This paper introduces a Gauss–Seidel fixed-point iteration approach for estimating the parameters of the Epanechnikov–Burr XII distribution (EBD) probability density function using maximum likelihood principles. The proposed method updates the shape parameter θ and the scale parameter α in an alternating manner based on explicitly derived fixed-point equations. Numerical experiments are conducted to investigate the convergence behavior of the algorithm and to evaluate its performance in comparison with standard numerical optimization techniques.

In this paper, we provide some generalizations of the concept of fusion frames following that evaluate their representability via a linear operator in Hilbert \(C^*\)-modules. We assume that \(\Upsilon _\xi\) is self-adjoint and \(\Upsilon _\xi(\frak{N} _\xi)= \frak{N} _\xi\) for all \(\xi \in \mathfrak{S}\), and show that if a \(g-\)fusion frame \(\{(\frak{N} _\xi, \Upsilon _\xi)\}_{\xi \in \mathfrak{S}}\) is represented via a linear operator \(\mathcal{T}\) on \(\hbox{span} \{\frak{N} _\xi\}_{ \xi \in \mathfrak{S}}\), then \(\mathcal{T}\) is bounded. Moreover, if \(\{(\frak{N} _\xi, \Upsilon _\xi)\}_{\xi \in \mathfrak{S}}\) is a tight \(g-\)fusion frame, then \(\Upsilon_\xi \) is not represented via an invertible linear operator on \(\hbox{span}\{\frak{N} _\xi\}_{\xi \in \mathfrak{S}}\), We show that, under certain conditions, a linear operator may also be used to express the perturbation of representable fusion frames. Finally, we investigate the stability of this type of fusion frames.

This study aims to explore the impact of various aspects of online customer experience on trust, satisfaction, and repurchase intention in the context of digital transformation. Data were collected from 466 valid survey responses and analyzed using SPSS and SmartPLS. The findings indicate that components of online customer experience, including aesthetic experience, customer experience with online employees, community experience, user experience with ai chatbots, and personalized online experience, all exert positive effects on trust and satisfaction, which serve as critical mediators that foster repurchase intention. Notably, the study highlights the prominent role of advanced technological factors such as AI chatbots and personalization in enhancing the quality of digital experiences. These insights not only contribute to updating the theoretical framework of consumer behavior in online environments but also provide valuable implications for both academic research and marketing practice in the era of artificial intelligence.

The modified degenerate Gamma function \(\Gamma^{*}\lambda(x)\) is defined for positive values of x; however, it is not defined for zero or negative values of \(x\). In this study, the concepts of neutrix and neutrix limit are employed to extend the definition of the modified degenerate Gamma function \(\Gamma^{*}\lambda(x)\) for all real values of \(x\). The results demonstrate that the established definitions and findings recover the classical results for Euler’s Gamma function \(\Gamma(x)\) as \(\lambda \rightarrow 0\) for all real values of \(x\). Additionally, explicit equations for \(\Gamma^{*}\lambda(0)\) and \(\Gamma^{*}_\lambda(-n)\), where n is a positive integer, are given.

In this paper, we study a class of second-order abstract differential equation problems of the elliptic type with operator coefficients with general Robin boundary conditions in a non-commutative setting, i.e the unbounded linear operator in the equation does not commute with the one that appears in the boundary conditions containing two spectral complex parameters. We study the case when the second member belongs to the Hölder space. We give necessary and sufficient conditions of compatibility to obtain a strict solution and also to ensure that the strict solution has the maximal regularity property. This paper is naturally the continuation of the ones studied in [16] and [10].

The major objective of this manuscript is to present another novel extension of metric spaces namely; mapping weighted D-complete metric space. In the process, the uniqueness of various strictures of common and common coupled fixed point theorems have been investigated and verified for two pairs of self commutative maps under influence other improved types of extended contractive conditions in these spaces. On the other hand, a novel extended notion of Hausdorff distance namely; Hausdorff ∇∗-distance has been defined in these spaces. Our second main results of various novel types of improved coincidence fixed point theorems have been obtained by applying the conception of generalized Hausdorff ∇∗-distance. Additionally, various practical implementations of the existence fixed point for two pairs of self commutative maps as a solution for certain non-linear Volterra integral equations have been presented and investigated in the framework of map weighted D-complete metric spaces.

