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A coupled cell system is a network of dynamical systems, or 'cells', coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can… (More)

Let M be a monoid (e.g. N, Z, or Z D), and A an abelian group. A M is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F : A M −→ A M that commutes with all shift maps. Let µ be a (possibly nonstationary) probability measure on A M ; we develop sufficient conditions on µ and F so that the sequence {F N µ} ∞ N=1… (More)

- Marcus Pivato
- 2010

Do you want a rigorous book that remembers where PDEs come from and what they look like? This highly visual introduction to linear PDEs and initial/boundary value problems connects the theory to physical reality, all the time providing a rigorous mathematical foundation for all solution methods. Readers are gradually introduced to abstraction – the most… (More)

- Marcus Pivato
- 2008

A right-sided, nearest neighbour cellular automaton (RNNCA) is a continuous transformation Φ : A Z −→A Z determined by a local rule φ : A {0,1} −→A so that, for any a ∈ A Z and any z ∈ Z, Φ(a) z = φ(a z , a z+1). We say that Φ is bipermutative if, for any choice of a ∈ A, the map A ∋ b → φ(a, b) ∈ A is bijective, and also, for any choice of b ∈ A, the map A… (More)

- MARCUS PIVATO
- 2006

Let M = Z D be a D-dimensional lattice, and let (A, +) be an abelian group. A M is then a compact abelian group under componentwise addition. A continuous function : A M −→ A M is called a linear cellular automaton if there is a finite subset F ⊂ M and non-zero coefficients ϕ f ∈ Z so that, for any a ∈ A M , (a) = f∈F ϕ f · σ f (a). Suppose that µ is a… (More)

- Marcus Pivato
- 2008

If A is a finite alphabet, U ⊂ Z D , and µ U is a probability measure on A U that " looks like " the marginal projection of a stationary stochastic process on A Z D , then can we " extend " µ U to such a process? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when… (More)

- Marcus Pivato
- 2008

If X is a discrete abelian group and A a finite set, then a cellular automaton (CA) is a continuous map F : A X −→ A X that commutes with all X-shifts. If φ : A −→ R, then, for any a ∈ A X , we define Σφ(a) = x∈X φ(a x) (if finite); φ is conserved by F if Σφ is constant under the action of F. We characterize such conservation laws in several ways, deriving… (More)

If M is a monoid, and A is an abelian group, then A M is a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F : A M −→ A M that commutes with all shift maps. If F is diffusive, and µ is a harmonically mixing (HM) probability measure on A M , then the sequence {F N µ} ∞ N =1 weak*-converges to the Haar measure on A M , in… (More)

Let A Z be the Cantor space of bi-infinite sequences in a finite alphabet A, and let σ be the shift map on A Z. A cellular automaton is a continuous, σ-commuting self-map Φ of A Z , and a Φ-invariant subshift is a closed, (Φ, σ)-invariant subset S ⊂ A Z. Suppose a ∈ A Z is S-admissible everywhere except for some small region we call a defect. It has been… (More)