Metrics for Learning in Topological Persistence

Acknowledgments We gratefully acknowledge Roel Neggers for providing the DALES simulation data. JLS acknowledges support by the DFG-funded transregional research collaborative TR32 on Patterns in Soil–Vegetation–Atmosphere Systems.Peer reviewedPublisher PD

The Structure of the Convective Boundary Layer as Deduced from Topological Invariants

We study the convective boundary layer (CBL) through low-order topological properties of updrafts and downdrafts, that is, based solely on the sign of the vertical velocity. The geometric representation of the CBL as a pair of two-dimensional cubical complexes, one each for updrafts and downdrafts, is exemplarily obtained from two simulations of the CBL, a realistic daily cycle and an idealized quasi-steady CBL growing into linear stratification. Each cubical complex is defined as a set of grid cells that have the same sign of vertical velocity, either positive or negative. Low-order topological invariants, namely the Betti numbers of the cubical complexes, are found to capture key aspects of the boundary-layer organization and evolution over the diurnal cycle. An unsupervised-learning algorithm is trained using the topological invariants in order to classify the spatio–temporal evolution of convection over a whole day. The successful classification of the CBL by using this approach illustrates the potential of such simplified representation of turbulent flow for data reduction and boundary-layer parametrization approaches

Kilohertz droplet-on-demand serial femtosecond crystallography at the European XFEL station FXE

X-ray Free Electron Lasers (XFELs) allow the collection of high-quality serial femtosecond crystallography data. The next generation of megahertz superconducting FELs promises to drastically reduce data collection times, enabling the capture of more structures with higher signal-to-noise ratios and facilitating more complex experiments. Currently, gas dynamic virtual nozzles (GDVNs) stand as the sole delivery method capable of best utilizing the repetition rate of megahertz sources for crystallography. However, their substantial sample consumption renders their use impractical for many protein targets in serial crystallography experiments. Here, we present a novel application of a droplet-on-demand injection method, which allowed operation at 47 kHz at the European XFEL (EuXFEL) by tailoring a multi-droplet injection scheme for each macro-pulse. We demonstrate a collection rate of 150 000 indexed patterns per hour. We show that the performance and effective data collection rate are comparable to GDVN, with a sample consumption reduction of two orders of magnitude. We present lysozyme crystallographic data using the Large Pixel Detector at the femtosecond x-ray experiment endstation. Significant improvement of the crystallographic statistics was made by correcting for a systematic drift of the photon energy in the EuXFEL macro-pulse train, which was characterized from indexing the individual frames in the pulse train. This is the highest resolution protein structure collected and reported at the EuXFEL at 1.38 Å resolution