Papers

The coalescent is a central framework in population genetics for modelling the ancestral relationships among sampled individuals through a genealogy, represented as a rooted and ranked binary tree. In this model, lineages coalesce at a rate inversely proportional to the effective population size, a time-varying quantity of primary interest. The bounded coalescent conditions genealogies on the time to the most recent common ancestor being bounded above by a fixed time. This model is useful in various contexts, such as phylodynamics of infectious diseases with known introduction times and single-cell lineage tracing in synthetic barcoding experiments. To our knowledge, there is no existing tool that infers variable effective population size trajectories under the bounded coalescent. We view estimation under the bounded coalescent as equivalent to estimation of the intensity function of an inhomogeneous point process. We provide an efficient algorithm for coalescent simulation under the bounded coalescent using point process methods, retaining the exactness of naive rejection sampling while substantially reducing computational cost and avoiding repeated numerical inversion of the bounded cumulative hazard. We then develop a Markov chain Monte Carlo procedure for posterior inference of effective population size trajectories that avoids discretization of the likelihood integrals. In simulations, conditioning on the bound reduces the median sum of squared errors in two of three settings, with less favourable results in the most rapidly varying setting. We illustrate the method using severe acute respiratory syndrome coronavirus 2 sequence data from Washington State.

In nuclear many-body physics, energy density functional (EDF) theory is one of the most powerful approaches for describing finite nuclei and nuclear matter. However, its predictive capability depends on calibrating model parameters to experimental and observational data. In this work, we investigate an alternative approach: Constraining the parameters with the continuous density profiles of finite nuclei obtained from ab initio calculations. We apply Bayesian analysis to infer the parameters of Skyrme EDF from the density profiles and binding energies of 16O, 40Ca, and 48Ca. We show that the data effectively constrain the parameters associated with the properties of uniform nuclear matter, whereas those governing non-uniform nuclear matter remain partially constrained and require additional input. Furthermore, using the inferred parameter distributions, we successfully predict the density profiles and binding energy of 208Pb, which is excluded from the training data. This demonstrates the predictive capability of the framework. In conclusion, these results establish Bayesian analysis of density profiles as a promising route for incorporating accurate ab initio results of light nuclei into EDF development and strengthening the connection between both approaches.

We prove that a lattice $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb C)$ has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and Leuzinger in 2008. The same characterization by quaternion algebras holds if the difference set of traces is not dense. Our method also gives a similar result for lattices in $\mathrm{SL}_d(\mathbb R)$, for every $d\ge3$: a lattice $\Gamma<\mathrm{SL}_d(\mathbb R)$ has positive trace gap if and only if all its traces are integers. Equivalently, after conjugation, it has finite index in the norm-one group of an order in a central simple algebra of degree $d$ over $\mathbb Q$ that splits over $\mathbb R$. We also discuss spectral consequences and other applications of the results and techniques.

Sparse activation reduces mixture-of-experts computation without eliminating the need to store all experts. We present Routide, a Swift/MLX runtime that executes the text path of a pinned public Qwen3.6-35B-A3B quantized checkpoint while keeping expert weights in iPhone storage and a byte-budgeted subset in memory. We characterize cache-policy sensitivity, numerical comparison boundaries, and measurement limits. Across five recorded 128-token workloads, fixed-route replay gives 0.00% demand hits with a 512 MiB LRU cache, 18.80% with seeded random eviction at the same budget, and 38.58% with 576 MiB LRU. The apparent capacity cliff is therefore a policy/workload interaction, not a universal memory requirement. Same-runtime Mac controls preserve generated sequences across eviction and asynchronous prefetch, including 2,560 exact token comparisons and 10,334 speculative loads. In contrast, complete resident-Python versus recorded-phone sequences disagree on all five tested cases, precluding a general numerical equivalence claim. Two separately scoped iOS 27 memory protocols observe sampled process-footprint peaks of 1.87-2.32 GiB on short prompts and 2.39-2.73 GiB on one longer prompt. We retain a thermal stopping event, negative timing comparisons, and a single qualified whole-device power estimate. These results establish bounded feasibility and identify limitations that a deployment claim must not hide.

