Papers

We study algebraic vector bundles on cluster varieties. Over a perfect field of characteristic different from two, every algebraic vector bundle is trivial on a cluster variety with principal coefficients at an acyclic seed of irreducible finite Dynkin type. The proof is based on integral contractions of cellular $\mathbb A^1$-chain complexes and successive freezing. We also construct smooth cluster varieties admitting indecomposable rank-two vector bundles, whose nontriviality is detected by quadratic Euler classes. The cellular calculation also gives a criterion for stable freeness and computes integral real cohomology and the images of cycle class maps.

Ego-trajectory prediction in urban roadway work zones remains an underexplored topic in autonomous-driving research, with limited dedicated data and systematic evaluation. Roadway work zones involve temporary changes in the lane layouts and connectivity of the roads, complicating interactions with surrounding vehicles and workers. These changes make future vehicle trajectories difficult to predict. To address these challenges, we collect WZDrive-Data, a multimodal dataset comprising 1,183 work-zone driving segments from 33 drivers, using a human-in-the-loop driving simulator with prerecorded worker animations and VR-controlled workers. Driver control and gaze signals are recorded from human participants, while vehicle trajectories and scene information are obtained from the simulator. Our proposed multimodal ego-trajectory predictor, WZDrive, achieves a mean minADE (K=6) of 1.09 m and minFDE (K=6) of 2.12 m. Compared with the best-performing state-of-the-art baseline for each metric in our benchmark, WZDrive reduces minADE (K=6) by 7.18% and minFDE (K=6) by 13.18%. This research addresses a gap in autonomous-driving research on urban work zones by combining human-in-the-loop data collection with multimodal learning from human driving signals and in the context of work zones.

Electronic properties of materials depend both on band energies and quantum geometry of electronic states. We relate bonding indices to the quantum geometric tensor (QGT), showing how chemical bonding dictates the momentum-space texture of bands, bridging chemical and physical descriptions of solids. We show the QGT decomposed into the three center bond index, weighted by their oriented area. Resolving these contributions by orbital pair reveals the donor–acceptor character that distinguish ionic, covalent, hypervalent, and metavalent bonding. We identify the bonding pattern for topology in materials: noncollinear three center bonds, which are the source of Berry curvature in solids. Bond fluctuation can be directly linked to electromagnetic response by resolving their energy spectrum, opening a quantitative route towards the design of physical properties of materials from bonding motifs.

We develop a neural transport framework for nonlinear, nonlocal aggregationdiffusion equations arising in chemotaxis, polymer dynamics, and related applications. Based on the Onsager variational principle, we reformulate the governing equation as a constrained optimal control problem, taking the velocity field as the control and evolving the density via the associated continuity equation. The velocity field is parameterized by a neural network trained on the variational objective evaluated along particle trajectories. By coupling the density and generalized score equations into a single system solved via the Euler method, the score function propagates efficiently for loss evaluation. Furthermore, we establish a rigorous error analysis that decomposes the total error into approximation, optimization, Monte Carlo sampling, and time-discretization components, and prove corresponding error bounds. Numerical experiments across several representative models demonstrate the accuracy and effectiveness of the proposed approach.

Parameter-efficient continual learning updates a restricted subset of adapter parameters to incorporate sequential tasks while limiting catastrophic forgetting. Gradient attribution is a common rule for selecting that subset, but it requires an additional backward pass over calibration data. We identify a parameter-pool confound in comparisons between initialization-time gradient masks and unrestricted random masks under standard zero-initialized LoRA ($B=0$): the gradient with respect to $A$ is identically zero ($\partial\mathcal{L}/\partial A=0$), so gradient masks select strictly from the $B$ matrix whereas unrestricted random masks draw from both $A$ and $B$. Such a comparison changes both the selection rule and the candidate parameter pool. Isolating the selection criterion by restricting both approaches to the same parameter pool, we find a negative result: a uniform random mask consistently achieves lower loss-based backward transfer than gradient attribution across seven seeds on Qwen2.5-3B. The effect is concentrated on the medical task, which contributes 80% of the aggregate; code and math move in the same direction, and legal shows no reliable difference. This advantage is consistent with a stability–plasticity trade-off under the tested task sequence, in which random masking slightly reduces single-task fitting. On the two discrete-option tasks (Medical and Legal), the mean accuracy-BWT difference is not statistically distinguishable from zero.

