will contain infinitely many prime terms — known as Mersenne primes. The Mersenne prime conjecture is related to a classical problem in number theory concerning perfect numbers. A whole number is said to be perfect if, like 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14, it is equal to the sum of all its divisors apart from itself. Euclid pointed out that 2k−1(2k − 1) is perfect whenever 2 − 1 is pr...

where a . 21, p . 0 and k [ N is fixed, has positive non-oscillatory solutions which converge to the positive equilibrium x 1⁄4 aþ 1: This result solves Open Problem 1 in Stević, 2005, On the recursive sequence xnþ1 1⁄4 aþ ðxpn21=xnÞ; Journal of Applied Mathematics and Computing 18(1–2), 229–234, as well as, Open Problem 1 in DeVault, Kent and Kosmala, 2003, On the recursive sequence xnþ1 1⁄4 p...

New theorems of asymptotical stability and uniformly asymptotical stability for nonautonomous difference equations are given in this paper. The classical Liapunov asymptotical stability theorem of nonautonomous difference equations relies on the existence of a positive definite Liapunov function that has an indefinitely small upper bound and whose variation along a given nonautonomous differenc...

The oscillatory behavior of some differential and difference equations have been investigated (see, for instance, [1], [3], [4], [5]). In recent years, the oscillations of discrete analogues of delay differential equations have been given [2], [7]. Furthermore, explicit conditions for the oscillation of difference equations with constant coefficients have been studied [6]. Erbe and Zhang [2] ha...

Abstract. The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix p...

In this paper we put into relation the index of an infinite aperiodic word and its recurrence function. With the use of this relation, we then give a new characterization of Sturmian words. As a byproduct, we give a new proof of theorem of Damanik and Lenz describing the index of a Sturmian word in terms of the continued fraction expansion of its slope.

We prove severaìautomatic' ergodicity results for cocycles on a discrete nonsingular ergodic equivalence relation on a probability space (X; S;) with values in virtually nilpotent groups. The hypotheses required for automatic ergodicity are invari-ance or quasi-invariance of the cocycles under asymptotically central automorphisms of the equivalence relation. If the cocycles have certain recurre...

The stability analysis for linear implicit m-th order difference equations is discussed. We allow the leading coefficient coefficient to be singular, i.e., we include the situation that the system does not generate an explicit recursion. A spectral condition for the characterization of asymptotic stability is presented and computable formulas are derived for the real and complex stability radii...

Corrigendum Corrigendum to " WKB (Liouville–Green) analysis of second order difference equations and applications " [J. There is a hypothesis which is missing from Theorem 2.3 in the above article [1]. The line above Eq. (2.34). 'For N 1 infinite assume that (2.30) and (2.31) hold for all n ≥ N , that' should read 'For N 1 infinite assume that (2.30) and (2.31) hold for all n ≥ N , that for eac...

To better understand the evolution of dispersal in spatially heterogenous landscapes, we study difference equation models of populations that reproduce and disperse in a landscape consisting of k patches. The connectivity of the patches and costs of dispersal are determined by a k × k column substochastic matrix S where Sij represents the fraction of dispersing individuals from patch j that end...