Papers

Given $p \in (1,\infty)$, let $\operatorname{X}=(\operatorname{X},\rho,\mu)$ be a metric measure space such that the measure $\mu$ is uniformly locally doubling and $\operatorname{X}$ supports a weak local $(1,p)$-Poincaré inequality. Let $(\operatorname{Y},\operatorname{d},\underline{y})$ be a complete pointed metric space. We prove that the equivalence class of a Borel map $ u:\operatorname{X} \to \operatorname{Y}$ (modulo coincidence $\mu$-a.e.) belongs to the Ambrosio–Reshetnyak–Sobolev class $W_{p}^{1}(\operatorname{X},\operatorname{Y})$ if and only if $h \circ u$ belongs to the Sobolev space $W_{p}^{1}(\operatorname{X})$ for every $1$-Lipschitz function $h:\operatorname{Y} \to \mathbb{R}$.

In this paper, we first study the singular value distribution of the unscaled matrices arising from Jacobi weighted histopolation. For the Jacobi weight with parameters $(\alpha,\beta)$, and using the Jacobi basis with parameters $(\alpha+1,\beta+1)$, we previously proved that, for $\alpha,\beta>1/2$ and quasi-uniform meshes, the growth of these matrices in the sense of singular value distribution is slower than any positive power of the logarithm of the matrix size. However, the numerical behavior of the unscaled matrices suggested a nontrivial singular value symbol, which we identify explicitly in this paper. Starting from the standard asymptotic formula for Jacobi polynomials, we derive an asymptotic representation of the weighted cell averages defining the entries of the histopolation matrices. This representation allows us to analyze the associated Gram sequence by means of GLT theory. We identify its GLT symbol and, consequently, we obtain the singular value distribution of the histopolation matrices for any $\alpha,\beta>-1$. We also characterize the set $\mathcal Z$ where the resulting symbol vanishes and compute its measure, showing that this set depends only on the mesh, while the Jacobi parameters affect the values of the symbol on its support. When this measure is positive, the singular value distribution implies the discrete asymptotic ill-posedness of the problem, while a comparison with the Toeplitz setting suggests exponential growth of the spectral condition number. When the measure of $\mathcal Z$ is zero, the problem is well-posed and the observed conditioning is algebraic in the matrix size. Finally, numerical experiments supporting the theoretical findings are reported and discussed, while conclusions are given in the final section of the current work together with a few open problems.

We introduce the notion of mixed configuration spaces of the $3$-ball. That is, we consider both distinct circles and distinct points in the interior of the $3$-ball and we prove that the projection onto the configuration space of circles of the $3$-ball is a locally trivial fibration. The key idea for defining the mixed configuration spaces is to study fixed points of orientation-preserving homeomorphism of the $3$-ball that leave invariant a trivial link of $n$ components in the interior of the $3$-ball. In particular, we prove that two fixed points are Nielsen equivalent if and only if the associated loop braids are conjugate by an element of a distinguished free subgroup of rank $n$. This result stands as a $3$-dimensional counterpart of the $2$-dimensional result where fixed points of homeomorphisms of the punctured disc are characterized in terms of braid elements of the classical Artin braid group.

Fine-grained image editing requires more than producing a visually plausible result: an editor must execute the requested attribute change precisely while leaving everything else intact. However, existing benchmarks leave a critical gap between realism and verifiability: benchmarks built on realistic images typically rely on human or vision–language model judgments, while deterministic evaluation has largely focused on synthetic shape canvases, with application-oriented extensions primarily limited to charts. This makes it difficult to determine precisely how much of a requested edit was executed, where unintended changes occurred, and whether small differences between models reflect genuine editing capability or evaluator uncertainty. To bridge this gap, we present VeriEdit-Bench, a benchmark for fine-grained, instruction-faithful image editing across realistic structured assets with deterministic, four-axis evaluation. Its 1,740 cases are compiled from the source code of 153 Scalable Vector Graphics (SVG) graphics, charts, web interfaces, and presentation slides. Controlled source-code edits preserve the original visual context while yielding exact target images, pixel-level edit masks, and explicit edit specifications, enabling reproducible scoring along four axes: edit fidelity, preservation, localization, and magnitude. Evaluating eleven editors, we find that even the strongest model remains far from full credit; rankings for the same recoloring operation reverse between charts and SVG graphics; and outputs with similar pixel-accuracy profiles can still differ substantially in localization and change magnitude. This decomposition yields graded, verifiable feedback and exposes model-specific capability and failure profiles that holistic scores or evaluator-dependent judgments may obscure.

