单纯形构造定理的一个证明

<正> 作为三角形在 n 维欧氏空间 E~n 中的推广,是 n 维单纯形 A_0A_1…A_n,这里 A_0,A_1,…,A_n 是单纯形的顶点,它们不在同一个 n-1维超平面上.n 维单纯形有 C_(n+1)~2条稜     

空间分割的存在性

许多组合论的书和文章讨论过超平面分割空间的问题,但一般给出的证明实际上只是得到了区域数的上界。本文将证明这个上界是可以达到的。 设m维欧氏空间R~m中n个超平面最多确     

n元椭球面的判定及所围椭球体的体积

给出n元二次曲面是椭球面的充要条件和所对应的n维椭球体的体积计算公式,并且条件和体积的计算中只用到曲面中的系数行列式,使判定和体积计算较为方便.     

关于垂足单形的一个猜想

设 E~n 中 n 维单形(?)={A_1,A_2,…,A_(n+1)}的顶点集为{A_1,A_2,…,A_(n+1)},有向体积为 V(?),以{A_1,A_(i-1),A_(i+1),…A_n)为顶点集的 n-1维单形(?)称为(?)的“侧面”(下文中(?)所在的 n-1维超平面也记为(?)),“侧面”(?)的 n-1维体积记为(?).自 E~n 中任意一点 M 向超平面(?),(?),…,(?)作垂线,垂足分别为 H_1,H_2,…,H_(n+1),则称顶点集是{H_1,H_2,…,H_(n+1)}的单形(?)_M 为 M 关于(?)的垂足单形,其 n 维有向体积     

多重超平面完备残差图

本文研究了任意维超平面完备残差图和多重超平面完备残差图,将Erd¨os、Harary和Klawe’s定义的平面残差图推广到任意维超平面.利用容斥原理以及集合的运算性质等方法,获得了任意维超平面完备残差图的最小阶和唯一极图,以及任意维超平面完备残差图的一个重要性质,同时获得了多重任意维超平面完备残差图的最小阶和唯一极图.     

关于m-HPK(n_1,n_2,n_3,n_4)-残差图

本文研究了3-维超平面完备残差图以及m重3-维超平面完备残差图.利用容斥原理以及集合的运算性质等方法,分别获得了3-维超平面完备残差图和m重3-维超平面完备残差图的最下阶以及二者的唯一极图,将文献[1]中定义的残差图从平面推广到超平面上.     

利用特征根研究二次曲面的切平面问题

利用特征根研究二次曲面的切平面问题,给出了平面为二次曲面的切平面的充要条件.     

关于m-HPK(n1,n2,n3,n4)-残差图

本文研究了3-维超平面完备残差图以及m 重3-维超平面完备残差图. 利用容斥原理以及集合的运算性质等方法, 分别获得了3-维超平面完备残差图和m 重3-维超平面完备残差图的最下阶以及二者的唯一极图, 将文献[1] 中定义的残差图从平面推广到超平面上.     

典型域扩充空间的上同调

<正> 在紧致Kahler流形中,最重要和最典型的是Grassmann流形.Grassmann流形G(m+n,n)由C~(m+n)中全体n维超平面组成,它可以实现为G(m+n,n)={是m×(m+n)矩阵,rank3=m     

区域分割的计数问题

首先讨论了k维欧氏空间被n 个(k-1)维超平面分割的区域计数问题.然后用角形线和锯齿形线分割平面,采用递归和余差公式两种方法,解决了平面区域分割的计数问题.最后对分割问题的条件与限制进行了讨论.     

Maximal subgroups of the orthogonal group

Let V be an n-dimensional regular quadratic space over a field K of characteristic not 2. Assume n 4. Let W be a regular hyperplane and v a nonzero vector orthogonal to W. Suppose every regular hyperplane in W is universal. If is an isometry of V not leaving W invariant, then , together with the isometries of W, generate the orthogonal group of V, with one exception.The work of the author was partially supported by NSERC Grant A-7862.     

The Nash modification and hyperplane sections on surfaces

We prove that the planar components of the tangent cone of a complex analytic surface at a point correspond to the base points of hyperplane sections by the Nash modification. This correspondence is then used to characterize domination relations between the normalized Nash modification and the normalized blow-up of a point.     

From local to global behavior in competitive Lotka-Volterra systems

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics.

The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point and the carrying simplex of the system lies to one side of its tangent hyperplane at , then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.

     

A Note on the Simplex with Odd Integral Edge-Lengths

In this paper the following result is proved: Theorem.In an n-dimensional Euclidean space,there exists an n-simplex such that (i)Its all edge-lengthsare odd integers; (ii)The distances from all its vertices to a hyperplane are integers,if and only if $n\equiv \pm (mod 8)$.     

Again Sharpening of a Negative ExponentGeometric Inequality in the Space E~n

DerinitionInageometricinequalitybyaftersimplify,ifturnupgeometricmetricinde-nominatorofafractor,butnoturnupgeometricmetricinnumerator,thencallednegativeex-ponentgeometricinequalitysuchgeometicinequality,isnegativeexponentgecmetricinequalityforshort-LetrandR.denotetheinradiusandcircumradiusofsimplexXinthen-dimensionalEuc-tideanspaceEw,respectively.LetPbearbitraryPOintinsidethe8implexX,andd,,d,,..',d. ,bethedistancefrompointPton lthen-l-dimensionalhyperplane,article[ljobtainanegativeexponent…     

Two inequalities for volumes of convex bodies

Precise upper and lower bounds are given for that portion of a convex body cut off from the latter, in n-dimensional space, by a hyperplane passing through its centroid; a bound is also given for the whole volume in terms of relative n-diameters of Bernstein type.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 99–106, January, 1969.     

Two Characterizations of Smooth Norms

In an arbitrary Minkowski space M, n2, let there be given an arbitrary finite set P of weighted points whose affine hull is n-dimensional. We show that the unit ball B of M has smooth boundary if and only if each median hyperplane (minimizing the sum of weighted distances with respect to P) is spanned by n affinely independent points from P. Moreover, B has the same property if and only if every center hyperplane (minimizing the maximal weighted distance with respect to P) has the same maximal distance to at least n+1 affinely independent points from P.     

关于高维蝴蝶定理

利用距离几何方法将古典的蝴蝶定理推广到n维欧氏空间一般二次超曲面情形,同时也得到高维蝴蝶定理的逆定理.     

多元积分中值定理的中间点(英文)

研究了多元球体上的积分中值定理的中间点的渐近性质,证明了当球体半径趋于0时,中间点近似落在过球体中心的切平面上.     

Subvarieties of small codimension in smooth projective varieties

Let X PN Cbe an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y.Assume that either mn/2 and X is a complete intersection or that m N2.We show deg(X)|deg(Y)and codim Y Y codimPN X,where Y is the linear span of Y.These bounds are sharp.As an application,we classify smooth projective n-dimensional quadratic varieties swept out by m[n/2]+1 dimensional quadrics passing through one point.