Papers

We answer in ZFC a question of N. Carlson by constructing a zero-dimensional Tychonoff space $X$, with exactly one non-isolated point, for which every point admits a decreasing local $\pi$-base and \[ \pi\chi(X)=\pi\chi_{\mathrm l}(X)=\omega <\pi\chi_{\mathrm c}(X)=\omega_1 <\pi\chi_{\mathrm d}(X)=\omega_2. \] A preliminary chain-refinement lemma shows that every inclusion-chain of subsets of a set has a reverse-well-ordered coinitial subchain of controlled cardinality. Thus the result is independent of whether a decreasing local $\pi$-base is understood as an inclusion-chain, a strictly decreasing transfinite sequence, or a weakly decreasing transfinite sequence. The construction combines a countable star filter, whose centred local $\pi$-character is at least the pseudointersection number $\mathfrak p$, with a matrix filter that gives the exact values $\omega_1$ and $\omega_2$.

In Recirculating Aquaculture Systems (RAS), precision feeding is critical for minimizing costs and improving fish welfare. However, existing methods lack cognitive alignment between fish behaviors and management knowledge, impeding translation into executable, interpretable feeding decisions. To address this, we propose THPL, a generative feeding decision framework tailored for rainbow trout (Oncorhynchus mykiss) in RAS. First, Fishsort extracts trajectories to establish an Activity Coefficient (AC) quantifying feeding intensity. Second, a Hierarchical Behavior Encoder (HBE) models individual temporal progression and collective dynamics using Temporal and Set Transformers, transforming trajectory tensors into dual-evidence representations of explicit physical and implicit soft tokens. Finally, these tokens are integrated with environmental parameters, metadata, and expert rules to fine-tune an LLM via LoRA, followed by counterfactual multimodal Direct Preference Optimization (mDPO) to reinforce causal reasoning. Results show that AC exhibits a statistically significant monotonic positive correlation with expert-annotated feeding intensity (Spearman $\rho = 0.925$, $p < 0.001$). Ablations indicate that decision accuracy improves from 33.33% (text-only baseline) to 93.33% with dual-evidence tokens, confirming that continuous spatiotemporal tokens provide necessary physical grounding for LLMs. Compared with standard LoRA, counterfactual mDPO elevates decision accuracy from 93.33% to 96.67%, advances METEOR from 58.10% to 85.30%, reduces Self-BLEU-2 from 58.79% to 52.88%, and increases Distinct-3 from 6.68% to 7.81%, suppressing templating and actuation biases while reinforcing causal consistency and operational safety. Overall, by integrating continuous kinematics with LLM reasoning, this study provides a novel decision support paradigm for precision aquaculture.

Sampling from the posterior is the central task of computational Bayesian inverse problems. The standard workhorse in Bayesian inference—Markov chain Monte Carlo (MCMC)—is sequential, yields correlated samples, and must be rerun for each new observation. Conditional flow matching offers an amortized alternative: a transport map, trained once on joint samples of parameter and data, that yields independent approximate posterior samples for any observation at negligible online cost, without new likelihood evaluations. We give a careful, MCMC-literate assessment of flow matching for PDE-based inverse problems with function-valued parameters. Exploiting the flow's tractable density, we derive computable accuracy estimates for the learned posterior in total variation distance and Kullback–Leibler divergence and, moreover, propose a hybrid sampler that is made asymptotically exact by Metropolization. We validate the accuracy estimates and demonstrate amortization in several numerical examples, including electrical impedance tomography and a likelihood-free Lotka–Volterra model. Flow matching reproduces posterior means and credible bands closely at a fraction of the forward solves required by MCMC, but its accuracy degrades for concentrated posteriors and for observations that are atypical under the prior.

Co-designing ML models and the accelerators that run them is an unusual reasoning task: architects must draw confident, high-stakes conclusions about systems that do not yet exist, and the pace of both model evolution and hardware cadence means the analysis burden grows every quarter. The evidence behind each decision–hundreds of gigabytes of fresh simulation sweeps over novel design points–is by construction absent from any LLM's pretraining corpus, and there is no external literature to retrieve; naive "chat-with-your-data" approaches hallucinate exactly where correctness matters most. We present Coco (Copilot for Codesign), an agentic platform deployed with TPU architects that accelerates the co-design lifecycle of setting up experiments, sweeping simulators, and deriving insights. Coco is built as four layers: (i) a datastore that automatically registers every simulation sweep into a normalized relational schema, so agents ground every number in a SQL query rather than scraping heterogeneous files; (ii) a library of tools with typed APIs that agents compose without human orchestration; (iii) agents that encode recurring analysis workflows–most notably iso-execution analysis, which compares systems at matched execution configurations, including swept-but-dominated points off the Pareto frontier; and (iv) a platform UX whose navigation state doubles as agent context. We report early deployment experience toward a reduction in time-to-simulation and time-to-insight, and argue that co-design is a distinct agentic domain: its data must be retrieved rather than memorized, its workflows are recurring but context-dependent, and expert adoption hinges on UX that balances IDE-style control with interactive exploration.

