Papers

We revisit the problem of approximating a linear time-invariant LTI model by a reduced-order model (ROM) in an $H_2$-optimal sense. A widely used method for computing such ROMs is the iterative rational Krylov algorithm (IRKA). However, in practice, the stability of the resulting ROM is not generally guaranteed. In this work, we develop a novel framework that can be used to obtain $H_2$-optimal models of single-input single-output LTI systems while enforcing stability of the ROM. We use Malmquist-Takenaka orthogonal expansions in the corresponding $H_2$-spaces and their connection to hyperbolic geometry to develop an equilibrium function. We show that the poles of an $H_2$-optimal ROM transfer function are defined by the zeros of this equilibrium function. The resulting nonlinear root-finding problem is solved using a Newton-type method formulated on the products of Poincaré disk manifolds, where updates are performed along hyperbolic geodesics ensuring that the stability constraint is satisfied at every iteration. We present numerical experiments on benchmark systems for both discrete- and continuous-time LTI MOR problems, including examples where IRKA produces unstable reduced models while the proposed method preserves stability throughout the iteration.

We study sampling from $\pi(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ is $\alpha$-strongly convex and $\beta$-smooth with condition number $\kappa=\beta/\alpha$. For this problem, the proximal sampler preserves the target exactly and recovers overdamped Langevin dynamics in the small-step limit, but its Gaussian resampling step discards directional information. Underdamped Langevin dynamics retains such information through momentum to accelerate exploration, motivating a proximal construction that combines target preservation with directional persistence. We introduce reflected proximal dynamics, which combines proximal sampling with an exact reflection of the auxiliary variable, conditional kinetic motion, and controlled refreshment. We develop two realizations: a flux-resampled underdamped half-turn and a persistent-momentum quarter-turn. From a warm start with constant $\chi^2$ divergence, both methods achieve total variation error $\varepsilon$ using $\widetilde O(\sqrt{\kappa}d^{1/4})$ expected oracle queries. We further remove the supplied warm-start assumption by combining either sampler with existing posterior samplers in a proximal initialization stage with an optimized coarse step size. Accounting for both initialization and sampling yields $\widetilde O(\kappa^{6/7}d^{1/4})$ expected oracle queries. In both regimes, the dependence on the target accuracy $\varepsilon$ is $\mathrm{polylog}(1/\varepsilon)$, making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.

The anomalous Hall effect (AHE) in unconventional magnets has attracted considerable attention. A giant AHE has recently been observed in the candidate $p$-wave magnet Gd$_3$(Ru$_{1-\delta}$Rh$_{\delta}$)$_4$Al$_{12}$ [Nature 646, 837(2025)], while its theoretical interpretation invokes a spontaneous magnetization to break time-reversal symmetry ($\mathcal{T}$), although the magnetic order itself already breaks $\mathcal{T}$. Consequently, the microscopic origin of the AHE in odd-parity magnets (OPMs) remains an open question. Here we establish the symmetry condition and microscopic mechanism for the AHE in OPMs. Since the OPM order is classified by the spin group whereas the AHE is governed by the magnetic group, the AHE is allowed only when the magnetic point group is ferromagnetic; conversely, it is forbidden whenever the effective time reversal $\mathcal{T}\tau$ (with $\tau$ a fractional translation), which is essential to the realization of the OPM, is preserved. Microscopically, OPMs generically host a hidden Zeeman field (HZF) rooted in this intrinsic $\mathcal{T}$ breaking, and its interplay with spin-orbit coupling (SOC) generates the AHE without any external field. We demonstrate this picture with models on the square, honeycomb, and kagome lattices. In particular, a square-lattice helimagnet with sublattice-resolved SOC maps exactly onto an anisotropic Rashba–Zeeman model, whose Zeeman term is nothing but the HZF and directly yields the AHE. Our results establish the symmetry condition for the AHE in OPMs and identify the HZF–SOC interplay as a general route to realize it.

