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Let $(M,d)$ be a complete metric space and let $\mathcal{F}(M)$ denote the Lipschitz-free space over $M$. We develop a ``Choquet theory of Lipschitz-free spaces'' that draws from the classical Choquet theory and the De Leeuw representation of elements of $\mathcal{F}(M)$ (and its bidual) by positive Radon measures on $\beta\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y) \in M \times M$, $x \neq y$. We define a quasi-order $\preccurlyeq$ on the positive Radon measures on $\beta\widetilde{M}$ that is analogous to the classical Choquet order. Rather than in the classical case where the focus lies on maximal measures, we study the $\preccurlyeq$-minimal measures and show that they have a host of desirable properties. Among the applications of this theory is a solution (given elsewhere) to the extreme point problem for Lipschitz-free spaces.

In a paper from 1987, Gruenhage defined a class of topological spaces that now bear his name, and used it to solve a problem of Talagrand on the existence of dense $G_\delta$ metrizable subsets of Gul'ko compact spaces. Gruenhage's paper became highly influential among researchers in renorming theory, a branch of Banach space theory. In this paper we survey Gruenhage and related spaces, and their interactions with renorming theory.

We prove that every element of a Lipschitz-free space admits an expression as a convex series of elements with compact support. As a consequence, we conclude that all extreme points of the unit ball of Lipschitz-free spaces are elementary molecules, solving a long-standing problem. We also deduce that all elements of a Lipschitz-free space with the Radon-Nikod\'ym property can be expressed as convex integrals of molecules. Our results are based on a recent theory of integral representation for functionals on Lipschitz spaces which draws on classical Choquet theory, due to the third named author.

Variable stars in stellar clusters can offer key constraints on stellar evolution and pulsation models, utilising estimates of host cluster properties to constrain stellar physical parameters. We present a catalogue of 86 luminous (F814W<19) variable stars in M31 clusters identified by mining the archival Panchromatic Hubble Andromeda Treasury (PHAT) survey using a combination of statistical analysis of sparse PHAT light curves and difference imaging. We determine the evolutionary phases and initial masses of these variable stars by matching them with theoretical isochrones generated using host cluster properties from the literature. We calculate the probability of PHAT photometry being blended due to the highly crowded nature of cluster environments for each cluster-variable star, using these probabilities to inform our level of confidence in the derived properties of each star. Our 86 cluster-variable stars have initial masses between 0.8--67 $M_{\odot}$. Their evolutionary phases span the main sequence, more evolved hydrogen- and helium-burning phases, and the post-asymptotic giant branch. We identify numerous candidate variable star types: RV Tauri variables, red supergiants and slowly pulsating B-type supergiants, along with Wolf Rayet stars, $\alpha$ Cygni and Mira variables, a classical Cepheid and a possible super-asymptotic giant. We characterise 12 cluster-variable stars at higher confidence based on their difference image quality and lower blending probability. Ours is the first systematic study of variable stars in extragalactic stellar clusters leveraging the superior resolution of the Hubble Space Telescope and demonstrating the unique power of stellar clusters in constraining the fundamental properties of variable stars., Comment: 36 pages, 18 figures; accepted for publication in ApJ

In this paper, we employ a Bayesian approach to assess the reliability of a critical component in the Mars Sample Return program, focusing on the Earth Entry System's risk of containment not assured upon reentry. Our study uses Gaussian Process modeling under a Bayesian regime to analyze the Earth Entry System's resilience against operational stress. This Bayesian framework allows for a detailed probabilistic evaluation of the risk of containment not assured, indicating the feasibility of meeting the mission's stringent safety goal of 0.999999 probability of success. The findings underscore the effectiveness of Bayesian methods for complex uncertainty quantification analyses of computer simulations, providing valuable insights for computational reliability analysis in a risk-averse setting.

The H\"usler-Reiss distribution describes the limit of the pointwise maxima of a bivariate normal distribution. This distribution is defined by a single parameter, $\lambda$. We provide asymptotic theory for maximum likelihood estimation of $\lambda$ under a block maxima approach. Our work assumes independent and identically distributed bivariate normal random variables, grouped into blocks where the block size and number of blocks increase simultaneously. With these assumptions our results provide conditions for the asymptotic normality of the Maximum Likelihood Estimator (MLE). We characterize the bias of the MLE, provide conditions under which this bias is asymptotically negligible, and discuss how to choose the block size to minimize a bias-variance trade-off. The proofs are an extension of previous results for choosing the block size in the estimation of univariate extreme value distributions (Dombry and Ferreria 2019), providing a potential basis for extensions to multivariate cases where both the marginal and dependence parameters are unknown. The proofs rely on the Argmax Theorem applied to a localized loglikelihood function, combined with a Lindeberg-Feller Central Limit Theorem argument to establish asymptotic normality. Possible applications of the method include composite likelihood estimation in Brown-Resnick processes, where it is known that the bivariate distributions are of H\"usler-Reiss form.

