Papers

Numerical studies suggest that quantum reservoir computing performs optimally near the edge of chaos, a behavior commonly attributed to a balance between memory retention and nonlinear information processing. Here we develop an explicit quantum reservoir model and rigorously analyze its chaotic regime and integrable point. In the chaotic regime, we show that Gaussian-unitary-ensemble statistics lead to rapid memory loss: learning performance decays exponentially with the storage time required by a temporal task. This decay limits the ability of chaotic reservoirs to learn long-range temporal dependencies. At the integrable point, described by a free-fermion model, we prove that amplitude encoding and a linear readout of fermionic two-point correlations cannot learn a representative class of nonlinear tasks coupling present and past inputs. Introducing interactions alleviates this obstruction. By identifying distinct limitations on memory retention in the chaotic regime and nonlinear processing at the integrable point, our analysis provides a theoretical basis for the observed edge-of-chaos advantage and guides the design of quantum reservoirs for temporal learning.

We present a unified, transformation-free linear regression framework tailored for ordinal compositional data, such as distributions across educational levels or aggregate Likert-scale survey responses. Traditional log-ratio approaches often obscure interpretability, ignore the inherent ordering of categories, and struggle with boundary zero values. To overcome these limitations, we propose a deterministic model constrained to the space of column-stochastic transformation matrices, naturally preserving the fundamental geometry of the simplex. By adopting the weighted 1-Wasserstein distance as the loss function, our method directly embeds the ordinal nature of the data into the estimation process. We provide a computationally efficient and globally optimal solution explicitly formulated as a Linear Programming (LP) problem. The framework systematically addresses four distinct regression scenarios, introducing a novel ordinal tensor product to rigorously handle interactions across multiple compositional predictors. Furthermore, we equip the model with a suitable regularization strategy and novel diagnostic tools, including a Wasserstein-based coefficient of determination ($R^2_W$) and an Order Preservation Index (OPI). Extensive simulation studies and two empirical applications demonstrate the robust predictive performance and interpretability of the proposed methodology.

Semantic-driven time-series generation offers a promising way to improve downstream learning in few-shot forecasting, but directly generating numerical sequences from language often fails to preserve the intended temporal structure. We propose VisualBridge, which uses time-series plots as a visual intermediate to bridge high-level temporal semantics and numerical sequences. An MLLM first converts plotted series into structured semantic representations, enabling explicit control over temporal properties such as trend, seasonality, and volatility. We then learn a semantic editing policy with downstream forecasting rewards, allowing the generation process to favor temporal patterns that are beneficial for the target task. The resulting sequences are further modeled by a temporal VAE to produce consistent multivariate augmentations. Experiments on standard public forecasting benchmarks demonstrate that VisualBridge improves few-shot forecasting over conventional augmentation methods, with ablations validating the roles of visual semantic grounding, learned semantic control, and VAE-based generation.

Solid-phase reactions offer a route to integrate superconductors into Ge/SiGe heterostructures, but forming high-quality superconducting layers near the buried Ge quantum well remains challenging. We investigate reactions of V, Ta, and Nb with elemental Ge and with the Si$_{0.2}$Ge$_{0.8}$ barrier of Ge/SiGe heterostructures at processing temperatures of 365-390 $^\circ$C. Cross-sectional electron microscopy and low-temperature electrical transport show that deposition on Ge at 390 $^\circ$C, in which approximately 2 nm metal increments are separated by growth stops for in-situ annealing, produces reactions throughout the deposited films and superconducting transitions at approximately 0.14 K for V-Ge, 2.1 K for Ta-Ge, and 2.8 K for Nb-Ge. The Nb-Ge layer has an out-of-plane critical field of approximately 0.8 T at 2.1 K. In the heterostructure samples, intermixing is confined to regions approximately 5-12 nm thick near the metal-SiGe interface, leaving a substantial residual metal layer. Microscopy and transport together suggest that this residual film dominates the measured superconducting response. These results establish low-temperature formation of superconducting refractory-metal-Ge layers while identifying the limited interfacial reaction as the principal obstacle to extending this approach toward the buried quantum well.

