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This article addresses the problem of boundary correspondence for a sequence of homeomorphisms that change the capacity of a condenser in a controlled way. To study the overall boundary behavior of these mappings, we introduce some capacity metrics in a sequence of domains with nondegenerate core. Completions with respect to these metrics add to the domains new points called boundary elements. As one of the consequences, we obtain not only sufficient conditions for the global uniform convergence of a sequence of homeomorphisms, but some applications to elasticity theory as well. [ABSTRACT FROM AUTHOR]

Given a nonempty compact set in a proper subdomain of the complex plane, we denote the diameter of and the distance from to the boundary of by and , respectively. The quantity is invariant under similarities and plays an important role in geometric function theory. In case has the hyperbolic distance , we consider the infimum of the quantity over compact subsets of with at least two points, where stands for the hyperbolic diameter of . Let the upper half-plane be . We show that is positive if and only if the boundary of is uniformly perfect and for all , with equality holding precisely when is convex. [ABSTRACT FROM AUTHOR]

We study the relationship between knotoids and knots in the direct product of the two-dimensional torus and an interval. Each knotoid on the sphere can be lifted to a knot of geometric degree 1 in the thickened torus. We prove that lifting is a bijection on the set of prime knotoids of complexity greater than 1. [ABSTRACT FROM AUTHOR]

We study the limit behavior of sequences of cyclic systems of ordinary differential equations that were invented for the mathematical description of multistage synthesis. The main construction of the article is the distribution function of initial data. It enables us to indicate necessary and sufficient existence conditions as well as completely describe the structure and all typical properties of the limits of solutions to the integro-differential equations of 'convolution' type to which the systems of cyclic synthesis are easily reduced. All notions, methods, and problems under discussion belong to such classical areas as real function theory, Euler integrals, and asymptotic analysis. [ABSTRACT FROM AUTHOR]

We define V ( α, β) ( α < 1 and β > 1), the new subclass of analytic functions with bounded positive real part, $$V\left( {\alpha ,\beta } \right): = \left\{ {f \in A:\alpha < \operatorname{Re} \left\{ {{{\left( {\frac{z}{{f\left( z \right)}}} \right)}^2}f'\left( z \right)} \right\} < \beta } \right\}$$ , and study some properties of V ( α, β). We also study the class U ( γ) ( γ > 0): $$u\left( \gamma \right): = \left\{ {f \in A:\left| {{{\left( {\frac{z}{{f\left( z \right)}}} \right)}^2}f'\left( z \right)} \right| - 1 < \gamma } \right\}$$ , where A is the class of normalized functions. [ABSTRACT FROM AUTHOR]

We construct the classes of functions analytic in the unit disk, other than lacunary, and possessing the following property: If a function belongs to one of these classes and the sum of the squares of its coefficients is unbounded then there exists a set of positive measure on the circle where the function has no radial limits. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H sG, where H sG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following Theorem. Let ℱ be a saturated formation containing the class of all supersoluble groups [Figure not available: see fulltext.] and let G be a group with E a normal subgroup of G such that G/E ∈ ℱ. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ℱ. [ABSTRACT FROM AUTHOR]

This article completes the authors’s series on stability in the Liouville theorem on the Heisenberg group. We show that every mapping with bounded distortion on a John domain of the Heisenberg group is approximated by a conformal mapping with order of closeness √ K − 1 in the uniform norm and with order of closeness K − 1 in the Sobolev L-norm for all p < C/ K−1. We construct two examples, demonstrating the asymptotic sharpness of our results. [ABSTRACT FROM AUTHOR]

We study the differentiability of mappings in the geometry of Carnot-Carathéodory spaces under the condition of minimal smoothness of vector fields. We introduce a new concept of hc-differentiability and prove the hc-differentiability of Lipschitz mappings of Carnot-Carathéodory spaces (a generalization of Rademacher’s theorem) and a generalization of Stepanov’s theorem. As a consequence, we obtain the hc-differentiability almost everywhere of the quasiconformal mappings of Carnot-Carathéodory spaces. We establish the hc-differentiability of rectifiable curves by way of proof. Moreover, the paper contains a new proof of the functorial property of the correspondence “a local basis ↦ the nilpotent tangent cone.” [ABSTRACT FROM AUTHOR]

This paper is the first of the author’s three articles on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness $$\sqrt {K - 1} $$ in the uniform norm and order of closeness K − 1 in the Sobolev norm L for all $$p < \tfrac{C}{{K - 1}}$$ . In the present article we study integrability of mappings with bounded specific oscillation on spaces of homogeneous type. As an example, we consider mappings with bounded distortion on the Heisenberg group. We prove that a mapping with bounded distortion belongs to the Sobolev class W , where p → ∞ as the distortion coefficient tends to 1. [ABSTRACT FROM AUTHOR]

In the class of univalent bounded normalized holomorphic functions in the unit disk, we give an asymptotic estimate for the coefficients when the uniform norm of the modulus of the function tends to infinity. [ABSTRACT FROM AUTHOR]

The concept of infinitesimal optimality is specified for discrete dynamic extremal problems. [ABSTRACT FROM AUTHOR]