Papers

Robotic manipulators operating over long durations often experience uneven joint degradation, which causes the weakest actuator to fail prematurely, leads to unplanned downtime, and results in significant operational losses. Traditional motion-planning algorithms do not account for joint health conditions and therefore tend to exacerbate this imbalance during extended operation. To address this challenge, this study introduces the RUL-aware RRT*, a motion-planning method that incorporates joint remaining useful life information into the planning process and adaptively adjusts joint usage in response to evolving health conditions. The method is evaluated across three representative scenarios, namely the Full Health Scenario (FHS), the Heterogeneous Degradation Scenario (HDS), and the Local Degradation Scenario (LDS). The results show that the RUL-aware RRT* effectively suppresses degradation imbalance, delays the emergence of bottleneck failures, and improves the long-term reliability of the robotic system during extended operation. These findings demonstrate that integrating health feedback into motion planning provides a practical and robust pathway for enhancing the durability and operational resilience of manipulators subject to continuous wear.

We present Windfoil, a GPU-friendly algorithm that treats rasterisation and differentiable vector graphics as two sides of the same problem by evaluating the box-filtered winding number of quadratic Bézier contours in closed form. We implement this in WebGPU, allowing it to run across a range of environments, including a web browser on a consumer laptop, and apply the system to real-time 2D rendering, high-resolution rasterisation for print media, and a differentiable renderer. We compare our renderer against Skia, a production-grade engine, and Slug, a popular GPU rasterisation algorithm for games and real-time applications, measuring fidelity to a reference box-filtered coverage. Our renderer matches the reference more closely than either, at performance comparable to Slug. We also compare our optimiser against DiffVG and Bézier Splatting, where it reaches equivalent or better reconstruction quality at a fraction of the per-step cost, scaling to tens of thousands of shapes at interactive rates.

This paper establishes sufficient conditions for finite-horizon solvability of fully coupled nonlinear mean-field forward–backward stochastic difference equations with dependence on unconditional first moments. The forward recursion uses conditional projections of the next backward state and its product with the innovation. Building on existing domination–monotonicity methods, we formulate deterministic matrix combinations in centered and mean coordinates and prove a continuation estimate uniform in the homotopy parameter. Global Lipschitz continuity and one active domination–coercivity direction yield a unique square-integrable adapted solution, an a priori bound, and coefficient stability. The active parameter can be normalized without imposing an additional smallness restriction on the original coefficients. A sign transformation handles the opposite monotonicity orientation. A nonlinear example with saturating state and mean interactions verifies the assumptions, including degenerate domination directions. A scalar mean-field LQ buffer-adjustment model illustrates the theorem: its Hamiltonian solution characterizes the unique open-loop optimizer. An exact finite scenario-tree calculation checks this characterization against direct quadratic optimization.

We present sparse dictionary differentiable predictive control (SD-DPC), a framework for learning sparse, interpretable feedback policies for nonlinear systems from data. A prediction model is first identified by rollout-based sparse identification of nonlinear dynamics (SINDy), building on gradient-based and multistep formulations. The policy is then parameterized as a sparse combination of dictionary functions and trained by differentiating a constrained finite-horizon predictive-control objective through this model, so that its terms are selected by closed-loop performance rather than by imitating a previously trained controller. The result is an explicit feedback law with only a handful of terms. Across three benchmark control problems, SD-DPC satisfies the constraints in almost all test scenarios, outperforms a policy distilled onto the same terms by up to an order of magnitude, and requires orders of magnitude less memory and online computation than an optimization benchmark, while admitting explicit sensitivity bounds.

The properties of galaxies that host active galactic nuclei (AGNs) offer clues to the conditions that enable accretion onto supermassive black holes. A galaxy's morphology can provide significant information on the evolutionary processes underway. By modeling the surface-brightness profiles of galaxies that host AGN, we can characterize physical properties of the host galaxy. With a sample of sources across a wide range of redshifts, we can investigate how both the supermassive black hole and galaxy evolve over time. Applying the best practices from previous work, we perform Sersic + point source fits of 96 X-ray-selected AGN in the Extended Groth Strip across up to 13 bands of James Webb Space Telescope and Hubble Space Telescope imaging using AstroPhot. We examine both parametric (Sersic) and nonparametric methods in tandem to best determine the physical properties of the host galaxy. After separating the light of the host galaxy and the AGN, we perform independent CIGALE model fits to photometric spectral energy distributions to estimate physical properties including the host galaxy stellar mass and star formation rate, and AGN disk luminosity. In the stellar-mass/star-formation-rate plane, the AGN host galaxies align with the population of massive non-AGN galaxies in the control sample across all redshifts. The AGN disk luminosities result in ratios of bolometric to X-ray flux within the expected range, 5-20:1. We provide an evaluation of the limitations of Sersic modeling to obtain galaxy properties and provide recommendations on best practices for multi-band AGN+host galaxy image decomposition with James Webb Space Telescope imaging.

