Papers

We classify all diffeomorphism-invariant theories of gravity which possess traced field equations of motion of second order in derivatives. We also prove in the affirmative a conjecture regarding the form of theories constructed algebraically from the Riemann tensor with second-order traced field equations and classify all polynomial quasi-topological gravities with second-order traced field equations.

Neural network-based predictive modeling with high-dimensional structured Gaussian targets requires an efficient and numerically stable, yet expressive approximation of the covariance matrix. We propose SCORE: a scalable framework, combining scoring rule training with an expressive covariance approximation learned in spectral space. For $d$-dimensional data, the learning task is decomposed into learning the marginal distributions and learning a structured correlation matrix, which enables dense dependencies with linear storage and $\mathcal{O}(d\log d)$ cost. We utilize the closed form Gaussian kernel score for training, which remains defined even for degenerate covariances and admits bounded gradients during optimization. We characterize kernel scores under invertible transforms and prove exact invariance under unitary transforms. At population level, our two-level objective recovers the true marginals and projects the target correlation onto the representable class; finite-sample PAC bounds show that the errors of the two stages enter additively. We evaluate our model on a variety of tasks with a commonly assumed Gaussian domain: Time-series forecasting, monocular depth estimation, and spatial weather prediction, showing improved performance at lower computational cost.

Few-shot learning aims to recognize novel categories from limited labeled examples. Recent studies incorporate textual semantics to compensate for limited visual observations and improve class representations. However, high image-text agreement may reflect both intrinsic object properties and incidental context, making support prototypes susceptible to contextual contamination. To address this problem, we propose DVLA-RL++, which extends DVLA-RL with complementary semantic purification (CSP) and counterfactual reinforcement-learning gating (CRG). Specifically, CSP generates intrinsic and nuisance descriptions from labeled supports and compares their agreement with each support token. An ambiguity-dependent rejection margin guides sparse evidence allocation, while an intrinsic semantic anchor fills the unassigned mass to provide a fallback when visual evidence is unreliable. CRG learns layer-wise semantic fusion strengths using a reward that balances recognition performance and nuisance exposure. An independently executed reference trajectory on the same episode provides a paired learning signal. Theoretical analysis relates retained evidence and anchor quality to prototype stability and establishes conditions for unbiased on-policy gradient estimation. Experiments on standard, fine-grained, and cross-domain benchmarks show state-of-the-art accuracy, with an average gain of 1.4% over DVLA-RL. The project page is available at https://peacelwh.github.io/TPAMI27-DVLA-RLpp/.

Particle filters are a standard tool for nonlinear state estimation, but their resampling step is discrete, preventing gradient-based learning in variational sequential Monte Carlo. We introduce Differentiable Systematic Resampling (DSR), a temperature-controlled relaxation of systematic resampling, that preserves the CDF-ordered, banded structure of systematic resampling while enabling full gradient flow. DSR converges to exact systematic resampling as the temperature vanishes, and we prove a pointwise exponential convergence rate for the induced bias. Compared to optimal-transport-based differentiable resampling, DSR avoids iterative solvers and has substantially lower computational overhead. Experiments on stochastic dynamical systems and real-world handwriting data show that DSR achieves comparable or superior filtering and dynamics learning performance.

We establish a conditional theory of genuinely non-perturbative global dynamics for the cutoff Boltzmann equation with hard potentials on the three-dimensional torus. We prove that any global mild solution satisfying \[ \sup_{t\ge0}\int_{\mathbb R^3}W(v)\sup_{x\in\mathbb T^3}F(t,x,v)\,dv<\infty, \] for a suitable stretched-exponential weight \(W\), enjoys uniform weighted \(L^\infty\) control, is unique in the dissipative mild class, and converges exponentially to the global Maxwellian. In particular, these conclusions hold for arbitrary-amplitude solutions admitting a uniform stretched-exponential velocity upper bound. The key mechanism is a *tail-to-coercivity principle*: uniform velocity-tail control generates positive lower bounds for the local density and nonlinear collision frequency, without imposing macroscopic coercivity a priori. A double-Duhamel positivity argument also generates a Gaussian lower bound at arbitrarily short positive times, allowing initial data with local vacuum. We establish a polynomially weighted counterpart under an additional spatially uniform lower velocity profile, obtaining analogous weighted \(L^\infty\) bounds, uniqueness, and exponential relaxation. Our results provide a low-regularity alternative to conditional theories requiring smoothness and pointwise macroscopic bounds. They reveal how a single velocity-tail condition controls large-data Boltzmann dynamics and suggest a route toward problems with physical boundary conditions.

