Papers

Answering a question of Noga Alon, we generalize a result of Janzer and Yip to show that every $k$-edge-coloring of $K_{r^k+1}$ contains a monochromatic subgraph with chromatic number at least $r+1$ and diameter $O(k^{3/2}r^{k/2}\log r)$.

AI-generated covariates from notes, conversations, images, and wearable streams can change the causal question when their roles are left unspecified. A generated feature may represent a treatment version, pre-action state, history, design variable, mediator, outcome proxy, observation process, or intercurrent event; these roles are not interchangeable. We formulate a causal type discipline for sequential experiments: a versioned representation map, a causal role classifier, a claim-status filter, and an estimand lock. The lock fixes a standardized proximal effect before generated covariates enter the analysis. Under audit correctness and standard identification assumptions, admissible role assignments preserve this estimand. We apply the established conditional-covariance characterization of compression bias to substitution of generated representations for design-relevant states. A standardized decomposition separates compression, conditional-law, and standardization drift. Further results cover mediator adjustment, post-action leakage, marker-intervention conflation, outcome-guided discovery, and state-measurement error. Cluster-level orthogonal estimators distinguish empirical and superpopulation targets under repeated sessions and missing outcomes. Simulations show that refinement helps when it retains design-relevant information, whereas design erasure, leakage, and same-data marker selection can produce bias or undercoverage. The framework places causal semantics and claim status before confirmatory inference with generated representations.

Approaches to incorporating human awareness into mobile robot decision-making mainly focus on collision avoidance in low-level motion planning, often overlooking the challenges posed by human presence and high-level behavior. To address this vacancy, we present HINT-Plan, a novel approach to integrate human intention prediction into robot task planning. HINT-Plan employs Vision Language Models (VLMs) to anticipate high-level human intentions from third-person image observations, convert them into goal states, and solve joint task-planning problems. To effectively enable scene awareness in context-rich environments, we use hierarchical Scene Graphs (SGs) as high-level representations of the environment, and translate environmental topology and actionable knowledge into formal planning language to ensure executable plans. Evaluated in a photorealistic simulation, HINT-Plan achieves an overall success rate of 69.71% in joint human-robot task planning, substantially outperforming the baselines by up to 35.29%, while also reducing functional conflicts. The results show the effectiveness of explicitly incorporating inferred human intentions into formal multi-agent task planning for proactive human-aware robot decision-making.

A MoonBoard is a standardized bouldering wall used in gyms around the world. Climbs up the wall limited to only a subset of holds are known as problems. We introduce PointGrade, a novel machine learning approach to predicting the difficulty of a MoonBoard problem. By sampling a point cloud from pre-scanned meshes of every hold, our model combines 3D object classification architecture with existing sequence-based approaches to difficulty grade prediction. Our method captures latent geometric information contained the climb, outperforming other work on the problem that neglect this data.

Downlink channel state information (CSI) feedback is a key bottleneck in frequency-division duplex (FDD) massive MIMO systems, as the user equipment (UE) must convey its estimated channel to the base station (BS) over a limited uplink (UL) budget. To improve CSI reconstruction accuracy under tight feedback constraints, we propose prior-aided masked vector quantization (PM-VQ), a learning-based separate source–channel coding (SSCC) feedback scheme conditioned on the average angle–delay power map—a compact representation of the channel second-order statistics available at both the UE and the BS. In PM-VQ, a prior-aided encoder maps the CSI to latent tokens, and a spatially-adaptive masking module (SAMM) scores and selects the most informative tokens within the feedback budget. The selected tokens are vector-quantized and fed back together with their positions, while an adaptive de-masking module (ADM) completes the latent representation at the BS before prior-conditioned decoding. To support variable-rate compression, a single model is trained over a range of selected-token counts, enabling operation across multiple feedback dimensions without retraining. We evaluate PM-VQ against three representative baselines on a Sionna-generated 3GPP TR~38.901 UMa dataset, focusing on the most challenging diffuse regime where channel energy is spread across many angle–delay coefficients. Simulation results show that PM-VQ achieves the lowest NMSE across all tested SNR levels and feedback dimensions in this regime. Moreover, the angle–delay power-map prior remains beneficial even when estimated from only a few channel realizations.

