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This paper studies the problem on the frequency of the event where a graph with a fixed number of vertices contains induced subgraphs that are isomorphic to distance graphs in spaces of definite dimensions. In particular, the case of plane and three-dimensional spaces is considered.

In this paper, we describe automorphisms of the lattice \( {\mathbb A} \) of all subalgebras of the semiring ℝ+[x] of polynomials in one variable over the semifield ℝ+ of nonnegative real numbers. It is proved that any automorphism of the lattice A is generated by an automorphism of the semiring ℝ+[x] that is induced by a substitution x ⟼ px for some positive real number p. It follows that the automorphism group of the lattice \( {\mathbb A} \) is isomorphic to the group of all positive real numbers with multiplication. A technique of unigenerated subalgebras is applied.

We define the length of a finite system of generators of a given algebra \( \mathcal{A} \) as the smallest number k such that words of length not greater than k generate \( \mathcal{A} \) as a vector space, and the length of the algebra is the maximum of the lengths of its systems of generators. In this paper, we obtain a classification of matrix subalgebras of length 1 up to conjugation. In particular, we describe arbitrary commutative matrix subalgebras of length 1, as well as those that are maximal with respect to inclusion.

Let \( G = A\mathop { * }\limits_C B \) be an amalgamated product of finite rank free groups A, B, and C. We introduce atomic measures and corresponding asymptotic densities on a set of normal forms of elements in G. We also define two strata of normal forms: the first one consists of regular (or stable) normal forms, and the second stratum is formed by singular (or unstable) normal forms. In a series of previous works about classical algorithmic problems, it was shown that standard algorithms work fast on elements of the first stratum and nothing is known about their work on the second stratum. In this paper, we give probabilistic and asymptotic estimates of these strata.

In this paper, we prove that if two incidence rings constructed by the same semiperfect ring and some two quasi-ordered sets are elementarily equivalent, then the given sets are elementarily equivalent.

In the paper, upper bounds for sums of the form $$ \sum\limits_{{0 < n \leq N}} {\psi \left( {f(n)} \right)}, $$ where f(x) is a polynomial and ψ(x) = x − [x] − 1/2, are obtained. The cases $$ f(x) = \frac{1}{\alpha }{x^2} + \beta x + \gamma \quad {\text{and}}\quad f(x) = \frac{1}{\alpha }{x^3} + \beta {x^2} + \gamma x + \delta, $$ where α is a large positive number, are considered. Weyl’s method and Popov’s arguments (Mat. Zametki, 18, 699–704 (1975)) are used. Bibliography: 10 titles.

In this paper, we present lower bounds for the number of vertices in a graph with a large chromatic number that does not contain small odd cycles. Bibliographyy 6 titles.

In this paper, we describe automorphisms of the semigroup G2(R) of nonnegative invertible matrices if R is a (not necessarily commutative) partially ordered ring without zero divisors with 1/n for some natural number n > 1.

The paper continues investigations of the well-known class T of typically real functions f(z) in the disk U = {z:|z

The paper suggests a new approach to deriving lower bounds for the Laplacian spectral radius and upper bounds for the smallest eigenvalve of the signless Laplacian of an undirected simple r-partite graph on n vertices, 2 ≤ r ≤ n. The approach is based on inequalities for the extreme eigenvalves of a block-partitioned Hermitian matrix, established earlier, and on the Rayleigh principle. Specific lower and upper bounds generalizing known results and extending them from r = 2 to r ≥ 2 are considered, and the cases where these bounds are sharp are described. Bibliography: 13 titles.

The paper concerns the operator determined by the kernel of the Aizenberg integral representation for holomorphic functions. A special class of domains such that this operator acts from $ {C^\alpha }\left( {\partial \Omega } \right) $ to $ {H^\alpha }\left( \Omega \right) $ is introduced. An example of a nonconvex domain that belongs to this class described. Bibliography: 4 titles.

The paper is concerned with special sets of continuants occurring in applications. For each of these sets, the problem of finding the maximal and minimal continuants is solved. Methods of finding these extrema are singled out and grouped. This results in the methods of basic permutations, quadratic irrationalities, unit variation, and majorizing inequalities. The last named one is the only method involving new ideas; the remaining ones were already examined in parts.

