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

Study of Chaos in the buck Converter

A dc-dc buck converter controlled by naturallysampled, constant-frequency pulsewidth modulation in continuous conduction mode gives rise to a great variety of behavior, depending on the values of the parameters of the circuit. We analyze the one-periodic and two-periodic orbits which cross the voltage ramp once per cycle, and we study their stability by computing the characteristic multipliers associated with each one. Subharmonics, bifurcations, and the presence of a strange attractor are also studied. A plot of the number of crossings in the ramp is drawn. This becomes a helpful tool for investigating the evolution of the trajectories when they are close to the attractor. When analytic computations are impossible, we resort to numerical algorithms to simulate the orbits.Peer Reviewe

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

On preserving passivity in sampled-data linear systems

Passivity is a well know phenomenon in several engineering areas. Due to its interesting properties, it is used in several areas of control engineering. Generally, this property is lost under direct discretization. In this work a new methodology which allows to preserve continuous-time passivity is presented. This methodology is based on choosing a proper output, which preserves the passivity structure, while keeping the continuous-time energy function. Analytic formulation and numerical examples, both open and close loop, are provided in the paper

Using optimal control to optimize the extraction rate of a durable non-renewable resource with a monopolistic primary supplier

The problem dealt with in this paper is that of optimizing the path of the extraction rate (and, consequently, the price) for the monopolistic owner of the primary sources of a totally or partially durable non-renewable resource (such as precious metals or gemstones) in a continuous-time frame, assuming that there is an upper bound on the extraction rate and with an interest rate equal to zero. The durability of the resource implies that, unlike the case of non-durable resources, at any time there is a stock of already-used amounts of the resource that are still potentially reusable, in addition to the resource available in the ground for extraction. The problem is addressed using the Maximum Principle of Pontryagin in the framework of optimal control theory, which allows identifying the patterns that the optimal policies can adopt. In this framework, the Hamiltonian is linear in the control input, which implies a bang-bang control policy governed by a switching surface. There is an underlying geometry to the problem that determines the solutions. It is characterized by the switching surface, its time derivative, the intersection point (if any) and the bang-bang trajectories through this point.Peer ReviewedPostprint (published version

El Estatuto tras la sentencia

Este trabajo analiza la fundamentación jurídica de la Sentencia del Tribunal Constitucional sobre el nuevo Estatuto de Autonomía de Cataluña (STC 31/2010, de 28 de junio) con el propósito de valorar la doctrina que contiene sobre la naturaleza y función de los Estatutos de Autonomía. Se trata de una cuestión de gran alcance porque se enfrenta a los límites materiales de una norma a la que la Constitución atribuye un papel esencial en la configuración de nuestro Estado de las Autonomías, y aquello que le confiere singularidad frente a otros modelos de organización territorial. Es asimismo la cuestión clave de la Sentencia, pues constituye la base sobre la cual se funda el conjunto del recurso de inconstitucionalidad: el ámbito material del Estatuto. El trabajo expone en primer lugar los planteamientos jurídicos de las partes personadas en el proceso, que reflejan distintas concepciones sobre las características normativas del Estatuto de Autonomía. En segundo lugar se analiza el razonamiento jurídico de la Sentencia sobre tal cuestión: el contenido posible de los Estatutos de Autonomía (FFJJ 4 y 5) viene determinado por su naturaleza (FJ 3) y su función constitucional (FJ 4), de los que se derivan unos límites cuantitativos y cualitativos a tal contenido (FJ 6). En tercer lugar se examina cómo el Tribunal aplica el criterio establecido en el anterior razonamiento para enjuiciar algunos de los principales preceptos impugnados. Ello se lleva a cabo mediante una operación que consiste en mantener el contenido de la práctica totalidad de los preceptos estatutarios impugnados sin declarar su inconstitucionalidad, utilizando de forma heterodoxa la técnica de las sentencias interpretativas. La conclusión pone de manifiesto que la fundamentación de la STC 31/2010 ha establecido una doctrina sobre la naturaleza y posición del Estatuto que reduce drásticamente su función constitucional, marginando aquellas características que le conferían una singularidad como norma, lo que a su vez conduce a rebajar al mínimo su trascendencia en la configuración del Estado Autonómico

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'

Leonhard Euler: quatre lliçons escollides

Aprofitant la commemoració durant el curs 2006-2007 del tricentenari del naixement d’Euler, es presenten quatre temes d’entre els seus treballs: els logaritmes, la fórmula d’Euler, el problema de Basilea i el disseny d’un cabrestant. La tria s’ha fet procurant que aquests temes siguin assequibles a estudiants de primer curs i que n’hi hagués algun de matemàtica aplicada.Peer Reviewe

Leonhard Euler: quatre lIiçons escollides

L'FME dedica el curs acadèmic 2006-2007 a la figura del matemàtic suís Leonhard Euler, una de les ments més importants de la història, comparable a Gauss o Arquímedes. La lliçó inaugural va anar a càrrec d'Enric Fossas, catedràtic i director de l'Institut d'Organització i Control de Sistemes Industrials de la UPCPostprint (published version