Managing sustainability in modern supply chains requires decision-making tools that can accommodate conflicting criteria, uncertain data, and evaluations that involve both positive and negative impacts. To address these challenges, this study develops a new decision-making framework based on bipolar complex Pythagorean fuzzy sets (BCPFSs). The model integrates bipolarity, complex-valued membership degrees, and Pythagorean structures to capture the nuanced interplay of economic, environmental, and social considerations. On this foundation, two aggregation operators—the bipolar complex Pythagorean fuzzy weighted averaging (BCPFWA) and weighted geometric (BCPFWG) operators—are introduced to synthesize multidimensional information while preserving uncertainty and dual evaluations. The applicability of the framework is demonstrated through a case study on green supply chain management (GSCM). Six supplier strategies, ranging from cost-oriented to fully balanced sustainability-focused approaches, are assessed against eight attributes including cost efficiency, product quality, carbon emissions, waste management, technological integration, and social responsibility. The analysis reveals that the balanced sustainability supplier emerges as the most effective choice, consistently ranked highest by both operators. Comparative results with conventional fuzzy aggregation approaches show that the proposed operators provide richer, more stable, and more interpretable rankings, especially when trade-offs between cost and sustainability are present. This research contributes to both theory and practice: it extends the scope of fuzzy decision-making by unifying multiple existing models as special cases, and it offers a practical toolset for organizations seeking resilient and environmentally responsible supply chain solutions. The findings demonstrate that BCPF-based aggregation can enhance strategic decision-making in contexts where sustainability and uncertainty are inseparably linked.

This study formulates an open problem on spatial localization for quasilinear parabolic equations of pLaplacian type on the half-space R+N, with smooth, nonnegative initial data and boundary conditions that grow unbounded over time. While localization has been widely studied under homogeneous or decaying boundary conditions, its persistence under unbounded boundary input remains unresolved. We introduce two forms of spatial localization: effective localization, where the solution remains within a finite spatial domain, and strict localization, where it vanishes beyond a fixed boundary. The problem examines how the structure of the nonlinearities A(v) and B(v), and the growth rate of the boundary function ϕ influence solution confinement or spread. The study extends classical localization theory to degenerate and singular diffusion processes with diffusion exponent p > 1, offering new insight into spatial confinement under persistent external forcing.

Recently, many researchers have been searching for a new family of distributions that improves the fit to a specific dataset and has attractive properties. In this paper, we present a flexible method for constructing continuous distributions by controlling their reliability. Our new technique incorporates a single parameter into the chosen continuous distribution, but it is a novel method and independent of any previously used methods. We present two special cases of this family. The new modified family offers significant improvements in fit and applicability in many fields. Furthermore, the maximum likelihood method is used to estimate the parameters in the proposed family. Real data analysis is performed to demonstrate the suitability of the new family.

The present paper is devoted to an in-depth study of ϕ-Ricci symmetric LP-Sasakian manifolds, which represent a significant class of Lorentzian para-Sasakian manifolds with rich geometric structures and distinctive curvature characteristics. The notion of ϕ-Ricci symmetry imposes specific constraints on the Ricci tensor in relation to the structure tensor ϕ, offering deeper insight into the curvature behavior and intrinsic geometry of these manifolds. This investigation aims to explore various fundamental properties of ϕ-Ricci symmetric LP-Sasakian manifolds, examining how these symmetry conditions influence their global and local geometric features. To support and illustrate the theoretical analysis, we construct an explicit example of a three-dimensional ϕ-Ricci symmetric LP-Sasakian manifold. Moreover, we study W1-flat LP-Sasakian manifold.