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At least 883 records · Page 49Linked to original sources
arXiv2026

Failure of semipositive metric extension in a smooth projective family

We construct a smooth projective family of rational surfaces, an effective line bundle on its total space, and a prescribed semipositively curved singular Hermitian metric on the central fibre that has no semipositive extension to any neighbourhood of that fibre. It has analytic singularities and is of minimal singularity type. The failure persists even when the restriction is only required to have the same singularity type as the prescribed metric. The family is obtained by blowing up four disjoint sections of a product, with three points becoming collinear on the central fibre. An integrable adjoint section on that fibre cannot extend because the corresponding adjoint systems vanish on every nearby fibre. This gives a negative answer to Păun's metric-extension question, listed as Question 38 by Dinew, Guedj, and Zeriahi.

math.AG↗
arXiv2026

Twisted Diophantine Approximation I: Asymptotic Theory

We establish an exact zero-one law for twisted Diophantine approximation with an arbitrary fixed real matrix and a positive non-increasing approximation function satisfying dyadic regularity. The criterion is the convergence or divergence of a dyadic series whose summands are Khintchine-Groshev block volumes divided by homogeneous lattice-point counts. An equivalent formulation is using all successive minima of the associated diagonal lattice trajectory, capturing homogeneous clustering in every direction. We also obtain an almost-sure zero-infinity law for inhomogeneous $ψ$-Lagrange constants, separate Hausdorff-measure criteria, and dimension results. Within this regularity class, our theorem recovers the Kurzweil and Fuchs-Kim criteria. At the critical exponent, this series determines whether the set of badly approximable shifts has zero or full Lebesgue measure.

math.NT↗
arXiv2026

Twisted Diophantine Approximation II: Uniform Theory

We develop a general method for uniform twisted metric Diophantine approximation with an arbitrary fixed real matrix. Using all successive minima of the associated diagonal lattice trajectory, we estimate lattice-point counts, accounting for clustering in every direction. Ratios of these counts give exact Hausdorff dimensions of sets of twisted $ψ$-Dirichlet vectors for a large class of functions $ψ$, including all except finitely many power functions {$T^{-τ}$, $τ>0$}. We obtain Hausdorff dimension formulae for the endpoint sets obtained by taking the intersection and union of the sets of twisted $cψ$-Dirichlet vectors over $c>0$, and prove that taking the intersections causes no dimension drop. Furthermore, we obtain Hausdorff dimensions of level sets of the uniform Diophantine exponent. In addition, we prove zero-full laws for general Hausdorff measures of $ψ$-Dirichlet sets. Applications recover the one-dimensional Kim-Liao formulae, strengthen Aggarwal's escape-of-mass upper bound, reproduce the metric criteria of Moshchevitin and the second author and give certain extensions of the above statements.

math.NT↗
arXiv2026

One from Infinity: Actualizing Futures from Pretrained World Models into Robot Actions

A pretrained video world model admits many plausible futures for a scene, but a robot must realize the exact task-conditioned one. To turn world models into executable robot policies, existing methods fine-tune the heavy world model backbone using large-scale robot data and computational resources. Challenging this status quo, we argue that the expensive part has already been paid in the world model pretraining since the representation space of a video world model lays out the diverse potential futures. In this case, what remains is to select the future that accomplishes the task and to read out the actions that realize it. We formalize this task as actualization, which learns a task-conditioned selection and realization on top of a prior supplied by a frozen world model. This can be solved by a tiny actualizer model. We implement RoboActualizer with as few as 60M parameters on top of a frozen world model encoder. The actualizer is composed of two lightweight DiT experts that jointly predict future latents and actions by flow matching. The model can be trained entirely on a single GPU with 32 GB peak memory. With up to 100x fewer trainable parameters than existing WAMs and VLAs, RoboActualizer reaches great performance on simulation benchmarks including LIBERO, LIBERO-Plus, RoboTwin 2.0 and five tasks on two real-world platforms, with a low latency of 39 ms that allows real-time control.

