Thematrix expression and relationships among several definitions of Boolean derivatives are given by using the Cheng product. We introduce several definitions of Boolean derivatives. By using the Cheng product, the matrix expressions of Boolean derivative are given, respectively. Furthermore, the relationships among different definitions are presented. The logical calculation is converted into ...

Simulation of nonisothermal, multiphase flow through fractured geothermal reservoirs involves the solution of a system of strongly nonlinear algebraic equations. The Newton-Raphson method used to solve such a nonlinear system of equations requires the evaluation of a Jacobian matrix. In this paper we discuss automatic differentiation (AD) as a method for analytically computing the Jacobian matr...

Simulation of nonisothermal, multiphase flow through fractured geothermal reservoirs involves the solution of a system of strongly nonlinear algebraic equations. The Newton-Raphson method used to solve such a nonlinear system of equations requires the evaluation of a Jacobian matrix. In this paper we discuss automatic differentiation (AD) as a method for analytically computing the Jacobian matr...

Optimizing the input covariance matrix of a multiple-antenna transmit system with partial channelstructure feedback is an important issue to fully exploit the channel capacity. Efficient design of the optimal input covariance matrix, however, remains unavailable although its eigenvector structure was clearly revealed in a recent publication. In this paper, we obtain an explicit derivative funct...

We derive iterative methods for computing the Fréchet derivative of the map which 4 sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has 5 orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices 6 as well as rectangular matrices having more rows than columns. Our derivation relies on a novel 7 identity that relates the ...

We propose integrable discretizations of derivative nonlinear Schrödinger (DNLS) equations such as the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reduc...

We derive iterative methods for computing the Fréchet derivative of the map which 6 sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has 7 orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices 8 as well as rectangular matrices having more rows than columns. Our derivation relies on a novel 9 identity that relates the ...

We derive iterative methods for computing the Fréchet derivative of the map which sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a novel identity that relates the Fréchet ...

We derive iterative methods for computing the Fréchet derivative of the map which sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a novel identity that relates the Fréchet ...

In this manuscript a new method is introduced for solving fractional differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use fractional-order Legendre wavelets and operational matrix of fractional-order integration. First the fractional-order Legendre wavelets (FLWs) are presented. Then a family of piecewise functions is proposed, based on whi...