We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once punctured torus and the 4–times punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic laminations, when it is endowed with the natural metric which, for small distances, is ...

A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel’s Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov-Hausdorff distance is zero. We establish a completeness theorem. As appli...

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we asso...

The Hausdorff dimension of a product X × Y can be strictly greater than that of Y , even when the Hausdorff dimension of X is zero. But when X is countable, the Hausdorff dimensions of Y and X × Y are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than “being countable” and stronger than “having Hausdorff dimension zero”. Fremlin asked whether it ...

Comparing DNA or protein sequences plays an important role in the functional analysis of genomes. Despite many methods available for sequences comparison, few methods retain the information content of sequences. We propose a new approach, the Yau-Hausdorff method, which considers all translations and rotations when seeking the best match of graphical curves of DNA or protein sequences. The comp...

The standard contact structure ξst on the unit sphere S 2n−1 = ∂D2n ⊂ Cn can be defined as the hyperplane field of complex tangencies. In other words, if we write J0 for the complex structure on (the tangent bundle of) C n, then ξst(p) = TpS 2n−1 ∩ J0(TpS 2n−1) for all p ∈ S2n−1. Conversely, given any contact structure ξ = kerα on S2n−1 (with α a 1-form such that α ∧ (dα)n−1 is a volume form de...

Intracerebral grafting techniques of fetal neural cells have been used essentially with two main types of lesion paradigms, namely damage to long projection systems, in which the source and the target are clearly separate, and damage to neurons that are involved in local circuits within a small (sub)region of the brain. With the first lesion paradigm, grafts placed homotopically (in the source)...

The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all homotopically self-contained. The left half of this statement essentially means that any functor that looks like it could be a tensor product (or product, or sma...

Hausdorff operators (Hausdorff summability methods) appeared long ago aiming to solve certain classical problems in analysis. Modern theory of Hausdorff operators started with the work of Siskakis in complex analysis setting and with the work of Georgakis and Liflyand-Móricz in the Fourier transform setting. While Hausdorff operators for power series are still studied mostly in dimension one, t...