In this paper, we obtain an upper estimate for the cardinality of a subset of the discrete torus over a field of three elements of which any four points do not form a nonsingular parallelogram.

Let α k and $ {\hat{\alpha }_k} $ denote respectively the maximum cardinality of a k-regular induced subgraph and the co-k-plex number of a given graph. In this paper, we introduce a convex quadratic programming upper bound on $ {\hat{\alpha }_k} $ , which is also an upper bound on α k . The new bound denoted by $ {\hat{\upsilon }_k} $ improves the bound υ k given in [3]. For regular graphs, we prove a necessary and sufficient condition under which $ {\hat{\upsilon }_k} $ equals υ k . We also show that the graphs for which $ {\hat{\alpha }_k} $ equals $ {\hat{\upsilon }_k} $ coincide with those such that α k equals υ k . Next, an improvement of $ {\hat{\upsilon }_k} $ denoted by $ {\hat{\vartheta }_k} $ is proposed, which is not worse than the upper bound ϑ k for α k introduced in [8]. Finally, some computational experiments performed to appraise the gains brought by $ {\hat{\vartheta }_k} $ are reported.

An important property of a scale-free network which allows one to communicate between its nodes is the connectivity. From the simulation results it is known that such networks are resistant to random damage. It is also known from experiments that usually disconnecting such a graph is obtained by isolating a vertex. In this paper, we deal with a problem of isolating a vertex both experimentally and theoretically. For the theoretical results, we use the LCD model defined by Bollobas and Riordan. Among others, we prove that even after random deleting of nα vertices (mα edges) for α < (d − 1)/d, where d is the out-degree, with high probability, there are no isolated vertices in the remaining graph.

The problem of separation of a regular factor from a matrix polynomial has been of interest to mathematicians for a long time because the solution of this problem gives a method for the determination of roots of matrix one-sided equations. Various methods of contemporary mathematics are used for the solution of this problem. In 1956, Lopatyns’kyi, in terms of systems of differential equations, gave necessary and sufficient conditions for representation of a matrix in the form of the product of regular factors with coprime determinants. Gochberg, Lancaster, and Rodman in [7] and Malyshev in [4] solve this problem by the methods of Jordan chains. In [3], Kazimirs’kyi proposes an essentially new approach to the problem of separation of a regular divisor from a matrix polynomial. This approach is based on the notions of a determining matrix and values of a matrix on a system of roots of a polynomial. In these terms, Kazimirs’kyi established necessary and sufficient conditions for the existence of these divisors and proposes a method for their determination. The obtained results hold for polynomial matrices over an algebraically closed field of characteristic zero. The specific feature of methods of solution and terms used in the representation of results do not enable one to generalize these methods for wider classes of fields. This generalization became possible after work [2] in which the Kazimirs’kyi results are represented in another form. In the present paper, we investigate polynomial matrices over an infinite field. Under certain restrictions on the canonical diagonal form of a divisor, necessary and sufficient conditions for its existence are established. It is worth noting that, after the solution of the problem of separation of a regular divisor by Kazimirs’kyi, the number of works devoted to regular divisors sharply dropped. Problems of description of nonassociative divisors of matrices [6, 10] and representation of matrices in the form of the product of several factors with preassigned characteristics [5, 8, 13] have attracted the attention of researchers. Let F be a field and let A(x) be a nonsingular (n × n) matrix over F[x] representable in the form of a matrix polynomial over the field F : A(x) = A k x k + A k−1 x k−1 +… + A 0 . The matrix A(x) is called monic if A k = E is an identity matrix of order k and regular if det A k ≠ 0. We say that the matrix A(x) is right regularizable if there exists an invertible matrix U (x) such that

In this paper, we discuss the algebraic independence and algebraic relations, first, for reciprocal sums of even terms in Fibonacci numbers \( \sum\limits_{n = 1}^\infty {F_{2n}^{ - 2s}} \), and second, for sums of evenly even and unevenly even types \( \sum\limits_{n = 1}^\infty {F_{4n}^{ - 2s}}, \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}} \). The numbers \( \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}}, \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 4}} \), and are shown to be algebraically independent, and each sum \( \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}} \left( {s \geqslant 4} \right) \) is written as an explicit rational function of these three numbers over \( \mathbb{Q} \). Similar results are obtained for various series of even type, including the reciprocal sums of Lucas numbers \( \sum\limits_{n = 1}^\infty {L_{2n}^{ - p}}, \sum\limits_{n = 1}^\infty {L_{4n}^{ - p}} \), and \( \sum\limits_{n = 1}^\infty {L_{4n - 2}^{ - p}} \).