In this paper, we derive the spin polarization and the spin Chern number in momentum and real spaces in terms of the Wilson loop in each spin sector. For these derivations, the spin matrix in the subspace of occupied or unoccupied states is generalized by the unitary rotation to prevent the spectral gap from closing. In the presence of a finite spin gap, the spin eigenvectors recompose the eigenstates of the Hamiltonian in two spin sectors. A dimensional reduction leads to the charge polarization in each spin sector in terms of the Wilson loop along the effective Brillouin zone (BZ) in one dimension (1D). A linear combination of the charge polarizations defines the spin polarization, which naturally quantifies the spin Chern number in momentum space. It is also shown that the spin Chern number is directly derived from the local Wilson loops on a discrete 2D BZ. The theoretical treatment is further generalized for real space in a supercell framework, which can deal with ordered and disordered systems in a unified manner. In this framework, the spin Chern number in real space is formulated in terms of the Wilson loop around the BZ boundary. These formulations are used to numerically evaluate topological phases in the Bernevig-Hughes-Zhang model. Numerical results show an equivalence of the spin Chern number and the ${\mathbb{Z}_2}$ index.

The object of interest is the maximum number, $h(n)$, of Hamilton cycles in an $n$-tournament. By considering the expected number of Hamilton cycles in various classes of random tournaments, we obtain new asymptotic lower bounds on $h(n)$. The best result so far is approximately $ 2.85584...$ times the expected number $g(n)$ of Hamilton cycles in a random $n$-tournament, and it is conjectured that $h(n)\sim cg(n)$ where $c\approx 2.855958$. The same statements hold for Hamilton paths.

Video generation is moving from the cloud to phones for privacy and offline use, and applications often ask for several clips at a time, so on-device video generation is a throughput problem before it is a latency problem. Contemporary mobile SoCs carry both a GPU and a neural processing unit (NPU) on shared DRAM, yet existing on-device video diffusion systems run the denoiser on one engine while the other sits idle or only decodes the finished latent afterwards. Heterogeneous designs built for mobile LLMs do not carry over: they synchronise the two engines at every operator and partition weight tensors between them instead of sharing one stored copy. We first measure the platform limits that decide what can run on this class of SoC: how large one buffer may be, whether both runtimes can map the same buffer, and how much activation memory the NPU toolchain lets one graph hold. Guided by these measurements, we store the base weights once in INT8 in the layout the NPU reads natively; the GPU consumes the same bytes directly as integer dot-product operands, so neither runtime rearranges weights on load. We then flatten the denoising trajectory into its full sequence of block executions and cut it once at a block boundary, so the GPU runs the prefix and the NPU the suffix. The cut is placed from measured rather than modelled rates, including the decoder that only the NPU can run, and taken while both engines run together because each engine's speed changes when the other is active. We present HeteroVDiff, which serves a multi-expert video diffusion model across the NPU and GPU with 48% less weight memory than giving each engine its own copy. Generating videos back to back on a Galaxy S25 with decoding inside the pipeline, HeteroVDiff sustains 1.35x the stream throughput of NPU-only execution, and on VBench-2.0 it matches FP32 in the total score over all 18 dimensions.

Bagging, which refers to the repeated subsampling of data and averaging the resulting outputs, is a common approach to reduce sampling variability of unstable learning and data-driven optimization procedures. However, uncertainty quantification for these bagged estimators, namely to generate confidence intervals for their target quantities, faces challenges in the simultaneous attainment of coverage validity while being computationally affordable. The latter refers to the practical consideration of overall resampling effort which, when using conventional bootstraps for instance, would lead to nested resampling procedures and thus can be multiplicatively large. In this paper, we develop an interval construction approach that is both asymptotically coverage-valid and computationally light in that it does not use much more than the resampling effort in the bagging scheme itself. Our approach builds on the recent so-called cheap bootstrap idea, a method designed to substantially reduce the number of resamples based on an integrative treatment between data noises and Monte Carlo resampling noises, and an in-depth derivation of the algorithmic configurations under which the Monte Carlo error from incomplete bagging is negligible. Our theory shows that our required computation budget is significantly smaller than existing methods in relation to the bag size and number of Monte Carlo subsamples. Numerical experiments in random forests, as well as an optimality bound estimation in stochastic optimization where bagging is useful, show that our proposed intervals achieve reliable coverage with low computation cost.

