Generalized WG inverse

2020 ◽  
pp. 2150072
Author(s):  
Sanzhang Xu ◽  
Hongxing Wang ◽  
Jianlong Chen ◽  
Xiaofeng Chen ◽  
Tiwei Zhao
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Form ◽  
Necessary And Sufficient

In this paper, we introduce a generalized inverse of a matrix, namely, the generalized WG inverse, which a generalization of the WG inverse. Several necessary and sufficient conditions such that a matrix to be generalized WG invertible are obtained. Moreover, the formulae of generalized WG inverse of a matrix are given. Finally, we give a algebraic form of the generalized WG inverse.

scholarly journals Controllability, Reachability, and Stabilizability of Finite Automata: A Controllability Matrix Method

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Yalu Li ◽  
Wenhui Dou ◽  
Haitao Li ◽  
Xin Liu
Keyword(s):  
Matrix Method ◽  
Finite Automata ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Form ◽  
Controllability Matrix ◽  
Necessary And Sufficient ◽  
Semitensor Product Of Matrices ◽  
Product Of Matrices ◽  
Inputs And Outputs

This paper investigates the controllability, reachability, and stabilizability of finite automata by using the semitensor product of matrices. Firstly, by expressing the states, inputs, and outputs as vector forms, an algebraic form is obtained for finite automata. Secondly, based on the algebraic form, a controllability matrix is constructed for finite automata. Thirdly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of finite automata by using the controllability matrix. Finally, an illustrative example is given to support the obtained new results.

scholarly journals Robust set stabilization of Boolean control networks with impulsive effects

2018 ◽  
Vol 23 (4) ◽  
pp. 553-567 ◽  
Author(s):  
Xiaojing Xu ◽  
Yansheng Liu ◽  
Haitao Li ◽  
Fuad E. Alsaadi
Keyword(s):  
Feedback Control ◽  
State Feedback ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
State Feedback Control ◽  
Impulsive Effects ◽  
Algebraic Form ◽  
Control Networks ◽  
Set Stabilization ◽  
Necessary And Sufficient

This paper addresses the robust set stabilization problem of Boolean control networks (BCNs) with impulsive effects via the semi-tensor product method. Firstly, the closed-loop system consisting of a BCN with impulsive effects and a given state feedback control is converted into an algebraic form. Secondly, based on the algebraic form, some necessary and sufficient conditions are presented for the robust set stabilization of BCNs with impulsive effects under a given state feedback control and a free-form control sequence, respectively. Thirdly, as applications, some necessary and sufficient conditions are presented for robust partial stabilization and robust output tracking of BCNs with impulsive effects, respectively. The study of two illustrative examples shows that the obtained new results are effective.

Synchronization in mutual-coupled temporal Boolean networks

2017 ◽  
Vol 40 (7) ◽  
pp. 2211-2216 ◽  
Author(s):  
Qiang Wei ◽  
Cheng-jun Xie
Keyword(s):  
Theoretical Analysis ◽  
Tensor Product ◽  
Sufficient Conditions ◽  
Boolean Networks ◽  
Necessary And Sufficient Conditions ◽  
Complete Synchronization ◽  
Logical Relationship ◽  
Algebraic Form ◽  
Algebraic Forms ◽  
Necessary And Sufficient

In this paper, we first propose a mutual-coupled temporal Boolean networks model and then investigate complete synchronization in mutual-coupled temporal Boolean networks. The mutual-coupled temporal Boolean networks model with logical relationship is converted into an algebraic form based on a semi-tensor product. Necessary and sufficient conditions are derived to realize synchronization based on the algebraic forms. An example illustrates the effectiveness of the theoretical analysis.

scholarly journals The outer inverse f(2)T,S of a homomorphism of right R-modules

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6459-6468
Author(s):  
Zhou Wang
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Outer Inverse ◽  
Basic Properties ◽  
Definition Of ◽  
Necessary And Sufficient

