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...

Using a distributional approach to integration in superspace, we investigate a Cauchy-Pompeiu integral formula in super Dunkl-Clifford analysis and several related results, such as Stokes formula, Morera's theorem and Painlevé theorem for super Dunkl-monogenic functions. These results are nice generalizations of well-known facts in complex analysis.

We state and prove a chain rule formula for the composition $T\left(u\right)$ of a vector-valued function $u\in W^{1,r}\left(\mathrm{\Omega };{\mathbb{R}}^M\right)$ by a globally Lipschitz-continuous, piecewise $C^1$ function $T$. We also prove that the map $u\to T\left(u\right)$ is continuous from $W^{1,r}\left(\mathrm{\Omega };{\mathbb{R}}^M\right)$ into $W^{1,r}\left(\mathrm{\Omega }\right)$ for the strong topologies of these spaces.

The notion of $\tilde{\ell }$-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of $\tilde{\ell }$-stable functions coincides with the class of C${}^{1,1}$ functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp. 383–387.

We give a complete characterization of closed sets $F\subset {ℝ}^2$ whose distance function $d_F:=\mathrm{dist}(·,F)$ is DC (i.e., is the difference of two convex functions on ${ℝ}^2$). Using this characterization, a number of properties of such sets is proved.

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 $f\in W^{k,p}\left(\Omega \right)$ possesses an $L^p$ derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space $W^{k,p}\left(\Omega \right)$. Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

We present a descriptive definition of a multidimensional generalized Riemann integral based on a concept of generalized absolute continuity for additive functions of sets of bounded variation.

We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.

Any given increasing ${[0,1]}^2\to [0,1]$ function is completely determined by its contour lines. In this paper we show how each individual uninorm property can be translated into a property of contour lines. In particular, we describe commutativity in terms of orthosymmetry and we link associativity to the portation law and the exchange principle. Contrapositivity and rotation invariance are used to characterize uninorms that have a continuous contour line.

In [2] a delta convex function on ${ℝ}^2$ is constructed which is strictly differentiable at $0$ but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class $C^1\left({ℝ}^2\right)$ which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.

Modifying Mawhin's definition of the GP-integral we define a well-behaved integral over n-dimensional compact intervals. While its starting definition is of Riemann type, we also establish an equivalent descriptive definition involving characteristic null conditions. This characterization is then used to obtain a quite general form of the divergence theorem.