Embedding heavy main-group elements at the core of rigid π-conjugated frameworks is an ideal strategy to achieve stability through geometric confinement, yet its realization has remained synthetically elusive. Here we show that a directed, regioselective trilithiation strategy enables controlled incorporation of heavy elements into triangulene frameworks. Using nitrogen-bridged macrocycles as precursors, antimony-, bismuth-, tin-, and lead-centered triangulenes were synthesized. Single-crystal X-ray analyses reveal exceptionally deep bowl-shaped geometries and one-dimensional columnar assemblies. Despite highly strained bonding environments, these compounds exhibit remarkable thermal and chemical stability, highlighting the effectiveness of entropic stabilization arising from the rigid structural confinement system. Moreover, the antimony-centered triangulene undergoes reversible and controllable redox transformation between trivalent and pentavalent states through chelation and coordination effects, demonstrating how topological embedding translates into distinct functional behavior. This work establishes a general synthetic platform for heavy-atom embedding in curved π-systems and opens opportunities for design of unique main-group architectures.

Abstract Background Drug repurposing offers a cost-effective alternative to de novo discovery, yet the relative contributions of model complexity, data volume, and feature modalities to knowledge graph–based repurposing remain poorly quantified under rigorous temporal validation. Methods We constructed a pharmacology knowledge graph from ChEMBL 36 comprising 5,348 entities (3,127 drugs, 1,156 proteins, 1,065 indications) and 20,015 edges across 4 relation types. We enforced a strict temporal split (training: $$\le $$ 2022; testing: 2023–2025) with biologically verified hard negatives mined from failed assays and clinical trials. We benchmarked five Knowledge Graph Embedding models (TransE, TransR, RotatE, ComplEx, DistMult; 0.78–0.81M parameters) and a Standard GNN (3.44M parameters) that incorporates drug chemical structure using a GAT encoder and ESM-2 embeddings evaluated by PR-AUC and Hits@ k on drug–protein and drug–indication link prediction. Scaling (0.78M–9.75M parameters; 25–100% data) and feature ablation studies isolated contributions of model capacity, graph density, and node feature modalities. Results Feature ablation revealed a counter-intuitive performance hierarchy. Removing the GAT-based drug structure encoder entirely from the GNN and retaining only topological embeddings combined with ESM-2 protein features improved drug–protein PR-AUC from 0.5631 to 0.5785, while simultaneously reducing VRAM usage from 5.30 GB to 353 MB. Additionally, replacing the GAT encoder with Morgan fingerprints further degraded performance (PR-AUC = 0.5286), indicating that explicit chemical structure representations can be not only redundant but detrimental for predicting pharmacological network interactions. Scaling the GNN beyond 2.44 M parameters yielded diminishing performance gains, whereas increasing training data consistently improved model performance with no observable ceiling. External validation confirmed 6 of the top 14 novel predictions (42.9%) as established therapeutic indications. Conclusions Drug pharmacological behavior can be accurately predicted using target-centric information and drug–network topology alone, without requiring explicit drug chemical structure representations. Model performance is substantially more sensitive to data volume and node density than to architectural complexity, with scaling in model size yielding limited returns relative to improvements in graph coverage. Consequently, state-of-the-art performance is achievable on budget hardware, with a model using only 352 MB VRAM on a consumer GPU.

Abstract Motivated by applications to spaces of embeddings and automorphisms of manifolds, we consider a tower of ‐categories of truncated right modules over a unital ‐operad . We study monoidality and naturality properties of this tower, identify its layers, describe the difference between the towers as varies and generalise these results to the level of Morita ‐categories. Applied to the ‐framed ‐operad, this extends Goodwillie–Weiss' embedding calculus and its layer identification to the level of bordism categories. Applied to other variants of the ‐operad, it yields new versions of embedding calculus, such as one for topological embeddings — based on — or one similar to Boavida de Brito–Weiss' configuration categories — based on . In addition, we prove a delooping result in the context of embedding calculus, establish a convergence result for topological embedding calculus, improve upon the smooth convergence result of Goodwillie, Klein and Weiss and deduce an Alexander trick for homology 4‐spheres.

