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This paper is concerned with the existence of badly approximable numbers in problems involving lacunary or sublacunary sequences. The principal results depend upon Peres–Schlag’s method.

In this paper, we present three algorithms: the first one solves zero-dimensional parametric homogeneous polynomial systems within single exponential time in the number n of unknowns; it decomposes the parameter space into a finite number of constructible sets and computes the finite number of solutions by parametric rational representations uniformly in each constructible set. The second algorithm factirizes absolutely multivariate parametic polynomials within single exponential time in n and in the upper bound d on the degree of the factorized polynomials. The third algorithm decomposes algebraic varieties defined by parametric polynomial systems of positive dimension into absolutely irreducible components uniformly in the values of the parameters. The complexity bound for this algorithm is double exponential in n. On the other hand, the lower bound on the complexity of the problem of resolution of parametric polynomial systems is double exponential in n. Bibliography: 72 titles.

In the paper, the theory of extremal length of vector measures is used to show that the generalized condenser capacity in the sense of Aikawa and Ohtsuka is related to the module of the family of surfaces separating the condenser’s plates and disjoint with a given set. It is proved that the system of polyhedral surfaces from the above family is sufficient for approximating the module of this family. Bibliography: 17 titles.

Let e be the fundamental unit of a field $$ Q\left( {\sqrt {d} } \right) $$ . In the paper, it is proved that e > d 3/2/ log2 d for almost all d such that N(e) = −1. Bibliography: 6 titles.

In this work, recursive expansions in Hilbert space H = L 2[0, 1] are considered. We discuss a related notion of frames in finite-dimensional spaces. We also suggest a constructive approach to extend an arbitrary basis to obtain a tight frame. The algorithm of extending is applied to bases of a special form, whose Gram matrix is circulant. A construction of a chain of nested subspaces $$ \left\{ {{V^n}} \right\}_{n = 1}^\infty $$ is given, and in its foundation lies an example of a function that can be expressed as a linear combination of its contractions and translations. The main result of the paper is the theorem that provides the uniform convergence of recursive Fourier series with respect to the chain $$ \left\{ {{V^n}} \right\}_{n = 1}^\infty $$ for continuous functions.

In this paper, we consider the functional differential equation with impulsive perturbations $$ \left\{ {\begin{array}{*{20}{c}} {{x^{\prime}}(t) = f\left( {t,{x_t}} \right),} \hfill & {t \geq {t_0},\quad t \ne {t_k},\quad x \in {\mathbb{R}^n},} \hfill \\ {\Delta x(t) = {I_k}\left( {t,x\left( {{t^{-} }} \right)} \right),} \hfill & {t = {t_k},\quad k \in {\mathbb{Z}^{+} }.} \hfill \\ \end{array} } \right. $$ Criteria on uniform asymptotic stability of sets are established for the above system using Lyapunov functions and the Razumikhin technique. Some examples are also discussed to illustrate the theorems.

In this paper, we investigate the structure of reachable sets for general contact sub-Lorentzian metrics on \( {\mathbb{R}^3} \). In some particular cases, the presented method leads to explicit formulas for functions describing reachable sets. We also compute the image under exponential mapping and prove that the sub-Lorentzian distance is continuous for the mentioned structures. All presented results concerning reachable sets can be directly applied to generic control affine systems in \( {\mathbb{R}^3} \) with a scalar input u and constraints |u| ≤ δ.

A complex n × n matrix A is said to be nonderogatory if the degree of its minimal polynomial is equal to the degree of the characteristic polynomial. The aim of the paper is to prove the following assertion: Let $ A\bar{A} $ be a nonderogatory matrix with real positive spectrum. Then A can be made real by a unitary congruence transformation if and only if A and $ \bar{A} $ are unitarily congruent. Bibliography: 5 titles.

Let A = D A + B be a block r × r, r ≥ 2, Hermitian matrix of order n, where D A is the block diagonal part of A. The main results of the paper are Theorems 2.1 and 2.2, which state the sharp inequalities $$ {\lambda_1}(A) \geqslant {\lambda_1}\left( {{D_A} + \xi B} \right)\,\,\,and\,\,\,{\lambda_n}(A) \leqslant {\lambda_n}\left( {{D_A} + \xi B} \right), - \frac{1}{{r - 1}} \leqslant \xi \leqslant 1, $$ and analyze the equality cases. Some implications of these results are indicated. As applications, matrices occurring in spectral graph theory are considered, and new lower bounds on the chromatic number of a graph are obtained. Bibliography: 7 titles.

