It is well known that the Grauert-Riemenschneider canonical sheaf $\mathcal{K}_X$ of holomorphic square-integrable $n$-forms is a central tool in $L^2$-theory for the $\overline\partial$-operator on a singular complex space $X$ of pure dimension $n$. It was shown a few years ago that a comprehensive $L^2$-theory requires also the study of the sheaf $\mathcal{K}_X^s$ of holomorphic square-integrable $n$-forms with a Dirichlet boundary condition at the singular set of $X$. In the present paper, we describe and classify the behaviour of $\mathcal{K}_X^s$ in isolated canonical Gorenstein singularities, and give applications to the $L^2$-theory for the $\overline\partial$-operator on such spaces.

Numerical software depends on fast, accurate implementations of mathematical primitives like sin, exp, and log. Modern superoptimizers can optimize floating-point kernels against a given set of such primitives, but a more fundamental question remains open: which new primitives are worth implementing in the first place? We formulate this as numerical library learning: given a workload of floating-point kernels, identify the mathematical primitives whose expert implementations would most improve speed and accuracy. Our key insight is that numerical superoptimizers already have the machinery well-suited to this problem. Their search procedures happen to enumerate candidate primitives, their equivalence procedures can generalize and deduplicate candidates, and their cost models can estimate counterfactual utility: how much the workload would improve if a given primitive were available. We present GrowLibm, which repurposes the Herbie superoptimizer as a numerical library learner. GrowLibm mines candidate primitives from the superoptimizer's intermediate search results, ranks them by counterfactual utility, and prunes redundant candidates. Across three scientific applications (PROJ, CoolProp, and Basilisk), GrowLibm identifies compact, reusable primitives that can be implemented effectively using standard numerical techniques. When Herbie is extended with these expert implementations, kernel speed improves by up to 2.2x at fixed accuracy, and maximum achievable accuracy also improves, in one case from 56.0% to 93.5%. We also prototype an LLVM matcher that recognizes learned primitives in optimized IR, recovering 26 replacement sites across five PROJ projections and improving end-to-end application performance by up to 5%.

We propose Uni-ArrayDPS, a novel diffusion-based refinement framework for unified multi-channel speech enhancement and separation. Existing methods for multi-channel speech enhancement/separation are mostly discriminative and are highly effective at producing high-SNR outputs. However, they can still generate unnatural speech with non-linear distortions caused by the neural network and regression-based objectives. To address this issue, we propose Uni-ArrayDPS, which refines the outputs of any strong discriminative model using a speech diffusion prior. Uni-ArrayDPS is generative, array-agnostic, and training-free, and supports both enhancement and separation. Given a discriminative model's enhanced/separated speech, we use it, together with the noisy mixtures, to estimate the noise spatial covariance matrix (SCM). We then use this SCM to compute the likelihood required for diffusion posterior sampling of the clean speech source(s). Uni-ArrayDPS requires only a pre-trained clean-speech diffusion model as a prior and does not require additional training or fine-tuning, allowing it to generalize directly across tasks (enhancement/separation), microphone array geometries, and discriminative model backbones. Extensive experiments show that Uni-ArrayDPS consistently improves a wide range of discriminative models for both enhancement and separation tasks. We also report strong results on a real-world dataset. Audio demos are provided at \href{https://xzwy.github.io/Uni-ArrayDPS/}{https://xzwy.github.io/Uni-ArrayDPS/}.

