Unloading containers in the courier, express and parcel industry is a physically demanding and labor-intensive work. Automatizing this process is an important step towards increasing the efficiency of parcel-handling systems. This work investigates the potential of reinforcement learning to learn a policy for item selection in container unloading scenarios. For that, a simulation environment is created and a masked deep Q-learning with a specially designed neural network architecture is implemented. The results indicate that the agent can learn to select items with an average success rate of 60 %, which is significantly better than a random policy at a random chance of 20 %. The findings suggest that RL could be a promising approach for automatizing item unloading tasks in the future.

This paper proposes a novel framework connecting fermionic second quantization and Dyck languages. By defining translations of creation and annihilation operators using bracket alphabets, the study establishes nullity criteria for expectation values of chains of second quantization operators. Those translations are designed to reveal sufficient conditions for the nullity of the expectation values relatively to one-determinant states or to the physical vacuum. The nullity criteria are purely syntactic, and simply reduce to the inspection of sequences of opening and closing brackets. Moreover, numbers and transformations in Dyck languages can be imported in the context of fermionic second quantization. One of these numbers, the depth, originally absent from second quantization, can be used to introduce more nullity criteria.

We study spike-and-slab priors for generalized linear models with possible grouped sparsity. The main result is an oracle Bernstein--von Mises theorem for the fractional posterior under supportwise likelihood assumptions. The proof develops sparse local asymptotic normality and Laplace approximation around support-specific pseudo-true centers, and combines them with fixed-prior mass, support penalization, recovery geometry, and beta-min separation to obtain contraction, support recovery, Gaussian mixture approximation, and collapse to the oracle Gaussian law. Model-entry verifications are given for Gaussian regression and for logistic, Poisson, probit, Gamma log-link, and negative-binomial log-link regression under stated sufficient conditions. The ordinary posterior is treated only through restricted Gaussian and canonical-link extensions, with coverage under additional active-dimension and moment conditions.

The observation of anomalously large polarization rotations in pump-probe experiments with circularly polarized light has recently challenged the conventional understanding of the inverse Faraday effect. The striking magnitude of these responses implies the generation of effective magnetic fields orders of magnitude larger than theoretical expectations, raising fundamental questions about the nature of light-induced time-reversal symmetry breaking. In this work we demonstrate that a static polarization rotation can originate entirely from the antisymmetric component of the third-order optical susceptibility, without generating a macroscopic magnetization of the material. We show that this light-induced Faraday effect is inherently dynamical, emerging when Kleinman symmetry breaks down. Using a minimal sp tight-binding model on a square lattice, we demonstrate that the light-induced Faraday response can be sizable even far from dissipative resonances. While the effect emerges at a purely electronic level, we show that resonant coupling with phonons can significantly enhance the pump-probe response.

A theorem of Ding and Geiges states that every closed, connected contact $3$-manifold can be obtained from the standard tight contact $3$-sphere by contact $(\pm1)$-surgery along a Legendrian link. The literature also contains some examples of contact Kirby moves, i.e. explicit operations on front projections of Legendrian surgery links that change the surgery link but preserve the contactomorphism type of the surgered manifold. Among the most commonly used are cancelling pairs and contact handle slides; however, these moves alone are not sufficient to relate all contact surgery diagrams of contactomorphic contact manifolds. In this article, we introduce two new families of contact Kirby moves, called lantern moves and chain moves, and use them to give a complete set of contact Kirby moves. More precisely, we show that two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of planar isotopies, Legendrian Reidemeister moves, insertions or removals of standard cancelling pairs, the two standard contact handle slides, the standard lantern move, and the standard chain move. All these moves are explicit diagrammatic operations in the front projection. The proof follows an approach initiated by Avdek through his ribbon-move framework, which is rooted in the Giroux correspondence, and combines it with a presentation by Gervais of the mapping class group. We also discuss several consequences of the main theorem, illustrating the effectiveness of the contact Kirby calculus by recovering the invariance of Gompf's $d_3$-invariant purely diagrammatically and by deriving the topological Kirby theorem from contact-geometric methods.

