关于球面上的Lagrange插值

1引 言 单位球面上的插值问题一直是三元插值问题中比较受关注的部分.近年来,球面上的 Lagrange插值问题已经得到了很好地解决.例如[1]中给出了构造单位球面上的Lagrange 插值适定结点组的一种方法：添加圆周法.[2]和[3]中研究了单位球面上的多项式插值问题,给出了构造单位球面上的插值适定结点组的另外两种方法.     

构造球面逼近算子的一种方法

借助于经典球面分析的Bochner-Riesz平均,Cesàro平均及有关球调和多项式的Gauss积分公式构造出了两类球面平移算子,并且以K-泛函为工具给出了逼近的上界估计.     

关于球面上的一种Hermite插值

本文旨在提出单位球面上的一种Hermite插值格式.为此,本文首先研究了沿球面同轴圆周组上的Hermite插值问题,给出了三种适定的插值泛函组.然后研究了球面上的Hermite插值问题,给出了球面上Hermite插值的一种叠加插值法,即添加圆周组法.进一步将二者结合,导出了一类球面上Hermite插值的适定插值泛函组.为了说明这类适定插值泛函组的构造方法,在本文最后还给出了构造球面上低次Hermite适定插值泛函组的一些具体例子和数值算例.     

球到球的标准等距极小浸入的构造

关于球到球的标准等距极小浸入,M.Docarmo 和 N.Wallach 在[1]中已证明了用球面调和函数可以构造出来.即在对应于某一特征值的球面调和函数空间里寻找一个正交基.但找到的基不一定是正交的,需将其正交化;而内积又是通过球面上的积分定义的,所以正交化过程的计算量非常大.本文将用球面带调和函数的性质,给出一种找基及其正交化的简便方法.     

关于四元射影空间的实超曲面

本文研究四元射影空间的实超曲面.从不同的角度刻划了由 Lawson 构造的一类广义超球面.     

不带Gilbert项的Landau-Lifshitz方程的一些精确爆破解（英文）

本文构造了柱坐标下多维Landau-Lifshitz方程的一些爆破解.在这些解中,存在两个球面S~2上有限初始能量的爆破解.对于非均质的各向同性的Landau-Lifshitz方程,给定有限能量的光滑初始条件,解能否发展出有限能量球面上的爆破解还是不清楚的.文章中的例子说明这种情形的爆破是可以发生的.     

空间形式中具有两个线性相关平均曲率函数的超曲面

在空间形式中, 我们构造了一类泛函, 其临界点包括极小与r 极小超曲面. 给出了临界超曲面的代数、微分和变分刻画. 我们证明了Simons 类不存在定理: 在单位球面中不存在稳定的临界超曲面. 同时证明了Alexandrov 类存在性定理: 在欧氏空间中球面是唯一的稳定的临界超曲面.     

具有时变时滞惯性四元Hopfield神经网络反周期解的动力学研究

周期性、反周期性和概周期性是时变神经网络的重要动态行为特性.本文在不将所研究的神经网络分解为实值系统的情况下,根据重合度理论中的延拓定理和不等式技巧,通过构造不同于现有平衡点稳定性研究的李雅普诺夫函数,研究了一类具有变时滞的惯性四元Hopfield神经网络的反周期解的动力学问题,给出了上述神经网络反周期解存在的一个新的判别条件.并通过构造李雅普诺夫函数论证了上述神经网络反周期解的指数稳定性.     

常曲率球面到单位球面的等距极小浸入与嵌入

本文主要估计了球面到单位球面的等距极小浸入的覆盖层数，从而导出一些有关嵌入的结果。同时，在本文证明过程中，给出了一个构造指定覆盖层数的等距极小浸入的方法．     

带马尔可夫跳的随机Hopfield神经网络的以分布渐近稳定性

讨论了带马尔可夫跳的随机Hopfield神经网络的以分布渐近稳定性.通过构造合适的Lyapunov函数,获得了判定带马尔可夫跳的随机Hopfield神经网络的以分布渐近稳定性的充分条件.     

The estimate for approximation error of spherical neural networks

Compared with planar hyperplane, fitting data on the sphere has been an important and an active issue in geoscience, metrology, brain imaging, and so on. In this paper, with the help of the Jackson‐type theorem of polynomial approximation on the sphere, we construct spherical feed‐forward neural networks to approximate the continuous function defined on the sphere. As a metric, the modulus of smoothness of spherical function is used to measure the error of the approximation, and a Jackson‐type theorem on the approximation is established. Copyright © 2011 John Wiley & Sons, Ltd.     

球面带形平移网络逼近的Jackson定理

研究了球面带型平移网络逼近阶用球面调和多项式的最佳逼近及光滑模的刻画问题．借助于球调和多项式的最佳逼近多项式和Riesz平均构造出了单位球面Sq上的带形平移网络,并建立了球面带形平移网络对Lp(Sq)中函数一致逼近的Jackson型定理．所得结果表明球面带形平移网络可以达到球调和多项式的逼近阶．     

Nonlinear dynamic system identification using recurrent neural network with multi-segment piecewise-linear connection weight

This paper introduces a new concept of the connection weight to the standard recurrent neural networks—Elman and Jordan networks. The architecture of the modified networks is the same as that of the original recurrent neural networks. However, unlike the original recurrent neural networks whose connection weight is a single real number, in the modified networks the weight of each connection is multi-valued, depending on the value of the input data involved. The backpropagation learning algorithm is also modified to suit the proposed concept. The modified networks have been benchmarked against the feedforward neural network and the original recurrent neural networks. The experimental results on twelve benchmark problems show that the modified networks are clearly superior to the other three methods.     

Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field

Fractional order quaternion-valued neural networks are a type of fractional order neural networks for which neuron state, synaptic connection strengths, and neuron activation functions are quaternion. This paper is dealing with the Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. The fractional order quaternion-valued neural networks are separated into four real-valued systems forming an equivalent four real-valued fractional order neural networks, which decreases the computational complexity by avoiding the noncommutativity of quaternion multiplication. Via some fractional inequality techniques and suitable Lyapunov functional, a brand new criterion is proposed first to ensure the Mittag-Leffler stability for the addressed neural networks. Besides, the combination of quaternion-valued adaptive and impulsive control is intended to realize the asymptotically synchronization between two fractional order quaternion-valued neural networks. Ultimately, two numerical simulations are provided to check the accuracy and validity of our obtained theoretical results.     

Cubature Formula for Spherical Basis Function Networks

Some mathematical models in geophysics and graphic processing need to compute integrals with scattered data on the sphere. Thus cubature formula plays an important role in computing these spherical integrals. This paper is devoted to establishing an exact positive cubature formula for spherical basis function networks. The authors give an existence proof of the exact positive cubature formula for spherical basis function networks, and prove that the cubature points needed in the cubature formula are not larger than the number of the scattered data.     

New results for global stability of a class of neutral-type neural systems with time delays

This paper studies the global convergence properties of a class of neutral-type neural networks with discrete time delays. This class of neutral systems includes Cohen–Grossberg neural networks, Hopfield neural networks and cellular neural networks. Based on the Lyapunov stability theorems, some delay independent sufficient conditions for the global asymptotic stability of the equilibrium point for this class of neutral-type systems are derived. It is shown that the results presented in this paper for neutral-type delayed neural networks are the generalization of a recently reported stability result. A numerical example is also given to demonstrate the applicability of our proposed stability criteria.     

Artificial neural networks in bankruptcy prediction: General framework and cross-validation analysis

In this paper, we present a general framework for understanding the role of artificial neural networks (ANNs) in bankruptcy prediction. We give a comprehensive review of neural network applications in this area and illustrate the link between neural networks and traditional Bayesian classification theory. The method of cross-validation is used to examine the between-sample variation of neural networks for bankruptcy prediction. Based on a matched sample of 220 firms, our findings indicate that neural networks are significantly better than logistic regression models in prediction as well as classification rate estimation. In addition, neural networks are robust to sampling variations in overall classification performance.     

离散时间型复值神经网络的全局指数周期性

复值神经网络是神经网络的一个分支,也是最近几年快速发展的一个领域,在图像处理、模式识别、联想记忆等方面有广泛的应用．目前,对于复值神经网络动力学方面的研究主要集中在稳定性上,对于离散时间型复值神经网络周期性的研究还几乎没有．首先将连续时间型复值神经网络模型离散化得到离散时间型复值神经网络模型,然后利用M矩阵理论、不等式技巧和Lyapunov方法,获得了全局指数周期性的一个充分条件,最后给出的具有仿真的数值例子验证了获得结果的有效性.     

一类递归小波神经网络的稳定性研究

在小波神经网络（WNNs）和递归神经网络（RNNs）的基础上,提出了一类递归小波神经网络（RWNNs）模型,它具有两种网络模型的优点A·D2根据Liapunov渐近稳定理论,对该模型的渐近稳定性进行了研究,并给出了相关的定理和公式．仿真结果表明该模型对非线性动态系统有良好的辨识效果．