Safety verification ensures that a system avoids undesired behaviour. Liveness complements safety, ensuring that the system also achieves its desired objectives. A complete specification of functional correctness must combine both safety and liveness. Proving with mathematical certainty that a system satisfies a safety property demands presenting an appropriate inductive invariant of the system, whereas proving liveness requires showing a measure of progress witnessed by a ranking function. Neural model checking has recently introduced a data-driven approach to the formal verification of reactive systems, albeit focusing on ranking functions and thus addressing liveness properties only.

Stellarators are magnetic confinement devices under active development to deliver steady-state carbon-free fusion energy. Their design involves a high-dimensional, constrained optimization problem that requires expensive physics simulations and significant domain expertise. Recent advances in plasma physics and open-source tools have made stellarator optimization more accessible. However, broader community progress is currently bottlenecked by the lack of standardized optimization problems with strong baselines and datasets that enable data-driven approaches, particularly for quasi-isodynamic (QI) stellarator configurations, considered as a promising path to commercial fusion due to their inherent resilience to currentdriven disruptions. Here, we release an open dataset of diverse QI-like stellarator plasma boundary shapes, paired with their ideal magnetohydrodynamic (MHD) equilibria and performance metrics. We generated this dataset by sampling a variety of QI fields and optimizing corresponding stellarator plasma boundaries. We introduce three optimization benchmarks of increasing complexity: (1) a singleobjective geometric optimization problem, (2) a "simple-to-build" QI stellarator, and (3) a multi-objective ideal-MHD stable QI stellarator that investigates trade-offs between compactness and coil simplicity. For every benchmark, we provide reference code, evaluation scripts, and strong baselines based on classical optimization techniques. Finally, we show how learned models trained on our dataset can efficiently generate novel, feasible configurations without querying expensive physics oracles.

Plus: Autonomous drones may have killed soldiers for the first time. Imagine tuning in to the opening kickoff of a World Cup match and seeing a player intentionally kick the ball out of bounds. You may question the logic of surrendering possession seconds into a game. If you were Jesse Davis, though, you'd know that this play could be a prime setup to score. Davis is a professor of computer science at KU Leuven in Belgium and head of its Sports Analytics Lab, which has been at the vanguard of a data awakening in soccer. Using AI and data analytics, his team has uncovered hidden tactical patterns and challenged long-held assumptions about how the game should be played.

Pope Leo XIV has described Barcelona's Sagrada Família as a masterpiece of stones, colours and light as he inaugurated its newest - and tallest - tower. The giant Tower of Jesus Christ, completed in February, has brought the church to a soaring height of 172.5m (566ft) - cementing it as the tallest church in the world. His visit to the iconic basilica also marks 100 years since the death of its architect, Antoni Gaudí. Among those attending the service were Spanish royals King Felipe VI and Queen Letizia, as well as Prime Minister Pedro Sánchez. The pope's week-long visit to Spain, which began on Saturday, is the first by a pope in some 15 years.

Rescuers work at the site of an apartment building hit by a Russian drone strike in Zaporizhzhia, Ukraine, on Saturday. Ukraine and Russia traded aerial attacks on Saturday as President Volodymyr Zelenskyy held what he called a special meeting on next steps with top aides. A Ukrainian drone struck the machine room building of one of power units at the Russian-occupied Zaporizhzhia nuclear power plant in southeastern Ukraine on Saturday afternoon, causing unspecified damage, Interfax reported, citing Rosatom Chief Executive Officer Alexey Likhachev. Core equipment wasn't damaged, he said. Ukraine's southern military command denied any strikes, saying its military personnel "act exclusively within the framework of international humanitarian law and are aware of the consequences of any actions against nuclear facilities." In a post on Facebook late Saturday, it added, "It is the Russian Federation that has illegally kept the Zaporizhzhia Nuclear Power Plant under military control since March 2022, turning a civilian nuclear facility into an element of military infrastructure."