We study the formation of $\mathcal{N}$-quark baryons during the confinement crossover of an $SU(\mathcal{N})$ gauge theory containing quarks in the fundamental representation, with masses around or below the confinement scale. We assume stepwise baryon formation through the successive fusion of pairs of quark-clusters, with color interaction rates obeying quadratic Casimir scaling. We include the complete set of reactions between $SU(\mathcal{N})$ antisymmetric clusters dressed with their specific spin-flavor multiplicities. We show that at large $\mathcal{N}$, baryon formation is hindered by a Casimir bottleneck, such that fusion of few-quark clusters in the initial steps of the baryon assembly chain is strongly outweighed by cluster destruction processes. We find that for $\mathcal{N}=12\,(18)$ the baryon-to-meson yield ratio is suppressed down to $10^{-12}\,(10^{-23})$ of its value at $\mathcal{N}=3$. We embed the mechanism in a vectorlike three-flavor model whose lightest baryon can be electrically neutral, while axial-anomaly-induced couplings and dimension-six operators induce decays of the unstable mesons and charged baryons safely before big-bang nucleosynthesis. The relic density of cosmologically stable neutral $\mathcal{N}$-baryons is thus set during confinement at values so small that, for sufficiently large $\mathcal{N}$, the baryon masses required to saturate the dark matter energy density exceed by several orders of magnitude the unitarity bound on annihilation cross sections.

We classify constant numerical solutions of the $\Z_2$-graded twisted reflection equation for the fundamental $GL$-type R-matrix and an arbitrary parity sequence, without assuming evenness or invertibility. The classification is based on a four-index locality principle established for this type of reflection equation. We determine the singular and invertible solutions and recover previously conjectured matrices for the minimal symmetric grading. Even invertible solutions comply with the recent classification of graded Satake diagrams. The classical stabilizer of a full-support lower-triangular non-even family is a Lie superalgebra whose even part is an electrical Lie algebra.

LLM compression reduces inference costs and memory requirements, but selecting a method and configuration remains largely empirical because comparable resource reductions can produce different capability losses. We systematically investigate capability scaling-down laws for LLM compression across pruning, quantization, and distillation. Our framework measures capability loss in mathematics, code generation, and question answering, and relates these measurements to model size, training stage, compression settings, data availability, and training exposure. We develop simple predictive relations and evaluate their accuracy, measurement efficiency, and generalization to unseen configurations and model states. Sharing the density response across pruning levels halves the configuration measurements needed to fit a pruning predictor: on new Pythia states, on pre-registered OLMo-2 test states and under Wanda pruning, the compact relation matches a regression fitted with all measurements on math and code to within 0.020 nats per token, with coefficients refitted for each setting. Controlled distillation experiments show that the cost of heavy data reuse recurs across question-answering distributions, while the net benefit depends on the evaluation distribution. We further evaluate the decision value of these predictions by comparing numerical selection with configuration medians and fixed method priorities. Independent evaluations across two model families show that selection captures most of the available cross-method benefit for question answering within the tested candidate sets, where a fixed method priority attains the same regret, with smaller opportunities for mathematics and code. These results clarify the predictive scope of capability scaling-down laws and their use in compression method selection. Our code is publicly available at: https://github.com/LabRAI/scaling_down_law.

