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We report on results from an experiment at the European XFEL where we measured the x-ray Thomson scattering (XRTS) spectrum of single crystal silicon with ultrahigh resolution. Compared to similar previous experiments, we consider a more complex scattering setup, in which the scattering vector changes orientation through the crystal lattice. In doing so, we are able to observe strong geometric dependencies in the inelastic scattering spectrum of silicon at low scattering angles. Furthermore, the high quality of the experimental data allows us to benchmark state-of-the-art TDDFT calculations, and demonstrate TDDFT's ability to accurately predict these geometric dependencies. Finally, we note that this experimental data was collected at a much faster rate than another recently reported dataset using the same setup, demonstrating that ultrahigh resolution XRTS data can be collected in more general experimental scenarios.

We prove an asymptotic formula for the second moment of $L$-functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight., Comment: 23 pages

We present 50-fs, single-shot measurements of the x-ray thermal diffuse scattering (TDS) from copper foils that have been shocked via nanosecond laser-ablation up to pressures above 135~GPa. We hence deduce the x-ray Debye-Waller (DW) factor, providing a temperature measurement. The targets were laser-shocked with the DiPOLE 100-X laser at the High Energy Density (HED) endstation of the European X-ray Free-Electron Laser (EuXFEL). Single x-ray pulses, with a photon energy of 18 keV, were scattered from the samples and recorded on Varex detectors. Despite the targets being highly textured (as evinced by large variations in the elastic scattering), and with such texture changing upon compression, the absolute intensity of the azimuthally averaged inelastic TDS between the Bragg peaks is largely insensitive to these changes, and, allowing for both Compton scattering and the low-level scattering from a sacrificial ablator layer, provides a reliable measurement of $T/\Theta_D^2$, where $\Theta_D$ is the Debye temperature. We compare our results with the predictions of the SESAME 3336 and LEOS 290 equations of state for copper, and find good agreement within experimental errors. We thus demonstrate that single-shot temperature measurements of dynamically compressed materials can be made via thermal diffuse scattering of XFEL radation., Comment: 17 pages, 9 figures in main article; 10 pages, 5 figures in supplementary material

Here we present a derivation of the hierarchical equations of motion (HEOM) for an overdamped Lorentz-Drude environment containing an undamped oscillator (LDUO-HEOM). The new approach avoids the artifact of superfluous damping inherent in underdamped spectral densities. We show that the new HEOM is a useful model for intramolecular vibrations in condensed phase molecules.

The higher-order nonlinear Schrodinger equation (Dysthe's equation in the context of water-waves) models the time evolution of the slowly modulated amplitude of a wave-packet in dispersive partial differential equations (PDE). These systems, of which water-waves are a canonical example, require the presence of a small-valued ordering parameter so that a multi-scale expansion can be performed. However, often the resulting system itself contains the small-ordering parameter. Thus, these models are difficult to interpret from a formal asymptotics perspective. This paper derives a parameter-free, higher-order evolution equation for a generic infinite-dimensional dispersive PDE with weak linear damping and/or forcing. Instead of focusing on the water-wave problem or another specific problem, our procedure avoids the complicated algebra by placing the PDE in an infinite-dimensional Hilbert space and Taylor expanding with Frechet derivatives. An attractive feature of this procedure is that it can be used in many different physical settings, including water-waves, nonlinear optics and any dispersive system with weak dissipation or forcing. The paper concludes by discussing two specific examples.

A common task in forensic biology is to interpret and evaluate short tandem repeat DNA profiles. The first step in these interpretations is to assign a number of contributors to the profiles, a task that is most often performed manually by a scientist using their knowledge of DNA profile behaviour. Studies using constructed DNA profiles have shown that as DNA profiles become more complex, and the number of DNA-donating individuals increases, the ability for scientists to assign the target number. There have been a number of machine learning algorithms developed that seek to assign the number of contributors to a DNA profile, however due to practical limitations in being able to generate DNA profiles in a laboratory, the algorithms have been based on summaries of the available information. In this work we develop an analysis pipeline that simulates the electrophoretic signal of an STR profile, allowing virtually unlimited, pre-labelled training material to be generated. We show that by simulating 100 000 profiles and training a number of contributors estimation tool using a deep neural network architecture (in an algorithm named deepNoC) that a high level of performance is achieved (89% for 1 to 10 contributors). The trained network can then have fine-tuning training performed with only a few hundred profiles in order to achieve the same accuracy within a specific laboratory. We also build into deepNoC secondary outputs that provide a level of explainability to a user of algorithm, and show how they can be displayed in an intuitive manner., Comment: 29 pages, 8 figures

