We study the amplitude-constrained additive white Gaussian noise channel. It is well known that the capacity-achieving input distribution for this channel is discrete and supported on finitely many points. The best known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $A$ and upper-bounded by a term of order $A^2$, where $A$ denotes the amplitude constraint. It was conjectured in [1] that the linear scaling is optimal. In this work, we establish a new lower bound of order $A\sqrt{\log A}$, improving the known bound and ruling out the conjectured linear scaling. To obtain this result, we quantify the fact that the capacity-achieving output distribution is close to the uniform distribution in the interior of the amplitude constraint. Next, we introduce a wrapping operation that maps the problem to a compact domain and develop a theory of best approximation of the uniform distribution by finite Gaussian mixtures. These approximation bounds are then combined with stability properties of capacity-achieving distributions to yield the final support-size lower bound.

Motivated by the work of Scully \textit{et al.} [ \textcolor{blue}{Proc. Nat. Acad. Sci. 115, 8131 (2018)}] and Camblong \textit{et al.}[ \textcolor{blue}{Phys. Rev. D 102, 085010 (2020)}], we investigate horizon-brightened acceleration radiation (HBAR) for freely falling two-level atoms in the geometry of a Bardeen regular black hole. Building on the quantum-optics approach to acceleration radiation and its near-horizon conformal quantum mechanics (CQM) structure, we show that the dominant physics is again governed by an inverse-square potential in the radial Klein-Gordon equation, with an effective coupling fixed by the Bardeen surface gravity. Using geodesic expansions and a near-horizon CQM reduction of the scalar field, we derive the excitation probability for atoms falling through a Boulware-like vacuum in the presence of a stretched-horizon mirror. The resulting spectrum is Planckian in the mode frequency, with a temperature determined by the Bardeen Hawking temperature. We analyze how the regular core parameter controls the strength of the radiation and demonstrate that the excitation probability is strongly suppressed as the geometry approaches the extremal (cold remnant) limit. Numerical results illustrate the dependence of the spectrum on the Bardeen parameter and on the atomic transition frequency.

The use of machine learning models in system identification has increased due to their ability to approximate complex nonlinear dynamics with high accuracy. However, often it is not clear how the performance of trained models scales with given resources such as data, compute, and model size. To allow for a better understanding of the scalability of the performance of machine learning models, we verify neural scaling laws (NSLs) in the context of system identification from input-state-output data using different evaluation metrics for accuracy and different system architectures, including input-affine and physics-informed port-Hamiltonian representations. Our verified NSLs can help to forecast performance improvements and guide model design or data acquisition.

We introduce two novel numerical approaches for computing Feynman integrals based on their complete monotonicity (CM) and Stieltjes properties. The first method uses that scalar Feynman integrals are CM, meaning that all their derivatives have a fixed sign, in the Euclidean kinematic region. This imposes strong constraints on the function space. Simultaneously, these integrals obey systems of linear differential equations with respect to kinematic parameters. By imposing that the solutions to these differential equations satisfy complete monotonicity across the Euclidean region, we develop an efficient and highly constraining numerical bootstrap method. We provide a proof of principle of the power of our approach by applying it to a class of multi-loop Feynman integrals with internal masses. The second method is based on a refinement of CM. We prove that Feynman integrals, within a certain range of parameters, such as dimension and propagator exponents, are not only CM but in fact Stieltjes functions. The latter can be described efficiently by Padé approximants that are known to converge in the cut complex plane. This means that these representations are valid also in analytically continued kinematics, such as physical scattering regions. These insights allow us to obtain rational approximations to Feynman integrals from minimal information, such as a Taylor expansion about a soft limit. We demonstrate the effectiveness of this method by applying it to a 20-loop banana-type Feynman integral. Finally, we comment on a number of extensions of these novel avenues for computing Feynman integrals.

