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The main goal of this paper is to prove the following statement. Let $$f = \sum\limits_{n \in E} {a_n z^n } $$ be a function holomorphic and of bounded characteristic in the unit disk $$\mathbb{D}$$ , where E is a Λ(1)-subset of ℤ+. Assume that f has a pseudocontinuation of bounded characteristic to the annulus {zɛℂ:1

Let B be a Blaschke product with simple zeros in the unit disk, let Λ be the set of its zeros, and let ϕ∈H∞. It is known that ϕ+BH∞ is a weak* generator of the algebra H∞/BH∞ if (for B that satisfy the Carleson condition (C)) and only if the sequence ϕ(Λ) is a weak* generator of the algebra l∞. In this paper, we show that for any Blaschke product B with simple zeros that does not satisfy condition (C), there exists B=B1·…·BN, where N ∈ℕ, and B1, …, BN are Blaschke products satisfying condition (C), there exists a function ϕ∈H∞ such that ϕ(Λ) is a weak* generator of the algebra l∞, and ϕ+BH∞ is not a weak* generator of the algebra H∞/BH∞. Bibliography: 12 titles.

The paper contains a brief survey of the author’s results on the diffeomorphism type of quotients of complex surfaces by anti-holomorphic involutions. The conjecture of complete decomposability is discussed, which says that if such a quotient is simply connected, then it is completely decomposable, i.e., is diffeomorphic to the connected sum of several copies of the projective plane (possibly, with reversed orientation) and the quadric. Bibliography: 11 titles.

Let C be the space of 2π-periodic continuous real functions with the uniform norm, let Hn be the set of trigonometric polynomials of order not more than n, let ω2(f) be the second continuity modulus for a function f∈C, and let Tn(f) be the best approximation polynomial of order n for f∈C. Set\(A_0 \left( f \right) = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi f \); U:C→C;\(C\left( {U,h} \right) = \mathop {\sup }\limits_{f \in C} ^{\frac{{\left\| {U\left( f \right) - f} \right\|}}{{\omega _2 \left( {f,h} \right)}}} \). In this paper, for h sufficiently large we find the values C(U,h) for some positive operators U. For example, C(A0,h) and C(T0,h) are found. For n=1,2,3 we find the values\(C\left( {U,\frac{\pi }{{n + l}}} \right)\) for some linear positive operators U:C→Hn. We establish relations between C(T0,h) and exact constants in the inequality ω2(f,h1)≤C(h1;h)ω2(f,h) for some h and h1 such that 0

In this paper, we characterize the Hermite polynomials via stochastic tools: the conditional mean value and the Wiener process.

Let k be a field, K/k a finite extension of it of degree n. We denote G=Aut(kK), Go=Aut(k K) and fix in K a basis ω1,...,ωn over k. In this basis, to any automorphism group of kK there corresponds a matrix group, which is denoted by the same symbol. Let G′≤G., In this paper, the conditions under which G′⊎Go is a maximal torus in G′ are studied. The calculation of NG′(G′⊎Go) is carried out, provided that thee conditions are fulfilled. The case G′=SL (kK) is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, G′⊎Go is a maximal torus in G′. Bibligraphy: 2 titles.

The present paper develops the ideas presented in [1]. Let o be a Dedeking ring, and let Λ be a finitely generated algebra over o. An integral representation in the broad sense of the ring Λ over o is a homomorphism of Λ to the endomorphism ring of a finitely generated module over o. A representation in the restricted sense is a representation by matrices over o. Thus, the problem of describing the integral representations over o is subdivided into the following two problems: the description of representations in the broad sense and the selection of them of representations in the restricted sense. It is proved that any representation of Λ by matrices over the field k of fractions of the ring o is equivalent over k to an integral representation in the broad sense. This fact simplifies the problem of describing the representations in the broad sense. A representation is equivalent to a representation in the restricted sense if its degree over k and the order of the ideal class group of the ring o are relatively prime, or if it is the direct sum of h copies of one and the same representation over k, where h is the exponent of the ideal class group of o. Bibliography:3 titles.

