This paper addresses the n-quaternion matrix equation AX - YB = C . We investigate the solutions to this equation, including the general solution, the Hermitian solution, and the anti-Hermitian solution. Firstly, we present the left real element representation and the right real element representation of the quaternion. Secondly, by combining the semi-tensor product of matrices, the n-quaternion matrix equation is transformed into a real linear system. Subsequently, the ℋ -representation is utilized to obtain its special solutions. Thirdly, numerical experiments are conducted to verify the effectiveness of the proposed method. Finally, the solving method of the n-quaternion matrix equation is applied to the encryption and decryption process of color images.
In this paper we introduce a conforming discontinuous Galerkin(CDG)finite element method for solving the heat equation.Unlike the rest of discontinuous Galerkin(DG)methods,the numerical-flux {▽uh·n} is not introduced to the computation in the CDG method.Addi-tionally,the numerical-trace {uh} is not the average(uh|T1+uh|T2)/2(or some other simple average used in other DG methods),but a lifted Pk+1 polynomial from the Pk solution uh on nearby four triangles in 2D,or eight tetrahedra in 3D.We show a two-order supercon-vergence in space approximation when using the CDG method with a backward Euler time discretization,on triangular and tetrahedral meshes,for solving the heat equation.Numerical tests are reported which confirm the theory.
The class of SDD _1 matrices is an important subclass of H-matrices. In this paper, we develop new two-sided determinant bounds for SDD _1 matrices by analyzing the structure of their Schur complements. Compared with existing bounds, our results are shown to be significantly tighter in most cases while remaining computable with similar effort.
Here, we consider the regular N-dimensional fractional Sturm-Liouville problems (N-D FSLPs) of order ν =(ν _1,ν _2,⋯ ,ν _N)∈ (0,1] . The FSLP is defined using a fractional version of the gradient operator in terms of left and right Caputo fractional derivatives (CFDs). We establish the essential properties of the eigenvalues (EVs) and eigenfunctions (EFs) of the FSLP using the principle of fractional variational calculus. To validate our theoretical study, we take the fractional Laplace equation as an example and demonstrate that the EVs exist for fractional order ν∈ (0,1] .
This paper re-explores the coupling between shape-from-shading (SfS) and multi-view stereo (MVS), i.e., reconstructing the geometry of a shaded surface from a set of views. We first provide intuitive evidence for the uniqueness of a solution based on epipolar geometry and Lambertian assumption arguments. Then, we explore several alternatives for approximating this solution, from the semi-Lagrangian approximation of the underlying Hamilton-Jacobi equation to the local optimisation of a neural implicit surface. We qualitatively and quantitatively evaluate the latter approach, which we denote SfS-NeuS, on several synthetic and real-world examples and show that it outperforms comparable state-of-the-art 3D reconstruction methods.