In this paper, we study the optimal control of a phase field tumor growth model of Cahn–Hilliard type with possibly singular potential in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We investigate the asymptotic behavior of the control problem as the relaxation parameter $\,\alpha\,$ approaches zero. It is shown that the corresponding state and adjoint state variables of the relaxed problem converge in a suitable sense to their counterparts for the unrelaxed regime where $\alpha=0$. Also, weak limits of optimal controls for the relaxed problem can be shown to be optimal controls for the unrelaxed case. Moreover, every fixed optimal control $\,u^*\,$ for the unrelaxed system turns out to be the limit of a family $\{u^*_\alpha\}_{\alpha>0}$ of minimizers of suitable optimal control problems for the relaxed systems, where the original cost functional is replaced by the so-called “adapted” cost functional. It is also shown that the variational inequalities satisfied by $u^*_\alpha$, which constitute the first-order necessary optimality conditions of the relaxed problems with adapted cost functional, converge to the corresponding variational inequality fulfilled by $u^*$ for the unrelaxed problem with the original (nonadapted) cost functional.

We introduce AgentPersonaBench (APB), a benchmark evaluating whether persona conditioning faithfully steers downstream agent behavior. While language models are increasingly deployed for persona-driven user simulation, existing benchmarks primarily evaluate conversational styling or self-reports rather than authentic behavioral fidelity. APB evaluates latent persona adherence one trait at a time, embedding each target trait within a complete synthetic profile without explicitly naming the trait or disclosing the test. Ground-truth adherence is verified strictly from observable actions across four interaction surfaces of increasing realism: survey, chat, web (interactive web environments), and app (desktop software environments). APB comprises 2,460 tasks spanning 867 traits, verified through automated audits and expert review. Our evaluation of 20 frontier model arms demonstrates that high-fidelity user simulation is already attainable: leading models achieve up to 84.7% full-pass adherence under unprompted conditions. At the same time, APB identifies clear behavioral boundaries: adherence drops across interaction modalities (only 37.9-64.3% pass all four surfaces), multi-attribute demands degrade retention, and competing model families exhibit pronounced behavioral divergence.

Large language models increasingly mediate tool use in Model Context Protocol (MCP) systems, where adversarial influence may enter through user instructions, tool schemas, tool outputs, or protocol messages. Existing benchmarks often evaluate deployed agents, conflating model susceptibility with guardrails, orchestration, and general task capability. We introduce COPEX (COntext Provider EXploitation), a controlled benchmark that isolates the model as an MCP client by fixing the surrounding agent stack and varying only the tool-selecting model. COPEX covers 25 attack types instantiated as 125 scenarios across four entry surfaces: model/agent, client, server/tool, and transport. Across nine models and 3,375 trials, the mean attack success rate is 64.4%, with surface-level means ranging from 58.3% to 71.4%. Some client- and transport-level attacks succeed partly outside the model's observation or control, separating system exposure from model susceptibility. Combined input and context scanning reduces mean attack success by 49.6% on an eight-attack defense subset relative to the undefended setting. The benchmark is available at https://github.com/inspire-center/copex.

