We characterize the set of discontinuity points of A-continuous quasi-continuous functions of the first class defined on hereditarily normal weakly pairwise attainable spaces. Besides, we obtain several sufficient properties of a subset of a normed space which guarantee an existence of a linearly continuous function with the given discontinuity point set.

For a function analytic in the unit disc the concepts of Gelfond-Leont'ev-Salagen and Gelfond-Leont'ev-Ruscheweyh derivatives of n-th order are introduced and the asymptotic behaviour of the maximal terms of their power development as n→+∞ is investigated.

The paper contains a description of algebraic basis of algebra of block-symmetric polynomials on the l 1 -sum of the copies of l 1

It is considered the system of differential equations that describes the dynamics of an infinite system of atoms on 2D{lattice. Results on existence of the periodic travelling waves are obtained.

The asymptotic behaviour of the Nevanlinna characteristic for loxodromic functions as well as the Nevanlinna type characteristic for elliptic functions are investigated.

A spectral problem generated by the Sturm-Liouville equation on the edges of a regular equilateral finite star-tree with the Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices is considered. The potential in the Sturm-Liouville equations on the edges is the same on each edge and symmetric with respect to the edge midpoint. The structure of the function whose set of zeros coincides with the spectrum of such a problem (characteristic function) is described.

We introduce L-idempotent analogues of topological vector spaces by means of domain theory, study their basic properties, and prove the existence of free (dually) continuous L-semi- modules over domains, (dually) continuous lattices and semilattices.

In this paper we investigate interplays between several type of the attainable spaces. We introduce the class of discreetly saturated spaces which contains all metrizable spaces and all hereditarily separable perfectly normal spaces and check that a first countable discreetly saturated space has all type of attainability. We also obtain that weakly discreetly attainable spaces are countably resolvable.

Let R∈(0,+∞] , f(z)=∑cn zn be an analytic function in the disk, T f (r) be the Nevanlinna characteristic, N f (r,α,β,a) be the integrated counting function of a -points of f in the sector 0<|z|≤r , α≤arg α z<β , and (ω n (ω)) be a~sequence of independent equidistributed on [0,1] random variables. Under some conditions on the growth of f it is proved that for random analytic function f ω (z)=∑e 2πiω n (ω) a n z n almost surely for every a∈C and all α<β≤α+2π the relation N f ω (r,α,β,a)∼β−α 2π T f ω (r) , r→R , holds outside some exceptional set E⊂(0,R) .

In the paper it is proved that the abelian clean Bezout ring is an elementary divisors ring, if and only if it is a duo-ring and shows that the projective-free right (left) Bezout ring is right (left) Hermite ring if his stable rank not more 2.

We introduce an involution map for the functional monad which generalizes the transversa- lity map for the inclusion hyperspace monad and the conjugation map for the capacity monad.