A Munchausen number is a mathematical curiosity: raise each digit to the power of itself, add them all up, and recover the original number. In the seminal paper on this topic, D. Van Berkel derived a bound on such numbers for any given radix, which means that they can be completely enumerated in principle. We present a simpler argument which yields a bound one half the size and show that a radically different approach would be required for further reductions.

In this paper, we apply the concept of cubic set to commutative ideals of $BCK$-algebras, and then characterize their basic properties. We discuss relations among cubic commutative ideals, cubic subalgebras, and cubic ideals of $BCK$-algebras. We provide a condition for a cubic ideal to be a cubic commutative ideal. We define inverse images of cubic commutative ideals and establish how the inverse images of a cubic commutative ideal becomes a cubic commutative ideal. We introduce products of cubic $BCK$-algebras. Finally, we discuss the relationships between (cubic) commutative ideals, implicative ideals, and positive implicative ideals in $BCK/BCI$-algebras.

The aim of this paper is to introduce and study strong forms of $\mu$-Lindelöfness in generalized topological spaces with a hereditary class, called $\mathcal{S} \mu\mathcal{H}$-Lindelöfness and $\mathbf{S}-\mathcal{S}\mu\mathcal{H}$-Lindelöfness. Interesting characterizations of these spaces are presented. Several effects of various types of functions on them are studied.

The purpose of this note is to prove some results in Bunke-Schick differential K-theory and topological $\mathbb{Z}/k\mathbb{Z}$ K-theory. The first one is an index theorem for the odd-dimensional geometric families of $\mathbb{Z}/k\mathbb{Z}$-manifolds. The second one is an alternative proof of the Freed-Melrose $\mathbb{Z}/k\mathbb{Z}$-index theorem in the framework of differential K-theory.

In this paper, we introduce a new concept of simultaneous remotal sets and farthest points in Banach spaces and we present various characterizations of such points in certain Banach spaces.

Let $Y$ be a nonempty subset of a set $X$ and let $T(X,Y)$ be the semigroup (under composition) of all functions $X\rightarrow X$ whose range is a subset of $Y$. We give necessary and sufficient conditions for elements in $T(X,Y)$ to be left and right magnifying.

In this paper, we give an extensive study of ideal topological spaces and introduce some new types of sets with the help of a local function. Several characterizations of these sets will also be discussed through this paper. Moreover, we obtain characterizations of $\Psi_{\omega}$-operator and $\omega$-codense.

It is well-known that Hausdorff dimension is not a topological invariant; that is, that two homeomorphic continua can have different Hausdorff dimension, although their topological dimension will be equal. We show that it is possible to take any continuum embeddable in $\mathbb{R}^n$ and embed it in such a way that its Hausdorff dimension is $n$. In doing so, we can obtain an arbitrarily high Hausdorff dimension for any nondegenerate continuum. As an example, we will give different embeddings of an arc whose Hausdorff dimension is any real number between $1$ and $\infty$, including an arc of infinite Hausdorff dimension.

In this paper we discuss how to construct piecewise linear chaotic maps with a prescribed probability distribution on a finite number of open intervals of equal length that form a partition of the unit interval. The idea and method of how to find such a map are given in [3]. But a formal proof is not given. In this paper we provide a formal proof.

A proper embedding of a graph $G$ in a pseudosurface $P$ is an embedding in which the regions of the complement of $G$ in $P$ are homeomorphic to discs and a vertex of $G$ appears at each pinchpoint of $P$; we say that a proper embedding of $G$ in $P$ is self dual if there exists an isomorphism from $G$ to its topological dual. We determine five possible graphs with 7 vertices and 13 edges that could be self-dual embeddable in the pinched sphere, and we establish, by way of computer-powered methods, that such a self-embedding exists for exactly two of these five graphs.

The Stern–Brocot tree is a method of generating or organizing all fractions in the interval \((0,1)\) by starting with the endpoints \(\frac{0}{1} \) and \(\frac{1}{1}\) and repeatedly applying the mediant operation: \(m\left( \frac{a}{b},\frac{c}{d} \right) =\frac{a+c}{b+d}\). A recent paper of Aiylam considers two generalizations: one is to apply the mediant operation starting with an arbitrary interval \(\left( \frac{a}{b},\frac{c}{d} \right)\) (the fractions must be non-negative), and the other is to allow arbitrary reduction of generated fractions to lower terms. In the present paper, we give simpler proofs of some of Aiylam's results, and we give a simpler method of generating just the portion of the tree that leads to a given fraction.