With the advanced capabilities of ground- and space-based telescopes, exoplanet science is beginning to characterize the physics and chemistry of exoplanet atmospheres. However, interpreting exoplanet observations requires sophisticated modeling tools that rely on laboratory data to provide critical inputs and constraints. In preparation for the influx of observational data that the coming decades will bring, laboratory experiments that simulate the diverse conditions expected in exoplanet atmospheres, surfaces and interiors are vital to advancing models and thereby our understanding of these worlds. Here we discuss the key areas where laboratory experiments are providing fundamental data for exoplanet models, particularly for low-mass planets from rocky worlds to sub-Neptunes. First, we present a series of experiments to measure outgassing and volatile solubilities that are essential for establishing the connection between low-mass exoplanet interiors and their observable atmospheres. We then discuss additional laboratory techniques that can be used to understand the interior-atmosphere connection and simulate the high pressure-high temperature conditions of exoplanet interiors. Next, we summarize the experimental methods used to constrain the spectroscopic properties of atmospheric gases and aerosols along with their formation and reaction mechanisms. We also discuss how similar methods can be used to constrain exoplanet surface compositions, which is important for interpreting observations of atmosphere-less worlds. Finally, we conclude by presenting several examples of astrobiology experiments that constrain how life can modify the atmosphere and surface of rocky exoplanets. Together, these laboratory efforts are crucial to maximizing the scientific yield of exoplanet observations in the coming decades.

We discuss first the discovery of the odderon by the TOTEM and D0 collaborations. We then describe the gap between jets measurements sensitive to the high gluon density regime and the possible observation of saturation phenomenon in Pb Pb interactions. We also mention the sensitivity to beyond standard model physics and to the production of axion-like particles via photon photon interactions.

Magnetospheric accretion is the paradigm for accretion in Classical T-Tauri Stars (CTTS). However, the standard, one-flow magnetospheric accretion model fails to replicate important characteristics such as the observed Balmer decrements. We address this limitation by adopting a model with two axisymmetric magnetospheric accretion flows of different accretion rates and geometries. We calculate the fluxes of the hydrogen $H_α$, $H_β$, and $H_γ$ lines of each flow with the magnetospheric accretion model and use Bayesian statistics to fit the Balmer line fluxes of 139 CTTS in the Orion OB1b subassociation, and in the Upper Scorpius, Lupus and Chamaeleon I star-forming regions. We find that the Balmer decrements and line fluxes can be fitted by two distinct but coexisting flows: a compact, high accretion rate flow, close to the star and narrow (mean inner radius $R_i \sim 2.9 R_*$ and mean width $ΔR \sim 0.7 R_*$), covering a few percent of the emitting area, and a more spread out flow, thicker ($ΔR \sim 1.2 R_*$), and larger ($R_i \sim 3.7 R_*$), with lower accretion rate, encompassing the rest of the emitting area. The two-flow model can also reproduce the empirical correlation between the luminosity in $H_α$ and the accretion luminosity. Overall, our findings suggest that a multicolumn approach provides a more accurate representation of the observed Balmer line emission, in agreement with results of numerical simulations.

Stochastic resetting -- the intermittent restart of random processes -- has profoundly reshaped first-passage theory, providing a mechanism to control and optimize completion times. While the influence of resetting on mean first-passage times is now well understood, its impact on absorption probabilities in confined domains remains comparatively unexplored. We present a complete analysis of the classical gambler's ruin problem under geometric resetting. At each time step, the walker is reset to its initial position with probability gamma, or otherwise performs a biased nearest-neighbor step. Our approach proceeds in three stages. First, we derive a renewal equation for the ruin probability q_z(gamma) by conditioning on the first step. Second, we develop a spectral representation on a weighted Hilbert space that diagonalizes the transition operator and yields explicit closed-form expressions. Third, this representation enables a precise critical-point analysis in state space. Our central result is a striking geometric invariance: when the domain size a is even, the ruin

We investigate the renormalization group flow and beta functions of Yang-Mills theory and adjoint QCD in a strong, stable, self-dual background field $F$. In deep UV, theory runs according to the standard beta function, $β_0$. Treating the background as a superselection sector, we find that the theory abelianizes below the scale $\sqrt{F}$ and remains strictly abelian in the deep infrared. In the intermediate weakly-coupled regime ($Λ_{\rm YM} \ll μ\lesssim \sqrt{F}$), the gauge coupling remarkably continues to run despite the absence of propagating charged degrees of freedom. Because all non-zero Landau levels decouple, this running is driven exclusively by exact zero modes, resulting in an integer-quantized beta function coefficient, $\widetilde β_0$. Finally, we conjecture that this abelian dynamics is governed by an emergent non-commutative effective field theory that is free of pathological UV/IR mixing.

