By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

Research and keep watch over of Boolean Networks provides a scientific new method of the research of Boolean keep an eye on networks. the elemental software during this procedure is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality might be expressed as a traditional discrete-time linear approach. within the mild of this linear expression, definite significant concerns pertaining to Boolean community topology – mounted issues, cycles, brief instances and basins of attractors – might be simply printed via a collection of formulae. This framework renders the state-space method of dynamic keep an eye on platforms acceptable to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean keep an eye on community makes it attainable to enquire easy regulate difficulties together with controllability, observability, stabilization, disturbance decoupling and so on.

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

**Sample text**

A question which then naturally arises is how to arrange three-dimensional data. A cubic matrix approach has been proposed for this purpose [1, 2] and has been used in some statistics problems [8–10], but, in general, has not been very successful. , on paper), (2) the conventional matrix product does not apply, hence some new product rules have to be produced, (3) it is very difficult to generalize this approach to even higher-dimensional cases. The basic idea concerning the semi-tensor product of matrices is that no matter what the dimension of the data, they are arranged in one- or two-dimensional form.

Its rows and columns are labeled by double index (i, j ), the columns are arranged by the ordered multi-index Id(i, j ; m, n), and the rows are arranged by the ordered multi-index Id(j, i; n, m). The element at position [(I, J ), (i, j )] is then I,J = w(I J ),(ij ) = δi,j 1, I = i and J = j, 0, otherwise. 16 1. Letting m = 2, n = 3, the swap matrix W[m,n] can be constructed as follows. Using double index (i, j ) to label its columns and rows, the columns of W are labeled by Id(i, j ; 2, 3), that is, (11, 12, 13, 21, 22, 23), and the rows of W are labeled by Id(j, i; 3, 2), that is, (11, 21, 12, 22, 13, 23).

Ik and arranged as follows: Let it , t = 1, . . , k, run from 1 to nt with the order that t = k first, then t = k − 1, and so on, until t = 1. ,βk if and only if there exists 1 ≤ j ≤ k such that αi = β i , i = 1, . . , j − 1, αj < βj . If the numbers n1 , . . , nk of i1 , . . , ik are equal, we may use Id(i1 , . . , ik ; n) := Id(i1 , . . , ik ; n, . . , n). If ni are obviously known, the expression of Id can be simplified as Id(i1 , . . , ik ) := Id(i1 , . . , ik ; n1 , . . , nk ).