Let \(X\in M_n(\mathbb C)\), \(n\geq2\), and let \[ \delta_2(X) = \sup_{\dim S=2}\delta(X_S), \] where \(X_S\) denotes the compression of \(X\) to \(S\), and \(\delta(X_S)\) is the diameter of the largest disk contained in the numerical range \(W(X_S)\). Bourin and Lee proved that every positive block matrix \[ M= \begin{pmatrix} A&X X^*&B \end{pmatrix} \] satisfies \[ \norm{M}\leq\norm{A+B}+\delta_2(X). \] In the same work, they conjectured that the inequality \[ \norm{M}\leq\norm{A+B} \] for every positive block matrix with prescribed off-diagonal block \(X\) characterizes essentially Hermitian matrices. We resolve this conjecture by proving the stronger exact formula \[ \sup_{\left(\begin{smallmatrix}A&X\\X^*&B\end{smallmatrix}\right)\geq0} \left\{ \norm{\begin{pmatrix}A&X\\X^*&B\end{pmatrix}} -\norm{A+B} \right\} = \delta_2(X). \] Thus the elliptical width is precisely the optimal norm defect associated with a fixed off-diagonal block. The reverse inequality is obtained by lifting an arbitrary two-dimensional compression of \(X\) to an explicit family of positive block matrices, yielding quantitative two-sided estimates with error of order \(t^{-1}\). As a consequence, the norm inequality above holds universally if and only if \(X\) is essentially Hermitian, proving Conjecture 3.3 of Bourin and Lee. We also construct an explicit \(3\times3\) normal example showing that the radius constant in their normal-off-diagonal eigenvalue estimate is optimal at the leading eigenvalue level. By direct-sum amplification of this construction, we further prove that the same radius constant is sharp at every eigenvalue level \(j\geq0\), thereby resolving their corresponding sharpness question.

We prove a Chernoff-exponent guarantee for the output of spectrally initialized batch variational EM after a prescribed iteration budget, without assuming global maximization of its variational objective. The iteration repeatedly estimates the entire block probability matrix and community proportions from the same sparse graph. We consider a fixed number of communities with proportions bounded away from zero and fixed, positive, distinct connectivity profiles; neither assortativity nor full rank is required. The algorithm uses simultaneous softmax updates without sample splitting or posterior thresholding. Its analysis must control the feedback between estimated parameters, soft labels and reused edges at an exponentially small risk scale. We establish a uniform one-step bound over data-dependent soft assignments whose random remainder has exponentially small expectation. Combined with an exponentially reliable regularized spectral initializer, this yields an unconditional expected misclassification rate bounded by $\exp\{-(1-o(1))J_n\}$ throughout the sparse, diverging-degree regime, where $J_n$ is the minimum nodewise Chernoff information. Matching lower bounds establish first-order logarithmic minimax optimality on local parameter spaces allowing unknown connectivity and varying community counts. The algorithm also attains the sharp first-order exact-recovery threshold. Numerical experiments illustrate refinement gains and sensitivity to initialization and imbalance.

Agents built on large language models (LLMs) build and run software through tool calls. A call reaches its program through several hops, and any hop can change the call without notice. When the changed call fails, the agent retries a correct call, which costs users time and money. Benchmarks and failure analyses do not see the change, because they read the call and its result but not what a hop received. We define intent-execution correspondence (IEC) as the property that the executed action matches the action the emitted call denotes under the tool contract. Our protocol observes what each hop received without executing the call, and names the first hop that changed it by the receiver's own parser. IntAct then delivers the call in a form that this hop cannot alter, or refuses the call. We build IEC-Bench from the changes observed in real-world use, with chains of dependent calls under the execution paths of 4 widely-used harnesses. In 47,828 shell calls within production sessions, Claude Code's Bash tool changes 12.0% of the calls that carry code, escape sequences, or long text. For 80.7% of the calls whose backslashes are changed, the wrong action runs without any reported error. All 10 measured harnesses change a call. Trajectory-based judgment attributes 95.1% of the production failures to the LLM, although the path caused more than half of them. On IEC-Bench, the path raises the token cost per passed task 2.4 times (up to 12.3 times). A hop that changes a call also hides the changes after it, so 55.1% of the failures on one path appear only after its first hop is repaired. IntAct, deployed in a commercial product, recovers 79.2% of the failures with a changed call. Harnesses should therefore be designed and tested hop-by-hop to ensure a correct call executes as intended or is refused.

We propose a topological model for capability changes during large-model training. The model represents knowledge states and preregistered capability probes as subcomplexes of a finite regular CW complex. We prove that attaching a single $n$-cell can only create a class in $\mathrm H_n$ or kill a class in $\mathrm H_{n-1}$. With the reliability threshold and the training checkpoint as two parameter axes, capability gains and losses make the spaces along the training axis nonnested. Union or intersection bridges produce zigzag persistence modules, and we distinguish checkpoint, transition, and bridge-sensitive classes. A finite four-stage example is computed by boundary matrices. The framework records algebraic changes under a chosen encoding, and a homological change is called a candidate capability emergence only after robustness tests, null-model comparisons, and independent behavioral validation.