World-action models should preserve the intended manipulation across equivalent descriptions of a spatial goal. We evaluate this consistency in 864 rollouts from three models across four RoboLab scenes, holding initial states and task goals fixed. Describing relations through the reference object rather than the object-to-move lowers the rate of reaching a stable placement by 12.2 percentage points on average. The effect of instruction rephrasing varies across spatial relations, so aggregate scores can hide sensitivity to wording. Standalone tests of the policies' language understanding identify relation reversals and show that adding images can disrupt otherwise correct text-only interpretations. Rollout analysis also distinguishes spatial placement from instruction following: moving the wrong object can satisfy the requested relation, and rates of stable placement fall from 43.2% at any time to 21.8% at episode end. These findings motivate a more thorough evaluation of world-action models' steerability, language understanding, and grounding. Evaluations should test consistency across equivalent instructions, verify that the intended object moved, and check whether the requested arrangement persists to the end of the rollout.

We give negative answers to two questions of Farjoun about localizations of groups. For every odd prime $p$, we construct a finite $p$-group $G$, an infinite non-nilpotent group $H$, and a local embedding $f\colon G\hookrightarrow H$. The associated localization satisfies $L_f(G)\cong H$, showing that localizations need not preserve nilpotence and that the localization map $G\to L(G)$ need not be surjective when $G$ is a finite $p$-group.

The Recursive Projection Aggregation (RPA) decoder for Reed-Muller codes uses, in its default implementation, all one-dimensional subspaces at each level of recursion. We restrict our attention to the case where the number of errors is $t<d_{\mathrm{min}}/2$. It is known that if the number of errors is $t<d_{\mathrm{min}}/2$, then $2t+1$ subspaces at each level of recursion are sufficient to correct these errors. We show that, for structured error patterns, the number of subspaces required for correction can be reduced further by appropriately selecting the subspaces used for projection. In particular, when errors occur as a single burst of length $t < d_{\mathrm{min}}/2$, three subspaces at each level of recursion suffice for correction. We obtain similar results for multiple bursts and other structured error patterns as well.

We study classical Bopp–Podolsky (BP) electrodynamics on four-dimensional de Sitter spacetime, focusing on its transverse retarded Green function and gauge-invariant electromagnetic response. Because de Sitter spacetime is maximally symmetric, its vector Green functions admit a covariant expansion in a finite basis of invariant bitensors, with scalar coefficients depending only on the invariant separation. For conserved sources, the fourth-order field operator factorizes into Maxwell and massive-vector operators, yielding the BP retarded kernel as the difference of their transverse Green functions. Using the invariant bitensor decomposition, we reduce the propagator equations to a hypergeometric master equation and determine the distributional structure of its solution. The common leading short-distance singularity of the Maxwell and massive-vector sectors cancels, leaving a logarithmic master singularity and a softened vector-potential wavefront. After one curl, the gauge-invariant field-strength kernel contains a null-cone delta distribution and a regular timelike tail. The smooth Maxwell master tail is annihilated by the curl, so the interior electromagnetic tail is generated entirely by the massive sector. We also derive an exact horizon-regular static potential, demonstrate the cancellation of the Coulomb divergence, recover the flat-space BP potential, and formulate the generalized Gauss law. On global de Sitter spacetime, the compactness of the spatial sections requires the total charge to vanish; we implement this constraint using an antipodal charge configuration with smooth factorized initial data. The analysis is classical and does not assume a quantum Lee–Wick contour or a preferred de Sitter vacuum.

