The orders of nonsingular derivations of Lie algebras of characteristic two

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-$p$ groups and Lie algebras. A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of positive characteristic p was initiated by Shalev and continued by the present author. In this paper we continue this study in the case of characteristic two. Among other results, we prove that any divisor n of 2^k-1 with $n^4>(2^k-n)^{3}$ belongs to N_2. Our methods consist of elementary arguments with polynomials over finite fields and a little character theory of finite groups.Comment: 11 page

Type inference in mathematics

In the theory of programming languages, type inference is the process of inferring the type of an expression automatically, often making use of information from the context in which the expression appears. Such mechanisms turn out to be extremely useful in the practice of interactive theorem proving, whereby users interact with a computational proof assistant to construct formal axiomatic derivations of mathematical theorems. This article explains some of the mechanisms for type inference used by the Mathematical Components project, which is working towards a verification of the Feit-Thompson theorem

Formal Proofs for Nonlinear Optimization

We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of $f$ over $K$. This method allows to bound in a modular way some of the constituents of $f$ by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations. Our implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table

Fibonacci Designs

A Metis design is one for which v=r+k+1. This paper deals with Metis designs that are quasi-residual. The parameters of such designs and the corresponding symmetric designs can be expressed by Fibonacci numbers. Although the question of existence seems intractable because of the size of the designs, the nonexistence of corresponding difference sets can be dealt with in a substantive way. We also recall some inequalities for the number of fixed points of an automorphism of a symmetric design and suggest possible connections to the designs that would be the symmetric extensions of Metis designs.Comment: 7 page

Modeling of particle segregation in a rotating drum

Mixing of granular solids is a processing step in a wide range of industries. The fundamental phenomena in granule mixing are still poorly understood, making it difficult to a priori predict the effectiveness of mixing processes. While mixing of granules is easy when the particles are homogeneous in size, shape and density and other properties, in practice they are not. With such a mixture, homogenizing is far more complex, since the heterogeneous particles tend to segregate, and special care has to be taken in the design of the mixing process to avoid this. In view of the practical need for better understanding and control of solids mixing, the work in this thesis has two closely coupled objectives. The first objective is to obtain a better understanding of segregation mechanisms. This insight should enable the enhancement of mixing and at the same time suppress segregation, or vice versa, namely the deliberate and controlled segregation of a mixture. The second objective is to provide guidelines for mixing operations that can be derived from insights extracted from the data on mixing behaviour at different rotational velocities and fill levels. From this perspective, we here report an extensive numerical study of mixing and segregation in a bed of bidisperse granules in a rotating horizontal drum, which is the simplest relevant geometry in industrial practice. Two types of segregation can occur: fast radial segregation during which smaller or denser particles accumulate along the axis of rotation; and slow axial segregation with fully segregated bands of small and large particles perpendicular to the rotating axis, with in general particle bands of large particles adjacent to the end walls. This thesis reports on both radial and axial segregation phenomena in a horizontally rotating drum. While visual observation of the particle bed was used as a qualitative observation technique to determine the degree of mixing/segregation, in parallel a more quantitative method was developed as well, which was based on calculating the entropy over the systems. By subdividing the system with a lattice, calculating the entropy of mixing in each cell of the lattice, and summarizing them over the system, a measure for the degree of overall segregation was obtained. By using different grids (a 3D mesh, a 2D set of slices perpendicular to the axis, or 2D bars parallel to the axis), different types of segregation could be distinguished. The radial segregation dynamics were investigated in semi-2D (very short) drums, which inhibits axial segregation. Diagrams were prepared that visualise the mixing behaviour as function of the Froude number (rotational speed) for systems with different bidisperse systems. It was found that while almost all systems showed radial segregation at low Fr (rolling regime), and most showed inverted radial segregation at high Fr (cataracting or centrifuging regime), at Fr ≈ 0.56 all systems became radially mixed. This could be understood by assuming a percolation mechanism. In the moving layer on top of the load, smaller particles percolate in between the moving larger particles, down to the centre of the load, as long as the motion is not too fast. The same phenomenon is inverted at high speeds. In between, the flowing layer is expanded in such a way that many large voids are present, which makes the percolation mechanism less selective on the particle size. The little segregation that occurs is negligible, since the two phenomena described above work in different directions. Surprisingly this transitional Fr number is the same for all investigated systems. Since axial segregation is always preceded by radial segregation, it is logical to also study axial segregation. This was done by studying longer drums, which allows axial segregation to develop along the axis. Axial segregation was found for most systems; its occurrence is mostly dictated by differences in size. It was found that for drums that have intermediate length, surprising dynamic behaviour results. The axial segregation developed with low and high frequency oscillations. While the low frequency oscillations could be understood as the development and migration of segregated areas in the system, the higher frequency oscillations, with a period of 10 to 20 revolutions, were not identified before. This oscillatory behaviour is probably coupled to the use of intermediately sized drums, as this behaviour has not been seen with very long drums. We ascribe the oscillations to the influence of the (vertical) end walls, which expose the adjacent particles to different forces than those particles inside the drum load. These differences induce an axial flow in the system. The particles adjacent to the vertical walls tend to be lifted higher than the particles far away from the vertical walls. This creates a concave profile of the load surface throughout the drum, inducing the particles (in the rolling regime) to follow a path away from the vertical walls towards the centre of the drum. Once past the centre, the particles flow back to the vertical walls in response to the locally convex bed profile. Even in this particular flow profile the percolation mechanism is of importance: smaller particles percolate through the flowing layer and end up deeper inside the bed, while the larger particles accumulate on top of the flowing layer and are conveyed back to the vertical walls. Due to the percolation of the small particles the final end configuration must clearly be a banding configuration of large-small-large particle bands. Prolonged rotation of the bed increases the concave form of the flowing layer. This induces fast oscillations and a sudden mixing of a part of the large particle band with the small particle band, giving fast mixing and leading to a configuration, in which a small-particle band is formed below the large-particles bands. Subsequently segregation into three bands (large-small-large) slowly occurs again, after which the asymmetry in the angel of repose further increases. The configuration, in which larger particles accumulate on top of the bed adjacent to the end walls, coincides with a minimum in energy dissipation, which is not present when the systems segregates radially or axially into three pure bands. The effect found implies that the end walls are important in the dynamics of axial segregation. This effect is studied further by varying the end wall properties. The above mentioned fast and slow oscillations vanish in systems that have smoother end walls, while also the rate of segregation decreases; nevertheless the same axially segregated three band (large-small-large) state of mixing resulted finally. Reducing the friction further to completely smooth end walls however changed the final configuration into a two-banded system. Systems with no end wall at all, simulated through periodic end walls, only gave radial segregation over the (considerable) simulated time span. We expect here that as long as there is still a driving force for axial segregation, the absence of the induction of axial flow by the end walls make the transition very slow or impossible. The formation of two axial bands lowers the energy dissipation by the bed, whereas neither radial segregation nor axial segregation into three bands reduced the power absorption at constant angular velocity. While the oscillatory behaviour is relevant in its own right, their study also allows shedding some light on the fundamental mechanisms underlying the segregation mechanisms, and especially the transition from radial to axial segregation. The fact that this is dependent on not only the properties of the granular materials, but also on the geometry and design of the drum, implies that these findings have relevance to the design and operation of processes. <br/