Showing papers in "Journal of Geometry in 2011"
TL;DR: In this paper, it was shown that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to the Euclidean plane, which is the case for any convex set.
Abstract: We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to Euclidean plane.
24 citations
TL;DR: In this article, a survey about q-analogues of some classical theorems in extremal set theory is presented, which are related to determining the chromatic number of the qanalogue of Kneser graphs.
Abstract: In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.
20 citations
TL;DR: In this article, the authors present several old and new results concerning substructures (e.g., partial spreads, ovoids, and tight sets) of polar spaces, and present several new results for polar spaces.
Abstract: We present several old and new results concerning substructures (e.g. partial spreads, ovoids, and tight sets) of polar spaces.
18 citations
TL;DR: In this paper, a method for generating new combinatorial sums by extending the concept of Riordan arrays to bi-infinite matrices was proposed, which can be used to deal with other problems, such as asymptotic approximation and combinative inversion.
Abstract: The aim of this work is to show how Riordan arrays are able to generate and close combinatorial identities, by means of the method of coefficients (generating functions). We also show how the same approach can be used to deal with other combinatorial problems, for instance asymptotic approximation and combinatorial inversion. Finally, we propose a method for generating new combinatorial sums by extending the concept of Riordan arrays to bi-infinite matrices.
15 citations
TL;DR: The canonical connection on a Reimannian almost product manifold is an analogue to the Hermitian connection on an almost hermitian manifold as mentioned in this paper, and it is shown in this paper that the canonical connection can be found on a class of almost product manifolds with non-integrable almost product structure.
Abstract: The canonical connection on a Reimannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Reimannian almost product manifolds with non-integrable almost product structure.
14 citations
TL;DR: In this article, the authors trace the history of geometries where Desargues' theorem is not valid and discuss work by Beltrami, Klein, Wiener, Peano, Moulton, and Hilbert.
Abstract: We trace the history of geometries where Desargues’ theorem is not valid. Roughly we cover the time from the middle of the nineteenth century until the first decade of the twentieth century, discussing work by Beltrami, Klein, Wiener, Peano, Moulton, and of course Hilbert.
11 citations
[...]
TL;DR: In this paper, the authors proved constructively that every rational semi-final in a finite-dimensional real vector space with respect to a subfield of the field of real numbers is contained in a complete (simplicial) rational semifan.
Abstract: In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this paper this result is proved constructively on use of techniques from polyhedral geometry.
11 citations
TL;DR: For each n between 1 and 6, it was shown in this article that a certain arrangement of n equal circles is the unique optimally dense packing on a standard triangular flat torus (the quotient of the plane by the lattice generated by two unit vectors with a 60◦ angle).
Abstract: For each n between 1 and 6, we prove that a certain arrangement of n equal circles is the unique optimally dense packing on a standard triangular flat torus (the quotient of the plane by the lattice generated by two unit vectors with a 60◦ angle). The packings of 1, 2, 3, 4 and 6 circles are based on either a toroidal triangular close packing or a toroidal triangular close packing with one circle removed. The packing of 5 circles is irregular. This proves two cases of a conjecture stronger than L. Fejes Toth’s conjecture about the strong solidity of the triangular close packing on the plane.
10 citations
TL;DR: In this article, the maximum number of colors in combinatorial structures under the assumption that no totally multicolored sets of a specified type occur is investigated, and problems and results are discussed.
Abstract: We discuss problems and results on the maximum number of colors in combinatorial structures under the assumption that no totally multicolored sets of a specified type occur.
10 citations
TL;DR: In this paper, it was shown that there are no helicoidal surfaces without parabolic points in the 3D Lorentz-Minkowski space satisfying the Laplace operator with respect to the second fundamental form.
Abstract: In this paper, we study helicoidal surfaces without parabolic points in the 3-dimensional Lorentz–Minkowski space under the condition Δ
II
r
i
= λ
i
r
i
where Δ
II
is the Laplace operator with respect to the second fundamental form and λ
i
is a real number. We prove that there are no helicoidal surfaces without parabolic points in the 3-dimensional Lorentz–Minkowski space satisfying that condition.
