Vol 38, No 2 (2017)
- Year: 2017
- Articles: 21
- URL: https://journals.rcsi.science/1995-0802/issue/view/12425
Article
Finite mean oscillation in upper regular metric spaces
Abstract
There are established some sufficient conditions for boundary homeomorphic extension in metric spaces in which the measure of a ball of radius ε is controlled from above by a wide class of functions depending on ε. We consider a class of mappings whose ring moduli are integrally majorated. These results involve a finite mean oscillation and asymptotic estimation of majorants.
Group actions, Teichmüller spaces and cobordisms
Abstract
We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism M whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary ∂M. In addition to the standard conformal ergodic action of a uniformhyperbolic lattice on the round sphere Sn−1 and its quasiconformal deformations in Sn, we present several constructions of unusual actions of such lattices on everywhere wild spheres (boundaries of quasisymmetric embeddings of the closed n-ball into Sn), on non-trivial (n − 1)-knots in Sn+1, as well as actions defining non-trivial compact cobordisms with complete hyperbolic structures in its interiors. We show that such unusual actions always correspond to discrete representations of a given hyperbolic lattice from “non-standard” components of its varieties of representations (faithful or with large kernels of defining homomorphisms).
Nonlinear matrix differential-algebraic boundary value problem
Abstract
We set forth solvability conditions and construction of the generalized Green’s operator for Noetherian boundary value problem for the matrix differential-algebraic equations and solvability conditions and the constructive scheme for constructing solutions of nonlinearNoetherian boundary value problem for matrix differential-algebraic equation. We show that the principal results in the theory of weakly nonlinear periodic oscillations remain valid for nonlinear Noetherian boundary value problem for matrix differential-algebraic equation. The study is illustrated by the periodic problem for a Duffing-type matrix differential equation.
On quasinearly subharmonic functions
Abstract
We recall the definition of quasinearly subharmonic functions, point out that this function class includes, among others, subharmonic functions, quasisubharmonic functions, nearly subharmonic functions and essentially almost subharmonic functions. It is shown that the sum of two quasinearly subharmonic functions may not be quasinearly subharmonic. Moreover, we characterize the harmonicity via quasinearly subharmonicity.
A theorem on generalized nonions and their properties for the applied structures in physics
Abstract
The central part of the paper consists of a theorem on generalized nonions governing dynamical systems modelling of special ternary, quaternary, quinary, senary, etc. structures, due to the third named author. Let Mn(C), n ≥ 2, be the set of n × n-matrices with complex entries. The theorem states that in Mn(C) there exists a basis such that PQ− λsQP = 0, s = 0, 1, 2,.., n − 1, where {P,Q,}, u, v are specified in Section 1, formulae (1) and (2).The particular cases n = 2, 3, 4 with other choices of u, v were discussed by James Joseph Sylvester (1883, 1884) and by Charles Sanders Peirce (1882).In particular, λ = j, j3 = 1, j ≠ 1, generates nonions. Before the section on the above theorem and its visualization on a two-sheeted Riemann surface, we give three physical motivations for the topic: controlled noncommutativity: Sylvester–Peirce approach vs. Max Planck approach (1900), supersonic flow of a ternary alloy in gas, and changing hexagonal to pentagonal structure in pentacene.
Weak regularity of degenerate elliptic equations
Abstract
Let φ: Ω → D be a conformal mapping of a bounded simply connected planar domain Ω onto the unit disc D ⊂ ℝ2. We prove existence and uniqueness in Ω of weak solutions of a degenerate Poisson equation for a hyperbolic weight h(z) = |φz′|2 in a corresponding two weighted Sobolev space W21 (Ω, h, 1).Here φz′ is a complex derivative. We also study weak regularity of the solutions in conformal regular domains. The domain Ω is a conformal regular domain [4] if (φ−1)w′ ∈ Lα(D) for some α > 2.
The domain of existence of a limit cycle of Liénard system
Abstract
In this paper, the Liénard system of differential equations is considered. The domain of location of the limit cycle of this system is estimated. The hypothesis under which the Liénard system has the unique stable limit cycle are also stated.
On unique determination of conformal type for domains in Euclidean spaces
Abstract
This survey is devoted to discussion of problems of the unique determination of conformal type for domains in Euclidean spaces. It consists of three parts. In the first of them, we consider results on the problem of the unique determination for (generally speaking) nonconvex domains in Rn, where n ≥ 4. The second part is devoted to study of problems on unique determination of convex polyhedral domains in three-dimensional Euclidean space by relative conformal moduli of boundary condensers. Finally, in the third part, we study problems of unique determination of 3-connected plane domains by relative conformal moduli of pairs of boundary components.