Dento-maxillofacial cone-beam CT (CBCT) reports may contain dozens of tooth-specific, anatomical, and spatial findings from a single 3D scan. Learning to generate such reports from limited clinical data is challenging because routine reports may not exhaustively document image findings, and a non-mention may reflect either absence or non-reporting. We present EviDent-CBCT, an evidence-bottlenecked framework designed for this incomplete supervision. An anatomy-aware network maps each CBCT scan to a discrete record of tooth-level, global, and tooth-IAC evidence. A dental-logic consistency projection reconciles incompatible evidence before a deterministic renderer and an image-blind local language model generate the report using only this record. For tooth-level evidence, reliability-aware training uses eligible non-mentions as reduced-weight negatives, while unreported global and tooth-IAC labels remain unknown. A metal-sensitive input channel preserves intensity cues from dental materials. Across three validation runs, EviDent-CBCT achieves $0.666\pm0.006$ merged evidence set-F1 and $0.402\pm0.003$ RadFact-Lite-Dental logical-F1, versus $0.371\pm0.018$ for the strongest controlled direct baseline. In the ODIN 2026 challenge, it ranked second in automated evaluation and third in blinded clinical Arena comparison on the hidden test set. These results support the discrete evidence record as an effective and auditable interface for CBCT report generation.

Security and privacy notifications, such as login alerts, spam email warnings, and cookie consent requests, play a critical role in shaping users' responses to digital risks. Yet most notifications overlook cognitive accessibility, limiting their effectiveness for people with intellectual and developmental disabilities (IDD). We investigate how adults with IDD perceive and respond to common security and privacy notifications across mobile and web applications. Through a formative user study with seven adults with IDD, we identify three factors shaping understanding and decision-making: (1) interpretation is influenced by task and interface context; (2) unfamiliar terms, both technical and non-technical, are grounded in everyday concepts; and (3) uncertainty about outcomes leads to hesitation, avoidance, diagnostic exploration, or support-seeking. These findings lead to three design implications: (1) address context-dependent language misunderstandings beyond jargon simplification; (2) make action-outcome connections transparent; and (3) enable interdependent decision-making. Together, these insights aim to inform the design of more cognitively accessible security and privacy notifications that better support safe and supported user action.

GraphRAG pipelines construct auxiliary structures during offline indexing–semantic summaries, hierarchical edges, and pre-computed scores–that determine how retrieval is prioritised at query time. Prior attacks target only instance-level components (nodes, edges, triples), overlooking these schema-level structures. We formalise Auxiliary Schema-Level Entity as a novel attack surface and propose the 3S Framework (Semantics, Structure, Scoring) for its systematic exploitation. Our Hop-Decayed Influence (HDI) attack identifies high-impact targets through query-aware influence propagation and corrupts their auxiliary structures post-indexing. Across two benchmarks (HotpotQA, 2WikiMultiHopQA) and two architectures (Microsoft GraphRAG, HippoRAG2), HDI achieves 88-94% attack success rate while modifying as few as 0.016% of auxiliary structures. Each modification affects up to 6.00 queries (Schema Leverage Ratio), demonstrating 1:N amplification unavailable to instance-level attacks. Manipulated structures evade perplexity and paraphrase defenses with over 99% evasion rate, as they remain linguistically coherent system-generated artifacts. These results reveal that auxiliary schema-level entities receive implicit trust without runtime validation, constituting a structural blind spot in current GraphRAG defenses. https://github.com/Jisung-Pacific/HDI-GraphRAG-Attack.