The paper is concerned with an application of limit theorems to the study of increasing permutations of stable random processes. By the increasing permutation of a function is meant the nondecreasing function with the same distribution. The trajectories of a random process may be approximated by step-functions, and then the continuity of the increasing permutation operator permits one to apply the Skorokhod invariance principle to obtain the distribution of the random process. The distribution function and the expected value of the increasing permutation of a stable random process are given explicitly. Also the univariate distributions of the increasing permutation of the Cauchy process are obtained. In various normed spaces the images of the unit balls with respect to the operator of increasing permutation are described. A separate section is devoted to the increasing permutations of higher-dimensional processes. Bibliography: 5 titles.

In this paper, we introduce a scheme of summation of independent random variables with random replacements. We consider a series of double arrays of identically distributed random variables that are row-wise independent, but such that neighboring rows contain a random common part of the repeating terms. By this-scheme we bscribe a model of strongly dependent noise. To investigate the sample mean of this noise, we consider the sum of random variables over the whole double array and its conditional variance with respect to replacements. For columns of the arrays we prove a covariance inequality. As a corollary of it, we demonstrate the law of large numbers for conditional variances. Bibliography: 4 titles.

This paper contains two results on the asymptotic behavior of uniform probability measure on partitions of a finite set as its cardinality tends to infinity. The first one states that there exists a normalization of the corresponding Young diagrams such that the induced measure has a weak limit. This limit is shown to be a δ-measure supported by the unit square (Theorem 1). It implies that the majority of partition blocks have approximately the same length. Theorem 2 clarifies the limit distribution of these blocks. The techniques used can also be useful for deriving a range of analogous results. Bibliography: 13 titles.

Let u1=1, u2=2, u3,... be the sequence of Fibonacci numbers. A Fibonacci partition of a natural number n is a partition of n into different Fibonacci numbers. In this paper it is proved that the set of Fibonacci partitions of a natural number, partially ordered with respect to refinement is the lattice of ideals of a multizigzag. On the basis of this theorem we obtain some results concerning the coefficients of the Taylor series of infinite products $$\prod\limits_{i = 1}^{ + \infty } {\left( {1 + zq^{ui} } \right) = 1 + \sum\limits_{k = 1}^{ + \infty } {a_k \left( z \right)q^k } } $$ where $$z = \pm 1, - \tfrac{1}{2} \pm i\tfrac{{\sqrt 3 }}{2}$$ , ±i. Bibliography: 6 titles.

The resutls of this paper show that the structure of sets mentioned in the title is not trivial. For example, it is shown that there exist countalbe sets of uniqueness for logarithmic potential, i.e., closed countable subsets E of the unit circle\(\mathbb{T}\) such that $$f \in C(\mathbb{T}),f|_E = 0,U^f |_E = 0 \Rightarrow f \equiv 0.$$

The aim of this paper is to show that the well-studied families of GEM and Poisson-Dirichlet measures may be obtained as distributions of normalized cycle lengths on the space of vitual pemutations (i.e., elements of a projective limit of symmetric groups). Two characterizations of Ewens distibutions are given. Bibliography: 9 titles.

M. Davis proved in the early 1950s that every recursively enumerable set has an arithmetic representation with a unique bounded universal quantifier, known today as the Davis normal form. Davis, H. Putnam, and J. Robinson showed in 1961 how the Davis normal form can be transformed into a purely existential exponential Diophantine representation which uses not only addition and multiplication, but also exponentiation. The present author eliminated the exponentiation in 1970 and thus obtained the unsolvability of Hilbert's tenth problem. The paper presents a new method for transforming the Davis normal form into the exponential Diophantine representation. Bibliography: 12 titles.

