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In quantum physics (QP) education, the use of representations such as diagrams and visual aids that connect to mathematical concepts is crucial. Research in representation theory indicates that combining symbolic-mathematical elements (e.g. formulae) with visual-graphical representations enhances conceptual understanding more effectively than representations that merely depict phenomena. However, common representations vary widely, and existing categorisation systems do not adequately distinguish between them in QP. To address this, we developed a new set of differentiation criteria based on insights from representation research, QP education, and specific aspects of the quantum sciences. We created a comprehensive category system for evaluating visual QP representations for educational use, grounded in Ainsworths (2006) DeFT Framework. Twenty-one experts from four countries evaluated this category system using four qubit representations: the Bloch sphere, Circle Notation, Quantum Bead, and the pie chart (Qake) model. This evaluation enabled us to assess the discriminative power of our criteria and the effectiveness of each representation in supporting the learning of QP concepts. It evaluated how well each representation conveyed quantum concepts such as quantum state, measurement, superposition, entanglement, and quantum technologies (X-, Z-, and H-gates) across 16 criteria. The results showed significant differences in the effectiveness of these representations, particularly in conveying key concepts like superposition and measurement. Additionally, expert ratings indicated notable variations in the potential of each representation to induce misconceptions, linked to differences in shape, measurement behaviour, and requirements for understanding entanglement. We also discuss considerations for developing new representations and suggest directions for future empirical studies.

In a previous paper, one of us interpreted mod 2 Dyer-Lashof operations as explicit A-module extensions between Brown-Gitler modules, and showed these A-modules can be topologically realized by finite spectra occurring as fibers of maps between 2-local dual Brown-Gitler spectra. Starting from these constructions, in this paper, we show that infinite families of these finite spectra are of chromatic type 2, with mod 2 cohomology that is free over A(1). Applications include classifying the dual Brown-Gitler spectra after localization with respect to K-theory., Comment: 26 pages. v2: minor revisions, with a much improved formulation of the main theorem

The spin structure functions of the proton and the deuteron were measured during the EG4 experiment at Jefferson Lab in 2006. Data were collected for longitudinally polarized electron scattering off longitudinally polarized NH$_3$ and ND$_3$ targets, for $Q^2$ values as small as 0.012 and 0.02 GeV$^2$, respectively, using the CEBAF Large Acceptance Spectrometer (CLAS). This is the archival paper of the EG4 experiment that summaries the previously reported results of the polarized structure functions $g_1$, $A_1F_1$, and their moments $\overline \Gamma_1$, $\overline \gamma_0$, and $\overline I_{TT}$, for both the proton and the deuteron. In addition, we report on new results on the neutron $g_1$ extracted by combining proton and deuteron data and correcting for Fermi smearing, and on the neutron moments $\overline \Gamma_1$, $\overline \gamma_0$, and $\overline I_{TT}$ formed directly from those of the proton and the deuteron. Our data are in good agreement with the Gerasimov-Drell-Hearn sum rule for the proton, deuteron, and neutron. Furthermore, the isovector combination was formed for $g_1$ and the Bjorken integral $\overline \Gamma_1^{p-n}$, and compared to available theoretical predictions. All of our results provide for the first time extensive tests of spin observable predictions from chiral effective field theory ($\chi$EFT) in a $Q^2$ range commensurate with the pion mass. They motivate further improvement in $\chi$EFT calculations from other approaches such as the lattice gauge method., Comment: 33 pages. 26 figures. Data table provided as supplementary material (30 pages latex and csv tables)

