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

The Selberg zeta function for convex co-compact Schottky groups

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on ${\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $\exp (C |s|^\delta)$ where $\delta$ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound $\exp (C |s|^{n+1})$, and it gives new bounds on the number of resonances (scattering poles) of $\Gamma \backslash {\mathbb H}^{n+1}$. The proof of this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider $\Gamma \backslash {\mathbb H}^{n+1}$ as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets

Some recent developments in quantization of fractal measures

We give an overview on the quantization problem for fractal measures, including some related results and methods which have been developed in the last decades. Based on the work of Graf and Luschgy, we propose a three-step procedure to estimate the quantization errors. We survey some recent progress, which makes use of this procedure, including the quantization for self-affine measures, Markov-type measures on graph-directed fractals, and product measures on multiscale Moran sets. Several open problems are mentioned.Comment: 13 page

Expression and Function of Serotonin 2A and 2B Receptors in the Mammalian Respiratory Network

Neurons of the respiratory network in the lower brainstem express a variety of serotonin receptors (5-HTRs) that act primarily through adenylyl cyclase. However, there is one receptor family including 5-HT2A, 5-HT2B, and 5-HT2C receptors that are directed towards protein kinase C (PKC). In contrast to 5-HT2ARs, expression and function of 5-HT2BRs within the respiratory network are still unclear. 5-HT2BR utilizes a Gq-mediated signaling cascade involving calcium and leading to activation of phospholipase C and IP3/DAG pathways. Based on previous studies, this signal pathway appears to mediate excitatory actions on respiration. In the present study, we analyzed receptor expression in pontine and medullary regions of the respiratory network both at the transcriptional and translational level using quantitative RT-PCR and self-made as well as commercially available antibodies, respectively. In addition we measured effects of selective agonists and antagonists for 5-HT2ARs and 5-HT2BRs given intra-arterially on phrenic nerve discharges in juvenile rats using the perfused brainstem preparation. The drugs caused significant changes in discharge activity. Co-administration of both agonists revealed a dominance of the 5-HT2BR. Given the nature of the signaling pathways, we investigated whether intracellular calcium may explain effects observed in the respiratory network. Taken together, the results of this study suggest a significant role of both receptors in respiratory network modulation

We are all one together : peer educators\u27 views about falls prevention education for community-dwelling older adults - a qualitative study

Background: Falls are common in older people. Despite strong evidence for effective falls prevention strategies, there appears to be limited translation of these strategies from research to clinical practice. Use of peers in delivering falls prevention education messages has been proposed to improve uptake of falls prevention strategies and facilitate translation to practice. Volunteer peer educators often deliver educational presentations on falls prevention to community-dwelling older adults. However, research evaluating the effectiveness of peer-led education approaches in falls prevention has been limited and no known study has evaluated such a program from the perspective of peer educators involved in delivering the message. The purpose of this study was to explore peer educators’ perspective about their role in delivering peer-led falls prevention education for community-dwelling older adults. Methods: A two-stage qualitative inductive constant comparative design was used.In stage one (core component) focus group interviews involving a total of eleven participants were conducted. During stage two (supplementary component) semi-structured interviews with two participants were conducted. Data were analysed thematically by two researchers independently. Key themes were identified and findings were displayed in a conceptual framework. Results: Peer educators were motivated to deliver educational presentations and importantly, to reach an optimal peer connection with their audience. Key themes identified included both personal and organisational factors that impact on educators’ capacity to facilitate their peers’ engagement with the message. Personal factors that facilitated message delivery and engagement included peer-to-peer connection and perceived credibility, while barriers included a reluctance to accept the message that they were at risk of falling by some members in the audience. Organisational factors, including ongoing training for peer educators and formative feedback following presentations, were perceived as essential because they affect successful message delivery. Conclusions: Peer educators have the potential to effectively deliver falls prevention education to older adults and influence acceptance of the message as they possess the peer-to-peer connection that facilitates optimal engagement. There is a need to consider incorporating learnings from this research into a formal large scale evaluation of the effectiveness of the peer education approach in reducing falls in older adults

The Hausdorff and dynamical dimensions of self-affine sponges : a dimension gap result

We construct a self-affine sponge in R 3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space

The Phrenic Component of Acute Schizophrenia – A Name and Its Physiological Reality

Decreased heart rate variability (HRV) was shown for unmedicated patients with schizophrenia and their first-degree relatives, implying genetic associations. This is known to be an important risk factor for increased cardiac mortality in other diseases. The interaction of cardio-respiratory function and respiratory physiology has never been investigated in the disease although it might be closely related to the pattern of autonomic dysfunction. We hypothesized that increased breathing rates and reduced cardio-respiratory coupling in patients with acute schizophrenia would be associated with low vagal function. We assessed variability of breathing rates and depth, HRV and cardio-respiratory coupling in patients, their first-degree relatives and controls at rest. Control subjects were investigated a second time by means of a stress task to identify stress-related changes of cardio-respiratory function. A total of 73 subjects were investigated, consisting of 23 unmedicated patients, 20 healthy, first-degree relatives and 30 control subjects matched for age, gender, smoking and physical fitness. The LifeShirt®, a multi-function ambulatory device, was used for data recording (30 minutes). Patients breathe significantly faster (p<.001) and shallower (p<.001) than controls most pronouncedly during exhalation. Patients' breathing is characterized by a significantly increased amount of middle- (p<.001), high- (p<.001), and very high frequency fluctuations (p<.001). These measures correlated positively with positive symptoms as assessed by the PANSS scale (e.g., middle frequency: r = 521; p<.01). Cardio-respiratory coupling was reduced in patients only, while HRV was decreased in patients and healthy relatives in comparison to controls. Respiratory alterations might reflect arousal in acutely ill patients, which is supported by comparable physiological changes in healthy subjects during stress. Future research needs to further investigate these findings with respect to their physiological consequences for patients. These results are invaluable for researchers studying changes of biological signals prone to the influence of breathing rate and rhythm (e.g., functional imaging)