In this paper we examine the impact of graph theory and more particularly the scale-free topology on Immune Network models. In the case of a simple but not trivial model we analyze network performances as long term selectivity properties, its computational capabilities as memory capacity, and relation with Neural Networks. A more advanced Immune Network model is conceptualized and it is developed a scaffold for further mathematical investigation.

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.