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.

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.

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".

The scope of this note is a self-contained presentation of a mathematical method that enables us to give an absolute upper bound for the difference of the Gini coefficients $$\left|G({\sigma }_1,\cdots ,{\sigma }_n)-G({\gamma }_1,\cdots ,{\gamma }_n)\right|,$$ where $({\gamma }_1,\cdots ,{\gamma }_n)$ represents the vector of the gross wages and $({\sigma }_1,\cdots ,{\sigma }_n)$ represents the vector of the corresponding super-gross wages that is used in the Czech Republic for calculating the net wage. Since (as of June 2019) ${\sigma }_i=100·⌈1.34{\gamma }_i/100⌉$, the study of the above difference seems to be somewhat inaccessible for many economists. However, our estimate based...

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 $x\in {ℝ}^m$ if there are numbers $f_{\alpha }\left(x\right)$, |α| ≤ n, such that $f(x+h)-{\sum }_{\left|\alpha \right|\le n}{f}_{\alpha }\left(x\right)h^{\alpha }/\alpha !$ is $O\left(h^u\right)$ 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....

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.