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Let K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by O K , and A denotes any O K -order in K [G]. The paper describes an algorithm that explicitly computes the Picard group Pic(A), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of O L in Pic(O K [G]) can be numerically determined.

Abstract. Preference algebra, an extension of the algebra of database relations, is a well-studied field in the area of personalized databases. It allows modelling user wishes by preference terms; they represent strict partial orders telling which database objects the user prefers over other ones. There are a number of constructors that allow combining simple preferences into quite complex, nested ones. A preference term is then used as a database query, and the results are the maximal objects according to the order it denotes. Depending on the size of the database, this can be computationally expensive. For optimisation, preference queries and the corresponding terms are transformed using a number of algebraic laws. So far, the correctness proofs for such laws have been performed by hand and in a point-wise fashion. We enrich the standard theory of relational databases to an algebraic framework that allows completely point-free reasoning about complex preferences. This black-box view is amenable to a treatment in first-order logic and hence to fully automated proofs using off-the-shelf verification tools. We exemplify the use of the calculus with some non-trivial laws, notably concerning so-called preference prefilters which perform a preselection to speed up the computation of the maximal objects proper.

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In this paper we present a framework for a novel kind of context-aware preference query composition whereby queries for the Preference SQL system are created. We choose a commercial e-business platform for outdoor activities as a use case and develop a context model for this domain within our framework. The suggested model considers explicit user input, domain-specific knowledge, contextual knowledge and location-based sensor data in a comprehensive approach. Aside from the theoretical background of preferences, the optimization of preference queries and our novel generator based model we give special attention to the aspects of the implementation and the practical experiences. We provide a sketch of the implementation and summarize our user studies which have been done in a joint project with an industrial partner. Category: Smart and intelligent computing