


Vol 52, No 1 (2018)
- Year: 2018
- Articles: 13
- URL: https://journal-vniispk.ru/0016-2663/issue/view/14575
Article
Duhamel Algebras and Applications
Abstract
We introduce Duhamel algebras and study their properties and applications. We prove that a Banach space of analytic functions on the unit disc that satisfy certain conditions is a Duhamel algebra and describe its closed ideals. These results substantially generalize and improve the main results of Wigley’s papers. Some other related questions are also discussed.



Results on the Colombeau Products of the Distribution x+−r−1/2 with the Distributions x−−k−1/2 and x−k−1/2
Abstract
Results on the products of the distribution x+−r−1/2 with the distributions x−−k−1/2 and x−k−1/2 are obtained in the differential algebra G(ℝ) of Colombeau generalized functions, which contains the space D′(ℝ) of Schwartz distributions as a subspace; in this algebra the notion of association is defined, which is a faithful generalization of weak equality in G(ℝ). This enables treating the results in terms of distributions again.



On the Distribution of Zero Sets of Holomorphic Functions
Abstract
Let M be a subharmonic function with Riesz measure νM in a domain D in the n-dimensional complex Euclidean space ℂn, and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ⊂ D, and satisfies |f| ⩽ expM on D. Then restrictions on the growth of νM near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.



Summation of Unordered Arrays
Abstract
An approach to the summation of unordered number and matrix arrays based on ordering them by absolute value (greedy summation) is proposed. Theorems on products of greedy sums are proved. A relationship between the theory of greedy summation and the theory of generalized Dirichlet series is revealed. The notion of asymptotic Dirichlet series is considered.



Brief Communications



Restricted Lie (Super)Algebras in Characteristic 3
Abstract
We give explicit formulas proving that the following Lie (super)algebras are restricted: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, simple subquotients over an algebraically closed field of characteristic 3 of these (super)algebras (under certain conditions), and deformed divergence-free Lie superalgebras of a certain type with any finite number of indeterminates in any characteristic.



Invariant Subspaces for Commuting Operators on a Real Banach Space
Abstract
It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator T ∈ A satisfies the condition



On Extrapolation Properties of Schatten–von Neumann Classes
Abstract
For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators.



On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure
Abstract
It is shown that, for any compact set K ⊂ ℝn (n ⩾ 2) of positive Lebesgue measure and any bounded domain G ⊃ K, there exists a function in the Hölder class C1,1(G) that is a solution of the minimal surface equation in G \ K and cannot be extended from G \ K to G as a solution of this equation.



Essential Spectrum of Schrödinger Operators on Periodic Graphs
Abstract
We give a description of the essential spectra of unbounded operators ℋq on L2(Γ) determined by the Schrödinger operators −d2/dx2 + q(x) on the edges of Γ and general vertex conditions. We introduce a set of limit operators of ℋq such that the essential spectrum of ℋq is the union of the spectra of limit operators. We apply this result to describe the essential spectra of the operators ℋq with periodic potentials perturbed by terms slowly oscillating at infinity.



On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Arithmetically Self-Similar Weight of a Generalized Cantor Type
Abstract
Spectral asymptotics of the Sturm–Liouville problem with an arithmetically self-similar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity property, which imposes significant restrictions on the self-similarity parameters of the weight. This work introduces a new method for estimating the eigenvalue counting function. This makes it possible to consider a much wider class of self-similar measures.



Dichotomy of Iterated Means for Nonlinear Operators
Abstract
In this paper, we discuss a dichotomy of iterated means of nonlinear operators acting on a compact convex subset of a finite-dimensional real Banach space. As an application, we study the mean ergodicity of nonhomogeneous Markov chains.



Monodromization and Difference Equations with Meromorphic Periodic Coefficients
Abstract
We consider a system of two first-order difference equations in the complex plane. We assume that the matrix of the system is a 1-periodic meromorphic function having two simple poles per period and bounded as Im z → ±∞. We prove the existence and uniqueness of minimal meromorphic solutions, i.e., solutions having simultaneously a minimal set of poles and minimal possible growth as Im z → ±∞. We consider the monodromy matrix representing the shift-byperiod operator in the space of meromorphic solutions and corresponding to a basis built of two minimal solutions. We check that it has the same functional structure as the matrix of the initial system of equations and, in particular, is a meromorphic periodic function with two simple poles per period. This implies that the initial equation is invariant with respect to the monodromization procedure, that is, a natural renormalization procedure arising when trying to extend the Floquet–Bloch theory to difference equations defined on the real line or complex plane and having periodic coefficients. Our initial system itself arises after one renormalization of a self-adjoint difference Schrödinger equation with 1-periodic meromorphic potential bounded at ±i∞ and having two poles per period.


