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 provides a scientific new method of the research of Boolean regulate networks. the basic software during this process is a unique matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality could be expressed as a standard discrete-time linear approach. within the gentle of this linear expression, convinced significant concerns referring to Boolean community topology – mounted issues, cycles, temporary instances and basins of attractors – may be simply printed by way of a suite of formulae. This framework renders the state-space method of dynamic keep watch over platforms appropriate to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean regulate community makes it attainable to enquire easy keep an eye on difficulties together with controllability, observability, stabilization, disturbance decoupling and so on.

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Xm S m and take their sum. i Step 2: Multiply m i=1 xi S by Y (which is a standard inner product). It is easy to check that this algorithm produces the same result. Now, in the first step it seems that we have (S 1 · · · S n ) × X. This calculation motivates a new algorithm, which is defined as follows. 5 Let T be an np-dimensional row vector and X a p-dimensional column vector. Split T into p equal blocks, named T 1 , . . , T p , which are 1 × n matrices. Define a left semi-tensor product, denoted by , as p T X= T i xi ∈ R n .

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 an algebra, the structure constants are always a set of 3-dimensional data. Next, we consider an s-linear mapping on an n-dimensional vector space. Let V be an n-dimensional vector space and let φ : V × V × · · · × V → R, satisfying (for s any 1 ≤ i ≤ s, α, β ∈ R) φ(X1 , X2 , . . , αXi + 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 , .

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