Gravitoelectromagnetism in the Weyl formalism is investigated through an analysis of the consequences of a restricted gauge symmetry acting on the tensor field $A_{μν}$. The propagator associated with the GEM field is explicitly derived and decomposed within the Barnes--Rivers formalism, revealing contributions from the spin-2, spin-1, and scalar spin-0 sectors. By coupling the theory to conserved sources, it is shown that only the spin-2 and scalar sectors contribute to physical processes, while the spin-1 component decouples. The resulting effective propagator can then be written in a compact metric form closely resembling the graviton propagator of linearized General Relativity. The role of gauge fixing is also analyzed by considering both Lorentz-like and de Donder-type gauge conditions. It is shown that the Lorentz-like gauge is consistent with the restricted gauge symmetry of the theory and leads to gauge-independent physical amplitudes, whereas the de Donder gauge introduces residual gauge-dependent scalar contributions, signaling an incompatibility with the underlying symmetry structure. The gauge symmetry is further extended to the fermionic and electromagnetic sectors through diffeomorphism-like transformations. In both cases, conserved currents are derived and shown to coincide with the corresponding energy-momentum tensors, implying that the GEM field couples to matter in the same manner as in linearized gravity. Finally, the associated Ward identities are verified, providing a nontrivial consistency check of the gauge structure and interaction vertices of the theory.

Prolog is a well-known declarative programming language commonly used in introductory courses on logic and reasoning. However, many students find Prolog challenging because it lacks the familiar debugging mechanisms found in imperative languages. In large classes, this difficulty is exacerbated by the challenge of providing timely and personalized feedback to students. In this work, we introduce ProDebug, the first tool to combine Large Language Models (LLMs) with spectrum-based and mutation-based techniques for automated debugging of Prolog assignments. ProDebug automatically identifies faults and proposes bug repairs for student Git submissions. Faults are detected using three approaches--spectrum-based, mutation-based, and LLM reasoning--while repairs are generated using mutation-based techniques and LLMs. Our evaluation on 1499 buggy student submissions from a bachelor's level programming class demonstrates the potential of automated, LLM-augmented feedback systems to scale support for declarative programming education.

Recent advances in RAG have shifted toward an agentic paradigm, where LLMs interact with retrieval systems over multiple turns and iteratively refine queries based on intermediate results. At the same time, LLMs have demonstrated a strong ability to construct structured queries that precisely express their information needs. However, contemporary RAG systems remain heavily focused on engineering complex retrieval backends, including dense, hybrid, and graph-based retrieval architectures. In this study, we argue that agentic RAG should delegate greater control to the LLM to steer the retrieval process, while relying on a lightweight retrieval interface that provides fine-grained control and faithfully executes the LLM's structured intent. Guided by this principle, we propose an agentic RAG framework that enables LLMs to formulate retrieval intents using logical expressions while simplifying the retrieval backend to an inverted-index-based system. Extensive experiments show that our framework matches a strong agentic hybrid baseline, while substantially reducing construction and serving cost. Moreover, we show that anchoring the retrieval process in logical queries substantially reduces hallucinations in generated responses.

We construct open Gromov-Witten invariants in genus zero for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians, which are weakly unobstructed by a bounding cochain. This uses the foundational work of \cite{HH25,HH26} and the algebraic framework of \cite{ST21}. We prove the open WDVV relations and show that these invariants are independent of the choice of almost complex structure and under Hamiltonian isotopy. We also prove a relation between open Gromov-Witten invariants of cobordant Lagrangians.

We present a combined analysis of recent CE$ν$NS data on germanium from two complementary experiments: COHERENT, which uses neutrinos from pion decay at rest, and CONUS+, which detects reactor antineutrinos. Exploiting the complementarity of these two datasets in a joint statistical analysis, we extract the germanium root-mean-square neutron radius and neutron skin with improved precision, disentangling spectral shape distortions from overall normalizations and reducing systematic uncertainties. We also determine the weak mixing angle at low momentum transfer, providing a test of the Standard Model in a less-explored kinematic regime. A key systematic uncertainty in CE$ν$NS ionization measurements is the nuclear quenching factor; we therefore present our results as a function of variations of the Lindhard model. For the nuclear form factor, we adopt the analytical Klein-Nystrand parametrization and benchmark it against predictions from the large-scale nuclear Shell Model, assessing the impact of nuclear structure uncertainties on our results. Our analysis demonstrates the power of combining datasets across different neutrino sources to maximize sensitivity to both nuclear and electroweak physics.