9 citations
TL;DR: In this article, the centroaffine space curve with vanishing centroidaffine curvatures is studied, and the authors classify centro affine space curves with constant centroid affine curvature into two classes.
Abstract: We study the geometric characters of a centroaffine space curve with vanishing centroaffine curvatures, and classify the centroaffine space curves with constant centroaffine curvatures, which are centroaffine homogeneous curves in \({\mathbb{R}^3}\). Moreover, we can find a centroaffine homogeneous surface on which such a space curve lies.
TL;DR: In this article, it was shown that all general planar Stewart Gough platforms with a type II DM self-motion can be classified into two so-called Darboux Mannheim (DM) types (I and II).
Abstract: Due to previous publications of the author, it is already known that one-parametric self-motions of general planar Stewart Gough platforms can be classified into two so-called Darboux Mannheim (DM) types (I and II). Moreover, the author also proved the necessity of three conditions for obtaining a type II DM self-motion. Based on this result we determine in the article at hand, all general planar Stewart Gough platforms with a type II DM self-motion. This is an important step in the solution of the famous Borel Bricard problem.
TL;DR: This work surveys recent results on the extendability of linear codes over finite fields with link to projective geometry and some applications to optimal linear codes problem.
Abstract: We survey recent results on the extendability of linear codes over finite fields with link to projective geometry and some applications to optimal linear codes problem.
[...]
TL;DR: In this article, it was shown that arcs in PG(2, qn) correspond to (qn + 1 − x)-cap in the case where x = 0, and if x = 1 or 2, this cap is contained in the intersection of n quadrics.
Abstract: In this article, we begin with arcs in PG(2, qn) and show that they correspond to caps in PG(2n, q) via the Andre/Bruck–Bose representation of PG(2, qn) in PG(2n, q). In particular, we show that a conic of PG(2, qn) that meets l∞ in x points corresponds to a (qn + 1 − x)-cap in PG(2n, q). If x = 0, this cap is the intersection of n quadrics. If x = 1 or 2, this cap is contained in the intersection of n quadrics and we discuss ways of extending these caps. We also investigate the structure of the n quadrics.
TL;DR: In this paper, the generalized area distance of a pair of planar curves is shown to be an improper indefinite affine sphere with singularities, and the singularity set of the improper affine spheres corresponds to the area evolute of the pair of curves, and this fact allows to describe a clear geometric picture of the former.
Abstract: Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefinite improper affine sphere in \({\mathbb {R}^3}\) is the generalized distance of a pair of planar curves. Considering this representation, the singularity set of the improper affine sphere corresponds to the area evolute of the pair of curves, and this fact allows us to describe a clear geometric picture of the former. Other symmetry sets of the pair of curves, like the affine area symmetry set and the affine envelope symmetry set can be also used to describe geometric properties of the improper affine sphere.
TL;DR: For a simple polygon P having k ≥ 1 reflex vertices, there exists a point q ∈ P such that every half-polygon that contains q contains nearly 1/2(k + 1) times the area of P.
Abstract: We prove that, for every simple polygon P having k ≥ 1 reflex vertices, there exists a point \({q \in P}\) such that every half-polygon that contains q contains nearly 1/2(k + 1) times the area of P. We also give a family of examples showing that this result is the best possible.
TL;DR: In this article, the authors of Stachel and Wallner (Sib Math J 45(4):785-794, 2004) defined a variant of the notion of confocality for the Euclidean space.
Abstract: Using selfadjoint regular endomorphisms, the authors of Stachel and Wallner (Sib Math J 45(4):785–794, 2004) defined, for an indefinite inner product, a variant of the notion of confocality for the Euclidean space. Our aim is to give a definition that is a common generalization of the usual confocality, and the variant in Stachel and Wallner (Sib Math J 45(4):785–794, 2004). We use this definition to prove a more general form of Ivory’s theorem.
TL;DR: A survey of recent results in the local theory of generalized quadrangles can be found in this paper, including a solution of a question of Payne which generalizes work of Havas et al.
