Том 58, № 3 (2017)
- Жылы: 2017
- Мақалалар: 18
- URL: https://journals.rcsi.science/0037-4466/issue/view/10421
Article
Quasiconformal extension of quasimöbius mappings of Jordan domains
Аннотация
We introduce the new class of Jordan arcs (curves) of bounded rotation which includes all arcs (curves) of bounded turning. We prove that if the boundary of a Jordan domain has bounded rotation everywhere but possibly one singular point then every quasimöbius embedding of this domain extends to a quasiconformal automorphism of the entire plane.
Spherical cubature formulas in Sobolev spaces
Аннотация
We study sequences of cubature formulas on the unit sphere in a multidimensional Euclidean space. The grids for the cubature formulas under consideration embed in each other consecutively, forming in the limit a dense subset on the initial sphere. As the domain of cubature formulas, i.e. as the class of integrands, we take spherical Sobolev spaces. These spaces may have fractional smoothness. We prove that, among all possible spherical cubature formulas with given grid, there exists and is unique a formula with the least norm of the error, an optimal formula. The weights of the optimal cubature formula are shown to be solutions to a special nondegenerate system of linear equations. We prove that the errors of cubature formulas tend to zero as the number of nodes grows indefinitely.
The removability problem for functions with zero spherical means
Аннотация
We study the functions on the punctured n-dimensional sphere having zero integrals over all admissible “hemispheres.” We find a condition for the point to be a removable set for this class of functions and show that the condition cannot be dropped or substantially improved.
The problem of determining the one-dimensional kernel of the electroviscoelasticity equation
Аннотация
We consider the problem of finding the kernel K(t), for t ∈ [0, T], in the integrodifferential system of electroviscoelasticity. We assume that the coefficients depend only on one spatial variable. Replacing the inverse problem with an equivalent system of integral equations, we apply the contraction mapping principle in the space of continuous functions with weighted norms. We prove a global unique solvability theorem and obtain a stability estimate for the solution to the inverse problem.
Integral representation and embedding theorems for n-dimensional multianisotropic spaces with one anisotropic vertex
Аннотация
We prove embedding theorems for the multianisotropic Sobolev spaces generated by the completely regular Newton polyhedron. Under study is the case of the polyhedron with one anisotropic vertex. We obtain a special integral representation of functions in terms of the tuple of multi-indices of the Newton polyhedron.
Simple finite-dimensional algebras without finite basis of identities
Аннотация
In 1993, Shestakov posed a problem of existence of a central simple finite-dimensional algebra over a field of characteristic 0 whose identities cannot be defined by a finite set (Dniester Notebook, Problem 3.103). In 2012, Isaev and the author constructed an example that gave a positive answer to this problem. In 2015, the author constructed an example of a central simple seven-dimensional commutative algebra without finite basis of identities. In this article we continue the study of Shestakov’s problem in the case of anticommutative algebras. We construct an example of a simple seven-dimensional anticommutative algebra over a field of characteristic 0 without finite basis of identities.
On the pronormality of subgroups of odd index in finite simple symplectic groups
Аннотация
A subgroup H of a group G is pronormal if the subgroups H and Hg are conjugate in 〈H,Hg〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSLn(q), PSUn(q), E6(q), 2E6(q), where in all cases q is odd and n is not a power of 2, and P Sp2n(q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp2n(q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.
The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp2n(q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2m or 2m(22k+1), this group has a nonpronormal subgroup of odd index. If n = 2m, then we show that all subgroups of P Sp2n(q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp2n(q) is still open when n = 2m(22k + 1) and q ≡ ±3 (mod 8).
C-distribution semigroups and C-ultradistribution semigroups in locally convex spaces
Аннотация
The main purpose of this paper is to study C-distribution semigroups and C-ultradistribution semigroups in the setting of sequentially complete locally convex spaces. We provide a few important theoretical novelties in this field and some interesting examples. Under consideration are stationary dense operators in a sequentially complete locally convex space.
A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations
Аннотация
We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.
Generalizations of the Wada representations and virtual link groups
Аннотация
We extend the Wada representations of the classical braid group to the virtual and welded braid groups. Using the resulting representations, we construct the groups of virtual links and prove that they are link invariants. We give some examples of calculating the groups of torus (virtual) links.
The extended large deviation principle for a process with independent increments
Аннотация
Considering a process with independent increments under the moment Cramér condition, we establish the extended large deviation principle in the space of functions without discontinuities of the second kind which is endowed with the Borovkov metric.
Some properties of solutions to a family of integral equations arising in the models of living systems
Аннотация
We consider the well-posedness problem of nonlinear integral and differential equations with delay which arises in the elaboration of mathematical models of living systems. The questions of existence, uniqueness, and nonnegativity of solutions to these systems on an infinite semiaxis are studied as well as continuous dependence of solutions on the initial data on finite time segments.
Algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups
Аннотация
We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UTn(Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UTn(Z) has no proper existentially closed subgroups.
The root class residuality of Baumslag–Solitar groups
Аннотация
Given a homomorphically closed root class K of groups, we find a criterion for a Baumslag–Solitar group to be a residually K-group. In particular, we establish that all Baumslag–Solitar groups are residually soluble and a Baumslag–Solitar group is residually finite soluble if and only if it is residually finite.
Minimal generalized computable enumerations and high degrees
Аннотация
We establish that the set of minimal generalized computable enumerations of every infinite family computable with respect to a high oracle is effectively infinite. We find some sufficient condition for enumerations of the infinite families computable with respect to high oracles under which there exist minimal generalized computable enumerations that are irreducible to the enumerations.