By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li
The Boolean community has turn into a robust device 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 offers a scientific new method of the research of Boolean keep an eye on networks. the basic device during this technique is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality might be expressed as a standard discrete-time linear procedure. within the mild of this linear expression, sure significant concerns referring to Boolean community topology – fastened issues, cycles, brief occasions and basins of attractors – will be simply published via a collection of formulae. This framework renders the state-space method of dynamic regulate structures acceptable to Boolean keep watch over networks. The bilinear-systemic illustration of a Boolean keep watch over community makes it attainable to enquire simple regulate difficulties together with controllability, observability, stabilization, disturbance decoupling, identity, optimum keep an eye on, and so on.
The publication is self-contained, requiring merely wisdom of linear algebra and the fundamentals of the regulate idea of linear platforms. It starts off with a brief creation to prepositional common sense and the strategies and homes of the STP and progressing through the (bi)linear expression of Boolean (control) networks to disturbance decoupling and decomposition of Boolean keep watch over structures. ultimately multi-valued good judgment 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 booklet will be downloaded from http://lsc.amss.ac.cn/~dcheng/.
Analysis and regulate of Boolean Networks may be a primary reference for researchers in structures biology, regulate, structures technology and physics. The booklet used to be built for a quick direction for graduate scholars and is acceptable for that objective. laptop 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
Nσ (k) ) if the indices i1 , . . , ik in the sequence are running in the following order: first, iσ (k) runs from 1 to nσ (k) , then iσ (k−1) runs from 1 to nσ (k−1) , and so on, until, finally, iσ (1) runs from 1 to nσ (1) . We now introduce some notation. Let a ∈ Z and b ∈ Z+ . As in the programming language C, we use a%b to denote the remainder of a/b, which is always nonnegative, and [t] for the largest integer that is less than or equal to t. 25] = −2. (−7)%3 = 2, 26 2 Semi-tensor Product of Matrices It is easy to see that a b + a%b.
Nk ) as (λ1 , . . 7) ⎩pj = [ pj +1 ], λj = pj %nj + 1, j = k − 1, . . , 1. nj +1 2. Multi-index to single index. From multi-index (λ1 , . . , λk ) in the order of Id(i1 , . . , ik ; n1 , . . , nk ) back to the single index, we have k−1 p= (λj − 1)nj +1 nj +2 · · · nk + λk . 8) j =1 The following example illustrates the conversion between different types of indices. 3, we may use different types of indices to label the elements. 1. Consider an element which is x11 in single-index form.
3 Let C ∈ Ms×t . Then, for any integer m > 0, we have W[s,m] W[m,t] = Im ⊗ C. 79) Finally, we consider how to express a matrix in stacking form and vice versa, via the semi-tensor product. 20 Let A ∈ Mm×n . 81) Vc (In ). Conversely, we can retrieve A from its row- or column-stacking form. 21 Let A ∈ Mm×n . Then A = Im ⊗ VrT (In ) Vr (A) = Im ⊗ VrT (In ) W[n,m] Vc (A). 82) As an elementary application of semi-tensor product, we consider the following example. 4 Properties of the Semi-tensor Product 47 Fig.