Using skein-theoretic methods, we study the SL2(C) character variety of the five-punctured sphere with arbitrary conjugacy class at each of the five punctures. We show that a Fricke-Klein-Vogt-type relation is realized as a symmetric determinantal hypersurface. We also give a realization of the character variety in terms of cluster variables. We explicitly derive the Poisson structure of the simple closed curves on the surface, and prove that cluster mutations induce automorphisms of the character variety.

We study the kernels of Lefschetz operators arising from complete intersections of nef classes. When the classes are semiample, we provide a complete characterization of mixed Lefschetz properties below a fixed degree – all obstructions are detected by irreducible subvarieties of relevant codimensions. We further show that the kernels in the next degree range vanish off the diagonal, while on the diagonal they admit bases consisting of fundamental classes of irreducible subvarieties annihilated by the Lefschetz operator. We also establish a kernel decoupling theorem for nef classes on complex Lefschetz modules. As an application, we obtain a kernel characterization for supercritical collections of nef classes, thereby proving a conjecture of Shenfeld-van Handel and Hu-Xiao.

Latent reasoning enables vision-language-action (VLA) models to transform multimodal observations into task-relevant internal states before generating continuous robot actions. While existing methods learn to generate or refine such states for each policy query, they discard successful reasoning after execution and therefore reconstruct similar computation from scratch. We present Reasoning and Flow Memory (FLOWMEM), a unified VLA model that turns successful latent computation into reusable reasoning experience. Rather than appending a fixed retrieved context, FLOWMEM dynamically retrieves and recomposes compatible latent fragments as the embodied context evolves, forming a reasoning route that follows the temporal structure and progress of successful computation. The route is then refined using current visual and proprioceptive evidence before it conditions action generation. Experiments on RoboMME and LIBERO-Plus show that FLOWMEM attains 48.0% and 77.3% success, outperforming memory-free policies by 1.7 and 4.1 percentage points, respectively. These results demonstrate the value of reusing successful latent computation for closed-loop VLA control.

Long-horizon mobile manipulation requires a robot to navigate multi-room environments and execute a sequence of manipulation skills under a single natural language instruction. Learning and evaluating these skills present three challenges: similar observations under a fixed task instruction may make skill selection ambiguous; even when a preceding skill succeeds, the robot state inherited by the next skill may deviate from its demonstrated starting states and affect execution; and task-level metrics hinder skill-specific diagnosis, while early failures leave later skills untested. We therefore introduce ManiUnit, a manipulation skill dataset and benchmark built from 50 BEHAVIOR-1K activities. Its dataset contains 137,899 segments across 21 skill types and 417 subtasks, and its benchmark contains 1,260 test instances. Correspondingly, ManiUnit pairs each segment with an explicit subtask instruction; measures sensitivity to perturbations of the robot's starting base position or joint configuration; and restores intermediate simulator states and defines local success conditions so that each skill can be evaluated without executing preceding stages. Evaluations of representative vision-language-action (VLA) policies show that similar aggregate scores can hide substantial per-skill differences. The tested starting-state perturbations also degrade execution: on the full benchmark, joint perturbations reduce success rates by approximately 56% relative to those from demonstrated starting states. On two long-horizon activities, a skill policy trained on ManiUnit segments achieves 78.7% local manipulation success, compared with 49.3% for a task policy trained on complete demonstrations. The trained skills further support complete-task execution on these activities, as coordinating the task and skill policies through a planner raises full-task success from 4.0% to 18.0%.

Finite element simulations carried out on meshes issued from real-world geometries - patient-specific anatomies, scanned industrial parts, fractured media - almost never satisfy the shape-regularity assumptions under which classical a posteriori error analysis is written. Practitioners are therefore left without a quantitative answer to a very legitimate question: how wrong is my simulation when my mesh is coarse and contains badly shaped elements? In this work, we study to which extent equilibrated flux estimators can provide such an answer. Indeed, they deliver a guaranteed and fully computable upper bound of the energy-norm discretization error, with a constant equal to one; and, crucially for the present study, its derivation uses no shape-regularity, quasi-uniformity or angle condition. The only geometric ingredient is the Poincaré constant on a single cell, which Payne and Weinberger showed to be at most hK /pi on any convex cell K, however distorted (hK being the cell size). Mesh quality affects the local efficiency - the sharpness of the estimator - but never its reliability, so that the failure mode under distortion is pessimism rather than false confidence.

