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Almost-sure growth rate of generalized random Fibonacci sequences

Élise Janvresse, Benoît Rittaud, Thierry de la Rue (2010)

Annales de l'I.H.P. Probabilités et statistiques

We study the generalized random Fibonacci sequences defined by their first non-negative terms and for n≥1, Fn+2=λFn+1±Fn (linear case) and ̃Fn+2=|λ̃Fn+1±̃Fn| (non-linear case), where each ± sign is independent and either + with probability p or − with probability 1−p (0<p≤1). Our main result is that, when λ is of the form λk=2cos(π/k) for some integer k≥3, the exponential growth of Fn for 0<p≤1, and of ̃Fn for 1/k<p≤1, is almost surely positive and given by ∫0∞log x dνk, ρ(x),...

An a b c d theorem over function fields and applications

Pietro Corvaja, Umberto Zannier (2011)

Bulletin de la Société Mathématique de France

We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u , v , in term of the degree of u , v , which is sharp whenever u , v have few distinct zeros and poles compared to their degree. This sharpens the “ a b c d -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface x a + y a + z c = 1 contains only finitely many rational or elliptic curves,...

An accurate approximation of zeta-generalized-Euler-constant functions

Vito Lampret (2010)

Open Mathematics

Zeta-generalized-Euler-constant functions, γ s : = k = 1 1 k s - k k + 1 d x x s and γ ˜ s : = k = 1 - 1 k + 1 1 k s - k k + 1 d x x s defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and γ ˜ (1) = ln 4 π , are studied and estimated with high accuracy.

Currently displaying 1241 – 1260 of 1970