In a previous paper (Comput. Methods Appl. Mech. Engrg 170, 223-237, 1999) we introduced a new method for computing the dominant Lyapunov exponent of a chaotic map by using spatial integration involving a matrix norm. We conjectured that this sequence of integrals decayed proportional to 1/n. We now prove this conjecture and derive a bound on the next term in the asymptotic expansion of the ter...

Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase space. A recent numerical study of spatially extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a nontrivial scaling behavior, D(L)~L(-γ), with the system size L. He...

Recent studies of correlations of intensity in databases of natural images revealed a remarkable property. The two point correlations are described in terms of power law behavior, with an exponent which seems to be robust. In the present Letter we consider the statistical meaning of that result. We study many individual images of one of the databases considered. We find that the same law charac...

In this article we study atomic and molecular decompositions in 2-microlocal Besov and Triebel–Lizorkin spaces with variable integrability. We show that, in most cases, the convergence implied in such decompositions holds not only in the distributions sense, but also in the function spaces themselves. As an application, we give a simple proof for the denseness of the Schwartz class in such spac...

In this Letter, we derive exponent inequalities relating the dynamic exponent z to the steady state exponent Γ for a general class of stochastically driven dynamical systems. We begin by deriving a general exact inequality, relating the response function and the correlation function, from which the various exponent inequalities emanate. We then distinguish between two classes of dynamical syste...

The asymptotics of the Schatten norms of finite Toeplitz matrices generated by functions with a Fisher-Hartwig singularity are described as the matrix dimension n goes to infinity. The message of the paper is to reveal some kind of a kink: the pth Schatten norm increases as n to the power 1/p before the singularity reaches a critical point and as n to an exponent depending on the singularity be...

We show how the theory of affine geometries over the ring Z/〈q−1〉 can be used to understand the properties of toric and generalized toric codes over Fq. The minimum distance of these codes is strongly tied to the collections of lines in the finite geometry that contain subsets of the exponent vectors of the monomials that are evaluated to produce the standard generator matrix for the code. We a...

Methods from scattering theory are introduced to analyze random Schrödinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the " thermodynamic limit " of the spectral shift function per u...

The well-known generalisation of hard Cmeans (HCM) clustering is fuzzy C-means (FCM) clustering where a weight exponent on each fuzzy membership is introduced as the degree of fuzziness. An alternative generalisation of HCM clustering is proposed in this paper. This is called fuzzy entropy (FE) clustering where a weight factor of the fuzzy entropy function is introduced as the degree of fuzzy e...