In this paper, we introduce the definition of the generalized inverse f(2)T,S, which is an outer inverse of the homomorphism f of right R-modules with prescribed image T and kernel S. Some basic properties of the generalized inverse f(2)T,S are presented. It is shown that the Drazin inverse, the group inverse and the Moore-Penrose inverse, if they exist, are all the generalized inverse f 2) T,S. In addition, we give necessary and sufficient conditions for the existence of the generalized inverse f(1,2)T,S.

scholarly journals The Generalized Inverse A(2)T, Sof a Matrix Over an Associative Ring

2007 ◽  
Vol 83 (3) ◽  
pp. 423-438 ◽  
Author(s):  
Yaoming Yu ◽  
Guorong Wang
Keyword(s):  
Associative Ring ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Arbitrary Matrix ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Explicit Expressions

AbstractIn this paper we establish the definition of the generalized inverse A(2)T, Swhich is a {2} inverse of a matrixAwith prescribed imageTand kernelsover an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverseand some explicit expressions forof a matrix A over an associative ring, which reduce to the group inverse or {1} inverses. In addition, we show that for an arbitrary matrixAover an associative ring, the Drazin inverse Ad, the group inverse Agand the Moore-Penrose inverse. if they exist, are all the generalized inverse A(2)T, S.

scholarly journals Lp,q-Label Coloring Problem with Application to Channel Allocation

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhenbin Liu ◽  
Yuqiang Wu
Keyword(s):  
Wireless Network ◽  
Channel Allocation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Form ◽  
Logical Operators ◽  
Coloring Problem ◽  
All Solutions ◽  
Necessary And Sufficient

In this paper, the Lp,q-coloring problem of the graph is studied with application to channel allocation of the wireless network. First, by introducing two new logical operators, some necessary and sufficient conditions for solving the Lp,q-coloring problem are given. Moreover, it is noted that all solutions of the obtained logical equations are corresponding to each coloring scheme. Second, by using the semitensor product, the necessary and sufficient conditions are converted to an algebraic form. Based on this, all coloring schemes can be obtained through searching all column indices of the zero columns. Finally, the obtained result is applied to analyze channel allocation of the wireless network. Furthermore, an illustration example is given to show the effectiveness of the obtained results in this paper.

scholarly journals On polynomialEPrmatrices

1992 ◽  
Vol 15 (2) ◽  
pp. 261-266
Author(s):  
Ar. Meenakshi ◽  
N. Anandam
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

This paper gives a characterization ofEPr-λ-matrices. Necessary and sufficient conditions are determined for (i) the Moore-Penrose inverse of anEPr-λ-matrix to be anEPr-λ-matrix and (ii) Moore-Penrose inverse of the product ofEPr-λ-matrices to be anEPr-λ-matrix. Further, a condition for the generalized inverse of the product ofλ-matrices to be aλ-matrix is determined.

scholarly journals The Forward Order Law for Least Squareg-Inverse of Multiple Matrix Products

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 277 ◽  
Author(s):  
Zhiping Xiong ◽  
Zhongshan Liu
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Schur Complement ◽  
Necessary And Sufficient Conditions ◽  
Matrix Product ◽  
The Core ◽  
Matrix Products ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Generalized Schur Complement

The generalized inverse has many important applications in the aspects of the theoretic research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the forward order laws for the generalized inverse of the matrix product. In this paper, by using the extremal ranks of the generalized Schur complement, we obtain some necessary and sufficient conditions for the forward order laws A 1 { 1 , 3 } A 2 { 1 , 3 } ⋯ A n { 1 , 3 } ⊆ ( A 1 A 2 ⋯ A n ) { 1 , 3 } and A 1 { 1 , 4 } A 2 { 1 , 4 } ⋯ A n { 1 , 4 } ⊆ ( A 1 A 2 ⋯ A n ) { 1 , 4 } .

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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