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Том 53, № 1 (2018)
- Год: 2018
- Статей: 9
- URL: https://journals.rcsi.science/1068-3623/issue/view/14079
Differential Equations
A Sub-Density Theorem of Sturm-Liouville Eigenvalue Problem with Finitely Many Singularities
Аннотация
We study the distribution of the Sturm-Liouville eigenvalues of a potential with finitely many singularities. There is an asymptotically periodical structure on this class of eigenvalues as described by the entire function theory. We describe the singularities of its potential function explicitly in its eigenvalue asymptotics.
1-5
On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power
Аннотация
A linear differential operator P(x, D) = P(x1,... xn, D1,..., Dn) = ∑αγα(x)Dα with coefficients γα(x) defined in En is called formally almost hypoelliptic in En if all the derivatives DνξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in En. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.
6-15
Functional Analysis
Groups of Invertible Binary Operations of a Topological Space
Аннотация
In this paper, continuous binary operations of a topological space are studied and a criterion of their invertibility is proved. The classification problem of groups of invertible continuous binary operations of locally compact and locally connected spaces is solved. A theorem on the binary distributive representation of a topological group is also proved.
16-20
On a Composition Preserving Inequalities between Polynomials
Аннотация
The Schur-Szegö composition of two polynomials \(f\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{z^j}} \) and \(g\left( z \right) = \sum\nolimits_{j = 0}^n {{B_j}{z^j}} \), both of degree n, is defined by \(f * g\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{B_j}{{\left( {\begin{array}{*{20}{c}} n \\ j \end{array}} \right)}^{ - 1}}{z^j}} \). In this paper, we estimate the minimum and the maximum of the modulus of f * g(z) on z = 1 and thereby obtain results analogues to Bernstein type inequalities for polynomials.
21-26
On Generalized Derivations and Centralizers of Operator Algebras with Involution
Аннотация
Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H and A(H) ⊆ B(H) be a standard operator algebra which is closed under the adjoint operation. Let F: A(H)→ B(H) be a linear mapping satisfying F(AA*A) = F(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H), where the associated linear mapping d: A(H) → B(H) satisfies the relation d(AA*A) = d(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H). Then F is of the form F(A) = SA − AT for all A ∈ A(H) and some S, T ∈ B(H), that is, F is a generalized derivation. We also prove some results concerning centralizers on A(H) and semisimple H*-algebras.
27-33
Integral Equations
One-parameter Family of Positive Solutions for a Nonlinear Integral Equation Arising in Physical Kinetics
Аннотация
The paper is devoted to the question of solvability of a Urysohn type nonlinear integral equation. This equation has an application in the kinetic theory of gases and can be derived from Boltzmann model equation. We prove an existence theorem of one-parameter family of positive solutions in the space of functions possessing linear growth at infinity. Moreover, for each member of this family we find an exact asymptotic formula at infinity. We obtain two-sided estimates for solution, as well as describe an iterative method for construction of solution.We conclude the paper by giving examples of functions that describe nonlinearity and satisfy the conditions of the main theorem.
34-40
On a Homogeneous Integral Equation with Two Kernels
Аннотация
The present paper is devoted to the finding conditions of nontrivial (non-zero) solvability of some classes of equations of the form \(S\left( x \right) = \int_0^\infty {{T_1}\left( {x - t} \right)S\left( t \right)} dt + \int_{ - \infty }^0 {{T_2}\left( {x - t} \right)S\left( t \right)} dt\) , x ∈ R, with respect to unknown function S. The asymptotic behavior of the solution S is also studied.
41-46
On Integral Equations the Kernels of Which are Homogeneous Functions of Degree (−1)
Аннотация
The present paper deals with integral equations the kernels of which are homogeneous functions of degree (−1). Factorization approach to such equations is developed. The constructed operator factorization is applied to the equation with a positive symmetric kernel. We prove that in the conservative case, both the homogeneous equation and the corresponding nonhomogeneous equation with a positive free term can possess positive solutions simultaneously.
47-55
Real and Complex Analysis
On an Equivalency of Rare Differentiation Bases of Rectangles
Аннотация
The paper considers differentiation properties of density bases formed of bounded open sets.We prove that two quasi-equivalent subbases of some density basis differentiate the same class of non-negative functions. Applications for bases formed of rectangles are discussed.
56-60