卷 221, 编号 5 (2017)
- 年: 2017
- 文章: 6
- URL: https://journals.rcsi.science/1072-3374/issue/view/14813
Article
Estimates of a product of the inner radii of nonoverlapping domains
摘要
In the geometric theory of functions of complex variables, we consider V.N. Dubinin’s extremal problem associated with estimates of a functional defined on a system of nonoverlapping domains and obtain a particular solution of this problem.
On the inner radii of symmetric nonoverlapping domains
摘要
One of the classical problems of the geometric theory of functions that concerns extreme decompositions of the complex plane is considered.
On recent advances in boundary-value problems in the plane
摘要
The survey is devoted to recent advances in nonclassical solutions of the main boundary-value problems such as the well-known Dirichlet, Hilbert, Neumann, Poincaré, and Riemann problems in the plane. Such solutions are essentially different from the variational solutions of the classical mathematical physics and based on the nonstandard point of view of the geometric function theory with a clear visual sense. The traditional approach of the latter is the meaning of the boundary values of functions in the sense of the so-called angular limits or limits along certain classes of curves terminated at the boundary. This become necessary if we start to consider boundary data that are only measurable, and it is turned out to be useful under the study of problems in the field of mathematical physics as well. Thus, we essentially widen the notion of solutions and, furthermore, obtain spaces of solutions of the infinite dimension for all the given boundary-value problems. The latter concerns the Laplace equation, as well as its counterparts in the potential theory for inhomogeneous and anisotropic media.
Convergence of skew Brownian motions with local times at several points that are contracted into a single one
摘要
Conditions of convergence in mean of skew Brownian motions with local times at several points that are contracted into a limit point are obtained. It is proved that the limit process is also a skew Brownian motion with local time at the limit point. A formula to calculate the coefficient of the local time of the limit process is given.
Spectral and pseudospectral functions of various dimensions for symmetric systems
摘要
The main object of the paper is a symmetric system Jy′ − B(t)y = ⋋∆(t)y defined on an interval Ι = [a, b) with the regular endpoint a. Let φ(⋅, λ) be a matrix solution φ(⋅, λ) of this system of an arbitrary dimension, and let \( \left( V\kern0.5em f\right)(s)={\displaystyle \underset{I}{\int }{\varphi}^{\ast}\left( t, s\right)\varDelta (t) f(t) d t} \) be the Fourier transform of the function f(⋅) ∈ LΔ2(I). We define a pseudospectral function of the system as a matrix-valued distribution function σ(·) of the dimension nσ such that V is a partial isometry from \( {L}_{\varDelta}^2(I)\kern0.5em \mathrm{t}\mathrm{o}\kern0.5em {L}^2\left(\sigma; \kern0.5em {\mathbb{C}}^{n_{\sigma}}\right) \) with minimally possible kernel. Moreover, we find the minimally possible value of nσ and parametrize all spectral and pseudospectral functions of every possible dimensions nσ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; Sakhnovich, Sakhnovich and Roitberg; Langer and Textorius.
On monogenic mappings of a quaternionic variable
摘要
Earlier [1], a new class of quaternionic so-called G-monogenic (differentiable in the meaning of Gâteaux) mappings was considered. In the present paper, we introduce quaternionic H-monogenic (differentiable in the sense of Hausdorff) mappings and establish a relation between G- and H-monogenic mappings. The equivalence of different definitions of a G-monogenic mapping is proved.