Traditional Semantic Table Interpretation (STI) methods rely primarily on the underlying table data to create semantic annotations. This year's SemTab challenge introduced the ``Metadata to KG'' track, which focuses on performing STI by using only metadata information, without access to the underlying data. In response to this new challenge, we introduce a new term: Column Vocabulary Association (CVA). This term refers to the task of semantic annotation of column headers solely based on metadata information. In this study, we evaluate the performance of various methods in executing the CVA task, including a Large Language Models (LLMs) and Retrieval Augmented Generation (RAG) approach, as well as a more traditional similarity approach with SemanticBERT. Our methodology uses a zero-shot setting, with no pretraining or examples passed to the Large Language Models (LLMs), as we aim to avoid a domain-specific setting. We investigate a total of 7 different LLMs, of which three commercial GPT models (i.e. gpt-3.5-turbo-0.125, gpt-4o and gpt-4-turbo) and four open source models (i.e. llama3-80b, llama3-7b, gemma-7b and mixtral-8x7b). We integrate this models with RAG systems, and we explore how variations in temperature settings affect performances. Moreover, we continue our investigation by performing the CVA task utilizing SemanticBERT, analyzing how various metadata information influence its performance. Initial findings indicate that LLMs generally perform well at temperatures below 1.0, achieving an accuracy of 100\% in certain cases. Nevertheless, our investigation also reveal that the nature of the data significantly influences CVA task outcomes. In fact, in cases where the input data and glossary are related (for example by being created by the same organizations) traditional methods appear to surpass the performance of LLMs.

We present the first homogeneous release of several thousand Type Ia supernovae (SNe Ia), all having spectroscopic classification, and spectroscopic redshifts for half the sample. This release, named the "DR2", contains 3628 nearby (z < 0.3) SNe Ia discovered, followed and classified by the Zwicky Transient Facility survey between March 2018 and December 2020. Of these, 3000 have good-to-excellent sampling and 2667 pass standard cosmology light-curve quality cuts. This release is thus the largest SN Ia release to date, increasing by an order of magnitude the number of well characterized low-redshift objects. With the "DR2", we also provide a volume-limited (z < 0.06) sample of nearly a thousand SNe Ia. With such a large, homogeneous and well controlled dataset, we are studying key current questions on SN cosmology, such as the linearity SNe Ia standardization, the SN and host dependencies, the diversity of the SN Ia population, and the accuracy of the current light-curve modeling. These, and more, are studied in detail in a series of articles associated with this release. Alongside the SN Ia parameters, we publish our force-photometry gri-band light curves, 5138 spectra, local and global host properties, observing logs, and a python tool to ease use and access of these data. The photometric accuracy of the "DR2" is not yet suited for cosmological parameter inference, which will follow as "DR2.5" release. We nonetheless demonstrate that the multi-thousand SN Ia Hubble Diagram has a typical 0.15 mag scatter., Comment: ZTF SN Ia DR2 release paper. Accepted for publication (Astronomy and Astrophysics). Reference paper for the ZTF SN Ia DR2 A&A Special Issue

The extraction of keypoints in images is at the basis of many computer vision applications, from localization to 3D reconstruction. Keypoints come with a score permitting to rank them according to their quality. While learned keypoints often exhibit better properties than handcrafted ones, their scores are not easily interpretable, making it virtually impossible to compare the quality of individual keypoints across methods. We propose a framework that can refine, and at the same time characterize with an interpretable score, the keypoints extracted by any method. Our approach leverages a modified robust Gaussian Mixture Model fit designed to both reject non-robust keypoints and refine the remaining ones. Our score comprises two components: one relates to the probability of extracting the same keypoint in an image captured from another viewpoint, the other relates to the localization accuracy of the keypoint. These two interpretable components permit a comparison of individual keypoints extracted across different methods. Through extensive experiments we demonstrate that, when applied to popular keypoint detectors, our framework consistently improves the repeatability of keypoints as well as their performance in homography and two/multiple-view pose recovery tasks., Comment: Accepted at ECCV 2024

For d/Deaf and hard of hearing (DHH) people, captioning is an essential accessibility tool. Significant developments in artificial intelligence (AI) mean that Automatic Speech Recognition (ASR) is now a part of many popular applications. This makes creating captions easy and broadly available - but transcription needs high levels of accuracy to be accessible. Scientific publications and industry report very low error rates, claiming AI has reached human parity or even outperforms manual transcription. At the same time the DHH community reports serious issues with the accuracy and reliability of ASR. There seems to be a mismatch between technical innovations and the real-life experience for people who depend on transcription. Independent and comprehensive data is needed to capture the state of ASR. We measured the performance of eleven common ASR services with recordings of Higher Education lectures. We evaluated the influence of technical conditions like streaming, the use of vocabularies, and differences between languages. Our results show that accuracy ranges widely between vendors and for the individual audio samples. We also measured a significant lower quality for streaming ASR, which is used for live events. Our study shows that despite the recent improvements of ASR, common services lack reliability in accuracy.

