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

By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

Research and regulate of Boolean Networks offers a scientific new method of the research of Boolean regulate networks. the elemental instrument during this technique is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality should be expressed as a traditional discrete-time linear approach. within the mild of this linear expression, definite significant matters relating Boolean community topology – mounted issues, cycles, temporary occasions and basins of attractors – will be simply printed through 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 simple keep watch over difficulties together with controllability, observability, stabilization, disturbance decoupling and so on.

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

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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). 41), we have (11) (12) (13) (21) (22) (23) ⎡ ⎤ 1 0 0 0 0 0 (11) ⎢ 0 0 0 1 0 0 ⎥ (21) ⎢ ⎥ ⎢ 0 1 0 0 0 0 ⎥ (12) ⎢ ⎥ W[2,3] = ⎢ .

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).

13) Mg = ⎣ ... ⎦. n n n n r11···1 · · · r11···kn · · · r1k2 ···kn · · · rk1 k2 ···kn Mg is called the payoff matrix of game g. 2 Semi-tensor Product of Matrices We consider the conventional matrix product first. 6 Let U and V be m- and n-dimensional vector spaces, respectively. Assume F ∈ L(U × V , R), that is, F is a bilinear mapping from U × V to R. Denote by {u1 , . . , um } and {v1 , . . , vn } the bases of U and V , respectively. We call S = (sij ) the structure matrix of F , where sij = F (ui , vj ), i = 1, .

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