抛物类二次曲线外切多边形中有向面积的定值定理及其应用

利用有向面积定值法,对抛物线外切多边形中的对角线三角形和切点三角形之间的关系进行研究,得到抛物类二次曲线外切n边形(n≥4)中有向面积的一个定值定理,并据此推出抛物线外切多边形中三线共点的点多达n(n-3)个,以及射影几何中著名的Brianchon定理等结论.     

The screensaver map: Dynamics on elliptic curves arising from polygonal folding

We study the repeated folding of a two-parameter family of quadrilaterals about their successively transformed diagonals by examining the evolution of the diagonal lengths. Successively mapped pairs of squared lengths lie on an elliptic curve on which folding acts as translation under the group law. We prove the rotation number attains all possible values and any value determines a unique curve in parameter space. For rational parameters we give an algorithm to determine if the folding map is periodic. This gives a partial explanation for the diversity and intricacy of the curves traced out by the paths of the vertices of the transformed quadrilaterals. This research was supported by a grant from NSERC.     

Understanding the hierarchical classification of quadrilaterals through the ordered relation according to diagonal properties

This article's aim is to suggest a supplementary learning environment to understand the hierarchical classification of quadrilaterals for high school or higher degree learners. Three diagonal properties, ‘being congruent’, ‘being perpendicular’ and ‘dividing each other in particular ratio,’ and all possible combinations of these properties, were used to construct the quadrilaterals in a dynamic geometry environment. According to the diagonal properties, 15 quadrilaterals could be constructed and an order relation was constituted on 16 quadrilaterals including the quadrilateral that did not have any diagonal property. The definition of order relation is ‘any quadrilateral Q_i is included by another quadrilateral Q_j, if and only if Q_i has all diagonal properties of Q_j.’ According to this relation, an ordered relation diagram was created, and it was found that this relation was not well ordered. After the dynamic geometry construction of each quadrilateral, observations about the diagonal properties of special quadrilaterals were noted. Furthermore, the conditions under which a quadrilateral can be concave are examined. This alternative approach to the construction of quadrilaterals provided an opportunity to define quadrilaterals with more economical and less confusing way than using angle and side properties. For example, ‘a Kite is a quadrilateral whose diagonals are perpendicular and at least one of the diagonals bisects the other’ and ‘a Trapezoid is a quadrilateral whose diagonals divide each other in same ratio.’     

Bounds on the cardinalities of families of nearly neighborly quadrilaterals

We show nearly neighborly families of nine quadrilaterals and conjecture that this is the maximum. Using techniques developed by J. Zaks for nearly neighborly tetrahedra, we show that a nearly neighborly family of quadrilaterals has at most 14 members. A nearly neighborly family of quadrilaterals is said to share a base line if all nearly neighborly quadrilaterals lie on the same side of the line and the line contains a side of each. These families form an important special case. We show that there are 35 inequivalent examples of nearly neighborly families of four quadrilaterals on a base line, but that it is impossible to have a family of five nearly neighborly quadrilaterals on a base line.This work was supported in part by the University of Illinois Research Board, UIUC, Urbana, IL.     

Mixed methods on quadrilaterals and hexahedra

We describe a new family of discrete spaces suitable for use with mixed methods on certain quadrilateral and hexahedral meshes. The new spaces are natural in the sense of differential geometry, so all the usual mixed method theory, including the hybrid formulation, carries over to these new elements with proofs unchanged. Because transforming general quadrilaterals into squares introduces nonlinearity and because mixed methods involve the divergence operator, the new spaces are more complicated than either the corresponding Raviart-Thomas spaces for rectangles or corresponding finite element spaces for quadrilaterals. The new spaces are also limited to meshes obtained from a rectangular mesh through the application of a single global bilinear transformation. Despite this limitation, the new elements may be useful in certain topologically regular problems, where initially rectangular grids are deformed to match features of the physical region. They also illustrate the difficulties introduced into the theory of mixed methods by nonlinear transformations.     

