Abstract. This paper chronicles the wide dispersal of Perron's 1907 result on positive matrices into many fields of science. The many proofs given during the last 93 years are categorized and critiqued (including Perron's original two proofs), and a more natural proof is presented. This simple-to-understand result of Perron provides a unequaled vehicle for taking students on a tour of many applied areas with some depth.

A nonlinear compensation force for stick-slip friction is developed to supplement a proportional + derivative control law applied to a one-degree-of-freedom mechanical system. Inertial control objects acted on by stick-slip friction are common mechanical components in mechanical servo systems and the conceptual model chosen for this investigation is a mass sliding on a rough surface. The choice of a discontinuous compensation force is motivated by the requirement that the desired reference be a unique equilibrium point of the system. The stick-slip friction force, modelled with a sticking force term and a slipping force term, generates discontinuous state derivatives. A Lyapunov function is introduced to prove global asymptotic stability of the desired reference using a modification of the direct method for discontinuous systems. Stability is verified numerically as well as experimentally. The nonlinear compensation force is robust with respect to the character of the slipping force which is assumed to lie within a piecewise linear band. Exact knowledge of the static friction force levels is not required, only upper bounds for these levels. Stability and control effectiveness is verified analytically, numerically and experimentally on a laboratory test stand.

The annihilation operator of the unforced quantum harmonic oscillator is chaotic.

One-dimensional acoustic response of ducts is a classical engineering problem. The acoustic response in a hard-walled duct with a dissipative end condition can be visualized as a combination of standing and propagating wave response. A modal decomposition based on the system eigenvalues derived here produces an infinite order state space model incorporating this behavior. This allows computation of system transient response as well as frequency response. The shapes of duct characteristic response derived here are in stark contrast to those previously available for ducts. It is shown that the traditionally employed sinusoidal responses cannot be used to compute duct response for dissipative ends. A comparison between the frequency response of a finite order truncation of the new state space model and a previous exact frequency response is included. The new transient response of the truncated state space model is demonstrated and truncation error investigated. High frequency behavior of the state space model is discussed.

Bacterial chromosomes frequently contain arrays of contiguous genes that group according to related metabolic roles. We propose that clustering of genes for metabolically related functions confers thermodynamic advantage to the organism based upon our protein immobility model (PIM) of intracellular diffusion. This thermodynamic effect provides the selection force argument that is missing from previous models of gene clustering. The PIM posits that clustered genes produce local clusters of enzymes in bacteria owing to the co-linearity of transcription and translation, and to the relative immobility of large proteins released into the cytosol. We maintain that the resulting physical proximity of enzymes for related pathway steps minimizes the steady state level of reaction step intermediates and thus conserves the energy and material required for rapid growth and maintenance. Support for this idea comes from in silico experiments using the PIM applied to a model metabolic pathway A --> B --> C. The metabolites A, B, and C are small molecules that diffuse freely in a cytosol crowded with macromolecules, whereas the large enzyme molecules, E1 and E2, tend to remain in the vicinity of their point of release. Modeling E1 as a source of B from A, and E2 as a sink for B, numerical experiments suggest that the steady state concentration of B in the cytosol increases approximately in proportion to the square of the distance of the E1 and E2 separation. A further model prediction is that the steady state concentration of B is influenced by the geometric effects of the spatial location and orientation of E1 relative to E2. These results suggest that: (i) gene clustering reduces the energy and material costs of enzyme reactions linked by metabolic intermediates; (ii) gene clusters near ori, the origin of replication, utilize the geometric effect to conserve free energy by further reducing the steady state concentration of the intermediate; (iii) gene organization on a chromosome influences the organism's capacity to accelerate into steady state growth and is, in turn, influenced by the abundance and frequency of access to nutrients.