Many scientific problems require accurate modeling of complex physical phenomena, such as fluid dynamics or climate modeling. These phenomena often result in high-dimensional and thus computationally expensive computational models that can limit their application to real-time and multi-query problems. Model-order reduction (MOR) is an approach that seeks to approximate full-order models (FOMs) using reduced-order models (ROMs), trading a minor reduction in accuracy for a major reduction in computational cost. In this work, we propose a non-intrusive MOR method using generative machine learning by means of denoising diffusion probabilistic models (DDPMs) for generating solutions of the dynamical systems under different instances of their parameters. Unlike conventional DDPMs, which often operate purely in the spatial domain, we aim to generate spatio-temporal solutions to improve quality and temporal coherence. In addition, we embed the DDPM in a latent space obtained by sequentially applying proper orthogonal decomposition and an autoencoder to reduce the data dimensionality and the computational cost of the DDPM. We test our approach on a parametrized 2D fluid flow around an obstacle. The numerical experiments demonstrate that our latent DDPM can (i) produce accurate and temporally coherent solutions, (ii) achieve strong generalization capabilities to scenarios involving unseen parameter values, and (iii) extrapolate in time beyond the training horizon.

Fuzzy hypervector spaces provide a structural framework for modeling graded uncertainty in algebraic systems with multivalued operations. In this paper, we introduce and investigate admissible mappings as a constructive mechanism for generating fuzzy subhyperspaces. These mappings represent fuzzy substructures through families of fuzzy points and allow a systematic characterization of generators. We prove that maximal admissible mappings generate fuzzy subhyperspaces in a manner analogous to bases in classical linear theory, and we present examples over real and finite hypervector spaces to clarify the role of maximality. The proposed approach contributes to the structural foundations of fuzzy modeling in hyperalgebraic settings and may support further developments in soft computing environments involving graded and non-deterministic structures.

Radar-based navigation is appealing for its robustness to adverse conditions involving airborne particles, such as precipitation, dust, fog, and smoke, that can cause lidar-based systems to fail. Recently, direct methods that retain and use the entire radar scan rather than sparse points have improved on-road global localization performance. However, they have yet to be deployed in off-road environments or in closed-loop systems. Additionally, even direct global maps may lose information: their global nature leads to a smoothing out of viewpoint-dependent radar artifacts, which can provide pose information when mapping and localization occur along similar trajectories. This paper introduces Direct Radar Teach & Repeat (DRT&R): a direct spinning radar-based navigation stack that maximizes the amount of retained information by combining direct radar processing with local mapping. DRT&R yields state-of-the-art (SOTA) localization performance in both on-road and off-road environments. Using 344 km of on-road data and 20 km of off-road data, DRT&R is able to localize to within 4 cm in most on-road and off-road conditions, and 12 cm in geometrically degenerate and sparse environments. DRT&R is also evaluated autonomously in closed loop with an MPC controller for more than 10 km using a Clearpath Warthog off-road vehicle, demonstrating that it runs in real time and achieves SOTA tracking performance for off-road radar navigation.

A species-resolved anisotropy-response scaling framework is used to investigate the structure and formation dynamics of the $f_0(980)$ in relativistic nuclear collisions. Published measurements in high-multiplicity p+Pb collisions at $\sqrt{s_{NN}}=8.16$~TeV exhibit broad scaling closure under a single-meson response construction, with a small effective final-state response, $\zeta_{f_0}\simeq0.04$, whereas a symmetric ${\rm K\bar K}$ constituent-response construction gives substantially poorer closure. An explicitly molecular ${\rm K\bar K}$ calculation also exhibits single-meson-like scaling but gives a substantially larger effective final-state response, $\zeta_{f_0}\simeq0.16$. Thus, single-meson scaling alone is not a unique structural discriminator. The combined response construction and $\zeta_{f_0}$ constraints characterize the measured $f_0(980)$ by a single-meson-like anisotropy response with weak final-state sensitivity, distinctly different from the molecular benchmark. These results establish anisotropy-response scaling as a new experimental probe of $f_0(980)$ formation dynamics and motivate systematic tests across collision systems and beam energies.

