By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

The Boolean community has develop into a strong device for describing and simulating mobile networks within which the weather behave in an on–off style. research and keep watch over of Boolean Networks provides a scientific new method of the research of Boolean regulate networks. the basic instrument during this method is a unique matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality may be expressed as a standard discrete-time linear approach. within the mild of this linear expression, convinced significant matters bearing on Boolean community topology – mounted 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 watch over networks. The bilinear-systemic illustration of a Boolean regulate community makes it attainable to enquire easy keep an eye on difficulties together with controllability, observability, stabilization, disturbance decoupling, id, optimum keep watch over, and so forth.

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

Analysis and regulate of Boolean Networks could be a basic reference for researchers in platforms biology, keep watch over, platforms technological know-how and physics. The ebook used to be built for a quick direction for graduate scholars and is acceptable for that function. machine scientists and logicians can also locate this ebook to be of curiosity.

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

**Example text**

2. A logical system is called a fuzzy logic if its logical variables may take any values from Df . 11 k-valued unary operators ¬p p k (p) i,k (p) 1 0 (k − 2)/(k − 1) 0 (k − 2)/(k − 1) .. 1/(k − 1) .. (k − 3)/(k − 1) .. 0 .. (i − 1)/(k − 1) .. (k − i)/(k − 1) .. (i − 2)/(k − 1) .. 1 .. 2/(k − 1) (k − 3)/(k − 1) 1/(k − 1) 0 1/(k − 1) (k − 2)/(k − 1) 0 0 0 1 1 0 3. A logical operator σ : Dks → Dk is an s-ary k-valued logical operator; a logical operator σ : Dfs → Df is an s-ary fuzzy logical operator.

Define a left semi-tensor product, denoted by , as p T X= T i xi ∈ R n . 6 and propose another algorithm. 6 continued) We rearrange the structure constants of F into a row as T : Vr (S) = (s11 , . . , s1n , . . , 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.

In the following example we give an example of 3-dimensional data. 1 Consider R3 , with its canonical basis {e1 , e2 , e3 }. Any vector X ∈ R3 may then be expressed as X = x1 e1 + x2 e2 + x3 e3 . When the basis is fixed, we simply use X = (x1 , x2 , x3 )T to represent it. From simple vector algebra we know that in R3 there is a cross product, ×, such that for any two vectors X, Y ∈ R3 we D. 1007/978-0-85729-097-7_2, © Springer-Verlag London Limited 2011 19 20 2 have X × Y ∈ R3 , defined as follows: ⎛⎡ e1 X × Y = det⎝⎣ x1 y1 e2 x2 y2 Semi-tensor Product of Matrices ⎤⎞ e3 x3 ⎦ ⎠ .