Том 57, № 5 (2016)
- Год: 2016
- Статей: 22
- URL: https://journals.rcsi.science/0037-4466/issue/view/10386
Article
Quasiconformality of the injective mappings transforming spheres to quasispheres
Аннотация
We prove that every injective mapping of a domain \(D \subset \overline {{\mathbb{R}^n}} \) transforming spheres Σ ⊂ D to K-quasispheres (the images of spheres under K-quasiconformal automorphisms of \(\overline {{\mathbb{R}^n}} \)) is K′-quasiconformal with K′ depending only on K and tending to 1 as K → 1. This is a quasiconformal analog of the classical Carathéodory Theorem on the Möbius property of an injective mapping of a domain D ⊂ Rn which sends spheres to spheres.
Theorems of comparison and stability with probability 1 for one-dimensional stochastic differential equations
Аннотация
We prove the comparison theorems for scalar stochastic differential equations in the case of different diffusion coefficients. Conditions are given of stability with probability 1 with respect to the trivial solution to stochastic differential equations with random coefficients. The results remain valid for deterministic analogs of stochastic differential equations with symmetric integrals.
Describing 4-paths in 3-polytopes with minimum degree 5
Аннотация
Back in 1922, Franklin proved that each 3-polytope with minimum degree 5 has a 5-vertex adjacent to two vertices of degree at most 6, which is tight. This result has been extended and refined in several directions. In particular, Jendrol’ and Madaras (1996) ensured a 4-path with the degree-sum at most 23. The purpose of this note is to prove that each 3-polytope with minimum degree 5 has a (6, 5, 6, 6)-path or (5, 5, 5, 7)-path, which is tight and refines both above mentioned results.
Lower semicontinuity of mappings with bounded (θ, 1)-weighted (p, q)-distortion
Аннотация
We prove that, under some extra conditions, the locally uniform limit of mappings with bounded (θ, 1)-weighted (p, q)-distortion is a mapping of bounded (θ, 1)-weighted (p, q)-distortion too. Moreover, we obtain the lower semicontinuity of the distortion coefficients.
On existence of a universal function for Lp[0, 1] with p∈(0, 1)
Аннотация
We show that, for every number p ∈ (0, 1), there is g ∈ L1[0, 1] (a universal function) that has monotone coefficients ck(g) and the Fourier–Walsh series convergent to g (in the norm of L1[0, 1]) such that, for every f ∈ Lp[0, 1], there are numbers δk = ±1, 0 and an increasing sequence of positive integers Nq such that the series ∑ k=0+∞δkck(g)Wk (with {Wk} theWalsh system) and the subsequence \(\sigma _{{N_q}}^{\left( \alpha \right)}\), α ∈ (−1, 0), of its Cesáro means converge to f in the metric of Lp[0, 1].
Commuting Krichever–Novikov differential operators with polynomial coefficients
Аннотация
Under study are some commuting rank 2 differential operators with polynomial coefficients. We prove that, for every spectral curve of the form w2 = z3+c2z2+c1z+c0 with arbitrary coefficients ci, there exist commuting nonselfadjoint operators of orders 4 and 6 with polynomial coefficients of arbitrary degree.
Some pathological examples of solutions to a Beltrami equation
Аннотация
We construct an example of a bounded solution to a uniformly elliptic Beltrami equation that has no nontangential limit values almost everywhere on the boundary of the unit disk and also an example of a solution to such an equation that is not identically zero and has zero nontangential limit values almost everywhere on the boundary of the unit disk. These examples show that, in the general case of the Hardy spaces of solutions to a uniformly elliptic Beltrami equation (and to more general noncanonical first-order elliptic systems), the usual statement of boundary value problems used for holomorphic and generalized analytic functions is ill-posed.
Subcomplex and sub-Kähler structures
Аннотация
We introduce the notion of subcomplex structure on a manifold of arbitrary real dimension and consider some important particular cases of pseudocomplex structures: pseudotwistor, affinor, and sub-Kähler structures. It is shown how subtwistor and affinor structures can give sub-Riemannian and sub-Kähler structures. We also prove that all classical structures (twistor, Kähler, and almost contact metric structures) are particular cases of subcomplex structures. The theory is based on the use of a degenerate 1-form or a 2-form with radical of arbitrary dimension.
The structure of slices over minimal logic
Аннотация
In [1], we introduced a classification of extensions of Johansson’s minimal logic J by means of slices and proved the decidability of the classification. In this article, we find sufficiently simple necessary conditions for the maximality of logics in the slices formulated in terms of frames. This makes it possible to describe an efficient procedure for computing the slice number of any finitely axiomatizable logic over J. The maximal logics of the upper slices are written down explicitly.
Holomorphic factorization of polynomials
Аннотация
We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial P(s, λ), s ∈ Cn, λ ∈ C, in a domain Ω ⊂ Cn which is connected with the ordering of the real part of the roots of the equation P(s, λ) = 0, s ∈ Ω.
On some classes of inverse problems with overdetermination data on spatial manifolds
Аннотация
We consider the question of well-posedness of the inverse problems of determining the righthand side (a source function) of a special form of a parabolic system of equations. The overdetermination data are the values of a solution and its normal derivatives on a system of surfaces in a spatial domain. In particular, the cross-sections of the domain can be used as these surfaces. Sharp conditions are presented for the data of the problem to ensure well-posedness.
Existence of radially symmetric solutions of the inhomogeneous p-Laplace equation
Аннотация
We consider the Dirichlet problem for the inhomogeneous p-Laplace equation with p nonlinear source. New sufficient conditions are established for the existence of weak bounded radially symmetric solutions as well as a priori estimates of solution and of the gradient of solution. We obtain an explicit formula that shows the dependence of the existence of these solutions on the dimension of the problem, the size of the domain, the exponent p, the nonlinear source, and the exterior mass forces.
On fully quasitransitive abelian groups
Аннотация
We describe fully quasitransitive torsion-free groups in the class of groups whose endomorphism ring is a skew field as well as in the class of groups that are direct sums of homogeneous groups. We prove the full transitivity of fully quasitransitive cohesive groups and the quasitransitive torsion-free groups coinciding with their pseudosocle and having p-rank ≤ 1 for each prime p.