The Word Error Rate (WER) is the common measure of accuracy for Automatic Speech Recognition (ASR). Transcripts are usually pre-processed by substituting specific characters to account for non-semantic differences. As a result of this normalisation, information on the accuracy of punctuation or capitalisation is lost. We present a non-destructive, token-based approach using an extended Levenshtein distance algorithm to compute a robust WER and additional orthographic metrics. Transcription errors are also classified more granularly by existing string similarity and phonetic algorithms. An evaluation on several datasets demonstrates the practical equivalence of our approach compared to common WER computations. We also provide an exemplary analysis of derived use cases, such as a punctuation error rate, and a web application for interactive use and visualisation of our implementation. The code is available open-source., Comment: Accepted in INTERSPEECH 2024

Computing ratios of normalizing constants plays an important role in statistical modeling. Two important examples are hypothesis testing in latent variables models, and model comparison in Bayesian statistics. In both examples, the likelihood ratio and the Bayes factor are defined as the ratio of the normalizing constants of posterior distributions. We propose in this article a novel methodology that estimates this ratio using stochastic approximation principle. Our estimator is consistent and asymptotically Gaussian. Its asymptotic variance is smaller than the one of the popular optimal bridge sampling estimator. Furthermore, it is much more robust to little overlap between the two unnormalized distributions considered. Thanks to its online definition, our procedure can be integrated in an estimation process in latent variables model, and therefore reduce the computational effort. The performances of the estimator are illustrated through a simulation study and compared to two other estimators : the ratio importance sampling and the optimal bridge sampling estimators.

In this study, we present an innovative fusion of language models and query analysis techniques to unlock cognition in artificial intelligence. The introduced open-source AI system seamlessly integrates a Chess engine with a language model, enabling it to predict moves and provide strategic explanations. Leveraging a vector database to achieve retrievable answer generation, our AI system elucidates its decision-making process, bridging the gap between raw computation and human-like understanding. Our choice of Chess as the demonstration environment underscores the versatility of our approach. Beyond Chess, our system holds promise for diverse applications, from medical diagnostics to financial forecasting. Our AI system is available at https://github.com/TheOpenSI/CoSMIC.git, Comment: This paper is accepted to Australasian Conference on Information Systems 2024

We present a resonant-mode, transverse-field, radio-frequency (rf) neutron spin flipper design that uses high-temperature superconducting films to ensure sharp transitions between uniform magnetic field regions. Resonant mode allows for low power, high frequency operation but requires strict homogeneity of the magnetic fields inside the device. This design was found to efficiently flip neutrons at 96.6$\pm 0.6\%$ at an effective frequency of 4 MHz with a beam size of $2.5~\times~2.5$~cm and a wavelength of 0.4 nm. The high frequency and efficiency enable this device to perform high-resolution neutron spectroscopy with comparable performance to currently implemented rf flipper designs. The limitation of the maximum frequency was found due to the field homogeneity of the device. We numerically analyze the maximum possible efficiency of this design using a Bloch solver simulation with magnetic fields generated from finite-element simulations. We also discuss future improvements of the efficiency and frequency to the design based on the experimental and simulation results.

Silicon presents itself as a high-capacity anode material for lithium-ion batteries with a promising future. The high ability for lithiation comes along with massive volume changes and a problematic voltage hysteresis, causing reduced efficiency, detrimental heat generation, and a complicated state-of-charge estimation. During slow cycling, amorphous silicon nanoparticles show a larger voltage hysteresis than after relaxation periods. Interestingly, the voltage relaxes for at least several days, which has not been physically explained so far. We apply a chemo-mechanical continuum model in a core-shell geometry interpreted as a silicon particle covered by the solid-electrolyte interphase to account for the hysteresis phenomena. The silicon core (de)lithiates during every cycle while the covering shell is chemically inactive. The visco-elastoplastic behavior of the shell explains the voltage hysteresis during cycling and after relaxation. We identify a logarithmic voltage relaxation, which fits with the established Garofalo law for viscosity. Our chemo-mechanical model describes the observed voltage hysteresis phenomena and outperforms the empirical Plett model. In addition to our full model, we present a reduced model to allow for easy voltage profile estimations. The presented results support the mechanical explanation of the silicon voltage hysteresis with a core-shell model and encourage further efforts into the investigation of the silicon anode mechanics.

