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For the perturbed Hamiltonian of a multifrequency resonance harmonic oscillator, a new approach to calculating the coefficients in the procedure of quantum averaging is proposed. The procedure of quantum averaging is transferred to the space of the graded algebra of symbols by using twisted product introduced in the paper. As a result, the averaged Hamiltonian is represented as a function of generators of the quantum symmetry algebra of the harmonic part of the Hamiltonian. The proposed method is applied to the spectral problem for the Hamiltonian of the cylindrical Penning trap. [ABSTRACT FROM AUTHOR]

In 1993, I. P. Shestakov posed the question of the existence of a central simple finite-dimensional algebra over a field of characteristic zero whose identities are not given by a finite set of identities. In 2012, I. M. Isaev and the author of the present paper constructed an example of a seven-dimensional central simple algebra over any field that does not have a finite basis of identities. In the present paper, we construct an example of a six-dimensional central simple algebra over a field of characteristic zero which has no finite basis of identities. [ABSTRACT FROM AUTHOR]

In the paper, it is proved that the radical of a relatively free associative algebra of countable rank over an infinite field of characteristic p > 0 is a nil ideal of bounded index if the complexity of the corresponding variety is less than p. Moreover, a description of a basis for trace identities for the matrix algebra M n over an infinite field of characteristic p > 0, n < p, is obtained in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of constructing semiclassical asymptotics for the tunnel splitting of the spectrum of an operator defined on an irreducible representation of the Lie algebra . It is assumed that the operator is a quadratic function of the generators of the algebra. We present coherent states and a unitary coherent transform that allow us to reduce the problem to the analysis of a second-order differential operator in the space of holomorphic functions. Semiclassical asymptotic spectral series and the corresponding wave functions are constructed as decompositions in coherent states. For some values of the system parameters, the minimal energy corresponds to a pair of nondegenerate equilibria, and the discrete spectrum of the operator has an exponentially small tunnel splitting of the levels. We apply the complex WKB method to prove asymptotic formulas for the tunnel splitting of the energies. We also show that, in contrast to the one-dimensional Schrödinger operator, the tunnel splitting in this problem not only decays exponentially but also contains an oscillating factor, which can be interpreted as tunneling interference between distinct instantons. We also show that, for some parameter values, the tunneling is completely suppressed and some of the spectral levels are doubly degenerate, which is not typical of one-dimensional systems. [ABSTRACT FROM AUTHOR]

Let be an embedded projective variety. The Lie algebra defined by the tangent directions to at smooth points is an interesting algebraic invariant of . In some cases, this algebra is isomorphic to the symbol algebra of a filtered system of distributions on a Fano manifold, which plays an important role in the theory of these manifolds. In addition, algebras defined by tangent directions are interesting on their own right. In this paper, we study the Lie algebra corresponding to a variety that is the projectivization of the orbit of the lowest weight vector of an irreducible representation of a complex semisimple Lie group. We describe these algebras in terms of generators and relations. In many cases, we can describe their structure completely. [ABSTRACT FROM AUTHOR]

In the paper, we consider the problem of uniform approximation of a continuous function defined on a compact metric space by elements of the sum of two algebras in the space of all continuous functions on . We prove a Chebyshev-type theorem for characterization of best approximation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of algebras satisfying certain conditions which has been named power algebras and we prove some fixed point theorems for a new class of contractions. We give new conditions for the existence and uniqueness of fixed points for this new class of mappings which are not a contraction in the usual algebra. Also, we introduce a new iterative algorithm for fixed-point problems in a Banach algebra. This results improve and extend the corresponding conclusions of the Ishikawa algorithm under weaker conditions and lead to stronger results. In conclusion, we give an application to nonlinear integral equations. [ABSTRACT FROM AUTHOR]

An algebra is called a binary Leibniz algebra if each of its two-generated subalgebras is a Leibniz algebra. In the present paper, we give a description of binary Leibniz algebras in terms of identities. As a consequence, we show that the variety of binary Leibniz algebras is not Schreier and that the freedom theorem fails to hold for this variety. [ABSTRACT FROM AUTHOR]

