In 1940 Paul Erdős introduced the ‘rational Hilbert space’, which consists of all vectors in the real Hilbert space 2 that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces E of the Banach spaces p ...

Let M t , v r ( n m ) 2 ? < be the collection of self-affine carpets with expanding matrix diag which are totally disconnected, possessing vacant rows and uniform horizontal fibers. In this paper, we introduce a notion structure tree metric space, thanks to new notion, completely characterize when two in Lipschitz equivalent.

We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that l-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Ni...

Rotation-torsion energy levels of acetaldehyde calculated from a spectroscopic Hamiltonian obtained from the fit of high-resolution microwave and far-infrared spectra, exhibit anomalously large splittings for some asymmetry doublets. It is shown that these anomalies are caused by chaos-enhanced quantum dynamical tunneling between classically disconnected regions of the phase space. A major cons...

We reformulate the notion of connectedness for compact metric spaces in a manner that may be implemented computationally. In particular, our techniques can distinguish between sets that are connected; have a nite number of connected components; have in nitely many connected components; or are totally disconnected. We hope that this approach will prove useful for studying structures in the phase...