Charge transport in polymers, such as polyacrylonitrile (PAN), is crucial for electronics and energy storage. For instance, PAN can transport cations e.g., Li+, by facilitating dynamic cation-nitrile coordination in batteries. However, little is known regarding the underlying role of complex reactive polymer configurations. Herein, we develop a deep-learning potential, trained on ab initio energies and forces of nonequilibrium reactive PAN configurations, to unravel the kinetics of PAN cyclization initiated by a nucleophile (OH- dissociated from LiOH) attacking the terminal nitrile carbon. We find, based on the reaction free-energetics, rates, and charge analysis, that the nucleophile attack producing the first ring is the rate-limiting step, which subsequently triggers Li+-coupled electron transfer along the PAN backbone, causing ~10,000 times faster sequential ring-formation of the remaining nitriles. PAN's extended configurations, where dipolar and H-bonding interactions are minimal, enable such rapid kinetics. By validating our computational findings with IR and NMR experiments, we establish a pathway for designing reactive polymers with enhanced charge transport for energy applications.

We aim to search for satellites and circumplanetary or circumsubstellar disks around directly imaged substellar companions, exploring their immediate environment to constrain the conditions for satellites and disk formation. We conducted a dedicated survey of twelve planets and brown dwarfs with VLT/SPHERE using a novel application of the star hopping technique. By building libraries of contemporaneous point spread function (PSF) references from nearby stars, we applied a frame-by-frame subtraction of each companion's flux using the Negative Fake Companion method (NEGFC). This approach mitigates temporal PSF variability and enhances sensitivity to faint circumplanetary features. We derived contrast curves, translated them into mass detection limits using evolutionary models, and constrained the dynamically stable regions through estimates of Hill radii from orbital fits. Our analysis yields stringent limits on the presence of massive satellites, generally excluding companions more massive than a few Jupiter masses at separations beyond 1-5 au, depending on each system's Hill radius. In most cases, no convincing point-like or extended residuals were found. However, we identify promising signals for three systems: extended residuals consistent with a circumplanetary disk around CT Cha b, tentative repeated residuals near TYC 8047-232-1 B that may trace a bound satellite companion of 3-6 MJup, and marginal residual signals at the location of the previously reported candidate around DH Tau b, whose interpretation, however, remains uncertain due to possible contamination by instrumental effects. These results confirm the power of star hopping in reducing PSF-related artifacts and provide some of the most stringent constraints to date on the mass and location of potential satellites and disks around directly imaged substellar companions.

Optimization algorithms can be interpreted through the lens of dynamical systems as the interconnection of linear systems and a set of subgradient nonlinearities. This dynamical systems formulation allows for the analysis and synthesis of optimization algorithms by solving robust control problems. In this work, we use the celebrated internal model principle in control theory to structurally factorize convergent composite optimization algorithms into suitable network-dependent internal models and core subcontrollers. As the key benefit, we reveal that this permits us to synthesize optimization algorithms even if information is transmitted over networks featuring dynamical phenomena such as time delays, channel memory, or crosstalk. Design of these algorithms is achieved under bisection in the exponential convergence rate either through a nonconvex local search or by alternation of convex semidefinite programs. We demonstrate factorization of existing optimization algorithms and the automated synthesis of new optimization algorithms in the networked setting.

Libor Šnobl and Pavel Winternitz classified all of the Lie algebras of dimension six and smaller. Using this classification, we formulated and proved structure constant formulas for the universal enveloping algebras of the nilpotent Lie algebras of dimension five and less.

