Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

On the word problem and the conjugacy problem for groups of the formF/V(R)

LetF be a free group with at most countable system of free generators, letR be its normal subgroup recursively enumerable with respect to , and let be a variety of groups that differs from and for which the corresponding verbal subgroupV of the free group of countable rank is recursive. It is proved that the word problem inF/V(R) is solvable if and only if this problem is solvable inF/R, and if , then there exists anR such, that the conjugacy problem inF/R is solvable, but this problem is unsolvable inF/V(R) for any Abelian variety (all algorithmic problems are regarded with respect to the images of under the corresponding natural epimorphisms). Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 3–9, January, 1997. Translated by M. I. Anokhin     

Sharp Jackson-Stechkin inequality inL2 for multidimensional spheres

In this paper we prove the Jackson-Stechkin inequalityE _n−1(f)<ω _n (f, 2τ _n ,λ),n≥1,m≥5,r≥1, f ∈L2( ),f ≢ const, which is sharp for eachn=2, 3, ...; hereE _n−1 (f) is the best approximation of a functionf by spherical polynomials of degree ≤n−1, ω _n (f, τ) is theτth modulus of continuity off based on the translations ,t ∈ ℝ,x ∈ , , is the measure of the unit Euclidean sphere , , andτ _n ,λ is the first positive zero of the Gegenbauer cosine polynomial (cost). Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 333–355, September, 1996. The present paper was discussed at Ural State University in a seminar headed by Professor Arestov. The author is grateful to Professor Arestov and Associate Professor Popov for useful conversations. This research was supported by the State Commission for Higher Education of the Russian Federation under grant No. 2-16-5-31 and by the Russian Foundation for Basic Research under grant No. 93-011-196.     

Estimate of fourier transforms with respect to the system of generalized eigenfunctions of the Schrödinger operator with Stummel-type potential

Suppose thatА is a nonnegative self-adjoint extension to { } of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

Perturbations of an abstract Euler-Poisson-Darboux equation

The stability of the uniform correctness of the Cauchy problem ,t>0,u(0)=u ₀,u′(0)=0 fork>0 with respect to perturbations of the operator is studied. Translated fromMatematickeskie Zameiki, Vol. 60, No. 3, pp. 363–369, September, 1996.     

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

A boundary value problem for Poisson’s two-media equation inL p /2 -spaces

In the paper we study a binding boundary value problem for two media for Poisson's equation μΔu=f(x) with solutions in the class , 1<p<∞, with the corresponding seminorm, where

A relative cohomological invariant for group pairs

We define a cohomological invariantE(G, S, M) whereG is a group,S is a non empty family of (not necessarily distinct) subgroups of infinite index inG andM is a -module ( is the field of two elements). In this paper we are interested in the special case where the family of subgroups consists of just one subgroup, andM is the -module . The invariant will be denoted byE(G, S). We study the relations of this invariant with other endse(G), e(G, S) ande(G,S)), and some results are obtained in the case whereG andS have certain properties of duality.     

The theory of multi-dimensional polynomial approximation

We consider the problem of polynomial approximation to a real valued functionf defined on a compact set . An approximation theorem is proven in terms of the newly defined modulus of approximation. It is shown to imply a multidimensional Jackson type theorem which is stronger than previously known results even for the interval [−1, 1]. A strong multidimensional Bernstein type inverse theorem is also proven. We allow quite general approximation quasi-norms including for 0<q≤∞. We have found that the space of polynomials ℙ on a compact setX induces a semimetric which encapsulates the local structure of ℙ. Any semimetric ρ equivalent to suffices for the rough theory presented here. Many examples of sets and their metrics are presented.     

Undecidability of the word problem in relatively free rings

An example of a series of varieties of rings with the finite basis property is constructed for which the word problem in the relatively free ring of rankn in the variety is decidable if and only ifn <p. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 582–594, April, 2000.     

Elementary properties of free extensions

For any partial groupoid , let Fr be the free extension of to a total groupoid. We show that implies and that the theory of Fr is uniformly recursive in the theory of . These results fail if “groupoid” is replaced by “semigroup”, “commutative semigroup”, “group”, “abelian group”, “semilattice”, “K-lattice” for any nontrivial varietyK of lattices, or “Boolean algebra”. Research supported in part by NSF Grant MCS78-01867. We thank the referee for his valuable comments. Presented by B. Jónsson.     

Factorizations of composition formations

It is proved that, if is a singly generated composition formation, where , then is a composition formation. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 389–395, March, 1999.     

On certain relations between the maximum modulus and the maximal term of an entire dirichlet series

For an entire Dirichlet series , sufficient conditions on the exponents are established such that the following relations hold outside a set of finite measure asx→+∞:

Sharpness of certain Campbell and Pommerenke estimates

The paper is concerned with the sharpness of some well-known estimates in universal linear-invariant families of regular functions. It is shown that the estimate of | arg ƒ′(z)|,z ∈ Δ = {z: |z| < 1} obtained by Pommerenke in 1964 is sharp; the extremal function is found. A lower estimate for the Schwarzian derivative in is obtained. For , a sharp estimate of order of the function ƒr(z)=ƒ(rz)/r withr ∈ (0, 1) is found; this estimate is applied to solve other problems. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 665–672, May, 1998.     

InfiniteB 2[g] sequences

We exhibit, for any integerg≥2, an infinite sequenceA ∈B ₂[g] such that . Furthermore, we obtain better estimates for small values ofg. For instance, we exhibit an infinite sequenceA ∈B ₂[2] such that Partially supported by Colciencias, Colombia and Universidad del Cauca.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

On residue complexes,dualizing sheaves and local cohomology modules

According to Grothendieck Duality Theory [RD], on each varietyV over a fieldk, there is a canonical complex of -modules, theresidue complex . These complexes satisfy (and are characterized by) functorial properties in the categoryV ofk-varieties. In [Ye] a complex is constructed explicitly (when the fieldk is perfect). The main result of this paper is that the two families of complexes, and , which carry certain additional data (such as trace maps…), are uniquely isomorphic. As a corollary we recover Lipman’s canonical dualizing sheaf of [Li], and we obtain formulas for residues of local cohomology classes of differential forms.     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means

Suppose that a lower triangular matrix μ:[μ _m ⁽ⁿ⁾ ] defines a conservative summation method for series, i.e.,