A randomized proof system for arithmetic is introduced. A proof of an arithmetical formula is defined as its derivation from the axioms of arithmetic by the standard rules of inference of arithmetic and also one more rule which we call the random substitution rule. Such proofs can be regarded as a special kind of interactive proof and, more exactly, as a special kind of the Arthur-Merlin proofs. The main result of the paper shows that a proof in arithmetic with the random substitution rule can be considerably shorter than an arithmetical proof of the same formula. Namely, there exists a set of formulas such that (i) these formulas are provable in arithmetic but, unless PSPACE=NP, do not have polynomially long proofs; (ii) these proofs have polynomially long proofs in arithmetic with random substitution (whatever random numbers appear) and the probability of error of these proofs is exponentially small. Bibliography: 10 titles.

Some algorithms are suggested for constructing pseudoinverse matrices and for solving systems with rectangular matrices whose entries depend on a parameter in polynomial and rational ways. The cases of one- and two-parameter matrices are considered. The construction of pseudoinverse matrices are realized on the basis of rank factorization algorithms. In the case of matrices with polynomial occurrence of parameters, these algorithms are the ΔW-1 and ΔW-2 algorithms for one- and two-parameter matrices, respectively. In the case of matrices with rational occurrence of parameters, these algorithms are the irreducible factorization algorithms. This paper is a continuation of the author's studies of the solution of systems with one-parameter matrices and an extension of the results to the case of two-parameter matrices with polynomial and rational entries. Bibliography: 12 titles.

In this paper, algorithms which realize some operations over scalar polynomials in one and two variables and their computer realization are suggested. The following operations are considered: 1) the computation of the GCD for given scalar polynomials and the decomposition of each polynomial into a product of two factors: the first factor is the GCD, and the second factors form a sequence of relatively prime polynomials; 2) the division of polynomials by their common divisor; 3) the decomposition of polynomials in two variables into irreducible factors; 4) the computation of the LCM for given scalar polynomials. Bibliography: 5 titles.

The paper deals with antiisomorphisms of endomorphism rings of modules close to free ones (those modules are strong generators, i.e., they contain free cyclic direct summands). A criterion for such antiisomorphisms to be induced by a Morita antiequivalence is obtained. Specifically, any antiisomorphism of endomorphism rings of progenerators (e.g., of matrix rings) is induced by a Morita antiequivalence. Bibliography: 9 titles.

Foundations of a general theory of polynomial operator interpolation in abstract spaces are laid. In the first part of the paper, the main results of this theory for Hilbert spaces are presented. In particular, a necessary and sufficient condition for the existence of an interpolational operator polynomial is obtained, and a structural description of these polynomials and of the interpolants preserving operator polynomials of the corresponding degree is given.

The problem of multiple interpolation in the class of entire functions of finite order was solved in [1]. The analogous problem for the class of functions of finite order in the upper half-plane C + = {z: Imz > 0} was discussed in [2], where necessary and sufficient conditions for its solvability were found. The difficulties that appear upon passage from the interpolation problem in classes of entire functions to classes of functions that are analytic in the half-plane C + are caused by the possibility of interpolation nodes accumulating at points of the real axis. The sufficient conditions found in [3] eliminate the possibility of such accumulation and make it possible to apply the methods and theory of entire functions. In the present paper we consider a problem more general than that of [2], eliminate the above-noted shortcomings, and find necessary and sufficient conditions for solvability of our problem. In its formulation, our problem is close to problems on free interpolation in H ~176 , since the values of the derivative of a function at interpolation nodes are subjected to some natural constraints. We will use ideas from [1, 2], as well as ideas due to Jones [3] for solution of problems on free interpolation in H ~176 Let Lo, ~]+ denote the class of functions that are analytic in C + and of order _ 1. Definition. A divisor D = {an, qn} (i.e., a set of different complex numbers an E C +, n = 1, 2,..., whose limit points all lie on the rea/axis, together with their multiplicities qn, qn >_ 1, is an integer) is said to be an interpolation divisor in the class Lo, oo] + if, for any sequence of numbers {bnk}, k = 1,..., q,, n = 1, 2,..., that satisfies the conditions (Irna,)~-~lb~kl In + In maxl

The main objective of this paper is stated in the title. In addition, a characterization of one-component inner functions is presented. Bibliography: 7 titles.

