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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 V 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: v2; 31 pages; added example due to Gabber disproving local finiteness in general; some other corrections

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: 16 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

In this paper we describe the number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits in finite-dimensional simple Lie algebras. Thanks to the results obtained earlier we need to solve the problem for the two largest rigid nilpotent orbits in Lie algebras of type ${\rm E}_8$. As a corollary we compute the number of small modules in the corresponding reduced enveloping algebras over algebraically closed fields of characteristic $p>5$., Comment: This version has been accepted for publication

This survey article has two components. The first part gives a gentle introduction to Serre's notion of $G$-complete reducibility, where $G$ is a connected reductive algebraic group defined over an algebraically closed field. The second part concerns consequences of this theory when $G$ is simple of exceptional type, specifically its role in elucidating the subgroup structure of $G$. The latter subject has a history going back about sixty years. We give an overview of what is known, up to the present day. We also take the opportunity to offer several corrections to the literature., Comment: 43 pages. To appear in the proceedings of Groups St Andrews 2022 in Newcastle

We describe some recent computer investigations with the `Constraint Logic Programming over Finite Domains' -- CLP(FD) -- library in the Prolog programming environment to search for new simple Lie algebras over the field $\GF(2)$ of $2$ elements. Motivated by a paper of Grishkov et. al., we specifically look for those with a `thin decomposition', and we settle one of their conjectures. We extrapolate from our results the existence of two new infinite families of simple Lie algebras, in addition to finding seven new sporadic examples in dimension $31$. We also better contextualise some previously discovered simple algebras, putting them into families which do not seem to have ever appeared in the literature, and give an updated table of those currently known., Comment: 47 pages

In order to reduce cobalt within the primary circuit of pressurised water reactors (PWRs), wear-resistant steels are being researched and developed. In particular interest is the understanding of galling mechanisms, an adhesive wear mechanism which is particularly prevalent in PWR valves. Here we show that large shear stresses and adhesive transfer occur during galling by exploiting the 2 wt per cent manganese difference between 304L and 316L stainless steels, even at relatively low compressive stresses of 50 MPa. Through these findings, the galling mechanisms of stainless steels can be better understood, which may help with the development of galling resistant stainless steels.

We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type $BC_n$; these exist only in characteristic $2$. We also give a formula for the dimensions of their irreducible modules., Comment: 22 pages; version accepted for publication in Transformation Groups; appendix added to show an alternative approach using a significant theorem in Conrad--Gabber--Prasad

Let $k'/k$ be a finite purely inseparable field extension and let $G'$ be a reductive $k'$-group. We denote by $G=\R_{k'/k}(G')$ the Weil restriction of $G'$ across $k'/k$, a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical $\RR_{u}(G_{\bar{k}})$ in terms of invariants of the extension $k'/k$, starting with the case $G'=\GL_n$ and applying these results to the case where $G'$ is a simple group., Comment: 10 pages; v2 deals with all split simple groups under mild constraints on the characteristic; to appear in Archiv der Mathematik

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalisation if $G$ is of type $D_n$. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups., Comment: 26 pages

We prove the existence and uniqueness of a new maximal subgroup of the algebraic group of type $E_8$ in characteristic $3$. This has type $F_4$, and was missing from previous lists of maximal subgroups produced by Seitz and Liebeck--Seitz. We also prove a result about the finite group $H={}^3\!D_4(2)$, that if $H$ embeds in $E_8$ (in any characteristic $p$) and has two composition factors on the adjoint module then $p=3$ and $H$ lies in this new maximal $F_4$ subgroup., Comment: 12 pages. Proc. Amer. Math. Soc., to appear

Beam-ACO, a modification of the traditional Ant Colony Optimization (ACO) algorithms that incorporates a modified beam search, is one of the most effective ACO algorithms for solving the Traveling Salesman Problem (TSP). Although adding beam search to the ACO heuristic search process is effective, it also increases the amount of work (in terms of partial paths) done by the algorithm at each step. In this work, we introduce a greedy variant of Beam-ACO that uses a greedy path selection heuristic. The exploitation of the greedy path selection is offset by the exploration required in maintaining the beam of paths. This approach has the added benefit of avoiding costly calls to a random number generator and reduces the algorithms internal state, making it simpler to parallelize. Our experiments demonstrate that not only is our greedy Beam-ACO (gBeam-ACO) faster than traditional Beam-ACO, in some cases by an order of magnitude, but it does not sacrifice quality of the found solution, especially on large TSP instances. We also found that our greedy algorithm, which we refer to as gBeam-ACO, was less dependent on hyperparameter settings.