Umpire Dan Bellino's baffling foul tip call on Seiya Suzuki renews calls for robot review in MLB Dakich: sports media has created an'industry' out of complaining about white athletes like Caitlin Clark Greg Sankey insists SEC is'strongest league' despite Big Ten winning three straight national championships Phillies look to upset Dodgers behind Zack Wheeler as Philadelphia's turnaround continues in LA Joey McGuire calls Steve Sarkisian's bluff, dares Texas to play Texas Tech in Week 1 Rams troublemaker WR Puka Nacua says he's a changed man after biting incident and stint in rehab Chiefs have no plans to release Rashee Rice and see jail time as a'life lesson' opportunity Dr Oz: Is this a flaw or a feature? Father Mike Schmitz: Pope Leo XIV wants this world view in line with humanity's good Pompeo warns Iran will rebuild nuclear facilities'the moment' it gets the chance Purple Heart recipient speaks out after Graham Platner's controversial remarks'Chipotle Karen' caught hurling burrito bowl at worker's face Oklahoma City is -162 on the moneyline and -3.5 favorites with the total set at 212.5 as of Friday afternoon Despite getting to a Game 7, the 2026 Western Conference Finals between the San Antonio Spurs and Oklahoma City Thunder haven't lived up to the Game 1 double-overtime instant classic. While the winning team has alternated over the past four games, the margin has been at least 13 points. Plus, Oklahoma City's flopping has been the biggest storyline of the conference finals, which is a bummer for us die-hard NBA fans. However, that will be mostly forgotten if the Spurs-Thunder series finale is another thriller.

Umpire Dan Bellino's baffling foul tip call on Seiya Suzuki renews calls for robot review in MLB Dakich: sports media has created an'industry' out of complaining about white athletes like Caitlin Clark Greg Sankey insists SEC is'strongest league' despite Big Ten winning three straight national championships Phillies look to upset Dodgers behind Zack Wheeler as Philadelphia's turnaround continues in LA Joey McGuire calls Steve Sarkisian's bluff, dares Texas to play Texas Tech in Week 1 Rams troublemaker WR Puka Nacua says he's a changed man after biting incident and stint in rehab Chiefs have no plans to release Rashee Rice and see jail time as a'life lesson' opportunity Diamondbacks fans catch same player's home run on back-to-back nights after showing up on the wrong date Dr Oz: Is this a flaw or a feature? Father Mike Schmitz: Pope Leo XIV wants this world view in line with humanity's good Pompeo warns Iran will rebuild nuclear facilities'the moment' it gets the chance Purple Heart recipient speaks out after Graham Platner's controversial remarks'Chipotle Karen' caught hurling burrito bowl at worker's face The Carolina Hurricanes are in the Eastern Conference Final for the second straight season and the fourth of Rod Brind'Amour's tenure behind the bench, and they've got the chance to close things out in Game 5 against the Montreal Canadiens. Of course, teams coming into Game 5 with a 3-1 lead are historically almost guaranteed to move on to the Stanley Cup Final; the Canes are not going to get ahead of their skis. Hurricanes forward Jackson Blake, who scored the OT-winner to sweep the Philadelphia Flyers and send Carolina to the conference final, talked about the need to focus on the game tonight and not start thinking ahead to the Western Conference Champion Vegas Golden Knights. It's exciting for sure, Blake said.

Reactor no 3 lost off-site power for about 24 hours after the attack. Reactor no 3 lost off-site power for about 24 hours after the attack. A drone strike that cut off external power to a nuclear reactor in the United Arab Emirates this week has revived concerns over the safety of nuclear plants during wartime. Reactor no 3 at the Barakah nuclear plant lost vital off-site power for about 24 hours after the attack on Sunday, forcing it to rely on emergency diesel generators. The UAE's defence ministry said on Tuesday that three drones targeting the plant had originated from Iraqi territory, suggesting a pro-Iranian proxy group was most likely to have been behind the strike.

NextEra Energy is seeking to acquire Dominion Energy in an all-stock deal valued at about $67bn, creating a massive power company as the energy needs of artificial intelligence (AI) drive demand higher in the United States. It is one of the biggest proposed mergers so far this year and would create the world's largest regulated electric utility business by market capitalisation, the companies said on Monday. The region has a fast-growing population and the world's biggest data centre hub, which is in Virginia. The deal will enable a swifter build-out of power infrastructure to deliver electricity to data centres proposing to connect to NextEra and Dominion, which total about 130 gigawatts of electricity demand, the companies' executives said. One gigawatt can power about 750,000 homes. The merger builds on NextEra's efforts to tap into surging demand for supplying electricity to data centres developed by Big Tech, largely for training and rolling out AI technologies.

Will Gulf states join war? What is the UAE's Barakah nuclear plant, nearly hit by a drone? A drone attack that caused a fire close to the Barakah Nuclear Energy Plant in the United Arab Emirates has raised further concerns about nuclear security and military escalation in the Gulf as discussions of peace between Iran and the United States hang in the balance. Barakah was the first nuclear power station to be built on the Arabian Peninsula. What is the Barakah Nuclear Energy Plant? Barakah is a nuclear energy plant located in Al Dhafra, the largest municipal region of the emirate of Abu Dhabi.