直觉模糊集和区间值模糊集的截集、分解定理和表现定理

该文给出了直觉模糊集和区间值模糊集的截集定义,建立了直觉模糊集和区间值模糊集的分解定理和表现定理.首先,将直觉模糊集的截集视为三值模糊集,给出了直觉模糊集的四类截集定义,指出这些截集是模糊集截集概念的推广且与模糊集的截集有完全一样的性质.其次,利用本文给出的截集概念,建立了直觉模糊集的分解定理和表现定理.指出直觉模糊集的每种截集都对应两种分解定理和表现定理,从而建立了直觉模糊集的八种分解定理和八种表现定理.最后,利用直觉模糊集的截集理论,给出了区间值模糊集的截集定义,并建立了区间值模糊集的八种分解定理和表现定理.这些工作为研究直觉模糊集和区间值模糊集建立了理论基础.     

区段定理和回路增益定理

本文提出了新的区段定理和回路增益定理.这两个定理包含和改进了文献[1]中的区段 定理、文献[2]中的钝性(passivity)定理、文献[3]中的一个重新阐述的钝性定理以及文献[1] 中的定理6、4、14.     

交互式进化计算的适应值噪声及收敛鲁棒性

噪声是影响进化计算(evolutionary computation,简称EC)算法性能的一个重要因素.对于传统EC中的噪声,已有许多研究成果,但交互式进化计算(interactive evolutionary computation,简称IEC)的噪声研究成果却较少.首先回顾了传统EC中噪声的定义、来源、类型及各种处理噪声的方法;其次,从IEC的理性用户观点出发,研究了IEC的适应值噪声及收敛鲁棒性.其中,空间的映射关系、个体间的占优关系以及IEC的收敛等是研究收敛鲁棒性的两个定理(强条件定理和弱条件定理)的基础.这两个定理表明,理性用户条件下的噪声不会影响算法全局收敛性.在这两个定理的基础上进一步得出了如下结论:有效的适应度尺度变换是弱条件定理的一部分,IEC中"真"适应值是用户偏好等.并以不满足弱条件定理,即破坏算法收敛性为依据,给出了IEC中适应值噪声的狭义定义.实验进一步验证了这两个定理.上述结论为进一步研究IEC作了必要的铺垫.     

基于Coq的杨忠道定理形式化证明

实现拓扑学定理的机器证明,是吴文俊院士生前的宿愿.杨忠道定理涉及一般拓扑学中的诸多基本概念,对深刻理解拓扑空间的本质有重要意义.该定理表明,拓扑空间中每一个子集的导集为闭集当且仅当此空间中的每一个单点集的导集为闭集,是一般拓扑学中的一个重要定理.基于定理证明辅助工具Coq,从公理化集合论机器证明系统出发,对一般拓扑学中的开集、闭集、邻域、凝聚点和导集等拓扑基本概念进行形式化描述,给出这些概念基本性质的形式化验证,建立了拓扑空间的形式化框架.在此基础上,实现基于Coq的杨忠道定理形式化证明.全部引理、定理和推论均完整给出Coq的形式化描述和机器证明代码,并在计算机上运行通过,体现了基于Coq的数学定理机器证明具有可读性、交互性和智能性的特点,其证明过程规范、严谨、可靠.杨忠道定理的形式化证明是一般拓扑学形式化内容的一个深刻体现.     

粗糙集的格刻画

刻画了粗糙集的格结构,提出了半序关系粗糙集的概念,并引进了极大理想集的概念,讨论了知识边界区域划分问题,提出了易变性定理和半序粗糙反链原理,提出了半序粗糙集的选择定理、表示定理和嵌紧定理等.     

随机中立型泛函微分方程的Lasalle定理

随机型的Lasalle定理是研究随机系统的稳定性的重要理论工具.本文应用It公式、半鞅收敛定理与ko lm ogorov-Cˇentsov定理等随机分析知识,以及H lder不等式等技巧,首次建立了一般随机中立型泛函微分方程的Lasalle定理,由此得到一些有用的随机稳定性判据.本文所建立的结果涵盖并推广了已有文献的结论.最后给出了一个例子说明本文结果的有效性.     

