We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, of the natural n...

The creation and propagation of jump discontinuities in the solutions of semilinear strictly hyperbolic systems is studied in the case where the initial data has a discrete set, {xi}~= 1, of jump discontinuities. Let S be the smallest closed set which satisfies: (i) S is a union of forward characteristics. (ii) S contains all the forward characteristics from the points {xi}~= 1" (iii) if two fo...

In Cantor’s original proof of the uncountability of the reals (not the diagonalization argument), he constructs, given any countable sequence of real numbers, a real number not in the sequence. When we apply this argument to a certain standard enumeration of the rationals, the real number we produce will necessarily be irrational. Using some planar geometry, including Pick’s theorem on the numb...

Let G be a topological group acting continuously on an infinite compact space X. Suppose the dynamical system (X, G) is minimal, i.e., suppose that every point in X has a dense G-orbit. We show that X is coabsolute with a Cantor space if G is ω-bounded. This generalizes a theorem of Balcar and B laszczyk [1].

We prove the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. This result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems. 2000AMS subject classification: 37K55, 35L05.

This document presents the formalization of introductory material from recursion theory — definitions and basic properties of primitive recursive functions, Cantor pairing function and computably enumerable sets (including a proof of existence of a one-complete computably enumerable set and a proof of the Rice’s theorem).

We discuss the isomorphism problem for both C* and smooth crossed products by minimal diffeomorphisms. For C* crossed products, examples demonstrate the failure of the obvious analog of the Giordano-PutnamSkau Theorem on minimal homeomorphisms of the Cantor set. For smooth crossed products, there are many open problems.

In this paper we define a new class of metric spaces, called multimodel Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz map from one multi-model Cantor set to another has constant Radon-Nikodym derivative on some clopen. We use this to obtain an invariant up to bilipschitz hom...

A ‘symbolic dynamical system’ is a continuous transformation Φ : X−→X of closed perfect subset X ⊆ A, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The function Φ induces a directed graph structure on V, whose geometry reveals information about the dynamical system (X ,Φ). The ‘dimension’ dim(V) is an exponent des...

In the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the ...