A ranked query is a query which returns the top-ranking elements of a set, sorted by rank, where the rank corresponds to some sort of preference function defined on the items of the set. This thesis investigates the problem of adding rank query capabilities to several index data structures on top of their existing functionality. Among the data structures investigated are suffix trees, range tre...

Most of the earlier work on clustering has mainly been focused on numerical data whose inherent geometric properties can be exploited to naturally define distance functions between data points. Working only on numeric values prohibits it from being used to cluster real world data containing categorical values. Recently, the problem of clustering categorical data has started drawing interest. Th...

In the polynomial approach to systems control, the static output feedback problem can be formulated as follows: given two polynomial matrices D(s) and N(s), nd a constant matrix K such that polynomial matrix D(s) + KN(s) is stable. In this paper, we show that solving this problem amounts to solving a linear matrix inequality with a non-convex rank constraint. Several numerical experiments illus...

This paper is concerned with the eigenvalue decay of the solution to operator Lya-punov equations with right-hand sides of finite rank. We show that the kth eigenvalue decays exponentially in √ k, provided that the involved operator A generates an exponentially stable continuous semigroup, and A is either self-adjoint or diagonalizable. Numerical experiments with discretizations of 1D and 2D PD...

Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Given a semistable vector bundle E over X , we show that its direct image F∗E under the Frobenius map F of X is again semistable. We deduce a numerical characterization of the stable rank-p vector bundles F∗L, where L is a line bundle over X .

The compressibility of Schur complement matrices is the essential ingredient for H-matrix techniques, and is well understood for Laplace type problems. The Helmholtz case is more difficult: there are several theoretical results which indicate when good compression is possible with additional techniques, and in practice sometimes basic H-matrix techniques work well. We investigate the compressib...

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximati...

A transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponential-time algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed k ≥ 3, our algorithms for hypergraphs of rank k, where the rank is the maximum size of a hyperedge, outperform the previous best. This also implies improved upper bounds on the maximu...

Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rankdeficient, but have off-diagonal blocks that are. Such matrices arise ...