Accurate thermal analysis of heterogeneous 2.5D/3D-IC packages is essential yet computationally prohibitive. A single full-package FEM simulation can take hours, while AI-based surrogates treat the entire stack as a monolithic prediction target and must be retrained whenever the die count or topology changes. To address this limitation, this work proposes Domain-Decomposed AI-Accelerated Iterative Solver for Thermal Analysis (DAIST), a composable thermal solver that decomposes the global package simulation into block-level subdomain problems, replaces subdomain solvers with neural operators, and couples them through iterative exchanges of interfacial temperature and heat flux. This local-to-global architecture eliminates the topology lock-in of monolithic models: block-level neural operators can be directly reused in unseen package assemblies without retraining. The iterative coupling strategy further provides a controllable accuracy-runtime tradeoff, where the iteration budget can be adjusted to trade accuracy for runtime. Evaluated on a multi-chiplet system and an advanced packaging system, DAIST achieves up to $178\times$ speedup over traditional FEM solvers with mean temperature errors of 0.068% and 0.323%, respectively, while demonstrating cross-topology reuse of block-level models across structurally distinct package assemblies.

Recent benchmarks rely on user simulators to evaluate AI agents in multi-turn interaction. While existing simulation techniques demonstrate surface fidelity to human style and behavior, ecologically valid interactive benchmarking also requires alignment in when and how agents fail across simulated and real user populations. We find that existing simulators lack outcome calibration: agreement with observed success rates and failure patterns when real users interact with the same agent. We introduce Calibrated User Embeddings (CUE), a framework that both encodes observed sessions and samples continuous representations, then decodes them into persona commands to steer LLMs to act as user simulators without training. Through this, we evaluate user-conditioned replay of past sessions and aggregate metric agreement when sampling novel personas for the same tasks. On $\tau^2$-Bench, CUEd simulators commit fewer simulator-attributed errors and more faithfully reproduce real-user agent failure modes, aggregate success rates, and outcomes for specific task-user pairs than other persona-based simulation methods. These gains coexist with competitive user fidelity as measured using metrics established in prior work. After being fit to mostly customer support interactions, the same CUEd simulators generalize to document creation, math tutoring, and casual conversation, and remain effective across different simulator LLMs without CUE retraining.

Coding agents are extending their reach into the physical world by writing and executing robot control programs. One might expect the agents to use the existing mature software stack that engineers have developed over decades to access sensors and control motion. Yet prior work primarily engineers complex custom harnesses to orchestrate agents for robot use, particularly by prescribing specialized workflows and providing bespoke interfaces. This raises the question: "Is such additional harness engineering necessary?" We introduce OpenRUA, a zero-abstraction harness that bypasses bespoke abstraction layers by providing off-the-shelf coding agents with only terminal access to the robot's native software interface ROS 2. OpenRUA employs a minimalist workspace-as-harness design, only offering ROS 2 documentation and basic tools while leaving the coding agent to organize its own work without orchestrating any agentic workflow. Within this workspace, OpenRUA recasts perception as file I/O and manipulation as coding. With Claude Code powered by Claude Opus 5, OpenRUA achieves success rates of 99.0% on CaP-Bench and 87.0% on LIBERO-PRO, demonstrating that an off-the-shelf coding agent can serve as a zero-shot visuomotor policy through the robot's native interface, without bespoke primitives or task-specific training. Under this minimalist design, further analysis reveals striking emergent behaviors of coding agents: (1) For perception, the agent spontaneously writes programs that process raw sensory inputs and derive metric measurements in 96.80% of episodes. (2) For manipulation, the agent spontaneously builds motion-control clients (e.g., gripper control) in 95.87% of episodes and closed-loop control programs (e.g., adjusting motion based on sensor feedback) in 50.13% of episodes. Our code is available at https://github.com/terminalworld/OpenRUA.

In error correction, the difference between a codeword and a received word, a candidate noise sequence, carries information on the confidence of the codeword as a proposed decoding. For additive channels, it is known that if noise sequences are ranked in decreasing order of likelihood, a lower rank correlates with higher confidence in a decoding. Here we substantially expand considerations by exploring the relationship between the likelihood ranks arising in list decoding and the posterior probability of decoding correctness, establishing that the ranks associated with subsequent list entries carry reliability information that is not obtainable from the first rank alone. Building on the recent development of Soft-Output Guessing Random Additive Noise Decoding, we first show that, under a random codebook model, soft-output expressions for list decoding coincide with the exact finite-blocklength conditional probabilities of decoding correctness. For hard-decision maximum likelihood decoding, we characterize the asymptotic reliability of the first list entry. We identify a decision function whose sign determines whether the posterior correctness probability of the first entry in the list converges to one or zero and whose absolute value gives the exponential rate of convergence. Perhaps surprisingly, the asymptotic reliability of the first list entry depends only on the first, second and last ranks. For list size one, the result recovers the previously known likelihood-rank threshold for reliable decoding. For list size two, a sufficiently late second entry can imply that the first entry is asymptotically correct even when its rank, viewed in isolation, would indicate otherwise.

