Papers

Motivated by the work of Druţu–Sapir, we introduce a definition of relatively hyperbolic TDLC groups that depends only on their Cayley-Abels graphs and show that this new definition is equivalent to one introduced by Arora–Martínez-Pedroja. Then, we prove that relative hyperbolicity for TDLC groups is invariant under quasiisometry.

We construct an exactly solvable extension of the spin-$1/2$ Kitaev honeycomb model in which conserved $\mathbb{Z}_2$ fluxes determine not only the signs but also the connectivity of nearest-neighbor Majorana hopping. At a tuned loop point, hopping survives only across opposite-flux plaquettes, so every vertex has active degree zero or two and the matter Hamiltonian fragments in each flux sector into independent Majorana rings and isolated zero modes. Exact ring spectra and bond counting then bound the matter energy over all local flux configurations and all four Wilson-loop sectors, and split it exactly into a frustrated triangular-lattice Ising term, whose extensively degenerate ground states are the fully packed loop coverings, and a non-negative Majorana residual. On admissible commensurate tori, the residual selects precisely the three translation-related $2/3$-vison crystals, which saturate the bound, and an exact fermion-parity identity shows that for antiferromagnetic coupling their vacua also survive projection, making them rigorous ground states of the spin model. Within a single crystal, a depth-one local unitary then maps the ground space onto that of a sheared square-lattice toric code, giving fourfold topological degeneracy. The model thus realizes exact nonperturbative gauge-matter feedback: the flux fixes where the Majorana fermions may move, and their zero-point energy selects in return a topologically ordered vison crystal with spontaneously broken translation symmetry.

Determining isotopic ratios on the surfaces of extraterrestrial objects can provide insight into their formation conditions and evolution. Recently, remote-sensing detections of the 4.14 {\mu}m HDO absorption band have enabled astronomers to estimate the D/H ratio on several icy objects in the outer solar system. Here, we quantify how ice phase and radiation processing affect the 4.14 {\mu}m absorption band. While the band is visible in H2O-ice samples with a crystalline component, we cannot detect it in amorphous H2O-ice. Furthermore, we find that radiation processing, using 10 keV electrons as a proxy, quickly makes this absorption feature undetectable at a rate consistent with the amorphization of an initially crystalline sample. Interestingly, we also find that recrystallizing the irradiated sample causes this band to reappear nearly to its original band depth. For the Saturnian satellite Mimas, we estimate that the HDO feature will decrease by a factor of two within ~1 x 10^4 yrs at typical depths probed by remote sensing spectroscopy. While we suspect that existing methods could still determine the D/H ratio from a surface composed entirely of crystalline H2O-ice, deriving it for surfaces with a significant amorphous fraction may lead to severe underestimation of the true D/H ratio. However, given the direct correlation between the amorphous fraction of the sample and the HDO band depth, we propose that one could still estimate the D/H ratio of a surface with mixed phases by determining the crystalline fraction of the surface H2O-ice using other absorption features.

We prove sharp metric entropy estimates for spaces of Kähler potentials. In complex dimension $n$, normalized potentials have Kolmogorov entropy of order $\e^{-n}$ in the background $L^1$ metric. On a polarized manifold, a relative-entropy sublevel has the same order in the Mabuchi–Darvas $d_1$ metric, including its full finite-energy closure. The upper bound is $C_X(1+B)^{n+1}\e^{-n}$ for entropy budget $B$. For toric potentials, the sharp exponent is $n/2$.

Despite strong performance on standard benchmarks, it remains unclear whether large language models (LLMs) evaluate social pragmatics in ways that align with human judgments. We evaluate LLM politeness judgments using two English-language datasets with complementary annotation formats: continuous human ratings and three-way categorical labels. Across the seven evaluated models, we find that inter-model agreement is stronger than model–human agreement. Strategy-level analyses suggest that model–human alignment is associated with explicit linguistic cues, while some rapport-building strategies occur more frequently in misaligned cases. In the categorical task, model predictions exhibit systematic neutral compression, characterized by the overproduction of Neutral labels and the underprediction of Impolite labels. This pattern persists when expert consensus is used as the reference on a diagnostic subset. Our findings highlight the need for pragmatic evaluations that go beyond aggregate agreement metrics by examining directional patterns of model–human disagreement across different human references.