Torque observations in reinforcement learning remain challenging because simulated and measured torque differ in scale, offset, and noise. In this paper, we propose a simple torque observation alignment method for robots with direct-drive (DD) actuators, in which motor current maps linearly to joint torque through a motor-type-specific torque constant K_tau. First, dynamometer calibration identifies K_tau* and corrects the scale mismatch between simulated and real torque. Second, the method uses delta_tau(t) = tau(t) - tau(t-1) as the observation in both domains to eliminate the constant offset instead of using the direct torque tau(t), which carries a domain-dependent bias. Third, Gaussian noise obtained from the dynamometer measurement data is injected during the learning process. To validate the proposed method, we train a teacher-student grasping policy entirely in simulation and deploy the distilled student on a multifingered DD gripper. The deployed policy performs proprioceptive grasping using only joint positions and torque differences. We conduct an ablation study comparing the proposed method with alternative alignment variants on nine in-distribution (ID) objects. The proposed method achieves 100% grasp success. These results demonstrate that the proposed alignment method improves the robustness of zero-shot policy transfer on the DD gripper against real-world torque-observation mismatches.

Networks of interfaces in materials and living systems evolve through motion and rearrangement of interfaces and junctions. Local equilibrium at a junction requires interfacial force balance. We show that in three dimensions the classical Herring condition for triple lines is necessary but insufficient: four triple lines meeting at a quadruple node may each satisfy Herring equilibrium, while the four conditions remain mutually incompatible. Such a frustrated node has no admissible local equilibrium geometry. In the semi-isotropic limit, the six interfacial energies must form the edge lengths of a non-degenerate tetrahedron, with constructibility determined by the Cayley-Menger determinant. This hidden compatibility constraint arises from three-dimensional geometry and applies broadly to foams, polycrystals, tissues and other interfacial networks.

Onboard vision-language models could enable satellites to answer queries directly, but exhaustive tiled inference over high-resolution imagery is slow and energy-intensive. We identify answer-invariant token redundancy (AITR): image tiles and vision tokens that can be removed without changing the final answer. We present Rift, a two-stage system that performs query-conditioned tile pruning followed by elastic prefill to reduce token budget. We evaluate it on LLaVA-1.5 7B running on Jetson AGX Orin. Compared with exhaustive tiled inference, Rift reduces energy by 78% and latency by 69%, while increasing accuracy from 45% to 73%.

In this paper, we propose RGBD20K, a novel dataset for facilitating the development of more robust and general RGB-D semantic segmentation by encompassing abundant categories and high-quality annotations. RGBD20K possesses several attractive properties: (1) Expanded Semantic Space. In particular, it covers 160 fine-grained categories, largely surpassing the category diversity of existing popular RGB-D benchmarks (e.g., NYUv2 with 40 classes and SUN RGB-D with 37 classes). With such enriched semantic coverage, we expect to promote the learning of more generalizable segmentation models. (2) Larger Scale. Compared with current benchmarks, RGBD20K offers 20,000 RGB-D image pairs, providing a substantially larger training resource that benefits the development of more powerful deep models. (3) High-Fidelity Annotation. We perform rigorous re-evaluation and correction of existing labels to resolve long-standing annotation noise, resulting in a clean and reliable ground-truth foundation. Furthermore, we propose a novel score-purified fusion (SPF) method, which achieves state-of-the-art performance across all evaluated benchmarks, demonstrating the effectiveness of our approach in leveraging high-quality multimodal information for RGB-D semantic segmentation. The dataset is here: https://github.com/ShaohuaDong2021/RGBD20K/.

Atmospheric observations are sparse, heterogeneous, and unevenly distributed, whereas many generative atmospheric models learn distributions over regularly gridded multivariate states. Once pretrained, diffusion models can supply atmospheric priors that can be combined with observation-derived likelihood factors in a Bayesian formulation. However, these observation sources differ substantially in geometry and sampling density, complicating the consistent use of their observations within a common inference framework. Here, we formulate this reconstruction problem as generative atmospheric super-resolution and introduce composable observation interfaces for conditioning a single pretrained 13-variable atmospheric diffusion model. The interfaces convert sparse radiosonde (R), clustered aircraft (A), and dense irregular surface-station (S) observations into source-specific likelihood factors that specify where observations constrain the gridded state, how residuals are counted under uneven sampling, and how strongly each source guides posterior sampling. We developed the aircraft and surface observation interfaces using 2019 observations and evaluated the selected interfaces throughout 2020 without further tuning. Compared with reconstructions conditioned only on radiosonde observations, the composed R+A+S interface reduces RMSE evaluated against ERA5 by $9.24\%$ across all 13 state variables over the CONUS domain. The aircraft and surface factors provide complementary improvements in upper-air and surface variables. The R+A+S combination also lowers the Continuous Ranked Probability Score (CRPS), while evaluations at held-out aircraft and surface-station observations show reduced prediction errors. Together, these results demonstrate a modular route for conditioning a pretrained atmospheric generative prior on heterogeneous in situ observations without retraining the underlying model.

