By David Eppstein

The concentration of this e-book is a mathematical constitution modeling a actual or organic process that may be in any of a few `states.' each one country is characterised via a suite of binary positive aspects, and differs from another neighbor country or states by way of only one of these characteristic. an easy instance of a `state’ is a partial resolution of a jigsaw puzzle, which might be reworked into one other partial answer or into the ultimate resolution simply by including or elimination a unmarried adjacent piece. The evolution of one of these approach through the years is taken into account. one of these constitution is analyzed from algebraic and probabilistic (stochastic) standpoints.

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**Example text**

14) T|U = {τ|U τ ∈ U}. We refer to the pair (S|U , T|U ) as the projection of (S, T) under U. 5 Lemma. (i) For each τ ∈ U, τ|U is a well-deﬁned function. (ii) (S|U , T|U ) is a token system. (iii) τ|U = τ˜|U for any τ ∈ U. Proof. (i) Suppose that Q, M ∈ [S], with Qτ = R, M τ = W , Q ∼ R and M ∼ W . We have to show that [R] = [W ]. 4 (with τ = m), Qτ = R and M τ = W yield {τ } = R \ Q = W \ M ⊆ R ∩ W . 15) that R W = Q M ⊆ T \ U ; so R ∼ W , and [R] = [W ] follows. (ii) This follows immediately from (i) and the fact that U = ∅.

1: for any two adjacent states Y and Z, we have1 |Y Z| = d(Y , Z) = 2. 4, this equation holds in fact for any medium. 5 generalizes that property. 1 Wellgradedness We begin by recalling some basic set-theoretical facts. We omit the proofs (cf. 5 at the end of this chapter). 1 We recall that d(P, Q) denotes the symmetric diﬀerence distance between two sets P and Q; cf. 5). 1 Lemma. For any three sets P , Q and S, we have P Consequently, if both P Q ⊆ (P S and S S) ∪ (S Q). 1) Q; thus d(P, Q) ≤ d(P, S) + d(S, Q), with moreover d(P, Q) = d(P, S) + d(S, Q) ⇐⇒ P ∩ Q ⊆ S ⊆ P ∪ Q.

4 is a medium. 20 Prove that a ﬁnite medium (S, T) can be given 2|T|/2 diﬀerent orientations. 21 Given an example of a non rooted medium with a countable number of tokens and such that each of its states has a ﬁnite positive content. Can we have a medium having some, but not all, of its states having ﬁnite positive contents? 22 To Axioms [Ma] and [Mb] of a medium, add the Axiom [Mc] stating that the tokens are commutative transformations. Is [Ma] consistent with [Ma] and [Mb]? What are the consequences of a token system satisfying [Ma], [Mb] and [Mc]?