The topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits has proven to be a powerful method, however knot theory can only be applied to three-dimensional systems. Still, the core principles upon which this approach is built--determinism and continuity--apply in any dimension. We propose an alternative framework in which these principles are enforc...

A system of di®erential forms will establish a topology and a topological structure on a domain of independent variables such that is possible to determine which maps or processes acting on the system are continuous. Perhaps the most simple topology is that generated by the existence of a single 1-form of Action, its Pfa® sequence of exterior di®erentials, and their intersections. In such a top...

By dimensional reduction, Einstein-Hermitian equations of n + 1 dimensional closed Kähler manifolds lead to vortex equations of n dimensional closed Kähler manifolds. A Yang-Mills-Higgs functional to unitary bundles over closed Kähler manifolds has topological invariance by adding the additional terms which have ghost fields. Henceforth we achieve the matter (Higgs field) coupled topological fi...

Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space and to a conditional variational principle. We use this to recover information on the topological entropy and Hausdorff dimension of the level sets. Our approa...

The theoretic ground of a local-topological method for defining a minimal attractor embedding dimension on the basis of matrix decomposition is proposed. The investigation of digital electrocardio signals using local-topological analysis of chaotic attractor trajectories is carried out. The computer confirmation of the theoretical results is presented.

To any germ X of a complex analytic variety with local ring OX one associates the topological Lie algebra Θ(X) = DerOX of vector fields on X. We show that isolated hypersurface singularities X of dimension at least 3 are uniquely determined up to isomorphism by the topological Lie algebra Θ(X).

Topological relations have found applications in many fields such as similarity analysis, spatial query, object matching, inconsistency detection, spatial association rule mining, spatial reasoning, and so on. This paper is concentrated on the hierarchical representation of the topological relations between a line and a region in IR. In this study, line-region relations are classified into basi...

We develop the theory of Krull dimension for S4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for a T1-space to have a finite modal Krull di...

The metric dimension of a compact metric space is defined here as the order of growth of the exponential metric entropy of the space. The metric dimension depends on the metric, but is always bounded below by the topological dimension. Moreover, there is always an equivalent metric in which the metric and topological dimensions agree. This result may be used to define the topological dimension ...

Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although they are well understood, a thermodynamic description of these materials remained elusive, firstly because the edges yield a non-extensive contribution to the thermodynamic potential, and secondly because topological field theories involve no...