A general framework for island systems

Abstract

The notion of an island defined on a rectangular board is an elementary combinatorial concept that occurred first in [3]. Results of [3] were starting points for investigations exploring several variations and various aspects of this notion. In this paper we introduce a general framework for islands that subsumes all earlier studied concepts of islands on finite boards, moreover we show that the prime implicants of a Boolean function, the formal concepts of a formal context, convex subgraphs of a simple graph, and some particular subsets of a projective plane also fit into this framework. We axiomatize those cases where islands have the property of being pairwise comparable or disjoint, or they are distant, introducing the notion of a connective island domain and of a proximity domain, respectively. In the general case the maximal systems of islands are characterised by using the concept of an admissible system. We also characterise all possible island systems in the case of connective island domains and proximity domains. 2010 Mathematics Subject Classification. 06A06.

Enhancing Broiler Weight Estimation through Gaussian Kernel Density Estimation Modeling

“…The Fourier series and transform methods, pioneered by Kronmal and Tarter (1968) [20] and Davis (1997) [21], respectively, characterize these data within the frequency domain to derive the density function. The orthogonal polynomials technique proposed by Schwartz (1967) [22], the splines technique by Wahba (1975) [23], and the general delta sequences technique by Foldes and Revez (1974) [24] each offer unique means of density estimation via specialized mathematical constructs [25].…”

CD-independent subsets in meet-distributive lattices

“…This result was soon followed by many related ones, due to Barát, Foldes, E. K. Horváth, G. Horváth, Lengvárszky, Németh, Pach, Pluhár, Pongrácz, Šešelja, Szabó, and Tepavčević. The results of these authors, written alone or in various groups, range from triangular boards to the continuous case and from lattice theory to combinatorics, see [6], [21], [24], [25], [26], [28], [29], [30], [32], [33], and some further papers not referenced here. Since [21] and [24] give good overviews on islands, we do not go into further historical details.…”

“…The results of these authors, written alone or in various groups, range from triangular boards to the continuous case and from lattice theory to combinatorics, see [6], [21], [24], [25], [26], [28], [29], [30], [32], [33], and some further papers not referenced here. Since [21] and [24] give good overviews on islands, we do not go into further historical details. However, we mention the following feature of this research field.…”

Islands: from coding theory to enumerative combinatorics and to lattice theory - overview and open problems

“…It is possible to generalize the notion of islands in such a way that formal concepts and prime implicants of Boolean functions are covered, the details can be read in [8].…”

“…This condition is not a sufficient one, hovewer the paper proves that maximal preisland systems and maximal admissible families are the same. An island domain .C; K/ is a connective island domain if 8A; B 2 C W .A \ B ¤ ¿ and B ª A/ H) 9K 2 K W A K Â A [ B: A result of [8] is that the following three conditions are equivalent for any island domain .C; K/: (i) .C; K/ is a connective island domain. (ii) Every system of pre-islands corresponding to .C; K/ is CD-independent.…”