Robot policies deployed in real environments naturally accumulate mixed-quality experience, including successful executions, partial progress, and failures. Although these rollouts provide valuable information for further learning, directly incorporating them into imitation learning may reinforce undesirable behaviors, while offline reinforcement learning often suffers from unreliable value estimation under sparse rewards and limited data coverage. We consider a practical post-deployment setting where learning relies only on naturally accumulated autonomous rollouts, without additional human corrections or exploratory interaction. To effectively exploit such experience, we propose Predictive Action Chunk Learning (PACL). PACL first learns a predictive chunk-level critic that evaluates temporally extended action sequences and augments temporal difference learning with future latent prediction, providing richer supervision for long-horizon value estimation. The learned critic then converts chunk-level Q-values into discrete quality conditions, which guide a diffusion actor to learn jointly from these mixed-quality experiences without treating all behaviors as equivalent supervision. At inference, the actor generates multiple action chunks and the critic selects the highest valued candidate. Experiments across simulated and real-world robot manipulation tasks show that PACL consistently improves the pretrained policy and outperforms strong imitation learning and offline reinforcement learning baselines.

The simulation of nonlinear ordinary differential equations on quantum computers is inherently challenging, as quantum gates are linear operators on qubit states. In this paper, we put forth a Koopman-von Neumann (KvN) operator based algorithm for solving nonlinear ordinary differential equations on a quantum computer which overcomes the innate limitations of quantum operations. In this approach, a Liouville probability density is embedded into a wavefunction, the evolution of which is governed by an operator dual to the Koopman operator. The trajectories corresponding to the nonlinear differential equations are reconstructed from the average value of Koopman observables evaluated through quantum measurements. We specifically evaluate the computational limitations of solving nonlinear equations within this Liouville embedding framework. An Ehrenfest-type estimate is derived that relates the reconstruction error to the covariance of the transported density, highlighting the competition between linear stretching and Hessian-induced folding.This leads to a key stability criterion, which depends on the local Ehrenfest-Reynolds number. We discuss the effects of measurement-induced errors including those due to Hadamard-test sampling, amplitude estimation, and bias due to covariance in probabilistic Grover-type inference on the evaluation of trajectories. The limiting bounds on the accuracy of quantum computations are numerically verified for the Lotka-Volterra system and for the quartic oscillator. We observe a rapid growth in computational errors when the Ehrenfest-Reynolds number approaches the predicted threshold. The bounds resulting from our analysis provide key guidelines for selecting the width of the initial Gaussian probability distribution associated with a system of nonlinear ordinary differential equations.

Low-rank adaptation (LoRA) has become a popular parameter-efficient fine-tuning method for large language models. A key challenge in LoRA is how to determine the rank of each adaptation matrix, as rank directly controls its capacity and efficiency. Existing adaptive-rank methods typically allocate ranks according to manually designed importance scores, which are not directly derived from an optimization objective. In this work, we propose $\ell_p$-LoRA, a principled rank-allocation method based on $\ell_p$ regularization with $0<p<1$, which is a classical sparsity-inducing technique in signal processing and statistics. Specifically, we regularize the energy of each rank-one LoRA component, encouraging redundant components to vanish while preserving important ones. We derive the corresponding proximal subproblem and reduce the matrix optimization to a two-dimensional problem, leading to an implicit thresholding criterion for identifying redundant components. Experiments on natural language understanding and question-answering tasks demonstrate that the proposed method achieves competitive performance with existing LoRA baselines.

Most of the memory of a 3D Gaussian Splatting model holds spherical-harmonic colour coefficients, yet each Gaussian is seen only from the narrow cone of directions of the training cameras. We turn this into a distortion metric that other compressors can adopt: a per-Gaussian observation Gram matrix, accumulated from viewing directions and blending weights, is the exact first-order map from coefficient changes to squared image error and needs only the model and the camera poses. Under it, degree reduction becomes a closed-form projection that generalises truncation, degree allocation a Lagrangian rate-distortion problem, and vector quantisation the matrix-weighted Lloyd algorithm, of which Compressed3D's quantiser is the scalar case. Swapped into Compressed3D with everything else unchanged, the metric raises PSNR by +0.49 dB before fine-tuning, with SSIM and LPIPS following, and at matched rate still gains +0.32 dB without a single training image. A training-free stack built on the metric alone is 15% smaller than the image-free GSICO at equal quality on Mip-NeRF 360.

