Papers

Recent advances in Physical AI have accelerated the use of foundation models in autonomous systems such as unmanned aerial vehicles (UAVs), which must perceive, reason, plan, and act in dynamic environments. Existing benchmarks assess physical perception, intuitive physics, embodied navigation, and collaborative reasoning, but rarely evaluate the agentic decision-making required for reliable autonomy. We introduce PhysAI-Bench, a benchmark for evaluating this capability. It contains 10,178 standardized decision instances automatically extracted from conversational traces of autonomous UAV missions. Each instance preserves mission context, temporal dependencies, physical constraints, Model Context Protocol (MCP) tool calls, Agent-to-Agent (A2A) interactions, sensor observations, and AI-native 6G network conditions, including latency, packet loss, throughput, edge load, and network slicing. We expose only information preceding each decision, preventing future-event leakage and approximating online decision-making. We evaluate 29 foundation models using a two-stage protocol. We select model-specific configurations from 12 combinations of zero-, three-, and five-shot prompting and four temperatures, tested in three runs on a 35-instance, human-verified development set. We then freeze each selected configuration and evaluate it in three runs on a fixed, episode-disjoint set of 500 instances. GPT-5.3 achieves the highest accuracy (52.00%), followed by GPT-5.2 (49.40%) and Grok~4.5 (49.07%). Few-shot prompting generally improves performance, while temperature has limited influence. The results demonstrate that reliable agentic decision-making in Physical AI remains an open challenge. The dataset is available at https://github.com/maferrag/physai-bench

The well-known open problem of Bell and Skandera asks whether a real-rooted polynomial $f(t)$ with positive integer coefficients and constant term one is the $f$-polynomial of a simplicial complex. Mu and Welker proved that if the recursive decomposition $f(t)=g(t)+th(t)$ satisfies the corresponding coefficient inequality $h_i<g_i$, then this open problem has an affirmative answer. Mu and Welker also conjectured that the real-rootedness of $f(t)$ implies that of $g(t)$ and $h(t)$. We give counterexamples to the conjecture of Mu and Welker for every degree at least three, and prove that the assertion holds in degrees $1$ and $2$. Moreover, each polynomial we construct is the $f$-polynomial of a simplicial complex.

PASS provide a flexible waveguide-based architecture for reconfiguring wireless propagation environments and creating geometry-dependent radiating apertures. This paper investigates a near-field multicast ISAC system enabled by a lossy multi-waveguide PASS, where a base station transmits a common message to multiple communication users while simultaneously sensing one or multiple targets. The PA positions along the waveguides and the feed-domain transmit covariance matrix are jointly designed to improve sensing accuracy under multicast communication constraints. We first develop a near-field multicast ISAC signal model that accounts for waveguide attenuation, equal-radiated-power operation, geometry-dependent free-space propagation, and monostatic sensing. Then, we derive the FIM for target parameter estimation and obtain a compact projected-Jacobian representation by exploiting the block-diagonal PA transfer structure. This representation reveals how the PA geometry and transmit covariance jointly affect the CRB. Based on this structure, we further characterize the per-waveguide and cross-waveguide FIM contributions, the loss-aperture tradeoff, the identifiability condition, and the communication-sensing phase conflict. To minimize the CRB, we formulate a joint PA-position and transmit-covariance optimization problem subject to a multicast rate constraint, a feed-power budget, and PA deployment constraints. An alternating optimization algorithm is developed, where the covariance subproblem is solved as a semidefinite program and the PA-position subproblem is handled by waveguide-wise block coordinate descent.