Halbach spheres provide a theoretically elegant means of generating highly homogeneous magnetic fields, but practical implementation is hindered by challenging fabrication and restricted interior access. This study examines discrete spherical Halbach configurations assembled from permanent magnets placed at the vertices of Platonic and Archimedean solids. Analytical calculations, numerical field simulations, and experimental measurements indicate that polyhedra with icosahedral symmetry achieve the most favorable balance among field strength, homogeneity, and interior accessibility. They produce exceptionally flat fourth-order central saddle points, resulting in a usable homogeneous field volume up to a factor of 260 larger than that of traditional Halbach disk or cylindrical arrays. Several magnet assemblies composed of cubical NdFeB magnets are fabricated and their three dimensional field distributions characterized, demonstrating homogeneous regions of up to several cubic centimeters with deviations below 1%. The findings establish discrete icosahedrally symmetric magnet arrays as practical, scalable building blocks for compact, highly homogeneous magnetic field sources suited to mobile magnetic resonance, and magnetophoretic applications.

We examine the observational discrepancies of two widely used models describing anisotropic (dark) matter distributions around a black hole, focusing on their photon spheres, shadow radii, and lensing observables. The models considered are the vacuum and Einstein cluster dark matter models, characterized by negative and zero radial pressure, respectively. The analysis reveals that these models display contrasting photon sphere behaviors. In particular, the Einstein cluster results in a more pronounced deviation in the shadow radius relative to the standard Schwarzschild black hole. Additionally, a distinctive lensing phenomenon associated with the matter halo is identified in both models.

We study extensions of the classic \emph{Line Cover} problem, which asks whether a set of $n$ points in the plane can be covered using $k$ lines. Line Cover is known to be NP-hard, and we focus on two natural generalizations. The first is \textbf{Line Clustering}, where the goal is to find $k$ lines minimizing the sum of squared distances from the input points to their nearest line. The second is \textbf{Hyperplane Cover}, which asks whether $n$ points in $\mathbb{R}^d$ can be covered by $k$ hyperplanes. We also study the more general \textbf{Projective Clustering} problem, which unifies both settings and has applications in machine learning, data analysis, and computational geometry. In this problem, one seeks $k$ affine subspaces of dimension $r$ that minimize the sum of squared distances from the given points in $\mathbb{R}^d$ to the nearest subspace. Our results reveal notable differences in the parameterized complexity of these problems. While Line Cover is fixed-parameter tractable when parameterized by $k$, we show that Line Clustering is W[1]-hard with respect to $k$ and does not admit an algorithm with running time $n^{o(k)}$ unless the Exponential Time Hypothesis fails. Hyperplane Cover has been known to be NP-hard since the 1980s, following work of Megiddo and Tamir, even for $d=2$, we show that it remains NP-hard even when $k=2$. Finally, we present an algorithm for Projective Clustering running in $n^{O(dk(r+1))}$ time. This bound matches our lower bound for Line Clustering and generalizes the classic algorithm for $k$-Means Clustering ($r=0$) by Inaba, Katoh, and Imai [SoCG 1994].

Stencil mask lithography is an advanced technique for fully in-situ fabricating Josephson junctions, which is increasingly being used for multi-terminal Josephson junctions. This study provides information on the optimal mask design and mask reliability. For this, 270 mask designs were systematically fabricated and investigated under scanning electron microscope. Reliable statements are made about mask yield, minimal dimensions, and systematic dependencies on the number of superconducting terminals. We find that stencil mask lithography can be used reliably for fabricating multi-terminal Josephson junctions, enabling lateral mask dimensions down to 40$\,$nm on average.

The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$, in function of the word norm of that element $g$. It has been studied for several classes of finitely generated groups, including free groups, linear groups and virtually abelian groups. This paper shows that if $G$ is virtually nilpotent, then $\text{RF}_G = \log^δ$ for some $δ\in \mathbb{N}\cup\{0\}$, with moreover an explicit formula for $δ$ in terms of Lie algebras. This implies in particular that it is an invariant of the complex Mal'cev completion, leading to the application that residual finiteness growth is a profinite invariant for virtually nilpotent groups.

Nitrogen-vacancy (NV) centers in diamond provide a versatile quantum sensing platform for biological imaging through magnetic field detection, offering unlimited photostability and the ability to perform long-term observations without photobleaching or phototoxicity. However, conventional stage-top incubators are incompatible with the unique requirements for NV widefield magnetometry to study cellular dynamics. Here, we present a purpose-built compact incubation platform that maintains precise environmental control of temperature, CO$_2$ atmosphere, and humidity while accommodating the complex constraints of NV widefield microscopy. The system employs a 3D-printed biocompatible chamber with integrated heating elements, temperature control, and humidified gas flow to create a stable physiological environment directly on the diamond sensing surface. We demonstrate sustained viability and proliferation of HT29 colorectal cancer cells over 90 hours of continuous incubation, with successful magnetic field imaging of immunomagnetically labeled cells after extended cultivation periods. This incubation platform enables long-term cultivation and real-time monitoring of biological samples on NV widefield magnetometry platforms, opening new possibilities for studying dynamic cellular processes using quantum sensing technologies.