Semi-inclusive charged jet spectra per trigger at STAR are presented binned by event activity (EA) as determined by the Beam Beam Counter (BBC) signal in the Au-going direction. The selected EA determination is motivated by correlations between the number of charged tracks in the Time Projection Chamber (TPC) ($|\eta|<1$) and EA ($\eta_\mathrm{EA}\in[-5,-2]$) which are also presented. The jet spectra per trigger at high EA are suppressed relative to the spectra at low EA. A PYTHIA investigation refutes that the suppression results from a trivial autocorrelation between jet kinematics and the acceptance of the EA and the TPC., Comment: Proceedings for Quark Matter 2019 from Small Systems Parallel Session

The galling behaviour of 316L stainless steel was investigated in both the unoxidised and oxidised states, after exposure in simulated PWR water for 850 hours. Galling testing was performed according to ASTM G196 in ambient conditions. 316L was found to gall by the wedge growth and flow mechanism in both conditions. This resulted in folds ahead of the prow and adhesive junction, forming a heavily sheared multilayered prow. The galling trough was seen to have failed through successive shear failure during wedge flow. Immediately beneath the surface a highly sheared nanocrystalline layer was seen, termed the tribologically affected zone (TAZ). It was observed that strain-induced martensite formed within the TAZ. Galling damage was quantified using Rt (maximum height - maximum depth) and galling area (the proportion of the sample which is considered galled), and it was shown that both damage measures decreased significantly on the oxidised samples. At an applied normal stress of 4.2 MPa the galled area was 14 % vs. 1.2 % and the Rt was 780 um vs. 26 um for the unoxidised and oxidised sample respectively. This trend was present at higher applied normal stresses, although less prominent. This difference in galling behaviour is likely to be a result of a reduction in adhesion in the case of the oxidised surface., Comment: 10 pages, 11 figures

The thermal effects of plastic and frictional heat generation and elevated temperature were examined along with the role of adhesion in the context of galling wear, using a representative crystal plasticity, normally loaded, sliding surface model. Galling frequency behaviour was predicted for 316L steel. Deformation of the surfaces was dominated by the surface geometry, with no significant effect due to variations in frictional models. Plastic and frictional heating were found to have a minimal effect on the deformation of the surface, with the rapid conduction of heat preventing any highly localised heating. There was no corresponding effect on the predicted galling frequency response. Isothermal, elevated temperature conditions caused a decrease in galling resistance, driven by the temperature sensitivity of the critical resolved shear stress. The extent of deformation, as quantified by the area of plastically deformed material and plastic reach, increased with temperature. Comparisons were made with literature results for several surface amplitude and wavelength conditions. Model results compared favourably with those in the literature. However, the reduction in predicted galling resistance with elevated temperature for a fixed surface was not as severe as observations in the literature, suggesting other mechanisms (e.g. phase transformations, surface coatings and oxides) are likely important., Comment: Submitted manuscript, 38 pages, 19 figures

These proceedings report preliminary measurements of correlations between mid-rapidity charged tracks and high-rapidity event activity (EA) at STAR for $\sqrt{s_\mathrm{NN}}= 200\, \mathrm{GeV}$ p+Au collisions taken in 2015. These correlations are intriguing because they inform the current debate over use of the Glauber model in `small' systems (here meaning p+A or d+A and denoted as `s+A') and have implications for calculating nuclear modification and quenching observables in these systems. The results support concerns about centrality binning in p+Au collisions, and as such motivate using ratios of semi-inclusive, as opposed to fully inclusive, jet spectra to look for jet enhancement or suppression., Comment: Presented at Hot Quarks 2018 - Workshop for young scientists on the physics of ultrarelativistic nucleus-nucleus collisions, Texel, The Netherlands, September 7-14 2018

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $\mathfrak{g}$. The first predicts the maximal dimension of simple $\mathfrak{g}$-modules and in this paper we apply the Lefschetz principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of $\mathfrak{gl}_n(k)$ whenever $k$ is an algebraically closed field of characteristic $p \gg n$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, a short proof of the first Kac--Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring $R \subseteq \mathbb{C}$, after base change to a field of large positive characteristic., Comment: 16 pages

Let $R$ be a commutative unital ring. Given a finitely presented affine $R$-group scheme $G$ acting on a separated scheme $X$ of finite type over $R$, we show that there is a prime $p_0$ such that for any $R$-algebra $k$ which is an algebraically closed field of characteristic $p\geq p_0$, the centraliser in $G_k$ of any closed subscheme of $X_k$ is smooth. When $X$ is not necessarily separated we show similarly that for any closed subscheme $Y \subseteq X$ there is a $p_1$ depending on $Y$ such that when $k$ has characteristic $p \geq p_1$ the normaliser of $Y$ in $G_k$ is smooth. We prove these results using the Lefschetz principle together with careful application of Gr\"obner basis techniques, and using a suitable notion of the complexity of an action. We apply our results to demonstrate that the Kostant-Kirillov-Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics. In particular, every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit., Comment: v2; 21 pages

A new algorithm is presented for solving the soft-margin Support Vector Machine (SVM) optimization problem with an $\ell^{1}$ penalty. This algorithm is designed to require a modest number of passes over the data, which is an important measure of its cost for very large data sets. The algorithm uses smoothing for the hinge-loss function, and an active set approach for the $\ell^{1}$ penalty., Comment: 13 pp, 1 figure