This paper critically evaluates the HESA (Higher Education Statistics Agency) Data Report for the Employer Justified Retirement Age (EJRA) Review Group at the University of Cambridge (\cite{CambridgeHESA2024}), identifying significant methodological flaws and misinterpretations. Our analysis reveals issues such as unclear application of data filters, inconsistent variable treatment, and erroneous statistical conclusions. The Report suggests that the EJRA increased job creation rates at Cambridge, but we show Cambridge consistently had lower job creation rates for Established Academic Careers compared to other Russell Group universities, both before and after EJRA implementation in 2011, with no evidence for a significant change in this deficit post implementation. This suggests that EJRA is not a significant factor driving job creation rates. Since other universities without an EJRA exhibit higher job creation rates, this suggests job creation can be sustained without such a policy. We conclude that the EJRA did not achieve its intended goal of increasing opportunities for young academics and may have exacerbated existing disparities compared to other leading universities. We recommend EJRA be abolished at Cambridge since it does not meet its aims and could be viewed as unlawful age discrimination.\newline \it{This version is a revision reflecting some of the comments made by members of the university EJRA review group in public discussion, see \cite{Holmes24}.}, Comment: This is a revision of our earlier paper

Let $\mathrm{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space $(M,d)$ that vanish at a point $0\in M$. We investigate its dual $\mathrm{Lip}_0(M)^*$ using the de Leeuw transform, which allows representing each functional on $\mathrm{Lip}_0(M)$ as a (non-unique) measure on $\beta\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y)\in M\times M$, $x\neq y$. We distinguish a set of points of $\beta\widetilde{M}$ that are "away from infinity", which can be assigned coordinates belonging to the Lipschitz realcompactification $M^{\mathcal{R}}$ of $M$. We define a natural metric $\bar{d}$ on $M^{\mathcal{R}}$ extending $d$ and we show that optimal (i.e. positive and norm-minimal) de Leeuw representations of well-behaved functionals are characterised by $\bar{d}$-cyclical monotonicity of their support, extending known results for functionals in $\mathcal{F}(M)$, the predual of $\mathrm{Lip}_0(M)$. We also extend the Kantorovich-Rubinstein theorem to normal Hausdorff spaces, in particular to $M^{\mathcal{R}}$, and use this to characterise measure-induced and majorisable functionals in $\mathrm{Lip}_0(M)^*$ as those admitting optimal representations with additional finiteness properties. Finally, we use de Leeuw representations to define a natural L-projection of $\mathrm{Lip}_0(M)^*$ onto $\mathcal{F}(M)$ under some conditions on $M$.

On August 9-10, 2023, a workshop was convened at the Pacific Northwest National Laboratory (PNNL) in Richland, WA that brought together a group of internationally recognized experts in metabolomics, natural products discovery, chemical ecology, chemical and biological threat assessment, cheminformatics, computational chemistry, cloud computing, artificial intelligence, and novel technology development. These experts were invited to assess the value and feasibility of a grand-scale project to create new technologies that would allow the identification and quantification of all small molecules, or to decode the molecular universe. The Decoding the Molecular Universe project would extend and complement the success of the Human Genome Project by developing new capabilities and technologies to measure small molecules (defined as non-protein, non-polymer molecules less than 1500 Daltons) of any origin and generated in biological systems or produced abiotically. Workshop attendees 1) explored what new understanding of biological and environmental systems could be revealed through the lens of small molecules; 2) characterized the similarities in current needs and technical challenges between each science or mission area for unambiguous and comprehensive determination of the composition and quantities of small molecules of any sample; 3) determined the extent to which technologies or methods currently exist for unambiguously and comprehensively determining the small molecule composition of any sample and in a reasonable time; and 4) identified the attributes of the ideal technology or approach for universal small molecule measurement and identification. The workshop concluded with a discussion of how a project of this scale could be undertaken, possible thrusts for the project, early proof-of-principle applications, and similar efforts upon which the project could be modeled.

Let $T$ be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrisable space. Let $\mathcal{M}^T$ be the non-empty set of all proper metrics $d$ on $T$ compatible with its topology, and equip $\mathcal{M}^T$ with the topology of uniform convergence, where the metrics are regarded as functions on $T^2$. We prove that the set $\mathcal{A}^{T,1}$ of metrics $d\in\mathcal{M}^T$ for which the Lipschitz-free space $\mathcal{F}(T,d)$ has the metric approximation property is a dense set in $\mathcal{M}^T$, and is furthermore residual in $\mathcal{M}^T$ when $T$ is zero-dimensional. We also prove that if $T$ is uncountable then the set $\mathcal{A}^T_f$ of metrics $d\in\mathcal{M}^T$ for which $\mathcal{F}(T,d)$ fails the approximation property is dense in $\mathcal{M}^T$. Combining the last statement with a result of Dalet, we conclude that for any `properly metrisable' space $T$, $\mathcal{A}^T_f$ is either empty or dense in $\mathcal{M}^T$., Comment: 12 pages, new results added

We introduce convex integrals of molecules in Lipschitz-free spaces $\mathcal{F}(M)$ as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of $\mathcal{F}(M)$ that are induced by Radon measures on $M$, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of $\mathcal{F}(M)$. In particular, we show that all elements of $\mathcal{F}(M)$ are convex series of molecules when $M$ is uniformly discrete and identify all extreme points of the unit ball of $\mathcal{F}(M)$ in that case., Comment: v3: Improved results from Sections 3 and 6

