Vorträge in der Woche 14.12.2015 bis 20.12.2015

Montag, 14.12.2015: Interconnection of singular values arising from tensor matricizations

André Uschmajew (Universität Bonn)

A higher-order tensor, when viewed as a hypermatrix or $d$-dimensional array, allows several possible matricizations, that is, reshapes into matrices. The simultaneous decay of singular values of some of these matricizations plays a crucial role for the approximability of the tensor in certain low-rank formats like the (hierarchical) Tucker format or the tensor train format. It is therefore an important and interesting question which simultaneous properties the singular values of different tensor matricizations actually can have. More concretely, does there exist a tensor with prescribed singular values for certain matricizations, as it is suggested from counting the degrees of freedom? Such questions turn out to be surprisingly hard in general. In this talk we will present some preliminary results regarding the singular values related to the Tucker matricizations. It is shown that the singular values for these matricizations cannot be totally independent from each other in general. However, numerical experiments suggest that this is at least locally the case, and an alternating projection method is proposed for constructing tensors with prescribed singular values (assuming their feasibility). Regarding the related problem of describing tensors that have the same singular values in all Tucker matricizations, it is known that orthogonal equivalence under multilinear matrix multiplication is a sufficient condition. It is fairly easy to see that it is not necessary. However, an explicit example of this phenomenon will be given, which seems a less trivial task. This is joint work with Wolfgang Hackbusch (MPI Leipzig).

Dienstag, 15.12.2015: Amenability II

Julien Sessler

Donnerstag, 17.12.2015: Multidomain spectral method for Schrödinger equations

Prof. Christian Klein (Institut de Mathématiques de Bourgogne)

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymp- totically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.

Donnerstag, 17.12.2015: Mathematikunterricht gestern, heute und morgen

StD Thomas Jurke (Kultusministerium Baden-Württemberg)

Das Fach Mathematik ist – wenn man verschiedenen Erhebungen glauben möchte – das Unter-richtsfach, das am stärksten polarisiert. Es steht bei unseren Schülerinnen und Schülern sowohl an der Spitze der beliebtesten als auch an der Spitze der unbeliebtesten Fächer. Wahrscheinlich war das schon immer so. Warum gibt es an unseren Schulen überhaupt Mathematikunterricht? Die Beantwortung dieser Frage ist nicht so einfach, wie es zunächst scheint. Die Art und Weise ihrer Beantwortung wirft sofort eine Reihe weiterer Fragen auf. Ist die bundesweite Vereinheitlichung der Anforderungen sinnvoll? Was bedeutet diese für die Prüfungsformate? Was bedeutet sie für die Frage des Einsatzes von Hilfsmitteln? Der Vortrag versucht eine Annäherung an die Beantwortung dieser und ähnlicher Fragen, die nur im Zusammenhang bewertet werden können.

Freitag, 18.12.2015: Semigroup methods for Gaver's system attended by a cold standby unit and a repairman with multiple vacations

Abdukerim Haji (Urumqi)