We survey recent results concerning the combinatorial size of subsets of groups. For a cardinal κ, according to its arrangement in a group G, a subset of G is distinguished as κ-large, κ-small, κ-thin, κ-thick, and P κ -small. By analogy with topology, there arise the following combinatorial cardinal invariants of a group: density, cellularity, resolvability, spread, etc. The paper consists of 7 sections: Ballean context, Amenability, Ideals, Partitions, Packings, Around thin subsets, and Colorings.

In this paper, we give an estimate on the growth of the scaling sequence of the Pascal adic transformation with respect to the sup-metric. We construct a special class of α-names of positive cumulative measure. The linear growth of its cardinality implies the logarithmic growth of the scaling sequence. Bibliography: 14 titles.

In this paper, we consider problems related to the decomposability of multivalued measure-preserving transformations (i.e., polymorphisms) generated by an action of two finite groups on a Lebesgue space. We give a general construction of such polymorphisms and prove a convenient decomposability criterion. In the case where both generating groups are of order 2, we use this criterion to further characterize the decomposability. In the last section, we present a method of constructing an approximate decomposition of polymorphisms that can be used for obtaining a decomposition in the usual sense. Bibliography: 6 titles.

This paper obtains an asymptotic formula for the number of solutions to the equation \( l_1^2 + { }l_2^2 + l_3^2 + l_4^2 = N \) in integers l1, l2, l3, l4 such that a < {ηlj} < b, where η is a quadratic irrational number, 0 ≤ a < b ≤ 1, j = 1, 2, 3, 4.

The paper is dedicated to the memory of my teacher Nikolay Alexandrovich Shanin, who in his public speeches always emphasized the significance of research related to computational efficiency. During the last 30 years, many mathematicians working in the mathematical theory of computer science have begun to understand this efficiency as the computability in a polynomial number of steps by a Turing machine (the class FP). Bibliography: 4 titles.

This paper concerns BMO-regularity and AK-stability for couples (X, Y) of quasi-Banach lattices of measurable functions on the measure space $ \left( {\mathbb{T},m} \right) \times \left( {\Omega, \mu } \right) $ , where $ \left( {\mathbb{T},m} \right) $ is the unit circle with Lebesgue measure. In an earlier work, S. Kislyakov introduced a weaker version of BMO-regularity and conjectured that this property is the same as the “strong” one in the case of couples of lattices having the Fatou property. Here we prove that these properties are indeed equivalent, thus verifying that BMO-regularity for couples is a self-dual property stable under division by a lattice. We also study another refinement of the AK-stability property and develop some techniques which allow us to slightly enlarge the class of weighted l p -valued lattices for which AK-stability implies BMO-regularity. Finally, we discuss some points which might be relevant to the yet unanswered question about the relationship between AK-stability and BMO-regularity in general. Bibliography: 15 titles.

The celebrated theorem established by Martin Davis, Hilary Putnam, and Julia Robinson in 1961 states that every effectively enumerable set of natural numbers has an exponential Diophantine representation. This theorem was improved by the author in two ways: However, it remains unknown whether these two improvements could be combined, that is, whether every effectively enumerable set has a single-fold (or at least finite-fold) Diophantine representation. In the paper, we discuss known results about single-fold exponential Diophantine representations, their applications, possible approaches to improving them to the case of genuine Diophantine representations, and what would follow if such improvement is impossible. Bibliography: 27 titles.

When attacking Diophantine counting problems by the circle method, the use of smoothly weighted counting functions has become commonplace to avoid technical difficulties. It can, however, be problematic to then recover corresponding results for the unweighted number of solutions. This paper looks at quadratic forms in four or more variables representing an integer. We show how an asymptotic formula for the number of unweighted solutions in an expanding region can be obtained despite applying a weighted version of the circle method. Moreover, by carefully choosing the weight, the resulting error term is made nontrivial. Bibliography: 9 titles.

In this paper, the structure of the set of threshold functions and complexity problems are considered. The notion of the signature of a threshold function is defined. It is shown that if a threshold function essentially depends on all of its variables, then the signature of this function is unique. The set of threshold functions is partitioned into classes with equal signatures. A theorem characterizing this partition is proved. The importance of the class of monotone threshold functions is emphasized. The complexity of transferring one threshold function specified by a linear form into another is examined. It is shown that in the worst case this transfer would take exponential time. The structure of the set of linear forms specifying the same threshold function is also examined. It is proved that for any threshold function this set of linear forms has a unique basis in terms of the operation of addition of linear forms. It is also shown that this basis is countable.

In this paper, we prove that every automorphism of a Chevalley group of type A l , D l , or E l , l ≥ 3, over a commutative local ring without 1/2 is standard, i.e., it is a composition of ring, inner, central, and graph automorphisms.