On the boundary behavior of mappings with finite distortion in the plane
Abstract
In terms of prime ends by Caratheodory, it is studied the boundary behavior of the socalled lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec as well as to finitely bi–Lipschitz mappings that a far–reaching extension of the known classes of isometric and quasiisometric mappings.
Convexity and Teichmüller spaces
Abstract
We provide a negative answer to Royden’s problem whether any finite dimensional Teichmüller space of dimension greater than 1 is biholomorhically equivalent to a bounded convex domain in complex Euclidean space.
We prove that any Teichmüller space T(0, n) of punctured spheres with a sufficiently large number n ≥ n0 > 4 of punctures cannot be mapped biholomorphically onto a bounded convex domain in ℂn−3.
Generic existence of nondegenerate homoclinic solutions
Abstract
The paper focuses on the homoclinic solutions of a general second order Hamiltonian system. By applying an abstract parametric transversality result, it is shown that generically the problem admits finitely many homoclinic solutions. These solutions are nondegenerate in the sense that they correspond to nondegenerate critical points of the associated functional and depend smoothly on the vector field.
An extension of monogenic functions and spatial potentials
Abstract
We obtain explicitly principal extensions of analytic functions of the complex variable into an infinite-dimensional commutative Banach algebra associated with the three-dimensional Laplace equation. We consider an extension of differentiable in the sense of Gâteaux functions with values in a topological vector space being an expansion of the mentioned algebra and its relations to spatial potentials.
Moduli, capacity, BV-functions on the Riemann surfaces
Abstract
Let R is a Riemann surface, glued from finitely or countably many domains in the extended complex plane so that the following conditions are satisfied: each point in R projects onto a point w = prW in one on the glued domains, each point in R has a neighbourhood which is a univalent disk, or multivalent disk with the unique ramification point at the centre of disk. We study elementary properties of functions of bounded variation and sets of finite perimeter in an open set Q ⊂ R {W ∈ R: W is a ramification point or prW = ∞}. Further, by using Ziemer’s technique, we obtain the main result
String theory and quasiconformal maps
Abstract
The phase space of the closed string theory may be identified with the space of smooth loops. This reduces the problem of quantization of string theory to the quantization of the space of smooth loops. In this paper we describe the solution of the latter problem obtained in a series of papers. But the symplectic form of string theory is correctly defined not only on the space of smooth loops but also on its Hilbert completion coinciding with the Sobolev space of half-differentiable functions. So it is reasonable to consider this space as the phase manifold of non-smooth string theory. There is a natural group associated with this Sobolev space, namely the group of quasisymmetric homeomorphisms of the circle acting by change of variable. Unfortunately, this action is not smooth. However, we are able to quantize the Sobolev space of half-differentiable functions provided with the action of the group of quasisymmetric homeomorphisms using methods of noncommutative geometry.
Some generalizations of the class of functions convex in one direction
Abstract
This paper concerns the extension of univalent conditions known for a ring domain to the case of a half-plane. We study the possibilities of applying Rahman and Brickman methods to functions analytic in the upper half-plane. Some generalizations of the functions convex in one direction are obtained. We also investigate the curvelinear half-planes, the boundary of which can be attained by a family of shift curves, and demonstrate the efficiency of the quasiconformal extension method. The subclasses of domains, whose boundaries are quasiconformal curves, are identified. Some sufficient conditions for univalence of functions analytic in these domains are established.
On local behavior of mappings with unbounded characteristic
Abstract
This article is a survey of the earlier results of the author. We study space mappings with branching that satisfy modulus inequalities. For classes of these mappings, we obtain several sufficient conditions for the normality of families. Moreover, we prove that the normal families of the so-called Q-mappings have the logarithmic order of growth in a neighborhood of a point. We obtain a result on the normal families of open discrete mappings f: D → C {a, b} from the class Wloc1,1 with finite distortion that do not take at least two fixed values a ≠ b in C whose maximal dilatation has a majorant of finite mean oscillation at every point. This result is an analog of the well-known Montel theorem for analytic functions.
A uniqueness theorem for the non-Euclidean Darboux equation
Abstract
A non-Euclidean analog of the generalized Darboux equation is considered. For the case where its solutions are radial functions of second variable we obtain a uniqueness result (Theorem 1) which deals with zero sets of these solutions. The example of the function in Theorem 2 of the paper shows that Theorem 1 cannot be essentially reinforced.
On positive definiteness of some radial functions
Abstract
We consider the functions hμ,ν introduced by Zastavnyi in 2002. The family of these functions is a subfamily of Buhmann’s functions and contains the families of functions introduced by Trigub in 1987 and Wendland in 1995. We investigate the problems of positive definiteness and smoothness at zero for the linear combinations β2εhμ,ν (x/β2) − β1εμ,ν (x/β1).