The goal of this thesis is to explore the properties of a certain class of sequences, rigid divisibility sequences, generated by the iteration of certain polynomials whose coefficients are algebraic integers. The main goal is to provide, as far as is possible, a classification and description of those polynomials which generate rigid divisibility sequences.

Let $\mathbb{P}_G([0,\infty))$ and $\mathbb{P}_G^{'}([0,\infty))$ be the sets of positive semidefinite definite matrices order $n$, respectively, with nonnegative entries, where some positions zero entries are restricted by a simple graph $G$ $n$ vertices. It is proved that for connected $n\geq 3$, set powers preserving semidefiniteness on precisely same as definiteness $\mathbb{P}_G^{'}([0,\in...

Definition 2.1. When a and b are integers, we say a divides b if b = ak for some k ∈ Z. We then write a | b (read as “a divides b”). Example 2.2. We have 2 | 6 (because 6 = 2 · 3), 4 | (−12), and 5 | 0. We have ±1 | b for every b ∈ Z. However, 6 does not divide 2 and 0 does not divide 5. Divisibility is a relation, much like inequalities. In particular, the relation 2 | 6 is not the number 3, e...

omputer 012.10.00 Abstract Many schema labelling approaches have been designed to facilitate querying of XML documents. The proposed algorithms are based on the fact that ancestor–descendant relationships among nodes can be quickly determined. Schema labelling is a family of technologies widely used in indexing tree, graph, or structured XML graph, in which a unique identifier is assigned to ea...

The function bk(n) is defined as the number of partitions of n that contain no summand divisible by k. In this paper we study the 2-divisibility of b5(n) and the 2and 3-divisibility of b13(n). In particular, we give exact criteria for the parity of b5(2n) and b13(2n).

Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical viewpoint. We exhibit calculations, together with a heuristic argument, to suggest that these sequences ...

We construct a random matrix model for the bijection Ψ between classical and free infinitely divisible distributions: for every d ≥ 1, we associate in a quite natural way to each ∗-infinitely divisible distribution μ a distribution Pμd on the space of d×d hermitian matrices such that Pμd ∗ Pνd = P μ∗ν d . The spectral distribution of a random matrix with distribution Pμd converges in probabilit...

We give necessary and sufficient conditions for existence and infinite divisibility of α-determinantal processes. For that purpose we use results on negative binomial and ordinary binomial multivariate distributions.

We clarify the relationship between higher étale wild kernels of a number field at the prime 2 and the Galois-coinvariants of Tate-twisted class groups in the 2cyclotomic tower of the field. We also determine the relationship between the étale wild kernel and the group of infinitely divisible elements of H(F, Z2(j + 1)){2}.

Let Y be a nonnegative random variable with mean μ and finite positive variance σ, and let Y , defined on the same space as Y , have the Y size biased distribution, that is, the distribution characterized by E[Y f(Y )] = μEf(Y ) for all functions f for which these expectations exist. Under a variety of conditions on the coupling of Y and Y , including combinations of boundedness and monotonicit...