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...

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...

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...

A graph G is said to be 2-divisible if for all (nonempty) induced subgraphs H of G, V (H) can be partitioned into two sets A,B such that ω(A) < ω(H) and ω(B) < ω(H). A graph G is said to be perfectly divisible if for all induced subgraphs H of G, V (H) can be partitioned into two sets A,B such that H[A] is perfect and ω(B) < ω(H). We prove that if a graph is (P5, C5)-free, then it is 2-divisibl...

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 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...

If F is known to be free, with generators gi, /3 may be constructed by setting j3g; = &», with bi so chosen that pbi = agi. Conversely, let F have the cited property and represent F as a quotient group Fo/Ro> where F0 is a free abelian group. Choose A—F and B = Fo in (1.1), let a be the identity, and p the given homomorphism of .Fo onto F with kernel Ro. Then, by (1.2), a=p/3 is an isomorphism,...

We examine the different ways of writing a cover of curves φ : C → D over a field K as a composition φ = φn ◦ φn−1 ◦ · · · ◦ φ1, where each φi is a cover of curves over K of degree at least 2 which cannot be written as the composition of two lower-degree covers. We show that if the monodromy group Mon(φ) has a transitive abelian subgroup then the sequence (deg φi)1≤i≤n is uniquely determined up...