Text-to-image generation enables users to explore several images generated from the same prompt. For these generated images to be useful, each one must reflect the user's preferences, measured by a learned reward, and differ visually from the others to maintain diversity. Existing methods are limited: they either address reward and diversity separately or combine them in one aggregate score, enabling high diversity to offset low rewards. In this paper, we address these limitations by formulating generation as satisficing: every image (candidate) must satisfy a reward floor and the batch of images must satisfy a diversity cutoff. The reward floor controls the balance between worst-candidate reward and batch diversity; we show that varying this floor defines a Pareto frontier. To traverse this frontier, we introduce SatisDive, a training-free inference-time method. SatisDive uses a batch-relative reward cutoff to distinguish lower- from higher-reward candidates, emphasizing reward improvement for candidates below the cutoff and diversity among candidates above it. On Pick-a-Pic, at matched DreamSim, SatisDive improves worst-candidate reward over FK steering by up to 0.43 with FLUX.1-dev as the base model and HPSv3 as the reward, and by up to 0.70 with SANA-1.6B as the base model and ImageReward as the reward. More broadly, across their overlapping DreamSim ranges, SatisDive's satisfaction-diversity curve Pareto-dominates FK steering's curve in each setting.

This paper addresses the following problem: given a planar embedding graph, compute a representation of the shortest-path trees rooted at all the boundary nodes of the graph. Klein gave an $O(n \log n)$ algorithm for this problem; the algorithm subsequently became an essential ingredient in dozens of algorithms, addressing problems ranging from distance oracles to edit distance. However, the correctness and analysis in that original paper is complicated and messy and hard to understand. In this paper, we give a simple and clear analysis.

Massively parallel GPU simulators train multi-robot policies in thousands of environments, and many fleets use private Fifth-Generation (5G) networks, where each robot's delay depends on its teammates' traffic. Network-in-the-loop training places a simulated 5G network inside this loop. However, GPU robot simulators reduce the network to an independent delay per message, while packet-level simulators run one scenario per CPU process and cannot keep pace with thousands of parallel environments. To bridge this gap, we present Isaac-Net, a GPU-batched 5G New Radio (NR) module that advances the uplink of thousands of environments in lockstep with Isaac Lab physics. Isaac-Net simulates every slot, the 0.5~ms interval in which the base station decides which robots transmit, for all environments at once. Extensive experiments confirm that its NR engine reproduces the median delay of ns-3 5G-LENA across loads, with a median delay 5–10% low on an unseen carrier and 9% high at 32 robots per environment in closed loop. The engine also reproduces the Age of Information (AoI), the age of each robot's newest delivered report, while an independent delay per message leaves the AoI tail about three times too light. In a configuration validated against 5G-LENA, Isaac-Net keeps the network in the loop for about one million robots on one GPU at 83% of the Isaac Lab rate without the network, measured under a random policy. Isaac-Net is open source at https://github.com/ZzZTripleZzZ/isaac-net

Human-computer interaction (HCI) is in the middle of a transition: large language models can now generate functional user interfaces (UIs) on demand from natural-language task descriptions. A user explains what they are trying to accomplish, and the system materializes a working interface to support it. This capability already exists in systems such as Claude and ChatGPT and continues to grow in fidelity as the underlying models improve. The next step along this trajectory is to move from interfaces that are merely generated to interfaces that are generated well. We propose a framework in which the dialogue between user and artificial intelligence (AI) becomes a Space to Think: a shared, structured cognitive workspace in which task decomposition produces an on-demand user interface as an extension of the user's thinking rather than as a separate artifact. Within this paradigm, classical HCI design knowledge (Nielsen's heuristics, Norman's affordance prescriptions, Web Content Accessibility Guidelines (WCAG) success criteria, cognitive-load constraints, and mixed-initiative principles) is encoded as skills: machine-readable skill.md files that the generating agent loads at runtime as software engineering tools. Skills turn HCI design knowledge into declarative, inspectable, version-controlled, and editable artifacts owned by the HCI community itself so that accessibility, learnability, and consistency become properties of a generative process rather than properties of a finished product. We outline a research agenda depicting a future where the HCI field transitions from today's design and knowledge heuristic checklist towards a future where the craft becomes machine-readable, executable, and open.