Co-stabilization, i.e., designing a single controller that stabilizes multiple systems, is a fundamental problem in robust and data-driven control. While any stabilizable linear system admits a stabilizing linear static state feedback controller, this equivalence does not extend to co-stabilization. In particular, there exist system collections that cannot be co-stabilized by linear static state feedback but can be co-stabilized using linear dynamic state feedback. In this paper, we study the role of controller memory in co-stabilization. We show that linear dynamic state feedback strictly enlarges the set of co-stabilizable systems compared to static feedback, for both scalar systems and high-dimensional examples. At the same time, we identify structural limitations that cannot be overcome even with dynamic controllers. We also develop a path-integral-based algorithm for computing co-stabilizing controllers for a finite set of systems. Numerical results demonstrate that increasing controller memory enlarges the feasible co-stabilization region. These results highlight controller architecture as a key structural factor in co-stabilization, with potential implications for reducing the sample complexity of learning-based control.

The high-recoil event reported by LUX–ZEPLIN motivates a neutral electroweak state with an inelastic splitting in the sub-MeV range. We propose supersymmetric 331 model from the supersymmetric $SU(6)$ GUT model, and we study the electroweakino sector relevant to this recoil. Above the $331$-breaking scale hypercharge is not an independent gauge factor, and the leading bino–wino cancellation is replaced by a relation between the physical $U(1)_X$ and $SU(3)_L$ gaugino invariants, $I_X=-I_L$. A singlet–adjoint gauge-kinetic source also correlates the colored invariant. We diagonalize the neutral gauge–Higgs fermion sector of the renormalizable model and determine the tree-level Higgsino splitting without introducing an effective bino mass by hand. The required solutions form a continuous locus in the gaugino and $331$-breaking parameters. At the reference point, the solution lies close to the leading sum rule, while variation of the $331$-breaking coupling moves the locus continuously. The same calculation tests the ordering of the additional neutral states, the gauge/gaugino contribution to the $331$-breaking scalar mass, and the gravitino ordering. We compare this construction with recent $SU(5)$, singlino-assisted, electroweak-multiplet, nuclear-response and solar-capture analyses of the LZ event.

Kinetic Ising models are widely used to describe binary opinion dynamics, but their microscopic validity has rarely been tested empirically. Here, we infer the transition probabilities governing opinion updates from a year-long online social network and show that they are accurately described by an Ising heat-bath dynamics. The inferred parameters admit a direct sociological interpretation: the external field quantifies intrinsic bias, the coupling strength measures social influence, and a persistence term captures temporal inertia. We further show that persistence is positively correlated with node degree, while both persistence and interaction strength are strongly correlated with global network heterogeneity and clustering. Using the inferred time-dependent parameters in Monte Carlo simulations on the empirical temporal networks, we accurately reproduce the observed response functions, flip probabilities, and macroscopic opinion dynamics. Within the Twitter climate debate, our results provide direct empirical support for a kinetic Ising description of online opinion formation and establish a quantitative link between microscopic social behavior and evolving network topology.

A rigorous implementation of energy-dependent ion- and electron-induced electron emission in a continuum-kinetic framework is used to reveal their effects on the scaling of plasma properties in the sheath with an applied bias potential to the walls. The approach comes with a novel methodology for modeling particle-induced electron emission (PIEE) that includes 1) improved fitting functions for the yield and spectra, 2) the use of SRIM and a summation of the Lindhard formula and a modified Bethe formula for obtaining accurate stopping powers, and 3) a binding energy correction to the PIEE spectra and ion-induced yield based on density functional theory (DFT) calculations. The emission models are implemented as a fully energy dependent and dynamic boundary condition. For this investigation, tungsten and graphite walls are studied for their relevance in magnetic fusion experiments. Equations are derived from fluid theory that predict the relative importance of ion- and electron-induced emission on the structure of the sheath. The simulations provide evidence for the theoretical predictions, showing that a transition from a classical to space-charge limited (SCL) sheath depends primarily on electron-induced emission. Furthermore, claims in previous literature of increased heat and particle loads to the walls due to electron emission are supported, however differing mechanisms for the increase are observed. The increased thermal and particle fluxes due to PIEE are primarily driven by collisional transfer of energy from emitted electrons in the presheath. Finally, quantitative predictions for the device modeled in this study coincide with previous modeling efforts and experimental measurements.