Molecular spectra observed in the interstellar medium are commonly analyzed using non-local thermodynamic equilibrium models, which requires accurate state-to-state collisional rate coefficients. Close-coupling calculations become prohibitively expensive for transitions from highly excited rotational states, therefore available datasets are often truncated at low initial states and temperatures; and since each new system also requires an accurate ab initio potential energy surface, many detected molecules have no collisional data. We employ two machine learning strategies to predict rotational state-to-state rate coefficients of molecules included in the train set beyond the range of initial states and temperatures computed so far, and predicting those of a molecule absent from training and validation sets. Feed-forward neural networks are trained on state-to-state de-excitation rate coefficients of 24 linear molecules colliding with para-H2. Scheme 1 includes all systems in the train+validation set, with 70%, 80%, and 90% initial-state boundaries and a 90% temperature boundary. Scheme 2 is trained on 22 systems described only by transition and molecular properties, and predicts two excluded systems, CS and CF+, whose astrophysical impact is explored with non-LTE calculations. For the 90% boundary, Scheme 1 predicts 99.4% of the test transitions within a factor of 2, and no prediction deviates from the reference value by more than a factor of 4.7. This model was used to complete 18 datasets. Scheme 2 reproduces the rate coefficients of two excluded systems within a factor of 3 for more than 92% of their transitions. Both schemes provide near-instantaneous complements to close-coupling calculations: Scheme 1 extends the range of a molecule already included in the training set with high accuracy, while Scheme 2 gives a reasonable first estimate for a molecule that has not been studied.

This paper details our submission to KnowledgeGraphEval 2026 Subtask 2 on directed Arabic relation extraction. We explored encoder-based and large language model approaches for predicting one of 40 relations or no_relation from an Arabic sentence and a subject-object pair. Our encoder systems use typed entity markers and separate relation-existence and relation-classification heads, while our LLM systems fine-tune Yehia and Qwen3.5 with QLoRA and ontology-constrained prediction. Our best system, an ensemble of Yehia-RE, Qwen-RE, and TypedPair, achieved a micro-F1 score of 94.06 and ranked 1st in Subtask 2. Yehia-RE (full-vocabulary) achieved 92.98, while the best encoder and Qwen-RE achieved 92.69 and 91.94 respectively. Our results highlight the importance of constrained prediction and complementary modeling approaches for Arabic relation extraction.

We interpret Transformer networks as learned approximations to a hybrid probabilistic-inference procedure that combines max-product Loopy Belief Propagation (LBP) with Expectation–Maximization (EM). In this view, a Transformer performs inference in partially observed Markov Random Fields (MRFs) whose structure and parameters are not given a priori but are inferred from data. We show that self-attention implements a differentiable approximation to max-product LBP on an input-conditioned, dynamically structured MRF, with attention weights encoding learned structural affinities. We further establish an EM correspondence at the level of depth: each layer updates latent representations in an E-step-like forward pass, while the transition $\theta^{(l)} \to \theta^{(l+1)}$ across layers acts as an implicit M-step. Building on this interpretation, we introduce meta-models, a design paradigm in which inputs generate instance-specific parameters, unifying mechanisms such as self-attention, mixture-of-experts, and gated units within a single framework. Empirical results support the proposed perspective.

We study the two-sample estimation of the powered Sliced Wasserstein functional under projected $r$-th moment assumptions. Let $N=n\wedge m$, where $n$ and $m$ are two sample sizes. We establish the minimax rates in the noncritical regimes. When $p<r<2p$, the minimax risk is of order $N^{-(r-p)/r}$, while for $r>2p$ it is of order $N^{-1/2}$. In both cases, the natural plug-in estimator is minimax rate-optimal. At the endpoint $r=p$, the minimax risk is bounded away from zero, so uniformly consistent estimation is impossible. At the critical order $r=2p$, we obtain the exact worst-case rate of the plug-in estimator and a lower bound for the minimax risk, leaving a logarithmic gap between the lower and upper bounds. The exact minimax rate in this critical case remains open.

