Том 102, № 1-2 (2017)

A nonclassical nonlinear third-order partial differential equation modeling nonstationary processes in a semiconductor medium is studied. Seven classes of exact solutions of this equation given as explicit, implicit, and quadrature formulas are constructed. It is shown that, among these solutions, there are some bounded globally with respect to time and some approaching infinity on finite time intervals. These results are of interest in view of the following property of initial boundary-value problems for the given equation. Here the method of energy estimates used in many papers to justify the boundedness or unboundedness of solutions does not provide sufficient conditions for the unboundedness of solutions on finite time intervals.

The Lomov regularization method [1] is generalized to integro-partial differential equations. It turns out that the regularization procedure essentially depends on the type of integral operator. The case in which the upper limit of the integral is not the differentiation variable is the most difficult one. It is not considered in the present paper. Only the case in which the upper limit of the integral operator coincides with the differentiation variable is studied. For such problems, an algorithm for constructing regularized asymptotics is developed.

The rate of ϕ-strong approximation of periodic functions by trigonometric polynomials constructed on the basis of interpolating polynomials with equidistant nodes is considered.

We prove that the affine-triangular subgroups are Borel subgroups of Cremona groups.

In the paper, we suggest a method for finding relations concerning series defining the Buchstaber formal group. This method is applied to the cases in which the exponent of the group is an elliptic function of level n = 2, 3, and 4. An algebraic relation for the series defining the universal Buchstaber formal group is also proved.

It is shown that every homogeneous separable torsion-free group is strongly invariant simple (i.e., has no nontrivial strongly invariant subgroups) and, for a completely decomposable torsion-free group, every strongly invariant subgroup coincides with some direct summand of the group. The strongly invariant subgroups of torsion-free separable groups are described. In a torsion-free group of finite rank, every strongly inert subgroup is commensurable with some strongly invariant subgroup if and only if the group is free. The periodic groups, torsion-free groups, and split mixed groups in which every fully invariant subgroup is strongly invariant are described.

We consider the spectral synthesis problem for the differentiation operator on the space of infinitely differentiable functions on a finite or infinite interval of the real line and the dual problem of local description of closed submodules in a special module of entire functions.

Two constructions on a space with two convexities, n-semiconvexity and n-biconvexity, are considered. Features of the sets corresponding to these constructions on the 2-sphere are studied. Their separation properties are considered.

The existence of periodic solutions of nonlinear equations generalizing logistic equations with delay is studied. The existence of sets of periodic solutions of two types is established. The stability and asymptotics of periodic solutions under changes of the parameters are studied.

We say that a group has an MF-property if it can be embedded in the group of unitary elements of the C*-algebra ΠM_n/⊕M_n. In the present paper we prove the MF-property for the Baumslag group \(\left\langle {a,b|{a^{{a^b}}} = {a^2}} \right\rangle \) and also some general assertions concerning this property.

The Gehring–Martin–Tan number and the Tan number are real quantities defined for two-generated subgroups of the group PSL(2,ℂ). It follows from the necessary discreteness conditions proved by Gehring and Martin and, independently, by Tan that, for discrete groups, these quantities are bounded below by 1. In the paper, we find precise values of these numbers for the majority of elementary discrete groups and prove that, for every real r ≥ 1, there are infinitely many elementary discrete groups with the Gehring–Martin–Tan number equal to r and the Tan number equal to r.

We consider the Cauchy problem for the heat equation in a cylinder C_T = X × (0, T) over a domain X in Rⁿ, with data on a strip lying on the lateral surface. The strip is of the form S × (0, T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S, we derive an explicit formula for solutions of this problem.

λ-convergent multiple Walsh–Paley series on a multidimensional dyadic group are studied. It is proved that, for all λ > 1, any arbitrary finite union of hyperplanes parallel to coordinate hyperplanes is a set of uniqueness for such series.

This article has been retracted at the request of the Editorial Board of the journal in accordance with the COPE guidelines. This article contains a significant amount of overlap with the following article by the same author: Naser Zamani, “Characterizations of Graded Distributive Modules,” Journal of Applied Mathematics (2008), Volume 5, Issue 16. The author concedes that the articles overlap significantly and apologizes for any inconvenience.