This paper contributes to the historical understanding of the developments surrounding the Levi-Civita parallel transport problem, exploring its connections with the local problem of isometric immersions and alternative proposals. Additionally, it highlights one of its remarkable applications: the geometric interpretation of Foucault’s pendulum precession. It also recalls how other geometric explanations of this phenomenon emerged in the context of Berry and Hannay phases.

E-commerce retailers bear substantial additional costs arising from high product return rates due to lenient return policies and consumers’ impulsive purchasing. This study aims to accurately predict product return behavior before payment, supporting proactive return management and reducing potential losses. Based on the Graph Transformer, we proposed a novel return prediction model, Returnformer, which focuses on capturing user–product connections represented in topological structures of bipartite graphs. The Returnformer first integrates global topological embeddings into original node features to alleviate structural information loss caused by graph partitioning. It then employs a Graph Transformer to capture long-range user–item dependencies within local subgraphs. In addition, a graph-level attention mechanism is introduced to facilitate the propagation of global return patterns across different subgraphs. Experiments on a real-world e-commerce dataset show that the Returnformer outperforms four machine learning models in terms of prediction accuracy, demonstrating superior performance compared to the state-of-the-art models. The proposed model enables retailers to identify potential return risks prior to payment, thereby supporting timely and proactive preventive interventions.

Mangrove ecosystems act as highly efficient carbon sinks and provide critical ecological services for climate regulation; however, they are increasingly threatened by anthropogenic pressures and climate change. Recent advances in annual satellite-based Earth Observation (EO) embedding (EM) datasets have enabled the integration of multi-source data fusion, improving land use and land cover analyses. Nevertheless, EM datasets have rarely been applied for the monitoring of mangrove ecosystems. Within this context, the present study compares the performance of EM with combined Sentinel-1, Sentinel-2, and Landsat 8/9 datasets (S2S1L8/9) through the integration of a pretrained ResNet152–U-Net architecture and Random Forest (RF) modeling to assess spatio-temporal patterns of mangrove intactness and degradation from 2017 to 2024. A ResNet152 encoder pretrained on ImageNet was employed to train a U-Net model using the Global Mangrove Dataset (2020) for the mapping of potential mangrove extent. The EM dataset outperformed the S2S1L8/9 combination, achieving validation Intersection over Union (IoU) scores of 0.85 and 0.84, and Dice coefficients of 0.89 and 0.88, respectively. Spatial comparison further indicated that EM yielded a lower Root Mean Square Error (RMSE) (6.51 ha) compared to S2S1L8/9 (7.27 ha). The RF model, trained on intact and non-intact mangrove samples across multiple years, confirmed the superior performance of EM in delineating intact mangrove areas along coastlines, with an overall accuracy of 0.97, an F1-score of 0.97, and a Matthews Correlation Coefficient of 0.93. Degraded mangrove areas were identified by masking intact regions, and gain–loss analysis revealed a decline of approximately 3% in 2021 and 2022 relative to the 2017 baseline. These findings demonstrate that EM provides a more accurate and spatially consistent approach than conventional multisource datasets for mapping mangrove intactness and degradation. By minimizing classification errors and improving coastline delineation, EM establishes a robust framework for large-scale monitoring of mangrove dynamics, supporting conservation planning, carbon accounting, and climate resilience strategies across diverse coastal and terrestrial systems at global scales.

TDAGENE: Inference of Gene Regulatory Network Based on Topological Data Analysis and Graph Attention Network for Single-Cell RNA Sequencing Data.

