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Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum\limits_{k=1}^{n}X_k a_k$ according to the arithmetic structure of vectors $a_k$. Recently, the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove multidimensional generalizations of the results Eliseeva and Zaitsev (2012). They are also the refinements of the results of Friedland and Sodin (2007) and Rudelson and Vershynin (2009)., Comment: 13 pages

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.

In this paper, we introduce a novel polynomial-time algorithm to compute graph invariants based on the idea of a modified random walk on graphs. Though not proved to be a full graph invariant yet, our method gives the right answer for the graph instances other well-known methods could not compute (such as special Furer gadgets and point-line incidence graphs of finite projective planes of higher degrees).

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.

In this paper we describe all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application we describe all two-term tilting complexes over Brauer star algebra and compute their endomorphism rings., 16 pages

In this paper, we study the problem on the existence of nonmeasurable automorphisms of finite-dimensional and infinite-dimensional Lie groups over the field of real numbers and also over non-Archimedean local fields. The nonmeasurability of automorphisms is considered relative to real-valued measures and also measures with values in non-Archimedean local fields. Their existence is proved and a procedure for their construction is given. Their application to the construction of nonmeasurable irreducible unitary representations is demonstrated.

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.

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 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.

In this paper, we consider a new class of random dynamical systems that contains, in particular, neural networks and complicated circuits. For these systems, we consider the viability problem: we suppose that the system survives only the system state is in a prescribed domain Π of the phase space. The approach developed here is based on some fundamental ideas proposed by A. Kolmogorov, R. Thom, M. Gromov, L. Valiant, L. Van Valen, and others. Under some conditions it is shown that almost all systems from this class with fixed parameters are unstable in the following sense: the probability P t to leave Π within the time interval [0, t] tends to 1 as t → ∞. However, it is allowed to change these parameters sometimes (“evolutionary” case), then it may happen that P t

In the late 1970s and early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NP-hard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved spaces. Bibliography: 50 titles.

In the paper, we prove that every automorphism of any adjoint Chevalley group of type B2 or G2 is standard, i.e., it is a composition of an “inner” automorphism, a ring automorphism, and a central automorphism.

In this paper, we introduce and study a new class of generalized metric spaces, which we call k*-metrizable spaces, and suggest various applications of such spaces in topological algebra, functional analysis, and measure theory.

We discuss modular forms as objects of computer algebra and as elements of certain p-adic Banach modules. We discuss a problem-solving approach in number theory, which is based on the use of generating functions and their connection with modular forms. In particular, the critical values of various L-functions of modular forms produce nontrivial but computable solutions of arithmetical problems. Namely, for a prime number we consider three classical cusp eigenforms $$f_j (z) = \sum\limits_{n = 1}^\infty {a_{n,j} e(nz)} \in \mathcal{S}_{k_j } (N_j ,\psi _j ) (j = 1,2,3)$$ of weights k 1, k 2, and k 3, of conductors N 1, N 2, and N 3, and of Nebentypus characters ψj mod N j . The purpose of this paper is to describe a four-variable p-adic L-function attached to Garrett’s triple product of three Coleman’s families $$k_j \mapsto \left\{ {f_{j,k_j } = \sum\limits_{n = 1}^\infty {a_{n,j} (k)q^n } } \right\}$$ of cusp eigenforms of three fixed slopes $$\sigma _j = v_p (\alpha _{p,j}^{(1)} (k_j )) \geqslant 0$$ , where $$\alpha _{p,j}^{(1)} = \alpha _{p,j}^{(1)} (k_j )$$ is an eigenvalue (which depends on k j ) of Atkin’s operator U = U p .

The paper presents an O(mn2nlog Z) deterministic algorithm for solving the mean payoff game problem, m and n being the numbers of arcs and vertices, respectively, in the game graph, and Z being the maximum weight (the weights are assumed to be integers). The theoretical basis for the algorithm is the potential theory for mean payoff games. This theory allows one to restate the problem in terms of solving systems of algebraic equations with minima and maxima. Also, in order to solve the mean payoff game problem, the arc reweighting technique is used. To this end, simple modifications, which do not change the set of winning strategies, are applied to the game graph; in the end, a trivial instance of the problem is obtained. It is shown that any game graph can be simplified by n reweightings. Bibliography: 16 titles.

In this paper, we prove that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly ordered associative ring on some specially defined subgroup coincides with the composition of an inner automorphism of the semigroup, an order-preserving automorphism of the ring, and a central homothety.

In this paper, we calculate the space ExtGL(n1)(Ln(λ), Ln(μ)), where GL(n) is the general linear group of degree n over an algebraically closed field of positive characteristic, Ln(λ) and Ln(μ) are rational irreducible GL(n)-modules with highest weights λ and μ, respectively, the restriction of Ln(λ) to any Levi subgroup of GL(n) is semisimple, λ is a p-restricted weight, and μ does not strictly dominate λ.

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f α : K n → K n depending on “time” α ∈ {K − 0} such that f α −1 = f −αM, the relation f α1 (x) = f α2 (x) for some x ∈ Kn implies α1 = α2, and each map f α has no invariant points. The neighborhood {f α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kn. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α1, υ, α2), s ≤ d, where αi + αi+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ a = f α1 × ⋯ × f αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems Ln(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system Ln(q). If K = Fq, the graphs L(n, Fq) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems Ln(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles.

Let A and B be (nn)-matrices. For an index set S ? {1, …, n}, denote by A(S) the principal submatrix that lies in the rows and columns indexed by S. Denote by S' the complement of S and define ?(A, B) = $$\mathop \sum \limits_S $$ det A(S) det B(S'), where the summation is over all subsets of {1, …, n} and, by convention, det A(Ø) = det B(Ø) = 1. C. R. Johnson conjectured that if A and B are Hermitian and A is positive semidefinite, then the polynomial ?(?A,-B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for n ? 3. Bapat also proved this result for any n with the condition that both A and B are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any n under the additional assumption that both A and B are matrices whose graph is a tree.

This paper proves that there are only finitely many subgroupsH of finite index in Sp(4,Z) such that coresponding quotient\(\mathcal{H}/H\) of the Siegel upper half space of rank two is not of general type.

The paper presents new two-sided bounds for the Perron root of a block-partitioned nonnegative matrix, improving Chistyakov’s bounds. The equality cases are analyzed. As an application, new conditions sufficient for a complex matrix to be a nonsingular H-matrix are obtained. Bibliography: 8 titles.