Homothetic transformations and geometric loci: properties of triangles and quadrilaterals

In this paper, we use geometric transformations to find some interesting properties related with geometric loci. In particular, given a triangle or a cyclic quadrilateral, the locus generated by the centroid or by the orthocentre (for triangles) or by the anticentre (for cyclic quadrilaterals) when one vertex moves on the circumcircle of the figure are considered. These loci are studied in paragraphs 3 and 4. By means of the homothetic transformations some properties of triangles and quadrilaterals are found. The study of these properties can be used, with profit, in a classroom activity supported by Dynamic Geometry System.     

曲边三角形及曲边四边形上的混合函数插值格式

一、引言 二元函数在标准三角形上的混合函数插值格式在许多文献,例如,Birkhofft,Barnhill,Gordon及Gregory等的文章中都有讨论。在三角形周边上对高阶偏导数进行插值,而且计算比较简单的是J.A.Gregory的文章中所给出的一种混合函数插值格式。这种格式是由简单函数的线性组合所构成的,而且格式是对称的,因此计算比较简便。但是J.A.Gregory只是对直边三角形给出了格式。本文企图推广Gregory的格式,给出曲边三角形上对高阶偏导数进行插值的插值格式。我们还进一步给出了曲边四边形上     

On conformal moduli of polygonal quadrilaterals

The change of conformal moduli of polygonal quadrilaterals under some geometric transformations is studied. We consider the motion of one vertex when the other vertices remain fixed, the rotation of sides, polarization, symmetrization, and averaging transformation of the quadrilaterals. Some open problems are formulated.     

On the three diagonals of a cyclic quadrilateral

If the side lengths of a non-degenerate cyclic quadrilateral are given, but not necessarily in cyclic order, then three diagonal lengths arise in the resulting three cyclic quadrilaterals, just as three possible pairs of supplementary angles arise as opposite vertices, and where the diagonals intersect, in each of the three configurations. We obtain a formula for the sum of the lengths of the three diagonals minus the sum of the four sides which enables us to deduce the geometric inequality that the sum of the side lengths is less than the sum of the lengths of the three diagonals. We obtain another formula when these lengths are replaced by their squares, and this yields a similar inequality. A proof of both formulas is given which uses algebraic geometry, but which proceeds by analysis of degenerate situations. Two alternative proofs of the linear version of the inequality (which implies the quadratic version) are supplied which use trigonometry and Lagrange multipliers respectively. An unusual feature of these results is that they refer not to one configuration, but rather concern three possible configurations.     

Partitioning linear spaces into complete quadrilaterals andN-gons

In this paper we find the set of integers for which there exists a linear space having all lines of degree four with the property that the set of lines can be partitioned into the classical configurations: complete quadrilaterals, triangles and 4-gons.     

A domain decomposition and mixed method for a linear parabolic boundary value problem

In this paper, we discuss the convergence of a domain decompositionmethod for the solution of linear parabolic equations in theirmixed formulations. The subdomain meshes need not be quasi-uniform;they are composed of triangles or quadrilaterals that do notmatch at interfaces. For the ease of computation, this lackof continuity is compensated by a mortar technique based onpiecewise constant (discontinuous) multipliers. It is shownthat the method on triangles, parallelograms or slightly distortedparallelograms is convergent at the expense of a half-orderloss of accuracy compared with mortar methods based on piecewiselinear multipliers.     

Vojta's Main Conjecture for blowup surfaces

In this paper, we prove Vojta's Main Conjecture for split blowups of products of certain elliptic curves with themselves. We then deduce from the conjecture bounds on the average number of rational points lying on curves on these surfaces, and expound upon this connection for abelian surfaces and rational surfaces.

     

Brianchon定理在二次曲线外切2n边形中的推广

利用有向面积方法,对二次曲线外切2n边形(n≥2)进行研究,得到二次曲线外切2n边形(n≥2)中有向面积的几个定值定理及其推论,从而把射影几何中著名的Brianchon定理推广到二次曲线外切2n边形的情形.     

