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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 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bm\theta\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bm\theta)= 1/n$ for almost every point $\vv x$ on $M$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent $w_0(\vv x)=1/n$ for almost all $\vv x\in \cM$ if and only if for any $\bm\theta\in\R^n$ the inhomogeneous exponent $w_0(\vv x,\bm\theta)=1/n$ for almost all $\vv x\in \cM$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered in \cite{Beresnevich-Velani-new-inhom}. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas of \cite{Beresnevich-Velani-new-inhom} while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory., Comment: Dedicated to A.O. Gelfond on what would have been his 100th birthday 13 pages

Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For ≥ 3, if every (r−2)-plane $$\overline {x_1 , \ldots ,x_{r - 1} } $$ , where the x i are generic points, also meets X in a point x r different from x 1,..., x r−1, then X is contained in a linear subspace L such that codim L X ≥ r − 2. In this paper, our purpose is to present another derivation of this result for r = 3 and then to introduce a generalization to nonequidimensional varieties. For the sake of clarity, we shall reformulate our problem as follows. Let Z be an equidimensional variety (maybe singular and/or reducible) of dimension n, other than a linear space, embedded into ℙr, where r ≥ n + 1. The variety of trisecant lines of Z, say V 1,3(Z), has dimension strictly less than 2n, unless Z is included in an (n + 1)-dimensional linear space and has degree at least 3, in which case dim V 1,3(Z) = 2n. This also implies that if dim V 1,3(Z) = 2n, then Z can be embedded in ℙ n + 1. Then we inquire the more general case, where Z is not required to be equidimensional. In that case, let Z be a possibly singular variety of dimension n, which may be neither irreducible nor equidimensional, embedded into ℙr, where r ≥ n + 1, and let Y be a proper subvariety of dimension k ≥ 1. Consider now S being a component of maximal dimension of the closure of $$\{ l \in \mathbb{G}(1,r)|\exists p \in Y, q_1 , q_2 \in Z\backslash Y, q_1 , q_2 ,p \in l\} $$ . We show that S has dimension strictly less than n + k, unless the union of lines in S has dimension n + 1, in which case dim S = n + k. In the latter case, if the dimension of the space is strictly greater than n + 1, then the union of lines in S cannot cover the whole space. This is the main result of our paper. We also introduce some examples showing that our bound is strict.

Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of extrema of curvature to the winding numbers of the curves (polygonal lines) and their evolutes is obtained. Also are considered higher-dimensional analogs of the four-vertex theorem for regular and shellable triangulations., Several changes in the last section. In the original version of this paper we claimed that any regular triangulation of a convex d-polytope has at least d ears. For a proof we used the same arguments as in Schatteman's paper [22]. Since this paper has certain gaps (see our paper [1]), the d -ears problem of a regular triangulation is still open

sp-Groups form an interesting and informative class of Abelian mixed groups. In this paper, we systematically study self-small sp-groups of finite rank and their endomorphism rings.

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of exceptional Lie type or Tits group.

A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class N κ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel $$ {\mathbf{N}}_{\omega }(z)\coloneq \frac{f(z)-\overline{f\left(\omega \right)}}{z-\overline{\omega}} $$ has κ negative squares on ℂ+. The generalized Stieltjes class $$ {\mathbf{N}}_{\kappa}^k\left(\kappa, k\in {\mathrm{\mathbb{Z}}}_{+}\right) $$ is defined as the set of functions f ϵ N κ such that z f ϵ N k . The full indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ consists in the following: Given κ, k ϵ ℤ+, and a sequence $$ \mathbf{s}={\left\{{s}_i\right\}}_{i=0}^{\infty } $$ of real numbers, to describe the set of functions $$ f\in {\mathbf{N}}_{\kappa}^k $$ , which satisfy the asymptotic expansion $$ f(z)=-\frac{s_0}{z}-\cdots -\frac{s_2n}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right)\kern1em \left(z=-y\in {\mathrm{\mathbb{R}}}_{-},y\uparrow \infty \right) $$ for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A [0;N] generated by $$ {\mathfrak{J}}_{\left[0;N\right]} $$ . The u-resolvent matrices of the operator A [0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ to be solvable and indeterminate are found. Explicit formulae for Pade approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.

It is well known that for a standard Brownian motion (BM) {B(t), t ≥ 0} with values in R d, its convex hull V (t) = conv{B(s), s ≤ t} with probability 1 for each t > 0 contains 0 as an interior point. We also know that the winding number of a typical path of a two-dimensional BM is equal to +∞. The aim of this paper is to show that these properties are not specifically “Brownian,” but hold for a much larger class of d-dimensional self-similar processes. This class contains, in particular, d-dimensional fractional Brownian motions and (concerning convex hulls) strictly stable Levy processes. Bibliography: 10 titles.

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, the number of ways for gluing together several polygons into a surface of genus g is investigated. An elementary proof of the formula for the generating function $$ {\mathrm{C}}_{\mathfrak{g}}^{\left[2\right]}(z) $$ of the number of gluings of a surface of genus g from two polygons is given. Moreover, a similar formula is proved for gluings of a surface of genus g from three polygons. As a consequence, a direct formula is obtained for the number of gluings of a torus from three polygons. Bibliography: 10 titles.