In the present paper, the embedding problem is considered for number fields with p-groups whose kernel is either of two groups with two generators α and β and with the following relations: (1) αρ=1, αρ=1, [α,β,β]=1, [α,β,α,α]=1, or (2) αρ=[α, β α], βρ=1, [α,β,β]=1. It is shown that for the solvability of the original embedding problem it is necessary and sufficient to have the solvability of the associated Abelian and local problems for all completions of the base fields. Bibliography: 7 titles.

In this paper, a family of weak constructive theories, containing arithmetic and a theory of natural-valued functions of natural arguments, is suggested. The functions are polynomially bounded and computable in time bounded by polynomials in their arguments. The languages of these theories contain functional constants for addition and multiplication and the equality predicate. Other functional constants are also allowed, provided that the corresponding functions satisfy the above conditions of polynomial boundedness. From the proofs of the theories considered witness functions for provable formulas, computable in polynomial time, can algorithmically be extracted. If an argument of a witness is a function, then the latter is used in the witness algorithm as an oracle. Bibliography: 2 titles.

This paper presents necessary and sufficient conditions under which isomorphism of endomorphism rings of additive groups of arbitrary associative rings with 1 implies isomorphism of these rings. For a certain class of Abelian groups, we present a criterion which shows when isomorphism of their endomorphism rings implies isomorphism of these groups. We demonstrate necessary and sufficient conditions under which an arbitrary ring is the endomorphism ring of an Abelian group. This solves Problem 84 in L. Fuchs’ “Infinite Abelian Groups.”

The notion of an extension is important in the study of partially ordered groups. In the present paper the notion of a lexicographic extension of a partially ordered group by an AO-group is studied. A result is obtained concerning an AO-group G which is a lexicographic extension of a directed subgroup of G.

We show that in some cases it is possible to get the Levy measure v for an infinitely divisible distribution μ as a limit of some signed measuresL N(μ) based on the convolution powers of μ. In the paper we give a collection of results in this area and apply the method to the investigation of the infinite divisibility of sub-stable random vectors onR n.

In the present paper, we consider the basic properties of α-generalized Poisson random walks taking place in a multidimensional space.

In this paper we propose convenient criteria for equalities or inequalities at levels of random variables.

Function spaces play an important role in complex analysis, in the theory of differential equations, in functional analysis and in almost every other branch of modern mathematics. In this paper we give and study the notion of clopen convergence. Also, we study the notion of clopen continuity and define new topologies on function spaces. These results generalize basic results of R. Arens, J. Dugundji and A. Di Concilio (see [1], [4], [2] and [3]).

The paper deals with the essentiality of the convexity condition in the well-known selection theorems due to E. Michael. It is proved that the convexity of the range of a lower semi-continuous multivalued mapping may be replaced by an essentially weaker assumption: each value of the multivalued mapping is a graph of a Lipschitzian function of n variables in a suitable coordinate system. Bibliography: 3 titles.

The paper contains the proof of the algorithmic unrecognizability for two properties of some relatively free soluble groups, as well as the proof of the lack in a certain sense of the recursive enumerability for one more property of such groups. It is also proved that the indecomposability in the class of soluble finitely based varieties of groups is unrecognizable. An example is given of a relatively free soluble group with solvable word problem and with unsolvable identity problem. A result is obtained involving truth-table degrees of the word problem for relatively free soluble groups. Bibliography: 22 titles.

In the paper, cubature formulas for calculation of an integral along the hypercube are constructed. These formulas are invariant under the group of all transformations of the hyperoctahedron and precise for all polynomials of degree less than or equal to 7. Bibliography: 2 titles

In this paper, we study the computation complexity of some algebraic combinatorial problems that are closely associated with the graph isomorphism problem. The key point of our considerations is a relation algebra which is a combinatorial analog of a cellular algebra. We present upper bounds on the complexity of central algorithms for relation algebras such as finding the standard basis of extensions and intersection of relation algebras. Also, an approach is described to the graph isomorphism problem involving Schurian relation algebras (these algebras arise from the centralizing rings of permutation groups). We also discuss a number of open problems and possible directions for further investigation. Bibliography: 18 titles.