With the growing urgency of carbon neutrality, sustainable intelligent transportation systems must support green path planning that simultaneously accounts for arbitrary origin–destination (OD) pairs, strict emission constraints, and large-scale concurrent requests. To address this challenge, this paper proposes a Constrained Reinforcement Learning for Shortest Path (CRL-SP) framework, which generates real-time, scalable, and emission-compliant routing solutions under hard carbon constraints. The framework adopts a dual-tower Universal Value Function Approximation architecture based on Deep Q-Networks (UVFA-DQN), integrating topological embeddings with contextual features, and is trained via a progressive strategy that combines demonstration-based behavior cloning with interactive reinforcement learning. Numerical experiments on benchmark networks demonstrate that CRL-SP consistently recovers optimal or near-optimal solutions, achieving path match rates and link overlap ratios typically above 0.85, while relative deviations in travel time and emissions remain close to zero. The proposed model preserves this level of performance under unseen and sparsely sampled emission budgets, and is able to generate feasible and structurally consistent paths even when classical constrained shortest-path algorithms fail due to insufficient budgets. In terms of computational efficiency, CRL-SP supports large-scale batch inference with throughput exceeding tens of thousands of paths per second, yielding orders-of-magnitude speedups over traditional label-setting methods. Overall, CRL-SP provides a practical and scalable learning-based solution for emission-constrained path planning in sustainable intelligent transportation systems.

Radical-Net: A chemistry-enhanced transformer for elementary radical reactions in pollutant chemistry.

Probabilistic modeling over the combinatorially large space of tree topologies remains a central challenge in phylogenetic inference. Previous approaches often necessitate pre-sampled tree topologies, limiting their modeling capability to a subset of the entire tree space. A recent advancement is ARTree, a deep autoregressive model that offers unrestricted distributions for tree topologies. However, its reliance on repetitive tree traversals and inefficient local message passing for computing topological node representations may hamper the scalability to large datasets. This paper proposes ARTreeFormer, a novel approach that harnesses fixed-point iteration and attention mechanisms to accelerate ARTree. By introducing a fixed-point iteration algorithm for computing the topological node embeddings, ARTreeFormer allows for fast vectorized computation, especially on CUDA devices. This, together with an attention-based global message passing scheme, significantly improves the computation speed of ARTree while maintaining great approximation performance. We demonstrate the effectiveness and efficiency of our method on a benchmark of challenging real data phylogenetic inference problems.

The brain's synaptic network, characterized by parallel connections and feedback loops, drives interaction pathways between neurons through a large system with infinitely many degrees of freedom. This system is best modeled by the graph C*-algebra of the underlying directed graph, the Toeplitz-Cuntz-Krieger (TCK) algebra, which captures the diversity of path-structured flow connectivity. Equipped with the gauge action, the TCK algebra defines an algebraic quantum system, and here we demonstrate that its thermodynamic properties provide a natural framework for describing the dynamic mappings of potential flow pathways within the network. Specifically, the KMS states of this system represent the stationary distributions of a non-Markovian stochastic process with memory decay, capturing how influence propagates along exponentially weighted paths, and yield global statistical measures of neuronal interactions. Applied to the C. elegans synaptic network, our framework reveals that neurolocomotor neurons emerge as the primary hubs of incoming path-structured flow at inverse temperatures where the entropy of KMS states peaks. This finding aligns with experimental evidence of the foundational role of locomotion in C. elegans behavior, suggesting that functional centrality may arise from the topological embedding of neurons rather than solely from local physiological properties. Our results highlight the potential of algebraic quantum methods and graph algebras to uncover patterns of functional organization in complex systems and neuroscience.

Let (M,g) be a connected, compact Riemannian manifold of dimensionan n. We demonstrate that, after a suitable normalization, a shrinking gradient Ricci soliton (M,g,f,λ) is trivial exactly when the mean value of f is less than or equal to n2. Moreover, we prove that a normalized non-steady gradient Ricci soliton (M,g,f,λ) is trivial if and only if its scalar curvature S satisfies the relation S=λf+n2. In addition, we establish that if (M,g,f,λ) admits an isometric immersion as a hypersurface in the Euclidean space, then the soliton must necessarily be of a shrinking type. In such a case, the constant λ and the mean curvature of M satisfy a certain inequality, with equality occurring precisely when M is isometric to a round sphere.

ABSTRACT In this paper, we study the problem of local isometric immersion of pseudospherical surfaces determined by the solutions of a class of third‐order nonlinear partial differential equations with the type . We prove that there are two subclasses of equations admitting a local isometric immersion into the three‐dimensional Euclidean space for which the coefficients of the second fundamental form depend on a jet of finite order of , and furthermore, these coefficients are universal, namely, they are functions of and , independent of . Finally, we show that the generalized Camassa–Holm equation describing pseudospherical surfaces has a universal second fundamental form.

