Volume 1, Issue 5

We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighbourhood of infinity or in the neighbourhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.

The conditions are given for passage of the limit through the double Denjoy integral defined by V.G. Chelidze.

The properties of solutions of the equation u″ ( t ) = p 1 ( t ) u ( τ 1 ( t )) + p 2 ( t ) u′ ( τ 2 ( t )) are investigated where p i : [ a , +∞[→ R ( i = 1, 2) are locally summable functions, τ 1 : [ a , +∞[→ R is a measurable function and τ 2 : [ a , +∞[→ R is a nondecreasing locally absolutely continuous one. Moreover, τ i ( t ) ≥ t ( i = 1, 2), p 1 ( t ) ≥ 0, , ε = const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [ a , +∞ [ form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .

The notion of a Sierpinski topological space is introduced and some properties of such spaces connected with Borel measures are considered.

Two-weighted inequalities are proved for anisotropic potentials. These estimates are used to obtain the refinements of the well-known imbedding theorems in the scale of weighted Lebesgue spaces.

A class of linear systems which after ordinary linear systems are in a certain sense the simplest ones, is associated with every algebraic function field. From the standpoint developed in the paper ordinary linear systems are associated with the rational function field.

In [Kratz and Stadtmüller, J. Approximation Theory 54: 326-337, 1988], for continuous functions f from the domain of certain discrete operators L n the inequalities are proved concerning the modulus of continuity of L n f . Here we present analogues of the results obtained for the Durrmeyer-type modification of L n . Moreover, we give the estimates of the rate of convergence of in Hölder-type norms

Weighted Zygmund type estimates are obtained for the continuity modulus of some convolution type integrals. In the case of fractional integrals this is strengthened to a result on isomorphism between certain weighted generalized Hölder type spaces.

The unquantified set theory MLSR containing the symbols ∪, \, =, ∈, R ( R ( x ) is interpreted as a rank of x ) is considered. It is proved that there exists an algorithm which for any formula Q of the MLSR theory decides whether Q is true or not using the space c | Q | 3 (| Q | is the length of Q ).

Perfect mappings from ℝ to ℝ and also mappings close to perfect ones are considered. Some of their properties are given.