In this paper an analogue of the formulas [D. M. Chibisov,Teor. Veroyatn. Primen.,30, 269–288 (1985);Izv. Akad. Nauk UzSSR,6, 23–30 (1982)] for the difference between the power of a given asymptotically efficient test and that of the most powerful test is justified for one-sample L-and R-tests, i.e., tests based on linear combinations of order statistics and linear rank statistics. This formula directly yields the Hodges-Lehmann deficiency of corresponding tests. A general theorem is stated which is applied to L-and R-tests. The explicit expressions given by this formula for L- and R-tests are also presented. The expression related to R-tests agrees with the one obtained in [W. Albers, P. J. Bickel, and W. R. Van Zwet,Ann. Statist.,4, 108–156 (1976);6, 1170–1171 (1978)]. We present here a nontechnical (heuristic) proof of these results.

The paper deals with random variables which are the values of independent identically distributed stochastic processes at random points in time. We obtain conditions for the weak convergence of their sums, at almost all points in time, to the same infinitely divisible distribution and describe the limit distribution for these sums. Also we obtain an analog of the Donsker theorem and limit theorems for empirical processes for such random variables.

The paper belongs to the line of inquiry concerning the description of overgroups of nonsplit maximal tori. It deals with the case where the ground field is the field of fractions of a Dedekind domain. Its principal result is a construction of maximal subgroups of the above-mentioned kind. Bibliography: 5 titles.

The paper considers the problem of computing the invariant polynomials of a general (regular or singular) one-parameter polynomial matrix. Two new direct methods for computing invariant polynomials, based on the Δ W and ∇ V rank-factorization methods, are suggested. Each of the methods may be regarded as a method for successively exhausting roots of invariant polynomials from the matrix spectrum. Application of the methods to computing adjoint matrices for regular polynomial matrices, to finding the canonical decomposition into a product of regular matrices such that the characteristic polynomial of each of them coincides with the corresponding invariant polynomial, and to computing matrix eigenvectors associated with roots of its invariant polynomials are considered. Bibliography: 5 titles.

This paper contains a description of multipliers in the spaces Fp,θl, FLp,θl, and BLp,∞l, 0

In this paper we give a description of the domain of normal attraction of a strictly semistable probability measure on a group G, which is a semidirect product of a compact group K and H=Rd. This result is a generalization of analogous results for the case of a stable limit law (see [3, 8, 9, 12]).

In this paper we give a characterization of the classMSS of all one-dimensional d.f.'s G satisfying the functional equation Gα(x)=G(L(x)), where α∈(0, 1] andL(x) is strictly monotone.

Considering an infinite string of i.i.d. random letters drawn from a finite alphabet X, we are interested in the number of letters needed until one or more given words are observed as a substring. Given a large subset of words of length n (typically the whole Xn), one can define the corresponding cover time as the maximum of the waiting times associated with each of the words considered, that is, the number of random letters needed until each word appears at least once. The symmetric case of uniformly distributed random letters has been studied throughly, but it is less well known how the results change if the distribution of random letters is no longer uniform. In the paper, a comparison is made between results in the symmetric and nonsymmetric cases, both known and new ones.

In this paper we establish asymptotic expansions (a.e.) under alternatives for the distribution functions of sums of independent identically distributed random variables (i.i.d.r.v.'s.), linear combinations of order statistics, and one-sample rank statistics (L- and R-statistics). The general Lemma from [V. E. Bening,Bull. Moscow State Univ., Ser. 15, 2 36–44 (1994)] is applied to these statistics. Section 1 contains the statement of the theorem, in Sec. 2 the theorems is proved; its proof involves four auxiliary lemmas, also contained in Sec. 2. Finally Sec. 3 contains the proofs of these lemmas.

In this paper, we prove an inverse of the law of the iterated logarithm for a randomly indexed sequence of random variables with indices assumed to be independent of the basic sequence. We also prove that the Borel-Cantelli version of the Anscombe condition, introduced by A. Gut, is necessary and sufficient for the complete convergence (in the Hsu-Robbins sense) of a randomly indexed sequence of random variables.