Predictive synthesis concerns the prospective modeling of how a targeted structure can be realized. The synthesis of metal-organic frameworks (MOFs) and covalent organic frameworks (COFs) intrinsically involves a multidimensional space of interdependent variables, posing a challenge for bench chemists to parameterize and predict. In this Perspective, we discuss the evolution from chemical-intuition-driven experimentation toward the integration of artificial intelligence (AI) and machine learning (ML) in the laboratory. By virtue of prior knowledge, uncertainty, and experimental failure in determining what should be attempted next, we envision predictive synthesis progressing from static recipe recommendation toward real-time control and transferable principles of framework formation. In this context, the significance lies in transforming trial-and-error practice into a means of measurable chemical reasoning, thereby accelerating crystalline materials discovery and advancing synthetic chemistry.

Forecasting rare, high-cost events in multivariate time-series systems is hard because their precursors are seldom observed, so a purely data-driven model has little to learn from. The engineers who run these systems, however, hold knowledge that precedes such events, information absent from the scarce labels. We call this a semantic prior and inject it into a self-supervised event forecaster through two switchable mechanisms, a covariate mechanism and an attention-bias mechanism, each the identity at its null, so one architecture serves every prior type with no per-dataset architecture search. A prior helps when it supplies information the observed window does not already contain: in a short context an informative prior beats no prior in-domain (pooled h-AUROC over the no-prior baseline: a periodicity clock +14%, p<0.001; a conservation residual +47% at a 0.25 label fraction), and a working prior transfers out-of-domain (+53%), substituting for feature transfer, while priors the window already recovers stay inert and the attention-bias mechanism shows no measurable gain. KITE is a switchable, attributable route to contextual conditioning, and a substrate for scaling toward cross-domain, pretrained time-series foundation world models that carry their governing structure explicitly.

Gradient-based attribution provides input-specific information about model parameters, but computing it via backpropagation for many samples is expensive, while static importance scores discard input dependence. We derive KPSA (Kernelized Parameter Sensitivity Atlas) from a block decomposition of the empirical neural tangent kernel. It measures normalized parameter-group sensitivity, embeds those sources in an explicit query space, and uses a lightweight kernel to estimate sensitivities for unseen inputs. We derive bounds that separate finite-source from irreducible representation error and give margin-based top-$k$ recovery. This paper studies two interpretability applications. Firstly, KPSA identifies parameter groups with high susceptibility to perturbations across vision, forecasting, and protein-modeling setups. Across a fivefold range of parameter budgets ($0.02$-$0.1\%$), its selected groups attain $39$-$85\%$ of the perturbation susceptibility of a first-order backpropagation oracle. In vision and forecasting, these parameter groups form sparse sensitivity circuits that extend across depth and vary systematically with the input. Secondly, we use KPSA on an activation-attribution problem for MLP features in transformers. It retains $81\%$ of the gap between a backpropagation oracle and a constant-sensitivity baseline on vision while increasing batched query-time throughput from 551 to 2,041 images/s and reducing peak GPU memory from 4.55 to 0.44GB. On the protein-modeling task, it recovers $68\%$ of this gap. An ablation of KPSA's kernel aggregation shows that nearest-neighbor retrieval in KPSA space can remain competitive. The empirics show that compact kernel atlases can store useful input-dependent parameter sensitivity while amortizing repeated gradient attribution, a step toward scalable conditional interpretability. Code can be found at https://github.com/Axym-Labs/kpsa .

For a finite abelian group $G$ acting linearly on the plane let $\mathcal{X}$ be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of $\mathcal{X}$ whose corresponding subschemes of the plane have a given representation type, denoted by $\mathcal{X}^{[\rho]}$. By means of a Grothendieck-Riemann-Roch theorem of Toën we give explicit representation theoretic formulas of $\dim(\mathcal{X}^{[\rho]})$ for all $\rho \in \operatorname{Rep}(G)$ which are similar to the McKay correspondence. We also give a classification of all such quotient stacks according to an associated quadratic form called the Euler form. For $[\mathbb{A}^2/\mu_n]$ given by the cyclic action $\zeta \cdot (x,y) = (\zeta x, \zeta^{-1} y)$, we give an interpretation of the corresponding Euler form in terms of border strips which may be removed by a diagram, an equational characterization of $n$-core partitions, and the number of significant arrows of a monomial ideal as defined in "Smooth and Irreducible Multigraded Hilbert Schemes" by D. Maclagan and G. Smith.

