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Information theory and computation theory can also be used to define an entropy for individual microstates in spatially extended systems [17, 28]. In a microscopic view, information or entropy quantified in terms of the Gibbs H-function is a globally conserved quantity due to Liouville’s theorem. A natural question to consider is to what extent this statement has a local analogue in spatially extended dynamical systems. This article explores this question for one-dimensional reversible or surjective cellular automata.
9 are not possible for the formal Bargmann-Fock representation. Again, this was one of our main motivations to consider a suitable convergence scheme for the Wick star product. 12. Let A be a ∗ -algebra with unit 1 and let G be a group acting on A by g : A −→ A. e. there exists a unitary (or more general: projectively unitary) representation U of G on the GNS pre-Hilbert space Hω . Then the states ωg with ∗ -automorphisms ωg (a) = (ω ◦ where ψg = Ug∗ ψ1 g )(a) = ψg , π(a)ψg , ∈ Hω are called coherent with respect to G.
F 5. f 6. f are : C ∞ (Cn ) −→ [0, +∞] enjoy the following p, p, m, ,R,S = |α| f m, ,R,S for α ∈ C. p, p, p, + g m, ,R,S ≤ f m, ,R,S + g m, ,R,S . p, p, p, m−1, ,R,S ≤ f m,2 ,R,S and f m−1, ,R,S √ p, p, 2m+2 S! f m+1,2 ,0,0 . m, ,0,S ≤ √ p, p, 2m+2 R! f m+1,2 +1,0,0 . m, ,R,0 ≤ √ √ m+2 p, p, 2m+3 R! 2 S! f m+2,4 +1,0,0 . m, ,R,S ≤ 1. ,R,S p, m,2 +1,R,S . ≤ f Proof. The first part is clear by a simple induction. For the second part the case m = 0 follows directly from Minkowski’s inequality. Then m > 0 is shown inductively by using again Minkowski’s inequality, for both cases of odd and even .