By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

Research and regulate of Boolean Networks offers a scientific new method of the research of Boolean keep watch over networks. the basic software during this procedure is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality may be expressed as a standard discrete-time linear process. within the gentle of this linear expression, convinced significant matters pertaining to Boolean community topology – fastened issues, cycles, brief instances and basins of attractors – could be simply printed by way of a suite of formulae. This framework renders the state-space method of dynamic keep an eye on 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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**Example text**

X1 yn , x2 yn , . . , xm yn )T . They both consist of {xi yj }. However, in X ⊗ Y the elements are arranged in the order of Id(i, j ; m, n), while in Y ⊗ X the elements are arranged in the order of Id(j, i; n, m). 1 we have Y ⊗ X = W[m,n] (X ⊗ Y ). 46) It is easy to check that XY = X ⊗ Y , so we have Y X = W[m,n] XY. 47) The following proposition comes from the definition. 6 1. Let X = (xij ) be a set of data arranged as a column vector by the ordered multiindex Id(i, j ; m, n). Then W[m,n] X is a column with the same data, arranged by the ordered multi-index Id(j, i; n, m).

1, respectively. We call this the extended logic. 12. There are several other types of 3-valued logic. 31). 13 of [3]. 13, one easily sees that the Kleene–Dienes logic is the same as the extended logic. Throughout this book, our default multivalued logic is the extended logic, unless otherwise stated. References 1. : An Algebraic Introduction to Mathematical Logic. Springer, New York (1975) 2. : Logic for Mathematicians. Cambridge University Press, Cambridge (1988). Revised edn. 3. : Fuzzy Logic and Neural Network.

In fact, the data should be arranged into one row. Different Id’s, corresponding to certain index permutations, cause certain permutations of the data. For convenience, we now present a brief introduction to the permutation group. 1 Multiple-Dimensional Data 25 group called the kth order permutation group. We use 1, . . , k to denote the k elements. If we suppose that k = 5, then S5 consists of all possible permutations of five elements: {1, 2, 3, 4, 5}. An element σ ∈ S5 can be expressed as ⎡ ⎤ 1 2 3 4 5 σ = ⎣ ↓ ↓ ↓ ↓ ↓ ⎦ ∈ S5 .