proposed detectors like EURO. The adiabatic oscillations of spherical Newtonian stars can be expanded into a complete set of modes consisting of purely radial, spheroidal (or poloidal) and toroidal (or axial) modes. These last modes are connected to `trivial' rotations of the equilibrium model and contain no energy. In a relativistic treatment these oscillations are damped through gravitational radiation. If the star is set into uniform rotation the Coriolis force acts as a restoring force for toroidal motions and the family of r-modes or Rossby waves appear. Recently it has been suggested by Andersson, Friedman and Morsink that these modes are generically unstable via the gravitational radiation induced CFS (Chandrasekhar-Friedman-Schutz) instability. One of the basic predictions of General Relativity is the dragging of inertial frames around a relativistically rotating body. This dragging also influences the pulsation spectra of relativistic stars and seemingly in a more dramatic way for toroidal oscillations. Kojima suggested that, if one calculates the r-mode frequencies using general relativity to lowest order in the angular velocity of the background star, a continuous part occurs in the spectrum, in contrast to the calculations from Newtonian theory, where the spectrum is`discrete'. Spectra containing continuous parts have been found in many cases in the past in the study of differentially (= `non-uniformly') rotating fluids. The continuous part in these cases was again seen for the r-modes together with many interesting phenomena such as the passage of low-order r-modes from the discrete into the continuous part as the differential rotation increases, and the presence of low order discrete p-modes in the middle of the continuous part for more rapidly rotating disks. The stars under consideration here have no differential rotation and the existence of a continuous part in the spectrum is attributed to the dragging of inertial frames due to general relativity. Jointly with K.Kokkotas from Aristotle University of Thessaloniki, I gave a rigorous proof of Kojima's suggestion . Note here that these values are in general not just eigenvalues as Kojima assumed, but mathematically analogous to spectral values associated with scattering states in quantum mechanics. Hence the associated `modes' cannot be assumed to satisfy usual physical boundary conditions. In the sequel, we dropped Kojima's slow motion assumption. The results here are still preliminary but very promising. They suggest that the continuous part is not an artifact of the slow motion approximation. But they also indicate a `discontinuous' dependence of the spectrum on the fact whether the underlying space is finite (accompanied by a boundary condition at a finite radius) or infinite. Of course, nearly every numerical calculation necessarily has to assume a finite space. If this feature persists it will have important consequences on the methods to compute these values. Furthermore, recent numerical calculations indicate that the existence of a pure discrete r-mode is in question. Results by Ruoff and Kokkotas indicate that r-modes satisfying the usual boundary conditions exist only for neutron stars with a very stiff equation of state or for stars which are non-relativistic. Elastodynamics Physical systems are mostly described by differential equations together with initial conditions and boundary conditions. For systems showing some kind of `memory', local differential equations are not appropriate. Particularly in the description of the damping behaviour of viscoelastic media, fractional differential equations are suggested by many authors. A main advantage of that approach compared to others is that the damping behaviour can be described with only few parameters by replacing integer derivatives in damping terms by fractional derivatives. The definition of the fractional derivative is far from being unique and many of these approaches use ad-hoc definitions given by Riemann and Liouville. Usually this leads to the introduction of a time t0 where the system `is turned on'. As a consequence, the description is not homogeneous in time whereas the system is clearly expected to show this behaviour. Another consequence is the fact that the whole history of the system for times smaller than t0 has to be described by initial conditions. It is very likely that this cannot be done by local initial conditions. On the other hand non-local initial conditions usually lack physical interpretation. From an analytical point of view further disadvantages are the loss of the important semigroup property of integer order derivatives and the fact that those approaches do not specify an admissible function space for the solutions. The last very much hinders a discussion of the uniqueness/stability of the solutions and related also the generalization to partial differential equations. Beginning from 1994, my colleague Siegmar Kempfle from the University of the Federal Armed Forces Hamburg (UniBW Hamburg) and I developed a functional analytic approach to such pseudo-differential equations which avoids these problems and especially addresses the physically important question of `causality'. Over time our approach attracted more and more international recognition resulting in invited talks and publications as well as invitations to act as referee in the subject for relevant scientific magazines. Also from the very beginning a main goal was to compare our results to experiment. Indeed such experiments have recently been made by I. Schaefer at the UniBW Hamburg using as reference example PFTE (Teflon). The comparison of the measured impulse and frequency responses of the rod with our calculated solutions have shown a good agreement, whereas `classical' models failed. Due to its success and as a further challenge we currently start to generalize our approach to partial differential equations with fractional time derivatives to describe wave propagation in viscoelastic materials. At present S. Kempfle, I. Schaefer at the UniBW Hamburg and I are applying for such a common project to the DFG. A major point in the project is again the comparison of the mathematical results to experiment. | 