We study decision problems under uncertainty, where the decision-maker has access to $K$ data sources that carry {\em biased} information about the underlying risk factors. The biases are measured by the mismatch between the risk factor distribution and the $K$ data-generating distributions with respect to an optimal transport (OT) distance. In this situation the decision-maker can exploit the information contained in the biased samples by solving a distributionally robust optimization (DRO) problem, where the ambiguity set is defined as the intersection of $K$ OT neighborhoods, each of which is centered at the empirical distribution on the samples generated by a biased data source. We show that if the decision-maker has a prior belief about the biases, then the out-of-sample performance of the DRO solution can improve with $K$ -- irrespective of the magnitude of the biases. We also show that, under standard convexity assumptions, the proposed DRO problem is computationally tractable if either $K$ or the dimension of the risk factors is kept constant.

Accurate segmentation of prostate tumours from PET images presents a formidable challenge in medical image analysis. Despite considerable work and improvement in delineating organs from CT and MR modalities, the existing standards do not transfer well and produce quality results in PET related tasks. Particularly, contemporary methods fail to accurately consider the intensity-based scaling applied by the physicians during manual annotation of tumour contours. In this paper, we observe that the prostate-localised uptake threshold ranges are beneficial for suppressing outliers. Therefore, we utilize the intensity threshold values, to implement a new custom-feature-clipping normalisation technique. We evaluate multiple, established U-Net variants under different normalisation schemes, using the nnU-Net framework. All models were trained and tested on multiple datasets, obtained with two radioactive tracers: [68-Ga]Ga-PSMA-11 and [18-F]PSMA-1007. Our results show that the U-Net models achieve much better performance when the PET scans are preprocessed with our novel clipping technique.

The Large Binocular Telescope, with its expansive collecting area, angular resolving power, and advanced optical design, provides a robust platform for development and operation of advanced instrumentation for astronomical research. The LBT currently hosts a mature suite of instruments for spectroscopy and imaging at optical through mid-infrared wavelengths, supported by sophisticated adaptive optics systems. This contribution summarizes the current state of instrumentation, including upgrades to existing instruments and commissioning of second generation instruments now in progress. The LBT is soliciting proposals for next generation instrument concepts, with participation open to consortium members and others interested in participation in the Observatory., Comment: 6 pages, Proc. SPIE 13096-4, Ground-based and Airborne Instrumentation for Astronomy X, Yokohama 2024

Diverse and realistic floor plan data are essential for the development of useful computer-aided methods in architectural design. Today's large-scale floor plan datasets predominantly feature simple floor plan layouts, typically representing single-apartment dwellings only. To compensate for the mismatch between current datasets and the real world, we develop \textbf{Modified Swiss Dwellings} (MSD) -- the first large-scale floor plan dataset that contains a significant share of layouts of multi-apartment dwellings. MSD features over 5.3K floor plans of medium- to large-scale building complexes, covering over 18.9K distinct apartments. We validate that existing approaches for floor plan generation, while effective in simpler scenarios, cannot yet seamlessly address the challenges posed by MSD. Our benchmark calls for new research in floor plan machine understanding. Code and data are open., Comment: ECCV 2024 (incl. Suppl. Mat.)

By prior work, we have many results related to distributed graph algorithms for problems that can be defined with local constraints; the formal framework used in prior work is locally checkable labeling problems (LCLs), introduced by Naor and Stockmeyer in the 1990s. It is known, for example, that if we have a deterministic algorithm that solves an LCL in $o(\log n)$ rounds, we can speed it up to $O(\log^*n)$ rounds, and if we have a randomized $O(\log^*n)$ rounds algorithm, we can derandomize it for free. It is also known that randomness helps with some LCL problems: there are LCL problems with randomized complexity $\Theta(\log\log n)$ and deterministic complexity $\Theta(\log n)$. However, so far there have not been any LCL problems in which the use of shared randomness has been necessary; in all prior algorithms it has been enough that the nodes have access to their own private sources of randomness. Could it be the case that shared randomness never helps with LCLs? Could we have a general technique that takes any distributed graph algorithm for any LCL that uses shared randomness, and turns it into an equally fast algorithm where private randomness is enough? In this work we show that the answer is no. We present an LCL problem $\Pi$ such that the round complexity of $\Pi$ is $\Omega(\sqrt n)$ in the usual randomized \local model with private randomness, but if the nodes have access to a source of shared randomness, then the complexity drops to $O(\log n)$. As corollaries, we also resolve several other open questions related to the landscape of distributed computing in the context of LCL problems. In particular, problem $\Pi$ demonstrates that distributed quantum algorithms for LCL problems strictly benefit from a shared quantum state. Problem $\Pi$ also gives a separation between finitely dependent distributions and non-signaling distributions.

