We consider a type of distance-regular graph $Γ=(X, \mathcal R)$ called a bilinear forms graph. We assume that the diameter $D$ of $Γ$ is at least $3$. Fix adjacent vertices $x,y \in X$. In our first main result, we introduce an equitable partition of $X$ that has $6D-2$ subsets and the following feature: for every subset in the equitable partition, the vertices in the subset are equidistant to $x$ and equidistant to $y$. This equitable partition is called the $(x,y)$-partition of $X$. By definition, the subconstituent algebra $T=T(x)$ is generated by the Bose-Mesner algebra of $Γ$ and the dual Bose-Mesner algebra of $Γ$ with respect to $x$. As we will see, for the $(x,y)$-partition of $X$ the characteristic vectors of the subsets form a basis for a $T$-module $U=U(x,y)$. In our second main result, we decompose $U$ into an orthogonal direct sum of irreducible $T$-modules. This sum has five summands: the primary $T$-module and four irreducible $T$-modules that have endpoint one. We show that every irreducible $T$-module with endpoint one is isomorphic to exactly one of the nonprimary summands.

For the BCS theory of superconductivity, the electron-phonon interaction is transformed to an attractive electron-electron interaction in the vicinity of the Fermi energy only. At the same time, its formal derivation using a unitary transformation reveals that the electrons attract one another whenever their energies do not differ more than the phonon energy $ω_\mathrm{D}$, independent of closeness to the Fermi energy. Consequently, the order parameter becomes finite even away from the Fermi level. Yet, for small interactions, its magnitude is usually small and can be safely ignored, justifying the BCS approximation. Intriguingly, we find that an accumulation of density-of-states at an energy $\varepsilon_\mathrm{peak}$ in proximity to the Fermi energy induces a significant order parameter magnitude around $\varepsilon_\mathrm{peak}$, which exceeds the one at $E_\mathrm{F}$ for moderate coupling strengths. This strong enhancement is heralded by the softening of an additional collective mode, which resembles a second phase transition. We predict measurable signatures in the thermodynamic and spectroscopic response of this unexpected phenomenon, guiding future experimental searches for it.

This paper introduces differential-geometric methods to study $n$-dimensional locally conformally flat spaces as submanifolds in $\mathbb{R}^{n+2}$. We derive explicit formulas relating intrinsic and ambient differential-geometric objects, including curvature tensors, the codifferential and laplacian operators. We apply this approach to Friedmann-Lemaître-Robertson-Walker (FLRW) spaces using newfound embedding formulas, obtaining new and simplified expressions for the photon propagator in four dimensions.

We survey the localization method for proving inequalities in high dimension, pioneered by Lovász and Simonovits (1993), and its stochastic extension developed by Eldan (2012). The method has found applications in a surprising wide variety of settings, ranging from its original motivation in isoperimetric inequalities to optimization, concentration of measure, and bounding the mixing rate of Markov chains. At heart, the method converts a given instance of an inequality (for a set or distribution in high dimension) into a highly structured instance, often just one-dimensional.

The Johnson-Nyquist noise associated with the intrinsic conductivity of the electron liquid, induces fluctuations of the electron density in charge-neutral graphene devices. In the presence of external electric and magnetic fields, the fluctuations of charge density and electric current induce a fluctuating hydrodynamic flow. We show that the resulting advection of charge produces a fluctuation contribution to the macroscopic conductivity of the system, $σ_{\mathrm{fl}}$, and develop a quantitative theory of $σ_{\mathrm{fl}}$. At zero magnetic field, $σ_{\mathrm{fl}}$ diverges logarithmically with the system size and becomes rapidly suppressed at relatively small fields. This results in giant magnetoresistance of the system.

