Papers

Context: The eccentric von Zeipel-Lidov-Kozai effect, which is widely applied to diverse astrophysical settings, can drive the inner binary to extremely high eccentricities, where short-range forces such as general relativity (GR) become prominent. Aims: Poincare surfaces of section show that GR effects reshape phase-space structures, giving rise to a widespread and large-scale chaotic sea. This work aims to uncover the underlying mechanism of GR-enabled chaos. Methods: The dynamical structures are studied within an adiabatic framework, where an adiabatic invariant is constructed. Phase portraits, defined by the level curves of this invariant, establish analytical boundaries for the chaotic domains. Numerical simulations are subsequently performed to map orbital flips, maximum eccentricities, and Fast Lyapunov Indicators (FLI) across the initial eccentricity-inclination parameter space. Results: Phase portrait analysis demonstrates that GR precession introduces an uncertainty zone near polar inclinations, where trajectories crossing this zone inherently become chaotic. Leveraging this framework, the chaotic boundaries across the full parameter space are analytically derived, yielding excellent agreement with numerical maps of flipping orbits, maximum eccentricity, and FLI. Conclusions: In the presence of short-range forces, flipping orbits accompanied by extreme eccentricity excitation are fundamentally chaotic, and the underlying mechanism of large-scale chaos originates from the periodic passage of trajectories through the uncertainty zone in phase space over secular timescales.

Attributing language model outputs to their internal computations is an open problem in interpretability. Existing methods, which use causal interventions, gradients, or learnable masks, either are infeasibly expensive or struggle to identify actual causally-important internal computations. We propose framing attribution as the problem of identifying nested subsets of internal components which minimise a downstream loss. To learn this task, we introduce Matryoshka Attribution (MAttr), a mask learning method that parametrises the mask with a simple differentiable sigmoid top-$k$ operator. We supervise training over all sparsities simultaneously by randomising $k$ over training, resulting in a learned ordering of components by attribution score. MAttr achieves number 1 on the official leaderboard of the Mechanistic Interpretability Benchmark (Mueller et al., 2025); our method identifies sparse and task-transferrable circuits across varying circuit bases. As a practical application, we show that MAttr can be trained with reinforcement learning to identify weight changes responsible for downstream behaviours in LLM finetuning. We train MAttr on refusal judge scores and find that restoring $1\%$ of Llama 3.1 8B Instruct's weights to their base model state is sufficient to remove refusals while maintaining capabilities. We view MAttr as a successful formulation of interpretability into a learnable objective that we can tackle with gradient descent, and encourage future work along these lines.

In aerospace systems, powered descent requires efficiently selecting a landing site while fine-scale hazards remain unresolvable until low altitude. This process presents a decision challenge since the actor must select a site and make corresponding actions before all information is known. To successfully solve this problem, an agent must reason over potential risks and make corrections as new observations are made. We introduce a lightweight model of altitude-limited information where each landing site is summarized by a mean score and a designed ambiguity proxy that contracts as the vehicle descends and senses within a cone-shaped footprint under an altitude-to-resolution schedule. Using this abstraction, we derive closed-form, risk-averse site scoring techniques (an entropic certainty-equivalent and a Gaussian Conditional Value at Risk surrogate) and pair them with greedy and exploratory planners to prioritize sites that are both high-value and robust to late-revealed terrain detail. These rollout-free heuristics improve lower-tail landing outcomes (1st percentile and certainty-equivalent) relative to mean-based baselines, with the largest gains when refinement occurs late and unresolved detail is large. We also demonstrate that these methods perform comparably to or better than Monte Carlo Tree Search baselines with orders-of-magnitude faster computation. Our results are supported by numerical simulations.

Flaky tests produce non-deterministic outcomes without code change, eroding CI confidence and delaying deliveries. While vocabulary-based machine learning prediction has proven effective for Java and JavaScript, no study has evaluated it for Swift, a language whose testing style is dominated by UI and asynchronous code. We collect 91 flaky and 22,349 stable tests from 15 open-source Swift projects via re-execution and commit-history mining, then train five classifiers (Random Forest, Decision Tree, Naive Bayes, SVM, KNN) on TF-IDF unigram+bigram features under stratified 5-fold cross-validation. Random Forest achieves the best performance (Precision = 0.92, F1 = 0.86, AUC = 0.95) and substantially outperforms trivial baselines, among them a vocabulary-threshold rule applied to the most informative tokens, confirming a genuine discriminative signal (MCC = 0.75 vs. 0.08 for the best baseline). Information-gain analysis reveals two complementary signal types. Flakiness markers appear predominantly in unstable tests and comprise concurrency primitives (async, await), expectation-based synchronisation (expectation, fulfill), error propagation (throws), and explicit timing dependence (timeout, wait, now). Stability markers, chiefly the assertion vocabulary of plainly synchronous tests (xctassertequal), count as evidence against flakiness. Error analysis shows that the model fails when flakiness is hidden in shared infrastructure outside the test body or when async constructs are used in a deterministic context, exposing the intrinsic limit of lexical prediction. These results extend vocabulary-based flakiness detection to the Swift ecosystem and characterise both its effectiveness and its boundaries.

