Papers

We study online covering in which each instantaneous service pays its purchase cost and one maximum waiting time, with no effect on future requests. For static realizable services, monotone subadditivity suffices for optimal competitive ratios; submodularity is unnecessary. A normalized monotone subadditive lower-bound oracle with realization factor $\rho$ yields ratios $\rho+1$ deterministically and $1/(1-e^{-1/\rho})$ randomly against an oblivious adversary. Exact batch optimization gives the optimal constants $2$ and $e/(e-1)$. The randomized algorithm uses one global threshold on a seed-independent virtual-height trajectory, whose active time is a lower bound on the offline optimum. Weighted vertex cover gives a strict separation from submodularity on a three-edge bipartite path, with polynomial-time batch implementations through min-cut and LP rounding. An offline consecutive-batch normal form also transfers static approximation guarantees to the offline problem.

On-policy reasoning models can learn from task rewards or teacher signals, but these sources differ in form and can favor conflicting updates, leaving unclear which should guide a given reasoning action. We introduce Expected Reasoning-Step Return (ERSR), which treats semantic reasoning steps as macro-actions and uses Monte Carlo student-policy rollouts to estimate the expected final task reward of student-generated and teacher-proposed actions in a common return space for step-level comparison. ERSR analysis reveals an outcome-dependent asymmetry: student actions are more beneficial than teacher replacements on successful trajectories, whereas teacher replacements become more beneficial on failed trajectories. We further show that student answer-probe gains track student-step ERSR utility and distinguish beneficial from harmful reasoning steps. Based on these findings, we propose \textbf{Return-Referenced On-Policy Learning (R$^2$OPL)}, which reinforces student reasoning on successful trajectories and distills teacher signals on failed ones, while using group success rate for difficulty scaling and student-probe gains for step-level modulation. Experiments across reasoning benchmarks and teacher–student configurations show that R$^2$OPL consistently outperforms strong baselines. ERSR training dynamics further show that R$^2$OPL jointly exploits substantial utility from both reward- and teacher-side signals, whereas existing hybrids often leave substantial residual utility in one branch.

In this article, we describe a method to obtain lower bounds for the $2k$-th moment of values of $L$-functions localised at the central point for all real $k > 0$. In particular, for the families of quadratic twists of self-dual irreducible unitary cuspidal automorphic representations of $\operatorname{GL}_D(\mathbb{A}_{\mathbb{Q}})$ with $D\in\{2,3,4\}$, our results are new in the range $0<k<\tfrac12$ when $D\in\{2,3\}$, and new for all $k>0$ when $D=4$.

Predicting whether a trained model will generalize under distribution shift remains difficult, especially when target-domain data are unavailable. We introduce STAMP (Semantic Temporal Augmented Model Prediction), a source-only, target-label-free criterion that estimates out-of-distribution (OOD) performance from paired source-domain images. STAMP computes the output-space correlation ratio $\eta^2=S_B/S_T$ by contrasting semantically stable pairs with random pairs: higher $\eta^2$ indicates that model outputs vary with semantic identity rather than nuisance variation. On 44 chest X-ray models spanning CNNs, ViTs, MetaFormers, foundation models, and SSL/VLM probes, temporal STAMP attains Spearman correlations of $0.844$–$0.855$ with macro AUROC on VinDr-CXR, CheXpert, and MIMIC-CXR; a class-matched variant improves single-class RSNA from $0.311$ to $0.663$. STAMP attains the best average source-only medical ranking and outperforms the target-domain ATC and AoTL estimators without any target data. On 27 ImageNet models, temperature-scaled STAMPTS attains $\rho{=}0.984$ on ObjectNet and $\rho\geq0.905$ on four additional distribution shifts, with partial correlations of $0.662$–$0.949$ after controlling for ImageNet accuracy. Requiring approximately 12s per model on one GPU, STAMP is a practical pre-deployment model-selection and auditing tool.

