On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order

The sporadic complete 12‐arc in PG(2, 13) contains eight points from a conic. In PG(2,q) with q>13 odd, all known complete k‐arcs sharing exactly ½(q+3) points with a conic 𝒞 have size at most ½(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (½(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (½(q+3)+4)‐arc in PG(2, 17), and two complete (½(q+3)+3)‐arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (½(q^r+3)+3)‐arc in PG(2,q^r) with r odd and q≡3 (mod 4) sharing ½(q^r+3) points with a conic, whenever PG(2,q) has a (½(q^r+3)+3)‐arc sharing ½(q^r+3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 25–47, 2010     

Polar Conic Set-Valued Map in Vector Optimization. Continuity and Derivability

In the context of vector optimization, several results are stated mainly about the continuity and the derivability of a conic set-valued map (the polar conic function) having a close relation with the positive efficient points, the ideal points and other distinguished elements of the efficient line. The contingent cone to the set of the general positive quasiefficient points at a point x ₀ is also related with the frontier of the dual cone of the image at x ₀ of the polar conic function. This work was partially supported by Grant SEJ2006–15401–C04–02 of Spanish Ministerio de Ciencia y Tecnología and Grant S-0505/tic/0230-D3 of Comunidad Autónoma de Madrid. The authors are grateful to the referees for suggestions which led to improving the paper.     

Equations of <Emphasis Type="Italic">G</Emphasis>-minimal conic bundles with degree 4

We consider the first nontrivial case of relatively G-minimal conic bundles which are G-minimal and have r = 4 singular fibers. Classification obtained gives explicit equations of minimal conic bundles (S, G) and an explicit action of the group G on the Picard group Pic(S) and on the surface S.     

On the intersection of a Hermitian curve with a conic

Let ${\mathcal{H}}${\mathcal{H}} be a Hermitian curve and let Γ be a conic of PG(2, q ²). In this paper we determine the possible intersection configurations between Γ and H{\mathcal{H}} under the hypotheses that Γ and H{\mathcal{H}} either share two points with the same tangent lines or contain a common Baer subconic. Moreover, the intersection configurations between a degenerate Hermitian curve and a conic sharing a Baer subconic are also determined.     

A very simple proof of Pascal’s hexagon theorem and some applications

In this article we present a simple and elegant algebraic proof of Pascal’s hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic.     

Algebras with involution that become hyperbolic over the function field of a conic

We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree 4 and give an example of such a division algebra with orthogonal involution of degree 8 that does not contain (Q,⁻), even though it contains Q and is totally decomposable into a tensor product of quaternion algebras.     

An analysis of cubic approximation schemes for conic sections

In this paper a piece of a conic section is approximated by a cubic or piecewise cubic polynomial. The main tool is to define the two inner control points of the cubic as an affine combination, defined by [0, 1], of two control points of the conic. If is taken to depend on the weightw of the latter, a function (w) results which is used to distinguish between different algorithms and to analyze their properties. One of the approximations is a piecewise cubic havingG ⁴ continuity at the break points.     

Birational models for varieties of Poncelet curves

We consider Poncelet pairs (S,C), whereS is a smooth conic andC is a degree—c plane curve having the Poncelet property with respect toS. We prove that forc>4 the projection (S,C)→C is generically one-to-one and use this to describe a birational model of the variety of Poncelet curves forc odd.     

Aperiodic invariant continua for surface homeomorphisms

We prove that if a homeomorphism of a closed orientable surface S has no wandering points and leaves invariant a compact, connected set K which contains no periodic points, then either K=S=\mathbbT²{K=S=\mathbb{T}^2} , or K is the intersection of a decreasing sequence of annuli. A version for non-orientable surfaces is given.     

Configurations of inscribed equilateral triangles

This paper explores equilateral triangles XYZ with vertices on sidelines of a given triangle ABC such that one side of XYZ is parallel to the corresponding side of ABC. There are six such triangles. They have many interesting properties which we investigate using trilinear coordinates. Our results improve and add to the earlier results of Blas Herrera Gómez about these configurations. We obtain new characterizations of several central points of the triangles and identify interesting pairs of triangles that are homologic (or perspective) and orthologic. The recognition of the Darboux cubic of a triangle is also accomplished in these configurations. Triples of circles intersecting in a point and six points on a conic also appear.      

Conic Set-Valued Maps in Vector Optimization

The paper deals with the properties of a conic set-valued function defined on the set of all ideal points of vector programming problems. The results here about the continuity and derivability of this conic set-valued map, can be used to get information about the sensitivity of the problem and the stability of the order associated to every ideal point. Furthermore, it is proved that certain contingent cones are determined by the ideal conic set-valued map.      

Small arcs in projective spaces

InPG(2,q), small completek-arcs, which have a large number of points in common with a conic have been constructed. This article generalizes these results. InPG(n,q), n 3, small completek-arcs, which havek–1 points in common with a normal rational curve, are constructed. For increasing values of the dimensionn, it is even possible to construct smaller complete arcs.Senior research assistant of the Belgian National Fund for Scientific Research NFWO     

A nonmonotone trust-region method of conic model for unconstrained optimization

In this paper, we present a nonmonotone trust-region method of conic model for unconstrained optimization. The new method combines a new trust-region subproblem of conic model proposed in [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231] with a nonmonotone technique for solving unconstrained optimization. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the method of [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231].     

Characterization of complete exterior sets of conics

Let be a set of exterior points of a nondegenerate conic inPG(2,q) with the property that the line joining any 2 points in misses the conic. Ifq1 (mod 4) then consists of the exterior points on a passant, ifq3 (mod 4) then other examples exist (at least forq=7, 11, ..., 31).Support from the Dutch organization for scientific Research (NWO) is gratefully acknowledged     

Equiangular lines in C (part II)

A subset S of a complex projective space is F-regular provided each two points of S have the same non-zero distance and each subset of three points of S has the same shape invariant. The aim of this paper is the determination for any odd integer r, of the largest integer n(r) such tht CP^r−1 contains an F-regular subset of n(r) points.It is established that n(r) ≤ 2r − 2 for any odd integer r and n(1 + 2^s) = 2^s+1 for any integer s.     

Geometric meanings of the parameters on rational conic segments

Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces.     

LDPC codes generated by conics in the classical projective plane

We construct various classes of low-density parity-check codes using point-line incidence structures in the classical projective plane PG(2,q). Each incidence structure is based on the various classes of points and lines created by the geometry of a conic in the plane. For each class, we prove various properties about dimension and minimum distance. Some arguments involve the geometry of two conics in the plane. As a result, we prove, under mild conditions, the existence of two conics, one entirely internal or external to the other. We conclude with some simulation data to exhibit the effectiveness of our codes.     

Una proprietà gruppale delle involuzioni planari che mutano in sé un'ovale di un piano proiettivo finito

Summary The purpose of the present paper is to prove the following theorem: Let Ω be an oval in the projective plane P of odd order n. If P admits a collineation group G wich maps Ω onto itself and is doubly transitive on Ω, then P is desarguesian, Ω is a conic and G contains all collineations in the little projective group PSL(2, n) of P wich leaves Ω invariant.

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Entrata in Redazione il 5 april 1977.