Standard deployment-ready object detectors for autonomous vehicles degrade in adverse weather and lighting conditions without being trained on extensive domain-specific data. While large-scale vision foundation models offer robust zero-shot generalization, their high computational cost makes them impractical for real-time deployment. To bridge this gap, we propose a foundation-guided auto-annotation pipeline that enhances standard detectors without architectural changes. We first benchmark three distinct models, YOLOv8, Co-DETR, and SAM3, on our custom real-world driving dataset spanning 25 unique operational scenarios across various route, weather, and lighting conditions. Based on our analysis, SAM3 demonstrates superior accuracy and resilience across all scenarios. Thus, we deploy it as an offline auto-annotator to generate pseudo-labels on the unannotated subset of our dataset. Fine-tuning the baseline YOLOv8 on these annotations yields a 16.04% higher overall mean Average Precision (mAP) and improves cross-environmental stability compared to the baseline model, highlighted by a 32.73% and 28.65% mAP increase in Residential Direct Sunlight and Highway Fog, respectively. These results demonstrate that standard detectors can achieve environmental resilience without the need for extensive manual annotation or architectural modifications.

Vector databases are increasingly deployed in distributed settings where different users, sites, or domains maintain vector data. Existing vector databases rely on a coordinator to record which shards store which parts of the vector space and to route each query to those shards. In a decentralized setting, peers may join, leave, or move data without a trusted node tracking every change, and outdated routing information can therefore send queries to the wrong peers or require contacting many peers, reducing vector retrieval recall and increasing network latency. We present Semord, a decentralized vector search overlay system that achieves high recall by routing each ANN query to a small set of relevant peers, without relying on a centralized coordinator. Semord addresses this problem by making semantic locality routable: 1) We propose VHash to place semantically related vectors near each other in the overlay key space while avoiding load imbalance, so that each query only needs to contact a small neighborhood of peers for distributed local ANN ranking. 2) We design VecDHT, a communication protocol that maintains decentralized routing, region metadata, churn resilience, and VHash updates under membership and workload changes. Our extensive experiments on a real testbed show that Semord improves recall by more than 15% and reduces contacted peers by over 60% compared with decentralized baselines. Semord also approaches the recall and latency of a centralized oracle baseline while reducing peak peer-local ANN index memory by more than 2X. Controlled large-scale simulations further show that Semord scales across real-world embedding workloads and remains robust under churn for scoped vector retrieval as a decentralized overlay.

Intermittency is ubiquitous in nature and often arises from interactions between processes evolving on different time scales. We investigate whether, and how, intermittency in a fast system can be transmitted to a slower system that, in isolation, would settle onto a periodic regime. We address this in a unidirectionally coupled (skew-product) setting using two models of increasing complexity: coupled Lorenz-63 systems and coupled Kuramoto-Sivashinsky equations, in which the fast driving subsystem exhibits type-I intermittency. Increasing the driver's burst frequency progressively displaces the response attractor from its unforced limit cycle, until saturation. The induced regimes are jointly controlled by the coupling strength $C$ and the time-scale separation $\tau$, which we explore through the variance of ensemble maxima, power spectra, extreme value statistics and phase-locking indices. In the Lorenz-63 model, $C$ mainly governs the loss of periodicity and the statistics of extremes, while $\tau$ mainly governs the reshaping of the response spectrum; in the Kuramoto-Sivashinsky model these roles are less separable. In both systems, the transition to the intermittency-dominated regime is marked by heavy-tailed (Fréchet-type) extremes, which give way to bounded statistics at stronger coupling. The response delay, estimated via time-lagged mutual information, decreases exponentially as the two time scales become comparable. We identify phase synchronization during the driver's laminar phases as the mechanism behind several regime transitions, with chaotic bursts repeatedly breaking an otherwise stable locking. These results shed light on how fast intermittent variability, such as atmospheric forcing, may shape slower climate components such as the ocean or sea ice.