We study the decay \(\eta_c\to p\bar{p}\eta\) by combining the SU(3) flavor structure of the primary production with final-state interactions in the \(p\eta\) and \(\bar{p}\eta\) subsystems, using the chiral unitary approach. The low-energy \(S_{11}\) dynamics, including the dynamically generated \(N^*(1535)\), is incorporated into the complete coupled-channel amplitude, and a phenomenological \(N^*(1895)\) contribution is included coherently. The \(p\eta\) and \(\bar{p}\eta\) invariant-mass distributions are well described, and the \(p\bar{p}\) distribution is predicted as a kinematic reflection of the \(\eta N\) dynamics. The results support the dynamical interpretation of the \(N^*(1535)\) and suggest possible additional \(S_{11}\) strength near \(1.9\)~GeV.

We study how the critical behavior of holographic superconductors controls the interior geometry of charged black holes. Although the exterior condensate vanishes at criticality, the critical zero mode develops logarithmic oscillations near the Reissner–Nordström Cauchy horizon, rendering the interior critical limit nonuniform. We show that the transmitted critical-mode amplitude, together with local Cauchy-horizon quantities, fixes both the leading Einstein–Rosen collapse $\Delta z_{\rm ER}$ width and the phase $\Theta_J$ in the subsequent Josephson regime. If the condensate obeys $t\equiv1-T/T_c\propto\epsilon^{2r}$, then $\Delta z_{\rm ER}\propto t^{1/r}$ and $\Theta_J\propto t^{-1/r}$. This Josephson phase is transferred to the first Kasner epoch, producing oscillatory Kasner exponents and an accumulation law $t_n\sim n^{-r}$ for successive zeros of the scalar velocity. The critical zero mode fixes the leading collapse and phase coefficients, which are quantitatively confirmed by fully nonlinear solutions in both the ordinary and tricritical cases. At low temperature, the interior instead depends on the infrared completion of the zero-temperature geometry. Thus near-critical scaling is governed by critical-mode transmission, whereas the low-temperature behavior is controlled by the infrared endpoint.

Feed-forward 3D models provide strong multi-view geometric priors, while on- line simultaneous localization and mapping (SLAM) relies mainly on local mea- surements and can accumulate drift over long sequences. Existing attempts to combine the two typically treat feed-forward predictions as an external geomet- ric state that is aligned or fused with the online estimate after the fact, which keeps broader multi-view evidence outside the optimizer that refines the SLAM state. We present F2SLAM, which instead converts feed-forward geometry di- rectly into optimization-native target-weight measurements attached to a persis- tent dense factor graph. A high-frequency stream maintains local tracking con- straints and graph connectivity, while a low-frequency stream uses wider multi- view context to selectively refresh existing measurements after a state-consistency check. Both streams constrain the same poses, inverse depths, and optional cam- era intrinsics through a single dense bundle adjustment. Experiments on multiple benchmarks demonstrate consistently strong trajectory estimation and improved dense reconstruction in both calibrated and uncalibrated settings. Notably, the uncalibrated configuration reduces the average ATE RMSE from 0.030 m for the strongest feed-forward baseline to 0.002 m on the Replica dataset.

Electron cotunneling in semiconductor quantum dots is a higher-order tunneling mechanism that connects initial and final charge states through energetically inaccessible virtual intermediate configurations. In quantum-dot arrays, such processes can produce simultaneous charge motion across multiple tunnel junctions and enable transitions that would otherwise be suppressed by Coulomb blockade. Here, building on an earlier proof-of-principle demonstration, we investigate radio-frequency (rf)-driven correlated cotunneling and its application to dispersive quantum-dot readout. Using both theoretical modeling and experimental measurements in two different foundry-fabricated silicon devices, we show that an external rf drive can induce correlated cyclic single-electron transitions, resulting in enhanced charge-polarization of the system. We develop a general theoretical framework for rf electron-cascade readout, which exploits cotunneling events in quantum-dot systems with three or more dots to produce a relative signal enhancement, and extend its experimental demonstration to a gate-sensed device architecture. Our results establish the operating principles of rf electron-cascade readout and provide a route to increased dispersive readout fidelity in semiconductor quantum computing architectures.

