The purpose of this article is twofold: (1) On one hand, we rigorously derive the Newton-Maxwell equation in the Coulomb gauge from first principles of quantum electrodynamics in agreement with the formal Bohr correspondence principle of quantum mechanics. (2) On the other hand, we establish the global well-posedness of the Newton-Maxwell system on energy-spaces under weak assumptions on the charge distribution. Both results improve the state of the art, and are obtained by incorporating semiclassical and measure theoretical techniques. One of the novelties is the use of quantum propagation properties in order to build global solutions of the Newton-Maxwell equation.
In this manuscript, we give a new proof of strong minimality of certain automorphic functions, originally results of Freitag and Scanlon (2017), Casale, Freitag, and Nagloo (2020), Blázquez-Sanz, Casale, Freitag, and Nagloo (2020). Our proof is shorter and conceptually different than those presently in the literature.
We study the existence and uniqueness of rank-based interacting systems of stochastic differential equations. These systems can be seen as modifications with state-dependent coefficients of the Atlas model in mathematical finance. The coefficients of the underlying SDEs are possibly discontinuous. We first establish strong well-posedness for a planar system with rank-dependent drift coefficients, and non-rank-dependent and non-uniformly elliptic diffusion coefficients. We then state weak well-posedness for two classes of high-dimensional rank-based interacting SDEs with elliptic diffusion coefficients. Finally, we address the positivity of solutions in the case where the diffusion coefficients vanish at zero.
Low-cost environmental sensors are increasingly used for real-time monitoring, but they often suffer from limitations such as accuracy, calibration drift, short lifespan, and sensitivity to environmental factors. These issues are particularly evident in airborne pollen monitoring devices like Beenose, which provide live high-frequency data yet frequently deviate from manual reference methods such as Hirst traps. To address some of these challenges, we propose a deep learning framework for anomaly correction in multivariate time series collected from Beenose sensors, using Hirst trap measurements as a free anomaly reference. Our approach extends LSTM-based autoencoders with a penalized loss function that explicitly integrates the reference data. The main contributions are: (i) the introduction of data augmentation strategies to address limited training data, and (ii) the adaptation of an LSTM-based autoencoder trained with a regularized loss function. Ablation experiments show that the main performance gain comes from the reference-based regularization term, while the architectural components play a complementary role.
