Non-convex optimization is a difficult and crucial task.

Shaping multi-megawatt loads, such as data centers, impacts generator dispatch on the electric grid, which in turn affects system CO2 emissions and energy cost. Substantiating the effectiveness of prevalent load shaping strategies, such as those based on grid-level average carbon intensity, locational marginal price, or marginal emissions, is challenging due to the lack of detailed counterfactual data required for accurate attribution. This study uses a series of calibrated granular ERCOT day-ahead direct current optimal power flow (DC-OPF) simulations for counterfactual analysis of a broad set of load shaping strategies on grid CO2 emissions and cost of electricity. In terms of annual grid level CO2 emissions reductions, LMP-based shaping outperforms other common strategies, but can be significantly improved upon. Examining the performance of practicable strategies under different grid conditions motivates a more effective load shaping approach: one that "cherry-picks" a daily strategy based on observable grid signals and historical data. The cherry-picking approach to power load shaping is applicable to any large flexible consumer on the electricity grid, such as data centers, distributed energy resources and Virtual Power Plants (VPPs).

Nonnegative matrix factorization (NMF) is a widely used tool for learning parts-based, low-dimensional representations of nonnegative data, with applications in vision, text, and bioinformatics. In clustering applications, orthogonal NMF (ONMF) variants further impose (approximate) orthogonality on the representation matrix so that its rows behave like soft cluster indicators. Existing algorithms, however, are typically derived from optimization viewpoints and do not explicitly exploit the conic geometry induced by NMF: data points lie in a convex cone whose extreme rays encode fundamental directions or "topics". In this work we revisit NMF from this geometric perspective and propose Cone Collapse, an algorithm that starts from the full nonnegative orthant and iteratively shrinks it toward the minimal cone generated by the data. We prove that, under mild assumptions on the data, Cone Collapse terminates in finitely many steps and recovers the minimal generating cone of $\mathbf{X}^\top$ . Building on this basis, we then derive a cone-aware orthogonal NMF model (CC-NMF) by applying uni-orthogonal NMF to the recovered extreme rays. Across 16 benchmark gene-expression, text, and image datasets, CC-NMF consistently matches or outperforms strong NMF baselines-including multiplicative updates, ANLS, projective NMF, ONMF, and sparse NMF-in terms of clustering purity. These results demonstrate that explicitly recovering the data cone can yield both theoretically grounded and empirically strong NMF-based clustering methods.

The discovery of Bell that there exist quantum correlations that cannot be reproduced classically is one of the most important in the foundations of quantum mechanics, as well as having practical implications. Bell's result was originally proven in a simple bipartite causal structure, but analogous results have also been shown in further causal structures. Here we study the only causal structure with six or fewer nodes in which the question of whether or not there exist quantum correlations that cannot be achieved classically was open. In this causal structure we show that such quantum correlations exist using a method that involves imposing additional restrictions on the correlations. This hence completes the picture of which causal structures of up to six nodes support non-classical quantum correlations. We also provide further illustrations of our method using other causal structures.

The independent control of multiple magnetic microrobots under a shared global signal presents critical challenges in biomedical applications such as targeted drug delivery and microsurgeries. Most existing systems only allow all agents to move synchronously, limiting their use in applications that require differentiated actuation. This research aims to design a controller capable of regulating the radial distance between micro-agents using only the angle ψof a global magnetic field as the actuation parameter, demonstrating potential for practical applications. The proposed cascade control approach enables faster and more precise adjustment of the inter-agent distance than a proportional controller, while maintaining smooth transitions and avoiding abrupt changes in the orientation of the magnetic field, making it suitable for real-world implementation. A bibliographic review was conducted to develop the physical model, considering magnetic dipole-dipole interactions and velocities in viscous media. A PID controller was implemented to regulate the radial distance, followed by a PD controller in cascade to smooth changes in field orientation. These controllers were simulated in MATLAB, showing that the PID controller reduced convergence time to the desired radius by about 40%. When adding the second controller, the combined PID+PD scheme achieved smooth angular trajectories within similar timeframes, with fluctuations of only \pm 5^\circ. These results validate the feasibility of controlling the radial distance of two microrobots using a shared magnetic field in a fast and precise manner, without abrupt variations in the control angle. However, the model is limited to a 2D environment and two agents, suggesting future research to extend the controller to 3D systems and multiple agents.

Energy system models are increasingly employed to guide long-term planning in multi-sectoral environments where decisions span electricity, heat, transport, land use, and industry. While these models provide rigorous quantitative insights, their outputs are often highly technical, making them difficult to interpret for non-expert stakeholders such as policymakers, planners, and the public. This communication gap limits the accessibility and practical impact of scenario-based modeling, particularly as energy transitions grow more complex with rising shares of renewables, sectoral integration, and deep uncertainties. To address this challenge, we propose the Renewable Energy Large Language Model (RE-LLM), a hybrid framework that integrates Large Language Models (LLMs) directly into the energy system modeling workflow. RE-LLM combines three core elements: (i) optimization-based scenario exploration, (ii) machine learning surrogates that accelerate computationally intensive simulations, and (iii) LLM-powered natural language generation that translates complex results into clear, stakeholder-oriented explanations. This integrated design not only reduces computational burden but also enhances inter-pretability, enabling real-time reasoning about trade-offs, sensitivities, and policy implications. The framework is adaptable across different optimization platforms and energy system models, ensuring broad applicability beyond the case study presented. By merging speed, rigor, and interpretability, RE-LLM advances a new paradigm of human-centric energy modeling. It enables interactive, multilingual, and accessible engagement with future energy pathways, ultimately bridging the final gap between data-driven analysis and actionable decision-making for sustainable transitions.

We are extending Popularity Bias Memorization theorem from arXiv:archive/2404.12008 in several directions. We extend it to arbitrary degree distributions and also prove both upper and lower estimates for the alignment with top-k singular hyperspace.

The notation φ ( n) = O ( g (n)) means that there is a constant c > 0 such that | φ ( n) | c| g ( n)| . "plain vallina" spectral initialization, its performance still suffers when the number of measurements Stages s1) and s2) are delineated next in reverse order.