We introduce a stretched-grid artificial intelligence (AI) weather forecasting model with 2-km resolution over the western United States and part of the Northeast Pacific and approximately 31-km resolution elsewhere globally. Forecasting over the western U.S. is challenging because complex topography and atmospheric rivers (ARs) strongly influence orographic precipitation. West-WRF AI 2-km builds on a global model pretrained with a 40-year European Centre for Medium-Range Weather Forecasts Reanalysis v5 (ERA5) dataset and is fine-tuned with the Center for Western Weather and Water Extremes (CW3E) 2-km regional reanalysis to produce autoregressive 6-hourly forecasts of precipitation and integrated vapor transport (IVT). Forecasts are evaluated over winters 2020-2023 using gridded precipitation observations, rain gauges, and AR Reconnaissance dropsondes and are benchmarked against coarser-resolution AI forecasts and regional and global numerical weather prediction (NWP) systems. West-WRF AI 2-km reproduces observed precipitation-intensity distributions, retains fine-scale spectral variability, and produces sharper narrow coastal precipitation bands and localized, terrain-sensitive extremes. Its broader-scale performance remains comparable to coarser-resolution configurations while preserving large-scale skill despite higher resolution. Dropsonde verification shows lower errors and improved categorical skill at the most extreme IVT threshold. Overall, West-WRF AI 2-km provides its greatest value for localized precipitation extremes and intense AR-related moisture transport.

Vision-based gesture control accepts commands from any hand in the camera field of view, which is unsafe in shared indoor spaces. This paper presents IGate, an identity-gated control stack that includes gesture control and face tracking, in which commands are admitted only when an enrolled operator is verified. The system performs few-shot user enrolment from 20 initial face frames, without prior user-specific training: verification compares an embedding of the current face crop against the enrolled template by cosine similarity, while face tracking uses proportional correction. Gesture control is achieved by classifying extracted hand landmarks using an RBF-SVM trained on a custom dataset. Additionally, a hierarchical finite-state machine handles mode selection, default, and fallback behaviours. The approach is tested on a DJI Tello EDU, each component evaluated offline and in-flight across 270 trials (149 flown). Face verification yields a 0.32% offline equal error rate versus 19.3% in-flight. Under hover-locked conditions, the RBF-SVM gesture classifier outperforms the geometric rule (0.850 vs. 0.651 accuracy), with 82% of this gap stemming from the depth channel. All logs and reproduction scripts will be released.

Large language models (LLMs) can improve solutions to verifiable scientific and algorithmic problems by spending additional computation at test time. Recent systems achieve strong results with increasingly elaborate evolutionary search harnesses or by updating model parameters during test-time training. We ask how much of this machinery is necessary. We introduce Hill Sampling, a form of hill-climbing optimization that repeatedly samples candidate programs from a frozen LLM, retains the best program found so far, and conditions all subsequent samples on that program. We evaluate the method on circle packing, sums and differences of sets, and Erdos' minimum-overlap problem using three open-weight models. Hill Sampling sets a new state of the art on circle packing among published methods, improves over the AlphaEvolve reference on Erdos' minimum-overlap problem, and achieves strong results on sums and differences of finite sets. The circle-packing and Erdos results require only hours of wall-clock time on eight NVIDIA H100 GPUs. We compare Hill Sampling against what is, to our knowledge, the largest application by parameter count of evolution strategies (ES) to LLM weights at test time. Surprisingly, when evaluating a method by the best program it generates, we find that learning model weights via ES is worse than invoking ES with a learning rate set to zero, i.e., using random weight-space perturbations to search for better models. Moreover, repeated sampling outperforms both ES methods, and Hill Sampling is the strongest of all. These results suggest a simple test-time compute allocation strategy: repeatedly sample edits to the best verified solution found so far before introducing additional complexity, such as adding archives, diversity mechanisms, evolutionary scaffolds, or test-time parameter learning.

Consumers can invest resources to understand and avoid their behavioral mistakes, and their incentives to do so depend on the market consequences of remaining naive. We incorporate this feedback between consumers' cognitive states and market outcomes into a general contracting model. Firms face a trade-off between the depth and reach of exploitation: deeper exploitation raises profit from a naive consumer but induces greater cognitive investment, promoting sophistication and shrinking the exploitable consumer base. This trade-off disciplines exploitation and can cause policies that benefit consumers when cognition is fixed to backfire when cognition is endogenous.

