Papers

We prove that, in every dimension, a bounded, non-constant entire solution of the Allen–Cahn equation whose zero set is contained in a half-space is a one-dimensional heteroclinic solution.

In this article, we prove a stability result for the identification of the coefficients appearing in a coupled relativistic Schrödinger system posed on a bounded domain of $\mathbb{R}^{1+n}$, where $n \geq 3$. More precisely, by utilizing input-output operator, where Neumann boundary data are available only on a subset of the boundary, and constructing an appropriate geometric optics solution, we derive a stability estimate of $\log$-$\log$ type for the matrix-valued first-order coefficients. This result is subsequently used to obtain a $\log$-$\log$-$\log$ type stability estimate for the zeroth-order matrix-valued potential.

Reliable individual monitoring is essential for dairy cattle welfare and early heat stress detection in Precision Livestock Farming (PLF). However, translating raw multi-camera videos into automated diagnostics remains challenging due to disconnected tracking and recognition pipelines and complex physical environments. Dense mutual occlusions and cross-camera viewpoint variations fragment identities, while environmental noise, including camera vibrations, and limited agricultural data obscure subtle heat stress cues such as rapid breathing and flank undulations. To bridge low-level visual features and high-level behavioral diagnostics, we propose CowCare, a unified framework combining viewpoint-robust tracking with spatiotemporal heat stress recognition. Its tracking module, CowTrack, maintains cross-camera identity consistency through a View-Decoupled Aggregation Network (VDA-Net). Instead of relying on static global representations, VDA-Net aggregates localized textures under heavy occlusion and uses a Viewpoint Decoupling Transformer (VDT) to suppress transient viewpoint variations. CowTrack further incorporates a causal online self-supervision mechanism for continuous adaptation to dynamic farm scenes. The resulting tracklets are subsequently processed by TSDF-Net, a fine-grained video understanding network. Its Temporal Stabilization and Aggregation Module (TSAM) suppresses high-frequency jitter, while a Diffusion Semantic Prior (DSP) uses frozen pre-trained diffusion semantics to enhance subtle behavioral signals and mitigate overfitting. Extensive evaluations on two real-world farm benchmarks demonstrate the framework's effectiveness: CowTrack achieves 83.48% HOTA and 87.72% IDF1 for global multi-object tracking on MVCow, and TSDF-Net achieves 87.72% Top-1 accuracy on HSCow. These results establish CowCare as a robust end-to-end framework for automated livestock health monitoring.

Automated decision makers, such as AI agents that control robots or network devices, must often satisfy hard cost constraints while maintaining competitive performance. In such settings, a cost measure, such as a safety violation rate, energy consumption, or a distortion level, must stay within a given budget in every episode of operation. Furthermore, a cumulative reward metric, accounting for per-step performance, must stay close to the reward of a meaningful reference policy. We formulate this setup as an episodic sequential decision problem in which the state sequence is arbitrary, and the reward and cost may depend on past actions within the same episode. We propose the online conformal episodic agent (OCEA), a policy that selects one candidate action per episode using a regret-controlled algorithm, and switches to a zero-cost action when the remaining budget cannot absorb the maximum cost of one more round. We prove that OCEA satisfies per-episode cost constraints for every state sequence and for every realization of its internal randomness, while achieving a sublinear expected policy regret with respect to the best action sequence in hindsight from a comparator class satisfying the same per-episode cost constraints. We apply OCEA to risk-averse control and online compression with distortion guarantees.

