A closed character formula for symmetric powers of irreducible representations

We prove a closed character formula for the symmetric powers $S^N V(\lambda)$ of a fixed irreducible representation $V(\lambda)$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(\lambda)$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(\lambda)$ and explain the connections between our character formula, vector partition functions and iterated partial fraction decomposition.Comment: 14 pages, 1 figure, published versio

Diffusion map particle systems for generative modeling

We propose a novel diffusion map particle system (DMPS) for generative modeling, based on diffusion maps and Laplacian-adjusted Wasserstein gradient descent (LAWGD). Diffusion maps are used to approximate the generator of the corresponding Langevin diffusion process from samples, and hence to learn the underlying data-generating manifold. On the other hand, LAWGD enables efficient sampling from the target distribution given a suitable choice of kernel, which we construct here via a spectral approximation of the generator, computed with diffusion maps. Our method requires no offline training and minimal tuning, and can outperform other approaches on data sets of moderate dimension

Deformations and stability in complex hyperbolic geometry

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-trivial (quasi-Fuchsian) deformations and point out that such flexible manifolds have a common feature being Stein spaces. While deformations of complex surfaces from our first class are induced by quasiconformal homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their complex analytic submanifolds) from another class are quasiconformally unstable, but nevertheless their deformations are induced by homeomorphisms

Algorithms for square-3PC(·, ·)-free Berge graphs

We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole, and at least two of the paths are of length 2. This class generalizes clawfree Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of complexity O(n7) to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class

Xcerpt: A Rule-Based Query and Transformation Language for the Web

This thesis investigates querying the Web and the Semantic Web. It proposes a new rulebased query language called Xcerpt. Xcerpt differs from other query languages in that it uses patterns instead of paths for the selection of data, and in that it supports both rule chaining and recursion. Rule chaining serves for structuring large queries, as well as for designing complex query programs (e.g. involving queries to the Semantic Web), and for modelling inference rules. Query patterns may contain special constructs like partial subqueries, optional subqueries, or negated subqueries that account for the particularly flexible structure of data on the Web. Furthermore, this thesis introduces the syntax of the language Xcerpt, which is illustrated on a large collection of use cases both from the conventional Web and the Semantic Web. In addition, a declarative semantics in form of a Tarski-style model theory is described, and an algorithm is proposed that performs a backward chaining evaluation of Xcerpt programs. This algorithm has also been implemented (partly) in a prototypical runtime system. A salient aspect of this algorithm is the specification of a non-standard unification algorithm called simulation unification that supports the new query constructs described above. This unification is symmetric in the sense that variables in both terms can be bound. On the other hand it is in contrast to standard unification assymmetric in the sense that the unification determines that the one term is a subterm of the other term.Diese Arbeit untersucht das Anfragen des Webs und des Semantischen Webs. Sie stellt eine neue regel-basierte Anfragesprache namens Xcerpt vor. Xcerpt unterscheidet sich von anderen Anfragesprachen insofern, als dass es zur Selektion von Daten sog. Pattern (,,Muster'') verwendet und sowohl Regelschliessen als auch Rekursion unterstützt, was sowohl zur Strukturierung größerer Anfragen als auch zur Erstellung komplexer Anfrageprogramme, und zur Modellierung von Inferenzregeln dient. Anfrage-Pattern können spezielle Konstrukte, wie partielle Teilanfragen, optionale Teilanfragen, oder negierte Teilanfragen, enthalten, die der besonders flexiblen Struktur von Daten im Web genügen. In dieser Arbeit wird weiterhin die Syntax von Xcerpt eingeführt, und mit Hilfe mehrerer Anwendungsszenarien sowohl aus dem konventionellen als auch aus dem semantischen Web erläutert. Ausserdem wird eine deklarative Semantik im Stil von Tarski's Modelltheorie beschrieben und ein Algorithmus vorgeschlagen, der eine rückwärtsschliessende Auswertung von Xcerpt durchführt und in einem prototypischen Laufzeitsystem implementiert wurde. Wesentlicher Bestandteil des Rückwärtsschliessens ist die Spezifikation eines nicht-standard Unifikations-Algorithmus, der die oben genannten speziellen Xcerpt-Konstrukte berücksichtigt. Diese Unifikation ist symmetrisch in dem Sinne, dass sie Variablen in beiden angeglichenen (,,unifizierten'') Termen binden kann. Andererseits ist sie im Gegensatz zur Standardunifikation assymmetrisch in dem Sinne, dass der dadurch geleistete Angleich den einen Term als ,,Teilterm'' des anderen erkennt