Structure-mapping forms analogies by aligning systems of relationally connected elements based on shared structure instead of surface features. We introduce a new structure-mapping algorithm: Structure-Mapping with Tight Bounds (SMTB) that is 5–15x faster than the structure-mapping engine (SME) and about 50% better at finding mappings in large nested domains. SMTB is part of the broader Cognitive Rule Engine (CRE) project, a flexible multi-language-compatible framework with an accessible Python interface to state-of-the-art C++ implementations of core algorithms commonly used in cognitive systems such as pattern matching, planning, and structure-mapping. CRE and SMTB are designed to work with a wide range of representation choices. Unlike SME, which biases higher-order correspondences in tree-like predicate logic, SMTB maximizes relational connectivity without privileging higher-order relations. This allows SMTB to work just as well over arbitrary relational graphs as it does in tree-like domains of nested predicate logic. We discuss situations where privileging "higher-orderness" in structure-mapping can cause issues, and illustrate how SMTB avoids failure modes that SME would encounter in these situations. We also provide an evaluation comparing SMTB to SME v4 over 5845 domain pairs from the SME corpus.

A controlled comparative study of a mesh-free Kolmogorov–Arnold Physics-Informed Neural Network (KAN-PINN) applied to nonlinear strain-limiting partial differential equations is presented. Three distinct training runs across varying material-parameter pairs $(\alpha, \beta)$ are evaluated within a fixed computational pipeline comprising an exact hard boundary ansatz, a robust pseudo-Huber residual loss, and a multi-stage Adam-to-L-BFGS optimizer schedule. A fundamental metric inversion is revealed by the analysis: while training loss reflects operator-specific residual scaling and constitutive stiffness, the operator-independent relative $L_2$ error demonstrates that intermediate parameter regimes achieve superior solution accuracy and enhanced generalization.

Robot manipulation policies must generalize across visual shifts while preserving scene context relevant to action. General-purpose vision encoders are not tailored to visuomotor control, while object-centric approaches often use segmentation masks as hard filters that discard potentially useful context. We propose RoboMP-DINOv2 (Robotics Mask-Prompted DINOv2), a full-scene vision encoder that treats masks as spatial prompts rather than visibility filters. It extracts dense DINOv2 features from the full observation, injects learned region-specific embeddings at masked locations, and jointly contextualizes prompted and unprompted tokens for action prediction. We further introduce masked-region color randomization (MCR) to improve appearance robustness, yielding RoboMP-DINOv2-MCR. Across seven simulated manipulation settings, RoboMP-DINOv2 achieves 60.7% success under spatial shifts and 59.7% under scene clutter, compared with 50.7% and 41.0% for a DINOv2-based Diffusion Policy. Under unseen object colors, RoboMP-DINOv2-MCR achieves 72.5% success versus 35.1% for the strongest color-randomized baseline. Additional experiments and representation analyses show improved robustness while preserving behaviorally relevant scene information. Code is available at https://github.com/han20192019/RoboMP_DINOv2.

Rapid intensification (RI) remains one of the most consequential and difficult aspects of tropical cyclone (TC) forecasting. Although full-physics numerical weather prediction models can represent the processes governing RI, resolving storm-environment interactions remains computationally expensive, while purely data-driven approaches often lack physical interpretability. We present FAST-ML, a hybrid framework that bridges data-driven efficiency with physical constraints. A physically informed dual-stream neural parameterization ingests 3D ERA5 fields to diagnose ventilation controls—environmental wind shear and mid-level entropy deficit. By optimizing these parameters end-to-end through a differentiable FAST intensity model, this architecture establishes a robust new paradigm for observation-driven parameter optimization, ensuring storm evolution remains strictly governed by thermodynamic principles. By better capturing the storm's continuous intensity evolution, FAST-ML improves upon its physical baseline, reducing ensemble CRPS across forecast lead times, with a reduction of approximately 31% at 60 h and nearly halving the RI false alarm ratio without sacrificing detection skill. In a 100-member ensemble configuration, FAST-ML produces intensity forecasts comparable to FNV3 for selected storms under the evaluated input configurations. Furthermore, zero-shot tests on selected Eastern Pacific storms provide encouraging evidence of cross-basin transferability. FAST-ML provides a modular intensity forecasting framework that can be coupled with externally supplied storm tracks and environmental fields. It demonstrates that observation-driven parameter learning within physically constrained dynamics simultaneously enhances accuracy, interpretability, and computational efficiency.

