By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li
The Boolean community has develop into a robust instrument for describing and simulating mobile networks within which the weather behave in an on–off style. research and keep an eye on of Boolean Networks provides a scientific new method of the research of Boolean keep watch over networks. the basic instrument during this procedure is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality will be expressed as a traditional discrete-time linear approach. within the mild of this linear expression, sure significant concerns referring to Boolean community topology – fastened issues, cycles, temporary instances and basins of attractors – may be simply published by means of a collection of formulae. This framework renders the state-space method of dynamic keep an eye on structures acceptable to Boolean keep watch over networks. The bilinear-systemic illustration of a Boolean keep watch over community makes it attainable to enquire uncomplicated keep an eye on difficulties together with controllability, observability, stabilization, disturbance decoupling, identity, optimum regulate, and so on.
The publication is self-contained, requiring basically wisdom of linear algebra and the fundamentals of the keep an eye on concept of linear structures. It starts with a quick advent to prepositional common sense and the innovations and houses of the STP and progressing through the (bi)linear expression of Boolean (control) networks to disturbance decoupling and decomposition of Boolean regulate structures. eventually multi-valued common sense is taken into account as a extra exact method of describing genuine networks and stochastic Boolean networks are touched upon. correct numerical calculations are defined in an appendix and a MATLAB® toolbox for the algorithms within the ebook will be downloaded from http://lsc.amss.ac.cn/~dcheng/.
Analysis and keep watch over of Boolean Networks could be a basic reference for researchers in platforms biology, regulate, platforms technology and physics. The publication was once constructed for a brief path for graduate scholars and is acceptable for that objective. machine scientists and logicians can also locate this publication to be of curiosity.
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Additional info for Analysis and Control of Boolean Networks: A Semi-tensor Product Approach
Let Y= 2 −1 × (−2) + 1 ⎤ 2 1 −1 3 2 −1 ⎦ , X = ⎣0 1 2 −1 1 1 2 × 1 = −3 4 . ⎡ Then Y= −1 2 . 3 2 ⎡ X ⎤ (21) × (−1) + (−13) × 3 (21) × 2 + (−13) × 2 Y = ⎣ (01) × (−1) + (2 − 1) × 3 (01) × 2 + (2 − 1) × 2 ⎦ (2 − 1) × (−1) + (11) × 3 (2 − 1) × 2 + (11) × 2 ⎡ ⎤ −5 8 2 8 = ⎣ 6 −4 4 0 ⎦ . 2 1. The dimension of the semi-tensor product of two matrices can be determined by deleting the largest common factor of the dimensions of the two factor matrices. For instance, Ap×qr Br×s Cqst×l = (A B)p×qs Cqst×l = (A B C)pt×l .
Sm1 , . . , smn ), called the structure matrix of F . This is a row vector of dimension mn, labeled by the ordered multi-index Id(i, j ; m, n). 16) Y. 16), but what is its advantage? 16) realized the product of 2-dimensional data (a matrix) with 1-dimensional data by using the product of two sets of 1-dimensional data. If, in this product, 2-dimensional data can be converted into 1-dimensional data, we would expect that the same thing can be done for higher-dimensional data. 14) because it allows the product of higher-dimensional data to be taken.
T. The structure matrix S of F can be constructed as follows. Its data are labeled by the ordered multi-index Id(i, j, k; m, n, t) to form an mnt-dimensional row vector as S = (s111 , . . , s11t , . . , s1n1 , . . , s1nt , . . , smn1 , . . , smnt ). Then, for X ∈ U , Y ∈ V , Z ∈ W , it is easy to verify that F (X, Y, Z) = S X Y Z. Observe that in a semi-tensor product, can automatically find the “pointer” of different hierarchies and then perform the required computation. 8 can be used for any multilinear mapping.