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A function from $\Bbb F_{2^n}$ to $\Bbb F_{2^n}$ is said to be {\em $k$th order sum-free} if the sum of its values over each $k$-dimensional $\Bbb F_2$-affine subspace of $\Bbb F_{2^n}$ is nonzero. This notion was recently introduced by C. Carlet as, among other things, a generalization of APN functions. At the center of this new topic is a conjecture about the sum-freedom of the multiplicative inverse function $f_{\text{\rm inv}}(x)=x^{-1}$ (with $0^{-1}$ defined to be $0$). It is known that $f_{\text{\rm inv}}$ is 2nd order (equivalently, $(n-2)$th order) sum-free if and only if $n$ is odd, and it is conjectured that for $3\le k\le n-3$, $f_{\text{\rm inv}}$ is never $k$th order sum-free. The conjecture has been confirmed for even $n$ but remains open for odd $n$. In the present paper, we show that the conjecture holds under each of the following conditions: (1) $n=13$; (2) $3\mid n$; (3) $5\mid n$; (4) the smallest prime divisor $l$ of $n$ satisfies $(l-1)(l+2)\le (n+1)/2$. We also determine the ``right'' $q$-ary generalization of the binary multiplicative inverse function $f_{\text{\rm inv}}$ in the context of sum-freedom. This $q$-ary generalization not only maintains most results for its binary version, but also exhibits some extraordinary phenomena that are not observed in the binary case., Comment: 25 pages

In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modelling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also by applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.

We present a combined strong and weak gravitational-lensing analysis of the massive galaxy cluster MACS J1423.8+2404 ($z=0.545$, MACS J1423 hereafter), one of the most dynamically relaxed and massive cool-core clusters discovered in the MAssive Cluster Survey at $z>0.5$. We combine high-resolution imaging from the Hubble Space Telescope (HST) in the F606W, F814W, and F160W pass-bands with spectroscopic observations taken as part of the KALEIDOSCOPE survey with the Multi-Unit Spectroscopic Explorer mounted on the Very Large Telescope. Our strong lensing analysis of the mass distribution in the cluster core is constrained by four multiple-image systems (17 individual images) within redshift range $1.779

Non--Contact Atomic Force Microscopy with CO--functionalized metal tips (referred to as HR-AFM) provides access to the internal structure of individual molecules adsorbed on a surface with totally unprecedented resolution. Previous works have shown that deep learning (DL) models can retrieve the chemical and structural information encoded in a 3D stack of constant-height HR--AFM images, leading to molecular identification. In this work, we overcome their limitations by using a well-established description of the molecular structure in terms of topological fingerprints, the 1024--bit Extended Connectivity Chemical Fingerprints of radius 2 (ECFP4), that were developed for substructure and similarity searching. ECFPs provide local structural information of the molecule, each bit correlating with a particular substructure within the molecule. Our DL model is able to extract this optimized structural descriptor from the 3D HR--AFM stacks and use it, through virtual screening, to identify molecules from their predicted ECFP4 with a retrieval accuracy on theoretical images of 95.4\%. Furthermore, this approach, unlike previous DL models, assigns a confidence score, the Tanimoto similarity, to each of the candidate molecules, thus providing information on the reliability of the identification. By construction, the number of times a certain substructure is present in the molecule is lost during the hashing process, necessary to make them useful for machine learning applications. We show that it is possible to complement the fingerprint-based virtual screening with global information provided by another DL model that predicts from the same HR--AFM stacks the chemical formula, boosting the identification accuracy up to a 97.6\%. Finally, we perform a limited test with experimental images, obtaining promising results towards the application of this pipeline under real conditions

We present a sample of over 100 highly X-ray luminous galaxy clusters at $z\sim$ 0.5-0.9, discovered by the extended Massive Cluster Survey (eMACS) in ROSAT All-Sky Survey (RASS) data. Follow-up observations of a subset at higher resolution and greater depth with the Chandra X-ray Observatory are used to map the gaseous intra-cluster medium, while strong-gravitational-lensing features identified in Hubble Space Telescope imaging allow us to constrain the total mass distribution. We present evidence of the exceptional gravitational-lensing power of these massive systems, search for substructure along the line of sight by mapping the radial velocities of cluster members obtained through extensive ground-based spectroscopy, and identify dramatic cases of galaxy evolution in high-density cluster environments. The available observations of the eMACS sample presented here provide a wealth of insights into the properties of very massive clusters ($\gtrsim 10^{15} M_\odot$) at $z > 0.5$, which act as powerful lenses to study galaxies in the very distant Universe. We also discuss the evolutionary state, galaxy population, and large-scale environment of eMACS clusters and release to the community all data and science products to further the understanding of the first generation of truly massive clusters to have formed in the Universe., Comment: Submitted to MNRAS, comments welcome! A version including the complete appendices is available at http://www.astro.dur.ac.uk/~ace/eMACS.pdf

