Dequantisation of the Dirac Monopole

Using a sheaf-theoretic extension of conventional principal bundle theory, the Dirac monopole is formulated as a spherically symmetric model free of singularities outside the origin such that the charge may assume arbitrary real values. For integral charges, the construction effectively coincides with the usual model. Spin structures and Dirac operators are also generalised by the same technique.Comment: 22 pages. Version to appear in Proc. R. Soc. London

Complex Bifurcation from Real Paths

A new bifurcation phenomenon, called complex bifurcation, is studied. The basic idea is simply that real solution paths of real analytic problems frequently have complex paths bifurcating from them. It is shown that this phenomenon occurs at fold points, at pitchfork bifurcation points, and at isola centers. It is also shown that perturbed bifurcations can yield two disjoint real solution branches that are connected by complex paths bifurcating from the perturbed solution paths. This may be useful in finding new real solutions. A discussion of how existing codes for computing real solution paths may be trivially modified to compute complex paths is included, and examples of numerically computed complex solution paths for a nonlinear two point boundary value problem, and a problem from fluid mechanics are given

HyperKhaler Metrics Building and Integrable Models

Methods developed for the analysis of integrable systems are used to study the problem of hyperK\"ahler metrics building as formulated in D=2 N=4 supersymmetric harmonic superspace. We show, in particular, that the constraint equation $\beta\partial^{++2}\omega -\xi^{++2}\exp 2\beta\omega =0$ and its Toda like generalizations are integrable. Explicit solutions together with the conserved currents generating the symmetry responsible of the integrability of these equations are given. Other features are also discussedComment: Latex file, 12 page

The effects of rheological layering on post-seismic deformation

We examine the effects of rheological layering on post-seismic deformation using models of an elastic layer over a viscoelastic layer and a viscoelastic half-space. We extend a general linear viscoelastic theory we have previously proposed to models with two layers over a half-space, although we only consider univiscous Maxwell and biviscous Burgers rheologies. In layered viscoelastic models, there are multiple mechanical timescales of post-seismic deformation; however, not all of these timescales arise as distinct phases of post-seismic relaxation observed at the surface. The surface displacements in layered models with only univiscous, Maxwell viscoelastic rheologies always exhibit one exponential-like phase of relaxation. Layered models containing biviscous rheologies may produce multiple phases of relaxation, where the distinctness of the phases depends on the geometry and the contrast in strengths between the layers. Post-seismic displacements in models with biviscous rheologies can often be described by logarithmic functions

Quantum model of interacting ``strings'' on the square lattice

The model which is the generalization of the one-dimensional XY-spin chain for the case of the two-dimensional square lattice is considered. The subspace of the ``string'' states is studied. The solution to the eigenvalue problem is obtained for the single ``string'' in cases of the ``string'' with fixed ends and ``string'' of types (1,1) and (1,2) living on the torus. The latter case has the features of a self-interacting system and looks not to be integrable while the previous two cases are equivalent to the free-fermion model.Comment: LaTeX, 33 pages, 16 figure

Adaptive Filtering for Large Space Structures: A Closed-Form Solution

In a previous paper Schaechter proposes using an extended Kalman filter to estimate adaptively the (slowly varying) frequencies and damping ratios of a large space structure. The time varying gains for estimating the frequencies and damping ratios can be determined in closed form so it is not necessary to integrate the matrix Riccati equations. After certain approximations, the time varying adaptive gain can be written as the product of a constant matrix times a matrix derived from the components of the estimated state vector. This is an important savings of computer resources and allows the adaptive filter to be implemented with approximately the same effort as the nonadaptive filter. The success of this new approach for adaptive filtering was demonstrated using synthetic data from a two mode system