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We investigate the transient dynamics of Block Coordinate Descent algorithms in valleys of the optimization landscape. Iterates converge linearly to a vicinity of the valley oor and then progress in a zig-zag fashion along the direction of the valley oor. When the valley sides are symmetric, the contraction factor to a vicinity of the valley oor appears to be no worse than 1/8, but without symmetry the contraction factor can approach 1. Progress along the direction of the valley oor is proportional to the gradient on the valley oor and inversely proportional to the narrowness" of the valley. We quantify narrowness using the eigenvalues of the Hessian on the valley oor and give explicit formulas for certain cases. Progress also depends on the direction of the valley with respect to the blocks of coordinates. When the valley sides are symmetric, we give an explicit formula for this dependence and use it to show that in higher dimensions nearly all directions give progress similar to the worst case direction. Finally, we observe that when starting the algorithm, the ordering of blocks in the first few steps can be important, but show that a greedy strategy with respect to objective function improvement can be a bad choice. [ABSTRACT FROM AUTHOR]

The undamped small-amplitude superimposed horizontal motion of a load supported symmetrically between identical isotropic hyperelastic springs, each subjected to an initial finite uniaxial static stretch, is reviewed in general terms. The small superimposed motion is discussed for the classical incompressible Mooney–Rivlin, James–Guth, and Gent models, as well as two limit classes of compressible Blatz–Ko material models, f = 0 and f = 1. The small-amplitude vibrational frequency is presented for each model, and the effects of limited extensibility are demonstrated for the James–Guth and Gent materials. Unstable equilibrium states are exhibited for the Blatz–Ko compressible foamed rubber material with f = 0 , while the others exhibit infinitesimally stable motion for an essentially arbitrary initial static stretch. It is shown that unstable equilibrium states exist for the general compressible Blatz–Ko model with 0 < f < 1 , and these states are characterized graphically in terms of the static stretch and the material Poisson ratio. The article concludes with a discussion of the Blatz–Ko and Gent–Thomas foamed rubber materials and their relation to the classical molecular-based uni-constant theory of elasticity. [ABSTRACT FROM AUTHOR]

As technological improvements continue to infiltrate and impact medical practice, it has become possible to non-invasively collect dense physiological time series data from individual patients in real time. These advances continue to improve physicians' ability to detect and to treat infections early. One important benefit of early detection and treatment of nascent infections is that it leads to earlier resolution. In response to current and anticipated advances in data capture, we introduce the Early Treatment Gain (ETG) as a measure to quantify this benefit. Roughly, we define the gain to be the limiting ratio: ETG = differential change in time of resolution / differential change in treatment time . We study the gain using standard dynamical models and demonstrate its use with time series data from Surgical Intensive Care Unit (SICU) patients facing ventilator associated pneumonia. The main conclusion from the mathematical modelling is that the ETG is always greater than one unless there is an effective immune response, in which case the ETG can be less than one. Using real patient time series data, we observe that the formula derived for a linear model can be applied and that this produces a ETG greater than one. [ABSTRACT FROM AUTHOR]

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.

Ventilator associated pneumonia is a common and costly complication in critically ill and injured surgical patients. The diagnosis of pneumonia remains problematic and non-specific. Using clinical criteria, a diagnosis of pneumonia is typically not made until an infection is well established. Semi-quantitative cultures of endotracheal aspirate and broncho-alveolar lavage are employed to improve the accuracy of diagnosis but are invasive and require time for culture results to become available. We report data that show that an inexpensive, rapid and non-invasive alternative may exist. In particular we show that: 1). Bio-aerosols evolved in the breath of ventilated patients and captured in the hygroscopic condenser humidifier filter of the ventilator circuit contain pathogenic micro-organisms. 2). The number (CFU/ml) and identity (Genus, species) of the pathogens in the aerosol samples can rapidly and inexpensively be determined by PCR. 3). Data from a convenience sample of filters correlate with clinical findings from standard microbiological methods such as broncho-alveolar lavage. The evaluation of the bacterial load evolved in exhaled breath by PCR is amenable to repeated sampling. Since increasing bacterial burden is believed to correlate with the establishment of infection, the use of quantitative PCR may provide a method to rapidly, inexpensively, and effectively detect and diagnose the early onset of pneumonia and identify pathogens involved.

Biologists have long observed periodic-like oxygen consumption oscillations in yeast populations under certain conditions, and several unsatisfactory explanations for this phenomenon have been proposed. These ‘autonomous oscillations’ have often appeared with periods that are nearly integer divisors of the calculated doubling time of the culture. We hypothesize that these oscillations could be caused by a form of cell cycle synchronization that we call clustering. We develop some novel ordinary differential equation models of the cell cycle. For these models, and for random and stochastic perturbations, we give both rigorous proofs and simulations showing that both positive and negative growth rate feedback within the cell cycle are possible agents that can cause clustering of populations within the cell cycle. It occurs for a variety of models and for a broad selection of parameter values. These results suggest that the clustering phenomenon is robust and is likely to be observed in nature. Since there are necessarily an integer number of clusters, clustering would lead to periodic-like behaviour with periods that are nearly integer divisors of the period of the cell cycle. Related experiments have shown conclusively that cell cycle clustering occurs in some oscillating yeast cultures.

In random differential equations with bounded noise minimal forward invariant (MFI) sets play a central role since they support stationary measures. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations.