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1-Lipschitz aggregation operators and quasi-copulas

Anna Kolesárová (2003)

Kybernetika

In the paper, binary 1-Lipschitz aggregation operators and specially quasi-copulas are studied. The characterization of 1-Lipschitz aggregation operators as solutions to a functional equation similar to the Frank functional equation is recalled, and moreover, the importance of quasi-copulas and dual quasi-copulas for describing the structure of 1-Lipschitz aggregation operators with neutral element or annihilator is shown. Also a characterization of quasi-copulas as solutions to a certain functional...

A note on the three-segment problem

Martin Doležal (2009)

Mathematica Bohemica

We improve a theorem of C. L. Belna (1972) which concerns boundary behaviour of complex-valued functions in the open upper half-plane and gives a partial answer to the (still open) three-segment problem.

Construction of aggregation operators: new composition method

Tomasa Calvo, Andrea Mesiarová, Ľubica Valášková (2003)

Kybernetika

A new construction method for aggregation operators based on a composition of aggregation operators is proposed. Several general properties of this construction method are recalled. Further, several special cases are discussed. It is also shown, that this construction generalizes a recently introduced twofold integral, which is exactly a composition of the Choquet and Sugeno integral by means of a min operator.

Extension to copulas and quasi-copulas as special 1 -Lipschitz aggregation operators

Erich Peter Klement, Anna Kolesárová (2005)

Kybernetika

Smallest and greatest 1 -Lipschitz aggregation operators with given diagonal section, opposite diagonal section, and with graphs passing through a single point of the unit cube, respectively, are determined. These results are used to find smallest and greatest copulas and quasi-copulas with these properties (provided they exist).

Homogeneous aggregation operators

Tatiana Rückschlossová, Roman Rückschloss (2006)

Kybernetika

Recently, the utilization of invariant aggregation operators, i.e., aggregation operators not depending on a given scale of measurement was found as a very current theme. One type of invariantness of aggregation operators is the homogeneity what means that an aggregation operator is invariant with respect to multiplication by a constant. We present here a complete characterization of homogeneous aggregation operators. We discuss a relationship between homogeneity, kernel property and shift-invariance...

McShane equi-integrability and Vitali’s convergence theorem

Jaroslav Kurzweil, Štefan Schwabik (2004)

Mathematica Bohemica

The McShane integral of functions f I → ℝ defined on an m -dimensional interval I is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.

On affinity of Peano type functions

Tomasz Słonka (2012)

Colloquium Mathematicae

We show that if n is a positive integer and 2 ℵ ₀ ≤ ℵ ₙ , then for every positive integer m and for every real constant c > 0 there are functions f ₁ , . . . , f n + m : ℝ ⁿ → ℝ such that ( f ₁ , . . . , f n + m ) ( ℝ ⁿ ) = ℝ n + m and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that ( f i ₁ , . . . , f i ₙ ) ( y ) = y + w for y ∈ x + ( - c , c ) × ℝ n - 1 .

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