We present a framework that maps harmonic vibrational systems onto effective multi-orbital tight-binding Hamiltonians, establishing a direct correspondence between phononic degrees of freedom and graph-based lattice models. Within this mapping, the Cartesian displacement components of each mass become internal orbitals, while elastic interactions generate effective onsite energies and hopping amplitudes determined by the geometry of the system. This representation enables the application of numerical and conceptual tools originally developed for electronic transport to the study of phononic systems. The method is first applied to a monoatomic chain containing an angular bend. For this benchmark system, analytical expressions for the polarization-resolved transmission and reflection probabilities are derived and compared against calculations performed using a recursive scattering-matrix method within the mapped tight-binding representation. Excellent agreement is obtained, validating the mapping and demonstrating its ability to describe geometry-induced mode conversion and the influence of evanescent states. We then investigate zigzag chains, where alternating bond orientations generate effective dimerized Hamiltonians reminiscent of the Su–Schrieffer–Heeger (SSH) model. Although the uniform zigzag chain develops a spectral gap, no localized edge states are observed. We show that modified boundary parameters generated by the phononic mapping suppress the edge-state formation expected from the ideal SSH picture. Introducing a geometric domain wall restores localized domain-wall states inside the gap, which give rise to resonant transmission channels across an otherwise insulating frequency window. The resonance energies remain robust against system-size variations and the corresponding transmission approaches unity.

Observations of collective signatures within jets of particles have been recently reported by measurements at the Large Hadron Collider at CERN laboratory, providing insights on the minimum conditions required for forming a Quark Gluon Plasma state. A quantitative comparison with data measurements on intrajet azimuthal anisotropies and predictions from PYTHIA event generator soft QCD model parameter tunes: Monash, CP5, and QCD-scheme CR is presented. Energetic jets with transverse momentum over 550 GeV from proton-proton collisions at 13 TeV are reconstructed. Results from predictions using the standard E-scheme (ES) and jet axis choice are compared with those from the Winner-Take-All (WTA) frame. Significant differences are present between predictions from WTA and ES frames at the lowest and highest intrajet charged particles multiplicities ($N_{\mathrm{ch}}$), with the QCD-scheme CR tune showing the smallest variation between frames. Predictions for $v_2$ at the highest $N_{\mathrm{ch}}$ are consistent with the data measurements at a 2.3$\sigma$ uncertainty level. These results set new guidance towards the search for intrajet collective effects and understanding of nonperturbative QCD dynamics originated from a single parton.

Natural load-bearing and transport networks are not assembled in a single step; they emerge through a temporally ordered process of growth, branching, reinforcement, and loop formation. Inspired by this developmental logic, this work introduces a morphogenetic graph-generation framework for mechanical lattices in which a discrete dot matrix provides potential nodes and the final architecture is created by sequential cross-layer and intra-layer growth. The same rule is visualized in two dimensions as a leaf-vein-like developmental sequence and implemented in three dimensions on a 3x3x3 nodal matrix containing 27 candidate nodes. A dataset of distinct three-dimensional lattices was evaluated by beam-based finite element analysis and represented directly as graphs. A graph convolutional neural network (GCNN) with three graph-convolution layers and dual global pooling learns the topology-property mapping and predicts effective compressive stiffness. Coupling the GCNN surrogate with rapid structural sampling enables inverse design: for a target stiffness of 1000 MPa, the selected design was predicted at 1042.43 MPa and validated by finite element analysis at 1027.49 MPa. Beyond straight members, the framework has also been extended to parameterized horseshoe-shaped curved beams made of nonlinear materials, enabling topology-geometry design toward prescribed deformation shapes. Our work provides a paradigm for augmenting the database of mechanical metamaterials, and the resulting perspective links biological morphogenesis, graph learning, and nonlinear shape programming in a unified generative design framework for architected materials.

