An Algorithmic Metatheorem for Directed Treewidth

The notion of directed treewidth was introduced by Johnson, Robertson, Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as a first step towards an algorithmic metatheory for digraphs. They showed that some NP-complete properties such as Hamiltonicity can be decided in polynomial time on digraphs of constant directed treewidth. Nevertheless, despite more than one decade of intensive research, the list of hard combinatorial problems that are known to be solvable in polynomial time when restricted to digraphs of constant directed treewidth has remained scarce. In this work we enrich this list by providing for the first time an algorithmic metatheorem connecting the monadic second order logic of graphs to directed treewidth. We show that most of the known positive algorithmic results for digraphs of constant directed treewidth can be reformulated in terms of our metatheorem. Additionally, we show how to use our metatheorem to provide polynomial time algorithms for two classes of combinatorial problems that have not yet been studied in the context of directed width measures. More precisely, for each fixed $k,w \in \mathbb{N}$, we show how to count in polynomial time on digraphs of directed treewidth $w$, the number of minimum spanning strong subgraphs that are the union of $k$ directed paths, and the number of maximal subgraphs that are the union of $k$ directed paths and satisfy a given minor closed property. To prove our metatheorem we devise two technical tools which we believe to be of independent interest. First, we introduce the notion of tree-zig-zag number of a digraph, a new directed width measure that is at most a constant times directed treewidth. Second, we introduce the notion of $z$-saturated tree slice language, a new formalism for the specification and manipulation of infinite sets of digraphs.Comment: 41 pages, 6 figures, Accepted to Discrete Applied Mathematic

Toward an ecological aesthetics: music as emergence

In this article we intend to suggest some ecological based principles to support the possibility of develop an ecological aesthetics. We consider that an ecological aesthetics is founded in concepts as “direct perception”, “acquisition of affordances and invariants”, “embodied embedded perception” and so on. Here we will purpose that can be possible explain especially soundscape music perception in terms of direct perception, working with perception of first hand (in a Gibsonian sense). We will present notions as embedded sound, detection of sonic affordances and invariants, and at the end we purpose an experience with perception/action paradigm to make soundscape music as emergence of a self-organized system

Toward an ecological conception of timbre

This paper is part of a series in which we had worked in the last 6 months, and, specifically, intend to investigate the notion of timbre through the ecological perspective proposed by James Gibson in his Theory of Direct Perception. First of all, we discussed the traditional approach to timbre, mainly as developed in acoustics and psychoacoustics. Later, we proposed a new conception of timbre that was born in concepts of ecological approach. The ecological approach to perception proposed by Gibson (1966, 1979) presupposes a level of analysis of perceptual stimulated that includes, but is quite broader than the usual physical aspect. Gibson suggests as focus the relationship between the perceiver and his environment. At the core of this approach, is the notion of affordances, invariant combinations of properties at the ecological level, taken with reference to the anatomy and action systems of species or individual, and also with reference to its biological and social needs. Objects and events are understood as relates to a perceiving organism by the meaning of structured information, thus affording possibilities of action by the organism. Event perception aims at identifying properties of events to specify changes of the environment that are relevant to the organism. The perception of form is understood as a special instance of event perception, which is the identity of an object depends on the nature of the events in which is involved and what remains invariant over time. From this perspective, perception is not in any sense created by the brain, but is a part of the world where information can be found. Consequently, an ecological approach represents a form of direct realism that opposes the indirect realist based on predominant approaches to perception borrowed from psychoacoustics and computational approach

G2G_2-monopoles with singularities (examples)

$G_2$-monopoles are solutions to gauge theoretical equations on $G_2$-manifolds. If the $G_2$-manifolds under consideration are compact, then any irreducible $G_2$-monopole must have singularities. It is then important to understand which kind of singularities $G_2$-monopoles can have. We give examples (in the noncompact case) of non-Abelian monopoles with Dirac type singularities, and examples of monopoles whose singularities are not of that type. We also give an existence result for Abelian monopoles with Dirac type singularities on compact manifolds. This should be one of the building blocks in a gluing construction aimed at constructing non-Abelian ones.Comment: Lett Math Phys (2016