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.

Alterations of the global water cycle and their effects on river structure, function and services

River structure and functioning are governed naturally by geography and climate but are vulnerable to natural and human-related disturbances, ranging from channel engineering to pollution and biological invasions. Biological communities in river ecosystems are able to respond to disturbances faster than those in most other aquatic systems. However, some extremely strong or lasting disturbances constrain the responses of river organisms and jeopardise their extraordinary resilience. Among these, the artificial alteration of river drainage structure and the intense use of water resources by humans may irreversibly influence these systems. The increased canalisation and damming of river courses interferes with sediment transport, alters biogeochemical cycles and leads to a decrease in biodiversity, both at local and global scales. Furthermore, water abstraction can especially affect the functioning of arid and semi-arid rivers. In particular, interception and assimilation of inorganic nutrients can be detrimental under hydrologically abnormal conditions. Among other effects, abstraction and increased nutrient loading might cause a shift from heterotrophy to autotrophy, through direct effects on primary producers and indirect effects through food webs, even in low-light river systems. The simultaneous desires to conserve and to provide ecosystem services present several challenges, both in research and management.

Intensity-moments characterization of general pulsed paraxial beams with the Wigner distribution function

On the basis of the space-time Wigner distribution function (STWDF), we use the matrix formalism to study the propagation laws for the intensity moments of quasi-monochromatic and polychromatic pulsed paraxial beams. The advantages of this approach are reviewed. Also, a least-squares fitting method for interpreting the physical meaning of the effective curvature matrix is described by means of the STWDF. Then the concept is extended to the higher-order situation, and what me believe is a novel technique for characterizing the beam phase is presented. (C) 1999 Optical Society of America [S0740-3232(99)001009-1].

Understanding function and function-to-form evolution

This is a report on a workshop held at Cambridge University Engineering Design Centre, 17-10 June 1992. This workshop was held to discuss the issue of 'function' and 'function-to-form' evolution in mechanical design. The authors organised this workshop as they felt that their understanding of these topics was incomplete and that discussions between researchers might help to clarify some key issues.

The topic chosen for the workshop proved to be a stimulating one. The term 'function' is part of a designer's daily vocabulary, however there is poor agreement about its definition. In order to develop computer systems to support product evolution, a precise definition is required. Further the value of 'function' and 'function-to-form' evolution as a good choice of workshop topic is evident from the lack of firm conclusions that resulted from the sessions. This lack of consensus made for lively discussion and left participants questioning many of their preconceived ideas.

Attendance at the workshop was by invitation only. A list of the participants (not all those invited could attend due to time and financial constraints) is given in Appendix 1.

Extended bidirectional reflectance distribution function for polarized light scattering from subsurface defects under a smooth surface

By introducing the scattering probability of a subsurface defect (SSD) and statistical distribution functions of SSD radius, refractive index, and position, we derive an extended bidirectional reflectance distribution function (BRDF) from the Jones scattering matrix. This function is applicable to the calculation for comparison with measurement of polarized light-scattering resulting from a SSD. A numerical calculation of the extended BRDF for the case of p-polarized incident light was performed by means of the Monte Carlo method. Our numerical results indicate that the extended BRDF strongly depends on the light incidence angle, the light scattering angle, and the out-of-plane azimuth angle. We observe a 180 degrees symmetry with respect to the azimuth angle. We further investigate the influence of the SSD density, the substrate refractive index, and the statistical distributions of the SSD radius and refractive index on the extended BRDF. For transparent substrates, we also find the dependence of the extended BRDF on the SSD positions. (c) 2006 Optical Society of America.

A note on the von Bertalanffy growth function concerning the allocation of surplus energy to reproduction

We propose an extended form of the von Bertalanffy growth function (VBGF), where the allocation of surplus energy to reproduction is considered. Any function can be used in our model to describe the ratio of energy allocation for reproduction to that for somatic growth. As an example, two models for energy allocation were derived: a step-function and a logistic function. The extended model can jointly describe growth in adult and juvenile stages. The change in growth rate between the two stages can be either gradual or steep; the latter gives a biphasic VBGF. The results of curve fitting indicated that a consideration of reproductive energy is meaningful for model extension. By controlling parameter values, our comprehensive model gives various growth curve shapes ranging from indeterminate to determinate growth. An increase in the number of parameters is unavoidable in practical applications of this new model. Additional information on reproduction will improve the reliability of model estimates.

Effects of rapid decompression and exposure to bright light on visual function in black rockfish (Sebastes melanops) and Pacific halibut (Hippoglossus stenolepis)

Demersal fishes hauled up from depth experience rapid decompression. In physoclists, this can cause overexpansion of the swim bladder and resultant injuries to multiple organs (barotrauma), including severe exophthalmia (“pop-eye”). Before release, fishes can also be subjected to asphyxia and exposure to direct sunlight. Little is known, however, about possible sensory deficits resulting from the events accompanying capture. To address this issue, electroretinography was used to measure the changes in retinal light sensitivity, flicker fusion frequency, and spectral sensitivity in black rockfish (Sebastes melanops) subjected to rapid decompression (from 4 atmospheres absolute [ATA] to 1 ATA) and Pacific halibut (Hippoglossus stenolepis) exposed to 15 minutes of simulated sunlight. Rapid decompression had no measurable influence on retinal function in black rockfish. In contrast, exposure to bright light significantly reduced retinal light sensitivity of Pacific halibut, predominately by affecting the photopigment which absorbs the green wavelengths of light (≈520–580 nm) most strongly. This detriment is likely to have severe consequences for postrelease foraging success in green-wavelength-dominated coastal waters. The visual system of Pacific halibut has characteristics typical of species adapted to low light environments, and these characteristics may underlie their vulnerability to injury from exposure to bright light.