Showing papers in "Journal of Geometry in 2003"

Zeros of Chromatic and Flow Polynomials of Graphs

Abstract: We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids.

On the spectrum of minimal blocking sets in PG(2, q)

Abstract: The spectrum problem for minimal blocking sets means that we wish to determine the possible cardinalities of minimal blocking sets. Besides surveying the results on this problem some new results (or new proofs) are given.

Harmonic and analytic functions on graphs

Abstract: Harmonic and analytic functions have natural discrete analogues. Harmonic functions can be defined on every graph, while analytic functions (or, more precisely, holomorphic forms) can be defined on graphs embedded in orientable surfaces. Many important properties of the "true" harmonic and analytic functions can be carried over to the discrete setting. We prove that a nonzero analytic function can vanish only on a very small connected piece. As an application, we describe a simple local random process on embedded graphs, which have the property that observing them in a small neighborhood of a node through a polynomial time, we can infer the genus of the surface.

Minding isometries of ruled surfaces in pseudo-Galilean space

Abstract: In this paper we study the Minding isometries of skew ruled surfaces in pseudo-Galilean space. The Minding isometries are isometries of ruled surfaces which are also generator-preserving. The obtained results can be easily transfered to the ruled surfaces in Galilean space.

Parallelisms of projective spaces

Abstract: Abstract. New and old results on parallelisms of projective spaces are surveyed.

Configurations of ovals

Abstract: We survey the known hyperovals in $ \mathrm{PG}(2,q) $ . We then survey the relationship of the study of configurations of ovals in $ \mathrm{PG}(2,q) $ called augmented fans to that of ovoids of $ \mathrm{PG}(3,q) $ , flocks of the quadratic cone of $ \mathrm{PG}(3,q) $ , flocks of a translation oval cone of $ \mathrm{PG}(3,q) $ and spreads of certain generalised quadrangles of order $ q $ . We also consider the configurations of ovals in $ \mathrm{PG}(2,q) $ called herds and their relationship with flocks of the quadratic cone of $ \mathrm{PG}(3,q) $ .

A variational characterization of canonical angles between subspaces

Abstract: Canonical angles between subspaces of a unitary space are characterized by a min-max property which involves inner products.

Antipodes sur un tétraèdre régulier

Abstract: The aim of this article is to compute the set of farthest points of an arbitrary point of the surface of a regular tetrahedron endowed with its intrinsic metric.

Generalized ($\kappa,\mu $)-contact metric manifoldswith $\|{grad}\kappa \| =$ constant

Abstract: We locally classify the 3-dimensional generalized ($\kappa,\mu$)-contact metric manifolds, which satisfy the condition $\Vert grad\kappa \Vert =$const. ($ ot=0$). This class of manifolds is determined by two arbitrary functions of one variable.

Contact hypersurfaces of Kaehler manifolds

Abstract: Contact hypersurfaces of a Kaehler manifold have been characterized and classified, assuming the second fundamental form to be Codazzi (in particular, parallel). We have also discussed the special cases when the ambient space is a (i) Calabi-Yau manifold and (ii) a complex space-form.

On 4-flattening theorems and the curves of Carathéodory, Barner and Segre

Abstract: We discuss three classes of closed curves in the Euclidean space $\mathbb{R}^{3}$ which have non-vanishing curvature and at least 4 flattenings (points at which the torsion vanishes). Calling these classes (de.ned below) Barner, Segre and Caratheodory, we prove that Barner $\subset$ (Segre $\cap$ Caratheodory). We also prove that (Segre)\ (Segre $\cap$ Caratheodory) and (Caratheodory)\(Segre $\cap$ Caratheodory) are open sets in the space of closed smooth curves with the C∞-topology. Finally, we define a class of closed curves containing the class of Segre curves and -based on contact topology considerations, as the Huygens principle- we establish the conjecture that any curve of our class has at least 4 flattenings.

Hyper-reguli and non-André quasi-subgeometry partitions of projective spaces

Abstract: A question raised by Ostrom on the existence of hyper-reguli that are not Andre hyper-reguli is completely determined for hyper-reguli of order $q^{t}$ where t is composite. Various new types of ‘generalized Andre replacements’ are constructed that produce many new classes of generalized Andre planes. In general, for t=ds, new non-Andre quasi-subgeometry partitions are constructed of ${\it PG}(s-1,q^{d})$ by quasi-subgeometries isomorphic to PG$(ds/e-1,q^{e})$, for various divisors e of d. When $d=2$, this produces new non-Andre sub-geometry partitions of PG $(2s-1,q^{2})$ by subgeometries isomorphic to PG $(2s-1,q)$ and PG$(s-1,q^{2})$.