Long-horizon compositional manipulation has become increasingly important for real-world robot deployment, where a single task involves multiple coordinated subtasks. Existing world-action models (WAMs) jointly predict short-horizon visual futures and actions, but typically lack explicit subtask-level reasoning. We propose Visual Goal-conditioned Action Reasoning (ViGAR), a hierarchical framework that factorizes manipulation into a visual subgoal planner and a subgoal executor. Given the current observation and global instruction, the subgoal planner predicts a visual subgoal for the next subtask. The subgoal executor then jointly generates future visual trajectories and actions conditioned on the predicted subgoal. Both components share a pretrained world-model representation, enabling task-level planning and action generation to benefit from common physical knowledge. Moreover, our framework naturally supports in-context learning: using a global goal image as context can induce different subtask decompositions and behaviors without parameter updates. On the RoboTwin Clean2Random benchmark, ViGAR achieves 82.00% and 67.02% success rates under the Clean and Random settings, respectively, surpassing the strongest baseline by 12.86 percentage points in average success rate. Real-world robot experiments on five compositional and two in-context learning tasks further confirm the effectiveness of ViGAR.

AN~Col is a shell Be star that has shown transient spectral features caused by a hot, evolved companion on a months-long orbit. If confirmed, this would place it among the growing category of post-mass-transfer spun-up Be + stripped subdwarf OB-type (sdOB) binaries. Characterizing the binary orbit enables obtaining dynamical masses, to provide a rare anchor for models of binary evolution and mass transfer. We present a full 3-dimensional orbital solution, dynamical masses, and dynamical parallax, for the newly discovered Be + sdOB binary AN~Col. As a prerequisite to obtain a reliable orbital solution, we need to characterize the disk-wide density oscillation cycle that gives rise to the dominant spectroscopic cycles in order to separate the weaker periodic orbital variations. We use near-IR interferometry time series from GRAVITY to confirm the companion and to map the relative astrometric orbit. With the knowledge of the orbital period, we identify spectral lines that trace the orbit of each of the two stellar components and we measure their radial velocities to obtain a double-lined spectroscopic orbit. Combining interferometry and spectroscopy, we derive the full 3-dimensional orbital solution. The disk has completed three full oscillation cycles since 2010, with a cycle length of ~5.1 yr. The ~87.7-day orbit is compatible with edge-on orientation, but the lack of eclipses in TESS light curves limits the orbital inclination to 88.4$^{\circ}$ or less. The resulting masses are $M_{\rm Be} = 8.39\pm0.67$ Msun and $M_{\rm sdOB} = 0.81\pm0.22$ Msun. As in other Be + sdOB binaries, the estimated efficiency of the past mass transfer is unexpectedly high. Detection of a weak \spec{He}{ii}{4686} absorption and other \ion{He}{i} lines from the companion indicates that its effective temperature is close to 30 kK, and that it is a slow rotator. [abstract continues but no space left...]

Finding an optimal behaviour policy within a given environment is a widely studied problem in domains as diverse as games, robotics, energy infrastructure, communication networks, and multi-agent systems. Numerous benchmarks have been developed to support such research, but the vast majority assume that the agent's embodiment (design) is fixed, focusing instead on policy learning alone. Lifting this assumption gives rise to a broader class of problems in which optimizing embodiment and policy separately is highly suboptimal. An agent's embodiment strongly shapes which control policies can be discovered, while the optimal embodiment is in turn defined by the policies it admits. To help the research community study this class of problems explicitly and systematically, we introduce Co-Design Gym - a suite of benchmark environments for jointly optimizing embodiment and policy. Our environments span domains such as robotic manipulation and locomotion, multi-robot cooperation, deformable and soft dynamics, video games, electricity grids, wireless networks, F1 racing, multi-agent warehouses, and optimal control, offering 20 environment families (domains), with over 85 distinct co-design presets in total. We further contribute a systematic evaluation of representative co-design algorithms, characterizing the current state of the art. Together, these contributions lay the groundwork for cumulative, comparable progress in co-design.

Consider a subcritical functional $Q$ associated with a $p$-Schrödinger operator defined on a domain $\Omega$ in noncompact Riemannian manifold. Employing the supersolution construction, we obtain a family of strictly positive critical Hardy weights for $Q$. The construction is based on the Green potential, with a strictly positive charge, which is assumed to vanish at infinity. An interesting new feature of this family, depending on a parameter $0\leq \varepsilon\leq 1$, is the transition from positive-criticality to null-criticality at $\varepsilon=0$. Moreover, we show that the best Hardy constant for this family is given by $c=[(p-1)/p]^{p-1}$, and that for $\varepsilon=0$ the corresponding Hardy weight is optimal. We furthermore, prove an analogous result on graphs.