Let $S$ denote a $k$-subset of $\mathbb{Z}$. Let $G$ be a (multiplicative) group of size $k$. Assume that there exists a function $\varphi\colon S \to G$ that is injective and that satisfies the following: If $x, y, z \in S$ and $z = xy$, then $\varphi(z) = \varphi(x) \varphi(y)$. In 1990, Forcade and Pollington conjectured that if $S = \{ 1, 2, \ldots, k \}$, then $G$ is abelian. This has been a long-standing conjecture highlighted by Richard K. Guy as an unsolved problem, and this problem appears to have remained open, with its open status having been noted by Caicedo et al. in 2021 [Electron. J. Combin.]. We succeed in proving the Forcade-Pollington conjecture in the affirmative and in full generality.

LLM-based program evolution relies on evaluation feedback to guide the iterative search for high-performing programs. However, evaluation is often computationally expensive, making it essential to allocate limited resources to candidates that can most effectively advance the search. Existing LLM-based methods typically rely on fixed allocation strategies throughout the search, potentially wasting resources on low-value candidates while overlooking promising ones. We propose EvoAlloc, a self-evolving resource-allocation agent that learns from search experience to revise its strategy for allocating computational resources across candidates. EvoAlloc periodically consolidates prior search and allocation outcomes into reusable experience, which informs subsequent strategy revisions. It further uses a counterfactual exploration mechanism to occasionally evaluate candidates denied resources by the allocator, revealing their outcomes to enrich its experience for future strategy updates. Across coding and agent-harness optimization benchmarks, EvoAlloc requires 59-82% fewer full evaluations and 61-89% fewer total LLM tokens to reach baseline-level performance. Moreover, under the same full-evaluation budget, EvoAlloc achieves 8.7-12.0% higher final performance.

Large language models (LLMs) can expose memorized training sequences under prefix-based extraction: given a prefix from a training example, the model may assign high probability to the original continuation. In deployed systems, however, prefixes are rarely evaluated in isolation. They often appear together with instructions, retrieved documents, or other task-specific context, as in retrieval-augmented generation (RAG). This motivates examining whether contextual conditioning mitigates memorization or merely changes the set of memorized samples that become extractable. We investigate this issue through paired item-level measurements of probabilistic suffix extraction. For each prefix-suffix pair, we score the target suffix under an empty prompt and under retrieved contexts of varying relevance, across three open-weight instruction-tuned models. We find that context does not simply erase memorization. Instead, extractable memorization consists of a context-robust core and a context-sensitive boundary. Many samples that are extractable without context remain extractable under the retrieved context, especially as the prefix length increases. At the same time, context mainly affects marginal samples near the extraction threshold: it suppresses some exposures, but also enables new ones that are missed by prefix-only evaluation. These findings qualify the view that RAG reduces memorization risk. Context can lower aggregate extraction by suppressing boundary cases, yet robustly extractable samples persist, and context-enabled extractability remains security-relevant.

The first demonstration of 4D vertex reconstruction at the $50\,\mathrm{ps}$ scale using a silicon pixel detector is reported. Hadronic interactions are produced by directing a $180\,\mathrm{GeV}/c$ mixed-hadron beam onto a fixed target upstream of the Timepix4 beam telescope at the CERN SPS. The resulting particles are reconstructed using 4D track and vertex fits that include time directly alongside the spatial parameters. The vertex time resolution follows the expected statistical scaling with the number of timing measurements, reaching $54\pm5\,\mathrm{ps}$ for vertices reconstructed from 16 timing-plane clusters. An irreducible component of $29.2\pm0.6\,\mathrm{ps}$ remains in the fitted resolution model, including sizeable contributions from the clock distribution and the time reference system rather than from the combination of the timing measurements. The observed scaling down to $54\,\mathrm{ps}$ without an unforeseen resolution floor above represents a significant experimental step towards precision-timing in future high-pile-up detectors.

