We explore the spatial frequency property of the far-field diffraction intensity pattern of an optical vortex after passing through an annular aperture in detail. The result reveals that the spatial spectrum consists of alternate bright and dark rings and, in particular, the number of the bright rings is nicely identical to the absolute value of the topological charge. Based on this property, w...

Some conditions equivalent to a strong quasi-divisor property (SQDP) for a partly ordered group G are derived. It is proved that if G is defined by a family of t-valuations of finite character, then G admits an SQDP if and only if it admits a quasi-divisor property and any finitely generated t-ideal is generated by two elements. A topological density condition in topological group of finitely g...

Let (X, d) be a compact metric space, f : X 7→ X be a continuous map satisfying a property we call almost specification (which is slightly weaker than the g-almost product property of Pfister and Sullivan), and φ : X 7→ R be a continuous function. We show that the set of points for which the Birkhoff average of φ does not exist (which we call the irregular set) is either empty or has full topol...

We found a quantum cohomology, homology of quantum Hall effect which arises as the invariant property of the Chern-Simons theory of quantum Hall effect and showed that it should be equivalent to the quantum cohomology which arose as the invariant property of topological sigma models. This isomorphism should be related with an equivalence between the supersymmetric-and quantization structures in...

A continuous map between topological f : X → Y is said to satisfy the path-lifting property if for any path p : [0, 1] → Y and any x ∈ f(p(0)) there exists a lifting p̃ of the path p with the intitial value x, i.e. there exists a path p̃ such that f ◦ p̃ = p and p̃(0) = x. Similarly, a smooth map between Riemannian manifolds f : X → Y is said to satisfy the rectifiable path-lifting property if the ...

The building block hypothesis implies that the epistatic property of a given problem must be connected well to the linkage property of the employed representation and crossover operator in the design of genetic algorithms. A good handling of building blocks has much to do with topological linkages of genes in the chromosome. In this paper, we provide a taxonomy of the approaches that exploit to...

Categories of lax (T, V )-algebras are shown to have pullbackstable coproducts if T preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax (T, V )-algebras, with T preserving inverse images. Moreov...

Let (X, d) be a compact metric space, f : X 7→ X be a continuous map satisfying a property we call almost specification (which is slightly weaker than the g-almost product property of Pfister and Sullivan), and φ : X 7→ R be a continuous function. We show that the set of points for which the Birkhoff average of φ does not exist (which we call the irregular set) is either empty or has full topol...

Several new characterizations of strongly irresolvable topological spaces are found and precise relationships are noted between strong irresolvability, hereditary irresolvability, and submaximality. It is noted that strong irresolvablity is a faint topological property, while neither hereditary irresolvablity nor submaximality are semitopological.