Cross-modal retrieval has been advanced by vision-language pre-trained models that encode images and texts into a shared dense embedding space. While dense representations effectively capture overall semantic similarity, they often obscure fine-grained visual-textual information needed for precise cross-modal matching. Recent methods introduce a learned sparse branch to complement dense matching with lexical evidence, but they rely on a redundant language-model token space and lack explicit grounding for sparse dimensions. We propose GRASP, a compact and grounded sparse learning framework that mines visual-textual concepts from the corpus. A lightweight sparse head is trained to predict concepts relevant to each image or text, yielding interpretable concept-level evidence that complements dense semantic matching. Extensive experiments show that GRASP improves retrieval accuracy over the state-of-the-art dense-sparse baselines while yielding a more compact and grounded sparse space.

When a language model explains an answer it has already given, does it reuse the computation that produced the answer or reconstruct a story from the answer alone? Attribution, transportability and recoverability are each compatible with causal use without establishing it. We propose an evidence standard: pair each positive statistic with a variable specific null that removes the tested variable's identity while matching relevant nuisance dimensions as far as possible, and audit unmatched dimensions. We apply this standard to a known cause. A cue naming a wrong option raises the rate of choosing that option by 64 to 68 percentage points across three models. Explanations mention the cue in 1.8 percent of items or fewer in three of four models tested. Three estimator classes yield favorable statistics, but none establishes causal sensitivity to the cue contrast under its own control in the three-model analysis. In the strongest case, a recovered cue direction reaches $R^2$ of 0.95 and exceeds a geometry matched random direction in all three seeds, while a direction fitted by the same pipeline with cue labels scrambled reproduces 61 to 76 percent of its effect at comparable realized edit magnitude. A fourth model passes one interchange endpoint, but unequal edit magnitudes and a contrast that changes both cue identity and cue-answer agreement limit its interpretation. These experiments leave causal access unresolved. They establish an evidentiary requirement: favorable mechanistic statistics must survive controls for variable identity and nuisance structure. Reusable controls separate generic from identity specific transport effects, fit null directions with scrambled labels, and audit realized intervention magnitudes.

Attention weights depend on token counts, which change with the representation of a signal. We study refinement symmetry: splitting a representation while preserving content, position, visible context, and total mass should preserve its contribution. Building on proportional and quadrature attention, we show that split invariance forces the local mass factor to be linear for any fixed positive attention kernel, provided that factor is nondecreasing. For changed representations, a physical coupling bounds attention error by separating feature change from weight reallocation. In Qwen2.5-Omni-7B, duplicating half the visual tokens threefold changes 255 of 3,586 MVBench answers under standard attention; measure weighting preserves every answer under matched visibility. Under natural frame resampling, it reduces distributional drift. At twofold merging of a frozen video encoding, a five-seed evaluation shows an all-partition-correct accuracy gain of 1.04 percentage points over global count weighting (average group mass) and 0.93 points over standard attention. The advantage over global count also holds on WorldSense but depends on the compression budget. The result is a representation principle with a measured benefit in robustness across partitions.

In this paper, we investigate the geometry of the Gieseker moduli space of semistable rank $2$ torsion-free sheaves on $\mathbb{P}^3$ with Chern classes $(c_1, c_2, c_3) = (-1, c_2, c_2^2-2)$ for $c_2 \geq 2$. We use the modular Serre correspondence to relate these moduli spaces to spaces of pairs and to suitable Hilbert schemes of one-dimensional subschemes in $\mathbb{P}^3$. For $c_2 \geq 4$, we prove that the moduli space is irreducible of dimension $c_2^2+3c_2+5$. For the case $c_2 = 3$, relying on a known geometric description of a relevant Hilbert scheme, we show that the moduli space $\mathcal{M}(-1,3,7)$ consists of exactly two irreducible components, namely the generic component of reflexive sheaves and a $T$-component, and we prove that their intersection is nonempty.

