[en] A Matrix Continued-Fraction method is used to study the localization length of the states at the band center of a two dimensional crystals with disorder given by the Anderson model. It is found that exponentially localized states which scale according to the work of Mac Kinnon and Kramer, becomes weakly localized as the disorder becomes weaker, and there is some critical disorder for which the localization length does not saturate with the width of the strips, this confirms the resuts found by Pichard and Sarma. Weakly localized states are also found in one dimension for w/v < or approx., 1. The transition from extended to localized states is also examined in one dimension. (author)

[en] Set q=exp(2πiτ), where τ is an irrational number, and let R_q be the radius of holomorphy of the Rogers-Ramanujan function G_q(z)=1+Σ_n=1^∞zⁿ(qⁿ²)/((1-q)...(1-qⁿ)). As is known, R_q≤1 and for each α element of [0,1] there exists q=q(α) such that R_q(α)=α. It is proved here that the function H_q(z)=G_q(z)/G_q(qz) is meromorphic not only in the disc =(|z|< R_q), but also in the disc D=(|z|<1), which is larger for R_q<1; and that the Rogers-Ramanujan continued fraction converges to H_q on compact subsets contained in DΩ_q, where Ω_q is the union of circles with centres at z=0 and passing through the poles of H_q. The convergence of the Rogers-Ramanujan continued fraction in the domain {|z|q,1/(2+|1+q|))}Ω_q was established earlier by Lubinsky

[en] A refined version of a classical theorem of Poincare on difference equations with coefficients having limits is established. Applications include several results that are similar to well-known theorems of Van Vleck, Thron-Waadeland, and Perron, but are proved under considerably weaker assumptions

No abstract available

[en] This is an investigation of the problem of the asymptotic distribution of the Frobenius numbers of n relatively prime integers. For n=3 virtually definitive results are obtained. For n>3 it is shown that the distributions appearing form a compact set. An essential role is played by the limit theorem for logarithms of denominators of continued fractions of random numbers.

[en] The analytic weighted Lipschitz algebra Λ_ω is the Banach algebra of all analytic functions on the unit disk D, that are continuous on D-bar and such that sup z,w element of D-bar z≠w |f(z) - f(w)| / w(|z - w|) < +∞, where ω is a modulus of continuity. We give a new characterization of outer functions in Λω, by their modulus in T. As application, we obtain a refinement of Shirokov's construction of outer functions in Λω vanishing on a given ω-Carleson set. We obtain also an extension of Havin-Shamoyan-Carleson-Jacobs Theorem to an arbitrary modulus of continuity. (author)

[en] The differences are described between the statistics of the periodic continued fractions of quadratic irrational numbers and the universal Gauss-Kuzmin statistics of the continued fractions of random real numbers. Besides being a survey of these differences, this article discusses the results of V.A. Bykovskii [Bykovsky] and his school, who have proved the author's 1993 conjectures on the similarities of these statistics.

[en] The Markov partitions constructed by Adler and Weiss and the pre-Markov partitions related to them are important in the investigation of the properties of an Anosov diffeomorphism of a 2-torus. A connection is established between the number of equivalence classes of the simplest pre-Markov partitions of a fixed diffeomorphism with respect to the natural equivalence and the continued fraction expressing the slope of the unstable direction of the matrix defining this diffeomorphism. Bibliography: 7 titles.

No abstract available

[en] For each real number, we obtain an asymptotic for the number of partial quotients in the continued fraction expansion that can be obtained from the first n terms of its β-expansion. A novelty of our approach is the use of methods of the theory of dynamical systems