Showing papers in "Journal of Geometry in 1986"
TL;DR: In this article, it was shown that each parabolic curve f in R2 produces a Laguerre plane if f and all its images under dilatations are cycles.
Abstract: We show that each parabolic curve f in R2 produces a Laguerre plane
$$\mathbb{L}(f)$$
if f and all its images under dilatations are cycles. Likewise, two hyperbolic curves f1,f2 produce a Minkowski planeM(f1,f2). We determine for which curves
$$\mathbb{L}(f)$$
is miquelian resp. ovoidal, and for which pairs f1,f2,M(f1,f2) is miquelian resp. satisfies the rectangle axiom, thus providing many examples of non-embeddable planes.
34 citations
TL;DR: In this article, a synthetic construction of the Figueroa planes, which avoids coordinates and groups, is presented, which is based on the same approach as the one described in this paper.
Abstract: The constructions of the Figueroa planes by Figueroa, Hering-Schaeffer and Dempwolff make essential use of the collineation groups. Here we give a synthetic construction of these planes, which avoids coordinates and groups.
24 citations
TL;DR: In this paper, the authors gave a class of semifields of order q4 for any prime power q = pr with p greater than 3, and showed that this class has left nucleus GF(q2), and right and middle nucleus GF (q).
Abstract: This paper gives a class of semifields of order q4 for any prime power q = pr with p greater than 3. It is shown that this class has left nucleus GF(q2), and right and middle nucleus GF(q). Although it is not proved, it is believed that all the semifields in this class are new.
17 citations
15 citations
TL;DR: In this article, it was shown that a convex reduced body is not necessarily of constant width as an example of equilateral triangle shows, and that convex reduction with a smooth boundary is not always of a constant width.
Abstract: Note that convex reduced body is not necessarily of constant width as an example of equilateral triangle shows. One could mention here that, dually, completeness (in the sense of Meissner) characterizes constant width. (I.e., a compact convex body is of constant width if and only if any greater body has a greater diameter, see [i, p.128-129].) However convex reduced body with a smooth boundary is of constant width. That was indicated by Heil [4, w in 1978 and proved by Groemer [2, w in 1983.
14 citations
TL;DR: In this article, the dimension of the kernel of a starshaped set has been studied and the following improved Krasnosel'skii theorems have been established: for each k and d, 0 ≤ k ≤ d, define f(d,k) = d+1,2d−2k+2 if 1 ≤ k ≥ d.
Abstract: We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 ≤ k ≤ d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d−2k+2} if 1 ≤ k ≤ d. Theorem 1. Let S be a compact, connected, locally starshaped set in Rd, S not convex. Then for a k with 0 ≤ k ≤ d, dim ker S ≥ k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S. Theorem 2. Let S be a nonempty compact set in Rd. Then for a k with 0 ≤ k ≤ d, dim ker S ≥ k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.
13 citations
TL;DR: In this paper, a solution to the Fomenko conjecture concerning surfaces with zero normal torsion in E4 was given. But this was only for surfaces with pointwise planar normal sections.
Abstract: In this paper, we classify surfaces in Em with pointwise planar normal sections. Our classification solves completely an open problem proposed in [4]. In particular, our result gives a solution to the Fomenko conjecture [6] concerning surfaces with zero normal torsion in E4.
12 citations
TL;DR: For arbitrary quadratic forms, including the cases of characteristic 2 and of infinite dimensions, several affine and projective-metric structures are considered, and the corresponding isomorphisms are determined as discussed by the authors.
Abstract: For arbitrary quadratic forms, including the cases of characteristic 2 and of infinite dimensions, several affine-metric and projective-metric structures are considered, and the corresponding isomorphisms are determined. As an application, a general fundamental theorem of the miquelian circle geometry is proved.
12 citations
TL;DR: In this article, a concept of topological projective geometry is defined, which in contrast to the definitions given in [Mi] and [SO] does not contain any dimensional restrictions.
Abstract: A concept of topological projective geometry is defined, which in contrast to the definitions given in [Mi] and [SO] does not contain any dimensional restrictions. Besides elementary properties it is shown in this paper that these topological geometries always possess a coordinatization over a uniquely determined topological division ring if the dimension is finite.
11 citations
10 citations
TL;DR: In this paper, it was shown that an isotropic submanifold in Em with pointwise planar normal sections is isometric to a symmetric space of rank one or to a Euclidean space.
Abstract: Submanifolds of Em with pointwise planar normal sections were studied in [1] and others. In the present paper, we will prove that an isotropic submanifold in Em with pointwise planar normal sections is isometric to a symmetric space of rank one or to a Euclidean space. Moreover we will determine such surfaces in Em with the above assumptions.
TL;DR: In this paper, a generalization of π-spaces, namely the (σ, n)-spaces with planes is presented, where the language of (n, d)-systems is introduced and studied by G. Tallini.
Abstract: Planar spaces with planes isomorphic to PG(d, Q) or to AG(d, Q), with d ⩾3, are presented, and a natural generalization of π-spaces, namely the (σ, n)-spaces, is also studied. For this purpose, we use the language of (n, d)-systems, which was introduced and studied by G. Tallini [13], and for which we give a brief sketch of the theory.
TL;DR: In this paper, it was shown that if G is the group PΓL(2,q) acting on the points of the projective line in the usual way, then for q>27 there is a set Λ of 5 points such that no non-trivial element of G fixes Λ.
Abstract: We show that if G is the group PΓL(2,q)(for q a prime-power) acting on the points of the projective line in the usual way, then for q>27 there is a set Λ of 5 points such that no non-trivial element of G fixes Λ
TL;DR: In this paper, a method of coordinatizing a B-oval is introduced which gives rise to a ternary ring structure called anoval ternaries ring (or OTR).
Abstract: In this paper, a method of coordinatizing a B-oval is introduced which gives rise to a ternary ring structure called anoval ternary ring (or OTR). By examining these OTR's, we can deduce sufficient conditions for a B-oval to be projective. The connections between OTR's and the Hall ternary rings, which arise from the coordinatization of a projective plane, are also examined.
TL;DR: In this paper, the authors classified all finite linear spaces with line degrees n and n-k having at most n 2+n+1 lines and showed that if n is large compared with k, then any such linear space can be embedded in a projective plane of order n−1 or n.
Abstract: In this paper we shall classify all finite linear spaces with line degrees n and n-k having at most n2+n+1 lines. As a consequence of this classification it follows: If n is large compared with k, then any such linear space can be embedded in a projective plane of order n−1 or n.
TL;DR: In this article, an isotropic counterpart to Holditch's theorem is presented, which is based on a chord of fixed length that moves with both endpoints along a curve which confines an unbounded convex domain.
Abstract: In this paper we present an isotropic counterpart to Holditch's theorem. The considerations are based on an isotropic Holditchmotion defined by a chord of fixed isotropic length that moves with both endpoints along a curve which confines an unbounded convex domain in the isotropic plane.
TL;DR: Affine planes with reflections on every line such that the 3-reflection theorem is valid for any three lines through one point are either euclidean planes or elliptic planes of characteristic 2.
Abstract: Affine planes with reflections on every line such that the 3-reflection theorem is valid for any three lines through one point are either euclidean planes or elliptic planes of characteristic 2. A slight weakening of the conditions yields to pseudo euclidean planes and hyperbolic planes of characteristic 2.
TL;DR: In this paper, it was shown that a finite group with a non-trivial partition π (G) and a set ǫ(G) of subgroups such that any two components of G are contained in at least one W with W e √ g and W ≠ G is a p-group or a Frobenius group.
Abstract: We prove: Let G be a finite group with a non-trivial partition π (G) and a set ɛ(G) of subgroups such that any two components of π (G) are contained in at least one W with W e ɛ (G) and W ≠ G; then G is a p-group or a Frobenius group.
TL;DR: In this article, the following Krasnosel'skii-type theorem is proved: Let S be a nonempty set in R2 whose closure cl S is convex and bounded, and for every 9 point subset T of cl S there correspond points p1 and p2 such that each point of T is clearly visible via S from at least one of p1 or p2.
Abstract: The following Krasnosel'skii-type theorem is proved: Let S be a nonempty set in R2 whose closure cl S is convex and bounded. Assume that for every 9 point subset T of cl S there correspond points p1 and p2 (depending on T) such that each point of T is clearly visible via S from at least one of p1 or p2. Then S is a union of two starshaped sets. The number 9 is best possible.
TL;DR: In this article, it was shown that every order function of a translation plane with kernel ≠ GF(2), GF(3) can he extended to an order function function of the projective closure.
Abstract: We show that every order function of a translation plane with kernel ≠ GF(2), GF(3) can he extended to an order function of the projective closure.
TL;DR: In this article, a Krasnosel'skii theorem for non-closed bounded sets in road networks was obtained, where f(d) = d2 − 2d+3 if d ≠ 3 and f(D)=2d+1 if d = 3.
Abstract: This work will be concerned with a Krasnosel'skii theorem for nonclosed bounded sets in Rd, and the following theorem will be obtained: For each d ⩾ 2, define f(d) = d2 − 2d+3 if d ≠ 3 and f(d)=2d+1 if d = 3. Let S be a nonempty bounded set in Rd, d ⩾ 2, and assume that cl S ∼ S is a finite union of convex components, each having closure a polytope. If every f(d) points of S see via S a common point, then there is a point p in cl S such that Bp ≡ s:s in S and (p,s]⊄ S is nowhere dense in S.
TL;DR: In this paper, it was shown that a power-associative loop with a finite distributive subloop lattice is monogenic and all its subloops are monogenic.
Abstract: The following facts are shown: A loop with a finite distributive subloop lattice is finite, monogenic and all its subloops are monogenic. Therefore, power-associative loops having finite distributive subloop lattices are cyclic groups. A loop G with its subloop lattice L(G) being a finite n-dimensional projective geometry is generated by at most n+1 elements. For all n ɛ IN, n≧4, there are power-associative loops whose subloop lattices are projective lines with n points. Furthermore, for a given projective planePn (desarguesian or non-desarguesian) of order n there exists a power-associative loopG with L(G) ∼-Pn.
TL;DR: In this article, it was shown that an inequality among parameters of a non-symmetric strongly resolvable design, conjectured by A Beutelspacher and U Porta, is valid in general.
Abstract: It is shown that an inequality among parameters of a non-symmetric strongly resolvable design, conjectured by A Beutelspacher and U Porta [1], is valid in general