We consider the construction and the properties of the Riemann solver for the hyperbolic system \begin{equation}\label{E:hyp0} u_t + f(u)_x = 0, \end{equation} assuming only that $Df$ is strictly hyperbolic. In the first part we prove a general regularity theorem on the admissible curves $T_i$ of the $i$-family, depending on the number of inflection points of $f$: namely, if there is only one inflection point, $T_i$ is $C^{1,1}$. If the $i$-th eigenvalue of $Df$ is genuinely nonlinear, by it is well known that $T_i$ is $C^{2,1}$. However, we give an example of an admissible curve $T_i$ which is only Lipschitz continuous if $f$ has two inflection points. In the second part, we show a general method for constructing the curves $T_i$, and we prove a stability result for the solution to the Riemann problem. In particular we prove the uniqueness of the admissible curves for \eqref{E:hyp0}. Finally we apply the construction to various approximations to \eqref{E:hyp0}: vanishing viscosity, relaxation schemes and the semidiscrete upwind scheme. In particular, when the system is in conservation form, we obtain the existence of smooth travelling profiles for all small admissible jumps of \eqref{E:hyp0}.

A variant of the lattice Boltzmann scheme is presented as a mesoscopic model of glassy behavior. A hierarchical density-dependent interaction potential is introduced, which allows the coexistence and competition of multiple density minima. We find that this competition allows to model geometrical frustration which produces disordered patterns with sharp density contrasts and no phase-separation. A way of modeling mechanical arrest of real glasses is also proposed and discussed.

We apply the lattice Boltzmann methods to study the segregation of binary fluid mixtures under oscillatory shear flow in two dimensions. The algorithm allows to simulate systems whose dynamics is described by the Navier-Stokes and the convection-diffusion equations. The interplay between several time scales produces a rich and complex phenomenology. We investigate the effects of different oscillation frequencies and viscosities on the morphology of the phase separating domains. We find that at high frequencies the evolution is almost isotropic with growth exponents 2/3 and 1/3 in the inertial (low viscosity) and diffusive (high viscosity) regimes, respectively. When the period of the applied shear flow becomes of the same order of the relaxation time T(R) of the shear velocity profile, anisotropic effects are clearly observable. In correspondence with nonlinear patterns for the velocity profiles, we find configurations where lamellar order close to the walls coexists with isotropic domains in the middle of the system. For particular values of frequency and viscosity it can also happen that the convective effects induced by the oscillations cause an interruption or a slowing of the segregation process, as found in some experiments. Finally, at very low frequencies, the morphology of domains is characterized by lamellar order everywhere in the system resembling what happens in the case with steady shear.

We present a fast implementation of a recently proposed speech compression scheme, based on an all-pole model of the vocal tract. Each frame of the speech signal is analyzed by storing the parameters of the complex damped exponentials deduced from the all-pole model and its initial conditions. In mathematical terms, the analysis stage corresponds to solving a structured total least squares (STLS) problem. It is shown that by exploiting the displacement rank structure of the involved matrices the STLS problem can be solved in a very fast way. Synthesis is computationally very cheap since it consists of adding the complex damped exponentials based on the transmitted parameters. The compression scheme is applied on a speech signal. The speed improvement of the fast vocoder analysis scheme is demonstrated. Furthermore, the quality of the compression scheme is compared with that of a standard coding algorithm, by using the segmental Signal-to-Noise Ratio.

We discuss a stochastic closure for the equation of motion satisfied by multiscale correlation functions in the framework of shell models of turbulence. We present a plausible closure scheme to calculate the anomalous scaling exponents of structure functions by using the exact constraints imposed by the equation of motion. We present an explicit calculation for fifth-order scaling exponent by varying the free parameter entering in the nonlinear term of the model. The same method applied to the case of shell models for Kraichnan passive scalar provides a connection between the concept of zero-modes and time-dependent cascade processes.

We present the results of a numerical investigation of three-dimensional decaying turbulence with statistically homogeneous and anisotropic initial conditions. We show that at large times, in the inertial range of scales: (i) isotropic velocity fluctuations decay self-similarly at an algebraic rate which can be obtained by dimensional arguments; (ii) the ratio of anisotropic to isotropic fluctuations of a given intensity falls off in time as a power law, with an exponent approximately independent of the strength of the fluctuation; (iii) the decay of anisotropic fluctuations is not self-similar, their statistics becoming more and more intermittent as time elapses. We also investigate the early stages of the decay. The different short-time behavior observed in two experiments differing by the phase organization of their initial conditions gives a new hunch on the degree of universality of small-scale turbulence statistics, i.e., its independence of the conditions at large scales.