A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a k-strong defining set, in which if less than k entries are deleted, the property of being a defining set is retained. Equivalently, a k-strong defining set intersects every Latin trade in the Latin square at least k times. In the addition table for integers modulo n, when n is even we determine the minimum size of a k-strong defining set for any k. For odd n we give a construction for a minimally 2-strong defining set. We furthermore give computational results for Latin squares of small orders. Crown Copyright (c) 2026 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

In this paper, we examine the issues caused by a mismatch in loss functions in typical predict and optimise problems with specific reference to the 3^rd Technical Challenge of the IEEE Computational Intelligence Society. In this competition, entrants were asked to forecast building energy use and solar generation at six buildings and six solar installations, and then use their forecast to optimize energy cost while scheduling classes and batteries over a month. We examine the possible effect of underforecasting and overforecasting and asymmetric errors on the optimisation cost. We explore the different nature of loss functions for the prediction and optimisation phase and propose to adjust the final forecasts for a better optimisation cost. We report that while there is a positive correlation between these two, more appropriate loss functions can be used to optimise the costs associated with final decisions. Our findings are of significant value in designing forecasting and decision-making systems.

Sample consumption for serial femtosecond crystallography with X-ray free electron lasers remains a major limitation preventing broader use in macromolecular crystallography. This drawback is exacerbated in time-resolved (TR) experiments, where the amount of sample required per reaction time point is multiplied by the number of time points investigated. To reduce this limitation, we demonstrate a segmented droplet generation strategy coupled to a mix-and-inject approach for TR studies at the European XFEL. The injector produces synchronized droplet trains that enable stable and reproducible injection of protein crystal slurries at significantly reduced flow rates. Using the human flavoenzyme NAD(P)H:quinone oxidoreductase 1 (NQO1) as a test system, we collected diffraction data after mixing with NADH at 0.3 s and 1.2 s delays. The segmented injection approach achieved up to 97% reduction in sample consumption compared with continuous-flow injection while maintaining data quality suitable for TR crystallography. Reproducible electron density features consistent with low-occupancy NADH binding illustrate both the feasibility and the current limits of studying dynamic redox enzymes using this approach. This work establishes segmented droplet generation as a sample-efficient and XFEL-compatible method for future time-resolved serial crystallography experiments.

Structural biology investigations using synchrotron radiation have a long history at the photon science facilities at DESY. Presently, EMBL and DESY operate state-of-the-art macromolecular crystallography and biological SAXS stations at the synchrotron radiation source PETRA III for the international user community. New experimental opportunities for experiments with femtosecond temporal resolution and for extremely small macromolecular crystals have become available with the advent of X-ray free-electron lasers (XFELs) such as the European XFEL. Within large international collaborations, groups at DESY and the European XFEL have contributed significantly to the development of experimental and data analysis methods to enable serial crystallography experiments at both XFELs and high-brilliance synchrotron radiation sources. The available portfolio of analytical infrastructure in photon science at DESY has attracted several campus partners to contribute to the development of instruments and methods and provide their own complementary experimental techniques, thereby establishing a fruitful scientific environment to make significant contributions to present and future societal challenges in the field of life sciences.

Streamlining and automating the operation of complex large-scale X-ray light sources like the European XFEL can minimize inefficiencies, thereby boosting their scientific outcome. To that end, advanced mathematical methods such as machine learning can be employed. In this paper, we define the core concepts related to the exploitation of such methods at European XFEL. We aim to empower scientists and operators through strict information quality control and easily explainable metrics. As methods based on machine learning may be error-prone, interpretable procedures and validation metrics are essential to gauge the advantage of using them and identify possible difficulties. To illustrate these principles, we present two selected applications. The first aims at preventing damage to X-ray imagers caused by the interaction of X-rays with a crystallizing liquid jet used as a sample delivery system. The second optimizes the in-plane calibration of the position of modules composing an imager.

