In this paper, we define asymptotically generalized Cantor sets in metric measure spaces by generalizing the notion of $\lambda$-similarity maps. We define the notion of $(\lambda, c, \nu)$-similarity maps, and extend the Moran theorem about the generalized Cantor set in $\mathbf{R}^d$ to this general setting. As an example, we construct generalized Cantor sets in Riemannian manifolds by using $(\lambda, c, \nu)$-similarity maps.

This paper is a continuation of [6]. We study partially ordered rings in terms of non-negative semi-cones and convex ideals, considering order-preserving homomorphisms, residue class rings, and certain product rings, etc.

Let $X$ be a continuum. For each natural number $n, F_n(X)$ is the $n^{th}$-symmetric product of $X$ and $X^n$ is the product of $X$ with itself $n$ times. In this paper we consider the problem of determining the continua $X$ such that $X^n$ can be embedded in $F_n(X)$. Moreover, we characterize finite graphs $X$ for which $X^2$ is embeddable in $F_2(X)$.

Schrödinger operators $T_0 = -\Delta + q(x)$ with rapidly oscillating complex-valued potentials $q(x)$ are considered. Each of such operators is sectorial and hence has Friedrichs extension. We prove that $T_0$ is essentially m-sectorial in the sense that the closure of $T_0$ coincides with its Friedrichs extension $T$. In particular, $T_0$ is essentially self-adjoint if the rapidly oscillating potential $q(x)$ is realvalued. Further, we prove $\sigma_{ess} (T) = [0, \infty)$ under somewhat stricter condition on the potentials $q(x)$.

We study the uniqueness of meromorphic functions when certain nonlinear differential polynomial sharing a nonzero polynomial having common poles and thus radically improve and extend some recent results due to of Wang-Lu-Chen [17], Sahoo [16] and Liu and Yang [14].

In this paper, we study transcendence of values of Mahler functions satisfying first-order rational difference equations of Mahler type with constant coefficients.

We study the construction of spacelike constant mean curvature (CMC) surfaces with mean curvature $0 \le H \lt 1$ in 3-dimensional de Sitter space $\mathbf{S}^{2, 1}$, by using Iwasawa splitting. We also study their singularities and create some criteria for them.

We describe the Hopf algebra quotients and Hopf images of the smash coproduct of a group algebra by the algebra of functions on a finite group.