Paolo A. Oliverio

A new discrete dynamical system of signed integer partitions

In Brylawski (1973) Brylawski described the covering property for the domination order on non-negative integer partitions by means of two rules. Recently, in Bisi et?al. (in press), Cattaneo et?al. (2014), Cattaneo et?al. (2015) the two classical Brylawski covering rules have been generalized in order to obtain a new lattice structure in the more general signed integer partition context. Moreover, in Cattaneo et?al. (2014), Cattaneo et?al. (2015), the covering rules of the above signed partition lattice have been interpreted as evolution rules of a discrete dynamical model of a two-dimensional p-n semiconductor junction in which each positive number represents a distribution of holes (positive charges) located in a suitable strip at the left semiconductor of the junction and each negative number a distribution of electrons (negative charges) in a corresponding strip at the right semiconductor of the junction. In this paper we introduce and study a new sub-model of the above dynamical model, which is constructed by using a single vertical evolution rule. This evolution rule describes the natural annihilation of a hole-electron pair at the boundary region of the two semiconductors. We prove several mathematical properties of such new discrete dynamical model and we provide a discussion of its physical properties.

Parallel and sequential dynamics of two discrete models of signed integer partitions

Abstract In this paper we complete and generalize some previous results concerning the computing of the sequential and parallel convergent time for two discrete dynamical system of signed integer partitions. We also refine the concept of parallel convergent time for a finite graded partially ordered set (briefly poset) X which is also a discrete dynamical model. To this aim we define the concept of fundamental sequence of X and we compute this sequence in two particularly important cases. In the first case, when X is the finite lattice S ( n , r ) of all the signed integer partitions a r , … , a 1 , b 1 , … , b n - r such that r ⩾ a r ⩾ ⋯ ⩾ a 1 ⩾ 0 ⩾ b 1 ⩾ ⋯ ⩾ b n - r ⩾ - ( n - r ) , where n ⩾ r ⩾ 0 and the unique part that can be repeated is 0. In the second case, when X is the sub-lattice S ( n , d , r ) of all the signed integer partitions of S ( n , r ) having exactly d non-zero parts. The relevance of the previous lattices as discrete dynamical models is related to their link with some unsolved extremal combinatorial sum problems.

Dominance order on signed integer partitions

Abstract In 1973 Brylawski introduced and studied in detail the dominance partial order on the set Par(m) of all integer partitions of a fixed positive integer m. As it is well known, the dominance order is one of the most important partial orders on the finite set Par(m). Therefore it is very natural to ask how it changes if we allow the summands of an integer partition to take also negative values. In such a case, m can be an arbitrary integer and Par(m) becomes an infinite set. In this paperwe extend the classical dominance order in this more general case. In particular, we consider the resulting lattice Par(m) as an infinite increasing union on n of a sequence of finite lattices O(m, n). The lattice O(m, n) can be considered a generalization of the Brylawski lattice. We study in detail the lattice structure of O(m, n).

Parallel Rank of Two Sandpile Models of Signed Integer Partitions

We introduce the concept of fundamental sequence for a finite graded poset X which is also a discrete dynamical model. The concept of fundamental sequence is a refinement of the concept of parallel convergence time for these models. We compute the parallel convergence time and the fundamental sequence when X is the finite lattice of all the signed integer partitions such that , where , and when X is the sublattice of all the signed integer partitions of having exactly d nonzero parts.

A Discrete Dynamical Model of Signed Partitions

We use a discrete dynamical model with three evolution rules in order to analyze the structure of a partially ordered set of signed integer partitions whose main properties are actually not known. This model is related to the study of some extremal combinatorial sum problems.

Dependency and accuracy measures for directed graphs

Abstract In this paper we use finite directed graphs (digraphs) as mathematical models to study two basic notions widely analyzed in granular computing: the attribute dependency and the approximation accuracy. To be more specific, at first we interpret any digraph as a Boolean information table, next we study the approximation accuracy for three fundamentals digraph families: the directed path, the directed cycle and the transitive tournament. We also introduce a new global average for the attribute dependency in any information table and we determine such number for any directed path. For the transitive tournament we provide a lower bound.

On surfaces with p g =2, q =1 and K 2 =5

We consider minimal surfaces of general type with pg=2, q=1 and K2=5. We provide a stratification of the corresponding moduli space

Indiscernibility structures induced from function sets : Graph and digraph case

\mathcal{M}

A natural extension of the Young partition lattice

and we give some bounds for the number and the dimensions of its irreducible components.

The adjacency matrix of a graph as a data table: a geometric perspective

Abstract A very general structure in mathematics is a function system, that is a structure 𝒯 = ⟨U𝒯, Ω𝒯, Λ𝒯 ⟩, where U𝒯 is a finite universe set and Ω𝒯 is a finite set of functions ai: U →Λ𝒯. In this paper we use a function system to develop a mathematical theory of the indiscernibility. More in detail, we first use a natural equivalence relation induced by any function subset A ⊆Ω𝒯 to introduce a complete lattice of set partitions of U𝒯. We prove several properties of this order structure and we develop two specific cases of study concerning directed and undirected graphs. Next, for finite function systems we introduce two approximation measures that have a deep similarity with the Lebesgue measure and with the conditional probability. Also in this case we provide two specific cases of study on directed and undirected graphs.