LLM agents increasingly control robots using VLA policies, perception models, and procedural knowledge. These modules are difficult to reuse because their interfaces, embodiment assumptions, and conditions of use differ, and their benefits on physical tasks remain unclear. We package them as embodied skills that extend agent skills with explicit embodiment assumptions and conditions of use, building 250 action, perception, and experience skills from 1,052 resources. On this basis we introduce EmbodiedSkillBench, which runs an agent in a closed observation-action loop and records success, actions, tokens, and wall-clock time. It supports three modes: native control, skill-only execution, and Agent-Directed Reuse (ADR), a mode we propose in which the agent keeps its native controls and decides for each skill whether to invoke, modify, or ignore it. Across four agents on 100 perturbed LIBERO-Pro tasks, we find that skills help an agent only as far as they exceed what it can already do. Skill-only execution lifts GPT-5.6-Sol from 11% to 63% and Kimi-K3 and Claude-Opus-5 by about 20 points, while cutting their token usage by 58% to 85%. Yet all four agents then end up between 61% and 64%, and GPT-6-Astra falls from 92% under native control to 61%. This suggests that as agents grow stronger, the role of embodied skills shifts from supplying missing capability to reducing cost: GPT-6-Astra already solves most tasks through native control, so restricting it to skills removes more than it adds. Under ADR, which lets it keep native control and call skills selectively, GPT-6-Astra recovers to 85%, within 7 points of native control, while using 14.1% fewer tokens and 11.9% less time. With Jev as a skill choosing motion directions and gripper actions, it reaches 81% with 50% fewer tokens and 57% less time. The code is available at https://github.com/HanZhi306/EmbodiedSkill-Bench.

Originally coined 100 years ago to represent a shear-reversible sol-gel transition, the range of the term "thixotropy" has gradually expanded to encompass time-dependent changes in viscosity. Since this risks confusing "thixotropy" with ordinary nonlinear viscoelastic behavior, in this work we seek simple, historically faithful, quantifiable, and experimentally testable definitions of "thixotropy" that avoid this confusion. By analysing both historical tradition and published experimental data, we arrive at three complementary definitions. Definition 0 provides an idealized, purely inelastic reference useful primarily for recognizing thixotropy in a constitutive equation. Definitions 1 and 2, which are more practical, require, respectively, unusually large strains to reach steady state, and unbounded growth of the material relaxation time during rest. Definition 1 can apply in either shear-rate or stress-controlled modes, with the latter encompassing the so-called "viscosity bifurcation." These criteria are tested across diverse materials that we identify as thixotropic based on their responses clearly exceeding their viscoelastic reference scales. The framework is deliberately phenomenological and does not require specific microscopic mechanisms, allowing diverse materials to be described within a common rheological language, as is typical for rheological terminology. We believe that these definitions provide a practical framework for assigning thixotropy a more definite place in rheological taxonomy.

Morphogenesis is driven by the interplay of genetic regulation and tissue mechanics. In vivo video data is limited, particularly under genetic mutations and environmental perturbations, making generalization beyond the most representative conditions in the dataset challenging. However, at tissue scales, the dynamics of Drosophila can be described as a continuum flow, motivating the usage of representations learned across other physical systems as priors for developmental dynamics. We investigate finetuning Walrus, a Continuum Dynamics Foundation Model pretrained across a broad set of simulated continuous systems, on tissue velocity fields from only 12 wild-type Drosophila blastoderm videos. We evaluate autoregressive prediction both in-distribution and zero-shot under unseen temperature and genetic perturbations. Zero-shot refers to evaluation on perturbation conditions never seen during wild-type finetuning, without any additional training. In-distribution, a Fourier Neural Operator model trained from scratch achieves best performance on VRMSE. However, Walrus (CRPS) is the strongest model under the most severe distribution shift of unseen samples of mutated embryos and temperature conditions. Interestingly, predictions capture the shape of the mean surface tissue velocity trajectory and its separation between 17{\deg}C and 27{\deg}C, including the timing of the peak at Germ Band Extension onset, a biologically meaningful milestone of embryo development. Our results provide preliminary support that representations learned from different dynamical systems can transfer to experimental morphogenesis in developmental regimes, provided they admit a continuum description. More broadly, our results motivate further study of physics foundation models as a new complementary tool for the simulation of morphogenesis in low-data regimes.