Spectroscopic techniques are essential for studying material properties, but the small cross-sections of some methods may result in low signal-to-noise ratios (SNRs) in the collected spectra. In this article we present methods, based on combining Bragg spectroscopy with photon counting and deconvolution algorithms, which increase the SNRs, making the spectra better suited to further analysis. We aim to provide a comprehensive guide for constructing spectra from camera images. The efficacy of these methods is validated on synthetic and experimental data, the latter coming from the field of high-energy density (HED) science, where x-ray spectroscopy is essential for the understanding of materials under extreme thermodynamic conditions.

Small-scale brightenigs are ubiquitous, dynamic and energetic phenomena found in the chromopshere. An advanced filter-detection algorithm applied to high-resolution observations from the Interface Region Imaging Spectrograph enables the detection of these brightenings close to the noise level. This algorithm also tracks the movement of these brightenings and extracts their characteristics. This work outlines the results of an in-depth analysis of a quiet-Sun dataset including a comparison of a brighter domain - associated with a super-granular boundary - to the quiescent inter-network domains. Several characteristics of brightenings from both domains are extracted and analysed, providing a range of sizes, durations, brightness values, travel distances, and speeds. The ``Active" quiet-Sun events tend to travel shorter distances and at slower speeds along the plane-of-sky than their ``True" quiet-Sun counterparts. These results are consistent with the magnetic field model of super-granular photospheric structures and the magnetic canopy model of the chromosphere above. Spectroscopic analyses reveal that BPs demonstrate blue-shift (as well as some bi-directionality) and that they may rise from the chromosphere into the TR. We believe these bright points to be magnetic in nature, are likely the result of magnetic reconnection, and follow current sheets between magnetic field gradients, rather than along magnetic field lines themselves., Comment: 18 pages, 11 figures

This study demonstrates that Large Language Models (LLMs) can transcribe historical handwritten documents with significantly higher accuracy than specialized Handwritten Text Recognition (HTR) software, while being faster and more cost-effective. We introduce an open-source software tool called Transcription Pearl that leverages these capabilities to automatically transcribe and correct batches of handwritten documents using commercially available multimodal LLMs from OpenAI, Anthropic, and Google. In tests on a diverse corpus of 18th/19th century English language handwritten documents, LLMs achieved Character Error Rates (CER) of 5.7 to 7% and Word Error Rates (WER) of 8.9 to 15.9%, improvements of 14% and 32% respectively over specialized state-of-the-art HTR software like Transkribus. Most significantly, when LLMs were then used to correct those transcriptions as well as texts generated by conventional HTR software, they achieved near-human levels of accuracy, that is CERs as low as 1.8% and WERs of 3.5%. The LLMs also completed these tasks 50 times faster and at approximately 1/50th the cost of proprietary HTR programs. These results demonstrate that when LLMs are incorporated into software tools like Transcription Pearl, they provide an accessible, fast, and highly accurate method for mass transcription of historical handwritten documents, significantly streamlining the digitization process., Comment: 29 Pages, 11 Tables, 2 Figures

The superiorization methodology (SM) is an optimization heuristic in which an iterative algorithm, which aims to solve a particular problem, is ``superiorized'' to promote solutions that are improved with respect to some secondary criterion. This superiorization is achieved by perturbing iterates of the algorithm in nonascending directions of a prescribed function that penalizes undesirable characteristics in the solution; the solution produced by the superiorized algorithm should therefore be improved with respect to the value of this function. In this paper, we broaden the SM to allow for the perturbations to be introduced by an arbitrary procedure instead, using a plug-and-play approach. This allows for operations such as image denoisers or deep neural networks, which have applications to a broad class of problems, to be incorporated within the superiorization methodology. As proof of concept, we perform numerical simulations involving low-dose and sparse-view computed tomography image reconstruction, comparing the plug-and-play approach to a conventionally superiorized algorithm, as well as a post-processing approach. The plug-and-play approach provides comparable or better image quality in most cases, while also providing advantages in terms of computing time, and data fidelity of the solutions., Comment: 20 pages, 6 figures

We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov-Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable--something that is impossible in the case of a constant delay.

