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 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

On extension of solutions of a simultaneous system of iterative functional equations

Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form $$\varphi(x) = h (x, \varphi[f_1(x)],\ldots,\varphi[f_m(x)]),$$ $$\varphi(x) = H (x, \varphi[F_1(x)],\ldots,\varphi[F_m(x)]),$$ to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008) 4, 531-541])

Persistently optimal policies in stochastic dynamic programming with generalized discounting

In this paper we study a Markov decision process with a non-linear discount function. Our approach is in spirit of the von Neumann-Morgenstern concept and is based on the notion of expectation. First, we define a utility on the space of trajectories of the process in the finite and infinite time horizon and then take their expected values. It turns out that the associated optimization problem leads to a non-stationary dynamic programming and an infinite system of Bellman equations, which result in obtaining persistently optimal policies. Our theory is enriched by examples.Stochastic dynamic programming, Persistently optimal policies, Variable discounting, Bellman equation, Resource extraction, Growth theory

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