Current stereo vision-based approaches for food portion estimation are hindered by the user burden of using a mobile phone to capture multiple images or placing a reference object in the scene for scale calibration. As a step towards passive dietary intake monitoring, we introduce a method that uses stereo video from a wearable egocentric device to reconstruct food items in true physical scale to accurately estimate volume. By utilizing FoundationStereo for zero-shot stereo matching, SAM 2 for food segmentation, and TSDF fusion over a set of keyframes, our method yields a visually and metrically accurate 3D mesh of any plated food item from a single stereo recording. We leverage the unique offerings of wearables, such as a factory-calibrated stereo baseline and camera poses estimated on-device via visual-inertial odometry (VIO), to perform fully automatic reconstruction without a scale reference object in the scene. We show strong initial volume estimation results across five recorded food items, achieving a mean absolute percentage error of 9.1%.

Stereo vision is increasingly used for robotic manipulation, with a growing body of work incorporating it into Vision-Language-Action (VLA) models. However, the scarcity of large-scale, real-world stereo datasets remains a fundamental bottleneck for scaling such policies. Recent generative monocular-to-stereo methods target settings such as VR content and autonomous driving. No prior work produces realistic synthetic stereo data for robotic manipulation, where stereo supervision has come predominantly from simulation. We propose StereoEngine, which introduces two complementary methods to synthesize stereo data for robotic manipulation. In the first, we warp a monocular (left) video into the right view using estimated depth and inpaint the disoccluded regions with a video diffusion model, lifting monocular demonstrations into stereo pairs without training on real stereo. In the second, we render a stereo pair in simulation and jointly generate both photorealistic views, producing consistent left-right data without any real video as input. StereoEngine outperforms existing baselines in generated stereo quality, and using its synthetic data for policy mid-training or task-specific fine-tuning substantially improves downstream bimanual manipulation. Project website: https://stereoengine.github.io

Indoor navigation to task-relevant targets remains difficult when the target is small, occluded, or visually indistinct. Meanwhile, smart appliances, mobile devices, and IoT tags emit Wi-Fi signals during everyday communication, providing a pervasive but underused spatial cue. We study visual-Wi-Fi navigation, where a robot reaches a signal-emitting device in an unmapped environment using onboard visual observations and Wi-Fi packets overheard from the device, without connecting to the device or modifying infrastructure. Realizing this requires pose-dependent Wi-Fi observations during closed-loop policy learning and reliable spatial cues from bearing measurements corrupted by indoor multipath propagation on a moving robot. We present WiFiSpaces, which extends the Habitat embodied simulator with online ray-traced Wi-Fi sensing, rendering Wi-Fi observations for a robot-mounted antenna array at continuous poses during policy rollouts. On top of it, we build ViFiNav, a localization-aware navigation framework: an online source-localization algorithm exploits robot motion to disambiguate and robustly aggregate bearings from multiple viewpoints into a confidence-aware source estimate, and a learned policy fuses this evolving estimate with depth observations and learns to act on it as it is refined. Experiments show that exploiting robot motion is key to accurate source estimates, and that fine-tuning on ray-traced Wi-Fi observations in WiFiSpaces improves the success rate of ViFiNav. The policy transfers zero-shot to unseen scene datasets, and we deploy it on a real robot with a Wi-Fi sensor built from a commodity router, without any real-world training.

Parents seeking childcare often face long waitlists. At the same time, vacancies at childcare centers can remain unfilled for extended periods. We trace this paradox to two persistent features of childcare services. First, families consider multiple child care options due to enrollment uncertainties, resulting in phantom listings that remain on waitlists even after families have found care elsewhere or no longer need a slot. Second, filling a slot is time-consuming: centers learn whether a family is actively seeking enrollment only after making an offer and waiting for a response. As families pursue alternatives while they wait and their children age, these delays and feedback effects amplify the mismatch between demand and available capacity. We study two operational remedies. First, we examine regular verification that removes phantom listings before offers are made, improving the efficiency of enrollment and reducing vacancies. Second, we study mixed-age classrooms, in which some slots designated for older children can be flexibly used by younger children. This flexibility increases utilization, shortens waits, and reduces wasted offers. Its benefits extend beyond conventional pooling: as children move between age groups, classrooms can be filled through internal flows rather than costly external search, further reducing enrollment delays. Together, verification and mixed use offer a way to expand effective capacity in early childhood classrooms, where regulatory and operational constraints make adding physical capacity particularly costly.