Explainability is increasingly required for perception models in intelligent vehicles, yet whether explanations remain faithful under driving domain shift is still poorly understood. This work audits post-hoc explanations of a fixed YOLOv8s pedestrian detector across PIE and JAAD using ROI-based D-Deletion, frozen confidence terciles, rank-based tests, bootstrap intervals, and Holm correction. The audit shows that deletion-based faithfulness is strongly coupled to detection strength at explanation time, with Spearman correlations between 0.70 and 0.82 for D-RISE, making naive confidence-stratified comparisons unreliable. After controlling for detection strength within fixed f0 bins, D-RISE faithfulness remains domain-dependent in the central f0 range, with PIE showing higher D-Deletion than JAAD and Holm-adjusted significance. A non-perturbative EigenCAM baseline is less faithful than D-RISE but also exhibits score coupling, suggesting that the effect is not specific to D-RISE and is related to the deletion-based evaluation setup. These results motivate confidence-controlled XAI audits for safety-critical perception under domain shift.

Benchmarks for agent-generated data work grade a pipeline by running it once against a fixed snapshot. ArrivalBench instead re-executes the pipeline an agent leaves behind under adversarial but replayable delivery schedules (late, duplicated, out-of-order and retried records) and requires its final state to equal a batch recomputation of the complete log. Because the oracle recomputes rather than classifies, a wrong table and a crash are distinct verdicts: a crash is visible to monitoring a team already runs, and a wrong table is not. On 40 tasks we built, our reimplementation of single-execution grading certifies 86-100% of the pipelines eleven models produce; re-executing the same artifacts finds 7.0-79.2% of the certified ones silently wrong. The gap is not produced by the repair loop: within the same model and task, pipelines repaired against the snapshot test fail replay about as often as those that passed it first time. In every model, idempotency hazards fail more often than ordering hazards. Separating a wrong answer from a crash also changes how interventions read: a hazard warning cuts one model's silent failure from 48.2% to 10.5% while raising its crash rate from 9.0% to 37.0%, so all-in failure moves only from 51.0% to 44.0%. All eleven arms were independently re-run, and rates moved by at most 5.9 points.

Dissipative dynamics can stabilize a quantum target without providing a reliable sequence of finite control operations. We derive checkable conditions for implementing such stabilization with imperfect controls, with bounds on convergence, errors during operations, and physical cost. The analysis includes pulses, idle intervals, and all simultaneously acting background dynamics. Using a common Lyapunov measure of deviation from the target, we require a contraction bound that remains uniformly below one over the prescribed uncertainty set after accounting for implementation errors. If all allowed paths preserve the target subspace throughout each operation, repeated cycles converge to that subspace from every initial state, even when errors vary between cycles. Comparison with these certified target-preserving paths gives an explicit long-time error bound under bounded leakage. Two single-qubit counterexamples show why a stabilizing average or a stabilizing generator obtained by unitary conjugation does not ensure these path conditions. The certification approach also supports a finite-library search. For a specified four-level cascade model with finite pulses, calibration and timing errors, dephasing, and bounded pumping, this search yields a sequence with an explicit preparation-time bound. For every initial state, it guarantees trace distance at most $0.1$ from the target at the end of preparation and throughout subsequent cycles.

Large language model (LLM)-driven evolutionary search is a powerful paradigm for automated program and algorithm discovery, yet existing systems are largely memoryless: each run explores from scratch, so agents repeatedly rediscover the same improvements and re-encounter the same dead ends. We introduce SEDIMA, a persistent hierarchical insight memory for evolutionary search agents. SEDIMA distills raw traces into natural-language insights, clusters them by semantic similarity using attention-weighted centroids, and retrieves relevant guidance to condition future mutations, accumulating transferable knowledge across runs and problems rather than within a single trajectory. As a drop-in module that leaves the search operators unmodified, SEDIMA improves average final performance by 5.5% on AlgoTune and 6.6% on ALE-Bench LITE under a fixed budget of 100 evaluated candidates. Under OpenEvolve, SEDIMA requires 32.3% fewer iterations on average to reach baseline-best performance across the five evaluated backbones.