We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case., Comment: To appear in JEMS; significant error noted by referee and corrected from previous version: Lemma 3.1 wrongly claimed Q_G(\lambda) was M-isotypic; see Rk 7.4(ii) in this version

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that the characteristic $p$ of $k$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie algebra $\mathfrak{g}={\rm Lie}(G)$. Specifically, we show that one of the following holds: $\mathfrak{m}={\rm Lie}(M)$ for some maximal connected subgroup $M$ of $G$, or $\mathfrak{m}$ is a maximal Witt subalgebra of $\mathfrak{g}$, or $\mathfrak{m}$ is a maximal $\it{\mbox{exotic semidirect product}}$. The conjugacy classes of maximal connected subgroups of G are known thanks to the work of Seitz, Testerman and Liebeck--Seitz. All maximal Witt subalgebras of $\mathfrak{g}$ are $G$-conjugate and they occur when $G$ is not of type ${\rm E}_6$ and $p-1$ coincides with the Coxeter number of $G$. We show that there are two conjugacy classes of maximal exotic semidirect products in $\mathfrak{g}$, one in characteristic $5$ and one in characteristic $7$, and both occur when $G$ is a group of type ${\rm E}_7$., Comment: This version is accepted for publication in Journal of the American Mathematical Society; 40 pages

Thin film solid state lithium-based batteries (TSSBs) are increasingly attractive for their intrinsic safety due to the use of a nonflammable solid electrolyte, cycling stability, and ability to be easily patterned in small form factors. However, existing methods for fabricating TSSBs are limited to planar geometries, which severely limits areal energy density when the electrodes are kept sufficiently thin to achieve high areal power. In order to circumvent this limitation, we report the first successful fabrication of fully conformal, 3D full cell TSSBs formed in micromachined silicon substrates with aspect ratios up to ~10 using atomic layer deposition (ALD) at low processing temperatures (at or below 250C) to deposit all active battery components. The cells utilize a prelithiated LiV$_2$O$_5$ cathode, a very thin (40 - 100 nm) LiPON-like lithium polyphosphazene (Li$_2$PO$_2$N) solid electrolyte, and a SnN$_x$ conversion anode, along with Ru and TiN current collectors. Planar all-ALD solid state cells deliver 37 {\mu}Ah/cm$^2${\mu}m normalized to the cathode thickness with only 0.02% per-cycle capacity loss for hundreds of cycles. Fabrication of full cells in 3D substrates increases the areal discharge capacity by up to a factor of 9.3x while simultaneously improving the rate performance, which corresponds well to trends identified by finite element simulations of the cathode film. This work shows that the exceptional conformality of ALD, combined with conventional semiconductor fabrication methods, provides an avenue for the successful realization of long-sought 3D TSSBs which provide power performance scaling in regimes inaccessible to planar form factor devices., Comment: 32 pages, 15 figures

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k' be a purely inseparable field extension of k of degree p^e and let G denote the Weil restriction of scalars R_{k'/k}(G') of a reductive k'-group G'. When G= \R_{k'/k}(G') we also provide some results on the orders of elements of the unipotent radical \RR_u(G_{\bar k}) of the extension of scalars of G to the algebraic closure \bar k of k., Comment: 19 pages; 2nd version a major revision; to be published in Mich. J. Math

We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras]., Comment: 4 pages

Single cell responses are shaped by the geometry of signaling kinetic trajectories carved in a multidimensional space spanned by signaling protein abundances. It is however challenging to assay large number (>3) of signaling species in live-cell imaging which makes it difficult to probe single cell signaling kinetic trajectories in large dimensions. Flow and mass cytometry techniques can measure a large number (4 - >40) of signaling species but are unable to track single cells. Thus cytometry experiments provide detailed time stamped snapshots of single cell signaling kinetics. Is it possible to use the time stamped cytometry data to reconstruct single cell signaling trajectories? Borrowing concepts of conserved and slow variables from non-equilibrium statistical physics we develop an approach to reconstruct signaling trajectories using snapshot data by creating new variables that remain invariant or vary slowly during the signaling kinetics. We apply this approach to reconstruct trajectories using snapshot data obtained from in silico simulations and live-cell imaging measurements. The use of invariants and slow variables to reconstruct trajectories provides a radically different way to track object using snapshot data. The approach is likely to have implications for solving matching problems in a wide range of disciplines., Comment: revised version, accepted for publication in Royal Society Open Science

Let G be a simple simple-connected exceptional algebraic group of type G_2, F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with \g=Lie(G). For each nilpotent orbit G.e of \g, we list the Jordan blocks of the action of e on the minimal induced module V_min of \g. We also establish when the centralisers G_v of vectors v\in V_min and stabilisers \Stab_G of 1-spaces \subset V_min are smooth; that is, when \dim G_v=\dim\g_v or \dim \Stab_G=\dim\Stab_\g., Comment: This contains corrections and should be used instead of the published version ; code now available at github.com/davistem/nilpotent_orbits_GAP