Empirical ramp fitting can assign weight to pure-noise features even when the population optimum ignores them. We quantify this gap for norm-constrained adversarial classification with Gaussian signal and noise. The variance cost relative to normalized signed mean separates into two factors: selecting observations inside the active margin window and the curvature induced by the norm constraint. Changing the tail variance leaves the activewindow probability unchanged but changes the second factor. With positive attack budget and a signal-only predictor of risk below one half, we prove a uniform quadratic tail-deletion bound, including at zero tail variance. Sufficiently accurate approximate global empirical minimizers admit exact fixeddimensional asymptotic covariances in the low-risk regime with isotropic principal covariance. For positive tail variance at most principal variance, the product exceeds one; an additional moment condition transfers it to expected excess ramp and robust classification risks. A wide window analysis characterizes when this ordering reverses. Controlled experiments test the decomposition, and a separate contamination study examines its scope outside the Gaussian training model.

This paper introduces a multi-temporal tabular dataset derived from satellite images to map burned areas in the Chapada dos Veadeiros National Park, in Goiás, Brazil, covering the years 2020 to 2022. The dataset contains blue, green, red, and near-infrared bands, as well as the BAI, EVI, GEMI, NDVI, and NDWI spectral indices from the WFI sensor on the CBERS-4A, CBERS-4, and AMAZONIA-1 satellites, organized into a regular grid. We applied the Random Forest classifier to develop and validate models based on samples labeled as totally burned, partially burned, and non-burned. Two classification approaches were tested: one combining burned and non-burned areas into binary classes and another distinguishing between totally burned (TB), partially burned (PB), and non-burned (NB) classes. Seven validation approaches assessed different post-classification combinations, focusing on accuracy, precision, recall, and intersection over union (IoU) metrics. Results showed higher IoU when TB, PB, and NB were used as individual classes and TB was reclassified as burned area (BA) while PB and NB were grouped as non-burned. Comparing the annual results of this approach to the MCD64A1 product, the errors of omission for the BA class were 22% in 2020, 28% in 2021 and 59% in 2022, while the errors of commission were 46%, 43% and 46%, respectively. The study highlights the utility of the WFI sensor for burned area mapping without inter-satellite spectral calibration and suggests further exploration with other machine learning algorithms to evaluate the dataset potential and limitations.

Patch-based autoregressive time-series forecasting often ties input representation, learned transitions, and recursive execution to one patch length. We ask which of these roles can be adjusted separately. A supporting atomic-encoding study finds greater sensitivity to model width than to atom grouping on the evaluated grid. Our main finding is that a frozen parent's recursive trajectory is easier to fit than the observed future with lightweight parallel exits. Autoregressive Trajectory Distillation (ATD) turns this into selectable ATD-1/2/4/8 execution, with ATD-1 exactly recovering the parent. On a paired four-data-set comparison, ATD-8 reaches $5.54\times$ end-to-end speedup with stable quality across widths. Fewer calls do not automatically remove the parent's existing forecast error: ATD improves trajectory fidelity in all 21 seed runs but forecast accuracy in only 15 against matched clean-future supervision. We further find a correctable residual projection along a train-selected periodic history direction. Spectrum Tangent applies this correction without adding neural parameters or Transformer calls. At horizon 720, it reduces mean squared error (MSE) and mean absolute error (MAE) by 2.54% and 2.33% over seven data sets and two output widths, while remaining $3.24\times$ faster than recursive inference. Level and shape projections sometimes disagree. Trajectory compressibility, the fidelity-accuracy mismatch, and the correction recur across three public AR parents. Together these results separate representation, transition, and execution as AR design axes. Code is available at https://github.com/RowanFFF/ATD-Spectrum-Tangent.

We prove finite time blow-up from smooth, radial, compactly supported initial data for the energy supercritical defocusing nonlinear Schrödinger equation $i\partial_tu+\Delta u-|u|^{p-1}u=0$ in $\mathbb R^d$, for $(d,p)\in\{(3,27),(4,9),(5,7)\}$. The solutions are asymptotically self-similar, their $L^\infty$ norm grows at the scaling rate $(T-t)^{-1/(p-1)}$, and their critical Sobolev norm $\dot H^{s_c}$ diverges. This closes the conjecture of Bourgain \cite{Bourgain2000} on global well-posedness and scattering for the defocusing energy supercritical equation in dimensions three and four. The proof relies on a shooting argument and is computer-assisted.