In the paper, some properties of a singly generated C*-subalgebra of the algebra of all bounded operators B( l2( X)) on the Hilbert space l2( X) with the generator T φ induced by a mapping φ of an infinite set X into itself are investigated. A condition on φ is presented under which the operator T φ is continuous, and it is proved that, if this is the case, then the study of these algebras can be reduced to that of C*-algebras generated by a finite family of partial isometries of a special form. A complete description of the C*-algebras generated by an injective mapping on X is given. Examples of C*-algebras generated by noninjective mappings are treated. [ABSTRACT FROM AUTHOR]

In the paper, we continue the study of the relatively free Grassmann algebra without unit over an infinite field of characteristic 2, which was initiated in previous works of the author. The main attention is paid here to the relationship between the -spaces of n-words, i.e., the -spaces generated by all monomials in containing each of their variables with multiplicity n. The results of this note will enable one to form a more complete picture of possible inclusions between the -spaces of r-and n-words for r > n. [ABSTRACT FROM AUTHOR]

The present paper is devoted to estimating the codimensions cn for the variety of associative algebras given by the identity [x1, ..., xl] = 0 for odd l. The characteristic of the ground field is different from 2 and 3. The construction of the so-called extended Grassmann algebra is applied very effectively. [ABSTRACT FROM AUTHOR]

Suppose that A and B are C*-algebras, A is separable, and B is stable. The elements of the group E1(A, B) in Connes—Higson E-theory are represented by *-homomorphisms from the suspension of A to the asymptotic algebra 21B. In the paper, an endofunctor M in the category of C* -algebras is constructed and a set of special homotopy classes of *-homomorphisms from A to MUB is defined so that this set endowed with the natural structure of an Abelian group coincides with E1(A,B). [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of predual spaces of JBW-algebras. It is proved that the predual space of a JBW-algebra is a strongly facially symmetric space if and only if this algebra is the direct sum of an Abelian algebra and an algebra of type I2. [ABSTRACT FROM AUTHOR]

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to explicitly construct canonical representatives in every strict isomorphism class of commutative formal groups over an arbitrary torsion-free ring. The case of an Z( p) -algebra is treated separately. We prove that, under natural conditions on a subring, the canonical representatives of formal groups over the subring agree with the representatives for the ring. Necessary and sufficient conditions for a mapping induced on strict isomorphism classes of formal groups by a homomorphism of torsion-free rings to be injective and surjective are established. [ABSTRACT FROM AUTHOR]

The paper focuses on the right and left eigenvectors of a network matrix that belong to the largest eigenvalue. It is shown that each of vector entries measures the walk centrality of the corresponding node’s position in the network’s link structure and of the positions of the node’s adjacent nodes; as a result, it indicates to which degree the node can be associated with the structure’s core, i.e., the structural coreness of the node. The relationship between the vectors’ coordinates and the position of the nodes, as well as the actual computation of the coordinates, is based on an iterative computational scheme known as the power method. The paper studies the method’s convergence for networks of different structure. Some possible applications are discussed. The paper also includes a numerical example dealing with a real network of 197 nodes and 780 links. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of solutions of a semilinear elliptic equation in the exterior of a compact set as $|x|\backslash to\; \backslash nomathbreak\; \backslash infty$. Such equations were considered by many authors (for example, Kondrat'ev, Landis, Oleinik, Veron, etc.). In the present paper, we study the case in which in the equation contains lower terms. The coefficients of the lower terms are arbitrary bounded measurable functions. It is shown that the solutions of the equation tend to zero as $|x|\backslash to\; \backslash nomathbreak\; \backslash infty$. [ABSTRACT FROM AUTHOR]

The paper deals with the Rees algebra R of a graded ideal I of a standard graded algebra A generated by a subset of generators of the maximal graded ideal of A. We compute the cohomology modules of R in terms of the cohomology modules of A and depth( R ) in terms of depth (A). [ABSTRACT FROM AUTHOR]