High-mass stars and star clusters form from the fragmentation of massive dense clumps driven by gravity, turbulence, and magnetic fields. The ALMAGAL project observed $\sim1000$ clumps at $\sim$1000\,au resolution, enabling a statistically significant characterization of this process across a large range of clump physical parameters and evolutionary stages. In this work, we investigated the spatial distribution of dense cores in the 514 massive, potentially cluster-forming, clumps hosting at least 4 cores, to trace fragmentation's initial conditions and early evolution. We used quantitative descriptors, evaluated against the clump bolometric luminosity-to-mass ratio as an indicator of evolution. Core separations were measured with the minimum spanning tree method (MST) and compared with the Jeans gravitational fragmentation theory. We used the $Q$ parameter and the mass segregation ratio, $Λ_{MSR}$, to evaluate whether cores have specific arrangements or differences depending on their masses. ALMAGAL cores are usually arranged in elliptical groups with an axis ratio $e\sim2.2$, but $e\geq$5 is also observed. A single characteristic core separation per clump is found in $\sim76$% of cases, but signatures of multiple fragmentation lengths not rare. Typical core separations are compatible with the clump-averaged thermal Jeans length, $λ^{th}_{J}$, though a population, typical of low-fragmented/young clumps, has wider separations with $l\approx3\timesλ^{th}_{J}$. The core separation decreases on average from $l\sim22000$ au in younger systems to $l\sim7000$ au in more evolved ones. Cores are typically distributed in fractal-type subclusters, with centrally concentrated patterns appearing only at later stages, but without a progressive evolutionary transition. Finally, mass segregation is found in 110 systems, with its occurrence increasing with evolution.

In the wideband paradigm of pulsar timing, the time of arrival of a pulsar pulse is measured simultaneously with the corresponding dispersion measure from a frequency-resolved integrated pulse profile. We present a new method for performing wideband measurements that rigorously accounts for measurement noise. We demonstrate this method using observations of PSR J2124$-$3358 made as part of the Indian Pulsar Timing Array experiment using the upgraded Giant Metre-wave Radio Telescope, and show that our method produces more realistic measurement uncertainty estimates compared to the existing wideband measurement method.

Context. Recent James Webb Space Telescope (JWST) discoveries have unveiled an abundance of faint and massive Active Galactic Nuclei (AGNs) at high redshifts (z=4-9), that surpass by 10 to 100 times the extrapolated bolometric (Bol) and ultraviolet (UV) luminosity functions (LF) from previous AGN campaigns. The two main models that are put forward to explain these observations correspond to light seeds (150 Msol) accreting in episodes of super Eddington, and heavy seeds ($10^3$ - $10^5$ Msol) growing at the Eddington limit. Future gravitational observatories like the Laser Interferometer Satellite Antenna (LISA) will help disentangle these models by reporting the BH-BH merger events from mid to high redshifts. Aims. This work aims to report the predicted merger rates in the heavy seed scenario in light of recent JWST data. In our models we explore (i) instantaneous merging between BHs, (ii) delayed merging after a dynamical timescale, as well as extreme spin configurations (a=0.99, a=-0.99) to bracket BH mass growth. Methods. We use Delphi, a semi-analytical model that tracks baryonic physics over a hierarchical evolution of dark matter halos through cosmic time within the first billion years of the Universe. We calibrate this model for it to simultaneously reproduce galaxy and JWST-AGNs observables. Results. We show reasonable agreement with the Bolometric Luminosity function at z=6, where BHs must accrete 10-100 times more gas than in previous works calibrated to pre-JWST data. However, we underpredict (overpredict) the bright end $10^45.5$ erg s$^-1$ (all luminosity range) at z=7 (z=5) by 1-3.2 dex (0.22-1.6 dex). Regarding BH-BH merger events, the instantaneous (delayed) models predict a total of 28.06 (19.61) yr$^-1$ for BHs at z>=5, which is within the range of merger rates reported in previous literature.

In this paper, we establish a condition on weighted graphs with finite measure that guarantees the validity of a global Poincaré inequality. This condition can be viewed as a discrete analogue of the criterion introduced by J. Boman in 1982 for Whitney cubes, which in turn characterizes the condition originally proposed by F. John in his seminal 1961 work.

[abridged] The previously inaccessible star formation tracer Pa$α$ can now be spatially resolved by JWST NIRCam slitless spectroscopy in distant galaxies up to cosmic noon. In the first study of its kind, we combine JWST NGDEEP NIRISS and FRESCO NIRCam slitless spectroscopy to provide the first direct comparison of spatially resolved dust-obscured (traced by Pa$α$) versus unobscured (traced by H$α$) star formation across the main sequence. We stack Pa$α$ and H$α$ emission-line maps, along with stellar continuum images at both wavelengths of 31 galaxies at 1<z<1.8 in three bins of stellar mass. Surface brightness profiles are measured and equivalent width (EW) profiles computed. Increasing Pa$α$ and H$α$ EW profiles with galactocentric radius across all stellar masses probed provide direct evidence for the inside-out growth of galaxies both via dust-obscured and unobscured star formation for the first time. For galaxies on the main sequence, a weakly positive ($0.1\pm0.1$) Pa$α$/H$α$ line profile as a function of radius is found at $8.8\leqslant\mathrm{log}(M_{*}/\mathrm{M}_{\odot})<9.9$ with a negative ($-0.4\pm0.1$) Pa$α$/H$α$ line profile found at $9.9\leqslant\mathrm{log}(M_{*}/\mathrm{M}_{\odot})<11.0$. Low mass galaxies ($7.7\leqslant\mathrm{log}(M_{*}/\mathrm{M}_{\odot})<8.8$) with high sSFRs are found to have a negative ($-0.5\pm0.1$) Pa$α$/H$α$ line profile gradient. Our results demonstrate that while inside-out growth via star formation is ubiquitous across the main sequence just after cosmic noon, centrally concentrated dust attenuation is not. Along with other recent work in the literature, our findings motivate future studies of resolved SFR profiles in large samples of individual cosmic noon galaxies across the main sequence, to understand the intrinsic scatter in spatially resolved star formation.

A massive ``dark companion'' -- such as an intermediate-mass black hole or other compact dark object -- orbiting the supermassive black hole at the Galactic Center can dynamically reshape the surrounding dark-matter spike. Through gravitational heating and angular-momentum exchange, the companion excavates a ``scoured'' region that lowers the inner density and suppresses the expected annihilation signal. We quantify this effect by computing the suppression of the dark-matter annihilation $J$-factor induced by such a companion, combining an analytic scouring-radius model with full numerical integrations of the modified density profile. We scan the parameter space of companion mass, orbital separation, system age, and spike slope, explicitly including the interplay with the annihilation plateau. \textcolor{black}{For canonical Gondolo--Silk spikes with $γ_{\rm sp} \gtrsim 2$, the scouring radius grows only weakly with injected energy because the binding energy is concentrated at small radii; however, once scouring occurs, the $J$-factor suppression can be substantial, for sufficiently massive or long-lived companions. In the shallower-spike regime with $γ_{\rm sp} \lesssim 2$, the same companion drives $r_{\rm scour}$ to much larger multiples of $R_{\rm core}$, and even a modest ${\sim}10^{4}\,M_\odot$ companion on a $\mathcal{O}(100)\,{\rm AU}$ orbit and $\mathcal{O}({\rm Gyr})$ age can suppress the annihilation flux by one to two orders of magnitude. Whether a steep spike is significantly suppressed therefore depends on the companion's mass and lifetime, not merely on the sign of $(2-γ_{\rm sp})$.} The numerical results are accurately captured (typically at the $\lesssim 10\%$ level) by a simple fitting formula in terms of a dimensionless scouring parameter that measures the ratio between the scoured region and the annihilation core. [...]

We report broadband dielectric spectra of the non-Kramers hexaaluminate PrMgAl\textsubscript{11}O\textsubscript{19}, revealing a pronounced interplay between permittivity and magnetization at cryogenic temperatures. The zero-field dielectric response follows a Barrett-type quantum-paraelectric form, while a broad dielectric anomaly near \SI{5}{K} shows a complex field dependence that mirrors the multi-hump behavior of the magnetic specific heat, evidencing robust magnetoelectric coupling. The inverse permittivity $\varepsilon'^{-1}(T,H)$ scales linearly with $M^2$, consistent with a biquadratic $P^2M^2$ term in a Landau framework. Fits yield temperature-dependent coupling constant $λ(T)$ that decreases with heating from ($1.07\pm0.01)\times10^{-4}\,μ_{\mathrm{B}}^{-2}$ (at 5\,K) to $(4.77\pm0.02)\times10^{-5}\,μ_{\mathrm{B}}^{-2}$ (at 10\,K), reflecting the thermal population of low-lying energy levels of Pr$^{3+}$. Consistently, the uniaxial thermal expansion develops an additional low-temperature hump below $\sim\SI{30}{K}$ that is progressively suppressed by magnetic field, recovering an approximately saturated response by \SI{9}{T}. These results identify PrMgAl\textsubscript{11}O\textsubscript{19} as a paradigmatic non-Kramers hexaaluminate where quantum paraelectricity and magnetoelectric interactions are intrinsically entangled, establishing hexaaluminates as a tunable platform for magnetoelectric physics in frustrated quantum materials.

Planet formation simulations consistently predict compact systems of numerous small planets in chains of mean motion resonances formed by planet-disk interaction, but transiting planet surveys have found most systems to be non-resonant and somewhat dynamically excited. A scenario in which nearly all of the primordial resonant chains undergo dynamical instabilities and collisions has previously been found to closely match many features of the observed planet sample. However, existing models have not been tested against new observations that show a steep decline in the resonant fraction as a function of stellar age on a timescale of ~100 Myr. We construct a simplified model incorporating Type I migration, growth from embryos, and N-body integrations continued to 500 Myr and use it to generate a synthetic planet population. Nearly all systems exit the disk phase in a resonant configuration but begin slowly diffusing away from the center of the resonance. Dynamical instabilities can arise on timescales of tens or hundreds of Myr, especially when systems formed in disks with a convergent migration trap. In this case, a secondary chain of smaller planets that remained at their birth location eventually breaks, destabilizing the inner resonant chain. We also show that the instability statistics are well modeled by a Weibull distribution, and use this to extrapolate our population to Gyr ages. The close match of our modeled systems to the observed population implies that the high resonance fraction predicted by this class of models is in fact consistent with the data, and the previously-reported overabundance of resonant systems was a consequence of comparing simulations of early evolution to mature Gyr-old systems. This result also suggests that instabilities triggered by disk dissipation or other very early mechanisms are unlikely to be consistent with observed young systems.

Recent cosmological bounds on the sum of neutrino masses, M_nu = sum m_nu, are in tension with laboratory oscillation experiments, making cosmological tests of neutrino free-streaming imperative. In order to study the scale-dependent clustering of massive neutrinos, we develop a fast linear response method, FAST-nu f, applicable to neutrinos and other non-relativistic hot dark matter. Using it as an accurate linear approximation to help us reduce the dynamic range of emulator training data, based upon a non-linear perturbation theory for massive neutrinos, we improve the emulator's accuracy at small M_nu and length scales by a factor of two. We significantly sharpen its momentum resolution for the slowest neutrinos, which, despite their small mass fraction, dominate small-scale clustering. Furthermore, we extend the emulator from the degenerate to the normal and inverted mass orderings. Applying this new emulator, Cosmic-Enu-II, to large halos in N-body simulations, we show that non-linear perturbation theory can reproduce the neutrino density profile in the halo outskirts, 2R_vir < r < 10R_vir , to better than 10%.

Statistical analysis on non-Euclidean spaces typically relies on distances as the primary tool for constructing likelihoods. However, manifold-valued data admits richer structures in addition to Riemannian distances. We demonstrate that simple, tractable models that do not rely exclusively on distances can be constructed on the manifold of symmetric positive definite (SPD) matrices, which naturally arises in brain connectivity analysis. Specifically, we highlight the manifold-valued Mahalanobis distribution, a parametric family that extends classical multivariate concepts to the SPD manifold. We develop estimators for this distribution and establish their asymptotic properties. Building on this framework, we propose a novel ANOVA test that leverages the manifold structure to obtain a test statistic that better captures the dimensionality of the data. We theoretically demonstrate that our test achieves superior statistical power compared to distance-based Fréchet ANOVA methods.

We investigate the environments of massive quiescent galaxies at 3 < z < 5 using the GAlaxy Evolution and Assembly (GAEA) theoretical model. We select galaxies with stellar mass ~10^10.8 Msun and specific star formation rate below 0.3x t_Hubble, yielding in a sample of about 5,000 galaxies within a simulated volume of ~685 Mpc. These galaxies have formation times that cover well the range inferred from recent observational data, including a few rare objects with very short formation time-scales and early formation epochs. Model high-z quiescent galaxies are alpha-enhanced and exhibit a wide range of stellar metallicity, in broad agreement with current observational estimates. Massive high-z quiescent galaxies in our model occupy a wide range of environments, from void-like regions to dense knots at the intersections of filaments. Quiescent galaxies in underdense regions typically reside in halos that collapsed early and grew rapidly at high redshift, though this trend becomes difficult to identify observationally due to large intrinsic scatter in star formation histories. The descendants of high-z massive quiescent galaxies display a broad distribution in mass and environment by z=0, reflecting the stochastic nature of mergers. About one-third of these systems remain permanently quenched, while most rejuvenation events are merger-driven and more common in overdense regions. Our results highlight the diversity of early quiescent galaxies and caution against assuming that all such systems trace the progenitors of present day most massive clusters.

We consider Kogut-Susskind fermions (also known as staggered fermions) in a $(3+1)$-dimensional Hamiltonian formalism and examine a chiral transformation and its associated chiral anomaly. The Hamiltonian of the massless Kogut-Susskind fermion has symmetry under the shift transformations in each space direction $S_k \, (k=1,2,3)$, and the product of the three shift transformations in particular (the odd shifts in general) may be regarded as a unitary discrete chiral transformation, modulo two-site translations. The hermitian part of the transformation kernel $Γ= i S_1 S_2 S_3$ can define an axial charge as $Q_A = (1/2)\sum_x χ^\dagger(x) \left(Γ+Γ^\dagger \right)χ(x)$, which is non-onsite, nonquantized, and commutative with the vector charge, analogous to $\tilde{Q}_A = (1/2) \sum_n ( χ^\dagger_n χ_{n+1} + χ^\dagger_{n+1} χ_{n} )$ for the $(1+1)$ dimensional Kogut-Susskind fermion. However, our $Q_A$ cannot be expressed in terms of any quantized charges in a generalized Onsager algebra. Although $Q_A$ does not commute with the fermion Hamiltonian in general when coupled to background link gauge fields, we show that they become commutative for a class of $U(1)$ link configurations carrying nontrivial magnetic and electric fields. We then verify numerically that the vacuum expectation value of $Q_A$ satisfies the anomalous conservation law of axial charge in the continuum two-flavor theory under an adiabatic evolution of the link gauge field.

Waning immunity and reinfection are critical features of many infectious diseases, but epidemiological models often fail to capture the intricate interaction between an individual's history of immunity and their current infection status; when they do, the approach is usually overly simplistic. We develop a novel dual-age structured model that simultaneously tracks immunity age (time since the last recovery from infection) and infection age (time since infection) to analyze epidemic dynamics under conditions of waning immunity and reinfection. The model is formulated as a system of age-structured partial differential equations that describe susceptible and infected populations stratified by both immunity and infection ages. We derive basic reproduction numbers associated with the model and numerically solve the system using a second-order Runge-Kutta scheme along the characteristic lines. We further extend the model to explore vaccination interventions, specifically targeting individuals according to their immunity age. Numerical results reveal that higher contact rates produce larger amplitude oscillations with longer interepidemic periods. The relationship between initial infection levels and long-term epidemic behavior is nonmonotonic. Vaccination efficiency depends critically on the viral load profile across immunity and infection age, with more pronounced viral load distributions requiring higher vaccination rates for disease elimination. Most efficient vaccination strategies begin with intermediate immunity ages rather than targeting only fully susceptible individuals. The structured dual-age framework provides a flexible approach to analyzing the dynamics of reinfection and evaluating targeted vaccination strategies based on the history of immunity.

The complex task of unraveling the assembly history of the Milky Way is in constant evolution with new substructures identified continuously. To properly validate and characterise the family of galactic progenitors, it is important to take into account all the effects that can shape the distribution of tracers in the Galaxy. First among the often overlooked actors of galactic dynamics is the rotating bar of the Milky Way that can affect orbital tracers in multiple ways. We want to fully characterise the effect of the rotating bar of the Milky Way on the distribution of galactic tracers, provide diagnostics helpful in identifying its effect and explore the implications for the search and identification of substructures. We use the in-house Orbital Integration Tool (OrbIT), built to include the full effect of the bar and exploit its multidimensional output to perform a complete dynamical characterisation of a large sample of carefully selected Milky Way stars with very precise astrometry. We identify conspicuous overdensities in several orbital parameter spaces and verify that they are caused by the bar-induced resonances. We also show how contamination by trapped tracers provides local density enhancements that mimic the clumping usually attributed to genuine substructures. We provide a new and expedite way of identifying resonant loci and, consequently, to estimate the contribution of stars trapped into orbital resonances to phase-space overdensities previously identified as candidate relics of past merging events. Among those analysed here, we found that the detections of Cluster 3 and Shakti seem to have gained a non-negligible boost from resonance-trapped stars. Nyx is the most extreme case, with 70% of assigned member stars lying on resonant orbit, strongly suggesting that it is not the genuine relic of a merger event but an overdensity caused by bar-induced resonances

We study the stellar mass function (SMF) of quiescent and star-forming galaxies and its dependence on morphology in 10 redshift bins at $0.2<z<5.5$ using the COSMOS2025 catalog built from $0.54 \, {\rm deg}^2$ JWST imaging from COSMOS-Web. Galaxies are selected by type using the $NUVrJ$ rest-frame color diagram and classified morphologically by bulge-to-total light ratio ($B/T$). The quiescent SMF shows rapid early build-up, with the most massive systems (${\rm log}(M_{\star}/{\rm M_{\odot}})\gtrsim11$) assembled by $z\sim1$ and evolving little since. The star-forming SMF evolves more slowly, following a mass-evolution scenario where galaxies grow via star formation and quench at the characteristic mass $\log(M^{*}/{\rm M}_{\odot})\sim10.6$. Bulge systems ($B/T>0.6$) dominate the quiescent SMF at ${\rm log}(M_{\star}/{\rm M_{\odot}})>10$ at all redshifts, while disks ($B/T<0.2$) dominate at ${\rm log}(M_{\star}/{\rm M_{\odot}})<9$. However, most bulge-dominated galaxies are star-forming, with their fraction increasing with redshift and decreasing mass, consistent with being progenitors of quiescent bulges. We find evidence for environmental quenching onset at $z\sim3$ from the upturn in the quiescent SMF at ${\rm log}(M_{\star}/{\rm M_{\odot}})<9.5$, contributed by disk-dominated galaxies consistent with satellite quenching that retains disk morphologies. Number densities of ${\rm log}(M_{\star}/{\rm M_{\odot}})>10$ quiescent galaxies are lower than recent literature by $0.1-0.7$ dex, but agree well with simulations at $2<z<3$. At $z>3$, simulations increasingly underpredict observations. Finally, we build an empirical model describing galaxy number density evolution by parametrizing quenching rates, baryon conversion efficiency, and bulge formation. Our model supports a scenario where star-forming galaxies grow central bulges before quenching in massive halos.

Theories of flavor operate at various scales. Recently it has been pointed out that in the context of modular flavor symmetries certain combinations of observables are highly constrained, or even uniquely fixed, by modular invariance and holomorphicity. We find that even in the absence of supersymmetry these combinations are surprisingly immune against quantum corrections. This applies, in particular, to the standard model (SM) and certain 2-Higgs doublet models (2HDMs).

All known proofs of the PCP theorem rely on multiple "composition" steps, where PCPs over large alphabets are turned into PCPs over much smaller alphabets at a (relatively) small price in the soundness error of the PCP. Algebraic proofs, starting with the work of Arora, Lund, Motwani, Sudan, and Szegedy use at least 2 such composition steps, whereas the "Gap amplification" proof of Dinur uses $Θ(\log n)$ such composition steps. In this work, we present the first PCP construction using just one composition step. The key ingredient, missing in previous work and finally supplied in this paper, is a basic PCP (of Proximity) of size $2^{n^ε}$, for any $ε> 0$, that makes $O_ε(1)$ queries. At the core of our new construction is a new class of alternatives to "sum-check" protocols. As used in past PCPs, these provide a method by which to verify that an $m$-variate degree $d$ polynomial $P$ evaluates to zero at every point of some set $S \subseteq \mathbb{F}_q^m$. Previous works had shown how to check this condition for sets of the form $S = H^m$ using $O(m)$ queries with alphabet $\mathbb{F}_q^d$ assuming $d \geq |H|$. Our work improves this basic protocol in two ways: First we extend it to broader classes of sets $S$ (ones closer to Hamming balls rather than cubes). Second, it reduces the number of queries from $O(m)$ to an absolute constant for the settings of $S$ we consider. Specifically when $S = (\{0,1\}^{m/c}_{\leq 1})^c$, we give such an alternate to the sum-check protocol with $O(1)$ queries with alphabet $\mathbb{F}_q^{O(c+d)}$, using proofs of size $q^{O(m^2/c)}$. Our new protocols use the notion of Macaulay bases to extend previously known protocols to these new settings with surprising ease. In doing so, they highlight why these notions from algebra may be of further use in complexity theory.