Multi-environment POMDPs (ME-POMDPs) generalize POMDPs by encoding a finite set of possible environments, capturing model uncertainty. The finite-horizon robust reward maximization problem for a ME-POMDP asks for a policy that maximizes reward in the worst environment, and recent work has shown this problem to be $\mathsf{PSPACE}$-complete. We expand the complexity landscape of ME-POMDPs by identifying which subclasses retain this $\mathsf{PSPACE}$-completeness. In particular, we take subclasses of POMDPs that are tractable in the single-environment case, and investigate the computational complexity of these classes when extended to the multi-environment setting. We first prove the $\mathsf{NP}$-completeness of multi-environment blind MDPs and deterministic POMDPs using a certain closure-under-mixture condition. We then prove $\mathsf{PSPACE}$-completeness for a range of subclasses that do not satisfy that closure condition, yielding new complexity results for multi-observation POMDPs. Finally, we show existential theory of the reals ($\exists \mathbb{R}$)-completeness for numerous ME-POMDP problems under time-dependent Markovian policies or finite-state controllers that are $\mathsf{NP}$-complete for standard POMDPs. These results reveal a subtle relationship between POMDP and ME-POMDP complexity that depends heavily on the subclasses involved.

In this follow-up to \cite{Sua25}, we extend the results to the coupled anomaly flow which incorporates the evolution of a Hermitian metric on a holomorphic vector bundle over a compact complex $3$-fold with general slope parameter $\alpha'$. This yields a similar sufficient smallness condition on the slope parameter $\alpha'$ in order to extend the flow on $[0, \tau)$ to $[0, \tau + \epsilon)$. We also augment the argument of \cite{Sua25} slightly to fix the peculiarity of the units in the condition for $\alpha'$ for the $k = 1$ case.

Language models are frequently aligned to human preferences via imperfect evaluators whose errors can be exploited during optimization. We train policies to search for comparisons on which an evaluator may reverse the underlying preference, providing candidates for additional supervision. Since human preferences are unavailable during policy training, we use disagreement among an ensemble of preference models as a proxy for prediction error. We formulate this search as a zero-sum Stackelberg game between two opposing policies, a Leader which tries to minimize disagreement, and a Follower which maximizes it. For a general class of finite contextual Stackelberg games, we establish equilibrium existence and, under regularization, uniqueness and a closed-form characterization. We introduce Stackelberg Mirror Descent-Ascent (SMIDA), a two-timescale algorithm with linear last-iterate convergence in the tabular setting, and develop a neural policy-gradient analogue. Experiments with a synthetic oracle support disagreement as an informative proxy for prediction error. On held out data the trained policies outperform reference policies and increase the fraction of comparisons that exceed a disagreement threshold and reverse the oracle's preference.

The spatial structures of chemical populations constrain Galactic disk evolution and its underlying physical processes. Gaia XP spectra provide one of the largest spectroscopic samples available. We use Gaia XP spectra to study Galactic disk mono-abundance populations (MAPs), measuring their scale heights as a function of radius and radial surface-mass-density profiles. We construct a red giant branch (RGB) sample and divide it into MAPs according to $[\alpha/{\rm M}]$ and $[{\rm M/H}]$. The sample covers a $6\times6$ kpc region centered on the Sun in the Galactic plane and extends to $|Z|<3.7$ kpc. We estimate XP completeness using a Gaia photometric parent sample and combine PARSEC isochrones with a Kroupa initial mass function (IMF) to calculate the selection function and convert observed stellar counts into intrinsic stellar mass densities. We fit vertical density profiles with an exponential model to derive scale heights and integrate the fitted profiles to obtain surface mass densities. The resulting profiles are generally consistent with those from APOGEE. All MAPs show flaring, with the high-$\alpha$ population exhibiting the strongest flaring and the metal-poor and sub-solar low-$\alpha$ populations showing the weakest. Surface-mass-density profiles differ significantly among MAPs: the high-$\alpha$ and metal-rich low-$\alpha$ populations show approximately exponential declines, whereas the other low-$\alpha$ populations exhibit plateau-like features, indicating that the Galactic disk cannot be described by a simple single-exponential profile. These differential structures suggest a complex Galactic disk assembly history likely regulated by birth conditions, secular dynamical heating, radial migration, and external perturbations.

