Jean-Pierre Eckmann

Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems

We show that values of the correlation dimension estimated over a decade from the Grassberger-Procaccia algorithm cannot exceed the value 2 log10N if N is the number of points in the time series. When this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present. The estimation of Lyapunov exponents is also discussed.

Curvature of co-links uncovers hidden thematic layers in the World Wide Web

Beyond the information stored in pages of the World Wide Web, novel types of “meta-information” are created when pages connect to each other. Such meta-information is a collective effect of independent agents writing and linking pages, hidden from the casual user. Accessing it and understanding the interrelation between connectivity and content in the World Wide Web is a challenging problem [Botafogo, R. A. & Shneiderman, B. (1991) in Proceedings of Hypertext (Assoc. Comput. Mach., New York), pp. 63–77 and Albert, R. & Barabási, A.-L. (2002) Rev. Mod. Phys. 74, 47–97]. We demonstrate here how thematic relationships can be located precisely by looking only at the graph of hyperlinks, gleaning content and context from the Web without having to read what is in the pages. We begin by noting that reciprocal links (co-links) between pages signal a mutual recognition of authors and then focus on triangles containing such links, because triangles indicate a transitive relation. The importance of triangles is quantified by the clustering coefficient [Watts, D. J. & Strogatz, S. H. (1999) Nature (London) 393, 440–442], which we interpret as a curvature [Bridson, M. R. & Haefliger, A. (1999) Metric Spaces of Non-Positive Curvature (Springer, Berlin)]. This curvature defines a World Wide Web landscape whose connected regions of high curvature characterize a common topic. We show experimentally that reciprocity and curvature, when combined, accurately capture this meta-information for a wide variety of topics. As an example of future directions we analyze the neural network of Caenorhabditis elegans, using the same methods.

Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures

Abstract:We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of Hörmander used in the study of hypoelliptic differential operators.

Universal properties of maps on an interval

We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points.

Decay properties and Borel summability for the Schwinger functions in

For the truncated Schwinger functions of theP(Φ)2 field theories, we show strong decrease in the separation of points. This shows uniqueness of theories withP of degree four. We also extend the domain of analyticity in the coupling constant. For theories withP of degree four, the combination of these two results gives Borel summability.

P(\phi)_{2}

AbstractNew bounds are given for the L2-norm of the solution of the Kuramoto-Sivashinsky equation

A global attracting set for the Kuramoto-Sivashinsky equation

\partial _t U(x,t) = - (\partial _x^2 + \partial _x^4 )U(x,t) - U(x,t)\partial _x U(x,t)

Entropy Production in Nonlinear, Thermally Driven Hamiltonian Systems

, for initial data which are periodic with periodL. There is no requirement on the antisymmetry of the initial data. The result is