We present beam measurements of the CHIME telescope using a radio calibration source deployed on a drone payload. During test flights, the pulsing calibration source and the telescope were synchronized to GPS time, enabling in-situ background subtraction for the full $N^{2}$ visibility matrix for one CHIME cylindrical reflector. We use the autocorrelation products to estimate the primary beam width and centroid location, and compare these quantities to solar transit measurements and holographic measurements where they overlap on the sky. We find that the drone, solar, and holography data have similar beam parameter evolution across frequency and both spatial coordinates. This paper presents the first drone-based beam measurement of a large cylindrical radio interferometer. Furthermore, the unique analysis and instrumentation described in this paper lays the foundation for near-field measurements of experiments like CHIME., Comment: Submitted to IEEE OJAP June 30, 2024

Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can be defined as a sheaf for the coherent topology on a certain category of compact Hausdorff spaces. In this case, the sheaf condition has a fairly simple explicit description, which arises from studying the relationship between the coherent, regular and extensive topologies. In this paper, we establish this relationship under minimal assumptions on the category, going beyond the case of compact Hausdorff spaces. Along the way, we also provide a characterization of sheaves and covering sieves for these categories. All results in this paper have been fully formalized in the Lean proof assistant., Comment: The Journal of Symbolic Logic, In press

The measurement of magnitudes with different filters in photometric surveys gives access to cosmological distances and parameters. However, for current and future large surveys like the ZTF, DES, HSC or LSST, the photometric calibration uncertainties are almost comparable to statistical uncertainties in the error budget of type Ia cosmology analysis, which limits our ability to use type Ia supernovae for precision cosmology. The knowledge of the bandpasses of the survey filters at the per-mill level can help reach the sub-percent precision for magnitudes. We show how a misknowledge of the bandpasses central wavelengths or of the presence of out-of-band leakages leads to biased cosmological measurements. Then, we present how to measure the filter throughputs at the required precision with a Collimated Beam Projector., Comment: 2 pages, 2 figures, contribution to the 2024 Cosmology session of the 58th Rencontres de Moriond

The 1S hyperfine splitting in hydrogen is measured to an impressive ppt precision and will soon be measured to ppm precision in muonic hydrogen. The latter measurement will rely on theoretical predictions, which are limited by knowledge of the proton polarizability effect $\Delta_\text{pol}$. Data-driven evaluations of $\Delta_\text{pol}$ have long been in significant tension with baryon chiral perturbation theory. Here we present improved results for $\Delta_\text{pol}$ driven by new spin structure data, reducing the long-standing tension between theory and experiment and halving the dominating uncertainty in hyperfine splitting calculations.

In the rapidly evolving interdisciplinary field of quantum information science and technology, a big obstacle is the necessity of understanding high-level mathematics to solve complex problems. Visualizations like the (dimensional) circle notation enable us to visualize not only single-qubit but also complex multi-qubit states, entanglement, and quantum algorithms. Current findings in educational research suggest that incorporating visualizations in settings of problem solving can have beneficial effects on students' performance and cognitive load compared to solely relying on symbolic problem solving content. In this pilot study, we aim to take a first step to identify in which contexts students benefit from the presentation of visualizations of multi-qubit systems in addition to mathematical formalism. We compare students' performance, time taken, and cognitive load when solving problems using the mathematical-symbolic Dirac notation alone with using it accompanied by the circle notation or the dimensional circle notation in single- and multi-qubit systems. Although little overall differences in students' performance can be detected depending on the presented representations, we observe that problem-solving performance is student- and context-dependent. In addition, the results indicate reduced cognitive load when participants are presented with visualization. The results are discussed with respect to relevant design aspects for future studies., Comment: 32(+20) pages, 16(+6) figures