In this article, we consider the stochastic nonlinear heat equation driven by Lévy space-time white noise in dimension one. For the spatial average of the solution, we prove quantitative and functional central limit theorems under $m_1+m_{2p}<\infty$ for some $p\in(1,\frac{3}{2})$. These results extend the Gaussian fluctuation theory for the parabolic Anderson model to the nonlinear setting. The main new feature is a minimum-type term in the second Malliavin derivative estimate caused by the nonlinear coefficient.

We prove that the language of palindromic reduced words in a Coxeter system is regular if and only if the Coxeter group is a direct product of a finite Coxeter group with finitely many copies of the infinite dihedral group.

Coordinated arm-hand motion planning is fundamental to dexterous robotic manipulation in complex and constrained environments. A straightforward solution is to decompose the problem into separate arm path planning and hand motion generation; however, this poses a dilemma: decomposition can miss feasible solutions that require coordinated arm-hand adaptation along the path. Alternatively, directly planning in the high-dimensional joint arm-hand configuration space captures such coupling but faces a substantially enlarged search space and nonconvex collision constraints. To characterize this coupling, we formulate feasible hand fibers that capture collision-free hand configurations for each arm configuration. Based on this formulation, we propose CAMP, a high-success and efficient cooperative arm-hand motion planner for constrained environments. CAMP constructs candidate trajectories through layered hand search with local arm relaxation, then compactly represents them using endpoint-preserving via-point movement primitives (VMPs) for coarse-to-fine joint optimization. Across six constrained simulation tasks, CAMP achieves 84.2-98.5% planning success, outperforming alternative planners with competitive efficiency. Ablation studies verify the contributions of arm relaxation, VMP representation, and coarse-to-fine optimization, while real-robot experiments demonstrate CAMP on constrained manipulation tasks. The project website is available at https://camp-armhand.github.io/.

Skilled humans can catch fast-moving objects softly by coordinating interception, velocity matching, and follow-through to mitigate impact. Learning such impact-aware catching with reinforcement learning (RL), however, is challenging, as the policy must achieve reliable interception and grasping while regulating the sensitive transition into contact. Moreover, even a capable privileged-state RL teacher may not provide ideal demonstrations for a deployable imitation-learning (IL) student: teacher failures limit task coverage, while small variations in pre-contact motion can produce substantially different impact and grasping outcomes. We characterize this phenomenon through interventional outcome sensitivity and introduce the outcome-sensitive window (OSW) to guide targeted demonstration construction. Building on this formulation, we propose Outcome-Sensitive Motion Search, which learns a task-conditioned manifold of successful OSW motions and performs local geodesic search to refine successful teacher rollouts and repair task conditions where the teacher fails. We then validate candidate motions through complete rollouts under a calibrated IL-student action-error model and retain only successful executions as demonstrations. Extensive simulation experiments demonstrate that our method effectively repairs task conditions where the teacher fails and enables the resulting IL policy to outperform the privileged RL teacher in both catching success and impact mitigation.

We obtain a large number of exact hyperbolic, trigonometric, and periodic solutions in terms of Jacobi elliptic functions as well as algebraic solutions with a power law tail of the integrable local Fokas-Lennels equation and integrable nonlocal Fokas-Lennels equation. Further, we consider a one-parameter family of generalized Fokas-Lenells equations and obtain a few of their exact solutions.

Quantum error-correcting decoders provide both corrections and confidence estimates for the logical-error risk of individual measurement records. Such confidence can guide postselection, but it depends on the accuracy of the decoder noise model. When the physical device drifts away from that model, confidence measures such as the logical gap can become miscalibrated. We show that this mismatch can itself provide useful information. On IBM quantum hardware, syndrome statistics drift across consecutive acquisition windows, and the resulting detector histories differ substantially from those predicted by the stationary detector error model used for decoding. We observe a similar hardware-model mismatch in independent Google surface-code memory data. We exploit this effect by assigning additional rejection weight to detector histories that are unlikely under the assumed model. For minimum-weight perfect matching (MWPM), the minimum weight $W$ provides an inexpensive proxy for syndrome surprisal. On held-out IBM repetition-code memories, likelihood-aware postselection reduces the retained logical-error probability (LEP) by $10.6\%$ at $d=7$ and $28.8\%$ at $d=9$, both at $15\%$ rejection. On Google ten-round $d=5$ XZZX surface-code memory data, it reduces retained LEP by $75.4\%$ at $90\%$ rejection. Applied to Google's magic-state cultivation experiment with the Tesseract decoder, it reduces tomography infidelity by $34.7\%$ with $30\%$ additional grafting-stage rejection, or increases accepted yield by $30.4\%$ at a target fidelity of $97.0\%$. Burst-noise simulations show that the advantage persists as code distance and syndrome-extraction volume increase. The method reuses quantities already produced during logical-gap decoding and requires only an additional fitting step.