In this note, we study a broad class of time-dependent and irreversible dynamics for quantum spin systems in both finite and infinite volume. We develop a simple and flexible strategy for expanding commutators of the form $[A,\tau_{s,t} \, B]$ in terms of self-avoiding paths of sets, where $A$ and $B$ are local operators and $\tau_{s,t}$ is the Heisenberg time evolution. This adaptive Dyson expansion requires no backwards time evolution and is thus also applicable to irreversible dynamics. Furthermore, we discuss how such an expansion directly implies Lieb-Robinson bounds. A central feature of our result is that the obtained Lieb-Robinson bounds are independent of any Hamiltonian and dissipative on-site terms of the generator.

Smart grids rely on distributed phasor measurement units (PMUs) and wide-area communication networks whose data-acquisition pipelines are vulnerable to stealthy false data injection (FDI) attacks that evade conventional bad-data detection (BDD). Moving target defense (MTD) counters this exposure by continuously remapping sensor identities, but existing schemes apply static or pre-scheduled randomization and are each constrained by a single failure mode: seed-based methods are fast yet fragile to key compromise, while machine learning (ML) based prediction methods are resilient yet incur prohibitive latency. Neither adapts its randomization to live threat evidence. This paper presents ROAM, a deep reinforcement learning (DRL) driven MTD framework in which a deep Q-network (DQN) agent continuously re-optimizes the sensor-to-ID mapping at the aggregation layer. At each decision step, the agent selects the deception coverage, flags the specific sensors to remap, and sets the split of the remapped sensors between seed-based and ML-prediction-based recovery, without modifying PMU hardware and while preserving IEEE C37.118 compatibility. On held-out BDD-compliant FDI scenarios across the IEEE 14-, 57-, and 118- bus systems, ROAM keeps the percentage of stealthy attacks below 3% while sustaining detection rates above 93%. Against an adaptive soft actor-critic (SAC) attacker that observes the defender's configuration, ROAM keeps stealthy attacks below 2.6% with detection above 92%. Hardware-in-the-loop testing on an OPAL-RT platform confirms real-time feasibility, with sub-18ms average shuffle execution and an average reshuffle blind window of 2.6% of the 3-s decision cycle.

The star formation efficiency per free-fall time ($\epsilon_{\rm ff}$) quantifies how efficiently giant molecular clouds (GMCs) convert gas into stars and is often observed to be orders of magnitude lower than expected for free-fall collapse. Observers typically estimate $\epsilon_{\rm ff}$ by mapping far-infrared (FIR) dust emission to infer gas surface densities ($\Sigma_{\rm gas}$) and counting embedded protostars, assuming simplified lifetimes and masses. Using the fiducial STARFORGE radiation-magnetohydrodynamics simulation of an $M_{\rm cl} = 2\times10^4 M_\odot$ GMC that self-consistently models star cluster formation and stellar feedback, we generate synthetic FIR observations and apply the same methodologies used in GMC surveys to investigate this discrepancy. We present the first $\epsilon_{\rm ff}-\Sigma_{\rm gas}$ analysis in a fully feedback-regulated GMC simulation that resolves the formation of stellar systems. Our synthetic measurements reproduce the low observationally inferred efficiencies, showing that feedback-regulated star formation naturally produces $\epsilon_{\rm ff} \sim 1-3\%$ without requiring extreme initial conditions. We also find that $\epsilon_{\rm ff}$ varies strongly over a GMC's lifetime, suggesting that much of the observed scatter reflects evolutionary sampling rather than intrinsic cloud-to-cloud differences. Finally, by comparing observational and simulation definitions, we show that methodological assumptions introduce systematic biases, with a transition near $\log \Sigma_{\rm gas} \approx 2.3 M_\odot {\rm pc}^{-2}$ that depends on resolution. Above this threshold, the smoothing of dense structure increases both the inferred free-fall time and enclosed gas, with the latter dominating and suppressing $\epsilon_{\rm ff}$. Below the threshold, the discrepancies are primarily driven by star formation rate assumptions.

