Wavelet and Linear Algebra

Abstract This paper addresses the solvability of the tensor system A⋉X = B, X ⋉C = D using the
semi-tensor product framework. Initially, we examine the associated tensor-vector system
under the semi-tensor product operation, deriving necessary and sufficient conditions for
the existence of solutions and propose explicit solution methods. Furthermore, we introduce suitable tensor partitioning techniques that enable the reduction of the tensor-vector
system to a linear system formulated via the t-product. This approach facilitates a comprehensive study of the solvability of the tensor system when the unknown tensor X is a
vector, matrix or a higher-order tensor under the semi-tensor product. We apply the STP
framework to model and control Boolean gene regulatory networks with pinning control
inputs. Several illustrative examples are provided to demonstrate the applicability and effectiveness of the proposed approaches. Our study includes the tensor equation A⋉X = B
and the matrix system A ⋉ X = B, X ⋉ C = D as special cases.

Abstract First, we define and investigate some new classes of interval tensors, called interval exceptionally regular tensors  ($ER-$tensor) and interval $wP-$tensors which is relevant to interval strictly semi-positive tensors. Also, we show that $ER-$tensor is a wide class of interval tensors, which includes many important structured tensors. Second, some classes of interval matrices are extended to interval tensors, such as interval $R(R_0)$-tensor and column sufficient interval tensor. We discuss their relationships with interval positive semi-definite tensors and some other structured interval tensors. In addition,  necessary and sufficient conditions for interval (strictly) copositive and interval $E_0$-tensors are presented and investigated. Finally, we extend the concept of the column sufficient interval matrix to the column sufficient interval tensor.