In this paper, we consider the quaternion matrix equation $X-A\hat{X}B=C$, and study its minimal norm least squares solution, j-self-conjugate least-squares solution and anti-j-self-conjugate least-squares solution. By the real representation matrices of quaternion matrices, their particular structure and the properties of Frobenius norm, we convert above least-squares problems into corresponding problems of real matrix equations. The final results of the expressions only involve real matrices, and thus, the corresponding algorithms only involve real operations. Compared with the existing results, they are more convenient and efficient, which are also illustrated by the last two numerical examples.

In this article, our interest is the quaternion matrix equation $ (AXB, DXE) = (C, F) $, and we study its minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution. By applying of the real representation matrices of quaternion matrices and relative properties, we convert the quaternion least squares problems with constrained variables into the corresponding real least squares problems with free variables, and then we obtain the solutions of corresponding problems. The final results can be expressed only by real matrices and vectors, and thus the corresponding algorithms only involves real operations and avoid complex quaternion operations. Therefore, they are portable and convenient. In the end, we give two examples to verify the effectiveness of the purposed algorithms.

This paper investigates robust set stability for the switched Boolean networks (SBNs) with arbitrary switching signal affected by one-bit function perturbation (OBFP). Firstly, the dynamics of these networks are converted into algebraic forms utilizing semi-tensor product (STP) method. Secondly, OBFP impact on the set stability of SBNs is divided into four cases. Then, by constructing a state set and defining an index vector, several necessary and sufficient conditions are provided to detect whether an SBN with arbitrary switching signal is still stable to the given set after OBFP. Finally, a biological example is proposed to demonstrate the effectiveness of the obtained theoretical results.