Geospatial health disproportionality remains a critical public health concern, as communities face heterogeneous illness risks due to varying exposures to adverse socioeconomic and environmental conditions. While statistical models have been adopted to identify risk factors, studies that account for the complex, non-linear dependencies and spatial regularities inherent in comorbid disease patterns are underdeveloped. In this work, we propose a novel spatially regularized variational autoencoder (VAE) to characterize and map the geospatial disproportion of childhood comorbidity in West Africa, focusing on diarrhea, fever, and acute respiratory infection (ARI). To model dependence between these conditions, this study integrates a bivariate Gumbel copula into the VAE framework, enabling flexible modeling of asymmetric dependence and quantification of joint and conditional morbidity risks. Additionally, covariate effects within the framework were quantified to facilitate epidemiological interpretation of risk factors. The proposed method was benchmarked against commonly used methods and applied to characterize comorbidity in West Africa using the Demographic and Health Survey data. Findings reveal pronounced spatial heterogeneity in the likelihood of comorbidity among West African children, with the strongest co-occurrence observed between fever and ARI. Household wealth, maternal education, and access to improved water sources were associated with the likelihood of comorbidity. These patterns highlight high-risk areas and underscore the need for targeted, location-specific public health interventions.

Let $n\in \mathbb N\cap[2,\infty)$. In this article, we show that there exists a bounded $C^1$ domain $Ω\subset \mathbb R^n$ such that, for any given $s\in(1,2)\setminus\{\frac32\}$, \begin{align*} \left[H_0^1(Ω),H^2(Ω)\cap H_0^1(Ω)\right]_{s-1} =H^s(Ω)\cap H_0^1(Ω)=H_0^s(Ω) \end{align*} with equivalent norms, but \begin{align*} \left[H_0^1(Ω),H^2(Ω)\cap H_0^1(Ω)\right]_{\frac12} \subsetneqq H^{\frac32}(Ω)\cap H_0^1(Ω), \end{align*} which provides a counterexample to Problem 3.3.19 of Kenig in [CBMS Regional Conf. Ser. in Math. 83, 1994]. As applications, we prove that for such a domain $Ω$ \begin{align*} H^2(Ω)\cap H_0^1(Ω)\subsetneqq D(-Δ_D) \end{align*} (the domain of the Dirichlet Laplacian operator $-Δ_D$ on $Ω$) and construct a solution of the homogeneous heat equation with zero Dirichlet boundary condition, which does not belong to $L^2((0,T);H^2(Ω)\cap H_0^1(Ω))$ for any given $T\in(0,\infty)$.

Random Telegraph Noise is a ubiquitous process manifesting across technology and the natural world. It is characterized by random jumps between two distinct states with Poissonian waiting times, and is the origin of 1/f noise. Understanding and characterizing this noise is critical for the reliable operation of micro-, nano- and quantum-technologies. In this work we probe random telegraph noise using a levitated microparticle sensor whose dynamics are driven almost entirely by this non-white source of noise. We observe a startling resonant behaviour, characterized by a thousand-fold increase in the underdamped sensor's position fluctuations, enabling us to measure the spectral properties of the noise over six decades of timescale. This work not only provides a unique way to probe random telegraph noise, but also demonstrates a platform for studying non-equilibrium stochastic dynamics in the presence of realistic non-white noise, with applications from biology to social behaviour.

Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $χ_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in [Discrete Math. 306 (16) (2006) 1997-2004] that every tree $G$ satisfies $χ_{r}(G)=\min\{r,Δ(G)\}+1$. It is known that every cycle graph $C_{n}$ with order $n$ has $χ_{L,r}(C_{n})=χ_{r}(C_{n})$. The main results are the following: $(1)$ If $G$ is a tree, then $χ_{L,r}(G)=\min\{r,Δ(G)\}+1$; $(2)$ Let $G$ be a unicyclic graph which is not isomorphic to the cycle $C_{n}$. If $n\neq 5$ and $r\geq3$, then $χ_{L,r}(G)=\min\{r,Δ(G)\}+1$; otherwise, $\min\{r,Δ(G)\}+1\leqχ_{L,r}(G)\leq\min\{r,Δ(G)\}+2$.

