By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

ISBN-10: 0857290967

ISBN-13: 9780857290960

ISBN-10: 0857290975

ISBN-13: 9780857290977

The Boolean community has turn into a robust software for describing and simulating mobile networks during 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 regulate networks. the elemental instrument during this process is a singular matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality could be expressed as a standard discrete-time linear procedure. within the mild of this linear expression, convinced significant concerns bearing on Boolean community topology – fastened issues, cycles, brief occasions and basins of attractors – may be simply published via a suite of formulae. This framework renders the state-space method of dynamic regulate structures appropriate to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean keep an eye on community makes it attainable to enquire simple keep an eye on difficulties together with controllability, observability, stabilization, disturbance decoupling, id, optimum regulate, and so forth.

The e-book is self-contained, requiring simply wisdom of linear algebra and the fundamentals of the keep an eye on conception of linear structures. It starts off with a brief advent to prepositional common sense and the techniques and homes of the STP and progressing through the (bi)linear expression of Boolean (control) networks to disturbance decoupling and decomposition of Boolean regulate structures. ultimately multi-valued common sense is taken into account as a extra particular means of describing actual 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 may be downloaded from http://lsc.amss.ac.cn/~dcheng/.

Analysis and keep watch over of Boolean Networks could be a primary reference for researchers in platforms biology, keep an eye on, platforms technological know-how and physics. The booklet was once built for a quick path for graduate scholars and is acceptable for that objective. computing device scientists and logicians can also locate this e-book to be of curiosity.

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**Extra resources for Analysis and Control of Boolean Networks: A Semi-tensor Product Approach**

**Example text**

35) k 4. Consider the set of real kth order homogeneous polynomials of x ∈ Rn and denote it by Bnk . Under conventional addition and real number multiplication, Bnk is a vector space. It is obvious that x k contains a basis (x k itself is not a basis because it contains redundant elements). Hence, every p(x) ∈ Bnk can be expressed k as p(x) = Cx k , where the coefficients C ∈ Rn are not unique. Note that here x = (x1 , x2 , . . , xn )T is a column vector. In the rest of this section we describe some basic properties of the semi-tensor product.

If we consider the conventional matrix product as a particular case of the semitensor product, then we have XY T ZW T = Y T (Z X) W T. 22) holds. Hence, when the conventional matrix product is extended to the semi-tensor product, the previous inconsistency disappears. The following two examples show how to use the semi-tensor product to perform multilinear computations. 12 1. 2) and {e1 , e2 , . . , en } a basis of V . For any two elements in this basis we calculate the product as n ei ∗ ej = k cij ek , k=1 i, j, k = 1, 2, .

It+1 , it , . . , tk ; n1 , . . , nt+1 , nt , . . , nk ). W[m,n] can be constructed in an alternative way which is convenient in some applications. Denoting by δni the ith column of the identity matrix In , we have the following. 7 W[m,n] = δn1 1 δm ··· δnn 1 δm ··· δn1 m δm ··· δnn m . δm For convenience, we provide two more forms of swap matrix: ⎡ T⎤ Im ⊗ δn1 ⎥ ⎢ .. W[m,n] = ⎣ ⎦ . 49) Im ⊗ δnn T and, similarly, 1 m . , . . 50) The following factorization properties reflect the blockwise permutation property of the swap matrix.

### Analysis and Control of Boolean Networks: A Semi-tensor Product Approach by Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

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