In this paper, it has been investigated three‐dimensional almost pure twin Walker manifolds equipped with a Walker metric , an almost product structure F satisfying the purity condition, and the associated twin metric . First, the curvature properties of the manifold are studied, and explicit expressions for the Ricci curvature tensor are obtained. Later, the necessary and sufficient conditions for the manifold to be Ricci‐flat are derived. Furthermore, the symmetry structure of the manifold is analyzed by determining its Killing and conformal vector fields through the solution of the corresponding systems of partial differential equations. In addition, the geodesic equations are derived and solved for certain special cases. The obtained results show that the qualitative behavior of geodesics strongly depends on the defining function f ( x , y , z ). Finally, graphical representations of selected geodesics are presented using Python‐based visualizations, illustrating the geometric behavior of the manifold.
This article introduces the notion of a Bianchini‐type g ‐coupling. It investigates the existence and uniqueness of strongly coupled coincidence points associated with these mappings in quasi‐partial metric spaces. In addition, several examples and an application are provided to verify and illustrate the usefulness of the obtained results.
The Mikhailov–Novikov–Wang (MNW) equation is a nonlinear dispersive equation of higher order that occurs in the theory of integrable systems and nonlinear wave propagation. In spite of analytic methods having been applied to build traveling‐wave solutions of this equation, little attention has been paid to systematically exploiting its integrability structure through polynomial first integrals and phase‐space reduction. Specifically, a geometrical understanding of the relationship of invariant manifolds to explicit analytic solutions has not been fully investigated. This study investigates the MNW equation from the perspective of dynamical systems. A traveling‐wave transformation is used to simplify the equation to a nonlinear ordinary differential system. This is then followed by the construction of a polynomial first integral, such that the system can be reduced to a lower dimensional invariant manifold, with explicit finding of the solutions in closed form. The improved Kudryashov method, the modified improved Kudryashov method, and the (1/ G ′ )‐expansion method are separately applied in order to further expand the solution space and to check analytical consistency. The analysis yields several exact solutions, such as kink‐type, antikink, smooth soliton, and singular wave profiles. The consistency between solutions obtained using various analytical frameworks indicates the strength of the proposed technique and emphasizes the integrability of the MNW equation. This work also provides a systematic technique that can be used with other higher order integrable nonlinear evolution equations.
Bounded data defined on the unit interval commonly arise in biostatistics, epidemiology, environmental sciences, economics, and risk modeling. Classical unit distributions are widely useful for modeling such data. Such models may provide limited flexibility when the data exhibit pronounced skewness or complex hazard rate behavior. This paper introduces a new three‐parameter distribution, called the q ‐unit Burr‐XII distribution, by incorporating the q ‐logarithmic function from nonextensive statistical mechanics into the unit Burr‐XII distribution. The deformation parameter q adds more control on the skewness and the shape of the distribution, consequently improving the ability to model diverse bounded data patterns. Several properties of the proposed model are investigated. Different methods are considered for estimating the model parameters, and their performance is assessed through a simulation study. To model bounded responses in the presence of covariates, a parametric quantile regression model is presented based on the proposed model. The methodology is illustrated using two real datasets, including a biomedical application involving android fat percentage as the response variable, with age, body mass index, gender, and physical activity level as explanatory variables. Comparative analyses against established alternatives, such as the beta, Kumaraswamy, unit Weibull, and unit Burr‐XII models, indicate that the proposed model provides a more flexible fit in several settings. These findings highlight the usefulness of the q‐unit Burr‐XII distribution as an alternative framework for modeling bounded data and quantile‐based regression analysis.
This paper investigates an operator constructed with Charlier–Poisson polynomials and analyzes its approximation behavior. We establish results on the convergence of the operator and study its rate of approximation by employing the modulus of continuity. In addition, we obtain the rate of approximation by using the Lipschitz class and Peetre’s K-functional. Later, we observe the graphical analysis for the given operator.