In the framework of three flavor neutrino oscillation, we demonstrate that the measures of entanglement can be expressed in terms of experimentally accessible appearance and disappearance probabilities. We explicitly show here that the genuine tripartite entanglement measure, i.e., the tangle vanishes identically for all flavors signifying that three flavor neutrino system form a W-type entangled state. Further, we investigate alternative measures of tripartite entanglement like the partial tangle and the concurrence fill which capture the total sharing of entanglement beyond pairwise correlations. In terms of bipartite and bi-partitioned entanglement measures, we derive the symmetric invariant and the concurrence fill, which quantify the distributed entanglement completely expressible in terms of flavor transition probabilities. These entanglement measures display distinct energy dependent patterns across the oscillation window which can be experimentally accessible in the long baseline experiments like DUNE providing an alternative quantum information perspective on flavor evolution. We use GLobal Long Baseline Experiment Simulator (\textsf{GLoBES}) simulations within the DUNE set up to investigate these tripartite entanglement measures in terms of neutrino energy and the length of the baseline. It is observed that, at the point of maximal mixing, these measures show near maximal entanglement between the muon and the tau flavor modes establishing entanglement monogamy. Within the DUNE set up, the wide band of energy and expected higher sensitivity to CP-violation at second oscillation maximum provide a unique advantage to explore the quantum correlation effects across a broader energy window.

Dynamic Treatment Regimes (DTRs) provide a systematic framework for optimizing sequential decision-making in chronic disease management, where therapies must adapt to patients' evolving clinical profiles. Inverse probability weighting (IPW) is a cornerstone methodology for estimating regime values from observational data due to its intuitive formulation and established theoretical properties, yet standard IPW estimators face significant limitations, including variance instability and data inefficiency. A fundamental but underexplored source of inefficiency lies in the strict alignment requirement between observed and target treatment trajectories, which fails to account for partial compatibility and discards substantial information from individuals with only minimal deviations from the regime. We propose two novel methodologies that relax the strict inclusion rule through flexible compatibility mechanisms. Both methods provide computationally tractable alternatives that can be easily integrated into existing IPW workflows, offering more efficient approaches to DTR estimation. Theoretical analysis demonstrates that both estimators preserve consistency while achieving superior finite-sample efficiency compared to standard IPW, and comprehensive simulation studies confirm improved stability. We illustrate the practical utility of our methods through an application to HIV treatment data from the AIDS Clinical Trials Group Study 175 (ACTG175).

Supported nanoparticle catalysts are widely used in the chemical industry. Computational modeling of supported nanoparticles based on density functional theory (DFT) often involves structural searches of stable local minimum energy configurations and molecular dynamics simulations at finite temperature. These are computationally demanding tasks that are intractable within DFT for large systems. In the last two decades, machine learning interatomic potentials (MLIPs) have been successfully used to substantially increase the size and time scales accessible to simulations approximating DFT accuracy. However, training reliable MLIPs is non-trivial as it requires many costly DFT calculations. Recently, several universal MLIPs (uMLIPs) have been developed, which are trained on large datasets that cover a wide range of molecules and materials. Here, we benchmark the accuracy and the efficiency of these uMLIPs in describing Cu nanoparticles supported on Al$_2$O$_3$ surfaces against our domain-specific DP-UniAlCu model. We find that the MACE-OMAT can reproduce reasonably well the low-energy structures found in global optimization at an energy accuracy comparable to DP-UniAlCu. Interestingly, the MatterSim-v1.0.0-1M model, which exhibits larger deviations in the binding energies, can find even more stable configurations than the other two models in some supported nanoparticle sizes, showing its capability in structure exploration. For MD simulations, MACE-OMAT and MatterSim-v1.0.0-1M can qualitatively reproduce the mean-squared displacements of Cu atoms (MSD$_\mathrm{Cu}$) predicted by DP-UniAlCu, albeit at roughly two orders of magnitude higher cost. We demonstrate that the uMLIPs can be very useful in simulating supported nanoparticles even without any fine-tuning, though their reduced efficiency remains a limiting factor for large-scale simulations.