This paper presents a detailed analysis of a gas-rich star-forming ultra-diffuse galaxy (UDG) as part of the ESO Large Programme 'Looking into the faintEst WIth MUSE (LEWIS)'. Among the UDGs in the LEWIS sample, UDG 6 is the only galaxy that hosts a significant amount of ionised gas with evidence of emission lines, suggesting recent star-forming activity. The main goal of this work is to constrain the formation history of this UDG by comparing its properties with the main formation scenarios proposed for this extreme class of galaxies. We adopted integral field spectroscopy from MUSE to derive the morphology and the structural properties of the stellar and gas components of UDG 6. We applied spectral fitting and Voronoi tessellation algorithms to the MUSE data-cube to derive the kinematics and properties of the gas and stellar component. Moreover, we derived the GCs populations' properties by applying a multi-band spectrophotometric analysis. We confirmed that UDG 6 is a member of Hydra I cluster. It is characterised by a regular and elongated shape and contains a significant dust content, a metal-poor ionised gas fraction and an underlying old-to-intermediate stellar component. Evidence of local and clumpy star-forming activity has been revealed through the analysis of emission line, and an arc-like tidal feature was discovered from unsharp masking analysis. UDG 6 might originate from a 'puffed-up dwarf' whose stellar content has been stretched out to larger radii, passively evolving into a more diffuse galaxy. Being located in a dynamically active region of the cluster, characterised by tidal features and stripping phenomena, we suggest that the environmental processes have played a role in shaping the properties of UDG 6. A tidal interaction with a nearby galaxy might have triggered recent star-formation activity, without dramatically altering the coherent gas rotation in UDG 6.

Fluid exchange between a soap film and its bounding menisci governs film drainage and stability, with direct implications for the lifetime of surface bubbles and liquid foams. Despite recent advances, a quantitative characterization of this coupling, associated with the phenomenon of marginal regeneration, remains incomplete. The volumetric flux per unit length of contact follows a well-established scaling law involving geometrical parameters such as the film height and the meniscus radius of curvature. However, the dimensionless prefactor of this relation - the flux coefficient - remains difficult to determine for vertical menisci because of the complex and intermittent flows occurring at the film-meniscus interface. Here, we quantify this flux into the meniscus generated by inserting a solid plate into a vertical soap film. We consider both vertical and inclined plates and further investigate the effects of plate inclination, height, and width. Focusing on the dynamics of the growth of the meniscus driven by liquid supplied by the film, we analyze both steady and transient regimes resulting from the interplay between capillary pressure, gravity, and liquid exchange. Combining experiments, numerical simulations, and theoretical modelling, we determine the flux coefficient using three independent methods and show that it remains constant over the range of parameters explored.

The edge-cloud computing continuum demands self-management mechanisms that scale across autonomous administrative domains while honouring tenant- and operator-specified data sovereignty. We present Neural Pub/Sub, a federated-broker autonomic substrate whose self-organising behaviour emerges from market-based price signals rather than centralised control. Its MAPE-K control loop closes over per-broker health and load monitoring, marginal-cost clearing-price analysis, placement planning over a polymatroidal feasibility region, federated cross-domain dispatch, and shared peer subscription summaries with bounded-staleness price signals. The Plan step is anchored in a Walrasian convergence proposition: under gross-substitutes valuations on tree and series-parallel service-dependency DAGs, decentralised price-based allocation matches the welfare of a centralised oracle. We evaluate the substrate on a 4-VM, 4-domain, 48-worker federated edge-cloud testbed (single data centre, 50 ms emulated WAN) in a 1005-run campaign augmented by a fair-process-count sharded-oracle comparator. The federated market dominates a single-process oracle by 2-4% with 45 of 45 per-seed wins (sign-test p ~ 2.8e-14, Hodges-Lehmann median -39.6 ms); against a four-shard centralised orchestrator at equal process count the gap stays within +/-1.5% across all nine (pipeline, load) cells. Round-robin completion rate collapses 98.8% -> 22.4% -> 3.3% across arrival rates 5/10/15 pps while the market preserves completion; the advantage decomposes into three Walrasian properties (information completeness, admission control, price discovery). Federation withstands broker death and network partition (completion rate >= 98.7% across 75 cells), and sovereignty enforcement adds no measurable runtime overhead across 60 governance-grid runs. Heterogeneous-domain stressors and cross-site WAN deployment remain future work.

