Using the Focal Reducer Spectrograph (FORS) at the Very Large Telescope (VLT) during the FORS commissioning time in December 1998 we took long slit spectra of the gravitational arc visible on images of the galaxy cluster 1E 0657 (z = 0.296). This cluster is one of the hottest (massive) cluster known so far and hence perfectly acts as a gravitational telescope magnifying the flux of background sources up to a factor of 17. Here we present the spectra of the gravitational arc (z = 3.23) and 4 additional high redshift objects (z=2.35 to 3.09), that also fall on the slit by chance coincidence. We briefly discuss the stellar contents of these galaxies and show first models of the observed spectra. Furthermore we point out the effectivity of using FORS in combination with available gravitational telescopes.

D. Mehlert S. Seitz R. P. Saglia T. L. Hoffmann I. Appenzeller R. Bender U Hopp R. -P. Kudritzki A. W. A. Pauldrach

Recent advances in the modelling of stellar winds driven by radiation pressure make it possible to fit many wind-sensitive features in the UV spectra of hot stars, opening the way for a hydrodynamically consistent determination of stellar radii, masses, and luminosities from the UV spectrum alone. It is thus no longer necessary to assume a theoretical mass-luminosity relation. As the method has been shown to work for massive O-stars, we are now able to test predictions from the post-AGB evolutionary calculations quantitatively for the first time. Here we present the first rather surprising consequences of using the new generation of model atmospheres for the analysis of a sample of central stars of planetary nebulae.

A. W. A. Pauldrach T. L. Hoffmann R. H. Méndez

Type Ia supernovae are an important tool for studying the expansion history of the universe. Advancing our yet incomplete understanding of the explosion scenario requires detailed and realistic numerical models in order to interpret and analyze the growing amount of observational data. Here we present first results of our new NLTE model calculations for the expanding atmospheres of type Ia supernovae that employ a detailed and consistent treatment of all important NLTE effects as well as line blocking and blanketing. The comparison of the synthetic spectra resulting from these models with observed data shows that the employed methods represent an important step towards a more realistic description of the atmospheres of supernovae Ia.

D. Sauer A. W. A. Pauldrach

Aims: The aim of this work is to verify whether turbulent magnetic reconnection can provide the additional energy input required to explain the up to now only poorly understood ionization mechanism of the diffuse ionized gas (DIG) in galaxies and its observed emission line spectra. Methods: We use a detailed non-LTE radiative transfer code that does not make use of the usual restrictive gaseous nebula approximations to compute synthetic spectra for gas at low densities. Excitation of the gas is via an additional heating term in the energy balance as well as by photoionization. Numerical values for this heating term are derived from three-dimensional resistive magnetohydrodynamic two-fluid plasma--neutral-gas simulations to compute energy dissipation rates for the DIG under typical conditions. Results: Our simulations show that magnetic reconnection can liberate enough energy to by itself fully or partially ionize the gas. However, synthetic spectra from purely thermally excited gas are incompatible with the observed spectra; a photoionization source must additionally be present to establish the correct (observed) ionization balance in the gas.

T. L. Hoffmann S. Lieb A. W. A. Pauldrach H. Lesch P. J. N. Hultzsch G. T. Birk

Let $F^\lambda_{\sigma} [G]$ be a crossed product of a group $G$ and the field $F$. We study the Lie properties of $F^\lambda_{\sigma} [G]$ in order to obtain a characterization of those crossed products which are upper (lower) Lie nilpotent and Lie $(n,m)$-Engel.

Adalbert Bovdi Alexander Grishkov

We present significant improvements of our approach in constructing detailed atmospheric models and synthetic spectra for hot luminous stars: 1. A sophisticated and consistent description of line blocking and blanketing. Our solution concept renders the line blocking influence on the ionizing fluxes (mainly the EUV and UV are affected) in identical quality as the synthetic high resolution spectra representing the observable region. Line blanketing is properly accounted for in the energy balance. 2. A considerably improved and enhanced atomic data archive providing the basis for a detailed multilevel NLTE treatment of the metal ions (from C to Zn) and an adequate representation of line blocking and the radiative line acceleration. 3. A revised inclusion of EUV and X-ray radiation produced by cooling zones originating from shock heated matter. This new tool not only provides an easy to use method for O-star diagnostics, whereby physical constraints on the properties of stellar winds, stellar parameters, and abundances can be obtained via a comparison of observed and synthetic spectra, but also allows the astrophysically important information about the ionizing fluxes of hot stars to be determined automatically. Results illustrating this are discussed by means of a basic model grid calculated for O-stars with solar metallicity. To further demonstrate the astrophysical potential of our new method we provide a first detailed spectral diagnostic determination of the stellar parameters, the wind parameters, and the abundances by an exemplary application to the O9.5Ia supergiant alpha Cam.