In this paper, we study perfect codes in bi-Cayley graphs, focusing on when two subgroups $H$ and $K$ of a group G can form a perfect code of the form $H_0 \cup K_1$. We first consider the problem for a general group G and then examine what happens for special classes of groups, particularly abelian groups. Along the way, we obtain several results, including a necessary and sufficient condition for normal subgroups to form a perfect code, as well as an equivalent condition for abelian groups in terms of their Sylow 2-subgroups. Finally, we study the same problem for dihedral groups, providing results for this important class of non-abelian groups.

Visual grounding is a central challenge for GUI agents, requiring an instruction to be mapped to a target within a crowded screenshot. Coordinate generation with single-point supervision reduces a spatially extended target to one position, despite the presence of many valid click locations. Region-based patch scoring alleviates this ambiguity, but its spatial decisions remain tied to visual-token footprints that may span the target, neighboring controls, and background. To address these limitations, we propose Seg2Click, which reformulates GUI grounding as executable-region prediction: identifying where a click can validly execute the given instruction. Using bounding-box supervision, Seg2Click predicts an executable-region mask and computes the final click point through a deterministic rule. This formulation captures the spatial extent of valid click locations and enables finer localization beyond the visual-token grid, making it well suited to small, densely arranged controls without requiring a verifier or zoom-in pass. Experiments across five backbone configurations from two model families demonstrate the effectiveness of Seg2Click. Seg2Click-9B achieves 92.06%, 94.10%, and 55.22% on ScreenSpot, ScreenSpot-v2, and ScreenSpot-Pro, respectively, while matched-backbone comparisons show improvements over both coordinate generation and patch-based grounding. Code will be available at https://github.com/jcwang0602/Seg2Click.

Understanding the relationship between generalization and hallucination requires clarifying the role of memorization. We study this relationship from an information-theoretic perspective through a Selector-Density Conditional Mutual Information (CMI) framework. In the sparse random-fact limit, a uniformly bounded information budget per fact forces a nonvanishing false-positive rate at recall bounded away from zero, even when all facts are observed during training. In a rules-and-facts setting, we decompose model-data mutual information into rule learning, atomic-fact memory, and excess memorization, revealing competition between rule learning and factual memory even without excess memorization. A joint budget bound quantifies the resulting constraint on atomic hallucination under perfect atomic recall. Moreover, an additive Gaussian model shows that maintaining high recall for atomic and composite facts incurs false positives even under optimal data mixing. Finally, controlled relational experiments hint how training exposure, rule complexity, and model capacity shape these trade-offs.

In-silico optimization of nucleic acids relies on predictive oracles that imperfectly capture biological function, so sequences optimized against one model may exploit model-specific artifacts and fail to generalize to independent models. Recent studies have shown that not all designers generalize equally well to independent models. In this work, we use agentic program search to iteratively propose, execute, evaluate, and revise code for nucleic-acid designers. We describe the successful strategies used to prompt the search. AutoNovo designers achieved the best observed mean among the evaluated methods on 3/3 DNA-design tasks in terms of in-silico performance, and 6/6 transfer tasks in terms of independent models. The designers come from a family of discovered nucleic acid designers rather than a single universal best designer. We plan to release the resulting designers. These results are the first steps in creating nucleic acid designers that maximize cross-model generalization, and thus improving the odds that in-silico designed sequences reflect real biology.

As large language models are increasingly adopted as multi-party communication facilitators, there is a growing demand for agents capable of balancing diverse group dynamics across contexts. However, existing LLM facilitators are often fine-tuned for task-specific objectives; generalized LLM-based facilitation methods that can adapt to different types of tasks and diverse group dynamics are underdeveloped. To address this gap, we present GUIDE (Generalized Unified Intervention for Dynamic Exchanges, a framework for developing generalized LLM-based facilitators across different multi-party contexts. GUIDE evaluates facilitation across both objective task outcomes and subjective metrics, including psychological safety and perceived friction. Within this framework, we study whether LLMs can self-learn generalized facilitation strategy offline to real-time interventions. We first collect multi-party human interactions guided by task-specific expert LLM facilitators (N=432 conversations). We then employ a self-learning method in which an LLM induces an actionable strategic playbook from conversations and iteratively refines it using feedback from an LLM judge aligned with human annotators (N=20), developing a single generalized facilitator that outperforms the three task-specific expert facilitators on held-out conversations 61.4% win rate. Finally, we evaluate these improvements empirically; in a real-time online empirical study, the self-learned generalized facilitator preserves facilitator performance and group productivity, improves speaking comfort and reduces interpersonal friction, while using 24.7% fewer turns and 29.0% fewer words than the expert prompts. These findings provide empirical evidence that LLMs can learn generalizable, task-agnostic social facilitation from task-specific human interactions, on-par with hand-crafted task-specific prompts across diverse group settings.