Predict+Optimize frameworks integrate forecasting and optimization to address real-world challenges such as renewable energy scheduling, where variability and uncertainty are critical factors. This paper benchmarks solutions from the IEEE-CIS Technical Challenge on Predict+Optimize for Renewable Energy Scheduling, focusing on forecasting renewable production and demand and optimizing energy cost. The competition attracted 49 participants in total. The top-ranked method employed stochastic optimization using LightGBM ensembles, and achieved at least a 2% reduction in energy costs compared to deterministic approaches, demonstrating that the most accurate point forecast does not necessarily guarantee the best performance in downstream optimization. The published data and problem setting establish a benchmark for further research into integrated forecasting-optimization methods for energy systems, highlighting the importance of considering forecast uncertainty in optimization models to achieve cost-effective and reliable energy management. The novelty of this work lies in its comprehensive evaluation of Predict+Optimize methodologies applied to a real-world renewable energy scheduling problem, providing insights into the scalability, generalizability, and effectiveness of the proposed solutions. Potential applications extend beyond energy systems to any domain requiring integrated forecasting and optimization, such as supply chain management, transportation planning, and financial portfolio optimization.

Phytochromes are biliprotein photoreceptors widespread amongst microorganisms and ubiquitous in plants where they control developmental processes as diverse as germination, stem elongation and floral induction through the photoconversion of inactive Pr to the Pfr signalling state. Here we report crystal structures of the chromophore-binding module of soybean phytochrome A, including ~2.2 Å XFEL structures of Pr and Pfr at ambient temperature and high resolution cryogenic structures of Pr. In the Pfr structure, the chromophore is exposed to the medium, the D-ring remaining α-facial following the likely clockwise photoflip. The chromophore shifts within its pocket, while its propionate side chains, their partners as well as three neighbouring tyrosines shift radically. Helices near the chromophore show substantial shifts that might represent components of the light signal. These changes reflect those in bacteriophytochromes despite their quite different signalling mechanisms, implying that fundamental aspects of phytochrome photoactivation have been repurposed for photoregulation in the eukaryotic plant.

A correction in the paper by Oberthür et al. [(2025). J. Synchrotron Rad. 32, 474–485] is made..

We report on recent developments that enable megahertz hard X-ray phase contrast imaging (MHz XPCI) experiments at the Single Particles, Clusters, and Biomolecules and Serial Femtosecond Crystallography (SPB/SFX) instrument of the European XFEL facility (EuXFEL). We describe the technical implementation of the key components, including an MHz fast camera and a modular indirect X-ray microscope system based on fast scintillators coupled through a high-resolution optical microscope, which enable full-field X-ray microscopy with phase contrast of fast and irreversible phenomena. The image quality for MHz XPCI data showed significant improvement compared with a pilot demonstration of the technique using parallel beam illumination, which also allows access to up to 24 keV photon energies at the SPB/SFX instrument of the EuXFEL. With these developments, MHz XPCI was implemented as a new method offered for a broad user community (academic and industrial) and is accessible via standard user proposals. Furthermore, intra-train pulse diagnostics with a high few-micrometre spatial resolution and recording up to 128 images of consecutive pulses in a train at up to 1.1 MHz repetition rate is available upstream of the instrument. Together with the diagnostic camera upstream of the instrument and the MHz XPCI setup at the SPB/SFX instrument, simultaneous two-plane measurements for future beam studies and feedback for machine parameter tuning are now possible.

Penicillin-binding protein 2a (PBP2a) is a transpeptidase responsible for the β-lactam resistance in methicillin-resistant Staphylococcus aureus (MRSA), posing significant challenges to antibiotic therapy. PBP2a's unique structural features, including its highly flexible active site and allosteric regulation, enable it to maintain catalytic activity even in the presence of β-lactam antibiotics. Despite extensive characterization using cryogenic crystallography, key questions remain about its dynamic properties and conformational changes under near-physiological conditions. Room-temperature crystallography methods, particularly serial femtosecond X-ray crystallography (SFX) at XFELs, provide a powerful approach to capture these dynamics. Unlike cryogenic conditions that can constrain protein flexibility, SFX enables the study of conformational variability and interaction networks closer to physiological states. Here, we present the first room-temperature structures of PBP2a obtained using SFX to uncover insights into the enzyme's flexibility, allosteric communication, and catalytic mechanisms. These findings are built upon optimized large-scale production and crystallization protocols for PBP2a, ensuring high-quality microcrystals suitable for SFX data collection. The room-temperature structures reveal novel interaction patterns, including unique salt bridge networks and dynamic structural elements absent in cryogenic studies. Furthermore, comparative analyses highlight how environmental conditions influence the conformational states of PBP2a, providing new perspectives on its resistance mechanisms. By integrating structural data from EuXFEL and LCLS, this study not only enhances our understanding of PBP2a's functional dynamics but also underscores the value of room-temperature crystallography in studying antibiotic resistance. These insights could guide the design of next-generation β-lactam antibiotics capable of overcoming PBP2a-mediated resistance. ### Competing Interest Statement The authors have declared no competing interest.