Let $p>5$ be an odd prime, put $n=(p-1)/2$, and let $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol. Define $D_p^{\pm}$ to be the following determinants \[ \begin{aligned} D_p^{\pm}&:=\det\left[(i\pm j)\left(\frac{i\pm j}{p}\right) \right]_{1\leq i,j\leq n}. \end{aligned} \] In this article, we give explicit formulas for these determinants modulo $p$ in terms of Bernoulli numbers. As a consequence of Reinhart's recent counterexample to the Ankeny–Artin–Chowla conjecture, we find a prime $p$ such that $D_p^{\pm} \equiv 0 \pmod p$. This gives a negative answer to a question of the Sun.

In undersampled inverse problems (such as sparse-view computed tomography), only a small number of measurements are collected, which reduces radiation exposure and acquisition time and cost, and can also address the inaccessibility of certain acquisition arrangements. Most learning-based methods for such problems require supervision in the form of fully sampled measurements and ground-truth images, which are costly or even infeasible to acquire. To overcome this issue, we propose Exact Learning for Inverse Problems from Partial Self-supervision (ELIPPS), an incomplete self-supervised training paradigm for undersampled inverse problems that requires neither ground-truth images nor fully sampled measurements. ELIPPS learns solely from incomplete forward measurements on a fixed incomplete supervision set. Our theory shows that when the data distribution is invariant under certain transformations, minimizing a masked empirical risk is equivalent to minimizing the full self-supervised risk, i.e. the risk against the complete, noise-free measurement. The equivalence is exact for arbitrary equivariant hypothesis classes and holds for signal-dependent noise such as the pre-log Poisson statistics of low-dose tomography, and for the log-transformed count model up to a quantifiable bias. When the underlying coverage condition is only approximately satisfied on a discrete grid, we give a stability estimate in terms of the associated frame constants and the irreducible error of the inverse problem. We realize ELIPPS for computed tomography by exploiting rotation and reflection invariance. In our experiments, ELIPPS substantially outperforms naive masked supervision and reaches the same order of accuracy as a reference model trained with full clean measurements.

Let $T$ be a bounded self-adjoint operator on a separable infinite-dimensional right quaternionic Hilbert space, and let $\sigdS(T)$ be the set of right eigenvalues of finite type. We prove that Weyl's theorem holds for $T$: $$\sigeS(T)=\sigS(T)\setminus\sigdS(T),$$ which completes the inclusion obtained in the general case. The proof proceeds through a single three-way criterion: a point of $\sigS(T)$ lies outside $\sigeS(T)$, and equally lies in $\sigdS(T)$, exactly when it is isolated with a spectral atom of finite rank. The natural route through Riesz projections is not available as it stands, because a Riesz projection need not be orthogonal; we show that for a self-adjoint operator it in fact is, the Riesz projection of an isolated spectral part coinciding with the corresponding spectral projection, so that the algebraic and geometric multiplicities agree. A $2\times2$ quaternionic matrix shows that this last identity fails without self-adjointness, and a normal operator with genuinely spherical $S$-spectrum shows what has to change in the normal case.

