Collocation and quadrature methods for Cauchy singular integral equations on an interval with variable coefficients are studied. Convergence rates are proved in weighted uniform and uniform norms.

Experimental studies on vehicular traffic provide data on quantities like density, flux, and mean speed of the vehicles. However, the diagrams relating these variables (the fundamental and \emph{speed} diagrams) show some peculiarities not yet fully reproduced nor explained by mathematical models. In this paper, resting on the methods of kinetic theory, we introduce a new traffic model which takes into account the heterogeneous nature of the flow of vehicles along a road.

One of the greatest challenges in biomedicine is to get a unified view of observations made from the molecular up to the organism scale. Towards this goal, multiscale models have been highly instrumental in contexts such as the cardiovascular field, angiogenesis, neurosciences and tumour biology. More recently, such models are becoming an increasingly important resource to address immunological questions as well.

Relative dispersion of tracers - i.e. very small, neutrally buoyant particles-, is particularly efficient in incompressible turbulent flows. Due to the non smooth behaviour of velocity differences in the inertial range, the separation distance between two trajectories, R(t)=X1(t)-X2(t) , grows as a power of time superdiffusively, R2(t)t3 , as first observed by L.F. Richardson [1]. This now well established result has no counterpart in the theory of heavy particle suspensions, namely finite-size particles with a mass density much larger that of the carrier fluid.

The authors study the convergence and the stability of a collocation and a discrete collocation method for Cauchy singular integral equations with weakly singular perturbation kernels in some weighted uniform norms. Uniform error estimates are also given. © 1997 Rocky Mountain Mathematics Consortium.