Structural changes in employment in old industrial regions of Ukraine

The formation of independent economy in Ukraine and its transition to the market is going under the low competitiveness of its main productive complexes and worn-out fixed capital. That will cause undesirable structural changes, which will be and already is associated with structural unemployment. These and similar issues are the main subject of the paper. The principle of full employment is being realized in Ukraine at present. A very low rate of unemployment, about 2-3 percent, has been shown by the official statistics of the country. It is quite obvious that there is a hidden unemployment and the large number of lay out people will appear soon, and first of all in basic industries. Eastern Ukraine, called Donbass, is exactly the region with such an economy. More than 30 percent of population of the region work for coal and steel industries. There are following tools elaborated in the course of the study and discussed in details in the paper to overcome the problem: * tax exemptions for priority sectors, for investment in socially important projects (employment, social infrastructure development, R&D), for specific geographical entities (underdeveloped districts, territories with high level of unemployment, settlements with ecological hazards); * setting up associations of enterprises to concentrate investment resources on socially important and economically profitable projects of the region's restructuring; * creation of the Fund for Regional Development and the Bank for Restructuring and Economic Development of Donbass which are aimed at financial support of investment projects of economic restructuring; * direct budget investments in different sectors, large industrial companies which are very important for the region.

On the numerical radius of operators in Lebesgue spaces

We show that the absolute numerical index of the space $L_p(\mu)$ is $p^{-1/p} q^{-1/q}$ (where $1/p+1/q=1$). In other words, we prove that $$\sup\{\int |x|^{p-1}|Tx|\, d\mu \, : \ x\in L_p(\mu),\,\|x\|_p=1\} \,\geq \,p^{-\frac{1}{p}} q^{-\frac{1}{q}}\,\|T\|$$ for every $T\in \mathcal{L}(L_p(\mu))$ and that this inequality is the best possible when the dimension of $L_p(\mu)$ is greater than one. We also give lower bounds for the best constant of equivalence between the numerical radius and the operator norm in $L_p(\mu)$ for atomless $\mu$ when restricting to rank-one operators or narrow operators.Comment: 14 page

Bindweeds or random walks in random environments on multiplexed trees and their asympotics

We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree. The term \textit{multiplexed} means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\}\times\{1,...,d\}$. This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term \textit{random environment} means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates. This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (\textit{i.e.} the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere