On blocking sets of inversive planes

Let S be a blocking set in an inversive plane of order q. It was shown by Bruen and Rothschild 1 that |S| ≥ 2q for q ≥ 9. We prove that if q is sufficiently large, C is a fixed natural number and |S = 2q + C, then roughly 2/3 of the circles of the plane meet S in one point and 1/3 of the circles of the plane meet S in four points. The complete classification of minimal blocking sets in inversive planes of order q ≤ 5 and the sizes of some examples of minimal blocking sets in planes of order q ≤ 37 are given. Geometric properties of some of these blocking sets are also studied. © 2004 Wiley Periodicals, Inc.     

Uniformly Distributed Distances – a Geometric Application of Janson's Inequality

denote the distances determined by n points in the plane. It is shown that , where the minimum is taken over all point sets with minimal distance . This bound is asymptotically tight. Received: September 4, 1997     

Minimal locally cyclic triangulations of the projective plane

A triangulation of a surface is locally cyclic if each cycle of length three in its 1-skeleton bounds a face. It is shown that any locally cyclic triangulation of the projective plane can be obtained by repeatedly using the vertex splitting operation and starting with one of five minimal locally cyclic triangulations.     

The steiner ratio for five points

It was conjectured by Gilbert and Pollak [5] that for any finite set of points in the Euclidean plane, the ratio of the length of a Steiner minimal tree to the length of a minimal spanning tree is at least 3/2. The present paper proves the conjecture for five points, using a formula for the length of full Steiner trees.     

On a proposed divide-and-conquer minimal spanning tree algorithm

In a recent paper published in this journal, R. Chang and R. Lee purport to devise anO(N logN) time minimal spanning tree algorithm forN points in the plane that is based on a divide-and-conquer strategy using Voronoi diagrams. In this brief note, we present families of problem instances to show that the Chang-Lee worst-case timing analysis is incorrect, resulting in a time bound ofO(N ² logN). Since it is known that alternate, trulyO(N logN) time algorithms are available anyway, the general utility of the Chang-Lee algorithm is questionable.This author's research is supported in part by the Washington State Technology Center and by the National Science Foundation under grants ECS-8403859 and MIP-8603879.     

On extendable planes of order ten

It is well known that the only way of extending a projective plane of order n (conceivable orders are 2, 4 and 10) is adjoining a set of hyperovals to the given projective plane. A converse is proved in this note. It is shown as a corollary that the existence of an extendable plane of order 10 is equivalent to the existence of a quasi-symmetric 2-(111, 12, 10)-design.     

Axisymmetric instability of pressure-driven flow in an annular channel at high Reynolds numbers

For the pressure-driven flow in an annular channel, its linear instability with respect to axisymmetric perturbations at high Reynolds numbers is investigated within the framework of the triple-deck theory. It is shown that the problem is reduced to that of the two-dimensional linear instability of the Poiseuille flow in a plane channel. The ratio of the inner to outer radii of the channel is found at which the instability is minimal.     

A Characterization of Quaternion Planes, Revisited

It is shown that the group PSL₂(H) cannot act effectively on any eight-dimensional stable plane. Together with previous results, this entails that every eight-dimensional stable plane admitting a nontrivial action of SL₂(H) embeds into the projective plane over Hamilton's quaternions H.     

Minimal Immersions of Kahler Manifolds into Euclidean Spaces

It is proved here that a minimal isometric immersion of a Kähler-Einsteinor homogeneous Kähler-manifold into an Euclidean spacemust be totally geodesic. As an application, it is shown thatan open subset of the real hyperbolic plane RH² cannot be minimallyimmersed into the Euclidean space. As another application, aproof is given that if an irreducible Kähler manifold isminimally immersed in a Euclidean space, then its restrictedholonomy group must be U(n), where n = dim_CM. 2000 MathematicsSubject Classification 53B25 (primary); 53C42 (secondary).     

4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups

 This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit non-solvable automorphism groups. It is shown that such a plane is either semi-classical or a single plane admitting the group SL(2, ). Various characterizations of this single Laguerre plane are obtained. Received October 17 2000; in revised form April 23 2001 Published online August 5, 2002     

Analytic formulas for full steiner trees

It was conjectured by Gilbert and Pollak [6] that, for any finite set of points in the Euclidean plane, the ratio of the length of a Steiner minimal tree to the length of a minimal spanning tree is at least . To date, this has been proved only for at most five points. In this paper, some analytic formulas for the length of full Steiner trees are considered. These provide an alternative proof of the conjecture for quadrilaterals, and the foundation for a possible approach for more complicated polygons.     