Adulteration of milk with water, urea, calcium compounds and starch is still a widespread food safety problem, especially in areas where there is no access to laboratory based chemical testing, and is a global threat to human food safety and security, especially for children and the elderly. We present the development of a reagent free screening method, based on droplet evaporative deposition method, optical microscopy, and quantitative image analysis, for consistent detection and classification of milk adulteration. Droplets of Single Toned ST, 3% fat, and Double Toned DT, 1.5% fat milk samples, adulterated with water, urea, calcium, and starch, respectively, at different concentrations were tested. Deposition patterns were characterized by image processing using radial intensity profile descriptors area under the curve, and edge decay slope and gray level co occurrence matrix GLCM texture features contrast, correlation, energy, homogeneity, and entropy. The descriptors exhibit consistent adulterant specific trends: water and urea adulteration resulted in increasingly smooth, more homogeneous deposits decreasing contrast, increasing homogeneity, while calcium and starch adulteration resulted in structurally rougher deposits increasing contrast, decreasing homogeneity. Urea was further distinguished in the two groups by a significant increase in homogeneity and entropy collapse at higher concentrations, whereas calcium and starch were distinguished by diverging area under curve AUC trends. Milk type ST vs. DT was resolved by a combined multivariate signature of the descriptors at baseline. Our findings show that a simple, two level feature based framework: first resolving milk type, then adulterant family, then specific adulterant identity can be realized entirely from optical microscopy data without additional chemical reagents.

The A* search is a fundamental path-finding algorithm in artificial intelligence. While admissible and consistent heuristics guarantee efficient performance by expanding each state at most once, modern search applications frequently employ powerful but inconsistent heuristics derived from machine learning, randomized evaluations, etc. A long-standing theoretical barrier to using these inconsistent heuristics is the risk of catastrophic node re-expansion, which yields a worst-case exponential time complexity of $\Omega(2^n)$. However, empirical observations contradict this pessimistic bound, demonstrating that inconsistent A* operates highly efficiently in practice. To bridge this significant gap between theory and practice, this paper presents the first smoothed analysis of the A* algorithm using inconsistent heuristics. We model typical real-world noise by applying slight random perturbations to the edge weights of worst-case search graphs. Our main result proves that the expected smoothed time complexity of inconsistent A* is bounded by a polynomial, specifically a total iteration number of $O(n^2 m \kappa)$, where $n$ is the number of nodes, $m$ is the number of edges, and $\kappa$ controls the scale of random perturbations. Furthermore, we also show that this result naturally extends to the functionally equivalent problem of Dijkstra's algorithm on negative-weight graphs.

Form Field Detection (FFD) is a fundamental component of document understanding systems, enabling applications ranging from large-scale industrial digitization to accessible form interaction for automated analysis. Unlike conventional object detection tasks, FFD is inherently challenging because fields are often defined by layout structure and whitespace rather than visible foreground content. Existing large-scale datasets frequently rely on heuristic annotation pipelines, resulting in noisy and inconsistent labels that hinder reliable evaluation. In this work, we introduce mini-CommonForms, a carefully curated FFD benchmark with consistent, high-quality annotations, and present a detailed evaluation of state-of-the-art detection approaches. The benchmark is designed to support reproducible research in document automation and accessibility-oriented applications. Dataset and code are available at https://github.com/moured/mini-commonforms

Remote source monitoring is a key use case for Internet of things (IoT) applications, calling on efficient communications protocols to ensure timely data delivery. In this paper, we consider a practically inspired scenario in which an IoT device reports sensor readings to a gateway over a correlated, lossy channel, receiving feedback on the transmission outcome only after some delay. For this setting, we investigate practical threshold-based reporting strategies for monitoring a Markov process, and evaluate two approaches: a reactive strategy where the node awaits feedback after an update, and a proactive strategy that preemptively transmits a second update. Leveraging Markov renewal processes, we provide an exact performance analysis in terms of the mean squared error (MSE). Our study characterizes the fundamental trade-off between estimation error and communication rate, revealing that proactive transmissions significantly improve MSE even under strict IoT rate constraints.

