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1-Lipschitz aggregation operators and quasi-copulas

Anna Kolesárová (2003)


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 chain rule formula for the composition of a vector-valued function by a piecewise smooth function

François Murat, Cristina Trombetti (2003)

Bollettino dell'Unione Matematica Italiana

We state and prove a chain rule formula for the composition T u of a vector-valued function u W 1 , r Ω ; R M by a globally Lipschitz-continuous, piecewise C 1 function T . We also prove that the map u T u is continuous from W 1 , r Ω ; R M into W 1 , r Ω for the strong topologies of these spaces.

A characterization of Sobolev spaces via local derivatives

David Swanson (2010)

Colloquium Mathematicae

Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function f W k , p ( Ω ) possesses an L p derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space W k , p ( Ω ) . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

A generalized sharp Whitney theorem for jets.

Charles Fefferman (2005)

Revista Matemática Iberoamericana

Suppose that, for each point x in a given subset E ⊂ Rn, we are given an m-jet f(x) and a convex, symmetric set σ(x) of m-jets at x. We ask whether there exist a function F ∈ Cm,w(Rn) and a finite constant M, such that the m-jet of F at x belongs to f(x) + Mσ(x) for all x ∈ E. We give a necessary and sufficient condition for the existence of such F, M, provided each σ(x) satisfies a condition that we call "Whitnet w-convexity".

Approximate and L p Peano derivatives of nonintegral order

J. Marshall Ash, Hajrudin Fejzić (2005)

Studia Mathematica

Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. L p , 1 ≤ p ≤ ∞) sense at x m if there are numbers f α ( x ) , |α| ≤ n, such that f ( x + h ) - | α | n f α ( x ) h α / α ! is O ( h u ) in the approximate (resp. L p ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or L p sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and f = g on Π....

Associative n -dimensional copulas

Andrea Stupňanová, Anna Kolesárová (2011)


The associativity of n -dimensional copulas in the sense of Post is studied. These copulas are shown to be just n -ary extensions of associative 2-dimensional copulas with special constraints, thus they solve an open problem of R. Mesiar posed during the International Conference FSTA 2010 in Liptovský Ján, Slovakia.

Baire category results for quasi–copulas

Fabrizio Durante, Juan Fernández-Sánchez, Wolfgang Trutschnig (2016)

Dependence Modeling

The aim of this manuscript is to determine the relative size of several functions (copulas, quasi– copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi– copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are obtained via a checkerboard approximation of quasi–copulas.

Caractère lipschitzien d'une distance associée à des champs de vecteurs engendrant une algèbre de Lie de rang maximal. Quelques conséquences

Rose-Marie Hervé, Michel Hervé (1990)

Annales de l'institut Fourier

La métrique attachée de façon naturelle à des champs de vecteurs C est susceptible de plusieurs définitions voisines ; on montre que, suivant la définition adoptée, elle peut avoir, ou ne pas avoir, un caractère localement lipschitzien qui a pour conséquence l’existence de points L -réguliers, pour certains opérateurs différentiels L , sur les frontières des boules pour la métrique.

Continuity of order-preserving functions

Boris Lavrič (1997)

Commentationes Mathematicae Universitatis Carolinae

Let the spaces 𝐑 m and 𝐑 n be ordered by cones P and Q respectively, let A be a nonempty subset of 𝐑 m , and let f : A 𝐑 n be an order-preserving function. Suppose that P is generating in 𝐑 m , and that Q contains no affine line. Then f is locally bounded on the interior of A , and continuous almost everywhere with respect to the Lebesgue measure on 𝐑 m . If in addition P is a closed halfspace and if A is connected, then f is continuous if and only if the range f ( A ) is connected.

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