On-policy self-distillation (OPSD) trains a question-only student with token-level feedback from a teacher given training-only privileged information (PI). OPSD therefore provides dense, on-policy supervision, and is free of a larger external teacher, but its effectiveness rests on how PI is designed and utilized. Our preliminary diagnostics suggest a significant gap between teacher utility and student learnability, where a small fraction of high-disagreement tokens dominate the distillation signal, and short teacher continuations at these positions further expose more explicit PI leakage than transferable correction cues, indicating a strong intent on injecting PI-conditioned shortcuts. We propose Adaptive On-Policy Self-Distillation (AOPSD), which adapts what information the teacher receives and how strongly its feedback influences learning. AOPSD encodes each solution as a reasoning DAG, orders problems by the student's evolving capability, and reveals only the affordable subgraph and its next frontier as PI. For high-disagreement tokens, AOPSD utilizes short teacher continuations as probes to encourage useful guidance while mitigating PI-conditioned shortcuts among teacher supervisions. On HMMT25, AIME24, AIME25, and BRUMo25, AOPSD achieves 72.5% Pass@8, which is 6.7 percentage points above OPSD and 4.2 above the strongest competing baseline while reducing 15 percentage points of training time at lower cost.

We establish a dimension-free vector-valued Poincaré-type inequality on the discrete cube for Banach spaces with Rademacher type $p$, $1\le p\le2$. As a consequence, we obtain $$ T_p^R(X)\le T_p^E(X)\le p\,T_p^R(X). $$ This gives an alternative proof that Rademacher type and Enflo type coincide, while improving the constant $\pi/\sqrt{2}$ in the quantitative estimate of Ivanisvili, van Handel, and Volberg \cite{IVV} to $p$.

ELBO-based reinforcement learning offers a sampler-agnostic approach to fine-tuning flow matching models with reward feedback. Timestep weighting in ELBO-based RL has large impact on performance, and it also provides a unified view (as we show in this work) to understand prediction losses heuristically chosen in prior work, yet it remains under-researched and is often chosen to inherit pretrain configs. We investigate impacts and dynamics of timestep weighting in ELBO-based RL. We show that effective weighting depends on both the reward landscape and stage of learning. (1) Through experiments on controlled CIFAR image generation, complemented by robotics, we investigate how weighting impacts reward-driven updates across noise levels. (2) Through gradient analysis, we reveal distinct patterns of cross-noise coordination across tasks and their evolution during training. These findings motivate the hypothesis that useful weighting depends on the gap between the policy's current behavior and the behavior favored by the reward. (3) Guided by this analysis, we study simple static weighting, budgeted profile selection, and dynamic schedules that improve performance beyond conventional target choices. Our results establish timestep weighting as an important design choice for flow-matching RL and motivate further research into methods that choose and adapt it throughout learning.

We consider a reversible diffusion with unknown drift, observed through trajectories, and construct samplers of its invariant law. The trajectories contain samples of this law, but a sampler built from them by density or score estimation inherits the errors of these estimates, in particular on the barriers between metastable states. The trajectories also contain the dynamics, and for a reversible diffusion the ratio between the law of the process and its invariant law solves the backward Kolmogorov equation, so that the eigenpairs of the generator propagate it in closed form. Backward Kolmogorov transport (BKT) evaluates this ratio with eigenpairs estimated from the trajectories and coefficients computed once from the initial particles, and moves the particles along the Wasserstein gradient flow of the Kullback-Leibler divergence. The particles do not interact, BKT defines a transport map, and no density, score or drift is estimated. With exact eigenpairs BKT transports the spectral projection of the initial law exactly when this projection is positive. For estimated eigenpairs we prove a transport identity in which truncation enters only through the initial law and estimation only through the residual of the eigenpairs, and a local stability bound in the 2-Wasserstein distance. With coefficients computed from the transported particles, the sampling error of the retained modes cancels, and a central limit theorem identifies the asymptotic variance. Experiments on Ornstein-Uhlenbeck processes, multi-well potentials, separable products and alanine dipeptide, including comparisons with particle methods based on density estimates, agree with these results, and with an accurate spectrum the errors of BKT reach or fall below those of independent samples.

