Separability generalizes Dirac's theorem

Separability Generalizes Dirac's Theorem

In our study of the extremities of a graph, we define a moplex as a maximal clique module the neighborhood of which is a minimal separator of the graph. This notion enables us to strengthen Dirac’s theorem (Dirac, 1961): ‘‘Every non-clique triangulated graph has at least two non-adjacent simplicial vertices’’, restricting the definition of a simplicial vertex; this also enables us to strengthen...

Formalizing the separability condition in Bell’s Theorem

The nonseparability of physical systems is often invoked in philosophical analyses of what has come to be known as Bell’s Theorem. Until recently, the formalization of the notion of separability was assumed to be unproblematic, equivalent to that of outcome independence (Jarrett incompleteness). Although this equivalence has been called into question, an alternative has not yet been specified w...

A Skorohod representation theorem without separability

Let (S, d) be a metric space, G a σ-field on S and (μn : n ≥ 0) a sequence of probabilities on G. Suppose G countably generated, the map (x, y) 7→ d(x, y) measurable with respect to G ⊗ G, and μn perfect for n > 0. Say that (μn) has a Skorohod representation if, on some probability space, there are random variables Xn such that Xn ∼ μn for all n ≥ 0 and d(Xn, X0) P −→ 0. It is shown that (μn) h...

On the Kurosh Theorem and Separability

A Survey on Skorokhod Representation Theorem without Separability

Let S be a metric space, G a σ-field of subsets of S and (μn : n ≥ 0) a sequence of probability measures on G. Say that (μn) admits a Skorokhod representation if, on some probability space, there are random variables Xn with values in (S,G) such that Xn ∼ μn for each n ≥ 0 and Xn → X0 in probability. We focus on results of the following type: (μn) has a Skorokhod representation if and only if J...