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.

J. S. Birman and W. W. Menasco [Topology, 33, 525–556 (1994)] introduced and studied a class of imbedded tori in closed braid complements that admit a standard tiling. K. Y. Ng [J. Knot Theory Ramifications, 7, 205–216 (1998)] showed that each essential torus in a closed braid complement that admits standard tiling possesses a staircase pattern. The combinatorial and geometric description of the tori from this class was partially given in [Fund. Math., 189, 195–226 (2006)]. In the present paper, we continue the study of the decomposition of tiled tori into building blocks and show that they are similar in a certain sense. In particular, for the class of geometric tiled tori described by Ng in [J. Knot Theory Ramifications, 7, 205–216 (1998)], we show that each torus of this type consists only of elementary blocks.

It was shown by Griggs and Wu that a graph of minimal degree 4 on n vertices has a spanning tree with at least $$ \frac{2}{5} $$ n leaves, which is asymptomatically the best possible bound for this class of graphs. In this paper, we present a polynomial time algorithm that finds in any graph with k vertices of degree greater than or equal to 4 and k′ vertices of degree 3 a spanning tree with $$ \left[ {\frac{2}{5} \cdot k + \frac{2}{{15}} \cdot k'} \right] $$ leaves.

In this paper, the classification of rings of quasi-endomorphisms of strongly indecomposable Abelian torsion-free groups of rank 4 with pseudosocles of rank 3 is obtained.

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, three-dimensional maximum mobile almost contact manifolds are considered. In a special frame, we have obtained the form of structural objects for the case of constant φ-analytic curvature H = −3 of the first and also second and third classes of the Tanno theorem. The basis vector field of the Lie algebra of infinitesimal automorphisms for each of the considered structures and their commutators are found.

The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that $$ \dot{V} $$ (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if $$ \mathop {V}\limits^{(m + {1})} $$ (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that $$ \dot{V} $$ (t, x) ≤ −c(∥x∥) can be weakened and replaced by that $$ \dot{V} $$ (t, x) ≤ 0 and − (− $$ \dot{V} $$ (tk, x)| + − $$ \ddot{V} $$ (tk, x)| + ⋯ + − $$ \mathop {V}\limits^{(m)} $$ (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]

The main objective of this paper is to study some qualitative behavior of the solutions of the two difference equations $$ {x_{n + 1}} = {{{a{x_n} - b{x_{n - k}}}} \left/ {{\left( {c{x_n} - d{x_{n - k}}} \right)}} \right.}\quad n = 0,\,1,\,2, \ldots, $$ and $$ {x_{n + 1}} = {{{a{x_{n - k}} + b{x_n}}} \left/ {{\left( {c{x_n} - d{x_{n - k}}} \right)}} \right.}\quad n = 0,\,1,\,2, \ldots, $$ where the initial conditions x − k, ⋯, x − 1, x0 are arbitrary positive real numbers and the coefficients a, b, c, and d are positive constants, while k is a positive integer number.

A curvilinear three-web formed by three pencils of circles is called a circle web. Generally speaking, the circle three-web is not regular, i.e., it is not locally diffeomorphic to a web formed by three families of parallel straight lines. In this paper, all regular circle three-webs are classified up to circular transformations. The main result is as follows: There exist 48 nonequivalent (with respect to circular transformations) types of regular three-webs. Five of them contain ∞3 nonequivalent webs each, 11 types contain ∞2 nonequivalent webs each, and 12 types contain ∞1 nonequivalent webs each; 5 webs admit a one-parameter group of automorphisms.

In this paper, we considered a power of a non-Hermitian random matrix and prove that the empirical distribution of its singular values converges to the Fuss–Catalan distribution almost surely. Bibliography: 3 titles.