The existence of Anderson localization, characterized by vanishing diffusion due to strong disorder, has been demonstrated in numerous ways. A systematic approach based on the Anderson quantum model of the Fermi gas in random lattices that can describe both diffusive and localized regimes has not yet been fully established. We build on a recent publication \cite{Janis:2025ab} and present a microscopic theory of disordered electrons that covers both the metallic phase with extended Bloch waves and the localized phase, where a propagating particle forms a quantum bound state with the hole left behind at the origin. The general theory provides a framework for constructing controlled approximations to one- and two-particle Green functions that satisfy the necessary conservation laws and causality requirements across the full range of disorder strength. It is used explicitly to derive a local, mean-field-like approximation for the two-particle irreducible vertices, enabling quantitative analysis of the solution's dynamic properties in both metallic and localized phases, including critical behavior at the mobility edge. A new instability line for the dynamical electron-hole correlation function of the metallic phase is introduced.

Skew-spectra allow us to extract non-Gaussian information by taking the square of a map and finding the power spectrum of this new map with the original map. This allows us to use much of the infrastructure of power spectra and avoid the intricacies of estimating three point statistics. In this paper we present the first extension of skew-spectra to arbitrary spin-$s$ fields, as a means to extract non-Gaussian information efficiently from cosmological data sets like cosmic shear or CMB polarization. We apply the formalism to weak lensing in the context of large scale structure, and discuss different ways of combining fields to build skew-spectra, all while avoiding the problems associated with mass-mapping. We provide plots of these new statistics for $Λ$CDM and vary cosmological parameters.

Signed networks provide a principled framework for representing systems in which interactions are not merely present or absent but qualitatively distinct: friendly or antagonistic, supportive or conflicting, excitatory or inhibitory. This polarity reshapes how we think about structure and dynamics in complex systems: a negative tie is not simply a missing positive one but a constraint that generates tension, and possibly asymmetry. Across disciplines, from sociology to neuroscience and machine learning, signed networks provide a shared language to formalise duality, balance, and opposition as integral components of system behaviour. This review provides a comprehensive and foundational summary of signed network theory. It formalises the mathematical principles of signed graphs and surveys signed-network-specific measures, including signed degree distributions, clustering, centralities, motifs, and Laplacians. It revisits balance theory, tracing its cognitive and structural formulations and their connections to frustration. Structural aspects of signed networks are examined, analysing key topics such as null models, node embeddings, sign prediction, and community detection. Subsequent sections address dynamical processes on and of signed networks, such as opinion dynamics, contagion models, and data-driven approaches for studying evolving networks. Practical challenges in constructing, inferring and validating signed data from real-world systems are also highlighted, and we offer an overview of currently available datasets. We also address common pitfalls and challenges that arise when modelling or analysing signed data. Overall, this review integrates theoretical foundations, methodological approaches, and cross-domain examples, providing a structured entry point and a reference framework for researchers interested in the study of signed networks in complex systems.

Histopolation is the approximation procedure that associates a degree $ d-1 $ polynomial $ p_{d-1} \in \mathscr{P}_{d-1} (I) $ with a locally integrable function $ f $ imposing that the integral (or, equivalently, the average) of $p$ coincides with that of $f$ on a collection of $ d $ distinct segments $s_i$. In this work we discuss unisolvence and conditioning of the associated matrices, in an asymptotic linear algebra perspective, i.e., when the matrix-size $d$ tends to infinity. While the unisolvence is a rather sparse topic, the conditioning in the unisolvent setting has a uniform behavior: as for the case of standard Vandermonde matrix-sequences with real nodes, the conditioning is inherently exponential as a function of $d$ when the monomial basis is chosen. In contrast, for an appropriate selection of supports, the Chebyshev polynomials of second kind exhibit a bounded conditioning. A linear behavior is also observed in the Frobenius norm.