How can the conceptual commitments of a literary work connect its creation, interpretation, and evaluation? We propose Literary Axioms, a conceptual framework in which a work develops a configuration of claims about experience and commitments to literary form. A literary axiom is a provisional organizing premise whose scope and textual realization can be examined. The framework develops the proposal by Qiang Liu that writers select and combine such premises, realize them through characters, events, language, and form, and make them available for reconstruction by readers. We distinguish an open axiom space, a selected configuration, its relational organization, and situated reader reconstructions. This yields an account of coherence as organized compatibility or conflict, originality as a change in claims, relations, or realization, and interpretive richness as a plurality of textually grounded reconstructions. Five dimensions connect this account to literary evaluation: coherence, contextual reach, distinctiveness, realization fidelity, and interpretive richness. We specify boundary cases and derive five empirical predictions with observations that would count against them. A concrete implementation contains 1455 Chinese language records, 1464 mappings to 149 works, and 472 typed relationships. A descriptive audit establishes the structural integrity of the artifact and identifies gaps in reuse, context coding, and provenance. The paper contributes a theory based framework, an inspectable implementation, and a research program. Its predictions await reader and creative writing studies.

The urban heat island (UHI) effect amplifies heat stress in cities, increasing heat-related illness, mortality, sleep disruption, and energy demand. Because UHI intensity varies at neighborhood scales, intra-city temperature mapping is essential for assessing public health impacts, yet existing monitoring networks are sparse. Crowdsourced weather stations provide dense, hourly observations but suffer from noise and systematic biases, including preferential sampling driven by where residents install sensors. We introduce two scalable Gaussian process frameworks which adjust for preferential sampling and enable statewide, high-resolution temperature prediction. The first integrates sampling correction directly within a sparse variational Gaussian process (SVGPR+PS). The second uses a two-stage estimator involving a log-Gaussian Cox process (LGCP+GP) that estimates the sampling intensity field and incorporates it into temperature prediction. Both methods rely on inducing-point approximations and variational inference to scale beyond traditional Gaussian processes. Applied to approximately 10,000 citizen stations across North Carolina, the models generate hourly temperature fields within minutes. Validation against independent urban heat campaign data shows that the preferential sampling correction improves predictive performance, increasing average afternoon correlations across four cities from 0.26 to 0.43. This scalable framework enables neighborhood-level UHI assessment and supports studies of heat exposure disparities statewide.

Exchange alone gives the electron gas a local field factor with a sharp peak just below 2kF. Quantum Monte Carlo finds none. Both spin channels are computed here from their lowest diagrams with a dynamically screened interaction and no fitting. Parallel spin correlation is the negative image of exchange and removes about half the peak; the calculated spin-resolved factors account for the QMC data at rs = 1 and 2, and at q = 0 for the compressibility and spin susceptibility obtained from the ground-state energy. The two classes of diagrams are the two terms of the kernel of Hedin's vertex equation, of which the second is routinely dropped.

Aerial tracking of autonomous ground vehicles can provide useful situational awareness in environments where fixed external perception infrastructure is unavailable. However, visual-only tracking from an unmanned aerial vehicle (UAV) is limited by camera field of view, viewpoint changes, occlusions, and temporary loss of target visibility. This paper presents a heterogeneous UAV–UGV methodology for aerial tracking of small-scale autonomous ground vehicles in simulation. The system is implemented in ROS~2 and Gazebo Sim using a Tello-like UAV equipped with a front-facing RGB camera and multiple LIMO-like unmanned ground vehicles (UGVs). Three approaches are evaluated: a training-free color-marker baseline, a YOLO-based visual baseline, and a proposed UAV–UGV fusion method. The proposed method uses the same visual detector as the YOLO baseline, but combines visual detections with UGV state information to maintain the target estimate during visual gaps. The methods are evaluated across four scenarios: single-car tracking, crossing trajectories, occlusion, and UAV-assisted awareness. Results over 600 simulation runs show that the proposed fusion method achieves nearly continuous track coverage across all scenarios while reducing localization error relative to the YOLO baseline. Compared with the visual-only YOLO approach, the fusion method reduces RMSE by 37.3%, 64.2%, 62.6%, and 61.8% in the four evaluated scenarios, respectively. The method also enables conflict-awareness lead times of 10.20~s in the crossing scenario and 9.07~s in the UAV-assisted awareness scenario. These results indicate that heterogeneous UAV–UGV fusion can improve aerial tracking continuity and robustness when visual observations are intermittent.

