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Том 231, № 1 (2018)
- Год: 2018
- Статей: 8
- URL: https://bakhtiniada.ru/1072-3374/issue/view/14918
Article
Extremal decomposition of the complex plane with restrictions for free poles
Аннотация
The problems of extremal decomposition with free poles on a circle are well known in the geometric theory of functions of a complex variable. One of such problems is the problem of maximum of the functional
where γ ∈ (0, n], B0, B1, B2,...,Bn, n ≥ 2, are pairwise disjoint domains in \( \overline{\mathrm{C}},{a}_0=0,\left|{a}_k\right|=1,k=\overline{1,n} \) are different points of the circle, r(B, a) is the inner radius of the domain B ⊂ \( \overline{\mathrm{C}} \) relative to the point a ∈ B. We consider a more general problem, in which the restriction \( \left|{a}_k\right|=1,k=\overline{1,n}, \) is replaced by a more general condition.
1-15
16-27
Estimates of approximative characteristics of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of many variables with given majorant of mixed continuity moduli in the space L∞
Аннотация
We obtain exact-order estimates of the approximation of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of several variables in the space L∞, by using operators of orthogonal projection, as well as linear operators subjected to some conditions.
28-40
Approximative properties of the Weierstrass integrals on the classes \( {W}_{\beta}^r{H}^{\alpha } \)
Аннотация
The work focuses on the solution of a problem of approximation theory. The task is to investigate approximative properties of the Weierstrass integrals on the classes \( {W}_{\beta}^r{H}^{\alpha } \). We obtain asymptotic equalities for the upper borders of defluxion of functions from the classes \( {W}_{\beta}^r{H}^{\alpha } \) from the Weierstrass integrals.
41-47
Stochastic differential equation in a random environment
Аннотация
Solutions of the Itô stochastic differential equation in a random environment are considered. The random environment is formed by the generalized telegraph process. It is proved that the initial problem is equivalent to a system of two stochastic differential equations with nonrandom coefficients. The first equation is the Itô equation, and the initial process is its solution. The second equation is an equation with Poisson process, and its solution is a generalized telegraph process. The theorems of existence and uniqueness of strong and weak solutions are proved.
48-69
On the global behavior of homeomorphisms of metric spaces
Аннотация
For the metric spaces, the homeomorphisms more general than conformal mappings are studied. It is proved that the families of ine indicated mappings are equicontinuous in their closure under definite conditions imposed on the boundaries of the corresponding domains.
70-82
Boundary triples for integral systems on finite intervals
Аннотация
Let P, Q, and W be real functions of bounded variation on [0, l], and let W be nondecreasing. The integral system
on a finite compact interval [0, l] was considered in [6]. The maximal and minimal linear relations Amax and Amin associated with the integral system (0.1) are studied in the Hilbert space L2(W). It is shown that the linear relation Amin is symmetric with deficiency indices n±(Amin) = 2 and Amax = \( {A}_{min}^{\ast }. \) Boundary triples for Amax are constructed, and the the corresponding Weyl functions are calculated.
83-100
On a product of the inner radii of symmetric multiply connected domains
Аннотация
The article is devoted to the study of a quite general problem of the geometric theory of functions on an extreme decomposition of the complex plane. The problem of maximum of the functional
where γ ∈ (0, 1], n ≥ 2, a0 = 0,\( \left|{a}_k\right|=1,k=\overline{1,n},\kern0.5em {a}_k\in {B}_k\subset \overline{\mathrm{C}},k=\overline{0,n},{\left\{{B}_k\right\}}_{k=0}^n \) are pairwise disjoint domains, \( {\left\{{B}_k\right\}}_{k=0}^n \) are symmetric domains with respect to the unit circle, and r(B, a) is the inner radius of the domain B ⊂ \( \overline{\mathrm{C}} \) relative to the point a ∈ B, is considered.
101-109