As digital media increasingly faces copyright threats, robust protection schemes are crucial. While gray-scale watermarking techniques are well-developed, color watermarking problems require more advanced solutions due to the complex spectral relationships among RGB channels. This paper introduces a novel color watermarking scheme based on the split quaternion matrix model, a 4D algebraic structure that preserves the linear spectral relationships among RGB channels, enhances computational efficiency, and improves color image processing. Watermark embedding employs a double split quaternion singular value decomposition (SVDSQ) process. The host image is divided into patches, and dominant singular values are extracted from each patch using SVDSQ. These values are assembled into a matrix, which undergoes a second SVDSQ to embed the watermark. This dual-layered method enhances the adaptability of the watermark embedding payload. Experimental results show that, under the same experimental conditions, our watermarking scheme demonstrates strong robustness against noise and geometric attacks, while maintaining a high peak signal-to-noise ratio (PSNR >= 35), compared to recent watermarking schemes. Despite introducing real part redundancy and requiring prior information for the watermark extraction, this approach advances color image processing using split quaternion matrices. Future work will focus on addressing these limitations and exploring new applications.

The introduction of quaternions in the quantum chemistry framework provides new ideas for solving some of the difficulties encountered in traditional approaches and is expected to expand the scope of quantum chemistry research. This paper investigates the eigenvalue problem of a Hermitian quaternion matrix by means of the complex representation of a quaternion matrix. This paper also gives an efficient algebraic algorithm for finding the eigenvalues and eigenvectors of a Hermitian quaternion matrix, which requires much less computational time than existing algorithms. This novel algorithm provides an algebraic method for quantum chemistry research and is expected to have potential applications in large-scale molecular structure prediction and reaction mechanism studies.

A commutative quaternion total least squares (CQTLS) problem is a method of solving overdetermined sets of linear equations AX approximate to B with errors in the matrices A and B . In the theoretical studies and numerical calculations of commutative quaternionic theory, the CQTLS problem is an extremely effective tool for the study of telecommunications, geodesy, and image processing theory. This paper, by means of the real representation of a commutative quaternion matrix, studies the CQTLS problem, derives necessary and sufficient conditions for the CQTLS problem has a commutative quaternion solution, and gives an algebraic algorithm for solving the CQTLS problem.

In recent years, the reduced biquaternion algebras have been widely used in color image processing problems and in the field of electromagnetism. This paper studies eigen-problems of reduced biquaternion matrices by means of a complex representation of a reduced biquaternion matrix and derives new algebraic algorithms to find the eigenvalues and eigenvectors of reduced biquaternion matrices. This paper also concludes that the number of eigenvalues of an n×n reduced biquaternion matrix is infinite. In addition, the proposed algebraic algorithms are shown to be effective in application to a color face recognition problem.

In the theoretical studies and numerical computations of split quaternionic quantum mechanics, the singular value decomposition theorem of split quaternion matrices plays an essential role. In general the problem of singular value decomposition over split quaternion algebra has hitherto remained tangential for split quaternion matrices. In this paper, by means of a real representation matrix of a split quaternion matrix, the singular value decomposition of split quaternion matrices is studied, the singular value decomposition theorem and the corresponding algorithm for split quaternion matrices are given. Finally, three applications for solving the split quaternion least squares problem and the wave separation problem are given based on the singular value decomposition over split quaternion algebra.

The eigenvalue problem of a Hermitian quaternion matrix plays a crucial role in quaternion quantum mechanics because it is closely related to the solution of Schrödinger equation. In this paper, a fast algorithm is proposed for finding the eigenvalues and corresponding eigenvectors of a Hermitian quaternion matrix based on the real representation of a quaternion matrix as well as the special structure and properties of a Hermitian quaternion matrix. Numerical experiments demonstrate that, compared with the existing computational methods for the eigenvalue problem of a Hermitian quaternion matrix, the proposed method in this paper not only greatly improves the computational efficiency, but also achieves better experimental results in terms of the corresponding computational errors.

By means of a complex representation of a commutative quaternion matrix, the singular value decomposition and the generalized inverse problems of a commutative quaternion matrix are studied, and the corresponding theorems and algorithms are given. In addition, based on the singular value decomposition and generalized inverse of a commutative quaternion matrix, the numerical experiments for solving the least squares problem and the color image watermarking problem are given. Numerical experiments illustrate the effectiveness and reliability of the proposed algorithms.

As a very effective research tool, the commutative quaternion least squares (LS) problems, especially the commutative quaternion total least squares (TLS) problems, have a wide range of potential applications in mathematical physics, telecommunications, and image processing. This paper studies the commutative quaternion TLS problem using the real and complex representation of a commutative quaternion matrix and gives two algorithms for solving the real and complex solutions of the commutative quaternion TLS problem in commutative quaternionic theory.

Unlike quaternions and split quaternions, reduced biquaternions satisfy the multiplication commutative rule and are commonly used in image processing, fuzzy recognition, image compression, Hopfield neural networks, and digital signal processing. However, although algebraic techniques have been developed for the diagonalisation of quaternion and split quaternion matrices, the diagonalisation of a reduced biquaternion matrix is yet to be studied. In this study, we derive sufficient and necessary conditions for the diagonalisation of a reduced biquaternion matrix and devise two numerical methods for the diagonalisation of a reduced biquaternion matrix. These methods were derived using complex and real representations in the reduced biquaternionic algebra.

Biquaternion algebra is an algebraic structure originating from a complex number and has mainly been used in quantum mechanics, special and general relativity, classical, relativistic, and covariant electrodynamics, and signal processing. In this paper, the problem of the diagonalization of a biquaternion matrix is studied, by means of a complex representation of a biquaternion matrix, and an algebraic algorithm for the diagonalization of a biquaternion matrix is presented. In addition, numerical examples demonstrate the effectiveness of the algebraic algorithm.