In many scientific applications, hypotheses are generated and tested continuously in a stream. We develop a framework for improving online multiple testing procedures with false discovery rate (FDR) control under arbitrary dependence. Our approach is two-fold: we construct methods via the online e-closure principle, as well as a novel formulation of online compound e-values that is defined through donations. This yields strict power improvements over state-of-the-art e-value and p-value procedures while retaining FDR control. We further derive algorithms that compute the decision at time $t$ in $O(\log t)$ time, and we demonstrate improved empirical performance on synthetic and real data.

We investigated if signatures of coronal mass ejections (CMEs) can be retrieved in the differential emission measure (DEM) from Sun-as-a-star extreme-ultraviolet (EUV) observations. We analyzed 16 major, eruptive (i.e., associated with CMEs) flares. For each flare we constructed light curves of the average intensity of full-disk images taken by the SDO/AIA mission in EUV channels centered at 94, 131, 171, 193, 211, and 335 Angstrom. We also corrected the light curves for the flare gradual phase. From the analysis of the light curves we find that all the studied flares exhibit dimmings, where the intensity decreases with respect to the pre-flare phase, mainly in the 171, 193, and 211 Angstrom channels. The dimmings in these channels become more pronounced upon applying the gradual-phase correction. Calculation of the DEM from the six AIA EUV channels shows that during all the observed dimmings, the DEM decreases with respect to its value in the pre-flare phase in the temperature range $10^{5.7}-10^{6.3}$ K. The signature of the dimming is more pronounced in the range $10^{5.7}-10^{6.0}$ K for the DEMs calculated with the original light curves, and in the range $10^{6.0}-10^{6.3}$ K for the DEMs calculated by taking into account the gradual-phase correction. For a sample event, we also calculated DEMs from EVE and spatially resolved AIA observations of the source region to assess the impact of spectral resolution and full-disk averaging. For both these cases the temperature range where the dimming in the DEM is more pronounced is similar to that resulting from the analysis of the spatially averaged AIA data; the magnitude of the dimming is similar for the EVE and larger for the spatially resolved AIA observations. Coronal dimmings associated with CMEs can be detected in Sun-as-a-star DEMs. The flare gradual phase can lead to an underestimation of the magnitude of the dimming.

Neutral-atom quantum simulation is susceptible to entanglement between the atom's internal electronic state and its center-of-mass position. In many alkali Rydberg platforms, the 'spin-motion coupling' is exacerbated by the free expansion required to avoid ponderomotive anti-trapping from optical fields. A recent proposal (arXiv:2505.01071) claims sufficiently excited Rydberg states could be trapped in a monochromatic, far-detuned, circularly polarized optical field by harnessing a large vector polarizability. We disprove the proposal through analytic calculation and measurement of the vector polarizability of the $54S$, $54P$, and $53D$ orbitals of Cesium. Regarding the optical angular frequency $ω$, we analytically derive that the scalar, vector, and tensor polarizabilities scale as $ω^{-2}$, $ω^{-3}$, and $ω^{-4}$, as opposed to the proposed scaling of $ω^{-2}$, $ω^{-1}$, and $ω^{-2}$. We refine the sum-over-states expression for vector and tensor polarizability to be numerically stable and predict negligible vector and tensor polarizabilities far detuned from resonances, in agreement with our measurements. However, we find vector polarizability can enhance a recent proposal for near-detuned attractive trapping. Furthermore, we evaluate the breakdown of the electric-dipole approximation and derive no effect stronger than ponderomotive repulsion. We conclude that an attractive, monochromatic, far-detuned optical trap for alkali Rydberg states is not possible, regardless of the beam geometry.