Attention-based rerankers score documents by aggregating query-to-document attention and subtracting a null-query calibration pass to remove positional and structural bias. Although widely used, this calibration assumes that the null pass removes irrelevant signal from each document. We show that modern prompt content, e.g. constraints, instructions, personas, and demonstrations can violate this assumption when it enters the scoring readout, making the null pass relevance-aware rather than null. We find that calibration is especially harmful when applied to prompts containing longer, more detailed instructions as the null-pass step removes relevant signal. Based on these findings, we propose interpolated null calibration, a training-free modification that controls how much of the instruction content enters the null baseline. It recovers attention-based reranking performance on instruction-heavy tasks where standard calibration fails, while preserving calibration's benefits when the null pass remains relevance-agnostic. On instruction heavy tasks, the recovered rankings surpass generative rerankers. We also show that in-context demonstrations improve attention-based reranking with little calibration interference, since demonstrations act only through the query pass and leave the null pass unchanged.

Next-day wildfire prediction requires models whose forecasts can be evaluated alongside the assumptions and historical evidence used in their computation. Although deep learning can learn spatial patterns from remote-sensing data, predictive performance alone does not establish physical fidelity or operational trustworthiness. This study investigates three modular augmentations for next-day active-fire prediction: wind- and slope-conditioned attention biases, physics-feature retrieval-augmented output correction, and fire conditioned dual-stream gating. The attention biases expose prescribed directional preferences, while the retrieval module selects historical tiles using a nine-dimensional environmental and fire-state descriptor and applies a learned correction to a frozen model's logits. The modules are evaluated across five backbones on the Next Day Wildfire Spread benchmark, using staged ablations, directional audits, retrieval perturbations, calibration measures, and computational comparisons. The three-seed mean F1 score and area under the precision–recall curve (AUC-PR) of a SwinUNETR model with all three augmentations are 0.4216 and 0.3673. Then, a mixed ensemble (two augmented architectures and one non-augmented architecture) model achieves 0.4292 and 0.3790. Benefits vary across architectures, and retrieval-related improvements in AUC-PR do not consistently translate into higher F1. The constructed wind bias aligns closely with input wind, but its alignment with observed next-day fire displacement is much weaker, distinguishing prior inspectability from predictive physical fidelity. The study contributes a framework for exposing and evaluating selected domain-informed components within wildfire prediction models. Together, the results presented show that predictive performance, operational trustworthiness, and computational practicality need not be competing objectives.

According to Christian tradition, Luke is credited with authoring a Gospel and the Acts of the Apostles, even if his name does not appear in either book, both originally written in Koine Greek. Several biblical scholars assume that both texts were written by a common author, while others deduce the presence of two authors. Different studies have been found to support either finding, some based on qualitative evaluation, while a few others consider the occurrence frequency differences between the two books. To propose an enhanced quantitative analysis, this study is grounded on two recent authorship attribution models. The Burrows' Delta, applied with eleven different feature sizes, demonstrates common authorship. An author verification model confirms this finding. The following experiments consider several stylistic representations, feature sizes, and distance functions to confirm that Luke is the true author of both books.

Labor strikes do not necessarily lead to complete institutional disruption, particularly when non-striking sectors maintain part of normal activities. We propose a minimal dynamical model to investigate how such residual activity affects the persistence of labor conflicts. The model couples strike participation, institutional response, and overall functionality through two competing mechanisms: a direct suppressive effect of residual activity on strike participation and an indirect effect arising from the reduced institutional pressure to resolve the conflict. Analytical calculations reveal a counterintuitive regime in which increasing residual activity increases, rather than decreases, stationary strike participation. We characterize this behavior through a strike-response susceptibility and derive an exact compensation threshold separating the conventional regime from a resilience-paradox regime. At the threshold, the two mechanisms exactly balance, making stationary strike participation independent of residual activity. Numerical results confirm the analytical predictions and illustrate the underlying mechanism. Our results show that institutional resilience can have an unintended dynamical consequence: by absorbing part of the disruption generated by a strike, continued operation may reduce the urgency of institutional response and thereby favor conflict persistence. This provides a simple example of how individually stabilizing processes can collectively generate counterintuitive outcomes in complex social systems.