cs.RO↗
arXiv2026

Eternal Sunshine of the Spotless Mind: Systematically Erasing LLM's Memories

We consider persistent LLMs that accumulate memories of their interactions with a user over time. Such LLMs maintain memories using external storage, which they can query to overcome the limitations of a fixed context window. Such systems have numerous practical applications, as they can draw on all past interactions when responding to user queries. In this paper, we ask whether LLMs can forget information shared with them upon a user's request. We find that current LLMs fail to delete such information---even when they claim to have forgotten it and even when operating with a limited context. To this end, we consider a new direction of study: Deletion of LLM Memories. We show that naively removing messages that match a user's deletion request is insufficient, since conversations naturally introduce message dependencies that cause information to persist. To correctly handle deletion requests, we propose the DeLLM framework. It dynamically constructs relevant context for each LLM query and maintains a provenance graph of messages to determine which ones must be removed during deletion. Our experiments show that DeLLM achieves a high deletion rate while maintaining utility.

cs.CL↗
arXiv2026

Sets of cardinality seven are not sum-dominant

We give a self-contained elementary proof of the known result that every set $A\subset\R$ of cardinality seven satisfies $|A+A|\le |A-A|$. The argument adapts Hegarty's method, using representation counts and the largest positive differences. An exact counting identity, two applications of the Cauchy--Schwarz inequality, and an analysis of a symmetric six-element set with one point added complete the proof, with no computer enumeration. This answers in the affirmative a question of Chu for the seven-element case.

math.CO↗
arXiv2026

FineART: Fine-grained Annotated Robotic Trajectory Dataset and Vision-Language-Action Model for Bimanual Manipulation

Robots operating in real-world environments must execute complex, multi-step bimanual tasks over long horizons rather than single, isolated actions. Current manipulation datasets struggle to support this capability: although single-arm datasets reach hundreds of thousands of trajectories, they typically provide only one high-level instruction per episode while the rare bimanual effort that does label subtasks annotates only a fraction of its hours. We present FineART, a densely annotated bimanual manipulation dataset of 40,543 episodes, 1,718 hours, and 533,913 subtasks across 151 tasks. We also introduce FineART-VLA, a vision-language-action policy that predicts its own next subtask, and show that mid-training it this way yields substantial gains. Specifically, success on a spatial disambiguation task increases from 32.0% to 100.0%, and step-by-step human subtask guidance lifts success on an unseen long-horizon task from 16.0% to 76.0%. Furthermore, after minimal fine-tuning on a new robot, the policy requires one-tenth the data of baselines without mid-training and generalizes zero-shot to completely unseen tasks on the new hardware. We open-source the full dataset, model weights, and training code.

cs.RO↗
arXiv2026

Support-Set Target Leakage in Relational Foundation Models during In-Context Learning: Model Dependence and Evaluation Reliability

Relational in-context learning (ICL) uses labeled support examples and their linked relational context to predict labels for new queries. This creates a failure mode when target-derived features are present in the support context but unavailable for the query. We study this setting as support-set target leakage. We construct 20 controlled target-derived features that vary in signal fidelity, representation, semantic transparency, coverage, and zero-, one-, and two-hop relational placement, and evaluate them across 13 RelBench tasks and five relational ICL configurations that vary the ICL head, message-passing depth, pretraining cohort, or relational encoder architecture. We evaluate matched 0-hop, 1-hop, and 2-hop leakage settings, together with a Full leakage condition containing all 20 leaker columns. Within the tested configurations, target-table (0-hop) and Full leakage produce the largest aggregate deviations from clean evaluation, while higher-hop effects are often weaker, consistent with differences in effective exposure associated with temporal reachability, sampling, and aggregation fidelity. Leakage effects are strongly task- and model-dependent and can reverse relative conclusions between model variants even when aggregate changes are small. For leaker detection, we compare an Integrated Gradients (IG)-based screening method with mutual information (MI) and leave-one-column-out (LOCO) on a common Baseline subset. Ranking quality is strongest in the high-impact 0-hop and Full leakage conditions, but detector-based removal does not consistently restore the clean evaluation. A four-task rel-salt case study further shows the same evaluation concern with native-schema leakage candidates from the original relational schema. These results identify the support/query information boundary as an important component of reliable relational ICL evaluation.

cs.AI↗
arXiv2026

Kähler--Ricci Shrinkers with Constant Scalar Curvature Are Rigid

Let $(M^{2m},g,f,J)$ be a complete gradient Kähler--Ricci shrinker satisfying \[ \operatorname{Ric}+\nabla^2 f=\frac12 g. \] We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer $k\in\{0,\ldots,m\}$ such that \[ R\equiv k \] and \[ (M^{2m},g,J)\cong \bigl(N^{2k}\times\mathbb C^{m-k}, g_N+g_{\mathrm{Euc}},J_N\oplus J_0\bigr), \] where $(N^{2k},g_N,J_N)$ is Kähler--Einstein and \[ \operatorname{Ric}_{g_N}=\frac12 g_N. \] As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition $\mathcal{L}_{\nabla f}\operatorname{Ric}=0$ implies radial flatness and hence rigidity.