A thorough understanding of the temperature dependence of semiconductor band gaps is essential for optimizing optoelectronic devices. In this respect, the origin of the pronounced temperature-induced gap bowing observed in low-temperature phases of formamidinium-methylammonium (FA-MA) lead iodide perovskites has remained elusive until now. By combining temperature and pressure-dependent photoluminescence measurements on a series of FA$_x$MA$_{1-x}$PbI$_3$ mixed-cation single crystals with $x\in[0,1]$, we unravel the origin of this bowing. Both thermal expansion as well as electron-phonon interaction effects are responsible. However, the latter is the leading term, driven by the activation of an anomalous electron-phonon coupling mechanism linked to mixed vibrational modes, which combine inorganic-cage phonons involving octahedral tilting with low-frequency FA librations, i.e., FA rattler modes. This occurs in the orthorhombic and (pseudo)tetragonal low-temperature phases, presumably featuring stripe domains with alternating octahedral tilt-axis patterns for FA concentrations between 20\% and 90\%. In this way, we have shed light on an intriguing behavior of lead halide perovskites that directly affects their optoelectronic properties.

Douyin Music, a large-scale platform with millions of daily users, adopts an immersive, feed-based discovery paradigm, where users passively explore music through continuous recommendations. While effective for passive music discovery, this paradigm restricts users to recommendation results and provides limited support for explicitly specifying listening intents. Unlike conventional search, where users express well-defined intents through explicit queries such as specific songs or artists, real-world active music discovery is often situational and colloquial, involving vague or underspecified requests. While LLMs enable natural language interaction, their direct use in music discovery remains limited by insufficient music-domain knowledge, lack of music-query collaborative reasoning, and shallow understanding of personalized preferences. To address these challenges, we introduce MuChator, an interactive MusicLLM-based framework that enables users to actively express situational music intents in natural language. MuChator incorporates three key components: (1) Music Knowledge Pre-training, a three-stage scheme that incrementally injects objective music knowledge, subjective music knowledge, and personalized music preferences into LLMs; (2) Context-aware Instruction Tuning, which constructs high-quality user-query-music triplets through an automated synthesis pipeline to align LLMs with active and situational user intents; and (3) Preference Alignment with Hybrid RM, which jointly models intent relevance, personalized preferences, and basic constraints, and is optimized using GRPO-based reinforcement learning. Extensive evaluations on industrial music recommendation datasets demonstrate that MuChator outperforms leading proprietary models, such as Gemini-3-Pro. The model has been deployed on Douyin Music App within ByteDance, with 46.49\% improvement of user active days in online A/B test.

We study an ecology-inspired model for a population of bounded size, whose dynamics is governed by random birth, death, and immigration events. Stochastic fluctuations in the number of individuals give rise to a succession of alternating active and vacant periods, where the population is respectively extant and extinct. Using both analytical and numerical techniques, we characterize the statistics of the two kinds of period, quantifying their duration and frequency, and the typical population sizes in active periods. In sharp contrast to the deterministic mean-field behavior, governed by logistic dynamics, active periods may exhibit pronounced bimodality: either short durations with very small populations, or much longer durations with population sizes close to the maximum. We also investigate how these results change when the population evolves on random networks of three classes: Erdős-Rényi, regular, and geographic. The main effect of the network structure is to induce population clustering, with individuals aggregated into localized groups. This, in turn, limits population growth and increases the frequency of vacant periods.

We investigate magnetoresistance phenomena associated with the magnetization hard-axis collapse in polycrystalline Co thin films. Transport measurements reveal that, for specific orientations of the applied magnetic field, the system exhibits distinct remanent resistance levels in both the in-plane longitudinal and transverse voltage responses. In particular, the planar Hall resistance shows multiple stable and reproducible levels at room temperature, enabling the identification of at least three remanent states that can be distinguished and used for information storage. These resistance levels originate from non-uniform magnetic configurations stabilized after the application and removal of the external magnetic field in the hard-axis region. Since this phenomenon remains largely unexplored, we present an incipient study addressing its potential implications from an applied-physics perspective. The observation of such behavior in polycrystalline Co thin films grown on Si substrates suggests a simple and low-cost platform for spintronic memory and sensing devices based on the remanent planar Hall effect.