The first open case of Hadwiger's conjecture states that every $K_7$-minor-free graph is 6-colorable. We prove that this is the case for $K_7^=$-minor-free graphs, where $K_7^=$ denotes the graph obtained from $K_7$ by deleting two independent edges. The proof is based on an independently interesting density result: Every 5-connected $K_7^=$-minor-free graph with $n\ge 6$ vertices has at most $4n-8$ edges.

Muon updates matrix-valued neural-network parameters by orthogonalizing a gradient-based momentum matrix. Its reliance on derivatives limits its use when gradients are unavailable or unreliable. We develop a derivative-free framework that constructs Muon-style updates from structured finite differences. Four variants are considered: full entrywise recovery, random low-rank surrogates, basis-aligned rank-one probing, and direct structured search. Exhaustive basis-aligned probing is equivalent, up to positive scaling before ideal polar orthogonalization, to coordinate finite differences. Matrix-regression experiments show that random rank-one probing can reduce the number of function evaluations substantially, at the cost of less accurate updates. Controlled noisy-gradient experiments on regression and a neural network illustrate when accurate function values can compensate for an unreliable gradient oracle. A small CartPole study further examines orthogonal rank-one probes under a fixed episode budget. These results support structured probing as a practical option for selected black-box problems; they do not establish a general convergence guarantee or an advantage over accurate, inexpensive gradients.

Deep Reinforcement Learning has demonstrated remarkable capability in quadrotor control, yet learned policies offer no guarantee of respecting safety constraints during training or deployment. We present CALOS (Control-Affine Lyapunov On-manifold Safety), a runtime safety layer that enforces attitude constraints on a quadrotor without modifying the underlying learning algorithm. CALOS formulates four tilt-angle inequalities and a Lyapunov descent condition as a single quadratic program whose solution is the minimum-norm correction to the nominal torque output of the policy. The quadratic program is solved exactly via active-set enumeration over the three-dimensional torque space, with a computational cost low enough to enforce constraints in real time across thousands of parallel simulation environments, as required by modern massively parallel Deep Reinforcement Learning training. Evaluated on trajectory-tracking tasks in NVIDIA Isaac Lab, CALOS reduces lateral tracking error by 55-60% relative to an unconstrained Proximal Policy Optimization baseline while achieving zero attitude-constraint violations on the training trajectory. By restricting exploration to safe regions of the state space, the safety layer also accelerates training convergence and improves data efficiency without producing suboptimal policies.

Behavioral cloning trains a policy offline on expert demonstrations, but deployment is closed loop: each action affects the observations the policy receives next. We study how much closed-loop driving competence a compact multimodal policy can acquire from offline demonstrations in the CARLA simulator. The policy uses five-frame histories of RGB images, LiDAR, vehicle telemetry, and lane waypoints to predict throttle, brake, and steering at 20 Hz. Demonstrations were collected in three stages, ending with a systematic route-generation procedure that enumerates spawn points and feasible maneuvers and verifies completed autopilot routes. The released 1.36 million parameter policy was trained on 236,882 windows, representing about 3.3 hours of driving from 448 captures. The resulting policy drives autonomously for hours on training and held-out routes. In our runs, it did so without collisions and also transferred qualitatively to an unseen CARLA town with different road geometry. We also observed recovery from large trajectory deviations, although we do not claim systematic recovery without controlled evaluation. We report offline metrics and distinguish measured results from qualitative closed-loop observations. We release the code, trained checkpoint, ONNX model, data sample, and an evidence audit for the reported claims.

We present dynamical measurements of dark matter halo masses at $z\approx7$ using galaxy pairs. Galaxy pairs in the early Universe are usually in their first infall, before substantial orbital evolution or phase mixing. Their relative velocities closely trace the halo mass, especially when the pair separation is near the halo virial radius. In the TNG100 simulation, we find that dynamical mass estimators can recover halo masses with an intrinsic scatter of $\sim$0.2 dex. We apply this method to 10 pair systems at z=6-8 with [O III] spectroscopy from the JWST Advanced Deep Extragalactic Survey (JADES). We infer the stellar-to-halo mass relation while marginalizing over projection effects, measurement uncertainties, and intrinsic scatter. We find a mean halo mass of $\log\,(M_{200}/M_\odot)=11.06\pm0.21$ at $\log\,(M_{*}/M_\odot)=9$. These galaxies do not appear to inhabit unusually massive halos. Our results instead indicate a high stellar-to-halo mass ratio, corresponding to an integrated star formation efficiency of $5^{+3}_{-2}\%$, about twice the TNG100 prediction, although the current statistical significance is limited. Future observations with larger samples will tighten these constraints and directly examine whether enhanced star formation efficiency drives the overabundance of luminous galaxies at cosmic dawn.

