We prove the universal approximation theorem by showing the equivalence of TFN and our model. Complex spherical harmonics are related to Clebsch-Gordan coefficients via [51, 3.7.72] We can therefore adapt Eq. (2) by substituting C To see this, we look at the result's real component null [ H To prove this theorem we first introduce a proposition by Villar et al. [57]. GemNet's variance varies strongly between layers and increases significantly after each block without scaling factors (top). We use 4 stacked interaction blocks and an embedding size of 128 throughout the model.

The process learns generalized Casimirs for energy and entropy guaranteed to be conserved and nondecreasing, respectively.

How is Russia replenishing its military? What is a'coalition of the willing'? How China forgot promises and'debts' to Ukraine How are Europe, the US pulling apart on Ukraine? Ukrainian drones attacked the Salavat petrochemical complex, controlled by energy giant Gazprom, in Russia's Bashkortostan, causing a fire, regional Governor Radiy Khabirov said on Wednesday via his Telegram channel. Ukraine's military struck two Russian oil distribution facilities in the Bryansk and Samara regions, as well as a military airfield in Russia-occupied Crimea, Kyiv's general staff said on Telegram.

Graphs are central to the chemical sciences, providing a natural language to describe molecules, proteins, reactions, and industrial processes. They capture interactions and structures that underpin materials, biology, and medicine. This primer, Graph Data Modeling: Molecules, Proteins, & Chemical Processes, introduces graphs as mathematical objects in chemistry and shows how learning algorithms (particularly graph neural networks) can operate on them. We outline the foundations of graph design, key prediction tasks, representative examples across chemical sciences, and the role of machine learning in graph-based modeling. Together, these concepts prepare readers to apply graph methods to the next generation of chemical discovery.

Conceptual spaces represent entities and concepts using cognitively meaningful dimensions, typically referring to perceptual features. Such representations are widely used in cognitive science and have the potential to serve as a cornerstone for explainable AI. Unfortunately, they have proven notoriously difficult to learn, although recent LLMs appear to capture the required perceptual features to a remarkable extent. Nonetheless, practical methods for extracting the corresponding conceptual spaces are currently still lacking. While various methods exist for extracting embeddings from LLMs, extracting conceptual spaces also requires us to encode the underlying features. In this paper, we propose a strategy in which features (e.g. sweetness) are encoded by embedding the description of a corresponding prototype (e.g. a very sweet food). To improve this strategy, we fine-tune the LLM to align the prototype embeddings with the corresponding conceptual space dimensions. Our empirical analysis finds this approach to be highly effective.

Large language models (LLMs) have shown promise in zero-shot and single step reasoning and decision making problems, but in long horizon sequential planning tasks, their errors compound, often leading to unreliable or inefficient behavior. We introduce LogicGuard, a modular actor-critic architecture in which an LLM actor is guided by a trajectory level LLM critic that communicates through Linear Temporal Logic (LTL). Our setup combines the reasoning strengths of language models with the guarantees of formal logic. The actor selects high-level actions from natural language observations, while the critic analyzes full trajectories and proposes new LTL constraints that shield the actor from future unsafe or inefficient behavior. LogicGuard supports both fixed safety rules and adaptive, learned constraints, and is model-agnostic: any LLM-based planner can serve as the actor, with LogicGuard acting as a logic-generating wrapper. We formalize planning as graph traversal under symbolic constraints, allowing LogicGuard to analyze failed or suboptimal trajectories and generate new temporal logic rules that improve future behavior. To demonstrate generality, we evaluate LogicGuard across two distinct settings: short-horizon general tasks and long-horizon specialist tasks. On the Behavior benchmark of 100 household tasks, LogicGuard increases task completion rates by 25% over a baseline InnerMonologue planner. On the Minecraft diamond-mining task, which is long-horizon and requires multiple interdependent subgoals, LogicGuard improves both efficiency and safety compared to SayCan and InnerMonologue. These results show that enabling LLMs to supervise each other through temporal logic yields more reliable, efficient and safe decision-making for both embodied agents.

Human skin provides a rich tactile sensing stream, localizing intentional and unintentional contact events over a large and contoured region. Replicating these tactile sensing capabilities for dexterous robotic manipulation systems remains a longstanding challenge. In this work, we take a step towards this goal by introducing DexSkin. DexSkin is a soft, conformable capacitive electronic skin that enables sensitive, localized, and calibratable tactile sensing, and can be tailored to varying geometries. We demonstrate its efficacy for learning downstream robotic manipulation by sensorizing a pair of parallel jaw gripper fingers, providing tactile coverage across almost the entire finger surfaces. We empirically evaluate DexSkin's capabilities in learning challenging manipulation tasks that require sensing coverage across the entire surface of the fingers, such as reorienting objects in hand and wrapping elastic bands around boxes, in a learning-from-demonstration framework. We then show that, critically for data-driven approaches, DexSkin can be calibrated to enable model transfer across sensor instances, and demonstrate its applicability to online reinforcement learning on real robots. Our results highlight DexSkin's suitability and practicality for learning real-world, contact-rich manipulation. Please see our project webpage for videos and visualizations: https://dex-skin.github.io/.