极大值函数的一类光滑逼近函数的性质研究

针对极大值函数的一类光滑逼近——凝聚函数,对其作进一步研究.指出凝聚函数的一阶导数对光滑参数取极限时恰好得到极大值函数的一个次梯度,从而凝聚函数不仅可以一致逼近极大值函数,而且该函数富含极大值函数的一阶信息,可很好的刻画极大值函数的一阶特征.进一步,对光滑逼近函数的光滑参数做简单分析,得到的结果揭示了光滑参数的变动对凝聚函数的影响.并分别以正值函数及绝对值函数为例,对所得到的结果给出几何说明.     

三角函数的推广

利用函数方程定义仿正切函数类,再用仿正切函数定义仿正、余弦函数类,并对这些函数类进行研究.所定义的函数类中的函数具有通常三角函数的性质及运算规律,故而可将其作为三角函数的某种推广.通过实例证实.了通常的正切函数其实是仿正切函数的一种解析延拓.     

强一般型代数体函数分担小函数的唯一性（英文）

代数体函数的唯一性理论是值分布论中的重要研究课题.由于代数体函数的多值性,代数体函数的唯一性的研究成果没有亚纯函数那么丰富.亚纯函数分担小函数的唯一性问题已经被彻底解决,代数体函数分担小函数的唯一性问题还值得深入研究.研究了强一般型代数体函数分担小函数的唯一性问题,得到几个唯一性定理,所得结果将亚纯函数分担小函数唯一性定理推广到强一般型代数体函数.     

抽象函数的若干题型及其解题策略

函数是高中数学的主干知识,学好函数知识对完成高中学业有着举足轻重的地位.其中的抽象函数由于其抽象性、隐蔽性、复杂性,在学习时使不少学生倍感困难.抽象函数是指只给出函数的某些性质而未给出函数解析式的函数.一般说来,这类函数大多是根据教材中某些具体函数的性质与结构特征,经过抽象、概括、升华而成的.可以说抽象函数问题是高中函数内容的一大难点,为了突破这一难点,笔者试图以抽象函数的若干题型的分析解答,来归纳、总结、提炼其解题策略,以求对同学们有所帮助.     

一种改进的Heaviside函数与Dirac函数的C-V图像分割研究

C-V模型中Heaviside函数和Dirac函数正则化逼近影响对目标图像的分割,根据Heaviside函数和Dirac函数的性质,提出了新的正则化Heaviside函数和Dirac函数.首先分析了C-V模型中正则化的Heaviside函数和Dirac函数在图像分割中所起的作用,在此基础上提出了新的正则化的Heaviside函数和Dirac函数,改进了C-V模型.实验结果表明,运用正则化的Heaviside函数和Dirac函数的图像分割效果较好.     

解抽象函数问题的三大策略

函数是中学数学的重点内容,而抽象函数问题又是函数内容的难点之一.抽象函数是指没有给出具体的函数解析式或图像,但给出了函数满足的一部分性质或运算法则.由于此类函数问题     

带等式约束的光滑优化问题的一类新的精确罚函数

罚函数方法是将约束优化问题转化为无约束优化问题的主要方法之一. 不包含目标函数和约束函数梯度信息的罚函数, 称为简单罚函数. 对传统精确罚函数而言, 如果它是简单的就一定是非光滑的; 如果它是光滑的, 就一定不是简单的. 针对等式约束优化问题, 提出一类新的简单罚函数, 该罚函数通过增加一个新的变量来控制罚项. 证明了此罚函数的光滑性和精确性, 并给出了一种解决等式约束优化问题的罚函数算法. 数值结果表明, 该算法对于求解等式约束优化问题是可行的.     

高考分段层出不穷

所谓分段函数即对于自变量x的不同取值范围时,有着不同的对应法则.它是一个函数,而不是多个函数.分段函数的定义域是各段自变量x的并集,值域是各段函数值对应的并集.由于部分学生对分段函数的认识既肤浅又模糊,以致于解题时常常出错或束手无策.     

基于二次多项式的积极集光滑化max函数及其在无约束minimax问题中的应用

本文基于分段二次多项式方程,构造了一种积极集策略的光滑化max函数.通过给出与光滑化max函数相关的分量函数指标集的直接计算方法,将分段二次多项式方程转化为一般二次多项式方程.利用二次多项式方程根的性质,给出了该光滑化max函数的稳定计算策略,证明了其具有一阶光滑性,其梯度函数具有局部Lipschitz连续性和强半光滑性.该光滑化max函数仅与函数值较大的分量函数相关,适用于含分量函数较多且复杂的max函数的问题.为了验证其效率,本文基于该函数构造了一种解含多个复杂分量函数的无约束minimax问题的光滑化算法,数值实验表明了该光滑化max函数的可行性及有效性.     

