Improvements to laser forming through process control refinements

Laser forming is a process that uses the energy of relatively high powered lasers to cause permanent deformation to components by inducing localised thermal stresses. It is envisaged that this material processing technique will find a number of commercial applications. This paper briefly discusses laser forming and the development of a basic process monitoring and control system used to overcome variability problems due to the complex nature of the lasers themselves and the manner in which they interact with material. It then goes on to show how the basic control system was modified, using increased feedback data sampling, time delays and a modified control algorithm which takes account of the forming rate in addition to the error. The effect of these developments is then illustrated by a series of tests which show the modifications significantly improve process tolerances.     

Deformations of schemes and other bialgebraic structures

There has long been a philosophy that every deformation problem in characteristic zero should be governed by a differential graded Lie algebra (DGLA). In this paper, we show how to construct a Simplicial Deformation Complex (SDC) governing any bialgebraic deformation problem. Examples of such problems are deformations of a Hopf algebra, or of an arbitrary scheme. In characteristic zero, SDCs and DGLAs are shown to be equivalent.

     

Presenting higher stacks as simplicial schemes

We show that an n$n$-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin n$n$-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin n$n$-stacks, Deligne–Mumford n$n$-stacks and n$n$-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in particular works for derived n$n$-stacks (replacing rings with simplicial rings). We exploit this to describe quasi-coherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich’s dg-schemes. As an application, we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks.