Dario Benedetto

Language Trees and Zipping

In this Letter we present a very general method for extracting 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.

A Non-Maxwellian Steady Distribution for One-Dimensional Granular Media

We consider a nonlinear Fokker–Planck equation for a one-dimensional granular medium. This is a kinetic approximation of a system of nearly elastic particles in a thermal bath. We prove that homogeneous solutions tend asymptotically in time toward a unique non-Maxwellian stationary distribution.

SOME CONSIDERATIONS ON THE DERIVATION OF THE NONLINEAR QUANTUM BOLTZMANN EQUATION.

In this paper we analyze a system of Nidentical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermis Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d≥2.

Data compression and learning in time sequences analysis

Abstract Motivated by the problem of the definition of a distance between two sequences of characters, we investigate the so-called learning process of a typical sequential data compression schemes. We focus on the problem of how a compression algorithm optimizes its features at the interface between two different sequences A and B while zipping the sequence A+B obtained by simply appending B after A. We show the existence of a scaling function (the “learning function”) which rules the way in which the compression algorithm learns a sequence B after having compressed a sequence A. In particular it turns out that there exists a cross-over length for the sequence B, which depends on the relative entropy between A and B, below which the compression algorithm does not learn the sequence B (measuring in this way the cross-entropy between A and B) and above which it starts learning B, i.e. optimizing the compression using the specific features of B. We check the scaling on three main classes of systems: Bernoulli schemes, Markovian sequences and the symbolic dynamic generated by a nontrivial chaotic system (the Lozi map). As a last application of the method we present the results of a recognition experiment, namely recognize which dynamical systems produced a given time sequence. We finally point out the potentiality of these results for segmentation purposes, i.e. the identification of homogeneous sub-sequences in heterogeneous sequences (with applications in various fields from genetic to time-series analysis).

ON THE WEAK-COUPLING LIMIT FOR BOSONS AND FERMIONS

In this paper we consider a large system of bosons or fermions. We start with an initial datum which is compatible with the Bose–Einstein, respectively Fermi–Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time t, agrees with the analogous expansion for the solution to the Uehling–Uhlenbeck equation. This paper follows the same spirit as the companion work,2 where the authors investigated the weak-coupling limit for particles obeying the Maxwell–Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

An example of mathematical authorship attribution

In this paper we discuss a novel mathematical approach to authorship attribution which we implemented recently to face a concrete problem of author recognition. The fundamental ideas for our methods came from statistical mechanics and information theory. We combine two approaches. Both of them use similarity measures between couples of texts as indicators of stylistic closeness: the first one is based on the comparison of frequencies of fixed length substrings (n-grams) throughout the texts; the second one relies on a suitable use of compression algorithms as relative entropy approximators, in the spirit of the so-called Ziv–Merhav theorem. The two methods were separately developed and then combined, together with a suitable and theoretically founded ranking analysis, to produce an original authorship attribution procedure that yielded very successful results on the specific problem to which it was applied. This ranking analysis could be of interest also in other application fields.

A hydrodynamic model arising in the context of granular media

Abstract In this note, we propose a formal argument identifying the hydrodynamic limit of a Fokker-Planck model for granular media appearing in [1]. More precisely, in the limit of large background temperature and vanishing friction, this hydrodynamic limit is described by the classical system of isentropic gas dynamics with a nonstandard pressure law (specifically, the pressure is proportional to the cube root of the density). Finally, some qualitative properties of the hydrodynamic model are studied.

The collapse phenomenon in one-dimensional inelastic point particle

Abstract We consider a one-dimensional system of n inelastic particles on a line, with coefficient of restitution 1−2e. We prove that if en≲ ln 2 no collapses are possible for any initial datum, and we exhibit explicit collapsing solutions for en≳π. For n=4 we construct a positive measure set of initial data which collapse in a finite time. For n=3 we also consider stochastic perturbations of the system and prove the occurrence of collapses with positive probability if e is sufficiently close to 1/2. Finally, we consider the limit n→∞ of the exact collapse for n particles, obtaining a collapsing measure solution concentrated into two hydrodynamic profiles.

Zipping out relevant information

Although the abundance of information and its accessibility represents an important cultural advance, it also introduces a new challenge: retrieving relevant information. However, the growing body of available data provides an ideal test bed for theoretical constructions and models. This opportunity has stimulated considerable interest from researchers in many different communities-physicists, mathematicians, economists, and statisticians, to name a few. In this spirit, we seek to discover the most suitable tools for examining large masses of data and extracting useful information from it. The information-theoretic method described in this article applies to any kind of corpora of character strings, independent of the type of coding behind them. The method has great potential for fields where human intuition might fail: DNA and protein sequences, geological time series, stock market data, medical monitoring, and so on.

A one dimensional Boltzmann equation with inelastic collisions

We consider the Boltzmann equation for inelastic particles on the line and prove some preliminary results on existence and uniqueness of the solutions. We also discuss some connections with another kinetic equation investigated by the same authors.