Steiner minimal trees for three points with one convex polygonal obstacle

The problem of constructing Steiner minimal trees in the Euclidean plane is NP-hard. When in addition obstacles are present, difficulties of constructing obstacle-avoiding Steiner minimal trees are compounded. This problem, which has many obvious practical applications when designing complex transportation and distribution systems, has received very little attention in the literature. The construction of Steiner minimal trees for three terminal points in the Euclidean plane (without obstacles) has been completely solved (among others by Fermat, Torricelli, Cavallieri, Simpson, Heinen) during the span of the last three centuries. This construction is a cornerstone for both exact algorithms and heuristics for the Euclidean Steiner tree problem with arbitrarily many terminal points. An algorithm for three terminal points in the presence of one polygonal convex obstacle is given. It is shown that this algorithm has the worst-case time complexityO(n), wheren is the number of extreme points on the obstacle. As an extension to the underlying algorithm, if the obstacle is appropriately preprocessed inO(n) time, we can solve any problem instance with three arbitrary terminal points and the preprocessed convex polygonal obstacle inO(logn) time. We believe that the three terminal points algorithm will play a critical role in the development of heuristics for problem instances with arbitrarily many terminal points and obstacles.     

Translation planes admitting a pair of index three homology groups

It is shown that a translation plane of order which admits two homology groups of order must in fact admit symmetric homology groups of this order. It is further shown that a plane admitting such symmetric index 3 homology groups is, with a finite number of exceptions, a generalized André plane. A list of the possibly exceptional orders is determined. Received 20 March 2000.     

The uniqueness of Hering's translation plane of order 27

It is shown that the following conjecture of Kallaher and Ostrom [2] is correct: Hering's translation plane of order 27 is the only translation plane of odd dimension over its kernel which has a collineation group isomorphic to SL(2, w) with w prime to 5 and to the characteristic, and having no affine perspectivity.     

Approximation by holomorphic functions on pseudoconvex domains with isolated degeneracies

Let D be a bounded domain with smooth boundary which is strictly pseudoconvex except at a finite number of points. It is shown that functions continuous on ¯D and holomorphic on D can be approximated uniformly on D by functions holomorphic on ¯D.Author partially supported by NSF grant MPS 75-07062. While carrying out this research, the author was a visitor at the University of Washington in Seattle.     

Structure of three-dimensional minimal parabolic surfaces in euclidean space

It is shown that the structure of a three-dimensional minimal parabolic surface is determined by the pair (V², γ), where V² is a minimal two-dimensional surface in Sⁿ and γ satisfies Δγ+2γ=0 (here Δ is the Laplace operator in ℝ⁴). It is also shown that the singularities of the surface are determined by zeros of γ. Bibliography: 9 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 20–38.     

Projective Bundles of Singular Plane Cubics

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard to handle, it has been shown in [Keb00] that there exists a partial resolution of singularities which transforms a bundle of possibly badly singular curves into a bundle of nodal and cuspidal plane cubics. In cases which are of interest for classification theory, the total spaces of th se bundles will clearly be projective. It is, however, generally false that an arbitrary bundle of plane cubics is globally projective. For that reason the question of projectivity and the study of moduli seems to be of interest, and the present work gives a characterization of the projective bundles.     

The steiner ratio conjecture is true for five points

The long-standing conjecture of Gilbert and Pollak states that for any set of n given points in the Euclidean plane, the ratio of the length of a Steiner minimal tree and the length of a minimal (spanning) tree is at least $\text{3}\text{2}$. This conjecture was shown to be true for n = 3 by Gilbert and Pollak, and for n = 4 by Pollak. Recently, Du, Yao and Hwang used a different approach to give a shorter proof for n = 4. In this paper we continue this approach to prove the conjecture for n = 5. Such results for small n are useful in obtaining bounds for the ratio of the two lengths in the general case.     

A characterization of the minimalbasis of the torus

D. König asks the interesting question in [7] whether there are facts corresponding to the theorem of Kuratowski which apply to closed orientable or non-orientable surfaces of any genus. Since then this problem has been solved only for the projective plane ([2], [3], [8]). In order to demonstrate that König’s question can be affirmed we shall first prove, that every minimal graph of the minimal basis of all graphs which cannot be embedded into the orientable surface f of genusp has orientable genusp+1 and non-orientable genusq with 1≦q≦2p+2. Then let f be the torus. We shall derive a characterization of all minimal graphs of the minimal basis with the nonorientable genusq=1 which are not embeddable into the torus. There will be two very important graphs signed withX ₈ andX ₇ later. Furthermore 19 graphsG ₁,G ₂, ...,G ₁₉ of the minimal basisM(torus, >₄) will be specified. We shall prove that five of them have non-orientable genusq=1, ten of them have non-orientable genusq=2 and four of them non-orientable genusq=3. Then we shall point out a method of determining graphs of the minimal basisM(torus, >₄) which are embeddable into the projective plane. Using the possibilities of embedding into the projective plane the results of [2] and [3] are necessary. This method will be called saturation method. Using the minimal basisM(projective plane, >₄) of [3] we shall at last develop a method of determining all graphs ofM(torus, >₄) which have non-orientable genusq≧2. Applying this method we shall succeed in characterizing all minimal graphs which are not embeddable into the torus. The importance of the saturation method will be shown by determining another graphG ₂₀≠G ₁,G ₂, ...,G ₁₉ ofM(torus, >₄).     

On the Bennequin Invariant and the Geometry of Wave Fronts

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J ⁺ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.