Self-propelled particles that only repel one another can still separate into a dense and a dilute phase, a process known as motility-induced phase separation (MIPS). In a clean system the domains coarsen like any conserved mixture: domains grow as $\ell(t)\sim t^{1/3}$. Living and synthetic swimmers, however, move through rough, porous, or patterned surroundings. Using large-scale simulations of active Brownian particles, we ask how frozen heterogeneity changes MIPS coarsening. We introduce disorder in two ways, as a quenched random force field acting on particle positions and as a quenched random torque field acting on their orientations. Both destroy Lifshitz-Slyozov growth. A transient power law with a disorder-dependent exponent gives way to activated dynamics in which the effective dynamic exponent grows without bound and domains grow at most logarithmically. The crossover obeys the scaling form known from the random-field Ising model, and the domain morphology depends on disorder strength, so superuniversality fails. The two kinds of disorder act through different mechanisms. Random forces trap particles in the sinks of a random drift field once the drift beats self-propulsion. Random torques act as a sign-random, quenched chirality that erodes persistence and lowers the local Péclet number toward its critical value. Our results show that quenched disorder is a relevant perturbation for the kinetics of active phase separation and connect MIPS coarsening to the physics of pinned interfaces in disordered magnets.

Transferring visual representations pretrained on human videos to robot manipulation requires learning reliable correspondences between human and robot demonstrations. However, paired demonstrations can differ in execution rate and in the proportion of non-key frames that do not directly reflect task progress. Frames at the same relative timestamp may therefore represent different task stages, which can cause correspondence learning to fail. To address these issues, we propose Frame-Level Temporal Alignment (FLTA), a framework that uses two temporal priors to adapt visual encoders pretrained on human videos for robot manipulation. It learns shared task-progress representations without frame-level correspondence annotations, allowing frames at different relative temporal positions to match. A global progress prior combines normalized temporal positions with visual similarity to construct a soft correspondence target. A local temporal order prior penalizes backward transitions while allowing stays and varying forward rates to accommodate execution-rate differences. With ResNet-50 and ViT encoders, our method achieves relative improvements of 46.93% and 65.96%, respectively, over the best baselines in average simulation success rates and achieves higher task success rates on real-world manipulation tasks. These results also suggest that effective human-robot adaptation depends less on the number of parameters updated than on which parameters are selected. Our project page is available at https://rtx5090ultra.github.io/FLTA-Project-Page/.

Determining whether two programs are functionally equivalent is central to code modernization, patch validation, refactoring, and code-generation evaluation. Yet the usual signals are incomplete: tests cover only finite inputs, textual similarity confuses implementation with behavior, and unconstrained LLM judgments are difficult to audit. Direct execution is often impossible when a program depends on an obsolete, licensed, unavailable, or unsafe environment. We introduce FEAgent, a selective equivalence assessor agent that combines typed program-graph evidence with differential surrogate execution. FEAgent first aligns public interfaces and behaviorally relevant graph anchors, then issues bounded queries over call-flow, control-flow, data-flow, type, import, and effect relations. Next, a branch-aware generator agent proposes discriminating inputs, and two blinded LLM surrogates independently predict source and target observables. Every claim and predicted divergence is recorded in an evidence ledger. A deterministic reconciler then returns EQUIVALENT, INEQUIVALENT, or UNCLEAR rather than forcing a verdict when paths are uncovered or evidence conflicts. We evaluate FEAgent on function-level equivalence and repository-level bug patches, where the existing oracle is a benchmark label or a passing test suite. Every disagreement with that oracle is adjudicated by direct execution, revealing errors in benchmark labels and behavioral divergences missed by unit-test-only scoring. On EquiBench, execution confirms FEAgent's disagreements with published labels on 216 of 1,200 evaluated pairs (18.0%); on SWE-bench Verified, 94 of 331 test-passing agent patches (28.4%) diverge from the reference patch. FEAgent thus serves as an audit layer between testing and formal verification, keeping its evidence reviewable and its uncertainty explicit without claiming a proof of equivalence.

