Talks

The best approximation of a matrix by a low-rank matrix can be obtained by the singular value decomposition. For large-sized matrices this approach is too costly. Instead one may use a block decomposition. Approximating the smaller
block matrices by low-rank matrices and agglomerating them into a new, coarser
block decomposition, one obtains a recursive method. The required computation work is O(rnm) where r is the desired rank and n x m is the size of the matrix. New estimates are presented for the errors A-B and M-A,
where A is the result of the recursive truncation applied to M, while B is the best approximation.

Homogenization theory comprises the study of heterogeneous
materials. In mathematical terms this goes along with the discussion of
differential equations with oscillatory coefficients and the behavior of the
respective solutions, when the oscillations become infinitely fast. The aim in homogenization theory is to show convergence of the solutions for infinitely fast oscillations and to find an effective equation satisfied by the limit. In a Hilbert space setting, we discuss homogenization of ordinary differential equations and give an operator-theoretic reason, when it is likely that the limit equation is of integro-differential type -- in contrast to the equation one started out with. We also discuss possible generalizations to non-autonomous and/or partial differential equations.

Bachelor-Vortrag

Bachelor-Vortrag

In this talk, we discuss our recent progress on the famous directed cycle double cover conjecture of Jaeger. We define the class of trigraphs and prove that a graph connections conjecture formulated on trigraphs implies general Jaeger's conjecture. In addition, we give supporting evidence for our conjecture. This is joint work with Martin Loebl.

Given an $n$-element set which contains a known number $d$ of unknown special elements, we are allowed to use an oracle which accepts queries of size $k$ and responds positively if at least one of the elements of the queried set is in our set of unknowns. The case of a single unknown element has been studied and solved in the past by Rényi (1961), Katona (1966) and more recently by Hosszu, Tapolcai and Wiener (2013). We generalise their results in both the adaptive (on-line) and non-adaptive (parallelised) case for general d. Our approach provides new links between separability and (hyper-)graph girth, as well as new bounds for the problem.
This is joint work with F. Benevides, D. Gerbner and C. Palmer.