A random variableX is geometrically infinitely divisible iff for every p ∈ (0, 1) there exists random variable Xp such that X d = ∑T (p) k=1 Xp,k, where Xp,k’s are i.i.d. copies of Xp, and random variable T (p) independent of {Xp,1, Xp,2, . . .} has geometric distribution with the parameter p. In the paper we give some new characterization of geometrically infinitely divisible distribution. The...

Based on congruences mod p and on properties of Bernoulli polynomials and Bernoulli numbers, several conditions are derived for x, k > 2 to satisfy the Diophantine equation 1k + 2k H-\(x 1 )* = xk . It is proved that ord2(x 3) = orà2k + 3 and that x cannot be divisible by any regular prime. Furthermore, by using the results of experiments with the above conditions on an SGI workstation it is pr...

In a recent article Pillai (1990, Ann. Inst. Statist. Math., 42, 157-161) showed that the distribution 1-E~(-x~), 0 < c~ _< 1; 0 _< x, where E~(x) is the Mittag-Lettter function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein fun...

This paper presents an off-line divisible e-cash scheme where a user can withdraw a divisible coin of monetary value 2 that he can parceled and spend anonymously and unlinkably. We present the construction of a security tag that allows to protect the anonymity of honest users and to revoke anonymity only in case of cheat for protocols based on a binary tree structure without using a trusted thi...

We investigate the semigroup structure of bosonic Gaussian quantum channels. Particular focus lies on the sets of channels which are divisible, idempotent or Markovian (in the sense of either belonging to one-parameter semigroups or being infinitesimal divisible). We show that the non-compactness of the set of Gaussian channels allows for remarkable differences when comparing the semigroup stru...

We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, . . .. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is ne...

We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, . . .. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is ne...