The paper deals with the existence of solutions for the dynamic equation on time scales $$ \begin{array}{cc}\hfill {u}^{\varDelta \varDelta \varDelta \varDelta}(t)= f\left( t, u\left(\sigma (t)\right),{u}^{\varDelta \varDelta}(t)\right),\hfill & \hfill t\in {\left[0,1\right]}_T,\hfill \end{array} $$ with the multipoint boundary conditions $$ \begin{array}{cccc}\hfill u(0)=0,\hfill & \hfill u\left(\sigma (1)\right)={\displaystyle \sum_{i=1}^{m-2}{a}_i u\left({\xi}_i\right),}\hfill & \hfill {u}^{\varDelta \varDelta}(0)=0,\hfill & \hfill {u}^{\varDelta \varDelta}\left(\sigma (1)\right)={\displaystyle \sum_{j=1}^{n-2}{b}_j{u}^{\varDelta \varDelta}\left({\eta}_j\right),}\hfill \end{array} $$ where T is a time scale [0, 1] T = {t ∈ T | 0 ≤ t ≤ 1}, a i > 0, i = 1, 2, …, m − 2, b j > 0, j = 1, 2, …, n − 2, 0 < ξ1 < ξ2 < … < ξ m−2 < ρ(1), and 0 < η 1 < η 2 < … < η n−2 < ρ(1). The existence result is given by using Green’s function, the method of upper and lower solutions, and the monotone iterative technique. We also give an example to illustrate our result.

The paper considers the lattice of fully invariant subgroups of the cotorsion hull when a separable primary group T is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation difined on the set of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.

We consider a differential equation $$ {u^{(n) }}(t)+p(t)\left| {u\left( {\sigma (t)} \right)\left| {^{{\mu (t)}}} \right.\mathrm{sign}\,u\left( {\sigma (t)} \right)=0,} \right. $$ where p ∈ L loc (R +; R +), μ ∈ C(R +; (0, +∞)), σ ∈ C(R +; R +), and σ(t) ≥ t for t ∈ R +. We say that the equation is almost linear if the condition lim t→+∞ μ(t) = 1 is satisfied. At the same time, if lim sup t→+∞ μ(t) ≠ 1 or lim inf t→+∞ μ(t) ≠ 1, then Eq. is called an essentially nonlinear differential equation. The oscillatory properties of almost linear differential equations have been extensively studied. In the paper, new sufficient (necessary and sufficient) conditions are established for a general class of essentially nonlinear functional differential equations to have Property A.

The classification problem for finite and infinite algebraic lattices has also been extensively addressed. A wild classification problem contains the classification problem of pairs of matrices up to simultaneous similarity. In this paper, we prove that the classification problem for graphs is wild by reducing the classification problem for finite 2-nilpotent p-groups to the classification problem for graphs.

It is known that the Girsanov exponent $ {{\mathfrak{z}}_t} $ , which is a solution of the Doleans-Dade equation $ {{\mathfrak{z}}_{{\, \mathfrak{t}}}}=1+\int\limits_0^t {{{\mathfrak{z}}_s}} \alpha (s)d{B_s} $ generated by a Brownian motion B t and a random process α(t) with $ \int\limits_0^t {{\alpha^2}(s)ds0, $$ holds. In this paper, we show that $ \int\limits_0^t {\alpha (s)d{B_s}} $ , can lie replaced by a purely discontinuous square integrable martingale M t with paths from the Skorokhod space $ {{\mathbb{D}}_{{\left[ {0,\infty } \right)}}} $ hailing jumps $ \alpha (s)\varDelta {M_t}>-1 $ . The method of proof differs from the original Benes proof. Bibliography: 13 titles.

The aim of this paper is to study the asymptotic properties and oscillation of the nth-order delay differential equation E $$ {{\left( {r(t){{{\left[ {{z^{{\left( {n-1} \right)}}}(t)} \right]}}^{\gamma }}} \right)}^{\prime }}+q(t)f\left( {x\left( {\tau (t)} \right)} \right)=0. $$ The results obtained are based on some new comparison theorems that reduce the problem of oscillation of an nth-order equation to the problem of oscillation of one or more first-order equations. We handle both cases $$ \begin{array}{*{20}{c}} {\int\limits^{\infty } {{r^{{-1/\gamma }}}(t)dt=\infty } } & {\mathrm{and}} & {\int\limits^{\infty } {{r^{{-1/\gamma }}}(t)dt

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.

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.

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.

In this paper we survey the Tychono. property for discrete groups. General results and relations with amenability, with discrete potential theory (random walks), and with the theory of weights and of characters are discussed. We then present a characterization of Tychonoff property for virtually abelian groups, for polycyclic groups, and for linear groups. We also consider the case of metabelian groups, and several examples are given.

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.

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.

We present two methods of constructing amenable (in the sense of Greenleaf) actions of nonamenable groups. In the first part of the paper, we construct a class of faithful transitive amenable actions of the free group using Schreier graphs. In the second part, we show that every finitely generated residually finite group can be embedded into a bigger residually finite group, which acts level-transitively on a locally finite rooted tree, so that the induced action on the boundary of the tree is amenable on every orbit. Bibliography: 25 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.

In this paper we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds up to isomorphism., Comment: 46 pages, AMS-LaTeX

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.