Solution of the Matrix Second Semi-Tensor Product Equation A ∘ l X ∘ l B=C

In this paper, the solution of the matrix second semi-tensor product equation A∘lX∘lB=Cis studied. Firstly, the solvability of the matrix-vector second semi-tensor product equation is investigated. At the same time, the compatibility conditions, the sufficient and necessary conditions and the specific solution methods for the matrix solution are given. Secondly, we further consider the solvability of the second semi-tensor product equation of the matrix. For each part, several examples are given to illustrate the validity of the results.

矩阵方程AXB=C的解的存在唯一性及其数值分析

利用Kronecker乘积,给出了矩阵方程AXB=C存在唯一解的充要条件,并对方程的解进行了扰动分析,最后给出了求良态方程近似解的一个数值方法及其应用实例.

Semi-tensor product of matrices approach to reachability of finite automata with application to language recognition

This paper investigates the transition function and the reachability conditions of finite automata by using a semitensor product of matrices, which is a new powerful matrix analysis tool. The states and input symbols are first expressed in vector forms, then the transition function is described in an algebraic form. Using this algebraic representation, a sufficient and necessary condition of the reachability of any two states is proposed, based on which an algorithm is developed for discovering all the paths from one state to another. Furthermore, a mechanism is established to recognize the language acceptable by a finite automaton. Finally, illustrative examples show that the results/algorithms presented in this paper are suitable for both deterministic finite automata （DFA） and nondeterministic finite automata （NFA）.

On Semi-tensor Product of Matrices and Its Applications

Abstract The left semi-tensor product of matrices was proposed in [2]. In this paper the right semi-tensor product is introduced first. Some basic properties are presented and compared with those of the left semi-tensor product.Then two new applications are investigated. Firstly, its applications to connection, an important concept in differential geometry, is considered. The structure matrix and the Christoffel matrix are introduced. The transfer formulas under coordinate transformation are expressed in matrix form. Certain new results are obtained. Secondly, the structure of finite dimensional Lie algebra, etc. are investigated under the matrix expression.These applications demonstrate the usefulness of the new matrix products.

Logic and logic-based control

This paper gives a matrix expression of logic. Under the matrix expression, a general description of the logical operators is proposed. Using the semi-tensor product of matrices, the proofs of logical equivalences, implications, etc., can be simplified a lot. Certain general properties are revealed. Then, based on matrix expression, the logical operators are extended to multi-valued logic, which provides a foundation for fuzzy logical inference. Finally, we propose a new type of logic, called mix-valued logic, and a new design technique, called logic-based fuzzy control. They provide a numerically computable framework for the application of fuzzy logic for the control of fuzzy systems.

Stability of Nonlinear Feedback Shift Registers with Periodic Input

The stability of Non-Linear Feedback Shift Registers(NFSRs)plays an important role in the cryptographic security.Due to the complexity of nonlinear systems and the lack of efficient algebraic tools,the theorems related to the stability of NFSRs are still not well-developed.In this paper,we view the NFSR with periodic inputs as a Boolean control network.Based on the mathematical tool of semi-tensor product(STP),the Boolean network can be mapped into an algebraic form.Through these basic theories,we analyze the state space of non-autonomous NFSRs,and discuss the stability of an NFSR with periodic inputs of limited length or unlimited length.The simulation results are provided to prove the efficiency of the model.Based on these works,we can provide a method to analyze the stability of the NFSR with periodic input,including limited length and unlimited length.By this,we can efficiently reduce the computational complexity,and its efficiency is demonstrated by applying the theorem in simulations dealing with the stability of a non-autonomous NFSR.

Contracted product of hypermatrices via STP of matrices

An equivalent definition of hypermatrices is introduced.The matrix expression of hypermatrices is proposed.Using permu-tation matrices,the conversion between different matrix expressions is revealed.The various kinds of contracted products of hypermatrices are realized by semi-tensor products(STP)of matrices via matrix expressions of hypermatrices.

一类带逻辑脉冲线性系统的最优控制问题

对固定的观测时间,通过逻辑形式判断是否对动态受控系统的运行进行干预,实现目标的优化,这是生产生活实际常见的问题,这一类问题可以用带逻辑判断脉冲受控系统的最优控制问题进行描述.本文主要研究这一类最优控制问题,首先运用矩阵半张量积方法,将带有逻辑判断的脉冲表达式转换为代数表达式,然后通过Banach压缩映像原理,证明了受控系统解的存在唯一性;进而使用Mazur定理证明最优控制的存在性;最后,给出控制取值集U为全空间或开集情形下的最优性条件.作为应用,论文还给出一个模拟仿真的实例.

Context-Aware System Modeling Based on Boolean Control Network

Boolean control network consists of a set of Boolean variables whose state is determined by other variables in the network. Boolean network is used for modeling complex system. In this paper, we have presented a model of a context-aware system used in smart home based on Boolean control networks. This modeling describes the relationship between the context elements (person, time, location, and activity) and services (Morning Call, Sleeping, Guarding, Entertainment, and normal), which is effective to logical inference. We apply semi tensor matrix product to describe the dynamic of the system. This matrix form of expression is a convenient and reasonable way to design logic control system.

非奇异M-矩阵及其逆矩阵Hadamard积最小特征值的新下界

针对非奇异M-矩阵及其逆矩阵Hadamard积的最小特征值问题,首先,回顾了已有文献应用矩阵的特征值存在域定理和逆矩阵元素的估计式;其次,结合M-矩阵Hadamard积的相关性质特征及不等式的构造、放缩技巧,给出了非奇异M-矩阵与其逆矩阵是双随机矩阵的Hadamard积的最小特征值下界τ(A°A^-1)的一个仅与A矩阵的元素相关的估计式,推广了已有文献的结果;最后,用数值例子表明所给估计式的下界比已有结果得到的下界更精确.

A matrix-based static approach to analysis of finite state machines

Traditional matrix-based approaches in the field of finite state machines construct state transition matrices,and then use the powers of the state transition matrices to represent corresponding dynamic transition processes,which are cornerstones of system analysis.In this study,we propose a static matrix-based approach that revisits a finite state machine from its structure rather than its dynamic transition process,thus avoiding the“explosion of complexity”problem inherent in the existing approaches.Based on the static approach,we reexamine the issues of closed-loop detection and controllability for deterministic finite state machines.In addition,we propose controllable equivalent form and minimal controllable equivalent form concepts and give corresponding algorithms.

Linearization of Multi-Valued Nonlinear Feedback Shift Registers

Nonlinear feedback shift registers(NFSRs) have been used in many stream ciphers for cryptographic security. The linearization of NFSRs is to describe their state transitions using some matrices. Such matrices are called their state transition matrices. Compared to extensive work on binary NFSRs, much less work has been done on multi-valued NFSRs. This paper uses a semi-tensor product approach to investigate the linearization of multi-valued NFSRs, by viewing them as logical networks. A new state transition matrix is found for a multi-valued NFSR, which can be simply computed from the truth table of its feedback function. The new state transition matrix is easier to compute and is more explicit than the existing results. Some properties of the state transition matrix are provided as well, which are helpful to theoretically analyze multi-valued NFSRs.

A minimum adequate set of multi-valued logic

An adequate set of k-valued logic is provided,which contains only two operators.It is also proved that this adequate set is of minimum size。