Let $M$ be a complete Kähler manifold of complex dimension $n\geq 1$ whose Kähler form is $d$-bounded in the sense of Gromov. We prove that its Bergman space, namely the Hilbert space of $L^2$ holomorphic $n$-forms, is infinite-dimensional. The proof combines Gromov's construction of auxiliary holomorphic line bundles, parameter-uniform $\db$ estimates, and $L^2$ methods with singular weights.

We study tangent flows of noncollapsed Ricci flow limit spaces in the sense of Fang-Li. We prove that the regular part of every negative time slice is geodesically convex: any minimizing geodesic that meets the regular part has its entire interior contained in it. We also identify each negative time slice with the metric completion of its regular Riemannian part, thereby identifying the negative-time tangent flow with its associated metric soliton.

Analytical greenhouse models traditionally rely on the gray approximation, simplifying radiative transfer but treating optical depth as an unconstrained parameter that yields unrealistic surface temperatures. We derive an exact, closed-form solution directly from Schwarzschild's equations for a non-gray atmosphere. While the gray approach is physically inadequate because it over-weights saturated bands that carry negligible net energy transport, we show that the non-gray formulation constrains column optical depth via the surface net radiative flux, naturally accounting for the atmospheric window without ad hoc corrections. The resulting effective optical depth penalizes saturated cores due to the negative spectral covariance between opacity and transmission. This shifts the computed surface equilibrium to an observationally consistent 288 K, yielding a purely radiative Transient Climate Response of +2.18 K under doubled \ce{CO2}, fully consistent with Intergovernmental Panel on Climate Change estimates. The framework eliminates any theoretical basis for greenhouse-effect saturation. While not intended to replace three-dimensional general circulation models, this work reconciles classical radiative transfer with modern climate estimates, demonstrating that proper flux-weighting aligns analytical theory with anthropogenic warming.

The standard Langevin noise formalism (LNF) is a well-established approach for macroscopic quantum electrodynamics in unbounded lossy media, and accordingly, its application to finite objects requires endowing the surrounding vacuum with a fictitious absorption that is taken to asymptotically vanish. Existing justifications for such a singular limiting procedure have been mainly restricted to phenomena governed by vacuum fluctuations and medium-induced interactions (Casimir-Polder forces, thermal radiation, quantum emitter dynamics, etc.), thereby strictly excluding scenarios that involve the scattering of external quantum radiation. Conversely, the modified Langevin noise formalism (MLNF) has recently been established as the complete canonical quantum theory for arbitrary macroscopic systems and electromagnetic processes, entirely bypassing the need for limiting procedures, treating incident quantum radiation fields as independent degrees of freedom, and exactly reducing to the LNF for unbounded media. In this work, we demonstrate that the LNF limiting procedure is universally valid by showing that it recovers any physical prediction of the MLNF across all regimes, including arbitrary quantum scattering processes. Specifically, we introduce a suitable mapping associating any MLNF state and observable with a corresponding entity within the LNF augmented by fictitious vacuum absorption. We then rigorously prove that, as the fictitious absorption tends to zero, every LNF matrix element constructed via this mapping exactly converges to its MLNF original, so that the limiting procedure establishes the asymptotic convergence of the LNF to the MLNF.

We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold $(M,J,\mathbb{T})$, and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial $\mathbb{T}$-invariant generalized Kähler metric with its symplectic form in $2\pi c_1(M,J)$, we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.

