A continuous open finite-to-one surjection f preserves Cantor rank. If the domain of f is a Hausdorff compact space, Cantor degree variations are bounded by the maximal size of the finite fibres. If the fibres are infinite, we show inequalities involving the maximal and minimal rank of the fibres. A closed preorder on a denumerable Hausdorff compact space is the intersection of clopen preoders....

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...

In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) hyperbolic basic set $C^2$-robust homoclinic tangency the largest codimension for any dimensional manifold, using blenders. The former implies that an analog Moreira's theorem on real line does not hold dimension. latter solves question posed by Barrientos A.Rai...

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set “cancels randomness” in the sense that some of its members, when added to Martin-Löf random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructiv...

in this paper‎, ‎we give a complete proof of theorem 4.1(ii) and a new‎ ‎elementary proof of theorem 4.1(i) in [li and shen‎, ‎on the‎ ‎intersection of the normalizers of the derived subgroups of all‎ ‎subgroups of a finite group‎, ‎ j‎. ‎algebra, ‎323  (2010) 1349--1357]‎. ‎in addition‎, ‎we also give a generalization of baer's theorem‎.

Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of poin...

A well known theorem of Cantor asserts that the cardinal of the power-set of a given set always exceeds the cardinal of the original set. An analogous result for sets having additional structure is the well known theorem that the set of initial segments of a well ordered set always has order type greater than the original set. These two theorems suggest that there should be a similar result for...

This paper focuses on the relationship between an $L$-subset and the system of level-elements induced by it, where the underlying lattice $L$ is a complete residuated lattice and the domain set of $L$-subset is an $L$-partially ordered set $(X,P)$. Firstly, we obtain the sufficient and necessary condition that an $L$-subset is represented by its system of level-elements. Then, a new representat...