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Multiplication, distributivity and fuzzy-integral. II

Wolfgang Sander, Jens Siedekum (2005)

Kybernetika

Based on results of generalized additions and generalized multiplications, proven in Part I, we first show a structure theorem on two generalized additions which do not coincide. Then we prove structure and representation theorems for generalized multiplications which are connected by a strong and weak distributivity law, respectively. Finally – as a last preparation for the introduction of a framework for a fuzzy integral – we introduce generalized differences with respect to t-conorms (which are...

Multiplication, distributivity and fuzzy-integral. III

Wolfgang Sander, Jens Siedekum (2005)

Kybernetika

Based on the results of generalized additions, multiplications and differences proven in Part I and II of this paper a framework for a general integral is presented. Moreover it is shown that many results of the literature are contained as special cases in our results.

Numerical experimentation and comparison of fuzzy integrals.

Manuel Jorge Bolaños, Luis Daniel Hernández, Antonio Salmerón (1996)

Mathware and Soft Computing

Sugeno and Choquet integrals have been widely studied in the literature from a theoretical viewpoint. However, the behavior of these functionals is known in a general way, but not in practical applications and in particular cases. This paper presents the results of a numerical comparison that attempts to be a basis for a better comprehension and usefulness of both integrals.

On ☆ - associated comonotone functions

Ondrej Hutník, Jozef Pócs (2018)

Kybernetika

We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper [1]. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to + -associatedness of functions (as proved by Boczek and Kaluszka), but also to their ☆ -associatedness with ☆ being an arbitrary strictly...

On asymptotic behaviour of universal fuzzy measures

Ladislav Mišík, János T. Tóth (2006)

Kybernetika

The asymptotic behaviour of universal fuzzy measures is investigated in the present paper. For each universal fuzzy measure a class of fuzzy measures preserving some natural properties is defined by means of convergence with respect to ultrafilters.

On nearly radial marginals of high-dimensional probability measures

Bo'az Klartag (2010)

Journal of the European Mathematical Society

Suppose that μ is an absolutely continuous probability measure on R n, for large n . Then μ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if n ≥ ( C / ε ) C d , then there exist d -dimensional marginals of μ that are ε -far from being sphericallysymmetric, in an appropriate sense. Here C > 0 is a universal constant.

On possibilistic marginal problem

Jiřina Vejnarová (2007)

Kybernetika

A possibilistic marginal problem is introduced in a way analogous to probabilistic framework, to address the question of whether or not a common extension exists for a given set of marginal distributions. Similarities and differences between possibilistic and probabilistic marginal problems will be demonstrated, concerning necessary condition and sets of all solutions. The operators of composition will be recalled and we will show how to use them for finding a T -product extension. Finally, a necessary...

On some contributions to quantum structures by fuzzy sets

Beloslav Riečan (2007)

Kybernetika

It is well known that the fuzzy sets theory can be successfully used in quantum models ([5, 26]). In this paper we give first a review of recent development in the probability theory on tribes and their generalizations – multivalued (MV)-algebras. Secondly we show some applications of the described method to develop probability theory on IF-events.

On some properties of squares of Sierpiński sets

Andrzej Nowik (2004)

Colloquium Mathematicae

We investigate some geometrical properties of squares of special Sierpiński sets. In particular, we prove that (under CH) there exists a Sierpiński set S and a function p: S → S such that the images of the graph of this function under π'(⟨x,y⟩) = x - y and π''(⟨x,y⟩) = x + y are both Lusin sets.

On the extension of D -poset valued measures

Beloslav Riečan (1998)

Czechoslovak Mathematical Journal

A variant of Alexandrov theorem is proved stating that a compact, subadditive D -poset valued mapping is continuous. Then the measure extension theorem is proved for MV-algebra valued measures.

On the extension of measures.

Baltasar Rodríguez-Salinas (2001)

RACSAM

We give necessary and sufficient conditions for a totally ordered by extension family (Ω, Σx, μx)x ∈ X of spaces of probability to have a measure μ which is an extension of all the measures μx. As an application we study when a probability measure on Ω has an extension defined on all the subsets of Ω.

On the g -entropy and its Hudetz correction

Beloslav Riečan (2002)

Kybernetika

The Hudetz correction of the fuzzy entropy is applied to the g -entropy. The new invariant is expressed by the Hudetz correction of fuzzy entropy.

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