Showing papers in "Journal of Geometry in 1982"
TL;DR: In this article, a translation plane of odd order q2 admits SL(2,q) (or PSL( 2,q)) as a collineation group and π is a Desarguesian, Hall, or Hering plane.
Abstract: Let π be a translation plane of odd order q2, where q=pr and p is a prime. If π admits SL(2,q) (or PSL(2,q)) as a collineation group then π is a Desarguesian, Hall, or Hering plane, or one of two Walker planes of order 25.
24 citations
TL;DR: In this paper, a base-free definition of a Veronese variety V(n,r) is given and also an illuminating description of its osculating primes from which can be deduced in a general form and without difficulty the phenomena of degeneracy in case of small characteristics.
Abstract: By means of linear algebra a base-free definition of a Veronese variety V(n,r) is given and also an illuminating description of its osculating primes from which can be deduced in a general form and without difficulty the phenomena of degeneracy in case of small characteristics. (Instance best known: For characteristic 2 all tangents of a conic are confluent.) The last section investigates special problems for the V(1,r) in characteristic p: So the osculating primes of a V(1,p) intersect its node in a V(1,p-2). Furthermore it becomes clearer why for 2
22 citations
21 citations
TL;DR: In this paper, the connection between linear transitive groups of collineations and the algebraic description of projective (or affine) planes has been established for Laguerre-planes.
Abstract: Analogous to the wellknown results about the connection between linear transitive groups of collineations and the algebraic description of projective (or affine) planes, we give some statements for Laguerre-planes. In particular we use automorphisms with fixpoints, which induce in the residual plane of a fixpoint dilatations, translations, shears or reflections. We apply these methods in order to characterize certain ovoidal Laguerre-planes.
17 citations
TL;DR: In this article, the authors constructed some compact hypersurfaces of constant scalar curvature in a sphere with cohomogeneity, where the curvature of the hypersurface is constant.
Abstract: We construct some compact hypersurfaces of constant scalar curvature in a sphere with cohomogeneity 1([2]).
14 citations
TL;DR: In this article, the dimension of an arbitrary nonempty set is at least k if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional neighborhood in S. In each case, the number 3 is best possible.
Abstract: Let S be an arbitrary nonempty set in Rd. The following results are true for every k, 0⩽k⩽d: the dimension of ker S is at least k if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional neighborhood in S. Similarly, ker S contains a k-dimensional e-neighborhood if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional e-neighborhood in S. In the plane, we have the following results concerning finitely starlike sets: for S an arbitrary nonempty set in R2, S is finitely starlike if every three points of cl S are clearly visible from a common point of S. In case S ⊂−R2 and int cl S∼S=∅, then S is finitely starlike if and only if every three points of S are visible from a common point of S. In each case, the number 3 is best possible.
14 citations
TL;DR: In this article, the authors generalized some results known in projective planes to symmetric BIBD's with λ⩾2 blocks and found bounds for their sizes.
Abstract: Blocking sets in symmetric BIBD's with λ⩾2 are defined and bounds for their sizes are found. Thus, some results known in projective planes are generalized to symmetric BIBD's.
12 citations
TL;DR: The concept of an ordered projective Hjelmslev plane was intuitively introduced by Hjelslev in "Einleitung in die allgemeine Kongruenglehre" as discussed by the authors.
Abstract: The concept of an ordered projective Hjelmslev plane was intuitively introduced by Hjelmslev in “Einleitung in die allgemeine Kongruenglehre” ([9], [10]).
10 citations
TL;DR: In this paper, the characterization theorem given in [2] for Lorentz transformations of the ℝ2 is carried over to the case of finite planes, where the authors show that the characterization is also applicable to finite planes.
Abstract: The characterization theorem given in [2] for Lorentz transformations of the ℝ2 is carried over to the case of finite planes.
9 citations
TL;DR: In this paper, the authors determine all collineation groups of finite translation planes of even order, which are generated by sufficiently large elementary abelian 2-subgroups all of whose involutions centralize a Baer subplane.
Abstract: We determine all collineation groups of finite translation planes of even order, which are generated by sufficiently large elementary abelian 2-subgroups all of whose involutions centralize a Baer subplane.
