Singular Perturbations and Order Reduction in Control Theory - An Overview

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / DAAB-07-72-C-0259U.S. Air Force / AFOSR 73-257

Nonlinear Spin Dynamics in Nuclear Magnets

A method is developed for solving nonlinear systems of differential, or integrodifferential, equations with stochastic fields. The method makes it possible to give an accurate solution for an interesting physical problem: What are the peculiarities of nonlinear spin dynamics in nonequilibrium nuclear magnets coupled with a resonator? Evolution equations for nuclear spins are derived basing on a Hamiltonian with dipole interactions. The ensemble of spins is coupled with a resonator electric circuit. Seven types of main relaxation regimes are found: free induction, collective induction, free relaxation, collective relaxation, weak superradiance, pure superradiance, and triggered superradiance. The initial motion of spins can be originated by two reasons, either by an imposed initial coherence or by local spin fluctuations due to nonsecular dipole interactions. The relaxation regimes caused by the second reason cannot be described by the Bloch equations. Numerical estimates show good agreement with experiment.Comment: 1 file, 47 pages, LaTe

Factors Associated with Revision Surgery after Internal Fixation of Hip Fractures

Background: Femoral neck fractures are associated with high rates of revision surgery after management with internal fixation. Using data from the Fixation using Alternative Implants for the Treatment of Hip fractures (FAITH) trial evaluating methods of internal fixation in patients with femoral neck fractures, we investigated associations between baseline and surgical factors and the need for revision surgery to promote healing, relieve pain, treat infection or improve function over 24 months postsurgery. Additionally, we investigated factors associated with (1) hardware removal and (2) implant exchange from cancellous screws (CS) or sliding hip screw (SHS) to total hip arthroplasty, hemiarthroplasty, or another internal fixation device. Methods: We identified 15 potential factors a priori that may be associated with revision surgery, 7 with hardware removal, and 14 with implant exchange. We used multivariable Cox proportional hazards analyses in our investigation. Results: Factors associated with increased risk of revision surgery included: female sex, [hazard ratio (HR) 1.79, 95% confidence interval (CI) 1.25-2.50; P = 0.001], higher body mass index (fo

On solving boundary value problems for multi-scale systems using asymptotic approximations and multiple-shooting

This paper describes a method for obtaining the numerical solution of certain singularly perturbed boundary value problems for linear systems of ordinary differential equations whose solutions involve dynamics with multiple time scales. A technique to decouple different time scales using a Riccati transformation is presented. For the slow modes, which provide smooth solutions, regular perturbation expansion and multiple shooting strategies are combined. Fast modes, which are assumed to be significant only in endpoint layers, are computed after appropriate stretchings have been made in the system. Advantages of this approach include adaptability, flexible use of output points, and automatic determination of layer thicknesses. Dedicated to Professor Germund Dahlquist, with the hope that he will long continue to make significant and singular perturbations to the numerical analysis of differential equations