This work studies the relationship between the multigrading in the cohomology of moment-angle complexes, introduced by V. Buchstaber and T. Panov in their computation of these cohomology groups, and the multigrading in the cohomology of the corresponding complements of coordinate subspace arrangements, arising from the mixed Hodge structure. It is shown that these multigradings coincide. This result refines a theorem from Yu. Eliyashev's 2013 dissertation. Furthermore, results on the mixed Hodge structure in the cohomology of complements of coordinate subspace arrangements are applied to the study of the same varieties over finite fields. In particular, various formulas are obtained for the number of $\mathbb{F}_q$-rational points and for the associated global $\zeta$-functions.

Harnessing rich modality features has recently gained prominence in multi-modal recommendation systems (MRSs). To delineate user modality preferences, existing works focus on modality fusion, which can be achieved via attentive summation or concatenation. In this work, we introduce the modality inhibitance effect - a phenomenon which limits existing MRSs when these models transit from dual-modal to tri-modal configurations resulting in significant performance degradation. In particular, apart from the modal-specific contamination encapsulated in the newly introduced modality, the inclusion of new features results in excessive informational redundancy, thereby severely impeding effective integration within current approaches. To address the modality inhibitance effect, we propose a new multi-scale modality wavelet (MMWave) multi-modal recommender, which aims to fully capitalize on essential modality features and suppress modal-specific noise. This is acheived by employing wavelet decomposition from a multi-scale (time & frequency) perspective, ensuring that only essential (non-redundant) features of each modality are extracted. We then account for the latent associations between item modality features by designing a new structure-aware multi-modal graph learning module that captures high-order latent structural information pertaining to items. To modulate the relative contributions of modality features and achieve effective integration, we then propose a new adaptive modality regulator component that employs a new cross-weight integration strategy. Experiments conducted on four real-world datasets validate the efficacy of our proposed model. The source code for this work has been made publicly available at https://github.com/kennethorq/MMWave.

We study the incompressible Oldroyd-B system on $\mathbb R^d$, $d\ge2$, in the regime in which the velocity is the only dissipated quantity: the stress tensor undergoes neither relaxation nor diffusion. In this regime the operator $\mathbb P\operatorname{div}$ acting on symmetric stresses has an infinite-dimensional kernel consisting of exact stationary states, so no decay estimate can hold for the full stress tensor and the existing sharp-decay theory for dissipative systems does not apply. We identify the active pair $U=(u,\Lambda^{-1}\mathbb P\operatorname{div}\tau)$ as the correct decaying observable and establish, in a critical hybrid Besov framework, a complete characterization of its large-time behavior. For $-d/2<\sigma_1<\sigma\le d/2-1$ we prove that $\|U(t)\|_{\mathcal H^{\sigma}}=O(t^{-(\sigma-\sigma_1)/2})$ holds if and only if the low-frequency part of $U_0$ belongs to $\dot B^{\sigma_1}_{2,\infty}$, and, for $\sigma<d/2-1$, that the two-sided estimate $\|U(t)\|_{\mathcal H^{\sigma}}\simeq t^{-(\sigma-\sigma_1)/2}$ holds if and only if the low-frequency part of $U_0$ belongs to a nondegenerate subclass $\mathfrak B^{\sigma_1}_{2,\infty}$ of $\dot B^{\sigma_1}_{2,\infty}$.

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\leq p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.