MACS J0600.1-2008 (MACS0600) is an X-ray luminous, massive galaxy cluster at $z_{\mathrm{d}}=0.43$, studied previously by the REionization LensIng Cluster Survey (RELICS) and ALMA Lensing Cluster Survey (ALCS) projects which revealed a complex, bimodal mass distribution and an intriguing high-redshift object behind it. Here, we report on the results of a combined analysis of the extended strong lensing (SL), X-ray, Sunyaev-Zeldovich (SZ), and galaxy luminosity-density properties of this system. Using new JWST and ground-based Gemini-N and Keck data, we obtain 13 new spectroscopic redshifts of multiply imaged galaxies and identify 12 new photometric multiple-image systems and candidates, including two multiply imaged $z\sim7$ objects. Taking advantage of the larger areal coverage, our analysis reveals an additional bimodal, massive SL structure which we measure spectroscopically to lie adjacent to the cluster and whose existence was implied by previous SL-modeling analyses. While based in part on photometric systems identified in ground-based imaging requiring further verification, our extended SL model suggests that the cluster may have the second-largest critical area and effective Einstein radius observed to date, $A_{\mathrm{crit}}\simeq2.16 \mathrm{arcmin}^2$ and $\theta_{\mathrm{E}}=49.7''\pm5.0''$ for a source at $z_{\mathrm{s}}=2$, enclosing a total mass of $M(<\theta_{\mathrm{E}})=(4.7\pm0.7)\times10^{14} \mathrm{M}_{\odot}$. These results are also supported by the galaxy luminosity distribution, the SZ and X-ray data. Yet another, probably related massive cluster structure, discovered in X-rays $5'$ (1.7 Mpc) further north, suggests that MACS0600 is part of an even larger filamentary structure. This discovery adds to several recent detections of massive structures around SL galaxy clusters and establishes MACS0600 as a prime target for future high-redshift surveys with JWST., Comment: Accepted for publication in MNRAS. Version updated to the accepted paper

In previous work we developed a new statistical method for calculating the individual activities of ions including the association of ions. Here we study multi-particle electrostatic interactions connected within higher cluster integrals and identify the ionization constants of the mass action law of associating ion clusters. In contrast to Bjerrum and Fuoss, our concept of association is not based on spatial criteria, but instead on the strength of interaction measured in powers of the Bjerrum parameter ($e^2 / D_0 k_{\text{B}} T a$ ; $a$ is contact) and defined by asymptotic properties of the cluster integrals. For ion pair formation our mass action constant is the classical counterpart of Planck's famous hydrogenic partition function. As a rule, the new association constants are smaller than traditional expressions, e.g., by Fuoss and Kraus, in the interesting regions of interaction parameters about fifty percent. Several examples including CaCl$_2$, MgCl$_2$, Na$_2$SO$_4$, K$_2$SO$_4$, LaCl$_3$ and a model of seawater are studied. For several associating electrolytes and seawater, reasonable agreement with experiments and Monte Carlo results is achieved., Comment: 20 pages, 6 figures, 1 table

We present a constraint on the abundance of supergiant (SG) stars at redshift z approx. 1, based on recent observations of a strongly lensed arc at this redshift. First we derive a free-form model of MACS J0416.1-2403 using data from the BUFFALO program. The new lens model is based on 72 multiply lensed galaxies that produce 214 multiple images, making it the largest sample of spectroscopically confirmed lensed galaxies on this cluster. The larger coverage in BUFFALO allows us to measure the shear up to the outskirts of the cluster, and extend the range of lensing constraints up to ~ 1 Mpc from the central region, providing a mass estimate up to this radius. As an application, we make predictions for the number of high-redshift multiply-lensed galaxies detected in future observations with JWST. Then we focus on a previously known lensed galaxy at z=1.0054, nicknamed Spock, which contains four previously reported transients. We interpret these transients as microcaustic crossings of SG stars and compute the probability of such events. Based on simplifications regarding the stellar evolution, we find that microlensing (by stars in the intracluster medium) of SG stars at z=1.0054 can fully explain these events. The inferred abundance of SG stars is consistent with either (1) a number density of stars with bolometric luminosities beyond the Humphreys-Davidson (HD) limit (L ~ $6\times10^5 L_{\odot}$) that is below 400 stars per sq. kpc, or (2) the absence of stars beyond the HD limit but with a SG number density of ~ 9000 per sq. kpc for stars with luminosities between $10^5$ and $6\times10^5$. This is equivalent to one SG star per 10x10 pc$^2$. We finally make predictions for future observations with JWST's NIRcam. We find that in observations made with the F200W filter that reach 29 mag AB, if cool red SG stars exist at z~1 beyond the HD limit, they should be easily detected in this arc, Comment: 24 pages & 18 figures