We present DisCo algebra—a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a magma-operad (non-associative partial combination, restricted to the image of disintegration) on the bare family $\mathcal{P}_n=\mathrm{Prob}([0,1]^n)$. No probability integral transform, no external summary functions, no enrichment of objects with conditional kernels. The disintegration side uses block-wise joint CDFs as intrinsic, measure-determined 1D summaries, yielding a lossy compression. The combination side reads the same summaries and reconstructs via regular conditional measures, yielding a lossless embedding on admissible inputs. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not—it defines a renormalization operator $R_{\mathcal{I}}=\text{Combine}\circ\text{Disintegrate}$. The information loss of this operator is exactly the probability curvature $\mathcal{K}_{\mathcal{I}}(\mu)$—the conditional KL divergence between the true conditional distribution and the conditionally independent product. This curvature measures the obstruction to lossless decomposition and reveals a hard limit on how much cross-block dependence can be captured by CDF-based scalar block summaries.

Transformer architectures have achieved strong results in natural language processing and computer vision, but their deployment is limited by high computational, memory and storage costs, motivating research into model compression techniques, such as pruning, while preserving performance. Prior work has shown substantial redundancy in large language models (LLMs), and redundancy in vision transformers has also been investigated, but it remains less systematically analyzed than in LLMs. In this work, we extend similarity-based redundancy analysis to vision models by measuring cosine similarity between layer inputs and outputs to identify transformations with minimal impact. We study redundancy patterns across several transformer architectures and find that deep vision transformers such as DINOv2 and SwinV2, like language models, exhibit significant redundancy. We then evaluate how this redundancy can be exploited to improve efficiency, analyzing the trade-off between inference speed and performance degradation under different pruning strategies. Our results show that moderate pruning can yield substantial acceleration with limited impact on accuracy, and that 20% of the DINOv2 blocks can be removed without any retraining, for a 1.24x throughput gain and a 20% lower energy per image at a relative accuracy loss below 1% on ImageNet-1K. We further propose a model selection pipeline that automatically identifies the best pruned configuration for a given dataset, reducing FLOPs by 13% to 27% on average across architectures with an average accuracy change between -0.3 and +1.0 points.

We study the exponential Green energies associated with a finite point configuration on a Riemann surface. We focus on studying optimal or close-to-optimal configurations, and show that they must be well-spread out on the surface. We study in particular the round sphere and obtain values for the minimal Green energy of a configuration. The first version of this paper was written with the help of ChatGPT.

Brain tumor classification from MRI scans represents a critical challenge in medical imaging, where accuracy directly impacts patient outcomes. This paper introduces GaborFlow, a novel hardware-software co-optimization framework that revolutionizes the design of the Gabor Filter accelerator for generating feature maps to enhance brain tumor detection. Unlike conventional approaches that rely on fixed filter parameters, our method employs trainable Gabor coefficients within a carefully optimized range, integrated with an EfficientNet-B0 backbone CNN architecture. The innovation lies in the synergistic combination of software optimization through Bayesian hyperparameter tuning and hardware efficiency through precision-optimized multipliers. Our GF baseline accelerator achieves remarkable results, with 98.6% classification accuracy, 28.08% power benefits, and 48.83% footprint savings compared to state-of-the-art implementations, demonstrating that medical AI can be both highly accurate and energy-efficient. The hardware design files are made freely available for easy adoption and further usage by the researchers and designers.

We prove a compact two-weight $T1$ theorem for the $\alpha$-fractional Riesz vector, $0\leq\alpha<n$, when one measure is supported on a compact regular $C^{1,\mathrm{Dini}}$ curve $\Gamma$. The measures may have common point masses. Under the global boundedness assumptions, we give a two-sided estimate for the essential norm in terms of Muckenhoupt and testing constants computed after restricting both measures to small cubes. Compactness is equivalent to the uniform vanishing of these local bounds as the cube side length tends to zero. The smooth truncations have a unique weak operator limit and converge in operator norm in the compact case. For arclength measure and distance power weights, we determine the compactness ranges for Green gradients on bounded tubes and for Newtonian gradients and Biot–Savart fields on the whole space. The whole-space operators in these ranges are compact but are not Hilbert–Schmidt. When $\alpha=n-1$ and both measures have positive continuous arclength densities $a,b$ on the same closed curve, we obtain the exact essential norm $\pi c_{n-1,n}\max_\Gamma\sqrt{ab}$.