We consider an R G-module A over a commutative Noetherian ring R. Let G be a group having infinite section p-rank (or infinite 0-rank) such that C G (A) = 1, A/C A (G) is not a Noetherian R-module, but the quotient A/C A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank, respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.

This paper studies functional methods for specification of Latin squares over the sets of n-dimensional Boolean vectors, n-dimensional vectors over an arbitrary finite prime field and over an arbitrary finite Abelian group. In conclusion, a method for constructing classes of nongroup Latin squares is presented.

In this paper, we prove that the cardinality of the set of all precomplete classes for definite automata is continuum.

In this paper, we prove that every automorphism of a Chevalley group of type Bl, l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., it is a composition of ring, inner, and central automorphisms.

We consider a class of discrete dynamical models allowing a quantum description. Our approach to quantization consists in the introduction of a gauge connection with values in an n-dimensional unitary representation of some group (of internal symmetries) Γ; elements of the connection are interpreted as amplitudes of quantum transitions. The standard quantization is a special case of this construction: Feynman’s path amplitude e i ∫ Ldt can be interpreted as a parallel transport with values in the (1-dimensional) fundamental representation of the group Γ= U(1). If we take a finite group as the quantizing group Γ, all our manipulations – in contrast to the standard quantization – remain within the framework of constructive discrete mathematics, requiring no more than the ring of algebraic integers. On the other hand, the standard quantization can be approximated by taking 1-dimensional representations of sufficiently large finite groups. The models considered in this paper are defined on regular graphs with transitive groups of automorphisms (space symmetries). The vertices of the graphs take values in finite sets of local states. The evolution of the models proceeds in discrete time steps. We assume that one-time-step quantum transitions are allowed only within neighborhoods of the graph vertices. Simple illustrations are given. An essential part of our study was carried out with the help of a program in C implementing computer algebra and computational group theory algorithms that we develop now. Bibliography: 4 titles.

This paper contains a review of the authors’ results in the theory of algorithm complexity. The results described concern methods for obtaining lower bounds (containing almost all exponential lower bounds on monotone complexity of monotone functions), synthesis of asymptotically optimal functional networks, minimization of Boolean functions, and the problem of solving Boolean equations.

The completeness problem for bases of the form Φ ∪ ν, where Φ ⊆ Pk and ν is a finite system of automaton functions, is considered. Previously, the problem for k = 2 was solved by the author; it was also shown that there is an algorithm for determining the completeness of the system Φ∪ν when [Φ] = Pk. The paper is concerned with the case where [Φ] is the maximal (precomplete) class in Pk. The problem of completeness for systems Φ ∪ ν is shown to be undecidable if Φ is embedded in a Slupecki class and algorithmically decidable if Φ contains the class preserving all constants. Thus, the bases in Pk, k ≥ 3, can be classified according to their ability to guarantee the decidability of the completeness problem for automaton functions.

In this paper, we prove that every automorphism of an (elementary) Chevalley group of type A l , D l , or E l , l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., is the composition of inner, ring, graph, and central automorphisms.

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p3 for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A4-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p2 + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.

The aim of this paper is to derive consequences of a result of Gotze and Zaitsev (2008). We show that the i.i.d. case of this result implies a multidimensional version of some results of Sakhanenko (1985). We establish bounds for the rate of strong Gaussian approximation of sums of i.i.d. R d -valued random vectors ξ j having finite moments E IIξ j IIγ, γ>2. Bibliography: 13 titles.

In this work, we investigate the partially ordered monoids S over which the class of free (over a poset) S-posets is axiomatizable. Similar questions for S-sets were considered in papers of V. Gould, S. Bulman-Fleming, and A. A. Stepanova.

Let a linear algebraic group G act on an algebraic variety X. Classification of all these actions, in particular birational classification, is of great interest. A complete classification related to Galois cohomologies of the group G was established. Another important question is reducibility, in some sense, of this action to an action of G on an affine variety. It has been shown that if the stabilizer of a typical point under the action of a reductive group G on a variety X is reductive, then X is birationally isomorphic to an affine variety \( \bar X \) with stable action of G. In this paper, I show that if a typical orbit of the action of G is quasiaffine, then the variety X is birationally isomorphic to an affine variety \( \bar X \).

In this paper, we present a survey of results on automorphisms of distance-regular graphs obtained at the department of algebra and topology of IMM UB RAS in the last five years. Also, we explain the Higman method of application of the character theory to the investigation of automorphisms of distance-regular graphs.

