# New PDF release: A Refinement of the Michelson-Morley Experiment

By Kennedy R.J.

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Let r > 0. A selection function Φ is called r-regular on A ⊂ L(G) if Φ(ϕ) is an r-regular subset of ϕ for all ϕ ∈ A. , on A = {ϕ ∈ L(G) : L(ϕ) = {0}}, where L(ϕ) denotes the periodicity lattice of ϕ (cf. 4). It is inspired by similar constructions in [5] and [30]. 6. SELECTION FUNCTIONS AND THINNING PROCEDURES Fix an enumeration (qn ) of the non-negative rational numbers contained in (0, 1) and let r > 0. 11). A point x ∈ ϕ is in Ψn,r (ϕ) if and only if the extended index function I˜ applied to (ϕ, x) is closer to qn than the extended index function applied to (ϕ, y) for all y ∈ ϕ such that d(x, y) ≤ r.

A factor graph Γ : L(G) → L(G) × L(G × G) is given by a pair of measurable, equivariant mappings Γ = (V, E), where V : L(G) → L(G) satisfies V (ϕ) ⊂ ϕ for all ϕ ∈ L(G), and E : L(G) → L(G × G) is such that E(ϕ) ⊂ V (ϕ) × V (ϕ) for all ϕ ∈ L(G). We call V the vertex mapping and E the edge mapping of Γ. Let us now shortly summarize some basic terminology from graph theory, which refers to a directed graph Γ = (V, E). An edge (x, y) ∈ E connects two vertices, the starting point x ∈ V and the endpoint y ∈ V .

Indeed, let T0 := idL(G) and then inductively Tn+1 (ϕ) := Φrn+1 (Tn (ϕ)), n ∈ N. 16) In the special case, where rn = cn for some c > 1, we will call this thinning procedure the c-exponential thinning procedure. 12. The thinning procedure (Tn ) defined above is (rn )-regular on the aperiodic, locally finite subsets A ⊂ L(G). 7. , equivariant mappings, that map a locally finite point set ϕ to a graph Γ(ϕ) with vertex set V (ϕ) ⊂ ϕ. Many of the recent results on graphs defined on point processes (cf.