Lobachevskii Journal of Mathematics

In this paper, we introduce the concept of A-statistical uniform integrability of sequences of random variables which is not only more general than the concept of uniform integrability, but is also weaker than the concept of uniform integrability. We also give some characterizations of A-statistical uniform integrability and prove a law of large numbers.

The paper deals with the normal extensions of cancellative commutative semigroups and the Toeplitz algebras for those semigroups. By the Toeplitz algebra for a semigroup S one means the reduced semigroup C*-algebra C_r^*(S). We study the normal extensions of cancellative commutative semigroups by the additive group ℤ_n of integers modulo n. Moreover, we assume that such an extension is generated by one element. We present a general method for constructing normal extensions of semigroups which contain no non-trivial subgroups. The Grothendieck group for a given semigroup and the group of all integers are involved in this construction. Examples of such extensions for the additive semigroup of non-negative integers are given. A criterion for a normal extension generated by an element to be isomorphic to a numerical semigroup is given in number-theoretic terms. The results concerning the Toeplitz algebras are the following. For a cancellative commutative semigroup S and its normal extension L generated by one element, there exists a natural embedding the semigroup C*-algebra C_r^*(S) into C_r^*(L). The semigroup C*-algebra C_r^*(L) is topologically ℤ_n-graded. The results in the paper are announced without proofs.

Processes of mass transfer (primarily swelling and hydration/dehydration metamorphic processes) between underground fluid and solid phase in porous media in various kinds of rocks (primarily, sedimentary rocks) is examined. Penetration of water in swelling porous media and coupled process of consolidation and physical-chemical interaction between fluid and solid phase are analyzed. It is shown that government equation in this case is non-linear equation of “diffusion—reaction” type. The equation describes the influence of mass exchange processes (for example swelling and metamorphic dehydration) on distribution of pressure in pores, stresses and deformations in rocks.

It is found the spectrum of a singular integral operator on the compound contour with elliptic function in the kernel. Explicit solution of the corresponding singular integral equation is constructed by applying its reduction to the boundary value problem on the Riemann surface.

We introduce a symmetry property for unit-regular rings as follows: a ∈ R is unit-regular if and only if aR ⊕ (a − u)R = R (equivalently, Ra ⊕ R(a − u) = R) for some unit u of R if and only if aR ⊕ (a − u)R =(2a − u)R (equivalently, Ra ⊕ R(a − u) = R(2a − u)) for some unit u of R. Let M and N be right R-modules and α, β ∈ Hom(M, N) such that α + β is regular. It is shown that αS ⊕ βS =(α + β)S, where S = End(M) if and only if Tα ⊕ Tβ = T(α + β), where T = End(N). We also introduce partial order α ≤^⊕β and minus partial order α ≤⁻β for any α, β ∈ Hom(M, N); they translate into module-theoretic language defined in a ring in [7] and [8]. We analyze some relationships between ≤^⊕ and ≤⁻ on the endomorphism rings of the modules M and N.

The problems of solvability and construction of solutions of a nonlocal boundary value problem for the second-order Fredholm integro-differential equation with degenerate kernel, integral conditions, spectral parameters and reflecting deviation are considered. Using the method of the degenerate kernel, the boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving systems of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed for each of five cases. The stability of the solution of the boundary value problem for given values in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed. For other cases, the absence of nontrivial solutions of the problem is proved.