Structure and enumeration of K4-minor-free links and link diagrams

We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimum number of crossings or edges in a projection of L' in L. Further, we obtain counting formulas and asymptotic estimates for the connected K4-minor-free link-diagrams, minimal K4-minor-free link-diagrams, and K4-minor-free diagrams of the unknot.Peer ReviewedPostprint (author's final draft

Root systems from Toric Calabi-Yau Geometry. Towards new algebraic structures and symmetries in physics?

The algebraic approach to the construction of the reflexive polyhedra that yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres reveals graphs that include and generalize the Dynkin diagrams associated with gauge symmetries. In this work we continue to study the structure of graphs obtained from $CY_3$ reflexive polyhedra. We show how some particularly defined integral matrices can be assigned to these diagrams. This family of matrices and its associated graphs may be obtained by relaxing the restrictions on the individual entries of the generalized Cartan matrices associated with the Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras. These graphs keep however the affine structure, as it was in Kac-Moody Dynkin diagrams. We presented a possible root structure for some simple cases. We conjecture that these generalized graphs and associated link matrices may characterize generalizations of these algebras.Comment: 24 pages, 6 figure

Linear lambda terms as invariants of rooted trivalent maps

The main aim of the article is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between $\alpha$-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the "easy" direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.Comment: accepted author manuscript, posted six months after publicatio

The grammar of developable double corrugations (for formal architectural applications)

This paper investigates the geometrical basis of regular corrugations, with specific emphasis on Developable Double Corrugations (DDCs), which form a unique sub-branch of Origami Folding and Creasing Algorithms. The aim of the exercise is three fold – (1) To define and isolate a ‘single smallest starting block’ for a given set of distinct and divergent DDC patterns, such that this starting block becomes the generator of all DDCs when different generative rules are applied to it. (2) To delineate those generic parameters and generative rules which would apply to the starting block, such that different DDCs are created as a result (3) To use the knowledge from points (1) and (2) to create a complete family of architectural forms and shapes using DDCs. For this purpose, a matrix of 12 underlying geometry types are identified and used as archetypes. The objective is to mathematically explore DDCs for architectural form finding, using physical folding as a primary algorithmic tool. Some DDCs have more degrees of freedom than others and can fit varied geometries, while others cannot. The discussion and conclusions involve - (a) identifying why certain DDCs are ideal for certain forms and not others, when all of them are generated using the same/or similar starting block(s), (b) discussing the critical significance of flat-foldability in this specific context and (c) what we can do with this knowledge of DDCs in the field of architectural research and practice in the future

Multiple Partitioning of Multiplex Signed Networks: Application to European Parliament Votes

For more than a decade, graphs have been used to model the voting behavior taking place in parliaments. However, the methods described in the literature suffer from several limitations. The two main ones are that 1) they rely on some temporal integration of the raw data, which causes some information loss, and/or 2) they identify groups of antagonistic voters, but not the context associated to their occurrence. In this article, we propose a novel method taking advantage of multiplex signed graphs to solve both these issues. It consists in first partitioning separately each layer, before grouping these partitions by similarity. We show the interest of our approach by applying it to a European Parliament dataset.Comment: Social Networks, 2020, 60, 83 - 10

Informe bibliomètric bimestral Campus Baix Llobregat. Base de dades Scopus. Juliol-agost 2018

Informe bibliomètric bimestral Campus Baix Llobregat. Base de dades Scopus. Data de la cerca 31/08/2018Postprint (author's final draft

Compression with wildcards: All exact, or all minimal hitting sets

Our main objective is the COMPRESSED enumeration (based on wildcards) of all minimal hitting sets of general hypergraphs. To the author's best knowledge the only previous attempt towards compression, due to Toda [T], is based on BDD's and much different from our techniques. Numerical experiments show that traditional one-by-one enumeration schemes cannot compete against compressed enumeration when the degree of compression is high. Our method works particularly well in these two cases: Either compressing all exact hitting sets, or all minimum-cardinality hitting sets. It also supports parallelization and cut-off (i.e. restriction to all minimal hitting sets of cardinality at most m).Comment: 30 pages, many Table