Monopoles from Rational Maps

The moduli space of charge k SU(2) BPS monopoles is diffeomorphic to the moduli space of degree k rational maps between Riemann spheres. In this note we describe a numerical algorithm to compute the monopole fields and energy density from the rational map. The results for some symmetric examples are presented.Comment: 8 pages, 2 figures. To appear in Phys. Lett.

Instanton Moduli and Topological Soliton Dynamics

It has been proposed by Atiyah and Manton that the dynamics of Skyrmions may be approximated by motion on a finite dimensional manifold obtained from the moduli space of SU(2) Yang-Mills instantons. 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 dimensional dynamics of sine-Gordon solitons by finite dimensional dynamics on a manifold obtained from instanton moduli. In the first example we use the moduli space of CP1 instantons and in the second example we use the moduli space of SU(2) Yang-Mills instantons. The metric and potential functions on these manifolds are constructed and the resulting dynamics is compared with the explicit exact soliton solutions of the sine-Gordon theory.Comment: uuencoded tex file, 27 pages including 4 figures, requires phyzzx macro. DAMTP 94-5

Non-Bogomolny SU(N) BPS Monopoles

For N>2 we present static monopole solutions of the second order SU(N) BPS Yang-Mills-Higgs equations which are not solutions of the first order Bogomolny equations. These spherically symmetric solutions may be interpreted as monopole anti-monopole configurations and their construction involves harmonic maps into complex projective spaces.Comment: 14 pages, 1 figur

Kink Chains from Instantons on a Torus

We describe how the procedure of calculating approximate solitons from instanton holonomies may be extended to the case of soliton crystals. It is shown how sine-Gordon kink chains may be obtained from CP1 instantons on a torus. These kink chains turn out to be remarkably accurate approximations to the true solutions. Some remarks on the relevance of this work to Skyrme crystals are also made.Comment: latex 17 pages, DAMTP 94-7

Image Transformations and Printing of Plaster Layers in Spiral Growth Manufacturing

Spiral growth manufacturing (SGM) is a high speed rapid manufacturing technique in which objects are built up, layer by layer, by simultaneously depositing, levelling and selectively consolidating thin powder layers onto a rotating build platform. The size and position of the jetted droplets are mapped by the position and greyscale level of pixels within an 8 bit greyscale bitmap image. This paper reports on the development of software in which mathematical algorithms apply geometric transformations to images in preparation for printing onto a rotating substrate. In support of this work, dimensional accuracy measurements of printed images and methods to correct radial print density variations are reported. The accuracy of printed images were found to be within ±0.2mm of their predicted size. The experimental work is briefly extended to the direct printing of plaster layers, formed by mixing two reactive ink solutions.Mechanical Engineerin

Spiral Growth Manufacturing (SGM) – A Continuous Additive Manufacturing Technology for Processing Metal Powder by Selective Laser Melting

Spiral growth manufacturing is a new innovative powder based rapid manufacturing technique. The innovation exists in the methodology in which powder layers are deposited. Unlike other pre-placed powder systems, the deposited layers move relative to the location at which they are processed. This is made possible by a rotating build drum into which powder is deposited, in spiralled layers, from a stationary hopper. With this configuration powder can be continuously deposited and levelled and simultaneously processed, eliminating delays in the build cycle. Stainless steel and cobalt-chrome powder is selectively melted using a 100W flash lamp pumped Nd:YAG laser. This paper reports on factors affecting build rate and on build strategies for creating a number of axis-symmetric thin and thick walled cylinders. Experimental results suggest that build rate for thin walled structures bonded to a substrate will ultimately be governed by tangential movements of the powder particles when frictional forces are not sufficient to accelerate the particles along a curved path, provided that enough laser power is available for melting. Even melt pool balling, which is evident when melting one layer at high speeds, diminishes in multiple layer builds due to re-melting and infilling.Mechanical Engineerin

Solitons, Links and Knots

Using numerical simulations of the full nonlinear equations of motion we investigate topological solitons of a modified O(3) sigma model in three space dimensions, in which the solitons are stabilized by the Hopf charge. We find that for solitons up to charge five the solutions have the structure of closed strings, which become increasingly twisted as the charge increases. However, for higher charge the solutions are more exotic and comprise linked loops and knots. We discuss the structure and formation of these solitons and demonstrate that the key property responsible for producing such a rich variety of solitons is that of string reconnection.Comment: 24 pages plus 14 figures in GIF forma

Automated Reasoning and Presentation Support for Formalizing Mathematics in Mizar

This paper presents a combination of several automated reasoning and proof presentation tools with the Mizar system for formalization of mathematics. The combination forms an online service called MizAR, similar to the SystemOnTPTP service for first-order automated reasoning. The main differences to SystemOnTPTP are the use of the Mizar language that is oriented towards human mathematicians (rather than the pure first-order logic used in SystemOnTPTP), and setting the service in the context of the large Mizar Mathematical Library of previous theorems,definitions, and proofs (rather than the isolated problems that are solved in SystemOnTPTP). These differences poses new challenges and new opportunities for automated reasoning and for proof presentation tools. This paper describes the overall structure of MizAR, and presents the automated reasoning systems and proof presentation tools that are combined to make MizAR a useful mathematical service.Comment: To appear in 10th International Conference on. Artificial Intelligence and Symbolic Computation AISC 201

Multi-soliton dynamics in the Skyrme model

We exhibit the dynamical scattering of multi-solitons in the Skyrme model for configurations with charge two, three and four. First, we construct maximally attractive configurations from a simple profile function and the product ansatz. Then using a sophisticated numerical algorithm, initially well-separated skyrmions in approximately symmetric configurations are shown to scatter through the known minimum energy configurations. These scattering events illustrate a number of similarities to BPS monopole configurations of the same charge. A simple modification of the dynamics to a dissipative regime, allows us to compute the minimal energy skyrmions for baryon numbers one to four to within a few percent.Comment: latex, 10 pages, plus 5 figures (as gif files

The moduli space metric for tetrahedrally symmetric 4-monopoles

The metric on the moduli space of SU(2) charge four BPS monopoles with tetrahedral symmetry is calculated using numerical methods. In the asymptotic region, in which the four monopoles are located on the vertices of a large tetrahedron, the metric is in excellent agreement with the point particle metric. We find that the four monopoles are accelerated through the cubic monopole configuration and compute the time advance. Numerical evidence is presented for a remarkable equivalence between a proper distance in the 4-monopole moduli space and a related proper distance in the point particle moduli space