Volume 9, Issue 1

The purpose of this paper is to consider the three-dimensional versions of the theory of electroelasticity for a transversally esotropic body. Applying the potential method and the theory of singular integral equations, the normality of singular integral equations corresponding to the boundary value problems of electroelasticity are proved and the symbolic matrix is calculated. The uniqueness and existence theorem for the basic BVP s of electroelasticity are given.

In this note we shall consider the following problem: which conditions should satisfy a function ℎ : (0, 1) → ℝ in order to guarantee the existence of a (regular) measure μ in with compact support and for some positive constants 𝑐 2 , and 𝑐 2 independent of γ ∈ Γ and 𝑟 ∈ (0,1)? The theory of self-similar fractals provides outstanding examples of sets fulfilling (♡) with ℎ(𝑟) = 𝑟 𝑑 , 0 ≤ 𝑑 ≤ 𝑛, and a suitable measure μ . Analogously, we shall rely on some recent techniques for the construction of pseudo self-similar fractals in order to deal with our more general task.

A Cauchy problem for a functional-differential inclusion of neutral type with a nonconvex right-hand side is investigated. Questions of the solvability of such a problem are considered, estimates analogous to the Filippov's estimates are obtained and the density principle is proved.

L. Zhizhiashvili proved that if for some 𝑝, 1 ≤ 𝑝 ≤ ∞, and α ∈ (0, 1), then the 𝐿 𝑝 -deviation of 𝑓 from its Cesàro mean is 𝑂(𝑛 α 𝑤(1/𝑛)) where 𝑤(·) is a modulus of continuity. In this paper we show that this estimation is non-amplifiable for 𝑝 = 1.

The asymptotic behavior as 𝑡 → ∞ of solutions of a nonlinear integro-differential equation is studied. The equation arises as a model describing the penetration of the electromagnetic field in to a substance.

We prove that any surjective homomorphism of Maltsev algebras is a Kan fibration.

Two symmetric invariant probability measures μ 1 and μ 2 are constructed such that each of them possesses the strong uniqueness property but their product μ 1 × μ 2 turns out to be a symmetric invariant probability measure without the uniqueness property.

Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigmacompact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey-continuous characteristic functional.

Linear dynamical systems are introduced in a general axiomatic way, and their development is carried out in great simplicity. The approach is closely related both with the classical transfer function approach and with the Willems behavioral approach.

A Bochner mean square deviation for random elements of 2-convex Banach lattices is introduced and investigated. Results, analogous to the law of large numbers for squares of random elements are proved in some classes of Köthe function spaces.

The paper is devoted to proving the existence and uniqueness of a strong solution of a mixed problem with integral boundary conditions for a certain singular parabolic equation. A functional analysis method is used. The proof is based on an energy inequality and on the density of the range of the operator generated by the studied problem.

It is shown that if 𝑀 1 , 𝑀 2 are 𝑟-maximal sets and 𝑀 1 ≡ 𝑄1–𝑁 𝑀 2 , then 𝑀 1 ≡ 𝑚 𝑀 2 . In addition, we prove that there exists a simultaneously 𝑄 1–𝑁 - and 𝑊-complete recursively enumerable set which is not 𝑠𝑄-complete.

The upper bound of dimension of vector spaces of generalized theta-series corresponding to some ternary quadratic forms is established. In a number of cases, the dimension of vector spaces of generalized theta-series is established and bases of these spaces are constructed.

The aim of this paper is to study, in the class of Hölder functions, a nonhomogeneous linear integral equation with coefficient cos 𝑥. Necessary and sufficient conditions for the solvability of this equation are given under some assumptions on its kernel. The solution is constructed analytically, using the Fredholm theory and the theory of singular integral equations.

Corrections are given for [Mania and Tevzadze, Georgian Math. J. 7: 765–792, 2000], most of which consist in replacing “ess sup” by “ess inf” in all the exposed formulas.