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Том 234, № 1 (2018)
- Год: 2018
- Статей: 10
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14954
Article
Separating transformation and extremal problems on nonoverlapping simply connected domains
Аннотация
We consider the well-known problem of maximum of the functional
where B0, …, Bn are pairwise disjoint domains in \( \overline{\mathrm{\mathbb{C}}} \), a0 = 0, |ak| = 1, \( k=\overline{1,n} \), are different points of the circle, γ ∈ (0, n], and r(B, a) is the inner radius of the domain \( B\subset \overline{\mathrm{\mathbb{C}}} \) relative to the point a. In the case of simply connected domains for n=2, 3, and 4, we have obtained the solution of this problem for the maximum interval of values of the parameter γ.
1-13
To the problem of extremal partition of the complex plane
Аннотация
We consider one of the classical problems of the geometric theory of functions of a complex variable on a maximum of the functional
where n ∈ ℕ, n ≥ 2, γ ∈ ℝ+, \( {A}_n={\left\{{a}_k\right\}}_{k=1}^n \) is a system of points such that |ak| = 1, a0 = 0, B0, B∞, \( {\left\{{B}_k\right\}}_{k=1}^n \) is a system of pairwise nonoverlapping domains, \( {a}_k\in {B}_k\subset \overline{\mathrm{\mathbb{C}}} \), \( k=\overline{0,n} \), \( \infty \in {B}_{\infty}\subset \overline{\mathrm{\mathbb{C}}} \), r(B, a) is the inner radius of the domain \( B\subset \overline{\mathrm{\mathbb{C}}} \) with respect to the point a ∈ B. We have analyzed this problem under some weaker restrictions on pairwise nonoverlapping domains.
14-20
Automorphisms of semigroups of k-linked upfamilies
Аннотация
A family \( \mathcal{A} \) of non-empty subsets of a set X is called an upfamily, if, for each set \( A\in \mathcal{A} \); any set B ⊃ A belongs to \( \mathcal{A} \). An upfamily \( \mathrm{\mathcal{L}} \) is called k-linked, if \( \cap \mathrm{\mathcal{F}}\ne \varnothing \) for any subfamily \( \mathrm{\mathcal{F}}\subset \mathrm{\mathcal{L}} \) of cardinality \( \left|\mathrm{\mathcal{F}}\right|\le k \). The extension Nk(X) consists of all k-linked upfamilies on X. Any associative binary operation ∗ : X × X → X can be extended to an associative binary operation ∗ : Nk(X) × Nk(X) → Nk(X). Here, we study automorphisms of the extensions of groups and finite monogenic semigroups. We also describe the automorphism groups of extensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups.
21-34
Bernstein–Walsh type inequalities in unbounded regions with piecewise asymptotically conformal curve in the weighted Lebesgue space
Аннотация
We have obtained the pointwise Bernstein–Walsh type estimation for algebraic polynomials in the unbounded regions with piecewise asymptotically conformal boundary, having exterior and interior zero angles, in the weighted Lebesgue space.
35-48
Recent progress in subset combinatorics of groups
Аннотация
We systematize and analyze some results obtained in the subset combinatorics of G groups presented in previous surveys [1–4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively to their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in G-spaces, new ideals in the Boolean algebra \( {\mathcal{P}}_G \) of all subsets of a group G and in the Stone– Čech compactification βG of G, and the combinatorial derivation.
49-60
The Cauchy–Stieltjes integrals in the theory of analytic functions
Аннотация
We study various Stieltjes integrals as Poisson–Stieltjes, conjugate Poisson–Stieltjes, Schwartz–Stieltjes, and Cauchy–Stieltjes ones and prove theorems on the existence of their finite angular limits a.e. in terms of the Hilbert–Stieltjes integral. These results hold for arbitrary bounded integrands that are differentiable a.e. and, in particular, for integrands of the class \( \mathcal{C}\mathrm{\mathcal{B}}\mathcal{V} \) (countably bounded variation).
61-72
On the Recursive Sequence \( {x}_{n+1}=\frac{x_{n-\left(k+1\right)}}{1+{x}_n{x}_{n-1}\dots {x}_{n-k}} \)
Аннотация
A solution of the following difference equation is investigated:
where x−(k+1); x−k; : : : ; x−1; x0 ???? (0;∞) and k = 0; 1; 2; : : : .
73-81
Factorization of generalized γ-generating matrices
Аннотация
The class of γ-generating matrices and its subclasses of regular and singular γ-generating matrices were introduced by D. Z. Arov in [8], where it was shown that every γ-generating matrix admits an essentially unique regular–singular factorization. The class of generalized γ-generating matrices was introduced in [14, 20]. In the present paper, subclasses of singular and regular generalized –generating matrices are introduced and studied. As the main result, a theorem of existence of the regular–singular factorization for a rational generalized γ-generating matrix is proved.
82-97
Order Estimates of Approximation Characteristics of Functions From the Anisotropic Nikol'skii–Besov Classes
Аннотация
We obtained the exact order estimates of deviations of functions from the anisotropic Nikol’skii–Besov classes \( {B}_{p,\theta}^r\left({\mathrm{\mathbb{R}}}^d\right) \) from their sections of the Fourier integral. The error of the approximation is evaluated in the metric of the Lebesgue space L∞(ℝd).
98-105
Limiting profile of solutions of quasilinear parabolic equations with flat peaking
Аннотация
The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representative
and with the following blow-up condition for the energy:
where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the condition
a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T.
106-116