Terms Of Submatrices Research Articles

Matrix-Based Generalization for Power-Mismatch Newton-Raphson Load Flow Computations With Arbitrary Number of Phases

The standard power-mismatch Newton method is still frequently used for computing load flow due to its simplicity and generality. In this paper, a matrix-based generalization for the usual power flow equations to an arbitrary number of phases is derived. The proposed equations enable computing power injections and the Jacobian matrix in terms of submatrices that compose the network admittance matrix. Besides the more compact representation, another advantage of the proposed generalization is execution time reduction compared to the standard scalar formulation. Simulations are carried out to demonstrate the time reduction achieved via the proposed equations.

Formulation of effective interaction in terms of the non-perturbative and analytic box

One of the useful and practical methods for solving quantum-mechanical many- body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so called box introduced by Kuo et. al. It has been one of the central problems how to calculate the box accurately' and efficiently. Introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With the transformed Hamiltonian we derive a recursion method of calculating the box non-perturbatively and analytically. The box given in this study corresponds to a non-perturbative solution for the energy-dependent. Hamiltonian interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. The present approach has a possibility of resolving many of the theoretical and practical difficulties encountered in the calculation of effective-interaction and/or Hamiltonian.

Formulation of an effective interaction in terms of renormalized vertices and propagators

One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so-called $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box introduced by Kuo et al. It has been one of the central problems how to calculate the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box can be expressed in two ways: One is in the form of a continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures derivation of an exact $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box given in this study corresponds to a nonperturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Z}$-box approach based on the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.

Generalization of the Chinese remainder theorem

Sufficient conditions for a system Ax = r to have an integral solution in the case of a basic matrix A in terms of submatrices and permanents of A are derived. Matrix A in the Chinese remainder theorem is a particular case of a basic matrix. The derivation can be extended to the case where the propositional formula that describes the sign scheme of A is a minimal unsatisfiable CNF.

Localized molecular orbitals in the Hückel model of perturbed alternant hydrocarbons and their relation to the charge–bond order matrix

AbstractAn interrelation is expected between the charge–bond order matrix and the relevant representation matrix of noncanonical (localized) molecular orbitals (NCMOs) of perturbed alternant hydrocarbons (PAHs), as was the case with parent AHs. Accordingly, a single procedure is developed to obtain these matrices directly on the basis of solution of respective noncanonical one‐electron problems in the framework of the simple Hückel model, i.e., of the commutation equation for the one‐electron density matrix and of the block‐diagonalization problem for the Hamiltonian matrix, the latter originating from the Brillouin theorem. The procedure consists of three principal steps: (i) an initial passing to the basis of NCMOs of parent AHs by means of the nonperturbative block diagonalization of the relevant common Hamiltonian matrix (H) represented in the basis of 2pz AOs of carbon atoms; followed by (ii) application of the noncommutative Rayleigh–Schrödinger perturbation theory to solve the above‐specified noncanonical problems; and (iii) subsequent retransformation of the power series obtained into the initial basis of 2pz AOs. Rederivation of the classical results concerning the charge and bond order redistributions in AHs due to perturbation (including the rule of the alternating polarity for one‐center perturbations) following from the above‐described procedure indicates that these results are actually part of the noncanonical theory of MOs for PAHs. The principal achievement of the study, however, consists of obtaining general algebraic expressions for the common NCMO representation matrix of PAHs in terms of submatrices (blocks) of the relevant charge–bond order matrix. These expressions permit not only investigation of the effect of perturbation on the shapes of localized MOs, but also demonstrate the relationship between the given reshaping of these MOs and the respective charge redistribution. For local perturbations of a specific Coulomb parameter, reshaping of a single NCMO is shown to reflect the rule of the alternating polarity, namely of NCMO whose principal AO coincides with the site of perturbation. Moreover, the overall reshaping pattern of NCMOs was found in line with predictions of the simple resonance theory about increased contributions of certain quinoidal structures to the electronic structures of PAHs due to perturbations. NCMOs of pyridine and biphenyl molecules are studied in detail as examples. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Signal flow graphs-Computer-aided system analysis and sensitivity calculations

Concepts which promise to extend many fundamental results of network theory to general systems are introduced. The basis for these extensions is the introduction of two matrices, the summing matrix S and the branching matrix B, which completely describe the topology of a signal flow graph. This leads to a formulation of system equations in terms of submatrices of the S- and B-matrices suitable for digital-computer programming. Consequently, many computer-aided circuit analysis and design programs can now be employed for the computer-aided analysis and design of systems representable by signal flow graphs. This formulation also leads to a straightforward algorithm for obtaining the system gain, an alternate to using Mason's gain formula. Furthermore, the power of this formulation, and its strong relation to network theory, is demonstrated by the derivation of a theorem similar to Tellegen's theorem in network theory. The theorem depends only on the topological properties of the summing and branching matrices and not on the functional relationships between the branch

The Solution of Linear Algebraic Equations

The rank of the coefficient matrix plays a dominant role in the theory of linear algebraic equations. It is not surprising, therefore, that a test for the rank of a matrix, that was a by-product of some work in dimensional analysis, proves to be an an admirable tool in this theory With its aid the consistency requirement assumes a simple and effective form, and the solution of both homogeneous and non-homogeneous systems is given explicitly in terms of submatrices.