Related papers: Excursions on Cantor-like Sets
The notion of unboundedly order converges has been recieved recently a particular attention by several authors. The main result of the present paper shows that the notion is efficient and deserves that care. It states that a vector lattice…
Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which…
A set $A\subseteq\mathbb N$ is called $complete$ if every sufficiently large integer can be written as the sum of distinct elements of $A$. In this paper we present a new method for proving the completeness of a set, improving results of…
In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to \'etale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in…
We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence $\Beta=(\beta_n)_{n\in\Z}$ of real numbers greater than one. We…
We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.
Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try…
It was shown by Antunovi\'{c}, Burdzy, Peres, and Ruscher that a Cantor function added to one-dimensional Brownian motion has zeros in the middle $\alpha$-Cantor set, $\alpha \in (0,1)$, with positive probability if and only if $\alpha \neq…
The notions of a proper dyadic subbase and an independent subbase was introduced by H. Tsuiki to investigate in {0, 1, bot}-sequence codings of topological spaces. We show that every separable metrizable space has a proper dyadic subbase…
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
Let G be a group and let O_G denote the set of left orderings on G. Then O_G can be topologized in a natural way, and we shall study this topology to answer three conjectures. In particular we shall show that O_G can never be countably…
In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group $G_2$. While he did not…
It has been shown that Cantor bubble Julia sets can appear in the dynamics of polynomials and their singular perturbations. In this paper, we present a criterion that guarantees the existence of Cantor bubble Julia sets for certain rational…
Given $\rho\in (0,1/4]$, the four corner Cantor set $E\subset \mathbb{R}^{2}$ is a self-similar set generated by the iterated function system \[ \left\{(\rho x, \rho y), \quad(\rho x, \rho y+1-\rho),\quad (\rho x+1-\rho, \rho y),\quad(\rho…
We give multiple descriptions of a topological universe of finitary sets, which can be seen as a natural limit completion of the hereditarily finite sets. This universe is characterized as a metric completion of the hereditarily finite…
As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the…
In various application fields, tensor type data are used recently and then a typical rank is important. Although there may be more than one typical ranks over the real number field, a generic rank over the complex number field is the…
Recently, many structured tensors are defined and their properties are discussed in the literature. In this paper, we introduce a new class of structured tensors, called exceptionally regular tensor, which is relevant to the tensor…
Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,\infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$\breve{C}$ech…
A discrete set in the Euclidian space is almost periodic, if the measure with the unite masses at points of the set is almost periodic in the weak sense. We prove the following result: if A is a discrete almost periodic set and the set A-A…