The outstanding opto-electronic properties of lead halide perovskites have been related to the formation of polarons. Nevertheless, the observation of the atomistic deformation brought about by one electron-hole pair in these materials has remained elusive. Here, we measure the diffraction patterns of single CsPbBr$_3$ quantum dots (QDs) with and without resonant excitation in the single exciton limit using serial femtosecond crystallography (SFX). By reconstructing the 3D differential diffraction pattern, we observe small shifts of the Bragg peaks indicative of a crystal-wide deformation field. Building on DFT calculations, we show that these shifts are consistent with the lattice distortion induced by a delocalized electron and a localized hole, forming a mixed large/small exciton polaron. This result creates a clear picture of the polaronic deformation in CsPbBr$_3$ QDs, highlights the exceptional sensitivity of SFX to lattice distortions in few-nanometer crystallites, and establishes an experimental platform for future studies of electron-lattice interactions.

The Outstanding optoelectronic properties of lead halide perovskites have been related to the formation of polarons. Nevertheless, the observation of the atomistic deformation brought about by one electron-hole pair in these materials has remained elusive. Here, we measure the transient structure of single CsPbBr3 quantum dots (QDs) after resonant excitation in the single exciton limit using serial femtosecond crystallography (SFX). By reconstructing the three-dimensional (3D) differential diffraction pattern and building on density functional theory (DFT) calculations, we assign the lattice distortion after photoexcitation to the combined presence of a delocalized electron and a localized hole, forming a mixed large/small exciton-polaron. This result creates a clear picture of the polaronic deformation in CsPbBr3 QDs and demonstrates the exceptional sensitivity of SFX to lattice distortions in few-nanometer crystallites. We plan to use this experimental platform for future studies of electron-lattice interactions.

Nanoparticles, exhibiting functionally relevant structural heterogeneity, are at the forefront of cutting-edge research. Now, high-throughput single-particle imaging (SPI) with x-ray free-electron lasers (XFELs) creates unprecedented opportunities for recovering the shape distributions of millions of particles that exhibit functionally relevant structural heterogeneity. To realize this potential, three challenges have to be overcome: (1) simultaneous parametrization of structural variability in real and reciprocal spaces; (2) efficiently inferring the latent parameters of each SPI measurement; (3) scaling up comparisons between $10^5$ structural models and $10^6$ XFEL-SPI measurements. Here, we describe how we overcame these three challenges to resolve the non-equilibrium shape distributions within millions of gold nanoparticles imaged at the European XFEL. These shape distributions allowed us to quantify the degree of asymmetry in these particles, discover a relatively stable `shape envelope' amongst nanoparticles, discern finite-size effects related to shape-controlling surfactants, and extrapolate nanoparticles' shapes to their idealized thermodynamic limit. Ultimately, these demonstrations show that XFEL SPI can help transform nanoparticle shape characterization from anecdotally interesting to statistically meaningful.

Detecting microsecond structural perturbations in biomolecules has wide relevance in biology, chemistry and medicine. Here we show how MHz repetition rates at X-ray free-electron lasers can be used to produce microsecond time-series of protein scattering with exceptionally low noise levels of 0.001%. We demonstrate the approach by examining Jɑ helix unfolding of a light-oxygen-voltage photosensory domain. This time-resolved acquisition strategy is easy to implement and widely applicable for direct observation of structural dynamics of many biochemical processes.

OaPAC is a recently discovered blue-light-using flavin adenosine dinucleotide (BLUF) photoactivated adenylate cyclase from the cyanobacterium Oscillatoria acuminata that uses adenosine triphosphate and translates the light signal into the production of cyclic adenosine monophosphate. Here, we report crystal structures of the enzyme in the absence of its natural substrate determined from room-temperature serial crystallography data collected at both an X-ray free-electron laser and a synchrotron, and we compare these structures with cryo-macromolecular crystallography structures obtained at a synchrotron by us and others. These results reveal slight differences in the structure of the enzyme due to data collection at different temperatures and X-ray sources. We further investigate the effect of the Y6W mutation in the BLUF domain, a mutation which results in a rearrangement of the hydrogen-bond network around the flavin and a notable rotation of the side chain of the critical Gln48 residue. These studies pave the way for picosecond-millisecond time-resolved serial crystallography experiments at X-ray free-electron lasers and synchrotrons in order to determine the early structural intermediates and correlate them with the well studied picosecond-millisecond spectroscopic intermediates.