On-policy self-distillation (OPSD) of large language models (LLMs) has demonstrated the ability to improve reasoning capabilities while preserving previously acquired knowledge. Despite substantial empirical success, the dynamics of OPSD in continual reasoning remain incompletely understood. Modeling LLM reasoning as search over a directed acyclic graph, we provide a unified theoretical analysis of both the dynamics of post-training—OPSD and supervised fine-tuning (SFT) in continual learning—and the impact of pre-training on subsequent performance. Our findings establish three key insights with an optimization guarantee: (i) OPSD with hints from correct outputs enables continual learning without forgetting by sparse yet effective gradient descent updates induced by the hint structure. (ii) SFT on correct reasoning paths can lead to catastrophic forgetting due to dense updates along the training paths, which overwrite the information previously acquired. (iii) Diversity in pre-training is crucial for enabling a post-trained model to reach a correct output when a rollout starts from an intermediate state. Our results, supported by theoretical analysis, show that reliable continual reasoning depends on how post-training updates interact with the reasoning structure established during pre-training.

How strongly do the remaining diffusion-sampling steps amplify a perturbation at a late latent state? Standard measurements answer this question with newly sampled isotropic noise, even though perturbations encountered during sampling have already been transformed by earlier steps. We compare these two cases directly. For each trajectory, we transport a centered perturbation from an earlier step to a late state, then replay its direction at the same magnitude as a newly sampled isotropic perturbation; both then undergo the same remaining updates. Across the samplers we study, median paired shaped-to-fresh angular-gain ratios range from $1.14$ to $2.35$. Shaped angular gain exceeds its matched fresh counterpart in every trajectory in the original main cohorts. The same pattern appears when endpoint change is measured by latent RMS. The effect also appears in perceptual feature representations: earlier-sampling directions cause larger feature changes at the endpoint, even when the corresponding pixel-space change is comparable. Permuting shaped directions across trajectories weakens the effect, including within class, indicating that the advantage depends on alignment with the receiving trajectory as well as on shared directional structure.

Patient world models used for clinical trial simulation can agree on transition kernels and arm-specific risks, yet disagree on the fraction of patients harmed by switching treatment—the counterfactual quantity that matters for intervention-aware reasoning. We audit this reliability gap in a two-stage shared-response SCM: a categorical intermediate health state is followed by common terminal care. Under independent stages, the sharp harm interval has closed-form endpoints for at most three intermediate states, with an exactness boundary at four states. Declared dependence and response-mismatch budgets yield calibrated outer bounds when stage independence or complete mediation is relaxed; in a symmetric three-state model the entire sensitivity frontier is sharp, $[0,\min\{1/2,1/3+(\rho+\delta)/2\}]$, and shows exactly how budgets erase the gain over endpoint-only bounds. Two eight-variable response LPs propagate interventional uncertainty for finite-sample audits. Exact witnesses verify attainability. On public clinical simulators (EpiCare; sepsis), native configurations show little resolved stage dependence and no additional joint-compatibility gain over pairwise transport—honest negative results for reliability claims. All experiments are locally reproducible; guarantees remain conditional on the stated causal model. The results provide a concrete protocol for deciding when a patient world model is safe to trust for counterfactual harm.

Data-driven mechanistic hypotheses are essential to scientific discovery because they explain how underlying processes produce observed phenomena. AI agents and AI scientists increasingly support scientific data analysis. However, their ability to turn empirical findings into mechanistic hypotheses remains insufficiently examined. To address this gap, we introduce MechHypoBench, the first benchmark for evaluating whether AI agents and AI scientists can generate such hypotheses from empirical data. It combines paper-derived mechanisms from 14 scientific fields with real-world datasets containing 17.98 million records. The construction retains the observational complexity of empirical data while providing a specified underlying mechanism. Agents analyze the observations and propose open-form hypotheses. We develop an evaluation framework that assesses open-form mechanistic hypotheses through their consequences under withheld conditions. Experiments with general agents and AI scientists reveal a substantial gap between generated hypotheses and the underlying mechanisms.