LLM-as-a-Judge is the standard paradigm for NLP evaluation, yet its systemic reliability remains poorly understood despite being widely treated as a deterministic ground truth. We present a comprehensive reliability audit, stresstesting six frontier models across four benchmarks, five prompt formats, two presentation orders, three sampling temperatures, and ten repetitions per condition. Our empirical analysis reveals severe vulnerabilities: verdicts change across identical replications at temperature zero, position-order swaps flip the majority of verdicts on challenging tasks, and the most deterministic judge achieves perfect consistency by trivially repeating incorrect verdicts, agreeing with ground truth only 51% of the time. To formalize these multi-faceted failure modes, we introduce the trustworthy verdict rate (T ), a unified metric capturing the joint probability that an evaluation is reproducible, order-invariant, and accurate. UsingT , we derive a theoretical upper bound on accuracy imposed by position bias and show that reliability is item-specific rather than modellevel. Finally, we demonstrate that shifting from pairwise win-rate to holistic rubric scoring improves trustworthiness more than any single-format prompting intervention, offering an actionable framework for robust NLP evaluation.

A massless particle on the circular null geodesic of a Schwarzschild or Kerr black hole radiates gravitational waves at all multipoles $\ell$ with a power that falls off only as $1/\ell$, so that the total instantaneous power is logarithmically divergent. We derive this result analytically from the Zerilli–Regge–Wheeler and Teukolsky equations with a null stress-energy tensor as source, and find that both the power radiated to infinity and the power absorbed by the horizon approach $\dot{E}_\ell\to g(a) \kappa E^2/(M^2\ell)$, where $E$ is the energy of the particle, $M$ the mass of the black hole and $a$ its spin, with $\kappa=0.063653$ and a spin factor $g(a)$, for which we obtain, for any spin, the closed form $g=27M^2(r_0-M)/[r_0^2(r_0+3M)]$ in terms of the radius $r_0$ of the Kerr light ring. The $1/\ell$ behavior has two causes: the near-resonance of the photon with the quasinormal modes, which prevents any exponential suppression at large $\ell$, and a cancellation in the source that is specific to massless particles, which reduces the power per multipole by a factor $\ell^2$ with respect to a generic source. At leading order half of the radiation falls into the black hole, and we compute the first correction to this equal split in the non-spinning case. The same analysis describes massive particles on orbits close to the light ring, where the mass turns the $1/\ell$ law into an exponential cut-off. We recover the dominant, even-parity term of the geodesic synchrotron radiation formulae for massive particles of Breuer, Ruffini, Tiomno and Vishveshwara (1973), but we find that their odd-parity term is too small by a factor of four. All results are checked against numerical solutions, up to $\ell=12800$ in Schwarzschild and $\ell=800$ in Kerr.

Continuing the Perception Test challenge series, we organised the fourth edition as a workshop at the European Conference on Computer Vision (ECCV) 2026 in Malmö, Sweden. This edition focused on spatial intelligence and featured four different tracks: unified multiple-choice videoQA and grounded videoQA from the original Perception Test benchmark, alongside two new tracks based on city-scale walking-tour videos (KilometerAudio and KilometerVision). In this report, we describe the new benchmarks used for the city-scale tracks and summarise the winning solutions across all tracks, including a generalist model that competed across all tracks with satisfactory performance. The winning solutions in the newly added city-scale tracks demonstrated that complex spatial and multimodal reasoning can be solved by expensive agentic pipelines, but remains difficult for multimodal models used standalone.

Let $p>3$ be a prime number. If $A$ and $B$ are two principally polarized superspecial abelian surfaces over $\mathbb{F}_{p^{2}}$ with $p^{2}$-Frobenius $[-p]$, we prove that there exists a separable polarized isogeny between them with multiplier at most $p/\sqrt{2}$. The multiplier bound is uniform in the pair and asymptotically optimal up to a small multiplicative constant. Given quaternionic coordinates, we provide a deterministic algorithm that satisfies this upper bound in polynomial time. For unrestricted multipliers, $\mathrm{KLPT}^{2}$ provides a heuristic algorithm with a multiplier upper bound of order $p^{6+o(1)}$ for arbitrary pairs and $p^{3+o(1)}$ when one polarization matrix is the identity matrix. Our algorithm's improvement is an explicit upper bound below $p$ with an unconditional deterministic guarantee.

In this work, we propose a mathematical description of the critical point of boundary phase transitions in 2+1D topological orders. Given two boundary phases, $\mathcal{C}_A$ and $\mathcal{C}_B$, with corresponding Lagrangian algebras $A$ and $B$ in a 2+1D topological order $\mathcal{C}$, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to $D = A \times_{M} B$, which is the fiber product of $A$ and $B$ over $A$-$B$-algebra $M$. We prove that $D$ is also a condensable algebra. Additionally, we present an alternative classification of condensable algebras in $Z(Vec_G^\omega)$ and reveal splitting phenomenon that arises during the condensation of certain algebras. We also perform explicit computations for the condensable algebras within several examples of $Z(Vec_G^\omega)$, covering both Abelian and non-Abelian groups $G$.

The locus $S_d(X)$ of $d$-secant lines to a projective variety $X$ plays an important role in the study of projective varieties via projections (Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010). For a projective scheme $X\subseteq\mathbb{P}^r$ satisfying $\textbf{N}_{d,2}$, we show that $S_d(X)\cup X$ is cut out set-theoretically by the $r$-minors of a matrix $M$ constructed from $d$-forms scheme-theoretically defining $X$ and their linear syzygies. In particular, $S_d(X)=\mathbb{P}^r$ if and only if every $r$-minor of $M$ vanishes. Using this, we find a singular threefold $X\subseteq\mathbb{P}^5$ with $S_4(X)\ne \mathbb{P}^5$ and $(I_X)_3=0$, and a normal fourfold $X\subseteq\mathbb{P}^7$ with $S_3(X)\ne \mathbb{P}^7$ and $(I_X)_2=0$. The nonexistence of such varieties in the smooth case is an open question raised by the second author and by Gruson-Peskine. Next, we consider the locus $S_{k,d}(X)$ of $k$-planes $L$ such that $L\cap X$ contains a hypersurface of degree $\ge d$ in $L$ so that $S_{1,d}(X)=S_d(X)$. The locus $S_{k,d}(X)\cup X$ is set-theoretically defined by the $(r+1-k)$-minors of $M$. As a result, we obtain determinantal equations vanishing on the $(q+1)$-secant variety $\sigma_{q+1}X$ from the equations of $\sigma_qX$ and their linear syzygies when $\sigma_qX$ satisfies $\textbf{N}_{q+1,2}$. Finally, we extend the $d$-secant lemma for lines to the locus $S_{k,d}(X)$ of $k$-planes. This yields a lower bound on the number of linear syzygies.

Helium nanodroplets are ideal testbeds for elucidating secondary processes induced by ionizing radiation in condensed matter. Here we study photoionization electron spectra of large He nanodroplets in the extreme ultraviolet region, where photoelectrons undergo multiple inelastic collisions with He atoms in the nanodroplets. The resulting characteristic electron energy-loss features are identified by their dependence on photon energy and nanodroplet size and by comparison with a simple numerical simulation. Conditions for minimizing electron scattering with the He nanodroplets are discussed.

Simulators based on differential equations are ubiquitous in science and engineering. They are often used in simulation-based inference to evaluate the posterior distribution of the input parameters based on real-world observations of the simulator outputs. However, inference becomes challenging when individual simulator evaluations are computationally expensive. In such cases, a Bayesian optimization-based active learning approach with Gaussian process surrogate models has been used to maximize the information obtained from a limited simulation budget. Recently, gradients of simulator outputs with respect to input parameters have become increasingly available, yet they are rarely exploited for inference. Even though we only need to learn the simulator input-output relationship, gradient information can provide an additional valuable signal to guide the active learning procedure. This is of particular interest in the case of expensive simulators, when sample efficiency is crucial. In this paper, we demonstrate how incorporating gradient information into the Gaussian process surrogate accelerates Bayesian optimization-based inference under a limited simulation budget. Our results show significant improvement in convergence speed from using gradient information. For reverse-mode differentiation, the inference efficiency gains are maintained when accounting for the additional computational cost. In contrast, for forward-mode differentiation, the inference speed-up does not outweigh the computational costs. These results indicate that gradient-enhanced surrogates are beneficial primarily in problems where the number of parameters exceeds the output dimensionality, where reverse-mode differentiation is efficient.

