等周问题的一个初等证明

本文把欧氏平面,半球面和非欧面之中,不含给定边界,含有给定边界和含有边界而且在其上给定端点这样三种等周问题,给以初等、统一的证明.其要点在于把它们的存在性和唯一性简明扼要地归结到下述初等引理,即一个给定四边边长的四边形的面积以四顶共圆时为其唯一的极大.     

谈仿射性质在初等几何中的应用

利用高等几何中的仿射性质和平行投影方法，可解决大量初等几何问题，平行投影是比较容易理解也比较直观的一种投影方法，在初等几何中也适用。适当运用这种方法，可以在解决问题时带来事半功倍的效果,通过对三角形、平行四边形、椭圆的相关命题的证明，实例说明高等几何对初等几何的指导作用。     

单连通区域上解析函数的插值问题

本文利用单位圆盘上Hardy空间插值问题的已知结论,用较初等的方法,对边界至少含有两个不同点的任意单连通区域,给出插值问题有解的充分必要条件。     

化二次射影几何问题为初等几何问题

众所周知,有关一次射影几何的某些问题可以把某条直线投射到无穷远后,化为初等几     

关于小伸缩商拟共形群的几个定理

本文研究了单位球Bn上小伸缩商拟共形群的离散性质，给出了一个收敛定理，并且证明了在一定限制条件下任意一个非初等非离散小伸缩商拟共形群含有一个二元生成的非初等非离散子群。     

多媒体计算机辅助函数教学的尝试

函数在中学数学教材中占有重要的地位,是最基本的概念之一．特别在高中教材里,开始系统、精确地介绍函数的概念并研究幂函数、指数函数、对数函数、三角函数及反三角函数等基本初等函数的概念、性质和内容．对于给定的某个初等函数的特性研究一般包括下列内容：（１）函...     

射影几何对初等几何指导一例

１　引言射影几何是高等师范院校数学专业必修的一门基础课,开设此课的目的之一,是因它对初等几何有着广泛的指导作用．因此,尝试用射影几何的知识去解决初等几何的问题,倘能行通,再着力用射影的观点将原题推广,以求得出更为普遍的新命题,……．这就为改进和提高几何课的教学质量提供一条途径．２　一道赛题由香港数理教育学会主办的１９９８年初中数学竞赛,加试的三道解答题中第二题是如下一道平面几何题［１］．图１ＡＤＢＭＯＱＰＮＣ已知Ｐ为ＡＢＣＤ内一点,Ｏ为ＡＣ与ＢＣ的交点,Ｍ、Ｎ分别为ＰＢ、ＰＣ的中点,Ｑ为ＡＮ与…     

蝴蝶定理及其推广

蝴蝶定理是初等几何中的近代名题，可称为数学殿堂里的一颗璀璨的珍珠．自1985年杜锡录教授介绍到我国以来，不少数学家、数学教育工作者对此作过研究．本文在给出蝴蝶定理的一个简洁证明的基础上研究其推广形式并加以证明．     

含开边界二维Stokes问题的Galerkin边界元解法

本文推导了含有开边界的二维有限域上Stokes问题的边界积分方程, 得出基于单层位势的第一类间接边界积分方程.对与之等价的边界变分方程用Galerkin边界元求解以得出单层位势的向量密度. 对于含有开边界端点的边界单元,采用特别的插值函数, 以模拟其固有的奇异性.论文用若干数值算例模拟了含有开边界的有限区域上不可压缩粘性流体的绕流.       

一类有理插值曲面模型及其可视化约束控制

本文构造一类新的基于函数值和偏导数值的双变量加权混合有理插值样条.与已有的有理插值样条相比,这类新的有理插值样条具有以下四方面的特性,其一,插值函数可以由简单的对称基函数来表示;其二,对任何正参数,插值函数满足C1连续,而且,在不限制参数取值的条件之下,插值曲面保持光滑;其三,插值函数不但含有参数,而且带有加权系数,增加了插值函数的自由度;其四,插值曲面的形状随着参数与加权系数的变化而变化.同时,本文讨论此类插值曲面的性质,包括基函数的性质、积分加权系数的性质和插值函数的边界性质.此类插值函数的优势在于,不改变给定插值数据的前提下,通过选择合适的参数和不同的加权系数,对插值区域内的任意点的函数值进行修改.因此可将其应用于曲面设计,根据实际设计需要,自由地修改曲面形状.数值实验表明,此类新的有理样条插值具有良好的约束控制性质.     

The Funk and Hilbert geometries for spaces of constant curvature

The goal of this paper is to introduce and to study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of the hyperbolic space $\mathbb H ^n$ H n and of the sphere $S^n$ S n . We highlight some striking similarities among the three cases (Euclidean, spherical and hyperbolic) which hold at least at a formal level. The proofs of the basic properties of the classical Funk metric on subsets of $\mathbb R ^n$ R n use similarity properties of Euclidean triangles which of course do not hold in the non-Euclidean cases. Transforming the side lengths of triangles using hyperbolic and circular functions and using some non-Euclidean trigonometric formulae, the Euclidean similarity techniques are transported into the non-Euclidean worlds. We start by giving three representations of the Funk metric in each of the non-Euclidean cases, which parallel known representations for the Euclidean case. The non-Euclidean Funk metrics are shown to be Finslerian, and the associated Finsler norms are described. We then study their geodesics. The Hilbert geometry of convex sets in the non-Euclidean constant curvature spaces $S^n$ S n and $\mathbb H ^n$ H n is then developed by using the properties of the Funk metric and by introducing a non-Euclidean cross ratio. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. This leads to a formulation and a discussion of Hilbert’s Problem IV in the non-Euclidean settings. Projection maps between the spaces $\mathbb R ^n, \mathbb H ^n$ R n , H n and the upper hemisphere establish equivalences between the Hilbert geometries of convex sets in the three spaces of constant curvature, but such an equivalence does not hold for Funk geometries.     

Circle Geometries Modeled in Projective Lines over {{\mathbb{R}}^2} -rings

In this paper, we consider classical circle geometries and connect them with places of planar Cayley–Klein geometries. There are, in principle, only three types of $ {{\mathbb{R}}^2} $ -ring structures and, thus, only three types of corresponding circle geometries. Thus, each generalization to non-Euclidean planes turns out to be just another representation of the classical Euclidean cases. We believe that even the Euclidean cases of circle geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle geometries might also be of interest in themselves. For example, among the planar Cayley–Klein geometries, the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated similarly to the isotropic Möbius geometry by suitable projections of the Blaschke cylinder.     

经典等周不等式的又一简单初等的证明

In this short note we will give another simple and elementary proof of the classical isoperimetric inequality in the Euclidean plane.     

Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model

This paper explores a deep transformation in mathematical epistemology and its consequences for teaching and learning. With the advent of non-Euclidean geometries, direct, iconic correspondences between physical space and the deductive structures of mathematical inquiry were broken. For non-Euclidean ideas even to become thinkable the mathematical community needed to accumulate over twenty centuries of reflection and effort: a precious instance of distributed intelligence at the cultural level. In geometry education after this crisis, relations between intuitions and geometrical reasoning must be established philosophically, rather than taken for granted. One approach seeks intuitive supports only for Euclidean explorations, viewing non-Euclidean inquiry as fundamentally non-intuitive in nature. We argue for moving beyond such an impoverished approach, using dynamic geometry environments to develop new intuitions even in the extremely challenging setting of hyperbolic geometry. Our efforts reverse the typical direction, using formal structures as a source for a new family of intuitions that emerge from exploring a digital model of hyperbolic geometry. This digital model is elaborated within a Euclidean dynamic geometry environment, enabling a conceptual dance that re-configures Euclidean knowledge as a support for building intuitions in hyperbolic space—intuitions based not directly on physical experience but on analogies extending Euclidean concepts.     

Curve Shortening and the Banchoff–Pohl Inequality in Symmetric Minkowski Geometries

In this paper the classical Banchoff–Pohl inequality, an isoperimetric inequality for nonsimple closed curves in the Euclidean plane, involving the square of the winding number, is generalized to symmetric Minkowski geometries. The proof uses the well-known curve shortening flow.     

Analytic inequalities,isoperimetric inequalities and logarithmic Sobolev inequalities

There is a simple equivalence between isoperimetric inequalities in Riemannian manifolds and certain analytic inequalities on the same manifold, more extensive than the familiar equivalence of the classical isoperimetric inequality in Euclidean space and the associated Sobolev inequality. By an isoperimetric inequality in this connection we mean any inequality involving the Riemannian volume and Riemannian surface measure of a subset α and its boundary, respectively. We exploit the equivalence to give log-Sobolev inequalities for Riemannian manifolds. Some applications to Schrödinger equations are also given.     

Constant mean curvature tori with spherical curvature lines in noneuclidean geometry

A previous result in Euclidean geometry [7] on H-tori with plane and spherical curvature lines is extended here to the two noneuclidean geometries. There result families of H-tori with only spherical curvature lines, which are explicitly representable by elliptic and theta functions (or ordinary integrals of elementary functions). Among the geometric properties, it is shown that the midpoints of the generating spheres vary on geodesics. The hyperbolic case is more similar to the Euclidean situation than the elliptic one. In elliptic geometry the constructed surfaces depend on one additional rational parameter and, as a limiting case, there are even countably many minimal tori of this type.     

Isoperimetric inequalities and conjugate points on Lorentzian surfaces

Recently, an isoperimetric inequality for a sector on the Minkowski 2-spacetime has been derived by the method of parallels and the relativistic Gauss-Bonnet formula. In the present paper, we derive an isoperimetric inequality for a sector on a Lorentzian surface with curvatureK ≤ C. As a sector can be modeled by a geodesic variation of a timelike geodesic, our isoperimetric inequality gives an upper bound for the spacelike boundary of a sector. As an application of our results, we give an elementary proof of the existence of conjugate points on a Lorentzian surface with curvatureK ≤ C < 0 and we obtain an upper bound for the (timelike) diameter of a globally hyperbolic Lorentzian surface withK ≤ C < 0 by comparison of sectors.     

A generalization of the Discrete Isoperimetric Inequality for Piecewise Smooth Curves of Constant Geodesic Curvature

Summary The discrete isoperimetric problem is to determine the maximal area polygon with at most <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>k$ vertices and of a given perimeter. It is a classical fact that the unique optimal polygon on the Euclidean plane is the regular one. The same statement for the hyperbolic plane was proved by K\'aroly Bezdek [1] and on the sphere by L\'aszl\'o Fejes T\'oth [3]. In the present paper we extend the discrete isoperimetric inequality for ``polygons' on the three planes of constant curvature bounded by arcs of a given constant geodesic curvature.