Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

Identifying Meaningful Places

Place identification refers to the process of analyzing sensor data in order to detect places, i.e., spatial areas that are linked with activities and associated with meanings. Place information can be used, e.g., to provide awareness cues in applications that support social interactions, to provide personalized and location-sensitive information to the user, and to support mobile user studies by providing cues about the situations the study participant has encountered. Regularities in human movement patterns make it possible to detect personally meaningful places by analyzing location traces of a user. This thesis focuses on providing system level support for place identification, as well as on algorithmic issues related to the place identification process. The move from location to place requires interactions between location sensing technologies (e.g., GPS or GSM positioning), algorithms that identify places from location data and applications and services that utilize place information. These interactions can be facilitated using a mobile platform, i.e., an application or framework that runs on a mobile phone. For the purposes of this thesis, mobile platforms automate data capture and processing and provide means for disseminating data to applications and other system components. The first contribution of the thesis is BeTelGeuse, a freely available, open source mobile platform that supports multiple runtime environments. The actual place identification process can be understood as a data analysis task where the goal is to analyze (location) measurements and to identify areas that are meaningful to the user. The second contribution of the thesis is the Dirichlet Process Clustering (DPCluster) algorithm, a novel place identification algorithm. The performance of the DPCluster algorithm is evaluated using twelve different datasets that have been collected by different users, at different locations and over different periods of time. As part of the evaluation we compare the DPCluster algorithm against other state-of-the-art place identification algorithms. The results indicate that the DPCluster algorithm provides improved generalization performance against spatial and temporal variations in location measurements.