发现空间目标并定位的模型

针对在一个圆柱体内用最少的红球、蓝球发现黄球的问题,采用多个椭圆覆盖圆柱顶面的方法,设计出利用18个球覆盖的方案,叙述了该方案的设计思路,并严格证明出方案的可行性.针对用最少的红球、蓝球发现并定位黄球的问题,设计出36个球的方案,严格证明出方案的可行性.     

发现并定位黄球

本文讨论了如何设置红球和蓝球的数量和位置,发现圆柱区域内的黄球并进行定位的问题.我们考虑了一对红球蓝球发现黄球并定位的问题,在此基础上进行扩展,基本解决了黄球的发现并定位的问题.在静止黄球发现问题中,采用了正三角形扩展和正六边形扩展两种方法.在静止黄球的定位问题中,我们结合正三角形和正六边形运用了旋转法和添点法进行扩展.在运动黄球的定位问题中,讨论了体积概率模型和时间概率模型,给出了两种模型的概率求解公式.在系统协同定位模型中,我们给出了发现定位分步模型和周期系统跟踪模型,其中后者在仿真中实现了大于80%的定位性能,该系统可以简单扩展为多目标快速定位问题.此外,文章讨论了精确测量和颜色切换模型,快速定位问题,多目标定位等问题.     

一道取球试题的拓展

2005年山东高考理科第19题是:袋中有黑球和白球共7个,从中任取2个球都是白球的概率为17,现有甲、乙两人从袋中轮流摸取1球,甲先取、乙后取,然后甲再取…取后不放回,直到两人中有一人取到白球时即终止,每一个球在每一次被取出的机会是等可能的,用ξ表示取球终止时所需要的取球次数.(Ⅰ)求袋中原有白球的个数.(Ⅱ)求随机变量ξ的概率分布.(Ⅲ)求甲取到白球的概率.而2005年浙江高考理科第18题是(部分抄录):袋子A和B中装有若干个均匀的红球和白球,从A中摸出一个红球的概率是13,从B中摸出一个红球的概率是p.(Ⅰ)从A中有放回地摸球,每次摸出一个…     

无穷维赋范线性空间上的填球问题（英文）

受Riesz引理及一个有趣问题"无穷维赋范线性空间上的闭单位球是否可以由有限个开单位球覆盖"的启发,本文得到一个有用的结果,利用这个结果可以给出Kottman定理的一个简单证明及填球数的上界估计.并藉由上界估计考虑了L~p(Ω空间上的填球问题.     

对一道组合题的再分析

翻开《中学生数学》2004年第9月上期第36 页上有王倩茹同学写的一篇《一类排列组合问题 的辨析》一文,为便于说明,现摘出题目及其解答 (取其认为正确的一解)如下: 有3个不同的红球,5个相同的白球,要从 中取出3个球,其中至少有一个红球的取法有多 少种?     

化“隐”为“显”“圆”来完“美”

1 问题提出 在江苏高考中,“圆”作为8个C级要求的知识点之一,是高考必考的知识点.纵观2008年至今的江苏高考方案,有关圆的试题的呈现时明时隐,有时明隐难辨.具体表现为:2008年13题(隐)、18题(明),2009年18题(明),2010年9题(明),2011年14题(明),2012年12题(明),2013年17题(2)(隐),2014年9题(明),2015年10题(明),2016年18题(明).对圆“显性”的考查,学生在求解时难度不大,若题目中“隐性”存在圆,如果不能充分挖掘题中隐含的信息,将圆化“隐”为“显”,则计算往往会非常繁琐,以致难以求解.笔者对圆的定义、性质、方程等方面展开阐述.     

多阶段协同拦截问题的制导方法研究（英文）

研究了多阶段协同拦截问题中多个飞行器分阶段发射协同拦截一个高机动目标.首先,给出了多阶段协同拦截问题的描述.由于测量噪声以及目标的不确定性,用一个动态分布描述目标的位置状态,该分布通过卡尔曼滤波得到.然后,基于最大拦截概率,将制导问题转化为覆盖控制问题以获得拦截器的期望位置.最后,基于传统的优化方法,给出了每个飞行器的制导控制律.     

不确定信息多目标线性优化的鲁棒方法

研究不确定信息的多目标线性优化问题,其数据不能精确给出但是属于一个给定的集合.首先,采用鲁棒方法把该问题转化为一个确定的多目标优化问题.然后,给出此问题解存在的充分条件.最后,通过实例验证了用鲁棒方法解决不确定信息的多目标线性优化问题的有效性.     

游戏公平性问题之慎思

2006年中考中这样一类概率问题成为命题的热点,计算事件发生概率的大小,判断游戏公平与否;若不公平,修改规则使游戏公平.对于此类问题中的很多题目,笔者心存疑虑:参考答案中所谓的“公平的游戏”,真的公平吗?下面结合具体例子来谈谈自己的一些看法.例1(2006年山西省临汾市中考题)小明和小乐做摸球游戏.一只不透明的口袋里放有3个红球和5个绿球,每个球除颜色外都相同,每次摸球都将袋中的球充分搅匀,从中任意摸出一个球,记录颜色后再放回,若是红球小明得3分,若是绿球小乐得2分,游戏结束时得分多者获胜.1)你认为这个游戏对双方公平吗?2)若你认…     