Multi-robot systems that combine the complementary capabilities of specialised platforms are critical for complex missions in challenging, inaccessible environments. Current deployments assign an operator or operator team to each robot, with collaboration relying on operator-to-operator communication to manually coordinate heterogeneous platforms with incompatible interfaces. This operator-dependent approach increases cognitive load and limits scalability across robots from different manufacturers. This study designs and evaluates a shared-visualisation collaborative teleoperation interface to improve operator coordination, situational awareness, and the task performance in multi-operator, multi-robot systems. A comparative user study with 16 participants compared three teleoperation conditions: co-located operators with a shared interface, remote operators with a shared interface, and co-located operators with individual interfaces. Participants controlled two simulated robots in a nuclear power plant digital twin to complete disaster response tasks. Performance metrics, physiological data, and behavioural measures were recorded to assess task performance, cognitive load, and operator awareness across conditions. The co-located shared-interface condition achieved the fastest mission completion (292.8 s), outperforming remote operation (351.6 s) and individual interfaces (385.1 s), with the fewest collisions (0.9 per trial) and highest situational awareness accuracy (91.1 %). Physical co-presence was associated with reduced cognitive load, with NASA-TLX scores increasing from 38.2 in co-located to 57.7 in remote operation, supported by consistent physiological indicators. These results demonstrate that co-presence and shared interfaces provide complementary benefits, supporting workload regulation and mutual awareness respectively in multi-operator, multi-robot teleoperation.

Nonlinear response models remain ubiquitous across disciplines with recruitment curves describing muscle response to neural stimulation. Conventional acquisition methods typically rely on static uniform sampling, requiring many stimuli, while adaptive threshold-hunting designs such as ML-PEST forfeit accurate characterization of the overall curve. We present a Bayesian adaptive design that uses posterior uncertainty over curve parameters to propose stimuli that maximize information gain across the full recruitment curve. We additionally implement a focused acquisition method that can target arbitrary subset of curve parameters, which we use to evaluate and interpret our design. In simulations, our design outperforms uniform sampling in recovering threshold while matching ML-PEST, and outperforms ML-PEST in predictive curve accuracy while matching uniform. It likewise matches our threshold-focused acquisition, suggesting our design doesn't trade off threshold estimation for recovering the curve. We further show data collected using our design yields greater power when detecting a change in threshold in intervention studies. Our design proposes the next stimulus within 5 seconds, making it feasible for transcranial magnetic stimulation (TMS) experiments. Finally, we validate our design in TMS experiments and provide a library for Python, called onMEP, for online acquisition of neural recruitment curves.

Steering candidate generation toward user-defined objectives is central to inverse materials design, yet remains difficult when objectives are black-box, changing, or non-differentiable, as in many simulation- and experiment-derived properties. Generative models offer a solution but typically require retraining or fine-tuning for each new objective, demanding labeled data and substantial compute, or rely on differentiable guidance that is brittle to gradient noise and undefined for many practical targets. We present DiffGA, a retraining-free framework that injects evolutionary search into the sampling trajectory of a retrained diffusion model, evolving partially denoised states under an arbitrary black-box oracle. Because oracle evaluation often dominates campaign cost, we couple DiffGA with an on-the-fly Bayesian surrogate that reuses representations from state-of-the-art machine-learned force fields (MLFFs). A Gaussian process is fit during the campaign on the force fields's graph embeddings and, once it demonstrably ranks an unseen generation, replaces expensive property evaluations under an uncertainty-aware acquisition rule, reducing oracle cost with negligible loss of accuracy. Across three crystal-design tasks using a frozen MatterGen backbone, DiffGA outperforms diffusion-only, compute-matched, purely evolutionary, and fine-tuned baselines in oracle-assessed rates of stable, unique, and novel hits. Rather than proposing scientific compounds, we use density functional theory to quantify MLFF oracle fidelity, showing that it is a faithful proxy in key regimes but systematically biased in others, delineating when proxy-driven generative discovery can be trusted. On a representative sample, DiffGA's top hits survive first principles re-evaluation as physically realistic hits, while the surrogate-accelerated pipeline preserves this fidelity at a fraction of the cost.

