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1

Studying Word Equations by a Method of Weighted Frequencies

Saarela A.

Fundamenta Informaticae

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2018

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Vol. 162, nr 2/3

223--235

EN

We briefly survey some results and open problems on word equations, especially on those equations where the right-hand side is a power of a variable. We discuss a method that was recently used to prove one of the results, and we prove improved versions of some lemmas that are related to the method and can be used as tools when studying word equations. We use the method and the tools to give new, simple proofs for several old results.

2

A note on invariant measures

Niemiec P.

Opuscula Mathematica

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2011

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Vol. 31, no. 3

425-431

EN

The aim of the paper is to show that if F is a family of continuous transformations of a nonempty compact Hausdorff space Ω, then there is no F-invariant probabilistic regular Borel measures on Ω iff there are φ1..., φp ∈ F (for some p ≥ 2) and a continuous function u: Ω, p ? R such that Σ σ ∈Spu(x&sigma(1),...,x&sigmap = 0 and lim infn?∞1/n Σ n-1/k=0 (u o Φk)9x1,...xp) ≥ 1 for each x1,...xp ∈ Ω, where Φ: Ωp ∋ (x1,...xp) ? (φ1(x1,..., φp(xp)∈ Ωp and Φk is the k-th iterate of Φ. A modified version of this result in case the family F generates an equicontinuous semigroup is proved.

3

On the compactness and countable compactness of 2R in ZF

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 4

293-302

EN

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements "2R is countably compact" and "2R is compact".

4

Countable compact scattered T2 spaces and weak forms of AC

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2006

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Vol. 54, no 1

75-84

EN

We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T2 topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T2 space is scattered iff it is metrizable. (3) If the real line R can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T2 space is scattered. (4) It is not provable in ZF+¬AC that there exists a countable compact T2 space which is dense-in-itself.

5

Tensor products in concrete categories

Jagiełło D.

Demonstratio Mathematica

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1999

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Vol. 32, nr 2

272-280

EN

In this paper we consider the notion of tensor multiplication in the concrete categories (by the concrete category we mean the category with fixed covariant faithful functor U : -- Ens). The reason of this choice is the observation of the constructions of tensor product in the categories of abelian groups, vector spaces or more generally in any variety (which are of course concrete). We modify this constructions to give the universal method of introduction the tensor multiplication in any concrete category. Moreover we are not restricted because many important categories are concrete. Our aim was the general overview on the tensor multiplication in order to apply it to objects in any category which fulfill suficient conditions. In order to do this we use the construction of tensor product via Freyd's representability theorern ([4], [1]). This allowed us to formulate the probIem in the language of theory of category. The main result of this work is theorem 2 which gives the conditions sufficient to existence the tensor product in the concrete category. As an exarnple of the nontrivial aplication of this theorem we give the proof of the existence of the tensor product in the category of compact spaces.