A hierarchical structure of transformation semigroups with applications to probability limit measures

The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels. This kernel hierarchy produces a set of tools that provides direct access to computations of interest in probability limit theorems; in particular, finding certain factors of idempotent limit measures. In addition, when considering transformation semigroups that arise naturally from edge colorings of directed graphs, as in the road-coloring problem, the hierarchy produces simple techniques to determine the rank of the kernel and to decide when a given kernel is a right group. In particular, it is shown that all kernels of rank one less than the number of vertices must be right groups and their structure for the case of two generators is described.Comment: 35 pages, 4 figure

Small Extended Generalized Quadrangles

We consider extensions of generalized quadrangles with parameters (s, t), and establish lower bounds (in terms of s and t) for the number of points, sometimes under additional hypotheses. We also study the structure of geometries attaining these bounds, give several constructions and some uniqueness proofs, and examine the question of further extensions

Foreword to Special Issue on the occassion of Jack van Lint's sixtieth birthday

Without abstract

Acceptor–donor–acceptor small molecules based on derivatives of 3,4-ethylenedioxythiophene for solution processed organic solar cells

Three simple semiconducting acceptor–donor–acceptor (A–D–A) small molecules based on an electron-rich (3,4-ethylenedioxythiophene) EDOT central core have been synthesised (DIN-2TE, DRH-2TE, DECA-2TE) and characterised. Organic photovoltaic (OPV) devices incorporating these materials have been prepared and evaluated. The physical properties of the molecules were characterised by TGA, DSC, UV/vis spectroscopy and cyclic voltammetry. The optical HOMO–LUMO energy gaps of the molecules in the solid state were in the range 1.57–1.82 eV, and in solution 1.88–2.04 eV. Electrochemical HOMO–LUMO energy gaps determined by cyclic voltammetry were found to be in the range 1.97–2.31 eV. The addition of 1% 1,8-diiodooctane (DIO) to photoactive blends of the A–D–A molecules and PC71BM more than doubled the power conversion efficiency (PCE) in the case of DRH-2TE:PC71BM devices to 1.36%

Steiner t-designs for large t

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008, ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in Computer Scienc

On the orders of primitive groups with restricted nonabelian composition factors

AbstractWe prove that, given c0, there exists c such that the following holds. If G is a primitive permutation group of degree n, no composition factor of which is an alternating group of degree greater than c0 or a classical group of dimension greater than c0, then ¦G¦⩽nc. In particular, if the nonabelian composition factors of G have bounded order, then ¦G¦ is polynomially bounded

On hyperovals of polar spaces

We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon E-3 has up to isomorphism a unique full embedding into the dual polar space DH(5, 4)