A search for signatures of a dark analog to quantum chromodynamics is performed. The analysis targets long-lived dark mesons that decay into standard-model particles, with a high branching fraction of the dark mesons decaying into muons. The dark mesons are formed by the hadronisation of dark partons, which are produced by a decay of the Higgs boson. The search is performed using a data set corresponding to an integrated luminosity of 41.6 fb$^{-1}$, which was collected in proton-proton collisions at $\sqrt{s}$ = 13 TeV by the CMS experiment at the CERN LHC in 2018 using non-prompt muon triggers. The search is based on resonant muon pair signatures. Machine-learning techniques are employed in the analysis, utilising boosted decision trees to discriminate between signal and background. No significant excess is observed above the standard model expectation. Upper limits on the branching fraction of the Higgs boson decaying to dark partons are determined to be as low as 10$^{-4}$ at 95% confidence level, surpassing and extending the existing limits on models with dark $\tildeω$ mesons for mean proper decay lengths of less than 500 mm and for $\tildeω$ masses down to 0.3 GeV. First limits are set for extended dark-shower models with two dark flavours that contain dark photons, probing their masses down to 0.33 GeV.

Measuring the ionizing photon production efficiency $ξ_{\mathrm{ion,0}}$ -- the ratio of ionizing photon output rate $Q_{\rm H^0}$ to UV continuum luminosity $L_{\rm UV}$ -- in galaxies at $z > 6$ is crucial for constraining their contribution to cosmic reionization. We present integrated and spatially resolved measurements of $ξ_{\mathrm{ion,0}}$ for 12 exceptionally bright ($M_\mathrm{UV} \sim -22$ mag) star-forming galaxies at $z \sim 7$ from the REBELS survey. These measurements are based on JWST NIRSpec/IFU PRISM spectroscopy, probing the rest-frame UV and optical regime. Notably, in 8 of the 12 galaxies, the spectral coverage includes H$α$, enabling self-consistent dust attenuation estimates in both the ionized gas and stellar continuum via the Balmer decrement and rest-UV slope, respectively. We find global $\logξ_{\mathrm{ion,0}}$ values ranging from $25.19\pm0.11$ to $25.61\pm0.11$, with a weighted mean of $25.44\pm0.15$, consistent with the canonical value of $\sim25.3$. Using a sample of 25 star-forming clumps within these galaxies, we explore local variations in LyC production efficiency, finding a broader range, from $24.52\pm0.21$ to $26.18\pm0.61$. We identify strong correlations between $ξ_{\mathrm{ion,0}}$ and specific star formation rate, star formation surface density, H$β$ equivalent width, and stellar mass. Clumps with the highest $ξ_{\mathrm{ion,0}}$ exhibit $\mathrm{EW}_0(\mathrm{H}β) \ge 150$ Angstrom, consistent with young stellar ages. From previous Ly$α$ measurements in three galaxies, we estimate a typical Ly$α$ escape fraction of $f_{\rm esc, Lyα} \sim 2\%$, suggesting similar or lower escape fractions for LyC photons. Combining this with our H$α$ measurements, we infer ionized bubble sizes $\sim 1$ pMpc, aligned with expectations from Ly$α$-detected systems and reionization models.

Hybrid metric-Palatini gravity unifies the metric and Palatini formalisms while preserving a propagating scalar degree of freedom, offering a compelling route to modified gravity consistent with current observations. Motivated by this success, we consider an extended framework -- the hybrid metric-Palatini scalar-tensor (HMPST) theory -- in which an additional scalar field $ϕ$ modulates the curvature couplings, enriching the dynamics and enabling nontrivial self-interactions through scalar potentials. We focus on the analytically tractable linear-$f(\hat{R})$ subclass and study its cosmological, strong-field, and weak-field regimes. In homogeneous and isotropic settings, we identify de Sitter and matter-dominated cosmological solutions describing accelerated expansion and early-universe behavior. For static, spherically symmetric configurations, the field equations yield analytic solutions generalizing the Janis-Newman-Winicour and Buchdahl metrics, including the Schwarzschild-de Sitter limit. In the weak-field regime, linearized perturbations around Minkowski space lead to Yukawa-type corrections to the gravitational potential, with an effective Newton constant $G_{\rm eff}$ and post-Newtonian parameter $γ$ that recover General Relativity for heavy or weakly coupled scalars. These results show that the linear-$f(\hat{R})$ HMPST subclass provides a consistent and unified description of gravity across cosmological, astrophysical, and Solar System scales, offering a fertile framework for connecting modified gravity to observations and effective field-theoretic extensions.

Static analysis tools provide a powerful means to detect security vulnerabilities by specifying queries that encode vulnerable code patterns. However, writing such queries is challenging and requires diverse expertise in security and program analysis. To address this challenge, we present QLCoder - an agentic framework that automatically synthesizes queries in CodeQL, a powerful static analysis engine, directly from a given CVE metadata. QLCode embeds an LLM in a synthesis loop with execution feedback, while constraining its reasoning using a custom MCP interface that allows structured interaction with a Language Server Protocol (for syntax guidance) and a RAG database (for semantic retrieval of queries and documentation). This approach allows QLCoder to generate syntactically and semantically valid security queries. We evaluate QLCode on 176 existing CVEs across 111 Java projects. Building upon the Claude Code agent framework, QLCoder synthesizes correct queries that detect the CVE in the vulnerable but not in the patched versions for 53.4% of CVEs. In comparison, using only Claude Code synthesizes 10% correct queries. QLCoder code is available publicly at https://github.com/neuralprogram/QLCoder.

It is established that black holes have entropy and behave as thermodynamical systems. Associating entropy to gravitational fields has not remained limited to black holes, necessitating the notion of the second law of thermodynamics in gravitating systems. There have been many ideas and attempts to prove the second law in gravitational systems starting from first principles. Within the covariant phase space formalism, we define gravitational entropy as the charge associated with the local boosts, detaching the gravitational entropy from horizons or trapped surfaces. Our definition encompasses and generalizes the existing notions of entropy. Using this definition for the Einstein gravity case, we compute variations of the entropy along the path of any causal observer and establish that the entropy variations are always non-negative if the matter content satisfies the strong energy condition integrated along any segment of the causal path.

We propose a level-set-based semi-Lagrangian method on graded adaptive Cartesian grids to address the problem of surface reconstruction from point clouds. The goal is to obtain an implicit, high-quality representation of real shapes that can subsequently serve as computational domain for partial differential equation models. The mathematical formulation is variational, incorporating a curvature constraint that minimizes the surface area while being weighted by the distance of the reconstructed surface from the input point cloud. Within the level set framework, this problem is reformulated as an advection-diffusion equation, which we solve using a semi-Lagrangian scheme coupled with a local high-order interpolator. Building on the features of the level set and semi-Lagrangian method, we use quadtree and octree data structures to represent the grid and generate a mesh with the finest resolution near the zero level set, i.e., the reconstructed surface interface. The complete surface reconstruction workflow is described, including localization and reinitialization techniques, as well as strategies to handle complex and evolving topologies. A broad set of numerical tests in two and three dimensions is presented to assess the effectiveness of the method.

The reduced cross section of the semiexclusive $(l,l'p)$ lepton scat tering process irrespective of the type of interaction is determined mainly by bound nucleon momentum distribution in target and nucleon final state interaction with residual nucleus. These cross sections can be identified with distorted nuclear spectral functions and therefore are similar up to Coulomb c orrections for neutrino and electron scattering on nuclei. In this article we exploit this similarity and use data with precise kinematics and large statistics for semiexclusive electron scattering on oxygen target to test models employed in the GENIE neutrino event generator. We find that these models can not reproduce well the measured reduced cross sections in all allowed kinematic region and the GENIE event generator needs to better describe both the nuclear ground states and nucleon final state interaction. The approach presented in this paper provides a great opportunity to test better the accuracy of nuclear models of quasielastic neutrino-nucleus scattering, employed in neutrino event generators.

We investigate vortex dipoles on surfaces of variable negative curvature, focusing on a catenoid of arbitrary throat radius as a concrete example. We construct the effective dynamical system including mutual and geometric self-interaction terms and show that the resulting Hamiltonian dynamics makes dipoles follow catenoid geodesics, in agreement with recent works, Gustafsson (J. Nonlinear Sci. 32, 62, 2022) and by Drivas, Glukhovskiy and Khesin (Int. Math. Res. Not. 2024, 14, 10880-10894). We utilize the symplectic structure to find a conserved momentum map J related to the U(1) symmetry along the azimuthal direction. We verify the conservation of both the Hamiltonian and this momentum for arbitrary throat radius. We then demonstrate direct and exchange scattering of classical vortices on the catenoid, and we contrast this with the collective rotational motion (with azimuthal drift) that arises for chiral pairs. Finally, we build a finite-dipole dynamical system on the catenoid and show that the self-propulsion terms emerge to leading order in the dipole size. This provides a concrete realization, on a curved minimal surface, of the intuitive statement that a finite dipole propels orthogonal to the dipole axis, with a speed modulated by curvature.

Causal discovery is the subfield of causal inference concerned with estimating the structure of cause-and-effect relationships in a system of interrelated variables, as opposed to quantifying the strength or describing the form of causal effects. As interest in causal discovery builds in fields such as ecology, public health, and environmental sciences where data are regularly collected with spatial and temporal structures, approaches must evolve to manage autocorrelation and complex confounding. As it stands, the few proposed causal discovery algorithms for spatiotemporal data require summarizing across locations, ignore spatial autocorrelation, and/or scale poorly to high dimensions. Here, we introduce our developing framework that extends time-series causal discovery to systems with spatial structure, building upon work on causal discovery across contexts and methods for handling spatial confounding in causal effect estimation. We close by outlining remaining gaps in the literature and directions for future research.

We consider expansions for the kernels of operator functions of second-order minimal operators on a curved background. We show that the terms of these expansions originate in the ultraviolet or infrared regions. We propose a systematic approach to obtaining ultraviolet terms using term-by-term integration of the DeWitt expansion of the heat kernel. We discuss two methods for regularizing infrared divergences arising at intermediate computational steps -- using analytic continuation and introducing a mass term -- and the relationship between them.

The electromagnetic production of a dilepton pair in the muon - ion scattering, usually denoted muon trident process, is investigated considering the feasibility of studying processes induced by muons at LHC using its far-forward detectors. The total and differential cross - sections are estimated taking into account of the Bethe-Heitler and bremsstrahlung channels, and predictions for the event rates expected for muon - tungsten ($μW$) collisions at the LHC energies are presented. Our results indicate that the observation of the muon trident process is feasible at FASER$ν$. In particular, our results indicate that the electromagnetic production of $τ^+ τ^-$ pairs in the $μW$ scattering can, in principle, be observed for the first time ever at the LHC. In addition, we present our predictions for the production of QED bound states. Finally, results for the trident process at the FASER$ν$2 detector are also presented.

Preliminary astrometric data from the fourth data release of the $Gaia$ mission revealed a 33 M$_{\odot}$ dark companion to a metal-poor red giant star, deemed $Gaia$ BH3. This system hosts both the most massive known stellar-origin black hole and the lowest-metallicity star yet discovered in orbit around a black hole. The formation pathway for this peculiar stellar-black hole binary system has yet to be determined, with possible production mechanisms that include isolated binary evolution and dynamical capture. The chemical composition of the stellar companion in $Gaia$ BH3 (hereafter \bhstar) can help constrain the potential formation mechanisms of this system. Here, we conduct the most comprehensive chemical analysis of \bhstar\ to date using high resolution spectra obtained by the Tull Coudé Spectrograph on the 2.7m Harlan J. Smith Telescope at McDonald Observatory to constrain potential formation mechanisms. We derived 29 elemental abundances ranging from lithium to thorium and find that \bhstar\ is an $α$-enriched ([$α$/Fe] = 0.41), r-I neutron-capture star ([Eu/Fe] = 0.57). We conclude that \bhstar\ shows no chemical peculiarities (defined as deviations from the expected chemical pattern of an r-I halo red giant) in any elements, which is in alignment with both the dynamical capture and isolated binary evolution formation scenarios. With an upper limit detection on Th, we use the Th/Eu chronometer to place limits on the cosmochronometric age of this system. These observations lay the groundwork for heavy-element chemical analysis for subsequent black hole and low-metallicity stellar binaries that will likely be found in $Gaia$ DR4.

In the March 2025 issue of Pour la Science, Jean-Paul Delahaye described a wonderful solution to the following problem: How many ways can you divide a 3 by 2n rectangle into two connected, congruent pieces? We show that this problem can be solved by the transfer matrix method, and demonstrate this by computing the generating function for the number of ways to divide a 4 by n rectangle into two connected, congruent parts.