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 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.     

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.     

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.     

An operator model for the oscillation problem of liquids on an elastic bottom

This paper deals with the problem of small oscillations in a liquid layer of finite depth under the assumption that the bottom is an elastic medium. The system of equations corresponding to the problem is written out and explained. The main aim of the paper is to recast these equations in the form

The Hopf algebra Rep U_{q} \widehat{\frak g \frak l}_\infty

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of in the limitN→∞. The resulting Hopf algebra Rep is a tensor product of its Hopf subalgebras Rep_a ,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a and the algebra of regular functions on the prounipotent proalgebraic group (resp., ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep _a and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.     

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.     

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let be the category of left unitary Λ-modules. A subcateqory of the category is said to be small if the pairwise nonisomorphic objects of form a set. The main result of this paper consists of the fact that for every small full subcategory , there exists a ring Γ such that is dual to a small full subcategory of the category . Some applications of this result are indicated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 66–73.     

Functional Identities on Upper Triangular Matrix Algebras

Let , be the algebra of upper triangular r×r matrices over field aF. We describe multilinear maps , where n≤r, such that [f(A,...,A), A]=0 for all , provided that |F| >n + 1. As an application of the results obtained, we describe linear automorphisms θ of such that [θ(A²), θ(A)]=0 for all . Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

An asymptotic method for calculating limits of maximal means

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

Estimates of the independence number of a hypergraph and the Ryser conjecture

For the Legendre-Sobolev orthonormal polynomials depending on the parametersM≥0,N≥0, weighted and uniform estimates on the orthogonality interval are obtained. It is shown that, unlike the Legendre orthonormal polynomials, the polynomials forM>0,N>0 decay at the rate ofn ^−3/2 at the points 1 and −1. The values of are calculated. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 871–880, December, 1997. Translated by N. K. Kulman     

Almost semirecursive sets

LetA be a subset of , and leta∉A. The setA is said to be almost semirecursive, if there is a two-place general recursive functionf such thatf(x, y)ε{x, y, a}∧({x, y}⊆A⇌f(x, y)εA) for all . Among other facts, it is proved that ifA and are almost semirecursive sets, thenA is a semirecursive set, and that there exists a wsr^*-set that is neither a wsr-nor an almost semirecursive set. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 188–193, August, 1999.     

Lie algebras determined by finite valued Auslander-Reiten quivers

Let г denote a connected valued Auslander-Reiten quiver, let ℒ(γ) denote the free abelian group generated by the vertex setγ ₀ and let ℒ(Γ) be the universal cover ofг with fundamental groupG. It is proved that whenγ is a finite connected valued Auslander-Reiten quiver,ℒ(γ) is a Lie subalgebra ofℋ(г), and is just the “orbit” Lie algebra ℒ( )/G, where ℋ (г)₁ is the degenerate Hall algebra ofг and ℒ( )/G is the “orbit” Lie algebra induced by .     

Analytic solutions of the Borel problem

LetF ⊂ ℂ be a dense-in-itself set that has a nonempty connected interior and contains the origin, and let be the space of infinitely differentiable complex-valued functions onF. For some classes of such setsF, we prove that for an arbitrary sequence of complex numbers there exists a functionf ε withf ⁽ⁿ⁾(0)=d _n,n=0, 1, 2, ..., and study the analyticity properties off. The functionf is constructed in the form of various function series, namely, a power series, a series of simple fractions, and an exponential series. Analytic solutions of the multidimensional Borel problem are also considered. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 525–538, April, 2000.     

Relationship between manifolds of solutions to generic and model generalized Cauchy-Riemann systems with a singular point

The unique solvability of a specific two-dimensional integral equation is proved. It is used to establish a one-to-one correspondence between the sets of continuous solutions to two systems of the form and defined in the same domainG (z=0∈G). Translated fromMatermaticheskie Zametki, Vol. 66, No. 2, pp. 302–307, August, 1999.     

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     

Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare with a generalized estimate including the least square estimate (LSE) , based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that (or the modified MLE ) is better than (or the modified estimate ). Further, we note that does not attain the bound for third order asymptotic median unbiased estimates.     

Martin boundary of unlimited covering surfaces

LetW be an open Riemann surface and ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ¹ (resp., ) the minimal Martin boundary ofW (resp., ). For ζ ∈ Δ, let ζ be the (cardinal) number of the set of pionts which lie over ζ and the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following: , where is the number of components of π^-1 M and π is the projection of ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.     

What we know about the second adjunction mapping

Let be an algebraic submanifold of complex projective space ℙ ^N . Assume thatn:=dim and let denote the restriction of the hyperplane section bundle to . The meromorphic map associated to fork≥1 ties together the pluricanonical maps of the surface sections of . Known results show that the behavior of is far better than one would expect from experience with the pluricanonical mappings of algebraic surfaces. In this article we discuss the known results on the structure of the mappings and describe the open problems. Conferenza tenuta il 5 giugno 1997