Stability of polar decompositions

Certain continuity properties of the factors in generalized polar decompositions of real and complex matrices are studied. A complete characterization is given of those generalized polar decompositions that persist under small perturbations in the matrix and in the scalar product. Connections are made with quadratic matrix equations, and with stability properties of certain invariant subspaces

A unified approach to Darboux transformations

We analyze a certain class of integral equations related to Marchenko equations and Gel'fand-Levitan equations associated with various systems of ordinary differential operators. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution. We show how this result provides a unified approach to Darboux transformations associated with various systems of ordinary differential operators. We illustrate our theory by deriving the Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a discrete eigenvalue is added to the spectrum.Comment: final version that will appear in Inverse Problem

Exact solutions to the focusing nonlinear Schrodinger equation

A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates.Comment: 60 pages, 18 figure

Efficacy and safety of a two-step method of skin preparation for peripheral intravenous catheter insertion: a prospective multi-centre randomised trial

We have developed a two-step procedure for preparing the skin before peripheral venous catheter (PVC) insertions. This procedure involves two successive swabbings with wipes soaked in alcoholic antiseptic. We investigated whether this two-step procedure was as effective and safe as the standard four-step procedure – washing with detergent, rinsing, drying, applying antiseptic – by carrying out a multicentre randomised equivalence study comparing the frequency of precursor signs of infection at the site of insertion for the two skin preparation procedures. The study was carried out over an eight-month period, and 248 PVC insertion sites were evaluated. The two-step procedure was used for 130 subjects and the standard procedure for 118. Taking into account all the confounding factors predisposing patients to the complications studied, the characteristics of the two groups of patients were found to be similar, with no significant differences noted. The incidence of precursor signs of infection was 11 % 24 hours after PVC insertion (27/248), 25 % at 48 hours (50/203) and at 29 % at 72 hours (34/119). Eleven patients had complications necessitating the withdrawal of the PVC: sensitivity of the insertion site, with redness and/or slight swelling and/or a palpable venous cord. No major complications were observed in this study. The frequency of local complications associated with PVCs reported in this study, whether simple or severe, was not affected by the skin preparation procedure used for PVC insertion (two-step or four-step procedure)