Parameter-efficient fine-tuning has hundreds of methods, usually compared by parameter count and benchmark score. We compare them as maps. A pretrained checkpoint is a base point in weight space, and a weight-space method is a smooth map from a small parameter space that starts at that point. These maps form a slice category in which the frozen model is initial and full fine-tuning terminal. Three invariants of the map do most of the work. Its image near the base point governs expressivity, through the first-order image, the dimension, a min-cut bound on rank and whether that point is smooth or an apex. Its fibres, with the optimizer's metric, govern training dynamics. Base change moves the base point and covers splitting initialisations, quantisation, restarts and merging. We prove, for example, that zero-initialised LoRA, and Householder reflection adaptation at its default start under stated rank and parity conditions, begin at the apex of a cone; that no exact initialisation lets LoRA reach an update of rank above twice its own; that exactly orthogonal methods never change a weight's singular values in exact arithmetic, however often they are merged; that under Adam with no epsilon, weight decay or clipping and one shared schedule, LoRA+ from the standard zero initialisation follows exactly the trajectory of plain LoRA with its scale multiplied by the learning-rate ratio; that Adam with shared hyperparameters respects only the signed permutations inside LoRA's continuous symmetry group of full-rank factors; and that the activation function decides when a scaling adapter folds into preceding weights. We classify 382 methods from 138 papers by seven coordinates read off their formulas, with 403 simulation arrows and 485 obstructions. Every numbered result is proved in full, and results due to others are credited where we re-derive them.

Feed-forward 3D foundation models reconstruct scenes from unposed images, but the quadratic cost of global attention limits their scalability to large view sets. We present Zebra3D, a layerwise sparse-linear hybrid attention framework that improves the efficiency of pretrained 3D foundation models without training from scratch. Across VGGT, HunyuanWorld-Mirror, and $\pi^3$, we observe a broadly consistent depth-wise pattern: early stages are relatively tolerant to linear replacement, while sensitive middle-to-late stages tend to exhibit more concentrated attention. Zebra3D exploits this structure by assigning linear-centered attention to tolerant stages and tile-sparse attention to sensitive stages, using landmark summaries to preserve unselected context. Closed-form affine fitting enables training-free conversion, while optional post-conversion training combines task supervision with output and relational feature distillation. Across all three backbones, Zebra3D achieves strong reconstruction and camera-pose accuracy at high attention sparsity, with training-free variants remaining close to their dense counterparts. End-to-end speedups increase with input view count, reaching up to $7.1\times$ at 1,024 views. After post-conversion training, Zebra3D-VGGT reaches 80.5% RealEstate10K AUC@30 at 94% mean attention sparsity, within 1.1 percentage points of dense VGGT fine-tuned on the same data and schedule.

We report a comprehensive muon spin rotation and relaxation (muSR) study of the hexagonal 6H-perovskites Ba3LnRu2O9 (Ln = Gd, Ho) to investigate the role of rare-earth anisotropy in the magnetic ground state of 4d-4f coupled systems. Ba3HoRu2O9 exhibits two successive magnetic orderings at TN1 = 46 K and TN2 = 10 K, leading to a complex magnetic ground state, whereas Ba3GdRu2O9, with Gd3+ (L = 0) S-state ions, shows primarily single magnetic ordering at TN = 14.5 K. Zero-field (ZF) and longitudinal-field (LF) muSR measurements reveal markedly different magnetic behavior despite their structural similarity. Ba3HoRu2O9 displays a multi-scale inhomogeneous magnetic state characterized by two-component exponential relaxation, broad local-field distributions, incomplete LF decoupling, and persistent spin dynamics. The muSR results identify two characteristic magnetic energy scales near 46 and 10 K, associated with correlated Ru-Ho interactions and quasistatic magnetic ordering, respectively. Redfield analysis further demonstrates strong local magnetic fields and distributed spin fluctuation timescales, confirming magnetic inhomogeneity. Machine-learning-assisted muon-site and dipolar-field calculations show that the broad field distribution naturally arises from muons sensing distinct coexisting magnetic domains at different structural sites. In contrast, Ba3GdRu2O9 exhibits a comparatively homogeneous antiferromagnetic ground state below 14.5 K, with Gaussian relaxation consistent with a narrow static field distribution. These results establish that the rare-earth ion in Ba3LnRu2O9 is a powerful tuning parameter, driving the ground state from homogeneous to complex, multi-scale inhomogeneous behavior through competing 4d-4f magnetic interactions.

