Showing papers in "Journal of Geometry in 1994"

Separation of two convex sets in convexity structures

Abstract: A convexity structure satisfies the separation propertyS4 if any two disjoint convex sets extend to complementary half-spaces. This property is investigated for alignment spaces,n-ary convexities, and graphs. In particular, it is proven that a) ann-ary convexity isS4 iff every pair of disjoint polytopes with at mostn vertices can be separated by complementary half spaces, and b) an interval convexity isS4 iff it satisfies the analogue of the Pasch axiom of plane geometry.

Classification of ovoids inPG(3, 32)

Abstract: A classification of the ovoids inPG(3, 32) is completed with the aid of a computer. The ovoids are examined in terms of which ovals can possibly appear as secant plane sections. A weak necessary condition for two ovals to appear together as plane sections of an ovoid surprisingly turns out to be sufficient to demonstrate that the only possible secant plane sections are translation ovals. A known result regarding ovoids with such plane sections then identifies the ovoids as either elliptic quadrics or Tits ovoids.

Ruled submanifolds with finite type Gauss map

Abstract: We show that a ruled submanifold with finite type Gauss map in a Euclidean space is a cylinder on a curve of finite type or a plane.

The Fermat-Torricelli point and isosceles tetrahedra

Abstract: An analytical, unifying approach to geometric properties of the Fermat-Torricelli point of four affinely independent points yields several characterizations of isosceles tetrahedra and, in particular, a characterization or regular tetrahedra within the set of isosceles tetrahedra by means of the solid angle sum.

Classification of hyperovals in PG(2, 32)

Abstract: In this paper, we describe an exhaustive computer search that demonstrates that there are precisely 6 isomorphism classes of hyperovals inPG(2,32). The six classes had previously been discovered, and it was known that any further hyperovals would have stabiliser groups of orders 1 or 2. As the techniques for finding hyperovals involved a mixture of group theory and computer search, an exhaustive search was regarded as the only feasible way to eliminate these final cases with small group.

Real hypersurfaces of quaternionic projective space satisfying $$\nabla _{U_i } A = 0$$

Abstract: We classify real hypersurfaces of quaternionic projective space satisfying $$ abla _{U_i } A = 0$$ , i=1,2,3.

Irregular hyperovals in PG (2, 64)

Abstract: Two irregular hyperovals in the Desarguesian projective planePG(2, 64) of order 64 are constructed. One has a collineation stabiliser of order 60, the other a stabiliser of order 15. It is a lso shown, with the aid of a computer, that there are no more (irregular) hyperova ls inPG(2, 64) stabilised by a collineation of order 5.

Zum Stufenaufbau des Parallelenaxioms

Abstract: Euclid's parallel postulate is shown to be equivalent to the conjunction of the following two weaker postulates: “Any perpendicular to one side of a right angle intersects any perpendicular to the other side” and “For any acute angle Oxy, the segmentPQ — whereP is a point onOx, Q a point onOy andPQ ⊥ Oy — grows indefinitely, i. e. can be made longer than any given segment”.

Intersection of arcs and normal rational curves in spaces of even characteristic

Abstract: We study arcs K in PG( n , q ), n ≥ 3, q odd, having many points common with a given normal rational curve L . In particular, we show that, if 0.09 q + 2.09 ≥ n ≥ 3, q large, then ( q + l)/2 is the largest possible number of points of K on L , improving on the bound given in [11], [12], [14]. When | K ∩ L | = ( q + l)/2, we show that the points of K ∩ L are invariant under a cyclic linear collineation of order ( q ± l)/2. The corresponding questions for q even are discussed in [13]. Introduction Let Σ = PG( n , q ) denote the n -dimensional projective space over the field GF( q ). A k-arc in Σ, with k ≥ n + 1, is a set K of k points such that no n + 1 points of K belong to a hyperplane of Σ. A point r of PG( n , q ) extends a k -arc K , in PG( n , q ), to a ( k + l)-arc if and only if K ∪ { r } is a ( k + l)-arc. A k -arc K of PG( n , q ) is complete if and only if K is not contained in a ( k + l)-arc of PG( n , q ). Otherwise, K is called incomplete .

Holomorphically projective transformations on complex Riemannian manifold

Abstract: The aim of the paper is to prove that if a complex Riemannian manifold with holomorphic characteristic connection is holomorphically projective equivalent to a locally symmetric space then it is a complex Riemannian manifold of pointwise constant holomorphic characteristic sectional curvature.

On the Chebyshev nets in a hypersurface of a Weyl space

Abstract: In [1], Zlatanov introduced the Chebyshev vector fields of the first and second kind and the geodesic vector fields for an n-dimensional net in the Weyl spaceWn. After having defined, in [2], the Chebyshev and geodesic curvatures of the lines of an arbitrary net,the b-nets and the c-nets, Tsareva and Zlatanov studied, among other things, some properties of the Chebyshev nets. In this paper, we consider an n-dimensional net in the hypersurfaceWn of the Weyl SpaceWn+1 and study some properties of the Chebyshev vector fields of the first and second kind and the geodesic vector fields of this net. Finally, two theorems concerning the b-nets and c-nets inWn are obtained.

Zum Drehvorgang der Darboux-Achse einer Raumkurve

Abstract: In one of his papers J. Hartl says that he decomposes the Darboux rotation for the Frenet frame of a space curve into two simultaneous rotations. What he does, however, is to decompose the rotation for the Darboux axis of the space curve into two simultaneous rotations. In the following approach this rotation for the Darboux axis is being described from a more general aspect.

Improvements of n-dimensional Euler inequality

Abstract: In this paper, a new upper bound for the volume of an n-dimensional simplex is given, which improves some previous results stated in [2] and [4].

Ein Axiomensystem für partielle affine Räume

Abstract: A partial linear space with parallelism is called partial affine space if it is embeddable in an affine space with the same pointset preserving the parallelism. These partial affine spaces will be characterized by a system of three axioms for partial linear spaces with parallelism.

A characterisation of thas maximal arcs in translation planes of square order

Abstract: In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of order q2. We characterise these as being exactly those (non-trivial) maximal arcs that are stabilised by an homology of order q− 1.

Translation planes admitting many homologies

Abstract: The translation planes with spreads inPG(3,2 r ) that admit >2 r +1 axes (coaxes) of homologies of orderu≠1 are classified.

The Apollonius problem in four-dimensional Laguerre planes

Abstract: The number of circles of a four-dimensional locally compact Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.

Weak stability of almost regular contact foliations

Abstract: We prove that on a compact manifold, a contact foliation obtained by a smallC1 perturbation of an almost regular contact flow has at least two closed characteristics. This solves the Weinstein conjecture for contact forms which areC1-close to almost regular contact forms.

How the elliptic integrals K and E arise from circles and points in the Minkowski plane

Abstract: Various “self-circumferences” of unit circles placed on a Minkowskian plane are expressed in terms of elliptic integrals of the first and second kinds. Landen's transformation is interpreted in this geometric context, and the self-circumferences are estimated asymptotically in the case of circles whose boundaries come very close to the origin.

A proof for Theodorus' theorem by drawing diagrams

Abstract: In Plato's dialogue Theaetetus, the mathematician Theodorus (ca. 470-400 BC) is said to have proved the irrationality of √3, √5, ..., √17 “by drawing diagrams”. In this paper such a proof is presented and compared with another, more arithmetical, reconstructed proof by W. Knorr. Moreover a hypothetical early theory of similar rectangles is extracted from Euclid's Elements, which could have served as a basis for the geometrical proof of Theodorus' theorem.

A cyclographic approach to the vertices of plane curves

Abstract: For a closed plane curveα the centers of distance functions are considered satisfying the condition that the function attains at least two absolute minima alongα. From the topology of the closure of this set some information on specific vertices of the curve can be obtained. If a simply closed curveα has a finite number of vertices only, then the structure of this set is given by a tree, composed from regular arcs, such that the end points of this tree belong to minimal osculating circles ofα, which entirely are located in the interior ofα. Furthermore, the curve can be reconstructed from this set as the envelope of a suitable family of circles. These relations are used to give alternative proofs and extensions of several known vertex theorems for closed curves.

COMPOSITION ALGEBRAS AND PG(m, 2)

Abstract: Neil A. Gordon, Trevor M. Jarvis, Johannes G. Maks, Ron Shaw The multiplication law for the non-associative algebra of Cayley numbers can be expressed (in a suitable basis) in the form exey -- (-1)f(x'Y)ex+y, x, y E V, where V = V(3,2) denotes the 3-dimensional vector space over GF(2) = $~ = {0,1}. All i=2-valued functions f which give rise to a Cayley algebra are determined, as are those which give rise to the algebra of split octonions. In the case of Cayley numbers the invariance group of f is a flag-transitive subgroup of GL(3,2) which is isomorphic to ~7 >~ ~3. In the course of some more general considerations a new proof of Hurwitz's theorem is obtained. 1. INTRODUCTION Recall that a real octonion algebra is a real composition algebra of dimension eight, that is, a real algebra ~4 with unit and equipped with a non-degenerate quadratic form Q such that Q(ala2) = Q(al)Q(a2) for all ai E Jg. Up to isomorphism there are only two such algebras: the Cayley octonion algebra of signature (8,0) and the split octonion algebra of neutral signature (4,4). Let V(3,2) denote a vector space of dimension three over the Galois field GF(2) = I= 2. It is well-known that V(3,2) may be used to label an orthonormal basis {ex} of the Cayley algebra such that exey = ~- ex§ for all x,y e V(3,2): see Freudenthal [1] and Bourbaki [2], Ch.III, Appendix, Prop.4, p.616. We adopt a projective geometry point of view by considering the seven non-zero elements of V(3,2) to be the seven points in the Fano plane PG(2,2). Three points x,y,z E PG(2,2) are collinear if and only if x + y + z = 0. Each line contains three points and, dually, through each point pass exactly three lines.

Determination of the automorphism group of a hyperbolic K-loop

Abstract: To each commutative Euclidean field (K, +, ·) there corresponds, via Hermitian matrices, a Minkowski-Space-Time-World (H,K, →), a hyperbolic space (H1+, C) and a K-loop (H1+, ⊕). The set H1+ of points of the hyperbolic space can be turned in a K-loop, and conversely, the incidence structure of the hyperbolic space can be reconstructed from the K-loop. Therefore we call such K-loops hyperbolic. The automorphism group of a hyperbolic K-loop will be determined. The main result is stated in Theorem 1.

On the extremality of the polynomials used for obtaining the best known upper bounds for the kissing numbers

Abstract: We prove that the polynomials used for obtaining the best known upper bounds for some kissing numbers (the maximum number of nonoverlapping unit spheres that can touch a unit sphere in n dimensions) are best between the polynomials of the same or lower degree. We give also some extremal polynomials we have obtained using a method proposed in [4]. The upper bounds obtained in this way are slightly better than these from [1]. However the improvements are not in the integer part for dimensionsn ≤ 18.

Some remarks on the theorem of Möbius

Abstract: H.S.M. Coxeter in his book “Introduction to Geometry” quotes a theorem of Mobius. In the paper two counterexamples are given. A corrected version of the theorem is stated and proved.