§0. Introduction Let X be a compact metric space and T : X → X is continuous transformation. Then the dynamics of T is a widely studied subject. In particular, h(T ) the entropy of T is a well understood object. Let Γ ⊂ X ×X be a closed set. Then Γ induces certain dynamics and entropy h(Γ). If X is a finite set then Γ can be naturally viewed as a directed graph. That is, if X = {1, ..., n} then...

We discuss the connection of additive set functions to dual function spaces by means of integrals. Isometrical isomorphisms are presented between: a) bounded contents and the dual of uniformly approximable functions, b) charges on Baire algebras and the duals of bounded, continuous functions over A-topological spaces, c) Hewitt measures on Baire σ-algebras and the duals of continuous functions ...

If all functions (N → N) → N are continuous then 0 = 1. This is a theorem of intensional (and hence of extensional) intuitionistic dependent-type theories, with existence in the formulation of continuity expressed as a Σ type via the Curry-Howard interpretation. With an intuitionistic notion of anonymous existence, defined as the propositional truncation of Σ, it is consistent that all such fun...

This note contains two main results. (1) (Discrete time) Suppose S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose the initial option prices agree with the Cox–Ross–Rubinstein binomial tree model.) Then S is a geometric simple random walk. (2) (Continuous time) Suppose S = S0eσX−σ 〈X〉/2 is a cont...

We consider an arbitrary linear elliptic first–order differential operator A with smooth coefficients acting on sections of a complex vector bundle E over a compact smooth manifold M with smooth boundary Σ. We describe the analytic and topological properties of A in a collar neighborhood U of Σ and analyze various ways of writing A U in product form; we discuss the sectorial projections of the ...

To each closed subset S of a finite dimensional Euclidean space corresponds a σ-ideal of sets J (S) which is σ-generated over S by the convex subsets of S. The set-theoretic properties of this ideal hold geometric information about the set. We discuss the relation of reducibility between convexity ideals and the connections between convexity ideals to other types of ideals, such as the ideals w...

We investigate some subtle and interesting phenomena in the du-ality theory of operator spaces and operator algebras. In particular, we give several applications of operator space theory, based on the surprising fact that certain maps are always weak *-continuous on dual operator spaces. For example , if X is a subspace of a C *-algebra A, and if a ∈ A satisfies aX ⊂ X and a * X ⊂ X, and if X i...