We establish a general inequality and optimal inequalities involving the normalized Casorati curvatures and the generalized normalized Casorati curvatures within the horizontal space of a Riemannian map from a Riemannian manifold to a nearly Kaehler manifold with a constant holomorphic sectional curvature. Riemannian maps serve as generalizations of isometric immersions and Riemannian submersions. Consequently, we extend these inequalities to encompass Riemannian submanifolds within nearly Kaehler manifolds with a constant holomorphic sectional curvature.

Isometric Immersions of Lightlike Manifolds in Lorentzian Products

We prove that a generic reservoir system admits a generalized synchronization that is a topological embedding of the input system's attractor. We also prove that for sufficiently high reservoir dimension (given by Nash's embedding theorem), there exists an isometric embedding generalized synchronization. The isometric embedding can be constructed explicitly when the reservoir system and source dynamics are linear.

On the Fundamental Theorem of Submanifold Theory and Isometric Immersions with Supercritical Low Regularity

Accelerated prediction of molecular properties for per- and polyfluoroalkyl substances using graph neural networks with adjacency-free message passing.

Reconstructing the attractors of complex nonlinear dynamical systems from available measurements is key to analyse and predict their time evolution. Existing attractor reconstruction methods typically rely on topological embedding and may produce poor reconstructions, which differ significantly from the actual attractor, because measurements are corrupted by noise and often available only for some of the state variables and/or their combinations, and the time series are often relatively short. Here, we propose the use of Homogeneous Differentiators (HD) to effectively de-noise measurements and more faithfully reconstruct attractors of nonlinear systems. Homogeneous Differentiators are supported by rigorous theoretical guarantees about their de-noising capabilities, and their results can be fruitfully combined with time-delay embedding, differential embedding and functional observability. We apply our proposed HD-based methodology to simulated dynamical models of increasing complexity, from the Lorenz system to the Hindmarsh–Rose model and the Epileptor model for neural dynamics, as well as to empirical data of EEG recordings. In the presence of corrupting noise of various types, we obtain drastically improved quality and resolution of the reconstructed attractors, as well as significantly reduced computational time, which can be orders of magnitude lower than that of alternative methods. Our tests show the flexibility and effectiveness of Homogeneous Differentiators and suggest that they can become the tool of choice for preprocessing noisy signals and reconstructing attractors of highly nonlinear dynamical systems from both theoretical models and real data. • We reconstruct dynamical attractors of complex nonlinear systems from noisy data. • We use Homogeneous Differentiators for efficient and faithful reconstruction. • The results are more accurate and much faster than with alternative methods. • The approach works successfully with various noise types and complex models. • We also reconstruct de-noised attractors from empirical data of neural activity.

We show that any closed oriented 3-manifold can be topologically embedded in some simply connected closed symplectic 4-manifold and that it can be made a smooth embedding after one stabilization.As a corollary of the proof, we show that the homology cobordism group is generated by Stein fillable 3-manifolds.We also find obstructions on smooth embeddings: there exist 3-manifolds that cannot smoothly embed in a way that appropriately respects orientations in any symplectic manifold with a weakly convex boundary. 57K43; 53D05Corollary 1.12 Given any 3-manifold Y there exists a Stein fillable 3-manifold Y 0 and a degree one map f W Y 0 !Y .The distinction between smooth and topological embeddings serves as a tool for detecting exotic structures on compact manifolds.If we encounter two homeomorphic 4-manifolds such that a 3-manifold embeds smoothly in one but not the other, then they are not diffeomorphic; they form an exotic pair.Corollary 1.13, which we will present shortly, was initially demonstrated by Akbulut [3] and subsequently proven by many others.However, we will offer an alternative proof stemming from the study of embeddings of 3-manifolds into 4-manifolds.Corollary 1.13There exist compact 4-manifolds with boundary X and X 0 such that b 2 .X / D b 2 .X 0 / D 1 that are homeomorphic but not diffeomorphic.