The application of Shapley values to high-dimensional, time-series-like data is computationally challenging - and sometimes impossible. For $N$ inputs the problem is $2^N$ hard. In image processing, clusters of pixels, referred to as superpixels, are used to streamline computations. This research presents an efficient solution for time-seres-like data that adapts the idea of superpixels for Shapley value computation. Motivated by a forensic DNA classification example, the method is applied to multivariate time-series-like data whose features have been classified by a convolutional neural network (CNN). In DNA processing, it is important to identify alleles from the background noise created by DNA extraction and processing. A single DNA profile has $31,200$ scan points to classify, and the classification decisions must be defensible in a court of law. This means that classification is routinely performed by human readers - a monumental and time consuming process. The application of a CNN with fast computation of meaningful Shapley values provides a potential alternative to the classification. This research demonstrates the realistic, accurate and fast computation of Shapley values for this massive task, Comment: 16 pages, 5 figures

Terwilliger algebras are a subalgebra of a matrix algebra that are constructed from association schemes over finite sets. In 2010, Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative. In that paper she gave five equivalent conditions for a Terwilliger algebra to be almost commutative. In this paper, we provide a classification of which groups result in an almost commutative Terwilliger algebra when looking at the group association scheme (the Schur ring generated by the conjugacy classes of the group). In particular, we show that all such groups are either abelian, or Camina groups. Following this classification, we then compute the dimension and non-primary primitive idempotents for each Terwilliger algebra of this form for the first three types of groups whose group association scheme gives an almost commutative Terwilliger algebra. The final case will be considered in a second paper., Comment: Submitted to Discrete Mathematics

The end of Moore's Law is driving cloud providers to offload virtualization and the network data plane to SmartNICs to improve compute efficiency. Even though individual OS control plane tasks consume up to 5% of cycles across the fleet, they remain on the host CPU because they are tightly intertwined with OS mechanisms. Moreover, offloading puts the slow PCIe interconnect in the critical path of OS decisions. We propose Tide, a new split OS architecture that separates OS control plane policies from mechanisms and offloads the control plane policies onto a SmartNIC. Tide has a new host-SmartNIC communication API, state synchronization mechanism, and communication mechanisms that overcome the PCIe bottleneck, even for $\mu$s-scale workloads. Tide frees up host compute for applications and unlocks new optimization opportunities, including machine learning-driven policies, scheduling on the network I/O path, and reducing on-host interference. We demonstrate that Tide enables OS control planes that are competitive with on-host performance for the most difficult $\mu$s-scale workloads. Tide outperforms on-host control planes for memory management (saving 16 host cores), Stubby network RPCs (saving 8 cores), and GCE virtual machine management (11.2% performance improvement)., Comment: About 11 pages

DNA profiles are made up from multiple series of electrophoretic signal measuring fluorescence over time. Typically, human DNA analysts 'read' DNA profiles using their experience to distinguish instrument noise, artefactual signal, and signal corresponding to DNA fragments of interest. Recent work has developed an artificial neural network, ANN, to carry out the task of classifying fluorescence types into categories in DNA profile electrophoretic signal. But the creation of the necessarily large amount of labelled training data for the ANN is time consuming and expensive, and a limiting factor in the ability to robustly train the ANN. If realistic, prelabelled, training data could be simulated then this would remove the barrier to training an ANN with high efficacy. Here we develop a generative adversarial network, GAN, modified from the pix2pix GAN to achieve this task. With 1078 DNA profiles we train the GAN and achieve the ability to simulate DNA profile information, and then use the generator from the GAN as a 'realism filter' that applies the noise and artefact elements exhibited in typical electrophoretic signal., Comment: 29 pages, 9 Figures

