Keywords: liquid alloys, microstratification, dispersed particles, spontaneous dispersion, criteria for true (entropic) spontaneous dispersion, mechanisms of quasi-spontaneous dispersion, methods and results for assessing size of dispersed particles.

In this paper we provide some important and significant observations on invo-regular rings appeared in Ann. Univ. Mariae Curie-Sklodowska Sect. A Mathematica (2018). In addition we exhibit that strongly invo-regular rings, invo-regular rings and quasi invo-regular rings all coincide with the well known and well characterized tripotent rings and all these rings further coincide with the well known unit regular rings.

This paper provides a new discovery that adds to the contemporary theory of modelling of non-stationary queues, by capturing analytically the behavior of the root parameter, for the G1/M/1 PSFFA model of non-stationary Ek/M/1 queue. Additionally, it has been also revealed that the resulting root parameter reduces to the PSFFA model of non-stationary queue. Potential applications of PSFFA to Computer engineering are provided. Some challenging open problems are addressed combined with closing remarks and future research directions.

Quasi-interpolation is a powerful tool for approximating functions using radial basis functions (RBFs) such as Gaussian kernels. This avoids solving large systems of equations as in RBF interpolation. However, quasi-interpolation with Gaussian kernels on compact intervals can have significant errors near the boundaries. This paper proposes a quasi-interpolation method with Gaussian kernels using Chebyshev points and boundary corrections to improve the approximation near the boundaries. The boundary corrections use a linear approximation of the function beyond the interval to estimate the truncation error and add correction terms. Numerical studies on test functions show that the proposed method reduces errors significantly near boundaries compared to quasi-interpolation without corrections, for both equally spaced and Chebyshev points. The convergence and accuracy with the boundary corrections are generally better with Chebyshev points compared to equally spaced points. The proposed method provides an efficient way to perform quasi-interpolation on compact intervals while controlling the boundary errors.

For many disciplines of science, all conceivable collisional cross sections and reactions must be precisely known. Although the recent decades were a trial of large-scale research to give these data, many essential atomic and molecular cross section data still missing, and the reliability of the existing cross sections has to be validated. In this paper, we present electron stripping, electron capture, and system breakdown cross sections in carbon (C5+) ions and lithium (Li2+) ions collisions with atomic hydrogen based on the Monte Carlo models of classical and quasi-classical trajectories. According to our expectation, the QCTMC results show higher results in comparison to standard CTMC data, emphasizing the role of Heisenberg correction constraint, especially in the low energy regime. On the Other hand, at high energy, the Heisenberg correction term has less influence as a projectile momentum increase. We present the total cross sections of electron stripping, electron capture, and system breakdown in C5+ ions and Li2+ ions collisions with atomic hydrogen in the impact energy range from 10 keV to 160 keV, which is of interest in astrophysical plasmas, atmospheric sciences, plasma laboratories, and fusion research.

Recommendation systems are crucial in navigating the vast digital market. However, user data’s dynamic and non-stationary nature often hinders their efficacy. Traditional models struggle to adapt to the evolving preferences and behaviours inherent in user interaction data, posing a significant challenge for accurate prediction and personalisation. Addressing this, we propose a novel theoretical framework, the Non-stationary Transformer, designed to capture and leverage the temporal dynamics within data effectively. This approach enhances the traditional transformer architecture by introducing mechanisms accounting for non-stationary elements, offering a robust and adaptable solution for recommendation systems. Our experimental analysis, encompassing deep learning and reinforcement learning paradigms, demonstrates the framework’s superiority over benchmark models. The empirical results confirm the efficacy of our proposed framework, which not only provides significant performance enhancements, approximately 8% in Logloss reduction and up to 2% increase in F1 score but also underscores its potential applicability across accumulative reward scenarios. These findings advocate adopting Non-stationary Transformer models to tackle the complexities of today’s recommendation tasks.