函数S-粗集与规律双向分离

函数S-粗集(function singular rough sets)是用R-函数等价类定义的,函数是一个规律,函数S-粗集具有规律特征.函数S-粗集推广了Z.Pawlak粗集.利用函数S-粗集,给出规律生成,规律分离的讨论,提出规律分离定理.给出的结果在投资分险规律估计中得到了应用.     

与GA-凸函数有关的两个函数

引入两个与GA-凸函数有关的函数,研究了它们的凸性、单调性及相互关系,获得了已有文献关于GA-凸函数的Hadamard型不等式的加细,也得到一些新的不等式.     

The representation of the generalized linear Euler sums with parameters

In this paper, by using residue method, we obtain the representations of some basic linear generalized Euler sums with parameters. Based on the linear generalized Euler sums with parameters, some new Euler sums are obtained and expressed in the closed forms. When the parameters of new Euler sums take special values, we can get some usual expressions of Euler sums. Moreover, the integrals of many special functions can be expressed as the Euler sums given in this paper.     

Multistability of neural networks with discontinuous activation function

In this paper, the multistability is studied for two-dimensional neural networks with multilevel activation functions. And it is showed that the system has n² isolated equilibrium points which are locally exponentially stable, where the activation function has n segments. Furthermore, evoked by periodic external input, n² periodic orbits which are locally exponentially attractive, can be found. And these results are extended to k-neuron networks, which is really enlarge the capacity of the associative memories. Examples and simulation results are used to illustrate the theory.     

The representation of Euler sums with parameters

In this paper, by choosing different kernel functions and base functions, we obtain some Euler sums with parameters. Moreover, we also obtain the new Euler sums with parameters by differentiating, limiting and elementary arithmetic. Thus, more Euler sums with parameters can be obtained. Furthermore, some Euler sums given in this paper are closed forms.     

Bilinear embedding for real elliptic differential operators in divergence form with potentials

We present a simple Bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials. The constants are dimension-free. The p-range of applicability of this estimate is (1,∞) for any real accretive (nonsymmetric) matrix A of coefficients.     

Some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables

Several inequalities for differentiable convex, wright-convex and quasi-convex mapping are obtained respectively that are connected with the celebrated Hermite-Hadamard integral inequality. Also, some error estimates for weighted Trapezoid formula and higher moments of random variables are given.     

Playability with and without capture

A two-player qualitative differential game with a given target is considered. The first player is assumed to have two objectives: to drive the system to the target in finite time (playability) and then to keep it there indefinitely (playability with capture). Both are to be accomplished against all possible control actions of the second player. Playability without capture implies that the second player cannot avoid the target, but can prevent capture by choosing controls to drive the system out of the target and perhaps keep it out indefinitely (playability with escape) against all possible control actions of the first player. Sufficient conditions for playability with and without capture and playability with escape are derived. Means of determining states from which each case may be attained are examined. Candidates for capturing and escape subtargets, to be used in conjunction with the sufficient conditions developed here, may be obtained from the conditions shown to be necessary on boundaries of the subtargets. An example closes the paper.This research was supported by a University of Queensland Travel Grant.     

Boundary integral equations on the sphere with radial basis functions: error analysis

Radial basis functions are used to define approximate solutions to boundary integral equations on the unit sphere. These equations arise from the integral reformulation of the Laplace equation in the exterior of the sphere, with given Dirichlet or Neumann data, and a vanishing condition at infinity. Error estimates are proved. Numerical results supporting the theoretical results are presented.     

目标属性为肯定和否定的多目标决策问题解的有效性条件

本文建立了一类目标属性为肯定和否定的多目标决策模型，定义了相应的有效性。并在此基础上，讨论了有效性条件。     

带新NCP函数的Lagrangian乘子方法

在经营管理、工程设计、科学研究、军事指挥等方面普遍存在着最优化问题,而实际问题中出现的绝大多数问题都被归纳为非线性规划问题之中。作为带等式、不等式约束的复杂事例,最优化问题的求解向来较为繁琐、困难。适当条件下,非线性互补函数(NCP)可以与约束优化问题相结合,其中NCP函数的无约束极小解对应原约束问题的解及其乘子。本文提出了一类新的NCP函数用于解决等式和不等式约束非线性规划问题,结合新的NCP函数构造了增广Lagrangian函数。在适当假设条件下,证明了增广Lagrangian函数与原问题的解之间的一一对应关系。同时构造了相应算法,并证明了该算法的收敛性和有效性。