We prove that isolated spectral clusters preserve their finite-group representation under equivariant spectral convergence. In particular, an irreducible limiting eigenspace cannot split into several nearby eigenvalues, while an isotypic limiting eigenspace imposes corresponding divisibility constraints on the multiplicities in the approximating cluster. We then construct a one-parameter equivariant Steklov homogenisation on closed orientable surfaces, allowing Laplace spectral representations to be transferred to Steklov spectral clusters. Applied to the hyperbolic surfaces of Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault, this yields first Steklov multiplicities 16 and 21 in genera 10 and 17, and at least 24 in genus 37. These examples give counterexamples to Jammes's conjecture on first Steklov multiplicity.

The strong Szegő theorem and the Fisher-Hartwig formula describe the asymptotic behavior of determinants obtained by restricting translationally invariant operators to large finite regions. In particular, Szegő's theorem states that, beyond the leading extensive contribution to such a determinant, there is a universal finite correction, $\sum_{r>0} rV_rV_{-r}$ where $V_r$ are the Fourier coefficients of the logarithm of the symbol. Although this formula appears throughout analysis and mathematical physics, the origin of its quadratic form and of the weight $r$ is often obscured in its standard analytic derivations. We develop an elementary picture in which the strong Szegő term arises from boundary leakage: paths contributing to the matrix exponential leave the finite interval and subsequently return. Counting the shortest such excursions immediately produces the factor $r$. We then show that an isolated boundary contributes exactly one half of the strong Szegő correction, while the two boundaries of a long interval decouple. For finite-range matrices this gives a rigorous and intuitive elementary derivation of the asymptotics, with extensions to sufficiently rapidly decaying interactions. A complementary free-fermion argument provides a physical interpretation of the same cancellation mechanism. Numerical examples illustrate the asymtotics in the well controlled regime, the Fisher Hartwig logarithmic correction. The boundary formulation does not itself require translation invariance and therefore suggests natural extensions to disordered systems, higher dimensions, and symbols with internal matrix structure. To illustrate this we numerically go beyond the formal applicability region of the Szegő formula, considering slow decay and disordered cases, and showing in some cases a robustness of the approach even for slowly decaying but disordered potentials.

Neural surrogates that forecast a physical field from sparse, noisy sensors read a window of nearby sensors and a history of past snapshots. Practitioners choose both by comparing fitted models with each other. Such comparisons cannot tell a model's shortfall from missing information. We measure the trade-off between window and history against the Bayes-optimal error of the same readings. An exact Gaussian analysis shows that noisy or sparse observation of a field with local dynamics makes the optimal forecast read distant sensors only through spatial correlation. On a forced linear field with a known optimum, trained models that read only the window come within 1% of it. We then train such models on viscous Burgers flow and the chaotic Kuramoto-Sivashinsky (KS) equation, with every deciding comparison fixed before any test evaluation. In every field except KS, eight snapshots remove a smaller fraction of the one-snapshot error as the lead time, the number of steps to the forecast target, grows. In KS, for windows up to one correlation length, they remove a larger fraction. This lead-time inversion persists across sensor densities, three architectures and a two-dimensional domain. It shrinks as damping lowers the Lyapunov exponent. Damping also removes the linear instability of KS. We therefore tie the inversion to instability and leave chaos and linear amplification unseparated. For surrogate design the results give two rules. Size the window in correlation lengths of the field. Split a fixed input budget between window and history for the lead time the model will serve, because the best split moves toward wider windows as the lead time grows. In Burgers flow the split chosen at lead time one ends with 13.5 times the best error at lead time eight.