Ultra-long user behavior sequences carry rich signals of stable and diverse preferences, yet industrial recommender systems typically truncate histories to a few hundred actions under strict latency and memory budgets, leaving long-term interests under-utilized. Users also pursue multiple heterogeneous intents across modalities such as news, Q&A, and short video, which sparse ID embeddings alone struggle to represent. We present Multi-interest Sequence Representation (MuSeR), a retrieval framework built on the deployed MGS system, which integrates three components: (i) hierarchical temporal compression, which retains recent actions at full resolution while progressively pooling older segments, so that per-user histories of $10^{4}$-$10^{5}$ interactions fit within a fixed serving budget; (ii) disentangled multi-query interest extraction with orthogonality regularization; and (iii) multimodal semantic alignment, which augments sparse item IDs with textual summaries distilled from a large language model. For industrial deployment, MuSeR further adopts asynchronous user-representation refresh with adaptive caching and hierarchical beam-search retrieval across heterogeneous hardware. On three public benchmarks and a large-scale industrial dataset, MuSeR consistently improves Recall@$K$ over strong long-sequence and multi-interest baselines. In online A/B tests on Baidu APP's homepage feed, discovery feed, and short-video scenarios, MuSeR yields +0.26% daily active users and +0.89% total session duration (both statistically significant, p<0.05), alongside reduced serving latency and cost. Rather than proposing a new modeling primitive, our contribution is a system-level integration that makes long-term, multi-interest, and multimodal modeling jointly deployable in a real-time production pipeline, together with the engineering practices required to sustain it.

Rigorous error analysis for numerical approximations of parabolic equations with hysteresis remains limited, even for the widely used Prandtl–Ishlinskii model of play type. In this work, we establish an $O(h+\tau)$ error bound for an implicit Euler $P_1$ finite element discretization. The analysis requires neither higher-order temporal regularity nor additional spatial regularity of the hysteresis variables. For the temporal discretization, we exploit a convex subgradient-flow structure in a weighted Hilbert space and estimates of residual monotone terms to obtain first-order convergence. For the spatial discretization, only the diffusive field is restricted to the finite element space, and a constraint-preserving comparison yields an $O(h)$ semidiscrete estimate. The analysis is developed for play-type Prandtl–Ishlinskii operators formulated directly on a spatial Hilbert space, encompassing the canonical pointwise extension as well as more general spatially structured constraints.

Let $\widetilde G$ be a nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In previous work, we introduced $\prod_{\rho/2}^s(\widetilde G)$, the set of genuine small representations with infinitesimal character $\rho/2$ in the simply laced setting, and showed that, when $G$ is split, these representations are exhausted by the Kazhdan–Patterson lift of the trivial representation. In this paper, we consider nonsplit real Spin groups of type $D_n$. For each $\rho/2$-compatible genuine central character, we determine the irreducible constituents of $\mathrm{Lift}_G^{\widetilde G}(\mathbb C)$. We also describe explicitly the relevant genuine small representations in terms of Langlands parameters. The nonsplit case exhibits a new phenomenon: the lift of the trivial representation need not exhaust $\prod\nolimits^s_{\rho/2}(\widetilde G)$. For the families considered here, equality holds only for $\widetilde{\mathrm{Spin}}(2p+2,2p)$; in the other two families the lift is a proper subset, and in one case it is zero.

In 1928 H. Cartan stated a conjecture about holomorphic curves in $\mathbb{P}^n$ parametrized by the unit disc, omitting $n+2$ hyperplanes in general position. He proved it for $n=2$. In 1996 the first-named author constructed counterexamples for all $n\geq 3$, and proposed a modified form of Cartan's conjecture which he proved for $n=3$. In this paper we prove this modified conjecture in all dimensions. As a byproduct we prove a quantitative version of the classical theorem that analytic functions are linearly dependent if and only if their Wronskian determinant is zero. We interpret our results in terms of the Kobayashi–Royden pseudometric on the complement of $n+2$ hyperplanes, and on smooth algebraic varieties in the algebraic torus.