A nullity condition for complex contact metric manifolds

Abstract: A nullity condition for real contact manifolds is defined by Blair, Koufogiorgos and Papantoniu. Lately, Boeckx classified such manifolds completely. In this paper, a nullity condition for complex contact manifolds is defined as follows: take a complex contact manifold whose vertical space is annihilated by the curvature. Then, apply an $\mathcal{H}$-homothetic deformation. In this way, we get a condition which is invariant under $\mathcal{H}$-homothetic deformations. A complex contact manifold satisfying this condition is called a complex (κ,μ)-space. Some curvature properties of complex (κ,μ)-spaces are studied and it is shown that, just as in the real case, the curvature tensor of a complex (κ,μ)-space is completely determined.

On the addition of convex sets in the hyperbolic plane

Abstract: Analogue to the definition $K + L := \bigcup_{x\in K}(x + L)$ of the Minkowski addition in the euclidean geometry it is proposed to define the (noncommutative) addition $K \vdash L := \bigcup_{0\, \leqsl\, \rho’\,\leqsl\, a(\varphi),0\,\leqsl\,\varphi\,<\, 2\pi}T_{\rho’}^{(\varphi)}(L)$ for compact, convex and smoothly bounded sets K and L in the hyperbolic plane $\Omega$ (Klein’s model). Here $\rho = a(\varphi)$ is the representation of the boundary $\partial$ K in geodesic polar coordinates and $T_{\rho}^{(\varphi)}$ is the hyperbolic translation of $\Omega$ of length $\rho$ along the line through the origin o of direction $\varphi$. In general this addition does not preserve convexity but nevertheless we may prove as main results: (1) $o \in$ int $K, o \in$ int L and K,L “horocyclic convex” imply the strict convexity of $K \vdash L$, and (2) in this case there exists a hyperbolic mixed volume $V_h(K,L)$ of K and L which has a representation by a suitable integral over the unit circle.

Simplices with congruent k-faces

Abstract: Let S be a non-degenerate simplex in $\mathbb{R}^{2}$ We prove that S is regular if, for some k $\in$ {1,,n-2}, all its k-dimensional faces are congruent On the other hand, there are non-regular simplices with the property that all their (n1)-dimensional faces are congruent

The correlations of finite Desarguesian planes, Part I: Generalities

Abstract: The polarities of Desarguesian planes have long been known. This author has undertaken to classify the correlations of finite Desarguesian planes in general. In [6] we have presented all the correlations with identity companion automorphism which are not polarities, of these planes. In this sequence of papers, we classify the correlations of planes of order $ p^{2^{i}(2n+1)}, n eq 0 $, with companion automorphism ( $p^{2^{i}t}$ ), p an odd prime, $ t eq 0 $. This represents a complete classification of the correlations of planes of odd nonsquare order (i = 0). Some of the correlations of planes of odd square order ($ t eq 0 $ ) are also covered by the present analysis.

Characterizations of quadric and Hermitian Veroneseans over finite fields

Abstract: In Hirschfeld and Thas [5] the most important characterizations of quadric Veroneseans are surveyed. However a few difficult cases were still open, in particular the even case. In [10, 11] Thas and Van Maldeghem not only solve all open cases, but they also generalize most of these characterizations in several ways: they do not restrict themselves to the quadric Veronesean of the plane PG\((2,q) \), they allow ovals instead of conics, and they also characterize projections of quadric Veroneseans. Further, Cooperstein, Thas and Van Maldeghem [1] contains some properties of Hermitian Veroneseans over finite fields and also these varieties and some of their projections are characterized. All these results on Veroneseans will be surveyed here.

Constant Minkowskian width in terms of double normals

Abstract: We extend the notion of a double normal of a convex body from smooth, strictly convex Minkowski planes to arbitrary two-dimensional real, normed, linear spaces in two different ways. Then, for both of these ways, we obtain the following characterization theorem: a convex body K in a Minkowski plane is of constant Minkowskian width iff every chord I of K splits K into two compact convex sets K1 and K2 such that I is a Minkowskian double normal of K1 or K2. Furthermore, the Euclidean version of this theorem yields a new characterization of d-dimensional Euclidean ball where d ≥ 3.