The memory usage and decoding latency of LLM inference grow rapidly with context length. To reduce these costs, key-value (KV) cache compression methods selectively retain cached states based on token importance or differences in attention patterns across heads. However, we discover that retrieval capability varies substantially with relative distance, even within the same attention head. To exploit this structure, we introduce Distance-KV, which learns a static KV retention pattern over the joint space of layers, attention heads, and relative distances. The pattern is learned offline with the language model frozen and reused across inputs to prune and compact the KV cache without online importance scoring. Across three backbone models and four long-context benchmarks, Distance-KV consistently achieves the best overall performance among competing KV cache compression methods, exceeding the strongest compression baseline by up to 9.3 points on RULER at 128K. On Llama-3.1-8B-Instruct at 128K, Distance-KV reduces KV cache memory by 65.4% and achieves a $1.66\times$ decoding speedup relative to Dense. Together, these results identify relative distance as an important structural dimension for understanding how LLMs retrieve information over long contexts and for designing more efficient inference methods.

Multi-agent coding has emerged as an increasingly active direction in software engineering, where complex development tasks are decomposed across multiple specialized agents working on different parts of the problem. Despite the shift from individual problem solving to distributed collaboration, multi-agent systems still lack a direct measure of collaboration and are largely evaluated through task-level outcomes inherited from single-agent coding, conflating individual coding capability with cross-agent coordination. We introduce AsynCodeBench, a dependency-centric benchmark for asynchronous multi-agent software engineering that represents each task with an explicit dependency graph and executable Dependency Checkers. Through this dependency-tracking process, we propose two complementary measures: Asynchronous Dependency Pass Rate (ADPR), which measures how many cross-agent dependencies are ultimately satisfied, and Dependency Resolution Step (DRS), which measures when each dependency first becomes satisfied during execution. AsynCodeBench comprises 19 tasks from real-world repositories, exposing 52 directed dependencies as explicit units for evaluating cross-agent collaboration. Experiments across model families, scales, and generations reveal a clear gap between coding and collaboration capability: improvements in coding performance do not necessarily translate into stronger collaboration, and task-level metrics can diverge substantially from dependency-level collaboration measures. Dependency-trajectory analysis further reveals that successful coordination often emerges not gradually, but through concentrated bursts in which many dependencies become resolved over a short portion of the execution trajectory, a pattern we term a hopping window.

Maximum common edge subgraph (MCES) matching finds a partial vertex correspondence between two labeled graphs that preserves as many labeled edges as possible. Molecular similarity search requires matching many graph pairs, making the cost of repeated queries important. The strongest baseline attains accurate MCES solutions but trains a separate network for each pair. We introduce Equivariant Neural Primal-Dual Assignment (ENPDA), which learns a shared matching policy and applies it to new pairs without further training, answering queries roughly three orders of magnitude faster and recovering its training cost after a few dozen queries. The policy recomputes exact objective marginals for candidate matches and learns corrections and step sizes that update their scores. Target prices respond to competition when several source vertices favor the same target. Four update rounds and a Hungarian projection produce a partial one-to-one matching. We prove per-pair guarantees that hold for any network parameters. In exact arithmetic, reordering either graph permutes the assignment and price states, the projected matching is one-to-one, and repaired prices give a valid MCES upper bound. Subtracting the preserved-edge count bounds the optimality gap; combined with structural caps, these certificates prove global optimality for 60 of 291 native test pairs. On three molecular benchmarks with disjoint train/validation/test splits, ENPDA improves over an analytic counterpart with the same update and projection budget by 7.4-8.6 accuracy points; after one second of refinement search, 2.5-3 points of the gain remain. Transferred without fine-tuning to edge-deletion tasks from social and protein graphs, the policy gains 9.1-17.6 points over the analytic counterpart. When output matchings must keep aromatic rings intact, ENPDA recovers more reference bonds than the baselines on all three datasets.

Table images preserve structural information that are often lost in text serialization, and reasoning over them requires locating relevant rows, columns, and cells step by step. Current multimodal large language models (MLLMs) encode the whole image once before reasoning, so they cannot pick up row-, column-, and cell-level evidence as the question unfolds. Encoder-side table structure and generic interleaved visual chain-of-thought still do not bind each reasoning step to that structure. We propose InterTab, an Interleaved structure-aware framework for CoT reasoning over Table images, interleaves chain-of-thought with tool calls that crop structure-aligned table regions. First, we build InterTab-22K, includes reasoning trajectories in which each step is tied to both a structural location and a bounding box. InterTab is trained in two stages: supervised structure-aware alignment (SSA) on InterTab-22K teaches the model to interleave reasoning with structure-aligned crops, and active localization optimization (ALO) further optimizes answer correctness, localization IoU, and output format, while penalizing missing or excessive tool calls. Experiments on nine table benchmarks show that InterTab improves the average accuracy of its backbone from 68.28% to 73.17% and achieves the best average performance among all compared methods. Code and data will be released soon.

Quantization-Aware Pre-Training (QAPT) can increase the inference efficiency of DNNs, but a problematic behaviour known as rounding boundary weight oscillation can introduce detrimental noise into the training process and significantly reduce convergence speed. While existing methods can reduce this detrimental noise, they either introduce additional hyperparameters or memory overhead, or cannot consistently improve model accuracy. In this work, we propose optimization with $\textbf{C}$onstrained $\textbf{E}$mpirical $\textbf{W}$eight dis$\textbf{T}$ribution (CEWT), the first hyperparameter-free memory-overhead-free oscillation suppression method that consistently improves QAPT performance: an optimizer post-update step that projects weights to the nearest point in weight space whose empirical distribution (histogram) matches a zero-mean Gaussian. Our key insight is many quantizers are designed with the implicit assumption that the to-be-quantized data are permutations of samples from a zero-mean Gaussian, and this assumption is not true during QAPT. By enforcing the zero-mean Gaussian prior as a hard constraint, CEWT can suppress this detrimental noise. Empirical results on various combinations of SOTA quantizers and hypersphere optimizers suggest, that with a geomean increase of 4% in training time, CEWT can consistently reduce the pre-training perplexity (by an average of 2.5 and up to 21 points) of low-precision (down to 1-bit activations and weights and up to 610M parameters) LLaMA/GPT models without introducing any hyperparameters or storage overhead. Code is available at https://github.com/1733116199/cewt

External memory lets language-model agents answer questions about histories too long for the answer model's context window. Updates create a harder problem than retrieving a recent fact: changing one relation can redirect a multi-hop question to records about an entity absent from the question. We study this update-dependent evidence selection problem and introduce the Contract Memory Compiler (CMC). Before seeing a question, CMC uses a language model to identify relations in the history and record where each one was stated. It applies later updates to determine the current relations, follows them from entities named in the question, and passes the corresponding original records to the answer model in one call. Thus the current state determines which evidence is read, rather than merely refreshing values in a previously selected context. To the best of our knowledge, CMC achieves state-of-the-art multi-hop accuracy on FactConsolidation, reaching 78.25% overall and 61.0% at 262K. With the extracted relations and answer model held fixed, selecting evidence before resolving updates reduces multi-hop accuracy to 21.50%. We also introduce MQuAKE-MemStream, a derived dataset of ordered memory streams built from MQuAKE-Remastered counterfactual cases.

Accurate one-step predictions do not ensure that a world model's rollouts retain the data distribution. We make the model's decoded stationary law explicit by learning a decoder of a fixed Gaussian reference and constraining the behaviour-averaged transition to preserve that reference. For controlled systems, a joint transition uses a conditional action chart to preserve behaviour occupancy without requiring invariance at each fixed action. Joint state–action rotations and parallel Gaussian noise give an exactly preserving transition with a tractable conditional density. We derive an absolute convergence bound from finite initialization banks and control departure from the reference through conditional action-space divergence. Across $216$ fitted pixel checkpoints on twelve control tasks, the occupancy model with a reference mixture retains every evaluated chain at $10^5$ steps in all $36$ task–seed cells, with a rollout-minus-reference energy-statistic difference of $-0.0002\pm0.0003$ (training-seed standard error). Each of the four nonpreserving comparison arms loses chains, although the Gaussian arm is more accurate at ten steps. An offline DreamerV3 reference also achieves better short-horizon accuracy. These results distinguish three properties of a world model: the distribution it approaches, the rate of approach, and the conditional dynamics it learns.

Multivariate Time Series Forecasting (MTSF) models that mix information across channels assume that the past of one channel carries information about the future of another. Yet they are evaluated on a small fixed set of standard datasets whose cross-channel structure is rarely examined. We ask two questions: "How can we reliably measure lagged, non-linear, and joint coupling in MTSF datasets?" and "Do the standard datasets actually have such coupling?" To answer the first, we test four candidate measures on synthetic datasets with planted ground-truth coupling: Granger Causality (GC), Transfer Entropy (TE), lagged Mutual Information (MI), and the CD gain, a model-based measure we introduce that compares a channel-dependent (CD) model to its channel-independent (CI) variant. Only lagged MI and the CD gain recover every planted coupling. For the second question, the answer is a definite no, as the standard datasets have a median of only 23% lagged-coupled channel pairs and a median CD gain of -4.9%, compared to 78% and +4.6% on chaotic ODE systems. We therefore propose MixBench-TS, a benchmark of 10 real-world datasets with a median of 55.5% lagged-coupled pairs and a median CD gain of +1.7%. Across six state-of-the-art models tuned under one protocol, CI models win on 10/10 (MSE) and 8/10 (MAE) standard datasets, but on only 3/10 and 2/10 MixBench-TS datasets. We recommend using our benchmark for evaluating new CD models. Moreover, we propose profiling new datasets with lagged MI and the CD gain before using them to evaluate multivariate models. Code and data are available at https://anonymous.4open.science/r/mixbench-ts-B027.

The compact four-dimensional manifold $\mathbb{CP}_2\# 3\overline{\mathbb{CP}_2}$ is known to admit a toric Kähler-Einstein metric $g_{\text{KE}}$, but the metric is not known in closed form, which makes it difficult to draw conclusions about its geometry. In this article, we use a combination of analytic and computer-assisted techniques to produce an approximate Einstein metric $g$ described explicitly herein, and also prove that the true Einstein metric $g_{\text{KE}}$ is close to $g$, where both the closeness and the topology are described explicitly. As an application, we prove bounds on the first invariant eigenvalue of the Laplace-Beltrami operator, and prove that this Kähler-Einstein metric does not have positive holomorphic sectional curvature everywhere.

Disc-type phase-change memory is a promising device architecture for analog in-memory computing, enabled by the unique physical properties of ultra-thin phase-change materials and the device geometry. In this study, we develop a comparative analytical framework using compact electro-thermal models to examine the state dependent readout and programming behavior of these devices. The framework provides insight into the fundamental physical mechanisms responsible for their enhanced functionality and highlights promising directions for further device optimization and performance improvement.

