Entropy of semiclassical measures for nonpositively curved surfaces

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. We follow the same main strategy as in the Anosov case (arXiv:0809.0230). We focus on the main differences and refer the reader to (arXiv:0809.0230) for the details of analogous lemmas.Comment: 20 pages. This note provides a detailed proof of a result announced in appendix A of a previous work (arXiv:0809.0230, version 2

Delocalization of slowly damped eigenmodes on Anosov manifolds

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with spectral parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a "strip" of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.Comment: 28 pages; compared with version 1, minor modifications, add two reference

Some open questions in "wave chaos"

The subject area referred to as "wave chaos", "quantum chaos" or "quantum chaology" has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability etc. After giving a rough account on "what is quantum chaos?", I intend to list some pending questions, some of them having been raised a long time ago, some others more recent

Semiclassical measures and the Schroedinger flow on Riemannian manifolds

In this article we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schroedinger equation at times scales $\alpha_{h}$ tending to infinity as the semiclassical parameter $h$ tends to zero (when $\alpha _{h}=1/h$ this is equivalent to consider solutions to the non-semiclassical Schreodinger equation). Some general results are presented, among which a weak version of Egorov's theorem that holds in this setting. A complete characterization is given for the Euclidean space and Zoll manifolds (that is, manifolds with periodic geodesic flow) via averaging formulae relating the semiclassical measures corresponding to the evolution to those of the initial states. The case of the flat torus is also addressed; it is shown that non-classical behavior may occur when energy concentrates on resonant frequencies. Moreover, we present an example showing that the semiclassical measures associated to a sequence of states no longer determines those of their evolutions. Finally, some results concerning the equation with a potential are presented.Comment: 18 pages; Theorems 1,2 extendend to deal with arbitrary time-scales; references adde

Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow

The main result of the paper is Egorov's theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.Comment: 23 pages, no figures. Completely revised and improved version of dg-ga/970301

Asymptotics for turbulent flame speeds of the viscous G-equation enhanced by cellular and shear flows

G-equations are well-known front propagation models in turbulent combustion and describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton-Jacobi equations with convex ($L^1$ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue $\bar H$ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed $s_T$. An important problem in turbulent combustion theory is to study properties of $s_T$, in particular how $s_T$ depends on the flow amplitude $A$. In this paper, we will study the behavior of $\bar H=\bar H(A,d)$ as $A\to +\infty$ at any fixed diffusion constant $d > 0$. For the cellular flow, we show that $$\bar H(A,d)\leq O(\sqrt {\mathrm {log}A}) \quad \text{for all $d>0$}.$$ Compared with the inviscid G-equation ($d=0$), the diffusion dramatically slows down the front propagation. For the shear flow, the limit \nit $\lim_{A\to +\infty}{\bar H(A,d)\over A} = \lambda (d) >0$ where $\lambda (d)$ is strictly decreasing in $d$, and has zero derivative at $d=0$. The linear growth law is also valid for $s_T$ of the curvature dependent G-equation in shear flows.Comment: 27 pages. We improve the upper bound from no power growth to square root of log growt

National inventory of emergency departments in Singapore

Background: Emergency departments (EDs) are the basic units of emergency care. We performed a national inventory of all Singapore EDs and describe their characteristics and capabilities. Methods: Singapore EDs accessible to the general public 24/7 were surveyed using the National ED Inventories instrument ( http://www.emnet-nedi.org). ED staff members were asked about ED characteristics with reference to calendar year 2007. Results: Fourteen EDs participated (100% response). All EDs were located in hospitals, and most (92%) were independent departments. One was a psychiatric ED; the rest were general EDs. Among general EDs, all had a contiguous layout, with medical and surgical care provided in one area. All but two EDs saw both adults and children; one ED was adult-only, and the other saw only children. Six were in the public sector and seven in private health-care institutions, with public EDs seeing the majority (78%) of ED patients. Each private ED had an annual patient census of 60,000. They received 98% of ambulances and had an inpatient admission rate of 30%. Two public EDs reported being overcapacity; no private EDs did. For both public and private EDs, availability of consultant resources in EDs was high, while technological resources varied. Conclusion: Characteristics and capabilities of Singapore EDs varied and were largely dependent on whether they are in public or private hospitals. This initial inventory establishes a benchmark to further monitor the development of emergency care in Singapore

Comprehensive analysis of the HEPN superfamily: identification of novel roles in intra-genomic conflicts, defense, pathogenesis and RNA processing

BACKGROUND: The major role of enzymatic toxins that target nucleic acids in biological conflicts at all levels has become increasingly apparent thanks in large part to the advances of comparative genomics. Typically, toxins evolve rapidly hampering the identification of these proteins by sequence analysis. Here we analyze an unexpectedly widespread superfamily of toxin domains most of which possess RNase activity. RESULTS: The HEPN superfamily is comprised of all α-helical domains that were first identified as being associated with DNA polymerase β-type nucleotidyltransferases in prokaryotes and animal Sacsin proteins. Using sensitive sequence and structure comparison methods, we vastly extend the HEPN superfamily by identifying numerous novel families and by detecting diverged HEPN domains in several known protein families. The new HEPN families include the RNase LS and LsoA catalytic domains, KEN domains (e.g. RNaseL and Ire1) and the RNase domains of RloC and PrrC. The majority of HEPN domains contain conserved motifs that constitute a metal-independent endoRNase active site. Some HEPN domains lacking this motif probably function as non-catalytic RNA-binding domains, such as in the case of the mannitol repressor MtlR. Our analysis shows that HEPN domains function as toxins that are shared by numerous systems implicated in intra-genomic, inter-genomic and intra-organismal conflicts across the three domains of cellular life. In prokaryotes HEPN domains are essential components of numerous toxin-antitoxin (TA) and abortive infection (Abi) systems and in addition are tightly associated with many restriction-modification (R-M) and CRISPR-Cas systems, and occasionally with other defense systems such as Pgl and Ter. We present evidence of multiple modes of action of HEPN domains in these systems, which include direct attack on viral RNAs (e.g. LsoA and RNase LS) in conjunction with other RNase domains (e.g. a novel RNase H fold domain, NamA), suicidal or dormancy-inducing attack on self RNAs (RM systems and possibly CRISPR-Cas systems), and suicidal attack coupled with direct interaction with phage components (Abi systems). These findings are compatible with the hypothesis on coupling of pathogen-targeting (immunity) and self-directed (programmed cell death and dormancy induction) responses in the evolution of robust antiviral strategies. We propose that altruistic cell suicide mediated by HEPN domains and other functionally similar RNases was essential for the evolution of kin and group selection and cell cooperation. HEPN domains were repeatedly acquired by eukaryotes and incorporated into several core functions such as endonucleolytic processing of the 5.8S-25S/28S rRNA precursor (Las1), a novel ER membrane-associated RNA degradation system (C6orf70), sensing of unprocessed transcripts at the nuclear periphery (Swt1). Multiple lines of evidence suggest that, similar to prokaryotes, HEPN proteins were recruited to antiviral, antitransposon, apoptotic systems or RNA-level response to unfolded proteins (Sacsin and KEN domains) in several groups of eukaryotes. CONCLUSIONS: Extensive sequence and structure comparisons reveal unexpectedly broad presence of the HEPN domain in an enormous variety of defense and stress response systems across the tree of life. In addition, HEPN domains have been recruited to perform essential functions, in particular in eukaryotic rRNA processing. These findings are expected to stimulate experiments that could shed light on diverse cellular processes across the three domains of life. REVIEWERS: This article was reviewed by Martijn Huynen, Igor Zhulin and Nick Grishi