We give a semigroup characterization of kaleidoscopical graphs. A connected graph Г (considered as a metric space with the path metric) is called kaleidoscopical if there is a vertex coloring of Г which is bijective on each unit ball.

We present a new concept of lower semicontinuity for vector-valued mappings, so-called lower epi-semicontinuity, which is closely related with the sequential closedness of epigraphs of these mappings. We consider the case when the mappings take values in a real Banach space Y partially ordered by a closed convex pointed cone L, and we make no additional assumptions on topological interior of the ordering cone. We also study the comparison of the lower epi-semicontinuity with the classical notion of lower semicontinuity and its generalizations.

The paper deals with the properties of regular subgraphs of linear extension graphs. We obtain a necessary and sufficient condition for regularity of linear extension graphs, prove a proposition on cardinality and degree partition of special linear extension graphs boundaries, give a characterization of a linear extension graphs class with 0 or 1 bump numbers linear extensions.

Let (r n ) be a positive nondecreasing sequence of finite genus tending to +∞ , and (η n (ω)) be a sequence of independent random variables such that η n (ω) are uniformly distributed on the circles |z|=r n . Then for almost all ω the following assertion holds: if f is an entire function of finite order with zeros at the points η n (ω) and only at them, then for every ε>0 we have lnM f (r)=o(T 3/2 f (r)ln 3+ε T f (r)) , r→+∞ , where M f (r) is the maximum modulus and T f (r) is the Nevanlinna characteristic of the function f .

It is proved that a right Bezout ring with stable range 2, over which transpose matrix to invertible matrix is invertible matrix, is the right Hermite ring.

Uniformly convergent sequences of meromorphic functions in the Caratheodory-Landau sense on annuli are considered. We prove that the sequences of their Nevanlinna type characteristics converge uniformly on intervals. The result is applied to the study of the Nevanlinna characteristics of Julia's exceptional functions.

We consider an abstract Cauchy problem for parabolic differential-operator equations in Hilbert spaces. Initial boundary value problems for parabolic equations are reduced to the Cauchy problem for a system of parabolic differential equations. It is proved that the solution of an initial boundary value problem for partial parabolic equation can be approximated by linear combinations of elementary solutions. Completeness of elementary solutions is also proved for differential-operator equations in abstract Hilbert spaces. The obtained abstract results are applied to differential equations.

Hahn’s results about Baire classifications the real values separately continuous functions are moved on the functions with values in arbitrary local convex spaces.

Using the fuzzy modification of the Prokhorov metric on the set of probability measures we prove the theorem on existence of simultaneous extensions of partial fuzzy metrics (for the Lukasiewicz t-norm) defined on closed subsets of a compact metrizable space.

Images of the faithful triangular representation of Kaloujnine's 2-groups are obtained.

We describe growth majorants of subharmonic in R m (m≥2) functions. To do this, we exceptionally reduce the problem to problems in the theory of positive monotonous functions.

The best possible asymptotic estimates for Lebesgue integral means $m_{p}(r,\log f), 1 \leq p$ of logarithms of analytic functions $f(z)$ in the unit disc in terms of their Nevanlinna characteristic $T(r,f)$ are obtained. We get sharp relation between the order of $T(r,f)$ and the order of $m_{p}(r,\log f)$ for an analytic function $f(z)$ of finite order $\alpha(f).$ This generalizes well-known results of L.~R.~Sons and C.~N.~Linden.

Paper treats special pseudo-Riemannian spaces | equidistant manifolds. Author found tensor characteristic necessary and sufficient condition for a space to admit exactly n - 2 equidistant vector field. Research was carried out locally, in a tensor form, without any limitation of signature of studied spaces.

We introduce some diadic analogue of the Choquet game and a class of diadic Baire spaces which is a subclass of Baire spaces and is wider then the class Choquet spaces. We prove that for any diadic Baire space X , a Banach space Y , a countable Asplund ∗ norming set E⊆Y ∗ and for every map φ:X→Y , such that zφ is quasi-continuous for any z∈E , the discontinuity point set C(φ) is residual.

Necessary and sufficient conditions existence of monic divisors with canonical diagonal form diag(1,…,1,φ(x),…,φ(x)) of polynomial matrices over infinite field established.

We establish relations between domains of convergence and absolute convergence of the multiple random Dirichlet series.