Data quality now matters as much as compute for training language models. Much training data comes from human annotation of text, and interpretive annotation has no ground truth that could settle what is "accurate". Two lines of work respond to this. One combines annotators into a "ground truth" and measures how well they agree with each other; the other treats their disagreement as a signal. Both compare different people at one point in time. We measure something else: how well one reader reproduces their own reading of the same text over time. One expert human reader and three LLM families segmented three Sumerian myths and labelled the causal function of each segment with one of seven states. Across runs months apart, the human cut the text in much the same places but named the segments differently, in every myth. The models show no such consistent pattern: their gap between the two layers is positive in some myths and negative in others, and its size varies. The human's label changes are not random: the runs go through much the same functions but start them one step apart, while model runs start them at the same places. We argue that this pattern is a usable measure of data quality and a contamination check: a "human" annotation whose labels are as stable as its boundaries, and whose functions start in sync, looks like a model's.

For a homogeneous $A$-hypergeometric system, let $K_v$ be the intersection of the negative support family for a fake exponent $v$, and let $C_w$ be the set of indices of variables that do not occur in any minimal monomial generator of $\operatorname{in}_w(I_A)$. We prove that $K_v=\operatorname{nsupp}(v)\cap C_w$ holds for every parameter and every fake exponent if and only if $\operatorname{Cone}(A_{C_w})$ meets the interior of $\operatorname{Cone}(A)$. We give equivalent criteria using the reduced Gröbner basis and the associated regular triangulation. When the condition fails, we construct an infinite family of fake exponents for which the equality fails.

Extreme winter weather has repeatedly disrupted gas-fired power generation in the United States, yet the plant-level data needed to systematically quantify outage risk remain proprietary. Using publicly available weather and electricity demand data together with anonymized generator contingency records from the North American Electric Reliability Corporation (NERC), we develop a three-stage Bayesian probabilistic framework for estimating winter-driven generator performance. Applied to New York State (2013–2022), the framework sequentially estimates: the hourly probability of a generator contingency event, the expected net available capacity conditioned on an event occurring, and the event duration. Colder conditions and higher electricity demand are associated with higher failure probability, lower retained capacity, and longer event duration. Under the most severe observed stress conditions, estimated mean hourly event probability reaches 24% , while expected mean net available capacity falls to 13% of nameplate rating. Full outage events have a median duration of 12.7 hours, while partial derating event duration increases from 2.4 to 7.1 hours with capacity loss severity. The proposed framework establishes a transferable baseline that utilities with access to plant-level records can directly extend to obtain more precise reliability estimates for operational planning and resource adequacy assessment.

The Einstein Probe (EP) has discovered a fast X-ray transient, EP260321a, whose optical counterpart was subsequently classified as a broad-lined Type Ic supernova. Its observed rise time of about 690 seconds is inconsistent with theoretical predictions for shock breakout (SBO) from a Wolf-Rayet star, which is typically regarded as the progenitor of envelope-stripped supernovae. In this paper, we propose that EP260321a arose from a SBO within extended circumstellar matter (CSM), where breakout photons were initially trapped, then diffused out of the unshocked CSM as the ejecta expanded and the opacity decreased. We modeled this photon diffusion process using both Monte Carlo simulations and an analytical approach to describe the temperature evolution. Through Markov Chain Monte Carlo sampling, we obtain the best-fit CSM parameters of log10(Mcsm/Msun) = -3.3 for the total CSM mass, Rin = 160 Rsun for the inner CSM radius, and Rout = 612 Rsun for the outer CSM radius. The presence of such a dense CSM indicates that the progenitor of EP260321a underwent episodic mass loss just days before the explosion, providing new observational constraints on the late-stage evolution of massive stars.

Reliable safety evaluation of autonomous vehicles (AVs) is essential to improving road safety, yet it depends critically on realistic simulation of rare crashes. Existing crash scenario generation methods can increase collision occurrence, but often fail to realistically reproduce how crashes evolve before impact or the distribution of crash types observed in the real world. Here, we present CrashSim, a human behavior-informed crash scenario generation framework that uses real-world crash priors to guide generative multi-agent traffic simulation for more reliable AV safety evaluation. These priors capture how real-world crashes evolve before impact and how different crash types are distributed, allowing limited crash data to guide realistic and scalable scenario generation across naturalistic driving contexts. We evaluate CrashSim against competing methods, showing that it more closely reproduces real-world pre-impact behavior, collision dynamics, collision geometry and crash-type distributions. We further use CrashSim to construct nuCrash dataset, containing over 4,000 crash and near-crash scenarios. Closed-loop evaluation of five AV planners shows that nuCrash more effectively exposes differences in planner safety capabilities than nuScenes. An LLM-assisted evaluation agent further analyzes planner failures to provide capability-level diagnoses and targeted improvement guidance. Together, CrashSim enables realistic and scalable crash generation for more informative AV safety evaluation.

