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Let k be a field, let G be a smooth affine k-group and V a finite-dimensional G-module. We say V is \emph{rigid} if the socle series and radical series coincide for the action of G on each indecomposable summand of V; say V is \emph{geometrically rigid} (resp.~\emph{absolutely rigid}) if V is rigid after base change of G and V to \bar k (resp.~any field extension of k). We show that all simple G-modules are geometrically rigid, though not in general absolutely rigid. More precisley, we show that if V is a simple G-module, then there is a finite purely inseparable extension k_V/k naturally attached to V such that V_{k_V} is absolutely rigid as a G_{k_V}-module. The proof for connected G turns on an investigation of algebras of the form K\otimes_k E where K and E are field extensions of k; we give an example of such an algebra which is not rigid as a module over itself. We establish the existence of the purely inseparable field extension k_V/k through an analogous version for artinian algebras. In the second half of the paper we apply recent results on the structure and representation theory of pseudo-reductive groups to gives a concrete description of k_V. Namely, we combine the main structure theorem of the Conrad--Prasad classification of pseudo-reductive G together with our previous high weight theory. For V a simple G-module, we calculate the minimal field of definition of the geometric Jacobson radical of \End_G(V) in terms of the high weight of G and the Conrad--Prasad classification data; this gives a concrete construction of the field k_V as a subextension of the minimal field of definition of the geometric unipotent radical of G. We also observe that the Conrad--Prasad classification can be used to hone the dimension formula for G we had previously established; we also use it to give a description of \End_G(V) which includes a dimension formula., Comment: 30 pages

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{g}$. We call a subvariety $\mathfrak{X}$ of the nilpotent cone $N \subset \mathfrak{g}$ monogamous if for every $e\in \mathfrak{X}$, the $\mathfrak{sl}_2$-triples $(e,h,f)$ with $f\in \mathfrak{X}$ are conjugate under the centraliser $C_G(e)$. Building on work by the first two authors, we show there is a unique maximal closed $G$-stable monogamous subvariety $V \subset N$ and that it is an orbit closure, hence irreducible. We show that $V$ can also be characterised in terms of Serre's $G$-complete reducibility., Comment: 15 pages

The galling mechanism of Tristelle 5183, an Fe-based hardfacing alloy, was investigated at elevated temperature. The test was performed using a bespoke galling rig. Adhesive transfer and galling were found to occur, as a result of shear at the adhesion boundary and the activation of an internal shear plane within one of the tribosurfaces. During deformation, carbides were observed to have fractured, as a result of the shear train they were exposed to and their lack of ductility. In the case of niobium carbides, their fracture resulted in the formation of voids, which were found to coalesce and led to cracking and adhesive transfer. A tribologically affected zone (TAZ) was found to form, which contained nanocrystalline austenite, as a result of the shear exerted within 30{\mu}m of the adhesion boundaries. The galling of Tristelle 5183 initiated from the formation of an adhesive boundary, followed by sub-surface shear in only one tribosurface, Following further sub-surface shear, an internal shear plane is activated. internal shear and shear at the adhesion boundary continues until fracture occur, resulting in adhesive transfer., Comment: 9 pages, 12 Figures

We develop and deploy a set of constraints for the purpose of calculating minimal sizes of lottery designs. Specifically, we find the minimum number of tickets of size six which are needed to match at least two balls on any draw of size six, whenever there are at most 70 balls., Comment: 21 pages; to appear in Constraints

Researchers and legislators alike continue the search for methods of drawing fair districting plans. A districting plan is a partition of a state's subdivisions (e.g. counties, voting precincts, etc.). By modeling these districting plans as graphs, they are easier to create, store, and operate on. Since graph partitioning with balancing populations is a computationally intractable (NP-hard) problem most attempts to solve them use heuristic methods. In this paper, we present a variant on the $k$-medoids algorithm where, given a set of initial medoids, we find a partition of the graph's vertices to admit a districting plan. We first use the $k$-medoids process to find an initial districting plan, then use local search strategies to fine-tune the results, such as reducing population imbalances between districts. We also experiment with coarsening the graph to work with fewer vertices. The algorithm is tested on Iowa and Florida using 2010 census data to evaluate the effectiveness., Comment: 25 pages, 7 figures, and 6 tables