Configurations arising from maximal arcs

We show that the existence of a maximal arc in a finite projective plane implies the existence of certain block designs and partial geometries. The block designs obtained have interesting resolvability properties.     

Some inherited maximal arcs in derived dual translation planes

In 1974 J. A. Thas constructed a class of maximal arcs in certain translation planes of orderq ². In this paper a new class of maximal arcs is constructed in certain derived dual translation planes that are inherited from the duals of the Thas maximal arcs. It is noted that some (but not all) of the maximal arcs are isomorphic to a class constructed by the author.The author gratefully acknowledges the support of an Australian Postgraduate Research Award.     

Tightness in subset bounds for coherent configurations

Association schemes have many applications to the study of designs, codes, and geometries and are well studied. Coherent configurations are a natural generalization of association schemes, however, analogous applications have yet to be fully explored. Recently, Hobart [Mich. Math. J. 58:231–239, 2009] generalized the linear programming bound for association schemes, showing that a subset Y of a coherent configuration determines positive semidefinite matrices, which can be used to constrain certain properties of the subset. The bounds are tight when one of these matrices is singular, and in this paper we show that this gives information on the relations between Y and any other subset. We apply this result to sets of nonincident points and lines in coherent configurations determined by projective planes (where the points of the subset correspond to a maximal arc) and partial geometries.     

Some maximal arcs in Hall planes

In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of square order, including the Desarguesian ones, but not the Hall planes. We construct a family of maximal arcs in the Hall planes inherited from the Thas maximal arcs in the Desarguesian planes. In particular, maximal arcs are shown to exist in all Hall planes of even order.The author gratefully acknowledges the support of an Australian Postgraduate Research Award.     

Generalized Quadrangles with an Abelian Singer Group

In this note we characterize thick finite generalized quadrangles constructed from a generalized hyperoval as those admitting an abelian Singer group, i.e., an abelian group acting regularly on the points. S. De Winter: The first author is a Research Assistant of the Fund for Scientific Research—Flanders (Belgium). K. Thas: The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).     

The solvability of some chemotaxis systems

In this paper, we study the local existence and uniqueness of classical solutions to a wide class of systems of chemotaxis equations. These systems are essentially quasi-linear strongly coupled partial differential equations. We also study the maximal interval of existence in time of solutions. The results are illustrated in application to a number of partial differential equation models arising in biology.     

Hypercomplex numbers in some geometries of two sets. I

The most important problem in the theory of phenomenologically symmetric geometries of two sets is that of classification of these geometries. In this paper, complexifying the metric functions of some known phenomenologically symmetric geometries of two sets (PSGTS) with the use of associative hypercomplex numbers, we find metric functions of new geometries in question. For these geometries, we find equations of the groups of motions and establish phenomenological symmetry, i.e., find functional relations between metric functions for certain finite number of arbitrary points. In particular, for one-component metric functions of PSGTS’s of ranks (2, 2), (3, 2), (3, 3), we find (n + 1)-component metric functions of the same ranks. For these metric functions, we find finite equations of the groups of motions and equations that express their phenomenological symmetry.     

Representation of convex geometries by circles on the plane

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an $n$-dimensional space equipped with a convex hull operator, by the result of Kashiwabara et al. (2005). Allowing circles rather than points, as was suggested by Czédli (2014), may presumably reduce the dimension for representation. This paper introduces a property, the Weak 2 × 3-Carousel rule, which is satisfied by all convex geometries of circles on the plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.     

On Af.Af* geometries

This paper is intended to be a first step towards the classification of finite flag-transitive geometries of rank 3 with affine planes and dual affine point-residues. We describe those of diameter 1. In the case of diameter > 1, we describe minimal quotients, assuming that the number of lines through two points is large enough.     