The ESO public survey VISTA Variables in the V\'ia L\'actea (VVV) surveyed the inner Galactic bulge and the adjacent southern Galactic disk from $2009-2015$. Upon its conclusion, the complementary VVV eXtended (VVVX) survey has expanded both the temporal as well as spatial coverage of the original VVV area, widening it from $562$ to $1700$ sq. deg., as well as providing additional epochs in $JHK_{\rm s}$ filters from $2016-2023$. With the completion of VVVX observations during the first semester of 2023, we present here the observing strategy, a description of data quality and access, and the legacy of VVVX. VVVX took $\sim 2000$ hours, covering about 4% of the sky in the bulge and southern disk. VVVX covered most of the gaps left between the VVV and the VISTA Hemisphere Survey (VHS) areas and extended the VVV time baseline in the obscured regions affected by high extinction and hence hidden from optical observations. VVVX provides a deep $JHK_{\rm s}$ catalogue of $\gtrsim 1.5\times10^9$ point sources, as well as a $K_{\rm s}$ band catalogue of $\sim 10^7$ variable sources. Within the existing VVV area, we produced a $5D$ map of the surveyed region by combining positions, distances, and proper motions of well-understood distance indicators such as red clump stars, RR Lyrae, and Cepheid variables. In March 2023 we successfully finished the VVVX survey observations that started in 2016, an accomplishment for ESO Paranal Observatory upon 4200 hours of observations for VVV+VVVX. The VVV+VVVX catalogues complement those from the Gaia mission at low Galactic latitudes and provide spectroscopic targets for the forthcoming ESO high-multiplex spectrographs MOONS and 4MOST., Comment: 17 pages, 11 figures (+ appendix). Accepted for publication in Astronomy and Astrophysics in section 14: Catalogs and data

Measuring Deeply Virtual Compton Scattering on the neutron is one of the necessary steps to understand the structure of the nucleon in terms of Generalized Parton Distributions (GPDs). Neutron targets play a complementary role to transversely polarized proton targets in the determination of the GPD $E$. This poorly known and poorly constrained GPD is essential to obtain the contribution of the quarks' angular momentum to the spin of the nucleon. DVCS on the neutron was measured for the first time selecting the exclusive final state by detecting the neutron, using the Jefferson Lab longitudinally polarized electron beam, with energies up to 10.6 GeV, and the CLAS12 detector. The extracted beam-spin asymmetries, combined with DVCS observables measured on the proton, allow a clean quark-flavor separation of the imaginary parts of the GPDs $H$ and $E$., Comment: 7 pages, 6 figures

Probabilistic programming (PP) is a programming paradigm that allows for writing statistical models like ordinary programs, performing simulations by running those programs, and analyzing and refining their statistical behavior using powerful inference engines. This paper takes a step towards leveraging PP for reasoning about data-aware processes. To this end, we present a systematic translation of Data Petri Nets (DPNs) into a model written in a PP language whose features are supported by most PP systems. We show that our translation is sound and provides statistical guarantees for simulating DPNs. Furthermore, we discuss how PP can be used for process mining tasks and report on a prototype implementation of our translation. We also discuss further analysis scenarios that could be easily approached based on the proposed translation and available PP tools.

Excitons in twisted bilayers of transition metal dichalcogenides have strongly modified dispersion relations due to the formation of periodic moir\'e potentials. The strong coupling to a light field in an optical cavity leads to the appearance of moir\'e polaritons. In this paper, we derive a theoretical model for the linear absorption spectrum of the coupled moir\'e polariton-phonon system based on the time-convolutionless (TCL) approach. Results obtained by numerically solving the TCL equation are compared to those obtained in the Markovian limit and from a perturbative treatment of non-Markovian corrections. A key quantity for the interpretation of the findings is the generalized phonon spectral density. We discuss the phonon impact on the spectrum for realistic moir\'e exciton dispersions by varying twist angle and temperature. Key features introduced by the coupling to phonons are broadenings and energy shifts of the upper and lower polariton peak and the appearance of phonon sidebands between them. We analyze these features with respect to the role of Markovian and non-Markovian effects and find that they strongly depend on the twist angle. We can distinguish between the regimes of large, small, and intermediate twist angles. In the latter phonon effects are particularly pronounced due to dominating phonon transitions into regions which are characterized by van Hove singularities in the density of states.