Network decision-making tasks, including adaptive bitrate streaming, load balancing, and congestion control, require policies to process heterogeneous states and actions while balancing near- and long-term consequences. Existing solutions generalize poorly across network tasks, require substantial task-specific redesign, and lack reliable evaluation of long- horizon action consequences. Methods that learn through online interaction additionally face unstable optimization. We present NetRL, a two-stage LLM-based reinforcement learning framework that combines supervised fine-tuning on demonstration trajectories with online policy optimization. A universal mixed-embedding interface combines textual task descriptions with precise numerical states, while a network action head produces valid decisions. At its core, NetGRPO evaluates candidate actions over multiple future trajectories, directs deeper search toward uncertain alternatives through confidence-aware budgeting, and performs group-relative, clipped, and KL-regularized updates. NetRL achieves 23.6% higher QoE than NetLLM, 12.2% lower job completion time than Genet, and 24.1% higher network utility than Orca, yielding state-of-the-art performance among the evaluated methods. Confidence-aware budgeting reduces end-to-end training time by 27.1% without reducing validation QoE. NetRL also converges stably and generalizes to unseen network traces without retraining. Code and model weights are available at the repository linked in this paper.

Let $p\geq11$ be a prime, and let $\bar\rho:\Gal(\bar\QQ_p/\QQ_p)\rightarrow\GL_2(\bar\FF_p)$ be a reducible and very generic representation. In this paper, we use ghost series \cite{Lghost2} to study the irreducible components of Kisin's potentially crystalline deformation rings of $\bar\rho$. For wildly ramified principal series inertial types and arbitrary regular Hodge–Tate weights, we distinguish the components by the crystabelline slopes on their generic fibers and determine the total number of irreducible components. Combining this with an explicit formula for the total Hilbert–Samuel multiplicity of the special fiber obtained from the Breuil–Mézard conjecture, we compute the Hilbert–Samuel multiplicity of the special fiber of each component. Every such multiplicity is one when $\bar\rho$ is non-split, and is either one or two when $\bar\rho$ is split, according to whether the generic fiber of the corresponding component parametrizes reducible or irreducible Galois representations.

Text-to-image models have made remarkable progress in visual fidelity and instruction following, yet complex compositional generation remains challenging. Existing agentic approaches address this limitation through iterative generation and editing, adapting their actions based on interaction history. However, they typically fix the continuation state (i.e., the image from which refinement proceeds) to the latest image. Since iterative editing is not necessarily monotonic, the agent may continue refining from the latest image even when it has already degraded. In this work, we introduce GenRevisit, an agentic framework that extends action selection to joint continuation-state and action selection. The agent can not only use interaction history to adjust editing instructions but also select an image candidate to edit, jointly deciding where to continue from and how to refine. We train GenRevisit with SFT followed by agentic reinforcement learning. To address the coarse credit assignment of final outcome reward in GRPO, which assigns the same trajectory-level advantage to all actions, we further propose Normalized Action Credit Assignment (NACA) to redistribute positive advantage according to each action's contribution to reducing the remaining objective. On GenEval2, GenRevisit achieves GM scores of 84.3 and 74.4 with Qwen-Image-2512 and Z-Image-Turbo, surpassing the strongest baselines using the same generators by 12.7 and 10.9 points respectively. GenRevisit achieves state-of-the-art performance on GenEval, GenEval++, and GenEval2 among baselines.

Commercial balancing wheels are closed systems: an internal controller keeps the board upright, and a rider steers by shifting weight and twisting the hips. We augment a one-wheel with a lead-screw weight shift for longitudinal motion and a reaction wheel for yaw. The reaction wheel stores limited momentum, so a turn is a sequence of torque pulses, and whether a pulse rotates the body depends on static friction at the tire contact. From a 24-pulse torque sweep on one surface we fit a two-part model that gives the probability that a pulse rotates the body, treating the breakaway torque as random, and the rotation angle when it does; at maximum torque the posterior engagement probability is 0.54 (90% credible interval 0.24-0.84). The weight shift is modeled as a rate- limited input with a measured speed gain and an assumed first-order response, which predicts stopping distances four times those of a direct-force model. These models drive an event-triggered supervisor that schedules pulses by expected heading improvement, brakes when the predicted braking path would leave the safe set, tightens margins for model uncertainty, and updates the friction estimate online. In simulation on 500 plants drawn from the fitted models, the robust supervisor completed 92.6% of missions with 1.8% safety violations and 1.6% safe aborts, versus 84.0% and 7.2% for a supervisor using mean models and 1.8% and 20.4% for a baseline that ignores friction. A video of the hardware experiments is available online (https://github.com/ASAS-Labs/x-wheel).