We consider the problem of dividing a set of indivisible goods among agents with additive valuations. This problem has been studied under various objectives in both the computer science and the operations research literature. Our main contribution is a novel dictator test using this problem, which can separate a dictator from any function sufficiently far from a dictator. We use this test to prove the following hardness results (assuming the unique games conjecture is true): (1) We show that it is NP-hard to approximate the max Nash welfare by a factor better than $\sqrt[3]{\frac{81}{65}} - \varepsilon \approx 1.0761$. This improves on the previous best known inapproximability factor of $\sqrt{\frac87} - \varepsilon \approx 1.069$. (2) We show that it is NP-hard to approximate the maximum budgeted allocation by a factor better than $\frac{243}{227} - \varepsilon \approx 1.07$. This improves on the previous best known inapproximability factor of $\frac{16}{15} - \varepsilon \approx 1.067$. (3) We show that it is NP-hard to approximate the max generalized assignment problem (GAP) by a factor better than $\frac{145}{129} - \varepsilon \approx 1.124$. This improves on the previous best known inapproximability factor of $\frac{11}{10} - \varepsilon \approx 1.10$.

Interstellar pickup ions originate from the neutral interstellar medium, are ionized in the heliosphere, and picked up by the solar wind. They initially form a torus-shaped velocity distribution function, which is generally believed to be transformed rapidly into an isotropic shell distribution by pitch-angle scattering. With the SupraThermal Electron Proton onboard Solar Orbiter we observe clear torus-shaped velocity distribution functions at an unprecedented one minute resolution. While these tori are variable on a time scale of one minute, they remain stable for over ten hours without signs of significant scattering. We conclude that they are populated by a huge fraction of the expected total number of pick-up ions injected in the past of the same solar wind stream.

The electrification of long-haul freight transport introduces several new challenges, such as the limited capacity and congestion at en-route charging infrastructure. To reduce waiting times during peak periods, this paper proposes a framework for coordinated charging scheduling. The approach employs a mixed-integer formulation to optimize charging-related costs across charging, operation, battery degradation, and congestion delay, considering a range of scenarios. The results demonstrate that coordinated scheduling yields substantial cost savings up to 36% compared to uncoordinated scheduling, particularly by reducing battery degradation and delay costs.

The goal of this note is to show that the left $χ$-coalgebra, which is an additional structure on one of the coefficients used in the construction of the cyclic operator for the cyclic sets that generalises the twisted nerve of a group by Loday is an instance of a general theory of left $χ$-coalgebras. It is also shown that in the general case, the resulting cyclic operator requires that the other coefficient needs to be equipped with a crossed $G$-sets structure.

The electron cyclotron emission (ECE) diagnostics suite at ITER utilizes a front-end quasi-optical (QO) system whose design is fundamentally constrained by a field-stop concept. The field-stop defines the Gaussian beam variation throughout the optical system and within the plasma, thereby setting the ECE sampling volume and spatial resolution. An in-situ hot calibration source, optimized using Gaussian beam transmission criteria, provides independent and absolute electron temperature measurements. The QO system extends beyond the front-end to include the polarization splitter unit (PSU), transmission lines, and switchyard, forming an integrated optical path to the ECE instruments. Misalignment between the front-end and PSU reduces the effective field-stop size, degrading spatial resolution and measurement fidelity. The oblique ECE view, a key feature of the ITER design, enhances sensitivity to non-thermal electron populations and complements the diagnosis of neoclassical tearing modes. Integrated QO design and plasma physics understanding are essential for reliable ITER ECE measurements.

We introduce dynamic and static formulations that formally extend unbalanced optimal transport from the space of positive densities to the space of Riemannian metrics. The first construction is based on a dynamic variational formulation in which the evolution of a Riemannian metric is driven by transport together with a source term. Choosing the $L^2$-metric to penalize the transport vector field and the Ebin metric to penalize the source component yields a new Riemannian metric on the manifold of Riemannian metrics, which we call the Wasserstein--Ebin metric. Our main result shows that the volume map defines a Riemannian submersion from the Wasserstein--Ebin metric to the Wasserstein--Fisher--Rao metric on the space of smooth densities. In addition, we construct a Riemannian submersion from the automorphism group of the tangent bundle onto the space of Riemannian metrics, providing a generalization of Otto's geometric description for the Wasserstein metric to the setting of the Wasserstein--Ebin metric. To propose a static formulation of unbalanced optimal Riemannian metric transport, we introduce two Kullback--Leibler-type divergences on the space of Riemannian metrics: one inspired by matrix information geometry, and another related, through the volume map, to the classical Kullback--Leibler divergence on densities. Establishing a link between the static and dynamic formulations remains an open direction for future work.

