Generalized quaternions ℍ_α , β are a class of four-dimensional non-commutative algebras that include Hamilton and split quaternions. They offer flexible options for modeling multichannel data. In the paper, we construct a matrix representation for generalized quaternion matrices. Using the matrix representation, we prove the existence of the singular value decomposition (SVD) for any generalized quaternion matrix and develop the corresponding algorithm GQSVD. Based on this, we derive explicit expressions for the Moore–Penrose inverse of a generalized quaternion matrix and the minimum-norm least-squares solution to AX=B . We also investigate several fundamental properties of the Frobenius norm and the induced spectral norm. Moreover, we apply the proposed algorithm to color image compression, geomagnetic signal denoising, and image encryption and decryption. The experimental results indicate that the choice of (α ,β ) provides additional flexibility for adapting the balance between reconstruction quality and computational efficiency to different application requirements.
In this work, we propose a new mesh-based reduction method for cell centered finite volume methods approximating a nonlinear diffusion problem. It is elaborated in the context of the TPFA (Two-Point Flux Approximation) scheme, which is well-known for the discretization of numerous mathematical and physical models like multiphase flows in complex porous media. The main point relies on the Red-Black partition of the domain using unstructured meshes. The key idea is to separate the degrees of freedom (DOFs) into two categories. Each one is characterized by a color: "Red" or "Black". We developed an approach such that all Red control volumes have only Black neighbors. As the nonlinear solver is based on the Newton–Raphson method, the Jacobian matrix is composed of 4 blocks. One block is a non-singular diagonal matrix. The Schur Complement is then easily computed. As a consequence, the Red DOFs can be eliminated without any fill-in. This improves greatly the solver performance. Numerical experiments validate these results by testing the reduced solver on linear, nonlinear, and heterogeneous problems. A gain of factor 2 in CPU time can be achieved between the reduced solver and the original one. More importantly, the efficiency is also enhanced when the Incomplete LU (ILU) factorization preconditioner is applied.
Melanoma remains one of the most serious and challenging forms of skin cancer, mainly because of treatment resistance and disease progression. In this work, we study melanoma treatment using a comparative optimal control approach. Starting from a mathematical model of melanoma dynamics involving CAR T-cells and tumor-associated macrophages (TAMs), we introduce three control functions representing radiotherapy, TAM-targeted therapeutic intervention, and CAR T-cell administration. We compare these strategies when they are applied alone or in combination, in order to evaluate their ability to control tumor progression while preserving the immune response. We first study the existence and local stability of equilibrium points under constant control rates. Then, we formulate an optimal control problem with time-dependent controls and perform numerical simulations to compare the different therapeutic strategies. Our results suggest that the combined use of radiotherapy, TAM-targeted therapeutic intervention, and CAR T-cell administration gives the best control of melanoma progression. When the simultaneous use of all controls is not feasible, the combination of radiotherapy and CAR T-cell administration appears more effective than the combination of TAM-targeted therapeutic intervention and CAR T-cell administration. Moreover, CAR T-cell administration plays a central role and, under certain model conditions, may be sufficient on its own to prevent melanoma progression.
We analyze a model that establishes a relationship between labor and capital while incorporating seasonality and periodicity effects. Our study focuses on investigating the existence and asymptotic behavior of the solutions, considering the seasonal and periodic variations in labor and capital. We propose a numerical scheme using a meshless method, and its convergence is demonstrated. Several numerical examples illustrate the theoretical results obtained.
Time-inhomogeneous stochastic processes are constructed starting from the pseudo-first-order kinetic model by adopting two different approaches. For constant carrying capacity, the transition probability density functions and the related conditional moments are determined. The case in which the stochastic processes are restricted by a reflecting condition in the zero-state is also analyzed. Particular attention is paid to the first-passage time problem through a constant boundary for time-homogeneous processes and for time-inhomogeneous processes with proportional intensity functions. Various numerical computations and simulations are performed to compare the proposed models.
