三维空间的分划问题

华罗庚教授在文[1]中提出如下分划问题：空间有n个平面，其中没有两个平面平行，没有三个平面相交于同一条直线，也没有四个平面过同一个点．求证：它们分．我们发现，正三棱柱的三个侧面所在平面完全满足上述条件，按上述公式计算应有V3=1个交点，E3=6段交线，S3=12片面，把空间完成F3=8分．可是实际上，这三个平面却没有交点，只有三条交线，9片面，把空间分成7块（如图）．因此这个分划问题有误．很自然地，我们会想到：问题1要得到华教授所述结论，n个平面应满足怎样的条件？问题2若依华教授所列条件，正确的点、线、面、块数该是多少…     

两类圆锥曲线的两个性质定理

笔者从高等数学的两个问题出发，研究了该问题的特性及结论．由此联想到中学数学中的一些几何问题，对此研究分析，得到了两个具有普遍性的几何命题．这两个命题对于在解题中启发学生思维，提供解题思路、减化运算量方面大有好处．现叙述如下：     

正三角形与其同心圆的若干性质

笔者近日发现数学通报《数学问题》栏目中的两个问题1283(2000．11期)、1287(2000．12期)有惊人的相似之处，这引起了笔者的兴趣．相似的问题预示着应有相似的解法．实际上两个问题的提供者给出的解答方法就是相同的，都是引进辅助角借助三角变换完成解答或证明，而过程繁琐，令读者望而生畏，失去阅读下去的勇气和兴趣。     

用截面平分四面体的一点补充

文[1]提出了两个有意思的问题：1)如何作一直线，使其平分已知三角形的面积，2)如何作一平面，使其平分已知四面体的体积．但文中对第一个问题只给出了中线和平行于底的平行线两种情况；对第二个问题也仅给出了对边棱中点构成的平面．其实这两个问题应该有很多方法，下面给出另外一种简单方法作为其补充．     

正整数的组成

正整数的组成曹松峰编译（河南省焦作市教委教研室４５４１５０）我们的讨论从解决下列三个问题开始．问题１．求出方程ｘ＋ｙ＋ｚ＋ｗ＝７的所有非负整数解的个数．问题２．五个书店共订购某种课本４００册，每个订户至少订６５册．有多少种订购方法？问题３．某个游乐场...     

关于乘法逆特征值问题有解的充分条件

1．引言本文讨论如下乘法逆特征值问题[1]有解的充分条件．问题MR．给定nxn实矩阵A—（a＊和n个实数人l，…，An，求实对角矩阵X一山x以xl，…，1＊使得＊A的特征值为人，…，入．关于此问题可解的充分条件，deOliveira［2］，何旭初和戴华［31等给出过一些结果，但由于都是把此问题作为一般代数特征值反问题的特例来处理，没有注意到乘法问题的独特性，因而得到的可解条件比较强．本文根据乘法问题的特点，运用［31，［4］中的技术及拓扑度理论给出一些新的条件，这些条件大大改进了［2，3，5］中的结果．2．主要结果首先弓l进几个记号…     

四阶幻方的几个有趣性质简

正方体是一种常见而且典型的几何模型．立体几何中所研究的很多边角关系都可以在正方体中直观的展示出来，比如很多同学对这样一个问题比较困惑：有没有四个面都是直角三角形的四面体？此问题若从常规角度出发，不易举证．如图1，构造正方体，不难发现，三棱锥A—BCD四个丽都是直角三角形．可见，借助正方体研究问题，可以弥补初学者空间想象能力的不足，给解题提供一定的依据．下面请看几个例子．     

双层线性规划的一个全局优化方法

用线性规划对偶理论分析了双层线性规划的最优解与下层问题的对偶问题可行域上极点之间的关系，通过求得下层问题的对偶问题可行域上的极点，将双层线性规划转化为有限个线性规划问题，从而用线性规划方法求得问题的全局最优解．由于下层对偶问题可行域上只有有限个极点，所以方法具有全局收敛性．     

一对一错号排列的一个公式

排列组合问题非常灵活多变，比较难以把握．例如，四人同室，他们各写一张贺年卡，先集中起来，然后每人从中拿一张别人送出的贺年卡，则四张贺年卡不同的分配方法有多少种？它的原型是下列“小球人盒”问题．将编号为1、2、3、4的4个小球装人编号为1、2、3、4的4个盒中，每盒装1个球，且球与盒的编号不同，问不同的装球方法有多少种．我们把这个问题叫做“一对一错号排列”问题．本问题，由于球与盒的个数不多，用常规法解，难度倒不是很大，但增加球与盒的个数后，情况就不一样了，仍用常规法解，难度将随之增大．因此，对于这类问题如果…     

关于调整气象观测站问题的数学模型

本文讨论了西安市第一届大学生数学模型竞赛题之一——调整气象观测站问题．若用主成份与逐步剔除法解决此问题，似可根据“损失信息量”的概念来确定减少测站个数．通过计算分析，在不损失信息量的条件下，可以去掉两个测站；在损失信息量为０．８５２％的条件下，可以去掉３个测站．这种算法，使节约设站开资，减少测站数量更具科学性．     

Blind source separation with nonlinear autocorrelation and non-Gaussianity

Blind source separation (BSS) is a problem that is often encountered in many applications, such as biomedical signal processing and analysis, speech and image processing, wireless telecommunication systems, data mining, sonar, radar enhancement, etc. One often solves the BSS problem by using the statistical properties of original sources, e.g., non-Gaussianity or time-structure information. Nevertheless, real-life mixtures are likely to contain both non-Gaussianity and time-structure information sources, rendering the algorithms using only one statistical property fail. In this paper, we address the BSS problem when source signals have non-Gaussianity and temporal structure with nonlinear autocorrelation. Based on the two statistical characteristics of sources, we develop an objective function. Maximizing the objective function, we propose a gradient ascent source separation algorithm. Furthermore, We give some mathematical properties for the algorithm. Computer simulations for sources with square temporal autocorrelation and non-Gaussianity illustrate the efficiency of the proposed approach.     