The paper refines the classical Ostrowski disk theorem and suggests lower bounds for the smallest-in-modulus eigenvalue and the smallest singular value of a square matrix under certain diagonal dominance conditions. A lower bound for the smallest-in-modulus eigenvalue of a product of m ≥ 2 matrices satisfying joint diagonal dominance conditions is obtained. The particular cases of the bounds suggested that correspond to the infinity norm are discussed separately and compared with some known results. Bibliography: 8 titles.

The paper studies the region of values Dm,n(T) of the system {f(z1), f(z2),..., f(zm), f(r1), f(r2),..., f(rn)}, where m ≥ 1; n > 1; zj, j = 1, ... m, are arbitrary fixed points of the disk U = {z: ¦z¦ 0 for Im z ≠= 0, z ∈ U. An algebraic characterization of the set Dm,n(T) in terms of nonnegative-definite Hermitian forms is provided, and all the boundary functions are described. As an implication, the region of values of f(z1) in the subclass of functions f ∈ T with prescribed values f(rj) (j = 1, 2, 3) is determined. Bibliography: 12 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.

The paper presents a shortcut introduction to the theory of hyperbolic virtual polytopes from the point of view of combinatorial rigidity. (It is assumed that the reader is acquainted with the notions of Laman graph, 3D lifting, and pointed tiling.) From this point of view, a hyperbolic virtual polytope is a stressed pointed graph embedded in the sphere S 2. The advantage of such a presentation is that it gives an alternative and most convincing proof of existence of hyperbolic virtual polytopes. Bibliography: 20 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.

This paper is a part of the project suggested by A. M. Vershik and the author and aimed to combine known results on the representation theory of the finite and infinite symmetric groups and a circle of results related to the quantum inverse scattering method and Bethe ansatz. In this first part, we consider the simplest spectral properties of a distinguished operator in the group algebra of the symmetric group, which we call the periodic Coxeter Laplacian. Namely, we study this operator in the two-row representations of symmetric groups and in the “ferromagnetic” asymptotic mode. Bibliography: 11 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.

In this paper, for any reduced Abelian group A whose torsion-free rank is infinite, we construct a countable set A(A) of Abelian groups connected with the group A in a definite way and such that for any two different groups B and C from the set A(A) the groups B and C are isomorphic but Hom(B,X) ≅ Hom(C,X) for any Abelian group X. The construction of such a set of Abelian groups is closely connected with Problem 34 from L. Fuchs’ book “Infinite Abelian Groups,” Vol. 1.

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.

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, 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.

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

Let $$ F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} $$ be an etale map of degree deg F = d. An etale map $$ G \in \mathbb{C}{\left[ {X,Y} \right]^2} $$ is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =(X + A(X, Y), Y + B(X, Y)) and G ◦ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every $$ {\mathbb{C}^2} $$ -automorphism F of degree d has a d-inverse approximation, namely, F −1. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.

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.

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.

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.

Let R’ = NT(m, S). The purpose of this paper is to investigate elementary equivalences UT(n,K) ≡ UT(m, S) and Λ(R) ≡ Λ(R’) for arbitrary associative coefficient rings with identity. The main theorem gives the description of such equivalences for n > 4. In addition, we investigate isomorphisms and elementary equivalence of Jordan niltriangular matrix rings.

The notion of a Gaschutz system of a finite soluble group was introduced by S. F. Kamornikov in 2008 (this is a set of complements of crowns of pairwise nonisomorphic non-Frattini factors of a chief series of the group). In the present paper, properties of Gaschutz systems are investigated. In particular, we calculate the number of Gaschutz systems in a finite soluble group and prove their conjugacy, obtain a connection between \( \mathfrak{N} \)-prefrattini subgroups and normalizers of Gaschutz systems, and investigate factorizations of the normalizer of a Gaschutz system.

The concept of the lower weakly solvable radical of Lie algebras is important in the study of Lie algebras. The purpose of this paper is to investigate the generalization of this concept to lattice ordered Lie algebras over partially ordered fields. Some results concerning properties of the lower weakly solvable l-radical of lattice ordered Lie algebras are obtained. Necessary and sufficient conditions for the l-prime radical of a Lie l-algebra to be equal to the lower weakly solvable l-radical of the Lie l-algebra are presented.

In this paper, it is proved that in a Coxeter group of extra large type every torsion-free, finitely generated subgroup is free.