Abstract: In this lecture, I will survey several recent results in the local theory of generalized quadrangles. Starting with a short introduction to the global automorphism theory, I will motivate as such the local viewpoint, and overview some of the most important local properties which are investigated nowadays. Recent results on skew translation quadrangles and forms will be described, including a solution of a question of Payne which generalizes work of Havas et al. (Finite geometries, groups, and computation, 2006; Adv Geom 26:389–396, 2006), and then I will mention parts of a classification of skew translation quadrangles which is being prepared by the author. Finally, I will consider conditions which are both global and local.
TL;DR: In this article, the same authors report on recent results concerning designs with the same parameters as the classical geometric designs AG(n, q) formed by the points and d-dimensional subspaces of the n-dimensional projective space over the field GF(q) with q elements.
Abstract: We report on recent results concerning designs with the same parameters as the classical geometric designs PG
d
(n, q) formed by the points and d-dimensional subspaces of the n-dimensional projective space PG(n, q) over the field GF(q) with q elements, where 1 ≤ d ≤ n−1. The corresponding case of designs with the same parameters as the classical geometric designs AG
d
(n, q) formed by the points and d-dimensional subspaces of the n-dimensional affine space AG(n, q) will also be discussed, albeit in less detail.
TL;DR: In this article, the acute triangulation with seven triangles can be proved for any triangle, consisting only of axioms A1-A15 in Pambuccian (Can. Math. Bull. 53, 534-541, 2010).
Abstract: Axiom A16 from Pambuccian (Can. Math. Bull. 53, 534–541, 2010) is shown to be superfluous as it depends on axioms A1–A15. This provides a surprisingly simple axiom system in which the acute triangulation with seven triangles can be proved for any triangle, consisting only of A1–A15 in Pambuccian (Can. Math. Bull. 53, 534–541, 2010).
TL;DR: In this paper, the Minkowski measures of asymmetry among all Reuleaux polygons of order n were investigated, and it was shown that all regular R- polygons have minimal asymmetry.
Abstract: In a previous paper, we showed that for regular Reuleaux polygons Rn the equality \({{\rm as}_\infty(R_n) = 1/(2\cos \frac\pi{2n} -1)}\) holds, where \({{\rm as}_\infty(\cdot)}\) denotes the Minkowski measure of asymmetry for convex bodies, and \({{\rm as}_\infty(K)\leq \frac 12(\sqrt{3}+1)}\) for all convex domains K of constant width, with equality holds iff K is a Reuleaux triangle. In this paper, we investigate the Minkowski measures of asymmetry among all Reuleaux polygons of order n and show that regular Reuleaux polygons of order n (n ≥ 3 and odd) have the minimal Minkowski measure of asymmetry.
TL;DR: In this paper, the authors presented a technique for building a new loop starting from the loops (K,+), (P,\widehat{+})} and satisfying suitable conditions, generalizing the construction presented in Zizioli.
Abstract: In this paper we present a technique for building a new loop starting from the loops (K,+), $${(P,\widehat{+})}$$
and (P, +) fulfilling suitable conditions, generalizing the construction presented in Zizioli (J Geom 95(1–2):173–186, 2009) where $${K=\mathbb{Z}_2}$$
or $${K=\mathbb{Z}_3}$$
and (P, +) is an abelian group. We investigate the dependence of the properties of the new loop on the corresponding properties of the initial ones (associativity, Bol condition, automorphic inverse property, Moufang condition), and we provide some examples.
TL;DR: In this paper, it was shown that the condition for local arc length parameterization along a real algebraic curve to extend meromorphically to the complex plane is quite restrictive: for curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.
Abstract: A singular flat geometry may be canonically assigned to a real algebraic curve Γ; namely, via analytic continuation of unit speed parameterization of the real locus \({\Gamma_\mathbb{R}}\) . Globally, the metric \({\rho=|Q|=|q(z)|dzd\bar{z}}\) is given by the meromorphic quadratic differential Q on Γ induced by the standard complex form dx2 + dy2 on \({\mathbb{C}^2=\{(x,y)\}}\) . By considering basic properties of Q, we show that the condition for local arc length parameterization along \({\Gamma_\mathbb{R}}\) to extend meromorphically to the complex plane is quite restrictive: For curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.