Risk regulation imposes directional constraints on scores; we adopt their strict per-input form – the score monotone non-decreasing in every exposure input – as a normative commitment. Deployed pipelines – monotone hand-crafted aggregates feeding sign-constrained gradient boosting – already satisfy it by composition, so constrained-versus-unconstrained comparisons price a guarantee the incumbent has for free. We instead hold admissibility fixed on both sides and measure what learning the aggregation is worth. Our instrument is a recurrent network whose state is classical risk statistics (an exponentially weighted moving average and a high-water mark with learned transforms), monotone by construction in every input and per MC-dropout sample. The central finding, by functional regression, is a subsumption boundary: a learned monotone channel reproduces the geometrically weighted separable family of hand-crafted statistics, one channel per member, to Spearman $\rho \ge 0.996$, approximates window statistics with measurable ceilings, and fails at consecutivity ($\rho = 0.924$) and time localization (0.628), both structural, and at the exposure floor (0.829), a learnability boundary. One explicit admissible basis repairs each failure (rank correlation 1.000). In or near the separable family, learned and engineered aggregation are substitutes, and the learned channel is never statistically behind at full sample size and specified capacity. Its advantages are incumbent-specific: a committed grid pays up to 0.019 AUC in decay regions it leaves uncovered (the learned channel stays within 0.004 of the strongest engineered consumer at every swept point); the highest-dimensional comparator degrades fastest with scarce data; and beyond the training support, grid-fed tree-ensemble scores go flat while a strictly increasing head keeps ranking. No single incumbent is dominated on all three axes.

The next generation of multimodal research agents must reason over long-lived research histories rather than short model completions. During a single task, an agent may repeatedly search the web, inspect visual evidence, revisit earlier hypotheses, and accumulate tens of thousands of tokens of multimodal context. Despite this trend, online RL for multimodal research agents remains largely confined to shorter contexts and interaction horizons. We push online RL training to 128k context and 75+ tool-interaction turns. To our knowledge, this is the first online multimodal deep-research RL study trained at 128k context, and the first trained with a 75 tool-turn horizon. Scaling to this regime exposes several practical limitations of conventional RL training. Early in training, weak policies make poor use of large interaction budgets, causing expensive rollouts with little reward improvement. Later, policy entropy can collapse before performance has saturated, prematurely ending useful learning. We introduce Long-MDR, a three-component training recipe designed specifically for this setting: On-Policy Distillation Warmup, Progressive Horizon Expansion, and Entropy-Triggered Rescue. Together, these techniques improve both the learning efficiency and stability of long-horizon RL, enabling continued gains in a regime where direct training is slow and costly. Code will be available at https://github.com/navileon/Long-MDR

In this paper, we investigate Bohr-type inequalities for certain integral transforms of some bounded holomorphic operator-valued functions defined on a shifted disc. More precisely, we consider functions $f\in H^{\infty}(\Omega_\gamma,\mathcal{B}(\mathcal{H}))$, where $\mathcal{B}(\mathcal{H})$ denotes the space of bounded linear operators on a complex Hilbert space $\mathcal{H}$, and study the Bohr phenomena associated with the series representations in \eqref{e1.2a} and \eqref{e2.1}. Moreover, we determine the sharp radii $R_\gamma>0$ for which the corresponding majorant series satisfy the required inequalities for all $|z|\leq R_\gamma$. One of our main results extends some recent result due to Kumar and Sahoo \cite{Kumar-Sahoo-Meditarr-2023} to a large extent, and the other one is new in the literature

The present article is a generalization of our paper \cite{GP21}, together with new applications. The original problem came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants cannot increase if we deform a given singularity. In \cite{GP21} we proved this for the completed fiber dimension under deformations with section. In general the nearby fiber contains however several singularities and in this article we prove that also the sum of the completed fiber dimensions behaves semicontinuous. This is well known for analytic or algebraic families. However, for families of power series the problem is more difficult, since the the modules defining the invariants are quasi-finite but not finite over the base space. In fact, in general the usual fibre dimension is not semicontinuous. However, if we pass to the completed fibres in a family of rings or modules over arbitrary Noetherian rings, we can prove that their dimension is semicontinuous. The proof is different from that given in \cite{GP21} and, though more general, even easier. Finally we apply this to prove the semicontinuity of several singularity invariants, such as the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections. We end with a problem concerning the Milnor number of an isolated complete intersection singularity.

