有限博弈的矩阵半张量积方法

矩阵半张量积被广泛地应用在有限博弈的研究中,例如:1)演化博弈; 2)势博弈; 3)有限博弈的向量空间分解; 4)基于势博弈的优化与控制; 5)合作博弈等.本文的目的,就是对上述各种应用做一个全面的介绍,包括其原理、主要成果、以及尚待解决的问题.     

公理化的矩阵半张量积

本文给出矩阵半张量积的一个公理化框架, 它包括矩阵–矩阵半张量积、矩阵–向量半张量积和向量–向量半张量积. 首先, 对目前通用的各类矩阵半张量积的基本性质与应用做一个综述性的回顾. 然后, 介绍一种新近出现的矩阵半张量积, 即保维数矩阵半张量积. 跟普通矩阵乘法一样, 它是多功能的, 即它可同时实现矩阵–矩阵乘积、矩阵–向量乘积和向量–向量乘积这3种功能. 最后, 本文介绍保维数矩阵半张量积的一些代数性质, 包括非方矩阵的Cayley-Hamilton定理, 非方矩阵的特征值、特征向量等.     

矩阵奇异值分解在计算技术中的应用

奇异值是矩阵的一个良好特征。本文通过对矩阵奇异值分解的分析，对计算机图像压缩和数字水印技术方面的应用方法进行了描述，同时提供了部分应用例子。     

Petri网信标和陷阱计算的矩阵半张量积方法

本文基于矩阵半张量积(semi-tensor product,STP)方法研究了普通Petri网(Petri nets,PNs)信标和陷阱的计算问题.首先,利用STP方法建立了两个矩阵方程,分别称为Petri网的信标方程(siphon equation,SE)和陷阱方程(trap equation,TE).其次,证明了计算Petri网的信标和陷阱分别等价于求信标方程(SE)和陷阱方程(TE)的非零解.同时,给出了计算Petri网所有信标和陷阱的算法.最后,实例和实验结果说明了本文方法的可行性与有效性.本文所提出的方法对于Petri网信标和陷阱的计算是非常有效的,它只涉及到矩阵的乘法运算.     

关于矩阵张量积计算的研究

利用矩阵张量积有关理论,讨论了矩阵张量积的计算问题,分析了算法的复杂性,并研究了并行算法及计算复杂性问题。     

基于受限区分矩阵的值约简方法*

一般启发式值约简算法中启发信息选取不够合理,获取规则的置信度不高,同时还需要多次遍历决策表,时间开销较大。针对上述问题,提出了一种基于加权平均的属性值重要度的概念,并利用受限区分矩阵构造了一种新的值约简方法。该方法无须多次遍历决策表,且不必考虑属性值恢复的问题,大大方便了规则摄取,有效地降低了计算的时间复杂度,且属性值约简更加合理,保证了最后获取的规则具有较高的置信度。最后通过真实的医学数据实验结果表明,该方法具有较好的约简效果。     

基于矩阵张量积的数据加密矩阵的构造

通过讨论二阶矩阵的可逆性,研究了基于矩阵张量积合成加密矩阵的构造问题,并用实例进行了加密实践。     

符号矩阵奇异值的计算机分解与广义逆计算

本文从符号计算(计算机代数)的观点,对数值矩阵计算理论中传统的奇异值分解算法及其相关的广义逆矩阵计算方法加以改进,使之适于处理符号矩阵且在计算机上实现.并指出其在矩阵理论及线性问题求解中的某些应用,还给出了一个有趣的物理实例.     

基于矩阵张量积的数据加密方案

1 引言在当今信息社会中,计算机技术的应用日趋深化,大量的数据交换越来越频繁,由此引起的数据安全也更加突出。作为保障数据安全的一种非常重要方法,数据加密技术已越来越受重视,对它的研究也就有着非常重要的现实意义。文[2,3]研究了基于最大秩距离码的密码系统,文[4]对此密码系统进行了进一步的讨论。1929年,美国的希尔(Hill)提出了一种代数加密体制,其思想是用代数编码方式来进行加密和解密,采用解方程的方法把明文变成密文。本文利用矩阵的张量积理论,提出一种代数加密方案。     

Parallel characterizations of a generalized Shapley value and a generalized Banzhaf value for cooperative games with level structure of cooperation

We present parallel characterizations of two different values in the framework of restricted cooperation games. The restrictions are introduced as a finite sequence of partitions defined on the player set, each of them being coarser than the previous one, hence forming a structure of different levels of a priori unions. On the one hand, we consider a value first introduced in Ref. [18], which extends the Shapley value to games with different levels of a priori unions. On the other hand, we introduce another solution for the same type of games, which extends the Banzhaf value in the same manner. We characterize these two values using logically comparable properties.     

The symmetric Banzhaf value for fuzzy games with a coalition structure

In this paper,a generalized form of the symmetric Banzhaf value for cooperative fuzzy games with a coalition structure is proposed.Three axiomatic systems of the symmetric Banzhaf value are given by extending crisp case.Furthermore,we study the symmetric Banzhaf values for two special kinds of fuzzy games,which are called fuzzy games with multilinear extension form and a coalition structure,and fuzzy games with Choquet integral form and a coalition structure,respectively.     