Tabular foundation models adapt to new datasets through in-context learning (ICL), and, like most transformers, they tie computational depth to the number of independently parameterized layers. We ask whether the ICL stage of such a model needs a distinct set of parameters for every layer, or only depth. We introduce Tab-Loop, which replaces the 12 untied ICL blocks of TabICLv2 with 4 tied blocks applied $k$ times and changes nothing else: the encoders, decoder, synthetic prior and training recipe are shared with an untied baseline trained for the same compute. At matched pretraining compute (18.2 EFLOP) and matched inference compute (12 block applications), Tab-Loop matches the baseline with 62% fewer parameters (10.4M vs.\ 27.6M): held-out log loss is 0.4583 vs.\ 0.4588, mean accuracy on 200 TALENT datasets is 84.10% vs.\ 83.92%, and Elo on 38 TabArena datasets is 1409 vs.\ 1396. A 4-block untied control with Tab-Loop's parameter count, which sees 32% more training tables for the same compute, trails both (83.46%; Elo 1343). Because weights are shared, depth becomes a choice made at inference. Accuracy rises with the number of loops through the trained range and holds beyond it, and continued training with deeper loops extends the useful range, reaching the baseline's stage-2 scores at 4–5$\times$ its inference compute. At this scale, the ICL stage of a tabular foundation model needs depth more than it needs distinct parameters.

While asynchronous execution enables VLA models to operate in real-time environments by predicting the next chunk during execution, it inevitably introduces boundary misalignment with the committed action stream. Existing delay-aware solutions address this at the prohibitive cost of extensive retraining, architectural modifications, or fixed-prefix inpainting that inflates computation when low latency is vital. To address these problems, we propose FreeBar, a training-free real-time inference plugin that enables robust asynchronous execution with only marginal additional inference cost by casting chunk alignment in flow-based VLAs as barrier-guided action-space constraint. Specifically, FreeBar introduces a lightweight Control Barrier Function (CBF)-inspired consistency constraint between consecutive action chunks and solves the resulting correction in closed form, avoiding costly backpropagation through the VLA action head. By progressively tightening the consistency bound during generation, FreeBar aligns generated chunks with the committed execution stream while preserving early-stage generative and within-chunk future action flexibility,. Across Kinetix, RoboCasa, and diverse real-world robotic platforms with multiple VLA backbones, FreeBar improves success rates over naive asynchronous execution by 10%–15% on average, with larger gains in highly dynamic scenarios. It also preserves real-time computational efficiency, decreasing inference latency per action chunk over 45% than RTC. Additionally, as a universal plugin, FreeBar remains robust across distinct latency environments, base policies, and hyperparameter settings.

