2,383 research outputs found

    The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions

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    The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius rr centered at the integer points, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this motion on time intervals of order 1/r1/r and in the limit as r0+r\to 0^+, in the case of two space dimensions

    On the distribution of free-path lengths for the periodic Lorentz gas III

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    In a flat 2-torus with a disk of diameter rr removed, let Φr(t)\Phi_r(t) be the distribution of free-path lengths (the probability that a segment of length larger than tt with uniformly distributed origin and direction does not meet the disk). We prove that Φr(t/r)\Phi_r(t/r) behaves like 2π2t\frac{2}{\pi^2 t} for each t>2t>2 and in the limit as r0+r\to 0^+, in some appropriate sense. We then discuss the implications of this result in the context of kinetic theory.Comment: 26 pages, 5 figures, to be published in Commun. Math. Phy

    Euclidean random matching in 2D for non-constant densities

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    We consider the 2-dimensional random matching problem in R2.\mathbb{R}^2. In a challenging paper, Caracciolo et. al. arXiv:1402.6993 on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent 2,2, between two samples of NN uniformly distributed points in the unit square is logN/2πN\log N/2\pi N plus corrections, while the expected value of the square of the Wasserstein distance between one sample of NN uniformly distributed points and the uniform measure on the square is logN/4πN\log N/4\pi N. These conjectures has been proved by Ambrosio et al. arXiv:1611.04960. Here we consider the case in which the points are sampled from a non uniform density. For first we give formal arguments leading to the conjecture that if the density is regular and positive in a regular, bounded and connected domain Λ\Lambda in the plane, then the leading term of the expected values of the Wasserstein distances are exactly the same as in the case of uniform density, but for the multiplicative factor equal to the measure of Λ\Lambda. We do not prove these results but, in the case in which the domain is a square, we prove estimates from above that coincides with the conjectured result.Comment: 14 pages, 3 figure

    Voluntary Societies and Urban Elites in 19th Century Italy

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    Paper given at European History [E-seminars

    Language Trees and Zipping

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    In this letter we present a very general method to extract information from a generic string of characters, e.g. a text, a DNA sequence or a time series. Based on data-compression techniques, its key point is the computation of a suitable measure of the remoteness of two bodies of knowledge. We present the implementation of the method to linguistic motivated problems, featuring highly accurate results for language recognition, authorship attribution and language classification.Comment: 5 pages, RevTeX4, 1 eps figure. In press in Phys. Rev. Lett. (January 2002

    On the complete phase synchronization for the Kuramoto model in the mean-field limit

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    We study the Kuramoto model for coupled oscillators. For the case of identical natural frequencies, we give a new proof of the complete frequency synchronization for all initial data; extending this result to the continuous version of the model, we manage to prove the complete phase synchronization for any non-atomic measure-valued initial datum. We also discuss the relation between the boundedness of the entropy and the convergence to an incoherent state, for the case of non identical natural frequencies

    Measuring complexity with zippers

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    Physics concepts have often been borrowed and independently developed by other fields of science. In this perspective a significant example is that of entropy in Information Theory. The aim of this paper is to provide a short and pedagogical introduction to the use of data compression techniques for the estimate of entropy and other relevant quantities in Information Theory and Algorithmic Information Theory. We consider in particular the LZ77 algorithm as case study and discuss how a zipper can be used for information extraction.Comment: 10 pages, 3 figure
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