Initialization Strategy for Nonlinear Systems

The Study Group was asked to provide some hints concerning a choice of initial values to be used for nonlinear algebraic systems. The group has considered the available options and outlined the pros and cons of various methods and provided some recommendations

A Medical Waste Sterilizer

Sterilization of medical waste is very important for the environment, as the exposition may result in various diseases because of viral or bacterial content of the waste. There are devices developed for this purpose aiming to sterilize the waste in a form called batch process, meaning that a certain amount of waste is placed into the system and subjected to a sterilization procedure for a while and removed from the system afterwards. The procedure is repeated by the next set of waste till the whole set is sterilized. Our aim, however, is to design a device, a rotating cylindrical container having tubular lights attached to the walls inside, through which the waste is exposed to ultraviolet light as it gets rotated and moved towards the exit. The process will continue till the whole set is fed into the system. Such a device would be more effective as compared to batch processing types. The study group is asked to develop a mathematical model to analyse the effect of the number and location of tubes which will lead to maximal exposure during certain amount of time, which also needs an estimate, the sample will reside in the device before it gets discharged

Gluing Borel-Smith functions and the group of endo-trivial modules

The aim of this paper is to describe the group of endo-trivial modules for a $p$-group $P$, in terms of the obstruction group for the gluing problem of Borel-Smith functions.Comment: (to appear) Bulletin of London Mathematical Societ

The Fine Moduli Space of Representations of Clifford Algebras

Given a fixed binary form $f(u,v)$ of degree $d$ over a field $k$, the associated \emph{Clifford algebra} is the $k$-algebra $C_f=k\{u,v\}/I$, where $I$ is the two-sided ideal generated by elements of the form $(\alpha u+\beta v)^{d}-f(\alpha,\beta)$ with $\alpha$ and $\beta$ arbitrary elements in $k$. All representations of $C_f$ have dimensions that are multiples of $d$, and occur in families. In this article we construct fine moduli spaces $U=U_{f,r}$ for the irreducible $rd$-dimensional representations of $C_f$ for each $r \geq 2$. Our construction starts with the projective curve $C \subset \mathbb{P}^{2}_{k}$ defined by the equation $w^d=f(u,v)$, and produces $U_{f,r}$ as a quasiprojective variety in the moduli space $\mathcal{M}(r,d_r)$ of stable vector bundles over $C$ with rank $r$ and degree $d_r=r(d+g-1)$, where $g$ denotes the genus of $C$.Comment: Final version. To appear in Int. Math. Res. Not. IMR