The small-scale turbulent dynamo is a key mechanism for amplifying galactic magnetic fields, yet the resulting field morphology remains poorly understood. Using 3D driven turbulence simulations across a range of compressibilities, characterised by Mach number, and Minkowski functionals, we quantitatively investigate the morphology of magnetic fields generated by the small-scale turbulent dynamo in both the exponentially growing kinematic stage and the statistically steady saturated stage. In both stages and across all Mach numbers, we find that the magnetic field departs significantly from a Gaussian random field. Magnetic structures are statistically less curved and more interconnected in the saturated stage than in the kinematic stage, with these morphological differences decreasing as compressibility increases. Our work provides a quantitative description of how density fluctuations in turbulence and the back-reaction of amplified magnetic fields via the Lorentz force together shape complex, non-Gaussian magnetic structures, and offers a valuable framework for comparing simulations with both analytical models and polarisation observations.

Second-order nonlinear optical materials enable frequency doubling of light (second-harmonic generation, SHG), which is essential for optoelectronic applications ranging from materials characterization to quantum technologies. However, comparing SHG performance across materials remains challenging as the second-order nonlinear susceptibility $χ^{(2)}$ spans several orders of magnitude and strongly depends on the band gap $E_g$. To address this, we empirically validate a theoretical upper bound on $χ^{(2)}$ using new databases of \textit{ab initio}-computed nonlinear optical (NLO) properties. We then formulate a normalized descriptor, $\hat{d}$, which expresses the NLO response of a material relative to the band gap-dependent physical limit. We show that $\hat{d}$ exhibits a similar distribution across a wide range of band gap energies. This universality supports the use of $\hat{d}$ as a robust, generalizable descriptor for data-driven and chemistry-informed machine learning models of NLO response, enabling accelerated materials discovery and optimization across broad application frequencies.

Understanding the complicated processes that regulate star formation and cause a galaxy to become quiescent is key to our comprehension of galaxy evolution. We used eight well resolved star-forming z$<$ 0.3 galaxies from the UVCANDELS survey, where a total of 10 HST bands including UV follow up in UVIS/F275W allow us to reconstruct the star formation histories (SFHs) of regions across each galaxy. This approach provides a powerful tool to explore the spatio-temporal connection between star formation and galaxy evolution. The spatial and temporal profiles of stellar mass and star formation rate surface density were obtained from the SFHs of these regions. We measure scaling relations and projected radial profiles of regions within each galaxy at the time of observation and at 1 Gyr lookback time, noting possible trends in the evolution. By comparing the change in star formation over time we can infer the timing and location of star formation and see early signs of star formation shut off before quenching occurs. We compared the star formation rate density -- stellar mass density scaling relations for individual galaxies as they evolve from 1 Gyr lookback time. The correlation lines pivot around a log-stellar mass surface density of 7.25 [$M_\odot$ $kpc^{-2}$] may be evidence of a self-regulating process on these scales. Radial profiles of galaxy Log sSFR show an overall decrease over 1 Gyr, but five galaxies show a greater change in Log sSFR at the outskirts than the center indicating a possible early onset of quenching in these galaxies.

Robust impact monitoring of near-Earth objects is an essential task of planetary defense. Current systems such as NASA's Sentry-II, the University of Pisa's CLOMON2, and ESA's Aegis have been highly successful, but independent approaches are essential to ensure reliability and to cross-validate predictions of possible impacts. We present NEOForCE (Near-Earth Objects' Forecast of Collisional Events), a new independent monitoring system for asteroid impact prediction. By relying on orbital solutions from DynAstVO at Paris Observatory and using an original methodology for uncertainty propagation, NEOForCE provides an alternative line of verification for impact assessments and strengthens the overall robustness of planetary defense. As other monitoring systems, NEOForCE samples several thousand virtual asteroids from the uncertainty region and integrates their orbits up to 100 years into the future. Instead of searching for close approaches of the virtual asteroids with the Earth, our system looks for times when the Earth comes close to the realistic uncertainty regions around them, which are mostly stretched along their osculating orbits. We also estimate the maximal impact probability, and only if this value is large enough do we continue to the next step. In this second step, we compute how the original asteroid orbit should be modified so that the new trajectory leads to an Earth impact, which allows us to confirm the possible collision and estimate the impact probability. We tested NEOForCE against NASA's Sentry-II system on five representative asteroids: 2000 SG344, 2005 QK76, 2008 JL3, 2023 DO and 2008 EX5. NEOForCE successfully recovered mostly all possible collisions reported by Sentry-II with impact probabilities above e-7, demonstrating the robustness of our approach. In addition, NEOForCE identified several potential impacts at the e-7 - e-6 level that Sentry-II did not report.

