Открытый доступ
Доступ предоставлен
Только для подписчиков
Том 220, № 5 (2017)
- Год: 2017
- Статей: 8
- URL: https://ogarev-online.ru/1072-3374/issue/view/14803
Article
On some properties of the orthogonal polynomials over a contour with general Jacobi weight
Аннотация
In the present work, we continue to study the growth of the orthogonal polynomials over a contour with a weight function in the weighted Lebesgue space, when the contour and the weight function have some singularities. The case where there is no interference of a weight function and a contour is studied. We consider a piecewise smooth contour with interior zero angles and investigate the case of more general contours.
533-553
1-D Schrödinger Operators with Local Interactions on a Discrete Set with Unbounded Potential
Аннотация
We study spectral properties of the one-dimensional Schrödinger operators \( {\mathrm{H}}_{\mathrm{X},\alpha, \mathrm{q}}:=-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+\mathrm{q}(x)+{\varSigma_x}_{{}_n}\in X{\alpha}_n\delta \left(x-{x}_n\right) \) with local interactions, d* = 0, and an unbounded potential q being a piecewise constant function, by using the technique of boundary triplets and the corresponding Weyl functions. Under various sufficient conditions for the self-adjointness and discreteness of Jacobi matrices, we obtain the condition of self-adjointness and discreteness for the operator HX,α,q.
554-583
584-590
On the regularization of a matrix differential-algebraic boundary-value problem
Аннотация
The conditions of regularization and the structure of generalized Green’s operator for a re-gularized linear matrix differential-algebraic boundary-value problem are found. To solve the problem of regularization of a generalized matrix differential-algebraic boundary-value problem, the original conditions of solvability and the structure of the general solution of a matrix equation of the Sylvester type are used.
591-602
On a Model Semilinear Elliptic Equation in the Plane
Аннотация
Assume that Ω is a regular domain in the complex plane ℂ, and A(z) is a symmetric 2×2 matrix with measurable entries, det A = 1, and such that 1/K|ξ|2 ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|2, ξ ∈ ℝ2, 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = eu in Ω and show that the well-known Liouville–Bieberbach function solves the problem under an appropriate choice of the matrix A(z). The proof is based on the fact that every regular solution u can be expressed as u(z) = T(ω(z)), where ω : Ω → G stands for a quasiconformal homeomorphism generated by the matrix A(z), and T is a solution of the semilinear weihted Bieberbach equation △T = m(w)e in G. Here, the weight m(w) is the Jacobian determinant of the inverse mapping ω−1(w).
603-614
Adapted statistical experiments
Аннотация
We study statistical experiments with a random change of time, which transforms a discrete stochastic basis in a continuous one. The adapted stochastic experiments are studied in continuous stochas-tic basis in the series scheme. The transition to limit by the series parameter generates an approximation of adapted statistical experiments by a diffusion process with evolution.
615-623
Solutions of some partial differential equations with variable coefficients by properties of monogenic functions
Аннотация
We study some partial differential equations, by using the properties of Gateaux differen-tiable functions on a commutative algebra. It is proved that components of differentiable functions satisfy some partial differential equations with coefficients related to properties of the bases of subspaces of the corresponding algebra.
624-632
Metric Properties of Orlicz–Sobolev Classes
Аннотация
The homeomorphisms of the Orlicz–Sobolev class Wloc1,φ under a condition of the Calderón type on φ in ℝn, n ≥ 3 are considered. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property, as well as various theorems on estimates of a distortion of the Euclidean distance are proved. In particular, the results hold for the homeomorphisms of the Sobolev classes Wloc1,p with p > n − 1.
633-642