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Dense orderings, partitions and weak forms of choice

Carlos González (1995)

Fundamenta Mathematicae

We investigate the relative consistency and independence of statements which imply the existence of various kinds of dense orders, including dense linear orders. We study as well the relationship between these statements and others involving partition properties. Since we work in ZF (i.e. without the Axiom of Choice), we also analyze the role that some weaker forms of AC play in this context

Densité et dimension

Patrick Assouad (1983)

Annales de l'institut Fourier

Une partie 𝒮 de 2 X est appelée une classe de Vapnik-Cervonenkis si la croissance de la fonction Δ 𝒮 : r Sup { | A | | A X , | A | = r } est polynomiale; ces classes se trouvent être utiles en Statistique et en Calcul des Probabilités (voir par exemple Vapnik, Cervonenkis [V.N. Vapnik, A.YA. Cervonenkis, Theor. Prob. Appl., 16 (1971), 264-280], Dudley [R.M. Dudley, Ann. of Prob., 6 (1978), 899-929]).Le présent travail est un essai de synthèse sur les classes de Vapnik-Cervonenkis. Mais il contient aussi beaucoup de résultats nouveaux,...

Descriptive set-theoretical properties of an abstract density operator

Szymon Gła̧b (2009)

Open Mathematics

Let 𝒦 (ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that { K 𝒦 ( ) : x K ( d + ( x , K ) = 1 o r d - ( x , K ) = 1 ) } is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.

Determinacy of adversarial Gowers games

Christian Rosendal (2014)

Fundamenta Mathematicae

We prove a game-theoretic dichotomy for G δ σ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis’ proof of Σ⁰₃ determinacy.

DFIS: A novel data filling approach for an incomplete soft set

Hongwu Qin, Xiuqin Ma, Tutut Herawan, Jasni Mohamad Zain (2012)

International Journal of Applied Mathematics and Computer Science

The research on incomplete soft sets is an integral part of the research on soft sets and has been initiated recently. However, the existing approach for dealing with incomplete soft sets is only applicable to decision making and has low forecasting accuracy. In order to solve these problems, in this paper we propose a novel data filling approach for incomplete soft sets. The missing data are filled in terms of the association degree between the parameters when a stronger association exists between...

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