The Geometry of the Newton Method on Non-Compact Lie Groups

An important class of optimization problems involve minimizing a cost function on a Lie group. In the case where the Lie group is non-compact there is no natural choice of a Riemannian metric and it is not possible to apply recent results on the optimization of functions on Riemannian manifolds. In this paper the invariant structure of a Lie group is exploited to provide a strong interpretation of a Newton iteration on a general Lie group. The paper unifies several previous algorithms proposed in the literature in a single theoretical framework. Local asymptotic quadratic convergence is proved for the algorithms considered.     

Energy of scattering solitons in the Ward model

We present the results derived in our paper [Proc. Royal. Soc. Lond. A 461 (2005) 1965], where we have shown that the energy density of a scattering-soliton solution in Ward's integrable chiral model is the same as the energy density of a static multi-lump solution of the ℂP ³ sigma model. This explains the quantization of the total energy in the Ward model. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

The Interaction Energy of Well-Separated Skyrme Solitons

We prove that the asymptotic field of a Skyrme soliton of any degree has a non-trivial multipole expansion. It follows that every Skyrme soliton has a well-defined leading multipole moment. We derive an expression for the linear interaction energy of well-separated Skyrme solitons in terms of their leading multipole moments. This expression can always be made negative by suitable rotations of one of the Skyrme solitons in space and iso-space. We show that the linear interaction energy dominates for large separation if the orders of the Skyrme solitons multipole moments differ by at most two. In that case there are therefore always attractive forces between the Skyrme solitons.     

Connections on discrete fibre bundles

A new approach to gauge fields on a discrete space-time is proposed, in which the fundamental object is a discrete version of a principal fibre bundle. If the bundle is twisted, the gauge fields are topologically non-trivial automatically.     

Vortices and Jacobian varieties

We investigate the geometry of the moduli space of N$N$ vortices on line bundles over a closed Riemann surface Σ$\Sigma$ of genus g>1$g>1$, in the little explored situation where 1≤N<g$1\le N<g$. In the regime where the area of the surface is just large enough to accommodate N$N$ vortices (which we call the dissolving limit), we describe the relation between the geometry of the moduli space and the complex geometry of the Jacobian variety of Σ$\Sigma$. For N=1$N=1$, we show that the metric on the moduli space converges to a natural Bergman metric on Σ$\Sigma$. When N>1$N>1$, the vortex metric typically degenerates as the dissolving limit is approached, the degeneration occurring precisely on the critical locus of the Abel–Jacobi map of Σ$\Sigma$ at degree N$N$. We describe consequences of this phenomenon from the point of view of multivortex dynamics.     

Pair production at strong coupling in weak external fields

The rate of electron-positron pair production in a weak external electric field at large values of e is computed. The method is extended to compute the rate of monopole-antimonopole pair production in a weak external magnetic field at large magnetic charge.     

Monopole pair production in a magnetic field

The rate of monopole pair production in a weak external magnetic field is computed by a semiclassical method. The result is analogous to that for electron-positron pair production in an electric field.     

On the high Rayleigh number heating of a fluid above a horizontal surface

From the observed behaviour of free convection at high Rayleigh number above a horizontal surface, a simple model of the temperature profile in such a region is constructed. This model is used to predict the gross properties of a layer of fluid which is heated from below. At the top boundary, the effect of replacing a rigid insulated surface by a stable layer of fluid (that is a temperature inversion) is considered. The behaviour of the region is found to depend strongly upon the boundary conditions at the base and at the top of the convection layer.