The Carina OB1 association, harboring the famous Carina Nebula near its center, is the most massive, well-studied OB association in our Galaxy. Containing several massive star clusters with different ages, it shows a complex spatio-temporal structure. We explore the formation history and kinematic evolution of the eight most massive clusters in Car OB1. We determine the cluster ages, stellar populations, the initial massive star content, and the number of supernova explosions since the time of each cluster's formation. Using Gaia DR3 astrometry, we determine the 3D configuration of the association and perform a traceback analysis of the cluster positions over the last 13 Myr. The traceback analysis places the birth location of the oldest (13 Myr) cluster, NGC 3293, in the Carina spiral arm. Approximately four Myr later, a series of 11 supernova explosions started in NGC 3293. We construct a possible scenario in which the expanding bubble driven by these supernova explosions interacted with the southern part of the original cloud complex and led to the sequential formation of the younger clusters in the central Carina Nebula. Our scenario suggests that the sequence of 16+/-4 supernova explosions in the association during the last 10 Myr played an important role in creating the massive clusters in Car OB1. However, the supernova feedback does not dominate the large-scale expansion of the association; the general expansion rather seems to reflect the initial kinematics of the original unbound cloud complex.

DETR-style detectors use one-to-one bipartite matching during training to assign object queries to ground-truth objects, enabling end-to-end set prediction without non-maximum suppression (NMS). However, without an explicit de-duplication procedure, multiple queries can still produce highly similar hypotheses for the same object, making training unstable and predictions less decisive. Inspired by the sequential ordering of NMS, we propose DETRNN, a plug-and-play module that turns unordered object queries into a competition-aware sequence for recurrent refinement. DETRNN builds an explicit confidence-and-similarity based order from prior predictions, then refines queries with an RNN along this order to model competition inside the decoder. This ordered recurrent refinement reduces redundant predictions, stabilizes optimization, and improves final detection accuracy. Experiments on multiple DETR-style detectors show consistent gains with comparable efficiency. Code repository: https://github.com/nigelwoo2008-coder/DETRNN.

Generative large language models (LLMs) inherit undesirable behaviors from pre-training, including demographic bias and toxic generation, that often emerge only after deployment and affect a small subset of inputs. A repair should eliminate the identified defect, preserve the model's overall functionality and, ideally, provide correctness guarantees. Existing approaches address this only partially: gradient-based fine-tuning lacks per-instance guarantees and becomes unstable with few defect samples; model editing assumes explicit knowledge replacement rather than behavioral correction; and constraint-based repair is largely restricted to discriminative models with unique target outputs. We present FORGE, a framework for targeted behavioral repair of generative language models that separates defect localization, weight editing, and behavioral verification into independent stages. Its core is a repair abstraction that converts localized defective generation into explicit optimization objectives, enabling verification-oriented repair techniques to operate on autoregressive generation. FORGE is editing-mechanism agnostic: we instantiate it with (1) a constraint-based quadratic optimization method that provides per-sample repair certificates and (2) a null-space projection editor that minimizes interference with the original model distribution, both under the same localization and verification protocol. On five open-source LLMs, FORGE consistently achieves larger reductions in bias and toxicity than gradient-based fine-tuning with minor perplexity degradation. The two backends exhibit complementary performance across architectures, which a lightweight causal probe traces to where toxicity-related signals concentrate. FORGE also remains effective with only a handful of defective examples, where conventional fine-tuning often oscillates or fails to converge.

