We consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A and an element f∈X. Under appropriate assumptions on the data the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence {un}⊂X which guarantee its convergence to the solution u. We then present several applications which provide the continuous dependence of the solution with respect to the data K, A and f, on one hand, and the convergence of an associate penalty problem, on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials which, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the anlysis of two nonlinear elastic constitutive laws.

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm-Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’ A special attention is given to the overlap between the two potentials along symmetric curves which represent two different forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes.

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.

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.