The Node.js ecosystem heavily relies on NPM packages, and software supply chain attacks targeting malicious NPM packages are rampant. Malicious code primarily triggers during package installation, import, and runtime. Traditional static analysis fails to understand code semantics; machine learning-based methods rely on feature extraction, which suffers from concept drift; existing LLM solutions suffer from high invocation costs, high data security risks, and poor performance. To overcome these limitations, we propose DistillGuard, a lightweight malicious NPM package detection framework that combines static graph semantic analysis and LLM knowledge distillation. DistillGuard first acquires multi-granular features through three static analysis modules. Then, it leverages online LLM to distill high-quality security knowledge and structured labels. Finally, it uses LoRa to efficiently fine-tune the open-source Qwen3-8B model to support offline deployment. Experiments show that DistillGuard achieves an accuracy of 95.3%, a precision of 99.4%, and an F1 score of 93.8%, outperforming state-of-the-art tools, improving the F1-score by 11.1 to 30.0 percentage points over the baselines. Our empirical research further reveals the stages of malicious attacks and the distribution of malicious behaviors. We also summarized eight typical API attack chains for malicious activities, providing practical insights for NPM supply chain security defense.

We develop and analyze HDG methods for two central problems in finite element exterior calculus, the Hodge-Dirac problem and the Hodge-Laplace problem, in arbitrary dimension $n$. Our analysis allows for equal-order polynomial spaces with $\mathcal{O}(1)$ penalty parameters, by contrast with previous work on the Hodge-Laplace problem requiring an underlying conforming complex and $\mathcal{O}(h)$ or $\mathcal{O}(h^{-1})$ penalties. For the Hodge-Dirac problem with degree-$r$ polynomials, our error estimates give optimal order-$(r+1)$ convergence in all form degrees under suitable regularity hypotheses. For the Hodge-Laplace problem, we prove optimal order-$(r+1)$ convergence for the $k$-form solution and order-$(r+\frac{1}{2})$ convergence for the $(k \pm 1)$-form auxiliary variables. We obtain improved order-$(r+1)$ auxiliary-variable estimates when $k=1$ and/or $k=n-1$, which in particular sharpens some recent HDG error estimates for the two-dimensional vector Poisson equation. Our analysis also encompasses cases of lower regularity, including solutions with reentrant-corner singularities on non-convex domains, and requires only mild mesh regularity conditions. The results are illustrated by numerical experiments in dimensions two and three.

The dissipative transverse-field Ising (TFI) model provides a paradigmatic setting for nonequilibrium quantum many-body physics. We show that a class of boundary-driven TFI models subject to dissipation at only one boundary possesses hidden time-reversal symmetry, which enables an exact construction of their nonequilibrium steady states. The solution admits a matrix-product representation and defines a nonequilibrium partition function from which steady-state observables can be evaluated efficiently. We use the exact solution to characterize the microscopic structure of the steady state through its z-magnetization and two-point correlations. A striking feature is that the local field at the dissipative boundary governs the spatial organization of the steady state throughout the chain. The steady state typically exhibits boundary-localized magnetization profiles and exponentially decaying correlations, whose characteristic length scales are set by the dissipative boundary. In the weak-driving limit, suitably tuned boundary fields can reorganize the NESS into a delocalized single-interface structure, giving rise to long-range correlations that decay linearly with distance.

Let $R$ be a normalized root system in $\mathbb{R}^d$ with reflection group $G$, let $k$ be a $G$-invariant multiplicity function, and let $\mathcal{F}_k$ be the associated Dunkl transform. We write $k_\alpha=k(\alpha)>0$ and $\mathrm{d}\mu_k(x)=w(x)\,\mathrm{d}x$, where $w(x)=\prod_{\alpha\in R_+}|\langle x,\alpha\rangle|^{2k_\alpha}$. We study observability, Hölder-type interpolation, and spectral inequalities for the Dunkl heat equation on $\mathbb{R}^d$. We establish a Bernstein inequality for ordinary derivatives and a Logvinenko–Sereda theorem with a spectral constant of the form $e^{C(1+N)}$ for functions whose Dunkl transforms are supported in $\overline{B(0,N)}$. We characterize observable sets as the measurable sets that are thick with respect to $\mu_k$, and prove the equivalence of the observability, Hölder-type interpolation, and spectral inequalities.