The analysis of US political language is usually based on the written form (e.g. presidential addresses) or posts broadcasted on various social networks. Oral production, however, which is even more frequent, can better reveal the style and mode of thinking of the speaker. This study covers this mode of linguistic communication by considering 19 candidates from the presidential elections between 1960 to 2024. Our main research objectives are to disclose the main trends hidden in those presidential debates. Do we observe a clear simplification of the US political language over time? Does Trump have poor language compared to the other candidates? Do unusual stylistic features occur only with a single, specific president? Moreover, can we detect a pattern explaining the success or failure of some nominees? Over time, this study demonstrates a significant reduction in political language complexity, a decrease of the mean sentence length, and a noteworthy decline of complex terms. Moreover, the emotional tone increases over the decades, while logical and rational thinking tends to lessen.

This paper investigates market competition among wireless service providers (SPs) that serve customers using shared spectrum. Prior work has analyzed such markets through models of competition with congestible resources, capturing both the congestion-sensitive nature of wireless spectrum and the effects of spectrum sharing on service quality. These models typically assume that the market demand function is known, enabling SPs to optimize pricing or quantity decisions under either Bertrand or Cournot competition. In contrast, we consider a setting in which the demand function is initially unknown and must be learned over time. We model this learning process using multi-agent reinforcement learning (MARL), allowing competing SPs to learn market dynamics while adapting their competitive strategies. Although MARL has shown strong performance in a variety of economic settings, recent work has demonstrated that it can also give rise to tacit collusion among self-interested agents. We therefore examine whether similar collusive behavior emerges in shared-spectrum markets and how its prevalence depends on the mode of competition (price versus quantity) and the choice of MARL algorithm. Our results provide insight into the interaction between learning dynamics, market structure, and spectrum sharing, with implications for both wireless market design and the deployment of learning-enabled decision-making systems.

Reference labels used to train medical image classification models are not always as certain as they may appear, and this uncertainty has implications on performance metrics. In this study, we propose a framework to analyze model performance for periventricular Fazekas score prediction that goes beyond conventional metrics. The Fazekas score is an ordinal visual rating scale used to assess the severity of white matter hyperintensities and is known to be affected by inter-rater variability. While the best Fazekas score prediction model achieved a Matthews correlation coefficient (MCC) of 0.70, performance varied across data splits and loss functions, making interpretation of model capabilities difficult. Rather than interpreting epistemic uncertainty of a model's prediction as an isolated scalar value, our approach of uncertainty mapping relates uncertainty to its position within the learned feature representation. This highlights regions of class-boundary transitions where cases appear more ambiguous and misclassifications are more likely. It also identifies potential label disagreement, including low-uncertainty misclassified cases that expert review found to be inconsistent with the original reference Fazekas score. Therefore, uncertainty mapping allows model behaviour to be examined in relation to class separation and potential model-label disagreement. Loss function choice also influenced the uncertainty profile, with some models showing clearer class separation and more localized uncertainty in ambiguous regions than others. These findings suggest that uncertainty mapping for Fazekas score predictions can support model interpretation and targeted dataset review when reference labels are affected by ambiguity/ inter-rater variability.

Early-career work both signals a worker's ability and builds long-run skills. We introduce an AI industry into a career signaling model. The industry sells tokens, which workers convert into output indistinguishable from their own. We compare three regimes: no token trade, purchases only, and open trade with side sales. With no token trade, market learns every type, signalling incentives create a rat race and workers work above the full information level. With token purchases only, lower types buy output but the market still learns every type. Buying by lower types raises the signal higher types must post, so the high types can work harder than before AI. Buyers work less with AI, even below the full-information level. With side sales and abundant AI, the signal no longer depends on effort and every type under-invests in skill. Workers gain from token trade only when welfare improves, never the reverse, and the difference is the AI industry's profit. In education, where output has no value, every worker loses in the separating equilibrium, but with valuable output and cheap tokens everyone gains. Token taxes, wage taxes, and watermarking have ambiguous effects on workers' payoffs and welfare, and their effects depend on whether workers can sell output on the side.

