Bivariate Binomial Moments and Bonferroni-Type Inequalities

We obtain bivariate forms of Gumbel’s, Fréchet’s and Chung’s linear inequalities for P(S ≥ u, T ≥ v) in terms of the bivariate binomial moments {S _{i, j} }, 1 ≤ i ≤ k,1 ≤ j ≤ l of the joint distribution of (S, T). At u = v = 1, the Gumbel and Fréchet bounds improve monotonically with non-decreasing (k, l). The method of proof uses combinatorial identities, and reveals a multiplicative structure before taking expectation over sample points.     

Boundary value problem with normal derivatives for a higher-order elliptic equation on the plane

For an elliptic operator of order 2l with constant (and only leading) real coefficients, we consider a boundary value problem in which the normal derivatives of order (k _j ?1), j = 1,..., l, where 1 ≤ k ₁ < ··· < k _l, are specified. It becomes the Dirichlet problem for kj = j and the Neumann problem for k _j = j + 1. We obtain a sufficient condition for the Fredholm property of which problem and derive an index formula.     

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t₀, t₁, …, t_N} and Ω_N = {x₀, x₁, …, x_N–1}, where x_j = (t_j + t_{j + 1})/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums S_{n, N}( f, x) with respect to the system of polynomials {p?_k,N(x)} _k=0 ^N–1 , forming an orthonormal system on nonuniform point systems Ω_N consisting of finite number N of points from the segment [–1, 1] with weight Δt_j = t_{j + 1}–t_j. We find the growth order for the Lebesgue function L_n,N (x) of the considered partial discrete Fourier sums S_n,N ( f, x) as n = O(δ _N ^?2/7 ), δ_N = max_{0≤ j≤N?1} Δt_j More exactly, we have a two-sided pointwise estimate for the Lebesgue function L_{n, N}(x), depending on n and the position of the point x from [–1, 1].     

The Upper Connected Vertex Detour Monophonic Number of a Graph

For any vertex x in a connected graph G of order n ≥ 2, a set S _x ? V (G) is an x-detour monophonic set of G if each vertex v ∈ V (G) lies on an x-y detour monophonic path for some element y in S _x . The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dm _x (G). A connected x-detour monophonic set of G is an x-detour monophonic set S _x such that the subgraph induced by S _x is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by cdm _x (G). A connected x-detour monophonic set S _x of G is called a minimal connected x-detour monophonic set if no proper subset of S _x is a connected x-detour monophonic set. The upper connected x-detour monophonic number of G, denoted by cdm⁺ _x (G), is defined to be the maximum cardinality of a minimal connected x-detour monophonic set of G. We determine bounds and exact values of these parameters for some special classes of graphs. We also prove that for positive integers r,d and k with 2 ≤ r ≤ d and k ≥ 2, there exists a connected graph G with monophonic radius r, monophonic diameter d and upper connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j,k,l and n with 2 ≤ j ≤ k ≤ l ≤ n - 3, there exists a connected graph G of order n with dm _x (G) = j,dm⁺ _x (G) = k and cdm⁺ _x (G) = l for some vertex x in G.     

Block partitions of sequences

Given a sequence A = (a ₁, …, a _n ) of real numbers, a block B of A is either a set B = {a _i , a _i+1, …, a _j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a _i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B ₁, …, B _k with |b _i ?b _j | ≤ 1 for every i, j. We extend this result in several directions.     

Hermite—Padé Approximants of the Mittag-Leffler Functions

The convergence rate of type II Hermite–Padé approximants for a system of degenerate hypergeometric functions {₁F₁(1, γ; λ_jz)} _j=1 ^k is found in the case when the numbers {λ_j} _j=1 ^k are the roots of the equation λ^k = 1 or real numbers and \(\gamma\in\mathbb{C}\;\backslash\left\{0,-1,-2,...\right\}\). More general statements are obtained for approximants of this type (including nondiagonal ones) in the case of k = 2. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.     

The Relaxed Edge-Coloring Game and <Emphasis Type="Italic">k</Emphasis>-Degenerate Graphs

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ?j and d≥2j+2 for 0≤j≤Δ?1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math. 307, 1767–1775, 2007). More broadly, we generalize the main result in Dunn, C. (Discret. Math. 307, 1767–1775, 2007) by showing that if G is k-degenerate with Δ(G)=Δ and j∈[Δ+k?1], then there exists a function h(k,j) such that Alice has a winning strategy for this game with r=Δ+k?j and d≥h(k,j).     

On criteria of estimation of the errors of cubature formulas

The paper considers cubature formulas for calculating integrals of functions f(X), X = (x ₁, …, x _n ) which are defined on the n-dimensional unit hypercube K ⁿ = [0, 1] ⁿ and have integrable mixed derivatives of the kind \(\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)\), 0 ≤ α _j ≤ 2. We estimate the errors R[f] = \(\smallint _{K^n } \) f(X)dX ? Σ _{k = 1} ^N c _k f(X(k)) of cubature formulas (c _k > 0) as functions of the weights c _k of nodes X(k) and properties of integrable functions. The error is estimated in terms of the integrals of the derivatives of f over r-dimensional faces (r≤n) of the hypercube K ⁿ : |R(f)| ≤ \(\sum _{\alpha _j } \) G(α _j )\(\int_{K^r } {\left| {\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)} \right|} \) dX _r , where coefficients G(α _j ) are criteria which depend only on parameters c _k and X(k). We present an algorithm to calculate these criteria in the two- and n-dimensional cases. Examples are given. A particular case of the criteria is the discrepancy, and the algorithm proposed is a generalization of those used to compute the discrepancy. The results obtained can be used for optimization of cubature formulas as functions of c _k and X(k).     

Minimum Difference Representations of Graphs

Define a k-minimum-difference-representation (k-MDR) of a graph G to be a family of sets \({\{S(v): v\in V(G)\}}\) such that u and v are adjacent in G if and only if min{|S(u)?S(v)|, |S(v)?S(u)|} ≥ k. Define ρ _min(G) to be the smallest k for which G has a k-MDR. In this note, we show that {ρ _min(G)} is unbounded. In particular, we prove that for every k there is an n ₀ such that for n > n ₀ ‘almost all’ graphs of order n satisfy ρ _min(G) > k. As our main tool, we prove a Ramsey-type result on traces of hypergraphs.     

Unexpected spectral asymptotics for wave equations on certain compact spacetimes

We study the spectral asymptotics of wave equations on certain compact spacetimes, where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime S¹×S². For the Laplacian on S¹×S², theWeyl asymptotic law gives a growth rate O(s^3/2) for the eigenvalue counting function N(s) = #{λ_j: 0 ≤ λ_j ≤ s}. For the wave operator, there are two corresponding eigenvalue counting functions: N^±(s) = #{λ_j: 0 < ±λ_j ≤ s}, and they both have a growth rate of O(s²). More precisely, there is a leading term π²s²/4 and a correction term of as^3/2, where the constant a is different for N^±. These results are not robust in that if we include a speed of propagation constant to the wave operator, the result depends on number theoretic properties of the constant, and generalizations to S¹ × S^q are valid for q even but not q odd. We also examine some related examples.     

The program iteration method in a game problem of guidance

Let a sequence of d-dimensional vectors n _k = (n _k ¹ , n _k ² ,..., n _k ^d ) with positive integer coordinates satisfy the condition n _k ^j = α _j m _k +O(1), k ∈ ?, 1 ≤ j ≤ d, where α ₁ > 0,..., α _d > 0 and {m _k } _k=1 ^∞ is an increasing sequence of positive integers. Under some conditions on a function φ: [0,+∞) → [0,+∞), it is proved that, if the sequence of Fourier sums \({S_{{m_k}}}\) (g, x) converges almost everywhere for any function g ∈ φ(L)([0, 2π)), then, for any d ∈ ? and f ∈ φ(L)(ln⁺ L) ^d?1([0, 2π) ^d ), the sequence \({S_{{n_k}}}\) (f, x) of rectangular partial sums of the multiple trigonometric Fourier series of the function f and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.     

Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices

Let {c _j } _j=0 ⁿ be a sequence of matrix moments associated with a matrix of measures supported on the unit circle, and let {P _j } _j=0 ⁿ be its corresponding sequence of monic matrix orthogonal polynomials. In this contribution, we consider a perturbation on the moments and find an explicit relation for the perturbed orthogonal polynomials in terms of {P _j } _j=0 ⁿ . We also obtain an expression for the corresponding second kind polynomials.     

Tensor products of complementary series of rank one Lie groups

We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.     

Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κ _k (G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λ _k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T ₁, T ₂, …, T _l in G such that S ? V (T _i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ ₃(G □ H) ? λ ₃(G) + λ ₃(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.     

Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials

For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent (over k) rational functions of the structure constants. (2) There exists an “algebraic normal form” to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis—namely, there are two finite systems of nonconstant polynomials on the space of structure constants, {f_i}i∈I and {b_j}j∈J, such that the ideal generated by the set {f_i}i∈I is prime and, for every tuple c of structure constants satisfying the property b_j(c) ≠ 0 for all j ∈ J, there exists a unique new basis of this algebra in which the tuple c′ of its structure constants satisfies the property f_i(c′) = 0 for all i ∈ I.     

A new characterization of symmetric group by NSE

Let G be a group and ω(G) be the set of element orders of G. Let k ∈ ω(G) and m _k (G) be the number of elements of order k in G. Let nse(G) = {m _k (G): k ∈ ω(G)}. Assume r is a prime number and let G be a group such that nse(G) = nse(S _r ), where S _r is the symmetric group of degree r. In this paper we prove that G ? S _r , if r divides the order of G and r ² does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.     

Metric projection onto subsets of compact connected two-dimensional Riemannian manifolds

The paper is focused on combinatorial properties of the metric projection P _E of a compact connected Riemannian two-dimensional manifold M ² onto its subset E consisting of k closed connected sets E _j . A point x ∈ M ² is called singular if P _E (x) contains points from at least three distinct E _j . An exact estimate of the number of singular points is obtained in terms of k and the type of the manifold M ². A similar estimate is proved for subsets E of a normed plane consisting of a finite number of connected components.     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

On Functional Versions of the Arc-Sine Law

Let X ₁,X ₂,… be a sequence of random variables. Let S _k =X ₁+???+X _k and assume that S _k /b _k converges in distribution for some numerical sequence (b _k ). We study the weak convergence of the random processes {Λ _n (z), z∈?}, where

$\Lambda_{n}(z)=\frac{1}{n}\sum_{k=1}^{n}I\left\{\frac{S_{k}}{b_{k}}\leq z\right\}.$

We consider the same problem when the normalized partial sums S _k /b _k are replaced by other functionals of the sequence (X _n ). In particular, we investigate the case of sample extremes in detail.

     

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.