We present the first strong-gravitational-lensing analysis of the galaxy cluster RXJ0437.1+0043 (RXJ0437; z = 0.285). Newly obtained, deep MUSE observations, Keck/MOSFIRE near-infrared spectroscopy, and Hubble Space Telescope SNAPshot imaging reveal 13 multiply imaged background galaxies, three of them (at z=1.98, 2.97, and 6.02, respectively) in hyperbolic umbilic (H-U) lensing configurations. The H-U images are located only 20 -- 50 kpc from the cluster centre, i.e., at distances well inside the Einstein radius where images from other lens configurations are demagnified and often unobservable. Extremely rare (only one H-U lens was known previously) these systems are able to constrain the inner slope of the mass distribution -- and unlike radial arcs, the presence of H-U configurations is not biased towards shallow cores. The galaxies lensed by RXJ0437 are magnified by factors ranging from 30 to 300 and (in the case of H-U systems) stretched nearly isotropically. Taking advantage of this extreme magnification, we demonstrate how the source galaxies in H-U systems can be used to probe for small-scale ($\sim 10^{9} M_{\odot}$) substructures, providing additional insight into the nature of dark matter., Comment: 17 pages, 13 figures, 3 tables (including 1 in an appendix). Accepted in MNRAS

We study the population of star-forming clumps in A521-sys1, a $\rm z=1.04$ system gravitationally lensed by the foreground ($\rm z=0.25$) cluster Abell 0521. The galaxy presents one complete counter--image with a mean magnification of $\rm \mu\sim4$ and a wide arc containing two partial images of A521-sys1 with magnifications reaching $\rm \mu>20$, allowing the investigations of clumps down to scales of $\rm R_{eff}<50$ pc. We identify 18 unique clumps with a total of 45 multiple images. Intrinsic sizes and UV magnitudes reveal clumps with elevated surface brightnesses, comparable to similar systems at redshifts $\rm z\gtrsim1.0$. Such clumps account for $\sim40\%$ of the galaxy UV luminosity, implying that a significant fraction of the recent star-formation activity is taking place there. Clump masses range from $\rm 10^6\ M_\odot$ to $\rm 10^9\ M_\odot$ and sizes from tens to hundreds of parsec, resulting in mass surface densities from $10$ to $\rm 10^3\ M_\odot\ pc^{-2}$, with a median of $\rm \sim10^2\ M_\odot\ pc^{-2}$. These properties suggest that we detect star formation taking place across a wide range of scale, from cluster aggregates to giant star-forming complexes. We find ages of less than $100$ Myr, consistent with clumps being observed close to their natal region. The lack of galactocentric trends with mass, mass density, or age and the lack of old migrated clumps can be explained either by dissolution of clumps after few $\sim100$ Myr or by stellar evolution making them fall below the detectability limits of our data., Comment: 17 pages, 12 figures, 3 tables + appendix (6 pages, 6 figures, 2 tables). Accepted for publication in MNRAS