New Maximal Arcs in Desarguesian Planes

Constructions are described of maximal arcs in Desarguesian projective planes utilizing sets of conics on a common nucleus in PG(2, q). Several new infinite families of maximal arcs in PG(2, q) are presented and a complete enumeration is carried out for Desarguesian planes of order 16, 32, and 64. For each arc we list the order of its stabilizer and the numbers of subarcs it contains. Maximal arcs may be used to construct interesting new partial geometries, 2-weight codes, and resolvable Steiner 2-designs.     

On maximal subgroups of free idempotent generated semigroups

We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister-Schreier type rewriting.     

Some classes of finite homomorphism-homogeneous point-line geometries

In 2006, P. J. Cameron and J. Ne?et?ril introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we classify finite homomorphism-homogeneous point-line geometries up to a certain point. We classify all disconnected point-line geometries, and all connected point-line geometries that contain a pair of intersecting proper lines (we say that a line is proper if it contains at least three points). In a way, this is the best one can hope for, since a recent result by Rusinov and Schweitzer implies that there is no polynomially computable characterization of finite connected homomorphism-homogeneous point-line geometries that do not contain a pair of intersecting proper lines (unless P=coNP).     

A comparative analysis of methods for constructing weak orders from partial orders

This paper compares three methods (sequel, cardinal, maximal) for constructing a weak order from a partial order on a finite set. The constructed weak orders include the partial order. To evaluate the methods, several different selection disciplines were used to stochastically generate partial orders from a fixed linear order. The error of a weak order which includes a generated partial order is a function of the number of ordered pairs added to the partial order to get the weak order which are the reverse of ordered pairs in the fixed linear order. In all cases, the sequel and cardinal mean errors were much lower than the maximal mean error. In most but not all cases, the cardinal mean error was lower than the sequel mean error.     

Characterizations of quadric and Hermitian Veroneseans over finite fields

In Hirschfeld and Thas [5] the most important characterizations of quadric Veroneseans are surveyed. However a few difficult cases were still open, in particular the even case. In [10, 11] Thas and Van Maldeghem not only solve all open cases, but they also generalize most of these characterizations in several ways: they do not restrict themselves to the quadric Veronesean of the plane PG, they allow ovals instead of conics, and they also characterize projections of quadric Veroneseans. Further, Cooperstein, Thas and Van Maldeghem [1] contains some properties of Hermitian Veroneseans over finite fields and also these varieties and some of their projections are characterized. All these results on Veroneseans will be surveyed here.     

Efficient numerical integration of arbitrarily broken cells using the moment fitting approach

The finite cell method is based on a fictitious domain approach, providing a simple and fast mesh generation of structures with complex geometries. However, this simplification leads to intersected cells where the standard Gauss quadrature does not perform well. To perform the numerical integration of these cells, we use the moment fitting approach that generates an individual quadrature rule for every broken cell. In this paper, we will perform a non-linear optimization approach to find the optimal position and number of the integration points. The findings show that the proposed method leads to efficient quadrature rules that require less integration points than other existing integration methods. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q²), Q⁻(2r+1,q), and W(2r−1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1,q), H(2r+1,q²), and Q⁺(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q⁻(4m+3,q) is a pseudo-ovoid of the ambient projective space.     

Constructions of intriguing sets of polar spaces from field reduction and derivation

The concepts of a tight set of points and an m-ovoid of a generalised quadrangle were unified recently by Bamberg, Law and Penttila under the title of intriguing sets. This unification was subsequently extended to polar spaces of arbitrary rank. The first part of this paper deals with a method of constructing intriguing sets of one polar space from those of another via field reduction. In the second part of this paper, we generalise an ovoid derivation of Payne and Thas to a derivation of intriguing sets.      

Core-free,rank two coset geometries from edge-transitive bipartite graphs

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge-transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively small vertex-stabilizers.     

Miklós–Manickam–Singhi conjectures on partial geometries

In this paper we give a proof of the Miklós–Manickam–Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific partial geometries that are counterexamples to the conjecture are described.