Algebraic Decoupling Research Articles

ПАКЕТ GAP ДЛЯ РОЗЩЕПЛЕННЯ ЛІНІЙНИХ СИСТЕМ

There are a number of problems, the solution of which requires the breakdown of the initial system of equations using algebraic decoupling methods. This means reducing the matrix of coefficients to block-diagonal (or block-triangular) form by means of substitution of variables. The main computational tasks when using such methods are finding the centralizer of several matrices or compiling the algebra generated by these matrices. For calculations, it is convenient to use the GAP computer algebra system because the system itself is designed for discrete algebra calculations. The problem is that the GAP program does not support calculations with real numbers. For practical problems you can try to replace them (with some accuracy) by rational numbers. At the same time, the decision may turn out to be excessively cumbersome. On the other hand, the advantage of GAP is the complete absence of rounding errors.

Geometrical method of decoupling

The computation of tunes and matched beam distributions are essential steps in the analysis of circular accelerators. If certain symmetries - like midplane symmetrie - are present, then it is possible to treat the betatron motion in the horizontal, the vertical plane and (under certain circumstances) the longitudinal motion separately using the well-known Courant-Snyder theory, or to apply transformations that have been described previously as for instance the method of Teng and Edwards. In a preceeding paper it has been shown that this method requires a modification for the treatment of isochronous cyclotrons with non-negligible space charge forces. Unfortunately the modification was numerically not as stable as desired and it was still unclear, if the extension would work for all thinkable cases. Hence a systematic derivation of a more general treatment seemed advisable. In a second paper the author suggested the use of real Dirac matrices as basic tools to coupled linear optics and gave a straightforward recipe to decouple positive definite Hamiltonians with imaginary eigenvalues. In this article this method is generalized and simplified in order to formulate a straightforward method to decouple Hamiltonian matrices with eigenvalues on the real and the imaginary axis. It is shown that this algebraic decoupling is closely related to a geometric "decoupling" by the orthogonalization of the vectors $\vec E$, $\vec B$ and $\vec P$, that were introduced with the so-called "electromechanical equivalence". We present a structure-preserving block-diagonalization of symplectic or Hamiltonian matrices, respectively. When used iteratively, the decoupling algorithm can also be applied to n-dimensional systems and requires ${\cal O}(n^2)$ iterations to converge to a given precision.

Optimal generation of single-qubit operation from an always-on interaction by algebraic decoupling

We present a direct algebraic decoupling approach to generate arbitrary single-qubit operations in the presence of a constant interaction by applying local control signals. To overcome the difficulty of undesirable entanglement generated by the untunable interaction, we derive local control fields that are designed to both drive the qubit systems back to unentangled states at the end of the time interval over which the desired single-qubit operation is completed. This approach is seen to be particularly relevant for the physical implementation of solid-state quantum computation and for the design of low-power pulses in NMR.

Coupled harmonic oscillator systems: Improved algebraic decoupling approach

AbstractWe present a novel explicit decoupling procedure that leads to exact solutions of the Schrödinger equation for N harmonic oscillators in the presence of general bilinear perturbing potentials. This generalizes previous work, which was restricted to orthogonal transformations with conditions imposed on the perturbing terms. It is shown that the formalism is sufficiently general to embrace both stationary‐state and resonance‐state problems and, in particular, we include exact model calculations of a free electron interacting with a lattice and a discussion of an application to an ideal ion trap. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Coupled harmonic oscillator systems: An elementary algebraic decoupling approach

We present simple explicit coordinate transformations which serve to decouple the Schrödinger equation for a pair of (not necessarily identical) harmonic oscillators in the presence of bilinear perturbing potentials. We derive general conditions for the decoupling, and give some examples of physical interest. These include the much studied example with just a static perturbation, the parallel problem with a dynamic coupling term, and the classic example of an isotropic two-dimensional oscillator in a transverse magnetic field, first solved by Fock (1928) by separation of variables.

Algebraic decoupling of variables for systems of ODEs of quasipolynomial form

A generalization of the reduction technique for ODEs recently introduced by Gao and Liu is given. It is shown that the use of algebraic methods allows the extension of the procedure to much more general flows, as well as the derivation of simple criteria for the identification of reducible systems.

A Fault Detection Isolation Observer Design by Hybrid Disturbance Decoupling Approach

Abstract A fault detection isolation observer developed so far requires that the number of unknown inputs (including the faults of no interest and disturbances to be algebraically rejected) must be less than that of outputs. The existence condition is hard to meet and restricts the practical use of a fault detection isolation observer. The development of the fault detection isolation observer with some mild existence condition has been an open problem. The purpose of this paper is to propose a fault detection isolation observer which gives a viable solution to the problem. The basic concept of this proposition is the use of the unknown disturbance modelling approach together with the algebraic decoupling approach. A numerical example shows the applicability of the proposed fault detection isolation observer.