We present an investigation of epsilon -entropy, h(epsilon), in dynamical systems, stochastic processes and turbulence, This tool allows for a suitable characterization of dynamical behaviours arising in systems with many different scales of motion. Particular emphasis is put on a recently proposed approach to the calculation of the epsilon -entropy based on the exit-time statistics. The advantages of this method are demonstrated in examples of deterministic diffusive maps, intermittent maps, stochastic self- and multi-affine signals and experimental turbulent data.

Casual mutations and natural selection have driven the evolution of protein amino acid sequences that we observe at present in nature. The question about which is the dominant force of proteins evolution is still lacking of an unambiguous answer. Casual mutations tend to randomize protein sequences while, in order to have the correct functionality, one expects that selection mechanisms impose rigid constraints on amino acid sequences.

The use of Euler polynomials and Euler numbers allows us to construct a quadrature rule similar to the well-known Euler--MacLaurin quadrature formula, using Euler (instead of Bernoulli) numbers, and even (instead of odd) order derivatives of a given function evaluated at the extrema of the considered interval. An expression of the remainder term and numerical examples are also given. © 2001, Heldermann Verlag. All rights reserved.

In this paper, we study the retrieval of water vapor profiles from simulated FORUM measurements. We show that the bias towards the a-priori introduced by the Optimal Estimation technique can be reduced by using larger errors for the a-priori. Reducing the strength of the a-priori may, however, cause unphysical oscillations in the resulting profiles because of the ill-conditioning of the retrieval problem. An a-posteriori regularization technique, the Iterative Variable Strength method, is thus applied to reduce the amplitude of the oscillations.

Thisarticleisconcernedwiththeasymptoticbehaviorofthetwo-dimensionalinviscid Boussinesq equations with a damping term in the velocity equation. Precisely, we provide the time- decay rates of the smooth solutions to that system. The key ingredient is a careful analysis of the Green kernel of the linearized problem in Fourier space, combined with bilinear estimates and interpolation inequalities for handling the nonlinearity.

We provide a framework to extend the relative entropy method to a class of diffusive relaxation systems with discrete velocities. The methodology is detailed in the toy case of the 1D Jin-Xin model under the diffusive scaling, and provides a direct proof of convergence to the limit parabolic equation in any interval of time, in the regime where the solutions are smooth. Recently, the same approach has been successfully used to show the strong convergence of a vector-BGK model to the 2D incompressible Navier-Stokes equations.

Internal waves describe the (linear) response of an incompressible sta- bly stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the slope has the same inclination as the group velocity.