Triangles with Three Rational Medians

We present a characterization of all rational sided triangles with three rational medians. It turns out that they each correspond to a point on a one-parameter family of elliptic curves. It is possible to show that the rank of this family is at least two and in fact some reasonably high rank curves appear among them.     

A Plasticity Principle of Convex Quadrilaterals on a Convex Surface of Bounded Specific Curvature

We derive the plasticity equations for convex quadrilaterals on a complete convex surface with bounded specific curvature and prove a plasticity principle which states that: Given four shortest arcs which meet at the weighted Fermat-Torricelli point their endpoints form a convex quadrilateral and the weighted Fermat-Torricelli point belongs to the interior of this convex quadrilateral, an increase of the weight corresponding to a shortest arc causes a decrease of the two weights that correspond to the two neighboring shortest arcs and an increase of the weight corresponding to the opposite shortest arc by solving the inverse weighted Fermat-Torricelli problem for quadrilaterals on a convex surface of bounded specific curvature. The invariance of the weighted Fermat-Torricelli point(geometric plasticity principle) and the plasticity principle of quadrilaterals characterize the evolution of quadrilaterals on a complete convex surface. Furthermore, we show a connection between the plasticity of convex quadrilaterals on a complete convex surface with bounded specific curvature with the plasticity of some generalized convex quadrilaterals on a manifold which is certainly composed by triangles. We also study some cases of symmetrization of weighted convex quadrilaterals by introducing a new symmetrization technique which transforms some classes of weighted geodesic convex quadrilaterals on a convex surface to parallelograms in the tangent plane at the weighted Fermat-Torricelli point of the corresponding quadrilateral. This geometric method provides some pattern for the variable weights with respect to the 4-inverse weighted Fermat-Torricelli problem such that the weighted Fermat-Torricelli point remains invariant. By introducing the notion of superplasticity, we derive as an application of plasticity the connection between the Fermat-Torricelli point for some weighted kites with the fundamental equation of P. de Fermat for real exponents in the two dimensional Euclidean space. By using as an initial condition to the 3 body problem the solution of the 3-inverse weighted Fermat-Torricelli problem we give some future perspectives in plasticity, in order to derive new periodic solutions (chronotrees). We conclude with some philosophical ideas regarding Leibniz geometric monad in the sense of Euclid which use as an internal principle the plasticity of quadrilaterals.     

Pseudospectral method for quadrilaterals

In this paper, we investigate the pseudospectral method on quadrilaterals. Some results on Legendre-Gauss-type interpolation are established, which play important roles in the pseudospectral method for partial differential equations defined on quadrilaterals. As examples of applications, we propose pseudospectral methods for two model problems and prove their spectral accuracy in space. Numerical results demonstrate the efficiency of the suggested algorithms. The approximation results and techniques developed in this paper are also applicable to other problems defined on quadrilaterals.     

Pseudospectral method for quadrilaterals

In this paper, we investigate the pseudospectral method on quadrilaterals. Some results on Legendre–Gauss-type interpolation are established, which play important roles in the pseudospectral method for partial differential equations defined on quadrilaterals. As examples of applications, we propose pseudospectral methods for two model problems and prove their spectral accuracy in space. Numerical results demonstrate the efficiency of the suggested algorithms. The approximation results and techniques developed in this paper are also applicable to other problems defined on quadrilaterals.     

四边形的等积三角剖分

1970年Monsky证明了著名的Richman猜想: 正方形不能剖分成奇数个面积相等的三角形。近年来Stein等人研究一类特殊类型的四边形的等积三角剖分问题，获得了许多重要结果。该文进一步研究四边形等积三角剖分的待解决问题。      

Propriétés arithmétiques d'une famille de surfaces K3

We study a family of K3 surfaces which have a big automorphism group. We begin with generalisations of Silverman's results: construction of canonical heights, density of rational points in one orbit,... We continue the study in estimating the density of rational points on the orbiting rational curves; this estimate is compatible with Batyrev–Manin conjecture. Moreover we settle, under more geometric hypothesis, the number of rational points of such surfaces of bounded height.