We prove strong hybrid subconvex bounds simultaneously in the $q$ and $t$ aspects for $L$-functions of selfdual $\mathrm{GL}_3$ cusp forms twisted by primitive Dirichlet characters. We additionally prove analogous hybrid subconvex bounds for central values of certain $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions. The subconvex bounds that we obtain are strong in the sense that, modulo current knowledge on estimates for the second moment of $\mathrm{GL}_3$ $L$-functions, they are the natural limit of the first moment method pioneered by Li and by Blomer. The method of proof relies on an explicit $\mathrm{GL}_3 \times \mathrm{GL}_2 \leftrightsquigarrow \mathrm{GL}_4 \times \mathrm{GL}_1$ spectral reciprocity formula, which relates a $\mathrm{GL}_2$ moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions to a $\mathrm{GL}_1$ moment of $\mathrm{GL}_4 \times \mathrm{GL}_1$ Rankin-Selberg $L$-functions. A key additional input is a Lindel\"of-on-average upper bound for the second moment of Dirichlet $L$-functions restricted to a coset, which is of independent interest., Comment: 46 pages

The choice of coordinate system in a bearings-only (BO) tracking problem influences the methods used to observe and predict the state of a moving target. Modified Polar Coordinates (MPC) and Log-Polar Coordinates (LPC) have some advantages over Cartesian coordinates. In this paper, we derive closed-form expressions for the target state prior distribution after ownship manoeuvre: the mean, covariance, and higher-order moments in LPC. We explore the use of these closed-form expressions in simulation by modifying an existing BO tracker that uses the UKF. Rather than propagating sigma points, we directly substitute current values of the mean and covariance into the time update equations at the ownship turn. This modified UKF, the CFE-UKF, performs similarly to the pure UKF, verifying the closed-form expressions. The closed-form third and fourth central moments indicate non-Gaussianity of the target state when the ownship turns. By monitoring these metrics and appropriately initialising relative range error, we can achieve a desired output mean estimated range error (MRE). The availability of these higher-order moments facilitates other extensions of the tracker not possible with a standard UKF., Comment: 11 pages, 6 figures. This work has been submitted to the IEEE for possible publication

Dense plasma environment affects the electronic structure of ions via variations of the microscopic electrical fields, also known as plasma screening. This effect can be either estimated by simplified analytical models, or by computationally expensive and to date unverified numerical calculations. We have experimentally quantified plasma screening from the energy shifts of the bound-bound transitions in matter driven by the x-ray free electron laser (XFEL). This was enabled by identification of detailed electronic configurations of the observed K{\alpha}, K\b{eta} and K{\gamma} lines. This work paves the way for improving plasma screening models including connected effects like ionization potential depression and continuum lowering, which will advance the understanding of atomic physics in Warm Dense Matter regime.

Mosaic crystals, with their high integrated reflectivities, are widely-employed in spectrometers used to diagnose high energy density systems. X-ray Thomson scattering (XRTS) has emerged as a powerful diagnostic tool of these systems, providing in principle direct access to important properties such as the temperature via detailed balance. However, the measured XRTS spectrum is broadened by the spectrometer instrument function (IF), and without careful consideration of the IF one risks misdiagnosing system conditions. Here, we consider in detail the IF of 40 $\mu$m and 100 $\mu$m mosaic HAPG crystals, and how the broadening varies across the spectrometer in an energy range of 6.7-8.6 keV. Notably, we find a strong asymmetry in the shape of the IF towards higher energies. As an example, we consider the effect of the asymmetry in the IF on the temperature inferred via XRTS for simulated 80 eV CH plasmas, and find that the temperature can be overestimated if an approximate symmetric IF is used. We therefore expect a detailed consideration of the full IF will have an important impact on system properties inferred via XRTS in both forward modelling and model-free approaches., Comment: 19 pages, 13 figures