Win measures, including the win ratio (WR), win odds (WO), net benefit (NB), and desirability of outcome ranking (DOOR), are increasingly used in randomized clinical trials with multiple hierarchical ordinal endpoints. In practice, however, one or more component endpoints may have missing data. The standard pairwise-comparison approach, which treats pairs with missing outcomes as ties, can produce biased estimates, even if the data are missing completely at random (MCAR). Although inverse probability of censoring weighting (IPCW) methods have been developed for censored survival endpoints, corresponding methods for addressing missing hierarchical ordinal endpoints are not yet available. To address this gap, we develop inverse probability weighting (IPW) and augmented IPW (AIPW) estimators for win measures with hierarchical ordinal endpoints subject to missing data, allowing missingness to depend on treatment assignment and baseline covariates. The IPW estimator corrects bias by reweighting complete observed outcomes using joint non-missingness probabilities involved in estimating the joint cell probabilities that define the win measures. The AIPW estimator additionally incorporates outcome modeling, improving efficiency and achieving double robustness. For inference, we derive closed-form variance estimators for both methods based on influence functions. Simulation studies show that the standard approach can be substantially biased, whereas the proposed IPW and AIPW estimators remain consistent with near-nominal coverage. Furthermore, the AIPW estimator is generally more efficient than IPW estimator. Applications to the SCOUT-CAP and ACTT-1 trials illustrate the practical utility of the proposed methods. An R package, WinMO, is provided for implementation.

In this paper, we present near-IR polarized images of the AB Aur disk at three epochs spanning 3.85 years with SPHERE/IRDIS, as well as Halpha images obtained with SPHERE/ZIMPOL at a single epoch. The purpose of this study is to analyze the dynamics of the entire disk and of the various structures in near-IR polarimetry, and to identify sources of Halpha emission to derive constraints on their mass accretion rate. The dynamical study in the near-IR shows that the disk globally follows Keplerian rotation, but we observe a departure from this behavior at radii smaller than ~60au. At the smallest radius of ~25au, we measure a deviation from Keplerian rotation as large as ~12deg over 3.85 years, demonstrating sub-Keplerian rotation. The two bright spirals within the millimeter cavity have different dynamic trends, and we discuss their possible link with the identified planet candidates. We also discuss the implications of the non-Keplerian behavior, and we posit that it could be related to interactions with multiple protoplanets orbiting out of the disk plane on elliptical orbits. Furthermore, the orbital analysis of the compact sources (labeled f1, f2, and f3) suggests that their orbital planes are significantly inclined with respect to the disk plane by several tens of degrees. The variability of the shadows suggests that they are produced by optically thick regions located within ~60au. For the photometric analysis in Halpha, we derive a flux of about 8.22x10^{-15} erg/s/cm^2 for the entire feature f1, but only 6.46x10^{-16} erg/s/cm^2 at the location of AB Aur b, consistent with non-detection. If f1 were a point source and the accretion remained constant for 1Myr, it would correspond to ~5-20 Jupiter masses according to the magnetospheric accretion model or ~6-10 Jupiter masses according to the boundary layer accretion model.

We use density-functional theory with a Hubbard correction to investigate Ti-centered electron polarons at neutral PbO-centered $180^\circ$ domain walls in tetragonal PbTiO$_{3}$. The Hubbard parameter for Ti $3d$ states is determined using the finite-size-corrected polaronic energy-level alignment procedure, yielding stable electron-polaron formation in bulk PbTiO$_3$ with a trapping energy of $-$0.06 eV. In the domain-wall supercell, the excess electron localizes on Ti and forms a Ti$^{3+}$ center with an occupied $d_{xy}$ orbital in-gap state. Comparison of bulk-like and near-wall Ti sites shows that their trapping energies differ by only about 0.01 eV, indicating that this neutral domain wall does not provide a significant thermodynamic driving force for electron-polaron segregation. While the Ising-like reversal of the out-of-plane polarization is preserved, the localized electron induces a finite transverse polarization component normal to the wall, enhancing a local Néel-like distortion that is strongest when the polaron is located at the wall. These results show that neutral $180^\circ$ domain walls in PbTiO$_3$ do not substantially alter the stability of Ti-centered electron polarons, but they can couple to the polaron-induced lattice distortion through a localized transverse polarization response.