Exploiting the fundamentally achromatic nature of gravitational lensing, we present a lens model for the massive galaxy cluster SMACSJ0723.3-7323 (SMACSJ0723, z=0.388) that significantly improves upon earlier work. Building on strong-lensing constraints identified in prior Hubble Space Telescope (HST) observations, the mass model utilizes 21 multiple-image systems, 17 of which were newly discovered in Early Release Observation (ERO) data from the James Webb Space Telescope (JWST). The resulting lens model maps the cluster mass distribution to an RMS spatial precision of 0.32'' and is publicly available. Consistent with previous analyses, our study shows SMACSJ0723.3-7323 to be well described by a single large-scale component centered on the location of the brightest cluster galaxy. However, satisfying all lensing constraints provided by the JWST data, the model point to the need for the inclusion of an additional, diffuse component west of the cluster. A comparison of the galaxy, mass, and gas distributions in the core of SMACSJ0723 based on HST, JWST, and Chandra data reveals a concentrated regular elliptical profile along with tell-tale signs of a recent merger, possibly proceeding almost along our line of sight. The exquisite sensitivity of JWST's NIRCAM reveals in spectacular fashion both the extended intra-cluster-light distribution and numerous star-forming clumps in magnified background galaxies. The high-precision lens model derived here for SMACSJ0723-7323 demonstrates the unprecedented power of combining HST and JWST data for studies of structure formation and evolution in the distant Universe., Comment: Updated to match the revised version submitted to ApJ - analysis, results and conclusions are unchanged. Link to the lensing outputs: https://github.com/guillaumemahler/SMACS0723-mahler2022

The pressure exerted by the ambient hot X-ray gas on cluster galaxies can lead to the presence of ram pressure stripped (RPS) galaxies, characterized by asymmetric shapes, and, in some cases, tails of blue stars and/or X-ray gas, with increased star formation. We searched for such galaxies in the cluster MS 0451.6-0305 at z~0.5, based on Hubble Space Telescope (HST) imaging covering a region of about 6x6 Mpc^2, an eight magnitude ground-based catalogue with photometric redshifts, and a spectroscopic redshift catalogue. We defined as cluster members a spectroscopic redshift sample of 359 galaxies within 4sigma_v of the mean cluster velocity, and a photometric redshift sample covering the [0.48,0.61] range. We searched for RPS galaxies and tested the error on their classification with a Zooniverse collaboration, and computed the phase space diagram for the spectroscopic sample. We ran the LePhare stellar population synthesis code to analyze and compare the properties of RPS and non-RPS galaxies. We find 56 and 273 RPS candidates in the spectroscopic and photometric redshift samples, respectively, distributed throughout the cluster and tending to avoid high density regions. The phase space diagram gives the percentages of virialized, backsplash, and infall galaxies. RPS galaxy candidates typically show rather high star formation rates, young ages, and relatively low masses. This study confirms that RPS galaxies host, on average, younger stellar populations and strongly form stars when compared with non-RPS counterparts. The fact that RPS candidates with spectroscopic and with photometric redshifts have comparable properties shows that large samples of such objects could be gathered based on multi-band photometry only, a promising result in view of future very large imaging surveys., Comment: Accepted for publication in Astronomy & Astrophysics

We consider the Berglund-H\"ubsch-Henningson-Takahashi duality of Landau-Ginzburg orbifolds with a symmetry group generated by some diagonal symmetries and some permutations of variables. We study the orbifold Euler characteristics, the orbifold zeta functions and the orbifold E-functions of such dual pairs. We conjecture that we get a mirror symmetry between these invariants even on each level, where we call level the conjugacy class of a permutation. We support this conjecture by giving partial results for each of these invariants.

REQUIEM-2D (REsolving QUIEscent Magnified galaxies with 2D grism spectroscopy) is comprised of a sample of 8 massive ($\log M_*/M_\odot > 10.6$) strongly lensed quiescent galaxies at $z\sim2$. REQUIEM-2D combines the natural magnification from strong gravitational lensing with the high spatial-resolution grism spectroscopy of \emph{Hubble Space Telescope} through a spectrophotometric fit to study spatially resolved stellar populations. We show that quiescent galaxies in the REQUIEM-2D survey have diverse formation histories manifesting as a gradient in stellar ages, including examples of (1) a younger central region supporting outside-in formation, (2) flat age gradients that show evidence for both spatially-uniform early formation or inside-out quenching, and (3) regions at a fixed radial distance having different ages (such asymmetries cannot be recovered when averaging stellar population measurements azimuthally). The typical dust attenuation curve for the REQUIEM-2D galaxies is constrained to be steeper than Calzetti's law in the UV and generally consistent with $A_V<1$. Combined together and accounting for the different physical radial distances and formation time-scales, we find that the REQUIEM-2D galaxies that formed earlier in the universe exhibit slow and uniform growth in their inner core, whereas the galaxies that formed later have rapid inner growth in their inner core with younger ages relative to the outskirts. These results challenge the currently accepted paradigm of how massive quiescent galaxies form, where the earliest galaxies are thought to form most rapidly. Significantly larger samples close to the epoch of formation with similar data quality and higher spectral resolution are required to validate this finding., Comment: Submitted to ApJ