Given a compact and connected metric space (continuum) X, we study topological and dynamical properties of the hyperspace of normal fuzzy sets F1(X) equipped with the Hausdorff, endograph or sendograph metric. Among the many results we show that it is contractible, path connected, locally contractible, locally path connected, locally simply connected and locally connected. For the endograph met...

We show that Bowen’s equation, which characterises the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered c...

We prove that the semigroup operation of a topological semigroup S extends to a continuous semigroup operation on its the Stone-Čech compactification βS provided S is a pseudocompact openly factorizable space, which means that each map f : S → Y to a second countable space Y can be written as the composition f = g ◦ p of an open map p : X → Z onto a second countable space Z and a map g : Z → Y ...

In this paper, two significant weaknesses of locally linear embedding (LLE) applied to computer vision are addressed: ‘‘intrinsic dimension’’ and ‘‘eigenvector meanings’’. ‘‘Topological embedding’’ and ‘‘multi-resolution nonlinearity capture’’ are introduced based on mathematical analysis of topological manifolds and LLE. The manifold topological analysis (MTA) method is described and is based ...

Let X be an uncountable compact metrizable space of topological dimension zero. Given any a ∈ [0,∞] there is a homeomorphism on X whose topological entropy is a.

In this paper, we study five-dimensional Dirac fermions of which the extra-dimension is compactified on quantum graphs. We find that there a nontrivial correspondence between matrices specifying boundary conditions at vertex graphs and zero-dimensional Hamiltonians in gapped free-fermion systems. Based correspondence, provide complete topological classification terms noninteracting fermionic ph...

Condensed matter systems with topological order and metamaterials with left-handed chirality have attracted recently extensive interests in the fields of physics and optics. So far the topological order and chirality of electromagnetic wave are two independent concepts, and there is no work to address their connection. Here we propose to establish the relation between the topological order in c...

Let G be a countable infinite amenable group and P be a polyhedron. We give a construction of minimal subshifts of P with arbitrarily mean topological dimension less than dimP .

The main goal of this paper is to show that the inductive dimension of a σ-compact metric space X can be characterized in terms of algebraical sums of connectivity (or Darboux) functions X → R. As an intermediate step we show, using a result of Hayashi [9], that for any dense Gδ set G ∈ R the union of G and some k homeomorphic images of G is universal for k-dimensional separable metric spaces. ...

Irreversible processes can be described in Open non-equilibrium thermodynamic systems, of topological dimension 4. By means of Continuous Topological evolution, such processes can cause local decay to Closed non-equilibrium thermodynamic states, of topological dimension 3. These topologically coherent, perhaps deformable, regions or states of one or more components appear to "emerge" as compact...