Most robots manipulate objects solely with their end effectors, whereas humans flexibly leverage different body parts, such as the forearm and elbow, especially when handling oversized objects. Learning such whole-arm manipulation is chal-lenging due to long-horizon sparse rewards, limited contact sens-ing, and the sim-to-real gap in contact and actuator dynamics. To address these challenges, we propose Current-Aligned Link Manipulation, a framework for learning long-horizon contact-rich manipulation using motor current as joint load related feedback. Three stage-specific policies first learn repositioning, grasping, and lifting using privileged simulation information, and a stage router sequences them to generate complete task demonstrations. For sim-to-real transfer, a causal current mapper predicts physical motor current from simulated joint histories, aligning the actuator current observation between simulation and hardware. A unified student policy then learns from these demonstrations using only deployable sensor observations and is further refined with DAgger. The task policies are trained entirely in simulation, and the final student is deployed on hardware. Experiments demonstrate 76.2% (762/1000 trials) complete-task success in simulation and 73.3% success (22/30 trials) on the physical robot for sequential oversized-object lifting.

AI-driven scientific discovery accelerates research by autonomously developing solutions and designs. Large language model (LLM) agents support this process through iterative generation and evaluation. Yet these iterations alone do not ensure cumulative progress or establish which directions to pursue next. Costly evaluation further constrains the scope of exploration. Neural architecture discovery brings these challenges together, coupling open-ended design with resource-intensive experimentation. We introduce EvoTreeNAD, a genealogy-guided evolutionary algorithm that constructs trainable architectures without a supplied seed or a hand-specified search space. Starting from an empty root, it grows a persistent genealogy in which each new node represents a complete architecture. Top-percentile values computed from each node and its descendants guide lineage selection. Using the selected design history, an Idea Agent proposes a variant and a Code Agent implements it. Each evaluated variant becomes a child node, expanding the genealogy while providing evidence for subsequent lineage selection. Our theoretical analysis establishes the existence of stationary variation regimes as the genealogy grows. Under specified variation assumptions, sustained top-percentile family values quantify the probability of generating high-reward architectures in these regimes. EvoTreeNAD discovers architectures that outperform the compared NAS and NAD baselines, achieving CIFAR-10/100 test errors of $2.05{\pm}0.06\%$ and $15.09{\pm}0.22\%$. On all six MedMNIST-v2 tasks, the discovered architectures surpass the strongest listed baselines. A controlled CIFAR-10 study further shows that EvoTreeNAD outperforms direct generation, best-of-$N$ greedy continuation, and full-family-mean routing.

Decentralized LLM-based multi-agent systems coordinate through local interactions, but an agent can remain responsive while its task-solving quality persistently degrades. Such gray failures require protecting current tasks before sufficient evidence exists to alter future routing, while still allowing recovered agents to rejoin. We introduce MeshHeal, a fully decentralized self-healing framework that couples ability-matched peer review across two timescales. At the fast timescale, an adaptive hierarchy escalates uncertain or low-scoring outputs from repeated single-reviewer evaluation to committee deliberation and, when needed, correction before use. At the slow timescale, a task- and ability-conditioned peer-relative detector aggregates scores to distinguish persistent degradation from ordinary output variation, trigger mandatory committee review, and eventually exclude degraded agents from ordinary routing; recovery probes provide fresh evidence for reintegration. To faithfully evaluate routing, we introduce Model-Backed MAS Evaluation, which ties ability assignments to execution models, since prompt-based ability assignments alone can leave routing errors hidden. Across BBH, MATH, and MMLU-Pro, MeshHeal achieves 0.839 degraded-phase accuracy using 51k total model tokens per task, versus the strongest baseline Symphony's 0.807 accuracy using 115k per task. Under staggered degradation and recovery, MeshHeal isolates degraded agents, keeps them excluded from ordinary task execution until recovery, and returns them to normal routing.

Large language model (LLM)-based multi-agent systems can automate alpha factor mining, but their reliance on external APIs limits control over cost, availability, and confidentiality. Long research loops also tend to revisit a few successful economic mechanisms that lead to research path collapse. To address these limitations, we propose AlphaDiverse, a framework that integrates a multi-agent alpha research system, diverse research path collection, and post-training for local agents. We let the research system generate complementary plan portfolios and vary research environments across loops to collect diverse research paths. Using these diverse traces, we warm-start local Planner and Realizer agents with supervised fine-tuning. Then, we propose a joint GRPO method to optimize both of them using predictive quality and diversity of contributions. Research feedback is confined to inner period data, while a frozen final model is evaluated on a later outer period data, thereby avoiding test-set tuning. Experiments across four Chinese stock universes show that AlphaDiverse can combine competitive prediction with broader exploration.

A multispecies, collisionless plasma is modeled by the relativistic Vlasov-Maxwell system. Assuming the plasma is globally neutral, we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the cancellation properties of the limiting spatial averages, as well as the masses and charges of the particles. In doing so, we establish a countably infinite number of asymptotic profiles for the charge and current density, electric and magnetic fields, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution. Our approach relies upon a refined field representation that features self-similar behavior away from the boundary of the light cone. In each case we establish a linear scattering result in $L^\infty$ for every particle distribution function, namely we show that they converge as $t \to \infty$ along the transported spatial characteristics at increasingly faster rates. In the case that charge cancellation does not occur, our methods can be further extended to demonstrate polyhomogeneous expansions with logarithmic corrections.

APOGEE delivers high-resolution, near-infrared spectroscopy and highly precise radial velocities (RVs), with extensive repeat observations spanning more than a decade. Long-term RV repeatability is nevertheless limited by instrumental zero-point systematics: exposure-to-exposure drift and fiber-dependent offsets that are only partially tracked by lamp-based wavelength calibration, and that can evolve as fibers are serviced or replaced. We infer and remove these zero points directly from the published per-visit RVs using an uber-calibration framework based on shared repeat observations. We solve sequentially for two sets of minimal calibration units: (plate, night) exposure-level offsets and time-dependent (fiber, observing year) offsets. We apply the method independently to APO and LCO DR17 and to APO DR19. After correction, the same-fiber repeatability within two days reaches a limiting precision of ~14 m/s, and the long-baseline single-visit precision improves by a factor of two from 98.2 to 44.7 m/s. We release per-visit, value-added RV catalogs for DR17 (APO+LCO) and DR19 (APO) containing 3,501,727 visits (3,336,141 fully corrected) of 930,519 unique stars, enabling more reliable RV variability, binary, and Galactic-dynamics studies.

We prove a Torelli theorem for full moduli stacks of vector bundles of rank at least three with fixed determinant on a smooth, proper, geometrically connected, tame Deligne–Mumford curve of coarse genus at least two over an arbitrary field $k$. We also prove a Torelli theorem for the full moduli stack of vector bundles of rank at least three of fixed-degree under an explicit compatibility hypothesis. The method is intrinsic to the stack: faithful special-linear linearizations recover Picard stacks, and determinants of tangent complexes recover the canonical principal polarization. This is the first part in a series of two papers.

We study weak solutions to degenerate parabolic double phase equations with growth exponents $2\le p<q$ and a modulating coefficient that is Hölder continuous with exponent $\alpha$. If the solution itself is Hölder continuous with exponent $\gamma$, we prove higher integrability of its gradient under the gap condition $q\le p+\alpha/(1-\gamma)$ together with $q<p+1$. This is the first gap bound of Hölder interpolative type for parabolic double phase problems, and it is the parabolic counterpart of the corresponding bound for elliptic problems. For $\alpha<1$ it allows exponents beyond the range $q\le p+\alpha$ known for bounded solutions.

Finite-domain normalized spatial measurements combine an incident field, geometric support, local response, and channel weighting. These operations can break an otherwise separable response, so accurate single-axis calibration alone does not establish separability of a multidimensional measurement. We represent normalized readouts as channel-weight expectations under a parameter-dependent effective measure. For fixed reception and channel operators, local sensitivity is the covariance between the channel weight and the source score. This representation distinguishes axis separability, parameter mixing, and local recoverability, and applies at the appropriate detector level to continuous-electrode position-sensitive detectors, Shack–Hartmann and pyramid wavefront sensors, back-focal-plane split detection, and finite pixel arrays. A quadrant photodetector (QPD) provides an analytically tractable case. Starting from the complete two-dimensional power integrals, we identify the spatial-moment origin of its lowest-order cross term. Axis calibration and system symmetry then constrain the inverse map, giving Axis-Anchored Cross-Residual Inversion (ACRI). Numerical tests of this QPD realization support the sensitivity relations and show that the constrained cross correction substantially reduces the off-axis bias retained by single-axis inversion. The results illustrate how the general measurement representation guides inverse construction, with performance bounded by the working domain, calibration state, and preserved symmetries.

This paper investigates the null controllability of the complex cubic Ginzburg–Landau equation with controls supported on general space-time measurable sets. Starting from a positive-measure control set, we use a slicing argument to obtain time slices of uniformly positive spatial measure. This yields uniform spectral inequalities on the slices, which are the key ingredient in constructing quantitative stabilizing feedbacks. A density-point argument provides a sequence of shrinking intervals accumulating at the density point, on which the active times have a uniform positive density. We then apply a time-iteration scheme with piecewise feedback laws and increasing decay rates. Iterating the resulting stabilization estimates shows that the closed-loop solution reaches the zero state at the selected density point. This provides a constructive approach to the local null controllability on measurable control sets for nonlinear parabolic equations.

Tool-using agents are continually updated with new interaction data. After each policy update, however, previously estimated action credits may become stale. Recomputing them from scratch can require many additional tool calls and environment interactions, making repeated updates increasingly expensive. We ask a simple question: when does historical action credit actually need to be updated? Our key observation is that a change in action value does not necessarily imply a change in the decision. Historical credit can still be useful as long as policy-induced drift is too small to overturn the existing action ranking. Building on this idea, we introduce pairwise branch sensitivity to capture how strongly a policy update affects the downstream regions that distinguish two candidate actions. We then derive a first-order anchored credit-transport estimator that updates historical credit using old interventional trajectories, and propose a Decision-Sufficient Credit Gate (DSC-Gate) that chooses whether to reuse, transport, or resample credit. Experiments show that branch sensitivity explains credit drift substantially better than global policy distance. With sufficient historical data, credit transport reduces estimation error, while its benefit to decision making is concentrated on updates that affect action-distinguishing branches. On a fully independent test set, DSC-Gate changes mean regret by only +0.00004 relative to a gap-based gate while reducing mean new tool steps from 472 to 286, a 39.4% reduction. We observe the same pattern after a real tool-agent parameter update. Overall, our results show that agents do not need to recompute action credit after every policy update: much of the historical evidence can be reused or cheaply corrected, reducing the additional interaction required to keep action decisions up to date.

Existing video deraining methods typically exploit neighboring frames through either explicit alignment or implicit spatiotemporal aggregation. Explicit alignment relies on accurate motion estimation, which can become unreliable under dense rain, while implicit aggregation avoids alignment but lacks explicit guidance on the directional and temporally coherent structure of rain. This leaves a gap between reliable temporal aggregation and explicit modeling of rain motion. To address these limitations, we propose FluidRain, a lightweight video derainer that uses divergence-free rain flow to guide Loop-in-Loop attention across scales and neighboring frames. Motivated by fluid mechanics, we model rain motion as a divergence-free image-space flow and use it to organize multi-scale and temporal aggregation. Specifically, FluidRain first estimates a rain-flow field for each frame and projects it onto the divergence-free subspace. The resulting flow steers window attention along rain streaks, enabling neighboring frames to be aggregated without explicit alignment. Since rain-flow structure is preserved across scales and nearby frames, Loop-in-Loop reuses the same attention operator across both dimensions, resulting in a three-frame model with only 0.80M parameters. Experiments on four benchmarks show that FluidRain remains competitive with substantially larger restoration models. We further examine how temporal evidence scales with different input views. To evaluate whether the model remains reliable when rain motion changes across frames, we introduce RainSyn-Gust, which injects controlled changes in rain-streak direction into existing benchmarks. We also develop a physics-based no-reference metric that evaluates real-rain removal without requiring clean targets.