On some nonself mappings

Let X be a Banach space, let K be a non–empty closed subset of X and let T : K → X be a non–self mapping. The main result of this paper is that if T satisfies the contractive–type condition (1.1) below and maps ?K (?K the boundary of K) into K then T has a unique fixed point in K.     

Coupled mode perturbation theory of range dependence

The conventional coupled mode solution is combined with perturbation theory to give a fast, accurate range-dependent normal mode solution for deep water acoustic propagation. Perturbation theory is used to calculate the new normal modes at each range step. The new modes are obtained as a linear combination of the modes for the previous step without requiring a numerical solution of the depth-separated wave equation. The process may be repeated for many steps and yields normal modes and eigenvalues which are sufficiently accurate for solution of practical problems in deep water. The method is applied to long-range propagation through oceanic fronts.     

On pth roots of stochastic matrices

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic pth root of a stochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of matrix pth roots, and in particular on the existence of stochastic pth roots of stochastic matrices. Our contributions include characterization of when a real matrix has a real pth root, a classification of pth roots of a possibly singular matrix, a sufficient condition for a pth root of a stochastic matrix to have unit row sums, and the identification of two classes of stochastic matrices that have stochastic pth roots for all p. We also delineate a wide variety of possible configurations as regards existence, nature (primary or nonprimary), and number of stochastic roots, and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix.     

Taming functionality: easy-to-handle chiral phosphiranes

Enantiopure chiral phosphiranes possessing a binaphthyl backbone demonstrate remarkable thermal stability, are highly resistant to air-oxidation and are effective ligands in catalytic asymmetric hydrosilylations.     

eHydrogenation: Hydrogen-free Electrochemical Hydrogenation

Hydrogenation reactions are staple transformations commonly used across scientific fields to synthesise pharmaceuticals, natural products, and various functional materials. However, the vast majority of these reactions require the use of a toxic and costly catalyst leading to unpractical, hazardous and often functionally limited conditions. Herein, we report a new, general, practical, efficient, mild and high-yielding hydrogen-free electrochemical method for the reduction of alkene, alkyne, nitro and azido groups. Finally, this method has been applied to deuterium labelling.