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Том 51, № 3 (2017)
- Год: 2017
- Статей: 10
- URL: https://journals.rcsi.science/0016-2663/issue/view/14571
Article
In Memoriam
161-161
The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications
Аннотация
The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 is described in terms of the gradient of its sigma function. As an application, solutions of the corresponding families of polynomial dynamical systems in C4 with two polynomial integrals are constructed. These systems were introduced by Buchstaber and Mikhailov on the basis of commuting vector fields on the symmetric square of algebraic curves.
162-176
On equivariant indices of 1-forms on varieties
Аннотация
Given a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group), its equivariant homological index and (reduced) equivariant radial index are defined as elements of the ring of complex representations of the group. We show that these indices coincide on a germ of a smooth complex analytic G-variety. This makes it possible to consider the difference between them as a version of the equivariant Milnor number of a germ of a G-variety with an isolated singular point.
177-184
Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions
Аннотация
In this paper we prove that for an arbitrary pair {T1, T0} of contractions on Hilbert space with trace class difference, there exists a function ξ in L1(T) (called a spectral shift function for the pair {T1, T0}) such that the trace formula trace(f(T1) − f(T0)) = ∫Tf′(ζ)ξ(ζ)dζ holds for an arbitrary operator Lipschitz function f analytic in the unit disk.
185-203
An analogue of the big q-Jacobi polynomials in the algebra of symmetric functions
Аннотация
It is well known how to construct a system of symmetric orthogonal polynomials in an arbitrary finite number of variables from an arbitrary system of orthogonal polynomials on the real line. In the special case of the big q-Jacobi polynomials, the number of variables can be made infinite. As a result, in the algebra of symmetric functions, there arises an inhomogeneous basis whose elements are orthogonal with respect to some probability measure. This measure is defined on a certain space of infinite point configurations and hence determines a random point process.
204-220
Brief Communications
221-224
Trace formulas for a discrete Schrödinger operator
Аннотация
The Schrödinger operator with complex decaying potential on a lattice is considered. Trace formulas are derived on the basis of classical results of complex analysis. These formulas are applied to obtain global estimates of all zeros of the Fredholm determinant in terms of the potential.
225-229
Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients
Аннотация
Let O ⊂ Rd be a bounded domain of class C1,1. Let 0 < ε - 1. In L2(O;Cn) we consider a positive definite strongly elliptic second-order operator BD,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (BD,ε − ζQ0(·/ε))−1 as ε → 0. Here the matrix-valued function Q0 is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L2(O;Cn)-operator norm and in the norm of operators acting from L2(O;Cn) to the Sobolev space H1(O;Cn) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q0(x/ε)∂tvε(x, t) = −(BD,εvε)(x, t).
230-235
Systems of dilated functions: Completeness, minimality, basisness
Аннотация
The completeness, minimality, and basis property in L2[0, π] and Lp[0, π], p ≠ 2, are considered for systems of dilated functions un(x) = S(nx), n ∈ N, where S is the trigonometric polynomial S(x) = Σk=0maksin(kx), a0am ≠ 0. A series of results are presented and several unanswered questions are mentioned.
236-239
Reduced synthesis in harmonic analysis and compact synthesis in operator theory
Аннотация
The notion of reduced synthesis in the context of harmonic analysis on general locally compact groups is introduced; in the classical situation of commutative groups, this notion means that a function f in the Fourier algebra is annihilated by any pseudofunction supported on f−1(0). A relationship between reduced synthesis and compact synthesis (i.e., the possibility of approximating compact operators by pseudointegral ones without increasing the support) is determined, which makes it possible to obtain new results both in operator theory and in harmonic analysis. Applications to the theory of linear operator equations are also given.
240-243