Let B(m, n) be a free periodic group of arbitrary rank m with period n. In this paper, we prove that for all odd numbers n ≥ 1003 the normalizer of any nontrivial subgroup N of the group B(m, n) coincides with N if the subgroup N is free in the variety of all n-periodic groups. From this, there follows a positive answer for all prime numbers n > 997 to the following problem set by S. I. Adian in the Kourovka Notebook: is it true that none of the proper normal subgroups of the group B(m, n) of prime period n > 665 is a free periodic group? The obtained result also strengthens a similar result of A. Yu. Ol’shanskii by reducing the boundary of exponent n from n > 1078 to n ≥ 1003. For primes 665 < n ≤ 997, the mentioned question is still open.

This paper considers semigroups of operators generated by pseudodifferential operators in weighted L p -spaces of vector functions on $$ {\mathbb{R}^n} $$ (or on a compact manifold without boundary). Sufficient conditions for a semigroup to be strongly continuous and analytic are obtained, conditions for it to be completely continuous are found, and the distribution of the eigenvalues of its infinitesimal generator is examined. Also, an integral representation that singles out the principal term of the semigroup as t → 0+ is established.

Let L p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm $$ {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} $$ , and let C = L ∞ be the space of continuous 2π-periodic functions with the norm $$ {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| $$ . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that $$ P(f) \leqslant M\left\| f \right\| $$ for every f ∈ C. By $$ \sum\limits_{k = 0}^\infty {{A_k}} (f) $$ denote the Fourier series of the function f, and let $$ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty $$ be a sequence of real numbers for which $$ \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) $$ is the Fourier series of a certain function f λ ∈ L p . The paper considers questions related to approximating the function f λ by its Fourier sums S n (f λ) on a point set and in the spaces L p and CP. Estimates for $$ {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} $$ and P(f λ − S n (f λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions f and f λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f. Bibliography: 11 titles.

The paper considers the capacity of a polycondenser and the module of a family of measures associated with the class of curves connecting the polycondenser’s plates. It is proved that these quantities are equal. Also a relation between the polycondenser’s capacity and the module of a family of vector measures associated with the class of functions admissible for the polycondenser’s capacity is established. The results obtained generalize the earlier results by M. Ohtsuka and H. Aikawa.

The paper states and studies a problem that is closely related to the problems mentioned in the title.

The paper constructs coordinate splines on a closed interval, provides realizations of the corresponding biorthogonal system, and constructs finite-dimensional spaces of splines (nonpolynomialin general) of the class C 1. Bibliography: 7 titles.

We study in this paper Hardy-type spaces on a measure space ( $$ \mathbb{T} $$ , m) × (Ω, µ), where ( $$ \mathbb{T} $$ , m) is the unit circle with Lebesgue measure. There is a characterization of analytic stability for real interpolation of weighted Hardy spaces on $$ \mathbb{T} $$ × Ω, a complete proof of which was present in the literature only for the case where µ is a point mass. Here this gap is filled, and a proof of the general case is presented. In a previous work by Kislyakov, certain results concerning BMO-regular lattices on ( $$ \mathbb{T} $$ × Ω, m × µ) were proved under the assumption that the measure µ is discrete. Here this extraneous assumption is lifted. Bibliography: 9 titles.

In 2005, the following question was posed by Duggal, Djordjevic, and Kubrusly: Assume that T is a contraction of the class C 10 such that I − T * T is compact and the spectrum of T is the unit disk. Can the isometric asymptote of T be a reductive unitary operator? In this paper, we give a positive answer to this question. We construct two kinds of examples. One of them are the operators of multiplication by independent variable in the closure of analytic polynomials in L 2(ν),where ν is an appropriate positive finite Borel measure on the closed unit disk. The second kind of examples is based on a theorem by Chevreau, Exner, and Pearcy. We obtain a contraction T satisfying all the needed conditions and such that I − T * T belongs to the Schatten–von Neumann classes $$ {\mathfrak{S}_p} $$ for all p > 1. We give an example of a contraction T such that I − T * T belongs to $$ {\mathfrak{S}_p} $$ for all p > 1, T is quasisimilar to a unitary operator and has “more” invariant subspaces than this unitary operator. Also, following Bercovici and Kerchy, we show that if a subset of the unit circle is the spectrum of a contraction quasisimilar to a given absolutely continuous unitary operator, then this contraction T can be chosen so that I − T*T is compact. Bibliography: 29 titles.

This paper contains new and known results on modules in which submodules are close to direct summands. The main results are presented with proofs.