The notion of coupled fixed point was introduced in 2006 by Bhaskar and Lakshmikantham. On the other hand, Filipovićet al. [M. Filipovićet al., 'Remarks on 'Cone metric spaces and fixed-point theorems of T-Kannan and T-Chatterjea contractive mappings',' Math. Comput. Modelling 54, 1467-1472 (2011)] proved several fixed and periodic point theorems for solid cones on cone metric spaces. In this paper we prove some coupled fixed-point theorems for certain T-contractions and study the existence of solutions of a system of nonlinear integral equations using the results of our work. The results of this paper extend and generalize well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

In the present paper, we give necessary and sufficient conditions for the boundedness of commutators of fractional maximal operator on Orlicz spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators. [ABSTRACT FROM AUTHOR]

In the present paper, the existence problem for nonconvertible (0, 1) matrices is solved completely. A similar result is obtained for the set of symmetric (0, 1)-matrices. [ABSTRACT FROM AUTHOR]

In the paper, an analog of the Procesi-Razmyslov theorem for the algebra of semiinvariants of representations of an arbitrary quiver with dimension vector (2, 2, ..., 2) is obtained. [ABSTRACT FROM AUTHOR]

For the Lipschitz mapping of a metric compact set into itself, there is a classical upper bound on topological entropy, namely, the product of the entropy dimension of the compact set by the logarithm of the Lipschitz constant. The Ghys conjecture is that, by varying the metric, one can approximate the upper bound arbitrarily closely to the exact value of the topological entropy. In the present paper, we obtain a criterion for the validity of the Ghys conjecture for an individual mapping. Applying this criterion, we prove the Ghys conjecture for hyperbolic mappings. [ABSTRACT FROM AUTHOR]

In the paper, the solvability of the word problem and the conjugacy problem is proved for a wide class of finitely presented groups defined by periodic defining relations of a sufficiently large odd degree. In the proof, we use a certain simplified version of the classification of periodic words and transformations of these words, which was presented in detail in the author’s monograph devoted to the well-known Burnside problem. The result is completed by the proof of an interesting result of Sarkisyan on the existence of a group, given by defining relations of the form E = 1, for which the word problem is unsolvable. This result was first published in abstracts of papers of the 13th All-Union Algebra Symposium in Gomel in 1975. [ABSTRACT FROM AUTHOR]

In this paper, we construct, for each n ∈ ℕ, a compact set X ⊂ ℂ (depending on n) such that the set of all polyanalytic polynomials of order n is not dense in C( X), but the set of all polyanalytic polynomials of order 2 n is already dense in C( X). [ABSTRACT FROM AUTHOR]

The theorem proved in this paper gives a congruence for the traces of powers of an algebraic integer for the case in which the exponent of the power is a prime power. The theorem implies a congruence in Gauss’ form for the traces of the sums of powers of algebraic integers, generalizing many familiar versions of Fermat’s little theorem. Applied to the traces of integer matrices, this gives a proof of Arnold’s conjecture about the congruence of the traces of powers of such matrices for the case in which the exponent of the power is a prime power. [ABSTRACT FROM AUTHOR]

As A. S. Belov proved, the partial sums of an even 2π-periodic function f expanded in a Fourier series with convex coefficients {α n }, are uniformly bounded below if the conditions a n = O( n −1), n → ∞, are satisfied; moreover, this assertion is no longer valid if the exponent −1 in this condition is replaced by a greater one. In this paper, we obtain analogs of these results for trigonometric polynomials of best approximation to the function f in the metric of L . [ABSTRACT FROM AUTHOR]

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed. [ABSTRACT FROM AUTHOR]

In the paper, the convergence properties of the Rogers--Ramanujan continued fraction $1+\backslash frac\{qz\}\{1+\backslash frac\{q^2z\}\{1+\backslash cdots\}\}$ are studied for $q=\backslash exp\; (2\backslash pi\; i\backslash tau)$, where $\backslash tau$ is a rational number. It is shown that the function $H\_q$ to which the fraction converges is a counterexample to the Stahl conjecture (the hyperelliptic version of the well-known Baker--Gammel--Wills conjecture). It is also shown that, for any rational $\backslash tau$, the number of spurious poles of the diagonal Padé approximants of the hyperelliptic function $H\_q$ does not exceed one half of its genus. [ABSTRACT FROM AUTHOR]