We construct an off-shell BRST-invariant gauge-fixed formulation of a nonlinear sigma model coupled to a non-Abelian gauge field. Starting from the dynamics of GBs and their consistent couplings to gauge and ghost fields, we construct the corresponding BRST-invariant quantum theory and provide a geometrical interpretation in terms of Killing vectors. A central point of our treatment is that the Lie-bracket closure of the target-space Killing vectors provides the geometric realization of the algebra entering the BRST differential. The physical state space is characterized by the cohomology of the nilpotent BRST charge, providing a consistent separation of physical and non-physical degrees of freedom. We explicitly derive the BRST charge and present the associated symmetry transformations. Finally, we analyze the structure of the physical Hilbert space and discuss the conditions under which BRST symmetry supports unitarity and constrains the quantum effective theory. Quantum statements are conditional on a BRST-preserving measure and regulator, and the four-dimensional model is treated as an effective field theory.

Video understanding is increasingly performed by multi-stage LLM agents that separate temporal grounding, visual observation, and reasoning. Yet these stages are typically evaluated on different benchmarks and distributions, making it difficult to determine where hallucinations originate. We first organize existing benchmarks around these stages and show that their scores provide inconsistent diagnostic signals: stronger stage-level performance does not reliably imply lower downstream hallucination, and even benchmarks targeting the same capability can disagree. We therefore introduce a causal stage-intervention protocol that overwrites individual stages while holding the downstream task fixed. Across 60,008 runs on three video-agent architectures, we find that grounding is the dominant source of downstream error, with roughly four times the causal impact of corrupting visual observations. Successful grounding depends primarily on locating the correct region rather than precise temporal overlap, explaining why standard mIoU metrics poorly predict downstream reliability. We further find that incorrect evidence is substantially more harmful than missing evidence. Finally, auditing existing benchmarks against these interventions reveals that their scores do not reliably predict causal cascade sensitivity and can fail under distribution shift. These results motivate intervention-based, stage-aware evaluation for trustworthy video agents.

Generative AI chatbots are increasingly embedded in everyday life, yet most large-scale studies describe global patterns. This paper presents an Australia-focused analysis of WildChat, a public dataset of real-world ChatGPT interaction logs. Using descriptive analysis and a multi-layer classification scheme, we analysed 37,845 conversations identified as Australian, examining language diversity, work relevance, interaction intent, topic distribution, turn-taking, temporal change, work activities, and Australia-related domains. Our findings show that the Australian subset is strongly action-oriented and comparatively work-oriented, with most interactions classified as doing and a majority of conversations classified as work-related. The dataset also shows multilingual use and a growing presence of self-expression over time. Australia-related conversations frequently invoke local institutions, laws, regulators, education systems, companies, cultural references, and public services. Finally, we outline implications for future research, including local AI evaluation, multilingual participation, context-aware design, and safeguards for everyday high-stakes domains.

In this paper, we study the asymptotic behavior of the extremal process of a standard one-dimensional branching Brownian motion (BBM) with drift $-\rho>-\sqrt2$ and absorbing barrier at level $-x$. We prove that the two-dimensional point process, with first component being the extremal process of the BBM and the second component being the running minimum of the BBM with drift, converges weakly to a decorated Poisson point process (DPPP) on $\mathbb{R} \times [0, \infty)$. This framework allows us to supply a common limit coupling of the extremal processes for different barrier levels and to explicitly derive the limit, as $t\to\infty$, of the extremal process of the killed BBM killed at level $-x$, demonstrating that the double limit, when $t\to\infty$ first and then $x\to\infty$, of the extremal process of the BBM killed at level $-x$ coincides with the limit of the extremal process of (un-killed) BBM.

