By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

The Boolean community has develop into a robust device for describing and simulating mobile networks within which the weather behave in an on–off model. research and regulate of Boolean Networks offers a scientific new method of the research of Boolean regulate networks. the elemental instrument during this method is a unique matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality should be expressed as a traditional discrete-time linear method. within the mild of this linear expression, definite significant concerns relating Boolean community topology – fastened issues, cycles, brief instances and basins of attractors – might be simply printed by way of a suite of formulae. This framework renders the state-space method of dynamic keep watch over structures appropriate to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean keep watch over community makes it attainable to enquire easy regulate difficulties together with controllability, observability, stabilization, disturbance decoupling, identity, optimum regulate, and so on.

The e-book is self-contained, requiring in simple terms wisdom of linear algebra and the fundamentals of the keep watch over thought of linear platforms. It starts with a quick creation to prepositional common sense and the ideas 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 good judgment is taken into account as a extra particular 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 publication may be downloaded from http://lsc.amss.ac.cn/~dcheng/.

Analysis and keep an eye on of Boolean Networks could be a primary reference for researchers in structures biology, regulate, platforms technology and physics. The booklet used to be built for a quick direction for graduate scholars and is acceptable for that function. laptop scientists and logicians can also locate this booklet to be of curiosity.

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

Example text

N, k = 1, . . , 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.

Jt ≤ n is the set of structure constants of φ. Structure constants of φ ∈ Tt s form a set of (s + t)-dimensional data. Next, we consider how to arrange higher-dimensional data. In linear algebra onedimensional data are arranged as a column or a row, called a vector, while twodimensional data are arranged as a rectangle, called a matrix. In these forms matrix computation becomes a very convenient and powerful tool for dealing with oneor two-dimensional data. A question which then naturally arises is how to arrange three-dimensional data.

Ik ) := Id(i1 , . . , ik ; n1 , . . , nk ). 3 1. Assume x = {xij k | i = 1, 2, 3; j = 1, 2; k = 1, 2}. If we arrange the data according to the ordered multi-index Id(i, j, k), they are x111 , x112 , x121 , x122 , x211 , x212 , x221 , x222 , x311 , x312 , x321 , x322 . If they are arranged by Id(j, k, i), they become x111 , x211 , x311 , x112 , x212 , x312 , x121 , x221 , x321 , x122 , x222 , x322 . 2. Let x = {x1 , x2 , . . , x24 }. If we use λ1 , λ2 , λ3 to express the data in the form ai = aλ1 ,λ2 ,λ3 , then under different Id’s they have different arrangements: (a) Using the ordered multi-index Id(λ1 , λ2 , λ3 ; 2, 3, 4), the elements are arranged as x111 x121 x131 ..

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