Symmetries of the free Schrödinger equation in the non-commutative plane

We study all the symmetries of the free Schr odinger equation in the non-commu- tative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schröodinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.

Understanding Flood Regime Changes in Europe: a state-of-the-art assessment

There is growing concern that flooding is becoming more frequent and severe in Europe. A better understanding of flood regime changes and their drivers is therefore needed. The paper reviews the current knowledge on flood regime changes in European rivers that has traditionally been obtained through two alternative research approaches. The first approach is the data-based detection of changes in observed flood events. Current methods are reviewed together with their challenges and opportunities. For example, observation biases, the merging of different data sources and accounting for nonlinear drivers and responses. The second approach consists of modelled scenarios of future floods. Challenges and opportunities associated with flood change scenarios are discussed such as fully accounting for uncertainties in the modelling cascade and feedbacks. To make progress in flood change research, we suggest that a synthesis of these two approaches is needed. This can be achieved by focusing on long duration records and flood-rich and flood-poor periods rather than on short duration flood trends only, by formally attributing causes of observed flood changes, by validating scenarios against observed flood regime dynamics, and by developing low-dimensional models of flood changes and feedbacks. The paper finishes with a call for a joint European flood change research network.