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A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.

A map (S,G) is a closed Riemann surface S with embedded graph G such that S \G is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s and contemporary authors are actively developing it. In the present paper, after recalling the concept of a circular map introduced by the author and Mednykh, a relationship between bipartite maps and circular maps is demonstrated via the concept of the duality of maps. In this way an enumeration formula for the number of bipartite maps with a given number of edges is obtained. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. The hypermaps are also known as dessins d’enfants (or Grothendieck’s dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of the present paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have n edges and are self-equivalent with respect to reversing the colors of vertices. Bibliography: 13 titles.

The quadratic Vieta formulas (x, y) ↦ (u, v) = (x + y, xy) in the complex geometry define a two-fold branched covering ℂ2 → ℂ2 ramified over the parabola u 2 = 4v. Thinking about topics considered in Arnold’s paper Topological content of the Maxwell theorem on multipole representation of spherical functions, I came to a very simple idea: in fact, these formulas describe the algebraic structure, i.e., addition and multiplication, of complex numbers. What if, instead of the field of complex numbers, we consider an arbitrary ring? Namely for an arbitrary ring A (commutative, with unity) consider the mapping Φ: A 2 → A 2 defined by the Vieta formulas (x, y) ↦ (u, v) = (x + y, xy). What kind of algebraic properties of the ring itself does this map reflect? At first, it is interesting to investigate the simplest finite rings A = ℤ m and A = ℤ k ×ℤ m . Recently, it has been very popular to consider graphs associated to rings (the zero-divisor graph, the Cayley graph, etc.). In the present paper, we study the directed graph defined by the Vieta mapping Φ.

In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).

The paper is devoted to the master thesis on “information theory” which was written by the author in 1956–57. The topic was suggested by his advisor A. A. Bobrov (a student of A. Ya. Khinchin and A. N. Kolmogorov), and the thesis was written under the influence of lectures by N. I. Gavrilov (a student of I. G. Petrovskii) on the qualitative theory of differential equations, which included the statement of Birkhoff’s theorem for ergodic dynamical systems. In the thesis, the author used the concept of Shannon entropy in the study of ergodic dynamical systems f(p, t) in a separable compact metric space R with an invariant measure μ (where μ(R) = 1) and introduced the notion of the (ϵ, T)-entropy of a system as a quantitative characteristic of the degree of mixing. In the work, not only partitions of R were considered, but also partitions of the interval (−∞,∞) into subintervals of length T > 0. In particular, f(p, T) was regarded as an automorphism S of X = R, and the (ϵ, T)-entropy is essentially the e-entropy of S. But, despite some “oversights” in the definition of the (ϵ, T)-entropy and many years that have passed, the author decided to publish the corresponding chapter of the thesis in connection with the following: 1) There is a number of papers that refer to this work in the explanation of the history of the concept of Kolmogorov’s entropy. 2) Recently, B. M. Gurevich obtained new results on the ϵ-entropy hϵ(S), which show that for two ergodic automorphisms with equal finite entropies their ϵ-entropies also coincide for all ϵ, but, on the other hand, there are unexpected nonergodic automorphisms with equal finite entropies, but different ϵ-entropies for some ϵ. This shows that the concept of ϵ-entropy is of scientific value.

The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link. In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.

An association scheme is said to be T-amorphous if it is antisymmetric and any tournament obtained by an appropriate merging of its classes is doubly regular. The goal of this paper is to study basic properties of T-amorphous schemes.

Reed–Muller codes are one of the most well-studied families of codes; however, there are till open problems regarding their structure. Recently a new ring-theoretic approach has emerged that provides a rather intuitive construction of these codes. This approach is centered around the notion of basic Reed–Muller codes. It is known that basic Reed–Muller codes ℳπ(m, k) over a prime field are powers of the radical RS of a corresponding group algebra and over a nonprime field there are no such equalities, except for trivial ones. In this paper, we consider the ideals ℜSℳπ(m, k) that arise while studying the inclusions of the basic codes and radical powers.

The paper presents a technique for the automatic calculation of Belyi functions for trees with weighted edges. Bibliography: 20 titles.

The main purpose of this paper is to give a complete description of isometry groups on the 4-dimensional simply connected nilpotent Lie groups. We distinguish between two geometrically distinct cases of degenerate and nondegenerate center of the group. Since Walker metrics appear as the underlying structure of neutral signature metrics on the nilpotent Lie groups with degenerate center, we find necessary and sufficient condition for them to locally admit the nilpotent group of isometries.

