Ukraine is preparing to present a revised peace plan to the White House, as it seeks to avoid making territorial concessions to Russia. Kyiv is set propose alternatives to the US after President Volodymyr Zelensky again ruled out surrendering land, saying he had no right to do so under Ukrainian or international law. He made the comments as he met European and Nato leaders on Monday, part of a collective push to deter the US from backing a peace deal which includes major concessions for Ukraine, and which allies fear would leave it vulnerable to a future invasion. Meanwhile, the city of Sumy in north-western Ukraine was left without power overnight after a Russian drone attack. The region's governor said more than a dozen drones had hit power infrastructure, the latest in Russia's nightly attacks.

Chernobyl radiation shield'lost safety function' after drone strike, UN watchdog says A protective shield covering the Chernobyl nuclear reactor in Ukraine can no longer provide its main containment function following a drone strike earlier this year, according to a UN watchdog. International Atomic Energy Agency (IAEA) inspectors found that the massive structure, built over the site of the 1986 nuclear disaster, had lost its primary safety functions including the confinement capability. In February, Ukraine accused Russia of targeting the power plant - a claim the Kremlin denied. The IAEA said repairs were essential to prevent further degradation of the nuclear shelter. However environmental expert Jim Smith told the BBC: It is not something to panic about.

What is in the 28-point US plan for Ukraine? 'Ukraine is running out of men, money and time' Can the US get all sides to end the war? Why is Europe opposing Trump's peace plan? IAEA flags damage to Chornobyl nuclear plant's protective shield in Ukraine A drone strike has damaged a protective shield at the Chornobyl nuclear plant in Ukraine, rendering it unable to contain the radioactive material from the 1986 explosion of the plant, the United Nations nuclear watchdog said. The International Atomic Energy Agency (IAEA) said on Friday that the shield can no longer perform its main safety function, following an inspection of the steel structure last week.

Then, the desired lemma is obtained.

Robotic systems navigating in real-world settings require a semantic understanding of their environment to properly determine safe actions. This work aims to develop the mathematical underpinnings of such a representation -- specifically, the goal is to develop safety filters that are risk-aware. To this end, we take a two step approach: encoding an understanding of the environment via Poisson's equation, and associated risk via Laplace guidance fields. That is, we first solve a Dirichlet problem for Poisson's equation to generate a safety function that encodes system safety as its 0-superlevel set. We then separately solve a Dirichlet problem for Laplace's equation to synthesize a safe \textit{guidance field} that encodes variable levels of caution around obstacles -- by enforcing a tunable flux boundary condition. The safety function and guidance fields are then combined to define a safety constraint and used to synthesize a risk-aware safety filter which, given a semantic understanding of an environment with associated risk levels of environmental features, guarantees safety while prioritizing avoidance of higher risk obstacles. We demonstrate this method in simulation and discuss how \textit{a priori} understandings of obstacle risk can be directly incorporated into the safety filter to generate safe behaviors that are risk-aware.

Text-to-image models have recently made significant advances in generating realistic and semantically coherent images, driven by advanced diffusion models and large-scale web-crawled datasets. However, these datasets often contain inappropriate or biased content, raising concerns about the generation of harmful outputs when provided with malicious text prompts. We propose Safe Text embedding Guidance (STG), a training-free approach to improve the safety of diffusion models by guiding the text embeddings during sampling. STG adjusts the text embeddings based on a safety function evaluated on the expected final denoised image, allowing the model to generate safer outputs without additional training. Theoretically, we show that STG aligns the underlying model distribution with safety constraints, thereby achieving safer outputs while minimally affecting generation quality. Experiments on various safety scenarios, including nudity, violence, and artist-style removal, show that STG consistently outperforms both training-based and training-free baselines in removing unsafe content while preserving the core semantic intent of input prompts. Our code is available at https://github.com/aailab-kaist/STG.

This paper presents an input-constrained nonlinear guidance law to address the problem of intercepting a stationary target in contested environments with multiple defending agents. Contrary to prior approaches that rely on explicit knowledge of defender strategies or utilize conservative safety conditions based on a defender's range, our work characterizes defender threats geometrically through engagement zones that delineate inevitable interception regions. Outside these engagement zones, the interceptor remains invulnerable. The proposed guidance law switches between a repulsive safety maneuver near these zones and a pursuit maneuver outside their influence. To deal with multiple engagement zones, we employ a smooth minimum function (log-sum-exponent approximation) that aggregates threats from all the zones while prioritizing the most critical threats. Input saturation is modeled and embedded in the non-holonomic vehicle dynamics so the controller respects actuator limits while maintaining stability. Numerical simulations with several defenders demonstrate the proposed method's ability to avoid engagement zones and achieve interception across diverse initial conditions.

Ensuring safety is a key aspect in sequential decision making problems, such as robotics or process control.

Then, the desired lemma is obtained.