18 research outputs found
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 âuniversal psychometricsâ 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 ATHernĂĄndez Orallo, J.; Dowe, DL.; HernĂĄndez 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 NÂș10
(re) nace aquĂ âla nueva A&Pâ. Ser no solo una revista de arquitectura para universitarios sino tambiĂ©n y mĂĄs 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 paĂs y en el extranjero, sino antes que nada, instituir un espacio propio en que podamos confrontar ideas, opiniones, posturas... y acercarnos un poco mĂĄs; en que podamos acordar, disentir, discutir, polemizar, refutar, reflexionar en voz alta. El policĂa de la Prefectura que controla las costas del puerto, el portuario que fundamenta su polĂtica de inversiones, el inversionista que sueña con lotear el Parque España, el industrial que de noche vierte toneladas de desechos tĂłxicos al rĂo, el pescador que asustado observa el color del pez en su anzuelo, el villero que lava su Ășnica camisa en el rĂo, "el ciudadano comĂșn" que no encuentra acceso al rĂo, el polĂtico que promete abrir Rosario al ParanĂĄ, el ferroviario que ya no es ferroviario, el ingeniero de Obras Sanitarias que todavĂa 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 nĂșmero de A&P.Universidad Nacional de Rosario. Facultad de Arquitectura, Planeamiento y Diseñ