Displaying similar documents to “An F σ semigroup of zero measure which contains a translate of every countable set”

Sets of extended uniqueness and σ -porosity

Miroslav Zelený (1997)

Commentationes Mathematicae Universitatis Carolinae

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We show that there exists a closed non- σ -porous set of extended uniqueness. We also give a new proof of Lyons’ theorem, which shows that the class of H ( n ) -sets is not large in U 0 .

Leaping convergents of Tasoev continued fractions

Takao Komatsu (2011)

Discussiones Mathematicae - General Algebra and Applications

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Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form p r n + i / q r n + i (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.

Vector integral equations with discontinuous right-hand side

Filippo Cammaroto, Paolo Cubiotti (1999)

Commentationes Mathematicae Universitatis Carolinae

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We deal with the integral equation u ( t ) = f ( ∫ I g ( t , z ) u ( z ) d z ) , with t ∈ I = [ 0 , 1 ] , f : 𝐑 n → 𝐑 n and g : I × I → [ 0 , + ∞ [ . We prove an existence theorem for solutions u ∈ L ∞ ( I , 𝐑 n ) where the function f is not assumed to be continuous, extending a result previously obtained for the case n = 1 .

Leaping convergents of Hurwitz continued fractions

Takao Komatsu (2011)

Discussiones Mathematicae - General Algebra and Applications

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Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent p r n + i / q r n + i (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms...

Hausdorff dimension of the maximal run-length in dyadic expansion

Ruibiao Zou (2011)

Czechoslovak Mathematical Journal

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For any x ∈ [ 0 , 1 ) , let x = [ ϵ 1 , ϵ 2 , ⋯ , ] be its dyadic expansion. Call r n ( x ) : = max { j ≥ 1 : ϵ i + 1 = ⋯ = ϵ i + j = 1 , 0 ≤ i ≤ n - j } the n -th maximal run-length function of x . P. Erdös and A. Rényi showed that lim n → ∞ r n ( x ) / log 2 n = 1 almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than log 2 n , is quantified by their Hausdorff dimension.