Giampiero Chiaselotti

A Method to Count the Positive 3-Subsets in a Set of Real Numbers with Non-Negative Sum

Let a1,?, anbe n real numbers with non-negative sum. We show that if n? 12 there exist at least ( n?1 2 ) subsets of { a1,? , an} with three elements which have non-negative sum.

Simple graphs in granular computing

Given a graph, we interpret its adjacency matrix as an information table. We study this correspondence in two directions. Firstly, on the side of graphs by applying to it standard techniques from granular computing. In this way, we are able to connect automorphisms on graphs to the so-called indiscernibility relation and a particular hypergraph built from the starting graph to core and reducts. On the other hand, new concepts are introduced on graphs that have an interesting correspondence on information tables. In particular, some new topological interpretations of the graph and the concept of extended core are given.

On the connection of hypergraph theory with formal concept analysis and rough set theory

We present a unique framework for connecting different topics: hypergraphs from one side and Formal Concept Analysis and Rough Set Theory from the other. This is done through the formal equivalence among Boolean information tables, formal contexts and hypergraphs. Links with generic (i.e., not Boolean) information tables are established, through so-called nominal scaling. The particular case of k-uniform complete hypergraphs will then be studied. In this framework, we are able to solve typical problems of Rough Set Theory and Formal Concept Analysis using combinatorial techniques. More in detail, we will give a formula to compute the degree of dependency and the partial implication between two sets of attributes, compute the set of reducts and define the structure of the partitions generated by all the definable indiscernibility relations.

On a Problem Concerning the Weight Functions

Let X be a finite set with n elements. A function f: X ??R such that ?x?Xf(x) ? 0 is called an -weight function. In 1988 Manickam and Singhi conjectured that, if d is a positive integer and f is an -weight function with n? 4 d there exist at least (n?1 d?1) subsets Y of X with |Y | =d for which ?y?Yf(y) ? 0. In this paper we study this conjecture and we show that it is true if f is a n -weight function and |{x?X: f(x) ? 0}| ?d?n 2 .

New results related to a conjecture of Manickam and Singhi

In 1988 Manickam and Singhi conjectured that for every positive integer d and every n>=4d, every set of n real numbers whose sum is non-negative contains at least (n-1d-1) subsets of size d whose sums are non-negative. In this paper we make use of Halls matching theorem in order to study some numbers related to this conjecture.

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.

Micro and macro models of granular computing induced by the indiscernibility relation

In rough set theory (RST), and more generally in granular computing on information tables (GRC-IT), a central tool is the Pawlaks indiscernibility relation between objects of a universe set with respect to a fixed attribute subset. Let us observe that Pawlaks relation induces in a natural way an equivalence relation ź on the attribute power set that identifies two attribute subsets yielding the same indiscernibility partition. We call indistinguishability relation of a given information table I the equivalence relation ź, that can be considered as a kind of global indiscernibility. In this paper we investigate the mathematical foundations of indistinguishability relation through the introduction of two new structures that are, respectively, a complete lattice and an abstract simplicial complex. We show that these structures can be studied at both a micro granular and a macro granular level and that are naturally related to the core and the reducts of I . We first discuss the role of these structures in GrC-IT by providing some interpretations, then we prove several mathematical results concerning the fundamental properties of such structures.

Rough Set Theory Applied to Simple Undirected Graphs

The incidence matrix of a simple undirected graph is used as an information table. Then, rough set notions are applied to it: approximations, membership function, positive region and discernibility matrix. The particular cases of complete and bipartite graphs are analyzed. The symmetry induced in graphs by the indiscernibility relation is studied and a new concept of generalized discernibility matrix is introduced.

Simple Undirected Graphs as Formal Contexts

The adjacency matrix of a graph is interpreted as a formal context. Then, the counterpart of Formal Concept Analysis (FCA) tools are introduced in graph theory. Moreover, a formal context is seen as a Boolean information table, the structure at the basis of Rough Set Theory (RST). Hence, we also apply RST tools to graphs. The peculiarity of the graph case, put in evidence and studied in the paper, is that both FCA and RST are based on a (different) binary relation between objects.