Stallings automata for free-times-abelian groups: intersections and index

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which—as it happens in the free group—is computable in the finitely generated case. This approach provides a neat geometric description of (even non-(finitely generated)) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals, respectively.The first author was partially supported by CMUP (UID/MAT/ 00144/2019), which is funded by FCT (Portugal) with national (MCTES) and European structural funds through the FEDER programs, under the partnership agreement PT2020. Parts of this project were developed during the participation of the first author in the “Logic and Algorithms in Group Theory” meeting held in the Hausdorff Research Institute for Mathematics (Bonn) in fall 2018. He was also partially supported by MINECO grant PID2019-107444GA-I00 and the Basque Government grant IT974-16. Both authors acknowledge partial support from the Spanish Agencia Estatal de Investigación, through grant MTM2017-82740-P (AEI/ FEDER, UE), and also from the Barcelona Graduate School of Mathematics through the “María de Maeztu” Program for Units of Excellence in R&D (MDM-2014-0445).Peer ReviewedPostprint (published version

On Collective Computation

Postprint (published version

Aprenentatge de la Programació al Citilab: Squeak i Scratch

Aquest article vol ser un recull de les primeres experiències al Citilab apropant la programació dels ordinadors a la gent del carrer. Sense cap requeriment previ, qualsevol pot inscriure's a algun dels cursos de la nostra oferta docent en programació. Aix o ha estat possible principalment gràcies a l'existència de programari adequat, tot ell realitzat en entorns Smalltalk: Scratch per a nens petits (i no tan petits!), BotsInc com a iniciació a l'Smalltalk i el mateix Squeak, implementació open source d'Smalltalk utilitzada en la creació del programari esmentat, per a programadors experimentats.Postprint (published version

Social insects and swarm intelligence

Most of the questions on the dynamics of systems of strongly interacting (simple) agents we can pose could be reframed, without any modification, in the context of social insects, since these are, without any doubt, a truly paradigm for complex systems. This chapter surveys some of the mathematical models that have been successfully used to analyze swarm behavior.Postprint (published version

Weighted Contrastive Divergence

Learning algorithms for energy based Boltzmann architectures that rely on gradient descent are in general computationally prohibitive, typically due to the exponential number of terms involved in computing the partition function. In this way one has to resort to approximation schemes for the evaluation of the gradient. This is the case of Restricted Boltzmann Machines (RBM) and its learning algorithm Contrastive Divergence (CD). It is well-known that CD has a number of shortcomings, and its approximation to the gradient has several drawbacks. Overcoming these defects has been the basis of much research and new algorithms have been devised, such as persistent CD. In this manuscript we propose a new algorithm that we call Weighted CD (WCD), built from small modifications of the negative phase in standard CD. However small these modifications may be, experimental work reported in this paper suggest that WCD provides a significant improvement over standard CD and persistent CD at a small additional computational cost

Pattern formation and optimization in army ant raids

Army ant colonies display complex foraging raid patterns involving thousands of individuals communicating through chemical trails. In this paper we explore, by means of a simple search algorithm, the properties of these trails in order to test the hypothesis that their structure reflects an optimized mechanism for exploring and exploiting food resources. The raid patterns of three army ant species, {em Eciton hamatum}, {em Eciton burchelli} and {em Eciton rapax}, are analysed. The respective diets of these species involve large but rare, small but common, and a combination of large but rare and small but common, food sources. Using a model proposed by Deneubourg and collaborators, we simulate the formation of raid patterns in response to different food distributions. Our results indicate that the empirically observed raid patterns maximise return on investment, that is, the amount of food brought back to the nest per unit of energy expended, for each of the diets. Moreover, the values of the parameters that characterise the three optimal pattern-generating mechanisms are strikingly similar. Therefore the same behavioural rules at the individual level can produce optimal colony-level patterns. The evolutionary implications of these findings are discussed.Postprint (published version

A Computational characterization of collective chaos

We suggest that the interaction of a Globally Coupled Map (GCM) with an individual element inside the system is, from a computational point of view, indistinguishable of a mu,epsilon-dependent noise in the turbulent region of the phase space. Therefore, we can use the framework of Computational Mechanics to give a measure that clearly separates the ordered from the turbulent phases. Furthermore, our procedure is able to detect a small ordered domain inside the turbulent phase. These results reinforce the view of GCMs as properly defined mean field models of complex nonlinear networks.Preprin

Stopping criteria in contrastive divergence: Alternatives to the reconstruction error

Restricted Boltzmann Machines (RBMs) are general unsupervised learning devices to ascertain generative models of data distributions. RBMs are often trained using the Contrastive Divergence learning algorithm (CD), an approximation to the gradient of the data log-likelihood. A simple reconstruction error is often used to decide whether the approximation provided by the CD algorithm is good enough, though several authors (Schulz et al., 2010; Fischer & Igel, 2010) have raised doubts concerning the feasibility of this procedure. However, not many alternatives to the reconstruction error have been used in the literature. In this manuscript we investigate simple alternatives to the reconstruction error in order to detect as soon as possible the decrease in the log-likelihood during learning.Peer ReviewedPostprint (published version

Neighborhood-based stopping criterion for contrastive divergence

Restricted Boltzmann Machines (RBMs) are general unsupervised learning devices to ascertain generative models of data distributions. RBMs are often trained using the Contrastive Divergence (CD) learning algorithm, an approximation to the gradient of the data log-likelihood (logL). A simple reconstruction error is often used as a stopping criterion for CD, although several authors have raised doubts concerning the feasibility of this procedure. In many cases, the evolution curve of the reconstruction error is monotonic, while the logL is not, thus indicating that the former is not a good estimator of the optimal stopping point for learning. However, not many alternatives to the reconstruction error have been discussed in the literature. An estimation of the logL of the training data based on annealed importance sampling is feasible but computationally very expensive. In this manuscript, we present a simple and cheap alternative, based on the inclusion of information contained in neighboring states to the training set, as a stopping criterion for CD learning.Peer ReviewedPostprint (author's final draft

Formulación numérica en elementos finitos de problemas de flujo multifásico no isotermo y transporte de solutos reactivos en medios porosos

En este artículo se presenta la formulación numérica del flujo multifásico no isotermo y del transporte reactivo de un sistema multicomponente de solutos en condiciones no isotermas. El modelo numérico ha sido implementado en un código (FADES-CORE c ?) el cual ha sido verificado y validado. En este artículo se describe el esquema de resolución utilizado y las distintas variantes del mismo. Para la discretización temporal se ha formulado un algoritmo de optimización del paso de tiempo autom´atico. Por último se incluye un ejemplo de validación del correcto funcionamiento del modelo.Peer Reviewe