Main Belt Comets (MBCs) exhibit sublimation-driven activity while occupying asteroid-like orbits in the Main Asteroid Belt. MBCs and candidates show stronger clustering of their longitudes of perihelion around 15{\deg} than other objects from the Outer Main Belt (OMB). This potential property of MBCs could facilitate the discovery of new candidates by observing objects in similar orbits. We acquired deep r-band images of 534 targeted asteroids using the INT/WFC between 2018 and 2020. Our sample is comprised of OMB objects observed near perihelion, with longitudes of perihelion between 0{\deg} and 30{\deg} and orbital parameters similar to knowns MBCs. Our pipeline applied activity detection methods to 319 of these objects to look for tails or comae, and we visually inspected the remaining asteroids. Our activity detection pipeline highlighted a faint anti-solar tail-like feature around 2001 NL19 (279870) observed on 2018 November 07, six months after perihelion. This is consistent with cometary activity but additional observations of this object will be needed during its next perihelion to investigate its potential MBC status. If it is active our survey yields a detection rate of $\sim$1:300, which is higher than previous similar surveys, supporting the idea of dynamical clustering of MBCs. If not, it is consistent with previously estimated abundance rates of MBCs in the OMB (<1:500)., Comment: 13 pages, 9 figures, accepted for publication in MNRAS

The Geometric Thin-Film equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon data, admits solutions that are globally defined for all time and are expressible as push-forwards of Borel measurable functions whose behaviour is governed by a set of ordinary differential equations (ODEs). The existence is first demonstrated in the special case of a finite weighted sum of delta functions whose centres evolve over time -- these are known as `particle solutions'. In the general case, we construct a convergent sequence of particle solutions whose limit yields a solution of the above form. Moreover, we demonstrate that all weak solutions constructed in this way are $1/2$-H\"older continuous in time and are uniquely determined by the initial conditions., Comment: 24 pages, 0 figures

Let $\|\cdot\|$ be a norm on $\mathbb{R}^N$ and let $M$ be a closed $C^1$-submanifold of $\mathbb{R}^N$. Consider the pointed metric space $(M,d)$, where $d$ is the metric given by $d(x,y)=\|x-y\|$, $x,y\in M$. Then the Lipschitz-free space $\mathcal{F}(M)$ has the Metric Approximation Property., Comment: 19 pages

We present a new self-contained convergence analysis of the particle method that can be applied to a range of PDEs, including the Camassa-Holm equation. It is a development of the analysis of Chertock, Liu and Pendleton, which used compactness properties of spaces of functions having bounded variation. In our analysis we establish solutions by applying a metric Arzel\`a-Ascoli compactness result to a space of measure-valued functions equipped with the bounded Lipschitz metric. All the convergence and regularity results of the previous analysis follow as a consequence and are computationally easier to establish., Comment: 14 pages, 0 figures

We present DEIMoS: Data Extraction for Integrated Multidimensional Spectrometry, a Python application programming interface (API) and command-line tool for high-dimensional mass spectrometry data analysis workflows that offers ease of development and access to efficient algorithmic implementations. Functionality includes feature detection, feature alignment, collision cross section (CCS) calibration, isotope detection, and MS/MS spectral deconvolution, with the output comprising detected features aligned across study samples and characterized by mass, CCS, tandem mass spectra, and isotopic signature. Notably, DEIMoS operates on N-dimensional data, largely agnostic to acquisition instrumentation; algorithm implementations simultaneously utilize all dimensions to (i) offer greater separation between features, thus improving detection sensitivity, (ii) increase alignment/feature matching confidence among datasets, and (iii) mitigate convolution artifacts in tandem mass spectra. We demonstrate DEIMoS with LC-IMS-MS/MS data to illustrate the advantages of a multidimensional approach in each data processing step.

Planets are observed to orbit the component star(s) of stellar binary systems on so-called circumprimary or circumsecondary orbits, as well as around the entire binary system on so-called circumbinary orbits. Depending on the orbital parameters of the binary system a planet will be dynamically stable if it orbits within some critical separation of the semimajor axis in the circumprimary case, or beyond some critical separation for the circumbinary case. We present N-body simulations of star-forming regions that contain populations of primordial binaries to determine the fraction of binary systems that can host stable planets at various semimajor axes, and how this fraction of stable systems evolves over time. Dynamical encounters in star-forming regions can alter the orbits of some binary systems, which can induce long-term dynamical instabilities in the planetary system and can even change the size of the habitable zone(s) of the component stars. However, the overall fraction of binaries that can host stable planetary systems is not greatly affected by either the assumed binary population, or the density of the star-forming region. Instead, the critical factor in determining how many stable planetary systems exist in the Galaxy is the stellar binary fraction - the more stars that are born as singles in stellar nurseries, the higher the fraction of stable planetary systems., Comment: 13 pages, 7 figures, 2 short appendices, accepted for publication in MNRAS. A video abstract created by Helena Gibbon is available here: https://youtu.be/76kANnTK9-s