Numerical characteristics of identities of nonassociative algebras are considered. A series of algebras with subexponential colength growth is constructed. [ABSTRACT FROM AUTHOR]

In this paper, some converses of Jessen's inequality, in terms of the time scale calculus, are proved. The obtained results are used to derive new converse inequalities for generalized means, power means and Hölder's inequality. [ABSTRACT FROM AUTHOR]

It was a question of Taibleson, open for a long time that the almost everywhere convergence of Fejér (or ( C, 1)) means of Fourier series of integrable functions with respect the character system of the group of 2-adic integers. This question was answered by Gá t in 1997. The aim of this paper is to investigate the maximal operator of the sup |σ|. Among other things, we prove that this operator is bounded from the Hardy space H to the Lebesgue space L if and only if 1/2 < p < ∞. The two-dimensional maximal operator is also discussed. [ABSTRACT FROM AUTHOR]

Earlier, for odd n, the author described unitary n × n matrices U with the following property: if H is a Hankel matrix, then U* HU is a Hankel matrix as well. In the present paper, it is shown that this description also holds for even n starting from n = 4. [ABSTRACT FROM AUTHOR]

Formulas for the regularized trace of the one-dimensional non-self-adjoint Dirac operator with L-potential are obtained. The cases of periodic and antiperiodic boundary conditions as well as of the Dirichlet boundary conditions are considered. The formulas are obtained by using the method of similar operators on the basis of results from the papers [1] and [2]. [ABSTRACT FROM AUTHOR]

In the paper,we study algebras having n bilinearmultiplication operations [InlineMediaObject not available: see fulltext.]: A× A → A, s = 1, ..., n, such that ( a[InlineMediaObject not available: see fulltext.] b) [InlineMediaObject not available: see fulltext.] c = a[InlineMediaObject not available: see fulltext.] ( b[InlineMediaObject not available: see fulltext.] c), s, r = 1,..., n, a, b, c ∈ A. The radical of such an algebra is defined as the intersection of the annihilators of irreducible A-modules, and it is proved that the radical coincides with the intersection of the maximal right ideals each of which is s-regular for some operation [InlineMediaObject not available: see fulltext.]. This implies that the quotient algebra by the radical is semisimple. If an n-tuple algebra is Artinian, then the radical is nilpotent, and the semisimple Artinian n-tuple algebra is the direct sum of two-sided ideals each of which is a simple algebra. Moreover, in terms of sandwich algebras, we describe a finite-dimensional n-tuple algebra A, over an algebraically closed field, which is a simple A-module. [ABSTRACT FROM AUTHOR]

The article focuses on the algebraic integer, a Pisot number. The Diophantine properties of Pisot numbers are explored, along with the evaluation of a graph with three-dimensional affine planes. The results of a computer simulation in the second minimal Pisot number of the roots of an equation are discussed.

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ℝ to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2 n-dimensional torus $\mathbb{T}^{2n} $. This reduction enables us to identify the Schwartz space on ℝ with the space of smooth sections Γ( T, ξ), compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of ℝ, we construct an index formula in accordance with the classical Atya-Singer formula. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to formulate and prove an L- L analog of Miyachi's theorem for connected nilpotent Lie groups with noncompact center for 2 ≤ p, q ≤ +∞. This allows us to solve the sharpness problem in both Hardy's and Cowling-Price's uncertainty principles. When G is of compact center, we show that the aforementioned uncertainty principles fail to hold. Our results extend those of [1], where G is further assumed to be simply connected, p = 2, and q = +∞. When G is more generally exponential solvable, such a principle also holds provided that the center of G is not trivial. Representation theory and a localized Plancherel formula play an important role in the proofs. [ABSTRACT FROM AUTHOR]

