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

By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

Research and keep watch over of Boolean Networks offers a scientific new method of the research of Boolean keep watch over networks. the basic instrument during this technique is a singular matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality may be expressed as a standard discrete-time linear method. within the mild of this linear expression, sure significant matters bearing on Boolean community topology – mounted issues, cycles, brief instances and basins of attractors – will be simply published through a collection of formulae. This framework renders the state-space method of dynamic keep an eye on structures appropriate to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean regulate community makes it attainable to enquire easy keep watch over difficulties together with controllability, observability, stabilization, disturbance decoupling and so forth.

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Xi + βYi , . . , Xs−1 , Xs ) = αφ(X1 , X2 , . . , Xi , . . , Xs−1 , Xs ) + βφ(X1 , X2 , . . , Yi , . . , Xs−1 , Xs ). 5) shows the linearity of φ with respect to each vector argument. Choosing a basis of V , {e1 , e2 , . . , en }, the structure constants of φ are defined as φ(ei1 , ei2 , . . ,is , ij = 1, 2, . . , n, j = 1, 2, . . , s. ,is | i1 , . . , is = 1, 2, . . , n}, uniquely determine φ. Conventionally, φ is called a tensor, where s is called its covariant degree. 22 2 Semi-tensor Product of Matrices It is clear that for a tensor with covariant degree s, its structure constants form a set of s-dimensional data.

For the remainder of this section we mainly consider k-valued logic. First, we define some unary operators. It is easy to see that there are k k unary operators. We define some which will be useful in the sequel: (1) “negation”, ¬, (2) “rotator”, k , (3) “i-confirmer”, i,k , i = 1, 2, . . , k. 8 1. Let p = i k−1 . Then ¬p = 2. Let p = i k−1 . (k − 1) − i . 35) Then kp = i,k p = i−1 k−1 , 1, i > 0, i = 0. 36) 3. p, p = 0, p = k−i k−1 , k−i k−1 . 11 shows the truth values of these unary operators.

8), we have p = (λ1 − 1)n2 n3 + (λ2 − 1)n3 + λ3 = 1 · 3 · 4 + 0 + 4 = 16. Hence x214 = x16 . 3. In the order of Id(λ2 , λ3 , λ1 ; 3, 4, 2), the data are arranged as x111 x113 .. x211 x213 x112 x114 x212 x214 x131 x133 x231 x233 x132 x134 x232 x234 . For this index, if we want to use the formulas for conversion between natural multi-index and single index, we can construct an auxiliary natural multiindex yΛ1 ,Λ2 ,Λ3 , where Λ1 = λ2 , Λ2 = λ3 , Λ3 = λ1 and N1 = n2 = 3, N2 = n3 = 4, N3 = n1 = 2. Then, bi,j,k is indexed by (Λ1 , Λ2 , Λ3 ) in the order of Id(Λ1 , Λ2 , Λ3 ; N1 , N2 , N3 ).

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