James M. McQuillan

Pencils of Hyperconics in Projective Planes of CharacteristicTwo

Let F be any field, finite or infinite, of characteristic 2. Put Π=PG(2,F). Let H1,H2 be hyperconics in Π. In this note we study the intersectionH1∩ H2. In particular we obtain canonical forms for H1,H2 in the cases where |H1∩ H2|=4,5,6. One interesting consequence is that the case |H1∩ H2|=6 can only occur if F contains a subfield of order 4. Related results concerning “pencils of hyperconics” are presented in Theorems 6 through 9. This work also leads to an extension to general fields of characteristic 2 of the well-known even intersection property for hyperovals in PG(2,4) which is pursued elsewhere ([2]).

Theoretical and computational properties of transpositions

Transposable genetic elements are prevalent across many living organisms from bacteria to large mammals. Given the linear primary structure of genetic material, this process is natural to study from a theoretical perspective using formal language theory. We abstract the process of genetic transposition to operations on languages and study it combinatorially and computationally. It is shown that the power of such systems is large relative to the classic Chomsky Hierarchy. However, we are still able to algorithmically determine whether or not a string is a possible product of the iterated application of the operations.

Strong vertex-magic and edge-magic labelings of 2-regular graphs of odd order using Kotzig completion

Abstract Let G be a 2-regular graph with 2 m + 1 vertices and assume that G has a strong vertex-magic total labeling. It is shown that the four graphs G ∪ 2 m C 3 , G ∪ ( 2 m + 2 ) C 3 , G ∪ m C 8 and G ∪ ( m + 1 ) C 8 also have a strong vertex-magic total labeling. These theorems follow from a new use of carefully prescribed Kotzig arrays. To illustrate the power of this technique, we show how just three of these arrays, combined with known labelings for smaller 2-regular graphs, immediately provide strong vertex-magic total labelings for 68 different 2-regular graphs of order 49.

Hyperovals and Fano configurations

LetF be any field, finite or infinite, of characteristic 2. Put π =PG(2,F). The classification of hyperovals is a difficult open problem. In this note we study the structure of the translation hyperovals and the hyperconics. We determine which quadrangles in a hyperconic have a given line as Fano line. The hyperconics and the translation hyperovals are similar with respect to containing quadrangles with certain Fano lines. We give two axioms satisfied by both. Remarkably, any hyperoval satisfying these must either be a hyperconic or else a translation hyperoval. It would be of great interest if one could find a way to slightly relax these conditions, or to piece together quadrangles with certain lines as Fano lines to obtain a new hyperoval.

Hyperovals in PG(2,4) and a self-dual code

Abstract The size and structure is given for a set of maximum cardinality of PG(2,4) hyperovals that pairwise intersect in exactly two points. A self-dual [42s,42s/2]-code is obtained from the geometry of PG(2,4s) for each s=1,2,…