On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

The isometries of the cut, metric and hypermetric cones

We show that the symmetry groups of the cut cone Cut_n and the metric cone Met_n both consist of the isometries induced by the permutations on , that is, for n ≥ 5. For n = 4 we have . This result can be extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, for n ≥ 5, where Hyp_n denotes the hypermetric cone.     

Implementing Brouwer’s database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on \(n\le 1300\) vertices. We have implemented most of the infinite families of graphs listed there in the open-source software Sagemath (The Sage Developers, http://www.sagemath.org), as well as provided constructions of the “sporadic” cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of n.     

Nitrogen derivatives of pinenes possessing the properties of plant retradants

New nitrogen derivatives possessing the activity of plant growth regulators have been synthesized from the readily available natural terpenoids α- and β-pinenes.     

Optimum shape of radiation-cooled ring fins

Conclusions Figure 6 shows that V decreases as x₀ increases, as is to be expected, since the heat flux per unit length of root at a fixed Q₀ then decreases.Figure 6 also shows that the fin of (2.21) is the best of those considered. Such fins have very sharp edges, so it is better to consider a fin of the form of (2.9), which results in not more than 7% increase in weight for the x₀ considered. A fin of the type of (2.1) increases the weight by not less than 25%.The line with circles in Fig. 6 is the optimum V(x₀) for a ring fin of constant thickness (rectangular cross-section), as derived from [8]. It is clear that such a fin increases the weight by over 90% relative to the optimal forms.     

Orientational instability in a nematic liquid crystal in a decaying Poiseuille flow

The results of studies of orientational dynamics and instability in an MBBA nematic liquid crystal in a decaying Poiseuille flow are considered. The experiments were made on a wedge cell with a gap width varying in a direction perpendicular to the flow. Confining surfaces ensured homeotropic adhesion of the nematic to the surface. Above a certain critical value of the initial pressure drop, a uniform orientational instability is observed, which corresponds to the emergence of the director from the plane of the flow. The dependence of the critical pressure drop on the local thickness of the liquid crystal layer and on the external destabilizing electric field is determined. Simulation of nematodynamics equations is carried out. The results of theoretical calculations are in qualitative and quantitative agreement with the experimental data.