Siberian Mathematical Journal最新文献

This paper studies the basic topological properties of weighted composition operatorson the weighted sequence spaces ( l^{p}(operatorname{w}) ), with ( 0<p<infty ),given by a weight sequence ( operatorname{w} ) of positive realssuch as boundedness, compactness, compactness of differences of two operators, formulas for their essential norms,and description of closed range operators.Previously these properties were studied by Luan and Khoi in the case of Hilbert space ( (p=2) ).Their methods can be also applied with some minor modifications to the case of Banach spaces ( l^{p}(operatorname{w}) )with ( p>1 ).They based essentially on using the dual spaces of continuous linear functionals and, consequently,cannot be applied to the quasi-Banach case ( (0<p<1) ). Moreover, some of them do not work evenin ( l^{1}(operatorname{w}) ).Motivated by these reasons, we develop a more universal approach that allows studyingthe whole scale of spaces ( {l^{p}(operatorname{w}):p>0} ).To this end, we establish the necessary and sufficient conditions for a linear operator to be compacton an abstract quasi-Banach sequence space. These conditions are new even in the case of Banach spaces.Moreover, we introduce the new characteristic, the ( omega )-essential norm of a continuous linear operator ( L )on a quasi-Banach space ( X ).This characteristic measures the distance in the operator metric, between ( L ) and the set of all ( omega )-compact operators on ( X ).Here an operator ( K ) is ( omega )-compact on ( X ) if ( K ) is compact and coordinatewise continuous on ( X ).We show that for ( l^{p}(operatorname{w}) ) with ( p>1 ) the essential and ( omega )-essential normsof a weighted composition operator coincide, whereas for ( 0<pleq 1 ) we do not know whether the same is true or not.Our main results for weighted composition operators in ( l^{p}(operatorname{w}) ) ( (0<p<infty) ) are as follows:We provide criteria for an operator to be bounded, compact, or closed range, and completely describe the pairs of operatorswith compact difference; as well as some exact formula for the ( omega )-essential norm. Some key aspects ofour approach can be used for other operatorsand scales of spaces.

We consider the Cauchy problem for the modified Korteweg-de Vries equationwith time-dependent coefficients and a self-consistent source in the class of rapidly decreasing functions.To solve the problem, we find Lax pairs and employ the inverse scattering method.Note that in the case under study the Dirac operator is not selfadjoint, and so the eigenvalues can be multiples.We find the equations describing the dynamics in time of the scattering data of a nonselfadjoint Dirac operatorwhose potential is a solution to the modified Korteweg-de Vries equation with time-dependent coefficientsand a self-consistent source in the class of rapidly decreasing functions.As a special case,we examine a loaded modified Korteweg-de Vries equation with a self-consistent source.The equations describe the dynamics in time of the scattering data of a nonselfadjoint Dirac operatorwhose potential is a solution to the loaded modified Korteweg-de Vries equation with variable coefficientsin the class of rapidly decreasing functions.We provide some examples that illustrate applications of the results.

We solve the simultaneous recovery of thermal conductivity and a high-frequency coefficient of a sourcein a one-dimensional initial-boundary value problem for the heat equation withDirichlet boundary conditions and an inhomogeneous initial conditionfrom some information on the partial asymptotics of a solution. We showthat the coefficients can be restored from some data on the asymptotics of a solution,which is constructed and justified.This article was inspired by Denisov’s research on a variety of inverseproblems without accounting for high-frequency oscillations.Also, we continue the research by Levenshtam and his students which firstlyaddressed the inverse problems for parabolic equations with high-frequency coefficients and developed the relevantmethodology. In contrast tothe previous research of the case that only the source function or its factors are unknown,we assume that the thermal conductivity and the factor of a source functionare unknown simultaneously.

Let ( G ) be the group of all limited permutations of the set of naturals.We prove that every countable locally finite group is isomorphic to some regularsubgroup of ( G ). Also, if a regular subgroup ( H ) of ( G ) contains an elementof infinite order then ( H ) has a normal infinite cyclic subgroup of finite index.

We introduce the notion of ITBM-constructive algebra, which is a generalization of the notion of constructive algebra,and study completions of such algebras.We obtain some criterion for the existence of completions for metrized algebrasand prove that each ITBM-constructive metrized algebrawhich has completion can be naturally extended to the ITBM-constructivecompletion. Using these results, we establish the existence ofITBM-constructive presentations for some particular algebras.

We prove the existence of a triple ( ({mathfrak{X}},G,H) ), where ( {mathfrak{X}} )is a class of finite groups consisting of groups of odd order which is complete(i.e., closed under subgroups, homomorphic images, and extensions),( G ) is a finite simple group, ( H ) is an ( {mathfrak{X}} )-maximal subgroup in ( G ),and ( H ) is not pronormal in ( G ).

We present the following constructions: extension of a trivial quandleby a group with a nontrivial abelian subgroup,a quandle over a noncommutative groupby an arbitrary antiautomorphism,a quandle presenting from a generalized Alexander quandlewith the replacement of the automorphism by a central antiautomorphism, anda quandle over an ( n )-dimensional moduledepending on ( n(n-1)-1 )parameters with ( ngeq 2 ) in a commutative unital ring.

We construct some linear nonhomogeneous differential operator ( mathcal{Q} ) on the Heisenberg groupwhose kernel is interconnected with the Lie algebra of the group of conformal mappings.In more detail, the kernel of ( mathcal{Q} ) coincides with first two coordinate functions of mappings ofthe Lie algebra of conformal mappings.We derive the integral representation formula andgive a coercive estimate for ( mathcal{Q} ).

The equivalence of the Haar system in a rearrangementinvariant space ( X ) on ( [0,1] ) and a sequence of pairwise disjoint functionsin some Lorentz space is known to imply that ( X=L_{2}[0,1] ) up to the equivalence ofnorms. We show that the same holds for the class of uniformdisjointly homogeneous rearrangement invariant spaces and obtain a fewconsequences for the properties of isomorphic embeddings of such spaces.In particular, the ( L_{p}[0,1] ) space with ( 1<p<infty ) is theonly uniform ( p )-disjointly homogeneous rearrangement invariant space on ( [0,1] )with nontrivial Boyd indices which has two rearrangement invariant representationson the half-axis ( (0,infty) ).

This article addresses the problem of boundary correspondencefor a sequence of homeomorphisms thatchange the capacity of a condenser in a controlled way.To study the overall boundary behavior of these mappings,we introduce some capacity metricsin 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 forthe global uniform convergence of a sequence of homeomorphisms,but some applications to elasticity theory as well.