Expander (Tanner) codes combine sparse graphs with local constraints, enabling linear-time decoding and asymptotically good distance--rate tradeoffs. A standard constraint-counting argument yields the global-rate lower bound $R\ge 2r-1$ for a Tanner code with local rate $r$, which gives no positive-rate guarantee in the low-rate regime $r\le 1/2$. This regime is nonetheless important in applications that require algebraic local constraints (e.g., Reed--Solomon locality and the Schur-product/multiplication property). We introduce \emph{Algebraic Expander Codes}, an explicit algebraic family of Tanner-type codes whose local constraints are Reed--Solomon and whose global rate remains bounded away from $0$ for every fixed $r\in(0,1)$ (in particular, for $r\le 1/2$), while achieving constant relative distance. Our codes are defined by evaluating a structured subspace of polynomials on an orbit of a non-commutative subgroup of $\mathrm{AGL}(1,\mathbb{F})$ generated by translations and scalings. The resulting sparse coset geometry forms a strong spectral expander, proved via additive character-sum estimates, while the rate analysis uses a new notion of polynomial degree and a polytope-volume/dimension-counting argument.

Precision agriculture promises higher yields and sustainability, but adoption is slowed by the high cost of cyber-physical systems (CPS) and the lack of systematic design methods. We present a cost-aware design space exploration (DSE) framework for multimodal drone-rover platforms to integrate budget, energy, sensing, payload, computation, and communication constraints. Using integer linear programming (ILP) with SAT-based verification, our approach trades off among cost, coverage, and payload while ensuring constraint compliance and a multitude of alternatives. We conduct case studies on smaller and larger-sized farms to show that our method consistently achieves full coverage within budget while maximizing payload efficiency, outperforming state-of-the-art CPS DSE approaches.

Causal discovery aims to infer causal relationships among variables from observational data, typically represented by a directed acyclic graph (DAG). Most existing methods assume independent and identically distributed observations, an assumption often violated in modern applications. In addition, many datasets contain a mixture of continuous and discrete variables, which further complicates causal modeling when dependence across samples is present. To address these challenges, we propose a de-correlation framework for causal discovery from dependent mixed data. Our approach formulates a structural equation model with latent variables that accommodates both continuous and discrete variables while allowing correlated Gaussian errors across units. We estimate the dependence structure among samples via a pairwise maximum likelihood estimator for the covariance matrix and develop an EM algorithm to impute latent variables underlying discrete observations. A de-correlation transformation of the recovered latent data enables the use of standard causal discovery algorithms to learn the underlying causal graph. Simulation studies demonstrate that the proposed method substantially improves causal graph recovery compared with applying standard methods directly to the original dependent data. We apply our approach to single-cell RNA sequencing data to infer gene regulatory networks governing embryonic stem cell differentiation. The inferred regulatory networks show significantly improved predictive likelihood on test data, and many high-confidence edges are supported by known regulatory interactions reported in the literature.

Quantum mode parameter estimation determines parameters governing the shape of electromagnetic modes occupied by a quantum state of radiation. Canonical examples, time delays and frequency shifts, underpin radar, lidar, and optical clocks. A comprehensive framework recently established that broad families of quantum states can attain the Heisenberg limit, surpassing any classical strategy. This raises a fundamental question: among all quantum-enhanced strategies, which is truly optimal? Answering this requires identifying physically meaningful resources governing each estimation task, so quantum states can be compared on equal footing. We show these resources are connected to the eigenmode basis of the generator of the relevant mode transformation. For time-shift estimation, whose generator is diagonal in the frequency domain, the pertinent resources are the mean frequency and bandwidth; analogous quantities emerge for other transformations. Our framework unifies two historically separate perspectives: the particle-number aspect and the mode-structure of quantum light, providing a coherent picture of quantum-enhanced sensing with multimode radiation. Within this unified framework, we derive a tight upper bound on the quantum Fisher information for multimode Gaussian states, expressed in terms of these natural resources, and analytically identify the optimal Gaussian states saturating it. These optimal states take a particularly transparent form in the generator eigenbasis, a structural simplicity reflecting the deep connection between the geometry of the mode transformation and the architecture of the optimal probe. We further demonstrate that multimode homodyne detection constitutes the optimal measurement, achieving this bound and completing the end-to-end characterization of optimal quantum metrology strategies for mode parameter estimation.

