Devastating impact of Minneapolis shooting on Trump is worse than expected: Poll reveals America's crushing verdict... and what he must do next Bodies are STILL in wreckage of private jet that crashed in Maine on Sunday, killing six including powerful lawyer's attorney wife School principal accused of shoplifting from Walmart using'stacking' method at self-checkout Melania's shock role in Trump's showdown with Kristi Noem revealed: MARK HALPERIN's fly-on-wall account of Oval Office meeting... and who is ACTUALLY taking the fall for Alex Pretti shooting I was barely eating but kept gaining weight. Then I discovered the'taboo' cancer doctors NEVER talk about. Now sex will never be the same... don't ignore these signs Harper Beckham, 14, puts on a stylish display in a fluffy coat and vintage Chanel bag in Paris with her family - after Nicola Peltz's heartbreaking comments about sister-in-law Devastating truth about Blind Side actor Quinton Aaron: More to this'than everyone is letting on', friends reveal... as co-star Sandra Bullock'monitors' situation The wild truth about my influencer sons, their psycho dad and how lawsuits nearly left them bankrupt - by Jake and Logan Paul's MOM Trump knifes'little Napoleon' Border Patrol commander over Minnesota mayhem as he declares: 'We'll de-escalate' Lost tomb of the mysterious'cloud people' unearthed after 1,400 years in'discovery of the decade' I've seen possessed children scream like beasts and strung up like puppets... these chilling exorcism cases PROVE hell is real There is a hidden battlefield within our world, where forces of light and darkness collide, believers say, in a conflict that sometimes spills into everyday life. In its most extreme form, the clash is described as possession: a person seemingly seized by demonic beings, their body overtaken, their voice and movements warped into something not quite human. For Anglican reverend Chris Lee, 43, this is not a theological abstraction but a reality he has lived with for nearly two decades.

Could Ukraine hold a presidential election right now? Will Europe use frozen Russian assets to fund war? How can Ukraine rebuild China ties? 'Ukraine is running out of men, money and time' At least four people were killed in a Russian drone attack on a passenger train in Ukraine's Kharkiv region, Ukrainian President Volodymyr Zelenskyy said on Telegram. Zelenskyy added that four people were still missing, and that two people were injured in the attack.

Devastating impact of Minneapolis shooting on Trump is worse than expected: Poll reveals America's crushing verdict... and what he must do next Bodies are STILL in wreckage of private jet that crashed in Maine on Sunday, killing six including powerful lawyer's attorney wife School principal accused of shoplifting from Walmart using'stacking' method at self-checkout Melania's shock role in Trump's showdown with Kristi Noem revealed: MARK HALPERIN's fly-on-wall account of Oval Office meeting... and who is ACTUALLY taking the fall for Alex Pretti shooting I was barely eating but kept gaining weight. Then I discovered the'taboo' cancer doctors NEVER talk about. Now sex will never be the same... don't ignore these signs Harper Beckham, 14, puts on a stylish display in a fluffy coat and vintage Chanel bag in Paris with her family - after Nicola Peltz's heartbreaking comments about sister-in-law Devastating truth about Blind Side actor Quinton Aaron: More to this'than everyone is letting on', friends reveal... as co-star Sandra Bullock'monitors' situation The wild truth about my influencer sons, their psycho dad and how lawsuits nearly left them bankrupt - by Jake and Logan Paul's MOM Trump knifes'little Napoleon' Border Patrol commander over Minnesota mayhem as he declares: 'We'll de-escalate' Lost tomb of the mysterious'cloud people' unearthed after 1,400 years in'discovery of the decade' Yet another reason to use the air fryer! If you're cooking this evening, a new study may encourage you to reach for the air fryer . Researchers from the University of Birmingham say that cooking even very fatty food in an air fryer produces far fewer air-pollution particles than other forms of frying.

It was hard to miss 33-year-old model and writer Abena Christine Jon'el's appearance at a recent major fashion show in Ghana. Walking the runway with her prosthetic leg wrapped in a colourful African print her appearance made a big impact. The Ghanaian-American was hoping to make a statement about the visibility of people with disabilities, building on years of work in the US and here in Ghana of speaking out on the issue. At two years old, Abena's life became defined by a challenge most adults would struggle to face. A large tumour had appeared on her right calf, the first sign of a rare, aggressive soft-tissue cancer, rhabdomyosarcoma.