Future low-altitude wireless networks (LAWNs) are expected to simultaneously support sensing, communication, and charging, resulting in tightly coupled multi-objective optimization problems with strong interdependencies among heterogeneous functions. However, existing multi-objective frameworks typically rely on complex problem-specific formulations and alternating optimization procedures, which suffer from high computational complexity and limited scalability in large-scale, highly dynamic deployments. In this paper, we propose a unified topology-aware (TA) framework that abstracts terrestrial and non-terrestrial devices as nodes, and models their interactions as edges, forming a bipartite graph representation of the LAWN. By leveraging this graph representation, we develop a low-complexity topology reconfiguration algorithm based on a unified resource-adjustment rule. Simulation results demonstrate that the proposed TA-based method consistently outperforms state-of-the-art approaches in multi-functional coordination efficiency, while maintaining robustness and scalability in scenarios involving massive terrestrial and airborne user equipments.

AI coding agents are untrusted system components, yet they require autonomy on the developer machines they run on. This contradiction is a security problem. Sandboxes provide an isolated environment, but for local macOS development, the existing local, open-source options for AI coding agents are few in number and cumbersome to use. I conducted a formative online user survey which indicates that fewer than 40% of AI coding agent users run their agents in a sandbox and identifies the top usability barriers hindering AI coding agent sandbox adoption. These findings are used to develop SideKernel: an open-source, local, microVM-based macOS sandbox for AI coding agents designed for usability. To evaluate SideKernel, I compiled a list of sandboxes available on the market and filtered it against five inclusion criteria. Then I performed a comparative analysis between SideKernel and the sandboxes that satisfy these criteria, across 23 capability tests derived from the usability barriers revealed by the user survey. I discovered that only a few sandboxes are similar to SideKernel, and that among those, Docker Sandboxes and SideKernel score highest on capability features related to usability. A secondary contribution of this paper is a survey of the existing solution space for local, open-source, microVM-based macOS sandboxes for AI coding agents.

Multi-party financial chatrooms are vital for sales-and-trading professionals, but their complexity makes manual recovery of missed trades infeasible: each Request for Quote (RFQ) is an event whose final price and trade outcome appear many messages after the RFQ-trigger message (the inquiry message), interleaved with concurrent RFQs from other participants. We cast this as event extraction (EE) over multi-party dialogue and present FinDialogLens, a hybrid LLM pipeline in which compact fine-tuned classifiers act as inference-time scaffolds: they detect RFQ-triggers and price/trade outcome metadata, an RFQ-Level Module segments per-event RFQ windows, and a Trade Engine fills argument roles. With GPT-4o, FinDialogLens reaches 92.1% and 94.3% accuracy on final price and trade outcome, respectively, outperforming full-chatroom CoT prompting methods; fine-tuned open-source LLMs with as few as 3B parameters achieve comparable performance with modest in-domain data. To make the LLM-based solution practical at scale, a difficulty-aware router balances cost and accuracy by allocating RFQs between a low-cost rule-based engine and the higher-performing LLM-powered Trade Engine, cutting LLM calls by 85% on final price while recovering half of the accuracy gap to FinDialogLens (GPT-4o), saving over $300/day at our 70,000-RFQ/day scale.

Shear transformations are the elementary events of plasticity in glasses and underpin many mesoscopic models of plastic flow, yet their atomic origin and energetics remain difficult to establish. Here we use a machine-learned interatomic potential to show that, in amorphous carbon as a representative network glass, a shear transformation corresponds to the breaking or formation of a single covalent bond. The simplicity of the associated reaction coordinate, the bond length, enables us to extract energy landscapes and barriers. Despite the intrinsic structural disorder, we find that all energy landscapes collapse onto a universal barrier shape. What remains bond-specific is a single scale, the bond-jump distance, which fixes the barrier height through a cubic law. Combined with the activation statistics of shear transformations, this yields a model that predicts the plastic flow stress from independently determined bond-level quantities. Our results provide microscopic support for phenomenological models of plastic flow in glasses and suggest a route towards a quantitative description of amorphous plasticity.

We study the exact entanglement dynamics of two qubits coupled to a common bosonic reservoir through a pure dephasing interaction. We ask how this dynamics changes when the qubits and the reservoir already share correlations before the qubit state is prepared. In that case the system and reservoir first interact for a long time and settle into a joint equilibrium state. Only afterward is the qubit pair projected onto the desired state by a local measurement. For an Ohmic spectral density we obtain the exact reduced two qubit dynamics. Using the Wootters concurrence we show that initial correlations always speed up entanglement sudden death relative to the factorized case, and that this gap grows quadratically with the coupling strength in the weak-to-moderate coupling regime. Finally we show that a symmetric sequence of periodic {\pi} pulses applied to both qubits extends the entanglement lifetime close to a factor of seven over the range of pulse numbers studied.

The operation of dynamically polarized targets in nuclear physics experiments relies on continuous tuning of the microwave frequency to compensate for radiation damage and evolving material properties, a task that is traditionally performed through manual trial-and-error by expert operators. This work presents a data-driven control framework that combines surrogate modeling with reinforcement learning to optimize the target polarization. Using operational data from the APOLLO cryogenic target system, we train and evaluate multilayer perceptron and Gaussian process regression models to predict polarization as a function of microwave frequency, beam current, and accumulated radiation dose. We show that Gaussian process-based models provide calibrated uncertainty estimates and reliably identify regions outside the training distribution, while MLPs exhibit limited sensitivity to distributional shift. To enable learning and control across multiple target samples, we introduce a Gaussian process approximation and embed the surrogate model within a standardized simulation environment. A reinforcement learning agent is trained using a lower-confidence-bound reward formulation that balances performance maximization against uncertainty. We are able to show an almost 2x improvement on the operators actions utilizing our RL agent.

Effective wildfire monitoring requires relating visual evidence to physical fire dynamics, yet real videos with synchronized physical annotations are scarce and high-fidelity 3D simulation is costly. We present a simulation-grounded vision-language model (VLM) framework that automatically converts 2D wildfire simulations into labeled video episodes. A fixed Blender mapping produces low-detail 3D proxies aligned with simulator terrain, fuel layout, fire activity, and wind cues; controllable video generation supplies richer appearance. The proxies are intermediate representations rather than finely rendered final scenes. Generated videos and simulator labels form reusable multimodal memory for a training-free multi-agent VLM system that retrieves reference episodes, reconciles visual and memory-based predictions, and produces structured wildfire reports. On held-out generated episodes, video memory achieves 51.5% exact four-tag accuracy, compared with 22.6% for direct VLM querying and 16-17% for text-only memory; the complete system achieves 77.3% accuracy on six simulator-derived report fields. Component ablations, cross-generator tests, and three real-UAV evaluations assess retrieval, reporting, generator changes, and observable monitoring tasks. The framework connects automatic simulation-to-proxy conversion with memory-based VLM reasoning under scarce real-world physical annotations.

In earlier work, we proved that the length polyhedron of an interval order has a unique Hilbert basis and that every element of this basis is a binary vector. The supports of these binary vectors form a set system whose intersection graph is a cocomparability graph. This leads naturally to the question of which graphs can arise as intersection graphs of such Hilbert-basis supports. We introduce Hilbert fragments, defined as induced subgraphs of these intersection graphs. Thus Hilbert fragments form a hereditary class of cocomparability graphs recording the intersection patterns among the supports of Hilbert-basis elements of interval-order length polyhedra. We characterize Hilbert fragments by an axiomatically defined vertex-labeling, called a track-labeling, and give applications illustrating how this characterization can reveal an underlying one-dimensional structure even when the Hilbert fragments themselves are not interval graphs.

This paper describes a mathematics interface intended for both educational applications and exploratory mathematics. Both learners and mathematics practitioners often use pen and paper or a whiteboard to perform equational reasoning, manipulate expressions, or construct proofs. These workflows lead to a variety of problems: transcription errors, tedious writing and notation, and/or unclear standards for proof justification. We extend the Hazel Prover, an equational-reasoning interface in the Hazel live programming environment, to add capabilities for exploratory mathematics, with varying levels of verbosity and automation aimed at both students and expert users. Students require more deliberate practice when learning mathematical concepts and correspondingly more verbose justifications, while experts may benefit from significant mathematical automation. With this in mind, our interface supports multiple levels of mathematical automation and simplification, grounded in a rewrite search architecture. For expert users, motivated by a gap in the accessibility of formal methods, we support proof export to the Rocq theorem prover, extending this rewrite search to proof tactics. This paper closes with several case studies covering elementary- to college-level mathematics.