We establish a nonasymptotic Gaussian approximation bound over centered Euclidean balls for sums of independent, not necessarily identically distributed, random vectors in \(\mathbb R^p\). Let \(S_n\) denote the sum, let \(\Sigma=\operatorname{Var}(S_n)\), and let \(G\sim N(0,\Sigma)\). Define the effective rank and the normalized squared spectral mass by \[ r(\Sigma)=\frac{\operatorname{tr}\Sigma}{\|\Sigma\|}, \qquad v(\Sigma)=\frac{\operatorname{tr}(\Sigma^2)}{\|\Sigma\|^2}. \] Under a uniform directional fourth-moment condition, covariance-balance assumptions, and the spectral conditions \(r(\Sigma)\le C_\Sigma v(\Sigma)\) and \(v(\Sigma)\ge6\), we prove \[ \sup_{z\ge0} \left| \mathbb P\{\|S_n\|^2\le z\} - \mathbb P\{\|G\|^2\le z\} \right| \le C\,\frac{r(\Sigma)^{3/2}}{n}. \] The constant depends only on the fixed parameters in the assumptions and is independent of the ambient dimension and individual small eigenvalues of the covariance matrix. The bound is uniform at the endpoint \(v(\Sigma)=6\), without a logarithmic loss. Its main feature is the combination of the \(n^{-1}\) rate with dependence on effective rank, including this endpoint. Consequently, the bound can vanish when \(r(\Sigma)=o(n^{2/3})\), even when \(p\) is much larger than \(n\). The same estimate extends to random elements of a separable Hilbert space with trace class covariance. The proof builds on the Fourier and symmetrization methods of \citet{GoetzeZaitsev2014}, combined with moment matching, truncation, and comparison estimates that retain the covariance structure.

Neural crest cells (NCCs) provide a well-established system for investigating collective migration during embryonic development and exhibit migratory behaviors with parallels to cancer invasion. Previous studies establish roles for co-attraction, contact inhibition of locomotion, polarity persistence, and tissue confinement in guiding NCC migration. However, it remains unclear whether NCC migration in confined environments occurs through spatial restriction alone or also through a repolarization response induced by contact with boundaries of confinement. By developing a computational model that couples self-generated chemical transport with contact-mediated repolarization and mechanical interactions, we show that cell-boundary contact can reorient cell polarity and sustain directional migration under confinement, whereas no persistent collective direction emerges without confinement. Moreover, suppressing contact-mediated repolarization from non-permissive confinement boundaries while preserving the mechanical interactions prevents sustained directional migration, suggesting that spatial restriction alone is insufficient in the model. Together, these findings establish an experimentally testable mechanism through which self-generated chemical signals and contact with confining tissues jointly regulate collective guidance of NCCs. These insights provide a basis for investigating developmental migration defects and potential strategies to regulate collective tissue invasion.

A stable matching in a bipartite preference system is a matching that has no unmatched pair whose endpoints both prefer each other to their current assignments. Rothblum gave a linear description of the stable marriage polytope, the convex hull of the incidence vectors of stable matchings. We prove that this polytope is box-totally dual integral. The proof uses the affine representation of the stable marriage polytope as an order polytope together with the matricial characterization of box-TDI polyhedra. Since Kiraly and Pap proved that Rothblum's description is totally dual integral, our result further implies that Rothblum's system is box-totally dual integral.

We calculate the spectrum of gravitational waves generated by a cosmological magnetic field with a narrow-band power spectrum. For a spectrum narrowly peaked at a comoving wavenumber $k_c$, we find that the gravitational wave signal peaks at $2k_c$ and has a sharp cutoff at this scale. We apply this result to an ultralight dark matter model of magnetogenesis operating around recombination, in which an axion-photon coupling generates a helical magnetic field by parametric resonance. We determine the conditions for the resulting gravitational waves to enter the frequency range of pulsar timing experiments. Although the required frequency can be reached for sufficiently large coupling, the corresponding amplitude is below the experimental sensitivity for physically allowed energy densities of the magnetic field. We generalize this conclusion to models of magnetogenesis operating at other epochs, estimating as $T \simeq 160 \mathrm{MeV}$ the temperature of a magnetogenesis process assumed to be the source of the stochastic gravitational wave background suggested by pulsar timing arrays. We place the lower bound $T \gtrsim 1$-$10 \mathrm{MeV}$ on such viable models.