Vision-language-action (VLA) policies enable diverse robotic manipulation but can fail during execution without recognizing their own errors. Human observers provide complementary signals, as unexpected robot behavior can trigger rapid vocal, facial, or verbal reactions before failure is completed. We introduce SocialVLA, a local, policy-agnostic social perception gateway that converts spontaneous human reactions into runtime intervention signals for VLA manipulation. SocialVLA combines causal paralinguistic audio detection, visual reaction recognition, explicit stop phrases, and robot-relevance estimation. An asynchronous first-event fusion mechanism triggers a VLA hold from the earliest sufficiently confident signal, while a separate speech channel captures verbal corrections for participant-directed continuation, restart, or instruction revision. We evaluate SocialVLA on physical Unitree G1 manipulation using 15 participants, with 238 annotated intervention-worthy episodes and 1.038 h of non-intervention behavior. Frozen offline replay achieves 54.6% recall and 69.5% precision, while unfiltered audio-video fusion reaches 64.3% recall. Relevance estimation reduces false-stop episodes from 100 to 57 and increases precision from 60.5% to 69.8%. In prospective deployment on an unseen 16th participant, the frozen system achieves 59.5% recall and 91.7% precision. Median detector-to-fusion latency is 47.9 ms, VLA-gate-to-physical-hold latency is 336 ms, and reaction-onset-to-hold latency is 1.021 s. These results demonstrate a complete local pathway from spontaneous social reaction to physical VLA interruption and participant-directed recovery.

Reinforcement learning from verifiable rewards (RLVR) usually optimizes answer correctness, yet useful language-model behavior also requires high-quality reasoning and concise responses. Existing multi-reward post-training methods typically scalarize rewards or combine specialists without explicitly protecting a reward priority order. This is problematic when trade-offs are asymmetric: conciseness, for example, should not improve at the cost of correctness. We introduce Lexicographic Multi-Objective On-Policy Distillation (LMOPD), a multi-teacher method for integrating reward-specialized policies under explicit priorities. For each student rollout, LMOPD selects the specialist for the first objective whose gate detects a deficiency, then locally projects its centered log-policy correction to remove components that oppose higher-priority specialists. We evaluate 30B-A3B mixture-of-experts transformer models in two- and four-expert settings on three math benchmarks, measuring retained specialist gains. With two experts, LMOPD's point estimates fully retain the accuracy and reasoning-quality gains while acquiring $46.9\%$ of the conciseness gain. With four experts, it retains $\approx90\%$ of both the accuracy gain and reasoning-correctness gain, compared to only $\approx57\%$ by the next best evaluated baseline. Matched four-expertablations show that lexicographic routing outperforms random routing and that projection further strengthens both top-priority capabilities. Across both scales, LMOPD preserves the highest-priority capabilities more effectively than the existing baselines we evaluate, demonstrating the value of explicit priorities for specialist integration.

This work establishes finite-time blow-up for the completely integrable defocusing Ablowitz-Ladik equation on both infinite and finite lattices. We first establish local well-posedness for arbitrary initial data in the space $\ell^\infty$ of uniformly bounded sequences, with no decay assumptions and under general boundary conditions. We then show that the unit circle acts as a separatrix for the lattice dynamics: sites with an initial modulus inside, on, or outside the unit circle maintain their modulus within their respective region for as long as the solution exists. As a consequence, initial data in the closure of the $\ell^\infty$ unit ball generate global-in-time solutions. On the other hand, the unit circle separatrix opens up the possibility for finite-time blow-up, which is restricted to sites with initial modulus outside the unit circle. We establish this scenario by constructing a class of initial data whose corresponding solutions are singular, with explicitly given exact blow-up times. We also construct solutions emanating from initial data with an arbitrarily large single-site excitation outside the unit circle that exist globally in time, a behavior driven by the convergence of the corresponding modulus to the unit circle equilibrium. In the presence of such solutions, along with other important classes of analytical, global-in-time solutions that remain outside the closed $\ell^\infty$ unit ball for all times, we examine the fundamental question of the dominant dynamics for general initial data outside the closed unit ball. By utilizing the constructed class of singular solutions as subsolutions for the general problem, we prove that the dynamics for the completely integrable defocusing Ablowitz-Ladik lattice is dominated by finite-time blow-up. We conclude with numerical studies of illustrative examples of general initial data

Recent advances in deep learning for time series prediction have amplified the need for reliable uncertainty quantification. Conformal prediction has gained attention as a distribution-free framework for constructing prediction intervals with coverage guarantees. However, its coverage guarantees rely on data exchangeability, an assumption generally violated in time series. Active research has focused on developing conformal prediction methods for time series that overcome this limitation. While deep sequence models, such as recurrent neural networks and Transformers, have often been used in conformal prediction for time series, limited work has systematically studied how deep sequence models can be utilized in conformal prediction for time series. In this work, we systematically investigate the use of deep sequence models in conformal prediction for time series through three approaches: conditional quantile regression, conditional quantile function estimation, and localized conformal prediction. We provide a theoretical analysis establishing asymptotic conditional coverage guarantees for all three approaches under suitable assumptions. Through comprehensive experiments on real-world datasets, we demonstrate the effectiveness of leveraging deep sequence models into conformal prediction for time series.