The sphere covering problem has broad applications in science however, exact solutions are computationally infeasible on curved, high-dimensional manifolds because distances must be evaluated using a position-dependent metric. We present $\texttt{isobank}$, a normalizing flow that learns an approximately isometric coordinate transformation, such that Euclidean distances in the latent space approximate metric-induced proper distances in the physical parameter space. This enables template placement using Euclidean distance checks in the latent space, avoiding repeated metric evaluations during placement. We evaluate $\texttt{isobank}$ in the context of gravitational wave template bank construction, where the sphere covering problem corresponds to the template placement problem. On a five-dimensional eccentric binary parameter space, $\texttt{isobank}$ reaches 97% coverage at a maximum mismatch of $\mu=0.03$ with 23% fewer templates than the state-of-the-art stochastic placement code $\texttt{mbank}$, which reaches 94% coverage. Coverage is measured on independent injections drawn uniformly over the masses, spins and eccentricity, using the exact waveform match rather than the metric approximation. Training the flow takes about 2.5h and placing the bank under 1h on 32 CPU cores. $\texttt{isobank}$ is publicly available on Github https://github.com/Alexandra-Wernersson/isobank

A battery-powered device, such as a delivery drone that works from a depot, must stop to recharge between jobs, and the time spent recharging delays every job that follows. Scheduling models that fix the duration of a recharge do not describe a device whose recharge takes longer when it acquires more energy. We study a single device that executes a batch of non-preemptive jobs with known execution times, energy demands, and optional deadlines. Under partial recharging the device may acquire any amount of energy between jobs; under complete recharging every recharge fills the battery. Four objectives and four relationships between execution time and energy demand give 32 variants. When acquiring $q$ units of energy takes $q$ time units, we show that 14 of the 16 partial-recharging variants are polynomial, and we give a tight 2-approximation for the average completion time, one of the two NP-hard variants. Under complete recharging, the four variants with equal energy demands are polynomial and the other 12 are NP-hard; for makespan we give a $5/4$-approximation. When each recharge also incurs a fixed setup time $h$, the equal-energy variants remain polynomial and the other 24 are strongly NP-hard if $h$ is part of the input. For these we give exact algorithms that are exponential only in the number of jobs, and approximation algorithms for makespan and, when the battery starts empty, for the weighted average completion time. Experiments on synthetic and trace-derived job sets compare the algorithms with exact optima. The model is offline and deterministic, and we have not validated the schedules on hardware.

The problem of segregation measurement is to quantitatively describe the interaction of a population's attributes (such as race, ethnicity, education level, or income) with a separating distinction (such as organizational rank, school district, or spatial location). This primarily expository article gives an opinionated tour of one mathematical approach to measuring segregation. We begin with some basic intuitions about diversity and unify these through the framework of convex functions on the probability simplex. Formalizing segregation as a local-to-global comparison of diversity measures, resulting in the general class of Jensen informations as segregation measurements. We describe and computationally illustrate three ways to incorporate spatial structure into segregation measurements: spatial smoothing, aggregation, and local Jensen information. We close with several suggestions for future work.

We refine the real enumerative invariants insensitive to changes in the real structure introduced in our previous paper on del Pezzo surfaces, and extend these refined invariants to arbitrary rational surfaces. The construction involves a choice of an auxiliary conic bundle together with a Pin-structure. We prove that the resulting invariants are independent of these auxiliary choices and establish recursive relations for their computation.

Large language models often solve a theorem forward yet fail to disprove a closely related false one: a falsification gap that supervised fine-tuning does not close and can actively worsen. We frame counterexample generation as constrained witness emission against a deterministic per-theorem Python verifier, and release SymCE, a corpus of 4,707 false undergraduate-algebra and real-analysis conjectures, each paired with executable verifiers. The verifier also serves as the reward function, making SymCE a training environment. Training Qwen3-4B with SFT followed by GRPO under this oracle reveals an imitation trap: counterexample-only SFT collapses true-theorem recognition from 0.27 to 0.00, while RLVR with a sparse outcome-only reward repairs this and exceeds the base, to 0.66. The collapse replicates across four seeds and on Gemma-3-4B. Sparse and dense rewards yield statistically indistinguishable in-domain success yet diverge by 33 points on a held-out calibration probe, a dissociation we trace to the partial-credit term. Our 4B model outperforms every evaluated 7B open-weights math specialist, remains competitive with six frontier commercial APIs, and transfers under unchanged prompting to GSM8K, MATH-500 and MMLU-college-math. A human audit of 177 verifier decisions finds 97.7% accuracy. Code, data, verifier modules and annotations: https://github.com/ce-rlvr/SymCE.

