Notes on spectral theory on Banach spaces

We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. We will try to find conditions under which the action of T is given by a series. This provides a Banach-space version of the well-known Hilbert-space result of E. Schmidt. Based on joint work/collaboration with Edmunds/Evans/Harris

Correlated D-meson decays competing against thermal QGP dilepton radiation

The QGP that might be created in ultrarelativistic heavy-ion collisions is expected to radiate thermal dilepton radiation. However, this thermal dilepton radiation interferes with dileptons originating from hadron decays. In the invariant mass region between the $\phi$ and $J/\Psi$ peak ($1\,$GeV$\lesssim M_{\ell^+ \ell^-} \lesssim 3 \,$GeV) the most substantial background of hadron decays originates from correlated D$\bar{\mathrm{D}}$-meson decays. We evaluate this background using a Langevin simulation for charm quarks. As background medium we utilize the well-tested UrQMD-hybrid model. The required drag and diffusion coefficients are taken from a resonance approach. The decoupling of the charm quarks from the hot medium is performed at a temperature of $130\,$MeV and as hadronization mechanism a coalescence approach is chosen. This model for charm quark interactions with the medium has already been successfully applied to the study of the medium modification and the elliptic flow at FAIR, RHIC and LHC energies. In this proceeding we present our results for the dilepton radiation from correlated D$\bar{\mathrm{D}}$ decays at RHIC energy in comparison to PHENIX measurements in the invariant mass range between 1 and 3 GeV using different interaction scenarios. These results can be utilized to estimate the thermal QGP radiation.Comment: 4 pages, 1 figur

Quality of non-compactness for Sobolev Embedding with one point non-compactness

This paper investigates instances of Sobolev embeddings characterized by local compactness at every point within their domain, except for a single point. We obtain the sharp conditions that distinguish compactness from non-compactness and observe that in the context of Sobolev embeddings, non-compactness occurring at only one point within the domain could give rise to an infinite-dimensional subspace where the embedding is invertible (i.e., not strictly singular). Furthermore, we establish lower bounds for the Bernstein numbers, entropy numbers, and the measure of non-compactness.Comment: 17 page

ROS3P---an accurate third-order Rosenbrock solver designed for parabolic problems

In this note we present a new Rosenbrock solver which is third--order accurate for nonlinear parabolic problems. Since Rosenbrock methods suffer from order reductions when they are applied to partial differential equations, additional order conditions have to be satisfied. Although these conditions have been known for a longer time, from the practical point of view only little has been done to construct new methods. {sc Steinebach cite{St95 modified the well--known solver RODAS of {sc Hairer and {sc Wanner cite{HaWa96 to preserve its classical order four for special problem classes including linear parabolic equations. His solver RODASP, however, drops down to order three for nonlinear parabolic problems. Our motivation here was to derive an efficient third--order Rosenbrock solver for the nonlinear situation. Such a method exists with three stages and two function evaluations only. A comparison with other third--order methods shows the substantial potential of our new method