This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of d-orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when d = 1. The underlying algebraic framework allowed a systematic derivation of the recurrence relations, difference equation, lowering and rising operators and generating functions which these polynomials satisfy., Fethi Bouzeffour, Hanen Ben Mansour, Ali Zaghouani., and Obsahuje bibliografii
A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M.Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n 6 x with k prime factors such that a fixed quadratic equation has exactly 2k solutions modulo n., Neha Prabhu., and Seznam literatury
Let $\Delta_{n,d}$ (resp. $\Delta_{n,d}'$) be the simplicial complex and the facet ideal $I_{n,d}=(x_1\cdots x_d,x_{d-k+1}\cdots
x_{2d-k},\ldots,x_{n-d+1}\cdots x_n)$ (resp. $J_{n,d}=(x_1\cdots x_d,x_{d-k+1}\cdots x_{2d-k},\ldots,x_{n-2d+2k+1}\cdots x_{n-d+2k},x_{n-d+k+1}\cdots x_nx_1\cdots x_k)$). When $d\geq2k+1$, we give the exact formulas to compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}^t$ for all $t\geq1$. When $d=2k$, we compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}$, and give lower bounds for the depth and Stanley depth of quotient rings $S/I_{n,d}^t$ for all $t\geq1$., Xiaoqi Wei, Yan Gu., and Obsahuje bibliografické odkazy
We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This affirmatively answers a question of Chung and Graham (2002) for the particular case of Cayley graphs of abelian groups, while in general the answer is negative., Yoshiharu Kohayakawa, Vojtěch Rödl, Mathias Schacht., and Obsahuje seznam literatury
Let $G$ be a locally compact group and let $1 \le p < \infty.$ Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space $L^p(G)$ in terms of the weights. Sufficient and necessary conditions for disjoint hypercyclic and disjoint supercyclic powers of weighted translations generated by aperiodic elements on groups will be given., Liang Zhang, Hui-Qiang Lu, Xiao-Mei Fu, Ze-Hua Zhou., and Obsahuje bibliografické odkazy
The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order n which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class Ω_{n} of doubly stochastic matrices (convex hull of n×n permutation matrices). An alternative description of this partial order is given. We define a class of special faces of Ω_{n} induced by permutation matrices, which we call Bruhat faces. Several examples of Bruhat faces are given and several results are presented., Richard A. Brualdi, Geir Dahl, Eliseu Fritscher., and Obsahuje seznam literatury
Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph G, a hereditary graph property P and l\geqslant 1 we define X{'_{P,l}} to be the minimum number of colours needed to properly colour the edges of G, such that any subgraph of G induced by edges coloured by (at most) l colours is in P. We present a necessary and sufficient condition for the existence of X{'_{P,l}} . We focus on edge-colourings of graphs with respect to the hereditary properties Ok and Sk, where Ok contains all graphs whose components have order at most k+1, and Sk contains all graphs of maximum degree at most k. We determine the value of X{'_{{S_k},l}}(G) for any graph G,k \geqslant 1, l\geqslant 1 and we present a number of results on X{'_{{O_k},l}}(G) ., Samantha Dorfling, Tomáš Vetrík., and Obsahuje seznam literatury
In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for g(n, l), the biggest number k guaranteeing that there exist l graphs on n vertices, each two having edit distance at least k. By edit distance of two graphs G, F we mean the number of edges needed to be added to or deleted from graph G to obtain graph F. This new extremal number g(n, l) is closely linked to the edit distance of graphs. Using probabilistic methods we show that g(n, l) is close to \frac{1} {2}\left( {\begin{array}{*{20}c} n // 2 // \end{array} } \right) for small values of l > 2. We also present some exact values for small n and lower bounds for very large l close to the number of non-isomorphic graphs of n vertices., Tomasz Dzido, Krzysztof Krzywdziński., and Obsahuje seznam literatury