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By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

Research and regulate of Boolean Networks provides a scientific new method of the research of Boolean keep watch over networks. the basic instrument during this procedure is a unique 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 procedure. within the mild of this linear expression, definite significant matters bearing on Boolean community topology – fastened issues, cycles, brief instances and basins of attractors – should be simply published by way of a suite of formulae. This framework renders the state-space method of dynamic keep an eye on platforms appropriate to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean regulate community makes it attainable to enquire simple regulate difficulties together with controllability, observability, stabilization, disturbance decoupling and so forth.

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73) 2. Let Z ∈ Rt be a column vector. Then ZA = W[m,t] AW[t,n] Z = (It ⊗ A)Z. 74) 3. Let X ∈ Rm be a row vector. Then T XT A = Vr (A) X. 75) 4. Let Y ∈ Rn be a row vector. Then AY = Y T Vc (A). 76) 5. Let X ∈ Rm be a column vector and Y ∈ Rn a row vector. Then XY = Y W[m,n] X. 19 Let A ∈ Mm×n and B ∈ Ms×t . 18 Assume ⎡ a A = 11 a21 a12 , a22 where m = n = 2, s = 3 and t = 2. Then ⎡ 1 0 0 ⎢0 0 1 ⎢ ⎢0 0 0 W[3,2] = ⎢ ⎢0 1 0 ⎢ ⎣0 0 0 0 0 0 b11 B = ⎣ b21 b31 0 0 0 0 1 0 0 0 1 0 0 0 ⎤ b12 b22 ⎦ , b32 ⎤ 0 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎦ 1 A.

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. Next, we define some binary operators. 9 Let p and q be two k-valued logical variables. 5 0 0 1 1 1 1 1 1 and their conjunction as p ∧ q = min(p, q). 9 is a natural generalization of Boolean logic. When k = 2, it is obvious that these definitions of disjunction and conjunction coincide with those in Boolean logic.

16) realized the product of 2-dimensional data (a matrix) with 1-dimensional data by using the product of two sets of 1-dimensional data. If, in this product, 2-dimensional data can be converted into 1-dimensional data, we would expect that the same thing can be done for higher-dimensional data. 14) because it allows the product of higher-dimensional data to be taken. Let us see one more example. 8 Let U , V , and W be m-, n-, and t -dimensional vector spaces, respectively, and let F ∈ L(U × V × W, R).

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