We establish lower bounds on the number of resonances for three-dimensional semiclassical Dirac operators with mass-type potentials. For a pair $\Dl_j=\Dl_0+\mbf{\beta}v_j$, $j=1,2$, satisfying the exterior analyticity, decay and non-degeneracy assumptions of the paper, we associate a relative distribution $\mu$ to the corresponding mass functions. We prove that if $E_0>1$ is a boundary point of $\supp\mu$, then every complex neighbourhood of $E_0$, and likewise of $-E_0$, contains in total at least $C\hbar^{-3}$ resonances of the two operators, counted with multiplicity. The proof transfers an analytic wave-front singularity of $\mu$ to the relative classical density of states and combines this with Khochman's local trace formula and a relative semiclassical trace formula. As a consequence, if $v$ is a single bounded mass-type potential, analytic with power decay in an exterior sector, with $\inf(1+v)>0$ and $\sup v>0$, then $\Dl=\Dl_0+\mbf{\beta}v$ has at least $C\hbar^{-3}$ resonances in every complex neighbourhood of $E_0=1+\sup v$ and of $-E_0$, for all sufficiently small $\hbar$. The exponent agrees with Khochman's upper bound, the two resonance families are related by spectral symmetry, and $E_0$ is necessarily a trapping energy. We also prove the relative semiclassical trace formula needed in the argument for general Hermitian matrix-valued perturbations.

Increasing the batch size during training is a common practice in large language model (LLM) pretraining, yet the theoretical justification behind its success is not well understood. Analyses of stochastic optimization often assume uniformly bounded stochastic gradient variance, yet recent evidence suggests that this assumption fails in many practical nonconvex problems. The Blum–Gladyshev (BG-$0$) noise model relaxes this assumption by allowing the variance to grow quadratically with the distance from initialization, suggesting that batch size schedulers can help by controlling the variance growth during training. However, this growth can be overly conservative in practice. We empirically investigate variance growth in LLM pretraining and observe that a generalized BG model with a tunable growth exponent provides a tighter description of practical noise behavior. Motivated by this observation, we introduce the generalized BG-$a$ noise model, which interpolates between bounded variance ($a=0$) and BG-$0$ noise ($a=2$). Under $L$-smoothness, we derive an information-theoretic lower bound with growth-dependent oracle complexity $\Omega(\epsilon^{-(4+a)})$ and establish a matching upper bound in $\epsilon$-dependence by increasing the batch size as the iterates move away from initialization. Finally, we propose an adaptive batch scheduler that controls variance growth through dynamic batch size adjustments during training. In pretraining OLMo2 models of up to 1B parameters on C4, our scheduler achieves a lower validation loss than both small and large batch training under matched token budgets, while using less than 10% of the iterations of small batch training.

We construct spectral-parameter solutions of the quantum Yang–Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients $k_2,k_4,k_6,k_8$ subject to the single relation $4k_8-k_4^2+k_2k_6=0$ – which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral $R$-operator built from multiplication and unit has Yang–Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary $9\times9$ $R$-matrix satisfying the quantum Yang–Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The $A_3$ Frobenius manifold is worked out explicitly.

Personalized large language models often require a complete adaptation state for each user. However, this paradigm scales poorly as the user population grows. We revisit this design through the lens of personalization capacity allocation: how much adaptation capacity can be shared across users, how the shared capacity should be composed, and how much must remain user-specific. We answer them through three complementary empirical analyses. We find that independent user adapters contain substantial cross-user reusable structure, that the utility of reusable directions reflects both user relevance and variation across queries, and that user histories provide transferable signals for compact individual correction. Motivated by these findings, we propose LINEUP. It learns a bank of reusable low-rank personalization factors, composes them through user-conditioned recall and query-dependent calibration, and restricts target-user adaptation to a tiny user code over a shared correction space. This design decouples expressive personalization capacity from per-user trainable state. Each target user optimizes only eight scalars, while all shared components remain fixed. By comparison, the evaluated private-LoRA configuration uses 4.19 million per-user parameters. Our theoretical analysis gives a finite-step, finite-history risk bound and sufficient conditions for user-code refinement to improve on history initialization. Across six tasks spanning personalized classification, prediction, and generation, LINEUP leads on all 12 metrics, each averaged over three independent runs (e.g., reducing LaMP-3 RMSE by 11.4% relative to the strongest baseline). It maintains advantages under limited history. These results show that rich personalization can be supported primarily by reusable, conditionally composed shared capacity, while independent user adaptation remains confined to a tiny correction state.

