It has been proposed by Atiyah and Manton [1] that the dynamics of Skyrmions in IR may be approximated by motion on a finite dimensional manifold obtained from the moduli space of SU(2) Yang-Mills instantons in IR. Motivated by this work we describe how similar results exist for other soliton and instanton systems. We describe in detail two examples for the approximation of the infinite dimensi...

The quotient space of all smooth and connected curves represented by a fixed number of boundary points is a finite-dimensional Riemannian manifold, also known as a shape manifold. This makes the preservation of locality a critically important issue when reducing the dimensionality of shapes on the manifold. We present a completely unsupervised clustering algorithm employing diffusion maps for l...

The loop space LP1 of the Riemann sphere consisting of all C or Sobolev W k,p maps S → P1 is an infinite dimensional complex manifold. The loop group LPGL(2, C) acts on LP1. We prove that the group of LPGL(2, C) invariant holomorphic line bundles on LP1 is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensi...

Array codes or two-dimensional codes in m-metric are subsets/subspaces of the space Matm×s(Fq), the linear space of all m × s-matrices with entries from a finite field Fq endowed with the m-metric [15]. In this paper, we obtain a lower bound over the number of parity checks of a two-dimensional m-code that corrects any two-dimensional array which has both clustered errors as well as errors in s...

This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as background notes to a series of seminars given at NTNU and subsequently at Sheffield. The topics covered are: the smooth structure of the space of smooth loops; constructions involving vector bundles; submanifolds and tubular neighbourhoods; and a short ...

Hessenberg decomposition is the basic tool used in computational linear algebra to approximate the eigenvalues of a matrix. In this article, we generalize Hessenberg decomposition to continuous matrix fields over topological spaces. This works in great generality: the space is only required to be normal and to have finite covering dimension. As applications, we derive some new structure results...