Complete characterization of convergence to equilibrium for an inelastic Kac model

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index $\alpha$ is determined by the so-called degree of inelasticity, $p>0$, of the particles: $\alpha=\frac{2}{1+p}$. This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index $\alpha$. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted

A CASE OF SYSTEMIC CANINE PROTOTHECOSIS IN ITALY

Introduction: Protothecosis is a rare disease of human and animals caused by aerobic, microscopic, unicellular, achlorophyllic algae of the genus Prototheca, family Chlorellaceae. The canine infections tend to be more severe, with systemic diffusion of the disease. In Italy prototheca mastitis in cattle is well known but very few cases of canine protothecosis have been reported. Materials and Methods: A 2-year-old, male mixed-breed dog was referred with a 30 days history of poorly localized pain and neurological signs. Magnetic resonance imaging revealed multifocal lesions compatible with meningoencephalitis . In the three weeks following MRI the neurologic signs remained stable but the dog developed anorexia and vomiting. An acute renal insufficiency was diagnosed. Because of this severe complication, the dog was euthanized upon request of the owner and a complete post mortem examination was performed. Results: Post-mortem examination and histopathological evalutation revealed multiple disseminated inflammatory foci associated with algae-like structures with cell walls positive with periodic-acid- Schiff (PAS) and metenamina silver stain. Morphological features were consistent with Prototheca. Conclusions: Protothecosis should always be included in the list of differential diagnoses of an animal with signs of multifocal lesions of the CNS, especially when an inflammatory process affecting multiple organs is present