Constructions of large translation nets with nonabelian translation groups

In this paper the first infinite series of translation nets with nonabelian translation groups and a large number of parallel classes are constructed. For that purpose we investigate partial congruence partitions (PCPs) with at least one normal component.Two series correspond to partial congruence partitions containing one normal elementary abelian component. The construction results by using some basic facts about the first cohomology group of the translation group G regarded as an extension of the normal component which itself is a group of central translations.The other series correspond to partial congruence partitions containing two normal nonabelian components. The constructions are based on the well known automorphism method which leads to so-called splitting translation nets. By investigating the Suzuki groups Sz(q), the protective unitary groups PSU(3, q ²) and the Ree groups R(q) as doubly transitive permutation groups, we obtain examples of nonabelian groups admitting a large number of pairwise orthogonal fixed-point-free group automorphisms.     

Groups Admitting a Kantor Family and a Factorized NormalSubgroup

We study the structure of a finite groupG admitting a Kantor family (F, F ^*) of type (s, t) and a nontrivial normal subgroupX which isfactorized byF F ^*. The most interesting cases, giving necessary conditions on the structure ofG and the parameterss andt, are those where a further Kantor family is induced inX, or where a partial congruence partition is induced in the factor groupG/X. Most of the known finite generalized quadrangles can be constructed as coset geometries with respect to a Kantor family. We show that the parameters of a skew translation generalized quadrangle necessarily are powers of the same prime. Furthermore, the structure of nonabelian groups admitting a Kantor family consisting only of abelian members is considered.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Streaming Algorithms for Line Simplification

We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p ₀,p ₁,p ₂,… in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k ²) additional storage.     

On completely free elements in finite fields

Letq>1 be a prime power,m>1 an integer,GF(q ^m) andGF (q) the Galois fields of orderq ^m andq, respectively. We show that the different module structures of (GF(q ^m), +) arising from the intermediate fields of the field extensionGF(q ^m) overGF (q), can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. The results can be generalized to finite cyclic Galois extensions over arbitrary fields.In 1986, D. Blessenohl and K. Johnsen proved that there exist elements inGF(q ^m) which generate normal bases inGF(q ^m) overany intermediate fieldGF(q ^d) ofGF(q ^m) overGF(q). Such elements are called completely free inGF(q ^m) overGF(q). Using our ideas, we give a detailed and constructive proof of the most difficult part of that theorem, i.e., the existence of completely free elements inGF(q ^m), overGF(q) provided thatm is a prime power. The general existence problem of completely free elements is easily reduced to this special case.Furthermore, we develop a recursive formula for the number of completely free elements inGF(q ^m) overGF(q) in the case wherem is a prime power.     

On Finite Elation Generalized Quadrangles with Symmetries

We study the structure of finite groups G which act as elationgroups on finite generalized quadrangles and contain a fullgroup of symmetries about some line through the base point.Such groups are related to the translation groups of translationtransversal designs with parameters depending on those of thequadrangles. Using results on the structure of p-groups which act as translationgroups on transversal designs and results on the index of theHughes subgroups of finite p-groups, we can show how restrictedthe structure of elation groups of finite generalized quadrangleswith symmetries is. One of our main results is that G is necessarily an elementaryabelian 2-group, provided that G has even cardinality. In particular,the elation generalized quadrangle coordinatized by G is a translationgeneralized quadrangle with G as translation group, that is,G contains full groups of symmetries about every line throughthe base point.     

Benchmarking the ACEnano Toolbox for Characterisation of Nanoparticle Size and Concentration by Interlaboratory Comparisons

ACEnano is an EU-funded project which aims at developing, optimising and validating methods for the detection and characterisation of nanomaterials (NMs) in increasingly complex matrices to improve confidence in the results and support their use in regulation. Within this project, several interlaboratory comparisons (ILCs) for the determination of particle size and concentration have been organised to benchmark existing analytical methods. In this paper the results of a number of these ILCs for the characterisation of NMs are presented and discussed. The results of the analyses of pristine well-defined particles such as 60 nm Au NMs in a simple aqueous suspension showed that laboratories are well capable of determining the sizes of these particles. The analysis of particles in complex matrices or formulations such as consumer products resulted in larger variations in particle sizes within technologies and clear differences in capability between techniques. Sunscreen lotion sample analysis by laboratories using spICP-MS and TEM/SEM identified and confirmed the TiO₂ particles as being nanoscale and compliant with the EU definition of an NM for regulatory purposes. In a toothpaste sample orthogonal results by PTA, spICP-MS and TEM/SEM agreed and stated the TiO₂ particles as not fitting the EU definition of an NM. In general, from the results of these ILCs we conclude that laboratories are well capable of determining particle sizes of NM, even in fairly complex formulations.