math.DG↗
arXiv2026

A unified structure-preserving framework for geometric flows with coupled orientation and curvature dependence

We develop a structure-preserving parametric finite element framework for geometric flows whose energy density couples the unit normal and the curvature. This class includes bending energies with orientation-dependent rigidity or spontaneous curvature. A common fully discrete formulation treats closed curves in two dimensions and closed surfaces in three dimensions, and accommodates the $L^2$ flow, curve or surface diffusion, and the area- or volume-constrained $L^2$ flow. The formulation couples the geometric and curvature updates so that their contributions satisfy a discrete energy inequality. It uses continuous piecewise linear elements, a surface energy matrix, and a mass-lumped projection of the curvature derivative of the density. For positive densities satisfying a directional condition and convexity in curvature, we prove energy dissipation without a time-step restriction. The diffusion and constrained flows also preserve the enclosed area or volume exactly. The analysis allows non-even anisotropies and nonseparable dependence on orientation and curvature. Numerical experiments exhibit approximately second-order convergence in the manifold distance and confirm the discrete structural properties. Shape relaxation under an anisotropic Helfrich-type energy illustrates the use of the framework for coupled directional and bending effects.

math.NA↗
arXiv2026

Sample Complexity of Equivariant Reinforcement Learning

Reinforcement learning (RL) is a powerful framework for robotic control, yet its practical application is often hindered by high sample complexity. This is particularly restrictive in physical domains where interaction data is costly. While the world often exhibits geometric and physical symmetries, standard RL algorithms typically fail to exploit this structure. In this paper, we demonstrate that exploiting group symmetries significantly reduces the sample complexity of RL. Focusing on finite-horizon Markov decision processes, we find that leveraging homomorphisms induced by group symmetries significantly reduces the theoretical upper and lower bounds on the number of environment interactions required to reach an optimal return. We further extend these bounds to continuous state and action spaces, providing corresponding sample-complexity guarantees under appropriate regularity assumptions. Beyond theory, we validate our findings through controlled experiments and demonstrate the advantages of symmetry-aware policy learning on high-dimensional continuous robotic simulations. Our results show that integrating symmetry into the learning pipeline yields substantial gains in sample efficiency and performance, offering a principled path toward more data-efficient robotics.

cs.CC↗
arXiv2026

Attributing extreme-event probability to a source variable

Information-flow theory quantifies directional coupling through the rate of change of a target's Shannon entropy, a bulk functional insensitive to the tail of the distribution. We ask instead how a source variable contributes to the probability that the target exceeds a threshold. For source-additive drift the source's share of the marginal probability current is exact, and it separates into a mean-forcing part and a conditional-excess part that vanishes under independence. An identity links the two descriptions: the Liang information flow is the density-weighted mean of the derivative of the specific source current, whereas the exceedance current is its level at the threshold. This explains why entropy-based coupling collapses in saturated regimes where the source contributes most to the extreme; the Rényi information flow, a higher-order expansion and a Fisher-normalised response fail for the same reason. The flux decomposition, although exact, does not attribute: its terms are gross transports that nearly cancel. The quantity that does attribute is an adjoint response built from the backward generator which, read as a relative change, is uniformly accurate across two decades of event probability. We map the operating envelope of the estimator under omitted drivers, hidden slow memory, state-dependent coupling and multiplicative noise. Under correct specification the attribution carries a reproducible shortfall of ten to twenty-five per cent; the misspecifications we test bias it upward by up to ninety per cent. Applied to the 2003 European and 2010 Russian heatwaves in reanalysis, the attributed contribution of soil moisture is strongly threshold dependent, rising from the per cent level at moderate thresholds to a factor of two at the rarest, so a contribution quoted without its threshold is underspecified.

physics.data-an↗
arXiv2026

Discrete-time polynomial systems with inputs and outputs: structure, reachability, observability, and minimal realizations

We study the realization problem for discrete-time input/output polynomial systems. These are formalized using tools from commutative algebra and algebraic geometry as systems whose state spaces are algebraic varieties, or more abstractly the set of k-points of an affine k-scheme, where k is an arbitrary infinite field. The input/output behaviors of such systems are described by "polynomial response maps" in which outputs are polynomial functions of past inputs. The main results show that every polynomial response map admits a canonical (quasi-reachable and algebraically observable) realization, which is unique up to isomorphism. The key to the approach is to linearize dynamics by considering a "dual" system in which states are functions defined on states. Finite dimensionality of the canonical realization, and its polynomiality, are characterized in terms of the space, algebra, and field of observables of the map, as well as in terms of algebraic input/output difference equations, and by a Jacobian rank criterion. A particular subclass consists of the maps that we call "bounded," defined by the property that their degree in the past inputs is uniformly bounded. Bounded maps are shown to be finitely realizable if and only if they are realizable by finite-dimensional state-affine systems, whose theory in turn reduces to that of rational formal power series. We also study the lattice of quasi-reachable realizations of a given map, including normal realizations. This work is an update of the PhD thesis written by the author in 1976; connections to recent work are briefly discussed in the last section.

eess.SY↗
arXiv2026

Agent-Based Evolutionary Dynamics for Mixed Autonomy Weaving Ramps

Existing models of mixed-autonomy weaving ramps characterize how altruistic connected and automated vehicles (CAVs) can improve traffic efficiency at the population level, but provide limited insight into how such behavior emerges from decentralized vehicle interactions or how it is affected by finite populations, heterogeneous preferences, and imperfect information. We develop an agent-based model of a macroscopic weaving-ramp framework in which individual vehicles adapt their lane choices using an evolutionary game-theoretic update rule and altruism-based objectives providing a microscopic interpretation of the original Wardrop model. We prove convergence of the decentralized dynamics to the unique equilibrium predicted by the macroscopic theory. Beyond reproducing aggregate equilibrium behavior, the framework enables the study of deployment-level questions that cannot be addressed by static analysis. Simulation results demonstrate close agreement with the macroscopic predictions while revealing how convergence rates, adaptation to changing traffic conditions, heterogeneous altruism levels among CAVs, and imperfect state information influence system performance and the distribution of altruistic burden across vehicles. These results provide a bridge between equilibrium traffic theory and decentralized mixed-autonomy deployment.

cs.MA↗
arXiv2026

Losing the name before the box: measuring and repairing what narrow fine-tuning costs a detector outside its deployment vocabulary

A detector pretrained on a broad corpus is fine-tuned on a narrow domain, its in-domain accuracy improves, and it ships. We ask what happens meanwhile to its coverage of objects the vocabulary never names, which in obstacle detection and inspection carry the risk. No in-domain test set holds an example of one. We give a longitudinal protocol: one pretrained checkpoint against its own fine-tuned descendants. It tracks held-out top-$K$ proposal coverage $C_τ$: of categories pretraining covered and the vocabulary omits, the share of boxes a detector's top $K$ regions still cover. The quantity is the open-world proposal literature's; the longitudinal reading is not. $C_τ$ falls while in-domain accuracy rises, on four architectures and three domains, by $5.12$ to $63.35$ points on boxes above $1024$ px$^2$. No in-domain number identifies the fall, and neither does detection average precision, which charges a missed and a misnamed box alike. On the one architecture scoring both, adaptation costs $87\%$ of the AP against a fifth of the coverage, and the naming goes first at all six depths of its freeze ladder, every run. What breaks is structured: three architectures sharing no pretraining run agree on which categories lose coverage, and those a model never learned do not lose any. A repair follows and needs no training: mixing a quarter of the pretrained state back, normalisation statistics included, raises coverage on every cell swept for at most $2.47$ points of in-domain accuracy. Seeing it costs one extra evaluation pass.

cs.RO↗
arXiv2026

Most A$_α$-eigenvalues of a tree are small

We study the distribution of $A_α$-eigenvalues of a tree. We extend the works \cite{Jacobs2021} and \cite{SIN2020} by proving that the number of $A_α$-eigenvalues, for $0\leq α\leq \frac{1}{2}$, less than or equal to the average degree $d_α= α(2- 2/n)$ is, at least, $\lceil \frac{n}{2} \rceil$ for a tree with $n$ vertices. Several counterexamples for $\frac{1}{2} < α\leq 1$ are exhibited. We also derive the same result for other well-known families such as the Deformed Laplacian and the matrix $B_β$.

math.CO↗