Displaying similar documents to “Characterizations of groups generated by Kronecker sets”

Baireness of C k ( X ) for ordered X

Michael Granado, Gary Gruenhage (2006)

Commentationes Mathematicae Universitatis Carolinae

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We show that if X is a subspace of a linearly ordered space, then C k ( X ) is a Baire space if and only if C k ( X ) is Choquet iff X has the Moving Off Property.

On infinite composition of affine mappings

László Máté (1999)

Fundamenta Mathematicae

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 Let F i = 1 , . . . , N be affine mappings of n . It is well known that if there exists j ≤ 1 such that for every σ 1 , . . . , σ j 1 , . . . , N the composition (1) F σ 1 . . . F σ j is a contraction, then for any infinite sequence σ 1 , σ 2 , . . . 1 , . . . , N and any z n , the sequence (2) F σ 1 . . . F σ n ( z ) is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any z n and any σ = σ 1 , σ 2 , . . . belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every σ = σ 1 , σ 2 , . . . Σ the composition (1) is a contraction....

A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

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1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) x = 1 / d ( x ) + 1 / ( d ( x ) d ( x ) ) + . . . + 1 / ( d ( x ) d ( x ) . . . d n ( x ) ) + . . . , where d j ( x ) , j 1 is a sequence of positive integers satisfying d₁(x) ≥ 2 and d j + 1 ( x ) d j ( x ) for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) l i m n d n 1 / n ( x ) = e . He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. d i m H x ( 0 , 1 ] : ( 2 ) f a i l s = 1 . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and d i m H to denote...

A note on evaluations of some exponential sums

Marko J. Moisio (2000)

Acta Arithmetica

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1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form S ( a , p α + 1 ) : = x q χ ( a x p α + 1 ) where χ is a non-trivial additive character of the finite field q , q = p e odd, and a * q . In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of p α + 1 . The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate...

Intersection topologies with respect to separable GO-spaces and the countable ordinals

M. Jones (1995)

Fundamenta Mathematicae

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Given two topologies, T 1 and T 2 , on the same set X, the intersection topologywith respect to T 1 and T 2 is the topology with basis U 1 U 2 : U 1 T 1 , U 2 T 2 . Equivalently, T is the join of T 1 and T 2 in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and ω 1 -compactness in this class of topologies. We demonstrate that the majority of his results...

Guidance properties of a cylindrical defocusing waveguide

Oldřich John, Charles A. Stuart (1994)

Commentationes Mathematicae Universitatis Carolinae

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We discuss the propagation of electromagnetic waves of a special form through an inhomogeneous isotropic medium which has a cylindrical symmetry and a nonlinear dielectric response. For the case where this response is of self-focusing type the problem is treated in [1]. Here we continue this study by dealing with a defocusing dielectric response. This tends to inhibit the guidance properties of the medium and so guidance can only be expected provided that the cylindrical stratification...

On the cardinality and weight spectra of compact spaces, II

Istvan Juhász, Saharon Shelah (1998)

Fundamenta Mathematicae

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Let B(κ,λ) be the subalgebra of P(κ) generated by [ κ ] λ . It is shown that if B is any homomorphic image of B(κ,λ) then either | B | < 2 λ or | B | = | B | λ ; moreover, if X is the Stone space of B then either | X | 2 2 λ or | X | = | B | = | B | λ . This implies the existence of 0-dimensional compact T 2 spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.