Semi-tensor product approach to networked evolutionary games

In this paper a comprehensive introduction for modeling and control of networked evolutionary games （NEGs） via semi-tensor product （STP） approach is presented. First, we review the mathematical model of an NEG, which consists of three ingredients： network graph, fundamental network game, and strategy updating rule. Three kinds of network graphs are considered, which are i） undirected graph for symmetric games; ii） directed graph for asymmetric games, and iii） d-directed graph for symmetric games with partial neighborhood information. Three kinds of fundamental evolutionary games （FEGs） are discussed, which are i） two strategies and symmetric （S-2）; ii） two strategies and asymmetric （A-2）; and iii） three strategies and symmetric （S-3）. Three strategy updating rules （SUR） are introduced, which are i） Unconditional Imitation （UI）; ii） Fermi Rule（FR）; iii） Myopic Best Response Adjustment Rule （MBRA）. First, we review the fundamental evolutionary equation （FEE） and use it to construct network profile dynamics （NPD）of NEGs.
To show how the dynamics of an NEG can be modeled as a discrete time dynamics within an algebraic state space, the fundamental evolutionary equation （FEE） of each player is discussed. Using FEEs, the network strategy profile dynamics （NSPD） is built by providing efficient algorithms. Finally, we consider three more complicated NEGs： i） NEG with different length historical information, ii） NEG with multi-species, and iii） NEG with time-varying payoffs. In all the cases, formulas are provided to construct the corresponding NSPDs. Using these NSPDs, certain properties are explored. Examples are presented to demonstrate the model constructing method, analysis and control design technique, and to reveal certain dynamic behaviors of NEGs.     

Matrix approach to simplification of finite state machines using semi‐tensor product of matrices

In this paper, we use a new mathematical tool, semi‐tensor product of matrices, to investigate the problem of simplification of finite state machines (FSMs) in a mathematical manner. First, based on the dynamic equations of state transition and output behavior which are developed recently, an algebraic criterion of k‐difference states is established. Second, using the criterion, a scheme is designed to construct the incompatible graphs of FSMs. Third, with the incompatible graphs and the method of searching internally stable sets of graphs proposed by the authors, a solution is proposed to obtain all of the compatible state set (CSS) of FSMs. Then, with the aid of the CSS, we investigate three kinds of structures of state space of FSMs, including compatible cover of state set (CCSS), representative set of state set (RSSS), and minimum representative set of state set (MRSSS); necessary and sufficient conditions are proposed to formulate the three kinds of structures. Finally, examples are given to exemplify minimum realizations of FSMs by these conditions.     

Non-regular feedback linearization of nonlinear systems via a normal form algorithm

In this paper, the problem of non-regular static state feedback linearization of affine nonlinear systems is considered. First of all, a new canonical form for non-regular feedback linear systems is proposed. Using this form, a recursive algorithm is presented, which yields a condition for single input linearization. Then the left semi-tensor product of matrices is introduced and several new properties are developed. Using the recursive framework and new matrix product, a formula is presented for normal form algorithm. Based on it, a set of conditions for single-input (approximate) linearizability is presented.     

具有切换概率分布的概率布尔网络的依分布稳定和镇定

布尔网络作为研究基因调控网络的一种重要模型,近年来引起了国内外很多学者的广泛关注.本文利用代数状态空间表示方法,研究具有切换概率分布的概率布尔网络的依分布稳定和镇定问题.首先,回顾针对切换布尔网络稳定性分析的现有的研究结果.其次,给出具有切换概率分布的概率布尔网络依分布稳定的定义,并利用矩阵的半张量积建立具有切换概率分布的概率布尔网络的代数表示.再次,基于该代数表示,建立具有切换概率分布的概率布尔网络的依分布稳定的充分必要条件.最后,给出具有切换概率分布的概率布尔控制网络镇定问题可解的充要条件,并给出相应的控制设计方法.     

Stability and stabilization of Boolean networks with impulsive effects

This paper investigates the stability and stabilization of Boolean networks with impulsive effects. After giving a survey on semi-tensor product of matrices, we convert a Boolean network with impulsive effects into impulsive discrete-time dynamics. Then, some necessary and sufficient conditions are given for the stability and stabilization of Boolean networks with impulsive effects. Finally, examples are provided to illustrate the efficiency of the obtained results.     

On finite potential games

A linear system, called the potential equation (PE), is presented. It is proved that a finite game is potential if and only if its potential equation has solution. Some properties of the potential equation are obtained. Based on these properties, a closed form solution of the PE is obtained. Moreover, a formula based on the solution of the PE is obtained to calculate the potential function. Finally, it is proved that a networked evolutionary game is potential if and only if its fundamental network game is potential. Some interesting examples are presented to illustrate the theoretical results.     

Disturbance decoupling control design for switched Boolean control networks

This paper investigates a kind of disturbance decoupling problems (DDPs) for switched Boolean control networks (BCNs), that is, the disturbance-independent output decoupling problem, via the semi-tensor product of matrices. By converting the dynamics of switched BCNs into an algebraic form and using the redundant variable separation technique, all possible state feedback controllers are designed for the disturbance-independent output decoupling problem. Then, based on the obtained state feedback disturbance decoupling controllers and the system’s output equation, a constructive procedure is proposed to design all possible output feedback disturbance decoupling controllers. The study of two illustrative examples shows that the new results obtained in this paper are effective in designing disturbance decoupling controllers for switched BCNs.