Britain is to develop new air defence weapons alongside the EU's four biggest military powers, deepening ties with the European defence sector. The project will invite manufacturers in the UK, Germany, France, Italy and Poland to submit plans to build low-cost missiles and autonomous drones. The allies are pledging a speedy process to build the weapons together, inspired by Ukraine's development of cheap drones to counter attacks from Russia. The UK's Ministry of Defence (MoD) says the programme will prioritise a lightweight, affordable surface-to-air weapon, with the first project to be delivered by next year. The plan, announced at a meeting of the five countries' defence ministers in the Polish city of Krakow, marks a boost to UK-Europe ties after the failure of talks last year over UK participation in the EU's new €150bn (£130bn) defence fund.

A.2 Proof of Theorem 1 Before showing properties of W defined by (3), we provide two lemmas as follows. Referring to Theorem 1.6 of [32], we have the following result for a sequence of random matrices. One-peer undirected graphs generated in Alg. 2 with s = 3 and u = 1, . . . From the node's perspective, an equivalent version of Alg. 2 is presented in Alg. 4. In the remainder In Alg. 4, we compute This yields the equivalence between Alg.2 and Alg. 4 for the case of v We first provide the following three lemmas. Averaging the above equations yields the result.

Aligning language models (LMs) to human preferences has emerged as a critical pursuit, enabling these models to better serve diverse user needs. Existing methods primarily focus on optimizing LMs for a single reward function, limiting their adaptability to varied objectives. Here, we propose $\textbf{multi-objective decoding~(MOD)}$, a decoding-time algorithm that outputs the next token from a linear combination of predictions of all base models, for any given weighting over different objectives.We exploit a common form among a family of $f$-divergence regularized alignment approaches (such as PPO, DPO, and their variants) to identify a closed-form solution by Legendre transform, and derive an efficient decoding strategy.Theoretically, we show why existing approaches can be sub-optimal even in natural settings and obtain optimality guarantees for our method.Empirical results demonstrate the effectiveness of the algorithm. For example, compared to a parameter-merging baseline, MOD achieves 12.8\% overall reward improvement when equally optimizing towards $3$ objectives. Moreover, we experiment with MOD on combining three fully-finetuned LMs of different model sizes, each aimed at different objectives such as safety, coding, and general user preference. Unlike traditional methods that require careful curation of a mixture of datasets to achieve comprehensive improvement, we can quickly experiment with preference weightings using MOD to find the best combination of models.

Federated learning (FL) faces challenges in ensuring both privacy and communication efficiency, particularly in resource-constrained environments such as Internet of Things (IoT) and edge networks. While sign-based methods, such as sign stochastic gradient descent with majority voting (SIGNSGD-MV), offer substantial bandwidth savings, they remain vulnerable to inference attacks due to exposure of gradient signs. Existing secure aggregation techniques are either incompatible with sign-based methods or incur prohibitive overhead. To address these limitations, we propose Hi-SAFE, a lightweight and cryptographically secure aggregation framework for sign-based FL. Our core contribution is the construction of efficient majority vote polynomials for SIGNSGD-MV, derived from Fermat's Little Theorem. This formulation represents the majority vote as a low-degree polynomial over a finite field, enabling secure evaluation that hides intermediate values and reveals only the final result. We further introduce a hierarchical subgrouping strategy that ensures constant multiplicative depth and bounded per-user complexity, independent of the number of users n.

The UK lacks a plan to defend itself from military attack, a committee of MPs has warned. In a highly critical report, the defence committee says the UK is over-reliant on US resources and that preparations to defend itself and overseas territories in the event of attack are nowhere near where they need to be. The committee's chair, Labour MP Tan Dhesi, said: Putin's brutal invasion of Ukraine, unrelenting disinformation campaigns, and repeated incursions into European airspace mean that we cannot afford to bury our heads in the sand. It comes as the Ministry of Defence (MoD) identified parts of the country where six or more new munitions factories could be built. In June, Defence Secretary John Healey announced plans to move the UK to war-fighting readiness, including £1.5bn to support the construction of new munitions factories, which will be built by private contractors.

Safe and efficient robot operation in complex human environments can benefit from good models of site-specific motion patterns. Maps of Dynamics (MoDs) provide such models by encoding statistical motion patterns in a map, but existing representations use discrete spatial sampling and typically require costly offline construction. We propose a continuous spatio-temporal MoD representation based on implicit neural functions that directly map coordinates to the parameters of a Semi-Wrapped Gaussian Mixture Model. This removes the need for discretization and imputation for unevenly sampled regions, enabling smooth generalization across both space and time. Evaluated on a large public dataset with long-term real-world people tracking data, our method achieves better accuracy of motion representation and smoother velocity distributions in sparse regions while still being computationally efficient, compared to available baselines. The proposed approach demonstrates a powerful and efficient way of modeling complex human motion patterns.

Modulations are a critical part of sound design and music production, enabling the creation of complex and evolving audio. Modern synthesizers provide envelopes, low frequency oscillators (LFOs), and more parameter automation tools that allow users to modulate the output with ease. However, determining the modulation signals used to create a sound is difficult, and existing sound-matching / parameter estimation systems are often uninterpretable black boxes or predict high-dimensional framewise parameter values without considering the shape, structure, and routing of the underlying modulation curves. We propose a neural sound-matching approach that leverages modulation extraction, constrained control signal parameterizations, and differentiable digital signal processing (DDSP) to discover the modulations present in a sound. We demonstrate the effectiveness of our approach on highly modulated synthetic and real audio samples, its applicability to different DDSP synth architectures, and investigate the trade-off it incurs between interpretability and sound-matching accuracy. We make our code and audio samples available and provide the trained DDSP synths in a VST plugin.