Graph ambiguity

In this paper, we propose a rigorous way to define the concept of ambiguity in the domain of graphs. In past studies, the classical definition of ambiguity has been derived starting from fuzzy set and fuzzy information theories. Our aim is to show that also in the domain of the graphs it is possible to derive a formulation able to capture the same semantic and mathematical concept. To strengthen the theoretical results, we discuss the application of the graph ambiguity concept to the graph classification setting, conceiving a new kind of inexact graph matching procedure. The results prove that the graph ambiguity concept is a characterizing and discriminative property of graphs. (C) 2013 Elsevier B.V. All rights reserved

Isometric Embeddings and Noncommutative Branes in Homogeneous Gravitational Waves

We characterize the worldvolume theories on symmetric D-branes in a six-dimensional Cahen-Wallach pp-wave supported by a constant Neveu-Schwarz three-form flux. We find a class of flat noncommutative euclidean D3-branes analogous to branes in a constant magnetic field, as well as curved noncommutative lorentzian D3-branes analogous to branes in an electric background. In the former case the noncommutative field theory on the branes is constructed from first principles, related to dynamics of fuzzy spheres in the worldvolumes, and used to analyse the flat space limits of the string theory. The worldvolume theories on all other symmetric branes in the background are local field theories. The physical origins of all these theories are described through the interplay between isometric embeddings of branes in the spacetime and the Penrose-Gueven limit of AdS3 x S3 with Neveu-Schwarz three-form flux. The noncommutative field theory of a non-symmetric spacetime-filling D-brane is also constructed, giving a spatially varying but time-independent noncommutativity analogous to that of the Dolan-Nappi model.Comment: 52 pages; v2: References adde

Quantale Modules, with Applications to Logic and Image Processing

We propose a categorical and algebraic study of quantale modules. The results and constructions presented are also applied to abstract algebraic logic and to image processing tasks.Comment: 150 pages, 17 figures, 3 tables, Doctoral dissertation, Univ Salern

The moduli space of matroids

In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.Comment: 83 page