A bubble bursting inside a drop resting on a substrate can eject jet droplets where a bubble at a bath would not. We compute how much liquid it ejects and how the ejected droplets are distributed in size. The volume ejected by the jet grows by more than an order of magnitude as the Bond number decreases from $0.5$ to $0.1$ and then saturates, and the larger the Bond number, the smaller the viscosity that suppresses ejection. At zero Bond number the substrate alone extends the ejection boundary to larger Ohnesorge numbers, contrary to expectation, and yet reduces the volume ejected at the same distance from it. Gravity sets how much liquid the jet carries, not how the jet fragments: scaled with its mean, the droplet-size distribution is independent of the Bond number and has an exponential tail, as for free hollow droplets. The simulations, built on a closed family of sessile equilibria, reproduce the measured velocity of the first droplet within 7\,%, while the volumes emitted in experiments are eight to ten times those computed for the same geometry.

Designing effective reward functions in RL typically requires substantial expertise and trial and error. Recent work automates this process with LLM-based systems that generate and iteratively improve reward code using policy feedback. However, these methods are often hard to compare because they differ in implementation details, feedback assumptions, and evaluation environments. To address this, we introduce a Benchmark for Iterative Reward Design (BIRD) that expresses existing methods in a unified configuration and evaluation space. This lets us compare algorithms directly, ablate individual design choices, and prototype new components under matched feedback conditions and policy-training budgets. Across MuJoCo, Meta-World, Assistax, and HumanoidBench, we identify a small set of simple design choices that consistently improve performance. Combining these choices yields significantly better performance than the evaluated methods from prior work. Our results highlight the strength of simple baselines and motivate further study of when additional algorithmic complexity improves iterative reward design. Code is available at https://github.com/safety-research/bird.

Learning contact-rich dexterous manipulation from human demonstrations provides a scalable source of interaction data, yet transferring such skills to robots remains challenging due to unreliable human–robot correspondence. Existing human-to-robot transfer methods typically rely on visual appearance or motion similarity, which may associate similar motions with different contact states and force patterns. Tactile dynamics provide interaction-aware cues to distinguish manipulation processes with similar motions but different contact states. We introduce tactile-guided optimal transport (TacOT), a framework for human-to-robot contact-rich manipulation. TacOT leverages action–tactile dynamic time warping to identify human–robot demonstration correspondences with consistent interaction dynamics and uses these correspondences to guide soft optimal transport alignment in a shared policy representation space. This enables human demonstrations to provide contact-rich supervision for robot policy learning without requiring predefined frame-level human–robot pairing. Across four real-world dexterous manipulation tasks, TacOT improves closed-loop success rates over action-guided OT by up to 17 points on in-distribution tasks and 20 points under targeted human-to-robot out-of-distribution transfer. Further analyses show that tactile-guided correspondence selects demonstration pairs with more consistent contact dynamics and produces latent representations that better reflect interaction-state evolution. These results demonstrate that tactile dynamics provide an effective semantic signal for establishing reliable human-to-robot correspondence in contact-rich dexterous manipulation.

