On the Lyapunov transformation for stable matrices

The matrices studied here are positive stable (or briefly stable). These are matrices, real or complex, whose eigenvalues have positive real parts. A theorem of Lyapunov states that A is stable if and only if there exists H ˃ 0 such that AH + HA* = I. Let A be a stable matrix. Three aspects of the Lyapunov transformation L_A :H → AH + HA* are discussed.

1. Let C₁ (A) = {AH + HA* :H ≥ 0} and C₂ (A) = {H: AH+HA* ≥ 0}. The problems of determining the cones C₁(A) and C₂(A) are still unsolved. Using solvability theory for linear equations over cones it is proved that C₁(A) is the polar of C₂(A*), and it is also shown that C₁ (A) = C₁(A^-1). The inertia assumed by matrices in C₁(A) is characterized.

2. The index of dissipation of A was defined to be the maximum number of equal eigenvalues of H, where H runs through all matrices in the interior of C₂(A). Upper and lower bounds, as well as some properties of this index, are given.

3. We consider the minimal eigenvalue of the Lyapunov transform AH+HA*, where H varies over the set of all positive semi-definite matrices whose largest eigenvalue is less than or equal to one. Denote it by ψ(A). It is proved that if A is Hermitian and has eigenvalues μ₁ ≥ μ₂…≥ μ_n ˃ 0, then ψ(A) = -(μ₁-μ_n)²/(4(μ₁ + μ_n)). The value of ψ(A) is also determined in case A is a normal, stable matrix. Then ψ(A) can be expressed in terms of at most three of the eigenvalues of A. If A is an arbitrary stable matrix, then upper and lower bounds for ψ(A) are obtained.

Estimation of fishing mortality and migration coefficient of the fishery of juvenile Macrobrachium rosenbergii from Kalu River in Maharashtra

The freshwater prawn, Macrobrachium rosenbergii breeds in estuaries and the juveniles after completion of their larval stage start their upward migration towards rivers. It is at this stage fishing of juveniles takes place in river mouths. Kalu River near Titwala, in Maharashtra is estimated based on data presented by Indulkar and Shirgur (1995) for 1991 and 1992 fishing seasons. The fishing mortality was estimated to be 1.50 and 1.28 for a fishing season of 3 months in 1991 and 1992 respectively, while the migration coefficient was computed to be 3.53 during the fishing season. As the average exploitation rate during the study period was only 0.24, the juveniles are not heavily fished and there is a scope for almost doubling the present catch to about 4 million seeds per fishing season.

The effect of dopants on the twist elastic constant and the rotational viscosity coefficient for nematics

The twist elastic constant, K2, and the rotational viscosity coefficient, γ1, are of importance when the response lime for the in-plane switching mode is studied. Since adding dopants is one technique to improve the response characteristics, the effect of dopants on these physical properties is significant. The effect on K2 and γ1 of adding alkyl(alkoxy) phenylcyclopentenones and alkyl(alkoxy) cyanobiphenyls to the base mixture ZLI-4792 together with their temperature dependence have been investigated using different temperature scales. The reduced temperature scale showed the effect of these dopants on K2 is small. On the other hand, the temperature dependence of γ1 depends on both the absolute temperature scale and the reduced temperature scale. Therefore, it is clear that the choice of temperature scale with which to compare γ1 for different systems raises fundamental questions which way not have a unique answer. 2000 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint.

Measurement of coefficient of consolidation during reconsolidation of liquefied sand

Saturated sands particularly at low relative density commonly exhibit rises in excess pore pressure when subjected to earthquake loading. The excess pore pressure can approach a maximum value, limited by the initial vertical effective stress. After the completion of earthquake shaking, these excess pore pressures dissipate according to the consolidation equation, which can be solved to produce a Fourier series solution. It will be shown by manipulation of this Fourier series that excess pore pressure traces provide a method for back-calculation of coefficient of consolidation Cv. This method is validated against dissipation curves generated using known values of C v and seen to be more accurate in the middle of the layer. The method is then applied to data recorded in centrifuge tests to evaluate Cv throughout the reconsolidation process following liquefaction conditions. C v is seen to fit better as a function of excess pore pressure ratio than effective stress for the stress levels considered. For the soil investigated, Cv is about three times smaller at excess pore pressure ratio of 0.9 compared to excess pore pressure ratio of 0. Copyright © 1996-2011 ASTM.

Effect of hanging coefficient on the efficiency of frame net

Comparative fishing experiments with frame nets of 0.4 and 0.5 hanging coefficients were conducted. Results indicate that net with hanging coefficient of 0.4 as more effective for better catch.

Lyapunov's stability and operational margin of grid-connected photovoltaic modules

Analyses of photovoltaic power generation based on Lyapunov's theorems are presented. The characteristics of the photovoltaic module and the power conditioning unit are analyzed in order to establish energy functions that assess the stability of solutions and define safe regions of operation. Furthermore, it is shown that grid-connected photovoltaic modules driven at maximum power may become unstable under normal grid transients. In such cases, stability can be maintained by allowing an operational margin defined as the energy difference between the stable and the unstable solutions of the system. Simulations show that modules cope well with grid transients when a sufficiently large margin is used.

Re-examination of the coefficient of determination (r2) using road materials engineering case studies

The analysis of scientific data is integral to materials engineering and science. The correlation between measured variables is often quantified by estimating the coefficient of determination or the r2 value. This is the recognised procedure for determining linear relationships. The authors review the derivation of the r2 value and derive an associated quantity, termed the relative deviation (RD), which is the ratio of the root mean square of the deviations about the fitted line to the root mean square of the deviations about the y bar line expressed as a percentage. The relative deviation has an advantage over the coefficient of determination in that it has greater numerical sensitivity to changes in the spread of data about the fitted line, especially when the scatter is small. In addition, the relative deviation is able to define, in percentage terms, the reduction in scatter when different independent variables are correlated with a common dependent variable. Four case studies in the materials field (aggregate crushing value, Atterberg limits, permeability and creep of asphalt) from work carried out at the Queensland Main Roads Department are presented to show the use of the new parameter RD.

Lyapunov-like conditions for the existence of zeno behavior in hybrid and lagrangian hybrid systems

Lyapunov-like conditions that utilize generalizations of energy and barrier functions certifying Zeno behavior near Zeno equilibria are presented. To better illustrate these conditions, we will study them in the context of Lagrangian hybrid systems. Through the observation that Lagrangian hybrid systems with isolated Zeno equilibria must have a onedimensional configuration space, we utilize our Lyapunov-like conditions to obtain easily verifiable necessary and sufficient conditions for the existence of Zeno behavior in systems of this form. © 2007 IEEE.