TL;DR: The authors survey the most important characterizations of quadric Veroneseans and Segre varieties of the last thirty years, including some very recent results, including a survey of the most recent results.
Abstract: We survey the most important characterizations of quadric Veroneseans and Segre varieties of the last thirty years, including some very recent results.
TL;DR: In this article, the authors overview some recent results on projective planes of Lenz-Barlotti class V. Mathematics Subject Classification (2010) 21E20, 51EA15.
Abstract: We overview some recent results on projective planes of Lenz- Barlotti class V. Mathematics Subject Classification (2010). 21E20, 51EA15.
TL;DR: In this article, it was shown that λd(k) is the minimum λ > 0 such that the following holds: for any finite family of closed balls in a finite family, such that every k elements of the family have a common line transversal.
Abstract: The number λd(k) is defined as the minimum λ > 0 such that the following holds: For any finite family \({\mathcal {F}=\{B_1,B_2, \ldots , B_n\}}\) of closed balls in \({{\mathbb{R}}^d}\) such that every k elements of \({\mathcal {F}}\) have a common line transversal, the elements of the blown up family \({\lambda\mathcal {F}=\{\lambda B_1,\lambda B_2, \ldots , \lambda B_n\}}\) have a common line transversal In this paper we show that \({\lambda_d(d+1)\leq4, \lambda_2(4)\leq 2\sqrt 2}\) and λ2(3) < 288
TL;DR: In this paper, it was shown that the normals to an ellipse can be determined by a quartic, and that the sum of normals remains constant as this point moves on one of a family of conjugate ellipses or hyperbolas.
Abstract: The squares of the normals to an ellipse are shown to be determined by a quartic. It follows from the properties of that quartic that the sum of the squares of the normals from a point remains constant as this point moves on one of a family of conjugate ellipses or hyperbolas. Likewise the sum of the squares of the normals to a hyperbola from any point on a conjugate ellipse is a constant.
TL;DR: In this article, it was shown that every sufficiently high dimensional Euclidean sphere admits an odd dimensional Riemannian submanifold M having the properties: (1) M is a homogeneous sub manifold with nonzero parallel mean curvature vector in the ambient sphere; (2) m is a Berger sphere; and (3) m has a Sasakian space form of constant φ -sectional curvature.
Abstract: We show that every sufficiently high dimensional Euclidean sphere admits an odd dimensional Riemannian submanifold M having the properties: (1) M is a homogeneous submanifold with nonzero parallel mean curvature vector in the ambient sphere; (2) M is a Berger sphere; (3) M is a Sasakian space form of constant $${\phi}$$
-sectional curvature. Note that our manifold M is diffeomorphic but not isometric to a Euclidean sphere.
TL;DR: In this article, specific constructions of heterogeneous hash families are developed using thwarts in transversal designs, and some open questions are posed, such as how to accommodate more columns in the hash family for specific distributions of numbers of symbols.
Abstract: Constructions that use hash families to select columns from small covering arrays in order to construct larger ones can exploit heterogeneity in the numbers of symbols in the rows of the hash family. For specific distributions of numbers of symbols, the efficacy of the construction is improved by accommodating more columns in the hash family. Known constructions of such heterogeneous hash families employ finite geometries and their associated transversal designs. Using thwarts in transversal designs, specific constructions of heterogeneous hash families are developed, and some open questions are posed.
TL;DR: In this article, the authors present a comparative study of two fundamental invariants of exotic R-4-spiders, i.e., the smooth compact submanifolds of R and the possibility of embedding R in some spin manifolds.
Abstract: We present a comparative study of two fundamental invariants of exotic \({\mathbb R^{4}}\)’s. The first invariant e(R) is defined intrinsically using the smooth compact submanifolds of R while the other γ(R) depends of the possibility of embedding R in some spin manifolds. We prove that γ is dominated by e on all exotic \({\mathbb R^{4}}\)’s.