The thermal evolution and interior structure of giant exoplanets are sensitive to the treatment of radiative opacity. At temperatures of ~2000 K, depletion of alkali metals can create a window of reduced opacity, potentially giving rise to deep radiative zones. While such zones have been discussed for Jupiter, their role in the evolution and characterisation of warm giant exoplanets has not been systematically investigated. We investigate how opacity windows and the resulting deep radiative zones affect the cooling, radius evolution, and the characterisation of interiors and atmospheres of giant exoplanets. We computed thermal evolution models for warm Jupiters spanning masses of 0.3 to 1.0 Jupiter masses with equilibrium temperatures of 200 to 800 K, with a parametrised reduction of the radiative opacity near ~2000 K. Deep radiative zones develop in moderately irradiated Jupiters older than ~4 Gyr even with unmodified opacities, and earlier and more extensively when the opacity is reduced. A deep opacity window accelerates the planetary cooling, reducing predicted radii by up to 5% and interior temperatures on the order of a few 10%. We show that this translates to a ~10 percentage point difference in inferred bulk metallicity. Deep radiative zones are likely common in warm giant exoplanets and could decouple atmospheric composition from bulk interior composition, complicating the interpretation of atmospheric observations. We suggest that the opacity treatment introduces significant uncertainties in atmospheric and interior characterisation.

Atomic structures of nanomaterials are inherently dynamic, continuously reshaped through interactions with chemical species and external stimuli. Such dynamics are further amplified as the size and dimensionality of nanomaterials are reduced. Despite advances in analytical methods, it remains challenging to capture structural dynamics of nanomaterials in reactive environments with both atomic spatial resolution and commensurate temporal resolution. Here, we directly visualize atomic-scale dynamics of gold (Au) nanocrystals in reactive liquid environments with millisecond-speed liquid cell electron microscopy (EM) and deep-learning denoising. We uncover reversible fluctuations in local crystallinity of Au nanocrystals dependent on the surrounding chemical environment. These transient fluctuations, driven by interactions at nanocrystal-liquid interfaces, critically influence dissolution kinetics and grain boundary relaxation. By overcoming the spatiotemporal limitations in conventional liquid cell EM, our findings provide insights into how transient nanoscale structures dictate the stability and reactivity of nanomaterials.

Milestone Driven Agile Execution is a hybrid management framework where the empirical control component of agile development is retained but the prioritization of the backlog is done according to a macro or strategic (milestone) plan that drives the execution of the project. MDAX is method agnostic, in the sense that the development approach is not embedded in the execution mechanism but in the plan that drives it. This allows organizations using it to choose the development approach that suites them most,

{Supernova (SN) siblings are powerful tools used to calibrate and improve distance measurement methods, and to make the systematic uncertainty to distances to their host galaxies considerably lower compared to other techniques.} {In this paper we present distance estimates to NGC6951, a galaxy that hosted the Type IIP SN~2020dpw, the Type Ib SN~2021sjt, and three other SNe.} {Photometric observations of the two objects were carried out using two 80cm RC telescopes located in Hungary, while spectra were obtained from the LCO and the WiseRep database. For the distance estimates, we applied the expanding photosphere method (EPM), which connects the observed angular radius ($θ$) of a SN to its physical radius and is related to the velocity of the photosphere ($v_{\rm ph}$). Although the EPM is mostly applied to derive the distance of Type IIP SNe, in the literature there are several examples of this technique being used for Type IIn and stripped-envelope SNe as well. Therefore, we made another attempt to infer the distance of the Type Ib SN~2021sjt by applying the EPM together with its Type IIP sibling SN~2020dpw. } {Our analysis resulted in a distance of $25.76 \pm 0.34 (\rm random) \pm 5.51$ (systematic) Mpc and $24.57 \pm 1.27 (\rm random) \pm 4.64$ (systematic) Mpc for SN~2020dpw and SN~2021sjt, respectively. Systematic errors were estimated with respect to the used dilution factor, the interstellar reddening, and the date of the explosion (which was fixed to a value between the last non-detection and the first detection for each object).} {The obtained distance values agree with each other and with the literature, which shows the validity of the methods used. In this way, new and perhaps improved distance estimates to NGC 6951 were obtained, and the applicability of the EPM for Type Ib SNe was tested.}