The limit behavior of Markov chains with discrete time and a finite number of states (MCDT) depending on the number n of its steps has been almost completely investigated [1–4]. In [5], MCDT with forbidden transitions were investigated, and in [6], the sum of a random number of functionals of random variables related by a homogeneous Markov chain (HMC) was considered. In the present paper, we continue the investigation of the limit behavior of the MCDT with random stopping time which is determined by a Markov walk plan II with a fixed number of certain transitions [7, 8]. Here we apply a method similar to that of [6], which allows us to obtain, together with some generalizations of the results of [6], a number of new assertions.

Plans of multidimensional random walks which allow the factorization of the transition probabilities are used in sequential analysis [1], in order to obtain unbiased estimates of some parameters of a number of distributions [2], and in testing the homogeneity hypothesis by the Kolmogorov-Smirnov test [3, 4]. Therefore, their complete description is of significant interest. In the present paper we obtain necessary and sufficient conditions which the one-step transition probabilities of such a plan should satisfy. In addition, we demonstrate that any discrete distribution can be generated (in a nonunique way) by a random walk which allows the factorization of the transition probabilities. Also we consider a series of examples. Note that although in the books [1, 3] random walks are used whose forms differ from those considered here, they can be reduced to the walks given here by coordinate transformation.

In the present paper we consider the relation between the distributions of probability of attaining the boundary point of a polynomial plan and that of a Poisson plan. This relation is used to compare the moments of the functions of the boundary point coordinates, the closedness and completeness conditions of plans, the classes of the unbiased estimates and the unbiased estimated functions of the unknown parameters.

The method of probability metrics for the investigation of limit theorems and of stability problems of the characterization of probability distributions has been described in [1, 2]. This method has obtained an exhaustive development in its application to the distributions of random variables. Precisely for them one has the largest amount of different probability metrics and inequalities connecting the values of these metrics. In this paper we introduce some probability metrics enabling us to determine the distances between the distributions of random closed sets, and we also consider their application to the limit theorem in the scheme of the union of random sets.

Consider the following stochastic model. Let the resource consumption in discrete time be represented in the form {sup {eta}}{sub k} = Y(k) + n{sub k} where y(k) are deterministic components of consumption and w{sub k} are independent Gaussian random variables with zero mean. Correspondingly, the resource supply is given (in discrete time) by the process {xi}{sub k} = z(k) - {tau}{sub k}, where z(k) are its deterministic components and {tau}{sub k} are quantities characterizing the undersupply of the resource associated with breakdowns, repairs, and so on; their structure will be described below in detail. This model describes a quite realistic system of transporting and storage of gas, oil-products, and so on [1]. The supply to the users is conducted over pipelines (main and local), the main deficits and losses being associated with the seasonal variations in consumption and accidents; to decrease the losses along the main pipelines, storage facilities are constructed which are used as compensators. The model considered in this paper describes the case of a terminal storage, i.e., of a district or an industrial region.

Necessary higher-order optimality conditions are given for nonlinear smooth nonstationary control systems, linear with respect to control and defined on a smooth finite-dimensional manifold; the admissible values of the control belong to a closed convex polyhedron, while the initial and final times and the initial and final points of the trajectory are not fixed. Unlike the previously known necessary optimality conditions, the paper does not require that the passage from one condition, of a given sequence of necessary conditions, to another one should be performed only when the preceding condition degenerates on a time interval of nonzero length.

An arbitrary Muckenhoupt A2-weight w2 on the special contour γρ (ρ ≥ 1/2) generates a function Yρ,w (λ, t), which for ρ=1, w2(z) ≡ 1 coincides with the exponential exp{iλt}. In the paper, with the aid of B. S. Pavlov's geometric approach, one obtains criteria for the unconditional basis property of families of functions of the form {yρ,w(λk,t):λk∈Λ} in the space L2(0, σ). The analytic foundation of the constructions is a generalization of M. M. Dzhrbashyan's certain results (power weight) to the case of arbitrary Muckenhoupt A2- weights.