In vision-based off-road navigation, a robot needs to know not only which obstacles to avoid but also which terrain is better. The first is handled by freespace detection or semantic segmentation. The second is usually answered with a traversability score, but no universal ground truth exists for such a score, so perception falls back on a predefined value per semantic class or a freespace confidence. These scores say what a region is, not which region a robot should prefer. We therefore formulate this preference as visual traversability ranking, an ordering of visible terrain that can be supervised by comparisons between two regions. Standard annotations do not label preference, but they imply its direction. We present TravPro, which converts these annotations into ordered region pairs and fits a small readout on frozen vision–language model (VLM) patch tokens to these pairs. The tokens are clustered once into a fixed prototype bank, and the readout learns a preference score per prototype. The readout is then applied to every patch and serves as a teacher that turns sparse comparisons into dense preference pseudo-labels without pixel-wise annotation. An RGB student distills these maps into a dense terrain-preference map together with a non-ground mask that excludes obstacles and background from the ranking. On five unseen domains, TravPro reaches a mean pairwise accuracy of 0.915 against 0.783 for the strongest baseline, producing an ordering sensitive to surface condition that a per-class value cannot represent. The same VLM and the same supervision yield no such ordering when the VLM is prompted and the supervision is used as dense targets; what matters is how they are used.

Let $\mathfrak R=\nabla(-\Delta_D)^{-1/2}$ be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space $\mathbb R^n_+$, and let $H^1_r(\mathbb R^n_+)$ denote the Hardy space of restrictions. We establish an endpoint theory for commutators with $\mathfrak R$ on $H^1_r(\mathbb R^n_+)$. For every $b\in \mathrm{BMO}(\mathbb R^n_+)$, the commutator $[b,\mathfrak R]$ extends boundedly from $H^1_r(\mathbb R^n_+)$ to $L^{1,\infty}(\mathbb R^n_+)$. We then prove the sharp strong-endpoint characterization \[ [b,\mathfrak R]:H^1_r(\mathbb R^n_+)\to H^1_r(\mathbb R^n_+) \quad\Longleftrightarrow\quad b\in \mathrm{BMO}^{\log}(\mathbb R^n_+). \] Moreover, the $\mathrm{BMO}^{\log}$ norm is quantitatively equivalent to the sum of the usual $\mathrm{BMO}$ norm and the $H^1_r$ operator norms of the component commutators $[b,\mathfrak R_j]$. The logarithmic condition records the distance to the boundary and is therefore intrinsic to the Dirichlet geometry.

Consider a constraint satisfaction problem $\mathbf{C}$ on finitely many independent random variables with dependency graph $G$. Let $p_a$ be the violation probability of a constraint $a\in \mathbf{C}$ and $N_G^2 (a)$ the set of constraints at distance one or two from $a$ in $G$. Suppose that, there exists $x\in (0,1)^{\mathbf{C}}$ such that, for a sufficiently small universal constant $c > 0$, and for all $a \in \mathbf{C}$, \[ p_a \leq c \cdot x_a \prod_{b\in N_G^2(a)}(1-x_b). \] Under the above analog of the asymmetric Lovász Local Lemma, we give an FPRAS for the probability that all constraints are satisfied, and an approximate sampler, running in polynomial expected time, for the product distribution conditioned on this event. The degree of the polynomial in the running time is independent of the domain sizes, constraint sizes, or degree of the dependency graph. Up to the choice of the constant $c$, our condition on $p_a$ matches known hardness results. Our work builds on the method of Liu, Wang, Yin, Zhang, and Zhou, who obtained an FPRAS for the probability of satisfaction in the setting of the symmetric Lovász Local Lemma. Our sampling result is new even in this special case.

Multispectral camera arrays capture image data in various spectral bands, enabling image acquisition beyond human perception. These systems are widely used in medical, agricultural, environmental, and remote sensing applications. However, not all recorded bands are needed for classification tasks, thus reducing them can lower hardware complexity and cost. The conditional filter band selection algorithm addresses this by selecting low-noise, non-redundant bands to minimize the number of filters and cameras. This paper improves the approach by introducing a second classification stage that estimates object material in addition to the object label. This information is merged by a decision-tree based band selection strategy. The proposed method achieves a 28.9% relative reduction in classification error on the SMM50 dataset compared to the state-of-the-art. Moreover, for the same classification accuracy, the required number of cameras is reduced from 7 to 4, demonstrating that the proposed approach improves performance while significantly lowering hardware requirements.

