3,357 research outputs found
On implicational bases of closure systems with unique critical sets
We show that every optimum basis of a finite closure system, in D.Maier's
sense, is also right-side optimum, which is a parameter of a minimum CNF
representation of a Horn Boolean function. New parameters for the size of the
binary part are also established. We introduce a K-basis of a general closure
system, which is a refinement of the canonical basis of Duquenne and Guigues,
and discuss a polynomial algorithm to obtain it. We study closure systems with
the unique criticals and some of its subclasses, where the K-basis is unique. A
further refinement in the form of the E-basis is possible for closure systems
without D-cycles. There is a polynomial algorithm to recognize the D-relation
from a K-basis. Thus, closure systems without D-cycles can be effectively
recognized. While E-basis achieves an optimum in one of its parts, the
optimization of the others is an NP-complete problem.Comment: Presented on International Symposium of Artificial Intelligence and
Mathematics (ISAIM-2012), Ft. Lauderdale, FL, USA Results are included into
plenary talk on conference Universal Algebra and Lattice Theory, June 2012,
Szeged, Hungary 29 pages and 2 figure
Geometric lattice structure of covering-based rough sets through matroids
Covering-based rough set theory is a useful tool to deal with inexact,
uncertain or vague knowledge in information systems. Geometric lattice has
widely used in diverse fields, especially search algorithm design which plays
important role in covering reductions. In this paper, we construct four
geometric lattice structures of covering-based rough sets through matroids, and
compare their relationships. First, a geometric lattice structure of
covering-based rough sets is established through the transversal matroid
induced by the covering, and its characteristics including atoms, modular
elements and modular pairs are studied. We also construct a one-to-one
correspondence between this type of geometric lattices and transversal matroids
in the context of covering-based rough sets. Second, sufficient and necessary
conditions for three types of covering upper approximation operators to be
closure operators of matroids are presented. We exhibit three types of matroids
through closure axioms, and then obtain three geometric lattice structures of
covering-based rough sets. Third, these four geometric lattice structures are
compared. Some core concepts such as reducible elements in covering-based rough
sets are investigated with geometric lattices. In a word, this work points out
an interesting view, namely geometric lattice, to study covering-based rough
sets
Bifurcation of hyperbolic planforms
Motivated by a model for the perception of textures by the visual cortex in
primates, we analyse the bifurcation of periodic patterns for nonlinear
equations describing the state of a system defined on the space of structure
tensors, when these equations are further invariant with respect to the
isometries of this space. We show that the problem reduces to a bifurcation
problem in the hyperbolic plane D (Poincar\'e disc). We make use of the concept
of periodic lattice in D to further reduce the problem to one on a compact
Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The
knowledge of the symmetry group of this surface allows to carry out the
machinery of equivariant bifurcation theory. Solutions which generically
bifurcate are called "H-planforms", by analogy with the "planforms" introduced
for pattern formation in Euclidean space. This concept is applied to the case
of an octagonal periodic pattern, where we are able to classify all possible
H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These
patterns are however not straightforward to compute, even numerically, and in
the last section we describe a method for computation illustrated with a
selection of images of octagonal H-planforms.Comment: 26 pages, 11 figure
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