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Dual generalized commutative quaternions have broad application prospects in many fields. Additionally, the matrix equation A X B = C has important applications in mathematics and engineering, especially in control systems, economics, computer science, and other disciplines. However, research on the matrix equation A X B = C over the dual generalized commutative quaternions remains relatively insufficient. In this paper, we derive the necessary and sufficient conditions for the solvability of the dual generalized commutative quaternion matrix equation A X B = C . Furthermore, we provide the general solution expression for this matrix equation, when it is solvable. Finally, a numerical algorithm and an example are provided to confirm the reliability of the main conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we develop an effective iterative algorithm to solve a generalized Sylvester tensor equation over quaternions which includes several well-studied matrix/tensor equations as special cases. We discuss the convergence of this algorithm within a finite number of iterations, assuming negligible round-off errors for any initial tensor. Moreover, we demonstrate the unique minimal Frobenius norm solution achievable by selecting specific types of initial tensors. Additionally, numerical examples are presented to illustrate the practicality and validity of our proposed algorithm. These examples include demonstrating the algorithm's effectiveness in addressing three-dimensional microscopic heat transport and color video restoration problems. [ABSTRACT FROM AUTHOR]

Many real‐world applications can be modelled as graphs or networks, including social networks and biological networks. The theory of algebraic combinatorics provides tools to analyze the functioning of these networks, and it also contributes to the understanding of complex systems and their dynamics. Algebraic methods help uncover hidden patterns and properties that may not be immediately apparent in a visual representation of a graph. In this paper, we introduce left and right inverse graphs associated with finite loop structures and two mappings, P‐edge labeling and V‐edge labeling, of Latin squares. Moreover, this work includes some structural and graphical results of the commutator subloop, nucleus, and loop isotopes of inverse property quasigroups. [ABSTRACT FROM AUTHOR]

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti‐) η$$ \eta $$‐Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti‐) η$$ \eta $$‐Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented. [ABSTRACT FROM AUTHOR]

In order to design a series of satisfied bathroom furniture to meet demands of the consumers, the materials, structures, functions and styles of bathroom cabinet had been discussed and analyzed through reviewing correlative literature and investigating the building material markets, home furnishing stores, home expos, construction expos etc. Some suggestions were provided combined with the modern level of technology and demand for bathroom cabinet design, in order to provide a reference for its future development. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation A X B   =   C , and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a 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 A X   =   C and X B   =   C . Finally, we provide a numerical example to demonstrate the findings of this paper. [ABSTRACT FROM AUTHOR]