The splitting length of a mixed Abelian group G is defined as the smallest positive integer n such that \( \mathop \otimes \limits^n G \) splits. The task of determining the splitting length of mixed Abelian groups was formulated by Irwin, Khabbaz, and Rayna. In this paper, a criterion for determining whether \( \mathop \otimes \limits^n G \) splits for countable mixed Abelian groups G of torsion-free rank 1 is found.

The concepts of prime Lie algebras and semiprime Lie algebras are important in the study of Lie algebras. The purpose of this paper is to investigate generalizations of these concepts to lattice ordered Lie algebras over partially ordered fields. Some results concerning the properties of l-prime and l-semiprime lattice ordered Lie algebras are obtained. A necessary and sufficient condition for a lattice ordered Lie algebra to be an l-prime Lie l-algebra is presented.

In the present paper, the technique of spectral sequences with A ∞ -structures in their terms is developed for differential algebras with filtrations. Applications of this technique to the multiplicative spectral sequences of fibrations are given. We show that if the base of fibration is connected and simply connected, then the structure graded A ∞ -algebra in the second term of the spectral sequence of a fibration is the tensor product of the cohomology A ∞ -algebra of the base and the cohomology A ∞ -algebra of the fibre of this fibration.

In this paper, we study Abelian groups that are small with respect to different classes of groups. Completely decomposable torsion free groups that are small with respect to an arbitrary class of torsion free groups are described completely. Direct products of groups small with respect to the class of slender groups are derived.

In this paper, it is shown that the dual \( \widetilde{\text{Qord}}\mathfrak{A} \) of the quasiorder lattice of any algebra \( \mathfrak{A} \) is isomorphic to a sublattice of the topology lattice \( \Im \left( \mathfrak{A} \right) \). Further, if \( \mathfrak{A} \) is a finite algebra, then \( \widetilde{\text{Qord}}\mathfrak{A} \cong \Im \left( \mathfrak{A} \right) \). We give a sufficient condition for the lattices \( \widetilde{\text{Con}}\mathfrak{A}{\text{,}} \widetilde{\text{Qord}}\mathfrak{A} \), and \( \Im \left( \mathfrak{A} \right) \). to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras.

An introduction to the theory of semifields is included in the first part of the article: basic concepts, initial properties, and several methods of investigating semifields are examined. Semifields with a generator, in particular bounded semifields, are considered. Elements of the theory of kernels of semifields are also included in the paper: the structure of principal kernels; the kernel generated by the element 2 = 1 +1; indecomposable and maximal spectra of semifields; properties of the lattice of kernels of a semifield. A fragment of arp-semiring theory, which is the basis of a new method in semifield theory, is also included in the first part. The second part of the work is devoted to sheaves of semifields and functional representations of semifields. Properties of semifields of sections of semifield sheaves over a zero-dimensional compact are described. Two structural sheaves of semifields, which are the analogs of Pierce and Lambek sheaves for rings, are constructed. These sheaves give isomorphic functional representations of arbitrary, strongly Gelfand, and biregular semifields. As a result, sheaf characterizations of strongly Gelfand, biregular, and Boolean semifields are obtained.

This review paper is devoted to the Jacobi bound for systems of partial differential polynomials. We prove the conjecture for the system of n partial differential equations in n differential variables which are independent over a prime differential ideal \(\mathfrak{p}\). On the one hand, this generalizes our result about the Jacobi bound for ordinary differential polynomials independent over a prime differential ideal \(\mathfrak{p}\) and, on the other hand, the result by Tomasovic, who proved the Jacobi bound for linear partial differential polynomials.

In 1998, E. Couselo, S. Gonzalez, V. Markov, and A. Nechaev defined the recursive codes and obtained some results that allowed one to conjecture the existence of recursive MDS-codes of dimension 2 and length 4 over any finite alphabet of cardinality q ∉ {2, 6}. This conjecture remained open only for q ∈ {14, 18, 26, 42}. It is shown in this paper that there exist such codes for q = 42. We used a new construction, that of pseudogeometry with clusters.

This paper considers proper families of functions, which are used in functional specification of Latin squares of large size over the set of n-dimensional binary vectors. Proper families of functions are studied from the viewpoint of the intrinsic structure of the corresponding graphs of essential dependence and their adjacency matrices. Various necessary and sufficient conditions for a binary matrix to be treated as the adjacency matrix of the graph of essential dependence of a proper family of functions are derived. Also, transformations of matrices are considered under which the indicated property is preserved. It is demonstrated that any directed graph without loops and multiple edges can be embedded as an induced subgraph into the graph of essential dependence of some proper family of functions. Moreover, such embedding is reasonably economical, and the functions of the resulting proper family inherit properties of the functions that realize the original graph as the graph of essential dependence.