Choosing a few robust algorithms or schedules can require evaluating every candidate under many adverse scenarios, even though an agent only needs to identify the winners. A predictor may suggest those winners, but cannot rule out a surprise in an unevaluated scenario. We introduce Certified Adaptive Selection (CAS), which returns the exact top-$k$ set as soon as remaining uncertainty cannot change it. CAS keeps guaranteed lower and upper bounds on Expected Shortfall (ES), the average loss in the worst scenarios. Its stopping test compares the worst admissible selected candidate with the best admissible excluded one; for information-disjoint Cartesian intervals this one-cut test is necessary and sufficient. Prediction has a separate role: selecting the next outcome to evaluate, never authorizing the decision. For empirical ES with exact unit-cost queries, we compute in polynomial time the fewest evaluations that could have proved the realized decision in hindsight. Our prediction-guided agent CAS-PG is exact for every predictor, including adversarial ones. On two public algorithm-selection matrices, CAS-PG certifies the exact top three after 630 of 3150 and 981 of 2200 replayed queries, or $1.14$ and $1.18$ times that hindsight minimum. Without historical outcomes, its transcript-only variant requires $1.28$ and $1.37$ times the minimum, outperforming the tested non-predictive certified policies. Separately, a prediction-free CAS policy avoids $93.0\%$ of exact recourse solves in synthetic energy scheduling. One certificate thus supports several acquisition policies and separately proved risk/cost extensions: prediction changes the work, not the justified choice.

Oscillation monitors have root-mean-square (RMS) energy detectors at their core. These detect oscillation in a predefined frequency band by comparing the signal power in that band with a threshold (usually set empirically). A key design decision is the choice of frequency bands. Traditionally, for monitoring the stability of wide-area electromechanical modes, finer bands were set below 1 Hz, while higher frequencies were covered by wider bands. However, this approach can yield poor detection in modern power systems with fast and largely unknown local dynamics. This letter proposes an algorithm to set frequency bands from the ambient noise spectrum to maximize the average detection performance.

We study the relationship between fair division with $\textit{subsidies}$ and $\textit{best-of-both-worlds}$ (BoBW) fairness for indivisible items. We prove a general reduction of BoBW existence to envy-freeability. Specifically, for any scale-free ex-post property $P$ (e.g., EF1, EFX, PO) and any scale-free valuation class $\mathcal{V}$ (e.g, additive, submodular, subadditive, monotone), if every instance in $\mathcal{V}$ admits a $P$-allocation that can be made envy-free using subsidies, then every such instance admits an ex-ante envy-free lottery supported on $P$-allocations. The proof uses linear programming duality. As applications, the reduction gives alternative proofs of BoBW existence results for additive goods, chores, and mixtures of goods and chores. The converse of our general reduction holds for two agents but fails for three: We provide a four-good counterexample with monotone submodular valuations where an ex-ante envy-free and ex-post EF1 lottery exists, but no allocation is simultaneously EF1 and envy-freeable. For lexicographic valuations, we prove simultaneous EFX, Pareto optimality, and envy-freeability (and hence BoBW guarantees for EFX+PO) for three agents, and show that even EFX and envy-freeability can be incompatible for four agents.

Audio source separation models must preserve fine spectral structure while keeping the cost of the separator tractable. Band-split architectures reduce this cost by manually partitioning the spectrum into predefined frequency bands and encoding each band as a token, but the same boundaries are used for every mixture. Because informative spectral structure varies with source content, fixed boundaries can merge discriminative components or spend tokens where little resolution is needed. We introduce Dynamic Band Splitting (DBS), an end-to-end framework that adapts boundary locations and band widths to each mixture under a fixed band-token budget. DBS predicts content-conditioned boundary scores within frequency blocks, selects a prescribed number of boundaries using hard Top-$K$, and applies content-dependent pooling to obtain compact band tokens. After separation, it smooths and expands these tokens to the original STFT grid, refines them along frequency, and decodes complex masks with a frequency-shared head. DBS improved separation over the corresponding static models on DnR v2, MUSDB18-HQ, and EchoSet, with DBS-BSRNN achieving relative gain of 29.0% in average SI-SDR on DnR v2. These results and boundary analyses on DnR v2 show that content-adaptive frequency partitioning benefits multiple separator backbones under a fixed band-token budget. Code and audio demos are available at https://cslikai.cn/Dynamic-Band-Splitting