A. W. A. Pauldrach T. L. Hoffmann M. Lennon

The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra. However, there does not exist a closed form representation of the pseudoinverse, which can be straightforwardly computed. Therefore, an algorithmic computation is necessary. An algorithmic computation can only be evaluated by also considering the underlying hardware, typically digital hardware, which is responsible for performing the actual computations step by step. In this paper, we analyze if and to what degree the pseudoinverse actually can be computed on digital hardware platforms modeled as Turing machines. For this, we utilize the notion of an effective algorithm which describes a provably correct computation: upon an input of any error parameter, the algorithm provides an approximation within the given error bound with respect to the unknown solution. We prove that an effective algorithm for computing the pseudoinverse of any matrix can not exist on a Turing machine, although provably correct algorithms do exist for specific classes of matrices. Even more, our results introduce a lower bound on the accuracy that can be obtained algorithmically when computing the pseudoinverse on Turing machines.

Holger Boche Adalbert Fono Gitta Kutyniok

Deep learning still has drawbacks in terms of trustworthiness, which describes a comprehensible, fair, safe, and reliable method. To mitigate the potential risk of AI, clear obligations associated to trustworthiness have been proposed via regulatory guidelines, e.g., in the European AI Act. Therefore, a central question is to what extent trustworthy deep learning can be realized. Establishing the described properties constituting trustworthiness requires that the factors influencing an algorithmic computation can be retraced, i.e., the algorithmic implementation is transparent. Motivated by the observation that the current evolution of deep learning models necessitates a change in computing technology, we derive a mathematical framework which enables us to analyze whether a transparent implementation in a computing model is feasible. We exemplarily apply our trustworthiness framework to analyze deep learning approaches for inverse problems in digital and analog computing models represented by Turing and Blum-Shub-Smale Machines, respectively. Based on previous results, we find that Blum-Shub-Smale Machines have the potential to establish trustworthy solvers for inverse problems under fairly general conditions, whereas Turing machines cannot guarantee trustworthiness to the same degree.

Holger Boche Adalbert Fono Gitta Kutyniok

The unwavering success of deep learning in the past decade led to the increasing prevalence of deep learning methods in various application fields. However, the downsides of deep learning, most prominently its lack of trustworthiness, may not be compatible with safety-critical or high-responsibility applications requiring stricter performance guarantees. Recently, several instances of deep learning applications have been shown to be subject to theoretical limitations of computability, undermining the feasibility of performance guarantees when employed on real-world computers. We extend the findings by studying computability in the deep learning framework from two perspectives: From an application viewpoint in the context of classification problems and a general limitation viewpoint in the context of training neural networks. In particular, we show restrictions on the algorithmic solvability of classification problems that also render the algorithmic detection of failure in computations in a general setting infeasible. Subsequently, we prove algorithmic limitations in training deep neural networks even in cases where the underlying problem is well-behaved. Finally, we end with a positive observation, showing that in quantized versions of classification and deep network training, computability restrictions do not arise or can be overcome to a certain degree.

Holger Boche Vit Fojtik Adalbert Fono Gitta Kutyniok

Charged boron vacancies (V$_\text{B}^-$) in hexagonal boron nitride (hBN) have emerged as a promising platform for quantum nanoscale sensing and imaging. While these primarily involve electron spins, nuclear spins provide an additional resource for quantum operations. This work presents a comprehensive experimental and theoretical study of the properties and coherent control of the nearest-neighbor $^{15}$N nuclear spins of V$_\text{B}^-$-ensembles in isotope-enriched h$^{10}$B$^{15}$N. Multi-nuclear spin states are selectively addressed, enabled by state-specific nuclear spin transitions arising from spin-state mixing. We perform Rabi driving between selected state pairs, define elementary quantum gates, and measure longer than 10~$\mu$s nuclear Rabi coherence times. We observe a two orders of magnitude nuclear g-factor enhancement that underpins fast nuclear spin gates. Accompanying numerical simulations provide a deep insight into the underlying mechanisms. These results establish the foundations for leveraging nuclear spins in V$_\text{B}^-$ center-based quantum applications, particularly for extending coherence times and enhancing the sensitivity of 2D quantum sensing foils.

Adalbert Tibiássy Charlie J. Patrickson Thomas Poirier James H. Edgar Bruno Lopez-Rodriguez Viktor Ivády Isaac J. Luxmoore