AppliedMath

Several studies have reported methods for signal similarity measurement. However, none of the reported methods consider temporal peak-shape features. In this paper, we formalize signal similarity using mathematical concepts and define a new distance function between signals that considers temporal peak-shape characteristics, providing higher precision than current similarity measurements. This distance function addresses latent geometric characteristics in quotient spaces that are not addressed by existing methods. We include an example of using this method on discrete MEG recordings, known for their high spatial and temporal resolution, which were recorded in neighborhoods of extreme points in a cross-area projection of brain activity.

This study introduces a novel hybrid framework that integrates machine learning (ML) with fractional-order differential equations (FDE) to enhance the prediction and clinical management of psoriasis, leveraging real-world data from the UCI Dermatology Dataset. By optimizing ML models, particularly the Voting Ensemble, to inform FDE parameters, and developing a user-friendly graphical user interface (GUI) for real-time diagnostics, the approach bridges computational efficiency with physiological realism, capturing memory-dependent disease progression beyond traditional integer-order models. Key findings reveal that the Voting Ensemble achieves a precision of 0.986 ± 0.007 and an AUC of 0.992 ± 0.005. At the same time, the fractional-order model, with an optimized order of 0.6781 and a mean square error (MSE) of 0.0031, accurately simulates disease trajectories, closely aligning with empirical trends for features such as Age and SawToothRete. The GUI effectively translates these insights into clinical tools, demonstrating probabilities ranging from 0% to 100% based on input features, supporting early detection and personalized planning. The framework’s robustness and potential for broader application to chronic conditions highlight its significance in advancing healthcare.

The geometric foundations of General Relativity are revisited here, with particular attention to its gauge invariance, as a key to understanding the true nature of spacetime. Beyond the common image of spacetime as a deformable “fabric” filling the Universe, curvature is interpreted as the dynamic interplay between matter and interacting fields, a view already emphasized by Einstein and Weyl but sometimes overlooked in the literature. Building on these tools, a Newtonian framework is reconstructed that captures essential aspects of cosmology, showing how classical intuition can coexist with modern geometric insights. This perspective shifts the focus from substance to relationships, offering a fresh magnifying glass through which to reinterpret gravitational dynamics and the large-scale structure of the Universe. The similarities of this approach with other recent, more ambitious ones carried out at the quantum level are quite remarkable.