By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li
Research and regulate of Boolean Networks provides a scientific new method of the research of Boolean regulate networks. the elemental software during this strategy is a singular matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality will be expressed as a traditional discrete-time linear method. within the mild of this linear expression, yes significant concerns pertaining to Boolean community topology – fastened issues, cycles, temporary instances and basins of attractors – will be simply printed via a collection of formulae. This framework renders the state-space method of dynamic keep an eye on structures acceptable to Boolean regulate networks. The bilinear-systemic illustration of a Boolean keep an eye on community makes it attainable to enquire uncomplicated keep an eye on 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
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. For instance, Ap×qr Br×s Cqst×l = (A B)p×qs Cqst×l = (A B C)pt×l . In the first product, r is deleted, and in the second product, qs is deleted.
Obviously, ψ is a tensor of covariant degree 4. Next, we consider a more general case. Let μ : V → R be a linear mapping on V , μ(ei ) = ci , i = 1, . . , n. Then, μ can be expressed as μ = c1 e1∗ + c2 e2∗ + · · · + cn en∗ , where ei∗ : V → R satisfies ei∗ (ej ) = δi,j = 1, 0, i = j, i = j. It can be seen easily that the set of linear mappings on V forms a vector space, called the dual space of V and denoted by V ∗ . Let X = x1 e1 + x2 e2 + · · · + xn en ∈ V and μ = μ1 e1∗ + μ2 e2∗ + · · · + μn en∗ ∈ V ∗ .
Hence the data arrangement is important for the semitensor product of data. 1), the elements of x are labeled by k indices. Moreover, suppose the elements of x are arranged in a row (or a column). It is said that the data are labeled by indices i1 , . . , ik according to an ordered multi-index, denoted by Id or, more precisely, Id(i1 , . . , ik ; n1 , . . , nk ), if the elements are labeled by i1 , . . , 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.