一位名师一道题

分析与解　这是一个操作型问题 ,而且操作的模式不同 ,每一步操作有多种选择 ,在处理这类问题时 ,应该抽出各操作之间的相同点 ,建立一个在操作过程中的不变量 .我们给不同颜色的球赋值 (这是寻找操作不变量时常用的方法 ) ,设每个白球、绿球、红球的分值分别为 1 ,2 ,3 .考虑盒子中所有球的分值的总和F ,则F的值在模 4的意义下 ,每次操作结果不变 .(a)注意到 ,最初F =2 0 0 0≡ 0(mod 4) ,于是 ,设最后剩下的 3个球中白、绿、红球数分别为x、y、z,则　x + y +z=3 ,且　x + 2 y+ 3z≡ 0 (mod 4) .所以　y+ 2z≡ 1 (mod 4) ,从而　y≠ 0 (…     

Variable target value subgradient method

Polyak's subgradient algorithm for nondifferentiable optimization problems requires prior knowledge of the optimal value of the objective function to find an optimal solution. In this paper we extend the convergence properties of the Polyak's subgradient algorithm with a fixed target value to a more general case with variable target values. Then a target value updating scheme is provided which finds an optimal solution without prior knowledge of the optimal objective value. The convergence proof of the scheme is provided and computational results of the scheme are reported.Most of this research was performed when the first author was visiting Decision and Information Systems Department, College of Business, Arizona State University.     

Numerical solution of an optimal investment problem with proportional transaction costs

This paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite-horizon optimal investment problem with proportional transaction costs. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable. The proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain, a characteristics method for time discretization to deal with the large gradients of the solution, a Newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle (free boundary) feature. Moreover, piecewise linear Lagrange finite elements for spatial discretization are considered. Numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in Dai and Yi (2009) [6].     

Bike sharing systems: Solving the static rebalancing problem

This paper deals with a new problem that is a generalization of the many to many pickup and delivery problem and which is motivated by operating self-service bike sharing systems. There is only one commodity, initially distributed among the vertices of a graph, and a capacitated single vehicle aims to redistribute the commodity in order to reach a target distribution. Each vertex can be visited several times and also can be used as a buffer in which the commodity is stored for a later visit. This problem is NP-hard, since it contains several NP-hard problems as special cases (the TSP being maybe the most obvious one). Even finding a tractable exact formulation remains problematic.This paper presents efficient algorithms for solving instances of reasonable size, and contains several theoretical results related to these algorithms. A branch-and-cut algorithm is proposed for solving a relaxation of the problem. An upper bound of the optimal solution of the problem is obtained by a tabu search, which is based on some theoretical properties of the solution, once fixed the sequence of the visited vertices. The possibility of using the information provided by the relaxation receives a special attention, both from a theoretical and a practical point of view. It is proven that to build a feasible solution of the problem by using the one obtained by the relaxation is an NP-hard problem. Nevertheless, a tabu search initialized with the optimal solution of the relaxation often shows that it is the optimal one.The algorithms have been tested on a set of instances coming from the literature, proving their effectiveness.     

允许长度估计误差的SBH最优重构问题及其算法

本文讨论了允许长度估计误差和杂交错误的更实际SBH(Sequencing by Hybridization)最优重构问题.通过对SBH谱集中k-tuple之间的相关信息的分析和最优重构性质的讨论,我们得到若干非最优解的删除法则和最优解的判定法则,并获得了一个能够极大地减少最优解重构随意性的动态规划计算方法.由此,我们给出了该SBH问题的一个新重构算法.该算法既允许SBH谱集含有一般杂交实验中可能出现的探针错配所产生的正错误,也允许目标DNA序列长度有估计误差,所以本文的算法具有更一般的适应性和实用性.模拟计算结果表明我们的算法也是十分有效的(即使在谱集有多达100%的正错误情况).     

A Survey of the Coupon Collector’s Problem with Random Sample Sizes

This paper surveys the coupon collector’s waiting time problem with random sample sizes and equally likely balls. Consider an urn containing m red balls. For each draw, a random number of balls are removed from the urn. The group of removed balls is painted white and returned to the urn. Several approaches to addressing this problem are discussed, including a Markov chain approach to compute the distribution and expected value of the number of draws required for the urn to contain j white balls given that it currently contains i white balls. As a special case, E[N], the expected number of draws until all the balls are white given that all are currently red is also obtained.      

A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB)

In this paper, we outline an impulse stochastic control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB) assuming the policyholder is allowed to withdraw funds continuously. We develop a numerical scheme for solving the Hamilton–Jacobi–Bellman (HJB) variational inequality corresponding to the impulse control problem. We prove the convergence of our scheme to the viscosity solution of the continuous withdrawal problem, provided a strong comparison result holds. The scheme can be easily generalized to price discrete withdrawal contracts. Numerical experiments are conducted, which show a region where the optimal control appears to be non-unique.     

实用下料优化问题模型建立及解法

“下料问题(cuttingstockproblem)”是把相同形状的一些原材料分割加工成若干个不同规格大小的零件的问题,此类问题在工程技术和工业生产中有着重要和广泛的应用.本文首先以材料最省为原则建立模型,采用分层基因算法模型求解出模型的解,若此结果不符合时间限制条件,则通过以客户时间需求为第一目标的分组抽样模型处理后,再借助分层基因算法给出该模型的最优解.     

A Fast and Stable Algorithm for Linear Parabolic Partial Differential Equations

In the paper, a two-grid finite element scheme is discussed for distributed optimal control governed by elliptic equations. With this new scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.