Let A = F_2[T]. The orthogonal multiplicity m(f) of a monic f in A of degree d >= 1 is the number of g in A with deg g < d, gcd(g,f) = 1, and all partial quotients of g/f of degree one. Blackburn (1998) asked how the proportion p(d) of monic polynomials of degree d with m(f) > 0 behaves as d tends to infinity. We attach to each prime divisor P of f a residue psi_P(f) in F_2 of the differential dT/(T(T+1)f), and prove two unconditional facts about this family: a reciprocity relation, namely that the psi_P(f) sum to zero whenever f is prime to T(T+1), and an exact local distribution, namely that psi_P vanishes for a proportion exactly (|P|-2)/(2(|P|-1)) of the admissible completions when P exactly divides f, and for exactly half of them when P^e exactly divides f with e >= 2. Granting that these local conditions pool independently, we obtain p(d) ~ L d^(-1/2) with L = (9/(4 sqrt(pi))) times the product over the monic irreducibles P of A of degree at least two of (1 + 1/(2|P|)) (1 - 1/|P|)^(1/2), equal to 1.2116120743... The same Euler product evaluated at a second parameter returns the exact identity that the m(f) of a given degree sum to 2^d, which fixes its normalisation. We give the exact value of 2^d p(d) for 1 <= d <= 33; the predicted values agree with these to within 5 10^(-5) for 27 <= d <= 33.

ReAct-based agents typically rely on a single LLM policy to propose actions, interact with the environment, and decide when a task is complete. This coupling makes action authorization and completion control difficult to enforce independently, allowing errors to propagate and unsupported completion claims to terminate execution. We introduce DeReAct, a modular agent architecture that externalizes two gating policies: a Critic that validates proposed actions before execution, and a Context Manager that reconstructs an environment-supported State and certifies task completion. Across GAIA and SWE-bench Verified, DeReAct improves Pass@1 most for weaker Brain models, with gains of 6.5–7.0 points for Qwen3-Coder-480B and 4.2–5.2 points for Claude Sonnet~4.5; gains diminish as Brain capability increases. Trajectory and ablation analyses show that external gating is effective when targeted failures are sufficiently prevalent and the gating policy is itself sufficient. With Claude Opus~4.5, Pass@1 remains comparable to ReAct, while DeReAct produces more evidence-complete and constraint-satisfying trajectories, indicating that completion control can trade earlier termination for stronger grounding. Overall, DeReAct improves weaker agents while retaining grounding benefits as models strengthen.

We study idealized maximal eventually different families through Van Douwen maximality, productivity, and ultrafilter maximality. We show that Van Douwen $\mathcal{J}$-MED families correspond to $(\mathcal{J}\times\emptyset)$-MAD families, and for coanalytic ideals containing the finite sets they are further reduced to infinite analytic $\mathcal{J}$-MAD families. This yields nonexistence results for analytic Van Douwen families for several standard ideals, together with a characterization of the finite case. We recall the definition of finite productivity and introduce finite-section and $\omega$-centered productivity. Finite-section productivity is equivalent to $\mathcal{U}^*$-maximality for some ultrafilter $\mathcal{U}$ extending $\mathcal{J}^*$, and these notions admit natural Stone-space characterizations. We construct Borel finitely productive families for uniformly weakly Ramsey ideals and Borel $\omega$-centered productive families for uniformly weakly $P^+$ ideals, while no analytic finite-section productive $\mathrm{Fin}$-MED family exists. Finally, for every Ramsey ultrafilter $\mathcal{U}$ and $1\leq\alpha<\omega_1$, there is no analytic $(\mathcal{U}^{\alpha})^*$-MED family, while under $V=L$, there is a $ \boldsymbol{\Sigma}^1_2 $ Ramsey ultrafilter $ \mathcal{U} $ where there is a coanalytic $ \mathcal{U}^* $-MED family.