We classify the linear solution spaces of the chordal Belavin–Polyakov–Zamolodchikov (BPZ) equations, which arise in Dubédat's commutation relations for multiple Schramm–Loewner evolutions (SLE). Without imposing growth conditions, we determine the subspaces selected by successive conformal Ward identities and their exact dimensions. For every $\kappa>0$ and BPZ spectral parameter $\lambda\in\mathbb{R}$, we determine the spectrum and Jordan structure of the translation operator. When $\lambda=0$ and $\kappa\in(0,8)$, we further determine all admissible scaling exponents and the dimensions of the corresponding translation-invariant solution spaces. A triangular change of derivative coordinates yields a first-order system of rational Knizhnik–Zamolodchikov type, allowing us to represent the Ward operators explicitly and reduce the classification to finite-dimensional linear algebra. For $2N$ boundary points and $\kappa\in(0,8)$, translation-invariant BPZ solutions with $\lambda=0$ and the homogeneity required by Möbius covariance automatically satisfy the third Ward identity and the usual power-law bound. This recovers, under weaker assumptions, the Catalan dimension formula previously established by Flores and Kleban. At $\kappa=8$, the third Ward identity imposes an additional constraint when $N\ge2$, but the full Möbius-covariant solution space still has Catalan dimension, establishing completeness of the partition functions constructed from uniform spanning trees. Finally, at $\kappa=4$ and $\lambda>0$, we construct an explicit basis of positive BPZ solutions and identify its elements as partition functions for level lines of a Gaussian free field with suitable harmonic means.

Multi-modal object re-identification exploits complementary RGB, near-infrared (NIR), and thermal-infrared (TIR) observations to retrieve target objects. However, existing methods commonly employ visual encoders optimized for global image-text alignment and select tokens using learned importance scores. Such designs fail to preserve fine-grained identity cues or explicitly account for token redundancy, resulting in underrepresented local evidence and duplicated tokens that lead to noisy and costly cross-modal interaction. To address this gap, we propose TRIM-ReID, a compact framework that unifies dense feature extraction, intra-modal token reduction, and inter-modal aligned interaction. Specifically, semantically rich and spatially coherent patch features are extracted by Dense Identity Representation (DIR), which leverages DINOv3 to preserve fine-grained identity information. We then introduce Token Diversity Mining (TDM) to identify complementary local evidence and construct compact modality-specific token sets by suppressing repetitive patches while preserving informative diversity. Retained tokens are subsequently fused by Modal Relational Interaction (MRI) to enable effective information exchange across modalities, while a triangular alignment loss explicitly regularizes their joint relationships to maintain cross-modal semantic consistency under independent token selection. Extensive experiments on RGBNT201, RGBNT100, and MSVR310 demonstrate that TRIM-ReID achieves state-of-the-art performance.

Static spherically symmetric asymptotically flat configurations in General Relativity with a minimally coupled scalar field typically possess naked singularities. We study a linear stability of these configurations against radial (monopole) perturbations in case of power-law scalar field potentials. We employ a new shooting method that extracts the eigenvalues corresponding to the exponentially growing modes of the perturbation. The problem is reduced to one second order “master” equation in a divergent form with a regularized effective potential. We confirm our previous results showing that instability modes exist for sufficiently small scalar charges of the initial static configurations.

Quantum networks will rely on intermediate nodes to distribute entanglement, but the same nodes may learn which users are being connected. We introduce a minimal two-call routing primitive in which a user, Alice, establishes an entangled link with one of several clients while testing whether the provider, Bob, has disturbed a coherently encoded request. The construction adapts the hidden-order mechanism of quantum private queries: one call contains a direct request, the other a coherent superposition of the desired and null addresses, and Alice keeps their order secret. A private matching qubit held by the selected client restores the coherent test state after a favorable Bell-measurement outcome, so that Alice can test the returned query without any further operation by Bob. The claim is deliberately limited. The protocol shows that a routing request can carry a physical cheating test, and it detects an explicit destination-reading attack with nonzero probability. It does not yet provide a quantitative information-disturbance tradeoff, composable security, or certification of an unmeasured output against arbitrary attacks. We also characterize the exact perfect-pass limit, in which Bob's residual state, averaged over the private matching states, is independent of the destination.

