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By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

Research and keep an eye on of Boolean Networks offers a scientific new method of the research of Boolean keep watch over networks. the basic instrument during this method is a unique matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality could be expressed as a standard discrete-time linear process. within the mild of this linear expression, sure significant concerns referring to Boolean community topology – fastened issues, cycles, temporary occasions and basins of attractors – might be simply printed via a suite of formulae. This framework renders the state-space method of dynamic keep watch over platforms acceptable to Boolean keep watch over networks. The bilinear-systemic illustration of a Boolean regulate community makes it attainable to enquire uncomplicated keep an eye on difficulties together with controllability, observability, stabilization, disturbance decoupling and so forth.

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Tk ; n1 , . . , nt+1 , nt , . . , nk ). W[m,n] can be constructed in an alternative way which is convenient in some applications. Denoting by δni the ith column of the identity matrix In , we have the following. 7 W[m,n] = δn1 1 δm ··· δnn 1 δm ··· δn1 m δm ··· δnn m . δm For convenience, we provide two more forms of swap matrix: ⎡ T⎤ Im ⊗ δn1 ⎢ ⎥ .. W[m,n] = ⎣ ⎦ . 49) Im ⊗ δnn T and, similarly, 1 m , . . , In ⊗ δm . 50) The following factorization properties reflect the blockwise permutation property of the swap matrix.

31) 36 2 Semi-tensor Product of Matrices 2. Let X ∈ Rn , Y ∈ Rq be column vectors and A ∈ Mm×n , B ∈ Mp×q . Then, (AX) (BY ) = (A ⊗ B)(X Y ). 32) Particularly, (AX)k = (A ⊗ · · · ⊗ A)X k . 33) k 3. Let X ∈ Rm , Y ∈ Rp be row vectors and A, B be matrices (as in 2. above). Then (XA) (Y B) = (X Y )(B ⊗ A). 34) Hence, (XA)k = Xk (A ⊗ · · · ⊗ A). 35) k 4. Consider the set of real kth order homogeneous polynomials of x ∈ Rn and denote it by Bnk . Under conventional addition and real number multiplication, Bnk is a vector space.

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.

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