Integrating Vision-Language Models (VLMs) into autonomous UAV navigation enables high-level semantic reasoning but introduces critical temporal bottlenecks that can destabilize flight control. This paper presents a systematic benchmark that evaluates the operational trade-offs among model scale, hardware constraints, and flight reliability. By subjecting the system to extensive stress tests on both a high-performance workstation and a constrained integrated-graphics laptop, we isolate the kinetic impact of inference latency against reasoning accuracy. Our findings reveal a performance dichotomy: lightweight models struggle with structural hallucinations that break deterministic state machines, while heavyweight models induce trajectory overcorrections due to inference lag. We demonstrate that a mid-weight architecture strikes the optimal balance, successfully rehabilitating navigation in degraded environments without compromising real-time stability across different hardware profiles.

The effective radii of young star clusters correlate with cluster mass, but the large scatter about that relation is not well understood. Theory predicts that a cluster's initial radius depends on the gas surface density of its natal environment, so the star formation environment should leave a measurable imprint on cluster structure. In a companion analysis of 5,105 clusters across 31 LEGUS galaxies, we found that galaxy-wide specific star formation rate (sSFR) carries no predictive power for cluster radius. Here we test whether that null result reflects the spatial scale on which the environment was measured. We cross-match the LEGUS cluster catalog of Brown & Gnedin with H-alpha maps from the PHANGS-MUSE survey in the four galaxies common to both surveys, and measure the local star formation rate surface density, Sigma_SFR, at the position of every cluster by aperture photometry. For the 325 clusters younger than 10 Myr with reliable radius and mass measurements, local Sigma_SFR significantly predicts effective radius (beta = 0.079, p = 2 x 10^-4) in a model containing mass and age, while the host galaxy's sSFR does not (beta = 0.042, p = 0.40) for those same clusters. The local model generalizes nearly three times better under cross-validation (CV R^2 = 0.061 versus 0.022). The effect operates among clusters within a single galaxy: it survives galaxy fixed effects (beta = 0.068, p = 0.001) and appears independently in each of the two galaxies large enough to fit alone. Including local Sigma_SFR also reduces the mass coefficient by 26%. The birth environment measurably shapes cluster structure, but only when it is measured on the scale on which clusters actually form.

Physical experiments frequently result in inverse problems, where the recorded response is used to deduce information about the sample, typically yielding only partial recovery. This work presents a reverse Monte Carlo modeling method for solving such problems involving statistical averaging, e.g. at the nanoscale. The proposed algorithm operates in an input subspace of the forward problem, with dimensions reduced from those of the original input space. The dimensionality reduction focuses on features that influence the response, as irrelevant information is lost in the process. We first study the performance of the method in tailored test cases that also reveal information loss when several input characteristics produce similar responses. Finally, we apply the method to a previously published simulated dataset of X-ray absorption, X-ray emission, and X-ray photoelectron spectra of the water molecule.

Let $G=(G_1,...,G_n)$ be a graph system on a common $n$-vertex set $V$. For $v\in V$, set \[ \delta(v)=\min_{i\in[n]}d_{G_i}(v), \qquad E_{\cap}=\bigcap_{i=1}^n E(G_i), \] and let $d^c(u,v)$ denote the length of a shortest rainbow $u$–$v$ path. We study Hamiltonicity conditions imposed only on pairs at small rainbow distance. Suppose that every $G_i$ is $2$-connected and that \[ \delta(u),\delta(v)<\frac n2, \qquad d^c(u,v)\le2 \quad\Longrightarrow\quad uv\in E_{\cap}. \] Our main result shows that any system satisfying this local Fan-type condition but containing no rainbow Hamilton cycle must satisfy \[ E_{\cap}=\varnothing, \qquad \sum_{v\in V}\delta(v)\le\frac{n^2-1}{2}. \] Consequently, either $E_{\cap}\ne\varnothing$ or $\sum_{v\in V}\delta(v)\ge\lceil n^2/2\rceil$ guarantees a rainbow Hamilton cycle. We further prove that the local Ore-type condition \[ \delta(u)+\delta(v)<n, \qquad d^c(u,v)\le2 \quad\Longrightarrow\quad uv\in E_{\cap} \] already forces a rainbow Hamilton cycle. Along the way, the Fan-type hypothesis yields a rainbow Hamilton path and a rainbow $(n-1)$-cycle. For diagonal systems, our conditions reduce to the classical Fan and local Ore conditions. The proofs combine rainbow rotation–extension with a double-counting argument in an auxiliary digraph. Keywords: graph system; rainbow Hamilton cycle; rainbow distance; Fan-type condition; Ore-type condition; rotation–extension.

Let $\mathbb{R}\langle X,Y\rangle$ be the free algebra on two symmetric letters. A polynomial is trace-positive if it has non-negative trace on every pair of real symmetric matrices, and it lies in the cone $\Theta$ if it is cyclically equivalent to a sum of Hermitian squares. Burgdorf and Klep proved that the two notions coincide in degree at most four; from degree six onwards they differ. We give a complete, explicit description of both cones on the six-dimensional space $Z$ of symmetric homogeneous sextics in two symmetric letters whose abelianization vanishes identically, and show that on $Z$ they coincide: for $f\in Z$, trace-positivity is equivalent to membership in $\Theta_3$, to non-negativity of the trace on pairs of $2\times2$ symmetric matrices, and to being cyclically equivalent to a sum of four Hermitian squares of cubics with vanishing abelianization. Membership in $\Theta$ reduces to a $4\times4$ positive semidefinite Gram matrix whose diagonal blocks are fixed by the content-$(4,2)$ and $(2,4)$ parts of $f$; each of those parts ranges over a two-dimensional cone with two linear facets, and the content-$(3,3)$ part is then confined to a rectangle whose sides are the geometric means $\sqrt{p_+q_+}$ and $\sqrt{p_-q_-}$ of the eigenvalues of those blocks. Each facet is certified by one of two families of pairs of $2\times2$ symmetric matrices. The abelianization hypothesis is natural: every bivariate degree-six example we are aware of – a trace-positive polynomial outside $\Theta$ – arises by lifting a non-negative, non-SOS polynomial in two variables to a cyclically sorted polynomial; and that lifting cannot operate when the abelianization vanishes. Symmetry of $f$ is essential: dropping it adds one antisymmetric direction, on which the trace vanishes identically, yet whose non-zero elements all lie outside $\Theta$.

Burde proved that the Lie algebra $\mathfrak{sl}_2(\mathbb{K})$, over a field $\mathbb{K}$ of characteristic $p > 2$, admits compatible left-symmetric algebra structures if and only if $p = 3$. Since the even part of the Lie superalgebra $\mathfrak{osp}(1|2)$ is isomorphic to $\mathfrak{sl}_2(\mathbb{K})$, we ask whether Burde's structures can be extended to compatible left-symmetric superalgebra structures on $\mathfrak{osp}(1|2)$. We verify Burde's left-symmetric structures on $\mathfrak{sl}_2(\mathbb{K})$, and then test their possible extensions to $\mathfrak{osp}(1|2)$. We show that no such extension exists in characteristic 3. Together with the obstruction coming from the even part for characteristic $0$ and $p > 3$, this gives non-existence for a field $\mathbb{K}$ of characteristic different from $2$.

This paper is an English translation of a contribution to the latest German edition (2017) of Hermann Weyl's classic text "Symmetry", edited and annotated by Erhard Scholz, Klaus Volkert, and myself. In this contribution I discuss various physical aspects of the notion of symmetry and place them in context from the perspective of theoretical physics. In particular, a distinction is drawn between symmetries of individual configurations (solutions of equilibrium equations or equations of motion) and symmetries of the laws themselves. Likewise, a distinction is made between those operations that alter physical states and so-called gauge symmetries, which leave physical states unchanged and merely represent a redundancy in the mathematical description. Besides gauge symmetries, I comment on the connection to conservation laws, symmetry breaking, and the philosophy of space.

