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The Choquet integral as Lebesgue integral and related inequalities

Radko MesiarJun LiEndre Pap — 2010


The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions...

On some geometric transformation of t-norms.

Erich Peter KlementRadko MesiarEndre Pap — 1998

Mathware and Soft Computing

Given a triangular norm T, its t-reverse T*, introduced by C. Kimberling (Publ. Math. Debrecen 20, 21-39, 1973) under the name invert, is studied. The question under which conditions we have T** = T is completely solved. The t-reverses of ordinal sums of t-norms are investigated and a complete description of continuous, self-reverse t-norms is given, leading to a new characterization of the continuous t-norms T such that the function G(x,y) = x + y - T(x,y) is a t-conorm, a problem originally studied...

Generated triangular norms

Erich Peter KlementRadko MesiarEndre Pap — 2000


An overview of generated triangular norms and their applications is presented. Several properties of generated t -norms are investigated by means of the corresponding generators, including convergence properties. Some applications are given. An exhaustive list of relevant references is included.

Transformations of copulas

Erich Peter KlementRadko MesiarEndre Pap — 2005


Transformations of copulas by means of increasing bijections on the unit interval and attractors of copulas are discussed. The invariance of copulas under such transformations as well as the relationship to maximum attractors and Archimax copulas is investigated.

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