8 citations
TL;DR: In this paper, the authors present a complete solution for the problem of finding a common intrinsic characterization of all projective metric spaces, which can be formulated algebraically in the Clifford algebra C(Q) of (V,K,Q), even if this algebra is commutative.
Abstract: If V is a vector space over a commutative field K, and if Q∶V → K is an arbitrary quadratic form ≠ 0, then the metric vector space (V,K,Q) determines a projective metric space (π(V,K),≡Q), consisting of the projective space π(V,K) and a congruence relation ≡Q induced by Q, By generalizing results of F.Bachmann, H.Karzel, R.Lingenberg, W.Nolte and U.Ott, in this paper we present a complete solution for the problem of finding a common intrinsic characterization of all projective metric spaces. The solution is based on a sharp three-reflection-theorem, which can be formulated algebraically in the Clifford algebra C(Q) of (V,K,Q), even if this algebra is commutative.
TL;DR: In this paper, the inverse implication in the condition above may be dropped, being a consequence of the direct one, and it is shown that the inverse implies that the condition may also be dropped.
Abstract: By a theorem of A. D. Alexandrov a bijection α: R
n
n
→Rn (n⩾3) satisfiyng d(a,b)=0⇔d (α(a),α(b))=0 ∀a, b ∃ Rn (d distance associated to the Minkowski metric) is a Lorentz transformation up to a dilatation. We prove that the inverse implication in the condition above may be dropped, being a consequence of the direct one.
TL;DR: In this paper, the authors give an answer if K = GF(p), p = 5,7,11: σ must be a bijective collineation in case p = 7,11; there are non-injective mappings in casep=5.
Abstract: Given a commutative field K we define d(A,B):= (a1−b1)2−(a2−b2)2 for A=(a1,a2), B=(b1,b2) e K2. Given moreover a fixed k e KO, W. Benz has asked for all mappings σ: K2→K2 such that d(A,B)=k implies
. This paper gives an answer if K=GF(p), p=5,7,11: σ must be a bijective collineation in case p = 7,11; there are non-injective mappings in case p=5. To obtain some of these results we have mads use of a computer.
TL;DR: In this paper, derived semifield planes admitting affine elations with more than one center are examined in detail and several new examples of such plantes are constructed, and a new characterization of the Hall planes of even order among derived semiield planes is also given.
Abstract: Derived semifield planes admitting non trivial affine elations with more than one centre are examined in detail and several new examples of such plantes are constructed. A new characterization of the Hall planes of even order among derived semifield planes is also given.
TL;DR: In this paper, a Krasnosel'skii-type theorem for non-closed bounded sets in R2 has been proved, where every 4 or fewer points in S see a common point via S. The number 4 is best possible.
Abstract: This work will be concerned with a Krasnosel'skii-type theorem for nonclosed bounded sets in R2, and the following result will be obtained: Let S be a nonempty bounded set in R2 whose complement consists of a finite number of locally compact components. Assume that every 4 or fewer points in S see a common point via S. Then for some point p in cl S, the set A ≡ {s ∶ s in S and (p,s]
$$
ot \subseteq$$
S} is nowhere dense in S. The number 4 is best possible.
TL;DR: In this article, a definition of a quadric model is given in the theory of the geometries studied by W.Benz, and the field and dimension of the model are determined.
Abstract: In this paper a definition of a quadric model is given in the theory of the geometries studied by W.Benz [1]. Field and dimension of the model are determined. It is shown which geometries are characterized by the existence of such a model.
TL;DR: In this paper, it was shown that the Desargues-theorems are valid in every semi-regular nearaffine space, whose dimension is equal to or larger than three.
Abstract: Nearaffine spaces were introduced by J. ANDRe [1] as a generalization of affine spaces. Desarguesian semi-regular nearaffine spaces can be described as spaces over nearfields (cf. [2]). We show that the Desargues-theorems are valid in every nearaffine space, whose dimension is equal to or larger than three.
TL;DR: A projective representation of the group of automorphisms of a geometry Σ(ℜℒ) over a kinematic algebra is given in this paper.
Abstract: There is a unique projective representation of the group of automorphisms of a geometry Σ(ℜℒ) [1] over a kinematic [3] algebra ℒ which is compatible with the quadric model [2].
