Onset of synchronization in networks of second-order Kuramoto oscillators with delayed coupling: Exact results and application to phase-locked loops

We consider the inertial Kuramoto model of $N$ globally coupled oscillators characterized by both their phase and angular velocity, in which there is a time delay in the interaction between the oscillators. Besides the academic interest, we show that the model can be related to a network of phase-locked loops widely used in electronic circuits for generating a stable frequency at multiples of an input frequency. We study the model for a generic choice of the natural frequency distribution of the oscillators, to elucidate how a synchronized phase bifurcates from an incoherent phase as the coupling constant between the oscillators is tuned. We show that in contrast to the case with no delay, here the system in the stationary state may exhibit either a subcritical or a supercritical bifurcation between a synchronized and an incoherent phase, which is dictated by the value of the delay present in the interaction and the precise value of inertia of the oscillators. Our theoretical analysis, performed in the limit $N \to \infty$, is based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to the kinetic equation satisfied by the single-oscillator distribution function. We check our results by performing direct numerical integration of the dynamics for large $N$, and highlight the subtleties arising from having a finite number of oscillators.Comment: 15 pages, 4 figures; v2: 16 pages, 5 figures, published versio

An Evaluation of the Principal\u27s Academies of Tennessee

The problem related to this study was to determine which components of the Tennessee Principal\u27s Administrator Academy are effective and which ineffective in influencing the principal\u27s performance. The purpose of this study was to evaluate the effects of the Principal\u27s Academy that influences the principal\u27s day-to-day job performance at the local school site. The study also attempted to determine if factors such as age, size of school, per pupil expenditure, number of teachers on the respondent\u27s staff, educational level, school setting, years in present position, years attending the academy had any effect on the administrator\u27s perceptions of the academy. Tennessee administrators were given the opportunity to respond to the questionnaire used to determine the effectiveness of the Principal\u27s Academy. Five research questions were answered, and seven hypotheses stated in null form were tested using the Kruskal-Wallis Anova for data involving more than two groups. The Mann-Whitney-Wilcoxan Rank Sum W Test was used to determine if there was any significant difference in the respondent\u27s perceptions of the academy as it related to the year they attended. All null hypotheses were retained except the hypothesis related to the year the respondents attended the academy. In years 1984 and 1985 there as a significant difference in the perceptions of the respondents; thus, the hypothesis was rejected. The key motivating factor other than to meet the state mandate of attendance is self-improvement. The collegiality and social network associated with the Principal\u27s Academy is valuable, and attending the Principal\u27s Academy is a factor in school administrators implementing school improvement strategies. Research should be conducted to develop an evaluation instrument that would be used to evaluate future principal\u27s academies

Optimum Average Diffusion Coefficient: An Objective Index in Description of Wood Drying Data

In the existing schemes for estimating average diffusion coefficients, the equations are approximate because of the use of only the first term in an infinite series and the subjective nature of the methods. The method described here takes into account all data points, and provides a systematic and objective way of analyzing wood drying data. Using the formula of the theoretical Fourier series solution, a series of theoretical Ē values, representing the fractional amounts of water in wood during drying, were coupled with experimental data, and the sum of squares minimized. The method sets up upper and lower expected bounds for diffusion coefficients, and then locates the optimum average diffusion coefficient by using a FORTRAN program based on the golden section search principle. Using data from a previous drying study on six hardwoods, it was found that the theoretical curves in the longitudinal direction fitted the data points satisfactorily. This suggests that diffusion coefficients in the longitudinal direction are virtually constant. This method, however, depends upon the assumption that the value of Ē at the surface drops immediately to zero as drying starts

The Effects of Contralateral Exercise On Patient Pain and Range of Motion

Clinicians are commonly tasked with treating patients suffering from range of motion (ROM) deficits, pain, or both. Treatment options intended to increase ROM, decrease pain and restore function are vast. Often interventions are used that focus on a singular symptom location (e.g., stretching and joint mobilizations) although a long-standing call exists to utilize global approaches to treating dysfunction. This review was conducted to determine whether evidence exists to support the use of contralateral exercise (CE)to treatROM deficits or pain