The empirical pseudopotential method (EPM) computes band structures with one diagonalization per $k$ point and no self-consistency, but its fixed atomic form factors do not transfer between crystal structures. To address this limitation, we introduce EpNet-Si, an EPM for silicon in which an equivariant network predicts each atom's plane-wave form factor from its local environment: a species potential plus monopole, dipole and quadrupole corrections, mixed by a learned router over atomic densities, together with an $s/p$ nonlocal projector. With 6{,}195 parameters, the model is trained through the eigensolver on reference density functional theory (DFT) band energies of only 77 cells. Among 231 distortions of the trained structure types, the 142 gapped cells have mean band and gap errors of 15 and 8\,meV; among 192 unseen structures, the 165 gapped cells have mean errors of 38 and 49\,meV. Physical properties derived from EpNet-Si, including effective masses, gap pressure coefficients, deformation potentials and the density of states at the Fermi level, agree with those obtained from the reference DFT calculations. These results suggest that the classical EPM, fitted to a small set of reference data, can be extended into a fast alternative for modern electronic band-structure calculations.

Richardson varieties provide geometric representatives for products of Schubert classes, with stratifications indexed by the subintervals of Bruhat intervals. We construct analogous geometric models for tilted Bruhat intervals for all split connected reductive groups. Our construction uses Richardson varieties in affine flag varieties and yields abstract tilted Richardson varieties whose isomorphism classes, together with their labeled stratifications, are independent of the choices involved. We obtain an explicit formula for the point-counting polynomials of their open parts in terms of increasing minimal weight paths, extending Dyer's formula for Kazhdan–Lusztig $R$-polynomials. Over the real numbers, their totally nonnegative parts are regular CW complexes homeomorphic to closed balls, which gives a natural geometric realization of the posets of tilted Bruhat subintervals in all Lie types. Over the complex numbers, we prove that their fundamental classes push forward under a continuous loop evaluation map to the Poincaré duals of the minimal-degree coefficients of quantum Schubert products.

We extend the packing results of Etayo, Hedenmalm, and Ortega-Cerda for complex tori to arbitrary compact Riemann surfaces. The construction uses two different approaches, one based on complex geometry of line bundles, and the other on Laplace designs. Both approaches give the same basic existence of close-to-optimal point configurations on the given surface. This version was written with the help of ChatGPT.

Prior-data fitted networks (PFNs) are transformers pretrained on simulated datasets to approximate Bayesian posterior predictive distributions. Standard PFN theory assumes a separable task prior, under which the feature distribution and the mechanism generating labels from features are independent. We show that common PFN priors are non-separable, even in simple settings. Consequently, standard pretraining targets a predictive distribution based on a task posterior updated using both the labeled context and the query features. We characterize its difference from the predictive based on the context posterior alone and show that this difference vanishes under posterior concentration. Under misspecification (i.e., when the true joint distribution lies outside the prior's support), however, prior non-separability can favor better feature fit at the expense of conditional label fit. We illustrate the consequences for feature attribution, Bayesian optimization, and martingale checks of Bayesian behavior. The same dependence also enables PFNs to learn from additional unlabeled observations. We demonstrate this with SemiTabICL, a PFN explicitly pretrained to use these observations, which improves accuracy over a matched supervised baseline. Our findings highlight the need for explicit rules on unlabeled data use in PFN evaluation and establish the dependence between feature and label distributions as a consequential prior design choice.

The average-reward setting for reinforcement learning is challenging because its Bellman operator is known to contract in span only when the underlying Markov chain is mixing. Whereas existing work typically assumes a contraction property, e.g., through environment mixing, we consider the setting where we are given a reward-agnostic $m$-step mixing policy, and we control the contraction and exploration properties of the learned policy by regularizing toward the mixing policy. We provide finite-time analysis of $Q$-learning algorithms that can be instantiated with multiple choices of regularization, with guarantees measured with respect to the regularized fixed point. Our framework exposes novel regularizer-dependent objects that control contraction, exploration, and operator bias. In the one-step setting, our rates match the best known rates for $Q$-iteration and synchronous $Q$-learning, and for $m$-step mixing we derive new rates. We also study $Q$-planning variants, and show that planning acts as time-localized learning, where the planning horizon $H$ trades off the bias from the boundary value against the sample complexity. Finally, we present numerical studies validating the analysis: (i) knowledge of a strong mixing policy improves convergence and exploration, and its advantage increases with bottleneck severity and, (ii) our theory predicts the convergence and bias rankings of regularizers from the $\alpha$-divergence family.