The optimal control of the solutions of the initial-finish problem for Sobolev-type equations is studied. The abstract results obtained in the paper are applied to the linear Hoff model on graphs. [ABSTRACT FROM AUTHOR]

Following the terminology introduced by V. V. Trofimov and A. T. Fomenko, we say that a self-adjoint operator $$\varphi :\mathfrak{g}* \to \mathfrak{g}$$ is sectional if it satisfies the identity ad a = ad x, $$x \in \mathfrak{g}*$$, where $$\mathfrak{g}$$ is a finite-dimensional Lie algebra and $$a \in \mathfrak{g}*$$ and $$\beta \in \mathfrak{g}$$ are fixed elements. In the case of a semisimple Lie algebra $$\mathfrak{g}$$, the above identity takes the form [ ϕx, a] = [ β, x] and naturally arises in the theory of integrable systems and differential geometry (namely, in the dynamics of n-dimensional rigid bodies, the argument shift method, and the classification of projectively equivalent Riemannian metrics). This paper studies general properties of sectional operators, in particular, integrability and the bi-Hamiltonian property for the corresponding Euler equation $$\dot x = ad_{\varphi x}^* x$$. [ABSTRACT FROM AUTHOR]

It is proved in the paper that the multiplication operation on an arbitrary E-algebra can be extended to an E-algebra A-morphism. As a corollary, it is proved that every May algebra defined by an E-algebra is a Cartan object in the category of May algebras. [ABSTRACT FROM AUTHOR]

The Hoffman-Singleton graph is the unique strongly regular graph with parameters (50, 7, 0, 1). A well-known hypothesis states that a distance-regular graph in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph has intersection array {50, 42, 1; 1, 2, 50} or {50, 42, 9; 1, 2, 42}. In the present paper, we prove this hypothesis under the assumption that a distance-regular graph is a Terwilliger graph and the graph diameter is at most 5. [ABSTRACT FROM AUTHOR]

In this paper, we obtain new algebraic identities between hypergeometric functions satisfying linear differential equations of arbitrary order as well as between the fundamental matrices of hypergeometric equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic analog of the Baker-Hausdorff formula for operators satisfying the canonical commutation relations. This formula can be used to obtain an exact solution to the problem of the dynamics of a quantum system interacting with the environment. [ABSTRACT FROM AUTHOR]

The paper is devoted to weighted spaces ℒ( G) on a locally compact group G. If w is a positive measurable function on G, then the space ℒ( G), p ≥ 1, is defined by the relation ℒ( G) = { f: fw ∈ ℒ p ( G)}. The weights w for which these spaces are algebras with respect to the ordinary convolution are treated. It is shown that, for p > 1, every sigma-compact group admits a weight defining such an algebra. The following criterion is proved (which was known earlier for special cases only): a space ℒ( G) is an algebra if and only if the function w is semimultiplicative. It is proved that the invariance of the space ℒ( G) with respect to translations is a sufficient condition for the existence of an approximate identity in the algebra ℒ( G). It is also shown that, for a nondiscrete group G and for p > 1, no approximate identity of an invariant weighted algebra can be bounded. [ABSTRACT FROM AUTHOR]

In this paper, an inequality between the L q -mean of the kth derivative of an algebraic polynomial of degree n ≥ 1 and the L 0-mean of the polynomial on a closed interval is obtained. Earlier, the author obtained the best constant in this inequality for k = 0, q ∈ [0,∞] and 1 ≤ k ≤ n, q ∈ {0} ∪ [1,∞]. Here a newmethod for finding the best constant for all 0 ≤ k ≤ n, q ∈ [0,∞], and, in particular, for the case 1 ≤ k ≤ n, q ∈ (0, 1), which has not been studied before is proposed. We find the order of growth of the best constant with respect to n as n → ∞ for fixed k and q. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate whether the roots of a biquadratic equation determined by a pair of real symmetric positive definite matrices of order 3 and a three-dimensional vector of parameters are real. We obtain the explicit representation of the discriminant of such a polynomial as the sum of at most two squares. [ABSTRACT FROM AUTHOR]