We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products, with an extra natural condition (namely, rotating structure vector field). We prove that, provided the surface is not totally umbilical, such families exist in both cases if, and only if, the ambient manifold is a product and the surface is minimal. In particular, there exists no associated families of surfaces with rotating structure vector field in the Heisenberg group.

In this paper, we give three different characterizations for the boundedness and compactness of composition operators between different weighted Dirichlet spaces in the unit disk.

We investigate the simplicity of the lowest eigenvalue for an eigenvalue problem involving the Finsler $p$-Laplace operator and a nonlocal term on a bounded domain subject to the homogeneous Dirichlet boundary condition.

Every affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object, its connection. Using this description of the local geometry of an affine homogeneous space we construct a variety $\mathfrak{M}(\,\mathfrak{g}\,V\,)$, which serves as a coarse moduli space for the local isometry classes of affine homogeneous spaces. Infinitesimal deformations of an isometry class of affine homogeneous spaces in this moduli space are \linebreak described by the Spencer cohomology of a comodule associated to a point in $\mathfrak{M}_\infty(\,\mathfrak{g}\,V\,)$. In an appendix we discuss the relevance of this construction to the study of locally homogeneous spaces.

Topological properties of compactly-fibered coset spaces are investigated. It is proved that for a compactly-fibered coset space $X$ with $Nag(X)\leq\tau$, the closure of a family of $G_{\tau}$-sets is also a $G_{\tau}$-set in $X$. We also show that the equation $\chi(X)=\pi\chi(X)$ holds for any compactly-fibered coset space $X$. A Dichotomy Theorem for compactly-fibered coset spaces is established: every remainder of such a space has the Baire property, or is $\sigma$-compact.

Let $\rm I\!E$ be a complete ultrametric space, let $\rm I\!E$ be a perfect complete ultrametric field and let $A$ be a Banach $\rm I\!E$-algebra which is either a full $\rm I\!E$-subalgebra of the algebra of continuous functions from $\rm I\!E$ to $\rm I\!E$ owning all characteristic functions of clopens of $\rm I\!E$, or a full $\rm I\!E$-subalgebra of the algebra of uniformly continuous functions from $\rm I\!E$ to $\rm I\!E$ owning all characteristic functions of uniformly open subsets of $\rm I\!E$. We prove that all maximal ideals of finite codimension of $A$ are of codimension $1$.

Let $R$ be a commutative ring. In this paper, we introduce and study the compressed essential graph of $R$, $EG_E(R)$. The compressed essential graph of $R$ is a graph whose vertices are equivalence classes of non-zero zero-divisors of $R$ and two distinct vertices $[x]$ and $[y]$ are adjacent if and only if $\ann(xy)$ is an essential ideal of $R$. It is shown if $R$ is reduced, then $EG_E(R)=\Gamma_E(R)$, where $\Gamma_E(R)$ denotes the compressed zero-divisor graph of $R$. Furthermore, for a non-reduced Noetherian ring $R$ with $3<|EG_E(R)|<\infty $, it is shown that $EG_E(R)=\Gamma_E(R)$ if and only if \begin{itemize} \item[(i)] $\Nil(R)=\ann(Z(R))$. \item[(ii)] Every non-zero element of $\Nil(R)$ is irreducible in $Z(R)$. \end{itemize}

In this paper, we introduce a family of drift exponentially fitted stochastic Runge-Kutta (DEFSRK) methods for multi-dimensional It\^{o} stochastic differential equations (SDEs). For the presented class of DEFSRK methods, the regions of mean-square stability (MS-stability) are obtained with reasonable results. Also, general order conditions for the coefficients and the random variables of the DEFSRK methods are extracted. Then, a set of order conditions for a subclass with stochastic weak second order is obtained. Some numerical examples are presented to establish the efficiency and accuracy of the new schemes.

Let $R$ be a commutative ring with identity. We denote by $\mathcal{D}\mathrm{iv}(R)$ the divided spectrum of $R$ (the set of all divided prime ideals of $R$). By a divspectral space, we mean a topological space homeomorphic with the subspace $\mathcal{D}\mathrm{iv}(R)$ of $\mathrm{Spec}(R)$ endowed with the Zariski topology, for some ring $R$. A divspectral set is a poset which is order isomorphic to $(\mathcal{D}\mathrm{iv}(R),\subseteq)$, for some ring $R$. The main purpose of this paper is to provide some topological (resp., algebraic) characterizations of of divspectral spaces (resp., sets).

Let $(X,T,\mu)$ be a Cantor minimal system and $[[T]]$ the associated topological full group. We analyze $C^*_\pi([[T]])$, where $\pi$ is the Koopman representation attached to the action of $[[T]]$ on $(X,\mu)$. Specifically, we show that $C^*_\pi([[T]])=C^*_\pi([[T]]')$ and that the kernel of the character $\tau$ on $C^*_\pi([[T]])$ coming from containment of the trivial representation is a hereditary $C^*$-subalgebra of $C(X)\rtimes\mathbb{Z}$. Consequently, $\ker\tau$ is stably isomorphic to $C(X)\rtimes\mathbb{Z}$, and $C^*_\pi([[T]]')$ is not AF. We also prove that if $G$ is a finitely generated, elementary amenable group and $C^ *(G)$ has real rank zero, then $G$ is finite.