Vol 39, No 2 (2018)
- Year: 2018
- Articles: 26
- URL: https://journals.rcsi.science/1995-0802/issue/view/12542
Article
Classes of Finite Solutions to the Inverse Problem of the Logarithmic Potential
Abstract
We obtain a new class of solutions to the inverse problems of the logarithmic potential in the form of a logarithmic function of a ratio of polynomials of the same degree. We give examples of finite solvability of the inverse problems.
Periodic Conjugation Problem for Linear Elliptic Equations of Second Order with Constant Coefficients
Abstract
We consider a periodic problem of conjugation on the real axis for a linear differential elliptic equation of second order with constant coefficients. There is known a representation of general solution of the equation in terms of analytic functions depending on affine connected variables. We introduce auxiliary analytic functions enabling us to reduce the problem to a system of two periodic Riemann boundary value problems. That problem was solved first by L. I. Chibrikova. We use her results for solving of our problem. We obtain its explicit solution and conditions of solvability, evaluate the index and defect numbers.
A Note on Separating Function Sets
Abstract
We study separating function sets. We find some necessary and sufficient conditions for Cp(X) or Cp2 (X) to have a point-separating subspace that is a metric space with certain nice properties. One of the corollaries to our discussion is that for a zero-dimensional X, Cp(X) has a discrete point-separating space if and only if Cp2 (X) does.
On 2-Absorbing Ideals in Commutative Rings with Unity
Abstract
In this paper, we prove some important results of 2-absorbing ideals in a commutative ring with 1 ≠ 0. The conceptof n-weakly prime ideal in a commutative ring with 1 ≠ 0 is introduced and prove number of results concerning to n-weakly prime ideals in commutative rings.
Ricci Solitons on Lorentzian Walker Manifolds of Low Dimension
Abstract
The Ricci soliton equation on four-dimensional conformally flat Lorentzian Walker manifolds is investigated, non-trivial solutions are found. New Ricci soliton metrics on three dimensional Lorentzian Walker manifolds are obtained.
Locally Convex Limit Spaces of Measurable Functions with Order Units and Its Duals
Abstract
We consider linear normed spaces of measurable functions dominated by positive measurable function powered by real positive parameter. Also, we consider its dual and predual, and we propose a method for constructing a limit spaces of these functional spaces taken by power parameter. We prove that these limit spaces are (LF)-spaces and also prove that the limit spaces presume the relation of duality, i.e., the limit space of predual spaces is predual for the limit space of dominated functions, and the limit space of duals is dual for it. Also, the limit space of predual spaces is embedded into the limit space of dual spaces.
The Least Root of a Continuous Function
Abstract
For each ε > 0 and each scalar real valued and continuous on a compact set Ω ⊂ Rn, ξ ∈ [a, b] function g(τ, ξ) such that g(τ, a) · g(τ, b) < 0 we construct a function gε(τ, ξ), for which the least root ξ(τ) of the equation gε(τ, ξ) = 0 continuously depends on τ, while |g(τ, ξ) − gε(τ, ξ)| < ε. We give examples illustrating the fact that in a general case assumptions are unimprovable.
About Functional Equation, Generated by Equilateral Triangle
Abstract
We considered 6-fold linear equation in the class of analytic functions outside the equilateral triangle and vanishing at infinity. We proposed a method of equivalent regularization, using the theory of boundary value problem of Carleman. Also, the paper is dedicated to applications to the problem of moments for entire functions of exponential type. In particular, we construct a system of functions biorthogonal whit piecewise-quasipolynomial weight to some power system on the three rays. The indicator of such functions is a piecewise-trigonometric function, with a period of 2π/3.
Some Transcendental Equations with Trigonometric and Hyperbolic Functions
Abstract
Eigenvalues of many spectral problems for differential equations are the roots of transcendental equations. In this article we prove the statements about the roots of some equations with trigonometric and hyperbolic functions. These equations have countable sets of real and pure imaginary roots.
Nontrivial Pseudocharacters on Groups and Their Applications
Abstract
In this paper questions of the existence of non-trivial pseudocharacters for different classes of groups and the most important applications of pseudocharacters are considered. We review the results obtained for non-trivial pseudocharacters of free group constructions. Pseudocharacter is the real functions f on a group G such that 1) |f(ab) − f(a) − f(b)| ≤ ε for some ε > 0 and for any a, b ∈ G and 2) f(xn) = nf(x) ∀n ∈ Z, ∀x ∈ G. Existence of non-trivial pseudocharacters implies the results for second group of bounded cohomologies and the width of verbal subgroups. Results of R.I. Grigorchuk, V.G. Bardakov, V.A. Fayziev and author on this topic are examined. Theorems about conditions of the existence of non-trivial pseudocharacters on such group objects as free products with amalgamation, HNN-extensions, group with one defining relation, anomalous products are given in the article.
