Uniqueness of limit cycles bounded by two invariant parabolas

In this paper we consider a class of cubic polynomial systems with two invariant parabolas and prove in the parameter space the existence of neighborhoods such that in one the system has a unique limit cycle and in the other the system has at most three limit cycles, bounded by the invariant parabolas.     

Napoleon revisited

Napoleon's Theorem can be neatly proved using a tessellation of the plane. The theorem can be generalized by using three similar triangles (instead of the three equilateral triangles) erected in different ways on the three sides of the triangle. Various interesting special cases occur.Dedicated to H. S. M. Coxeter on the occasion of his 80th birthday.     

Infinitely Many Hypohamiltonian Cubic Graphs of Girth 7

The trivalent Coxeter graph of order 28 is the only known hypohamiltonian cubic graph of girth 7. In this paper we will construct an infinite family of hypohamiltonian cubic graphs of girth 7 and cyclic connectivity 6. The existence of cyclically 7-edge-connected hypohamiltonian cubic graphs other than the Coxeter graph, however, remains open.     

Torsion Free Commutator Subgroups of Generalized Coxeter Groups

In this note we consider generalized Coxeter groups and we study the problem of when their commutator subgroup is torsion free. As a consequence we describe all (i) Coxeter groups, (ii) triangle groups and (iii) index two orientation preserving subgroups of the finite co-volume hyperbolic Coxeter tetrahedra, for which the commutator subgroup is torsion free.     

A generalized Malfatti problem

Malfatti?s problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangle?s sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly.     

Almost all triangle-free triple systems are tripartite

A triangle in a triple system is a collection of three edges isomorphic to {123,124,345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all triangle-free triple systems with vertex set [n] are tripartite. Our proof uses the hypergraph regularity lemma of Frankl and R?dl [13], and a stability theorem for triangle-free triple systems due to Keevash and the second author [15].     

Each maximal planar graph with exactly two separating triangles is Hamiltonian

A classical result of Whitney states that each maximal planar graph without separating triangles is Hamiltonian, where a separating triangle is a triangle whose removal separates the graph. Chen [Any maximal planar graph with only one separating triangle is Hamiltonian J. Combin. Optim. 7 (2003) 79-86] proved that any maximal planar graph with only one separating triangle is still Hamiltonian. In this paper, it is shown that the conclusion of Whitney's Theorem still holds if there are exactly two separating triangles.     

Hyperbolic Coxeter groups of large dimension

We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinbergs theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension n, then n 61.Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular their existence in high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means.     

On the cubic of Napoleon

This paper explores properties of an interesting cubic from the Euler pencil of analagmatic cubics associated to a triangle. Our arguments are algebraic and use a combination of trilinear coordinates and complex numbers. We study its basic properties with emphasis on ways how to recognise this curve in various geometric configurations. Since this cubic is closely tied with the so called Napoleon configuration of a scalene triangle, we named it the Napoleon cubic.     

ON THE BERLINSKII''''S THEOREM FOR CUBIC SYSTEMS

1 IntroductionIn [1]-[31, the Berlinskii's theorem of the distribution of critical pointsfor quadratic differential systems is extended to the general n--th differentialsystems with nZ finite critical points. For cubic systems with 9 critical pointswhich must be all elementary, i.e., saddles or antisaddles, there are possiblysix kinds of distributions of critical points, such as 6 -- 3, 5 -- 4, 5 -- 3 1, 4 4 1, 4 -- 3 2 and 3 -- 3 3. For example, "5 -- 3 1" means that 5 outmostcriti…     

Placement of the Desargues configuration on a cubic curve

Hilbert and Cohn-Vossen once declared that the configurations of Desargues and Pappus are by far the most important projective configurations. These two are very similar in many respects: both are regular and self-dual, both could be constructed with ruler alone and hence exist over the rational plane, the final collinearity in both instances are automatic and both could be regarded as self-inscribed and self-circumscribed p9lygons (see [1, p. 128]). Nevertheless, there is one fundamental difference between these two configurations, viz. while the Pappus-Brianchon configuration can be realized as nine points on a non-singular cubic curve over the complex plane (in doubly infinite ways), it is impossible to get such a representation for the Desargues configuration. In fact, the configuration of Desargues can be placed in a projective plane in such a way that its vertices lie on a cubic curve over a field k if and only if k is of characteristic 2 and has at least 16 elements. Moreover, any cubic curve containing the vertices of this configuration must be singular.This research of all the three authors was supported by the HSERC of Canada.     

The Word Problem for Pride Groups

Pride groups are defined by means of finite (simplicial) graphs, and examples include Artin groups, Coxeter groups, and generalized tetrahedron groups. Under suitable conditions, we calculate an upper bound of the first order Dehn function for a finitely presented Pride group. We thus obtain sufficient conditions for when finitely presented Pride groups have solvable word problems. As a corollary to our main result, we show that the first order Dehn function of a generalized tetrahedron group, containing finite generalized triangle groups, is at most cubic.     

On a special case of the intersection of quadric and cubic surfaces

The intersection curve between two surfaces in three-dimensional real projective space RP³ is important in the study of computer graphics and solid modelling. However, much of the past work has been directed towards the intersection of two quadric surfaces. In this paper we study the intersection curve between a quadric and a cubic surface and its projection onto the plane at infinity. Formulas for the plane and space curves are given for the intersection of a quadric and a cubic surface. A family of cubic surfaces that give the same space curve when we intersect them with a quadric surface is found. By generalizing the methods in Wang et al. (2002) [6] that are used to parametrize the space curve between two quadric surfaces, we give a parametrization for the intersection curve between a quadric and a cubic surface when the intersection has a singularity of order 3.     

Conditions for coincidence of two cubic Bézier curves

This paper presents a necessary and sufficient condition for judging whether two cubic Bézier curves are coincident: two cubic Bézier curves whose control points are not collinear are coincident if and only if their corresponding control points are coincident or one curve is the reversal of the other curve. However, this is not true for degree higher than 3. This paper provides a set of counterexamples of degree 4.     

Two row mixed-integer cuts via lifting

Recently Andersen et al. [1], Borozan and Cornuéjols [6] and Cornuéjols and Margot [9] have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible non-negative integer variables (giving the two row mixed-integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In the other cases (maximal lattice-free triangles with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the latter family of triangles, we present sufficient conditions to yield an extreme inequality for the two row mixed-integer infinite-group problem.     

The Construction of Basis Functions for Curved Elements in the Finite Element Method

Part basis (wedge) functions are constructed for plane triangularelements with two straight and one curved side, using ideasfrom three dimensional geometry. Such elements are desirablefor regions with curved boundaries or interfaces. Lagrange interpolatingfunctions which are in turn (i) linear, (ii) quadratic, and(iii) cubic along the straight sides of the triangle are considered.Curved elements which had previously been dealt with using isoparametricco-ordinates appear as special cases of this more general geometricaltreatment. The ideas can be extended in a straightforward mannerto deal with tetrahedral elements with three plane and one curvedsurface.     

一种构造三次PH曲线的几何方法

通过给出始末两点以及对应的切线与弦线,利用三次PH曲线控制多边形的边与角之间的几何关系,通过加入辅助线,用几何方法求出控制多边形的弦长,从而构造出满足初始条件的控制多边形.在此基础上求出满足条件的三次PH曲线,并给出了数值实例.     

Special matchings and Coxeter groups

In the recent paper [Adv. Applied Math., 38 (2007), 210–226] it is proved that the special matchings of permutations generate a Coxeter group. In this paper we generalize this result to a class of Coxeter groups which includes many Weyl and affine Weyl groups. Our proofs are simpler, and shorter, than those in [loc. cit.] All authors are partially supported by EU grant HPRN-CT-2001-00272. Received: 30 October 2006     

Paare und Tripel halbkonzentrischer Kreise der euklidischen Ebene

In [1], semi-concentric circles have been defined and considered, also their special nets and pencils.—The isochordal curve and the isogonal curve of two circles, the isochordal lines and the isogonal points of three circles show many differences according to whether the two or three circles are semiconcentric or not. Results for semi-concentric circles are developed here, while results for other circles have been given in [2] and [5].     

三角形的九点二次曲线

在任意三角形内,三边中点,三高的垂足,以及连接顶点与垂心的三线段的中点,都在同一圆上,此圆即为三角形九点圆.三角形的九点圆是欧氏几何中著名的优美定理,被称为欧拉圆和费尔巴哈圆.本文试图把垂心改换为平面内的任意点,相应地把三条高线改换为过每个顶点各一条的共点直线组时,则将把三角形的九点圆有趣地推广为三角形的九点二次曲线.并具体讨论在不同的区域内得到的九点二次曲线.