Generalized relativistic pseudopotentials (GRPP) of atomic cores implying the use of different potentials for atomic electronic shells with different principal quantum numbers give rise to accurate and reliable relativistic electronic structure models of atoms, molecules, clusters, and solids. These models readily incorporate the effects of Breit electron-electron interactions and one-loop quantum electrodynamics effects. Here we report the computational procedure for evaluating one-electron integrals of GRPP over contracted Gaussian functions. This procedure was implemented in a library of routines named LIBGRPP, which can be integrated into existing quantum chemistry software, thus enabling the application of various methods to solve the many-electron problem with GRPPs. Pilot applications to electronic transitions in the ThO and UO2 molecules using the new library and intermediate-Hamiltonian Fock space relativistic coupled cluster method are presented. The results clearly demonstrate that rather economical tiny-core GRPP models can exceed in accuracy relativistic all-electron models defined by Dirac-Coulomb and Dirac-Coulomb-Gaunt Hamiltonians.

This paper presents a comprehensive approach to enhancing submersible safety through mathematical modeling and decision-making frameworks. We develop models for submersible location prediction, emergency preparedness, and scenario extrapolation, including a Seawater Density Model, Submersible Mechanical Model, Bayesian Searching Model, and Extended Kalman Filter. Sensitivity analysis confirms the robustness of these models under varying conditions, making them adaptable for different marine environments.In the first section, we focus on accurately locating a submersible after communication loss. The Seawater Density Model employs a hyperbolic tangent function to model depth-dependent density in the Ionian Sea, while the Submersible Mechanical Model simulates underwater dynamics using the 4th order Runge-Kutta method. Uncertainties are addressed using an Extended Kalman Filter, enhancing accuracy, as shown by trajectory and Mean Squared Error (MSE) comparisons.In the second section, we develop a Bayesian Searching Model to efficiently locate a missing submersible. This model iteratively updates location probabilities using a bimodal Gaussian distribution. The search zone is discretized into grids, and simulations demonstrate the model's ability to effectively narrow down the search area and improve detection success.In the third section, we adapt the models to different environments, such as the Caribbean Sea. A warning and obstacle avoidance system is introduced to manage multiple submersibles in close proximity, dynamically adjusting their paths to avoid collisions.Reliability analysis demonstrates the robustness of the models against changes in density and sudden environmental shifts, showing minimal deviation from neutral buoyancy. These results confirm the reliability and adaptability of the models for enhancing submersible safety across various marine settings.

Since the seminal work of Kolmogorov, probability theory has been based on measure theory, where the central components are so-called probability measures, defined as measures with total mass equal to 1. In Kolmogorov’s theory, a probability measure is used to model an experiment with a single outcome that will belong to exactly one out of several disjoint sets. In this paper, we present a different basic model where an experiment results in a multiset, i.e. for each of the disjoint sets we get the number of observations in the set. This new framework is consistent with Kolmogorov’s theory, but the theory focuses on expected values rather than probabilities. We present examples from testing Goodness-of-Fit, Bayesian statistics, and quantum theory, where the shifted focus gives new insight or better performance. We also provide several new theorems that address some problems related to the change in focus.

This research delves into the intricate challenges confronting the agricultural sector, with a specialized focus on mitigating infections in tomato crops, particularly powdery mildew induced by the Leveillula Taurica pathogen. Tomatoes, renowned for their nutritional richness, are vital to global food security. However, conventional methodologies for disease detection exhibit both laborious processes and limited accuracy. In response to these challenges, this study advocates for an innovative fusion of hyperspectral imaging and deep learning methodologies to detect crop disease and its severity. The systematic workflow commenced with the curation of a dataset, involving the acquisition of live images through OpenCV, followed by conversion to RGB format and subsequent feature extraction utilizing a pre-trained visual geometry group (VGG-16) model for enhanced analysis. Sequentially, RGB images were transformed into simulated hyperspectral images (SHSI) leveraging a Neural Network generator model, offering a distinctive viewpoint on spectral information. This novel approach transcends conventional constraints by delivering a three-dimensional perspective, seamlessly integrating spatial and spectral dimensions for holistic data acquisition. The SHSI is further transmuted into a 3D visualization cube comprehensive grasp of spatial and spectral aspects encompassing spectral, spatial, and Haralick features. The research concludes with severity detection, categorized as low, moderate, or high, employing a Gaussian Mixture Model (GMM) and K-means for visualization.