Superconductivity in twisted graphene is probed by tunneling spectroscopy and superfluid stiffness, two observables that access the same order parameter from complementary perspectives. We show that a finite-momentum pair-density-wave (PDW) state, consistent with reported Kekulé signatures, reconciles substantial low-energy tunneling weight with an approximately $T^2$ suppression of the low-temperature superfluid stiffness. The PDW order produces a Bogoliubov Fermi surface and finite zero-bias conductance. The same gapless quasiparticles also enter the geometric superfluid response, yielding a low-temperature stiffness suppression that persists in the flat-band limit. We further predict that, under density or displacement-field tuning, enhanced residual zero-bias conductance should accompany reduced low-temperature stiffness, providing a direct experimental link between tunneling spectroscopy and phase rigidity in twisted graphene.

Topological semimetals offer a rich platform for exploring massless fermion physics and realizing exotic properties with potential technological applications. GdPS, a magnetic semiconductor derived from the nodal-line semimetal ZrSiS family, exhibits a field-induced insulator-to-metal transition driven by exchange splitting. This transition is accompanied by an unusual, isotropic, and gigantic negative magnetoresistance, attributed to negligible magnetic anisotropy resulting from the weak spin-orbit coupling of half-filled Gd3+ 4f orbitals and light S atoms. In this work, we investigate Se substitution, which is expected to enhance spin-orbit coupling. Indeed, we observe slightly increased magnetic anisotropy in magnetotransport. Moreover, Se substitution suppresses the field-induced insulator-to-metal transition, likely due to an enlarged band gap that demands a higher exchange splitting to close. These findings provide deeper insights into the interplay between spin-orbit coupling, magnetic anisotropy, and transport behavior in GdPS, offering guidance for future materials design for desired functionalities.

We study semilinear elliptic equations \begin{equation*} \begin{cases} -Δu = f(u) & \text{in } Ω, \\ \partial_νu = 0 & \text{on } \partialΩ, \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in bounded domains. A classical result by Casten-Holland and Matano shows that stable nonconstant solutions cannot exist in convex domains, although unstable spatial patterns may still occur. In this paper we investigate rigidity properties of classical solutions without imposing stability assumptions and aim to identify structural conditions on the nonlinearity ensuring that all solutions are constant. We prove that every classical solution of the Neumann problem is constant provided the nonlinearity satisfies a suitable `monotonicity' condition, which includes the cases where the nonlinearity has a fixed sign or changes sign in a controlled way around one of its zeros. This yields a rigidity result depending solely on the structure of the nonlinearity and does not require convexity assumptions on the domain. We also discuss the sharpness of our assumptions by constructing examples of nonlinearities for which nonconstant solutions exist. In particular, inspired by the approach of Lin-Ni-Takagi, we consider exponential-type nonlinearities in dimension $N=2$, and show that when a parameter crosses a critical threshold, the associated Neumann problem admits nontrivial and nonconstant solutions for sufficiently small diffusion.