Code switching moves an encoded qubit between two codes whose transversal gates together are universal, but the switching operation itself is usually not transversal and can spread a single fault into an uncorrectable error. We study switching between two permutation-invariant codes on the same qubits, one with transversal Clifford gates and one with transversal gates from the non-Clifford binary icosahedral group. Both are compatible with transversal $X$, $Z$, and $F = HS^\dagger$, and the encoders with this property form a small linear space. We show that its codes of distance at least three are cut out by at most two quadratic equations, and that all of them have the same error geometry in the sense that their single-qubit error states have the same inner products. It follows that a switching unitary commuting with qubit permutations and with the shared transversal gates at every instant, and tolerating one single-qubit fault at an unknown time, exists if and only if the two codes are joined by a continuous path of distance-three codes. Such a switch can be made error transparent, so that a fault at any moment reaches the target code as the same single-qubit error. For real encoders, the distance-three codes are the solutions of a single quadratic equation, and on thirteen qubits, they form an explicit circle through two Clifford codes and an icosahedral code. We construct the switching unitary along this circle as a collective operator, leaving its synthesis into physical control open, and verify that its recovery infidelity per fault is at machine precision, whereas two natural alternatives leave average infidelities of $0.13$ and $0.85$. A survey over odd qubit numbers up to $29$ lists where such switches exist.

Surrogate modelling is important to enhance the computational efficiency of uncertainty quantification for physical processes involving random parameters. The advantage of Fourier neural operator (FNO) compared to conventional surrogates is its cross-resolution prediction capability. In the current study, a cross-resolution surrogate based on FNO is built for two-phase transient Darcy flows in porous media with random parameters. A multiresolution Sobolev learning method is proposed for FNO to enhance the cross-resolution prediction accuracy by incorporating first and second-order gradients in the loss function using training samples of different resolutions. The gradients are approximated by resolution-related finite difference schemes compatible with the grid for numerical simulation. Experimental results demonstrate the effectiveness of the new method in training FNO for building cross-resolution surrogates.

We revisit the Gentzen-style monadic translation of Gödel's System T. The translation is parametrized by a nucleus, a monad-like structure that need not satisfy the monad laws. A fundamental theorem of logical relations provides a uniform correctness argument for its instances. By choosing suitable nuclei, we obtain majorants, moduli of pointwise and uniform continuity, internal dialogue trees, and functionals of general bar recursion, all represented by terms of T. The internal dialogue-tree application gives a simpler construction, with correctness established directly using Church-encoded terms. We also extend the translation and its fundamental theorem to sums and finite lists. Finally, we develop the Kuroda-style translation and its continuity instance, comparing the moduli obtained with those of the Gentzen-style translation.

We prove that no single rank ordered adversary strategy is globally optimal for the prediction with expert advice problem with six or more experts, in both the geometric stopping and finite time horizon settings. The proof is based on establishing a leading order correction when one expert moves far ahead of the others. This allows us to connect optimal strategies between $n$ and $j<n$ experts and utilize recent results on the exact optimality set for the five expert problem.

GUI grounding is a critical capability for GUI agents powered by vision-language models, helping them execute user instructions by locating the corresponding elements in screenshots. Single-step grounding struggles with small elements and dense layouts, motivating multi-step visual search. However, existing approaches commonly rely on textual reasoning misaligned with visual space or costly multi-round interactions with external visual tools. To make multi-step visual search an explicit spatial process within the model, we propose ReLaViS, which performs Recurrent Latent Visual Search in a single interaction round. At each step, a spatial search head uses the hidden state to query the screenshot's visual tokens, producing a spatial search distribution that explicitly represents the search focus. This distribution then aggregates the visual tokens into latent visual evidence, which is recurrently fed back as the next input embedding to condition subsequent search. We further introduce a GUI-aware coarse-to-fine inductive bias through trajectories constructed from flat element annotations, supervising search from the global interface through intermediate element groups to the target. Built on Qwen2.5-VL-7B, ReLaViS improves ScreenSpot-Pro accuracy by 3.1 percentage points to 56.3% with only a 3.5% increase in inference FLOPs and outperforms the matched single-step baseline on all five benchmarks.

Consider the nearest-neighbor antiferromagnetic Ising model on $\mathbb{Z}^d$, $d\ge2$, at inverse temperature $\beta$ and in a uniform external field $h$. Let $\beta_c(d)$ be the critical inverse temperature of the zero-field Ising model. We prove that for every $\beta>\beta_c(d)$, the Gibbs state is not unique whenever $|h|$ is sufficiently small. This extends a previous result of Dobrushin (1968) from sufficiently large $\beta$ to every $\beta>\beta_c(d)$.