A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.

HADES is a versatile magnetic spectrometer aimed at studying dielectron production in pion, proton and heavy-ion induced collisions. Its main features include a ring imaging gas Cherenkov detector for electron-hadron discrimination, a tracking system consisting of a set of 6 superconducting coils producing a toroidal field and drift chambers and a multiplicity and electron trigger array for additional electron-hadron discrimination and event characterization. A two-stage trigger system enhances events containing electrons. The physics program is focused on the investigation of hadron properties in nuclei and in the hot and dense hadronic matter. The detector system is characterized by an 85 % azimuthal coverage over a polar angle interval from 18• to 85• , a single electron efficiency of 50 % and a vector meson mass resolution of 2.5 %. Identification of pions, kaons and protons is achieved combining time-of-flight and energy loss measurements over a large momentum range. This paper describes the main features and the performance of the detector system.

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the following objects: the propagator (time-dependent operator which solves the Cauchy problem for the dynamic equation), the spectral function (sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive λ) and the counting function (number of eigenvalues between zero and a positive λ). We derive explicit two-term asymptotic formulae for all three. For the propagator "asymptotic" is understood as asymptotic in terms of smoothness, whereas for the spectral and counting functions "asymptotic" is understood as asymptotic with respect to λ → +∞.Mathematics Subject Classification (2010). Primary 35P20; Secondary 35J46, 35R01.

We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature and study the resulting system of Euler-Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with metric of a pp-wave and parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely Weyl) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in Minkowski space. We discuss the possibility of using this nonRiemannian solution as a mathematical model for the graviton or the neutrino.