We study the exponential map on the group of real two-by-two matrices with determinant one, endowed with a left-invariant Riemannian metric. Using the Euler-Poincaré equation and its linearization, we determine the conjugate and Maxwell times and identify a domain on which the exponential map is a global diffeomorphism onto the complement of the Maxwell stratum. This provides an alternative proof that the first Maxwell time coincides with the cut time. We obtain an explicit matrix description of the cut locus and characterize the globally minimizing geodesics. Furthermore, we analyze the first Maxwell-time function, determine its unique global minimum, and show that the injectivity radius is $\frac{\pi \sqrt{14}}{4}$. Finally, we characterize the intersection of the cut and conjugate loci and prove that every geodesic is equi-optimal.

We establish a family of sharp higher order isoperimetric type inequalities for smooth compact surfaces in Euclidean spaces of arbitrary codimension. These inequalities involve arbitrarily high derivatives of the boundary curve, together with the mean curvature vector of the surface. For example, the first and second order inequalities yield the sharp two-sided estimate for the isoperimetric deficit: \begin{align*} & \frac{\pi}{R}\int_{\partial M}|x-R\nu|^2\, ds +2\pi\int_M x\cdot H\, dA \le L^2-4\pi A\\ \le& \frac{\pi}{R}\int_{\partial M}|x-R\nu|^2\, ds +2\pi\int_M x\cdot H\, dA +\frac{\pi}{3R}\int_{\partial M}|x+R^2\kappa|^2\, ds, \end{align*} where $R=\frac{L}{2\pi}$ and the origin is chosen to be the centroid of the boundary. For minimal surfaces, equality in the lower bound holds precisely for flat disks, whereas the second inequality admits non-round equality cases. Our results extend that of Carleman, Sachs, Chakerian, and Li-Schoen-Yau. A distinctive feature of our work is the unexpected rigidity structure arising at higher orders. At the first and second orders, equality curves admit decompositions into circular Fourier modes supported in mutually orthogonal planes. Remarkably, this phenomenon fails at higher orders. We establish a multiplicative factorization of the unit tangent map into elementary planar rotations acting on a fixed unit vector, providing a characterization of equality cases at arbitrary order. As a result, we construct a third order equality curve in $\mathbb R^3$ whose Fourier modes cannot be decomposed into mutually orthogonal circular modes.

Participant-disjoint cross-validation prevents direct participant overlap, but it may still produce optimistic estimates when acquisition site and diagnosis are structurally dependent. We examined Parkinson's disease classification using 159 pressure-walkway summary features from 133 WearGait-PD participants, including 65 participants with Parkinson's disease and 68 controls. The cohort was non-factorial: two sites contained only one diagnostic group, whereas VA-Seattle was the only site containing both groups. In repeated participant-level nested cross-validation, site alone achieved an area under the receiver operating characteristic curve (AUC) of 0.955 plus or minus 0.003, exceeding the gait-feature model (0.882 plus or minus 0.014). Under the evaluated logistic-regression pipelines, adding gait features to site, age, and sex produced no measurable improvement in pooled discrimination. More site-controlled evaluations produced lower and imprecise estimates: VA-Seattle same-site AUC was 0.668 (95% bootstrap confidence interval 0.477 to 0.836), and the JHOC + Bayview-to-VA-Seattle transport stress test achieved 0.576 (0.375 to 0.770). Supporting within-diagnosis site-prediction and site-conditioned permutation analyses showed that the gait representation retained substantial acquisition-site structure. These findings indicate that participant-disjoint pooled validation did not isolate disease-related discrimination from site-associated variation. They encourage careful examination of site-label dependence and, where possible, support for pooled results through within-site or independent validation before broader claims are made about generalization across institutions.