Fuzzy cognitive maps (FCMs) have garnered significant attention for modeling and analyzing complex systems, owing to their ability to effectively handle uncertainty and nonlinear relationships. Interval-valued fuzzy cognitive maps (IVFCMs), as an extension of FCMs, further enhance this capability by incorporating interval-valued fuzzy numbers to better represent system uncertainty and complexity. Despite their potential, most existing research on IVFCMs has focused on applications, with limited advancement in their theoretical foundations. This study established a comprehensive theoretical framework for IVFCMs, addressing critical gaps in their mathematical basis and dynamic behavior. The key contributions are as follows: (1) Formalization of the inference mechanism, including a rigorous definition of state updates using interval-valued fuzzy operations, and a proof that IVFCMs reduce to conventional FCMs as interval widths approach zero; (2) establishment of convergence guarantees by deriving sufficient stability conditions through spectral radius analysis and the Banach fixed-point theorem for sigmoid activation functions; (3) the proposal of novel centrality measures, introducing interval-valued degree and closeness centrality, supported by an optimized Dijkstra algorithm for efficient identification of key nodes; and (4) empirical validation through a time series prediction case study, demonstrating the model's superior performance in managing uncertainty and noise across different datasets. Overall, this work addressed fundamental theoretical challenges related to inference, convergence, and structural analysis in IVFCMs, while also demonstrating their practical utility in complex system modeling.

Complex interval-valued intuitionistic fuzzy sets not only consider uncertainty and periodicity semantics at the same time but also choose to express the information value with an interval value to give experts more freedom and make the solution to the problem more reasonable. In this study, we used the interval quaternion number space to generalize and extend the utility of complex interval-valued intuitionistic fuzzy sets, analyze their order relation, and offer new operations based on interval quaternion numbers. We proposed a new score function and correlation coefficient under interval quaternion representation. We applied the interval quaternion representation and correlation coefficient to a multi-criterion decision making model and applied the model to enterprise decision-making.

With the continuous development of the fuzzy set theory, neutrosophic set theory can better solve uncertain, incomplete and inconsistent information. As a special subset of the neutrosophic set, the single-valued neutrosophic set has a significant advantage when the value expressing the degree of membership is a set of finite discrete numbers. Therefore, in this paper, we first discuss the change valuesof single-valued neutrosophic numbers when treating them as variables and classifying these change values with the help of basicoperations. Second, the convergence of sequences of single-valued neutrosophic numbers are proposed based on subtractionand division operations. Further, we depict the concept of single-valued neutrosophic functions (SVNF) andstudy in detail their derivatives and differentials. Finally, we develop the two kinds of indefinite integrals of SVNFand give the relevant examples.

In this paper, we propose a direct method to solve the dual fuzzy matrix equation of the form $\mathbf{A}\tilde{\mathbf{X}}+\tilde{\mathbf{B}}=\mathbf{C}\tilde{\mathbf{X}}+\tilde{\mathbf{D}}$ with $\mathbf{A}$, $\mathbf{C}$ matrices of crisp coefficients and $\tilde{\mathbf{B}}$, $\tilde{\mathbf{D}}$ fuzzy number matrices. Extended solution and algebraic solution of the dual fuzzy matrix equations are defined and the relationship between them is investigated. This article focuses on the algebraic solution and a necessary and sufficient condition for the unique algebraic solution existence is given. By algebraic methods we not need to transform a dual fuzzy matrix equation into two crisp matrix equations to solve. In addition, the general dual fuzzy matrix equations and dual fuzzy linear systems are investigated based on the generalized inverses of the matrices. Especially, the solution formula and calculation method of the dual fuzzy matrix equation with triangular fuzzy number matrices are given and discussed. The effectiveness of the proposed method is illustrated with examples.

The interval-valued intuitionistic fuzzy set (IVIFS), which is an extension of the intuitionistic fuzzy set (IFS), characterizes the membership and non-membership degree with the number of intervals. This paper begins with an introduce to order relation, which is embedding. Based on the embedding, we have proposed a separation measure for intervals, and used the coimplication function and interval width to construct it. Then, considering epistemic interpretation of IVIFSs, we generalized the measure of the interval values to IVIFS, and obtained the IVI-separation measure, which can compare the accuracy of two elements on IVIFSs. At the same time, a special construction method was provided based on aggregate functions. Finally, we studied the separation measure of intersection, union, and complement operations on IVIFSs.

The complex fuzzy environment is an innovative tool to deal with fuzzy situations in different mathematical problems. Aiming at the concept of complex intuitionistic fuzzy subgroups, this paper has introduced cut-subsets of complex intuitionistic fuzzy sets, and studied the relationship among the cut-subsets and complex intuitionistic fuzzy subgroups, complex intuitionistic fuzzy Abel subgroups, and complex intuitionistic fuzzy cyclic subgroups. Further, we gave the left and right cosets of complex intuitionistic fuzzy subgroups, defined complex intuitionistic fuzzy normal subgroups, and discussed some of their algebraic properties. Based on this thought, we proposed a new concept of $({\alpha }_{1,2},{\beta }_{1,2})$-complex intuitionistic fuzzy subgroups, and proved that an $({\alpha }_{1,2},{\beta }_{1,2})$-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. At the same time, $({\alpha }_{1,2},{\beta }_{1,2})$-complex intuitionistic fuzzy normal subgroups and their cosets were introduced. Finally, we established a general homomorphism of complex intuitionistic fuzzy subgroups, and studied the relationship between the image and pre-image of complex intuitionistic fuzzy subgroups and complex intuitionistic fuzzy normal subgroups, respectively, under group homomorphism.

In this paper, the concepts of the $\alpha -$pseudo-differentiability and the pseudo-Stieltjes integrability are proposed, and the corresponding transformation theorems and Newton–Leibniz formula are established. The obtained results provide a framework for analyzing nonlinear differential equations.