yet another application of the ergodic theorem and a factorization lemma (Lemma 4.3) show that the limit of the space averages equals the probability average at a point in
Hence multiplying by the Hasse invariant, if necessary, it follows from Theorem 1 that every odd dihedral representation as above also comes from a classical modular form of level
The main goal of this paper is to give closed formulas for the Arakelov-Green function G and the Faltings delta-invariant δ of a compact Riemann surface.. Both G and δ are
This argument relies on some deep and technical results: the localization theorem in K -theory proved by Thomason [12], the bivariant Riemann-Roch Theorem [3, 5.11.11], and
The proof of the following theorem known till now is based on the martingale theory (see e.g. We give a \pure dyadic analysis" proof for it.. Theorem 5. [AVD]), then can be
As Ω in Lemma 3.1 becomes large, we can have a considerable but finite number of fundamental solutions belonging to different classes that satisfy the bounds.. We will see that
Each countable group G admits a left invariant topology τ such that (G, τ ) is a nonmetrizable countable almost discrete topologically homogeneous extremally disconnected space with
