Random billiards with wall temperature and associated Markov chains

By a random billiard we mean a billiard system in which the standard specular reflection rule is replaced with a Markov transition probabilities operator P that, at each collision of the billiard particle with the boundary of the billiard domain, gives the probability distribution of the post-collision velocity for a given pre-collision velocity. A random billiard with microstructure (RBM) is a random billiard for which P is derived from a choice of geometric/mechanical structure on the boundary of the billiard domain. RBMs provide simple and explicit mechanical models of particle-surface interaction that can incorporate thermal effects and permit a detailed study of thermostatic action from the perspective of the standard theory of Markov chains on general state spaces. We focus on the operator P itself and how it relates to the mechanical/geometric features of the microstructure, such as mass ratios, curvatures, and potentials. The main results are as follows: (1) we characterize the stationary probabilities (equilibrium states) of P and show how standard equilibrium distributions studied in classical statistical mechanics, such as the Maxwell-Boltzmann distribution and the Knudsen cosine law, arise naturally as generalized invariant billiard measures; (2) we obtain some basic functional theoretic properties of P. Under very general conditions, we show that P is a self-adjoint operator of norm 1 on an appropriate Hilbert space. In a simple but illustrative example, we show that P is a compact (Hilbert-Schmidt) operator. This leads to the issue of relating the spectrum of eigenvalues of P to the features of the microstructure;(3) we explore the latter issue both analytically and numerically in a few representative examples;(4) we present a general algorithm for simulating these Markov chains based on a geometric description of the invariant volumes of classical statistical mechanics

Billiards in a general domain with random reflections

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain ${\mathcal D} \subset {\mathbb R}^d$ until it hits the boundary and bounces randomly inside according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord "picked at random" in ${\mathcal D}$, and we study the angle of intersection of the process with a $(d-1)$-dimensional manifold contained in ${\mathcal D}$.Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains

Rationing tests for drug-resistant tuberculosis - who are we prepared to miss?

BACKGROUND: Early identification of patients with drug-resistant tuberculosis (DR-TB) increases the likelihood of treatment success and interrupts transmission. Resource-constrained settings use risk profiling to ration the use of drug susceptibility testing (DST). Nevertheless, no studies have yet quantified how many patients with DR-TB this strategy will miss. METHODS: A total of 1,545 subjects, who presented to Lima health centres with possible TB symptoms, completed a clinic-epidemiological questionnaire and provided sputum samples for TB culture and DST. The proportion of drug resistance in this population was calculated and the data was analysed to demonstrate the effect of rationing tests to patients with multidrug-resistant TB (MDR-TB) risk factors on the number of tests needed and corresponding proportion of missed patients with DR-TB. RESULTS: Overall, 147/1,545 (9.5%) subjects had culture-positive TB, of which 32 (21.8%) had DR-TB (MDR, 13.6%; isoniazid mono-resistant, 7.5%; rifampicin mono-resistant, 0.7%). A total of 553 subjects (35.8%) reported one or more MDR-TB risk factors; of these, 506 (91.5%; 95% CI, 88.9-93.7%) did not have TB, 32/553 (5.8%; 95% CI, 3.4-8.1%) had drug-susceptible TB, and only 15/553 (2.7%; 95% CI, 1.5-4.4%) had DR-TB. Rationing DST to those with an MDR-TB risk factor would have missed more than half of the DR-TB population (17/32, 53.2%; 95% CI, 34.7-70.9). CONCLUSIONS: Rationing DST based on known MDR-TB risk factors misses an unacceptable proportion of patients with drug-resistance in settings with ongoing DR-TB transmission. Investment in diagnostic services to allow universal DST for people with presumptive TB should be a high priority

Vascular response after implantation of biolimus A9-eluting stent with bioabsorbable polymer and everolimus-eluting stents with durable polymer. Results of the optical coherence tomography analysis of the BIOACTIVE randomized trial

AbstractBackgroundIn BIOACTIVE study, we evaluated vascular responses after the implant of biolimus A9-eluting stent (BES; BioMatrixTM) and the everolimus-eluting stent (EES; XIENCE VTM). In this study, we present the optical coherence tomography analysis (OCT) 6 months post-intervention.MethodsPatients were randomized to treatment with BES (n = 22) or EES (n = 18). The primary outcome was the frequency of non-covered, poorly positioned struts by OCT.ResultsOCT was performed in 26 patients (BES: n = 15; EES: n = 11) and 749 tomographic images and 7,725 stent struts were analyzed. BES and EES showed similar luminal and stent areas. Neointimal hyperplasia area, neointimal thickness and the percentage of in-stent obstruction (8.44 ± 5.10% vs. 9.21 ± 6.36%; p = 0.74) were similar. The rates of not covered struts (BES: 2.10 ± 3.60% vs. ESS: 2.46 ± 2.15%, p = 0.77) and poorly positioned struts (BES: 0.48 ± 1.48% vs. EES 0.44 ± 1.05%, p = 0.94) were similarly low. The frequency of frames with signs consistent with peri-strut inflammatory infiltrate was low and similar between BES (15.53 ± 20.77%) and EES (11.70 ± 27.51%; p = 0.68).ConclusionsThe second-generation drug-eluting stents BES and EES were equally effective at suppressing the neointimal formation after 6 months, with favorable vascular responses. The frequency of frames with peri-strut infiltrate signals per patient was low, and lower than that observed historically with first-generation drug-eluting stents

Un análisis exploratorio de la relación entre pobreza multidimensional y conflicto armado : el caso de Antioquia en Colombia

RESUMEN: Este artículo analiza la relación entre pobreza y conflicto armado en Antioquia, Colombia. El análisis de la pobreza está enmarcado en el enfoque de las capacidades de Sen, el cual conforma la base conceptual del índice de pobreza multidimensional (IPM) desarrollado por Alkire y Foster. El IPM es calculado con información derivada de la base de datos Sisbén, la cual se utiliza para seleccionar la población atendida por los programas de asistencia social del Gobierno colombiano. Este artículo consideró tres dimensiones de pobreza: estándares de vida, salud, y educación. El conflicto armado fue medido por medio de datos de conteo acerca de la ocurrencia de eventos de violencia registrados entre 1996 y 2010 en cada municipalidad de Antioquia. Luego, la relación entre la pobreza y el conflicto armado se analizó mediante métodos exploratorios y no paramétricos como las distribuciones kernel. Los resultados sugieren que el IPM es robusto con respecto a la elección del umbral de pobreza multidimensional. Los mapas de caja y bigotes sugieren que los pobres están localizados en las regiones periféricas de Antioquia. Las distribuciones kernel muestran que las áreas más afectadas por el conflicto, usualmente, tienen altos niveles de pobreza multidimensional.ABSTRACT: This paper analyses the relationship between poverty and armed conflict in Antioquia, Colombia. The poverty analysis it uses is framed according to Sen’s capability approach, which constitutes the conceptual basis for the Multidimensional Poverty Index (MPI) subsequently developed by Alkire and Foster. The MPI is measured employing data from the government database SISBEN, which is used by the Colombian authorities to identify beneficiaries of social assistance programmes. The paper considers three poverty dimensions: living standards, health, and education. Armed conflict is measured using count data on violent events recorded for every municipality in the Department of Antioquia between 1996 and 2010. The relationship between poverty and armed conflict is then analysed using exploratory and non-parametric methods such as kernel distributions. Results suggest that the MPI is robust when compared multidimensional cutoffs. The MPI box-plot maps suggest that poor people are located in Antioquia’s peripheral areas. Kernel distributions show that areas most affected by conflict tend to show higher levels of multidimensional poverty