X-ray multi-projection imaging (XMPI) is an emerging experimental technique for the acquisition of rotation-free, time-resolved, volumetric information on stochastic processes. The technique is developed for high-brilliance light-source facilities, aiming to address known limitations of state-of-the-art imaging methods in the acquisition of 4D sample information, linked to their need for sample rotation. XMPI relies on a beam-splitting scheme, that illuminates a sample from multiple, angularly spaced viewpoints, and employs fast, indirect, X-ray imaging detectors for the collection of the data. This approach enables studies of previously inaccessible phenomena of industrial and societal relevance such as fractures in solids, propagation of shock waves, laser-based 3D printing, or even fast processes in the biological domain. In this work, we discuss in detail the beam-splitting scheme of XMPI. More specifically, we explore the relevant properties of X-ray splitter optics for their use in XMPI schemes, both at synchrotron insertion devices and XFEL facilities. Furthermore, we describe two distinct XMPI schemes, designed to faciliate large samples and complex sample environments. Finally, we present experimental proof of the feasibility of MHz-rate XMPI at the European XFEL. This detailed overview aims to state the challenges and the potential of XMPI and act as a stepping stone for future development of the technique.

The European X-ray Free Electron Laser (European XFEL) is a cutting-edge user facility that generates per second up to 27,000 ultra-short, spatially coherent X-ray pulses within an energy range of 0.26 to more than 20 keV. Specialized instrumentation, including various 2D X-ray detectors capable of handling the unique time structure of the beam, is required. The one-megapixel AGIPD (AGIPD1M) detectors, developed for the European XFEL by the AGIPD Consortium, are the primary detectors used for user experiments at the SPB/SFX and MID instruments. The first AGIPD1M detector was installed at SPB/SFX when the facility began operation in 2017, and the second one was installed at MID in November 2018. The AGIPD detector systems require a dedicated infrastructure, well-defined safety systems, and high-level control procedures to ensure stable and safe operation. As of now, the AGIPD1M detectors installed at the SPB/SFX and MID experimental end stations are fully integrated into the European XFEL environment, including mechanical integration, vacuum, power, control, data acquisition, and data processing systems. Specific high-level procedures allow facilitated detector control, and dedicated interlock systems based on Programmable Logic Controllers ensure detector safety in case of power, vacuum, or cooling failure. The first 6 years of operation have clearly demonstrated that the AGIPD1M detectors provide high-quality scientific results. The collected data, along with additional dedicated studies, have also enabled the identification and quantification of issues related to detector performance, ensuring stable operation. Characterization and calibration of detectors are among the most critical and challenging aspects of operation due to their complex nature. A methodology has been developed to enable detector characterization and data correction, both in near real-time (online) and offline mode. The calibration process optimizes detector performance and ensures the highest quality of experimental results. Overall, the experience gained from integrating and operating the AGIPD detectors at the European XFEL, along with the developed methodology for detector characterization and calibration, provides valuable insights for the development of next-generation detectors for Free Electron Laser X-ray sources.

Nanoparticles, exhibiting functionally relevant structural heterogeneity, are at the forefront of cutting-edge research. Now, high-throughput single-particle imaging (SPI) with X-ray free-electron lasers (XFELs) creates opportunities for recovering the shape distributions of millions of particles that exhibit functionally relevant structural heterogeneity. To realize this potential, three challenges have to be overcome: (1) simultaneous parametrization of structural variability in real and reciprocal spaces; (2) efficiently inferring the latent parameters of each SPI measurement; (3) scaling up comparisons between 105 structural models and 106 XFEL-SPI measurements. Here, we describe how we overcame these three challenges to resolve the nonequilibrium shape distributions within millions of gold nanoparticles imaged at the European XFEL. These shape distributions allowed us to quantify the degree of asymmetry in these particles, discover a relatively stable "shape envelope" among nanoparticles, discern finite-size effects related to shape-controlling surfactants, and extrapolate nanoparticles' shapes to their idealized thermodynamic limit. Ultimately, these demonstrations show that XFEL SPI can help transform nanoparticle shape characterization from anecdotally interesting to statistically meaningful.