On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function

We derive new results about properties of the Witten zeta function associated with the group SU(3), and use them to prove an asymptotic formula for the number of n-dimensional representations of SU(3) counted up to equivalence. Our analysis also relates the Witten zeta function of SU(3) to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.Comment: To appear in Acta Arithmetic

Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross-Pitaevskii Equation

We rigorously analyze the bifurcation of stationary so called nonlinear Bloch waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be expressed as finite sums of quasi-periodic functions, and which in a formal asymptotic expansion are obtained from solutions of the so called algebraic coupled mode equations. Here we justify this expansion by proving the existence of NLBs and estimating the error of the formal asymptotics. The analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In addition, we illustrate some relations of NLBs to other classes of solutions of the GP equation, in particular to so called out--of--gap solitons and truncated NLBs, and present some numerical experiments concerning the stability of these solutions.Comment: 32 pages, 12 figures, changes: discussion of assumptions reorganized, a new section on stability of the studied solutions, 15 new references adde

A knowledge server for reasoning about temporal constraints between classes and instances of events

An outstanding example of early Reformation dress, notice the geometric fabric design, the fur-trimmed collar of the coat, and the decorative shir

Pattern Discovery from Event Data

Events are ubiquitous in real-life. With the rapid rise of the popularity of social media channels, massive amounts of event data, such as information about festivals, concerts, or meetings, are increasingly created and shared by users on the Internet. Deriving insights or knowledge from such social media data provides a semantically rich basis for many applications, for instance, social media marketing, service recommendation, sales promotion, or enrichment of existing data sources. In spite of substantial research on discovering valuable knowledge from various types of social media data such as microblog data, check-in data, or GPS trajectories, interestingly there has been only little work on mining event data for useful patterns. In this thesis, we focus on the discovery of interesting, useful patterns from datasets of events, where information about these events is shared by and spread across social media platforms. To deal with the existence of heterogeneous event data sources, we propose a comprehensive framework to model events for pattern mining purposes, where each event is described by three components: context, time, and location. This framework allows one to easily define how events are related in terms of conceptual, temporal, and spatial (geographic) relationships. Moreover, we also take into account hierarchies for contexts, time, and locations of events, which naturally exist as useful background knowledge to derive patterns at different levels of abstraction and granularity. Based on this framework, we focus on the following problems: (i) mining interval-based event sequence patterns, (ii) mining periodic event patterns, and (iii) extracting semantic annotations for locations of events. Generally, the first two problems consider correlations of events whereas the last one takes correlations of event components into account. In particular, the first problem is a generalization of mining sequential patterns from traditional data, where patterns representing complex temporal relationships among events can be discovered at different levels of abstraction and granularity. The second problem is to find periodic event patterns, where a notion of relaxed periodicity is formulated for events as well as for groups of events that co-occur. The third~problem is to extract semantic annotations for locations on the basis of exploiting correlations of contexts, time, and locations of events. For the three problems above, we respectively propose novel and efficient approaches. Our experiments clearly indicate that extracted patterns and knowledge can be well utilized in various useful tasks, such as event prediction, semantic search for locations, or topic-based clustering of locations

Dynamic context adaptation in multimedia documents

ABSTRACT Multimedia documents are collections of media objects, synchronized by means of sets of temporal and spatial constraints. Any multimedia document definition is valid as long as the referred media objects are available and the constraints are satisfiable. Document validity depends on the context in which the document has to be presented. In this paper, we introduce a framework to characterize context adaptation, in the presence of both physical and user oriented context requirements. We define semantically equivalent presentation fragments as alternative to undeliverable ones. In the absence of equivalence, undeliverable media are replaced with candidates that minimize the loss of information/quality in the presentation