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.