Russian drone strike killed 12 Ukrainian coal miners hours after President Volodymyr Zelenskyy announced postponement of U.S.-brokered peace talks with Russia until February.

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' A Russian drone strike on a bus carrying miners in Ukraine's Dnipropetrovsk region killed at least 12 people, according to officials. Ukrainian Deputy Prime Minister Denys Shmyhal denounced the strike as a "cynical and targeted" attack on energy workers.

The Kennedy Center in Washington DC will be closed for a two-year renovation beginning in July, President Donald Trump has announced. In a post on Truth Social on Sunday, Trump said the centre would close on 4 July this year in honor of the 250th Anniversary of our Country. The move follows several artists cancelling performances at the storied institution after it was recently renamed as the Trump Kennedy Center. Shortly after taking office, the president fired several of the board members at the centre and replaced them with allies, who then voted to make Trump chairman of the board. The new board renamed the institution the Donald J Trump and the John F Kennedy Memorial Center for the Performing Arts in December.

Three West African juntas have turned to Russia. The US has declared a stark policy shift towards three West African countries which are battling Islamist insurgents and whose military governments have broken defence ties with France and turned towards Russia. The state department announced that Nick Checker, head of its Bureau of African Affairs, would visit Mali's capital Bamako to convey the United States' respect for Mali's sovereignty and chart a new course in relations, moving past policy missteps. It adds that the US also looks forward to co-operating with Mali's allies, neighbouring Burkina Faso and Niger, on shared security and economic interests. Absent from the agenda is the longstanding American concern for democracy and human rights.

Americans pulling into a Starbucks drive thru might think they are being served by a friendly staff member. But at some locations, the voice listening to the order is actually an AI robot. Behind the counter inside the store, baristas can lean on a virtual personal assistant to recall recipes or manage schedules. In the back of the shop, a scanning tool has taken on the painstaking process of counting the inventory, relieving staff of one of retail's most tedious chores, in a bid to fix the out-of-stock gaps that have frustrated the firm. The new technology is part of the hundreds of millions of dollars the 55-year-old coffee giant has been investing as it tries to win back customers after several years of struggling sales.

Large language models (LLMs) process and predict sequences containing text to answer questions, and address tasks including document summarization, providing recommendations, writing software and solving quantitative problems. We provide a mathematical framework for LLMs by describing the encoding of text sequences into sequences of tokens, defining the architecture for next-token prediction models, explaining how these models are learned from data, and demonstrating how they are deployed to address a variety of tasks. The mathematical sophistication required to understand this material is not high, and relies on straightforward ideas from information theory, probability and optimization. Nonetheless, the combination of ideas resting on these different components from the mathematical sciences yields a complex algorithmic structure; and this algorithmic structure has demonstrated remarkable empirical successes. The mathematical framework established here provides a platform from which it is possible to formulate and address questions concerning the accuracy, efficiency and robustness of the algorithms that constitute LLMs. The framework also suggests directions for development of modified and new methodologies.

Overparameterized models have recently challenged conventional learning theory by exhibiting improved generalization beyond the interpolation limit, a phenomenon known as benign overfitting. This work introduces Adaptive Benign Overfitting (ABO), extending the recursive least-squares (RLS) framework to this regime through a numerically stable formulation based on orthogonal-triangular updates. A QR-based exponentially weighted RLS (QR-EWRLS) algorithm is introduced, combining random Fourier feature mappings with forgetting-factor regularization to enable online adaptation under non-stationary conditions. The orthogonal decomposition prevents the numerical divergence associated with covariance-form RLS while retaining adaptability to evolving data distributions. Experiments on nonlinear synthetic time series confirm that the proposed approach maintains bounded residuals and stable condition numbers while reproducing the double-descent behavior characteristic of overparameterized models. Applications to forecasting foreign exchange and electricity demand show that ABO is highly accurate (comparable to baseline kernel methods) while achieving speed improvements of between 20 and 40 percent. The results provide a unified view linking adaptive filtering, kernel approximation, and benign overfitting within a stable online learning framework.

The inference of conditional distributions is a fundamental problem in statistics, essential for prediction, uncertainty quantification, and probabilistic modeling. A wide range of methodologies have been developed for this task. This article reviews and compares several representative approaches spanning classical nonparametric methods and modern generative models. We begin with the single-index method of Hall and Yao (2005), which estimates the conditional distribution through a dimension-reducing index and nonparametric smoothing of the resulting one-dimensional cumulative conditional distribution function. We then examine the basis-expansion approaches, including FlexCode (Izbicki and Lee, 2017) and DeepCDE (Dalmasso et al., 2020), which convert conditional density estimation into a set of nonparametric regression problems. In addition, we discuss two recent generative simulation-based methods that leverage modern deep generative architectures: the generative conditional distribution sampler (Zhou et al., 2023) and the conditional denoising diffusion probabilistic model (Fu et al., 2024; Yang et al., 2025). A systematic numerical comparison of these approaches is provided using a unified evaluation framework that ensures fairness and reproducibility. The performance metrics used for the estimated conditional distribution include the mean-squared errors of conditional mean and standard deviation, as well as the Wasserstein distance. We also discuss their flexibility and computational costs, highlighting the distinct advantages and limitations of each approach.

We propose a variational tail bound for norms of random vectors under moment assumptions on their one-dimensional marginals. A simplified version of the bound that parametrizes the ``aggregating distribution'' using a certain pushforward of the Gaussian distribution is also provided. We apply the proposed method to reproduce some of the well-known bounds on norms of Gaussian random vectors, and also obtain dimension-free tail bounds for the Euclidean norm of random vectors with arbitrary moment profiles. Furthermore, we reproduce a dimension-free concentration inequality for sum of independent and identically distributed positive semidefinite matrices with sub-exponential marginals, and obtain a concentration inequality for the sample covariance matrix of sub-exponential random vectors. We also obtain a tail bound for the operator norm of a random matrix series whose random coefficients may have arbitrary moment profiles. Furthermore, we use coupling to formulate an abstraction of the proposed approach that applies more broadly.

We study continual mean estimation, where data vectors arrive sequentially and the goal is to maintain accurate estimates of the running mean. We address this problem under user-level differential privacy, which protects each user's entire dataset even when they contribute multiple data points. Previous work on this problem has focused on pure differential privacy. While important, this approach limits applicability, as it leads to overly noisy estimates. In contrast, we analyze the problem under approximate differential privacy, adopting recent advances in the Matrix Factorization mechanism. We introduce a novel mean estimation specific factorization, which is both efficient and accurate, achieving asymptotically lower mean-squared error bounds in continual mean estimation under user-level differential privacy.