The ability of vision-language models (VLMs) to abstain from unanswerable questions is as important as their ability to answer answerable ones accurately. Recently, several benchmarks have emerged to evaluate and improve VLM abstention, but they have substantial limitations. First, samples often contain shortcut cues in images or questions that reveal answerability, while an explicit "unanswerable" option further prevents accurate assessment of spontaneous abstention. Second, as training data, they generally provide only binary labels without fine-grained explanations for deeper supervision. To address these limitations, we introduce Visual Answerability Diagnosis with Rationales (VAD-R), a benchmark constructed through a two-stage pipeline of shortcut filtering and quality verification to prevent answerability leakage. Each example is annotated with step-by-step rationales and causal evidence-gap labels. Evaluation of state-of-the-art open- and closed-source VLMs on VAD-R reveals limited spontaneous abstention, with average recall rates of only 11.4% and 16.3%, respectively. Probing analyses show that hidden-state representations in certain layers can effectively distinguish answerability, yet this distinction fails to manifest in final responses. Motivated by this observation, we introduce Rep2Act, a representation-to-action alignment method that translates latent answerability awareness into explicit abstention decisions. Rep2Act improves action accuracy on VAD-R from 56.67% to 86.33% for Qwen2.5-VL-3B and from 59.33% to 88.67% for Qwen2.5-VL-7B. On the out-of-distribution TUBench, Rep2Act achieves an average F1 score of 53.3% with only a 3B model, surpassing the closed-source GPT-4 Turbo and GPT-4o by 16.2% and 1.1%, respectively.

Soft state representations assign states nonnegative class weights that sum to one, but agreement with reference labels does not guarantee accurate prediction of the next weight vector. We derive lower confidence bounds on the minimum population root-mean-square prediction error over linear predictors with a prescribed spectral-norm limit. The bounds compare within-class successor variation with retained within-class input variation, with an alternative based on reference-label deviation. They are estimated from independent transition pairs without fitting a prediction matrix to the evaluation data. A bound above a chosen error tolerance excludes that accuracy for every admissible predictor at the stated confidence level; a bound of zero does not establish whether an accurate predictor exists. For coordinates constructed from Kernel Affine Hull Machine reconstruction scores, we relate the reference class's affinity advantage over competitors to label agreement. The class-centered population bound equals the optimum in a four-state example. Using 117,000 previously generated evaluation datasets from fixed four-state designs, we compare the finite-sample certificates with an empirical-risk-minimization baseline. For one Van der Pol representation selected before evaluation, the label-based bound is positive in all 64 independent evaluation datasets at the fitted matrix's spectral norm. Separately, exact matrix propagation under deterministic dynamics yields a Koopman interpretation of the coordinate-function span. The certificates help identify forecasting targets that cannot be reached by refitting a fixed representation within a specified predictor class and evaluation distribution.

In 1975, Kaplansky conjectured that every five-dimensional division algebra over a sufficiently large finite field is a field or a twisted field. We disprove this conjecture. For every prime power $q=p^e\equiv1\pmod3$ and every $n\ge5$ with $\gcd(n,6)=1$, we construct semifields of order $q^n$. For fixed $q,n$, the family represents $\varphi(n)$ isotopy classes if $p\equiv1\pmod3$, and $\varphi(n)/2$ if $p\equiv2\pmod3$, where $\varphi$ is Euler's totient function. Using new isotopy invariants and the structural properties of our construction, we prove that none of these semifields is isotopic to a finite field or an Albert's generalized twisted field. In particular, the five-dimensional specialization gives infinitely many pairwise nonisotopic counterexamples to Kaplansky's conjecture over arbitrarily large finite fields. More generally, for each prime dimension $n\ge5$, the examples occur over arbitrarily large fields in every characteristic other than three, contradicting the classification asserted by Menichetti in 1996, whose proof contains gaps.

Little is known about the dynamics of Toeplitz operators on the Bergman space, and even in the case of harmonic symbols, few results are available. In this paper, we study the dynamical properties of Toeplitz operators with symbols of the form $a \bar{z} + p (z)$ on the Bergman space, where $a\neq 0$ and $p$ is analytic on the closed unit disk. We obtain some necessary conditions and some sufficient conditions for such operators to be hypercyclic, weakly mixing, mixing, chaotic, or frequently hypercyclic. As an application, we obtain a complete characterization of hypercyclicity for Toeplitz operators with harmonic linear polynomial symbols. We also show that Toeplitz operators with the same symbol may exhibit different hypercyclic behavior on the Hardy and Bergman spaces.

The energy spectrum and mass composition of cosmic rays are measured by two complementary classes of instruments: space- and balloon-borne detectors that resolve individual elements up to sub-PeV energies, and ground-based air-shower observatories that extend the reach to beyond $10^{11}$ GeV but resolve only broad mass groups. The Global Spline Fit (GSF) is a data-driven model that combines both into a single, self-consistent description of the flux of all elements from hydrogen to nickel. The flux of four leading mass groups is parametrized by cubic basis splines, imposing smoothness but no astrophysical expectation on the spectral shape, while the energy-scale offsets of the participating experiments are cross-calibrated within their quoted systematic uncertainties as part of the fit. The model is defined at the local interstellar spectrum, with the effect of solar modulation at the lowest energies accounted for in the fit. The fit uses the most precise recent data, among them the elemental spectra from AMS-02, CALET, and DAMPE, the knee-region measurements of LHAASO, and the fluorescence-based composition of the Pierre Auger Observatory, and describes about one thousand data points with $\chi^2/\mathrm{ndf}\approx 1.3$, or 0.83 after de-weighting localized disagreements between data sets, demonstrating that the global body of cosmic-ray data is consistent once energy scales are aligned. The model delivers the flux, the mass composition, and their full covariance, along with a compact reduced representation for uncertainty propagation. We further provide the cosmic-ray nucleon flux, which differs from widely used parametrizations by 20-50% over four decades in energy. This exceeds the remaining model uncertainty several times over and has direct consequences for atmospheric neutrino and muon flux predictions.

We introduce curve band depth (CBD), a band-based data depth for samples of unparameterized planar curves. CBD is motivated by band depth and modified band depth for functional data, but targets trajectory data. Unlike the halfspace-based curve depth of \citet{de2021depth} and the curve stabbing depth of \citet{durocher2023csd}, CBD is defined through a geometric band region generated by two curves, and measures the arc-length proportion of a target curve lying inside such bands. We develop a CBD family consisting of an integral version (int-CBD), an infimal version (inf-CBD), and a fast-walk variant (FW-CBD). The fast-walk band is a narrower band construction contained in the global convex-combination band. We establish boundedness, vanishing at infinity, and similarity invariance for these constructions, together with a Borel-measurability result for the induced depth maps under a mild measurability assumption. A length-penalized variant is proposed for samples with heterogeneous curve lengths. We implement the methods via arc-length sampling and polygonal approximations, and evaluate them through classification of overlapping handwriting data and MNIST-derived digit curves, online-signature screening on MOBISIG, and an exploratory clustering task based on decomposed band contributions.

We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the $L^\infty\cap H^1_{\rm loc}$ class. First, in the plane and the half-plane, every such flow without stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation. The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that $\nabla P\in L^q$ for some $1\le q\le 2$. Combined with standard elliptic estimates for the pressure, this result yields a concise new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to $H^1_{\rm loc}$. The pressure serves as a unifying quantity throughout the analysis.

We establish Morrey-type conditions ensuring that a finite signed Radon measure belongs to the dual of the energy space of solutions to nonlinear parabolic equations of $p$-Laplacian type. More precisely, for the parabolic cylinders $Q_{r,r^p}(z):=B_r(x)\times(t-r^p,t+r^p)$, we prove that \[ |\mu|\bigl(Q_{r,r^p}(z)\cap(\Omega\times (0,T))\bigr) \leq Mr^{n+p-\vartheta}, \qquad \vartheta<p, \] implies duality, and this threshold is sharp in general. More generally, for cylinders with time length proportional to $r^q$, $q>1$, the sufficient threshold is $\vartheta<\min\{p,q\}$. Finally, we discuss the resulting variational, energy, and renormalized solution theories for nonlinear parabolic equations with measure data, relating our conclusions to the existing literature.