arXiv subjects

Sônia Pinto-de-Carvalho

Publications and source records attributed to Sônia Pinto-de-Carvalho.

5 recordsLinked to original sources

arXiv2025

Optical Billiards

This paper investigates the dynamics of optical billiards, a generalization of classic billiards, where light rays travel within a refractive medium and reflect elastically at the boundary. Inspired by studies of acoustic modes in rapidly rotating stars, optical billiards are approached using Riemannian geometry, treating the refractive index as a metric. The discrete dynamics is explored through geodesics, curvature properties, highlighting similarities and differences with classical billiards. Examples include geodesic circular billiards and refractive media with zero Gaussian curvature, revealing integrability conditions and singular behaviors. The study establishes foundational results for optical billiards, emphasizing their dependence on boundary convexity and refractive index.

math.DS↗

arXiv2023

Symmetric periodic orbits in symmetric billiards

In this text we study billiards on ovals and investigate some consequences of a rotational symmetry of the boundary on the dynamics. As it simplifies some calculations, the symmetry helps to obtain the results. We focus on periodic orbits with the same symmetry of the boundary which always exist and prove that in general half of them are elliptic and Moser stable and the other half are hyperbolic with homo(hetero)clinic intersections.

math.DS↗

arXiv2022

Break-up of resonant invariant circles in perturbations of the geodesic circular billiard on surfaces of constant curvature

We study the non-persistence of horizontal invariant circles for geodesically convex perturbations of the geodesic circular billiard on surfaces of constant curvature and show that the result obtained by Ramírez-Ros for the planar case, remains true for billiards on surfaces with constant curvature.

math.DS↗

arXiv2022

A Random Billiard Map in the Circle

We consider a random billiard map, the one in which the standard specular reflection rule is replaced by a random reflection given by a Markov operator. We exhibit an invariant measure for random billiards on general tables. In the special case of a circular table we show that almost every (random) orbit is dense in the boundary as well as in the circular ring formed between the circle boundary and the random caustic. We additionally prove Strong Knudsen's Law for a particular case of families of absolutely continuous measures with respect to Liouville.

math.DS↗

arXiv2017

Modeling the prevalence of Schistosoma mansoni infection in an endemic population

We investigate theoretically if treatment alone can reduce the schistosomiasis's prevalence in an infected population, in a long-lasting sustainable way. We use a non-linear system of ordinary differential equations (a SI system combined with a logistic population growth) which describes the time evolution of the non-infected and infected populations, in terms of the recovering, infection, and demographic rates. Our model leads to the conclusion that the only way to eliminate this endemic disease is to implement public health policies aimed at both treatment and environment. We apply our model to the endemic area of Virgem das Graças, in Brazil, where the prevalence of Schistosiamisis in 2001 was greater than 50\%. The epidemiological data are extracted from a longitudinal study carried on the region between 2001 and 2010 and the demographic parameter from official Brazilian population data. When these estimated parameters are entered, our model gives a limit prevalence for Virgem das Graças of 11\%, which is still significantly high even though treatment in accordance to the government regulations is systematically performed. This estimative reinforces once again, that in order to eliminate Schistomiasis, public health policies aimed at treatment, sanitation, including snail control, and health education programs are mandatory.

q-bio.PE↗