Devastating impact of Minneapolis shooting on Trump is worse than expected: Poll reveals America's crushing verdict... and what he must do next Bodies are STILL in wreckage of private jet that crashed in Maine on Sunday, killing six including powerful lawyer's attorney wife School principal accused of shoplifting from Walmart using'stacking' method at self-checkout Melania's shock role in Trump's showdown with Kristi Noem revealed: MARK HALPERIN's fly-on-wall account of Oval Office meeting... and who is ACTUALLY taking the fall for Alex Pretti shooting I was barely eating but kept gaining weight. Then I discovered the'taboo' cancer doctors NEVER talk about. Now sex will never be the same... don't ignore these signs Harper Beckham, 14, puts on a stylish display in a fluffy coat and vintage Chanel bag in Paris with her family - after Nicola Peltz's heartbreaking comments about sister-in-law Devastating truth about Blind Side actor Quinton Aaron: More to this'than everyone is letting on', friends reveal... as co-star Sandra Bullock'monitors' situation The wild truth about my influencer sons, their psycho dad and how lawsuits nearly left them bankrupt - by Jake and Logan Paul's MOM Trump knifes'little Napoleon' Border Patrol commander over Minnesota mayhem as he declares: 'We'll de-escalate' Lost tomb of the mysterious'cloud people' unearthed after 1,400 years in'discovery of the decade' READ MORE: How AI cops will be used to patrol Britain's streets Scientists have uncovered four entirely new personality types that all ChatGPT users fall into. According to experts from the University of Oxford and the Berlin University Alliance, every chatbot user has a unique personality type, each with their own motivations. Some truly tech-savvy users fall into the category of ' AI enthusiasts'.

Winners will emerge from the Artificial Intelligence (AI) boom, but there will be carnage along the way, the boss of a US tech giant has warned. Chuck Robbins, chairman and chief executive of Cisco Systems, told the BBC the technology will be bigger than the internet, but the current market is probably a bubble and some companies won't make it. Cisco, one of the world's leading technology companies, is behind some of the critical IT infrastructure enabling day-to-day use of AI. Robbins said some jobs will be changed, or even eliminated, by AI, particularly in areas like customer services where companies will need fewer people, but urged workers to embrace, not fear, the technology. His comments follow a series of warnings over the recent surge in investment in AI, with some claiming the sector amounts to a bubble set to burst, rocking markets and bankrupting companies.

Traditional data-driven methods, effective for deterministic systems or stochastic differential equations (SDEs) with Gaussian noise, fail to handle the discontinuous sample paths and heavy-tailed fluctuations characteristic of Lévy processes, particularly when the noise is state-dependent. To bridge this gap, we establish nonlocal Kramers-Moyal formulas, rigorously generalizing the classical Kramers-Moyal relations to SDEs with multiplicative Lévy noise. These formulas provide a direct link between short-time transition probability densities (or sample path statistics) and the underlying SDE coefficients: the drift vector, diffusion matrix, Lévy jump measure kernel, and Lévy noise intensity functions. Leveraging these theoretical foundations, we develop novel data-driven algorithms capable of simultaneously identifying all governing components from data and establish convergence results and error analysis for the algorithms. We validate the framework through extensive numerical experiments on prototypical systems. This work provides a principled and practical toolbox for discovering interpretable SDE models governing complex systems influenced by discontinuous, heavy-tailed, state-dependent fluctuations, with broad applicability in climate science, neuroscience, epidemiology, finance, and biological physics.

We propose an (offline) multi-dimensional distributional reinforcement learning framework (KE-DRL) that leverages Hilbert space mappings to estimate the kernel mean embedding of the multi-dimensional value distribution under a proposed target policy. In our setting, the state-action variables are multi-dimensional and continuous. By mapping probability measures into a reproducing kernel Hilbert space via kernel mean embeddings, our method replaces Wasserstein metrics with an integral probability metric. This enables efficient estimation in multi-dimensional state-action spaces and reward settings, where direct computation of Wasserstein distances is computationally challenging. Theoretically, we establish contraction properties of the distributional Bellman operator under our proposed metric involving the Matern family of kernels and provide uniform convergence guarantees. Simulations and empirical results demonstrate robust off-policy evaluation and recovery of the kernel mean embedding under mild assumptions, namely, Lipschitz continuity and boundedness of the kernels, highlighting the potential of embedding-based approaches in complex real-world decision-making scenarios and risk evaluation.

The deployment of Artificial Intelligence in high-risk domains, such as finance and healthcare, necessitates models that are both fair and transparent. While regulatory frameworks, including the EU's AI Act, mandate bias mitigation, they are deliberately vague about the definition of bias. In line with existing research, we argue that true fairness requires addressing bias at the intersections of protected groups. We propose a unified framework that leverages Mixed-Integer Optimization (MIO) to train intersectionally fair and intrinsically interpretable classifiers. We prove the equivalence of two measures of intersectional fairness (MSD and SPSF) in detecting the most unfair subgroup and empirically demonstrate that our MIO-based algorithm improves performance in finding bias. We train high-performing, interpretable classifiers that bound intersectional bias below an acceptable threshold, offering a robust solution for regulated industries and beyond.

We propose a framework to construct practical kernel-based two-sample tests from the family of $f$-divergences. The test statistic is computed from the witness function of a regularized variational representation of the divergence, which we estimate using kernel methods. The proposed test is adaptive over hyperparameters such as the kernel bandwidth and the regularization parameter. We provide theoretical guarantees for statistical test power across our family of $f$-divergence estimates. While our test covers a variety of $f$-divergences, we bring particular focus to the Hockey-Stick divergence, motivated by its applications to differential privacy auditing and machine unlearning evaluation. For two-sample testing, experiments demonstrate that different $f$-divergences are sensitive to different localized differences, illustrating the importance of leveraging diverse statistics. For machine unlearning, we propose a relative test that distinguishes true unlearning failures from safe distributional variations.