Resolving how multiple superconducting energy scales evolve under magnetic field is important for distinguishing multiband, anisotropic, and surface-sensitive superconducting responses. Here, we investigate the superconducting state of PdTe using low-temperature soft point-contact spectroscopy in a sealed $^3$He immersion environment. The resulting Andreev-reflection spectra reveal two characteristic superconducting energy scales near 0.5 and 1.1 meV that are not adequately described by a single-gap BTK model. Both energy scales are progressively suppressed upon warming toward the same superconducting transition temperature $T_c$. In contrast, their magnetic-field evolution reveals two distinct characteristic fields near 4.5 and 7.5 kG. The lower scale is associated with the loss of the zero-resistance superconducting state, whereas the higher-energy spectroscopic response persists to substantially larger fields. Independent field-dependent gap analysis yields suppression scales consistent with those identified from the zero-bias conductance. These results reveal a field-resolved hierarchy of superconducting energy scales in PdTe and indicate superconducting responses with distinct magnetic-field robustness.

Randomized-withdrawal trials re-randomize induction responders to continued treatment or withdrawal. Baseline characteristics are often reported for the induction population but unavailable for responders, limiting population-adjusted indirect comparisons of maintenance outcomes. We describe re-randomization-probability weighting, which combines induction-population calibration with inverse weighting by known maintenance assignment probabilities. Coding induction non-response as failure defines durable response: response at both induction and maintenance. The design weights reproduce induction-population covariate totals in randomization expectation, allowing population adjustment using reported induction baselines. Conditional maintenance probabilities can also be estimated, although different induction treatments generate different responder populations. A synthetic three-trial demonstration reduced durable-response bias from 0.106 to 0.001 and from 0.055 to 0.001 in two comparisons. Full-procedure simulations showed interval coverage of 91.5% - 94.3%; stress scenarios illustrated bias from omitted covariates and incorrect assignment weights. The method connects established design weighting and aggregate dilution approaches to population-adjusted comparison across randomized-withdrawal and treat-through designs. It requires compatible durable-response endpoints and credible transport and modeling assumptions. Withdrawal after different induction regimens is not automatically a common comparator. Where no valid anchor exists, unanchored comparison is appropriate only if its demanding adjustment assumptions are defensible.

Vision-based tactile sensors provide rich contact information, but processing high-resolution images can be costly for resource-constrained platforms such as space robots. This work investigates whether a compact representation of tactile motion can classify object rotation across different gravity conditions. Dense optical flow from a simulated GelSight Mini is aggregated over a 7x9 grid into 126 features and used to classify the direction of load-induced rotation under Earth, Mars, Moon, and orbital gravity. Gravity causes a small but significant shift in these features, accounting for 1.6% of their variance (R2 = 0.016). Despite its small magnitude, this shift affects models trained only on Earth data: XGBoost accuracy decreases from 94.4% on Earth to 75.9% in orbit. In contrast, a single model trained across all four gravity domains achieves 96.3% overall accuracy and 95.1%-97.0% across individual domains, without using gravity as an input. The representation can also be reduced to 40 features while retaining 95.7% accuracy, with XGBoost requiring only 0.14 ms per inference. These findings show that Earth-gravity performance alone is insufficient to establish the transferability of tactile perception for space robotic manipulation, highlighting the need to account for gravity-induced domain shifts during training and validation.

For every quartic Salem number $\beta$, we prove that the equal-weight Bernoulli convolution $\nu_{\beta^{-1}}$ has Hausdorff dimension strictly less than one. The main step converts large Fourier coefficients at geometrically separated frequencies into a deficit in the Shannon entropy of a finite convolution. A positive trigonometric product gives the entropy estimate, while the algebraic norm controls the number of distinct atoms in each spatial cell. For the two quartic Salem numbers in $(1,2)$, reciprocal differences at suitable return times produce the required Fourier coefficients. A finite block cover then gives the dimension bound. This proof is independent of numerical computation. Separate exact calculations give $\dimH\nu_{\beta_1^{-1}}<1-1.4\cdot10^{-23}$ and $\dimH\nu_{\beta_2^{-1}}<1-7.8\cdot10^{-22}$, where $\beta_1<\beta_2$ are the two parameters in $(1,2)$. The accompanying verification programs use rational interval arithmetic and the Python standard library.