伊藤-泊松型随机微分方程的线性二次控制

本文研究伊藤-泊松型随机微分方程的线性二次控制问题,利用动态规划方法、伊藤公式等技巧,通过解HJB方程,我们得到了随机Riccati方程及另外两个微分方程,求出控制变量,解决了线性二次最优控制最优问题.     

Robust stability for genetic regulatory networks with linear fractional uncertainties

In this paper, the asymptotic stability analysis problem for a class of delayed genetic regulatory networks (GRNs) with linear fractional uncertainties and stochastic perturbations is studied. By employing a more effective Lyapunov functional and using a lemma to estimate the derivative of the Lyapunov functional, some new sufficient conditions for the stability problem of GRNs are derived in terms of linear matrix inequality (LMI). Finally, two numerical examples are used to demonstrate the usefulness of the main results and less conservatism of the derived conditions.     

多层防御模式下武器目标分配决策的群体智能优化算法

武器目标分配(WTA)是军事运筹学中经典的NP完全问题,迄今为止未找到求精确解的多项式时间算法.针对武器数量、布防空间、运行维护成本以及人力资源等多约束下的多层防御WTA问题,采用粒子群优化(PSO)和蚁群优化(ACO)两种群体智能算法求解.给出了PSO和ACO算法实现方案,通过一个算例评估两个算法的性能.结果表明,两种算法都能给出高质量的近似最优解,对求解WTA问题是有效的.PSO在解的质量、算法鲁棒性和计算效率方面均优于ACO.     

Determining an optimum location for an undesirable facility in a workroom environment

This paper deals with the problem of placing an undesirable but necessary piece of equipment, process or facility into a working environment. Locating a piece of equipment that produces contaminants or creates stresses for nearby workers, placing a storage facility for flammable materials or locating hazardous waste in the workroom environment, are all typical examples of the undesirable facility location problem. The degree of undesirability between an existing facility or worker and the new undesirable entity is reflected through a weighting factor. The problem is formally defined to be the selection of a location within the convex region that maximizes the minimum weighted Euclidean distance with respect to all existing facilities. A ‘Maximin’ model is formulated and two solution procedures introduced. A geometrical approach and an algorithmic approach are described in detail. An example is provided for each solution procedure and the computational efficiency of the algorithm is discussed and illustrated.     

On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets

In the paper, we consider a problem of convex Semi-Infinite Programming with an infinite index set in the form of a convex polyhedron. In study of this problem, we apply the approach suggested in our recent paper [Kostyukova OI, Tchemisova TV. Sufficient optimality conditions for convex Semi Infinite Programming. Optim. Methods Softw. 2010;25:279–297], and based on the notions of immobile indices and their immobility orders. The main result of the paper consists in explicit optimality conditions that do not use constraint qualifications and have the form of criterion. The comparison of the new optimality conditions with other known results is provided.     

Satisfactory graph partition,variants, and generalizations

The Satisfactory Partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [M. Gerber, D. Kobler, Algorithmic approach to the satisfactory graph partitioning problem, European Journal of Operational Research 125 (2000) 283–291] and studied further by other authors. In this paper we first review some applications and related problems. Then, we survey structural, complexity, and approximation results obtained for Satisfactory Partition and for some of its variants and generalizations. A list of open questions concludes this survey.     

Extracting post-nonlinear signal with specific kurtosis range

Blind source extraction (BSE) is an important technique to extract a desired source from the mixed signals and the post-nonlinear (PNL) mixture is more realistic model in many situations. In this paper, we address the problem of extracting the source of interest from the PNL mixture. First, the prior knowledge about the desired source, such as its normalized kurtosis range, can be treated as a constraint and incorporated into the contrast function. Therefore, BSE from the PNL mixture can be formulated a constrained optimization problem. Second, the inverse of the unknown nonlinear function is approximated by the multi-layer perceptions (MLP) network because neural network can uniformly approximate any continuous function if there is sufficient number of neurons in the hidden layer. Finally, the source of interest can be extracted from the PNL mixture by minimizing the constrained optimization problem with standard gradient descent method. Extensive computer simulations and experiments demonstrate the validity of our algorithm.     

On the use of Kronecker's algorithm in the generalized rational interpolation problem

Kronecker's algorithm can be used to solve the generalized rational interpolation problem. In order to present the algorithm, rational forms are used here instead of too restrictive rational fractions. The proposed algorithm is reliable as soon as the functionals that characterize the problem satisfy two precise conditions. These conditions are fulfilled in the modified Hermite rational interpolation problem and, as a consequence, in the special case of the Cauchy problem and of the Padé approximation problem. This reliability covers two properties: on one hand, every rational form resulting from the algorithm is a solution of the problem whereas, on the other hand, every solution of the problem is found by the algorithm (with the exception of a possible reduction of the rational form). However, if the algorithm yields a non-reduced rational form, then the corresponding rational fraction is not a solution of the problem.     

On density of transitive operator algebras

In this paper we study the transitive algebra question by considering the invariant subspace problem relative to von Neumann algebras. We prove that the algebra (not necessarily ∗) generated by a pair of sums of two unitary generators of L(F_∞) and its commutant is strong-operator dense in B(H). The relations between the transitive algebra question and the invariant subspace problem relative to some von Neumann algebras are discussed.