Showing papers in "Journal of Geometry in 1992"
TL;DR: In this article, a new hyperoval is constructed in the Desarguesian projective plane with a full collineation stabiliser of order 3 and it is shown that the stabilizer is either trivial or has order 2 with the generator fixing no point on the hyperoval.
Abstract: A new hyperoval is constructed in the Desarguesian projective planePG(2, 32). It has a full collineation stabiliser of order 3. It is also shown, with the aid of a computer, that the collineation stabiliser of any further new hyperoval is either trivial or has order 2 with the generator fixing no point on the hyperoval.
30 citations
TL;DR: In this article, the authors investigate c-sets in 2-designs, with particular regard to sets of type (0,n) in projective planes, and associate a Hadamard design to a hyperoval of a projective plane of even order q and investigate some properties of its lines.
Abstract: In this paper, we investigatec-sets in 2-designs, with particular regard to sets of type (0,n) in projective planes. In particular, we associate a Hadamard design to a hyperoval of a projective plane of even orderq and we investigate some properties of its lines. This gives information on the order of the projective plane.
18 citations
17 citations
TL;DR: In this article, two overconstrained mechanisms are presented that both are related to regular polyhedra in the Euclidean 3-space, and both allow at least two types of constrained motions.
Abstract: Two overconstrained mechanisms are presented that both are related to regular polyhedra in the Euclidean 3-space. The first example, the HEUREKA-polyhedron, is a modification of BUCKMINSTER-FULLERS Jitterbug [1]. The spherical joints at the vertices of 8 regular triangles are replaced by particular cardan joints. A 15m high model of this polyhedron was exhibited at the national research exposition of Switzerland 1991 in Zurich. Further the GRUNBAUM-framework is discussed. Here the 10 regular tetrahedra inscribed to a regular pentagon-dodecahedron are linked together at the common vertices. This framework allows at least two types of constrained motions. The first was found by R. Connelly [2]. These motions preserve the fivefold symmetry with respect to any face axis. Motions of the two second type preserve the symmetry with respect to any vertex axis.
17 citations
11 citations
11 citations
TL;DR: In this article, planar groups of stable planes are introduced and information about their size and their structure is derived, and a special case are the stabilizers of quadrangles in compact connected projective planes (i.e., automorphism groups of locally compact connected ternary fields).
Abstract: (Semi-) planar groups of stable planes are introduced, and information about their size and their structure is derived. A special case are the stabilizers of quadrangles in compact connected projective planes (i.e. automorphism groups of locally compact connected ternary fields).
11 citations
TL;DR: In this paper, the authors considered whether there exist finite translation planes with arbitrarily small kernels admitting nonsolvable collineation groups, and they showed that there exist translation planes of dimension >N and orderq3 admitting GL(2,q) as a group.
Abstract: In this article, the question is considered whether there exist finite translation planes with arbitrarily small kernels admitting nonsolvable collineation groups. For any integerN, it is shown that there exist translation planes of dimension >N and orderq3 admittingGL(2,q) as a collineation group.
11 citations
TL;DR: In this article, chain geometries over local alternative algebras are introduced, and the isomorphisms between them are determined and embedded into a chain geometry over associative graphs.
Abstract: In this paper, chain geometries over local alternative algebras are introduced. We determine the isomorphisms between them, and embed them into chain geometries over associative algebras.
9 citations
TL;DR: In this article, a spread having the propertyE2 is built up of reguli which have two lines in common, and the authors apply this construction to the real 3-space to produce an example of a spread which has the property E2 and is different from an elliptic linear line congruence.
Abstract: Let πpp be a pappian projective 3-space and b∁e a set of lines of πpp; we define:-a lineg of πpp has theproperty R with respect to C, if all lines of ∁ meetingg form a regulus-∁ has theproperty E2, if there exists a pencil of lines containing only lines having the propertyR with respect to ∁. A spread having the propertyE2 is built up of reguli which have two lines in common. In the present paper, we point out a method of constructing a spread of πpp which has the propertyE2. This construction is applied to the real 3-space to produce an example of a spread which has the property E2 and is different from an elliptic linear line congruence; the discussed example represents a 4-dimensional translation plane with a 6-dimensional collineation group. Finally, the propertiesE2 andR are used to characterize the elliptic linear line congruences of a pappian 3-space.
8 citations
TL;DR: In this paper, it was shown that there exist exactly ten translation planes of order A9 which admit a quaternion group consisting of homologies with affine axis, up to isomorphism.
Abstract: We show that there exist, up to isomorphism, exactly ten translation planes of order A9 which admit a quaternion group consisting of homologies with affine axis.
TL;DR: In this article, it was shown that every left-noetherian AH-ring is left-artinian and every maximal linearly independent subset of a free left R-module V is a basis of V if and only if the H-ring R is left noetherian or V is finitely generated.
Abstract: First it will be shown that every left-noetherian AH-ring is left-artinian. For an AH-ring R every finite linearly independent subset of a free left R-module V can be completed to a basis of V. But a maximal linearly independent subset of a free left R-module V need not be a basis of V. For an H-ring R, every maximal linearly independent subset of a free left R-module V is a basis of V if and only if the H-ring R is left-noetherian or V is finitely generated.
TL;DR: In this paper, a generalization of Pfalzgraf's q-simplex condition Simq is presented, where configurational conditions (Schliesungsaussagen) of a noncommutative space can be derived from pairs (Δ, δ, Δ) of digraphs where Δ is a partial digraph of δ.
Abstract: Configurational conditions (Schliesungsaussagen) of a noncommutative space will be developped from pairs (Δ,\(\tilde \Delta \)) of digraphs where Δ is a partial digraph of\(\tilde \Delta \). In this way we obtain an extensive generalization of Pfalzgraf 's q-simplex-conditions Simq.
TL;DR: To any spatial point-reflection geometry, there corresponds a determined commutative kinematic space as discussed by the authors, which corresponds to a determined set of points in the point field.
Abstract: To any spatial point-reflection geometry there corresponds a determined commutative kinematic space.
TL;DR: In this article, a result on enclosed (k, n)-arcs in PG(2,q) was used to characterise all minimal blocking sets of sizes =q + n in PG 2,q with at least twonsecants.
Abstract: A result on enclosed (k,n)-arcs in PG(2,q) is used to characterise all minimal blocking sets of sizes =q +n in PG(2,q) with at least twon-secants. This also permits a slight extension of the bounds on blocking sets established by Blokhuis and Brouwer.
TL;DR: In this article, the concept of locally regular coloured graph was introduced as a generalization to any dimension of the regularity for maps on surfaces of W. Threlfall, and it was shown that locally regular colored graphs can be obtained from the classical spherical, euclidean and hyperbollic tessellations.
Abstract: In this work we introduce the concept of locally regular coloured graph as a generalization to any dimension of the concept of regularity for maps on surfaces of W. Threlfall. We prove that locally regular coloured graphs can be obtained from the classical spherical, euclidean and hyperbollic tessellations. Finally we describe locally regular coloured graphs on spherical 3-manifolds.
TL;DR: In this article, the authors constructed a class of ovals in all Figueroa planes of characteristic not two, which coincides with the Ovali di Roma of Cherowitzo.
Abstract: We construct a class of ovals in all Figueroa planes of characteristic not two. This class coincides with the Ovali di Roma of Cherowitzo in the case the Figueroa plane is finite. We also construct a polarity whose set of absolute points is the set of points of the oval.
TL;DR: In this paper, a map of OSS elements of an affine Cayley-Klein (CK-) plane into the Lie algebra A4(δ 2 ) of the aequiform transformations of the given plane A2(ℜ, ǫ 2) was studied.
Abstract: In part I we have studied a map of ”osculating elements” of an affine Cayley-Klein (CK-) plane into the Lie algebra A4(ɛ2) of the aequiform transformations A4(ɛ2) of the given plane A2(ℜ, ɛ2). If we use the real projective space P3(ℜ) over A4(ɛ2) each osculating element defines a straight line in P3(ℜ). We now give a one parameter motion in A4(ɛ2) and study second order properties and their analogon in the Lie algebra and P3(ℜ), respectively. We show that the wellknown relationship between the points of the moving frame and the osculating circles of the point paths in the fixed frame may be interpreted as part of a quadratic map Γ of certain straight Lines of P3(ℜ). An analogous result holds for the curvature of pairs of envelopes; the mapV induced in P3(ℜ) than is contained in a cubic relationship of straight lines.
TL;DR: In this paper, it was shown that the intersection of F(E) and an arbitrary plane through P and Q belongs to a CASSINI curve, and the carrier set of all area elements in 3-space having normal vectors parallel to PQ and a fixed illumination intensity E>0.
Abstract: Let the line segment PQ in E3 be a light source with classical LAMBERT distribution of luminous intensity. Further on, let F(E) be the carrier set of all area elements in 3-space having normal vectors parallel to PQ and a fixed illumination intensity E>0. It is shown that the intersection of F(E) and an arbitrary plane through P and Q belongs to a CASSINI curve.
TL;DR: In this article, a C1-class of plane closed curves is considered, which contains all ovals and rosettes, and the class is divided into some subclasses for which greatest lower bounds of numbers of antipodal and arc-antipodal sets are determined.
Abstract: A C1-class of plane closed curves is considered, which contains all ovals and rosettes. This class is divided into some subclasses for which greatest lower bounds of numbers of antipodal and arc-antipodal sets are determined.
TL;DR: In this paper, the classification of bipatterns into henomeric and homeomeric types is discussed, where the set of motifs forms two transitivity classes under the symmetry group.
Abstract: A bipattern in the plane is like a pattern, in the sense of Grunbaum and Shephard's book, Tiling and Patterns, but the set of motifs forms two transitivity classes under the symmetry group. In this paper we discuss the classification of such bipatterns into henomeric and homeomeric types.
TL;DR: In this article, the notion of compatibility between multiple-valued orderings and epimorphisms of projective planes has been studied and its algebraic counterpart has been investigated using the machinery of lifting orderings via places.
Abstract: We codify the notion of compatibility between multiple-valued orderings and epimorphisms of projective planes and investigate its algebraic counterpart. Using the machinery of lifting orderings via places, we construct a wide variety of proper, multiple-valued ordered projective planes.
TL;DR: In this paper, the simplicial simplicial 4-arrangement formed by mirror symmetry and possibly the hyperplane at infinity with the 24 facet hyperplanes of the 24-cell T was shown to be simplicial in each dimension.
Abstract: Let l = A
4
1
(49) be the 4-arrangement in real projective 4-space that includes all 24 facet hyperplanes of the 24-cellT, its 24 hyperplanes of mirror symmetry, and the hyperplane at infinity In this paper we show that ℝ is theonly simplicial 4-arrangement formed by including some of the hyperplanes of mirror symmetry and possibly the hyperplane at infinity with the 24 facet hyperplanes ofT Only one other simplicial 4-arrangement is known that does not lie in a natural sequence of arrangements that are simplicial in each dimension Two of the simplicial 3-arrangements induced by X were not previously in the literature, so 19 joinirreducible simplicial 3-arrangements are currently known
TL;DR: In this article, the number of nonisomorphic semifield planes of order p4 associated to Boerner-Lantz semifields is determined, where p4 is the largest number of semidefinite planes.
Abstract: We determine the number of nonisomorphic semifield planes of order p4 associated to the Boerner-Lantz semifields.
TL;DR: In this paper, it was shown that in most cases X must contain involutory perspectivities and α e Aut(P) is a perspectivity of odd prime order inducing a nontrivial automorphism on X.
Abstract: LetP be a projective plane and X a finite group of Lie-type in Aut(P). Suppose α e Aut(P) is a perspectivity of odd prime order inducing a nontrivial automorphism on X. We show that in most cases X must contain involutory perspectivities.
TL;DR: In this paper, the volume of a cone with vertex a submanifold P of a Riemannian or Kaehler manifold M is estimated as a function of the mean curvature of P and the sectional curvature.
Abstract: We give some estimates for the volume of a cone with vertex a submanifold P of a Riemannian or Kaehler manifold M. The estimates are functions of bounds of the mean curvature of P and the sectional curvature of M. They are sharp on cones having a basis which is contained in a tubular hypersurface about P in a space form or in a complex space form.
TL;DR: In this article, it was shown that the diameter of an extended partial geometry (EpGα) with α ≥ 2 is bounded by [s/2] −ϕ + 4, where ϕ is the index of the geometry.
Abstract: In [9] we proved that the diameter Δ of an Extended partial Geometry (EpGα), of order (s,t) with α ≥2 is bounded by [s/2] −ϕ + 4, where ϕ is the index of the geometry (i.e. the minimum positive antiflag number). An example of EpG2(s,1) with ϕ=α + 1 = 3 attaining the bound is given by a truncated Ds+2 Coxeter complex (see Example 1.2 for a different simple description).