Showing papers in "Journal of Geometry in 1972"

Geometric and pseudo-geometric graphs (q 2 +1,q+1,1)

Abstract: Let θ0 be a particular vertex of a strongly regular graph G with parameters v, n1, p 11 1 p 11 1 . Let A be the adjacency matrix of G, and B the submatrix of A whose rows correspond to the vertices of G adjacent to θ0 and whose columns correspond to the vertices of G nonadjacent to θ0. Then the designD(θ0) with incidence matrix B has the parameters v′=n1 b′=v-n1−1, r′=n1−p11/1−1, k′ = p 11 2 . In this paper we study the connection between G andD(θ0) when the graph G is geometric or pseudo-geometric (q2+1,q+1,1).

Projektive Darstellung der Moulton-Ebenen

Abstract: The Moulton planes can be characterized as 2-dimensional topological projective planes having a 4-dimensional collineation group, which fixes exactly one nonincident point-line-pair a∋w. We give a representation of these geometries on the real protective plane such that a and W coincide with the origin and the line of infinity. This representation shows that the collineation groups of nonisomorphic Moulton planes act differently, although they are isomorphic as topological groups.

Eine Charakterisierung der Ovoidalen Kettengeometrien

Abstract: A common characterization of the ovoidal Möbius-, Laguerre- and Minkowski-planes is given.

Über Sechseckgewebe auf Flächen

Abstract: Let F be a surface of negative curvature in a 3-dimensional Euclidean space. We consider the “Kurvengewebe” C on F, consisting of the lines of curvature and of a family of asymptotic lines of F. An integral formula is proved for the curvature of C, and surfaces are investigated for which C is a “Sechseckgewebe”.

Zur Darstellung Miquelscher Möbiusräume

Abstract: In this paper we prove the following theorem (shown by J.TITS [7] in the case of finite dimension): LetO be a set of at least 2 points of a projective space π of dimension ≥ 2 and suppose that the intersections ofO and the planes of π are either empty or one point sets or Pascalian ovals. ThenO is an ellipsoid (nondegenerate quadric of index 1). As a corollary we obtain that every Miquelian Mobius-space is the geometry of the plane sections of an ellipsoid.

Varietes lineaires et Automorphismes des Espaces affins sur un presque-corps non planaire

Abstract: Two proceeding papers were devoted to the description of all linear subspaces and of the automorphism group of the “affine” n-space over a planar nearfield.

Zur Konstruktion geschlitzter Inzidenzgruppen

Abstract: In the first part a derivation method for incidence-groups is developed. The connections to the usual derivations of near-rings are shown in part 2 and examples are constructed. We characterize those finite slit incidence-groups with special affine kernel which are derivations of abelian incidence-groups. Applying these results to special classes of finite incidence-groups we show that there are 2-sided non-abelian finite incidence-groups with abelian affine kernel. The class of derivations of abelian finite incidence-groups contains all splitting a-2-sided incidence-groups with abelian kernel and no splitting kernel-2-sided non-2-sided incidence-group with abelian kernel. In the last part the a-2-sided incidence-groups are algebraically described. The results were partly communicated at the “Conference on Geometry” in March 1971 at the University of Waterloo, Ontario, Canada.

Non-existence of collineations equivalent to some loop laws

Abstract: If in a translation plane there is a set of collineations whose existence is necessary and sufficient for the validity of the multiplicative left-inversive law, the weak-inversive law, the anti-automorphicinverse law, or the left Bol law in one of the coordinatizing right quasifields, then the plane is a Moufang plane. The same holds also for the Moufang law, if the coordinatizing quasifield is not the Hall system of order 9.

Affine mappings in the geometries of algebras

Abstract: Modules and algebras over a commutative ring R and the geometry associated with an algebra are studied. The non-homogeneous members of a module, for which no maximal ideal of R contains every co-ordinate, the generators of a module, for which polynomials in the co-ordinates take arbitrary values and the non-singular members of an algebra are investigated in relation to the group G of non-singular R-linear transformations. Particular attention is given to bi-quadratic extensions of fields. Geometrical isomorphisms are proved to be exactly those transformations that can be written as the composition of a translation, a member of G and a co-ordinate-wise extension to the algebra of an automorphism of R.

Eigenschaften von vollstaendigen Systemen orthogonaler lateinischer Quadrate, die bestimmte affine Ebenen repraesentieren

Abstract: A set of n-1 mutually orthogonal Latin squares of order n is a model of an affine plane with exactly n points on a line and every affine plane with n points on a line can be represented by n-1 mutually orthogonal Latin squares ([1]) In this paper we investigate properties of finite planes through the complete set of mutually orthogonal Latin squares representing the plane and mainly — vice versa — properties of the squares representing a fixed plane The results are based on the geometrical configurations which hold in the planes For presumed definitions and theorems which are not specially referred to see [4], [7], [3] or [6]

Elations and homologies in collineation groups of finite projective planes

Abstract: Let G be a collineation group of a finite projective plane. The action of G on the centers and axes of non-identity elations and homologies is discussed. There are several results on the possible numbers of orbits of centers, axes, and center-axis pairs of homologies and elations of a particular order. Several results on the generation of homologies or elations by other homologies or elations reveal additional information on the structures formed by the centers and axes. Some sets of sufficient conditions for the centers and axes to form Desarguesian subplanes are given.