We present a personalized Korean visual speech recognition (VSR) system and quantify, on the nine-camera OLKAVS corpus, the gap between the population-level benchmark score and an individual user's error. A video-only Conformer initialized from English-trained weights attains 9.95 - 12.19% character error rate (CER) under the corpus protocol against the published 26.64, and 19.00 - 21.52 on unseen wording. Per speaker, CER spans 1.0 to 52.2%, with seen wording lowering CER by 7.0 - 9.0 points and professional delivery and spontaneous speech raising it by 8.5 - 10.5 and 12.7 points. A low-rank adapter with 4.6% of the parameters, trained on 4 to 29 minutes of the user's frontal video, lowers the CER of twelve high-error speakers by 2.13 to 3.58 points, transfers to every camera without loss, and keeps 85% of the full fine-tuning gain at 12% of its cost to other speakers. Cameras above the mouth plane add about six CER points as a constant offset that training on all views keeps small.

Crystal structure determination requires the phases of scattered waves – yet diffraction measures only their intensities. Direct methods exploit phase invariants but become less reliable as diffraction information diminishes. Learned phase prediction has lowered the resolution barrier, yet remains primarily confined to centrosymmetric crystals with binary phases. We introduce PhiGen, a generative reformulation of traditional direct methods that learns origin-independent phase relationships for binary and continuous phasing. Across 210 space groups, including groups absent from training, it recovered high-quality maps for 99.0% of centrosymmetric structures and invariant-consistent phases for 92.8% of non-centrosymmetric structures. From simulated 3 Å zeolite powder data, the network recovered framework maps for 84.2% of held-out structures, versus 1.0% for Superflip. For experimental ZSM-25 and TNU-9, generated phases seeded high-resolution phase extension. These results suggest a route to structure determination from low-resolution, incomplete, and overlapped diffraction data.

This paper is the final installment in our series on the geometric theory of complex Monge-Ampere equations. We study singular Kahler metrics on compact normal Kahler spaces with klt singularities whose volume densities lie in $L^p$ for some $p>1$. We show that a lower bound for the Ricci current in the sense of pluripotential theory is equivalent to a synthetic Ricci lower bound in the sense of RCD theory. As a consequence, every Kahler class on a compact normal Kahler space with klt singularities admits an RCD structure, which makes differential geometric analysis available on singular complex spaces. As an application, we show that the fundamental group of a compact Kahler Calabi-Yau space with klt singularities is almost Abelian. We further establish compactness results for singular Kahler-Einstein spaces.

We study a multidimensional nonlinear filtering model whose coefficients depend on a given observation-adapted predictable process and whose observation drift may grow linearly in both the state and the random input. Due to the unboundedness of the observation drift, a global reference measure is not available. To overcome this hurdle, a localized entropy argument is adapted to prove the stopped likelihood to be a uniformly integrable martingale at each control-energy stopping level. The stopped Zakai equation, and hence, the stopped filtering equation is derived. The global filtering equation is then established by de-localization. The uniqueness of the solution to the stopped Zakai equation is obtained by a duality backward stochastic partial differential equation. This uniqueness then propagates to that of the global filtering equation through the stopped ones. Finally, a stability result is established in $W_1$-distance of measures.

Cross-embodiment learning has shown that a single model, such as a vision-language-action (VLA) model, can learn state representations and manipulation skills that can be applied across heterogeneous robots to accomplish various tasks. We hypothesize that the same holds for safety enforcement. The reasoning required to satisfy a safety constraint, such as detecting an obstacle, recognizing that it should be avoided, and selecting a safe abstract action, is largely shared across robots. What differs across embodiments is how the abstract safe action is realized: morphology, kinematics, and dynamics determine which actions are safe and feasible. Consequently, the same action can be safe for one robot and unsafe for another. This is especially important for generalist manipulation policies that operate in a common end-effector action space without explicitly capturing how safety depends on the robot's morphology and kinematics. We propose embodiment-conditioned safety filtering, in which a Hamilton-Jacobi reachability-based value function and its corresponding safety-maximizing policy are shared across robots. Using a morphology-aware latent representation of the robot and its environment, we perform Hamilton-Jacobi reachability analysis directly in latent space so that the learned safety concepts can generalize across embodiments while remaining explicitly conditioned on each robot's morphology and kinematics. We evaluate our approach across five bimanual robot embodiments and five manipulation tasks with whole-body collision-avoidance constraints. Our results show that a single policy, jointly trained across five manipulation tasks and four embodiments, exhibits zero-shot generalization to a held-out embodiment, reducing the nominal policy's collision rate. They also show that training using more embodiments improves generalization.