The classical Hurwitz theorem claims that there are exactly four normed algebras with division: the real numbers (ℝ), complex numbers (ℂ), quaternions (ℍ), and octonions (𝕆). The length of ℝ as an algebra over itself is zero; the length of ℂ as an ℝ-algebra equals one. The purpose of the present paper is to prove that the lengths of the ℝ-algebras of quaternions and octonions equal two and three, respectively.

In this paper, the functorial property of the inverse image for transitive Lie algebroids is proved and also there is proved the functorial property for all objects that are necessary for building transitive Lie algebroids due to K. Mackenzie—bundles L of finite-dimensional Lie algebras, covariant connections of derivations ▽, associated differential 2-dimensional forms Ω with values in the bundle L, couplings, and the Mackenzie obstructions. On the base of the functorial properties, a final object for the structure of transitive Lie prealgebroid and for the universal cohomology class inducing the Mackenzie obstruction can be constructed.

A homomorphic encryption allows specific types of computations on ciphertext and generates an encrypted result that matches the result of operations performed on the plaintext. Some classic cryptosystems, e.g., RSA and ElGamal, allow homomorphic computation of only one operation. In 2009, C. Gentry suggested a model of a fully homomorphic algebraic system, i.e., a cryptosystem that supports both addition and multiplication operations. This cryptosystem is based on lattices. Later M. Dijk, C. Gentry, S. Halevi, and V. Vaikuntanathan suggested a fully homomorphic system based on integers. In a 2010 paper of A. V. Gribov, P. A. Zolotykh, and A. V. Mikhalev, a cryptosystem based on a quasigroup ring was constructed, developing an approach of S. K. Rososhek, and a homomorphic property of this system was investigated. An example of a quasigroup for which this system is homomorphic is given. Also a homomorphic property of the ElGamal cryptosystem based on a medial quasigroup is shown.

The paper is devoted to the popularization of one of the topics at the border between analysis and number theory that is related to polynomial with integer coefficients.

In this paper, we prove the paragraded version of the Wedderburn–Artin theorem. Following the methods known from the abstract case, we first prove the density theorem and observe the matrix rings whose entries are from a paragraded ring. However, in order to arrive at the desired structure theorem, we introduce the notion of a Jacobson radical of a paragraded ring and prove some properties which are analogous to the abstract case. In the process, we study the faithful and irreducible paragraded modules over noncommutative paragraded rings and prove the paragraded version of the well-known Schur lemma.

Let S + and S + ′ be the Schwartz spaces of rapidly decreasing functions and tempered distributions on ℝ+ , respectively. Let P(S + ′ ) be the space of continuous polynomials over S + ′ and let P ′ (S + ′ ) be its strong dual. These spaces have representations in the form of Fock type spaces $$ \varGamma \left({S}_{+}\right):=\underset{n\in {\mathbb{Z}}_{+}}{\oplus}\left({\oplus}_{s,\mathrm{p}}^n{S}_{+}\right)\kern2em \mathrm{and}\kern2em \varGamma \left({S}_{+}^{\prime}\right):=\underset{n\in {\mathbb{Z}}_{+}}{\times}\left({\oplus}_{s,\mathrm{p}}^n{S}_{+}^{\prime}\right), $$ respectively. In the present paper, the Gâteaux differentiability of elements of the spaces P(S + ′ ), P ′ (S + ′ ), Γ(S +), and Γ(S + ′ ) is investigated. The relationship between the Gâteaux derivative, the operators of creation and annihilation in the Fock type spaces, and the differentiations on Γ(S +) and Γ(S + ′ ) is established.

The paper presents a full description of idempotent elements of the semigroup of binary relations BX(D), which are defined by semilattices of the class Σ3(X, 8). For the case where X is a finite set and Z7 = O, we derive formulas for calculating the number of idempotent elements of the respective semigroup.

The minimality theorem states, in particular, that on cohomology H(A) of a dg algebra there exists sequence of operations mi : H(A)⊗i → H(A), i = 2, 3, . . . , which form a minimal A ∞ -algebra (H(A), {m i }). This structure defines on the bar construction BH(A) a correct differential dm so that the bar constructions (BH(A), d m ) and BA have isomorphic homology modules. It is known that if A is equipped additionally with a structure of homotopy Gerstenhaber algebra, then on BA there is a multiplication which turns it into a dg bialgebra. In this paper, we construct algebraic operations Ep,q : H(A) ⊗p ⊗H(A) ⊗q → H(A), p, q = 0, 1, 2, . . ., which turn (H(A), {m i }, {E p,q }) into a B ∞ -algebra. These operations determine on BH(A) correct multiplication, so that (BH(A), d m ) and BA have isomorphic homology algebras.

