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This paper proposed an efficient spectral-Galerkin approximation and error estimation for the fourth-order Steklov resource problem. Firstly, an appropriate Sobolev space was introduced and the weak form of the original problem, along with its corresponding discrete formulation, was derived. Secondly, the existence and uniqueness of the weak solution and the approximation solution were proven based on the Lax-Milgram lemma. Furthermore, error estimates for the approximation solution were demonstrated by using the approximation properties of the orthogonal projection operators. Additionally, a set of appropriate basis functions in the approximation space was constructed and the matrix form of the discrete scheme based on tensor product was derived. Finally, some numerical examples were given to show the effectiveness of the algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

By utilizing the concepts and theories of singular value, tensor product, tensor sum and the connection of eigenvalue with determinant of matrices, another proof on theorem 2 in literature [1] was given, the singular value and spectral decomposition and the determinant of the matrix suited to A* = A² were investigated, the conclusions were also obtained: (A⊗B)* = (A⊗B)² on tensor product about two matrices A and B which satisfied the above condition, the relationship of eigenvalues of A* with A, the relationship between eigenvectors, and spectral decomposition formular of the matrice A. [ABSTRACT FROM AUTHOR]

Recently, ALIZADEH et al proposed a modification of the Harary index in which the contributions of vertex pairs were weighted by the product of their degrees. It is named multiplicatively weighted Harary index and defined as: , where δG(u) denotes the degree of the vertex u in the graph G and dG(u, v) denotes the distance between two vertices u and v in the graph G. In this paper, the explicit formulae for the multiplicatively weighted Harary index of tensor product G× Kr, the strong product G☒Kr and the wreath product G1 ◦ G2 in terms of other graph invariants including additively weighted Harary index, Harary index, the first and the second Zagreb indices and the first and the second Zagreb coindices, are obtained, where Kr is the complete graph. Additionally, we apply our results to compute the multiplicatively weighted Harary index of open fence and closed fence graphs.

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Separable operators and PPT operators on a finite dimensional tensor product space are introduced. Then, general forms of separable self-adjoint operators, separable projective operators and separable positive operators are obtained. The relationship between separable operators and PPT operators are proved, and some examples which are PPT operators but not separable operators are given. Lastly, separable maps and PPT maps are introduced and discussed, and it is proved that every separable map maps a PPT operator as a PPT operator. [ABSTRACT FROM AUTHOR]