By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

The Boolean community has turn into a strong software 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 elemental software during this technique is a singular matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality may be expressed as a standard discrete-time linear process. within the gentle of this linear expression, yes significant concerns touching on Boolean community topology – mounted issues, cycles, brief occasions and basins of attractors – could be simply printed via a collection of formulae. This framework renders the state-space method of dynamic keep watch over platforms appropriate to Boolean keep watch over networks. The bilinear-systemic illustration of a Boolean regulate community makes it attainable to enquire easy regulate difficulties together with controllability, observability, stabilization, disturbance decoupling, id, optimum regulate, and so forth.

The e-book is self-contained, requiring basically wisdom of linear algebra and the fundamentals of the keep watch over conception of linear structures. It starts with a quick advent to prepositional common sense and the options and houses of the STP and progressing through the (bi)linear expression of Boolean (control) networks to disturbance decoupling and decomposition of Boolean keep watch over platforms. eventually multi-valued common sense is taken into account as a extra unique means of describing actual networks and stochastic Boolean networks are touched upon. appropriate numerical calculations are defined in an appendix and a MATLAB® toolbox for the algorithms within the e-book could be downloaded from http://lsc.amss.ac.cn/~dcheng/.

Analysis and keep watch over of Boolean Networks may be a basic reference for researchers in structures biology, regulate, platforms technological know-how and physics. The publication used to be constructed for a quick direction for graduate scholars and is appropriate for that goal. laptop scientists and logicians can also locate this publication to be of curiosity.

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**Sample text**

5. So far, the semi-tensor product is a generalization of the matrix product from the equal dimension case to the multiple dimension case. 9 1. Let X = [2 −1 1 2], Y = [−2 1]T . Then X 2. 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.

The following conversion formulas can easily be proven by mathematical induction. 1 Let S be a set of data with n = ki=1 ni elements. The data are labeled by single index as {xi } and by k-fold index, by the ordered multi-index Id(λ1 , . . , λk ; n1 , . . , nk ), as S = {sp | p = 1, . . ,λk | 1 ≤ λi ≤ ni ; i = 1, . . , k}. We then have the following conversion formulas: 1. Single index to multi-index. Defining pk := p − 1, the single index p can be converted into the order of the ordered multi-index Id(i1 , .

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. Unlike the conventional matrix product, which can generally treat only one- or two-dimensional data, the semi-tensor product of matrices can be used to deal with any finite-dimensional data. Next, we give a general definition of semi-tensor product.