The infinite cyclohedron and its automorphism group

Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degree 2 central extension of Thompson's infinite finitely presented simple group T with the cyclic group of order 2. These results are inspired by a similar recent analysis by the first author of the automorphism group of an infinite-dimensional associahedron.Comment: 18 pages, 8 figure

Leonhard Euler: quatre lliçons escollides

Aprofitant la commemoració durant el curs 2006-2007 del tricentenari del naixement d'Euler, es presenten en aquest article quatre temes dels seus treballs: els logaritmes, la fórmula d'Euler, el problema de Basilea i el disseny d'un cabrestant. La tria dels temes s'ha fet amb el criteri que il.lustri de manera assequible part del treball d'Euler i que inclogui un tema proper a les aplicacions.On the occasion of the 300th anniversary of Leonhard Euler’s birth this paper presents four topics from his work: the logarithms, Euler’s formula, the Basel problem and masting of ships. The choice of the topics aims to give a plain illustration of part of Euler’s work and to include a topic on applied mathematics

A black-box model for neurons

We explore the identification of neuronal voltage traces by artificial neural networks based on wavelets (Wavenet). More precisely, we apply a modification in the representation of dynamical systems by Wavenet which decreases the number of used functions; this approach combines localized and global scope functions (unlike Wavenet, which uses localized functions only). As a proof-of-concept, we focus on the identification of voltage traces obtained by simulation of a paradigmatic neuron model, the Morris-Lecar model. We show that, after training our artificial network with biologically plausible input currents, the network is able to identify the neuron's behaviour with high accuracy, thus obtaining a black box that can be then used for predictive goals. Interestingly, the interval of input currents used for training, ranging from stimuli for which the neuron is quiescent to stimuli that elicit spikes, shows the ability of our network to identify abrupt changes in the bifurcation diagram, from almost linear input-output relationships to highly nonlinear ones. These findings open new avenues to investigate the identification of other neuron models and to provide heuristic models for real neurons by stimulating them in closed-loop experiments, that is, using the dynamic-clamp, a well-known electrophysiology technique.Peer ReviewedPostprint (author's final draft

The optimum speed of wheel - rail's system

El concepte de velocitat òptima és un paràmetre bàsic per arribar a definir el programa d'explotació més convenient d'un servei ferroviari. Basat en l'anànisis de costos i beneficis permet donar resposta a quina ha de ser la velocitat més adequada a considerar d'acord amb un determinat entorn competencial i econòmic. Aquest concepte va ser introduït pels serveis ferroviaris francesos d'alta velocitat posant de manifest que l'objectiu no havia de ser el circular tan ràpid com fos possible sinó com comercialment fos necessari

A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems

Piecewise smooth dynamical systems make use of discontinuities to model switching between regions of smooth evolution. This introduces an ambiguity in prescribing dynamics at the discontinuity: should it be given by a limiting value on one side or other of the discontinuity, or a member of some set containing those values? One way to remove the ambiguity is to regularize the discontinuity, the most common being either to smooth out the discontinuity, or to introduce a hysteresis between switching in one direction or the other across the discontinuity. Here we show that the two can in general lead to qualitatively different dynamical outcomes. We then define a higher dimensional model with both smoothing and hysteresis, and study the competing limits in which hysteretic or smoothing effect dominate the behaviour, only the former of which correspond to Filippov's standard `sliding modes'

Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation $x\mapsto x^{-1}$, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.Comment: 10 page

Can There be Art Without an Artist?

Generative Adversarial Network (GAN) based art has proliferated in the past year, going from a shiny new tool to generate fake human faces to a stage where anyone can generate thousands of artistic images with minimal effort. Some of these images are now ``good'' enough to win accolades from qualified judges. In this paper, we explore how Generative Models have impacted artistry, not only from a qualitative point of view, but also from an angle of exploitation of artisans --both via plagiarism, where models are trained on their artwork without permission, and via profit shifting, where profits in the art market have shifted from art creators to model owners or to traders in the unregulated secondary crypto market. This confluence of factors risks completely detaching humans from the artistic process, devaluing the labor of artists and distorting the public perception of the value of art