In this study, we expand a 2D sine map via adding the discrete memristor to introduce a new 3D fractional-order sine-based memristor map. Under commensurate and incommensurate orders, we conduct an extensive exploration and analysis of its nonlinear dynamic behaviors, employing diverse numerical techniques, such as analyzing Lyapunov exponents, visualizing phase portraits, and plotting bifurcation diagrams. The results emphasize the sine-based memristor map’s sensitivity to fractional-order parameters, resulting in the emergence of distinct and diverse dynamic patterns. In addition, we employ the sample entropy ($SampEn$) method and $C_0$ complexity to quantitatively measure complexity, and we also utilize the 0–1 test to validate the presence of chaos in the proposed fractional-order sine-based memristor map. Finally, MATLAB simulations are be executed to confirm the results provided.

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euclidean operator radius. The research broadens its scope to analyze how n-tuple operators, when used as parts of $2\times 2$ operator matrices, illustrate inequalities connected to the Euclidean operator radius. By using the Euclidean numerical radius and Euclidean operator norm for n-tuple operators, the study introduces a range of new inequalities. These inequalities not only set limits for the addition, multiplication, and Euclidean numerical radius of n-tuple operators but also help in establishing inequalities for the Euclidean operator radius. This process involves carefully examining the Euclidean numerical radius inequalities of $2\times 2$ operator matrices with n-tuple operators. Additionally, a new inequality is derived, focusing specifically on the Euclidean operator norm of $2\times 2$ operator matrices. Throughout, the research keeps circling back to the idea of finding and understanding symmetries in linear operators and matrices. The paper highlights the significance of symmetry in mathematics and its impact on various mathematical areas.

Green supply chain management (GSCM) has become essential for organizations seeking to balance environmental sustainability, regulatory compliance, and economic resilience. However, selecting appropriate green supply chain strategies constitutes a complex multicriteria decision-making (MCDM) problem due to diverse sustainability practices, conflicting objectives, dynamic market conditions, and significant uncertainty in expert evaluations. To address these challenges, this study proposes an intelligent multicriteria group decision-making (MCGDM) framework to assess 15 GSCM strategies across 15 environmental, operational, economic, and regulatory criteria. The framework employs complex fractional orthopair fuzzy sets $\left(C_{FOFS}\right)$ to model uncertainty, expert hesitation, and complex-valued judgments. Expert weights are determined using the analytic hierarchy process (AHP), while criteria weights are derived objectively through the entropy method. A modified technique for order preference by similarity to the ideal solution (TOPSIS) is applied to obtain a robust ranking of alternatives. Evaluations from five multidisciplinary experts ensure practical relevance and validity. The results indicate enhanced uncertainty modeling, improved ranking stability, and greater interpretability compared with conventional fuzzy and deterministic approaches. The proposed framework provides a transparent and effective decision support tool for strategic GSCM planning.

In this study, we introduce novel concepts within the framework of fuzzy bipolar b-metric spaces, focusing on various mappings such as ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-contractive mappings, which are essential for quantifying distances between dissimilar elements. We establish fixed-point theorems for these mappings, demonstrating the existence of invariant points under certain conditions. To enhance the credibility and applicability of our findings, we provide illustrative examples that support these theorems and expand the existing knowledge in this field. Furthermore, we explore practical applications of our research, particularly in solving integral equations and fractional differential equations, showcasing the robustness and utility of our theoretical advancements. Symmetry, both in its traditional sense and within the fuzzy context, is fundamental to our study of fuzzy bipolar b-metric spaces. The introduced contractive mappings and fixed-point theorems expand the theoretical framework and offer robust tools for addressing practical problems where symmetry is significant.

The concept of A-curves is considered as a novel application of real field extensions in solving some algebraic vectorial equations defined by Euclidean norms. In this paper, we present a novel insight through the classification of A-curves by illustrating many new semi-module isomorphisms between the direct product of weak fuzzy complex numbers with itself and the direct product of classical Euclidean vector spaces multiplied by itself. These isomorphisms will give us a full classification of A-curves that are related to weak fuzzy complex ring. Also, we provide many examples to explain the contribution of our work.

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