The algebra of split quaternions is a recently increasing topic in the study of theory and numerical computation in split quaternionic mechanics. This paper, by means of a real representation of a split quaternion matrix, studies the problem of canonical forms of a split quaternion matrix and derives algebraic techniques for finding the canonical forms of a split quaternion matrix. This paper also gives two applications for the right eigenvalue and diagonalization in split quaternionic mechanics.

The quaternionic Schrödinger equation ∂/∂ t|f⟩ =-A|f⟩ is the crucial part of the study of quaternionic quantum mechanics and plays indispensable roles in related fields. One of the practical and special cases that has received more attention from mathematicians and physicists is that A is a Hermitian quaternion matrix. The problem can be equivalent to a Hermitian quaternion right eigenvalue problem Aα =αλ by discretization. This paper, by means of a complex representation method, studies the Hermitian quaternion Schrödinger equation problem, and proposes a novel algebraic method (complex structure-preserving method) for right eigenvalue problems of Hermitian quaternion matrices. Moreover, the complex structure-preserving method is superior and formally simple compared to previous methods, and numerical experiments also demonstrate the effectiveness of the method.

We consider a retrospective inverse heat conduction problem with nonstationary inhomogeneous Dirichlet boundary conditions. It is approximated by a Crank–Nicolson scheme that has the second order of approximation both in the spatial variable and in time. It is proposed to use the iterative method of conjugate gradients to determine the solution of the resulting system of linear algebraic equations. Examples are given of reconstructing smooth, nonsmooth, and discontinuous initial conditions, including the introduction of a “noise” characteristic, typical of additional conditions of inverse problems, and its smoothing using the Savitzky–Golay filter.

This paper, by means of two matrix representations of a commutative quaternion matrix, studies the relationship between the solutions of commutative quaternion equality constrained least squares (LSE) problems and that of complex and real LSE problems and derives two algebraic methods for finding the solutions of equality constrained least squares problems in commutative quaternionic theory.

In the theoretical explorations and numerical computations in reduced biquaternionic electromagnetics, the reduced biquaternion total least squares (RBTLS) problem is an extremely effective tool for the study of reduced biquaternionic electromagnetics and electromagnetic field theory. This paper studies for the first time the RBTLS problem by means of a complex representation of a reduced biquaternion matrix, derives the necessary and sufficient conditions for the RBTLS problem to have a reduced biquaternion solution, and gives an efficient algorithm for solving the RBTLS problem. Finally, numerical examples are presented to demonstrate the efficiency of the proposed algorithm.

In the theoretical explorations and numerical computations of split quaternionic mechanics, a common and extremely effective tool for the study of quantum mechanics and quantum field theory is the split quaternion equality constrained least squares (LSESQ) problem. This paper for the first time studies the generalized singular value decomposition of split quaternion matrices (GSVDSQ) based on the $$2\times 2$$ isomorphic representation of split quaternion matrices and obtains a GSVDSQ theorem. In addition, this paper proves the necessary and sufficient conditions for the LSESQ problem to have solutions and gives an efficient method for solving the LSESQ problem. Finally, two numerical examples are presented to demonstrate the efficiency of the proposed method.

In this paper, we present an Online Generalized Multiscale Finite Element Method(Online GMsFEM) for heat and mass transfer problem in heterogeneous media with artificial ground freezing pipes. The mathematical model of the process is based on the classical Stefan model, which describes heat transfer with a phase transition and takes into account filtration in a porous medium. The model is described by a system of equations for temperature and pressure. For fine grid solution, we use a finite element method using the fictitious domain method. To derive a solution on the coarse grid, we use a model reduction procedure based on Online GMsFEM. Online version of GMsFEM allows to us to take less number of offline multiscale basis functions. In our approach, we use decoupled offline basis functions constructed with snapshot space and based on spectral problems. This is the standard approach of basis construction. To take into account artificial ground freezing pipes, we compute an additional basis functions on the offline stage. For the accurate approximation of phase change we add online multiscale basis functions. We construct online basis that minimizes error by values of local residuals. Online procedure is significantly improves the accuracy of standard GMsFEM. We present numerical results in two-dimensional domain with layered heterogeneity. To investigate accuracy of the method, we present results with different number of offline and online basis functions. The presented results show that Online GMsFEM can produce solution with high accuracy and requires small computational resources.

In theoretical studies and numerical computations of three-dimensional electromagnetics, Maxwell’s equations play important roles. With the breakthroughs made by physicists in the field of high-dimensional electromagnetics, it has become possible to use split quaternion algebraic representations and solve some classical Maxwell’s equations. Especially, Maxwell’s equations with a discrete double-curl operator, which can be discretized into a generalized eigen-problem for a Hermitian matrix pencil. This paper studies Maxwell’s equations and the generalized eigen-problem of the matrix pencil over split quaternion algebra and proposes an efficient method for solving Maxwell’s equations with a discrete double-curl operator based on an isomorphic mapping. In addition, this paper obtains an algebraic method for the split quaternion generalized eigen-problem. Finally, numerical experiments show the feasibility of the proposed method in this paper.

A mathematical model of heat transfer between the permafrost foundation of a building with a system of freezing pipes and atmospheric air is constructed and numerically implemented. As a mathematical model of heat transfer in the building foundation, the Stefan problem in the enthalpy statement with smoothed coefficients and a smeared temperature source is used. The heat transfer in the system of pipes is described by a one-dimensional convection–conduction equation. Numerical results obtained on model problems showed that the proposed numerical method significantly reduces the computational complexity of the problem.