Asymptotic Behavior of Certain Ducci Sequences Asymptotic Behavior of Certain Ducci Sequences

Asymptotic Behavior of Certain Ducci Sequences

The Ducci map is considered applied to vectors in R. It is shown that for certain starting vectors of odd length, the corresponding sequence of iterates asymptotically approaches a periodic vector, but never becomes periodic itself. On the other hand, for any starting vectors not of this type, it is shown that the corresponding Ducci sequence becomes eventually periodic. A simple method for det...

Ducci-sequences and cyclotomic polynomials

We study Ducci-sequences using basic properties of cyclotomic polynomials over F2. We determine the period of a given Ducci-sequence in terms of the order of a polynomial, and in terms of the multiplicative orders of certain elements in finite fields. We also compute some examples and study links between Duccisequences, primitive polynomials and Artin’s conjecture on primitive roots. © 2005 Els...

Ducci Sequences in Higher Dimensions

A Ducci sequence is a sequence of n-tuples of integers obtained by iterating the map (a1, . . . , an) !→ (|a1 − a2|, |a2 − a3|, . . . , |an − a1|). In this paper, we generalize the concept of Ducci sequences to sequences of d-dimensional arrays, extend some of the basic results on Ducci sequences to this case, and point out some new phenomena that occur. Our main tool is the description of Ducc...

Asymptotic Behavior of Multivariate Reward Processes with Nonlinear Reward Functions

Asymptotic Behavior of Weighted Sums of Weakly Negative Dependent Random Variables

Let be a sequence of weakly negative dependent (denoted by, WND) random variables with common distribution function F and let be other sequence of positive random variables independent of and for some and for all . In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, randomly weighted sums and randomly indexed weighted sums of heavy...