A continuous cryogenic-pellet injector has been used to realize long-pulse high-performance plasmas in Wendelstein 7-X (W7-X). A hydrogen ice pellet deposits particles directly in the confined region. If the particles are deposited sufficiently far inside the plasma, they produce a density gradient that suppresses ion-temperature-gradient turbulence and improves plasma confinement. However, after multiple pellets have been injected and the plasma density and temperature have increased, the plasma performance begins to saturate. In this work, we analyze multiple pellet-injection experiments conducted in 2024 and 2025, during which the injected pellet sizes varied unintentionally. This analysis reveals a positive correlation between pellet size and the quasi-steady-state stored energy of W7-X plasmas. Although this pellet-size dependence can be understood qualitatively from pellet-ablation physics, the measured deposition position differs quantitatively from the neutral-gas-shielding (NGS) model prediction. This discrepancy suggests significant inward transport of the pellet cloud. The trend identified here suggests that injection of even larger pellets could further improve plasma performance.

Stochastic Gradient Langevin Dynamics (SGLD) reduces the cost of Langevin-based sampling by replacing full-dataset drift evaluations with mini-batch approximations, but the resulting subsampling error may offset this computational saving. We study this trade-off for stochastic differential equations with finite-sum drifts and compare the computational cost of SGLD with that of the Euler-Maruyama (EM) method. For a prescribed mean-square accuracy $\varepsilon^2$, we derive complexity estimates that explicitly track the dependence on the dataset size $m$, mini-batch size $s$, and accuracy parameter $\varepsilon$. The resulting comparison reveals distinct parameter regimes in which either method is preferable. In particular, EM can have lower leading-order cost only in a small-data, aggressive-subsampling regime, whereas SGLD is favoured over most of the remaining parameter space. In the practically relevant regime $s \ll m$, the transition between the two methods occurs at the scale $m \asymp \varepsilon^{-1}$. We complement the theoretical analysis with numerical experiments based on a Gaussian Bayesian inference model, which examine the predicted cost regimes together with the underlying discretisation error and variance estimates.

A lot of expectations have been put for years on integrating domain expert knowledge in image classification models. Several approaches have been explored, Concept Bottleneck Models (CBMs) opened up a new avenue of research leading to many variants, and more recently to Concept-based Embedding Models (CEMs). CBM consider binary encoding of each concept, while CEM expands this idea by embedding each concept through two vectors. However in real-life scenarii, domain experts' knowledge is usually organized in concepts determined by various attributes, each attribute encoded either with numerical values, or range of values, or binary values, or categorical values. In this work, we first finetune an image feature extractor for classifying attributes representing the downstream object classes, where the class attributes have been specified by experts under various encoding formats. A classification head is then learnt from these various attributes to categorize target objects. We experimentally show that it improves the classification for three datasets: Kaggle fish dataset, AWA2 and a more challenging new wood charcoal dataset. We then propose an automatic selection of potential missclassified data. In this second step, experts are asked for those data to eventually modify the predicted attributes in order to improve the classification.

Deep neural networks that are independently trained to similar performance can be connected by low-loss parametric curves in weight space, a phenomenon known as Mode Connectivity (MC). This geometric property underpins practical techniques such as weight averaging, model ensembling, and model merging. We argue that low-loss connectivity is an incomplete geometric criterion: it controls loss only along a one-dimensional trajectory while leaving the surrounding weight-space neighborhood unconstrained, so the optimized curve may traverse sharp ridges that become fragile under distribution shift. We therefore reformulate mode connectivity as a neighborhood-robust path optimization problem, seeking a curve whose entire local neighborhood maintains low loss. We propose Sharp Mode Connectivity (SMC), which applies a first-order sharpness-aware approximation to the resulting minimax functional, enforcing flatness along the entire curve rather than only on it. We derive a practical optimization algorithm for connectivity paths under this sharpness-aware objective. Under severe blur corruptions from CIFAR-10-C, SMC achieves up to 6.09% absolute accuracy improvement over standard MC. Remarkably, SMC produces negative loss barriers, meaning that models obtained at interior points of the optimized path can outperform the average endpoint loss. These results, validated across ResNet-18, VGG16-BN, and ViT-Tiny on CIFAR-10 and ImageNet-100, establish path-wise flatness as a practical principle for robust weight-space interpolation.

