On the Gibbs phenomenon 1: Recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function

It is well known that the Fourier series of an analytic or periodic function, truncated after 2N+1 terms, converges exponentially with N, even in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. Here, we show that the first 2N+1 Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation (in the maximum norm) can be constructed

On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function

AbstractIt is well known that the Fourier series of an analytic and periodic function, truncated after 2N+1 terms, converges exponentially with N, even in the maximum norm. It is also known that if the function is not periodic, the rate of convergence deteriorates; in particular, there is no convergence in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. In this paper we show that the first 2N+1 Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation (in the maximum norm) can be constructed. The proof is a constructive one and makes use of the Gegenbauer polynomials Cλn(x). It consists of two steps. In the first step we show that the first m coefficients of the Gegenbauer expansion (based on Cλn(x), for 0⩽n⩽m) of any L2 function can be obtained, within exponential accuracy, provided that both λ and m are proportional to (but smaller than) N. In the second step we construct the Gegenbauer expansion based on Cλn, 0⩽n⩽m, from the coefficients found in the first step. We show that this series converges exponentially with N, provided that the original function is analytic (though nonperiodic). Thus we prove that the Gibbs phenomenon can be completely overcome

Rhythmogenic neuronal networks, pacemakers, and k-cores

Neuronal networks are controlled by a combination of the dynamics of individual neurons and the connectivity of the network that links them together. We study a minimal model of the preBotzinger complex, a small neuronal network that controls the breathing rhythm of mammals through periodic firing bursts. We show that the properties of a such a randomly connected network of identical excitatory neurons are fundamentally different from those of uniformly connected neuronal networks as described by mean-field theory. We show that (i) the connectivity properties of the networks determines the location of emergent pacemakers that trigger the firing bursts and (ii) that the collective desensitization that terminates the firing bursts is determined again by the network connectivity, through k-core clusters of neurons.Comment: 4+ pages, 4 figures, submitted to Phys. Rev. Let

Universal psychometrics: measuring cognitive abilities in the machine kingdom

We present and develop the notion of ‘universal psychometrics’ as a subject of study, and eventually a discipline, that focusses on the measurement of cognitive abilities for the machine kingdom, which comprises any (cognitive) system, individual or collective, either artificial, biological or hybrid. Universal psychometrics can be built, of course, upon the experience, techniques and methodologies from (human) psychometrics, comparative cognition and related areas. Conversely, the perspective and techniques which are being developed in the area of machine intelligence measurement using (algorithmic) information theory can be of much broader applicability and implication outside artificial intelligence. This general approach to universal psychometrics spurs the re-understanding of most (if not all) of the big issues about the measurement of cognitive abilities, and creates a new foundation for (re)defining and mathematically formalising the concept of cognitive task, evaluable subject, interface, task choice, difficulty, agent response curves, etc. We introduce the notion of a universal cognitive test and discuss whether (and when) it may be necessary for exploring the machine kingdom. On the issue of intelligence and very general abilities, we also get some results and connections with the related notions of no-free-lunch theorems and universal priorsWe thank the anonymous reviewers for their comments. This work was supported by the MEC-MINECO projects CONSOLIDER-INGENIO CSD2007-00022 and TIN 2010-21062-C02-02, GVA project PROMETEO/2008/051, the COST -European Cooperation in the field of Scientific and Technical Research IC0801 ATHernández Orallo, J.; Dowe, DL.; Hernández Lloreda, MV. (2014). Universal psychometrics: measuring cognitive abilities in the machine kingdom. Cognitive Systems Research. 27:50-74. https://doi.org/10.1016/j.cogsys.2013.06.001S50742

Community detection in graphs

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.Comment: Review article. 103 pages, 42 figures, 2 tables. Two sections expanded + minor modifications. Three figures + one table + references added. Final version published in Physics Report

A&P : Revista de la Facultad de Arquitectura, Planeamiento y Diseño Nº10

(re) nace aquí “la nueva A&P”. Ser no solo una revista de arquitectura para universitarios sino también y más que eso, ser una revista de universitarios. De nuestros Alumnos, docentes egresados...; de nuestra Facultad de Arquitectura; de nuestra UNR. Integrar no solo el caudal de textos sobre temas de arquitectura que se editan en el país y en el extranjero, sino antes que nada, instituir un espacio propio en que podamos confrontar ideas, opiniones, posturas... y acercarnos un poco más; en que podamos acordar, disentir, discutir, polemizar, refutar, reflexionar en voz alta. El policía de la Prefectura que controla las costas del puerto, el portuario que fundamenta su política de inversiones, el inversionista que sueña con lotear el Parque España, el industrial que de noche vierte toneladas de desechos tóxicos al río, el pescador que asustado observa el color del pez en su anzuelo, el villero que lava su única camisa en el río, "el ciudadano común" que no encuentra acceso al río, el político que promete abrir Rosario al Paraná, el ferroviario que ya no es ferroviario, el ingeniero de Obras Sanitarias que todavía es ingeniero de Obras Sanitarias, El intendente, el planificador , el aduanero, el historiador... estos son algunos de los personajes cuyos destinos se entrecruzan en las historias de nuestra CIUDAD Y SU RELACION CON EL RIO, tema central de este primer número de A&P.Universidad Nacional de Rosario. Facultad de Arquitectura, Planeamiento y Diseñ