Talks

Bachelorvortrag

A linear operator $T$ on a Banach space is called Tadmor-Ritt if its spectrum is contained in the closed unit disc and the resolvent satisfies $C(T)=\sup_{|z|>1} \|(z-1)R(z,T)\|<\infty$. Such operators can be seen as discrete analog for sectorial operators.
We prove corresponding $H^{\infty}$-functional calculus estimates, which generalize and improve results by Vitse. Moreover, they are in conformity with the best so-far known power-bound for Tadmor-Ritt operators in terms of the constant $C(T)$.
We furthermore show the effect of having discrete square function estimates on the derived estimates.

We say a graph $G$ is universal for a set of graphs $\mathcal{H}$ if for each $H\in\mathcal{H}$ we have $H\subset G$. There are several results stating that the random graph $G(n,p)$ is universal for various classes of graphs $\mathcal{H}$, for appropriate functions $p=p(n)$. In order for $p$ not to be very close to one, we need the graphs in $\mathcal{H}$ to be quite sparse. There are then (at least) three natural graph classes one could consider: trees, graphs with bounded degree, and graphs with bounded degeneracy. I will outline the current state of knowledge (mainly due to other people) and sketch one or two proofs

Bachelor-Vortrag

After the introduction of random matrices to nuclear physics by Eugene Wigner in 1955, random quantum systems have grown in popularity. Wigner's idea was to consider families of Hamiltonians that underlie a certain probability distribution to describe overly complicated systems. Of particular interest are, of course, the spectra of these Hamiltonians. In this talk we consider random, in general non-self-adjoint, tridiagonal operators on the Hilbert space of square-summable sequences. To model randomness, we use an approach by Davies that eliminates all probabilistic arguments.

Despite the rising interest, not much is known about the spectra of non-self-adjoint random operators. The Feinberg-Zee random hopping matrix reveals this in a beautiful manner. The boundary of its spectrum appears to be fractal, but a proof has not been found yet. While we can not give a proof either, we present a reason why this is very plausible. Certain tridiagonal operators share remarkable symmetries that allow us to enlarge known subsets of the spectrum by sizeable amounts. In some cases like the Feinberg-Zee random hopping matrix, this implies that the spectrum contains an infinite sequence of Julia sets.

One of the most important and frequently used preconditioning techniques for solving symmetric positive definite systems is based on computing the approximate inverse factorizations. It is also a well-known fact that such factors can be computed column-wise by the orthogonalization process applied to the unit basis vectors provided that we use a non-standard inner product induced by the positive definite system matrix A. In this contribution we consider the classical Gram-Schmidt algorithm (CGS), the modified Gram-Schmidt algorithm (MGS) and also yet another variant of sequential orthogonalization, which is motivated originally by the AINV preconditioner and which uses oblique projections.

The orthogonality between computed vectors is crucial for the quality of the preconditioner constructed in the approximate inverse factorization. While for the case of the standard inner product there exists a complete rounding error analysis for all main orthogonalization schemes, the numerical properties of the schemes with a non-standard inner product are much less understood. We will formulate results on the loss of orthogonality and on the factorization error for all previously mentioned orthogonalization schemes.

This contribution is joint work with Jiří Kopal (Technical University Liberec), Miroslav Tůma and Alicja Smoktunowicz (Warsaw University of Technology).

Die von Peter Sonneveld erdachten Methoden, allen voran die neueste, IDR(s), können zur approximativen Eigenwertberechnung linearer Operatoren herangezogen werden. Im Gegensatz zu klassischen Krylovraumverfahren, welche Tridiagonal- oder Hessenbergmatrizen berechnen, berechnen Sonneveld-Methoden Büschel aus einer Band-Hessenbergmatrix und einer oberen Band-Dreiecksmatrix, von denen einige Eigenwerte bekannt sind. Basierend auf einer trivialen Beobachtung präsentieren wir Wege, die anderen Eigenwerte stabil zu berechnen.

Abstract:
We study the Decomposition Conjecture posed by Barát and Thomassen (2006), which states that for every tree T there exists a natural number k_T such that, if G is a k_T-edge-connected graph and |E(T)| divides |E(G)|, then G admits a decomposition into copies of T. This conjecture was verified for stars, some bistars, paths whose length is a power of 2, and paths of length 3. We verify the Decomposition Conjecture for paths of length 5. In this talk I will discuss the ideas behind the proof of this result.

Die von Peter Sonneveld erdachten Methoden, allen voran die neueste, IDR(s), können zur approximativen Eigenwertberechnung linearer Operatoren herangezogen werden. Im Gegensatz zu klassischen Krylovraumverfahren, welche Tridiagonal- oder Hessenbergmatrizen berechnen, berechnen Sonneveld-Methoden Büschel aus einer Band-Hessenbergmatrix und einer oberen Band-Dreiecksmatrix, von denen einige Eigenwerte bekannt sind. Basierend auf einer trivialen Beobachtung präsentieren wir Wege, die anderen Eigenwerte stabil zu berechnen.