The Turing degrees of ∆2 sets as well as the degrees of sets in certain upper segments of the Ershov hierarchy are characterized. In particular, it is shown that, up to Turing equivalence and admissible indexing, the noncomputable ∆2 sets are exactly the decision problems of countable partial orders of tree shape with branches of positive finite length only.

We investigate the possibility of the existence of nonsparse strongly summable ultrafilters on certain abelian groups. In particular, we show that every strongly summable ultrafilter on the countably infinite Boolean group is sparse. This answers a question of Hindman, Steprāns and Strauss.

The color component of the urban environment is most flexible and sensitive to changes. Of course, it easier repaint façades than change street pattern. Especially with arsenal tools offered by modern technologies.As lighting streets squares changes, facades also become variable, turning into gigantic screens before our eyes. Will we be able keep technological break-through within reasonable et...