Abridged. We aim to extend Paper I to the eccentric secondary orbit case and now analyse the elliptic restricted three-body problem (ERTBP). Our secondary masses include Ceres, Mercury, Earth, and Neptune so that we can help infer the presence of both large asteroids and planets through our analysis. We study dust particles on both planar (2D-ERTBP) and non-coplanar (3D-ERTBP) orbits. We computed families of 1/1 resonant symmetric and asymmetric periodic orbits in the 2D-ERTBP and 3D-ERTBP. Doing so produces novel results in their own right and diagnostic tools for precise guidance for our next steps. Based on them, we generated detailed phase portraits. Finally, we run suitably seeded $N$-body simulations to compare with the periodic orbits, and to provide observational links with debris close to WDs. New families of asymmetric periodic orbits, called $E^A_{41}$, $E^A_{42}$, and $E^A_{43}$ in the high-eccentricity 2D-ERTBP regime were discovered. They are generated by the new bifurcation point, $B_T^4$, along the new $A_2$ family, showcased in Paper I. Moreover, new isolated families of asymmetric periodic orbits, namely $E^A_{44}$, $E^A_{45}$, and $E^A_{46}$ were discovered. Families of short- and long- period asymmetric orbits starting from $L_4$, called $E^A_S$ and $E^A_L$, were also computed. In both the 2D- and 3D-ERTBP regimes, highly eccentric motion is exhibited for the dust particles in the majority of the families of either symmetric or asymmetric periodic orbits. The $N$-body simulations reveal that over ten years, the orbital period deviations of the dust are higher than in the circular restricted three-body problem (Paper I); in some architectures, Mercury-mass planets can regularly generate deviations exceeding $\sim$100 s, which potentially aids observational inferences of hidden asteroids or planets orbiting WDs.

We give checkable positive-definiteness criteria for the limiting neural tangent kernel (NTK) and high-probability finite-width gradient-descent guarantees for scalar- or vector-valued physics-informed neural networks (PINNs) with linear constraints. The constraints may be strong-form differential rows of any fixed finite order, including coupled systems, with any linear initial or boundary conditions; weak-form residual and boundary functionals; or finite-measure nonlocal observations such as integral, nonlocal-diffusion, and Dirac-data rows, all in any dimension. For each class, positive definiteness of the limiting NTK is equivalent to a rank condition on a finite coefficient, functional, or moment matrix of the fixed design: a certificate computed before training that detects structural zero modes. The model hypotheses are those of an ordinary two-layer PINN, a smooth nonpolynomial activation with bounded symmetric initialization, and are met by standard choices such as $\tanh$ with uniform initialization. Given a certificate, explicit width and step-size conditions ensure, with high probability, that the empirical NTK retains at least half the limiting gap and that full-batch gradient descent decreases the training loss geometrically at every iteration. Controlled experiments check the certificates and the finite-width mechanism.

Comprehensive 3D scene understanding for autonomous driving requires modeling geometry, semantics, and motion. However, camera-based occupancy and scene flow prediction are sensitive to unreliable spatial and temporal aggregation, caused by semantically incompatible image features, misaligned historical observations, and incomplete voxel structures. To address this issue, we propose SelectOccFlow, a selective spatiotemporal aggregation framework that progressively refines contextual evidence across image, temporal, and voxel domains. To obtain semantically compatible image evidence, we design Semantic-Guided Sampling (SGS) to regulate feature sampling with semantic priors. Since reliable image evidence alone cannot resolve temporal inconsistency, we then present State-Conditioned Temporal Aggregation (SCTA) to selectively retrieve historical evidence according to voxel states. To further enhance the structural completeness of voxel representations, we introduce Extent-Aware Spatial Aggregation (ESA), which exploits directional structural support to refine foreground geometry. Experiments on OpenOcc demonstrate that SelectOccFlow achieves a state-of-the-art OccScore of 44.9, improving the previous best by +4.2%. It also maintains competitive occupancy performance on Occ3D-nus and improves the mean OccScore under nuScenes-C corruptions by +11.1%, demonstrating improved robustness to visual corruptions. The source code will be made publicly available at https://github.com/muchen1021/SelectOccFlow.

In this paper we devise and analyze Lie Trotter splitting schemes for non-autonomous evolution equations, following two approaches: 1) A fully non-autonomous scheme, motivated by the potential use of explicit solutions of some subproblems. 2) A splitting scheme relying on the concept of Magnus integrators, where we freeze the operators at the beginning of each time interval as this suffices for first order convergence of the Lie-Trotter integrator. We derive first order convergence in a general setting and apply the analysis to the Bloch-Torrey equation of magnetic resonance imaging, where a decomposition into three subproblems is well-motivated by the physical role of the underlying operators. Numerical tests, complementing the time stepping scheme by a finite element space discretization, illustrate the theoretical findings.