The Neumann spectral problem for an elliptic operator of the second order with singularly perturbed coefficients is considered. The asymptotic behavior of eigenvalues and eigenfunctions is studied. The problem describes the eigenmodes of a composite material with a finite number of heavy and soft inclusions. The number-by-number convergence of the eigenvalues and the corresponding eigenspaces is established. The limit eigenvalue problem involves a non-local boundary condition which arises from the non trivial coupling of the inclusions.

Two conjectures concerning the number of maximum modulus points of an entire and a meromorphic function on the circle z:|z|=r are presented and discussed.

We introduce the notion of a right H 1 -compositor and prove that for a hereditarily Baire metrizable space X , a normal space Y and a mapping f:X→Y the following conditions are equivalent: (i) f is piecewise continuous; (ii) f is k -continuous; (iii) f is G δ -measurable; if, moreover, Y is perfect, then (i)--(iii) are equivalent to: (iv) f is a right H 1 -compositor.

We consider a question on description of the set of characters of the algebra of bounded type symmetric analytic functions on L ∞ [0;1] and establish some connection with the trigonometric moment problem.

Let G be an infinite Abelian group. We give a complete characterization of those finitely generated subgroups of G which are the von Neumann radicals for some Hausdorff group topologies on G. It is proved that every infinite finitely generated Abelian group admits a complete Hausdorff minimally almost periodic group topology. The latter result resolves a particular case of Comfort’s problem.

A subset A of a metric space (X,d) is called thick if, for every r>0 , there is a∈A such that B d (a,r)⊆A, where B d (a,r). We show that if (X,d) is unbounded and has no asymptotically isolated balls then, for each r>0 , there exists a partition X=X 1 ∪X 2 such that B d (X 1 ,r) and B d (X 2 ,r) are not thick.

Notes of the Problem Session which has been held on the section of Banach Spaces during the International conference dedicated to the 120-th anniversary of Stefan Banach in Lviv (Ukraine), September 17–21, 2012.

In the paper we describe the structure of AH -completions and H -completions of the discrete semilattices (N,min) and (N,max) . We give an example of an H -complete topological semilattice which is not AH -complete. Also for an arbitrary infinite cardinal λ we construct an H -complete topological semilattice of cardinality λ which has 2 λ many open-and-closed continuous homomorphic images which are not H -complete topological semilattices. The constructed examples give a negative answer to Question 17 in the paper J. W. Stepp, Algebraic maximal semilattices, Pacific J. Math., 58 (1975), no.1, 243--248.

We prove that the Bing plane B is σ -metrizable but not strongly σ -metrizable. Also we show that for c -connected topological spaces X and Y each separately continuous function f:X×Y→B is constant.

We consider the M θ /G/1/m queue with two service modes (basic and postthreshold) with the distribution functions of service time F(x) and F ˜ (x) respectively. The postthreshold mode is used in conjunction with blocking of input flow if at the beginning of service of the next customer the number of customers in the system ξ(t) satisfies the condition ξ(t)>h 2 . Return to the basic mode and stop blocking carried out at the beginning of service of the next customer, if ξ(t)≤h 1 , where h 1 ≤h 2 . Laplace transforms for distributions of the number of customers in the system on the busy period and for the busy time distribution function are found. The mean duration of the busy time is found, and formulas for the stationary distribution of number of customers in the system and stationary characteristics of queue are obtained. The case of m=∞ is considered.

We prove the existence and uniqueness of the solutions of the optimal control problems for systems described by evolution equations degenerated at the initial moment. A characterization is given for the solutions of the considered problems. We also consider in detail the case of the final observation and obtain a set of correlations that characterize the optimal controls.

Let a∈C , f(z)=∑c n z n be an analytic function in the disk |z|<1 such that S f (r):=(∑|c n | 2 r 2n ) 1 2 →+∞ , r→1 , 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 , N f (r,a)=N f (r,0,2π,a) , and E⊂(0,1) be a set such that supE=1 . It is proved that if N f (r,a)∼lnS f (r) , E∋r→1 , then N f (r,α,β,a)∼β−α 2π lnS f (r) , E∋r→1 , uniformly in α<β≤α+2π .

We obtain criterions for delta-subharmonic function to belong to the class of functions of finite (γ,ε) )-type in a half-plane. These criterions are formulated in terms of Fourier coefficients of a function.