TL;DR: In this paper, the authors presented three characterizations of euclidean spaces based on four-point properties in which the embedded quadruples contain a linear triple and some three of the distances determined by the four points are equal.
Abstract: Characterizations of generalized euclidean spaces by means of euclidean four-point properties.state that every metric space which is complete, and which contains a metric line joining each two of its points is a generalized euclidean space if and only if each quadruple from a certain class of quadruples of the space is congruent with a quadruple of points in a euclidean space. It is known that it suffices to consider only quadruples containing a linear triple, or quadruples in which one of the linear points is a metric midpoint of the other two. Another class of four-point properties involves quadruples which contain a linear triple and a point equidistant from two of the linear points. The present paper presents three characterizations of euclidean spaces based on four-point properties in which the embedded quadruples contain a linear triple and some three of the distances determined by the four points are equal.
TL;DR: The automorphism group of the group generated by all the affine reflections in a Desarguesian plane is isomorphic to the full collineation group of a plane.
Abstract: The automorphism group of the group generated by all the affine reflections in a Desarguesian plane is isomorphic to the full collineation group of the plane.
TL;DR: In this article, it was shown that if the nearaffine space is not necessaryly semi-regular, then it is possible to construct a desarguesian semi-normally regular near-affine spaces over nearfields.
Abstract: Ten years ago J. ANDRE introduced the concept of nearaffine space. He described desarguesian semi-regular nearaffine spaces as spaces over nearfields (cf. [3]). We show that this is also possible, if the nearaffine space is not necessaryly semi-regular.
TL;DR: In this paper, a series of inequalities among the parameters of strongly resolvable designs are proved and by means of examples it is shown that these inequalities are best possible, and the best possible.
Abstract: We prove a series of inequalities among the parameters of a strongly resolvable design. By means of examples it is shown that these inequalities are best possible.
TL;DR: In this paper, it was shown that the ℒ-trace of the system of chains can be characterized by constant curvature, which allows affine interpretation of non-affine points.
Abstract: In the case of certain geometries Σ(ℜℒ) [1] over kinematic algebras [5],[6] the ℒ-trace of the system of chains can be characterized by constant curvature. The curvature allows an affine interpretation of non-affine points.
TL;DR: Finsler spaces with generalized metric are defined as C∞ manifolds, endowed with a Finslerian connection and a tensor field of type (O,2).
Abstract: Finsler spaces with generalized metric are defined as C∞ — manifolds, endowed with a Finslerian connection and a Finslerian tensor field of type (O,2). For this field, both the symmetric and the antisymmetric parts are non-degenerate, and the covariant h- and v-derivations vanish.
TL;DR: For projective reflection groups of a special type, this paper proved that every such group is isomorphic to a group of orthogonal transformations, i.e., every group of reflection groups is a special kind of transformation.
Abstract: For projective reflection groups of a special type we prove the following theorem: Every such group is isomorphic to a group of orthogonal transformations.
TL;DR: A set of arithmetical invariants for each vertex of a graph was defined in this article, and these invariants are expressed by the eigenvalues and eigenvectors of the graph.
Abstract: A set of arithmetical invariants for each vertex of a graph was defined in [1]. In this paper these invariants are expressed by the eigenvalues and eigenvectors of the graph.
TL;DR: Geradenkongruenzen im dreidimensionalen elliptischen Raum sind unter anderem von BLASCHKE [3] und MULLER [18] untersucht worden.
Abstract: Geradenkongruenzen im dreidimensionalen elliptischen Raum sind unter anderem von BLASCHKE [3] und MULLER [18] untersucht worden. In der vorliegenden Arbeit werden Integralinvarianten geschlossener Kongruenzflachen als Integralinvarianten der Kongruenz gedeutet. Weiter werden zwei Resultate von STEPHANIDIS [21] uber geschlossene Geradenkongruenzen vom euklidischen in den elliptischen Raum ubertragen und dualisiert. Es zeigt sich, das mit den eingefuhrten Begriffen eine Ubereinstimmung mit euklidischen Resultaten erzielt werden kann. Durch Dualisierung lassen sich im Elliptischen neue Resultate erzielen, deren euklidische Analoga bisher nicht bekannt sind.