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.     

The symmetry axiom in Minkowski planes

We give a synthetic proof that in a symmetric Minkowski plane the rectangle axiom (G) holds. Dedicated to Professor Helmut Karzel     

Arcs in Minkowski planes

The aim of this paper is to study some properties of k-arcs in Minkowski planes focalizing the attention on problems of existence and completness.Work done under the auspicies of G.N.S.A.G.A. supported by 40% grants of M.U.R.S.T.In memoriam Giuseppe Tallini     

A perimeter-based angle measure in Minkowski planes

Measuring angles in the Euclidean plane is a well-known topic, but for general normed planes there exists a variety of different concepts. These can be of a special kind, e.g. also preserving special orthogonality types. But these concepts are no angle measures in the sense of measure theory since they are not additive. This motivates us to define a new angle measure for normed planes that is in fact a measure in the sense of measure theory. Furthermore, we look at related types of rotation and reflection.     

A remark on the introduction of coordinates in Minkowski planes

Every Minkowski parallel-translation plane in the sense of ARTZY [1] satisfies the quadrangles axiom G of BENZ [4, p. 299]. It follows that the class of all parallel-translation planes coincides with the class of all Minkowski planes over a Tits-nearfield. The results can be extended to Minkowski geometries without the tangency property (called B^*-geometries in [4]).     

Geometric algebra of strictly convex Minkowski planes

Imposing geometric or group-theoretical conditions on left reflections or the group \({\mathfrak{G}}\) generated by them, we obtain many characterizations of the Euclidean plane and of Radon planes within the framework of strictly convex Minkowski planes. In particular, Bachmann’s view of geometry provides a rich source of pertinent conditions on \({\mathfrak{G}}\) . A special role in characterizing the Euclidean plane and Radon planes is played by the shape of the locus of images of a point x under the set of left reflections in lines having a point distinct from x in common.     

Weakly Regular and Regular sets in Minkowski planes

A Regular (respectively Weakly Regular) set for an incidence structure Q is a set of points such that the identity is the only automorphism of Q which maps onto itself (respectively which fixes pointwise). In this work Weakly Regular and Regular sets in Minkowski planes are investigated.Work done within the activity of G.N.S.A.G.A. of C.N.R. and supported by 40 % grants of M.U.R.S.T.G.Rinaldi thanks Fondazione Francesco Severi and Banca Popolare dell'Etruria e del Lazio for the prize which allowed this research.     

On the Minkowski planes constructed by Artzy and Groh

The Minkowski planes constructed by R. Artzy and H. Groh [1] are characterized among the locally, connected and finite dimensional Minkowski planes as strongly semi-(p, w)-transitive Minkowski planes (see Theorem 2). The types of the Artzy-Groh planes in the typification of the Minkowski planes by M. Klein are determined (see Proposition 4). The second author was supported by a DAAD scholarship for a research visit at TU München. He sincerely thanks the Zentrum Mathematik der TU München for their hospitality.     

Minkowski planes admitting automorphism groups of small type

We prove that a Minkowski plane with an automorphism group of type 5¹ is of order 5 and, if it is of type 4 or 7 it is of order 3 or 5. Received 5 January 1999.     

A classification of Minkowski planes over half-ordered fields

This paper concerns a construction of Minkowski planes over half-ordered fields [5] and [20]. Solving various functional equations the Klein-Kroll types of these Minkowski planes are determined with respect toG- andq-translations and (p, q)-homotheties. Examples for some of the resulting types are given.     

Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle

Consider a system {φ _n } of polynomials orthonormal on the unit circle with respect to a measuredμ, withμ′>0 almost everywhere. Denoting byk _n the leading coefficient ofφ _n , a simple new proof is given for E. A. Rakhmanov's important result that lim _n→∞,k _n /k _n+1=1; this result plays a crucial role in extending Szegö's theory about polynomials orthogonal with respect to measuresdμ with logμ′∈L ¹ to a wider class of orthogonal polynomials.