This paper studies the antichain polynomials of $[k]\times P$, where $P$ is a connected minuscule poset. We give a formula for the number of antichains, counted by size, of an arbitrary poset. Using this formula, we present necessary and sufficient conditions for the palindromicity of antichain polynomials for two infinite families of connected minuscule posets. We show that, for every connected minuscule poset $P$, if the antichain polynomial of $[k]\times P$ is palindromic, then it has only real and strictly negative zeros. This result, in particular, gives an affirmative answer to Ding-Dong's conjecture about $\gamma$-positivity of the antichain polynomial of $[k]\times P$. By constructing a bijection between antichains and labeled Dyck paths, we also give a layer-refined enumeration of the antichains in $[2]\times[m]\times[n]$. We establish a relation between such antichains and Clar covers of the hexagonal flakes $O(2,m,n)$, thus prove a conjectured determinantal formula for a family of Zhang-Zhang polynomials. Real-rootedness and stability results are also obtained when the shortest chain has length at most two. Finally, we present infinitely many connected Peck posets whose antichain polynomials are not unimodal, disproving the log-concavity conjecture of Ding and Dong.

This paper is the continuation of our earlier work on Holder estimates for Kahler potentials on compact normal Kahler spaces. We establish a geometric counterpart for analytic Holder regularity of Kolodziej. If a singular metric $\omega_\phi$ has Holder continuous potentials with respect to a smooth background metric, then its distance function is Holder continuous with respect to a smooth background distance. The result holds on normal Kahler spaces and yields compactness of the metric completion as well as its identification with the underlying complex space if $\omega_\phi$ is a Kahler current. Under this positivity assumption, our estimates establish the Holder equivalence between the intrinsic canonical Kahler metrics and the extrinsic smooth metrics on Kahler-RCD spaces, particularly on smoothable Kahler-Einstein spaces.

Recurrent invariant solutions provide a dynamical framework for interpreting turbulence, but their computation becomes increasingly difficult in spatially extended flows, where close whole-domain recurrences are rare. We study two-dimensional Kolmogorov flow in a [Lx,Ly] = [2pi,12pi] domain with Re = 10 using a distributed reduced-order model composed of a patch-based autoencoder and a neural ordinary differential equation. Near- recurrence candidates are refined by latent multiple shooting, decoded to physical space, screened under the full Navier-Stokes dynamics and supplied to a full-state Newton- Krylov solver. Fourteen candidates converge to relative periodic orbits (RPOs), with periods 3.65 < T <16.45. The catalogue contains both domain-filling states and RPOs with recurrent activity localised to a restricted cross-stream region. The weakly modulated surroundings generally remain finite-amplitude and distinct from the laminar solution. Selected localised RPOs persist under continuation in Reynolds number, and recurrent states can be reconverged after truncating substantial portions of their surroundings. In addition, the model trained at Ly = 12pi generates convergent RPO seeds at Ly = 6pi and 8pi without retraining. These results demonstrate that distributed learned dynamics can provide useful initial conditions for exact coherent states searches in extended domains and reveal recurrent solutions with strongly localised temporal dynamics.

LLM-based retrievers and rerankers have advanced passage ranking, yet both paradigms interact with the corpus in a single pass and commit to the resulting candidate set, leaving relevant documents permanently unrecoverable once missed. We introduce Seek, Self-Evaluative Exploration for Knowledge Retrieval, a training-free framework that addresses this limitation through iterative corpus interaction at test time. At each round, an LLM generates pseudo-passages conditioned on accumulated relevance feedback, a retriever surfaces fresh candidates, and a dedicated assessor assigns graded relevance judgments that guide subsequent rounds. On TREC Deep Learning, Seek matches trained rerankers in ranking quality while consistently improving Recall@100 over single-pass BM25. On the reasoning-intensive BRIGHT benchmark, Seek with Qwen2.5-7B achieves an 82% relative gain over BM25, surpassing all trained baselines, and Seek with GPT-4.1 reaches 37.4 average nDCG@10, exceeding the strongest baseline by 37%.