We present a detailed chemo-dynamical analysis of the chemically peculiar very metal-poor ([Fe/H]$=-2.22\pm 0.11$) Milky Way (MW) halo star, 2MASS J00101758$-$1735387 (hereafter J0010-1735), discovered by the $R$-Process Alliance. We derive abundances for 45 elements, including 28 neutron-capture elements, and confirm that \thestar\ is highly $r$-process-enhanced, with [Eu/Fe]=$+1.30$. The light, $\alpha$, and Fe-peak element abundances ratios relative to Fe are all sub-solar, aside from C, Si, and most extremely, Ca. We investigate three possible formation scenarios based on the star's chemical abundances and orbital kinematics: whether it formed in the in-situ MW halo, in a MW dwarf galaxy satellite, or in a satellite of the Magellanic Cloud system. Using the position of \thestar\ in the [Mg/Mn]–[Al/Fe] plane, the abundance pattern of both the light and heavy elements, and the kinematics, we conclude that J0010-1735 most plausibly originated in a now-disrupted ultra-faint dwarf galaxy that was accreted into the MW halo. We examine potential chemical enrichment scenarios of J0010-1735, including one similar to the $r$-process-enhanced UFD Reticulum II, or enrichment via a sub-Chandrasekhar Type Ia SNe, potentially supplemented by a Ca-rich gap transient and by a separate neutron-rich site that facilitated $r$-process enrichment. A nuclear cosmochronometric age of $12.5\pm4.2$ Gyr is derived from the [Th/Eu] ratio, consistent with an ancient origin in one of the earliest star-forming systems in the universe.

The linear viscoelastic behavior of a polymer shows an equivalence of loading frequency and temperature, the time-temperature equivalence (TTE). Horizontal shifting extends moduli measured over limited ranges to much wider ranges, through the glass transition. Multimaterial 3D printing has created digitally mixed polymers, a new class in which elastomeric and glassy resins are mixed voxel by voxel in prescribed ratios. Composition shifts the location and breadth of the glass transition and increases the rubbery modulus. Here we extend TTE to a time-temperature-composition equivalence (TTCE). Across all measured temperatures, frequencies, and compositions, the storage modulus collapses onto one master curve. Composition translates the transition and stretches its breadth, both linearly. The phase angle and relaxation spectrum superpose under the same operations, so the family shares one distribution of relaxation times. A single closed-form TTCE gives the storage and loss moduli of the family, guiding graded and architected designs for elasticity and dissipation.

Let $E$ be the exterior algebra of an $n$-dimensional vector space over a field, let $F$ be a graded free $E$-module with $r$ generators of degrees $d_1 \le \dots \le d_r$, and let $M$ range over the monomial submodules of Amata and Crupi. The Hilbert function of $F/M$ is a shifted sum of $f$-vectors of simplicial complexes; the companion paper determines the convex hull of such sums, and here we count them. For $r = 1$ the count is Linusson's (1999); we show that no new recursion is needed for general $r$. Using two statistics of an $f$-vector – its top nonvanishing degree and the initial degree of the corresponding ideal – we prove that the number of achievable Hilbert functions is a product of $r-1$ transfer matrices, one per generator-degree gap, with entries differences of Linusson's refined counts $E^p(n,k)$. The rank enters only as the length of a matrix product. The same holds for the proper variant. Consequences: a linear recurrence in $r$ for constant gaps, with exact exponential asymptotics $K \lambda^r$ when the gap is positive; polynomiality in $r$ when all gaps vanish; a threshold beyond which the count is a plain power; and polynomiality in the number of vertices.