It was known that the canonical bundle of a smooth toroidal compactification of a noncompact finite-volume ball quotient is nef in dimension $n\geq3$ and ample in dimension $n\geq6$. We prove that the canonical bundle is ample in dimension $n\geq4$ and show that this dimension bound is sharp. We construct infinitely many pairwise noncommensurable noncompact Picard modular threefolds whose smooth projective toroidal compactifications have non-ample canonical bundle. Each compactification contains a smooth rational curve meeting the interior and having canonical degree zero. More generally, in every dimension $n\geq2$, we construct infinitely many pairwise noncommensurable noncompact Picard modular varieties whose smooth toroidal compactifications contain smooth rational curves meeting the interior. In particular, such curves persist in arbitrarily high dimension, even when the canonical bundle is ample.

Developing general-purpose numerical methods for non-Markovian quantum state diffusion has long been a challenge in the simulation of open quantum systems. This paper proposes a deterministic polynomial chaos Galerkin method, which is easy to implement, accurate, and efficient. By combining a low-rank decomposition of bath correlations with Galerkin projection onto a polynomial chaos basis, the method yields a deterministic formulation compatible with standard numerical solvers for partial differential equations. The reduced density matrix is obtained from a single deterministic solve at a cost comparable to that of computing one stochastic hierarchy trajectory. Replacing an ensemble of $N_{\mathrm{traj}}$ trajectories therefore reduces the computational cost by approximately a factor of $N_{\mathrm{traj}}$ and eliminates statistical sampling error. Benchmark comparisons show that the proposed method achieves smaller errors at lower computational cost than the tested stochastic hierarchy methods. Applications involving nonquadratic potentials, diffraction and interference, and coupled three-dimensional dynamics demonstrate its ability to handle challenging problems.

Realistic household simulation must capture not only diverse environments but also the lived-in object arrangements and spatial constraints that shape robot motion and interaction. Existing resources often trade off scale, real-world correspondence, and interaction readiness, leaving a gap in faithful, interactive replicas of how real homes are actually arranged. To this end, we introduce LIVIN, a benchmark for spatial and embodied intelligence built on digital twins of 30 diverse lived-in homes. These replicas preserve observed room layouts, furniture configurations, and everyday belongings. To construct them, we design a human-in-the-loop workflow comprising instance recognition, architectural reconstruction, and object generation and placement, with intermediate results reviewed and corrected by humans against the source observations at each stage. We evaluate four tasks in LIVIN: 3D detection, 3D reconstruction, navigation, and loco-manipulation. Our evaluations show that current methods remain challenged by the dense object arrangements, occlusions, limited free space, and constrained interaction regions found in realistic lived-in homes. We hope LIVIN will help advance embodied AI in real-world homes, from spatial understanding to robotic interaction, and ultimately bring embodied intelligence into everyday home environments.

Multiple candidate flight paths between the same vertiports provide flexibility for managing noise exposure and cruise separation in on-demand urban air mobility (UAM). This paper introduces the fleet scheduling problem with alternative flight paths (FSP-AFP). We develop a time-expanded fleet-flow formulation for a finite aircraft fleet, in which each service or empty repositioning flight is assigned a planar path from a predefined set of alternatives connecting its origin and destination vertiports. These path choices are jointly optimized with cruise altitudes, departure times, and repositioning movements, while maintaining aircraft flow balance and enforcing geometric separation between flights cruising concurrently at the same altitude. The objective combines flight costs, service departure delays, and population-weighted noise exposure. A tailored branch-and-cut algorithm incorporates three complementary clique constructions based on a request-augmented conflict graph, exploiting physical flight conflicts, request exclusivity, and request-route relationships. Computational experiments show that, compared with a baseline implemented using Gurobi's native lazy-constraint functionality, the proposed algorithm generally obtains stronger lower bounds and finds feasible solutions for all test instances, while generating fewer cuts on average and achieving a 43.16-fold reduction in the mean number of processed search nodes. An operational case study shows that increasing the number of candidate planar routes per vertiport pair from one to four reduces the weighted objective by 6.71% and modeled noise exposure by 15.14% under the cruise-separation constraints. Further increases in the number of candidate routes yield only marginal improvement in the objective value. These findings show the value of coordinating alternative flight paths, scheduling, and fleet circulation in UAM.