Volume 8, Issue 3

The basic plane boundary value problems of statics of the elastic mixture theory are considered when on the boundary are given: a displacement vector (the first problem), a stress vector (the second problem); differences of partial displacements and the sum of stress vector components (the third problem). A simple method of deriving Fredholm type integral equations of second order for these problems is given. The properties of the new operators are established. Using these operators and generalized Green formulas we investigate the above-mentioned integral equations and prove the existence and uniqueness of a solution of all the boundary value problems in a finite and an infinite domain.

The use of Euler polynomials and Euler numbers allows us to construct a quadrature rule similar to the well-known Euler–MacLaurin quadrature formula, using Euler (instead of Bernoulli) numbers, and even (instead of odd) order derivatives of a given function evaluated at the extrema of the considered interval. An expression of the remainder term and numerical examples are also given.

First, the basic properties of mean dilatation (MD-) numbers for linear operators acting from a finite-dimensional Hilbert space are investigated. Among other results, in terms of first and second order MD-numbers, a characterization of isometries is obtained and a dimension-free estimation of the p -th order MD-number by means of the first order MD-number is established. After that asymptotic MD-numbers for a continuous linear operator acting from an infinite-dimensional Hilbert space are introduced and it is shown that in the case of an infinite-dimensional domain the asymptotic p -th order MD-number, rather unexpectedly, is simply the p -th power of the asymptotic first order MD-number (Theorem 3.1).

The purpose of the paper is reducing the three-dimensional Signorini problem to a variational inequality which occurs on the two-dimensional boundary of a domain occupied by an elastic anisotropic body. The uniqueness and existence theorems for the solution of the boundary variational inequality are proved and a boundary element procedure together with an abstract error estimate is described for the Galerkin numerical approximation.

We consider the behaviour of measure zero subsets of a vector space under the operation of vector sum. The question whether the vector sum of such sets can be nonmeasurable is discussed in connection with the measure extension problem, and a certain generalization of the classical Sierpiński result [Fund. Math. 1: 105–111, 1920] is presented.

We deal with differentiable cells defined by solutions to certain linear elliptic systems of first order. It turns out that in some cases families of such cells attached to a given submanifold may be described by Fredholm operators in appropriate function spaces. Using the previous results of the author on the existence of elliptic Riemann–Hilbert problems for generalized Cauchy–Riemann systems, we indicate some classes of systems which give rise to non-linear Fredholm operators of such type.

Let ω conformally map the unit circle on a plane singly-connected domain D bounded by a simple rectifiable curve. It is shown that for the function lg ω ′ to be represented in the unit circle by a Cauchy type A -integral with density arg ω ′, it is necessary and sufficient that D be a Smirnov domain. In particular, for this representation to be done by a Cauchy–Lebesgue type integral with the same density, it is necessary and sufficient that the function lg ω ′ belong to the Hardy class H 1 .

Necessary and sufficient conditions, which govern a trace inequality for one-sided potentials in the “diagonal” case, are established. An application to the existence of positive solutions of a certain nonlinear integral equation is presented.

This paper deals with holomorphic functions from Bergman spaces B p in the disk and provides the existence of deformations (variations) which do not increase the norm of functions and preserve some other prescribed properties. Admissible variations are constructed (for even integer p ≥ 2) using special quasiconformal maps of the complex plane (in line with a new approach to variational problems for holomorphic functions in Banach spaces).

Frequency responses are introduced as objects that are much finer than transfer functions, and their investigation is carried out via convolutional codes. In particular, a canonical one-to-one correspondence is established between frequency responses and equivalence classes of AR-models. These new objects are important because the study of linear dynamical systems can to a great extent be reduced to the study of them.

Boundary integral equations of elasticity theory in a plane domain with a peak at the boundary are considered. Solvability and uniqueness theorems as well as results on the asymptotic behaviour of solutions near the peak are obtained.

Problems for pluriregular equations in Clifford analysis are solved effectively. They are related to the equations called polyharmonic, polywave, polyheat, harmonic-heat, harmonic-wave, wave-heat and harmonic-wave-heat equations which are considered here for the first time.

Conditions for the boundedness are established, and the norms of Volterra type one-dimensional integral operators with fixed singularities of first order in the kernel are calculated in the space L 2 with weight. Integral equations of second order, containing the said operators, are investigated. Conditions for the solvability and solution formulas are given.