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In this paper, we provide a systematic characterization of finite BZMVdM-algebra by using semi-tensor product of matrices. The abstract operation law about logic of the finite algebra is transformed into the simple operation of concrete logical matrices. In addition, we study some properties of BZMVdM-algebra, such as homomorphism, isomorphism, and the product of the BZMVdM-algebra. Through logical matrix operation, the direct verifiable conditions for detecting the above properties are given. [ABSTRACT FROM AUTHOR]

This paper investigates the controllability and set controllability of periodically switched Boolean control networks (PSBCNs) via the semi-tensor product. Firstly, the algebraic form of switching signal is given and a parallel system is formed by combining the switching signal and the state. Then the controllability problem of PSBCNs is transformed into the set controllability problem of the new system, further, the criterion for controllability is obtained. Secondly, the set controllability matrix of PSBCNs is constructed, and the necessary and sufficient condition of set controllability for PSBCNs is presented. In addition, a more general PSBCN is considered and the criteria for its controllability and set controllability are proposed. Finally, a practical example is given to illustrate the obtained results, which shows that our new results are correct and effective. [ABSTRACT FROM AUTHOR]

Unlike real or complex numbers, quaternion multiplication does not satisfy the commutative law. This makes it impossible to generalize some conclusions on semi‐tensor product to quaternion directly, which is valid on real or complex number fields. In this paper, we define semi‐tensor product of quaternion matrices and study the properties of semi‐tensor product of quaternion matrices, and the conclusion is applied to the color digital image restoration model. [ABSTRACT FROM AUTHOR]

In this paper, we use semi-tensor product of quaternion matrices, L -representation of quaternion matrices, and GH -representation of special quaternion matrices such as quaternion (anti)-centrosymmetric matrices to solve the special solutions of quaternion matrix equation. Based on semi-tensor product of quaternion matrices and the structure matrix of the multiplication of quaternion, we propose the vector representation operation conclusion of quaternion matrices, and study the different matrix representations of quaternion matrices. Then the problem of the quaternion matrix equation is transformed into the corresponding problem in the real number fields by using vector representation and L -representation of quaternion matrices, combined with the special structure of (anti)-centrosymmetric matrices, the independent elements are extracted by GH -representation method, so as to reduce the number of variables to be calculated and improve the calculation accuracy. Finally, the effectiveness of the method is verified by numerical examples, and the time comparison with the two existing algorithms is carried out. The algorithm in this paper is also applied in a centrosymmetric color digital image restoration model. [ABSTRACT FROM AUTHOR]

In this paper, we consider the split quaternion matrix equation X − A f (X) B = C , f (X) ∈ { X , X H , X i H , X j H X k H } . The H representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, H representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, by using the special structure of the real representation of quaternion matrices, the properties of the Moore–Penrose inverse of quaternion matrices and the Kronecker product of matrices, we study least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C, respectively. First we study the special structures of quaternion bihermitian and skew bihermitian matrices. Then the problem of solving the least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C can be transformed into particular real linear system using these special structures. We deduce the form of all least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C and propose real structure-preserving algorithms to compute the minimal norm least-squares bihermitian and skew bihermitian solution. All the computation process only involve real arithmetic. Numerical examples are provided to verify the effectiveness of our algorithms. Particularly, we also propose the conditions and computation method of bihermitian and skew bihermitian solutions of quaternion matrix equation AXB = C. [ABSTRACT FROM AUTHOR]

The traditional set stability of Boolean networks (BNs) refers to whether all the states can converge to a given state subset. Different from the existing results, the set stability investigated in this paper is whether all states in a given initial set can converge to a given destination set. This paper studies the set stability and set stabilization avoiding undesirable sets of BNs and Boolean control networks (BCNs), respectively. First, by virtue of the semi-tensor product (STP) of matrices, the dynamics of BNs avoiding a given undesirable set are established. Then, the set reachability and set stability of BNs from the initial set to destination set avoiding an undesirable set are investigated, respectively. Furthermore, the set stabilization of BCNs from the initial set to destination set avoiding a given undesirable set are investigated. Finally, a design method for finding the time optimal set stabilizer is proposed, and an example is provided to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]