Hans Cuypers

Symplectic geometries, transvection groups, and modules

We show that any connected partial linear space in which there is a line with at least four points and that has the property that any pair of intersecting lines is contained in a subspace isomorphic to a symplectic plane is isomorphic to the geometry of hyperbolic lines in some symplectic geometry. As a corollary to this result we obtain a characterization of the subgroups of the symplectic groups that are generated by transvection subgroups. Also a characterization of the natural modules for these groups is obtained.

Locally polar geometries with affine planes

Abstract In this paper we consider partial linear spaces containing a set of subspaces isomorphic to affine planes, such that the lines and these affine planes on a fixed point form a non-degenerate polar spaces of rank at least 2. We obtain a complete classification, provided that the rank is at least 3.

Interactive Mathematical Documents on the Web

This paper deals with our work on interactive mathematical documents. These documents accomodate various sources, users, and mathematical services. Communication of mathematics between these entities is based on the OpenMath standard and Java technology. But, for the management of the communication, more protocols and tools are needed. We describe an architecture that serves as a framework for our work on interactive documents, and we report on what we have implemented so far.

The dual of Pasch's axiom

We consider partial linear spaces that satisfy the dual of Paschs axiom. We give a uniform proof of some old and new characterizations of partial linear spaces and graphs related to projective spaces and the hyperbolic lines of symplectic spaces. Furthermore, we use these results to classify a class of groups that are generated by a conjugacy class of subgroups of order 3.

Finite locally generalized quadrangles with affine planes

We classify all finite connected partial linear spaces containing a setAof subspaces isomorphic to an affine plane, such that the lines and affine planes inAon a fixed but arbitrary point form a generalized quadrangle. It appears that these spaces are standard quotients of affine polar spaces of rank 3.

On a generalization of Fischer spaces

In this paper generalized Fischer spaces are defined and some results on their classification are obtained.

On the embeddability of polar spaces

We show that every nondegenerate polar space of rank at least 4 with at least three points on each line can be embedded in a projective space. Together with some results from [9] and [12], this provides a particularly elementary proof that any such polar space is of classical type. Our methods involve the use of geometric hyperplanes and work equally well for spaces of finite or infinite rank.

Interactive learning and mathematical calculus

A variety of problems in mathematical calculus can be solved by recursively applying a finite number of rules. Often, a generic solving strategy can be extracted and an interactive exercise system that emulates a tutor can be implemented. In this paper we show how software developed by us can be used to realize this interactivity. In particular, an implementation of a generic exercise for computing the derivative of elementary functions is presented.

The hyperplanes of the

Abstract. We give a complete description of the geometric hyperplanes of the 759-point near hexagon belonging to M24.

M_24

In [4] Huang has given a characterization of the bilinear forms graphsHq (n, d) using the weak 4-vertex conditions and a condition on the parametersn, d andq. We show that the weak 4-vertex condition can be replaced by a condition on the eigenvalues of the adjacency matrix of the graph. Furthermore, we show that the conditions on the parameters can be weakened.