高精度伺服系统低速问题研究

采用相平面法、代数分析法和李雅普诺夫稳定性定理得出了三阶系统克服低速滞滑爬 行的判定定理.通过该判定定理得到系统参数与低速滞滑爬行的关系,在满足常规稳定控制器 参数选择范围的基础上,给出了为消除低速滞滑爬行,PID控制器参数还应满足的条件.该判定 定理简单实用,对于高精度伺服系统的调试具有很好的指导意义.     

NTRU中多项式的逆问题

在NTRU公钥密码体制中,一个多项式是否有逆多项式是一个很重要的问题.本文介绍了NTRU公钥密码体制,给出了NTRU中多项式是否有逆的判定定理,并对所提出的定理进行了相应的证明.最后我们用例子来说明怎样运用该判定定理,给出了求解多项式逆的算法.     

离散Hopfield神经网络的 稳定性研究1)

推广了前人关于离散Hopfield神经网络的稳定性定理及周期为2极限环的存在定理,并从理论上给出了新的严格的证明.进一步,提出了关于部分变元稳定和部分变元为极限环的新概念,并给出了判别定理.最后给出了几个有趣的例子,揭示这类网络渐近行为的复杂性.     

时间尺度上约束Birkhoff系统的Noether对称性与守恒量

研究时间尺度上约束Birkhoff系统的Noether对称性.基于时间尺度上Pfaff Birkhoff原理,建立了时间尺度上带乘子形式的约束Birkhoff方程.〖JP2〗给出了时间尺度上的Noether等式,定义了时间尺度上约束Birkhoff系统Noether对称性.〖JP〗提出并证明了时间尺度上约束Birkhoff系统的Noether定理,该定理揭示了时间尺度上Noether对称性与守恒量之间的关系.给出定理的两个特例：时间尺度上Birkhoff系统和经典约束Birkhoff系统的Noether定理.文末给出算例以说明方法和结果的有效性.     

Global asymptotic stability of delay BAM neural networks with impulses via nonsmooth analysis

In this paper, we study global asymptotic stability of delay bi-directional associative memory (BAM) neural networks with impulses. We obtain a sufficient condition of ensuring existence and uniqueness of equilibrium point for delay BAM neural networks with impulses basing on nonsmooth analysis. And we give a criteria of global asymptotic stability of the unique equilibrium point for delay BAM neural networks with impulses using Lyapunov method. At last, we present examples to illustrate that our results are feasible.     

一类时延反馈神经网络的稳定性及吸引域的估计

反馈型神经网络由于具有极为丰富的动力学行为和整体计算能力（如优化、联想、振荡和混饨）而倍受关注，近几年的研究表明，当网络的时延足够小时，具有延时的对称Hopfield型神经网络和无时延情况一样也是全局稳定的．本文通过构造适当Lyapunov泛函的方法，对一类具有时延的反馈型神经网络平衡点的渐近稳定性进行了分析，得到了平衡点渐近稳定的充分条件：要检验一个有时间延迟的反馈型神经网络的稳定性，只要测试一个特定矩阵的定性性质或一个特定不等式即可．最后我们也提供了一种估计网络渐近稳定平衡点吸引域的方法．     

Robust stability of uncertain fuzzy Cohen–Grossberg BAM neural networks with time-varying delays

In this paper, the Takagi–Sugeno (TS) fuzzy model representation is extended to the stability analysis for uncertain Cohen–Grossberg type bidirectional associative memory (BAM) neural networks with time-varying delays using linear matrix inequality (LMI) theory. A novel LMI-based stability criterion is obtained by using LMI optimization algorithms to guarantee the asymptotic stability of uncertain Cohen–Grossberg BAM neural networks with time varying delays which are represented by TS fuzzy models. Finally, the proposed stability conditions are demonstrated with numerical examples.     

