Periodic boundary conditions on the pseudosphere

We provide a framework to build periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary conditions in the Euclidean plane, we introduce all the needed mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane. We stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems and we illustrate how to implement such periodic boundary conditions in two examples, the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices.Comment: 30 pages, minor corrections, accepted to J. Phys.

Combinatorial optimization in networks with Shared Risk Link Groups

International audienceThe notion of Shared Risk Link Groups (SRLG) captures survivability issues when a set of links of a network may fail simultaneously. The theory of survivable network design relies on basic combinatorial objects that are rather easy to compute in the classical graph models: shortest paths, minimum cuts, or pairs of disjoint paths. In the SRLG context, the optimization criterion for these objects is no longer the number of edges they use, but the number of SRLGs involved. Unfortunately, computing these combinatorial objects is NP-hard and hard to approximate with this objective in general. Nevertheless some objects can be computed in polynomial time when the SRLGs satisfy certain structural properties of locality which correspond to practical ones, namely the star property (all links affected by a given SRLG are incident to a unique node) and the span 1 property (the links affected by a given SRLG form a connected component of the network). The star property is defined in a multi-colored model where a link can be affected by several SRLGs while the span property is defined only in a mono-colored model where a link can be affected by at most one SRLG. In this paper, we extend these notions to characterize new cases in which these optimization problems can be solved in polynomial time. We also investigate the computational impact of the transformation from the multi-colored model to the mono-colored one. Experimental results are presented to validate the proposed algorithms and principles

Wallpaper Maps

A wallpaper map is a conformal projection of a spherical earth onto regular polygons with which the plane can be tiled continuously. A complete set of distinct wallpaper maps that satisfy certain natural symmetry conditions is derived and illustrated. Though all of the projections have been published before, the family had not been characterized as a whole. Some wallpaper maps generalize to one-parameter subfamilies in which the sphere is pre-transformed by a conformal automorphism

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

On Chemical Activity Foundations of a classificatory approach to the study of chemical combination

Estructura y actividad son conceptos centrales y sin embargo ambiguos de la química que admiten una variedad de denotaciones distintas. La popular (aún si raramente explícita) identificación de estructura química con una teoría atómica de la constitución de la materia que da cuenta de su comportamiento durante el cambio químico entra en conflicto con perspectivas contemporáneas en la historia y la filosofía de la química. Al mismo tiempo, trivializa el concepto de actividad química, privándolo de cualquier dimensión teórica relevante. En contra de esta perspectiva definimos estructura y actividad como aproximaciones epistémicas opuestas, caracterizadas por otorgar prioridades ontológicas inversas a los conceptos de relación y propiedad, que interactúan en la teoría química. La prioridad dada a las relaciones como determinantes de las propiedades de las sustancias característica de la teoría de la actividad química motiva una matematización peculiar de esta aproximación epistémica, usando los formalismos de la teoría de categorías, el análisis formal de conceptos, y el análisis de redes. El formalismo matemático resultante permite o sugiere una reconstrucción exitosa de constructos químicos clave tales como ácido, base, y función orgánica, provee los fundamentos de una metodología para el análisis de similitud entre sustancias vistas como objetos en una red de reacciones químicas, y revela el potencial de la actividad química como una auténtica teoría de la combinación química, complementaria a la teoría estructural y perfectamente capaz de construir conocimiento químico por sus propios medios. / Abstract. Structure and activity are central yet ambiguous concepts of chemical science, being susceptible to a variety of distinct denotations. The popular (though seldom explicit) attachment of chemical structure to an atomic constitution theory that accounts for the behavior of matter as it undergoes chemical change conicts with contemporary perspectives in the history and philosophy of chemistry and trivializes the concept of chemical activity, depriving it of any relevant theoretical dimension. Against this perspective, we define structure and activity as opposing epistemic approaches that interplay in chemical theory and are characterized by reverse ontological priorities ascribed to the concepts of property and relation. The prioritization of relations as determinants of substance properties characteristic of chemical activity theory motivates a peculiar mathematization of this epistemic approach, using the formalisms of category theory, formal concept analysis, and network analysis. The resulting mathematical formalism allows or suggests the possibility of a successful reconstruction of key chemical constructs such as acid, base, and organic function; provides the foundation of a methodology for the analysis of similarity in chemical reaction networks; and unveils the potential of chemical activity as a fully-eshed theory of chemical combination, complementary to structure theory and readily capable of constructing chemical knowledge by its own means.Doctorad