Quantum field theory predicts a nonlinear response of the vacuum to strong electromagnetic fields of macroscopic extent. This fundamental tenet has remained experimentally challenging and is yet to be tested in the laboratory. A particularly distinct signature of the resulting optical activity of the quantum vacuum is vacuum birefringence. This offers an excellent opportunity for a precision test of nonlinear quantum electrodynamics in an uncharted parameter regime. Recently, the operation of the high-intensity laser ReLaX provided by the Helmholtz International Beamline for Extreme Fields (HIBEF) has been inaugurated at the High Energy Density (HED) scientific instrument of the European XFEL. We make the case that this worldwide unique combination of an x-ray free-electron laser and an ultra-intense near-infrared laser together with recent advances in high-precision x-ray polarimetry, refinements of prospective discovery scenarios, and progress in their accurate theoretical modelling have set the stage for performing an actual discovery experiment of quantum vacuum nonlinearity., Comment: 34 pages, 21 figures

We study the problem of privacy-preserving $k$-means clustering in the horizontally federated setting. Existing federated approaches using secure computation, suffer from substantial overheads and do not offer output privacy. At the same time, differentially private (DP) $k$-means algorithms assume a trusted central curator and do not extend to federated settings. Naively combining the secure and DP solutions results in a protocol with impractical overhead. Instead, our work provides enhancements to both the DP and secure computation components, resulting in a design that is faster, more private, and more accurate than previous work. By utilizing the computational DP model, we design a lightweight, secure aggregation-based approach that achieves four orders of magnitude speed-up over state-of-the-art related work. Furthermore, we not only maintain the utility of the state-of-the-art in the central model of DP, but we improve the utility further by taking advantage of constrained clustering techniques.

We present new exclusion bounds obtained at the European X-ray Free Electron Laser facility (EuXFEL) on axion-like particles (ALPs) in the mass range 10^{-3} eV < m_a < 10^4 eV. Our experiment exploits the Primakoff effect via which photons can, in the presence of a strong external electric field, decay into axions, which then convert back into photons after passing through an opaque wall. While similar searches have been performed previously at a 3^rd generation synchrotron, our work demonstrates improved sensitivity, exploiting the higher brightness of X-rays at EuXFEL., Comment: 7 pages, 6 figures

We study commutative Schur rings over the symplectic groups Sp$(n,2)$ containing the class $\mathcal C$ of symplectic transvections. We find the possible partitions of $\mathcal C$ determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp$(n,2)$., Comment: 28 pages; one figure

We establish two new variants of arithmetic quantum ergodicity. The first is for self-dual $\mathrm{GL}_2$ Hecke-Maass newforms over $\mathbb{Q}$ as the level and Laplace eigenvalue vary jointly. The second is a nonsplit analogue wherein almost all restrictions of Hilbert (respectively Bianchi) Hecke-Maass cusp forms to the modular surface dissipate as their Laplace eigenvalues grow., Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:2103.05634

Using a novel ultrahigh resolution ($\Delta E \sim 0.1\,$eV) setup to measure electronic features in x-ray Thomson scattering (XRTS) experiments at the European XFEL in Germany, we have studied the collective plasmon excitation in aluminium at ambient conditions, which we can measure very accurately even at low momentum transfers. As a result, we can resolve previously reported discrepancies between ab initio time-dependent density functional theory simulations and experimental observations. The demonstrated capability for high-resolution XRTS measurements will be a game changer for the diagnosis of experiments with matter under extreme densities, temperatures, and pressures, and unlock the full potential of state-of-the-art x-ray free electron laser (XFEL) facilities to study planetary interior conditions, to understand inertial confinement fusion applications, and for material science and discovery.

It is a folklore result in arithmetic quantum chaos that quantum unique ergodicity on the modular surface with an effective rate of convergence follows from subconvex bounds for certain triple product $L$-functions. The physical space manifestation of this result, namely the equidistribution of mass of Hecke-Maass cusp forms, was proven to follow from subconvexity by Watson, whereas the phase space manifestation of quantum unique ergodicity has only previously appeared in the literature for Eisenstein series via work of Jakobson. We detail the analogous phase space result for Hecke-Maass cusp forms. The proof relies on the Watson-Ichino triple product formula together with a careful analysis of certain archimedean integrals of Whittaker functions., Comment: 31 pages