Let $\delta \in [0,1/2]$ and $Q\geq 1$. Given a compact $C^{\infty}$-curve $\mathcal{C}$ in $\mathbb{R}^n$, denote by $N_{\mathcal{C}}(\delta, Q)$ the number of pairs $(\mathbf{a}, q)\in \mathbb{Z}^n\times [Q/2,Q]$ such that the rational point $\mathbf{a}/q$ is $\delta/q$-close to $\mathcal{C}$. We investigate the range of $\delta$ in terms of $Q$ which is necessary for the heuristic $N_{\mathcal{C}}(\delta, Q)\asymp\delta^{n-1}Q^2$ to be correct for the moment curve. For a nondegenerate curve $\mathcal{C}\subset \mathbb{R}^n$ and for sufficiently large $Q$, Hickman and the second author previously established that $N_{\mathcal{C}}(\delta, Q)\lesssim_{\nu} \delta^{n-1}Q^2$ for all $\delta\in[Q^{\alpha(n+1)+\nu},1/2)$ and any $\nu>0$. Here $\alpha(n)=-4n^{-2}+O(n^{-3})$ was explicitly computed. We show that this range is surprisingly close to being sharp in the asymptotic sense as $n\to\infty$, and can at most be extended to $\delta\in[Q^{\alpha(n)},1/2)$. In particular, the sharp exponent can only be about $O(n^{-3})$ better, even though it was widely believed before that an improvement of $O(n^{-1})$ should be possible. The complementary upper bound in this remaining range has been recently established by Gan–Guo–Oh.

The emerging of large language models (LLMs) has posed significant challenges to the thermal management of data center. Intense GPU computation for LLMs results in localized hotspots. Moreover, spiking thermal loads during training and inference bursts make real-time cooling response more difficult to predict and control. Thermal-aware capacity planning of data center requires massive expensive high-fidelity CFD simulations. AI models can perform real-time prediction for unseen designs. However, existing works either have large prediction error, or have over-simplified assumptions for data center operations. This work presents an AI-driven framework that can perform thermal-aware capacity planning for a real-world data center in seconds. The embedded AI model learns from numerous key parameters (rack power, server power, server placement, HVAC settings etc.), and provides temperature prediction within milliseconds. This AI model is tested against high-fidelity CFD simulations, and results show that for unseen data center designs, model can achieve high accuracy with 10000X speedup. Driven by the AI model, the authors design the thermal-aware capacity planning framework. This framework can help data center designers and operators instantaneously optimize both workload distribution and HVAC cooling efficiency.

We model directed motion of cells that aggregate into a cluster in the presence of chemotaxis or fluid transport. Starting from a stochastic agent-based model (ABM) of Galante & Levy (2013), Chavy-Waddy & Kolokolnikov (2016), we consider two model variants, with (I) directed motion of individuals, or (II) implicit individual bias in one direction. In both cases, we derive fourth order nonlinear PDEs representing the continuum limits of the models as a generalization of Chavy-Waddy & Kolokolnikov (2016). An exact analytic solution then describes cluster density profile as well as the group velocity, and we identify the regimes where the travelling cluster exists. Excellent agreement is observed between the discrete models and the exact solutions of the PDEs in both models, including the existence and speed of the travelling cluster.

Contextual bandits require balancing nonlinear reward modeling with online efficiency. Tree ensembles and neural methods capture nonlinearities but require periodic retraining and large replay buffers. Linear models update efficiently per observation with O(1) memory, but are fundamentally restricted to linear reward structures. We propose Residual Quantization (RQ) as a representation layer to bridge this gap. An offline-trained RQ codebook maps continuous contexts into discrete centroid assignments across multiple levels, set dynamically through a shadow mechanism. This enables a spectrum of additive bandit algorithms that achieve nonlinear expressivity with strictly bounded memory. Across 13 datasets, RQ variants beat their non-RQ counterparts on 11 of 13 datasets, often by wide margins, while matching doubling-retrain XGBoost and neural baselines using up to 1000 times less memory.