凸多边形的最优切割策略

本文研究的是在一个平面区域内切割出一个预定的凸多边形的最优策略问题 .首先应用动态规划建立模型 ,然后 ,证明了优化变换的两个准则 ,最后 ,我们对极先切割边进行了讨论 ,得出了简明的最优切割策略     

一个网格多边形问题

在网格上画凸多边形,凸多边形的顶点都在格点上,当然这样的凸多边形可以画出无数个. 如果我们限定凸多边形的边只能是一个网格的一条边或是它的一条对角线,也就是说凸多边形的边只有两种长度1和√2,那么能画出多少种不同的凸多边形呢?     

凸多边形置入问题求解的近似算法

本文在引进了挤压弹性势能另一等价定义的基础上,把求凸多边形间的嵌入深度转化为求最小碰撞距离.文中着重研究了凸多边形初始碰撞问题,得到了求嵌入深度的快速算法,据此构造出了求解凸多边形置入问题的快速近似算法.应用本文所提出的算法,不仅可解决求凸多边形间嵌入深度的实质性困难,而且可实现凸多边形几何布局问题求解自动化.     

与给定线段相交的定长线段的运动测度公式及其应用

本文研究了凸多边形的运动测度的表达式,利用线段将已知其包含测度的凸多边形分划成其它的凸多边形,通过计算出与给定线段相交的定长线段的运动测度公式,从而得到某些凸多边形的运动测度的具体表达式,并把它应用到几何概率问题中,获得一个新的几何概率结果.     

关于平面凸多边形的一个Bonnesen型不等式

本文探索了关于平面凸多边形的Bonnesen型不等式.利用分析方法,先构造一个解析函数的不等式,进而得到了一个关于平面凸多边形的Bonnesen型不等式.     

两凸多边形交集面积的计算机算法

为了求解两个平面凸多边形的交集面积,对原有的六步法进行了改进.在求交点的过程中,通过判断线段之间是否排斥和跨立来确定线段之间是否相交;在交集顶点的排序过程中,通过各点相对于起始点的方位角余弦值大小对交集顶点进行排序.整个求面积的过程易于实现,时间复杂度为O(mn),其中,m和n分别为两个凸多边形的顶点数目.     

真假四边形市长风波

在一个名叫几何的王国里,有一座繁荣的小城—凸多边形城,这里住着各种各样的多边形.城中有一个四边形家族,成员非常多,如不规则的四边形一家,平行四边形一家,梯形一家等等.其中平行四边形一家地位特殊,因为这个凸多边形城市的市长便是那大名鼎鼎、无人不晓的平行四边形.平行四边形处事公平,执政已有多年,在它的带领下,凸多边形城市才发展成为一片欣欣向荣的景象.平日里,平行四边形市长最喜     

数学奥林匹克讲与练（23）

177．用不在多边形内部相交的对角线将凸多边形割分成若干三角形．求证：可用三种颜色给凸多边形的顶点着色,使割分所得的每个三角形的三个顶点的颜色均不相同．     

最短内接折线问题

本文对具有固定边界点的最短内接折线问题、退化的问题和非凸多边形的问题进行了讨论,并给出了有效的组合优化求解方法.我们还提出了通过固定内点的最短内接折线问题,并对特殊情况给出了一些结果.     

一元选择题两组

本专栏提供“四里挑一”的选择题,要求口答,故称一望而解.如果动笔演算,就得自省其“双基”不过硬了.一、凸多边形选择题(初中)1.一个凸多边形除了一个内角之外,其余各角之和为1500°。那么这个内角为     

基于光滑化方法求解非线性l1问题

讨论了求解非线性l1问题的一种新的光滑函数法.通过对非线性l1问题模型的转化,将该问题化为一个不可微优化问题,据此提出了基于BFGS迭代的非线性l1问题的光滑函数法,介绍了非线性l1问题的光滑函数的有关性质、算法步骤及其收敛性.数值仿真显示了提出的光滑函数方法可以避免数值计算的溢出,具有一定的有效性.     

