We prove existence of solutions of a new free boundary problem described by a system of degenerate parabolic equations. The problem arises in petroleum engineering and concerns fluid flows in diatomite rocks. The unknown functions represent the pressure of the fluid and a damage parameter of the porous rock. These quantities are not necessarily continuous on the free boundary, which considerably complicates the mathematical analysis.
We construct a model of traffic flow with sources and destinations on a road s network. The model is based on a conservation law for the density of traffic and on semilinearequations for traffic-type functions, i. e. functions describing paths for cars.
In the application of Pad\'{e} methods to signal
processing a basic problem is to take into account the effect of
measurement noise on the computed approximants. Qualitative
deterministic noise models have been proposed which are consistent
with experimental results. In this paper the Pad\'{e} approximants
to the $Z$-transform of a complex Gaussian discrete white noise
process are considered. Properties of the condensed density of the
Pad\'{e} poles such as circular symmetry, asymptotic concentration
on the unit circle and independence on the noise variance are
proved. An analytic model of the condensed density of the Pad\'{e}
poles for all orders of the approximants is also computed. Some
Montecarlo simulations are provided.