This study explores the vocabulary richness of oral political communication. A model explaining the lexicon growth is proposed by subdividing the whole vocabulary into terms generated by general and specialized glossaries.

Levene's test for homoscedasticity is a standard procedure used to evaluate whether multiple groups of independent observations share a common variance. Because Levene's test relies on an Analysis of Variance (ANOVA) applied to transformed absolute or squared deviations, it structurally mirrors the statistical properties of ANOVA itself. Let $\mu_j$ and $\sigma^2_j$ denote the expectation and variance of the observations in group $j$. It was previously established that for a given significance level $\alpha$, ANOVA can result in a logical contradiction: failing to reject the global null hypothesis $H_0: \mu_1 = \mu_2 = \mu_3$ while simultaneously rejecting the localized hypothesis $H_0': \mu_1 = \mu_2$ with the same or higher confidence. In this paper, we show that Levene's test directly inherits this same “paradox” regarding group variances $\sigma^2_j$. We provide theoretical reasoning and a numerical illustration of this inconsistency, demonstrating how the addition of a well-behaved third group can dilute the test statistic and mask a significant localized variance discrepancy.

A central goal of autonomous reinforcement learning is continuous policy training without external resets. However, existing paradigms largely depend on underlying environmental reversibility, a property absent in real world manipulation, where events such as pushing objects off tables or spilling granular substances cannot be undone. We introduce REVERSAL-BENCH, a benchmark that controls reversibility via a continuous parameter $\rho \in [0, 1]$ and provides a reset oracle, a ground-truth verification mechanism to test state recoverability across eight manipulation settings in five physics engines. Evaluating a broad spectrum of policy architectures, including standard actor-critic algorithms, safe RL, and specialized reset-free frameworks, reveals a sharp reversibility cliff: reset-free agents are consistently absorbed into irrecoverable states as $\rho$ increases, whereas episodic agents maintain steady learning. We see this failure mode across autonomous reset-free baselines and constrained RL. Because reset-free agents lack external resets, any transition into an irrecoverable state results in permanent absorption, leaving the agent trapped where further learning halts. We show that this absorption phenomenon persists in full physics simulations under learned manipulation policies. By evaluating against geometrically identical reversible counterparts, we confirm that this breakdown is causally driven by irreversibility rather than obstacle complexity. We release the benchmark suite, a large multi-simulator dataset labeled with recoverability and a reset oracle. We also evaluate a safety shield that intervenes before irreversible failures occur, showing that while recoverability can be predicted accurately, active recovery primarily succeeds only when the agent can physically steer clear of the trap

We define a diagonal-inversion statistic on labeled Dyck paths carrying both decorated rises and decorated contractible valleys. This gives an explicit candidate for a bivariate refinement of the univariate Theta conjecture of D'Adderio, Iraci, and Vanden Wyngaerd, recently proved by D'Adderio, Pagaria, and the authors of this work. The univariate conjecture can be recovered from our bivariate version by setting $q=1$. The new conjecture recovers the rise and valley versions of the Delta conjecture when either decoration parameter vanishes, and provides a combinatorial interpretation of the symmetric function $\Theta_{e_l} \Theta_{e_k}\nabla e_{n-k-l}$, which is also conjectured to be the Frobenius characteristic of a certain graded module of diagonal coinvariants with two sets of commuting variables and two sets of anticommuting variables. In support of the Theta conjecture and its touching refinement, we prove its Schröder case, that is, the scalar product of the symmetric function side with $e_{n-d} h_d$ matches the combinatorial side restricted to Schröder paths. The proof uses a finer combinatorial argument than the previously known cases, leveraging two Gaussian product identities.