Equation of state measurements at Jovian or stellar conditions are currently conducted by dynamic shock compression driven by multi-kilojoule multi-beam nanosecond-duration lasers. These experiments require precise design of the target and specific tailoring of the spatial and temporal laser profiles to reach the highest pressures. At the same time, the studies are limited by the low repetition rate of the lasers. Here, we show that by the irradiation of a thin wire with single beam Joule-class short-pulse laser, a converging cylindrical shock is generated compressing the wire material to conditions relevant for the above applications. The shockwave was observed using Phase Contrast Imaging employing a hard X-ray Free Electron Laser with unprecedented temporal and spatial sensitivity. The data collected for Cu wires is in agreement with hydrodynamic simulations of an ablative shock launched by a highly-impulsive and transient resistive heating of the wire surface. The subsequent cylindrical shockwave travels towards the wire axis and is predicted to reach a compression factor of 9 and pressures above 800 Mbar. Simulations for astrophysical relevant materials underline the potential of this compression technique as a new tool for high energy density studies at high repetition rates.

Resonant inelastic x-ray scattering (RIXS) is a widely used spectroscopic technique, providing access to the electronic structure and dynamics of atoms, molecules, and solids. However, RIXS requires a narrow bandwidth x-ray probe to achieve high spectral resolution. The challenges in delivering an energetic monochromated beam from an x-ray free electron laser (XFEL) thus limit its use in few-shot experiments, including for the study of high energy density systems. Here we demonstrate that by correlating the measurements of the self-amplified spontaneous emission (SASE) spectrum of an XFEL with the RIXS signal, using a dynamic kernel deconvolution with a neural surrogate, we can achieve electronic structure resolutions substantially higher than those normally afforded by the bandwidth of the incoming x-ray beam. We further show how this technique allows us to discriminate between the valence structures of Fe and Fe$_2$O$_3$, and provides access to temperature measurements as well as M-shell binding energies estimates in warm-dense Fe compounds.

Detecting anomalous behaviour of satellites is an important goal within the broader task of space situational awareness. The Two Line Element (TLE) data published by NORAD is the only widely-available, comprehensive source of data for satellite orbits. We present here a filtering approach for detecting anomalies in satellite orbits from TLE data. Optimal proposal particle filters are deployed to track the state of the satellites' orbits. New TLEs that are unlikely given our belief of the current orbital state are designated as anomalies. The change in the orbits over time is modelled using the SGP4 model with some adaptations. A model uncertainty is derived to handle the errors in SGP4 around singularities in the orbital elements. The proposed techniques are evaluated on a set of 15 satellites for which ground truth is available and the particle filters are shown to be superior at detecting the subtle in-track and cross-track manoeuvres in the simulated dataset, as well as providing a measure of uncertainty of detections.

Two parties with private data sets can find shared elements using a Private Set Intersection (PSI) protocol without revealing any information beyond the intersection. Circuit PSI protocols privately compute an arbitrary function of the intersection - such as its cardinality, and are often employed in an unbalanced setting where one party has more data than the other. Existing protocols are either computationally inefficient or require extensive server-client communication on the order of the larger set. We introduce Practically Efficient PSI or PEPSI, a non-interactive solution where only the client sends its encrypted data. PEPSI can process an intersection of 1024 client items with a million server items in under a second, using less than 5 MB of communication. Our work is over 4 orders of magnitude faster than an existing non-interactive circuit PSI protocol and requires only 10% of the communication. It is also up to 20 times faster than the work of Ion et al., which computes a limited set of functions and has communication costs proportional to the larger set. Our work is the first to demonstrate that non-interactive circuit PSI can be practically applied in an unbalanced setting.

We report on experiments where solid-density Mg plasmas are created by heating with the focused output of the Linac Coherent Light Source x-ray free-electron-laser. We study the K-shell emission from the Helium and Lithium-like ions using Bragg crystal spectroscopy. Observation of the dielectronic satellites in Lithium-like ions confirms that the M-shell electrons appear bound for these high charge states. An analysis of the intensity of these satellites indicates that when modelled with an atomic-kinetics code, the ionisation potential depression model employed needs to produce depressions for these ions which lie between those predicted by the well known Stewart-Pyatt and Ecker-Kroll models. These results are largely consistent with recent Density Functional Theory calculations.