A Sufficient Condition for the Existence of Large Empty Convex Polygons

   Abstract. Let P be a set of points in general position in the plane. We say that P is k -convex if no triangle determined by P contains more than k points of P in the interior. We say that a subset A of P in convex position forms an empty polygon (in P ) if no point of P \ A lies in the convex hull of A . We show that for any k,n there is an N=N(k,n) such that any k -convex set of at least N points in general position in the plane contains an empty n -gon. We also prove an analogous statement in R ^d for each odd d≥ 3 .     

平面凸集存在最小凸生成集的充要条件

本文在平面上解决了StevenRLay在 [1 ]中提出的开放性问题“什么样的凸集存在唯一的最小凸生成子集” ,给出并证明了“平面上的凸集存在唯一的最小凸生成子集”的一个充要条件 .同时证明了En 中的开集一定不存在最小凸生成集 .     

关于平面19-点集的空凸分划问题

主要研究了平面上处于一般位置的19-点集,根据其凸包边数的不同,分别讨论了其所含空凸多边形的个数,得出G(19)≤5.在此基础上,对平面上处于一般位置的n-点集得出G(n)≤[11n/42],从而改进了G(n)的上界.     

调和解存在的一个充分条件

给出具有两个实自变数的未知函数的线性偏微分方程（组）调和函数解存在的一个充分条件，及在此条件下方程（组）调和函数解的简化求法．     

关于凸函数的一个分析性质的充要条件

凸函数是一类重要的函数,其等价命题很多,但没有揭示其分析性质,本文给出凸函数的一个分析性质的充要条件.     

Minimum Distance Between the Faces of Two Convex Polyhedra: A Sufficient Condition

The problem of finding the Euclidean distance between two convex polyhedra can be reduced to the combinatorial optimization problem of finding the minimum distance between their faces. This paper presents a global optimality criterion for this problem. An algorithm (QLDPA) for the fast computation of the distance between convex and bounded polyhedra is proposed as an application of it. Computer experiments show its fast performance, especially when the total number of vertices is large.     

奇异P-Laplace方程存在正解的充分必要条件

讨论了一维奇异P-Laplace方程{φp（u′））′ f(t,u)=0,t∈（0,1)；u(0=u(1)=0存在C^1[0，1]或C[0，1]正解的一个充分必要条件．用到的方法主要有上下解方法和Schaude，不动点定理．     

关于余直径的一个充分条件

本文给出了一个关于长圈和长路的新的充分条件．主要结果是：设在３－连通图Ｇ中，任一对距离为２的顶点ｕ，ｖ，都满足ｍａｘ｛ｄ（ｕ），ｄ（ｖ）｝≥ｍ/２，那么ｄ^＊（Ｇ）≥ｍｉｎ｛ｎ－１，ｍ－２｝．     

一类广义凸函数的充要条件

本文给出了广义梯度的广义单调性定义,借助这些定义给出了一类广义凸函数的充要条件。     

二元函数梯度存在的一个充要条件

本文首先拓展了二元函数方向导数的计算公式,并给出了该公式成立的充分必要条件,在此基础上也给出了二元函数梯度存在的一个充要条件.     

Mi群的一个充分条件

In this note, we give a sufficient condition for Mi-group. In particular, we show that if a finite group G is the semidirect product of two subgroups with coprime orders, in which one is a Sylow tower group and its Sylow subgroups are all abelian, and the other is an Mi-group and all of its proper subgroups are also Mi-groups, then G is an Mi-group.     

A Sufficient Condition for Mi-groups

In this note, we give a sufficient condition for Mi-group. In particular, we show that if a finite group G is the semidirect product of two subgroups with coprime orders, in which one is a Sylow tower group and its Sylow subgroups are all abelian, and the other is an Mi-group and all of its proper subgroups are also Mi-groups, then G is an Mi-group.     

Asymptotic Enumeration of Convex Polygons

A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice ^dtaking at least one step in every dimension. A polygon on ^dis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonsp_{n, d}of length 2nwithd(n)→∞ is asymptoticallywherer=r(n, d) is the unique solution ofr coth r=2n/d−1andb(r)=d(r coth r−r²/sinh² r). The convergence is uniform over alld?ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1−d⁻¹)^2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon.     

切锥包含的一个充分条件

本文用Ekeland，L变分原理及分析中的几何方法研究了抽象的广义方向导数和广义梯度中的切锥包含问题，给出了一个切锥包含定理．     

A Necessary and Sufficient Condition for the Existence of a Heterochromatic Spanning Tree in a Graph

We prove the following theorem. An edge-colored (not necessary to be proper) connected graph G of order n has a heterochromatic spanning tree if and only if for any r colors (1≤r≤n−2), the removal of all the edges colored with these r colors from G results in a graph having at most r+1 components, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors.     

二次系统存在抛物线奇异环的充要条件

In this paper . we obtained a necessary and sufficient condition for the existence of parabolic separatrix cycles in a quadratic system, and drawed its all possible phase-portraits. Further we obtain an accurate parameter interval that a Ⅲ type system exist ltmit cycles.     

一类四阶奇异微分方程正解存在的充分必要条件

通过Green函数给出了一对具体的上下解,对一类奇异微分方程边值问题做了研究,得到正解存在的充分必要条件.     

一类奇异次线性边值问题正解存在的充分必要条件

本文研究一类奇异次线性边值问题正解的存在性，得到Ｃ［０，１］正解和Ｃ１［０，１］正解存在的充分必要条件，也得到正解的唯一性．     

Circle Numbers of Regular Convex Polygons

The circle number function is extended here to regular convex polygons. To this end, the radius of the polygonal circle is defined as the Minkowski functional of the circumscribed polygonal disc, and the arc-length of the polygonal circle is measured in a generalized Minkowski space having the rotated polar body as the unit disc.