We study primordial black hole (PBH) formation in a minimally coupled $f(T)$ teleparallel cosmology that generates a transient departure from standard radiation domination. The model is constructed so that modified-gravity effects are negligible at early and late times, but become dynamically relevant over a finite epoch, during which an effective torsion component reduces the total equation-of-state parameter below 1/3.We show that this transient softening lowers the collapse threshold for overdensities at horizon re-entry, leading to an exponential enhancement of PBH formation. In addition, the modified background alters the relation between temperature and horizon mass, producing a localized feature in the PBH mass function. For representative parameters, PBHs with asteroid-scale masses can account for a significant fraction, or even the entirety, of dark matter for perturbation amplitudes $σ^2 \sim \mathcal{O}(10^{-3})$, while remaining consistent with current constraints. Our results demonstrate that modified gravity alone can efficiently generate PBHs, without requiring ad hoc modifications of the radiation sector.

The development of intense high-energy radiation sources and the improvement of techniques for detecting charged fragments have made possible experiments on multiple ionization of a molecule with registration of the momentum and charge of dissociation products in coincidence. This technique allows to determine (`fix') a molecular geometry at the time of fragmentation and called fixed-in-space molecule. In this work, the dynamics the water molecule dissociation fragments resulting from interaction with intense X-ray radiation has been studied. The charge distribution of oxygen ions was calculated, Newton diagrams were constructed for fragments - protons and the oxygen ion - for various charge states of the latter, and the released kinetic energy was evaluated. Calculations were performed using the [1]code for parameters close to [2]. The predictions for the different pulse parameters are done.

Micro-segmentation as a core requirement of zero trust architecture (ZTA) divides networks into small security zones, called micro-segments, thereby minimizing impact of security breaches and restricting lateral movement of attackers. Existing approaches for Industrial Internet of Things (IIoT) networks often remain centralized, static, or difficult to interpret. These limitations are critical in IIoT, where devices are heterogeneous, communication behavior evolves over time, and raw data sharing across sites is often undesirable. Accordingly, we propose EFAH-ZTM, an Explainable Federated Autoencoder-Hypergraph framework for Zero Trust micro-segmentation in IIoT networks. The framework includes a trained federated DNAE that learns behavioral embeddings from distributed clients. kNN-based and Manifold-based hypergraphs capture higher-order relationships among device-flow instances. To generate micro-segments, MiniBatch KMeans and HDBSCAN clustering techniques are applied on the spectral embeddings, while an operational risk score that combines reconstruction error and structural outlierness drives allow/block policy decisions. Trustworthiness of the policy decision is improved through feature-level explanations using LIME and SHAP. Experiments on the WUSTL-IIoT-2021 dataset show that HDBSCAN achieved the strongest structural quality, while the manifold-based hypergraph produces the best oracle-aligned security efficacy that reaches a purity of 0.9990 with near-zero contamination. Similarly, the explainability module also showed high fidelity and stability, with surrogate classifier having an accuracy of 0.9927 and stable explanations across runs. Moreover, an ablation analysis shows that the federated learning preserves competitive segmentation quality relative to centralized training, and the hypergraph modeling significantly improves structural separation and risk stratification.

Full waveform inversion is an ill-posed inverse problem whose solution non-uniqueness -- i.e., arising from band-limited, finite-aperture, noisy data -- calls for uncertainty quantification to avoid overconfident geological interpretations. Bayesian inference addresses this need by characterizing the solution as a posterior distribution rather than a single point estimate. Sampling from this distribution, however, remains computationally challenging: Markov chain Monte Carlo and non-amortized variational inference require repeated wave equation solves, while amortized variational inference approaches that avoid repeated solves rely on training data that are inherently scarce in geoscience and face unresolved generalization challenges in high dimensions. To address these limitations, we integrate Stein variational gradient descent with the alternating direction method of multipliers under a dual augmented Lagrangian formulation. By fixing the wave operator at a background model that is updated between frequency batches, it need only be factorized once per particle per frequency, eliminating per-iteration refactorization and reducing the total cost to that of a handful of deterministic inversions while inheriting the favorable conditioning of extended-space formulations. Applied to the Marmousi~II model, the proposed method provides well-calibrated uncertainty estimates and achieves inversion quality comparable to that of the standard augmented Lagrangian SVGD at a fraction of the computational cost.