An n × n sign pattern A is said to be potentially nilpotent if there exists a nilpotent real matrix B with the same sign pattern as A. Let D_{n,r} be an n × n sign pattern with 2 \geqslant r \geqslant n such that the superdiagonal and the (n, n) entries are positive, the (i, 1) (i = 1,..., r) and (i, i − r + 1) (i = r + 1,..., n) entries are negative, and zeros elsewhere. We prove that for r \geqslant 3 and n \geqslant 4r − 2, the sign pattern D_{n,r} is not potentially nilpotent, and so not spectrally arbitrary., Yanling Shao, Yubin Gao, Wei Gao., and Obsahuje seznam literatury
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
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