The Directed Feedback Vertex Set problem (DFVS) asks whether a digraph can be made acyclic by deleting at most $k$ vertices. Whether DFVS admits a polynomial kernel parameterized by $k$ is a major open problem in kernelization, even for planar digraphs. We resolve the planar case by giving a deterministic kernel with $O(k^{66}\log^2 k)$ vertices and arcs. Our algorithm proceeds in three stages. First, we apply structural reduction rules to the input digraph, bounding the number of directed faces and some special vertices. Second, we pass to the planar dual, where vertex deletion corresponds to adding groups of reverse arcs to make each weakly connected component strongly connected. The structural bounds in the first stage yield a small retained vertex set in the dual. We then compress the dual instance by identifying vertices with the same distance records from this retained vertex set. The main technical contribution is a directed-cut argument showing that this identification preserves feasibility. Finally, we transform the polynomial-size dual instance back into an instance of Planar Directed Feedback Vertex Set via a $3$-CNF encoding and a planar graph construction.

Inferring network topology from noisy node observations is a central problem in graph signal processing. In this paper, we consider Laplacian-constrained graph estimation for Gaussian Markov random fields, focusing on the underdetermined regime in which the number of samples is smaller than the number of graph nodes. Existing approaches often formulate the problem as a sparsity-regularized maximum-likelihood estimation problem. While effective, such methods typically require iterative optimization and are often computationally demanding, particularly under Laplacian constraints. Instead, we propose a non-iterative estimator of graph Laplacians that uses effective resistance for regularization, and evaluate the method using a simple sparsification procedure. Experiments show that with some trade-off in edge and weight recovery on the considered dataset, computational cost for moderately sized graphs can be substantially reduced.

We study closed range and essential norms of bounded composition operators $C_\varphi$ on weighted Dirichlet spaces $\mathcal{D}_{\alpha}$. For $-1<\alpha<0$, we first consider maps whose images are obtained by removing a compact subset from a simply connected subdomain of $\mathbb{D}$. In this setting, closed range is equivalent to the Reverse Carleson property and the Reverse Carleson Disk Condition for the counting measure $\mu_{\varphi,\alpha}$, uniform weighted area density of the image, $\mathbb{T}\subset\overline{\varphi(\mathbb{D})}$, and the inclusion of an outer annulus in the image. We also obtain a closed range characterization under the Uniform Tail Condition. For all $\alpha>-1$, we establish two-sided essential norm estimates in terms of local averages of the generalized Nevanlinna counting function and an exact formula using boundary tail integrals. If the normalized counting density has vanishing oscillation at the boundary, we obtain exact formulas in terms of its boundary limsup, Berezin transform, and local averages.

Open-world deployment requires humanoid robots to cross highly heterogeneous terrain safely, with perception that simultaneously provides wide coverage, local accuracy, and redundancy against sensor failure. Existing approaches struggle to satisfy all three: one forward depth camera or nearby height sampling covers too little; odometry-corrected elevation maps drift under aggressive motion and miss thin vertical structures; image-level encoding costs grow with camera count. We present UniPoint, a humanoid whole-body locomotion framework built on multi-source point-level sensor fusion. Measurements from a 360{\deg} light detection and ranging (LiDAR) sensor and two depth cameras are early-fused into one base-frame point set. Voxelization resamples it to a fixed number of tokens encoded by linear self-attention and proprioception-queried cross-attention, decoupling forward cost from sensor count. The point set retains standing thin barriers; a single-modality failure removes only part of the tokens, so the policy degrades gracefully. A single training run with terrain-aware rewards, perception-degradation injection, and domain randomization produces one policy for all eight terrain types, deployed on an onboard RK3588 without fine-tuning. On a DR02 humanoid, 20 trials at each of nine real-world settings over seven terrain types validate the policy on 70-cm-high platforms, 100-cm gaps, thin barriers, and sparse or narrow footholds; it also generalizes zero-shot outdoors.