Collinear interaction of vortex pairs with different strengths - criteria for leapfrogging

We formulate a system of equations that describes the motion of four vortices made up of two interacting vortex pairs, where the absolute strengths of the pairs are different. Each vortex pair moves along the same axis in the same sense. In much of the literature, the vortex pairs have equal strength. The vortex pairs can either escape to in finite separation or undergo a periodic leapfrogging motion. We determine an explicit criterion in terms of the initial horizontal separation of the vortex pairs given as a function of the ratio of their strengths, to describe a periodic leapfrogging motion when interacting along the line of symmetry. In an appendix we also contrast a special case of interaction of a vortex pair with a single vortex of the same strength in which a vortex exchange occurs

The QRD and SVD of matrices over a real algebra

Recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool. In this article we show how to generalise the QR decomposition and SVD to a wide class of real algebras, including all finite-dimensional semi-simple algebras, (twisted) group algebras and Clifford algebras. Two approaches are described for computing the QRD/SVD: one Jacobi method with a generalised Givens rotation, and one based on the Artin-Wedderburn theorem.Comment: Uses elsarticle.cl

Mathematical modelling of a viscida network

We develop a general model to describe a network of interconnected thin viscous sheets, or viscidas, which evolve under the action of surface tension. A junction between two viscidas is analysed by considering a single viscida containing a smoothed corner, where the centreline angle changes rapidly, and then considering the limit as the smoothing tends to zero. The analysis is generalized to derive a simple model for the behaviour at a junction between an arbitrary number of viscidas, which is then coupled to the governing equation for each viscida. We thus obtain a general theory, consisting of partial differential equations and algebraic conservation laws, for a system of viscidas connected at junctions. This approach provides a framework to understand the fabrication of microstructured optical fibres containing closely spaced holes separated by interconnected thin viscous struts. We show sample solutions for simple networks with and or 3. We also demonstrate that there is no uniquely defined junction model to describe interconnections between viscidas of different thicknesses

Mathematical modelling of a viscida network

We develop a general model to describe a network of interconnected thin viscous sheets, or viscidas, which evolve under the action of surface tension. A junction between two viscidas is analysed by considering a single viscida containing a smoothed corner, where the centreline angle changes rapidly, and then considering the limit as the smoothing tends to zero. The analysis is generalized to derive a simple model for the behaviour at a junction between an arbitrary number of viscidas, which is then coupled to the governing equation for each viscida. We thus obtain a general theory, consisting of partial differential equations and algebraic conservation laws, for a system of viscidas connected at junctions. This approach provides a framework to understand the fabrication of microstructured optical fibres containing closely spaced holes separated by interconnected thin viscous struts. We show sample solutions for simple networks with and or 3. We also demonstrate that there is no uniquely defined junction model to describe interconnections between viscidas of different thicknesses