Abstract. We use geometric methods to study two natural twocomponent generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff(S 1 ) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm equation, giving explicit examples of large subspaces of positive curvature.

Wearable sensor systems which allow for remote or self-monitoring of health-related parameters are regarded as one means to alleviate the consequences of demographic change. This paper aims to summarize current research in wearable sensors as well as in sensor-enhanced health information systems. Wearable sensor technologies are already advanced in terms of their technical capabilities and are frequently used for cardio-vascular monitoring. Epidemiologic predictions suggest that neuropsychiatric diseases will have a growing impact on our health systems and thus should be addressed more intensively. Two current project examples demonstrate the benefit of wearable sensor technologies: long-term, objective measurement under daily-life, unsupervised conditions. Finally, up-to-date approaches for the implementation of sensor-enhanced health information systems are outlined. Wearable sensors are an integral part of future pervasive, ubiquitous and person-centered health care delivery. Future challenges include their integration into sensor-enhanced health information systems and sound evaluation studies involving measures of workload reduction and costs.

As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potentialshown that the Schrödinger operators H t = − + W t have spectrum (surface states) in the spectral gaps of H 0 , for suitable t ∈ (0, 1). We also discuss the density of these surface states as compared to the density of the bulk. Our approach is variational and it is first applied to the well-known dislocation problem (Korotyaev (2000(Korotyaev ( , 2005 [15,16]) on the real line. We then proceed to the dislocation problem for an infinite strip and for the plane. In Appendix A, we discuss regularity properties of the eigenvalue branches in the one-dimensional dislocation problem for suitable classes of potentials.

Abstract. As a model for an interface in solid state physics, we consider two real-valued potentials V (1) and V (2) on the cylinder or tube S = R × (R/Z) where we assume that there exists an interval (a 0 , b 0 ) which is free of spectrum of −∆ + V (k) for k = 1, 2. We are then interested in the spectrum of Ht = −∆ + Vt, for t ∈ R, where Vt(x, y) = V (1) (x, y), for x > 0, and Vt(x, y) = V (2) (x + t, y), for x < 0. While the essential spectrum of Ht is independent of t, we show that discrete spectrum, related to the interface at x = 0, is created in the interval (a 0 , b 0 ) at suitable values of the parameter t, provided −∆ + V (2) has some essential spectrum in (−∞, a 0 ]. We do not require V (1) or V (2) to be periodic. We furthermore show that the discrete eigenvalues of Ht are Lipschitz continuous functions of t if the potential V (2) is locally of bounded variation.

Abstract. In this paper, we study two-component versions of the periodic Hunter-Saxton equation and its µ-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group Diff(S) with a space of scalar functions on S we show that both equations are locally well-posed. The main result of the paper is that the sectional curvature associated with the 2HS is constant and positive and that 2µHS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in [11].