P-adic Spaces of Continuous Functions II

Athanasios Katsaras

Displaying similar documents to “P-adic Spaces of Continuous Functions II”

P-adic Spaces of Continuous Functions I

Athanasios Katsaras (2008)

Annales mathématiques Blaise Pascal

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Properties of the so called $θ_{o}$ -complete topological spaces are investigated. Also, necessary and sufficient conditions are given so that the space $C (X, E)$ of all continuous functions, from a zero-dimensional topological space $X$ to a non-Archimedean locally convex space $E$ , equipped with the topology of uniform convergence on the compact subsets of $X$ to be polarly barrelled or polarly quasi-barrelled.

On ergodicity of some cylinder flows

Krzysztof Frączek (2000)

Fundamenta Mathematicae

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We study ergodicity of cylinder flows of the form $T_{f} : T \times ℝ \to T \times ℝ$ , $T_{f} (x, y) = (x + α, y + f (x))$ , where $f : T \to ℝ$ is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that $D^{k} f$ is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of $D^{k} f$ have some good properties, then $T_{f}$ is ergodic. Moreover, there exists $ε_{f} > 0$ such that if $v : T \to ℝ$ is a function with zero integral such that $D^{k} v$ is of bounded...

Parametrized Cichoń's diagram and small sets

Janusz Pawlikowski, Ireneusz Recław (1995)

Fundamenta Mathematicae

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We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^{w} \times 2^{w}$ and continuous functions $e, f : w^{w} \to w^{w}$ such that • N is $G_{δ}$ and $N_{x} : x \in w^{w}$ , the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^{w}$ ; • M is $F_{σ}$ and $M_{x} : x \in w^{w}$ is a basis for the ideal of meager subsets of $2^{w}$ ; • $\forall x, y N_{e (x)} \subseteq N_{y} \Rightarrow M_{x} \subseteq M_{f (y)}$ . From this we derive that for a separable metric space X, •if for all Borel (resp. $G_{δ}$ ) sets...

Inessentiality with respect to subspaces

Michael Levin (1995)

Fundamenta Mathematicae

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Let X be a compactum and let $A = (A_{i}, B_{i}) : i = 1, 2, . . .$ be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed $F_{i}$ separating $A_{i}$ and $B_{i}$ the intersection $(\cap F_{i}) \cap Y$ is not empty. So A is inessential on Y if there exist closed $F_{i}$ separating $A_{i}$ and $B_{i}$ such that $\cap F_{i}$ does not intersect Y. Properties of inessentiality are studied and applied to prove: Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on...

Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem

Ramez Sami (1999)

Fundamenta Mathematicae

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We prove the following theorem: Given a⊆ω and $1 \leq α < ω_{1}^{C K}$ , if for some $η < ℵ_{1}$ and all u ∈ WO of length η, a is $Σ_{α}^{0} (u)$ , then a is $Σ_{α}^{0}$ . We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: $Σ_{1}^{1}$ -Turing-determinacy implies the existence of $0$ .

A note on Tsirelson type ideals

Boban Veličković (1999)

Fundamenta Mathematicae

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Using Tsirelson’s well-known example of a Banach space which does not contain a copy of $c_{0}$ or $l_{p}$ , for p ≥ 1, we construct a simple Borel ideal $I_{T}$ such that the Borel cardinalities of the quotient spaces $P (ℕ) / I_{T}$ and $P (ℕ) / I_{0}$ are incomparable, where $I_{0}$ is the summable ideal of all sets A ⊆ ℕ such that $\sum_{n \in A} 1 / (n + 1) < \infty$ . This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.

Growth of the product $\prod_{j = 1}^{n} (1 - x^{a_{j}})$

J. P. Bell, P. B. Borwein, L. B. Richmond (1998)

Acta Arithmetica

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We estimate the maximum of $\prod_{j = 1}^{n} | 1 - x^{a_{j}} |$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_{j}$ is $j^{k}$ or when $a_{j}$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_{j}$ is j. In contrast we show, under fairly general conditions, that the maximum is less than $2^{n} / n^{r}$ , where r is an arbitrary positive number. One consequence...

Bing maps and finite-dimensional maps

Michael Levin (1996)

Fundamenta Mathematicae

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Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map $g : X \to 𝕀^{k}$ such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open. Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map $g : X \to 𝕀^{k}$ such that dim (f × g) = 1. We improve this result of Sternfeld...

Sierpiński's hierarchy and locally Lipschitz functions

Michał Morayne (1995)

Fundamenta Mathematicae

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Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and $α < ω_{1}$ then f ○ g ∈ $B_{α} (Z)$ for every $g \in B_{α} (Z) \cap^{Z} I$ if and only if f is continuous on I, where $B_{α} (Z)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes $S_{α} (Z) (α > 0)$ in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally...

Theta functions of quadratic forms over imaginary quadratic fields

Olav K. Richter (2000)

Acta Arithmetica

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1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function $ϑ_{Q} (z) : = \sum_{g \in ℤ^{n}} e x p π i^{t} g Q g z$ , Im z > 0, is a modular form of weight n/2 on $Γ_{0} (N)$ , where N is the level of Q, i.e. $N Q^{- 1}$ is integral and $N Q^{- 1}$ has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier...

On the restricted Waring problem over $_{2^{n}} [t]$

Luis Gallardo (2000)

Acta Arithmetica

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1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, $_{16}$ , each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is...

On $B_{2 k}$ -sequences

Martin Helm (1993)

Acta Arithmetica

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Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a $B_{r}$ -sequence A satisfies $l i m i n f_{n \to \infty} (A (n) / (n^{1 / r})) = 0$ . The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3]. Here we present a different, very short proof...

Borel partitions of unity and lower Carathéodory multifunctions

S. Srivastava (1995)

Fundamenta Mathematicae

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We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in $A (ℰ \otimes ℬ (X))$ into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential...