Vol 225, No 6 (2017)
- Year: 2017
- Articles: 14
- URL: https://journals.rcsi.science/1072-3374/issue/view/14854
Article
Interaction of the Hecke–Shimura Rings and Zeta Functions
Abstract
An automorphic structure on a Lie group consists of the Hecke–Shimura ring of an arithmetic discrete subgroup and a linear representation of the ring on an invariant space of automorphic forms given by Hecke operators. The paper is devoted to interactions (transfer homomorphisms) of the Hecke–Shimura rings of integral symplectic groups and integral orthogonal groups of integral positive definite quadratic forms. Bibliography: 10 titles.
Two-Sided Estimates for Some Functionals in Terms of the Best Approximations
Abstract
Let C be the space of continuous 2π-periodic functions. For some integrals of the form
where ωr(f, t) is the modulus of continuity of order r of a function f in C, two-sided bounds in terms of the best approximations by trigonometric polynomials are established.
Critical Values and Moduli of the Derivative of a Complex Polynomial at its Zeros
Abstract
Under some constraints on the critical values of an algebraic polynomial with complex coefficients, a sharp inequality for the product of certain powers of moduli of its derivative at its zeros is established. The equality is attained for a suitable Chebyshev polynomial of the first kind.
Equality of the Capacity and Module of a Condenser on a Sub-Finsler Space
Abstract
In this paper, the capacity and module of a condenser, and also some functional classes on a sub-Finsler space are defined. Their general properties are studied, and the equality of the capacity and module of a condenser is proved.
Simplex-Module Algorithm for Expansion of Algebraic Numbers in Multidimensional Continued Fractions
Abstract
The simplex-module algorithm for expansion of algebraic numbers α = (α1, . . . , αd) in multidimensional continued fractions is suggested. The method is based on minimal rational simplices s, where α ∈ s, and Pisot matrices Pα, for which \( \widehat{\alpha} \) = (α1, . . . , αd, 1) is an eigenvector. A multidimensional generalization of the Lagrange theorem is proved.
ON the p-Harmonic Robin Radius in the Euclidean Space
Abstract
For p > 1, the notion of the p-harmonic Robin radius of a domain in the space ℝn, n ≥ 2, is introduced. In the case where the corresponding part of the boundary degenerates, the Robin–Neumann radius is considered. The monotonicity of the p-harmonic Robin radius under some deformations of a domain is proved. Some extremal decomposition problems in the Euclidean space are solved. The definitions and proofs are based on the technique of moduli of curve families. Bibliography: 23 titles.
On an Extremal Metric Approach to Extremal Decomposition Problems
Abstract
Applications of the method of modules of curve families to extremal problems of geometric function theory are presented. For these problems, the associated quadratic differentials have a large number of free poles. Bibliography: 13 titles.
Condensers and Equivalent Open Sets on a Riemann Surface
Abstract
Condensers on the compact closure of an open set on a Riemann surface are studied. The equality of the condenser capacity and module is proved. The definition of NED-sets on a Riemann surface is given, and it is proved that the NED-sets do not affect the condenser module. Also a criterion of equivalence of open sets on a Riemann surface is established.
Lattice Points in Many-Dimensional Balls
Abstract
Let Pk(n) be the difference between the number of points of the integer lattice contained in the ball \( {y}_1^2+\cdots {y}_k^2\le n \) and the volume of this ball. The paper investigates the asymptotic behavior of
the sums
Modules of Space Configurations and Removable Sets
Abstract
The sufficiency of the family of broken lines in computing the module of a configuration is established. Also it is proved that the sets that are removable for the condenser module are also removable for the configuration module.