A Note on Flips in Diagonal Rectangulations

Rectangulations are partitions of a square into axis-aligned rectangles. A number of results provide bijections between combinatorial equivalence classes of rectangulations and families of pattern-avoiding permutations. Other results deal with local changes involving a single edge of a rectangulation, referred to as flips, edge rotations, or edge pivoting. Such operations induce a graph on equivalence classes of rectangulations, related to so-called flip graphs on triangulations and other families of geometric partitions. In this note, we consider a family of flip operations on the equivalence classes of diagonal rectangulations, and their interpretation as transpositions in the associated Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This complements results from Law and Reading (JCTA, 2012) and provides a complete characterization of flip operations on diagonal rectangulations, in both geometric and combinatorial terms

La exigua, desigual y menguante financiación de la universidad pública española (2009-2015)

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A new meta-module for efficient reconfiguration of hinged-units modular robots

We present a robust and compact meta-module for edge-hinged modular robot units such as M-TRAN, SuperBot, SMORES, UBot, PolyBot and CKBot, as well as for central-point-hinged ones such as Molecubes and Roombots. Thanks to the rotational degrees of freedom of these units, the novel meta-module is able to expand and contract, as to double/halve its length in each dimension. Moreover, for a large class of edge-hinged robots the proposed meta-module also performs the scrunch/relax and transfer operations required by any tunneling-based reconfiguration strategy, such as those designed for Crystalline and Telecube robots. These results make it possible to apply efficient geometric reconfiguration algorithms to this type of robots. We prove the size of this new meta-module to be optimal. Its robustness and performance substantially improve over previous results.Peer ReviewedPostprint (author's final draft

Long Proteins with Unique Optimal Foldings in the H-P Model

It is widely accepted that (1) the natural or folded state of proteins is a global energy minimum, and (2) in most cases proteins fold to a unique state determined by their amino acid sequence. The H-P (hydrophobic-hydrophilic) model is a simple combinatorial model designed to answer qualitative questions about the protein folding process. In this paper we consider a problem suggested by Brian Hayes in 1998: what proteins in the two-dimensional H-P model have unique optimal (minimum energy) foldings? In particular, we prove that there are closed chains of monomers (amino acids) with this property for all (even) lengths; and that there are open monomer chains with this property for all lengths divisible by four.Comment: 22 pages, 18 figure

A new meta-module design for efficient reconfiguration of modular robots

This is a post-peer-review, pre-copyedit version of an article published in Autonomous Robots. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10514-021-09977-6We propose a new meta-module design for two important classes of modular robots. The new metamodule is three-dimensional, robust and compact, improving on the previously proposed one. It applies to socalled “edge-hinged” modular robot units, such as MTRAN, SuperBot, SMORES, UBot, PolyBot and CKBot, as well as to so-called “central-point-hinged” modular robot units, which include Molecubes and Roombots. The new meta-module uses the rotational degrees of freedom of these two types of robot units in order to expand and contract, as to double or halve its length in each of the two directions of its three dimensions, therefore simulating the capabilities of Crystalline and Telecube robots. Furthermore, in the edge-hinged case we prove that the novel meta-module can also perform the scrunch, relax and transfer moves that are necessary in any tunnelingbased reconfiguration algorithm for expanding/contracting modular robots such as Crystalline and Telecube. This implies that the use of meta-meta-modules is unnecessary, and that currently existing efficient reconfiguration algorithms can be applied to a much larger set of modular robots than initially intended.This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922. I.P. was supported by the Austrian Science Fund (FWF): W1230. V.S. and R.S. were supported by projects MINECO MTM2015-63791-R and Gen. Cat. 2017SGR1640. R.S. was also supported by MINECO through the Ram´on y Cajal program.Peer ReviewedPostprint (published version

Segmenting trajectories: A framework and algorithms using spatiotemporal criteria

In this paper we address the problem of segmenting a trajectory based on spatiotemporal criteria. We require that each segment is homogeneous in the sense that a set of spatiotemporal criteria are fulfilled. We define different such criteria, including location, heading, speed, velocity, curvature, sinuosity, curviness, and shape. We present an algorithmic framework that allows us to segment any trajectory into a minimum number of segments under any of these criteria, or any combination of these criteria. In this framework, a segmentation can generally be computed in O(n log n) time, where n is the number of edges of the trajectory to be segmented. We also discuss the robustness of our approach.Peer ReviewedPostprint (published version

Caminos de desviación mínima local

Un camino que conecta dos puntos s y t en el plano es de desviación mínima respecto de un conjunto de puntos S si la mayor de las distancias entre un punto del camino y S es la menor posible entre todos los caminos que conectan s y t. En este trabajo estudiamos los caminos de desviación mínima que satisfacen además que todo subcamino es también de desviación mínima, a los que llamamos caminos de desviación mínima local.Postprint (published version

Empty triangles in good drawings of the complete graph

A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at most once. In any good drawing, the edges of three pairwise connected vertices form a Jordan curve which we call a triangle. We say that a triangle is empty if one of the two connected components it induces does not contain any of the remaining vertices of the drawing of the graph. We show that the number of empty triangles in any good drawing of the complete graph Kn with n vertices is at least n.Peer ReviewedPostprint (author’s final draft