离散Hopfield双向联想记忆神经网络的稳定性分析

首先将离散Hopfield双向联想记忆神经网络转化成一个特殊的离散Hopfield网络 模型.在此基础上,对离散Hopfield双向联想记忆神经网络的全局渐近稳定性和全局指数稳 定性进行了新的分析.证明了神经网络连接权矩阵在给定的约束条件下有唯一的而且是渐近 稳定的平衡点.利用Lyapunov方程正对角解的存在性得到了几个判定平衡点为全局渐近稳 定和全局指数稳定的充分条件.这些条件可以用于设计全局渐近稳定和全局指数稳定的神经 网络.所做的分析扩展了以前的稳定性结果.     

Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays

This paper presents a sufficient condition for the existence, uniqueness and global asymptotic stability of the equilibrium point for bidirectional associative memory (BAM) neural networks with distributed time delays. The results impose constraint conditions on the network parameters of neural system independently of the delay parameter, and they are applicable to all continuous nonmonotonic neuron activation functions. It is shown that in some special cases of the results, the stability criteria can be easily checked. Some examples are also given to compare the results with the previous results derived in the literature.     

New delay-dependent exponential stability criteria of BAM neural networks with time delays

In this paper, the global exponential stability is investigated for the bi-directional associative memory networks with time delays. Several new sufficient conditions are presented to ensure global exponential stability of delayed bi-directional associative memory neural networks based on the Lyapunov functional method as well as linear matrix inequality technique. To the best of our knowledge, few reports about such “linearization” approach to exponential stability analysis for delayed neural network models have been presented in literature. The method, called parameterized first-order model transformation, is used to transform neural networks. The obtained conditions show to be less conservative and restrictive than that reported in the literature. Two numerical simulations are also given to illustrate the efficiency of our result.     

Stability analysis of discrete-time recurrent neural networks based on standard neural network models

In order to conveniently analyze the stability of various discrete-time recurrent neural networks (RNNs), including bidirectional associative memory, Hopfield, cellular neural network, Cohen-Grossberg neural network, and recurrent multiplayer perceptrons, etc., the novel neural network model, named standard neural network model (SNNM) is advanced to describe this class of discrete-time RNNs. The SNNM is the interconnection of a linear dynamic system and a bounded static nonlinear operator. By combining Lyapunov functional with S-Procedure, some useful criteria of global asymptotic stability for the discrete-time SNNMs are derived, whose conditions are formulated as linear matrix inequalities. Most delayed (or non-delayed) RNNs can be transformed into the SNNMs to be stability analyzed in a unified way. Some application examples of the SNNMs to the stability analysis of the discrete-time RNNs shows that the SNNMs make the stability conditions of the RNNs easily verified.     

Global robust stability of bidirectional associative memory neural networks with multiple time delays.

This correspondence presents a sufficient condition for the existence, uniqueness, and global robust asymptotic stability of the equilibrium point for bidirectional associative memory neural networks with discrete time delays. The results impose constraint conditions on the network parameters of the neural system independently of the delay parameter, and they are applicable to all bounded continuous nonmonotonic neuron activation functions. Some numerical examples are given to compare our results with the previous robust stability results derived in the literature.     

时滞标准神经网络模型及其应用

提出一种新的神经网络模型---时滞标准神经网络模型(DSNNM),它由线性动力学系统和有界静态时滞非线性算子连接而成.利用不同的Lyapunov泛函和S方法推导出DSNNM全局渐近稳定性和全局指数稳定性的充分条件,这些条件可表示为线性不等式(LMI)形式.大多数时滞(或非时滞)动态神经网络(DANN)稳定性分析或神经网络控制系统都可以转化为DSNNM,以便用统一的方法进行稳定性分析或镇定控制.从DSNNM应用于时滞联想记忆(BAM)神经网络的稳定性分析以及PH中和过程神经控制器的综合实例,可以看出,得到的稳定性判据扩展并改进了以往文献中的稳定性定理,而且可将稳定性分析推广到非线性控制系统的综合.     

Exponential periodicity and stability of delayed neural networks

In this paper, exponential periodicity and stability of delayed neural networks is investigated. Without assuming the boundedness and differentiability of the activation functions, some new sufficient conditions ensuring existence and uniqueness of periodic solution for a general class of neural systems are obtained. The delayed Hopfield network, bidirectional associative memory network, and cellular neural network are special cases of the neural system model considered.