Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics and others. Due to its application in control theory, the matrix equation AXB=C has been extensively studied. However, there is currently limited information on matrix equation AXB=C over the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of the dual quaternion matrix equation AXB=C, and present the expression for the general solution when it is solvable. As an application, we derive the ϕ-Hermitian solutions for the dual quaternion matrix equation AXAϕ=C, where the ϕ-Hermitian extends the concepts of Hermiticity and η-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.

In this paper, we establish the necessary and sufficient conditions for solving a dual split quaternion matrix equation AXB=C, and present the general solution expression when solvability is achieved. As an application, we delve into the necessary and sufficient condition for the existence of Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations AX=C and XB=C. Finally, we provide a numerical example to demonstrate the findings of this paper.

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. To illustrate our main findings, we present two numerical algorithms and examples in this paper.

We tested selecting data randomly or proportionally in class imbalanced sample. Collecting data into the training and test set according to the initial ratio of QSOs, galaxies and stars were rec-ommended. We experimented using the original imbalanced data or introducing the class balance technologies: SMOTE, SMOTEENN, SMOTETomek, ADASYN, BorderlineSMOTE1, Border-lineSMOTE2, and RandomUndersampling. The SMOTEENN performed the best in the Sample 1. The LightGBM, CatBoost, XGBoost, and RF were compared when adopting the SMOTEENN using the petroMag_u, petroMag_g, petroMag_r, petroMag_i, petroMag_z, J, H, Ks, W1, W2, W3, W4 magnitudes as features. All of the precisions or recalls exceeded 0.94. The RF cost a little more time than the other three algorithms, but resulted in the best evaluating indicators. Utilizing the SMOTEENN +RF technology, the precision, recall and f1-score for QSOs (galaxies, stars) could achieve 0.98 (0.99, 0.98), 0.99 (0.96, 1.00), 0.98 (0.97, 0.99) respectively in Sample 1. Utilizing the SMOTEENN +RF technology, the precision, recall and f1-score for QSOs (galaxies, stars) could achieve 0.94 (0.96, 0.96), 0.98 (0.90, 0.97), 0.96 (0.93, 0.97) using the petroMag_u, petroMag_g, petroMag_r, petroMag_i, petroMag_z, W1, W2, W3, W4 magnitudes as features.

We investigate and discuss in detail the structure of the restricted singular value decomposition for a tensor triplet under t-product (T-RSVD). The algorithm is provided with a numerical example illustrating the main result. For applications, we consider color image watermarking processing with T-RSVD.