On the commutation of generalized means on probability spaces

Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be probability spaces in each of which there is at least one measurable set whose measure is strictly between $0$ and $1$. We say that $(f,g)$ is a $(\lambda, \mu)$-switch if, for every $\mathscr{L} \otimes \mathscr{M}$-measurable function $h: X \times Y \to \mathbf{R}$ for which $h[X\times Y]$ is contained in a compact subset of $I$, it holds $$f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;d\mu\right)\right)d \lambda\right)\! = g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;d\lambda\right)\right)d \mu\right)\!,$$ where $f^{-1}$ is the inverse of the corestriction of $f$ to $f[I]$, and similarly for $g^{-1}$. We prove that this notion is well-defined, by establishing that the above functional equation is well-posed (the equation can be interpreted as a permutation of generalized means and raised as a problem in the theory of decision making under uncertainty), and show that $(f,g)$ is a $(\lambda, \mu)$-switch if and only if $f = ag + b$ for some $a,b \in \mathbf R$, $a \ne 0$.Comment: 9 pages, no figures. Fixed minor details. Final version to appear in Indagationes Mathematica

Real Economic Convergence in the EU Accession Countries

The paper aims to assess the real economic convergence among eight CEE countries that accessed the EU, as well as their convergence with the EU. Two aspects of convergence are analysed: (a) income convergence as a tendency to close the income gap; (b) cyclical convergence as a tendency to the conformity of business cycles. Income convergence is analysed in terms of ? and ? coefficients using regression equations between GDP per capita levels and GDP growth rates. Cyclical convergence is analysed using industrial production indexes and industrial confidence indicators. The analysis covers the period 1993-2004. The main findings may be summarised as follows: 1) CEE countries converge between themselves and towards the EU as regards the income level; 2) CEE countries reveal a good cyclical synchronisation with the EU; cyclical conformity within the region is better seen when the group is split into three subgroups: (a) Czech Republic, Slovakia and Slovenia, (b) Hungary and Poland, (c) the Baltic states. Both types of economic convergence are strongly affected by the dependence on the EU markets, including trade and capital flows.Economic Convergence, Economic Growth, Business Cycles, Economic Integration

Explicit solutions of the invariance equation for means

Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in a recent paper of P. Kahlig and J. Matkowski, and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M, we give explicit formula for a rich family of pairs of M-complementary means. We prove that this method cannot be extended for higher dimension. Some examples are given and two open questions are proposed

On some properties of solutions of a functional equation

W pracy dowodzi się twierdzenia które orzeka, że jeżeli dane funkcje f i h są monofoniczne (względnie monotoniczne i wypukłe) to rozwiązanie równania (1) jest monofoniczne (wypukłe)