Recently, there has been an explosion of interest in the study of Schrijdinger operators and Jacobi matrices with almost periodic potential (see, e.g., the review [ 161). The general belief is that generically the spectrum is a Cantor set, i.e., a nowhere dense perfect, closed set. Since it is easy to prove that the spectrum is closed and perfect (see, e.g., [2]), the key is to prove that the s...

A novel method to detect and correct inaccuracies in a set of unconstrained dense correspondences between two images is presented. Starting with a robust, general-purpose dense correspondence algorithm, an initial pose estimate and dense 3D scene reconstruction are obtained and bundle-adjusted. Reprojection errors are then computed for each correspondence pair, which is used as a metric to dist...

We study the space H(d) of continuous Z-actions on the Cantor set, particularly questions on the existence and nature of actions whose isomorphism class is dense (Rohlin’s property). Kechris and Rosendal showed that for d = 1 there is an action on the Cantor set whose isomorphism class is residual; we prove in contrast that for d ≥ 2 every isomorphism class in H(d) is meager. On the other hand,...

A permutation in a digraph $G=(V, E)$ is bijection $f:V \rightarrow V$ such that for all $v \in we either have $f$ fixes $v$ or $(v, f(v)) E$. derangement $G$ does not fix any vertex. Bucic, Devlin, Hendon, Horne and Lund proved digraph, the ratio of derangements to permutations at most $1/2$. Answering question posed by Lund, show set possible ratios digraphs dense interval $[0, 1/2]$.

The notions of thin and very thin dense subsets of a product space were introduced by the third author, and in this article we also introduce the notion of a slim dense set in a product. We obtain a number of results concerning the existence and non-existence of these types of small dense sets, and we study the relations among them.

We continue to study topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology τ , which was started in [B-K 1], [B-D-K 1; 2], [B-D-M], and [M]. We prove that the set of periodic homeomorphisms is τ -dense in Homeo(X) and deduce from this result that the topological group (Homeo(X), τ) has the Rokhlin property, i.e., there exists a...