We study spectral graph neural networks built from Hermite polynomials and propose HermNet, a simple model that combines a nodewise predictor with normalized Hermite propagation. Its sparse recurrence requires neither eigendecomposition nor a learned basis. We distinguish the basic model from optional coordinate calibration, response normalization and Gaussian derivative regularization. Hermite and other complete polynomial bases span the same degree-bounded filter space, but their coordinates can produce different optimization behavior under limited training budgets. We analyze this behavior through spectral signal energy, label sampling, changes in learned features and the bias–variance trade-off of regularization. Controlled synthetic experiments identify a regime in which plain HermNet outperforms matched polynomial-basis alternatives, including with a jointly trained nonlinear predictor. Curvature regularization further improves HermNet when the same functional penalty is available to every comparator. Fixed-predictor controls support the advantage under short training budgets, but longer training removes the plain-model lead. Matched real-data comparisons show accuracy deficits, and architectural and numerical studies identify further limits. Together, the analysis and experiments clarify when Hermite propagation is useful and how calibration and regularization affect its performance.

Let $X$ be a compact Kahler manifold. The analytic stability theorem of Kolodziej for complex Monge-Ampere equation states that for any Kahler metrics $\omega$ and $\omega'$ in the same cohomology class, if their volume measures are bounded in $L^p(X)$ (for some $p>1$) and close in $L^1(X)$, then their Kahler potentials are close in $L^\infty(X)$. In this paper, we establish the geometric stability for complex Monge-Ampère equations that $L^1$-closeness of volume measures implies $L^\infty$-closeness for the induced distance functions by $\omega$ and $\omega'$. Consequently, we prove that any non-smooth Kahler current with volume measure bounded in $L^p$ (for some $p>1$) and Ricci current bounded below induces a unique metric space, which turns out to be a compact RCD space homeomorphic to $X$ itself.

Identifying the minimal set of nodes whose removal breaks a complex network apart, also referred as the network dismantling problem, is a highly non-trivial task with applications in multiple domains. Whereas network dismantling has been extensively studied over the past decade, research has primarily focused on the formulations of the optimization problem for single-layer networks, neglecting that many, if not all, real networks display multiple layers of interdependent interactions. In such networks, the optimization problem is fundamentally different as the effect of removing nodes propagates within and across layers in a way that can not be predicted using a single-layer perspective. Here, we propose a dismantling algorithm named MultiDismantler, which leverages multiplex network representation and deep reinforcement learning to optimally dismantle multi-layer interdependent networks. MultiDismantler is trained on small synthetic multiplex graphs; when applied to large, real and synthetic networks, it displays exceptional dismantling performance, clearly outperforming all existing methods that rely on a single-layer approach to network dismantling. We show that MultiDismantler is effective in guiding strategies for the containment of diseases in social networks characterized by multiple layers of social interactions. Also, we show that MultiDismantler is useful in the design of protocols aimed at delaying the onset of cascading failures in interdependent critical infrastructures.

Visual Navigation Models (VNMs) enable robots to navigate from egocentric visual observations without geometric localization and planning, but long-range navigation still requires pre-built maps. This paper presents the Remote Visual Navigation Model (ReVNM), which uses a single remote surveillance camera to serve as both an observation source and an implicit environmental map for visual navigation. While the use of remote cameras could eliminate the need for pre-built maps as well as onboard vision processing, their limited field of view instead of egocentric observations makes it hard to achieve collision-free navigation. The lack of existing data with diverse remote viewpoints, which are crucial for training robust VNMs, further complicates the challenge. In this work, we propose a learning-by-synthesis approach to address this two-fold challenge. Our ReVNM extends a state-of-the-art VNM architecture with an exocentric-to-egocentric (exo2ego) module that predicts an egocentric depth observation from remote-camera observations. This helps the VNM to plan a path while considering obstacles in front of the robot. Trained only on randomly generated worlds with diverse obstacle layouts and camera viewpoints, ReVNM can generalize well to real robot navigation without additional fine-tuning. Experiments in both simulation and real-world environments confirmed the effectiveness of the proposed approach.