Property C'', strong measure zero sets and subsets of the plane

Janusz Pawlikowski

Displaying similar documents to “Property C'', strong measure zero sets and subsets of the plane”

The Zahorski theorem is valid in Gevrey classes

Jean Schmets, Manuel Valdivia (1996)

Fundamenta Mathematicae

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Let Ω,F,G be a partition of $ℝ^{n}$ such that Ω is open, F is $F_{σ}$ and of the first category, and G is $G_{δ}$ . We prove that, for every γ ∈ ]1,∞[, there is an element of the Gevrey class Γγ which is analytic on Ω, has F as its set of defect points and has G as its set of divergence points.

For almost every tent map, the turning point is typical

Henk Bruin (1998)

Fundamenta Mathematicae

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Let $T_{a}$ be the tent map with slope a. Let c be its turning point, and $μ_{a}$ the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, $ʃ g d μ_{a} = l i m_{n \to \infty} \frac{1}{n} \sum_{i = 0}^{n - 1} g (T_{a}^{i} (c))$ . As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.

Spaces of upper semicontinuous multi-valued functions on complete metric spaces

Katsuro Sakai, Shigenori Uehara (1999)

Fundamenta Mathematicae

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Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by $U S C C_{B} (X)$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify $φ \in U S C C_{B} (X)$ with its graph which is a closed subset of X × ℝ. The space $U S C C_{B} (X)$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then $U S C C_{B} (X)$ is homeomorphic...

Gauss sum for the adjoint representation of $G L_{n} (q)$ and $S L_{n} (q)$

Yeon-Kwan Jeong, In-Sok Lee, Hyekyoung Oh, Kyung-Hwan Park (2000)

Acta Arithmetica

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Operators on C(ω^α) which do not preserve C(ω^α)

Dale Alspach (1997)

Fundamenta Mathematicae

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It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from $C (ω^{ω^{α}})$ onto itself such that if Y is a subspace of $C (ω^{ω^{α}})$ which is isomorphic to $C (ω^{ω^{α}})$ , then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from $C (ω^{ω^{α}})$ onto itself there is a subspace of $C (ω^{ω^{α}})$ which is isomorphic to $C (ω^{ω^{α}})$ on which the operator is an isomorphism.

The homotopy groups of the L2 -localization of a certain type one finite complex at the prime 3

Yoshitaka Nakazawa, Katsumi Shimomura (1997)

Fundamenta Mathematicae

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For the Brown-Peterson spectrum BP at the prime 3, $v_{2}$ denotes Hazewinkel’s second polynomial generator of $B P_{*}$ . Let $L_{2}$ denote the Bousfield localization functor with respect to $v_{2}^{- 1} B P$ . A typical example of type one finite spectra is the mod 3 Moore spectrum M. In this paper, we determine the homotopy groups $π_{*} (L_{2} M \land X)$ for the 8 skeleton X of BP.

The exceptional set of Goldbach numbers (II)

Hongze Li (2000)

Acta Arithmetica

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1. Introduction. A positive number which is a sum of two odd primes is called a Goldbach number. Let E(x) denote the number of even numbers not exceeding x which cannot be written as a sum of two odd primes. Then the Goldbach conjecture is equivalent to proving that E(x) = 2 for every x ≥ 4. E(x) is usually called the exceptional set of Goldbach numbers. In [8] H. L. Montgomery and R. C. Vaughan proved that $E (x) = O (x^{1 - Δ})$ for some positive constant Δ > 0 $. I n [3] C h e n a n d P a n p r o v e d t h a t o n e c a n t a k e Δ > 0.01 . I n [6], w e p r o v e d t h a t E (x) = O (x^{0.921})$ . In this paper we prove the following...

An ordinal version of some applications of the classical interpolation theorem

Benoît Bossard (1997)

Fundamenta Mathematicae

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Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space $E_{1}$ with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space $E_{2}$ with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of $E_{1}$ and $E_{2}$ can be controlled by the Szlenk index of E, where the...

Dugundji extenders and retracts on generalized ordered spaces

Gary Gruenhage, Yasunao Hattori, Haruto Ohta (1998)

Fundamenta Mathematicae

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For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an $L_{c h}$ -extender (resp. $L_{c c h}$ -extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous $L_{c h}$ -extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology...

Asymptotic aspects of the Diophantine equation $p^{k} x^{n k} - z^{k} = l$

Wolfgang Jenkner (2000)

Acta Arithmetica

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A combinatorial approach to partitions with parts in the gaps

Dennis Eichhorn (1998)

Acta Arithmetica

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Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let $p_{k, m} (j, n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p *_{k, m} (j, n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p *_{k, m} (j, n) = p_{k, m} (j, n)$ .

Inverse limit of M -cocycles and applications

Jan Kwiatkowski (1998)

Fundamenta Mathematicae

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For any m, 2 ≤ m < ∞, we construct an ergodic dynamical system having spectral multiplicity m and infinite rank. Given r > 1, 0 < b < 1 such that rb > 1 we construct a dynamical system (X, B, μ, T) with simple spectrum such that r(T) = r, F*(T) = b, and $C (T) / w c l T^{n} : n \in ℤ = \infty$

A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

W. Jurkat, D. Nonnenmacher (1994)

Fundamenta Mathematicae

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Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^{n}$ , we are led to an integration process over quite general sets $A \subseteq q ℝ^{n}$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.

Homotopy orbits of free loop spaces

Marcel Bökstedt, Iver Ottosen (1999)

Fundamenta Mathematicae

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Let X be a space with free loop space ΛX and mod two cohomology R = H*X. We construct functors $Ω_{λ} (R)$ and ℓ(R) together with algebra homomorphisms $e : Ω_{λ} (R) \to H * (Λ X)$ and $ψ : ℓ (R) \to H * (E S^{1} \times_{S^{1}} Λ X)$ . When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms.

Exact m-covers and the linear form $\sum_{s = 1}^{k} x_{s} / n_{s}$

Zhi-Wei Sun (1997)

Acta Arithmetica

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Normal subspaces in products of two ordinals

Nobuyuki Kemoto, Tsugunori Nogura, Kerry Smith, Yukinobu Yajima (1996)

Fundamenta Mathematicae

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Let λ be an ordinal number. It is shown that normality, collectionwise normality and shrinking are equivalent for all subspaces of ${(λ + 1)}^{2}$ .

A polarized partition relation and failure of GCH at singular strong limit

Saharon Shelah (1998)

Fundamenta Mathematicae

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The main result is that for λ strong limit singular failing the continuum hypothesis (i.e. $2^{λ} > λ^{+}$ ), a polarized partition theorem holds.