We investigate polarized synchrotron emission from an optically thin accretion flow around a rotating braneworld black hole, focusing on how higher-dimensional gravity modifies photon propagation and polarization transport. We integrate the null-geodesic and parallel-transport equations for the polarization four-vector and, within a common emissivity and magnetic-field prescription, examine the effects of tidal charge, black-hole spin, observer inclination, and prograde or retrograde orbital motion. We analyze the linearly polarized intensity, electric-vector position angle (EVPA), Stokes parameters, the joint $Q$–$U$ polarization-state space, and the cumulative net linear polarization degree, and compare representative braneworld configurations with Kerr and Kerr–Newman spacetimes. Within the parameter range considered, tidal charge and spin do not generate a qualitatively new large-scale polarization morphology. Instead, they primarily reorganize the polarized intensity, orientation, and polarization-state distribution in the strong-field region near the shadow boundary and photon ring. Viewing inclination controls the dominant projected morphology, while high inclinations enhance projection and Doppler asymmetries and make the distinction between prograde and retrograde flows more pronounced. The Stokes components and their joint state space expose local bending, stretching, and trajectory concentration that are less apparent in intensity images. The cumulative net linear polarization degree decreases with increasing cumulative radius, and larger tidal charge or spin lowers its amplitude for the representative models considered. Together, these complementary polarization observables provide a potential route for separating spacetime, viewing-geometry, and accretion-flow effects and for constraining braneworld tidal charge with horizon-scale polarimetry.

A piezoelectric transducer on a pitching and plunging aerofoil with a trailing edge flap turns the limit cycle that follows flutter into electrical power, and two modelling changes to that harvester have been studied separately. Replacing incompressible strip theory by the compressible Euler equations reduces the stabilising effect of a transducer mounted on the pitch, where the electrical time constant meets the flutter frequency, and raises the harvested power, and letting the section react against a fuselage free to translate, instead of rigid ground, does the same two things by about the same amounts. Here both changes are put in one model, which needs no new mesh motion in the flow solver, since the support's translation displaces the wall exactly as the plunge does. On the harvested power the two published corrections simply multiply. On the flutter boundary they more than compound, and at strong enough coupling they reverse the sign of the transducer's effect at the load resistance where both companion papers harvest, although neither change alone reverses it. The free support widens the band of resistance over which the two aerodynamic models disagree in sign, carrying its upper edge out by about a factor of three. Above the resistance where the electrical pole meets the flutter frequency, where the transducer acts as an added stiffness, the product of the published corrections predicts the shift to within seven parts in a thousand. Below it, where the transducer acts as a damper whose effect depends on phase, the product is wrong by a factor of up to fourteen, and on the pitch mounting by a sign. The same transducer lowers the boundary while flattening the growth of the instability above it. Whether corrections found one change at a time may be multiplied therefore depends on the quantity and, for the flutter boundary, on the regime of the transducer.

Helmke and Shayman showed that, for generic targets, least-squares distance to the manifold of rank-r matrices is a nondegenerate Morse function whose indices are governed by Bruhat-cell codimensions. We develop the analogous theory for Monarch matrices, products of a fixed permutation matrix with two block-diagonal factors. The parametrization's gauge dimension, as well as the codimensions and closure relations of the associated Bruhat-type cells, is determined by a common transfer-matrix invariant, although these quantities are generally distinct. In the equal-block case, we prove the Monarch-Schur theorem, characterizing the permutations that minimize gauge freedom and maximize manifold dimension, and derive a general lower bound for arbitrary partitions. We further establish generic second-order nondegeneracy, a Morse-index formula, and an explicit focal set of measure zero. Finally, we show that the unregularized gradient flow lacks coercivity because of unbounded flat directions. A quartic regularizer restores global convergence, selects a canonical minimum-norm exact solution when one exists, and otherwise yields a well-defined best-approximation image. The regularizer preserves the Monarch architecture and provides a geometric interpretation of weight decay in structured factorized layers.

We study nonlinear mean-curvature spectral equations for hypersurfaces in Euclidean, spherical and hyperbolic space. For an admissible analytic response $F$ and ambient sectional curvature $c$, we prove minimality when the spectral parameter satisfies $\sigma\ge nc$, and strict mean-curvature bounds at gradient points when $\sigma<nc$. The class includes every power $F(t)=t^{p-1}$, $p>1$, and positive mixtures satisfying an explicit width condition. In particular, the result proves Chen's conjecture in Euclidean codimension one and biharmonic minimality in constant negative curvature, without completeness or holonomicity. A common real-analytic continuation and focal-trace argument treats all three geometries. Explicit umbilical models show that the unconditional spectral minimality range is sharp. Independently, we prove quantitative local rigidity, modulo ambient isometries, of compact nonminimal geodesic spheres in all three geometries and proper equal-radius Clifford hypersurfaces in spheres, for smooth laws with $F,F'>0$ near the model amplitude.