This work introduces and analyzes B-spline approximation spaces defined on general geometric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the parameter domain and are mapped to the physical domain via a geometric parametrization. Both the univariate approximation spaces and the geometric mapping are built using maximally smooth B-splines. We construct two such spaces, employing either the sparse-grid combination technique or the hierarchical subspace decomposition of sparse-grid tensor products, and we prove their mathematical equivalence. Furthermore, we derive approximation error estimates and inverse inequalities that highlight the advantages of sparse-grid tensor products. Specifically, under suitable regularity assumptions on the solution, these spaces achieve the same approximation order as standard tensor product spaces while using significantly fewer degrees of freedom. Additionally, our estimates indicate that, in the case of non-tensor-product domains, stronger regularity assumptions on the solution -- particularly concerning isotropic (non-mixed) derivatives -- are required to achieve optimal convergence rates compared to sparse-grid methods defined on tensor-product domains.

We present a new analysis of cosmic dipole anisotropy using gamma-ray bursts (GRBs) as high-redshift standardizable candles. GRBs are ideal probes for testing the cosmological principle thanks to their high luminosity, wide redshift range, and nearly isotropic sky coverage. For the first time, we employ the luminosity-time (L-T) relation, known in the literature as the bidimensional X-ray Dainotti relation, corrected for redshift evolution, to standardize a sample of 176 long GRBs detected by \textit{Swift}. We test for dipolar modulations in the GRB Hubble diagram using both the Dipole Fit Method and a new approach introduced here, the Anisotropic Residual Analysis Method. Both methods yield consistent results: a dipole amplitude of $A_d \simeq 0.6 \pm 0.2$ pointing towards (RA, DEC) $\approx (134^\circ \pm 30^{\circ}, -36^\circ \pm 21^{\circ})$ (equatorial coordinates). As shown in the Appendix, this corresponds to a boost velocity of the observer with respect to the GRB rest-frame in the antipodal direction from the dipole direction. Extensive isotropy tests and 20,000 Monte Carlo simulations confirm that the detected signal cannot be explained by chance alignments or by the angular distribution of the GRB sample. We also show how, by incorporating a dipole term, residual correlations are eliminated, showing that the dipole model provides a better fit than standard isotropic $Λ$CDM.

We explore the observational prospects for detecting gravitational lensing induced by cosmological matter currents, a relativistic correction to the standard density lensing effect arising from the motion of matter. We propose to isolate this contribution by cross-correlating the weak-lensing convergence field with a reconstructed cosmic momentum field inferred from galaxy redshift surveys. Using numerical simulations, we demonstrate that this reconstructed momentum field is uncorrelated with the density lensing signal, enabling a clean separation of the gravitomagnetic component. We then forecast the detectability of this signal for upcoming wide-field galaxy and weak-lensing surveys, showing that a statistically significant detection may be achievable under realistic observational conditions. Such a measurement would provide the first direct probe of the large-scale cosmic momentum field, offering a novel test of general relativity and Lorentz invariance on cosmological scales.

Causal mediation analysis in cluster-randomized trials (CRTs) is essential for explaining how cluster-level interventions affect individual outcomes, yet it is complicated by interference, post-treatment confounding, and hierarchical covariate adjustment. We develop a Bayesian nonparametric framework that simultaneously accommodates interference and a post-treatment confounder that precedes the mediator. Identification is achieved through a multivariate Gaussian copula that replaces cross-world independence with a single dependence parameter, yielding a built-in sensitivity analysis to residual post-treatment confounding. For estimation, we introduce a nested common atoms enriched Dirichlet process (CA-EDP) prior that integrates the Common Atoms Model (CAM) to share information across clusters while capturing between- and within-cluster heterogeneity, and an Enriched Dirichlet Process (EDP) structure delivering robust covariate adjustment without impacting the outcome model. We provide formal theoretical support for our prior by deriving the model's key distributional properties, including its partially exchangeable partition structure, and by establishing convergence guarantees for the practical truncation-based posterior inference strategy. We demonstrate the performance of the proposed methods in simulations and provide further illustration through a reanalysis of a completed CRT.