Circle geometry in affine Cayley-Klein planes

Some theorems from inversive and Euclidean circle geometry are extended to all affine Cayley-Klein planes. In particular, we obtain an analogue to the first step of Clifford’s chain of theorems, a statement related to Napoleon’s theorem, extensions of Wood’s theorem on similar-perspective triangles and of the known fact that the three radical axes of three given circles are parallel or have a point in common. For proving these statements, we use generalized complex numbers. Supported by a grant D01-761/24.10.06 from the Ministry of Education and Sciences, and by a grant 108/2007 from Sofia University.     

Extending AG(4, 2) to S(4, {5, 6}, 17)

A construction of S(4, {5, 6}, 17) is given using the geometry of AG(4, 2) and PG(3, 2). © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 113–117, 1999     

Symmetries of parabolic geometries

We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly |1|-graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of Weyl structures to discuss more interesting |1|-graded geometries which can carry a symmetry in a point with nonzero curvature. More concretely, we discuss the number of different symmetries which can exist at the point with nonzero curvature.     

Key Distribution Patterns Using Tangent Circle Structures

The problem of key management in a communications network is of primary importance. A key distribution pattern is an incidence structure which provides a secure method of distributing keys in a large network reducing storage requirements. It is of interest to find explicit constructions for key distribution patterns. In O'Keefe [5–7], examples are shown using the finite circle geometries (Minkowski, Laguerre and inversive planes); in Quinn [12], examples are constructed from conics in finite projective and affine planes. In this paper, we construct some examples using the finite tangent-circle structures, introduced in Quattrocchi and Rinaldi [10] and we give a comparison of the storage requirements.     

Symmetric planes with non-classical tangent translation planes

We construct symmetric planes associated with an arbitrary locally compact connected nearfield . If is a proper nearfield, i.e. {;;}, then the tangent translation plane of this symmetric plane is not classical. All previously known examples of symmetric planes have classical tangent translation planes.Herrn Professor Dr. H. Salzmann zum 65. Geburtstag gewidmet     

Pathological projective planes: Associate affine planes

In [7] the author showed the existence of projective plane pathological with respect to the collineation groups of its sub and quotient planes. Similar pathologies are obtainable with respect to collineation groups of associated affine planes. (i.e. the affine planes obtained by distinguishing a line as the line at infinity) as expressable in the following theorem.     

Toroidal circle planes that are not Minkowski planes

We construct first examples of circle planes on the torus that are no Minkowski planes, but satisfy the same axiom of joining as flat Minkowski planes. The circle planes constructed by us form a special class ofhyperbola structures (see [4]) or(B*)-Geometrien (see [2]).This research was supported by a Feodor Lynen Fellowship and an ARC International Research Fellowship.     

Hankel planes

Starting from an (m+1)×(n+1) matrix A one can construct (m+p+1)×(n+1)(p+1) block Toeplitz matrices , p≥0, based on the rows of A. The connections between the ranks of the two matrices is studied by comparing the corresponding vector spaces of row relations R and R(p). A main tool are the Hankel matrices with rows in R. The dimension of R(p) is determined in terms of geometric invariants attached to the Hankel matrices with rows in R. The study of Hankel r-planes of P^m, for r≥1, turns out to be very useful and interesting in itself since they constitute a subvariety of the Grassmannian G(r,m).     

Closure planes

We introduce a simple algebraic method for constructing infinite affine (and projective) planes from an infinite set of finite planes of prime power order stemming from a “root” plane. The construction uses finite fields and infinite extensions of finite fields in a critical way. We obtain a classical-looking result which states that if the construction succeeds over the algebraic closure of a finite field, then both the infinite plane and the original root plane must be Desarguesian. The Lenz–Barlotti types for these planes are then linked to the Lenz–Barlotti type of the root plane. Examples are then given. These show that under suitable conditions, the method can yield infinitely many non-isomorphic infinite planes. These examples are of Lenz–Barlotti types II.1 and V.1.     

Barbilian planes

An axiomatization of a class of planes generalizing projective planes is given. With a Moufang condition, these planes have coordinates from an arbitrary alternative ring in whichab=1 impliesba=1. The new techniques include covering planes, homotopy, tangent bundles, and the Lie ring of sections of the tangent bundle, although there is no topological, differential, or algebraic geometric structure.This research was supported in part by NSF grant DMS 850697-01.     

Nearaffine planes

In this paper we develop a theory for nearaffine planes analogous to the theory of ordinary affine translation planes. In a subsequent paper we shall use this theory to give a characterization of a certain class of Minkowski planes.     

Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

Affine geometries and sets of equiorthogonal frequency hypercubes

Equiorthogonal frequency hypercubes are one particular generalization of orthogonal latin squares. A complete set of mutually equiorthogonal frequency hypercubes (MEFH) of order n and dimension d, using m distinct symbols, has (n − 1)^d/(m − 1) hypercubes. In this article, we prove that an affine geometry of dimension dh over 𝔽_m can always be used to construct a complete set of MEFH of order m^h and dimension d, using m distinct symbols. We also provide necessary and sufficient conditions for a complete set of MEFH to be equivalent to an affine geometry. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 435–441, 2000     

Distance-Regular (0,α)-Reguli

We introduce distance-regular (0,α)-reguli and show that they give rise to (0,α)-geometries with a distance-regular point graph. This generalises the SPG-reguli of Thas [14] and the strongly regular (α,β)-reguli of Hamilton and Mathon [9], which yield semipartial geometries and strongly regular (α,β)-geometries, respectively. We describe two infinite classes of examples, one of which is a generalisation of the well-known semipartial geometry T_n^*(B) arising from a Baer subspace PG(n, q) in PG(n, q²). Research Fellow supported by the Flemish Institute for the Promotion of Scientific and Technological Research in Industry (IWT), grant no. IWT/SB/13367/Tonesi Research assistant of the Fund for Scientific Research Flanders (FWO-Vlaanderen).     

Homogeneous planes

A projective plane is calledlocally p-desarguesian if every quadrangle generates a desarguesian subplane of orderp. Several properties of such planes are investigated, with special respect to the set of Latin squares associated with the plane. These results are related with a part of Gleason's theorem.In memoriam Giuseppe TalliniResearch partially supported by OTKA T 14302     

Collineations of V-Spaces

V-spaces are special non-planar ³-spaces in the sense of Betten. The group of collineations of a given V-space is a Lie group of dimension at least 4. We will determine all V-spaces with a large collineation group. If the V-space is generated by a function f, then dim 5 implies f(x)=c|x| ^p for some constants c>0, p>1.