By George Elmer Forsythe, Michael A. Malcolm, Cleve B. Moler
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1) synchronizes if the following two conditions are satisfied: (1) limt→∞ ui − uj = 0 for all i, j, (2) There exists an n by n matrix U ∈ Ws such that (U ⊗ V )(G(t) ⊗ D(t) − I ⊗ Y (t)) 0 for all t. Proof: We prove this via Lyapunov’s second method. Construct the Lyapunov function g(x) = 12 xT (U ⊗ V )x.
2 Schematic diagram of a neural network. Transportation networks An excellent example of a man-made network is the transportation network, for instance a network of roads connecting cities, or the network of (passenger or freight) airplane flights between cities. Analyzing and understanding such networks is of great utility to modern society and several problems regarding such networks are notoriously hard. For instance, in the Traveling Salesman Problem (TSP) which is NP-complete, the goal is to find the shortest path which starts at one city, visits all cities once and return to the starting city.
However, for non-regular and the eigenvalues of L and L ˜ and L will have different properties. 3 this means that λ2 (L) ≤ 2n n 2 r(L) ≤ n 2 = 8(n−1) kn2 8 k(n+1) 8 n 8 1 n− n , n even 8 ≤ , n odd n− 1 n 8 , n even ≤ , n odd k(n + 1) The reason for studying the two cases is that C2k gives better bounds 1 when k grows slower than n 3 and the construction in [Atay et al. (2006)] 1 gives better bounds when k grows faster than n 3 . In either case, λ2 , r → 0 as n → ∞. 1). 37. In conclusion, whereas λ2 and r → 0 as n → ∞ for Construction 1, they remain bounded from below for Construction 2.
Computer methods for mathematical computations by George Elmer Forsythe, Michael A. Malcolm, Cleve B. Moler