Association schemes from ovoids in \( PG(3,q) \)

Abstract: We discuss three proofs of a conjecture of De Caen and Van Dam on the existence of some four-class association scheme on the set of unordered pairs of distinct points of the projective line \( \textit{PG}(1,4^f) \), where\( f\geq 2 \) is an integer. Our emphasis is on the proof using inversive planes and ovoids in \( PG(3,q) \).

Conclaves of planes in PG (4,2) and certain planes external to the Grassmannian ${\cal G}_{1,4,2} \subset$ PG(9,2)

Abstract: We show that in $\operatorname{PG}(4,2)$ there exist octets $\mathcal{P} _{8}=\{\pi_{1},\,\ldots\,,\pi_{8}\}$ of planes such that the 28 intersections $\pi_{i}\cap\pi_{j}$ are distinct points. Such conclaves (see [6]) $\mathcal{P}_{8}$ of planes in $\operatorname{PG}(4,2)$ are shown to be in bijective correspondence with those planes $P$ in $\operatorname{PG}(9,2)$ which are external to the Grassmannian $\mathcal{G}_{1,4,2}$ and which belong to the orbit $\operatorname{orb}(2\gamma)$ (see [4]). The fact that, under the action of $\operatorname{GL}(5,2),$ the stabilizer groups $\mathcal{G}_{\mathcal{P}_{8}}$ and $\mathcal{G}_{P}$ both have the structure $2^{3}:(7:3)$ is thus illuminated. Starting out from a regulus-free partial spread $\mathcal{S}_{8}$ in $\operatorname{PG}(4,2)$ we also give a construction of a conclave of planes $P\in\operatorname{orb}(2\gamma)\subset\operatorname{PG}(9,2).$

An upper bound for the minimum weight of dual codes of Figueroa planes

Abstract: In this note we give an explicit construction for words of weight 2q 3 - q 2 - q in the dual p-ary code of the Figueroa plane of order q 3, where q > 2 is any power of the prime p. When p is odd this then allows us, for the Figueroa planes, to improve on the previously known upper bound of 2q 3 for the minimum weight of the dual p-ary code of any plane of order q 3. The construction is the same as one that applies to desarguesian planes of order q 3 as described in [3].

Symmetric incidence groupoids

Abstract: In [7] point-reflection geometries were studied which can be derived from commutative kinematic spaces without involutory elements. But the class of point-reflection geometries is larger. For example, elliptic planes with their reflections cannot be derived from commutative kinematic spaces. Here we investigate a larger class of reflection geometries.

A new geometrical construction for the near hexagon withparameters $(s,t,T_2)=(2,5, \lbrace 1, 2 \rbrace)$

Abstract: We provide purely geometrical proofs for the uniqueness of the near hexagon with parameters $(s,t,T_2)=(2, 5, \{1, 2\})$. As a by-product we find a new construction for the near hexagon. Generalizing this new construction, we obtain an infinite class $\mathcal{C}$ of incidence structures.In [4] it will be proved that all members of $\mathcal{C}$ are near polygons.

Hyperovals in Steiner systems

Abstract: In this paper, we show that the basic necessary condition for the existence of a (k; 0, 2)-set in an S(2, 4, v) is also sufficient. It solves a problem posed by de Resmini [6] and we also prove some asymptotic results concerning the existence of hyperovals in Steiner systems with large block size. The results are generally applicable to designs with maximal arcs.

Almost Hermitian 4-manifolds of pointwise constant anti-holomorphic sectional curvature

Abstract: We show that a 4-dimensional almost Hermitian manifold (M, J, g) is of pointwise constant anti-holomorphic sectional curvature if and only if (M, J, g) is self-dual with J-invariant Ricci tensor and K 1212 = 0, where K is the complexification of the Riemannian curvature tensor.

Some integral geometric results in the simply isotropic space

Abstract: The measurability with respect to the group of the motions in the three dimensional simply isotropic space I 3 (1) of sets of geometric elements is studied and also some Crofton type formulas are given.

Harmonicity and gauge transformations in dimension 3

Abstract: Let (M, η) be a contact manifold. Consider a particular class of gauge transformations for the contact form η. In this note we study the harmonicity of the identity map of the manifold M, of dimension three, endowed with the Webster metric g and with another metric $\widetilde{g}$ obtained from g after a gauge transformation.