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Vol 59, No 3 (2018)
- Year: 2018
- Articles: 18
- URL: https://journals.rcsi.science/0037-4466/issue/view/10467
Article
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Integro-Local Limit Theorems for Compound Renewal Processes under Cramér’S Condition. I
Abstract
We obtain integro-local limit theorems in the phase space for compound renewal processes under Cramér’s moment condition. These theorems apply in a domain analogous to Cramér’s zone of deviations for random walks. It includes the zone of normal and moderately large deviations. Under the same conditions we establish some integro-local theorems for finite-dimensional distributions of compound renewal processes.
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On the Centralizer Dimension and Lattice of Generalized Baumslag–Solitar Groups
Abstract
A generalized Baumslag–Solitar group (a GBS group) is a finitely generated group G acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups. Each GBS group is the fundamental group π1(A) of some labeled graph A. We describe the centralizers of elements and the centralizer lattice. Also, we find the centralizer dimension for GBS groups if A is a labeled tree.
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Construction and Study of Exact Solutions to A Nonlinear Heat Equation
Abstract
We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.
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Maximal Surfaces on Five-Dimensional Group Structures
Abstract
For the classes of the mappings Lipschitz in the sub-Riemannian sense and taking values in the Heisenberg group we introduce some suitable notions of variation of an argument and the corresponding increment of the area functional and derive several basic properties of maximal surfaces on the five-dimensional sub-Lorentzian structures.
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A Computably Enumerable Partial Ordering Without Computably Enumerable Maximal Chains and Antichains
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A One-Dimensional Schrödinger Operator with Square-Integrable Potential
Abstract
We study the spectral properties of a one-dimensional Schrödinger operator with squareintegrable potential whose domain is defined by the Dirichlet boundary conditions. The main results are concerned with the asymptotics of the eigenvalues, the asymptotic behavior of the operator semigroup generated by the negative of the differential operator under consideration. Moreover, we derive deviation estimates for the spectral projections and estimates for the equiconvergence of the spectral decompositions. Our asymptotic formulas for eigenvalues refine the well-known ones.
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On Approximation Characteristics of Some Classes of Functions of Small Smoothness
Abstract
We find the exact order of the entropy numbers and the Kolmogorov widths of a class of functions defined by a condition on the uniform norm of blocks of the Fourier series taken over a lacunar sequence of indices. This result generalizes a result by B. S. Kashin and V. N. Temlyakov.
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Phaseless Inverse Problems That Use Wave Interference
Abstract
We consider the inverse problems for differential equations with complex-valued solutions in which the modulus of a solution to the direct problem on some special sets is a given information in order to determine coefficients of this equation; the phase of this solution is assumed unknown. Earlier, in similar problems the modulus of the part of a solution that corresponds to the field scattered on inhomogeneities in a wide range of frequencies was assumed given. The study of high-frequency asymptotics of this field allows us to extract from this information some geometric characteristics of an unknown coefficient (integrals over straight lines in the problems of recovering the potential and Riemannian distances between the boundary points in the problem of the refraction index recovering). But this is physically much more difficult to measure the modulus of a scattered field than that of the full field. In this connection the question arises how to state inverse problems with the full-field measurements as a useful information. The present article is devoted to the study of this question. We propose to take two plane waves moving in opposite directions as an initiating field and to measure the modulus of a full-field solution relating to interference of the incident waves. We consider also the problems of recovering the potential for the Schrödinger equation and the permittivity coefficient of the Maxwell system of equations corresponding to time-periodic electromagnetic oscillations. For these problems we establish uniqueness theorems for solutions. The problems are reduced to solving some well-known problems.
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On Splittings, Subgroups, and Theories of Partially Commutative Metabelian Groups
Abstract
We consider two splittings of a partially commutative metabelian group G. The universal theories and splittings of G are compared. We prove that all nilpotent subgroups of G are abelian and give description of the Fitting subgroup of G.
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Homological Resolutions in Problems About Separating Cycles
Abstract
We study the homological cycles that separate a set of divisors in a complex-analytic manifold. A generalization of the notion of separating cycle is proposed for the case of a collection of closed sets in an arbitrary real manifold.
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Betti Numbers of Small Covers and Their Two-Fold Coverings
Abstract
We compute the Betti numbers of two-fold coverings of small covers with some special properties; in particular, we use the results of Davis and Januszkiewicz on the cohomology of small covers. It turned out that their proof contains some gap that we describe in detail and fill in.
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Finite Groups Whose n-Maximal Subgroups Are Modular
Abstract
Let G be a finite group. If Mn< Mn−1< · · · < M1< M0 = G with Mi a maximal subgroup of Mi−1 for all i = 1,..., n, then Mn (n > 0) is an n-maximal subgroup of G. A subgroup M of G is called modular provided that (i) 〈X,M ∩ Z〉 = 〈X,M〉 ∩ Z for all X ≤ G and Z ≤ G such that X ≤ Z, and (ii) 〈M,Y ∩ Z〉 = 〈M,Y 〉 ∩ Z for all Y ≤ G and Z ≤ G such that M ≤ Z. In this paper, we study finite groups whose n-maximal subgroups are modular.
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