Measuring Stiffness

Many problems have been proposed for the testing and evaluation of mathematical software for the solution of the initial value problem for a system of ordinary differential equations. The stiffness of a given problem is of great significance for testing, but the concept is at best somewhat vague and the few quantitative measures of stiffness being used are of limited value. A relatively precise and quantitative way is proposed to answer the questions: How does the stiffness vary in the course of the integration of a given problem? Is a given problem A stiffer than a given problem B, and if so, by how much?     

需求价格弹性与边际收益研究

研究特定需求价格弹性条件下需求函数的确定问题和特定边际收益条件下需求函数的确定问题,并给出了有关的计算公式.研究结果表明,为了实现既定的需求价格弹性或边际收益,对应的需求函数有无穷多个.     

On Shunkov groups with a strongly embedded subgroup

The problem of reconstructing a locally Euclidean metric on a disk from the geodesic curvature of the boundary given in the sought metric is considered. This problem is an analog and a generalization of the classical problem of finding a closed plane curve from its curvature given as a function of the arc length. The solution of this problem in our approach can be interpreted as finding a plane domain with the standard Euclidean metric whose boundary has a given geodesic curvature.     

基于模糊收益率的组合投资模型

本文考虑了收益率为模糊数的投资组合选择问题,利用模型约束简化方差约束,建立了投资组合选择的模糊线性规划模型,然后引进模糊期望把模糊线性规划问题化为普通参数线性规划问题,最后给出了一个数值算例.     

Solving a chemical batch scheduling problem by local search

In this paper the following chemical batch scheduling problem is considered: a set of orders has to be processed on a set of facilities. For each order a given amount of a product must be produced by means of chemical reactions before a given deadline. The production consists of a sequence of processes whereby each process has to be performed by one facility out of a given subset of facilities allowed for this process. The processing times depend on the choice of the facility and the processing is done in batch mode with given minimum and maximum sizes. The problem is to assign the processes to the facilities, splitting them into batches, and scheduling these batches in order to produce the demands within the given deadlines. For the scheduling part of the problem we present an approach based on the following steps. First, a procedure to calculate the minimum number of batches needed to satisfy the demands is presented. Based on this, the given problem is modeled in two different ways: as a general shop scheduling problem with set-up times or as scheduling problem with positive time-lags. Finally, a two-phase tabu search method is presented which is based on the two different formulations of the problem. The method is tested on some real world data. This revised version was published online in June 2006 with corrections to the Cover Date.     

Synthesis of minimal linear stabilizers

We consider the stabilizer synthesis problem for linear stationary control dynamical systems; attention is mainly focused on the investigation of possibilities to reduce the order of such stabilizers. The problem is considered in two settings. The first setting deals with the construction of stabilizers of given order (in particular, the minimum possible order); no conditions are imposed on the spectrum of the closed system. For scalar (single-input-single-output) control systems, we suggest an approach to the solution of this problem and obtain necessary and sufficient conditions for the existence of a stabilizer of a given order. The second problem is that of synthesizing a stabilizer of minimum order with given dynamic properties (a given spectrum or a given distribution of the spectrum of the closed system, in particular, with a guaranteed convergence rate of the closed system). For this problem, we suggest two approaches that permit one to obtain an upper bound for the dimension of such stabilizers.     

Partial inverse maximum spanning tree in which weight can only be decreased under $$l_p$$-norm

The maximum or minimum spanning tree problem is a classical combinatorial optimization problem. In this paper, we consider the partial inverse maximum spanning tree problem in which the weight function can only be decreased. Given a graph, an acyclic edge set, and an edge weight function, the goal of this problem is to decrease weights as little as possible such that there exists with respect to function containing the given edge set. If the given edge set has at least two edges, we show that this problem is APX-Hard. If the given edge set contains only one edge, we present a polynomial time algorithm.     

On Oka’s extra-zero problem and examples

After the solution of Cousin II problem by Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. By the secondly named author, some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was given. The purpose of the present paper is to give a complete solution of this problem with examples and some new questions.     

葡萄酒质量的评价模型

针对2012年“高教社杯”全国大学生数学建模竞赛A题“葡萄酒的评价”进行建模,首先介绍了问题的背景;然后分析了对几个问题的理解,并给出部分问题的具体模型和解答结果;接着分析了竞赛论文中普遍存在的问题。最后提出值得进一步研究的几个问题。