A linear bounded operator G acting from a Banach space X into a Banach space Y is a Daugavet center if every linear bounded rank-1 operator T:X→Y fulfills ∥G+T∥=∥G∥+∥T∥ . We prove that G:X→Y is a~Daugavet center if and only if for every separable subspaces X 1 ⊂X and Y 1 ⊂Y there exist separable subspaces X 2 ⊂X and Y 2 ⊂Y such that X 1 ⊂X 2 , Y 1 ⊂Y 2 , G(X 2 )⊂Y 2 and the restriction G| X 2 :X 2 →Y 2 of G is a Daugavet center. We apply this fact to study the set of G -narrow operators.

In the paper the role of an initial object is played by a couple (L,L 0 ) of closed linear relations in a Hilbert space H , such that L 0 ⊂L . Each closed linear relation L 1 (M 1 ) such that L 0 ⊂L 1 ⊂L (respectively L ∗ ⊂M 1 ⊂L ∗ 0 ) is said to be a proper extension of L 0 (L ∗ ) . In the terms of abstract boundary operators i.e. bounded linear operator U(V) acting from L(M) to G (G is an auxiliary Hilbert space) such that the null space of U(V) contains L 0 (L ∗ ) , criteria of mutual adjointness for mentioned above relations L 1 and M 1 are established.

In this paper we give differential equations characterizing timelike and spacelike curves lying on hyperbolic sphere H 3 0 and Lorentzian sphere S 3 1 in the Minkowski space-time E 4 1 . Furthermore, we give the integral characterizations of these curves in E 4 1

We consider subharmonic functions in the exterior of a compact set in a finite-dimensional space that grows near the compact. We assume that the compact has some generalized convex property. We get an integral condition on function's Riesz measure and check its accuracy.

Let F be a field, A be a vector space over F , GL(F,A) be the group of all automorphisms of the vector space A . If B≤A then denote by Core G (B) the largest G -invariant subspace of B . A subspace B is called almost G -invariant if dim F (B/Core G (B)) is finite. In this paper we described the case where every subspace of A is almost G -invariant.

We prove analogues of the classical Wiman inequality for entire Dirichlet series f(z)=∑ +∞ n=0 a n e zλ n with arbitrary positive exponents (λ n ) such that sup{λ n :n≥0}=+∞ .

The Dirichlet spectral problem for an elliptic operator of the fourth order with singularly perturbed coefficients is considered. The problem describes the eigenmodes of a plate with finite number of the stiff and light-weight inclusions of an arbitrary shape. The asymptotic behavior of eigenvalues and eigenfunctions is studied. The number-by-number convergence of the eigenvalues and the corresponding eigenspaces is established. The limit eigenvalue problem involves a non-local boundary conditions. Justification of the asymptotic formulas is based on the norm resolvent convergence of a family of unbounded self-adjoint operators.

The purpose of this paper is to consider some generalizations of the class of functions having zero integrals over balls of a fixed radius. We obtain an analog of John's uniqueness theorem for weighted spherical means on sphere.

We consider unique admissible quivers, i. e. quivers of Gorenstein exponent matrices. It is proved that admissible quiver with a loop at each vertex is unique if and only if it is a simple cycle, and that there are different from the simple cycles unique quivers with any number of vertices.

The special type mappings between Riemannian spaces with almost quaternion structure are studied. The fundamental theorems of theory of these mappings are proved.

We study asymptotics of eigenvalues, eigenfunctions and norming constants of singular energy-dependent Sturm-Liouville equations with complex-valued potentials. The analysis essentially exploits the integral representation of solutions, which we derive using the connection between the problem under study and a Dirac system of a special form.

Let M k be {the} set of k -valued meromorphic in G={z:r 0 ⩽|z|} functions with {a}~branch point of order k−1 {at} ∞ ; let E ∗ be a set of circles {with finite} sum of radii. Denote M ∗ (r,f)=max|f(z)|, z∈{te iθ :0⩽θ⩽2kπ, r 0 ⩽t⩽r}∖E ∗ , f∈M k ; m(r,f)=1 2πk ∫ 2πk 0 ln + |f(re iθ )|dθ . If f∈M k is a solution of the equation P(z,f,f ′ )=0 and P is a polynomial in all variables then either |f(re iθ )|≤r ν , re iθ ∈G∖E ∗ , ν>0 or m(r,f) has growth order ρ⩾1 2k , and the following equality holds lnM ∗ (r,f)=(c+o(1))r ρ , c≠0, r→+∞.

It is shown that a commutative Bezout ring R of stable range 2 is an elementary divisor ring if and only if for each ideal I every finitely generated projective R/I -module is a direct sum of principal ideals generated by idempotents.

We shown that for each lower continuous finite valued mapping from metrizable topological space X in Sorgenfrey line the set of points of upper continuous is residual in X .