Algebraic methods for the solution of some linear matrix equations

The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method

Exact solution of some linear matrix equations using algebraic methods

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given

Algebraic methods for the study of some linear matrix equations

AbstractAn algebraic viewpoint permits the formulation of necessary and sufficient conditions for the existence of a unique solution to some linear matrix equations. The theory developed is then used to express the solution in a finite series form and to prove a stability theorem

The Fantastic Four: A plug 'n' play set of optimal control pulses for enhancing nmr spectroscopy

We present highly robust, optimal control-based shaped pulses designed to replace all 90{\deg} and 180{\deg} hard pulses in a given pulse sequence for improved performance. Special attention was devoted to ensuring that the pulses can be simply substituted in a one-to-one fashion for the original hard pulses without any additional modification of the existing sequence. The set of four pulses for each nucleus therefore consists of 90{\deg} and 180{\deg} point-to-point (PP) and universal rotation (UR) pulses of identical duration. These 1 ms pulses provide uniform performance over resonance offsets of 20 kHz (1H) and 35 kHz (13C) and tolerate reasonably large radio frequency (RF) inhomogeneity/miscalibration of (+/-)15% (1H) and (+/-)10% (13C), making them especially suitable for NMR of small-to-medium-sized molecules (for which relaxation effects during the pulse are negligible) at an accessible and widely utilized spectrometer field strength of 600 MHz. The experimental performance of conventional hard-pulse sequences is shown to be greatly improved by incorporating the new pulses, each set referred to as the Fantastic Four (Fanta4).Comment: 28 pages, 19 figure

Some open problems in matrix theory arising in linear systems and control

Control theory has long provided a rich source of motivation for developments in matrix theory. Accordingly, we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control. The problems discussed include robust stability, matrix exponentials, induced norms, stabilizability and pole assignability, and nonstandard matrix equations. A substantial number of references are included to acquaint matrix theorists with problems and trends in this application area.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30236/1/0000630.pd

Decrease in selected temperature after intracranial dopamine injections in goldfish

Goldfish (Carassius auratus) (40-80g) were injected with dopamine into the forebrain to study the possible involvement of this amine in central temperature regulation in these fish. Dopamine caused a decrease in selected temperature after injection into the rostral nucleus preopticus periventricularis (NPP). This effect was dependent on the dose of dopamine administered. Doses of 25, 50, 100 and 250 ng were used, injected in a volume of 0.2μ1. Injections in regions adjacent to the NPP elicited hypothermic effects only at the higher dosages. These effects were not consistent. Injections in caudal regions of the NPP elicited no effect. The effects of dopamine were blocked by haloperidol, a selective antagonist of dopamine. It is suggested that dopamine acts on central thermoregulatory neurons, present in the rostral NPP, in the mediation of thermoregulatory behavior. Further, it is suggested that this action is mediated via dopaminergic receptors