In this paper we give a full description of regular elements of the semigroup that are defined by semilattices of the class, when Z7 ∩ Z6 ≠ ∅. Formulas are derived by means of which the number of regular elements of the semigroup is calculated when it is a finite set.

The classes of right and left γ-generating matrices were introduced by D. Z. Arov in the 1980s. These matrices play an important role in the description of solutions of the indeterminate Nehari problem. In the present paper, the classes of the so-called generalized right and left γ-generating matrices, being resolvent matrices of the Nehari–Takagi problem, are introduced. For matrices of these classes, some factorization theorems are proved, and the connection between the class of generalized γ-generating matrices and the class of generalized j pq -inner matrix valued functions is found. Subclasses of singular, regular, and strongly regular generalized -generating matrices are introduced and studied.

The paper is devoted to the development of the theory of lower Q-homeomorphisms relative to a p-modulus in ℝ n , n ≥ 2. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property is proved, various theorems on estimates of distortion of the Euclidean distance are given, an estimate of the ball image measure is established, and, as a consequence, an analog of the Ikoma–Schwartz lemma is proved.

The paper continues the investigation of the lengths of quasi-commuting matrix pairs; specifically, it considers the problem of realizability of different positive integers as values of the length function for quasi-commuting matrix pairs.

The paper provides a short overview of a series of articles devoted to C*-algebras generated by a self-mapping on a countable set. Such an algebra can be seen as a representation of the universal C*-algebra generated by a family of partial isometries satisfying a set of conditions. These conditions are determined by the initial mapping.

The paper generalizes results by E. Janvresse, T. de la Rue, and Y. Velenik and results by the second author on the fluctuations in the ergodic sums for the Pascal adic transformation in the case of an arbitrary ergodic invariant measure and arbitrary cylinder function.

In this paper, we give new upper bounds for the chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete integer number d, the chromatic number of ℤn with critical distance \( \sqrt{2d} \) has a polynomial growth in n with exponent less than or equal to d (sometimes this estimate is sharp). The same statement is true not only in the Euclidean norm, but also in any lp norm. Moreover, we have given concrete estimates for some small dimensions as well as upper bounds for the chromatic number of ℚpn, where by ℚp we mean the ring of all rational numbers having denominators not divisible by some prime numbers.

Many invariants of knots rely upon smoothing the knot at its crossings. To compute them, it is necessary to know how to count the number of connected components the knot diagram is broken into after the smoothing. In this paper, it is shown how to use a modification of a theorem of Zulli together with a modification of the spectral theory of graphs to approach such problems systematically. We give an application to counting subdiagrams of pretzel knots which have one component after oriented and unoriented smoothings.

The aim of the paper is to study a service system model introduced by I. Kaj and M. Taqqu. A limit theorem for the process of integral workload on the service system is proved. This theorem generalizes the corresponding result of I. Kaj and M. Taqqu, because the weak convergence in the Skorokhod space is established.

The paper presents estimates of small deviation probabilities of the sum $$ {\displaystyle \sum_{j\ge 1}{\leftthreetimes}_j{X}_j} $$ , where {⋋j} are positive numbers and {Xj} are i.i.d. positive random variables satisfying weak restrictions at zero and infinity. Bibliography: 16 titles.

In this paper, we study mod-retractable modules, CSL-modules, fully Kasch modules, and their interrelations. Right fully Kasch rings are described. It is proved that for a module M of finite length, the following conditions are equivalent. (1) In the category σ(M), every module is retractable. (2) In the category σ(M), every module is coretractable. (3) M is a CSL-module. (4) Ext 1 (S 1 , S 2) = 0 for any two simple nonisomorphic modules S 1 , S 2 ∈ σ(M). (5) M is a fully Kasch module.

Let L be the Laplace matrix of a weighted digraph. The aim of the paper is to establish a simple way for computing any coefficient of the characteristic polynomial of L as a constant sign sum over the incoming spanning forests. The idea is to express L as the product of generalized (weighted) incidence matrices. It turns out that the minors of them can be studied in terms of the tree-like structure of the digraph. This makes it possible to compute the minors of L.

In 2007, A. L. Smirnov formulated an interesting conjecture on the generalized rings introduced and studied by N. V. Durov. In this paper, the conjecture is proved.

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 this paper, we introduce a new integral operator, which we call a quasi-convolution. The kernel of this operator is analogous to the kernel of the operator of usual convolution but has a variable scale. Classical inversion by the usual Fourier transform is not possible in this case, but we give examples of inversion for special cases in dimensions one and two. We also give applications to the problem of recognition of images in medicine, where quasi-convolutions are connected with the transformation of coded images. Bibliography: 7 titles.