Existence of Solutions of Anisotropic Elliptic Equations with Variable Exponents in Unbounded Domains
Abstract
We consider a class of anisotropic elliptic differential equations of second order with divergent form and variable exponents. The corresponding elliptic operators are pseudo-monotone and coercive. We obtain solvability conditions for the Dirichlet problem in unbounded domains Ω ⊂ ℝn, n ≥ 2. The proof of existence of solutions is free of restrictions on growth of data for |x| → ∞.
Embedding Theorems for Interpolation Spaces BLp,qs,k
Abstract
We consider the space obtained from spaces Triebel–Lizorkin with help interpolation by the real method. Embedding theorems for these spaces are studied. It is shown that the space of traces belong to the same class of spaces.
On the Group of Continuous Automorphisms of Some Profinite Groups
Abstract
We prove some conditions on a given abstract group G, such that the group Autc(\(\hat G\)) of the continuous automorphisms of the profinite completion \(\hat G\) of G endowed with the congruence subgroup topology, is profinite. Also, for a given abstract group G, if Autc(\(\hat G\)) is profinite, then we establish relations betweenG, Aut(G), \(\widehat {Aut(G)}\), and Autc(\(\hat G\)) when each of these groups is endowed with appropriate topology. Finally, we applied the obtained results to the class of one-relator groups given by the presentation Gmn = 〈a, b; [am, bn] = 1〉 (m > 1,n > 1).
Mathematical Model of Qualitative Properties of Exciton Diffusion Generated by Electron Probe in a Homogeneous Semiconductor Material
Abstract
The qualitative properties of the two-dimensional mathematical model of excitons diffusion excited by an electron beamin a semiconductormaterial are investigated. For the studied model proved continuous dependence of the solution from the input data. It is shown that the model can be applied to estimate the diffusion coefficient and the mobility of excitons on the results of experimental measurements. In the simulation are used parameters that are typical for gallium nitride.
Twice Periodic Solutions of a Nonlinear Elliptic Second-Order Systems
Abstract
The problem of the existence of twice periodic solutions of a nonlinear elliptic system of second-order equations on a plane whose main periods are given is studied. Conditions for the existence of solutions with prescribed poles, as well as with prescribed zeros and poles, are obtained.
Behavior of the Singular Integral Along the Real Axis with the Density Vanishing at Infinity Near Infinity
Abstract
We study the behavior of singular integral along the real axis in the neighborhood of the point at infinity, when its density satisfy Hölder condition in any finite part of the real axis and is continuous function in the neighborhood of the point at infinity, which decreases with the order of decreasing same as the lower then the minus first power of logarithm of the absolute value of the coordinate of the point of the real axis, when we move this point unlimitedly from the origin.
The Groups of Basic Automorphisms of Complete Cartan Foliations
Abstract
For a complete Cartan foliation (M,F) we introduce two algebraic invariants g0(M,F) and g1(M,F) which we call structure Lie algebras. If the transverse Cartan geometry of (M,F) is effective then g0(M,F) = g1(M,F). Weprove that if g0(M,F) is zero then in the category of Cartan foliations the group of all basic automorphisms of the foliation (M,F) admits a unique structure of a finite-dimensional Lie group. In particular, we obtain sufficient conditions for this group to be discrete. We give some exact (i.e. best possible) estimates of the dimension of this group depending on the transverse geometry and topology of leaves. We construct several examples of groups of all basic automorphisms of complete Cartan foliations.
A Necessary Condition for The Residual Nilpotence of HNN-Extensions
Abstract
Let G be amultiple HNN-extension of a group A, and let all its associated subgroups be properly contained in some locally nilpotent subgroup of A. We prove that if G is residually nilpotent, then all the associated subgroups are p′-isolated in A for some prime p. Moreover, if q is a prime such that G is residually a q′-torsion-free nilpotent group, then p = q.
Discrete Pseudo-Differential Operators and Boundary Value Problems in a Half-Space and a Cone
Abstract
We consider a certain class of discrete pseudo-differential operators and related equations in a sharp convex cone and describe their invertibility conditions in L2 spaces. For this purpose we introduce a concept of periodic wave factorization for elliptic symbol and show its applicability for the studying. For a half-space case we consider the Laplace equation and describe a solution of the discrete Dirichlet problem.