A dynamical approach to von Neumann dimension

International audienceLet $\Gamma$ be an amenable group and $V$ be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of $\ell^2(\Gamma;V)$ (with respect to $\Gamma$) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a $\Gamma$-invariant linear subspaces $Y$ of $\ell^p(\Gamma;V)$ a real positive number \dlp Y (which is the von Neumann dimension when $p=2$). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective $\Gamma$-equivariant linear map of finite-type from $\ell^p(\Gamma;V) \to \ell^p(\Gamma; V')$ if $\dim V > \dim V'$. A generalization of the Ornstein-Weiss lemma is developed along the way

The Generalized Graetz Problem in Finite Domains

We consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous “extended” and “conjugated” Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework includes both heterogeneous and possibly anisotropic diffusion tensors. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of L2 . In the case of finite domains a similar basis can be exhibited, and the mode’s amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis includes both the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multireflection image method. Error estimates are provided when numerically truncating the spectrum to a finite number of modes. Numerical examples are validated for reference configurations and provided in nontrivial cases. Our methodology shows how to map the solution of stationary convection-diffusion problems in finite three-dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost

Connectedness of spheres in Cayley graphs

We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order n² to the sphere of radius n. We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs

Time delay and Calabi invariant in classical scattering theory

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H_0,H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud-Wigner formula of quantum mechanics. Using recent results of V. Buslaev and A. Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincar\'e scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincar\'e ball.Comment: 22 page

Uniform stability estimates for the discrete Calderon problems

In this article, we focus on the analysis of discrete versions of the Calderon problem in dimension d \geq 3. In particular, our goal is to obtain stability estimates for the discrete Calderon problems that hold uniformly with respect to the discretization parameter. Our approach mimics the one in the continuous setting. Namely, we shall prove discrete Carleman estimates for the discrete Laplace operator. A main difference with the continuous ones is that there, the Carleman parameters cannot be taken arbitrarily large, but should be smaller than some frequency scale depending on the mesh size. Following the by-now classical Complex Geometric Optics (CGO) approach, we can thus derive discrete CGO solutions, but with limited range of parameters. As in the continuous case, we then use these solutions to obtain uniform stability estimates for the discrete Calderon problems.Comment: 38 pages, 2 figure

Shape optimization for the generalized Graetz problem

We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion equation. The problem boils down to the optimization of generalized eigen values on a two phases domain. Shape sensitivity analysis is performed with respect to the evolution of the interface between the fluid and solid phase. In particular physical settings, counterexamples where there is no optimal domains are exhibited. Numerical examples of optimal domains with different physical parameters and constraints are presented. Two different numerical methods (level-set and mesh-morphing) are show-cased and compared

A Small but Efficient Collaboration for the Spiral2 Control System Development

http://accelconf.web.cern.ch/AccelConf/ICALEPCS2013/papers/tucobab01.pdfThe Spiral2 radioactive ion beam facility to be commissioned in 2014 at Ganil (Caen) is built within international collaborations. This also concerns the control system development shared by three laboratories: Ganil has to coordinate the control and automated systems work packages, CEA/IRFU is in charge of the "injector" (sources and low energy beam lines) and the LLRF, CNRS/IPHC provides the emittancemeters and a beam diagnostics platform. Besides the technology Epics based, this collaboration, although being handled with a few people, nevertheless requires an appropriate and tight organization to reach the objectives given by the project. This contribution describes how, started in 2006, the collaboration for controls has been managed both from the technological point of view and the organizational one, taking into account not only the previous experience, technical background or skill of each partner, but also their existing working practices and "cultural" approaches. A first feedback comes from successful beam tests carried out at Saclay and Grenoble; a next challenge is the migration to operation, Ganil having to run Spiral2 as the other members are moving to new projects

Amenability of groups and GG-sets

This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure. The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collaboration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups

Background Light in Potential Sites for the ANTARES Undersea Neutrino Telescope

The ANTARES collaboration has performed a series of {\em in situ} measurements to study the background light for a planned undersea neutrino telescope. Such background can be caused by $^{40}$K decays or by biological activity. We report on measurements at two sites in the Mediterranean Sea at depths of 2400~m and 2700~m, respectively. Three photomultiplier tubes were used to measure single counting rates and coincidence rates for pairs of tubes at various distances. The background rate is seen to consist of three components: a constant rate due to $^{40}$K decays, a continuum rate that varies on a time scale of several hours simultaneously over distances up to at least 40~m, and random bursts a few seconds long that are only correlated in time over distances of the order of a meter. A trigger requiring coincidences between nearby photomultiplier tubes should reduce the trigger rate for a neutrino telescope to a manageable level with only a small loss in efficiency.Comment: 18 pages, 8 figures, accepted for publication in Astroparticle Physic

Community psychiatric nurses and the care co-ordinator role: squeezed to provide ‘limited nursing’.

Background: The Care Programme Approach (CPA) is the key policy underpinning community-focused mental health services but has been unevenly implemented and is associated with increased inpatient bed use. The care co-ordinator role is central to the CPA and is most often held by Community Psychiatric Nurses (CPNs), but there has been little research into how this role is conducted or how it impacts on the work of CPNs and their ability to meet the needs of service users. Aim: The study aimed to identify and illuminate the factors that either facilitated or constrained the ability of CPNs, in their role as care co-ordinators, to meet service users’ and carers’ needs. Methods: A multiple case study of seven sectorised community mental health teams was employed over two years using predominantly qualitative methods of participant observation, semi-structured interviews and document review. Findings: Additional duties and responsibilities specifically associated with the care co-ordinator role and multidisciplinary working, combined with heavy workloads, combined to produce ‘limited nursing’, whereby CPNs are unable to provide evidence-based psychosocial interventions that are recognised to reduce relapse amongst people with severe mental illness. Conclusions: The role of the CPA care co-ordinator was not designed to support the provision of psychosocial interventions. Consequently, CPNs in the co-ordinator role faced with competing demands are unable to provide the range of structured, evidence-based interventions required. This may partially account for the increased inpatient bed use associated with the CPA